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+[    {        "title": "2402.11876v1.Hausdorff_dimension_of_random_attractors_for_a_stochastic_delayed_parabolic_equation_in_Banach_spaces.pdf",        "content": "arXiv:2402.11876v1  [math.AP]  19 Feb 2024Hausdorff dimension of random attractors for a stochastic de layed\nparabolic equation in Banach spaces\nWenjie Hu1,2, Tom´ as Caraballo3,4∗, Yueliang Duan5.\n1The MOE-LCSM, School of Mathematics and Statistics, Hunan Norm al University,\nChangsha, Hunan 410081, China\n2Journal House, Hunan Normal University, Changsha, Hunan 4100 81, China\n3Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Facultad de Ma tem´ aticas,\nUniversidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain\n4Department of Mathematics, Wenzhou University,\nWenzhou, Zhejiang Province 325035, China\n5Department of Mathematics, Shantou University, Shantou 51506 3, China\nAbstract\nThe main purpose of this paper is to give an upper bound of Hausdorff dimension of random at-\ntractors for a stochastic delayed parabolic equation in Banach spa ces. The estimation of dimensions\nof random attractors are obtained by combining the squeezing pro perty and a covering lemma of\nfinite subspace of Banach spaces, which generalizes the method es tablished in Hilbert spaces. Unlike\nthe existing works, where orthogonal projectors with finite rank s applied for proving the squeezing\nproperty of stochastic partial differential equations in Hilbert spa ces, we adopt the state decompo-\nsition of phase space based on the exponential dichotomy of the th e linear deterministic part of the\nstudied SDPE to obtain similar squeezing property due to the lack of s mooth inner product geom-\netry structure. The obtained dimension of the random attractor s depend only on the spectrum of\nthe linear part and the random Lipschitz constant of the nonlinear t erm, while not relating to the\ncompact embedding of the phase space to another Banach space a s the existing works did.\nKey words Hausdorff dimension, random dynamical system, random attra ctors, random delayed\ndifferential equation, stochastic delayed parabolic equat ion\n1 Introduction\nFor infinite dimensional random dynamical systems (RDSs), t he existence of random attractors can\nreduce the essential parts of the random flows to random compa ct sets. Furthermore, if the attractors\nhave finite Hausdorff dimension or fractal dimension, then th e attractors can be described by a finite\nnumber of parameters and hence the limit dynamics of the infin ite dimensional RDSs are likely to be\nstudied by the concepts and methods of finite dimensional RDS s. The study of random attractors for\nRDSs dates back to the pioneer works [12, 13, 23], where Craue l, Flandoli and Schmalfuß, amongst\nothers, generalized the concept of global attractors of infi nite dimensional dissipative systems and es-\ntablished the basic framework of random attractors for infin ite dimensional RDSs. Since then, the\nexistence, dimension estimation and qualitative properti es of random attractors for various stochastic\n∗Corresponding author. E-mail address: caraball@us.es (To m´ as Caraballo)\n1evolution equations have been investigated by many researc hers. For example, for the stochastic partial\ndifferential equations (SPDEs) without time delay, Caraball o et al. [6], Gao et al. [24] and Li and Guo\n[33] explored the existence of global attractors on bounded domains. In [2], [40] and [43], the authors\nobtained the existence of global attractors on unbounded do mains. For SPDEs with delay, the existence\nof random attractors and their qualitative properties have been extensively and intensively studied in\n[3, 7, 9, 26, 27, 32, 39] and the references therein.\nCriteria for the finiteHausdorff dimensionality of attracto rs for deterministic fluiddynamics models\nhave been derived by Douady and Oesterle [20], which was late r generalized by Constantin, Foias and\nTemam [11] and [44]. Then, it was further extended to the stochastic case in [14] and [35], where the\nRDS is first linearized and the global Lyapunov exponents of t he linearized mapping is then examined.\nDebussche showed that the random attractors of many RDSs hav e finite Hausdorff dimension by an\nergodicity argument in [18] and furthergave a precise bound on the dimension by combining the method\nof linearization andLyapunov exponents in[19]. [15, 21, 29 , 30, 42] considered thefractal dimensionality\nofrandomsetsandrandomattractors. Intherecentworks[38 ]and[43], theauthorsprovedthefiniteness\nof fractal dimension of random attractor for SPDEs with line ar multiplicative white noise by extending\nthe idea of [18] to obtain the existence of random exponentia l attractors.\nDespite the fact that the finite Hausdorff dimension and fract al dimension of attractors for abstract\nRDSs and applications to SPDEs in Hilbert spaces have been ex tensively and intensively studied, to\nour best knowledge, the estimation of dimensions of stochas tic differential equations with delays, i.e.,\nthe stochastic partial functional differential equations (S PFDEs) have not been extensively studied\nsince their natural phase space are Banach spaces. The lack o f smooth inner product causes the\nexisting methods can not be directly applied. Therefore, in our recent work [25], we extend the method\nestablished in [37] to recast SPFDEs into an auxiliary Hilbe rt space and adopt the method established\nin [19] to give upper bound of the Hausdorff and fractal dimens ions of a delayed reaction-diffusion\nequation by requiring the nonlinear term to be twice continu ous and the first derivative satisfies certain\nconditions. Nevertheless, the natural phase space for SPFD Es are Banach spaces and recast SPFDEs\ninto Hilbert spaces are unnatural and the conditions are qui te strong and hence one naturally wonders\nwhat can we say about the Hausdorff dimension of random attrac tors for SPFDEs in their natural phase\nspace, i.e. the Banach spaces? Here, we extend the squeeze me thod in [18] for estimating dimensions\nof SPDEs in Hilbert spaces to SPFDEs in Banach spaces by combi ng the squeezing property and the\ncovering of finite subspace of Banach spaces. Unlike [18], wh ere orthogonal projector in Hilbert space\nand variational technique are adopted to obtain squeezing p roperty, we prove similar squeezing property\nby semigroup approach and a phase space decomposition based on the exponential dichotomy of linear\npart.\nItshouldbepointed out that in [8] and [36], theauthors prov ed theexistence of randomexponential\nattractors and uniform exponential attractors of random dy namical systems in Banach spaces based on\na smoothing property of the systems, which indicates the fini te fractal dimensionality of the systems.\nHowever, the fractal dimension obtained by smoothing prope rty may depend on the choice of another\nembedding space, which may vary from space to space. Further more, the dimension estimation depends\non the entropy number between two spaces for which is general ly quite difficult to obtain an explicit\nbound. Here, the dimension obtained by our method only depen ds on the inner characteristics of the\nstudied equation while not depend the entropy number betwee n two spaces.\nWe consider the following stochastic delayed parabolic equ ation with additive noise\n\n\n∂u\n∂t(x,t) = ∆u(x,t)−µu(x,t)−σu(x,t−τ)+F(u(x,t))+f(u(x,t−τ))\n+/summationtextm\nj=1gj(x)dωj(t)\ndt,t>0,x∈ O,\nu0(x,s) =φ(x,s),−τ≤s≤0,x∈ O,\nu0(x,t) = 0,−τ≤t,x∈∂O,(1.1)\n2whereO ⊆RNis a bounded open domain with smooth boundary ∂O,{ωj}m\nj=1are mutually indepen-\ndent two-sided real-valued Wiener process on an appropriat e probability space to be specified below.\nEquation (1.1) can model many processes from chemistry or ma thematical biology. For instance, it\ncan be used to describe the evolution of mature populations f or age-structured species, where ∆ u(x,t)\nandF(u(x,t))−µu(x,t) represent the spatial diffusion and the death rate of mature i ndividuals,\nf(u(x,t−τ))−σu(x,t−τ) represents birth rate,/summationtextm\nj=1gj(x)dωj(t)\ndtstands for the random perturbations\nor environmental effects.\nWe organize the remaining part of this paper as follows. In Se ction 2, we review some results\nabout the existence of random attractors in the existing lit erature and prove the squeezing property of\nthe RDS generated by (1.1) on the random attractors. Then, we give upper bound of the Hausdorff\ndimension estimation for (1.1) in Section 3. At last, we summ arize the paper and point out some\npotential directions for future research in Section 4.\n2 Squeezing property\nThis section is concerned about the squeezing property of th e RDS generated by (1.1) on the random\nattractors. In order to adopt the RDS theory, as a first step, w e follow the idea of [16] to transform\nthe stochastic (1.1) into a random delayed equation, i.e., a path-wise deterministic delayed equation.\nThe same idea has been adopted by many authors when dealing wi th random attractors or invariant\nmanifolds for various stochastic evolution equations, suc h as [17, 26, 27, 32].\nWe consider the canonical probability space (Ω ,F,P) with\nΩ ={ω= (ω1,ω2,...,ω m)∈C(R;Rm) :ωi(0) = 0}\nandFis the Borel σ-algebra induced by the compact open topology of Ω ,whilePis the corresponding\nWiener measure on (Ω ,F). Then, we identify Wwith\nW(t,ω) = (ω1(t),ω2(t),...,ω m(t)) fort∈R.\nMoreover, we define the time shift by\nθtω(·) =ω(·+t)−ω(t),t∈R.\nThen,/parenleftbig\nΩ,F,P,{θt}t∈R/parenrightbig\nis a metric dynamical system. Consider the stochastic stati onary solution of\nthe one dimensional Ornstein-Uhlenbeck equation\ndzj+µzjdt= dωj(t),j= 1,...,m, (2.1)\nwhich is given by\nzj(t)/defineszj(θtωj) =−µ/integraldisplay0\n−∞eµs(θtωj)(s)ds, t∈R. (2.2)\nProposition 4.3.3 in [1] implies that there exists a tempere d function 0 <r(ω)<∞such that\nm/summationdisplay\nj=1|zj(ωj)|2≤r(ω), (2.3)\nwherer(ω) satisfies, for P-a.e.ω∈Ω,\nr(θtω)≤eµ\n2|t|r(ω), t∈R. (2.4)\n3Combining (2.3) with (2.4), we obtain that for P-a.e.��∈Ω,\nm/summationdisplay\nj=1|zj(θtωj)|2≤eµ\n2|t|r(ω), t∈R. (2.5)\nMoreover, we have\nm/summationdisplay\nj=1|zj(θξωj)|2≤eµτ\n2r(ω), (2.6)\nfor anyξ∈[−τ,0] andP-a.e.ω∈Ω. Putting z(θtω) =/summationtextm\nj=1gjzj(θtωj), we have\ndz+µzdt=m/summationdisplay\nj=1gjdωj.\nTake the transformation v(t) =u(t)−z(θtω). Then, simple computation gives\ndv(x,t)\ndt= ∆v(x,t)−µv(x,t)−σv(x,t−τ)−σz(θt−τω)+F(v(t,x)+z(θtω))+f(v(x,t−τ)+z(θt−τω))+∆z(θtω)\n(2.7)\nwith boundary condition\nv(x,t) = 0 for ( x,t)∈∂O×(0,∞) (2.8)\nand initial condition\nv(x,ξ,ω) =ψ(x,ξ,ω)/definesφ(x,ξ)−z(θ−τω) for (x,ξ)∈ O ×[−τ,0]. (2.9)\nLetH=L2(O) be the space of square Lesbegue integrable functions on Owith its usual norm /bardbl·/bardblH\nand inner product ( ·,·)H,A= ∆ with the domain D(A) =H2(D)∩H1\n0(D). LetX=C([−τ,0],H)\nbe the space of continuous functions from [ −τ,0] toHendowed with the supremum norm /bardblφ/bardbl=\nsupθ∈[−r,0]/bardblφ(θ)/bardblHfor anyφ∈Xand define ut∈Xbyut(ξ) =u(t+ξ) forξ∈[−τ,0]. Define\nL:X/ma√sto→Handf:X/ma√sto→HbyLφ=σφ(−τ),f(φ) =f(φ(−τ)) respectively. Then (2.7) can be written\nas the following abstract SPFDE in H\ndv(t)\ndt=Av(t)−µv(t)−Lvt−Lz(θt+·ω)+F(v(t)+z(θtω))+f(vt+z(θt+·ω))+Az(θtω),(2.10)\nwhereθt+·ωis defined as θt+ξωforξ∈[−τ,0].\nWefollow [18], [32]and[39]tomakethefollowingassumptio nsonFandfthroughouttheremaining\npart of this paper.\nHypothesis A1 F:R→Ris a polynomial of odd degree with negative leading coefficien t\nF(u) =2p−1/summationdisplay\nk=1akuk,a2p−1<0.\nwe assume for simplicity that p≤2 ifn= 3.fis Lipschitz continuous with 0being a fixed point, that\nis,f(0) =0and\n/bardblf(φ)−f(ϕ)/bardblH≤Lf/bardblφ−ϕ/bardbl\nfor anyφ,ϕ∈X.\nForthepathwisedeterministicproblem(2.7)-(2.9)underH ypothesisA1, itfollowsfromtheGalerkin\nmethodasin[4]thatfor P-a.e.ω∈Ωandψ∈X, ithasauniquesolution v(·,ω,ψ)∈C/parenleftbig\n[−τ,∞),L2(O)/parenrightbig\n∩\n4L2\nloc/parenleftbig\nτ,∞;H1\n0(O)/parenrightbig\n. Moreover, this solution is continuous in ψ∈Xandvt(·,·,ψ) : (Ω,F)→(X,B(X))\nis measurable. By the uniqueness of the solution and semigro up property of v, we can see that the\nsolutionvof (2.7)-(2.9) generates a continuous RDS. Let u(t,ω,φ) =v(t,ω,ψ) +z(θtω), thenuis\nthe solution of (1.1) with initial condition φ. We now define a mapping Φ : R+×Ω×X→Xby\nΦ(t,ω,φ) =ut(·,ω,φ) =vt(·,ω,ψ) +z(θt+.ω), whereut(ξ,ω,φ) =u(t+ξ,ω,φ) forξ∈[−τ,0]. Appar-\nently, Φ is an RDS on Xassociated with problem (1.1).\nIt follows from [39, Theorem 5.1] and [32, Theorem 4.1] that, if Hypothesis A1 holds, then (1.1)\nadmits a pullback random attractor A(ω)⊂ BX(0,c+cr(ω)), the ball centered at 0 with radius c+cr(ω)\ninX, wherecis constant defined in [32, Lemma 3.1]. Subsequently, we prov e the squeezing property\nof on the attractor A(ω). Unlike the previous works [7, 9, 32, 39], we obtain the resu lts by using the\nsemigroup approach instead of variation techniques as we wo rk in Banach spaces. It follows from [41,\nTheorem 2.6, Chapter 2] that the following linear equation\n/braceleftbiggd˜v(t)\ndt=A˜v(t)−µ˜v(t)−L˜vt,\n˜v0=φ(2.11)\nadmits a unique global solution ˜ vφ(t) : [−τ,∞)→H. Define a compact semigroup S(t) :X→Xby\nS(t)φ= ˜vφ\nt(·) and denote by ASits infinitesimal generator.\nLet 0< µ1≤µ2≤ ··· ≤µm≤ ···,µm→+∞asm→+∞be eigenvalues of the following\neigenvalue problem on O:\n−∆u(x) =µu(x), u(x)|x∈∂O= 0, x∈ O, (2.12)\nwith corresponding eigenfunctions {em}m∈N. SinceASis compact, it follows from Theorem 1.2 (i) in\n[41] that the spectrum of ASare point spectra, which we denote by ̺1> ̺2>···with multiplicity\nn1,n2,···. Moreover, it follows from [41] that the characteristic val ues̺1>̺2>···of the linear part\nASare the roots of the following characteristic equation\nµ2\nm−/parenleftBig\nλ+µ−σe−λτ/parenrightBig\n= 0,m= 1,2,···. (2.13)\nWe introduce the following state decomposition results of t he linear part ASof (2.11) established\nin [41]. For any given ̺m<0,m≥1, there is a\nkm=n1+n2+···+nm (2.14)\ndimensional subspace XU\nkmsuch that\nX=XU\nkm/circleplusdisplay\nXS\nkm\nis the decomposition of Xby̺m. LetPkmandQkmbe the projection of XontoXU\nkmandXS\nkm\nrespectively, that is XU\nkm=PkmX,XS\nkm= (I−Pkm)X=QkmX. It follows from the definition of Pkm\nandQkmthat\n/bardblQkmS(t)x/bardbl ≤Ke̺mt/bardblx/bardbl, t≥0, (2.15)\nwhereKis a positive constant. Moreover, there exists a positive co nstantMsuch that\n/bardblS(t)φ/bardbl ≤Me̺1t/bardblφ/bardbl, (2.16)\nfor anyt≥0.\nFor the later use, we extend the domain of S(t) to the following space of some discontinuous\nfunctions\nˆC=/braceleftbigg\nφ: [−τ,0]→H;φ|[−r,0)is continuous and lim\nθ→0−φ(θ)∈Hexists/bracerightbigg\n(2.17)\n5and introduce the following variation of constants formula established in [41, Section 4.2, Theorem 2.1]\nvt(·,ω,φ) =S(t)φ+/integraldisplayt\n0S(t−s)X0[−Lz(θs+·ω)+F(v(s)+z(θsω))+f(vs+z(θs+·ω))+Az(θsω)]ds\n(2.18)\nfor anyt≥0, whereX0: [−τ,0]→B(X) is bounded linear operator on Xgiven byX0(θ) = 0 if\n−r≤θ<0 andX0(0) =Id.\nRemark 2.1. In general, the solution semigroup defined by (2.18)have no definition at discontinuous\nfunctions and the integral in the formula is undefined as an in tegral in the phase space. However, if\ninterpreted correctly, one can see that (2.18)does make sense. Details can be found in [5] Pages 144\nand 145.\nSubsequently, we prove the squeezing property of the RDS Φ.\nTheorem 2.1. LetPbe the finite dimensional projection PkmofXontoXU\nkm,K,M,̺ mandρ1being\ndefined by (2.15)and(2.16)respectively and assume Hypothesis A1 holds, then we have\n/bardblP[Φ(t,ω,ϕ)−Φ(t,ω,ψ)]/bardbl ≤Me(MLf+̺1)t+/integraltextt\n0R(θsω)ds/bardblϕ−ψ/bardbl (2.19)\nand\n/bardbl(I−P)[Φ(t,ω,ϕ)−Φ(t,ω,ψ)]/bardbl ≤[Ke̺mt+KM/radicalbig\n2(̺1−̺m)e[(MLf+̺1)t+/integraltextt\n0MR(θsω)ds+/integraltextt\n0R2(θsω)ds]\n+KMLfe[(MLf+̺1)t+/integraltextt\n0MR(θsω)ds]\n̺1−̺m]/bardblϕ−ψ/bardbl(2.20)\nfor anyt≥0andϕ,ψ∈ A(ω), whereR(θsω) =/summationtext2p−1\nk=1ak(c+(c+1)r(θsω))k−1withr(ω)being defined\nby(2.3)andcbeing a constant.\nProof.For anyφ,χ∈ A(ω), denote by y=φ−χandwt(·,ω) = Φ(t,ω,φ)−Φ(t,ω,χ) =ut(·,ω,φ)−\nut(·,ω,χ) =vt(·,ω,ϕ)−vt(·,ω,ψ), whereφ=ϕ+z(θ−τω) andχ=ψ+z(θ−τω) and hence y=φ−χ.\nThen it follows from (2.18) that\nwt(·,ω) =S(t)y+/integraldisplayt\n0S(t−s)X0[F(v(s,ω,ϕ)+z(θsω))−F(v(s,ω,ψ)+z(θsω))]ds\n+/integraldisplayt\n0S(t−s)X0[f(vs(·,ω,ϕ)+z(θs+·ω))−f(vs(·,ω,ψ)+z(θs+·ω))]ds.(2.21)\nTaking projection I−Pon both sides of (2.21) leads to\n/bardbl(I−P)wt(·,ω)/bardbl=/bardbl(I−P)S(t)y+/integraldisplayt\n0(I−P)S(t−s)X0[F(v(s,ω,ϕ)+z(θsω))−F(v(s,ω,ψ)+z(θsω))]ds\n+/integraldisplayt\n0(I−P)S(t)X0[f(vs(·,ω,ϕ) +z(θs+·ω))−f(vs(·,ω,ψ)+z(θs+·ω))]ds/bardbl\n≤Ke̺mt/bardbly/bardbl+/bardbl/integraldisplayt\n0(I−P)S(t−s)X0[F(v(s,ω,ϕ))−F(v(s,ω,ψ))]ds/bardbl\n+LfM/integraldisplayt\n0e̺1t/bardbl(I−P)ws/bardblds.\n(2.22)\n6Since we have assumed that Fis a polynomial with order 2 p−1, then we have\n/bardblF(v(t,ω,ϕ))−F(v(t,ω,ψ))/bardblH=/bardbl2p−1/summationdisplay\nk=1ak(vk(t,ω,ϕ)−vk(t,ω,ψ))/bardblH\n=/bardbl(v(t,ω,ϕ)−v(t,ω,ψ))2p−1/summationdisplay\nk=1akk−1/summationdisplay\nj=0(vk−1−j(t,ω,ϕ)vj(t,ω,ψ))/bardblH\n≤/bardbl(v(t,ω,ϕ)−v(t,ω,ψ))/bardblH2p−1/summationdisplay\nk=1akk−1/summationdisplay\nj=0/bardbl(vk−1−j(t,ω,ϕ)/bardbl/bardblvj(t,ω,ψ))/bardblH\n≤/bardbl(v(t,ω,ϕ)−v(t,ω,ψ))/bardblH2p−1/summationdisplay\nk=1ak(c+(c+1)r(θtω))k−1,\n(2.23)\nwhere the last inequality follows from the fact that v(t,ω,ϕ)∈ A(θtω)−z(θtω),v(t,ω,ψ)∈ A(θtω)−\nz(θtω) andA(θtω)+z(θtω)⊂ BX(0,c+(c+1)r(θtω)). Incorporating (2.23) into (2.22) gives\n/bardbl(I−P)wt(·,ω)/bardbl ≤Ke̺mt/bardbly/bardbl+M/integraldisplayt\n0R(θsω)e̺1(t−s)/bardbl(I−P)ws(·,ω)/bardblds\n+LfM/integraldisplayt\n0e̺1(t−s)/bardbl(I−P)ws(·,ω)/bardblds,(2.24)\nwhereR(θsω) =/summationtext2p−1\nk=1ak(c+(c+1)r(θsω))k−1. Multiplying both sides of (2.24) by e−̺1tgives\ne−̺1t/bardbl(I−P)wt(·,ω)/bardbl ≤Ke(̺m−̺1)t/bardbly/bardbl+M/integraldisplayt\n0R(θsω)e−̺1s/bardbl(I−P)ws(·,ω)/bardblds\n+LfM/integraldisplayt\n0e−̺1s/bardbl(I−P)ws(·,ω)/bardblds.(2.25)\nBy applying the Gronwall inequality, we have\ne−̺1t/bardbl(I−P)wt(·,ω)/bardbl ≤K/bardbly/bardbl[e(̺m−̺1)t+M/integraldisplayt\n0(R(θsω)+Lf)e(̺m−̺1)se/integraltextt\n0M(R(θrω)+Lf)drds]\n≤K/bardbly/bardbl{e(̺m−̺1)t+MeMLft+M/integraltextt\n0R(θsω)ds[/integraldisplayt\n0R(θsω)e(̺m−̺1)sds+Lf\n̺1−̺m]}\n≤K/bardbly/bardbl{e(̺m−̺1)t+MeMLft+M/integraltextt\n0R(θsω)ds[(/integraldisplayt\n0R2(θsω)ds)1\n2(/integraldisplayt\n0e2(̺m−̺1)sds)1\n2+Lf\n̺1−̺m]}\n≤K{e(̺m−̺1)t+MeMLft+M/integraltextt\n0R(θsω)ds[e/integraltextt\n0R2(θsω)ds1/radicalbig\n2(̺1−̺m)+Lf\n̺1−̺m]}/bardbly/bardbl,\n(2.26)\nwhere the third inequality follows from the H¨ older’s inequ ality and the last inequality follows from the\nfact that√x≤ex. (2.26) indicates that\n/bardbl(I−P)wt(·,ω)/bardbl ≤[Ke̺mt+KM/radicalbig\n2(̺1−̺m)e[(MLf+̺1)t+/integraltextt\n0MR(θsω)ds+/integraltextt\n0R2(θsω)ds]\n+KMLfe[(MLf+̺1)t+/integraltextt\n0MR(θsω)ds]\n̺1−̺m]/bardbly/bardbl.(2.27)\n7Subsequently, we prove the first part. Since S(t,σ)y=P(t)S(t,σ)y+(I−P(t))S(t,σ)y, we have\n/bardblPwt(·,ω)/bardbl=/bardblPS(t)y+/integraldisplayt\n0PS(t−s)X0[F(v(s,ω,ϕ)+z(θsω))−F(v(s,ω,ψ)+z(θsω))]ds\n+/integraldisplayt\n0PS(t−s)X0[f(vs(·,ω,ϕ)+z(θs+·ω))−f(vs(·,ω,ψ)+z(θs+·ω))]ds/bardbl\n≤Me̺1t/bardbly/bardbl+M/integraldisplayt\n0(R(θsω)+Lf)e̺1(t−s)/bardblPws(·,ω)/bardblds.(2.28)\nMultiplying both sides of (2.28) by e−̺1tgives rise to\ne−̺1t/bardblPwt(·,ω)/bardbl ≤M/bardbly/bardbl+M/integraldisplayt\n0(R(θsω)+Lf)e−̺1s/bardblPws(·,ω)/bardblds. (2.29)\nAgain, it follows from the Gronwall inequality that\ne−̺1t/bardblPwt(·,ω)/bardbl ≤MeM/integraltextt\n0(R(θsω)+Lf)ds/bardbly/bardbl, (2.30)\nimplying that\n/bardblPwt(·,ω)/bardbl ≤Me(MLf+̺1)t+/integraltextt\n0MR(θsω)ds/bardbly/bardbl. (2.31)\nThis proves the first part.\n3 Hausdorff dimensions of random attractors\nIn this section, we study the Hausdorff dimension of the attra ctorsA(ω) for the RDS generated by\n(1.1). The Hausdorff dimension of the random attractor A(ω)⊂Xis\ndH(A(ω)) = inf{d:µH(A(ω),d) = 0}\nwhere, ford≥0,\nµH(A(ω),d) = lim\nε→0µH(A(ω),d,ε)\ndenotes the d-dimensional Hausdorff measure of the set A(ω)⊂X, where\nµH(A(ω),d,ε) = inf/summationdisplay\nird\ni\nand the infimum is taken over all coverings of A(ω) by balls of radius ri/lessorequalslantε. It can be shown that\nthere exists dH(A(ω))∈[0,+∞] such that µH(A(ω),d) = 0 ford > dH(A(ω)) andµH(A(ω),d) =∞\nford<dH(A(ω)).dH(A) is called the Hausdorff dimension of the Hausdorff dimension ofA(ω).\nFor a finite dimensional subspace Fof a Banach space X, denote by BF\nr(x) the ball in Fof center\nxand radius r, that isBF\nr(x) ={y∈F|/bardbly−x/bardbl ≤r}. It is proved in [34] that the following covering\nlemma of balls in finite dimensional Banach spaces is true.\nLemma 3.1. For every finite dimensional subspace Fof a Banach space X, we have\nN/parenleftbig\nr1,BF\nr2/parenrightbig\n≤m2m/parenleftbigg\n1+r2\nr1/parenrightbiggm\n, (3.1)\nfor allr1>0,r2>0, wherem= dimFandN/parenleftbig\nr1,BF\nr2/parenrightbig\nis the minimum number of balls needed to\ncoverBF\nr2by the ball of radius r1calculated in the Banach space X.\n8We can now establish the upper bound for the Hausdorff dimensi on of the random attractors for\nthe random dynamical system Φ( t,ω,φ) in the Banach space Xunder Hypothesis A1.\nTheorem 3.1. Assume Hypothesis A1 holds and there exists 0<α<2such that\n(αM+2K+2KMLf\n̺1−̺m+2KM/radicalbig\n2(̺1−̺m))e(MLf+̺1+2E(R(θtω))+2E(R2(θtω)))t0<1.(3.2)\nThen, the Hausdorff dimension of the global attractor A(ω)of(1.1)satisfies\nd<−lnkm−kmln(2+4\nα)\nln(αM+2K+2KML f\n̺1−̺m+2KM√\n2(̺1−̺m))+(MLf+̺1+2E(R)+2E(R2))t0,(3.3)\nwherekmis the dimension of PXdefined by (2.14),K,M,̺ mandρ1being defined by (2.15)and(2.16)\nrespectively and R(θtω)is defined in Theorem 2.1.\nProof.SinceA(ω) is a compact subset of X, for any 0 < ε <1, there exist r1,...,rNin (0,ε] and\n˜u1,...,˜uNinXsuch that\nA(ω)⊂N/uniondisplay\ni=1B(˜ui,ri), (3.4)\nwhereB(˜ui,ri) represents the ball in Xof center ˜uiand radius ri. Without loss of generality, we can\nassume that for any i\nB(˜ui,ri)∩A(ω)/ne}ationslash=∅, (3.5)\notherwise, it can be deleted from the sequence ˜ u1,...,˜uN. Therefore, we can choose ui,i= 1,2,···,N\nsuch that\nui∈B(˜ui,ri)∩A(ω), (3.6)\nand\nA(ω)⊂N/uniondisplay\ni=1(B(ui,2ri)∩A(ω)). (3.7)\nIt follows from (2.19) and (2.20) that for any u∈B(ui,2ri)∩A(ω), we have\n/bardblP[Φ(t0,ω,u)−Φ(t0,ω,ui)]/bardbl ≤2Me(MLf+̺1)t0+/integraltextt0\n0MR(θsω)ds/bardblu−ui/bardbl. (3.8)\nand\n/bardbl(I−P)[[Φ(t0,ω,u)−Φ(t0,ω,ui)]]/bardbl ≤2[Ke̺mt+KM/radicalbig\n2(̺1−̺m)e[(MLf+̺1)t+/integraltextt\n0MR(θsω)ds+/integraltextt\n0R2(θsω)ds]\n+KMLfe[(MLf+̺1)t+/integraltextt\n0MR(θsω)ds]\n̺1−̺m]/bardblu−ui/bardbl.\n(3.9)\nBy Lemma 3.1, for any α>0, we can find y1\ni,...,yni\nisuch that\nBPX/parenleftBig\nPΦ(t0,ω,ui),2Me(MLf+̺1)t0+/integraltextt0\n0MR(θsω)dsri/parenrightBig\n⊂ni/uniondisplay\nj=1BPX/parenleftBig\nyj\ni,αMe(MLf+̺1)t0+/integraltextt0\n0MR(θsω)dsri/parenrightBig\n(3.10)\n9with\nni≤km2km/parenleftbigg\n1+2\nα/parenrightbiggkm\n, (3.11)\nwherekmis the dimension of PXand we have denoted by BPX(y,r) the ball in PXof radiusrand\ncentery.\nSet\nuj\ni=yj\ni+(I−P)Φ(t0,ω,ui) (3.12)\nfori= 1,...,N,j = 1,...,n i. Then, for any u∈B(ui,2ri)∩A(ω), there exists a jsuch that\n/vextenddouble/vextenddouble/vextenddoubleΦ(t0,ω,u)−uj\ni/vextenddouble/vextenddouble/vextenddouble≤/vextenddouble/vextenddouble/vextenddoublePΦ(t0,ω,u)−yj\ni/vextenddouble/vextenddouble/vextenddouble+/bardbl(I−P)Φ(t0,ω,u)−(I−P)Φ(t0,ω,ui)/bardbl\n≤(αMe(MLf+̺1)t0+/integraltextt0\n0MR(θsω)ds+2(Ke̺mt0+KMLfe[(MLf+̺1)t0+/integraltextt\n0MR(θsω)ds]\n̺1−̺m)\n+2KM/radicalbig\n2(̺1−̺m)e[(MLf+̺1)t0+/integraltextt0\n0MR(θsω)ds+/integraltextt0\n0R2(θsω)ds])ri\n≤(αM+2K+2KMLf\n̺1−̺m+2KM/radicalbig\n2(̺1−̺m))e[(MLf+̺1)t0+/integraltextt0\n0MR(θsω)ds+/integraltextt0\n0R2(θsω)ds])ri\n(3.13)\nwith\nni≤km(2+4\nα)km(3.14)\nDenote byη=αM+2K+2KML f\n̺1−̺m+2KM√\n2(̺1−̺m)andC(θt0ω) =e[(MLf+̺1)t0+/integraltextt0\n0MR(θsω)ds+/integraltextt0\n0R2(θsω)ds],\nthen we have\nΦ(t0,ω,B(ui,2ri))∩A(θt0ω)⊂ni/uniondisplay\nj=1B/parenleftBig\nuj\ni,ηC(θt0ω)ri/parenrightBig\n. (3.15)\nThanks to the invariance of A(ω), i.e.,A(θt0ω) = Φ(t0,ω,A(ω)), we have\nA(θt0ω)⊂N/uniondisplay\ni=1ni/uniondisplay\nj=1B/parenleftBig\nuj\ni,ηC(θt0ω)ri/parenrightBig\n. (3.16)\nThis implies that, for any d≥0,\nµH(A(θt0ω),d,ηC(θt0ω)ε)≤N/summationdisplay\ni=1ni/summationdisplay\nj=1ηdrd\ni≤km(2+4\nα)kmηdCd(θt0ω)N/summationdisplay\ni=1rd\ni, (3.17)\nwe deduce, by taking the infimum over all the coverings of A(ω) by balls of radii less than ε,\nµH(A(θt0ω),d,ηε)≤km(2+4\nα)kmηdCd(θt0ω)µH(A(ω),d,ε). (3.18)\nApplying the formula recursively for ktimes and by adopting the fact/integraltext(k−1)t0\n0[R(θrω)+R2(θsω)]dr+/integraltextkt0\n(k−1)t0[R(θrω)+R2(θsω)]ds=/integraltextkt0\n0[R(θrω)+R2(θsω)]ds, we have\nµH/parenleftBig\nA(θkt0ω),d,˜C(ω)(ηε)k/parenrightBig\n≤[km(2+4\nα)kmηd]k˜Cd(ω)µH(A(ω),d,ε), (3.19)\n10where˜C(ω) =e[(MLf+̺1)kt0+/integraltextkt0\n0MR(θsω)ds+/integraltextkt0\n0R2(θsω)ds]. Thanks to the ergodicity of θt,t∈R, for\nalmost allω∈Ω, we have\n1\nt[/integraldisplayt\n0R(θsω)ds+/integraldisplayt\n0R2(θsω)ds]→E(R(θtω))+E(R2(θtω)) (3.20)\nwhent→ ∞. Therefore, there exist k0(ω) such that for all k≥k0(ω), we have\n/integraldisplaykt0\n0R(θsω)ds+/integraldisplaykt0\n0R2(θsω)ds]≤2(E(R(θtω))+E(R2(θtω)))kt0. (3.21)\nimplying that\n˜C(ω)≤e(MLf+̺1+2E(R(θtω))+2E(R2(θtω)))kt0 (3.22)\nIncorporating (3.22) into (3.19) gives\nµH/parenleftBig\nA(θkt0ω),d,˜C(ω)(ηε)k/parenrightBig\n≤[km(2+4\nα)km(ηe(MLf+̺1+2E(R(θtω))+2E(R2(θtω)))t0)d]kµH(A(ω),d,ε),\n(3.23)\nHence, if\nd<−lnkm−kmln(2+4\nα)\nln(αM+2K+2KML f\n̺1−̺m+2KM√\n2(̺1−̺m))+(MLf+̺1+2E(R(θtω))+2E(R2(θtω)))t0,(3.24)\nthen\nkm(2+4\nα)km(ηe(MLf+̺1+2E(R(θtω))+2E(R2(θtω)))t0)d<1. (3.25)\nThus, by taking k→ ∞, we have (ηε)k→0 and (3.19) leads to\nµH(A(θkt0ω),d,(ηε)k˜Cd(ω))→0. (3.26)\nBy the ergodicity of θt,t∈R, we have\nµH(A(ω),d,ε)→0. (3.27)\nfor almost all ω∈Ω. This completes the proof.\nRemark 3.1. By(3.3), we can see estimation of dHdepends on the parameter α. If we take α↑2and\nassume2M+2K+2KML f\n̺1−̺m+2KM√\n2(̺1−̺m))e(MLf+̺1+2E(R(θtω))+2E(R2(θtω)))t0<1, then for all α∈(0,2),\nwe haveαM+2K+2KML f\n̺1−̺m+2KM√\n2(̺1−̺m))e(MLf+̺1+2E(R(θtω))+2E(R2(θtω)))t0<1and hence we obtain the\nestimation\ndH≤−lnΛ−Λln4\nln(2M+2K+2KML f\n̺1−̺m+2KM√\n2(̺1−̺m))+(MLf+̺1+2E(R(θtω))+2E(R2(θtω)))t0,(3.28)\nwhich is independent of α.\n114 Conclusions\nIn this paper, we give an upper bound of Hausdorff dimension of random attractors for a stochastic\ndelayed parabolic equation in Banach spaces that depends on ly on the inner characteristics of the\nequation, while not relating to the compact embedding of the phase space to another Banach space as\nthe existing works did. Similar procedures can also be emplo yed to study the fractal dimension. In\n[28], the authors proposed the Lyapunov dimension and gave t he famous Kplan-York formula, which\nwas proved to be correct for Navis-Stokes equation in Hilber t space in [14]. In [11], Constantin, Foias\nand Temam also showed the Lyapunov dimension can control the Hausdorff dimension and gave better\nestimation ofHausdorffdimensionthanthesqueezemethod, i .e., themethodweusedhere. Furthermore,\n[22] showed the dimension of chaotic attractor of delayed Ma ckey-Glass equation satisfies the Kplan-\nYork formula by numerical simulations and Ledrappier and Yo ung [31] showed the Kplan-York formula\nwas mathematically correct for random semiflow generated by stochastic ordinary differential equation,\nwhich heuristically inspire us to believe the Kplan-York fo rmula also holds for random delay differential\nequations in Banach spaces. 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Equ., 263 (2017), 6347-6383.\n[44] Zhou S., Zhao C., Wang Y.: Finite dimensionality and upper semicontin uity of compact kernel sections of\nnon-autonomous lattice systems. Discrete Cont. Dyn. Syst.-A, 2 1 (2008), 1259-1277.\n14"    },    {        "title": "2402.11888v1.The_Volterra_lattice__Abel_s_equation_of_the_first_kind__and_the_SIR_epidemic_models.pdf",        "content": "arXiv:2402.11888v1  [nlin.SI]  19 Feb 2024The Volterra lattice, Abel’s equation of the\nfirst kind, and the SIR epidemic models\nAtsushiNobe\nFaculty of Political Science and Economics, Waseda University,\n1-6-1 Nishiwaseda, Shinjuku, Tokyo 169-8050, Japan\ne-mail:nobe@waseda.jp\nAbstract\nThe Volterra lattice, when imposing non-zero constant boun dary values, admits\nthe structureof a completely integrable Hamiltonian syste m if the system size is suffi-\nciently small. Such a Volterra lattice can be regarded as an e pidemic model known as\nthe SIR model with vaccination, which extends the celebrate d SIR model to account\nfor vaccination. Upon the introduction of an appropriate va riable transformation,\nthe SIR model with vaccination reduces to an Abel equation of the first kind, which\ncorresponds to an exact differential equation. The equipoten tial curve of the exact\ndifferential equation is the Lambert curve. Thus, the general solution to the ini-\ntial value problem of the SIR model with vaccination, or the V olterra lattice with\nconstant boundary values, is implicitly provided by using t he Lambert W function.\n1 Introduction\nThe Volterra lattice is a simultaneous system of infinitely many first-o rder differential\nequations that pertain to the nodes on a one-dimensional infinite lat tice [1, 2, 3]. By\nimposing an appropriate boundary condition, the Volterra lattice re duces to a completely\nintegrableHamiltonianflowonafinite-dimensional Poissonmanifold. Sig nificant boundary\nconditions that contribute to complete integrability include the perio dic boundary and the\nopen-end boundary. When the Volterra lattice imposes either the p eriodic or the open-end\nboundary, it admits a bi-Hamiltonian structure on the Poisson manifo ld, thereby providing\na sufficient number of conserved quantities [4, 5, 6] .\nInthis article, we consider another boundaryconditionthat contr ibutes tothe complete\nintegrabilityoftheVolterralattice: theboundarynodesareassign edconstantvalues, which\nare not necessarily zero. Although such a Volterra lattice has a Pois son structure, and\nhence, is a Hamiltonian flow on a finite-dimensional Poisson manifold, it d oes not admit a\nbi-Hamiltonian structure. Thus, the Volterra lattice with constant boundary values is, in\ngeneral, not a completely integrable Hamiltonian system. Neverthele ss, if the system size\n1is sufficiently small, the Volterra lattice exhibits complete integrability b ecause a sufficient\nnumber of conserved quantities immediately follow from the Hamiltonia n. If the system\nsize is two, the Volterra lattice with constant boundary values redu ces to an integrable\nepidemic model called the SIR model with vaccination [7], which is an exte nsion of the\nSIR model [8] under the influence of vaccination and is abbreviated t o the SIRv model. If\nthe system size is three, the Volterra lattice with constant bounda ry values also exhibits\nthe complete integrability; however, its significance as an epidemic mo del is unclear.\nUpon the introduction of an appropriate variable transformation, the SIRv model is\ntransformed into an exact differential equation via a first-order n onlinear differential equa-\ntion of degree three called Abel’s equation of the first kind [9, 10]. The exact differential\nequation thus obtained possesses the potential arising from the s ymplectic structure of\nthe Volterra lattice. Moreover, the invariant curve of the SIRv mo del, or the equipoten-\ntial curve of the Abel equation, is the Lambert curve. Hence, the general solution to the\nSIRv model, or the Volterra lattice with constant boundary values, is implicitly provided\nin terms of the Lamber W function [11]. In addition, an integrable discr etization of the\nSIRv model, possessing exactly the same conserved quantity as th e continuous model, is\nachieved through a geometric construction utilizing the Lambert cu rve [12].\nThis article is organized as follows. In $2, we introduce the Volterra lattice and briefly\nreview its Poisson structure. Then we show that the Volterra lattic e with constant bound-\nary values is a completely integrable Hamiltonian flow on the Poisson man ifold if the\nsystem size is either two or three. We moreover show that the two- dimensional Volterra\nlattice can be regarded as an epidemic model called the SIRv model. In $3, we investigate\nAbel’s equation of the first kind and show that it reduces to an exact differential equation\nif some conditions are satisfied. We also show that the Abel equation can be related with\nthe SIRv model. The equipotential curve of the exact differential e quation is the Lambert\ncurve, thereby the general solution to the initial value problems of the SIRv model, or the\ntwo-dimensional Volterra lattice with constant boundary values, is implicitly provided in\nterms of the Lambert W function. We devote to concluding remarks in$4.\n2 The Volterra lattice\nLetpa natural number. The following simultaneous system of inifinitely man y first-order\ndifferential equationsthatpertaintothenodesonaone-dimension al infinitelatticeiscalled\nthe Volterra lattice or the Lotka-Volterra system [1, 2, 3, 4, 5]\n˙ai=ai/parenleftBiggp/summationdisplay\nj=1ai+j−p/summationdisplay\nj=1ai−j/parenrightBigg\n(i∈Z), (1)\nwhereai=ai(t) is a differentiable function in tassigned to the i-th node, and “˙” denotes\nthe derivative with respect to t. Throughout this article, we assume p= 1, which achieves\nthe simplification of (1):\n˙ai=ai(ai+1−ai−1) (i∈Z). (2)\n22.1 Boundary conditions\nLet us consider the differentiable L-dimensional manifold\nV=RL(a1,a2,...,a L),\nwhere (a1,a2,...,a L) stands for the local, and hence, the global coordinates. The Volt erra\nlattices equipped with the following two boundary conditions are know n as completely\nintegrable Hamiltonian systems on the phase space V[4, 5, 6].\n(I) Periodic boundary ( ai=aL+i).\na1=aL+1a2 aL−1aL=a0\n(II) Open-end boundary ( a0=aL= 0).\na1 aL−1aL= 0 a0= 0\nThe exterior nodes a−1,a−2,...andaL+1,aL+2,...are not uniquely determined, even\nwhen the values of the inner nodes a0,a1,...,a Lare assigned. Actually, since\n0 = ˙a0=a0(a1−a−1) = 0(a1−a−1),\n0 = ˙aL=aL(aL+1−aL−1) = 0(aL+1−aL−1)\nholds, the nodes a−1andaL+1are arbitrary, thereby we inductively arrive at the\nclaim. Since a0=aL= 0 and the exterior nodes are arbitrary, the Volterra lattice\nwith the open-end boundary is a Hamiltonian system on the phase spa ceV=RL−1,\nmaking it a subsystem of the one with the periodic boundary (I) on V=RL.\nIn this article, we consider the third boundary condition with which th e Volterra lattice\n(2) is completely integrable Hamiltonian system on V=RMfor certain system sizes M:\n(III) Constant boundary ( a0=α,aM+1=β,α,β∈R\\{0}).\na1 aMaM+1=β a0=α\n3Contrary to (II), the exterior nodes a−1,a−2,...andaM+2,aM+3,...are uniquely\ndetermined as rational functions of the inner nodes a0,a1,...,a M+1via (2):\nai−1=ai+1−˙ai\nai(i <0),\nai+1=ai−1+˙ai\nai(i > M+1).\nIn particular, the nearest exterior nodes are reflective with resp ect to the boundary\nnodesa0andaM+1,\na−1=a1andaM+2=aM,\nsincea0/\\e}atio\\slash= 0,aM+1/\\e}atio\\slash= 0 and ˙ a0= ˙aM+1= 0. Hence, the system with the constant\nboundary is not a subsystem of the one with the periodic boundary ( I).\nNumerical examination implies that the Volterra lattice (2) with the co nstant bound-\nary (III) is, in general, not completely integrable. Nevertheless, t he Volterra lattice with\nthe constant boundary (III) exhibits the complete integrability if t he system size Mis\nsufficiently small, since it has a Poisson structure on V=RMfor anyM.\n2.2 Poisson structures\nBefore reviewing the Poisson structure, we introduce the two-fie ld form of the Volterra\nlattice. Let a2k=xkanda2k−1=yk. Then the Volterra lattice (2) reduces to the system\nof first-order ODEs called the two-field form\n˙xk=xk(yk+1−yk), (3)\n˙yk=yk(xk−xk−1) (4)\nfork∈Z.\nFirst remark the bi-Hamiltonian structure of the Volterra lattice (3 –4) on the 2 N-\ndimensional phase space\nV=R2N(x1,...,x N,y1,...,y N)\nwith the periodic boundary (I). We adopt the following convention: t he Poisson brackets\nare defined by writing down all non-vanishing brackets between the coordinate functions.\nWe say that two Poisson brackets on Vare compatible if their arbitrary linear combination\nis also a Poisson bracket on V.\nProposition 2.1 ([6]).Suppose the periodic boundary condition (I). The relations\n{xk,yk}2=xkyk,\n{xk,yk+1}2=−xkyk+1\n4define a quadratic Poisson bracket {·,·}2onV. The flow (3–4) is a Hamiltonian system on\n(V,{·,·}2) with the Hamiltonian function\nH1(x,y) :=N/summationdisplay\nk=1(xk+yk).\nAlso the relations\n{xk,yk}3=xkyk(xk+yk),\n{xk,yk+1}3=−xkyk+1(xk+yk+1),\n{xk,xk+1}3=−xkyk+1xk+1,\n{yk,yk+1}3=−ykxkyk+1\ndefine a cubic Poisson bracket {·,·}3onVcompatible with {·,·}2. The flow (3–4) is a\nHamiltonian system on ( V,{·,·}3) with the Hamiltonian function\nH0(x,y) :=1\n2N/summationdisplay\nk=1(logxk+logyk).\nThese Poisson structures are also valid for the open-end boundar y (II).\nLetF(V) be the set of smooth real-valued functions on the M-dimensional manifold\nV. ForF,G∈ F(V) and a Poisson bracket {·,·}onV, we have\n{F,G}=M/summationdisplay\ni,j=1Aij∂F\n∂ai∂G\n∂aj,\nwhereAij={ai,aj}forms the skew-symmetric M×Mmatrix as a coordinate represen-\ntation of the Poisson tensor.\nThe assertion in Proposition 2.1 can be confirmed as follows. For the c ubic Poisson\nbracket{,}3, we have\n{H0,xk}3=1\n2({logxk−1,xk}3+{logxk+1,xk}3+{logyk,xk}3+{logyk+1,xk}3)\n=1\n2/parenleftbigg\n−xk−1ykxk\nxk−1+xkyk+1xk+1\nxk+1−xkyk(xk+yk)\nyk+xkyk+1(xk+yk+1)\nyk+1/parenrightbigg\n=xk(yk+1−yk).\nFor the quadratic bracket {,}2, we also have\n{H1,xk}2={yk,xk}2+{yk+1,xk}2=xk(yk+1−yk).\nWe similarly obtain\n{H0,yk}3={H1,yk}2=yk(xk−xk−1).\n5Thus (3–4) defines two compatible Hamiltonian flows on V=R2N:\n˙xk={H1,xk}2={H0,xk}3,\n˙yk={H1,yk}2={H0,yk}3.\nMeanwhile, suppose the Volterra lattice (3–4) to have the constan t boundary (III). If\nwe consider the phase space\nV=R2N(x1,...,x N,y1,...,y N)\nof 2N-dimension then (3–4) reduces to\n˙y1=y1(x1−α), (5)\n˙xk=xk(yk+1−yk) for k= 1,2,...,N−1, (6)\n˙yk=yk(xk−xk−1) for k= 2,3,...,N, (7)\n˙xN=xN(β−yN). (8)\nWhereas, if the phase space is of (2 N−1)-dimension,\nV=R2N−1(x1,...,x N−1,y1,...,y N),\nthen\n˙y1=y1(x1−α), (9)\n˙xk=xk(yk+1−yk) for k= 1,2,...,N−1, (10)\n˙yk=yk(xk−xk−1) for k= 2,3,...,N−1, (11)\n˙yN=yN(β−xN−1). (12)\nLet us consider the following function H01(x,y) inxkandyk\nH01(x,y) :=N/summationdisplay\nk=1(xk+yk)−αN/summationdisplay\nk=1logxk−βN/summationdisplay\nk=1logyk. (13)\nIfM= 2Nthen the boundaries are x0=αandyN+1=β, and the derivative of H01with\nrespect to talways vanishes\n˙H01(x,y) =N/summationdisplay\nk=1(xkyk+1−xk−1yk)−αN/summationdisplay\nk=1(yk+1−yk)−βN/summationdisplay\nk=1(xk−xk−1)\n=βxN−αy1−α(β−y1)−β(xN−α) = 0.\nTherefore, H01is a conserved quantity of the system (5–8) for arbitrary α,β.\nIfM= 2N−1 then the boundaries are x0=αandxN=β, butH01is, in general, not\na conserved quantity; the derivative of H01with respect to tdoes not vanish:\n˙H01(x,y) =xN−1yN−αy1+yN(β−xN−1)−α(yN−y1)−β(β−α)\n=(yN−β)(β−α).\n6However, this computation suggests that if α=βthenH01is still a conserved quantity.\nMoreover, it implies that H01is divided into two conserved quantities G1andG2asH01=\nG1−αG2ifα=β, where\nG1(x,y) :=N/summationdisplay\nk=1(xk+yk)−αN/summationdisplay\nk=1logxk,\nG2(x,y) :=N/summationdisplay\nk=1logyk.\nNow, investigate the Poisson structure of the Volterra lattice (3– 4) equipped with the\nconstant boundary (III), or (5–8) and (9–12). We easily see tha t the Hamiltonians H0and\nH1in Proposition 2.1 are no longer conserved when the constant bound ary condition (I\nII) is imposed. However, we know that the function H01(see (13)) is still a conserved\nquantity of (5–8) for arbitrary α,β, and is of (9–12) when α=β. We then obtain the\nfollowing proposition concerning the Poisson structure of the Volte rra lattice with the\nconstant boundary (III).\nProposition 2.2. Suppose the constant boundary condition (III). If V=R2N, the flow\n(5–8) is the Hamiltonian system on the Poisson manifold ( V,{·,·}2) with the Hamiltonian\nfunction H01for anyα,β. Similarly, if V=R2N−1andα=β, the flow (9–12) is also the\nHamiltonian system on ( V,{·,·}2) withH01.\nProof.We have\n{H01,logxk}2=−xkyk\nxk+xkyk+1\nxk+βxkyk\nxkyk−βxkyk+1\nxkyk+1=yk+1−yk,\n{H01,logyk}2=xkyk\nyk−xk−1yk\nyk−αxkyk\nxkyk+αxk−1yk\nxk−1yk=xk−xk−1\nfor non-boundary nodes xkandyk. IfV=R2N, these are also valid for the boundary\nnodesy1andxNfor any boundary values α,β, since we have\n{H01,logy1}2=x1y1\ny1−αx1y1\nx1y1=x1−α,\n{H01,logxN}2=−xNyN\nxN+βxNyN\nxNyN=β−yN.\nWhereas, if V=R2N−1andα=β, at the boundary, we also have {H01,logy1}2=x1−α\nand\n{H01,logyN}2=−xN−1yN\nyN+αxN−1yN\nxN−1yN=α−xN−1=β−xN−1.\nHence the Hamiltonian flow\nd\ndtlogxk={H01,logxk}2=yk+1−yk,\nd\ndtlogyk={H01,logyk}2=xk−xk−1\n7is equivalent to the Volterra lattice (3–4) for both V=R2Nwith arbitrary α,βand\nV=R2N−1withα=β.\nRemark 1. In the limit as α,β→0, the constant boundary condition (III) reduces to the\nopen-end one (II), and H01consistently approaches H1, the Hamiltonian of the Volterra\nlattice with the open-end boundary (II) with respect to the P oisson bracket {,}2.\nFor the Poisson bracket {,}2on the 2N-dimensional phase space\nV=R2N(x1,...,x N,y1,...,y N),\nwe have the skew-symmetric 2 N×2NmatrixA= (Aij) of the Poisson tensor that possess\nthe following non-zero entries\nA2k−1,2k=−xkykand A2k,2k−1=xkykfork= 1,2,...,N,\nA2k,2k+1=−xkyk+1and A2k+1,2k=xkyk+1fork= 1,2,...,N−1.\nThe matrix Ais non-degenerate, thereby we also have a symplectic structure o nV.\nIn order to capture the symplectic structure, we introduce new c oordinate variables\nxk= logy1+logy2+···+logyk,\npk= logxk\nfork= 1,2,...,N. Then we have\n{pk,xk}2={logxk,logyk}2= 1\nfork= 1,2,...,Nand\n{pk,xk+1}2={logxk,logyk}2+{logxk,logyk+1}2= 0\nfork= 1,2,...,N−1. Hence x1,...,xN,p1,...,pNform canonically conjugate coordinates\non the symplectic manifold ( V=R2N,Ω) possessing the symplectic form\nΩ =N/summationdisplay\nk=1dpk∧dxk=N/summationdisplay\nk=1k/summationdisplay\nℓ=1dxk∧dyℓ\nxkyℓ. (14)\nThe symplectic manifold ( V,Ω) is called the canonical phase space of the Hamiltonian flow\n(5–8).\nIn these canonical coordinates, the hamiltonian H01is represented by\nH01(p,x) =N/summationdisplay\nk=1epk+N/summationdisplay\nk=1exk−xk−1−αN/summationdisplay\nk=1pk−βxN,\n8where we assume x0= 0. Hence we have\n∂H01\n∂pk=epk−α=xk−α,\n∂H01\n∂xk=exk−xk−1−exk+1−xk=yk−yk+1,\n˙xk=k/summationdisplay\nℓ=1˙yℓ\nyℓ=k/summationdisplay\nℓ=1(xℓ−xℓ−1) =xk−α,\n˙pk=˙xk\nxk=yk+1−yk.\nTherefore, the Volterra lattice (5–8) is canonically represented b y the Hamilton equations\n˙xk=∂H01\n∂pk, (15)\n˙pk=−∂H01\n∂xk(16)\nfork= 1,2,...,N.\nThus, we obtain:\nProposition 2.3. Suppose the constant boundary condition (III). The flow (5–8) is a\nHamiltonian system on the symplectic manifold ( V=R2N,Ω) with the Hamiltonian func-\ntionH01with respect to the symplectic form Ω given by (14).\nFor the Poisson bracket {,}2on the (2 N−1)-dimensional phase space\nV=R2N−1(x1,...,x N−1,y1,...,y N),\nthe skew-symmetric (2 N−1)×(2N−1) matrix of the Poisson tensor is degenerate, thereby\nit does not define a symplectic structure on V.\nIt should be remarked that the Hamiltonian H01and the functions Gi∈ F(V) (i= 1,2)\nare in involution with respect to the Poisson bracket {,}2onV=R2N−1,\n{H01,G1}2={H01,G2}2= 0.\nIndeed, we have H01=G1−αG2and\n{G1,G2}2=N/summationdisplay\nk=1{xk−αlogxk,logyk}2+N−1/summationdisplay\nk=1{xk−αlogxk,logyk+1}2\n=N/summationdisplay\nk=1(xk−α)−N−1/summationdisplay\nk=1(xk−α)\n=xN−α= 0,\nwhere we use the assumption xN=β=α. Thus, if α=β, there exist at least two func-\ntionally independent conserved quantities for the Hamiltonian flow (9 –12) on the Poisson\nmanifold ( V=R2N−1,{,}2).\n92.3 Complete integrability\nBy virtue of the above observation, we easily find two completely inte grable Hamiltonian\nsystems equipped with the constant boundary (III).\nThe first one is the case where M= 2NandN= 1. The Hamiltonian H01achieves\na sufficient number of conserved quantities since the phase space Vis of two-dimension.\nThe Hamiltonian flow (5–8) then reduces to\n˙x1=x1(β−y1), (17)\n˙y1=y1(x1−α), (18)\nand the Hamiltonian\nH01(x1,y1) =x1+y1−αlogx1−βlogy1\nto the conserved quantity.\nIfα >0 and as β→0, (17–18) is nothing but the SIR epidemic model [8], which is\nknown as an integrable dynamical system crucial for mathematical analysis on the spread\nof infectious diseases. Moreover, if α >0 andβ <0, (17–18) is an integrable extension\nof the SIR model under the influence of vaccination, called the SIRv model [7]. In the\nnext section, we will investigate (17–18) precisely, and reveal the relation with an exact\ndifferential equation via Abel’s equation of the first kind. We moreove r provide the general\nsolution to the initial value problem of (17–18) in terms of the Lamber t W function.\nAs mentioned earlier, the symplectic structure of (17–18) is explicit ly given as\nΩ =dp∧dx=dx1∧dy1\nx1y1, (19)\nwhere the canonically conjugate coordinates xandpare given by\nx= logy1andp= logx1.\nThe Hamiltonian in the canonically conjugate coordinates,\nH01(p,x) =ep+ex−αp−βx,\nsolves Hamilton’s canonical equations (15–16) of motion with N= 1.\nThe case where M= 2N−1 andN= 2 includes the second completely integrable sys-\ntem. In this three-dimensional system, we moreover assume α=β. Then the Hamiltonian\nflow (9–12) reduces to the system\n˙y1=y1(x1−α), (20)\n˙x1=x1(y2−y1), (21)\n˙y2=y2(α−x1), (22)\n10which possesses two conserved quantities:\nG1(x1,y1,y2) =y1+x1+y2−αlogx1,\nG2(x1,y1,y2) = logy1+logy2.\nHowever, having these conserved quantities is sufficient for (20–2 2) to exhibit complete\nintegrability, since the phase space Vis of three-dimension.\nWe can eliminate y2from (20–22) by employing the conserved quantity G2. LetG2be\na constant log ℓ(ℓ >0). Then we have y1y2=ℓ. By replacing y2withℓ/y1, (20–21) and\nG1respectively reduce to\n˙y1=y1(x1−α),\n˙x1=x1/parenleftbiggℓ\ny1−y1/parenrightbigg\nand\nG1(x1,y1) =y1+x1+ℓ\ny1−αlogx1.\nThis is a two-dimensional completely integrable system possessing th e conserved quantity\nG1, but it differs slightly from (17–18), or the SIRv model. In the limit as ℓ→0, it\napproaches the SIR model. However, its significance as an epidemic m odel has remained\nunclear.\n3 Abel’sequationofthefirstkind andtheSIRv model\nIn the previous section, we observed that the SIRv model is a comp letely integrable Hamil-\ntonian system on the symplectic manifold ( V=R2,Ω), where Ω is the symplectic form\n(19). Meanwhile, in this section, we investigate the SIRv model from another perspective\non integrability through a first-order nonlinear differential equatio n of degree three known\nas Abel’s equation of the first kind [10]. Abel’s equation of the first kind is a generaliza-\ntion of the Riccati equation, and similar to the Riccati equation, it ad mits the completely\nintegrability under certain conditions.\n3.1 Abel’s equation of the first kind\nLet us consider the following first-order nonlinear ODE of degree th ree pertaining to φ=\nφ(x)\ndφ\ndx=f0+f1φ+f2φ2+f3φ3, (23)\nwheref0,f1,f2andf3are meromorphic functions in x. The equation (23) is called Abel’s\nequation of the first kind if f3/\\e}atio\\slash≡0, whereas the Riccati equation if f3≡0 andf2/\\e}atio\\slash≡0.\nHereafter we assume f3/\\e}atio\\slash≡0, which makes (23) Abel’s equation of the first kind. We\nhave the following lemma [10].\n11Lemma 3.1. Iff0,f1,f2andf3satisfy\nf0≡0, (24)\nd\ndxlogf2\nf3=f1 (25)\nthen(23)reduces to an exact differential equation.\nProof.First introduce a new dependent variable ϕ=ϕ(x) such that\nφϕ= exp/parenleftbigg/integraldisplay\nf1dx/parenrightbigg\n=:F(x).\nIff0≡0, (23) reduces to the following ODE concerning ϕ:\nϕdϕ\ndx+f2Fϕ+f3F2= 0. (26)\nNext consider a function ̟inxandϕ:\n̟(x,ϕ) =e−(1+kϕ)−kG,\nwhere\nG(x) :=/integraldisplay\nf2Fdx=/integraldisplay\nf2exp/parenleftbigg/integraldisplay\nf1dx/parenrightbigg\ndx\nandkis the integration constant of (25):\nk=f2\nf3F.\nBy multiplying the integrating factor ̟, (26) reduces to the exact differential equation\ndΨ(x,ϕ) = 0 (27)\npossessing the potential\nΨ(x,ϕ) = (1+ kϕ)̟(x,ϕ).\nIndeed, we compute\ndΨ(x,ϕ) =−k2̟/bracketleftbig/parenleftbig\nf2Fϕ+f3F2/parenrightbig\ndx+ϕdϕ/bracketrightbig\n,\nwhich asserts the equivalence of (26) and (27).\n12The solution to the exact differential equation (27) is provided by us ing the Lambert\nW function W(z). Through the relationship w=W(z), this function parametrizes the\nfollowing non-algebraic curve on the ( z,w)-plane, known as the Lambert curve [11]:\n(wew−z= 0).\nThe Lambert W function W(z) is single-real-valued on [0 ,∞), while it is double-real-\nvalued on [ −e−1,0) and is not defined on ( −∞,−e−1) as a real function. We denote the\nbrach such that −1≤W(z)<0 byW0(z), and the one such that W(z)<−1 byW−1(z).\nHence the function\nW(z) =/braceleftBigg\nW0(z) (−e−1≤z <0),\nW(z) (z≥0)\nis real analytic on [ −e−1,∞). The Taylor series of W(z) about 0 is\n∞/summationdisplay\nn=0(−1)n+1nn−1\nn!zn\nwith the radius e−1of convergence.\nProposition 3.2. Let\nϕ(x) =−1\nk/bracketleftbig\nW/parenleftbig\n−Ψ0ekG/parenrightbig\n+1/bracketrightbig\n(28)\nusing the Lambert W function, where Ψ0is the initial value of the potential Ψ. Thenϕ\nsolves the initial value problem of the exact differential eq uation(27), which is equivalent\nto(26)with imposing (25).\nProof.First note that, given initial value x0ofx, the solution curve, or the equipotential\ncurve, of the exact differential equation (27) is given by\nΨ(x,ϕ) = (1+ kϕ)̟(x,ϕ) = Ψ(x0,ϕ(x0)) = Ψ0. (29)\nSince̟(x,ϕ) =e−(1+kϕ)−kG, we have\n−(1+kϕ)e−(1+kϕ)=−Ψ0ekG. (30)\nThe solution curve is the Lambert curve on the ( z,w)-plane by imposing\nz=−Ψ0ekGandw=−(1+kϕ).\nIt immediately follows\n−(1+kϕ) =W/parenleftbig\n−Ψ0ekG/parenrightbig\n(31)\nfrom (30). Thus the solution to the exact differential equation (27 ) is given by (28).\n13Corollary 3.3. Let\nφ(x) =F\nϕ=−kF\nW(−Ψ0ekG)+1. (32)\nThenφsolves the initial value problem of Abel’s equation (23)of the first kind with imposing\n(24)and(25).\nRemarkthatthereal-valuedLambertWfunction W(z)isdefinedonlyfor z∈[−e−1,∞)⊂\nR. Moreover, W(z) is double-valued and has the branch point only at z=−e−1. Given\ninitial value x0ofx, the initial value Ψ0of the potential Ψ( x,ϕ) at (x0,ϕ(x0)) is uniquely\ndetermined. Therefore, the solutions (28) and (32) potentially po ssess the branch point\nonly atxthat satisfies\nΨ0=e−kG(x)−1. (33)\nThe branches of W(z) in (28) and (32) are uniquely determined as follows.\n(1) If Ψ0>0 then the solution curve ϕ=ϕ(x) is restricted to the region\nS:={(x,ϕ)|kϕ >−1}\non the (x,ϕ)-plane, since ̟is always positive (see (29)). In this case, Wtakes negative\nvalue (see (31)), therefore, we have Ψ0≤e−kG−1, sinceW(z) is defined only on z≥ −e−1.\nThus Ψ0obeys\n0<Ψ0≤e−kG−1.\nMoreover, there exists exactly a branch point xwhen Ψ0>0 (see (33)), thereby ϕ(x)\nis double-valued on S. Let\nS−:={(x,ϕ)| −1< kϕ≤0} ⊂ S.\nSinceϕ(x) is given by (28), we choose the branch W0such that W0≥ −1 for (x,ϕ)∈ S−\nto give the solution (28). Whereas, we choose another branch W−1such that W−1<−1\nfor (x,ϕ)∈ S+to give (28), where we let\nS+:=S\\S−={(x,ϕ)|kϕ >0}.\n(2) If Ψ0<0 then the solution curve is contained in\nT:={(x,ϕ)|kϕ <−1}.\nFor such an initial value Ψ0of Ψ,Win (28) necessarily takes positive value. This is\nconsistent with the positivity of the independent variable −Ψ0ekG(x)ofW, because Wis\npositive single-valued on the positive real axis. Hence, for ( x,ϕ)∈ T, the solution (28) is\nuniquely given by the single-valued Wdefined on [0 ,∞).\n(3) Finally, if Ψ0= 0 then ( x,ϕ) is on the boundary between SandT. Hence,\nkϕ(x)≡ −1, thereby Ψ ≡0. This gives a constant solution to (26), which is a special\nsolution contained in (28).\nAbove discussion is summarized into the following:\n14Proposition 3.4. Assume(24)and(25)to be satisfied in order that Abel’s equation (26)\nof the first kind reduces to the exact differential equation (27). LetΨ0be the initial value\nof the potential Ψto(27).\n(1) IfΨ0>0, the Lambert W function Win the solution (28)to(26)is either\n(a) the branch W0for(x,ϕ)∈ S−or\n(b) the branch W−1for(x,ϕ)∈ S+,\nboth of which are defined on [−e−1,0).\n(2) IfΨ0≤0thenWin(28)is the single-valued W function defined on [0,∞).\n3.2 The SIRv model and its general solution\nNow, we relate Abel’s equation (23) of the first kind to the SIRv mode l (17—18), following\nthe approach of [9]. Introduce a new variable t=t(x) as an antiderivative of φ(x):\nt=/integraldisplay\nφ(x)dx.\nThen we find\ndx\ndt=1\ndt\ndx=1\nφand\nd2x\ndt2=d\ndt1\nφ=−1\nφ2dx\ndtdφ\ndx=−1\nφ3dφ\ndx.\nIt follows that we have\nφ=1\nx′anddφ\ndx=−x′′\n(x′)3,\nwhere we denote the derivative with respect to tby′for simplicity.\nThen Abel’s equation (23) of the first kind reduces to the second or der ODE\nx′′+f0(x′)3+f1(x′)2+f2x′+f3= 0.\nDividing by x, we obtain\n(logx)′′+f2(logx)′+f0\nx(x′)3+xf1+1\nx2(x′)2+f3\nx= 0. (34)\nLet us consider the following system of first-order ODEs\nx′=−xy+ax+b, (35)\ny′=xy+cy+d, (36)\n15wherea,b,c,d∈R. From the first equation (35), we have\ny=a+b\nx−(logx)′.\nBy substituting this into the second equation (36), we get\n(logx)′′−c(logx)′−x(logx)′+b\nx2+a(x+c)+b\nx(x+c)+d= 0.\nComparing this with (34), we see that, if\nf0= 0, f1=−1\nx, f2=−x−c,and\nf3=a(x+c)x+b/parenleftbigg\nx+c+1\nx/parenrightbigg\n+dx,\nthesystem (35–36) ofODEsreduces to Abel’sequation(23)ofthe first kind, which satisfies\n(24). Moreover, if b=d= 0, the condition (25) is also satisfied:\nd\ndxlogf2\nf3=d\ndxlog1\n−ax=−1\nx=f1,\nthereby (35–36) reduces to the exact differential equation (27) .\nThus, weobtainthefollowingpropositionthatclaimsarelationbetwee nAbel’sequation\nof the first kind and the SIRv model, thereby the integrability of the SIRv model via the\nexact differential equation.\nProposition 3.5. Let\nf0= 0, f1=−1\nx, f2=γ−x,andf3=ν(γ−x)x, (37)\nwhereγandνare real numbers, in particular, ν/\\e}atio\\slash= 0, since we assume f3/\\e}atio\\slash≡0. Then,(23)\nreduces to\ndφ\ndx=−1\nxφ+(γ−x)φ2+ν(γ−x)xφ3. (38)\nThe Abel equation (38)of the first kind is equivalent to\nx′=−xy−νx, (39)\ny′=xy−γy, (40)\nwhich is (35–36) with imposing\na=−ν, b= 0, c=−γ,andd= 0.\nMoreover, (39–40) attributes to the exact differential equa tion(27).\n16Suppose xandyrespectively stand for the populations of the susceptible and the\ninfected of an infectious disease. Also let γandνrespectively represent the transmission\nrate and the vaccination rate. Then the system (39–40) of ODEs g overns the spread of the\ninfectious diseases under the influence of vaccination, and is called t he SIRv model [7, 12].\nThus, when we call (39–40) the SIRv model, we assume γandνto be positive, since they\nrespectively stand for the rates of transmission and vaccination.\nThe discussion in the previous section leads to the following propositio n concerning the\nintegrability of the SIRv model on the Poisson mmanifold.\nProposition 3.6. Let\nx0=α=γ, x 1=x, x 2=β=−ν,andy1=y.\nThen the SIRv model (39–40) is the completely integrable Vol terra lattice (17–18) with the\nconstant boundary (III) on the Poisson manifold (V=R2,{,}2)possessing the Hamilto-\nnian\nH01(x,y) =x+y−γlogx+νlogy. (41)\nThe SIRv model (39–40) is also the Hamiltonian flow arising fr om the symplectic structure\nΩ =dx∧dy\nxy\non the symplectic manifold (V,Ω).\nRemark 2. Suppose ν= 0against our assumption f3=ν(γ−x)x/\\e}atio\\slash≡0. Then (39–\n40) reduces to the original SIR mode. Unfortunately, the dis cussion above, based on Abel’s\nequation of the first kind, is not valid for the SIR model. The S IR model can be investigated\nusing the Riccati equation instead of the Abel equation [12] .\nThe SIRv model attributes to Abel’s equation (38) of the first kind e mploying the\ncoefficients (37), which reduces to the exact differential equation (27). Hence the SIRv\nmodel (39–40) and the Volterra lattice (17–18) are exactly solved as follows.\nWe have\nF= exp/integraldisplay\nf1dx=CF\nx,\nG=/integraldisplay\nf2Fdx=CF(γlogx−x)+CG,\nwhereCFandCGare integration constants. Since eCGacts on̟, or Ψ, as a constant\nmultiplication, it does not affects the exact differential equation dΨ = 0. Hence, we can\nassumeCG= 0 without loss of generality. Moreover, since we define the consta ntkto be\nthe integration constant\nk=f2\nf3F=1\nνCF,\n17we may assume CF= 1 by putting k= 1/ν. Thus we obtain\nF=1\nx, G=γlogx−x,andk=1\nν.\nThen the equation (26) concerning ϕreduces to\nϕdϕ\ndx+γ−x\nxϕ+νγ−x\nx= 0. (42)\nThis attributes to the exact differential equation\ndΨ(x,ϕ) =d/bracketleftbigg/parenleftbigg\n1+1\nνϕ/parenrightbigg\n̟(x,ϕ)/bracketrightbigg\n= 0\nwith employing the integrating factor given by\n̟(x,ϕ) = exp/bracketleftbigg\n−/parenleftbigg\n1+1\nνϕ/parenrightbigg\n−1\nν(γlogx−x)/bracketrightbigg\n=x−γ\nνex\nν−ϕ\nν−1.\nThe potential is\nΨ(x,ϕ) =/parenleftbigg\n1+1\nνϕ/parenrightbigg\n̟(x,ϕ) =−1\nνexp/parenleftbiggH01\nν/parenrightbigg\n, (43)\nwhereH01is the Hamiltonian (41). Thus the potential Ψ is, of course, conserv ed through\nthe evolution. In addition, the exact differential equation dΨ = 0 is equivalent to Hamil-\nton’s canonical equation of motion,\ndH01(p,x) =∂H01\n∂pdp+∂H01\n∂xdx= 0,\non the symplectic manifold ( V=R2,Ω).\nRemark that, for any initial values x0,y0∈R, the initial value Ψ0of the potential is\nnegative when ν >0. Hence, if ν >0, by virtue of Proposition 3.4, the solution (28) to\n(42) is given by the single-valued Lambert W function Won [0,∞). Explicitly, we have\nϕ(x) =−ν/bracketleftbigg\nW/parenleftbigg1\nνxγ\nνe1\nν(H0\n01−x)/parenrightbigg\n+1/bracketrightbigg\n,\nwhereH0\n01:=H01(x0,y0). Similarly, the solution (32) to Abel’s equation (38) of the first\nkind is also provided by the single-valued W:\nφ(x) =−1\nνx/bracketleftbigg\nW/parenleftbigg1\nνxγ\nνe1\nν(H0\n01−x)/parenrightbigg\n+1/bracketrightbigg.\n18Since the independent variable tis the antiderivative of φ, the solution to the SIRv\nmodel (39–40) is implicitly given by\nt=−/integraldisplaydx\nνx/bracketleftbigg\nW/parenleftbigg1\nνxγ\nνe1\nν(H0\n01−x)/parenrightbigg\n+1/bracketrightbigg. (44)\nMoreover, since we have\ny=−ν−x′\nx\nform (39), we obtain\ny=νW/parenleftbigg1\nνxγ\nνe1\nν(H0\n01−x)/parenrightbigg\n, (45)\nifxsatisfies (44).\nWe summarize the discussion above into the following:\nTheorem 3.7. TheSIRvmodel(39–40) reducesto the exactdifferentialequa tiondΨ(x,ϕ) =\n0for the potential (43)by introducing ϕ= (logx)′. The solution to the initial value prob-\nlem of the SIRv model is implicitly provided by (44)and(45), whereH0\n01is the initial value\nof the conserved quantity. Remark that Wis the single-valued Lambert W function defined\non[0,∞), since we assume ν >0.\nFor the Volterra lattice (17–18), we have\nk=−1\nβ, ϕ=˙x1\nx1=β−y1,and Ψ =1\nβexp/parenleftbigg\n−H01\nβ/parenrightbigg\n.\nRemark that we assume β/\\e}atio\\slash= 0. If and only if β <0, Ψ0<0. Hence, according to\nProposition 3.4, ( x1,ϕ(x1)) = (x1,β−y1) is contained in T, thereby ( x1,y1) is in\n/tildewideT:={(x1,y1)|y1> β}.\nThe solution ( x1,y1) to the Volterra lattice (17–18) is given by the single-valued Lamber t\nW function.\nWhereas, ifandonlyif β >0,Ψ0>0. Hence, accordingtoProposition3.4,( x1,ϕ(x1)) =\n(x1,β−y1) is contained in S=S−∪S+, thereby ( x1,y1) is in\n/tildewideS=/tildewideS−∪/tildewideS+,\n/tildewideS−:={(x1,y1)|0< y1≤β},\n/tildewideS+:={(x1,y1)|y1> β}.\nThe solution ( x1,y1) to the Volterra lattice (17–18) is given by the branch W0of the\nLambert W function if ( x1,y1)∈/tildewideS−, while by the branch W−1if (x1,y1)∈/tildewideS+.\nThus, we obtain:\n19Theorem 3.8. The general solution to the initial value problem of the Volt erra lattice\n(17–18) is implicitly provided by\nt=/integraldisplaydx1\nβx1/bracketleftbigg\nW/parenleftbigg\n−1\nβx−α\nβ\n1e1\nβ(x1−H0\n01)/parenrightbigg\n+1/bracketrightbigg,\ny1=−βW/parenleftbigg\n−1\nβx−α\nβ\n1e1\nβ(x1−H0\n01)/parenrightbigg\n,\nwhereH0\n01is the initial value of the Hamiltonian H01. Ifβ <0, the function Wis the\nsingle-valued Lambert W function, whereas, if β >0,Wis the branch W0of the Lambert\nW function when (x1,y1)∈/tildewideS−, and is the branch W−1when(x1,y1)∈/tildewideS+.\n4 Concluding remarks\nWe investigate the complete integrability of the Volterra lattice with im posing the con-\nstant boundary condition (III) in terms of the Poisson structure {,}2and the symplectic\nstructure Ω on the phase space V=RM. We then find that the Volterra lattice has the\nsymplectic structure Ω if M= 2Nfor any boundary values α,β, and hence it achieves the\ncomplete integrability if M= 2. Such a Volterra lattice is nothing but the SIRv model, an\nintegrable extension of theSIR epidemic model under theinfluence o f vaccination. Wheres,\nifM= 2N−1 andα=β, the Voltera lattice has the Poisson structure {,}2, and hence it\nalso admits the complete integrability if M= 3 and α=β. While such a Volterra lattice\ncan also be seen as an integrable extension of the SIR model, its signifi cance as an epidemic\nmodel has not yet been revealed, as far as the author knows.\nMeanwhile, upon the introduction of an appropriate variable transf ormation, the SIRv\nmodel is transformed into Abel’s equation of the first kind, which att ributes to an exact\ndifferential equation. The potential of the exact differential equa tion is, of course, the\nconserved quantity of the SIRv model, i.e., the Hamiltonian H01of the Volterra lattice on\nthe symplectic manifold ( R2,Ω). Thus, the exact differential equation is equivalent to the\nHamiltonian flow on ( R2,Ω) with the Hamiltonian H01. In addition, the invariant curve of\nthe SIRv model, or the equipotential curve of the exact differentia l equation, is provided\nby the Lambert curve. Thus, we implicitly obtain the general solution to the initial value\nproblem of the SIRv model, or the Volterra lattice on ( R2,Ω), in terms of the Lambert W\nfunction.\nThe integrable discretization of completely integrable systems has b een extensively\nstudied for several decades. Regarding SIR epidemic models, ther e has been enthusiastic\ninvestigation into integrable discretization, resulting in the discover y of several discrete\nmodels that possess complete integrability (see [13, 14, 15, 16]). I n particular, an inte-\ngrable discretization of the SIRv model, which preserves the same c onserved quantity as\nthe continuous model, has been achieved through a geometric cons truction utilizing its in-\nvariant curve. Although the process of geometric discretization w as omitted in this article\n20due to space limitations, it has been thoroughly documented in [12]. I nterested readers\nare encouraged to refer to this article for further insight into the geometric discretization\nof the SIRv model.\nReferences\n[1] Kac, M. and van Moerbeke, P., Some probabilistic aspects of scat tering theory, in\nArthurs, A.M. (ed.), Functional integration and its applications , Clarendon Press,\n(1975) 87–96.\n[2] Kac,M.andvanMoerbeke, P., Onanexplicitly solublesystem ofnon linear differential\nequations related to certain Toda lattices, Adv. Math. ,16(1975) 160–169.\n[3] Moser, J, Finitely many mass points on the line under the influence o f an exponential\npotential–an integrable system, in Moser, J. (ed.), Dynamical systems, theory and\napplications , Lecture Notes in Physics, 38, Springer–Verlag, (1975) 467–497.\n[4] Bogoyavlenskij, O.I., Algebraic constructions of integrable dyna mical systems–\nextensions of the Volterra lattice, Russian Math. Surveys ,46(1991), 1–64.\n[5] Bogoyavlenskij, O.I., Integrable Lotka-Volterra Systems, Regul. Chaotic Dyn. ,13\n(2008), 543–556.\n[6] Suris, Y.B., The Problem of Integrable Discretization: Hamiltonian App roach,\nProgress in Mathematics 219, Birkh¨ auser, 2003.\n[7] Hethcote, H.W., Asymptotic behavior and stability in epidemic models ,Mathematical\nProblems in Biology , Lecture Notes in Biomathematics 2, Springer–Verlag, (1974)\n83–92.\n[8] Kermack, W. and McKendrick, A., A contribution to the mathemat ical theory of\nepidemics, Proc. Roy. Soc. Lond A ,115(1927) 700–721.\n[9] Harko, T., Lobo, F. and Mak, M., Exact analytical solutions of the Susceptible-\nInfected-Recovered (SIR) epidemic model and of the SIR model w ith equal death\nand birth rate, Appl. Math. Comput. ,236(2014) 184–194.\n[10] Mancas, S. and Rosu, H., Integrable Abel equations and Vein’s e quation, Math. Meth.\nAppl. Sci. ,39(2016) 1376–1387.\n[11] Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffery, D.J. and Knuth, D.E., On the\nLambert W functions, Adv. Compt. Math. ,5(1996) 329–360.\n[12] Nobe, A., Exact solutions to SIR epidemic models via integrable disc retization, arXiv:\n2303.17198 (2023).\n21[13] Willox, R., Grammaticos, B., Carstea, A.S. and Ramani, A., Epidemic d ynamics:\ndiscrete-time and cellular automaton models Physica A ,328(2003) 13–22.\n[14] Satsuma, J., Willox, R., Ramani, A., Grammaticos, B. and Carstea, A.S., Extending\nthe SIR epidemic model, Physica A ,336(2004) 369–375.\n[15] Sekiguchi, M. Ishiwata, E.andNakata,Y., Dynamicsofanultra- discreteSIRepidemic\nmodel with time delay, Math. Biosci. Eng. ,15(2018) 653–666.\n[16] Tanaka, Y. and Maruno, K., Integrable discretizations of the S IR model,\narXiv:2209.08549 (2022).\n22"    },    {        "title": "2402.11899v1.A_study_of_the_radio_spectrum_of_Mrk_421.pdf",        "content": "arXiv:2402.11899v1  [astro-ph.HE]  19 Feb 2024Draft version February 20, 2024\nTypeset using L ATEX default style in AASTeX62\nA study of the radio spectrum of Mrk 421\nJee Won Lee ,1Sang-Sung Lee ,1,2Jeffrey Hodgson ,3Algaba Juan-Carlos ,4Sang-Hyun Kim ,1,2\nWhee Yeon Cheong ,1,2Hyeon-Woo Jeong ,1,2andSincheol Kang1\n1Korea Astronomy and Space Science Institute, 776 Daedeok-d aero, Yuseong-gu, Daejeon 34055, Republic of Korea\n2University of Science and Technology, 217 Gajeong-ro, Yuse ong-gu, Daejeon 34113, Korea\n3Department of Physics and Astronomy, Sejong University, 20 9 Neungdong-ro, Gwangjin-gu, Seoul, South Korea\n4Department of Physics, Faculty of Science, Universiti Mala ya, 50603 Kuala Lumpur, Malaysia\nSubmitted to ApJ\nABSTRACT\nWe present the results of a spectral analysis using simultan eous multifrequency (22, 43, 86, and\n129 GHz) very long baseline interferometry (VLBI) observat ions of the Korean VLBI Network (KVN)\non BL Lac object, Markarian 421 (Mrk 421). The data we used was obtained from January 2013 to\nJune 2018. The light curves showed several flux enhancements with global decreases. To separate the\nvariable and quiescent components in the multifrequency li ght curves for milliarcsecond-scale emission\nregions, we assumed that the quiescent radiation comes from the emission regions radiating constant\noptically-thin synchrotron emissions (i.e., a minimum flux density with an optically thin spectral\nindex). The quiescent spectrum determined from the multifr equency light curves was subtracted from\nthe total CLEAN flux density, yielding a variable component i n the flux that produces the time-\ndependent spectrum. We found that the observed spectra were flat at 22-43 GHz, and relatively steep\nat 43-86 GHz, whereas the quiescent-corrected spectra are s ometimes quite different from the observed\nspectra (e.g., sometimes inverted at 22-43 GHz ). The quiesc ent-corrected spectral indices were much\nmore variable than the observed spectral indices. This spec tral investigation implies that the quiescent-\nspectrum correction can significantly affect the multifrequ ency spectral index of variable compact radio\nsources such as blazars. Therefore, the synchrotron self-a bsorption B-field strength (BSSA)can be\nsignificantly affected because BSSAis proportional to the fifth power of turnover frequency.\nKeywords: BL Lacertae objects: individual (Mrk 421)—galaxies:active — quasars: relativistic jets—\nradio continuum: galaxies\n1.INTRODUCTION\nActive Galactic Nuclei (AGN) are the central regions of gala xies that exhibit unusually bright luminosity across\nmultiwavelengths, from radio to gamma rays. It is thought th at the mechanism responsible for this luminosity is a\nsupermassive black hole ( 106−1010M⊙)at the center of the AGN, which accretes enormous amounts of m atter from\nits surrounding environment. A relativistic jet is formed b y the accreted matter ( Ulrich et al. 1997 ).\nA blazar is a subclass of AGN that exhibits extremely energet ic and highly variable observational behavior. The\nsource is among the most luminous and powerful objects, and f eatures a relativistic jet that points directly toward\nus with a very small viewing angle. This alignment causes a bl azar to appear as a point-like source in the sky that\nexhibits rapid and dramatic variability in its brightness ( Urry & Padovani 1995 ). Blazars encompass both BL Lac\nobjects and flat-spectrum radio quasars (FSRQs). While FSRQ s and BL Lac objects share similar characteristics,\nsuch as being highly variable and bright, their spectra diffe r. The spectrum of a BL Lac object has weak or no broad\nemission, unlike that of an FSRQ.\nCorresponding author: Sang-Sung Lee\nsslee@kasi.re.kr2 Lee et al.\nMarkarian 421 (Mrk 421), is one of the most extensively studi ed high-synchrotron-peaked blazars and one of the clos-\nest sources, located at a redshift z=0.031. Its high luminos ity enables us to investigate its emission ( Katarzyński et al.\n2003;Charlot et al. 2006 ;Horan et al. 2009 ;Acciari et al. 2011 ;Aleksić et al. 2015 ;Arbet-Engels et al. 2021 ). Its\nsource Mrk 421 was the first extragalactic source detected in the TeV energy bands, as reported by Punch et al. (1992)\nusing the Whipple 10-m Cherenkov telescope. Mrk 421 is class ified as a BL Lac object.\nAcciari et al. (2011) reported no correlation between TeV γ-ray and optical/radio light curves obtained from obser-\nvations performed from December 2005 to May 2008 with VERITA S/Whipple gamma-ray data and Metsahovi and\nUMRAO radio data. On the other hand, when focusing on the very high energy (VHE) E > 100 GeV, they found\na strong correlation between TeV gamma-ray and X-rays and su ggested that this can be explained by the one-zone\nsynchrotron self-Compton model ( Zhu et al. 2016 ). In a multiwavelength study conducted on Mrk 421 over 5.5-y r\nperiod, from December 2012 to April 2018, Arbet-Engels et al. (2021) performed several cross-correlation analyses.\nThey found that the TeV and X-ray light curves were very well c orrelated, with a lag of less than 0.6 days. The GeV\nand 15 GHz radio light curves also showed correlation, with a lag of over 30-100 days. However, no correlation was\nobserved between the TeV and the radio light curves. They sug gested that the variability at high energy may be due\nto the leptonic mechanism, given the short time scale of vari ability.\nAlthough extensive multiwavelength studies on Mrk 421 have been extensively conducted to date, research on its\nradio spectrum has been limited. In particular, simultaneo us multifrequency very long baseline interferometry (VLBI )\nimaging observations with high time resolution have not bee n performed yet. In this study, we present various analyses\nof the radio spectrum of Mrk 421 to explain the correlation be tween the spectrum and the flux density variations.\nIn this paper, we report results from simultaneous multifre quency VLBI observations of Mrk 421 carried out using\nthe Korea VLBI Network (KVN) at 22, 43, and 86, from January 20 13 to June 2018. The paper is organized as follows.\nIn Section 2, we present descriptions of the observations for both the KV N and all archival data used in this paper.\nSection 3and4describe the results and various analyses of flux density var iability at multiple frequencies, respectively.\nFinally, Section 5includes a discussion on the results and analysis.\n2.OBSERVATIONS AND DATA CALIBRATION\n2.1.KVN observations\nWe conducted simultaneous multifrequency VLBI imaging mon itoring observations of Mrk 421 using the Korean\nVLBI Network (KVN) at 22, 43, 86, and 129 GHz as part of the iMOG ABA(Interferometric Monitoring of Gamma-ray\nBright active galactic Nuclei) project ( Lee et al. 2016 ,2017b ;Algaba et al. 2018 ;Lee et al. 2020 ;Kang et al. 2021 ).\nThe KVN is a VLBI network system consisting of three 21-meter radio telescopes located in Seoul (KVN Yonsei),\nUlsan (KVN Ulsan), and Jeju (KVN Tamna), Korea. The monitori ng observations were primarily carried out every\nmonth. However, we excluded the 129 GHz data from the analysi s because there were no valid measurements, higher\nthan 3-σof errors. The observation period was from January 2013 to Ju ne 2018, except for maintenance periods from\nJune to August. Measurements were made in 41 epochs over a per iod of 5.5 years. At 22, 43, and 86 GHz, 40 epochs,\n33 epochs, and 21 epochs were measured, respectively. The ob serving frequencies were 21.700-21.764, 43.400-43.464,\n86.800-86.864, and 129.300-129.364 GHz, with a total bandw idth of 256 MHz. The angular resolutions of the KVN\nwere 6, 3, and 1.5 mas at 22, 43, and 86 GHz, respectively. We us ed a left circular polarization observation mode at a\nrecording rate of 1 Gbps. For more details of the KVN observat ions, see Lee et al. (2016,2017a).\n2.2.Data calibration\nThe observed data were processed using the DiFX software cor relator in Daejeon, Korea ( Deller et al. 2007 ;Lee et al.\n2015). The DiFX correlator generated a cross-correlation funct ion spectrum with a resolution of 0.125 MHz and an\naccumulation period of one second. After the correlation, t he data was calibrated with the AIPS (Astronomy Image\nProcessing System) software package, provided by the Natio nal Radio Astronomy Observatory (NRAO), following\nstandard procedures, including phase and amplitude calibr ation, fringe fitting, and bandpass calibration. We used the\nKVN pipeline developed by Hodgson et al. (2016) to perform the data calibration. Sensitivity at high frequ encies was\nimproved by transferring phase solutions from lower to high er frequencies (i.e., FPT) as described by Rioja & Dodson\n(2011) and Algaba et al. (2015). Amplitude calibration was conducted using the system tem peratures measured\nduring the observations. The system temperatures were adju sted for atmospheric opacity based on sky-tipping curve\nmeasurements conducted every hour at each radio telescope. Renormalization of the fringe amplitudes was conducted to\ncorrect the amplitude distortion due to quantization, and l osses from quantization and re-quantization were correcte diMOGABA: Mrk 421 3\nFigure 1. CLEAN maps of Mrk 421 obtained in Epoch 14 (April 23, 2014). Th e left, middle, and right panels indicate source\nstructures at 22, 43, and 86 GHz, respectively. The source sh ows a compact core-dominated structure at all frequencies. The\ncontour level starts from 3 times the noise and increases by a factor of√\n2. The beam size is indicated by the filled ellipse in\nthe bottom left corner.\n(Lee et al. 2015 ). After the amplitude calibration and the re-quantization loss calibration, the uncertainty of the\namplitude calibration was within 5 % at 22 and 43 GHz and 10-30 % at 86 and 129 GHz.\n2.3.Imaging and modelfitting\nAfter the phase and amplitude calibration, we used DIFMAP to make a CLEAN contour map. To make a CLEAN\nmap with the phase self-calibration, for the first time we per formed the startmod command. This uses a point source\nmodel of 1 Jy to conduct the initial phase self-calibration. Then, the visibility data at 22, 43, and 86 GHz were averaged\nat 30-second intervals. This averaging time aligned with th e typical coherence timescales of KVN observations. We\nflagged bad amplitude and phase data points and outliers in th e vplot. Then, the repetitive CLEANing and phase\nself-calibration within the central emitting regions were performed until the noise root-mean-square (rms) level was\nno longer significantly lowered. The standard CLEANing and s elf-calibration within the central emitting regions were\nconducted until no significant flux density was added compare d to the image rms level.\nSince Mrk 421 is a compact source observed on mas-scales in th e KVN observations, finding the best model is\nstraightforward. Figure 1displays the CLEAN contour maps of Mrk 421 at 22, 43, and 86 GHz for a representative\nepoch (well observed at all frequencies) showing compact co re-dominated components. After making the CLEAN\ncontour map, we computed the image quality factor ξr. This is the ratio of the image rms noise and its mathematical\nexpectation, ξr=Sr/Sr,exp, whereSris the maximum absolute flux density in the residual image, an dSr,expis the\nexpectation of Sr. For a more comprehensive understanding of this image quali ty evaluation scheme, please refer4 Lee et al.\n56250 56500 56750 57000 57250 57500 57750 58000 58250\nMJD0.10.20.30.40.50.60.70.8Clean Flux [Jy]22 GHz (KVN)\n43 GHz (KVN)\n86 GHz (KVN)\nFigure 2. Light curves of Mrk 421. Yellow, green, and sky blue dots indi cate 22, 43, and 86GHz, respectively.\ntoLobanov et al. (2006) and Lee et al. (2016,2017a ). Theξrvalues derived in this paper range from 0.65 - 0.97,\nsuggesting that the images sufficiently represent the struct ure identified in the visibility data.\nTo establish the model of the source, we fit the model in the uv-data. In this process, we used a single circular\ntwo-dimensional Gaussian model to obtain the model paramet ers, the information of the model, such as flux, size, and\ndegree of the major axis. The model fitting was repeated until reduced χ2approached one, and model fitting stops\nwhen the reduced χ2does not decrease anymore.\nTable 1lists the fitted parameters, including the observing freque ncies, restoring beam size Bmaj,min, position angle\nBPA, total CLEAN flux density SKVN, peak flux density Spin units of Jy per beam, rms in the image σ, the dynamic\nrange (ratio of peak to rms) of the image D, and image quality f actorξr.\n3.RESULTS\n3.1.Multifrequency light curves\nWe present the results of the multifrequency radio observat ions of Mrk 421. Figure 2displays the multifrequency\nlight curves of Mrk 421 at 22, 43, and 86 GHz observed between J anuary 16, 2013 (MJD 56308) and June 1, 2018\n(MJD 58270). During the 5.5-year observation period, the flu x density gradually decreases and several small flux\nenhancements are shown (which we call flares). The light curv es show a similar trend at all frequencies. Although\nwe can’t show the 15 GHz light curve of the OVRO in this study, t he 15 GHz light curve exhibits flux enhancements\npeaking at approximately MJD 56400, 56900, and 57200 (see Fi gure 1 in Arbet-Engels et al. 2021 ). In the KVN light\ncurves, only the flare at MJD 56900 is seen due to the relativel y sparse data points. Because of that, the flare at MJD\n56900 looks like a large flare. But the flare is very sharp indee d.iMOGABA: Mrk 421 5\nTable 1 . Image parameters\nEpoch MJD Band BmajBminBPASKVN Sp σ D ξ r\n(mas) (mas) (◦) (Jy) (Jy/beam) (mJy/beam)\n(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)\n2013 Jan 16 56308 K 0.300 0.189 −89.9 0.530 0.523 3 193 0.73\n2013 Jan 16 56308 Q 0.148 0.094 −85.2 0.508 0.496 6 89 0.72\n2013 Jan 16 56308 W 0.077 0.045 −72.6 0.278 0.281 11 26 0.75\n2013 Apr 11 56393 K 0.299 0.182 −82.7 0.435 0.435 4 103 0.79\n2013 Apr 11 56393 Q 0.147 0.096 −78.7 0.404 0.401 7 55 0.72\n2013 Apr 11 56393 W 0.076 0.043 −83.1 0.230 0.218 16 13 0.70\n2013 May 07 56419 K 0.300 0.180 −77.9 0.471 0.470 5 87 0.75\n2013 May 07 56419 Q 0.149 0.093 −77.2 0.408 0.404 11 35 0.68\n2013 Oct 15 56580 K 0.303 0.176 −77.2 0.507 0.505 5 95 0.82\n2013 Oct 15 56580 Q 0.151 0.089 −77.5 0.465 0.436 10 42 0.76\n2013 Oct 15 56580 W 0.077 0.043 −85.8 0.306 0.217 26 8 0.78\n2013 Nov 19 56615 K 0.304 0.179 −81.0 0.447 0.446 3 144 0.72\n2013 Nov 19 56615 Q 0.153 0.086 −84.2 0.390 0.369 5 79 0.80\n2013 Nov 19 56615 W 0.078 0.043 87.3 0.245 0.208 8 26 0.73\n2013 Dec 24 56650 K 0.283 0.170 79.6 0.476 0.472 6 84 0.74\n2013 Dec 24 56650 Q 0.146 0.086 79.5 0.471 0.446 7 65 0.83\n2013 Dec 24 56650 W 0.070 0.042 82.9 0.301 0.285 9 33 0.82\n2014 Jan 27 56684 K 0.298 0.187 −83.5 0.399 0.398 4 113 0.74\n2014 Jan 27 56684 Q 0.148 0.098 −79.6 0.378 0.368 5 75 0.71\n2014 Feb 28 56716 K 0.376 0.164 −63.9 0.394 0.393 5 76 0.72\n2014 Feb 28 56716 Q 0.176 0.086 −63.6 0.406 0.400 9 47 0.81\n2014 Feb 28 56716 W 0.113 0.037 −62.1 0.198 0.216 11 20 0.74\n2014 Mar 22 56738 K 0.309 0.172 −85.6 0.379 0.379 4 108 0.73\n2014 Mar 22 56738 Q 0.150 0.091 −85.7 0.353 0.353 6 61 0.83\n2014 Mar 22 56738 W 0.082 0.041 89.8 0.268 0.253 10 26 0.80\n2014 Apr 22 56769 K 0.304 0.175 −79.9 0.407 0.407 2 171 0.83\n2014 Apr 22 56769 Q 0.148 0.091 −78.4 0.389 0.384 3 131 0.79\n2014 Apr 22 56769 W 0.078 0.042 −79.7 0.302 0.286 8 37 0.85\n2014 Sep 01 56901 K 0.300 0.180 −80.1 0.723 0.720 7 99 0.83\n2014 Sep 01 56901 Q 0.142 0.095 −82.1 0.615 0.616 6 110 0.70\n2014 Sep 01 56901 W 0.078 0.043 −76.9 0.510 0.481 27 18 0.92\n2014 Oct 29 56959 K 0.330 0.160 86.2 0.430 0.428 5 79 0.74\n2014 Oct 29 56959 Q 0.174 0.075 −84.3 0.415 0.404 7 56 0.79\n2014 Nov 28 56989 K 0.334 0.164 −76.9 0.496 0.492 8 64 0.83\n2014 Nov 28 56989 Q 0.169 0.084 −70.2 0.509 0.508 9 57 0.70\n2014 Dec 25 57016 K 0.324 0.162 −84.6 0.464 0.462 6 78 0.87\n2014 Dec 25 57016 Q 0.148 0.090 −82.8 0.376 0.384 10 37 0.89\n2015 Feb 23 57076 K 0.312 0.170 −80.3 0.427 0.425 4 109 0.93\nTable 1 continued on next page6 Lee et al.\nTable 1 (continued)\nEpoch MJD Band BmajBminBPASKVN Sp σ D ξ r\n(mas) (mas) (◦) (Jy) (Jy/beam) (mJy/beam)\n(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)\n2015 Feb 23 57076 Q 0.149 0.094 −78.8 0.368 0.365 3 115 0.77\n2015 Feb 23 57076 W 0.078 0.042 −77.7 0.233 0.220 7 34 0.84\n2015 Mar 26 57107 K 0.312 0.168 −84.2 0.398 0.396 4 111 0.98\n2015 Mar 26 57107 Q 0.147 0.091 −81.9 0.361 0.353 2 144 0.80\n2015 Mar 26 57107 W 0.079 0.041 −83.6 0.210 0.191 6 34 0.78\n2015 Apr 30 57142 K 0.323 0.163 −77.8 0.458 0.456 7 69 0.85\n2015 Apr 30 57142 Q 0.149 0.088 −81.9 0.407 0.404 4 110 0.84\n2015 Apr 30 57142 W 0.082 0.040 −78.0 0.304 0.288 11 25 0.79\n2015 Sep 24 57289 K 0.316 0.171 −69.6 0.528 0.528 6 95 0.84\n2015 Sep 24 57289 Q 0.156 0.083 −77.2 0.418 0.420 10 43 0.74\n2015 Oct 23 57318 K 0.347 0.165 80.8 0.438 0.436 9 46 0.81\n2015 Nov 30 57356 K 0.293 0.181 −86.9 0.396 0.396 4 96 0.89\n2015 Nov 30 57356 Q 0.141 0.096 −87.0 0.361 0.363 4 83 0.78\n2015 Nov 30 57356 W 0.076 0.043 −86.2 0.235 0.218 8 27 0.85\n2015 Dec 28 57384 K 0.300 0.175 −85.1 0.375 0.375 3 125 0.83\n2015 Dec 28 57384 Q 0.147 0.091 −81.1 0.344 0.343 2 144 0.78\n2015 Dec 28 57384 W 0.080 0.041 −85.3 0.217 0.202 6 36 0.79\n2016 Jan 13 57400 K 0.303 0.174 −88.6 0.337 0.337 2 139 0.71\n2016 Jan 13 57400 Q 0.147 0.091 −84.3 0.332 0.321 2 135 0.79\n2016 Jan 13 57400 W 0.080 0.041 89.5 0.191 0.164 6 26 0.84\n2016 Mar 01 57448 K 0.303 0.173 −85.4 0.417 0.417 3 123 0.90\n2016 Mar 01 57448 Q 0.149 0.089 −85.1 0.398 0.392 3 119 0.81\n2016 Mar 01 57448 W 0.077 0.042 −86.5 0.279 0.259 9 29 0.81\n2016 Apr 24 57502 K 0.301 0.185 −81.8 0.464 0.465 5 88 0.94\n2016 Apr 24 57502 Q 0.152 0.091 −80.9 0.427 0.410 3 117 0.90\n2016 Apr 24 57502 W 0.078 0.044 −83.8 0.245 0.251 6 44 0.77\n2016 Oct 18 57679 K 0.330 0.168 80.8 0.428 0.428 5 87 0.71\n2016 Nov 27 57719 K 0.310 0.183 −88.6 0.365 0.365 4 86 0.89\n2016 Dec 28 57750 K 0.290 0.184 −87.4 0.321 0.321 3 95 0.84\n2016 Dec 28 57750 Q 0.148 0.088 89.6 0.309 0.304 4 76 0.85\n2017 Mar 28 57840 K 0.372 0.172 −62.2 0.339 0.340 2 181 0.65\n2017 Mar 28 57840 W 0.097 0.042 -61.7 0.165 0.149 4 37 0.85\n2017 Apr 19 57862 K 0.310 0.170 −86.0 0.345 0.344 5 65 0.82\n2017 Apr 19 57862 Q 0.166 0.082 84.9 0.312 0.290 8 35 0.80\n2017 May 21 57894 K 0.316 0.174 83.9 0.341 0.341 5 68 0.78\n2017 May 21 57894 Q 0.168 0.081 80.3 0.295 0.282 4 68 0.75\n2017 Jun 17 57921 K 0.310 0.174 −77.3 0.356 0.355 5 75 0.93\n2017 Jun 17 57921 Q 0.164 0.080 −80.2 0.315 0.302 3 101 0.78\n2017 Sep 19 58015 K 0.321 0.180 −78.4 0.364 0.365 7 53 0.83\nTable 1 continued on next pageiMOGABA: Mrk 421 7\nTable 1 (continued)\nEpoch MJD Band BmajBminBPASKVN Sp σ D ξ r\n(mas) (mas) (◦) (Jy) (Jy/beam) (mJy/beam)\n(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)\n2017 Sep 19 58015 Q 0.165 0.083 −78.9 0.312 0.308 3 98 0.80\n2017 Oct 21 58047 K 0.307 0.176 85.4 0.338 0.338 3 133 0.86\n2017 Oct 21 58047 Q 0.162 0.084 79.7 0.321 0.310 3 120 0.80\n2017 Nov 04 58061 K 0.330 0.185 −68.5 0.320 0.320 4 87 0.87\n2017 Nov 04 58061 Q 0.178 0.086 −69.0 0.310 0.301 4 79 0.77\n2017 Nov 22 58079 K 0.293 0.183 −84.4 0.299 0.298 4 82 0.85\n2017 Nov 22 58079 Q 0.156 0.085 −89.1 0.290 0.281 4 80 1.00\n2017 Dec 06 58093 K 0.296 0.178 −88.3 0.278 0.275 5 51 0.97\n2018 Jan 13 58131 Q 0.158 0.084 88.2 0.308 0.302 2 159 0.77\n2018 Jan 13 58131 W 0.075 0.043 −85.3 0.203 0.189 6 31 0.78\n2018 Mar 25 58202 K 0.300 0.184 −79.8 0.295 0.290 11 27 0.82\n2018 Mar 25 58202 W 0.085 0.042 −73.3 0.177 0.224 32 7 0.71\n2018 Apr 19 58227 K 0.307 0.174 −88.7 0.363 0.362 6 60 0.89\n2018 Apr 19 58227 Q 0.157 0.083 89.6 0.331 0.321 3 95 0.84\n2018 Apr 19 58227 W 0.082 0.040 88.8 0.239 0.244 11 22 0.95\n2018 May 26 58264 K 0.303 0.181 83.9 0.333 0.333 7 47 0.84\n2018 May 31 58269 K 0.331 0.171 80.0 0.327 0.327 8 41 0.79\nNote — Column designation: 1 - Date; 2 - modified Julian date; 3 - obs erving frequency band: K - 22 GHz band;\nQ - 43 GHz band; W - 86 GHz band; D - 129 GHz band; 4-6 - restoring b eam: 4 - major axis; 5 - minor axis;\n6 - position angle of the major axis; 7 - total CLEAN KVN flux den sity; 8 - peak flux density; 9 - off-source RMS\nin the image; 10 - dynamic range of the image; 11 - quality of th e residual noise in the image (i.e., the ratio of the\nimage root-mean-square noise to its mathematical expectat ion).\n4.ANALYSIS\n4.1.Radio spectrum\nThe simultaneous multifrequency monitoring KVN observati ons enable us to investigate source spectra accurately\nwithout the time uncertainty. The multifrequency spectral analyses of variable blazars using arcsecond-scale observ a-\ntions by Kim et al. (2022) and Jeong et al. (2023) show that variable spectral properties can be revealed by s ubtracting\noff the quiescent spectrum in the multifrequency light curve s. We calculated the quiescent spectrum by fitting the four\nlocal minima obtained on MJD 56716, 57107, 57400, and 57840 - 58093 shown in Figure 3. The fourth local minimum\nwe used is obtained on MJD 58093 at 22 GHz, MJD 58079 at 43 GHz, a nd MJD 57840 at 86 GHz. The quiescent\nspectrum is obtained as 0.263 ±0.045 Jy, 0.271 ±0.021 Jy, and 0.156 ±0.027 Jy at 22, 43, and 86 GHz, respectively, by\nextrapolating each of the three frequencies to a common time (MJD 58269), as shown in Figure 4.\nThe mas-scale spectra of Mrk 421 are shown in Figure 5. Out of the 41 epochs, data existed for at least two\nfrequencies for 33 epochs. The observed spectrum is shown in black, and the quiescent-corrected spectrum (the\nobserved spectrum by subtracting the quiescent spectrum) i s shown in red. The spectra generally appear to be flat\nbetween 22 and 43 GHz, and seem to be relatively steep between 43 and 86 GHz. The quiescent-corrected spectra\nare sometimes consistent with the observed spectra and some times quite different. Unlike the observed spectra, the\nquiescent-corrected spectra generally show inverted spec tra between 22 and 43 GHz. Although we were not able to\nprecisely calculate the turnover frequency, νc, of Mrk 421 in this analysis, we clearly find that the quiescen t-corrected\nspectrum becomes steeper between 22 and 43 GHz and flatter bet ween 43 and 86 GHz than the observed spectrum.\nThis implies that the quiescent emission correction for the source spectrum in the mas-scale may be important to\nstudy the intrinsic spectral properties of the variable emi ssion regions.8 Lee et al.\n20 30 40 50 60 70 80\nF requency (GHz)0.150.200.250.300.350.40Flux density (Jy)MJD 56716\nMJD 57107\nMJD 57400\nMJD 57840\nFigure 3. Plot of local minima on MJD 56716, 57107, 57400, and 57840 - 58 093.\n3 × 101\n4 × 101\n6 × 101\nF requency (GHz)1.4 × 10−11.6 × 10−11.8 × 10−12 × 10−12. 2 × 10−12.4 × 10−12.6 × 10−12.8 × 10−1Minimum Flux (Jy)\nFigure 4. The minimum flux density at each frequency, by fitting the four local minima obtained on MJD 56716, 57107, 57400,\nand 57840 - 58093, and extrapolating each of the three freque ncies to a common time (MJD 58269).\n4.2.Spectral index\nThe spectral index αis defined as Sν∝να, whereνis the observing frequency, and Sνis the flux density. We\ncalculated αfor the data pairs 22 and 43 GHz, α22/43, and 43 and 86 GHz, α43/86(see Figure 6). The range of observed\nα22/43was−0.34 - 0.04 with a mean α22/43of−0.12, and α43/86was−1.04 -−0.27 with a mean α43/86of−0.63.\nTheα22/43was relatively flat, and α43/86was steep. As before, we calculated the quiescent-correcte d spectral indices.\nThe range of the quiescent-corrected spectral indices betw een 22 and 43 GHz αcor,22/43was−1.7 - 0.04 with a mean\nαcor,22/43of−0.57. For 43 and 86 GHz, the range of the quiescent-corrected spectral indices αcor,43/86was−1.69 -\n−0.47 with a mean αcor,43/86of−0.33.\nIn Figure 6, we see that the 22 and 43 GHz quiescent-corrected spectral i ndices are consistently lower (steeper) than\nthe observed spectral indices, although the errors are larg e. We noticed that the values of α22/43were steeper when\nthe flares were around the peaks on MJD 56901 and MJD 57289, whe reas those were flatter at local minima (MJD\n56716 and MJD 57400).\nThe 43 and 86 GHz quiescent-corrected spectral indices show ed more variability with spectral indices that were\nsometimes consistent, sometimes lower, and sometimes high er than the observed spectral indices, although the errors\nwere large. To further investigate the effect of correcting f or the quiescent spectrum, we plotted the spectral indices\nversus flux density. In Figures 7-8, a) and b) indicate the observed spectral index, and c) and d) indicate theiMOGABA: Mrk 421 9\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56308\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56393\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56419\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56580\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56615\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56650\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56684\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56716\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56738\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56769\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56901\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56959\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)56989\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57016\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57076\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57107\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57142\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57289\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57318\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57356\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57384\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57400\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57448\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57502\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57679\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57719\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57750\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57840\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57862\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57894\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)57921\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)58015\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)58047\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)58061\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)58079\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)58093\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)58131\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)58202\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)58227\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)58264\n101\n102\nF requency (GHz)10−310−210−1100Flux density (Jy)58269\nFigure 5. Spectrum of Mrk 421. Black and red symbols represent the obse rved spectrum and quiescent-corrected spectrum,\nrespectively.\nquiescent-corrected spectral index versus flux density, re spectively. To analyze the relation between the spectral in dex\nand flux density quantitatively, we computed the Spearman co rrelation coefficient ρand probability pfor each case.\nThe values of ρofα22/43for 22 GHz and 43 GHz flux density were −0.24 and 0.04, respectively. The values of ρof\nαcor,22/43for 22 GHz and for 43 GHz flux density were 0.4 and 0.66, respect ively.. The values of ρofα43/86for 43 GHz\nand for 86 GHz were −0.11 and 0.6, respectively. The values of ρofαcor,43/86for 43 GHz and for 86 GHz were −0.44\nand 0.33, respectively.\nThe original 22 and 43 GHz spectral indices were uncorrelate d with the flux densities. For the quiescent-corrected\nspectral indices (c and d), the spectral indices and flux dens ities showed a significant correlation (See, Figure 7). In\nFigure 8, there is no correlation between α43/86and the 43 GHz flux density, but the α43/86is correlated with 86 GHz\nflux density. The α43/86increases with the 86 GHz flux density. In the quiescent-corr ected case, we now see an\nanti-correlation between the corrected spectral indices v s. 43 GHz flux densities. The quiescent-corrected spectral10 Lee et al.\n56250 56500 56750 57000 57250 57500 57750 58000 58250\nMJD−2.0−1.5−1.0−0.50.00.51.01.52.0Spectral indexα\ncor , 22/43\nα\n22/43\n56250 56500 56750 57000 57250 57500 57750 58000 58250\nMJD−2.0−1.5−1.0−0.50.00.51.01.52.0Spectral indexα\ncor , 43/ 86\nα\n43/ 86\nFigure 6. Spectral index αof Mrk 421. The top panel is α22/43and the bottom panel is α43/86. The red symbols indicate the\nquiescent-corrected spectral index. A horizontal line is a guideline of zero.\nindices were less correlated with the 86 GHz flux densities th an before.\n5.DISCUSSION\nFigures 7and8show that correcting for the quiescent spectrum can signific antly affect the measurements of the\nspectral indices. This is of particular importance for calc ulations that require measurements of the synchrotron self -\nabsorption frequency, for example, the magnetic field stren gth,BSSA∝ν5\nc. This means that even small errors in\nmeasuring νccan lead to significant errors when measuring the magnetic fie ld strength. This suggests that magnetic\nfield strength estimations based on multifrequency data sho uld be carefully obtained with a proper determination of\nthe optically thin quiescent spectrum of compact radio sour ces even on mas-scales.\nTo test the previous work, which did not consider correcting the quiescent spectrum, we re-analyzed the multifre-\nquency spectral properties of OJ 287 ( Lee et al. 2020 ). The minimum flux densities of OJ 287 were 2.58, 2.27, 1.83,\nand 1.36 at 22, 43, 86, and 129 GHz, respectively. The spectra l index between 86 and 129 GHz was −0.73. We used\nthis optically thin spectral index to correct the observed s pectrum. We subtracted the minimum flux density obtained\nfrom the optically thin spectral index at each frequency for 8 epochs analyzed in Lee et al. (2020). Figure 9shows the\nspectra of OJ 287. The black and grey symbols indicate the obs erved and the quiescent-corrected spectra, respectively.\nAfter subtracting the optically thin spectrum, the spectru m was fitted with a curved power-law function ( Lee et al.\n2016,2017b ,2020). Then, we obtained the turnover frequency and the peak flux d ensity. The turnover frequencies ob-iMOGABA: Mrk421 11\n3 × 10−1\n4 × 10−1\n6 × 10−1\nFlux\n22 (Jy)−2−1012α\n22/43\nρ = -0. 24\np =0.179a)\n3 × 10−1\n4 × 10−1\n6 × 10−1\nFlux\n43 (Jy)−2−1012α\n22/43\nρ =0.04\np =0.816b)\n10−1\nFlux\n22 (Jy)−2−1012α\nco r, 22/43\nρ =0.4\np =0.023c)\n10−2\n10−1\nFlux\n43 (Jy)−2−1012α\nco r, 22/43\nρ =0.66\np =3.8176e -05d)\nFigure 7. Spectral index as a function of the flux density. The left and r ight panels indicate 22 and 43GHz, respectively. a)\nand b) indicate αvs. flux density. c) and d) indicate the αcorvs. flux density. ρand p values indicate coefficient and probability,\nrespectively.\ntained from the quiescent-corrected spectra were in the ran ge of 28-80 GHz, and the peak flux densities were 0.7-3.3 Jy.\nThe turnover frequencies were larger by a factor of approxim ately 2.5 larger than the ones obtained from the observed\nspectra. Because of the BSSA∝ν5\nc, the synchrotron self-absorption magnetic field strength m ight be underestimated\ninLee et al. (2020). This could also potentially change the comparison result between the equipartition magnetic field\nstrength and synchrotron self-absorption magnetic field st rength.\n6.SUMMARY\nWe have presented the results of simultaneous multifrequen cy observations at 22, 43, and 86 GHz with the KVN.\nThe light curves showed a global decaying and several small r adio flares. To separate the variable component and\nthe quiescent component in the light curves, we subtracted t he quiescent component, assuming the minimum flux\ndensity. We found that the observed spectra were flat at 22-43 GHz, and relatively steep at 43-86 GHz, whereas\nthe quiescent-corrected spectra are sometimes quite differ ent from the observed spectra (e.g., sometimes inverted at\n22-43 GHz ). This means that the turnover frequency, νcis shifted to a higher frequency. Based on the assumption\nof the synchrotron self-absorption magnetic field strength estimation, BSSA∝ν5\nc. Therefore, to properly estimate the\nmagnetic field strength, subtracting the quiescent spectru m in the KVN light curves is important.12 Lee et al.\n3 × 10−1\n4 × 10−1\n5 × 10−1\n6 × 10−1\nFlux\n43 (Jy)−2.0−1.5−1.0−0.50.00.51.01.52.0α\n43/ 86\nρ = -0.11\np =0.644a)\n2 × 10−1\n3 × 10−1\n4 × 10−1\nFlux\n86 (Jy)−2.0−1.5−1.0−0.50.00.51.01.52.0α\n43/ 86\nρ =0.6\np =0.006b)\n10−1\nFlux\n43 (Jy)−4−2024α\nco r, 43/ 86\nρ = -0.44\np =0.063c)\n10−1\nFlux\n86 (Jy)−4−2024α\nco r, 43/ 86\nρ =0.33\np =0.173d)\nFigure 8. Spectral index as a function of the flux density. The left and r ight panels indicate 43 and 86GHz, respectively. a)\nand b) indicate αvs. flux density. c) and d) indicate the αcorvs. flux density. ρand p values indicate coefficient and probability,\nrespectively.\nWe thank the anonymous reviewer for valuable comments and su ggestions that helped to improve the paper. We are\ngrateful to all staff members in KVN who helped to operate the a rray and to correlate the data. The KVN is a facility\noperated by the Korea Astronomy and Space Science Institute . The KVN operations are supported by KREONET\n(Korea Research Environment Open NETwork) which is managed and operated by KISTI (Korea Institute of Science\nand Technology Information). This work was supported by the National Research Foundation of Korea (NRF) grant\nfunded by the Korea government (MIST) (2020R1A2C2009003).\nREFERENCES\nAcciari, V. A., Aliu, E., Arlen, T., et al. 2011, ApJ, 738, 25.\ndoi:10.1088/0004-637X/738/1/25\nAleksić, J., Ansoldi, S., Antonelli, L. A., et al. 2015, A&A,\n576, A126. doi:10.1051/0004-6361/201424216\nAlgaba, J.-C., Zhao, G.-Y., Lee, S.-S., et al. 2015, Journal\nof Korean Astronomical Society, 48, 237.\ndoi:10.5303/JKAS.2015.48.5.237Algaba, J.-C., Lee, S.-S., Kim, D.-W., et al. 2018, ApJ, 852,\n30. doi:10.3847/1538-4357/aa9e50\nArbet-Engels, A., Baack, D., Balbo, M., et al. 2021, A&A,\n647, A88. doi:10.1051/0004-6361/201935557\nCharlot, P., Gabuzda, D. C., Sol, H., et al. 2006, A&A, 457,\n455. doi:10.1051/0004-6361:20054078\nDeller, A. T., Tingay, S. J., Bailes, M., et al. 2007, PASP,\n119, 318. doi:10.1086/513572iMOGABA: Mrk421 13\n101\n102\nF requency (GHz)10−1100101Flu  density (Jy)\nMJD=56308\n101\n102\nF requency (GHz)10−1100101Flu  density (Jy)\nMJD=56738\n101\n102\nF requency (GHz)10−1100101Flu  density (Jy)\nMJD=56769\n101\n102\nF requency (GHz)10−1100101Flu  density (Jy)\nMJD=56960\n101\n102\nF requency (GHz)10−1100101Flu  density (Jy)\nMJD=57356\n101\n102\nF requency (GHz)10−1100101Flu  density (Jy)\nMJD=57384\n101\n102\nF requency (GHz)10−1100101Flu  density (Jy)\nMJD=57400\n101\n102\nF requency (GHz)10−1100101Flu  density (Jy)\nMJD=57448\nFigure 9. The spectra of OJ287. 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M. & Padovani, P. 1995, PASP, 107, 803.\ndoi:10.1086/133630\nZhu, Q., Yan, D., Zhang, P., et al. 2016, MNRAS, 463,\n4481. doi:10.1093/mnras/stw2346"    },    {        "title": "2402.11906v1.Non_monotonous_shear_rate_dependence_of_dielectric_relaxation_frequency_of_a_nematic_liquid_crystal_revealed_by_rheo_dielectric_spectroscopy.pdf",        "content": "Non-monotonous shear rate dependence of dielectric relaxation frequency of a\nnematic liquid crystal revealed by rheo-dielectric spectroscopy\nK. Anaswara Das, M. Praveen Kumar and Surajit Dhara∗\nSchool of Physics, University of Hyderabad,\nHyderabad-500046, India\n(Dated: February 20, 2024)\nDielectric relaxation of materials provides important information on the polarisation dynamics\nat different time scales. We study the dielectric relaxation of a nematic liquid crystal under steady\nrotational shear and simultaneously measure the viscosity. The dielectric anisotropy of the nematic\nis positive and the applied electric field is parallel to the velocity gradient direction with a magnitude\nlarger than the Freedericksz threshold field. The complex dielectric constant and effective viscosity\ndecrease rapidly with increasing shear rate. The dielectric relaxation frequency exhibits a non-\nmonotonous shear rate dependence, first decreasing but beyond a critical shear rate increasing.\nOur experiments suggest the emergence of collective dipolar relaxation under the influence of the\ncompeting effects of hydrodynamic and dielectric torques.\nI. INTRODUCTION\nThe structures and properties of complex fluids such as\npolymeric systems, colloidal suspensions and liquid crys-\ntals have been studied under simultaneous shear flow and\nelectric fields [1, 2]. In some fluids, the viscosity is en-\nhanced significantly due to the application of an electric\nfield. This effect is known as the electrorheological effect\nand is very important for applications in electromechan-\nical devices [3–6]. In nematic liquid crystals (LCs), com-\nposed of axially polar molecules, under the application of\nsufficient electric field the effective viscosity is enhanced\nand this effect is due to the change in the orientation of\nthe director (average aligning direction of the molecules)\nwith respect to the shear flow direction [7–14]. In uniaxial\nnematic LCs, there are two principal dielectric constants,\nnamely ϵ||andϵ⊥, where the subscripts refer components\nin relation to the director [17]. The dielectric anisotropy\n(∆ϵ=ϵ||−ϵ⊥) of the nematic LCs composed of polar\nmolecules could be zero, positive or negative, depending\non the orientation of the permanent dipole moments with\nrespect to the long molecular axis [18].\nBoth the dielectric constants are frequency dependent\nand the corresponding relaxation frequencies f||andf⊥\nare related to the time scales of rotation of the dipoles\nalong the short and long molecular axes, respectively [18].\nFor axially polar molecules, the relaxation frequency f||\nis much smaller than f⊥and it is the opposite in the\ncase of transversely polar molecules [18]. Usually, these\nrelaxation frequencies are measured in cells composed of\nparallel electrodes in which the director is aligned either\nperpendicular (homeotropic) or parallel (homogeneous)\nto the electrodes. In cells the nematic LCs are in the\nquiescent state, hence such measurements provide equi-\nlibrium relaxation times which are important for device\napplications.\n∗surajit@uohyd.ac.inA nematic LC is far away from the equilibrium when\nsubjected to a steady shear flow and depending on the\nsystem it may exhibit three dynamic modes, namely flow-\naligned, wagging and tumbling [19–24]. In flow aligned\nnematics, the director forms a steady state with a tilt\nangle θL(Lesli angle) with respect to the flow direction\nwhereas, in the case of wagging, the director oscillates\nand for tumbling motions, the director makes full rota-\ntions in the shear plane [25–29]. In a flow-aligned nematic\nwith positive dielectric anisotropy (∆ ϵ >0) the applica-\ntion of sufficient electric field perpendicular to the flow di-\nrection induces an apparent change in viscosity [17]. The\nchange in viscosity due to the change in the director ori-\nentation can be measured easily from the rheo-dielectric\nstudies at a frequency much below the dielectric relax-\nation frequency [11, 13–16]. However, so far it is not well\nunderstood if the shear flow affects the dielectric relax-\nation frequencies of the nematic LCs [30, 31]. Since the\nshear rate is much slower than the equilibrium rotation\nof the dipoles ( ˙ γ≪τ−1, relaxation time), one would ex-\npect that the shear flow should not affect the dielectric\nrelaxation frequency. The rheo-dielectric studies at low\nelectric fields on some cyanobiphenyl nematic LCs show\nno measurable effect of shear flow on the dielectric relax-\nation frequency [30].\nIn this paper, we apply a sufficiently high electric field\nthat orients the director of the nematic along the velocity\ngradient direction (perpendicular to the shear flow direc-\ntion) and study the frequency dispersion of the effective\ndielectric constant of a nematic liquid crystal at various\nshear rates. We measure the real and imaginary parts of\nthe dielectric constants and analyse their frequency de-\npendence at varying voltage and shear rates. Our study\nshows evidence of a collective dielectric relaxation mode\nin the intermediate shear rate range. We discuss the pos-\nsible origin of the collective dipolar relaxation.arXiv:2402.11906v1  [cond-mat.soft]  19 Feb 20242\nII. EXPERIMENT\nWe worked on a nematic liquid crystal mixture, known\nas E7. It is a mixture of several cyanobiphenyl, cyan-\noterphenol and triphenyl compounds at some specific\ncompositions [32]. It was obtained from Grand Winton\nInc, and used without further purification. At 5◦C the\nparallel and perpendicular components of the dielectric\nconstant are given by ϵ||= 19 .6 and ϵ⊥= 4.5, respec-\ntively and the dielectric anisotropy is positive (∆ ϵ=\nϵ||−ϵ⊥= 15.1) (see Fig.S1, SM)[33]. The nematic\nto isotropic phase transition temperature is 61◦C and\nit exhibits a broad nematic temperature range (61◦C to\n−10◦C). The rheo-dielectric measurements were made by\na strain-controlled rheometer (Anton Paar MCR 501) us-\ning parallel-plate geometry with a plate diameter of 50\nmm. The plate gap was fixed at 80 µm and no alignment\nlayer was used in this experiment. The bottom plate was\nfixed while the top plate was rotated at different shear\nrates. The bottom plate was attached with a Peltier tem-\nperature controller for controlling the temperature with\nan accuracy of 0 .1◦C. A hood was used to cover the mea-\nsuring plates for temperature uniformity. An LCR meter\n(Agilent E4980A) was connected to the plates for rheo-\ndielectric measurements. The applied electric field was\ngreater than the Freedericksz threshold field ( E≫Eth)\nand the frequency was increased from 100 Hz to 1.0 MHz.\nThe experiments were performed at different shear rates\nand the frequency-dependent real and imaginary parts of\nthe dielectric constant were measured by the LCR me-\nter. All the measurements were made at 5◦C. Before the\nmeasurement, the sample was pre-sheared for 5 minutes\nat a fixed shear rate ( ˙ γ=100 s-1). A schematic diagram\nof the experimental setup is shown in Fig.1.\nFIG. 1. Experimental setup for rheo-dielectric measurements.\nIII. RESULTS AND DISCUSSION\nWe measured the frequency dispersion of the effective\ndielectric constants at a fixed electric field and different\nshear rates (Fig.2). At room temperature the complete\nrelaxation mode is not observed due to the limiting fre-\nquency range of the LCR meter (100 Hz to 2 MHz) hence,\n(a)\n(b)FIG. 2. Plots of (a) real ( ϵ′\neff) and (b) imaginary ( ϵ′′\neff) com-\nponents of the effective dielectric constant with frequency f,\nat different shear rates and temperature T=5◦C. The applied\nelectric field E= 2.5×105V/m, which is 10 times higher than\nthe Freedericksz threshold field, Eth= 2.5×104V/m.\nall the experiments were performed at a fixed tempera-\nture of 5◦C. Figure 2(a,b) shows the dielectric disper-\nsion of the real ( ϵ′\neff) and imaginary ( ϵ′′\neff) parts of the\neffective dielectric constant at different shear rates. The\napplied electric field was Eth= 2.5×105V/m, which\nis 10 times larger than the Freedericksz threshold field\n(Eth= 2.5×104V/m). In the low-frequency region,\n(f <104Hz), at zero shear rate (i.e., quiescent nematic),\nϵ′\neffis independent of frequency and decreases with in-\ncreasing shear rate. On the other hand, ϵ′′\neffdecreases\nwith the increasing frequency in the low-frequency region\n(f < 103Hz). It varies as ϵ′′\neff∝f−1, which suggests\nthe influence of the conductivity (direct current) on the\nimaginary component of the dielectric constant.\nFigure 3(a) and (b) shows the variation of ϵ′\neffand\nϵ′′\neffobtained from Fig.1 with the shear rate at a fixed\nfrequency f= 4 kHz. Both decrease exponentially\nwith increasing shear rate. For example, at ˙ γ= 0 s-1,\nϵ′\neff= 20 .3. This value is nearly equal to the paral-\nlel component of the dielectric constant measured in a\nhomeotropic cell i.e.,ϵ||≃19.6 (see Fig.S1, SM)[33]. It\nsuggests that the director is almost perpendicular to the\nconfining plates and parallel to the electric field direc-\ntion. With increasing shear rate, ϵ′\neffdecreases rapidly\nand eventually becomes constant i.e,ϵ′\neff≃6, when the\nshear rate is increased to ˙ γ= 200 s-1. This dielectric con-\nstant at this shear rate is slightly larger than the value3\n(a) (b)\n(c)(d)\nFIG. 3. Variation of (a) ϵ′\neffand (b) ϵ′′\neffwith shear rate\nat frequency f= 4 kHz (obtained from Fig.1). (c) Time-\ndependent effective viscosity ηeffatE= 2.5×105V/m and\nat different shear rates. (d) Variation of ηeffat different shear\nrates obtained from the figure (c). Shear-dependent director\norientations are shown in the insets.\nmeasured in a homogeneous (planar) cell i.e.,ϵ⊥≃4.5\n(see Fig.S1, SM)[33]. Hence, these results demonstrate\nthat initially ( ˙ γ= 0 s-1) the director is perpendicular to\nthe plates i.e.,parallel to the velocity gradient direction\nand it gradually tilts in the shear plane towards the shear\nflow direction. Further, the effective viscosity ηeffof the\nliquid crystal was measured simultaneously with the di-\nelectric dispersion measurements. Figure 3(c) shows that\nthe viscosity is independent of time but it decreases with\nincreasing shear rate. Figure 3(d) shows the variation\nof the viscosity at different shear rates. At ˙ γ= 1 s-1,\nηeff= 0.7 Pa s and it reduces to about 0 .11 Pa s when\n˙γ= 100 s-1. At ˙ γ= 1 s-1, the effective viscosity can be\nconsidered as the Miesowicz viscosity for the director ori-\nentation being parallel to the velocity gradient direction\ni.e.,ηeff≃η1and at the highest shear rate ( ˙ γ= 100 s-1),\nηeff≃η3, where η3is the Miesowicz viscosity with the\ndirector being nearly parallel to the velocity direction.\nOne important observation is the change of dielectric\nrelaxation frequency with shear rate (Fig. 2(b)). It is\napparent that not only does the peak height of the ϵ′′\neff\ndecrease, but the frequency at which the peak occurs,\nso-called the dielectric relaxation frequency also changes\nwith shear rate. To obtain the actual dielectric relax-\nation frequency ( fr), we fitted the dielectric data to the\nHavriliak-Negami relaxation function [36],\nϵ∗(f) =ϵ∞+∆ϵ\n[1 + ( i2πfτ)α]β−iσ0\nϵ02πf(1)\nwhere ∆ ϵis the dielectric strength, ϵ∞is the dielectricpermittivity at the high-frequency limit, σ0is the conduc-\ntivity and τis the relaxation time. The corresponding\nrelaxation frequency is given by fr= 1/2πτ(see Fig.S2,\nSM [33]). The exponents αandβdescribe the asym-\nmetry and broadness of the corresponding spectra and\nare found to vary in the range of 0.8 to 1 (see Table S1,\nSM [33]).\n(a)\n(b) (c)\nFIG. 4. (a) Variation of the effective relaxation frequency\nfrwith shear rate at E= 2.5×105V/m. Dependence of (b)\nminimum relaxation frequency fmin\nrand (c) the critical shear\nrate ˙γc. The standard deviation of the relaxation frequency\nat zero shear rate is about 2 kHz.\nFigure 4(a) shows the variation of the relaxation fre-\nquency frwith the shear rate ˙ γat an applied field of\nE= 2.5×105V/m. The relaxation frequency decreases\nfrom 320 kHz to 116.7 kHz when ˙ γis increased from 0\nto 23 s-1. It shows a pronounced minimum at ˙ γ= 23 s-1,\nabove this, frincreases to about 400 kHz, when ˙ γis fur-\nther increased to 200 s-1. At ˙γ= 0 s-1, the director is par-\nallel to the field direction, hence the effective relaxation\nfrequency frcorresponds to the rotation of the longitudi-\nnal (axial) components of the dipole moments about the\nshort axis i.e.,fr∼f||\nr∼320 kHz, where f||\nris the relax-\nation frequency measured in homeotropic cell (quiescent\nnematic)(see Fig.S1, SM) [33]. The relaxation frequency\nof the transverse dipole moments about the long axis\nin the homogeneous cell (quiescent nematic) is f⊥\nr>2\nMHz (see Fig.S1, SM) [33] which is much larger than f||\nr.\nHence, one would have expected the relaxation frequency\nfrto increase continuously from 320 kHz to 400 kHz with4\nincreasing shear rate. Thus, the minimum relaxation fre-\nquency ( fmin\nr) at a particular shear rate is rather unex-\npected and can not be explained simply by considering\nthe tilting of the director. In this context, two possible\ndirector dynamics should be mentioned. Any effect of\nelectrohydrodynamic instability on the dielectric relax-\nation is ruled out as both the dielectric and conductivity\nanisotropies (∆ ϵand ∆ σ) are positive [37]. Further, E7 is\na flow-aligning nematic hence the tumbling and wagging\nof the director is absent [38]. Hence, the reduction of re-\nlaxation frequency, in other words, the slowing down of\nthe relaxation time ( τ) could hint at a possible collective\ndipolar relaxation. One can expect that the collective re-\nsponse should also contribute to enhancing the effective\ndielectric constant. However, such an effect can not be\nobserved because of the counter effect of tilting of the di-\nrector in the shear plane that tends to reduce the effective\ndielectric constant.\nWe performed experiments at a few different electric\nfields ( E≫Eth) at the same temperature (see Fig.S3\nand S4, SM [33]) and observed that the minimum re-\nlaxation frequency fmin\nrdecreases almost linearly with\nthe increasing field (Fig. 4(b)). The critical shear rate\nγcat which the relaxation frequency is minimum is also\nfound to increase with the field (Fig. 4(c)). We measured\nthe dielectric relaxation frequency at a few electric fields\nin a homogeneous cell of thickness 11 .3µm(quiescent\nnematic). It is observed that the relaxation frequency\ndecreases almost linearly with increasing electric field as\nexpected (see Fig.S5, SM [38]). Hence, the decrease of\nfmin\nrwith increasing field under shear flow can be at-\ntributed to the increasing tilt angle of the molecules with\nrespect to the field direction. Figure 4(c) demonstrates\nthe competing effects, i.e., the high shear rate is required\nat a larger electric field for the collective response to be\nobserved (see later discussion).\nBased on the above results we propose possible director\nconfigurations at different shear rates. In our experiment,\nthe bottom plate is fixed and the top plate is rotated\nwith different shear rates. At zero shear rate ( ˙ γ= 0 s-1),\nthe molecules are aligned perpendicular to the plates as\nthe applied electric field is very much greater than the\nFreedericksz threshold field (see Fig. 5(a)). At a high\nshear rate ( ˙ γ≫˙γc), the hydrodynamic torque due to the\nshear flow is much larger than the dielectric torque due\nto the applied electric field. As a result, the molecules\nalign nearly parallel to the shear flow direction except\nat the centre as shown in Fig.5(c) (see later discussion).\nAt the critical shear rate ( ˙ γ∼˙γc) these two torques\nare comparable and the director attains an intermediate\norientation ( π/2≤θ≤θL).\nFurther, the shear rate experienced by fluid elements in\nparallel plate geometry depends on their position r(dis-\ntance from the centre of plates) and is given by ˙ γ=rω/h ,\nwhere his the gap between the plates and ωis the angular\nfrequency. Conventionally, in parallel plate systems, the\nshear rate at the rim ( r=a) is considered for the calcu-\nlation of stress and viscosity of Newtonian fluids, whereais the radius of the plate. In complex fluids like liquid\ncrystals, the shear rate, and consequently the hydrody-\nnamic torque very much depends on rand is not the same\nthroughout the plate. The hydrodynamic torque is zero\nat the centre and it increases from the centre to the edge\nof the plates. Near the critical shear rate (0 < γ∼γc),\nthe director at the centre is vertical and, as one moves to-\nwards the perimeter, the hydrodynamic torque increases\ngradually and the director tilts continuously towards the\nshear flow direction, creating a half skyrmion-like direc-\ntor deformation as shown in Fig.5(b). Such a complex\ndirector distortion can give rise to flexoelectric polarisa-\ntion [17, 37] which can couple to the orientational fluc-\ntuations of the director. As a result, it can give rise to a\ncollective polarisation mode. However, such polarization\nis expected to relax at a much lower frequency [34] hence\nthe influence of flexoelectric polarization can be ignored\nin our experiments. The complex director configuration\nin the low shear rate range (0 < γ≤γc) can give rise to\na cooperative dipolar motion due to secondary coupling\nof the steric interactions of the molecules to the electric\npolarization and consequently reduces the dielectric re-\nlaxation frequency. The secondary effect is very different\nfrom the primary statistical correlation function associ-\nated with the molecular dipole reorientations to polar-\nization, which occurs at a much higher frequency [34].\nHowever, theoretical / simulation studies are required to\nunderstand the results quantitatively.\n(a)(b) (c)\n(d) (e) (f)\nFIG. 5. Top view (xy-plane) of the director at electric field\nE≫Eth(along z-axis) and different shear rates, (a) ˙ γ= 0,\n(b) ˙γ∼˙γcand (c) ˙ γ≫˙γc. Side views of the director in the\nvertical mid-plane (zy-plane) at the corresponding shear rates\nare shown in (d-f). The rotation of the top plate is indicated\nby blue arrows. Note that the director is twisted radially\noutward forming a half skyrmion-like structure in (b,e) and\nthe structure is almost uniform except at the centre in figure\n(c,f).\nFurther, a few comments are in order. First, two di-\nmensionless numbers, that characterize the flow are Deb-\norah number De=τ˙γ, and Ericksen numbers Er=\nη˙γ\nK/h2[39], where τis the characteristic relaxation time\nof the material, his the gap between the two plates, η\nis the effective viscosity and Kis the mean elastic con-\nstant. In the experimental shear rate range ( ˙ γ=1-200\ns-1),Er= 103∼105andDe= 10−2∼2 (see Table S2,5\nSM) [33]. It suggests liquid-like response and the viscous\neffects dominate over the elastic effects. Second, liquid\ncrystals contain finite impurity ions which can move to-\nwards the opposite electrodes under ac electric field and\ncreate space charge polarization. The electrode charg-\ning time τ=λDL/2D, where λDis the Debye screening\nlength, Lis the gap between the electrodes and Dis\nthe diffusion constant of the ions. Taking D∼10−11\nm2s-1[40],λD∼0.1µm[41],L= 80 µm, the estimated\nelectrode polarisation frequency is about 2.5 Hz which is\nfar below the experimental frequency range. Third, since\nthe maximum applied electric field is not very strong (far\nbelow the dielectric breakdown) the dielectric response is\nassumed to be in the linear regime.\nIV. CONCLUSION\nThe effective dielectric constant and shear viscosity at\nan electric field higher than the Freedericksz threshold\nfield of a nematic liquid crystal decrease very rapidly un-\nder rotational shear due to the tilting of the director in\nthe shear plane from the field to the shear flow direction.The dielectric relaxation frequency shows a pronounced\nminimum at a critical shear rate. The results suggest the\ndevelopment of a cooperative dipolar relaxation in the\nlow shear rate range. These results are important for ap-\nplications of liquid crystals as well as polar liquids in elec-\ntromechanical devices. Considering the competing effects\nof the hydrodynamic and dielectric torques, we proposed\na simple physical model that demonstrates the director\nconfiguration at different shear rates qualitatively. In\nessence, our findings uncover cooperative dipolar relax-\nation under in the presence of transverse electric and flow\nfields that reduce dielectric relaxation frequency. We fo-\ncussed on an apolar nematic LC, however rheo-dielectric\nstudies on liquid crystals with macroscopic polarisations\nsuch as ferroelectric nematic and ferroelectric smectics\nLCs are promising for new relaxation dynamics.\nV. ACKNOWLEDGMENTS\nAcknowledgments : SD acknowledges financial sup-\nport from SERB (SPR/2022/000001). AD acknowledges\nDST for fellowship (DST/WISE-PhD/PM/2023/57).\nWe thank Dr. Arun Roy for the useful discussions.\n[1] R. G. Larson, The Structure and Rheology of Complex\nFluids (Oxford University Press, New York, 1999).\n[2] M. Parthasarathy, and D. J. Klingenberg, Materials Sci-\nence and Engineering: R: Reports, RI7, 57 (1996).\n[3] W. W. Winslow, Appl. Phys., 20, 1137 (1967).\n[4] D. R. Gamota, J. Rheol., 35, 399 (1991).\n[5] S. Fukayama, J. Mol. Liq., 90131 (2001).\n[6] D. J. Klingenberg,, J. Chem.Phys., 91, 7888 (1989).\n[7] T. Carlsson and K. Skarp, Mol. Cryst. Liq. Cryst., 78,\n157 (1981).\n[8] S. Skarp, T. Carlsson, S. T. Lagerwall and B. Stebler,\nMol. Cryst. Liq. Cryst., 66, 199 (1981).\n[9] K-L Tse and A. D. Shine, J. Rheol., 39, 1021 (1995).\n[10] I. Yang and A. D. Shine, J. Rheol., 361079 (1992).\n[11] K. Negita, J. Chem. Phys. 105, 7837 (1996).\n[12] H. Watanabe, Rheologica Acta, 37, 6 (1998).\n[13] P. Patrico, C. R. Leal, L.F.V. Pinto, A. Boto and M.T.\nCidade, Liq. Cryst., 39, 25 (1999).\n[14] M.T. Cidade, G. Pereira, A. Bubnov, V. Hamplova, M.\nKaspar and J. P. Casquilho, Liq. Cryst., 39, 191 (2012).\n[15] J. Ananthaiah, Rasmita Sahoo, M. V. Rasna and Surajit\nDhara, Phy. Rev. E 89, 022510 (2014).\n[16] J. Ananthaiah, M. Rajeswari, V. S. S. Sastry, R.\nDabrowski and Surajit Dhara, Euro. Phy. J. E 34, 74\n(2011).\n[17] P. G. de Gennes, The Physics of Liquid Crystals (Oxford\nUniversity Press, Oxford, England, 1974).\n[18] W. H. de Jeu, Physical Properties of Liquid Crystals , 2nd\ned. (Cambridge University Press, Cambridge, 1992).\n[19] F. M. Leslie, J. Phys. D: Appl Phys 9, 925 (1976).\n[20] F. M. Leslie, Adv. Liq. Cryst. 4, 1 (1979).\n[21] F. M. Leslie, Theory and flow phenomena in nematic\nliquid crystals. Theory and Applications of Liquid Crys-tals. Springer, Berlin Heidelberg New York, pp 235-254\n(1987).\n[22] Andreas M. Menzel, Phys. Rep., 554, 1 (2015).\n[23] G. Rien¨ acker, S. Hess, Physica A., 267, 321 (1999).\n[24] V. V. Belyaev, Viscosity of Nematic Liquid Crystals, 1st\ned. (Cambridge International Science, Cambridge, 2011).\n[25] A. Archer, R. G. Larson, J. Chem. Phys., 103, 3108\n(1995).\n[26] D. J. Ternet, R. G. Larson, L. G. Leal, Rheol. Acta, 38,\n183 (1999).\n[27] W. R. Burghardt, G. G. Fuller, Macromolecules, 24, 2546\n(1991).\n[28] G. Rien¨ acker, S. Hess, Physica, A 315, 537 (2002).\n[29] Y.G. Tao, W. K. den Otter and W. J. Briels, 86, 56005\n(2009).\n[30] H. Watanabe, T. Sato, M. Hirose, K. Osaki, and M- Yao,\nRheol. Acta, 37, 519 (1998).\n[31] H. Watanabe, T. Sato, M. Matsumiya, T. Inoue and K.\nOsaki, Nihon Reoroji Gakkaishi, 27, 121 (1999).\n[32] https://shop.synthon-chemicals.com/en/LIQUID-\nCRYSTALS/LIQUID-CRYSTALS-MIXTURES/Liquid-\ncrystal-mixture-E7.html.\n[33] Supplementary material presents additional results of di-\nelectric relaxation at different electric fields, also mea-\nsurements in homogeneous and homeotropic cells.\n[34] B. I. Outram and S. J. Elston, Phys. Rev. E Phys. Rev.\nE88, 012506 (2013).\n[35]\n[36] S. Havriliak and S. Negami, J. Polymer, 8, 161 (1967).\n[37] L. M. Blinov and V. G. Chigrinov, Electrooptic Effects\nin Liquid Crystal Materials (Springer, Berlin, 1994).\n[38] D. J. Ternet, R. G. Larson, L. G. Leal, Rheol. Acta 38,\n183 (1999).6\n[39] R. G. Larson and D. W. Mead, Liq. Cryst. 15, 151 (1993).\n[40] S. Hern` andez-Navarro, P. Tierno, J. Ign´ es-Mullol and F.\nSagu´ es, Soft Matter 9, 7999 (2013).[41] O. D. Lavrentovich, Curr. Opin. Colloid Interface Sci.,\n21, 97 (2016)."    },    {        "title": "2402.11926v2.Lax_Wendroff_Flux_Reconstruction_on_adaptive_curvilinear_meshes_with_error_based_time_stepping_for_hyperbolic_conservation_laws.pdf",        "content": "LAX-WENDROFF FLUX RECONSTRUCTION ON ADAPTIVE\nCURVILINEAR MESHES WITH ERROR BASED TIME STEPPING FOR\nHYPERBOLIC CONSERVATION LAWS\nA P REPRINT\nArpit Babbar\nCentre for Applicable Mathematics\nTata Institute of Fundamental Research\nBangalore – 560065\narpit@tifrbng.res.inPraveen Chandrashekar∗\nCentre for Applicable Mathematics\nTata Institute of Fundamental Research\nBangalore – 560065\npraveen@math.tifrbng.res.in\nFebruary 21, 2024\nABSTRACT\nLax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for\nsolving hyperbolic conservation laws. This work extends the LWFR scheme to solve conservation\nlaws on curvilinear meshes with adaptive mesh refinement (AMR). The scheme uses a subcell\nbased blending limiter to perform shock capturing and exploits the same subcell structure to obtain\nadmissibility preservation on curvilinear meshes. It is proven that the proposed extension of LWFR\nscheme to curvilinear grids preserves constant solution (free stream preservation) under the standard\nmetric identities. For curvilinear meshes, linear Fourier stability analysis cannot be used to obtain an\noptimal CFL number. Thus, an embedded-error based time step computation method is proposed\nfor LWFR method which reduces fine-tuning process required to select a stable CFL number using\nthe wave speed based time step computation. The developments are tested on compressible Euler’s\nequations, validating the blending limiter, admissibility preservation, AMR algorithm, curvilinear\nmeshes and error based time stepping.\nKeywords Hyperbolic conservation laws ·Lax-Wendroff flux reconstruction ·Curvilinear grids ·Admissibility\npreservation and shock Capturing ·Adaptive mesh refinement ·Error based time stepping\n1 Introduction\nLax-Wendroff method is a single step method for time dependent problems in contrast to method of lines approach\nwhich combines a spatial discretization scheme with a Runge-Kutta method in time. The Lax-Wendroff idea has\nbeen used for hyperbolic conservation laws to develop single step finite volume and discontinuous Galerkin methods\n[31,30,50,8,11]. Another approach to develop high order, single-stage schemes is based on ADER schemes [ 41,12].\nFlux Reconstruction (FR) method introduced by Huynh [ 18] is a finite-element type high order method which is\nquadrature-free. The key idea in this method is to construct a continuous flux approximation and then use collocation\nat solution points which leads to an efficient implementation that can exploit optimized matrix-vector operations and\nvectorization capabilities of modern CPUs. The continuous flux approximation requires a correction function whose\nchoice affects the accuracy and stability of the method [ 18,43,44,42]; by properly choosing the correction function and\nsolution points, the FR method can be shown to be equivalent to some discontinuous Galerkin and spectral difference\nschemes [18, 42].\nIn [2], a Lax-Wendroff Flux Reconstruction (LWFR) scheme was proposed which used the approximate Lax-Wendroff\nprocedure of [ 50] to obtain an element local high order approximation of the time averaged flux and then performs\n∗Corresponding authorarXiv:2402.11926v2  [math.NA]  20 Feb 2024APREPRINT - FEBRUARY 21, 2024\nthe FR procedure on it to perform evolution in a single stage. The numerical flux was carefully constructed in [ 2]\nto obtain enhanced accuracy and linear stability based on Fourier stability analysis. In [ 3], a subcell based shock\ncapturing blending scheme was introduced for LWFR based on the work of subcell based scheme of [ 17]. To enhance\naccuracy, [ 3] used Gauss-Legendre solution points and performed MUSCL-Hancock reconstruction on the subcells.\nSince the subcells used in [ 3] were inherently non-cell centred, the MUSCL-Hancock scheme was extended to non-cell\ncentred grids along with the proof of [ 4] for admissibility preservation. The subcell structure was exploited to obtain a\nprovably admissibility preserving LWFR scheme by careful construction of the blended numerical flux at the element\ninterfaces.\nIn this work, the LWFR scheme of [2] is further developed to incorporate three new features:\n1. Ability to work on curvilinear, body-fitted grids\n2. Ability to work on locally and dynamically adapted grids with hanging nodes\n3. Automatic error based time step computation\nCurvilinear grids are defined in terms of a tensor product polynomial map from a reference element to the physical\nelement. The conservation law is transformed to the coordinates of the reference element and then the LWFR procedure\nis applied leading to a collocation method that has similar structure as on Cartesian grids. This structure also facilitates\nthe extension of the provably admissibility preserving subcell based blending scheme of [ 3] to curvilinear grids. The\nFR formulation on curvilinear grids is based on its equivalence with the DG scheme, see [ 23], which also obtained\ncertain metric identities that are required for preservation of constant solutions, that is, free stream preservation. See\nreferences in [ 23] for a review of earlier study of metric terms in the context of other higher order schemes like finite\ndifference schemes. The free stream preserving conditions for the LWFR scheme are proven to be the same discrete\nmetric identities as that of [ 23]. The only requirement for the required metric identities in two dimensions is that the\nmappings used to define the curvilinear elements must have degree less than or equal to the degree of polynomials used\nto approximate the solution.\nIn many problems, there are non-trivial and sharp solution features only in some localized parts of the domain and\nthese features can move with respect to time. Using a uniform mesh to resolve small scale features is computationally\nexpensive and adaptive mesh refinement (AMR) is thus very useful. In this work, we perform adaptive mesh refinement\nbased on some local error or solution smoothness indicator. Elements with high error indicator are flagged for refinement\nand those with low values are flagged for coarsening. A consequence of this procedure is that we get non-conformal\nelements with hanging nodes which is not a major problem with discontinuous Galerkin type methods, except that\none has to ensure conservation is satisfied. For discontinuous Galerkin methods based on quadrature, conservation is\nensured by performing quadrature on the cell faces from the refined side of the face [ 38,47]. For FR type methods\nwhich are of collocation type, we need numerical fluxes at certain points on the element faces, which have to computed\non a refined face without loss of accuracy and such that conservation is also satisfied. For the LWFR scheme, we use\nthe Mortar Element Method [ 25,26] to compute the solution and fluxes at non-conformal faces. The resulting method\nis conservative and also preserves free-stream condition on curvilinear, adapted grids.\nThe choice of time step is restricted by a CFL-type condition in order to satisfy linear stability and some other non-linear\nstability requirements like maintaining positive solutions. Linear stability analysis can be performed on uniform\nCartesian grids only, leading to some CFL-type condition which depends on wave speed estimates. In practice these\nconditions are then also used for curvilinear grids but they may not be optimal and may require tuning the time\nstep for each problem by adding a safety factor. Thus, automatic time step selection methods based on some error\nestimates become very relevant for curvilinear grids. Error based time stepping methods are already developed for\nODE solvers; and by using a method of lines approach to convert partial differential equations to a system of ordinary\ndifferential equations, error-based time stepping schemes of ODE solvers have been applied to partial differential\nequations [ 5,20,45] and recent application to CFD problems can be found in [ 32,34]. The LWFR scheme makes use\nof a Taylor expansion in time of the time averaged flux; by truncating the Taylor expansion at one order lower, we can\nobtain two levels of approximation, whose difference is used as a local error indicator to adapt the time step. As a\nconsequence the user does not need to specify a CFL number, but only needs to give some error tolerances based on\nwhich the time step is automatically decreased or increased.\nThe rest of the paper is organized as follows. In Section 2, we review notations and the transformation of conservation\nlaws from curved elements to a reference cube following [ 23,22]. In Section 3, the LWFR scheme of [ 2] is extended\nto curvilinear grids. In Section 3.1, we review FR on curvilinear grids and use it to construct LWFR on curvilinear\ngrids in Section 3.2. Section 3.3 shows that the free stream preservation condition of LWFR is the standard metric\nidentity of [ 23]. In Section 4, the admissibility preserving subcell limiter for LWFR from [ 3] is reviewed and extended\nto curvilinear grids. In Section 5, the Mortar Element Method for treatment of non-conformal interfaces in AMR of [ 25]\nis extended to LWFR. In Section 6, error-based time stepping methods are discussed; Section 6.1 reviews error-based\n2APREPRINT - FEBRUARY 21, 2024\ntime stepping methods for Runge-Kutta and Section 6.2 introduces an embedded error-based time stepping method for\nLWFR. In Section 7, numerical results are shown to demonstrate the scheme’s capability of handling adaptively refined\ncurved meshes and benefits of error-based time stepping. Section 8 gives a summary and draws conclusions from the\nwork.\n2 Conservation laws and curvilinear grids\nThe developments in this work are applicable to a wide class of hyperbolic conservation laws but the numerical\nexperiments are performed on 2-D compressible Euler’s equations, which are a system of conservation laws given by\n∂\n∂t\nρ\nρu\nρv\nE\n+∂\n∂x\nρu\np+ρu2\nρuv\n(E+p)u\n+∂\n∂y\nρv\nρuv\np+ρv2\n(E+p)v\n=0 (1)\nHere, ρ, pandEdenote the density, pressure and total energy per unit volume of the gas, respectively and (u, v)are\nCartesian components of the fluid velocity. For a polytropic gas, an equation of state E=E(ρ, u, v, p )which leads to\na closed system is given by\nE=p\nγ−1+1\n2ρ(u2+v2) (2)\nwhere γ > 1is the adiabatic constant. For the sake of simplicity and generality, we subsequently explain the\ndevelopment of the algorithms for a general hyperbolic conservation law written as\nut+∇x·f(u) =0 (3)\nwhere u∈Rpis the vector of conserved quantities, f(u) = (f1, . . . ,fd)∈Rp×dis the corresponding physical flux,\nxis in domain Ω⊂Rdand\n∇x·f=dX\ni=1∂xifi (4)\nLet us partition ΩintoMnon-overlapping quadrilateral/hexahedral elements Ωesuch that\nΩ =M[\ne=1Ωe\nThe elements Ωeare allowed to have curved boundaries in order to match curved boundaries of the problem domain Ω.\nTo construct the numerical approximation, we map each element Ωeto a reference element Ωo= [−1,1]dby a bijective\nmapΘe: Ωo→Ωe\nx= Θ e(ξ)\nwhere ξ= (ξi)d\ni=1are the coordinates in the reference element, and the subscript ewill usually be suppressed. We will\ndenote a d-dimensional multi-index as p= (pi)d\ni=1. In this work, the reference map is defined using tensor product\nLagrange interpolation of degree N≥1,\nΘ(ξ) =X\np∈Nd\nNbxpℓp(ξ) (5)\nwhere\nNd\nN={p= (p1, . . . , p d) :pi∈ {0,1, . . . , N },1≤i≤d} (6)\nand{ℓp}p∈Nd\nNis the degree NLagrange polynomial corresponding to the Gauss-Legendre-Lobatto (GLL) points\n{ξp}p∈Nd\nNso that Θ(ξp) =bxpfor all p∈Nd\nN. Thus, the points {ξp}p∈Nd\nNare where the reference map will be\nspecified and they will also be taken to be the solution points of the Flux Reconstruction scheme throughout this work.\nThe functions {ℓp}p∈Nd\nNcan be written as a tensor product of the 1-D Lagrange polynomials {ℓpi}N\npi=0of degree N\ncorresponding to the GLL points {ξpi}N\npi=0\nℓp(ξ) =dY\ni=1ℓpi(ξi), ℓ pi(ξi) =NY\nk=0,k̸=iξi−ξpk\nξpi−ξpk(7)\nThe numerical approximation of the conservation law will be developed by first transforming the PDE in terms of the\ncoordinates of the reference cell. To do this, we need to introduce covariant and contravariant basis vectors with respect\nto the reference coordinates.\n3APREPRINT - FEBRUARY 21, 2024\nDefinition 1 (Covariant basis) The coordinate basis vectors {ai}d\ni=1are defined so that ai,ajare tangent to {ξk=\nconst}where i, j, k are cyclic. They are explicitly given as\nai= (ai,1, . . . , a i,d) =∂x\n∂ξi, 1≤i≤d (8)\nDefinition 2 (Contravariant basis) The contravariant basis vectors\b\nai\td\ni=1are the respective normal vectors to the\ncoordinate planes {ξi= const }3\ni=1. They are explicitly given as\nai= (ai\n1, . . . , ai\nd) =∇xξi, 1≤i≤d (9)\nThe covariant basis vectors aican be computed by differentiating the reference map Θ(ξ). The contravariant basis\nvectors can be computed using [23, 22]\nJai=J∇ξi=aj×ak (10)\nwhere (i, j, k )are cyclic, and Jdenotes the Jacobian of the transformation which also satisfies\nJ=det\u0014∂x\n∂ξ\u0015\n=ai·(aj×ak) ( i, j, k )cyclic\nThe divergence of a flux vector can be computed in reference coordinates using the contravariant basis vectors as [ 23,22]\n∇x·f=1\nJdX\ni=1∂\n∂ξi(Jai·f) (11)\nConsequently, the gradient of a scalar function ϕbecomes\n∇ϕ=1\nJdX\ni=1∂\n∂ξi[(Jai)ϕ] (12)\nWithin each element Ωe, performing change of variables with the reference map Θe(11), the transformed conservation\nlaw is given by\neut+∇ξ·˜f=0 (13)\nwhere\neu=Ju, ˜fi=Jai·f=dX\nn=1Jai\nnfn (14)\nThe flux ˜fis referred to as the contravariant flux.\nThe vectors {Jai}d\ni=1are called the metric terms and the metric identity is given by\ndX\ni=1∂(Jai)\n∂ξi=0 (15)\nThe metric identity can be obtained by reasoning that the gradient of a constant function is zero and using (12) or\nthat a constant solution must remain constant in (13). The metric identity is crucial for studying free stream stream\npreservation of a numerical scheme.\nRemark 1 The equations for two dimensional case can be obtained by setting (Θ(ξ))3=x3(ξ) = ξ3so that\na3= (0,0,1).\n3 Lax-Wendroff Flux Reconstruction (LWFR) on curvilinear grids\nThe solution of the conservation law will be approximated by piecewise polynomial functions which are allowed to be\ndiscontinuous across the elements. In each element Ωe, the solution is approximated by\nbuδ\ne(ξ) =X\npue,pℓp(ξ) (16)\nwhere the ℓpare tensor-product polynomials of degree Nwhich have been already introduced before to define the map\nto the reference element. The hat will be used to denote functions written in terms of the reference coordinates and the\ndelta denotes functions which are possibly discontinuous across the element boundaries. Note that the coefficients ue,p\nare the values of the function at the solution points which are GLL points.\n4APREPRINT - FEBRUARY 21, 2024\nξξR\n2\nξL\n2ξL\n1 ξR\n1\nΘe(ˆξ1,ˆξ2)ˆnR,1ˆnL,1ˆnR,2\nˆnL,2ˆnR,1ˆnL,1ˆnR,2\nnL,2∂ΩR\no,i∂ΩR\ni\nFigure 1: Illustration of reference map, solution point projections, reference and physical normals\n3.1 Flux Reconstruction (FR)\nRecall that we defined the multi-index p= (pi)d\ni=1(6)where pi∈ {0,1. . . , N }. Let i∈ {1, . . . , d }denote a\ncoordinate direction and S∈ {L, R}so that (S, i)corresponds to the face ∂ΩS\no,iin direction ion side Swhich has\nthe reference outward normal bnS,i, see Figure 1. Thus, ∂ΩR\no,idenotes the face where reference outward normal is\nbnR,i=eiand∂ΩL\no,ihas outward unit normal bnL,i=−bnR,i.\nThe FR scheme is a collocation scheme at each of the solution points\b\nξp= (ξpi)d\ni=1, pi= 0, . . . , N\t\n. We will thus\nexplain the scheme for a fixed ξ=ξpand denote ξS\nias the projection of ξto the face S=L, R in the ithdirection\n(see Figure 1), i.e.,\n(ξS\ni)j=(ξj, j ̸=i\n−1, j =i, S=L\n+1, j =i, S=R(17)\nThe first step is to construct an approximation to the flux by interpolating at the solution points\n(˜fδ\ne)i(ξ) =X\np(Jai·f)(ξp)ℓp(ξ) (18)\nwhich may be discontinuous across the element interfaces. In order to couple the neighbouring elements and\nensure conservation property, continuity of the normal flux at the interfaces is enforced by constructing the\ncontinuous flux approximation using the FR correction functions gL, gR[18]. We construct this for the contravariant\nflux˜fδ(18) by performing correction along each direction i,\n(˜fe(ξ))i= (˜fδ\ne(ξ))i+ ((˜fe·bnR,i)∗−˜fδ\ne·bnR,i)(ξR\ni)gR(ξpi)−((˜fe·bnL,i)∗−˜fδ\ne·bnL,i)(ξL\ni)gL(ξpi) (19)\nwhere ˜fe·bnS,i(ξS\ni)denotes the trace value of the normal flux in element Ωeand(˜fe·bni)∗(ξS\ni)denotes the numerical\nflux. We will use Rusanov’s numerical flux [37] which for the face (S, i)is given by\n(˜fe·bnS,i)∗=˜f∗(u−\nS,i,u+\nS,i,bnS,i) =1\n2[(˜fδ·bnS,i)++ (˜fδ·bnS,i)−]−λS,i\n2(u+\nS,i−u−\nS,i) (20)\nThe(˜fδ·nS,i)±andu±\nS,idenote the trace values of the normal flux and solution from outer, inner directions respectively;\nthe inner direction corresponds to the element Ωewhile the outer direction corresponds to its neighbour across the\ninterface (S, i). The λS,iis a local wave speed estimate at the interface (S, i). For compressbile Euler’s equations (1),\nthe wave speed is estimated as [33]\nλ= max( |v−|,|v+|) + max( |c−|,|c+|), v±=v·n±, c±=p\nγp±/ρ±\nwhere nis the physical unit normal at the interface. The FR correction functions gL, gRin the degree N+ 1polynomial\nspace PN+1are a crucial ingredient of the FR scheme and have the property\ngL(−1) = gR(1) = 1 , g L(1) = gR(−1) = 0\nReference [ 18] gives a discussion on how the choice of correction functions leads to equivalence between FR and\nvariants of DG scheme. In this work, the correction functions known as g2orgHUfrom [ 18] are used since along with\n5APREPRINT - FEBRUARY 21, 2024\nGauss-Legendre-Lobatto (GLL) solution points, they lead to an FR scheme which is equivalent to a DG scheme using\nthe same GLL solution and quadrature points. Once the continuous flux approximation is obtained, the FR scheme is\ngiven by\nduδ\ne,p\ndt+1\nJe,p∇ξ·˜fe(ξp) =0,∀p (21)\nwhere Je,pis the Jacobian of the transformation at solution points xe,p. The FR scheme is explicitly written as\nduδ\ne,p\ndt+1\nJe,p∇ξ·˜fδ\ne(ξ)\n+1\nJe,pdX\ni=1((˜fe·bnR,i)∗−˜fδ\ne·bnR,i)(ξR\ni)g′\nR(ξpi)−((˜fe·bnL,i)∗−˜fδ\ne·bnL,i)(ξL\ni)g′\nL(ξpi) =0(22)\n3.2 Lax-Wendroff Flux Reconstruction (LWFR)\nThe LWFR scheme is obtained by following the Lax-Wendroff procedure for Cartesian domains [ 2] on the transformed\nequation (13). With undenoting the solution at time level t=tn, the solution at the next time level can be written\nusing Taylor expansion in time as\nun+1=un+N+1X\nk=1∆tk\nk!∂(k)\ntun+O(∆tN+2)\nwhere Nis the solution polynomial degree. Then, use ut=−1\nJ∇ξ·˜ffrom (13) to swap a temporal derivative with a\nspatial derivative and retaining terms upto O(∆tN+1)\nun+1=un−1\nJN+1X\nk=1∆tk\nk!∂(k−1)\nt (∇ξ·˜f)\nShifting indices and writing in a conservative form\nun+1=un−∆t\nJ∇ξ��˜F (23)\nwhere ˜Fis a time averaged approximation of the contravaraint flux ˜fgiven by\n˜F=NX\nk=0∆tk\n(k+ 1)!∂k\nt˜f≈1\n∆tˆtn+1\ntn˜fdt (24)\nWe first construct an element local order N+ 1approximation ˜Fδ\neto˜F(Section 3.2.1)\n˜Fδ\ne(ξ) =X\np˜Fe,pℓp(ξ)\nand which will be in general discontinuous across the element interfaces. Then, we construct the\ncontinuous time averaged flux approximation by performing a correction along each direction i, analogous to the\ncase of FR (19), leading to\n(˜Fe(ξ))i= (˜Fδ\ne(ξ))i+ ((˜Fe·bnR,i)∗−˜Fδ\ne·bnR,i)(ξR\ni)gR(ξpi)−((˜Fe·bnL,i)∗−˜Fδ\ne·bnL,i)(ξL\ni)gL(ξpi)(25)\nwhere, as in (20), the numerical flux (˜Fe·bnS,i)∗is an approximation to the time average flux and is computed by a\nRusanov-type approximation,\n(˜Fe·bnS,i)∗=1\n2[(˜Fδ·bnS,i)++ (˜Fδ·bnS,i)−]−λS,i\n2(U+\nS,i−U−\nS,i) (26)\nwhere Uis the approximation of time average solution given by\nU=NX\nk=0∆tk\n(k+ 1)!∂k\ntu≈1\n∆tˆtn+1\ntnudt (27)\n6APREPRINT - FEBRUARY 21, 2024\nThe computation of dissipative part of (26) using the time averaged solution instead of the solution at time tnwas\nintroduced in [ 2] and was termed D2 dissipation. It is a natural choice in approximating the time averaged numerical\nflux and doesn’t add any significant computational cost because the temporal derivatives of uare already available\nwhen computing the local approximation ˜Fδ. The choice of D2 dissipation reduces to an upwind scheme in case of\nconstant advection equation and leads to enhanced Fourier CFL stability limit [2].\nThe Lax-Wendroff update is performed following (21) for (23)\nun+1\ne,p=un\ne,p−∆t\nJe,p∇ξ·˜Fe(ξp)\nwhich can be explicitly written as\nun+1\ne,p=un\ne,p−∆t\nJe,p∇ξ·˜Fδ\ne(ξp)\n−∆t\nJe,pdX\ni=1((˜Fe·bnR,i)∗−˜Fδ\ne·bnR,i)(ξR\ni)g′\nR(ξpi)−((˜Fe·bnL,i)∗−˜Fδ\ne·bnL,i)(ξL\ni)g′\nL(ξpi)(28)\nBy multiplying (28) by quadrature weights Je,pwpand summing over p, it is easy to see that the scheme is conservative\n(see Appendix A) in the sense that\nun+1\ne=un\ne−∆t\n|Ωe| dX\ni=1ˆ\n∂ΩR\no,i(˜Fe·bnR,i)∗dSξ+ˆ\n∂ΩL\no,i(˜Fe·bnL,i)∗dSξ!\n, (29)\nwhere the element mean value ueis defined to be\nue=1\n|Ωe|X\npue,pJe,pwp (30)\n3.2.1 Approximate Lax-Wendroff procedure\nWe now illustrate how to approximate the time average flux at the solution points ˜Fe,pwhich is required to construct\nthe element local approximation ˜Fδ\ne(ξ)using the approximate Lax-Wendroff procedure [ 50]. For N= 1,(24) requires\n∂t˜fwhich is approximated as\n∂t˜fδ(ξp) =˜f(up+ ∆t(ut)p)−˜f(up−∆t(ut)p)\n2∆t(31)\nwhere element index eis suppressed as all these operations are local to each element. The time index is also suppresed\nas all quantities are used from time level tn. Theutabove is approximated using (13)\n(ut)p=−1\nJp∇ξ·˜fδ(ξp) (32)\nwhere ˜fδ\neis the cell local approximation to the flux ˜fgiven in (18). For N= 2, (24) additionally requires ∂tt˜f\n∂tt˜fδ(ξp) =1\n∆t2\u0014\n˜f\u0012\nup+ ∆t(ut)p+∆t\n2(utt)p\u0013\n−2˜f(up) +˜f\u0012\nup−∆t(ut)p+∆t\n2(utt)p\u0013\u0015\nwhere the element index eis again suppressed. We approximate uttas\n(utt)p=−1\nJp∇ξ·∂t˜fδ(ξp) (33)\nThe procedure for other degrees will be similar and the derivatives ∇ξare computed using a differentiation matrix. The\nimplementation can be made efficient by accounting for cancellations of ∆tterms. Since this step is similar to that on\nCartesian grids, the reader is referred to Section 4 of [2] for more details.\n3.3 Free stream preservation for LWFR\nSince the divergence in a Flux Reconstruction (FR) scheme (22) is computed as the derivative of a polynomial, the\nfollowing metric identity is required for our scheme to preserve a constant state\ndX\ni=1∂\n∂ξiIN(Jai) =0 (34)\n7APREPRINT - FEBRUARY 21, 2024\nwhere INis the degree Ninterpolation operator defined as\nIN(f)(ξ) =X\npℓp(ξ)f(ξp) (35)\nThe study of free-stream preservation was made in [ 23] showing that satisfying (34) gives free stream preservation.\nHowever, it was also shown that the identities impose additional constraints on the degree of the reference map Θ. The\nremedy given in (34) is to replace the metric terms Jaiby a different degree Napproximation IN(Jai)so that (34)\nreduces to\ndX\ni=1∂\n∂ξiININ(Jai) =dX\ni=1∂\n∂ξiIN(Jai) =0 (36)\nIn [23], choices of INlike the conservative curl form were proposed which ensured (36) without any additional\nconstraints on the degree of the reference map Θ. Those choices are only relevant in 3-D as, in 2-D, they are equivalent\nto interpolating Θto a degree Npolynomial before computing the metric terms which is the choice of INwe make in\nthis work by defining the reference map as in (5).\nIn this section, we show that the identities (34) are enough to ensure free stream preservation for LWFR. Throughout this\nsection, we assume that the mesh is well-constructed [23] which is a property that follows from the natural assumption\nof global continuity of the reference map.\nDefinition 3 Consider a mesh where element faces in reference element Ωoare denoted as\b\n∂ΩS\no,i\t\nfor coordinate\ndirections 1≤i≤dandS=L/R chosen so that the corresponding reference normals {bnS,i}arebnR,i=eiand\nbnL,i=−bnR,iwhere {ei}d\ni=1is the Cartesian basis, see Figure 1. The mesh is said to be well-constructed if the\nfollowing is satisfied\ndX\nm=1(IN(Jam)+−IN(Jam)−)(bnS,i)m=0∀1≤i≤d, s =L, R (37)\nwhere ±are used to denote trace values from Ωoor from the neighbouring element respectively.\nRemark 2 From (10), the identity (37) can be seen as a property of the tangential derivatives of the reference map\nat the faces and is thus obtained if the reference map is globally continuous. Also, since the unit normal vector of an\nelement at interface iis given by Jai/\r\rJai\r\r, (37) also gives us continuity of the unit normal across interfaces.\nAssuming the current solution is constant in space, un=c, we will begin by proving that the approximate time\naveraged flux and solution satisfy\n˜Fδ=˜fδ(c),U=uδ=c (38)\nFor the constant physical flux f(c), the contravariant flux ˜fwill be\n˜fi=IN(Jai)·f(c) =dX\nn=1IN(Jai\nn)fn(c)\nUsing (32), we obtain at each solution point\nut=−1\nJ∇ξ·˜fδ=−1\nJdX\ni=1∂ξi(˜fδ)i\n=−1\nJdX\ni=1∂ξi(IN(Jai)·f(c)) =−1\nJ dX\ni=1∂ξiIN(Jai)!\n·f(c)\n=0\nwhere the last equality follows from using the metric identity (34). For polynomial degree N= 1, recalling (31), this\nproves that\n∂t˜fδ=˜f(u+ ∆tut)−˜f(u−∆tut)\n2∆t=0\nThus, we obtain\n˜Fδ=˜fδ+∆t\n2∂t˜fδ=˜fδ,U=uδ+∆t\n2∂tuδ=uδ\n8APREPRINT - FEBRUARY 21, 2024\nBuilding on this, for N= 2, by (33),\nutt=−1\nJ∇ξ·∂t˜fδ=0\nwhich will prove ∂tt˜fδ=0and we similarly obtain the following for all degrees\n˜Fδ=NX\nk=0∆tk\n(k+ 1)!∂k\nt˜fδ=˜fδ=\b\nJai·f(c)\td\ni=1(39)\nU=NX\nk=0∆tk\n(k+ 1)!∂k\ntuδ=uδ=c (40)\nTo prove free stream preservation, we argue that the update (28) vanishes as the volume terms involving divergence of\n˜Fδand the surface terms involving trace values and numerical flux vanish. By (39), the volume terms in (28) are given\nby\n1\nJ∆t dX\ni=1∂ξiIN(Jai)!\n·f(c)\nand vanish by the metric identity (36). By (40), the dissipative part of the numerical flux (26) is computed with the\nconstant solution un=cand will thus vanish. For the central part of the numerical flux, as the mesh is well-constructed\n(Definition 3), the trace values are given by\n(˜Fδ·bni)+=dX\nm=1(IN(Jam)·f(c))+(bni)m=dX\nm=1(IN(Jam)·f(c))−(bni)m= (˜Fδ·bni)−\nThus, the numerical flux agrees with the physical flux at element interfaces, making the surface terms in (28) vanish.\n4 Shock capturing and admissibility preservation\nThe LWFR scheme (28) gives a high order method for smooth problems, but most practical problems involving\nhyperbolic conservation laws consist of non-smooth solutions containing shocks. In such situations, using a higher\norder method is bound to produce Gibbs oscillations, as is stated in Godunov’s order barrier theorem [ 15]. The cure\nis to non-linearly add dissipation in regions where the solution is non-smooth, with methods like artificial viscosity,\nlimiters and switching to a robust lower order scheme; the resultant scheme will be non-linear even for linear equations.\nIn this work, we use the blending scheme for LWFR proposed in [ 3] for Gauss-Legendre solution points. In order to be\ncompatible with Trixi.jl [33] and make use of this excellent code, we introduce LWFR with blending scheme for\nGauss-Legendre-Lobatto solution points, which are also used in Trixi.jl . As in [ 3], the blending scheme has to be\nconstructed to be provably admissibility preserving (Definition 4). The rest of this section consists of terminologies for\nadmissibility preservation and the admissibility preserving blending scheme, some of which is a review of [3].\n4.1 Admissibility preservation\nFor the Euler’s equations, since negative density and pressure are nonphysical, an admissible solution is one that\nbelongs to the admissible set {u:ρ(u), p(u)>0}. Since the admissibility preservation approach used in this work\ncan be used for more general problems, we introduce the general notations here.\nLetUad⊂Rpdenote the convex set in which physically correct solutions of the general conservation law (3)must\nbelong; we suppose that it can be written in terms of Kconstraints as\nUad={u∈Rp:pk(u)>0,1≤k≤K} (41)\nwhere each admissibility constraint pkis concave if pj>0for all j < k . For Euler’s equations, K= 2andp1, p2are\ndensity, pressure functions respectively; the density is clearly a concave function of uand if the density is positive\nthen it can be easily verified that the pressure is also a concave function of the conserved variables. The admissibility\npreserving property, also known as convex set preservation property , of the conservation law can be written as\nu(·, t0)∈ Uad =⇒ u(·, t)∈ Uad, t > t 0 (42)\nand thus we define an admissibility preserving flux reconstruction scheme as follows.\n9APREPRINT - FEBRUARY 21, 2024\nFe\u00001\n2Fe+1\n2\nxe\u00001\n2xe+1\n2f1\n2f3\n2f5\n2f7\n2\nGLL nodes\nFR element\nSubcells\nFigure 2: Subcells used by the lower order scheme for degree N= 4\nDefinition 4 The flux reconstruction scheme is said to be admissibility preserving if\nun\ne,p∈ Uad,∀e,p =⇒ un+1\ne,p∈ Uad,∀e,p\nwhere Uadis the admissibility set of the conservation law.\nTo obtain an admissibility preserving scheme, we exploit the weaker admissibility preservation in means property.\nDefinition 5 The flux reconstruction scheme is said to be admissibility preserving in the means if\nun\ne,p∈ Uad,∀e,p =⇒ un+1\ne∈ Uad,∀e\nwhere Uadis the admissibility set of the conservation law and uedenotes the element mean (30).\n4.2 Blending scheme\nIn this section, we explain the blending procedure which obtains admissibility preservation in means property for LWFR\nscheme on curvilinear grids using Gauss-Legendre-Lobatto solution points. The procedure is very similar to that of [ 3]\nfor Cartesian grids where Gauss-Legendre solution points were used.\nLet us write the LWFR update equation (23) as\nuH,n+1\ne =un\ne−∆t\n∆xeRH\ne (43)\nwhere ueis the vector of nodal values in the element Ωe. Suppose we also have a lower order, non-oscillatory scheme\navailable to us in the form\nuL,n+1\ne =un\ne−∆t\n∆xeRL\ne (44)\nThen a blended scheme is given by\nun+1\ne= (1−αe)uH,n+1\ne +αeuL,n+1\ne =un\ne−∆t\n∆xe[(1−αe)RH\ne+αeRL\ne] (45)\nwhere αe∈[0,1]must be chosen based on some local smoothness indicator. If αe= 0, then we obtain the high order\nLWFR scheme, while if αe= 1then the scheme becomes the low order scheme that is less oscillatory. In subsequent\nsections, we explain the details of the lower order scheme and the design of smoothness indicators.\n4.2.1 Blending scheme in 1-D\nLet us subdivide each element Ωe= [xe−1\n2, xe+1\n2]intoN+ 1subcells associated to the solution points {xe\np, p=\n0,1, . . . , N }of the LWFR scheme. Thus, we will have N+ 2subfaces within each element Ωedenoted by {xe\np+1\n2, p=\n−1, . . . , N }where xe\n−1\n2=xe−1\n2=xe\n0,xe\nN+1\n2=xe+1\n2=xe\nN. For maintaining a conservative scheme, the pthsubcell\nis chosen so that\nxe\np+1\n2−xe\np−1\n2=wp∆xe, 0≤p≤N (46)\nwhere wpis the pthquadrature weight associated with the solution points, and ∆xe=xe+1\n2−xe−1\n2. Figure 2 gives an\nillustration of the subcells for degree N= 4case.\nThe low order scheme is obtained by updating the solution in each of the subcells by a finite volume scheme,\nuL,n+1\ne,0=un\ne,0−∆t\nw0∆xe[fe\n1\n2−Fe−1\n2]\nuL,n+1\ne,p =un\ne,p−∆t\nwp∆xe[fe\np+1\n2−fe\np−1\n2], 1≤p≤N−1\nuL,n+1\ne,N =un\ne,N−∆t\nwN∆xe[Fe+1\n2−fe\nN−1\n2](47)\n10APREPRINT - FEBRUARY 21, 2024\nξ(2,2)\nξ0Θe(ˆξ1,ˆξ2)\nxe,0xe,(2,2)\nFigure 3: Subcells in a curved element\nThe fluxes fe\np+1\n2=f(un\np,un\np+1)are first order numerical fluxes and Fe+1\n2is the blended numerical flux which is\na convex combination of the time averaged numerical flux (26) and a lower order flux fe+1\n2=f(un−\ne+1\n2,un+\ne+1\n2) =\nf(ue,N,ue+1,0). The same blended numerical flux Fe+1\n2is used in the high order LWFR residual (43, 28); see\nRemark 1 of [ 3] for why it is crucial to do so to ensure conservation. In this work, Rusanov’s flux [ 37] will be used for\nthe inter-element fluxes and for fluxes in the lower order scheme. The element mean value obtained by the low order\nscheme, high order scheme and the blended scheme satisfy the conservative property\nuL,n+1\ne =uH,n+1\ne =un+1\ne=un\ne−∆t\n∆xe(Fe+1\n2−Fe−1\n2) (48)\nThe inter-element fluxes Fe±1\n2are used both in the low and high order schemes at xe+1\n2=xe\nN, xe−1\n2=xN\n0respectively,\nwhere both schemes have a solution point and an element interface. It has to be chosen carefully to balance accuracy,\nrobustness and ensure admissibility preservation. The following natural initial guess is made and then further corrected\nto enforce admissibility, as explained in Section 4.3\nFe+1\n2= (1−αe+1\n2)FLW\ne+1\n2+αe+1\n2fe+1\n2, αe+1\n2∈[0,1] (49)\nwhere FLW\ne+1\n2is the high order inter-element time-averaged numerical flux of the LWFR scheme (26) andfe+1\n2is a\nlower order flux at the face xe+1\n2shared between FR elements and subcells. The coefficient αe+1\n2is given byαe+αe+1\n2\nwhere αeis the blending coefficient computed with a smoothness indicator (Section 4.2.3). The reader is referred to\nSection 3.2 of [3] for more details.\nRemark 3 The contribution to RL\ne,RH\neof the flux Fe+1\n2has coefficients given by∆t\nwN∆xe,∆t\n∆xeg′\nR(ξN)respectively, as\ncan be seen from (47, 28). Since we use g2correction functions with Gauss-Legendre-Lobatto solution points, we have\nfrom equivalence of FR and DG, g′\nR(ξN) =ℓN(1)/wN= 1/wN. Thus, the coefficient is the same for both higher and\nlower order residuals and we add the contribution without a blending coefficient. This is different from the case of\nGauss-Legendre solution points used in the blending scheme of [3].\n4.2.2 Blending scheme for curvilinear grids\nThe subcells for a curved element will be defined by the reference map, as shown in Figure 3. As in Appendix B.3 of [ 17],\nthe finite volume formulation on subcells is obtained by an integral formulation of the transformed conservation law (13).\nIn the reference element, consider subcells Cpof size wp=Qd\ni=1wpiwith solution point (ξpi)d\ni=1corresponding to\nthe multi-index p= (pi)d\ni=1where pi∈ {0,1. . . , N }. Fix a subcell C=Cparound the solution point ξ= (ξpi)d\ni=1\nand denote ξL/R\ni as in (17). Integrate the conservation law on the fixed subcell C\nˆ\nCJutdV+ˆ\nC∇ξ·˜fdV=0\nNext perform one point quadrature in the first term and apply Gauss divergence theorem on the second term to get\nJe,pdup\ndtwp+ˆ\n∂C˜f·bndA=0 (50)\n11APREPRINT - FEBRUARY 21, 2024\nwherebnis the reference normal vector on the subcell surface. Now evaluate this surface integral by approximating\nfluxes in each direction with numerical fluxes\nˆ\n∂C˜f·bndA=dX\ni=1wp\nwpi[(˜fδ\nC·bnR,i)∗(ξR\ni) + ( ˜fδ\nC·bnL,i)∗(ξL\ni)],bnR,i=ei,bnL,i=−ei (51)\nThe explicit lower order method using forward Euler update is given by\nun+1\np=un\np−∆t\nJe,pdX\ni=11\nwpi[(˜fδ\nCp·bnR,i)∗(ξR\ni) + ( ˜fδ\nCp·bnL,i)∗(ξL\ni)] (52)\nFor the subcells whose interfaces are not shared by the FR element, the fluxes are computed, following [17], as\n(˜fδ\nCp·bnR,i)∗(ξR\ni) =∥(nR,i)p∥f∗\u0012\nup,upi+,(nR,i)p\n∥(nR,i)p∥\u0013\n(˜fδ\nCp·bnL,i)∗(ξL\ni) =∥(nL,i)p∥f∗\u0012\nupi−,up,(nL,i)p\n∥(nL,i)p∥\u0013\n(pi±)m=\u001apm m̸=i\npi±1m=i(53)\nwhere (ns,i)pis the normal vector of subcell Cpin direction iand side s∈ {L, R}. The numerical fluxes (53) are\ntaken to be Rusanov’s flux (20)\n˜f∗(u−,u+,n) =1\n2[(f·n)(u+) + (f·n)(u−)]−λ\n2(u+−u−) (54)\nAt the interfaces shared by FR elements, the first order numerical flux is computed by setting u±in(54) to element\ntrace values as in (20). However, the lower order residual needs to be computed using the same inter-element flux as the\nhigher order scheme at interfaces of the Flux Reconstruction (FR) elements. Thus, for example, for an element Ωeat\nsolution point ξ=ξpwithp=0, the subcell update will be given by\nun+1\ne,0=un\ne,0−∆t\nJe,pdX\ni=11\nwpi[(˜fδ\nC0·bnR,i)∗(ξR\ni) + ( ˜Fδ\ne·bnL,i)∗(ξL\ni)] (55)\nwhere (˜Fδ\ne·bni)∗(ξL\ni)is the blended numerical flux and is computed by taking a convex combination of the lower\norder flux chosen as in (20) and the time averaged flux (26). An initial guess is made as in 1-D (49) and then further\ncorrection is performed to ensure admissibility, as explained in Section 4.3.2. Other subcells neighbouring the element\ninterfaces will also use the blended numerical fluxes at the element interfaces and thus have an update similar to (55).\nThen, by multiplying each update equation of each subcell pbywpand summing over p, the conservation property is\nobtained\nuL,n+1\ne =X\npuL,n+1\ne,pwp=un\ne−∆t\n|Ωe| dX\ni=1ˆ\n∂ΩR\no,i(˜Fδ\ne·bnR,i)∗dSξ+ˆ\n∂ΩL\no,i(˜Fδ\ne·bnL,i)∗dSξ!\n(56)\nSince we also have the conservation property in the higher order scheme (29), the blended scheme will be conservative,\nanalogous to the 1-D case (48).\nThe expressions for normal vectors on the subcells needed to compute (53) are taken from Appendix B.4 of [ 17] where\nthey were derived by equating the high order flux difference and Discontinuous Galerkin split form. We directly state\nthe normal vectors here, denoting (nR,i)pas the outward normal direction in subcell Cpalong the positive idirection\n(nR,i)p=IN(Jai)pi|0+piX\nl=0wl∂ξiIN(Jai)pi|l, (pi|l)m=\u001apmm̸=i\nplm=i.\nwhere {wl}N\nl=0are quadrature weights corresponding to solution points, INis the approximation operator for metric\nterms (36), and (nL,i)pcan be obtained by the relation (nL,i)p=−(nR,i)pi−, where pi−was defined in (53).\n12APREPRINT - FEBRUARY 21, 2024\nFree-stream preservation. To show the free stream preservation of the lower order scheme with the chosen normal\nvectors, we consider a constant initial state u=cand show that the finite volume residual will be zero. A constant\nstate implies that the time average of the contravariant flux will be the contravariant flux itself (38). Thus, all numerical\nfluxes including element interface fluxes are first order fluxes like in (53) and the residual at pin direction iis given by\nf(c)\nwpi·((nR,i)p+ (nL,i)p) =f(c)\nwpi·(IN(Jai)pi|0+piX\nl=0wl∂ξiIN(Jai)pi|l−IN(Jai)pi|0−pi−1X\nl=0wl∂ξiIN(Jai)pi|l)\n=f(c)·∂ξiIN(Jai)p\nThe residuals in other directions give similar terms and summing them gives\nf(c)·dX\ni=1∂\n∂ξiIN(Jai)p=0\nby the metric identities, thus satisfying the free stream preservation condition.\n4.2.3 Smoothness indicator\nThe smoothness of the numerical solution is assessed by writing the degree Npolynomial within each element in terms\nof an orthonormal basis like Legendre polynomials and then analyzing the decay of the coefficients [ 29,21,17]. For a\nsystem of PDE, the orthonormal expansion of a derived quantity q(u)is used; a good choice for Euler’s equations is the\nproduct of density and pressure [17] which depends on all the conserved quantities.\nLetq=q(u)be the quantity used to measure the solution smoothness. With {Lj}N\nj=0being the 1-D Legendre\npolynomial basis of degree N, taking tensor product gives the degree NLegendre basis\nLp(ξ) =dY\ni=1Lpi(ξi), p i∈ {0,1, . . . , N }\nThe Legendre basis representation of qcan be obtained as\nqh(ξ) =X\npˆqpLp(ξ),ξ∈Ωo, ˆqp=ˆ\nΩoq(uδ(ξ))Lj(ξ)dξ\nThe Legendre coefficients {ˆqp}are computed using the quadrature induced by the solution points,\nˆqp=X\nqq(ue,q)Lp(ξq)wq\nDefine\nSK=X\np,pi≤Kˆq2\np\nwhich the measures the \"energy\" in qh. Then the energy contained in highest modes relative to the total energy of the\npolynomial is computed as follows\nE= max\u001aSN−SN−1\nSN,SN−1−SN−2\nSN−1\u001b\nIn 1-D, this simplifies to the expression of [17, 3]\nE= max(\nˆq2\nNPN\nj=0ˆq2\nj,ˆq2\nN−1PN−1\nj=0ˆq2\nj)\nTheNthLegendre coefficient ˆqNof a function which is in the Sobolev space H2decays as O(1/N2)(see Chapter\n5, Section 5.4.2 of [ 10]). We consider smooth functions to be those whose Legendre coefficients ˆqNdecay at rate\nproportional to 1/N2or faster so that their squares decay proportional to 1/N4[29]. Thus, the following dimensionless\nthreshold for smoothness is proposed in [17]\nT(N) =a·10−c(N+1)4\n13APREPRINT - FEBRUARY 21, 2024\nwhere parameters a=1\n2andc= 1.8are obtained through numerical experiments. To convert the highest mode energy\nindicator Eand threshold value Tinto a value in [0,1], the logistic function is used\n˜α(E) =1\n1 + exp( −s\nT(E−T))\nThe sharpness factor swas chosen to be s= 9.21024 so that blending coefficient equals α= 0.0001 when highest\nenergy indicator E= 0. In regions where ˜α= 0or˜α= 1, computational cost can be saved by performing only the\nlower order or higher order scheme respectively. Thus, the values of αare clipped as\nαe:=\n\n0,if˜α < α min\n˜α, ifαmin≤˜α≤1−αmin\n1,if1−αmin<˜α\nwithαmin= 0.001. Finally, since shocks can spread to the neighbouring cells as time advances, some smoothening of\nαis performed as\nαfinal\ne= max\nE∈Ee\u001a\nαe,1\n2αE\u001b\n(57)\nwhere Eedenotes the set of elements sharing a face with Ωe.\n4.3 Flux limiter for admissibility preservation\nWe first review the flux limiting process for admissibility preservation from [ 3] for 1-D and then do a natural extension\nto curvilinear meshes. The first step in obtaining an admissibility preserving blending scheme is to ensure that the lower\norder scheme preserves the admissibility set Uad. This is always true if all the fluxes in the lower order method are\ncomputed with an admissibility preserving low order finite volume method. But the LWFR scheme uses a time average\nnumerical flux and maintaining conservation requires that we use the same numerical flux at the element interfaces for\nboth lower and higher order schemes (Remark 1 of [ 3]). To maintain accuracy and admissibility, we carefully choose a\nblended numerical flux Fe+1\n2as in (49) but this choice may not ensure admissibility and further limitation is required.\nOur proposed procedure for choosing the blended numerical flux will give us an admissibility preserving lower order\nscheme. As a result of using the same numerical flux at element interfaces in both high and low order schemes, element\nmeans of both schemes are the same (Theorem 1). A consequence of this is that our scheme now preserves admissibility\nof element means and thus we can use the scaling limiter of [48] to get admissibility at all solution points.\nThe theoretical basis for flux limiting can be summarized in the following Theorem 1.\nTheorem 1 Consider the LWFR blending scheme (45) where low and high order schemes use the same numerical flux\n(˜Fδ\ne·bni)∗(ξs\ni)at every element interface and the lower order residual is computed using the first order finite volume\nscheme (52). Then the following can be said about admissibility preserving in means property (Definition 5) of the\nscheme:\n1.element means of both low and high order schemes are same, and thus the blended scheme (45) is admissibility\npreserving in means if and only if the lower order scheme is admissibility preserving in means;\n2.if the blended numerical flux (˜Fδ\ne·bni)∗(ξs\ni)is chosen to preserve the admissibility of lower-order updates at\nsolution points adjacent to the interfaces, then the blending scheme (45) will preserve admissibility in means.\nProof By (29, 56), element means are the same for both low and high order schemes. Thus, admissibility in means\nof one implies the same for the other, proving the first claim. For the second claim, note that our assumptions imply\nuL,n+1\ne,p given by (52, 55) are in Uadfor all p. Therefore, we obtain admissibility in means property of the lower order\nscheme by (56) and thus admissibility in means for the blended scheme. 2\n4.3.1 Flux limiter for admissibility preservation in 1-D\nFlux limiting ensures that the update obtained by the lower order scheme will be admissible so that, by Theorem 1,\nadmissibility in means is obtained. The procedure of flux limiting will be explained for the element Ωe= [xe−1\n2, xe+1\n2].\nThe lower order scheme is computed with a first order finite volume method so that admissibility is already ensured for\ninner solution points; i.e., we already have\nuL,n+1\ne,p∈ Uad, 1≤p≤N−1\n14APREPRINT - FEBRUARY 21, 2024\nThe admissibility for the first ( p= 0) and last solution points ( p=N) will be enforced by appropriately choosing\nthe inter-element flux Fe+1\n2. The first step is to choose a candidate for Fe+1\n2which is heuristically expected to give\nreasonable control on spurious oscillations while maintaining accuracy in smooth regions, e.g.,\nFe+1\n2= (1−αe+1\n2)FLW\ne+1\n2+αe+1\n2fe+1\n2, αe+1\n2=αe+αe+1\n2\nwhere fe+1\n2is the lower order flux at the face e+1\n2shared between FR elements and subcells, and αeis the blending\ncoefficient (45) based on an element-wise smoothness indicator (Section 4.2.3).\nThe next step is to correct Fe+1\n2to enforce the admissibility constraints. The guiding principle of this approach is to\nperform the correction within the face loops, minimizing storage requirements and additional memory reads. The lower\norder updates in subcells neighbouring the e+1\n2face with the candidate flux are\n“un+1\n0=un\ne+1,0−∆t\nw0∆xe+1(fe+1\n3\n2−Fe+1\n2)\n“un+1\nN=un\ne,N−∆t\nwN∆xe(Fe+1\n2−fe\nN+1\n2)(58)\nTo correct the interface flux, we will again use the fact that low order finite volume flux fe+1\n2=f(u−\ne+1\n2,u+\ne+1\n2)\npreserves admissibility, i.e.,\n“ulow,n+1\n0 =un\ne+1,0−∆t\nw0∆xe+1(fe+1\n3\n2−fe+1\n2)∈ Uad\n“ulow,n+1\nN =un\ne,N−∆t\nwN∆xe(fe+1\n2−fe\nN+1\n2)∈ Uad\nLet{pk,1≤1≤K}be the admissibility constraints (41) of the conservation law. The numerical flux is corrected by\niterating over the admissibility constraints as explained in Algorithm 1\nAlgorithm 1 Computation of blended flux Fe+1\n2\nInput: FLW\ne+1\n2,fe+1\n2,fe+1\n3\n2,fe\nN+1\n2,un\ne+1,0,un\ne,N, αe, αe+1\nOutput: Fe+1\n2\nαe+1\n2=αe+αe+1\n2\nFe+1\n2←(1−αe+1\n2)FLW\ne+1\n2+αe+1\n2fe+1\n2▷Heuristic guess to control oscillations\n“un+1\n0←un\ne+1,0−∆t\nw0∆xe+1(fe\n3\n2−Fe+1\n2) ▷FV inner updates with guessed Fe+1\n2\n“un+1\nN←un\ne,N−∆t\nwN∆xe(Fe+1\n2−fe\nN+1\n2)\n“ulow,n+1\n0 =un\ne+1,0−∆t\nw0∆xe+1(fe+1\n3\n2−fe+1\n2) ▷FV inner updates with fe+1\n2\n“ulow,n+1\nN =un\ne,N−∆t\nwN∆xe(fe+1\n2−fe\nN+1\n2)\nfork= 1 : Kdo ▷Correct Fe+1\n2forKadmissibility constraints\nϵ0, ϵN←1\n10pk(“ulow,n+1\n0 ),1\n10pk(“ulow,n+1\nN )\nθ←min\u0012\nminj=0,N\f\f\f\fϵj−pk(“ulow,n+1\nj))\npk(“un+1\nj)−pk(“ulow,n+1\nj)\f\f\f\f,1\u0013\nFe+1\n2←θFe+1\n2+ (1−θ)fe+1\n2▷FV inner updates with guessed Fe+1\n2\n“un+1\n0←un\ne+1,0−∆t\nw0∆xe+1(fe+1\n3\n2−Fe+1\n2)\n“un+1\nN←un\ne,N−∆t\nwN∆xe(Fe+1\n2−fe\nN+1\n2)\nend for\nBy concavity of pk, after the kthiteration, the updates computed using flux Fe+1\n2will satisfy\npk(“un+1\nj) =pk(θ(“un+1\nj)prev+ (1−θ)“ulow,n+1\nj )\n≥θpk((“un+1\nj)prev) + (1 −θ)pk(“ulow,n+1\nj )\n≥ϵj, j = 0, N(59)\n15APREPRINT - FEBRUARY 21, 2024\nsatisfying the kthadmissibility constraint; here (“un+1\nj)prevdenotes “un+1\njbefore the kthcorrection and the choice of\nϵj=1\n10pk(“ulow,n+1\nj )is made following [ 35]. After the Kiterations, all admissibility constraints will be satisfied and\nthe resulting flux Fe+1\n2will be used as the interface flux keeping the lower order updates and thus the element means\nadmissible. Thus, by Theorem 1, the choice of blended numerical flux gives us admissibility preservation in means. We\nnow use the scaling limiter of [ 48] to obtain an admissibility preserving scheme as defined in Definition 4. An overview\nof the complete residual computation of Lax-Wendroff Flux Reconstruction scheme can be found in Algorithm 3.\n4.3.2 Flux limiter for admissibility preservation on curved meshes\nConsider the calculation of the blended numerical flux for a corner solution point of the element, see Figure 3. A corner\nsolution point is adjacent to interfaces in all ddirections, making its admissibility preservation procedure different from\n1-D. In particular, let us consider the corner solution point p=0and show how we can apply the 1-D procedure in\nSection 4.3.1 to ensure admissibility at such points. The same procedure applies to other corner and non-corner points.\nThe lower order update at the corner is given by (55)\n“un+1\ne,0=un\ne,0−∆t\nJe,pdX\ni=11\nwpi[(˜fδ\nC0·bnR,i)∗(ξR\ni) + ( “Fδ\ne·bnL,i)∗(ξL\ni)] (60)\nwherebni=eiis the reference normal vector on the subcell interface in direction i,(˜fδ\nC0·bnR,i)∗denotes the lower order\nflux(51) at the subcell C0surrounding ξ0,(“Fδ\ne·bnL,i)∗(ξL\ni)is the initial guess candidate for the blended numerical\nflux. Pick ki>0such thatPd\ni=1ki= 1and\n“ulow,n+1\ni :=un\ne,0−∆t\nkiwpiJe,p[(˜fδ\nC0·bnR,i)∗(ξR\ni) + ( ˜f·bnL,i)∗(ξL\ni)], 1≤i≤d (61)\nsatisfy\n“ulow,n+1\ni ∈ Uad, 1≤i≤d (62)\nwhere (˜f·bni)∗(ξL\ni)is the first order finite volume flux computed at the FR element interface.\nThe{ki}that ensure (62) will exist provided the appropriate CFL restrictions are satisfied because the lower order\nscheme using the first order numerical flux at element interfaces is admissibility preserving. The choice of {ki}should\nbe made so that (62) is satisfied with the least time step restriction. However, we make the trivial choice of equal ki’s\nmotivated by the experience of [ 3], where it was found that even this choice does not impose any additional time step\nconstraints over the Fourier stability limit. After choosing ki’s, we have reduced the update to 1-D and can repeat the\nsame procedure as in Algorithm 1 where for all directions i, the neighbouring element is chosen along the normal\ndirection. After the flux limiting is performed following the Algorithm 1, we obtain (“Fδ\ne·bnL,i)∗(ξL\ni)such that\n“un+1\ni:=un\ne,0−∆t\nkiwpiJe,p[(˜fδ\nC0·bnR,i)∗(ξR\ni) + ( “Fδ·bnL,i)∗(ξL\ni)]∈ Uad (63)\nThen, we will get\ndX\ni=1ki“un+1\ni=“un+1\ne,0∈ Uad (64)\nalong with admissibility of all other corner and non-corner solution points where the flux (“Fδ·bni)∗(ξR\ni)is used.\nFinally, by Theorem 1, admissibility in means (Definition 30) is obtained and the scaling limiter of [ 48] can be used to\nobtain an admissibility preserving scheme (Definition 4).\n5 Adaptive mesh refinement\nAdaptive mesh refinement helps resolve flows where the relevant features are localized to certain regions of the physical\ndomain by increasing the mesh resolution in those regions and coarsening in the rest of the domain. In this work, we\nallow the adaptively refined meshes to be non-conforming, i.e., element neighbours need not have coinciding solution\npoints at the interfaces (Figure 4a). We handle the non-conformality using the mortar element method first introduced\nfor hyperbolic PDEs in [25].\nIn order to perform the transfer of solution during coarsening and refinement, we introduce some notations and operators.\nDefine the 1-D reference elements\nI0= [−1,0], I 1= [0,1], I = [−1,1],Nd\n1={0,1}d(65)\n16APREPRINT - FEBRUARY 21, 2024\nand the bijections ϕs:Is→Ifors= 0,1as\nϕ0(ξ) = 2 ξ+ 1, ξ∈I0, ϕ 1(ξ) = 2 ξ−1, ξ∈I1 (66)\nso that the inverse maps ϕ−1\ns:I→Isare given by\nϕ−1\n0(ξ) =ξ−1\n2, ξ∈I, ϕ−1\n1(ξ) =ξ+ 1\n2, ξ∈I (67)\nDenoting the 1-D solution points and Lagrange basis for Ias{ξp}N\np=0and{ℓp(ξ)}N\np=0respectively, the same for Is\nare given by\b\nϕ−1\ns(ξp)\tN\np=0and{ℓp(ϕs(ξ))}N\np=0respectively. We also define −´\nto be integration under quadrature at\nsolution points. Thus,\n−ˆ\nIu(ξ)dξ=NX\np=0u(ξp)wp,−ˆ\nIsu(ξ)dξ=NX\np=02u(ϕ−1\ns(ξp))wp\nIn order to get the solution point values of the refined elements, we will perform interpolation. All integrals in this\nsection are approximated by quadrature at solution points which are the degree NGauss-Legendre-Lobatto points. The\ninterpolation operator from Ito{Is}s=0,1is given by VΞsdefined as the Vandermonde matrix corresponding to the\nLagrange basis\n(Vs)pq=ℓq(ϕ−1\ns(ξp)), 0≤p, q≤N, s = 0,1 (68)\nFor the process of coarsening, we also define the L2projection operators {Ps}s=0,1which projects a polynomial u\ndefined on the Lagrange basis of Isto the Lagrange basis of Ias\n−ˆ\nIPs(u)(ξ)ℓi(ξ)dξ=−ˆ\nIsu(ξ)ℓi(ξ)dξ 0≤i≤N\nApproximating the integrals by quadrature on solution points, we obtain the matrix representations corresponding to the\nbasis\nPs\npq=1\n2wq\nwpℓp(ϕ−1\ns(ξq)) 0 ≤i, j≤N, s = 0,1 (69)\nwhere{wp}N\np=0are the quadrature weights corresponding to solution points. The transfer of solution during coarsening\nand refinement is performed by matrix vector operations using the operators (68, 69). Thus, the operators (68, 69) are\nstored as matrices for reference element at the beginning of the simulation and reused for the adaptation operations in\nall elements. Lastly, we introduce the notation of a product of matrix operators {Ai}d\ni=1acting on b= (bp)p∈Nd\nN=\n(bp1p2p3)p∈Nd\nNas\n(Aib)p=X\nq∈Nd\nN dY\ni=1(Ai)piqi!\nbq (70)\n5.1 Solution transfer between element and subelements\nCorresponding to the element Ωe, we denote the 2dsubdivisions as (Figure 4b)\nΩes= Θ e dY\ni=1Isi!\n,∀s∈Nd\n1\nwhere Isare defined in (65). We also define ϕs(ξ) = ( ϕsi(ξi))d\ni=1so that ϕsis a bijection between ΩesandΩe.\nRecall that {ℓp}p∈Nd\nNare Lagrange polynomials of degree Nwith variables ξ= (ξi)d\ni=1. Thus, the reference solution\npoints and Lagrange basis for Ωesare given by\b\nϕ−1\ns(ξp)\t\np∈Nd\nNand{ℓp(ϕs(ξ))}p∈Nd\nN, respectively. The respective\nrepresentations of solution approximations in Ωe,Ωesin reference coordinates are thus given by\nue(ξ) =X\nq∈Nd\nNℓq(ξ)ue,q,ues(ξ) =X\nq∈Nd\nN���q(ϕs(ξ))ues,q (71)\n17APREPRINT - FEBRUARY 21, 2024\n(a)\nΩe\nueRefine and interpolate\nCoarsen and projectΩe1\nue1Ωe2\nue2\nΩe3\nue3Ωe4\nue4\n(b)\nFigure 4: (a) Neighbouring elements with hanging nodes (b) Illustration of refinement and coarsening\n5.1.1 Interpolation for refinement\nAfter refining an element Ωeinto child elements {Ωes}s∈Nd\n1, the solution uehas to be interpolated on the solution\npoints of child elements to obtain {ues}s∈Nd\n1. The scheme will be specified by writing ues,qin terms of ue,q, which\nwere defined in (71). The interpolation is performed as\nues,p=X\nq∈Nd\nNℓq(ϕ−1\ns(ξp))ue,q=X\nq∈Nd\nN dY\ni=1ℓqi(ϕ−1\nsi(ξpi))!\nue,q\n=X\nq∈Nd\nN dY\ni=1(VΞsi)piqi!\nue,q(72)\nIn the product of operators notation (70), the interpolation can be written as\nues= dY\ni=1VΞsi!\nue\n5.1.2 Projection for coarsening\nWhen 2delements are joined into one single bigger element Ωe, the solution transfer is performed using L2projection\nof{ues}s∈Nd\n1intoue, which is given by\nX\ns∈Nd\n1−ˆ\nΩesuesℓp(ξ)dx=−ˆ\nΩeueℓp(ξ)dx,∀p∈Nd\nN (73)\nSubstituting (71) into (73) gives\nX\ns∈Nd\n1X\nq∈Nd\nN−ˆ\nΩesℓp(ξ)ℓq(ϕs(ξ))ues,qdx=X\nq∈Nd\nN−ˆ\nΩeℓp(ξ)ℓq(ξ)ue,qdx (74)\n18APREPRINT - FEBRUARY 21, 2024\nNote the 1-D identities\n−ˆ\nIℓp(ξ)ℓq(ξ)dξ=δpqwp\n−ˆ\nIsℓp(ξ)ℓq(ϕs(ξ))dξ=1\n2−ˆ1\n−1ℓp(ϕ−1\ns(ξ))ℓq(ξ)dξ=1\n2ℓp(ϕ−1\ns(ξq))wq=Ps\npqwp\nwhere the projection operator {Ps}s=0,1is defined in (69). Then, by change of variables, we have the following\n−ˆ\nΩeℓp(ξ)ℓq(ξ) =Je,pdY\ni=1wpiδpiqi,−ˆ\nΩesℓp(ξ)ℓq(ϕs(ξ)) =Je,pdY\ni=1wpiPsi\npiqi(75)\nUsing (75) in (74) and dividing both sides by Je,pgives\nue,p=X\ns∈Nd\n1X\nq∈Nd\nN dY\ni=1Psi\npiqi!\nues,q=X\ns∈Nd\n1 dY\ni=1Psi!\nues\nwhere the last equation follows using the product of operators notation (70).\n5.2 Mortar element method (MEM)\n5.2.1 Motivation and notation\nWhen the mesh is adaptively refined, there will be elements with different refinement levels sharing a face; in this work,\nwe assume that the refinement levels of those elements only differs by 2 (Figure 4a). Since the neighbouring elements\ndo not have a common face, the solution points on their faces do not coincide (Figure 5). We will use the Mortar\nElement Method (MEM) for computing the numerical flux at all the required points on such a face, while preserving\naccuracy and the conservative property (29). There are two steps to the method\n1.Prolong ˜Fδ·bnS,i,US,i,uS,i(26) from the neighbouring elements to a set of common solution points known\nas mortar solution points (Figure 5a).\n2.Compute the numerical flux at the mortar solution points as in (26) and map it back to the interfaces (Figure 5b).\nIn Sections 5.2.2, 5.2.3, we will explain these two steps through the specific case of Figure 5 and we first introduce\nnotations for the same.\nConsider the multi-indices s∈Nd−1\n1={0,1}d−1and the interface in right (positive) i= 1 direction of element\nΩe, denoted as Γ(Figure 5). We assume that the elements neighbouring Ωeat the interface Γare finer and thus\nwe have non-conforming subinterfaces {Γs}s∈Nd−1\n1which, by continuity of the reference map, can be written as\nΓs= Θ e({1} ×Qd−1\ni=1Isi) = Θ e({1} ×ϕ−1\ns(Id−1)). Thus, in reference coordinates, ϕs(66) is a bijection from Γs\ntoΓ. The interface Γcan be parametrized as y=γ(η) = Θ e(1,η)forη∈Id−1and thus the reference variable of\ninterface is denoted η=γ−1(y). The subinterfaces can also be written by using the same parametrization so that\nΓs=n\nγ(η) :η∈Qd−1\ni=1Isio\n. For the reference solution points on Γbeing{ηs}s∈Nd−1\n1, the solution points in Γs\nare respectively given by\b\nϕ−1\ns(ηp)\t\np∈Nd−1\n1and for {ℓp(η)}p∈Nd−1\nNbeing Lagrange polynomials in Γ, the Lagrange\npolynomials in Γsare given by {ℓp(ϕs(η))}p∈Nd−1\n1respectively. Since the solution points between ΓandΓsdo not\ncoincide, they will be mapped to common solution points in the mortars Ξsand then back to Γ,Γsafter computing the\ncommon numerical flux. The solution points in Ξsare actually given by\b\nϕ−1\ns(ηp)\t\np∈Nd−1\n1, i.e., they are the same as\nΓs. The quantities with subscripts Ξ−\ns,Ξ+\nswill denote trace values from larger, smaller elements respectively.\n5.2.2 Prolongation to mortars\nWe will explain the prolongation procedure for a quantity Fwhich could be the normal flux ˜Fδ·bnS,i, time average\nsolution US,ior the solution uS,i. The first step of MEM of mapping of solution point values from solution points\nat element interfaces Γ,Γsto solution points at mortars Ξ−\ns,Ξ+\nsis known as prolongation. The prolongation of\n{Fδ\nΓs}s∈Nd−1\n1from small elements Γsto mortar values {FΞ+\ns}s∈Nd−1\n1is the identity map since both have the same\nsolution points, and the prolongation of Fδ\nΓfrom the large interface Γto the{F−\nΞs}s∈Nd−1\n1is an interpolation to the\n19APREPRINT - FEBRUARY 21, 2024\nΩe\nueΓ\nΩe1\nue1Ωe2\nue2\nΩe3\nue3Ωe4\nue4u−Γ, U−\nΓ, F−\nΓΞ1\nΞ2u−\nΞ2, U−Ξ2, F−Ξ2u−\nΞ1, U−\nΞ1, F−\nΞ1\nu+\nΞ2, U+\nΞ2, F+\nΞ2u+\nΞ1, U+\nΞ1, F+\nΞ1\nF+\nΓ2F+\nΓ1\nΓ2Γ1\n(a)\nΩe\nueΓ\nΩe1\nue1Ωe2\nue2\nΩe3\nue3Ωe4\nue4FΓΞ1\nΞ2FΞ1\nFΞ2FΓ2FΓ1\nΓ2Γ1\n(b)\nFigure 5: (a) Prolongation to mortar and computation of numerical flux FΞ1,FΞ2, (b) Projection of numerical flux to\ninterfaces\nmortar solution points. Accuracy is maintained by the interpolation as the mortar elements are finer. Below, we explain\nthe matrix operations used to perform the interpolation.\nThe prolongation of {Fδ\nΓs}s∈Nd−1\n1to the mortar values {FΞ+\ns}s∈Nd−1\n1is the identity map. The {F−\nΞs}s∈Nd−1\n1in Lagrange\nbasis are given by\nFΞ−\ns(η) =X\np∈Nd−1\nNℓp(ϕs(η))FΞ−\ns,p,η∈d−1Y\ni=1Isi (76)\nThe coefficients {F−\nΞs,p}p∈Nd−1\nNare computed by interpolation\nFΞ−\ns,p=FΓ−(ϕ−1\ns(ηp)) =X\nq∈Nd−1\nNℓq(ϕ−1\ns(ηp))Fδ\nΓ(ηq) =X\nq∈Nd−1\nN d−1Y\ni=1ℓqi(ϕ−1\nsi(ηpi))!\nFδ\nΓ(ηq)\n=X\nq∈Nd−1\nN d−1Y\ni=1(Vsi)piqi!\nFδ\nΓ(ηq) (77)\n20APREPRINT - FEBRUARY 21, 2024\nwhere the interpolation operators {VΞs}s=0,1were defined in (68). Using the product of operators notation (70), we\ncan compactly write (77) as\nFΞ−\ns= d−1Y\ni=1Vsi!\nFδ\nΓ (78)\nThe same procedure is performed for obtaining UΞ±\ns,uΞ±\ns. The numerical fluxes\b\nF∗\nΞs\t\ns∈Nd−1\n1are then computed as\nin (26).\n5.2.3 Projection of numerical fluxes from mortars to faces\nIn this section, we use the notation F∗:= (˜Fe·bnS,i)∗to denote the numerical flux (26). In the second step of MEM,\nthe numerical fluxes {F∗\nΞs}s∈Nd−1\n1computed using values at {Ξ±\ns}s∈Nd−1\n1are mapped back to interfaces Γs,Γ. Since\nthe solution points on Γsare the same as those of Ξ±\ns, the mapping from {F∗\nΞs}s∈Nd−1\n1to{F∗\nΓs}s∈Nd−1\n1is the identity\nmap. In order to maintain the conservation property, an L2projection is performed to map all the fluxes {F∗\nΞs}s∈Nd−1\n1into one numerical flux F∗\nΓon the larger interface.\nAnL2projection of these fluxes to F∗\nΓonΓis performed as\nX\ns∈Nd−1\n1−ˆ\nΓsF∗\nΞsℓp=−ˆ\nΓF∗\nΓℓp,∀p∈Nd−1\nN (79)\nwhere integrals are computed with quadrature at solution points. As in (76), we write the mortar fluxes as\nF∗\nΞs(η) =X\nq∈Nd−1\n1ℓq(ϕs(η))F∗\nΞs,q,η∈Ξs\nF∗\nΓ(η) =X\nq∈Nd−1\n1ℓq(η)F∗\nΓ,q,η∈Γ\nThus, the integral identity (79) can be written as\nX\ns∈Nd−1\n1X\nq∈Nd−1\nN−ˆ\nΓsℓp(η)ℓq(ϕs(η))F∗\nΞs,q=X\nq∈Nd−1\np−ˆ\nΓℓp(η)ℓq(η)F∗\nΓ,q,∀p∈Nd−1\nN (80)\nUsing the identities (80), the equations (80) become\nX\ns∈Nd−1\n1X\nq∈Nd−1\nN d−1Y\ni=1wpiPsi\npiqi!\nF∗\nΞs,qJS\ne,p=wpF∗\nΓ,pJS\ne,p\nwhere JS\ne,pis the surface Jacobian, given by ||(Ja1)e,p||in this case ((6.29) of [ 22]). Then, dividing both sides by\nJS\ne,pwpgives\nF∗\nΓ,p=X\ns∈Nd−1\n1X\nq∈Nd−1\nN d−1Y\ni=1Psi\npiqi!\nF∗\nΞs,q=X\ns∈Nd−1\n1 d−1Y\ni=1Psi!\nF∗\nΞs(81)\nwhere the last identity is obtained by the product of operators notation (70). Note that the identity (79) implies\nX\ns∈Nd−1\n1−ˆ\nΓsF∗\nΞsv=−ˆ\nΓF∗\nΓv, v ∈PN\nThen, taking v= 1 shows that the total fluxes over an interface Γare the same as over {Γs}s∈Nd−1\n1and thus the\nconservation property (29) of LWFR is maintained by the LWFR scheme.\nRemark 4 (Freestream and admissibility preservation under AMR) Under the adaptively refined meshes, free\nstream preservation and provable admissibility preservation are respectively ensured.\n21APREPRINT - FEBRUARY 21, 2024\n1.When refining/coarsening, there are two ways to compute the metric terms - interpolate/project the metric terms\ndirectly or interpolate/project the reference map Θat solution points and use the newly obtained reference map\nto recompute the metric terms. The latter, which is the approach taken in this work, can lead to violation of free\nstream preservation as we can have (IN)eL(Jai)̸= (IN)eR(Jai)where ΩeLandΩeSare two neighbouring\nlarge and small elements respectively. Thus, the interface terms may not vanish in the update equation (28) with\nconstant unleading to a violation of free stream preservation. This issue only occurs in 3-D and is thus beyond\nthe scope of this work, but some remedies are to interpolate/project the metric terms when refining/coarsening\nor to use the reference map Θ∈PN/2, as explained in [ 24]. Another solution has been studied in [ 27] where a\ncommon finite element space with mixed degree N−1andNis used with continuity at the non-conformal\ninterfaces. Since this work only deals with problems in 2-D, we always have (IN)eL(Jai) = (IN)eR(Jai)\nensuring that the interface terms in (28) vanish when u=c. Further, since the metric terms are recomputed in\nthis work, the volume terms will vanish by the same arguments as in Section 3.3. Thus, free stream preservation\nis maintained even with the non-conformal, adaptively refined meshes.\n2.The flux limiting explained in Section 4.3 ensures admissibility in means (Definition 5) and then uses the\nscaling limiter of [ 48] to enforce admissibility of solution polynomial at all solution points to obtain an\nadmissibility preserving scheme (Definition 4). However, the procedure doesn’t ensure that the polynomial is\nadmissible at points which are not the solution points. Adaptive mesh refinement introduces such points into\nthe numerical method and can thus cause a failure of admissibility preservation in the following situations: (a)\nmortar solution values {u−\nΞs}obtained by interpolation as in (76) are not admissible, (b) mean values {ues}\nof the solution values {ues}obtained by interpolating from the larger element as in (72) are not admissible.\nSince the scaling limiter [ 48] can be used to enforce admissibility of solution at any desired points, the remedy\nto both the issues is further scaling; we simply perform scaling of solution point values {u−\nΞs},{ues}with the\nadmissible mean value ue. This will ensure that the mortar solution point values and the mean values {ues}\nare admissible.\n5.3 AMR indicators\nThe process of adaptively refining and coarsening the mesh requires a solution smoothness indicator. In this work, two\nsmoothness indicators have been used for adaptive mesh refinement. The first is the indicator of [ 17], explained in\nSection 4.2.3. The second is Löhner’s smoothness indicator [ 28] which uses the central finite difference formula for\nsecond derivative, which is given by\nαe= max\np∈Nd\nNmax\n1≤i≤d\f\fq(upi+)−2q(up) +q(upi−)\f\f\n\f\fq(upi+)−q(up)\f\f+\f\fq(up)−q(upi−)\f\f+fwave(\f\fq(upi+)\f\f+ 2|q(up)|+\f\fq(upi−)\f\f)\n(pi±)m=\u001apm, m ̸=i\npi±1, m =i\nwhere {up}p∈Nd\nNare the degrees of freedom in element Ωeandqis a derived quantity like the product of density and\npressure used in Section 4.2.3. The value fwave= 0.2has been chosen in all the tests.\nOnce a smoothness indicator is chosen, the three level controller implemented in Trixi.jl [32] is used to determine\nthe local refinement level. The mesh begins with an initial refinement level and the effective refinement level is\nprescribed by how much further refinement has been done to the initial mesh, The mesh is created with two thresholds\nmed_threshold andmax_threshold and three refinement levels base_level ,med_level andmax_level . Then,\nwe have\nlevel e=\n\nbase_level , α e≤med_threshold\nmed_level ,med_threshold ≤αe≤max_threshold\nmax_level ,max_threshold ≤αe\nBeyond these refinement levels, further refinement is performed to make sure that two neighbouring elements only\ndiffer by a refinement level of 1.\n22APREPRINT - FEBRUARY 21, 2024\n6 Time stepping\nThis section introduces an embedded error approximation method to compute the time step size ∆tfor the single stage\nLax-Wendroff Flux Reconstruction method. A standard way to compute the time step size ∆tnis to use [2, 3]\n∆tn=Csmin\ne,p|Je,p|\nσ(une,p)CFL( N) (82)\nwhere the minimum is taken over all elements {Ωe}e,Jeis the Jacobian of the change of variable map, σ(un\ne)is the\nlargest eigenvalue of the flux jacobian at state un\ne, approximating the local wave speed, CFL( N)is the optimal CFL\nnumber dependent on solution polynomial degree NandCs≤1is a safety factor. In [ 2], a Fourier stability analysis of\nthe LWFR scheme was performed on Cartesian grids, and the optimal CFL numbers were obtained for each degree N\nwhich guaranteed the stability of the scheme. However, the Fourier stability analysis does not apply to curvilinear grids\nand formula (82) need not guarantee L2stability which may require the CFL number to be fine-tuned for each problem.\nAlong with the L2stability, the time step has to be chosen so that the scheme does not give inadmissible solutions. An\nerror-based time stepping method inherently minimizes the parameter tuning process in time step computation. The\nparameters in an error-based time stepping scheme that a user has to specify are the absolute and relative error tolerances\nτa, τr, and they only affect the time step size logarithmically. In particular, because of the weak dependence, tolerances\nτa=τr= 10−6worked reasonably for all tests with shocks; although, it was possible to enhance performance by\nchoosing larger tolerances for some problems. Secondly, if inadmissibility is detected during any step in the scheme or\nif errors are too large, the time step is redone with a reduced time step size provided by the error estimate. The scheme\nalso has the capability of increasing and decreasing the time step size.\nWe begin by reviewing the error-based time stepping scheme for the Runge-Kutta ODE solvers from [ 32,34] in\nSection 6.1 and explain our extension of the same to LWFR in Section 6.2.\n6.1 Error estimation for Runge-Kutta schemes\nConsider an explicit Runge-Kutta method used for solving ordinary differential equations by evolving the numerical\nsolution from time level nton+ 1. For error-estimation, the method is constructed to have an embedded lower order\nupdatebun+1, as described in equation (3) of [ 32]. The difference in the two updates, un+1−bun+1, gives an indication\nof the time integration error, which is used to build a Proportional Integral Derivative (PID) controller to compute the\nnew time step size,\nf∆tn+1=κ(εβ1/k\nn+1εβ2/k\nnεβ3/k\nn−1)∆tn (83)\nwhere for qbeing the order of main method, ˆqbeing the order of embedded method, we have\nk= min( q,ˆq) + 1\nandβiare called control parameters which are optimized for the particular Runge-Kutta scheme [ 32]. For mbeing the\ndegrees of freedom in u, we pick absolute and relative tolerances τa, τrand then error approximation is made as\nεn+1=1\nwn+1, w n+1=\n1\nMMX\ni=1 \nun+1\ni−bun+1\ni\nτa+τrmax\b\f\fun+1\ni\f\f,|˘u|\t!2\n1\n2\n(84)\nwhere the sum is over all degrees of freedom, including solution points and conservation variables. The tolerances are\nto be chosen by the user but their influence on the scheme is logarithmic, unlike the CFL based scheme (82).\nThe limiting function κ(x) = 1 + tan−1(x−1)is used to prevent sudden increase in time step sizes. For normalization,\nPETSc uses˘u=bun+1while OrdinaryDiffEq.jl uses˘u=un. Following [ 32], if the time step factor f∆tn+1/∆tn≥\n0.92, the new time step is accepted and used in the next level as ∆tn+1=f∆tn+1. If not, or if admissibility is violated,\nevolution is redone with time step size ∆tn=f∆tn+1computed from (83).\n6.2 Error based time stepping for Lax-Wendroff flux reconstruction\nConsider the LWFR scheme (28) with polynomial degree Nand formal order of accuracy N+ 1\nun+1\ne,p=un\ne,p−∆t\nJe,p∇ξ·˜Fδ\ne(ξp)−Ce,p\n23APREPRINT - FEBRUARY 21, 2024\nwhere Ce,pcontains contributions at element interfaces. In order to construct a lower order embedded scheme without\nrequiring additional inter-element communication, consider an evolution where the interface correction terms Ce,pare\nnot used, i.e., consider the element local update\nun+1\nloc,e,p=un\ne,p−∆t\nJe,p∇ξ·˜Fδ\ne(ξp) (85)\nTruncating the locally computed time averaged flux ˜Fδ\ne(24) at one order lower\ncFδe=N−1X\nk=0∆tk\n(k+ 1)!∂k\nt˜fδ\ne (86)\nwe can consider another update\n\\un+1\nloc,e,p=un\ne,p−∆t\nJe,p∇ξ·cFδe(ξp) (87)\nwhich is also locally computed but is one order of accuracy lower. We thus use un+1\ne=un+1\nloc,eandbun+1\ne=\\un+1\nloc,ein\nthe formula (84) along with ˘u=bun+1; then we use the same procedure of redoing the time step sizes as in Section 6.1.\nThat is, after using the error estimate (84) to compute f∆tn+1(83) we redo the time step if f∆tn+1/∆tn≥0.92or if\nadmissibility is violated; otherwise we set ∆tn+1to be used at the next time level. The complete process is also detailed\nin Algorithm 4. In this work, we have used the control parameters β1= 0.6, β2=−0.2, β3= 0.0for all numerical\nresults which are the same as those used in [32] for BS 3(2)3 F, the third-order, four-stage RK method of [7]. We tried\nthe other control parameters from [ 32] but found the present choice to be either superior or only slightly different in\nperformance, measured by the number of iterations taken to reach the final time.\nAlgorithm 2 High level overview of element residual computation of order N+ 1including error approximation using\nun+1\nloc,[un+1\nloc\nforeineachelement(mesh) do\nComputen\n∂k\nt˜fδ\neoN−1\nk=0using the approximate Lax-Wendroff procedure (31) to obtaincFδe(86)\nCompute\\un+1\nloc,eusingcFδe(87)\nCompute ∂N+1\nt˜fδ\neusing the approximate Lax-Wendroff procedure (31) to obtain ˜Fδ\ne(24)\nCompute un+1\nloc,eusingFδ\neas in (85)\ntemporal_error[e]=X\ni\nun+1\nloc,e,i−\\un+1\nloc,e,i\nτa+τrmaxn\f\f\fun+1\nloc,e,i\f\f\f,\f\f\f\\un+1\nloc,e,i\f\f\fo\n2\nwhere the sum is over dofs in e\nCompute and add local contribution of Fδ\neto the residual (28)\nend for\nAlgorithm 3 High level overview of LWFR residual (Within time integration)\nCompute {αe}(Section 4.2.3)\nAssemble cell residual (Algorithm 2)\nforΓineachinterface(mesh) do\nCompute FLW\nΓ,fΓand blend them into FΓ(Algorithm 1)\nend for\nforeineachelement(mesh) do\nAdd contribution of numerical fluxes to residual of element e(common to high, low residual, see Remark 3)\nend for\nUpdate solution\nApply positivity limiter\n24APREPRINT - FEBRUARY 21, 2024\nAlgorithm 4 Lax-Wendroff Flux Reconstruction at a high level to explain error based time stepping\nInitialize t←0, time step number n←0, and initial state u0\nInitialize PID controller with ε0←1, ε−1←1\nInitialize ∆t0=f∆twith a user supplied value\nInitialize accept_step ←false\nwhile t < T do\nifaccept_step then\naccept_step ←false\nt←˜t\n∆tn+1←f∆t\nn←n+ 1\nelse\n∆tn←f∆t\nend if\nift+ ∆tn>final_time then\n∆tn←final_time −t\nend if\nun→un+1with∆tn(Algorithm 3, 2) computing temporal_error and checking admissibility\nwn+1←\u0012\n1\nMP\netemporal_error[e]\u00131\n2\n▷ M is the total dofs\nwn+1←max{wn+1,10−10} ▷To avoid division by zero\nεn+1←1\nwn+1\ndt_factor ←κ(εβ1/k\nn+1εβ2/k\nnεβ3/k\nn−1) ▷ κ(x) = 1 + tan−1(x−1)\nf∆t←dt_factor ·∆tn\nifdt_factor ≥accept_safety && no inadmissibility then\naccept_step ←true\nelse\naccept_step ←false\nend if\nifaccept_step then\n˜t←t+ ∆tn\nif˜t≈final_time then\n˜t←final_time\nend if\nApply callbacks ▷Analyze and postprocess solution, AMR\nPositivity correction for AMR (Remark 4)\nend if\nend while\n7 Numerical results\nThe numerical experiments are performed on 2-D Euler’s equations (1). Unless specified otherwise, the adiabatic\nconstant γwill be taken as 1.4in the numerical tests, which is the typical value for air. The CFL based time stepping\nschemes use the following formula for the time step (see 2.5 of [34], but also [19, 32])\n∆tn=2\nN+ 1CCFLmin\ne,p \n1\n|Je,p|dX\ni=1˜λi\ne,p!\n, C CFL≤1 (88)\nwhere {˜λi\ne,p}d\ni=1are wave speed estimates computed by the transformation\n˜λi\ne,p=dX\nn=1(Jai\nn)e,pλi\ne,p\nwhere{Jai}d\ni=1are the contravariant vectors (2)andλi\ne,pis the absolute maximum eigenvalue of f′\ni(ue,p). For Euler’s\nequations with velocity vector v={vi}and sound speed c,λi=|vi|+c. The CCFLin(88) may need to be fine-tuned\n25APREPRINT - FEBRUARY 21, 2024\ndepending on the problem. Other than the convergence test (Section 7.2.2), the results shown below have been generated\nwith error-based time stepping (Section 6.2). The scheme is implemented in a Julia package TrixiLW.jl written\nusing Trixi.jl [33,40,39] as a library. Trixi.jl is a high order PDE solver package in Julia [6] and uses the\nRunge-Kutta Discontinuous Galerkin method; TrixiLW.jl uses Julia ’s multiple dispatch to borrow features like\ncurved meshes support and postprocessing from Trixi.jl .TrixiLW.jl is not a fork of Trixi.jl but only uses it\nthrough Julia ’s package manager without modifying its internal code. The setup files for the numerical experiments in\nthis work are available at [1].\n7.1 Results on Cartesian grids\n7.1.1 Mach 2000 astrophysical jet\nIn this test, a hypersonic jet is injected into a gas at rest with a Mach number of 2000 relative to the sound speed in the\ngas at rest. Following [ 16,49], the domain is taken to be [0,1]×[−0.5,0.5], the ambient gas in the interior has state\nuadefined in primitive variables as\n(ρ, u, v, p )a= (0.5,0,0,0.4127)\nand inflow state ujis defined in primitive variables as\n(ρ, u, v, p )j= (5,800,0,0.4127)\nOn the left boundary, we impose the boundary conditions\nub=\u001aua, ify∈[−0.05,0.05]\nuj, otherwise\nand outflow conditions on the right, top and bottom of the computational domain.\nThe simulation is performed on a uniform 5122element mesh. This test requires admissibility preservation to be\nenforced to avoid solutions with negative pressure. This is a cold-start problem as the solution is constant with zero\nvelocity in the domain at time t= 0. However, there is a high speed inflow at the boundary, which the standard wave\nspeed estimate for time step approximation (89) does not account for. Thus, in order to use the CFL based time stepping,\nlower values of CCFL(88) have to be used in the first few iterations of the simulations. Once the high speed flow has\nentered the domain, the this value needs to be raised since otherwise, the simulation will use much smaller time steps\nthan the linear stability limit permits. Error based time stepping schemes automate this process by their adaptivity and\nability to redo the time steps. The simulation is run till t= 10−2and the log scaled density plot for degree N= 4\nsolution obtained on the uniform mesh is shown in Figure 6a. For an error-based time stepping scheme, we define the\neffective CCFLas\nCCFL:= ∆tn\"\n2\nN+ 1min\ne,p \n1\n|Je,p|dX\ni=1˜λi\ne,p!#−1\n(89)\nwhich is a reverse computation so that its usage in (82) will get ∆tnchosen in the error-based time stepping scheme (Al-\ngorithm 4). In Figure 6b, time tversus effective CCFL(89) is plotted upto t= 10−5to demonstrate that the scheme\nautomatically uses a smaller CCFLof∼10−3at the beginning which later increases and stabilizes at ∼10−1. Thus, the\nerror based time stepping is automatically doing what would have to be manually implemented for a CFL based time\nstepping scheme which would be problem-dependent and requiring user intervention.\n7.1.2 Kelvin-Helmholtz instability\nThe Kelvin-Helmholtz instability is a common fluid instability that occurs across density and tangential velocity\ngradients leading to a tangential shear flow. This instability leads to the formation of vortices that grow in amplitude\nand can eventually lead to the onset of turbulence. The initial condition is given by [35]\n(ρ, u, v, p ) =\u00121\n2+3\n4B,1\n2(B−1),1\n10sin(2πx),1\u0013\nwithB= tanh(15 y+ 7.5)−tanh(15 y−7.5)in domain Ω = [−1,1]2with periodic boundary conditions. The initial\ncondition has a Mach number M≤0.6which makes compressibility effects relevant but does not cause shocks to\ndevelop. Thus, a very mild shock capturing scheme is used by setting αe= min {αe, αmax}(Section 4.2.3) where\nαmax= 0.002. The same smoothness indicator of Section 4.2.3 is used for AMR indicator with parameters from\nSection 5.3 chosen to be\n(base_level ,med_level ,max_level ) = (4 ,0,8), (med_threshold ,max_threshold ) = (0 .0003,0.003)\n26APREPRINT - FEBRUARY 21, 2024\n0.0 0.2 0.4 0.6 0.8 1.0\nt 1e5\n103\n102\n101\nCCFLError based scheme\n(a) (b)\nFigure 6: Mach 2000 astrophysical jet (a) Density plot (b) Effective CCFL\n(a) (b)\nFigure 7: Kelvin-Helmholtz instability at t= 3using polynomial degree N= 4(a) density plots, (b) adaptively refined\nmesh\nwhere base_level = 0refers to a 2×2mesh. This test case, along with indicators’ configuration was taken from the\nexamples of Trixi.jl [32]. The simulation is run till t= 3using polynomial degree N= 4. There is a shear layer at\ny=±0.5which rolls up and develops smaller scale structures as time progresses. The results are shown in Figure 7\nand it can be seen that the AMR indicator is able to track the small scale structures. The simulation starts with a mesh\nof1024 elements which steadily increases to 13957 at the final time; the mesh is adaptively refined or coarsened at\nevery time step. The solution has non-trivial variations in small regions around the rolling structures which an adaptive\nmesh algorithm can capture efficiently, while a uniform mesh with similar resolution would require 262144 elements.\n7.1.3 Double mach reflection\nThis test case was originally proposed by Woodward and Colella [ 46] and consists of a shock impinging on a wedge/ramp\nwhich is inclined by 30 degrees. An equivalent problem is obtained on the rectangular domain Ω = [0 ,4]×[0,1]\nobtained by rotating the wedge so that the initial condition now consists of a shock angled at 60 degrees. The solution\n27APREPRINT - FEBRUARY 21, 2024\n(a)\n(b)\nFigure 8: Double Mach reflection with solution polynomial degree N= 4att= 0.2(a) Density plot, (b) Adaptively\nrefined mesh at final time\nconsists of a self similar shock structure with two triple points. Define ub=ub(x, y, t )in primitive variables as\n(ρ, u, v, p ) =(\n(8,8.25 cos(π\n6),−8.25 sin(π\n6),116.5),ifx <1\n6+y+20t√\n3\n(1.4,0,0,1), ifx >1\n6+y+20t√\n3\nand take the initial condition to be u0(x, y) =ub(x, y,0). With ub, we impose inflow boundary conditions at the left\nside{0} ×[0,1], outflow boundary conditions both on [0,1/6]× {0}and{4} ×[0,1], reflecting boundary conditions\non[1/6,4]× {0}and inflow boundary conditions on the upper side [0,4]× {1}.\nThe setup of Löhner’s smoothness indicator (5.3) is taken from an example of Trixi.jl [32]\n(base_level ,med_level ,max_level ) = (0 ,3,6), (med_threshold ,max_threshold ) = (0 .05,0.1)\nwhere base_level = 0corresponds to a 16×5mesh. The density solution obtained using polynomial degree N= 4\nis shown in Figure 8 where it is seen that AMR is tracing the shocks and small scale shearing well. The initial mesh\nconsists of 80 elements and is refined in first iteration in the vicinity of the shock to get 2411 elements. In later iterations,\nthe mesh is refined and coarsened in each iteration, and the number of elements keeps increasing up to 7793 elements at\nthe final time t= 0.2. In order to capture the same effective refinement, a uniform mesh will require 327680 elements.\n7.1.4 Forward facing step\nForward facing step is a classical test case from [ 14,46] where a uniform supersonic flow passes through a channel\nwith a forward facing step generating several phenomena like a strong bow shock, shock reflections and a Kelvin-\nHelmholtz instability. It is a good test for demonstrating a shock capturing scheme’s capability of capturing small scale\nvortex structures while suppressing spurious oscillations arising from shocks. The step is simulated in the domain\nΩ = ([0 ,3]×[0,1])\\([0.6,3]×[0,0.2])and the initial conditions are taken to be\n(ρ, u, v, p ) = (1 .4,3,0,1) inΩ\nThe left boundary condition is taken as an inflow and the right one is an outflow, while the rest are solid walls. The\ncorner (0.6,0.2)of the step is the center of a rarefaction fan and can lead to large errors and the formation of a spurious\n28APREPRINT - FEBRUARY 21, 2024\n(a)\n(b)\nFigure 9: Mach 3 flow over forward facing step at time t= 3using solution polynomial degree N= 4with Löhner’s\nindicator for mesh refinement. (a) Density plot (b) Adaptively refined mesh\nboundary layer, as shown in Figure 7a-7d of [ 46]. These errors can be reduced by refining the mesh near the corner,\nwhich is automated here with the AMR algorithm.\nThe setup of Löhner’s smoothness indicator (5.3) is taken from an example of Trixi.jl [32]\n(base_level ,med_level ,max_level ) = (0 ,2,5), (med_threshold ,max_threshold ) = (0 .05,0.1)\nThe density at t= 3obtained using polynomial degree N= 4and Löhner’s smoothness indicator (5.3) is plotted in\nFigure 9. The shocks have been well-traced and resolved by AMR and the spurious boundary layer and Mach stem\ndo not appear. The simulations starts with a mesh of 198 elements and the number peaks at 6700 elements during the\nsimulation then and decreases to 6099 at the final time t= 3. The mesh is adaptively refined or coarsened once every\n100 time steps. In order to capture the same effective refinement, a uniform mesh will require 202752 elements.\n7.2 Results on curved grids\n7.2.1 Free stream preservation\nIn this section, free stream preservation is tested for meshes with curved elements. Since we use a reference map of\ndegree Nin(5), free stream will be preserved following the discussion in Section 3.3. We numerically verify the\nsame for the meshes taken from Trixi.jl which are shown in Figure 10. The mesh in Figure 10a consists of curved\nboundaries and only the elements adjacent to the boundary are curved, while the one in Figure 10b is a non-conforming\nmesh with curved elements everywhere, and is used to verify that free stream preservation holds with adaptively refined\nmeshes. The mesh in Figure 10b is a 2-D reduction of the one used in Figure 3 of [ 36] and is defined by the global map\n(ξ, η)7→(x, y)from [0,3]2→Ωdescribed as\nx=ξ+3\n8cos\u0012π\n22ξ−3\n3\u0013\ncos\u0012\n2π2y−3\n3\u0013\n, y =η+3\n8cos\u00123π\n22ξ−3\n3\u0013\ncos\u0012π\n22η−3\n3\u0013\nThe free stream preservation is verified on these meshes by solving the Euler’s equation with constant initial data\n(ρ, u, v, p ) = (1 ,0.1,−0.2,10)\nand Dirichlet boundary conditions. Figure 10 shows the density at time t= 10 which is constant throughout the domain.\n29APREPRINT - FEBRUARY 21, 2024\n(a) (b)\nFigure 10: Density plots of free stream tests with mesh and solution polynomial degree N= 6att= 10 on (a) mesh\nwith curved boundaries, (b) mesh with refined curved elements\n7.2.2 Isentropic vortex\nThis is a test with exact solution taken from [ 17] where the domain is specified by the following transformation from\n[0,1]2→Ω\nx(ξ, η) =\u0012\nξLx−AxLysin(2πη)\nηLy+AyLxsin(2πξ)\u0013\nwhich is a distortion of the square [0, Lx]×[0, Ly]with sine waves of amplitudes Ax, Ay. Following [ 17], we choose\nlength Lx=Ly= 0.1and amplitudes Ax=Ay= 0.1. The boundaries are set to be periodic. A vortex with radius\nRv= 0.005is initialized in the curved domain with center (xv, yv) = ( Lx/2, Ly/2). The gas constant is taken to\nbeRgas= 287 .15and specific heat ratio γ= 1.4as before. The free stream state is defined by the Mach number\nM0= 0.5, temperature T0= 300 , pressure p0= 105, velocity u0=M0p\nγRgasT0and density ρ0=p0\nRgasT0. The\ninitial condition u0is given by\n(ρ, u, v, p ) = \nρ0\u0012T\nT0\u00131\nγ−1\n, u0\u0012\n1−βy−yv\nRve−r2\n2\u0013\n, u0βx−xv\nRve−r2\n2, ρ(x, y)RgasT!\nT(x, y) =T0−(u0β)2\n2Cpe−r2, r =p\n(x−xv)2+ (y−yv)2/Rv\nwhere Cp=Rgasγ/(γ−1)is the heat capacity at constant pressure and β= 0.2is the vortex strength. The vortex\nmoves in the positive xdirection with speed u0so that the exact solution at time tisu(x, y, t ) =u0(x−u0t, y)where\nu0is extended outside Ωby periodicity. We simulate the propagation of the vortex for one time period tp=Lx/u0and\nperform numerical convergence analysis for degree N= 3in Figure 11b, showing optimal rates in grid versus L2error\nnorm for all the conserved variables.\n7.2.3 Supersonic flow over cylinder\nSupersonic flow over a cylinder is computed at a free stream Mach number of 3with the initial condition\n(ρ, u, v, p ) = (1 .4,3,0,1)\nSolid wall boundary conditions are used at the top and bottom boundaries. A bow shock forms which reflects across the\nsolid walls and interacts with the small vortices forming in the wake of the cylinder. The setup of Löhner’s smoothness\n30APREPRINT - FEBRUARY 21, 2024\n16232264212822562\nNumber of elements108\n106\n104\n102\n100102L2 error\nDegree N=3\nv1\nv2\nE\nO(M4)\n(a) (b)\nFigure 11: Convergence analysis for isentropic vortex problem with polynomial degree N= 3. (a) Density plot, (b) L2\nerror norm of conserved variables\nindicator (5.3) is taken from an example of Trixi.jl [32]\n(base_level ,med_level ,max_level ) = (0 ,3,5), (med_threshold ,max_threshold ) = (0 .05,0.1)\nwhere base_level = 0refers to mesh in Figure 12a. The flow consists of a strong a shock and thus the positivity\nlimiter had to be used to enforce admissibility. The flow behind the cylinder is highly unsteady, with reflected shocks\nand vortices interacting continuously. The density profile of the numerical solution at t= 10 is shown in Figure 12 with\nmesh and solution polynomial degree N= 4using Löhner’s indicator (5.3) for AMR. The AMR indicator is tracing the\nshocks and the vortex structures forming in the wake well. The initial mesh has 561 elements which first increase to\n63000 elements followed by a fall to 39000 elements and then a steady increase to the peak of 85000 elements from\nwhich it steadily falls to 36000 elements by the end of the simulation. The mesh is refined or coarsened once every 100\ntime steps. In order to capture the same effective refinement, a uniform mesh will require 574464 elements.\n7.2.4 Inviscid bow shock upstream of a blunt body\nThis test simulates steady supersonic flow over a blunt body and is taken from [ 17] which followed the description\nproposed by the high order computational fluid dynamics workshop [ 9]. The domain, also shown in Figure 13 consists\nof a left and a right boundary. The left boundary is an arc of a circle with origin (3.85,0)and radius 5.9extended till\nx= 0on both ends. The right boundary consists of (a) the blunt body and (b) straight-edged outlets. The straight-edged\noutlets are {(0, y) :|y|>0.5}extended till the left boundary arc. The blunt body consists of a front of length 1and\ntwo quarter circles of radius 0.5. The domain is initialized with a Mach 4 flow, which is given in primitive variables by\n(ρ, u, v, p ) = (1 .4,4,0,1) (90)\nThe left boundary is set as supersonic inflow, the blunt body is a reflecting wall and the straight edges at x= 0are\nsupersonic outflow boundaries. Löhner’s smoothness indicator (5.3) for AMR is set up as\n(base_level ,med_level ,max_level ) = (0 ,1,2), (med_threshold ,max_threshold ) = (0 .05,0.1)\nwhere base_level = 0refers to mesh in Figure 13a. Since this is a test case with a strong bow shock, the positivity\nlimiter had to be used to enforce admissibility. The pressure obtained with polynomial degree N= 4 is shown\nin Figure 13 with adaptive mesh refinement performed using Löhner’s smoothness indicator (5.3) where the AMR\nprocedure is seen to be refining the mesh in the region of the bow shock. The initial mesh (Figure 13a) has 244 elements\nwhich steadily increases to ∼1600 elements till t≈1.5and then remains nearly constant as the solution reaches steady\nstate. The mesh is adaptively refined or coarsened at every time step.\n31APREPRINT - FEBRUARY 21, 2024\n(a)\n(b)\n(c)\nFigure 12: Mach 3 flow over cylinder using solution and mesh polynomial degree N= 4att= 10 (a) Initial mesh, (b)\nadaptively refined mesh at final time, (c) density plot at final time\n32APREPRINT - FEBRUARY 21, 2024\n(a) (b) (c) (d)\nFigure 13: Mach 4 flow over blunt body using polynomial degree N= 4showing (a) initial mesh, (b) adaptively refined\nmesh, (c) pressure plot, (d) Mach number plot\n(a) (b)\nFigure 14: Meshes for transonic flow over NACA0012 airfoil. (a) Initial mesh (b) adaptively refined mesh\n7.2.5 Transonic flow over NACA0012 airfoil\nThis is a steady transonic flow over the symmetric NACA0012 airfoil. The initial condition is taken to have Mach\nnumber M0= 0.85and it is given in primitive variables as\n(ρ, u, v, p ) =\u0012p0\nT0R, U0cosθ, U0sinθ, p0\u0013\nwhere p0= 1, T0= 1, R= 287 .87, θ=π/180,U0=M0c0and sound speed c0=p\nγp0/ρ0. The airfoil is of length\n1unit located in the rectangular domain [−20,20]2and the initial mesh has 728 elements. We run the simulation with\nmesh and solution polynomial degree N= 6using Löhner’s smoothness indicator (5.3) for AMR with the setup\n(base_level ,med_level ,max_level ) = (1 ,3,4), (med_threshold ,max_threshold ) = (0 .05,0.1)\nwhere base_level = 1refers to the mesh in Figure 14a . In Figure 14, we show the initial and adaptively refined\nmesh. In Figure 15, we show the Mach number and compare the coefficient of pressure Cpon the surface of airfoil with\nSU2 [ 13] results, seeing reasonable agreement in terms of the values and shock locations. The AMR procedure is found\nto steadily increase the number of elements till they peak at ∼4200 and decrease to stabilize at ∼3750 ; the region of the\nshocks is being refined by the AMR process. The mesh is adaptively refined or coarsened once every 100 time steps. In\norder to capture the same effective refinement, a uniform mesh will require 186368 elements.\n33APREPRINT - FEBRUARY 21, 2024\n0.0 0.2 0.4 0.6 0.8 1.0\nx1.0\n0.5\n0.0\n0.5\n1.0CpM=0.85,AoA=1°\nLWFR\nSU2\n(a) (b)\nFigure 15: Transonic flow over airfoil using degree N= 6 on adapted mesh (a) Mach number (b) Coefficient of\npressure on the surface of the airfoil\nCFL based (88)Error Based (Alg. 4)\n(Successful + failed)Ratio\ntolE=1e-6CFL\ntolE=1e-6Best tolECFL\nBest tolE\nFF Step (7.1.4) 57064557661457\n(7661453 + 4)0.745646355 (5e-6)\n(5646355 + 5)1.01\nCylinder (7.2.3) 1529064871262\n(871124 + 138)1.755657170 (5e-6)\n(651118+6052)2.327\nBlunt body (7.2.4) -4200\n(3800 + 400)-4200 (1e-6)\n(3800 + 400)-\nNACA0012 (7.2.5) 68568284778674\n(4778651 + 23)1.434778674 (1e-6)\n(4778651 + 23)1.43\nTable 1: Number of time steps comparing error and CFL based methods\n7.3 Performance comparison of time stepping schemes\nIn Table 1, we show comparison of total time steps needed by error (Algorithm 4) and CFL (88) based time stepping\nmethods for test cases where non-Cartesian meshes are used. The total time steps give a complete description of the\ncost because our experiments have shown that error estimation procedure only adds an additional computational cost of\n∼4%. The relative and absolute tolerances τa, τrin(84) are taken to be the same, and denoted tolE . The iterations\nwhich are redone because of error or admissibility criterion in Algorithm 4 are counted as failed (shown in Table 1 in\nred) while the rest as successful (shown in Table 1 in blue). The comparisons are made between the two time stepping\nschemes as follows - the constant CCFLin(88) is experimentally chosen to be the largest which can be used without\nadmissibility violation while error based time stepping is shown with tolE = 1e-6 and the best tolerance for the\nparticular test case (which is either 1e-6 or5e-6 ). Note that the choice of tolE = 1e-6 is made in all the results\nshown in previous sections. A poor quality (nearly degenerate) mesh (Figure 13b) was used in the flow over blunt\nbody (Section 7.2.4) and thus the CFL based scheme could not run till the final time t= 10 without admissibility\nviolation for any choice of CCFL. However, the error-based time stepping scheme is able to finish the simulation by its\nability to redo time steps; although there are many failed time steps as is to be expected. The error-based time stepping\nscheme is giving superior performance with tolE = 1e-6 for the supersonic flow over cylinder and transonic flow\nover airfoil (curved meshes tests) with ratio of total time steps being 1.755 and 1.43 respectively. However, for the\nforward facing step test with a straight sided quadrilateral mesh, error based time stepping with tolE = 1e-6 takes\nmore time steps than the fine-tuned CFL based time stepping. However, increasing the tolerance to tolE = 5e-6 gives\nthe same performance as the CFL based time stepping. By using tolE = 5e-6 , the performance of supersonic flow\ncylinder can be further obtained to get a ratio of 2.327. These results show the robustness of error-based time stepping\nand even improved efficiency in meshes with curved elements.\n34APREPRINT - FEBRUARY 21, 2024\n8 Summary and conclusions\nThe Lax-Wendroff Flux Reconstruction (LWFR) of [ 2] has been extended to curvilinear and dynamic, locally adapted\nmeshes. On curvilinear meshes, it is shown that satisfying the standard metric identities gives free stream preservation\nfor the LWFR scheme. The subcell based blending scheme of [ 3] has been extended to curvilinear meshes along with\nthe provable admissibility preservation of [ 3] based on the idea of appropriately choosing the blended numerical flux [3]\nat the element interfaces. Adaptive Mesh Refinement has been introduced for LWFR scheme using the Mortar Element\nMethod (MEM) of [ 25]. Fourier stability analysis to compute the optimal CFL number as in [ 2] is based on uniform\nCartesian meshes and does not apply to curvilinear grids. Thus, in order to use a wave speed based time step computation,\nthe CFL number has to be fine tuned for every problem, especially for curved grids. In order to decrease the fine-tuning\nprocess, an embedded errror-based time step computation method was introduced for LWFR by taking difference\nbetween two element local evolutions of the solutions using the local time averaged flux approximations - one which is\norder N+ 1and the other truncated to be order N. This is the first time error-based time stepping has been introduced\nfor a single stage evolution method for solving time dependent equations. Numerical results using compressible Euler\nequations were shown to validate the claims. It was shown that free stream condition is satisfied on curvilinear meshes\neven with non-conformal elements and that the LWFR scheme shows optimal convergence rates on domains with\ncurved boundaries and meshes. The AMR with shock capturing was tested on various problems to show the scheme’s\nrobustness and capability to automatically refine in regions comprising of relevant features like shocks and small scale\nstructures. The error based time stepping scheme is able to run with fewer time steps in comparison to the CFL based\nscheme and with less fine tuning.\nAcknowledgments\nThe work of Arpit Babbar and Praveen Chandrashekar is supported by the Department of Atomic Energy, Government\nof India, under project no. 12-R&D-TFR-5.01-0520.\nAdditional data\nThe animations of the results presented in this paper can be viewed at\nwww.youtube.com/playlist?list=PLHg8S7nd3rfvI1Uzc3FDaTFtQo5VBUZER\nA Conservation property of LWFR on curvilinear grids\nIn order to show that the LWFR scheme is conservative, multiply (28) withJe,pwpand sum over p∈Nd\nNto get, using\nthe exactness of quadrature\nun+1\ne=un\ne−∆t\n|Ωe|ˆ\nΩo∇ξ·˜Fδ\ne(ξ)dξ\n−∆t\n|Ωe|ˆ\nΩodX\ni=1((˜Fe·bnR,i)∗−˜Fδ\ne·bnR,i)(ξR\ni)g′\nR(ξpi)−((˜Fe·bnL,i)∗−˜Fδ\ne·bnL,i)(ξL\ni)g′\nL(ξpi)dξ\n(91)\nwhere ξs\niare as defined in (17). Then, note the following integral identities that are an application of Fubini’s theorem\nfollowed by fundamental theorem of Calculus\nˆ\nΩo∂ξi·˜Fδ\ne(ξ)dξ=ˆ\n∂ΩL\no,i[˜Fδ\ne·bnL,i]dSξ+ˆ\n∂ΩR\no,i[˜Fδ\ne·bnR,i]dSξ\nˆ\nΩo((˜Fe·bnR,i)∗−˜Fδ\ne·bnR,i)(ξR\ni)g′\nR(ξpi)dξ=ˆ\n∂ΩR\no,i[(˜Fe·bnR,i)∗−˜Fδ\ne·bnR,i]dSξ\nˆ\nΩo((˜Fe·bnL,i)∗−˜Fδ\ne·bnL,i)(ξL\ni)g′\nL(ξpi)dξ=−ˆ\n∂ΩL\no,i[(˜Fe·bnL,i)∗−˜Fδ\ne·bnL,i]dSξ\nwhere ∂ΩS\no,iis as in Figure 1 and we used gL(−1) = gR(1) = 1 ,gL(−1) = gR(1) = 0 . Then substituting these\nidentities into (91) gives us the conservative update of the cell average (29).\n35APREPRINT - FEBRUARY 21, 2024\nReferences\n[1]A. B ABBAR AND P. C HANDRASHEKAR ,Extension of lwfr to adaptive, curved meshes with error based time\nstepping .https://github.com/Arpit-Babbar/JCP2024 , 2024.\n[2] A. B ABBAR , S. K. K ENETTINKARA ,AND P. 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J AMESON ,An extended range of stable-\nsymmetric-conservative Flux Reconstruction correction functions , Computer Methods in Applied Mechanics and\nEngineering, 296 (2015), pp. 248–272. bibtex: Vincent2015.\n37APREPRINT - FEBRUARY 21, 2024\n[45] J. W ARE AND M. B ERZINS ,Adaptive finite volume methods for time-dependent p.d.e.s. , in Modeling, Mesh\nGeneration, and Adaptive Numerical Methods for Partial Differential Equations, I. Babuska, W. D. Henshaw,\nJ. E. Oliger, J. E. Flaherty, J. E. Hopcroft, and T. Tezduyar, eds., New York, NY , 1995, Springer New York,\npp. 417–430.\n[46] P. W OODWARD AND P. C OLELLA ,The numerical simulation of two-dimensional fluid flow with strong shocks ,\nJournal of Computational Physics, 54 (1984), pp. 115–173.\n[47] O. Z ANOTTI , F. F AMBRI ,AND M. D UMBSER ,Solving the relativistic magnetohydrodynamics equations with\nADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement , Monthly\nNotices of the Royal Astronomical Society, 452 (2015), pp. 3010–3029.\n[48] X. Z HANG AND C.-W. S HU,On maximum-principle-satisfying high order schemes for scalar conservation laws ,\nJournal of Computational Physics, 229 (2010), pp. 3091–3120.\n[49] X. Z HANG AND C.-W. S HU,On positivity-preserving high order discontinuous galerkin schemes for compressible\neuler equations on rectangular meshes , Journal of Computational Physics, 229 (2010), pp. 8918–8934.\n[50] D. Z ORÍO , A. B AEZA ,AND P. M ULET ,An Approximate Lax–Wendroff-Type Procedure for High Order Accurate\nSchemes for Hyperbolic Conservation Laws , Journal of Scientific Computing, 71 (2017), pp. 246–273. bibtex:\nZorio2017.\n38"    },    {        "title": "2402.11933v1.SLADE__Detecting_Dynamic_Anomalies_in_Edge_Streams_without_Labels_via_Self_Supervised_Learning.pdf",        "content": "SLADE: Detecting Dynamic Anomalies in Edge Streams without\nLabels via Self-Supervised Learning\nJongha Lee\nKAIST\njhsk777@kaist.ac.krSunwoo Kim\nKAIST\nkswoo97@kaist.ac.krKijung Shin\nKAIST\nkijungs@kaist.ac.kr\nABSTRACT\nTo detect anomalies in real-world graphs, such as social, email,\nand financial networks, various approaches have been developed.\nWhile they typically assume static input graphs, most real-world\ngraphs grow over time, naturally represented as edge streams. In\nthis context, we aim to achieve three goals: (a) instantly detecting\nanomalies as they occur, (b) adapting to dynamically changing\nstates, and (c) handling the scarcity of dynamic anomaly labels.\nIn this paper, we propose SLADE (Self-supervised Learning for\nAnomaly Detection in Edge Streams) for rapid detection of dy-\nnamic anomalies in edge streams, without relying on labels. SLADE\ndetects the shifts of nodes into abnormal states by observing devia-\ntions in their interaction patterns over time. To this end, it trains a\ndeep neural network to perform two self-supervised tasks: (a) min-\nimizing drift in node representations and (b) generating long-term\ninteraction patterns from short-term ones. Failure in these tasks\nfor a node signals its deviation from the norm. Notably, the neural\nnetwork and tasks are carefully designed so that all required opera-\ntions can be performed in constant time (w.r.t. the graph size) in\nresponse to each new edge in the input stream. In dynamic anomaly\ndetection across four real-world datasets, SLADE outperforms nine\ncompeting methods, even those leveraging label supervision.\n1 INTRODUCTION\nThe evolution of web technologies has dramatically enhanced hu-\nman life. Platforms such as email and social networks have made\npeople communicating with diverse individuals and accessing help-\nful information easier. Additionally, e-commerce has enabled people\nto engage in economic activities easily. However, as convenience\nhas increased, many problems have emerged, such as financial\ncrimes, social media account theft, and spammer that exploited it.\nMany graph anomaly detection techniques [ 25,27] have been\ndeveloped for tackling these problems. These techniques involve\nrepresenting the interactions between users as a graph, thereby\nharnessing the connectivity between users to effectively identify\nanomalies. However, graph anomaly detection in real-world sce-\nnarios poses several challenges, as discussed below.\nC1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection:C1) Time Delay in Detection: While most graph anomaly detec-\ntion methods assume static input graphs, real-world graphs evolve\nover time with continuous interaction events. In response to con-\ntinuous interaction events, it is important to quickly identify anom-\nalies. Delaying the detection of such anomalies can lead to increas-\ning harm to benign nodes as time passes. However, employing static\ngraph-based methods repeatedly on the entire graph, whenever an\ninteraction event occurs, inevitably leads to significant delays due\nto the substantial computational expenses involved. To mitigate\ndelays, it is necessary to model continuous interaction events asedge streams and employ incremental computation to assess the ab-\nnormality of each newly arriving edge with detection time constant\nregardless of the accumulated data size.\nMany studies [ 4,12] have developed anomaly detection methods\nfor edge streams, leveraging incremental computation. However, as\nthese methods are designed to target specific anomaly types (e.g.,\nburstiness), lacking learning-based components, they are often\nlimited in capturing complex ones deviating from targeted types.\nC2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States:C2) Dynamically Changing States: In web services, users can\nexhibit dynamic states varying over time. That is, a user’s behavior\ncan be normal during one time period but abnormal during another.\nFor example, a normal user’s account can be compromised and\nthen manipulated to disseminate promotional messages. As a result,\nthe user’s state transitions from normal to abnormal. Such a user\ncan be referred to as a dynamic anomaly, and detecting dynamic\nanomalies presents a greater challenge compared to the relatively\neasier task of identifying static anomalies.\nAddressing this challenge can be facilitated by tracking the evo-\nlution of node characteristics over time, and to this end, dynamic\nnode representation learning [ 28,38] can be employed. However,\nexisting dynamic node representation learning methods require\nlabel information for the purpose of anomaly detection, which is\ntypically scarce, as elaborated in the following paragraph.\nC3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels:C3) Lack of Anomaly Labels: Deep neural networks, such as graph\nneural networks, have proven effective in detecting complex anom-\nalies within graph-structured data. However, they typically rely on\nlabel supervision for training, which can be challenging to obtain,\nespecially for dynamic anomalies. Assuming the absence of anom-\naly labels, various unsupervised training techniques have been\ndeveloped for anomaly detection in static graphs [10, 23].\nHowever, to the best of our knowledge, existing methods cannot\nsimultaneously address all three challenges. They either lack the\nability to detect dynamic anomalies, lack incremental detection\nsuitable for edge streams, or rely on labels.\nIn this work, we propose SLADE (Self-supervised Learning for\nAnomaly Detection in Edge Streams) to simultaneously tackle all\nthree challenges. Its objective is to incrementally identify dynamic\nanomalies in edge streams, without relying on any label supervision.\nTo achieve this, SLADE learns dynamic representations of nodes\nover time, which capture their long-term interaction patterns, by\ntraining a deep neural network to perform two self-supervised tasks:\n(a) minimizing drift in the representations and (b) generating long-\nterm interaction patterns from short-term ones. Poor performance\non these tasks for a node indicates its deviation from its interac-\ntion pattern, signaling a potential transition to an abnormal state.\nNotably, our careful design ensures that all required operations are\nexecuted in constant time (w.r.t. the graph size) in response to each\nnew edge in the input stream. Our experiments involving dynamicarXiv:2402.11933v1  [cs.LG]  19 Feb 2024anomaly detection by 9 competing methods across 4 real-world\ndatasets confirm the strengths of SLADE , outlined below:\n•Unsupervised: We propose SLADE to detect complex dynamic\nanomalies in edge streams, without relying on label supervision.\n•Effective: In dynamic anomaly detection, SLADE shows an av-\nerage improvement of 12.80% and 4.23% (in terms of AUC) com-\npared to the best-performing (in each dataset) unsupervised and\nsupervised competitors, respectively.\n•Constant Inference Speed: We show both theoretically and\nempirically that, once trained, SLADE requires a constant amount\nof time per edge for dynamic anomaly detection in edge streams.\n2 RELATED WORK\nIn this section, we provide a concise review of graph-based anomaly\ndetection methods, categorized based on the nature of input graphs.\n2.1 Anomaly Detection in CTDGs\nAcontinuous-time dynamic graph (CTDG) is a stream of edges\naccompanied by timestamps, which we also term an edge stream .\nUnsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs:Unsupervised Anomaly Detection in CTDGs: The majority of\nunsupervised anomaly detection techniques applied to CTDGs aim\nto identify specific anomaly types. Sedanspot [ 12] focuses on de-\ntecting (a) bursts of activities and (b) bridge edges between sparsely\nconnected parts of the input graph. MIDAS [ 4] aims to spot bursts of\ninteractions within specific groups, and F-FADE [ 6] is effective for\nidentifying sudden surges in interactions between specific pairs of\nnodes and swift changes in the community memberships of nodes.\nLastly, AnoGraph [ 5] spots dense subgraph structures along with\nthe anomalous edges contained within them. These methods are\ngenerally efficient, leveraging incremental computation techniques.\nNonetheless, as previously mentioned, many of these approaches\nlack learnable components and thus may encounter challenges in\nidentifying complex anomaly patterns, i.e., deviations from normal\npatterns in various aspects that may not be predefined.\nRepresentation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs:Representation Learning in CTDGs: Representation learning in\nCTDGs involves maintaining and updating representations of nodes\nin response to each newly arriving edge, capturing evolving pat-\nterns of nodes over time. To this end, several neural-network archi-\ntectures have been proposed. JODIE [ 20] utilizes recurrent-neural-\nnetwork (RNN) modules to obtain dynamic node representations.\nDyrep [ 33] combines a deep temporal point process [ 1] with RNNs.\nTGAT [ 38] leverages temporal encoding and graph attention [ 35]\nto incorporate temporal information during neighborhood aggrega-\ntion. Based on temporal smoothness, DDGCL [ 32] learns dynamic\nnode representations by contrasting those of the same nodes in\ntwo nearby temporal views. TGN [ 28] utilizes a memory module\nto capture and store long-term patterns. This module is updated\nusing an RNN for each node, providing representations that encom-\npass both temporal and spatial characteristics. Many subsequent\nstudies [ 36,37] have also adopted memory modules. Assuming a\ngradual process where memories for nodes do not show substan-\ntial disparities before and after updates, DGTCC [ 13] contrasts\nmemories before and after updates for training.\nFor the purpose of anomaly detection, dynamic node represen-\ntations can naturally be used as inputs for a classifier, which is\ntrained in a supervised manner using anomaly labels. Recently,more advanced anomaly detection methods based on representa-\ntion learning in CTDGs have been proposed. SAD [ 31] combines a\nmemory bank with pseudo-label contrastive learning, which, how-\never, requires anomaly labels for training.\n2.2 Anomaly Detection in Other Graph Models\nIn this subsection, we introduce anomaly detection approaches\napplied to other graph models.\nAnomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs:Anomaly Detection in Static Graphs: Many approaches [ 10,11,\n22] have been developed for anomaly detection in a static graph ,\nwhich does not contain any temporal information. Among them, we\nfocus on those leveraging graph self-supervised learning, which ef-\nfectively deals with the absence of anomaly labels. ANEMONE [ 14]\ncontrasts node representations obtained from (a) features alone\nand (b) both graph topology and features, identifying nodes with\nsubstantial differences as anomalies. DOMINANT [ 10] aims to re-\nconstruct graph topology and attributes, using a graph-autoencoder\nmodule, and anomalies are identified based on reconstruction error.\nThese approaches assume a static graph, and their extensions to\ndynamic graphs are not trivial, as node representations need to\nevolve over time to accommodate temporal changes.\nAnomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs:Anomaly Detection in DTDGs: Adiscrete-time dynamic graph\n(DTDG) corresponds to a sequence of graphs occurring at each time\ninstance, also referred to as a graph stream . Formally, a DTDG is a\nset\b\nG1,G2,···,G𝑇\t\nwhereG𝑡=\b\nV𝑡,E𝑡\t\nis the graph snapshot\nat time𝑡, andV𝑡andE𝑡are node- and edge-set in G𝑡, respectively\n[15]. Several methods have been developed for detecting anom-\nalies, with an emphasis on anomalous edges, in a DTDG, which is\na sequence of graphs at each time instance. Netwalk [ 39] employs\nrandom walks and autoencoders to create similar representations\nfor nodes that frequently interact with each other. Then, it identifies\ninteractions between nodes with distinct representations as anoma-\nlous. AddGraph [ 40] constructs node representations through an\nattentive combination of (a) short-term structural patterns within\nthe current graph (and a few temporally-adjacent graphs) captured\nby a graph neural network (spec., GCN [ 18]) and (b) long-term\npatterns captured by an RNN (spec., GRU [ 7]). These node repre-\nsentations are then employed to evaluate the anomalousness of\nedges. Instead, transformers [ 34] are employed in TADDY [ 24] to\nacquire node representations that capture both global and local\nstructural patterns. Note that these methods designed for DTDGs\nare less suitable for time-critical applications when compared to\nthose for designed CTDGs, as discussed in Section 3. Moreover,\ntechnically, these methods are trained to distinguish edges and non-\nedge node pairs, while our proposed method contrasts long-term\nand short-term patterns for training.\n3 PROBLEM DESCRIPTION\nIn this section, we introduce notations and, based on them, define\nthe problem of interest, dynamic node anomaly detection in CTDGs.\nNotations:Notations:Notations:Notations:Notations:Notations:Notations:Notations:Notations:Notations:Notations:Notations:Notations:Notations:Notations:Notations:Notations: A continuous-time dynamic graph (CTDG) G=(𝛿1,𝛿2,\n···) is a stream (i.e., continuous sequence) of temporal edges with\ntimestamps. Each temporal edge 𝛿𝑛=(𝑣𝑖,𝑣𝑗,𝑡𝑛)arriving at time 𝑡𝑛\nis directional from the source node𝑣𝑖to the destination node 𝑣𝑗. The\ntemporal edges are ordered chronologically, i.e., 𝑡𝑛≤𝑡𝑛+1holds for\n2each𝑛∈{1,2,···}. We denote byV(𝑡)=Ð\n(𝑣𝑖,𝑣𝑗,𝑡𝑛)∈G∧𝑡𝑛≤𝑡{𝑣𝑖,𝑣𝑗}\nthe temporal set of nodes arriving at time 𝑡or earlier.\nProblem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description:Problem Description: We consider a CTDG G=(𝛿1,𝛿2,···),\nwhere each temporal edge 𝛿𝑛=(𝑣𝑖,𝑣𝑗,𝑡𝑛)indicates a behavior\nof the source node 𝑣𝑖towards the destination node 𝑣𝑗at time𝑡𝑛.\nThat is, the source node represents the “actor” node. We aim to ac-\ncurately classify the current dynamic status of each node, which is\neither normal orabnormal . We address this problem in an unsuper-\nvised setting. That is, we do not have access to the dynamic states\nof any nodes at any time as input. In our experimental setups, the\nground-truth dynamic states are used only for evaluation purposes.\nReal-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios:Real-world Scenarios: Due to its substantial impact, this problem\nhas been explored in many previous studies, but in supervised\nsettings [ 20,28,33,38]. For instance, in web services, a normal\nuser’s account can be compromised and then exploited to circulate\npromotional messages. In such a case, the user’s state transitions\nfrom normal to abnormal. Detecting such transitions promptly is\ncrucial to minimize the inconvenience caused by the dissemination\nof promotional messages.\nWhy CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?:Why CTDGs?: Anomaly detection methods designed for DTDGs\n(refer to Section 2.2) process input on a per-graph basis, where\nedges need to be aggregated over time to form a graph. Thus, they\nare susceptible to delays in predicting the current state of nodes\nupon the arrival of an edge. In theory, methods designed for static\ngraphs can be applied to our problem by re-running them when-\never a new edge arrives. However, this straightforward application\nmakes their time complexity per edge (super-)linear in the graph\nsize, causing notable delays. However, CTDG-based methods, in-\ncluding our proposed one, process the arriving edge, whenever it\narrives, typically with constant time complexity regardless of the ac-\ncumulated graph size. As discussed in Section 1, this highlights the\nadvantages of CTDG-based methods in time-critical applications,\nincluding (dynamic) anomaly detection.\nComparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection:Comparison with Edge Anomaly Detection: In many scenar-\nios, the dynamic state of an actor node can be equated with the\nanomalousness (or maliciousness) of its behavior. In other words,\nan actor node is assumed to be in the abnormal state if and only\nif it performs anomaly (or malicious) behavior. In such cases, this\ntask shares some similarities with detecting anomalous edges. How-\never, it should be noticed that the predicted current node states are\nused for predicting the anomalousness of future interactions (see\nSection 4.3) rather than assessing the anomalousness of interac-\ntions that have already occurred. Hence, for effective dynamic node\nanomaly detection, it is important to consider node-wise behavioral\ndynamics over time.\n4 PROPOSED METHOD: SLADE\nIn this section, we present SLADE (Self-supervised Learning for\nAnomaly Detection in Edge Streams), our proposed method for\nunsupervised dynamic anomaly detection in CTDGs.\nThe underlying intuition is that nodes in the normal state tend\nto exhibit structurally and temporally similar interaction patterns\nover time [ 2,3], while those in the abnormal state do not because\nrepeating similar abnormal actions increases the risk of detection.\nMotivated by this idea, we consider two key assumptions:•A1. Stable Long-Term Interaction Patterns: Nodes in the\nnormal state tend to repetitively engage in similar interactions\nover a long-term period. This stable long-term interaction pattern\nexhibits minimal variation within short-time intervals.\n•A2. Potential for Restoration of Patterns: It would be feasible\nto accurately regenerate the long-term interaction patterns of the\nnodes in the normal state using recent interaction information.\nThey account for both structural and temporal aspects of normal\nnodes. While A1focuses on temporal aspects, A2takes a further\nstep by specifying the extent of structural similarities over time.\nUpon these assumptions, SLADE employs two self-supervised\ntasks for training its model (i.e., deep neural network) for maintain-\ning and updating a dynamic representation of each node, which we\nexpect to capture its long-term interaction pattern.\n•S1. Temporal Contrast: This aims to minimize drift in dynamic\nnode representations over short-term periods (related to A1).\n•S2. Memory Generation: This aims to accurately generate dy-\nnamic node representations based only on recent interactions\n(related to A2).\nBy being trained for S1andS2, the model is expected to learn\nnormal interaction patterns satisfying A1andA2. Once the model\nis trained, SLADE identifies nodes for which the model performs\npoorly on S1andS2, as these nodes potentially deviate from pre-\nsumed normal interaction patterns.\nSpecifically, to obtain dynamic representations, SLADE employs\nneural networks in combination with memory modules (see Sec-\ntion 4.1). The memories (i.e., stored information), which reflect the\nnormal patterns of nodes, are updated and regenerated to minimize\nour self-supervised losses related to S1andS2(see Section 4.2).\nLastly, these dynamic representations are compared with their past\nvalues and momentary representations to compute anomaly scores\nfor nodes (see Section 4.3). The overview of SLADE is visually pre-\nsented in Figure 1, and the following subsections provide detailed\ndescriptions of each of its components.\n4.1 Core Modules of SLADE\nIn order to incrementally compute the dynamic representation of\neach node, SLADE employs three core modules:\n•Memory Module: These time-evolving parameter vectors rep-\nresent the long-term interaction patterns of each node, i.e., how\na node’s interaction has evolved over time.\n•Memory Updater: This neural network captures evolving char-\nacteristics of nodes’ interaction patterns. It is employed to update\nthe memory (i.e., stored information).\n•Memory Generator: This neural network is used to generate\nthe memory of a target node from its recent interactions.\nBelow, we examine the details of each module in order.\nMemory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module:Memory Module: InSLADE , the dynamic representation of each\nnode, which represents its long-term interaction patterns, is stored\nand updated in a memory module introduced by Rossi et al . [28] .\nSpecifically, the memory module consists of a memory vector 𝒔𝑖\nfor each node 𝑣𝑖, and each 𝒔𝑖captures the interactions of the node\n𝑣𝑖up to the current time. When each node 𝑣𝑖first emerges in the\ninput CTDG, 𝒔𝑖is initialized to a zero vector. As 𝑣𝑖participates in\n3𝒔𝑗\n𝒔𝑙Previous memory 𝒔𝑖’\nMessage 𝒎𝑖Current memory 𝒔𝑖\nGenerated memory ො𝒔𝑖\nRecent neighbors aggregationTGATMemory Updater\nMemory GeneratorTemporal Contrast Memory Generation\nPrevious memory 𝒔𝑖’\nCurrent memory 𝒔𝑖\nGenerated memory ො𝒔𝑖\nAnomaly score( 𝑣𝑖, 𝑡)\n(c) Self -supervised Learning (d) Anomaly Scoring𝑣𝑙𝑣𝑗\n𝑣𝑘𝑣𝑖𝑡𝑖𝑗\n𝑡𝑘𝑖𝑡𝑖𝑙(a) Input Edge Stream…\n𝑣𝑖𝑡𝑖𝑗𝑣𝑗\n𝑣𝑖𝑡12𝑣1𝑣2\n𝑡23𝑣2𝑣3\nRaw message 𝒓𝑖Cosine Similarity\n𝑣𝑖𝑣𝑗\nInteraction𝑣1…𝑣𝑖𝑣𝑗\nMemory module\nMemory masking\n𝒔𝑘𝑣𝑖MLPGRU\n(b) Query at time tFigure 1: Overview of SLADE , whose objective is to measure the anomaly score of a query node at any time. For each newly\narriving edge, SLADE updates the memory vector of each endpoint using GRU. Given a query node, SLADE masks the memory\nvector of the node and approximately regenerates it based on its recent interactions using TGAT. Then, it measures the anomaly\nscore of the query node based on the similarities (1) between previous and current memory vectors (related to S1) and (2)\nbetween current and generated memory vectors (related to S2). SLADE aims to maximize these similarities for model training.\ninteractions, 𝒔𝑖is continuously updated by the memory updater,\ndescribed in the following subsection.\nMemory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater:Memory Updater: Whenever a node participates in a new inter-\naction, the memory updater gradually updates its memory vector,\naiming to represent its stable long-term interaction pattern. First,\neach interaction is transformed into raw messages [ 28]. Each raw\nmessage consists of the encoded time difference between the most\nrecent appearance of one endpoint and the present time, along with\nthe memory vector of the other endpoint. For instance, upon the\narrival of a temporal edge (𝑣𝑖,𝑣𝑗,𝑡𝑖𝑗)at time𝑡𝑖𝑗, with𝑣𝑖and𝑣𝑗\nhaving memory vectors 𝒔′\n𝑖and𝒔′\n𝑗, respectively, the raw messages\nfor the source and destination nodes are created as follows:\n𝒓𝑖=[𝒔′\n𝑗||𝜙(𝑡𝑖𝑗−𝑡−\n𝑖)],𝒓𝑗=[𝒔′\n𝑖||𝜙(𝑡𝑖𝑗−𝑡−\n𝑗)], (1)\nwhere||denotes a concatenate operator, and 𝑡−\n𝑖denotes the time\nof the last interaction of 𝑣𝑖before the interaction time 𝑡𝑖𝑗. Follow-\ning [9], as the time encoding function 𝜙(·), we use\n𝜙(𝑡′)=𝑐𝑜𝑠\u0012\n𝑡′·[𝛼−0\n𝛽||𝛼−1\n𝛽||···||𝛼−𝑑𝑡−1\n𝛽]\u0013\n, (2)\nwhere·denotes the inner product, and 𝑑𝑡denotes the dimension\nof encoded vectors. Scalars 𝛼,𝛽and𝑑𝑡are hyperparameters.\nThen, the raw message 𝒓𝑖is converted to a message 𝒎𝑖, employ-\ning an MLP and then used to update from 𝒔′\n𝑖to𝒔𝑖, as follows:\n𝒔𝑖=GRU(𝒎𝑖,𝒔′\n𝑖)where 𝒎𝑖=MLP(𝒓𝑖). (3)\nHere 𝒔𝑖is the memory vector for node 𝑣𝑖after time𝑡𝑖𝑗, persisting\nuntil a new interaction involving node 𝑣𝑖occurs (see Appendix D.2\nfor exploration of alternatives to GRU [ 7]). We also maintain 𝒔′\n𝑖(i.e.,\nthe previous value of the memory vector) for its future usage.\nMemory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator:Memory Generator: The memory generator aims to restore the\nmemory vectors, which represent existing long-term interaction\npatterns, based on short-term interactions. The generated vectors\nare used for training and anomaly scoring, as described later.The first step of generation is to (temporarily) mask the memory\nvector of a target node at current time 𝑡to a zero vector. Then, the\nmemory vectors of its at most 𝑘latest (1-hop) neighbors and the\ncorresponding time information of their latest interactions are used\nas inputs of an encoder. As the encoder, we employ TGAT [ 38],\nwhich attends more to recent interactions (see Appendix D.2 for\nexploration of alternatives). That is, for a target node 𝑣𝑖and the\ncurrent time 𝑡, we generate its memory vector as follows:\nˆ𝒔𝑖=MultiHeadAttention (𝒒,K,V),where 𝒒=𝜙(𝑡−𝑡),\nandK=V=\u0002\ns𝑛1||𝜙(𝑡−𝑡′\n𝑛1),···,s𝑛𝑘||𝜙(𝑡−𝑡′\n𝑛𝑘)\u0003\n.\nHere{𝑛1,...,𝑛𝑘}denote the indices of the neighbors of the target\nnode𝑣𝑖,{𝑡′𝑛1,...,𝑡′𝑛𝑘}denote the timestamps of the most recent in-\nteractions with them before 𝑡, and ˆs𝑖denotes the generated memory\nvector of𝑣𝑖at𝑡. As the memory vector has been masked, only time\ninformation is used for query component 𝒒.\n4.2 Training Objective and Procedure\nFor the proposed temporal contrast task ( S1) and memory genera-\ntion task ( S2),SLADE is trained to minimize two loss components:\n•Temporal Contrast Loss: It encourages the agreement between\nprevious and updated memory vectors.\n•Memory Generation Loss: It encourages the similarity between\nretained and generated memory vectors.\nBelow, we describe each self-supervised loss component and then\nthe entire training process in greater detail.\nTemporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss:Temporal Contrast Loss: ForS1, we aim to minimize drift in\ndynamic node representations within short time intervals. For the\nmemory vector 𝒔𝑖of each node 𝑣𝑖, we use the previous memory\nvector 𝒔′\n𝑖as the positive sample and the memory vectors of the\nother nodes as negative samples. Specifically, for each interaction\n4at time𝑡of𝑣𝑖, we aim to minimize the following loss:\nℓ𝑐(𝑣𝑖,𝑡)=−logexp(𝑠𝑖𝑚(𝒔𝑖(𝑡+),𝒔′\n𝑖(𝑡+)))\nÍ|V(𝑡+)|\n𝑘=1exp(𝑠𝑖𝑚(𝒔𝑖(𝑡+),𝒔𝑘(𝑡+))), (4)\nwhere 𝒔𝑖(𝑡+)is the current memory vector 𝒔𝑖right after processing\nthe interaction at 𝑡(denoted as𝑡+to distinguish it from 𝑡right before\nprocessing the interaction), 𝒔′\n𝑖(𝑡+)is the previous memory vector,\nand𝑠𝑖𝑚is the cosine similarity function, i.e., 𝑠𝑖𝑚(𝒖,𝒗)=∥𝒖·𝒗∥\n∥𝒖∥·∥𝒗∥.\nMemory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss:Memory Generation Loss: ForS2, we aim to accurately generate\ndynamic node representations from recent interactions (i.e., short-\nterm patterns). For each node 𝑣𝑖, we expect its generated memory\nvector ˆ𝒔𝑖to be matched well with the (temporally masked) memory\nvector 𝒔𝑖, relative to other memory vectors. Specifically, for each\ninteraction at time 𝑡of𝑣𝑖, we aim to minimize the following loss:\nℓ𝑔(𝑣𝑖,𝑡)=−logexp(𝑠𝑖𝑚(ˆ𝒔𝑖(𝑡+),𝒔𝑖(𝑡+)))\nÍ|V(𝑡+)|\n𝑘=1exp(𝑠𝑖𝑚(ˆ𝒔𝑖(𝑡+),𝒔𝑘(𝑡+))), (5)\nwhere 𝒔𝑖(𝑡+)is the current memory vector 𝒔𝑖after processing the\ninteraction at 𝑡, and ˆ𝒔𝑖(𝑡+)is the generated memory vector ˆ𝒔𝑖. We\npropose a novel self-supervised learning task that encourages a\nmodel to learn the normal temporal pattern of data.\nBatch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training:Batch Processing for Efficient Training: For computational ef-\nficiency, we employ batch processing for training SLADE , which\nhas been commonly used [ 28,31,38]. That is, instead of process-\ning a single edge at a time, multiple edges are fed into the model\nsimultaneously. To this end, a stream of temporal edges is divided\ninto batches of a fixed size in chronological order. Consequently,\nmemory updates take place at the batch level, and for these updates,\nthe model uses only the interaction information that precedes the\ncurrent batch. Note that a single node can be engaged in multiple\ninteractions within a single batch, leading to multiple raw mes-\nsages for the node. To address this, SLADE aggregates the raw\nmessages into one raw message using mean pooling and continues\nwith the remaining steps of memory update. In order for the tem-\nporal contrast in Eq. (4)reflects A1, which emphasizes minimizing\nthe difference between memory vectors before and after an update\nwithin a “short” time span, the batch size should not be excessively\nlarge. Therefore, it is crucial to establish an appropriate batch size,\ntaking both this and efficiency into consideration.\nFinal Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective:Final Training Objective: For each batch 𝐵with temporal edges\nE𝐵, the overall temporal contrast loss is defined as follows:\nL𝑐=1\n|E𝐵|∑︁\n(𝑣𝑖,𝑣𝑗,𝑡)∈E𝐵𝜔𝑐𝑠ℓ𝑐(𝑣𝑖,𝑡)+𝜔𝑐𝑑ℓ𝑐(𝑣𝑗,𝑡), (6)\nwhere𝜔𝑐𝑠and𝜔𝑐𝑑are hyperparameters for weighting each source\nnode and each destination node, respectively. Similarly, the overall\nmemory generation loss for𝐵is defined as follows:\nL𝑔=1\n|E𝐵|∑︁\n(𝑣𝑖,𝑣𝑗,𝑡)∈E𝐵𝜔𝑔𝑠ℓ𝑔(𝑣𝑖,𝑡)+𝜔𝑔𝑑ℓ𝑔(𝑣𝑗,𝑡), (7)\nwhere𝜔𝑔𝑠and𝜔𝑔𝑑hyperparameters for weighting each source\nnode and each destination node, respectively.\nThe final lossLfor𝐵is the sum of both losses, i.e., L=L𝑐+L𝑔.\nNote that, since anomaly labels are assumed to be unavailable,\nSLADE is trained with the assumption that the state of all nodes\nappearing in the training set is normal, irrespective of their actual\nstates. However, SLADE still can learn normal patterns effectively,given that they constitute the majority of the training data, as\ndiscussed for various types of anomalies in Section 5.1.\n4.3 Anomaly Scoring\nAfter being trained, SLADE is able to measure the anomaly score\nof any node at any given time point. SLADE measures how much\neach node deviates from A1andA2by computing the temporal\ncontrast score and the memory generation score, which are\nbased on A1andA2, respectively. Below, we describe each of them.\nTemporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score:Temporal Contrast Score: This score is designed to detect anoma-\nlous nodes that deviate from A1, and to this end, it measures the ex-\ntent of abrupt changes in the long-term interaction pattern. Specif-\nically, the temporal contrast score 𝑠𝑐𝑐(𝑣𝑖,𝑡)of a node𝑣𝑖at time𝑡\n(spec., before processing any interaction at 𝑡) is defined as the cosine\ndistance between its current and previous memory vectors:\n𝑠𝑐𝑐(𝑣𝑖,𝑡)=1−𝑠𝑖𝑚(𝒔𝑖(𝑡),𝒔′\n𝑖(𝑡)). (8)\nMemory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score:Memory Generation Score: In order to identify anomalous nodes\ndeviating from A2, this score measures the degree of deviation of\nshort-term interaction patterns from long-term interaction patterns.\nSpecifically, the memory generation score 𝑠𝑐𝑔(𝑣𝑖,𝑡)of a node𝑣𝑖at\ntime𝑡is defined as the cosine distance between its current and\ngenerated memory vectors:\n𝑠𝑐𝑔(𝑣𝑖,𝑡)=1−𝑠𝑖𝑚(ˆ𝒔𝑖(𝑡),𝒔𝑖(𝑡)). (9)\nFinal Score:Final Score:Final Score:Final Score:Final Score:Final Score:Final Score:Final Score:Final Score:Final Score:Final Score:Final Score:Final Score:Final Score:Final Score:Final Score:Final Score: The final anomaly score is a combination of both\nscores, i.e., the final score𝑠𝑐(𝑣𝑖,𝑡)of a node𝑣𝑖at time𝑡is defined as\n𝑠𝑐(𝑣𝑖,𝑡)=(𝑠𝑐𝑐(𝑣𝑖,𝑡)+𝑠𝑐𝑔(𝑣𝑖,𝑡))/4, (10)\nwhere it is normalized to fall within the range of [0,1]. Nodes in\nthe normal state will have scores closer to 0, while those in the\nabnormal state will have scores closer to 1.\nNote that in our notation, 𝑠𝑐(𝑣𝑖,𝑡)denotes the anomaly score\nbefore observing or processing any interaction at time 𝑡, if such\nan interaction exists. We use this score to measure the potential\nabnormality of node 𝑣𝑖’s current state and also that of its subsequent\naction (which can occur at time 𝑡or later).\n5 DISCUSSION AND ANALYSIS\nIn this section, we discuss how SLADE deals with various types of\nanomalies. Then, we analyze the time complexity of SLADE .\n5.1 Discussion on Anomaly Types\nBelow, we discuss how SLADE can detect anomalies of various\ntypes, without any prior information about the types. In Section 6.2\n(RQ4), we empirically confirm its effectiveness for all these types.\nT1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies:T1) Hijacked Anomalies: This type involves a previously nor-\nmal user’s account being compromised at some point, exhibiting\nmalicious behaviors that deviate from the user’s normal pattern.\nSLADE , motivated by such anomalies, contrasts short-term and\nlong-term interaction patterns to spot them, as discussed above.\nT2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies:T2) New or Rarely-interacting Anomalies: Many deep learning-\nbased detection models struggle with anomalies involving (1) newly\nintroduced or (2) rarely interacting nodes due to limited data for\nlearning their normal behaviors. SLADE assigns a higher anomaly\nscore (both contrast and generation scores) to such nodes because\n5their memory vectors undergo substantial changes until their long-\nterm patterns are established. We find it advantageous to pay atten-\ntion to such nodes because, in some of the real-world datasets we\nused, including Wikipedia, Bitcoin-alpha, and Bitcoin-OTC, new\nand rarely interacting nodes are more likely to engage in anomalous\nactions than those with consistent interactions.1\nT3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies:T3) Consistent Anomalies: SLADE can also effectively identify\nanomalies exhibiting interaction patterns that are consistent over\ntime but deviating from those of normal nodes. Due to the limited\ncapacity (i.e., expressiveness) of the neural networks used in it,\nSLADE prioritizes learning prevalent patterns (i.e., those of normal\nnodes) over less common abnormal ones. As a result, this can lead\nto a violation of A2, causing SLADE to assign high anomaly scores,\nespecially memory generation scores, to such nodes.\n5.2 Complexity Analysis\nWe analyze the time complexity of SLADE “in action” after be-\ning trained. Specifically, we examine the cost of (a) updating the\nmemory in response to a newly arrived edge, and (b) calculating\nanomaly scores for a query node.\nMemory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update:Memory Update: Given a newly arriving edge, SLADE updates\nthe memory vector of each endpoint using GRU. The total time com-\nplexity is dominated by that of GRU, which is O(𝑑𝑠2+𝑑𝑠𝑑𝑚)[29],\nwhere𝑑𝑠and𝑑𝑚indicate the dimensions of memory vectors and\nmessages respectively.\nAnomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring:Anomaly Scoring: Given a query node, SLADE first aims to gener-\nate its memory vector using TGAT with input being a zero-masked\nmemory of the query node and the memory vectors of its at most\n𝑘most recent neighbors. Thus, the time complexity of obtaining\ngenerated memory is O(𝑘𝑑𝑠2)[41] if we assume, for simplicity,\nthat the embedding dimension of the attention layers is the same\nas the dimension 𝑑𝑠of the memory vectors. Then, SLADE com-\nputes the similarities (a) between the current memory and previous\nmemory vectors and (b) between current and generated memory\nvectors, takingO(𝑑𝑠2)time. As a result, the overall time complexity\nofSLADE for anomaly scoring of a node is O(𝑘𝑑𝑠2).\nThe time complexity for both tasks is O(𝑘𝑑𝑠2+𝑑𝑠𝑑𝑚), which\nisconstant with respect to the graph size (i.e., the numbers of\naccumulated nodes and edges). If we assume that anomaly scoring\n(for each endpoint) is performed whenever each edge arrives, the\ntotal complexity becomes linear in the number of accumulated\nedges, as confirmed empirically in Section 6.2 (RQ2).\n6 EXPERIMENTS\nIn this section, we review our experiments for answering the fol-\nlowing research questions:\n•RQ1) Accuracy: How accurately does SLADE detect anomalies,\ncompared to state-of-the-art competitors?\n•RQ2) Speed: Does SLADE exhibit detection speed constant with\nrespect to the graph size?\n•RQ3) Ablation Study: Does every component of SLADE con-\ntribute to its performance?\n•RQ4) Type Analysis: CanSLADE accurately detect various\ntypes of anomalies discussed in Section 5.1?\n1We suspect that new or inactive accounts are typically used to perform anomalous\nactions since the cost of being suspended is low for such accounts.6.1 Experiment Details\nIn this subsection, we describe datasets, baseline methods, and eval-\nuation metrics that are used throughout our experiments. Then, we\nclarify the implementation details of the proposed method SLADE .\nDatasets:Datasets:Datasets:Datasets:Datasets:Datasets:Datasets:Datasets:Datasets:Datasets:Datasets:Datasets:Datasets:Datasets:Datasets:Datasets:Datasets: We assess the performance of SLADE on four real-world\ndatasets: two social networks (Wikipedia and Reddit [ 20]) and two\nonline financial networks (Bitcoin-alpha and Bitcoin-OTC [ 19]).\nThe Wikipedia dataset records edits made by users on Wikipedia\npages. In this context, when a user is banned after a specific edit, the\nuser’s dynamic label is marked as abnormal. The Reddit dataset con-\nsists of posts made by users on subreddits. The user’s dynamic label\nindicates whether the user is banned after the specific post. The\nBitcoin-alpha and Bitcoin-OTC datasets are fundamentally struc-\ntured as trust-weighted signed networks. Within these datasets,\nBitcoin alpha or OTC members assign ratings to other members,\nranging from -10 (total distrust) to +10 (complete trust). We uti-\nlize these ratings with temporal information to identify anomalous\nnodes and assign dynamic labels to users. Further details of the\ndataset processing and the statistics are provided in Appendix A.\nAcross all datasets, we assess the performance of our method and\nthe baseline models by utilizing the final 15% of the dataset in\nchronological order as the test set.\nBaselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric:Baselines Methods and Evaluation Metric: We extensively com-\npareSLADE with several baseline methods capable of anomaly de-\ntection in CTDGs under inductive settings (i.e., without further opti-\nmization for test sets). For four rule-based methods (SedanSpot [ 12],\nMIDAS-R [ 4], F-FADE [ 6], Anoedge-l [ 5]), since they do not require\nany representation learning process, we evaluate their performance\nin test sets without model training. For five neural network-based\nmodels (JODIE [ 20], Dyrep [ 33], TGAT [ 38], TGN [ 28], SAD [ 31]),\nwe train each model by using train sets and do hyperparameter tun-\ning with validation sets. At last, we evaluate them by using test sets.\nIn scenarios where a model requires node or edge features, but these\nattributes are absent from the dataset, the model uses zero vectors\nas substitutes. A detailed description of these baselines is provided\nin Appendix B. To quantify the performance of each model, we use\nArea Under ROC (AUC) as an evaluation metric. For extra results\nin terms of Average Precision (AP), refer to Appendix D.1.\nImplementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details:Implementation Details: In our experiments, we use two ver-\nsions of our method, SLADE andSLADE -HP. For SLADE , as label\nutilization is not possible in its original setting, we do not conduct\nhyperparameter tuning on the validation dataset split. Instead, we\nuse the same hyperparameter combination across all datasets. We\nadopt a batch size of 100and an initial learning rate of 3×10−6.\nWe also fix a memory dimension to 256, a message dimension to\n128, and a max number of most recent neighbors (node degree) to\n20. For the loss function, we set 𝜔𝑐𝑠=1,𝜔𝑐𝑑=1,𝜔𝑔𝑠=0.1, and\n𝜔𝑔𝑑=0.1across all datasets. We train the model on the training\nset for 10 epochs and then evaluate its performance on the test set.\nAnother version of SLADE isSLADE -HP, where we tune the\nhyperparameters of SLADE by using the validation split of each\ndataset, just as we tune the hyperparameters of “all” baseline meth-\nods, including the rule-based ones. Refer to Appendix C.3 for details\nregarding the hyperparameter search space of SLADE -HP.\nFor the baseline methods, we assess the performance on the test\nset using two hyperparameter settings: (a) those recommended by\n6Table 1: AUC (in %) in the detection of dynamic anomaly\nnodes. The first four methods are rule-based models, and the\nothers are based on representation learning. For each dataset,\nthe best and the second-best performances are highlighted in\nboldface and underlined, respectively. In most cases, SLADE\nandSLADE -HP perform best, even when compared to models\nthat rely on label information. For results in terms of Average\nPrecision (AP), refer to Appendix D.1.\nMethod Wikipedia Reddit Bitcoin-alpha Bitcoin-OTC\nSedanSpot [12] 82.88±1.54 58.97±1.85 69.09±0.76 71.71±0.73\nMIDAS [4] 62.92±3.53 59.94±0.95 64.57±0.11 62.16±1.99\nF-FADE [6] 44.88±0.00 49.79±0.05 53.57±0.00 51.12±0.00\nAnoedge-l [5] 47.38±0.32 48.47±0.46 62.52±0.24 65.99±0.19\nJODIE [20] 85.75±0.17 61.47±0.58 73.53±1.26 69.13±0.96\nDyrep [33] 85.68±0.34 63.42±0.62 73.30±1.51 70.76±0.70\nTGAT [38] 83.24±1.11 65.12±1.65 71.67±0.90 68.33±1.23\nTGN [28] 87.47±0.22 67.16±1.03 69.90±0.99 76.23±0.23\nSAD [31] 86.15±0.63 68.45±1.27 68.56±3.17 64.67±4.97\nSLADE 87.75±0.68 72.19±0.60 76.32±0.28 75.80±0.19\nSLADE -HP 88.68±0.39 75.08±0.50 76.92±0.36 77.18±0.27\nFigure 2: AUC (in %) when varying the test start ratio. For\nlearning-based methods, temporal edges preceding the test\nstart ratio in the dataset are employed for training. If vali-\ndation is needed, the last 10% of the training set is used for\nvalidation. Note that SLADE performs best in most cases.\nthe authors, and (b) those leading to the best validation performance\nfor each specific dataset. Subsequently, we conduct a performance\nevaluation on the test set using these two distinct settings and\nreport the best-performing outcome.\nThe reported results have been obtained by averaging across\n10 separate runs, each with different random initializations of the\nmodels. A further description of the implementation and hyperpa-\nrameters of SLADE and baselines are provided in Appendix C.\n6.2 Experimental Results\nRQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy:RQ1) Accuracy: As shown in Table 1, SLADE andSLADE -HP\nsignificantly outperform other baseline methods in most of the\ndatasets. There are two notable observations in this analysis.\nFirst, regarding the unsupervised learning competitors (i.e., SedanS-\npot, MIDAS, F-FADE, and Anoedge-l), SLADE consistently outper-\nforms them across all datasets, achieving performance gains of\nup to 20.44% (in the Reddit dataset) compared to the second-best\nFigure 3: The left figure shows the linear increase of the\nrunning time of SLADE with respect to the number of edges\nin the Reddit dataset. The right figure shows the trade-off\nbetween detection speed and accuracy (with standard devia-\ntions) in the Wikipedia dataset provided by the competing\nmethods. The baseline methods with AUC scores below 60%\nare excluded from consideration to enhance the clarity of per-\nformance differences between the methods. SLADE exhibits\nconstant processing time per edge (as proven in Section 5.2),\noffering the best trade-off between speed and accuracy. For a\ntraining-time comparison, refer to Appendix D.3.\nperforming unsupervised model. This result demonstrates that real-\nworld anomalies exhibit complex patterns hard to be fully captured\nby fixed rule-based approaches. As a result, the necessity for more\nintricate representation models becomes evident.\nSecond, interestingly, while SLADE does not utilize any label\ninformation, even for hyperparameter tuning, it still outperforms\nall supervised baseline models, on all datasets except for the Bit-\ncoinOTC dataset. This result implies that the patterns of nodes in\nthe normal state in real-world graphs closely adhere to A1andA2,\nandSLADE effectively captures such patterns. Furthermore, it is\nevident that measuring the extent to which nodes deviate from the\npatterns provides crucial information for anomaly detection.\nIn addition, we measure the performances of the considered\nmethods while varying the proportion of the training split (or equiv-\nalently the test split). Specifically, we utilize the first T%of the\nedges as a train set and assess each model by using the remaining\n(100−T) %of edges. We refer to Tas atest start ratio . As shown\nin Figure 2, SLADE outperforms all baseline methods in most of\nthe settings, across various test start ratios. Moreover, as seen in\nthe performances of SLADE between the 40% and 80% ratios in\nthe Wikipedia dataset, using just half of the dataset leads to only a\nmarginal performance degradation (spec., 3.2%).\nRQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action:RQ2) Speed in Action: To empirically demonstrate the theoretical\ncomplexity analysis in Section 5.2, we measure the running time of\nSLADE “in action” after being trained on the Reddit dataset while\nvarying the number of edges. Anomaly scoring is performed (for\neach endpoint) only when each edge arrives. As depicted in the\nleft subplot of Figure 3, the running time of SLADE is linear in the\nnumber of edges, being aligned with our analysis.\nAdditionally, we compare the running time and AUC scores of\nSLADE with those of all considered methods. As shown in the\nright plot of Figure 3, while SLADE is about 1.99×slower than\nSedanSpot, which is the most accurate rule-based approach, SLADE\ndemonstrates a significant improvement of 5.88% in AUC scores,\ncompared to SedanSpot. Furthermore, SLADE is about 4.57×faster\nthan TGN, which is the second most accurate following SLADE .\nRQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study:RQ3) Ablation Study: The ablation study is conducted across the\nfour datasets to analyze the necessity of the used self-supervised\n7Table 2: Comparison of AUC (in %) of SLADE and its variants\nthat use a subset of the proposed components (i.e., tempo-\nral contrast lossL𝑐, memory generation loss L𝑔, temporal\ncontrast score 𝑠𝑐𝑐, and memory generation score 𝑠𝑐𝑔).SLADE\nwith all components performs the best overall, showing the\neffectiveness of each.\nL𝑐L𝑔𝑠𝑐𝑐𝑠𝑐𝑔 Wikipedia Reddit Bitcoin-alpha Bitcoin-OTC\n✓ -✓ - 85.69±1.17 48.89±1.53 76.38±0.48 73.87±0.23\n-✓ -✓ 87.57±0.35 61.77±0.12 74.84±0.40 75.80±0.26\n✓ ✓ ✓ - 87.42±1.06 48.51±1.56 76.45±0.52 74.10±0.25\n✓ ✓ -✓ 84.64±0.75 71.33±0.56 75.24±3.50 75.70±0.21\n✓ ✓ ✓ ✓ 87.75±0.68 72.19±0.60 76.32±0.28 75.80±0.19\nTable 3: AUC (in %) in the detection of dynamic anomaly\nnodes in the two synthetic datasets. Since anomalies are in-\njected only into the test sets, the comparison is limited to\nunsupervised methods. F-FADE, which consistently achieves\nan AUC value below 0.5, is omitted from the table. For results\nin terms of Average Precision (AP), refer to Appendix D.1.\nDataset SedanSpot [12] MIDAS [4] Anoedge-l [5] SLADE\nSynthetic-Hijack 77.13±2.21 81.80±1.26 58.87±2.47 98.08±1.02\nSynthetic-New 78.05±1.68 82.63±0.07 61.86±2.41 98.38±1.09\nlosses (Eq (4)and Eq (5)) and two anomaly detection scores (Eq (8)\nand Eq (9)). To this end, we utilize several variants of SLADE , where\ncertain scores or self-supervised losses are removed from SLADE .\nAs evident from Table 2, SLADE , which uses all the proposed self-\nsupervised losses and scores, outperforms its variants in most of\nthe datasets (Wikipedia, Reddit, and Bitcoin-OTC). Moreover, even\nin the Bitcoin-alpha dataset, the performance gap between SLADE\nand the best-performing variant is within the standard deviation\nrange. Notably, the generation score greatly contributes to accurate\nanomaly detection in most of the dataset, providing an empirical\nperformance gain of up to 48.81% (on the Reddit dataset).\nRQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis:RQ4) Type Analysis: We demonstrate the effectiveness of SLADE\nin capturing various types of anomalies discussed in Section 5.1.\nTo this end, we create synthetic datasets by injecting anomalies to\nthe Email-EU [ 26] dataset, which consists of emails between users\nin a large European research institution. Specifically, we create\nanomalies that repetitively send spam emails to random recipients\nwithin a short time interval, mimicking spammers, and based on\nthe timing of spamming, we have two different datasets:\n•Synthetic-Hijack : Accounts of previously normal users start\ndisseminating spam emails after being hijacked at a certain time\npoint and continue their anomalous actions. We further catego-\nrize each anomaly as Type T1 during its initial 20 interactions\nafter being hijacked, and as Type T3 after that.\n•Synthetic-New : New anomalous users appear and initiate spread-\ning spam emails from the beginning and continue their anoma-\nlous actions. We further categorize each anomaly as Type T2\nduring its first 20 interactions, and as Type T3 afterward.\nIn them, all anomalies are injected only into the test set (i.e., final\n10%of the dataset in chronological order), ensuring that they remain\nunknown to the model during training. Consequently, the task\ninvolves detecting these anomalies in an unsupervised manner, and\nSLADE is compared only with the unsupervised baseline methods.\nFigure 4: (a) and (b) show the distribution of anomaly scores\nassigned by SLADE to instances of each node type in the two\nsynthetic datasets (visualization is based on Gaussian kernel\ndensity estimation). (c) and (d) show the anomaly scores at\neach time period. Note that in all figures, SLADE clearly\ndistinguishes anomalies from normal nodes. For results from\nseveral baseline methods, refer to Appendix D.4.\nAs shown in Table 3, SLADE performs best, achieving perfor-\nmance gains up to 19.9% in the Synthetic-Hijack dataset and 19.06%\nin the Synthetic-New dataset. These results reaffirm the limitations\nof traditional unsupervised methods in capturing anomalies beyond\ntheir targeted types. In contrast, as it can learn normal patterns from\ndata,SLADE successfully identifies various types of anomalies.\nFurthermore, we provide qualitative analysis of how SLADE as-\nsigns scores to the normal and anomalies of each type. Figures 4(a)\nand (b) show that SLADE clearly separates anomaly score distribu-\ntions of all anomaly types ( T1,T2, and T3) from the distribution\nof the normal ones. Figure 4(c) shows that the anomaly scores of\nhijacked anomalies ( T1) increase shortly after being hijacked. Fig-\nure 4(d) shows that SLADE successfully assigns high anomaly scores\nto new or rarely interacting anomalies ( T2). Consistent anomalies\n(T3) receive high anomaly scores in both cases.\n7 CONCLUSION\nWe proposed SLADE , a novel self-supervised method for dynamic\nanomaly detection in edge streams, with the following strengths:\n•SLADE does not rely on any label supervision while being able\nto capture complex anomalies (Section 4).\n•SLADE outperforms state-of-the-art anomaly detection methods\nin the task of dynamic anomaly detection in edge streams. SLADE\nachieves a performance improvement of up to 12.80% and 4.23%\ncompared to the best-performing unsupervised and supervised\nbaseline methods, respectively (Section 6.2).\n•SLADE demonstrates a constant time complexity per edge, which\nis theoretically supported (Section 5.2).\n8REFERENCES\n[1]Odd Aalen, Ornulf Borgan, and Hakon Gjessing. 2008. Survival and event history\nanalysis: a process point of view . 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IEEE\nTransactions on Knowledge and Data Engineering (2023).\n9A APPENDIX: DATASET DETAILS\nA.1 Real-world Datasets\nWe use 4 real-world datasets for a dynamic anomaly detection task;\ntwo social network datasets (Wikipedia and Reddit)2[20] and two\nfinancial network datasets (Bitcoin-alpha and Bitcoin-OTC)3[19].\nBasic descriptive statistics of each dataset are provided in Table 4.\nBelow, we provide a detailed description of each dataset, focusing on\nthe process of creating the ground-truth dynamic anomaly labels.\nIn Wikipedia and Reddit datasets, which are social networks\nbetween users, a user’s dynamic label at time 𝑡indicates the user’s\nstate at time 𝑡. Specifically, if a user is banned by administrators\nat time𝑡, the label of the user at time 𝑡is marked as abnormal .\nOtherwise, the user has a label of normal . Note that these labels\nare inherently given in the original datasets. Therefore, we utilize\nthese dynamic labels as anomalous labels of users for our task.\nBitcoin-alpha and Bitcoin-OTC datasets are temporal weighted-\nsigned networks between users. Here, the weight of each directional\nedge indicates how much the source user trusts the destination user.\nSpecifically, for each edge, its weight, which lies between -10 (total\ndistrust) to +10 (total trust) is given, together with the time of the\ninteraction. Note that, in our experiments, the edge weights are\nutilized to create ground-truth dynamic labels, and they are not\nincluded in the inputs for the anomaly detection methods. While\nthe used social networks inherently contain dynamic node states\n(i.e., banned), such information of an individual user is not pro-\nvided in this case. Therefore, we assign a dynamic label to each\nnode according to the weights of the interactions the node is in-\nvolved in. Note that since the overall Bitcoin transaction systems\nare anonymized, it is not reliable to determine the state of a user\naccording to a single interaction.\nHence, to obtain reliable labels, we adopt a two-stage labeling\nprocess: for each user, first, we define the (1) overall state of the\nuser, then define (2) the dynamic state of the user based on the\noverall state. We further describe this process.\n(1):Defining overall state: For each user, if the sum of the scores\na user has received throughout the entire edge stream is smaller\nthan zero, the overall state of the user is decided to be abnormal .\nOtherwise, the overall state of the user is labeled as normal .\n(2):Defining dynamic state: For each user that has the normal\noverall state, the dynamic label of the user is assumed to be\nalways normal . On the other hand, for each user that has the\nabnormal overall state, when the user becomes the destination\nnode of a negative-weighted interaction (i.e., receiving a neg-\native trust score from other users), the dynamic label of the\nuser at the corresponding interaction time is assumed to be\nabnormal . Otherwise, if a user receives a positive trust score\nfrom another user, we assume the dynamic label of the user\nat that time is assumed to be normal . Note that these created\ndynamic node labels are utilized as ground-truth dynamic node\nlabels of our datasets.\n2http://snap.stanford.edu/jodie/#datasets\n3https://snap.stanford.edu/dataAs mentioned above, the weight of each edge determines the\nlabel of a destination node, not a source node4. Regarding node\nand edge features, all of the used baseline neural network-based\nanomaly detection models require feature information of nodes\nand edges. On the other hand, in Bitcoin-alpha and -OTC datasets,\nsuch information is not given. Following prior works that used\nsuch models in the corresponding datasets [ 16,30], we utilize zero\nvectors for node and edge features of such datasets.\nRegarding data splits, for all the datasets, we fix the chronological\nsplit with 70 % for training, 15% for validation, and the last 15 %\nfor testing, which is a common setting that is widely used in many\nexisting works [28, 31, 38].\nA.2 Semi-Synthetic Datasets\nNext, we describe the details of the Synthetic-Hijack and Synthetic-\nNew datasets, which we utilize in section 6.2. These datasets are\nvariants of the real-world email communication dataset: Email-\nEU [21]. Nodes indicate users, and an edge (𝑣𝑖,𝑣𝑗,𝑡)indicates an\nemail between users 𝑣𝑖and𝑣𝑗at time𝑡. Since the dataset does not\nhave ground-truth anomalous interactions, we inject anomalies\ndiscussed in Section 5.1 ( T1,T2, and T3). To this end, we first\nchoose the time interval where interactions have actively occurred\n(spec., interactions occur within (4.06×107,4.54×107)timestamp)\nsince our method and baselines can easily detect anomalies that are\ninjected in the later timestamp than 4.54×107. Then, we set the\ntimestamp region where the last 10%(in chronological order) in-\nteractions occur as an evaluation region. After, we select candidate\naccounts to perform anomalous actions:\n•Synthetic-Hijack: Randomly select 10 nodes ( ≈1%of the nor-\nmal nodes) from nodes that are normal before the final 10% of\nthe dataset and do not appear in the evaluation region.\n•Synthetic-New: Make 10 new nodes ( ≈1%of the normal nodes)\nthat have never appeared before the evaluation region.\nSubsequently, we inject anomalies equivalent to approximately 1%\nof the total normal interactions into the evaluation region (see the\nnext paragraph for details). Each of these anomalous actions is\nrepresented as temporal edge (𝑣𝑘,𝑣𝑙,𝑡𝑚), where at time 𝑡𝑚, the\nanomalous source node 𝑣𝑘from the selected candidates sends a\nspam email to some normal node 𝑣𝑙. In this case, the dynamic label\nof the node 𝑣𝑘at time𝑡𝑚is anomalous.\nWe further elaborate on how we make each anomalous edge. We\nconsider the characteristics of spammers. We assume that spam-\nmers tend to target an unspecified majority with multiple spam\nemails in a short time interval. Based on this assumption, (1) we\nsample a timestamp 𝑡in the evaluation region uniformly at random,\n(2) pick one anomalous node from the candidate pool, regarding\nit as the source node, (3) randomly choose 10 normal nodes as\ndestinations, and (4) make 10 edges by joining the selected source\nnode with each selected destination nodes. (5) Lastly, we assign\neach edge a timestamp 𝑡±𝛼, where𝛼is sampled uniformly at\nrandom from[0,300]. This process ((1) - (5)) is repeated until we\nhave approximately 1% of anomalies relative to the total normal\ninteractions for the Synthetic-Hijack and Synthetic-New.\n4For the sake of consistency with other datasets, the directions are reversed in\nthe Bitcoin datasets, enabling the scoring and evaluation to be performed for source\nnodes.\n10Table 4: Statistics of datasets used in our experiments. The count of anomalies refers to the number of edges where the dynamic\nstate of the actor node is abnormal, and the ratio represents the proportion of these abnormal edges in relation to the total\nnumber of edges.\nWikipedia Reddit Bitcoin-alpha Bitcoin-OTC Synthetic-Hijack Synthetic-New\n# Nodes 9,227 10,984 3,783 5,881 986 996\n# Edges 157,474 672,447 24,186 35,592 333,200 333,200\n# Features 172 172 0 0 0 0\n# anomalies 217 366 1,520 4,344 3,290 3,290\nanomalies ratio 0.14% 0.05% 6.28% 12.20% 0.99% 0.99%\nanomaly type post ban edit ban unreliable user unreliable user spammer spammer\nTable 5: Hyperparameter settings for all baseline methods of main experiments and SLADE on each dataset.\nMethod Wikipedia Reddit Bitcoin-alpha Bitcoin-OTC\nSedanSpot\n(sample size, # random walkers, restart probability)(5000,200,0.9) ( 20000,100,0.9) ( 20000,100,0.9) ( 5000,200,0.5)\nMIDAS\n(number of hash functions, CMS size, decay factor)(3,1024,0.1) ( 3,1024,0.5) ( 4,256,0.9) ( 2,256,0.9)\nF-FADE\n(batch size, memory size, embedding size)(100,100,100) ( 300,200,200) ( 100,400,100) ( 100,100,100)\nAnoedge-l\n(edge threshold, bucket size, decay factor)(50,128,0.5) ( 50,64,0.1) ( 50,128,0.9) ( 50,128,0.5)\nJODIE\n(pretraining batch size, batch size, node degree)(300,300,20) ( 200,100,10) ( 200,100,10) ( 200,100,10)\nDyrep\n(pretraining batch size, batch size, node degree)(200,200,10) ( 200,100,20) ( 200,100,10) ( 100,300,20)\nTGAT\n(pretraining batch size, batch size, node degree)(200,100,20) ( 200,100,20) ( 200,100,20) ( 300,300,10)\nTGN\n(pretraining batch size, batch size, node degree)(200,100,20) ( 200,100,10) ( 200,100,10) ( 200,100,10)\nSAD\n(batch size, anomaly alpha, supervised alpha)(100,0.1,0.01) ( 256,0.1,0.0005) ( 300,0.001,0.001) ( 200,0.01,0.1)\nSLADE\n(batch size,𝜔𝑔𝑠,𝜔𝑔𝑑)(100,0.1,0.1) ( 100,0.1,0.1) ( 100,0.1,0.1) ( 100,0.1,0.1)\nSLADE -HP\n(batch size,𝜔𝑔𝑠,𝜔𝑔𝑑)(300,10,10) ( 100,0.1,1) ( 300,1,10) ( 300,10,1)\nB APPENDIX: BASELINE METHOD DETAILS\nAmong the four experiments mentioned earlier, the last type anal-\nysis experiment utilizes only unsupervised baselines. we provide\nindividual baselines for the main experiments (RQ1, RQ2, RQ3) and\nthe type analysis experiment (RQ4).\nB.1 Details of Baselines in Main Experiments\nWe compare the performances of our proposed method and the nine\nbaseline methods in detecting dynamic anomalies in edge streams.\nThe used baseline methods can be categorized as below:\n•Rule-based: SedanSpot [ 12], MIDAS [ 4], F-FADE [ 6], and Anoedge-\nl [5]\n•Neural network-based: JODIE [ 20], Dyrep [ 33], TGAT [ 38],\nTGN [28], and SAD [31]\nFor rule-based approaches, although the training phase is not\nnecessary, some level of hyperparameter tuning can greatly in-\ncrease the performance of them in our task. For a fair comparison,\nfor each dataset, the optimal hyperparameter setting of each rule-\nbased method is determined using the validation set. The selected\nhyperparameter setting is used to evaluate the corresponding modelin the test set. There are several details regarding some of the used\nbaseline methods:\n•Anoedge-l: Bhatia et al . [5] propose several versions of\nAnoedge. Among them, we adopt Anoedge-l as our baseline\nmethod since it exhibits the best performance among them\nin our task.\n•MIDAS-R: Bhatia et al . [4] propose several versions of MI-\nDAS. Among them, we adopt MIDAS-R as our baseline\nmethod since it outperforms the others in our preliminary\nstudy.\n•F-FADE: This method processes a stream every minute,\nwhile interactions in the used datasets occur at a much dense\ntime interval (i.e., tens of interactions on original datasets\noccur within a minute). Due to this characteristic, F-FADE\nalways underperforms all other used methods, specifically,\ncannot capture any anomalies. For F-FADE, we modify each\ndataset by adjusting the time units to large intervals.\nIn neural network-based methods, except for our proposed method,\nafter selecting the hyperparameter settings based on its validation\nset, there are two strategies for utilizing a given dataset (training\n11Table 6: Hyperparameter settings for rule-based baseline\nmethods and SLADE in type analysis.\nMethod Synthetic-(∗)\nSedanSpot\n(sample size, # random walkers, restart probability)(10000,50,0.15)\nMIDAS\n(number of hash functions, CMS size, decay factor)(2,1024,0.5)\nF-FADE\n(batch size, memory size, embedding size)(100,100,100)\nAnoedge-l\n(edge threshold, bucket size, decay factor)(50,32,0.9)\nSLADE\n(batch size,𝜔𝑔𝑠,𝜔𝑔𝑑)(100,0.1,0.1)\nset and validation set) for training the final representation model\nwith the selected hyperparameter settings:\n•S1. Using Both: This indicates using both training and validation\nsets to train the final representation model with the selected\nhyperparameter settings. In this case, the model may utilize more\ninformation during training, while the model gets vulnerable to\nthe overfitting issue.\n•S2. Training Set Only: This indicates using only the training set\nto train the final representation model with the selected hyper-\nparameter settings. In this case, the model utilizes the validation\nset only for early stopping, but we do not fully utilize the given\ndataset during training.\nFor each model, except for SAD (since the method inherently uti-\nlizes a validation dataset during model training), we utilize both\nstrategies and report the higher test set evaluation performance\nbetween them.\nB.2 Details of Baselines in Type Analysis\nExperiments\nWe compare the performances of our proposed method and four\nunsupervised baseline methods in type analysis. The unsupervised\nbaseline methods are as follows: SedanSpot [ 12], MIDAS [ 4], F-\nFADE [ 6], and Anoedge-l [ 5]. In the Synthetic-hijack and Synthetic-\nNew datasets, anomalies exist only in the test set, making it impos-\nsible to conduct a validation. Thus, for each model, we utilize the\nhyperparameter combinations the respective paper has reported.\nIn cases where baseline settings in the previous work vary across\ndatasets, we use the combination that has shown good performance\nin our main experiments (Section 6.2).\nC APPENDIX: IMPLEMENTATION DETAILS\nIn this section, we introduce some details about our implementation,\nincluding the experiment environment and hyperparameters of the\nused models.\nC.1 Experiments Environment\nWe conduct all experiments with NVIDIA RTX 3090 Ti GPUs (24GB\nVRAM), 256GB of RAM, and two Intel Xeon Silver 4210R Processors.C.2 Details of Baselines in Main Experiments\nAs mentioned in Section 6.1, we tune most hyperparameters of\neach baseline method by conducting a full grid search on the vali-\ndation set of each dataset. For other hyperparameters, we strictly\nfollow the setting provided in their official code, because it leads\nto better results than grid searches. The selected hyperparameter\ncombination of each model is reported in Table 5.\nRule-based Methods. Hyperparameter search space of each rule-\nbased method is as below:\n•Sedanspot : sample size among (5000,10000,20000), number of\nrandom walkers among (100,200,300), and restart probability\namong(0.1,0.5,0.9)\n•MIDAS-R : number of hash functions among (2,3,4), CMS size\namong(256,512,1024), and decay factor among (0.1,0.5,0.9)\n•F-FADE : batch size among (100,200,300), limited memory size\namong(100,200,400), and embedding size among (100,200,300)\n•Anoedge-l : edge threshold among (25,50,100), bucket size among\n(64,128,256), and restart probability among (0.1,0.5,0.9)\nNeural Network-based Methods. We train all models with the\nAdam optimizer [ 17]. The hyperparameter search space of each\nneural network-based method is as below:\n•JODIE : self-supervised batch size among (100, 200, 300), super-\nvised batch size among (100,200,300), and node degree between\n(10,20)\n•Dyrep : self-supervised batch size among (100, 200, 300), super-\nvised batch size among (100,200,300), and node degree between\n(10,20)\n•TGAT : self-supervised batch size among (100, 200, 300), super-\nvised batch size among (100,200,300), and node degree between\n(10,20)\n•TGN : self-supervised batch size among (100, 200, 300), super-\nvised batch size among (100,200,300), and node degree between\n(10,20)\n•SAD : batch size among (100,200,300), anomaly alpha among\n(0.1,0.01,0.001)for deviation loss, and supervised alpha among\n(0.1,0.01,0.001)for supervised loss\nC.3 Details of Baseline Methods in Type\nAnalysis Experiments\nAs mentioned in B.2, We use hyperparameter settings from pre-\nvious studies or combinations for good performance on our main\nexperiments (Section 6.2). The selected hyperparameter setting of\neach model is reported in Table 6.\nC.4 Detailed Hyperparameters of SLADE and\nSLADE -HP\nWe train both SLADE andSLADE -HP using the Adam optimizer,\nwith a learning rate of 3×10−6and a weight decay of 10−4. We\nfix the dropout probability and the number of attention heads in\nTGAT to 0.1 and 2, respectively. In addition, we fix the scaling scalar\nof temporal encoding (Eq (2) in the main paper) to 𝛼=𝛽=10,\nthe weight of each loss in L𝑐(Eq (7) in the main paper) to 𝜔𝑐𝑠=\n𝜔𝑐𝑑=1, and the dimension of a time encoding to 256, which is\nequivalent to the memory dimension. As mentioned in Section 6.1 of\nthe main paper, we tune several hyperparameters of SLADE -HP on\n12Table 7: AP (in %) in the detection of dynamic anomaly nodes.\nThe first method randomly assigns anomaly scores from 0 to\n1, and the next four methods are rule-based models, and the\nothers are based on representation learning. For each dataset,\nthe best and the second-best performances are highlighted\nin boldface and underlined, respectively.\nMethod Wikipedia Reddit Bitcoin-alpha Bitcoin-OTC\nRandom Guess 0.22±0.05 0.09±0.01 8.96±0.53 6.37±0.28\nSedanSpot [12] 0.66±0.06 0.09±0.00 12.99±0.06 16.14±0.01\nMIDAS [4] 0.41±0.02 0.12±0.00 12.99±0.06 16.14±0.01\nF-FADE [6] 0.18±0.00 0.09±0.00 6.34±0.00 8.80±0.00\nAnoedge-l [5] 0.18±0.01 0.09±0.00 12.52±0.86 17.03±0.11\nJODIE [20] 1.40±0.02 0.25±0.05 14.31±0.40 15.79±0.11\nDyrep [33] 1.53±0.02 0.14±0.01 8.41±0.58 16.14±1.74\nTGAT [38] 1.48±0.14 0.27±0.12 12.19±0.13 18.81±0.63\nTGN [28] 1.30±0.04 0.19±0.01 9.65±0.40 19.87±1.75\nSAD [31] 2.77±0.91 0.28±0.06 12.55±0.58 14.33±0.88\nSLADE 1.24±0.14 0.23±0.02 15.40±0.19 20.21±0.28\nSLADE -HP 1.56±0.14 0.30±0.05 14.86±0.08 20.46±0.06\nTable 8: AP (in %) in the detection of dynamic anomaly nodes\nin the two synthetic datasets. In both datasets, SLADE per-\nforms best compared to unsupervised methods in AP.\nMethod Synthetic-Hijack Synthetic-New\nRandom Guess 6.61±0.07 6.63±0.01\nSedanSpot [12] 13.95±1.53 15.17±0.13\nMIDAS [4] 17.96±2.56 18.29±3.18\nF-FADE [6] 6.57±0.00 6.58±0.00\nAnoedge-l [5] 8.20±0.47 8.40±0.63\nSLADE 69.69±7.33 74.46±6.16\nthe validation set of each dataset, as we tune the hyperparameters\nof all baselines. The search space is as follows:\n•SLADE -HP: batch size among(100,200,300), the weight in mem-\nory generation loss L𝑔(Eq (8) in the main paper) for a source\nnode (𝜔𝑔𝑐) among(0.1,1,10), and for a destination node ( 𝜔𝑔𝑑)\namong(0.1,1,10).\nOur final hyperparameter settings for SLADE andSLADE -HP are\nalso reported in Table 5.\nD APPENDIX: ADDITIONAL EXPERIMENTS\nD.1 Performance Evaluation using Average\nPrecision\nIn this subsection, we evaluate each model with the Average Preci-\nsion (AP) metric regarding RQ1 and RQ4. Note that AP has different\ncharacteristics from that of the Area Under the ROC Curve (AUC).\nWhile AUC focuses on the overall distinguishability between two\ndifferent classes, AP focuses on how well a model assigns higher\nscores for positive samples than for negative samples in terms of\nprecision and recall.We utilize the same hyperparameter settings from Section C.2\nand C.3, and include a random guess model that randomly assigns\nanomaly scores between 0 and 1 for comparison. As shown in\nTable 7, SLADE -HP performs the best or second best in all real-\nworld graph datasets, demonstrating the effectiveness of SLADE -\nHP in our task regarding AP also. Furthermore, as shown in Table 8,\nSLADE significantly outperforms other methods in both synthetic\ndatasets.\nD.2 Variants in Model Architecture\nThere are two major neural network components in SLADE : (1)\nGRU [8], which updates the memory of each node, and (2) TGAT [38],\nwhich generates the memory of a target node. To demonstrate the\neffectiveness of each module in dynamic anomaly detection in edge\nstream, we compare the performances of SLADE and its several\nvariants where the memory updater and the memory generator are\nreplaced by other neural network architectures.\nMemory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):Memory Updater Variant (Instead of GRU):\n•SLADE -MLP : In this variant, we use MLP instead of the GRU\nmodule to update the memory of each node. Specifically, the\nmemory update procedure (Eq (3) in the main paper) is replaced\nby the one below:\n𝒔𝑖=MLP([𝒎𝑖||𝒔𝑖]). (11)\nNote that in this variant, 𝒎𝑖and 𝒔𝑖are treated as if they are\nindependent, and thus it cannot capture the temporal dependency\nbetween them.\nMemory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):Memory Generator Variants (Instead of TGAT):\n•SLADE -GAT : This variant calculates attention scores based only\non the currently given memory information, as follows:\nˆ𝒔𝑖=MultiHeadAttention (𝒒,K,V),𝒒=𝒔𝑖, (12)\nK=V=\u0002\n𝒔𝑛1,...,𝒔𝑛𝑘\u0003\n,\nwhere{𝑛1,...,𝑛𝑘}denote the indices of the neighbors of the tar-\nget node𝑣𝑖. Note that this variant cannot incorporate temporal\ninformation in its attention mechanism.\n•SLADE -SUM : In this variant, we use a modified temporal graph\nsum [28] for message passing, as follows:\nˆ𝒔𝑖=W2([¯𝒔𝑖||𝜙(𝑡−𝑡)]),\n¯𝒔𝑖=ReLU(𝑘∑︁\n𝑗=1W1(𝒔𝑛𝑗||𝜙(𝑡−𝑡′\n𝑛𝑗))), (13)\nW1,W2∈R𝑑𝑠×2𝑑𝑠,\nwhere{𝑛1,...,𝑛𝑘}denote the indices of the neighbors of the tar-\nget node𝑣𝑖,{𝑡′𝑛1,...,𝑡′𝑛𝑘}denote the times of the most recent\ninteractions with them, and the weights of each linear layer are\ndenoted as W1andW2, respectively. Note that this variant does\nnot use any attention mechanism, i.e., all neighbors have the\nsame importance.\nExperimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results:Experimental Results: We compare the performances of SLADE\nand the introduced three variants. As shown in Table 9, SLADE\nachieves the best performance in three out of four datasets. This\nresult demonstrates the necessity of GRU and TGAT, i.e., the impor-\ntance of modeling temporal dependency in memory updating and\n13Table 9: AUC (in %) in the detection of dynamic anomaly\nnodes. The first model is a memory updater variant, and\nthe next two models are memory generator variants. For\neach dataset, the best and the second-best performances are\nhighlighted in boldface and underlined, respectively. Across\nall datasets, SLADE consistently demonstrated the best or\nsecond-best performance compared to the other variants.\nMethod Wikipedia Reddit Bitcoin-alpha Bitcoin-OTC\nSLADE -MLP 85.97±1.17 55.35±3.34 74.68±1.35 75.42±0.65\nSLADE -GAT 86.86±0.44 67.15±1.85 75.67±0.57 74.24±0.25\nSLADE -SUM 88.43±0.44 70.89±0.53 75.84±0.26 75.42±0.15\nSLADE 87.75±0.68 72.19±0.60 76.32±0.28 75.80±0.19\nFigure 5: The left figure shows the increase in the running\ntime of SLADE with respect to the number of edges in the\nReddit dataset. The right figure shows the training time\nand AUC score (with standard deviations) in the Wikipedia\ndataset of the competing learning-based methods.\ntemporal attention in memory generation. Hyperparameter setting\nofSLADE variants is fixed as that of the original SLADE . Specif-\nically, SLADE -MLP and SLADE -GAT consistently underperform\nSLADE andSLADE -SUM, demonstrating the importance of utiliz-\ning temporal information and temporal dependency in detecting\ndynamic anomalies. Furthermore, while SLADE -SUM outperformsSLADE in the Wikipedia dataset, its performance gain is marginal\n(within the standard deviation).\nD.3 Model Training Speed\nIn this subsection, we analyze the training time of SLADE .\nWe first empirically demonstrate the training time of SLADE on\nthe Reddit dataset with varying edge counts, where the detailed\nsetting is the same as that described in Section 6.2 of the main\npaper. As shown in the left plot of Figure 5, the training runtime of\nSLADE is almost linear in the number of edges. Since the number\nof edges is much greater than that of nodes ( 67×), the overall scale\nis dominated by the number of edges.\nFurthermore, we compare the empirical training time of SLADE\nagainst that of other neural network-based methods, which require\nthe training procedure. As shown in the right plot of Figure 5,\nin terms of the empirical training speed, SLADE is competitive\ncompared to other methods. Specifically, among six neural network-\nbased dynamic anomaly detection methods, SLADE exhibits the\nsecond-fastest training time, showing the best anomaly detection\nperformance.\nD.4 Type Analysis of Baselines\nWe conduct an additional analysis regarding two baseline meth-\nods (MIDAS and SedanSpot), which ranked second- and third-best\nposition in our synthetic data experiments ( RQ4 ). First, as shown\nin the first row of Figure 6 (a) and (b), we verify that the score\ndistribution of the normal class and that of the consistent anomaly\nclass ( T3) largely overlaps in both models. Moreover, as shown in\nthe second row of Figure 6(a) and (b), MIDAS and SedanSpot assign\nrelatively high scores when anomalies of T1andT2occur, but not\nas distinctly as SLADE . These two results indicate that previous\nunsupervised anomaly detection baselines fail to detect anomalies\nif anomalies do not align with the targeted anomaly pattern, leading\nto high false negatives. These results emphasize the necessity of\nlearning-based unsupervised anomaly detection models ( SLADE ),\nwhich autonomously learn normal patterns, and are capable of\ndetecting a wide range of anomalies by finding cases that deviate\nfrom the learned normal patterns.\n14(a) Type Analysis of MIDAS\n (b) Type Analysis of SedanSpot\nFigure 6: For both (a) and (b), the above figures show the distribution of anomaly scores predicted by each baseline for each\nclass in the two synthetic datasets (utilizing kernel density estimation). The below figures show the anomaly scores predicted\nby each baseline over time for each class in both datasets. We display the anomaly scores of MIDAS on a log scale.\n15"    },    {        "title": "2402.11964v1.Tagged_particle_behavior_in_a_harmonic_chain_of_direction_reversing_active_Brownian_particles.pdf",        "content": "Tagged particle behavior in a harmonic chain of\ndirection reversing active Brownian particles\nShashank Prakash\nRaman Research Institute, Bengaluru 560080, India\nEmail: shashankp@rrimail.rri.res.in\nUrna Basu\nS. N. Bose National Centre for Basic Sciences, Kolkata 700106, India\nEmail: urna@bose.res.in\nSanjib Sabhapandit\nRaman Research Institute, Bengaluru 560080, India\nEmail: sanjib@rri.res.in\nAbstract. We study the tagged particle dynamics in a harmonic chain of direction\nreversing active Brownian particles, with spring constant k, rotation diffusion\ncoefficient DR, and directional reversal rate γ. We exactly compute the tagged particle\nposition variance for quenched and annealed initial orientations of the particles. For\nwell-separated time scales, k−1, D−1\nRandγ−1, the strength of spring constant krelative\ntoDRandγgives rise to different coupling limits and for each coupling limit there\nare short, intermediate, and long time regimes. In the thermodynamic limit, we show\nthat, to the leading order, the tagged particle variance exhibits an algebraic growth tν,\nwhere the value of the exponent νdepends on the specific regime. For a quenched initial\norientation, the exponent νcrosses over from 3 to 1 /2, via intermediate values 5 /2 or\n1, depending on the specific coupling limits. On the other hand, for the annealed\ninitial orientation, νcrosses over from 2 to 1 /2 via an intermediate value 3 /2 or 1\nfor strong coupling limit and weak coupling limit respectively. An additional time\nscale tN=N2/kemerges for a system with a finite number of oscillators N. We\nshow that the behavior of the tagged particle variance across tNcan be expressed in\nterms of a crossover scaling function, which we find exactly. Finally, we characterize\nthe stationary state behavior of the separation between two consecutive particles by\ncalculating the corresponding spatio-temporal correlation function.\nContents\n1 Introduction 3\n2 Model and Results 4arXiv:2402.11964v1  [cond-mat.stat-mech]  19 Feb 20242\n3 Variance of the position of a tagged particle 6\n4 Quenched initial orientation 9\n4.1 Short-time regime ( t≪τ1) (R-I) . . . . . . . . . . . . . . . . . . . . . . . 12\n4.2 Long-time regime ( t≫τ3) (R-IV) . . . . . . . . . . . . . . . . . . . . . . 12\n4.3 Strong-coupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14\n4.4 Weak coupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n4.5 Moderate-coupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n5 Annealed initial orientation 21\n5.1 Short-time regime ( t≪τ1) (R-I) . . . . . . . . . . . . . . . . . . . . . . . 22\n5.2 Long-time regime ( t≫τ2) (R-IV) . . . . . . . . . . . . . . . . . . . . . . 23\n5.3 Intermediate regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\n5.3.1 Strong-coupling limit [ k≫(DR,2γ)] (R-II) . . . . . . . . . . . . . 23\n5.3.2 Weak-coupling limit [ k≪(DR,2γ)] (R-III) . . . . . . . . . . . . . 24\n6 Finite-size effects 25\n7 Statistics of the separations 27\n8 Conclusions 35\nAppendix A Computation of effective noise correlation 37\nAppendix B Computation of Cn(β) 38\nAppendix C Some useful integrals and sums 403\n1. Introduction\nA system of interacting particles undergoing Newtonian dynamics results in a diffusive\nmotion for a tagged particle with the mean-squared displacement growing linearly with\ntime [1, 2, 3, 4, 5]. The tagged particle dynamics is still diffusive for a collection\nof interacting diffusive particles, in dimensions d≥2, albeit with a lower diffusivity.\nHowever, in one-dimensional interacting diffusive systems, the tagged particle dynamics\nshow a subdiffusive behavior. Systems with symmetric dynamics generically show a√\nt\ngrowth for mean-squared displacement, which, for example, has been observed for hard-\nparticle diffusive gases [6, 7, 8, 9, 10], symmetric simple exclusion process [11, 12, 13, 14,\n15], random average process [16, 17], and systems with harmonic interactions [18, 19],\nto name a few. On the other hand, systems with asymmetric dynamics give rise to\nsubdiffusive behavior with different exponents [20, 21, 22, 23, 24, 25]. This classical\nproblem of tagged particle motion in one-dimensional interacting systems is still of much\ninterest and various aspects of it are still being investigated. Examples include the study\nof typical and large deviations of the displacement fluctuations [26, 27, 28, 29, 30, 31, 32],\nduality relations [33], effects of pinning [34], disorder [35], and memory [36, 37], and\nentropy production [38].\nThough much has been known about the passive systems, the study of tagged\nparticle dynamics in an interacting system of active particles is still nascent. Active\nparticles self-propel by extracting energy from the surroundings at the individual level,\ngenerating a directed motion with a typical persistence time [39, 40, 41, 42, 43, 44].\nTheoretical attempts to characterize active motion often use minimal statistical models,\nwhere the position of an active particle evolves via an overdamped Langevin equation\n˙r(t) =v(t). Different stochastic dynamics of the self-propulsion velocity v(t) leads to\ndifferent models for active particles. Run-and-tumble particle (RTP) [45, 46, 47], active\nOrnstein-Uhlenbeck process (AOUP) [48], and active Brownian particle (ABP) [49, 50]\nare some of the basic models for which the single particle dynamics have been extensively\nstudied [51, 52, 53, 54, 55, 56, 57]. While at times much larger than the persistence\ntime these models generically show a diffusive behavior, the signatures of activity are\nmanifest at short-times, where the dynamics is strongly non-diffusive. This non-diffusive\nbehavior is expected to affect the dynamics of a tagged particle in interacting active\nsystems. Indeed, the variance of the position of a tagged particle, in a chain of such\nactive particles, shows a crossover [58, 59, 60] from a superdiffusive behavior, tµwith\nµ >1/2, to the subdiffusive behavior√\nt, typical of passive systems.\nRecently, it has been found that an active particle dynamics, namely, direction\nreversing active Brownian particle (DRABP), where the self-propulsion velocity v(t) has\nmultiple time-scales, display several novel features, in the intermediate time regimes, due\nto the interplay between these time-scales [61, 62]. It is natural to ask, how the presence\nof the additional time-scales changes the tagged particle behavior in an interacting\nsystem of such particles. In this paper, we comprehensively characterize the tagged\nparticle dynamics in a harmonic chain of DRABPs with periodic boundary conditions,4\nby studying the position-variance for quenched and annealed initial orientations. We find\nthat the presence of an additional time-scale leads to the emergence of new dynamical\nregimes with distinct super-diffusive growth of the variance [see Tables. 1-2]. We\nalso study the effect of the finite size of the chain on the variance and compute the\nscaling function that describes the crossover across the finite-size time scale. While\nthe fluctuations of the tagged particle grow with time, the separations between the\nadjacent tagged particles reach a stationary state, which we characterize analytically by\ncomputing the spatio-temporal correlation among the separation variables.\nThe paper is organized as follows: In section 2, we define the model and present\na summary of the main results. The basic setup of the calculation is presented in\nSec. 3. Sections 4, and 5 are devoted to the detailed derivation of the tagged particle\nvariance for quenched and annealed initial orientations, respectively. The main results\nare summarized in Tables. 1-2 for the convenience of the reader. We explore the finite-\nsize effects in Sec. 6, and finally in Sec. 7 we investigate the statistics of the separations\nbetween two consecutive particles.\n2. Model and Results\nThe position vector ⃗ r(t) of a single direction reversing active Brownian particle\n(DRABP) evolves via the Langevin equation,\n˙r(t) =v0σ(t) ˆn(t), (1)\nwhere v0denotes the self-propulsion speed of the particle and σ(t) is a dichotomous noise\nthat alternates between ±1 with a constant rate γ. The orientation vector ˆ n(t) undergoes\na Brownian motion on the unit sphere. In two dimensions, ˆ n(t) = (cos θ(t),sinθ(t)),\nwhere the orientation angle θ(t) undergoes a Brownian motion with diffusion constant\nDR. It has been shown that the position fluctuations of a DRABP shows distinctly\ndifferent behavior in different dynamical regimes characterized by the two active time-\nscales D−1\nRandγ−1[61]. From a strongly non-diffusive and anisotropic behavior at\nearly-times t≪(γ−1, D−1\nR), the DRABP crosses over to an effective diffusive behavior\nat late-times t≫(γ−1, D−1\nR). In fact, in the limit v0→ ∞ and ( DR+ 2γ)→ ∞ , while\nkeeping\nDeff=v2\n0\n2(DR+ 2γ), (2)\nfixed (which corresponds to the late-time regime), the DRABP typically behaves\nlike a passive Brownian particle, with a Gaussian position distribution,\nP(⃗ r, t) =1\n4πDefftexp\u0012\n−|⃗ r|2\n4Defft\u0013\n. (3)\nMost interestingly, for γ≫DR, a non-trivial scaling distribution emerges in the\nintermediate regime γ−1≪t≪D−1\nR.5\nIn this work, we consider a periodic chain of Nharmonically coupled direction\nreversing active Brownian particles, where each particle is uniquely identified by an\nindex α= 0,1,2, . . . , N −1. In particular, we focus on the fluctuations of x-components\nof positions {xα(t)}. The potential energy is given by,\nU(x0, x1, . . . x N−1) =k\n2N−1X\nα=0(xα−xα+1)2, (4)\nwith the periodic boundary condition xN=x0. The presence of the harmonic coupling\nintroduces another time-scale k−1, in addition to the two active time-scales D−1\nRand\nγ−1. The position of the α-th particle evolves via the Langevin equation,\n˙xα(t) =k[xα+1(t) +xα−1(t)−2xα(t)] +ξα(t), (5)\nwhere,\nξα(t) =v0σα(t) cosθα(t), (6)\ndenotes the active noise. The dichotomous noises {σα(t)}independently alternate\nbetween ±1 with a constant rate γ, and the orientations {θα(t)}undergo independent\nBrownian motions,\n˙θα(t) =p\n2DRηα(t), (7)\nwhere {ηα(t)}denote independent Gaussian white noises with ⟨ηα(t)⟩= 0, and\n⟨ηα(t)ηβ(t′)⟩=δα,βδ(t−t′). We consider the initial condition xα(0) = 0 and σα(0) = 1\nfor all αand initial orientations {θα(0)}to be either quenched or annealed. For the\nquenched case, we set the initial orientations of all the particles to a specific value θ0,\ni.e.,{θα(0) = θ0}. On the other hand, the initial orientation of each particle is drawn\nindependently from a uniform distribution in [0 ,2π] for the annealed case.\nThe objective of this work is to characterize the fluctuations of the position xα(t)\nof a tagged particle through its mean and variance. The presence of the three distinct\ntime scales leads to a set of different dynamical regimes, each characterized by a distinct\nbehavior of the position variance, ranging from subdiffusive to superdiffusive. These\nleading order dynamical behaviors, in each regime, also depend on the initial orientations\nof the particles. Tables 1 and 2 present a summary of the distinct dynamical behaviors,\nin thermodynamic limit, i.e., N→ ∞ , for the quenched and annealed initial orientations,\nrespectively. In particular, we find that, at short-time regime, i.e., at times much smaller\nthan all the time scales of the system, ⟨x2\nα(t)⟩c∼t3, reminiscent of the short time regime\nof the independent DRABP. On the other hand, at late times, i.e., at a time much larger\nthan all the time scales, ⟨x2\nα(t)⟩c∼√\nt, similar to the behavior of a tagged particle in\na harmonic chain of Brownian particles [18]. Furthermore, we study the effect of finite-\nsize on the position variance. An additional time scale, tN=N2/k, appears due to\nthe finiteness of the system, and we observe a crossover in variance from subdiffusive\nbehavior for t≪tNto diffusive behavior for t≫tN. This crossover is captured by6\na scaling behavior of the variance ⟨x2\nα(t)⟩c=Deffp\nt/k f (t/tN), where the crossover\nfunction is given by (74) exactly.\nWe also investigate the statistics of the separations between the adjacent particles\nyα(t) =xα+1(t)−xα(t), which eventually reaches a stationary state. In the passive\nlimit, i.e., when DR→ ∞ , and v0→ ∞ , keeping Defffixed, the stationary state of {yα}\nis given by the Boltzmann distribution,\nP({yα})∝exp\"\n−k\n2DeffN−1X\nα=0y2\nα#\n, (8)\nfor a thermodynamically large system. For finite DR, i.e., in the active regime,\nthe system reaches a nonequilibrium steady state which is no longer given by the\nabove Boltzmann distribution. We characterize the signatures of activity in this\nstationary state by computing the spatio-temporal two-point correlation, C(β, τ) =\nlimt→∞⟨y0(t)yβ(t+τ)⟩.We find that, in the thermodynamic limit, N→ ∞\nC(β, τ) =∞X\nn=0˜Cn(β)τn\nn!, (9)\nwhere the coefficients ˜Cn(β) can be computed explicitly [see Sec. 7]. Moreover, in the\nlarge activity limit, the equal-time spatial correlation decays exponentially,\nC(β,0) =Deff\nke−|β|/√µ\n2√µ,with µ=k\nDR+ 2γ. (10)\nOn the other hand, for large τ≫(DR+ 2γ)−1, we have,\nC(β, τ) =Deff\n2k√\nπke−β2/4kτ\n√τ. (11)\nIn the following sections, we provide detailed derivations of the results mentioned\nabove.\n3. Variance of the position of a tagged particle\nIn this section, we set up the formalism to compute the variance of the position xαof a\ngiven tagged particle with a fixed α. To compute the variance,\n\nx2\nα(t)\u000b\nc=\nx2\nα(t)\u000b\n−\nxα(t)\u000b2, (12)\nit is convenient to go to the normal modes of the harmonic chain. The decoupled normal\nmodes satisfy a set of Nfirst-order differential equations,\n˙˜xs(t) =−as˜xs(t) +˜ξs(t),with as= 4ksin2\u0010πs\nN\u0011\n, (13)7\nt≪τ1≪τ2≪τ3 τ1≪t≪τ2≪τ3 τ1≪τ2≪t≪τ3 τ1≪τ2≪τ3≪t\nShort-time regime Early-intermediate regime Late-intermediate regime Long-time regime\nτ1=k−1\nτ2=D−1\nR\nτ3=γ−1Strong Couplingτ1=k−1\nτ2=γ−1\nτ3=D−1\nRR-II\nt5/2\n[see (41)]R-III√\nt\n[see (47)]\nτ1=D−1\nR\nτ2=γ−1\nτ3=k−1Weak Couplingτ1=γ−1\nτ2=D−1\nR\nτ3=k−1R-VI\nt\n[see (53)]R-V\nt\n[see (52)]\nτ1=D−1\nR\nτ2=k−1\nτ3=γ−1R-VII\nt\n[see (52)]Moderate Couplingτ1=γ−1\nτ2=k−1\nτ3=D−1\nRR-I\nt3\n[see (33)]\nR-VIII\nt\n[see (53)]R-IX√\nt\n[see (47)]R-IV\n√\nt\n[see (37)]\nTable 1: Tabular representation of the leading order behavior of the variance in the\ndifferent dynamical regimes starting with quenched initial orientation ( θ0̸=π/2). The\nnine distinct regimes are indicated by R-I, R-II, . . . , R-IX. The first column indicates\nthe relative time-scales. The second column defines the three different time scales and\nthe topmost row indicates the specific regime considered. The equations describing the\ntheoretical predictions of the variance in the corresponding regimes are also mentioned.\nwhere {˜xs(t);s= 0,1, . . . N −1}denote the discrete Fourier transformation (DFT) of\n{xα(t)}and{˜ξs(t)}denotes the DFT of the active noise (6). We note that any arbitrary\n{fα(t)}and its DFT {˜fs(t)}with respect to α, are related by,\n˜fs(t) =1\nNN−1X\nα=0exp\u0012\n−i2πsα\nN\u0013\nfα(t) and fα(t) =N−1X\ns=0exp\u0012i2πsα\nN\u0013\n˜fs(t). (14)\nThe set of equations (13) can be formally solved to obtain,\n˜xs(t) = ˜xs(0)e−ast+e−astZt\n0east1˜ξs(t1)dt1, (15)\nwhere ˜ xs(0) denotes the DFT of the initial position profile. For annealed initial\norientation, cos θα(0) has a symmetric distribution, resulting in\nxα(t)\u000b\n= 0, for\ninitial position {xα(0) = 0 }. On the other hand, for the quenched initial orientation,8\nτ1≪τ2 t≪τ1≪τ2 τ1≪t≪τ2 τ1≪τ2≪t\nShort-time regime Intermediate regime Long-time regime\nτ1=k−1\nτ2= (DR+ 2γ)−1Strong CouplingR-II\nt3/2\n[see (66)]\nτ1= (DR+ 2γ)−1\nτ2=k−1Weak CouplingR-I\nt2\n[see (62)]R-III\nt\n[see (67)]R-IV\n√\nt\n[see (65)]\nTable 2: Tabular representation of the leading order behavior of the variance in the\ndifferent dynamical regimes starting with annealed initial orientation. The four distinct\nregimes are indicated by R-I, R-II, R-III, and R-IV. The first column indicates the\nrelative time-scales. The second column defines the two time-scales and the topmost\nrow indicates the specific regime considered. The equations describing the theoretical\npredictions of the variance in the corresponding regimes are also mentioned.\n{θα(0) = θ0}with{xα(0) = 0 }, taking the average over (15) and inverting it back via\n(14), we get,\nxα(t)\u000b\n=v0tcosθ0e−(DR+2γ)t. (16)\nUsing (14), the variance of the tagged particle position (12) can be expressed as,\n\nx2\nα(t)\u000b\nc=N−1X\ns=0N−1X\ns′=0exp\"\ni2πα(s−s′)\nN#\n\n˜xs(t)˜x∗\ns′(t)\u000b\nc, (17)\nwhere\n˜xs(t)˜x∗\ns′(t)\u000b\nc=\n˜xs(t)˜x∗\ns′(t)\u000b\n−\n˜xs(t)\u000b\n˜x∗\ns′(t)\u000b\n, (18)\nis the correlation of the Fourier modes. Here, ˜ x∗\ns′(t) denotes the complex conjugate of\n˜xs′(t) and\n...\u000b\ndenotes the statistical average over the active noise.\nThe correlation\n˜xs(t)˜x∗\ns′(t)\u000b\nccan be expressed in terms of the correlations of the\nFourier transforms of the active noise, using Eq. (15),\n\n˜xs(t)˜x∗\ns′(t)\u000b\nc=e−(as+as′)tZt\n0dt1Zt\n0dt2east1+as′t2\n˜ξs(t1)˜ξ∗\ns′(t2)\u000b\nc. (19)\nThe noise correlation in Fourier space, in turn, could be written in terms of the\ncorrelation of the noise in real space,\n\n˜ξs(t1)˜ξ∗\ns′(t2)\u000b\nc=1\nN2N−1X\nα=0N−1X\nα′=0exp\"\n−i2πα(s−s′)\nN#\n\nξα(t1)ξα(t2)\u000b\nc. (20)9\nWe note that the active noise ξα(t) for different particles are independent and\nthe two-time correlation, G(t1, t2) =\nξα(t1)ξα(t2)\u000b\ncis independent of αfor both the\nquenched and annealed initial conditions we have considered here. In particular, for the\nquenched initial orientation [see Appendix A for the detailed calculation], we have,\nG(t1, t2) =v2\n0\n2h\ne−(DR+2γ)|t1−t2|+ cos 2 θ0e−\u0000\nDR(t1+t2+2 min[ t1,t2])+2γ|t1−t2|\u0001\n−2 cos2θ0e−(DR+2γ)(t1+t2)i\n. (21)\nOn the other hand, for the annealed initial orientation [see Appendix A for a detailed\ncalculation],\nG(t1, t2) =v2\n0\n2e−(DR+2γ)|t1−t2|. (22)\nWe can express the noise auto-correlation in the Fourier space in (20), in terms of\nG(t1, t2), yielding,\n\n˜ξs(t1)˜ξ∗\ns′(t2)\u000b\nc=G(t1, t2)\nNδs,s′. (23)\nUsing (23) in (19) we get,\n\n˜xs(t)˜x∗\ns′(t)\u000b\nc=δs,s′\nNe−2asttZ\n0dt1tZ\n0dt2eas(t1+t2)G(t1, t2). (24)\nUsing the above equation in (17), we get a simplified expression for the variance of the\nposition of the tagged particle,\n\nx2\nα(t)\u000b\nc=N−1X\ns=0\n\f\f˜xs(t)\f\f2\u000b\nc. (25)\nIn the next two sections, we explicitly compute this variance for the quenched and\nannealed initial orientations.\n4. Quenched initial orientation\nFor the quenched initial orientation {θα(0) = θ0}, using (21) and (24) in (25), we get,\n\nx2\nα(t)\u000b\nc=v2\n0\nNN−1X\ns=0\u00141\n2(DR+ 2γ) \n1−e−2ast\nas+e−(DR+2γ)te−ast−1\nDR+ 2γ+as+e−ast(e−(DR+2γ)t−e−ast)\nDR+ 2γ−as!\n+cos 2θ0\n2(DR−2γ) \ne−4DRt−e−2ast\n2DR−as+e−(DR+2γ)te−ast−e−4DRt\n3DR−2γ−as+e−ast(e−ast−e−(DR+2γ)t)\nDR+ 2γ−as!\n−cos2θ0(e−(DR+2γ)t−e−ast)2\n(DR+ 2γ−as)2\u0015\n,\n(26)10\n100101102103\nt10−1100101102103/angbracketleftx2\nα(t)/angbracketrightcDR= 0.15\nDR= 0.015\nDR= 0.001\nFigure 1: Comparison between the theoretical expression for variance in (26) (shown by\nthe solid lines) with numerical simulation for different values of DRkeeping k= 1 and\nγ= 0.1 fixed. The symbols indicate the data obtained from numerical simulations with\nN= 4 and v0= 1.\nwhere asis defined in (13). The above equation is exact, and for any finite N, the\nposition variance of the tagged particle can be computed by numerically evaluating the\nsum. In figure 1 we compare this exact result with numerical simulations for a set of\ndifferent parameters and find excellent agreement, as expected.\nIt is particularly interesting to consider the limit of thermodynamically large system\nsize, i.e., N→ ∞ . In this limit, setting 2 πs/N =q, the sum in (26) can be converted\ninto an integral, which is most conveniently expressed as,\n\nx2\nα(t)\u000b\nc=v2\n0Zπ\n−πdq\n2π\u00141\n2(DR+ 2γ) \n1−e−2bqt\nbq+e−(DR+2γ)te−bqt−1\nDR+ 2γ+bq+e−bqt(e−(DR+2γ)t−e−bqt)\nDR+ 2γ−bq!\n+cos 2θ0\n2(DR−2γ) \ne−4DRt−e−2bqt\n2DR−bq+e−(DR+2γ)te−bqt−e−4DRt\n3DR−2γ−bq+e−bqt(e−bqt−e−(DR+2γ)t)\nDR+ 2γ−bq!\n−cos2θ0(e−(DR+2γ)t−e−bqt)2\n(DR+ 2γ−bq)2\u0015\n,\n(27)11\nwhere, we have defined,\nbq= 4ksin2\u0010q\n2\u0011\n. (28)\nThe first term and part of the second term in (27) can be integrated exactly, giving,\nB1(t)≡v2\n0\n2(DR+ 2γ)Zπ\n−πdq\n2π1−e−2bqt\nbq= 2Deffh\nI0(4kt) +I1(4kt)i\nt e−4kt, (29)\nand,\nB2≡v2\n0\n2(DR+ 2γ)Zπ\n−πdq\n2π1\u0002\n(D+ 2γ) +bq\u0003=Deffp\n(DR+ 2γ)(DR+ 2γ+ 4k), (30)\nwhere In(z) denotes the modified Bessel function of first kind and Deffis defined in (2).\nThis simplifies (27) to,\n\nx2\nα(t)\u000b\nc=B1(t)−B2+v2\n0Zπ\n−πdq\n2π\u0014e−bqt\n2(DR+ 2γ)(\ne−(DR+2γ)t\nDR+ 2γ+bq\n+e−(DR+2γ)t−e−bqt\nDR+ 2γ−bq)\n+cos 2θ0\n2(DR−2γ)(\ne−4DRt−e−2bqt\n2DR−bq+e−(DR+2γ)te−bqt−e−4DRt\n3DR−2γ−bq\n−e−bqt(e−(DR+2γ)t−e−bqt)\nDR+ 2γ−bq)\n−cos2θ0(e−(DR+2γ)t−e−bqt)2\n\u0002\nDR+ 2γ−bq\u00032\u0015\n. (31)\nWe now analyze the above equation in various dynamical regimes emerging\nfrom the interplay of the time-scales D−1\nR,γ−1and k−1. For any given distinct\nvalues of {D−1\nR, γ−1, k−1}, there are three different time-scales, the smallest τ1=\nmin(D−1\nR, γ−1, k−1), the largest τ3= max( D−1\nR, γ−1, k−1), and τ2, given by the third\none. We consider the most interesting scenario where the time-scales are well separated,\nsuch that τ1≪τ2≪τ3. Correspondingly, for each ordering of DR,γandk, there are\nfour dynamical regimes,\n(i) Short-time regime: t≪τ1\n(ii) Early-intermediate regime : τ1≪t≪τ2\n(iii) Late-intermediate regime : τ2≪t≪τ3\n(iv) Long-time regime: t≫τ3.\nTo analyse the behaviour of the tagged particle variance in the various regimes, it\nis convenient to recast (31) using a change of variable z2=bqt, as,\n\nx2\nα(t)\u000b\nc=B1(t)−B2+v2\n0\n2π√\nktZ√\n4kt\n−√\n4ktdz\u0014e−(DR+2γ)te−z2\n2(DR+ 2γ)\u0002\n(DR+ 2γ) + (z2/t)\u0003\n+e−(DR+2γ)te−z2−e−2z2\n2(DR+ 2γ)\u0002\n(DR+ 2γ)−(z2/t)\u0003+cos 2θ0(e−4DRt−e−2z2)\n(DR−2γ)(4DR−2(z2/t))\n+cos 2θ0(e−(DR+2γ)te−z2−e−4DRt)\n2(DR−2γ)\u0002\n(3DR−2γ)−(z2/t)\u0003+cos 2θ0(e−2z2−e−(DR+2γ)te−z2)\n2(DR−2γ)\u0002\n(DR+ 2γ)−(z2/t)\u0003\n−cos2θ0(e−(DR+2γ)t−e−z2)2\n\u0002\n(DR+ 2γ)−(z2/t)\u00032\u00151p\n1−z2/(4kt). (32)12\nBefore the intermediate regimes, we first discuss the short-time and long-time\nregimes, where the dynamical behaviors are, in fact, independent of the ordering of\nthe time-scales.\n4.1. Short-time regime (t≪τ1)(R-I)\nIn this regime the time tis much smaller than all the time-scales present in the system.\nHence, expanding (31) in Taylor series of t, we get the leading order behavior of the\ntagged particle variance,\n\nx2\nα(t)\u000b\nc=2v2\n0t3\n3\u0002\nDRsin2θ0+ 2γcos2θ0\u0003\n+O(t4). (33)\nNote that, this leading order behavior is independent of kand is the same as that of a\nsingle DRABP in the short-time regime ( t≪(k−1, D−1\nR, γ−1) [61], since the particles do\nnot feel the effect of the harmonic coupling. However, the effect of coupling shows up\nin the next order correction as,\nv2\n0\n12\u0014\n(DR−2γ)[7DR+ 6(k+γ)] cos 2 θ0−3(DR+ 2γ)[DR+ 2(k+γ)]\u0015\nt4. (34)\nIn figure 2 we compare the numerical simulation for the short-time behavior for different\norders of time scale with the asymptotic behavior (33) and get excellent agreement.\n4.2. Long-time regime (t≫τ3)(R-IV)\nIn the long-time regime, tis much larger than all the time-scales of the system, and\nconsequently, we have, DRt≫1, γt≫1 as well as kt≫1. Using these limits in (32),\nwe get,\n\nx2\nα(t)\u000b\nc=B1(t)−B2+v2\n0Z√\n4kt\n−√\n4ktdz e−2z2\n2π√\nkt\u0014cos 2θ0\n2(DR−2γ)\u0002\n(DR+ 2γ)−(z2/t)\u0003\n−cos 2θ0\n2(DR−2γ)(2DR−(z2/t))−1\n2(DR+ 2γ)\u0002\n(DR+ 2γ)−(z2/t)\u0003\n−cos2θ0\u0002\n(DR+ 2γ)−(z2/t)\u00032\u00151p\n1−z2/(4kt). (35)\nMoreover, the leading order behaviour of B1(t) in (29) is given by,\nB1(t) =Deff√\nπk\u0010√\n2t−1\n16k√\n2t\u0011\n+O(t−3/2). (36)\nClearly, the integral in (35) is dominated by contributions from near z= 0. In\nfact, to get the leading order contribution, it suffices to set z2/t→0, and extend the\nlimits of the integral to ±∞. Carrying out the resulting Gaussian integral and using13\n100101102103104\nt10−410−2100102104106/angbracketleftx2\nα(t)/angbracketrightct R-VI\nt3R-It5/2\nR-III\nt3\nR-Ik= 10−3,DR= 10−5,γ= 10−5\nk= 10−6,DR= 10−6,γ= 5×10−4\nFigure 2: The crossover behavior of the variance from the short-time regime to the\nearly-intermediate regime for quenched initial orientation. The solid lines are obtained\nby numerically integrating (31). The dotted lines correspond to the asymptotic behavior\ngiven by (33), (40), and (53). The symbols indicate the data obtained from numerical\nsimulations with N= 500 and v0= 1.\nthe large-time behaviour of B1(t), we finally get the long-time behaviour of the variance\nof the tagged particle position,\n\nx2\nα(t)\u000b\nc=Deffr\n2t\nπk−Deffp\n(DR+ 2γ)(DR+ 2γ+ 4k)\n−Deff√\n2πkt\"\n1\n16k+4DR+ (DR−2γ) cos 2 θ0\n4DR(DR+ 2γ)#\n+O(t−3/2). (37)\nNote that, similar to the short-time regime, the leading order behavior in the long-time\nregime is also independent of the ordering of the time-scales. As expected, the leading√\ntgrowth at late-times is the same as that of a tagged particle in a passive harmonic\nchain [18]. The correction terms, however, carry signatures of activity. In figure 3 we\nillustrate the√\ntgrowth in the long-time regime obtained from the numerical simulation\nalong with the analytical prediction (37).\nIn the following, we explore the behavior of the tagged particle in the remaining\ntwelve dynamical regimes. It is convenient to group them according to the relative14\n100101102103104\nt10−1100101102103/angbracketleftx2\nα(t)/angbracketrightc\nt1/2\nR-IV\nt1/2\nR-IIIt1/2R-IV\ntR-Vk= 1.0,DR= 5×10−4,γ= 0.8\nk= 5×10−4,DR= 1.0,γ= 0.8\nFigure 3: The crossover behavior of the variance from the late-intermediate regime to\nlong-time regime for quenched initial orientation for k≪DRandk≫DRwith fixed γ.\nThe solid lines are obtained by numerically integrating (31). The dotted lines indicate\nthe asymptotic behavior given by (37), (47), and (54). The symbols indicate the data\nobtained from numerical simulations with N= 500 and v0= 1.\nstrength of the coupling kwith respect to the activity parameters. We consider the three\nscenarios (i) Strong-coupling limit [ k≫(DR, γ)], (ii) Weak-coupling limit [ k≪(DR, γ)]\nand (iii) Moderate-coupling limit [min( DR, γ)≪k≪max( DR, γ)] below.\n4.3. Strong-coupling limit [k≫(DR, γ)]\nThe strong-coupling limit refers to the scenarios when the coupling strength kis larger\nthan both the activity parameters DRandγ. Correspondingly, we have,\nτ1=k−1≪ τ2= min( D−1\nR, γ−1)≪ τ3= max( D−1\nR, γ−1). (38)\nIn the intermediate regimes, t≫τ1and the leading order behavior of the tagged particle\nvariance can be obtained by taking the limit kt→ ∞ in (32). In this limit, the integrals15\ncan be performed explicitly [see (C.1)-(C.5) in Appendix C], yielding,\n⟨xα(t)2⟩=B1(t)−B2+Deff\n2p\n(DR+ 2γ)kt(\n√\nterfc\u0010p\n(DR+ 2γ)t\u0011\n+(DR+ 2γ)3/2cos 2θ0\n(DR−2γ)\nerf\u0010p\n(2γ−3DR)t\u0011\np\n(2γ−3DR)−erfi\u0000√4DRt\u0001\n√2DR\n√\nt e−4DRt\n+\"\u0012\n1−cos 2θ0(DR+ 2γ)\n(DR−2γ)\u0013h\nerfi\u0010p\n(DR+ 2γ)t\u0011\n−erfi\u0010p\n2(DR+ 2γ)t\u0011i\n+ cos2θo\u0012\u0014\n(DR+ 2γ)t+1\n2\u0015\nerfi\u0010p\n(DR+ 2γ)t\u0011\n−\u0014\n(DR+ 2γ)t+1\n4\u0015\nerfi\u0010p\n2(DR+ 2γ)t\u0011\u0013#\n√\nt e−2(DR+2γ)t\n+2t√πp\n2(DR+ 2γ) cos2θo\u0010\n1−√\n2e−(DR+2γ)t\u0011\n+O\u00121\nkt\u0013)\n. (39)\nThe leading behaviors in the early and late-intermediate regimes can be obtained by\ntaking the appropriate limits in the above equations.\nEarly intermediate regime (τ1≪t≪τ2≪τ3) (R-II).– We obtain the leading order\nbehavior in the early-intermediate regime by considering DRt≪1 and γt≪1 in (39)\nas,\n⟨xα(t)2⟩ ≃16\n15(√\n2−1)[DRsin2θ0+ 2γcos2θ0]v2\n0t5/2\n√\nπk. (40)\nClearly, for arbitrary initial orientation and any finite DRandγ, the leading order\nbehavior of the tagged particle variance ⟨xα(t)2⟩ ∼t5/2. Note that, γ= 0 corresponds\nto the ABP case, where t5/2behavior has been observed for θ0=π/2 [58].\nIn the special case γ= 0 and θ0= 0, the leading order term vanishes and we need\nto consider the subleading corrections to (40), which come from two sources: (i) the\nnext order terms in DRtandγtfrom (39) and (ii) subleading corrections in 1 /(kt) in\nthe expansion of [1 −z2/(4kt)]−1/2in (32) in the limit DRt≪1 and γt≪1. This results\nin,\n⟨xα(t)2⟩=v2\n0√τ1√π τ2t5/2\"\nL1+L2t\nτ2+15√\n2\n96L1τ1\nt+···#\n, (41)\nwhere τ1,τ2, and τ3are defined in (38) and the leading order coefficient L1has been\nalready obtained in (40). It is more conveniently expressed as,\nL1=16(√\n2−1)\n15×\n\nsin2θ0+ 2τ2\nτ3cos2θ0forDR> γ\nτ2\nτ3sin2θ0+ 2 cos2θ0forDR< γ.(42)16\nThe coefficient L2, appearing in the subleading term, is given by,\nL2=64(√\n2−1)\n105×\n\n\u0012\n1−2τ2\nτ3\u0013\"\n1 +7−√\n2\n2τ2\nτ3cos2θ0#\nforDR> γ\n\u0012τ2\nτ3−2\u0013\"\nτ2\nτ3+7−√\n2\n2cos2θ0#\nforDR< γ.(43)\nNote that, for any given values of DRandγ, depending on the relative strength of\nt/τ2andτ1/t, the two subleading terms could be of the same order. Interestingly, for\nthe special case γ= 0 and θ0= 0, the leading order behavior is given by\n⟨xα(t)2⟩=64(√\n2−1)\n105v2\n0D2\nR√\nπkt7/2, (44)\nwhich is consistent with the result obtained earlier for the ABP case with θ0= 0 [58].\nThet5/2growth of variance in the early intermediate regime for the strong coupling\nlimit is shown in figure 2 and figure 4 .\nLate intermediate regime (τ1≪τ2≪t≪τ3) (R-III).– Next, we consider the late-\nintermediate regimes τ1≪τ2≪t≪τ3. For τ2=D−1\nRandτ3=γ−1, i.e., DR≫γ, the\nleading order behavior can be obtained by taking the limit DRt≫1,γt≪1 in (39),\nas,\n\nx2\nα(t)\u000b\nc≃v2\n0√\nt\nDR√\n2πk. (45)\nThe subleading corrections to (45) can be obtained by considering the next order terms\nin (39). This results in,\n\nx2\nα(t)\u000b\nc=v2\n0√\nt\nDR√\n2πk\"\n1−rπτ2\n2t+O(τ2/t)#\n. (46)\nNote that, in this case, the tagged particle variance becomes independent of the initial\norientation θ0due to the large rotational diffusion constant. Incidentally, the above\nequation can also be obtained by considering the limit γ≪DR≪kin (37).\nOn the other hand, for τ2=γ−1andτ3=D−1\nR, i.e., γ≫DR, we have DRt≪1,\nγt≫1. Considering these limits in (39), the leading order behavior is given by,\n\nx2\nα(t)\u000b\nc≃v2\n0cos2θ0\nγ√\n2πk√\nt. (47)\nAgain the subleading corrections to (47) can be obtained by considering the next order\nterms in (39) which results in,\n\nx2\nα(t)\u000b\nc=v2\n0cos2θ0\nγ√\n2πk√\nt\"\n1−1\n2rπτ2\nt−4\n3(1−tan2θ0)t\nτ3+O(τ2/t)#\n. (48)17\nClearly, for arbitrary initial orientation, irrespective of the relative strength of DRand\nγ, the leading order behavior of the tagged particle variance ⟨xα(t)2⟩ ∼t1/2, except for\nthe special case γ= 0 and θ0=π/2, where this leading order term vanishes. In that\ncase, we need to consider the subleading corrections, ⟨xα(t)2⟩ ∼t3/2.\nNote that, in strong coupling limit, the subleading contributions coming from the\nexpansion of [1 −z2/(4kt)]−1/2in (32) is O(τ1/t)≪O(τ2/t). The crossover behavior of\nthe variance from the early-intermediate regime to the late-intermediate regime and the\nlate-intermediate regime to long-time behavior in the strong-coupling limit is shown in\nfigure 4 and figure 3 respectively.\n4.4. Weak coupling limit [k≪(DR, γ)]\nThe weak-coupling limit refers to the scenarios when the coupling strength kis smaller\nthan both the activity parameters DRandγand correspondingly the largest time-scale\nτ3=k−1. The other two time-scales are τ1= min( D−1\nR, γ−1),τ2= max( D−1\nR, γ−1). In\nthis case, we can expand (31) in powers of t/τ3,τ2/τ3andτ1/τ2. This expansion can be\nmost conveniently expressed in terms of the parameters k,DR,γas,\n\n|xα(t)|2\u000b\nc=B1(t)−B2+Deff∞X\nn=0∞X\nm=0A(m+n)(−4kt)n\nn!(\ne−(DR+2γ)t\nDR+ 2γ\u0012−4k\nDR+ 2γ\u0013m\n+2(DRsin2θ0−2γcos2θ0)\nD2\nR−4γ2(e−(DR+2γ)t−2n)\u00124k\nDR+ 2γ\u0013m\n−(DR+ 2γ) cos 2 θ0\nDR−2γ\u00142n−1\nDR\u00122k\nDR\u0013m\n−e−(DR+2γ)t\n3DR−2γ\u00124k\n3DR−2γ\u0013m\u0015\n+2(m+ 1) cos2θ0\nDR+ 2γ\u0000\n2e−(DR+2γ)t−2n\u0001\u00124k\nDR+ 2γ\u0013m)\n+Deff∞X\nm=0A(m)(\nDR+ 2γ\nDR−2γe−4DRtcos 2θ0\u00141\n2DR\u00122k\nDR\u0013m\n−1\n3DR−2γ\u00124k\nDR+ 2γ\u0013m\u0015\n−2(m+ 1) cos2θ0e−2(DR+2γ)t\nDR+ 2γ\u00124k\nDR+ 2γ\u0013m)\n,\n(49)\nwhere A(m) is a constant, given by\nA(m) =1\n2πZπ\n−πdqsin2m(q/2) =Γ\u0000\nm+1\n2\u0001\n√π m!. (50)\nAlthough the sum over min (49) can be performed explicitly [see (C.6)-(C.7)\nin Appendix C], the sum over nis hard to evaluate in a closed form. Nevertheless,\n(49) is exact for all parameters. In fact, the expansion as a power series in kt, as given\nin (49), is particularly useful in the intermediate regime of the weak-coupling limit,\nwhere kt≪1. In this limit, taking exp [ −(DR+ 2γ)t]→0, (49) reduces to a much18\nsimpler form\n\n|xα(t)|2\u000b\nc=B1(t)−B2−2Deff\nDR−2γ∞X\nn=0∞X\nm=0A(m+n)(−8kt)n\nn!\n×(\u0014DR\nDR+ 2γ+mcos2θ0\u0015\u00124k\nDR+ 2γ\u0013m\n+(DR+ 2γ) cos 2 θ0\n4DR\u00122k\nDR\u0013m)\n+e−4DRtDeff\n2DRDR+ 2γ\nDR−2γcos 2θ0∞X\nm=0A(m)\u0014\u00122k\nDR\u0013m\n−2DR\n3DR−2γ\u00124k\nDR+ 2γ\u0013m\u0015\n.(51)\nUnlike the strong-coupling limit, the behavior of the tagged particle variance in the\nintermediate regime in the weak-coupling limit depends on the relative strength of DR\nandγ. Hence we choose a particular order of DR, γand then examine (51) to get the\nleading order behavior of the position variance. In the following, we analyze the early\nintermediate and the late intermediate regimes separately.\nEarly intermediate regime (τ1≪t≪τ2≪τ3).– For DR≫γ, we have τ1=D−1\nR\nandτ2=γ−1(R-V). Considering the limit DRt≫1,γt≪1 and kt≪1 in (51) we get\nthe leading order behavior for this case as,\n\nxα(t)2\u000b\nc=v2\n0t\nDR\u0014\n1−(6 + cos 2 θ0)\n4DRt−2kt+···\u0015\n. (52)\nOn the other hand, for γ≫DR, we have τ1=γ−1andτ2=D−1\nR(R-VI).\nConsidering the limit DRt≪1,γt≫1 and kt≪1 in (51) we get the leading order\nbehavior for this case as,\n\nxα(t)2\u000b\nc=v2\n0t\nγcos2θ0\u0014\n1−3\n4γt−2kt−DRt(1−tan2θ0) +···\u0015\n. (53)\nInterestingly, for θ0=π/2, the tagged particle behaves ballistically at the leading order,\ni.e.,\nxα(t)2\u000b\nc≃v2\n0(DR/γ)t2.\nLate intermediate regime (τ1≪τ2≪t≪τ3) (R-V).– In this regime, both DRt≫1\nandγt≫1, while kt≪1. Considering these limits in (51), the leading order behavior\nof the variance can be written as,\n\nxα(t)2\u000b\nc≃2Defft, (54)\nindependent of the relative strength of DRandγ. The sub-leading corrections, however,\ndepend on the relative strength of DRandγ. In particular, for DR≫γthe tagged\nparticle behavior is the same as that in the early intermediate regime, given by (52).\nOn the other hand, for γ≫DR, we have,\n\nxα(t)2\u000b\nc=v2\n0t\n2γ\u0014\n1−3 cos2θ0\n2γt−2kt+···\u0015\n. (55)19\n100101102103104\nt10−310−1101103/angbracketleftx2\nα(t)/angbracketrightc√\nt R-IV\nt5/2\nR-II√\nt\nR-IV\ntR-VIIk= 1,DR= 5×10−4,γ= 5×10−4\nk= 5×10−4,DR= 5.0,γ= 10−6\nFigure 4: The crossover behavior of the variance from the early-intermediate regime to\nthe late-intermediate regime for quenched initial orientation. The solid lines are obtained\nby numerically integrating (31). The dotted lines correspond to the asymptotic behavior\ngiven by (37), (40), and (52). The symbols indicate the data obtained from numerical\nsimulations with N= 500 and v0= 1.\nNote that, in this regime, Deff, given by (2), becomes\nDeff→\n\nv2\n0\n2DRforγ≪DR\nv2\n0\n4γforγ≫DR.(56)\nFigure 5 shows the crossover of the variance from the early intermediate regime to the\nlate intermediate regime when γ≫DR, in the weak-coupling limit.\n4.5. Moderate-coupling limit [min( DR, γ)≪k≪max( DR, γ)]\nThe moderate-coupling refers to the scenarios when klies between the two activity\nparameters DRandγ. Consequently the coupling strength kdetermines the interme-\ndiate time scale, τ2=k−1, while the other two time-scales are τ1= min( D−1\nR, γ−1) and\nτ3= max( D−1\nR, γ−1).20\n100101102103104\nt100101102103/angbracketleftx2\nα(t)/angbracketrightctR-V\n√\nt\nR-IX\ntR-VI, R-VIIIk= 1×10−6,DR= 1×10−3,γ= 2.0\nk= 1×10−3,DR= 1×10−6,γ= 2.0\nFigure 5: The crossover behavior of the variance from the early-intermediate regime to\nthe late-intermediate regime for quenched initial orientation. The solid lines are obtained\nby numerically integrating (31). The dotted lines correspond to the asymptotic behavior\ngiven by (47), (52), and (53). The symbols indicate the data obtained from numerical\nsimulations with N= 100 and v0= 1.\nEarly intermediate regime (τ1≪t≪τ2≪τ3).– The behavior of the tagged par-\nticle in the early intermediate regime can be most conveniently obtained from (51),\nwhich was obtained in the limit kt≪1 and ( DR+ 2γ)t≫1. Therefore, the scenarios\ncorresponding to both the relative orders DR≫γ(R-VII) and γ≫DR(R-VIII) in\nthe moderate coupling limit are the same as the corresponding scenarios in the weak\ncoupling limit. Namely,\nxα(t)2\u000b\ncis given by (52) and (53), for DR≫γandγ≫DR,\nrespectively. Figure 4 and 5 show this early intermediate regime behavior, for DR≫γ\nandγ≫DR, respectively.\nLate intermediate regime (τ1≪τ2≪t≪τ3).– In this regime, kt≫1, and\nhence, the leading order behavior of the variance can be extracted from (39). Therefore,\nthe scenarios corresponding to both the relative orders DR≫γandγ≫DRin the\nmoderate coupling limit are the same as the corresponding scenarios in the strong\ncoupling limit. Namely, the leading order behavior of\nxα(t)2\u000b\ncis given by (47) and\n(45), for γ≫DRandDR≫γ, respectively. However, the subleading corrections differ\nfrom those obtained in the strong coupling limit and can be extracted directly from (32)21\nby considering the appropriate limits. For DR≫γ(R-IV), we get\n\nx2\nα(t)\u000b\nc=v2\n0√\nt\nDR√\n2πk\"\n1−1\n32kt−r\n2πk\nDR1√DRt+···#\n, (57)\nwhile for γ≫DR(R-IX), we get,\n\nx2\nα(t)\u000b\nc=v2\n0cos2θ0\n2γr\n2t\nπk\"\n1−s\nπk\n8γ2t−1\n32kt+4(1−2 cos2θ0)\n3 cos2θ0DRt+···#\n.(58)\nInterestingly, for θ0=π/2, leading order behaviour changes from√\nttot3/2. Figures 4\nand 5 show the crossover behavior from the early-intermediate regime to the late-\nintermediate regime in the moderate-coupling limit, for DR≫γand γ≫DR,\nrespectively.\n5. Annealed initial orientation\nIn this section, we consider the scenario where the initial orientation {θα(0)}are\nindependently drawn from a uniform distribution of [0 ,2π]. As a result, G(t1, t2) is\ngiven by (22). Putting (22) in (24) and using (25) we get,\n\nx2\nα(t)\u000b\nc=Deff\nNN−1X\ns=0\u00141−e−2ast\nas+e−(DR+2γ)te−ast−1\u0000\nDR+ 2γ+as\u0001+e−ast(e−(DR+2γ)t−e−ast)\u0000\nDR+ 2γ−as\u0001\u0015\n,(59)\nwhere asis defined in (13). Equation (59) can be converted into an integral in the large\nNlimit as,\n\nx2\nα(t)\u000b\nc=DeffZ2π\n0dq\n2π\u00141−e−2bqt\nbq+e−(DR+2γ)te−bqt−1\u0000\nDR+ 2γ+bq\u0001+e−bqt(e−(DR+2γ)t−e−bqt)\u0000\nDR+ 2γ−bq\u0001\u0015\n,\n(60)\nwhere bqis given by (28). As before, the first and part of the second terms can be\nintegrated exactly, given by (29) and (30) respectively. This simplifies (60) to,\n\nx2\nα(t)\u000b\nc=B1(t)−B2+DeffZ2π\n0dq\n2π\u0014e−(DR+2γ)te−bqt\n\u0000\nDR+ 2γ+bq\u0001+e−bqt(e−(DR+2γ)t−e−bqt)\u0000\nDR+ 2γ−bq\u0001\u0015\n(61)\nNote that, here the parameters DRand γappear together as ( DR+ 2γ).\nConsequently, we have two distinct time scales ( DR+ 2γ)−1andk−1, in contrast to\nthe quenched initial condition, where there are three distinct time scales set by D−1\nR,\nγ−1andk−1. In the following, we consider the three dynamical regimes emerging when\nthe two scales, τ1= min(( DR+ 2γ)−1, k−1) and τ2= max(( DR+ 2γ)−1, k−1), are well\nseparated:\n1. Short-time regime: t≪τ1\n2. Long-time regime: t≫τ2\n3. Intermediate regime: τ1≪t≪τ222\n100101102103104\nt100101102103104105106/angbracketleftx2\nα(t)/angbracketrightc\nt2R-It3/2\nR-IIR-IIItk= 2×10−3,DR= 1×10−5,γ= 5×10−6\nk= 1×10−6,DR= 4×10−3,γ= 5×10−4\n103104105106\nt3/2\nR-IIt\nR-III\nFigure 6: The crossover behavior of the variance from short-time regime to intermediate\nregime for annealed initial orientation. The solid lines are obtained by numerically\nintegrating (31). The dotted lines correspond to the asymptotic behavior given by (62),\n(66), and (67). The symbols indicate the data obtained from numerical simulations with\nN= 500 and v0= 1.\n5.1. Short-time regime (t≪τ1)(R-I)\nAt very short times, the particles do not feel the effect of the interaction. Consequently,\neach particle independently moves ballistically with the speed v0along its initial\norientation θα(0), which is drawn randomly from (0 ,2π). Hence, the average\ndisplacement is zero, and the variance\nxα(t)2\u000b\nc≈v2\n0t2\ncos2θα(0)\u000b\n=v2\n0t2/2. However,\nthe effect of the interaction is expected to show up in the subleading corrections. In\nfact, the leading order behavior and the subleading corrections can be systematically\nextracted by expanding (61) in power series of t,\n\nxα(t)2\u000b\nc=v2\n0t2\n2\u0014\n1−1\n3(DR+ 2γ+ 6k)t+O(t2)\u0015\n. (62)\nInterestingly, in the short-time regime, the position fluctuation in the quenched case\nis smaller than the annealed case by a factor proportional to t/τ2[see (33)]. Figure 6\nshows the t2growth of the variance at short-times.\nTo analyze the behavior of the variance in the remaining two regimes it is convenient23\nto recast (61) using a change of variable z2=bqt,\n\nxα(t)2\u000b\nc=B1(t)−B2+Deff\n2π√\nktZ√\n4kt\n−√\n4ktdz e−z2\u0014e−(DR+2γ)t\n(DR+ 2γ) +z2/t\n+e−(DR+2γ)t−e−z2\n(DR+ 2γ)−z2/t\u00151p\n1−z2/4kt. (63)\n5.2. Long-time regime (t≫τ2)(R-IV)\nIn the long-time regime, tis much larger than both the time-scales of the system.\nTherefore, setting e−(DR+2γ)t→0 in (63), we get,\n\nxα(t)2\u000b\nc=B1(t)−B2−Deff\n2π√\nktZ√\n4kt\n−√\n4ktdze−2z2\n\u0002\n(DR+ 2γ)−z2/t\u00031p\n1−z2/(4kt).(64)\nThe integral in (64) is dominated by the contribution from near z= 0. Consequently,\none can expand the integrand in powers of z2/[(DR+ 2γ)t] and z2/(4kt) and set the\ndomain of integral from −∞ to∞. Finally, we get the variance of the tagged particle\nin the long-time regime,\n\nx2\nα(t)\u000b\nc=Deffr\n2t\nπk−Deffp\n(DR+ 2γ)(DR+ 2γ+ 4k)\n−Deff√\n2πkt\"\n1\n16k+1\n2(DR+ 2γ)#\n+O(t−3/2). (65)\nFigure 7 shows the√\ntbehavior in the long-time regime.\nThe system eventually forgets the initial orientation, and the leading order behavior\nin the long time regime\nx2\nα(t)\u000b\nc≈Deffq\n2t\nπkfor both annealed and quenched initial\norientations. The difference due to the initial orientations shows up in the subleading\ncorrections [cf. (37) and (65)].\n5.3. Intermediate regime\nThe behavior of the tagged particle in the intermediate regime ( τ1≪t≪τ2) depends\non whether the coupling strength kis larger or smaller than the activity parameter\n(DR+ 2γ). In the strong coupling limit [ k≫(DR+ 2γ)], the two time-scales are\nτ1=k−1andτ2= (DR+ 2γ)−1, whereas in the weak coupling limit [ k≪(DR+ 2γ)],\nτ1= (DR+ 2γ)−1andτ2=k−1. In the following, we discuss the two limits separately.\n5.3.1. Strong-coupling limit [k≫(DR,2γ)](R-II) - In this limit, the intermediate\nregime is characterized by kt≫1 and ( DR+ 2γ)t≪1. The leading order behavior can\nbe obtained from (63) as,\n\nx2\nα(t)\u000b\nc=2(√\n2−1)v2\n0t3/2\n3√\nπk\"\n1− \n1−√\n2\n3!\n(DR+ 2γ)t−3√\n2\n32kt+···#\n. (66)\nFigures 6 and 7 show this t3/2growth in the variance.24\n100101102103104\nt10−1100101102103/angbracketleftx2\nα(t)/angbracketrightc\nt\nR−III√\nt\nR−IV\nt3/2R−II√\nt R−IVk= 1×10−3,DR= 1.0,γ= 1.0\nk= 2.0,DR= 5×10−3,γ= 2.5×10−3\nFigure 7: The crossover behavior of the variance from short-time regime to intermediate\nregime for annealed initial orientation. The solid lines are obtained by numerically\nintegrating (31). The dotted lines correspond to the asymptotic behavior given by (65),\n(66), and (67). The symbols indicate the data obtained from numerical simulations with\nv0= 1 and N= 500 for weak-coupling limit, while N=800 for strong-coupling limit.\n5.3.2. Weak-coupling limit [k≪(DR,2γ)](R-III) - On the other hand in the weak-\ncoupling limit, the intermediate regime is characterized by kt≪1 and ( DR+ 2γ)t≫1.\nThe leading order behavior can be obtained from (61) as,\n\nx2\nα(t)\u000b\nc= 2Defft\u0014\n1−1\n(DR+ 2γ)t−2kt+···\u0015\n(67)\nFigures 6 and 7 show this linear growth in variance.\nApart from the long-time regime, where the behavior is anyway expected to be\nindependent of the initial condition, the tagged particle variance in all the other\ndynamical regimes shows rather different behavior for the quenched and annealed initial\nconditions. In particular, the effect of the initial condition is drastically felt in the\nintermediate dynamical regime of the strong coupling limit, where randomizing the\ninitial orientation changes the behavior to t3/2from the t5/2observed for a quenched\ninitial condition. Moreover, at short-times also, the position variance of the tagged\nparticle with the annealed initial orientation shows a t2growth, as opposed to the t3\ngrowth for a quenched initial orientation.25\n6. Finite-size effects\nThere is an additional time scale tNassociated with any finite system of size N. This\nfinite-size time-scale tNdiverges in the thermodynamic limit N→ ∞ . Therefore, for a\nharmonic chain of NDRABPs, the behaviors we studied in earlier sections are expected\nto hold for t≪tN, for a large but finite N. On the other hand, for t≫tNone expects\na different behavior due to finite size effects. In this section, we study the finite-size\neffect and crossover behavior across tN, of the tagged particle variance.\nWe start with the exact expression for the position variance given by (26), which\nholds for an arbitrary value of N. We assume that the system size is large enough so\nthat tNis the largest time scale, i.e., {D−1\nR, γ−1, k−1} ≪ tN. Hence to study the finite\nsize effect and the cross-over behavior across tN, we study the behavior in the regime\n{D−1\nR, γ−1, k−1} ≪t. For t≫ {D−1\nR, γ−1}and large N, (26) simplifies to,\n\nx2\nα(t)\u000b\nc=Deff\nNN/2X\ns=−N/2\u00141−e−2ast\nas−1\nDR+ 2γ+as−e−2ast\nDR+ 2γ−as\n+(DR+ 2γ)e−2ast\nDR+ 2γ−as\u0012cos 2θ0\n2DR−as−2 cos2θ0\nDR+ 2γ−as\u0013\u0015\n, (68)\nwhere asis defined in (13). Note that, we have shifted the limits of summation using the\nfact that the sum in (26) is symmetric about N/2. Now, since t≫k−1, the summation\nin (68) is dominated by contributions from terms with s≪N. Hence, we can use the\napproximation e−2ast≃e−8π2s2t/tN, where the finite-size time-scale tN=N2/k. Clearly,\nfort≫tN,e−8π2s2t/tN→0 for s̸= 0. Then, from (68), we get, for t≫tN,\n\nx2\nα(t)\u000b\nc=Deff\nN\u0014\n2t+cos 2θ0\n2DR−1 + 2 cos2θ0\n(DR+ 2γ)\u0015\n+v2\n0\nNN/2X\ns=1\u00141\nas(DR+ 2γ+as)\u0015\n.(69)\nHence, for t≫tN, the leading order behavior of the variance grows linearly with time,\nas\nx2\nα(t)\u000b\nc≃2tDeff/N, which is consistent with earlier studies done on the harmonic\nchain of ABP, RTP, and AOUP [58].\nOn the other hand, for t≪tN, we cannot ignore e−8π2s2t/tN. However, since kt≫1,\nthe summation for terms containing e−4π2s2t/tNis still dominated by s≪N. Therefore\nEq. (26) becomes,\n\nx2\nα(t)\u000b\nc=Deff\nNN/2X\ns=−N/2\u00141−e−8π2s2t/tN\n4π2s2t/tN−1\nD+ 2γ+as\n+cos 2θ0e−8π2s2t/tN\n2DR−(2 cos2θ0+ 1)e−8π2s2t/tN\n2(DR+ 2γ)\u0015\n.(70)\nIn the thermodynamic limit N→ ∞ , (37) is recovered by converting the sum into an\nintegral by taking q= 2πs/N .26\n10−210−1100101\nt0.61.01.52.54.06.0\n2√z\n/radicalBig\n2\nπ/angbracketleftx2\nα(t)/angbracketrightc√\nk\nDeff√\ntN= 40\nN= 60\nN= 80\nN= 400\nFigure 8: Comparison between the theoretical expression for scaling function (74) (solid\nblack line) with numerical simulation for different values of N, keeping k= 1, DR= 1,\nγ= 1 and v0= 1 fixed.\nTo summarize, from (37) and (69), we have the leading order behavior of the\nvariance as\n\nx2\nα(t)\u000b\nc≃Deff×\n\nq\n2t\nπkfort≪tN\n2t\nNfort≫tN.(71)\nThus, in the scaling limit, t→ ∞ ,N→ ∞ , keeping t/tNfixed, suggests a scaling form,\n\nx2\nα(t)\u000b\nc=Deffr\nt\nkf\u0012t\ntN\u0013\n, (72)\nwhere the crossover function must have the limiting behavior,\nf(z)→(p\n2/πasz→0\n2√z asz→ ∞ .(73)\nIn fact, by taking the scaling limit in Eq. (70), we obtain the full scaling function as,\nf(z) =1\n4π2√z∞X\ns=−∞1−e−8π2s2z\ns2. (74)27\nIn figure 8 we compare this exact scaling function with the same obtained from numerical\nsimulations for different values of Nand find an excellent agreement.\nThe limiting behavior of f(z), mentioned in (73), can be obtained from the above\nequation. To see the large zbehavior, it is useful to separate the s= 0 term and recast\nthe above equation as,\nf(z) = 2√z+1\n12√z−1\n2π2√z∞X\ns=1e−8π2s2z\ns2, (75)\nwhere we have usedP∞\ns=1s−2=π2/6 and the symmetry of the summand for s→ −s.\nClearly, the first term gives the limiting behavior mentioned in (73), and the second\nterm is the leading order correction. The higher-order corrections can be systematically\nobtained from the series.\nOn the other hand, in the limit z≪1, with a change of variable 2 πs√z=u, the\nsummation in (74) can be converted to an integral as,\nf(z)z→0− − →1\n2πZ∞\n−∞1−e−2u2\nu2du=r\n2\nπ, (76)\nwhich is consistent with (73). However, it is not straightforward to obtain the corrections\nto the above limiting behavior systematically from (74). For this, one can use the Poisson\nsummation formula [63], to rewrite the sum in (74) as,\nf(z) =r\n2\nπ+∞X\nm=1\"\n2r\n2\nπexp\u0012\n−m2\n8z\u0013\n−m√zerfc\u0012m\n2√\n2z\u0013#\n, (77)\nwhich approaches the constantp\n2/π, asz→0. Furthermore, using the asymptotic\nexpansion of erfc( x), we have a systematic series expansion about z= 0,\nf(z) =r\n2\nπ\"\n1 + 8 z∞X\nm=11\nm2exp\u0012\n−m2\n8z\u0013∞X\nn=0(−4z)n\nm2n(2n+ 1)!!#\n. (78)\nTo summarize, we find that for t≫N2/kthe tagged particle performs a center of\nmass diffusion with an effective diffusion constant Deff/N. The exact scaling function\nf(z) — governing the crossover of the variance from sub-diffusive Deffq\n2t\nπkto diffusive\n2 (Deff/N)tbehavior across the finite-size time-scale — is equivalently given by (75),\n(77), and (78), each suitable for evaluating the function in different limiting scenarios.\n7. Statistics of the separations\nFor a chain of passive particles, while the center of mass diffuses freely in the absence\nof any global confining potential, the separations between the adjacent particles reach\nan equilibrium state. In fact, for a chain of Npassive particles with free boundary\nconditions (i.e., when x0andxN−1are only coupled to x1andxN−2respectively), at28\na temperature T, the N−1 separations, yα=xα+1−xα, with α= 0,1, . . . , N −2\neventually reach an equilibrium state given by the product measure,\nP({yα})∝exp\"\n−k\n2kBTN−2X\nα=0y2\nα#\n. (79)\nOn the other hand, with a periodic boundary condition, the Nseparations must satisfyPN−1\nα=0yα= 0, and hence are not independent. While this global constraint destroys the\nproduct measure of the equilibrium state for any finite N, one expects that in the limit\nN→ ∞ , the product measure is restored for any finite subset of {yα}. In particular,\nfor any α,\nP(yα) =r\nk\n2πkBTexp\u0014\n−k y2\nα\n2kBT\u0015\n, asN→ ∞ . (80)\nIt is interesting to ask how activity affects this behavior, which we explore in this section.\nFollowing (5), the equation of motion for the separation for the active chain is given\nby,\n˙yα=−k(2yα−yα+1−yα−1) +ηα(t), (81)\nwhere,\nηα(t) =ξα+1(t)−ξα(t). (82)\nSince the stationary state is independent of the initial condition, for simplicity, we set\n{yα(0) = 0 }for all α. Moreover, here, we are using the quenched initial orientation\n{θα(0) = θ0}and{σα(0) = 1 }for all αin the noise ηαabove. We begin by computing\ntwo-point spatio-temporal correlation function,\nyα(t)yβ(t+τ)\u000b\n. To proceed, we perform\nDFT [defined in (14)] on (81) with respect to α, so that we get Ndecoupled first order\ndifferential equations,\n˙˜ys(t) =−as˜ys(t) + ˜ηs(t),with, a s= 4ksin2\u0010πs\nN\u0011\n, (83)\nwhere {˜ys(t)}and{˜ηs(t)}(fors= 0,1, . . . N −1) denote the DFT of {˜yα(t)}and{˜ηα(t)}\nrespectively. Since {yα(0) = 0 }, we have {˜ys(0) = 0 }. The solution of (83) is given by,\n˜ys(t) =e−astZt\n0east1˜ηs(t1)dt1. (84)\nUsing (14), the spatio-temporal correlation can be expressed as,\n\nyα(t)yβ(t+τ)\u000b\n=N−1X\ns=0N−1X\ns′=0exp\"\ni2π(sα−s′β)\nN#\n\n˜ys(t)˜y∗\ns′(t+τ)\u000b\n. (85)\nNow, the correlation in the Fourier space,\n˜ys(t)˜y∗\ns′(t+τ)\u000b\ncan be obtained using (84),\n\n˜ys(t)˜y∗\ns′(t+τ)\u000b\n=e−(ast+as′(t+τ))Zt+τ\n0dt2Zt\n0dt1east1+as′t2\n˜ηs(t1)˜η∗\ns′(t2)\u000b\n, (86)29\nwhere,\n\n˜ηs(τ1)˜η∗\ns′(τ2)\u000b\n=1\nN2N−1X\nα,α′=0ei2π\nN(s′α′−sα)\nηα(τ1)ηα′(τ2)\u000b\n. (87)\nHere we note that,\n\nηα(τ1)ηα′(τ2)\u000b\n=\nξα+1(τ1)ξα′+1(τ2)\u000b\n+\nξα(τ1)ξα′(τ2)\u000b\n−\nξα+1(τ1)ξα′(τ2)\u000b\n−\nξα(τ1)ξα′+1(τ2)\u000b\n= [2δα,α′−δα+1,α′−δα−1,α′]G2(τ1, τ2), (88)\nwhere G2(τ1, τ2)≡\nξα(τ1)ξα(τ2)\u000b\nis given by,\nG2(τ1, τ2) =v2\n0\n2h\ne−(DR+2γ)|t2−t1|+ cos 2 θ0e−\u0000\nDR(t2+t1+2 min[ t1,t2])+2γ|t2−t1|\u0001i\n. (89)\nSubstituting (88) on (87) and performing the summations over α, α′, we get\n\n˜ηs(τ1)˜η∗\ns′(τ2)\u000b\n=4\nNsin2\u0010πs\nN\u0011\nG2(τ1, τ2)δs,s′. (90)\nTo calculate the spatio-temporal correlation, we substitute (90) in (86), and using\n(85) we get,\n\nyα(t)yβ(t+τ)\u000b\n=v2\n0\n2NkN−1X\ns=0cos (2 πs(α−β)/N)as\nas−DR−2γ\"\ne−as(t+2τ)(2γ−3DR+as) cos 2 θ0\n(as−2DR)(as−3DR+ 2γ)\n+ase−(DR+2γ)τ−(DR+ 2γ)e−asτ−e−(DR+2γ)(2t+τ)+ (as+DR+ 2γ)e−as(2t+τ)\nas(as+DR+ 2γ)\n+\u0002\n(DR−2γ)e−4DRt−asτ−(as−2DR)\u0000\ne−(DR+2γ)(2t+τ)−e−(2γτ+DR(4t+τ))\u0001\u0003\ncos 2θ0\n(as−2DR)(as−3DR+ 2γ)#\n.\n(91)\nHere we note that the imaginary part of\nyα(t)yβ(t+τ)\u000b\nturns out to be zero,\nas expected. To obtain the steady-state behavior of the correlation function, we take\nthe limit t→ ∞ in (91). Furthermore, since the correlation function depends only on\n|α−β|, we set α= 0 without loss of generality. This yields,\nC(β, τ) := lim\nt→∞\ny0(t)yβ(t+τ)\u000b\n=v2\n0\n2NkN−1X\ns=1cos (2 πsβ/N )\u0000\nase−(DR+2γ)τ−e−asτ(DR+ 2γ)\u0001\n(as−DR−2γ)(as+DR+ 2γ). (92)\nIn figures 9 and 10, we compare the steady-state behavior of the above-mentioned\ncorrelation function (92) with numerical simulation for varying τandβrespectively\nand find very good agreement.\nNext, we compute exactly the variance C(0,0), the equal-time spatial correlation\nC(β,0), and the spatio-temporal correction C(β, τ) in the thermodynamic limit N→ ∞ .30\n−8−6−4−2 0 2 4 6 8\nβ−0.050.000.050.100.15C(β,τ)\nτ= 0\nτ= 5\nτ= 10\nτ= 15\nτ= 20\nFigure 9: Comparison between the theoretical expression for C(β, τ) (solid lines) (92)\nwith the numerical simulation (symbols) for different values of τ, keeping k= 2.0,\nDR= 0.1,γ= 0.01,v0= 1 and N= 20 fixed.\n•Variance: Since the mean separation vanishes, i.e., ⟨yβ(t→ ∞ )⟩= 0, the variance is\ngiven by C(0,0). Substituting β= 0 and τ= 0 in (92) we get,\nC(0,0) =Deff\nNkN−1X\ns=11\n1 + 4 µsin2(πs/N ), (93)\nwhere Deffis given by (2) and\nµ=k\n(DR+ 2γ), (94)\ndenotes the ratio of the active time scale and the trap time scale. For any finite N,\nthe stationary state variance of yαcould be calculated by performing the summation in\n(93).\nNote that in the passive limit, v0→ ∞ and ( DR+2γ)→ ∞ while keeping Defffinite\nandµ→0. Consequently, in the passive limit, C(0,0) = ( Deff/k)(1−1/N)→Deff/k\nasN→ ∞ . Although the potential U({yα}) =k\n2PN−1\nα=0y2\nαsuggests a product measure\nequilibrium state,\nPeq({yα})∝exp [−U({yα})/Deff], (95)\nthe global constraintPN−1\nα=0yα= 0 weakly breaks the product measure, giving rise31\n0 5 10 15 20 25 30\nτ−0.20.00.20.40.60.8C(β,τ)\nβ= 0\nβ= 1\nβ= 5\nβ= 8\nFigure 10: Comparision between the theoretical expression for C(β, τ) (solid lines) (92)\nwith the numerical simulation (symbols) for different values of β, keeping k= 0.5,\nDR= 0.1,γ= 0.1,v0= 1 and N= 10 fixed\nto the 1 /Ncorrection in the variance. As expected, the correction disappears in the\nthermodynamic limit N→ ∞ .\nFor any finite activity µ, the summation over sin (93) can be converted to an\nintegration over q= 2πs/N in the large Nlimit. Then, we have,\nC(0,0) = lim\nt→∞\nN→∞\ny2\nα(t)\u000b\n=Deff\n2πkZ2π\n0dq\n1 + 4 µsin2(q/2)=Deff\nk√1 + 4 µ, (96)\nwhich implies that the typical fluctuation of the separations decreases with increasing\nactivity. It is interesting to compare the above result with that of a single DRABP in a\nharmonic trap of strength k, where the variance is given by Deff/(k(1 +µ)), which has\nthe same passive limit. It appears that, for large activity, the presence of the interaction\nallows for larger fluctuations.\n•Spatio-temporal correlation: Let us now consider the general spatio-temporal\ncorrelation function C(β, τ) in the thermodynamic limit. In the limit N→ ∞ the\nsummation in the expression (92) can be converted to an integral by taking q= 2πs/N ,\nC(β, τ) =v2\n0\n4πkZπ\n−πdqcos (βq)\u0014(DR+ 2γ)e−bqτ−bqe−(DR+2γ)τ\n(DR+ 2γ)2−b2\nq\u0015\n, (97)32\n100101\nτ0.10.20.3C(β,τ)N= 15\nN= 20\nN= 25\nFigure 11: Comparison between the integral expression for C(β, τ) (blue solid line) (97)\nwith the summation expression (symbols) (92) for different values of N, keeping k= 1,\nDR= 0.1,γ= 0.1,β= 1 and v0= 1. For any finite N, the integral (97) has a correction\n−Deff/(k N), we have added Deff/(k N) to the summation (92) in this plot.\nwhere bqis given by (28). Figure 11 shows how the summation expression (92)\napproaches the integral expression as Nis increased.\nThe integral is hard to compute explicitly for an arbitrary value of τ. Therefore,\nwe expand the integrand in Taylor’s series as a power of τand carry out the integral at\neach order. It turns out that it can be recast in the following form,\nC(β, τ) =∞X\nn=0Cn(β)\u0002\n(DR+ 2γ)τ\u0003n\nn!, (98)\nwhere, the coefficients Cn(β) are given below. First, for n= 0, we have,\nC0(β)≡C(β,0) =Deff\n2πkZπ\n−πdqcosβq\n4µsin2q/2 + 1(99)\n=Deff\nk(1 + 4 µ)3˜F2\u00121\n2,1,1; 1−β,1 +β;4µ\n1 + 4 µ\u0013\n, (100)\nwhere µis defined by (94). For a given β, the above expression of C0(β) has a series\nexpansion in powers of µ, starting from µβ. Next, it is straightforward to see that the33\ncoefficient of τis zero in the integrand in (97), resulting in C1(β) = 0. For n≥2, we\nhave,\nCn(β) =−(−1)nv2\n0\n4πkZπ\n−πdqcos (βq)Sn(q), (101)\nwhere,\n(DR+ 2γ)nSn(q) = (DR+ 2γ)bq(bq)n−1−(DR+ 2γ)n−1\nb2\nq−(DR+ 2γ)2. (102)\nAfter further algebraic manipulation, the integral can be performed [see Appendix B],\nwhich yields\nC2(β) =C0(β)−Deff\nkδβ,0, (103)\nand for n≥3,\nCn(β) =\n\n1 + (−1)n\n2C0(β)−(−1)n+βDeff\nkn−2X\nl=β1 + (−1)l+n\n2\u00122l\nl−β\u0013\nµlforβ≤n−2\n1 + (−1)n\n2C0(β) for β > n −2.\n(104)\nFrom the first line of (104), it is apparent that the summation over lcontributes only\nwhen l+nis even. For example, for n= 3, and 4 we have,\nC3(β) =Deff\nk(2δβ,0−δβ,1)µ,\nC4(β) =C0(β)−Deff\nk\u0002\nδβ,0+ (6δβ,0−4δβ,1+δβ,2)µ2\u0003\n.(105)\nIn fact, the summation over lin (104) can also be carried out, giving a closed-form\nexpression for Cn(β) for n≥3 and β≤n−2 as,\nCn(β) =1 + (−1)n\n2C0(β)\n−(−1)n+βDeff\n2k(\nµβ\u0014\n2F1\u0012\nβ+1\n2, β+ 1; 2 β+ 1; 4 µ\u0013\n+ (−1)n+β\n2F1\u0012\nβ+1\n2, β+ 1; 2 β+ 1;−4µ\u0013\u0015\n+µn−1\u00122(n−1)\nn−β−1\u0013\u0014\n3F2\u0012\n1, n−1\n2, n;n−β, n+β;−4µ\u0013\n−3F2\u0012\n1, n−1\n2, n;n−β, n+β; 4µ\u0013\u0015)\n. (106)\nFor any finite τ, the correlation function C(β, τ) can be evaluated up to arbitrary\naccuracy by taking sufficient number of coefficients Cn(β) in (98). However, this\nexpansion is not suitable for extracting the asymptotic large τbehavior of C(β, τ).34\n100101100110120130140150160\n(a)C(β,τ)\nτβ= 1\nβ= 2\nβ= 3\n10110210310−310−210−1\nt−1/2(b)C(β,τ)\nτβ= 1\nβ= 3\nβ= 5\nβ= 8\nFigure 12: Behaviour of C(β, τ) for different values of βat (a) short-times with k= 0.01,\nDR= 0.005, γ= 0.0025, and at (b) late-times with k= 1, DR= 0.5,γ= 0.25. The\nsymbols are obtained by numerically integrating (97). The dotted lines in (a) plot (98),\nkeeping the first three terms. On the right panel (b), the dotted lines plot the function\ngiven by (107). The dash-dotted line indicates the τ−1/2power law tail. We have taken\nv0= 1 for both the plots.\nThis can instead be done by taking τ≫ {(DR+ 2γ)−1, k−1}, where e−(DR+2γ)t→0\nande−bqτ→e−kq2τin (97). Consequently, the denominator in (97) can be further\napproximated by ( DR+ 2γ)2−b2\nq→(DR+ 2γ)2. This yields,\nC(β, τ) =Deff\n2πkZπ\n−πdqcos (βq)e−kq2τ=Deff\n2k√\nπke−β2/(4kτ)\n√τ. (107)\nIn figure 12, we compare the short-time and long-time behavior obtained from (98),\nby taking first three terms, and (107) respectively, with the exact integral expression\n(97) for spatio-temporal correlation.\n•Spatial correlation: The equal-time correlation C(β,0) is given by (100). It is\ninteresting to investigate the behavior of C(β,0) in the strongly active limit µ≫1.\nHowever, it is not straightforward to obtain the behavior of (100) in this limit. Hence,\nwe extract the limiting behavior from the integral in (99), which is dominated by the\ncontribution from the region |q| ≤1/√µ. Therefore, for large µ, the integral can be\napproximated by\nC(β,0)≃Deff\n2πkZ∞\n−∞dqcosβq\nµq2+ 1=Deff\nkexp (−|β|/√µ)\n2√µ. (108)\nOn the other hand, in the passive limit µ→0, we get from (99),\nC(β,0) =Deff\nkδβ,0, (109)35\n0 2 4 6 8 10\nβ10−710−610−510−410−310−210−1C(β,0)µ= 0.5\nµ= 1.0\nµ= 2.0\nFigure 13: Comparison between the exact expression (100) with the approximate\nexpression (108) for C(β,0) for different values of µ, keeping k= 1, Deff= 1, and\nv0= 1 fixed. The solid line shows the approximate expression, while the symbols shows\nthe exact expression evaluated at integer values ( β= 0,1, . . . , 10).\nconsistent with the product measure equilibrium state (95) obtained in the\nthermodynamic limit N→ ∞ . Interestingly, although (108) is derived using the limit\nµ≫1, taking µ→0 in (108) we get (109), which is also consistent with (96) in large N\nlimit. We see in the thermodynamic limit, the spatial correlation decays exponentially\nwith βin the active limit. In figure 13 we compare the approximate expression (108)\nwith the exact expression (100) evaluated numerically and find very good agreement for\nlarge µ.\n8. Conclusions\nWe investigate the dynamical behavior of a harmonic chain of direction reversing active\nBrownian particles, by characterizing the variance of the position of the tagged particle\nand the statistics of separation between two consecutive particles. There are three\nintrinsic time scales in the system ( k−1, D−1\nR, γ−1), given by the coupling strength k,\nrotational diffusion constant DR, and the direction reversal rate γ. The interplay of these\ntime-scales gives rise to multiple dynamical regimes [see Tables 1 and 2], characterized\nby qualitatively different behavior of the fluctuations, which we analyze in this paper. In36\nthe long times tmuch larger than the intrinsic time-scales, the system behaves similar to\na Rouse polymer— a harmonic chain of Brownian particles—exhibiting a√\ntsubdiffusive\nbehavior of the tagged particle variance. However, in all other dynamical regimes, the\nsignature of activity becomes apparent, giving rise to anomalous behavior tνwith a\nrange of values for νin the different regimes. In particular, we consider two different\nscenarios— one in which all the particles start with a quenched initial orientation θ0, and\nanother where initial orientations of the particles are chosen randomly from a uniform\ndistribution in [0 ,2π], independent of each other.\nFor the case of quenched initial orientation and well-separated time-scales, three\nlimiting scenarios emerge, depending on the relative strengths of the coupling time scale\nk−1, and the active time scales D−1\nRandγ−1. We refer to them as strong coupling\n[k≫(DR, γ)], weak coupling [ k≪(DR, γ)], and moderate coupling [ γ≪k≪DR\norDR≪k≪γ] limits. For a given order of coupling, there are four dynamical\nregimes, which we refer to as short-time, early-intermediate, late-intermediate, and\nlong-time regimes [see Table 1]. The short-time ( t≪ {k−1, D−1\nR, γ−1}) and long-\ntime ( t≫ {k−1, D−1\nR, γ−1}) behaviors are found to be independent of the coupling\nlimits. In the short-time regime, the position variance has a superdiffusive t3growth,\nsimilar to that of a single DRABP, as expected. On the other hand, the tagged particle\nexhibits the universal√\ntsubdiffusive behavior observed in single-file systems at long\ntimes. Contrarily, in the intermediate dynamical regimes, we observe different growth\nexponents in the three coupling limits. In the strong coupling limit, the variance grows\nast5/2during the early-intermediate regime, followed by a√\ntbehavior in the late-\nintermediate regime. The interplay of rotational diffusion and direction reversal gives\nrise to t5/2behavior during the early intermediate regime, which is also observed in\nactive Ornstein–Uhlenbeck particle (AOUP) in the intermediate regime of the strong\ncoupling limit [58]. In the moderate coupling limit, the variance grows as tin the\nearly intermediate regime, which crosses over to a subdiffusive√\ntgrowth in the late\nintermediate regime. In the weak coupling limit, irrespective of the relative order of\nactivity time scale, the variance grows linearly with time [see Table 1].\nFor annealed initial orientation, interestingly, DRandγappear together giving rise\nto one activity time-scale ( DR+ 2γ)−1in addition to the coupling time scale k−1. In\nthis case, there are only two limiting scenarios possible, namely, the strong coupling\nlimit, given by k≫(DR+ 2γ), and the weak coupling limit, given by k≪(DR+ 2γ).\nEvidently, there are three dynamical regimes in each coupling limit [see Table 2]. The\nshort-time ( t≪ {k−1,(DR+ 2γ)−1}) and long-time ( t≫ {k−1,(DR+ 2γ)−1}) behaviors\nare again independent of the coupling limits. In the short-time regime, the position\nvariance of the tagged particle grows as t2, similar to that of an independent DRABP\nwith annealed initial orientation. On the other hand, as expected, at long times the\nvariance grows subdiffusively as√\nt. The behavior in the intermediate regime depends on\nthe relative ordering of the time-scales— in the strong coupling limit, the variance grows\nsuperdiffusively as t3/2, while in the weak coupling limit, a linear growth is observed.\nWe also explore the finite-size effect on the tagged particle position variance. For37\na finite but large system size N, there is an additional time scale tN=N2/kand the\nfinite size effects appear for t≫tN. In this regime, the tagged particle exhibits center\nof mass motion with variance growing linearly with time, as 2( Deff/N)t. The crossover\nfrom the thermodynamic limit t≪tN, to the finite size dominated regime t≫tNis\ncaptured by a scaling behavior of the variance Deffp\nt/k f (t/tN), where the crossover\nfunction f(z) goes to a constant as z→0 and it diverges sublinearly as√zfor large z.\nFinally, we analyze the statistics of the separation yαbetween two consecutive\nparticles αandα+ 1. For a thermodynamically large harmonic chain of passive\nBrownian particles, the separation variables reach an equilibrium state given by the\nBoltzmann distribution, which has a Gaussian product measure. We find that,\nfor a harmonic chain of DRABPs, the activity breaks this product measure in the\nnonequilibrium stationary state, giving rise to nontrivial spatio-temporal correlations\namong the separation variables. First, we derive a series expansion of the spatio-\ntemporal correlation function C(β, τ) = lim t→∞⟨y0(t)yβ(t+τ)⟩in powers of τ. We\nalso show that for large τ,C(β, τ)∼e−β2/(4kτ)/√τ. Futhermore, we show that for large\nactivity, µ=k/(DR+ 2γ)≫1, the spatial correlation C(β,0) decays exponentially.\nAppendix A. Computation of effective noise correlation\nIn this Appendix, we provide the detailed calculation for the auto-correlation of the\neffective noise ξα(t) defined in (6). Since the σαandθαprocesses are independent, it\nsuffices to compute the correlations of σα(t) and cos ( θα(t)) separately and take the\nproduct to get correlation of ξα(t). The propagator for the θα(t) and σ(t) processes are\ngiven by,\nPθ(θα, t|θ0,0) =1√4π D Rtexp\u0014\n−(θα−θ0)2\n4DRt\u0015\n, (A.1)\nand\nPσ(σα, t|σ0,0) =1\n2\u0014\n1 +σασ0e−2γt\u0015\n. (A.2)\nUsing the above propagators, we get the mean and the auto-correlation of the noise for\nboth quenched and annealed initial orientations below.\n(i)Quenched initial orientation : From (A.1) and (A.2), we respectively get,\n\ncosθα(t)\u000b\n=Z∞\n−∞cosθα√4π D Rtexp\u0014\n−(θα−θ0)2\n4DRt\u0015\ndθα= cos θ0e−DRt, (A.3)\nand\n\nσα(t)\u000b\n=X\nσα=±1σα\n2\u0002\n1 +σαe−2γt\u0003\n=e−2γt. (A.4)38\nTherefore, from (6), we have,\n\nξα(t)\u000b\n=v0\ncosθα(t)\u000b\nσα(t)\u000b\n=v0cosθ0e−(2γ+DR)t. (A.5)\nThe two-time auto-correlations can also be calculated similarly, yielding,\n\ncosθα(t1) cosθα(t2)\u000b\n=e−DR|t1−t2|+ cos 2 θ0e−DR(t1+t2+2 min[ t1,t2]), (A.6)\nσα(t1)σα(t2)\u000b\n=e−2γ|t2−t1|, (A.7)\nConsequently,\n\nξα(t1)ξα(t2)\u000b\n=v2\n0\n2\u0002\ne−(DR+2γ)|t1−t2|+ cos 2 θ0e−DR(t1+t2+2 min[ t1,t2])−2γ|t1−t2|\u0003\n.(A.8)\n(ii)Annealed initial orientation : To find the correlations for annealed initial orientations,\nwe average over θ0with respect to a uniform distribution in [0 ,2π] in (A.5) and (A.8).\nNoting ⟨cosθ0⟩=⟨cos 2θ0⟩= 0,\n\nξα(t)\u000b\n= 0 and\nξα(t1)ξα(t2)\u000b\n=v2\n0\n2e−(DR+2γ)|t1−t2|. (A.9)\nFrom (A.5), (A.8), and (A.9), it is clear that G(t1, t2)≡\nξα(t1)ξα(t2)\u000b\n−\nξα(t1)\u000b\nξα(t2)\u000b\n, is given by (21) and (22) for the quenched and annealed initial\norientations respectively.\nAppendix B. Computation of Cn(β)\nIn (98), we have expressed the spatio-temporal correlation function of the separation\nvariables C(β, τ) in a power series of τwith coefficients Cn(β). The goal of this appendix\nis to evaluate Cn(β), starting from (101). These coefficients involve Sn(q), defined in\n(102), can be expressed as\nanSn(q) =n−2X\nr=0br+1\nqan−r−1\nbq+a,with a=DR+ 2γ. (B.1)\nWriting br+1\nq= (bq+a−a)r+1and performing binomial expansion in powers of bq+a\nandawe get,\nanSn(q) =n−2X\nr=0r+1X\nm=0(−1)r+1−m\u0012r+ 1\nm\u0013\n(bq+a)m−1an−m. (B.2)\nSeparating the m= 0,1 terms and performing another binomial expansion of ( bq+a)m−1\nform > 1 we get,\nanSn(q) =−(−a)n\nbq+a·1 + (−1)n\n2+(−1)nan−1\n4\u0002\n1 + (2 n−1)(−1)n\u0003\n−n−2X\nr=1r+1X\nm=2m−1X\nl=0(−1)r−m\u0012r+ 1\nm\u0013\u0012m−1\nl\u0013\nbl\nqan−l−1. (B.3)39\nNow, inserting the above expression of Sn(q) in (101) and carrying out the integration\noverqwe get,\nCn(β) =1 + (−1)n\n2C0(β)−\u0014\nn−1−(−1)n\n2\u0015Deff\n2kδβ,0+˜Sn, (B.4)\nwhere we used,\nZπ\n−πdqcos (βq)bl\nq=\n\n(−1)β(2l)! 2πkl\n(l−β)!(l+β)!forl≥β,\n0 for l < β,(B.5)\nand\n˜Sn=Deff\nkn−2X\nr=1r+1X\nm=2m−1X\nl=0(−1)n+β+r−m\u0012r+ 1\nm\u0013\u0012m−1\nl\u0013\u00122l\nl−β\u0013\nµl. (B.6)\nTo evaluate this sum, it is useful to change the order of the summations, which yields,\n˜Sn= (−1)n+βDeff\nkn−2X\nl=0\u00122l\nl−β\u0013\nµln−2X\nr=1r+1X\nm=2(−1)r+m\u0012r+ 1\nm\u0013\u0012m−1\nl\u0013\n, (B.7)\nwhere\u0000m−1\nl\u0001\n= 0 for m < l + 1. Furthermore, it is convenient to separate the l= 0 term\nas,\n˜Sn= (−1)n+βDeff\nk\"n−2X\nl=1\u00122l\nl−β\u0013\nµln−2X\nr=1r+1X\nm=l+1(−1)r+m\u0012r+ 1\nm\u0013\u0012m−1\nl\u0013\n+δβ,0n−2X\nr=1r+1X\nm=2(−1)r+m\u0012r+ 1\nm\u0013#\n. (B.8)\nThe summations over randmcan be performed explicitly, yielding,\n˜Sn= (−1)n+β+1Deff\nk\"n−2X\nl=11 + (−1)l+n\n2\u00122l\nl−β\u0013\nµl+1\n4δβ,0\u0000\n1−(2n−3)(−1)n\u0001#\n.\n(B.9)\nCombining (B.4) and (B.9) we get,\nCn(β) =1 + (−1)n\n2\u0014\nC0(β)−Deff\nkδβ,0\u0015\n−(−1)n+βDeff\nkn−2X\nl=11 + (−1)l+n\n2\u00122l\nl−β\u0013\nµl,\n(B.10)\nwhere,\u00002l\nl−β\u0001\n= 0 for l < β . Note that the term containing the summation over lcon-\ntributes only for n≥3. For n= 2, we have (103) whereas for n≥3 we get (104) in the\nmain text.40\nAppendix C. Some useful integrals and sums\nFor the convenience of the readers, we present a collection of integrals and summations,\nthat appear in the main text, below.\nZ∞\n−∞dze−(z2+a2)\na2+z2=π\na(1−erf(a)), (C.1)\nZ∞\n−∞dz(e−a2e−z2−e−2z2)\na2−z2=π\nae−2a2h\nerfi(a)−erfi(√\n2a)i\n, (C.2)\nZ∞\n−∞dze−a2−e−2z2\na2−2z2=−π√\n2ae−a2erfi(a), (C.3)\nZ∞\n−∞dze−(z2+b2)−e−a2\na2−b2−z2=π√\nb2−a2e−a2erf\u0010√\nb2−a2\u0011\n, (C.4)\nZ∞\n−∞dz\"\ne−a−e−z2\na−z2#2\n=−πe−2a\na3/2\"\n(2a+ 1) erfi\u0000√a\u0001\n−\u0012\n2a+1\n2\u0013\nerfi\u0010√\n2a\u0011#\n−√\n2π\na\u0010\n1−√\n2e−a\u0011\n,(C.5)\n∞X\nm=0Γ\u0000\nm+1\n2\u0001\n√π m!am=1√1−a, (C.6)\n∞X\nm=0Γ\u0000\nm+n+1\n2\u0001\n√π(m+n)!am=Γ(n+ 1/2)√π n!2F1\u0012\n1, n+1\n2; 1 + n;a\u0013\n. (C.7)\nIntegrals (C.1)-(C.5) are used in Sec. 4.3 to derive (39) in the strong coupling limit,\nby taking appropriate limit in (32). Summations (C.6)-(C.7) is used in Sec. 4.4, i.e.,\nweak coupling limit, to evaluate the sum in (49).\nReferences\n[1] Jepsen D W 1965 J. 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E 104L012601\n[62] Santra I, Basu U and Sabhapandit S 2021 Soft Matter 1710108\n[63] Zygmund A 2002 Trigonometric Series v. 1 (Cambridge University Press)"    },    {        "title": "2402.11970v1.Port_Hamiltonian_modeling_and_control_of_a_curling_HASEL_actuator.pdf",        "content": "Port-Hamiltonian modeling and control of\na curling HASEL actuator⋆\nNelson Cisneros ,Yongxin Wu ,Kanty Rabenorosoa ,\nYann Le Gorrec\nFEMTO-ST institute, UBFC, CNRS, Besan¸ con, France.(emails:\nnelson.cisneros@femto-st.fr, yongxin.wu@femto-st.fr,\nrkanty@femto-st.fr, yann.le.gorrec@femto-st.fr)\nAbstract: This paper is concerned with the modeling and control of a curling Hydraulically\nAmplified Self-healing Electrostatic (HASEL) actuator using the port-Hamiltonian (PH) ap-\nproach. For that purpose, we use a modular approach and consider the HASEL actuator as an\ninterconnection of elementary subsystems. Each subsystem is modeled by an electrical compo-\nnent consisting of a capacitor in parallel with an inductor connected through the conservation\nof volume of the moving liquid to a mechanical structure based on inertia, linear, and torsional\nsprings. The parameters are then identified, and the model is validated on the experimental\nsetup. Position control is achieved by using Interconnection and Damping Assignment-Passivity\nBased Control (IDA-PBC) with integral action (IA) for disturbance rejection. Simulation results\nshow the efficiency of the proposed controller.\nKeywords: Soft actuator, HASEL actuator, Port-Hamiltonian systems, IDA-PBC design.\n1. INTRODUCTION\nIn recent years, one of the most interesting technologies\nthat have been developed for soft robotic applications\nis the Hydraulically Amplified Self-healing Electrostatic\n(HASEL) actuator (Acome et al., 2018).\nHASEL actuators blend the advantages of Dielectric Elas-\ntomer Actuators (DEAs) and fluid-driven soft actuators,\ncombining the convenience of electrical control, excellent\nelectromechanical performances, extensive design flexibil-\nity, and various actuation modes (Rothemund et al., 2021).\nThere are different types of HASEL actuators, such as\npeano, planar, elastomeric donut, quadrant donut, high-\nstrain peano, and curling actuators. Some interesting\napplications of HASEL actuators can be found in the\nliterature: a soft gripper for aerial object manipulation\n(Kim and Cha, 2021), an actuator powering a robotic\narm (Acome et al., 2018), an electro-hydraulic rolling soft\nwheel (Ly et al., 2022), a peano actuator for enhanced\nstrain, load, and rotary motion (Tian et al., 2022) and soft-\nactuated joints based on the hydraulic mechanism used in\nspider legs (Kellaris et al., 2021).\nIn this paper, we consider, as a benchmark the design,\nmodeling, and control of a simple curling HASEL actuator\nthat can be used as a basic element in more complex\nrobotic structures such as robotic hands or soft grip-\npers. The considered curling HASEL actuator is based\non a strain limiting layer that is added to the traditional\nHASEL mechanism (Acome et al., 2018) to change its mo-\ntion from linear to angular deformation (Rothemund et al.,\n2021). It is important to derive a reliable model represent-\ning the system’s dynamics to control the actuator. Many\n⋆This work is supported by the EIPHI Graduate School (contract\nANR-17-EURE-0002).papers dealing with the modeling of HASEL actuators\nhave been recently proposed in the literature. In (Volchko\net al., 2022), Dynamic Mode Decomposition with Control\n(DMDc) is applied to derive a linear model, approximating\nthe system’s dynamics. In (Hainsworth et al., 2022), a non-\nlinear reduced-order mass-spring-damper (MSD) model\nfor a linear HASEL actuator is proposed. However, these\nworks do not take into account the non-linear behavior\n(such as the drift effect) or electrical dynamics of HASEL\nactuators in the model, which can make the control design\nmore challenging and difficult to implement in a real-world\napplication.\nPort-Hamiltonian (PH) formulations are particularly well\nadapted to represent multi-physical systems. The PH ap-\nproach is then an excellent candidate to represent the\nconsidered dynamics of the HASEL actuator. Intercon-\nnection and Damping Assignment-Passivity Based Control\n(IDA-PBC) serves as a highly effective tool for generating\nasymptotically stabilizing controllers for Port-Controlled\nHamiltonian (PCH) models. (Ortega et al., 2002). Pre-\nvious works used PHS to model soft robots with energy\nshaping and IDA-PBC controllers, showing good results.\nIn (Franco et al., 2021b) and (Franco et al., 2021a), energy\nshaping controllers are used to control the position of\na soft continuum manipulator with a large number of\ndegrees of freedom (DOF). In (Ayala et al., 2022), the\nIDA-PBC method has been successfully used to control\na nonlinear Cosserat rod model using an early lumping\napproach. More recently, in (Yeh et al., 2022), a PH for-\nmulation of a one DOF of HASEL planar actuator with\nposition control using an IDA-PBC with Integral Action\n(IA) has been proposed. Compare to (Yeh et al., 2022) we\nconsider here an actuator with bending motion instead of\nlinear deformation. This introduces nonlinearities in thearXiv:2402.11970v1  [math.DS]  19 Feb 2024interconnection matrix. Furthermore, we aim to capture\nthe end position drift effect. The main contributions of\nthis paper are:\n•We modeled a curling HASEL actuator using the\nport-Hamiltonian approach to capture the actuator’s\nelectrical and mechanical dynamics.\n•We identified the model compared with experimental\ndata and validated it with different input voltages.\n•We designed an IDA-PBC controller with integral\naction to control the end point position of the curling\nHASEL actuator and reject the input disturbances.\nThis paper is organized as follows: Section 2 presents the\nexperimental setup of the curling HASEL actuator and\nits modeling under the PH framework. The parameter\nidentification is detailed in Section 3. In Section 4, the\ncontroller design is presented. Section 5 presents the sim-\nulation results, and the conclusions are given in Section\n6.\n2. CURLING HASEL ACTUATOR AND ITS PH\nMODELING\nIn this section, we first introduce the experimental curling\nHASEL actuator that is used as a benchmark. We de-\nscribe the setup and its working principle with reasonable\nhypotheses that will be used for modeling purposes. We\nthen derive the PH model for this actuator.\n2.1 Experimental setup\nThe experimental setup is shown in Fig. 1. To measure the\nposition, we use a profile laser sensor, Keyence LJ-V7080.\nWe use the high voltage amplifier Trek model 610E. The\nHASEL actuator used in this work comes from Artimus\nRobotics . We use a dSPACE card CLP1104 to receive\nand send signals from/to the laser position sensor and\nhigh voltage amplifier. The pictures of Fig. 1 (right) show\nFig. 1. Experimental setup laser sensor and curling\nHASEL.the deformation of the actuator without (upper figure)\nand with an applied voltage (lower figure). Applying high\nvoltage, the actuator can achieve a horizontal displacement\nof approximately 3 cm.\n2.2 Curling HASEL actuator description and hypothesis\nThe curling HASEL actuator consists of a polymer shell\nfilled with dielectric liquid and half covered by a pair of\nelectrodes attached to a strain-limiting layer to get the\nbending motion. When an electric field is applied to the\nelectrodes, it creates Maxwell stress acting on the shell\nthat pushes the dielectric liquid inside the shell. This\nhydraulic pressure changes the shape of the shell and,\nfrom the strain-limiting layer, induces the bending of the\nactuator (Rothemund et al., 2021).\nWe consider that the actuator depth is uniform, so we\nproceed to a two-dimensional analysis. We model the\nactuator using interconnected subsystems. We separate\neach subsystem model into a chamber and a shell (cf Fig.\n2).\nFig. 2. Basic subsystem. Left: electrodes are totally un-\nzipped. Right: Electrodes are partially zipped when\nvoltage is applied. The shell is deformed.\nThe chamber is the area between the electrodes whilst\nthe shell receives the dielectric liquid when the electrodes\nare zipped. The total volume (the shell’s volume plus\nthe chamber’s volume) is considered constant, and the\ndielectric liquid is incompressible. The bending of the\nbottom film is modeled as a torsional spring. The top\nfilm of the shell is considered to be elongable and contains\nmechanical energy. The elongation is modeled as a linear\nspring. The shell will take on a specific shape based on the\nvolume transferred from the chamber to the shell, resulting\nfrom the zipping of the electrodes, which takes place\nwhen high voltage is applied. The electrodes are modeled\nas a variable capacitor. The model considers a variable\nlength of the zipped electrodes. The distance between the\nunzipped electrodes part is considered constant, see Fig.\n2.\n2.3 Geometric relations\nIn this part, we present the relations existing between\nthe angle θand the length of the zipped electrodes le.\nThen, we derive the actuator position h(θ) from θ, i.e.,\nthe displacement of the end position of the actuator.\nIt is crucial to obtain a relation between θandlebecause\nthe electrode’s capacitance depends on le. Therefore, it\nallows us to relate the electrical charge, that depends\non the capacitance, with the derivative of the electricalenergy respecting the angle θ, joining the electrical and\nthe mechanical part.\nWe represent the chamber as a rectangular area and the\nshell is modeled as two symmetric triangles. Fig. 2 shows\nthe model variables of a basic subsystem.\nThe area inside the shell is equal to:\nAs=1\n4lpLvsin(δ1) (1)\nwith\nδ1=π+θ\n2−sin−1\u0012Lv\nlpsin\u0012π−θ\n2\u0013\u0013\n. (2)\nThe total area is:\nAT=As+Xh(Le−le), (3)\nwhere Lvandlpare the lengths of the bottom and the top\nfilm. Xhis the height of the chamber and Leis the length\nof the chamber.\nThe zipped electrodes length is then:\nle=Le−1\nXh\u0012\nAT−Lvlp\n4sin(δ1)\u0013\n. (4)\n2.4 Curling HASEL port-Hamiltonian model\nThis section presents the port-Hamiltonian model for the\ncurling HASEL actuator. PH formulations consist in using\nthe energy variables as state variables and in writing the\ndynamics of the system on the form (van der Schaft, 2000):\n˙x= [J(x)−R(x)]∂H\n∂x(x) +g(x)u;\ny=gT(x)∂H\n∂x(x),(5)\nwhere J(x) =−JT(x) is the interconnection matrix,\nR(x) = RT(x)≥0 is the dissipation matrix and His\nthe total energy of the system (Hamiltonian).\nBy combining basic subsystems, we can represent the\noverall dynamic behavior of the HASEL actuator. In what\nfollows we consider four subsystems (cf Fig. 3) but the\nmodel can be extended to n∈Nsubsystems. We consider\nthat the subsystems share the same input voltage. The\ntotal energy of the system is given by:\nH(θ, lp, p, ϕ, Q ) =Hθ(θ) +Hlp(lp) +Hg(θ)+\nHp(p) +Hϕ(ϕ, Q)+HQ(Q, ϕ, θ, l p),(6)\nwhere\nHθ=1\n2nX\ni=1Kbiθ2\ni=1\n2θTKbθ (7)\nis the potential energy where Kb= diag[ Kb1Kb2. . . K bn]\nis the stiffness matrix and θ= [θ1θ2. . . θ n] represents\nthe angular vector of each subsystem, the second term is\nthe potential energy related to the linear springs:\nHlp=1\n4nX\ni=1Ki(lpi−Lpi)2=1\n4(lp−Lp)TK(lp−Lp),(8)hUin\nac1\na1\nm1\nm2\nm3\nm4θ1\nθ2\nθ3\nθ4hvert\nFig. 3. Four interconnected subsystems. The same voltage\nis applied to the entire system.\nwhere K= diag[ K1K2. . . K n] and lT\np= [lp1lp2. . . l pn],\nthe third term is the total potential energy related to\ngravity:\nHg=nX\ni=1Hgi, (9)\nand the fourth term the kinetic energy:\nHp=1\n2pTM−1p, (10)\nwhere Mis the matrix of inertia and pis the vector of\nangular momentum pT= [p1p2. . . p n].\nThe electrical energy has two components: the energy\nrelated to the magnetic flux that allows us to represent\nthe drift effect and the energy related to the charge. The\ninductor discharges the capacitor over time.\nHϕ=1\n2nX\ni=1ϕ2\ni\nLi=1\n2ϕTL−1ϕ (11)\nThe energy stored in the capacitor is:\nHQ=1\n2nX\ni=1Q2\ni\nCsi=1\n2QTC−1Q, (12)\nwhere ϕT= [ϕ1ϕ2. . . ϕ n] is the magnetic flux, L\nis the inductance of the equivalent electric circuit L=\ndiag[L1L2. . . L n].C= diag[ Cs1Cs2. . . C sn] is the\ncapacitance of the equivalent electric circuit and Q=\n[Q1Q2. . . Q n]Tis the charge. The capacitance of a\nsubsystem is Csi=C1i+C2iwhere C1i=ϵ0ϵrwlei\n2tis\nthe capacitance of the zipped part and C2i=ϵ0ϵrw(Le−lei)\n2t+Xhthe capacitance of the unzipped part.\nThe input gain is gaT= [ga1ga2... ga n]. To capture\nthe system’s nonlinearities, the input gain is a nonlinear\nfunction that depends on the angular position gai=\nγ1cos (γ2θi).\nThe conductance of the equivalent electric circuit is ¯R=\ndiag[1\nR11\nR2...1\nRn]. The damping coefficient of the system\nisb= diag[ b1b2... b n]. The resistance associated to\nthe inductance is rL= diag[ rL1rL2... rLn]. We definethe term d= diag\u0010\n2As\nlp\u0011\n. The proposed port-Hamiltonian\nmodel of the curling HASEL actuator is then:\n\n˙θ\n˙lp\n˙p\n˙ϕ\n˙Q\n\n|{z}\n˙x=\n0 0 I0 0\n0 0 d0 0\n−I−d−b0 0\n0 0 0 −rLI\n0 0 0 −I−¯R\n\n| {z }\nJ−R\n∇θH\n∇lpH\n∇pH\n∇ϕH\n∇QH\n\n|{z}\n∇xH+\n0\n0\n0\n0\n¯Rga(θ)\n\n|{z}\ngUin; (13)\ny= (¯Rga(θ))TC−1Q|{z }\ngT∇xH.\nThe output y=ieis the current that is power-conjugated\nto the input voltage. The energy balance equation can be\nwritten as:\n∂H\n∂t=−∂HT\n∂xR∂H\n∂x+yTu;\n∂H\n∂t≤yTu=ieUin.(14)\nThe displacement for ninterconnected subsystems is com-\nputed as\nh(θ) = (Lv+Le)\nnX\ni=1sin(iX\nj=1θj)\n (15)\n3. MODEL IDENTIFICATION AND VALIDATION\nWe identify the key parameters of the system using the\nexperimental data obtained from the experimental setup\nof Fig. 2.1. The Levenberg–Marquardt algorithm is used to\nfind the parameters Kb,b,L,γ1andγ2. The identification\nresults are shown in Fig. 4 with a fitting of 90 .7%. Two\ndatasets are used to validate the obtained parameters, one\nwith negative inputs and another with positive inputs, as\nshown in Fig. 5.\n0 5 10 15\nTime (seconds)01234h [cm]2 3 422.53\n(a)\n0 5 10 15\nTime (seconds)-6-4-20Voltage [KV] (b)\nFig. 4. (4a) Model identification, fitness: 90.7%. (4b) Input\nsignal.\nThe fitting between the model and the experimental data\nis computed using the normalized root mean squared error\n(NRMSE):\nfit(i) =∥xref(:)−xdata(:)∥\n∥xref(:)−(xref(:))∥(16)\nwhere ∥.∥is the 2-norm of a vector.\nOne can see the identified model can cope with the\nmain dynamics of the considered Curling HASEL actuator\nwith a fitting comprised between 85% and 89 .33%. The\nidentified parameters are listed in Table 1.\n0 5 10 15\nTime (seconds)012345h [cm]\n6 8 102.53(a)\n0 5 10 15\nTime (seconds)-6-4-20Voltage [KV] (b)\n0 5 10\nTime (seconds)-2024h [cm]4 6 822.53\n(c)\n0 5 10\nTime (seconds)0246Voltage [KV] (d)\nFig. 5. (5a) Model validation, negative input fitness:\n85.46% (5b) Input signal. 5c) Model validation, pos-\nitive input fitness: 89.33% (5d) Positive input signal.\nWe can observe the model’s behavior in response to a\nvariation of 10% around the nominal values.\nSymbol Value Units Definition\nLp 0.015 m Length of top film\nLv 0.015 m Length of bottom film\nLe 0.015 m Length of electrodes\nXh 0.002 m Chamber high\nm 0.047 kg Mass\nϵr 2.2 F/m Relative permittivity\nϵ0 8.854x10−12F/m Vacuum permittivity\nw 0.05 m Actuator width\nt 18x10−6 m Film thickness\nRi 10 Ω Resistance\nrL 20 Ω Resistance\nL 150 F Inductance\nK 400 N/m Spring constant\nKb 0.202 Nm/rad Torsional spring constant\nb 0.0199 kgs Damping\nγ1 104.33 - Gain parameter\nγ2 7.67 - Gain parameter\nTable 1. Model parameters.\n4. POSITION CONTROL DESIGN\nIn this work, we aim to control the endpoint position of the\ncurling HASEL actuator (denoted by h). To this end, we\npropose an IDA-PBC design method. This method aims\nto find a state feedback control law β(x) to map the open-\nloop system to a desired closed-loop system of the form:\n˙x= (Jd−Rd)∇xHd (17)\nwith the desired interconnection and damping matrices Jd,\nRdand the desired energy function Hdin the closed-loop\nsystem. The control scheme is shown in Fig. 6.\nThe desired equilibrium points x∗= [θ∗, l∗\np,0, ϕ∗, Q∗]T\nwhich can be computed from the desired endpoint position\nh∗(solving (15)) and the state variables x= [θ, lp, p, ϕ, Q ]T\nare the controller ( β(x)) inputs. We define as desired\ninterconnection and dissipation matrices:IDA-PBC\nControllerβ(x) x\nh(θ)h x∗\nx∗(h∗)h∗\n(J−R)∇xH+β(x)θCurling HASEL\nFig. 6. Closed-loop scheme with the controller β(x) inputs\nare the state variables xand the desired values x∗.\nThe system input is the necessary voltage computed\nby the controller. h(x) (15) is the function that allows\nus to find the final position as a function of each link\nangle.\nJd−Rd=\n0 0 J13 0α1\n0 0 J23 0α2\n−J13−J23−r33J43α3\n0 0 −J430α4\n−α1−α2−α3−α4−r55\n, (18)\nwhere J13,J23,J43,α1,α2,α3andα4are the control de-\nsign parameters to be determined. The desired closed loop\nenergy function is defined from the desired equilibrium\nposition of the actuator as:\nHd= (θ−θ∗)T˜Kb(θ−θ∗) + (lp−l∗\np)T˜K(lp−l∗\np)\n+pTM−1p+ (ϕ−ϕ∗)T˜Kϕ(ϕ−ϕ∗)\n+(Q−Q∗)T˜KQ(Q−Q∗)(19)\nThe derivative of Hdwith respect to xis given by:\n∇xHd=\n˜Kb(θ−θ∗)\n˜K(lp−l∗\np)\nM−1p\n˜Kϕ(ϕ−ϕ∗)\n˜KQ(Q−Q∗)\n. (20)\nTo get the state feedback matching the closed-loop system\nwith a desired PH system ˙ x= (Jd−Rd)∇xHddefined\nabove we need to solve the following matching equation:\ng⊥[J−R]∇xH=g⊥[Jd−Rd]∇xHd, (21)\nwith g⊥is a full rank annihilator of the input matrix g.\nWe choose the annihilator as follows:\ng⊥=\n1 0 0 0 0\n0 1 0 0 0\n0 0 1 0 0\n0 0 0 1 0\n. (22)\nWe find J13,J23,J43as a function of α1,α2,α3andα4.\nJ13=diag((diag( M−1p))−1(M−1p−α1˜KQ(Q−Q∗))); (23)\nJ23=diag((diag( M−1p))−1(dM−1p−α2˜KQ(Q−Q∗))); (24)\nr33=diag((diag( M−1p))−1(∇θH+d∇lpH+bM−1p (25)\n+α3˜KQ(Q−Q∗)−J13˜Kb(θ−θ∗)−J23˜K(lp−l∗\np)\n+J43˜Kϕ(ϕ−ϕ∗))).\nJ43=diag((diag( M−1p))−1(rLL−1ϕ−C−1Q+α4˜KQ(Q−Q∗));\n(26)\nWe obtain the control law considering the next design\nparameters α1=I,α2= 0,α3=Iandα4= 0.\nβ(x) = ( ¯RgaT¯Rga)−1¯RgaT(−˜Kb(θ−θ∗)−M−1p\n−r55˜KQ(Q−Q∗) +L−1ϕ+ (¯RC−1Q)),(27)Given fixed values for l∗\npandθ∗from the desired endpoint\nposition h∗, we can determine Q∗from the model at steady\nstate.\n4.1 Disturbance rejection using Integral Action\nIn this subsection, we propose to improve the robustness of\nthe controller (27) to two types of disturbances acting on\nthe actuator using a structure-preserving integral action.\nThe first one is the unknown mass load, which can be\nregarded as the unactuated external force disturbance\n(du). The other one is the disturbance on the actuated\ninput ( da) i.e. the input voltage. Thus, the disturbed\nclosed-loop system with the previous proposed IDA-PBC\ncontrol law (27) can be written as:\n˙Q\n˙θ\n˙lp\n˙p\n˙ϕ\n= [Jd−Rd]\n∇QHd\n∇θHd\n∇lpHd\n∇pHd\n∇ϕHd\n+\nda\n0\n0\ndu\n0\n, (28)\nwhere the desired interconnection and the damping matrix\nare defined as:\nJd(x) :=\"\nJaa(x)Jau(x)\n−JT\nau(x)Juu(x)#\n=\n0−α1−α2−α3−α4\nα10 0 −J13 0\nα20 0 −J23 0\nα3J13J23 0−J43\nα4J13 0J43 0\n; (29)\nRd(x) :=\"\nRaa(x)Rau(x)\nRT\nau(x)Ruu(x)#\n=\nr550 0 0 0\n00 0 0 0\n00 0 0 0\n00 0r330\n00 0 0 0\n. (30)\nUsing the method described in (Ferguson et al., 2017) we\nchoose the new closed-loop Hamiltonian as:\nHcl=Hd+Kint\n2(Q−xc)2(31)\nand the new closed loop system can be derived as:\n\n˙xa\n˙xu\n˙xc\n= (Jcl−Rcl)\n∇xaHcl\n∇xuHcl\n∇xcHcl\n+\nda\n0\n0\ndu\n0\n0\n. (32)\nThe structure-preserving Integral Action (IA) controller is\nthen given by:\n˙xc=−Rc2(x)∇xaH+ (Jau+Rau)∇xuH,\nuint= [−Jaa+Raa+Jc1(x)−Rc1(x)−\nRc2(x)]∇xaH+ [Jc1(x)−\nRc1(x)]Kint(xa−xc) + 2Rau∇xuH;(33)\nwhere uintis the output of the IA controller. xcis the\nIA controller state. The actuated state is the charge Qm\nwhilst the unactuated states are the angle θ, the length lp,\nthe angular momentum p, and the magnetic flux ϕ.\nFrom Jau=−[α10α30],Jc1=Rc2=Rau= 0 and Rc1=\nr55we obtain the control law:\nuint=−r55Kint(Q−xc);\n˙xc=−(α1˜Kb(θ−θ∗) +α3M−1p).(34)\nwhere the design parameter Kintis chosen as a vector of\ndimensions 1 ×n. The dimension of the controller β(x)IDA-PBC\nController(J−R)∇xH+β(x)+dβ(x) x∗\nx\nIntegral\nactionh(θ)h\ndθ x∗(h∗)h∗\n+\n+Curling HASEL\nFig. 7. Closed-loop scheme with IA.\nis 1×1. The control scheme is shown in Fig. 7 and the\ninterconnection and damping matrix are given by:\nJcl:=\n0Jau0\n−JT\nauJuu0\n0 0 0\n;Rcl:=\"r550r55\n0r330\nr550r55#\n. (35)\n5. NUMERICAL SIMULATION\nIn this section, we validate the proposed control meth-\nods with numerical simulations. The parameters of the\nactuator are given in Table. 1. We implement the IDA-\nPBC controller (27) with the Integral action controller\n(34) to achieve the desired endpoint position of the curling\nactuator. To show the different closed loop dynamics per-\nformances, we vary the tuning parameter ˜Kbwhile keeping\nthe rest of the tuning parameters constant. The controller\nparameters values are r55= diag([0 .1 0.1 0.1 0.1]) and\nKint= [0.5 11 1 .2 0.5]. One can observe the endpoint\nregulation to the desired position in Fig. 8 using the IDA-\nPBC method and the rejection of external disturbances in\nFig. 9 with the IDA-PBC+IA method.\nFig. 8 represents the actuator displacement when ˜Kbis\ntuned, while the parameter ˜KQis fixed to a constant value.\nThe desired endpoint position is h∗(θ) = 2 cm. From\nsimulation results shown in Fig. 8, the response time of\nthe closed-loop system decreases when ˜Kbincreases, since\n˜Kbcan be seen as the actuator’s stiffness in the closed-loop\nsystem.\n0 1 2 3 4\nTime (seconds)012h [cm]S e t p o i n t\nK b = 1 0\nK b = 8\nK b = 6\nK b = 4\nK b = 2\nFig. 8. Position control keeping constant the parameters\n˜KQ= 1000, the set-point equal to 2cm and varying\nthe tuning parameter related with the desired angle\n˜Kb.\nFig. 9 shows the actuator displacement to different desired\nset points. The external unactuated disturbance du=\n−0.04Nm is added at 3s during 2s , and the actuated\ndisturbance da=−30V is added at 7s during 2s (0.3%\nof full scale according to the amplifier specifications). The\nproposed controller with IA can reject the disturbances,\nwhile the controller without IA can not reject these dis-\nturbances. From the simulation results shown in Fig. 9(b),\nit is seen that the applied controlled voltage on the closed\n0 2 4 6 8 10 12\nTime (seconds)00.511.522.5h [cm]\n3 4 51.9822.022.042.06(a)\n0 2 4 6 8 10 12\nTime (seconds)0510Voltage [KV]\n(b)\nFig. 9. (9a) Position control ˜KQ= 1000, ˜Kb= 10 and\nKint= [0 .5 11 1 .2 0.5]. The simulation presents\ndisturbances duanddaat 3sand 7 srespectively. (9b)\nIDA-PBC and IDA-PBC+IA control signals.\nloop actuator always remains below 10kV, which aligns\nwith the physical consistency of the experimental setup.\n6. CONCLUSION\nIn this paper we use the port-Hamiltonian framework\nto model and control a curling HASEL actuator. The\nactuator’s dynamics is divided into two components. The\nmechanical part of the actuator is characterized by linear\nand torsional springs, while the deformable capacitor and\nthe inductor represent the electrical part of the system, the\ncoupling being done through the conservation of volume\nof the overall system. This model can cope with the main\ndynamic behavior of the actuator with nonlinearities such\nas the drift effect. An IDA-PBC controller with the integral\naction is proposed to control the position of the actuator. It\nis shown that the actuator endpoint position follows the set\npoint and that we can adjust the dynamic performances\nby varying the tuning parameters of the controller. The\nuse of integral action has improved the robustness of the\nclosed-loop system against external disturbances.\nThe perspectives of this work are to implement and\nvalidate the proposed IDA-PBC controller with IA in\nthe experimental setup. Furthermore, we intend to model\nand control more complex structures and soft robots\nbased on HASEL actuators (e.g., scorpion, fish, human\nhands-inspired designs) with the interconnection of basic\nsubsystems.\nREFERENCES\nAcome, E., Mitchell, S.K., Morrissey, T.G., Emmett, M.B., Ben-\njamin, C., King, M., Radakovitz, M., and Keplinger, C. (2018).\nHydraulically amplified self-healing electrostatic actuators with\nmuscle-like performance. Science , 359(6371), 61–65.Ayala, E.P., Wu, Y., Rabenorosoa, K., and Le Gorrec, Y. (2022).\nEnergy-based modeling and control of a piezotube actuated opti-\ncal fiber. IEEE/ASME Transactions on Mechatronics , 1–11.\nFerguson, J., Donaire, A., Ortega, R., and Middleton, R.H. (2017).\nNew results on disturbance rejection for energy-shaping controlled\nport-Hamiltonian systems.\nFranco, E., Casanovas, A.G., Tang, J., y Baena, F.R., and Astolfi, A.\n(2021a). Position regulation in cartesian space of a class of inex-\ntensible soft continuum manipulators with pneumatic actuation.\nMechatronics , 76, 102573.\nFranco, E., Garriga-Casanovas, A., Tang, J., y Baena, F.R., and\nAstolfi, A. (2021b). Adaptive energy shaping control of a class of\nnonlinear soft continuum manipulators. IEEE/ASME Transac-\ntions on Mechatronics , 27(1), 280–291.\nHainsworth, T., Schmidt, I., Sundaram, V., Whiting, G.L.,\nKeplinger, C., and MacCurdy, R. (2022). Simulating electro-\nhydraulic soft actuator assemblies via reduced order modeling.\nIn2022 IEEE 5th International Conference on Soft Robotics\n(RoboSoft) , 21–28. IEEE.\nKellaris, N., Rothemund, P., Zeng, Y., Mitchell, S.K., Smith, G.M.,\nJayaram, K., and Keplinger, C. (2021). Spider-inspired electrohy-\ndraulic actuators for fast, soft-actuated joints. Advanced Science ,\n8(14), 2100916.\nKim, S. and Cha, Y. (2021). Double-layered electrohydraulic actua-\ntor for bi-directional bending motion of soft gripper. In 2021 18th\nInternational Conference on Ubiquitous Robots (UR) , 645–649.\nLy, K., Mayekar, J.V., Aguasvivas, S., Keplinger, C., Rentschler,\nM.E., and Correll, N. (2022). Electro-hydraulic rolling soft wheel:Design, hybrid dynamic modeling, and model predictive control.\nIEEE Transactions on Robotics .\nOrtega, R., van Der Schaft, A., Maschke, B., and Escobar, G. (2002).\nInterconnection and damping assignment passivity-based control\nof port-controlled Hamiltonian systems. Automatica , 38(4), 585–\n596.\nRothemund, P., Kellaris, N., Mitchell, S.K., Acome, E., and\nKeplinger, C. (2021). HASEL artificial muscles for a new gener-\nation of lifelike robots—recent progress and future opportunities.\nAdvanced Materials , 33(19), 2003375.\nTian, Y., Liu, J., Wu, W., Liang, X., Pan, M., Bowen, C., Jiang, Y.,\nSun, J., McNally, T., Wu, D., et al. (2022). Peano-hydraulically\namplified self-healing electrostatic actuators based on a novel\nbilayer polymer shell for enhanced strain, load, and rotary motion.\nAdvanced Intelligent Systems , 2100239.\nvan der Schaft, A. (2000). L2-gain and passivity techniques in\nnonlinear control . Springer.\nVolchko, A., Mitchell, S.K., Morrissey, T.G., and Humbert, J.S.\n(2022). Model-based data-driven system identification and con-\ntroller synthesis framework for precise control of siso and miso\nHASEL-powered robotic systems. In 2022 IEEE 5th International\nConference on Soft Robotics (RoboSoft) , 209–216. IEEE.\nYeh, Y., Cisneros, N., Wu, Y., Rabenorosoa, K., and Gorrec, Y.L.\n(2022). Modeling and position control of the HASEL actuator\nvia port-Hamiltonian approach. IEEE Robotics and Automation\nLetters , 7(3), 7100–7107."    },    {        "title": "2402.12000v2.Thinking_Outside_the_Black_Box__Insights_from_a_Digital_Exhibition_in_the_Humanities.pdf",        "content": "Thinking\nOutside\nthe\nBlack\nBo x:\nInsights\nfr om\na\nDigital\nExhibition\nin\nthe\nHumanities\nSebastian\nBarzaghi\n1\n,\nAlice\nBordignon\n2\n,\nBianca\nGualandi\n3\n,\nSilvio\nPeroni\n4\n1\nDepar tment\nof\nCultur al\nHeritage,\nUniversity\nof\nBologna,\nVia\nDegli\nAriani,\n1,\nRavenna,\nItaly\n-\nsebastian.barzaghi2@unibo.it\n2\nDepar tment\nof\nClassical\nPhilology\nand\nItalian\nStudies,\nUniversity\nof\nBologna,\nVia\nZamboni,\n32,\nBologna,\nItaly\n-\nalice.bor dignon2@unibo.it\n3\nDepar tment\nof\nClassical\nPhilology\nand\nItalian\nStudies,\nUniversity\nof\nBologna,\nVia\nZamboni,\n32,\nBologna,\nItaly\n-\nbianca.gualandi4@unibo.it\n4\nDepar tment\nof\nClassical\nPhilology\nand\nItalian\nStudies,\nUniversity\nof\nBologna,\nVia\nZamboni,\n32,\nBologna,\nItaly\n-\nsilvio.per oni@unibo.it\nABSTRACT\nOne\nof\nthe\nmain\ngoals\nof\nOpen\nScience\nis\nto\nmake\nresearch\nmore\nreproducible.\nThere\nis\nno\nconsensus,\nhowever,\non\nwhat\nexactly\n“reproducibility”\nis,\nas\nopposed\nfor\nexample\nto\n“replicability”,\nand\nhow\nit\napplies\nto\ndifferent\nresearch\nfields.\nAfter\na\nshort\nreview\nof\nthe\nliterature\non\nreproducibility/replicability\nwith\na\nfocus\non\nthe\nhumanities,\nwe\ndescribe\nhow\nthe\ncreation\nof\nthe\ndigital\ntwin\nof\nthe\ntemporary\nexhibition\n“The\nOther\nRenaissance”\nhas\nbeen\ndocumented\nthroughout,\nwith\ndifferent\nmethods,\nbut\nwith\nconstant\nattention\nto\nresearch\ntransparency,\nopenness\nand\naccountability.\nA\ncareful\ndocumentation\nof\nthe\nstudy\ndesign,\ndata\ncollection\nand\nanalysis\ntechniques\nhelps\nreflect\nand\nmake\nall\npossible\ninfluencing\nfactors\nexplicit,\nand\nis\na\nfundamental\ntool\nfor\nreliability\nand\nrigour\nand\nfor\nopening\nthe\n“black\nbox”\nof\nresearch.\nKEYWORDS\nReproducible\nresearch,\nreplicability ,\ntransparency ,\nOpen\nScience,\nCultural\nHeritage,\ndigital\ntwin\n1.\nINTRODUCTION\nIn\nthis\ncontribution,\nwe\naim\nto\nanchor\nthe\ndiscussion\naround\nopen\nand\nreproducible\nresearch\nin\nthe\nArts\nand\nHumanities\nby\npresenting\nas\na\ncase\nstudy\nthe\ncreation\nof\nthe\ndigital\ntwin\nof\nthe\ntemporary\nexhibition\n“The\nOther\nRenaissance:\nUlisse\nAldrovandi\nand\nthe\nWonders\nof\nthe\nWorld”\n1\n,\ncurrently\nunder\ndevelopment\nwithin\nthe\nPNRR\nProject\nCHANGES,\nand\nspecifically\nits\nSpoke\n4\n–\nVirtual\ntechnologies\nfor\nmuseums\nand\nart\ncollections\n[1].\nThe\noriginal\nexhibition,\nheld\nin\nPoggi\nPalace\nMuseum\n(Bologna,\nItaly)\nbetween\nDecember\n2022\nand\nMay\n2023,\nconsisted\nof\nmore\nthan\n200\nobjects,\nmostly\nbelonging\nto\nthe\nnaturalist\nUlisse\nAldrovandi\nand\nnever\nexhibited\nbefore.\nThe\ncreation\nof\nthe\ndigital\ntwin\n–\nvia\nthe\nacquisition,\nprocessing,\nmodelling,\nexport,\nmetadata\ncreation,\nand\nupload\nof\nthe\n3D\nmodels\nto\na\nweb-based\nframework\n–\nwas\ndocumented\nthroughout\nin\na\nstructured\nmanner\nin\norder\nto\nmake\nthe\nentire\nprocess\ntransparent\nand\nreproducible.\nIndeed,\nno\nreproducibility\nis\npossible\nwithout\ntransparency,\nor\nthe\ncareful\nand\ncomplete\ndocumentation\nof\nall\nrelevant\naspects\nof\nthe\nstudy\n[6,\np.5].\n2.\nTHEORETICAL\nBACKGROUND\nAND\nRELA TED\nWORKS\nGoodman\nand\ncolleagues\n[6]\nsuggest\nwe\nshould\ntalk\nabout\nthree\ntypes\nof\n“reproducibility”:\n(i)\nmethods\nreproducibility\n,\ni.e.\nthe\nability\nto\nexactly\nreproduce\na\nstudy\nby\nusing\nthe\nsame\nraw\ndata\nand\nthe\nsame\nmethodologies\nto\nobtain\nthe\nsame\nresults,\n(ii)\nresults\nreproducibility\n–\nalso\nreferred\nto\nas\nreplicability\n–\ni.e.\nthe\nability\nto\nobtain\nthe\nsame\nresults\nfrom\nan\nindependent\nstudy\nusing\nthe\nsame\nmethodologies\nas\nthe\noriginal\nstudy,\nand\n(iii)\ninferential\nreproducibility\n,\ni.e.\n“the\ndrawing\nof\nqualitatively\nsimilar\nconclusions\nfrom\neither\nan\nindependent\nreplication\nof\na\nstudy\nor\na\nreanalysis\nof\nthe\noriginal\nstudy”.\nIn\nexplaining\nhow\nthis\ndiffers\nfrom\nthe\ntwo\ncategories\npreviously\ndescribed,\nthe\nauthors\nadd\nthat\nscientists\nmight\n“draw\nthe\nsame\nconclusions\nfrom\ndifferent\nsets\nof\nstudies\nand\ndata\nor\ncould\ndraw\ndifferent\nconclusions\nfrom\nthe\nsame\noriginal\ndata,\nsometimes\neven\nif\nthey\nagree\non\nthe\nanalytical\nresults”\n[6,\np.4].\nThe\nreasons\ncan\nbe\na\npriori\n,\nsuch\nas\na\ndifferent\nassessment\nof\nthe\nprobability\nof\nthe\nhypothesis\nbeing\nexplored,\nor\ncan\nbe\nlinked\nto\ndifferent\nchoices\nabout\nhow\nto\nanalyse\nand\nreport\ndata.\nThis\nthird\ntype\nof\nreproducibility\n–\nwhich\nis\nalso\nthe\nmost\nimportant\naccording\nto\n1\nhttps://site.unibo.it/aldrovandi500/en/mostra-l-altro-rinascimentothe\nauthors\n–\nmight\nbe\nthe\nmost\ncommon\nwhen\ntalking\nabout\nresearch\nreproducibility\nin\nthe\nhumanities.\nPeels\nand\nBouter\n[12]\nlook\nat\nhow\nthese\nconcepts\ncan\nbe\napplied\nto\nthe\nhumanities,\na\nfield\nthat\nhas\noften\nbeen\noverlooked\nwhen\ntalking\nabout\nreproducible\nresearch.\nThey\nprefer\nthe\nterms\n“replicability”\nand\n“replication”,\nand\nagain\nthey\ndefine\nthree\ndifferent\nlevels:\n(i)\nreanalysis\n,\nthat\nis\nGoodman\net\nal.’s\nmethods\nreproducibility\n,\n(ii)\ndirect\nreplication\n,\nwhere\nthe\nsame\nstudy\nprotocol\nis\napplied\nto\nnew\ndata,\nand\n(iii)\nconceptual\nreplication\n,\nwhere\nresearch\ndata\nare\nnew\nand\nthe\nstudy\nprotocol\nis\nmodified\n[12].\nThese\ndefinitions\ndo\nnot\nperfectly\noverlap\nwith\nthose\nseen\nbefore\nbut\nwhat\nis\ncrucial\nto\nnote\nhere\nis\nthat\nthe\nauthors\nfind\nthat,\nwhile\nreplication\ncan\ntake\nvarious\nforms\nacross\nthe\nhumanities,\nit\nis\nnot\nfundamentally\ndifferent\nfrom\nreplication\nin\nthe\nbiomedical,\nnatural,\nand\nsocial\nsciences\nand\ncan\nbe\nachieved\nby\npre-registering\nthe\nstudies,\nand\ndocumenting\nand\nsharing\nmethodologies\nand\ndata\n[12].\nThanks\nto\nspecific\nfunding\nfrom\nthe\nresearch\nfunder\nNWO\n2\n,\na\ngroup\nof\nDutch\nresearchers\nconducted\nreplication\nstudies\nin\na\nnumber\nof\ndisciplines,\nincluding\nthe\nhumanities,\nand\nrecently\npublished\na\nset\nof\nrecommendations\nand\nlessons\nlearned\n[3].\nThey\nfound\nthat\nin\nall\ncases\nreplication\nstudies\nhelp\ncorroborate\nthe\nfindings\nof\nthe\noriginal\nstudies\n(e.g.,\nextending\nthe\nnumber\nof\nsources\nor\nusing\na\nmore\nstate-of-the-art\napproach)\nand\ncan\nprovide\na\nmore\nthorough\nunderstanding\nof\nthe\nrelevant\nresearch\nfield\nand\nthe\navailable\nmethodological\nchoices\n[3,\npp.6-7].\nThey\nalso\nnote,\nhowever,\nthat\n“even\nexperienced,\nhighly\nconscientious\nresearchers\noften\nfind\nit\ndifficult\nto\ndocument\ntheir\nprotocols\nin\nenough\ndetail\nto\nsupport\ndirect\nreplication”\n[3,\np.8].\nThose\nopposing\nthe\napplication\nof\nthe\nreproducibility\nor\nreplicability\ncategories\nacross\nall\nresearch\nareas\ncite\nthe\nfact\nthat,\nin\nseveral\nhumanities\ndomains,\nresearchers\nmay\nlack\ncontrol\nover\nthe\nexperimental\nconditions\nof\nthe\noriginal\nstudy,\nor\nhave\ndifferent\nviewpoints\nthat\nproduce\ndifferent\ndata\nand\ninterpretations\n[9,\npp.12-13;\n13,\np.7].\nCarefully\ndocumenting\nthe\noriginal\nstudy\ndesign,\ndata\ncollection\nand\nanalysis,\nand\nreflecting\non\nall\npossible\ninfluencing\nfactors\nis\nfundamental\nfor\nreliability\nand\nrigour\nbut\ndoes\nnot\nautomatically\nensure\nreplicability\n[13,\np.10].\nIndeed,\naccording\nto\nthis\nview,\nto\nrequire\nreplicability\nof\nall\nepistemic\ncultures\nis\nharmful\nand\nimposes\n“universal\npolicies\nthat\nfail\nto\naccount\nfor\nlocal\n(epistemic)\ndifferences”,\nultimately\ndenying\nauthority\n–\nand\nrelated\nrewards\n–\nto\nresearchers\nin\nthe\nhumanities.\nOn\nthe\nother\nhand,\nthe\n“umbrella\nof\nOpen\nScience”\nis\nwide\nenough\nand\nits\n“accountability\ntoolbox”\nis\nbig\nenough\nto\ndevelop\nplural\nmethods\nfor\nassessing\nthe\nquality\nof\ndiverse\nresearch\npractices\n[13,\np.12].\nLeaving\nthis\ndiscussion\naside\nfor\nnow,\nin\nan\nincreasingly\nopen\nresearch\nenvironment,\nwell-defined\npractices\nare\nessential\nfor\nensuring\ntransparency,\nreliability,\nand\nequitable\naccess\nto\nresearch\noutcomes.\nIn\nthe\ncase\nof\nthe\ndigital\ntwin\nof\n“The\nOther\nRenaissance”\nexhibition,\nwe\nlooked\nfor\nsome\noperational\nindications\non\nhow\nto\nachieve\nthis\ngoal\nin\nthe\nliterature\nproduced\nin\nresearch\nfields\nrelevant\nto\nthe\nproject.\nWilson\net\nal.\n[16]\noutline\na\nset\nof\nrecommendations\nfor\nscientific\ncomputing,\napplicable\nacross\ndifferent\ndisciplines\nand\nat\nvarying\nlevels\nof\ncomputational\nexpertise.\nRegarding\ndata\nmanagement\npractices,\ntheir\nsuggestions\nfocus\non\nthe\nimportance\nof\nincremental\ndocumentation\nand\ndata\ncleaning.\nIn\nparticular,\nthey\nadvocate\nfor\ncontinuous\nretention\nof\nraw\ndata,\nrobust\nbackup\nstrategies,\ndata\nmanipulation\nfor\nimproving\nmachine\nand\nhuman\nreadability\nand\nfacilitating\nanalysis,\nmeticulous\nrecording\nof\nthe\nsteps\nused\nto\nprocess\ndata,\nusing\nmultiple\ntables\nin\na\nway\nthat\neach\nrecord\nin\none\ntable\nis\ninterlinked\nwith\nits\nrespective\nrepresentation\nin\nanother\ntable\nvia\na\nunique\nand\npersistent\nidentifier,\nand\nusing\nrepositories\nthat\nissue\nDOIs\nto\nthe\nvarious\ndata\nartefacts\nused\nand\nproduced\nfor\neasy\naccess\nand\ncitation.\nIn\nthe\narchaeological\ncontext,\nKaroune\nand\nPlomp\n[8]\nidentify\nthree\ndistinct\nlevels\nof\nworkflow\nto\nmake\nresearch\nactivities\nreproducible,\ndepending\non\nthe\ncomputational\nskills\nrequired\nto\ncarry\nout\nsuch\nactivities.\nPublic\naccess\nto\nresearch\nmaterials\nand\nmethods\nis\nfacilitated\nby\nthe\nfirst\nlevel,\nwhich\nconsists\nof\ntransparent\nrecording\nthrough\ndocumentation,\nrequiring\nonly\nthe\ncreation\nand\nmaintenance\nof\na\nwritten\nrecord\nof\neach\nanalysis\nstep,\ndone\nin\na\nformat\nthat\nallows\nother\npeers\nto\nread,\ncomprehend,\nand\nreplicate\nthe\nwork\ndone,\nwhile\nrequiring\nthe\nleast\namount\nof\ncomputational\nexpertise.\nOutputs\nat\nthis\nlevel\nof\nworkflow\nusually\ninclude\ndocuments\ndescribing\nthe\nmethods\nand\nprocesses,\nraw\ndata\nfiles,\nand\nanalysis\noutput\nfiles.\nEnsuring\nversion\ncontrol\nthrough\na\nshared\nfile\nnaming\nsystem\nand/or\nsoftware\nwith\nhistory\ntracking\nis\nalso\na\ncommon\ncharacteristic\nof\ntransparent\nrecording\nsince\nit\nfacilitates\nthe\ndocumentation\nof\nthe\nprocess\nas\na\nwhole.\n3.\nMAKING\nTHE\nDIGITISA TION\nPROCESS\nMORE\nTRANSP ARENT\nTo\nensure\na\nsolid\nbasis\nfor\ntransparency\nand\nreplicability,\nour\napproach\nclosely\nfollowed\nthe\naforementioned\nsets\nof\nbest\npractices,\nin\nline\nwith\nthe\nindications\nlisted\nin\nthe\nData\nManagement\nPlan\nof\nthe\nproject\n[7].\nThe\ndigitisation\nworkflow\ninvolved\ncreating\ntwo\ndatasets\nas\nGoogle\nSheet\nfiles\nshared\nbetween\nthe\nteam\nmembers:\none\n(Object\nTable,\nor\nOT)\nfor\nstoring\ncatalogue\ndescriptions\nof\nthe\nphysical\nobjects\nin\nthe\ncollection,\nthe\nother\n(Process\nTable,\nor\nPT)\nfor\nstoring\ndata\nabout\nthe\ndigitisation\nprocess.\nAfter\ndefining\nthe\nstructure\nof\nthe\ntables,\nthe\nvariables\nrepresented\nby\ntheir\nheadings,\nand\nthe\nexpected\nrepresentation\nfor\neach\nvalue,\nthe\ndata\nwere\npopulated\nin\nparallel\nby\nthe\nteam\nmembers.\nOn\nthe\none\nhand,\nthe\nOT\nwas\npopulated\nwith\ndata\ngleaned\nfrom\nofficial\nmuseum\nrecords\nand\npreliminary\nnotes\nrelated\nto\nthe\nexhibition\n2\nhttps://www .nwo.nl/en/researchprogrammes/replication-studiesobjects,\nand\nthus\nwas\nstructured\naround\na\ncataloguing\ndescription\nof\neach\nobject\n(e.g.\n“title”,\n“author”,\nand\nso\non).\nWhere\npossible,\ncontrolled\ndata\nvalues\n(e.g.\npeople\nnames,\nterms\nused\nfor\nobject\ntypes,\netc.)\nwere\naligned\nwith\nexisting\nvocabularies\n(such\nas\nWikiData\n3\n)\nand\nauthority\nlists\n(like\nVIAF\n4\nand\nULAN\n5\n).\nOn\nthe\nother\nhand,\nthe\nPT\nwas\npopulated\nwith\ndata\ninserted\nby\nthe\nresearchers\nduring\nthe\nacquisition\nof\nthe\nobjects\nand\nthe\ncreation\nof\ntheir\n3D\nmodels\nand,\nthus,\nwas\nstructured\naround\nthe\nsteps\ninvolved\nin\nthe\noverall\ndigitisation\nprocess\nand\ntheir\nrelevant\nattributes.\nOverall,\nthe\nsteps\ninclude\nan\ninitial\nacquisition\nactivity\nfor\ncapturing\nanalogue\nmaterials\nand\nrealising\ntheir\npreliminary\ndigital\nrepresentations,\nand\na\nseries\nof\nsubsequent\nactivities\n(\nprocessing\n,\nmodelling\n,\nexport\n,\nmetadata\ncreation\n,\nand\nupload\n)\nwhich\ninvolved\nthe\nuse\nof\ntools\nfor\nrefining\nand\npublishing\nthe\n3D\nmodels\nas\nusable,\nfully\ndescribed\nscientific\nobjects.\nIn\nturn,\nthese\nactivities\nwere\nrepresented\nas\na\nset\nof\ninformation\nthat\nincluded:\nthe\norganisation\nresponsible\nfor\nthe\nactivity,\nthe\npeople\nresponsible\nfor\nactually\ncarrying\nout\nthe\nactivity,\nthe\ntechnique\nand/or\ntools\nused\nto\nperform\nthe\nactivity,\nand\nthe\ntimespan\nin\nwhich\nthe\nactivity\nwas\ncarried\nout.\nThis\npreliminary\nwork\nresulted\nin\nthe\ncreation\nof\na\nrecord\nof\nthe\nentire\ndigitisation\nprocess.\nGoogle\nSheets\nand\nMicrosoft\nExcel\nin\nthe\nMicrosoft\nOffice\n365\nplatform\nwere\nstrong\nfacilitators\nfor\ndata\nretention,\nbackup\nand\nversioning\n6\n.\nMoreover,\nshared\nformatting\npractices\non\nelements\nsuch\nas\ndates\nand\nnames\nwere\nessential\nfor\npreparing\nthe\ndata\nfor\nthe\nsubsequent\nphases\nof\nthe\nproject.\nAt\nthe\nend\nof\nthis\nstage,\neach\nobject\nhad\nits\nmetadata,\nrelated\ndigitisation\nphases\nwith\ntheir\nfeatures,\nand\nunique\nidentifiers\nthat\nallowed\nthe\ntwo\ndatasets\nto\nbe\nlinked\nto\neach\nother.\nAs\ninformation\nwas\nadded\nto\nboth\ndatasets,\nmore\nwork\nwent\ninto\ngetting\nthem\nready\nto\nbe\npublished\nas\nmachine-readable\nrepresentations\nof\nthe\nentire\nphysical\ncollection,\nits\ndigital\ncounterpart,\nand\nthe\nprocedure\nthat,\nfrom\nthe\nformer,\nproduced\nthe\nlatter.\nThe\nResource\nDescription\nFramework\n(RDF)\n7\nwas\nselected\nas\na\nformal\ndata\nrepresentation\nfor\nenabling\ntransparent\ndata\npublishing.\nHowever,\nin\norder\nto\ntransform\nthe\ncurrent\ndata\ninto\nRDF\nstatements,\nthe\ntable\nstructures\nfirst\nhad\nto\nbe\nmapped\nto\ndata\nmodels\nthat\ncould\nexpress\nand\ndeepen\nthe\nsemantics\nof\nthe\ndata\nabout\ncultural\nheritage\nand\ndigitisation\nactivities.\nWe\nchose\nto\nreuse\nthe\nCIDOC\nConceptual\nReference\nModel\n(CIDOC\nCRM)\n8\n[4]\nto\nrepresent\nthe\ndata\ndetailing\nthe\nphysical\nand\ncontextual\nattributes\nof\nthe\ncollection\nobjects,\nand\nits\nextension\nCRM\nDigital\n(CRMdig)\n9\n[5]\nto\ndepict\nthe\nstages\nof\nthe\ndigitisation\nworkflow.\nThe\nSimplified\nAgile\nMethodology\nfor\nOntology\nDevelopment\n(SAMOD)\n[14],\na\nmethodology\nto\nquickly\ncreate\nsemantic\nmodels\nthat\nare\nsupported\nby\nrich\ndocumentation\nand\ntest\ncases,\nwas\nused\nto\ndraw\nthe\nneeded\nconceptual\nconstructs\nfrom\nCIDOC\nCRM\nand\nCRMdig\nand\npack\nthem\ninto\ntwo\ndata\nmodels.\n4.\nMAKING\nINTERPRET ATION\nMORE\nTRANSP ARENT\nOne\nof\nthe\nmain\ngoals\nof\ncultural\nheritage\ndigitisation\nis\nthe\nselection\nof\nspecific\nelements\nof\nreality\nto\nstore\ndigitally.\nThe\nselection\nprocess\ninvolves\na\ndeliberate\nhuman\nchoice\nabout\nthe\nphysical,\ngeometric,\nchromatic,\nmechanical,\nand\nstylistic\ncharacteristics\nof\nthe\nobjects\nto\ndigitise.\nThese\naspects\nare\nrecorded\ninside\na\n“grid\nof\ninformation”,\nsuch\nas\nvectors,\nimages,\n3D\nmodels,\ndatabases,\nand\ntables,\namong\nothers\n[2,\np.127].\nAccording\nto\nthis\nlogic,\na\ndigital\ntechnology\nsurvey\nis\nexpected\nto\napproximate\nreality\nbased\non\nsome\npredetermined\nfeatures\nselected\nat\nthe\noutset\nof\nthe\nsurvey\nproject.\nThe\nquantity\nand\nquality\nof\nthe\ndata\nobtained\nduring\nthe\nsurvey\nsignificantly\nimpact\nhow\naccurate\nthe\ndigitisation\nwill\nbe.\nIn\nthis\ncontext,\na\ndigital\nreplica\nis\ndefined\nas\nan\napproximate,\naesthetically\nconvincing\ncopy\nof\na\ncultural\nsite\nor\nartefact\n[2,\np.127].\nIn\nour\ncase\nstudy,\nthe\nmain\naim\nwas\nto\nobtain\nthe\ndigital\nversion\nof\nthe\nexhibition’s\nexperience,\nstarting\nfrom\nthe\ncreation\nof\nits\ndigital\ntwin\n10\n,\nlinking\nto\nthe\ndigital\nassets\nof\nthe\nvarious\nobjects\n(3D\nand\nmultimedia)\nin\nthe\ncollections,\nenriched\nby\nmetadata,\ncatalogued\nand\naccessible\nonline\nusing\ndifferent\ndevices\n[1,\np.2].\nOur\napproach\nfor\ncreating\nthe\ndigital\ntwin\nof\nAldrovandi’s\nexhibition\nincluded\nin\nthe\nfirst\nplace\nthe\nimplementation\nof\nvarious\nsetups\nand\ninstruments\nto\ncreate\nmorphologically\nprecise\nmodels\nwith\nhighly\ndetailed\ntextures.\nPhotogrammetry\nand\nstructured\nlight\nscanner\n(SLS)\nacquisition\ntechniques\nhave\nbeen\nused\nto\nobtain\nthe\ndigital\nrepresentation\nof\neach\nitem.\nThe\nchoice\nof\nthese\nmethodologies\nhas\nbeen\ninfluenced\nby\ncontextual\nfactors\n(such\nas\nlimited\ntime\nand\navailable\nspace),\n10\nSince\ncultural\nheritage\nmay\nbe\nintangible\nor\ntemporary ,\nNiccolucci\net\nal.\n[11]\nsuggest\nseparating\nthe\ndata\nexchange\ndimension\nfrom \nthe\nrepresentation\ndimension\nfor\ndigital\ntwins.\nThis\nreconceptualisation\nrethinks\ndata\nflows\nand\nbi-directionality\nas\npossible\nand\nas\nnot \nmandatory\nrequirements\nfor\ndigital\ntwins\nof\ncultural\nheritage\nartefacts\nor\nlandscapes,\nopening\nthe\npossibility\nfor\naccurate\ndigital\nmodels \n(i.e.\ndigital\nreplicas)\nto\nevolve\ndynamically\ninto\na\nfully\ndeveloped\ndigital\ntwin.\n9\nhttp://www .ics.forth.gr/isl/CRMdig/\n8\nhttp://www .cidoc-crm.or g/cidoc-crm/\n7\nhttps://www .w3.or g/TR/rdf1 1-concepts/\n6\nSince\ntransparent\nrecording\ndoes\nnot\ninvolve\nany\ncomputational\ncode,\nproprietary\nsoftware\nlike\nGoogle\nSheets\nis\nacceptable\nas\nlong \nas\nit\nincludes\nfeatures\nlike\nversioning\nand\nexporting\noutputs\nto\nopen\nformats\n(e.g.,\n.txt,\n.rtf,\n.pdf)\n[7].\n5\nhttps://www .getty .edu/research/tools/vocabularies/ulan/\n4\nhttps://viaf.or g/\n3\nhttps://www .wikidata.or g/materials,\nand\nthe\nobjects’\nsize.\nWe\nprovided\ndocumentation\nabout\nthe\nchallenges\nfaced\nand\nrelated\nsolutions\nadopted\nin\nthe\nacquisition\nand\nprocessing\nphase.\nThe\ndocumentation\nof\nthe\nrisks\n(e.g.\nacquisition\nof\nnon-Lambertian\nmaterials,\nlimited\nobject’s\nmobility,\netc.)\nand\nthe\nsolutions\nadopted\n(e.g.\ncross\npolarisation\ntechniques,\nspecific\nsetup\nschemas,\netc.)\npermits\nothers\nto\nretrace\nand\nrepeat,\nat\nleast\nin\ntheory,\nthe\nactions\ninvolved\nin\na\ncertain\nresearch\neffort,\nproducing\nnew\ndata\n[15,\np.2].\nConcerning\nscanner\nacquisitions,\nwe\ndefined\nsome\ncommon\nlimits\nregarding\ntexture\nfinal\nresolution,\nand\nwe\ndecided\non\na\nspecific\nrange\nfor\ngeometry\ncomplexity.\nDuring\nthe\nentire\nprocess,\nopen\ntechnologies\nand\nsoftware\nwere\nemployed\nto\nmaximise\nthe\nworkflow’s\nre-adoption\nfor\nthe\ncreation\nof\na\nvirtual\nexhibition\nin\ndifferent\nsettings.\nHowever,\nfor\nsome\nspecific\ntasks\n(e.g.\nraw\ndata\nelaboration),\nproprietary\nsoftware\nwas\nrequired\nsince\nopen-source\nsoftware\nfails\nto\nproduce\nsatisfactory\nresults.\nDocumenting\nprocessing\ndecisions\nmade\nfor\nextra\ntransparency\nshould\nbe\na\npart\nof\nthe\nscientific\nworkflow\nand\ncultural\nheritage\npreservation.\nThis\ncan\nbe\ndone,\nas\nproposed\nby\nMoore\net\nal.\n[10],\nby\nextracting\na\nprocessing\nreport\nfrom\nthe\nphotogrammetry\nsoftware.\nMetashape\n11\nand\n3DF\nZephyr\n12\n,\nthe\nmain\nsoftware\nused\nfor\nthe\nphotogrammetric\nprocessing\nphase,\nprovide\nthis\noption.\nThe\nfunction\nhas\nnot\nbeen\ndeveloped\nyet\nfor\nthe\nopen-source\nalternative\nMeshroom\n13\n,\nwhose\nimplementation\nin\nthis\nproject\nis\nunder\ntest.\nHowever,\nsoftware\nfor\nprocessing\nphotogrammetric\ndata\nis\nconsidered\nmore\nopen\nand\ntransparent\ncompared\nto\nsoftware\nused\nfor\nscanned\ndata\nelaboration.\nScanned\ndata\nwere\nelaborated\nusing\ndifferent\nversions\nof\nArtec\nStudio\n14\n,\nwhich\nhas\nproven\nto\nbe\na\n“black\nbox”\nfor\nthose\nwho\ndo\nnot\nown\nthe\nsoftware\nand\nthe\nlicence\nrequired\nto\nuse\nit,\nallowing\nraw\ndata\nexport\nonly\nin\nproprietary\nformats\nand\nwithout\nproviding\nany\nprocessing\nreport.\nRegarding\nmodelling\ninterventions,\nto\nguarantee\ntransparency\nconcerning\nthe\nmanipulation\nof\nthe\nsource\ndata\nwe\nprovided\ndifferent\nderivative\nversions\nfor\neach\n3D\nmodel.\nLevel\n0\nrepresents\nthe\nrough\nresult\nobtained\nby\nthe\nacquisition\nsoftware,\nwhile\nlevel\n1\nincludes\nthe\nfinal\nhigh-definition\nmodel,\nwhere\ngeometry\nissues\nhave\nbeen\nfixed\nand\nlacking\nparts\nhave\nbeen\nreconstructed.\nThe\ncomparison\nbetween\nthese\nversions\nenables\none\nto\nidentify\nwhich\nparts\nwere\nmodelled\nand\nwhich\nparts\nbelong\nto\nlevel\n0.\nLevel\n2\ninstead\nincludes\nthe\noptimised\nmodel\nfor\nweb\npublication.\nFinally,\nwe\nused\nas\nmany\nstandard\nand\ninteroperable\nformats\nas\npossible\nfor\nthe\ngenerated\ndata\nto\nfacilitate\ntheir\nreuse\non\ndifferent\nplatforms.\nSpecifically,\nwe\nused\nglTF,\nglb,\nobj,\nand\nmtl\nfor\n3D\nmodels;\ntiff,\njpg,\nraw,\nand\npng\nfor\nimages;\nmp4\nand\nmov\nfor\nvideos;\nand\nmp3\nfor\naudios.\n5.\nDISCUSSION\nAND\nCONCLUSIONS\nWe\nhave\ndescribed\nhow\nthe\ndigitisation\nprocess\nof\nthe\nexhibition\n“The\nOther\nRenaissance”\nhas\nbeen\ndocumented\nthroughout,\nwith\ndifferent\nmethods,\nbut\nwith\nconstant\nattention\nto\nresearch\ntransparency,\nopenness\nand\naccountability.\nSince\nany\nreality-capture\nor\nsource-based\nmodel\nis\naffected\nby\nthe\nlens\nof\ninterpretation\n(of\na\nhuman\nor\nsoftware),\ntracking\nsteps\nfor\nthe\ncreation\nof\na\n3D\nmodel\nis\nessential\nto\ngive\ntransparency\nto\nthese\ninterpretations,\nfacilitating\nthe\nrepeatability\nof\nthe\ncreation\nprocess\n[10].\nFurthermore,\ndata\nrelating\nto\nthe\ndigitisation\nprocess\ncan\noftentimes\nbe\ncaptured\nonly\nonce,\nwhile\nthe\nprocess\nis\nongoing,\nand\nit\nis\ntherefore\ncrucial\nto\nretain\nas\nmuch\ninformation\nas\npossible,\nstructure\nit\nappropriately\nand\nmake\nit\navailable\nin\nan\nopen\nand\nmachine-readable\nformat\nto\nprovide\na\nrecord\nof\nthe\nentire\nphysical\ncollection,\nits\ndigital\ncounterpart,\nand\nthe\nprocedure\nthat,\nfrom\nthe\nformer,\nproduced\nthe\nlatter.\nA\nparticularly\ninteresting\naspect\nis\nthe\ntemporary\nnature\nof\n“The\nOther\nRenaissance”\nexhibition.\nAt\nthe\ntime\nof\nwriting,\nthe\nexhibition\nconcluded\nmore\nthan\n6\nmonths\nago,\nobjects\non\nloan\nhave\nbeen\nlong\nreturned,\nand\nthe\nrooms\nwhere\nthe\nexhibition\ntook\nplace\nhave\nchanged\nuse.\nThe\nsame\nmethodologies\ncannot\nbe\napplied\nto\nthe\nsame\ndata\n–\nGoodman\net\nal.’s\nmethods\nreproducibility\nor\nPeels\nand\nBouter’s\nreanalysis\n–\nbecause\nthe\nphysical\ncollection\ndoes\nnot\nexist\nin\nits\noriginal\nform\nanymore.\nWhat\nis\npossible,\nhowever,\nis\nthat\nthe\nmethodologies\ndescribed\nare\napplied\nto\nnew\ndata\n(different\ncultural\nheritage\nobjects,\nexhibitions,\netc.).\nAdditionally,\nthe\ncareful\ndocumentation\nof\nthe\nresearch\nprocess\nmakes\nit\npossible\nfor\nothers\nto\njudge\nthe\nrelationship\nbetween\nthe\ndigital\ntwin\nand\nthe\nphysical\ncollection,\na\npiece\nof\ninformation\nthat\nis\ncrucial\nfor\nscientific\nscrutiny\nbut\nthat\nwould\notherwise\nhave\nbeen\nirremediably\nlost\non\nthe\nday\nthe\ntemporary\nexhibition\nclosed.\nDocumenting\nthe\nproject\nworkflow\nin\nthis\nmanner\nis\nnot\nsimple:\nit\nrequires\ncareful\nplanning,\nspecific\ncompetencies,\nand\nit\nis\nextremely\ntime-consuming.\nThese\nefforts\nmust\nbe\nrewarded\nin\nthe\nacademic\nsetting,\nif\na\nculture\nof\naccountability,\ndata\ncuration\nand\nopen,\nreproducible\nresearch\nis\nto\nbecome\nthe\nnorm.\nInitiatives\nlike\nCoARA\n15\nare\nindeed\nnudging\nthe\nscientific\ncommunity\nin\nthis\ndirection,\nbut\nwhile\nsome\npractices\n–\nsuch\nas\nthe\npublication\nof\nresearch\ndata\n“as\nopen\nas\npossible”\nand\naccording\nto\nFAIR\nprinciples\n–\nare\ngarnering\nincreasing\nattention,\nthe\nfocus\nmust\nbe\nkept\non\nmethodologies,\ntoo,\nand\non\nthe\nneed\nof\ncarefully\ndocumenting\neach\nstep\nof\na\nresearch\nproject.\nFurther,\nas\nnoted\nby\nPeels\n15\nhttps://coara.eu/\n14\nhttps://www .artec3d.com/it/3d-software/artec-studio\n13\nhttps://alicevision.or g/\n12\nhttps://www .3dflow .net/it/\n11\nhttps://www .agisoft.com/and\nBouter\n[12],\nguidelines\non\nhow\nto\nreport\nstudy\nprotocols,\nmethodologies\nand\nprocedures\nare\nneeded,\nand\nthis\nis\nperhaps\nespecially\ntrue\nin\nthe\nhumanities.\nThe\nestablishment\nof\nprinciples,\nlike\nFAIR,\nand\ndiscipline-specific\nrecommendations\non\nhow\nto\nmanage\nand\ndocument\nresearch\ndata\nin\na\ntransparent\nand\ntraceable\nmanner\nis\na\ngreat\nfirst\nstep\nin\nthis\ndirection.\nFAIR\nprinciples,\nsupposedly\ndiscipline-agnostic,\nare\nbeing\ndiscussed\nand\nadapted\nto\nthe\ndifferent\nresearch\ncultures,\nData\nManagement\nPlans\nare\nbecoming\nincreasingly\ncommon,\nand\ntemplates\nand\nonline\ntools\nare\nbeing\nproduced\nto\nhelp\nresearchers\nfill\nthem\nout\nin\na\nstructured\nand\nmachine-actionable\nmanner.\nThere\nis\nstill\nmore\nto\nbe\ndone,\nand\nmore\nexplicit\nattention\nneeds\nto\nbe\ndevoted\nto\nresearch\nmethodologies\nand\nhow\nto\ndocument\nthem\nin\nsufficient\ndetail.\nWe\nrecognise\nthat\nthe\ndebate\naround\nthe\ndefinition\nof\nreproducibility\nand\nreplicability,\nand\nwhether\nthese\nterms\nshould\nbe\napplied\nto\nresearch\nas\na\nwhole,\nacross\nall\ndisciplines,\nis\nnot\nsettled\n[3;6;9;12;13].\nHowever,\nthere\nseems\nto\nbe\nan\nagreement\non\nthe\nfact\nthat\nresearch\ncan\nbe\nreproducible\nin\nvarying\ndegrees,\nfrom\nan\n“ideal”\ncomputational\nreproducibility\nall\nthe\nway\nto\nfields\nwhere\nmultiple\ninterpretations\nof\na\ncertain\nphenomenon\ncoexist.\nReplication\nhere\nmay\nhelp\n“filter\nout\nfaulty\nreasoning\nor\nmisguided\ninterpretations,\ndraw\nattention\nto\nunnoticed\ncrucial\ndifferences\nin\nstudy\nmethods”\n[12]\nbut\nit\nis\nnot\nalways\npossible\nto\nascertain\nwhich\ninterpretation\nis\ncorrect.\nCircling\nback\nto\nthe\ndefinition\nof\ninferential\nreproducibility\n[6]\nand\nthe\ncritique\nof\nthe\nconcept\nof\nreplicability\nin\na\nhumanities\ncontext\n[9;13],\nthe\nresearchers’\ndifferent\nviewpoints,\ntheoretical\nbackground\nor\nprevious\nassessments\nalways\nhave\na\nbearing\non\nhow\nthe\nstudy\nis\nconducted\nand\nhow\nthe\nresults\nare\ninterpreted.\nA\ncareful\ndocumentation\nof\nthe\nstudy\ndesign,\ndata\ncollection,\nand\nanalysis\ntechniques\nhelp\nreflect\nand\nmake\nexplicit\nall\npossible\ninfluencing\nfactors,\nand\nis\na\nfundamental\ntool\nfor\nreliability\nand\nrigour\nand\nfor\nopening\nthe\n“black\nbox”\nof\nresearch.\nACKNOWLEDGEMENTS\nThis\nwork\nhas\nbeen\npartially\nfunded\nby\nProject\nPE\n0000020\nCHANGES\n-\nCUP\nB53C22003780006,\nNRP\nMission\n4\nComponent\n2\nInvestment\n1.3,\nFunded\nby\nthe\nEuropean\nUnion\n-\nNextGenerationEU.\nREFERENCES\n[1]\nBalzani,\nRoberto,\nSebastian\nBarzaghi,\nGabriele\nBitelli,\nFederica\nBonifazi,\net\nal.\n2024.\n«Saving\nTemporary\nExhibitions\nin\nVirtual\nEnvironments:\nThe\nDigital\nRenaissance\nof\nUlisse\nAldrovandi\n–\nAcquisition\nand\nDigitisation\nof\nCultural\nHeritage\nObjects».\nDigital\nApplications\nin\nArchaeology\nand\nCultural\nHeritage\n32:\ne00309.\nhttps://doi.or g/10.1016/j.daach.2023.e00309\n.\n[2]\nDemetrescu,\nEmanuel,\nEnzo\nd’Annibale,\nDaniele\nFerdani,\ne\nBruno\nFanini.\n2020.\n«Digital\nReplica\nof\nCultural\nLandscapes:\nAn\nExperimental\nReality-Based\nWorkflow\nto\nCreate\nRealistic,\nInteractive\nOpen\nWorld\nExperiences».\nJournal\nof\nCultural\nHeritage\n41:\n125–41.\nhttps://doi.or g/10.1016/j.culher .2019.07.018\n.\n[3]\nDerksen,\nMaarten,\nStephanie\nMeirmans,\nJonna\nBrenninkmeijer ,\nJeannette\nPols,\nAnnemarijn\nde\nBoer ,\nHans\nVan\nEyghen,\nSurya\nGayet,\net\nal.\n2024.\n«Replication\nStudies\nin\nthe\nNetherlands:\nLessons\nLearned\nand\nRecommendations\nfor\nFunders,\nPublishers\nand\nEditors,\nand\nUniversities».\nhttps://doi.or g/10.31219/osf.io/bj8xz\n.\n[4]\nDoerr ,\nMartin,\nChristian-Emil\nOre,\ne\nStephen\nStead.\n2007.\n«The\nCIDOC\nConceptual\nReference\nModel\n-\nA\nNew\nStandard\nfor\nKnowledge\nSharing»,\n51–56.\nhttps://doi.or g/10.13140/2.1.1420.6400\n.\n[5]\nDoerr ,\nMartin,\ne\nMaria\nTheodoridou.\ns.d.\n«CRMdig:\nA\nGeneric\nDigital\nProvenance\nModel\nfor\nScientific\nObservation».\n[6]\nGoodman,\nSteven\nN.,\nDaniele\nFanelli,\ne\nJohn\nP.\nA.\nIoannidis.\n2016.\n«What\ndoes\nresearch\nreproducibility\nmean?»\n8\n(341).\n[7]\nGualandi,\nBianca,\ne\nSilvio\nPeroni.\n2023.\n«Data\nManagement\nPlan:\nFirst\nVersion».\nhttps://zenodo.or g/records/7977103\n.\n[8]\nKaroune,\nEmma,\ne\nEsther\nPlomp.\n2022.\n«Removing\nBarriers\nto\nReproducible\nResearch\nin\nArchaeology».\nhttps://doi.or g/10.5281/ZENODO.7320029\n.\n[9]\nLeonelli,\nSabina.\n2018.\n«Rethinking\nReproducibility\nas\na\nCriterion\nfor\nResearch\nQuality».\nIn\nResearch\nin\nthe\nHistory\nof\nEconomic\nThought\nand\nMethodology .\nIncluding\na\nSymposium\non\nMary\nMorgan:\nCuriosity ,\nImagination,\nand\nSurprise,\n36B:129–46.\nEmerald\nPublishing\nLimited.\nhttps://doi.or g/10.1 108/S0743-41542018000036B009\n.\n[10]\nMoore,\nJennifer ,\nAdam\nRountrey ,\ne\nHannah\nScates\nKettler .\ns.d.\n3D\nData\nCreation\nto\nCuration:\nCommunity\nStandar ds\nfor\n3D\nData\nPreservation\n.\nALA.\nhttps://www .alastore.ala.or g/content/3d-data-creation-curation-community-standards-3d-data-preservation\n.\n[11]\nNiccolucci,\nFranco,\nBéatrice\nMarkhof f,\nMaria\nTheodoridou,\nAchille\nFelicetti,\ne\nSorin\nHermon.\n2023.\n«The\nHeritage\nDigital\nTwin:\nA\nBicycle\nMade\nfor\nTwo.\nThe\nIntegration\nof\nDigital\nMethodologies\ninto\nCultural\nHeritage\nResearch».\nOpen\nResear ch\nEurope\n3:\n64.\nhttps://doi.or g/10.12688/openreseurope.15496.1\n.\n[12]\nPeels,\nRik,\ne\nLex\nBouter .\n2018.\n«The\nPossibility\nand\nDesirability\nof\nReplication\nin\nthe\nHumanities».\nPalgrave\nCommunications\n4\n(1):\n95.\nhttps://doi.or g/10.1057/s41599-018-0149-x\n.\n[13]\nPenders,\nHolbrook,\ne\nDe\nRijcke.\n2019.\n«Rinse\nand\nRepeat:\nUnderstanding\nthe\nValue\nof\nReplication\nacross\nDifferent\nWays\nof\nKnowing».\nPublications\n7\n(3):\n52.\nhttps://doi.or g/10.3390/publications7030052\n.[14]\nPeroni,\nSilvio.\n2017.\n«A\nSimplified\nAgile\nMethodology\nfor\nOntology\nDevelopment».\nIn\nOWL:\nExperiences\nand\nDirections\n–\nReasoner\nEvaluation\n,\na\ncura\ndi\nMauro\nDragoni,\nMaría\nPoveda-V illalón,\ne\nErnesto\nJimenez-Ruiz,\n55–69.\nLecture\nNotes\nin\nComputer\nScience.\nCham:\nSpringer\nInternational\nPublishing.\nhttps://doi.or g/10.1007/978-3-319-54627-8_5\n.\n[15]\nRahal,\nRima-Maria,\nHanjo\nHamann,\nHilmar\nBrohmer ,\ne\nFlorian\nPethig.\n2022.\n«Sharing\nthe\nRecipe:\nReproducibility\nand\nReplicability\nin\nResearch\nAcross\nDisciplines».\nResear ch\nIdeas\nand\nOutcomes\n8:\ne89980.\nhttps://doi.or g/10.3897/rio.8.e89980\n.\n[16]\nWilson,\nGreg,\nJennifer\nBryan,\nKaren\nCranston,\nJustin\nKitzes,\nLex\nNederbragt,\ne\nTracy\nK.\nTeal.\n2017.\n«Good\nEnough\nPractices\nin\nScientific\nComputing».\nPLOS\nComputational\nBiology\n13\n(6):\ne1005510.\nhttps://doi.or g/10.1371/journal.pcbi.1005510\n."    },    {        "title": "2402.12003v1.Quantum_K_theory_of_IG_2__2n_.pdf",        "content": "arXiv:2402.12003v1  [math.AG]  19 Feb 2024QUANTUM K-THEORY OF IG(2,2n)\nV. BENEDETTI, N. PERRIN AND W. XU\nAbstract. We prove that the Schubert structure constants of the quantu mK-theory rings of\nsymplectic Grassmannians of lines have signs that alternat e with codimension and vanish for\ndegrees at least 3. We also give closed formulas that charact erize the multiplicative structure\nof these rings, including the Seidel representation and a Ch evalley formula.\nContents\n1. Introduction 1\n2. Preliminaries and proof strategy 4\n3. Replacing stable maps by evaluations 8\n4. Large degrees 14\n5. Degree 2 16\n6. Degree 1 22\n7. Non rationally connected cases 26\n8. Seidel representation 29\n9. Chevalley formula 30\nReferences 33\n1.Introduction\nTo a complex projective manifold Xone can attach various algebraic structures capturing\ndifferent kinds of information. For instance, cohomology rings enco de the behavior of (classes\nof) subvarieties: how they intersect (ordinary or equivariant coh omology) or how they are\nconnected by rational curves (quantum cohomology). In this pap er we are interested in K-\ntheory rings, which in turn encode information about coherent (or locally free) sheaves: how\nthey form short exact sequences and the behavior of their tenso r product. It turns out that\nclassicalK-theory can be seen as a refinement of classical cohomology - via th e Chern character\nhomomorphism. Moreover,onecan define aquantum K-theoryringwhich isboth adeformation\nof the classical K-theory ring and a refinement of the quantum cohomology ring. Ind eed, if\nquantum cohomology is an intersection theory of the space of genu s zero stable maps to X,\nquantumK-theory, which was introduced by [ Giv00], deals with the K-theory of this space\nand requires a more careful study of stable maps with reducible sou rces. Quantum cohomology\nand K-theory have gathered some attention in the context of hom ogeneous spaces where it\ngeneralizes classical Schubert calculus. We will focus on this case.\nFrom now on, X=G/PXwill be a rational projective homogeneous space. It is nowadays\nclassical that the transitive action of GonXimplies many positivity properties in cohomology\nand its various generalizations (see [ Gra01,Bri02,Buc02,Mih06,GK09,AGM11,BCMP22 ]\n2020Mathematics Subject Classification. Primary 14N35, 19E08; Secondary 14N15, 14M15, 14E08.\nKey words and phrases. Quantum K-theory, Gromov–Witten invariants, symplectic G rassmannians, Seidel\nrepresentation, Chevalley formula, rational connectedne ss.\n12 V. BENEDETTI, N. PERRIN AND W. XU\nand references therein). From previous positivity results and firs t explicit computations, it was\nsoon conjectured that positivity should also hold in quantum K-theory. To state a precise\nconjecture, we introduce a few notations. Schubert varieties in Xare indexed by WXand for\nu∈WX, letXube the Schubert variety with codim X(Xu) =ℓ(u) (see Subsection 2.1). Denote\nbyOu= [OXu] its class in QK( X), the (small) quantum K-theory ring of X. These classes\nform a basis for QK( X) and their products are defined by structure constants as follow s:\nOu⋆Ov=/summationdisplay\nd/summationdisplay\nw∈WXNw,d\nu,vqdOw,\nwhere the first sum runs over all effective classes d∈H2(X,Z) (see Subsection 2.2for more\ndetails). It can be proved (see [ BCMP13 ,Kat18,ACT22]) that this sum is actually finite.\nThe structure constants Nw,d\nu,vare expected to satisfy the following positivity property (see\n[LM06,BM11,BCMP18 ]):\nConjecture 1.1. Foru,v,w∈WXandd∈H2(X,Z), we have\n(−1)ℓ(uvw)+/integraltext\ndc1(TX)Nw,d\nu,v≥0.\nThis conjecture has been proven in a few cases: for minuscule spac es in [BCMP22 ] and for\nthe point/hyperplane incidence varieties in [ Ros22] and [Xu21]. In this paper, we consider the\ncase where X= IG(2,2n) is the variety of isotropic lines in a symplectic vector space. Our firs t\nresult is:\nTheorem 1.2. Conjecture 1.1is true forX= IG(2,2n).\nWe also obtain several more precise results on the structure of QK (X).\nProposition 1.3 (see Corollary 2.12and Remark 2.16).ForX= IG(2,2n)andu,v∈WX, in\nthe product Ou⋆Ov:\n(1) Terms with qdoccur only for d∈[0,2];\n(2) The powers of qthat occur form an interval.\nAlthough these properties also hold for the product [ Xu]⋆[Xv] in quantum cohomology,\nthe maximum power of qappearing in Ou⋆Ovis sometimes greater than that appearing in\n[Xu]⋆[Xv]. See Remark 7.2for more details.\nIn addition, we prove the following closed formula for multiplying with th e unique Schubert\ndivisor class. Recall that Schubert varieties in X= IG(2,2n) are indexed by pairs ( p1,p2) of\nintegers with 1 ≤p1<p2≤2nandp1+p2/\\⌉}atio\\slash= 2n+1. Explicitly, let ( Ek)k∈[1,2n]be an isotropic\nflag and set\nXp1,p2:={x∈X|Vx∩Ep1/\\⌉}atio\\slash= 0 andVx⊂Ep2},\nwhereVxis the 2-dimensional subspace in C2nrepresenting x∈X. Fora < b≤2nsuch that\na+b/\\⌉}atio\\slash≡1 mod 2n, we recursively set\nOa,b=\n\n[OXa,b] if 1 ≤a<b≤2n\nqOb,a+2nifa≤0,\n0 otherwise .\nTheorem 1.4 (see Theorem 9.5).The product O2n−2,2n⋆Oq1,q2equals:\n(1)Oq1−1,q2ifq1=q2−1;\n(2)Oq1−1,q2+Oq1,q2−1−Oq1−1,q2−1ifq1<q2−1andq1+q2/\\⌉}atio\\slash= 2n+2,2n+3;\n(3)Oq1−1,q2+Oq1,q2−1−Oq1−1,q2−2−Oq1−2,q2−1+Oq1−2,q2−2ifq1<q2−1andq1+q2=\n2n+3;\n(4)2Oq1−1,q2−1+Oq1−2,q2+Oq1,q2−2−2Oq1−2,q2−1−2Oq1−1,q2−2+Oq1−2,q2−2whenq1+q2=\n2n+2andq1/\\⌉}atio\\slash= 2, n;QUANTUM K-THEORY OF IG(2 ,2n) 3\n(4.1)2Oq1−1,q2−1+Oq1,q2−2−2Oq1−1,q2−2−Oq1−2,q2−1+Oq1−2,q2−2whenq1= 2, q2= 2n;\n(5)2Oq1−1,q2−1+Oq1−2,q2−2Oq1−2,q2−1−Oq1−1,q2−2+Oq1−2,q2−2whenq1=n, q2=n+2.\nFor quantum cohomology, Seidel [ Sei97] proved that there is a representation of π1(Aut(X))\nin the group QH( X)×\nlocof invertibles in QH( X)loc, the localization of quantum cohomology\nQH(X) where he quantum parameters are inverted. This representatio n was made explicit first\nfor cominuscule spaces and then in general in [ CMP07,CMP09,CP23]. Later, in [ BCMP22 ]\nthese results were extended to the quantum K-theory of cominuscule spaces. We prove that the\nSeidel representation in quantum K-theory also exists for X= IG(2,2n). In this case, the group\nπ1(Aut(X)) has order 2 and Seidel representation maps the unit to O2n−1,2n= 1∈QK(X) and\nthe unique non-trivial element to On−1,n= [OXn−1,n]∈QK(X).\nTheorem 1.5 (see Theorem 8.3).We have On−1,n⋆Op1,p2=Op1−n,p2−n.\nA geometric version of the above formula was conjectured in [ BCP23] for all projective ho-\nmogeneous spaces, and we prove it for X= IG(2,2n) in Theorem 1.6. We call a homoge-\nneous space Y=G/PYcominuscule if the unipotent radical of the parabolic subgroup PYis\nabelian. Let Wbe the Weyl group of Gand letwYbe the minimal length representative of\nthe longest Weyl group element modulo WY, the Weyl group of PY. A classical result states\nthatπ1(Aut(X))≃Wcomin={1} ∪ {wY∈W|Ycominuscule }. ForG= Sp2n, this group\nis{1,wLG(n,2n)}, where LG( n,2n) is the Lagrangian Grassmannian of maximal isotropic sub-\nspaces. For u,w∈W, letdmin(u,w) be the smallest power of qappearing in the quantum\ncohomology product [ Xu]⋆[Xw], and forA,B⊆X, let Γ d(A,B) be the locus swept out by\ndegreedcurves inXmeetingAandB.\nTheorem 1.6 (see Theorem 8.1).Letu∈W,w∈Wcomin, then\nΓdmin(u,w)(Xw0w,Xu) =w−1.Xwu.\nLet us sketch a few steps in our strategy for proving the positivity result (Theorem 1.2).\nFollowing [ BCMP13 ] and [Mih22], we prove in Section 2that to compute the quantum prod-\nuctOu⋆Ov, we only need to deal with irreducible source curves or source curv es with two\ncomponents: a main component of degree d−1 and a tail of degree 1. To compute structure\nconstants in K-theory, which are defined by pushforwards, we apply a result of K oll´ ar [Kol86]\n(see Theorem 2.10), which requires rational connectedness of general fibers of re strictions of the\nevaluation map. A key ingredient in proving rational connectedness of these fibers is Propo-\nsition3.13, which says that it suffices to prove rational connectedness of th e general fibers of\nrestrictions of the projection X3→X. Then, using a result of Brion [ Bri02] we reduce Theorem\n1.2to the statement that the image in Xof stable maps as above - i.e. the curve neighborhoods\n- have rational singularities and nice dimensional properties. We exp lain these reductions in\nSection3, and then prove rational connectedness results in Sections 4(d≥3),5(d= 2), and 6\n(d= 1).\nWhile proving Proposition 3.13, we also obtain the following result of general interest:\nProposition 1.7 (see Proposition 3.3).LetAbe a projective, irreducible and normal variety\nandg:A→B,h:B→Cbe surjective morphisms. If the general fibers of the maps gandh\nare rationally connected, so are the general fibers of the com positionh◦g.\nHowever, for maps with non-irreducible sources, there are a few e xplicit cases for which\nrational connectedness does not hold. We deal with these cases in Section7. We use the\ndefinition of structure constants to explicitly compute the produc t and conclude the proof of\nTheorem 1.2. Seidel representationis discussed in Section 8, and we provethe Chevalley formula\nin Section 9.\nVarieties will alwaysbe assumedtobe complex, offinite type and redu ced, but not necessarily\nirreducible.4 V. BENEDETTI, N. PERRIN AND W. XU\nAcknowledgements. The authors are partially supported by CATORE ANR-18-CE40-002 4.\nThe first two authors are also partially supported by FanoHK ANR-2 0-CE40-0023.\n2.Preliminaries and proof strategy\nInthissectionweintroducesomenotationsandproveourmainresu ltsassumingsomerational\nconnectedness results. The proof of these results will be provide d in Sections 4,5,6and7.\n2.1.Schubert varieties. LetZbe a projective rational variety homogeneous under the action\nof a reductive group G. LetB⊂Gbe a Borel subgroup and let T⊂Bbe a maximal torus.\nDenote by B−the Borel subgroup opposite to Bdefined by B∩B−=T. The variety Z\ncontains a unique B-invariant point, and we denote the parabolic subgroup stabilizing th is\npoint byPZ⊂Gand the point itself by 1 .PZ. We identify Zwith the quotient G/PZ.\nLetW=NG(T)/Tbe the Weyl group of ( G,T), letWZbe the Weyl group of PZand let\nWZ⊂Wbe the subset of minimal representatives of the cosets in W/WZ. Each element u∈W\ndefines aT-fixedpoint u.PZ∈Z, theSchubert cells andopposite Schubert cells are◦Zu=Bu.PZ\nand◦Zu=B−u.PZ, respectively, and the Schubert varieties andopposite Schubert varieties are\nZu=◦ZuandZu=◦Zu, respectively. These varieties and cells depend only on the coset uWZin\nW/WZ, and foru∈WZwe have dim( Zu) = codim(Zu,Z) =ℓ(u).\nLet≤denote the Bruhat order on W. Foru,v∈WZwe then have v≤u⇔Zv⊂Zu⇔\nZu∩Zv/\\⌉}atio\\slash=∅. In this case the intersection Zv\nu=Zu∩Zvis called a Richardson variety . This\nvariety is reduced, irreducible, rational, and dim( Zv\nu) =ℓ(u)−ℓ(v) [Ric92]. The Poincar´ e dual\nelement of u∈WZisu∨=w0uw0,Z∈WZ, which satisfies Zu∨=w0.Zuwherew0∈Wand\nw0,Z∈WZare the longest elements.\nFor a variety Y, denote by K(Y) itsK-homology group and by χY(F) the sheaf Euler-\ncharacteristic of F ∈K(Y). We set Ou= [OZu]∈K(Z) andOv= [OZv]∈K(Z). Then\n(Ou)u∈WZand (Ov)v∈WZare two basis of K(Z). The following result was proved in [ Bri02],\nand we will use it in Sections 7and9.\nTheorem 2.1. ForY⊂Za Cohen-Macaulay closed subvariety and for g∈Ggeneral, we have\n[OY] =/summationdisplay\nw∈WZχY∩gZw(OY∩gZw)[OZw(−∂Zw)].\nWhenY=Zu, we obtain:\nCorollary 2.2. Foru∈WZ, we have\nOu=/summationdisplay\nw∈WZ:w≤u[OZw(−∂Zw)].\n2.2.Quantum K-theoryand Curve neighborhoods. Foranyeffectivedegree d∈H2(Z,Z),\nletMd,n:=M0,n(Z,d) be the moduli space of genus zero degree dstable maps with nmarked\npoints and Md=Md,3(we always assume n≥3 ifd= 0). Let ev i:Md→Zfori∈[1,3]\nbe the evaluation map at the corresponding marked point. Let d= (d0,···,dr) be a sequence\nof effective curve classes and set |d|=/summationtextr\ni=0di. LetMdbe the closure in Mdof maps from a\nchain of irreducible curves of degrees ( d0,···,dr) such that the first two marked points lie on\nthe first component and the third marked point lies on the last compo nent. The varieties Md\nare irreducible with rational singularities (see [ Tho98,KP01,FP97]).\nGivenK-theoryclasses ( Fi)i∈[1,n]inK(Z), define the n-pointedK-theoreticGromov–Witten\ninvariant\n/a\\}⌊ra⌋k⌉tl⌉{tF1,···,Fn/a\\}⌊ra⌋k⌉tri}htd=χMd,n/parenleftBiggn/productdisplay\ni=1ev∗\ni(Fi)/parenrightBigg\n.QUANTUM K-THEORY OF IG(2 ,2n) 5\nThe quantum K-theory ofZis defined to be QK( Z) =K(Z)⊗Q[q] as a module. Let Zu\nandZvbe Schubert and opposite Schubert varieties. We write ( O∨\nu)u∈WZfor the dual basis\nof (Ou)u∈WZfor the Euler pairing in K(Z). Following Givental [ Giv00], we may define an\nassociative Q[q]-bilinear product ⋆on QK(Z) via\nOu⋆Ov=/summationdisplay\nd≥0/summationdisplay\nw∈WZκw,d\nu,vqdOw,\nwith\nκw,d\nu,v=/summationdisplay\n(−1)r/a\\}⌊ra⌋k⌉tl⌉{tOu,Ov,O∨\nw1/a\\}⌊ra⌋k⌉tri}htd0r/productdisplay\ni=1/a\\}⌊ra⌋k⌉tl⌉{tOwi,O∨\nwi+1/a\\}⌊ra⌋k⌉tri}htdi,\nwhere the sum is over all decompositions d=d0+···+drwithdi>0 fori>0 and all elements\nwi∈WZfori∈[1,r] where we set wr+1=w. Note that this product is well-defined since the\nabove sum is finite (see [ BCMP13 ,Kat18,ACT22]).\nWe reformulate Conjecture 1.1as follows:\nConjecture 2.3. Foru,v,w∈WZ, we have\n(−1)ℓ(uvw)+/integraltext\ndc1(TZ)κw,d\nu,v≥0.\nRemark 2.4. We have the equality\nNw,d\nu,v=κw∨,d\nu∨,v,\nwithu∨=w0uw0,Zandℓ(u∨) = dimZ−ℓ(u). In particular, the constants Nw,d\nu,vandκw,d\nu,vare\nconjectured to share the same positivity properties.\nThe Grassmannian X= IG(2,2n) of isotropic lines for a symplectic form ωonC2nis homo-\ngeneous for the symplectic group G= Sp2n. One of our main results is a proof of Conjecture\n2.3forX= IG(2,2n). From now on we assume that Z=X= IG(2,2n). Forx∈X, we denote\nbyVx⊂C2nthe 2-dimensional subspace corresponding to x. Note that Pic( X)∼=Zso that we\ncan assume that the degree dof any stable map is a non-negative integer.\n2.3.A first reduction. Givennsubvarieties Ai⊂Xfori∈[1,n], define the n-point degree\ndGromov–Witten variety asMd,n+1(A1,···,An) = ev−1\n1(A1)∩···∩ev−1\nn(An) and then-point\ndegreedcurve neighborhood as Γd(A1,···,An) = ev n+1(Md,n+1(ev−1\n1(A1)∩ ··· ∩ev−1\nn(An)).\nForn= 2, we extend this definition to non-irreducible curves and drop the reference to n: set\nMd(A,B) =Md,3(A,B)∩Mdand Γd(A,B) = ev 3(Md(A,B)).\nWe will mainly considerthese definitions for Schubert varieties and pa irs ofopposite Schubert\nvarieties. In particular, it follows from [ BCMP13 , Proposition 3.2] that Γ d(Xu) and Γ d(Xv) are\nSchubert varieties. Define u(d),v(−d)∈WXviaXu(d)= Γd(Xu) andXv(−d)= Γd(Xv). For\nopposite Schubert varieties Xu,Xv⊂X, we have the Gromov–Witten varieties Md(Xu,Xv) =\nev−1\n1(Xu)∩ev−1\n2(Xv) andMd(Xu,Xv) =Md(Xu,Xv)∩Mdwith surjective evaluation maps\nevd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv).\n[BCMP13 , Corollaries 3.1 and 3.3] and the fact that Mdis irreducible with rational singularities\nimply that the Gromov–Witten varieties Md(Xu,Xv) are irreducible with rational singularities.\nIt follows from [ BCMP13 , Lemma 5.1] and the projection formula that the product in QK( X)\ncan be reformulated using the above maps via the formula\nOu⋆Ov=/summationdisplay\nd≥0/summationdisplay\nd,|d|=d(−1)rqd(ev3)∗[OMd(Zu,Zv)].\nWe prove that, in the above formula, for the quantum product we o nly need to consider\ncurves of degree d=dord= (d−1,1). We start with the following geometric result on curves.6 V. BENEDETTI, N. PERRIN AND W. XU\nLemma 2.5. Letx,y∈Xbe general points. Then there exists a conic as well as a chain of\ntwo lines through xandy.\nProof.LetV=Vx⊕Vybe the four dimensional subspace of C2nspanned by xandy. Becausex\nandyare general, the symplectic form has rank 4 on V. ThenX∩Gr(2,V) is a smooth quadric\nof dimension 3 containing xandy, and the result follows. /square\nCorollary 2.6. Any point in Γd(Xu)can be joint to Xuby a chain of dlines.\nRemark 2.7. This result is true for any cominuscule variety [ BCMP13 , Lemma 4.7]and for any\ncoadjoint non-adjoint variety [ CP11, Proposition 3.15]. However, it is not true for homogeneous\nspaces in general (see [ CP11, Proposition 3.10] for the case of adjoint varieties for which this\ndoes not hold).\nWe include a proof of [ Mih22, Corollary 2.5] for completeness.\nProposition 2.8. Assume that for all d≥0, any point in Γd(Xu)can be joint to Xuby a chain\nofdlines. Then\nκw,d\nu,v=/a\\}⌊ra⌋k⌉tl⌉{tOu,Ov,O∨\nw/a\\}⌊ra⌋k⌉tri}htd−/summationdisplay\nκ/a\\}⌊ra⌋k⌉tl⌉{tOu,Ov,O∨\nκ/a\\}⌊ra⌋k⌉tri}htd−1/a\\}⌊ra⌋k⌉tl⌉{tOκ,O∨\nw/a\\}⌊ra⌋k⌉tri}ht1.\nProof.Define\nSd−d0=/summationdisplay\n(−1)rr/productdisplay\ni=1/a\\}⌊ra⌋k⌉tl⌉{tOwi,O∨\nwi+1/a\\}⌊ra⌋k⌉tri}htdi\nwhere the sum is over all decompositions d−d0=d1+···+drwithdi>0 fori >0 and all\nelementswi∈WXfori∈[1,r]. We have\nκw,d\nu,v=/summationdisplay\nw1∈WXd/summationdisplay\nd0=0/a\\}⌊ra⌋k⌉tl⌉{tOu,Ov,O∨\nw1/a\\}⌊ra⌋k⌉tri}htd0Sd−d0,\nthus we only need to prove that Sd−d0= 0 ford−d0≥2. As an easy consequence of [ BCMP13 ,\nProposition 3.2], we have /a\\}⌊ra⌋k⌉tl⌉{tOu,O∨\nw/a\\}⌊ra⌋k⌉tri}htd=δu(d),w. Hence,\nSd−d0=/summationdisplay\n(−1)rr/productdisplay\ni=1δwi(di),wi+1.\nForanyiandw, Γi(Xw) = Γ1(Γ1(···Γ1(Xw)···)), whereΓ 1isrepeated itimesintheright-hand\nside. By induction on i1andi2, this implies that w(i) = (w(i1))(i2) for anyi1+i2=i. Thus,\nfor any decomposition as above, we have ( ···(w1(d1))(d2)···)(dr) =w1(d1+···+dr). This\nultimately implies that/producttextr\ni=1δwi(di),wi+1=δw1(d1+···+dr),w=δw1(d−d0),wand is independent of\nthe decomposition. We thus have that Sd−d0= 0 ford−d0≥2. /square\nTo simplify notation, we set ( Ou⋆Ov)d=/summationtext\nd,|d|=d(−1)r(ev3)∗[OMd(Xu,Xv)] so that\nOu⋆Ov=/summationdisplay\nd≥0qd(Ou⋆Ov)d.\nProposition 2.8, the projection formula, and [ BCMP13 , Lemma 5.1] imply the following.\nCorollary 2.9. (Ou⋆Ov)d= (evd\n3)∗[OMd(Xu,Xv)]−(evd−1,1\n3)∗[OMd−1,1(Xu,Xv)].QUANTUM K-THEORY OF IG(2 ,2n) 7\n2.4.Proof outline. In this subsection, we prove Theorem 1.2assuming some rationality prop-\nerties that allow us to use the following theorem, which is a consequen ce of results of Koll´ ar\n[Kol86], see for example [ BCMP13 , Proposition 5.2]. We will prove these rationality properties\nin the next sections.\nTheorem 2.10. Iff:A→Bis a projective morphism of varieties with cohomologically trivial\ngeneral fiber (e.g. rationally connected) and such that AandBhave rational singularities, then\nf∗[OA] = [OB]inK-theory.\nWe will prove the following theorem in Sections 4,5and6(see Theorems 4.1,5.1and6.1).\nA dominant projective morphism f:A→BwithAirreducible is called GRCF (Generically\nRationally Connected Fibers) if a general fiber of fis rationally connected (see Definition 3.2).\nTheorem 2.11. Ford≥0, we have\n(1) The map evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)isGRCF,\n(2) Assume that evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)is birational, then\n(a)evd−1,1\n3(u,v) :Md−1,1(Xu,Xv)→Γd−1,1(Xu,Xv)is birational;\n(b)Γd−1,1(Xu,Xv)⊂Γd(Xu,Xv)is a divisor.\n(3) Assume that evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)is not birational, then Γd(Xu,Xv) =\nΓd−1,1(Xu,Xv)has rational singularities.\n(4) Ifd≥3, then the map evd−1,1\n3(u,v) :Md−1,1(Xu,Xv)→Γd−1,1(Xu,Xv)isGRCF, and\nΓd(Xu,Xv) = Γd−1,1(Xu,Xv)has rational singularities.\nCorollary 2.12. We have (Ou⋆Ov)d= 0ford≥3.\nProof.Corollary 2.9and Theorems 2.10,2.11imply the equalities\n(Ou⋆Ov)d= (ev3)∗[OMd(Xu,Xv)]−(ev3)∗[OMd−1,1(Xu,Xv)] = [OΓd(Xu,Xv)]−[OΓd−1,1(Xu,Xv)] = 0\nford≥3. /square\nFord∈[1,2], the map evd−1,1\n3(u,v) :Md−1,1(Xu,Xv)→Γd−1,1(Xu,Xv) may fail to be\nGRCF. To deal with these cases, we need to give a more explicit descr iption of the Schubert\nvarietiesXuandXv. Choose a basis ( ei)i∈[1,2n]ofC2nsuch thatω(ei,ej) =δi,2n+1−jfor\ni∈[1,n]. DefineEp=/a\\}⌊ra⌋k⌉tl⌉{tei|i∈[1,p]/a\\}⌊ra⌋k⌉tri}htandEp=/a\\}⌊ra⌋k⌉tl⌉{tei|2n+1−i∈[1,p]/a\\}⌊ra⌋k⌉tri}ht. It is a classical result\n(see for example [ BKT09]) thatWXis in bijection with pairs of integers p1< p2such that\np1+p2/\\⌉}atio\\slash= 2n+1. Explicitly, there exist integers p1<p2withp1+p2/\\⌉}atio\\slash= 2n+1 andq1<q2with\nq1+q2/\\⌉}atio\\slash= 2n+1 such that\nXu=Xp1,p2={x∈X|Vx∩Ep1/\\⌉}atio\\slash= 0 andVx⊂Ep2},\nXv=Xq1,q2={x∈X|Vx∩Eq1/\\⌉}atio\\slash= 0 andVx⊂Eq2}.\nWe will also use the corresponding Schubert cells\n◦Xu={x���X|dim(Vx∩Ep1) = 1 andVx⊂Ep2},\n◦Xv={x∈X|dim(Vx∩Eq1) = 1 andVx⊂Eq2}.\nWe setOp1,p2=Ou= [OXu] and define\nδp=/braceleftbigg0 ifp1+p2<2n+1\n1 ifp1+p2>2n+1andδq=/braceleftbigg0 ifq1+q2<2n+1\n1 ifq1+q2>2n+1.\nNotethatδp= 0ifandonlyif E⊥\np1⊇Ep2, andsimilarlyfor δq. We havedim Xu=p1+p2−3−δp,\nand similarly for Xv.\nConsider the following conditions on the pairs ( p1,p2) and (q1,q2):\n(C1) p1+q1= 2n=p2=q28 V. BENEDETTI, N. PERRIN AND W. XU\n(C2)/braceleftbigg\np1+q2= 2n=p2+q1,\np2−p1=q2−q1≥2 and max( δp,δq) = 1.\nThe following is proved in Sections 5and6, see Theorems 5.1and6.1. We denote by (C d)\nthe condition ( C1) ford= 1 and ( C2) ford= 2.\nTheorem 2.13. Assume that d∈[1,2].\n(1) If(Cd)holds, then evd−1,1\n3(u,v) :Md−1,1(Xu,Xv)→Γd−1,1(Xu,Xv)is generically\nfinite of degree 2.\n(2) Otherwise, the map evd−1,1\n3(u,v) :Md−1,1(Xu,Xv)→Γd−1,1(Xu,Xv)isGRCF.\nUsing this result, we compute ( Ou⋆Ov)qwhen (Cd), ford∈[1,2], does not hold.\nCorollary 2.14. Assume that neither (C1)nor(C2)holds, then\n(1)(Ou⋆Ov)d= 0if and only if evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)is not birational.\n(2)(−1)ℓ(uvw)+/integraltext\ndc1(TX)κw,d\nu,v≥0.\nProof.(1) This follows from Corollary 2.9and Theorems 2.10,2.11and2.13.\n(2) If the map evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv) is birational, then evd−1,1\n3(u,v) :\nMd−1,1(Xu,Xv)→Γd−1,1(Xu,Xv) is also birational by Theorem 2.11. By a result of Brion (see\n[BCMP22 , Theorem 8.11]) and that Md(Xu,Xv) andMd−1,1(Xu,Xv) have rational singulari-\nties, we have that the coefficients cwanddwin the expansions\n(ev3)∗[OMd(Xu,Xv)] =/summationdisplay\nwcwOwand (ev 3)∗[OMd−1,1(Xu,Xv)] =/summationdisplay\nwdwOw\nsatisfy (−1)ℓ(w)−ccw≥0 and (−1)ℓ(w)−ddw≥0, wherecanddare the codimensions of\nΓd(Xu,Xv) and Γ d−1,1(Xu,Xv) inX. The result follows from this, Corollary 2.9, and The-\norem2.11.(2).(b). /square\nWhen (Cd) holds for d∈[1,2], we explicitly compute the product Ou⋆Ov, whereOq1,q2p1,p2=\nOv\nu= [OXvu].\nProposition 2.15 (see Proposition 7.1).The following holds in QK(X):\n(1) If(C1)holds, then Ou⋆Ov=Ov\nu−q+qO2n−2,2n.\n(2) If(C2)holds, then\nOp1,p2⋆Oq1,q2=qOp2+q2−2n,2n\np1+q1,2n−q2+q2O2n−2,2n.\nIt is easy to verify that the above products satisfy Conjecture 2.3, proving Conjecture 2.3in\nthe remaining cases, thus completing the proof of Theorem 1.2.\nRemark 2.16. Assume that d∈[1,2] and neither ( C1) nor (C2) holds. Then ( Ou⋆Ov)d/\\⌉}atio\\slash= 0 if\nand only if evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv) is birational. If ev2\n3(u,v) is birational, then by\nLemma5.6, we havep1+q1≤2n−1, which implies that ev1\n3(u,v) is birational (see Section 6).\nTogether with Corollary 2.12and Proposition 2.15, this implies that the powers of qappearing\ninOu⋆Ovform an interval. The same argument implies that this interval property also holds\nin quantum cohomology.\n3.Replacing stable maps by evaluations\nFor proving the GRCF results stated in Theorems 2.11and2.13, we need a reduction step\nreplacing stable maps by the images of marked points.QUANTUM K-THEORY OF IG(2 ,2n) 9\n3.1.Some results on general fibers. A property is true for the general fiber of fif it is true\nforf−1(b) forbin a dense open subset of B. We start with the following classical result.\nLemma 3.1. Letf:A→Bbe a dominant morphism between irreducible varieties and le t\nΩ⊂Abe open and dense. Then Ωmeets any component of a general fiber of f.\nProof.SetW=A\\Ω. Iff(W) is not dense in B, then forb∈B\\f(W), we havef−1(b)⊂Ω.\nOtherwise, by the Fiber Dimension Theorem let U⊂Bbe the open subset such that any\ncomponent of f−1(b) has dimension dim A−dimBand any component of f−1(b)∩Whas\ndimension dim W−dimBforb∈U. Then, since dim W <dimA, any component of f−1(b)\nmeets Ω for b∈U. /square\nFor convenience, we introduce the following notations.\nDefinition 3.2. Letf:A→Bbe a dominant morphism between irreducible varieties.\n(1)fis called GIF (Generically Irreducible Fibers) if a general fiber of fis irreducible.\n(2)fis called GRCF (Generically Rationally Connected Fibers) if a general fib er offis\nrationally connected.\nProposition 3.3. Consider the following commutative diagram of dominant mor phisms between\nirreducible varieties, with Aprojective and normal:\nA\nf\n/d32/d32❆❆❆❆❆❆❆\ng\n/d15/d15\nBh/d47/d47C.\n(1) Assume that gandhareGIF, thenfisGIF.\n(2) Assume that gandhareGRCF, thenfisGRCF.\nProof.(1) SinceAis normal, the general fiber of fis normal by [ Bri02, Lemma 3], so it is\nenough to prove that this general fiber is connected. Let UB⊂Bbe an open dense subset\nsuch thatg−1(b) is irreducible for b∈UBand letUC⊂Cbe an open dense subset such that\nh−1(c) is irreducible and g−1(UB) meets any irreducible component of f−1(c) forc∈UC(use\nLemma3.1). Letc∈UCandf−1(c) =E1/coproducttext···/coproducttextEnbe its decomposition into connected\ncomponents. Since Ais projective, g(Ei)⊂h−1(c) is a closed subset for all i∈[1,n] and\nh−1(c) =∪n\ni=1g(Ei). Sinceh−1(c) is irreducible, we get that g(Ei) =h−1(c) for somei∈[1,n].\nUp to reordering the indices, we may assume that g(E1) =h−1(c). For any i∈[1,n] and\nanya∈Ei∩g−1(UB), we have g(a)∈g(E1). Therefore, in the decomposition g−1(g(a)) =\n(g−1(g(a))∩E1)/coproducttext···/coproducttext(g−1(g(a))∩En), we have ( g−1(g(a))∩E1)/\\⌉}atio\\slash=∅ /\\⌉}atio\\slash= (g−1(g(a))∩Ei).\nBut sinceg(a)∈UB, the fiberg−1(g(a)) is irreducible, thus Ei=E1andf−1(c) is connected\nand irreducible.\n(2) From (1), we have that the general fiber of fis irreducible. We also have a map g:\nf−1(c)→h−1(c) with base and general fibers rationally connected. The result follo ws from\n[GHS03, Corollary 1.3]. /square\nLemma 3.4. Letf:A→Bbe a dominant morphism between irreducible varieties. The\nfollowing conditions are equivalent:\n(1)fisGRCF.\n(2)f|ΩisGRCFfor any dense open subset Ω⊂A.\n(3)f|ΩisGRCFfor some dense open subset Ω⊂A.\nRemark 3.5. Here GRCF can be replaced by the property of having generically rat ional fibers.10 V. BENEDETTI, N. PERRIN AND W. XU\nProof.Assume that fis GRCF and let Ω ⊂Abe a dense open subset. Let U⊂Bbe a dense\nopen subset such that any component of f−1(b) meets Ω for b∈U. Thenf−1(b)∩Ω is dense\ninf−1(b) and therefore has to be rationally connected, proving the implicat ion (1)⇒(2). The\nimplication (2) ⇒(3) is clear. Assume that f|Ωis GRCF for some dense open subset Ω ⊂A.\nLetU⊂Bbe a dense open subset such that any component of f−1(b) meets Ω for b∈U. Then\nf−1(b)∩Ω is dense in f−1(b), thusf−1(b) is irreducible and rationally connected, proving the\nlast implication. /square\n3.2.Rational curves passing through points. We prove some lemmas on the geometry of\ncurves inXuseful for our reduction step proved in Section 3.3.\nLemma 3.6. Letx,y∈Xbe two points in general position and d≥4, thenΓd(x,y) =X=\nΓd−1,1(x,y).\nProof.Fixz∈Xgeneral and let V=Vx⊕Vy. Sincexandyare general, we have dim V= 4\nand the symplectic form has rank 4 on V. Pickt∈Xgeneral such that Vt⊂V. SetW=\nVt+Vz. Then we have dim W= 4 and the symplectic form has rank 4 on W. We thus have\nx,y,t∈IG(2,V) andt,z∈IG(2,W). Furthermore, IG(2 ,V) and IG(2,W) are smooth quadrics\nof dimension 3, therefore there exists a conic passing through x,yandt, and a conic as well as\na chain of two lines passing through tandz, proving the claim. /square\nLemma 3.7. Letx,y∈Xbe two points in general position and let z∈X. We have the\nequivalence:\nz∈Γ3(x,y)⇐⇒dim(Vx+Vy+Vz)≤5⇐⇒z∈Γ2,1(x,y).\nProof.Ifz∈Γ3(x,y), then by the Ker/Span technique (see [ Buc03,BKT03]), we must have\ndim(Vx+Vy+Vz)≤5. Assumethatthisconditionholds, itisenoughtoprovethat z∈Γ2,1(x,y).\nLetV=Vx⊕Vy. Sincexandyare general, we have dim V= 4 and the form has rank 4 on\nV. Since dim( V+Vz)≤5, there exists a non-zero vector v∈V∩Vz. Pickw∈Vz⊥∩V\\/a\\}⌊ra⌋k⌉tl⌉{tv/a\\}⌊ra⌋k⌉tri}ht\nand define t∈XviaVt=/a\\}⌊ra⌋k⌉tl⌉{tv,w/a\\}⌊ra⌋k⌉tri}ht. We have Vt⊂Vthusx,y,t∈IG(2,V). Since IG(2 ,V) is a\nsmooth quadric, there is a conic passing through x,y,t. On the other hand, we have v∈Vt∩Vz\nthustandzare on a line, proving the claim. /square\nLemma 3.8. Letx,y∈Xbe two points in general position and let z∈X. We have the\nequivalence:\nz∈Γ2(x,y)⇐⇒dim(Vx+Vy+Vz)≤4.\nProof.Ifz∈Γ2(x,y), then by the Ker/Span technique (see [ Buc03,BKT03]), we must have\ndim(Vx+Vy+Vz)≤4. Assumethat thisconditionholds, weprove z∈Γ2(x,y). LetV=Vx⊕Vy.\nSincexandyare general, we have dim V= 4 and the symplectic form has rank 4 on V. We\nhaveVz⊂Vthusz∈Gr(2,V)∩X. Since Gr(2 ,V)∩Xis a smooth quadric of dimension 3,\nthere is a conic passing through x,y,z. /square\n3.3.Forgetting maps. Recall the definition of Mdand the evaluation maps ev i:Md→Xfor\ni∈[1,3]. Let us start by fixing some notation on evaluation maps.\nNotation 3.9 (More evaluation maps) .For 1≤i≤3 define evd\n(i):Md→Xiby evd\n(i)=\n/producttexti\nj=1evj. Set Γ(i)\nd= evd\n(i)(Md). We have surjective morphisms:\nevd\n(i):Md→Γ(i)\ndandπd\n3: Γ(3)\nd→X,QUANTUM K-THEORY OF IG(2 ,2n) 11\nwherethelatteristheprojectionontothelastcomponent. SetΓ(3)\nd(Xu,Xv) = ev (3)(Md(Xu,Xv)).\nThe maps evd\n(3)andπd\n3induce surjective maps\nevd\n(3)(u,v) :Md(Xu,Xv)→Γ(3)\nd(Xu,Xv) andπd\n3(u,v) : Γ(3)\nd(Xu,Xv)→Γd(Xu,Xv).\nProposition 3.10. The map evd\n(3):Md→Γ(3)\ndisGRCF.\nProof.We will prove that evd\n(3)has rational general fibers and is birational if and only if d≤2.\nFord= 0, the map ev iis an isomorphism for i∈[1,3]. Ford >0, by Lemma 3.4, we can\nrestrict to the open subset of Mdgiven by stable maps of the form f:P1→Xof degreedwhich\nare birational onto their image. A degree 1 map P1→Xis an isomorphism onto its image, and\n3 general points in Γ(3)\n1determine a line. In particular, for d= 1, the map evd\n(3)is birational. A\nbirational degree 2 map P1→Xis an isomorphism onto its image. For ( x,y,z)∈Γ(3)\n2general,\nwe have dim( Vx+Vy+Vz) = 4 = Rk( ω|Vx+Vy+Vz) by Lemma 3.8. SetE=Vx+Vy+Vz, then\nx,y,z∈IG(2,E) which is a smooth quadric of dimension 3. Any conic passing though x,y,zis\ncontained in IG(2 ,E) and there is therefore a unique such conic. In particular, for d= 2, the\nmap evd\n(3)is birational.\nAssumed≥3. LetPk⊂C2n[s,t] be the subspace of homogeneous polynomials of degree k\nwith coefficients in C2nand let (p1,p2,p3) = (0,1,∞)∈(P1)3. Fora=⌊d\n2⌋andb=⌈d\n2⌉, we set\nx= (x1,x2,x3)∈X3and\nZ=\n\n(P,Q,x)∈Pa×Pb×X3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleFor [s,t]∈P1,/braceleftbigg(P∧Q)(s,t)/\\⌉}atio\\slash= 0,\nω(P,Q)(s,t) = 0,\nFori∈[1,3],[(P∧Q)(pi)] =xi\n\n.\nWe have a dominant map Z→Md, (P,Q,x)/mapsto→([s:t]/mapsto→[P(s,t)∧Q(s,t)]) with general fibers\nirreducible of constant dimension 4. In particular, dim Z=d(2n−1)+ 4n−1 and there is a\nunique irreducible component Z◦ofZof dimension dim Z. The component Z◦dominatesMd.\nLetπ:Z◦→Γ(3)\ndbe the projection on the last factors. By Lemma 3.1, the general fiber of π\ndominates the general fiber of evd\n(3). We therefore only need to prove that πis GRCF.\nLetY={(P,x)∈Pa×X3|P(s,t)/\\⌉}atio\\slash= 0 for all [ s,t]∈P1andP(pi)∈Vxifori∈[1,3]}.\nProjections induce maps p:Z→Y,p◦:Z◦→Yand ev P:Y→X3such thatπ= evP◦p◦.\nNote that ev−1\nP(x) is an open subset of a vector space and is therefore rational.\nWe first prove that Yis irreducible. Let S={P∈Pa|P(s,t)/\\⌉}atio\\slash= 0 for all [ s,t]∈P1}. The\nprojection on the first factor induces a map σ:Y→S, and we have a cartesian diagram:\nYθ/d47/d47\nσ\n/d15/d15Fl3\npr2\n/d15/d15\nSτ/d47/d47(P2n−1)3,\nwhereτ:S→(P2n−1)3is defined by τ(P) = ([P(p1)],[P(p2)],[P(p3)]), Fl = {(x,y)∈X×\nP2n−1|y⊂Vx}is the incidence variety between XandP2n−1, and the map θ:Y→Fl3is\ndefined byθ(P,x) = (x,[P(p1)],[P(p2)],[P(p3)]). Since pr2islocallytrivial with fibers( P2n−3)3,\nthe same is true for σ. Since furthermore Sis an open subset of Pa, we get that Yis irreducible\nof dimension 2 n(a+1)+3(2n−3).\nWe now consider the fibers of the map p:Z→Y. These fibers are given by an open subset\nin the set of solutions of the linear system on Qgiven by\n(1)/braceleftbiggω(P,Q)(s,t) = 0 for all [ s,t]∈P1,\nQ(pi)∈Vxifori∈[1,3].12 V. BENEDETTI, N. PERRIN AND W. XU\nNote that this linear system is generically of rank\ndimPb−(dimZ−dimp(Z))≤d−2+3(2n−2) = dim Pb−(dimZ−dimY).\nLet (ei)i∈[1,2n]be the standard basis of C2nso that the symplectic form is given by ω(ei,ej) =\nδj,2n+1−ifori∈[1,n], letP(s,t) =sae1+tae2nandletVx1=/a\\}⌊ra⌋k⌉tl⌉{te1,e2/a\\}⌊ra⌋k⌉tri}ht,Vx2=/a\\}⌊ra⌋k⌉tl⌉{te1+e2n,e2+e2n−1/a\\}⌊ra⌋k⌉tri}ht\nandVx3=/a\\}⌊ra⌋k⌉tl⌉{te2n−1,e2n/a\\}⌊ra⌋k⌉tri}ht. Then (P,x)∈Y. ForQ(s,t) =sbe2+tbe2n−1, we have ( P,Q,x)∈\np−1(P,x) and an easy check proves that the fiber p−1(P,x) is given by an open subset of the\nsolutions of a linear system of rank d−2+3(2n−2).\nLetY◦⊂Ybe the open subset where the system ( 1) has maximal rank d−2 +3(2n−2).\nThen the map p:p−1(Y◦)→Y◦is locally trivial with fiber an open subset of a vector space of\ndimension dim Z−dimY. This implies that Z◦is the unique component of Zdominating Y\nand that the map p:Z◦∩p−1(Y◦)→Y◦is locally trivial with fiber an open subset of a vector\nspace of dimension dim Z−dimY. We also have dimΓ(3)\nd= dimπ(Z◦) = dimev P(Y).\nPickx∈Γ(3)\ndgeneral. Note that ev−1\nP(x) is an opensubset ofavectorspaceand thus rational.\nSetF=π−1(x). Any irreducible component of Fmeetsp−1(Y◦) non-trivially by Lemma 3.1\nand is of dimension dim Z−dimΓ(3)\nd. LetF′be any such irreducible component. We have\na mapp:F′→ev−1\nP(x) whose general fiber over the image has dimension dim Z−dimY=\ndimF′−dimev−1\nP(x). In particular, p:F′→ev−1\nP(x) is dominant. On the other hand, since\nev−1\nP(x) meetsY◦, the mapp:F→ev−1\nP(x) is a locally trivial fibration with rational fibers over\na dense open subset of ev−1\nP(x). Therefore, Fmust be rational.\nBy Lemma 3.7, we have Γ(3)\n3dominatesX2with fibers of dimension at most 2 n. Hence,\ndimΓ(3)\n3≤2(4n−5)+2n= 10n−10. By Lemma 3.6, we have dimΓ(3)\nd= 3(4n−5) ford≥4.\nTherefore, dim Z−dimΓ(3)\n3≥6 and dimZ−dimΓ(3)\nd=d(2n−1)−8n+14≥10 whend≥4.\nThis implies that when d≥3, the fibers of evd\n(3)are of dimension at least 2. /square\nRecall that Md−1,1=Md−1×XM0,2(X,1)and considerthe map ev (4):Md−1,1→X4defined\nby ev(4)= ev1×ev2×evs×ev3, where ev sis the map induced by the fiber product structure\nand is obtained by evaluation at the singular point of the curve. Defin e Γ(4)\nd−1,1= ev(4)(Md−1,1).\nWe have a commutative diagram\n(2) Md−1,1\nev(4)\n/d41/d41❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚\nevd−1\n(3)×id\n/d15/d15\nΓ(3)\nd−1×XM0,2(X,1)id×ev2/d47/d47Γ(4)\nd−1,1.\nLemma 3.11. The projection pr : Γ(4)\nd−1,1→Γ(3)\nd−1,1isGRCF.\nProof.We need to discuss cases depending on the degree d. Ifd >4, then by Lemma 3.6,\npr−1(x,y,z) = Γ1(z) is a Schubert variety thus rational. If d= 1, then pr−1(x,y,z) ={x}. For\nd= 2 and (x,y,z) general in Γ(3)\n1,1, thenE=Vx+Vy+Vzhas dimension 4 and the restriction ω|E\nhas rank 4. Therefore, x,y,zare points on a 3-dimensional quadric Q, given byQ= IG(2,E),\nwithxandyon a line (xy) andzgeneral. It is easy to check that pr−1(x,y,z) = (xy)∩z⊥is a\nunique point on Q. Ford= 3 and (x,y,z) general in Γ(3)\n2,1, thenVx+Vy+Vzhas dimension 5, the\nrestrictionω|Vx+Vyhas rank 4 and the intersection Vz∩(Vx+Vy) has dimension 1 (by Lemma\n3.7). It is easy to check that pr−1(x,y,z) ={t∈X|(Vx+Vy)∩Vz⊂Vt⊂Vx+Vy} ≃P1.\nFinally, for d= 4, we have Γ(3)\n3,1=X3(by Lemma 3.6). For (x,y,z)∈X3general, the fiber is\ngiven by pr−1(x,y,z) ={t∈X|Vt=/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}htwitha∈Vx+Vy,b∈Vzandω(a,b) = 0}, which is\na general hyperplane section of P3×P1in its Segre embedding and therefore rational. /squareQUANTUM K-THEORY OF IG(2 ,2n) 13\nCorollary 3.12. The map evd−1,1\n(3):Md−1,1→Γ(3)\nd−1,1isGRCF.\nProof.All maps in ( 2) are surjective and Md−1,1is irreducible, projective and normal. Further-\nmore, id ×ev2is birational and the vertical map is GRCF by Proposition 3.10. By Proposition\n3.3, the map ev (4)is therefore GRCF. Finally, use Lemma 3.11. /square\nWe conclude with the main reduction result of this section.\nProposition 3.13. We have the following implications.\n(1) Ifπd\n3(u,v)isGRCF, thenevd\n3(u,v)isGRCF;\n(2) Ifπd−1,1\n3(u,v)isGRCF, thenevd−1,1\n3(u,v)isGRCF.\nProof.We have the following diagrams of surjective maps\nMd(Xu,Xv)\n/d39/d39❖❖❖❖❖❖❖❖❖❖❖❖\n/d15/d15\nΓ(3)\nd(Xu,Xv) /d47/d47Γd(Xu,Xv)andMd−1,1(Xu,Xv)\n/d40/d40◗◗◗◗◗◗◗◗◗◗◗◗◗◗\n/d15/d15\nΓ(4)\nd−1,1(Xu,Xv) /d47/d47Γd−1,1(Xu,Xv).\nSince theG-orbit ofXu×Xvis dense in X2and the evaluation maps are G-equivariant, the\ngeneral fibers of the vertical maps are general fibers of the map s\nevd\n(3):Md→Γ(3)\ndand evd−1,1\n(4):Md−1,1→Γ(4)\nd−1,1.\nFurthermore, Md(Xu,Xv) andMd−1,1(Xu,Xv) are projective, irreducible and normal, thus the\nresult follows from Propositions 3.3and3.10, Lemma 3.11, and Corollary 3.12. /square\nCorollary 3.14. IfΓd(x,y) = Γd−1,1(x,y)/\\⌉}atio\\slash=∅forx,y∈Xin general position, then we have\nΓd(Xu,Xv) = Γd−1,1(Xu,Xv). If furthermore πd\n3(u,v)isGRCF, thenevd−1,1\n3(u,v)isGRCF.\nProof.SinceMd(Xu,Xv) andMd−1,1(Xu,Xv) are irreducible, the same is true for Γ(3)\nd(Xu,Xv)\nand Γ(3)\nd−1,1(Xu,Xv). Let Ω be the open subset of X2such that Γ d(x,y) = Γd−1,1(x,y)/\\⌉}atio\\slash=∅for\n(x,y)∈Ω. Since the G-orbit ofXu×Xvis dense in X2and the evaluation maps are G-\nequivariant, we may assume Ω ∩(Xu×Xv)/\\⌉}atio\\slash=∅. Then\n{(x,y,z)∈Xu×Xv×X|(x,y)∈Ω andz∈Γd(x,y)}\n={(x,y,z)∈Xu×Xv×X|(x,y)∈Ω andz∈Γd−1,1(x,y)}\nis a dense open subset of both Γ(3)\nd(Xu,Xv) and Γ(3)\nd−1,1(Xu,Xu), proving that Γ(3)\nd(Xu,Xv) =\nΓ(3)\nd−1,1(Xu,Xv). We deduce that Γ d(Xu,Xv) = Γ d−1,1(Xu,Xv) andπd\n3(u,v) =πd−1,1\n3(u,v).\nThe last assertion follows from Proposition 3.13. /square\n3.4.Results on curve neighborhoods. Before going on, let us prove some preliminary re-\nsults on curve neighborhoods. More precisely, the next result will b e useful for proving that\nΓd(Xu,Xv) and Γ d−1,1(Xu,Xv) have rational singularities when they are equal and dis small.\nProposition 3.15. LetVkbe anyk-dimensional subspace of C2n. The subvariety {z∈X|\nVz∩Vk/\\⌉}atio\\slash= 0} ⊂Xhas rational singularities.\nProof.LetS=G(k,2n) and fors∈S, denote by Vs⊂C2nthe corresponding subspace. Define\nY={(z,s)∈X×S|Vz∩Vs/\\⌉}atio\\slash= 0}. The map Y→XisG-equivariant and therefore locally\ntrivial. Furthermore, the fiber of this map is the Schubert variety {s∈S|Vs∩Vz/\\⌉}atio\\slash= 0}and\ntherefore has rational singularities. In particular Yhas rational singularities. On the other\nhand, the fibers of the map Y→Sare of the form {z∈X|Vz∩Vs/\\⌉}atio\\slash= 0}, and it is easy to\ncheck that these varieties have constant dimension k+ 2n−4 (a birational model is given by14 V. BENEDETTI, N. PERRIN AND W. XU\n{(a,z)∈P(Vs)×X|a⊂Vz⊂a⊥}, which is a locally trivial P2n−3-bundle over P(Vs)). In\nparticular,byMiracleFlatness, theprojection Y→Sisflat. Finally, thefamily( Ys)s∈Sadmitsa\nglobalresolutionofsingularitiesgivenby the variety ˜Y={(l,z,s)∈P2n−1×X×S|l⊂Vz∩Vs}.\nWe can thus apply [ Elk78, Theorem 3], and we deduce that Yshas rational singularities for any\ns∈S. /square\nOut next result will apply when evd\n3is birational, giving a useful relationship among curve\nneighborhoods for degree ( d−1,1) curves and degree dcurves.\nLemma 3.16. Let us assume that d≤2and that evd\n3(u,v)andevd−1,1\n3(u,v)are birational.\nThenΓd−1,1(Xu,Xv)is a divisor inside Γd(Xu,Xv).\nProof.Recall that Γ d−1,1(Xu,Xv) and Γ d(Xu,Xv) are irreducible. We thus only need to com-\npute the codimension of the former in the latter. By the birationality assumption, the di-\nmension of Γ d(Xu,Xv) and Γ d−1,1(Xu,Xv) are equal to the dimension of Md(Xu,Xv) and\nMd−1,1(Xu,Xv). Note that Md−1,1is a union of boundary divisors (see [ FP97]) an is therefore\na divisor in Md. Now by Kleiman-tBertini applied to ev 1×ev2:Md→X2, we get that if\nMd−1,1(Xu,Xv) is non-empty it is a divisor in Md(Xu,Xv). /square\n4.Large degrees\nThe aim of this section is to prove the following result.\nTheorem 4.1. Ford≥3, we have\n(1) The map evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)isGRCFand never birational;\n(2) The map evd−1,1\n3(u,v) :Md−1,1(Xu,Xv)→Γd−1,1(Xu,Xv)isGRCF;\n(3)Γd(Xu,Xv) = Γd−1,1(Xu,Xv)has rational singularities.\nProof.The results follow from Propositions 3.10,3.13,4.2,4.4and Lemmas 3.6and4.3./square\nIn all cases of the Theorem above our strategy will be to use Propo sition3.13(and this\nstrategy will be used also for degree two and degree one curves). Since the case of degree d≥4\ncurves is much easier, let us deal with it separately.\nProposition 4.2. Ifd≥4, then the morphisms evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)and\nevd−1,1\n3(u,v) :Md−1,1(Xu,Xv)→Γd−1,1(Xu,Xv)areGRCF.\nProof.By assumption and Lemma 3.6, we have Γ(3)\nd(Xu,Xv) = Γ(3)\nd−1,1(Xu,Xv) =Xu×Xv×X\nand Γd(Xu,Xv) = Γd−1,1(Xu,Xv) =X. The fibers of πd\n3(u,v) andπd−1,1\n3(u,v) are isomorphic\ntoXu×Xvandarethusrationallyconnected(evenrational). Theresultfollow sfromProposition\n3.13. /square\nNotice that when d≥4, Γd(Xu,Xv) = Γd−1,1(Xu,Xv) =Xis smooth. We now focus on\ndegreed= 3 curves. We start with a general statement describing the image of the evaluation\nmap for cubics. Recall that XuandXvas incidence varieties can be defined as Xu={V2∈\nX|dim(V2∩Ep1)≥1 andV2⊂Ep2}andXv={V2∈X|dim(V2∩Eq1)≥1 andV2⊂Eq2}\nand let◦Xu⊂Xuand◦Xv⊂Xvbe the Schubert cells.\nLemma 4.3. We have Γ2,1(Xu,Xv) = Γ3(Xu,Xv) ={z∈X|dim(Vz∩(Ep2+Eq2))≥1}\nand this variety has rational singularities.\nProof.The first equality is a consequence of Lemma 3.7and of the fact that the Gorbit of\nXu×Xvis dense in X2. The fact that {z∈X|dim(Vz∩(Ep2+Eq2))≥1}has rational\nsingularities is a consequence of Proposition 3.15. So let us prove the second equality. Assume\nthatz∈Γ3(Xu,Xv), then there exists x∈Xu,y∈XvandV5∈Gr(5,2n) such that Vx+Vy+QUANTUM K-THEORY OF IG(2 ,2n) 15\nVz⊂V5. In particular dim( Vz∩(Vx+Vy))≥1 andVx+Vy⊂Ep2+Eq2, proving the inclusion\nof the LHS in the RHS. Assume now that z∈Xsatisfies dim( Vz∩(Ep2+Eq2))≥1 and pick a\nnon-zeroa∈Vz∩(Ep2+Eq2). Writea=a′+a′′witha′∈Ep2anda′′∈Eq2. Forzgeneral,\nwe may assume that a′/\\⌉}atio\\slash∈Ep1anda′′/\\⌉}atio\\slash∈Eq1.\nAssume first that p1/\\⌉}atio\\slash= 1/\\⌉}atio\\slash=q1. ThenEp1∩(a′)⊥andEq1∩(a′′)⊥are non-trivial, and we\nmay pickb′∈Ep1∩(a′)⊥andb′′∈Eq1∩(a′′)⊥. Then set Vx=/a\\}⌊ra⌋k⌉tl⌉{ta′,b′/a\\}⌊ra⌋k⌉tri}htandVy=/a\\}⌊ra⌋k⌉tl⌉{ta′′,b′′/a\\}⌊ra⌋k⌉tri}ht. We\nhavex∈Xuandy∈Xvand dim(Vx+Vy+Vz)≤5 thusz∈Γ3(x,y)⊂Γ3(Xu,Xv). Ifp1= 1\nthenp2<2n. We have Ep2⊂E⊥\np1andEp1∩(a′)⊥=Ep1, so we may proceed as above. The\nsame works if q1= 1. /square\nIn order to prove Theorem 4.1, we are left to prove the following result.\nProposition 4.4. The morphisms ev3\n3(u,v) :M3(Xu,Xv)→Γ3(Xu,Xv)andev2,1\n3(u,v) :\nM2,1(Xu,Xv)→Γ2,1(Xu,Xv)areGRCF.\nWe want to use Proposition 3.13, but we also want to avoid dealing with “badly behav-\ning” curves. Because of Lemma 3.4, we can restrict to studying the fibers of π3\n3(u,v)|U3:\nΓ(3)\n3(Xu,Xv)→Γ3(Xu,Xv), where U3⊂Γ(3)\n3(Xu,Xv) is a well-chosen dense open subset.\nWe postpone the proof of Proposition 4.4to the construction of U3and the description of its\nproperties.\nFor (x,y,z)∈Γ(3)\n3(Xu,Xv), consider the following conditions:\n(1)Vx∩Vy= 0,\n(2) dim(Vx+Vy+Vz) = 5,\n(3)Vx∈◦XuandVy∈◦Xv,\n(4)Vz∩(Vx+Vy)/\\⌉}atio\\slash⊆[(Vx∩Ep1)+Vy]∪[Vx+(Vy∩Eq1)].\nCondition ( 1) is a non-empty open condition, and by Lemma 3.7,\n{(x,y,z)∈Γ(3)\n3(Xu,Xv)|(Vx,Vy,Vz) satisfies ( 1)}=\n={(x,y,z)∈Xu×Xv×X|Vx∩Vy= 0,dim(Vx+Vy+Vz)≤5}.\nThus, (2) and (3) are non-empty open conditions, and\n{(x,y,z)∈Γ(3)\n3(Xu,Xv)|(Vx,Vy,Vz) satisfies ( 1)-(3)}=\n={(x,y,z)∈◦Xu×◦Xv×X|Vx∩Vy= 0,dim(Vx+Vy+Vz) = 5}.\nFor (x,y,z) in the latter set, dim( Vx∩Ep1) = dim(Vy∩Eq1) = 1, and ( Vx∩Ep1) +Vyand\nVx+(Vy∩Eq1) are both 3-dimensional subspaces of the 4-dimensional space Vx+Vy, so (4) is\na non-empty open condition and\nU3:={(x,y,z)∈Γ(3)\n3(Xu,Xv)|(Vx,Vy,Vz) satisfies ( 1)-(4)}\nis an open dense subset of Γ(3)\n3(Xu,Xv).\nProposition 4.5. Forz∈Γ3(Xu,Xv)general,(π3\n3(u,v))−1(z)∩U3is unirational.\nProof.Set\nE◦=/braceleftbiggEp2\\Ep1 ifEp2⊆E⊥\np1\nEp2\\(Ep1∪E⊥\np1) otherwise,\nE◦=/braceleftbigg\nEq2\\Eq1 ifEq2⊆(Eq1)⊥\nEq2\\(Eq1∪(Eq1)⊥) otherwise.\nWe have that E◦andE◦are open subsets in Ep2andEq2, respectively. Consider the map\np:Ep2×Eq2→Ep2+Eq2defined by ( a′,a′′)/mapsto→a′+a′′. Note that, since p2≥2,q2≥2, and16 V. BENEDETTI, N. PERRIN AND W. XU\nE•andE•are opposite flags, this map is surjective with fiber isomorphic to Ep2∩Eq2. The\nrestrictionp◦:E◦×E◦→Ep2+Eq2is therefore dominant with fibers over an element in the\nimage isomorphic to a non-empty open subset of Ep2∩Eq2. Fix a general zin\nΓ3(Xu,Xv) ={z∈X|dim(Vz∩(Ep2+Eq2))≥1}\n(see Lemma 4.3); we have that p−1\n◦(Vz) = (E◦×E◦)∩p−1(Vz) is a non-empty open subset of a\nvector space.\nDefine\nP:={((a′,a′′),b,c)∈p−1\n◦(Vz)×(C2n)2|b∈Ep1∩(a′)⊥\\0, c∈Eq1∩(a′′)⊥\\0}\nand set\nP◦={((a′,a′′),b,c)∈ P |dim/a\\}⌊ra⌋k⌉tl⌉{ta′,a′′,b,c/a\\}⌊ra⌋k⌉tri}ht= 4,dim(Vz+/a\\}⌊ra⌋k⌉tl⌉{ta′,a′′,b,c/a\\}⌊ra⌋k⌉tri}ht) = 5}.\nNotethat P◦isanopensubsetof P. Lemma 4.6impliesthat P◦isnon-empty. ThenProposition\n4.5follows from Lemma 4.6and Lemma 4.7below. /square\nLemma 4.6. Forz∈Γ3(Xu,Xv)general, the map P◦→(π3\n3(u,v))−1(z)∩ U3defined by\n((a′,a′′),b,c)/mapsto→(x,y)withVx=/a\\}⌊ra⌋k⌉tl⌉{ta′,b/a\\}⌊ra⌋k⌉tri}htandVy=/a\\}⌊ra⌋k⌉tl⌉{ta′′,c/a\\}⌊ra⌋k⌉tri}htis surjective.\nProof.Let (x,y,z)∈(π3\n3(u,v))−1(z)∩U3. Then by assumptions ( 1) and (2), dim(Vz∩(Vx+\nVy)) = 1. Let a∈Vz∩(Vx+Vy) be a non-zero element and write a=a′+a′′witha′∈Vxand\na′′∈Vy. By assumption ( 1), this decomposition is unique.\nWe claim that ( a′,a′′)∈p−1\n◦(Vz). Clearly, we have a′∈Ep2anda′′∈Eq2. By assumption\n(4), we have a′/\\⌉}atio\\slash∈Ep1anda′′/\\⌉}atio\\slash∈Eq1. Furthermore, if Ep2/\\⌉}atio\\slash⊆E⊥\np1, then since x∈◦Xu, we have\nVx∩E⊥\np1=Vx∩Ep1anda′/\\⌉}atio\\slash∈E⊥\np1. By a symmetric argument, if Eq2/\\⌉}atio\\slash⊆(Eq1)⊥, we have\na′′/\\⌉}atio\\slash∈(Eq1)⊥. We thus have ( a′,a′′)∈p−1\n◦(Vz).\nLetb∈Vx∩Ep1andc∈Vy∩Eq1be non-zero elements. By assumption ( 4), we have\nVx=/a\\}⌊ra⌋k⌉tl⌉{ta′,b/a\\}⌊ra⌋k⌉tri}htandVy=/a\\}⌊ra⌋k⌉tl⌉{ta′′,c/a\\}⌊ra⌋k⌉tri}ht, thusb∈(a′)⊥andc∈(a′′)⊥. Furthermore, by assumptions ( 1)\nand (2), we have (( a′,a′′),b,c)∈ P◦mapping to ( x,y), proving the result. /square\nLemma 4.7. The setPis unirational.\nProof.Consider the map P →p−1\n◦(Vz),((a′,a′′),b,c)/mapsto→(a′,a′′). We first prove that this map\nis surjective. Note that the choice of cis independent of the choice of b, so by symmetry, we\ncan concentrate on the map (( a′,a′′),b)/mapsto→(a′,a′′). We only need to prove that Ep1∩(a′)⊥is\nalways non-zero. If p1>1 the result follows. If p1= 1, thenEp2⊂E⊥\np1thusa′∈E⊥\np1and\nEp1∩(a′)⊥=Ep1, proving the result. Now, by construction of E◦andE◦, this map is the\ncomposition of vector bundles (with zero section removed), provin g the statement. /square\nFinally, we can prove Proposition 4.4.\nProof of Proposition 4.4.The result about degree d= 3 curves follows from Proposition 4.5,\nLemma3.4and Proposition 3.13. The result about degree (2 ,1) curves follows from Proposi-\ntion4.5and Corollary 3.14. /square\nWe now deal with small degree (i.e. degree two and one) curves.\n5.Degree 2\nRecall that there existsintegers p1<p2withp1+p2/\\⌉}atio\\slash= 2n+1andq1<q2withq1+q2/\\⌉}atio\\slash= 2n+1\nsuch that\nXu={x∈X|Vx∩Ep1/\\⌉}atio\\slash= 0 andVx⊂Ep2}\nXv={x∈X|Vx∩Eq1/\\⌉}atio\\slash= 0 andVx⊂Eq2}.QUANTUM K-THEORY OF IG(2 ,2n) 17\nRecall also the definition of δpandδqand condition ( C2)\n/braceleftbigg\np1+q2= 2n=p2+q1,\np2−p1=q2−q1≥2 and max( δp,δq) = 1.\nIn this section we prove the following result.\nTheorem 5.1. Assume that d= 2, we have\n(1) The map evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)isGRCF,\n(2) Ifevd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)is not birational, then we have Γd(Xu,Xv) =\nΓd−1,1(Xu,Xv)has rational singularities.\n(3) If(C2)does not hold, then evd−1,1\n3(u,v) :Md−1,1(Xu,Xv)→Γd−1,1(Xu,Xv)isGRCF.\n(4) If(C2)holds, then the map evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)is not birational,\nandevd−1,1\n3(u,v) :Md−1,1(Xu,Xv)→Γd−1,1(Xu,Xv)is generically finite of degree 2.\nProof.By Proposition 3.13, to prove part ( 1), it suffices to prove\nπ2\n3(u,v) : Γ(3)\n2(Xu,Xv)→Γ2(Xu,Xv)\nis GRCF. We are going to construct a birational morphism f:W →Γ(3)\n2(Xu,Xv) from an\nirreducible variety W, and prove that the general fiber of π=π2\n3(u,v)◦fis unirational. More\nprecisely, part ( 1) follows from Proposition 3.13, Lemma 3.4, Lemma 5.4, Lemma 5.5, and\nProposition 5.8.\nPart (2) follows from Lemmas 5.6,5.7, and Corollary 5.12.\nByProposition 3.13, toprove( 3), it sufficestoprove π1,1\n3(u,v) : Γ(3)\n1,1(Xu,Xv)→Γ1,1(Xu,Xv)\nis GRCF, which is Proposition 5.11.\nWhen(C2)holds,byCorollary 5.12, wehaveΓ 1,1(Xu,Xv)=X. Moreover,dim M1,1(Xu,Xv) =\ndimX. Part (4) follows from Lemma 5.6, Corollary 3.12, Proposition 5.11, and Lemma 3.1./square\nDefine the following open subsets of projective spaces:\n•Ifδp= 0, setP(Ep1)◦=P(Ep1) andP(Ep2)◦=P(Ep2).\n•Ifδp= 1, setP(Ep1)◦=P(Ep1)\\P(E⊥\np2) andP(Ep2)◦=P(Ep2)\\P(E⊥\np1).\n•Ifδq= 0, setP(Eq1)◦=P(Eq1) andP(Eq2)◦=P(Eq2).\n•Ifδq= 1, setP(Eq1)◦=P(Eq1)\\P(Ep2⊥) andP(Eq2)◦=P(Eq2)\\P(Eq1⊥).\nFor simplicity, set\nY=P(Ep1)◦×P(Eq2)◦×P(Eq1)◦×P(Ep2)◦.\nFor any element ([ a1],[a2],[b1],[b2],z)∈Y×X, consider the following conditions:\n(1)ω(a1,b2) = 0 =ω(a2,b1).\n(2)Vz∩/a\\}⌊ra⌋k⌉tl⌉{ta1,a2/a\\}⌊ra⌋k⌉tri}ht /\\⌉}atio\\slash= 0/\\⌉}atio\\slash=Vz∩/a\\}⌊ra⌋k⌉tl⌉{tb1,b2/a\\}⌊ra⌋k⌉tri}ht.\n(3)Vz∩/a\\}⌊ra⌋k⌉tl⌉{ta1,b2/a\\}⌊ra⌋k⌉tri}ht= 0 =Vz∩/a\\}⌊ra⌋k⌉tl⌉{tb1,a2/a\\}⌊ra⌋k⌉tri}ht.\n(4)ω(a1,b1)/\\⌉}atio\\slash= 0/\\⌉}atio\\slash=ω(a2,b2).\n(5) dim( /a\\}⌊ra⌋k⌉tl⌉{ta1,a2,b1,b2/a\\}⌊ra⌋k⌉tri}ht) = Rk(ω|/angbracketlefta1,a2,b1,b2/angbracketright) = 4.\nRemark 5.2. Conditions ( 2) and (3) imply that\ndim(Vz∩/a\\}⌊ra⌋k⌉tl⌉{ta1,a2/a\\}⌊ra⌋k⌉tri}ht) = 1 = dim( Vz∩/a\\}⌊ra⌋k⌉tl⌉{tb1,b2/a\\}⌊ra⌋k⌉tri}ht).\nSetW={([a1],[a2],[b1],[b2],z)∈Y×X|(1)-(5) are satisfied }.Note that Wis locally closed\ninY×X.\nDefinep:Ep2×Eq1→Ep2+Eq1andq:Ep1×Eq2→Ep1+Eq2.\nRemark 5.3. Denote the second projection from Wbyπ′:W →X. By Remark 5.2, the fiber\nπ′−1(z) can be identified with {[a1,a2],[b1,b2]∈P(q−1(Vz))×P(p−1(Vz))|([a1],[a2],[b1],[b2])∈\nYand satisfies ( 1),(4),(5)}.18 V. BENEDETTI, N. PERRIN AND W. XU\nLemma 5.4. The subset W ⊂Y×Xis irreducible and dimW= dimM2(Xu,Xv).\nProof.Note that conditions ( 3)-(5) are open. Define\nS={([a1],[a2],[b1],[b2])∈Y|(1),(4) and (5) are satisfied }\nand letp:S →P(Ep1)◦×P(Eq2)◦be the projection on the first two factors. The map pis\nsurjective with fibers of constant dimension p2+q1−2−δp−δq. Since these fibers are given\nby linear conditions, we get that Sis irreducible of dimension p1+p2+q1+q2−4−δp−δq.\nWe now consider the map p′:W → S obtained by projection on the first four factors. Fix\n([a1],[a2],[b1],[b2])∈ Swe look for zin the fiber of p′. By conditions ( 2) and (3) the space Vz\nmeets/a\\}⌊ra⌋k⌉tl⌉{ta1,a2/a\\}⌊ra⌋k⌉tri}htand/a\\}⌊ra⌋k⌉tl⌉{tb1,b2/a\\}⌊ra⌋k⌉tri}htin dimension exactly 1. Given [ a]∈P(/a\\}⌊ra⌋k⌉tl⌉{ta1,a2/a\\}⌊ra⌋k⌉tri}ht)\\{[a1],[a2]}, condition\n(4) implies that there is a unique zin the fiber such that a∈Vz. This implies that p′is a locally\ntrivialC×-bundle over Sand thus is irreducible of dimension p1+p2+q1+q2−3−δp−δq=\ndimM2(Xu,Xv). /square\nLemma 5.5. The mapf:W →Γ(3)\n2(Xu,Xv)given byf([a1],[a2],[b1],[b2],z) = (x,y,z)with\nVx=/a\\}⌊ra⌋k⌉tl⌉{ta1,b2/a\\}⌊ra⌋k⌉tri}htandVy=/a\\}⌊ra⌋k⌉tl⌉{ta2,b1/a\\}⌊ra⌋k⌉tri}htis well-defined and birational.\nProof.Since dim( /a\\}⌊ra⌋k⌉tl⌉{ta1,a2,b1,b2/a\\}⌊ra⌋k⌉tri}ht) = 4, we have that VxandVyhave dimension 2 and are isotropic\nby the conditions ω(a1,b2) =ω(a2,b1) = 0. This implies that x∈Xuandy∈Xvand since\nVz⊂Vx+Vywe have that ( x,y,z)∈Γ(3)\n2(Xu,Xv) by Lemma 3.8.\nConversely, for ( x,y,z) generalin Γ(3)\n2(Xu,Xv), we may assume that x∈◦Xuandy∈◦Xv. We\nthus have dim( Vx∩Ep1) = 1 so that [ a1] is uniquely given by /a\\}⌊ra⌋k⌉tl⌉{ta1/a\\}⌊ra⌋k⌉tri}ht=Vx∩Ep1. We also have that\n[b1] is given by /a\\}⌊ra⌋k⌉tl⌉{tb1/a\\}⌊ra⌋k⌉tri}ht=Vy∩Eq1. Furthermore, we may also assume that Vx∩Vz=Vy∩Vz= 0\nthus (/a\\}⌊ra⌋k⌉tl⌉{ta1/a\\}⌊ra⌋k⌉tri}ht+Vz)∩Vyhas dimension 1 and [ a2] is fixed by the condition /a\\}⌊ra⌋k⌉tl⌉{ta2/a\\}⌊ra⌋k⌉tri}ht= (/a\\}⌊ra⌋k⌉tl⌉{ta1/a\\}⌊ra⌋k⌉tri}ht+Vz)∩Vy.\nBy the same argument [ b2] is fixed by the condition /a\\}⌊ra⌋k⌉tl⌉{tb2/a\\}⌊ra⌋k⌉tri}ht= (/a\\}⌊ra⌋k⌉tl⌉{tb1/a\\}⌊ra⌋k⌉tri}ht+Vz)∩Vx. /square\nConsider the restriction π:W →Γ2(Xu,Xv) of the projection π′.\nLemma 5.6. The mapπis birational if and only if one of the following holds\n(1)p1+q2<2nandp2+q1<2n,\n(2)p1+q2= 2n,p2+q1<2nandmax(δp,δq) = 1,\n(3)p1+q2<2n,p2+q1= 2nandmax(δp,δq) = 1.\nProof.We only prove the if part, the converse will follow from the proof of P roposition 5.8.\nNote that if ( p1+q2= 2n,p2+q1<2n) or if (p1+q2<2n,p2+q1= 2n), we also prove the\nonly if condition. Assume first that p1+q2<2nandp2+q1<2n. Then, for z∈Γ2(Xu,Xv)\ngeneral, by Lemma 3.8, we have dim( Vz∩(Ep1+Eq2)) = 1 = dim( Vz∩(Ep2+Eq1)). This\ndefines unique elements [ a1+a2]∈P(Vz∩(Ep1+Eq2)) and [b2+b1]∈P(Vz∩(Ep2+Eq1)) with\n([a1],[a2],[b1],[b2],z)∈ W, proving part (1).\nAssume that p1+q2<2nandp2+q1= 2n(the casep1+q2= 2nandp2+q1<2nis\nsimilar). Then, for zgeneral, we have dim( Vz∩(Ep1+Eq2)) = 1 and this defines a unique\nelement [a1+a2]∈P(Vz∩(Ep1+Eq2)). For any [ b]∈P(Vz), we have a unique decomposition\nb=b1+b2with [b2]∈P(Ep2) and [b1]∈P(Eq1). If max(δp,δq) = 0 then the conditions\nω(a1,b2) =ω(a2,b1) = 0 are satisfied, and we get a fiber of dimension 1 parametrized by P(Vz).\nOtherwise, at least one of these (linear) conditions is non-trivialan d there is a unique [ b]∈P(Vz)\nsatisfying these conditions, proving part (2) and (3). /square\nLemma 5.7. Whenπis not birational,\nΓ2(Xu,Xv)⊆ {z∈X|Vz∩(Ep1+(Ep2∩Eq2)+Eq1)/\\⌉}atio\\slash= 0}.QUANTUM K-THEORY OF IG(2 ,2n) 19\nProof.Inthis case, byLemma 5.6, eitherp1+q2≥2norp2+q1≥2n. Without lossofgenerality,\nwe assume p1+q2≥2n. Note that if zis the image of a generic ( x,y,z) in Γ2(Xu,Xv), then\nVx⊂Ep2,Vy∩Eq1/\\⌉}atio\\slash= 0 andVz⊂Vx+Vy, so 0/\\⌉}atio\\slash=Vz∩(Vx+(Vy∩Eq1))⊂Vz∩(Ep2+Eq1)./square\nThe reverse inclusion will follow from the proof of Proposition 5.8.\nProposition 5.8. The general fiber of πis unirational.\nProof.In the cases of Lemma 5.6, the map is birational. The remaining cases are as follows.\nNote that the discussion proves the only if part of Lemma 5.6.\nCase 1:p1+q2≥2nandp2+q1>2n. Consider the map ga:W → {([a1],[a2],z)∈P(Ep1)◦×\nP(Eq2)◦×X|a1/\\⌉}atio\\slash∈Vz∩/a\\}⌊ra⌋k⌉tl⌉{ta1,a2/a\\}⌊ra⌋k⌉tri}ht /\\⌉}atio\\slash= 0 andω(a1,a2)/\\⌉}atio\\slash= 0}obtained by projection, and let W◦be\nthe open subset in Wwhere this map has fibers of minimal dimension. It suffices to prove th at\nF=W◦∩π−1(z) is unirational of positive dimension for z∈Γ2(Xu,Xv) general. Consider\nthe restriction ga:F→P(q−1(Vz)). A general fiber of this map is an open subset of the\nprojective space {[b1,b2]∈P(p−1(Vz))|ω(a1,b2) = 0 =ω(a2,b1)}. This projective space has\ndimension 1 + p2+q1−2n−δp−δq≥0, thus the map is dominant. Furthermore, we know\nthat fibers of this map have constant dimension. It follows that Fis unirational of dimension\np1+p2+q1+q2−4n+2−δp−δq≥1.\nCase 2:p1+q2>2nandp2+q1≥2n. This is similar to Case 1.\nCase 3:p1+q2= 2n=p2+q1. In this case, min( δp,δq) = 0, and for fixed z∈Γ2(Xu,Xv), we\nhavedimp−1(Vz) = dimq−1(Vz) = 2. Therefore, π−1(z)isbirationalto P1×P1ifmax(δp,δq) = 0\nor an open subset of a hyperplane section of bidegree (1 ,1) inP1×P1if max(δp,δq) = 1. We\nare done in the former case, in the latter case, we only need to prov e that the defining form\nis non-degenerate. We may assume p1≤q1which implies p1+ 2≤q1, and we exhibit a Vz\nfor which the form is non-degenerate. Set Vz=/a\\}⌊ra⌋k⌉tl⌉{teq1+e2n+1−q1,eq2+e2n+1−q2/a\\}⌊ra⌋k⌉tri}htifp2≥q1\nandVz=/a\\}⌊ra⌋k⌉tl⌉{tep2+e2n+1−p2,eq2+e2n+1−q2/a\\}⌊ra⌋k⌉tri}htifp2≤q1. Both cases are similar, so we only deal\nwith the latter. We have q−1(Vz) =/a\\}⌊ra⌋k⌉tl⌉{t(0,ep2+e2n+1−p2),(0,eq2+e2n+1−q2)/a\\}⌊ra⌋k⌉tri}htandp−1(Vz) =\n/a\\}⌊ra⌋k⌉tl⌉{t(ep2,e2n+1−p2),(e2n+1−q2,eq2)/a\\}⌊ra⌋k⌉tri}ht. The condition ω(a1,b2) = 0 is always trivial and the condition\nω(a2,b1) = 0 is induced by a matrix of the form\n/parenleftbigg\n±1 0\n0±1/parenrightbigg\nproving the result in this case.\nCase 4:p1+q2<2nandp2+q1>2n. Note that dim W ≥p1+q2+ 2n−3. By Lemma 5.7,\ndimΓ2(Xu,Xv)≤p1+q2+ 2n−4. Therefore, fibers of πare positive dimensional. For\nz∈Γ2(Xu,Xv) general, by Lemma 3.8, we have dim( Vz∩(Ep1+Eq2)) = 1 and therefore a\nunique element [ a1+a2]∈P(Vz∩(Ep1+Eq2)).\nFor ([a1],[a2],[b1],[b2],z)∈π−1(z) the elements [ a1] and [a2] are fixed as above. Therefore,\nπ−1(z) is birational to the linear space\n{[b1,b2]∈P(p−1(Vz))|ω(a1,b2) =ω(a2,b1) = 0}.\nCase 5:p1+q2>2nandp2+q1<2n. This is similar to Case 4.\nCase 6:p1+q2= 2n,p2+q1<2nand max(δp,δq) = 0. This follows from the proof of Lemma\n5.6part (2).\nCase 7:p1+q2<2n,p2+q1= 2nand max(δp,δq) = 0. This is similar to Case 6. /square\nCorollary 5.9. Whenπis not birational, we have\nΓ2(Xu,Xv) ={z∈X|Vz∩(Ep1+(Ep2∩Eq2)+Eq1)/\\⌉}atio\\slash= 0}\nandΓ2(Xu,Xv)has rational singularities.20 V. BENEDETTI, N. PERRIN AND W. XU\nProof.Whenp1+q2≥2n, we haveEp1+(Ep2∩Eq2)+Eq1=Ep2+Eq1; whenp2+q1≥2n,\nwe haveEp1+(Ep2∩Eq2)+Eq1=Ep1+Eq2. The result follows from Lemma 5.7and the proof\nof Proposition 5.8. The rational singularities statement follows from Proposition 3.15./square\nLemma 5.10. We have\nΓ1,1(Xu,Xv)⊆ {z∈X|Vz∩(Ep1+(Ep2∩Eq2)+Eq1)/\\⌉}atio\\slash= 0}.\nProof.Note that there is no assumption on πhere. Let ( x,y,z,t)∈Γ(4)\n1,1(Xu,Xv) be general,\nthenVt⊂Vx+Vy,Vt∩Vz/\\⌉}atio\\slash= 0 andVx+Vy⊆Ep1+(Ep2∩Eq2)+Eq1, soVz∩(Ep1+(Ep2∩\nEq2)+Eq1)/\\⌉}atio\\slash= 0, proving the result. /square\nThe reverse inclusion will follow from the proof of the following Propos ition.\nProposition 5.11. (1)Γ1,1(Xu,Xv)is non-empty if and only if p2+q2>2n.\n(2) If(C2)does not hold, then π1,1\n3(u,v)is GRCF.\n(3) If(C2)holds, then π1,1\n3(u,v)is generically finite of degree 2.\nProof.Note that if p2+q2≤2n, then Γ 1,1(Xu,Xv) =∅. Now assume p2+q2>2n.\nLetz∈Γ1,1(Xu,Xv) be general. We look for a∈Vz∩(Ep1+ (Ep2∩Eq2) +Eq1)\\0 with\ndecompositions a=a[1]+a[2]+a′\n[1], wherea[1]∈Ep1\\0,a[2]∈(Ep2∩Eq2)\\(Ep1∪Eq1),\na′\n[1]∈Eq1\\0, and\n(3) ω(a[2],a[1]) = 0,\n(4) ω(a[2],a′\n[1]) = 0.\nThen forVx=/a\\}⌊ra⌋k⌉tl⌉{ta[1],a[2]/a\\}⌊ra⌋k⌉tri}htandVy=/a\\}⌊ra⌋k⌉tl⌉{ta′\n[1],a[2]/a\\}⌊ra⌋k⌉tri}ht, we have that ( x,y,z)∈(π1,1\n3(u,v))−1(z). Without\nloss of generality, we assume\np2+q1≥p1+q2.\nLet\nm= max(0,p1+q2−2n).\nWritea= (a1,···,a2n) in the basis ( ei)i∈[1,2n]. Then\na[1]= (a1,···,ap1−m,ap1−m+1+λp1−m+1,···,ap1+λp1,0,···,0),\na′\n[1]= (0,···,0,a2n+1−q1+µ2n+1−q1,···,ap2+µp2,ap2+1,···,a2n),\nanda[2]=a−a[1]−a′\n[1], where the variables λiandµjsatisfy equations ( 3) and (4).\nCase 1:p2+q1>2n. This condition guarantees that there is at least one variable µp2. We first\ncompute the coefficients c1\np2andc2\np2ofµp2in equations ( 3) and (4), respectively. We get\nc1\np2=\n\n0 if p2<2n+1−p1\n−(a2n+1−p2+λ2n+1−p2) if 2n+1−p1<p2≤q2\n−a2n+1−p2 if 2n+1−p1<p2>q2\nand\nc2\np2=\n\n±a2n+1−p2ifp2<2n+1−p1\n−λ2n+1−p2if 2n+1−p1<p2≤q2\n0 if 2 n+1−p1<p2>q2.\nIfp1+p2<2n+1, then equation ( 3) is trivial, and we can solve equation ( 4) inµp2as long as\na2n+1−p2/\\⌉}atio\\slash= 0, proving the result in this case. Assume p1+p2>2n+1. Ifq1+q2<2n+ 1,\nthen equation ( 4) is trivial, and we can solve equation ( 3) inµp2as long as a2n+1−p2/\\⌉}atio\\slash= 0\nora2n+1−p2+λ2n+1−p2/\\⌉}atio\\slash= 0, proving the result in this case. We may therefore also assume\nq1+q2>2n+1. Sincep2+q1≥p1+q2, we get 2(p2+q1)≥p1+p2+q1+q2≥4n+4 thusQUANTUM K-THEORY OF IG(2 ,2n) 21\np2+q1≥2n+ 2. In particular p2−1≥2n+1−q1. We compute the coefficients c1\np2−1and\nc2\np2−1ofµp2−1in equations ( 3) and (4), respectively:\nc1\np2−1=/braceleftbigg−(a2n+2−p2+λ2n+2−p2) ifp2−1≤q2\n−a2n+2−p2 ifq2<p2−1\nand\nc2\np2−1=/braceleftbigg\n−λ2n+2−p2ifp2−1≤q2\n0 if q2<p2−1.\nIfp2≤q2, the minor associated to the variables µp2−1andµp2in the equations is equal to\n(a2n+2−p2+λ2n+2−p2)λ2n+1−p2−(a2n+1−p2+λ2n+1−p2)λ2n+2−p2so that we can solve the\nequations in this case and the result follows. If p2=q2+ 1, then the minor is equal to\na2n+1−p2λ2n+2−p2and the result follows. Otherwise, q2< p2−1. Notice that in this case\nc1\np2=−a2n+1−p2whilec2\np2= 0. So it suffices to find a coefficient isuch thatc2\niis different\nfrom zero; indeed, if we do so, then the determinant of the matrix w hose rows are ( c1\np2,c1\ni) and\n(c2\np2,c2\ni) is different from zero, and we can solve the equations with respect toµp2andµi. Note\nthat fori=q2we get\nc2\nq2=/braceleftbigg\n−λ2n+1−q2ifp1+q2≥2n+1\na2n+1−q2ifp1+q2≤2n,\nproving the result in this case.\nCase 2:p2+q1≤2n. In this case, we have Ep1⊕(Ep2∩Eq2)⊕Eq1⊆C2n. For any a∈\nVz∩(Ep1+(Ep2∩Eq2)+Eq1)\\0, we can write a=a[1]+a[2]+a′\n[1]in a unique way.\nSub-case 2.1: p1+q2<2n. ThenEp1+(Ep2∩Eq2)+Eq1/subsetnoteql C2n. Forz∈Γ1,1(Xu,Xv) general,\nwethus have Vz∩(Ep1+(Ep2∩Eq2)+Eq1) =/a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}htwitha/\\⌉}atio\\slash= 0. Forzgeneral, wehave a[1]∈Ep1\\0,\na[2]∈Ep2∩Eq2\\(Ep1∪Eq1), anda′\n[1]∈Eq1\\0. This implies the equalities Vx=/a\\}⌊ra⌋k⌉tl⌉{ta[1],a[2]/a\\}⌊ra⌋k⌉tri}htand\nVy=/a\\}⌊ra⌋k⌉tl⌉{ta′\n[1],a[2]/a\\}⌊ra⌋k⌉tri}ht, thus the map is birational.\nSub-case 2.2: p1+q2= 2n=p2+q1and (C2) does not hold. Wehave C2n=Ep1⊕(Ep2∩Eq2)⊕\nEq1. Either max( δp,δq) = 0, and the fiber over z∈Γ1,1(Xu,Xv) general is isomorphic to P(Vz).\nOtherwise, p2−p1=q2−q1= 1. Without loss of generality, we may assume p1≤q1. Then\np1≤n−2,p2≤n−1,q1≥n+ 1, andq2≤n+ 2. In particular δp= 0 andδq= 1. We\nwant to find a∈Vzsuch thatω(a[2],a′\n[1]) = 0. We have Ep2∩Eq2=/a\\}⌊ra⌋k⌉tl⌉{tep2/a\\}⌊ra⌋k⌉tri}htthusa′\n[1]∈e⊥\np2. Set\nE:=Ep1⊕(Ep2∩Eq2)⊕(Eq1∩e⊥\np2) which is of codimension 1 in C2nand forz∈Xgeneral,\nwe haveVz∩E=/a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}htwitha/\\⌉}atio\\slash= 0. This implies that the map is birational.\nSub-case 2.3: ( C2) holds. Again, we have C2n=Ep1⊕(Ep2∩Eq2)⊕Eq1. Without loss of\ngenerality, we may assume that p1≤q1. Our assumptions imply q1≥p1+ 2,q2≥p2+ 2,\nδp= 0 andδq= 1. Therefore, ( 3) is trivial. We claim that for z∈Xgeneral, equation ( 4)\nis a non-degenerate bilinear condition on P(Vz). Consider Vz=/a\\}⌊ra⌋k⌉tl⌉{tb,c/a\\}⌊ra⌋k⌉tri}ht, whereb=b1+b2+b′\n1\nandc=c1+c2+c′\n1for someb1,c1∈Ep1,b2,c2∈Ep2∩Eq2, andb′\n1,c′\n1∈Eq1. Then for\na=λb+µc∈Vzgeneral,a[2]=λb2+µc2,a′\n[1]=λb′\n1+µc′\n1, and equation ( 4) is of the form\nλ2ω(b2,b′\n1)+µ2ω(c2,c′\n1), proving the claim. /square\nCorollary 5.12. Assume that Γ1,1(Xu,Xv)is non-empty, then we have\nΓ1,1(Xu,Xv) ={z∈X|Vz∩(Ep1+(Ep2∩Eq2)+Eq1)/\\⌉}atio\\slash= 0}.\nIn particular, it has rational singularities.\nProof.This follows from Lemma 5.10, the proof of Proposition 5.11, and Proposition 3.15./square22 V. BENEDETTI, N. PERRIN AND W. XU\n6.Degree 1\nRecall Condition ( C1):p1+q1= 2n=p2=q2. In this section we prove the following result.\nTheorem 6.1. Assume that d= 1, we have\n(1) The map evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)isGRCF,\n(2) If the map evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)is not birational, then Γd(Xu,Xv) =\nΓd−1,1(Xu,Xv)has rational singularities.\n(3) If(C1)does not hold, then evd−1,1\n3(u,v) :Md−1,1(Xu,Xv)→Γd−1,1(Xu,Xv)isGRCF,\nand it is birational as soon as evd\n3(u,v)is.\n(4) If(C1)holds, then the map evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)is not birational,\nandevd−1,1\n3(u,v) :Md−1,1(Xu,Xv)→Γd−1,1(Xu,Xv)is generically finite of degree 2.\nIn fact, we will explicitly describe when ev1\n3(u,v) is birational. Consider the following condi-\ntion:\n(L1)/braceleftbigg\np1+q1≤2n−1+min(δp,δq) ifp2/\\⌉}atio\\slash= 2norq2/\\⌉}atio\\slash= 2n\np1+q1≤2n−1 if p2=q2= 2n.\nWe will see that condition ( L1) is the one that ensures that ev1\n3(u,v) is birational.\nProof of Theorem 6.1.By the proof of Proposition 3.10(respectively Corollary 3.12) whend=\n1, the map ev1\n(3)(u,v) (resp. ev0,1\n(3)(u,v)) is birational.\nIf (L1) holds then by Lemmas 6.3,6.4, and Lemma 3.1,π1\n3(u,v) is birational. If ( L1) does\nnot hold then by Lemmas 6.3,6.5and Lemma 3.1,π1\n3(u,v) is GRCF with positive dimensional\nfibers. Thus, by Proposition 3.13the map ev1\n3(u,v) is GRCF; moreover, it is birational if ( L1)\nholds, and it has positive dimensional fibers otherwise. This shows pa rt (1). We deduce part\n(2) from Lemmas 6.2and6.9.\nIf (L1) holds then by Lemmas 6.6,6.7, and Lemma 3.1,π0,1\n3(u,v) is birational. If neither\n(L1) nor (C1) hold then by Lemmas 6.6,6.8and Lemma 3.1,π0,1\n3(u,v) is GRCF with positive\ndimensional fibers. Thus, if ( C1) does not hold, by Proposition 3.13the map ev0,1\n3(u,v) is\nGRCF; moreover it is birational if ( L1) holds, and it has positive dimensional fibers otherwise.\nThis shows part ( 3). Finally, item ( 4) is a consequence of Lemma 6.9. /square\nWe start with a preliminary result about line neighborhoods. Through out this section on\ndegree 1 curves, we will use the following addition map\np:Ep1×Eq1→Ep1+Eq1⊂C2n.\nLemma 6.2. If neither (L1)nor(C1)hold then\nΓ1(Xu,Xv) = Γ0,1(Xu,Xv) ={z∈X|dim(Vz∩Ep2∩Eq2)≥1}\nhas rational singularities.\nProof.The statement about rational singularities is a consequence of Pro position 3.15. Recall\nalso the inclusion Γ 0,1(Xu,Xv)⊆Γ1(Xu,Xv). Denote the RHS of the above formula by A.\nConsiderx∈Xu,y∈Xvsuch that there exists a line passing through x,y. This means that\ndim(Vx∩Vy)≥1. Apointzbelongs to this line ifand only if Vz∩(Vx∩Vy)/\\⌉}atio\\slash= 0(andVz⊂Vx+Vy\nwheneverx/\\⌉}atio\\slash=y). This proves the inclusion Γ 1(Xu,Xv)⊆A.\nConversely, we prove that Ais contained in Γ 0,1(Xu,Xv). Since Γ 0,1(Xu,Xv) is closed, it is\nsufficient to show that a general point z∈Abelongs to Γ 0,1(Xu,Xv).\nIfp2=q2= 2nandp1+q1≥2n+1, thenA=X. Sincezis general, we may assume that\nVzis not contained in Ep1nor inEq1, and thus contains a one-dimensional subspace V1⊂Vz\nsuch thatV1∩Ep1=V1∩Eq1= 0. Consider any point ( a,b)∈p−1(V1\\0)∩((V⊥\n1∩Ep1)×Eq1)QUANTUM K-THEORY OF IG(2 ,2n) 23\n(such (a,b) always exists since dim( p−1(V1))≥2, dimEp1≥2 andV⊥\n1is a hyperplane in Ep1).\nThena∧b/\\⌉}atio\\slash= 0,a⊥V1andb⊥V1(sincea+b∈V1), and the point x∈Xsuch thatVx=/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}ht\nbelongs to Xu∩Xvand is on a line passing through z(since 0/\\⌉}atio\\slash=V1⊂Vx∩Vz). This shows\nthatz∈Γ0,1(Xu,Xv).\nWe now assume that p2/\\⌉}atio\\slash= 2norq2/\\⌉}atio\\slash= 2nand fixz∈Ageneral. Consider V1⊂Vz∩Ep2∩Eq2.\nSincezis general in A, we can assume that V1∩Ep1=V1∩Eq1= 0. Ifp1+q1≥2n+ 1\nthen construct a,bwitha∧b/\\⌉}atio\\slash= 0 as above so that zis on a line passing through x∈Xu∩Xv\nwithVx=/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}ht. We have z∈Γ0,1(Xu,Xv). Ifp1+q1= 2nthen, since ( L1) does not\nhold, we can assume that δp= 0. Since V1⊂Ep2we deduce that V⊥\n1∩Ep1=Ep1and one\ncan still find ( a,b)∈p−1(V1) witha∧b/\\⌉}atio\\slash= 0 anda⊥V1; again, one deduces as above that\nz∈Γ0,1(Xu,Xv). /square\n6.1.The case of ev1\n3(u,v).In this subsection, we prove all the results necessary to obtain pa rt\n(1) of Theorem 6.1. DefineE◦:=Ep2∩Eq2. Recall that◦Xuis the affine cell in Xu. Define\nǫ1:= max(1,p1+q1−2n+2),ǫ2:= max(1,p2+q2−4n+2), and\nU1={(x,y,z)∈Γ(3)\n1(Xu,Xv)|x,y,zdistinct,x∈◦Xu,y∈◦Xv,ω|Vx+Vy/\\⌉}atio\\slash= 0,\nVx∩Vy⊆E◦\\(Ep1∪Eq1),dim(Vz∩E◦)≤ǫ2,dim(Vz∩(Ep1+Eq1))≤ǫ1}.\nNote that U1is a dense open subset of Γ(3)\n1(Xu,Xv). The condition dim( Vz∩(Ep1+Eq1))≤ǫ1\nis essentially saying that as soon as Ep1+Eq1/\\⌉}atio\\slash=C2n, dim(Vz∩(Ep1+Eq1))≤1. Similarly, the\ncondition dim( Vz∩E◦)≤ǫ2implies that as soon as p2/\\⌉}atio\\slash= 2norq2/\\⌉}atio\\slash= 2nthen dim(Vz∩E◦)≤1.\nRecall the definition of π1\n3(u,v) : Γ(3)\n1(Xu,Xv)→Γ1(Xu,Xv) from Definition 3.9.\nLemma 6.3. Assume that q2/\\⌉}atio\\slash= 2nand letz∈Γ1(Xu,Xv)be a general point.\n(1) If(L1)holds then U1∩π1\n3(u,v)−1(z)is a single point.\n(2) If(L1)does not hold, then U1∩π1\n3(u,v)−1(z)is rationally connected and positive di-\nmensional.\nProof.Since a triple of distinct points ( x,y,z) inU1∩π1\n3(u,v)−1(z) belong to the same line, we\nhave dim(Vx∩Vy∩Vz) = 1 and dim( Vx+Vy+Vz) = 3. But then Vx∩Vyis also contained\ninVz. SetW=Vx∩Vy∩Vz. Forzgeneral we may assume that dim( Vz∩E◦) = 1, thus W\nis uniquely defined as the intersection Vz∩E◦. Denote by a,btwo vectors generating Vx∩Ep1\nandVy∩Eq1, respectively (here we are using the condition x∈◦Xu,y∈◦Xvin the definition\nofU1). SinceVx∩Vy⊂E◦\\(Ep1∪Eq1) we deduce that W∩Ep1= 0 andW∩Eq1= 0 and\nthataandbdo not belong to W. However, since dim( Vx+Vy+Vz) = 3,/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}ht∩Vz/\\⌉}atio\\slash= 0. Thus,\nthere exist ( α,β)/\\⌉}atio\\slash= (0,0) such that αa+βb∈Vz. The line generated by αa+βbis contained\ninW′:=Vz∩(Ep1+Eq1).\nIfp1+q1≤2n−1,W′is a line, and it uniquely determines /a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}ht ⊂Vxand/a\\}⌊ra⌋k⌉tl⌉{tb/a\\}⌊ra⌋k⌉tri}ht ⊂Vy. We\nobtain that Vx=W+/a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}htandVy=W+/a\\}⌊ra⌋k⌉tl⌉{tb/a\\}⌊ra⌋k⌉tri}ht, i.e.,π1\n3(u,v)−1(z)∩U1={(x,y,z)}.\nAssume that p1+q1≥2n, which implies that W′=Vz. Let us also suppose that δp≤δq(the\nother case can be treated similarly). Recall the definition of the map p:Ep1×Eq1→Ep1+Eq1,\nand define the set\nW:={(c,d)∈p−1(Vz)|c∈W⊥,dim(W+/a\\}⌊ra⌋k⌉tl⌉{tc/a\\}⌊ra⌋k⌉tri}ht) = dim(W+/a\\}⌊ra⌋k⌉tl⌉{td/a\\}⌊ra⌋k⌉tri}ht) = 2,c+d/\\⌉}atio\\slash= 0}.\nThis is an open subset of a linear subspace of p−1(Vz). Notice that since Vz⊂W⊥, we also\nhaved∈W⊥. The space Whas dimension at least 2 as soon as ( L1) does not hold, and it\nhas dimension 1 if p1+q1= 2nand 1 =δp=δq. For each point ( c,d)∈ W,Vx=W+/a\\}⌊ra⌋k⌉tl⌉{tc/a\\}⌊ra⌋k⌉tri}ht,\nandVy=W+/a\\}⌊ra⌋k⌉tl⌉{td/a\\}⌊ra⌋k⌉tri}ht, we have (x,y,z)∈π1\n3(u,v)−1(z)∩U1. Via this map Wdominates the fiber\nπ1\n3(u,v)−1(z)∩U1, which is therefore always unirational, and birational if ( L1) holds. /square24 V. BENEDETTI, N. PERRIN AND W. XU\nLemma 6.4. Assume that q2=p2= 2nand letz∈Γ1(Xu,Xv)be general. If (L1)holds then\nU1∩π1\n3(u,v)−1(z)is a single point.\nProof.Notice that in this case Vz∩E◦=Vzis not a line. However, the first steps of the proof\nof Lemma 6.3still hold if one takes Vx∩Vy∩Vzas the definition of W; then define a,b,α,β,W′\nas in the proof of Lemma 6.3.\nSincep1+q1≤2n−1,W′is a line, and it uniquely determines /a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}ht ⊂Vxand/a\\}⌊ra⌋k⌉tl⌉{tb/a\\}⌊ra⌋k⌉tri}ht ⊂Vy.\nMoreover, we have that Wis uniquely defined by W= Ker(ω|/angbracketlefta/angbracketright+Vz=/angbracketleftb/angbracketright+Vz) (here we use\nthatω|Vx+Vy/\\⌉}atio\\slash= 0). We obtain that Vx=W+/a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}htandVy=W+/a\\}⌊ra⌋k⌉tl⌉{tb/a\\}⌊ra⌋k⌉tri}ht, i.e.,{(x,y,z)}=\nπ1\n3(u,v)−1(z)∩U1. /square\nLemma 6.5. Assume that q2=p2= 2nand letz∈Γ1(Xu,Xv)be general. If (L1)does not\nhold, then U1∩π1\n3(u,v)−1(z)is rationally connected and positive dimensional.\nProof.Ifp1= 2n−1 thenXu=Xand ifq1= 2n−1 thenXv=X; in both cases the result\npresentsno difficulties, soweassume p1,q1≤2n−2. Let (x,y,z)be apoint in U1∩π1\n3(u,v)−1(z).\nThenW:=Vx∩Vy∩Vzis a line inVz. Notice that X= Γ1(Xu,Xv). This follows from Lemma\n6.2if (C1) does not hold, otherwise it follows from Lemma 6.9below. For zinXgeneral, we\nhaveVz∩(Ep1∪Eq1) = 0. With our assumption, p:Ep1×Eq1→C2nis surjective. Thus\nthere exist ( a,b)∈p−1(Vz) such that ω(a,b)/\\⌉}atio\\slash= 0,Vx=/a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}ht+WandVy=/a\\}⌊ra⌋k⌉tl⌉{tb/a\\}⌊ra⌋k⌉tri}ht+W. Consider\nall points (a,b)∈p−1(Vz) such that ω(a,b)/\\⌉}atio\\slash= 0,b⊥Wand dim( /a\\}⌊ra⌋k⌉tl⌉{ta,W/a\\}⌊ra⌋k⌉tri}ht) = dim( /a\\}⌊ra⌋k⌉tl⌉{tb,W/a\\}⌊ra⌋k⌉tri}ht) = 2.\nThen since a+b∈Vz,W⊂VzandVzis isotropic, the condition b⊥Wimpliesa⊥W. Setting\nVx=/a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}ht+WandVy=/a\\}⌊ra⌋k⌉tl⌉{tb/a\\}⌊ra⌋k⌉tri}ht+W, we get that ( x,y,z)∈ U1. Thus the fiber over zofπ1\n3(u,v)|U1\nis dominated by an open subset of A:={(a,b,w)∈p−1(Vz)×Vz|b⊥w}.\nLet pr2be the second projection from Ep1×Eq1. There exists V′⊂Eq1a (general) 2-\ndimensional subspace such that pr2(p−1(Vz)) = (Ep1∩Eq1)+V′. Consider the condition b⊥w\non ((Ep1∩Eq1) +V′)×Vz. Sinceq1≤2n−2, we have V′∩Vz= 0. By genericity of z, the\nsymplectic form defines a general bilinear form on V′×Vz, and the same genericity holds when\nthe form is restricted to (( Ep1∩Eq1)+V′)×Vz; the zero locus of this form is thus unirational.\nThis shows that Ais unirational since it is a locally trivial fibration over such a zero locus\nwith rational fiber over ( b,w) equal to {a∈Ep1|a+b∈Vz}. There exists a birational map\nP(A) :={([a,b],[w])∈P(p−1(Vz))×P(Vz)|ω(b,w) = 0} → U 1∩π1\n3(u,v)−1(z), ([a,b],[w])/mapsto→\n([/a\\}⌊ra⌋k⌉tl⌉{ta,w/a\\}⌊ra⌋k⌉tri}ht],[/a\\}⌊ra⌋k⌉tl⌉{tb,w/a\\}⌊ra⌋k⌉tri}ht],z) which is dominant; by composing with the projection A→P(A) we deduce\nthatU1∩π1\n3(u,v)−1(z) is unirational. Since ( Ep1∩Eq1) +V′is at least two-dimensional, the\ngeneral fiber of P(A)→P(Vz) is non-empty, and thus P(A) - as well as U1∩π1\n3(u,v)−1(z) - is\npositive dimensional. /square\n6.2.The case of ev0,1\n3(u,v).In this subsection we prove all the necessary results to obtain\npart (3) of Theorem 6.1. We will use the same definition of E◦as in the previous subsection.\nDefine\nU0,1:={(x,x,z)∈Γ(3)\n0,1(Xu,Xv)|x/\\⌉}atio\\slash=z,x∈◦Xu∩◦Xv,\nVx∩Ep1∩Eq1=Vz∩Vx∩Ep1=Vz∩Vx∩Eq1= 0}.\nThis is a non-empty open hence dense subset of Γ(3)\n0,1(Xu,Xv) that we will use in the following\nlemmas.\nLemma 6.6. Assume that q2/\\⌉}atio\\slash= 2nand letz∈Γ0,1(Xu,Xv)be general.\n(1) If(L1)holds, then U0,1∩π0,1\n3(u,v)−1(z)is a single point.\n(2) If(L1)does not hold, then U0,1∩π0,1\n3(u,v)−1(z)is rationally connected.QUANTUM K-THEORY OF IG(2 ,2n) 25\nProof.SetW:=Vx∩Vzfor (x,x,z)∈ U0,1. We have W∈P(A) withA=E◦∩(Ep1+Eq1).\nMoreover, for z∈Γ0,1(Xu,Xv) general, the line Wis uniquely determined by the intersection\nVz∩A. Consider the addition map\np′: (E◦∩Ep1)×(E◦∩Eq1)→E◦∩(Ep1+Eq1)\nandB:=p′−1(W)∩pr−1\n1(W⊥), where pr1is the first projection from ( E◦∩Ep1)×(E◦∩Eq1).\nThenBis a vector space. Moreover, via pr1,Bcan be realized as a locally trivial fibration over\npr1(B), so the projectivization Pof this fibration inside P(E◦∩Ep1)×P(E◦∩Eq1) is rational.\nNoticethat since W /∈P(Ep1)∪P(Eq1)by thedefinition of U0,1andsince weareintersectingwith\npr−1\n1(W⊥), it follows that if ( a,b)∈Bthen automatically 0 /\\⌉}atio\\slash=a∈W⊥and 0/\\⌉}atio\\slash=b∈W⊥. Let us\ndenote by P◦the open subset in Pdominated by points ( a,b) such that a/\\⌉}atio\\slash∈Eq1,b/\\⌉}atio\\slash∈Ep1. Then\nU0,1∩π0,1\n3(u,v)−1(z) is dominated by points ( x,x,z) such that Vx=/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}htwith [(a,b)]∈ P◦.\nThis means that P◦dominates U0,1∩π−1\n0,1(z), which is therefore unirational. Notice finally that\nif (L1) holds then the set Phas dimension equal to zero (the line Wuniquely defines the lines\n/a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}htand/a\\}⌊ra⌋k⌉tl⌉{tb/a\\}⌊ra⌋k⌉tri}ht). /square\nLemma 6.7. Assume that q2=p2= 2nand letz∈Γ0,1(Xu,Xv)be general. If (L1)holds\nthenU0,1∩π0,1\n3(u,v)−1(z)is a single point.\nProof.SetW:=Vx∩Vzfor (x,x,z)∈ U0,1andA:=Ep1+Eq1. We haveW∈P(Ep1+Eq1).\nMoreover, for general z∈Γ0,1(Xu,Xv) the lineWis uniquely determined by the intersection\nVz∩A. Notice that in this case p′=p, wherep′was defined in the proof of Lemma 6.6. The\nspaceB:=p−1(W) is one-dimensional since p1+q1≤2n−1. Let (a,b)∈Bwitha/\\⌉}atio\\slash= 0 and\nb/\\⌉}atio\\slash= 0. Then U0,1∩π0,1\n3(u,v)−1(z) is given by the single point ( x,x,z) such that Vx=/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}ht./square\nLemma 6.8. Assume that q2=p2= 2nand letz∈Γ0,1(Xu,Xv)be general. If neither (L1)\nnor(C1)hold, then U0,1∩π0,1\n3(u,v)−1(z)is rationally connected and positive dimensional.\nProof.As for irreducible lines, the case when XuorXvare equal to Xis trivial, so we assume\nq1,p1≤2n−2. Let (x,x,z) be a point in U0,1∩π0,1\n3(u,v)−1(z). ThenW:=Vx∩Vzis a\none-dimensional subspace in Vz. Notice that X= Γ0,1(Xu,Xv) by Lemma 6.2and forzinX\ngeneral,Vz∩(Ep1∪Eq1) = 0. With our assumption, the map p:Ep1×Eq1→C2nis surjective\nand not injective. In particular, there exist ( a,b)∈p−1(W) such that Vx=/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}ht=/a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}ht+W.\nConversely, consider all points ( a,b)∈p−1(W) such that b⊥Wand dim( /a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}ht+W) =\ndim(/a\\}⌊ra⌋k⌉tl⌉{tb/a\\}⌊ra⌋k⌉tri}ht+W) = 2, and a/\\⌉}atio\\slash∈Eq1,b/\\⌉}atio\\slash∈Ep1. Then since a+b∈Vz,b⊥WandVzis isotropic,\nwe have that a⊥W. SettingVx=/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}htwe get that ( x,x,z)∈ U0,1. This proves that the fiber\noverzofπ0,1\n3(u,v)|U0,1is dominated by an open subset of B:={(a,b,w)∈p−1(Vz)×Vz|b⊥\nw,a+b∈ /a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}ht}.\nLet pr2:Ep1×Eq1→Eq1be the second projection. Since p1+q1>2n, the intersection\nEp1∩Eq1is positive dimensional and contained in pr2p−1(/a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}ht) for anyw∈Vz. We claim that\nthere is no 0 /\\⌉}atio\\slash=w∈Vzsuch that pr2p−1(/a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}ht)⊂w⊥. Indeed, this would imply the inclusion\nEp1∩Eq1⊂w⊥and therefore the inclusion w⊂Vz∩(Ep1∩Eq1)⊥. This space is trivial except\nforp1+q1= 2n+ 1 for which /a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}htwould be uniquely determined. However, by genericity of\nz, thiswwill in general not satisfy the hyperplane condition pr2p−1(/a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}ht)⊂w⊥, proving the\nclaim.\nFrom the claim we deduce that everynon-zerow∈Vzdefines a (p1+q1−2n)-dimensional\naffine space w⊥∩pr2p−1(/a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}ht). This implies that Bis a locally trivial fibration over Vzwith\nfiber isomorphic to w⊥∩pr2p−1(/a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}ht), thus it is rational. There exists a birational map\nP(B) :={([a,b],[w])∈P(p−1(Vz))×P(Vz)|ω(b,w) = 0,a+b∈ /a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}ht} → U 0,1∩π0,1\n3(u,v)−1(z),\n([a,b],[w])/mapsto→([/a\\}⌊ra⌋k⌉tl⌉{ta,w/a\\}⌊ra⌋k⌉tri}ht],[/a\\}⌊ra⌋k⌉tl⌉{tb,w/a\\}⌊ra⌋k⌉tri}ht],z) which is dominant; by composing with the projection B→26 V. BENEDETTI, N. PERRIN AND W. XU\nP(B) we deduce that U0,1∩π0,1\n3(u,v)−1(z) is unirational. Since ( Ep1∩Eq1) is at least one-\ndimensional, the general fiber of P(B)→P(Vz) is non-empty, and thus P(B) - as well as\nU0,1∩π0,1\n3(u,v)−1(z) - is positive dimensional. /square\n6.3.The case of condition (C1).For lines, condition ( C1) holds exactly when ev0,1\n3(u,v) is\ngenerically finite but not birational. More precisely, the following resu lt holds.\nLemma 6.9. If(C1)holds, then Γ1(Xu,Xv) = Γ0,1(Xu,Xv) =Xandev0,1\n3(u,v)is generically\nfinite of degree 2.\nProof.By the proof of Corollary 3.12whend= 1, the map ev0,1\n(3)(u,v) is birational. Thus, it\nis sufficient to prove that π0,1\n3(u,v) is generically finite of degree 2, and by Lemma 3.1we are\nreduced to prove that, for z∈Xgeneral, U0,1∩π0,1\n3(u,v)−1(z) consists of two points. Then,\nthe first part of the proof goes as in the proof of Lemma 6.8: in this case the fiber over zof\nπ0,1\n3(u,v)|U0,1is{([a,b],w)∈P(p−1(Vz))×P(Vz)|b⊥wanda∧b/\\⌉}atio\\slash= 0,a+b∈ /a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}ht}. We again\nneed to look at elements 0 /\\⌉}atio\\slash=w∈Vzsuch that pr2p−1(/a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}ht)⊂w⊥. This condition is not empty\nanymore. In fact, it is a generic degree two hyperplane condition on Vz. Thus, there exist\nexactly two lines /a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}htand/a\\}⌊ra⌋k⌉tl⌉{tw′/a\\}⌊ra⌋k⌉tri}htthat satisfy it. Each of these two lines defines, modulo scalars,\na unique couple ( a,b) and (a′,b′), and therefore Vx=/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}htandVx′=/a\\}⌊ra⌋k⌉tl⌉{ta′,b′/a\\}⌊ra⌋k⌉tri}ht. We deduce that\nU0,1∩π0,1\n3(u,v)−1(z) ={(x,x,z),(x′,x′,z)}.\nThe fact that Γ 1(Xu,Xv) = Γ0,1(Xu,Xv) =Xis a consequence of the fact that throughout\ntheproof, the point zwasassumedtobe generalin X, andwehaveprovedthat π0,1\n3(u,v)−1(z)/\\⌉}atio\\slash=\n∅. /square\nThis was the last result needed to prove Theorem 6.1. Now we study the generically finite\ndegree 2 map above, as well as the similar one appearing for curves o f degree two, from the\npoint of view of quantum K-theory.\n7.Non rationally connected cases\nIn this section we prove positivity in QK( X) when the map evd−1,1\n3is not GRCF (which is,\nin cases ( C1) and (C2)). The reason why we need to single out these cases is the fact tha t we\nwould like to use Corollary 2.9, but we cannot apply Theorem 2.10since evd−1,1\n3(u,v) is not\nGRCF. We prove the following result.\nProposition 7.1 (see Proposition 2.15).The following holds in QK(X):\n(1) If(C1)holds, then Ou⋆Ov=Ov\nu−q+qO2n−2,2n.\n(2) If(C2)holds, then we have\nOp1,p2⋆Oq1,q2=qOp2+q2−2n,2n\np1+q1,2n−q2+q2O2n−2,2n.\nRemark 7.2. When (Cd) holds, by Theorems 5.1and6.1, fibers of evd\n3(u,v) is positive di-\nmensional. By the projection formula, this implies that the coefficient ofqdin [Xu]⋆[Xv] is\n0 in quantum cohomology. Proposition 7.1shows that in this case, the maximum power of\nqappearing in Ou⋆Ov, which isqd, is greater than the maximum power of qappearing in\n[Xu]⋆[Xv].\nTo compute the quantum product when ( C1) or (C2) holds, we will use Proposition 2.8. Our\nstrategy is to prove that the computation of the quantum produc tOu⋆Ovcan be essentially\nreduced to the computation of the class of OΓd−1(Xu,Xv). So we start with results express-\ning the class of OΓd−1(Xu,Xv)in terms of Schubert classes. For instance, for d= 1, we have\nΓd−1(Xu,Xv) =Xv\nu, and the following result expresses OXv\nuin terms of Schubert classes when\n(C1) holds.QUANTUM K-THEORY OF IG(2 ,2n) 27\nProposition 7.3. Letp∈[1,n],Xu={z∈X|Vz∩Ep/\\⌉}atio\\slash= 0}andXv={z∈X|Vz∩E2n−p/\\⌉}atio\\slash=\n0}.\n(1) Ifp<n, then\nOv\nu=Op,2n−p+2p−1/summationdisplay\nk=1Ok,2n−k−2Op−1,2n−p−3p−2/summationdisplay\nk=1Ok,2n−1−k+p−2/summationdisplay\nk=1Ok,2n−2−k.\n(2) Ifp=n, then\nOv\nu= 2n−1/summationdisplay\nk=1Ok,2n−k−On−1,n−3n−2/summationdisplay\nk=1Ok,2n−1−k+n−2/summationdisplay\nk=1Ok,2n−2−k.\nProof.LetXw=Xr1,r2, we compute χY∩gXw(OY∩gXw) forY=Xv\nuand use Theorem 2.1. We\nclaim that the following holds:\nχY∩gXw(OY∩gXw) =\n\n0 ifr1+r2<2n+1,\n0 ifr1+r2>2n+1 andr2≤2n−p,\n2 ifr1+r2= 2n+2 andr2>2n+1−p ,\n1 ifr1+r2= 2n+2 andr2= 2n+1−p,\n1 ifr1+r2>2n+2 andr2>2n−p.\nThe result then follows from Corollary 2.2.\nWe are left to prove the claim. Set Z=Y∩gXw. We have\nZ={z∈X|Vz∩Ep/\\⌉}atio\\slash= 0/\\⌉}atio\\slash=Vz∩E2n−p,Vz∩g.Er1/\\⌉}atio\\slash= 0 andVz⊂g.Er2}.\nFirst note that if z∈Z, thenEp∩g.Er2/\\⌉}atio\\slash= 0, thus we must have r2+p>2n. If this condition\nis satisfied, we have Vz⊂(Ep∩g.Er2)⊕(E2n−p∩g.Er2), thusg.Er1∩((Ep∩g.Er2)⊕(E2n−p∩\ng.Er2))/\\⌉}atio\\slash= 0. These subspaces are of dimension r1andp+r2−2n+2n−p+r2−2n= 2r2−2n\nare in general position in g.Er2, thus we also have r1+2r2−2n>r2,i.e.r1+r2>2n, proving\nthe first two cases.\nAssume that r1+r2≥2n+2. SetE=Ep∩g.Er2,F=E2n−p∩g.Er2andL=g.Er1. Let\nz∈Z, thenVz=/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}htfor somea∈E\\0 andb∈F\\0. Furthermore, there exists scalars\nλ,µ∈Csuch thatλa+µb∈L. Forzin a dense open subset of Z, we haveλ/\\⌉}atio\\slash= 0/\\⌉}atio\\slash=µ(otherwise\naorblies inL). Conversely, if c∈L∩(E⊕F) has a decomposition c=a+bwitha∈E\\0\nandb∈F\\0, thenVz=/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}htdefines a point z∈Zas soon asω(a,b) = 0. This proves that Z\nis birational to the following variety\nZ′={[c]∈P(L∩(E⊕F))|c=a+b,a∈E\\0,b∈F\\0 andω(a,b) = 0}.\nAssume that r1+r2≥2n+2 andr2= 2n+1−p, then dimE= 1 andE=/a\\}⌊ra⌋k⌉tl⌉{ta/a\\}⌊ra⌋k⌉tri}ht. We then have\nZ′={[c]∈P(L∩(E⊕F))|ω(c,a) = 0}=P(L∩(E⊕F)∩E⊥) proving the result in this case.\nFinally, assume that r1+r2≥2n+2 andr2>2n+1−p, then dimE >1. The condition\nc=a+bwithω(a,b) = 0 defines a quadratic form on P(L∩(E⊕F)), proving the result in this\ncase as well. Note that for r1+r2= 2n+2, we have P(L∩(E⊕F))≃P1so that the quadric\nis the union of two points. /square\nWe consider the case d= 2 and assume that ( C2) holds, i.e., we have p1+q2= 2n=p2+q1,\np2−p1=q2−q1≥2, and max( δp,δq) = 1. We may assume p1+p2≤q1+q2, the other case\nbeing symmetric. In this case, we have p1+p2≤2n−2,δp= 0,q1+q2≥2n+2, andδq= 1.\nNote that condition ( C2) implies the decomposition C2n=Ep1⊕(Ep2∩Eq2)⊕Eq1.\nLemma 7.4. If(C2)holds, then\nΓ1(Xu,Xv) ={z∈X|Vz∩(Ep2∩Eq2)/\\⌉}atio\\slash= 0/\\⌉}atio\\slash=Vz∩(Ep1+Eq1)}.28 V. BENEDETTI, N. PERRIN AND W. XU\nProof.Letz∈Γ1(Xu,Xv). Then there exists x∈Xuandy∈XvwithVx∩Vy/\\⌉}atio\\slash= 0 and\nVx∩Vy⊂Vz⊂Vx+Vy. In particular this proves the inclusion from left to right. Converse ly,\nletzin the right-hand side of the equality. Let a∈Vz∩(Ep2∩Eq2) andb∈Vz∩(Ep1+Eq1)\nwitha/\\⌉}atio\\slash= 0/\\⌉}atio\\slash=b. SinceEp2∩Eq2∩(Ep1+Eq1) = 0, we have Vz=/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}ht. Writeb=bp+bq\nwithbp∈Ep1andbq∈Eq1. Ifbp/\\⌉}atio\\slash= 0/\\⌉}atio\\slash=bq, then since ω(a,b) = 0 and δp= 0, we have\nω(a,bp) = 0 =ω(a,bq) and setting Vx=/a\\}⌊ra⌋k⌉tl⌉{ta,bp/a\\}⌊ra⌋k⌉tri}htandVy=/a\\}⌊ra⌋k⌉tl⌉{ta,bq/a\\}⌊ra⌋k⌉tri}ht, we havex∈Xu,y∈Xv, and\nzlies on the line joining xandy. Ifbq= 0, thenz∈Xuand sinceq1≥p1+2, we may choose\nc∈Eq1∩a⊥\\0 so that setting Vy=/a\\}⌊ra⌋k⌉tl⌉{ta,c/a\\}⌊ra⌋k⌉tri}ht, we havey∈Xv, proving the result. Finally, if\nbp= 0, thenz∈Xv. Choose any c∈Ep1\\0; sinceδp= 0, setting Vx=/a\\}⌊ra⌋k⌉tl⌉{ta,c/a\\}⌊ra⌋k⌉tri}ht, we havex∈Xu,\nproving the result. /square\nSetA=Ep1+Eq1,B=Ep2∩Eq2,p= dimA, andq= dimB. When ( C2) holds,\nwe havep=p1+q1,q=p2+q2−2n= 2n−p, andC2n=A⊕B=Ep⊕Eq. Define\nXλ={z∈X|Vz∩Ep/\\⌉}atio\\slash= 0},Xµ={z∈X|Vz∩Eq/\\⌉}atio\\slash= 0}, andXµ\nλ=Xλ∩Xµ.\nLemma 7.5. If(C2)holds, then OΓ1(Xu,Xv)=OXµ\nλ.\nProof.DefineZ1={([a],[b])∈P(Ep)×P(Eq)|ω(a,b) = 0}andZ0={([a],[b])∈P(A)×\nP(B)|ω(a,b) = 0}. We haveisomorphisms Z1→Xµ\nλandZ0→Γ1(Xu,Xv) givenby ([ a],[b])/mapsto→\n/a\\}⌊ra⌋k⌉tl⌉{ta,b/a\\}⌊ra⌋k⌉tri}ht.\nDefine the linear isomorphism ψ:C2n→C2nvia\nψ(ek) =\n\nek fork∈[1,p1]\nek+p2−p1fork∈[p1+1,p1+q1]\nek−q1fork∈[p1+q1+1,2n].\nFort∈Cdefineϕt=tIdC2n+ (1−t)ψ. We have ϕ0(Ep) =Aandϕ0(Eq) =B, while\nϕ1(Ep) =Epandϕ1(Eq) =Eq. Define\nZ={([a],[b],t)∈P(Ep)×P(Eq)×C|ω(ϕt(a),ϕt(b)) = 0}.\nThe formωt(a,b) =ω(ϕt(a),ϕt(b)) induces a bilinear form Ω t:Ep×Eq→C. This form\nhas maximal rank for t= 1 and rank 2 n−(p1+p2)≥2 fort= 0. Let U⊂Cbe the\nopen subset defined by U={t∈C|det(ϕt)/\\⌉}atio\\slash= 0 and Rk(Ω t)≥2}. Note that 0 ,1∈U.\nLetπ:Z→Cbe the third projection. We have π−1(0) =Z0andπ−1(1) =Z1. We have\na morphism e:Z→X,([a],[b],t)/mapsto→ /a\\}⌊ra⌋k⌉tl⌉{tϕt(a),ϕt(b)/a\\}⌊ra⌋k⌉tri}htwhich restricts to an isomorphism onto its\nimage onZt:=π−1(t). We therefore only need to prove that the map π−1(U)→Uis flat. Since\nZis defined by a unique non-trivial equation, any irreducible componen t ofZhas dimension\np+q−2. By Miracle Flatness, we only need to prove that the fibers of πhave dimension at\nmostp+q−3 which follows from the fact that Rk(Ω t)≥2. /square\nLemma 7.6. We have Γ1(Xv) ={W2∈X|dim(W2∩Eq2)≥1}.\nProof.IfW2∈Γ1(Xv), then there exists V2∈Xvsuch that dim( W2∩V2)≥1. SinceV2⊂Eq2,\nwe must have dim( W2∩Eq2)≥1. Therefore, Γ 1(Xv)⊆ {W2∈X|dim(W2∩Eq2)≥1}.\nFor the reverse containment, let W2∈X\\Xvsuch that dim( W2∩Eq2)≥1. Letu∈(W2∩\nEq2)\\0. IfW2∩Eq1= 0, letv∈(Eq1\\0)∩/a\\}⌊ra⌋k⌉tl⌉{tu/a\\}⌊ra⌋k⌉tri}ht⊥; otherwise, dim( W2∩Eq1) = dim(W2∩Eq2) = 1\nand letv∈(Eq2\\W2)∩ /a\\}⌊ra⌋k⌉tl⌉{tu/a\\}⌊ra⌋k⌉tri}ht⊥. By construction, dim /a\\}⌊ra⌋k⌉tl⌉{tu,v/a\\}⌊ra⌋k⌉tri}ht= 2 and /a\\}⌊ra⌋k⌉tl⌉{tu,v/a\\}⌊ra⌋k⌉tri}ht ∈Xv. Moreover,\nL={V2|dimV2= 2 and /a\\}⌊ra⌋k⌉tl⌉{tu/a\\}⌊ra⌋k⌉tri}ht ⊂V2⊂W2+/a\\}⌊ra⌋k⌉tl⌉{tv/a\\}⌊ra⌋k⌉tri}ht}is a line in Xconnecting W2andXv./square\nProof of Proposition 7.1.From Corollary 2.6and Proposition 2.8we deduce that\nκw,d\nu,v=/a\\}⌊ra⌋k⌉tl⌉{tOu,Ov,O∨\nw/a\\}⌊ra⌋k⌉tri}htd−/summationdisplay\nκ/a\\}⌊ra⌋k⌉tl⌉{tOu,Ov,O∨\nκ/a\\}⌊ra⌋k⌉tri}htd−1/a\\}⌊ra⌋k⌉tl⌉{tOκ,O∨\nw/a\\}⌊ra⌋k⌉tri}ht1.QUANTUM K-THEORY OF IG(2 ,2n) 29\nNote that /a\\}⌊ra⌋k⌉tl⌉{tOκ,O∨\nw/a\\}⌊ra⌋k⌉tri}ht1= 0 except for Γ 1(Xκ) =Xw.So fixingκwith this property, we get\nκw,d\nu,v=/a\\}⌊ra⌋k⌉tl⌉{tOu,Ov,O∨\nw/a\\}⌊ra⌋k⌉tri}htd−/summationdisplay\nκ,Γ1(Xκ)=Xw/a\\}⌊ra⌋k⌉tl⌉{tOu,Ov,O∨\nκ/a\\}⌊ra⌋k⌉tri}htd−1.\nBy Corollary 2.12, it suffices to compute κw,d\nu,vfor 0≤d≤2. Furthermore, by Theorems 2.10\nand2.11, for any degree r, we have\nOΓr(Xu,Xv)=/summationdisplay\nη/a\\}⌊ra⌋k⌉tl⌉{tOu,Ov,O∨\nη/a\\}⌊ra⌋k⌉tri}htrOη.\nIf (C1) holds, then Γ 1(Xu,Xv) =Xby Lemma 6.9; if (C2) holds, then Γ 2(Xu,Xv) =Xby\nCorollary 5.9, andOΓ1(Xu,Xv)=OXµ\nλby Lemma 7.5. The result then follows from Proposi-\ntion7.3and Lemma 7.6. /square\n8.Seidel representation\nForXsmooth and projective, Seidel introduced in [ Sei97] a representation of π1(Aut(X))\ninto the group of invertibles of QH( X)locwhere the quantum parameters are inverted. This\nrepresentation was computed explicitly in [ CMP09] (see also [ CP23]). Recently these results\nwere extended in the quantum K-theory of cominuscule spaces in [ BCP23]. We prove that these\nresults extend to X= IG(2,2n) for the quantum K-theory QK( X).\nNote thatπ1(Aut(X)) =Z/2Z. This group can be realized as a subgroup of the Weyl group\nas follows. Fix G= Sp2nandT⊂B⊂Ga maximal torus and a Borel subgroup. Let Wbe the\nWeyl group associated to ( G,T) andw0be the longest element for the length induced by B. For\nYprojective and homogeneous under G= Sp2n, letPYbe the parabolic subgroup containing a\nfixed Borel subgroup Bsuch thatY≃G/PY. For any such Y, letwYbe the minimal length\nrepresentative of w0inW/WPY. We callY=G/PYcominuscule if the unipotent radical of PY\nis abelian. We have\nπ1(Aut(X))≃Wcomin={1}∪{wY∈W|Ycominuscule }={1,wLG(n,2n)}.\nWe first prove the following geometric result. For u,w∈W, letdmin(u,w) be the smallest power\nofqappearing in the quantum product [ Xu]⋆[Xw]. By [CMP09], we have [ Xw0w]⋆[Xu] =\nqdmin(u,w)[Xwu].\nTheorem 8.1. Letu∈W,w∈Wcomin, thenΓdmin(u,w)(Xw0w,Xu) =w−1.Xwu.\nThis result is a special case of a conjecture stated in [ BCP23] (for more recent results on this\ntopic we refer to [ Tar23]). The statement is easily true for w= 1 so letw∈Wcomin\\{1}. We\nhaveXw={x∈X|Vx⊂En}andXw0w={x∈X|Vx⊂En}. We provethe following explicit\nresult which implies Theorem 8.1. LetXu=Xp1,p2:={x∈X|Vx∩Ep1/\\⌉}atio\\slash= 0 andVx⊂Ep2}\nwithp1<p2andp1+p2/\\⌉}atio\\slash= 2n+1.\nProposition 8.2. Letw∈Wcomin\\{1}and setd=dmin(u,w), we have:\n(1) Ifp2≤n, thend= 2andΓd(Xw0w,Xu) =w−1.Xp1+n,p2+n.\n(2) Ifp1≤n<p2, thend= 1andΓd(Xw0w,Xu) =w−1.Xp2−n,p1+n.\n(3) Ifp1>n, thend= 0andΓd(Xw0w,Xu) =w−1.Xp1−n,p2−n.\nProof.Note that we have w−1=wand thatwacts as follows on ( ei)i∈[1,2n]:\nw(ei) =/braceleftbiggei+nfori≤n\nei−nfori>n.\nNote also that d=dmin(u,w) = min{d|Γd(Xw0w,Xu)/\\⌉}atio\\slash=∅}.\n(1) Assume p2≤n, note that this implies δp= 0. Fore≤1 and for any z∈Γe(Xw0w,Xu),\nwe haveVz∩En∩Ep2/\\⌉}atio\\slash= 0. But the condition p2≤nimplies the vanishing En∩Ep2= 030 V. BENEDETTI, N. PERRIN AND W. XU\nproving the equality Γ e(Xw0w,Xu) =∅. Forz∈Γ2(Xw0w,Xu) general, there exists x∈Xw0w\nandy∈Xusuch thatVz⊂Vx+Vyby Lemma 3.8. In particular Vz⊂En⊕Ep2and\nVz∩(En⊕Ep1)/\\⌉}atio\\slash= 0. Since w−1.En+p1=En⊕Ep1andw−1.En+p2=En⊕Ep2, we get the\ninclusion Γ 2(Xw0w,Xu)⊂w−1.Xwu. Conversely, for z∈w−1.Xwugeneral, then Vz⊂En⊕Ep2\nandVz∩(En⊕Ep1)/\\⌉}atio\\slash= 0. Pick a basis ( a,b) ofVzsuch thata∈Vz∩(En⊕Ep1). Write\na=a1+a2andb=b1+b2witha1,b1∈En,a2∈Ep1andb2∈Ep2. Then setting Vx=/a\\}⌊ra⌋k⌉tl⌉{ta1,b1/a\\}⌊ra⌋k⌉tri}ht\nandVy=/a\\}⌊ra⌋k⌉tl⌉{ta2,b2/a\\}⌊ra⌋k⌉tri}ht, we havex∈Xw0w,y∈XuandVz⊂Vx+Vy, proving the result.\n(2) Assume p1≤n<p2. Ifz∈Xw0w∩Xu, thenVz∩En∩Ep1/\\⌉}atio\\slash= 0, but since En∩Ep1= 0,\nwe get Γ 0(Xw0w,Xu) =∅. Forz∈Γ1(Xw0w,Xu) general, there exists x∈Xw0wandy∈Xu\nsuch that dim( Vx+Vy) = 3 andVx∩Vy⊂Vz⊂Vx+Vy. In particular Vz⊂En⊕Ep1and\n0/\\⌉}atio\\slash=Vz∩Vx∩Vy⊂Vz∩(En∩Ep2). Sincew−1.En+p1=En⊕Ep1andw−1.Ep2−n=En∩Ep2,\nwe get the inclusion Γ 1(Xw0w,Xu)⊂w−1.Xwu.\nConversely, for z∈w−1.Xwugeneral, we have Vz⊂En⊕Ep1andVz∩(En∩Ep2)/\\⌉}atio\\slash= 0. Pick\na basis (a,b) ofVzsuch thata∈Vz∩(En∩Ep2). Writeb=b1+b2withb1∈Enandb2∈Ep1.\nWe haveω(a,b2) =ω(a,b)−ω(a,b1) and since a,b∈Vzanda,b1∈En, both terms vanish.\nSettingVx=/a\\}⌊ra⌋k⌉tl⌉{ta,b1/a\\}⌊ra⌋k⌉tri}htandVy=/a\\}⌊ra⌋k⌉tl⌉{ta,b2/a\\}⌊ra⌋k⌉tri}ht, we havex∈Xw0w,y∈Xu, andVx∩Vy⊂Vz⊂Vx+Vy,\nproving the result.\n(3) We have Γ 0(Xw0w,Xu) =Xu\nw0w. An element z∈Xlies inXu\nw0wif and only if we have\nVz⊂En∩Ep2andVz∩(En∩Ep1)/\\⌉}atio\\slash= 0. The result follows since w−1.Ep1−n=En∩Ep1and\nw−1.Ep2−n=En∩Ep2. /square\nTheorem 8.3. Letu∈W,w∈Wcomin, thenOu⋆Ow=qdmin(u,w)Owu.\nProof.Note that Ow0w=Owand that the maps evd\n3(w0w,u) :Md(Xw0w,Xu)→Γd(Xw0w,Xu)\nand evd−1,1\n3(w0w,u) :Md−1,1(Xw0w,Xu)→Γd−1,1(Xw0w,Xu) have general fibers which are ei-\nther empty or rationally connected. Moreover, whenever the for mer map is not birational,\nwe have Γ d(Xw0w,Xu) = Γ d−1,1(Xw0w,Xu) and that they have rational singularities (Theo-\nrem2.11). Theseand Corollary 2.9imply that the powersof qappearingin Ou⋆Owarethe same\nas those appearing in [ Xw0w]⋆[Xu]. That is to say, Ou⋆Ow=qd(Ou⋆Ow)dford=dmin(u,w).\nThe result then follows from Theorem 8.1and the fact that Γ dmin(u,w)−1,1(Xw0w,Xu) =∅since\nΓdmin(u,w)−1(Xw0w,Xu) =∅by Proposition 8.2. /square\n9.Chevalley formula\nIn this section we obtain a quantum Chevalley formula for QK( X) (see Theorem 9.5). We\nbegin by recalling the classical Chevalley formula.\n9.1.Classical Chevalley formula. Proposition 9.1below is a specialization of [ LP07], where\nwe extend the notation Oa,bby setting\nOa,b=/braceleftBigg\n[OXa,b] 1≤a<b≤2nanda+b/\\⌉}atio\\slash= 2n+1\n0 otherwise .\nProposition 9.1 (Classical Chevalley formula) .The product O2n−2,2n·Oq1,q2inK(X)is equal\nto:\n(1)Oq1−1,q2ifq1=q2−1;\n(2)Oq1−1,q2+Oq1,q2−1−Oq1−1,q2−1ifq1<q2−1andq1+q2/\\⌉}atio\\slash= 2n+2,2n+3;\n(3)Oq1−1,q2+Oq1,q2−1−Oq1−1,q2−2−Oq1−2,q2−1+Oq1−2,q2−2ifq1<q2−1andq1+q2=\n2n+3;\n(4)2·Oq1−1,q2−1+Oq1−2,q2+Oq1,q2−2−2·Oq1−2,q2−1−2·Oq1−1,q2−2+Oq1−2,q2−2when\nq1+q2= 2n+2andq1/\\⌉}atio\\slash=n;QUANTUM K-THEORY OF IG(2 ,2n) 31\n(5)2· Oq1−1,q2−1+Oq1−2,q2−2· Oq1−2,q2−1− Oq1−1,q2−2+Oq1−2,q2−2whenq1=nand\nq2=n+2.\nProof.We sketch a geometric proof different from the proof in [ LP07]. LetXu=X2n−2,2nand\nXv=Xq1,q2. SinceOu· Ov=Ov\nu, we apply Theorem 2.1toY:=Xv\nu. Letg∈Gbe general,\nXw=Xr1,r2, andZ=Y∩gXw=Xu∩Xv∩gXw. AssumeZis non-empty, then the dimension\nofXv∩gXwmust be at least one, which implies\n(5) either/braceleftBigg\nq1+r2≥2n+2\nq2+r1≥2n+1or/braceleftBigg\nq1+r2≥2n+1\nq2+r1≥2n+2.\nLetF1=Eq1∩gEr2andF2=Eq2∩gEr1. Forx∈Zgeneral, we have\ndim(Vx∩(F1+F2)∩E2n−2) = 1 andVx∩F1∩F2=Vx∩F1∩E2n−2=Vx∩F2∩E2n−2= 0.\nTherefore, the vector space Vxis spanned by nonzero vectors v1∈F1andv2∈F2such that\nv1+v2∈V:=E2n−2∩(F1+F2)\\(F1∪F2) andω(v1,v2) = 0. Consider the map given by\naddition\np:F1×F2→F1+F2.\nThenZis dominated by\nW:={(v1,v2)∈p−1(V)|ω(v1,v2) = 0}.\nNote that if dim F1≥3, then for any v2∈F2\\F1, we have dim( v2⊥∩F1)≥2 and/parenleftbig\n/a\\}⌊ra⌋k⌉tl⌉{tv2/a\\}⌊ra⌋k⌉tri}ht+(v2⊥∩F1)/parenrightbig\n∩E2n−2/\\⌉}atio\\slash= 0. This implies that Wis a locally trivial fibration over F2\\F1\nwith the fiber being an open subset of the set ofsolutions to a syste m of linear equations. Hence,\nZis unirational. Similarly, Zis unirational if dim F2≥3.\nNow assume dim Fi≤2 fori= 1,2, which implies q1+r1≤2n+1. The remaining cases are\nas follows.\nIfq1+r1= 2n+1,thenq2+r2= 2n+3. Thisimplies q2−q1=r2−r1= 1andmin {δq,δr}= 0.\nHence, the condition ω(v1,v2) = 0 is trivial and Wis a locally trivial fibration over F2\\F1with\nthe fiber being an open subset of the set of solutions to a system of linear equations. Hence, Z\nis unirational.\nIfq1+r1≤2nand min{δq,��r}= 0, thenF1∩F2=∅, which implies that p(W)∼=W; then\nZis birational to P(E2n−2∩(F1+F2))\\(P(F1)∪P(F2)).\nOtherwise, q1+q2=r1+r2=q1+r2=q2+r1= 2n+ 2. In this case, the condition\nω(v1,v2) = 0 is non-trivial and Zis birational to the vanishing set of a quadric on P1∼=\nP(E2n−2∩(F1+F2)).\nIn summary, χZ(OZ) = 0 if ( 5) does not hold; assuming ( 5), we have\nχZ(OZ) =/braceleftBigg\n2 ifq1+q2=r1+r2=q1+r2=q2+r1= 2n+2\n1 otherwise .\nThe rest follows from Corollary 2.2. /square\n9.2.Geometric properties of stable curves meeting a Schubert di visor.Let Γd(Xv) =\nev3(ev−1\n2(Xv)), where ev 1,ev2,ev3:Md→Xare the evaluation maps. From now on Xu=\nX2n−2,2nwill be a Schubert divisor and Xv=Xq1,q2.\nProposition 9.2. LetXube a Schubert divisor. We have\n(1)Γd(Xu,Xv) = Γd(Xv)for alld>0.\n(2)Γ0,1(Xu,Xv) = Γ1(Xv\nu).\n(3)Γd−1,1(Xu,Xv) = Γd(Xu,Xv) =Xfor alld>1.32 V. BENEDETTI, N. PERRIN AND W. XU\n(4) The general fibers of the maps evd\n3(u,v) :Md(Xu,Xv)→Γd(Xu,Xv)andevd−1,1\n3(u,v) :\nMd−1,1(Xu,Xv)→Γd−1,1(Xu,Xv)are rationally connected for all d≥0except for\nev0,1\n3(u,v)whenq1= 2andq2= 2n, whose general fibers consist of two points.\nProof.Part(1)followsfromthe factthatwhen d>0, adegreedcurvemustmeetthedivisor Xu.\nFor part ( 2), note that for a degree 0 curve meeting XuandXvis a point in Xu∩Xv=Xv\nu.\nFor Part ( 3), note that when d >1, Γd−1,1(Xu,Xv)⊇Γ1,1(Xu,Xv)⊇Γ1,1(Xv), where the\nlast containment follows from the fact that a degree 1 curve must m eetXu; by (1), we have\nΓd(Xu,Xv) = Γ d(Xv)⊇Γ2(Xv). Finally, Γ 1,1(Xv) = Γ 2(Xv) =Xby Lemma 2.5. Notice\nthat sinceXu=X2n,2n−2, (C2) is never satisfied and ( C1) is satisfied exactly when q1= 2 and\nq2= 2n; then part ( 4) follows from Theorem 2.11. /square\nNote that dim M1(Xu,Xv) = dimX+ 2n−1−1−codimXv= dimX+ 2n−1−1−\n(4n−2−q1−q2+δq) = dimX−2n+q1+q2−δq, and dimΓ 1(Xu,Xv) = dimΓ 1(Xv) =\ndimX−(4n−2−q2−2n+1) = dim X−2n+1+q2ifq2≤2n−1, and dimΓ 1(Xu,Xv) = dimX\nifq2= 2n. Therefore, as expected, dim M1(Xu,Xv) = dimΓ 1(Xu,Xv) if and only if ( L1) holds,\ni.e. if and only if\n(6) q1= 1, q2≤2n−1\nLemma 9.3. Γ1(Xv\nu) = Γ1(Xv)unlessq1= 1, in which case Γ1(Xv\nu) ={W2∈X|dim(W2∩\nEq2∩(E2n−2⊕E1))≥1}.\nProof.The fact that Γ 1(Xv\nu) = Γ1(Xv) is a consequence of Lemma 6.2whenq2/\\⌉}atio\\slash= 2nandq1>1\norq2= 2nandq1>2, and of Lemma 6.9whenq2= 2nandq1= 2. Therefore, let us assume\nthatq1= 1 andq2≤2n−1. Note that\nXv\nu={V2∈X|dim(V2∩E2n−2)≥1, E1⊂V2⊆Eq2}.\nSetA={W2∈X|dim(W2∩(E2n−2⊕E1)∩Eq2)≥1}. To see that Γ 1(Xv\nu)⊆A, let\nW2∈Γ1(Xv\nu); then dim( W2∩V2)≥1 for some V2∈Xv\nu. Note that V2=E1⊕(V2∩E2n−2∩\nEq2)⊆(E2n−2⊕E1)∩Eq2and it follows that Γ 1(Xv\nu)⊆A.\nFor Γ1(Xv\nu)⊇A, letW2∈A\\Xv\nuandw∈W2∩(E2n−2⊕E1)∩Eq2\\0. Ifw∈E1, then let\nz∈E2n−2∩Eq2\\0, andL={V2| /a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}ht ⊂V2⊂W2⊕/a\\}⌊ra⌋k⌉tl⌉{tz/a\\}⌊ra⌋k⌉tri}ht}is a line inXcontainingW2and the\npoint/a\\}⌊ra⌋k⌉tl⌉{tw,z/a\\}⌊ra⌋k⌉tri}htisinXv\nu. Ifw/\\⌉}atio\\slash∈E1, wecanwrite w=w1+w2, wherew1∈E1andw2∈E2n−2∩Eq2.\nSinceq1= 1 andq2≤2n−1,w1⊥w2. Therefore, L={V2| /a\\}⌊ra⌋k⌉tl⌉{tw/a\\}⌊ra⌋k⌉tri}ht ⊂V2⊂W2+/a\\}⌊ra⌋k⌉tl⌉{tw1,w2/a\\}⌊ra⌋k⌉tri}ht}is a\nline in X containing W2and the point /a\\}⌊ra⌋k⌉tl⌉{tw1,w2/a\\}⌊ra⌋k⌉tri}htis inXv\nu. /square\nRemark 9.4. The above lemma implies that [ OΓ1(Xvu)] = [Oq2−1,2n] whenq1= 1.\n9.3.Quantum Chevalley formula. By Corollary 2.9, Theorem 2.10and Proposition 9.2,\n(Ou⋆Ov)d=/braceleftBigg\n0d≥2\nOv\nud= 0\nWe thus only need to compute ( Ou⋆Ov)dwhend= 1. When ( C1) holds, i.e. when q1= 2 and\nq2= 2n, we can use Propositions 7.1and9.1to deduce:\nO2n−2,2n⋆O2,2n= 2·O1,2n−1+O2,2n−2−2·O1,2n−2−qO2n−1,2n+qO2n−2,2n.\nIf (L1) holds, i.e. if q1= 1,q2≤2n−1, because of Propositions 9.1,9.2and Lemmas 7.6,\n9.3, we have\nO2n−2,2n⋆O1,q2=O1,q2−1+qOq2,2n−qOq2−1,2n.\nIn all other cases, ( Ou⋆Ov)1= 0.QUANTUM K-THEORY OF IG(2 ,2n) 33\nLet us denote by\nOa,b=\n\n[OXa,b] 1≤a<b≤2nanda<b,\nq[OXb,2n]a= 0, b<2n,\n0 otherwise .\nTheorem 9.5 (Quantum Chevalley formula) .InQK(X), the product O2n−2,2n⋆Oq1,q2equals:\n(1)Oq1−1,q2ifq1=q2−1;\n(2)Oq1−1,q2+Oq1,q2−1−Oq1−1,q2−1ifq1<q2−1andq1+q2/\\⌉}atio\\slash= 2n+2,2n+3;\n(3)Oq1−1,q2+Oq1,q2−1−Oq1−1,q2−2−Oq1−2,q2−1+Oq1−2,q2−2ifq1<q2−1andq1+q2=\n2n+3;\n(4)2·Oq1−1,q2−1+Oq1−2,q2+Oq1,q2−2−2·Oq1−2,q2−1−2·Oq1−1,q2−2+Oq1−2,q2−2when\nq1+q2= 2n+2andq1/\\⌉}atio\\slash= 2, n;\n(4.1)2·Oq1−1,q2−1+Oq1,q2−2−2·Oq1−1,q2−2−Oq1−2,q2−1+Oq1−2,q2−2whenq1= 2, q2= 2n;\n(5)2·Oq1−1,q2−1+Oq1−2,q2−2·Oq1−2,q2−1−Oq1−1,q2−2+Oq1−2,q2−2whenq1=n, q2=n+2.\nReferences\n[ACT22] D. 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T ¸arigradschi Curve neighborhoods of Seidel pro ducts in quantum cohomology Preprint\narXiv:2309.05985.\n[Tho98] J. F. Thomsen Irreducibility of M0,n(G/P,β).Internat. J. Math , 9(3): 367–376, 1998.\n[Xu21] W. Xu Quantum K-theory of Incidence Varieties Prepri nt arXiv: 2112.13036.\nUniversit´ e Cˆ ote d’Azur, CNRS, Laboratoire J.-A. Dieudon n´ e, Parc Valrose, F-06108 Nice Cedex 2, France.\nEmail address :vladimiro.benedetti@univ-cotedazur.fr\nCentre de Math´ ematiques Laurent Schwartz (CMLS), CNRS, ´Ecole polytechnique, Institut Polytechnique de\nParis, 91120 Palaiseau, France.\nEmail address :nicolas.perrin.cmls@polytechnique.edu\nDepartment of Mathematics, 225 Stanger Street, McBryde Hal l, Virginia Tech University, Blacksburg, VA\n24061USA.\nEmail address :weihong@vt.edu"    },    {        "title": "2402.12049v2.Reinforcement_Learning_for_Optimal_Execution_when_Liquidity_is_Time_Varying.pdf",        "content": "Reinforcement Learning for Optimal Execution\nwhen Liquidity is Time-Varying\nAndrea Macrì1∗\nand Fabrizio Lillo1,2†\n1Scuola Normale Superiore, Pisa, Italy\n2Dipartimento di Matematica University of Bologna, Bologna, Italy\nFebruary 21, 2024\nAbstract\nOptimal execution is an important problem faced by any trader. Most solutions are based on\nthe assumption of constant market impact, while liquidity is known to be dynamic. Moreover,\nmodels with time-varying liquidity typically assume that it is observable, despite the fact that, in\nreality, it is latent and hard to measure in real time. In this paper we show that the use of Double\nDeep Q-learning, a form of Reinforcement Learning based on neural networks, is able to learn\noptimal trading policies when liquidity is time-varying. Specifically, we consider an Almgren-Chriss\nframework with temporary and permanent impact parameters following several deterministic and\nstochastic dynamics. Using extensive numerical experiments, we show that the trained algorithm\nlearns the optimal policy when the analytical solution is available, and overcomes benchmarks and\napproximated solutions when the solution is not available.\n1 Introduction\nThe problem of optimally executing financial trades has been extensively studied for at least two\ndecades. Building on the seminal papers of Bertsimas and Lo (1998) [5] and Almgren and Chriss\n(2000) [3], the concept of dividing large sell (buy) orders into smaller trades, considering risks and\nrewards, has been widely applied and extended through mathematical optimisation. Starting with the\nseminal, yet simplistic, Almgren-Chriss setting, which assumes an Arithmetic Brownian motion for\nmid-pricemovementswithlinearpermanentandtemporarypriceimpacts,theliteraturehasprogressed\nto more complex models, notable examples are [17, 16, 10, 12, 11, 6, 7, 15, 28]. Since these approaches\nmake use of specific market impact models, which might or might not accurately describe real data,\nthe non-parametric approach based on Reinforcement Learning (RL) has increasingly been used to\nsolve optimal liquidation problems. As described by [33], RL models an agent that sequentially learns\noptimal policies in a generalised environment, optimising a reward function with respect to future\ndiscounted cumulative rewards based on actions taken at each time step. Q-learning is one of the\nmost widely used RL algorithms. Q-learning in tabular form, was firstly applied by [26] in the context\nof optimal execution. In their setting a tabular Q-agent has to decide how many stocks to sell within\na certain time window using limit orders. The amount and the execution price at which these stocks\nare sold depend on the time left for execution, on the inventory left to be executed, and on market\nprices. Ref. [19] adopts a hybrid approach, both analytical and RL-based, where, starting from\nthe optimal execution à la Almgren-Chriss, a stylised agent, still modelled with a classic tabular Q-\nLearning algorithm, is able to modify its trade execution schedule to minimise a reward function that\ncorresponds to the Implementation Shortfall.\nMore recently, due to the recent advancements in machine learning methods, the lookup table has\nbeen replaced by deep neural nets, thus Deep Q-Learning (DQL) has been widely applied to various\nfinancial problems, including market making [1, 24], portfolio optimisation [35], trading [23, 34, 8, 20]\nand optimal trade execution [27, 13, 22, 21, 31].1For what concerns optimal execution problems, a\nmajor recent contribution is the paper by Ning et al. (2021) [27], where a Double Deep Q-Network\n∗andrea.macri@sns.it\n†fabrizio.lillo@sns.it\n1For a more comprehensive overview on the state of the art on RL methods in finance please refer to [32] and [18]\n1arXiv:2402.12049v2  [q-fin.TR]  20 Feb 2024(DDQN) algorithm for optimal execution is introduced, overcoming the bias in Q-value estimation\nfound in traditional DQL. This technique, incorporating two neural networks (Q-network and target\nQ-network), improves policy estimation accuracy. Ref. [27] considers a Almgren-Chriss framework\nwithout permanent impact and with a constant (across time and stocks) temporary impact. When\napplied to real data, they found that DDQN outperforms the Time-Weighted Average Price (TWAP)\nstrategy in seven out of nine stocks considered. Other notable contributions include [21], where DQL\ndevises a trading strategy tested on various stock indexes, and [13], comparing DDQN and Proximal\nPolicy Optimisation (PPO) algorithms on artificial data. PPO and DDQN outperform TWAP when\nit is not optimal and perform similarly to a slice-and-dice strategy when it is.\nOur paper considers the application of RL for optimal execution to a more realistic environment.\nIn particular, we consider a setting where liquidity, as described by the temporary and permanent\nimpact coefficient, are not constant but time varying. It is well known that liquidity is a dynamic\nand latent variable. Some papers (for example [4] and [14]) have considered optimal execution with\ntime varying liquidity, but the underlying assumption is that the model’s parameters are known, while\ntheir real time estimation is typically very complicated and noisy as shown in for instance in [9] and\n[25]. Thus estimating the impact parameters and simultaneously adjusting the trading strategy is very\nchallenging and here, in a simulated environment, we test if RL can be successfully used. To this end,\nwe consider an Almgren-Chriss market where the permanent and temporary coefficients are linear but\ntime-varying. We consider different dynamics, from very simple and deterministic ones to the case\nwhen they follow a stochastic process. We find that when a closed form solution is known, the RL\nalgorithm finds very similar strategies with similar costs. More interestingly, when the solution is not\nknown or is known only in some regime parameters of low variability, the RL algorithm outperforms\nthe benchmarks. In summary, the aim is to train a ‘model robust’ agent capable of adapting its\ntrading strategy based on the current liquidity profiles in the market. This entails leveraging minimal\ninformation, represented as features to the neural networks, to agnostically select the correct model\nand selling schedule to follow.\nThe paper is organised as follows: Section 2 introduces baseline models and the DDQL algorithm,\nSection 3 discusses the main results, and finally Section 4 provides conclusions and outlines further\nresearch directions.\n2 Methods\nWe will consider the problem of a trader who wants to unwind a portfolio of q0stocks within a time\nwindow [0,T]. The goal is to unwind the portfolio optimally using arrival price as benchmark, i.e.\nminimising the difference between the initial and final portfolio values. We assume that the trading\ninfluences the stock price via permanent and temporary impacts. The magnitude and dynamics of\nthese impacts will change over the trading experiments that will be presented. Initially, we assume\nprice dynamics to follow the classic Almgren-Chriss [3] model, and then we test these results against\nthose obtained by an agent modelled with a DDQL algorithm. We will further modify and complicate\nmarket dynamics to test RL’s capabilities in more complex environments. This section introduces the\nmain tools used in the experiments, namely baseline financial models and the DDQL algorithm used\nto model our agent.\n2.1 Market impact models\nBaseline Model The Almgren and Chriss [3] (A&C) model is used as the baseline model. We\nassume a trader with an initial inventory q0stocks, the trader needs to unwind this portfolio within\na time window [0,T], divided into Nsub-intervals of length τ=T\nN. Settingt= 1,2,...,Nthe model\nreads:\nSt=St−1−g/parenleftbiggvt\nτ/parenrightbigg\nτ+στ1\n2ξ\n˜St=St−1−h/parenleftbiggvt\nτ/parenrightbigg (1)\nwhereStis the mid-price for the stock at time t.Stevolves because of a diffusion part ξ(a draw from\na standard normal random variable) multiplied by the price volatility σ. The number of shares sold\n2during the interval [t−1,t]of lengthτisvt. The mid-price Stis impacted by the permanent impact\ntermgt/parenleftbigvt\nτ/parenrightbig, assumed to be linear and constant: g/parenleftbigvt\nτ/parenrightbig=κvt\nτ. The price received by the trader, ˜St, is\nequal to the mid-price impacted by a temporary impact term also assumed to be linear and constant\nin time:h(vt\nτ) = ˜αvt\nτ, in what follows we use and model directly the quantity α=˜α\nτ. The aim is to\nfind an optimal quantity to sell vtfor each sub-interval t, minimising the Implementation Shortfall\n(IS) cost functional given by:\nIS=S0q0−N/summationdisplay\nt=1˜Stvt (2)\nSinceISis stochastic, A&C assumes a mean variance optimisation (or equivalently a CARA utility\nmaximiser) characterised by a risk aversion parameter λ. A&C shows that the optimal inventory\nholdingqtfor each time t= 1,...Nis:\nq∗\nt=q0sinh(ω(T−t))\nsinh(ωT)\nwhereωsolves 2(cosh(ωτ)−1) =λσ2\n2˜ατ2.\nInthispaperthetrader(oragent)isassumedtoberiskneutralwith λ= 0, thusarisk-neutral agent\nin discrete time adopts the TWAP strategy, selling their inventory at a constant rate v∗=/parenleftbigq0\nN,...,q0\nN/parenrightbig\nor equivalently holding q∗\nt= (N−t)q0\nNstocks for each time t= 1,...N.\nTime-dependent impacts baseline model We then consider a different baseline model by as-\nsuming that the permanent and the temporary impacts are time-varying . This is a more realistic\nbaseline model assumption, since in real-market situations liquidity is variable and fluctuates. In fact,\nas discussed in [9], price impacts have strong intraday time-dependency since they react quite quickly\nto changing market conditions. Impact dynamics usually have a deterministic and a stochastic com-\nponent. In this baseline we consider deterministic dynamics, while in the next one we will consider a\nstochastic dynamics for the impacts.\nIn general, if both impacts are time-varying, the functional to be minimised is similar to the one\non the right-hand side of (2) but with time-varying impacts, i.e.:\nN/summationdisplay\nt=1˜Stvt=N/summationdisplay\nt=1(St−2−κt−1vt−1)vt−N/summationdisplay\nt=1αt(vt)2(3)\nIn our context, the time varying permanent impact is g(vt\nτ) =κtvt\nτ. We consider here the simplest\ndeterministic dynamics for κt, namely we consider a linear temporal dependence, κt=κ0±βκ×t. The\ntime varying temporary impact is assumed to be similarly specified, i.e., h(vt) =αtvt, with dynamics\nαt=α0±βα×t. Since impacts must be positive, we choose the parameters in such a way that both\nκt>0andαt>0,∀t. Fig. 2 below shows the impact parameter dynamics used in our experiments.\nThe optimal schedule in this scenario might be found via standard quadratic optimisation tech-\nniques. However, when the dynamics of the impacts are not known, blindly applying optimisation\nmethods may lead to inefficient trading schedules. We test the DDQL results against those obtained\nvia canonical methods as if the dynamics were known from the beginning. The aim is to test whether\nthe algorithm has learnt the correct parameter specification for both the impacts without prior knowl-\nedge of their deterministic dynamics, in a quest to obtain a ‘robust’ optimal execution schedule in an\nagnostic way.\nStochastic impacts baseline model Lastly, we consider an environment with stochastic impacts.\nThe use of both permanent and temporary stochastic impacts allows for more freedom when it comes\nto model liquidity. Thus, both the temporary and the permanent impacts are assumed to follow a\nsquare-root mean reverting process. We will choose the parameters in such a way that the impacts\nare almost surely positive, we will further assume positive correlation between the permanent and the\ntemporary impacts. The stochastic impacts model was proposed Barger and Lorig (2019) [4], where\nanapproximate optimal execution solution is found via a Taylor approximation around the long run\n3mean values for the processes considered. According to [4], the model is:\ndSt=−κtνtdt+σdWt\ndκt=λκ(θκ−κt)dt+σκ√κtdB(1)\nt\n˜St=St−1−αtνt\ndαt=λα(θα−αt)dt+σα√αtdB(2)\nt\n<B(1)\nt,B(2)\nt>=ω(4)\nwhereStis the stock price, ˜Stis the execution price, λκandθκare, respectively, the rate of mean\nreversion and the long run mean at which the permanent impact reverts to, λαandθαhave same\nmeaning but for the temporary impact dynamics.\nAt timet, the trading velocity νt, that is the instantaneous trading volume for a sell program is\nfound, assuming that α0=θα, κ0=θκ, as:\nνt=/parenleftbigg1\nN−t+λα(θα−αt)\n2αt+ (N−t)λκ(θκ−κt)\n6κt/parenrightbigg\n×qt (5)\nEq. (5) can be seen as a perturbation of the TWAP solution, which is optimal in the constant impact\ncase. In what follows we will consider a discrete-time version of the model. We will assume that the\ntrader trades within each of the Ntime-steps t, a quantity vt=νtτof stocks. The price and the\nimpacts’ paths over the Ndiscrete time-steps are simulated using the model in Eq. (4).\n2.2 DDQL algorithm\nDeep Reinforcement Learning is a machine learning paradigm where an agent, modelled with neural\nnetworks, learns how to interact with an environment. This is achieved by taking actions and receiving\nfeedback in the form of rewards or penalties. The primary objective of the agent is to maximise the\ncumulative reward over time by discovering an effective policy or strategy.\nWe denote the sets of all possible states, actions, rewards, and policies as S,A,RandΠrespec-\ntively. In this context, the algorithm, given the environment’s state st∈S, must decide which action\nat∈Ato perform. This action modifies the environment to a subsequent state st+1∈Sand results\nin receiving a reward rt∈R. The ultimate goal is to learn an optimal policy π∗∈Πthat maximises\nthe sum of discounted future rewards, denoted as/summationtext∞\nt=0γkrt, whereγ∈(0,1]is the discount factor.\nIn this paper, we employ Deep Reinforcement Learning to find the optimal execution schedule in\nvarious time-varying impact settings. Specifically, we adopt the Double Deep Q-Learning (DDQL)\nalgorithm, providing a model-agnostic solution to the optimal execution problem. Q-values, repre-\nsenting the quality of taking actions in each state of the environment, are obtained through neural\nnetworks known as Q-nets. We use Double Deep Q-learning, thus employing two neural networks,\nnamely the main Q-net ( Qmain) for action selection and the target Q-net ( Qtgt) for state evaluation.\nOverall, the algorithm undergoes an exploration phase, where it explores possible rewards by taking\nrandom actions in evolving states. Subsequently, it gradually transitions to an exploitation phase,\nleveraging learned information on rewards using Qmainto select actions. To maintain stability, the\nweights for Qtgtare periodically synchronised with those of Qmain.\nThis is achieved through a training routine that updates the main net using past states, actions,\nand rewards randomly sampled from a memory. The training occurs over Mepisodes, followed by a\ntest phase on an additional Bepisodes to evaluate the DDQL agent’s performance post-exploration.\n2.2.1 Algorithm setup for numerical experiments\nInordertotackletheproblemofoptimalexecutionwithDDQL,weconsideranagentwhowantstosell\na portfolio of q0stocks within a time-window [0,T]. During this time window, the agent totally sells\nits inventory and this is called an episode. The episodes are assumed to be divided into Ntime-steps\nand the time increment for the simulations is τ=T\nN. In this context, the agent decides the quantity\nof stocksvtto sell in each of the ttime-steps in which the episode is divided. The mid-price at which\nthe agent trades is modeled according to Eq. (1), but depending on the frameworks considered, the\ndynamics for the impact parameters change.\n4Over theMtrain episodes the algorithm learns, via exploration-exploitation, the best trading\nstrategy and changes the weights of the Q-nets accordingly. The architecture of the algorithm in the\ntraining phase is thus divided into two phases: an exploration and anexploitation phase, managed by\nthe parameter ϵ∈(0,1]that will decrease during training and initially set as ϵ= 1. Depending on the\nphase, the way actions vtare chosen changes. When the agent is exploring, it will randomly select sell\nactionsvtin order to explore different states and rewards in the environment, alternatively, when the\nagent is exploiting, it will use Qmainto selectvt.\nAction selection and reward function. At the beginning of each time interval and for each\nepisode, the agent has limited knowledge of the environment where it trades. This means that it\nhas access to states stmade either by tuples of ( qt,t) or (qt,t,St−1) (i.e. quantity, time and possibly\nmid-price), depending on the considered feature setup.\nGiven the current value of ϵ, a drawζfrom a uniform distribution determines whether to perform\nexploration or exploitation. Specifically, with probability ϵthe agent chooses to explore, and the sell\nactionvtis sampled from a binomial distribution with number of trials equal to qt(i.e. the inventory\nleft at the beginning of a sub-interval) and probability of success is1\nN−t. In this way, in the exploration\nphase the TWAP is selected on average. Alternatively (with probability 1−ϵ), the agent chooses the\nQ-optimal action, i.e. the action that maximises the Q-value from Qmain, doing exploitation of what\nlearnt in the exploration phase. Notice that the agent cannot sell more than the remaining inventory\nand, in addition, no buy actions can be performed in the selling program.\nIn this way, the agent randomly explores a large number of states and possible actions to perform.\nOnce every mactions taken during training episodes, ϵis multiplied by a constant c<1such that\nϵ←ϵ×c, in this way for large number of episodes M,ϵconverges to zero and the algorithm gradually\nstops exploring and starts to greedily exploit what it has learned in terms of weights Qmain. In\nformulae, the action decision rule for each time step tin the training phase is:\nϵ∈(0,1), ζ∼U(0,1)\nvt=/braceleftigg\n∼Bin(qt,1\nN−t),ifζ≤ϵ\nargmaxv′∈[0,qt]QM(st,v′|θmain),else(6)\nSubsequently, at each t, once that the action is chosen according to Eq. (6), the reward is calculated\nas follows:\nrt=St−1vt−αv2\nt (7)\nOverall, the rewards for every t∈[1,N]are:\nr1=S0v1−αv2\n1\nr2= (S1+κv1+N(0,σ))v2−αv2\n2\n...\nrN= (SN−1+κvN−1+N(0,σ))vN−αv2\nN(8)\nThus, for each time step tthe agent sees the reward from selling vtshares, and stores the state of\nthe environment, st, where the sell action was chosen along with the reward and the next state st+1\nwhere the environment evolves. As said, the aim of the agent is to cumulatively minimise such reward,\nsuch that the liquidation value of the inventory happens as close as possible to the initial value of the\nportfolio. At the end of the episode, the reward per episode is −q0S0+/summationtextN\nt=1St−1vt−αv2\nt.\nTraining scheme The states st, actionsvt, rewardsrtand subsequent future states st+1obtained\nby selling a quantity vtin statest, form atransition tuple that is stored into a memory of maximum\nlengthL. As soon as the memory contains at least btransitions, the algorithm starts to train the\nQ-nets. In order to train the nets, the algorithm samples random batches of length bfrom the memory,\nand for each sampled transition jit calculates:\nyj\nt(θtgt) =/braceleftigg\nrj\nt ift=N;\nrj\nt+γQtgt(sj\nt+1,v∗|θtgt)else(9)\n5In Eq. (9),rj\ntis the reward for the time step considered in the transition j,sj\nt+1is the subsequent state\nreachedint+1, knownsinceitisstoredinthesametransition j, whilev∗= argmaxvQmain(sj\nt,v|θmain).\nγis a discount factor that accounts for risk-aversion2. The loss to be minimised is the mean squared\nerror between the target y(θtgt)and the values obtained via the Q-main network. Thus, it is:\nL(θmain,θtgt) =1\nbb/summationdisplay\nℓ=1/parenleftig/bracketleftig\nyℓ\nt(θtgt)−Qmain(sℓ\nt,vℓ\nt|θmain)/bracketrightig/parenrightig2\nθ∗\nmain= argmin\nθmainL(θmain,θtgt)\nThe algorithm then uses back propagation and gradient descent to update the weights of the main\nnetwork. This sub-routine is repeated for each random batch of transitions sampled from the memory.\nFinally, as stated above, once every mactionsϵis decreased by a factor c<1and the algorithm sets\nQtgt=Qmain.\nAfter training, other B<Mtrading episodes are fed to the algorithm in order to exploit what\nlearnt during training, this is the testing phase and the actions are now selected using just Qmain. The\nresults shown below are those obtained in the test phase only.\nThe features of the Q-nets are ( qt,t,vt) or (qt,t,St−1,vt) and are normalised in the domain [−1,1]\nusing the procedure suggested in [27], normalised mid-prices ¯Stare obtained via min-max normalisa-\ntion. In our setup we use fully connected feed-forward neural networks with 5layers, each with 30\nhidden nodes activation functions are leakyReLu, and finally we use the ADAM optimiser for optimi-\nsation. The parameters used to calibrate the algorithm are reported in Table 1, the training algorithm\nis reported in Algorithm 1.\nTable 1: Fixed parameters used in the DDQL algorithm. The parameters not shown in the table\nchange depending on the experiments and are reported accordingly.\nDDQL parameters Model parameters\nNN layers 5 Mtrain eps. 10.000 Nintervals 10\nHidden nodes 30 Btest eps. 5.000 σ-stock 0.00001\nADAM lr 0.0001 mreset rate 100acts. S0-stock 10$\nBatch sizeb 32 cdecay rate 0.995 q0inventory 20\nLmemory\nlength15.000 γ-discount 1\n2We set aγ= 1since we model a risk neutral agent\n6Algorithm 1 Training of Double Deep Q-Learning agent for optimal execution\nRequire:\nInitialise with random weights Qmainand make a copy Qtgt;\nInitialise the memory with max length L. Setϵ= 1,bbatch size, Mtrain episodes;\nSet market dynamics parameters;\nfori in M do\nSetSi\n0=S0;\nfort in N do\nst←(qt,t); ▷or(qt,t,Si\nt−1)if mid-price considered\nvt←/braceleftigg\nsample Bin (qt,1\nN−t)with probability ϵ\nargmaxv′∈[0,qt]Qmain(st,v′|θmain)with probability (1−ϵ);\nrt←Si\nt−1vt−αv2\nt;\nSi\nt−1→Si\nt; ▷GenerateSi\ntfromSi\nt−1\nst+1←(qt+1,t+ 1); ▷or(qt+1,t+ 1,Si\nt)\nMemory←(st,rt,vt,st+1); ▷Memory storing\nifLength of memory ≥bthen\nforj inbdo\nSample a batch of (sj\nt,rj\nt,vj\nt,sj\nt+1)from memory;\nv∗= argmaxvQmain(sj\nt,v|θmain);\nyj\nt(θtgt) =/braceleftigg\nrj\nt ift=N;\nrj\nt+γQtgt(sj\nt+1,v∗|θtgt)else\nend for\nθ∗\nmain= argminθmainL(θmain,θtgt)via gradient descent with loss to minimise:\nL(θmain,θtgt) =1\nbb/summationdisplay\nℓ=1/parenleftig/bracketleftig\nyℓ\nt(θtgt)−Qmain(sℓ\nt,vℓ\nt|θmain)/bracketrightig/parenrightig2\nifLength of memory = Lthenhalve the length of memory\nend if\nend if\nAftermiterations decay ϵ=ϵ×c;\nAftermiterationsθtgt←θmain;\nend for\nend for\n73 Experiments and results\nIn this section we present the results of our numerical investigations. The aim of the experiments is\nto study whether the DDQL agent is able to find model-robust optimal execution strategies without\nany form of knowledge of the underlying impact model, but just based on simple sets of information.\nSince the algorithm does not know neither the parameters nor the model used for the dynamics of the\nimpacts we adopt a model agnostic approach to find optimal execution strategies. In this context, we\nstudy whether the DDQL agent is able to learn the optimal strategy (i) when the optimal solution\nis known but the environment is not known to the agent (the impact can be equally increasing or\ndecreasing) and (ii) when the optimal solution is not known in closed form but in an approximate\nform and the environment is still not known to the agent. In this sense, we investigate the use DDQL\nas a tool to develop ‘robust’ strategies. In a way, the goal is to use DDQL as a robust non-parametric\ntechnique, able to agnostically detect price impacts’ magnitude and dynamics in the market by using\nas few information as possible.\nNotice that the number of trading steps per episode is set to N= 10and the agent has to to sell\nq0= 20shares. After the test phase is complete, the average IS is calculated for each test episode.\nThus, inthetestphasewemeasureforeachepisodetheISforboththestrategies: theDDQNsuggested\nstrategy and the optimal one if impacts dynamics and parameters were known from the beginning.\nThe performance difference between the two approaches, for each test episode, is quantified as:\n∆P&L=Cagent−Cb\nCb\nwhereC=/summationtextN\nt=1vt˜St, the cash generated by the trading. Notice that Cbis the cash calculated for the\nbaseline model, while Cagentis the cash obtained using DDQL strategies. In the tables below we report\nthe average values and the standard deviation over the whole test set.\n3.1 Constant impact\nWe first consider mid-price dynamics generated according to the A&C model of Eq. (1). The impacts\nare constant and the coefficients are set equal to κ= 0.001andα= 0.002. When the market behaves\nas in Eq. (1), the TWAP is the optimal solution, i.e. the strategy with minimal costs is the one where\nq0is sold equally over the time steps. We want to check that the DDQL is able to recover a similar\nstrategy also by comparing the expected cost with the one of a TWAP. This would imply that DDQL\nunderstands that the impact parameters are fixed and trades accordingly with minimal discrepancy\ncompared to the closed-form solution’s costs. We do not expect that including price as feature should\nimprove the DDQL strategy performance.\nFeatures Q,T The results with the costs are reported in Table 2. We notice that, if the agent has\nknowledge of just qtandt, it retrieves an IS values very similar to the one that would be obtained with\nthe TWAP strategy. The ∆P&Lshows a difference of just −0.4basis points and a standard deviation\nof only two basis points. The average number of stocks sold per time-step and level of inventory is\nshown in the left panel of Fig. 1. It can be observed that the agent has learned a strategy consistent\nwith ‘slice and dice’ trading.\nFeatures Q,T,S If one includes also the price among the features, we observe that the DDQL agent\nis able to retrieve the same cost. The ∆P&Lis in the order of −0.25basis points, implying that the\noptimal cost has been learnt and reproduced by the agent. The average optimal trading per level of\ninventoryqt, time-step t, and normalised price level ¯Sare reported in the right panel of Fig. 1.\nThe strategy does slightly vary depending on the price level. It might be that during training\nsome states did bear a lower cost. However, the algorithm overall does not perform any better than a\nnoisy TWAP strategy, as expected. Moreover, it appears that allowing for more information in such\na simple environment does not significantly add any value. The strategy adjusts depending on the\nprice level considered to be optimal and does not outperform the previous case. This aligns with the\ntheoretical findings in [30], proving that dynamical strategies (which can depend on price) are also\nstatically optimal.\n8Constant Impact\nA&C DDQL ∆P&L ∆P&L\nfeatures E[IS]E[IS]avg. std.dev.\nQ,T 0.2607 0.2698 -0.455 2.5\nQ,T,S 0.2607 0.2652 -0.225 1.6\nTable 2: Constant impact. Comparison between the costs of the DDQL agent and of the A&C model.\n∆P&L values are reported in basis points.\nFigure 1: Constant impact. Left panel: Average number of stocks sold per time-step tand inventory\nlevelqt. Right panel: Average number of stocks sold per time-step t, inventory level qt, and normalised\nprice ¯S.\n3.2 Deterministically time-varying impact\nHaving established that the DDQL agent is able to recover the optimal solution when impacts are\nconstant, we test the algorithm with two different, yet deterministic, impacts patterns. It is widely\nrecognised that liquidity has intraday patterns3thus a trend in impact parameter is observed in\ncertain parts of the trading day, e.g. in the first part of the trading day or around important market\nannouncements. Ifoneknowsinadvancethefunctionalformofthedynamicsoftheimpactparameters,\nthe optimal execution is again a minimisation of a cost functional and can be solved via quadratic\noptimisation (although the well posedness of the problem is far from trivial, see [29]). However, the\nDDQL agent has no knowledge either about the existence of a pattern or whether the pattern shows\nan increasing or decreasing trend and the parameters for the impact have to be implicitly estimated\nby the agent employing DDQL. In making no assumption on market’s dynamics the nature of the\nproblem becomes more involved.\nAs explained in Section 2.1, we assume impacts follow a linear deterministic trend. We consider\neither an increasing or a decreasing impact, and in order to test the appropriateness of the parameters\nlearnt by the algorithm, we calculate the optimal strategy using quadratic optimisation4and then we\n3see for example [9] for a thorough discussion on this topic\n4We solved the problem using the .optimise() method in SciPy and applying Quadratic programming to Eq. (3).\n9simulate the optimal strategy and we measure its cost. The resulting IS is then compared with the\none obtained with the DDQL strategy. We also use as point of comparison the IS of a TWAP strategy,\nwhich is obviously not optimal. This serves as an additional benchmark of the performance of the\nDDQL agent in addition to the theoretically optimal one.\nFigure 2: Time-varying dynamics of the temporary and permanent impact\n3.2.1 Testing and training with increasing impacts\nThe impacts dynamics is g(vt\nτ) =κtvt\nτ, whereκt=κ0+βκ×tandh(vt) =αtvt, whereαt=α0+βα×t\nand the parameters for the impacts are reported in Table 3. The parameters are chosen such that the\naverage value of the impact parameter during the trading period are equal to the fixed impacts in the\nexperiments of the previous section. Fig. 3 shows the optimal solution for these parameters (green\nline, the red line is the TWAP reference). As expected, given that impacts are increasing, it is better\nto trade more than in a TWAP at the beginning to exploit the initial lower trading costs.\nTable 3: Parameters used in the increasing time varying setting.\nparameters value\nκ0 0.0001\nα0 0.0001\nβκ 0.0002\nβα 0.0004\nκ5=κ 0.001\nα5=α 0.002\nFeatures Q,T As shown in Table 4, when only price and time are used as features, the DDQL\nalgorithm is not able to really beat the TWAP strategy in terms of costs. In fact, the ∆P&Lbetween\nthe DDQL and the TWAP strategy is almost zero, i.e. ∆P&L=−3.20e−5. As a consequence, the\n∆P&Lbetween the DDQL and the theoretical simulated costs is significantly negative (see Table 4).\nLooking at the blue line in Fig. 3 it appears that the DDQL agent with two features learns to trade\nmore at the beginning, but not enough. A similar conclusion is obtained by looking at the full strategy\n(see left panel of Fig. 4). Looking at the ∆P&Lvalue and at the average quantity sold per inventory\nlevel and time-step, it can be concluded that market parameters have been noisily estimated leading\nto a strategy as good as a TWAP strategy.\nFeatures Q,T,S Adding the mid-price to the features leads to better results in terms of ∆P&L.\nIn fact, as reported in Table 4 the simulated IS for the DDQL agent performs better than the one\nobtained with TWAP. Comparing our results with the theoretical solution, adding the mid-price as a\nfeature lowers the ∆P&Lto just 2basis points of difference. Although not performing as good as the\ntheoretical solution, adding a feature significantly increases the learning capabilities of the algorithm.\nThis conclusion is supported by the average inventory holding and average trading schedule reported\nin Fig. 3 and right panel of Fig. 4, respectively. It can be seen that the agent has learnt to trade\ntoo aggressively within the first half of the execution window. This is compatible with the increasing\n10nature of the underlying impact parameters. We conclude that the agent has estimated the trend in\nthe impacts and has found a trading strategy accordingly.\nIncreasing Impact\nTheo. DDQL ∆P&Lvs. Theo. ∆P&Lvs. Theo. TWAP\ninputs E[IS]E[IS] avg. std.dev. E[IS]\nQ,T 0.1449 0.2401 -4.76 1.1 0.2326\nQ,T,S 0.1449 0.1944 -2.42 1.1 0.2326\nTable 4: Increasing impact. Comparison between the costs of the DDQL agent, of the theoretically\noptimal solution, and of a TWAP strategy. ∆P&L values are reported in basis points.\nFigure 3: Increasing impact. Average inventory holdings per time step with different features, com-\npared to the optimal inventory holding if the strategy was known from the beginning, and to the\ninventory holding if the TWAP strategy was used\n11Figure 4: Increasing impact. Left panel: Average number of shares sold per time-step tand inventory\nlevelqt. Right panel: Average number of shares sold per time-step t, inventory level qtand normalised\nprice ¯S\n3.2.2 Testing and training with decreasing impacts\nThe impacts dynamics is g(vt\nτ) =κtvt\nτ, whereκt=κ0−βκ×tandh(vt) =αtvt, whereαt=α0−βα×t\nand the parameters for the impacts are reported in Table 5. As before, the parameters are chosen such\nthattheaveragevalueoftheimpactparameterduringthetradingperiodareequaltothefixedimpacts\nin the experiments of the previous section. Fig. 5 shows the optimal solution for these parameters\n(green line, the red line is the TWAP reference). Since impacts are decreasing, it is optimal to trade\nless than in a TWAP at the beginning to avoid the initial high trading costs.\nTable 5: Parameters used in the decreasing time varying setting. The parameters are chosen such\nthat the average value corresponds to the fixed impacts in the experiment above.\nparameters value\nκ0 0.002\nα0 0.004\nβκ 0.0002\nβα 0.0004\nκ5=κ 0.001\nα5=α 0.002\nFeatures Q,T Similarly to the increasing impact case, when the features are inventory and time,\nthe DDQL algorithm has a sub-optimal performance. In fact, as reported in Table 6 its IS is higher\nthan the theoretical optimum but the learnt strategy still outperforms the TWAP. As in the previous\ncase, the algorithm manages to understand the direction of the trend. As it can be seen in Fig. 5\n(blue line) and in the left panel of Fig. 6 the average actions per inventory level and time-step seem\nto be consistent with an increasing cost of execution, since the algorithm trades more toward the end\nof the period when costs are lower. It can be concluded that although correct in the estimation of the\ntrend, the algorithm did not manage to calibrate its own actions to the magnitude of the impacts.\nFeatures Q,T,S When including the mid-price among the features, the algorithm’s performance\ngets fairly better. In fact, as it can be noticed in Table 6, when the algorithm features the mid-price\n12does better than TWAP by lengths and performs slightly worse than the optimal trading strategy.\nAgain, adding the mid-price as a feature adds more power to the DDQL agent when it comes of\nestimating magnitude and direction of the impacts. This is confirmed by the average actions chosen\nper level of inventory, time step and normalised mid-price ¯S, see Fig. 5 (orange line) and the right\npanel of Fig. 6. The strategy is again skewed to the right hand side of the heat-map implying that\nthe algorithm has perfectly understood the direction and magnitude of the impacts.\nDecreasing\nTheo. DDQL ∆P&Lvs. Theo. ∆P&Lvs. Theo. TWAP\ninputs E[IS]E[IS] avg. std.dev. E[IS]\nQ,T 0.2566 0.3080 -2.58 0.03 0.3588\nQ,T,S 0.2566 0.2877 -1.51 0.5 0.3588\nTable 6: Decreasing impact. Comparison between the costs of the DDQL agent, of the theoretical\nsolutions, and of a TWAP strategy. ∆P&L values are reported in basis points.\nFigure 5: Average inventory holdings per time step with different features, compared to the optimal\ninventory holding if the strategy was known from the beginning, and to the inventory holding if the\nTWAP strategy was used\n13Figure 6: Decreasing impact. Left panel: Average number of shares sold per time-step tand inventory\nlevelqt. Right panel: Average number of shares sold per time-step t, inventory level qtand normalised\nprice ¯S\n3.2.3 Mixed train with increasing and decreasing impacts\nWhile the previous results were obtained by using the same dynamics in the train and test phase,\nhere we consider a more realistic situation where the train phase contains different scenarios and in\nthe test phase only one of them is actually realised. More specifically, the DDQL agent is now trained\nwith 10,000trajectories using increasing impacts and 10,000trajectories using decreasing impacts.\nIt is subsequently tested on 5,000trajectories with either increasing or decreasing impacts trends.\nIt is worth to study such experimental setup since it is functional for an eventual deploy in real\nmarket situations of an algorithm based on this RL paradigm. In fact, if the algorithm is used in\na ‘production’ setting, and especially without any prior knowledge of the data at hand, one would\ntrain such an algorithm with several scenarios for the stocks intended to be sold and then, using\nsuch strategy, one would like the algorithm to be able to detect the specific liquidity pattern and\nsimultaneously adapt the trading strategy. We mimic this situation by training with both regimes,\nthen we test with either of the two, in order to assess whether the DDQL agent is able to recognise\nthe direction and the magnitude of the impacts at considered.\nIn this case the TWAP IS, used also above, becomes the natural benchmark. In fact, since the\ntraining set includes both increasing and decreasing impact while the test realises only one of them,\nthe obvious choice for a agnostic trader would be to use a TWAP strategy. For this reason, in the\ntables we report also the ∆P&LTWAPwith respect to the TWAP strategy. The standard ∆P&Lis\nalso reported but should be considered as a kind of unfeasible benchmark, since its calculation requires\nto know whether the impacts in the test set is increasing or decreasing.\nIncreasing impacts - Features Q,T Using as features only inventory and time, the agent is again\nnot able to find the optimal strategy. As reported in Table 7 the performance in terms of ∆P&Lis\nstill poor and the overall IS is similar to the one obtained with a TWAP strategy. As it can be seen\nin the left panel of Fig. 10 the strategy achieved by the agent does not appear in line with the one\nthat would be expected with such impact dynamics.\nIncreasing impacts - Features Q,T,S When the mid-price is included into the features, the\nperformance of the DDQL agent is dramatically better. In fact, as reported in Table 7 the performance\nin terms of ∆P&Lis almost zero, meaning that the agent performs as good as the optimal solution\n14in terms of costs. In turn, this implies that the agent has correctly estimated both the direction and\nthe magnitude of the impacts used in the test phase. In fact, as it can be seen in the right panel of\nFig. 10, the strategy found, conditioned to the price levels, is in line with the increasing magnitude in\nthe impact dynamics.\nIncreasing Impact\nTheo. DDQL ∆P&Lvs. Theo. ∆P&Lvs. Theo. TWAP ∆P&LTWAP\ninputs E[IS]E[IS] avg. std.dev. E[IS] avg.\nQ,T 0.1449 0.2554 -5.34 1.1 0.2326 -0.92\nQ,T,S 0.1449 0.1319 0.65 0.9 0.2326 5.2\nTable 7: Increasing impact. Comparison between DDQL agent, theoretical solutions and TWAP\nstrategy. ∆P&L values are reported in basis points.\nDecreasing impacts - Features Q,T If just the inventory and the time-step form the state seen\nby the agent, the estimation of the parameters is noisy as in the other cases. Looking at Table 8 it\ncan be seen that the resulting strategy performs slightly worse than the TWAP and it under-performs\nthe optimal selling schedule suggested via optimisation of the cost functional. This result is in line\nwith the strategy outlined in the left panel of Fig. 9.\nDecreasing impacts - Features Q,T,S If the mid-price features the states seen by the agent the\nperformance is fairly good if compared to the previous case. In fact, adding the price to the features\nof the algorithm increases the estimation power of DDQL and delivers a performance in line with the\noptimal one. This can be seen in Table 8 where the ∆P&Lis very low, suggesting that the impacts\ndynamics are estimated quite efficiently leading to a strategy consistent with the decreasing nature of\nthe impacts’ dynamics in the test set. This result is confirmed by the average action chosen per level\nof normalised mid-price, inventory and time-step shown in the right panel of Fig. 9.\nDecreasing Impact\nTheo. DDQL ∆P&Lvs. Theo. ∆P&Lvs. Theo. TWAP ∆P&LTWAP\ninputs E[IS]E[IS] avg. std.dev. E[IS] avg.\nQ,T 0.2566 0.3696 -5.62 0.32 0.3588 -0.51\nQ,T,S 0.2566 0.2456 0.86 0.03 0.3588 6.5\nTable 8: Decreasing impact. Comparison between DDQL agent, theoretical solutions and TWAP\nstrategy. ∆P&L values are reported in basis points.\n3.3 Stochastic impacts\nLastly, we consider stochastic impact coefficients. This experiment allows us to consider dynamics\nthat are much more complicated and gives the opportunity to thoroughly test the capabilities of the\nDDQL agent to identify optimal strategies. As an analytical point of comparison, we use the third\nmodel of Section 2.1 for which an approximated solution can be obtained as a Taylor expansion on the\nlong run means of the temporary and permanent impact dynamics as outlined in Barger and Lorig\n(2019) [4]. In this regime, the resulting optimal execution of Eq. (5) is in fact a perturbation of the\nTWAP strategy that accounts for deviation of the impacts with respect to their long run mean values.\nThe aim of the experiment is to test whether the agent is able to replicate, or even beat, the\nperformance that would be obtained if the parameters for Eq. (5) were known. In fact, the difficulty\nin considering stochastic impacts when implementing an optimal execution strategy lies not just in the\n15correct equation to be used, but in the correct parameters estimation for the impact model considered.\nBy using the DDQL agent we aim at tackling the problem in a model agnostic way, i.e. we do not\nestimate or make assumptions on either the model or the impacts’ magnitudes. We let the agent learn\nthem via exploration-exploitation.\nThe parameters chosen for the simulations are reported in Table 9. We assume that both impacts\nhover over their long run averages and are correlated, with correlation coefficient5ω= 0.9. Thus, the\nparameters are chosen such that the Feller condition holds and the values for the impact coefficients\nare almost surely positive6. We implement two types of experiments, we test for strong mean reversion\n(λ(high)\nαandλ(high)\nκ) and for weak mean reversion ( λ(low)\nαandλ(low)\nκ) parameters, leaving long run levels\nand volatility of the impacts untouched. Thus testing for either strong or weak memory properties in\nthe dynamics for the impacts.\nIn the experiments, for each train and test episode we simulate one path for αtand one for κt.\nSample paths for both αtandκtare reported in Fig 8.\nTable 9: Parameters used for the simulation of stochastic impacts. The parameters are chosen such\nthat the starting values correspond to the fixed impacts in the A&C experiment above.\nParameter Value Parameter Value\nλ(low)\nα 1θκ 0.001\nλ(low)\nκ 1θα 0.002\nλ(high)\nα 5σκ 0.002\nλ(high)\nκ 5σα 0.002\nω 0.9\nFeatures - Q,T When the agent can only access the level of inventory and time-steps as features,\nwhen considering the λ(low)\nα,κsetup, the resulting IS is similar to the one obtained using Eq. (5) as shown\nin Table 10. The ∆P&Lis, in fact, slightly positive but less than 2 basis points. The performance\nincreases in the case of λ(high)\nα,κsetup, as shown in Table 11 it seems that since the impact dynamics\nmean revert faster to their long-run average the agent, trained in this way, takes advantage and trades\naccordingly, thus obtaining lower IS and better ∆P&L.\nFeatures - Q,T,S Adding the price to the set of features does bring significant benefits to the\noptimal selling schedule. Considering the λ(low)\nα,κsetup, the ∆P&Lis now significantly positive, due to\nthe increased amount of cash generated by training a DDQL agent with these features. Thus, adding\nthe price to the set of features greatly improves the trading performance overall as seen in Table 10.\nWhen theλ(high)\nα,κsetup is considered, the performance of the agent is still extensively better than the\napproximated solution, but in line with those obtained with just inventory and time-step as features\nas shown in Table 11. We interpret this to be the result of the increased mean reversion rate, the\nagent learns that the dynamics tend to revert faster to the long run average and takes advantage\nwith its trading. Notably, the agent trained in this way has no notion of the stochastic dynamics of\nthe impacts parameters used in this setting, the algorithm in fact, learnsthe model and, implicitly,\nits parameters from the rewards obtained at each time-step and for each train episode. Thus, it is\ncapable of performing better than an approximate closed-form solution, with the advantage that the\nparameters used to obtain the sale schedule are neither assumed nor estimated in any way, they are\nlearnedimplicitly and directly from the few features observed by the agent and fed to the neural nets.\n5This high value reflects the fact that different liquidity measures are typically highly correlated. We expect the same\nhappens for temporary and permanent impact\n6It is worth noticing that in the space of solutions of [4] both buys and sells are allowed for a sell program, while\nour RL approach considers only sells. In order to provide a fair comparison between the two methods, we selected the\nparameters of the model in such a way that, using the [4] solution, the fraction of buys in a sell program is negligible.\nThis parameter choice has been achieved via extensive numerical simulations.\n16Figure 7: Stochastic impact. Left panel: three sample paths for αtwithλ(low)\nα. Right panel: three\nsample paths for κtwithλ(low)\nκ. Solid lines: three different realisations for the temporary and perma-\nnent impacts. Yellow area is the standard deviation around the average value of the impact.\nFigure 8: Left panel: sample paths for αtwithλ(high)\nα. Right panel: sample paths for κtλ(high)\nκ.\nSolid lines: three different realisations for the temporary and permanent impacts. Yellow area is the\nstandard deviation around the average value of the impact.\nStochastic Impact: λ(low)\nα,κ\nTheo. Theo. DDQL DDQL ∆P&L ∆P&L\ninputs E[IS]std.dev. [IS]E[IS]std.dev. [IS]avg. std.dev.\nQ,T 0.3129 0.63 0.2789 0.011 1.8 3.19\nQ,T,S 0.3129 0.63 0.2572 0.007 2.5 3.27\nTable 10: Stochastic impact. Comparison between DDQL agent and theoretical solutions. ∆P&L\nvalues are reported in basis points.\n17Stochastic Impact: λ(high)\nα,κ\nTheo. Theo. DDQL DDQL ∆P&L ∆P&L\ninputs E[IS]std.dev. [IS]E[IS]std.dev. [IS]avg. std.dev.\nQ,T 0.5017 1.83 0.3200 0.0084 9.2 73.3\nQ,T,S 0.5017 1.83 0.3158 0.0112 9.4 73.4\nTable 11: Stochastic impact. Comparison between DDQL agent and theoretical solutions. ∆P&L\nvalues are reported in basis points.\n4 Conclusions\nIn this paper we developed a DDQL algorithm able to learn the optimal liquidation strategy when\nthe liquidity is time-varying. By considering an Almgren-Chriss linear market impact model, we in-\nvestigated several deterministic and stochastic dynamics of the temporary and the permanent impact\ncoefficient. These include constant, linearly varying, and square-root mean reverting stochastic pro-\ncesses for the parameters. We also investigate the case when coefficients are linearly varying but the\nslope can be either positive or negative with equal probability.\nWe find that, when considering settings where the optimal solution is known, the algorithm is able\nto learn strategies with an expected cost comparable with the optimal one. More interestingly, when\nwe consider settings where the optimal solution is not available or is available in approximate form,\nwe find that the DDQL algorithm finds better solutions. This indicates that RL could be successfully\nused in real settings where liquidity is naturally latent and time-varying.\nThere are several possible extensions of our work. First of all, we considered a linear impact\nmodel, whereas it is known that temporary market impact is non-linear as shown in [2]. Second, more\nsophisticated market impact models, such as the Transient Impact Model by [7] and its extension\nby [6], better describes the reaction of prices to trades. Finally, other dimensions of liquidity, e.g.\nthe bid-ask spread, are important components of transaction cost and are known to be time varying\nwith non-trivial dynamics. It is certainly interesting to explore if RL based approaches are capable to\nfind optimal trading schedules also in these more complex and realistic environments. We leave these\ninteresting questions for future research.\nAcknowledgements\nF.L. acknowledges partial support by the European Program scheme ‘INFRAIA-01-2018-2019: Re-\nsearch and Innovation action’, grant agreement #871042 ’SoBigData++: European Integrated Infras-\ntructure for Social Mining and Big Data Analytics’.\nReferences\n[1] Jacob Abernethy and Satyen Kale. “Adaptive market making via online learning”. In: Advances\nin Neural Information Processing Systems 26 (2013).\n[2] Robert Almgren et al. “Direct estimation of equity market impact”. In: Risk18.7 (2005), pp. 58–\n62.\n[3] Almgren, R. and Chriss, N. “Optimal execution of portfolio transactions��. In: Journal of Risk 3\n(2000), pp. 5–39.\n[4] Weston Barger and Matthew Lorig. “Optimal liquidation under stochastic price impact”. In:\nInternational Journal of Theoretical and Applied Finance 22.02 (2019), p. 1850059.\n[5] Dimitris Bertsimas and Andrew W Lo. “Optimal control of execution costs”. 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In: Proceedings of the 23rd international conference on Machine learning . 2006,\npp. 673–680.\n19[27] Brian Ning, Franco Ho Ting Lin, and Sebastian Jaimungal. “Double deep q-learning for optimal\nexecution”. In: Applied Mathematical Finance 28.4 (2021), pp. 361–380.\n[28] Anna A Obizhaeva and Jiang Wang. “Optimal trading strategy and supply/demand dynamics”.\nIn:Journal of Financial markets 16.1 (2013), pp. 1–32.\n[29] Palmari, G., Lillo, F., and Eisler, Z. “Optimal execution in a time varying environment: well\nposedness and price manipulation”. In: in preparation (2024).\n[30] Alexander Schied, Torsten Schöneborn, and Michael Tehranchi. “Optimal basket liquidation for\nCARA investors is deterministic”. In: Applied Mathematical Finance 17.6 (2010), pp. 471–489.\n[31] Matthias Schnaubelt. “Deep reinforcement learning for the optimal placement of cryptocurrency\nlimit orders”. In: European Journal of Operational Research 296.3 (2022), pp. 993–1006.\n[32] Shuo Sun, Rundong Wang, and Bo An. “Reinforcement learning for quantitative trading”. In:\nACM Transactions on Intelligent Systems and Technology 14.3 (2023), pp. 1–29.\n[33] Richard S Sutton and Andrew G Barto. “Reinforcement learning: an introduction MIT Press”.\nIn:Cambridge, MA 22447 (1998).\n[34] Haoran Wei et al. “Model-based reinforcement learning for predictions and control for limit order\nbooks”. In: arXiv preprint arXiv:1910.03743 (2019).\n[35] Pengqian Yu et al. “Model-based deep reinforcement learning for dynamic portfolio optimiza-\ntion”. In: arXiv preprint arXiv:1901.08740 (2019).\n20A Figures\nFigure 9: Mixed train, test with decreasing impact. Top panel: average number of stock sold per time-\nsteptand inventory level qt. Bottom panel: average number of stock sold per time-step t, inventory\nlevelqtand normalised price ¯S\n21Figure 10: Mixed train, test with increasing impact. Top panel: average number of stock sold per\ntime-steptand inventory level qt. Bottom panel: average number of stock sold per time-step t,\ninventory level qtand normalised price ¯S\n22"    },    {        "title": "2402.12110v1.The_Complexity_of_Geodesic_Spanners_using_Steiner_Points.pdf",        "content": "The Complexity of Geodesic Spanners using\nSteiner Points\nSarita de Berg /envel⌢pe\nDepartment of Information and Computing Sciences, Utrecht University, The Netherlands\nTim Ophelders /envel⌢pe\nDepartment of Information and Computing Sciences, Utrecht University, the Netherlands\nDepartment of Mathematics and Computer Science, TU Eindhoven, the Netherlands\nIrene Parada /envel⌢pe\nDepartment of Mathematics, Universitat Politècnica de Catalunya, Spain\nFrank Staals /envel⌢pe\nDepartment of Information and Computing Sciences, Utrecht University, The Netherlands\nJules Wulms /envel⌢pe\nDepartment of Mathematics and Computer Science, TU Eindhoven, the Netherlands\nAbstract\nA geometric t-spannerGon a setSofnpoint sites in a metric space Pis a subgraph of the\ncomplete graph on Ssuch that for every pair of sites p,qthe distance inGis a mostttimes the\ndistanced(p,q)inP. We call a connection between two sites in the spanner a link. In some settings,\nsuch as when Pis a simple polygon with mvertices and a link is a shortest path in P, links can\nconsist of Θ(m)segments and thus have non-constant complexity. The total spanner complexity is a\nrecently-introduced measure of how compact a spanner is. In this paper, we study what happens\nif we are allowed to introduce kSteiner points to reduce the spanner complexity. We study such\nSteiner spanners in simple polygons, polygonal domains, and edge-weighted trees.\nSurprisingly, we show that Steiner points have only limited utility. For a spanner that uses k\nSteiner points, we provide an Ω(nm/k )lower bound on the worst-case complexity of any (3−ε)-\nspanner, and an Ω(mn1/(t+1)/k1/(t+1))lower bound on the worst-case complexity of any (t−ε)-\nspanner, for any constant ε∈(0,1)and integer constant t≥2. These lower bounds hold in all\nsettings. Additionally, we show NP-hardness for the problem of deciding whether a set of sites in a\npolygonal domain admits a 3-spanner with a given maximum complexity using kSteiner points.\nOn the positive side, for trees we show how to build a 2t-spanner that uses kSteiner points and\nof complexity O(mn1/t/k1/t+nlog(n/k)), for any integer t≥1. We generalize this result to forests,\nand apply it to obtain a 2√\n2t-spanner in a simple polygon or a 6t-spanner in a polygonal domain,\nwith total complexity O(mn1/t(logk)1+1/t/k1/t+nlog2n).\n2012 ACM Subject Classification Theory of computation →Computational Geometry\nKeywords and phrases spanner, simple polygon, polygonal domain, geodesic distance, complexity\nFunding Tim Ophelders : Partially supported by the Dutch Research Council (NWO) under project\nno. VI.Veni.212.260.\nIrene Parada : is a Serra Húnter fellow and was supported by a Margarita Salas fellowship funded by\nthe Ministry of Universities of Spain and the European Union (NextGenerationEU).\nAcknowledgements Work initiated at the 2023 AGA Workshop in Otterlo. We would like to thank\nWouter Meulemans for his contributions in the initial phase of the project.\n1 Introduction\nConsider a set Sofnpointsitesin a metric space P. In applications such as (wireless)\nnetwork design [ 3], regression analysis [ 20], vehicle routing [ 14,30], and constructing TSP\ntours [6], it is desirable to have a compact network that accurately captures the distancesarXiv:2402.12110v1  [cs.CG]  19 Feb 20242 The Complexity of Geodesic Spanners using Steiner Points\nbetween the sites in S. Spanners provide such a representation. Formally, a geometric\nt-spannerGis a subgraph of the complete graph on S, so that for every pair of sites p,q\nthe distance dG(p,q)inGis at mostttimes the distance d(p,q)inP[28]. The quality of\na spanner can be expressed in terms of the spanning ratio tand a term to measure how\n“compact” it is. Typical examples are the sizeof the spanner, that is, the number of edges\nofG, its weight (the sum of the edge lengths), or its diameter. Such spanners are well\nstudied [ 4,8,11,19]. For example, for point sites in Rdand any constant ε >0one can\nconstruct a (1 +ε)-spanner of size O(n/εd−1)[29]. Similar results exist for more general\nspaces [10,21,23]. Furthermore, there are various spanners with other desirable spanner\nproperties such as low maximum degree, or fault-tolerance [7, 26, 27, 29].\nWhen the sites represent physical locations, there are often other objects (e.g. buildings,\nlakes, roads, mountains)thatinfluencetheshortestpathbetweenthesites. Insuchsettings, we\nneed to explicitly incorporate the environment. We consider the case where this environment\nis modeled by a polygon Pwithmvertices, and possibly containing holes. The distance\nbetween a pair of points p,q∈Pis then given by their geodesic distance : the length of a\nshortest path between pandqthat is fully contained in P. This setting has been considered\nbefore. For example, Abam, Adeli, Homapou, and Asadollahpoor [ 1] present a geodesic\n(√\n10+ε)-spanner of size O(nlog2n)whenPis a simple polygon, and a geodesic (5 +ε)-\nspanner of size O(n√\nhlog2n)when the polygon has h > 1holes. Abam, de Berg, and\nSeraji [2] even obtained a (2 +ε)-spanner of size O(nlogn)whenPis actually a terrain. To\navoid confusion between the edges of Pand the edges ofG, we will from hereon use the term\nlinksto refer to the edges of G.\nAs argued by de Berg, van Kreveld, and Staals [ 16], each link in a geodesic spanner may\ncorrespond to a shortest path containing Ω(m)polygon vertices. Therefore, the spanner\ncomplexity , defined as the total number of line segments that make up all links in the spanner,\nmore appropriate measures how compact a geodesic spanner is. In this definition, a line\nsegment that appears in multiple links is counted multiple times: once for each link it\nappears in. The above spanners of [ 1,2] all have worst-case complexity Ω(mn), hence de\nBerg, van Kreveld, and Staals present an algorithm to construct a 2√\n2t-spanner in a simple\npolygon or a 6t-spanner in a polygon with holes with complexity O(mn1/t+nlog2n), for\nany integer t>1.1These complexity bounds are still relatively high. De Berg, van Kreveld,\nand Staals [ 16] also show that these results are almost tight. In particular, for sites in a\nsimple polygon, any geodesic (3−ε)-spanner has worst-case complexity Ω(nm), and for any\nconstantε∈(0,1)and integer constant t≥2, a(t−ε)-spanner has worst-case complexity\nΩ(mn1/(t−1)+n).\nProblem Statement. A very natural question is then if we can reduce the total complexity\nof a geodesic spanner by allowing Steiner points . That is, by adding an additional set S\nofkvertices inG, each one corresponding to a (Steiner) point in P. For the original sites\np,q∈Swe still require that their distance in Gis at mostttimes their distance in P(i.e.\ndG(p,q)≤td(p,q)), but the graph distance from a Steiner point p′∈Sto any other site\nis unrestrained. Allowing for such Steiner points has proven to be useful in reducing the\nweight [5,18] and size [ 26] of spanners. In our setting, it allows us to create additional\n“junction” vertices, thereby allowing us to share high-complexity subpaths. See Figure 1\n1De Berg, van Kreveld, and Staals [ 16] claim that the refinement by Abam, de Berg, and Seraji [ 2] can be\napplied to obtain a (2t+ε)-spanner of the same complexity (increased by a constant factor dependent\nonε). However, some details of how this refinement influences the complexity are missing.S. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 3\nFigure 1 A spanner in a simple polygon that uses two Steiner points (red squares). By adding\nthe two Steiner points, the spanner has a small spanning ratio and low complexity, as we no longer\nneed multiple links that pass through the middle section of P.\nfor an illustration. Indeed, if we are allowed to turn every polygon vertex into a Steiner\npoint, Clarkson [ 13] shows that, for any ε>0, we can obtain a (1 +ε)-spanner of complexity\nO((n+m)/ε). However, the number of polygon vertices mmay be much larger than the\nnumber of Steiner points we can afford. Hence, we focus on the scenario in which the number\nof Steiner points kis (much) smaller than m.\nOur Contributions. Surprisingly, we show that in this setting, Steiner points have only\nlimitedutility. Firstofall, weshowthatcomputingaminimumcardinalitysetofSteinerpoints\nfor sites in a polygonal domain that allow for a 3-spanner of a certain complexity is NP-hard.\nMoreover, we show that there is a set of nsites in a simple polygon with m= Ω(n)vertices\nfor which any (2−ε)-spanner (with k < n/ 2Steiner points) has complexity Ω(mn2/k2).\nSimilarly, we give a Ω(mn/k )andΩ(mn1/(1+t)/k1/(1+t))lower bound on the complexity of\na(3−ε)- and (t−ε)-spanner with kSteiner points. Hence, these results dash our hopes for\na near linear complexity spanner with “few” Steiner points and constant spanning ratio.\nThese lower bounds actually hold in a more restricted setting. Namely, when the metric\nspace is simply an edge-weighted tree that has mvertices, and the nsites are all placed in\nleaves of the tree. In this setting, we show that we can efficiently construct a spanner whose\ncomplexity is relatively close to optimal. In particular, our algorithm constructs a 2t-spanner\nof complexity O(mn1/t/k1/t+nlog(n/k)). The main idea is to partition the tree into k\nsubtrees of roughly equal size, construct a 2t-spanner without Steiner points on each subtree,\nand connect the spanners of adjacent trees using Steiner points. The key challenge that we\ntackle, and one of the main novelties of the paper, is to make sure that each subtree contains\nonly a constant number of Steiner points. We carefully argue that such a partition exists, and\nthat we can efficiently construct it. Constructing the spanner takes O(nlog(n/k) +m+K)\ntime, where Kis the output complexity. This output complexity is either the size of the\nspanner (O(nlog(n/k))), in case we only wish to report the endpoints of the links, or the\ncomplexity, in case we wish to explicitly report the shortest paths making up the links. An\nextension of this algorithm allows us to deal with a forest as well.\nThis algorithm for constructing a spanner on an edge-weighted tree turns out to be the\ncrucial ingredient for constructing low-complexity spanners for point sites in polygons. In\nparticular, given a set of sites in a simple polygon P, we use some of the techniques developed\nby de Berg, van Kreveld, and Staals [ 16] to build a set of trees whose leaves are the sites,\nand in which the distances in the trees are similar to the distances in the polygon. We then\nconstruct a 2t-spanner with kSteiner points on this forest of trees using the above algorithm,\nand argue that this actually results into a 2√\n2t-spanner with respect to the distances in the\npolygon. The main challenge here is to argue that the links used still have low complexity,4 The Complexity of Geodesic Spanners using Steiner Points\neven when they are now embedded in the polygon. We prove that the spanner (with respect\nto the polygon) has complexity O(mn1/t(logk)1+1/t/k1/t+nlog2n), and can be constructed\nin timeO(nlog2n+mlogn+K). Fork=O(1)our spanner thus matches the result of de\nBerg, van Kreveld, and Staals [ 16]. Finally, we extend these results to polygonal domains,\nwhere we get a similar complexity 6t-spanner.\nOrganization. We start with our results on edge-weighted trees in Section 2. To get a feel\nfor the problem, we first establish lower bounds on the spanner complexity in Section 2.1. In\nSection 2.2 we present the algorithm for efficiently constructing a low complexity 2t-spanner.\nWe extend this result to a forest in Section 2.3. In Section 3, we then show how to use these\nresults to obtain a 2√\n2t-spanner for sites in a simple polygon P. In Section 4 we further\nextend our algorithms to the most general case in which Pmay even have holes. In Section 5\nwe show that computing a minimum cardinality set of Steiner points with which we can\nsimultaneously achieve a particular spanning ratio and maximum complexity is NP-hard. In\nSection 6 we mention some open questions that remain.\n2 Steiner spanners for trees\nIn this section, we consider spanners on an edge-weighted rooted tree T. We allow only\npositive weights for the edges. The goal is to construct a t-spanner on the leaves of the tree\nthat useskSteiner points, i.e. the set of sites Sis the set of leaves. The Steiner points are\nnot restricted to the vertices of T, but can lie anywhere on the tree. We denote by nthe\nnumber of leaves and by mthe number of vertices in T. The complexity of a link between\ntwo sites (or Steiner points) p,q∈Tis the number of edges in the shortest path π(p,q), and\nthe distance d(p,q)is equal to the sum of the weights on this (unique) path. We denote\nbyT(v)the subtree of Trooted at vertex v. For an edge e∈Twith upper endpoint v1(the\nendpoint closest to the root) and lower endpoint v2, we denote by T(e) :=T(v2)∪{e}the\nsubtree oferooted atv1.\n2.1 Complexity lower bounds\nIn this section, we provide several lower bounds on the worst-case complexity of any (t−ε)-\nspanner that uses kSteiner points, where tis an integer constant and ε∈(0,1). When Steiner\npoints are not allowed, any (2−ε)-spanner in a simple polygon requires Ω(n2)edges [1]\nandΩ(mn2)complexity. If we allow a larger spanning ratio, say (3−ε)or even (t−ε), the\nworst-case complexity becomes Ω(mn)orΩ(mn1/(t−1)), respectively [ 16]. As the polygons\nused for these lower bounds are very tree-like, these bounds also hold in our tree setting.\nNext, we show how much each of these lower bounds is affected by the use of kSteiner points.\n▶Lemma 1. For any constant ε∈(0,1), there exists an edge-weighted tree Tfor which any\n(2−ε)-spanner using k<n/ 2Steiner points has complexity Ω(mn2/k2).\nProof.The treeTin our construction is a star of ‘pitchforks’, with long handles and short\nteeth, that attach at the handles. See Figure 2 for an illustration of the construction.\nPrecisely,Tconsists of 2kpitchforks, that have handles of weight 2, a comb of weight- 1teeth,\nand all other edges have much smaller weight, defined later. All pitchforks have an (almost)\nequal number of teeth, at the end of which are the sites, and the top of each comb has high\ncomplexity, as in Figure 2(b), leading to Θ(n/k)sites and Θ(m/k)vertices per pitchfork.\nFor a sitepin one pitchfork, consider the distance to a site qin the same pitchfork. First,\nobserve that for every ε, we can set the remaining weights to be small enough such thatS. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 5\n1\n2\n\u0002(m\nk)\n2kpitchforks\n\u0002(n\nk) sites\n\u0002(m\nn)\nh\nw\n(a)\n(b)\nFigure 2 (a) Our construction for an Ω(mn2/k2)lower bound on the complexity of any (2−ε)-\nspanner. (b)A more detailed version of the comb of a pitchfork highlighted in the orange disk, which\nis also used for our Ω(mn1/(t+1)/k1/(t+1))lower bound on the complexity of any (t−ε)-spanner.\n\u0002(m\nk)\n2k+ 1 combs\n\u0002(n\nk) sites\nw\nh\nFigure 3 Our construction for an Ω(nm/k )lower bound on the complexity of any (3−ε)-spanner.\nd(p,q)approaches 2. It follows that the path from ptoqin any (2−ε)-spannerGdoes not\nvisit any other site, as this would result in dG(p,q)violating the spanning ratio.\nTo analyze the complexity of a (2−ε)-spannerG, consider the at least kpitchforks that\ndo not contain a Steiner point (a possible Steiner point at the center of the star is not part\nof any pitchfork). Inside each of these pitchforks, the complete graph on the sites must be\na subgraph ofG. We thus have Ω(n2/k2)links of complexity Ω(m/k). As there are≥kof\nthese pitchforks, this leads to a Steiner spanner with Ω(mn2/k2)complexity. ◀\n▶Lemma 2. For any constant ε∈(0,1), there exists an edge-weighted tree Tfor which any\n(3−ε)-spanner using k<n/ 2Steiner points has complexity Ω(mn/k ).\nProof.We generalize the construction for the lower bound on the complexity of any (3−ε)-\nspanner [ 16], but instead of a polygon we work with a more restrictive setting of an edge-\nweighted tree. In the (3−ε)lower bound, the polygon consists of ‘combs’ with long thin\nteeth, and we use a congruent structure in our trees. See Figure 3 for an illustration of\nthe construction. Our tree Tconsist of a sequence of 2k+ 1of these combs, which are\nconnected by their short handles in series using high complexity paths, which we will refer\nto as corridors. Let wbe the total weight of the corridors. Remember that the spanner\nwill be constructed on the leaves of T, which are at the bottom of each tooth of each comb.\nFor each tooth we define the edge weight as h≫w. All other edges have negligible weight.\nWhenwapproaches 0, the distance between any two sites p,qapproaches 2h. So, in any\n(3−ε)-spannerGthere can be at most one other site on the path from ptoq.6 The Complexity of Geodesic Spanners using Steiner Points\nLetM= Θ(m/k)denote the complexity of a single corridor. For any particular Steiner\npoint, the set of sites that can be reached by a path of complexity < M/ 2cannot lie in\ndifferent combs. Hence, after placing kSteiner points, at least k+ 1of the 2k+ 1combs\nhave no site that lies within complexity <M/ 2of any Steiner point.\nConsider the k+ 1combs that have no path of complexity <M/ 2to any Steiner point.\nLetS1,...,Sk+1be the sets of sites that lie in these k+ 1combs, respectively. Observe that\n|Si|= Θ(n/k). We say that a link with one endpoint at a site in Sileaves its comb if its\nother endpoint is either a site in S\\Si, or a Steiner point. Any link that leaves its comb has\ncomplexity Ω(M).\nWe will show that there are Ω(n)links that leave their comb. There are two cases:\n(1) every site in/uniontext\niSihas a link that leaves its comb, or (2) for some iand some site p∈Si,\nthe sitephas no link that leaves its comb. In case (1), there are at least1\n2/summationtextk+1\ni=1|Si|=\nΩ((k+ 1)n\n2k+1) = Ω(n)links that leave a comb. In case (2), consider a site qin/uniontext\nj̸=iSj.\nThe path from qtopcan visit at most one other site. Since phas no link that leaves\nits comb,qmust have a link to a site in Si, which leaves its comb. There are at least/summationtext\nj̸=i|Sj|= Ω(kn\n2k+1) = Ω(n)such links. In both cases, we have Ω(n)links of complexity\nΩ(M) = Ω(m/k), so the total complexity is Ω(mn/k ). ◀\n▶Lemma 3. For any constant ε∈(0,1)and integer constant t≥2, there exists an edge-\nweighted tree Tfor which any (t−ε)-spanner using k < nSteiner points has complexity\nΩ(mn1/(t+1)/k1/(t+1)).\nBefore we prove Lemma 3, we first discuss a related result in a simpler metric space.\nLetϑnbe the 1-dimensional Euclidean metric space with npointsv1,...,vnon thex-axis at\n1,2,...,n. A link (vi,vj)has complexity|i−j|. Dinitz, Elkin, and Solomon [ 17] give a lower\nbound on the total complexity of any spanning subgraph of ϑn, given that the link-radius is\nat mostρ. The link-radius (called hop-radius in [ 17])ρ(G,r)of a graphGwith respect to a\nrootris defined as the maximum number of links needed to reach any vertex in Gfromr.\nThe link-radius of Gis then minr∈Vρ(G,r). The link-radius is bounded by the link-diameter,\nwhich is the minimum number of links that allow reachability between any two vertices.\n▶Lemma 4 (Dinitz et al. [ 17]).For any sufficiently large integer nand positive integer\nρ<logn, any spanning subgraph of ϑnwith link-radius at most ρhas complexity Ω(ρ·n1+1/ρ).\nProof of Lemma 3. Consider the tree construction illustrated in Figure 2(b). This edge-\nweighted tree Thas the shape of a comb of width wand height hwithnteeth separated by\ncorridors of complexity M= Θ(m/n)each. Each leaf at the bottom of a comb tooth is a site.\nAny spanning subgraph of ϑnof complexity Cand link-diameter ρis in one-to-one\ncorrespondence with a (ρ+ 1−ε)-spanner of complexity C·m/ninT[16]. Lemma 4 then\nimplies that any (t−ε)-spanner has worst-case complexity Ω(mn1/(t−1)).\nWhen a setSofkSteiner points is introduced, we consider the ≤k+ 1setsS0,...,Sk\nof consecutive sites that have no Steiner point between them; see Figure 4(a). We can\nreplace any link (p,q),p,q∈S∪Sfor whichπ(p,q)intersects a Steiner point sby two\nlinks: one from ptosand one from stoq. This increases the complexity by only a constant\nfactor. From now on, we thus assume there are no such links. For any (t−ε)-spannerGon\nS=S0∪···∪Sk, we can improve the complexity of G, while increasing the spanning ratio\nbetween sites in Siby at most 2.\n▷Claim 5. LetCibe the complexity of the subgraph of Ginduced by Siand the at most two\nSteiner points sℓandsrboundingSifrom the left and right, respectively. We can construct\na(t+ 2−ε)-spannerG′\nionSithat has complexity at most Ci.S. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 7\n(a)\n(b)\ns1\ns2\np`\nS1\nS2\nS0\nFigure 4 (a) The setsSidefined by the two (red square) Steiner points. (b)For any spanner\nonS1, every link to a Steiner point can be replaced by a link of smaller complexity, while increasing\nthe spanning ratio by at most one. Here, the dashed links can be replaced by the green links.\nProof of Claim. Letpℓandprdenote the leftmost and rightmost site in Si, respectively.\nWe replace each link (p,sℓ)(or(p,sr)),p∈Si, by the link (p,pℓ)(resp. (p,pr)). If there is a\nlink(sℓ,sr), it is replaced by the link (pℓ,pr). We claim this increases the spanning ratio\namong sites in Siby at most 2. Any path in Gbetween sites p,q∈Sithat visits either sℓ\nand/orsrcorresponds to a path via prand/orpℓinG′\ni. The length of the path increases by\n2hwhen visiting prorpℓ, so by at most 4 hwhen visiting both. As d(p,q)≥2h, the spanning\nratio increases at most two. ◀\nThe changes in the spanner in the proof of the previous claim only decrease the complexity\nof the subspanner on Si. Notice also that if we apply them to each of the sets Si, each\nlink ofGis changed by only one of the subspanners G′\ni. Thus, we consider the minimum\ncomplexity of any (t+ 2−ε)-spanner on these sites. By applying Lemma 4, we find that the\nworst-case complexity of any (t+ 2−ε)-spanner on these |Si|sites is Ω(m/n·|Si|1+1/(t+1)).\nThe complexity of Gis at least the sum of the complexities of these G′\nispanners over all Si:\nm\nnk/summationdisplay\ni=0Ω/parenleftig\n|Si|1+1/(t+1)/parenrightig\n, wherek/summationdisplay\ni=0|Si|= Θ(n).\nThis sum is minimal whenever |Si|= Θ(n/k)for alli∈0,...,k, which one can see by\napplying a logarithmic transformation and induction. So,\nm\nnk/summationdisplay\ni=0Ω/parenleftig\n|Si|1+1/(t+1)/parenrightig\n≤m\nnk/summationdisplay\ni=0Ω/parenleftig\n(n/k)1+1/(t+1)/parenrightig\n= Ω/parenleftig\nmn1/(t+1)/k1/(t+1)/parenrightig\n. ◀\n2.2 A low complexity Steiner spanner\nIn this section, we describe how to construct low complexity spanners for edge-weighted\ntrees. The goal is to construct a 2t-spanner of complexity O(mn1/t/k1/t+nlogn)that uses\nat mostkSteiner points. We first show that the spanner construction for a simple polygon\nof [16] can be used to obtain a low complexity spanner for a tree (without Steiner points).\n▶Lemma 6 (de Berg et al. [ 16]).For any integer t≥1, we can build a 2t-spanner for Tof\nsizeO(nlogn)and complexity O(mn1/t+nlogn)inO(nlogn+m)time.\nProof.De Berg, van Kreveld, and Staals consider constructing a spanner for some weighted\n1-dimensional space, which is related to a spanner in a polygon. They essentially show how8 The Complexity of Geodesic Spanners using Steiner Points\nto construct a 2t-spanner of complexity O(mn1/t+nlogn)on a shortest path tree. Their\napproach also applies to any edge-weighted rooted tree as we defined here, and we briefly\nsketch the construction.\nFirst, an ordering on the sites is defined based on an in-order traversal of the tree. Then,\nwe find an edge esuch that roughly half of the sites lie in each subtree after removing e. Fix\nN=n1/t. We partition the sites into Θ(N)groups based on the ordering. Each group is\nagain partitioned into Θ(N)groups, etc. At each level, we select the site that is closest to e\nas the group center cfor each group, and add a link to the spanner from cto the center of\nits parent group. This ensures that the spanning ratio holds for any pair of sites p,qwithp\nin the one subtree and qin the other. To also obtain good paths between sites in the same\nsubtree, we recurse on both subtrees. ◀\nSpanner construction. To construct a Steiner spanner G, we start by partitioning the sites\ninksetsS1,...Skby an in-order traversal of the tree. The first ⌈n/k⌉sites encountered are\ninS1, the second⌈n/k⌉inS2, etc. After this, the sites are reassigned into knew disjoint\nsetsS′\n1,...S′\nk. For each of these sets, we consider a subtree T′\ni⊆Twhose leaves are the\nsetS′\ni. There are four properties that we desire of these sets and their subtrees.\n1.The size of S′\niisO(n/k).\n2.The treesT′\nicoverT, i.e./uniontext\niT′\ni=T.\n3.The treesT′\niare disjoint apart from Steiner points.\n4.Each treeT′\nicontains only O(1)Steiner points.\nAs we prove later, these properties ensure that we can construct a spanner on each subtree T′\ni\nto obtain a spanner for T. We obtain such sets S′\niand the corresponding trees T′\nias follows.\nWe color the vertices and edges of the tree Tusingkcolors{1,...,k}in two steps. In\nthis coloring, an edge or vertex is allowed to have more than one color. First, for each set Si,\nwe color the smallest subtree that contains all sites in Siby colori. Note that after this step\nthere are no uncolored vertices that have an incident descendant edge that is colored. In\nthe next step, we color the remaining uncolored edges and vertices. These edges and their\n(possibly already colored) upper endpoints are colored using a bottom-up approach. We\nassign each uncolored edge and its upper endpoint the color with the lowest index ithat is\nassigned also to its lower endpoint.\nAfter coloring T, we place a Steiner point siat the root of tree Tiformed by all edges\nand vertices of color ifori∈{1,...,k}. Observe that it may happen that more than one\nSteiner point is assigned to some vertex. Slightly abusing our notation, we also denote by si\nthe vertex that the Steiner point siis placed at.\nFor each Steiner point si, we define a subtree T′\ni⊆T. The sites in T′\niwill be the set S′\ni.\nThe treeT′\niis a subtree of T(si). Whensiis the only Steiner point at the vertex, then\nT′\ni=T(si)\\/uniontext\nj(T(sj)\\{sj})forsja descendant of si. In other words, we look at the tree\nrooted atsiup to and including the next Steiner points, see Figure 5(a). When siis not\nthe only Steiner point at the vertex, we include only subtrees T(e)ofsi(up to the next\nSteiner points) that start with an edge ethat has color iand no color j >i. See Figure 5(b).\nWheneversihas the lowest or highest index of the Steiner points at si, we also include all\nT(e′)that start with an edge e′of colorj <iorj >i, respectively. This generalizes the\nscheme for when siis the only Steiner point at the vertex.\nBy creating T′\niin this way, siis not a leaf of T′\ni. We therefore adapt T′\niby adding an\nedge of weight zero between the vertex at siand a new leaf corresponding to si. On each\nsubtreeT′\ni, we construct a 2t-spanner using the algorithm of Lemma 6. These kspanners\nconnect at the Steiner points, which we formally prove in the spanner analysis.S. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 9\n(a)\n(b)\nT0\n1\nT0\n3\nT0\n2\nT0\n1\nT0\n2\nT0\n3\nT0\n4\nFigure 5 The treeTiis the subtree whose edges and vertices have color i. A Steiner point (square)\nis placed at the root of Ti. The shaded areas show the trees T′\ni. The examples show the case when\nthe Steiner points are (a)at different vertices or (b)share a vertex.\nAnalysis. To prove thatGis indeed a low complexity 2t-spanner forG, we first show that\nthe four properties stated before hold for S′\niandT′\ni. We often apply the following lemma,\nthat limits the number of colors an edge can be assigned by our coloring scheme.\n▶Lemma 7. An edge can have at most two colors.\nProof.First, observe that an edge can receive more than one color only in the first step of\nthe coloring. Suppose for contradiction that there is an edge einTthat has three colors\ni<j <ℓ . Letvbe the lower endpoint of e. Then there must be three sites pi∈Si,pj∈Sj,\npℓ∈SℓinT(v). Because these sets are defined by an in-order traversal, pimust appear\nbeforepjin the traversal. Similarly, pjappears before pℓ. Additionally, there must be a\nsitep′\nj∈SjinT\\T(v), otherwise the color jwould not be assigned to e. The sitep′\njmust\nappear before pior afterpℓin the traversal. In the first case, pimust be in Sjas it appears\nbetween two sites in Sj. In the second case, we find pℓ∈Sj, also giving a contradiction. ◀\n▶Lemma 8. The size of S′\niisO(n/k).\nProof.Each setS′\nican contain sites from an original set Sj,j̸=i, only if there is a site\np∈SjinT(si)and the Steiner point sjlies at or above si. Lemma 7 implies that there are\nat most two such sets that have a Steiner point above si. We claim that there can be at most\none such set Sjthat has a site p∈S′\nifor whichsi=sj. Note that j <iby definition of T′\ni.\nSuppose there is another set Sxfor which there is also a site q∈S′\niandsx=si. Consider\nthe first edge eande′on the paths π(si,p)andπ(si,q), respectively. Both eande′have\ncolori, otherwise porqwould not be in S′\ni. Assi=sj=sx,emust also have color jande′\nmust also have color x. Lemma 7 now implies that e̸=e′. However, there must be a site of\nSiin bothT(e)andT(e′), which is impossible as j <iandx<i. There are thus at most\nthree setsSj,j̸=i, from which a site is in T′\ni. As each set S1,...,Skcontains at most⌈n/k⌉\nsites, it follows that |S′\ni∩S|≤4⌈n/k⌉=O(n/k). ◀\n▶Lemma 9. The treesT′\nicoverT, i.e./uniontext\niT′\ni=T.10 The Complexity of Geodesic Spanners using Steiner Points\nProof.We prove the lemma by induction. Let vbe a vertex of Twhere a Steiner point is\nplaced, then the induction hypothesis is that T(v)is covered by all subtrees T′\nicorresponding\nto Steiner points in T(v).\nAs the base case, consider a vertex vat which there is at least one Steiner point and for\nwhich there are no Steiner points in T(v)\\{v}. If there is a single Steiner point siatv, then\nT′\ni=T(v). Next, we consider the case that there is more than one Steiner point at v. Lete\nbe an edge incident to vinT(v)andxbe the smallest color of e. Ifsx̸=vandehas no\nadditional color, then T(e)is included in T′\nj, wheresjis the lowest or highest numbered\nSteiner point at v. Ifedoes have an additional color y, thensy=v, as no site of Sycan\nappear inT\\T(v). It follows that T(e)is included in T′\ny. Finally, consider the case that\nsx=v. ThenT(e)is either included in T′\nx, or in a tree T′\ny, whereyis the potential other\ncolor ofe. We conclude that all subtrees of vare included in some T′\ni, and thus that in the\nbase case the hypothesis holds.\nLetvbe a vertex of Tthat has a Steiner point and for which there is at least one Steiner\npoint inT(v)\\{v}. We assume the induction hypothesis holds for all Steiner points in\nT(v)\\{v}. The exact same argument as for the base case tells us that each edge eincident\ntovinT(v)is included in some T′\nifor a Steiner point siatv. By definition, these subtrees\ninclude all of T(v)up to the next Steiner points. As the induction hypothesis holds for all\nSteiner points in T(v)belowv, we conclude that T(v)is covered by the subtrees.\nFinally, we need to show there is a Steiner point at the root rofT, as this would imply\nT(v)is covered. Because there are no uncolored edges or vertices after the coloring is\ncomplete, root rhas at least one color i. This implies that there is a Steiner point siatr.◀\n▶Lemma 10. The treesT′\niare disjoint apart from Steiner points.\nProof.Suppose there is a non-Steiner vertex v∈Tfor whichv∈T′\niandv∈T′\nj,i<j. The\nSteiner points siandsjare both above vinTby definition. If si=sj, then the first edge e\nofπ(si,v)has colorsiandj. However, as i<j,T(e)would not be included in T′\ni, which\nis a contradiction. If si̸=sj, assume that siis belowsjinT, the proof for sjbelowsiis\nsimilar. The path π(v,sj)must visitsi, which contradicts that v∈T′\nj. The same argument\nproves that an edge e∈Tcan also be contained in at most one subtree T′\ni. ◀\n▶Lemma 11. There are at most five Steiner points in T′\ni.\nProof.By definition siis inT′\ni, so we want to show that there are at most four other Steiner\npoints inT′\ni. Note that a Steiner point can occur in T′\nionly if its path to sidoes not\nencounter any other Steiner point. We first consider the number of Steiner points in subtrees\nT(e)for which the edge eis an edge incident to siinT′\nithat does not have color i, and then\nconsider the number of Steiner points in subtrees for which edoes have color i.\nLetT′\ni[j]be the subtree of T′\nirooted atsithat is the union of T(e)∩T′\nifor all edges e\nincident to siof colorj̸=iand not of color ias well, see Figure 6(a). We argue that this\nsubtree is non-empty for at most two colors j. Consider such an edge e. Becauseedoes not\nhave coloriande∈T′\ni, it must be that sjis abovesiinT. Thus, the parent edge of siinT\nmust also be colored j. By Lemma 7, the parent edge of siinTcan be assigned at most two\ncolors, soT′\ni[j]is non-empty for at most two colors.\nNext, we prove that T′\ni[j]contains at most one Steiner point other than si. We assume\nthati<j, the proof for i>jis symmetric. Assume for contradiction that T′\ni[j]contains\ntwo Steiner points sxandsy,x<y; see Figure 6(b). As shown before, there is a site of Sj\ninT\\T(si). Asi<j, this implies that i<x<y<j . Lete′be the first edge on π(si,sx)S. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 11\ne\n(a)\n(b)\nsx\nsy\nsi\nT0\ni\nsi\nT0\ni\ne\nT0\ni[j]\ne0\nsc\nFigure 6 Notation used in Lemma 11. In T′\ni[j]are all subtrees that start with an edge of color j.\nthat is not on π(si,sy), i.e. the first edge after the paths diverge. Let cbe a color of e′\nand letvandwbe the upper and lower endpoint of e′. The tree T(w)does not contain\nany sites of Sj, as these appear in the traversal after the sites of Sx. It follows that Scis\nbeforeSjin the in-order traversal, in other words i<c<j . The parent edge of sicannot\nbe coloredc, as a site of Scwould then appear either before a site in Sior after a site in Sj\nin the in-order traversal. It follows that scis on the path π(v,si). Ifsc̸=si, this contradicts\nthe assumption that this path does not contain a Steiner point. If sc=si, theni<cimplies\nthat the subtree starting with an edge of color jis in notT′\ni, which is a contradiction. We\nconclude that there are at most two Steiner points in the subtrees T′\ni[j]in total for all j̸=i.\nWhat remains is to show that the subtrees whose edge incident to sihas coloricontain\nat most two Steiner points. The proof is similar to the proof for T′\ni[j]. We assume for\ncontradiction that there are three such Steiner points sx,sy,sz, withx<y <z . Assume\nthatπ(si,sy)diverges later from π(si,sx)than fromπ(si,sz). The other case follows by\nsymmetry. Consider the first edge e′onπ(si,sy)after which the path to sxdiverges. Let c\nbe a color of e′. The in-order traversal implies that x<c<z . Because there is no Steiner\npoint on the path from sytosi, the Steiner point scmust lie at or above si. However, sc\ncannot be above sias this would imply that a site of Scappears before a site of Sxor after a\nsite ofSzin the traversal, which is a contradiction. It follows that si=sc. Consider the first\nedgeeonπ(si,sy)and letvbe its lower endpoint. This edge has colors iandc. So, there\nmust be a site p∈ScinT\\T(v). When both sxandszare inT(v), thenpmust appear\nbefore a site in Sxor after a site in Szin the traversal, which is a contradiction. If sz/∈T(v)\nandsx∈T(v), thenx<cagain implies that pmust appear after the sites in T(v)in the\ntraversal. Also, there must be a site q∈Siin the subtree that contains sz, as by definition\nthe first edge of this subtree has color i. This leads to a contradiction, as pappearing after q\nin the traversal would imply pappears between two sites in Sc, andpappearing before q\nwould imply qappears between two sites in Si. Finally, consider the case where sxandsz\nare both not in T(v). As both of the subtrees that contain sxandszstart with an edge of\ncolori, there must be a site of Siin both subtrees. However, this implies that one of these\nsites appears before the sites in Syand the other after, which is again a contradiction. So,\nthere are at most two Steiner points in all subtrees whose first edge has color i.\nWe conclude that there are at most five Steiner points in T′\ni. ◀\nWe are now ready to prove that our algorithm computes a spanner with low complexity.\n▶Lemma 12. The spannerGis a2t-spanner for Tof sizeO(nlog(n/k))and complexity\nO(mn1/t/k1/t+nlog(n/k)).12 The Complexity of Geodesic Spanners using Steiner Points\nProof.To bound the size and complexity of the spanner, we first consider the number of\nleavesniand vertices miin each subproblem T′\ni. Asniis equal to|S′\ni|plus the number of\nSteiner points in T′\ni, Lemmata 8 and 11 imply that ni=O(n/k) + 5 =O(n/k). Lemma 10\nstates the subtrees T′\niare disjoint apart from their shared Steiner points, so/summationtextmi=O(m).\nUsing the size of a 2t-spanner according to Lemma 6, we obtain the following size for G.\nk/summationdisplay\ni=1O/parenleftign\nklog/parenleftign\nk/parenrightig/parenrightig\n=k·O/parenleftign\nklog/parenleftign\nk/parenrightig/parenrightig\n=O/parenleftig\nnlog/parenleftign\nk/parenrightig/parenrightig\n.\nSimilarly, we obtain the following bound on the complexity of G.\nk/summationdisplay\ni=1O/parenleftbigg\nmi/parenleftign\nk/parenrightig1/t\n+n\nklog/parenleftign\nk/parenrightig/parenrightbigg\n=O/parenleftbiggmn1/t\nk1/t+nlog/parenleftign\nk/parenrightig/parenrightbigg\n.\nWhat remains is to show that Gis a2t-spanner. Let p,q∈Sbe two leaves in T. Ifp,q∈S′\ni\nfor somei∈{1,...,k}then the shortest path π(p,q)is contained within T′\ni. The 2t-spanner\nonT′\niimplies that dG(p,q)≤2td(p,q). If there is no such set S′\nithat contains both sites,\nconsider the sequence of vertices v1,...,vℓwhereπ(p,q)exits some subtree T′\ni. Letv,wbe\ntwo consecutive vertices in this sequence. Without loss of generality, assume that w∈T(v),\nand letsxbe the Steiner point at vfor whichw∈T′\nx(Lemmata 9 and 10 imply sxexists).\nThen the 2t-spanner on T′\nxensures that dG(v,w)≤2td(v,w). It follows that dG(p,q)≤\ndG(p,v1) +dG(v1,v2) +···+dG(vℓ,q)≤2t(d(p,v1) +d(v1,v2) +···+d(vℓ,q)) = 2td(p,q).◀\n▶Theorem13. LetTbe a tree with nleaves andmvertices, and t≤1be any integer constant.\nFor any integer 1≤k≤n, we can build a 2t-spannerGforTusing at most kSteiner points\nof sizeO(nlog(n/k))and complexity O(mn1/t/k1/t+nlog(n/k))inO(nlog(n/k) +m+K)\ntime, where Kis the output size.\nProof.Lemma 12 proves the spanning ratio, size, and complexity bound of G. What remains\nis to show thatGcan be constructed in O(nlog(n/k) +m+K)time.\nThe construction of Gconsists of three parts, partitioning the sites into sets S1,...,Sk\nto place the Steiner points, obtaining the trees T′\n1,...,T′\nk, and computing a 2t-spanner for\neachT′\niusing Lemma 6. The first part of the construction starts with an in-order traversal\nofTto obtain the sets S1,...,SkinO(m)time. Since the sites are leaves of T, we can\nuse leaf-to-root paths, to first find the smallest subtree that contains all sites of Sifor each\ncolori: Find the root rof the smallest subtree for Sias the lowest vertex shared by the\nleaf-to-root paths from the first and last site encountered in the in-order treewalk for each Si,\nand color every vertex and edge in the subtree rooted at r. Since every edge can get at\nmost two colors, this takes O(m)time. Then proceed in bottom-up fashion to color the\nremaining uncolored edges and vertices. Once a color jis no longer advanced upwards from a\nvertexv, that is, when another color is smaller and is used to color the edge upwards from v\nin the bottom-up approach, we can set the Steiner point sjat vertexv. Every edge of Tis\nconsidered only once in this phase of the process, and hence it takes O(m)time.\nWe find the assignment of vertices and edges to the trees T′\n1,...,T′\nkwith another in-order\ntreewalk of T, which makes use of a stack to remember the index of the tree we are currently\nassigning to. At the root of Tthere must be a Steiner point siso start the in-order traversal\nwith current index I=i. Assign every vertex and edge we encounter on the walk to tree T′\nI,\nand whenever a vertex with Steiner points is encountered, push the index Ion the stack and\nsetI=jto be the index of the encountered Steiner point sj. Now proceed the treewalk,S. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 13\nassigning edges and vertices to T′\nI. When the in-order traversal returns to sjafter having\nwalked over all descendant edges, we have completed T′\njand returned to the vertex that\nis a leaf of T′\ni. We can pop the topmost index off the stack, which will be i, and continue\nassigning to T′\ni.\nIn case we encounter a vertex vwith multiple Steiner points for the first time, push the\ncurrent index Ion the stack and set I=gcorresponding to the index of the Steiner point\nat this vertex with the lowest index. Once the in-order treewalk returns to vand has to\ncontinue walking on a descendant edge with color h > g, we have completed T′\ngand can\nsafely setIto the color of the next lowest indexed Steiner point at v. Once assigned to I, the\nhighest index of the Steiner points at such a vertex vis used until the treewalk has walked\nover all descendant edges: All trees T′\nxrooted atvhave been completed, and the index of\nthe tree which had vas a leaf can be popped off the stack. Once the in-order traversal is\nfinished, every vertex and edge has been assigned to a tree T′\nxaccording to the index I=x\nthat was maintained at that point in the traversal. Since an index is pushed to the stack only\nwhen the root of another tree T′\nyis encountered, and every push is matched to a single pop\noperation, when going into and coming out of a subtree at said root, we use O(k)constant\ntime interactions with the stack. This procedure hence takes O(m+k) =O(m)time.\nFinally, we use Lemma 6 to assess the running time for the computation of the 2t-spanners\nonT′\n1,...,T′\nk. Recall that Lemmata 8, 10, and 11 imply for each T′\nithatni=O(n/k)and\nthat/summationtext(mi) =m. The total running time to compute the 2t-spanners is then\nk/summationdisplay\ni=1O/parenleftign\nklogn\nk+mi/parenrightig\n=k·O/parenleftign\nklog/parenleftign\nk/parenrightig/parenrightig\n+O(m) =O/parenleftig\nnlog/parenleftign\nk/parenrightig\n+m/parenrightig\n.\nThus, computing the spanners asymptotically dominates the total running time. ◀\n2.3 Extending the tree spanner to a forest\nIn this section, we extend our tree spanner to a spanner for a forest F. We denote the trees\ninFbyT1,...,Tℓand byS(Ti)the sites inTi. AsFis disconnected, we cannot require all\nsites inSto have a path between them in the spanner. Instead, we say that Gis at-spanner\nforFifGis at-spanner for every Ti.\nLetk′=⌊k/2⌋. To construct a spanner on F, we partition the sites into k′setsS1,...Sk′\nusing an in-order traversal of F. In particular, we perform an in-order traversal of T1, then\ncontinue with an in-order traversal of T2, etc. As before, we assign the first ⌈n/k′⌉sites we\nencounter to S1, the second⌈n/k′⌉sites toS2, and so on. A set Sican be distributed over\nmany trees, however, only the first and last tree that has sites in Sican also contain sites\nfrom another set Sj. For a treeTwhose sites are all assigned to the same set, i.e. S(T)⊆Si\nfor somei∈{1,...,k′}, we construct a spanner on Tusing the algorithm of Lemma 6 and\nadd these edges to G. These trees thus do not contain any Steiner points. For a tree T\nwhose sites are not all assigned to the same set, i.e. S(T)⊈Sifor alli∈{1,...,k′}, letK\nbe the number of sets Sifor whichS(T)∩Si̸=∅. We construct the spanner of Theorem 13\nonTusingKSteiner points and add the edges to G. Because the intersection Si∩T(S)is\nnon-empty for a set Siin at most two trees, the total number of Steiner points we place is at\nmost 2k′≤k.\n▶Theorem 14. LetFbe a forest with nleaves and mvertices, and t≤1be any integer\nconstant. For any integer k≤n, we can build a 2t-spannerGforFusing at most kSteiner\npoints of size O(nlog(n/k))and complexity O(mn1/t/k1/t+nlog(n/k))inO(nlog(n/k) +\nm+K)time, where Kis the output size.14 The Complexity of Geodesic Spanners using Steiner Points\nProof.Clearly, the spanner Gconstructed by the above procedure is a 2t-spanner for\nT1,...,Tℓand thus a 2t-spanner forF. To bound the size and complexity of G, and the\nrunning time of the algorithm constructing G, we first consider the trees whose sites are all\nin a single set Si, and then consider the other trees. Let MinandMoutdenote the total\nnumber of vertices in all trees whose sites are in a single set, and the total number of vertices\nin the other trees, respectively. Thus m=Min+Mout.\nLetaidenote the number of trees whose sites are fully contained within the set Si. Let\nni,jandmi,jdenote the number of sites and vertices in the j-th subtree whose sites are\ncontained in Si. Note that/summationtext\njni,j≤⌈n/k′⌉and/summationtext\ni,jmi,j=Min. We obtain the following\nbound on the number of links of the spanner restricted to these trees by Lemma 6\nk′/summationdisplay\ni=1ai/summationdisplay\nj=1ni,jlogni,j≤k′/summationdisplay\ni=1ai/summationdisplay\nj=1ni,jlog/ceilingleftign\nk′/ceilingrightig\n≤log/ceilingleftign\nk′/ceilingrightigk′/summationdisplay\ni=1/ceilingleftign\nk′/ceilingrightig\n=O/parenleftig\nnlog/parenleftign\nk/parenrightig/parenrightig\n.(1)\nSimilarly, Lemma 6 implies the following bound on the complexity of Grestricted to these\ntrees\nk′/summationdisplay\ni=1ai/summationdisplay\nj=1(mi,jn1/t\ni,j+ni,jlogni,j)≤/ceilingleftign\nk′/ceilingrightig1/tk′/summationdisplay\ni=1ai/summationdisplay\nj=1mi,j+O/parenleftig\nnlog/parenleftign\nk/parenrightig/parenrightig\n=O/parenleftbiggMinn1/t\nk1/t+nlog/parenleftign\nk/parenrightig/parenrightbigg\n. (2)\nLastly, the algorithm of Lemma 6 is used to construct the 2t-spanners on these trees, which\nresults in a total running time of\nk′/summationdisplay\ni=1ai/summationdisplay\nj=1ni,jlogni,j+mi,j≤Min+k′/summationdisplay\ni=1ai/summationdisplay\nj=1ni,jlog/ceilingleftign\nk′/ceilingrightig\n≤O/parenleftig\nnlog/parenleftign\nk/parenrightig\n+Min/parenrightig\n.(3)\nNext, we consider the trees whose sites are not contained within a single set. Let Tjbe\nsuch a tree. Furthermore, let nj,mj, andkjdenote the number of sites, vertices, and Steiner\npoints inTj. Askjis equal to the number of sets for which Si∩Tj̸=∅, we haveni≤kj⌈n/k′⌉.\nTheorem 13 states that the size of Grestricted toTjisO/parenleftig\nkjn\nk′log/parenleftbign\nk′/parenrightbig/parenrightig\n, the complexity is\nO/parenleftig\nmjn1/t\nk′1/t+kjn\nk′log/parenleftbign\nk′/parenrightbig/parenrightig\n, and the time to compute this part of GisO/parenleftig\nkjn\nk′log/parenleftbign\nk′/parenrightbig\n+mj/parenrightig\n.\nRecall that/summationtext\njkj≤kand/summationtext\njmj=Mout. Summing over all trees Tjwhose sites are\nnot contained within a single set, we obtain the following expressions for the total size and\ncomplexity of their spanners, and the running time for constructing these 2t-spanners.\n/summationdisplay\njO/parenleftbiggkjn\nk′log/parenleftign\nk′/parenrightig/parenrightbigg\n=O/parenleftbiggk\nk′·nlog/parenleftign\nk′/parenrightig/parenrightbigg\n=O/parenleftig\nnlog/parenleftign\nk/parenrightig/parenrightig\n. (4)\n/summationdisplay\njO/parenleftbiggmjn1/t\nk′1/t+kjn\nk′log/parenleftign\nk′/parenrightig/parenrightbigg\n=O/parenleftbiggMoutn1/t\nk1/t+nlog/parenleftign\nk/parenrightig/parenrightbigg\n. (5)\n/summationdisplay\njO/parenleftbiggkjn\nk′log/parenleftign\nk′/parenrightig\n+mj/parenrightbigg\n=O/parenleftbiggk\nk′·nlog/parenleftign\nk′/parenrightig\n+Mout/parenrightbigg\n=O/parenleftig\nnlog/parenleftign\nk/parenrightig\n+Mout/parenrightig\n.(6)\nSumming Equations (1) and (4) we obtain the stated bound on the size of G. By summing\nEquations (2) and (5), and noting that m=Min+Mout, we obtain the complexity of G.\nTo obtain total running time for constructing G, recall that the algorithm consists of two\nsteps: First, an in-order traversal of F, which takes O(m)time. Second, constructing the\n2t-spanners for the trees of F, which takes time equal to the sum of Equations (3) and (6). ◀S. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 15\n3 Steiner spanners in simple polygons\nWe consider the problem of computing a t-spanner using kSteiner points for a set of npoint\nsites in a simple polygon Pwithmvertices. We measure the distance between two points in\np,qinPby their geodesic distance, i.e. the length of the shortest path π(p,q)fully contained\nwithinP. A link (p,q)in the spanner is the shortest path π(p,q), and its complexity is\nthe number of segments in this path. Lower bounds for trees straightforwardly extend to\npolygonal instances. Again, we aim to obtain a spanner of complexity close to the lower bound.\n▶Lemma 15. The lower bounds of Lemmata 1, 2, and 3 also hold for simple polygons.\nProof.As the trees used for the lower bounds are planar, we can construct a simple polygon P\nthat has the tree as free space and place the sites at the locations of the leaves of the tree.\nThe complexity of paths and distances between points in Pare the same as in the tree. ◀\nSpanner construction. Next, we describe how to obtain a low-complexity spanner in a\nsimple polygon using at most kSteiner points. In our approach, we combine ideas from [ 2]\nand [16] with the forest spanner of Theorem 14. We first give a short overview of the approach\nto obtain a low complexity 2√\n2t-spanner [ 16], and then discuss how to combine these ideas\nwith the forest spanner to obtain a low complexity Steiner spanner.\nWe partition the polygon Pinto two subpolygons PℓandPrby a vertical line segment\nλsuch that roughly half of the sites lie in either subpolygon. For the line segment λ, we\nthen consider the following weighted 1-dimensional space. For each site p∈S, letpλbe the\nprojection ofp: the closest point on λtop. The (weighted 1-dimensional distance) between\ntwo sitespλ,qλis defined as dw(pλ,qλ) :=d(p,pλ) +d(pλ,qλ) +d(q,qλ). In other words,\nthe sites in the 1-dimensional space are weighted by the distance to their original site in\nP. For this 1-dimensional space we construct a t-spannerGλ, and for each link (pλ,qλ)in\nGλwe add the link (p,q)to the spannerG. Finally, we process the subpolygons PℓandPr\nrecursively. De Berg, van Kreveld, and Staals [ 16] show that this gives a√\n2t-spanner in\na simple polygon. To obtain a low complexity spanner, they construct the 1-dimensional\nspanner using the approach of Lemma 6, which has a spanning ratio of 2t. This results in a\n2√\n2t-spanner of complexity O(mn1/t+nlog2n).\nIn our case, we require information on the paths from the sites to their projection instead\nof only their distance to decide where to place the Steiner points. This information is\ncaptured in the shortest path tree SPTλof the segment λ, which is the union of all shortest\npath from the vertices of Pto their closest point on λ. Additionally, we include all sites in S\nin this shortest path tree. The segment λis split into multiple edges at the projections of the\nsites. See Figure 7 for an example. The tree is rooted at the lower endpoint of λ. Observe\nthat there are O(m+n)vertices in SPTλ.\nWe adapt the algorithm to build a spanner in Pas follows. Instead of computing a\n1-dimensional spanner directly in each subproblem, we first collect the shortest path trees of\nall subproblems. Let SPTi,jdenote the shortest path tree of the j-th subproblem at the i-th\nlevel of the recursion. We exclude all vertices from SPTi,jthat have no site as a descendant.\nThis ensures that all leaves of the tree are sites. Let F=∪i,jSPTi,jbe the forest consisting\nof all trees. A site in Sor vertex of Pcan occur in multiple trees SPTi,j, but they are seen\nas distinct sites and vertices in the forest F. We call a tree SPTi,jlarge if 0≤i≤logkand\nsmall otherwise. In other words, the trees created in the recursion up to level logkare large.\nWe then partition Finto two forestsFsandFℓcontaining the small and large trees. For\neach tree inFswe directly apply the 2t-spanner of Lemma 6 that uses no Steiner points to\nobtain a spannerGs. For the forestFℓwe apply Theorem 14 to obtain a 2t-spannerGℓforFℓ.16 The Complexity of Geodesic Spanners using Steiner Points\n\u0015\np\np\n\u0015\nSPT i;j\nFigure 7 The shortest path tree of λinP′and its SPT i,j. The grey nodes and edges are not\nincluded in SPT i,j, but can be assigned to a T′\nias indicated by the colored backgrounds. The\nsquares show the Steiner points in SPT i,jandP′. The sites in P′are colored as the trees T′\ni.\nLetGF=Gs∪Gℓ. A Steiner point in GFcorresponds to either a vertex of Por a point on λ.\nLetSdenote the set of Steiner points. To obtain a spanner Gin the simple polygon, we add\na link (p,q),p,q∈S∪S, toGwhenever there is a link in GFbetween (a copy of) pandq.\n▶Lemma 16. The graphGis a2√\n2t-spanner for the sites SinPof sizeO(nlog2n).\nProof.Because the spanner GFis also a spanner for the weighted 1-dimensional space,\nLemma 2 of [ 16] directly implies that Gis a2√\n2t-spanner. The size of Gis the same\nas the size ofGF. Each site in Soccurs in one SPTi,jfor each level i. There are thus\nO(nlogk)sites inFℓandO(n(logn−logk))sites inFs. Theorem 14 then implies that the\nsize ofGℓisO(nlogklog(nlogk/k)) =O(nlog2n). Lemma 6 implies that the size of Gsis/summationtextO(logn)\ni=logk2i·O/parenleftbig\nn/2ilog(n/2i)/parenrightbig\n=O(nlog2n). It follows that the size of GisO(nlog2n).◀\nComplexity analysis. To bound the complexity of the links in G, we have to account for the\ncomplexity of links generated by both GsandGℓ. Bounding the complexity of Gsis relatively\nstraightforward, but to bound the complexity of Gℓwe first prove a lemma on the structure\nof a shortest path in Pbetween sites inFℓ.\nLetTbe a tree inFℓand letP′be the corresponding subpolygon of the subproblem.\nWe consider the shortest path between two sites that are assigned to the same subtree T′\ni\nofTby the forest algorithm. It can be that this shortest path uses vertices of P′that were\nexcluded from T, as they had no site as a descendant. For the analysis, we do include these\nvertices inTand assign them to subtrees T′\njas in Section 2.2; see Figure 7. The following\nlemma states that the complexity of a shortest path between two sites in the same subtree is\nbounded by the number of vertices in the subtree.\n▶Lemma 17. A shortest path π(p,q)inPbetween sites p,q∈T′\niuses vertices in T′\nionly.\nProof.Assume for contradiction that ris a highest vertex in Tused byπ(p,q)that is not\ninT′\ni. First, consider the case that ris the root of T. Recall that this means ris the bottom\nendpoint of λ, and thus lies on the boundary of P. Asris onπ(p,q), it must be that pand\nqlie in different subpolygons, and at least one of them lies below the horizontal line through\nr. This implies that si=r, which is a contradiction.\nNext, consider the case that ris not the root of T. Letr′be the parent of r. Ifr′is inT′\ni,\nthen it must be a leaf. We consider the following partition of P′. Recall that rλdenotesS. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 17\nr\u0015\n\u0015\nr0\nr\nA\nB\n\u0019r\nFigure 8 The extended shortest path πrseparates the polygon into P′\nr=A∪BandP′\n¬r.\nr0\nr0\np\nr\nr0\np\np\nr\nr\nCase (i)\nCase (ii)\nCase (iii)\nFigure 9 The tree subcases for p∈Bbased on the structure of the polygon around r′.\nthe closest point on λtor. We extend the shortest path π(r,rλ)to the boundary of P′by\nextending the first and last line segments of the path to obtain a path πr, see Figure 8. Let\n∂P′denote the boundary of P′. We define P′\nrto be the closed polygon bounded by ∂P′and\nπrthat contains the polygon edges incident to r, andP′\n¬r:=P′\\P′\nr. Becauseris a reflex\nvertex ofP′,P′\nris well-defined. Without loss of generality, we assume that P′\nrcontains the\npart ofλaboverλ, as in Figure 8. If both pandqare inP′\n¬r, thenr /∈π(p,q). It follows\nthatpand/orqare inP′\nr. Without loss of generality, assume that p∈P′\nr.\nWe distinguish two cases based on the location of p, see Figure 8. Either p∈A, where\nA⊂P′\nris bounded by the extension segment starting at rand∂P′, orp∈B, whereB⊂P′\nr\nis bounded by π(r,rλ), the extension segment starting at rλ, and∂P′.\nIfp∈A, thenpis a descendant of rinT. Aspandqare inT′\niandris not, it must be\nthatqis also a descendant of r. It follows that q∈A, but this means that ris not a reflex\nvertex onπ(p,q), which contradicts it being a shortest path.\nIfp∈B, the previous paragraph implies that q /∈A. Additionally, q /∈Bas well, as r\nwould then not be a reflex vertex in π(p,q). It follows that q∈P′\n¬r. Next, we make a\ndistinction on whether r′is a vertex of Por not. First, assume that r′is not a vertex of P,\nand thusr′∈λ. Becausep∈B,pλmust be at or above r′. Becauseq∈P′\n¬r,qλmust be\nbelowr′. This implies that the path in Tfromptoqvisitsr′, which contradicts p,q∈T′\ni.\nNext, we assume that r′is a vertex of P. We distinguish three different subcases based on\nthe shape of the polygon around r′, see Figure 9, and show each case results in a contradiction.\n(i) The edges of Pincident tor′are inP′\n¬r.Asris onπ(p,q),qmust be a descendant\nofr′. It follows that the Steiner point siis located on the path in Tfromqtor′, sopis\nalso a descendant of r′. It follows that pis on the segment rr′. However, for rto be on\nπ(p,q),qmust then be in A, which is a contradiction.\n(ii) The edges of Pincident tor′are inP′\nr, andr′is onπ(p,pλ).In this case, pis a\ndescendant of r′. This again implies that qis a descendent of r′, which contradicts q∈P′\n¬r.18 The Complexity of Geodesic Spanners using Steiner Points\n(iii) The edges of Pincident tor′are inP′\nr, andr′is not onπ(p,pλ).Thepathπ(p,q)\neither intersects the boundary of Btwice, which is not allowed as both are shortest paths,\nor visitsr′as well. However, this implies that q∈A, which is a contradiction. ◀\nNext, we use Lemma 17 to show that Ghas low complexity.\n▶Lemma 18. The spannerGhas complexity O(mn1/t(logk)1+1/t/k1/t+nlog2n)\nProof.To bound the complexity of the links in Ggenerated byGswe apply Lemma 6 directly.\nAs the algorithm of Lemma 6 is essentially the algorithm to obtain a low complexity spanner\nin a polygon using the shortest path tree, the complexity bound of Lemma 6 also holds in\nthe simple polygon setting. Using/summationtext2i\nj=0mi,j=O(m), wheremi,jdenotes the number of\nvertices in SPTi,j, we obtain a complexity of\nO(logn)/summationdisplay\ni=logk2i/summationdisplay\nj=0O/parenleftbigg\nmi,j/parenleftign\n2i/parenrightig1/t\n+n\n2ilog/parenleftign\n2i/parenrightig/parenrightbigg\n=O(mn1/t)O(logn)/summationdisplay\ni=logk1\n2i/t+O(nlog2n)\n=O/parenleftbiggmn1/t\nk1/t+nlog2n/parenrightbigg\n.\nForFℓ, the algorithm of Lemma 6 is used as a subroutine on every subtree T′\ni. Lemma 17\nimplies that the complexity bound of Theorem 14 also holds for links in P. Recall that the\nnumberofsitesin FℓisO(nlogk). Avertexof Pcanoccurinatmosttwosubproblemsateach\nlevel of the recursion that partitions P, thus the number of vertices in FℓisO((m+n)logk).\nAs thensites are equally divided over all subproblems at level i, the complexity of the links\ninGgenerated byGℓgiven by Theorem 14 is improved to\nO/parenleftbiggmlogk(nlogk)1/t\nk1/t+nlogklog/parenleftbiggnlogk\nk/parenrightbigg/parenrightbigg\n=O/parenleftbiggmn1/t(logk)1+1/t\nk1/t+nlog2n/parenrightbigg\n.◀\nRunning time. We use a similar algorithm to de Berg, van Kreveld, and Staals [ 16] to\nefficiently compute our spanner. We briefly sketch the algorithm here. We first preprocess\nthe polygon: We build a shortest path data structure [ 12,22], and the horizontal and vertical\ndecomposition [ 12,25] inO(m)time. Next, for each site we find the trapezoid (in the\nhorizontal and vertical decomposition) that contains it. For each trapezoid, we sort the sites\nwithin onx-coordinate. This preprocessing takes O(n(logn+ logm))time.\nThen, we recursively construct all shortest path trees SPTi,j. We find a vertical separ-\natorλsuch that at most 2n/3sites lie in either subpolygon in O(n+m)time using the vertical\ndecomposition ([ 16, Lemma 10]), and construct the shortest path tree SPTλinO(m+nlogm)\ntime using the horizontal decomposition ([ 16, Lemma 11]). The time to construct the shortest\npath trees SPTi,jfor allO(logn)levels is thus O(mlogn+nlogmlogn).\nFinally, we construct the spanners on FsandFℓ. Theorem 13 and Theorem 14 state that\nthe running time to construct these spanners is asymptotic in their size plus the number of\nvertices in the forest. As there are O((m+n)logn)vertices inF, the total running time is\nO(nlog2n+mlogn), which dominates the shortest path tree construction time.\n▶Theorem 19. LetSbe a set of npoint sites in a simple polygon Pwithmvertices,\nandt≥1be any integer constant. For any integer 1≤k≤n, we can build a geodesic\n2√\n2t-spanner using at most kSteiner points of size O(nlognlog(n/k))and complexity\nO(mn1/t(logk)1+1/t/k1/t+nlog2n)inO(nlog2n+mlogn+K)time, where Kis the\noutput size.S. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 19\n4 Steiner spanners in polygonal domains\nWhenever the polygon may contain holes, i.e. the environment is a polygonal domain P, the\nspanner construction in the previous section no longer suffices. In particular, we may need\na different type of separator, and shortest paths in Pare no longer restricted to vertices\nin some subtree (Lemma 17 does not hold). De Berg, van Kreveld, and Staals [ 16] run\ninto similar problems when generalizing their low complexity spanner for a simple polygon\nto a polygonal domain, and solve them as follows. There are two main changes in their\nconstruction. First, the separator is no longer a line segment, but consists of at most three\nshortest paths that partition the domain into two subdomains PrandPℓ. They call this a\nbalanced separator. They then construct a spanner Gλon the 1-dimensional space containing\nthe projections of the sites for each shortest path in the separator. Second, the links that are\nincluded in the spanner are no longer shortest paths, but consist of at most three shortest\npaths. Essentially, for each link (pλ,qλ)inGλ, the link that is added to Gis the shortest\npath between pandqinSPTλ. In contrast to the simple polygon, using a 1-dimensional\nspanner with spanning ratio tresults in a spanning ratio in Pof3t[16].\nTo construct a low complexity spanner using at most kSteiner points, we use our simple\npolygon approach with the adaptions of [ 16]. This means the number of trees, and thus the\nnumber of sites and vertices in the trees, increases by a constant factor, as we create at most\nthree shortest path trees at each level. To bound the complexity, we can no longer apply\nLemma 17. However, the links that are added to Gare shortest paths in the shortest path\ntree. Therefore, the bound on the complexity of GFdirectly translates to a bound on the\ncomplexity ofG. As in the simple polygon case, we thus obtain a spanner of size O(nlog2n)\nand complexity O(mn1/t(logk)1+1/t/k1/t+nlog2n).\nDe Berg, van Kreveld, and Staals [ 16] show that a balanced separator can be computed\ninO(n2logm+nmlogm)time, and that computing the highest level separator dominates\nthe computation of the lower level separators. After constructing the separator, the shortest\npath tree SPTλof a shortest path λin the separator can be constructed in O((n+m)logm)\ntime [24]. The time to construct all shortest path trees is thus dominated by the time to\nconstruct the balanced separators. As in the simple polygon case, we can build the spanner\nonFinO(nlog2n+mlogn)time.\n▶Theorem 20. LetSbe a set of npoint sites in a polygonal domain Pwithmver-\ntices, andt≥1be any integer constant,. For any integer 1≤k≤n, we can build a\ngeodesic 6t-spanner using at most kSteiner points of size O(nlognlog(n/k))and complexity\nO(mn1/t(logk)1+1/t/k1/t+nlog2n)inO(n2logm+nmlogm+K)time, where Kis the\noutput size.\n5 NP-hardness\nGiven a polygonal domain Pwithmvertices and a set Sofnsites inPas input, the\nSteiner Spanner problem asks whether there exist a spanner with spanning ratio tthat\nuses at most kSteiner points and has complexity at most C. In this section, we show\nthat this problem is NP-hard already for t= 3using a reduction from Vertex Cover in\npenny graphs, which is an NP-complete problem [ 9]. Recall that a vertex cover of a graph\nG= (V,E)is a set of vertices W⊆Vsuch that each edge in Ehas at least one incident\nvertex inW. The Vertex Cover problem cover asks whether a given graph has a vertex\ncover of size at most k.Penny graphs are contact graphs of unit disks. Equivalently, they\nare the graphs that admit a penny graph drawing : a straight-line drawing in the plane in20 The Complexity of Geodesic Spanners using Steiner Points\nwhich no two edges cross, all edges are unit-length, and the angle between any two edges\nemanating from the same vertex is at least π/3.\nTo show that the Steiner Spanner problem is NP-hard, we first show in Lemma 21\nthatVertex Cover remains hard on penny graphs without vertices of degree one.\n▶Lemma 21. Vertex Cover restricted to penny graphs without degree-one vertices is\nNP-complete, even if we are provided with a corresponding penny graph drawing.\nProof.Clearly the problem is in NP. We reduce from Vertex Cover restricted to penny\ngraphs, which has been shown to be NP-complete [ 9]. LetG= (V,E)be a penny graph. We\nconstruct a penny graph G′= (V′,E′)as follows. Repeatedly remove a degree one vertex\nand its neighbor until there are no degree-one vertices left. Let Rbe the set of vertices\nremoved from G, i.e.R=V\\V′. We show that Ghas a vertex cover Wof size at most k\nif and only if G′has a vertex cover W′of size at most k′, wherek′=k−|R|/2. Consider\na degree-1 vertex uofGand letvbe its only neighbor. If there is a vertex cover of Gof\nsizekthat includes u, we can produce a vertex cover of size at most kthat includes vand\nnotu. Thus, there is always a solution for Vertex Cover that includes vand notu, and\nany vertex cover of G−{u,v}can be extended to a vertex cover of Gby adding a single\nvertexu. Iterating this argument yields the desired result. We remark that the fact that we\ncan iteratively remove degree-1 vertices in this way is a standard reduction rule for finding a\nkernel for Vertex Cover [15]. ◀\nTo obtain a reduction from Vertex Cover on penny graphs to Steiner Spanner , we\nrequire that there are no small cycles in the graph. Define an s-subdivided penny graph to be\na graph obtained from a penny graph by replacing each edge by a path of sedges. Observe\nthat ans-subdivided penny graph has degree-one vertices if and only if the original penny\ngraph has degree-one vertices. In Lemma 22 we show that Vertex Cover remains hard on\n3-subdivided penny graphs without vertices of degree one.\n▶Lemma 22. Vertex Cover restricted to 3-subdivided penny graphs without degree-one\nvertices is NP-complete, even if we are provided with a corresponding penny graph drawing.\nProof.Again, NP membership is trivial. We reduce from Vertex Cover restricted to penny\ngraphs without degree-one vertices, which is NP-complete by Lemma 21. Let G= (V,E)\nbe a penny graph without degree-one vertices, and let G′= (V′,E′)be the corresponding\n3-subdivided penny graph. Let kbe the size of a minimal vertex cover of G, andk′be the\nsize of a minimal vertex cover of G′. It suffices to show that k′=k+|E|.\nWe can extend any vertex cover WofGto a vertex cover of G′by adding a single vertex\nto the cover for each subdivided edge e∈E:Wcontains at least one endpoint of e, soW\nalready covers the first or last edge of the path of three edges that replace einG′. The\nremaining two edges that replace ecan be covered by adding a single vertex to the cover.\nThe resulting vertex cover of G′has size|W|+|E|, sok′≤k+|E|.\nFor the reverse direction, we show that there exists a vertex cover W′ofG′, in which for\neach path (u,u′)−(u′,v′)−(v′,v)that replaces an edge (u,v)ofG, exactly one of u′andv′\nlie inW′. Indeed, at least one must lie in W′to cover the edge (u′,v′), and if both lie in W′,\nwe maintain a vertex cover by replacing v′byv. It follows that W′contains at least one of u\nandvfor each edge (u,v)ofG, and hence W′∩Vis a vertex cover for Gof size|W′|−|E|,\nsok′≥k+|E|. Therefore k′=k+|E|, which completes the proof. ◀\nNext, we show that Steiner Spanner is NP-hard by a reduction from Vertex Cover\non3-subdivided penny graphs without degree-one vertices. We construct a polygonal domain\nand a set of sites as follows, see Figure 10 for an example.S. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 21\n\"\nFigure 10 Constructing a Steiner Spanner instance from a penny graph with subdivided edges.\nConstruction. LetG= (V,E)be a 3-subdivided penny graph without degree-one vertices.\nBased on a penny graph drawing of the corresponding ‘unsubdivided’ graph, we obtain\na penny graph drawing of Gin which all ‘subdivided edges’ are straight, see Figure 10\n(left). We construct a polygonal domain whose free space consists of a small neighborhood\naround the vertices and edges of such a drawing. Near each vertex vofG, our polygonal\ndomain contains a corridor that splits into two corridors of length ε <1/8and place a\nsite at the end of each corridor, see Figure 10 (right). Let ph(v)andpℓ(v)denote these\nsites. These corridors are ‘carved’ into a hole adjacent to the vertex. The corridor to ph(v)\nhas a high complexity of 3M, while the corridor of pℓ(v)has a lower complexity of M,\nwithM > 26n. Additionally, we include a convex chain of Mvertices between the two\ncorridors, such that the shortest path π(ph(v),pℓ(v))has complexity 5M, while the shortest\npath frompℓ(v)to the entrance has a complexity of just M. We denote by δ < ε/ 3the\ndifference between the length of the shortest path π(ph(v),pℓ(v))and the path via v, i.e.\nd(ph(v),pℓ(v)) =d(ph(v),v) +d(pℓ(v),v)−δ= 2ε−δ. The set of sites Shas sizen:= 2|V|.\nSuppose that we want to decide whether Ghas a vertex cover of size k. Then the\nconstructed polygonal domain and sites form an instance of Steiner Spanner in which we\nallowkSteiner points, and require a spanning ratio of t= 3and a maximum complexity of\nC=M·(3n−2k+ 1/2). We show that this is a Yes-instance if and only if Ghas a vertex\ncover of size at most k. As all coordinates of Pand the value of Ccan be chosen such that\nthey are polynomial in n, this implies that the Steiner Spanner problem is NP-hard.\nVertex cover implies spanner. Suppose there is a vertex cover Wof size at most kforG.\nFor each vertex v∈W, we place a Steiner point s(v)at the entrance of the corridors of v.\nSites that belong to a vertex v∈Wlie at distance≤εfrom the Steiner point s(v). The\nremaining sites in Sbelong to a vertex adjacent to a vertex in W, and hence lie at distance\n≤1 +εfrom a Steiner point.\nTo obtain a spanner Gon the setS, we create, for each vertex v∈V, a link inGfrom its\nsitepℓ(v)in its low complexity corridor, to its closest Steiner point, breaking ties arbitrarily.\nFor the site ph(v)in the high complexity corridor of v, we add a link to s(v)ifv∈W, and\ntopℓ(v)otherwise. We also add a link between every pair of Steiner points if that link does\nnot visit another Steiner point. Let p,q∈Sbe two sites. To prove that Ghas a spanning\nratio of 3, we consider three cases: (i) pandqbelong to the same vertex, (ii) pandqbelong\nto two adjacent vertices, or (iii) pandqbelong to two non-adjacent vertices.\n(i)pandqbelong to the same vertex v.We haved(p,q) = 2ε−δ > ε. If there is no\nSteiner point at v, i.e.v /∈W, then the link (p,q)is inG. Otherwise, both links (p,s(v))\nand(q,s(v))of lengthεare inG. Hence,dG(p,q)≤2ε≤3ε<3d(p,q).22 The Complexity of Geodesic Spanners using Steiner Points\n(ii)pandqbelong to adjacent vertices. Thend(p,q) = 1+2ε, and there must be a Steiner\npoint at the entrance of the corridor of porqor both. Without loss of generality,\nassume there is a Steiner point s(v)at the entrance of p. Fromqthere is a path of\nlength≤1 + 3ε−δto its closest Steiner point. This Steiner point must have a path of\nlength≤2tos(v). Thus, the length of the path in the spanner connecting pandqis at\nmostdG(p,q)≤1 + 3ε−δ+ 2 +ε≤3 + 4ε<3d(p,q).\n(iii)pandqbelong to non-adjacent vertices. The distance between pandqisλ+ 2εfor\nsome integer λ≥2. There is a path of length at most λ+ 2between the Steiner points p\nandqare connected to. Thus, dG(p,q)≤1 + 3ε−δ+λ+ 2 + 1 + 3ε−δ≤4 +λ+ 6ε. As\nλ≥2,dG(p,q)≤3d(p,q).\nThus,Gis a 3-spanner. Finally, we need to bound the complexity of the spanner. We\nfirst consider the links from a site ph(v)for allv∈V. Such a link either has complexity\n3Mor5M, depending on whether there is a Steiner point s(v). The total complexity for\nallv∈Vis thus 5M(n/2−k) + 3Mk=M(5\n2n−2k). A link from a site pℓ(v)to a Steiner\npoint has complexity as most M+ 1. These links thus have complexity at most (M+ 1)·n/2\nin total. As the maximum degree of a penny graph is six, and any vertex of degree greater\nthan two is adjacent to only degree two vertices, the number of links from a Steiner point\nto other Steiner points can be at most six. As each such link has complexity at most two,\nthe total complexity of these links is at most 12k. The total complexity is thus at most\nM(5\n2n−2k) +Mn/ 2 +n/2 + 12k≤M(3n−2k) + 13n≤M(3n−2k+ 1/2), asM > 26n.\nThus, our spanner adheres to all of the posed conditions.\nSpanner implies vertex cover. Suppose thatGis a3-spanner on SinPthat uses at most\nkSteiner points and has complexity at most C. We show that Gthen has a vertex cover of\nsize at most k.\n▶Observation 23. At leastkvertices are within distance 2εof a Steiner point.\nProof.For any vertex v, we haved(ph(v),pℓ(v)) = 2ε−δ. Letπbe a minimum length path\ninGthat connects ph(v)topℓ(v). BecauseGis a3-spanner, we have ∥π∥=dG(ph(v),pℓ(v))≤\n3d(ph(v),pℓ(v))≤6ε−3δ <6ε. Hence, the first half of πlies within distance 3εofph(v),\nand the second half lies within distance 3εofpℓ(v). Therefore, the entirety of πlies within\ndistance 2εofv, and ifph(v)andpℓ(v)are not connected by a direct link, there must\nexist a Steiner point within distance 2εofv. Suppose for a contradiction that there are at\nmostk−1verticesvthat are within distance 2εof a Steiner point. For those vertices v,\nthe path inGconnecting ph(v)topℓ(v)has complexity at least 4M, and these paths are\ndisjoint for different vertices. For the (at least n/2−k+ 1) verticesvthat are not near a\nSteiner point,Gcontains a direct link from ph(v)topℓ(v)of complexity 5M. Additionally,\nthere exists a link that connects either ph(v)orpℓ(v)to another site to which a corridor\nofvadds at least Mcomplexity. The total complexity of the spanner is then at least\n(n/2−k+ 1)(5M+M) + (k−1)4M=M(3n−2k+ 2)>C. ◀\n▶Observation 24. There is no link between any pair of sites p,qthat belong to different\nvertices and do not have a Steiner point within distance 2ε.\nProof.Letvandwbe the vertices that pandqbelong to, respectively. Suppose that there\nwould be such a link. This link has complexity ≥2M. Additionally, there must be a link\nfrom bothpandqto its counterpart in the other corridor of vorwof complexity 5M. As the\nspanner is connected, there must also be a path in the spanner from pandqto every otherS. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 23\nu\nv\nw\nFigure 11 There are no 3- or 4-cycles in G.\nsite. There must thus be another link of complexity ≥Mleaving a corridor of vorw. To\nconnect all other sites we still require at least (n/2−2−k)5M+k3M+(n/2−2)Mcomplexity.\nThe total complexity is thus ≥M(1 + 2(5 + 1) + 3 n−2k−10−2) =M(3n−2k+ 1)>C,\nwhich is a contradiction. ◀\nTo obtain a vertex cover W, we assign a vertex to Wif it has a Steiner point within\ndistance 2ε. Observation 23 tells us that |W|≤k. To prove that Wis a vertex cover,\nassume for contradiction that there is an uncovered edge (u,v)∈E. Consider the two sites\npℓ(u)andpℓ(v). First, note that there are no 3- and 4-cycles in Gby construction, thus\nthe orange edges in Figure 11 do not exist. Also, Observation 24 implies that there is no\ndirect edge between sites of uandv, and thuspℓ(u)andpℓ(v)both have a link to a Steiner\npoint. Suppose the link between pℓ(u)and the Steiner point it is connected to does not visit\nv, and the link between pℓ(v)and the Steiner point it is connected to does not visit u. Then\ndG(pℓ(u),pℓ(v))≥ε+2(1−2ε)+1+2(1−2ε)+ε= 5−6ε. This contradicts the spanning ratio,\nasd(pℓ(u),pℓ(v)) = 1 + 2ε. Next, suppose without loss of generality that the link of pℓ(v)\ndoes visitu(and the link of pℓ(u)does not visit v). In that case, consider another neighbor\nw̸=uofv. Because the distance from vto the Steiner point is at least 2−2ε, and there are\nno short cycles, we have that dG(pℓ(v),pℓ(w))≥ε+ 2(2−2ε) +ε= 4−2ε>3d(pℓ(v),pℓ(w)).\nThis is again in contradiction to the spanning ratio. We conclude that Wis a vertex cover\nforG.\n▶Theorem 25. Steiner Spanner is NP-hard. More precisely, it is NP-hard to decide\nwhether a polygonal domain with nsites admits a 3-spanner using at most kSteiner points\nand complexity at most C=M·(3n−2k+ 1/2)withM > 26n.\n6 Future work\nOn the side of constructing low-complexity spanners, an interesting direction for future work\nwould be to close the gap between the upper and lower bounds, both with and without\nusing Steiner points. We believe it should be possible to increase the n1/(t+1)term ton1/t\n(or evenn1/(t−1)) in the analysis of our construction from Lemma 3. On the side of the\nhardness, many interesting open questions remain, such as: Is the problem still hard in a\nsimple polygon? Can we show hardness for other spanning ratios and/or a less restricted\ncomplexity requirement? Is the problem even in NP?\nReferences\n1Mohammad Ali Abam, Marjan Adeli, Hamid Homapour, and Pooya Zafar Asadollahpoor.\nGeometric spanners for points inside a polygonal domain. In Proc. 31st International Sym-\nposium on Computational Geometry, SoCG , volume 34 of LIPIcs, pages 186–197. Schloss\nDagstuhl - Leibniz-Zentrum für Informatik, 2015.24 The Complexity of Geodesic Spanners using Steiner Points\n2Mohammad Ali Abam, Mark de Berg, and Mohammad Javad Rezaei Seraji. Geodesic spanners\nfor points on a polyhedral terrain. SIAM J. Comput. , 48(6):1796–1810, 2019.\n3Khaled M. Alzoubi, Xiang-Yang Li, Yu Wang, Peng-Jun Wan, and Ophir Frieder. Geometric\nspanners for wireless ad hoc networks. IEEE Trans. Parallel Distributed Syst. , 14(4):408–421,\n2003.\n4Sunil Arya, Gautam Das, David M. Mount, Jeffrey S. Salowe, and Michiel H. M. Smid.\nEuclidean spanners: short, thin, and lanky. In Proc. 27th Annual ACM Symposium on Theory\nof Computing, STOC , pages 489–498. ACM, 1995.\n5Sujoy Bhore and Csaba D. Tóth. Light Euclidean Steiner spanners in the plane. In Proc. 37th\nInternational Symposium on Computational Geometry, SoCG , volume 189 of LIPIcs, pages\n15:1–15:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.\n6Glencora Borradaile and David Eppstein. Near-linear-time deterministic plane Steiner spanners\nfor well-spaced point sets. Comput. Geom. , 49:8–16, 2015.\n7Prosenjit Bose, Joachim Gudmundsson, and Michiel H. M. Smid. Constructing plane spanners\nof bounded degree and low weight. Algorithmica , 42(3-4):249–264, 2005.\n8Prosenjit Bose and Michiel H. M. Smid. On plane geometric spanners: A survey and open\nproblems. Comput. Geom. , 46(7):818–830, 2013.\n9Márcia R. Cerioli, Luérbio Faria, Talita O. Ferreira, and Fábio Protti. A note on maximum\nindependent sets and minimum clique partitions in unit disk graphs and penny graphs:\ncomplexity and approximation. RAIRO Theor. Informatics Appl. , 45(3):331–346, 2011.\n10T.-H. Hubert Chan, Anupam Gupta, Bruce M. Maggs, and Shuheng Zhou. On hierarchical\nrouting in doubling metrics. ACM Trans. Algorithms , 12(4):55:1–55:22, 2016.\n11T.-H. Hubert Chan, Mingfei Li, Li Ning, and Shay Solomon. New doubling spanners: Better\nand simpler. SIAM J. Comput. , 44(1):37–53, 2015.\n12Bernard Chazelle. Triangulating a simple polygon in linear time. Discret. Comput. Geom. ,\n6:485–524, 1991.\n13Kenneth L. Clarkson. Approximation algorithms for shortest path motion planning. In Proc.\n19th Annual ACM Symposium on Theory of Computing, STOC , pages 56–65. ACM, 1987.\n14Vincent Cohen-Addad, Arnold Filtser, Philip N. Klein, and Hung Le. On light spanners,\nlow-treewidth embeddings and efficient traversing in minor-free graphs. In Proc. 61st IEEE\nAnnual Symposium on Foundations of Computer Science, FOCS , pages 589–600. IEEE, 2020.\n15Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin\nPilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms . Springer, 2015.\n16Sarita de Berg, Marc J. van Kreveld, and Frank Staals. The complexity of geodesic spanners.\nInProc. 39th International Symposium on Computational Geometry, SoCG , volume 258 of\nLIPIcs, pages 16:1–16:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023.\n17Yefim Dinitz, Michael Elkin, and Shay Solomon. Shallow-low-light trees, and tight lower\nbounds for Euclidean spanners. In Proc. 49th Annual IEEE Symposium on Foundations of\nComputer Science, FOCS , pages 519–528, 2008.\n18Michael Elkin and Shay Solomon. Narrow-shallow-low-light trees with and without Steiner\npoints. SIAM J. Discret. Math. , 25(1):181–210, 2011.\n19Michael Elkin and Shay Solomon. Optimal Euclidean spanners: Really short, thin, and lanky.\nJ. ACM, 62(5):35:1–35:45, 2015.\n20Lee-Ad Gottlieb, Aryeh Kontorovich, and Robert Krauthgamer. Efficient regression in metric\nspaces via approximate Lipschitz extension. IEEE Trans. Inf. Theory , 63(8):4838–4849, 2017.\n21Lee-Ad Gottlieb and Liam Roditty. An optimal dynamic spanner for doubling metric spaces.\nInProc. 16th Annual European Symposium on Algorithms, ESA , volume 5193 of LNCS, pages\n478–489, 2008.\n22Leonidas J. Guibas and John Hershberger. Optimal shortest path queries in a simple polygon.\nJ. Comput. Syst. Sci. , 39(2):126–152, 1989.\n23Sariel Har-Peled and Manor Mendel. Fast construction of nets in low-dimensional metrics and\ntheir applications. SIAM J. Comput. , 35(5):1148–1184, 2006.S. de Berg, T. Ophelders, I. Parada, F. Staals, J. Wulms 25\n24John Hershberger and Subhash Suri. An optimal algorithm for Euclidean shortest paths in\nthe plane. SIAM J. Comput. , 28(6):2215–2256, 1999.\n25David G. Kirkpatrick. Optimal search in planar subdivisions. SIAM J. Comput. , 12(1):28–35,\n1983.\n26Hung Le and Shay Solomon. Truly optimal Euclidean spanners. In Proc. 60th IEEE Annual\nSymposium on Foundations of Computer Science, FOCS , pages 1078–1100. IEEE Computer\nSociety, 2019.\n27Christos Levcopoulos, Giri Narasimhan, and Michiel H. M. Smid. Efficient algorithms for\nconstructing fault-tolerant geometric spanners. In Proc. 13th Annual ACM Symposium on the\nTheory of Computing, STOC , pages 186–195. ACM, 1998.\n28Joseph S. B. Mitchell and Wolfgang Mulzer. Proximity algorithms. In Handbook of Discrete\nand Computational Geometry (3rd Edition) , chapter 32, pages 849–874. Chapman & Hall/CRC,\n2017.\n29Giri Narasimhan and Michiel H. M. Smid. Geometric Spanner Networks . Cambridge University\nPress, 2007.\n30Jan Remy, Reto Spöhel, and Andreas Weißl. On Euclidean vehicle routing with allocation.\nComput. Geom. , 43(4):357–376, 2010."    },    {        "title": "2402.12157v1.Nuclear_matter_radii_from_molecular_rotations_using_ultra_high_resolution_spectroscopy.pdf",        "content": "Nuclear matter radii from molecular rotations using ultra-high-resolution spectroscopy\nMichail Athanasakis-Kaklamanakis1, 2,∗and Gerda Neyens2,†\n1Experimental Physics Department, CERN, CH-1211 Geneva 23, Switzerland\n2KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium\n(Dated: February 20, 2024)\nThe rotational constant parametrizes the relative spacing between a molecule’s rotational energy\nlevels. It depends on the molecule’s classical moments of inertia, which, in all studies, are expressed\nby treating the constituent nuclei as point masses separated by the bond length. We point out that\ntreating the finite nuclear size leads to a correction to the rotational constant at the Hz level, which\nis resolvable with recently developed ultra-high-resolution molecular spectrometers. Nuclear-model-\nindependent measurements of nuclear matter radii can thus be envisioned in the future using such\napparatuses, advancing beyond the existing hadronic scattering experiments and further developing\nthe intersection of nuclear and molecular physics. In the present time, it appears that the computa-\ntional ability of ab initio quantum chemistry might be the limiting factor to the technical readiness\nof the approach. To test the premises of the proposed method, we call for benchmark experiments\nusing HD+that are feasible with state-of-the-art experiment and theory.\nINTRODUCTION\nMolecules are composed of two or more atoms joined\nby chemical bonds that couple the motion of the con-\nstituent nuclei. Even diatomic molecules, which have the\nsimplest geometry, possess vibrational and rotational de-\ngrees of freedom. As a result, their spectra exhibit a\nvibrational structure for each electronic level, and a rota-\ntional structure within each vibrational sub-level of each\nelectronic level.\nThe rotational spectra of diatomic molecules have been\nextensively studied for more than a century [1], and it\nis well-understood that the spacing between successive\nlines in a rotational spectrum is intimately linked to the\nmoments of inertia of the molecule. Invariably, the point-\nnucleus approximation is used to express the molecular\nmoments of inertia and thus to extract the rotational\nconstants [1–4]. This approximation is certainly justi-\nfied due to the large difference in scale between nuclear\nradii ( ∼10−15m) and the internuclear distance in a typ-\nical diatomic molecule ( ∼10−10m). Corrections due to\ntreating the finite nuclear size can thus be expected to\nbe extremely small, and so far considered negligible.\nIn the last few years, however, advances in microwave\nand infrared spectrometers with an unprecedented res-\nolution [5] and absolute accuracy [6] have been con-\nstructed, being capable of determining transition energies\nand thus rotational constants with sub-Hz precision and\nkHz accuracy. Moreover, trap-based laser spectroscopy\nof light molecular ions has also demonstrated the con-\ntrol of statistical and systematic errors at the sub-kHz\nlevel [7, 8]. Therefore, it is necessary to revisit the point-\nnucleus approximation in the molecular moment of iner-\ntia and consider the correction due to the full treatment\nof the finite nuclear size.\nIn light of these recent technical developments, here\nwe point out that the emergent technology in molecu-\nlar spectroscopy is sensitive to the matter radii of theconstituent nuclei. We firstly provide the expression for\nthe rotational constant in diatomic molecules where the\nconstituent nuclei are treated as spheres of finite size.\nGiven the resulting expression, we discuss how mea-\nsurements of nuclear matter radii can be performed via\nultra-high-resolution rotational spectroscopy of diatomic\nmolecules with state-of-the-art and next-generation spec-\ntroscopic techniques. Compared to the existing and\nestablished experimental methods for measurements of\nthe nuclear matter radius, the proposed approach of-\nfers a nuclear-model-independent determination of nu-\nclear matter radii, with potentially higher precision and\nsignificantly smaller isotope-dependent systematic uncer-\ntainties.\nMOLECULAR ROTATIONS\nDepending on the size of the diatomic molecule, pure\nrotational spectra, which are composed of transitions\nbetween rotational levels within the same electronic-\nvibrational state, are observed between the microwave\nand millimeter-wave regions of the electromagnetic spec-\ntrum. The rotational lines of a diatomic molecule, char-\nacterized by the rotational quantum number J, follow a\ncharacteristic, diverging pattern at the first order, which\nis given by the expression [1–3]:\nE(J) =BvJ(J+ 1) (1)\nwhere Bvis called the rotational constant and it gen-\nerally shows a dependence on the vibrational quantum\nnumber v. A schematic example of the effect of changes\nin the rotational constant on the spacing of rotational\nlines is shown in Fig. 1. The values of the rotational con-\nstant for different values of vare related to an equilibrium\nrotational constant Bethrough the expression [2]:arXiv:2402.12157v1  [physics.atom-ph]  19 Feb 20242\nFIG. 1. Top: Conceptual comparison between a point-nucleus\n(left) and a finite-nucleus (right) representation of a diatomic\nmolecule. Bottom: Schematic effect of a change in the ro-\ntational constant on the relative spacing of three successive\nrotational lines.\nBv=Be−αe\u0012\nv+1\n2\u0013\n(2)\nwhere αeis a constant that depends on the parameters of\nthe Morse potential in the vibrational term of the molec-\nular Hamiltonian [2], and Behas the expression:\nBe=h\n8π2cIe(3)\nIn Eq. 3, Ieis the equilibrium moment of inertia. That\nis, the moment of inertia of the molecule assuming that\nthe constituent nuclei are placed at a fixed distance from\neach other that is equal to the equilibrium bond length\nre.\nIn the point-nucleus approximation, the two nuclear\nmasses MAandMBhave an infinitely small radius and\nare separated by a distance re. Therefore, the point-\nnucleus moment of inertia is expressed as Ip\ne=µr2\newhere\nµis the reduced molecular mass µ=MAMB\nMA+MBand thus\nBp\ne=h\n8π2c1\nµr2e(4)\nis the rotational constant for the point-nucleus case.\nTo arrive at the molecular moment of inertia when\ntreating the nuclei as spheres of radius RAandRB, re-\nspectively, the parallel axis theorem can be used. Con-\nsider nucleus Aas placed at a distance dAfrom the cen-\nter of mass of the molecule and nucleus Bas placed\nat a distance dB. Due to the symmetry of a diatomic\nmolecule, the molecular center of mass will lie along the\ninternuclear axis, and by definition of the bond length,\nre=dA+dB.\nThe moment of inertia of a solid sphere about any axis\ngoing through its center is:\nIsphere =2\n5MR2(5)where Mis the sphere’s mass and Rits radius. From the\nparallel axis theorem, the moment of inertia of a solid\nsphere about an axis at a distance dfrom its center is:\nI=Isphere +Md2(6)\nFor two nuclei rotating about the same axis in the same\ndirection, the total moment of inertia of the molecule is\nequal to the sum of the moments of inertia of the nuclei.\nTherefore, using Eqs. 5 and 6, the moment of inertia of\nthe diatomic molecule is:\nIs\ne=2\n5MAR2\nA+MAd2\nA+2\n5MBR2\nB+MBd2\nB\n⇔Is\ne=2\n5\u0000\nMAR2\nA+MBR2\nB\u0001\n+MAd2\nA+MBd2\nB(7)\nThe last two terms are equivalent to the point-nucleus\nmoment of inertia Ip\ne. This can be seen by considering the\nmolecule’s center of mass as defining the zero coordinate\nof the internuclear axis. The locations of nuclei AandB\nare thus:\ndA=−MBre\nMA+MB(8)\ndB=MAre\nMA+MB(9)\nTherefore:\nMAd2\nA+MBd2\nB=MAM2\nBr2\ne\n(MA+MB)2+MBM2\nAr2\ne\n(MA+MB)2\n=MAMBMB+MBMAMA\n(MA+MB)2r2\ne\n=MAMB(MA+MB)\n(MA+MB)2r2\ne=MAMB\nMA+MBr2\ne\nAnd thus:\nMAd2\nA+MBd2\nB=µr2\ne (10)\nEq. 7 is therefore equal to:\nIs\ne=2\n5\u0000\nMAR2\nA+MBR2\nB\u0001\n+Ip\ne (11)\nAt the point-nucleus limit, where RA=RB= 0, Is\ne=\nIp\ne, as expected. The equilibrium rotational constant for\na diatomic molecule can thus be expressed as:\nBe=h\n8π2c1\n2\n5(MAR2\nA+MBR2\nB) +µr2e(12)3\nNUCLEAR MATTER RADII\nAs per Eq. 11, the moment of inertia of a diatomic\nmolecule when treating the finite nuclear size has a cor-\nrection that depends on the constituent nuclear masses\nand radii. Fig. 2 shows how the correction scales as a\nfunction of the equilibrium bond length refor three di-\natomic molecules of different mass. Evidently, the cor-\nrection to the rotational constant, quantified as the dif-\nference between the point-nucleus and the finite-nucleus\nrotational constants Bp\neandBs\ne, is in the order of a few\ntens of Hz for both the lightest possible molecule (H 2)\nand the actinide molecule ThF+.\nWith sub-kHz resolution, the state of the art in molec-\nular spectroscopy is already capable of determining the\nrotational constants of diatomic molecules with a Hz- or\nsub-Hz-level uncertainty, and further improvements can\nbe expected in the future. Therefore, it is worthwhile\nconsidering how such a precision can be utilized to mea-\nsure nuclear matter radii.\nThe size of a nucleus is most often discussed in terms of\ntwo observables; the mean-squared nuclear charge radius\n⟨r2⟩ch, which is determined solely by the proton distribu-\ntion, and the mean-squared nuclear matter radius ⟨r2⟩m,\nwhich is determined by both the proton and the neutron\ndistributions.\nThe charge radius can be measured through a vari-\nety of complementary experimental techniques. Muonic-\natom X-ray spectroscopy [9] and electron scattering\nexperiments [10, 11] have provided measurements of\nthe absolute nuclear charge radii of stable isotopes for\ndecades [12]. Laser spectroscopy, on the other hand, pro-\nvides high-precision measurements of the relative change\nin root-mean-squared (rms) nuclear charge radius be-\ntween isotope pairs in a nuclear-model-independent man-\nner, through the isotope shift [13]. By combining abso-\nlute measurements in stable isotopes and relative mea-\nsurements in radioactive isotopes, the charge radii of a\ngreat amount of nuclei far from nuclear stability have\nbeen measured [14]. The study of nuclear charge radii\nhas thus become a cornerstone of modern nuclear struc-\nture research, being pursued to investigate the emergence\nof new magic numbers [15, 16], the disappearance and\ninversion of odd-even staggering [17, 18], signatures of\nrare nuclear effects [19], and more. Recently, muonic-\natom and electron-scattering experiments on radioactive\nisotopes were demonstrated for the first time [20, 21],\nopening a new pathway towards absolute charge radius\ndetermination in unstable nuclei.\nMeasurements of the nuclear matter radius [22, 23],\non the other hand, are significantly more challenging.\nSince the pioneering experiments by Rutherford and his\ngroup [24], charged-particle scattering experiments have\nundergone continuous development and to this day are\nthe most widespread approach to measuring the dis-\nFIG. 2. Calculation of correction (in Hz) to the rotational\nconstant due to treating the finite size of the nuclei as opposed\nto the point-nucleus approximation. The nuclear matter radii\nused to calculate the corrections were approximated as equal\nto the charge radii and taken from Ref. [12]. Nuclear masses\nwere taken from the 2020 Atomic Mass Evaluation [35] and\nequilibrium bond lengths were taken from the NIST Chem-\nistry WebBook.\ntribution of nuclear matter [25], together with matter-\nantimatter annihilation approaches [26]. Extracting the\nnuclear matter radius from hadronic scattering requires\nmodeling of the underlying hadron dynamics, knowledge\nof which remains incomplete. Therefore, nuclear mat-\nter radii from hadronic scattering have large systematic\nuncertainties that are isotope-dependent and mostly un-\ncontrolled [27]. So far, only the PREX-2 [28] experiment\nhas come close to extracting the nuclear matter radius\nat the 1% level in statistical and systematic uncertain-\nties, based on the determination of the neutral weak form\nfactor from elastic electron scattering, thus avoiding the\nuncontrolled systematics of hadronic models.\nAs a result, while knowledge of the nuclear matter ra-\ndius and the neutron distribution is highly important in\nnuclear physics, for instance to explore the formation\nof neutron skins [28], the equation of state in neutron\nstars [29, 30], the structure of nuclear halos [31, 32],\nand the limits of the nuclear landscape [33, 34], their\nexperimental study requires large-scale and costly reac-\ntion spectrometers at specialized facilities. Accessing the\nnuclear matter radius via ultra-high-resolution rotational\nspectroscopy of diatomic molecules thus offers an alter-\nnative experimental tool of great value, with a reduced\ncost, potential for lower statistical uncertainties, and con-\ntrolled, isotope-independent systematic uncertainties.\nThe moment of inertia in Eq. 11 defines an equilibrium\nrotational constant Be(Eq. 12) for a diatomic molecule\ncomprised of realistic, finite-size nuclei. The equilib-4\nrium rotational constant, however, is not directly mea-\nsurable in the laboratory. Instead, the vibrational-state-\ndependent constants Bvare measured, which are related\ntoBevia Eq. 2.\nThe constant αein Eq. 2 is related to the parameters of\nthe internuclear potential used in the molecular Hamilto-\nnian. For the most commonly used Morse potential, αe\ndepends on the three Morse potential parameters α,D,\nandreas [2]:\nαe=3h2ωe\n16π2µr2eD\u00121\nαre−1\nα2r2e\u0013\n(13)\nwhere ωe=α\n2πcq\n2D\nµis the harmonic frequency of the\nelectronic state.\nConsider a diatomic molecule with atoms AandB\nfor which the atomic masses of both nuclei are known\nwith high precision and accuracy (a precision of 10−9\natomic mass units or better [35]), and the nuclear matter\nradius for atom Bhas been measured in a scattering\nexperiment. Combining Eqs. 2, 12, and 13, it is seen\nthat for each value of v, an equation with RA,α,D, and\nreas unknown is constructed.\nTherefore, a value for RAcan be extracted if Bvis\nmeasured for enough different values of v. For the Morse\npotential and alternative functions [36], which are de-\nscribed by three parameters, Bvwould have to be mea-\nsured for at least v= 0−3. With four values of Bv,\na set of four equations and four unknowns is formed.\nAdding further measurements of Bvfor higher values of\nvwill help reduce the uncertainty in the extracted pa-\nrameters by constraining the linear form of Eq. 2. While\nspectroscopy of excited vibrational states is not typi-\ncal for cryogenic buffer-gas-cooled spectrometers, vibra-\ntional relaxation through buffer-gas collisions is signifi-\ncantly more inefficient than rotational and translational\ncooling [37], which makes such spectrometers compatible\nwith the proposed approach. For ion trap experiments,\nexcited vibrational states can be studied via optical or\ninfrared excitations of overtone vibrational transitions.\nA significant aspect of the approach proposed here is its\nversatility, as practically no electronic-state requirements\nare imposed on the system or technique chosen for the\nmeasurement of the rotational constant. As a result, the\nmeasurements can be performed in either neutral or ionic\nsystems, in the ground or metastable states.\nHIGHER-ORDER CORRECTIONS\nBeyond the correction due to the finite nuclear size con-\nsidered here, other corrections to the molecular rotational\nconstant are to be expected at the level of precision that\nis necessary to extract the nuclear matter radius, such as\nquantum electrodynamics (QED) corrections. Hyperfine\neffects and the influence of the nuclear charge radius onthe bond length may also need to considered at this level\nof precision.\nThe presently described method proposes extracting\nthe nuclear matter radius by determining Bewith very\nhigh precision, and then using the relation between the\nmoment of inertia and Beto extract the matter radius.\nThe extraction of Becan be done by measuring Bvfor\nseveral vibrational states of the electronic ground state,\nand then performing a fit as per Eq. 2. In a fit of Bvver-\nsus\u0000\nv+1\n2\u0001\n, the y-intercept gives Beand the slope is re-\nlated to the parameters of the vibrational potential used\nin the molecular Hamiltonian. One of these parameters\nis the equilibrium bond length re, knowledge of which is\nalso necessary to use Eq. 12 and extract the nuclear mat-\nter radii. Depending on the vibrational potential used\nin the Hamiltonian, recould be extracted from the fit’s\nslope.\nSo far it has been considered that if Bvis measured for\na sufficient number of vibrational states with high pre-\ncision, then Beandrecould be extracted and provide\nthe nuclear matter radius of interest without the need\nto calculate any properties of the molecule with ab initio\nquantum chemistry. However, at the level of precision\ndiscussed here, there are several points that need consid-\neration.\nMost importantly, it is not clear to what degree the\nrelation between BeandBvin Eq. 2 is exact at this level\n– in fact, it is certainly not fully linear. The descrip-\ntion of this relation might depend to some extent on the\npotential energy function used, and it is known that the\nmost commonly used Morse potential energy function for\ndiatomic molecules has limited accuracy, leading to the\ndevelopment of more elaborate potential energy functions\n(see Refs. [36, 38, 39] and references therein). However,\nnone of these potentials have been held to the test of\nultra-high-precision spectroscopy so far.\nTherefore, the validity of Eq. 2 at the precision re-\nquired by this method remains to be confirmed. It is ex-\npected that additional terms will need to be considered\nin Eq. 2 at higher order of\u0000\nv+1\n2\u0001\n. These corrections\nwould make Eq. 2 non-linear, but would not affect the\nextraction of Bethrough the y-intercept of a fit between\nBvand\u0000\nv+1\n2\u0001\n; the higher-order terms do not need to be\nunderstood at an analytical, first-principles level to ex-\ntract Befrom the y-intercept. Nevertheless, these terms\nwill impact the ability to extract re, and theoretical and\nexperimental investigations into the applicability of Eq. 2\nat very high precision are still necessary.\nMost likely, extracting the nuclear matter radius from\nBewill require at least partly input from ab initio cal-\nculations; for instance, via a highly precise calculation of\nre. In this case, further corrections need to be consid-\nered, such as the influence of the nuclear charge radii on\nre, as well as radiative and relativistic corrections.\nEspecially for the correction due to the finite nuclear\ncharge radius, there are indications that the correction5\ncan be at the kHz level or higher. Within the Dunham\nparametrization of the rovibrational energy levels of di-\natomic molecules [40], the rotational constant Bis ex-\npressed as the k, l= 0,1 Dunham parameter Y01. While\nearly studies considered that the Dunham parameters\nneed only be scaled by the reduced molecular mass upon\nisotopic substitution of the constituent nuclei [1, 40], it\nhas long been noticed that this relation is incomplete and\ninstead the nuclear mass and nuclear charge radius need\nto be taken into account in an explicit manner [41, 42].\nRecently, we showed that the nuclear radius and mass\ncorrections to the Dunham expansion can be viewed as\nthe molecular equivalent of the more familiar atomic field\nand mass shifts and utilized directly in a King plot with\natomic data [43].\nDespite the extensive evidence of the validity of the\nnuclear mass and field corrections to the Dunham pa-\nrameters, however, they are not routinely considered in\nthe expression of the molecular rotational constant in\nspectroscopic studies. Instead, their influence is typi-\ncally ”absorbed” into an effective equilibrium bond dis-\ntance that is extracted from the measured spectra us-\ning the definition in Eq. 4. With current and planned\nadvances in the accuracy and precision of state-of-the-\nart spectroscopic equipment, these corrections need to be\ntaken into consideration in a systematic and consistent\nmanner. Especially for the accurate extraction of nuclear\nmatter radii, as proposed here, an accurate value for the\nequilibrium bond length reneeds to be determined, and\nthese corrections must therefore be fully treated. Further\ncorrections beyond the nuclear mass and field shifts also\nneed to be explored.\nTo assess the magnitude of these corrections to re,ab\ninitio molecular theory has to be employed. However,\nthe correction to the rotational constant due to the mat-\nter radius decreases for heavier molecules (Fig. 2), and\ntypically so does the computational precision of quantum\nchemistry as well. Meanwhile, higher-order corrections to\nthe bond length might increase in significance for heav-\nier molecules, such as the correction due to the nuclear\ncharge radius and QED effects.\nWhile QED corrections to electronic transition en-\nergies have been calculated for heavy molecules, even\nup to the theoretical superheavy molecule E(120)F [44],\nthe computational accuracy and precision remain limited\nconsidering the requirements of our proposed method.\nProgress in ab initio relativistic calculations relevant to\nmulti-electron systems is ongoing, but the computational\nprecision and accuracy are still many orders of magnitude\nfrom the level required by the method presented here.\nAs a result, only light molecules appear accessible by\nthe currently proposed method in the present time. This\ndoes not eliminate the near-term nuclear-physics impact\nof the proposed method, since the accurate and precise\ndetermination of the nuclear matter radius in light nuclei\ncan be used as a benchmark of a large number of emerg-ingab initio nuclear theory techniques [45, 46], whose ap-\nplicability is often limited to light nuclei. This research\npotential also includes direct benchmarks of large-scale\nlattice quantum chromodynamics calculations using the\nstructure of light nuclei [47–49].\nPROPOSED CASES\nHD+\nExtracting the nuclear matter radius via the method\nproposed here introduces a systematic uncertainty de-\npending on the values used for the matter radii of the\nother nuclei in the molecule. As a result, diatomic\nmolecules are highly preferable. Specifically, monohy-\ndrides are ideal, since the nuclear matter and charge radii\nof1H are identical, and progress is ongoing towards re-\nsolving the proton radius puzzle [50].\nThe hydrogen compounds remain the subject of sev-\neral high-precision experimental and theoretical stud-\nies [7, 8, 51–62]. HD+, in particular, possessing only\none electron, is an ideal probe to test the quantum-\nmechanical description of a three-body system at the\npart-per-trillion level [7, 8, 60], without electron corre-\nlation effects. As a result, it also provides an excellent\nsystem to investigate the magnitude of the higher-order\ncorrections required for the reliable extraction of nuclear\nmatter radii. As per Fig. 2, the finite-radius correction in\nH2is expected to be in the order of 102Hz, owing to the\nshort bond length of the system, and so the nuclear mat-\nter radius should be accessible with the precision already\ndemonstrated with optical spectroscopy.\nMajor breakthroughs have been achieved in the last\ndecade on the vibrational-rotational structure of HD+[8,\n51, 53, 60–62], including high-precision investigations of\nthe potential energy surface in the hydrogen compounds.\nThe dissociation energy of molecular hydrogen was the\nfirst observable for which QED effects were explicitly cal-\nculated in a molecule [63]. More recently, the theoretical\nvalue for the dissociation energy of H 2[64] was success-\nfully benchmarked by a highly precise measurement that\nachieved an uncertainty of less than 5 kHz [65].\nFor several of the higher-order corrections mentioned\nin the previous section, results have already been re-\nported for HD+. In their recent work on the precision\nspectroscopy of HD+[7], Alighanbari et al. provided an\nuncertainty breakdown for the ab initio calculation of\na rotational transition frequency in HD+that included\nQED and finite-size corrections and reached a theoretical\nuncertainty of 18 Hz. These ab initio calculations were\nsuccessfully benchmarked for several hyperfine compo-\nnents of the rotational transition, confirming the validity\nof the presented theoretical results at the ∼101-Hz level.\nImportantly, the relativistic and finite-charge-radii cor-\nrections to the rotational transition frequency in HD+6\nwere calculated to be the most dominant, having a rel-\native contribution of 10−5to the total transition fre-\nquency [7]. For a larger system, containing a heavier\nnucleus with a larger charge and charge radius, both cor-\nrections would be expected to increase in absolute and\nrelative magnitude even further, and thus their accurate\ncalculation is critical.\nCorrections of even higher order than those considered\nby Alighanbari et al. [7], that is, going beyond three-loop\nradiative corrections, can be expected to contribute less\nthan 100 Hz to the rotational transition frequency, while\nthe finite-matter-radius correction to the rotational con-\nstant is estimated to be a few hundred Hz (see Fig. 2). As\nthe quantity of interest for our method is the rotational\nconstant Be, rather than individual rotational transition\nfrequencies, the theoretical uncertainty of interest can be\nexpected to be even better than the 18 Hz reported in\nRef. [7]. The reason is that corrections that shift indi-\nvidual rotational transitions the same way will be signif-\nicantly suppressed when determining the rotational con-\nstant, as the latter is determined from the relative dif-\nference between rotational transitions (see for example\nRefs. [66, 67], where order-of-magnitude higher precision\nis achieved for the rotational constant compared to the\nindividual transition frequencies for light molecules).\nRefs. [7, 8, 62] also demonstrate the control of system-\natic shifts in the spectroscopy of rotational and rovibra-\ntional lines to the sub-kHz level. These shifts are likely\nto also be suppressed when translated into a rotational-\nconstant uncertainty, allowing the matter radius to be\nextracted without the further need for an order of mag-\nnitude improvement in the control of systematics. How-\never, this has to be investigated directly. It is also im-\nportant to note that systematic shifts were controlled to\nthis level in ion trap experiments, and a similar level of\ncontrol remains to be demonstrated in the more univer-\nsally applicable cryogenic buffer-gas cell approaches of\nRefs. [5, 6].\nOverall, with statistical and systematic uncertainties\nalready under control at the required level, both in ex-\nperiment and theory, a campaign could be pursued with\nHD+to measure multiple overtone rovibrational transi-\ntions to different upper vibrational states, and determine\nBvfor a range of v(for instance, such measurements were\nperformed in Ref. [68] for neutral HD, but at a much\nlower precision than required for this method). These\nmeasurements can then be used to test the linearity of\nBvas a function of\u0000\nv+1\n2\u0001\nat the Hz level, and test the\nvalidity of Eq. 2. As a next step, the consistency of Eq. 12\ncan be tested with this molecule to extract the matter ra-\ndius of the deuteron and compare it to the literature value\nfrom reaction experiments and nucleon-nucleon potential\ncalculations [69].Calcium radii\nOnce the premises of the proposed method are tested\nwith the spectroscopy of HD+, a measurement of the nu-\nclear matter radii of40,48Ca could also be pursued. The\ncalcium isotopic chain ( Z= 20) is of central interest in\nnuclear structure research, as it has a magic proton num-\nber and two stable isotopes with a magic neutron number\n(atN= 20,28), thus being doubly magic. The nuclear\nground states of the doubly magic40,48Ca are more en-\nergetically stable compared to nuclei in their vicinity, as\nseen through their binding energy, two-neutron separa-\ntion energy, and transition energy and cross-section to\ntheir first excited states [70]. These observations have\nbeen acknowledged since the early development of the\nnuclear shell model and they form a staple of nuclear\nphysics knowledge to this day.\nDespite an excess of 8 neutrons, the nuclear charge\nradius of48Ca is astonishingly similar to that of40Ca [15].\nThis is surprising and indicates either a deviation from\nthe constancy of nucleon density within the nucleus as\nobserved throughout the nuclear chart [70], or much more\nlikely the formation of a thick neutron skin in48Ca.\nTo explore these conclusions, knowledge of the nuclear\nmatter radii is necessary, which have been measured with\nproton elastic scattering [71–73] and collisions with a car-\nbon target [74]. While different studies point to the for-\nmation of a considerably thicker neutron skin in48Ca\ncompared to40Ca, there is noticeable statistical scatter\nbetween the different values reported for the rms nuclear\nmatter radii of40,48Ca that originates from the differ-\nent nuclear-model assumptions employed in each anal-\nysis. On the other hand, the measured electric dipole\npolarizability of48Ca implies a small neutron skin [75].\nLastly, Tanaka et al. [74] reported an abrupt and promi-\nnent increase in the rms nuclear matter radius beyond\nN= 28, which requires an independent validation.\nTo extract the matter radius of calcium nuclei using\nthe current method, calcium monohydride (CaH) would\nbe the most reasonable choice due to the precise knowl-\nedge of the proton radius. CaH, which is the lightest\npossible calcium molecule, is significantly heavier than\nthe hydrogen compounds discussed above. As a result, it\ncannot be reasonably expected that the theoretical pre-\ncision achieved for HD+can also be achieved for CaH in\nthe present time. We thus also call for the development\nof precision calculations of this and other light calcium\nmolecules, to assess the readiness level of applying the\ncurrently proposed method to CaH and other calcium\nmolecules for the extraction of nuclear matter radii.\nCONCLUSION\nBy including the finite nuclear size in the expression\nof the equilibrium rotational constant, we point out that7\nultra-high-resolution molecular spectroscopy techniques\nare already sensitive to the influence of nuclear matter\nradii on the rotational structure of diatomic molecules.\nWith further improvements in calculations of the molec-\nular bond length, ultra-high-resolution molecular spec-\ntroscopy can offer an alternative route to measuring nu-\nclear matter radii, at a much reduced cost and with lower\nsystematic uncertainties as compared to hadronic scat-\ntering experiments.\nThe main limitation of the proposed method appears\nto be the precision and accuracy of ab initio molecu-\nlar calculations that might be required to determine the\nequilibrium bond length re, which is involved in the ex-\npression that relates the nuclear matter radii and the\nequilibrium rotational constant Be. The necessary preci-\nsion and accuracy of such calculations has been demon-\nstrated in the case of HD+, and achieving similar perfor-\nmance in calculations of heavier systems is necessary.\nTo test the validity of our proposal at the required level\nof precision, we propose that high-precision spectroscopy\nof HD+is pursued. Measurements and ab initio calcu-\nlations at the part-per-trillion level have already been\nachieved for this molecule, and existing experimental se-\ntups can be used to test the premises of our proposed\nmethod. We also propose the spectroscopy of CaH to\ndetermine the nuclear matter radii of40,48Ca as a case of\nhigh nuclear physics importance [74, 75].\nACKNOWLEDGMENTS\nWe would like to thank Alexander A. Breier (Tech-\nnische Universit¨ at Berlin) and Shane G. 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Theoretically, we studied the equivariant Hopf bifurcation around the positive\nconstant steady-state solution. Standing and rotating waves have been investigated through the theory of\nisotropic subgroups and Lyapunov-Schmidt reduction. The existence conditions, the formula for the peri-\nodic direction and the periodic variation of bifurcation periodic solutions are obtained. Numerically, we\nselect appropriate parameters and conduct numerical simulations to illustrate the theoretical results and re-\nveal quite complicated dynamics on the disk. Different types of rotating and standing waves, as well as\nmore complex spatiotemporal patterns with random initial values, are new dynamic phenomena that do not\noccur in one-dimensional intervals.\nKeywords: Predator-prey model, Predator-taxis, Disk, Standing wave, Rotating wave\n1arXiv:2402.12163v1  [math.DS]  19 Feb 2024I. INTRODUCTION\nIn order to survive and reproduce, organisms will respond to the surrounding environment, such\nas staying away from the environment that hinders their own growth and approaching the environ-\nment that is conducive to them. This phenomenon of directed movement is called chemotaxis and\nresearch on it can be traced back to [1]. In predator-prey models, prey- or predator-taxis can ef-\nfectively characterize the predator’s directional movement toward habitats with high prey density\nor the prey’s behavior of perceiving predation risks to avoid predators, respectively. Many studies\nhave focused on the global existence, boundedness, or stability of solutions in models with prey-\ntaxis [2–7] or predator-taxis [8–10]. Besides, based on the Hopf bifurcation theory for quasilinear\nreaction-diffusion systems [11], the existence of center manifolds [12], and the principle of lin-\nearized stability [13], there are many studies exploring patterns induced by taxis and bifurcations,\nthrough center manifold reduction [14–17] or Lyapunov-Schmidt reduction method [18–20].\nThe above studies mostly focus on one-dimensional intervals. Nevertheless, in real life, the\ninteraction between predators and prey frequently occurs in two-dimensional spatial domains. Ex-\namples include lakes [21], irregularly shaped protected areas [22], and circular Petri dishes, which\nare often chosen as experimental arenas for many studies on the interaction between miniature\npredators and prey [23, 24]. Establishing a predator-prey model on a circular domain can better\ndepict real-world situations. Hence, we consider the Rosenzweig-MacArthur predator-prey model\nwith predator-taxis and time delay on a disk D={(r,θ): 0≤r≤R,0≤θ≤2π},\n\n\nut=d1∆rθu+χ∇rθ·(u∇rθv)+u\u0000\n1−u\nK\u0001\n−αuv\nu+1,(r,θ)∈D,t>0,\nvt=d2∆rθv−dv+αuτv\nuτ+1, (r,θ)∈D,t>0,\n∂ru(·,R,θ) =∂rv(·,R,θ) =0, θ∈[0,2π),(1)\nwhere u=u(r,θ,t),v=v(r,θ,t)represent the density of prey and predator at radius r, angle θand\ntime t, respectively, and uτ=u(r,θ,t−τ). The parameters d1,d2,χ,K,α,dare all positive and\ndefined in [16], in which the authors investigated the same model on a one-dimensional interval\n[0,lπ].\nWith the aid of the phase portraits of the normal form with O(2)symmetry [25], the equiv-\nariant bifurcation theory [26, 27], and various wave solutions induced by equivariant bifurcations\n[28, 29], we plan to further explore the joint effect of time delay, predator-taxis and O(2)symme-\ntry. Compared to results in [16], spatially inhomogeneous periodic solutions on a disk, including\nstanding and rotating waves are found. The new wave solutions generated in circular domains\n2TABLE I: Notations\nSymbols Descriptions\nY=L2(D) the Lebesgue space of integrable functions defined on D\nH1\n0(D) H1\n0(D) ={(u,v)∈H1(D)|∂ru=∂rv=0,θ∈[0,2π)}\nX X =H2(D)∩H1\n0(D)\nXC,YC the complexification of XandY\nC([−τ,0],XC)the Banach space of continuous mappings form [−τ,0]toXCwith supremum norm\nλp,λnm[28, 29] the eigenvalues of −∆rθonDwith homogeneous Neumann boundary conditions\nˆφc\np,ˆφc\nnm,ˆφs\nnm[28, 29] the normalised eigenfunctions corresponding to λp,λnm\nhave greater practical significance and can offer enhanced insights into characterizing population\ndynamics within high-dimensional spatial domains. In fact, the disk is a very simple abstraction of\na two-dimensional compact simple-connected region, where many methods or ideas can be tested\non it before applying to more complicated domains. The research methodologies employed to\nanalyze the properties of these wave solutions in this article can also be further extended to more\ncomplex domains.\nThe rest of the paper is organized as follows. In Section II, we analyze the equivariant Hopf bi-\nfurcation around the positive steady state through Lyapunov-Schmidt reduction. In Section III, we\nsimulated spatially inhomogeneous periodic solutions, including standing waves, rotating waves,\nand more complex periodic spatiotemporal patterns with random initial values.\nII. EQUIVARIANT HOPF BIFURCATION\nIn this section, we will study the constant steady states of (1) through a linearization analysis\nand analyze the equivariant Hopf bifurcation around the steady state through Lyapunov-Schmidt\nreduction [19, 26, 27, 30]. Firstly, we need to introduce some notations in Table I.\nIn [16], the authors obtained that the system defined on the interval [0,lπ]undergoes spa-\ntially inhomogeneous periodic solution at τ=τk\nn, around the unique positive constant steady state\n(u∗,v∗) =\u0010\nd\nα−d,(K−u∗)(1+u∗)\nKα\u0011\n, under assumptions (H1)-(H3).\n3(H1)0<K<1+2u∗,α>d,0<u∗<K,χ>0.\n(H2)Cnm−Bnm<0.\n(H3)χ>χ∗.\nIn fact, for U= (u,v)T,Uτ= (uτ,vτ)T, the linearized system of system (1) around (u∗,v∗)is\n∂U\n∂t=A0U+AτUτ, (2)\nwhere\nA0=\nd1∆rθ+a11χu∗∆rθ−d\n0 d2∆rθ\n,Aτ=\n0 0\na210\n,\nwith a11=1−2u∗\nK−αv∗\n(u∗+1)2,a21=αv∗\n(u∗+1)2. On a disk, the linearized system (2) restricted to\nspan\bˆφc\np,ˆφc\nnm,ˆφs\nnm\t\nis equivalent to a sequence of functional differential equations (FDEs) on R,\nwith a sequence of characteristic equations. Due to the symmetry, some of them have the following\nform\nΓ(γ,λp,χ) =γ2+Apγ+Bpe−γτ+Cp=0,p=0,1,2,···, (3)\nand the others are in the following form [28, 29]\nΓ(γ,λnm,χ) =\u0002\nγ2+Anmγ+Bnme−γτ+Cnm\u00032=0,n>0,m>0, (4)\nwith Anm= (d1+d2)λnm−a11,Bnm=a21(χu∗λnm+d),Cnm= (d1λnm−a11)d2λnm, and the\nexpressions for Ap,Bp,Cpcan be obtained analogously. In this article, we are more concerned\nabout the phenomenon caused by symmetry, so we will study the problem in the case of n>\n0,m>0. To distinguish the symbols with [16], we record the critical value of the bifurcation\nparameter τasτk\nnm, and Eq.(4) has a pair of repeated purely imaginary roots γ=±iω∗(ω∗>0),\nwithω∗=r\u0010\n(2Cnm−Anm)+p\n(2Cnm−Anm)2−4(Cnm−Bnm)\u0011\n/2, under the same assumptions\n(H1)-(H3). Besides, the transversality conditions hold, i.e. Re {γ′(τk\nnm)}̸=0. That is to say, system\n(1) will generate equivariant Hopf bifurcation at τ=τk\nnmon the disk, by [16, 28, 30].\nIn the following content, we will study the equivariant Hopf bifurcation around (u∗,v∗)and\nfind a periodic solution with a period near T=2π\nω∗. For preparation work, let\nCT(C1\nT) ={F(t,·,·):R→XC|F(t+T,·,·) =F(t,·,·)\nand is continuous (differentiable )with respect to t}.\n4ForF(t)∈CT(C1\nT), note that ∥F(t)∥0=max t∈[0,T]{∥F(t)∥X2\nC}(∥F(t)∥1=max{∥F(t)∥0,∥F′(t)∥0}).\nThen,CTandC1\nTare Banach spaces with norms ∥F(t)∥0and∥F(t)∥1, respectively. In fact, CTis\na Banach representation of the group S1, i.e.\nκ·F(t) =F(t+κ),κ∈S1,F∈C2π.\nFor all F,GinCT, define the inner product (·,·)T:CT×CT→Ras(F,G)T=1\nTRT\n0⟨F(t),G(t)⟩dt,\nwhere ⟨u(r,θ),v(r,θ)⟩=RR\nDru(r,θ)¯v(r,θ)drdθis the inner product weighted ronYC.\nNormalizing of the period, let ρ∈(−1,1)andU(t) = ( u((1+ρ)t),v((1+ρ)t))T, then system\n(1) can be transformed into\n(1+ρ)dU(t)\ndt=A0U(t)+AτU(t−(1+ρ)τ)+H(Ut,τ), (5)\nwhereH(φ,τ) =1\n2B(φ,φ) +1\n6C(φ,φ,φ) +o(∥φ∥3), for φ∈X2\nC.The above transformation\nsimplifies the study of periodic solutions with time Tand its similar times of system (1). Finding\nT\n1+ρ-periodic solutions of system (1) is equivalent to solving F(U,τ,ρ) =0, where F:C1\nT×\nR2→CTis defined by\nF(U,τ,ρ) =−(1+ρ)dU(t)\ndt+A0U(t)+AτU(t−(1+ρ)τ)+H(Ut,τ). (6)\nObviously, FisO(2)×S1-equivariant, i.e. for (δ,κ)∈O(2)×S1,\n(δ,κ)F(U,τ,ρ) =F\u0012\nU\u0012\nr,θ+δ\nn,t+κ\nω∗(1+ρ)\u0013\n,τ,ρ\u0013\n,(δstands f or a rotation )\nor\n(δ,κ)F(U,τ,ρ) =F\u0012\nU\u0012\nr,−θ,t+κ\nω∗(1+ρ)\u0013\n,τ,ρ\u0013\n,(δstands f or a re f lection )\nLetLτknm:XC→YCbe the first derivative of Fwith respect to U(t)at(U,τ,ρ) = ( 0,τk\nnm,0).\nClearly, the elements of Ker Lτknmcorrespond to the T-periodic solutions of the linearized equation\nLτknmU=0. LetL∗\nτknmbe the adjoint operator of Lτknmwith(·,·)that is a kind of bilinear form,\ndefined in [28, 30]. Then we have the following results.\nLemma II.1. Let\nV1= (1,a21eiω∗τknm\nd2λnm+iω∗)T,V2=−d2λnm+iω∗\n(d2λnm+iω∗)eiω∗τknm−d2λnm+iω∗(1,d2λnm+iω∗\n−a21eiω∗τknm).\nForϑ∈[−τ,0], define\n{ϕ1(ϑ),ϕ2(ϑ),ϕ3(ϑ),ϕ4(ϑ)}={eiω∗ϑV1ˆφc\nnm,e−iω∗ϑ¯V1ˆφc\nnm,eiω∗ϑV1ˆφs\nnm,e−iω∗ϑ¯V1ˆφs\nnm},\n5and\n{ϕ∗\n1(ϑ),ϕ∗\n2(ϑ),ϕ∗\n3(ϑ),ϕ∗\n4(ϑ)}={eiω∗ϑV2ˆφc\nnm,e−iω∗ϑ¯V2ˆφc\nnm,eiω∗ϑV2ˆφs\nnm,e−iω∗ϑ¯V2ˆφs\nnm}.\nThen\n(i)KerLτknmis spanned by {ϕ1,ϕ2,ϕ3,ϕ4}andKerL∗\nτknmis spanned by\b\nϕ∗\n1,ϕ∗\n2,ϕ∗\n3,ϕ∗\n4\t\n.\n(ii) (ϕ∗\nj,ϕj) =1, for j =1,2,3,4, and (ϕ∗\ni,ϕj) =0, for i̸=j.\nObviously, Lτknmis a Fredholm operator with index 0 [18, 31], and according to [30], we can\nget a space decomposition by O(2)×S1-invatiant subspaces of CT:\nCT=KerLτknm⊕RanLτknm,C1\nT=KerLτknm⊕W,\nwhere W= (KerL∗\nτknm)⊥∩C1\nT. Next, we aim to obtain the bifurcation mapping corresponding\ntoF(U,τ,ρ) =0 through Lyapunov-Schmidt reduction. Define two project operators P:CT→\nRanLτknmandI−P:CT→KerLτknm. Then, PandI−PareO(2)×S1-equivariant. According\nto the above space decomposition, F(U,τ,ρ) =0 can be decomposed into the following two\nequivalent equations:\nF1(V,W,τ,ρ) =PF(V+W,τ,ρ) =0,\nF2(V,W,τ,ρ) = ( I−P)F(V+W,τ,ρ) =0,\nwhere U=V+W,V∈KerLτknm,W∈W. Due to F1(0,0,τk\nnm,0) =0,DWF2(0,0,τ,0) =\nPLτknm=Lτknm, andLτknm\f\f\f\nWis invertible, by the implicit function theorem, we get a continuously\ndifferentiable O(2)×S1-equivariant map W:Lτknm×R2→Wand satisfies W(0,τk\nnm,0) =0 and\nF1(V+W(V,τ,ρ),τ,ρ) =0. Replacing WinF2(V,W,τ,ρ)byW, we have\nFW\n2(V,τ,ρ) = ( I−P)F(V+W(V,τ,ρ),τ,ρ) =0. (7)\nIt is easy to see FW\n2isO(2)×S1-equivariant and satisfies FW\n2(0,τk\nnm,0) =0 and DVFW\n2(0,τk\nnm,0) =\n0. Therefore, by [30], the periodic solution with time period close to2π\nω∗of system (1) corresponds\nto the solution of (7).\nHowever, finding periodic solutions of system (1) is closely related to the O(2)×S1symmetry,\nwhich often requires a subspace with simple eigenvalues to study. Therefore, we can discuss based\non the isotropy subgroup theory. By [27, 30, 32], the maximal isotropy subgroup of O(2)×S1\nisfSO(2)andZ2⊕ZC\n2. Obviously, the fSO(2)-symmetric solutions correspond to rotating wave\nsolutions, and satisfy that\n(¯θ,¯θ)u(r,θ,t) =u\u0012\nr,θ+¯θ\nn,t+¯θ\nω∗\u0013\n,¯θ∈S1, (8)\n6(¯θ,−¯θ)u(r,θ,t) =u\u0012\nr,θ+¯θ\nn,t−¯θ\nω∗\u0013\n,¯θ∈S1, (9)\nwhich correspond to the two possible senses of rotation: clockwise and counterclockwise. Note\nthat Fix (fSO(2),\nKerLτknm) =span{ψ,¯ψ}withψ=ϕ3,¯ψ=ϕ2orψ=ϕ1,¯ψ=ϕ4, and ψ∗=ϕ∗\n3orψ∗=ϕ∗\n1. It\nis easy to see that dimFix (fSO(2),KerLτknm) =2. Analogously, the Z2⊕ZC\n2-symmetric solutions\ncorrespond to a standing wave solutions, and satisfy that\nκu(r,θ,t) =u(r,−θ,t),(π,π)u(r,θ,t) =u\u0012\nr,θ+π\nn,t+T\n2\u0013\n,\nκ(π,π)u(r,θ,t) =u\u0012\nr,−θ−π\nn,t+T\n2\u0013\n.(10)\nNote that Fix (Z2⊕ZC\n2,KerLτknm) =span{ψ,¯ψ}withψ=ϕ1+ϕ3,¯ψ=ϕ4+ϕ2, andψ∗=ϕ∗\n1+\nϕ∗\n3.\nInspired by [27, 30], we simplify the problem once again and consider a new restriction map-\npingM: Fix(Σ,KerLτknm)×R2→KerLτknm,Σ=fSO(2)orZ2⊕ZC\n2,\nM(V,τ,ρ) =FW\n2(V,τ,ρ).\nThen,M(V,τ,ρ) =0 is equivalent to the one-dimensional bifurcation map G(z,τ,ρ):C×R2→\nC\nG(z,τ,ρ) = (ψ∗,M(zψ+¯z¯ψ,τ,ρ))T, (11)\nwhere z=⟨ψ∗,V⟩. Now, our problem is reduced to finding solutions of (11). Clearly, G(z,τ,ρ)\nisS1-equivariant, and satisfies Gz(0,τk\nnm,0) =0,G¯z(0,τk\nnm,0) =0. Therefore, there exist two\nfunctions g1,g2:R3→R, such that\nG(z,τ,ρ) =g1(|z|2,τ,ρ)z+g2(|z|2,τ,ρ)iz. (12)\nSinceGz(0,τk\nnm,0) =0, then g1(0,τk\nnm,0) =g2(0,τk\nnm,0) =0. Let z=˜ρei˜θ, thenG(z,τ,ρ) =0 is\nequivalent to either solving ˜ρ=0 org1(˜ρ2,τ,ρ)z=0 and g2(˜ρ2,τ,ρ)z=0. Notice that\nGτ(z,τk\nnm,0) =zγ′(τk\nnm)+O(|z|2),\nGρ(z,τk\nnm,0) =−iω∗z+O(|z|2).\nTherefore,\ndet\ng1\nτ(0,τk\nnm,0)g1\nρ(0,τk\nnm,0)\ng2\nτ(0,τk\nnm,0)g2\nρ(0,τk\nnm,0)\n=det\nRe{γ′(τk\nnm)}Im{γ′(τk\nnm)}\n0 −ω∗\n̸=0.\n7The implicit function theorem implies that there exists two functions τ=τ(˜ρ2)andρ=ρ(˜ρ2)\nsatisfying τ(0) =τk\nnm,ρ(0) =0 and\ng1(˜ρ2,τ(˜ρ2),ρ(˜ρ2)) = 0,g2(˜ρ2,τ(˜ρ2),ρ(˜ρ2)) = 0, (13)\nfor all sufficient small ˜ρ. Therefore, we have the following equivariant Hopf bifurcation theorem\nfor system (1).\nTheorem II.2. For every maximal isotropy subgroup Σ⊂O(2)×S1in which Σ=fSO(2)orZ2⊕\nZC\n2,Fix(Σ,X) ={U∈X|δ·U=U,δ∈Σ}is of dimension two, system (1) has a branch of periodic\nsolutions emanating from (u∗,v∗)atτ=τk\nnm, whose spatiotemporal symmetry can be completely\ncharacterized by Σ, corresponding to a branch of clockwise (counterclockwise) rotating waves of\nthe form (8) (form (9)) or a branch of standing waves of the form (10).\nIn what follows, we consider the bifurcation direction and the monotonicity of the period. We\nuse\ng21= (ψ∗,C(ψ,ψ,¯ψ))T+2(ψ∗,B(ψ,W11))T+(ψ∗,B(¯ψ,W20))T, (14)\nto denotes the coefficient of the term z2¯zin the Taylor expansion of G(z,τk\nnm,0). We still need to\ncompute W11andW20, which denote the coefficient of the terms z¯zandz2in the Taylor expansion\nofW(zψ+¯z¯ψ,τk\nnm,0), respectively. In fact,\nW11=−L−1\nτknmPB(ψ,¯ψ),W20=−L−1\nτknmPB(ψ,ψ).\nNamely, B(ψ,¯ψ),B(ψ,ψ)∈RanLτknm, thus, the projection operator Pon them acts as the iden-\ntity. Following the definition of Lτknm, we have\nW11=−\nd1∆rθ+a11χu∗∆rθ−d\n0 d2∆rθ\n−1\nB(ψ,¯ψ),\nW20=−\nd1∆rθ+a11χu∗∆rθ−d\na21e−2iω∗τknm d2∆rθ\n−1\nB(ψ,¯ψ).\nBy [26, 30, 33, 34], we summarize the following results.\nTheorem II.3. There exists a branch of Σ-symmetric periodic solutions ( Σ=fSO(2)orZ2⊕ZC\n2),\nparameterized by τ, bifurcating from the positive constant steady state of (1). Moreover,\n(i)τ′(0) =Re{g21}\nRe{γ′(τknm)}determines the direction of the bifurcation. The bifurcation is supercritical\n8(subcritical), if τ′(0)<0(>0).\n(ii)ρ′(0) =Im{γ′(τknm)¯g21}\nRe{γ′(τknm)}determines the period of the bifurcation periodic solutions. The period\nis greater (smaller) than2π\nω∗, ifρ′(0)>0(<0).\nIII. NUMERICAL SIMULATIONS\nFixing d1=0.1,d2=0.2,α=1,K=6,d=0.1,R=10, at the unique positive constant\nsteady solution (u∗,v∗) = ( 4,1.67), partial bifurcation curves on the χ−τplane are shown in\nFigure 1. Selecting (χ,τ) = ( 0.38,9.88)(Case 1) or (χ,τ) = ( 0.46,9.6)(Case 2), the standing\nwave (see Figure 2 and Figure 3), counterclockwise (see Figure 4) and clockwise rotating wave\n(see Figure 5) can be found. For simplicity, patterns of the clockwise rotating wave under Case 1\nand the counterclockwise rotating wave under Case 2 have not been listed. Besides, more complex\nperiodic spatiotemporal patterns (see Figure 6) with random initial value can be found.\nThrough numerical calculations, we obtain that under Case 1, ω∗=0.1567 ,τ0\n11=9.8270 .τ′(0)≈\n−0.4085, ρ′(0)≈0.1390 for ψ=eiω∗ϑV1ˆφc\n11, and τ′(0)≈ −0.0883 ,ρ′(0)≈0.0402 for ψ=\neiω∗ϑV1ˆφc\n11+eiω∗ϑV1ˆφs\n11. Under Case 2, ω∗=0.1938 ,τ0\n21=9.5520 .τ′(0)≈ −0.0631, ρ′(0)≈\n0.0467 for ψ=eiω∗ϑV1ˆφs\n21, andτ′(0)≈−0.4942 ,ρ′(0)≈0.1697 for ψ=eiω∗ϑV1ˆφc\n21+eiω∗ϑV1ˆφs\n21.\nBy Theorem II.3, four bifurcation periodic solutions under two cases are both supercritical the\nperiod of them are both greater than2π\nω∗.\nFIG. 1: Partial bifurcation curves on the χ−τplane. Parameters are\nd1=0.1,d2=0.2,α=1,K=6,d=0.1,R=10.\n9(a)\n (b)\nFIG. 2: System (1) generates standing wave that has a fixed axis with (χ,τ) = ( 0.38,9.88).\nInitial values are u(t,r,θ) =u∗(1+0.1·cost·cos2πr\nR·cosθ),v(t,r,θ) =\nv∗(1+0.1·cost·cos2πr\nR·cosθ),t∈[−τ,0). (a): u; (b): v.\n(a)\n (b)\nFIG. 3: System (1) generates standing wave that has two fixed axes with (χ,τ) = ( 0.46,9.6).\nInitial values are u(t,r,θ) =u∗(1+0.1·cost·cos2πr\nR·cos2θ),v(t,r,θ) =\nv∗(1+0.1·cost·cos2πr\nR·cos2θ),t∈[−τ,0). (a): u; (b): v.\nRemark III.1. On a one-dimensional interval, the solution often quickly converges to a coskx-\nlike spatially inhomogeneous periodic solution [14, 16]. However, the two-dimensional domain\nyields a more significant process of aggregation and dispersion, ultimately presenting periodic\npatterns with complex spatial structures. The standing wave, rotating wave, and more complex\nspatiotemporal patterns under different initial conditions on a disk are new interesting phenomena.\nIV . CONCLUSION\nIn this paper, we mainly investigated the equivariant Hopf bifurcation bifurcating from the\npositive equilibrium of (1). Methods of Lyapunov-Schmidt reduction and isotropic subgroups\nwere combined to explore the interaction between symmetry, time delay, and taxis on a disk.\n10(a)\n (b)\nFIG. 4: System (1) generates counterclockwise rotating wave with (χ,τ) = ( 0.38,9.88). The\npatterns at four different time within a period are selected to show the periodic changes in\npopulation distribution. Initial values are u(t,r,θ) =u∗(1+0.1·cost·cos2πr\nR·sinθ),\nv(t,r,θ) =v∗(1+0.1·cost·cos2πr\nR·cosθ),t∈[−τ,0). (a): u; (b): v.\n(a)\n (b)\nFIG. 5: System (1) generates clockwise rotating wave with (χ,τ) = ( 0.46,9.6). The patterns at\nfour different time within a period are selected to show the periodic changes in population\ndistribution. Initial values are u(t,r,θ) =u∗(1+0.1·cost·cos2πr\nR·cos2θ),v(t,r,θ) =\nv∗(1+0.1·cost·cos2πr\nR·sin2θ),t∈[−τ,0). (a): u; (b): v.\nWe have done numerical simulations and spatially inhomogeneous periodic solutions were found,\nincluding standing waves, rotating waves, and more complex patterns.\nThe time delay and taxis can induce the generation of spatially inhomogeneous periodic solu-\n11(a)\n(b)\nFIG. 6: System (1) generates rich transient patterns within a short term and periodic patterns with\ncomplex spatial structures after long-term evolution under random initial value and\n(χ,τ) = ( 0.46,9.6). The patterns at three different time within a period are selected to show the\nperiodic changes in population distribution. (a): u; (b): v.\ntions, while specific forms of standing and rotating waves will be generated on the disk. It is worth\nmentioning that the algorithm in this paper can also be extended to other fields, such as chemistry,\nmechanics, nonlinear optics, etc. This extension allows for the exploration of the properties of\nstanding and rotating waves, offering effective control in diverse applications.\n[1] E. F. Keller and L. A. Segel, Journal of Theoretical Biology 30, 225 (1971).\n[2] M. Chen, Nonlinear Dynamics 111, 19527 (2023).\n[3] W. Choi and I. Ahn, Applied Mathematics Letters 98, 256 (2019).\n[4] X. He and S. Zheng, Applied Mathematics Letters 49, 73 (2015).\n[5] H. Jin and Z. Wang, Journal of Differential Equations 262, 1257 (2017).\n[6] E. Giricheva, Nonlinear Dynamics 95, 875 (2019).\n12[7] S. Wu, J. Shi, and B. Wu, Journal of Differential Equations 260, 5847 (2016).\n[8] M. Chen and R. Wu, Zeitschrift für Naturforschung A 78, 315 (2023).\n[9] S. Wu, J. Wang, and J. Shi, Mathematical Models and Methods in Applied Sciences 28, 2275 (2018).\n[10] C. Yoon, Applicable Analysis 101, 5557 (2022).\n[11] H. Amann, in Delay Differential Equations and Dynamical Systems (Springer Berlin Heidelberg,\nBerlin, Heidelberg, 1991) pp. 53–63.\n[12] G. Simonett, Differential Integral Equations 8, 753 (1995).\n[13] A. K. Drangeid, Nonlinear Analysis-Theory Methods & Applications 13, 1091 (1989).\n[14] X. Dong and B. Niu, International Journal of Bifurcation and Chaos 33, 2350152 (2023).\n[15] T. Ma, X. Meng, T. Hayat, and A. Hobiny, Nonlinear Dynamics 108, 4581 (2022).\n[16] Q. Shi and Y . Song, Applied Mathematics Letters 131, 108062 (2022).\n[17] W. Zuo and Y . Song, Journal of Dynamics and Differential Equations 33, 1917 (2021).\n[18] J. Gao, S. Guo, and L. Ma, Nonlinear Dynamics 108, 4207 (2022).\n[19] C. Liu and S. Guo, Nonlinearity 35, 4283 (2022).\n[20] H. Qiu, S. Guo, and S. Li, International Journal of Bifurcation and Chaos 30, 2050022 (2020).\n[21] Z. Maciej Gliwicz, J. Slon, and I. Szynkarczyk, Freshwater Biology 51, 823 (2006).\n[22] Q. J. A. Khan and A. F. Ghaleb, Journal of Theoretical Biology 223, 171 (2003).\n[23] J. A. Banfield-Zanin and S. R. Leather, Insects 7, 49 (2016).\n[24] H. A. Meyer, H. E. Larsen, N. O. Akobi, and G. Broussard, Zoological Journal of the Linnean Society\n188, 860 (2020).\n[25] S. A. van Gils and J. Mallet-Paret, Proceedings of the Royal Society of Edinburgh Section A: Mathe-\nmatics 104, 279 (1986).\n[26] S. Guo, Journal of Differential Equations 317, 387 (2022).\n[27] X. Qu and S. Guo, Zeitschrift für angewandte Mathematik und Physik 74, 76 (2023).\n[28] Y . Chen, X. Zeng, and B. Niu, “Equivariant Hopf bifurcation in a class of partial functional differential\nequations on a circular domain,” (2023), arXiv:2305.05979 [math.DS].\n[29] Y . Chen, X. Zeng, and B. Niu, “Spatiotemporal patterns induced by Turing-Hopf interaction and\nsymmetry on a disk,” (2023), arXiv:2309.06133v2 [math.DS].\n[30] S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations (Springer-Verlag, New\nYork, 2013).\n[31] M. Qu and C. Zhang, Journal of Applied Analysis & Computation 11, 1371 (2020).\n13[32] M. Golubitsky, I. Stewart, and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory:\nVolume II (Springer-Verlag, New York, 1989).\n[33] T. Faria, Transactions of the American Mathematical Society 352, 2217 (2000).\n[34] J. Wu, Theory and Applications of Partial Functional Differential Equations (Springer-Verlag, New\nYork, 1996).\n14"    },    {        "title": "2402.12172v1.Large__N__Schur_index_of___mathcal_N_4__SYM_from_semiclassical_D3_brane.pdf",        "content": "Large NSchur index of N“4SYM\nfrom semiclassical D3 brane\nM. Beccariaa, A. Cabo-Bizeta,\naUniversità del Salento, Dipartimento di Matematica e Fisica Ennio De Giorgi ,\nand I.N.F.N. - sezione di Lecce, Via Arnesano, I-73100 Lecce, Italy\nE-mail: matteo.beccaria@le.infn.it, acbizet@gmail.com\nAbstract\nWe consider the refined Schur superconformal index of 4d N“4UpNqSYM and the\nfirst term of its giant-graviton expansion, first predicted in arXiv:2001.11667 using indirect\nsuperconformal algebra considerations and analytic continuation of fugacities. This correction\nis the leading non-perturbative correction to the index at large Nand we reproduce it from\nthe semiclassical partition function of quantum D3 brane wrapped on S1ˆS3in a twisted\nmodification of the AdS 5ˆS5string background, depending on the index R-symmetry fugacity.\nOur calculation does not exploit directly supersymmetry. It is based on the determination of\nthe partition function of the various bosonic and fermionic fluctuations on the wrapped brane\nwhose action is conformal with specific constant holonomies along thermal cycle. We show how\nthose partition functions may be obtained by adapting the operator counting method of Cardy\nto the twisted background.arXiv:2402.12172v1  [hep-th]  19 Feb 2024Contents\n1 Introduction 1\n1.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.2 Plan of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n2N“4Superconformal index 6\n2.1 The N“4Maxwell multiplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2.2 Schur index of N“4UpNqSYM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n3 D3-brane index from analytic continuation 8\n3.1 Schur specialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n3.2 Unrefined index and uÑ1limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n4 D3-brane in twisted AdS 5ˆS5and unflavored index 12\n4.1 Classical D3 action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n4.2 Wrapped brane fluctuation one-loop determinant and brane index . . . . . . . . . . . 13\n4.3 Computation of the single particle partition functions . . . . . . . . . . . . . . . . . . 14\n4.3.1 Scalar fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14\n4.3.2 Fermionic fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16\n4.3.3 Gauge field fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n4.4 Unflavored Schur index from twisted N“4fields on S1\nβˆS3. . . . . . . . . . . . . 18\n4.5 Supersymmetric Casimir energy on the wrapped brane . . . . . . . . . . . . . . . . . 19\n5 Flavored Schur index 20\n5.1 Single particle partition functions of twisted fields in S1\nβˆS3. . . . . . . . . . . . . 21\n5.2Z“0brane contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\n5.3X“0brane contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\n5.4 Supersymmetric Casimir energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\nA Special functions 25\nB Doubly wrapped D3 brane index in closed form and analytic continuation 26\n1 Introduction\nThe AdS/CFT correspondence establishes a duality between a boundary conformal gauge theory\nand a string dual where N„1{gsat constant ’t Hooft planar coupling, with Nbeing the gauge\ngroup rank and gsthe string coupling. A general puzzle appearing at finite Nis that the number\nof boundary gauge theory states at fixed conserved charges decreases due to trace relations while\nnon-perturbative states in the gravity theory become lighter at increasing gs. A way out of this\n1apparent contradiction is to call on some kind of stringy exclusion principle [1, 2].\nA context where this issue can be addressed is the study of finite Ncorrections to the super-\nconformal index [3] of a supersymmetric gauge theory with gravity dual. These are corrections that\nin many instances take the form of a giant graviton-like expansions [4] and originate from gauge\ntheory states with R-charge of order N. Let us denote by INpqqthe superconformal index with q\ncollectively standing for the set of fugacities. Finite Neffects admit the (typical) representation1\nINpqq“IKKpqq«\n1`8ÿ\nk“1qkNIGG\nkpqqff\n, (1.1)\nwhere IKKpqq”I8pqqis the NÑ8limit of the index and the key property of the quantities IGG\nkpqq\nis that they do not depend on N, at least at generic points in fugacity space. Thus, (1.1) encodes\nthe full finite Ndependence of the index as a sum of corrections that are non-perturbative at large\nN.\nThe naming of the expansion is due to the fact that finite Ncorrections to the large Nindex\nare captured on gravity side by certain wrapped D-branes BPS large “giant” configurations. This\ninterpretation was developed on a quantitative level in the works of Imamura and collaborators\n[5, 6, 7, 8, 9, 10, 11, 12, 13] (see also [14]) where a systematic analysis of finite Neffects in\nsuperconformal index from wrapped branes was carried out. In that approach, the factor IKKpqq\nis the contribution to the index from supergravity Kaluza-Klein BPS states (hence the label). The\n“classical” factor qkNcomesfromtheclassicalactionofthewrappedbrane,with kbeingatopological\nwrapping number. The remaining non-trivial pieces IGG\nkpqqrepresent the index of fluctuation modes\non the wrapped brane, a non-trivial superconformal theory with BPS reduced supersymmetry.2\nThe relation between (1.1) and a possible manifestation of a stringy exclusion principle lies\nin the fact that contributions at increasing khave typically alternating signs. As an illustrative\nparadigmatic example, the1\n2-BPS index of 4d N“4U(N) SYM theory on S1ˆS3takes the form\n(1.1) with ( Ris a Cartan of the SUp4qR-symmetry and pqqNis the Pochhammer symbol)\nISYM\nNpqq“Trp´1qFqR“1\npqqN,pIkpqq“p´ 1qkqkpk`1q{2\npqqk, (1.2)\nwhere the trace is restricted to1\n2-BPS states. The factor p´1qkinsidepIkpqqsuggests the emergence\nof an extra bulk grading [19, 20]. As remarked in [21], this grading is closely related to the fact that\nthe giant graviton expansion from the bulk is an Euclidean functional integral calculation where\nhigher order terms are saddles with kcoinciding giants. The bulk grading leads to cancellations that\nimplement the exclusion principle and are necessary for the index to reproduce black hole physics\nas shown recently in [22].3The extra grading emerging in the giant graviton expansions suggests\nthat open strings on D-branes are presumably not dual to physical states of the boundary gauge\ntheory. As recently pointed out in [20], they are objects that cancel out the redundancies among\n1Forinstance, theterm qkNmaybereplacedby qakNwithsomeconstant a. Also, theindex kisoftenamulti-index.\nIn the present discussion, these details are not relevant.\n2Working on the gauge theory side, apart from AdS/CFT, the expansion (1.1) was shown to arise counting\ninvariants in any unitary matrix model [15]. The expansion can be derived by free-fermion methods and interpreted\nas an instanton expansion in gauge theory in [16]. The first and second terms in this free-fermion expansion were\nderived in [15] and [17]. The expansion obtained by this approach is exact but does not agree term by term with\nthe wrapped D-brane expansion. This is not a contradiction because at mathematical level the index itself does not\ndetermine pIkpqqin (1.1), see for instance [18].\n3Outside the context of giant graviton expansions, it is known that states in the index with charge of order N2\ncan reproduce the entropy of supersymmetric black holes in AdS 5[23].\n2states at finite Nor, in other words, are dual to massive auxiliary ghost states on the boundary.\nTechnically, this holds for normalizable states, while other states associated with open strings on\nD3 giants are dual to gauge theory physical operators with R-charge of order N[24].4\nFrom the above considerations, a key achievement in this context is the ability to evaluate the\nbrane indices IGG\nkpqqin (1.1) on the gravity side of any given model. This is interesting already\nat leading wrapping k“1because the superconformal theory of fluctuations on the BPS wrapped\nD-branes (breaking a fraction of the original supersymmetry) is non-trivial and depends on the\nwrapping geometry.\nIn the approach of Imamura et al., brane indices are obtained by heavily relying on the unbroken\nsuperconformalsymmetryofthewrappedbranewhichiscleverlymappedtotheboundarysymmetry\nalgebra. Thisapproachprovidesthebraneindices IGG\nkpqqintermsoftheindexofasimplerboundary\nSCFT by analytic continuation in fugacity space.\nRecently, the analysis of (leading) finite Ncorrections to refined superconformal indices was\nreconsidered by direct semiclassical analysis of branes wrapped in a twisted string background\nwith peculiar mixing terms in the metric and supporting forms. The twist terms implement the\ngeometrization of fugacities present in the definition of the index.\nThis programme has been accomplished for the superconformal index of the 6d p2,0qtheory on\nS1\nβˆS5(βbeing the length of the thermal circle) whose finite Ncorrections are reproduced by\nsemiclassical M2 brane wrapped on S1ˆS2in a twisted M-theory background AdS 7ˆS4[25]. A\nsimilar analysis has been performed for the “mirror” case of the index of 3d N“8supersymmetric\nlevel-one UpNqˆUpNqABJM theory on S1\nβˆS2whose finite Ncorrections are computed by\nsemiclassical M5 brane wrapped on S1ˆS5in twisted AdS 4ˆS7background [26]. To recall, the\nleading non-perturbative correction to the unrefined index in the two cases had the following form\n(see Appendix A for notation)\nIp2,0q\nNpqq“Ip2,0q\nKKpqq„\n1´q\np1´qq2qN`Opq2Nqȷ\n, q “e´β, (1.3)\nIABJM\nNpq1q“IABJM\nKKpq1q„\n1´1\n6N3ˆq1\npq1, q1q78`. . .˙\nqN`Opq12Nqȷ\n, q1“e´1\n2β.(1.4)\nThe correction in (1.3) has the form (1.1), with the leading brane index IGG\nk“1pqqbeing a rational\nfunction of q. The result for ABJM in (1.4) is different in two respects. First, the dependence on\nqis no more rational, but involves instead an infinite product. Second, the brane index has an\nextra Ndependence apart from that in the classical factor qN. The correction indeed has an overall\nN3factor plus subleading corrections in 1{Ndenoted by dots in (1.4). This is a common feature\nof unrefined indices. As we mentioned before, the expansion (1.1) – with Nindependent brane\nindices IGG\nkpqq– holds at generic points in fugacity space. Additional polynomial dependence on N\nmay occur in unrefined indices and, more generally, in refined indices at special varieties defined by\nalgebraic relations between fugacities. This is a form of “wall-crossing” phenomenon [4, 19, 27]. As\nexplained in [26], in the gravity calculation extra powers of Ncome from zero modes of fluctuations\non the wrapped brane which are removed (or regularized) when generic fugacities are switched on.\n4The relation between giant brane configurations and the exclusion principle was already in the original studies\nof giant gravitons that were discovered by considering large angular momentum BPS particles moving in sphere part\nofAdS nˆSm. At high momentum they become spherical branes (due to the role of the form fields supporting the\ngeometry ) expanding in Smwith the stringy exclusion principle being same as condition that brane radius cannot\nexceed sphere one [2].\n31.1 Summary of results\nIn this paper, we apply the same strategy back to the simplest and more familiar case of 4d N“4\nUpNqSYM theory where finite Ncorrections to the index on S1\nβˆS3originate from D3 branes\nwrapped on S1ˆS3in twisted AdS 5ˆS5. In particular, we will focus on the refined Schur\nindex for which a wealth of useful results are available [28, 29, 30, 27] and that was treated by\nthe symmetry analytic continuation approach in [7]. One motivation for extending the analysis in\n[25, 26] to the N“4SYM case is that recently a similar (but complementary) analysis of the\n1\n2-BPS index appeared in [21]. There, the superconformal index is computed by localization of the\nwrapped brane fluctuations, i.e.exploiting supersymmetry from the start. In particular, and by\nconstruction, one keeps only contributions surviving supersymmetry cancellations. Thus, gapped\nsectors can be dropped, including non-constant modes on S3, reducing the problem to the study of\na supersymmetric quantum mechanical particle moving in two dimensions in a constant magnetic\nfield.\nInstead, our analysis will be more direct and will be based only on the conformal properties\nof the fluctuations. The brane index calculation will thus require a generalization (in presence of\nvarious fugacities) of the operator counting method in [31, 32] as in our previous applications in\n[25, 26]. For the unrefined Schur index of N“4UpNqSYM in S1\nβˆS3, we will reproduce the\nexpansion of the exact result in [29], i.e.the leading wrapping correction to the index\nISYM\nNpqq“IKKpqq„\n1´pN`2qqN`1`Opq2Nqȷ\n, (1.5)\nwhich is similar to the 6d p2,0qcase in (1.3). Again, the enhancement factor of Noriginates\nfrom zero modes of fluctuations and is absent in the flavored (or refined) Schur index ISYM\nNpq, uq\ndepending on an additional fugacity udual to certain R-charges. In this case, finite Ncorrections\nhave a dependence on Nfully encoded in the classical factor qNwith the brane indices being N\nindependent quantities. The detailed generalization of (1.5) reads\nISYM\nNpq, uq“IKKpq, uq„\n1`ˆ\nuN`3Gpq, uq´u´N´1Gpq, u´1q˙qN`1\n1´u2`Opq2Nqȷ\n.(1.6)\nwhere the function\nGpq, uq“´p1´u2q2\nu2pu´1q;u´1, qq2\n8\npu´2, u´1qq8pu2, u´1qq8, (1.7)\nis smooth for uÑ1and admits a Taylor expansion around q“0.\nAs a byproduct, the analysis of the index in terms of a twisted partition function allows one\nto discuss the supersymmetric Casimir energy [33] of the fluctuations on the wrapped brane. This\nquantity ED3\ncappears in the general relation\nZ1pS1\nβˆS3q“e´βED3\ncpI1pqq. (1.8)\nwhere Z1is the one-loop partition function of fluctuations and pI1pqqthe associated leading brane\nindex. The resulting value of ED3\ncwill be checked to match the regularized sum over fluctuation\nmodes of the zero point combination of Cartan charges defining the index. In particular, for the\nSchur index, we will get ED3\ncpuq“0for generic u, and ED3\ncp1q“´ 1in the unrefined limit.\nTo frame our analysis in a more general perspective, let us summarize the current status of\nfinite Ncorrections to superconformal index at large N. As we overviewed, there are three possible\napproaches available so far:\n4a) In the first approach, one uses unbroken superconformal symmetry to predict the brane index\nas the analytic continuation in fugacity space of a simpler boundary SCFT index. This was\nstarted in [5] by examining the index of S-fold theories and subsequently applied to many\nother models.\nb) The second (recent) approach exploits supersymmetry to localize fluctuations on the wrapped\nbrane. In this framework, the analysis is reduced to considering ungapped sectors, like the\nsupersymmetric quantum mechanical model of [21] for the1\n2-BPS index of N“4SYM.\nc) The third approach, adopted here and previously in [25, 26], directly exploits only conformal\nsymmetry on the brane worldvolume. For each fluctuation, one determines its associated\npartition function on a background with suitable twists related to the index. The different\ncontributions are finally combined in supersymmetric way (fluctuations belong to a multiplet\nof the preserved superconformal symmetry).\nIt is certainly useful to have several independent techniques available. However, as we explained,\nthe most interesting physical problems raised by (1.1) require going beyond the leading wrapping\ncontribution in order to analyze the specific cancellations among higher order contributions. It\nis thus important to critically assess the methods in (a), (b), and (c) from this point of view.\nAs a general comment, at wrapping order k, fluctuations on the wrapped brane are expected to\nbelong to a Upkqgauge theory, living on the stack of kbranes [4]. In method (a), higher wrapping\ncontributions were computed by introducing ad hocrules to handle the holonomy integrals of the\nUpkqtheory. Finding a general rule seems to be an open problem, see [12, 13] and [19] for recent\ndevelopments. In method (b), k-wrapping contributions to the1\n2-BPS index were (not surprisingly)\ncomputed by promoting the supersymmetric quantum mechanical model to a matrix one with fields\nbeing Upkqmatrices. This promising approach worked well in the1\n2-BPS index of N“4SYM.\nIt remains to see how the localization approach extends to other models and for refined indices.\nFinally, method (c) has been applied so far only to the leading wrapping contribution to the index.\nAs we mentioned, it has still to be extended to the direct analysis of fluctuations for the multiply\nwrapped brane by considering suitable conformal Upkqgauge theories for each fluctuation. We hope\nthat the interplay between the above-mentioned approaches will eventually lead to some consistent\nmethod to compute at any higher order the finite Ncontributions to superconformal indices.\n1.2 Plan of the paper\nIn Section 2 and 2.2 we discuss the algebraic aspects of superconformal index of N“4theories in 4d\nand its Schur specialization for N“4UpNqSYM. In Section 3 we summarize the results of [7], i.e.\nthe approach based on exploiting analytic continuation of the preserved superconformal symmetry\non the wrapped D3 branes. In Section 4 we introduce a twist in AdS 5and study the fluctuations\nof D3 branes wrapped on a S1ˆS3inside deformed AdS 5ˆS5. The explicit spectrum of scalar,\nfermionic, and gauge fluctuations on the brane are determined. All fluctuations turn out to be\nconformal fields on S1ˆS3with constant holonomy along the thermal cycle. By computing the\nassociated partition functions using an extensions of the operator counting method, we reproduce\nin Section 4.4 the leading wrapping finite Ncontribution to the unrefined index. The Ndependent\nenhancementin(1.5)isexplainedintermsofbranefluctuationszeromodes, similartowhathappens\nin the ABJM index [26]. We also discuss the supersymmetric Casimir energy of the system of\nfluctuations on the wrapped D3 brane. Finally, in Section 5, we deal with the refined index with\nan extra fugacity coupled to R-charges by adding extra twists in S5.\n52N“4Superconformal index\nFor a N“4theory on RˆS3with PSUp2,2|4qsymmetry the superconformal index [3] is defined\nas (notation is u“pu1, u2, u3q)\nIpq, y,uq“TrBPSrp´1qFqH`J`¯Jy2JuR1\n1uR2\n2uR3\n3s, u 1u2u3“1, (2.1)\nwhere His the dilatation operator, J,¯Jleft and right-handed spins5andRIare three Cartan\ngenerators of the R-symmetry sup4q. In terms of traceless RI\nJgenerators of SUp4q, with RI\nI“0,\nwe have explicitly R1“1\n2pR1\n1`R2\n2´R3\n3´R4\n4q,R2“1\n2pR1\n1´R2\n2`R3\n3´R4\n4q,R3“\n1\n2pR1\n1´R2\n2´R3\n3`R4\n4q. The BPS condition depends on the specific supercharge used to define\nthe index and can be taken to read\nH´2¯J´2R1\n1“0. (2.2)\nIt is convenient to change basis of fugacities and write\nIpq, y, u, vq“TrBPSrp´1qFqH`J`¯Jy2JuR1´R2vR2´R3s, (2.3)\nu“u1, v“u1u2. (2.4)\nFor an arbitrary N“2superconformal theory there is a corresponding chiral algebra and the Schur\nindex [34] is a specialization of the superconformal index giving the chiral algebra vacuum character\n[35]. In our conventions, the Schur limit is y“v“16,i.e.the trace\nIpq, uq“TrBPSrp´1qFqH`J`¯JuR1´R2s. (2.5)\n2.1 The N“4Maxwell multiplet\nTheN“4Maxwellmultipletisasinglefreevectormultipletwithcomponents Fab, λIa, ϕIJ,¯λI9a,¯F9a9b,\nwhere I, JareSUp4qRindices and ϕIJis in the 2-antisymmetric representation (with a reality\ncondition). This multiplet transforms as a short representation of the superconformal algebra and\nits index is obtained from the plethystic exponential\nIMaxwellpq, y, u, vq“PErpIMaxwellpq, y, u, vqs“exp„8ÿ\nn“11\nnpIMaxwellpqn, yn, un, vnqȷ\n,(2.6)\nwith the single particle index [36]\npIMaxwellpq, y, u, vq“qpu`vu´1`v´1q´pq2y`qy´1q´q2pu´1`uv´1`vq`2q3\np1´q2yqp1´qy´1q.(2.7)\nIn the unflavored Schur limit y“u“v“1, we get\npIMaxwellpqq”pIMaxwellpq,1,1,1q“2q´2q2\n1´q2“2q\n1`q, (2.8)\nwhere´2q2comes from the unflavored indices of a N“2vector multiplet and 2qfrom a N“2\nhypermultiplet.\n5They are suitable generators of the group SUp2qˆSUp2q, being the RˆS3version of the Lorentz group in R1,3.\n6Often, the index is defined tracing qH`¯J¯y2Jand Schur limit is ¯y“q1\n2. In our notation ¯y“q1\n2yand the Schur\nlimit corresponds to y“1. The Schur index gets contributions only from states obeying the additional BPS condition\nH`2J`2R4\n4“0and this allows to replace H`J`¯J“H´R3.\n62.2 Schur index of N“4UpNqSYM\nFor a gauge theory with multiplets in the representations Rof the gauge group, the index is [37, 3]\nINpq, y, u, vq“ż\nDUź\nRexp„8ÿ\nn“11\nnpIRpqn, yn, un, vnqχRpUnqȷ\n, (2.9)\nwhere integration is over Haar measure of gauge group and χis a representation character. For\nN“4SYM with gauge group UpNq, we need to consider a Maxwell multiplet in the adjoint of\nUpNq. In the unflavored case, this gives the expression [29]\nISYM\nNpqq“q´N2{4ηpτq3N\nπNN!żπ\n0dNαś\niăjϑ1pαi´αj, qq2\nś\ni,jϑ4pαi´αj, qq, (2.10)\nwhere αare gauge group holonomies and definitions of elliptic functions are in Appendix A.\nThe index may be evaluated at finite Nexpanding in powers of qand integrating over α. For\nthe lowest values of N, this gives\nISYM\n1pqq“1`2q`q2`2q3`2q4`¨¨¨ ,\nISYM\n2pqq“1`2q`4q2`4q3`6q4`¨¨¨ ,\nISYM\n3pqq“1`2q`4q2`8q3`9q4`¨¨¨ ,\nISYM\n4pqq“1`2q`4q2`8q3`14q4`¨¨¨ .(2.11)\nThe exact result was derived in [29] from (2.10) and takes the giant graviton expansion form\nISYM\nNpqq“1\nϑ4p0q8ÿ\nk“0p´1qk„ˆN`k\nN˙\n`ˆN`k´1\nN˙ȷ\nqkN`k2. (2.12)\nThe coefficients of the expansions (2.11) stabilize for NÑ8and correspond to the series\nIKKpqq”ISYM\n8pqq“1\nϑ4p0q“8ź\nn“1p1´q2nq´1p1´q2n´1q´2“8ź\nn“11`qn\n1´qn,(2.13)\nwhose explicit first terms are\nIKKpqq“1`2q`4q2`8q3`14q4`18q5`28q6`¨¨¨ . (2.14)\nThegeneratingfunction IKKpqqreproducestheindexofBPSKaluza-Kleinmodesofthesupergravity\nmultiplet in AdS 5ˆS5[3]. Including systematically the large Nexponentially suppressed terms in\n(2.12), we can write the large Nexpansion of the unflavored index as\nISYM\nNpqq“IKKpqq„\n1´pN`2qqN`1`1\n2pN`1qpN`4qq2N`4`Opq3Nqȷ\n.(2.15)\nThe flavored Schur index is obtained starting from the following replacement in the Maxwell single\nparticle index\npIMaxwellpqq“2q´2q2\n1´q2ÑpIMaxwellpq, uq”pIMaxwellpq,1, u,1q“qpu`u´1q´2q2\n1´q2.(2.16)\n7The first term in the numerator comes from the N“2hypermultiplet and is due to the extra\nfugacity uR1where R1is a R-symmetry Cartan taking values ˘1on the complex scalar and its\nconjugate in that multiplet. In this case, the large NKaluza-Klein factor takes the form7\nIKKpq, uq“PE„uq\n1´uq`u´1q\n1´u´1q´q2\n1´q2ȷ\n. (2.17)\nForu‰1, we don’t have a simple explicit expression for the index ISYM\nNpq, uqat finite Ngeneralizing\n(2.12) and a derivation of the large Nexpansion (2.15) is non-trivial starting from the localization\nformula (2.9).\n3 D3-brane index from analytic continuation\nIn the approach of Imamura et al., see in particular [5, 10], the general index (2.1) for N“4UpNq\nSYM can be computed at large Nas the triple sum\nISYM\nNpq, y,uq“IKKpq, y,uq8ÿ\nk“0pqu1qk1Npqu2qk2Npqu3qk3NID3\nkpq, y,uq,k“pk1, k2, k3q.(3.1)\nNotice the slight change of notation pIkÑID3\nk,cf.(1.1), to emphasize the specific case under\nconsideration. The N-independent factor IKKpq, y,uqis the NÑ8index,i.e.the index counting\ngauge invariant operators at large- N, ignoring trace relations, and agrees with the index from BPS\nKaluza-Klein supergravity modes on AdS 5ˆS5. The triple sum in (3.1) represents a sum over D3\nbranes wrapped kItimes, I“1,2,3, over the cycle ZI“0inS5,i.e.|Z1|2`|Z2|2`|Z3|2“1. The\ngenerators RIin (2.1) are associated with phase rotations in the ZIplane. The classification of the\nrelevant BPS wrapped branes was analyzed in [38] where it was shown how to build supersymmetric\ncycles in backgrounds AdS nˆSmthat generalize giant gravitons. In particular, for D3 branes in\nAdS 5ˆS5, any analytic constraint fpZ1,Z2,Z3q“0defines a supersymmetric configurations which\nis1\n2,1\n4, or1\n8-BPS depending on the number of arguments of f. The simplest case fpZ1q“0gives a\n1\n2-BPS solution and multiple zeroes are associated with a number of concentrical spherical giants,\nsee [39] for a discussion of the related geometries.8\nThe factorś\nIpquIqkINin (3.1) comes from the classical action of the wrapped brane. Finally,\nthe index ID3\nkrefers to the gauge theory on the wrapped D3 brane which is a Gk“Upk1qˆUpk2qˆ\nUpk3qquiver gauge theory with three bifundamental hypermultiplets. This index is by construction\nindependent on N. Thus, the expansion (3.1) has the form of a giant graviton expansion with each\nterm having the weight qkN`δk, where k“k1`k2`k3is the total wrapping and δkis a non-negative\ninteger.\nAsconjecturedin[10], thebraneindex ID3\nkisgivenbya Gkholonomyintegralsimilarinstructure\nto (2.9), i.e.schematically,\nID3\nkpq, y,uq“ż\nDUPErID3,vec\nkpq, y,uqχvecpUqsPErID3,hyp\nkpq, y,uqχhyppUqs,(3.2)\n7Foru“1, we get IKKpq,1q “PE„\n2q\n1´q´q2\n1´q2ȷ\n“PE„\n1\n2ř8\nk“1p3´p´ 1qkqqkȷ\n“ś8\nk“1´\n1\n1´qk¯1\n2p3´p´1qkq\n“\np1´qq´2p1´q2q´1p1´q3q´2p1´q4q´1p1´q5q´2p1´q6q´1¨¨¨, in agreement with (2.13).\n8The sum over three cycles ZI“0in (3.1) deserves a comment. Looking for simplicity at wrapping one, we need\nto consider the family of configurations aX`bY`cZ“0. It is parametrized by CP2and we need to sum over\nfluctuations around a representative in each of the three covering patches ZI‰0.\n8where ID3,vec\nk,ID3,hyp\nkare the contributions from vector and hypermultiplets in the quiver gauge\ntheory. In (3.2), integration over U“pU1, U2, U3qwith UIPUpkIqis a schematic way to represent\nthe integration over holonomies that implements singlet projection. Similarly, the factors χvecpUq\nandχhyppUqrepresent suitable group characters, cf.(2.9). Details of the integration cycles are a\nkey issue not completely understood and recently discussed in [12]. Here, we will focus on total\nwrapping k“1where holonomies are trivial and also quiver hypermultiplets are absent. Hence,\nafter the relabeling pZ1,Z2,Z3qÑpX,Y,Zqfor simpler notation, the expansion (3.1) reads\nISYM\nNpq, y,uq“IKK„\n1`qNˆ\nu1ID3\nXpq, y,uq`u2ID3\nYpq, y,uq`u3ID3\nZpq, y,uq˙\n`Opq2Nqȷ\n,(3.3)\nwhere ID3\nX”ID3\n1,0,0“ID3,vector\n1,0,0,etc.The vector multiplet index on D3 is again the plethystic\nexponential of a single particle index conjectured in [5, 10] to be9\npID3\nXpq, y,uq“1´p1´q´1u´1\n1qp1´q2yqp1´qy´1q\np1´qu2qp1´qu3q, (3.4)\nUsing (2.4), this reads, cf.Eq. (4.35) of [5],\npID3\nXpq, u, vq“1\nqu´qpy`1\nqyq1\nu´qp1\nv`v\nuq`q2py`1\nqyq`q22\nu´q3\np1´q{vqp1´qv{uq. (3.5)\nThe expressions for pID3\nY,pID3\nZare similarly obtained from (3.4) by permuting u1, u2, u3.\nThe index (3.5) is computed in two ways, both relying on the broken superconformal symmetry\non the D3 brane. By a direct computation, see Appendix A of [10], the modes on the brane\nare organized in multiplets of the reduced superalgebra (the wrapped brane breaks half of the\n32 supersymmetries and preserves a SUp2|2qˆSUp2|2qsuperalgebra) and the index is evaluated\nby summing the separate contributions with the result (3.5). Alternatively, one can rely on the\nautomorphism between the supersymmetry algebra on the boundary and on the wrapped D3-brane.\nThis gives the analytic continuation relation\npID3\nXpq, y, u, vq“pIMaxwellpq1, y1, u1, v1q,\nq1“q2\n3u´1\n3, y1“q´1\n3u´1\n3v, u1“q´5\n3u´2\n3, v1“q´1\n3yu´1\n3,(3.6)\nthat expresses the wrapped brane index in terms of the index of the Maxwell multiplet living on the\nboundary. Starting from (2.7) and using (3.6), one gets again (3.5). This second method is somehow\nimpressivesinceitreducesthewrappedbraneindextoacomputationinthefreeboundaryconformal\ntheory. Similar analytic continuations have been exploited in later works on finite Ncorrection to\nthe index in different models, like the M-brane indices considered in [8].\n3.1 Schur specialization\nThe refined Schur index is obtained by specializing y“v“1. From (3.6) we get\npID3\nXpq, uq“pIMaxwellpq2\n3u´1\n3, q´1\n3u´1\n3, q´5\n3u´2\n3, q´1\n3u´1\n3q. (3.7)\n9The definition of the index in [10] has fugacities qH`¯J¯y2JuR1\n1uR2\n2uR3\n3. To match our definition of the index we\nneed to identify ¯y“q1\n2y.\n9Fory“vthe following scaling invariance holds\npIMaxwellpq, y, u, yq“pIMaxwellpλ´1q, λ2y, λu, λ2yq. (3.8)\nHence, taking λ“pq´1\n3u´1\n3q´1\n2“pquq1\n6we can write pID3\nXpq, uqin terms of the Maxwell Schur index\npID3\nXpq, uq“pIMaxwellppquq´1\n6q2\n3u´1\n3,pquq1\n6q´5\n3u´2\n3q“pIMaxwellpq1\n2u´1\n2, q´3\n2u´1\n2q.(3.9)\nUsing (2.7), or taking the Schur limit in (3.5), this is\npID3\nXpq, uq“1\nuq´2\nuq`q2\n1´1\nuq. (3.10)\nAs discussed in [7], the general formula (3.1) simplifies to\nISYM\nNpq, uq“IKKpq, uq8ÿ\nk1,k2“0puqqk1Npu´1qqk2Nq2k1k2ID3\nX,k1pq, uqID3\nY,k2pq, uq. (3.11)\nIn (3.11), we recognize the index of a system of k1D3 branes and k2D3 branes wrapped on the two\n3-cycles X“0andY“0. Also, due to the Weyl reflection relation\nID3\nY,kpq, uq“ID3\nX,kpq, u´1q, (3.12)\none can compute (3.11) from the X“0index ID3\nX,kalone. As in (3.9), this is related by analytic\ncontinuation to the Schur index of the N“4SYM theory with gauge group Upkq\nID3\nX,kpq, uq“ISYM\nkpq1\n2u´1\n2, q´3\n2u´1\n2q. (3.13)\nIn particular, the leading contribution to the giant-brane expansion is\nISYM\nNpq, uq“IKKpq, uq„\n1`qNˆ\nuNID3\nXpq, uq`u´NID3\nXpq, u´1q˙\n`Opq2Nqȷ\n, (3.14)\nwhere\nID3\nXpq, uq”ID3\nX,1pq, uq“PErpID3\nXpq, uqs. (3.15)\nUsing expression (3.10), this evaluates to\nID3\nXpq, uq“´uq\n1´uq8ź\nn“1r1´pu´1qqns2\nr1´u´2pu´1qqn´1sr1´u2pu´1qqn`1s, (3.16)\nor, in terms of q-Pochhammer functions (A.4),\nID3\nXpq, uq“´ uqp1´u2qpu´1q, u´1qq2\n8\npu´2, u´1qq8pu2, u´1qq8. (3.17)\nThis expression is exact and its expansion in powers of qgives\nID3\nXpq, uq“u3\n1´u2q`p1´u2qq2`ˆ1\nu3´u3˙\nq3`ˆ1\nu6´1\nu2`1´u4˙\nq4`¨¨¨ .(3.18)\nNotice that one cannot use relation (3.13) to obtain an expansion like (3.18) for k“1, unless Ikpq, uq\nis known in exact form and not as a series in q. This is because of negative powers of qin (3.13),\n10requiring resummation if the latter identity is naively applied to a series in q.10For this reason, in\n[7], the expansion (3.18) (and its generalization to ną1) was obtained by applying the analytic\ncontinuation relation (3.13) inside the gauge holonomy integral expression of ISYM\nk. After that, the\nintegration over holonomies was evaluated by a conjectured ad hoc choice of the contributing poles.\nThis is a major difficulty for next-to-leading ( i.e.higher wrapping) contributions that we do not\ndiscuss here. Nevertheless, in Appendix B, we present a direct application of analytic continuation\nfor the k“2brane index starting from the exact expression of the SYM index and avoiding pole\nselection rules at all.\n3.2 Unrefined index and uÑ1limit\nLet us show how the (leading) giant graviton expansion of the unflavored index in (2.15) is repro-\nduced in the uÑ1limit. This comes from a non-trivial combination of the two terms in (3.14)\nwhich are separately singular for uÑ1, but with a smooth sum.\nWe isolate the singular uÑ1contribution to ID3\nXpq, uqby writing\nID3\nXpq, uq“u3\n1´u2q Gpq, uq, Gpq, uq“1\n1´uq8ź\nn“1r1´pu´1qqns2\nr1´u´2pu´1qqnsr1´u2pu´1qqn`1s,(3.19)\nwhere Gpq, uqis smooth for uÑ1. The value at u“1is\nGpq,1q“1\n1´q8ź\nn“11´qn\n1´qn`1“1. (3.20)\nAlso, we can compute the derivative with respect to uatu“1as follows. Taking logarithm, we\nget\nlogGpq, uq“´ logp1´uqq`8ÿ\nn“1„\n2 logp1´pq{uqnq´logp1´u´2pq{uqnq´logp1´u2pq{uqn`1qȷ\n,\n(3.21)\nand differentiation with respect to ugives after some rearrangement\nBuGpq,1q\nGpq,1q“q\n1´q`8ÿ\nn“1„\npn´2qqn\n1´qn´pn´1qqn`1\n1´qn`1ȷ\n“0,Ñ B uGpq,1q“0.(3.22)\nHence,\nuNID3\nXpq, uq`u´NID3\nXpq, u´1q“„uN`3\n1´u2Gpq, uq`u´N´3\n1´u´2Gpq,´uqȷ\nq\n“´rGpq,1qpN`2q`B uGpq,1qsq`Opu´1q“´ qpN`2q`Opu´1q.(3.23)\nreproducing the enhancement in (2.15).\n10To explain, if we have fpqq“1\n1´qwhich has regular expansion in q, we can apply qÑ1{qand obtain fpq´1q“\n´q\n1´qwhich also has a regular expansion in q. Of course, this is not visible (in particular in more complicated cases)\nif we apply qÑ1{qto1`q`q2`q3`¨¨¨.\n114 D3-brane in twisted AdS 5ˆS5and unflavored index\nAs we have seen, the large Nnon-perturbative corrections to the Schur index are expressed in terms\nof the single quantity pID3\nXpq, uqin (3.10) by using (3.11) and (3.12). Here, we want to reproduce\npID3\nXpq, uqby a genuine one-loop computation of the partition function on the wrapped D3 brane with\na twisted metric that geometrizes the J,¯Jand Cartan charge terms in (2.5). A similar approach\nwas used in [25, 26] as we reviewed in the Introduction.\nStarting from the unflavored case u“1, the aim is thus to reproduce the remarkably simple\nexpression11\npID3\nXpqq”pID3\nXpq,1q“1\nq´2q`q2\n1´q“1\nq`1´q , (4.1)\nby considering the semiclassical expansion of the D3 brane action on the background (scaling away\ncommon radius)12\nds2“ds2\nĆAdS 5`ds2\nS5,\nds2\nĆAdS 5“dx2`sinh2x drS3`cosh2x dτ2, τ“τ`β ,\ndrS3“dψ2\n1`sin2ψ1dψ2\n2`sin2ψ1sin2ψ2pdψ3`iκdτq2,\nds2\nS5“dw2`cos2w dS 3`sin2w dz2.(4.2)\nThe mixing between ψ3andτcorresponds to the Cartan combination H`J`¯Jin (2.4). The\nphysical (supersymmetric) value of the parameter κisκ“1, but it will be useful to keep it as a\nfree parameter for the time being.\nIn a static gauge, the D3 coordinates are ξ“pξ1, . . . , ξ4q”p τ, pPS3q,i.e.the brane wraps\nS1\nβˆS3where S1\nβis inĆAdS 5and has length β, while S3ĂS5. The fluctuating coordinates are\npx, ψi, w, zqwith background values x“w“z“0. These transverse fluctuations are 4 in AdS 5\nand 2 in S5for a total of 6 scalar fluctuations. To these we will add fermions θand the gauge field\nAon the brane. At semiclassical level, the partition function is13\nZ“ż\nDXDADθ e´SrX,A,θs“Z1e´Sclp1`OpT´1\n3qq, (4.3)\nZ1“e´F, F“1\n2ÿ\nap´1qFalog det ∆ a, (4.4)\nwhere alabels the fluctuation fields (bosonic and fermionic) and ∆a“´∇2`¨¨¨are the associated\n2nd order operators for quadratic fluctuations. They are defined on S1\nβˆS3with the induced metric\non the brane worldvolume. All fields (including fermions) are periodic on the “thermal” circle S1\nβ\nfor consistency with supersymmetry.\n4.1 Classical D3 action\nThe classical D3 brane action Sclgives the (unrefined) overall factor qNin (3.3) and its evaluation is\nstraightforward. After a suitable scaling of the gauge field, the bosonic part of the supersymmetric\n11Simplicity of (4.1), i.e.reduction to a finite number of modes on the wrapped brane, suggests a treatment by\nlocalization as in [21].\n12Here, the coordinate yshould not be confused with the fugacity in the index definition.\n13Recall that Sclhas an explicit overall factor of the D3 brane tension T3.\n12D3 brane action is the sum of the Born-Infeld and the Wess-Zumino actions [40] (we have no\nbackground antisymmetric field Babhere). In Euclidean notation,\nSDBI“T3ż\nd4ξa\ndetpGab`2πFabq, S WZ“´ż\nC4, (4.5)\nwith D3 tension\nT3“N\n2π2. (4.6)\nThe 4-form on S5is\nC4“iT3cos4w dz^volS3. (4.7)\nThe classical action is thus the sum of\nScl,DBI“T3ˆβˆVolpS3q“N\n2π2βp2π2q“Nβ , S cl,WZ“0. (4.8)\nAs expected, the total Scl“Nβcorresponds to the factor qNin (3.3), where as usual we denote\nq“e´β.\n4.2 Wrapped brane fluctuation one-loop determinant and brane index\nFor each separate fluctuation, the free energy in (4.4) splits into the \"Casimir\" part linear in βand\nthe genuine thermal part which is exponentially suppressed at large β, seee.g.[41],\nFpβq“β Ec`¯Fpβq, (4.9)\nwhere Ecis the Casimir contribution, the dominant term at low temperature βÑ`8. Denoting\nbyEnthe eigenvalues of the fluctuation operator on S3, and by dnthe associated multiplicity, one\nhas the general expression for the thermal part of the free energy\n¯Fpβq“ÿ\nndnlogp1´e´βEnq. (4.10)\nand the zeta-function regularized value of the Casimir energy\nEc“1\n2ÿ\nndnEn“1\n2ζEp´1q, ζ Epsq“ÿ\nndnE´s\nn. (4.11)\nThe associated one-loop partition function Z1in (4.3) is then\nZ1“e´βEcZpqq, (4.12)\nwith\nZpqq“e´¯F“expˆÿ\nkdk8ÿ\nn“11\nne´nβE k˙\n“PErpZpqqs, (4.13)\nwhere the single particle partition function is\npZpqq“ÿ\nndnqEn. (4.14)\nNotice that the zeta-function ζEpsqin (4.11) admits the following expression in terms of pZpqq\nζEpsq“1\nΓpsqż8\n0dβ βs´1pZpe´βq. (4.15)\n13In the present context, the partition function Z1is computed on a background with a twist adapted\nto the index and Ecis the so-called supersymmetric Casimir energy [33] (of fluctuations on the D3\nbrane). The expected precise relation between Z1and the index is\nZ1“e´βEcID3pqq. (4.16)\nComparing with (4.12) gives the key identification\nZpqq“ID3pqq, (4.17)\nthat clearly holds also at the level of single particle partition function and index. Thus, in our case,\nwe want to checkÿ\nfluctuationspZpqq“pID3\nXpqq“1\nq`1´q , (4.18)\nwhere we used (4.1).14\n4.3 Computation of the single particle partition functions\n4.3.1 Scalar fluctuations\nLet us begin by studying the bosonic scalar fluctuations by expanding at quadratic order the total\nbosonic action SDBI`SWZ.\nAdS 5sector It is convenient to introduce new variables XI,I“1, . . . , 4in terms of x, ψi\nX1“sinhxcosψ1, X 2“sinhxsinψ1cosψ2, (4.19)\nX3“sinhxsinψ1sinψ2cosψ3, X 4“sinhxsinψ1sinψ2sinψ3. (4.20)\nOne can check that\ndx2`sinh2xpdψ2\n1`sin2ψ1dψ2\n2`sin2ψ1sin2ψ2dψ2\n3q“dX2\nIloomoon\nflat´sinh2x dx2looooomooooon\nquartic,(4.21)\nand thus, at quadratic order,\nds2\nĆAdS 5“p1`X2\nI´κ2pX2\n3`X2\n4qqdτ2`dX2\nI`2iκpX3dX4´X4dX3qdτ . (4.22)\nDenoting by a“2,3,4the indices of the S3brane coordinates, we have\nds2\nĆAdS 5“r1`X2\n1`X2\n2`p1´κ2qpX2\n3`X2\n4q` 9X2\nI`2iκpX39X4´X49X3qspdξ1q2`BaXIBbXIdξadξb\n`off diagonal terms . (4.23)\nThe DBI action, at quadratic order in the fluctuations, reads\nSDBI“T3ż\nd4ξ?\nG, ds2“Gijdξidξj, (4.24)\n14We remark that in the unflavored case the three wrapped configurations on X“0,Y“0,Z“0are completely\nequivalent.\n14and using\nδa\nGp0q`Gp1q“a\nGp0q`1\n2a\nGp0qpGp0qqijGp1q\nij`¨¨¨ , (4.25)\nwe get (denoting by d3ξtheS3coordinates),\nδSDBI“1\n2βT3ż\nd3ξ?grgabBaXIBbXI`X2\n1`X2\n2`p1´κ2qpX2\n3`X2\n4q` 9X2\nI`2iκpX39X4´X49X3qs.\n(4.26)\nWe now rescale fluctations by pβT3q1{2, and setBτÑi2π\nβn, with nPZ,i.e.expand in periodic\nmodes on the thermal circle. The fields X1, X2are scalar fields on S3with common squared mass\nX1, X2:M2“1`n2\nβ, n β“2πn\nβ, n“0,˘1,˘2, . . . . (4.27)\nNotice that the squared mass of a conformally coupled scalar on S1ˆSdis1\n4pd´1q2that is 1 for\nS3. This is precisely what we have for n“0.\nThe fields X3, X4can be written in complex basis ϕ“X3`iX4?\n2,¯ϕ“X3´iX4?\n2and the expansion\nof the DBI action gives for them\nδSpX3,X4q\nDBI“βT3ż\nd3ξ?grgabBa¯ϕBbϕ`¯ϕpp1´κ2qϕ`2κ9ϕ´:ϕqs. (4.28)\nThis is a complex scalar on S3with mass\nX3, X4:M2“1`pnβ`iκq2. (4.29)\nIn this case, the conformally coupled scalars have an imaginary shift in nβin the Kaluza-Klein mass\nterm, similar to what happens in the M5 brane case studied in [26].\nS5sector In the sphere sector we parametrize fluctuations by setting\nw“arccos1´1\n4pY2\n1`Y2\n2q\n1`1\n4pY2\n1`Y2\n2q, z“arctanY2\nY1, (4.30)\nso that\ndw2`sin2w dz2“dY2\n1`dY2\n2\nr1`1\n4pY2\n1`Y2\n2qs2, (4.31)\nand\nds2\nS5“rp1´YcYcqgab`BaYcBbYcsdξadξb`p 9Yc9Ycqpdξ1q2`off diagonal terms .(4.32)\nThis gives the following action from DBI at quadratic order\nδSDBI“1\n2βT3ż\nd3ξ?grgabBaYcBbYc`9Yc9Yc´3YcYcs. (4.33)\nThe Wess-Zumino term is\nSWZ“´iT3ż\ncos4w dz^volS3“´iT3ż\nr1´2pYcYcqsY1dY2´Y2dY1\nYcYc^volS3,(4.34)\n15and thus the contribution to the fluctuations action is\nδSWZ“βT3ż\nd3ξ?gp4i Y19Y2q. (4.35)\nThe total δS“δSDBI`δSWZreads\nδS“1\n2βT3ż\nd3ξ?grgabBaYcBbYc`9Yc9Yc´3YcYc`8i Y19Y2s. (4.36)\nIntroducing again complex combinations η“Y1`iY2?\n2,¯η“Y1´iY2?\n2, we get\nδS“βT3ż\nd3ξ?grgabBa¯ηBbη`¯ηp´3η`49η´:ηqs. (4.37)\nThis corresponds to a complex field ηwith the following squared mass\nM2“1`pnβ`2iq2. (4.38)\n4.3.2 Fermionic fluctuations\nAt quadratic order in the fermions, the Euclidean fermionic action of the D3 brane is decoupled\nfrom the gauge fields and reads [42]\nSF“1\n2T3ż\nd4ξa\ndetgΘp1´ΓD3qΓαDαΘ, (4.39)\nwhere Θisadoubletof10dpositivechiralityMajorana-Weylspinors, Γα“BαxmΓm,Dα“BαxmDm\nare pullbacks, and ΓD3is a projector required by κ-symmetry invariance. The 10d covariant deriva-\ntive is\nDm“∇m`1\n16{Fp5qΓmbpiσ2q, (4.40)\nwith iσ2acting in doublet space. Let us use Latin letters for 10d indices 0,1, . . . , 9, Greek letters for\nD3 brane worldvolume indices 1,2,3,4, and underlined indices for flat tangent directions ˆ0,ˆ1, . . ..\nIn particular15\nΓD3“εα1...α4\n4!?detgΓα1...α4“Γˆ0ˆ1ˆ2ˆ3ˆ4. (4.41)\nThe two spinors in the doublet Θ“pθ1, θ2qobey Γ10θi“θiwith Γ10“Γˆ0...ˆ9. We fix κ-symmetry\nbyp1´Γ10bσ3qΘ“0orθ2“0and we will denote θ”θ1. UsingtΓD3,Γαu“0, we get\nSF“1\n2T3ż\nd4ξa\ndetg¯θΓαˆ\nDα`1\n16ΓD3{F5Γα˙\nθ . (4.42)\nOur coordinates are\nds2\nĆAdS 5“dx2`sinh2x drS3`cosh2x dτ2, τ“τ`β ,\ndrS3“dψ2\n1`sin2ψ1dψ2\n2`sin2ψ1sin2ψ2pdψ3`iκdτq2,\nds2\nS5“dw2`cos2w dS 3`sin2w dz2.(4.43)\n15Γα“BαXmEm\nmΓm, where Em\nnis the vielbein.\n16with ξαbeing τand the coordinates of S3. We label them as\nXmx ψ 1, ψ2, ψ3τ“ξ1w S 3“pξ2, ξ3, ξ4qz\nm 0 6 ,7,8 4 5 1 ,2,3 9\nOn the classical solution, that has in particular x“0, we remark the non-zero spin-connection\nΩˆ4ˆ0\n4“coshxthat will play an important role. The induced covariant derivative is\nDα“BαXmDm“BαXmˆ\nBm`1\n4Ωab\nmΓab˙\n. (4.44)\nForm“1,2,3,4we haveBαXm“δm\nαon the classical solution, so\nDα“Bα`1\n4Ωab\nαΓab. (4.45)\nForα“1,2,3on classical solution we obtain just the covariant derivative on S3. For α“4\nD4“B4`1\n2Γˆ4ˆ0. (4.46)\nHence,\nSF“1\n2T3ż\nd4ξa\ndetg¯θp{D`Mqθ . (4.47)\nwhere\nM“Γˆ41\n2Γˆ4ˆ0Γˆ4´1\n16ΓD3Γα{F5Γα“1\n2Γˆ0ˆ4´1\n16ΓD3Γα{F5Γα. (4.48)\nAround the classical solution Γα“Γα. Using the 5-form background expression, we get\n{F5“´4pΓˆ0ˆ6ˆ7ˆ8ˆ4`Γˆ5ˆ1ˆ2ˆ3ˆ9q. (4.49)\nThus,\n3ÿ\nα“1Γα{F5Γα“´4ˆ3p´Γˆ0ˆ6ˆ7ˆ8ˆ4`Γˆ5ˆ1ˆ2ˆ3ˆ9q, Γˆ4{F5Γˆ4“´4pΓˆ0ˆ6ˆ7ˆ8ˆ4´Γˆ5ˆ1ˆ2ˆ3ˆ9q. (4.50)\nThe chirality constraint gives\nΓˆ0ˆ1ˆ2ˆ3ˆ4ˆ5ˆ6ˆ7ˆ8ˆ9θ“θ,Γˆ0ˆ6ˆ7ˆ8ˆ4ˆ5ˆ1ˆ2ˆ3ˆ9θ“´θ,Γˆ5ˆ1ˆ2ˆ3ˆ9θ“´Γˆ4ˆ8ˆ7ˆ6ˆ0θ“Γˆ0ˆ6ˆ7ˆ8ˆ4θ , (4.51)\nandΓα{F5Γα“16Γˆ0ˆ6ˆ7ˆ8ˆ4. The mass term is therefore\nM“1\n2Γˆ0ˆ4´Γˆ0ˆ1ˆ2ˆ3ˆ4Γˆ0ˆ6ˆ7ˆ8ˆ4“1\n2Γˆ0ˆ4`Γˆ1ˆ2ˆ3ˆ6ˆ7ˆ8“1\n2Π1`Π2, (4.52)\nwith commuting Π1,2with Π2\n1,2“´I.\nIn conclusion, expanding fermions in Fourier modes along the thermal cycle, B1Ñinβ, and\ndiagonalizing the two projectors, we obtain the Dirac operators\nDF“iΓi∇i`M, M“nβ`iν, ν“˘1\n2,˘3\n2. (4.53)\nIf needed, the squared fermionic operator is\n∆1\n2“´∇2\nS3`1\n4R`M2“´∇2\nS3`3\n2`M2, (4.54)\nwhere∇S3containsthespinorconnectionandweusedthescalarcurvature RpS1ˆS3q“RpS3q“6.\n174.3.3 Gauge field fluctuations\nAt quadratic order, the gauge field decouples from the other fluctuations and its action is the\nMaxwell one S“´1\n4ş\nd4ξ?gF2onS1\nβˆS3. At this order, the background twist is not seen. So, we\ndon’t need any shift in its mass after expansion in S1\nβmodes. As expected from supersymmetry, the\nfield strength Fabtogether with the 6 scalars and 4 Majorana-Weyl fermions is part of an Abelian\nN“4Maxwell multiplet.\n4.4 Unflavored Schur index from twisted N“4fields on S1\nβˆS3\nFor the fields in the N“4Maxwell multiplet on S1\nβˆS3, including values of the shifts in the mass\nterm for scalars and fermions we found the values summarized in Table 1.\nfluctuations X1, X2X3, X4Y1, Y2 ψ V a\nν 0 1 21\n2,1\n2,3\n2,3\n20\nTable 1: Shifts for the S1\nβreduced mass term of conformally coupled fluctuations on S1\nβˆS3.\nThe associated single particle partition functions can be obtained for ν“0by using conformal\nsymmetry (and the absence of conformal anomaly in the S1ˆS3case) and mapping the problem\nto operator counting on R4[31]. The νą0case corresponds to a simple dressing by the universal\nfactor1\n2pqν`q´νq, see [26] for the completely analogous 6d calculation. This gives\npZϕpq;νq“qp1´q2q\np1´qq4qν`q´ν\n2,\npZψpq;νq“4q3{2p1´qq\np1´qq4qν`q´ν\n2,\npZVpqq“2q2p1´qqp3´qq\np1´qq4.(4.55)\nTo recall, in (4.55) the common denominator p1´qq4corresponds to the application in all possible\nways of the four derivatives Bµto the fields ϕ, ψ, V. For scalars, the numerator q´q3correspond to\nthe elementary field ϕwith a subtraction of the vanishing equation of motion B2ϕ“0. For the four\nMajorana-Weyl fermions, the numerator q3{2´q5{2corresponds similarly to the elementary field ψ\nwith subtraction of {Bψ“0. For the gauge field, the terms in 2q2p1´qqp3´qq“6q2´8q3`2q4\nhave a similar origin. We have six gauge invariant fields Fµν, the subtraction of vanishing 4`4\nterms fromBµFµν“0andBµF‹\nµν“0, and the double subtraction (thus with positive sign) of the\nscalar quantities BµBνFµν“BµBνF‹\nµν“0.\nUsing data in Table 1, we get (including the fermion sign)\npZpqq“pZscalarspqq`pZfermionspqq`pZVpqq“1`1\nq´q , (4.56)\nwith\npZscalarspqq“2rpZϕpq; 0q`pZϕpq; 1q`pZϕpq; 2qs,\npZfermionspqq“´ 2rpZψpq;1\n2q`pZψpq;3\n2qs.(4.57)\n18The equality in (4.56) is in agreement with (4.18). The constant term in (4.56) leads to a formal\ndivergence in the index after plethystic exponentiation. It originates from an effective total of two\nbosonic zero modes. Recall that each bosonic/fermionic zero mode contributes a constant term ˘1\n2\nto the total Zpqq. From the expansion of expressions (4.55), one finds the zero mode counting in\nTable 2.\nfluctuations X1, X2X3, X4Y1, Y2Vaψp1{2qψp3{2qtotal\n#zero modes 0 2 8 0 0 ´8 2\nTable 2: Number of zero modes for the various bosonic and fermionic fluctuations. Fermionic\nfluctiations are labeled with their shift ψpνqand the negative sign is the p´1qFto be taken into\naccount in total.\nAs in the M5 case discussed in [26], with the standard normalization of the D3 brane path\nintegral, each zero mode is expected to contribute a factor of T1{2\n3„?\nN. This explains the\npeculiar “wall-crossing” enhancement factor of Nin (2.15), from the total of 2 zero modes. This\nfactor is fully regularized in the index expression with non-trivial fugacity u, as discussed in next\nsection.\n4.5 Supersymmetric Casimir energy on the wrapped brane\nUsing (4.15) for the partition functions in (4.55), we compute the following supersymmetric Casimir\nenergies16\nEϕ\ncpνq“1´10ν4\n240, Eψ\ncpνq“´17´120ν2`80ν4\n960, EV\nc“11\n120, (4.58)\nthat reduce to the standard values for ν“0, see for instance Table 2 in [43] (there, the minus sign\nfor fermions is not incorporated in Ec). The total is\nED3\nc“2Ec,ϕp0q`2Ec,ϕp1q`2Ec,ϕp2q´2Ec,ψp1\n2q´2Ec,ψp3\n2q`Ec,V“´1,(4.59)\nwhich is of course also immediately obtained using the total single particle function in (4.56). This\nsupersymmetric Casimir energy is equivalently obtained as, cf.(2.1),\nED3\nc“1\n2ζD3p´1q, ζD3psq“ÿ\np´1qFpH`J`¯Jq´s. (4.60)\nwhere the sum is over the single particle states of the modes on the wrapped D3 brane and should\nnot be confused with the same quantity for N“4SYM (hence the “D3” label). In more details,\nthe states are derived in Appendix A of [10] that we summarize in Table 3 for the convenience of\nthe reader.\n16Strictly speaking, what is relevant here is the sum over all fields belonging to the Maxwell multiplet. Still, it may\nbe of interest to give the individual fluctuation contribution to the total.\n19state H J ¯J R ¯R\n¯ϕ ℓ´1 0 0ℓ\n2ℓ\n2\n¯χ ℓ´1\n201\n2ℓ\n2ℓ´1\n2\nF´ℓ 0 0ℓ\n2ℓ´2\n2\nψ ℓ´1\n21\n20ℓ´1\n2ℓ\n2\nw ℓ1\n21\n2ℓ´1\n2ℓ´1\n2\n¯ψ ℓ`1\n21\n20ℓ´1\n2ℓ´2\n2state H J ¯J R ¯R\nF`ℓ 0 0ℓ´2\n2ℓ\n2\nχ ℓ`1\n201\n2ℓ´2\n2ℓ´1\n2\nϕ ℓ`1 0 0ℓ´2\n2ℓ´2\n2\nTable 3: Single particle states on the wrapped D3 brane and their quantum numbers. A value of J\ncorresponds of course to a full SUp2qmultiplet with 2J`1values of Jz, and the same for ¯J, R, ¯R.\nThese fields are the modes on the Z“0brane. They are obtained decomposing the Maxwell\nmultiplet with respect to the unbroken supersymmetry on the brane wrapped on S1ˆS3. In\nparticular, they are organized in multiplet of the bosonic algebra SUp2qJˆSUp2q¯JˆSUp2qRˆ\nSUp2q¯R(we omit Up1qfactors not relevant for our discussion). The index ℓ“0,1,2, . . .represents\nthe angular momentum after Kaluza-Klein expansion in S3spherical harmonics. Summing over the\nstates in the table we get\nζD3psq“Sℓ“0psq`Sℓ“1psq`8ÿ\nℓ“2Sℓpsq, S ℓpsq“ÿ\nfixed ℓp´1qFpH`J`¯Jq´s,(4.61)\nwhere Sℓpsqis a finite sum and we need to split ℓ“0andℓ“1because representations with\nnegative Ror¯Rshould be omitted. From Table 3 we compute\nζD3p´1q“´ 1`0`lim\nsÑ´18ÿ\nℓ“2rpℓ´1q´s´2ℓ´s`pℓ`1q´ss“´ 1`lim\nsÑ´1p1´2´sq“´ 2.(4.62)\nPlugging this in (4.60) we match (4.59). Of course this had to work a priori, but the aim of the\nabove discussion is to emphasize the origin of the supersymmetric Casimir energy as a regularized\nsum of the combination of HandJ,¯Jappearing in the index, see also [44] for wrapped M5 brane\nin the context of the 6d p2,0qtheory. Notice that the peculiar 1{qterm in (4.56), or “ ´1” term in\n(4.62), comes from the ℓ“0mode of the scalar ¯ϕ. This negative energy state is not a tachyon and\nis compatible with unbroked supersymmetry due to the non-trivial geometry of the wrapped brane.\n5 Flavored Schur index\nIn our approach, the introduction of flavor fugacities in the superconformal index does not pose\nconceptual problems. Still, it is instructive to work out the details and, in particular, how the extra\nfugacities are included in the operator counting partition functions (4.55). Consistently with the\nconstraint in (2.1), we take\npu1, u2, u3q“pu, u´1,1q. (5.1)\nAs a first remark, the effect of the twist related to the J,¯Jcharges in (2.5) does not interfere with\ntheu-dependent twists in the S3part of the brane worldvolume for the configuration Z“0. Indeed,\n20as discussed in [26] in the case of the M5 brane system, in a static gauge the νshift does not depend\nonuthat can be absorbed in the non-fluctuating coordinates of S3. This will not be true for the\nX“0andY“0configurations and will lead to a udependent correction to the shift ν.\nApart from the set of shifts ν, what remains to be found is the suitable u-dependent generaliza-\ntion of the single particle partition functions in (4.55) obtained in principle from the study of the\nN“4Maxwell multiplet fields on the 4d space S1\nβˆrS3whererS3is obtained from S5with twisted\nmetric\nds2\nrS5“3ÿ\ni“1„\ndn2\ni`n2\nipdφi`iαidτq2ȷ\n,3ÿ\ni“1n2\ni“1, (5.2)\nwhere the three mixing coefficients αiare related to uibyqαi“uiand the wrapped configurations\nX“0,Y“0,Z“0correspond to n1“0,n2“0,n3“0, respectively.\nA second remark concerns the classical factors uN,u´Nin (3.14). Given a wrapping with\nni“0, the 4-form (4.7) requires dzÑdz`iα dτwhere zis identified with the angle φiandα”αi.\nEvaluation of the classical D3 action reproduces those factors from the Wess-Zumino part Scl\nWZ.\n5.1 Single particle partition functions of twisted fields in S1\nβˆS3\nWhat remains to be done is to compute the modified pZpq;νqÑpZpq, u;νqin (4.55). Again, this has\na simple operator counting derivation. Before discussing that, let us work out explicitly the result\nfor a scalar field to show the structure of the expected result. By a relabeling of coordinates we can\nwrite the twisted rS3metric (in all configurations) as\nds2\nrS3“dn2\n1`dn2\n2`n2\n1pdφ1`iαdτq2`n2\n2pdφ2`iα1dτq2, n1\n2`n2\n2“1.(5.3)\nChanging coordinates, this gives the metric of S1\nβˆrS3as\nds2\n1\nβˆrS3“dτ2`dχ2`cos2χpdφ1`iαdτq2`sin2χpdφ2`iα1dτq2. (5.4)\nA convenient basis for the eigenfunctions of the Laplacian on the standard S3is (see, e.g., [45])\nΦp,r1,r2“fp,r1,r2pχqeipr1`r2qφ1eipr2´r1qφ2, r 1, r2“´1\n2p, . . . ,1\n2p , p“0,1, ... . (5.5)\nUsing this representation for the wave-functions after a redefinition of φ1, φ2that momentarily\nabsorbs τin the mixing terms, we get the scalar free energy17\nF“1\n2log detp´∇2`1q\n“1\n28ÿ\np“0p{2ÿ\nr1,r2“´p{2log”`\nnβ`iαpr1`r2q`iα1pr1´r2q˘2`ppp`2q`1ı\n.(5.6)\nThe corresponding thermal partition function and the single-particle partition function are then\n¯F“8ÿ\np“0p{2ÿ\nr1,r2“´p{2logr1´e´βpp`1`pαpr1`r2q`α1pr1´r2qqs (5.7)\n17For the standard Laplacian on S3we have λp“ppp`2qanddp“pp`1q2and for a conformally coupled scalar\nwe need to add 1 to λp.\n21and\npZϕpq, u, u1q“8ÿ\np“0p{2ÿ\nr1,r2“´p{2qp`1`αpr1`r2q`α1pr1´r2q“q´q3\np1´u qqp1´u´1qqp1´u1qqp1´u1´1qq,\n(5.8)\nwhere we denoted\nu“qα, u1“qα1. (5.9)\nThis expression admits a simple derivation from operator counting in RˆrS3where we assign the\nweights u, u´1to two vector indices and u1, u1´1to the other two. For a scalar field, the numerator\nof (5.8) corresponds to ϕand the subtraction of the equations of motion B2ϕ. Both are scalar and\nno factors u, u1are necessary. The denominator factors corresponds to derivatives Bµand they have\none vector index with the above weights. For a spinor, one finds the same effect in derivatives plus\ntheu, u1factors associated with the spinor indices (same for ψand{Bψ)\npZψpq, u, u1q“pu1{2`u´1{2qpu11{2`u1´1{2qq3{2p1´qq\np1´u qqp1´u´1qqp1´u1qqp1´u1´1qq.(5.10)\nFinally, for the gauge field, the result is\npZVpq, u, u1q“c1pu, u1qq2´c2pu, u1qq3`2q4\np1´u qqp1´u´1qqp1´u1qqp1´u1´1qq,\nc1pu, u1q“2`pu`u´1qpu1`u1´1q\nc2pu, u1q“2pu`u´1`u1`u1´1q.(5.11)\nHere, the q2term in the numerator corresponds to six Fµνfields with the u, u1weights carried by\nthe two vector indices\nfield F12F13 F14 F23 F24 F34\nweight 1uu1uu1´1u´1u1u´1u1´11\nwith total equal to c1pu, u1q. The q3term is the subtraction of overcounted derivatives on the\nvanishing vanishing BµFµν“0andBµF‹\nµν“0. These have one vector index and give total c2pu, u1q.\nFinally, the 2q4term accounts for the overcounted derivatives BµBνFµν“BµBνF‹\nµν“0. These have\nno free indices and thus no u, u,1weights.\nUsing the above considerations we can discuss now separately the contribution from the Z“0\nbrane, symmetric under uÑu´1, and those of the X“0,Y“0branes, related by uÑu´1,cf.\n(3.12).\n5.2 Z“0brane contribution\nLet us consider the D3 brane wrapped on Z“0. Its single particle index reads18\npID3\nZpq, uq“q´1´1´pu`u´1qq`3q2´q3\np1´uqqp1´u´1qq. (5.12)\n18Use the general expression (3.4) after permutation of fugacities or take y“v“η“1in the last equation in\nEq. (4.35) of [5].\n22This configuration does not contribute the Schur index, consistently with cf.(3.14). This is due to\nthe fermionic term ´1in the expansion of pID3,Z“0\npID3\nZpq, uq“q´2`u`u´1´1`Opqq, (5.13)\nthat gives a zero after plethystic, see [5] for more discussions. Nevertheless, it is instructive to derive\n(5.12) – thus at the level of single particle index which is non-vanishing. One can check that the\ngeneralization of (4.56) is\npZscalars\nZpq, uq`pZfermions\nZpq, uq`pZV\nZpq, uq“pID3\nZpq, uq, (5.14)\nwith\npZscalars\nZpq, uq“2rpZϕ\nZpq, u; 0q`pZϕ\nZpq, u; 1q`pZϕ\nZpq, u; 2qs,\npZfermions\nZpq, uq“´ 2rpZψ\nZpq, u;1\n2q`pZψ\nZpq, u;3\n2qs,(5.15)\nand\npZϕ\nZpq, u;νq“qp1´q2q\np1´uqq2p1´u´1qq2qν`q´ν\n2,\npZψ\nZpq, u;νq“pu`2`u´1qq3{2p1´qq\np1´uqq2p1´u´1qq2qν`q´ν\n2,\npZV\nZpq, uq“pu2`4`u´2qq2´4pu`u´1qq3`2q4\np1´uqq2p1´u´1qq2,(5.16)\nwhich are the single particle partition functions in (5.8), (5.10), and (5.11), with u1“u´1, dressed\nby the universal factor taking into account the νshifts in Table 1 for scalars and fermions.19\n5.3 X“0brane contribution\nIn this case, we have to set u1“1and the single particle partition functions to be used are\npZϕ\nXpq, u;νq“qp1´q2q\np1´uqqp1´u´1qqp1´qq2qν`q´ν\n2,\npZψ\nXpq, u;νq“2pu1{2`u´1{2qq3{2p1´qq\np1´uqqp1´u´1qqp1´qq2qν`q´ν\n2,\npZV\nXpq, uq“2pu`1`u´1qq2´2pu`2`u´1qq3`2q4\np1´uqqp1´u´1qqp1´qq2.(5.17)\nFor this configuration, in the analysis of the fluctuations in S5, we have a non-trivial mixing in the\nWZ term that will correct the value of ν. This is due to dzÑdz`iαdτwith qα“uin\nds2\nrS5“dw2`cos2w dS 3`sin2wpdz`iαdτq2,\nC4“iT3cos4wpdz`iαdτq^volrS3.(5.18)\nRepeatingthecalculationinsection4.3.1withthereplacement gabѯgabpuq(thetwisted S3metric),\nwe have now\nds2\nrS5“rp1´YcYcq¯gab`BaYcBbYcsdξadξb\n`r 9Yc9Yc`2iαpY19Y2´Y29Y1q´α2YcYcspdξ1q2`off diagonal terms ,(5.19)\n19For the Z“0brane, there is no WZ contribution nor DBI mixing u-dependent contribution.\n23that gives the following quadratic expansion of the DBI action\nδSDBI“1\n2βT3ż\nd3ξ?¯gr¯gabBaYcBbYc`9Yc9Yc`4i α Y 19Y2´p3`µ2qYcYcs.(5.20)\nThe WZ term is\nSWZ“´iT3ż\ncos4wpdz`iαdτq^volrS3“´iT3ż\np1´2YcYcqˆY1dY2´Y2dY1\nYcYc`iαdτ˙\n^volrS3,\n(5.21)\nand thus\nδSWZ“βT3ż\nd3ξ?¯gp4iY19Y2´2αYcYcq. (5.22)\nThe total δS“δSDBI`δSWZis\nδS“1\n2βT3ż\nd3ξ?¯gr¯gabBaYcBbYc`9Yc9Yc´p1`αqp3`αqYcYc`4ip2`αqY19Y2s.(5.23)\nIntroducing η“Y1`iY2?\n2,¯η“Y1´iY2?\n2, we get finally\nδS“βT3ż\nd3ξ?grgabBa¯ηBbη`¯ηp´p1`µqp3`µqη`2pµ`2q9η´:ηqs. (5.24)\nThis corresponds to a squared mass for the complex field ηgiven by the expressions\nM2“1`pnβ`ip2`αqq2. (5.25)\nA similar αdependent correction to νis also present in fermions and the generalization of relation\n(4.56) reads now\npZscalars\nXpq, uq`pZfermions\nXpq, uq`pZV\nXpq, uq“pID3\nXpq, uq, (5.26)\nwith\npZscalars\nXpq, uq“2rpZϕ\nXpq, u; 0q`pZϕ\nXpq, u; 1q`pZϕ\nXpq, u; 2`αqs,\npZfermions\nXpq, uq“´ 2rpZψ\nXpq, u;1\n2`αq`pZψ\nXpq, u;3\n2`αqs,(5.27)\nThe r.h.s. in (5.26) was given in (3.10) and equality is easily checked using the explicit expressions\n(5.17).\nWe remark that in this case the small qexpansion of the single particle brane index reads\npID3\nXpq, uq“1\nuq`1\nu2`Opqq, (5.28)\nand the term 1{u2admits a finite plethystic (with singular uÑ1limit) regularizing the zero mode\ncontribution discussed in the unflavored case.\n245.4 Supersymmetric Casimir energy\nIt is interesting to look at the supersymmetric Casimir energy on the wrapped brane in presence\nof the fugacity u‰1. Let us consider as an illustrative example the Z“0configuration. We can\nwrite the single particle index in (5.12) as20\npID3\nZpq, uq“1\nq`u`u´1´1´q`p1´uq28ÿ\np“1p1`u2pqqp\nup`1. (5.29)\nThe associated zeta-function on the brane, cf.(4.60), is then\nζD3psq“p´ 1q´s´1`p1´uq28ÿ\np“1p1`u2pqp´s\nup`1“p´ 1q´s´1`pu`u´1´2qpLispuq`Lispu´1qq.\n(5.30)\nSetting s“´1and using Li ´1pzq“z\np1´zq2gives\nζD3p´1q“´ 2`pu`u´1´2qˆu\np1´uq2`u´1\np1´u´1q2˙\n“0,Ñ ED3\nc“0.(5.31)\nThis vanishing implies that the two limits sÑ´1anduÑ1do not commute. A similar vanishing\nis obtained for the X“0(and Weyl reflected Y“0) configurations.\nAcknowledgements\nWe thank Arkady A. Tseytlin, Yosuke Imamura, and Ji Hoon Lee for useful discussions related to\nvarious aspects of this work. MB and ACB are supported by the INFN grants GSS and GAST.\nA Special functions\nWe collect in this appendix the definition of the special functions appearing in the text and some\nuseful identities.\nDedekind eta function\nηpτq“q1\n128ź\nk“1p1´q2kq, q“eiπτ. (A.1)\nTheta functions\nϑ1pz, qq“2q1{4sinz8ź\nk“1p1´q2kqp1´2q2kcosp2zq`q4kq,\nϑ2pz, qq“2q1{4cosz8ź\nk“1p1´q2kqp1`2q2kcosp2zq`q4kq,\nϑ3pz, qq“8ź\nk“1p1´q2kqp1`2q2k´1cosp2zq`q4k´2q,\nϑ4pz, qq“8ź\nk“1p1´q2kqp1´2q2k´1cosp2zq`q4k´2q.(A.2)\n20Notice that q1gets two contributions, one from the sum and one from ´qbefore it.\n25q-Pochhammer\npa, qq8“8ź\nk“0p1´a qkq. (A.3)\nNotice that we can write the Dedekind function in (A.1) as\nηpτq“q1\n12pq2, q2q8. (A.4)\nB Doubly wrapped D3 brane index in closed form and analytic\ncontinuation\nIn this appendix we discuss the evaluation of the doubly wrapped D3 brane index ID3\nX,2pq, uq. We\nwill get it by computing the exact refined index for N“4Up2qSYM and applying (3.13) to the\nresult. This has to be compared with the approach in [7] where analytic continuation is done at\nthe level of the single particle index, thus requiring some ad hoc procedure to evaluate holonomy\nintegrals.\nThe refined Schur index in N“4UpNqSYM is given by the holonomy integral\nISYM\nNpq, uq“1\nN!¿\n|zn|“1Nź\nn“1dzn\n2πiznź\nn‰mˆ\n1´zn\nzm˙\nPE„\npIMaxwellpq, uqNÿ\nn,m“1zn\nzmȷ\n,(B.1)\nwhere,cf.(2.16),\npIMaxwellpq, uq“qpu`u´1q´2q2\n1´q2, (B.2)\nandznare gauge fugacities for the Maxwell multiplet in adjoint representation. In the N“2case\none of the two integrations trivializes and we can set from the start z1Ñz,z2Ñ1. This gives\nISYM\n2pq, uq“¿\n|z|“1dz\n2πiFpz;q, uq, (B.3)\nwith\nFpz;q, uq“1\n2zPE„\n2´p1´q\nuqp1´quq\np1´q2qpz`2`z´1qȷ\n. (B.4)\nAfter some rearrangement this can be written\nFpz;q, uq“´pq2;q2q4\n8p1\nz;q2q2\n8pz;q2q2\n8\n2p1´zq2pq\nu;q2q28pqu;q2q28pq\nuz;q2q8pqu\nz;q2q8pqz\nu;q2q8pquz;q2q8.(B.5)\nThe index of the D3 brane wrapped ktimes over X“0is given by (3.13). The problem is that if\nwe start with a q-series for ISYM\nkpq, uq, the transformation\nqÑq1{2u´1{2, uÑq´3{2u´1{2, (B.6)\nwill convert it into a series with an infinite number of singular terms 1{qnmixed with 1{umand\nthese have to be resummed before re-expanding at small qand|u|ă1. The strategy adopted in\n[7] was based on applying (B.6) to Fpz;q, uq, expand in small qand then deform the integration\n26contour oftznuin order to pick the relevant poles, no more in the circle |z|“1. In addition, when\ntaking the qexpansion of Fpz;q1{2u´1{2, q´3{2u´1{2qsome spurious poles collapse at z“0and have\nto be subtracted by hand. Here, we adopt a direct approach and evaluate (B.3) in closed form.\nThe result will be expressed in terms of elliptic functions and the continuation (B.6) will be easy to\nperform exploiting their symmetries. After that, the small qexpansion can be safely done.\nAs a first step, we use the triple product representation\nϑ4pu, q1{2q“pq, qq8px, qq8pq{x, qq8, x“q1{2e2iu, (B.7)\nand the shift relation\nϑ1pu, qq“ie´iuq1{4ϑ4pu´π\n2τ, qq, (B.8)\nwhere our convention is q“eiπτas in (A.1). If we parametrize\nz“e2πix, u“e2πiU, (B.9)\nwe get\nISYM\n2pq, uq“1\n2qηpτq6\nϑ4pπU, qq2¿dz\n2πizϑ1pπx, qq2\nϑ4pπx´πU, qqϑ4pπx`πU, qq. (B.10)\nThis integral has been evaluated in [46] (or using the methods of [30], who treated the very similar\nSUp2qcase) and the results reads ( ϑ1\napu, qq“B uϑapu, qq)\nISYM\n2pq, uq“1\n2qηpτq3\nϑ4pπU, qqϑ1\n4pπU, qq\nϑ1p2πU, qq. (B.11)\nAs a check we can write in uÑ1limit (or UÑ0)\nISYM\n2pq, uq“1\nϑ4p0, qqp1´4q3`9q8´16q15`25q24`¨¨¨q“1\nϑ4p0, qq8ÿ\nk“0p´1qkpk`1q2qkpk`2q,(B.12)\nin agreement with Eq.(21) of [29]. To this expression, we can apply the analytic continuation (B.6)\nand get, from (3.13),\nID3\nX,2pq, uq“ISYM\n2pq1{2u´1{2, q´3{2u´1{2q“1\n2pu\nqq3{8pq\nu,q\nuq3\n8ϑ1\n4p2πU`3π\n2τ1,?q?uq\nϑ1p4πU`3πτ1,?q?uqϑ4p2πU`3π\n2τ1,?q?uq,\n(B.13)\nwhere τ1is such that?q{?u“eiπτ1. Now, we use the general formula\nϑ1pu`mπτ, qq“p´ 1qmq´me´ip2mu`mpm´1qπτqϑ1pu, qq, (B.14)\nto get\nϑ1pu`3πτ, qq“´ q´9e´6iuϑ1pu, qq. (B.15)\nAlso, from (B.8) and (B.14), one easily obtains\nϑ4pu`3π\n2τ, qq“´ ie´3iuq´9{4ϑ1pu, qq. (B.16)\nDifferentiating with respect to ugives also\nϑ1\n4pu`3π\n2τ, qq“´ e´3iuq´9{4r3ϑ1pu, qq`iϑ1\n1pu, qqs. (B.17)\n27From these relations, we obtain the following closed formula for the 2-wrapped D3 brane index\nID3\nX,2pq, uq“i\n2q33{8u63{8pq\nu,q\nuq3\n83ϑ1p2πU,?q?uq`iϑ1\n1p2πU,?q?uq\nϑ1p2πU,?q?uqϑ1p4πU,?q?uq(B.18)\nExpanding this in powers of qgives the following terms\nID3\nX,2pq, uq“u10p2´u2qq4\np1´u2qp1´u4q`u5p2´u4qq5`p2`2u6´u12qq6\n`ˆ2\nu5`3u7´u15˙\nq7`ˆ\n3`2\nu10`3u8´u18˙\nq8`ˆ2\nu15`1\nu3`4u9´u21˙\nq9\n`ˆ\n3`2\nu20`4\nu6`4u10´u24˙\nq10`¨¨¨ , (B.19)\nin full agreement with the third line in Eq. (18) of [7].\nReferences\n[1] J. M. Maldacena and A. Strominger, AdS3Black Holes and a Stringy Exclusion Principle ,\nJHEP12(1998) 005 [ hep-th/9804085 ].\n[2] J. McGreevy, L. Susskind and N. Toumbas, Invasion of the Giant Gravitons from\nAnti-de Sitter Space ,JHEP06(2000) 008 [ hep-th/0003075 ].\n[3] J. Kinney, J. M. Maldacena, S. Minwalla and S. Raju, An Index for 4 Dimensional Super\nConformal Theories ,Commun. Math. Phys. 275(2007) 209 [ hep-th/0510251 ].\n[4] D. Gaiotto and J. H. 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Sauer,‡and Jacob Kongsted†\n†Department of Physics, Chemistry and Pharmacy, University of Southern Denmark,\nCampusvej 55, DK–5230 Odense M, Denmark\n‡Department of Chemistry, University of Copenhagen, DK-2100 Copenhagen Ø.\n¶Department of Chemistry, Technical University of Denmark, Kemitorvet Building 207,\nDK-2800 Kongens Lyngby, Denmark.\nE-mail: reinholdt@sdu.dk\nAbstract\nWe explore Davidson methods for obtaining excitation energies and other linear\nresponse properties within quantum self-consistent linear response (q-sc-LR) theory.\nDavidson-type methods allow for obtaining only a few selected excitation energies with-\nout explicitly constructing the electronic Hessian since they only require the ability to\nperform Hessian-vector multiplications. We apply the Davidson method to calculate\nthe excitation energies of hydrogen chains (up to H 10) and analyze aspects of statis-\ntical noise for computing excitation energies on quantum simulators. Additionally,\nwe apply Davidson methods for computing linear response properties such as static\npolarizabilities for H 2, LiH, H 2O, OH–, and NH 3, and show that unitary coupled clus-\nter outperforms classical projected coupled cluster for molecular systems with strong\n1arXiv:2402.12186v1  [physics.chem-ph]  19 Feb 2024correlation. Finally, we formulate the Davidson method for damped (complex) linear\nresponse, with application to the nitrogen K-edge X-ray absorption of ammonia, and\ntheC6coefficients of H 2, LiH, H 2O, OH–, and NH 3.\n1 Introduction\nQuantum computing is an emerging technology that may solve certain computational prob-\nlems significantly faster than classical computer architectures. One of the most promising\nareas of application is within quantum chemistry, where quantum computers have been pro-\nposed for solving the electronic Schr¨ odinger equation.1–3However, current available quantum\nhardware, i.e., near-term noisy intermediate-scale quantum (NISQ) devices, are fairly lim-\nited in their usefulness due to relatively small qubit counts and their susceptibility to noise,\nwhich prevents meaningful execution of deep quantum circuits required for the most promis-\ning quantum algorithms.4,5\nThe variational quantum eigensolver (VQE) approach6,7attempts to circumvent this\nproblem with a hybrid quantum-classical workflow in which quantum and classical processors\ncooperate. In this approach, the quantum processor evaluates relatively shallow quantum\ncircuits, mainly involving preparing quantum states and evaluating expectation values over\ncertain quantum mechanical operators (most notably the Hamiltonian) while the classical\ncomputer is used for obtaining the fermionic Hamiltonian and updating the parameters of the\nprepared wave function. The VQE method estimates the energy from the Rayleigh quotient\nas\nE(θ) =D\nΨ(θ)\f\f\fˆH\f\f\fΨ(θ)E\n⟨Ψ(θ)|Ψ(θ)⟩(1)\nwhere the wave function depends on a set of parameters (or angles) θ. The quantum processor\nis used to prepare a state |Ψ(θ)⟩and to measure terms of the Hamiltonian. The classical\nprocessor is then used to collect and post-process the obtained results and to update the\nparameters θ.\n2Much attention has been paid to obtaining ground-state energies with the VQE approach.\nHowever, many molecular properties require a description of the excited states. For example,\nabsorption spectra can be obtained from excitation energies and oscillator strengths of a\nmolecular system. Several approaches for obtaining excitation energies have emerged in\nrecent years.8–15In this work, we will focus on the quantum linear response (qLR) method,\nwhich is similar to the quantum equation of motion (qEOM) of Ollitrault et al..11The qEOM\nmethod is an extension of the well-known classical EOM16–18approach that can potentially\nleverage near-term, noisy quantum hardware to obtain excitation energies of molecules. After\npreparing and optimizing a ground state using VQE, the method uses a quantum processor\nto measure individual matrix elements of an EOM secular equation. The LR equations are\nthen solved on a classical computer. The qEOM method was refined by Asthana et al. with\nthe q-sc-EOM19method, which introduced a self-consistent excitation operator manifold\noriginally developed by Prasad et al.20in the context of self-consistent propagator theories.\nThese operators satisfy the vacuum annihilation condition and were shown to provide more\naccurate excitation energies, ionization potentials, and electron affinities than the original\nqEOM approach.19The self-consistent operators have also been applied to obtain linear\nresponse properties within the self-consistent quantum linear response q-sc-LR approach.21\nRecently, some of us have also explored the feasibility of other operator transformations for\nqLR formalisms.14\nWhen targeting larger molecules, the explicit construction of the Hessian can become a\ncomputational bottleneck in terms of quantum resources since many matrix elements need\nto be measured. Eventually, the classical processing costs may also become significant when\nthe cost of matrix diagonalization becomes prohibitive. The same problem is faced in many\nclassical linear response approaches, where Davidson-type methods22are commonly applied\nto overcome this issue. Resorting to Davidson-type approaches allows us to obtain excita-\ntion energies (and linear response properties) without explicitly constructing the electronic\nHessian, as they only require the ability to perform Hessian-vector multiplications.\n3The Davidson method22has already been explored with q-sc-EOM by Kim and Krylov,23\nwhere it was used for the simulation of excitation energies of H 2, H4, LiH, and (frozen-core)\nH2O molecules in an STO-3G basis. In the work by Kim and Krylov, elements of the subspace\nmatrix are directly measured (without the addition of ancillary qubits), which entails the\nevaluation of transition matrix elements of the UCC Hamiltonian between guess vectors and\nexcited Slater determinants. The implementation in Ref. 23 operates and manipulates state\nvectors on a classical computer. At first glance, this might appear problematic since the\ndimension of state vectors scales exponentially with the number of qubits. However, by\nusing sparse vectors and truncating the excitation level in the EOM expansion, the number\nof classically stored elements is only polynomial.\nIn this work, we use an approach that does not require explicitly computing transition\nmatrix elements of the UCC Hamiltonian. Instead, we formulate the Davidson method in\nterms of Hessian-vector products, which only require gradients over excitation operators.\nWe investigate using Davidson methods for obtaining excitation energies and implement\na Davidson solver for linear response properties within the quantum self-consistent linear\nresponse unitary coupled cluster approach.\nThe rest of the paper is organized as follows. In section 2, we briefly review the fun-\ndamental theory describing the VQE method for unitary coupled cluster. Next, we outline\nthe central equations in q-sc-LR and formulate the associated Davidson method in terms\nof Hessian-vector multiplications for both excitation energies in section 2.1, linear response\nproperties in section 2.2, and damped response theory in 2.3. In section 3, we provide com-\nputational details. In section 4.1, we report on the excitation energies of hydrogen chains\n(up to H 10) and analyze aspects of the statistical noise in section 4.2. In section 4.3, we\ncompute static polarizabilities for H 2, LiH, H 2O, OH–, and NH 3, and show that unitary\ncoupled cluster outperforms classical projected coupled cluster for molecular systems with\nstrong static correlation. Finally, in section 4.4, we use our formulation of the Davidson\nmethod for damped (complex) linear response, with application on the nitrogen K-edge X-\n4ray absorption of ammonia, while we compute C6coefficients in section 4.5 (which entail\ncomputations of the real electric dipole polarizability at imaginary frequencies) of H 2, LiH,\nH2O, OH–, and NH 3.\n2 Theory\nThe unitary coupled cluster wave function is parametrized as\n|ΨUCC⟩=eˆT(θ)−ˆT†(θ)|0⟩ (2)\nwith the cluster operator, ˆT, being defined as:\nˆT(θ) =X\nnˆTn(θ), (3)\nwhere nis the excitation order of the excitation operator ˆTnandθare the circuit parameters.\nIn this work, the coupled cluster expansion is truncated at single and double excitations and\nuses spin-adapted operators of the form:24–26\nˆT1(θ) =X\niaθa\ni1√\n2ˆEai (4)\nˆT2(θ) =X\ni≥j,a≥bθab\nij1\n2p\n(1 +δab) (1 + δij)\u0010\nˆEaiˆEbj+ˆEajˆEbi\u0011\n+X\ni≥j,a≥bθ′ab\nij1\n2√\n3\u0010\nˆEaiˆEbj−ˆEajˆEbi\u0011\n(5)\nHere, ˆEpq= ˆa†\np,αˆaq,α+ ˆa†\np,βˆaq,βis the singlet excitation operator. To find the parameters of\nthe ground state, the electronic energy is minimized with respect to θ, which is obtained\nfrom the electronic Hamiltonian:\nˆH=X\npqhpqˆEpq+1\n2X\npqrsgpqrsˆepqrs. (6)\n5Here, ˆ epqrs=ˆEpqˆErs−δqrˆEpsis the two-electron excitation operator, hpqare one-electron\nmolecular orbital integrals, and gpqrsare two-electron molecular orbital integrals. In the VQE\nmethod,6the circuit parameters are obtained by variational minimization of the energy\nEgs= min\nθD\nΨUCC(θ)\f\f\fˆH\f\f\fΨUCC(θ)E\n. (7)\nAfter a ground state circuit has been prepared and optimized, excitation energies and\nassociated transition properties can be obtained with the quantum linear response approach.\nIn qLR, excitation energies are obtained by classically solving a generalized eigenvalue prob-\nlem that takes the form\nE[2]Xk=E0kS[2]Xk, (8)\nwhere kindexes each of the Nexcitation energies. The generalized eigenvalue problem has\nthe following block structure\n\nA B\nB∗A∗\n\nYk\nZk\n=E0k\nΣ ∆\n−∆∗−Σ∗\n\nYk\nZk\n. (9)\nThe individual blocks contain elements defined as\nAij=D\nΨ\f\f\fh\nˆG†\ni,h\nˆH,ˆGjii\f\f\fΨE\n, (10)\nBij=D\nΨ\f\f\fh\nˆGi,h\nˆH,ˆGjii\f\f\fΨE\n, (11)\nΣij=D\nΨ\f\f\fh\nˆG†\ni,ˆGji\f\f\fΨE\n, (12)\n∆ij=D\nΨ\f\f\fh\nˆGi,ˆGji\f\f\fΨE\n. (13)\nThe ˆGoperators are the excitation operators, which in this work are spin-adapted just\nas the singles and doubles operators in the wavefunction parameterization.This parame-\nterization restricts the calculated excitation energies to be singlet excitations and reduces\n6the dimensionality of the generalized eigenvalue problem compared to using spin-conserving\noperators.\nBy replacing the naive excitation operators, ˆG, with their self-consistent manifold\nˆGsc=UˆGU†(14)\nthe overlap matrix becomes diagonal ( ∆=0andΣ=I), and the off-diagonal blocks of the\nHessian become zero ( B=0). This is known as q-sc-EOM or q-sc-LR19,21and simplifies\nEq. (9) to a regular eigenvalue problem\nAY k=E0kYk. (15)\nThe matrix elements of the eigenvalue problem are evaluated using a quantum computer.\nAs written, one needs to measure every element of the Amatrix, after which the eigen-\nvalues can be obtained by classical diagonalization. However, when only a few eigenvalues\n(and associated eigenvectors) of Eq. (15) are required, evaluating the complete matrix is\nnot strictly necessary. Instead, eigenvalues can be obtained with Davidson methods,22,27,28\nwhich only entail the action of Hessian-vector multiplication for arbitrary trial vectors. Such\nmethods are widespread28in classical quantum chemistry for obtaining excitation energies\nand solving linear response equations.\nThe Hessian-vector product σb=Abfor a trial vector bcan be approximated using the\ngradient gin a central finite-difference approach as29\nσb=Ab≈g(θ,+hb)−g(θ,−hb)\n2h, (16)\nwhere his a small positive parameter. The linear transformation in Eq. (16) is a central part\nof matrix-free algorithms that can extract a few selected excitation energies of a molecule. To\nperform the Hessian-vector multiplication in Eq. (16), we compute gradients of the excitation\n7parameters for the states |Ψ(θ, hb)⟩and|Ψ(θ,−hb)⟩. Explicitly, we compute gradients in\ntheΘparameters on states that are defined as\n|Ψ(θ,Θ)⟩=U(θ)U(Θ)|0⟩, (17)\nwithΘ=±hb. On noiseless simulators, such a finite difference approach can work without\nissues for small values of the displacement parameter h. Realizing more efficient or noise-\nresilient implementations, e.g., through clever contractions of the trial vectors or deriving\nparameter-shift rules, will be part of future research. We note that a similar approach\nwas proposed by Parrish et al. to compute state-averaged VQE (SA-VQE) Hessian-vector\nproducts.30\n2.1 The Davidson algorithm for excitation energies\nThe Davidson method solves the eigenvalue problem (Eq. (15)) using only Hessian-vector\ntransformations. In this method, the Amatrix is projected onto a subspace B. The projected\nHessian is diagonalized, and the subspace is expanded iteratively until convergence. The goal\nof the algorithm is to extract a small number ( K) of eigenvalues and associated eigenvectors\nof the full eigenvalue problem.\nAn initial set of normalized guess vectors B={b1,b2,···,bL}with L≤Nis generated.\nThe subspace dimension ( L) is typically much smaller than the full-space dimension ( N).\nOur implementation generates initial guess vectors based on orbital energy differences.31The\nHessian-vector products σbare then computed for each vector binBusing Eq. (16) and\nare collected in the matrix σB. Next, the projected Hessian matrix is evaluated as\nAB=B†AB=B†σB, (18)\nAs the dimension of ABis much smaller than that of the full Hessian, it is readily diagonalized\nusing classical linear algebra routines to obtain a set of eigenvectors zk(collected in the matrix\n8Z) and eigenvalues λk, of which only the Klowest are kept. The (approximate) eigenvector\nmatrix Xof the excitation vectors are followingly constructed as\nX=BZ (19)\nNext, residuals for the Kdesired roots are computed as\nrk= (A−λkI)Xk (20)\n=ABZk−λkIXk\n=σBZk−λkIXk\nIf the norm of the residual vectors is below some pre-defined threshold, the algorithm termi-\nnates, yielding λkas the eigenvalues and Xkas the corresponding eigenvectors. Otherwise,\ncorrection vectors are computed as\nδki= (λk−Aii)−1rki (21)\nIn the equation above, Aiiis a diagonal element of the Hessian or some suitable approxi-\nmation. We use a simple CI-inspired approximation using orbital energy differences in our\nimplementation. If the norm of a correction vector is above some pre-defined threshold, the\nvector is normalized, orthogonalized to the other trial vectors, and appended to B.\n2.2 Linear response with the Davidson method\nIn frequency-dependent linear response theory with self-consistent operators, the following\nequation for a perturbation Bwith frequency ωis solved\n\u0000\nE[2]−ωS[2]\u0001\nXB(ω) =VB. (22)\n9Here, VBis the property gradient, ωis the frequency, and the elements of the Hessian and\ngeneralized overlap matrices are defined as in Eq. (9). An element of property gradient can\nbe evaluated as a derivative\nVB,i=D\nΨ\f\f\fh\nˆB,ˆGii\f\f\fΨE\n=∂\n∂GiD\nΨ(θ,0)\f\f\fˆB\f\f\fΨ(θ,0)E\n(23)\nWhen applying self-consistent operators, the full linear system of Eq. (22) reduces to two\nindependent (half-dimension) equations\n(A−ωI)YB(ω) =VB, (24)\n(A+ωI)ZB(ω) =−V∗\nB. (25)\nSolving these response equations gives response vectors YBandZBwhich can be used to\ncompute linear response properties as\n⟨⟨A;B⟩⟩ω=VA·YB(ω) +V∗\nA·ZB(ω). (26)\nStatic properties are recovered in the limit ω→0.\nThe Davidson method can be straightforwardly generalized from solving the eigenvalue\nproblem (Eq. (9)) to solving the linear equation in Eq. (22). We use the same setup of\ndefining a subspace B, onto which the linear response equation is projected\n(AB−ωI)XB,B(ω) =VB,B. (27)\nHere, the projected Hessian and property vector are defined as\nAB=B†AB (28)\nVB,B=B†VB (29)\n10The solution vector can be lifted from the subspace to the full space as\nXB(ω) =BXB,B(ω). (30)\nThe general flow of the linear response Davidson routine is almost identical to the eigen-\nvalue case. Given an initial subspace B, the projected linear equation (Eq. (27)) is solved to\nobtain the subspace solution vector XB,B(ω). The residual is then computed as\nr=VB−(A−ωI)XB\n=VB−(A−ωI)BXB,B\n=VB−(ABXB,B−ωIBXB,B)\n=VB−(σBXB,B−ωIXB) (31)\nThe solution vector is returned if the residual norm is below some tolerance. Otherwise, a\nnew trial vector is added to the subspace from the preconditioned residual correction vector\nδi= (Aii−ω)−1ri (32)\nwhich is normalized, orthogonalized to B, and appended to the subspace vectors. When\napplying the self-consistent operator manifold, we note that the generalized overlap matrix\nis a unit matrix, while in a more general case, one would also need to project the generalized\noverlap matrix. One of the major ingredients in the Davidson solver is, therefore, still the\ncomputation of Hessian-vector products with arbitrary trial vectors, for which the approach\nin Eq. (16) remains applicable. The gradient vector, preconditioned as in Eq. (32), is often\nused as a start vector in the iterative procedure to solve the non-homogeneous linear response\nequations.\nLinear response functions can be used to obtain many molecular properties. In this work,\n11we focus on the polarizability, which is equal to (minus) the dipole–dipole response function\nαij(ω) =−⟨⟨ˆµi; ˆµj⟩⟩ω, (33)\nwith iandjreferring to cartesian components.\n2.3 Damped response\nThe linear response function in Eq. (26) has poles (divergences) at frequencies corresponding\nto excitation energies, which can easily be seen from the spectral representation of linear\nresponse properties\n⟨⟨A;B⟩⟩ω=−X\nk̸=0\nD\n0\f\f\fˆA\f\f\fkED\nk\f\f\fˆB\f\f\f0E\nωk−ω+D\n0\f\f\fˆB\f\f\fkED\nk\f\f\fˆA\f\f\f0E\nωk+ω\n, (34)\nwhere divergences happen when ω=±ωk. Linear response theory can also be formulated\nin a framework that is convergent at all frequencies, with the approach known as resonant-\nconvergent damped response theory, also known as the complex polarization propagator.32–38\nIn damped response theory, an imaginary damping parameter ( iγ) is introduced, and the\npolarizabilities become complex-valued but finite at all frequencies. In the sum-over-states\nrepresentation, the introduction of γleads to\n⟨⟨A;B⟩⟩ω,γ=−X\nk̸=0\nD\n0\f\f\fˆA\f\f\fkED\nk\f\f\fˆB\f\f\f0E\nωk−(ω+iγ)+D\n0\f\f\fˆB\f\f\fkED\nk\f\f\fˆA\f\f\f0E\nωk+ (ω+iγ)\n, (35)\nwhich remains finite even at ω=±ωk. Practically speaking, the extension of regular\nfrequency-dependent linear response theory to damped response theory entails replacing\nthe real frequency ωwith ω+iγ, where γis a real parameter. The classical parts of the\nDavidson method can be carried out without issue in complex algebra, with the only require-\nment that transposes are replaced by the complex adjoint. This approach leads to complex\n12trial vectors, which can be decomposed into purely real and imaginary parts as\nb=w+iu. (36)\nA Hessian-vector multiplication with a complex trial vector can then be computed as\nσb=σw+iσu. (37)\nThe extension to damped response does not lead to any new types of “quantum” operations\nsince the standard Hessian-vector multiplication can be applied. We note that as an alter-\nnative to complex algebra, one could also decompose the solution vector as in Eq. (36) and\nintroduce this decomposition in the linear response equations, which allows the problem to\nbe formulated as a regular (real) problem with twice the size.\nWith a complex linear response code, several interesting properties can be extracted.\nAbsorption cross-sections can be extracted from the imaginary part of the complex dipole–\ndipole polarizability as39\nσ(ω) =ω\nϵ0cIm(αiso(ω)), (38)\nwhere the isotropic polarizability is the average of the diagonal components, αiso=1\n3(αxx+\nαyy+αzz).\nSeveral other molecular properties can also be extracted. For example, long-range orientation-\naveraged dipole-dipole dispersion C6coefficients between two systems, AandB, can be\nevaluated using the Casimir-Polder relation40,41\nCAB\n6=3ℏ\nπZ∞\n0αA\niso(iω)αB\niso(iω)dω, (39)\nwhere αA\niso(iω) is the isotropic polarizability of system Aevaluated at the purely imaginary\nfrequency iω. Following the procedure outlined in Ref. 41, the integration in Eq. (39) can\n13be performed using Gauss-Legendre quadrature by mapping the [0 ,∞] range to [ −1,1] via\na change of variables\nω=ω0\u00121−t\n1 +t\u0013\n(40)\nwhere an appropriate value of ω0is 0.3.41The interval transformation gives the Casimir-\nPolder integral as\nCAB\n6=3ℏ\nπZ1\n−12ω0\n(1 +t)2αA\niso(iω(t))αB\niso(iω(t))dt (41)\nThe integral is then evaluated with Gauss-Legendre quadrature as\nCAB\n6≈3ℏ\nπX\nkwk2ω0\n(1 +tk)2\b\nαA\niso(iω(tk))αB\niso(iω(tk))\t\n, (42)\nwhere wkandtkare quadrature nodes and weights, respectively, and ω(tk) =ω0(1−tk)/(1+\ntk). We use a 12-point Gauss-Legendre quadrature to approximate the integral.\n3 Computational Details\nExcitation energies of hydrogen chains in a ladder-like configuration (see Figure 1) were\ncomputed with an STO-3G42,43basis with UCCSD. We modified the existing UCCSD im-\nplementation from Pennylane44(version 0.34.0) to support spin-adapted (singlet) fermionic\nexcitation operators and to enable the computation of gradients of excitation operators. Our\ncode is publically available on Github.45The ground-state UCCSD wave functions were op-\ntimized using the SLSQP optimizer46from SciPy.47Quantum circuits were simulated using\nthe Pennylane Lightning state-vector simulator, with gradients evaluated using the adjoint\ndifferentiation48approach, unless otherwise noted. Hessian-vector products (Eq. (16)) were\nperformed with a step size h= 10−6for the noise-free simulations. One- and two-electron\nintegrals and Hartree-Fock wave functions were obtained with PySCF.49FCI calculations of\nthe excitation energies were carried out using the Dalton program.50All calculations were\ncarried out on a desktop computer equipped with an Intel i9-7980XE CPU.\n14Geometries of LiH, OH–, and H 2O were optimized at the FCI/6-31G51–53level of the-\nory using the Dalton program.50The geometry of NH 3was obtained from MP2/cc-pVTZ54\ncalculations with the Orca program.55All geometries are available in the supporting informa-\ntion. UCCSD calculations of the polarizabilities were carried out with the same settings as\ndescribed above. All calculations were carried out as full-space calculations, i.e., we did not\napply any frozen-core approximation. The CCSD and FCI polarizabilities were computed\nwith the Dalton program. A development version of Dalton37was used for the damped\nresponse CCSD calculations of the polarizability at imaginary frequencies used to obtain the\nC6coefficients.\n4 Results and Discussion\n4.1 Excitation energies\nTable 1 shows the lowest excitation energy and associated timings on H 2nhydrogen-ladder\nchains, using either FCI, UCCSD with explicit construction of the Hessian, or UCCSD using\nthe Davidson solver. The dimension of the subspace for the converged excitation energies\nis also reported. For the subspace approaches, the two lowest roots were converged. The\ngeometry of the hydrogen chains is sketched in Figure 1.\nFigure 1: The geometry of the H 2nchains, shown for 2 n= 6.\nThe lowest excitation energy varies considerably and somewhat unpredictably with the\nsize of the hydrogen chain. For example, the excitation energy for H 6is 14.14 eV, which\n15suddenly drops to 2.98 eV with H 8, only to return to 10.54 eV for H 10. The UCCSD excitation\nenergies computed with the explicit construction or the Davidson solver are identical to\nconvergence thresholds for all the considered excitations. The UCCSD excitation energies\ngenerally agree well with the FCI results, with an exact or near-exact agreement for the\nsmaller hydrogen chains. The largest error relative to FCI (0.37 eV deviation) occurs for H 8,\nbut the trend in the excitation energy is well reproduced. Regarding timings, the Davidson\nmethod offers clear improvements over the full diagonalization approach, with calculations\nthat are an order of magnitude faster for the larger systems. The speed-up in timings is\nalmost entirely driven by the need to evaluate fewer Hessian-vector products. Formally, the\nfull diagonalization approach can be seen as evaluating a Hessian-vector product for each\nbasis vector of the matrix in Eq. (15). When parametrizing the excitation operator manifold\nwith single and double excitations, the matrix dimension grows asymptotically as N4with\nthe system size. In contrast, the required dimensions of the UCCSD and FCI subspace\nappear to level out at a size of around 13–14. We note that the FCI calculations are carried\nout using the Dalton program, which uses a Davidson-like28method to extract excitation\nenergies. For FCI, we report the dimension as half the subspace size since the FCI response\nsolver in Dalton uses paired trial vectors.\nTable 1: Excitation energies, timings, and matrix dimensions of the (reduced) Hessian for\nH2n/STO-3G chains. For FCI, we report half the dimension of the reduced space.aNot\ndetermined. The calculation would take an estimated 16 days.\n∆Eexc(eV) Time (seconds) Dimension\nFCI UCCSDaUCCSDbUCCSD UCCSD FCI UCCSD UCCSD\n2n Full diag. Davidson Full diag. Davidson Full diag. Davidson\n219.44 19.44 19.44 0.0450 0.0431 2 2 2\n410.08 10.09 10.09 6.85 4.97 2 14 10\n614.14 14.19 14.19 196 43.46 8 54 12\n82.98 3.35 3.35 9930 950 13 152 14\n1010.54 –a10.45 –a56400 13 350 14\n164.2 Sampling noise\nNext, we consider the effects of sampling noise in determining excitation energies, using H 2\nas a minimal example. For this set of tests, we used the parameter-shift rule for evaluating\ngradients in Eq. (16) since the adjoint differentiation approach does not permit the modeling\nof shot noise. When using spin-adapted singlet excitation operators, the dimension of the\nHessian is 2 ×2 for H 2/STO-3G, corresponding to one single and one double excitation.\nIn this small example, the Davidson method terminates trivially in a single iteration when\nwe use the initial trial vectors b1= (1,0) and b2= (0,1) to compute the Hessian. In the\npresence of sampling noise, a potentially important new complication emerges, namely that\nthe hermiticity of the Hessian is destroyed. We consider three options to tackle the non-\nhermiticity of ABin our algorithms: 1) using only the upper triangle of the Hessian (assigning\nAB,ji=AB,ij), 2) solving the eigenvalue with a non-hermitian Hessian, and 3) symmetrizing\nthe Hessian as ˜AB=1\n2\u0010\nAB+A†\nB\u0011\n. Additionally, the finite difference approach used to form\nthe Hessian-vector products (Eq. (16)) becomes sensitive to noise for small values of h.\nWe consider the effects of the hermiticity treatment and the size of the finite-difference\nstep in Figure 2, where we have computed excitation energies with each of the three ap-\nproaches to solving the eigenvalue problem for different values of the finite difference step\nparameter h, plotting the distribution ( N= 100) for the absolute error in the lowest excita-\ntion energy. For small values of h, the finite difference scheme is completely unreliable due to\nthe sampling errors, leading to extremely large errors in the excitation energies. Increasing h\ninitially decreases the error in the excitation energy until truncation errors in the finite dif-\nference scheme become significant. The optimal balance for every tested shot count amount\nappears surprisingly large at around h= 0.1, where, with 107shots, the mean absolute error\nbecomes 0.062 eV. By increasing hfurther (to the regime of truncation/discretization error),\nthe sampling errors are almost completely avoided, as reflected by the hardly visible error\nbars at h= 1.0. The truncation errors lead to significant misestimation of the parameter\nderivatives and, ultimately, excitation errors of more than 5 eV with 107shots. We note\n17that the optimal choice of hand error characteristics will likely be system-dependent. The\ntradeoff between sampling/truncation errors could be avoided if a parameter-shift rule was\ndeveloped to evaluate the Hessian-vector product; however, as discussed by Parrish et al.,\nthe practical implementation of such a scheme is non-trivial.30The choice of restoring the\nHessian’s hermiticity turns out to be less consequential, with all three approaches having\ncomparable performance. When sampling is deeply insufficient (low h), the Hessian is so\npoorly described that all methods yield poor results. On the other hand, around the optimal\nh, the hermiticity violations become small, making all the approaches nearly equivalent.\nFrom Figure 2, we find that using large step values his desirable since statistical noise\nerrors can be suppressed. Therefore, we investigated whether more elaborate finite difference\nschemes could allow larger step sizes. To examine this, we computed excitation energies of H 2\nwith higher-order numerical derivatives, namely 5-, 7-, 9, and 11-point symmetric stencils.\nWe first consider the noise-free case, shown in Figure 3. All the finite difference schemes can\nachieve errors below 10−6eV for the excitation energy with a proper choice of h. The region\nof stability widens when using the higher-order stencils. In particular, it is possible to use\nsignificantly larger step sizes without compromising the quality of the computed excitation\nenergy. Supposing we can accept an error of 10−3eV in the excitation energy, the largest\npermissible step is around h= 0.012 with the central difference scheme. The 5-point stencil\nallows a larger h= 0.13, while the 7-, 9, and 11-point stencils allow up to h= 0.28,h= 0.40\nandh= 0.50. These larger step sizes are well into the region of significantly diminished\nstatistical noise errors from Figure 2.\nNext, we apply the higher-order finite difference stencils in a setting with sampling noise.\nThe higher-order schemes require 4, 6, 8, and 10 gradient evaluations compared to just two\nfor the central difference (”3-point”) scheme. Thus, we scale the shot count by the gradient\nevaluation count to provide fair comparisons for the shot count costs of the different finite\ndifference approaches, i.e., we allow half the amount of shots per gradient evaluation for the\n5-point scheme compared to the central difference scheme. As shown in Figure 4, the higher-\n18106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\n106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\n104 shots\n106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\n106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\n106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\n105 shots\n106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\n106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\n106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\n106 shots\n106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\n106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\n106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\n107 shots\n106\n105\n104\n103\n102\n101\n100\nh103\n100103106Error in E (eV)\nFigure 2: Absolute error in the lowest excitation energy of H 2/STO-3G for a fixed number of\nshots (104– 107) as a function of the central difference step parameter ( h). The distributions\nare taken across N= 100 samples, shown as a box plot. The orange lines indicate the mean\nerror. The noise-free result is used as a reference. For the non-hermiticity treatment, the\nleft panels use the upper triangular part of the Hessian, the middle panels solve the non-\nhermitian problem, and the right panels use a symmetrized Hessian.\n19104\n103\n102\n101\n100\nh108\n106\n104\n102\n100Excitation energy error (eV)\nCentral\n5-Point\n7-Point\n9-Point\n11-PointFigure 3: Absolute error in the lowest excitation energy of noiseless H 2/STO-3G as a function\nof the finite difference step parameter for various finite difference schemes. The black line\nindicates a target error of 10−3eV.\n20order finite difference schemes allow for stability at significantly larger step parameters h\neven in the presence of statistical noise. The optimal h, in the sense of the lowest error in\nthe excitation energy, gradually increases from h= 0.1 with the central difference scheme to\nh= 0.5 with the 9-point stencil. On balance, this means that the mean absolute error in the\nexcitation energy can be decreased by applying the higher-order difference schemes. With a\n106shot budget, the best mean absolute deviation decreases from 0 .087 eV with the central\ndifference approach to 0 .062 and 0 .036 eV with the 5- and 7-point stencils, then increases\nslightly to 0 .040 and 0 .044 eV with the, 9- and 11-point stencils. Evidently, the increased\nstatistical error due to the lower shot count in the higher-order finite-difference schemes\nbecomes more significant than the associated decrease in truncation error. For the present\nsystem, the optimal balance thus appears to be the 7-point stencil with a h= 0.5. Again,\nwe note that, in general, the optimal balance will likely be somewhat system-dependent.\n0.010.050.10.20.51.0\nh103\n102\n101\n100101Error in E1 (eV)\nCentral\n0.010.050.10.20.51.0\nh103\n102\n101\n100101Error in E1 (eV)\n5-Point\n0.010.050.10.20.51.0\nh103\n102\n101\n100101Error in E1 (eV)\n7-Point\n0.010.050.10.20.51.0\nh103\n102\n101\n100101Error in E1 (eV)\n9-Point\n0.010.050.10.20.51.0\nh103\n102\n101\n100101Error in E1 (eV)\n11-Point\nFigure 4: Absolute error in the lowest excitation energy of H 2/STO-3G for a fixed number\nof shots (106) as a function of the finite difference step parameter ( h) for different finite\ndifference schemes. The distributions are taken across N= 100 samples. The orange lines\nindicate the mean error. The noise-free result is used as a reference. Each method is allowed\na fixed shot budget: the 5, 7, 9, and 11-point schemes use1\n2,1\n3,1\n4, and1\n5the amount of shots\nfor each gradient evaluation compared to the central difference scheme.\n214.3 Static polarizabilities\nWe now move on to consider the calculation of more general response properties (in a noise-\nfree setting). Table 2 shows static polarizabilities of selected small molecules computed using\nFCI, UCCSD, or conventional CCSD. All methods agree for the H 2molecule. As expected,\nthe double excitations in UCCSD/CCSD recover the FCI solution for this two-electron sys-\ntem. UCCSD and CCSD also have a good reproduction of the FCI polarizabilities of the\nequilibrium geometries of LiH, water, the hydroxide ion, and ammonia. The performance\nof UCCSD and CCSD is comparable for the equilibrium geometries, with mean absolute\ndeviations of 0.0038 and 0.0033 a.u. for UCCSD and CCSD, respectively. We also con-\nsider (symmetrically) stretched versions of these small molecules to generate systems with a\nmore challenging electronic structure. These partially dissociated systems introduce stronger\nstatic correlation, which can be difficult to model. Nevertheless, we find that the stretched\nLiH and OH–systems can be described well by both UCCSD and CCSD. This can perhaps\nbe rationalized from the observation that the dissociation in these systems involves just a\nsingle σbond (one pair of electrons). Since both methods include double excitations, they\ncan describe this single bond-breaking correctly. On the other hand, the stretched water\nand ammonia systems prove more challenging. UCCSD manages a mean absolute deviation\n(MAD) of 0.014 a.u. for water, which is considerably better than the CCSD result (0.84 a.u.).\nThe stretched ammonia system is even more challenging, with the CCSD polarizabilities be-\ning qualitatively wrong, having two negative components in the ground-state polarizability.\nThe UCCSD polarizability is qualitatively correct for this system, managing a MAD of 0.21\na.u. The exact ground-state polarizability is always non-negative, as seen from the sum-over-\nstates expression of the polarizability (see Eq. (34)) since both the denominators (excitation\nenergies) and numerators (squared transition moments) are positive. Excited-state polariz-\nabilities can be negative. This could indicate that the CCSD wave function has converged\nto an excited state. However, we did not find any negative excitation energies with CCSD,\nalthough complex eigenvalues appeared in the solution of the CCSD linear response equa-\n22tions. Additionally, we also characterized the stability of the HF ground-state wave function,\nwhich revealed an instability towards a UHF solution with several negative eigenvalues, i.e.,\na triplet instability.\nTable 2: Polarizabilities (in a.u.), for selected molecules (STO-3G basis) computed with FCI,\nUCCSD, or CCSD.\nFCI UCCSD CCSD\nαxx αyy αzz αxx αyy αzz αxx αyy αzz\nH2 0.000 0.000 2.775 0.000 0.000 2.775 0.000 0.000 2.775\nLiH 22.801 22.801 12.400 22.788 22.788 12.377 22.793 22.793 12.387\nLiH (2 Re)69.779 69.779 98.258 69.841 69.841 98.385 69.821 69.821 98.281\nH2O 0.056 5.395 2.000 0.056 5.393 2.000 0.056 5.399 1.992\nH2O (2Re)0.012 5.207 2.858 0.012 5.191 2.832 0.007 3.216 2.341\nOH−0.035 0.035 4.325 0.035 0.035 4.325 0.035 0.035 4.325\nOH−(2Re)0.054 0.054 37.421 0.054 0.054 37.421 0.054 0.054 37.421\nNH 3 4.980 4.980 1.160 4.977 4.977 1.158 4.979 4.979 1.152\nNH 3(2Re)7.349 7.349 3.755 7.655 7.606 3.690 -0.935 -0.935 2.388\n4.4 Dynamic polarizabilities\nNext, we turn to compute resonant-convergent frequency-dependent molecular properties\nwith damped response theory. Figure 5 shows the imaginary (corresponding to absorption)\nand real (corresponding to dispersion) parts of the damped dipole–dipole polarizability of\nammonia (in a STO-3G basis). The spectra were computed in the energy range correspond-\ning to the nitrogen K-edge X-ray absorption. We used a damping factor γ= 0.004556 a.u.\n(1000 cm−1). The UCCSD absorption spectrum matches the FCI result qualitatively well,\nwith a lower-intensity pre-edge feature followed by a more intense peak. With UCCSD, the\nintense peak is blue-shifted by about 0.4 eV, while the lower-intensity peak is shifted by\nabout 0.6 eV. Due to the rather small basis set employed, any comparisons to experiments\nare not necessarily meaningful. However, the transitions at least occur in the correct general\nregion for nitrogen K-edge X-ray absorption. We should also point out that the density of\nstates in the region is quite small when employing such a minimal basis set expansion, with\nonly three excited states within the plotted energy region. Thus, it would likely be more\n23computationally efficient to compute (targeted) excitation energies and associated oscillator\nstrengths to obtain the spectrum. We also show spectra generated from a Lorentzian convo-\nlution of CCSD excitation energies and oscillator strengths. As it turns out, the conventional\nCCSD spectra perform better than the UCCSD ones, with a smaller shift of 0.08 eV relative\nto the FCI spectrum. Clearly, one should not always expect UCCSD to outperform classical\nCCSD. However, ammonia has a relatively simple electronic structure at the equilibrium ge-\nometry, and as shown with the static polarizabilities in Table 2, the UCCSD method works\nwell in more complicated systems, including systems with multiconfigurational character.\n409 410 411 412 413 4140100002000030000() (a.u.)\nFCI\nUCCSD\nCCSD\n409 410 411 412 413 414\nEnergy (eV)0.5\n0.00.5Re(iso()) (a.u.)\n FCI\nUCCSD\nFigure 5: Imaginary and real parts of the isotropic polarizability of NH 3/STO-3G near\nthe nitrogen K-edge X-ray absorption energy region. For the imaginary part, we plot the\nabsorption cross-section (see Eq. (38)). The spectra are computed using damped response\ntheory with FCI or UCCSD. We also show CCSD spectra generated from excitation energies\nand oscillator strengths with a Lorentzian broadening.\n244.5 Dispersion coefficients\nTable 3 shows computed C6coefficients for the same set of molecules used in Table 2 with\nthe same STO-3G basis. We compare C6coefficients computed with UCCSD to coefficients\nevaluated with an FCI wave function. The C6coefficients vary considerably, from relatively\nsmall in magnitude (0.62 a.u. for H 2) to much more substantial values (409 a.u. for stretched\nLiH). Nevertheless, the UCCSD C6coefficients generally reproduce the reference FCI results\nvery well, with relative errors below 0.1% at equilibrium geometries. The stretched geome-\ntries are more challenging, particularly for water and ammonia, where we find relative errors\nof up to 9%. These systems were also challenging cases for the static polarizabilities. The\nCCSD C6coefficients are reproduced well for the equilibrium geometries, but the method\nfails to give even qualitatively accurate results for the stretched water and ammonia systems,\nwhere the C6coefficients are either strongly overestimated (water, +55%) or underestimated\n(ammonia, −99.9%). Thus, while CCSD might give a good description of the dispersion co-\nefficients for the equilibrium geometries, the method gives qualitatively incorrect results for\nsome of the stretched systems. Meanwhile, the UCCSD parametrization works well across\nboth equilibrium and stretched geometries since it is able to handle systems with strong\nstatic correlation or multiconfigurational character.\nTable 3: CAA\n6coefficients (in a.u.) for selected molecules (STO-3G basis) computed with\nFCI, UCCSD, or CCSD. Relative deviations (in percent) are given in parentheses.\nFCI UCCSD CCSD\nH2 0.62 0.62 (0.00) 0.62 (0.00)\nLiH 50.84 50.79 ( −0.09) 50.81 ( −0.05)\nLiH (2 Re)408.71 409.24 (0.13) 409.27 (0.14)\nH2O 3.79 3.79 ( −0.02) 3.79 ( −0.05)\nH2O (2Re) 3.04 3.10 (2.12) 4.71 (54.98)\nOH−1.28 1.28 (0.00) 1.28 (0.00)\nOH−(2Re)23.16 23.16 (0.00) 23.16 (0.00)\nNH 3 9.02 9.02 (0.02) 9.01 ( −0.08)\nNH 3(2Re)13.01 14.27 (9.66) 0.01 ( −99.91)\n255 Conclusion\nIn this work, we have formulated matrix-free (Davidson) methods for obtaining excitation\nenergies and linear response properties with the q-sc-LR method. Following a ground-state\nVQE calculation, our approach evaluates Hessian-vector products with arbitrary trial vectors\nwith a finite difference scheme using gradients of the excitation operators. In a noise-free\nsetting, we show that the Davidson method reproduces excitation energies of the full diag-\nonalization approach numerically exactly but with a significantly reduced cost, mainly from\nneeding to evaluate far fewer Hessian-vector multiplications. We analyze some of the effects\nof sampling noise on calculating the Hessian vector products and find that statistical noise\nnecessitates surprisingly large steps to handle the statistical noise. We explore the use of\nhigher-order finite difference schemes and demonstrate that the increased stability at high\nstep sizes allows for much larger step sizes, alleviating the sensitivity to statistical noise. We\ncompute linear response properties in a noise-free setting with the Davidson method. We\nshow that static UCCSD polarizabilities match FCI reference results well. We also find that\nthe UCCSD method outperforms classical projected CCSD for strongly correlated, stretched\nsystems such as H 2O and NH 3. Using damped response theory, we probe the nitrogen K-edge\nX-ray absorption of ammonia and find that the UCCSD spectra match the FCI reference\nresult well with a shift of at most 0.6 eV.\nAcknowledgement\nWe acknowledge the financial support of the Novo Nordisk Foundation for the focused re-\nsearch project Hybrid Quantum Chemistry on Hybrid Quantum Computers (HQC)2, grant\nnumber NNFSA220080996.\n26Supporting Information Available\nGeometries of H 2, LiH, H 2O, OH–, and NH 3inxyzformat.\nReferences\n(1) Bauer, B.; Bravyi, S.; Motta, M.; Chan, G. K.-L. Quantum Algorithms for Quantum\nChemistry and Quantum Materials Science. Chem. Rev. 2020 ,120, 12685–12717.\n(2) Cao, Y.; Romero, J.; Olson, J. P.; Degroote, M.; Johnson, P. D.; Kieferov´ a, M.;\nKivlichan, I. D.; Menke, T.; Peropadre, B.; Sawaya, N. P. D.; Sim, S.; Veis, L.; Aspuru-\nGuzik, A. Quantum Chemistry in the Age of Quantum Computing. Chem. Rev. 2019 ,\n119, 10856–10915.\n(3) Motta, M.; Rice, J. E. 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H. Gaussian basis sets for use in correlated molecular calculations. I. The\natoms boron through neon and hydrogen. J. Chem. Phys. 1989 ,90, 1007–1023.\n(55) Neese, F. Software update: The ORCA program system—Version 5.0. WIREs Comput\nMol Sci 2022 ,12, e1606.\n34TOC Graphic\n35"    },    {        "title": "2402.12209v1.Some_Riemannian_properties_of___mathbf_SU_n___endowed_with_a_bi_invariant_metric.pdf",        "content": "arXiv:2402.12209v1  [math.DG]  19 Feb 2024SOME RIEMANNIAN PROPERTIES OF SU n\nENDOWED WITH A BI-INVARIANT METRIC\nDONATO PERTICI AND ALBERTO DOLCETTI\nDipartimento di Matematica e Informatica, Viale Morgagni 6 7/a, 50134 Firenze, ITALIA\ndonato.pertici@unifi.it, http://orcid.org/0000-0003-4 667-9568\nalberto.dolcetti@unifi.it, http://orcid.org/0000-0001 -9791-8122\nAbstract. We study some properties of SUnendowed with the Frobenius\nmetricφ, which is, up to a positive constant multiple, the unique bi- invariant\nRiemannian metric on SUn. In particular we express the distance between\nP,Q∈SUnin terms of eigenvalues of P∗Q; we compute the diameter of\n(SUn,φ) and we determine its diametral pairs; we prove that the set o f all\nminimizing geodesic segments with endpoints P,Qcan be parametrized by\nmeans of a compact connected submanifold of sun, diffeomorphic to a suitable\ncomplex Grassmannian depending on PandQ.\nContents\nIntroduction 1\n1. Preliminary facts 4\n2. About the distance in ( SUn,φ) of any matrix from In 6\n3. The set Θ( Q) of minimizing logarithms of any special unitary matrix Q 11\n4. About diameter and diametral pairs of ( SUn,φ) 13\n5. Some geometrical properties of the Riemannian manifold ( SUn,φ) 15\nReferences 16\nKeywords. Special unitary group, Frobenius metric, bi-invariant met rics on a compact\nLie group, minimizing geodesics, homogeneous space, compl ex Grassmannian, diameter,\ndiametral pairs, generalized principal logarithm;\nMathematics Subject Classification (2020): 53C35, 15B30, 22E15.\nGrants: supported by GNSAGA-INdAM and by MUR-PRIN project “Multili near Alge-\nbraic Geometry”.\nIntroduction\nLetGbe a closed subgroup of the unitary group Unand letg⊆unbe its Lie algebra.\nIt is natural to consider on GtheFrobenius (orHilbert-Schmidt )metricφinduced on G\nby the Euclidean metric of Cn2(identified with gln(C): the Lie algebra of complex square\nmatrices of order n). SinceG⊆Un, the metric φis bi-invariant on G. Furthermore, if\n12 DONATO PERTICI AND ALBERTO DOLCETTI\nGis also absolutely simple and connected (e.g. the classical groupsSOnforn≥3 and\nn/\\e}atio\\slash= 4,SUnforn≥2 andSpnforn≥1), every bi-invariant (0 ,2)-tensor on G(such as\nthe Killing metric) is a constant multiple of φ(see, for instance, [Dolcetti-Pertici 2023,\nProp.1.8]). In particular, all bi-invariant (0 ,2)-tensors on such groups are Riemannnian\nor anti-Riemannian metrics and have the same geodesics and t he same isometries.\nIn [Dolcetti-Pertici 2023, Thm.2.3] we determined all isom etries of any absolutely simple,\ncompact, connected, real Lie group endowed with anybi-inva riantmetric; from that result,\nwe deduced the list of all isometries of SOn(n≥3,n/\\e}atio\\slash= 4),SUn(n≥2) andSpn(n≥1),\nendowed with any bi-invariant metric ([Dolcetti-Pertici 2 023, Thm.2.5(c),(d)]). Further-\nmore, bymeansofsuitabledifferentmethods, wedeterminedt helistofall isometries alsoof\nSO(4) and ofU(n) (n≥2), all endowed with the Frobenius metric ([Dolcetti-Perti ci 2023,\nThm.3.5, Thm.4.7]).\nOn the other hand, in [Pertici-Dolcetti 2022], we studied th e Riemannian properties of\nmany suitable subgroups Gof the unitary group Un(we called them SVD-closed) endowed\nwith the Frobenius metric φ. In particular, we expressed the distance between points an d\nwe parametrized the set of all minimizing geodesic segments of (G,φ) with arbitrary\nendpointsP,Q, by means of a set, denoted by g–plog(P∗Q), consisting of generalized\nprincipal logarithms of P∗Qbelonging to g; we proved that this last set is non-empty, it is\na union of finitely many compact submanifolds of the Lie algeb raunand that each of these\nsubmanifolds is diffeomorphic to a suitable homogeneous spa ce ([Pertici-Dolcetti 2022,\nThm.5.7 and Thm.6.5]). For a lot of families of SVD-closedsu bgroups ofUn(among them\nUn,SOn,Spnand others) we also determined the (Frobenius) diameter of e ach subgroup\nand the structure of each connected component of g–plog(P∗Q) as a homogeneous space\n([Pertici-Dolcetti 2022, Prop.6.7 and §7,§8]).\nMost classical subgroups of Unare SVD-closed with the exception of SUn, whenn≥3.\nFor this reason, in this paper we try to obtain similar result s for (SUn,φ). In particular,\nwe are able to express the distance between points, to comput e the diameter of ( SUn,φ),\nto determine its diametral pairs (i.e. the pairs of points wh ose distance is equal to the\ndiameter) and to parametrize the set of all minimizing geode sic segments with endpoints\nP,Q, by means of a compact connected submanifold of sun, which in general is not equal\ntosun–plog(P∗Q)), but it is always diffeomorphic to a suitable complex Grass mannian\ndepending on PandQ(Theorem 5.1 and Theorem 5.2). Since we already studied gene r-\nalized principal logarithms of some different types of matri ces in [Dolcetti-Pertici 2018],\n[Pertici-Dolcetti 2022], [Pertici 2023], we also determin e the conditions on Q∈SUn, so\nthat the set sun–plog(Q) is non-empty (Proposition 3.3).SOME RIEMANNIAN PROPERTIES OF SUn 3\nThe euclidean metric on gln(C) can be also restricted tothe Lie group GLn(C)of invertible\ncomplex matrices; this restriction is a Riemannian metric o nGLn(C), but it is not bi-\ninvariant. On the other hand GLn(C) can be endowed, in a natural way, with a semi-\nRiemannian metric, which, on the contrary, is bi-invariant : the so-called trace metric g,\ndefined, at the identity matrix, by g(X,Y) :=Re(Tr(XY)) (the real part of the trace of\nthe product XY) for every X,Y∈gln(C). We remark that, for any closed subgroup G\nofUn, the restriction to Gof the trace metric gagrees with the anti-Riemannian metric\n−φ(the opposite of the Frobenius metric of G) and hence it determines on Gthe same\nRiemannian structure of φ.\nThe restrictions of the trace metric define Riemannian struc tures on the space of pos-\nitive definite hermitian matrices and on the space of positiv e definite symmetric real\nmatrices. Such manifolds have a remarkable interest in many frameworks of both pure\nand applied mathematics (see for instance, among many other s, [Bridson-Haefliger 1999],\n[Moakher 2005], [Bhatia 2007], [Barbaresco 2008], [Moakhe r-Z´ era¨ ı 2011],\n[Nielsen-Bhatia 2013], [Nielsen-Barbaresco 2019], [Dolc etti-Pertici 2019],\n[Dolcetti-Pertici 2021], [Cruceru-Becigneul-Ganea 2021 ], [Boumal 2023],\n[Criscitiello-MartinezRubio-Boumal 2023], [Nielsen 202 3a], [Nielsen 2023b],\n[Zhang-Zhang-Sra 2023], [Nieuwboer 2024]). We have also st udied the restrictions of the\ntrace metric on other submanifolds of GLn(C), where they induce semi-Riemannian (but\nnot Riemannian) structures (see [Dolcetti-Pertici 2015], [Dolcetti-Pertici 2019],\n[Dolcetti-Pertici 2020]).\nThe present paper is organized as follows.\nIn Section 1, we recall the main notations and the basic facts used in this paper. We\nintroduce the Frobenius metric φon any closed connected subgroup GofUn, having\ng⊆unas Lie algebra. In particular, we observe that (Proposition 1.5):\n(i) for every P,Q∈G, the (Frobenius) distance d(P,Q) is equal to the minimum of the\n(Frobenius) norm /⌊ard⌊lX/⌊ard⌊lφ, whereX∈gis a logarithms of P∗Q;\n(ii) the map: X/ma√sto→γ(t) :=Pexp(tX) (0≤t≤1) is a bijection from the set\n{X∈g: exp(X) =P∗Qand/⌊ard⌊lX/⌊ard⌊lφ=d(P,Q)}onto the set of minimizing geodesic\nsegments of ( G,φ) with endpoints PandQ.\nNextSections are devotedtostudysome Riemannian properti es ofthegroup SUnendowed\nwith the Frobenius metric φ.\nAim of Section 2 is to find an explicit expression of the functi on\nm(Q) := min {/⌊ard⌊lX/⌊ard⌊l2\nφ:X∈sunand exp(X) =Q}=d(In,Q)2for any matrix Q∈SUn\n(Inis the identity matrix); the value of m(Q) is obtained in terms of the eigenvalues of Q\n(Proposition 2.8).\nIn Section 3, for every Q∈SUn, we analyse the set4 DONATO PERTICI AND ALBERTO DOLCETTI\nΘ(Q) :={X∈sun: exp(X) =Q,/⌊ard⌊lX/⌊ard⌊l2\nφ=m(Q) =d(In,Q)2}, which (as stated above)\nparametrizes the set of minimizing geodesic segments of ( SUn,φ) with endpoints Inand\nQ. We prove that Θ( Q) is a compact connected submanifold of sun, diffeomorphic to\na suitable complex Grassmannian (Proposition 3.2). Furthe rmore we also determine the\ngeometric structure of sun–plog(Q) (Proposition 3.3).\nIn Section 4, we obtain some fundamental results concerning the diameter of ( SUn,φ) and\nits diametral pairs (Proposition 4.1).\nIn the last Section 5, by means of the Propositions proved in S ections 2, 3, 4, we collect\nand prove the main results of this paper: the already mention ed Theorems 5.1 and 5.2.\n1.Preliminary facts\n1.1.Notations-Definitions. a) We denote by Cthe field of complex numbers, by i\nits imaginary unit, by Rthe field of real numbers and by Zthe ring of integers. The\ninteger part of any x∈Ris denoted by ⌊x⌋. We denote by |w|andew:=+∞/summationtext\ni=0wi\ni!,\nrespectively, the modulus and the exponential of an arbitra ry complex number w, while,\nfor anyz∈C\\{0}, we denote by arg( z)∈(−π,π] theprincipal value of the argument ofz\nand by log( z) := ln|z|+arg(z)itheprincipal value of the logarithm ofz. Clearly log( z) is\nthe unique complex logarithm of z, whose imaginary part lies in the interval ( −π,π]. Note\nthat, when |z|= 1, we have log( z) = arg(z)i, so, in particular, log( −1) =πi.Of course,\nif|z|=|w|= 1, we have z=wif and only if arg( z) = arg(w) (i.e. log(z) = log(w)).\nb)Wedenoteby GLn(C) (withn≥1) theLiegroupofinvertiblecomplexsquarematrices\nof ordern, bygln(C) its Lie algebra consisting of all n×ncomplex matrices and by Inthe\nidentity matrix of GLn(C). For every A∈gln(C),AT,A,A∗:=ATandA−1(provided\nthatAis invertible)are respectivelytranspose, conjugate, adj oint andinverseofthematrix\nA, tr(A) is its trace, det( A) is its determinant, while exp( A) :=+∞/summationtext\ni=0Ai\ni!∈GLn(C) denotes\nthe exponential of A. It is clear that, if λ1,···,λnare theneigenvalues of A∈gln(C),\ntheneλ1,···,eλnare theneigenvalues of exp( A). For convenience, we agree that a\ncomplex number ηis aneigenvalue of multiplicity 0 of a matrix A∈gln(C) ifηis not a\nusual eigenvalue of A, i.e. ifηis not a root of the characteristic polynomial of A.\nFor any integer n≥1, we denote by Un:={A∈gln(C) :AA∗=In}the unitary group\nof degreenand by un:={X∈gln(C) :X=−X∗}its Lie algebra of skew-hermitian\nmatrices; we also denote by SUn:={A∈Un: det(A) = 1}the special unitary group\nof degreenand by sun:={X∈un: tr(X) = 0}its Lie algebra. Finally, for any\nU∈Un, we denote by AdU:gln(C)→gln(C) the map defined by AdU(X) :=UXU∗(for\neveryX∈gln(C)) and we still denote by AdUthe restriction of this map to any subset of\ngln(C); so we can write both AdU:SUn→SUnandAdU:sun→sun.Note thatAdU\ncommutes with the exponential map.SOME RIEMANNIAN PROPERTIES OF SUn 5\nc) LetGbe a closed connected subgroup of GLn(C) having g⊂gln(C) as its Lie algebra.\nIfB∈G, A∈gand exp(A) =B, we say that Ais ag–logarithm ofB. Moreover, if Ais\nag–logarithm of Bsuch that −π≤Im(λ)≤πfor every eigenvalue λofA, we say that\nAis ageneralized principal g–logarithm ofB.\nWe denote by g–plog(B) the set of all generalized principal g–logarithms of B∈G.\nRecall that, if Gis also compact, the exponential map exp : g→Gis surjective (see, for\ninstance, [Sepanski 2007, Thm.5.12, p.102]).\nd) IfB1,···,Btare square matrices (of various orders), thent/circleplustext\nj=1Bj=B1⊕ ··· ⊕Bt\ndenotes the block diagonal square matrix with B1,···,Bton its diagonal.\nClearly we have exp(t/circleplustext\nj=1Bj) =t/circleplustext\nj=1exp(Bj) , for every B1,···,Bt.\nIfS1,...,Stare sets of square matrices, thent/circleplustext\nj=1Sj=S1⊕···⊕S tdenotes the set of all\nmatricest/circleplustext\nj=1Bj,withBj∈ Sjfor everyj.\n1.2.Lemma. Letn1,···,ntbe positive integers such that n=t/summationtext\nj=1njand letλ1,···,λt\nbe distinct complex numbers. Set A:=t/circleplustext\nj=1λjInj.Then a matrix B∈Uncommutes with\nAif and only if B∈t/circleplustext\nj=1Unj.\nFor a proof of the Lemma, see for instance [Pertici 2023, Lemm a1.3].\n1.3.Remark-Definition. We denote by φtheFrobenius (orHilbert-Schmidt )positive\ndefinite real scalar product ongln(C) defined by φ(A,B) := Re(tr(AB∗)), and we denote\nby/⌊ard⌊lA/⌊ard⌊lφ:=/radicalbig\nφ(A,A) =/radicalbig\ntr(AA∗) the related Frobenius norm . Note that, if A∈un,\nthen/⌊ard⌊lA/⌊ard⌊l2\nφ=−tr(A2). Since the eigenvalues of the skew-hermitian matrix Aare purely\nimaginary, we also get /⌊ard⌊lA/⌊ard⌊lφ=/radicalbig\n−tr(A2) =/radicalBigg\nn/summationtext\nj=1|λj|2, whereλ1,···,λnare then\n(possibly repeated) eigenvalues of A. For any arbitrary closed subgroup GofUn, we still\ndenote byφthe Riemannian metric on G, obtained by restriction of the Frobenius scalar\nproduct of gln(C). It is easy to check that the metric φ(called the Frobenius metric ofG)\nis bi-invariant on Gand we have φA(X,Y) =−tr(A∗XA∗Y), for any pair X,Yof tangent\nvectors toGatAand for any A∈G. We denote by d:=d(G,φ)the distance on Ginduced\nbyφand byδ(G,φ) := sup{d(P,Q) :P,Q∈G}thediameter ofGwith respect to d. Of\ncourseδ(G,φ) =d(A,B)<+∞, for someA,B∈G, becauseGis compact.\nIfA,B∈Gandd(A,B) =δ(G,φ), we say that AandBarediametral in (G,φ) or,\nequivalently, that they form a diametral pair of (G,φ). Sinceφis bi-invariant on the\ncompact Lie group G⊂Un,it is clear that every point of Gbelongs at least to a\ndiametral pair of ( G,φ). For some results on diametral pairs in real orthogonal gro ups,\nsee [Dolcetti-Pertici 2018, Prop.4.18].6 DONATO PERTICI AND ALBERTO DOLCETTI\n1.4.Proposition. LetGbe a closed subgroup of Unand letg⊆unbe its Lie algebra. Then\n(G,φ)is a globally symmetric Riemannian manifold with non-negat ive sectional curvature,\nwhose Levi-Civita connection agrees with the 0-connection of Cartan-Schouten of G. The\ngeodesics of (G,φ)are the curves t/ma√sto→Pexp(tX) (witht∈R), for every X∈gand\nP∈G; furthermore (G,φ)is a totally geodesic submanifold of (Un,φ).\nFor a proof of Proposition 1.4, we refer, for instance, to [Al exandrino-Bettiol 2015, §2.2].\n1.5.Proposition. LetGbe a closed connected subgroup of Unand letg⊆unbe its\nLie algebra. Then, for every P,Q∈G, the (Frobenius) distance d(P,Q)is equal to the\nminimum of the set { /⌊ard⌊lX/⌊ard⌊lφ:Xis ag–logarithm of P∗Q}.\nFurthermore, the map: X/ma√sto→γ(t) :=Pexp(tX) (0≤t≤1)is a bijection from the\nset{X:Xis ag–logarithm of P∗Qwith/⌊ard⌊lX/⌊ard⌊lφ=d(P,Q)}onto the set of minimizing\ngeodesic segments of (G,φ)with endpoints PandQ.\nProof.Anygeodesic segment γ, joiningPandQ, canbeparametrizedby γ(t) =Pexp(tX)\n(t∈[0,1]), withX∈g,exp(X) =P∗Q, and its length is/radicalbig\n−tr(X2) =/⌊ard⌊lX/⌊ard⌊lφ; so we con-\nclude by means of the classical Hopf-Rinow theorem. /square\n1.6.Remark. LetGbe as in Proposition 1.5. We resume some results obtained in\n[Pertici-Dolcetti 2022].\nIn Thm. 6.5, we proved that, if Gis SVD-closed too (for definition see §4 of that paper)\nandP,Q∈G, then\na)d(P,Q) =/radicalBigg\nn/summationtext\nj=1|log(µj)|2, whereµ1,···,µnare theneigenvalues of P∗Q;\nb)g–plog(P∗Q) ={X:Xis ag–logarithm of P∗Qwith/⌊ard⌊lX/⌊ard⌊lφ=d(P,Q)}and so\ng–plog(P∗Q) parametrizes the set of minimizing geodesic segments of ( G,φ) with end-\npointsPandQ.\nIn Prop.6.7 and Rem.6.8, we computed the diameter of suitabl e families of SVD-closed\nsubgroups of Un, among them all classical subgroups of Unwith the exception of SUn.\nFor the same families of SVD-closed subgroups GofUn, in Sections 7 and 8, we proved\nthat, for any matrix M∈G, the set g–plog(M) is a disjoint union of finitely many simply\nconnected compact submanifolds of un, each of them is diffeomorphic to a symmetric\n(homogeneous) space, we explicitly determined.\nAs remarked in [Pertici-Dolcetti 2022, Rem.3.7], SUnis not SVD-closed as soon as n≥3\nand therefore the Riemannian manifold ( SUn,φ) was excluded from that study. Hence,\nthe goal of the present paper is to research for similar resul ts in case of ( SUn,φ).\n2.About the distance in (SUn,φ)of any matrix from In\n2.1.Remarks-Definitions. a) Fix a matrix Q∈SUn(n≥2), and letµ1···,µnbe the\n(possibly repeated) eigenvalues of Q, so we have µ1µ2···µn= 1; it is also known thatSOME RIEMANNIAN PROPERTIES OF SUn 7\n|µ1|=|µ2|=···|µn|= 1,so log(µj) = arg(µj)i, forj= 1,···,n. Up to reordering the\neigenvalues, we can assume −π<arg(µ1)≤arg(µ2)≤ ··· ≤arg(µn)≤π.\nSince 1 =n/producttext\nj=1elog(µj)=e(arg(µ1)+···+arg(µn))i,we also haven/summationtext\nj=1arg(µj) = 2kπ, withk∈Z.\nWe also denote by s=s(Q)≥0 the multiplicity of −1 as an eigenvalue of Q.\nWhens≥1 we have arg( µj) =πif and only if n−s+ 1≤j≤n, while ifs= 0 we\nhave arg(µj)/\\e}atio\\slash=πfor everyj= 1,···,n.\nb)Withthesamenotationsasabove, theeigenvaluesofthein verseQ∗ofQareµ1,···,µn.\nHence, ifs= 0 we have arg( µj) =−arg(µj) for every j= 1,···,n, while ifs≥1 we\nhave arg(µj) =−arg(µj) for every j= 1,···,n−sand arg(µj) = arg(µj) =πfor\neveryj=n−s+ 1,···,n. In particular, in any case we have s(Q∗) =s(Q) =sand\n−π <arg(µn−s)≤ ··· ≤ arg(µ1)<π; moreover, if s≥1 we have arg( µj) =πfor every\nj=n−s+1,···,n.\nc) We setζ(Q) :=1\n2πn/summationtext\nj=1arg(µj).\nFrom (a),ζ(Q) is an integer dependent only on the principal arguments of t he eigenvalues\nofQ, while, from (b), it is easy to check that we have ζ(Q∗) =s(Q)−ζ(Q).\nFrom this last equality it is easy to deduce that s(Q)−⌊n\n2⌋ ≤ζ(Q)≤ ⌊n\n2⌋and also that\nthe assumption ζ(Q)≥ζ(Q∗) impliesζ(Q)≥0.\n2.2.Remark-Definition. Consider any integer ζsuch that −⌊n\n2⌋ ≤ζ≤ ⌊n\n2⌋(n≥2);\nwe say that an n-tuple (α1,···,αn) of real numbers is ζ-admissible if\n−π<α 1≤α2≤ ··· ≤αn≤πandn/summationtext\nj=1αj= 2πζ .\nAs seen in Remarks-Definitions 2.1 (a) and (c), the eigenvalu esµ1,···,µnof any matrix\nQofSUncan be reordered so that the n-tuple (arg( µ1),···,arg(µn)) isζ(Q)-admissible,\nbeingζ(Q) the integer belonging to/bracketleftbig\n−⌊n\n2⌋,⌊n\n2⌋/bracketrightbig\ndefined in Remarks-Definitions 2.1 (c).\nConversely, it is clear that, if ( α1,···,αn) is anyζ-admissible n-tuple of real numbers\n(withζ∈/bracketleftbig\n−⌊n\n2⌋,⌊n\n2⌋/bracketrightbig\n∩Z), there is at least one matrix Q∈SUnwhoseneigenvalues\nareµ1=eα1i,···,µn=eαni; so we have α1= arg(µ1),···,αn= arg(µn) andζ=ζ(Q).\n2.3.Remarks-Definitions. a) Letm:SUn→Rbe the function defined by\nm(Q) := inf{/⌊ard⌊lX/⌊ard⌊l2\nφ:X∈sunand exp(X) =Q}\nand, for any Q∈SUn, let\nΘ(Q) :={X∈sun: exp(X) =Q,/⌊ard⌊lX/⌊ard⌊l2\nφ=m(Q)}.\nFrom Proposition 1.5, it follows that, for every Q∈SUn, the mapX/ma√sto→ /⌊ard⌊lX/⌊ard⌊l2\nφhas an\nabsolute minimum on the set of sun–logarithms of Q; so, for any Q∈SUn,we can write\nm(Q) = min{/⌊ard⌊lX/⌊ard⌊l2\nφ:X∈sunand exp(X) =Q}=d(In,Q)2(wheredis the distance\ninduced by the Frobenius metric of SUn), so that the set Θ( Q) is non-empty.\nNow, fixQ∈SUn(n≥2) and denote µ1···,µntheneigenvalues of Q, so that the\nn-tuple (arg( µ1)···,arg(µn)) isζ(Q)-admissible, with ζ(Q)∈/bracketleftbig\n−⌊n\n2⌋,⌊n\n2⌋/bracketrightbig\n∩Z. We have\n(*)m(Q)≥n/summationtext\nj=1/parenleftbig\narg(µj)/parenrightbig2.8 DONATO PERTICI AND ALBERTO DOLCETTI\nIndeed by definition we have: m(Q)≥min{/⌊ard⌊lX/⌊ard⌊l2\nφ:X∈unand exp(X) =Q}and this\nlast is equal ton/summationtext\nj=1/parenleftbig\narg(µj)/parenrightbig2(by [Pertici-Dolcetti 2022, Rem.5.4 and Prop.5.5]).\nNote that, if Xis asun–logarithm of Q, then theneigenvalues of Xare of the form\n(arg(µ1)+2k1π)i,···,(arg(µn)+2knπ)i, for some integers k1,···,knsuch that\nn/summationtext\nj=1(arg(µj)+2kjπ) = 2π/parenleftbig\nζ(Q)+n/summationtext\nj=1kj/parenrightbig\n= 0, i.e.n/summationtext\nj=1kj=−ζ(Q).\nConversely, if h1,···,hn∈Zandn/summationtext\nj=1hj=−ζ(Q) (i.e.n/summationtext\nj=1(arg(µj) + 2hjπ) = 0), it is\neasy to determine a matrix X0∈sunsuch that exp( X0) =Q, whoseneigenvalues are\n(arg(µ1) +2h1π)i,···,(arg(µn)+ 2hnπ)i, and therefore /⌊ard⌊lX0/⌊ard⌊l2\nφ=n/summationtext\nj=1/parenleftbig\narg(µj) +2hjπ/parenrightbig2.\nHence we get m(Q) = min{n/summationtext\nj=1/parenleftbig\narg(µj)+2kjπ/parenrightbig2:k1,···kn∈Zandn/summationtext\nj=1kj=−ζ(Q)}.\nLetWQ:={k:= (k1,···,kn)∈Zn:n/summationtext\nj=1kj=−ζ(Q)}and denote by ψQ:WQ→Rthe\nmap defined by ψQ(k) =ψQ(k1,···,kn) =n/summationtext\nj=1/parenleftbig\narg(µj)+2kjπ/parenrightbig2; then we can write\nm(Q) = min{ψQ(k) :k∈WQ}.\nNext, it will be also useful to consider the map ∆Q:WQ→Zdefined by\n∆Q(k) := max {k1,···,kn}−min{k1,···,kn}, withk= (k1,···,kn)∈WQ, and the set\nZQ:={k∈WQ: ∆Q(k)≤1}.\nNote thatZQ/\\e}atio\\slash=∅. Indeed, if ζ=ζ(Q)≥0, then (0,···,0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn−ζ,−1,···,−1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nζ)∈ZQ, while if\nζ=ζ(Q)<0, then (1,···,1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n−ζ,0,···,0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn+ζ)∈ZQ.\nb) Note that the map m:Q/ma√sto→m(Q) depends only on the ζ(Q)-admissible n-tuple\n(arg(µ1),···,arg(µn)), sowecanunambiguouslywrite m(arg(µ1),···,arg(µn)) :=m(Q).\nFrom this fact and from Remark-Definition 2.2, in the sequel w e will also be able to con-\nsidermasafunctiononlyofanarbitrary ζ-admissible n-tuple(α1,···,αn)ofrealnumbers\nsuch thatζ∈/bracketleftbig\n−⌊n\n2⌋,⌊n\n2⌋/bracketrightbig\n∩Z.\nc) Keeping in mind Remarks 2.1 (b), it is easy to check that we h avem(Q∗) =m(Q), for\nanyQ∈SUn, orequivalently, m(α1,···,αn−s,π,···,π/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\ns) =m(−αn−s,···,−α1,π,···,π/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\ns),\nfor anyζ-admissible n-tuple (α1,···,αn−s,π,···,π/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\ns), withαn−s<π, s= 0,1,···,nand\nζ∈/bracketleftbig\n−⌊n\n2⌋,⌊n\n2⌋/bracketrightbig\n∩Z.Also note that, if Ψ is the automorphism of the vector space sun\ndefined by Ψ( X) :=−X(for anyX∈sun), we have Ψ/parenleftbig\nΘ(Q∗)/parenrightbig\n= Θ(Q).\nd) The map Q/ma√sto→m(Q) =d(Q,In)2is continuous on the compact Lie group SUn, so\nit has a maximum δnonSUn. Since (SUn,φ) is a homogeneous Riemannian manifold,\nit is clear that δn= max{m(Q) :Q∈SUn}agrees with the square of the diameter\nofSUnwith respect to d, i.e.δ(SUn,φ) =√δn.Keeping in mind (b), (c) above,\nRemarks 2.1 (c) and Remark-Definition 2.2, we deduce that δnis the maximum of the set\n/braceleftbig\nm(α1,···,αn) : (α1,···,αn)∈Rnis anyζ-admissible n-tuple, with ζ∈/bracketleftbig\n0,⌊n\n2⌋/bracketrightbig\n∩Z/bracerightbig\n.SOME RIEMANNIAN PROPERTIES OF SUn 9\n2.4.Remark. For anyQ∈SUnandU∈Un,we havem/parenleftbig\nAdU(Q)/parenrightbig\n=m(Q) and\nΘ/parenleftbig\nAdU(Q)/parenrightbig\n=AdU/parenleftbig\nΘ(Q)/parenrightbig\n.\nThe equality m/parenleftbig\nAdU(Q)/parenrightbig\n=m(Q) holds because the function m(Q) depends only on\nthe eigenvalues of Q,while the second equality Θ/parenleftbig\nAdU(Q)/parenrightbig\n=AdU/parenleftbig\nΘ(Q)/parenrightbig\ncan be easily\ndeduced from the first one, remembering that the map AdU: (SUn,φ)→(SUn,φ) is an\nisometry that commutes with the exponential map.\n2.5.Lemma. LetQbe a matrix of SUn(n≥2), whoseneigenvalues are µ1,···,µn.\nDenote by s(Q)the multiplicity of −1as an eigenvalue of Qand let\nζ(Q) =1\n2πn/summationtext\nj=1arg(µj), m(Q) = min{/⌊ard⌊lX/⌊ard⌊l2\nφ:X∈sunandexp(X) =Q},\nΘ(Q) ={X∈sun: exp(X) =Q,/⌊ard⌊lX/⌊ard⌊l2\nφ=m(Q)}.\nThen the following facts are equivalent:\ni)sun–plog(Q)/\\e}atio\\slash=∅;\nii)ζ(Q)∈ {0,1,···,s(Q)};\niii)m(Q) =n/summationtext\nj=1/parenleftbig\narg(µj)/parenrightbig2.\nMoreover, if any of the above conditions holds, then we have Θ(Q) =sun–plog(Q).\nProof.The equivalence between (i) and (iii) and the final assertion follow directly from\nthe proof of [Pertici-Dolcetti 2022, Prop.5.5(b)].\nSets:=s(Q) andζ:=ζ(Q). We can assume arg( µ1)≤arg(µ2)≤ ··· ≤arg(µn); so we\nhaveµn−s+1=µn−s+2=···=µn=−1, whileµ1,···,µn−sare all different from −1.\nAssume now (i) and fix a matrix X∈sun–plog(Q) so that the neigenvalues of Xare:\narg(µ1)i,···,arg(µn−s)i,−πiwith multiplicity k≥0 andπiwith multiplicity s−k≥0,\nfor somek∈ {0,···,s}. The condition tr( X) = 0 impliesn−s/summationtext\nj=1arg(µj) + (s−2k)π= 0.\nSinceζ=1\n2πn/summationtext\nj=1arg(µj) =1\n2π/parenleftbign−s/summationtext\nj=1arg(µj)+sπ/parenrightbig\n, we obtain ζ=k∈ {0,1,···,s}and so\n(i) implies (ii).\nAssume now (ii) (i.e. ζ∈ {0,1,···,s}). By Remarks-Definition 2.3 (a), we have\nm(Q) = min{n/summationtext\nj=1/parenleftbig\narg(µj)+2kjπ/parenrightbig2:k1,···kn∈Zandn/summationtext\nj=1kj=−ζ}.\nChooseh1=···hn−ζ= 0 andhn−ζ+1=hn−ζ+2=···=hn=−1, so thatn/summationtext\nj=1hj=−ζ.\nSinceζ≤s, wegetn/summationtext\nj=1/parenleftbig\narg(µj)+2hjπ/parenrightbig2=n−ζ/summationtext\nj=1/parenleftbig\narg(µj)/parenrightbig2+ζ(−π)2=n/summationtext\nj=1/parenleftbig\narg(µj)/parenrightbig2. This\ngivesm(Q)≤n/summationtext\nj=1/parenleftbig\narg(µj)/parenrightbig2. The equality follows from the inequality (*) of Remarks-\nDefinitions 2.3(a). Thus (ii) implies (iii) and the proof is c omplete. /square\n2.6.Lemma. With the same notations as in Remarks-Definitions 2.3 (a), we have\nψQ(k)>m(Q)for everyk∈WQ\\ZQ. Therefore m(Q) = min{ψQ(k) :k∈ZQ}.\nProof.Remembering that m(Q) = min{ψQ(k) :k∈WQ}, it suffices to prove that, if\nk∈WQ\\ZQ, then there exists l∈WQsuch thatψQ(k)>ψQ(l).10 DONATO PERTICI AND ALBERTO DOLCETTI\nLetk= (k1,···,kn)∈WQ\\ZQ, i.e.k1,···,kn∈Z,n/summationtext\nj=1kj=−ζ(Q) and ∆Q(k)≥2.\nFixtwoindices r,t∈ {1,···,n}suchthatkt= max{k1,···,kn}andkr= min{k1,···,kn}.\nFrom the definition we have kt−kr= ∆Q(k)≥2.\nWe definelj:=kjfor everyj∈ {1,···,n} \\ {r,t},lr:=kr+ 1 ,lt:=kt−1 and\nl:= (l1,···,ln)∈Zn. Sincen/summationtext\nj=1lj=n/summationtext\nj=1kj=−ζ(Q),we havel∈WQ.\nThe inequality ψQ(k)>ψQ(l) is equivalent to\n(arg(µr)+2krπ)2+(arg(µt)+2ktπ)2>(arg(µr)+2lrπ)2+(arg(µt)+2ltπ)2and the latter\nis equivalent to the inequality 2( kt−kr)π >2π+ arg(µr)−arg(µt), which is satisfied\nsince 2(kt−kr)π≥4πand arg(µr)−arg(µt)<2π.Hence we have ψQ(k)>ψQ(l) and\nthis concludes the proof. /square\n2.7.Lemma. With the same notations as in Remarks-Definitions 2.3 (a), le t\nk= (k1,···,kn)∈ZQ.\nIfζ(Q)≥0, thenkhasζ(Q)entries equal to −1, while the remaining ones are equal to\n0. In particular, if ζ(Q) = 0, thenZQconsists of the unique element (0,···,0).\nProof.Setζ:=ζ(Q). Since ∆Q(k)≤1, there exists an integer Hand a non-empty set\nJ⊆ {1,···,n}suchthatkj=Hfor everyj∈Jandki=H−1for everyi∈ {1,···,n}\\J.\nLetχ≥1 be the cardinality of J. Then−ζ=n/summationtext\nj=1kj=χH+(n−χ)(H−1) =n(H−1)+χ.\nRemembering Remarks 2.1 (c), we get n(H−1) =−ζ−χ≥ −⌊n\n2⌋−n>−2n: soH≥0.\nSinceζ≥0, we have n(H−1) =−ζ−χ≤ −1, henceH <1 and soH= 0.Thereforek\nnecessarily has ζentries equal to −1, while the remaining ones are equal to 0. /square\n2.8.Proposition. LetQbe a matrix of SUn(n≥2) whoseneigenvalues are µ1,···,µn\nordered so that we have arg(µ1)≤arg(µ2)≤ ··· ≤arg(µn)and assume that\nζ(Q) =1\n2πn/summationtext\nj=1arg(µj)≥0.Letm(Q) = min{/⌊ard⌊lX/⌊ard⌊l2\nφ:X∈sunandexp(X) =Q}and\nΘ(Q) ={X∈sun: exp(X) =Q,/⌊ard⌊lX/⌊ard⌊l2\nφ=m(Q)}.\nFinally, let X0be anysun–logarithm of Qwhoseneigenvalues are :\n(arg(µ1)+2h1π)i,···,(arg(µn)+2hnπ)i(withh1,···,hn∈Zandn/summationtext\nj=1hj=−ζ(Q)).\na) Ifζ(Q) = 0, thenm(Q) =n/summationtext\nj=1/parenleftbig\narg(µj)/parenrightbig2;\nfurthermore, X0∈Θ(Q)if and only if hj= 0for everyj∈ {1,···,n}.\nb) Ifζ(Q)≥1,thenm(Q) =n−ζ(Q)/summationtext\nj=1/parenleftbig\narg(µj)/parenrightbig2+n/summationtext\nj=n−ζ(Q)+1/parenleftbig\n2π−arg(µj)/parenrightbig2;\nfurthermore, if µn−ζ(Q)/\\e}atio\\slash=µn−ζ(Q)+1,we haveX0∈Θ(Q)if and only if\nhj= 0for everyj∈ {1,···,n−ζ(Q)}andhl=−1for everyl∈ {n−ζ(Q)+1,···,n};\nwhile, ifµn−ζ(Q)=µn−ζ(Q)+1, we have X0∈Θ(Q)if and only if\nhr= 0for everyr∈ {1,···,n−ζ(Q)−1}such thatµr/\\e}atio\\slash=µn−ζ(Q),\nht=−1for everyt∈ {n−ζ(Q)+2,···,n}such thatµt/\\e}atio\\slash=µn−ζ(Q),\nandhm= 0orhm=−1for every index msuch thatµm=µn−ζ(Q)SOME RIEMANNIAN PROPERTIES OF SUn 11\n(with the constraint that the equalityn/summationtext\nj=1hj=−ζ(Q)is satisfied ).\nProof.Setζ:=ζ(Q).\nIfζ= 0,the formula for m(Q) follows directly from Lemma 2.5. Clearly, if\nh1=···=hn= 0 wehave /⌊ard⌊lX0/⌊ard⌊l2\nφ=m(Q),andsoX0∈Θ(Q).Conversely, if X0∈Θ(Q)\nthen, by Lemma 2.6, ( h1,···,hn)∈ZQ; hence, by Lemma 2.7, we obtain hj= 0 for\neveryj∈ {1,···,n}and the proof of (a) is complete.\nNow letζ≥1,and denote k0:= (0,···,0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn−ζ,−1,···,−1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nζ)∈ZQ⊂WQ.Remembering\nRemarks-Definitions 2.3 (a), we want to prove that the minimu m of the map ψQonWQ\nis reached at k0.By Lemma 2.6, this minimum is reached only on ZQ, while, by Lemma\n2.7, everyk∈ZQhasζentries equal to −1 and the remaining ones equal to 0. Let\nk= (k1,···,kn)∈ZQand assume that there are two indices 1 ≤i < r≤nsuch\nthatki=−1 andkr= 0; then we define a new element l= (l1,···,ln)∈ZQwith\nlj=kjfor anyj/\\e}atio\\slash=i,r, l i= 0, lr=−1, and we obtain ψQ(k)≥ψQ(l).Indeed this\ninequality is equivalent to/parenleftbig\narg(µi)−2π/parenrightbig2+/parenleftbig\narg(µr)/parenrightbig2≥/parenleftbig\narg(µi)/parenrightbig2+/parenleftbig\narg(µr)−2π/parenrightbig2,\nand the latter inequality is equivalent to arg( µi)≤arg(µr),which is satisfied since i<r.\nAlso note that we have ψQ(k) =ψQ(l) if and only if arg( µi) = arg(µr),i.e. if and\nonly ifµi=µr.Repeating, if necessary, on lthe same operation performed on k, after\na finite number of steps we obtain ψQ(k)≥ψQ(k0), for anyk∈ZQ.It follows that\nwe havem(Q) =ψQ(k0) =n−ζ/summationtext\nj=1/parenleftbig\narg(µj)/parenrightbig2+n/summationtext\nj=n−ζ+1/parenleftbig\n2π−arg(µj)/parenrightbig2.We also obtain\nthat, ifµn−ζ/\\e}atio\\slash=µn−ζ+1,the equality ψQ(k) =ψQ(k0) holds if and only if k=k0;\nwhile, ifµn−ζ=µn−ζ+1,we haveψQ(k) =ψQ(k0) if and only if kr= 0 for every\nr∈ {1,···,n−ζ−1}such thatµr/\\e}atio\\slash=µn−ζ,kt=−1 for every t∈ {n−ζ+ 2,···,n}\nsuch thatµt/\\e}atio\\slash=µn−ζ, andkm= 0 orkm=−1 for every msuch thatµm=µn−ζ.Thus\nthe second part of statement (b) has also been proved. /square\n2.9.Remark. Note that, if 0 ≤ζ(Q)≤s(Q), the formulas in part (a) and in part (b) of\nProposition 2.8, reduce to formula (iii) of Lemma 2.5.\n3.The set Θ(Q)of minimizing logarithms of any special unitary matrix Q\n3.1.Remark. From Proposition 2.8 and with the same notations and hypothe ses, we get:\ni) ifζ(Q) = 0,then Θ(Q) agrees with the set of all sun–logarithms of Qwhosen\neigenvalues are : arg( µ1)i,arg(µ2)i,···,arg(µn−1)i,arg(µn)i;\nii) ifζ(Q)≥1,then Θ(Q) agrees with the set of all sun–logarithms of Qwhosen\neigenvalues are the following:\narg(µ1)i,arg(µ2)i,···,arg(µn−ζ(Q)−1)i,arg(µn−ζ(Q))i,/parenleftbig\narg(µn−ζ(Q)+1)−2π/parenrightbig\ni,\n/parenleftbig\narg(µn−ζ(Q)+2)−2π/parenrightbig\ni,···,/parenleftbig\narg(µn−1)−2π/parenrightbig\ni,/parenleftbig\narg(µn)−2π/parenrightbig\ni.12 DONATO PERTICI AND ALBERTO DOLCETTI\n3.2.Proposition. LetQbe a matrix of SUn(n≥2) whoseneigenvalues are µ1,···,µn\nordered so that we have arg(µ1)≤arg(µ2)≤ ··· ≤arg(µn)and assume that\nζ(Q) =1\n2πn/summationtext\nj=1arg(µj)≥0.Letm(Q) = min{/⌊ard⌊lX/⌊ard⌊l2\nφ:X∈sunandexp(X) =Q}and\nΘ(Q) =/braceleftbig\nX∈sun: exp(X) =Q,/⌊ard⌊lX/⌊ard⌊l2\nφ=m(Q)/bracerightbig\n.\nIfζ(Q)≥1, we also set ν1:=max/braceleftbig\nj∈ {1,···,n−ζ(Q)}:µn−ζ(Q)+1−j=µn−ζ(Q)/bracerightbig\nandν2:=max/braceleftbig\nj∈ {1,···,ζ(Q)}:µn−ζ(Q)+j=µn−ζ(Q)+1/bracerightbig\n.\na) Ifζ(Q) = 0, thenΘ(Q)is a single point.\nb) Ifζ(Q)≥1andµn−ζ(Q)/\\e}atio\\slash=µn−ζ(Q)+1, thenΘ(Q)is a single point.\nc) Ifζ(Q)≥1andµn−ζ(Q)=µn−ζ(Q)+1, thenΘ(Q)is a compact submanifold of sun\ndiffeomorphic to the symmetric spaceU(ν1+ν2)\nUν1⊕Uν2and therefore also to the complex Grass-\nmannian Gr(ν2;C(ν1+ν2)).\nProof.Setβ:=µn−ζ(Q).We denote by S1the set consisting of all distinct eigenvalues\nλofQsuch that arg( λ)<arg(β) and byS2the set consisting of all distinct eigenvalues\nϕofQsuch that arg( ϕ)>arg(β) (S1andS2can also be empty); for every eigenvalue ǫ\nofQ,we also denote by m(ǫ)≥1 the multiplicity of ǫas an eigenvalue of Q.If we set\nJ:=/parenleftbig/circleplustext\nλ∈S1λIm(λ)/parenrightbig\n⊕/parenleftbig\nβIm(β)/parenrightbig\n⊕/parenleftbig/circleplustext\nϕ∈S2ϕIm(ϕ)/parenrightbig\n,it is a well-known fact that there exists\nU∈Unsuch thatQ=AdU(J); so, by Remark 2.4 , we can assume that\nQ=J=/parenleftbig/circleplustext\nλ∈S1λIm(λ)/parenrightbig\n⊕/parenleftbig\nβIm(β)/parenrightbig\n⊕/parenleftbig/circleplustext\nϕ∈S2ϕIm(ϕ)/parenrightbig\n.\nNow, ifζ(Q) = 0 as in (a) or if ζ(Q)≥1 andβ=µn−ζ(Q)/\\e}atio\\slash=µn−ζ(Q)+1as in (b), we set\n/hatwideJ:=/parenleftbig/circleplustext\nλ∈S1arg(λ)iIm(λ)/parenrightbig\n⊕/parenleftbig\narg(β)iIm(β)/parenrightbig\n⊕/parenleftbig/circleplustext\nϕ∈S2(arg(ϕ)−2π)iIm(ϕ)/parenrightbig\n.\n(Here we agree that, if S1orS2are empty, the terms corresponding to them do not appear\nin all previous direct sums.)\nNote that exp( /hatwideJ) =J=Q.By Remark 3.1, a matrix X∈sunbelongs to Θ( Q) if and only\nif there isR∈Unsuch thatX=AdR(/hatwideJ) andQ= exp(X) =AdR(exp(/hatwideJ)) =AdR(Q),\ni. e. if and only if X=AdR(/hatwideJ) for some R∈Unsuch thatRQ=QR.By Lemma 1.2, a\nmatrixR∈Uncommutes with Qif and only if R∈/parenleftbig/circleplustext\nλ∈S1Um(λ)/parenrightbig\n⊕Um(β)⊕/parenleftbig/circleplustext\nϕ∈S2Um(ϕ)/parenrightbig\n;\nthis implies that Ralso commutes with /hatwideJ,and soX=AdR(/hatwideJ) =/hatwideJ.Hence/hatwideJis the\nunique element of Θ( Q).This proves (a) and (b).\nNow, assume as in (c): ζ(Q)≥1 andβ=µn−ζ(Q)=µn−ζ(Q)+1,so thatm(β) =ν1+ν2.\nIf we set /tildewideJ:=\n/parenleftbig/circleplustext\nλ∈S1arg(λ)iIm(λ)/parenrightbig\n⊕/parenleftbig\narg(β)iIν1/parenrightbig\n⊕/parenleftbig\n(arg(β)−2π)iIν2/parenrightbig\n⊕/parenleftbig/circleplustext\nϕ∈S2(arg(ϕ)−2π)iIm(ϕ)/parenrightbig\n,\nwe have exp( /tildewideJ) =J=Q; as in the proof of parts (a) and (b) and using Lemma 1.2 and\nRemark 3.1 again, we can prove that a matrix X∈sunbelongs to Θ( Q) if and only if\nX=AdR(/tildewideJ) for some R∈Uncommuting with Q,i.e. if and only if X=AdR(/tildewideJ) for\nsomeR∈G:=/parenleftbig/circleplustext\nλ∈S1Um(λ)/parenrightbig\n⊕U(ν1+ν2)⊕/parenleftbig/circleplustext\nϕ∈S2Um(ϕ)/parenrightbig\n.Since the map ( R,X)/ma√sto→AdR(X)\ndefines a left action of the compact Lie group Gonsun,we conclude that Θ( Q) agrees\nwith the orbit of /tildewideJwith respect to this action, while the corresponding isotro py subgroupSOME RIEMANNIAN PROPERTIES OF SUn 13\nat/tildewideJis/hatwideG:=/parenleftbig/circleplustext\nλ∈S1Um(λ)/parenrightbig\n⊕Uν1⊕Uν2⊕/parenleftbig/circleplustext\nϕ∈S2Um(ϕ)/parenrightbig\n.Hence we conclude that Θ( Q)\nis a compact submanifold of sundiffeomorphic toG\n/hatwideG∼=U(ν1+ν2)\nUν1⊕Uν2(see, for instance,\n[EoM-Orbit]). Since it is known thatU(ν1+ν2)\nUν1⊕Uν2is a symmetric space diffeomorphic to\nthecomplex Grassmannian Gr(ν2;C(ν1+ν2)),theproof of theProposition is complete. /square\n3.3.Proposition. LetQbe a matrix of SUn(n≥2) whoseneigenvalues are µ1,···,µn.\nDenote by s(Q)the multiplicity of −1as an eigenvalue of Qand let\nζ(Q) =1\n2πn/summationtext\nj=1arg(µj).Then\na) ifζ(Q)>s(Q)orζ(Q)<0,the setsun–plog(Q)is empty;\nb) if0≤ζ(Q)≤s(Q),the setsun–plog(Q)is a compact submanifold of sundiffeomorphic\nto the complex Grassmannian Gr(ζ(Q);Cs(Q));in particular, sun–plog(Q)is a single\npoint if and only if ζ(Q) = 0orζ(Q) =s(Q).\nProof.The proof follows directly from Lemma 2.5 and Proposition 3. 2. /square\n4.About diameter and diametral pairs of (SUn,φ)\n4.1.Proposition. Letm:SUn→Rbe the map defined in Remarks-Definitions 2.3 (a)\nand letδn= max{m(Q) :Q∈SUn}(n≥2).Then we have\nδn=\n\nnπ2ifnis even\n(n−1\nn)π2ifnis odd;\nfurthermore a matrix Q∈SUnsatisfiesm(Q) =δnif and only if\n\nQ=−In ifnis even\nQ=e(n−1)πi\nnInorQ=e−(n−1)πi\nnIn ifnis odd.\nProof.As noted in Remarks-Definitions 2.3 (d), δnagrees with the absolute maximum\nof the map ( α1,···,αn)/ma√sto→m(α1,···,αn), where (α1,···,αn) varies over the set of all\nζ-admissible n-tuples of real numbers, with ζ∈/bracketleftbig\n0,⌊n\n2⌋/bracketrightbig\n∩Z.Now, fix any ζ-admissible\nn-tuple (β1,···,βn) such that m(β1,···,βn) =δn(where1\n2πn/summationtext\nj=1βj=ζ∈/bracketleftbig\n0,⌊n\n2⌋/bracketrightbig\n∩Z).\nFirst of all, we will prove that we necessarily have ( β1,···,βn) = (π,···,π/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn) whenn\nis even and ( β1,···,βn) = ((n−1)π\nn,···,(n−1)π\nn/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nn) whennis odd, and then we will\nprove all the statements of the Proposition. The proof will b e done in several steps.\ni) Ifζ≥1,at least one of the following two conditions is necessarily t rue:\n(ia)β1=β2=···=βn−ζ=βn−ζ+1;\n(ib)βn−ζ=βn−ζ+1=···=βn−1=βn;\nwhile, ifζ= 0,condition (ib) is obviously always satisfied.\nIn the case ζ≥1,if both conditions (ia) and (ib) are false, then the sets\nJ1:=/braceleftbig\ni∈ {1,···,n−ζ}:βi<βn−ζ+1/bracerightbig\n, J2:=/braceleftbig\nj∈ {n−ζ+1,···,n}:βj>βn−ζ/bracerightbig\nare\nboth non-empty, and therefore we can define h:= maxJ1, k:= minJ2.Note that, from14 DONATO PERTICI AND ALBERTO DOLCETTI\ndefinitions of handk, we haveβh< βh+1andβk−1< βk; so we can choose ǫ>0 such\nthatβh+ǫ < β h+1andβk−1< βk−ǫ. Now we consider the new n-tuple (γ1,···,γn),\nwhereγr:=βrwhenr/\\e}atio\\slash=h,k, γ h:=βh+ǫ, γk:=βk−ǫ; we haven/summationtext\nj=1γj=n/summationtext\nj=1βj, so the\nn-tuple (γ1,···,γn) is alsoζ−admissible. Since βk−βh<2π, by Proposition 2.8 (b), we\ngetm(γ1,···,γn)−δn=γ2\nh+(2π−γk)2−/bracketleftbig\nβ2\nh+(2π−βk)2/bracketrightbig\n= 2ǫ/parenleftbig\nǫ+2π−(βk−βh)/parenrightbig\n>0,\nand this is contrary to the definition of δn. So we conclude that (ia) or (ib) is true.\nii) If condition (ia) holds (with ζ≥1), thenβ1=β2=···=βn−1=βn.\nIn fact, otherwise, there is an index r∈/braceleftbig\nn−ζ+1, n−ζ+2,···,n−1,n/bracerightbig\n, such that\nβr+1>βr=βr−1=···=βn−ζ+1=βn−ζ=···=β1.Then, after setting w:=βr+1and\nz:=βr=···=β1(so that we have rz= 2πζ−w−n/summationtext\nj=r+2βj), we choose ǫ>0 such that\nw−r·ǫ>z+ǫ,and we consider the ζ-admissible n-tuple (ϕ1,···,ϕn) defined by\nϕj:=z+ǫforj= 1,···,r, ϕ r+1:=w−r·ǫandϕh:=βhforh=r+2,···,n.\nTaking into account Proposition 2.8 (b), we obtain m(ϕ1,···,ϕn)−δn=\n(n−ζ)/bracketleftbig\n(z+ǫ)2−z2/bracketrightbig\n+(r−n+ζ)/bracketleftbig\n(2π−z−ǫ)2−(2π−z)2/bracketrightbig\n+(2π−w+rǫ)2−(2π−w)2=\nǫ2(r+r2)+2ǫ/bracketleftbig\n(n−ζ)z−(r−n+ζ)(2π−z)+r(2π−w)/bracketrightbig\n=ǫ2(r+r2)+2ǫ/bracketleftbig\nrz+2π(n−ζ)−rw/bracketrightbig\n=\nǫ2(r+r2)+2ǫ/bracketleftbig\n2πn−(r+1)w−n/summationtext\nj=r+2βj/bracketrightbig\n.Since (r+1)w+n/summationtext\nj=r+2βj≤nπ,then we get\nm(ϕ1,···,ϕn)−δn≥ǫ2(r+r2)+2ǫnπ >0,and this is impossible, bearing in mind the\ndefinition of δn. So we conclude that (ii) holds.\niii) If condition (ib) holds (now also with ζ≥0), thenβ2=β3=···=βn−1=βn.\nOtherwise, we have K:=/braceleftbig\nj∈ {2,3,···,n−ζ−1}:βj< βn−ζ/bracerightbig\n/\\e}atio\\slash=∅,and so, called\nt:= maxK,we fix a positive real number ǫsuch that −π<β 1−ǫ, βt+ǫ<βt+1=βn−ζ\nand we consider the ζ-admissible n-tuple (σ1,···,σn) defined by σi:=βifori/\\e}atio\\slash= 1,t,\nσ1:=β1−ǫandσt:=βt+ǫ.From Proposition 2.8, we obtain m(σ1,···,σn)−δn=\n(β1−ǫ)2−β2\n1+(βt+ǫ)2−β2\nt= 2ǫ(ǫ+βt−β1)>0,sinceβ1≤βt.Again, this contradicts\nthe definition of δn, so we conclude that (iii) is true.\niv) From (i), (ii), (iii), we deduce that, whatever the value ofζ∈/bracketleftbig\n0,⌊n\n2⌋/bracketrightbig\n∩Z, we\nnecessarily have −π<β 1≤β2=β3=···=βn≤π.\nSettingx0:=β2=···=βn,we haveβ1= 2πζ−(n−1)x0,whereζ∈/bracketleftbig\n0,⌊n\n2⌋/bracketrightbig\n∩Z,\n−π <2πζ−(n−1)x0≤x0≤π; these inequalities imply2ζπ\nn≤x0<(2ζ+1)π\nn−1if\n0≤ζ≤ ⌊n\n2⌋−1 and2ζπ\nn≤x0≤πifζ=⌊n\n2⌋.\nIn other words, necessarily the pair ( x0,ζ) belongs to the set Λ defined by\nΛ :=/braceleftbig\n(x,ϑ) :x∈[2ϑπ\nn,(2ϑ+1)π\nn−1), ϑ∈/bracketleftbig\n0,⌊n\n2⌋−1/bracketrightbig\n∩Zorx∈[2ϑπ\nn,π], ϑ=⌊n\n2⌋/bracerightbig\n.\nConversely, itiseasytocheckthat, forany( x,ϑ)∈Λ, then-tuple/parenleftbig\n2πϑ−(n−1)x,x,···,x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn−1/parenrightbig\nisϑ-admissible, with ϑ∈/bracketleftbig\n0,⌊n\n2⌋/bracketrightbig\n∩Z. Hence, if we define\nFϑ(x) :=m/parenleftbig\n2πϑ−(n−1)x,x,···, x/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn−1/parenrightbig\n,we haveδn=Fζ(x0)≥Fϑ(x),for any(x,ϑ)∈Λ.\nFrom Proposition 2.8, by means of easy calculations we obtai n :SOME RIEMANNIAN PROPERTIES OF SUn 15\nFϑ(x) =n(n−1)x2−4πϑnx+4π2ϑ(ϑ+1),for every (x,ϑ)∈Λ.\nFor anyfixed ϑ∈/bracketleftbig\n0,⌊n\n2⌋−1/bracketrightbig\n∩Z,thequadratic function Fϑreaches its absolute minimum\nat the point xϑ=2ϑπ\nn−1∈[2ϑπ\nn,(2ϑ+1)π\nn−1),and we have\nxϑ−2ϑπ\nn=2ϑπ\nn(n−1)<π\nn−1=(2ϑ+1)π\nn−1−xϑ; hence, for every x∈[2ϑπ\nn,(2ϑ+1)π\nn−1),\nthere exists /hatwidex∈[2ϑπ\nn,(2ϑ+1)π\nn−1) such that Fϑ(/hatwidex)>Fϑ(x); this implies that ϑ/\\e}atio\\slash=ζ,for\neveryϑ∈/bracketleftbig\n0,⌊n\n2⌋−1/bracketrightbig\n∩Z,and so we conclude that it must necessarily be ζ=⌊n\n2⌋. The\nfunctionFζ(x) (withζ=⌊n\n2⌋) is strictly decreasing on the interval/bracketleftbig2⌊n\n2⌋\nnπ,π/bracketrightbig\n,so it\nhas its unique maximum at the point2⌊n\n2⌋π\nn.We conclude that it must necessarily be\nx0=(n−1)π\nnwhennis odd, and x0=πwhennis even.\nSo, with an easy calculation, we obtain that we necessarily h aveβ1=β2=···=βn=π\nwhennis even, and β1=β2=···=βn=(n−1)π\nnwhennis odd.\nv) Since all matrices of SUnare diagonalizable, any n-tuple (λ,···,λ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn) (withλ∈[0,π]\nandλn\n2π∈/bracketleftbig\n0,⌊n\n2⌋/bracketrightbig\n∩Z) corresponds only to the matrix eλiIn∈SUn. Therefore, taking\ninto account Remarks-Definitions 2.3 (c), from (iv) we get th at we have m(Q) =δnif and\nonly ifQ=−Inwhennis even, and Q=e(n−1)πi\nnInorQ=e−(n−1)πi\nnInwhennis odd.\nHence, by means of Proposition 2.8, it is easy to check that δn=nπ2whennis even,\nandδn= (n−1\nn)π2whennis odd; so the proof is complete. /square\n5.Some geometrical properties of the Riemannian manifold (SUn,φ)\n5.1.Theorem. Letdbe the distance induced on SUnby the Frobenius metric φand let\nP,Q∈SUn.With the same notation of Remarks 2.1 (c), without loss of gen erality we\ncan assume ζ(P∗Q)≥ζ(Q∗P) (so thatζ(P∗Q)≥0by Remarks 2.1 (c)); we denote by\nµ1,···,µntheneigenvalues of P∗Qordered so that arg(µ1)≤arg(µ2)≤ ··· ≤arg(µn),\nand we set ζ:=ζ(P∗Q) =1\n2πn/summationtext\nj=1arg(µj)≥0.Then the following statements hold:\na)d(P;Q) =\n\n/radicalBigg\nn/summationtext\nj=1/parenleftbig\narg(µj)/parenrightbig2ifζ= 0,\n/radicalBigg\nn−ζ/summationtext\nj=1/parenleftbig\narg(µj)/parenrightbig2+n/summationtext\nj=n−ζ+1/parenleftbig\n2π−arg(µj)/parenrightbig2ifζ≥1;\nb) there exists a unique minimizing geodesic segment of (SUn,φ)with endpoints Pand\nQif and only if either ζ= 0 orζ≥1andµn−ζ/\\e}atio\\slash=µn−ζ+1;\nc) ifζ≥1andµn−ζ=µn−ζ+1,\nafter setting Θ(P∗Q) =/braceleftbig\nX∈sun: exp(X) =P∗Q,/⌊ard⌊lX/⌊ard⌊lφ=d(P,Q)},\nν1:=max/braceleftbig\nj∈ {1,···,n−ζ}:µn−ζ+1−j=µn−ζ/bracerightbig\n,\nν2:=max/braceleftbig\nj∈ {1,···,ζ}:µn−ζ+j=µn−ζ+1/bracerightbig\n,16 DONATO PERTICI AND ALBERTO DOLCETTI\nthe map:X/ma√sto→γ(t) :=Pexp(tX) (0≤t≤1)is a bijection from Θ(P∗Q)onto the set\nof minimizing geodesic segments of (SUn,φ)with endpoints PandQ, andΘ(P∗Q)is a\ncompact submanifold of sundiffeomorphic to the complex Grassmannian Gr(ν2;C(ν1+ν2)).\nProof.Sinceφis bi-invariant, we have/radicalbig\nm(P∗Q) =d(In,P∗Q) =d(P,Q); so part (a)\nfollows from Proposition 2.8, while parts (b) and (c) follow from Propositions 1.5 and\n3.2. /square\n5.2.Theorem. a) The diameter of (SUn,φ)is\nδ(SUn,φ) =\n\n√nπ ifnis even/radicalbigg\nn−1\nnπ ifnis odd.\nb) Ifnis even and Pis any matrix of SUn, then−Pis the unique diametral point of\nPin(SUn,φ)and the set of minimizing geodesic segments joining Pand−Pcan be\nparametrized by the complex Grassmannian Gr(n\n2;Cn).\nc) Ifnis odd and Pis any matrix of SUn, thenPhas precisely two diametral points\nP+,P−in(SUn,φ),withP+=e(n−1)πi\nnPandP−=e−(n−1)πi\nnP,and the sets of\nminimizing geodesic segments joining PwithP+andPwithP−can both be parametrized\nby the complex Grassmannian Gr(n−1\n2;Cn).\nProof.The Theorem follows easily from Proposition 4.1 and Theorem 5.1, taking into\naccount that, in case of diametral pairs, the values of the co nstants in part (c) of Theorem\n5.1 are :ζ=ν2=⌊n\n2⌋, ν1=n−⌊n\n2⌋. /square\nReferences\n[Alexandrino-Bettiol 2015] ALEXANDRINO Marcos M., BETTIO L Renato G., Lie Groups and\nGeometric Aspects of Isometric Actions , 2015, Springer, Cham.\n[Barbaresco 2008] BARBARESCOFr´ ed´ eric, “Innovative too lsforradarsignalprocessingbased on\nCartan’s geometry of SPD matrices & information geometry”, IEEE Radar Conference, Rome,\nItaly, 2008 , pp. 1-6.\n[Bhatia 2007] BHATIA Rajendra, Positive Definite Matrices , Princeton University Press, Prince-\nton.\n[Boumal 2023] BOUMAL Nicolas, An Introduction to Optimization on Smooth Manifolds , Cam-\nbridge University Press.\n[Bridson-Haefliger 1999] BRIDSON Martin R., HAEFLIGER Andr ´ e,Metric Spaces of Non-\nPositive Curvature , GMW 319, Springer-Verlag, Berlin.\n[Criscitiello-MartinezRubio-Boumal 2023] CRISCITIELLO Christopher, MARTINEZ-RUBIO\nDavid, BOUMAL Nicolas, “Open problem: polynomial linearly -convergent method for geodesi-\ncally convex optimization?”, Proceedings of Machine Learning Research , vol. 195:1–7.\n[Cruceru-Becigneul-Ganea 2021] CRUCERU Calin, BECIGNEUL Gary, GANEA Octavian-\nEugen, “Computationally tractable Riemannian manifolds f or graph embeddings”, Proceedings\nof the AAAI Conference on Artificial Intelligence , 35(8), 7133-7141.SOME RIEMANNIAN PROPERTIES OF SUn 17\n[Dolcetti-Pertici 2015] DOLCETTI Alberto, PERTICI Donato , “Some differential properties of\nGLn(R) with the trace metric”, Riv. Mat. Univ. Parma (N.S.), Vol. 6(2) (2015), 267–286.\n[Dolcetti-Pertici 2018] DOLCETTI Alberto, PERTICI Donato , “Skew symmetric logarithms and\ngeodesics on On(R)”,Adv. Geom. , 18(4): 495–507.\n[Dolcetti-Pertici 2019] DOLCETTI Alberto, PERTICI Donato , “Differential properties of spaces\nof symmetric real matrices”, Rendiconti Sem. Mat. Univ. Pol. Torino Vol. 77(1) (2019), 25–43.\n[Dolcetti-Pertici 2020] DOLCETTI Alberto, PERTICI Donato , “Real square roots of matrices:\ndifferential properties in semi-simple, symmetric and orth ogonal cases”, Riv. Mat. Univ. Parma\n(N.S.), Vol. 11(2) (2020), 315-333.\n[Dolcetti-Pertici 2021] DOLCETTI Alberto, PERTICI Donato , “Elliptic isometries of the mani-\nfold of positive definite real matrices with the trace metric ”,Rend. Circ. Mat. 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Thesis, QuSoft, Amsterdam, https://ir.cwi.nl/pub/33761/33761D1.pdf (ac-\ncessed on 1 February 2024).\n[Pertici 2023] PERTICI Donato, “Real logarithms of semi-si mple matrices”, in Boll. Unione Mat.\nItal.2023, 16(4): 649–666 .18 DONATO PERTICI AND ALBERTO DOLCETTI\n[Pertici-Dolcetti 2022] PERTICI Donato, DOLCETTI Alberto , “SVD-closed subgroups of the\nunitary group: generalized principal logarithms and minim izing geodesics”, arXiv:2212.11716v1,\nURL: https://arxiv.org/abs/2212.11716v1 (accessed on 18 may 2023).\n[Sepanski 2007] SEPANSKI Mark R., Compact Lie Groups , 2007, GTM 235, Springer, New York.\n[Zhang-Zhang-Sra 2023] ZHANG Peiyuan, ZHANG Jingzhao, SRA Suvrit, “Sion’s minimax the-\norem in geodesic metric spaces and a Riemannian extragradie nt algorithm”, SIAM J. OPTIM. ,\nVol. 33, No. 4 (2023), pp. 2885-2908."    },    {        "title": "2402.12213v1.Representation_formulas_and_far_field_behavior_of_time_periodic_incompressible_viscous_flow_around_a_translating_rigid_body.pdf",        "content": "arXiv:2402.12213v1  [math.AP]  19 Feb 2024Representation formulas and far-field behavior of\ntime-periodic incompressible viscous flow around a\ntranslating rigid body\nThomas Eiter∗†Ana Leonor Silvestre‡\nFebruary 20, 2024\nAbstract\nThispaper is concerned with integral representations and a symptotic expansions\nof solutions to the time-periodic incompressible Navier-S tokes equations for fluid\nflowintheexteriorofarigidbodythatmoveswithconstantve locity. Usingthetime-\nperiodicOseenfundamentalsolution, wederiverepresenta tion formulasforsolutions\nwith suitable regularity. From these formulas, the decompo sition of the velocity\ncomponent of the fundamental solution into steady-state an d purely periodic parts\nand their detailed decay rate in space, we deduce complete in formation on the\nasymptotic structure of the velocity and pressure fields.\nMSC2020: 35Q30, 76D05, 76D07, 35B10, 35A08, 35C15, 35C20\nKeywords: Time-periodic Navier-Stokes flows, exterior domains, time-periodic Oseen\nfundamental solution, integral representation, asymptotic exp ansion.\nContents\n1 Introduction 2\n2 Notations and fundamental solutions 3\n2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n2.2 Steady-state Stokes and Oseen fundamental solutions . . . . . . . . . . . . 4\n2.3 Time-periodic fundamental solutions . . . . . . . . . . . . . . . . . . . . . 6\n2.4 Anisotropic convolution estimates . . . . . . . . . . . . . . . . . . . . . . . 7\n∗Institute of Mathematics, University of Kassel, Heinrich-Plett Str . 40, 34132 Kassel, Germany\n†Weierstrass Institute for Applied Analysis and Stochastics, Mohre nstraße 39, 10117 Berlin, Germany.\nEmail:thomas.eiter@wias-berlin.de\n‡CEMAT and Department of Mathematics, Instituto Superior T´ ecn ico, Universidade de Lisboa, Av.\nRovisco Pais 1, 1049-001 Lisboa, Portugal.\nEmail:ana.silvestre@math.tecnico.ulisboa.pt\n13 Main results 9\n4 The linearized time-periodic problem 14\n4.1 Representation formulas for solutions . . . . . . . . . . . . . . . . . . . . . 14\n4.2 Asymptotic expansion of the velocity and pressure fields . . . . . . . . . . 20\n5 Proofs of the main results 25\n1 Introduction\nWe are interested in time-periodic incompressible viscous flow around a translating rigid\nbodyO. In a reference frame attached to the solid, the velocity v=v(t,x) and pressure\np=p(t,x) of the fluid satisfy\n\n\n∂tv+v·∇v−ν∆v+∇p−ζ·∇v=finT×Ω,\n∇·v= 0 in T×Ω,\nv=vbonT×Σ,\nlim\n|x|→∞v(t,x) = 0 for t∈T,(1.1)\nwhere Ω ⊂R3is an exterior domain and T:=R/TZis the torus group related to the\nperiodT>0 of the motion. The fluid has constant viscosity ν >0 and is subject to a\nT-periodic external force f. The body Oundergoes a translational motion with constant\nvelocityζ∈R3\\ {0}, and the fluid motion on its surface Σ := ∂Ω is prescribed via\nT-periodic boundary values vb, which may describe a periodic boundary motion or flux\nthrough the boundary.\nThe stationary counterpart of (1.1) is a classical mathematical pr oblem with well-\nknown properties, which have been studied, for example, in [1,15– 21]. In particular, the\nvelocity field exhibits different asymptotic behavior inside and outside a wake region cre-\nated by the translational motion of O. This behavior is inherited from the steady Oseen\nfundamental solution. From the point of view of applications, a prec ise description of\nthe far-field behavior of the flow can be very useful in numerical st udies, for which a\nbounded computational domain is defined, and consequently, appr opriate boundary con-\nditions must be imposed on the outer boundary of the truncated do main. Such boundary\nconditions should be consistent with the asymptotic decay of solutio ns, as was shown\nin [2,3].\nThe study of the time-periodic exterior problem (1.1) is more recent , see [6,9–12,22–\n24,27–29] for instance. In this paper, we exploit the mathematica l properties of time-\nperiodic Oseen fundamental solution, based on the results of [10,1 1,26], wherein a modern\nmathematical setting was developed for the time-periodic Stokes a nd Oseen problems and\nextended to the exterior Navier-Stokes equations. The first ste p of our analysis is the\nintegral representation obtained in Theorem 3.2 for ( v,p), where we followed [30]. Then,\nemploying the decomposition of the velocity component of the funda mental solution into\n2steady-stateandpurelyperiodiccontributions, wededucecomple teasymptoticexpansions\nfor the steady and purely periodic parts of the velocity and the pre ssure fields, the main\nresults being stated in Theorem 3.3.\nComparing our results with the cases Ω = R3, see [6,10,11], or with the simpler case\nvb=ζ, addressed for example in [8], we found that the time-periodic bound ary datavb\nsignificantly influence the far-field behavior of the velocity and of th e pressure of the flow.\nMore precisely, in Theorem 3.6 we show that, if the total flux throug h the boundary is\ntime-independent, the purely periodic part of the velocity as well as the pressure show\na faster decay rate, the same as for the whole space problem and t he corresponding\nfundamental solutions. However, if the flux is time-dependent, bo th quantities decay at\na slower rate.\nStructure of the paper. Section 2 contains the notations for functions spaces, differen-\ntial operators and distributions that are relevant for the mathem atical formulation and\nanalysis of the problem. We also recall the steady and time-periodic O seen fundamental\nsolutions and some results for their convolution with suitably decayin g functions, and we\nprepare a general result on estimates for certain convolution inte grals. In Section 3, we\npresent and discuss the main results of the paper. The linearized tim e-periodic problem\nis addressed in Section 4. Firstly, we deduce integral representat ion formulas for the\nvelocity and pressure of the linear problem using the time-periodic Os een fundamental\nsolution. Subsequently, a detailed analysis of the far-field behavior of solutions is carried\nout, exploiting the decomposition of the velocity fields into a time-inde pendent part and\na time-dependent part with mean value zero. The proofs of our main results are given in\nSection 5.\n2 Notations and fundamental solutions\n2.1 Notations\nThe standard basis of R3will be denoted by {e1,e2,e3}. Forx∈R3\\ {0}, we define\nthe unit vector ˆ x=x/|x|, where|·|denotes the Euclidean norm. For R >0 we set\nBR(x):={y∈R3||x−y|< R}andBR:=BR(0). The symbol Ω always denotes an\nexterior domain in R3, that is a domain that is the complement of a compact set in R3.\nThe unit outer normal at Σ = ∂Ω is denoted by n=n(x).\nThe gradient of a vector field vis defined by ( ∇v)ij=∂ivj=∂vj\n∂xi, i,j= 1,2,3,\nand∇ ·v=∂vi\n∂xidenotes its divergence. Here and in what follows, Einstein summation\nconvention is used. We adopt this notation for the divergence of a t ensor field F= (Fij)\nand define ∇ ·F=∂Fji\n∂xjei. The notation T(v,q) is used for the stress tensor T(v,q) :=\n2νD(v)−pI, where D(v) =1\n2/parenleftbig\n∇v+(∇v)⊤/parenrightbig\nandI∈R3×3is the identity matrix.\nForq∈[1,∞] andm∈N,Lq(D) andWm,q(D) denote classical Lebesgue and Sobolev\nspaces on a domain D⊆Rn, with norms /ba∇dbl·/ba∇dblq,Dand/ba∇dbl·/ba∇dblm,q,D, respectively. If q∈[1,∞),\nbyWm−1\nq,q(∂D) we indicate the trace space for Wm,q(D)-functions on the (sufficiently\nsmooth) boundary ∂DofD, equipped with the norm /ba∇dbl./ba∇dblm−1\nq,q,∂D.WhenDis an exterior\n3domain, it is convenient to consider the homogeneous Sobolev space s defined by\nDm,q(D) :={u∈L1\nloc(D);Dαu∈Lq(D) for any multi-index αwith|α|=m}\nand equipped with seminorm |u|m,q,D=/parenleftBig/summationtext\n|α|=m/ba∇dblDαu/ba∇dblq\nq,D/parenrightBig1/q\n, whereq∈[1,∞). The\nspaceDm,q\n0(D) is the closure of C∞\n0(D) with respect to the norm |·|m,q,D. The dual space\n(Dm,q\n0)′(D) will be denoted by D−m,q′(D), whereq′:=q/(q−1).\nIfXisaBanachspaceassociatedwiththespacevariable, Lr(T;X)denotesthespaceof\nall Bochner-measurable functions u:T→Xsuch that /ba∇dblu/ba∇dblLr(T;X):=/parenleftbig/integraltext\nT/ba∇dblu(t)/ba∇dblr\nXdt/parenrightbig1\nr=\n/parenleftBig\n1\nT/integraltextT\n0/ba∇dblu(t)/ba∇dblr\nXdt/parenrightBig1\nr<∞,forr∈[1,∞), and/ba∇dblu/ba∇dblL∞(T;X):= esssupt∈T/ba∇dblu(t)/ba∇dblX<∞,\nforr=∞. Note that we equip Twith the normalized Lebesgue measure. For r,q∈\n(1,∞) we further denote the space of boundary traces of functions in W1,r(T,Lq(D)3)∩\nLr(T,W2,q(D)3) by\nTr,q(T×∂D) :=/braceleftbig\nvb=v|T×∂D/vextendsingle/vextendsinglev∈W1,r(T,Lq(D)3)∩Lr(T,W2,q(D)3)/bracerightbig\n.(2.1)\nThe spaceTr,q(T×∂D) can be identified with a real interpolation space, but this property\nwill not be used in what follows. Moreover, by C(T;X) we denote the space of continuous\nfunctions from TtoX.\nAs in [10,11,26], we will denote the Dirac delta distributions on R3,T, andZbyδR3,\nδTandδZ, respectively. When studying the whole space problem, it will be form ulated in\nthe locally compact abelian group G:=T×R3, and the Dirac delta distribution on G,\nδG, will be used to define the fundamental solution to time-periodic pro blems. By S′(R3),\nS′(G) andS′(ˆG), where ˆG:=Z×R3, we denote the spaces of tempered distributions\nonR3,GandˆG, see [11] for a precise definition. Moreover, 1 Tdenotes the constant 1\nfunction on T. In the context of the exterior problem, δΣwill denote the Dirac delta\ndistribution on Σ, as defined, for example, in [4].\nAdditional notations will be introduced as they are needed.\n2.2 Steady-state Stokes and Oseen fundamental solutions\nIt is well known that\nE(x) =1\n4π|x|(2.2)\nis the fundamental solution of the Laplace operator in R3, that is, −∆E =δR3inS′(R3).\nThe fundamental solution of the classical Stokes system in R3is the pair (Lste\n0,P)∈\nS′(R3)3×3×S′(R3)3given by (see, for example, [21])\n(Lste\n0,P)(x) =/parenleftbigg1\n8πν|x|(I+ ˆx⊗ˆx),1\n4π|x|2ˆx/parenrightbigg\n.\nIn particular, the pressure component satisfies\nP(x) =−∇E(x), (2.3)\n4and one directly deduces the estimate\n∀α∈N3\n0∃C >0∀x/\\e}atio\\slash= 0 :/vextendsingle/vextendsingleDα\nxP(x)/vextendsingle/vextendsingle≤C|x|−2−|α|. (2.4)\nThe fundamental solution (Lste\nζ,P)∈S′(R3)3×3×S′(R3)3of the 3D Oseen system has\nthe same pressure part (2.3), and the velocity component is given b y (see [16,21,30])Lste\nζ(x) =1\n4πν|x|exp/parenleftbigg\n−sζ(x)\nν/parenrightbigg\nI−1−exp(−sζ(x)/ν)\n8π|x|sζ(x)(I−ˆx⊗ˆx)\n+|ζ|\n16π1−exp(−sζ(x)/ν)−exp(−sζ(x)/ν)sζ(x)/ν\nsζ(x)2/parenleftBig\nˆx+ˆζ/parenrightBig\n⊗/parenleftBig\nˆx+ˆζ/parenrightBig\n,(2.5)\nwheresζ(x) := [|ζ||x|+(ζ·x)]/2.It is subject to the pointwise estimate (see [16])\n∀α∈N3\n0∀ε>0∃C >0∀|x| ≥ε:/vextendsingle/vextendsingleDα\nx\nLste\nζ(x)/vextendsingle/vextendsingle≤C/bracketleftbig\n|x|/parenleftbig\n1+sζ(x)/parenrightbig/bracketrightbig−1−|α|/2.(2.6)\nObserve that this estimate is anisotropic in space. In particular, th e decay rate depends\non the direction ζ, and reflects the occurrence of a wake region in the flow behind the\nbody. Hence, there is a significant difference between the velocity c omponents of the\nStokes and Oseen fundamental solutions.\nThe following lemma yields estimates on convolutions ofLste\nζwith functions with\nsuitable decay. Throughout this work we will use the notation\nlog+(r):= max{1,logr}forr>0. (2.7)\nLemma 2.1. Letg∈L∞(R3)andA∈[2,∞),B∈[0,∞),M >0such that\n|g(x)| ≤M(1+|x|)−A(1+sζ(x))−B.\nThen there exists C=C(A,B,ζ)>0with the following properties:\n1. IfA+min{1,B}>3, then\n/vextendsingle/vextendsingle|Lste\nζ|∗g(x)/vextendsingle/vextendsingle≤CM/bracketleftbig\n(1+|x|)/parenleftbig\n1+sζ(x)/parenrightbig/bracketrightbig−1.\n2. IfA+min{1,B}>3andA+B≥7/2, then\n/vextendsingle/vextendsingle|∇Lste\nζ|∗g(x)/vextendsingle/vextendsingle≤CM/bracketleftbig\n(1+|x|)/parenleftbig\n1+sζ(x)/parenrightbig/bracketrightbig−3/2.\n3. IfA+min{1,B}= 3andA+B≥7/2, then\n/vextendsingle/vextendsingle|∇Lste\nζ|∗g(x)/vextendsingle/vextendsingle≤CM/bracketleftbig\n(1+|x|)/parenleftbig\n1+sζ(x)/parenrightbig/bracketrightbig−3/2log+|x|.\n4. IfA+B <3, then\n/vextendsingle/vextendsingle|∇Lste\nζ|∗g(x)/vextendsingle/vextendsingle≤CM(1+|x|)−(A+B)/2/parenleftbig\n1+sζ(x)/parenrightbig−(A+B−1)/2.\n5Proof.These are special cases of [25, Theorems 3.1 and 3.2], see also [6, The orem 3.1].\nWe next provide similar convolution estimates for ∇P in a particular case. However,\nsince∇P is strongly singular at the origin, we exclude a neighborhood from th e domain\nof integration.\nLemma 2.2. Letg∈L∞(Ω)andM >0such that\n|g(x)| ≤M/parenleftbig\n(1+|x|)(1+sζ(x))/parenrightbig−2.\nLetR >0andχ:=χR3\\BRbe the characteristic function of the exterior to the ball BR\ncentered at the origin. Then there exists C=C(R,ζ)>0such that\n/vextendsingle/vextendsingle/parenleftbig\nχ∇P/parenrightbig\n∗g(x)/vextendsingle/vextendsingle≤CM|x|−2min/braceleftbig\n1,(1+sζ(x))−2log+|x|+|x|−1log+|x|+(1+sζ(x))−1/bracerightbig\n.\nProof.We use (2.4) and the decay of gto estimate\nI(x):=/vextendsingle/vextendsingle/parenleftbig\nχ∇P/parenrightbig\n∗g(x)/vextendsingle/vextendsingle≤M/integraldisplay\nR3\\BR(x)|x−y|−3/parenleftbig\n(1+|y|)(1+sζ(y))/parenrightbig−2dy\n≤CM/integraldisplay\nR3(1+|x−y|)−3/parenleftbig\n(1+|y|)(1+sζ(y))/parenrightbig−2dy.\nFollowing the calculations in [25, Section 2], we obtain\nI(x)≤CM/parenleftbig\n|x|−2(1+sζ(x))−2log+|x|+|x|−3log+|x|+|x|−2(1+sζ(x))−1/parenrightbig\n.\nAlternatively, we can omit the anisotropic contributions and directly estimate\nI(x)≤CM/integraldisplay\nR3(1+|x−y|)−3(1+|y|)−2dy≤CM|x|−2,\nwhich gives an improved estimate along {x∈Ω|sζ(x) = 0}={x∈Ω|x=αζ, α > 0}.\nIn summary, we conclude the asserted estimate.\n2.3 Time-periodic fundamental solutions\nIn the time-periodic case, the fundamental solutions can be writte n as solutions to a\nsystem of partial differential equations on the group G=T×R3. Following [10,11,26],\nthe fundamental solution of the Stokes ( ζ= 0) or Oseen ( ζ/\\e}atio\\slash= 0) equations is a pair\n(L\nζ,Q)∈S′(G)3×3×S′(G)3satisfying\n/braceleftBigg\n∂t\nL\nζ−ν∆L\nζ+∇Q−(ζ·∇)L\nζ=IδGinG,\n∇·L\nζ= 0 inG.(2.8)\nThe pressure component is given by\nQ =δT⊗P, (2.9)\n6that is, we formally have Q( t,x) =δT(t)P(x), where P is the pressure of the fundamental\nsolution in the steady cases, given in (2.3). The velocity component c an be split in the\nformL\nζ= 1T⊗Lste\nζ+Lper\nζ, (2.10)\nwithLste\nζthe velocity steady part of the fundamental solution, already pre sented in the\nprevious section, andLper\nζthe purely periodic part ofL\nζ, which is given byLper\nζ=F−1\nG/bracketleftBigg\n1−δZ(k)\n|ξ|2+i/parenleftbig2πk\nT−ζ·ξ/parenrightbig/parenleftBig\nI−ˆξ⊗ˆξ/parenrightBig/bracketrightBigg\n,\nwhereFG:S′(G)→S′(ˆG),ˆG:=Z×R3, is the Fourier transform on the group G. For\ndetails on the definition of the Fourier transform on Gand the spaces S′(G) andS′(ˆG) of\ntempered distributions, we refer to [11]. We recall the following deca y properties ofLper\nζ.\nLemma 2.3. Letq∈[1,∞). Then\n∀α∈N3\n0∀ε>0∃C >0∀|x| ≥ε:/vextenddouble/vextenddoubleDα\nx\nLper\nζ(·,x)/vextenddouble/vextenddouble\nLq(T)≤C|x|−3−|α|.(2.11)\nProof.See [10, Theorem 1.1].\nIn comparison with the pointwise estimate (2.6) for the steady-sta te fundamental\nsolution, the estimate (2.11) is of higher order and homogeneous in s pace. Therefore, the\noccurrence of a wake region behind the body is mainly reflected in the steady-state part\nof the velocity field.\nThe following lemma can be used to derive pointwise estimates for conv olutions of the\nfundamental solutionLper\nζand suitably decaying functions.\nLemma 2.4. Letg∈L∞(T×R3)andA,M∈(0,∞)such that\n|g(t,x)| ≤M(1+|x|)−A.\nThen for any ε>0there exists C=C(A,ζ,T,ε)>0such that\n∀|x| ≥ε:/vextendsingle/vextendsingle/bracketleftbig\n|∇Lper\nζ|∗Gg/bracketrightbig\n(t,x)/vextendsingle/vextendsingle≤CM(1+|x|)−min{A,4}(2.12)\nand, ifA>3,\n∀|x| ≥ε:/vextendsingle/vextendsingle/bracketleftbig\n|Lper\nζ|∗Gg/bracketrightbig\n(t,x)/vextendsingle/vextendsingle≤CM(1+|x|)−3. (2.13)\nProof.See [6, Theorem 3.3].\n2.4 Anisotropic convolution estimates\nIn the next lemma we study convolutions of functions with anisotrop ic decay, which\nis expressed by sζ(x) := [|ζ||x|+(ζ·x)]/2. The characteristic function of the exterior\ndomain Ω will be denoted by χΩ.\n7Lemma 2.5. Letζ∈R3\\ {0}andφ∈L1(T×(R3\\ {0}))such that for some ε >0,\nα,β≥0andp∈(1,∞]\n∃C >0∀|x| ≥ε:/ba∇dblφ(·,x)/ba∇dblLp(T)≤C|x|−α(1+sζ(x))−β.\nLetR>0such that Σ⊂BR. Letp′:=p/(p−1), where∞′:= 1, and letfbe given by\n1.f=ˆfχΩwithˆf∈Lp′(T×Ω),suppˆf⊂T×BR, or\n2.f=ˆfδΣwithˆf∈Lp′(T×Σ),\nand define,\nΛ(t):=\n\n/integraldisplay\nΩˆf(t,y)dyiff=ˆfχΩ,\n/integraldisplay\nΣˆf(t,y)dS(y)iff=ˆfδΣ,Ξ(t):=\n\n/integraldisplay\nΩyˆf(t,y)dyiff=ˆfχΩ,\n/integraldisplay\nΣyˆf(t,y)dS(y)iff=ˆfδΣ,\nand\nM:=\n\n/ba∇dblˆf/ba∇dblLp′(T×Ω)iff=��fχΩ,\n/ba∇dblˆf/ba∇dblLp′(T×Σ)iff=ˆfδΣ.\nForS >R+εthere exists C=C(α,β,ε,R,S,ζ )>0such that\n∀|x| ≥S:|[φ∗Gf](t,x)| ≤C|x|−α(1+sζ(x))−βM.\nIfψ∈C1(T×(R3\\{0}))is such that |∇ψ|=φ, then there is C=C(α,β,ε,R,S,ζ )>0\nsuch that\n∀|x| ≥S:/vextendsingle/vextendsingle[ψ∗Gf](t,x)−/bracketleftbig\nψ(·,x)∗TΛ/bracketrightbig\n(t)/vextendsingle/vextendsingle≤C|x|−α(1+sζ(x))−βM.\nIfξ∈C2(T×R3\\{0})such that |∇2ξ|=φ, then there is C=C(α,β,ε,R,S,ζ )>0such\nthat\n∀|x| ≥S:/vextendsingle/vextendsingle[ξ∗Gf](t,x)−/bracketleftbig\nξ(·,x)∗TΛ/bracketrightbig\n(t)+/bracketleftbig\n∇ξ(·,x)∗TΞ/bracketrightbig\n(t)/vextendsingle/vextendsingle≤C|x|−α(1+sζ(x))−βM.\nIn particular, if ψ,ξandfare time-independent, then also ΛandΞare time-\nindependent and the previous estimates reduce to\n|[φ∗R3f](x)| ≤C|x|−α(1+sζ(x))−βM,/vextendsingle/vextendsingle[ψ∗R3f](x)−Λψ(x)/vextendsingle/vextendsingle≤C|x|−α(1+sζ(x))−βM,/vextendsingle/vextendsingle[ξ∗R3f](x)−Λξ(x)+Ξ·∇ξ(x)/vextendsingle/vextendsingle≤C|x|−α(1+sζ(x))−βM.\nProof.We only treat the case f=ˆfχΩhere. Forf=ˆfδΣwe can proceed along the same\nlines by replacing volume integrals over Ω with boundary integrals over Σ.\n8For|x| ≥S >R+ε≥R≥ |y|andθ∈[0,1] we have\n|x−θy| ≥ |x|−θ|y| ≥(1−θR/S)|x| ≥(1−R/S)|x| ≥S−R>ε,\n(1+2|ζ|R)(1+sζ(x−θy))≥1+2|ζ||θy|+sζ(x−θy)≥1+sζ(x).\nInparticular, thisyields |x|α(1+sζ(x))β≤C|x−θy|α(1+sζ(x−θy))β. Since supp ˆf⊂BR,\nwe thus obtain\n/vextendsingle/vextendsingle[φ∗Gf](t,x)/vextendsingle/vextendsingle≤C/integraldisplay\nΩ|x−y|−α(1+sζ(x−y))−β/parenleftbigg/integraldisplay\nT|ˆf(s,y)|p′ds/parenrightbigg1/p′\ndy\n≤C|x|−α(1+sζ(x))−βM.\nFurthermore, we use the fundamental theorem of calculus to ded uce\n/vextendsingle/vextendsingle[ψ∗Gf](t,x)−/bracketleftbig\nψ(·,x)∗T/bracketrightbig\nΛ(t)/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nT/integraldisplay\nΩ/parenleftbig\nψ(t−s,x−y)−ψ(t−s,x)/parenrightbigˆf(s,y)dyds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nT/integraldisplay\nΩ/integraldisplay1\n0y·∇ψ(t−s,x−θy)ˆf(s,y)dθdyds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤CR/integraldisplay\nΩ/integraldisplay1\n0|x−θy|−α(1+sζ(x−θy))−β/parenleftbigg/integraldisplay\nT|ˆf(s,y)|p′ds/parenrightbigg1/p′\ndθdy\n≤C|x|−α(1+sζ(x))−βM.\nSimilarly, from Taylor’s Theorem we deduce\n/vextendsingle/vextendsingle[ξ∗Gf](t,x)−/bracketleftbig\nξ(·,x)∗TΛ(t)/bracketrightbig\n+/bracketleftbig\n∇ξ(·,x)∗TΞ/bracketrightbig\n(t)/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nT/integraldisplay\nΩ/integraldisplay1\n0(1−θ)(y⊗y) :∇2ξ(x−θy)ˆf(s,y)dθdyds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤CR2/integraldisplay\nΩ/integraldisplay1\n0|x−θy|−α(1+sζ(x−θy))−β/parenleftbigg/integraldisplay\nT|ˆf(s,y)|p′ds/parenrightbigg1/p′\ndθdy\n≤C|x|−α(1+sζ(x))−βM,\nwhich completes the proof.\n3 Main results\nIn what follows, we always assume that 0 ∈R3\\Ω andζ/\\e}atio\\slash= 0. To formulate suitable\nintegrability properties of solutions, we recall the projections PandP⊥, which decompose\na functionw∈L1\nloc(T×Ω) into a steady-state part w0and apurely periodic part w⊥:\nw0(x):=Pw(x) :=/integraldisplay\nTw(t,x)dt, w ⊥(t,x):=P⊥w(t,x) :=w(t,x)−Pw(x).\n9Firstly, we consider weak solutions to (1.1). Since we are dealing with in compressible\nflows, we will take test functions in C∞\n0,σ(T×Ω), which is the subspace of C∞\n0(T×Ω)3\nconstituted by divergence-free functions (with respect to the s pace variable).\nAssumingf∈L2(T;D−1,2(Ω)3) andvb∈L∞(T;W1/2,2(Σ)3), a function v∈L1\nloc(T×\nΩ)3is called a weak solution to (1.1) if\ni.∇v∈L2(T×Ω)3×3,Pv∈L6(Ω)3,P⊥v∈L∞(T;L2(Ω)3),\nii.∇·v= 0 inT×Ω,v=vbonT×Σ,\niii. the identity\n/integraldisplay\nT/integraldisplay\nΩ/bracketleftbig\n−v·∂tϕ+∇v:∇ϕ−ζ·∇v·ϕ+(v·∇v)·ϕ/bracketrightbig\ndxdt=/integraldisplay\nT/integraldisplay\nΩf·ϕdxdt\nholds for all ϕ∈C∞\n0,σ(T×Ω).\nForthederivationoftheasymptotic decayratesof vandp, we havetoensureincreased\nregularity of weak solutions. To this end, we additionally assume one o f the following\nconditions:\n∃κ,ρ∈(1,∞) with2\nρ+3\nκ<1 :P⊥v∈Lρ(T;Lκ(Ω)3), (3.1)\n∃κ,ρ∈(1,∞) with2\nρ+3\nκ<2 :∇P⊥v∈Lρ(T;Lκ(Ω)3×3). (3.2)\nBoth assumptions increase the regularity of weak solutions as was s hown in [7].\nNow, suppose that the exterior domain Ω ⊂R3has aC2-boundary. To quantify the\nregularity of the data, recall the trace classes Tr,q(T×Σ) defined in (2.1).\nTheorem 3.1. Letf∈Lr(T;Lq(Ω)3)andvb∈Tr,q(T��Σ)for allq,r∈(1,∞). Letvbe\na weak time-periodic solution to (1.1)that satisfies (3.1)or(3.2). Then\n∀s2∈(1,∞), s1∈(4\n3,∞], s0∈(2,∞] :Pv∈D2,s2(Ω)3∩D1,s1(Ω)3∩Ls0(Ω)3,(3.3)\n∀q,r∈(1,∞) :P⊥v∈W1,r(T;Lq(Ω)3)∩Lr(T;W2,q(Ω)3), (3.4)\nand there exists a pressure field p∈L1\nloc(T×Ω)such that\n∀s2∈(1,∞) :Pp∈D1,s2(Ω),∀q,r∈(1,∞) :P⊥p∈Lr(T;D1,q(Ω)) (3.5)\nand(1.1)is satisfied in the strong sense.\nProof.The result was shown in [7] for vb∈C(T;C2(Σ)3)∩C1(T;C(Σ)3). In the proof\ngiven there, the regularity assumptions on the data are derived fr om the time-periodic\nmaximal regularity results in [13] for the linearized system, where bo undary data in the\nclassTr,q(T×Σ) were studied. This allows for the stated generalization.\n10In this framework, we can derive (implicit) representation formulas for time-periodic\nweak solutions to (1.1). In particular, we deduce two different repr esentation formulas for\nthe velocity field.\nTheorem 3.2. Letf∈Lr(T;Lq(Ω)3)andvb∈Tr,q(T×Σ)for allq,r∈(1,∞). Letvbe\na weak time-periodic solution to (1.1)that satisfies (3.1)or(3.2). Then\nv(t,x) =/integraldisplay\nT×Ω\nL\nζ(t−s,x−y)[f(s,y)−v(s,y)·∇v(s,y)]dsdy\n+/integraldisplay\nT×Σ\nL\nζ(t−s,x−y)[T(v,p)(s,y)n(y)+(ζ·n(y))vb(s,y)] dsdS(y)\n+/integraldisplay\nT×Σvb(t,y)·[D(L\nζei)(t−s,x−y)n(y)] dsdS(y)ei\n−/integraldisplay\nΣ[vb(t,y)·n(y)]P(x−y)dS(y),(3.6)\nv(t,x) =/integraldisplay\nT×Ω\nL\nζ(t−s,x−y)f(s,y)dsdy\n−/integraldisplay\nT×Ω[∇(L\nζei)(t−s,x−y)] :v(s,y)⊗v(s,y)dsdyei\n+/integraldisplay\nT×Σ\nL\nζ(t−s,x−y)[T(v,p)(s,y)n(y)+(ζ·n(y))vb(s,y)] dsdS(y)\n−/integraldisplay\nT×Σ\nL\nζ(t−s,x−y)(vb(s,y)·n(y))vb(s,y)dsdS(y)\n+/integraldisplay\nT×Σvb(t,y)·[D(L\nζei)(t−s,x−y)n(y)] dsdS(y)ei\n−/integraldisplay\nΣ[vb(t,y)·n(y)]P(x−y)dS(y)(3.7)\nfor all(t,x)∈T×Ω, and the pressure field p, associated to vby Theorem 3.1, satisfies\np(t,x) =p∞(t)+/integraldisplay\nΩP(x−y)·[f(t,y)−v(t,y)·∇v(t,y)]dy\n+/integraldisplay\nΣP(x−y)·T(v,p)(t,y)n(y)dS(y)\n+/integraldisplay\nΣP(x−y)·/bracketleftbig\n(ζ·n(y))vb(t,y)+(vb(t,y)·n(y))ζ/bracketrightbig\ndS(y)\n+/integraldisplay\nΣ2νvb(t,y)·∇P(x−y)n(y)dS(y)\n+/integraldisplay\nΣE(x−y)[∂tvb(t,y)·n(y)] dS(y)(3.8)\nfor all(t,x)∈T×Ωand some function p∞∈L1(T).\n11Based on these representation formulas and the decay propertie s of fundamental solu-\ntions, we will derive asymptotic expansions for ( v,p), in which the two functions\nΦ(t) :=/integraldisplay\nΣvb(t,y)·n(y)dS(y),Ψ(t) :=/integraldisplay\nΣvb(t,y)·n(y)ydS(y) (3.9)\nwill appear. The scalar function Φ denotes the total flux of the flow through the bound-\nary Σ. To specify the decay of the remainder terms, we use the fun ction log+defined\nin (2.7).\nTheorem 3.3. Letfandvbbe as in Theorem 3.2 with suppfcompact. Let vbe a weak\nsolution to (1.1)that satisfies (3.1)or(3.2). Then the velocity field vsatisfies\nv(t,x) =/bracketleftbigL\nζ(·,x)∗TF/bracketrightbig\n(t)+Φ(t)P(x)−Ψ(t)·∇P(x)+R(t,x),(3.10)\nwhere\nF(t):=/integraldisplay\nΩf(t,y)dy+/integraldisplay\nΣ/bracketleftbig\nT(v,p)(t,y)n(y)+/parenleftbig\nζ·n(y)−vb(t,y)·n(y)/parenrightbig\nvb(t,y)/bracketrightbig\ndS(y),\nand there exists C >0such that\n|R0(x)| ≤C/bracketleftbig\n|x|/parenleftbig\n1+sζ(x)/parenrightbig/bracketrightbig−3/2log+|x|, (3.11)\n|∇R0(x)| ≤C/bracketleftbig\n|x|/parenleftbig\n1+sζ(x)/parenrightbig/bracketrightbig−2log+/parenleftbigg|x|\n1+sζ(x)/parenrightbigg\n, (3.12)\n|R⊥(t,x)| ≤C|x|−3, (3.13)\n|∇R⊥(t,x)| ≤C|x|−7/2(1+sζ(x))−1/2, (3.14)\nfor all(t,x)∈T×Ω. The pressure field p, associated to vby Theorem 3.1, satisfies\np(t,x) =p∞(t)+Φ′(t)E(x)+/bracketleftbig\nF(t)+ζΦ(t)+Ψ′(t)/bracketrightbig\n·P(x)+R(t,x) (3.15)\nfor somep∞∈L1(T), and there exists C >0such that\n|R(t,x)| ≤C|x|−2min/braceleftbig\n1,(1+sζ(x))−2log+|x|+|x|−1log+|x|+(1+sζ(x))−1/bracerightbig\n(3.16)\nfor all(t,x)∈T×Ωwith|x|sufficiently large.\nTheorem 3.3 yields new insights into the decay properties of the veloc ity and pressure\nfields. First of all, due to the decomposition (2.10) of the velocity com ponent of the\nfundamental solution, we obtain separate asymptotic expansions for the steady-state and\npurely periodic parts of v.\nCorollary 3.4. In the situation of Theorem 3.3, the velocity field v=v0+v⊥satisfies\nv0(x) =Lste\nζ(x)F0+Φ0P(x)+R0(t,x), (3.17)\nv⊥(t,x) =/bracketleftbigLper\nζ(·,x)∗TF⊥/bracketrightbig\n(t)+Φ⊥(t)P(x)−Ψ⊥(t)·∇P(x)+R⊥(t,x) (3.18)\nfor all(t,x)∈T×Ω, whereR=R0+R⊥satisfies (3.11)–(3.14).\n12Combining the decay properties of the fundamental solutions with t hose of the (faster\ndecaying) remainder terms, we deduce the following result.\nCorollary 3.5. In the situation of Theorem 3.3, there exists C >0such that the velocity\nfieldv=v0+v⊥and the pressure psatisfy\n|v0(x)| ≤C/bracketleftbig\n|x|/parenleftbig\n1+sζ(x)/parenrightbig/bracketrightbig−1,|∇v0(x)| ≤C/bracketleftbig\n|x|/parenleftbig\n1+sζ(x)/parenrightbig/bracketrightbig−3/2,(3.19)\n|v⊥(t,x)| ≤C|x|−2, |∇v⊥(t,x)| ≤C|x|−3, (3.20)\n|p(t,x)−p∞(t)| ≤C|x|−1(3.21)\nfor all(t,x)∈T×Ωwith|x|sufficiently large.\nThe asymptotic properties of vand∇vwere already studied in [6] in the case of time-\nperiodic flow in the whole space Ω = R3. Comparing these decay rates with those given in\nCorollary 3.5, we observe that the steady-state parts v0and∇v0decay at the same rate,\nwhilev⊥and∇v⊥decay faster when Ω = R3. This difference is due to the presence of the\nboundary, more precisely, the time-dependent boundary data. I f the total flux Φ defined\nin (3.9) is time-independent, that is, if Φ ≡Φ0, then the asymptotic expansion simplifies\nand the decay rates for the purely periodic velocity field and the pre ssure increase as\nfollows.\nTheorem 3.6. In the situation of Theorem 3.3, assume ∂tΦ≡0, that is, Φ≡Φ0. Then\nvandpsatisfy\nv(t,x) =/bracketleftbigL\nζ(·,x)∗TF/bracketrightbig\n(t)+P(x)Φ0−∇P(x)Ψ(t)+R(t,x), (3.22)\np(t,x) =p∞(t)+/bracketleftbig\nF(t)+Φ0ζ+Ψ′(t)/bracketrightbig\n·P(x)+R(t,x) (3.23)\nfor somep∞∈L1(T)and forFas in Theorem 3.3. Moreover, R0satisfies (3.11)and\n(3.12),R⊥satisfies\n|R⊥(t,x)| ≤C|x|−4, (3.24)\n|∇R⊥(t,x)| ≤C|x|−4, (3.25)\nandRis subject to (3.16). Additionally, ∇v⊥has the asymptotic expansion\n∂jv⊥(t,x) =/bracketleftbig\n∂j\nLper\nζ(·,x)∗TFlin\n⊥/bracketrightbig\n(t)−Ψ⊥(t)∂j∇P(x)+R∂j\n⊥(t,x) (3.26)\nwith\nFlin(t):=/integraldisplay\nΩf(t,y)dy+/integraldisplay\nΣ/bracketleftbig\nT(v,p)(t,y)n(y)+/parenleftbig\nζ·n(y)/parenrightbig\nvb(t,y)/bracketrightbig\ndS(y),\n/vextendsingle/vextendsingleR∂j\n⊥(t,x)/vextendsingle/vextendsingle≤C|x|−9/2(1+sζ(x))−3/2\nforj= 1,2,3and all(t,x)∈T×Ω. In particular, there is C >0such that\n|v0(x)| ≤C/bracketleftbig\n|x|/parenleftbig\n1+sζ(x)/parenrightbig/bracketrightbig−1,|∇v0(x)| ≤C/bracketleftbig\n|x|/parenleftbig\n1+sζ(x)/parenrightbig/bracketrightbig−3/2,(3.27)\n|v⊥(t,x)| ≤C|x|−3, |∇v⊥(t,x)| ≤C|x|−4, (3.28)\n|p(t,x)−p∞(t)| ≤C|x|−2(3.29)\nfor all(t,x)∈T×Ωwith|x|sufficiently large.\n13For constant total flux, we thus obtain the same decay rates for the velocity field as\nin the case Ω = R3(without boundary), see [6]. Moreover, the steady-state part a nd\nthe purely periodic part of the velocity field and the respective grad ients as well as the\npressure decay with the same rate as the corresponding fundame ntal solutions.\nAlthough the decay of the term ∇R⊥given in (3.25) is faster than that in (3.14), it\nis of the same order as the decay of ∇Lper\nζ. Therefore, (3.22) does not yield a proper\nasymptotic expansion for ∇v⊥since a leading-order term is not identified. Instead, if one\nconsiders only the linear boundary contributions that appear in Flin\n⊥for the leading term,\nthen one arrives at (3.26), which properly identifies the leading term of an asymptotic\nexpansion for ∇v⊥.\nObviously, the assumption of constant total flux is also satisfied if w e consider a time-\nindependent forcing fand boundary data vb. In this case we have v(t,x) =v0(x), and\nfromTheorem3.6werediscover thewell-known asymptoticexpansio nanddecayestimates\nfor the velocity field associated to steady flow past a body, see [1,1 7,19,21].\nIn the previous theorems, the assumption that |x|is large is actually not necessary\nfor the decay estimates of the velocity field v, its gradient ∇vand the corresponding\nremainders since these are bounded due to the regularity from (3.3 ) and (3.4) and the\nembedding theorem from [7, Theorem 3]. However, bondedness wit h respect to time does\nnot follow for the pressure p, so that the assumption cannot be omitted completely.\nThe proofs of these results will be provided in Section 5.\n4 The linearized time-periodic problem\nIn the present section, we derive representation formulas and st udy the asymptotic be-\nhavior of the linearized time-periodic problem that is obtained from th e Navier-Stokes\nequations when the convective term v· ∇v=∇ ·(v⊗v) is neglected or replaced with\n∇ ·Ffor a tensor field F. At this stage, Fis know and satisfies suitable summability\nproperties.\n4.1 Representation formulas for solutions\nAssume Ω ⊂R3is an exterior domain with C2-boundary. For the whole section, let\nf∈Lr(T;Lq(Ω)3),F∈Lr(T;W1,q(Ω)3×3) andvb∈Tr,q(T×Σ) for some q∈(1,2) and\nr∈(1,∞), and consider a solution ( v,p) to the (linear) time-periodic Oseen problem\n\n\n∂tv−ν∆v+∇p−ζ·∇v=f−∇· FinT×Ω,\n∇·v= 0 in T×Ω,\nv=vb onT×Σ,\nlim\n|x|→∞v(t,x) = 0 for t∈T.(4.1)\n14Note that the existence of a solution ( v,p) withv=v0+v⊥and\nv0∈D2,q(Ω)3∩L2q/(2−q)(Ω)3,\nv⊥∈W1,r(T;Lq(Ω)3)∩Lr(T;W2,q(Ω)3),\np∈Lr(T;D1,q(Ω))\nwas shown in [14, Theorem 4.7]. Moreover, the velocity field vis unique in this class,\nwhile the pressure pis unique up to addition with a space-independent function.\nIn order to derive the integral representation formulas for the v elocity and pressure,\nwe first transform the exterior problem (4.1) to a whole-space pro blem. To do so, we\nextend the velocity and pressure to R3\\Ω in the following way:\n/tildewidev(t,x) :=/braceleftBigg\n0 ifx∈R3\\Ω,\nv(t,x) ifx∈Ω,/tildewidep(t,x) :=/braceleftBigg\n0 ifx∈R3\\Ω,\np(t,x) ifx∈Ω.(4.2)\nIn a first step we assume F≡0. The next lemma is proved along the same lines as in the\nsteady case [30].\nLemma 4.1. Under the above assumptions on the data and if F≡0, the extended fields\n(4.2)satisfy\n\n∂t/tildewidev−ν∆/tildewidev+∇/tildewidep−ζ·∇/tildewidev=F+ν∇ginG,\n∇·/tildewidev=g inG,\nlim\n|x|→∞/tildewidev(t,x) = 0 fort∈T,(4.3)\nwhere\ng:=−(vb·n)δΣ, (4.4)\nF:=χΩf+T(v,p)nδΣ+(ζ·n)vbδΣ+ν∇·[(n⊗vb+vb⊗n)δΣ].(4.5)\nProof.The distributional gradients of /tildewidev∈S′(G)3and/tildewidep∈S′(G) are given by\n∇/tildewidev=χΩ∇v−n⊗vbδΣ,∇/tildewidep=χΩ∇p−pnδΣ, (4.6)\nand for the time derivative, we have\n∂t/tildewidev=χΩ∂tv. (4.7)\nFrom (4.6), we obtain ∇·/tildewidev=χΩ∇·v−(n·vb)δΣ=−(vb·n)δΣ=ginGwithgdefined\nin (4.4). Moreover,\nζ·∇/tildewidev=χΩ(ζ·∇v)−(ζ·n)vbδΣ, (4.8)\n2D(/tildewidev) =∇/tildewidev+(∇/tildewidev)⊤= 2χΩD(v)−vb⊗nδΣ−n⊗vbδΣ, (4.9)\nand therefore\n∇·T(/tildewidev,/tildewidep) =2ν∇·(χΩD(v))−ν∇·(vb⊗nδΣ+n⊗vbδΣ)−∇·(/tildewidepI)\n=χΩ(ν∆v−∇p)−T(v,p)nδΣ−ν∇·[(n⊗vb+vb⊗n)δΣ].\n15From−ν∆/tildewidev+∇/tildewidep=−∇·T(/tildewidev,/tildewidep)+ν∇(∇·/tildewidev) and the previous relations, we also get\n−ν∆/tildewidev+∇/tildewidep=−χΩ(ν∆v−∇p)+T(v,p)nδΣ+ν∇g\n+ν∇·[(n⊗vb−vb⊗n)δΣ].(4.10)\nAdding the identities (4.7), (4.10) and (4.8) side by side, then replacin g the expression\nχΩ(∂tv−ν∆v+∇p−ζ·∇v) withχΩfyields the first equation of system (4.3) with F\ngiven by (4.5).\nLet us consider the solution (˜ v,˜p) of whole space problem (4.3). In general, ∇·˜v=g/\\e}atio\\slash=\n0 and therefore the integral representation of (˜ v,˜p) using the fundamental solution (L\nζ,Q)\ncannot be applied directly to (4.3). In what follows, the component- wise convolution is\ntaken over R3or overG. Since, by (2.3),\n∇·(P∗R3g) = (−∆E)∗R3g=δR3∗R3g=g,\nwe will decompose the velocity component of the solution of (4.3) as\n˜v= (˜v−P∗R3g)+P∗R3g=: ˜u+P∗R3g,\nwhere ˜usatisfies∇·˜u= 0 and\n∂t˜u−ν∆˜u−ζ·∇/tildewideu\n=∂t˜v−ν∆˜v−ζ·∇/tildewidev−∂t(P∗R3g)+ν∆(P∗R3g)+ζ·∇(P∗R3g)\n=−∇˜p+F+2ν∇g+∇(P∗R3(ζg))+∇(E∗R3(∂tg)).\nFrom\n/braceleftBigg\n∂t˜u−ν∆˜u−ζ·∇/tildewideu+∇(˜p−2νg−P∗R3(ζg)−E∗R3(∂tg)) =FinG,\n∇·˜u= 0 inG,\nwe immediately obtain the pressure in terms of a convolution in Gwith Q (recall (2.9))\nas\n˜p−P∗R3(ζg)−2νg−E∗R3(∂tg) = Q∗GF+p∞= P∗R3F+p∞,\nwherep∞∈L1\nloc(G) such that p∞(t,·) is a polynomial for each t∈T. This can readily\nbe verified via Fourier transform, see [5, Lemma 5.2.5]. In virtue of th e decay of the con-\nvolutions and the integrability properties of ∇pand the convolutions, these polynomials\nmust be constant, so that p∞∈L1\nloc(T) =L1(T). Therefore,\n˜p=p∞+P∗R3/bracketleftbig\nF+ζg/bracketrightbig\n+E∗R3(∂tg)+2νg, (4.11)\nwheregandFare given by (4.4) and (4.5), respectively, and ∂tg=−(∂tvb·n)δΣ.\nThe velocity component of a solution to the time-periodic whole-spac e problem (4.3)\nis unique up to addition by a polynomial in space, see [5, Lemma 5.2.5], so t hat the decay\n16of the convolution and the integrability properties of vyield the identity ˜ u=L\nζ∗GF,\nthat is,\n˜u= (1T⊗Lste\nζ)∗G[fχΩ+T(v,p)nδΣ+(ζ·n)vbδΣ]\n+ν(1T⊗Lste\nζ)∗G∇·[(n⊗vb+vb⊗n)δΣ]\n+Lper\nζ∗G[fχΩ+T(v,p)nδΣ+(ζ·n)vbδΣ]\n+νLper\nζ∗G∇·[(n⊗vb+vb⊗n)δΣ],\nwhere the first two convolutions, which are time-independent, can be written as\n(1T⊗Lste\nζ)∗G[fχΩ+T(v,p)nδΣ+(ζ·n)vbδΣ](x)\n=/integraldisplay\nT/integraldisplay\nΩ\nLste\nζ(x−y)f(t,y)dydt\n+/integraldisplay\nT/integraldisplay\nΣ\nLste\nζ(x−y)[T(v,p)(t,y)n(y)+(ζ·n)(y)vb(t,y)] dS(y)dt\n=/integraldisplay\nΩ\nLste\nζ(x−y)[Pf(y)] dy\n+/integraldisplay\nΣ\nLste\nζ(x−y)[PT(v,p)n(y)+(ζ·n)(y)Pvb(y)] dS(y)\n=:/tildewideu(1)\n0(x)+/tildewideu(2)\n0(x),\nand\nν(1T⊗Lste\nζ)∗G∇·[(n⊗vb+vb⊗n)δΣ](x)\n=−ν/integraldisplay\nT/integraldisplay\nΣvb(t,y)·/bracketleftbig\n∇y/parenleftbig\n(Lste\nζ)(x−y)ei/parenrightbig\nn(y)/bracketrightbig\neidS(y)dt\n−ν/integraldisplay\nT/integraldisplay\nΣn(y)·/bracketleftbig\n∇y/parenleftbigLste\nζ(x−y)ei/parenrightbig\nvb(t,y)/bracketrightbig\neidS(y)dt\n=−/integraldisplay\nT/integraldisplay\nΣvb(t,y)·D(Lste\nζ(x−y)ei)n(y)dS(y)dtei\n=−/integraldisplay\nΣPvb(y)·D(Lste\nζ(x−y)ei)n(y)dS(y)dtei=:/tildewideu(3)\n0(x).\nThe time-periodic terms can be written asLper\nζ∗G[fχΩ+T(v,p)nδΣ+(ζ·n)vbδΣ]\n=/integraldisplay\nT×Ω\nLper\nζ(t−s,x−y)f(s,y)dsdy\n+/integraldisplay\nT×Σ\nLper\nζ(t−s,x−y)[T(v,p)(s,y)n(y)+(ζ·n)(y)vb(s,y)] dsdS(y)\n=:/tildewideu(1)\n⊥(t,x)+/tildewideu(2)\n⊥(t,x),\nand\nνLper\nζ∗G∇·[(n⊗vb+vb⊗n)δΣ]\n=−/integraldisplay\nT/integraldisplay\nΣvb(t,y)·D(Lper\nζ(t−s,x−y)ei)n(y)dS(y)dsei=:/tildewideu(3)\n⊥(t,x).\n17For the velocity vin the exterior domain, the previous results yield the formula\nv(t,x) = ˜u(t,x)−/integraldisplay\nΣvb(t,y)·n(y)P(x−y)dS(y),(t,x)∈T×Ω.\nConcerning the pressure term, we restrict ˜ pfrom (4.11) to Ω using the fact that the\nsupport ofgis contained in Σ. For ( t,x)∈T×Ω we thus obtain\np(t,x) =p∞(t)+[P∗R3(F+ζg)](t,x)+[E∗R3(∂tg)](t,x)\n=p∞(t)+/integraldisplay\nΩf(t,y)(t,y)·P(x−y)dy\n+/integraldisplay\nΣP(x−y)·T(v,p)(t,y)n(y)dS(y)+/integraldisplay\nΣζ·n(y)P(x−y)·vb(t,y)dS(y)\n+2ν/integraldisplay\nΣvb(t,y)·∇P(x−y)n(y)dS(y)+/integraldisplay\nΣζ·P(x−y)vb(t,y)·n(y)dS(y)\n+/integraldisplay\nΣE(x−y)∂tvb(t,y)·n(y)dS(y)\n=:p∞(t)+/tildewidep(t,x)+/integraldisplay\nΣE(x−y)∂tvb(t,y)·n(y)dS(y).\nWith the above notations for /tildewideu(i)\n0(x),/tildewideu(i)\n⊥(t,x),i= 1,2,3, and/tildewidep(t,x), we thus proved:\nTheorem 4.2. In the case F≡0, the solution (v,p)of problem (4.1)admits the repre-\nsentation\nv(t,x) =/tildewideu(1)\n0(x)+/tildewideu(2)\n0(x)+/tildewideu(3)\n0(x)\n+/tildewideu(1)\n⊥(t,x)+/tildewideu(2)\n⊥(t,x)+/tildewideu(3)\n⊥(t,x)\n−/integraldisplay\nΣvb(t,y)·n(y)P(x−y)dS(y),(4.12)\np(t,x) =p∞(t)+/tildewidep(t,x)+/integraldisplay\nΣE(x−y)∂tvb(t,y)·n(y)dS(y) (4.13)\nfor all(t,x)∈T×Ωand somep∞∈L1(T).\nNow we address the case F/\\e}atio\\slash≡0. Later on, the tensor field Fwill be replaced with the\nnonlinear term v⊗v.\nLemma 4.3. Consider the problem (4.1)withf≡0andvb≡0. Under the stated\nassumptions on F, the extended fields /tildewidevand/tildewidepdefined in (4.2)satisfy\n\n\n∂t/tildewidev−ν∆/tildewidev+∇/tildewidep−ζ·∇/tildewidev=T(v,p)nδΣ−∇·(χΩF)−(n⊤F)δΣinG,\n∇·/tildewidev= 0 inG,\nlim\n|x|→∞/tildewidev(t,x) = 0 fort∈T.(4.14)\n18Proof.In this case, the distributional gradients of /tildewidevand/tildewidepare simply given by\n∇/tildewidev=χΩ∇v,∇/tildewidep=χΩ∇p−pnδΣ,\nso that from the proof of Lemma 4.3 with f≡0 andvb≡0, we get ∇·/tildewidev=χΩ∇·v= 0\ninGand\n∂t/tildewidev−ν∆/tildewidev+∇/tildewidep−ζ·∇/tildewidev=χΩ(∂tv−ν∆v+∇p−ζ·∇v)+T(v,p)nδΣinG.\nReplacingχΩ(∂tv−ν∆v+∇p−ζ·∇v) with−χΩ(∇·F) and using the relation for the\ndistributional divergence\n∇·(χΩF) =χΩ∇·F−(n⊤F)δΣ\nyields the first equation of (4.14).\nAs before, from the whole space problem (4.14), we derive represe ntation formulas.\nTheorem 4.4. IfF∈L∞(T×R3)3×3and|F(t,x)| ≤M(1 +|x|)−A, for some A≥2,\nthen the solution (v,p)to problem (4.1)withvb≡0andf≡0admits the representation\nv(t,x) =/integraldisplay\nΣ\nLste\nζ(x−y)/integraldisplay\nT/bracketleftbig\nT(v,p)(t,y)n(y)−n(y)⊤F(t,y)/bracketrightbig\ndtdS(y)\n−/integraldisplay\nΩ∇/parenleftbigLste\nζei/parenrightbig\n(x−y) :/integraldisplay\nTF(t,y)dtdyei\n+/integraldisplay\nT/integraldisplay\nΣ\nLper\nζ(t−s,x−y)/bracketleftbig\nT(v,p)(t,y)n(y)−n(y)⊤F(t,y)/bracketrightbig\ndS(y)dt\n−/integraldisplay\nT/integraldisplay\nΩ∇/parenleftbigLper\nζei/parenrightbig\n(t−s,x−y) :F(t,y)dydtei,(4.15)\np(t,x) =p∞(t)+/integraldisplay\nΣP(x−y)·T(v,p)n(y)dS(y)−/integraldisplay\nΩ(∇·F)(t,y)·P(x−y)dy(4.16)\nfor all(t,x)∈T×Ωand somep∞∈L1(T).\nProof.Consider the problem (4.14). Taking into account that ∇·/tildewidev= 0 inG, we obtain\nfrom the same uniqueness argument as above that\n˜p=p∞+Q∗G/bracketleftbig\nT(v,p)nδΣ−∇·(χΩF)−(n⊤F)δΣ/bracketrightbig\n=p∞+P∗R3/bracketleftbig\nT(v,p)nδΣ−(n⊤F)δΣ/bracketrightbig\n−(∇P)∗R3(χΩF)\nfor somep∞∈L1(T), and\n˜v=L\nζ∗G/bracketleftbig\nT(v,p)nδΣ−∇·(χΩF)−(n⊤F)δΣ/bracketrightbig\n= (1T⊗Lste\nζ)∗G/bracketleftbig\nT(v,p)nδΣ−(n⊤F)δΣ/bracketrightbig\n−/bracketleftbig/parenleftbig\n1T⊗∇(Lste\nζei)/parenrightbig\n∗G(χΩF)/bracketrightbig\nei\n+Lper\nζ∗G/bracketleftbig\nT(v,p)nδΣ−(n⊤F)δΣ/bracketrightbig\n−/bracketleftbig\n∇(Lper\nζei)∗G(χΩF)/bracketrightbig\nei.\nFrom here onwards, the arguments are similar to those of Theorem 4.2 and lead to the\nrepresentations (4.15) and (4.16).\n194.2 Asymptotic expansion of the velocity and pressure fields\nFirst, we consider the case F≡0. Here, we use the functions Φ and Ψ defined in (3.9),\nwhich depend on the boundary data vb.\nTheorem 4.5. Let(v,p)be the solution of system (4.1)withF≡0, and such that f\nhas compact support. The asymptotic expansion at infinity of the velocity field vis of the\nform\nv(t,x) =/bracketleftbigL\nζ(·,x)∗TFlin/bracketrightbig\n(t)−Φ(t)P(x)+Ψ(t)·∇P(x)+R(t,x), (4.17)\nwhere\nFlin(t):=/integraldisplay\nΩf(t,y)dy+/integraldisplay\nΣ[T(v,p)(t,y)n(y)+(ζ·n)(y)vb(t,y)]dS(y),(4.18)\nand there is C >0such that\n|Dα\nxR0(x)| ≤C/bracketleftbig\n|x|(1+sζ(x))/bracketrightbig−3/2−|α|/2,|Dβ\nxR⊥(t,x)| ≤C|x|−4−|β|(4.19)\nfor allα,β∈N3\n0with|β| ≤2, and all (t,x)∈T×Ωwith|x|sufficiently large. In\nparticular, asymptotic expansions for the steady-state pa rtv0and the purely periodic part\nv⊥ofvare given by\nv0(x) =Lste\nζ(x)Flin\n0−Φ0P(x)+R0(x), (4.20)\nv⊥(t,x) =/bracketleftbigLper\nζ(·,x)∗TFlin\n⊥/bracketrightbig\n(t)−Φ⊥(t)P(x)+Ψ⊥(t)·∇P(x)+R⊥(t,x).(4.21)\nThe asymptotic expansion at infinity of the pressure is of the form\np(t,x) =p∞(t)+Φ′(t)E(x)+/bracketleftbig\nFlin(t)+ζΦ(t)−Ψ′(t)/bracketrightbig\n·P(x)+R(t,x) (4.22)\nfor somep∞∈L1(T)and\n|Dα\nxR0(x)| ≤C|x|−3−|α|,|Dβ\nxR⊥(t,x)| ≤C|x|−3−|β|(4.23)\nfor allα,β∈N3\n0with|β| ≤1, and all (t,x)∈T×Ωwith|x|sufficiently large.\nProof.We derive (4.20) and (4.21) from the representation formula for vin Theorem 4.2.\nFor the steady-state part v0=Pvwe have\nv0(x) =/tildewideu(1)\n0(x)+/tildewideu(2)\n0(x)+/tildewideu(3)\n0(x)−/integraldisplay\nΣPvb(y)·n(y)P(x−y)dS(y).\nSince|∇Lste\nζ(x)| ≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−3/2for|x| ≥ε, from Lemma 2.5 we obtain\n/vextendsingle/vextendsingle/tildewideu(1)\n0(x)+/tildewideu(2)\n0(x)−Lste\nζ(x)Flin\n0/vextendsingle/vextendsingle≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−3/2\n20for|x|sufficiently large. Lemma 2.5 further yields\n/vextendsingle/vextendsingle/tildewideu(3)\n0(x)/vextendsingle/vextendsingle≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−3/2\nfor|x|sufficiently large. Similarly, using Lemma 2.5 and the identity |∇P(x)|=C|x|−3,\nwe deduce /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΣPvb(y)·n(y)P(x−y)dS(y)−P(x)Φ0/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|x|−3\nfor|x|sufficiently large. Collecting the last three estimates, we conclude (4 .20) with the\nasserted estimate of the remainder R0. Since the purely periodic part v⊥=P⊥vsatisfies\nv⊥(t,x) =/tildewideu(1)\n⊥(t,x)+/tildewideu(2)\n⊥(t,x)+/tildewideu(3)\n⊥(t,x)−/integraldisplay\nΣP⊥vb(t,y)·n(y)P(x−y)dS(y),\nwe obtain in the same way that\n/vextendsingle/vextendsingle/tildewideu(1)\n⊥(t,x)+/tildewideu(2)\n⊥(t,x)−/bracketleftbigLper\nζ(·,x)∗TFv,⊥/bracketrightbig\n(t)/vextendsingle/vextendsingle≤C|x|−4,\n/vextendsingle/vextendsingle/tildewideu(3)\n⊥(t,x)/vextendsingle/vextendsingle≤C|x|−4,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΣP⊥vb(t,y)·n(y)P(x−y)dS(y)−P(x)Φ⊥(t)+∇P(x)Ψ⊥(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|x|−4\nfor large |x|. Summarizing, we obtain (4.21) with |R⊥(t,x)| ≤C|x|−4for large |x|.\nNow let us turn to the asymptotic expansion (4.22) of the pressure . We define\np(1)(t,x) =/integraldisplay\nΩf(t,x)·P(x−y)dy,\np(2)(t,x) =/integraldisplay\nΣP(x−y)·T(v,p)n(y)dSy+/integraldisplay\nΣζ·n(y)P(x−y)·vb(t,y)dSy\n+/integraldisplay\nΣζ·P(x−y)vb(t,y)·n(y)dSy,\np(3)(t,x) =/integraldisplay\nΣE(x−y)∂tvb(t,y)·n(y)dSy,\nso thatp(t,x) =p(1)(t,x) +p(2)(t,x) +p(3)(t,x) +p∞(t). Arguing as above, we use\n|∇P(x)| ≤C|x|−3forx/\\e}atio\\slash= 0 to obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglep(1)(t,x)−P(x)·/integraldisplay\nΩf(t,y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|x|−3,\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglep(2)(t,x)−P(x)·/integraldisplay\nΣ[T(v,p)(t,y)n(y)+(ζ·n)(y)vb(t,y)] dSy−Φ(t)P(x)·ζ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|x|−3,\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglep(3)(t,x)−Φ′(t)E(x)+Ψ′(t)·P(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|x|−3\nfor|x|sufficiently large. Collecting these estimates, we arrive at (4.22) with remainder\ntermRsatisfying (4.23) for α=β= 0.\n21To obtain (4.19) for |α|,|β| ≤2 and (4.23) for |α|,|β| ≤1, we can argue in the same\nway. For example, for the pressure pwe thus have\n∂jR(t,x) =∂jp(t,x)−Φ′(t)∂jE(x)−/bracketleftbig\nFlin(t)+ζΦ(t)−Ψ′(t)/bracketrightbig\n·∂jP(x)\nforj= 1,2,3, and we can repeat the previous argument. Moreover, since ( v0,p0) satisfy\nthesteady Oseen equationsand fhascompact support, elliptic regularityimplies smooth-\nness of (v0,p0) far from the boundary. Therefore, the above argument is admis sible for\nderivatives of R0andR0of arbitrary order.\nRemark 4.6. By assuming more time regularity of the data fandvb, we could ensure\nhigher-order regularity of the full solution ( v,p) far from the boundary, that is, not only\nof the steady part. Then the previous argument also yields decay r ates for higher-order\nderivatives of the corresponding remainder terms.\nIn the next theorem we treat the homogeneous boundary value pr oblem containing\nsolely a forcing of the form ∇·F. Note that specific decay rates are assumed for Fand\n∇·F.\nTheorem 4.7. In system (4.1)assumevb≡0andf≡0. IfF∈L∞(T;W1,∞(Ω)3×3)\nwith\n|F(t,x)| ≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−2,|P⊥F(t,x)| ≤C|x|−A(4.24)\nfor someA>0, then the asymptotic expansion at infinity of the velocity fie ldvis of the\nform\nv(t,x) =L\nζ(·,x)∗T/bracketleftbigg/integraldisplay\nΣ/parenleftbig\nT(v,p)n+n⊤F/parenrightbig\ndS/bracketrightbigg\n(t)+R(t,x), (4.25)\nand there is C >0such that\n|R0(x)| ≤C/bracketleftbig\n|x|(1+sζ(x))/bracketrightbig−3/2log+|x|,|R⊥(t,x)| ≤C|x|−min{A,4}(4.26)\nfor all(t,x)∈T×Ωwith|x|sufficiently large. In particular, asymptotic expansions fo r\nthe steady-state part v0and the purely periodic part v⊥are given by\nv0(x) =Lste\nζ(x)/integraldisplay\nΣ/bracketleftbig\nPT(v,p)(t,y)n(y)+Pn(y)⊤F(t,y)/bracketrightbig\ndS(y)+R0(x), (4.27)\nv⊥(t,x) =/bracketleftbiggLper\nζ(·,x)∗T/integraldisplay\nΣ/bracketleftbig\nP⊥T(v,p)(t,y)n(y)+P⊥n(y)⊤F(t,y)/bracketrightbig\ndS(y)/bracketrightbigg\n(t)+R⊥(t,x).\n(4.28)\nAdditionally, if\n|(∇·F)(t,x)| ≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−5/2, (4.29)\nthen the asymptotic expansion at infinity of the pressure is o f the form\np(t,x) =p∞(t)+/parenleftbigg/integraldisplay\nΣ/bracketleftbig\nT(v,p)(t,y)n(y)+n(y)⊤F(t,y)/bracketrightbig\ndS(y)/parenrightbigg\n·P(x)+R(t,x),(4.30)\n22for somep∞∈L1(T)and\n|R(t,x)| ≤C|x|−2min/braceleftbig\n1,(1+sζ(x))−2log+|x|+|x|−1log+|x|+(1+sζ(x))−1/bracerightbig\n(4.31)\nfor all(t,x)∈T×Ωwith|x|sufficiently large.\nProof.WeusetherepresentationformulasfromTheorem4.4andsplit vintoseveral parts.\nThen\nu(t,x) :=L\nζ∗G[(T(v,p)nδΣ](t,x) =/bracketleftbiggL\nζ(·,x)∗T/integraldisplay\nΣT(v,p)ndS/bracketrightbigg\n(t)+Ru(t,x)\nis treated as in the previous lemma. For the part w, which involves F, it is convenient to\ndecompose w=wΩ+wΣwith\nwΩ(t,x) :=/integraldisplay\nT/integraldisplay\nΩ[∇(L\nζei)(t−s,x−y)] :F(s,y)dydsei\n=/integraldisplay\nΩ/bracketleftbig\n∇/parenleftbigLste\nζei/parenrightbig\n(x−y)/bracketrightbig\n:/integraldisplay\nTF(s,y)dsdyei\n+/integraldisplay\nT/integraldisplay\nΩ/bracketleftbig\n∇(Lper\nζei)(t−s,x−y)/bracketrightbig\n:F(s,y)dydsei\n=:wΩ\n0(x)+wΩ\n⊥(t,x),(4.32)\nwΣ(t,x) :=/integraldisplay\nT/integraldisplay\nΣ\nL\nζ(x−y,t−s)n(y)⊤F(s,y)dS(y)ds\n=/integraldisplay\nΣ\nLste\nζ(x−y)/integraldisplay\nTn(y)⊤F(s,y)dsdS(y)\n+/integraldisplay\nT/integraldisplay\nΣ\nLper\nζ(t−s,x−y)n(y)⊤F(s,y)dS(y)ds\n=:wΣ\n0(x)+wΣ\n⊥(t,x).(4.33)\nCombining the decay properties (4.24) with Lemma 2.1 and Lemma 2.4, w e conclude\n|wΩ\n0(x)| ≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−3/2log+|x|,\n|wΩ\n⊥(t,x)| ≤C|x|−min{A,4}.\nIf we define\nRwΣ(t,x):=wΣ(t,x)−/bracketleftbiggL\nζ(·,x)∗T/integraldisplay\nΣn⊤FdS/bracketrightbigg\n(t),\nthen Lemma 2.5 implies\n/vextendsingle/vextendsingleRwΣ\n0(x)/vextendsingle/vextendsingle≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−3/2,\n/vextendsingle/vextendsingleRwΣ\n⊥(t,x)/vextendsingle/vextendsingle≤C|x|−4\nfor all (t,x)∈T×Ω with|x|sufficiently large. Since we have R=Ru−wΩ−RwΣ,\nestimate (4.26) follows.\n23Now let us address the pressure term. We write (4.16) in the form p=p−qwith\nq(t,x):=/integraldisplay\nΩP(x−y)·(∇·F)(t,y)dy.\nAs in the proof of Theorem 4.5 we obtain\n/vextendsingle/vextendsingleRp(t,x)/vextendsingle/vextendsingle≤C|x|−3(4.34)\nfor all (t,x)∈T×Ω, where we split R=Rp−Rqwith\nRq(t,x) =q(t,x)−P(x)·/integraldisplay\nΣn(y)⊤F(t,y)dS(y).\nFixR∗>0 such that Σ ⊂BR∗and letχ∈C∞\n0(R3) be a smooth function such that\nχ(x) = 1 for |x| ≤R∗/2 andχ(x) = 0 for |x| ≥R∗. Then\nq(t,x) =/integraldisplay\nΩχ(x−y)P(x−y)·[∇·F(t,y)]dy\n+/integraldisplay\nΩ[1−χ(x−y)]∇P(x−y) :F(t,y)dy\n−/integraldisplay\nΩ∇χ(x−y)⊗P(x−y) :F(t,y)dy\n+/integraldisplay\nΣχ(x−y)P(x−y)·/bracketleftbig\nn(y)⊤F(t,y)/bracketrightbig\ndy\n=:q1(t,x)+q2(t,x)+q3(t,x)+q4(t,x).\nFor the first term, we use P( x) =C|x|−2and directly estimate\n/vextendsingle/vextendsingleq1(t,x)/vextendsingle/vextendsingle≤/integraldisplay\nBR∗(x)C|x−y|−2/parenleftbig\n|y|(1+sζ(y))/parenrightbig−5/2dy≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−5/2\nfor|x| ≥2R∗. For the second term, Lemma 2.2 implies\n/vextendsingle/vextendsingleq2(t,x)/vextendsingle/vextendsingle≤C|x|−2min/braceleftbig\n1,(1+sζ(x))−2log+|x|+|x|−1log+|x|+(1+sζ(x))−1/bracerightbig\n.\nFor the third term, we consider |x| ≥2R∗and deduce\n/vextendsingle/vextendsingleq3(t,x)/vextendsingle/vextendsingle≤/integraldisplay\nBR∗(x)\\BR∗/2(x)|x−y|−2/parenleftbig\n|y|(1+sζ(y))/parenrightbig−2dy≤/parenleftbig\n|x|(1+sζ(x))/parenrightbig−2.\nInvoking Lemma 2.5 for the last term, we conclude\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleq4(t,x)−P(x)·/integraldisplay\nΣF(t,y)n(y)dS(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|x|−3\nfor|x|large. In virtue of R=Rp−Rqand (4.34), we finally arrive at (4.31).\n245 Proofs of the main results\nThe representation formulas for the nonlinear case given in Theore m 3.2 can be deduced\nfrom the linear case as follows.\nProof of Theorem 3.2. The weak solution vto the nonlinear problem (1.1) solves the\nlinear problem (4.1) if we replace Fwithv⊗vsince we then have f−∇·F=f−v·∇v.\nBy Theorem 3.1 we deduce v· ∇v∈Lq(T×Ω)3for anyq∈(1,∞). Therefore, the\nrepresentation formulas (3.6), (3.7) and (3.8) directly follow from c ombining those for the\nlinear case given in Theorem 4.2 and Theorem 4.4.\nTo prove Theorem 3.3, the linear terms in the representation formu las can be handled\nas in the proof of Theorem 4.5. However, new difficulties arise from th e nonlinear terms.\nTo treat them, we begin with the following decay property of the velo city gradient. For\nthe whole section, we choose a radius R∗>0 such that Σ ⊂BR∗.\nLemma 5.1. For allε>0there exists C >0such that for all R>R ∗it holds\n/integraldisplay\nT/integraldisplay\nΩ\\BR|∇v|2dxdt≤CR−1+ε. (5.1)\nProof.For the case Ω = R3, this estimate was shown in [22, Lemma 5.2]. In the present\ncase of an exterior domain Ω, it follows in exactly the same way by using the increased\nregularity stated in Theorem 3.1.\nFrom (5.1) we can derive a first pointwise estimate of the remainder t erms for the\nvelocity field. This estimate serves as a starting point for an iterativ e procedure that\nincreases the decay rate step by step.\nProof of Theorem 3.3. We begin with the treatment of the velocity field and follow the\napproach from [22] and [6]. Instead of repeating these calculations here in detail, we\nexplain the major steps and the differences that are due to the occ urrence of boundary\nterms.\nTo separate the linear from the nonlinear contributions in the repre sentation for-\nmula (3.6), we let v=u−wwith\nw(t,x):=/integraldisplay\nT×Ω\nL\nζ(t−s,x−y)[v(s,y)·∇v(s,y)]dsdy.\nThenusatisfies\nu(t,x) =/tildewideu(1)\n0(x)+/tildewideu(2)\n0(x)+/tildewideu(3)\n0(x)\n+/tildewideu(1)\n⊥(t,x)+/tildewideu(2)\n⊥(t,x)+/tildewideu(3)\n⊥(t,x)\n−/integraldisplay\nΣvb(t,y)·n(y)P(x−y)dS(y)\nfor (t,x)∈T×Ω, where the terms on the right-hand side are exactly the same as in\nthe representation formula (4.12) for the linear case. Decomposin gu=u0+u⊥into\n25steady-state and purely periodic part, and arguing as in the proof of Theorem 4.5, we\nobtain that the associated remainder term\nRu(t,x):=u(t,x)−/bracketleftbigL\nζ(·,x)∗TFlin/bracketrightbig\n(t)+P(x)Φ(t)−∇P(x)Ψ(t),\nwhere\nFlin(t):=/integraldisplay\nΩf(t,y)dy+/integraldisplay\nΣ/bracketleftbig\nT(v,p)(t,y)n(y)+/parenleftbig\nζ·n(y)/parenrightbig\nvb(t,y)/bracketrightbig\ndS(y),\nsatisfies the same estimates as in the linear case, that is,\n|Dα\nxRu\n0(x)| ≤C/bracketleftbig\n|x|(1+sζ(x))/bracketrightbig−3/2−|α|/2,|Dα\nxRu\n⊥(t,x)| ≤C|x|−4−|α|(5.2)\nfor|α| ≤2 and(t,x)∈T×Ω with|x|large. In particular, in virtue ofthe decay properties\nof the fundamental solutions, this implies\n|Dα\nxu0(x)| ≤C/bracketleftbig\n|x|(1+sζ(x))/bracketrightbig−1−|α|/2, (5.3)\n|Dα\nxu⊥(t,x)| ≤C|x|−2−|α|. (5.4)\nNext we derive estimates of w, which will lead to an estimate of the actual remainder\ntermR=Ru−Rw, where\nRw(t,x):=w(t,x)−/integraldisplay\nT\nL\nζ(t−s,x)/integraldisplay\nΣ(vb(s,y)·n(y))vb(s,y)dS(y)ds.\nTo do so, we decompose w=w0+w⊥such that\nw0(x) =Lste\nζ∗R3/bracketleftbig\nχΩ/parenleftbig\nv0·∇v0+P(v⊥·∇v⊥)/parenrightbig/bracketrightbig\n(x), (5.5)\nw⊥(t,x) =Lper\nζ∗G/bracketleftbig\nχΩ/parenleftbig\nv0·∇v⊥+v⊥·∇v0+P⊥(v⊥·∇v⊥)/parenrightbig/bracketrightbig\n(t,x).(5.6)\nExploiting the decay properties ofL\nζand Lemma 5.1, one can argue as in [22] to first\nshow that\n∀ε>0∃C >0 :|w0(t,x)| ≤C|x|−1+ε(5.7)\nand\n∀ε>0∃C >0 :|w⊥(t,x)| ≤\nC/parenleftbig\n|x|−3/2+|x|−2/5+ε(/ba∇dblv0/ba∇dblL∞(B|x|/2)+/ba∇dblv⊥/ba∇dblL∞(T×B|x|/2))/parenrightbig\n.(5.8)\nCombining (5.7) with (5.3), we conclude\n∀ε>0∃C >0 :|v0(t,x)| ≤C|x|−1+ε. (5.9)\nSimilarly, (5.4) leads to\n∀ε>0∃C >0 :|v⊥(t,x)| ≤C/parenleftbig\n|w⊥(t,x)|+|x|−2/parenrightbig\n. (5.10)\n26We use these inequalities to further estimate (5.8) and to obtain\n∀ε>0∃C >0 :|w⊥(t,x)| ≤C/parenleftbig\n|x|−2/5+ε/ba∇dblw⊥/ba∇dblL∞(T×B|x|/2)+|x|−7/5+ε/parenrightbig\n.(5.11)\nSincev⊥∈L∞(T×Ω) by Theorem 3.1, we have w⊥∈L∞(T×Ω) due to (5.4). This\nallows to derive a first estimate of w⊥from (5.11), We can use this estimate to derive a\nnew bound of the right-hand side of (5.11), which improves the estim ate ofw⊥. Iterating\nthis procedure and combining it with (5.10), we arrive at\n∀ε>0∃C >0 :|v⊥(t,x)|+|w⊥(t,x)| ≤C|x|−7/5+ε. (5.12)\nTo improve these decay rates, we turn to the representation (3.7 ), from which we\ndeduce that w=wΩ+wΣwithwΩandwΣdefined as in (4.32) and (4.33) for F:=v⊗v,\nrespectively. First of all, Lemma 2.5 implies the estimates\n|wΣ\n0(x)| ≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−1, (5.13)\n|wΣ\n⊥(t,x)| ≤C|x|−3(5.14)\nfor|x|large. To obtain a decay rate for wΩ\n⊥, we use (5.9) and (5.12), which imply\n|v0⊗v⊥+v⊥⊗v0+P⊥(v⊥⊗v⊥)|(t,x)≤C/parenleftbig\n|x|−12/5+ε+|x|−14/5+ε/parenrightbig\n.\nNow Lemma 2.4 yields the estimate\n∀ε>0∃C >0 :|wΩ\n⊥(t,x)| ≤C|x|−12/5+ε. (5.15)\nIn virtue of (5.10) and (5.14), this yields the (optimal) estimate\n|v⊥(t,x)| ≤C|x|−2(5.16)\nfor all (t,x)∈T×Ω with|x|large. ForwΩ\n0we proceed similarly. From (5.9) and (5.16)\nwe deduce\n|v0⊗v0+P(v⊥⊗v⊥)|(x)≤C/parenleftbig\n|x|−2+ε+|x|−4/parenrightbig\n,\nso that Lemma 2.1 yields\n∀ε>0∃C >0 :|wΩ\n0(x)| ≤C|x|−1+ε(1+sζ(x))−1/2+ε. (5.17)\nCombining this estimate with (5.3) and (5.13), we conclude\n∀ε>0∃C >0 :|v0(x)| ≤C|x|−1+ε(1+sζ(x))−1/2+ε, (5.18)\nwhich leads to the improved pointwise estimate\n|v0⊗v0+P(v⊥⊗v⊥)|(x)≤C/parenleftbig\n|x|−2+ε(1+sζ(x))−1+ε+|x|−4/parenrightbig\n.\nRepeating the above argument and using Lemma 2.1 again, we deduce\n∀ε>0∃C >0 :|w0(x)| ≤C|x|−3/2+ε(1+sζ(x))−1+ε, (5.19)\n27and with (5.3) and (5.13) we conclude the (optimal) decay rate\n|v0(x)| ≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−1(5.20)\nfor allx∈Ω. Invoking now (5.16) and (5.20), we obtain\n|v0⊗v0+P(v⊥⊗v⊥)|(x)≤C/parenleftbig/parenleftbig\n|x|(1+sζ(x))/parenrightbig−2+|x|−4/parenrightbig\n≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−2\nand\n|v0⊗v⊥+v⊥⊗v0+P⊥(v⊥⊗v⊥)|(t,x)≤C/parenleftbig\n|x|−3(1+sζ(x))−1+|x|−4/parenrightbig\n≤C|x|−3.\nSincevis a solution to the linear problem (4.1) together with F=v⊗v, the asymptotic\nexpansion (3.10) with the asserted remainder estimates (3.11) and (3.13) now follows from\nTheorem 4.5 and Theorem 4.7.\nNow we turn to the asymptotic expansion of ∇vand derive estimates of ∇w. From\nthe formulas (5.5) and (5.6) we deduce\n∂jw0(x) =∂j\nLste\nζ∗/bracketleftbig\nχΩ/parenleftbig\nv0·∇v0+P(v⊥·∇v⊥)/parenrightbig/bracketrightbig\n(x), (5.21)\n∂jw⊥(t,x) =∂j\nLper\nζ∗/bracketleftbig\nχΩ/parenleftbig\nv0·∇v⊥+v⊥·∇v0+P⊥(v⊥·∇v⊥)/parenrightbig/bracketrightbig\n(t,x) (5.22)\nforj= 1,2,3. We use the integrability properties from Theorem 3.1 and the esti-\nmates (5.18) and (5.16) to proceed analogously to [6] and to conclud e the first estimate\n|∇w0(x)|+|∇w⊥(t,x)| ≤C|x|−1,\nand thus\n|∇v0(x)|+|∇v⊥(t,x)| ≤C|x|−1\nin virtue of (5.3) and (5.4). Together with (5.18) and (5.16), this est imate implies\n/vextendsingle/vextendsinglev0·∇v0+P(v⊥·∇v⊥)/vextendsingle/vextendsingle(x)≤C/parenleftbig\n|x|−2(1+sζ(x))−1+|x|−3/parenrightbig\n≤C|x|−2(1+sζ(x))−1/2,/vextendsingle/vextendsinglev0·∇v⊥+v⊥·∇v0+P⊥(v⊥·∇v⊥)/vextendsingle/vextendsingle(t,x)≤C/parenleftbig\n|x|−2(1+sζ(x))−1+|x|−3+|x|−3/parenrightbig\n≤C|x|−2.\nReturning now to (5.21) and (5.22), we can use Lemma 2.1 and Lemma 2 .4 to deduce\n|∇w0(x)| ≤C|x|−5/4(1+sζ(x))−3/4,\n|∇w⊥(t,x)| ≤C|x|−2,\nand thus\n|∇v0(x)| ≤C|x|−5/4(1+sζ(x))−3/4,\n|∇v⊥(t,x)| ≤C|x|−2,\n28due to (5.3) and (5.4). Again, this gives an improved estimate on the n onlinear terms,\nand repeating this argument once more, we finally arrive at the (opt imal) estimate\n|∇v0(x)| ≤C|x|−3/2(1+sζ(x))−3/2,\n|∇v⊥(t,x)| ≤C|x|−3.(5.23)\nTo derive estimates (3.12) and (3.14) for the corresponding remain der terms, we introduce\nanother decomposition of Rw. Letχ∈C∞\n0(R3) a smooth function such that χ(x) = 1 for\n|x| ≤R∗/2 andχ(x) = 0 for |x| ≥R∗. We decompose ∂jRw=I+J,j= 1,2,3, with\nI(t,x):=/integraldisplay\nT×Ωχ(x−y)∂j\nL\nζ(t−s,x−y)[v(s,y)·∇v(s,y)]d(s,y),\nJ(t,x):=/integraldisplay\nT×Ω[1−χ(x−y)]∂j\nL\nζ(t−s,x−y)[v(s,y)·∇v(s,y)]d(s,y)\n−/integraldisplay\nT∂j\nL\nζ(t−s,x)/integraldisplay\nΣ(vb(s,y)·n(y))vb(s,y)dS(y)ds.\nBy (5.20), (5.16) and (5.23), we have\n/vextendsingle/vextendsinglev0·∇v0+P(v⊥·∇v⊥)/vextendsingle/vextendsingle(x)≤C|x|−5/2(1+sζ(x))−5/2,/vextendsingle/vextendsinglev0·∇v⊥+v⊥·∇v0+P⊥(v⊥·∇v⊥)/vextendsingle/vextendsingle(t,x)≤C|x|−7/2(1+sζ(x))−3/2.\nSince∂j\nL\nζ∈L1\nloc(T×R3) (see [10,21]) and χ∂j\nL\nζhas compact support, Lemma 2.5\nimplies\n|I0(x)| ≤C|x|−5/2(1+sζ(x))−5/2,\n|I⊥(t,x)| ≤C|x|−7/2(1+sζ(x))−3/2\nfor|x|sufficiently large. For J0we usev·∇v=∇ ·(v⊗v) and integration by parts to\nobtain the decomposition J0=J1\n0+J2\n0+J3\n0with\n|J1\n0(x)| ≤/integraldisplay\nΩ[1−χ(x−y)]/vextendsingle/vextendsingle∂j∇Lste\nζ(x−y)/vextendsingle/vextendsingle/vextendsingle/vextendsinglev0(y)⊗v0(y)+P(v⊥⊗v⊥)(y)/vextendsingle/vextendsingledy,\n|J2\n0(x)| ≤/integraldisplay\nΩ|∇χ(x−y)|/vextendsingle/vextendsingle∂j\nLste\nζ(x−y)/vextendsingle/vextendsingle/vextendsingle/vextendsinglev0(y)⊗v0(y)+P(v⊥⊗v⊥)(y)/vextendsingle/vextendsingledy,\n|J3\n0(x)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΣ[1−χ(x−y)]∂j\nLste\nζ(x−y)P((vb·n)vb)(y)dy\n−∂j\nLste\nζ(x)/integraldisplay\nΣP/parenleftbig\n(vb·n)vb/parenrightbig\n(y)dS(y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nNow consider |x| ≥2R∗. For|y| ≤R∗, we then have |x−y| ≥ |x| − |y| ≥ |x|/2≥R∗.\nUsing Lemma 2.5, we deduce\n/vextendsingle/vextendsingleJ3\n0(x)/vextendsingle/vextendsingle≤/parenleftbig\n|x|(1+sζ(x))/parenrightbig−2\n29for|x|sufficiently large. Similarly, for J2\n0we deduce\n|J2\n0(x)| ≤C/integraldisplay\nBR∗(x)\\BR∗/2(x)/vextendsingle/vextendsingle∂j\nLste\nζ(x−y)/vextendsingle/vextendsingle/parenleftbig\n|y|(1+sζ(y))/parenrightbig−2dy≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−2\nfor large |x|since∂j\nLste\nζ(x−y)∈L1\nloc(R3). To estimate J1\n0, we use (2.6) and the decay\nestimates (5.20) and (5.16) to conclude\n|J1\n0(x)| ≤C/integraldisplay\nΩ\\BR∗/2(x)/parenleftbig\n|x−y|(1+sζ(x−y))/parenrightbig−2/parenleftbig\n|y|(1+sζ(y))/parenrightbig−2dy\n≤C/parenleftbig\n|x|(1+sζ(x))/parenrightbig−2log+/parenleftbigg|x|\n1+sζ(x)/parenrightbigg\nfor|x| ≥2R∗, where we used [6, Lemma 3.5]. Collecting all estimates for ∂jRw\n0=\nI0+J1\n0+J2\n0+J3\n0and combining them with (5.2), we obtain (3.12). Finally, we estimate\nJ⊥directly, which yields\n|J⊥(t,x)| ≤C/parenleftbig/integraldisplay\nΩ\\B1(x)|x−y|−4|y|−7/2(1+sζ(y))−3/2dy+|x|−4/parenrightbig\n≤C|x|−7/2(1+sζ(x))−1/2\nfor|x| ≥2R∗due to (2.11). Together with the estimate of I⊥and (5.2), this gives (3.14).\nThe results for the pressure, which can be decomposed into a comp onent associated\nwithvbandfand another one associated with F=v⊗v, are now obtained by combining\n(4.22)–(4.23) and (4.30)–(4.31).\nThe statements of Corollary 3.4 and Corollary 3.5 are now direct cons equences.\nProof of Corollary 3.4. Fromthe identityL\nζ= 1T⊗Lste\nζ+Lper\nζwe directly conclude (3.18)\nfrom (3.10) by taking the purely periodic part. Taking the steady-s tate part, we nearly\narrive at a similar formula, which only differs from (3.17) by the term Ψ 0·∇P(x). Since\n|∇P(x)| ≤C|x|−3, this term can be absorbed into the steady-state remainder R0.\nProof of Corollary 3.5. The asserted estimates were already shown in the process of prov -\ning Theorem 3.2. However, they are also direct consequences of th e asymptotic expan-\nsions (3.17), (3.18) and (3.15) combined with the decay properties o f the fundamental\nsolutions from (2.6), (2.11) and (2.4), respectively.\nFinally, we consider the case of constant total flux.\nProof of Theorem 3.6. First of all, note that ∂tΦ = 0 is equivalent to Φ ≡Φ0, and thus\nto Φ⊥= 0. Therefore, the asymptotic expansions from Theorem 3.3 and C orollary 3.4\nsimplify, and we obtain (3.22) and (3.23). Moreover, these formulas imply the improved\npointwise estimates (3.27)–(3.29) in virtue of the decay properties of the fundamental\nsolutions from (2.11) and (2.4). In particular, this yields\n|v0⊗v⊥+v⊥⊗v0+P⊥(v⊥⊗v⊥)|(t,x)≤C/parenleftbig\n|x|−4(1+sζ(x))−1+|x|−6/parenrightbig\n≤C|x|−4,\n30and we can conclude the remainder estimate (3.24) for R⊥from a combination of Theo-\nrem 4.5 and Theorem 4.7 as above. Moreover, we now have\n/vextendsingle/vextendsinglev0·∇v⊥+v⊥·∇v0+P⊥(v⊥·∇v⊥)/vextendsingle/vextendsingle(t,x)≤C|x|−9/2(1+sζ(x))−3/2,\nso that, similarly to above,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nT×Ω[1−χ(x−y)]∂j\nLper\nζ(t−s,x−y)P⊥[v(s,y)·∇v(s,y)]d(s,y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/integraldisplay\nΩ\\B2(x)|x−y|−4|y|−9/2(1+sζ(y))−3/2dy≤ |x|−9/2(1+sζ(x))−3/2\nfor|x| ≥2R∗. Therefore, the estimate of J⊥in the proof of Theorem 3.3 can be replaced\nwith\n|J⊥(t,x)| ≤C/parenleftbig\n|x|−9/2(1+sζ(x))−3/2+|x|−4/parenrightbig\n≤C|x|−4,\nand in virtue of (5.2) and the previous estimate of I⊥, we obtain (3.14).\nTo derive the asymptotic expansion (3.26), observe that R∂j\n⊥=∂jRu\n⊥−∂jw⊥. Due to\n∂jw(t,x) =I(t,x)+/integraldisplay\nT×Ω[1−χ(x−y)]∂j\nL\nζ(t−s,x−y)[v(s,y)·∇v(s,y)]d(s,y),\nthe previous integral estimate and the estimate of I⊥yield\n/vextendsingle/vextendsingleR∂j\n⊥(t,x)/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle∂jRu\n⊥(t,x)/vextendsingle/vextendsingle+|∂jw⊥(t,x)| ≤ |x|−9/2(1+sζ(x))−3/2\nby (5.2), which completes the proof of Theorem 3.6.\nReferences\n[1] K. Babenko, On stationary solutions of the problem of flow past a body of a viscous\nincompressible fluid, Math. USSR, Sb. 20(1973), 1–25.\n[2] S. B¨ onisch, V. Heuveline, and P. Wittwer, Adaptive boundary co nditions for exterior\nflow problems, J. Math. Fluid Mech. 7(2005), 85–107.\n[3] P. Deuring and S. Kraˇ cmar, Exterior stationary Navier-Stoke s flows in 3D with non-\nzero velocity at infinity: approximation by flows in bounded domains, Math. Nachr.\n269-270 (2004), 86-115.\n[4] Y.V. 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Fluid Mech. 22(7) (2020), 1-19.\n33"    },    {        "title": "2402.12246v1.Unconditional_quantum_MAGIC_advantage_in_shallow_circuit_computation.pdf",        "content": "Unconditional quantum MAGIC advantage in shallow circuit computation\nXingjian Zhang,1, 2, 3, 4, ∗Zhaokai Pan,1,†and Guoding Liu1,‡\n1Center for Quantum Information, Institute for Interdisciplinary In-\nformation Sciences, Tsinghua University, Beijing 100084, China\n2Hefei National Research Center for Physical Sciences at the Microscale and School of\nPhysical Sciences, University of Science and Technology of China, Hefei 230026, China\n3Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Informa-\ntion and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China\n4QICI Quantum Information and Computation Initiative, Department of Com-\nputer Science, The University of Hong Kong, Pokfulam Road, Hong Kong\n(Dated: February 20, 2024)\nQuantum theory promises computation speed-ups than classical means. The full power is be-\nlieved to reside in “magic” states, or equivalently non-Clifford operations — the secret sauce to\nestablish universal quantum computing. Despite the celebrated Gottesman-Knill Theorem stating\nthat magic-free computation can be efficiently simulated by a classical computer, it is still question-\nable whether “magic” is really magical. Indeed, all the existing results establish its supremacy for\nefficient computation upon unproven complexity assumptions or queries to black-box oracles. In this\nwork, we show that the magic advantage can be unconditionally established, at least in a shallow\ncircuit with a constant depth. For this purpose, we first construct a specific nonlocal game inspired\nby the linear binary constraint system, which requires the magic resource to generate the desired\nnonlocal statistics or quantum “pseudo telepathy.” For a relation problem targeting generating\nsuch correlations between arbitrary nonlocal computation sites, we construct a shallow circuit with\nbounded fan-in gates that takes the strategy for quantum pseudo telepathy as a sub-routine to solve\nthe problem with certainty. In contrast, magic-free counterparts inevitably require a logarithmic\ncircuit depth to the input size, and the separation is proven optimal. As by-products, we prove that\nthe nonlocal game we construct has non-unique perfect winning strategies, answering an open prob-\nlem in quantum self-testing. We also provide an efficient algorithm to aid the search for potential\nmagic-requiring nonlocal games similar to the current one. We anticipate our results to enlighten\nthe ultimate establishment of the unconditional advantage of universal quantum computation.\nI. INTRODUCTION\nStarting from Richard Feynman’s proposal of simulat-\ning physics with quantum means [1], it has been an ap-\npealing quest to exploit phenomena unique to quantum\ntheory to boost computation. A series of results, such\nas Shor’s factoring algorithm [2] and Grover’s search [3],\nstrengthen the belief in the power of quantum computa-\ntion. Notwithstanding the prosperity in the zoo of quan-\ntum algorithms, it is still intriguing to answer the fol-\nlowing questions: Does quantum theory indeed bring a\ncomputation advantage over classical means, and if yes,\nwhat is the origin of such power? A well-known state-\nment that seems to respond to both questions is quantum\n“magic” [4]. The so-called quantum magic states are be-\nyond the reach of stabilizer circuits, the ones initialized\nin the computational basis state and composed of only\nClifford gates and Pauli measurements. The Gottesman-\nKnill Theorem shows that stabilizer circuits can be per-\nfectly simulated by classical computers in a polynomial\ntime of the input size, deemed as efficient [5, 6]. On the\nother hand, attempts from the simulation field suggest\na strong relevance between the quantity of magic and\n∗zxj24@hku.hk\n†pzk20@mails.tsinghua.edu.cn\n‡lgd22@mails.tsinghua.edu.cnthe extent of quantum advantage [7–13]. Quantum al-\ngorithms richer in magic are often more difficult for a\nclassical computer to handle.\nIndeed, considering the structure of quantum state\nspace, magic states, or equivalently non-Clifford oper-\nations, are indispensable for a complete picture [4, 14].\nIn contrast, whether they indeed bring an advantage in\ncomputation remains validating. As the quantum state\nspace dimension grows exponentially in the number of\ncomponents, a naive yet natural thinking is that quan-\ntum theory may provide efficient parallel computing that\nexecutes super-polynomially or even exponentially faster\nthan any classical counterparts in solving specific prob-\nlems. Definitely, we need to look into algorithms that re-\nsort to the resource of quantum magic states. An uncon-\nditional proof of super-polynomial quantum advantage\nin computation is one of the ultimate goals for quan-\ntum and computer scientists. There are pretty good rea-\nsons to hold such a tenet, which shall validate a series of\nverisimilar tales in the complexity theory [15, 16]. Unfor-\ntunately, explorations in this direction to date have not\ngot rid of assumptions of unproven hardness for classical\nalgorithms, such as factoring a large number in Shor’s\nalgorithm [2], or the reliance of queries to a black-box\noracle as in Grover’s search [3], where the construction\nof the oracle may be hard work.\nTo firmly establish the quantum advantage, one may\nalternatively start from a more restrictive regime in com-arXiv:2402.12246v1  [quant-ph]  19 Feb 20242\nplexity. Instead of defining “efficient” as a polynomially\ngrowing time, a notable regime is the set of shallow cir-\ncuits [17, 18], where the circuit depth, or equivalently\nthe computation time, is restricted to a constant irrele-\nvant to the problem size. The consideration of quantum\nshallow circuits was partly attributed to an experimen-\ntal perspective, as it is relatively simpler to deal with\nsystem decoherence within a fixed time [19]. More im-\nportantly, theorists have rich toolkits from quantum in-\nformation theory to aid the investigations. A particular\ninstrument is quantum nonlocality, one of the most dis-\ntinguishing properties of quantum theory [20]. As shown\nby the renowned Bell theorem [21], entanglement leads to\npurely quantum correlations between nonlocal observers\nbeyond the scope of classical physics [22]. In specific\ncases, nonlocal observers can synchronize their behaviors\nperfectly without signaling to each other, where entangle-\nment brings them quantum “pseudo-telepathy” [23]. On\nthe contrary, any attempt to simulate such correlations\nby classical parties necessarily requires communication.\nOne can translate quantum nonlocality into a computa-\ntion task to generate nonlocal statistics among distant\ncomputing sites [24, 25]. While classical circuits require\na growing time with respect to the input size to scramble\nthe information, quantum shallow circuits are competent\nto the task, bringing an unconditional advantage.\nDespite the recent progress in shallow circuits, a vague\nquestion arises: Is “magic” indispensable for the full\npower of quantum computation in the low-complexity\nregime? Indeed, among all the existing explorations of\nquantum shallow circuits, the essential ingredient for the\nquantum advantage — long-range entanglement, can be\ngenerated with Clifford circuits without using the magic\nstate [26]. On the other hand, though not rigorous, with\nour experiences in the complexity theory such as the\npadding argument [27]1, we may be inclined to think of\na collapse of the power of universal quantum computa-\ntion if magic makes no difference in the low-complexity\nregime. Moreover, from an experimental consideration,\nwhile deep quantum circuits are still challenging at the\nmoment, preparing magic states and implementing a few\nlayers of non-Clifford gates are becoming easy [28, 29].\nIt would be interesting if we could devise computation\ntasks where quantum magic plays a role.\nIn this work, we unconditionally confirm that quan-\ntum magic brings an advantage, at least in a shallow cir-\ncuit. For this purpose, we construct a quantum pseudo-\ntelepathy correlation requiring magic resources and prove\nstrict upper bounds on the correlation strength of solely\nmagic-free operations. Then, we translate such nonlocal-\nity into a shallow circuit computation task.\n1In brief, the padding argument is a tool to prove that two com-\nplexity classes are equal assuming their counterparts in a low-\ncomplexity regime are equal. Nonetheless, note that confirming\na separation between two complexity classes does not guarantee\na separation between their counterparts in a higher complexity.II. MAGIC-NECESSARY BI-\nNARY CONSTRAINT SYSTEM\nOur starting point is a special nonlocal game originat-\ning from the linear binary constraint system (BCS) [32].\nA BCS comprises a set of Boolean functions, namely con-\nstraints, over binary variables vi. We take the variable\nvalues over {+1,−1}for later convenience. For a lin-\near BCS, the constraints are given by multilinear func-\ntions of the variables, namely in the form ofQ\ni∈Sjvi=\ncj∈ {+1,−1}, where Sjdefines a subset of the variables.\nGiven a linear BCS, consider a corresponding nonlocal\ngame with two parties, Alice and Bob. In each round of\nthe game, a referee picks a constraint from the BCS la-\nbelled by cjand a variable i∈ Sj. The referee asks Alice\nto assign values to the variables satisfying the constraint\nand Bob to output a value for vi. The nonlocal play-\ners win the game if and only if Alice gives a satisfying\nassignment for the constraint, and her assignment to vi\ncoincides with Bob’s. Alice and Bob cannot communi-\ncate with each other once the game starts. Nevertheless,\nthey can agree on a game strategy in advance.\nNow, we discuss the potential strategies of the nonlocal\nplayers. If the linear BCS has a solution where a fixed\nvalue assignment to the variables satisfies all the con-\nstraints, then Alice and Bob can win the associated non-\nlocal game. Actually, this is the only way that the nonlo-\ncal players restricted to classical means can win the game\nperfectly [32]. Notwithstanding, even if a fixed satisfy-\ning assignment to the BCS does not exist, the nonlocal\nplayers may still win the game perfectly. In the quantum\nworld, Alice and Bob can pre-share an entangled state.\nInstead of fixing the values of all the variables in each\nconstraint all at once, the players measure proper observ-\nables with respect to the referee’s questions and assign\nthe measurement results to the variables. For a valid joint\nmeasurement of multiple observables on Alice’s side, the\nobservables corresponding to the variables in each con-\nstraint need to be compatible. Due to the intrinsic ran-\ndomness in quantum measurements, an observable may\ntake different outcomes in each constraint, hence assign-\ning a different value to the same variable. Such flexibility\nbrings an advantage over classical means. A famous ex-\nample is the Mermin-Peres nonlocal game [30, 31], as\nshown in Fig. 1(b). The underlying BCS does not have\na fixed satisfying assignment and thus no classical win-\nning strategy for this nonlocal game. On the other hand,\nit can be won perfectly with a quantum strategy, where\nquantum entanglement brings Alice and Bob “pseudo-\ntelepathy” as if they knew what was going on on the\nother side via a “spooky action” [23].\nWhile seemingly different from the classical strategies,\nthe existence of a quantum perfect winning strategy is\nalso closely related to the properties of the underlying\nlinear BCS. Instead of taking values over the binary field,\nwe generalize the BCS to an operator-valued set of func-\ntions, where the scalar variables are replaced with Her-\nmitian operators Aiwith eigenvalues {+1,−1}, and the3\nAlice Bobj i\n|ψ⟩AB\n{vk}s.t.Q\nk∈Sjvk=cj vi\n(a)v1, Z⊗I v2,I⊗Z v3, Z⊗Z\nv4,I⊗X v5, X⊗Iv6, X⊗X\nv7, Z⊗X v8, X⊗Z v9, Y⊗Y\n(b)u\nvst\n(c)\nFIG. 1: (a) A general BCS nonlocal game. Given a BCS, the input or the question to Alice is a constraint indexed by\njand the question to Bob is a variable indexed by i, with i∈ Sj. Alice outputs an assignment to the variables {vk}\nsatisfying the constraint jand Bob outputs a value for vi. The players win the game if and only if Alice’s assignment\ntovimatches Bob’s. Alice and Bob cannot communicate once the game starts. Nevertheless, they can pre-share a\nquantum state |ψ⟩ABand take quantum measurements. (b) The Mermin-Peres BCS, presented as a graph [30, 31]a.\nThe BCS has 9 variables v1,···, v9and 6 constraints. Each straight line represents a constraint, where a single line\nstands for cj= 1, e.g., v1v2v3= 1, and a double line stands for cj=−1, e.g., v3v6v9=−1. This BCS has an operator-\nvalued solution, as denoted by the operators next to the variables. Note that each operator is a tensor product of\nPauli matrices. (c) A subgraph with four vertices to illustrate the BCS in Eq. (2). Each vertex vcorresponds to one\nvariable av, and there are four such variables in the subgraph. Each undirected edge e= (u, v) corresponds to three\nvariables xuv, yuv, zuv. With six edges in the subgraph, there are six variables for each kind. Every two disjoint edges\ne1= (u, v), e2= (s, t) correspond to two kinds of variable buv|st=bst|uvandcuv|st, cst|uv. The subgraph has three sets\nof disjoint edges, denoted by black solid lines, blue dashed lines, and orange dotted lines, respectively. Consequently,\nthere are three variables of the kind of buv|stand six variables of the kind of cuv|st.\naIt is also well-known by “magic square,” while we avoid using this name as it may be confused with the term quantum\n“magic” [4]. It is amusing that the perfect winning strategy of the magic square game does not require any quantum magic.\nconstraints are given in terms ofQ\ni∈SjAi=cjIwithI\nan identity operator of a finite dimension. Correspond-\ning to a valid joint measurement of Alice, here, the oper-\nators in each constraint need to be compatible and thus\nsimultaneously measurable. The existence of a quantum\nperfect winning strategy is equivalent to the operator-\nvalued BCS having a solution [32]. Suppose the solu-\ntion to the operator-valued BCS is given by a set of d-\ndimensional operators, {Ai}i, then the perfect winning\nstrategy in the corresponding nonlocal game goes as fol-\nlows: Alice and Bob first share a maximally entangled\nstate, |Φ+⟩=Pd−1\ni=0|ii⟩/√\nd. Afterward, Alice measures\nthe observables {Ai}i, and Bob measures the observables\n{AT\ni}ito assign values to the variables, where T denotes\nthe operator transpose. As the observables satisfy the\noperator-valued BCS, Alice’s measurement results natu-\nrally satisfy the constraint. Also, as the maximally en-\ntangled state has the property\n\nΦ+\f\fAt⊗AT\nt\f\fΦ+\u000b\n=1\ndtr\u0000\nA2\nt\u0001\n= 1, (1)\nthe assignments of Alice and Bob to the same variable\nthus coincide.\nAmong quantum strategies for the nonlocal game,\nthere are also different levels of capabilities. Instead of\nhaving access to all quantum states and operations, if\nthe nonlocal players are constrained from sharing certain\nquantum resources, in general, the probability of winningthe nonlocal game becomes limited. Here, we are inter-\nested in the difference made by quantum magic [4]. A\nrestricted class of quantum strategies is the set of only\nClifford operations, which map Pauli operators to them-\nselves by conjugation actions [5] and do not create quan-\ntum magic [4]. Specifically, Alice and Bob can apply\nClifford operations to a state initialized in |0⟩before the\nnonlocal game to create entanglement. Afterward, they\neach take a share of the state and apply only Pauli-string\nmeasurements to the state for the game. For simplicity,\nwe call it a Clifford strategy. If the nonlocal game has a\nClifford strategy that wins perfectly, then the underlying\nBCS has a Pauli-string solution, and vice versa . Notably,\nwe have the following general results. We present their\nproofs in Appendix B 2.\nTheorem 1. Given a linear BCS with lvariables and m\nconstraints, there exists a classical algorithm that finishes\ninpoly( l, m)steps to determine whether the BCS has a\nPauli-string operator-valued solution. If the answer is\naffirmative, the algorithm returns one such solution.\nTheorem 2. Suppose a linear BCS does not have a\nPauli-string solution. Then for its associated nonlocal\ngame, if Alice and Bob are restricted to Clifford strate-\ngies, either Alice fails to give satisfying assignments for\nall the constraints, or there exists one pair of questions\n(cj, vi), where the probability that Alice and Bob’s assign-\nments to vicoincide does not exceed 1/2.4\nIn the literature, various linear BCS nonlocal games\nhave been proposed to exhibit quantum pseudo-\ntelepathy, while Clifford operations suffice for the task.\nIt was conjectured that whenever a linear BCS nonlocal\ngame has a perfect winning strategy, it is either a Clifford\nor a classical one [33]. Recent group embedding results\nevidence the falseness of the above conjecture [34, 35].\nHere, we directly present a linear BCS nonlocal game\nto disprove the conjecture. To make it illustrative, we\nstate the underlying BCS in the language of graph the-\nory. Consider an undirected complete graph G= (V, E)\nwith nvertices. An undirected graph indicates that for\nany two connected vertices, uandv, the tuples ( u, v)\nand ( v, u) are the same and represent the same edge.\nThe BCS contains the following variables:\n1. Each vertex v∈Vcorresponds to one variable av.\n2. Each undirected edge, denoted by e= (u, v)∈E,\ncorresponds to three variables xuv, yuv, and zuv.\n3. Every two disjoint edges, denoted by e1= (u, v)∈\nEande2= (s, t)∈E, where s, t, u , and vare\ndifferent vertices, correspond to\n(a) one variable be1e2≡buv|st, where be1e2=\nbe2e1;\n(b) two variables ce1e2≡cuv|standce2e1≡cst|uv,\nwhere ce1e2̸=ce2e1in general.\nFor clarity, we express the variables with respect to the\nunderlying vertices and denote the BCS (nonlocal game)\nsize with the number of vertices. The smallest non-trivial\nBCS is defined on a graph with four vertices, as shown\nin Fig. 1(c). Based on these variables, the BCS contains\nthe following constraints,\nauavyuv= 1,∀(u, v)∈E,\nxuvyuvzuv= 1,∀(u, v)∈E,\nxuvxstbuv|st= 1,∀(u, v),(s, t)∈E,\nxuvzstcuv|st= 1,∀(u, v),(s, t)∈E,\nbuv|stbvs|utbsu|vt= 1,∀u, v, s, t ∈V,\ncuv|stcvs|utcsu|vt= 1,∀u, v, s, t ∈V,\nY\nv∈Vav=−1.(2)\nThis family of BCS’s exhibits a hierarchy among classical,\nClifford, and general quantum resources, as shown by the\nfollowing theorem.\nTheorem 3. For the nonlocal game defined through the\nBCS in Eq. (2),\n1. when n= 4, it has a Clifford strategy to win per-\nfectly, but it does not have a perfect-winning clas-\nsical strategy;\n2. when n∈2N+ 5 = {5,7,9,···}, it has a perfect-\nwinning classical strategy;3. when n∈2N+ 6 = {6,8,10,···}, it has strategies\nthat exploit quantum magic to win perfectly, but it\ndoes not have a perfect-winning Clifford strategy or\nclassical strategy.\nIn Appendix B, we present a group-theoretic method\nto determine the perfect-winning strategies. For later\nconvenience, we slightly modify the underlying BCS by\nintroducing ( n−3) additional variables and decompos-\ning the constraintQ\nv∈Vav=−1 into an equivalent set\nof (n−2) constraints, each consisting of three variables.\nWe shall explain the modification in Appendix C. In the\nassociated nonlocal game, denote the set of questions for\nAlice as QA, corresponding to the set of constraints, and\ndenote the set of questions for Bob as QB, corresponding\nto the variables. By construction of the BCS nonlocal\ngame, the two sets are correlated with each other. Sup-\npose the size of the underlying BCS is n, then there are\n3|QA|sets of questions to the players, with\n|QA|= 2\u0012n\n2\u0013\n+ 14\u0012n\n4\u0013\n+n−2. (3)\nAs a corollary of Theorem 2, for the nonlocal game with\nn∈2N+ 6, with uniformly distributed random ques-\ntions, the winning probabilities of all Clifford and classi-\ncal strategies can be upper-bounded by\npCli≤1−1\n6|QA|. (4)\nBefore moving on, we briefly discuss some features of\nthe nonlocal game we construct. As explained above,\nperfect winning strategies are given by the solutions to\nthe operator-valued version of the BCS. The most no-\ntable feature is that the operator-valued BCS of Eq. (2)\nwith n∈2N+6 has more than one inequivalent solutions\nand thus non-unique quantum strategies to win the cor-\nresponding nonlocal game perfectly. In particular, the\ncorresponding quantum strategies for the nonlocal game\ntake maximally entangled states with different dimen-\nsions and different measurements. This property sharply\ndiffers from common quantum nonlocal games, even in-\ncluding the ones with the “weak-form” self-testing prop-\nerty [36, 37]. While the measurements can be non-unique\nfor the latter, they require a unique entangled state with\na fixed dimension up to local isometry for the optimal\nquantum strategy [38]. Below, we present a realization\nofavandxuvin the above BCS when n= 8. Labelling\nthe vertices from 1 to 8,\nav=I8−2evv, v= 1,···,8,\nx1v=I8−e11−evv+e1v+ev1, v= 2,···,8,(5)\nwhereI8is an eight-dimensional identity operator, and\neijdenotes an elementary matrix, of which the element\nin the i’th row and j’th column is one and all the other\nelements are zero. The other operators can be deter-\nmined via av’s and x1v’s. Among all the perfect winning5\nstrategies, we can specify the one that consumes the least\namount of entanglement. With respect to the BCS size\nn, the number of e-bits for the strategy, namely, the two-\nqubit maximally entangled state ( |00⟩+|11⟩)/√\n2, scales\nat a rate of O(logn).\nIII. “MAGIC” COMPUTATIONAL AD-\nVANTAGE IN SHALLOW CIRCUITS\nThe nonlocal game shown in Theorem 3 separates the\ncapabilities between a generic quantum world and the\nmagic-free world to generate correlations. Now, we trans-\nlate correlation generation into a computation task of a\nrelation problem2and show the advantage of magic. As a\nreminder, the computation task is a single-user one. “Al-\nice” and “Bob” now refer to parts of the circuit, which is\nmerely for an intuitive thinking. In particular, one should\nnot consider the task as a distributed computation.\nBriefly speaking, a relation problem randomly selects\nan input bit string zinfrom a set and asks the compu-\ntation to output a bit string zout, such that zoutalways\nsatisfies a certain relation with respect to zin. Given a\nnonlocal game with size n, we can define a relation prob-\nlem, Rn\nN, by embedding it into a one-dimensional grid,\nas shown in Fig. 2. One can imagine that two experi-\nmentalists, Alice and Bob, each hold Ncomputing sites\nand collaborate to solve Rn\nN. We use the capital letter\nNfor the number of computing sites to distinguish it\nfrom the underlying nonlocal game size n. The input\nofRn\nNrandomly specifies one site of Alice and Bob, re-\nspectively, and picks up questions in the nonlocal game,\nwhich we encode as bit strings α∈ QA, β∈ QB. Among\nthe valid outputs of Rn\nN, the specified two sites are re-\nquired to output the correct answer with respect to the\nquestions, ( rA\nj,rB\nk) =f(α, β), denoted as a function of\nthe questions, with a non-negligible probability.\nIn a circuit comprising K-bounded fan-in gates, where\neach gate can act on at most Kinputs, the value K\nmimics the light speed for information scrambling [39].\nFurthermore, if the circuit is shallow, where the circuit\ndepth is a constant independent of the problem size, it\nrestricts the “time” for information scrambling; hence,\nmany sites in the circuit are “space-like” separated from\neach other. Alice and Bob must be capable of playing\nthe nonlocal game between such sites to compute the re-\nlation problem successfully. Suppose the nonlocal game\ndefining the relation problem cannot be won perfectly\nwithout a particular resource. In that case, the players\n2Strictly speaking, our problem not only requires the algorithm\nto return an output that satisfies certain criteria, but also to re-\nturn one specific output with a non-negligible probability. This\ndiffers from the standard relation problem, where it suffices for\nthe algorithm to return a valid output. A more accurate state-\nment of our results can be given in terms of a sampling problem.\nNevertheless, we shall keep using the term of a relation problem,\nwhich is more intuitive.must communicate to exchange information and generate\nthe desired correlation, which takes “time.” Therefore, a\nshallow circuit with bounded fan-in gates should fail. On\nthe contrary, things become different if the nonlocal game\ncan be won perfectly: first, entanglement can be created\nbetween certain sites and then distributed between two\narbitrary sites via entanglement swapping with bounded\nfan-in quantum gates in constant steps [25]; after shar-\ning entanglement, the two specified sites apply the per-\nfect winning strategy of the nonlocal game to output the\ncorrect answers of Rn\nN. Note that quantum gates act-\ning on near-neighbor qubits, or geometrically local gates,\nsuffice to complete the two steps. A subtle issue is the\nso-called bit and phase flips in entanglement swapping,\nwhere a subsystem of |Φ+⟩may experience a rotation of\na Pauli operator. One can embed the circuit outcomes in\nthese cases to the allowed relation for the computation\nproblem.\nApplying the Mermin-Peres nonlocal game to this\nlogic, an unconditional advantage of quantum shallow\ncircuits over their classical counterparts has been estab-\nlished [25]. Considering the relation problem Rn\nNdefined\nwith the new BCS nonlocal game in Eq. (2), one can fur-\nther expect an unconditional separation between generic\nquantum shallow circuits and those without magic. Here,\nwe specify the exact meaning of these circuits. As shown\nin Fig. 3, a generic gate transforms a number of classical\nbits and qubits. The classically controlled quantum gate\nand the measurement can be both unified in this way. In\nour definition, the fan-in includes both classical bits and\nqubits. We call a generic quantum shallow circuit com-\nprising general bounded fan-in gates a QNC0circuit [40]3.\nIf there is no qubit, the circuit degenerates into a clas-\nsical one, called an NC0circuit [17, 41]. For the circuit\nwithout magic, the initial quantum state must be magic-\nfree, which can be taken as |0⟩without loss of generality,\nthe classically controlled quantum gates must be Clifford\ngates, and the measurements must be Pauli measure-\nments. In this way, there is no magic in the quantum\npart in any step of this circuit, and we name such circuit\nClifNC0.\nThe following theorem illustrates the separation be-\ntween QNC0andClifNC0circuits in solving the BCS rela-\ntion computation problem. We fix n= 8 here, while the\nresult can be generalized to a general fixed value of n.\nWe provide the details in the Appendix C.\nTheorem 4. Given the relation problem, R8\nN, and a con-\nstant integer K≥14, the following holds:\n•The relation problem can be perfectly solved by\na constant-depth quantum circuit with K-bounded\nfan-in geometrically local gates.\n•Any circuit containing only K-bounded fan-in\nmagic-free gates that solves the relation problem\n3Like previous works in the field, we overuse the complexity classes\nnotations for the associated circuits.6\nInput of R8\nN:\nAn instance of ( j, k, α, β ) defining\nαi=(\nα,ifi=j,\n⊥,ifi̸=j,\nβi=(\nβ,ifi=k,\n⊥,ifi̸=k,\nwhere 1 ≤j < k≤N, α∈ QA, β∈ QB.\nValid outputs of R8\nN:\nBit strings {rA\ni(l)}iand{rB\ni(l)}i, l∈ {1,2,3},\nCase I: Long-range nonlocal game\nY\nj<i≤krA\ni(l) = +1 ,∀l∈ {1,2,3},\nY\nj≤i<krB\ni(l) = +1 ,∀l∈ {1,2,3},\n(rA\nj,rB\nk) =f(α, β),\nwhere rA\ni= (rA\ni(1), rA\ni(2), rA\ni(3))∈ {± 1}3and similar to rB\ni,\nf(α, β) is the satisfying assignment in the nonlocal game.\nCase II: Entanglement swapping flips\nThere exists l∈ {1,2,3}, such that\nY\nj<i≤krA\ni(l) =−1,orY\nj≤i<krB\ni(l) =−1.\nAdditional requirement:\nBoth cases occur with a non-negligible probability,\nfor all (j, k, α, β )\n(a)\n (b)\nFIG. 2: (a) Description of the shallow-circuit computation task R8\nN. (b) A quantum shallow circuit that realizes R8\nN.\nThe odd-valued sites can be regarded as held by Alice, and the even-valued sites can be regarded as held by Bob.\nEach site consists of a set of classical wires to receive the input of R8\nNand quantum wires initialized in |0⟩for three\nqubits. Throughout this work, we denote the classical systems by double wires and quantum systems by single wires.\nFirst, Alice and Bob prepare three pairs of Einstein-Podolski-Rosen (EPR) state, ( |00⟩+|11⟩)/√\n2, between their\nadjacent sites (2 i−1,2i), i∈N+, as shown by the blue boxes. Next, given an input instance ( j, k, α, β ), Alice and Bob\nperform concatenated Bell-state measurements (BSM) between their adjacent sites (2 j,2j+ 1),···,(2k−2,2k−1),\nas shown by the white boxes. In the end, Alice and Bob perform the perfect-winning strategy for the nonlocal game\nat sites 2 j−1 and 2 kwith respect to the questions αandβ, respectively, as shown by the orange boxes.\nR8\nNwith probability larger than (1 + pCli)/2with\npCliis given in Eq. (4), where the gates can be\nnon-geometrically local, must have a circuit depth\nat least increasing logarithmically with respect to N.\nCombined with previous results [24, 25], Theorem 4\nshows the hierarchy of the circuit power:\nNC0⊂ClifNC0⊂QNC0. (6)\nIn the second part of the theorem, the logarithmic separa-\ntion is tight. That is, a magic-free circuit with bounded\nfan-in gates can solve the problem, of which the depthgrows logarithmically. For a straightforward solution, all\nthe computing sites send their input to a fixed ancilla,\nwhich performs the nonlocal game calculation and sends\nback the result to the specified sites. As entanglement\ndoes not help spread the input information faster, using\nmagic-free entanglement does not improve the result.\nIV. CONCLUSIONS\nIn summary, we discover a family of BCS nonlocal\ngames that requires quantum magic to win perfectly and7\n(a)\n (b)\n (c)\nFIG. 3: (a) A general K-bounded fan-in gate. In general, it acts on ncbits and nqqubits, with nc+nq≤K.\nWhen nc= 0, it becomes a normal quantum gate characterized by a unitary operation. When nq= 0, it becomes a\nnormal classical gate. (b) The classically controlled quantum gate. The classical input icontrols whether to apply the\nquantum gate Uito the quantum input state |ψ⟩. In a ClifNC0circuit, we allow such classically controlled gate with\nUirestricted to a Clifford operation. (c) A general quantum measurement characterized by a positive operator-valued\nmeasure (POVM). The classical system acts as a register to record the measurement result. In a ClifNC0circuit, each\nmeasurement element Eiis a mixture of stabilizer states. Equivalently, for a magic-free quantum input state |ψ⟩,\nthe post-measurement state |ϕi⟩in a ClifNC0circuit still does not contain quantum magic. In this work, we simply\nconsider projective measurements. Both (b) and (c) can be unified under the general description of (a).\ntranslate it into an unconditional proof of magic compu-\ntation advantage. We believe other nonlocal games exist\nthat can demonstrate magic advantage. For instance, fol-\nlowing the method of embedding a general group into a\nBCS in Ref. [35], one can obtain candidate BCS nonlocal\ngames. In Appendix D, we review the procedure. Com-\nbined with Theorem 1, one can efficiently check whether\nit has a perfect-winning Clifford strategy. Besides those\nbased on the BCS, one may look for other types of non-\nlocal games. Note that to prove the computation advan-\ntage, the winning probability of all Clifford strategies in\nthe nonlocal game, which can exploit infinite-dimensional\nquantum systems, must be upper-bounded. To perform\nthe relation problem calculation in a realistic experiment,\none shall further consider the noise and loss. For this\npurpose, noise-tolerant methods, such as error correction\nand mitigation in a shallow circuit, need to be developed.\nWhile not being the purpose of this work, by applying the\n“game gluing” technique [42, 43], one can combine games\nwith different sizes in the family and construct a nonlo-\ncal game with a strict separation between the winning\nprobabilities of classical, Clifford, and general quantum\nstrategies. We consider this may be of an independent\ninterest to some research. This work takes the first step\nin proving the computation necessity of quantum magic\nunconditioned on any complexity assumption. We hope\nour results can inspire further explorations in this direc-\ntion, eventually going beyond the regime of shallow cir-\ncuits and solidifying the “magic” of universal quantum\ncomputation.\nACKNOWLEDGMENTS\nWe acknowledge Zhengfeng Ji and Honghao Fu for the\ninsightful discussions on the binary constraint systems\nand the group embedding results, Qi Zhao, Zhaohui Wei,\nand Yilei Chen for leading us to the consideration of non-\nClifford quantum operations, Yu Cai for introducing to us\nthe results in Ref. [43, 44], Yuwei Zhu, Boyang Chen, andYuxuan Yan for helpful discussions on the group repre-\nsentation theory and quantum magic. We express special\nthanks to Tian Ye for his vital and generous help during\nearly stages of this project in a weekly discussion. All\nthe authors acknowledge support by the National Natu-\nral Science Foundation of China Grant No. 12174216 and\nthe Innovation Program for Quantum Science and Tech-\nnology Grant No. 2021ZD0300804. XZ acknowledges ad-\nditional support by the National Natural Science Foun-\ndation of China through Grant No. 11975222 and support\nby the Hong Kong Research Grant Council through grant\nnumber R7035-21 of the Research Impact Fund.\nAll the authors contributed equally to this work.8\nAppendix A: Preliminaries\nIn this section, we review preliminary concepts for this work. We assume readers are familiar with the basic notions\nof linear algebra, graph theory, and group theory, and the basic description of quantum systems.\nIn the Appendix, we overuse some letters such as nwhen expressing the total number of items or labelling the\nvariables; nevertheless, their meaning can be specified from the context.\n1. Quantum magic and non-Clifford operations\nWe first briefly review the concept of quantum magic and related notions. For simplicity, we only consider the\nn-qubit system based on the Pauli group. Nevertheless, the results can be easily generalized to systems with a prime\ndimension by using the Weyl-Heisenberg algebra. Readers who are interested in the topic may refer to the Ph.D.\nthesis of Gottesman [5] and its following works for a more in-depth discussion.\nLet us start with the definition of the Pauli observables on a single qubit:\nI=\u0012\n1 0\n0 1\u0013\n, σ x=\u0012\n0 1\n1 0\u0013\n, σ y=\u0012\n0−i\ni0\u0013\n, σ z=\u0012\n1 0\n0−1\u0013\n, (A1)\nwhereIis the identity operator, and the other three observables σx,σy, and σzare always named nontrivial Pauli\nobservables. The n-qubit Pauli group is defined as the set of operators\nPn={±1,±i} × {I, σx, σy, σz}⊗n, (A2)\ntogether with the operator multiplication. Sometimes, one may omit the phase {±1,±i}and consider the n-qubit\nprojective Pauli group:\nP′\nn={I, σx, σy, σz}⊗n. (A3)\nThe Clifford group is defined as the normalizer of the Pauli group Pn:\nCn={C∈Un|∀P∈Pn, CPC†∈Pn}, (A4)\nwhereUnis the n-qubit unitary group. Operators in the Clifford group are called Clifford operations or gates. A\nhighly related concept is the stabilizer state, which is generated by applying Clifford gates on the computational basis\nstates, or equivalently the eigenstates of σ⊗n\nz. If a state cannot be prepared in this way or by mixing stabilizer states,\nthe state is said to contain quantum “magic” [4].\nIt is well known that if a quantum circuit only contains Clifford gates with the initial state being a computational\nbasis states and measurements as Pauli observables, the measurement results can be simulated by a polynomial-size\nclassical circuit. This is the famous Gottesman-Knill Theorem [5, 6]. Thus, this kind of circuit is deemed to be\n“easy” and contains no “magic.” Any circuit starting non-stabilizer states, applying non-Clifford gates, or measuring\nnon-Pauli observables might contain magic. As magic-free circuits can be simulated classically, magic is regarded as\nan indispensable resource for universal quantum computation.\n2. General binary constraint systems\nFor completeness, we first review the definition of a binary constraint system (BCS). A BCS consists of nbinary\nvariables v1,···, vnandmconstraints c1,···, cm, where each ciis a Boolean equation with respect to a subset of\nvj’s. Note that BCS with general Boolean constrains can describe general systems of equations [42]. In a linear\nBCS, all the constraints are given by addition over Z2, or the parity operation over Boolean variables ranging in\nZ2={0,1}. In the literature, such a BCS is also called a parity BCS. For convenience of a quantum generalization,\nit is equivalent to define the BCS over sign variables ranging in {+1,−1}. In this case, a Boolean function can be\nequivalently given by a multilinear function of a subset of variables on R. For a linear BCS, the parity constraint\nbecomes a product of the variables. In accordance with the notations in the main text, we mainly use the sign\nvariables and denote each constraint cjas a multilinear function of a set of variables {vi:i∈ Sj}, namely in the form\nofQ\ni∈Sjvi=cj∈ {+1,−1}. Nevertheless, it is sometimes more convenient to use the Boolean variables to represent\na BCS. In correspondence to the sign variables,\nvivj= +1⇔vi⊕vj= 0,\nvivj=−1⇔vi⊕vj= 1,(A5)9\nwhere the LHS are the notations using sign variables and the RHS are the notations using Boolean variables. We\nshall specify the notations if we resort to the Boolean variables.\nIf a BCS has a satisfying assignment, namely a fixed assignment to the variables that satisfies all the constraint,\nwe say it has a classical solution. Note that if all the constraints are cj= +1, the BCS can be trivially satisfied by\nassigning all the variables to be +1. With respect to the BCS size, searching for a classical solution to a general BCS\nisNP-hard. On the other hand, the problem is in Pfor linear BCS, where one can apply Gaussian elimination or the\nreplacement method to efficiently solve the system.\nThe quantum generalization of a BCS is an operator-valued constraint system. The variables vj’s are replaced with\nlinear operators Aj’s acting on a Hilbert space Hwith a finite dimension, such that\n1. Each Ajis Hermitian with eigenvalues in {1,−1}, i.e., Aj=A†\njandA2\nj=Ifor all j.\n2.Aj’s satisfy all the constraints with cireplaced with ciI, where Iis the identity operator on H.\n3. If AiandAjappear in the same constraint, they commute with each other, i.e., AiAj=AjAi.\nIf there exists a Hilbert space Hwith dimension dand a set of linear operators following the above requirements,\nwe say the BCS has a d-dimensional quantum satisfying assignment, or simply a quantum solution. As a side\nremark, the requirement that the operator variable acts on a finite-dimensional Hilbert space can be relaxed in several\ndirections, including allowing an infinite dimension and limits of finite-dimensional systems. We do not discuss such\ngeneralizations and refer readers to Ref. [35, 45] for a more detailed definition.\nIf a BCS has a quantum solution, one can apply quantum measurements to realize it in an experiment, where\nthey prepare independently and identically many copies of a quantum state and measure the observables in each\nconstraint. For each constraint, the measurement results shall satisfy the relation in the fashion of classical variables.\nNote that the requirement for a quantum solution guarantees the validity of a joint measurement for each constraint.\nDue to the intrinsic randomness in quantum measurements, the same variable may take different values in different\nconstraints. It is worth mentioning that measuring the set of observables on any state, including a maximally mixed\nstate, generates the desired statistics for the constraints. The quantum satisfying assignment is also called a state-\nindependent contextuality of the observables [46].\nAs the dimension can be arbitrary, searching for a quantum solution is undecidable [34, 35]. A helpful way of\nthinking is to regard the BCS as a group presentation statement [32, 47].\nDefinition 1 (Group presentation) .Given a set S, letF(S)be the free group on SandRa set of words on S, and\ndenote the quotient group of F(S)by the smallest normal subgroup containing each element in Ras⟨S:R⟩. A group\nGis said to have the presentation ⟨S:R⟩if it is isomorphic to ⟨S:R⟩.\nIn the group presentation, the elements in Sare called generators and the elements in Rare called relators. Given\na linear BCS, one can regard the constraints as a group presentation.\nDefinition 2 (Solution group of a linear BCS) .Given a linear BCS with nbinary variables {vi}andmconstraints\n{cj}, the solution group of the BCS is defined as the group with the following presentation:\nΓ ={{J, gi:i∈[n]}:{g2\ni=e,∀i∈[n],\nJ2=e,\n∀j∈[m], gkgl=glgk,∀k, l∈ Sj,\nJgi=giJ,∀i∈[n],\nY\ni∈Sjgi=Jχ(cj=−1),∀j∈[m]}},(A6)\nwhere the group element Jcorresponds to −1in the BCS, edefines the identity operator of the group, and χ(·)is the\nindicator function that takes the value 1if the argument is true and 0if the argument is false.\nNote that if J=e, the solution group is trivial, as the assignment of all the group elements to be esatisfies all the\nrelators. On the contrary, should J̸=e, the solution group is non-trivial, and the group presentation corresponds to\na valid operator-valued solution to the underlying BCS, as stated by the following lemma.\nLemma 1 ([43, 47]) .Given a linear BCS that defines a solution group with the group element Jcorresponding to −1, if\nJis non-trivial in some finite-dimensional representation of the solution group, then the BCS has a finite-dimensional\nquantum satisfying assignment. The converse is also true.10\nThe irreducible representation of the generators in the solution group determines the quantum realization [37, 47].\nHere, a representation of the group refers to a group homomorphism of the group to a set of unitary operators on\na Hilbert space, and an irreducible representation refers to a representation that does not have a non-trivial group-\ninvariant subspace [48]. For a classical solution, it can be described by the Abelian group, where all the group elements\ncommute with each other.\nAs an example of linear BCS, we review the Mermin-Peres BCS, also widely known as the “magic-square” system [30,\n31]. Note that one should not mistake the name “magic” with the quantum resource of magic states. The Mermin-\nPeres BCS involves n= 9 variables and m= 6 constraints. In terms of sign variables ranging in {+1,−1}, the BCS\nis defined as follows:\nv1v2v3= 1,\nv4v5v6= 1,\nv7v8v9= 1,\nv1v4v7= 1,\nv2v5v8= 1,\nv3v6v9=−1.(A7)\nThis BCS does not have a classical solution. On the other hand, it has a unique quantum solution over the Pauli\ngroup. We denote the operator that corresponds to viasAiin accordance with the notations above. The quantum\nsolution is given as follows [43, 44]:\nA1=σz⊗I,\nA2=I⊗σz,\nA3=σz⊗σz,\nA4=I⊗σx,\nA5=σx⊗I,\nA6=σx⊗σx,\nA7=σz⊗σx,\nA8=σx⊗σz,\nA9=σy⊗σy,(A8)\nwhere ⊗stands for the tensor product operation.\nGiven a BCS, one can define an associated nonlocal game [32]. In the nonlocal game, there are two cooperating\nplayers, Alice and Bob, who cannot communicate with each other once the game starts. With respect to a probability\ndistribution, a referee randomly selects one constraint, cs, and one variable, vt, contained in the constraint. In our\nwork, we always take the probability distribution to be uniform. The referee send sto Alice and tto Bob. Then,\nAlice returns an assignment to each variable viincsthat satisfies the constraint and Bob returns an assignment to\nvariable vt. They win the game if and only if the assignments of Bob and Alice to vtare the same. This type of\nnonlocal game extends the well-known Clauser-Horne-Shimony-Holt (CHSH) game [49], where the underlying BCS\ninvolves two binary variables and two multi-linear constraints,\nv1v2= 1,\nv1v2=−1.(A9)\nThis BCS game is equivalent to the CHSH game in the sense of the probability distribution that Alice and Bob can\nachieve. Note that this BCS does not have either a classical solution or a quantum solution. Still, quantum strategies\nfor the game can bring a higher winning probability than classical ones.\nTo maximize the winning probability, Alice and Bob can agree on a strategy for playing the game. We call a\nstrategy is perfect if it wins with probability 1. We say Alice and Bob apply a classical strategy if they can access only\nshared and local randomness. In quantum theory, Alice and Bob can pre-share entanglement and apply local quantum\noperations. In general, when a BCS does not have a solution, it is possible that Alice assigns different values to the\nsame variable upon different questions of constraint. However, it brings limited advantage. For classical strategies,\nusing basic linear algebra analysis, it is not hard to prove that a BCS game has a perfect classical strategy if and\nonly if the corresponding BCS has a solution. It follows that to decide whether a general BCS game has a perfect\nclassical strategy is in NP-hard for general Boolean constraints and in Pfor linear constraints with respect to the\nBCS size. In the above examples, the CHSH game has a maximal winning probability of 3 /4, and the Mermin-Peres11\ngame has a maximal winning probability of 8 /9. In the quantum case, if the BCS does not have an operator-valued\nsolution, then there does not exist a perfect winning strategy for the associated nonlocal game, and vice versa . This\nfact is first proved in Ref. [32]. If the BCS has a quantum solution, it is linked to a perfect quantum winning\nstrategy in a one-to-one correspondence. Suppose the quantum solution to the BCS is given by observables {Ai}i\nacting on a d-dimensional system. Alice and Bob first share a maximally entangled state, |Φ+⟩=Pd−1\ni=0|ii⟩/√\nd.\nWhen the nonlocal game starts, upon receiving the constraint cs, Alice measures the observables Aibelonging to the\nconstraint, and upon receiving the variable vt, Bob measures the transpose of the observable At, denoted by AT\nt. The\nmeasurement statistics satisfy the winning condition.\nFor the well-known existing BCS that are solvable, it either has a classical solution, which corresponds to an Abelian\ngroup, or a quantum solution of Pauli strings as in the case of the Mermin-Peres magic square, which corresponds to\nthe Pauli group. In Ref. [33], the author provides an efficient algorithm to determine perfect quantum solutions to\na special type of linear BCS, where each variable shows up in exactly two constraints. Moreover, if such a BCS has\na solution, the solution is necessarily given by Pauli strings, i.e., the solution is in the Pauli group. Following this\nresult, it was conjectured that any linear BCS with an operator-valued satisfying assignment belongs to either of the\ntwo cases [33]. As mentioned in the main text, this conjecture has been suggested false [34, 35]. In the next section,\nwe directly disprove this conjecture with a specific linear BCS.\nAppendix B: Group-Theoretic Analysis of the Binary Constraint System\nIn this section, we apply group-theoretic tools to analyse the properties of the proposed linear BCS in the main\ntext.\n1. A binary constraint system over the permutation group\nWe first review the BCS proposed in this work. Given an undirected complete graph G= (V, E) with nvertices, it\ndefines the following variables:\n1. Each vertex v∈Vcorresponds to one variable av.\n2. Each undirected edge, denoted by e= (u, v)∈E, corresponds to three variables xuv, yuv, zuv.\n3. Every two disjoint edges, denoted by e1= (u, v)∈Eande2= (s, t)∈E, where s, t, u, v are different vertices,\ncorrespond to\n(a) one variable buv|st≡bst|uv, where be1e2=be2e1;\n(b) two variables ce1e2≡cuv|standce2e1≡cst|uv, where ce1e2̸=ce2e1in general.\nBased on these variables, the BCS contains the following constraints,\nauavyuv= 1,∀(u, v)∈E,\nxuvyuvzuv= 1,∀(u, v)∈E,\nxuvxstbuv|st= 1,∀(u, v),(s, t)∈E,\nxuvzstcuv|st= 1,∀(u, v),(s, t)∈E,\nbuv|stbvs|utbsu|vt= 1,∀(u, v),(s, t)∈E,\ncuv|stcvs|utcsu|vt= 1,∀(u, v),(s, t)∈E,\nY\nv∈Vav=−1.(B1)\nFor a nontrivial BCS, the system size is at least n= 4. If the BCS has a solution, then the other variables can be\ngenerated by a′s and x′s. For convenience, we label the vertices with natural numbers from 1 to n. For each vertex\nvand each edge ( u, v), u̸=v, we can use transpositions between elements in the set [2 n] to represent the generators,\nwhere av= (2v−1 2v) and xuv= (2u−1 2v−1)(2u2v). Here, ( i j) represents a transposition between iandj. We\nhave the following results for the BCS.\nTheorem 5. For any BCS of the above form with n∈2N+ 5 = {5,7,9,···}, it has a classical solution.12\nProof. When n∈2N+ 5, the BCS can be satisfied by assigning all the variables av’s to be −1 and all the other\nvariables to be 1.\nTheorem 6. When n= 4, this BCS has a two-qubit Pauli-string solution. On the other hand, the BCS does not\nhave a classical solution or a single-qubit Pauli solution in this case.\nProof. The BCS having no classical solution can be directly checked by solving the BCS on the binary field. By using\nthe fact that there is no state-independent contextuality in a qubit system, one can prove that the BCS does not have\na single-qubit Pauli solution either [50–52]. Later, we prove this statement under the context of linear BCS.\nFor the former statement, here is one construction of the two-qubit Pauli-string solution. We abbreviate the Pauli\noperators I, σx, σy, σzasI, X, Y, Z , respectively, and omit the tensor-product operator in the expressions.\na1 a2 a3 a4\n−ZZ II ZI IZ\nx12 x13 x14 x23 x24 x34\n−Y Y XI IX II II ZZ\ny12 y13 y14 y23 y24 y34\n−ZZ−IZ−ZI ZI IZ ZZ\nz12 z13 z14 z23 z24 z34\n−XX−XZ−ZX ZI IZ II\nb12|34b13|24b14|23\nXX XI IX\nc12|34c13|24c14|23c34|12c24|13c23|14\n−Y Y XZ ZX Y Y−XZ−ZX\nTABLE I: A two-qubit Pauli-string solution for the BCS when n= 4.\nTheorem 7 (A special case of the results in Ref. [50–52]) .If a linear BCS has a single-qubit operator-valued solution,\nthen it has a classical solution.\nProof. Two-dimensional matrices have such a special property: Suppose A, B∈C2×2and [ A, B] =AB−BA= 0.\nAt least one of the following cases would happen (1) A=cIfor some c∈C; (2) B=cIfor some c∈C; (3) A=cB\nfor some c∈C. Therefore, all two-dimensional matrices that are not proportional to the identity matrix Ican be\nclassified into different equivalence classes, with elements in a class proportional to each other.\nGiven a single-qubit operator-valued solution, denote the first equivalent class as {c1O, c2O,···|c1, c2,···=±1}\nwhere O̸=±IandO2=I. Substituting {c1O, c2O,···|c1, c2,···=±1}with{c1I, c2I,···|c1, c2,···=±1}and not\nchanging other variables also provides a solution. This is because (1) variables in different equivalent classes do not\nshow up in the same constraint; (2) the constraints among the variables proportional to Iand in the first equivalent\nclass are not violated after the substitution. Similarly, we can set all variables proportional to Ito give a solution.\nThe proportional coefficient is a valid classical solution.\nTheorem 8. For any BCS of the above form with n∈2N+6 = {6,8,10,···}, it does not have a Pauli-string solution.\nProof. Later we shall present a general method to determine whether a general linear BCS has a Pauli-string solution,\nwhere the current result can be regarded as a special case. Nevertheless, here we present a graph-based proof specific\nto this BCS, which is more illustrative.\nWe first specify the commutation properties of Pauli strings. The Pauli group elements are either anti-commuting,\nlike{X, Z}=XZ+ZX= 0, or commuting, like [ X⊗X, Z⊗Z] = (X⊗X)(Z⊗Z)−(Z⊗Z)(X⊗X) = 0. Therefore, if\nwe swap any two operators in a multiplication of some Pauli-string operators, the operator value of the multiplication\nwould at most differ with a sign.\nNow we prove the theorem by contradiction. Assume the BCS described in the theorem has a Pauli-string solution.\nWithout loss of generality, with respect to the correspondence between the BCS variables and vertices in a fully con-\nnected undirected graph in Fig. 1(c), let us consider the sets of variables and constraints corresponding to a subgraph13\nwith five vertices, labeled with 1 through 5. For the quadrangle {1,2,4,5}, we have the equation b12|45b24|15b41|25=I.\nSubstituting all the b-type variables by xuvxstbuv|st=I, we have\nx12x45x24x15x41x25=I. (B2)\nSimilarly, for the quadrangles {1,3,4,5}and{2,3,4,5}, we have the equations\nx13x45x34x15x41x35=I. (B3)\nx23x45x34x25x42x35=I. (B4)\nNext, by multiplying the left and right sides of Eqs. (B2)(B3) and (B4), respectively, swapping the order of the\nvariables, and eliminating the adjacent two variables that are the same, we get\nx12x13x23x45=±I, (B5)\nwhere “ ±” denotes that either case would happen. In this step, we have used the commutation properties of Pauli\nstrings. In other words,\nx45∈ {± x12x13x23}. (B6)\nAs a reminder, xuvandxvurepresent the same variable. Note that there is nothing special about the choice of x45\namong the variables, and a similar result can be obtained with an arbitrary specification of a subgraph with five\nvertices. For instance, we can get\nx46∈ {± x12x13x23}. (B7)\nCombining the above two equations, we have x45=±x46. For n≥6, following the above procedure and enumerating\nall such identities, one shall find that all the x-type variables differ from each other up to a sign, i.e., xuv∈ {± x}for\nall (u, v)∈E. Thus we can assume that xuv=x′\nuvxwhere x′\nuv∈ {± 1}.\nFollowing the specification of the x-type variables, for any four distinct vertices u, v, s, t ∈V, we have the following\nexpressions:\nbuv|st=xuvxst=x′\nuvx′\nstI,\ncuv|st=xuvzst=xuvxstyst=x′\nuvx′\nstyst=x′\nuvx′\nstasat.(B8)\nConsequently,\nbuv|stbvs|utbsu|vt=x′\nuvx′\nstx′\nvsx′\nutx′\nsux′\nvt= 1,\ncuv|stcvs|utcsu|vt=x′\nuvx′\nstx′\nvsx′\nutx′\nsux′\nvtasatauatavat= 1.(B9)\nNote that all the a-type variables commute with each other, since they simultaneously appear in the last equation of\nthe BCS. We hence derive that auavasat= 1 for all four distinct vertices u, v, s, t ∈V, which is equivalent to\nau=avasat. (B10)\nBy applying a similar argument as for the x-type variables, we shall find that all the a-type variables are identical, i.e.,\nav≡a,∀v∈V. This contradicts the constraintQ\nv∈Vav=−1 when n, the number of vertices, is even. Therefore,\nthe BCS of the above form with n∈2N+ 6 does not have a Pauli-string solution.\nTheorem 9. For any BCS of the above form with n≥4, label the vertices from 1ton. The BCS has a solution over\nthe centralizer group of element J= (1 2)(3 4) ···(2n−1 2n)in the permutation group S2n.\nProof. Consider x(u,v)= (2u−1 2v−1)(2u2v),av= (2v−1 2v), and all the other variables generated by them.\nOne can easily check that the assignment satisfies the constraints. As Jdoes not map to the identity element of the\ngroup, this is a non-trivial quantum solution following Lemma 1. The solution group corresponds to the centralizer\nofJin the permutation group, given by\nCJ=Sn⋉ Zn\n2, (B11)\nwhich is the semi-product of the permutation group Sngenerated by x(u,v), and an Abelian group Zn\n2generated by\nav.14\nNow we solve the irreducible representations of the solution, which gives the quantum realizations. We have the\nfollowing theorem.\nTheorem 10. An irreducible representation of Sn⋉Zn\n2can be labeled by\u0000\nm, θ(m), ρ(n−m)\u0001\nwhere mis an integer in\n[0, n],θ(m)andρ(n−m)are two irreducible representations of permutation groups SmandSn−m, respectively. Given\nlabel\u0000\nm, θ(m), ρ(n−m)\u0001\n, we first get an irreducible representation of group Sm×Sn−m⋉ Zn\n2, which is given by\nϕ(m,θ,ρ ):Sm×Sn−m⋉ Zn\n2→Mdimθ(m)×dimρ(n−m)(R)\nx∈Sm, y∈Sn−m, z∈Zn\n2,(x, y, z )7→θ(m)(x)⊗ρ(n−m)(y)⊗φ(m)(z).(B12)\nHere, φ(m)is an irreducible representation of Zn\n2such that for any element z= (z1, z2,···, zn)∈Zn\n2where ∀j, zj∈\n{0,1},\nφ(m)(z) =mY\nj=1(−1)zj. (B13)\nThe irreducible representation of Sn⋉Zn\n2labeled by\u0000\nm, θ(m), ρ(n−m)\u0001\n, denoted as Φ(m,θ,ρ ), is the induced representation\nofϕ(m,θ,ρ ). Specifically, one first finds the left coset of Sm×Sn−m⋉ Zn\n2inSn⋉ Zn\n2, given by\nn\ng1Sm×Sn−m⋉ Zn\n2, g2Sm×Sn−m⋉ Zn\n2,···, g(n\nm)Sm×Sn−m⋉ Zn\n2o\n, (B14)\nwheren\ng1, g2,···, g(n\nn)o\nare representative elements and g1is the identity. Then, the induced representation Φ(m,θ,ρ )\nis defined on the bases\b\ngi⃗ ej|1≤i≤\u0000n\nm\u0001\n,1≤j≤dimϕ(m,θ,ρ )\t\nwhere ⃗ ejis a basis of the representation space of\nϕ(m,θ,ρ ). That is, for any element g∈Sn⋉ Zn\n2, suppose that ggi∈gσ(i)Sm×Sn−m⋉ Zn\n2and set hi=g−1\nσ(i)ggiwhere\nσis a permutation on\b\n1,2,···,\u0000n\nm\u0001\t\n, then\nΦ(m,θ,ρ )(g) =\u0000\nΠσ⊗Idimϕ(m,θ,ρ )\u0001 nM\ni=1ϕ(m,θ,ρ )(hi)!\n. (B15)\nHere, Πσis a permutation matrix defined on the computational basis |1⟩,|2⟩,···,\f\f\u0000n\nm\u0001\u000b\nand transforms |i⟩to|σ(i)⟩.\nProof. This theorem is a direct corollary of Proposition 25 in [48]. To get an irreducible representation of CJ=Sn⋉Zn\n2,\nwe start from the irreducible representation of Zn\n2. Note that Zn\n2=⟨a1⟩ × ⟨a2⟩ × ··· × ⟨ an⟩is generated by ntwo-\norder elements a1, a2,···, an. Any irreducible representation of Zn\n2can be labeled by a vector with length n, like\n(1,−1,···,1), denoting the values that ngenerators would be mapped to in the representation. Meanwhile, we\ncall two irreducible representations equivalent if they can be mutually transformed via Sn. In other words, two\nirreducible representations are equivalent if and only if the corresponding vectors have the same number of -1. To\nobtain an irreducible representation of CJ, we only need to consider inequivalent irreducible representations of Zn\n2\nunder the transformation of Sn. Without loss of generality, we choose these irreducible representations as (1 ,1,···,1),\n(−1,1,···,1), (−1,−1,···,1), and ···, (−1,−1,···,−1) and label them with the number of -1, that is, {0,1,···, n}.\nFor a number m∈ {0,1,···, n}, we get an irreducible representation of Zn\n2, denoted as φ(m), mapping the generators\naitoφ(m)(ai) = (−1)1i≤m. Then, we consider a subgroup of Sn, such that any element hin this subgroup satisfies\n∀z∈Zn\n2,\nφ(m)(gzg−1) =φ(m)(z), (B16)\nor equivalently, 1 ≤i≤n,\nφ(m)(gaig−1) =φ(m)(ai). (B17)\nClearly, this subgroup must be Sm×Sn−m. Then, one can define the irreducible representation of Sm×Sn−m⋉ Zn\n2\nas\nϕ(m,θ,ρ ):Sm×Sn−m⋉ Zn\n2→Mdimθ(m)×dimρ(n−m)(R)\nx∈Sm, y∈Sn−m, z∈Zn\n2,(x, y, z )7→θ(m)(x)⊗ρ(n−m)(y)⊗φ(m)(z),(B18)\nwhere θ(m)andρ(n−m)are two irreducible representations of permutation groups SmandSn−m, respectively. Note\nthat ϕ(m,θ,ρ )is a well-defined group homomorphism due to the condition of Eq. (B16). Proposition 25 in [48] tells\nus that any irreducible representation of Sn⋉ Zn\n2can be constructed by the induced representation of ϕ(m,θ,ρ )by\ntraversing m,θ, and ρ. Proof is done.15\nFor a perfect strategy of the non-local game, the element Jmust be mapped to a non-identity element. Note that\nany element in CJcommutes with J. Via Theorem 10, one can obtain the following result:\nΦ(m,θ,ρ )(J) =φ(m)(J)Idim Φ\n=φ(m) nY\ni=1ai!\nIdim Φ\n=nY\ni=1φ(m)(ai)Idim Φ\n= (−1)mod ( m,2)Idim Φ,(B19)\nwhere dim Φ =\u0000n\nm\u0001\ndimϕ=\u0000n\nm\u0001\ndimθ(m)dimρ(n−m)≥\u0000n\nm\u0001\n. Thus, Φ(m,θ,ρ )(J) corresponds to a perfect measurement\nstrategy if and only if mis odd. The smallest dimension of the quantum system for a perfect strategy is nwhen\nm= 1 or m=n−1 and dim θ(m)= dim ρ(n−m)= 1.\nThe quantum realization of the BCS in Eq. (B1) is not unique. The underlying reason is that unlike the Pauli\ngroup, the permutation group has more than one inequivalent irreducible representations [48]. For n= 8, which is the\nsmallest size for a non-trivial result where there is not a Pauli-string solution to the BCS, we consider the case where\nm= 1 and θandρare both trivial representations, in which the dimension of the quantum system is 8. It implies\nthat the corresponding non-local game can be realized with only 3 EPR pairs. The representations of the generators\nare given by the following:\na1=\n−1 0···0\n0 1···0\n............\n0 0···1\n, a2=\n1 0 ···0\n0−1···0\n............\n0 0 ···1\n,···, a8=\n1 0··· 0\n0 1··· 0\n............\n0 0··· − 1\n, (B20)\nx12=\n0 1 0 ···0\n1 0 0 ···0\n0 0 1 ···0\n...............\n0 0 0 ···1\n, x13=\n0 0 1 ···0\n0 1 0 ···0\n1 0 0 ···0\n...............\n0 0 0 ···1\n,···, x18=\n0 0 0 ···1\n0 1 0 ···0\n0 0 1 ···0\n...............\n1 0 0 ···0\n. (B21)\nFrom the expression of the generators, we can see that the measurement observables do not belong to the Pauli group\nand need magic to realize. Following the same derivation, one can prove that the smallest non-trivial irreducible\nrepresentation of the solution to the BCS defined over nvertices requires an n-dimensional system. Thus, we obtain\nan upper bound of the smallest number of qubits to win the non-local game.\nCorollary 1. The smallest number of qubits to win the associated nonlocal game of Eq. (B1) is2 logn.\n2. Capabilities of Clifford strategies in the nonlocal game\nIn proving the “magic” advantage in shallow circuit quantum computation, we need to specify the capabilities of\nClifford strategies in the nonlocal BCS game. Thanks to the algebraic structure of BCS, we can use mature techniques\nfrom linear algebra to obtain quantitative results.\nSuppose the players in a nonlocal game are restricted to Clifford operations only, or that they do not have access\nto quantum magic resources. In this case, the most general strategy they can apply to playing the nonlocal game is\nas follows:\n•Before the nonlocal game starts:\n1. Alice and Bob prepares an n-qubit state and initialize it in |0⟩.\n2. Alice and Bob apply joint Clifford operations and Pauli-string measurements to the state and evolve it into\nan entangled state ρAB, where the subscripts denote the subsystems they each will hold in the game.\n•After the nonlocal game starts: Alice and Bob each applies Pauli-string measurements to their own quantum\nsystem.16\nIn our discussions, we allow an arbitrarily large n. Using a convexity argument, we know that a mixed state does\nnot bring any advantage to Alice and Bob in winning the nonlocal game, and we can hence take ρABas a pure state\n|ψ⟩without loss of generality. By further applying the Schmidt decomposition result, a pure state can be written as\n|ψ⟩=d−1X\ni=0αi|ii⟩, (B22)\nwhere ∀i, αi≥0, andPd−1\ni=0α2\ni= 1. Note that the maximally entangled state, which is\n\f\fΦ+\u000b\n=d−1X\ni=01√\nd|ii⟩, (B23)\ncan be prepared by applying control-NOT operations to |0⟩, which is a Clifford operation. Therefore, a general\nbipartite entangled state |ψ⟩shared by Alice and Bob can only be linked with |Φ+⟩with a Clifford operation, i.e.,\n|ψ⟩=UC|Φ+⟩.\nBased upon the above observations, we discuss the capabilities of Clifford operations in playing a parity BCS\nnonlocal game. In demonstrating the magic advantage, we are interested in the parity BCS that do not have a\nPauli-string quantum satisfying assignment. We have the following result for these instances.\nTheorem 11. Suppose a parity BCS does not have a satisfying assignment with Pauli-string observables. Then, for\nany Clifford strategy, there exist a constraint labelled by csand a variable in it labelled by vt, where the probability\nthat the assignments of Alice and Bob to vtunder the constraint csare identical does not exceed 1/2.\nProof. In the first part of the proof, we prove the case where Alice and Bob initially share a maximally entangled\nstate|Φ+⟩in Eq. (B23) of an arbitrary dimension and then generalize the result to general Clifford strategies. In the\nBCS nonlocal game, without loss of generality, upon receiving the constraint labelled by s, Alice shall measure a set of\ncommuting Pauli-string observables that return a satisfying assignment to the constraint, since a failure in satisfying\nthe constraint results in a loss in the nonlocal game. On the other hand, the observables that she measures for the\nsame variable, e.g., vt, in different constraints can be different. To specify her strategy, we denote the observable\nAlice measures for vairable vtin the constraint csasA(s)\nt. On Bob’s side, we denote the observable he measures for\nvariable vtasBt.\nNow, suppose Alice and Bob initially share the maximally entangled state |Φ+⟩. Assume there exists a Clifford\nstrategy, such that ∀s, t,\n\nΦ+\f\fA(s)\nt⊗Bt\f\fΦ+\u000b\n>0. (B24)\nSince Alice and Bob apply a Clifford strategy, A(s)\ntandBtare both Pauli strings, so is A(s)\ntBT\nt. Using the property\nof|Φ+⟩, the left-hand side of the above equation equals ⟨Φ+|A(s)\ntBT\nt⊗I|Φ+⟩= tr\u0010\nA(s)\ntBT\nt\u0011\n/d, where dis the system\ndimension. Since A(s)\ntBT\ntis a Pauli string, we have tr\u0010\nA(s)\ntBT\nt\u0011\n/d∈ {0,±1}. According to our assumption, we\nconclude that A(s)\nt=BT\ntand⟨Φ+|A(s)\nt⊗Bt|Φ+⟩= 1 for all s, t. Besides,\n\nΦ+\f\fA(s1)\ntA(s2)\nt⊗I\f\fΦ+\u000b\n=\nΦ+\f\f(A(s1)\nt⊗I)(A(s2)\nt⊗I)\f\fΦ+\u000b\n=\nΦ+\f\f(A(s1)\nt⊗I)(A(s2)\nt⊗I)\f\fΦ+\u000b\nΦ+\f\f(I⊗Bt)(I⊗Bt)\f\fΦ+\u000b\n≥\nΦ+\f\fA(s1)\nt⊗Bt\f\fΦ+\u000b\nΦ+\f\fA(s2)\nt⊗Bt\f\fΦ+\u000b\n>0,(B25)\nwhich holds for all s1, s2andt. In the third line, we apply the Cauchy-Schwarz inequality. The only value that the\nabove equation can take is hence 1, indicating that A(s1)\nt=A(s2)\nt. Thus we can omit the superscript ( s).\nSince the linear BCS does not have a satisfying assignment with Pauli-string observables, we can use the com-\nmutation properties of Pauli operators and the substitution method and derive an expression At1At2···Atm=−I\nfor a set of variables that leads to a contradiction of I=−I. The proof of this statement shall be given in Corol-\nlary 2 in Appendix D 2. Therefore, for any Clifford strategy, there exists a particular pair of inputs s, tsuch that\n⟨Φ+|A(s)\nt⊗Bt|Φ+⟩ ≤0. Consequently,\nPr[win |s, t] =1\n2\u0010\n1 +\nΦ+\f\fA(s)\nt⊗Bt\f\fΦ+\u000b\u0011\n≤1\n2. (B26)17\nTherefore, the average winning probability of the game is\nPr[win] =X\ns′,t′Pr[win |s′, t′] Pr[s′, t′]≤1−1\n2Pr[s, t]. (B27)\nIn the second part of the proof, we use the definition of Clifford operations that map a Pauli-string observable to a\nPauli-string observable. For any initial state |ψ⟩that Alice and Bob may share in advance, it is linked with |Φ+⟩via\na Clifford operation UC. Then for any Pauli-string observables At, Bt,\n⟨ψ|At⊗Bt|ψ⟩=\nΦ+\f\fU†\nC(At⊗Bt)UC\f\fΦ+\u000b\n≡\nΦ+\f\fA′\nt⊗B′\nt\f\fΦ+\u000b\n, (B28)\nwhere A′\ntandB′\ntare also Pauli-string observables that adapt to the systems of Alice and Bob, respectively. Then,\neither {A′\nt}fails in yielding a satisfying assignment to one of the constraints, or the proof dates back to the first part.\nThis finishes the proof.\nTheorem 11 links the upper bound of the winning probability of the BCS nonlocal game by Clifford strategies with\nthe game size. Consider the BCS game we construct through Eq. (B1). We denote the set of questions for Alice as\n˜QA, namely the BCS constraints, and the set of questions for Bob as ˜QB, namely the BCS variables. In the following\ntables, we give the expressions to calculate the set sizes and provide the concrete numbers for the BCS with n= 8.\nFor the BCS game of size nwith uniformly distributed questions, one may obtain a direct upper bound on the average\nwinning probability of Clifford strategies as 1 −1/(2[3(|˜QA| −1) + n]), noticing that the constraintQ\nv∈Vav=−1\ncomprises nvariables, while each of the other constraints consist of three variables.\nconstraint format aa′y= 1xyz= 1xx′b= 1 xzc= 1 bb′b′′= 1cc′c′′= 1Qa=−1total\nexpression\u0000n\n2\u0001 \u0000n\n2\u0001\u0000n\n4\u0001\n·3\u0000n\n4\u0001\n·3·2\u0000n\n4\u0001\u0000n\n4\u0001\n·4 1 |˜QA|\nnumber 28 28 210 420 70 280 1 1037\nTABLE II: Number of constraints in the BCS game.\nvariable type axyz b c total\nexpression n\u0000n\n2\u0001\u0000n\n2\u0001\u0000n\n2\u0001\u0000n\n4\u0001\n·3\u0000n\n4\u0001\n·3·2|˜QB|\nnumber 8282828 210 420 722\nTABLE III: Number of variables in the BCS game.\nAppendix C: 1D Magic BCS Relation Problem\nIn this part, we introduce the relation problem in detail by embedding the BCS nonlocal game into a one-dimensional\ngrid. We will prove Theorem 4, showing this problem can be solved by a generic constant-depth quantum circuit with\nonly bounded fan-in gates, while any magic-free circuit requires a circuit depth that increases at least logarithmically\nto the input size. For simplicity, we consider the non-trivial BCS game with size n= 8, of which the quantum\nrealization requires the least number of qubits, namely three pairs of qubits4. One can consider other values of n,\nwhere the proofs are similar.\nBefore we commence, we slightly modify the underlying BCS defined in Eq. (B1). In the BCS, all the constraints\nconsists of three variables except forQ\nv∈Vav=−1, which consists of nvariables. For this n-variable constraint, we\ncan introduce ( n−3) new variables and turn it into an equivalent set of ( n−2) constraints with three variables each.\nThat is, we introduce new variables a12, a123,···, a1···n−2and convert the constraint as\na1a2a3···an=−1⇐⇒\n\na1a2a12= 1\na12a3a123= 1\n···\na1···n−3an−2a1···n−2= 1\na1···n−2an−1an=−1.(C1)\n4Note that n= 6 is the smallest-sized BCS in this family that\nrequires a “magical” realization. Nevertheless, if one encode theinformation into qubits, it also requires at least three qubits.18\nNote that the commutation requirement between variables in each constraint of the BCS is preserved in this conversion.\nDenote the set of constraints in the modified BCS as QA. Suppose the original BCS with size nconsists of |˜QA|\nconstraints. Then, in the modified BCS nonlocal game, there are 3 |QA|sets of questions, with |QA|=|˜QA|+n−3.\nFor the case of n= 8, by applying Theorem 11, we have the following result.\nLemma 2. In the modified BCS game with n= 8, suppose the questions are picked up uniformly at random. Then,\nthe maximal winning probability for all Clifford strategies is upper-bounded by\npCli≤1−1\n6252. (C2)\nNow we introduce the relation problem RN, which is labeled with a number Nrepresenting the problem size. One\ncan assume that two players, Alice and Bob, collaborate with each other to solve RN. The input and output of RN\nare given as follows.\n1. Input: in each round, Alice and Bob are given a question,\nq= (α1,···, αN, β1,···, βN), αi∈ QA∪ {⊥} , βi∈ QB∪ {⊥} , (C3)\nwhere qstands for “question”, αiis the input on Alice’s side at site i∈[N], and βiis the input on Bob’s side\nat site i∈[N].QAconsists of the set of constraints in the BCS, and QBconsists of the set of variables in the\nBCS. Here, {⊥}represents a null input.\n2. Output: in each round, Alice and Bob need to return a reaction to the question,\nr= (rA\n1,···,rA\nN,rB\n1,···,rB\nN), (C4)\nwhere rstands for “reaction”, rA\ni= (rA\ni(1), rA\ni(2), rA\ni(3))∈ {± 1}3is the output on Alice’s side at site i, and\nsimilarly on Bob’s side.\nNow, we define the 1D magic BCS relation problem RN. In the computation task, Alice and Bob are promised to\nreceive an instance given by a tuple ( j, k, α, β ),1≤j < k≤N, which defines the input as\nαi=(\nα,ifi=j,\n⊥,ifi̸=j,βi=(\nβ,ifi=k,\n⊥,ifi̸=k.(C5)\nThat is, we require the sites jon Alice’s side and kon Bob’s side to play the BCS nonlocal game with questions α\nandβ, respectively. Alice and Bob are required to give an output satisfying either of the following requirements:\n1. For any l∈ {1,2,3},\nY\nj<i≤krA\ni(l) = +1 ,\nY\nj≤i<krB\ni(l) = +1 ,(C6)\nand\n(rA\nj,rB\nk) =f(α, β), (C7)\nwhere fis the relation defined by the BCS nonlocal game.\n2. There exists l∈ {1,2,3}such that,\nY\nj<i≤krA\ni(l) =−1, (C8)\nor\nY\nj≤i<krB\ni(l) =−1. (C9)19\nIn addition, we require that case 1 occurs with a probability no smaller than a positive constant value p∈(0,1/64]. By\nconstant, we mean that pcannot be negligibly small, where there exists a sequence of positive values {p1,···, pt,···}\nsuch that p= lim t→∞pt= 0+. Note that if k=j, Alice and Bob are required to output ( rA\nj,rB\nk) =f(α, β).\nBelow, we show the 1D magic relation problem RNcan be solved by a QNC0circuit but cannot be solved by any\nClifNC0circuit. We first show that there exists a shallow circuit with generic bounded fan-in quantum gates that\nperfectly completes this task. Now consider the following strategy:\n1. As shown in Fig. 2(b) (1D magic relation problem), Alice and Bob share 3 Npairs of EPR states, |Φ+⟩⊗3N,\nwhere |Φ+⟩= (|00⟩+|11⟩)/√\n2, and arrange them in three layers, denoted by q2i−1(l), q2i(l), i∈[N], l∈ {1,2,3},\nwhere qubits q2i−1(l) and q2i(l) reside in the state |Φ+⟩. Alice holds the qubits q2i−1(l) and Bob holds the qubits\nq2i(l).\n2. Except for qubits q2j−2(l), q2j−1(l) and q2k(l), q2k+1(l), perform an entanglement swapping operation between\npairs of EPRs with a BSM on qubits q2i(l) and q2i+1(l). Denote the Bell state measurement results on the pair\nof adjacent qubits as ( rA\ni(l), rB\ni(l)).\n3. On the three pairs of qubits q2j−1(l) and q2k(l), Alice and Bob perform the measurements corresponding to the\nwinning strategy in the BCS nonlocal game and obtain outputs ( rA\nj(l), rB\nk(l)).\n4. Take an arbitrary measurement on the qubits that are not measured and record the measurement results with\nrespect to the site indices.\nBy construction, this strategy naturally meets the problem requirements, and the first requirement defined through\nEq. (C6) and (C7) is met with probability 1 /64. One can also see the reason that we modify the underlying BCS\ngame: Alice needs to output an assignment to all the variables that appear in the constraint. As Alice can only output\n3 bits in the relation problem, we need to decompose the original n-variable constraint into smaller ones. Note that\nat the end of the entanglement swapping operations, qubits q2j−1(l) and q2k(l) reside in |Φ+⟩with probability 1 /4\nfor each l∈ {1,2,3}. Measuring the observables in the winning strategy of the BCS nonlocal game thus results in\nthe desired statistics. Moreover, the above strategy can be realized in a constant depth with finite fan-in operations\nand a computational basis measurement. Nevertheless, there is a minimal fan-in size of the gates to realize the above\nstrategy. The following theorem gives a sufficient gate fan-in size.\nTheorem 12. Suppose the quantum computation circuits act on qubits or bits. Then, the fan-in size K= 14 is\nsufficient for the above strategy.\nProof. Based on Table II and III, in the modified BCS nonlocal game for the relation probelm R8\nN, the number of\nvariables is 727 + 1 <210, and the number of constraints is 1042 + 1 <211, where we also need to consider the null\ninput ⊥. Thus, all possible inputs αi, βiat site i∈[N] can be encoded as 11 bits.\nNext, we prove that all the operations in Fig. 2 can be implemented by K-bounded fan-in gates with K= 14. Note\nthat all constant input size boolean function can be computed in NC0, so we only care about the classically controlled\nquantum gates [Fig. 3(b)] and quantum measurements [Fig. 3(c)]. We analyze the algorithm step by step:\n1. EPR preparation: No classical bit is involved. The quantum circuit involves simply one or two-qubit quantum\ngates, hence K≥0 + 2 = 2 in this step.\n2. BSM: First determine whether the input is ⊥through classical computing. Then perform BSM if it so and do\nnothing if not. This step requires K≥1 + 2 = 3.\n3. Nonlocal game: 11 classical bits are needed to control 3-qubit quantum gates. This step requires K≥11+3 = 14.\nTherefore, all the operations in Fig. 2 can be implemented using Kbounded fan-in gates with K= 14. This finishes\nthe proof.\nAs a side remark, note that a classically controlled quantum gate with a constant number of classical control bits\ncan be decomposed into compositions of single-bit classically controlled quantum gates within a constant depth [53].\nThus, actually K= 4 is sufficient, albeit a compromise of a few circuit layers.\nBelow, we prove the hardness of the problem for a circuit with only bounded fan-in classical gates and Clifford\ngates, including classically controlled Clifford gates and constant-weight Pauli-string measurements. The bounded\nfan-in classically controlled quantum gates require the numbers of classical input bits and the controlled qubits to\nbe both finite. This circuit allows intermediate measurements, and the measurement results can be used to control\nsubsequent quantum gates. The constant-weight Pauli measurement is measuring a Pauli observable on a constant\nnumber of qubits. As a constant-weight Pauli measurement is equivalent to implementing a constant-depth Clifford20\ngate followed by the computational basis measurement on the first qubit and implementing the inverse of the Clifford\ngate, we can take all measurements as computational basis measurements, or σzmeasurements.\nNow, consider a magic-free circuit with depth D, and the gates within it have fan-in bounded by K, which means\nthe total number of input classical bits and qubits of the gates is no larger than K. Denote the qubit or bit at site\naasiaand suppose E1is the gate of the first layer that contains iaas an input. Then, supp( E1(ia)) determines a\nset of qubits and bits after the first layer of the circuit that may be affected by ia. Similarly, we can consider the\nqubits and bits that may be affected in the next layer of the circuit. Denote the gates in each circuit layer are given\nby{E1,E2,···,ED}. In the end, we call the set of qubits and bits\nL→\nC(ia) = supp( ED(supp( ···supp(E2(supp( E1(ia))))))) (C10)\nthe forward light cone of ia.\nThe backward light cone of an output bit or qubit, ob, at site bcan be defined with the reverse of the forward light\ncone of the input and given by\nL←\nC(ob) :={ia|ob∈L→\nC(ia)}. (C11)\nThe backward light cone of an output set, O, is defined as\nL←\nC(O) :=[\no∈OL←\nC(o). (C12)\nNote that if a depth- Dquantum circuit, C, only comprises gates with fan-in bounded by K, then\n|L←\nC(o)| ≤KD, (C13)\nand\n|L←\nC(O)| ≤ |O|KD. (C14)\nNote that the input of the relation problem RNis classical and given by q= (α1,···, αN, β1,···, βN). Before\nacting gates, there are also an arbitrary number of classical ancillas with value 0 and quantum ancillas at state |0⟩,\nwhich do not contain any input information. With the circuit evolution, the input information will spread among\nclassical bits and also qubits via classically controlled quantum gates. Nonetheless, due to the gates being bounded\nfan-in and the circuit is at constant depth, the input information cannot spread a lot. The output of RNcan be\nread out from the classical bits after constant-depth circuit evolution. Without loss of generality, we can assume the\noutput of RNis read out from the first 6 nbits at the last step of the circuit, which is r= (rA\n1,···,rA\nN,rB\n1,···,rB\nN) as\nmentioned above. Using the idea of information not spreading a lot, we will show that with high probability, an input\nvalue αj(βk) is independent of the output rB\nk(rA\nj), as presented in Lemma 4. Before proving it, we first present the\nfollowing lemma.\nLemma 3. LetCbe a circuit with classical inputs and an arbitrary number of quantum and classical ancillas, which\ncomprises gates, including classical gates, classical controlled quantum gates, and measurements, with fan-in upper\nbounded by K, depth D, and Moutput bits. Then, the following holds:\nLetObe a fixed subset of output bits, and suppose Iis a randomly chosen subset of input bits such that\nPr[v∈I]≤q (C15)\nfor every input bit v. Then\nPr[O∩L→\nC(I)̸=∅]≤q|O|2|O|KD. (C16)\nProof.\nPr[O∩L→\nC(I)̸=∅] =X\nP⊆O,P̸=∅Pr[O∩L→\nC(I) =P]\n≤X\nP⊆O,P̸=∅Pr[I∩L←\nC(P)̸=∅]\n≤X\nP⊆O,P̸=∅X\nv∈L←\nC(P)Pr[v∈I]\n≤2|O||O|KDq.(C17)21\nLemma 4. Consider a depth- Dquantum-classical circuit composed of gates, including classical gates, classical con-\ntrolled quantum gates, and measurements, of fan-in at most K. The input of the circuit q= (α1,···, αN, β1,···, βN)\nis determined by a tuple (j, k, α, β )with 1≤j < k ≤Nand given by Eq. (C5). We denote the set of all possible\ninputs as S. The output of the circuit is given by r= (rA\n1,···,rA\nN,rB\n1,···,rB\nN). Define the event EC⊂Sin which\nthe input parameters satisfy\nrA\nj∩L→\nC(βk) =∅andrB\nk∩L→\nC(αj) =∅. (C18)\nUnder a uniform choice of input from S, the event ECoccurs with probability Pr[EC]≥1−48KD\nN.\nProof. We consider a random input from the set S, which is constructed by a randomly generated tuple ( j, k, α, β ).\nNote that jandkare two different numbers uniformly and randomly picked from {1,2,···, N}while αandβare\nrandomly and uniformly picked from QAandQB, respectively. By considering the input bit set as I=QAon the\nsitejin Lemma 3, we have\nPr\u0002\nv∈ QA\u0003\n= Pr\n1≤i≤N[i=j] =1\nN. (C19)\nBased on Lemma 3, we obtain\nPr\u0002\nrB\nk∩L→\nC(αj)̸=∅\u0003\n≤2|rB\nk||rB\nk|KD1\nN. (C20)\nNote that rB\nkonly has 3 output bits, then\nPr\u0002\nrB\nk∩L→\nC(αj)̸=∅\u0003\n≤24KD\nN. (C21)\nSimilarly, we get\nPr\u0002\nrA\nj∩L→\nC(βk)̸=∅\u0003\n≤24KD\nN. (C22)\nThus,\nPr[EC] = Pr\u0002\nrA\nj∩L→\nC(βk) =∅ ∩rB\nk∩L→\nC(αj) =∅\u0003\n= 1−Pr\u0002\nrA\nj∩L→\nC(βk)̸=∅ ∪rB\nk∩L→\nC(αj)̸=∅\u0003\n≥1−Pr\u0002\nrA\nj∩L→\nC(βk)̸=∅\u0003\n−Pr\u0002\nrB\nk∩L→\nC(αj)̸=∅\u0003\n≥1−48KD\nN.(C23)\nWith Lemma 4, we are able to achieve our ultimate goal to prove the hardness of RNfor magic-free shallow circuits.\nThe main idea is that with high probability, the event defined in Lemma 4 would happen, and if this event happens,\nthe magic-free circuit cannot give a correct output as the nonlocal game requires magic to win with certainty.\nTheorem 13. LetCbe a depth- Dcircuit with classical input values and classical and quantum ancillas, which only\ncomprises Clifford gates with fan-in upper bounded by K, including classically controlled gates and Pauli measurements.\nNow, consider the classical input q= (α1,···, αN, β1,···, βN)determined by Eq. (C5) with αjandβkselected\nuniformly at random from QAandQB. Then for any constant value p∈(0,1), the average probability that Coutputs\nr= (rA\n1,···,rA\nN,rB\n1,···,rB\nN)such that (1) randqsatisfy the requirements in cases 1 and 2, and (2) outputs case 1\nwith probability no smaller than p, is at most48KD\nN+pCli, with pCligiven in Lemma 2. To meet the requirements\nwith a success probability larger than (1 +pCli)/2, the circuit depth requirement is Θ(log N).22\nProof. The average success probability of Cto output a correct relation between randqis\nPr[success] = Pr[case 1] Pr[success |case 1] + Pr[case 2] Pr[success |case 2]\n= Pr[case 1](Pr[ EC|case 1] Pr[success |case 1 , EC] + Pr\u0002\nEC\nC|case 1\u0003\nPr\u0002\nsuccess |case 1 , EC\nC\u0003\n)\n+ Pr[case 2] Pr[success |case 2]\n≤Pr[case 1](Pr[ EC|case 1] Pr[success |case 1 , EC] + 1−Pr[EC|case 1]) + Pr[case 2]\n=1−Pr[case 1] Pr[ EC|case 1](1 −Pr[success |case 1 , EC])\n=1−Pr[EC](1−Pr[success |case 1 , EC])\n=1−Pr[EC] + Pr[ EC] Pr[success |case 1 , EC]\n≤48KD\nN+ Pr[success |case 1 , EC].(C24)\nHere, ECis the event defined in Lemma 4 and EC\nCis the complementary set of EC. From the above inequality, notice\nthat the upper bound on the average success probability is irrelevant with the probability for case 1 to occur, i.e., the\nvalue of p. Now, we only need to investigate the value of Pr[success |case 1 , EC].\nWhen case 1 happens, the success condition is making the inputs αjandβkand the outputs rA\njandrB\nksatisfy the\nrelation defined by the BCS game, i.e., Eq. (C7). Also, when EChappens, the output rA\njonly depends on αjand\ndoes not depend on βk. And the reverse is true for rB\nk. It reduces to the case that Alice and Bob are trying to win the\nBCS non-local game without classical communication. As the circuit only comprises zero input states, Clifford gates,\nand Pauli measurements, it means that Alice and Bob need to win this non-local game with Pauli measurements and\nmagic-free states, whose winning probability is upper bounded by pCli. That is, Pr[success |case 1 , EC]≤pCli. Then,\nwe conclude that the average success probability of Cto output a correct relation between randqis\nPr[success] ≤48KD\nN+pCli. (C25)\nTo output the correct relation with a success probability larger than (1 + pCli)/2, the circuit depth Dhas a lower\nbound as below.\n48KD\nN+pCli≥1 +pCli\n2⇔D≥logN+ log1−pCli\n96\nlogK= Ω(log N). (C26)\nOn the other hand, as stated in the main text, there is a classical circuit with circuit depth O(logN) that solves the\nproblem. Therefore, the bound on the circuit depth for magic-free circuits to solve the problem is tight. This finishes\nthe proof.\nAppendix D: Finding Potential Magic-Necessary Linear Binary Constraint Systems\nIn this section, we discuss how to find other instances of linear BCS that necessarily requires magic for a perfect\nquantum solution. It is difficult to develop a general procedure for this target. Instead, we provide a “guess-and-test”\nprocedure: (1) First, obtain a potential BCS, and (2) Second, verify whether the BCS has a solution over the Pauli\ngroup. For the first step, one can use the group embedding results in Ref. [35]. Building on the group-theoretic\nresults including Ref. [32, 47], Ref. [35] provides an efficient procedure to embed a group into a BCS, which is a group\nhomomorphism of the original group to a non-trivial BCS solution group; hence the procedure constructs a BCS that\nnecessarily has a (quantum) solution. In particular, the output solution group inherits representation properties of\nthe original group. However, such a BCS may have a classical solution or a quantum solution over the Pauli group,\nwhere magic is absent. For this issue, we develop an efficient classical algorithm to decide whether a linear BCS has\na Pauli-string solution. Besides aiding the search for non-trivial linear BCS, we hope this result can help explore the\ndecidability problems for general BCS and nonlocal games [45].\n1. Slofstra’s group embedding procedure\nA group Gis said to be embedded into group Kif there exists an injective group homomorphism ϕ:G→K. One can\npose additional requirements to the group embedding to guarantee the inheritance of group representation properties,\nsee Definition 10 and 14 in Ref. [35] for example. This is also one of the core issues in the group embedding procedure23\nin Ref. [35]. For our purpose of finding magic-necessary BCS, we do not need to consider this issue. Nevertheless,\nwe faithfully review the group embedding results of Ref. [35] here and leave the problem of simplifying the procedure\nfor the current problem to future work. For convenience of stating the group embedding results, we use the Boolean\nvariables and parity constraints instead of the sign variables and multilinear constraints in this subsection.\nAs we now use the Boolean variable notation, a BCS can be compactly written as M⃗ v=⃗ c, where Mis an m×n\nBoolean matrix, ⃗ v= (v1,···, vn)Tis the vector of variables, and ⃗ c= (c1,···, cm)Tis the vector of constraints. The\nnon-zero elements in the j’th row of Mdefines the set of variables presented in the j’th constraint, Sj. We first define\nseveral classes of groups, including a restatement of the solution group with the current notations.\nDefinition 3 (Solution group of a linear BCS using Boolean variables) .Given a linear BCS with nbinary variables\n{vi}andmconstraints {cj}specified by M⃗ v=⃗ c, the solution group of the BCS is defined as the group with the\nfollowing presentation:\nΓ(M,⃗ c) ={{J, gi:i∈[n]}:{g2\ni=e,∀i∈[n],\nJ2=e,\n∀j∈[m], gkgl=glgk,∀k, l∈ Sj,\nJgi=giJ,∀i∈[n],\nY\ni∈Sjgi=Jcj,∀j∈[m]}},(D1)\nwhere the group element Jcorresponds to −1in the BCS, and edefines the identity operator of the group.\nDefinition 4 (Linear-plus-conjugacy group) .Given a linear BCS with nbinary variables {vi}andmconstraints\n{cj}specified by M⃗ v=⃗ c, andC ⊆[n]×[n]×[n]with [n] ={1,···, n}, the linear-plus-conjugacy group Γ(M,⃗ c,C)is\ndefined as\nΓ(M,⃗ c,C)≡ ⟨Γ(M,⃗ c) :gigjgi=gk,∀(i, j, k )∈ C⟩, (D2)\nwhere the relators gigjgi=gkare additionally posed to the solution group Γ(M,⃗ c).\nDefinition 5 (Homogeneous linear-plus-conjugacy group) .Given an m×nBoolean matrix Mwhere the set of non-\nzero elements in the j’th row given by Sj, andC ⊆[n]×[n]×[n], the homogeneous linear-plus-conjugacy group Γ(M,C)\nis defined as\nΓ0(M,C) ={{J, gi:i∈[n]}:{g2\ni=e,∀i∈[n],\nJ2=e,\n∀j∈[m], gkgl=glgk,∀k, l∈ Sj,\nJgi=giJ,∀i∈[n],\nY\ni∈Sjgi=e,∀j∈[m],\ngigjgi=gk,∀(i, j, k )∈ C}} .(D3)\nDefinition 6 (Extended homogeneous linear-plus-conjugacy group) .Given an m×nBoolean matrix M,C0⊆\n[n]×[n]×[n],C1⊆[l]×[n]×[n], and Lanl×llower-triangular matrix with non-negative integer entries, the extended\nhomogeneous linear-plus-conjugacy group EΓ(M,C0,C1, L)is defined as\nEΓ0(M,C0,C1, L)≡ ⟨Γ0(M,C0), h1,···, hl:higjh−1\ni=gk,∀(i, j, k )∈ C1,\nhihjh−1\ni=hLij\nj,∀i > j∩Lij>0⟩,(D4)\nwhere Lijrefers to the element on the i’th row and j’column of matrix L.\nWith the above definition, Ref. [35] proves the following embedding results:\nTheorem 14 ([35]).Suppose a group Ghas a presentation in the form of an extended homogeneous linear-plus-\nconjugacy group, given by EΓ(M,C0,C1, L). Then there are the following group embedding results:\n1. There exists a group embedding of EΓ0(M,C0,C1, L)into a homogeneous linear-plus-conjugacy group (Proposi-\ntion 33 in Ref. [35]):\nEΓ0(M,C0,C1, L)→Γ0(M′,C), (D5)\nwhere Γ0(M′,C)is a homogeneous linear-plus-conjugacy group.24\n2. The extended Γ0(M′,C)can be transformed into a linear-plus-conjugacy group:\nΓ0(M′,C)×Z2= Γ(M′,0,C). (D6)\n3. By adding relations involving a group element J̸=einto M′, which extends matrix M′into Nand adds\nnon-homogeneous linear constraints of Jto some group elements, extend the linear-plus-conjugacy group:\nΓ(M′,0,C)→Γ(N,⃗ c,C′), (D7)\nwhere ⃗ c̸= 0, with some entries equal to J.\n4. There exists a group embedding of Γ(N,⃗ c,C′)into a linear group (Proposition 27 in Ref. [35]):\nΓ(N,⃗ c,C)→Γ(N′,⃗ c′), (D8)\nwhich defines a BCS solution group that has a non-trivial group element J̸=e.\nThe proof is constructive, thus one can derive the concrete groups in each step. In brief, as long as a group can\nbe presented in the form of Def. 6, an extended homogeneous linear-plus-conjugacy group, it can be converted into a\nBCS solution group with a series of group embeddings and a proper construction of non-trivial relators with respect\nto a group element J̸=e, which is set to correspond to −1 in the BCS. As promised by Lemma 1, the underlying\nBCS of the solution group has an (operator-valued) solution.\n2. Efficient algorithm for finding perfect Pauli-string solutions to linear BCS\nNext, we provide an efficient algorithm to determine the existence of a Pauli-string solution to a general linear BCS.\nCombined with Slofstra’s group embedding procedure, one can guess and test BCS instances to search for a potential\nBCS that has a non-trivial solution group other than the Pauli group.\nWe first present some fundamental properties of Pauli-string observables.\nLemma 5. Suppose A1, A2,···, Anare Pauli-string observables. For i, j∈[n], define Cij=AiAjAiAjas the\ncommutator between AiandAj. Then, Cij’s have the following properties:\n1.Cij∈ {±I}. Specifically, Cij=Iwhen AiAj−AjAi= 0, and Cij=−Iwhen AiAj+AjAi= 0;\n2.Cij=CjiandCii=I;\n3.AiAj=CijAjAi.\nThe proof of Lemma 5 is straightforward, and we leave it to the readers as an exercise. According to the first\nproperty, Cijis always proportional to Iand thus commute with all the Ai’s. We can treat Cij’s as numbers ±1\nfor simplicity. The second property shows that for a group of n Ai’s, the number of independent commutators Cij’s\namong them is at most n(n−1)/2. The third property shall be vital for our later discussions, as it allows us to swap\ntwo adjacent variables AiandAjin a product of Pauli strings up to an additional coefficient Cij. Together with the\nfirst property, for a product of Pauli-string variables, we can arbitrarily rearrange their order up to a change in the\nsign.\nGiven a linear BCS, we first determine if it has a classical solution, i.e., Ai∈ {± 1}for all i∈[n]. This is equivalent\nto solving a system of linear equations over Z2, which can be done in poly( n) steps through, for example, the Gaussian\nelimination method. Going back to determine if the BCS has a Pauli-string solution, if we hope to apply a similar\nprocedure, the only obstacle is that the variables might not commute with each other. Nevertheless, thanks to the\nnice properties of Pauli strings in Lemma 5, we can do the same thing as finding a classical solution with at most a\ndifference in sign, which we record as a sign variable Ci. Should the BCS have a Pauli-string solution, at the end of\nthe elimination, we can express each variable Aias a product of some variables, which we term the “free variables,”\nmultiplied by a plus or minus sign Ci∈ {± 1}. We use the terminology “free variables” as they are allowed to take\nany value, while the remaining variables depend on their values. Now we give the rigorous statement and prove it.\nLemma 6. For a linear BCS with nvariables A1,···, Anandmconstraints, if it has a Pauli-string solution, then\nthere exists a set of free variables {Aik}k, such that each variable in the BCS can be represented in the form of\nAi=CiAi1Ai2···Aik···, where Ci∈ {± 1}andAik’s are arranged with the subscript kfrom small to large. This\nresult can be obtained in poly( n, m)steps.25\nProof. When finding an operator-valued solution to a linear BCS, we require the variables in the same constraint to be\ncompatible, as discussed in Appendix A 2. Now we prove a stronger statement that does not rely on this requirement.\nWe prove the lemma by mathematical induction on m. The statement holds when m= 1, where A1A2···An=±1.\nClearly, A1=±A2···An, and we can take A2···Anas free variables. We use the first property in Lemma 5 when\nthere is a need to change the order of two variables, which finishes in poly( n) steps using a sorting algorithm. Now\nassume the statement holds for m=k. When m=k+ 1, without loss of generality, suppose the first constraint is the\nadded constraint with A1A2···Al=±1. Thus, A1=±A2···Al. Substitute this expression into the other equations\nand simplify them using Lemma 5, which finishes in poly( n, m) steps using Gaussian elimination. Then, we get a\nset of kconstraints. According to the induction hypothesis, every variable Aican be represented as a product of\nfree variables up to a sign. So we can plug the free variables into the added constraint A1=±A2···Aland get the\nexpression for A1in terms of free variables. Therefore, the statement holds for m=k+ 1.\nNow, we get the expression for each variable in terms of a set of free variables, resulting in a linear BCS over Ai’s\nand additional sign variables Ci’s.\nWe further consider the conditions in the original linear BCS and eliminate all the non-free variable Ai’s. Using\nLemma 6, we find a set of free variables and use them to express all the other variables. Then, we substitute the\nexpressions into the original BCS and obtain a new BCS containing the free variables and Ci’s. By the definition of\nthe free variable, every free variable occurs for an even number of times in each constraint of the new BCS by this\nstep, otherwise, it is determined by the other variables through their commutators. By further applying Lemma 5, we\ncan get rid of all the Aivariables and obtain a set of equations of Ci’s and Ckl’s where k, lare the commutators of the\nfree variables AkandAl. In addition, if AiandAjappear in the same constraint of the original BCS, they commute\nwith each other, i.e., AiAjAiAj= 1. For each pair of AiandAj, by applying Lemma 6, replace them in the equation\nAiAjAiAj= 1 via their expressions in terms of the free variables and obtain the other equations of Ckl’s where k, l\nare indices of free variables. In the end, we convert the original linear BCS to an equivalent set of equations of Ci’s\n(i∈[n]) and Ckl’s (k, lare indices of free variables), which is just a system of linear equations over Z2. Note that\nall the procedures are simply substitutions and the order swaps between variables, which finish in poly( n, m) steps.\nShould the original BCS have a Pauli-string solution, we can efficiently solve the newly derived linear equations and\nget a set of valid values for Ci’s and Ckl’s.\nNote that by this step, we have not finished solving the original BCS over the Pauli group, as we have not determined\nthe operator values of Ai’s. The following lemma gives a systematic method to assign legitimate Pauli-string values\nfor all the free variables, and hence all the variables in the original BCS.\nLemma 7. Given the commutator relation among nvariables, i.e., Cij=AiAjAiAj=±1, the following procedure\ngives a Pauli-string assignment to the variables that satisfy the commutator relation:\n1. Construct a matrix S∈Zn×n\n2, where\nSij=(\n0,ifAiAjAiAj=I,\n1,ifAiAjAiAj=−I;(D9)\n2. Expand the matrix Sas\nS=X\ni<jSijEij, Sij∈ {0,1}, (D10)\nwhere Eij=eij+eji, with eijdenoting the elementary matrix in Zn×n\n2;\n3. For Sij̸= 0, letEijcorrespond to a set of noperators in the Pauli group,\nEij⇔ {Mij\nk}k∈[n], Mij\nk=\n\nσx,ifk=i,\nσz,ifk=j,\nI,otherwise ,(D11)\nwith0corresponding to a set of nidentity operators for completeness;\n4. Link Swith a set of nPauli strings via the following equation,\nS⇔\n\nAk=O\ni<j:Sij=1Mij\nk\f\f\f\fk∈[n]\n\n, (D12)\nwhere the tensor product operation is performed in ascending order with respect to the tuple (i, j).26\nWhile cumbersome, the assignment can be directly checked. Later we take the Mermin-Peres’ magic square BCS as\nan example to exhibit the entire procedure. As a side note, the correspondence between traceless symmetric matrices\noverZn×n\n2and Pauli strings was implicitly used in Lemma 7 in Ref. [42].\nNow we summarize the results for determining the Pauli-string solution to a linear BCS.\nTheorem 15. For a linear BCS with nvariables and mconstraints, there exists a classical algorithm that determines\nwhether it has a perfect quantum strategy on the Pauli group in poly( n, m)steps.\nThe procedures can be summarized as follows:\n1. Solve the BCS as if it is a classical one, with a recording of the commutator and sign changes in each step. Get\nan expression for each variable in terms of a set of free variables.\n2. Substitute the expressions into the original BCS. Get a system of linear functions of Ci’s and Ckl’s and solve\nthem over Z2.\n3. Assign a Pauli string to every free variable. Then derive the operator values of all the variables according to\nthe expressions in step 1.\nOn the other hand, if there is not a Pauli-string solution to the linear BCS, we shall come to a contradiction\nsomewhere in the procedures.\nCorollary 2. Suppose a linear BCS does not have a satisfying assignment with Pauli-string observables. On the one\nhand, we can use the substitution method of solving a BCS and obtain a relation for a subset of variables,\nAt1At2···Atk=−I. (D13)\nOn the other hand, by posing the commutation properties of Pauli strings to the BCS variables, the left-hand side can\nbe eliminated to I, resulting in a contradiction.\nThe proof of this corollary is similar to Lemma 6. In brief, in the algorithm for finding Pauli-string solutions,\nwe only use two operations throughout the process: (1) substitution of expressions, and (2) swapping two variables\nin an expression according to Lemma 5. If the algorithm cannot find a Pauli-string solution, it must result in a\ncontradiction. As we keep the right-hand side of each formula to be ±I, the contradiction is thus the form of the\nstatement in the corollary.\nAs an example, we apply the algorithm to find a Pauli-string solution to the Mermin-Peres magic square BCS. The\noriginal BCS is given by\nA1A2A3= 1,\nA4A5A6= 1,\nA7A8A9= 1,\nA1A4A7= 1,\nA2A5A8= 1,\nA3A6A9=−1.(D14)\nAfter the first step, we find that the BCS variables can be determined by a set of free variables {A5, A6, A8, A9}:\nA1=C1A5A6A8A9,\nA2=C2A5A8,\nA3=C3A6A9,\nA4=C4A5A6,\nA7=C7A8A9.(D15)\nSubstituting these expressions into the original BCS, we obtain the set of equations\nA1A2A3=C1C2C3C59C56C58C69C89= 1,\nA4A5A6=C4C56= 1,\nA7A8A9=C7C89= 1,\nA1A4A7=C1C4C7C59C68C56C58C69C89= 1,\nA2A5A8=C2C58= 1,\nA3A6A9=C3C69=−1.(D16)27\nUsing the commutation conditions between variables in the same constraint as the original BCS, we have the equations\nA1A2A1A2=C69= 1,\n···\nA4A7A4A7=C59C68C56C58C69C89= 1,\n···\nA8A9A8A9=C89= 1.(D17)\nSolving Eq. (D16) and (D17) over Z2, we have\nC1=C2=C4=C7= 1,\nC3=−1,\nC56=C58=C69=C89= 1,\nC59=C68=−1.(D18)\nWith respect to the commutators among the free variables, construct the matrix S:\nS=\nS55S56S58S59\nS65S66S68S69\nS85S86S88S89\nS95S96S98S99\n=\n0 0 0 1\n0 0 1 0\n0 1 0 0\n1 0 0 0\n=E59+E68. (D19)\nWe map the basis matrices E59andE68as\nE59⇔ {M59\n5, M59\n6, M59\n8, M59\n9}\n≡ {σx,I,I, σz},\nE68⇔ {M68\n5, M68\n6, M68\n8, M68\n9}\n≡ {I, σx, σz,I},(D20)\nand hence assign the free variables as\nA5=M59\n5⊗M68\n5=σx⊗I,\nA6=M59\n6⊗M68\n6=I⊗σx,\nA8=M59\n8⊗M68\n8=I⊗σz,\nA9=M59\n9⊗M68\n9=σz⊗I,(D21)\nand the other variables are then determined as\nA1=−σy⊗σy,\nA2=σx⊗σz,\nA3=−σz⊗σx,\nA4=σx⊗σx,\nA7=σz⊗σz.(D22)\nThis solution is equivalent to the solution in Eq. (A8) in the sense of a unitary transformation, or a relabelling of the\nvariables.\n[1] R. P. Feynman, Int. J. Theor. Phys. 21(1982).\n[2] P. W. Shor, SIAM J. 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Chuang, Quantum computation and quantum information (Cambridge university press, 2010)."    },    {        "title": "2402.12277v1.Correlations_equalities_and_some_upper_bounds_for_the_coupling_constant_implying_area_decay_of_Wilson_loop_for__Z_3__lattice_gauge_theories.pdf",        "content": "arXiv:2402.12277v1  [hep-lat]  19 Feb 2024Correlations equalities and some upper bounds for the coupl ing constant implying\narea decay of Wilson loop for Z3lattice gauge theories\nA. L. Mota∗\nDepartamento de Ciˆ encias Naturais, Universidade Federal de S˜ ao Jo˜ ao del Rei,\nC.P. 110, CEP 36301-160, S˜ ao Jo˜ ao del Rei, Brazil\nF. C. S´ a Barreto†\nDepartamento de F´ ısica, Universidade Federal de Minas Ger ais,\nC.P. 110, CEP 31270-901, Belo Horizonte, MG, Brazil\nCorrelation identities are obtained for Z3lattice gauge theory where the bonds of the plaquettes\nare decorated by generalized three-state Ising variables. Making use of correlation inequalities we\nobtain the area decay of the Wilson loop observable in a range of the coupling parameter larger\nthan those obtained from mean field theory considerations.\nI. INTRODUCTION\nCorrelation inequalities combined with exact identities are useful in ob taining rigorous results in statistical mechan-\nics. Among the various questions that are resolved by them one is th e decay of the correlation functions. The decay\nof the correlation functions give information about the critical cou plings of the statistical mechanics model. In this\npublication, the method will be applied to study the Z3lattice gauge problem where the bond variables are decorated\nby generalized three-state Ising variables en2πi/3, n=0,1,2. The Z3discrete gauge symmetry is relevant to many fields\nof physics, such as, for instance, in the study of confinement/de confinement transition in particle physics [1–6], in\ndark matter models, where the Z3symmetry ensures stability of dark matter particles [7–10], and, of course, in the\nstudy of lattice gauge theories and their transitions [11–16].\nIn this work, we firstly obtain correlation identities for the Z3gauge lattice, which are the gauge version of the\nspin identities obtained previously for the spin one Blume-Capel mode l ([17–20]). We then look for the area decay of\nWilson’s loop, which is a criterion for confinement. We apply the first an d second Griffiths inequalities [21, 22, 24–27]\nand Newman’s inequalities [25, 28] to the correlation identities to obta in the coupling parameter which gives the area\ndecay of the Wilson loop in d=3 and d=4 dimensions. In both dimensions t he coupling parameters obtained for the\narea decay of the Wilson loop improve over the mean field results.\nWe consider the well known Wilson loop observable of a Z3pure gauge Zdlattice theory with Wilson action defined\nas\n/an}b∇acketle{tW(C)/an}b∇acket∇i}ht= lim\nΛ→Zd/an}b∇acketle{tW(C)/an}b∇acket∇i}htΛ (1)\nwhere/an}b∇acketle{tW(C)/an}b∇acket∇i}htis the finite lattice Gibbs ensemble average. The Wilson action Boltzman n factor [29–31],is given by,\neβJ/summationtext\nP⊂ΛχP(2)\nwhere P denotes the unit squares (plaquettes) of Λ and χP=S1S2S3S4. We let Sbdenote the bond variables which\ntake values e0= 1,e2πi/3,e4πi/3. W(C) is the product of Sbalong the perimeter of the planar rectangle C of area\nA. The gauge coupling constant is βJ, 0< βJ < ∞. Area decay of /an}b∇acketle{tW(C)/an}b∇acket∇i}htis a criterion for confinement. For the\nZ2lattice gauge theory it is known that area decay occurs for small βJand for sufficiently large βJperimeter decay\noccurs for d≥3 [30, 31]. For d= 2,/an}b∇acketle{tW(C)/an}b∇acket∇i}hthas area decay for all βJby explicit calculation [31]. We take free\nboundary conditions and note that Griffiths first and second inequa lities apply ([32, 37–39]), and especially for the\nPotts model see ([26, 27]), and therefore imply the existence of th e thermodynamic limit [34]. For d = 3 and 4 we\nobtain lower bounds βLon the area decay of /an}b∇acketle{tW(C)/an}b∇acket∇i}hti,.e. for all β < βL,/an}b∇acketle{tW(C)/an}b∇acket∇i}hthas area decay, using correlation\nidentities and Griffiths inequalities. The correlation identities are a gau ge version of the identities presented before\n[18–20] and employed to obtain lower than mean field upper bounds on the critical temperature for the Blume - Capel\nspin 1 systems. The procedure is similar to that employed previously [ 33] to obtain the Z2lattice gauge correlation\nidentities as a version of Callen’s identities for the Ising spin 1/2 syste ms. For completeness, in section II, we give the\nmean field lower bound βM(βM< βL) using a decoupling and Griffiths inequality argument [35, 36].\n∗motaal@ufsj.edu.br\n†fcsabarreto@gmail.com2\nII. MEAN FIELD LOWER BOUNDS\nFor definiteness assume that C lies in the ( x1,x2) plane. Consider a bond b fixed in the lower left-hand corner of C.\nReplaceβbyβλ,λǫ[0,1], in the action for the 2( d−1) plaquettes (call them P1,P2,...,P2(d−1)) that have one bond\nin common with b. Denote the corresponding expectation by /an}b∇acketle{tW(C)/an}b∇acket∇i}htλ, and let J= 1, without loss of generality.\nIntegrating ( d/dλ)/an}b∇acketle{tW(C)/an}b∇acket∇i}htλgives, noting that /an}b∇acketle{tW(C)/an}b∇acket∇i}ht0= 0,\n/an}b∇acketle{tW(C)/an}b∇acket∇i}ht=/integraldisplay1\n0dλd\ndλ/an}b∇acketle{tW(C)/an}b∇acket∇i}htλ=β/integraldisplay1\n0dλ2(d−1)/summationdisplay\ni=1(/an}b∇acketle{tW(C)χPi/an}b∇acket∇i}htλ−/an}b∇acketle{tW(C)/an}b∇acket∇i}htλ/an}b∇acketle{tχPi/an}b∇acket∇i}htλ)\n≤β/integraldisplay1\n0dλ2(d−1)/summationdisplay\ni=1/an}b∇acketle{tW(C)χPi/an}b∇acket∇i}htλ≤β2(d−1)/summationdisplay\ni=1/an}b∇acketle{tW(C)χPi/an}b∇acket∇i}ht, (3)\nIn the previous we used Griffith’s first (second) inequality in the first (second) equality. Each term on the right\ncorresponds to a modified contour determined by the bonds of the variables of W(C)χPiwhich enlarges or diminishes\nC by one plaquette. We repeat the argument proceeding along succ essive rows of plaquettes enclosed by C. After A\napplications we arrive at\n/an}b∇acketle{tW(C)/an}b∇acket∇i}ht ≤βA(sum of [2( d−1)]Aterms) (4)\nEach term is non-negative and bounded above by 1 giving /an}b∇acketle{tW(C)/an}b∇acket∇i}ht ≤[β2(d−1)]A. Therefore, for each β,\nβ⊂(0,(2(d−1))−1),\n/an}b∇acketle{tW(C)/an}b∇acket∇i}ht ≤e−|ln(2β(d−1))|A. (5)\nIII. CORRELATION IDENTITIES FOR THE Z3LATTICE GAUGE MODEL\nWe define the action for the Z3lattice gauge model as\n−H=βJ/summationdisplay\nP⊂ΛχP (6)\nwhere,χPis the square plaquette and Siis the bond variable assuming values e0= 1,e2πi/3≡σ, ande4πi/3=σ2. Let\nSD=Si1...SiDdenote a product of distinct bond variables and for a fixed bond b oc curring in SDgive a numerical\nordering 1,2,... to the 2(d- 1) plaquettes that have one bond in commo n with b. We have,\n/angbracketleftBig\nSD/angbracketrightBig\n=1\nZ/summationdisplay\nSSDeβJ/summationtext\nP⊂ΛχP(7)\nwhere,\nZ=/summationdisplay\nSeβJ/summationtext\nP⊂ΛχP(8)\nLet us consider the bond b, with Sb, and the plaquette χbwhich contains the bond b. Let S(b)\nDbe the product SD\nwith the bond b deleted. We have,\n/angbracketleftBig\nSD/angbracketrightBig\n=1\nZ·/bracketleftBig/summationdisplay\nSS(b)\nD/parenleftBig/summationtext\nSbSbeβJχb\n/summationtext\nSbeβJχb/parenrightBig\n·eβJ/summationtext\nP⊂ΛχP/bracketrightBig\n(9)\nLet\nχb=Sb/summationdisplay\nneig.(b)SkSlSm (10)\nSumming over Sb, in (9) and introducing ∇=∂\n∂xthrough ea∇f(x) =f(x+a), we get\n/an}b∇acketle{tSD/an}b∇acket∇i}ht=/angbracketleftBigg\nS(b)\nD\neαb+2e−1\n2αbcos/parenleftBig√\n3\n2αb+2π\n3/parenrightBig\neαb+2e−1\n2αbcos/parenleftBig√\n3\n2αb/parenrightBig\n/angbracketrightBigg\n=/angbracketleftBigg\nS(b)\nD/productdisplay\nneig(b)e(βJSkSlSm)∇/angbracketrightBigg\n·f(x)|x=0 (11)3\nwhere,\nαb=βJ/summationdisplay\nneig.(b)SkSlSm (12)\nand\nf(x) =ex+σeσx+σ2eσ2x\nex+eσx+eσ2x=ex+2e−1\n2xcos/parenleftBig√\n3\n2x+2π\n3/parenrightBig\nex+2e−1\n2xcos/parenleftBig√\n3\n2x/parenrightBig. (13)\nExpanding the exponential in (11) and using S3n\ni=S3\ni= 1,S3n+1\ni=Si, andS3n+2\ni=S2\ni, forn= 1,2,..., we\nobtain,\n/angbracketleftBig\nSD/angbracketrightBig\n=/angbracketleftBig\nS(b)\nD/productdisplay\nneig(b)/bracketleftBig\nc0(βJ∇)+SkSlSms1(βJ∇)+S2\nkS2\nlS2\nms2(βJ∇)/bracketrightBig/angbracketrightBig\n·f(x)|x=0, (14)\nwith\nc0(x) =ex+eσx+eσ2x\n3;s1(x) =ex+σ2eσx+σeσ2x\n3; ands2(x) =ex+σeσx+σ2eσ2x\n3. (15)\nThe functions c0(x),s1(x), ands2(x) were named by their similarity to the hyperbolic cosh( x) and sinh( x) that\nappear in the equivalent treatment of the Z2lattice gauge [33]. The following properties, derived from Eqs. (13) a nd\n(15) , are usefull in the evaluation of /an}b∇acketle{tSD/an}b∇acket∇i}htin the next section:\nf(σx) =σ2f(x), f(σ2x) =σf(x), (16)\n1+σ+σ2= 0, σ3= 1, (17)\nand\nca\n0(βJ∇)sb\n1(βJ∇)sc\n2(βJ∇)f(x)|x=0/braceleftBig/ne}ationslash= 0 if (b+2c+1) = 3n\n= 0 otherwise ,(18)\nfora,b,c,andnpositive integers.\nIn contrast with the Z2case [33], where it is given by the analytical function tanh(x),f(x) (Eq. (13)) is non-\nanalytical for ex+2e−1\n2xcos/parenleftBig√\n3\n2x/parenrightBig\n= 0, i.e., at x≈ −1.869812799. However, as we shall see, this point is above both\nd= 3 and d= 4 upper limits for βJ.\nIV. EXACT CORRELATION IDENTITIES APPLIED TO THE d= 3, AND d= 4LATTICES (SQUARE\nPLAQUETTES)\nApplying Eq. (14), for d= 3 and d= 4, after some algebra, we obtain, in subsections (IVA) and (IVB) , with\nk=βJ,ρ2= 2+σ,ρ3= 3+σ,ρ5= 5+σ, andℜ{z}stands for the real part of the complex argument z:\nA. d=3\n/an}b∇acketle{tSD/an}b∇acket∇i}ht=A1/summationdisplay\ni/an}b∇acketle{tSDχ2\ni/an}b∇acket∇i}ht+A2/summationdisplay\ni<j/an}b∇acketle{tSDχiχj/an}b∇acket∇i}ht+A3/summationdisplay\ni<j<k/an}b∇acketle{tSDχiχ2\njχ2\nk/an}b∇acket∇i}ht\n+A4/summationdisplay\ni<j<k<l/an}b∇acketle{tSDχiχjχkχ2\nl/an}b∇acket∇i}ht+A5/summationdisplay\ni<j<k<l/an}b∇acketle{tSDχ2\niχ2\njχ2\nkχ2\nl/an}b∇acket∇i}ht, (19)\nwith4\nA1=4\n9/parenleftbigg1\n3f(4k)+f(k)−f(−2k)+2ℜ{ρ3f(ρ3k)}/parenrightbigg\n≥0, (20)\nA2=2\n3/parenleftbigg1\n3f(4k)+f(−2k)−4\n3ℜ{σf(ρ3k)}/parenrightbigg\n≥0, (21)\nA3=4\n3/parenleftbigg1\n3f(4k)−f(k)+2\n3ℜ{σf(ρ3k)}/parenrightbigg\n≤0, (22)\nA4=4\n9/parenleftbigg1\n3f(4k)+f(k)−f(−2k)+2\n3ℜ{σρ3f(ρ∗\n3k)}/parenrightbigg\n≤0, (23)\nA5=1\n9/parenleftbigg1\n3f(4k)+4f(k)+2f(−2k)+8\n3ℜ{σf(ρ3k)}/parenrightbigg\n≥0. (24)\nAll inequalities hold for 0 ≤k≤0.924906, where f(−2k) is analytical, except Eq. (20), that holds for 0 ≤k≤\n0.740736. Above this point we get A1<0. As we shall show, the upper bound for the coupling constant lies in the\ninterval of validity of Eqs. (20)-(24).\nB. d=4\n/an}b∇acketle{tSD/an}b∇acket∇i}ht=A1/summationdisplay\ni/an}b∇acketle{tSDχ2\ni/an}b∇acket∇i}ht+A2/summationdisplay\ni<j/an}b∇acketle{tSDχiχj/an}b∇acket∇i}ht+A3/summationdisplay\ni<j<k/an}b∇acketle{tSDχiχ2\njχ2\nk/an}b∇acket∇i}ht\n+A4/summationdisplay\ni<j<k<l/an}b∇acketle{tSDχiχjχkχ2\nl/an}b∇acket∇i}ht+A5/summationdisplay\ni<j<k<l<m/an}b∇acketle{tSDχiχjχkχlχm/an}b∇acket∇i}ht+A6/summationdisplay\ni<j<k<l/an}b∇acketle{tSDχ2\niχ2\njχ2\nkχ2\nl/an}b∇acket∇i}ht\n+A7/summationdisplay\ni<j<k<l<m/an}b∇acketle{tSDχiχjχ2\nkχ2\nlχ2\nm/an}b∇acket∇i}ht+A8/summationdisplay\ni<j<k<l<m<n/an}b∇acketle{tSDχiχjχkχlχ2\nmχ2\nn/an}b∇acket∇i}ht\n+A9/summationdisplay\ni<j<k<l<m<n/an}b∇acketle{tSDχiχ2\njχ2\nkχ2\nlχ2\nmχ2\nn/an}b∇acket∇i}ht, (25)\nwhere the A coefficients are given by\nA1=2\n81(f(6k)+15f(3k)−10f(−3k)+2ℜ{10(2+σ)f(ρ2k)+(5+σ)f(ρ5k)+5(2+ σ)f(2ρ2k)})≥0,(26)\nA2=5\n81(f(6k)+6f(3k)+8f(−3k)+2ℜ{−4(1+2σ)f(ρ2k)+2(1−σ)f(ρ5k)−(2+7σ)f(2ρ2k)})≥0,(27)\nA3=20\n81(f(6k)−3f(3k)−f(−3k)+2ℜ{−(7+2σ)f(ρ2k)+(2+σ)f(ρ5k)+(1+2 σ)f(2ρ2k)})≤0,(28)\nA4=20\n81(f(6k)−3f(3k)−f(−3k)+2ℜ{(5+7σ)f(ρ2k)−(1+2σ)f(ρ5k)−(2+σ)f(2ρ2k)})≤0,(29)\nA5=2\n81(f(6k)+15f(3k)−10f(−3k)+2ℜ{−10(1+2σ)f(ρ2k)−(4+5σ)f(ρ5k)−5(1−σ)f(2ρ2k)})≤0,(30)5\nA6=5\n81(f(6k)+6f(3k)+8f(−3k)+2ℜ{(8+4σ)f(ρ2k)+(2+4 σ)f(ρ5k)+(2σ−5)f(2ρ2k)})≥0,(31)\nA7=20\n81(f(6k)−3f(3k)−f(−3k)+2ℜ{(2−5σ)f(ρ2k)+(σ−1)f(ρ5k)+(1−σ)f(2ρ2k)})≥0,(32)\nA8=5\n81(f(6k)+6f(3k)+8f(−3k)+2ℜ{4(σ−1)f(ρ2k)−2(2+σ)f(ρ5k)+(7+5 σ)f(2ρ2k)})≥0,(33)\nA9=2\n81(f(6k)+15f(3k)−10f(−3k)+2ℜ{10(σ−1)f(ρ2k)+(4σ−1)f(ρ5k)−5(1+2σ)f(2ρ2k)})≤0.(34)\nAll inequalities, numerically checked, hold for 0 ≤k≤0.616604, where f(−3k) is analytical, except Eq. (26), that\nholds for 0 ≤k≤0.540037, and Eq. (32), that holds for 0 ≤k≤0.558283. Again, above these points we get A1<0\nandA7<0 respectively. As we shall show, the upper bound for the coupling c onstant lies in the domain of validity\nof Eqs. (26)-(34).\nV. APPLICATION OF THE CORRELATION INEQUALITIES\nIn order to derive the coupling bounds for the area decay we use th e same procedure as in the mean field derivation\n(section 2, equation (4)). Applying the same procedure to equatio ns (19) and (25), we will prove the following\nstatements.\nLetβbe such that,\n(a) 4(A1+A2+A5)<1. Then, for d= 3\n/an}b∇acketle{tW(C)/an}b∇acket∇i}ht ≤e−|ln(4(A1+A2+A5)|A. (35)\nWe numerically evaluate the condition 4( A1+A2+A5)<1, resulting in k <0.440761. It is worth to notice that\ninequalities (20)-(24) are valid within the interval 0 ≤k≤0.440761.\n(b) 6(A1+A2+A6+A7+A8)<1. Then, for d= 4\n/an}b∇acketle{tW(C)/an}b∇acket∇i}ht ≤e−|ln(6(A1+A2+A6+A7+A8)|A. (36)\nNumerically evaluating the condition 6( A1+A2+A6+A7+A8)<1 we obtain k <0.101220, the upper bound for\nthe area decay. All inequalities expressed by Eqs. (26)-(34) hold in the domain 0 ≤k≤0.101220.\nLet us prove result (a). For b ǫC,\n/an}b∇acketle{tW(C)/an}b∇acket∇i}ht=A1/summationdisplay\ni/an}b∇acketle{tW(C)χ2\ni/an}b∇acket∇i}ht+A2/summationdisplay\ni<j/an}b∇acketle{tW(C)χiχj/an}b∇acket∇i}ht+A3/summationdisplay\ni<j<k/an}b∇acketle{tW(C)χiχ2\njχ2\nk/an}b∇acket∇i}ht\n+A4/summationdisplay\ni<j<k<l/an}b∇acketle{tW(C)χiχjχkχ2\nl/an}b∇acket∇i}ht+A5/summationdisplay\ni<j<k<l/an}b∇acketle{tW(C)χ2\niχ2\njχ2\nkχ2\nl/an}b∇acket∇i}ht\n≤A1/summationdisplay\ni/an}b∇acketle{tW(C)χ2\ni/an}b∇acket∇i}ht+A2/summationdisplay\ni<j/an}b∇acketle{tW(C)χiχj/an}b∇acket∇i}ht+A5/summationdisplay\ni<j<k<l/an}b∇acketle{tW(C)χ2\niχ2\njχ2\nkχ2\nl/an}b∇acket∇i}ht (37)\nIn the last inequality we have considered the fact that A3andA4are negatives, A1,A2, andA5are positives,\nand/an}b∇acketle{tW(C)χPiχPjχPk/an}b∇acket∇i}htis positive by Griffith’s first inequality([26, 27]). For the other functio ns we make use of\nthe inequalities which are based on /an}b∇acketle{tS2\nASB/an}b∇acket∇i}ht ≤ /an}b∇acketle{tSB/an}b∇acket∇i}ht, whereS2\nA=/producttextS2\niandSB=/producttextSi([26, 27, 40, 41]). At each\napplication of the equality we pass to an inequality by dropping the ter ms; after A steps we arrive at\n/an}b∇acketle{tW(C)/an}b∇acket∇i}ht ≤(A1+A2+A5)A(sum of 4Aterms) (38)\nwhere each term is less than one.\nTo prove result (b) for d=4 we proceed as previously for the result (a) for d=3, using equation (25). We can drop\nthe terms in favor of an inequality. At each stage we have six terms f rom each A1,A2,A6,A7, andA8terms.6\nVI. FINAL COMMENTS\nWe have presented the application of the three-state generalized Ising variables to the lattice gauge system, where\nthe variables occupy the bonds of the lattice and are coupled by a fo ur-spin interaction (a plaquette). We have\npresented the derivation of correlation identities, which are exact in all dimensions and we made use of correlation\ninequalities for the generalized Potts model to obtain the upper bou nds for the area decay of Wilson’s loop. The\ncoupling constants obtained for those bounds or behaviors are ca lculated for d=3 and d=4. 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II I: A mean-field bound for binary correlations Communications in\nMathematical Physics 6121–127\n[40] Braga G A, Ferreira S and Barreto F S 1993 Correlation Ine qualities and Upper Bounds on the Critical Temperature for\nSome Spin Models Brazilian Journal of Physics 23343-355\n[41] Braga G A, Ferreira S and Barreto F S 1994 Upper bounds on t he critical temperature for the two-dimensional Blume-\nEmery-Griffiths model Journal of statistical physics 76819–834"    },    {        "title": "2402.12290v1.A_Lower_Bound_for_Estimating_Fréchet_Means.pdf",        "content": "arXiv:2402.12290v1  [math.ST]  19 Feb 2024A Lower Bound for Estimating Fr´ echet Means\nShayan Hundrieser∗\ns.hundrieser@math.uni-goettingen.de\nBenjamin Eltzner†\nbenjamin.eltzner@mpinat.mpg.de\nStephan F. Huckemann∗\nhuckeman@math.uni-goettingen.de\nFebruary 20, 2024\nAbstract\nFr´ echet means, conceptually appealing, generalize the Euclidean e xpectation to gen-\neral metric spaces. We explore how well Fr´ echet means can be est imated from in-\ndependent and identically distributed samples and uncover a fundam ental limitation:\nIn the vicinity of a probability distribution Pwith nonunique means, independent of\nsample size, it is not possible to uniformly estimate Fr´ echet means be low a precision\ndetermined by the diameter of the set of Fr´ echet means of P. Implications were pre-\nviously identified for empirical plug-in estimators as part of the phen omenon finite\nsample smeariness . Our findings thus confirm inevitable statistical challenges in the\nestimation of Fr´ echet means on metric spaces for which there exis t distributions with\nnonunique means. Illustrating the relevance of our lower bound, ex amples of extrinsic,\nintrinsic, Procrustes, diffusion and Wasserstein means showcase e ither deteriorating\nconstants or slow convergence rates of empirical Fr´ echet mean s for samples near the\nregime of nonunique means.\nKeywords : Intrinsic means, Extrinsic means, Procrustes means, Geom etric statistics,\nSpherical statistics, Smeariness, Wasserstein barycente r\nMSC 2020 subject classification : primary 62F10, 62H12; secondary 60D05\n1 Introduction\nWith the dawn of information age, richness and complexity of data have soared to unprece-\ndented levels, emphasizing the relevance of taking into acc ount the underlying geometry\nof data. Whereas earlier methods of data analysis implicitl y assumed data to be arranged\nin (flat) Euclidean spaces, recent techniques acknowledge a nd incorporate (non-flat) non-\nEuclidean aspects of data. This paradigm shift has lead to th e development of a plethora\n∗Institute for Mathematical Stochastics, University of G¨ o ttingen, Goldschmidtstraße 7, 37077 G¨ ottingen\n†Max Planck Institute for Multidisciplinary Sciences, Am Fa ßberg 11, 37077 G¨ ottingen\n1of statistical methods for non-Euclidean data which have pr oven instrumental in a mul-\ntitude of disciplines, including directional statistics ( Mardia et al. ,2000), shape analysis\n(Bookstein ,1991;Kendall ,1989;Kendall et al. ,2009;Small ,2012;Dryden and Mardia ,\n2016), medical imaging ( Arsigny et al. ,2006;Pennec et al. ,2006), and phylogenetic tree\nanalysis ( Billera et al. ,2001;Huson and Bryant ,2006;Garba et al. ,2021), to name a few.\nMarron and Dryden (2021) give a recent account of theory and application.\nA key tool to summarize the central tendency of a dataset invo lves location descriptors.\nIn classical settings where data is assumed to reside on Rm, their expected value or their\nsample average (i.e., the Euclidean sample mean) are the mos t commonly used measure\nof centrality. Generalizing this concept to a metric spaces (X,d),Fr´ echet (1948) proposed\nto consider for a random variable X∼Pminimizers of the expected squared distance,\nM(P)=argmin\nx∈XEX∼P[d2(x,X)].\nThe, possibly empty, collection of such minimizers is terme d theFr´ echet mean (set) . If\nit comprises a single element M(P)={¯P}, we speak of uniqueness of the Fr´ echet mean.\nMoreover, in case (X,d)is a Riemannian manifold equipped with geodesic distance th is\nnotion is referred to as intrinsic mean , whereas if(X,d)is embedded as a submanifold\nofRmand equipped with the Euclidean distance, the respective me an is also known as\nextrinsic means .\nIn applications, the measure Pis typically unknown and instead only independent and\nidentically distributed (i.i.d.) samples X1,...,X n∼Pare available. This gives rise to the\nempirical measure Pn∶=1\nn∑n\ni=1δXiand the associated empirical Fr´ echet mean (set) ,\nM(Pn)=argmin\nx∈X1\nnn\n/summation.disp\ni=1d2(x,Xi).\nA first observation about these empirical estimators is that under mild assumptions on\nthe metric space and Pthey are almost surely consistent as the sample size ntends to\ninfinity ( Ziezold ,1977;Bhattacharya and Patrangenaru ,2003;Huckemann ,2011b ;Sch¨ otz ,\n2022;Evans and Jaffe ,2024). Building up on this insight, various statistical questio ns\nsurrounding the performance and uncertainty of these estim ators for increasing sample\nsize arise.\nMost work devoted to this subject relies on a critical assump tion: uniqueness of the\nFr´ echet mean for the probability distribution P. This topic has been the subject of a long\nline of research. A first set of contributions operates under the assumption of Xbeing\na Riemannian manifold equipped with a geodesic metric and un der considerable concen-\ntration of measures ( Karcher ,1977;Kendall ,1990;Le,2001;Groisser ,2005;Afsari ,2011).\nBroadly speaking, under such assumptions the minimization problem for the Fr´ echet mean\nclosely mimics that for the Euclidean mean, which are, if exi stent, always unique. Another\nregime imposes that the curvature of the space is non-positi ve (Sturm ,2003;Bhattacharya\nand Patrangenaru ,2003;Afsari ,2011). More recently, uniqueness of means has also been\nconfirmed without imposing concentration conditions on the measures, e.g., on the circle\n(Hotz and Huckemann ,2015). Moreover, for empirical measures based on i.i.d. observa -\ntions from a continuous distribution on a manifold, a.s. uni queness has been confirmed\nbyArnaudon and Miclo (2014). Beyond these settings, non-uniqueness of Fr´ echet means\nrather appears to be an exceptional case since a slight pertu rbation of the measure P\nsuffices to ensure uniqueness (Lemma 2.4).\n2Upon imposing uniqueness of Fr´ echet means, several articl es contributed to the sta-\ntistical analysis of Fr´ echet sample means. While initial w orks focused on distributional\nlimits ( Bhattacharya and Patrangenaru ,2005;Huckemann ,2011a ;Bhattacharya and Bhat-\ntacharya ,2012;Barden et al. ,2013;Bhattacharya and Lin ,2017;Eltzner et al. ,2021b ),\nmore recently there has been growing interest in convergenc e rates ( Sch¨ otz ,2019a ,b;\nAhidar-Coutrix et al. ,2020;Le Gouic et al. ,2022) and concentration properties ( Romon\nand Brunel ,2023;Escande ,2023). Most of these results impose concentration assump-\ntions, as alluded to above, of the measures in conjunction wi th curvature conditions on\nthe behavior of the Fr´ echet function near its minimum, and c onfirm a parametric O(n−1/sl⊗sh.left2)\nconvergence rate. This perspective has expanded over time, with more articles exploring\nthe statistical behavior of empirical Fr´ echet means for di stributions which are supported\non larger domains ( McKilliam et al. ,2012;Hotz and Huckemann ,2015;Pennec ,2019;\nEltzner and Huckemann ,2019;Eltzner ,2022). A notable finding in this context has been\nthe phenomenon of smeariness , which asserts slower than parametric rates of convergence\nfor empirical Fr´ echet means.\nAlthough smeariness might be interpreted as an anomaly, affli cting merely boundary\ncases of distributions, its repercussions are nonetheless significant, impairing the efficacy\nof empirical plug-in estimators for intrinsic means across practical data sets ( Tran et al. ,\n2021). Such deterioration in performance spurred Hundrieser et al. (2020) to devise a\ndiagnostic tool aimed at quantifying this loss in performan ce of Fr´ echet sample means\nfrom non-smeary distributions in finite sample regimes. Upo n assuming uniqueness of\nFr´ echet population and sample means, M(P)={¯P}andM(Pn)={¯Pn}for¯P,¯Pn∈X,\ntheir analysis relies on the behavior of the variance modulation (Pennec ,2019) of the\nFr´ echet sample mean as a function of the sample size n,\nmn∶=nE[d2(¯Pn,¯P)]\nE[d2(X,¯P)](1.1)\nIntuitively, if this quantity is large, the performance of t he empirical mean is much worse\ncompared to what would be expected from the Euclidean settin g (where mn≡1) and\nmimics the finite sample behavior from a sample mean of a smear y distribution. For\nthis reason, estimators with a large variance modulation ar e said to be affected by finite\nsample smeariness . This effect is particularly dominant over empirical intrins ic means\nin data from circular or spherical distributions with (near ly) full support ( Hundrieser\net al. ,2020;Eltzner et al. ,2021a ) and extends to broader Riemannian manifolds that\nexhibit positive curvature ( Pennec ,2019). While these findings were mainly developed for\nintrinsic means on manifolds, alternative notions of means , such as extrinsic means or the\nrecently proposed diffusion means ( Hansen et al. ,2021;Eltzner et al. ,2023), seem less\nsusceptible to non-standard rates akin to smeariness, but n evertheless can still exhibit a\npronounced deterioration in the constant of the convergenc e rate.\nContribution of this work. This loss in statistical efficacy raises the question: Is it\npossible to construct estimators for Fr´ echet means which a re not affected by finite sample\neffects of smeariness? The central finding of this paper (Theor em2.5) lies in uncovering\na fundamental limitation: We show that uniformly in the vici nity of distributions with a\nnon-unique Fr´ echet mean, independent of the sample size, t he estimation error for Fr´ echet\nmeans cannot be improved below a certain precision. This inh erent lower bound cannot\nbe overcome by whatever procedure estimating Fr´ echet mean s, with striking consequences\n3for testing based on Fr´ echet means (Remark 2.9). Revisiting prior work and establishing\nnovel insights into extrinsic and intrinsic means, Procrus tes means, diffusion means, and\nWasserstein barycenters, we exemplarily illustrate the im pact of the lower bound manifest-\ning in deteriorating constants or slow convergence rates ne arby the regime of population\nmeasures with non-unique means.\nOutline. The work is structured as follows. Section 2contains the main results of this\nwork. Section 3details the impact on various notions of means where the more involved\narguments are deferred to the appendix. Finally, Section 4discusses methods to detect\nand amend for this nearby non-unique means.\nNotation. Throughout this work, (X,d)denotes a metric space with its power set P(X).\nGiven a nonempty set A⊆Xits diameter is diam (A)∶=supx,y∈Ad(x,y). The collection of\nBorel probability measures on XisP(X), and the total variation distance between two\nmeasuresP,Q∈P(X)is denoted by TV (P,Q). Further, if the (generalized) Fr´ echet mean\nofPexists and is unique, we write ¯Pfor the unique element in M(P).\n2 Main Result\nThe central finding of this work is stated for the subsequent g eneral notion of a mean.\nDefinition 2.1 (Fr´ echetρ-means).Letρ∶X×X→Rbe a Borel measurable function and\nletP∈P(X)be a probability measure. Assuming ρ(x,⋅)∈L1(P)for eachx∈X, its\nFr´ echetρ-function is defined as\nFP∶X→R, x↦EX∼P[ρ(x,X)].\nFurther, its Fr´ echetρ-mean (set) is defined as the collection of minimizers\nM(P)∶=argmin\nx∈XFP(x)∈P(X).\nMoreover, the Fr´ echet ρ-mean is said to be honest if for every x∈ X the function\nρ(⋅,x)∶X→Ris uniquely minimized by x.\nRemark 2.2 (Honesty) .ThatMis honest is equivalent to M(δx)={x}for everyx∈X.\nThis means that Malways recovers the position of a Dirac measure at a single po int and\ndoes not exhibit a systematic bias under strong concentrati on.\nRemark 2.3 (Relation to standard Fr´ echet mean) .The standard Fr´ echet mean arises by\ntakingρ∶=d2wheredis the metric onX. In particular, Fr´ echet means are honest.\nThe framework of Fr´ echet ρ-means captures a large selection of notions of means on\ngeometric structures, see Section 3for a number of examples, and extends the standard\nEuclidean mean. The latter would arise by choosing X=Rmandρ(x,y)=/parall⟩l.alt1x−y/parall⟩l.alt12−/parall⟩l.alt1y/parall⟩l.alt12.\nFor this setting, it is known that a unique minimizer exists i f the measure Padmits a\nfinite (first) moment (see Sturm (2003)). Otherwise, the associated Fr´ echet function may\nnot admit finite values and the Euclidean mean is empty.\nFor a general metric space Xand a given measurable function ρ, the Fr´ echet ρ-mean\nalso does not necessarily exist, and moreover, if it does, it may not be unique. Our\nmain results assess effects induced by non-uniqueness, which we formalize by the following\nassumption.\n4Assumption (NU). The Fr´ echet ρ-meanMis honest and P∈P(X)is a probability\nmeasure such that M(P)is nonempty and non-unique, i.e., /divid⟩s.alt0M(P)/divid⟩s.alt0>1.\nThe following result confirms that the condition of non-uniq ueness of Fr´ echet ρ-means\nis a property, unstable under perturbations of measures.\nLemma 2.4 (Instability of non-uniqueness) .Under Assumption (NU), it follows for all\nP∈P(X), allt∈(0,1]and allx∈M(P)that\nM(Px,t)={x}\nwith the perturbed measure Px,t∶=(1−t)P+tδx.\nProof. By Assumption (NU) , the Fr´ echet ρ-function fulfills for ˜ x∈X/slash.right{x}that\nFPx,t(˜x)=(1−t)FP(˜x)+tρ(˜x,x)\n>(1−t)min\nx′∈XFP(x′)+tmin\nx′′∈Xρ(x′′,x)\n=(1−t)FP(x)+tρ(x,x)=F˜Px,t(x),\nwhere the strict inequality is due to Mbeing honest.\nThis suggests non-uniqueness of Fr´ echet ρ-means to be rather exceptional. Nonetheless,\nin the immediate vicinity of a distribution with a non-uniqu e mean, a significant loss in\nthe performance of estimating population mean based on i.i. d. observations manifests.\nTheorem 2.5 (Uniform lower bound) .For given sample size n∈N, under Assump-\ntion(NU)it follows for all P∈P(X), allε∈(0,1)and allp≥1that\ninf\nˆPnsup\n0<t≤ε\nx∈M(P)EPx,t/brack⟩tl⟩ft.alt1dp(ˆPn(X1,...,X n),¯Px,t)/brack⟩tright.alt≥1\n2/par⟩nl⟩ft.alt4diam(M(P))\n2/par⟩nright.alt4p\n, (2.1)\nwhere the infimum is taken over all (measurable) estimators ˆPn∶Xn→Xbased on i.i.d.\nrandom variables X1,...,X n∼Px,t∶=(1−t)P+tδx. Further, for all η>0arbitrarily small\nit holds that\ninf\nˆPnsup\n0<t≤ε\nx∈M(P)PPx,t/par⟩nl⟩ft.alt4d(ˆPn(X1,...,X n),¯Px,t)≥diam(M(P))−η\n2/par⟩nright.alt4≥1\n2. (2.2)\nA key insight from Theorem 2.5is that the uniform error in estimating the Fr´ echet\nρ-mean does not improve as the sample size increases. This imp lies that under Assump-\ntion(NU)every method of estimating the respective mean based on i.i.d. mea surements\nwill be affected by deteriorating constants or slower converg ence rates (see Section 3for\nexamples of both manifesting for various notions of means). More precisely, for every\nestimation procedure there is a probability distribution f or which it does not realize a\nstatistical accuracy below a precision determined by diam (M(P)): Choosing p=2 and\nnoting that 1/slash.l⟩ft9<1/slash.l⟩ft8 the assertion of the corollary below follows at once from ( 2.1) above.\nCorollary 2.6. Under Assumption (NU)for every estimator ˆP∶⋃n∈NXn→Xand every\nsample size n∈Nthere exists a probability measure PˆP,n∈P(X)for which the Fr´ echet\nρ-meanM(PˆP,n)={¯PˆP,n}is unique such that for independent X1,...,X n∼PˆP,nit follows\nEPˆP,n/bracketleft.alt2d2(ˆP(X1,...,X n),¯PˆP,n)/bracketright.alt2≥diam(M(P))2/slash.left9.\n5Remark 2.7 (Comparison with Tran et al. (2021)).The work by Tran et al. (2021) asserts\nthat nearby a smeary probability distribution, i.e., a dist ribution for which the empirical\nFr´ echet means admit slower than parametric convergence ra tes, there exists a probability\nmeasure for which the variance modulation ( 1.1) of the empirical Fr´ echet mean can be\narbitrary large for some sample sizes but stays bounded. Thi s phenomenon has been\ncalledfinite sample smeariness byHundrieser et al. (2020). In addition, Corollary 2.6\nasserts that nearby the regime of non-uniqueness this pheno menon cannot be overcome\nby employing another estimation technique.\nRemark 2.8 (Complementary contributions) .Previous contributions by Sch¨ otz (2019b );\nAhidar-Coutrix et al. (2020);Le Gouic et al. (2022) on the convergence behavior and\nbyEscande (2023);Romon and Brunel (2023) on concentration properties of empirical\nFr´ echet means toward its population counterpart impose ce rtain strictly positive curva-\nture conditions on the behavior of the Fr´ echet function nea r its minimum. In particular,\nsuch assumption ensures that the Fr´ echet population mean i s unique. Our work is con-\ncerned with the complementary regime where non-uniqueness occurs and establishes a\nlower bound which does not improve as the sample size increas es. Insofar, our work\nconfirms the necessity of separability conditions to infer u niform convergence results.\nProof of Theorem 2.5.We first establish the lower bound ( 2.2) in probability for which we\nuse LeCam’s two-point method ( Tsybakov ,2009, Section 2). To this end, note by Assump-\ntion(NU) that diam(M(P))>0 and takex,y∈M(P)such thatd(x,y)≥diam(M(P))−η.\nThen, for each t∈(0,ε]it follows by Lemma 2.4that ¯Px,t=xand ¯Py,t=y, and thus\nd(¯Px,t,¯Py,t)≥diam(M(P))−η.\nMeanwhile, the total variation distance between ˜Px,tand ˜Py,tis bounded for each t∈(0,ε]\nby TV(˜Px,t,˜Py,t)≤2t.Consequently, for t∗∈(0,ε]to be specified later, it follows from\nInequality (2.9) of Tsybakov (2009) that the left-hand side of ( 2.2) is lower bounded by\ninfˆPnmax\nz∈{x,y}PPz,t∗/parenleft.alt4d(ˆPn(X1,...,X n),¯Pz,t∗)≥diam(M(P))−η\n2/parenright.alt4\n≥infˆΨnmax\nz∈{x,y}PPz,t∗/parenleft.alt1ˆΨn(X1,...,X n)≠z/parenright.alt1, (2.3)\nwhere the infimum in the second line is taken over all (measura ble) tests ˆΨn∶Xn→{x,y}.\nUsing Theorem 2.2(i) and LeCam’s inequalities (Lemma 2.3) i n conjunction with the\ntensorization property of the Hellinger distance (p. 83, Pr operty (iv)) of Tsybakov (2009)\nit follows that the right-hand side of ( 2.3) is lower bounded by\n1\n2/parenleft.alt11−TV(˜P⊗n\nx,t∗,˜P⊗n\ny,t∗)/parenright.alt1≥1−/radical.alt1\n1−(1−2t∗)n\n2. (2.4)\nHence, by choosing t∗>0 arbitrarily small, we conclude the validity of inequality (2.2).\nFor the lower bound ( 2.1) in expectation we use Markov’s inequality with ( 2.2). This\nyields forp≥1 andη∈(0,diam(M(P)))the left-hand side of ( 2.1) to be lower bounded by\n/parenleft.alt4diam(M(P))−η\n2/parenright.alt4p\ninf\nˆPnsup\n0<t≤ε\nx∈M(P)PPx,t/parenleft.alt4d(ˆPn(X1,...,X n),¯Px,t)≥diam(M(P))−η\n2/parenright.alt4\n≥/parenleft.alt4diam(M(P))−η\n2/parenright.alt4p1\n2.\n6Choosingη>0 arbitrarily small yields the assertion.\nRemark 2.9 (Impact on testing) .The proof of Theorem 2.5(Inequalities ( 2.3) and ( 2.4))\nalso reveals that under Assumption (NU) every test ˆΨn∶XN→{x,y}, to assess if the\nFr´ echet mean is located at xory, experiences a considerable loss in performance if the\ndata stems from a probability distribution that is close to Pwithx,y∈M(P). In fact,\nuniformly over the collection {Pz,t/divides.alt0z∈{x,y},t∈(0,ε)}no test performs better than a\nrandom coin flip.\n3 Slow Rates and Deteriorating Constants: Examples\nThe previous section highlighted inherent statistical cha llenges in estimating Fr´ echet means\nnearby the regime of non-uniqueness. For illustration, in t his section, we revisit and\nexpand beyond the literature specific examples of Fr´ echet m eans, demonstrating slow\nrates ordeteriorating constants in the convergence behavior of empirical plug-in estimator s.\nThroughout these examples, we always consider i.i.d. obser vationsX1,...,X n∼Pfrom a\nprobability distribution P, which serve to define the empirical measure Pn∶=1\nn∑n\ni=1δXi.\nGiven a measurable and integrable function ρas in Definition 2.1, this leads to a notion\nMof a Fr´ echet ρ-mean with population (resp. empirical) mean M(P)(resp.M(Pn)) and\npopulation (resp. empirical) Fr´ echet functions FP(resp.Fn∶=FPn).\n3.1 Extrinsic Means on Manifolds\nLetXbe a smooth manifold (e.g. Lee 2010 for a textbook) that is smoothly embedded\nviaj∶X→Rmin the Euclidean space and equipped with Euclidean distance dE∶X×X→\n[0,∞),(x,y)↦/parall⟩l.alt1j(x)−j(y)/parall⟩l.alt1. Then, the extrinsic mean setMEis defined as the Fr´ echet\nmean set onXwith respect to ρ=d2\nE.\nGiven a probability distribution P∈P(X),Hendriks and Landsman (1996,1998) ob-\ntained its mean location , which Bhattacharya and Patrangenaru (2003) called its extrinsic\nmean , thus by minimizing the Euclidean distance under constrain ing to the manifold.\nHence, the extrinsic mean set is given by the orthogonal proj ection of the Euclidean mean\nEj#P[X]in ambient space (of the pushforward measure j#P) to the embedded manifold\nME(P)=argmin\nx∈X/parall⟩l.alt2j(x)−Ej#P[X]/parall⟩l.alt2.\nIf nonvoid (e.g. if the manifold is compact), uniqueness of MEtherefore occurs if and\nonly ifEj#P[X]is located at a position y∈Rmfor which there exists a unique orthogonal\nprojectionx∈X, i.e.,/parall⟩l.alt1j(x)−y/parall⟩l.alt1</parall⟩l.alt1j(x′)−y/parall⟩l.alt1for allx′∈X. The collection of all such\npoints is termed non-focal points , whereas elements of its complement are called focal\npoints .\nIn consequence, Theorem 2.5finds application for a distribution Pif the Euclidean\nmean of the embedded distribution is a focal point.\nA simple example arises in the context of spheres Sm∶={x∈Rm+1/divid⟩s.alt0/parall⟩l.alt1x/parall⟩l.alt12=1}form≥1\nwhich can be embedded into Rm+1via the inclusion map. Here, every point Rm/slash.right{0}is non-\nfocal. Since the distance minimizing projection is given by the normalization x↦x/slash.l⟩ft/parall⟩l.alt1x/parall⟩l.alt1,\n7we obtain the following characterization of extrinsic mean sets on the sphere,\nME∶P(Sm)↦P(Sm), P↦⎧⎪⎪⎨⎪⎪⎩{EP[X]/slash.l⟩ft/parall⟩l.alt1EP[X]/parall⟩l.alt1} ifEP[X]≠0,\nSmifEP[X]=0.\nHence, by Theorem 2.5, the empirical extrinsic mean will likely perform statisti cally worse\nif the Euclidean mean EP[X]is close to the origin. Indeed, since the orthogonal project ion\nto the tangent space of the sphere at x∈Smis explicitly given by\nRm→{z∈Rm∶zTx=0},y↦(Idn+1−xxT)y\nthedeteriorating constants can be directly observed in the following, see also Bhattacharya\nand Patrangenaru (2005).\nTheorem 3.1 (Hendriks et al. 1996 , CLT extrinsic spherical mean) .LetP∈P(Sm)be\na probability distribution such that EP[X]≠0. Then, the extrinsic mean ME(P)={¯P}\nis unique and for i.i.d. random elements X1,...,X n∼Pit follows, as n→∞, for a\nmeasurable selection of extrinsic sample means ¯Pn∈ME(Pn)that\n√n(¯Pn−¯P)D/leftrightline/leftrightline→N(0,CΣCT).\nHerein, the asymptotic covariance is determined by matrice sΣ=(Cov[Xi\n1,Xj\n1])m+1\ni,j=1and\nC=(Idm+1−¯P¯PT)/slash.l⟩ft/parall⟩l.alt1EP[X]/parall⟩l.alt1.\nFrom this distributional limit it is apparent that the conve rgence rate of the empirical\nextrinsic spherical mean will always be of parametric order . However, the variance of the\nlimit distribution is reciprocally linked to the proximity of the Euclidean mean EP[X]to\nthe origin, an observation which aligns with Theorem 2.5.\n3.2 Intrinsic Means on Manifolds\nA notable conceptual drawback of extrinsic means is that the y depend on the embedding.\nThe following alternative notion of a mean only depends on in trinsic properties: Let X\nbe a Riemannian manifold (e.g. Lee 2018 for a textbook) with its intrinsic (also called\ngeodesic) distance dIinduced by the Riemannian metric. The intrinsic mean MIis then\ndefined as the Fr´ echet mean on Xwith respect to ρ=d2\nI.\nIn the following we focus on two Riemannian manifolds with po sitive (global) curvature,\nthe circle and the sphere, which feature probability distri butions with nonunique means\n(e.g. the uniform distribution) and where the implications of Theorem 2.5have been ob-\nserved for distributional limits via slower than parametri c convergence rates or increasing\nasymptotic variances.\nThe circle is the space S=[−π,π)with−πandπidentified and equipped with the usual\n(intrinsic) arc length distance dI(x,y)=min{/divid⟩s.alt0x−y/divid⟩s.alt0,2π−/divid⟩s.alt0x−y/divid⟩s.alt0}.An extensive analysis of\nintrinsic circular means has been carried out by Hotz and Huckemann (2015). According to\ntheir findings, for a probability measure Pand a mean element a∈MI(P)the antipodal\npointq(a)∶=a+π1[−π,0)(a)−π1[0,π)(a)does not admit positive mass. Further, if P\nfeatures a continuous density near q(a), then it is dominated by1\n2π, the density of the\nuniform distribution ( Hotz and Huckemann ,2015, Theorem 1). Moreover, if the density\n8is equal to1\n2πon a neighborhood near q(a), thenais not a unique intrinsic mean ( Hotz\nand Huckemann ,2015, Corollary 1).\nThe following details and extends general central limit the orems from McKilliam et al.\n(2012);Hotz and Huckemann (2015), highlighting an intimate relation between the asymp-\ntotic law and the behavior of the underlying probability mea sure near the antipode of the\nintrinsic population mean.\nProposition 3.2 (CLT intrinsic circular mean) .LetP∈P(S)be a probability distribution\nwith a unique intrinsic mean MI(P)={0}, assume that Pfeatures a continuous density on\n(π−δ,π)∪[−π,−π+δ)with respect to the arc length measure for some δ>0, and denote by\nσ2∶=EX∼P[X2]the Euclidean variance. Further, for n∈Nconsider i.i.d. random variables\nX1,...,X n∼Pand a measurable selection of intrinsic sample means ¯Pn∈ˆMI(Pn)\n(i)In casef(−π)=limx↘−πf(x)=limx↗πf(x)<1\n2π, it follows as n→∞that\n√n¯PnD/leftrightline →N/par⟩nl⟩ft.alt20,σ2/par⟩nl⟩ft.alt11−2πf(−π)/par⟩nright.alt1−2/par⟩nright.alt2.\n(ii)In casef(−π+ε)=1\n2π−G′(ε)+o(G′(ε))andf(π−ε)=1\n2π−G′(ε)+o(G′(ε)),\nfor a non-negative, locally at 0strictly convex, continuously differentiable function\nG∶[0,δ)→R≥0fulfillingG(0)=G′(0)=0, it follows as n→∞that\n√nsign(¯Pn)2πG(/divid⟩s.alt0¯Pn/divid⟩s.alt0)D/leftrightline →N/par⟩nl⟩ft.alt10,σ2/par⟩nright.alt1.\nSimilar to the distributional limit for empirical extrinsi c means (Theorem 3.1), here the\nasymptotic variance increases as the antipodal density f(−π)tends to1\n2π. Further, if the\ndensity attains the value1\n2πand is sufficiently regular near the antipode, the convergenc e\nrate can be slower than parametric, exemplified by the follow ing two types of smeariness.\nExample 3.3 (Power smeariness) .Forr>0 consider the probability measure, continuous\nwith respect to arc length,\ndPpow,r(x)=(π−1)r+π\nπr+π1[−1/slash.left2,1/slash.left2](x)dx+1\n2π⋅/par⟩nl⟩ft.alt21−/par⟩nl⟩ft.alt1π−x/par⟩nright.alt1r/par⟩nright.alt21[π−1,π)(x)dx\n+1\n2π⋅/par⟩nl⟩ft.alt21−/par⟩nl⟩ft.alt1π+x/par⟩nright.alt1r/par⟩nright.alt21[−π,−π+1](x)dx.\nSymmetry and Hotz and Huckemann (2015, Proposition 1) yield a unique intrinsic popula-\ntion mean at MI={0}withG(ε)=εr+1/slash.l⟩ft(2π(r+1))in[0,1). Hence, by Proposition 3.2(ii),\nfor intrinsic sample means ¯Pnbased onX1,...,X n∼Ppow,r, asn→∞, it follows that\nn1/slash.left(2r+2)¯Pn\nhas a nontrivial limiting distribution, so that Ppow,rispower smeary of order r. Notably,\nthe special case of power smeariness with integer order r∈{0,1,2,...}was also derived\nbyHotz and Huckemann (2015).\nExample 3.4 (Logarithmic smeariness) .Forr>0 and suitable cr>0 consider the prob-\nability measure, continuous with respect to arc length,\ndPlog,r(x)=cr⋅1[−1/slash.left2,1/slash.left2](x)dx+1[π−1/slash.left2,π)(x)max{1−G′\nr(π−x),0}\n2πdx\n+1[−π,−π+1/slash.left2](x)max{1−G′\nr(π+x),0}\n2πdx,\n9whereGr(x)=exp(−1/slash.l⟩ft/divid⟩s.alt0x/divid⟩s.alt0r)1(0,∞)(x)is a smooth function. Symmetry and Hotz and\nHuckemann (2015, Proposition 1) imply the intrinsic population mean to be un iqueMI=\n{0}. Then, by the strong law of Ziezold (1977) and Proposition 3.2(ii) it follows (see “Proof\nof Example 3.4” in the appendix for a complete argument) for the intrinsic s ample means\n¯Pnbased onX1,...,X n∼Plog,r, asn→∞, that we have a threefold limiting behavior\n(i)¯Pna.s./leftrightline/leftrightline→ 0,\n(ii)/par⟩nl⟩ft.alt1log√n/par⟩nright.alt11/slash.leftr¯PnD/leftrightline/leftrightline→ZwithZ∼1\n2(δ−1+δ1),\n(iii)r/par⟩nl⟩ft.alt1log√n/par⟩nright.alt1(1+r)/slash.leftr/par⟩nl⟩ft.alt2¯Pn−sign(¯Pn)\n/parenleft.alt1log√n/parenright.alt11/slash.leftr/par⟩nright.alt2D/leftrightline/leftrightline→ sign(Z′)⋅log/divid⟩s.alt0Z′/divid⟩s.alt0withZ′∼N/par⟩nl⟩ft.alt30,E[X2\n1]\n4π2/par⟩nright.alt3.\nThe Sphere of dimension m≥2, recall Section 3.1, is now viewed as the Riemannian\nimmersion Sm={x∈Rm+1/divid⟩s.alt0/parall⟩l.alt1x/parall⟩l.alt12=1}, inducing the (spherical) intrinsic metric dI(x,y)=\narccos−1⟨x,y⟩forx,y∈Sm. Further, with the standard unit column vectors e1,...,em+1\nofRm+1,ξ=e1denotes the north pole , coming with the inverse exponential chart\nexp−1\nξ∶Sm/slash.right{−ξ}→Rm, x↦(e2,...,em+1)T(x−⟨x,ξ⟩ξ)arccos⟨x,ξ⟩\n/parall⟩l.alt1x−⟨x,ξ⟩ξ/parall⟩l.alt1.\nThe topic of uniqueness of intrinsic means on the multivaria te sphere is not as thoroughly\nstudied as for the univariate setting. An important observa tion in this context is that if the\nunderlying probability measure Pis rotationally symmetric on Sm, then by Bhattacharya\nand Patrangenaru (2003, Example 2.2) the intrinsic mean MI(P)will be the union of\nparallel(m−1)-dimensional spheres or poles of the rotation axis.\nAn example of a family of distributions close to the regime of non-unique intrinsic\nmeans is detailed in Section 4 of Eltzner and Huckemann (2019). They considered the\nparametric family of probability distributions Pα∈P(Sm)indexed by α∈(0,1), wherePα\nis the mixture of a uniform distribution on the lower half-sp hereLm={x∈Sm/divid⟩s.alt0x1≤0}\nweighted by αand a Dirac measure at the north pole ξweighted by(1−α). Then,\naccording to Eltzner and Huckemann (2019, Lemma 4.1 and its proof) it follows that the\nintrinsic mean is given by MI(Pα)={ξ}if and only if\nα≤1\n1+γmwithγm∶=√π\n2Γ(m+1\n2)\nΓ(m+2\n2),\nwhereΓ∶(0,∞)→(0,∞)denotes the Gamma-function. They obtain this characteriza tion\nby explicitly computing the Fr´ echet population function. In particular, for α<1/slash.l⟩ft(1+γm)\nthe Hessian of the Fr´ echet population function parametriz ed in the exponential chart\nat the north pole is positive definite, whereas at the boundar y to non-uniqueness, α=\n1/slash.l⟩ft(1+γm), all second and third order derivatives vanish while the fou rth-order derivative\nforms a diagonal fourth-order tensor with positive entries . The following result establishes\nthe distributional limit for intrinsic sample means for the different regimes in α; it is a\nconsequence of Eltzner and Huckemann (2019, Theorems 2.11 and 4.3).\nExample 3.5 (CLT intrinsic spherical mean) .TakePα∈P(Sm)as above for m≥2 and\nα≥1/slash.l⟩ft(1+γm). Further, for n∈Nconsider i.i.d. random variables X1,...,X n∼Pαand a\nmeasurable selection of intrinsic sample means ¯Pn∈ˆMI(Pn).\n(i) In case α<1/slash.l⟩ft(1+γm), it follows as n→∞that\n√nexp−1\nξ(¯Pn)D/leftrightline →N(0,Σ(α)),\n10whereΣ(α)∈Rm×mis a symmetric positive definite matrix with identical eigen values\nλ1(α)=⋅⋅⋅=λm(α)which fulfill λ1(α)≍(1−α(1+γm))−2forα↗1/slash.l⟩ft(1+γm).\n(ii) In case α=1/slash.l⟩ft(1+γm), it follows as n→∞that\nn−1/slash.left6exp−1\nξ(¯Pn)D/leftrightline →H,\nwhereH=(H1,...,Hm)is a real-valued random variable such that (H3\n1,...,H3\nm)is\ndistributed according to centered multivariate normal wit h positive definite covari-\nance.\nThe asymptotic fluctuation aligns with our main result (Theo rem 2.5) and confirms\na significant (worsening of the variance) deterioration of c onstants as the parameter α\napproaches 1/slash.l⟩ft(1+γm). In particular, at the boundary α=1/slash.l⟩ft(1+γm)a slower than\nparametric convergence rate manifests. In the language of E xample 3.3this corresponds\nto power smeariness of order r=2.\nNotably, by Stirling’s formula, it follows for m→∞thatγm→∞, which suggests that\nin high-dimensional settings issues related to non-unique ness will occur with even a very\nsmall contribution of mass near the antipode of the north pol e. This aligns with the well-\nknowncurse of dimensionality , which suggests that estimation of parameters becomes\nincreasingly challenging as the ambient dimension increas es. Moreover, on high dimen-\nsional spheres, non-unique intrinsic population means and the associated deterioration in\nestimation capabilities is not just limited to assuming mas s antipodal to the mean, but can\nalso occur if the underlying probability measure is not conc entrated in a half sphere. For\na detailed account on this and resulting effects of smeariness we refer to Eltzner (2022).\n3.3 Procrustean Means for Shape Spaces\nProcrustes means have been introduced by Gower (1975) for landmark based shape spaces.\nThe latter describe geometrical objects in Rmdetermined by placing m+1≤k∈Nland-\nmarks at corresponding locations on each object, amounting to data matrices in Rm×k,\nmodulo the action of a suitable group. For the group of simila rity transformations, ac-\ncounting for translation amounts to data matrices in Rm×(k−1), accounting for size yields\nunit size data matrices in Sm×(k−1)−1⊂Rm×(k−1)and, accounting for common rotation of\nall landmarks by the group SO(m)of rotations in Rmyields the shape space\nΣk\nm∶={[x]∶x∈Sm×(k−1)−1},[x]={gx∶g∈SO(m)}\nof equivalence classes, as detailed, e.g. in Dryden and Mardia (2016). TheProcrustes\nmetric\ndP([x],[y])∶=min\ng∈SO(m)dR(x,gy), x,y∈Sm×(k−1)−1,\nusually considered, is the quotient metric from the residua l quasi-metric\ndR(x,y)=/radical.alt1\n1−tr(xTy)2, x,y∈Sm×(k−1)−1,\n(e.g. Huckemann (2012)); it is symmetric, satisfies the triangle inequality but dR(x,−x)=0.\nThe corresponding Fr´ echet ρ-meansM(P),P∈P(Σk\nm),ρ=d2\nP, are called Procrustes\nmeans . Means similarly constructed with respect to different group actions, are Pro-\ncrustean .\n11The planar shape spaces Σk\n2carry the canonical structure of complex projective spaces\nCPk−2, Riemannian manifolds of real dimension 2 (k−2), inducing the Procrustes metric.\nFor higher dimensions ( m>2) the shape spaces are stratified spaces, nonmanifolds com-\nprising manifolds of varying dimensions with highest dimen sional manifold stratum Σ∗\nwhich is open and dense in Σk\nm, due to the principal orbit theorem (Bredon ,1972, Chapter\nIV). Notably, while the manifold stability theorem (means of random shapes assuming Σ∗\nwith positive probability lie on Σ∗themselves, see ( Huckemann ,2012, Corollary 1)) does\nnot hold for Procrustes means in general ( Huckemann ,2012, Section 4.4), it seems to hold\nfor most applications. Moreover ( Huckemann ,2012, Theorem 3), for every x∈[x]∈Σ∗\nthere is a measurable liftψx∶Σ∗→Sm×(k−1)−1in optimal position to x, i.e.\n[ψx([x′])]=[x′]anddP([x],[x′])=dR(x,ψx([x′]))for all[x′]∈Σ∗,\nthat is locally smooth near [x]. Hence, a random shape Z∼P∈P(Σk\nm)a.s. assuming\nΣ∗can be lifted to Sm×(k−1)−1in optimal position to ξ∈ζ∈M(P), ifM(P)⊂Σ∗– this\nis the case for most realistic applications. Since for any ot herξ′∈Sm×(k−1)−1and random\nX∶=ψξ(Z)onSm×(k−1)−1,\nE[dR(ξ′,X)2]≥E[dR(ξ′,ψξ′([X]))2]=E[dP([ξ′],[X])2]\n≥E[dP([ξ],[X])2]=E[dR(ξ,X)2],\nξis a Fr´ echet ρ-mean (with ρ=d2\nR) ofψξ♯P, called a residual mean .\nVectorizing via\nSm×(k−1)−1→Sm(k−1)−1\n(x1,...,xj)=x↦vec(x)=(xT\n1,...,xT\nk)T,\ndue toE[d2\nR(x,X)]=1−vec(x)TE[vec(X)vec(X)T]vec(x), the residual mean set of a\nrandom configuration matrix X∈Sm×(k−1)−1is nonempty and given by the inverse vector-\nization of the intersection of Sm×(k−1)−1with the eigenspace to the largest eigenvalue of\nE[vec(X)vec(X)T].\nFinally, with the tangent space TξSm×(k−1)−1={y∈Rm×(k−1)∶tr(yTξ)=0}ofSm×(k−1)−1\natξ∈Sm×(k−1)−1, introduce a local chart of Sm×(k−1)−1nearξvia\nϕξ∶x↦x−tr(xTξ)ξ\ntr(xTξ)\nforxin the hemisphere determined by tr (xTξ)>0. Notably, if ϕξ(x)=y, thenx=y+ξ\n/parallel.alt1y+ξ/parallel.alt1.\nProposition 3.6 (CLT Procrustes means) .LetZ1,...,Z n,Zi.i.d.∼P∈P(Σk\nm)be random\nshapes, 2≤m<k, that a.s. assume Σ∗and suppose that ¯P∈Σ∗is the unique Procrustes\nmean ofP. Further suppose that ¯Pnis a measurable selection of the Procrustes mean\nof the empirical law Pn=1\nn∑n\ni=1δZi. Then with ξ∈¯P, the above local chart ϕξand the\nmeasurable lift ψξ, smooth near ξ, from above, we have\n√nvec/par⟩nl⟩ft.alt1ϕξ○ψξ(¯Pn)/par⟩nright.alt1D/leftrightline →N(0,H−DH−)\nwithN=m(k−1),D=cov[vec(ψξ(Z))vec(ψξ(Z))T]=∑N+1\nJ=1λjvj,λ1>λ2≥...≥λN+1,\nvec(ξ)=v1,...,vN+1∈SN, andH−=1\n2∑N+1\nj=2(λ1−λj)−1vjvT\nj.\n12The proof is deferred to the appendix. Notably, as detailed i n that proof, the columns\nof the generalized inverse H−of the Hessian matrix span the tangent space of Σ∗at¯P.\nWhen the Procrustes mean ceases to be unique, e.g. if λ2→λ1, due to symmetries in the\ndistribution P, say, then the “constants above deteriorate”.\n3.4 Diffusion Means on Manifolds\nNotably in this section, tdenotes (diffusion) time, and not perturbation scale as in the\nrest of this paper.\nExtending upon the idea of means on manifolds which shall be i ndependent of the\nembedding, Hansen et al. (2021) and Eltzner et al. (2023) recently proposed a concept of\na mean on geodesically and stochastically complete, connec ted Riemannian manifolds. The\nso-called diffusion mean MD,t(P)for diffusion time tare defined as maximum likelihood\nestimates for the heat kernel, the unique symmetric solutio n of the heat equation for a\npoint mass as initial condition at time t=0. Diffusion means for time tare location\nstatistics that can be treated in the framework of Fr´ echet ρ-means by selecting ρt(x,y)as\nthe negative log-likelihood at point yof a Brownian motion process at time tstarted at\nx. For diffusion time t→0 the diffusion mean set converges to a subset of the intrinsic\nmean set. It is an open question whether there are examples wh ere this subset is proper.\nFor circles, spheres and real projective spaces the limit fo rt→∞yields a subset of the\nextrinsic mean set, as recently shown by D¨ usberg (2024), and, he provided examples where\nthis subset is proper.\nOnRm,Smand hyperbolic spaces Hm(seeLee(2018)), diffusion means are honest, since\nin these cases the log-likelihood ρt(x,y)can be expressed for every t>0 as a monotone\ndecreasing function of the geodesic distance as shown by Alonso-Or´ an et al. (2021). It\nseems reasonable to expect that similar monotonicity holds for compact symmetric spaces\nyielding honest diffusion means, for instance, on real and com plex projective spaces and\non Lie groups. As such, they can be treated in the framework of Section 2.\nIn general, diffusion means for fixed values of tcan exhibit smeariness and non-uniqueness\nsimilarly to intrinsic means. However, for increasing t, smeariness and non-uniqueness\nseems less likely, as observed, when diffusion means approach extrinsic means. Especially,\nTheorem 4.8 in Eltzner et al. (2023) shows that simultaneous estimation of the diffusion\nmeanMD,t(P)and the diffusion time trestricts the possibility of full smeariness to a very\nrestricted set of probability measures and allows at most fo r directional smeariness other-\nwise. Moreover, numerical simulations also suggest that effe cts of finite sample smeariness\nfor a given data set, i.e., a large variance modulation ( 1.1), are mitigated by additionally\noptimizing over the time t. An explanation for this behavior is given by Theorem 2.5\n(theretdenotes perturbation scale not diffusion time): As the set of p robability distribu-\ntions exhibiting non-unique diffusion means is closely tied t o the time parameter t>0,\noptimizing over tmakes it less likely to approach the regime of non-uniquenes s.\nWhile diffusion means have only been recently developed with m any fundamental prop-\nerties still open, the above consideration show that diffusio n means may also suffer from\ndeteriorating constants and lower rates, leaving the preci se picture open for future research.\n133.5 Wasserstein Barycenters on Euclidean Spaces\nFor the space of Euclidean probability measures P2(Rm)with finite second moment we\nuse the 2-Wasserstein distance W2, defined for two probability measures µ,ν∈P2(Rm)as\nW2(µ,ν)∶=/par⟩nl⟩ft.alt4inf\nπ∈Π(µ,ν)/integral.disp/parall⟩l.alt1x−y/parall⟩l.alt12dπ(x,y)/par⟩nright.alt41/slash.left2\n,\nwhereΠ(µ,ν)denotes the collection of probability measures on R2mwhose marginals on\nRmcoincide with µandν, i.e.,\nΠ(µ,ν)∶=/brac⟩l⟩ft.alt4π∈P(Rm×Rm)/divid⟩s.alt4π(A×Rm)=µ(A)for allA∈B(Rm)\nπ(Rm×B)=ν(B)for allB∈B(Rm)/brac⟩right.alt4.\nJointly,(X,d)∶=(P2(Rm),W2)defines the 2-Wasserstein space overRm, seeVillani (2003,\n2008),Panaretos and Zemel (2020) for comprehensive monographs. A notable feature of\nthe Wasserstein metric is that it equips P2(Rm)with a geometry that is consistent with\nthe Euclidean geometry: For instance, the Wasserstein dist ance between two multivariate\nGaussians of identical covariance coincides with the differe nce between their expectations.\nThe Wasserstein barycenter, denoted by MW, is the Fr´ echet mean in the Wasserstein\nspace for a probability measure P. Theoretical foundations on existence and uniqueness\nwere first established by Agueh and Carlier (2011) and extended by Le Gouic and Loubes\n(2017). Specifically, a Wasserstein barycenter exists if Pis a Borel measure that admits a\nfinite second moment ( Le Gouic and Loubes ,2017, Theorem 2), i.e., if Eµ∼P[W2\n2(µ,δ0)]<\n∞, denoted by P∈P2(P2(Rm)). Moreover, if Pis finitely supported and if one of the\nsupport points is a measure that is absolutely continuous wi th respect to the Lebesgue\nmeasure, then the Wasserstein barycenter is unique ( Agueh and Carlier ,2011, Propositions\n2.3 and 3.5). In contrast, if Pis supported on the collection of measures with finite and\ndiscrete support, uniqueness can fail, see Example 3.9.\nGiven i.i.d. random measures µ1,...,µn∼Pwith corresponding empirical measure\nPn∶=1\nn∑n\ni=1δµi,Le Gouic et al. (2022) established under certain structural assumptions\nonPparametric convergence rates (1 /slash.l⟩ftnfor the expectation of the squared distance) of\nempirical Wasserstein barycenters ¯Pn∈MW(Pn)to the respective population Wasserstein\nbarycenter. To formalize their result, the following defini tion is necessary.\nDefinition 3.7. A convex function ϕ∶Rm→Ris calledα-strongly convex forα>0 if it is\ndifferentiable and for every x,y∈Rm,\n⟨∇ϕ(x),x−y⟩≥ϕ(x)−ϕ(y)+α\n2/parall⟩l.alt1x−y/parall⟩l.alt12.\nMoreover,ϕis calledβ-smooth forβ>0 if it is differentiable and for every x,y∈Rm,\n⟨∇ϕ(x),x−y⟩≤ϕ(x)−ϕ(y)+β\n2/parall⟩l.alt1x−y/parall⟩l.alt12.\nThese two regularity properties are deeply rooted in the opt imization literature since\nan objective function which fulfills both will exhibit linea r convergence rates of gradient\ndescent methods ( Karimi et al. ,2016). Notably, if a convex, differentiable function is\nα-strongly convex and β-smooth, then necessarily α≤β.\n14Theorem 3.8 (Le Gouic et al. 2022 , Corollary 16, sufficiency of condition β−α<1).\nLetP∈P2(P2(Rm))be a probability distribution with a unique Wasserstein bar ycenter\nMW(P)={¯P}. Let 0<α≤βand suppose that every µ∈supp(P)is the push-forward\nof¯Pby the gradient of an α-strongly convex and β-smooth function ϕ, i.e.,µ=(∇ϕ)#¯P.\nThen, ifβ−α<1, every empirical Wasserstein barycenter ¯Pn∈MW(Pn)fulfills\nE[W2\n2(¯Pn,¯P)]≤1\nn4Eµ∼P[W2\n2(µ,¯P)]\n(β−α−1)2.\nThe result by Le Gouic et al. (2022) leaves open whether the requirement β−α<1 is\na genuine restriction or an artifact of their proof. Indeed, asβ−αapproaches one, the\nunderlying constant blows up to infinity. To analyze necessa ry conditions on the difference\nβ−αto guarantee parametric convergence rates of empirical Was serstein barycenters we\nutilize Corollary 2.6and construct a measure Pwhich leads to non-unique means.\nExample 3.9 (Non-unique Wasserstein barycenter) .Define the points in the two-dimensional\nEuclidean plane (see Figure 1)\nA=(0,1)tB=(1,0)tE∶=(−1/slash.l⟩ft2,1/slash.l⟩ft2)tF∶=(1/slash.l⟩ft2,1/slash.l⟩ft2)t\nand define mixtures of Dirac measures,\nµ∶=1\n2(δA+δ−A), ν∶=1\n2(δB+δ−B), P∶=1\n2(δµ+δν).\nThen, by Le Gouic and Loubes (2017, Theorem 8) the Wasserstein barycenter is given by\nMW(P)=/brac⟩l⟩ft.alt31\n2(δE+δ−E),1\n2(δF+δ−F)/brac⟩right.alt3.\nNecessary and sufficient conditions for the existence of an α-convex and β-smooth\nfunctionϕwithβ≥α>0 such that ξ=∇ϕ#[1\n2(δE+δ−E)]forξ∈{µ,ν,1\n2(δE+δ−E)}\nwere derived by Taylor et al. (2017, Theorem 4). Invoking their analysis and utilizing\nCorollary 2.6, we arrive at the following proposition.\n−1 1\n−11A\n−AB −BE\n−EF\n−F\nFigure 1: Depicting the points from Example 3.9:E(resp.F) represents the midpoint between\nAand−B(resp.AandB). The set of Wasserstein barycenters for the probability measur e\n1\n2(δ1\n2(δA+δ−A)+δ1\n2(δB+δ−B))is given by{1\n2(δE+δ−E),1\n2(δF+δ−F)}.\n15Proposition 3.10 (Necessity of condition β−α≤2 for fast rates) .In the setting of\nExample 3.9, the following assertions hold.\n(i)Everyα-strongly convex and β-smooth function ϕ∶R2→Rwithβ>α≥0such that\nµ=∇ϕ#[1\n2(δE+δ−E)]fulfillsβ−α≥2.Further, if αis positive, then β−α>2.\n(ii)For everyδ>0there exists a choice for β≥α>0withβ−α≤2+δsuch that for\neveryξ∈{µ,ν,1\n2(δE+δ−E)}there exists an α-strongly convex and β-smooth function\nϕwhich fulfills ξ=ϕ#[1\n2(δE+δ−E)].\n(iii)For every estimator ˆP∶⋃n∈NP2(Rm)n→P2(Rm)and every sample µ1,...,µnof size\nn∈Nthere exists a probability measure PˆP,n∈{(1−t)P+tδξ/divid⟩s.alt0ξ∈M(P),t∈(0,1)}\nwith a unique Wasserstein barycenter MW(PˆP,n)={¯PˆP,n}such that\nE/bracketleft.alt2W2\n2(ˆP(µ1,...,µn),¯PˆP,n)/bracketright.alt2≥diam(MW(P))2/slash.left9=1/slash.left9.\nThe above proposition highlights that the upper bound on the differenceβ−α<2 is\nnecessary to infer parametric convergence rates for the empirical Was serstein barycenter.\nIn particular, this degenerate behavior cannot be overcome by whatever estimator in the\nregimeβ−α>2 without imposing additional assumptions on the underlyin g measures.\nIt remains open to investigate whether the analysis by Le Gouic et al. (2022) is optimal\nor whether it can be extended to the regime 1 ≤β−α<2. In the introduction of their\npaper, they state that the bounds for the support of probabil ity measures on spheres Sm\namounts to an open ball of radiusπ\n4, which is by a factor of 2 short of the bound ofπ\n2\ngiven by Afsari (2011). For this reason, we conjecture that their result extends t o the\nregime 1≤β−α<2.\nConjecture 3.11. With the notation of Theorem 3.8,\nE[W2\n2(¯Pn,¯P)]≤C\nnEµ∼P[W2\n2(µ,¯P)]\n(β−α−2)2,\nfor all 1≤β−α<2 with a constant C>0 that does not depend on P.\nIn addition, it would be worthwhile to explore sharp converg ence rates for the empirical\nWasserstein barycenter beyond this specific range. A notabl e contribution in this context\nwas made by Kroshnin (2021, Theorem 4.6.1) who provides an example where Pis con-\ncentrated on the collection of finitely supported probabili ty measures and the empirical\nbarycenters achieves a logarithmic convergence rate at bes t.\n4 Discussion\nThis work sheds light on the statistical challenges involvi ng (generalized) Fr´ echet means\nnearby the regime of non-uniqueness, demonstrating that es timating Fr´ echet means with\nhigh accuracy is not possible (Theorem 2.5) and that all testing procedures will experience\nlarge errors of first or second kind (Remark 2.9). For practical contexts this raises two\npoints. Firstly, how to identify the presence of non-unique ness effects, and secondly, how\nto proceed when such effects cannot be dismissed.\nWhen the objective is to determine a location descriptor (a F r´ echetρ-mean for given ρ)\nfor a data set, nearby the regime of non-unique means, it is re asonable to anticipate\n16that the distribution of an appropriate estimator (such as t he sample mean) based on\ni.i.d. observations is concentrated around the respective population means. As the sample\nsize increases and assuming the estimator is consistent, th is distribution tends to become\nmultimodal, a characteristic that can be assessed using the test by Eltzner (2020). If this\ntest does not reject nonuniqueness, a single location descr iptor likely is not sufficient to\nsuitably describe the data set. Instead, it is reasonable to perform cluster analysis to\npartition the data into structured sub-samples. For each su b-sample the Fr´ echet mean is\nthen to be calculated. These Fr´ echet means then provide a mo re accurate representation\nof location descriptors.\nMoreover, when employing Fr´ echet means for testing purpos es between datasets, unless\nnon-uniqueness can be ruled out, we suggest conducting a sam ple splitting procedure to\nassess if the respective test maintains the correct signific ance level between split samples\nfrom both datasets. If the effective level strongly deviates f rom the nominal level, then,\neither recalibrate, for instance by bootstrapping, or cons ider an alternative testing method.\nThe latter, however, amounts to multiple testing bias, warr anting appropriate correction\nmethods. We leave a comprehensive analysis of such procedur es as future research.\nAcknowledgements: S. Hundrieser gratefully acknowledges support by the Deuts che\nForschungsgemeinschaft (DFG, German Research Foundation ) as part of DFG RTG 2088,\nB. Eltzner and S.F. 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Springer.\nTsybakov, A. B. (2009). Introduction to Nonparametric Estimation . Springer Series in\nStatistics. Springer.\nVillani, C. (2003). Topics in optimal transportation , volume 58 of Graduate Studies in\nMathematics . American Mathematical Society.\nVillani, C. (2008). Optimal transport: old and new , volume 338 of A Series of Compre-\nhensive Studies in Mathematics . Springer.\nZiezold, H. (1977). On expected figures and a strong law of lar ge numbers for random\nelements in quasi-metric spaces. In Transactions of the Seventh Prague Conference\non Information Theory, Statistical Decision Functions, Rand om Processes and of the\n1974 European Meeting of Statisticians , pages 591–602. Springer.\nA Deferred Proofs\nProof of Proposition 3.2.Assertion(i)has been shown by McKilliam et al. (2012);Hotz\nand Huckemann (2015). For Assertion(ii), expand the empirical Fr�� echet function based\non the arc length distance for 0 <x<δ,\nFn(x)=1\nn/summation.disp\nXj∈[x−π,π)(Xj−x)2+1\nn/summation.disp\nXj<x−π(Xj+2π−x)2\n=1\nnn\n/summation.disp\nj=1(Xj−x)2+4π\nn/summation.disp\nXj<x−π(Xj−x+π)\nIn consequence, with the Euclidean mean Xn=1\nn∑j=1Xj, almost surely,\n1\n2gradFn(x)=x−Xn−2π\nn/summation.disp\nXj<x−π1. (A.1)\nFurther, upon rewriting the expectation,\nE⎡⎢⎢⎢⎢⎣1\nn/summation.disp\nXj<x−π1⎤⎥⎥⎥⎥⎦=/integral.dispx−π\n−πf(t)dt=/integral.dispx−π\n−π/par⟩nl⟩ft.alt31\n2π−G′(π+t)+o(G′(π+t))/par⟩nright.alt3dt\n=x\n2π−G(x)+o/par⟩nl⟩ft.alt1G(x)x/par⟩nright.alt1,\nwhere the last equality is due to the properties of G, we infer for the sum of Bernoulli\nvariables\n1\nnn\n/summation.disp\nj=11{Xj<x−π}=x\n2π−G(x)+o/par⟩nl⟩ft.alt1G(x)x/par⟩nright.alt1+Op⎛\n⎝/radical.alt1\n/divid⟩s.alt0x−G(x)/divid⟩s.alt0√n⎞\n⎠.\n21In conjunction with the analogue argument for −δ<x<0 we thus obtain from ( A.1) that\n√n\n2gradFn(x)=√n(sign(x)2πG(/divid⟩s.alt0x/divid⟩s.alt0)−Xn)+o/par⟩nl⟩ft.alt1G(x)x/par⟩nright.alt1+Op⎛\n⎝/radical.alt1\n/divid⟩s.alt0x+G(/divid⟩s.alt0x/divid⟩s.alt0)/divid⟩s.alt0√n⎞\n⎠.(A.2)\nThis yields Assertion (ii)since the above left-hand side vanishes by definition at x=¯Pn,\nit holds ¯Pn=op(1)due to Ziezold (1977), and by the classical Euclidean central limit\ntheorem which asserts√nXnD/leftrightline →N/par⟩nl⟩ft.alt10,σ2/par⟩nright.alt1.\nProof of Example 3.4.Since the density of Plog,ron(−π,−π+1/slash.l⟩ft2)∪(π−1/slash.l⟩ft2,π)is strictly\nsmaller than1\n2πwe see that cr>1\n2π. Hence, by Hotz and Huckemann (2015, Propostion\n1) there is a unique Fr´ echet population mean Pin[−1/slash.l⟩ft2,1/slash.l⟩ft2], which, by symmetry, is\nMI={0}. This implies the preliminary claim. Moreover, the strong l aw by Ziezold (1977)\nyields(i).\nNext, plug in x=¯Pnin (A.2) as done in the proof of Theorem 3.2, to obtain\n√n⋅sign(¯Pn)exp(−1/slash.l⟩ft/divid⟩s.alt0¯Pn/divid⟩s.alt0r)=√n\n2πXn+oP(1).\nSubjecting both sides above to y↦sign(y)⋅log(/divid⟩s.alt0y/divid⟩s.alt0)1R∖{0}(y)gives\nsign(¯Pn)/par⟩nl⟩ft.alt4log√n−1\n/divid⟩s.alt0¯Pn/divid⟩s.alt0r/par⟩nright.alt4=sign(Xn)log/divid⟩s.alt4√n\n2πXn+oP(1)/divid⟩s.alt4\n=sign(Xn)log/divid⟩s.alt4√n\n2πXn/divid⟩s.alt4+oP(1),\nwhere for the second equality we used that2π√nXn=OP(1)and log/par⟩nl⟩ft.alt1/divid⟩s.alt01+oP(1)/divid⟩s.alt0/par⟩nright.alt1=oP(1).\nSettingLn=sign(Xn)log/divid⟩s.alt2√n\n2πXn/divid⟩s.alt2, we proceed to\nsign(¯Pn)log√n−Ln=sign(¯Pn)+op(/divid⟩s.alt0̂P/divid⟩s.alt0r)\n/divid⟩s.alt0¯Pn/divid⟩s.alt0r.\nThis yields whenever /divid⟩s.alt0Ln/divid⟩s.alt0<log√n, and the probability for this event tends to 1, that\nsign(¯Pn)/divid⟩s.alt0¯Pn/divid⟩s.alt0r=1+op(/divid⟩s.alt0¯Pn/divid⟩s.alt0r)\nsign(¯Pn)log√n−Ln=1+op(/divid⟩s.alt0¯Pn/divid⟩s.alt0r)\nsign(¯Pn)log√n∞\n/summation.disp\nk=0/par⟩nl⟩ft.alt4sign(¯Pn)Ln\nlog√n/par⟩nright.alt4k\n.\nHence, since ¯Pn=oP(1)due to(i)we conclude\n/divid⟩s.alt0¯Pn/divid⟩s.alt0r=1\nlog√n+sign(¯Pn)Ln\n(log√n)2+oP/par⟩nl⟩ft.alt41\n(log√n)2/par⟩nright.alt4.\nBy symmetry of Plog,r, this yields Assertion (ii). It further implies\n¯Pn=sign(¯Pn)/par⟩nl⟩ft.alt41\nlog√n+sign(¯Pn)Ln\n(log√n)2+oP/par⟩nl⟩ft.alt41\n(log√n)2/par⟩nright.alt4/par⟩nright.alt41\nr\n=sign(¯Pn)\n(log√n)1\nr/par⟩nl⟩ft.alt41+1\nrsign(¯Pn)Ln\nlog√n+oP/par⟩nl⟩ft.alt41\nlog√n/par⟩nright.alt4/par⟩nright.alt4.\nWith the definition of Lnabove, we thus obtain Assertion (iii).\n22Proof of Proposition 3.6.With the notation of Proposition 3.6, let ˆv1∶=vec(ψξ○¯Pn),\nXi∶=vec(ψξn○Zi),i=1,...,n andX∶=vec(ψξ○Z). Further, let Dn∶=1\nn∑n\ni=1XiXT\ni=\n∑N+1\nj=1ˆλjˆvjˆvT\nj, with ˆλ1≥...≥ˆλN+1and ˆv1,..., ˆvN+1such thatvT\n1ˆv1≥0. In fact, by the\nstrong law of ( Huckemann ,2011b , Theorem A.3) we may assume that vT\n1ˆv1>0 forn\nsufficiently large a.s. and thus ˆ y∶=ϕv1(ˆv1)∈Tv1SNis well defined. Since the empirical\nFr´ echetd2\nR-function, in the local coordinate y∈Tv1SNis given by\nFn/par⟩nl⟩ft.alt1vec−1○ϕ−1\nv1(y)/par⟩nright.alt1=1−(y+v1)T\n/parall⟩l.alt1y+v1/parall⟩l.alt1Dny+v1\n/parall⟩l.alt1y+v1/parall⟩l.alt1=∶Gn(y),\nwe have, a.s. for nsufficiently large and suitable ̃ybetween ˆyand 0 that\n0=gradGn(ˆy)=gradGn(0)+HessGn(̃y)ˆy.\nAs shown in Huckemann and Eltzner (2019), the central limit theorem of Eltzner and\nHuckemann (2019) is applicable, yielding the distributional limit\n√nˆy→N/par⟩nl⟩ft.alt10,(HessyG(0))−D(HessyG(0))−/par⟩nright.alt1,\nwhereD=E[XXT],G(y)=1−(y+v1)T\n/parallel.alt1y+v1/parallel.alt1Dy+v1\n/parallel.alt1y+v1/parallel.alt1is the population Fr´ echet function in\nthe local coordinate and (HessyG(0))−is any generalized inverse of Hess yG(0)(justified\nbelow). Since\n−1\n2gradyG(y)=Dv1+y\n/parall⟩l.alt1y+v1/parall⟩l.alt12−(y+v1)TD(y+v1)\n/parall⟩l.alt1y+v1/parall⟩l.alt14(y+v1)\n−1\n2HessyG(y)=D\n/parall⟩l.alt1y+v1/parall⟩l.alt12−2D(y+v1)(y+v1)T+(y+v1)(y+v1)TD\n/parall⟩l.alt1y+v1/parall⟩l.alt14\n+4(y+v1)TD(y+v1)\n/parall⟩l.alt1y+v1/parall⟩l.alt16(y+v1)(y+v1)T−(y+v1)TD(y+v1)Idm+1\n/parall⟩l.alt1y+v1/parall⟩l.alt14\nwe have thus\nHessyG(0)=2N+1\n/summation.disp\nj=2(λ1−λj)vjvT\nj.\nSince, moreover a.s., for nsufficiently large, ˆ yis in the tangent space spanned by v2,...,vN+1,\nany generalized inverse can be taken, yielding the assertio n.\nProof of Proposition 3.10.We prove Assertions (i)and(ii)by invoking Taylor et al. (2017,\nTheorem 4 and Corollary 1), which states that a function ˜ ϕ∶{E,−E}→Rcan be extended\nto anα-strongly convex and β-smooth function ϕ∶R2→Rforβ>α≥0 and such that\n∇ϕ(E)=Aand∇ϕ(−E)=−Aif and only if\n˜ϕ(E)−˜ϕ(−E)−(−A)t(E+E)\n≥1\n2(1−α/slash.l⟩ftβ)/par⟩nl⟩ft.alt31\nβ/parall⟩l.alt12A/parall⟩l.alt12+α/parall⟩l.alt12E/parall⟩l.alt12−2α\nβ(−2A)t(−2E)/par⟩nright.alt3,and\nϕ(−E)−ϕ(E)−(A)t(−E−E)\n≥1\n2(1−α/slash.l⟩ftβ)/par⟩nl⟩ft.alt31\nβ/parall⟩l.alt12A/parall⟩l.alt12+α/parall⟩l.alt12E/parall⟩l.alt12−2α\nβ(2A)t(2E)/par⟩nright.alt3.\n23Since the right-hand sides are identical for both inequalit ies, we may assume without\nloss of generality that ˜ ϕ(E)−˜ϕ(−E)=∶κ≤0 and only continue with the first inequality.\nComputing the terms on both sides then yields that\n1≥κ+1≥β\n2(β−α)/par⟩nl⟩ft.alt31\nβ/parall⟩l.alt12A/parall⟩l.alt12+α/parall⟩l.alt12E/parall⟩l.alt12−2α\nβ(−2A)t(−2E)/par⟩nright.alt3\n=1\n2(β−α)/par⟩nl⟩ft.alt1/parall⟩l.alt12A/parall⟩l.alt12+αβ/parall⟩l.alt12E/parall⟩l.alt12−2α(−2A)t(−2E)/par⟩nright.alt1\n=1\n2(β−α)(4+2αβ−4α)=1\n2(β−α)(4+2α(β−2)).\nPerforming a change of variables γ=β−α≥0 thus yields the following inequality\nγ≥2+(β−γ)(β−2)\nSolving this inequality via the quadratic formula, we obtai n the set of solutions\n/brac⟩l⟩ft.alt4(γ,β)∈R2\n≥0/divid⟩s.alt41\n2/par⟩nl⟩ft.alt2−/radical.alt1\nγ2−4+γ+2/par⟩nright.alt2≤β≤1\n2/par⟩nl⟩ft.alt2/radical.alt1\nγ2−4+γ+2/par⟩nright.alt2,γ≥2/brac⟩right.alt4. (A.3)\nThe minimal value for γis given by 2, and it leads to β=2, which implies that α=0.\nConsequently, if α>0, thenβ−α=γ>2.\nMeanwhile, imposing the assumption ∇ϕ(E)=−Aand∇ϕ(−E)=Awould lead to\n˜ϕ(E)−˜ϕ(−E)−(A)t(E+E)\n≥1\n2(1−α/slash.l⟩ftβ)/par⟩nl⟩ft.alt31\nβ/parall⟩l.alt1−2A/parall⟩l.alt12+α/parall⟩l.alt12E/parall⟩l.alt12−2α\nβ(2A)t(−2E)/par⟩nright.alt3,\nwhich is equivalent to\n−1≥κ−1≥1\n2(β−α)(4+2αβ+4α),\nand thus has no solutions for β>α≥0. From these considerations we infer Assertion (i).\nTo show Assertion (ii)we first choose γ=2+max(δ,1/slash.l⟩ft2). Then, infer by ( A.3) there\nexists anα-strongly convex and β-smooth function ϕµ∶Rm→Rsuch thatϕµ(E)=ϕµ(−E)\nwithβ=(−/radical.alt1\nγ2−4+γ+2)/slash.l⟩ft2≤3 andα=β−γ>0, and which fulfills µ=∇ϕµ\n#[1\n2(δE+δ−E)].\nLikewise, since AtE=BtE, there also exists an α-strongly convex and β-smooth function\nϕν∶Rm→Rwhich fulfills ν=∇ϕν\n#[1\n2(δE+δ−E)]. Finally, for the remaining case, we\nconsider the function ϕI∶Rm→R,x↦1\n2/parall⟩l.alt1x/parall⟩l.alt12which satisfies that1\n2(δE+δ−E)=∇ϕI\n#[1\n2(δE+\nδ−E)]. In particular, ϕIis 1-strongly convex and 1-smooth, and since β≤3 andα<1, the\nvalidity of Assertion (ii)follows.\nFinally, Assertion (iii)follows from Corollary 2.6after noting that\ndiam(MW(P))2=W2\n2/par⟩nl⟩ft.alt31\n2(δE+δ−E),1\n2(δF+δ−F)/par⟩nright.alt3=1\nsince/parall⟩l.alt1E−F/parall⟩l.alt12=/parall⟩l.alt1E+F/parall⟩l.alt12=1.\n24"    },    {        "title": "2402.12324v2.Investigating__ω_z___Parametrizations_in_Horava_Lifshitz_Gravity__Observational_Constraints_with_Covariance_Matrix_Simulation.pdf",        "content": "Investigating ω(z)Parametrizations in Horava-Lifshitz Gravity: Observational\nConstraints with Covariance Matrix Simulation\nHimanshu Chaudhary,1,∗Ujjal Debnath,2,†Shibesh Kumar Jas\nPacif,1,‡Niyaz Uddin Molla,2,§G.Mustafa,3, 4,¶and S. K. Maurya5,∗∗\n1Pacif Institute of Cosmology and Selfology (PICS), Sagara, Sambalpur 768224, Odisha, India\n2Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711 103, India\n3Department of Physics, Zhejiang Normal University, Jinhua 321004, Peoples Republic of China\n4New Uzbekistan University, Mustaqillik ave. 54, 100007 Tashkent, Uzbekistan\n5Department of Mathematical and Physical Sciences, College of Arts and Sciences,\nUniversity of Nizwa, Nizwa 616, Sultanate of Oman.\nThis study investigates accelerated cosmic expansion using various cosmological models within\nHorava-Lifshitz gravity, including BBCCFHKO, Seljak, ASSS, PADE-I, PADE-II, and BAZS, utiliz-\ning Equation of State Parametrization. In order to constrain the cosmological parameters of all the\nmodels, we incorporate 24 Baryon Acoustic Oscillation points, 30 Cosmic Chronometer points, 40\nType Ia Supernovae points, 24 quasar Hubble diagram points, and 162 Gamma Ray Bursts points,\nalong with the latest Hubble constant measurement (R22). We treat rdas a free parameter, aiming\nto extract H0andrdusing late-time datasets and obtain optimal fitting values for each parameter\nin every model. The benefits of treating rdas a free parameter include reduced bias, improved\nprecision, and enhanced dataset compatibility. The obtained values of H0andrdare compared to\nthe ΛCDM model, demonstrating consistency with previous estimates from the Planck and SDSS\nstudies. We reduce errors by simulating random correlations in the covariance matrix. The Akaike\nInformation Criterion (AIC) and Bayesian Information Criterion (BIC) favour all models, including\nthe ΛCDM model having the lowest AIC. Validation using the reduced chi-square statistic indicates\nsatisfactory fits for all models. While ΛCDM remains favored, extensions warrant further investiga-\ntion. This study underscores the importance of exploring alternative cosmological models to deepen\nour understanding of the universe’s fundamental properties and evolution. Continued refinement of\nmodels is essential for advancing cosmological research.\nI. INTRODUCTION\nUnderstanding the mysterious properties of dark\nenergy (DE) is a major challenges to modern cosmology.\nThe equation of state (EoS) of dark energy is essential\nfor defining its features and behaviour as time progresses.\nCosmological parameterizations of the dark energy equa-\ntion of state ( ω) are often used in model construction\nto characterise this mysterious component ’DE’ in a\nsimpler and manageable manner. The dark energy\nequation of state (EoS) is important because it helps\nmeasure the connection between pressure and density\nof dark energy, offering understanding of its dynamic\nchanges and influence on the universe’s expansion.\nEarly parametrization systems such as Chevallier-\nPolarski-Linder (CPL), Barboza-Alcaniz (BA), and\nJassal-Bagla-Padmanabhan (JBP) have been extensively\nstudied. A concise summary of various parametrization\napproaches may be seen in references [1, 2]. Cosmolo-\ngists are continuously refining and extending different\nparametrization approaches in dark energy studies in\n∗himanshuch1729@gmail.com\n†ujjaldebnath@gmail.com\n‡shibesh.math@gmail.com\n§niyazuddin182@gmail.com\n¶gmustafa3828@gmail.com\n∗∗sunil@unizwa.edu.omorder to understand the complex nature of this mys-\nterious component. Upcoming observational missions\nlike the James Webb Space Telescope and extensive\ngalaxy surveys will provide more accurate data, allowing\nresearchers to better quantify and improve these pa-\nrameterizations. Cosmological parametrizations of dark\nenergy EoS will continue to be essential for exploring the\nmysteries of the universe’s accelerating expansion as our\nknowledge advancements. The fundamental principle\nunderlying this methodology involves the examination\nof a specific evolutionary scenario without making prior\nassumptions about any particular dark energy (DE)\nmodel. The objective is to identify the characteristics\nof the enigmatic component responsible for propelling\ncosmic acceleration. This approach is referred to as the\nmodel-independent method, where the model parameters\nare estimated using available observational dataset.\nNevertheless, this approach has its limitations. Firstly,\nseveral parametrizations face challenges related to\ndivergence. Secondly, it’s conceivable that some\nparametrization schemes could inadequately grasp\nnuanced understandings of dark energy’s true essence\ndue to limitations imposed by the assumed parametric\nstructure. To comprehend the cosmic acceleration\nof the Universe, numerous investigations have been\nundertaken, employing viable parametrizations of the\nEquation of State (EoS) [3, 4].Inspired by the param-\neterizations of the Equation of State (EoS),researchersarXiv:2402.12324v2  [astro-ph.CO]  22 Feb 20242\nhave thoroughly investigated the parametrization of the\ndeceleration parameter in the available literature. This\nemphasis is rooted in the transition of the Universe’s\nevolution, progressing from an initial deceleration phase\nto a subsequent era of late-time acceleration. As a\nresult, the complete cosmological model must include a\ntransition from a deceleration stage to an acceleration\nstage of expansion to fully represent the whole evolution\nof the Cosmos.The deceleration parameter, defined as\nq=−a¨a\n˙a2, assumes a pivotal role in elucidating the cosmic\ndynamics, with a(t) representing the customary scale\nfactor.This parameter plays a crucial role in discerning\nwhether the Universe is presently experiencing accelera-\ntion ( q <0) or deceleration ( q >0). Recent theoretical\nmodels have been developed to explore the extensive\nevolutionary history of the Universe by parameterizing\nq(z) as a function of the scale factor ( a(t)), time ( t),\nor redshift ( z) [5–10]. Likewise, numerous alternative\nparametrization schemes for various cosmological pa-\nrameters—both physical and geometrical—have been\nthoroughly examined in the literature over the past\ndecades, acknowledging the endeavor to elucidate the\nobserved or theorized cosmic history [11–33].A signif-\nicant challenge in Einstein’s gravitational theory is\nthe occurrence of ultraviolet (UV) divergence, often\nreferred to as the UV completion problem. To address\nthis challenge, Stelle (1976) [34] proposed a solution by\nintroducing higher-order curvature invariants into the\naction of different gravitational theories. Nevertheless,\nthis strategy faced challenges due to the emergence\nof ghost degrees of freedom attributed to higher-order\nderivatives in these theories. Horava (2009) [35] effec-\ntively addressed this issue by introducing additional\nterms with higher-order spatial curvature components\nin the action, leading to what is now recognized as\ndeformed Horava-Lifshitz gravity. Under high-energy\nconditions, Horava-Lifshitz gravity is formulated by\nrelinquishing Lorentz symmetry through a Lifshitz-type\nrescaling process [36].\nxi→b xi, t→tibz(i= 1,2,3, ......, d ) (1)\nHere, the variable zdenotes the exponent associated\nwith dynamical scaling, and drepresents the spacetime\ndimension, where bis an arbitrary constant term.\nLorentz symmetry is disrupted when z̸= 1 and is only\nrestored when z= 1. Modified Horava-Lifshitz gravity\nhas been thoroughly investigated in scientific literature\n[37–43]. Wei [44] explored the Friedmann equation\nwithin the deformed Horava-Lifshitz gravity framework\nfor the FLRW Universe.Jawad et al. [45] conducted a\ncomprehensive study, introducing a novel holographic\ndark energy (HDE) within the modified f(R) Horava-\nLifshitz gravity framework. Chen et al. [46] explored\nthe deformed Horava-Lifshitz gravity model to investi-\ngate quasi-normal modes of black holes. Additionally,\nSheykhi et al. [47] delved into ghost dark energy models\nwithin the context of deformed Horava-Lifshitz gravity.\nThe Horava-Lifshitz gravity model has been extensivelyinvestigated by various authors[48–54].\nHorava-Lifshitz gravity aims to solve the quantum\ngravity problem and overcome the lack of renormaliz-\nability in standard general relativity, ultimately aiming\nto provide a UV-complete theory. This framework\nhas generated significant attention but has also faced\nchallenges and criticisms, particularly about its viability\nas a quantum theory and the presence of ghost insta-\nbilities. Researchers are still studying and improving\nthe Horava-Lifshitz gravity framework, particularly its\neffects on cosmology, black hole physics, and its larger\nimpact on our awareness of basic interactions in the\nuniverse. It is crucial to mention that the scientific\ncommunity continuously argues and discusses the feasi-\nbility and consequences of Horava-Lifshitz gravity. The\nscientific understanding of this theoretical framework\nmay change as further research and experimental data\nbecome available. Jawad et al. [55] investigated the\ndynamical stability and cosmographic characteristics of\nsome cosmic models under deformed Horava-Lifshitz\ngravity, matching their results with observational data.\nThey studied cosmographic parameters and thermody-\nnamics with regard to Einstein’s gravity and deformed\nHorava-Lifshitz gravity associated with Kaniadakis\nHDE [56]. Additionally, they explored the cosmological\nconsequences of Sharma-Mittal holographic dark energy\n(SMHDE), analysing cosmological parameters and\nthermodynamics [57].\nA comprehensive understanding of the universe’s\nevolution hinges upon scrutinizing an array of cosmic\nphenomena, including cosmic microwave background\n(CMB) radiation, large-scale structure (LSS), baryon\nacoustic oscillations (BAO), weak lensing, and a few\nmore. Recent observational data highlight that the\nuniverse has traversed two distinct phases of cosmic\nacceleration. The first phase, known as inflation,\ninitiated the accelerated expansion of the universe.\nThe second phase, commencing roughly 6 billion years\npost-Big Bang, ushered in an extended period of cos-\nmic acceleration that persists to this day. Across its\nevolutionary trajectory, the universe has transitioned\nthrough varied epochs: from a radiation-dominated era,\ncharacterized by the motion of photons, to a matter\ndominated phase where free electrons combined with\nnuclei to form particles, culminating in the onset of the\ndark energy (DE) phase, intimately tied to accelerated\nexpansion. To probe the constraints on the universe’s\nexpansion rate, a crucial avenue lies in examining the\nHubble parameter, denoted as H(z), as a function of\nredshift.\nMotivated by the preceding discourse on late-time\nacceleration, this study explores deep into the realm\nof Horava-Lifshitz gravity, specifically targeting certain\nparametrizations of the dark energy equation of state\n(EoS) to shed light on this cosmic mystery. Our primary3\nobjective is to meticulously evaluate the compatibility\nof these parametrizations with observational data, thus\nderiving rigorous constraints on crucial cosmic param-\neters. Harnessing recent measurements of the Hubble\nparameter, H(z), extracted from cosmic chronometers,\nalongside a diverse array of datasets spanning Type Ia\nSupernovae, Gamma-Ray Bursts (GRB), Quasars, and\nuncorrelated Baryon Acoustic Oscillations (BAO), our\nanalysis offers a comprehensive survey of the cosmic\nterrain. Additionally, we aim to probe the ongoing\ncosmological tensions such as H0and the sensitivity of\nrd. The paper is structured as follows: In Section II, we\npresent the basic equations of Horava Lifshitz Gravity.\nIn Section III, we describe the dark energy models\nusing Equation of State Parametrization. Moving on\nto Section IV, we focus on the methodology employed\nto constrain the crucial parameters of each dark energy\nmodel. Our study’s outcomes are detailed in Section\nV, where we present the results obtained. Finally, in\nSection VI, we conclude the paper, offering discussions\non the implications and significance of our findings.II. HORAVA-LIFSHITZ GRAVITY AND BASIC\nEQUATIONS\nWe assume the Arnowitt-Deser-Misner decomposition\n([58–60]), which is provided as in the metric form:\nds2=gij\u0000\ndxi+Nidt\u0001\u0000\ndxj+Njdt\u0001\n−N2dt2.(2)\nIn the above equation, Nis representing the lapse func-\ntion, Niis providing the shift vector, gijis denoting\nthe spatial metric. For the present analysis, the scaling\ntransformation in the framework of coordinates reads as\nt→l3tandxi→lxi. The action of the HL gravity has\ntwo major components, namely, the potential term and\nkinetic term. The HL gravity action within the scope of\npotential term and kinetic term is defined as:\nSg=Sv+Sk=Z\ndtd3x√gN(Lv+Lk),\nwhere, the kinetic term is defined as\nSk=Z\ndtd3x√gN\"\n2\u0000\nKijKij−λK2\u0001\nκ2#\n,\nwhere, The extrinsic curvature is provided as\nKij=˙gij−∆iNj−∆jNi\n2N.\nThe number of invariants, while working with the La-\ngrangian, Lv, can be reduced due to its symmetric prop-\nerty [61–63]. This symmetry is referred to as detailed\nbalance. BY using this detailed balance, the action can\nbe revised as\nSg=Z\ndtd3x√gN\"\n2\u0000\nKijKij−λK2\u0001\nκ2+κ2CijCij\n2ω4−κ2µϵijkRi,j∆jRl\nk\n2ω2√g\n+κ2µ2RijRij\n8−κ2µ2\n8(3λ−1)\u001a(1−4λ)R2\n4+ ΛR−3Λ2\u001b\u0015\n, (3)\nwhere Cij=ϵijk∆k\u0010\nRj\ni−R\n4δj\ni\u0011\n√g.\nis the known as the Cotton tensor. Here the covariant\nderivatives can be determined by the spatial metric\ngijϵijk, which is the antisymmetric unit tensor. Here,\nthe parameter λis the dimensionless constant and the\nother parameters κ,ωandµare also the constants.\nHorava obtained a gravitational action by consid-ering the temporal dependency of the lapse func-\ntion N(t). For FRW metric with the assumptions\nN= 1, gij=a2(t)γij, Ni= 0 and\nγijdxidxj=dr2\n1−kr2+r2dΩ2\n2,\nand by taking the variations of Nandgij, we can obtain\nthe two Field equations which are given below [64, 65]:4\nH2=κ2ρ\n6 (3λ−1)+κ2\n6 (3λ−1)\u00143κ2µ2k2\n8 (3λ−1)a4+3κ2µ2Λ2\n8 (3λ−1)\u0015\n−κ4µ2Λk\n8 (3λ−1)2a2, (4)\n˙H+3H2\n2=−κ2p\n4 (3λ−1)−κ2\n4 (3λ−1)\u00143κ2µ2k2\n8 (3λ−1)a4+3κ2µ2Λ2\n8 (3λ−1)\u0015\n−κ4µ2Λk\n8 (3λ−1)2a2. (5)\nwhere k= 0,±1 represent the flat, closed and open\nuniverse respectively. In the field equations mentioned\nabove, the term proportional to1\na4is treated as the ”Dark\nradiation” term [59, 60] and Λ denotes the cosmological\nconstant. Also, H=˙a\narepresents the Hubble parame-\nter. Now we assume that the universe is composed of the\ndark matter (DM) and dark energy (DE). So the energy\ndensity ρand pressure pof the fluid can be written as\nρ=ρm+ρdandp=pm+pd, respectively. Assume\nthat the DM and DE are separately conserved. So their\nconservation equations are respectively given by\n˙ρm+ 3H(ρm+pm) = 0 , (6)\nand\n˙ρd+ 3H(ρd+pd) = 0 . (7)Since the dark matter has negligible pressure (i.e., pm∼\n0), so the equation (6) yields ρm=ρm0a−3. Let us de-\nnote the equation of state (EoS) parameter by w(z) =\np/ρ, so from the equation (7), we obtain the DE density\nρd=ρd0e3R1+w(z)\n1+zdz. Here, ρm0andρd0represent the\npresent values of the DM density and DE density respec-\ntively. We can set, Gc=κ2\n16π(3λ−1)with the condition\nκ4µ2Λ\n8(3λ−1)= 1, the aforementioned field equations may be\nderived from the detailed balance in the following forms:\nH2=8πGc\n3(ρm+ρd) +\u0012k2\n2Λa4+Λ\n2\u0013\n−k\na2,(8)\n˙H+3\n2H2=−4πGcpd−\u0012k2\n4Λa4−3Λ\n4\u0013\n−k\n2a2.(9)\nUsing the dimensionless parameters Ω i0≡8πGc\n3H2\n0ρi0,\nΩk0=−K\nH2\n0, ΩΛ0=Λ\n2H2\n0, we obtain\nH2(z) =H2\n0\u0014\nΩm0(1 +z)3+ Ωk0(1 +z)2+ Ω Λ0+Ω2\nk0(1 +z)4\n4ΩΛ0+ Ωd0e3R1+w(z)\n1+zdz\u0015\n, (10)\nwith\nΩm0+ Ωd0+ Ωk0+ Ω Λ0+Ω2\nk0\n4ΩΛ0= 1. (11)\nIII. MODELS UNDER CONSIDERATION WITH\nDARK ENERGY EQUATION OF STATE\nPARAMETRIZATIONS\nHere, we explore a few interesting parametriza-\ntions of dark energy equation of state consisting\nof three model parameters family of parametrization\nas follows: Bessett-Brownstone-Cardoso-Cortes-Fantaye-\nHlozek-Kotze-Okouma (BBCCFHKO) parametrization\n[66], Seljak parametrization [67], Alam-Sahni-Saini-\nStarobinski (ASSS) parametrization [68, 69], PADE-I\nparametrization [70–72], PADE-I parametrization [70–\n72] and Barboza-Alcaniz-Zhu-Silva (BAZ) parametriza-\ntion [73]. We thoroughly investigate and analyze these\nmodels to understand their role in shaping the Universe’s\ndynamics, particularly concerning late-time cosmic accel-eration. The chosen parametrizations cover a spectrum\nof theoretical scenarios, allowing us to comprehensively\nstudy how different EoS formulations impact the over-\nall behavior of the gravitational theory under examina-\ntion. By considering these diverse models, we aim to\nunderstand the nuanced effects of various dark energy\nparametrizations on the large-scale structure of the uni-\nverse. This section serves as the theoretical foundation\nfor our subsequent analysis, where we assess these mod-\nels against observational constraints derived from Baryon\nAcoustic Oscillations (BAO) data. Our approach con-\ntributes to advancing our understanding of the relation-\nship between dark energy and gravity within the Horava-\nLifshitz framework and establishes the groundwork for\nevaluating the empirical viability of these models through\nrigorous comparison with observational data.5\nA. BBCCFHKO Model\nThe ”BBCCFHKO” (Bessett-Brownstone-Cardoso-\nCortes-Fantaye-Hlozek-Kotze-Okouma) parametrizationis given by the EoS [66]\nw(z) =w0+w1z+w2z2. (12)\nIn this parametrization, we get the solution as\nρd=ρd0(1 +z)3(1+w0−w1+w2)e3(w1−2w2)z+3\n2w2z(1+2z),(13)\nSo from equation (10), we obtain\nH2(z) = H2\n0\u0014\nΩm0(1 +z)3+ Ωk0(1 +z)2+ Ω Λ0+Ω2\nk0(1 +z)4\n4ΩΛ0\n+\u0012\n1−Ωm0−Ωk0−ΩΛ0−Ω2\nk0\n4ΩΛ0\u0013\n(1 +z)3(1+w0−w1+w2)e3(w1−2w2)z+3\n2w2z(1+2z)\u0015\n, (14)\nB. SELJAK Model\nThe another parametrization [67] is given by the EoS\nw(z) =w0+w1z\n1 +z+w2z2\n(1 +z)2. (15)The solution is\nρd=ρd0(1+z)3(1+w0+w1+w2)e−3z\n2(1+z)2[2w1(1+z)+w2(2+3z)].\n(16)\nSo from equation (10), we obtain\nH2(z) = H2\n0\u0014\nΩm0(1 +z)3+ Ωk0(1 +z)2+ Ω Λ0+Ω2\nk0(1 +z)4\n4ΩΛ0\n+\u0012\n1−Ωm0−Ωk0−ΩΛ0−Ω2\nk0\n4ΩΛ0\u0013\n(1 +z)3(1+w0+w1+w2)e−3z\n2(1+z)2[2w1(1+z)+w2(2+3z)]\u0015\n. (17)\nC. ASSS Model\nIn terms of EoS, the ”ASSS” (Alam-Sahni-Saini-\nStarobinski) parametrization [68, 69] is seen as\nw(z) =−1 +w1(z+ 1) + 2( z+ 1)2w2\n3 [w0+w1+w1z+w2(z+ 1)2].(18)In this particular instance, we are able to get the solution,\nρd=ρd0\u0012w0+w1(z+ 1) + w2(1 +z)2\nw0+w1+w2\u0013\n, (19)\nthen from equation (10), we obtain\nH2(z) = H2\n0\u0014\nΩm0(1 +z)3+ Ωk0(1 +z)2+ Ω Λ0+Ω2\nk0(1 +z)4\n4ΩΛ0\n+\u0012\n1−Ωm0−Ωk0−ΩΛ0−Ω2\nk0\n4ΩΛ0\u0013 \u0012w0+w1(1 +z) +w2(1 +z)2\nw0+w1+w2\u0013\u0015\n. (20)\nD. PADE-I Model\nThe parametrization ”PADE-I” must be assumed in a\nspecific form: [70–72]\nw(z) =w0z+w1(1 +z)\n1 +w2(1 +z). (21)\nThe solution is obtained in this situation,\nρd=ρd0(1 +z)3(w0+w2)\nw2[1 +w2(1 +z)]−3(w0+w1w2)\nw2(1+w2),(22)\nthen from equation (10), we obtain6\nH2(z) = H2\n0\u0014\nΩm0(1 +z)3+ Ωk0(1 +z)2+ Ω Λ0+Ω2\nk0(1 +z)4\n4ΩΛ0\n+\u0012\n1−Ωm0−Ωk0−ΩΛ0−Ω2\nk0\n4ΩΛ0\u0013\n(1 +z)3(w0+w2)\nw2[1 +w2(1 +z)]−3(w0+w1w2)\nw2(1+w2)\u0015\n. (23)\nE. PADE-II Model\nThe ”PADE-II” parametrization is assumed as [70–72]\nw(z) =w0+w1log(1 +z)\n1 +w2log(1 +z). (24)In this case, we get the solution\nρd=ρd0(1 +z)3(w1+w2)\nw2[1 +w2log(1 +z)]3(−w1+w0w2)\nw2\n2 ,(25)\nthen from equation (10), we obtain\nH2(z) = H2\n0\u0014\nΩm0(1 +z)3+ Ωk0(1 +z)2+ Ω Λ0+Ω2\nk0(1 +z)4\n4ΩΛ0\n+\u0012\n1−Ωm0−Ωk0−ΩΛ0−Ω2\nk0\n4ΩΛ0\u0013\n(1 +z)3(w1+w2)\nw2[1 +w2log(1 +z)]3(−w1+w0w2)\nw2\n2\u0015\n. (26)\nF. BAZS Model\nThe ”BAZS” (Barboza-Alcaniz-Zhu-Silva)\nparametrization [73] is given by the EoS\nw(z) =w0+w11\n(1 +z)β. (27)\nIn this case, the solution becomes\nρd=ρd0(1 +z)3(1+w0)e3\nβw1[1−(1+z)−β], (28)\nthen from equation (10), we obtain\nH2(z) = H2\n0\u0014\nΩm0(z+ 1)3+ Ωk0(1 +z)2+ Ω Λ0+Ω2\nk0(1 +z)4\n4ΩΛ0\n+\u0012\n1−Ωm0−Ωk0−ΩΛ0−Ω2\nk0\n4ΩΛ0\u0013\n(1 +z)3(1+w0)e3\nβw1[1−(1+z)−β]\u0015\n. (29)\nIV. METHODOLOGY\nIn our investigation, we carefully picked a specific set\nof recent Baryon Acoustic Oscillation (BAO) measure-\nments extracted from multiple galaxy surveys, with a\nprimary focus on data from the Sloan Digital Sky Survey\n(SDSS) [74–79]. To enrich our dataset, we augmented\nit with valuable contributions from other surveys such\nas the Dark Energy Survey (DES) [80], the Dark En-\nergy Camera Legacy Survey (DECaLS) [81], and 6dFGS\nBAO [82]. Recognizing the potential for interdependen-cies among our selected data points, we diligently ad-\ndressed this concern. We carefully curated a subset to\nmitigate highly correlated points, fully appreciating the\nimportance of managing possible correlations within our\nchosen dataset. To ensure a comprehensive assessment of\nsystematic errors, we employed mock datasets generated\nfrom N-body simulations to precisely calculate covari-\nance matrices. Obtaining accurate covariance matrices\namong measurements derived from diverse observational\nsurveys presented a significant challenge. To address a\nsignificant challenge, we utilized covariance analysis as7\noutlined in reference [83]. Specifically, we adopted the\napproach where the diagonal elements of the covariance\nmatrix, denoted as Cii, were set equal to the square of the\n1σerrors, σ2\ni. In order to introduce correlations among\nour selected subset of data, we augmented the covariance\nmatrix by incorporating non-diagonal elements while pre-\nserving its symmetry. This was achieved by randomly\npairing data points, resulting in a total of twelve pairs.\nFor each pair, the non-diagonal elements were determined\nusing the expression Cij= 0.5σiσj, where σiandσjrep-\nresent the 1 σerrors associated with data points iandj,\nrespectively. Through this methodology, we effectively\nsimulated correlations within 56% of the BAO dataset\nunder consideration. This approach enabled us to cap-\nture the interdependence among data points, providing\na more comprehensive representation of the underlying\nstructure within the dataset. By incorporating these cor-\nrelations, we aimed to enhance the accuracy and reliabil-\nity of our analyses, thereby improving our understanding\nof the phenomenon under investigation. Figs. 1 and\n2 display posterior distributions of some parameters for\neach cosmological model, with and without a randomly\ntested covariance matrix comprising twelve components,\nreveal notable constraints. Interestingly, the distribution\nassociated with the covariance matrix, ranging from null\nto twelve components, shows a negligible impact. It ap-\npears almost indistinguishable from the dataset lacking\ncorrelations. This suggests that the inclusion of a covari-\nance matrix with twelve components has minimal influ-\nence on the posterior distributions, indicating a limited\neffect on the overall analysis outcome. To constrain our\ncosmological model parameters, we expanded our dataset\nby including various additional measurements. Firstly,\nwe incorporated thirty independent Hubble parameter\nmeasurements derived from cosmic chronometers (CC)\ndetailed in [84–87]. Additionally, our study integrates\nrecent Pantheon Type Ia Supernovae samples [88], 24\nbinned quasar [89] distance modulus data, 162 Gamma-\nRay Bursts (GRBs) [90], and the latest Hubble constant\nmeasurement (R22) as an additional prior [91]. To an-\nalyze these datasets, we adopted a nested sampling ap-\nproach using the Polychord package [92], enabling thor-\nough exploration of parameter space for determining op-\ntimal fit values. Additionally, we utilized the GetDist\npackage [93] to present our findings clearly and informa-\ntively, enhancing the interpretation and visualization of\nthe analysis outcomes. This comprehensive methodology\nensures a robust understanding of cosmological param-\neters while providing insights into the universe’s funda-\nmental properties.\nV. RESULTS\nFigs 3, 4, 5, 6, 7, and 8 depict the 68% and 95% con-\nfidence levels for pivotal cosmological parameters across\nthe BBCCFHKO, SELJAK, ASSS, PADE-I, PADE-II,\nand BAZS models, respectively. The best-fit values foressential parameters of each dark energy model are high-\nlighted in the Table I. In these models, incorporating the\nR22 prior into the Joint dataset results in a deviation of\nthe optimal fitting value for H0from the findings in [94],\nbut it shows a closer alignment with the SNIe sample in\n[91]. Conversely, without the R22 priors and using the\nJoint dataset alone, the estimated value of H0aligns\nmore closely with [94]. These findings indicate that\ninclusion of R22 prior diverges from [94], aligns with [91],\nshowing prior choice’s impact on cosmological parameter\nestimation. The determined values for the matter\ndensity parameter (Ω m0) seem lower compared to those\ndocumented in [94]. Additionally, in the BBCCFHKO,\nSELJAK, ASSS, and PADE-II models, the dark energy\ndensity parameter (Ω d0) appears higher than the values\ncompared in [94]. However, for the PADE-I and BAZS\nmodels, Ω d0is lower than the value documented in [94].\nHowever, this observation aligns with findings from\nalternative studies [95, 96]. Turning our attention to\nthe Baryon Acoustic Oscillations (BAO) scale, which\nis a fundamental scale in cosmology derived from the\ncosmic microwave background. This scale originates\nfrom a significant event known as the drag epoch ( zd),\nmarking a pivotal moment when baryons and photons\ntransitioned into distinct entities. The BAO scale ( rd)\nis essentially a measure of the cosmic sound horizon at\nzd, which reflects the separation distance between these\nbaryons and photons. To calculate rd, we consider the\nintegral of the ratio between the speed of sound ( cs) and\nthe Hubble parameter ( H) across a range of redshifts,\nstarting from zdand extending indefinitely. The speed of\nsound, cs, is determined by the fluctuations in pressure\n(δpγ) and density ( δρBand δργ) within the photon\nand baryon components. Mathematically, this simplifies\nto1√\n3(1+R), where Rrepresents the ratio of baryon\ndensity fluctuations to photon density fluctuations\n(R≡δρB\nδργ=3ρB\n4ργ). Observational data, as documented\nin [94], has provided us with the precise redshift of the\ndrag epoch, zd= 1059 .94±0.30. In the context of a\nΛCDM model, the BAO scale rdhas been estimated to\nbe 147 .09±0.26 megaparsecs (Mpc) based on measure-\nments by [94]. Fig. 9 show the posterior distribution\nfor the rd−H0contour plane of each Models. Figs 9a,\n9b, 9c, 9d, 9e, and 9f depict the posterior distributions\nfor the rd−H0parameters across different cosmological\nmodels, namely BBCCFHKO, Seljak, ASSS, PADE-I,\nPADE-II, and BAZS. In the BBCCFHKO model,\nthe BAO scale derived solely from BAO datasets is\n148.814±10.558 Mpc. Upon including the R22 prior\nexclusively into the BAO dataset, the sound horizon at\nthe drag epoch becomes 138 .907±2.129 Mpc. For the\nJoint dataset, the BAO scale is 145 .843±2.47 Mpc.\nIntegrating the R22 prior into the full dataset yields\nrd= 142 .742±1.658 Mpc. Moving to the Seljak Model,\nthe estimated BAO scale from BAO datasets stands at\n150.606±9.903 Mpc. When incorporating the R22 prior\nexclusively into the BAO dataset, the sound horizon at8\n0.20.30.4\nm0\n0.60.70.80.91.0d0\n0.60.70.80.91.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22\n(a) BBCCFHKO Model\n130140150160170180\nrd(Mpc)0.60.70.80.91.0d0\n0.60.70.80.91.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (b) BBCCFHKO Model\n0.20.30.4\nm0\n607080H0\n607080\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (c) BBCCFHKO Model\n130140150160170180\nrd(Mpc)607080H0\n607080\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22\n(d) BBCCFHKO Model\n0.200.250.30\nm0\n0.60.70.80.91.0d0\n0.60.70.80.91.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (e) SELJAK Model\n130140150160170\nrd(Mpc)0.60.70.80.91.0d0\n0.60.70.80.91.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (f) SELJAK Model\n0.200.250.30\nm0\n6065707580H0\n6065707580\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22\n(g) SELJAK Model\n130140150160170\nrd(Mpc)6065707580H0\n6065707580\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (h) SELJAK Model\n0.200.250.30\nm0\n0.60.81.0d0\n0.60.81.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (i) ASSS Model\n130140150160170\nrd(Mpc)0.60.81.0d0\n0.60.81.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22\n(j) ASSS Model\n0.200.250.30\nm0\n6065707580H0\n6065707580\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (k) ASSS Model\n130140150160170\nrd(Mpc)6065707580H0\n6065707580\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (l) ASSS Model\nFIG. 1: Posterior distributions for some parameters of each BBCCFHKO, SELJAK and ASSS cosmological model,\nwith and without a random covariance matrix of fourteen components, show negligible differences.9\n0.200.250.300.35\nm0\n0.50.60.70.80.91.0d0\n0.50.60.70.80.91.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22\n(a) PADE-I\n130140150160170\nrd(Mpc)0.50.60.70.80.91.0d0\n0.50.60.70.80.91.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (b) PADE-I\n0.200.250.300.35\nm0\n6065707580H0\n6065707580\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (c) PADE-I\n130140150160170\nrd(Mpc)6065707580H0\n6065707580\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22\n(d) PADE-I\n0.200.250.30\nm0\n0.60.81.0d0\n0.60.81.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (e) PADE-II\n130140150160170\nrd(Mpc)0.60.81.0d0\n0.60.81.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (f) PADE-II\n0.200.250.30\nm0\n6065707580H0\n6065707580\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22\n(g) PADE-II\n130140150160170\nrd(Mpc)6065707580H0\n6065707580\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (h) PADE-II\n0.200.250.30\nm0\n0.00.20.40.60.81.0d0\n0.00.20.40.60.81.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (i) BAZS Model\n130140150160170180\nrd(Mpc)0.00.20.40.60.81.0d0\n0.00.20.40.60.81.0\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22\n(j) BAZS Model\n0.200.250.30\nm0\n607080H0\n607080\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (k) BAZS Model\n130140150160170180\nrd(Mpc)607080H0\n607080\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 (l) BAZS Model\nFIG. 2: Posterior distributions for some parameters of each PADE-I, PADE-II and BAZS cosmological model, with\nand without a random covariance matrix of fourteen components, show negligible differences.10\nMCMC Results\nModel Parameter Priors BAO BAO + R22 Joint Joint + R22\nH0 [50,100] 70.256500±9.206346\n±5.502574 73.798696±2.543240\n±1.315504 69.837362±2.478047\n±1.204873 71.508817±1.614354\n±0.882681\nΛCDM Model Ωm0 [0.,1.] 0.270161±0.033280\n±0.013827 0.268255±0.043787\n±0.016790 0.274526±0.021367\n±0.009858 0.270609±0.022243\n±0.009678\nΩd0 [0.,1.] 0.725842±0.019262\n±0.011842 0.727203±0.024379\n±0.013626 0.721607±0.015585\n±0.007421 0.724854±0.016354\n±0.007274\nH0 [50,100] 68.426943±8.262138\n±4.922346 72.832662±1.852478\n±0.978457 69.966349±2.453491\n±1.403864 71.566891±1.341813\n±0.762010\nΩm0 [0.2,0.4] 0.289440±0.051495\n±0.031695 0.294937±0.058819\n±0.035537 0.278219±0.042599\n±0.028154 0.276926±0.043337\n±0.026488\nΩd0 [0.,1.] 0.799161±0.156086\n±0.108963 0.801391±0.171115\n±0.108760 0.861751±0.170117\n±0.103561 0.879090±0.150723\n±0.101620\nΩko [-0.1,0.1] 0.031966±0.097225\n±0.052741 0.022485±0.109242\n±0.059968 0.032869±0.116070\n±0.056449 0.032075±0.110630\n±0.049019\nBBCCFHKO Model ω0 [-1.,0.] −0.624406±0.366695\n±0.281325 −0.639231±0.354148\n±0.275344 −0.642233±0.321776\n±0.218169 −0.635815±0.314020\n±0.177811\nω1 [-1.,0.] −0.568919±0.420143\n±0.338673 −0.541905±0.441567\n±0.316165 −0.708950±0.279111\n±0.232536 −0.696101±0.297620\n±0.241898\nω2 [0.,0.5] 0.231295±0.213285\n±0.144360 0.254938±0.237581\n±0.133146 0.141631±0.135189\n±0.109658 0.138169±0.132888\n±0.113812\nH0 [50,100] 67.570047±8.228193\n±4.426642 72.781051±1.577453\n±0.899423 69.969977±2.586602\n±1.195973 71.620076±1.838642\n±0.715705\nΩm0 [0.2,0.35] 0.270552±0.034910\n±0.020419 0.269227±0.040219\n±0.019539 0.272473±0.033661\n±0.019950 0.270349±0.032824\n±0.018722\nΩd0 [0.5,1.] 0.744259±0.139284\n±0.118148 0.731970±0.124553\n±0.093721 0.835570±0.175374\n±0.114432 0.850709±0.165740\n±0.120153\nΩko [-0.1,0.1] 0.024766±0.105995\n±0.054668 0.024753±0.108174\n±0.048594 0.030884±0.094382\n±0.053635 0.036509±0.094101\n±0.051719\nSeljak Model ω0 [-1.,0.] −0.619122±0.374815\n±0.292328 −0.522760±0.458760\n±0.366946 −0.720375±0.265354\n±0.177656 −0.683386±0.278581\n±0.181528\nω1 [-1.,0.] −0.526962±0.458712\n±0.349414 −0.507130±0.466838\n±0.337217 −0.628424±0.358016\n±0.273138 −0.656706±0.336872\n±0.267715\nω2 [0,0.6] 0.247283±0.222169\n±0.161265 0.259705±0.240217\n±0.168164 0.238212±0.229324\n±0.156251 0.250947±0.231240\n±0.165445\nH0 [50,100] 67.680446±7.538381\n±4.702061 72.836766±1.884950\n±0.946895 69.793952±2.369460\n±1.063678 71.662231±1.286199\n±0.716641\nΩm0 [0.2,0.35] 0.260808±0.042585\n±0.019669 0.259219±0.034831\n±0.018256 0.270434±0.043839\n±0.022909 0.273596±0.038656\n±0.021547\nΩd0 [0.5,1.] 0.799803±0.186228\n±0.127045 0.807418±0.195904\n±0.138517 0.837586±0.217965\n±0.122737 0.821951±0.206378\n±0.123496\nΩko [-0.1,0.1] 0.041301±0.101066\n±0.051750 0.042300±0.094267\n±0.048454 0.016689±0.098524\n±0.046355 0.005594±0.084284\n±0.048281\nASSS Model ω0 [-1.,0.] −0.560449±0.408024\n±0.313702 −0.558606±0.414154\n±0.320142 −0.516510±0.466596\n±0.333040 −0.535597±0.442261\n±0.304625\nω1 [-1.,0.] −0.492119±0.471376\n±0.349576 −0.531309±0.450181\n±0.330018 −0.581720±0.410717\n±0.316176 −0.577589±0.406522\n±0.267491\nω2 [0.,0.6] 0.236802±0.228116\n±0.167579 0.247260±0.240740\n±0.162766 0.231355±0.217095\n±0.147438 0.260183±0.240016\n±0.147068\nH0 [50,100] 66.025899±9.075107\n±5.936379 72.941499±2.039997\n±1.070336 70.629511±2.704907\n±1.586378 72.418384±1.656293\n±0.933325\nΩm0 [0.2,0.4] 0.283701±0.070063\n±0.048922 0.255756±0.051195\n±0.033569 0.259798±0.033025\n±0.017857 0.246491±0.035532\n±0.019549\nΩd0 [0.4,1.] 0.734041±0.201313\n±0.142372 0.651614±0.133251\n±0.089020 0.642887±0.063867\n±0.027701 0.626553±0.066973\n±0.030257\nΩko [-0.1,0.1] 0.040170±0.085523\n±0.042545 0.031959±0.101338\n±0.048253 0.050643±0.069751\n±0.035440 0.054660±0.078051\n±0.038649\nPADE-I Model ω0 [-1.,0.] −0.494820±0.481410\n±0.363307 −0.509057±0.459600\n±0.350669 −0.325604±0.593855\n±0.221918 −0.371933±0.531032\n±0.314965\nω1 [-1.,0.] −0.471795±0.505651\n±0.370141 −0.514447±0.459083\n±0.334769 −0.501879±0.482091\n±0.351944 −0.479321±0.484860\n±0.321307\nω2 [0,0.03] 0.008268±0.007879\n±0.005663 0.009277±0.008877\n±0.006652 0.009696±0.008325\n±0.006643 0.004194±0.003486\n±0.002624\nH0 [50,100] 66.758695±7.419786\n±4.349298 72.861725±1.772759\n±0.920733 69.917720±2.364213\n±1.211341 71.559541±1.508022\n±0.718749\nΩm0 [0.2,0.35] 0.260914±0.051509\n±0.025055 0.264088±0.046485\n±0.022752 0.274505±0.039172\n±0.021232 0.274154±0.035940\n±0.019187\nΩd0 [0.,1.] 0.646429±0.084392\n±0.064433 0.673373±0.113472\n±0.077355 0.796735±0.186163\n±0.099636 0.809347±0.235715\n±0.110739\nΩko [-0.1,0.1] 0.006890±0.100043\n±0.070703 0.016671±0.097239\n±0.057671 0.029121±0.112661\n±0.052494 0.021928±0.097401\n±0.056613\nPADE-II Model ω0 [-1.,0.] −0.374352±0.579174\n±0.269573 −0.513235±0.481246\n±0.350679 −0.649674±0.321647\n±0.234693 −0.712952±0.265427\n±0.187883\nω1 [-1.,0.] −0.457804±0.493061\n±0.293144 −0.472406±0.471996\n±0.341449 −0.578139±0.400614\n±0.320368 −0.570732±0.418398\n±0.329160\nω2 [0.,1.] 0.589030±0.547250\n±0.315785 0.483573±0.453860\n±0.329326 0.446287±0.417365\n±0.326408 0.429395±0.409674\n±0.334776\nH0 [50,100] 66.537252±7.130054\n±4.693854 72.598747±1.660270\n±0.865182 69.874826±2.230762\n±1.292056 71.627710±1.238652\n±0.675046\nΩm0 [0.2,0.35] 0.259047±0.048039\n±0.021051 0.252782±0.047063\n±0.026481 0.271466±0.036481\n±0.018520 0.261734±0.043431\n±0.021092\nΩd0 [0.,1.] 0.528183±0.477438\n±0.259582 0.454640±0.394814\n±0.235195 0.508623±0.423490\n±0.249546 0.426611±0.383105\n±0.262024\nΩko [-0.1,0.1] 0.027318±0.104775\n±0.052819 0.012952±0.104601\n±0.057618 0.024104±0.095459\n±0.047064 0.024989±0.085770\n±0.042263\nBAZS Model ω0 [-1.,0.] −0.465337±0.524355\n±0.280183 −0.387889±0.548894\n±0.273483 −0.492112±0.463180\n±0.301665 −0.485236±0.479421\n±0.298413\nω1 [-1.,.0.] −0.490789±0.481041\n±0.338560 −0.444481±0.492149\n±0.302268 −0.621446±0.358811\n±0.283349 −0.642880±0.340487\n±0.256972\nβ [0.,1.] 0.544368±0.506087\n±0.323472 0.575541±0.511054\n±0.309603 0.511286±0.484442\n±0.325781 0.647869±0.576987\n±0.308959\nTABLE I: Summary of the MCMC Results.11\n607080\nH00.00.20.42\n1.0\n0.5\n0.01\n1.0\n0.8\n0.6\n0.4\n0.2\n0.00\n0.1\n0.00.1ko\n0.60.70.80.91.0d0\n0.200.250.300.350.40m0\n0.20.30.4\nm0\n0.60.70.80.91.0\nd0\n0.1\n0.00.1\nko\n1.0\n0.5\n0.0\n0\n1.0\n0.5\n0.0\n1\n0.00.20.4\n2\nBAO\nBAO + R22\nJOINT\nJOINT + R22\nFIG. 3: The figure illustrates the posterior distribution of various observational data measurements using the\nBBCCFHKO Model, highlighting the 1 σand 2 σregions.\nthe drag epoch is 139 .066±1.943 Mpc. For the Joint\ndataset, rdis 145 .677±2.516 Mpc. Including the R22\nprior in the complete dataset gives rd= 142 .459±1.694\nMpc. In the ASSS Model, the BAO scale estimated from\nBAO datasets is 150 .441±10.485 Mpc. Incorporating\nthe R22 prior exclusively into the BAO dataset leads to\na sound horizon at the drag epoch of 139 .187±1.869\nMpc. For the Joint dataset, rdis 145 .782±2.412Mpc. Including the R22 prior in the full dataset\nresults in rd= 142 .128±1.359 Mpc. For the PADE-I\nModel, the BAO scale estimated from BAO datasets\nis 151 .466±10.145 Mpc. Including the R22 prior\nexclusively in the BAO dataset alters the sound horizon\nat the drag epoch to 143 .601±6.898 Mpc. For the Joint\ndataset, rdis 146 .039±2.129 Mpc. Including the R22\nprior in the full dataset yields rd= 144 .687±2.06412\n60657075\nH00.00.20.40.62\n1.0\n0.5\n0.01\n1.0\n0.5\n0.00\n0.1\n0.00.1ko\n0.60.70.80.91.0d0\n0.200.250.300.35m0\n0.220.260.300.34\nm0\n0.60.70.80.91.0\nd0\n0.1\n0.00.1\nko\n1.0\n0.5\n0.0\n0\n1.0\n0.5\n0.0\n1\n0.00.20.40.6\n2\nBAO\nBAO + R22\nJOINT\nJOINT + R22\nFIG. 4: The figure illustrates the posterior distribution of various observational data measurements using the\nSELJAK Model, highlighting the 1 σand 2 σregions.\nMpc. In the case of the PADE-II Model, the BAO scale\nestimated from BAO datasets is 151 .172±9.731 Mpc.\nIntroducing the R22 prior exclusively into the BAO\ndataset results in a sound horizon at the drag epoch\nof 138 .548±2.268 Mpc. For the Joint dataset, rdis\n145.671±2.267 Mpc. Including the R22 prior in the full\ndataset produces rd= 142 .563±1.590 Mpc. Finally, in\nthe BAZS Model, the BAO scale estimated from BAOdatasets is 151 .879±11.025 Mpc. Incorporating the\nR22 prior exclusively into the BAO dataset leads to\na sound horizon at the drag epoch of 138 .938±2.265\nMpc. For the Joint dataset, rdis 145 .895±2.398 Mpc.\nIncluding the R22 prior in the full dataset results in\nrd= 142 .576±1.535 Mpc. Table II presents the best-fit\nvalues of rdandrd/rfidfor the ΛCDM, BBCCFHKO,\nSELJAK Model, ASSS, PADE-I, PADE-II, and BAZS13\n607080\nH00.00.20.40.62\n1.0\n0.5\n0.01\n1.0\n0.5\n0.00\n0.10\n0.05\n0.000.050.10ko\n0.60.81.0d0\n0.200.250.30m0\n0.210.240.270.300.33\nm0\n0.60.81.0\nd0\n0.1\n0.00.1\nko\n1.0\n0.5\n0.0\n0\n1.0\n0.5\n0.0\n1\n0.00.20.4\n2\nBAO\nBAO + R22\nJOINT\nJOINT + R22\nFIG. 5: The figure illustrates the posterior distribution of various observational data measurements using the ASSS\nModel, highlighting the 1 σand 2 σregions.\nmodels. The findings indicate that when incorporating\nthe joint dataset, the obtained value of rdin each dark\nenergy model closely agrees with the values obtained\nin [94]. However, when combining the R22 priors with\nthe joint dataset, the obtained value of rdaligns closely\nwith [97]. [98] introduces an insightful perspective by\nemploying Binning and Gaussian methods to amalga-\nmate measurements of 2D BAO and SNIa data. It’simportant to recognize that the outcomes we derive\nare closely tied to the initial range of priors we make\nabout the BAO scale ( rd) and the Hubble constant\n(H0). These priors significantly impact the values we\nestimate, particularly when considering the rd−H0\ncontour plot. An interesting observation emerges when\nwe omit the R22 prior from our analysis: in such cases,\nthe results for H0and rdtend to better align with14\n607080\nH00.000.010.022\n1.0\n0.5\n0.01\n1.0\n0.5\n0.00\n0.10\n0.05\n0.000.050.10ko\n0.60.81.0d0\n0.200.250.300.350.40m0\n0.20.30.4\nm0\n0.60.81.0\nd0\n0.1\n0.00.1\nko\n1.0\n0.5\n0.0\n0\n1.0\n0.5\n0.0\n1\n0.000.010.02\n2\nBAO\nBAO + R22\nJOINT\nJOINT + R22\nFIG. 6: The figure illustrates the posterior distribution of various observational data measurements using the\nPADE-I Model, highlighting the 1 σand 2 σregions.\nfindings from the Planck and SDSS experiments. When\nassessing various cosmological models, we employ both\nthe Akaike Information Criterion (AIC) [95] and the\nBayesian Information Criterion (BIC) for evaluation\npurposes. The AIC and BIC are statistical measures\nused to assess the goodness of fit of a model to a given\ndataset while penalizing for the number of parameters\nin the model. The AIC is calculated using the formula:AIC = −2 ln (Lmax) + 2 k+2k(2k+1)\nNtot−k−1. Here, Lmax\nrepresents the maximum likelihood of the data, Ntotis\nthe total number of data points, and kis the number\nof parameters in the model. When the total number\nof data points ( Ntot) is large, the expression simplifies\nto AIC ≃ −2 ln (Lmax) + 2 k, which is the conventional\nform of the AIC criterion. This criterion provides a\ntrade-off between the goodness of fit and the complexity15\n607080\nH00.00.51.02\n1.0\n0.5\n0.01\n1.0\n0.5\n0.00\n0.1\n0.00.1ko\n0.50.60.70.80.91.0d0\n0.200.250.30m0\n0.200.250.30\nm0\n0.50.60.70.80.91.0\nd0\n0.1\n0.00.1\nko\n1.0\n0.5\n0.0\n0\n1.0\n0.5\n0.0\n1\n0.00.51.0\n2\nBAO\nBAO + R22\nJOINT\nJOINT + R22\nFIG. 7: The figure illustrates the posterior distribution of various observational data measurements using the\nPADE-II Model, highlighting the 1 σand 2 σregions.\nof the model. On the other hand, the BIC is defined as:\nBIC = −2 ln (Lmax) +klnNtot. The BIC also evaluates\nthe goodness of fit while penalizing for the number\nof parameters, but it imposes a stronger penalty for\nadditional parameters compared to the AIC, particularly\nfor smaller sample sizes. Applying these criteria, we\ncompute the AIC and BIC for the standard ΛCDM,\nBBCCFHKO, Model 2, ASSS, PADE-I, PADE-II,and BAZS models. The obtained values for ΛCDM,\nBBCCFHKO, Model 2, ASSS, PADE-I, PADE-II, and\nBAZS models are, respectively, AIC = [277.38, 283.43,\n283.89, 284.51, 279.92, 284.33, 284.11] and BIC =\n[277.59, 280.61, 285.01, 284.79, 280.61, 285.01, 284.79].\nWhile the ΛCDM model appears to fit the data best\naccording to the lowest AIC, our collective analysis\nusing both AIC and BIC indicates that all tested models16\n60657075\nH00.00.51.0\n1.0\n0.5\n0.01\n1.0\n0.5\n0.00\n0.1\n0.00.1ko\n0.00.20.40.60.81.0d0\n0.200.220.240.260.280.300.32m0\n0.200.250.30\nm0\n0.00.51.0\nd0\n0.1\n0.00.1\nko\n1.0\n0.5\n0.0\n0\n1.0\n0.5\n0.0\n1\n0.00.51.0\nBAO\nBAO + R22\nJOINT\nJOINT + R22\nFIG. 8: The figure illustrates the posterior distribution of various observational data measurements using the BAZS\nModel, highlighting the 1 σand 2 σregions.\nhave some level of support from the existing data.\nThis suggests that none of the models can be outright\ndismissed based solely on the available evidence. When\ncomparing several alternative models - namely BBC-\nCFHKO, Model 2, ASSS, PADE-I, PADE-II, and BAZS\n- with the standard ΛCDM model, it’s important to\nnote that both proposed extensions encompass ΛCDM\nas a subset, differing only by 4 degrees of freedom.This aspect allows for the application of conventional\nstatistical tests. Our evaluation primarily employs the\nreduced chi-square statistic, denoted as χ2\nred=χ2/Dof,\nwhere ”Dof” represents the degrees of freedom of the\nmodel, and χ2signifies the weighted sum of squared\ndeviations from the expected values. For an equal\nnumber of iterations or observations across all models\nconsidered, the reduced chi-square statistic typically17\n60 70 80\nH0130140150160170180rd(Mpc)\nBAO\nBAO + R22\nJOINT + BAO\nJOINT + R22\n(a) BBCCFHKO Model\n60 70 80\nH0130140150160170180rd(Mpc)\nBAO\nBAO + R22\nJOINT + BAO\nJOINT + R22 (b) SELJAK Model\n60 70 80\nH0130140150160170180rd(Mpc)\nBAO\nBAO + R22\nJOINT + BAO\nJOINT + R22\n(c) ASSS Model\n60 70 80\nH0120130140150160170180rd(Mpc)\nBAO\nBAO + R22\nJOINT + BAO\nJOINT + R22 (d) PADE-I Model\n60 70 80\nH0130140150160170180rd(Mpc)\nBAO\nBAO + R22\nJOINT + BAO\nJOINT + R22\n(e) PADE-II Model\n55 60 65 70 75\nH0130140150160170180rd(Mpc)\nBAO\nBAO + R22\nJOINT + BAO\nJOINT + R22 (f) BAZS Model\nFIG. 9: The figure displays posterior distributions in rd−H0contour plane for diverse models, indicating 1 σand 2 σ\nconfidence.18\nrdandrd/rfidResults\nModel Parameter Priors BAO BAO + R22 Joint Joint + R22\nΛCDM Model rd(Mpc) [100.,200.] 145.807497±13.994434\n±9.060325 138.345762±4.939114\n±2.451777 145.811932±4.545176\n±2.347736 142.591327±3.951717\n±1.850175\nrd/rfid [0.9,1.1] 0.970661±0.068973\n±0.055989 0.930500±0.028238\n±0.021272 0.974064±0.057975\n±0.033510 0.947662±0.043021\n±0.028335\nBBCCFHKO Model rd(Mpc) [100.,200.] 148.814998±16.582710\n±10.558702 138.907129±4.009763\n±2.129761 145.843487±4.492890\n±2.470309 142.742429±3.385970\n±1.658470\nrd/rfid [0.9,1.1] 0.989411±0.087169\n±0.066947 0.932168±0.030151\n±0.023582 0.969308±0.053797\n±0.033283 0.952184±0.044764\n±0.023003\nSELJAK Model rd(Mpc) [100.,200.] 150.606839±16.032449\n±9.803797 139.066177±4.439605\n±1.943819 145.677334±5.156838\n±2.516764 142.459979±3.058618\n±1.694561\nrd/rfid [0.9,1.1] 0.998524±0.090764\n±0.069893 0.935228±0.034343\n±0.025663 0.969599±0.052968\n±0.033049 0.949586±0.040924\n±0.024000\nASSS Model rd(Mpc) [100.,200.] 150.411981±15.324384\n±10.485714 139.187248±3.501246\n±1.869135 145.782783±4.524498\n±2.412215 142.128325±3.010876\n±1.359288\nrd/rfid [0.9,1.1] 0.998603±0.092230\n±0.069854 0.932016±0.030153\n±0.021813 0.973786±0.053130\n±0.031887 0.948004±0.040423\n±0.025361\nPADE-I Model rd(Mpc) [100.,200.] 151.466362±16.674149\n±10.145834 143.601827±11.117678\n±6.898812 146.039121±4.321081\n±2.129653 144.687932±3.219709\n±2.064777\nrat [0.9,1.1] 1.004339±0.098625\n±0.064118 0.959966±0.057692\n±0.040284 0.969620±0.051280\n±0.027867 0.964376±0.045962\n±0.029856\nPADE-II Model rd(Mpc) [100.,200.] 151.172327±16.517274\n±9.731894 138.548556±4.577948\n±2.268212 145.671084±4.673106\n±2.267762 142.563665±3.198850\n±1.590137\nrd/rfid [0.9,1.1] 1.000740±0.092296\n±0.071722 0.931616±0.030019\n±0.021949 0.969216±0.051181\n±0.029315 0.953345±0.048027\n±0.029542\nBAZS Model rd(Mpc) [100.,200.] 151.879480±17.774221\n±11.025360 138.938830±4.242102\n±2.265621 145.895106±4.798483\n±2.398445 142.576916±2.570619\n±1.535535\nrd/rfid [0.9,1.1] 1.008048±0.097299\n±0.071163 0.933363±0.032343\n±0.024337 0.971685±0.054300\n±0.028294 0.950665±0.045818\n±0.026567\nTABLE II: Summary of rdandrd/rfidParameters of each Models.\nconverges close to 1, expressed as:χ2\nDof ΛCDM≈0.961646,\nχ2\nDofBBCCFHKO≈0.968758,χ2\nDofModel 2≈0.972228,\nχ2\nDofASSS≈0.972683,χ2\nDofPADE −I≈0.955944,\nχ2\nDofPADE −II≈0.969216 andχ2\nDofBAZS≈0.971214.\nThis comparison allows us to gauge the goodness of\nfit for each model, with values near 1 indicating a\nreasonable fit to the data.\nVI. CONCLUSIONS\nIn conclusion, our study focused on investigating the\naccelerated cosmic expansion in the late universe by\nemploying the BBCCFHKO, Seljak, ASSS, PADE-I,\nPADE-II, and BAZS Models within the context of\nHorava-Lifshitz gravity. This framework offers an alter-\nnative perspective to general relativity by incorporating\nanisotropic scaling at ultraviolet scales. Utilizing Equa-\ntion of State Parametrization, we derived the Hubble\nparameters for each dark energy model scenario. In\nthis study, we analyze 24 Baryon Acoustic Oscillation\n(BAO) points, along with 30 uncorrelated data points\nobtained from Cosmic Chronometers methods. We also\nincorporate 40 points from Type Ia supernovae, 24 points\nfrom the Hubble diagram for quasars, and a substantial\ndataset consisting of 162 points sourced from Gamma\nRay Bursts. Furthermore, our study includes the latest\nmeasurement of the Hubble constant conducted byR22. Our primary objective was to minimize inherent\nerrors in the posterior distribution, which could result\nfrom potential correlations among measurements. To\nachieve this, we implemented a method to simulate the\nintroduction of random correlations into the covariance\nmatrix. Following this, we determine the optimal fitting\nvalues for every cosmological parameter within each dark\nenergy model. Our primary objective was to extract H0\nandrdusing late-time datasets. By treating the sound\nhorizon rdas a free parameter instead of relying on prior\ninformation from the Cosmic Microwave Background\n(CMB), we gain several benefits. Firstly, it reduces\nbias by sidestepping potentially restrictive CMB priors,\nensuring a more impartial estimation of rdand other\ncosmological parameters. Secondly, it enhances precision\nby allowing late-time data, such as Baryon Acoustic\nOscillations (BAO) or Type Ia Supernovae (SNIe), to\ndirectly inform the measurement of rd, leading to more\naccurate results. Moreover, this approach promotes\ncompatibility between datasets from different cosmic\nepochs, facilitating a more holistic understanding of\nthe Universe’s evolution. We determine H0and rd\nfor each dark energy model relative to the ΛCDM\nmodel as follows: In the ΛCDM model, our analysis\nyields H0= 69 .828145 ±1.009964 km /s/Mpc and\nrd= 146 .826212 ±2.201612 Mpc. In the BBCCFHKO\nmodel, we obtain H0= 69.966349 ±1.403864 km /s/Mpc\nand rd= 145 .843487 ±2.470309 Mpc. Sim-\nilarly, for Model 2, the results indicate\nH0 = 69 .969977 ±1.195973 km /s/Mpc and\nrd= 145 .677334 ±2.516764 Mpc. In the ASSS model,\nour findings show H0= 69.793952 ±1.063678 km /s/Mpc\nandrd= 145 .782783 ±2.412215 Mpc. For the PADE-I\nmodel, we obtain H0= 70.629511 ±1.586378 km /s/Mpc19\nand rd= 146 .039121 ±2.129653 Mpc. In\nthe case of PADE-II, the results indicate\nH0 = 69 .917720 ±1.211341 km /s/Mpc and\nrd= 145 .671084 ±2.267762 Mpc. In the BAZS model,\nour findings reveal H0= 69.874826 ±1.292056 km /s/Mpc\nandrd= 145 .895106 ±2.398445 Mpc. Importantly,\nour assessments underscore that the values of H0and\nrdbased on low-redshift measurements align with\nearly Planck and SDSS estimations. Our evaluation\nof various cosmological models, including the standard\nΛCDM model and several proposed extensions, employs\nboth the Akaike Information Criterion (AIC) and the\nBayesian Information Criterion (BIC). The AIC and\nBIC values computed for each model provide insights\ninto their relative goodness of fit and complexity. No-\ntably, while the ΛCDM model exhibits the lowest AIC,\nindicating a relatively better fit, all models, including\nthe extensions, receive support from the collective AIC\nand BIC results. This suggests that none of the models\ncan be decisively dismissed based solely on the available\ndata. Furthermore, comparing the extended models to\nthe ΛCDM model reveals that they are nested within it,\ndiffering by a small number of degrees of freedom. Theevaluation, utilizing the reduced chi-square statistic,\nconfirms that all models demonstrate reasonable fits\nto the data, with values close to 1 indicating goodness\nof fit. These findings underscore the importance of\nconsidering alternative cosmological models beyond\nthe standard framework. While the ΛCDM model\nremains favored due to its simplicity and good fit, the\nextensions present viable candidates worthy of further\ninvestigation. 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Arturo Ure˜ na-L´ opez,3,§and Ricardo Becerril4,¶\n1Facultad de Ciencias de la Universidad Aut´ onoma del Estado de M´ exico (UAEM´ ex.),\nInstituto Literario No. 100, C.P. 50000, Toluca, Estado de M´ exico, M´ exico\n2Instituto de Astronom´ ıa, Universidad Nacional Aut´ onoma de M´ exico, AP 70-264, Ciudad de M´ exico 04510, M´ exico\n3Departamento de F´ ısica, DCI, Campus Le´ on, Universidad de Guanajuato, 37150, Le´ on, Guanajuato, M´ exico.\n4Instituto de F´ ısica y Matem´ aticas, Universidad Michoacana de San\nNicol´ as de Hidalgo. Edif. C-3, 58040 Morelia, Michoac´ an, M´ exico\n(Dated: February 20, 2024)\nWe investigated the possibility that compact stars could be described by a fermion-boson star\nwith a quartic self-interaction in the boson sector. Specifically, by varying the polytropic constant\nKand adiabatic index Γ in the polytropic equation of state, the boson mass µ, and the self-\ninteraction parameter Λ, we construct equilibrium configurations of these mixed-stars with total\nmass compatible with the mass constraints obtained from observational data of the collaborations\nNICE, NICER/XMN-Newton, and LIGO. Our work confirms that the addition of a boson sector\neases the comparison of neutron star models with gravitational events related to compact objects\nand that in such a case observations may have preference for a positive self-interaction in the boson\nsector.\nI. INTRODUCTION\nObjects composed of a combination of fermions and\nbosons were first studied by Henriques and Liddle in [1–\n3] (see also [4, 5]), and their properties have been fur-\nther explored in recent work [6–8]. These objects are\nknown as fermion-boson stars (FBS) and are solutions of\nthe Einstein-Klein-Gordon-hydrodynamic (EKGH) sys-\ntem of equations[9].\nThe boson sector of FBS consists of a massive scalar\nfield that is minimally coupled to gravity, with a boson\nmass µand a self-interaction parameter Λ in the scalar\npotential [10–13]. The fermion sector is usually modeled\nwith a polytropic equation of state (EOS) of the form\nP(ρ) =KρΓ[7], where Kand Γ take values that corre-\nspond to masses and compactness in the range of neutron\nstars (NS). The stability of FBS has been verified for a\nwide range of parameters, allowing for a variety of masses\nand sizes in the resulting objects [7, 8].\nOn the other hand, the successful detection of gravi-\ntational waves by the LIGO and Virgo scientific collab-\norations (LVC) has significantly widened the window to\nobserve our universe and to understand gravity itself, but\nsome of the gravitational wave events have brought new\ntheoretical challenges and opportunities. The GW190814\nevent [14], for example, involved a black hole (BH) with a\nmass of 22 .2M⊙−24.3M⊙and a spin of ≤0.07, accompa-\nnied by a compact object with a mass of 2 .5M⊙−2.67M⊙.\nThis binary source of a gravitational wave has the small-\nest mass ratio ever measured for a similar system, at\n∗svaldeza@uaemex.mx\n†aosorio@astro.unam.mx\n‡jmdavilad@uaemex.mx\n§lurena@ugto.mx\n¶ricardo.becerril@umich.mx0.112+0.008\n−0.009. The nature of the second object is unknown,\nand there are no constraints on its radius, so it could be\neither a very light black hole or a very heavy neutron star\n(NS), which would make it the heaviest NS ever observed.\nThe peculiarities of events GW170817 and GW190814\nhave been used to set limits on the maximum mass and\nradii of generic neutron stars [15–17], which in turn have\nbeen used to constrain the parameters of polytropic fluids\nin models [18]. More recent constraints on the proper-\nties of neutron stars have been found from observations\nof the NICER telescope [19–21]. In [22], it was already\nsuggested that an FBS, with a boson mass of the order of\nµ∼10−9eV, could help explain the constitution of the\nself-gravitating objects behind the occurrence of these\nevents, without self-interaction in the boson sector (i.e.\nΛ = 0).\nIn this paper, we investigate the possibility that the\nsecondary component of the event GW190814 is a FBS\nwith a quartic self-interaction in the boson sector. For\nthis, we have kept in mind four facts: (a) the mass of the\nsecondary object for the LIGO gravitational wave sig-\nnal GW190814 lies in the interval [2 .5,2.67]M⊙; (b) for\nRezzolla – Nathanail simulations for neutron stars, its\nmass is located in the interval [2 .0,2.3]M⊙; (c) for the\nNICER collaborations its corresponding mass interval is\n[1.95,2.2]M⊙forPSRJ0030+0451 and [1 .2,1.65]M⊙for\nPSJ0740+6620 ; (d) ultra-light bosons have been proposed\nas candidates to be dark matter in our universe [23–34],\nand then these new events could give hints about their\npossible existence. We constructed mixed star configura-\ntions selecting Γ and Kin such a way that our numerical\nresults yield total masses that match the bounds men-\ntioned in (a), (b), and (c) above. Furthermore, we also\nconsider the constraint of the sound speed cs<1 (in\nnatural units) in the fluid sector of the FBS.\nThis paper is structured as follows. Section II out-\nlines the equations of motion of the coupled Einstein-arXiv:2402.12328v1  [gr-qc]  19 Feb 20242\nKlein-Gordon-hydrodynamic system and the boundary\nconditions that allow us to construct equilibrium config-\nurations of FBS with a given quartic self-interaction in\nthe boson sector. We also discuss the general numerical\nresults in the case of FBS with total masses and radii\nsimilar to those of NS and the influence of the boson\nself-interaction in those quantities.\nIn Sec. III we present the comparison of our models\nwith observational data of selected gravitational events,\nand discuss the general properties of the boson sector\nthat can help the fluid star to be in more agreement with\nthe data. Finally, in Sec. IV, we provide a summary of\nour results and final remarks about our study.\nII. EQUILIBRIUM CONFIGURATIONS\nWe model the boson sector of the FBS of interest with\na complex scalar field ϕthat has a scalar potential V(ϕ).\nThe fermion sector is represented by a perfect fluid with\nrest-mass density ρ, pressure P, internal energy ϵand\n4-velocity uµ. The (relativistic) equations of motion for\nthis system, expressed in geometrical units with G=c=\nℏ= 1, are as follows:\nGµν= 8π\u0010\nT(ϕ)\nµν+T(f)\nµν\u0011\n,□2ϕ−ϕdV(ϕ)\nd(|ϕ|2)= 0,(1a)\n∇µT(f)µν= 0,∇µ(ρuµ) = 0 ,(1b)where the stress-energy tensors of the bosonic and\nfermionic components are, respectively,\nTϕ\nµν=1\n2(∂µϕ∂νϕ∗)−1\n2gµν(∂αϕ∗∂αϕ+ 2V),(1c)\nTf\nµν= [ρ(1 +ϵ) +P]uµuν+Pgµν. (1d)\nThe boson sector is endowed with a quartic scalar poten-\ntial of the form\nV(ϕ) =µ2\n2|ϕ|2+λ\n4|ϕ|4, (2)\nthat represents an ensemble of boson particles with mass\nµand a self-interaction parameter λ, which may be either\npositive or negative.\nWe are interested in equilibrium configurations, so we\nassume a static and spherically symmetric metric, with\nthe line element ds2=−α2(r)dt2+a2(r)dr2+r2dΩ2.\nThe scalar field is expressed in the standard harmonic\nform ϕ(t, r) =ϕ(r)e−iωt, with ωbeing the characteristic\nfrequency of the solution, while all fermionic variables\ndepend only on the radial coordinate r. To simplify the\nequations, we introduce a set of new variables: Ω = ω/µ,√\n4πϕ→ϕ, Λ = λ/(4πµ2), 4πρ→ρand 4 πP→P, with\nwhich Eqs. (1) explicitly become,\na′=a\n2\u00121−a2\nr+a2r\u0014\u0012Ω2\nα2+ 1 +Λ\n2ϕ2\u0013\nµ2ϕ2+Φ2\na2+ 2ρ(1 +ϵ)\u0015\u0013\n, (3a)\nα′=α\n2\u0012a2−1\nr+a2r\u0014\u0012Ω2\nα2−1−Λ\n2ϕ2\u0013\nµ2ϕ2+Φ2\na2+ 2P\u0015\u0013\n, (3b)\nfor the metric variables αanda, whereas for the field and\nfluid variables we get\nϕ′= Φ, (3c)\nΦ′=\u0012\n1−Ω2\nα2+ Λϕ2\u0013\na2µ2ϕ−\u00122\nr+α′\nα−a′\na\u0013\nΦ,(3d)\nP′=−α′\nα[ρ(1 +ϵ) +P]. (3e)\nHere, a prime denotes derivatives with respect to the\nradial coordinate rand the internal energy is defined as\nϵ=P/(ρ(Γ−1)).\nThe corresponding boundary conditions that guaranty\nregularity at the origin and asymptotic flatness at infinity\nare\na(0) = 1 , α(0) = 1 lim\nr→∞a(r) = 1 , (4a)\nϕ(0) = ϕ0,Φ(0) = 0 ,lim\nr→∞ϕ(r) = 0 , (4b)\nρ(0) = ρ0, P (0) = KρΓ\n0,lim\nr→∞P(r) = 0 .(4c)The total mass of the equilibrium configurations, MTis\nobtained using the Schwarzschild definition,\nMT= lim\nr→∞r\n2\u0014\n1−1\na2(r)\u0015\n. (5)\nThe numerical results from all the equations are di-\nmensionless and given in code units. We use the standard\nconvention for NS studies, where MT= 1 is equal to one\nsolar mass M⊙in physical units. To obtain the total\nphysical mass Mand the radius 99% R99of the equi-\nlibrium configurations, we use the following expressions,\nM= 1M⊙MT, R 99≃1.47 km r99, (6)\nwhere r99is the value of rcontaining 99% of the total\nmass MTbut given in code units. A typical solution of a\nNS is then obtained without a scalar field ( ϕ0= 0) and\nfor appropriate values of fluid quantities ρ0,Kand Γ.3\nFor the mixed case of a FBS, we only need to solve\nthe equations of motion (3) for different values of the\ncentral field density ϕ0, the boson mass µand the self-\ninteracting parameter Λ. In particular, the boson mass\nµin geometrical units is related to its physical value µB\nby\nµ= 0.75(µBc2/10−10eV). (7)\nWe solved the equations of motion of FBS for three\npairs of values Γ and Knamely {(Γ,K)}={(2.8, 5.6×\n104), (2.85, 7 ×104), (2.9, 9×104)}; for each single\npair we considering the following intervals for their free\nparameters: 0 ≤ϕ0≤0.2, 0.006≤ρ0≤0.06 and −50≤\nΛ≤501.\nThe general behavior of the resulting configurations, in\nterms of their rescaled physical total mass Mand the 99%\nradius R99, is shown in Fig. 1 for the cases Λ = 0 ,50,−50\nwith Γ = 2 .8 and K= 5.6×104. In all cases, the mass of\nthe boson is µ= 1, which according to Eq. (7) means that\nµBc2≃1.33×10−10eV. This is the expected mass for\nboson stars with a total mass of the order of solar masses\nand a radius in the range of tens of kilometers [35].\nEach of the curves in the graphs represents a family of\nequilibrium configurations with a fixed value of ϕ0, and\nfor which the central mass density ρ0is changed from\nbottom to top in each curve. Furthermore, as indicated\nin the graphs, the total area covered by the set of curves\nis limited by the extreme values of the central field ϕ0,\nwhose corresponding curves have been highlighted. It\nshould be noted that the curve ϕ0= 0 is purely fermionic\nand represents the standard NS for the chosen fluid pa-\nrameters.\nOne last important feature is the stability of the equi-\nlibrium configurations. As previously studied in [6], see\nalso [8], there is a standard method for determining the\nstability of FBS in terms of their number of bosons and\nfermions, which is slow and long. However, given that\nour FBS are fermion-dominated, we can take a shortcut\nby following the standard method for fermionic (fluid)\nstars.\nFor a given ϕ0, we increase the central density ρ0until\nthe maximum mass of the star is reached. All configu-\nrations with a central density higher than this should be\nunstable. It has been demonstrated in [7] that this short-\nened method is consistent with the standard one for the\nsmall values of ϕ0discussed here. For the sake of com-\npleteness, the configurations at the stability limit are also\nshown in Fig. 1.\nWhen Λ = 0, the total mass and radius of stable\nFBS are in the ranges of 0 .73< M/M ⊙<3 and\n10< R 99/km<22, respectively. The presence of the\nself-interaction parameter increases the area of the re-\ngion covered by the curves for Λ = 50 and decreases it\n1In physical units, the central density is of the order 1 .1≲\nρ0/ρnuc≲11, where the nuclear density is ρnuc= 2.66×\n1014g/cm3(ρnuc= 4.3×10−4in geometrical units).\n10 11 12 13 14 15 16\nR99 [km]0.51.01.52.02.5M [M]\n=0\n0=0\n0=0.1\n0=0.2\nStability\n10 12 14 16 18 20 22\nR99 [km]0.51.01.52.02.5M [M]\n=50\n0=0\n0=0.1\n0=0.2\nStability\n10 11 12 13 14 15 16\nR99 [km]0.51.01.52.02.5M [M]\n=50\n0=0\n0=0.1\n0=0.2\nStabilityFIG. 1. The top, middle and bottom panels of the figure show\nthe equilibrium configurations of FBS for different values of\nthe self-interaction parameter Λ = 0 ,50,−50, respectively, in\nterms of their physical total mass Mand 99% radius R99.\nThe cases Λ = 50 ,−50 are superimposed on the case Λ = 0 to\ncompare their coverage of the parameter space. For reference,\nthe configurations corresponding to the special values for the\nvariation of the central field ϕ0= 0,0.1,0.2 are highlighted in\neach plot, while the black line represents the stability bound-\nary. Further details can be found in the text.4\nfor Λ = −50, while the mass and radius ranges remain\nsimilar for the three cases shown in Fig. 1. There is some\noverlap of the regions in the different cases, which im-\nplies that the self-interaction parameter has some degree\nof degeneracy with the central density for certain values\nof the mass and radius of the configurations.\nFor a positive self-interaction parameter, the mass of\na boson star MBis given by MB≲0.165Λ1/2M⊙/µ[10]\nin the so-called Thomas-Fermi limit Λ ≫1, and then\nMB≲8M⊙for Λ = 50. This shows that the boson part\nof the FBS can contribute more to the mass of the star\nfor the same value of ϕ0, and then the bundle of curves\nin the middle panel of Fig. 1 can cover a larger portion\nof the parameter space.\nOn the contrary, for a negative self-interaction param-\neter the upper bound on the total mass in the Thomas-\nFermi limit |Λ| ≫1 isMB<1.4|Λ|−1/2M⊙/µ[36], so\nthatMB<0.2M⊙for Λ = −50. Being the mass contri-\nbution of the boson star this small, a narrow bundle of\ncurves close to the anchor fermion star is to be expected,\nas the one shown in the bottom panel of Fig. 1.\nIII. COMPARISON WITH OBSERVATIONS\nOur goal in this section is to find regions in this param-\neter space for which the values of the total mass Mand\nradius R99, of the resultant configurations, are compati-\nble with those of NS as inferred from theoretical models\nand the NICER’s observations.\nFigures 2 and 3 display the masses and radii of FBS ob-\ntained in physical units, superimposed on the properties\nof objects detected by different observational events. The\ngreen and red regions represent the density distributions\ndetermined by the NICER Collaboration for PSR J0740\n+ 6620 [19, 20] and PSRJ0030 + 0451 [21], respectively.\nThe yellow and cyan regions indicate the constraints on\nNS mass values obtained from simulations of a large num-\nber of polytropic models conducted by Rezzolla et al.\n[15] and Nathanail et al. [16] respectively. Whereas the\ngreen band represents the overlap region between both\nsimulations. The pink region corresponds to the mass\nassociated with the second compact object detected in\nGW190814 [14] by LIGO, whose source is unknown.\nThe case with fixed value Λ = 10 is presented in Fig. 2\nin terms of the central field strength ϕ0. In all instances,\nthe value ϕ0= 0 corresponds to the NS (white curve),\nwhich we also referred to before as the anchor fermion\nstar, constructed from three sets of values of the fluid\nparameters (Γ , K): (2.80,5.6×104) (top row), (2 .85,7×\n104) (middle row) and (2 .90,9×104) (bottom row), as\nindicated in each graph. Likewise, three values of the\nboson mass were considered for the solution of the boson\npart of the star: µ= 0.5 (left column), µ= 1.0 (middle\ncolumn) and µ= 2.0 (right column).\nAs noted in [22], as we increase the value of ϕ0, we can\ncover a larger area and agree with the regions suggested\nby the different observations. Notice that the NS pro-file mainly determines the mass range of the whole set\nof equilibrium configurations, while the scalar field helps\nto extend the range in the size of the FBS, as already\npointed out in the discussion of Figs. 1. In particular, the\nvariation on the field strength ϕ0seems to give the re-\nquired degree of freedom (width) to cover the confidence\nregions of PSRJ0030 + 0451 and PSRJ0740 + 6620. It\ncan be seen that the best options appear to be those with\n(Γ, K) = (2 .85,7×104),(2.90,9×104) and with the boson\nmass µof the order of unity ( µBc2≃10−10eV).\nIn Fig. 3, we present the masses and radii of FBS ob-\ntained with the central value of the scalar field fixed at\nϕ0= 0.1, and for different values of the self-interaction\nparameter Λ in the range ( −50,50). Given that the cen-\ntral scalar field is non-zero, the anchor configuration in\nthese cases is not an NS but one FBS with Λ = 0, and\nthen configurations with Λ >0 (Λ <0) have smaller\nradii (larger radii) in comparison to the anchor FBS.\nIn contrast to the cases in Fig. 2, the best scenarios\nseem to be those in the column µ= 2, for the three sets\nof values of the fluid parameters. The values of Λ provide\nthe extra width in the parameter space to cover the den-\nsity regions suggested by the observations, since Λ can\nalso influence the mass and radius of the configurations.\nIt is clear from the curves in Figs. 2 and 3 that the\nmasses and radii associated with the numerical config-\nurations are capable of crossing all the areas suggested\nby the observations. To illustrate this in more detail,\nwe have selected three cases with Λ = 10, ϕ0= 0.1125,\nK= 5.6×104, Γ = 2 .8, and µ= 1, for which the corre-\nsponding three FBS configurations have masses and radii\nthat match the NICER and LIGO data.\nThese cases are represented by the red stars in Fig. 2,\nand their internal morphology is presented in Fig. 4. The\ngray color map represents the boson density, whereas the\nrest-mass density of fluid is represented in the blue-red\nscale. In general, we see that the contribution of the\nboson sector decreases as the configuration is more mas-\nsive and compact, as a result of the concentration in-\nduced by the gravitational force of the fluid component,\nwhich is also the dominant one. The central sound speed\nof the FBS is also shown, which was calculated using\nc2\ns=Pγ(γ−1)/[Pγ−ρ(γ−1)]. Even for the most ex-\ntreme case with M= 2.56M⊙, the sound speed remains\nbelow unity.\nAs mentioned above, the secondary component of the\nrecently reported event GW190814 is of unknown nature,\nbut its extreme mass indicates that it could be a light BH\nor a very heavy NS. However, our numerical results in\nFigs. 2 and 3 make it plausible to identify this secondary\ncomponent with an FBS with a quartic self-interaction.\nTo verify this more carefully, we have performed a de-\ntailed survey in the parameter space {Γ, K, µ, Λ}to de-\ntermine the regions corresponding to an FBS with a fixed\ntotal mass of the order of M≃2.16M⊙. The results are\nshown in Fig. 5, for the total radius Ras a function of the\nself-interaction parameter Λ (top panel) and as a func-\ntion of the central field density ϕ0(bottom panel). The5\n0.00 0.05 0.10 0.15 0.20φ\nR99[km]1.01.21.41.61.82.02.22.42.6M[M⊙]GW190814\nPSR_J0740+6620PSR J0030+0451Rezzolla et al .2018Nathanail et al .2021µ=0.5\nR99[km]Γ=2.80\nK=5.6×104µ=1.0\nR99[km]µ=2.0\nR99[km]1.01.21.41.61.82.02.22.42.6M[M⊙]\nR99[km]Γ=2.85\nK=7×104\nR99[km]\n9 10 11 12 13 14 15\nR99[km]1.01.21.41.61.82.02.22.42.6M[M⊙]\n9 10 11 12 13 14 15\nR99[km]Γ=2.90\nK=9×104\n9 10 11 12 13 14 15\nR99[km]\nFIG. 2. We show a comparison between plots of mixed stars masses vs radii (blue-white curves) with Λ = 10 and the\nobservational data obtained from NICER (red and green regions) and LIGO (pink horizontal band) collaborations as well\nas from simulations performed by Rezzolla (green-yellow band) and Nathanail (cyan band). Each curve was constructed\nnumerically taking a fixed value for ϕ0and varying the central density ρ0in the range [0 .006,0.06]. Each row corresponds to a\ngiven set of fluid parameters (Γ , N) as indicated on the labels, and each column gives the results for a fixed value of the boson\nmass µ. See the text for more details.\nvalues of the polytropic parameters are the same as those\nconsidered in Figs. 2 and 3 above.\nIn general, we note that the presence of the scalar field\nϕ, in terms of its central density or its self-interaction,\nreduces the size of the FBS, with the total radius reach-\ning values on the order of R99≃12 km. The most ex-\ntreme cases are reached for the case µ= 2, together with\nthe highest values of ϕ0and Λ. For these extreme cases\nshown in Fig. 5, the compactness of the FBS is of theorder of C= 0.22, which is approximately 16% higher\nthan the anchor NS alone.\nAnother quantity of interest is the central sound speed\ncsof the FBS with M= 2.16M⊙, which we show in\nFig. 6. Similarly to the case of the compactness of the\nstar, we see that the presence of the scalar field ϕand its\nself-interaction Λ increases the value of cs, although in\nall cases the top sound speed is of the order of cs≃0.9.6\n−50 −25 0 25 50Λ\nR99[km]1.01.21.41.61.82.02.22.42.6M[M⊙]GW190814\nPSR_J0740+6620PSR J0030+0451Rezzolla et al .2018Nathanail et al .2021µ=0.5\nR99[km]Γ=2.80\nK=5.6×104µ=1.0\nR99[km]µ=2.0\nR99[km]1.01.21.41.61.82.02.22.42.6M[M⊙]\nR99[km]Γ=2.85\nK=7×104\nR99[km]\n9 10 11 12 13 14 15\nR99[km]1.01.21.41.61.82.02.22.42.6M[M⊙]\n9 10 11 12 13 14 15\nR99[km]Γ=2.90\nK=9×104\n9 10 11 12 13 14 15\nR99[km]\nFIG. 3. Comparison between families of mass-radius diagrams of mixed stars (multicolored curves) with ϕ0= 0.1 and the\nobservational data obtained from NICER (red and green regions) and LIGO (pink horizontal band) collaborations as well\nas from simulations performed by Rezzolla (green-yellow band) and Nathanail (cyan band). Each curve was constructed\nnumerically taking a fixed value for Λ and varying ρ0in the range [0 .006,0.06]. Each row corresponds to a given set of fluid\nparameters (Γ , N) as indicated on the labels, and each column gives the results for a fixed value of the boson mass µ. See the\ntext for more details.\nIV. CONCLUSIONS AND FINAL REMARKS\nWe have constructed FBS configurations with a self-\ninteraction term in the boson sector and considered\ndifferent values of their physical parameters to match\nthe data for NS obtained from NICER, Rezzolla, and\nNathanail collaborations as well as from the LIGO grav-\nitational wave signal GW190814. For the fluid param-\neters, we picked values that yielded NS masses in thevicinity of the values reported for different observed ob-\njects, and then varied the boson parameters, mass and\nself-interaction, to find their influence in the mass, size,\nand morphology of the corresponding FBS.\nIn general terms, the variation of the boson parameters\nallows our models to cover a wider area in the parameter\nspace of the observed objects, which means that the ad-\ndition of a boson sector to a given NS configuration could\nhelp explain the presence of different types of object. Ac-7\n−30−20−10 0 10 20 30\nx[km]−30−20−100102030y[km]log10ρφ\nPSR J0030+0451\nM= 1.4M⊙R= 12.9 kmCs= 0.66\n−20−10 0 10 20 30\nx[km]PSR_J0740+6620\nM= 2.1M⊙R= 13.3 kmCs= 0.80\n−20−10 0 10 20 30\nx[km]log10ρ\nGW190814\nM= 2.56M⊙R= 12.1 kmCs= 0.99\n−14−12−10−8−6−4−2 0\n−3.75−3.50−3.25−3.00−2.75−2.50\nFIG. 4. Two dimensional morphology of representative fermion-boson stars to model three observational constraints. The left\npanel shows an FBS with mass M= 1.4M⊙and size R99= 12.9km, which meets the NICER observational constraints for\nthe pulsar PSRJ0030+0451 . The middle panel is an FBS with mass M= 2.1M⊙and radius R= 13.3kmto model the pulsar\nPSRJ0740+6620 , satisfying the mass estimated by theory and NICER observations. The right panel shows the morphology\nof the FBS to model the event GW190814 with mass M= 2.56M⊙and radius R99= 12.1km. The fermion component is\nrepresented by the rest-mass density, and the boson component by the scalar field density, both in logarithmic scale. The three\nFBS are modeled by the same family of solutions (see the red stars in Fig. 2), by fixing the self-interaction parameter Λ = 10,\nthe central strength of the scalar field ϕ0= 0.1125, the equations for the state parameters Γ = 2 .8 and Knew= 5.6×104, and\nthe mass of the boson µ= 1. See the text for more details.\n−50−25 0 25 50\nΛ12.012.513.013.514.014.5R99[km]\nΓ = 2.80\nΓ = 2.85\nΓ = 2.90φ0= 0.1M= 2.16M⊙\nµ= 0.5\nµ= 1.0µ= 1.5\nµ= 2.0\n0.00 0.02 0.04 0.06 0.08 0.10 0.12\nφ012.012.513.013.514.014.5R99[km]\nΓ = 2.80\nΓ = 2.85\nΓ = 2.90Λ = 10M= 2.16M⊙\nµ= 0.5\nµ= 1.0µ= 1.5\nµ= 2.0\nFIG. 5. The size of the FBS as function of the self-interaction parameter (top panel) and the scalar field strength (bottom\npanel) for star with total mass M= 2.16M⊙. Changes in EoS are represented by black, blue, and red colors, while line styles\nshow the effects of varying the boson mass µ. See the text for more details.\ncording to our results, the best options seem to have a\nboson mass of order µ≃1 in our units, which means that\nthe total mass of the corresponding boson star is of the\nsame order of magnitude as that of the selected NS (or\nanchor NS, as we call it in the main text). Similarly, there\nis a preference for positive values of the self-interactionparameter, as a negative value of it makes it difficult to\nreconcile our theoretical results with observations.\nOur work confirms and extends the results presented\nin [22], in that the addition of a boson sector eases the\ncomparison of NS models with different observations.\nWe have already mentioned the preference for the value8\n−50−25 0 25 50\nΛ0.650.700.750.800.850.900.951.00cs\nΓ = 2.80\nΓ = 2.85\nΓ = 2.90φ0= 0.1M= 2.16M⊙\nµ= 0.5\nµ= 1.0µ= 1.5\nµ= 2.0\n0.00 0.02 0.04 0.06 0.08 0.10 0.12\nφ00.650.700.750.800.850.900.951.00cs\nΓ = 2.80\nΓ = 2.85\nΓ = 2.90Λ = 10M= 2.16M⊙\nµ= 0.5\nµ= 1.0µ= 1.5\nµ= 2.0\nFIG. 6. Sound speed at the center of the FBS as function of the self-interaction parameter (top panel), and as function of the\namplitude at the star center. In all models, we fixed the mass of the FBS to M= 2.16M⊙, the corresponding sizes are reported\nin Fig. 5. The colors of the lines represent the three equations of states (Γ = 2 .8, K= 5.6×104), (Γ = 2 .85, K= 7×104), and\n(Γ = 2 .9, K= 9×104), respectively. The line styles correspond to different choices for the boson mass, µ= 0.5,1.0,1.5,2.0.\nSee the text for more details.\nµ≃1, which is not surprising given the typical masses\nand sizes of compact objects suggested by the observa-\ntions we used. The interesting new result is that the\nself-interaction, the next-to-leading order parameter in\nthe description of the boson sector, in particular for the\nscalar field potential, should be positive.\nSome other comments are in turn. Our anchor NS\nwas modeled as a polytropic fluid, for which the param-\neter values were selected according to previous studies of\nNS [18]. However, our study could also consider other\ntypes of equation of state for the fermion sector, such as\nthose chosen in [22], for which we would expect the same\nqualitative results obtained so far: preference of data for\nµ≃1 and Λ ≳0.\nUltralight bosons have been proposed as candidates\nfor dark matter in our universe, and a boson with mass\nµBc2∼10−10eV would certainly qualify as ultralight,\nalthough such a mass would be 10 orders of magni-\ntude larger than the preferred astrophysical value of\nµBc2∼10−22eV [31, 34]. Our new result Λ ≳0, if\ntaken at face value, would mean that there is an extra\nconstraint on dark matter models with axion-like bosons\n(with negative self-interaction), at least for the cases for\nwhich µBc2∼10−10eV, as there are differentiable conse-\nquences at astrophysical and cosmological scales for the\nformation of cosmic structure from the sign of the self-interaction in the potential [37–40].\nAs already noted in [22], for comparison with obser-\nvations, we just took the confidence regions obtained\nby other groups that considered models quite different\nfrom ours to fit the data. In this respect, our constraints\nshould be considered conservative and more like a proof-\nof-concept for FBS with a self-interaction. A separate\nstudy with a direct comparison with raw data from the\ndifferent observations would be necessary to have a more\naccurate judgment of the appropriateness of FBS and\nthe role of their self-interaction in each of the reported\ngravitational events. 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We show that the transient evaporation\nprocess can be subdivided into three major phases (A, B, and C) based on the evaporation\nrate. Phase A is the fastest where edge evaporation dominates and leads to the formation\nof a gelated three phase contact line. Gelation is a result of sol-gel phase transition that\noccurs due to the accumulation of RBCs (red blood cells) as they get transported due\nto outward capillary flow generated during drop evaporation. The intermediate phase\nB consists of gelation front propagating radially inwards due to a combined effect of\noutward capillary flow and drop height reduction evaporating in CCR (constant contact\nradius) mode leading to a formation of a wet gel phase. We unearth that the gelation of\nthe entire droplet occurs in Phase B and the gel formed contains trace amounts of water\nthat is detectable from our experiments. Phase C is the final slowest stage of evaporation,\nwhere the wet gel transforms into dry-gel and leads to the formation of dessication stress\nconcentrations resulting in the formation of various kinds of crack patterns in the dried\nblood drop precipitate. Using profilometry and scanning electron microscopy we observe\nradial cracks in the thicker region of the precipitate and mud flat type cracks are observed\nin the center part of the evaporating droplet where the drop thickness is relatively small\nand curvature is negligible. We also study the evaporation of bacteria laden droplets to\nsimulate bacterial infection in human blood and show that the drop evaporation rate and\nfinal dried residue pattern does not change appreciably within the parameter variation\nof the bacterial concentration typically found in bacterial infection of living organisms.\nKey words: Authors should not enter keywords on the manuscript, as these must\nbe chosen by the author during the online submission process and will then be added\nduring the typesetting process (see http://journals.cambridge.org/data/relatedlink/jfm-\nkeywords.pdf for the full list)\n1. Introduction\nIn this study we investigate the mechanics of sessile blood droplet evaporation. Under-\nstanding the behavior of sessile blood droplet during evaporation is essential in forensic\nscience, biotechnology, and the development of advanced diagnostic tools (Chen et al.\n†Email address for correspondence: sbasu@iisc.ac.inarXiv:2402.12334v1  [physics.flu-dyn]  19 Feb 20242 Roy et al.\n2016). We also observe evaporation phenomena in different biological fluids in various\nnatural phenomena like blood clots dessication in wounds (Sobac & Brutin 2011; Brutin\net al. 2011; Laux et al. 2016), biofilm formations (Wilking et al. 2013), drying of tears\n(Traipe-Castro et al. 2014) to name a few. Apart from forensics and disease diagnostics,\nanalysis of dessicated biological samples are also important in several applications like\narcheological studies (Wilson et al. 2013), microarray applications for DNA, RNA,\nproteins, phenotype studies (Smalyukh et al. 2006). Analyzing the characteristics of\nsessile blood droplets during evaporation can provide valuable insights into the source,\nage, and composition of the blood sample. The physics of sessile blood drop evaporation\nfalls at the intersection of various disciplines like fluid mechanics, solid mechanics, soft\nmatter, statistical physics, surface chemistry, colloidal physics, biology to name a few\nand knowledge of the evaporation and precipitate formation still remains in its infancy\n(Lanotte et al. 2017). Blood is a complex fluid consisting of formed cellular elements\nsuspended in blood plasma (Hoffbrand & Steensma 2019). The various cellular elements\nlike erythrocytes (red blood cells), leukocytes (white blood cells) and the thrombocytes\n(platelets) forms approximately 45-50% of the blood volume with RBCs having the largest\npercentage among the cellular elements. Blood plasma on the other hand is an amber\ncolor liquid component of the blood which makes approximately 50-55% of the total blood\nvolume. 94-95% of the blood plasma by volume is essentially water with the remaining\n5-6% being dissolved proteins, electrolytes, glucose, clotting factors, hormones, carbon\ndioxide and oxygen. We can therefore approximate blood as a colloidal suspension of\nRBCs in plasma to first order accuracy. The morphology of dried residues of blood drops\nhence depends primarily on the deposition of RBCs, the major solute component of\nblood plasma. The evaporation/dessication of biofluids like blood, saliva, mucus, urine\nhas been explored majorly in sessile droplet and thin films on solid substrates due to the\nsimplicity of the experimental setup (Hu & Larson 2002). The major observations were\nthe formation of a ring like deposit at the contact line and a complex set of crack patterns\nin dessication drops and films. The different patterns formed in evaporating drops and\nfilms of biological fluids resembled patterns found in various colloidal dessicating drops\nand films hinting towards some unifying underlying laws that were common to biofluids,\ncolloids and pure systems (Tarasevich et al. 2011; Deegan et al. 1997, 2000; Dufresne\net al. 2003; Allain & Limat 1995; Denkov et al. 1992; Adachi et al. 1995). Previous\nstudies by scientists and engineers using model fluids like mucin revealed that both\ndeposition and crack formation depends strongly on the particle concentration and types\nof dissolved salts, macro-molecules, proteins and surfactants (Nguyen & Stebe 2002).\nThis work explores blood drop evaporation physics in depth and tests whether microbial\nbacteria present in blood can affect the evaporation process and dried precipitate residue.\nVarious kinds of optical diagnostics methods are used to study complex biological\nfluids in different conditions and have shown that pathological changes of solute and\nsolvent composition affects the morphology of the dried residues in evaporating drops\nand films (Yakhno et al. 2005; Shatokhina et al. 2004; Esmonde-White et al. 2014).\nUpto the recent decade, dessication studies on biological fluids have majorly focused on\nsolutions, but the evaporation physics of drying suspensions are relatively sparse. The\nseminal work of Brutin et al. (Brutin et al. 2011) and Chen et al. being some exception.\nBlood is a concentrated colloidal suspension with particle volume fraction / Hematocrit\n(Ht) ofHt∼40−45% in healthy individuals [ref]. Based on volume ratio, blood can\nbe thought of comprising of two major components, RBCs ( VRBCs /V∼40−45%) and\nplasma ( Vplasma /V∼60−55%) where VRBCs ,Vplasma represents the respective volumes\nin a total blood volume of V. Plasma is the protein (fibrinogen, albumin) laden aqueous\nsalt solution suspending medium for the micron size deformable RBCs in corpuscularBlood drop evaporation 3\nphase. RBCs are typically biconcave shape at rest. The very high concentration of\nRBCs makes visualization of evaporating blood droplets very challenging in general.\nBrutin and his collaborators performed blood drop evaporation analysis using optical\nmicroscopy and mass measurements. The droplet evaporation mechanism, dessication\ntime and morphology of dried blood residues were unearthed by Brutin et al. (Brutin et al.\n2011) using experimental techniques. However a comprehensive quantitative model of the\nvarious processes that occurs during blood drop evaporation are relatively unexplored in\ndetail. This work tries to fill some of the corresponding research gap by unearthing the\nfundamental mechanisms quantitatively. In general, the evaporation and the dried residue\nof a dessicating biological fluid such as blood depends on the ambient enviromental factors\n(like temperature and humidity), substrate characteristics (like chemical composition,\nwettability, adhesion), and droplet geometry, fluid mechanical, physicochemical, ther-\nmofluidic, and biophysical properties (like density, viscosity, surface tension, pathology).\nThe dried blood drop dominant pattern consists of a thick rim and a center region of\nrelatively small thickness. The rim thickness is caused due to the transport of RBCs\nby capillary flow induced by sessile drop evaporation. The variation of thickness from\nthe drop center to the outer rim region in the radial direction causes crack patterns of\ndifferent morphologies. Radial cracks are formed in the rim region whereas mud-flat type\nof cracks are formed in the center and outer periphery of the dried residues. Evaporating\ncolloidal and polymeric sessile drops in analogous forms a skin layer at the air-liquid\ninterface which eventually undergoes a buckling.\nCurrent literature on blood drop evaporation are primarily based on observational\ncharacteristics of drop shape, mass, and crack patterns (Choi et al. 2020; Mukhopadhyay\net al. 2020; Iqbal et al. 2020; Bahmani et al. 2017). However, fundamental mechanistic\nunderstanding of blood drop dessication process is far from complete and requires\nthorough quantitative analysis. This work serves as a bridge to understand the mechanics\nof blood drop evaporation using ideas from colloidal dessicating drops. We decipher\nthe various phases and the allied mechanisms of blood drop drying using high fidelity\ncolor optical imaging, quantitative theoretical modelling and experimental micro/nano\ncharacterization. It is important to note that the scope of the current work is to\nunderstand the various mechanisms and phases related to blood droplet drying physics\nin a quantitative manner using experimental and theoretical methods. However, full\nscale numerical simulation of the various processes involved is beyond the scope of the\npresent work. The current manuscript is structured as follows. Section 2 discusses the\nmaterials and methods in several subsections. Blood sample preparation, experimental set\nup and dried residue characterization is discussed in section 2.1, 2.2 and 2.3 respectively.\nSection 3.1 introduces the key results as a global overview. Section 3.3 discusses the\nvarious stages of blood drop evaporation. Section 3.4 discusses the quantitative modelling\nand the generalized mechanics of blood drop evaporation. Section 3.5 presents the\ncharacterization of dried blood residues and discusses the cracking mechanics due to\ndessication stresses. Section 4 concludes the manuscript.\n2. Materials and Methods\n2.1.Blood sample preparation\nBlood samples were collected from three healthy volunteers (1 female, 2 male) in\nthe age group (25-40 years) with proper consent. 10 ml volume of whole blood was\ndrawn in K3-EDTA vacutainer containers to prevent the blood from clotting. 2-4 ml\nof the collected blood were sent for several blood characterization tests relevant to4 Roy et al.\nFigure 1. Schematic of the experimental set up and sample top, side and bottom view images\nof dried blood drop precipitate. Scale bar depicted in white represents 1 mm.\nthe mechanics of the various processes during evaporation like CBC (Complete Blood\nCount), PT (Prothrombin Time), and ESR (Erythrocyte Sedimentation Rate). Overnight\ngrown stationary phase cultures of Klebsiella pneumoniae MH1698 were taken, and their\nabsorbance was measured at OD600nm. Various bacterial concentrations in the range\nofO(10−109)CFU/mL of the bacterial culture were pelleted down at 6000 rpm for 6\nminutes. The pellets were washed twice with phosphate-buffered saline (PBS) pH 7.0\nand finally resuspended in 500 µL EDTA whole blood. For confocal microscopy studies,\nS. Typhimurium expressing Green Fluorescent Protein (GFP) (pFV-GFP) (STM-GFP)\nstrains were used.\n2.2.Experimental Set up\nFigure 1 shows a schematic representation of the experimental setup. The experimental\nsetup consists of a 2 −5µL blood droplet evaporating on a a clean glass slide of dimensions\n75×25×1 mm3(procured from Blue Star). The blood droplet was placed gently on the\nglass slide using a micropipette. All the evaporation experiments were performed at\na relative humidity of 45 ±3% and a ambient temperature of 25 ±2◦Cmeasured using\nTSP-01 sensor, Thorlabs. The complete blood droplet evaporation process is recorded\nusing color optical diagnostics from three views, i.e., top, side and bottom view imaging.\nThe top and bottom view were operated in reflection mode whereas shadowgraphy was\nperformed using side view imaging. The color images of all the corresponding views\n(top, bottom and side) were recorded using 3 DSLR camera (Nikon D5600) coupled\nwith a Navitar zoom lens assembly (2X lens ×4X adapter tube). 5W LED light sourceBlood drop evaporation 5\n(Holmarc) and 50W mercury lamp were used to provide uniform illumination for the\nthree imaging views. A spatial resolution of 1pixel/ µm were used for the optical color\nimaging. Consecutive images were captured using a time delay of 10 seconds. Typical\nsample top, side and bottom view images are shown in figure 1 respectively. The scale\nbar in white denotes 1mm. The instantaneous geometrical parameters and features were\nextracted using image processing techniques applied to the side, top and bottom view\nimages using open source software ImageJ and python programming language.\n2.3.Dried residue characterization\nScanning Electron Microscopy (SEM) was employed to analyze the surface character-\nistics of the deposits, revealing the microcrack patterns through high-resolution imaging.\nUltra55 FE- SEM Karl Zeiss EDS instrument is used. SEM images showed the detailed\ncrack patterns and stacking of RBCs on the surface. Prior to the SEM examination,\nthe deposit surfaces were desiccated for 24 hours and then coated with a thin layer\nof gold, approximately 10 nm thick, to facilitate SEM and profilometric studies. For\nassessing the deposits geometrical features, a non-contact optical profilometry method\nwas utilized. The measurements were conducted using a Taylor Hobson 3D surface and\nfilm thickness optical profiler. Additionally, the TalySurf CCI, a commercial optical\nmetrology software, was used for processing the data. Zeiss LSM 880 NLO upright multi-\nphoton confocal microscope was used to image the dried sessile blood droplet precipitate\nat 10X magnification. ZEN Black software (Carl Zeiss) was used to obtain the Z stacks\nimages to study the bacterial deposition in the precipitates.\n3. Results and Discussions\n3.1.Global overview\nFigure 2(a) depicts the schematic representation of an evaporating sessile droplet.\nFigure 2(b) represents the normalized volume regression of the evaporating droplet for\nvarious blood samples (pure and bacteria laden). Figure 2(c) represents the top, side\nand bottom view image sequence of the evaporation process. In this work we study\ntransient sessile blood drop evaporation (figure 2(a)) with pure EDTA whole blood\nand bacteria laden EDTA whole blood with varying levels of bacterial concentration\ntheoretically and experimentally using lubrication theory, color optical imaging and\nmicro/nano characterization. We further show that the transient evaporation dynamics\ncan be subdivided into three phases (A, B, and C) based on the evaporation rate (figure\n2(b)). Phase A (figure 2(b,c) t/t∗= 0−0.2) is the fastest where edge evaporation\ndominates and leads to the formation of a gelation three phase contact line front\npropagating radially inwards. This occurs due to initial outward capillary flow generated\ndue to drop evaporation that transports the RBCs radially outwards. The intermediate\nphase B (figure 2(b,c) t/t∗= 0.2−0.7) consists of gelation of the entire droplet due to\nthe propagating gelation front due to the radial RBCs deposition and the reduction in\ndroplet height. Gelation in phase A and B occurs due to the sol-gel phase transition.\nThe sol-gel phase transition occurs when the solute concentration (RBCs here) becomes\nlarger than a critical concentration. We unearth that the gelation of the entire droplet\noccurs in Phase B and at the end of Phase B, the gel formed still contains small traces\nof water that is detectable from our experiments. Phase C (figure 2(b,c) t/t∗= 0.7−1.0)\nis the final slowest stage of evaporation, where the wet gel formed during Phase A and\nPhase B transforms into dry-gel due to a very slow evaporation process. The drying\ndroplet results in high azimuthal stress concentrations resulting in the formation of radial6 Roy et al.\nFigure 2. (a) Schematic of the coordinate system, composition of the blood and initial condition\nof sessile blood drop evaporation. (b) Non dimensional volume ( V/V 0) regression plotted as a\nfunction of non dimensional time ( t/t∗) for whole blood + EDTA, whole blood + EDTA +\n106CFU/ml KP (Klebsiella pneumoniae) bacteria and whole blood + EDTA + 109CFU/ml\nKP bacteria respectively. The different phases of sessile blood drop evaporation depicted as A,\nB and C respectively. (c) Top view, side view and bottom view time sequence images of the\nevaporation process. Scale bar for the top, side and bottom view represents 1 mm respectively.\nThe timestamps are in non dimensional units ( t/t∗).\ncracks. Mud flat cracks are observed in the center part of the evaporating droplet where\nthe drop thickness is relatively small and curvature is negligible. We further show that\nthe drop evaporation rate does not change appreciably within the parameter variation\nof the bacterial concentration typically found in blood of living organisms (figure 2(b)).\n3.2.Coordinate system and drop evaporation physics\nFigure 2(a) shows the schematic of the evaporating droplet and the associated coordi-\nnate system. Axisymmetric cylindrical coordinate system ( r−z) is used for quantitative\nanalysis of the evaporation process. Here rrepresents the radial coordinate and z\nthe axial coordinate, the vertical axis along which the drop interface profile would be\nsymmetric. For small droplets, the droplet interface geometry could be approximated\nby a spherical cap geometry with a particular contact angle ( θ), contact radius ( R)\nand central height ( h(0, t)) for a particular droplet volume and substrate. Isothermal\nsessile drop evaporation for thin droplets in the size range of mm (i.e., R∼O(10−3)m andBlood drop evaporation 7\nh < R ) can be modelled to high degree of accuracy by a diffusion limited model. The\nsize where reaction dominated evaporation phenomena begins is approximately at length\nscales of the order of 100nm which is considerably very small with respect to the typical\nsize scale we have in our experiments, therefore conforming to diffusion limited drop\nevaporation. Drop evaporation in general can occur in various modes like CCR (constant\ncontact radius) and CCA (constant contact angle) depending on the wettability of the\nsubstrate. For hydrophilic substrate with initial acute contact angle as is the present\ncase, drop evaporation generally occurs in CCR mode of evaporation. In CCR mode of\nevaporation the droplet three phase contact line is pinned throughout the evaporation\nprocess while the contact angle decreases with time. For the present case of evaporating\nwhole blood droplets, the evaporation occurs in CCR mode as can be observed from\nthe image snapshots in figure 2(c). From figure 2(b) we can observe the entire CCR\nmode of evaporation process can be further subdivided into three phases A, B, and C\nbased on descending evaporation rate. Figure 2(b) represents the graph of normalized\nvolume ratio ( V/V 0) with normalized time ( t/t∗) for the evaporating drop. Here Vdenotes\nthe instantaneous drop volume and V0denotes the initial drop volume. Here tdenotes\nthe instantaneous time and t∗denotes the complete evaporation time scale (defined\nexperimentally as the time instant where no significant changes are observed in drop\nmorphologies in all the three views). From figure 2(b,c) we can observe that phase A\ncorresponds to t/t∗= 0−0.2, phase B corresponds to t/t∗= 0.2−0.7 and phase C\ncorresponds to t/t∗= 0.7−1.0.\n3.3.Stages of blood drop evaporation\n3.3.1. Phase A\nFigure 3 shows the schematic representation of various processes that occurs during\nPhase A of blood droplet evaporation. Figure 3(a) shows the initial configuration of\nthe evaporating blood droplet. To first approximation, blood can be approximated as\na binary suspension (RBCs suspended in plasma). The evaporation occurs in CCR\nmode with a constant contact radius R. For small drops the droplet shape can be\napproximated by a spherical cap and the evaporation flux along the droplet interface\nincreases with radial coordinate r, with very large value at the three phase contact\nline. The non-uniform evaporation flux along the drop interface generates a radially\noutward capillary flow that causes the RBCs to be transported towards the three phase\ncontact line. Figure 3(b,c,d) shows the magnified view near the three phase contact line\natt/t∗= 0,0.1,0.2 respectively. At initial time of drop evaporation, a very thin precursor\nfilm exist at the outer edge of the droplet which slowly solidifies as a plasma layer. The\nouter radial capillary flow (shown in blue) causes the RBCs to accumulate resulting\nin the formation of a gel phase. After the intial gelation of the three phase contact\nline, a gelation front radially propagating inward is observed (blue line). Further the\nRBCs concentration increases at the outer periphery and leads to the reduction of partial\npressure of water vapor according to Raoult’s law thereby reducing the evaporation flux\nat the drop interface near the three phase contact line. A small dotted circle representing\na control volume (CV) is shown figure 3(b). Figure 3(e) shows the zoomed in view of\nthe CV. The blood suspension inside the CV is initially in sol phase. However, due to\nthe outward capillary flow, the RBCs concentration inside the control volume increases\nleading to a phase transition forming a wet gel phase. Figure 3(g) shows the reduction of\nevaporation flux Jat radial location r=R(at three phase contact line) due to Raoult’s\nlaw triggered by sol-gel phase transition. Figure 3(h) shows the corresponding rise in\nRBCs concentration cat the three phase contact line r=Rdue to the radially outward8 Roy et al.\nmovement of RBCs caused by the capillary flow generated due to CCR mode of drop\nevaporation. Sol-gel transition occurs at the three phase contact line as the concentration\nbecomes greater than a critical gelation concentration cg. Figure 4(a) shows the top, side\nand bottom image sequence for Phase A of the evaporating blood droplet in the top,\nmiddle and bottom panel. The constant contact pinned radius is denoted by Rand\nthe gelation radius by rg. The drop contact angle represnted by θ, and drop center\nheight by h=h(0, t). Figure 4(b) shows the evolution of various normalized geometrical\nparameters G(t) like normalized contact angle ( θ/θ0) and normalized contact radius\nR/R 0confirming CCR mode of drop evaporation. Figure 4(c) also shows the evolution\nof various normalized geometrical parameters G(t) like normalized droplet center height\n(h/h 0) and normalized gelation radius ( rg/rg0) during phase A. It can also be observed\nthat the droplet height reduction is faster in comparison to the gelation front propagation.\nFigure 4(d,e) shows the schematic representation of the evaporation phase A at t/t∗= 0\nandt/t∗= 0.2 respectively. Further figure 4(d,e) also depicts the change in evaporation\nflux schematically due to increase in RBCs concentration caused due to radial outward\ncapillary flow induced by the evaporating whole blood droplet.\n3.3.2. Phase B\nPhase B of the evaporation process spans from t/t∗= 0.2−0.7 and has the longest\nduration. In this phase the gelation front propagates further inwards. As the droplet\nevaporates in CCR mode, the three phase contact line is pinned, but the droplet contact\nangle and its corresponding maximum height decreases monotonically. As the droplet\nevaporates, the outward capillary flow causes a depletion of RBCs in center allowing\nthe droplet interface to smoothly change its curvature. Figure 5 shows a schematic\nrepresentation of the various processes occuring during phase B of evaporation. Figure\n5(a) shows a 3D schematic of the evaporation process in Phase B at t/t∗= 0.3,0.5,0.7.\nFigure 5(b) shows a magnified view of the droplet interface undergoing monotonic\ncurvature ( k) change from k < 0, to k= 0, to k > 0 at t/t∗<0.5,t/t∗∼0.5−0.6,\nandt/t∗>0.6 respectively. The corresponding side view images are also shown. Figure\n5(c) represents the corresponding top and bottom view images. Figure 6(a) shows the top,\nside and bottom view image sequence during phase B of evaporation. The timestamps\nare normalized time ( t/t∗). Figure 6(b) shows the droplet non dimensional geometrical\nparameters G(t) evolution as a function of normalized time. Various parameters like\nnormalized contact angle ( θ/θ0), normalized height h/h 0, normalized contact radius\n(R/R 0), and normalized gelation radius ( rg/rg0) is plotted in black, green, red, and blue\nrespectively. Figure 6(c) shows a schematic representation of the starting ( t/t∗= 0.2)\nand ending state ( t/t∗∼0.67−0.7) of phase B respectively. Rdenotes the contact radius\nandrgdenotes the gelation front radius. It is important to note the distinct color change\nin the top view at the end of phase B ( t/t∗= 0.67) signifying the completion of gelation\nof the entire droplet. At the end of phase B the entire droplet has undergone a sol-gel\nphase transition to a final wet gel form. The wet gel phase contains very small (trace)\namounts of liquid water that evaporates in the next phase C. We can also observe from\nthe side view image panel, that there is negligible change in droplet geometry signifying\nthe completion of gelation for the entire droplet.\n3.3.3. Phase C\nFigure 7 shows a schematic representation of phase C. Phase C is the final stage\n(t/t∗= 0.7−1.0) of the evaporation process with very small evaporation rate. In this\nphase the wet gel from the previous phase goes further dessication and transforms to dry\ngel. Schematic of wet gel and dry gel state is shown in Figure 7(a) and 7(b) respectively.Blood drop evaporation 9\nFigure 3. Schematic representation of the various processes occurring in Phase A of blood\ndroplet evaporation. (a) Initial configuration of the sessile blood droplet at t/t∗= 0 depicting\nthe evaporative flux and the radial outward capillary flow inside the evaporating droplet. (b,c,d)\nMagnified view of the outer edge of the droplet depicting the precursor film and the outward\ntransport of RBCs towards the edge at t/t∗= 0 (b), t/t∗= 0.1 (c), and t/t∗= 0.2 (d)\nrespectively. The blue vertical line shows the gelation front propagating radially inwards. (e) A\nsmall control volume (CV) near the three phase contact line depicting the sol phase in which\nRBCs are present inside the CV and getting transported across the surface of the CV. (f) Wet\ngel phase in the CV. (g) Schematic graph depicting the decrease of evaporative flux at the outer\nedge of the droplet due to RBCs accumulation. (h) Schematic depicting the evolution of average\nconcentration inside the CV.10 Roy et al.\nFigure 4. (a) Top view, side view and bottom view time sequence image snapshots of Phase A\nat different non dimensional time instants t/t∗= 0,0.05,0.1,0.15,0.2. (b,c) Non dimensional\ngeometrical drop parameters G(t) (normalized contact angle ( θ/θ0), normalized contact radius\n(R/R 0), normalized drop height ( h/h 0), and normalized gelation radius ( rg/rg0)) evolution as\na function of non dimensional time t/t∗. (d) Schematic representing the initial configuration of\nthe evaporating droplet ( t/t∗= 0). (e) Schematic representing the evaporating droplet at the\nend of evaporation phase A ( t/t∗= 0.2).Blood drop evaporation 11\nFigure 5. (a) 3D Schematic representation of blood droplet evaporation in Phase B from\nt/t∗= 0.3−0.7. (b) Schematic and side view data showing the curvature change at the drop\ncenter during evaporation process in Phase B. Corresponding top and bottom view data are\ndepicted in vertical panels.12 Roy et al.\nFigure 6. (a) Top view, side view and bottom view time sequence image snapshots of Phase B at\ndifferent non dimensional time instants t/t∗= 0.2,0.3,0.4,0.5,0.6,0.67. (b) Non dimensional\ngeometrical drop parameters G(t) (normalized contact angle ( θ/θ0), normalized drop height\n(h/h 0), normalized contact radius ( R/R 0) and normalized gelation radius rg/rg0) evolution as\na function of non dimensional time t/t∗. (c) Schematic representing the evaporating droplet at\nthe beginning of phase B ( t/t∗= 0.2) and end of phase B ( t/t∗∼0.67−7) respectively.\nHWC denotes comparatively high water content in comparsion to low water content\n(LWC) in the dry gel state. Figure 7(c) shows a typical side view image during phase\nC of the evaporation process. Figure 7(d) depicts reduction in evaporation flux as the\nsolute concentration increases further during the dessication process. Figure 7(e) shows a\nschematic of the top view of the evaporating drop in phase C ( t/t∗= 0.7−1.0) depicting\nthe formation of radial cracks, shrinkage of trace amounts of water and dried blood\nresidue. Figure 8(a) shows the image sequence of the top, side and bottom view of the\ndessicating drop respectively. Notice the distinct color change in top and bottom view as\nthe drop transforms from wet gel to dry gel. From both the top and bottom view we can\nobserve the evaporation process of the trace amounts of water present at the beginning\nof phase C. Dessication stresses are developed as the drop undergoes further evaporation\nleading to the formation of various types of cracks. Radial cracks are observed in the\nthicker outer rim region whereas mudflat types of cracks are observed in the center andBlood drop evaporation 13\nFigure 7. Schematic side view representing Phase C of evaporation process depicting (a)\nwet gel pahse at t/t∗= 0.7, and (b) dry gel phase at t/t∗= 1.0 respectively. HWC and LWC\ndenotes high and low relative water content in the wet and dry gel phase respectively. (c) Typical\nside view during Phase C of evaporation. (d) Schematic representing the evaporation flux time\nevolution at the drop center. (e) Schematic top view representation of the evaporating drop in\nphase C ( t/t∗= 0.7−1.0) depicting the radial cracks, entrapped water and dried blood residue.\nthe three phase contact line region. Figure 8(b) denotes the non dimensional geometric\nparameter evolution as a function of non dimensional time. Figure 8(c) denotes the\nschematic of the end of state of evaporation process in phase C.\n3.4.Generalized Mechanics of Blood drop evaporation\nThe initial capillary flow initiated due to the non uniform evaporation flux over the\ndrop surface can be modelled using Stokes flow in cylindrical coordinates (refer to figure\n2(a)). We follow the analysis of Yuri et al. (Tarasevich et al. 2011) very closely for\ncomputing the dynamics of drop evaporation. From the radial component of the Stokes14 Roy et al.\nFigure 8. (a) Top view, side view and bottom view time sequence image snapshots of Phase\nC at different non dimensional time instants t/t∗= 0.7,0.75,0.8,0.85,0.95,:1.0. (b) Non\ndimensional geometrical drop parameters G(t) (normalized contact angle, normalized drop\nheight, normalized contact radius and normalized gelation radius) evolution as a function of\nnon dimensional time t/t∗. (c) Schematic side view representation at the end stage of Phase C.\nequation with thin film approximation (lubrication approximation) we have\n∂p\n∂r=η∂2u\n∂z2(3.1)\nFrom the axial component of the Stokes equation we have\n∂p\n∂z= 0 (3.2)\nwhere pis the pressure, ris the radial coordinate, ηis the drop viscosity, uis the radial\nvelocity, and zis the axial coordinate. Assuming cylindrical symmetry about the vertical\naxis, the continuity equation can be written as\n1\nr∂(ru)\n∂r+∂w\n∂z= 0 (3.3)Blood drop evaporation 15\nFigure 9. (a) Height averaged non dimensional concentration field plotted as\na function of non dimensional radial coordinate at various non dimensional time\ninstants ¯t= 0 ,100,200,300,400. (b) Non dimensional evaporation flux plotted as\na function of non dimensional radial coordinate at various non dimensional time\ninstants ¯t= 0 ,100,200,300,400. (c) Non dimensional drop height variation as a\nfunction of non dimensional radial coordinate at various non dimensional time instants\n¯t= 0,100,200,300,400. (d) Theoretical and experimental comparison using profilometry of\nthe dried blood residue.\nThe boundary condition for pressure field on the drop surface z=h(r, t) is given by\np|z=h(r,t)=−σk (3.4)\nwhere σis the surface tension and kis the curvature of the drop interface given by\nk=1\nr∂\n∂r\u0012\nr∂h\n∂r\u0013\n(3.5)\nFurther, at the drop interface z=h(r, t) we have\n∂u\n∂z\f\f\f\nz=h(r,t)= 0 (3.6)\nowing to the fact that the radial velocity uhas negligible variation in the axial direction\nz. The no slip and no penetration boundary condition at the surface of the substrate\nz= 0 ensures\nu|z=0= 0 (3.7)\nand\nw|z=0= 0 (3.8)16 Roy et al.\nFigure 10. Non dimensional drop height variation as a function of non dimensional radial\ncoordinate at various non dimensional time instants (a) ¯t= 0−250, (b) ¯t= 300 −550, (c)\n¯t= 600 −850, (d) ¯t= 900 −1150, (e) ¯t= 1150 −1200, (f) magnified view of non dimensional\nheight profile near ¯ r= 0−0.2 for time instants ¯t= 1150 −1200, (g) ¯t= 1200 −1450. (h)\nTheoretical and experimental comparison of the drop center height evolution.\nUsing equation (3.3) - (3.8) the radial velocity ucan be written as\nu=−σ\nη∂\n∂r\u00121\nr∂\n∂r\u0012\nr∂h\n∂r\u0013\u0013\u0012z2\n2−hz\u0013\n(3.9)\nThezcomponent of fluid velocity is given by\nw=σ\nr∂\n∂r\u0012r\nη∂\n∂r\u00121\nr∂\n∂r\u0012\nr∂h\n∂r\u0013\u0013\u0012z3\n6−hz2\n2\u0013\u0013\n(3.10)\nThe drop height average radial velocity < u > is given as\n< u > =1\nhZh\n0udz=h2\n3σ\nη∂\n∂r\u00121\nr∂\n∂r\u0012\nr∂h\n∂r\u0013\u0013\n(3.11)\nUsing the continuity equation for the conservation of mass of the volatile component\n(here solvent majorly is the water content of the blood drop), the average radial velocity\nfrom equation (3.11), the dynamical equation for the drop height is given as\n∂h\n∂t=−1\nr∂(rh < u > )\n∂r−J\nρ(3.12)\nwhere Jis the evaporation flux and ρis the density of the droplet. Similarly, using the\ncontinuity equation for the non-volatile component, i.e. solute (conservation of mass for\nthe solute; here the major fraction of the ), the dynamics of the height averaged soluteBlood drop evaporation 17\nconcentration < c > is given as\n∂(h < c > )\n∂t=−1\nr∂\n∂r(rh < c >< u > ) (3.13)\nFor small drop size where gravity is negligible compared to surface tension effects, the\ndrop shape can be approximated by a spherical cap model. However the spherical cap\nmodel leads to singularities in the evaporation flux and flow rate near the contact line.\nAs a result, we replace the initial drop shape by a paraboloid of revolution without the\nloss of generality as a initial condition. The intial drop shape is therefore given by\nh|t=0=hf+h0\u0012\n1−\u0010r\nR\u00112\u0013\n(3.14)\nwhere hfis the drop height at the edge that corresponds to the precursor plasma film.\nh|r=R=hf (3.15)\nThe sum of hfandh0represents the central drop height thickness, i.e., at r= 0. Owing\nto the constant contact radius (CCR) mode of drop evaporation, the three phase contact\nline of the drop is pinned and hence the radial velocity vanishes. The radial velocity also\ngoes to zero at r= 0 due to vanishing curvature at the center. Therefore the boundary\ncondition for the height average radial velocity < u > to solve equation (3.12) and (3.13)\nsimultaneously is given by\n< u > |r=0=< u > |r=R= 0 (3.16)\nThe symmetry condition for the drop height profile gives\n∂h\n∂r\f\f\f\nr=0= 0 (3.17)\nSimilarly, the symmetry condition for the height averaged solute concentration is given\nby\n∂ < c >\n∂r\f\f\f\nr=0= 0 (3.18)\nThe initial height averaged solute concentration field is given by a known function f(r)\nof radial coordinate\n< c > (r,0) =f(r) (3.19)\nWe solve the equations for the drop height, solute concentration and evaporation flux in\nnon-dimensional coordinate space. The drop height is normalized with respect to initial\ndrop height h0ashf≪h0, i.e., ¯h=h/h 0. The radial coordinate ris normalized with\nrespect to drop contact radius R, i.e., ¯ r=r/R. The height average radial velocity < u >\nis normalized with respect to the viscous velocity scale uc=η0/ρh 0, i.e. ¯ u=< u > /u c.\nHere η0is the viscosity of the solvent. The time scale tis normalized by a reference\ntime scale R/u c, i.e., ¯t=tuc/R. The vapor flux Jis normalized by a reference flux\nJc=k∆T/Lh 0, i.e., ¯J=J/Jc; where kis the thermal conductivity of the liquid, ∆Tis\nthe difference between substrate and the saturation temperature and Lis the latent heat\nof vaporization. The height averaged solute concentration < c > was normalized by the\ngelation concentration cg, i.e, ¯< c > =< c > /c g. Using the above normalized variables\nequation (3.12) and (3.13) can be non-dimensionalized and written as a vector equation\nwith ¯Fand ¯fexpressed as a column vector\n∂¯F\n∂¯t+ ¯u∂¯F\n∂¯r=¯f (3.20)18 Roy et al.\nwhere\n¯F=\u0012¯h\n¯< c >\u0013\n(3.21)\n¯f= \n−¯h\n¯r∂\n∂¯r(¯r¯u)−E¯J\nE¯<c> ¯J\n¯h!\n(3.22)\nand\n¯u=¯h2\n3Ca∂\n∂¯r\u00121\n¯r∂\n∂¯r\u0012\n¯r∂h\n∂¯r\u0013\u0013\n(3.23)\nHere E=k∆T/ϵη 0LandCa=ηuc/ϵ3σrepresents the evaporation and capillary number\nrespectively, where ϵ=h0/R. The evaporative mass flux is modelled using techniques\nfrom heat transfer analysis and also assuming that the vapor density vanishes at the\nsol-gel propagating front; i.e., the evaporative flux goes to zero as the concentration\nreaches a critical value of ¯< c > = 1 ( < c > =Cg) during sol-gel phase transition at the\npropagating front. The dimensionless evaporation flux is given as\n¯J(¯r,¯t) =1−¯< c >2\nk+¯h(3.24)\nwhere kis a dimensionless non equilibrium parameter. The range of kis such that k→∞\nfor non-volatile liquids and k→0 for volatile liquids.\nThe system of differential equation for ¯hand ¯< c > given by equation (3.20) is\ncomputed using finite difference methods for various values of EandCacorresponding to\nwhole blood droplets used in our experiments. Whole blood drop of 3 −5µL drop volume\nwith contact radius ( R), drop height ( h0) and contact angle ( θ0) were used for the current\nexperiments. The initial condition for the nondimensional concentration profile required\nto solve equation (3.20) is given by\n¯< c > (r,0) = 2 −¯c0+2(¯c0−1)\n1 +ew(¯r−1)(3.25)\nwhere wis a measure of a characteristic length over which the concentration of col-\nloidal particles increases rapidly. From the above distribution it is important to note\nthat ¯< c > (0,0)≃0 and ¯< c > (1,0) = 1 suggesting the initial concentration at the\nedge of the droplet is equal to the gelation concentration. Figure 9(a) shows the non\ndimensional average concentration profile for various non dimensonal time instants\n¯t= 0,100,200,300,400. Figure 9(b) shows the evolution of the evaporation flux profile\nalong the drop interface for various non dimensonal time instants ¯t= 0,100,200,300,400.\nFigure 9(c) shows the non dimensional drop height profile as a function of non dimensional\nradial coordinate with non dimensional time as a parameter. Figure 9(d) compares the\nfinal dried residue droplet thickness profile with the steady state theoretical prediction.\nThe theoretical height profile (red dotted line) conforms with the experimental curve\nobtained from profilometry (black). Figure 10(a)-(g) shows the non dimensional height\nprofile variation as a function of non dimensional radial coordinate with non dimensional\ntime as a parameter in the range ¯t= 0−1450. Figure 10(h) shows the droplet center\nheight variation as a function of non dimensional time t/t∗. The theoretical curve in red\n(labelled a the.) agrees with the experimental variation (black square).\n3.5.Characterization of dried blood residues\nFigure 11 (a) shows the SEM (scanning electron microscope) image of the dried residue\nof whole blood drop. Various kinds of cracks are observed in general ranging from radialBlood drop evaporation 19\nFigure 11. Blood dried residue characterization using (a) SEM. (b) Radial flake size\ndistribution of the dried whole blood drop residue. (c) Cracks in dried whole blood precipitate\ncracks at the thickest portion to mudflat type of cracks at the drop center and close to\nthe droplet outer edge. The cracks are a result of dessication stress that forms during the\nphase transition from wet gel to dry gel. The cracks in general intersect each other either\nat right angles or 120◦. The various cracks distributes the dried residue into several flakes.\nFigure 11 (b) shows the radial size distribution of the flakes plotted as an area fraction\nratio. Here Afdenotes the area of a particular flake with a radial coordinate rf.Adenotes\nthe area of the entire dried residue and Rdenotes the drop contact radius. The flake size\nis highest between rf/R= 0.6−0.8, i.e., in the rim region. The flake size is smaller in the\ndrop center and the outer edge of the droplet. We observe that the crack thickness, flake\nsize are therefore proportional to the thickness of the dried residue. Regions having higher\nthickness will have larger flake size in general. Figure 11(c) depicts the various kinds of\ncracks formed as a result of dessication stress while transforming from wet gel to dry gel\nin phase C of blood drop evaporation. Figure 12 depicts the micro/nano characterization\nfor dried residue of pure plasma. We also observe that bacteria laden blood droplets\ndo not show significant difference in terms of evaporation characteristics from pure\nwhole blood droplets within the concentration range typically found in living organisms\nsimO(10−109) CFU/ml. This is due to the fact that the RBCs are approximately one\norder ( simO(10)) bigger in length scale and hence approximately two order ( simO(102))\nbigger in terms of area ratio. At the maximum bacterial concentration of 109CFU/ml20 Roy et al.\nFigure 12. Blood plasma dried residue characterization\n.\nFigure 13. Dried residue characterization using confocal microscopy for (a) Plasma and (b)\nWhole blood droplet. The red scale bar represents 500 µm.\n.Blood drop evaporation 21\nwill give approximately similar number density as RBCs. However, owing to the area\nratio, the bacteria will be uniformly distributed throughout the plasma protein matrix\nwith highest fraction embedded and packed between the curvatures of the RBCs. For\nbacteria to have appreciable effects in the corresponding evaporation physics the packed\ncell volume has to be affected significantly by the presence of bacteria. This can only\noccur at significant higher bacterial concentration that can affect the PCV (packed cell\nvolume) of the RBCs significantly. The distribution of bacteria in the dried blood residue\ncan be seen from confocal microscopy and SEM of the dried residue as shown in Figure\n13. Figure 13 (a), 13 (b) shows the bacterial deposition in the dried residue of blood\nplasma and whole blood droplet using confocal microscopy.\n4. Conclusion\nIn conclusion, we study the mechanics of sessile whole blood drop evaporation using\ndirect experimental visualization and theoretical methods like color image time sequence\nanalysis, lubrication analysis and micro/nano characterization. We identified that blood\ndrop evaporation in general can be subdivided into three phases (A,B,C) based on the\nevaporation rate. Phase A is the fastest and consists of the gelation of the three phase\ncontact line due to the radial transport of RBCs due to the outward capillary flow.\nThe gelation occurs due to sol-gel phase transition. Phase B consists of intermediate\nevaporation rate in which the gelation front moves radially inward. The radially inward\ngelation front propagation along with the droplet height reduction causes the phase\ntransition of the entire droplet into wet gel state. The wet gel phase changes to a dry gel\nphase in Phase C of droplet evaporation. The evaporation flux reduces significantly as the\npartial pressure of water vapor is reduced significantly due to Raoult’s law. As the wet gel\ntransitions to dry gel phase, the dried residue precipitate undergoes dessication stresses\nthat leads to the formation of different kinds of cracks on the dried precipitate. The\ncrack characteristics depends highly on the local thickness of the precipitate and hence\na one to one map exists between precipitate thickness and the corresponding cracks that\nare formed. We also show that the presence of bacteria does not change the evaporation\nphysics and the corresponding cracks\nDeclaration of Interests\nThe authors declare no conflict of interest.22 Roy et al.\nREFERENCES\nAdachi, Eiki, Dimitrov, Antony S & Nagayama, Kuniaki 1995 Stripe patterns formed on\na glass surface during droplet evaporation. Langmuir 11(4), 1057–1060.\nAllain, C & Limat, L 1995 Regular patterns of cracks formed by directional drying of a\ncollodial suspension. Physical review letters 74(15), 2981.\nBahmani, Leila, Neysari, Mahdi & Maleki, Maniya 2017 The study of drying and pattern\nformation of whole human blood drops and the effect of thalassaemia and neonatal\njaundice on the patterns. 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SPIE."    },    {        "title": "2402.12340v1.Simple_Mechanisms_for_Utility_Maximization__Approximating_Welfare_in_the_I_I_D__Unit_Demand_Setting.pdf",        "content": "arXiv:2402.12340v1  [cs.GT]  19 Feb 2024Simple Mechanisms for Utility Maximization: Approximatin g\nWelfare in the I.I.D. Unit-Demand Setting\nKira Goldner∗Taylor Lundy†\nFebruary 20, 2024\nAbstract\nWe investigate the objective of utility maximization from the perspec tive of Bayesianmech-\nanism design, initiating this direction, and focus on the unit-demand s etting where values are\ni.i.d. across both items and buyers. We take the approach of develop ing simple, approximately\noptimal mechanisms, targeting the simplest benchmark of optimal w elfare. We give a (1 −1/e)-\napproximation when there are more items than buyers, and an O(log(n/m))-approximation\nwhen there are more buyers than items, which is tight up to constan t factors. We also charac-\nterizecomplexitiesinthissettingthatdefyourintuitionfromthewelf areandrevenueliterature,\nand motivate why coming up with a better benchmark than welfare is a hard problem itself.\n1 Introduction\nUtility maximization—also sometimes called consumer surp lus, residual surplus, money burning,\nor a setting with ordeals—is best motivated by a social servi ce provider who wishes to allocate\ngoods whose demand far exceeds their supply (or budget), suc h as medications and vaccines. In\nsuch cases, it would beinequitable to gate-keep these goods by charging a price. Instead, acquiring\nthese items is often made laborious by requiring consumers t o wait in long lines, fill out forms,\ntravel physical distances, or even try other medications fir st. These ordealsact like payments\nby ensuring that only those who have high value for the good ar e actually able to obtain it. As\nwith social welfare maximization, the social service provi der has the option to use (non-monetary)\npayments as a tool in order to allocate items to those with hig her need; in contrast, in this setting,\ndoing so comes at a cost to the objective function, as neither consumer nor seller benefit from\nthese ordeals. Crucially, this yields a trade-off in the obje ctive function between allocation and\npayment—is it more important to give the goods to the right pe ople, potentially using expensive\npayments that decrease utility to do so, or to minimize payme nts, allowing the allocation to be\nmore random? Despite sitting between social welfare maximi zation and payment minimization,\nintuition from both regimes can fail, requiring clever solu tions somewhere in between.\n∗Boston University; goldner@bu.edu . Supported by NSF Award CNS-2228610 and a Shibulal Family Ca reer\nDevelopment Professorship.\n†University of British Columbia; tlundy@cs.ubc.ca . Supported by an NSERC Discovery Grant, a DND/NSERC\nDiscovery Grant Supplement, a CIFAR Canada AI Research Chai r (Alberta Machine Intelligence Institute), awards\nfrom Facebook Research and Amazon Research, and DARPA award FA8750-19-2-0222, CFDA #12.910 (Air Force\nResearch Laboratory).\n1Themultidimensional settingis onlymorecomplex. Much lik erevenuemaximization, whilethe\nsingle-dimensionalsettingissolvedwithaclosed-formth eoryfortheoptimalmechanism[Hartline and Roughgarden \n2008], in the multidimensional setting, the problem become s unwieldy. Nothing is known here on\nthe structure of the optimal mechanisms. And, unlike revenu e, the single-bidder multidimensional\nproblem does not give us a foothold, as the tension between effi cient allocation and using pay-\nments comes only from limited supply between agents. In fact , we do not even have strong upper\nboundson expected optimal utility beyond the trivial bound of social welfare. Finding the optimal\nmechanism for the general multidimensional setting is like ly a herculean and intractable task, and\nthe resulting mechanism is likely to be highly complex. Theo rem 1 gives some insight on how\ncomplex this mechanism may be, including the fact that the utility-optimal mechanism sometimes\nmustthrow away an item just to guarantee optimality! We therefore follow in the footsteps of the\nrevenue maximization literature and attempt to find a simple mechanism that can approximate\nthe utility of the optimal mechanism.\nSummary. We initiate the study of multidimensional Bayesian utility maximization . We inves-\ntigate simple, approximately-optimal mechanisms for this setting. In particular, the motivation\nfor this paper is to answer the following question.\nQuestion 1. What approximation to optimal utility can we obtain and by wh at simple mecha-\nnisms when buyers have unit-demand valuations?\nFavorites Mechanisms: We use the same underlying principle for all of our results: e ach buyer\npoints to their favorite item, and then allocation is handle d per-item among those who declare it\ntheir favorite via some fixed single-item mechanism. In the m ore-items-than-bidders regime, the\nsingle-item mechanism assigns items uniformly at random am ong those competing for the same\nfavorite. In the more-bidders-than-items regime, we use a k nown single-dimensional prior-free\nmechanism from Hartline and Roughgarden [2008] to allocate each individual item.\nResults. In this paper, we approximate optimal utility when there are nbuyers,mitems, and\nbuyer values for items are drawn i.i.d. across both items and buyers. We achieve a (1 −1/e)-\napproximation when there are more items than buyers ( m≥n) and an O(logn\nm)-approximation\nwhen there are more buyers than items ( n > m), which is tight up to constant factors. Together,\nthis implies a Θ(1+logn\nm) gap between optimal utility and social welfare. See Sectio n 4 for details.\nNote two potential drawbacks of our work: (1) we approximate the upper bound of social\nwelfare, and (2) we focus only on the highly symmetric settin g of i.i.d. across both items and\nbuyers. While these may at first glance appear to create an ove rly restrictive setting, in Section 3\nwe demonstrate the complexities of even slight relaxations , where our machinery fails, and why\nmeaningful extensions will require much innovation. This s ection includes several characterization\nresultsontheunit-demandBayesian utility-maximization problemingeneral, aswell aslimitations\nof various approaches. We include here informal statements of our results:\n•(Theorem 1) The utility-optimal unit-demand mechanism may not allocate all items.\n•(Theorem 2) Contrary to revenue, the optimal utility of the c opies setting is not an upper\nbound on every deterministic unit-demand mechanism.\n2•(Theorem 3) Relaxing i.i.d. to uniformly distributed favor ite items violates BIC in the\nPrior-Free-Favorites mechanism.\nTheseresultsmotivate whywestudytheexactsettingwedoan dwhyit isworthyof study: why\nwe aim for approximation, why we settle for social welfare as an upper bound, and why we cannot\nrelax i.i.d. Section 4 then resolves Question 1 in this setti ng with a Θ(1+logn\nm)-approximation.\n1.1 Related Work\nUtilitymaximizationhasbeenwell-exploredfromthepubli cfinanceliteratureineconomics[McGee,\n2003] and the strategic queuing literature in operations [H assin and Haviv, 2003]. Ordeals have\nalsobeenwell-studied in economics, bothas amethodtoelic it information withoutusingmonetary\npayments, andintheefficiency-equity trade-offwhenusingin -kindtransfers[Nichols and Zeckhauser,\n1982]. However, utility maximization and ordeals have mini mal prior work from the algorithmic\nmechanism design perspective.\nIn 2008, Hartline and Roughgarden [2008] completely resolv ed the question of optimal utility\nmaximizationinthesingle-dimensionalBayesiansetting. Anearlierpaperby[Chakravarty and Kaplan,\n2006] also studies when, for the single-dimensional settin g of multi-unit unit-demand, the utility-\noptimal mechanism gives randomly gives items away for free. Condorelli [2013] produces a similar\ntheory by extending Myerson [1981] to characterize optimal mechanisms for settings both with and\nwithout money, and parameterizes these results in an agent’ s willingness-to-pay, which does not\nhave to be equal to an agent’s value. Hartline and Roughgarde n [2008] also include a prior-free\nmechanism which, for nbidders and midentical items, achieves an O(1+logn\nm)-approximation\nof utility to social welfare, which they prove is tight. They further introduce an idea for a single-\ndimensional prior-free benchmark and provide a prior-free constant-approximation to that bench-\nmark.\nThe best (and only) known multidimensional approximation i sO(log|Ω|) where Ω is the\nspace of possible allocation outcomes [Fotakis et al., 2016 ]. In the unit-demand setting, this is\nO(mlogn). The mechanism used is a combination of VCG and uniformly ra ndom free allocation\nusing ideas from the above prior-free mechanism. We discuss in Section 3.3 why these ideas are\nnot enough, and how the idea of Favorites is crucial. In addit ion, Fotakis et al. [2016] give an algo-\nrithmic non-constructive result that achieves the same uti lity approximation while simultaneously\nguaranteeing constant welfare, although they note that run ning VCG with probability 1/2 will do\nthis as well.\nRecent work has studied variations on ordeals and money burn ing, but most of these settings\nare single-dimensional. Motivated by health insurance, Es saidi et al. [2020] investigate whether,\nin equilibrium, limiting entry of sellers into a two-sided m arket improves consumer surplus or\nnot. Dworczak et al. [2023] studies a different but related pro blem, the allocation of (divisible)\nmoney, and characterizes when using payments via ordeals to better allocate the money yields\nmore utility than ordeal-free equal payments to all agents. Patel and Urgun [2022] consider the\ninterplay between verification, costly only tothedesigner , andordeals, costly toboth, inthesingle-\ndimensionali.i.d. setting. Lundy et al.[2019]alsostudyt hepowerofverification, butinthesetting\nofallocating indivisibleresourcestofoodbanksusingwai t timesasanordeal. Ganesh and Hartline\n[2023] study the question of pen-testing by reinterpreting it as utility maximization; this is for a\nsingle-parameter environment with feasibility constrain ts.\n3A few recent works foray into multidimensional settings. As hlagi et al. [2023] study the objec-\ntive function that is a weighted sum of allocative efficiency a nd social welfare, which can include\nutility, forthesettingofunit-demandagentsfortotally- ordered objects, restrictedtoi.i.d. Thisset-\ntingwas introducedin[Fiat et al., 2016] andproven tobeinb etween single- andmulti-dimensional\nenvironments by several complexity metrics. Braverman et a l. [2016] study a truly multidimen-\nsional problem, the allocation of heterogeneous healthcar e services to unit-demand patients where\nwait times serve as ordeals, but they focus on the computatio n of efficient equilibria. Yang [2022]\nand Akbarpour et al. [2023] study a multidimensional ordeal setting that is quite different from\nwhat we study. The mechanism designer can allocate either a g ood (which has positive value) or\nan ordeal (which has negative value) and can also use monetar y payments (which are transferable\nand do not impact the objective function). They derives cond itions under which it is optimal\nto only allocate the good and not an ordeal, and under which on e ordeal mechanism dominates\nanother.\nNote that while there is ample work on mechanism design withoutmoney, it is not relevant to\nthis work, as we are particularly exploring the setting with payments that hurt the objective, and\nthe trade-off of using them.\nSimple-Yet-Approximately-Optimal Mechanisms for Seller Revenue. Whilesingle-dimensional\nBayesian revenue maximization is solved in closed-form by M yerson’s Nobel-prize-winning the-\nory [Myerson, 1981], in the multidimensional setting, even selling two heterogeneous goods to a\nsingle additive buyer can be intractable, requiring infinit ely many randomized options to extract\noptimalrevenue[Manelli and Vincent,2006;Daskalakis et a l.,2017]. Asaresultofthisintractabil-\nity, a large body of work has instead focused on approximatio n. In particular, this work has de-\nsignedsimplemechanisms that yield constant-factor approximations to t he optimal revenue in a\nvariety of settings [Cai et al., 2016; Chawla et al., 2010; Ba baioff et al., 2014; Cai and Zhao, 2017;\nEden et al., 2017; Cai et al., 2019; Yao, 2015; Chawla and Mill er, 2016; Rubinstein and Weinberg,\n2015], developing a barrage of techniques and insights alon g the way. There are two main steps\nrequired in these approaches: finding an upper bound on the op timal revenue, and then deter-\nmining which simple mechanisms can be used to approximate th is benchmark. For most of these\nsettings, the mechanisms are some variation on either selli ng all of the items separately, or selling\nthem all together in one grand bundle.\nThis work. The goal of this work is to take this approach for the setting o f multi-dimensional\nBayesian utility maximization. In particular, we focus on w hat the right simple mechanisms may\nbe for this objective, and begin with i.i.d. unit-demand.\n2 Preliminaries\nWe consider the setting where the mechanism designer wishes to sellmheterogeneous items to n\nbidders. Agent i’s valuation is a vector vi∈Rm, wherevijdenotes agents i’s private value (or\nwillingness-to-pay) for item j. Valuevijis drawn independently from a known prior distribution\nvij∼Fij, and in particular, we assume that values are independently and identically distributed\n(i.i.d.) across both items and bidders. That is, that there e xists some type distribution Fsuch\nthatFij=Ffor alli,j. We overload notation and let F(·) denote the CDF and f(·) denote the\nPDF of the distribution. Let F=Fn×mdenote the joint distribution. We use vto denote the\nvector of all agents’ types, v= (v1,...,vn)∈Rn×m.\n4A mechanism Mis comprised of an allocation rule x:Rn×m→[0,1]n×mand a payment rule\np:Rn×m→Rnwherexi(v) andpi(v) are the respective allocation and payment for agent iunder\ntypev. Note that xi(v) is a vector where xij(v) denotes the probability that bidder ireceives\nitemjunder type profile v, andpi(v) is a non-negative number.\nAgents are unit-demand , wanting at most one item. Formally, we would write vi(S) =\nmaxj∈Svij; agentionly gets value from his favorite item among those allocated . However, it\nis without loss to restrict attention to allocation rules th at allocate at most one item to each\nagent, that is, have the feasibility constraint/summationtext\njxij(v)≤1 for all agents iand allv. Then agent\ni’s expected utility is/summationtext\njxij(v)vij−pi, which we rewrite as xi(v)vi−pi(v), where the first term\nrepresents the dot product of two vectors.\nWe define the expected social welfare, utility, and revenue o bjectives:\nUtility =Ev∼F/bracketleftBigg/summationdisplay\nixi(v)vi−pi(v)/bracketrightBigg\nWelfare =Ev∼F/bracketleftBigg/summationdisplay\nixi(v)vi/bracketrightBigg\nRevenue =Ev∼F/bracketleftBigg/summationdisplay\nipi(v)/bracketrightBigg\nand observe by linearity of expectation that\nUtility =Welfare −Revenue .\nA mechanism is Bayesian incentive-compatible (BIC) if, in e xpectation over the other agents’\nreports, it incentivizes truthful reporting:\nEv−i∼F−i[xi(vi,v−i)vi−pi(vi,v−i)]≥Ev−i∼F−i/bracketleftbig\nxi(v′\ni,v−i)vi−pi(v′\ni,v−i)/bracketrightbig\n∀i,vi,v′\ni\nwhere the subscript −iindexes the vector everywhere but i.\nFor an objective, we let Optdenote the optimal amount attainable by any feasible, truth ful\nmechanism, that is,\nOptUtility= max\n(x,p)∈PEv∼F/bracketleftBigg/summationdisplay\nixi(v)vi−pi(v)/bracketrightBigg\n.\nwherePdenotes the set of feasible, BIC, and individually rational mechanisms. Namely for\nfeasibility/summationdisplay\njxij(v)≤1∀i,vand/summationdisplay\nixij(v)≤1∀j,v\nand for individual rationality\nEv−i∼F−i[xi(vi,v−i)vi−pi(vi,v−i)]≥0∀i,vi.\nWe now introduce the mechanism that we use for our main result s paired with two different\nsingle-item mechanisms.\nDefinition 1 (M-Favorites) .Each bidder announces their favorite item j, the item for which\nthey drew the highest value. Then, for the given BIC single-i tem mechanism M, for each item j,\nMis run on the bidders who reported item jas their favorite.\nIn particular, we will focus on two variants.\n5•Random-Favorites: Mallocates uniformly at random among all participants (and c harges\nnothing). That is, each item is allocated uniformly at rando mly among one of the bidders\nwho declares it their favorite.\n•Prior-Free-Favorites: Misthesingle-itemprior-freemechanismgiveninHartline a nd Roughgarden\n[2008] (detailed in Section 3.4).\nWe now prove that this mechanism is truthful in the i.i.d. set ting.\nLemma 1. TheM-Favorites mechanism is Bayesian incentive-compatible (B IC).\nProof.We must prove that it is in every agent’s best interest to repo rt their favorite item. This\nfollows from the fact that each bidder’s value for each item vijis drawn i.i.d. This implies that\nthe same expected number of bidders will report each item as t heir favorite. This means that, for\nany reported value, independent of the item, the expected pr obability of allocation across items\nis the same. Finally, as Mis truthful, the allocation obtained by participating in it is monotone\nnon-decreasing in the bidder’s report. Therefore, the bidd er will achieve the highest utility by\nparticipating in the mechanism for the item for which they ha ve the highest value, that is, their\nfavorite item. Since this allocation is BIC1, there exist payments that implement it.\n3 Complexities of General Unit-Demand\nInthissection, wemotivatewhyourmainresultsinSection4 arerestrictedtocomparetotheupper\nbound of optimal social welfare and to the i.i.d. setting. We do so by describing the complexities\nof maximizing expected utility in the general unit-demand s etting.\n3.1 Optimal Multidimensional Mechanisms\nWe present a surprising lack of structure on the utility-optimal mechanism for the unit-demand\nsetting, and why this motivates our choice to study simplici ty and approximation. We begin by\nreviewing what is known from the single-dimensional settin g.\nSingle-dimensional Mechanisms. In the single-parameter setting, the optimal mechanism\nalwaysallocates the items, it’s just a question of to who and for how much. The mechanism takes\nan analogous structure to the revenue-optimal mechanism, m aximizing “ironed” virtualwelfare,\nwhere for utility maximization, a buyer’s virtual value is θi(vi) =1−Fi(vi)\nfi(vi). To iron into ¯θi, this\nfunction is averaged until it is monotone non-decreasing in vi. Then, if there is one item, give it\nto the bidder with the highest ironed virtual value ¯θi(vi).\nWhat this mechanism concretely looks like depends highly on thehazard rate of the buyer’s\ntype distribution,fi(·)\n1−Fi(·). In one extreme case, when the hazard rate is non-decreasing (monotone\nhazard rate, or MHR), ¯θiis fully averaged, resulting in a mechanism that allocates u niformly at\nrandom and charges no payments. In the other extreme, when th e hazard rate is non-increasing\n(anti-MHR), θidoes not need to be averaged as it is already monotone non-dec reasing, and the\nmechanism looks like a second-price auction but in this virt ual space.\n1It is worth noting that this procedure is not DSIC. The fact th at the allocation is the same across items only\nholds in expectation.\n6Multidimensional Mechanisms. From this characterization, we can already expect the opti-\nmal multidimensional mechanism to have cases that make it lo ok like VCG, requiring payments,\nand cases that make it look like random allocations with no pa yments. However, we conclude that\ngetting a handle on the structure of this mechanism may be hig hly difficult.\nDespite the goal being to maximize utility—a dual goal of giv ing away all of the items most\nefficiently and charging as minimal payments as possible—it t urns out we can’t even rule out the\npossibility that the optimal mechanism does not allocate an item in order to better allocate the\nother items and avoid using payments. Mathematically, this means that we cannot represent the\noptimal allocation as a convex combination of deterministi c allocations.\nTheorem 1. If we are maximizing utility over all allocations and payment s, it isnotwithout\nloss to restrict attention to convex combinations of determ inistic allocations:/summationtext\nixij(v) = 1∀j,v.\nThere are instances where it maximizes utility not to allocat e an item jin every instance.\nNote that this applies to the general Bayesian utility maxim ization problem, and is not re-\nstricted to the i.i.d. setting—our construction will not be i.i.d.\nProof.We construct a setting in which the optimal mechanism must so metimes discard an item.\nConsider a setting with two unit-demand bidders and two item s. Bidder 1 has values ( v11,v12) =\n(1,3) deterministically. Bidder 2 has two potential type profil es with equal probability,\n(v21,v22) =/braceleftBigg\n(c,c+1) w.p. 1 /2\n(1,4) w.p. 1 /2.\nThe efficient allocation is x2(1,4) = (0,1) andx2(c,c+1) = (1 ,0), allocating the other item\nto bidder 1, yielding expected welfare 0 .5c+4. However, we must charge a payment of 1 to the\ntype (1,4) to prevent the bidder with type ( c,c+ 1) from misreporting, and this hurts utility.\nWe now show that any convex combination of deterministic all ocations loses an additive1\n2utility\ncompared to welfare.\nAssume our mechanism isa convex combination of deterministic allocations. First o bserve\nthat because we have an equal number of bidders and items, for each type profile, not only do\nthe allocation probabilities sum to 1 for each item across bi dders, but they also must sum to 1 for\neachbidderacross items. As v1is fixed, we’ll write our input to x2(·) andp2(·) asv2rather than\nv. Now, for some probabilities 0 ≤α,β≤1, letx2(c,c+1) = (α,1−α) andx2(1,4) = (β,1−β)\n(where bidder 1 is allocated whatever bidder 2 is not by assum ption). Then the resulting welfare\nfrom these allocations is\nWelfare = 0.5\nαc+(1−α)(c+1)/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\nbidder 2+(1−α)+3α/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nbidder 1\n+0.5\nβ+4(1−β)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nbidder 2+(1−β)+3β/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nbidder 1\n\n= 0.5(c+2+α)+0.5(5−β).\nThe payment we must charge bidder 2 to prevent misreporting i sp2(1,4) =α−β, giving utility\nUtility = 0.5(c+2+α)+0.5(5−α)\n= 0.5c+3.5.\n7Alternatively, we could offer the allocation x2(1,4) = (0,c\nc+1). Notice that the type ( c,c+1) is\nindifferent between this allocation and the allocation x= (1,0), and therefore requires no payment\nfor incentive-compatibility. This allocation achieves th e efficient outcome with probabilityc\nc+1and\ntherefore is arbitrarily close to welfare as cgrows large.\nThe construction in this proof actually highlights importa nt intuition about preventing mis-\nreporting via payments versus allocations. If the additive gap between an agent’s favorite type\nand allocated type is small, it is best for utility to maintai n incentive-compatibility with small\npayments, i.e., money burning. In contrast, if the ratio bet ween these types is small (as in the\nproof), we can best maintain incentive-compatibility with a fractional allocation, i.e., very rarely\n“burning” a good.\nThis result evidences complexity in optimal mechanisms, an d combined with the environment’s\nsimilaritiestorevenuemaximization, providesaconvinci ngargumentthatdesigningutility-optimal\nmechanisms is extremely likely to be intractable. As a resul t, we focus on approximation.\n3.2 Finding An Upper Bound on Optimal Utility\nApproximation to an unknownquantity like optimal expected utility requires a good (small) upper\nbound. The most naive upper bound on utility is optimal social welfare , which is what our result\napproximates, and thus also bounds the gap between optimal u tility and optimal welfare in the\nBayesian setting.\nHowever, one might wonder if there’s a nice upperbound on opt imal utility that is tighter than\nwelfare. Ouransweris, perhaps,butcertainlynotonethati seasytofind. Forinstance, therevenue\nmaximization literatureprovidesmanydifferentapproaches , buttothebestofourknowledge, none\nof them can easily be adapted to utility to obtain a better upp er bound. To illustrate this point,\nwe briefly discuss the difficulties in using two of these upper b ound approaches: the copies setting\nand the Lagrangian duality flow-based upper bound.\nThe Copies Setting. A commonly used upper bound on revenue is the copies setting, first\nintroduced in Chawla et al. [2007]. Each multidimensional b idder is split into mcopies who\npossess the same value as the original agent for item j, but are onlyinterested in item j. This is\nnow a single-dimensional setting. The copies setting is qui te useful for bounding optimal revenue;\nfor instance, the revenue of every deterministic unit-dema nd mechanism is upper-bounded by the\noptimal (single-dimensional) revenue in the copies settin g [Chawla et al., 2010]. However, this is\nnot the case for utility. While splitting each bidder into co pies eliminates incentive issues across\nitems, it also increases the competition for each item, whic h drives down utility. We formalize this\nintuition below.\nTheorem 2. There exists a distribution Fand deterministic mechanism Msuch that the expected\nutility of MoverFis greater than that of single-item optimal utility in the co pies setting induced\nbyF.\nProof.Consider a setting with 2 items and 2 bidders whose values are drawn i.i.d. from any anti-\nMHR distribution. In the induced copies setting, because Fis anti-MHR, the optimal per-item\nmechanism is a Vickrey (second-price) auction.\nConsider the deterministic multidimensional mechanism Vickrey-Favorites :M-favorites (Def-\ninition 1) with Vickrey as our per-item mechanism. For each v alue profile in which both bidders\n8have the same favorite item, this mechanism achieves the sam e utility as the copies setting. How-\never, when the bidders have different favorite items, the Vick rey-Favorites mechanism allocates\nefficiently whilecharging a payment of 0, whilethecopies set ting must still charge aprice, resulting\nin less utility.\nDuality-Based Upper Bounds Another technique used with great success in recent years is\na Lagrangian-duality-based approach to formulating an upp er bound on optimal revenue. This\nmethod formulates the optimization problem as a linear prog ram, relaxes certain constraints using\nLagrange multipliers, and takes its dual program. Within th e dual program, any feasible dual\nvariables result in a dual objective which upper bounds the o ptimal primal. Cai et al. [2016] rein-\nterpret these feasibility constraints as a network flow prob lem, and use this structure to construct\ndual variables that lead to a nice (tighter) upper bound on re venue.\nWhile the dual of the utility maximization linear program ca n also be expressed as a network\nflow problem, it lacks some of the key properties used in the re venue maximization approach\nto construct the “right” dual variables. Importantly, the s ingle-item revenue maximization flow\nproblem has an optimal solution that can be generalized to th e multi-item setting. The single-item\nutility-maximization dual, however, has countless trivia l optimal solutions to the flow problem.\nThen to create a useful flow, one must work in the much more comp lex multi-item setting at the\nonset. Furthermore, as expected by this single-item intuit ion, analogues of and variations on the\nmulti-item revenue flow yield upper bounds for utility that a re even greater than social welfare.\nAn Open Question. The above motivates a very important and large open question for this\ncommunity, likely to require technical innovation: what is a tractable upper bound on optimal\nutility? Is there a framework in the spirit of [Cai et al., 201 6] to producesuch upper boundsacross\nvarious multidimensional environments?\n3.3 Candidate Simple Mechanisms\nWith an upper bound chosen, the question remains: what simpl e mechanism yields utility that\napproximates social welfare? We now review natural candida tes.\nAs discussed in Section 3.1, at its extremes, the optimal sin gle-dimensional mechanism looks\nlike a uniformly random free allocation or a second-price au ction. For this reason, we take the mul-\ntidimensional analogues of these two mechanisms as candida tes for natural “simple” mechanisms\nthat we may wish to consider when approximating utility—(1) a uniformly random free allocation\nof goods to buyers, and (2) the VCG mechanism [Vickrey, 1961; Clarke, 1971; Groves, 1973], the\nwelfare-optimal mechanism that is the multidimensional an alogue of the second-price auction.\nRandomly giving the items away for free has clear utility ben efits, as it is the payment-\nminimizing allocation. However, with multiple items, it is easy to imagine an outcome where\ntwo of the random winners who would prefer to exchange items. Then a clear improvement over\nthis mechanism would allow agents to express preferences ov er which item they are in contention\nfor.\nIncontrast, VCGnot onlyallows agents toexpresspreferenc es, butit ensuresthat theitems are\nallocated in an efficient way. When agents prefer different item s, VCG can actually allocate these\nitems for free, yielding utility that matches welfare. Wher e VCG does poorly is when multiple\nbidders prefer the same item, VCG charges potentially very l arge payments in order to ensure\nthe efficient allocation. In fact, Fotakis et al. [2016] show t hat VCG’s utility cannot better than\n9|Ω|-approximate social welfare where Ω is the space of allocati ve outcomes, but their proof uses\nsingle-minded agents (with complementarities) to show thi s.\nThis motivates a mechanism that allows bidders to express th eir preferred item while handling\n“collisions” in buyer preferences in a way that does not hurt utility too much. Our simple mech-\nanism essentially must be of the form where buyers express th eir favorite item, and allocation is\ndetermine from there. We define a family of mechanisms which w e callM-Favorites (Definition 1),\nwhich allow agents to report their favorite items, and then u ses some fixed mechanism Mto re-\nsolve conflicts between bidders with the same favorite. A hug e added benefit is the tractability\nthis brings by allowing us to decompose the problem into many single-item problems. Moreover,\nwe have the property that if items are distributed i.i.d., th enM-Favorites is BIC whenever Mis\nBIC (Lemma 1).\n3.4 Complications Beyond i.i.d.\nHaving settled on the M-Favorites mechanism, we now reexamine the two places where we use\nthe i.i.d. assumption in our main results. First, the incent ive-compatibility guarantees of the\nM-Favorites mechanism proven in Lemma 1 relied highly on symm etry. Second, some of our\napproximation analysis does as well, but we suspect known te chniques may extend it beyond i.i.d.\nIn this section, we take a closer look at how the truthfulness of theM-Favorites mechanism\nrelies on the i.i.d. assumption. At first glance, it seems to b e only leveraging the fact that each\nof “single-item” mechanisms is identical and has the same nu mber of participants in expectation.\nA natural weaker condition would be to just assert this direc tly—that is, a setting in which, for\neach bidder, the probability that any given item is their fav orite is uniform across items. We call\nthis setting the uniform favorites setting.\nWhile the Random-Favorites mechanism used in Theorem 4 rema ins BIC in the uniform\nfavorites setting—an agent’s utility depends only on one’s value and the expected number of\ncompetitors—the Prior-Free-Favorites mechanism used in T heorem 5 does not.\nTheorem 3. In the uniform favorites setting, the Prior-Free-Favorites mechanism is not guaran-\nteed to be BIC.\nWe review the Prior-Free mechanism of Hartline and Roughgar den [2008]:\n•Rename the bids participating in this single-dimensional m echanism v1> ... > v n′.\n•Choosejuniformly at random from {0,...,log(n′)}.\n•Run av2j+1-lottery (with vn′+1:= 0): choose a bidder uniformly at random from those who\nreport at least v2j+1, then allocate to this bidder and charge price v2j+1.\nProof.We construct an example for the uniform favorites setting th at violates BIC. Consider n\nbidders and 2 items, where each bidder draws their values i.i .d. from the following item distribu-\ntions:\nvi1∼U(0,1) and vi2∼U{0,1}.\nThis is a uniform favorites setting, as each buyer prefers ea ch item with probability 0 .5.\nConsider an agent ℓwith values vℓ= (1−ǫ,1). If they report truthfully that they prefer item\n2, that’s the item they will be placed in contention for. Assu ming there are n′agents total who\n10prefer item 2, then the agent’s expected utility is1\n1+log(n′)(1\nn′). To see this, observe that ∀n′>1\nandj/ne}ationslash= log(n′), the agent receives 0 utility as v2j+1= 1.\nOn the other hand, if the agent misreports v′\nℓ= (1−ǫ,0), they are placed in contention\nfor item 1. In this case, in the case where j= log(n′), the agent gets utility (1 −ǫ)1\nn′, and in\nthe case j/ne}ationslash= log(n′), they gain some strictly positive expected utility that de pends only on n′.\nTherefore, there exists ǫ >0 such that this misreport provides strictly higher utility than truthful\nreporting.\nAs even this slight weakening of i.i.d. can cause a violation of BIC for M-favorites depending\nonM, we restrict our attention to the i.i.d. setting.\n4 Main Result: Approximating Utility via Welfare\nHaving motivated thestudyof approximating expected utili ty via social welfare in thei.i.d. setting\nusing a favorites mechanism, we are ready for our main result s. In this section, we prove a bound\non the gap between the optimal welfare and the optimal utilit y in the unit-demand setting. The\nbest known boundon this gap is that optimal utility is a log( |Ω|)-approximation to optimal welfare\n[Fotakis et al., 2016], where Ω is the set of possible outcome s. With nagents and mitems the\nnumber of possible outcomes is nm, meaning this bound is approximately O(mlogn).\nOur results give two new tighter bounds for the i.i.d. unit-d emand setting:\n•in the case where m≥n, we give a constant 1 −1\nebound, and\n•in the case where m < n, we give an O(logn\nm) bound, which is tight up to constant factors,\nimplying an overall Θ(1+logn\nm)-bound.\nWe use the M-Favorites mechanism for both cases: separating bidders by their favorite item\ninto a single-item setting. However, we use different single- dimensional mechanisms Min each\nregime to allocate the items: Random-Favorites in the m≥nregime and Prior-Free-Favorites in\nthen > mcase.\nWe remark that runningeither mechanism does not requirekno wledge of the priordistribution,\nso these mechanisms are actually prior-independent, altho ugh still restricted to the i.i.d. setting\nin order for BIC and the approximation analyses to hold.\nIt may also appear that these regimes could be unified via the P rior-Free-Favorites approach.\nWe point out in Section 4.2 where the analysis breaks for n < m, renderingboth regimes necessary.\n4.1 More Items than Bidders\nIn this section we prove a constant bound between optimal soc ial welfare and optimal utility using\nthe Random-Favorites mechanism.\nTheorem 4. Random-Favorites achieves utility at least to/parenleftbig\n1−1\ne/parenrightbig\nWelfare whenm≥n.\nProof.We begin with the case of an equal number of bidders and item ( m=n). In the Random-\nFavorites mechanism, an item is notallocated only when it is not any agent’s favorite item, whic h\noccurs with probability (1 −1/m)n. Asm=n, then the item isallocated with probability\n1−/parenleftbigg\n1−1\nm/parenrightbiggn\n≥1−1\ne.\n11Because m=n, if an item goes unallocated, then so does an agent, and vice v ersa. Hence, the\nprobability that an item is allocated is equal to the probabi lity than an agent is allocated.\nAsmgrows larger than n, the probability that an agentis allocated only increases: with more\nitems, the probability that agents collide on their favorit e is less likely, and this is the only event\nin which an agent is not allocated. Then for all m≥n, the probability that an agent is allocated\nin the Random-Favorites mechanism is at least 1 −1/e.\nAn agent’s value for their favorite item is, by definition, at least as high as their value for the\nitem they receive in the social-welfare-optimal allocatio n. Hence we guarantee utility that is a\n1−1\nefraction of the optimal welfare.\nAs an open problem, we include a variation on Random-Favorit es for future work to analyze,\nwhich should perform even better in the m≥nregime:\nDefinition 2 (Iterative Random-Favorites) .The procedure operates in rounds. In each round,\nevery bidder competes for their favorite of the remaining it ems. Each item is allocated uniformly\nat random to one of the competing bidders—those who would sel ect the item as their favorite.\nThe allocated items and bidders are removed and the next roun d proceeds the same way. This\nprocedure is repeated until there are no bidders left to allo cate to.\n4.2 More Bidders than Items\nIn this section, we prove a logarithmic bound between utilit y and social welfare in the case where\nthere are more bidders than items, i.e., m < n. We do this using the Prior-Free-Favorites mecha-\nnism.\nTheorem 5. In a setting with nunit-demand i.i.d. bidders and mi.i.d. items, when n > m,\nthe Prior-Free-Favorites mechanism achieves at least an O(logn\nm)-approximation to optimal social\nwelfare.\nProof.The proof follows three steps: we first show that for any item jthat the gap between the\nutility achieved and the highest value amongst all bidders w hose favorite item is jis logarithmic in\nthenumberofbidderswhohave that item astheir favorite. Se cond, welower boundtheprobability\nthat the bidder with the highest value for jamong all bidders has jas their favorite item. Then,\nwe combine these facts to prove the claim.\nFirst focus on a single item and assume there are n′bidders participating in the mechanism\nfor this item (i.e. n′bidders reported item jas their favorite item). Without loss of generality, we\norder the bidders by their values v1≥v2...≥vn′. We are now in a single item setting. Theorem\n5.2 from Hartline and Roughgarden [2008] gives the existenc e of a prior free mechanism in the\nsingle item setting which guarantees utility at leastv1\n2(1+log( n′)).\nConsider the agent iwith the highest value v∗\nijforjamongallagents. We now bound the\nprobability that v1, the highest value competing for j, is notv∗\nij. This occurs when agent iis\ncompeting for a different favorite item k, that is, vik> v∗\nij.\nWhat is the probability that iprefers some other item k, that is, vik> v∗\nij? Consider n+ 1\ndraws from the type distribution: ndraws for each bidder’s value for item j, and one draw for vik.\nLabel one of these draws vikuniformly at random. Of the remaining ndraws for bidders’ values\nfor itemj,v∗\nijmust be the largest, by definition. Using this sampling proce ss, the probability that\nvik> v∗\nijis1\nn+1, hence the probability that vik≤v∗\nijis 1−1\nn+1. There are at most m−1 other\n12items which jcould prefer, so the probability none of them are preferred i s (1−1\nn+1)m−1and we\nknowm < nand therefore the probability is at least\n/parenleftbigg\n1−1\nn+1/parenrightbiggm−1\n≥/parenleftbigg\n1−1\nn/parenrightbiggn−1\n≥1\ne.\nTherefore, per item j, by Jensen’s inequality, we yield expected utility at least\nEn′/bracketleftbigg1\ne·v∗\nij\n2(1+log( n′))/bracketrightbigg\n≥v∗\nij\nEn′[2e(1+log( n′))].\nFinally,En′[1 + log( n′)]≤1 + log(En′[n′]), again by Jensen’s inequality. The number of total\nbidders including agent iisEn′[n′]≤n\nm+ 1, so the per-item utility is at leastv∗\nij\n2e(1+log(n\nm+1)).\nSumming across items and using the fact that v∗\nij= max ivijwe get that the expected utility of\nPrior-Free-Favorites is at least a log fraction of the optim al social welfare:\nE[Utility]≥E[Welfare ]\n2e(1+log(n\nm+1)).\nRemark. Notethatthisanalysiscannottobedirectlyusedtoalsogiv eaconstant approximation\nwhenn < m—that is, we need our separate analysis of the n < mregime. This analysis uses a\nbound on the probability that an item’s “favorite” agent is a lso that agent’s favorite item. When\nthe number of itemsgrows large, this probability tends to 0.\nTightness. Proposition5.1inHartline and Roughgarden[2008]provide samatchinglowerbound\nof Ω(1+logn\nm), showing that this approximation to welfare is tight even i n the single item case.\nCorollary 6. In the i.i.d. unit-demand setting, the gap between optimal ut ility and optimal social\nwelfare is\nOptWelfare/OptUtility= Θ(1+logn\nm).\nAcknowledgements\nThe idea for this research direction originated jointly wit h Anna Karlin out of conversations with\nMark Braverman about [Braverman et al., 2016]. 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URL\nhttp://arxiv.org/abs/1406.3278 .\n16"    },    {        "title": "2402.12371v1.Computing_Enclosing_Depth.pdf",        "content": "Computing Enclosing Depth∗\nBernd G¨ artner1, Fatime Rasiti2, and Patrick Schnider3\n1Department of Computer Science, ETH Z¨ urich, gaertner@inf.ethz.ch\n2Department of Mathematics, ETH Z¨ urich, frasiti@student.ethz.ch\n3Department of Computer Science, ETH Z¨ urich,\npatrick.schnider@inf.ethz.ch\nAbstract\nEnclosing depth is a recently introduced depth measure which gives a lower bound\nto many depth measures studied in the literature. So far, enclosing depth has only been\nstudied from a combinatorial perspective. In this work, we give the first algorithms to\ncompute the enclosing depth of a query point with respect to a data point set in any\ndimension. In the plane we are able to optimize the algorithm to get a runtime of O(nlogn).\nIn constant dimension, our algorithms still run in polynomial time.\n1 Introduction\nMedians play an important role in statistics. In contrast to the mean value of some given data,\nthe median depends only on the order of the data points and not on their exact positions.\nHence, it is robust against outliers. As data sets are multidimensional in many cases, we are\ninterested in an extension of the term ’median’ to higher dimensions. Since there is no clear\norder of the data points, there are various generalizations of the median to higher dimensions\n[4,12,13]. In order to define the median of some data, the notion of depth of a query point\nhas been introduced. A median is then a query point with the highest depth. Many depth\nmeasures only depend on the relative positions of the data points, just like the median, making\nthem again robust against outliers.\nAfter the first depth measure was introduced by Tukey [ 22] (and is therefore known as\nTukey depth), Donoho and Gasko [ 9] established the idea of a multidimensional median as a\ndeepest point relative to the data points. Various depth measures with different properties have\nsince been introduced, such as simplicial depth [11] and convex hull peeling depth [4].\nDepth measures are an important tool in Computer Science for example in geometric\nmatching, pattern matching, clustering [ 8,10,19] and shape fitting applications [ 2]. Since\ndepth measures give a way to compute medians of data points they also find applications in\nStatistics such as data visualization [22] and regression analysis [16, 21].\nDefinition 1 (Depth measure) .Letd∈Nand(Rd)Sbe the family of all finite point sets in\nRd. A depth measure is a function D: (Rd)S×Rd→R≥0,(S, q)7→D(S, q). In particular, the\nfunction Dassigns to a given finite point set Sand a query point qa value, which describes\nhow deep the query point qlies within the data set S.\nAssume we are given a data set S. Consider all hyperplanes spanned by the points of S.\nThis arrangement A of hyperplanes divides Rdinto connected components of Rd\\A. We call\nthese connected components cells. A depth measure where all points in a cell have the same\ndepth is called combinatorial .\n∗This work is based on the Master thesis of the second author.\n1arXiv:2402.12371v1  [cs.CG]  19 Feb 2024Aloupis et al. [ 4] used the fact that simplicial depth is a combinatorial depth measure to\ncompute the simplicial median for d= 2 in O(n4) time. Sachini [ 15] modified this algorithm\nto compute the simplicial depth for the whole plane in O(n4) time. For the case d= 3 there\nare various algorithms that compute the simplicial depth of a single query point in O(n2) [7].\nCheng and Ouyang [ 7] discussed an extension of this algorithm to compute simplicial depth in\nR4inO(n4) time. Afshani et al. [ 1] later introduced methods to compute simplicial depth in\nO(ndlogn) time for d >4. This bound was improved by Pilz et al. [14] to O(nd−1).\nAnother well studied combinatorial depth measure is Tukey depth, also known as halfspace\ndepth. For the case d= 2, Aloupis et al. [ 3] gave a worst case lower bound of Ω( nlogn) for\ncomputing the Tukey depth of an arbitrary query point qwith respect to a given point set S\nof size n. In fact, the Tukey depth of a query point relative to a point set of size ncan be\ncomputed in O(nlogn) time [ 17]. There are different approaches to compute the Tukey depth\nin different dimensions [ 6,5,18]. The algorithm of Rousseeuw et al. [ 18] to compute the Tukey\ndepth of a query point in Rdford >2 has a run time of O(nd−1logn).\nStudying more general families of combinatorial depth measures, Schnider introduced the\nnotion of enclosing depth , which turns out to be a natural lower bounds for many combinatorial\ndepth measures [ 20]. In this work, we will focus on enclosing depth and provide algorithms to\ncompute it.\nDefinition 2 (k-enclosing) .LetSbe a point set of size (d+ 1)kinRdandqa query point. If\nScan be partitioned into d+ 1pairwise disjoint subsets S1, ..., S d+1, each of size k, such that\nfor any transversal p1∈S1, ..., p d+1∈Sd+1the point qlies in the convex hull of p1, ..., p d+1,\nthen we say that S1, ..., S d+1k-encloses the point q. (Figure 1)\nS1\nS2\nS3\np1\np2\np3\np0\n1\np0\n2\np0\n3\nq\nFigure 1: The set S=S1∪S2∪S35-encloses the query point qinR2.\nDefinition 3 (Enclosing Depth) .LetSbe a finite point set in Rdandqbe a query point. The\nenclosing depth of qwith respect to Sis the maximum ksuch that there exist subsets S1, ..., S d+1\nofSwhich k-enclose q. We denote it by ED(S, q). (Figure 2)\nIn this work, we present algorithms to compute the enclosing depth of a query point qwith\nrespect to a data set SinR2(Section 2) as well as in general dimension (Section 3).\n2 The planar case\nBefore describing our algorithm, we introduce some combinatorial lemmas that will be helpful\nin proving the correctness of our algorithm. For these lemmas, we will assume that the query\npoint qis the origin and that the data point set Slies on the unit circle. Combinatorially, this\nis not a restriction, as the following lemma shows.\n2S1\nS2\nS3\np1\np2\np3\np0\n1\np0\n2\np0\n3\nqFigure 2: The enclosing depth of the query point qis at least 5.\nLemma 1. LetS={s1, . . . , s n} ⊂R2be a data point set and q∈R2a query point such that\nS∪ {q}is in general position. Denote by S′={s′\n1, . . . , s′\nn}the point set defined by centrally\nprojecting each point in Sto a circle of unit radius with center q. Then qis in the convex hull\nofsi, sj, skif and only if it is in the convex hull of s′\ni, s′\nj, s′\nk.\nProof. Assume that qis not in the convex hull of a, b, c . Then there is a line ℓthrough qhaving\nall of a, b, c on the same side. This is invariant under central projection from q.\nLet now S={s1, . . . , s n}on the unit circle be ordered in counter-clockwise direction. By\naninterval [sa, sb] we denote all the points in Sthat lie between saandsb, that is,\n[sa, sb] =\u001a{s∈S|sa≤s≤sb} sa≤sb\n{s∈S|s≥saors≤sb}sa> sb\nIn the following, we write indices modulo n, that is, si=si−nfori≥n. We show that in\norder to find k-enclosing sets we can restrict our attention to intervals and their endpoints.\nLemma 2. Leta1, a2, b1, b2, c1, c2∈Ssuch that the intervals [a1, a2],[b1, b2]and[c1, c2]are\npairwise disjoint and for every choice of a∈ {a1, a2},b∈ {b1, b2},c∈ {c1, c2}the origin lies\nin the convex hull of a, b, c . Then for every choice a∈[a1, a2],b∈[b1, b2],c∈[c1, c2]the origin\nlies in the convex hull of a, b, c .\nProof. Assume for the sake of contradiction that the convex hull of a, b, c does not contain\nthe origin and let ℓbe a line through the origin that has all of a, b, c on the same side ℓ+. As\nthe intervals are pairwise disjoint, one of a1ora2must also be in ℓ+. Thus the convex hull\nofai, b, cdoes not contain the origin for some i∈ {1,2}. Repeating this argument two more\ntimes, we find that one of the 8 triangles spanned by the endpoints does not contain the origin,\nwhich is a contradiction to the assumptions of the lemma.\nWe want to restrict the considered intervals even further. To this end, we define for each\npoint s∈Sitsopposite neighbors s(r)ands(ℓ)as the last point in Sbefore −sand the first\npoint in Safter−s, respectively, see Figure 3 for an illustration.\nLemma 3. LetS1= [si, si+k],S2andS3be subsets of Sthat(k+1)-enclose the origin. Denote\nsj=s(r)\niandsm=s(ℓ)\ni+k. Then S1,S′\n2:= [sj−k, sj]andS′\n3:= [sm, sm+k]also(k+ 1)-enclose\nthe origin.\nProof. See Figure 4 for an illustration of the proof. We first note that (up to relabeling) we\nmust have that S2⊂[si+k+1, sj] and S3⊂[sm, si−1]. Indeed, both lines through siand the\norigin, as well as through si+kand the origin must separate S2andS3, as otherwise there would\n3s\n\u0000s\ns(`)\ns(r)Figure 3: The opposite neighbors s(r)ands(ℓ)of a point s∈S.\nbe a choice a∈S1,b∈S2,c∈S3whose convex hull does not contain the origin. Thus, we can\nwrite S2= [sb, sb+k] fori+k+ 1≤b≤j−kand similarly S2= [sc−k, sc] form+k≤c≤i−1.\nIn particular both si, sb, scandsi+k, sb, sccontain the origin.\nWe claim that si, sb, smalso contains the origin. Assume it does not. Then either the line\nℓithrough siand the origin or the line ℓbthrough sband the origin must separate scandsm.\nBy construction, ℓihas both scandsmon the same side, so it must be ℓb. But this would\nimply that m > c , which is a contradiction. A symmetric argument also shows that si+1, sb, sm\ncontains the origin. Analogously it can be shown that si, sj, scas well as si+1, sj, sccontains\nthe origin. Finally, by construction si, sj, smas well as si+k, sj, smcontains the origin. The\nstatement now follows from Lemma 2 and the fact that b≤j−kandm+k≤c.\nDescription of the algorithm: We are given a set Sofndata points in the plane and\na query point q. In a pre-processing step we first sort the points radially around q, giving a\ncounter-clockwise order s1, . . . , s nonSand then for each s∈Swe compute s(r)ands(ℓ)using\nbinary search. For the main part of the algorithm we run the following subroutine, which for a\ngiven integer kchecks whether there are three sets that ( k+ 1)-enclose q: for each si∈S, with\nsj=s(r)\niandsm=s(ℓ)\ni+kthe subroutine checks whether all 8 triangles a, b, c fora∈[si, si+k],\nb∈[sj−k, sj],c∈[sm, sm+k] contain qand the intervals are pairwise disjoint, returning TRUE\nif this holds for some si, and FALSE otherwise. By doing a binary search over the values of\nk∈ {0, . . . , n }we find the largest value kfor which the subroutine returns true and return\n(k+ 1).\nTheorem 4. The above algorithm computes the enclosing depth of qwith respect to S⊂R2in\ntime O(nlogn).\nProof. We first show the correctness of the algorithm. It follows from Lemma 2 that if the\nsubroutine returns TRUE then the considered intervals are indeed ( k+ 1)-enclosing. On the\nother hand, if there are ( k+ 1)-enclosing sets, by Lemma 3 the subroutine will find them.\nAs for the runtime, we can sort in time O(nlogn). After this, we perform 2 nbinary searches,\neach taking O(logn) time, thus the total runtime of the pre-processing step is O(nlogn). For\nthe runtime of the subroutine, we notice that for each choice of sithe required checks can be\ndone in time O(1), so the runtime of the subroutine is O(n). As we call it for O(logn) many\nvalues of kwe get the desired runtime.\n4si\nsi+k\nsj\nsm\nsc\nsc\u0000k\nsb\nsb+k\nS1\nS2\nS3Figure 4: An illustration of the proof of Lemma 4.\n3 Higher dimensions\nOur algorithm in general dimension is based on the following observation:\nLemma 5 (Lemma 20 in [ 20]).LetS1, . . . , S d+1⊂S⊂Rdbe point sets which enclose a point q,\nwhere S∪ {q}is in general position. Then there are d+ 1closed halfspaces H−\n1, . . . , H−\nd+1such\nthat each H−\nicontains qon its boundary, H−\ni∩S=Sifor each iandH−\n1∪. . .∪H−\nd+1=Rd.\nDenoting by H+\nithe complement of H−\niwe show below that we get Si⊂T\nj̸=iH+\njandT\niH+\ni={q}. As we also show later, given enclosing sets S1, . . . , S d+1we can rotate the\nhalfspaces H+\nito get closed halfspaces H′\niwhose boundaries contain qandd−1 points of\nSand for which H′\ni∩S=H+\ni∩SandT\niH′\ni={q}. On the other hand, given halfspaces\nH′\n1, . . . , H′\nd+1, each boundary containing qandd−1 points of SwithT\niH′\ni={q}, defining\nSi:=T\nj̸=i(H′\nj∩S) we will show that for every transversal p1∈S1, ..., p d+1∈Sd+1the point q\nlies in the convex hull of p1, ..., p d+1. Combining these facts, we get the following strengthening\nof Lemma 5.\nLemma 6. LetS1, . . . , S d+1⊂S⊂Rdandq∈Rdsuch that S∪{q}is in general position. Then\nS1, . . . , S d+1enclose qif and only if there are d+ 1halfspaces H′\n1, . . . H′\nd+1whose boundaries\ncontain qandd−1points of S, and for which Si⊂T\nj̸=iH+\njandT\niH+\ni={q}.\nLet us now formally prove all the steps outlined above. We say that d+ 1 halfspaces in\nRdare in general position if for any k≤dof them their boundary hyperplanes intersect in a\n(d−k)-dimensional affine subspace. This is equivalent to saying that the normal vectors of the\nbounding hyperplanes are affinely independent. The halfspaces H−\n1, . . . , H−\nd+1given by Lemma\n5 are in general position, as an investigation of the proof in [ 20] shows. In fact, this also follows\nfrom Lemma 9. Our first lemma relates the union of halfspaces to their intersection.\nThe following is well-known, but we state it as it will be helpful for us later.\nLemma 7. LetH1, . . . , H d+1bed+ 1closed halfspaces in Rdin general position, all of which\ncontain a point qon their boundary. ThenS\niHi=Rdif and only ifT\niHi={q}.\n5The next lemma relates the hyperplanes H−\niof Lemma 5 to their complements H+\ni.\nLemma 8. LetS=S1∪. . .∪Sd+1and let H−\n1, . . . , H−\nd+1be closed halfspaces in general position\nwhose boundaries all contain q. Denote by H+\n1, . . . H+\nd+1the closures of their complements.\nWe have that H−\ni∩S=Sifor every i∈ {1, . . . , d + 1}andS\niH−\ni=Rdif and only ifT\nj̸=iH+\nj∩S=Sifor every i∈ {1, . . . , d + 1}andT\niH+\ni={q}.\nProof. We first show thatT\niH+\ni={q}if and only ifT\niH−\ni={q}. Assume thatT\niH+\ni={q}.\nBy definition we have q∈T\niH−\ni. It remains to show that there is no other point in the\nintersection. For this, we note that the intersection of d+ 1 closed halfspaces in general position\ninRdis a convex set and as all boundary hyperplanes contain qit is either only qor a cone\nwith apex q. However, if it is a cone then any point in the interior of the cone must lie in the\ninterior of H−\nifor every i, so it does not lie in H+\nifor any i, which means thatS\niH+\ni̸=Rd.\nBy Lemma 7, this is a contradiction toT\niH+\ni={q}. The other direction is analogous. In\nparticular, by Lemma 7 we get thatS\niH−\ni=Rdif and only ifT\niH+\ni={q}.\nIt remains to show that H−\ni∩S=Sifor every i∈ {1, . . . , d +1}if and only ifT\nj̸=iH+\nj∩S=\nSi. This follows immediately from the fact that H+\niis the complement of H−\ni.\nThe next lemma shows that the conditionT\niH+\ni={q}is an immediate consequence of\nenclosingness.\nLemma 9. Lets1, . . . , s d+1, qbed+ 2points in Rdin general position such that qlies in the\nconvex hull of S={s1, . . . , s d+1}. Let H+\n1, . . . , H+\nd+1be closed halfspaces whose boundaries\ncontain qand such thatT\nj̸=iH+\nj∩S=sifor every i∈ {1, . . . , d + 1}. ThenT\niH+\ni={q}.\nProof. Consider the simplex spanned by s1, . . . , s d+1and denote by Fithe facet spanned by all\npoints except si. Note that H+\nicontains Fiand that the H+\nimust be in general position. This\nimplies thatS\niH+\ni=Rd. The statement not follows from Lemma 7.\nWe are now ready to show that we can rotate the H+\nito not only contain qon the boundary\nbut also ( d−1) points of S.\nLemma 10. LetS1, . . . , S d+1⊂S⊂Rdbe point sets which enclose a point q, where S∪ {q}\nis in general position. Then there are d+ 1closed halfspaces H+\n1, . . . , H+\nd+1such that each H+\ni\ncontains qandd−1points of Son its boundary, Si⊆T\nj̸=iH+\ni∩Sfor each iandT\niH+\ni={q}.\nProof. We start with the halfspaces H+\n1, . . . , H+\nd−1, which are the complements of the halfspaces\ngiven by Lemma 5. Assume the boundary of H+\nicontains qbut fewer than d−1 points of S. In\nparticular, it contains fewer than dpoints, so we can choose a ( d−2)-dimensional axis of rotation\nthrough qand all points of Sit contains. We rotate H+\niaround this axis until it hits some other\npoint, resulting in a rotated halfspace with one more point on the boundary. Also note that\nno point has been removed from H+\niduring the rotation, so we maintain Si⊆T\nj̸=iH+\ni∩S\nfor each i. Further it follows from Lemma 9 thatT\niH+\ni={q}. The lemma now follows by\nrepeating this process until all boundaries contain the required number of points.\nFinally, we show the other direction of Lemma 6.\nLemma 11. LetSbe a set of points in Rdandqa query point, such that S∪ {q}is in general\nposition. Let H+\n1, . . . , H+\nd+1be closed halfspaces such that each H+\nicontains qandd−1points\nofSon its boundary andT\niH+\ni={q}. Let Si=T\nj̸=iH+\ni∩S. Then S1, . . . , S d+1enclose q.\nProof. For each H+\niconsider the ( d−2)-dimensional rotation axis spanned by the d−1 points\non its boundary and rotate a small amount such that qis not in H+\nianymore. The resulting\nhalfspaces are still in general position but they now have an empty intersection and thus their\ncomplements intersect in a simplex ∆ containing q. Let vibe the vertex of ∆ that is the\nintersection of all bounding hyperplanes except the one of H+\ni. ThenT\nj̸=iH+\niis a cone Ci\nwith apex viwhich contains Si. As we have that for every choice p1∈C1, . . . , p d+1∈Cd+1\nthat the convex hull of p1, . . . , p d+1contains ∆ it follows that Then S1, . . . , S d+1enclose q.\n6Description of the algorithm: For each choice of d−1 points in S, consider the two\nhalfspaces defined by the hyperplane through qand these d−1 points. This defines a set H\nof 2\u0000n\nd−1\u0001\nhalfspaces. For any d+ 1 halfspaces H1, . . . , H d+1∈ Hfirst check ifT\niHi={q}. If\nnot, continue with the next choice, otherwise count for each ithe number of points of SinT\nj̸=iHjand denote by kthe smallest of the d+ 1 numbers. In the end, return the largest such\nkencountered over all choices of d+ 1 halfspaces in H.\nTheorem 12. The above algorithm computes the enclosing depth of qwith respect to S⊂Rd\nin time O(nd2).\nProof. The correctness follows from Lemma 6. As for the runtime, we have |H| ∈ O(nd−1) out\nof which we choose sets of d+ 1 elements, so there are O(n(d−1)(d+1) sets of halfspaces that\nwe consider. For each set the check and the counting can be done in time O(n) so the total\nruntime is O(n(d−1)(d+1)+1) =O(nd2).\n4 Conclusion\nWe have given two algorithms to compute the enclosing depth of a query point qwith respect\nto a data point set S, one for the plane and one in general dimension. The planar algorithm\nmatches the runtimes of computing many other depth measures. For some measures, such as\nTukey depth, matching lower bounds for the computation time have been shown [ 3]. It would\nbe interesting to adapt these lower bounds to enclosing depth.\nIn higher dimension, many depth measures can be computed in time O(nd−1), which is\nsignificantly faster than the runtime of our algorithm. We believe that enclosing depth can be\ncomputed more efficiently as well, but some additional ideas are likely required for this.\nFinally, there are other natural algorithmic problems for depth measures even in the plane,\nsuch as computing the depth of the entire plane or finding a deepest query point. Using our\nalgorithm and the fact that the arrangement spanned by the lines through all pairs of data\npoints has O(n4) cells we can solve both those problems in O(n5logn), but we again believe\nthat this is not optimal.\nReferences\n[1]Peyman Afshani, Donald R. Sheehy, and Yannik Stein. Approximating the simplicial\ndepth in high dimensions. In Proceedings of 32nd European workshop on Computational\nGeometry , pages 39–42, 2001.\n[2]Dror Aiger, Haim Kaplan, and Micha Sharir. Duality-based approximation algorithms for\ndepth queries and maximum depth. arXiv preprint , page arXiv:2006.12318, 2020.\n[3]Greg Aloupis, Carmen Cort´ es, Francisco G´ omez, Michael Soss, and Godfried Toussaint.\nLower bounds for computing statistical depth. Computational Statistics & Data Analysis ,\n40(2):223–229, 2002.\n[4]Greg Aloupis, Stefan Langerman, Michael Soss, and Godfried Toussaint. Algorithms for\nbivariate medians and a fermat-torricelli problem for lines. Computational Geometry , 2003.\n[5]David Bremner, Dan Chen, John Iacono, Stefan Langerman, and Pat Morin. Output-\nsensitive algorithms for tukey depth and related problems. Statistics and Computing , 18(3):\n259–266, 2008.\n[6] Dan Chen, Pat Morin, and Uli Wagner. Absolute approximation of tukey depth: Theory\nand experiments. Computational Geometry , 46(5): 566–573, 2013.\n[7]Andrew Y. Cheng and Ming Ouyang. On algorithms for simplicial depth. In Proceedings\nof the 13th Canadian Conference of Computation Geometry (CCCG) , pages 53–56, 2001.\n7[8]Andreas Christmann, Paul Fischer, and Thorsten Joachims. Classification based on the\nsupport vector machine, regression depth, and discriminant analysis. page 225–230, 2002.\n[9]David L. Donoho and Miriam Gasko. Breakdown properties of location estimates based on\nhalfspace depth and projected outlyingness. The Annals of Statistics , 20(4): 1803-1827\n,1992.\n[10]Rebecka J¨ ornsten. Clustering and classification based on the l1 data depth. Journal of\nMultivariate Analysis , 90(1): 67-89, 2004.\n[11]Regina Y. Liu. On a notion of data depth based on random simplices. The Annals of\nStatistics , 18(1): 405–414, 1990.\n[12]Regina Y. Liu, Jesse M. Parelius, and Kesar Singh. Multivariate analysis by data depth:\ndescriptive statistics, graphics and inference. Ann. Statist. , 27(3): 783–858, 06.1999.\n[13]Karl Mosler. Springer Berlin Heidelberg, Berlin, Heidelberg. Depth Statistics. , pages 17–34,\n06.1999.\n[14]Alexander Pilz, Emo Welzl, and Manuel Wettstein. From crossing-free graphs on wheel\nsets to embracing simplices and polytopes with few vertices. Discrete & Computational\nGeometry , 64(3): 1067–1097, 2020.\n[15]Sachini Kanchana Rajapakse. Computing Batched Depth Queries and the Depth of a Set\nof Points. Faculty of Graduate Studies of The University of Manitoba Winnipeg , pages\n20–26, 2022.\n[16]Peter J. Rousseeuw and Mia Hubert. Regression depth. Journal of the American Statistical\nAssociation , 94(446): 388-402, 1999.\n[17]Peter J. Rousseeuw and Ida Ruts. Bivariate location depth. Journal of the Royal Statistical\nSociety. Series C (Applied Statistics) , 45(4): 516–526, 1996.\n[18]Peter J. Rousseeuw and Anja Struyf. Computing location depth and regression depth in\nhigher dimensions. Statistics and Computing , 8(3): 193–203, 1998.\n[19]Ida Ruts and Peter J. Rousseeuw. Computing depth contours of bivariate point clouds.\nComputational Statistics and Data Analysis , 23(1): 153–168, 1996.\n[20]Patrick Schnider. Enclosing depth and other depth measures. Combinatorica , pages 1–23,\n2023.\n[21]Patrick Schnider and Pablo Sober´ on. Combinatorial depth measures for hyperplane\narrangements. In 39th International Symposium on Computational Geometry (SoCG 2023) .\nSchloss Dagstuhl-Leibniz-Zentrum f¨ ur Informatik, 2023.\n[22] John W. Tukey. Mathematics and the picturing of data. 1975.\n8"    },    {        "title": "2402.12392v1.A_Regression_Mixture_Model_to_understand_the_effect_of_the_Covid_19_pandemic_on_Public_Transport_Ridership.pdf",        "content": "A Regression Mixture Model to understand the\neffect of the Covid-19 pandemic on Public\nTransport Ridership\nHugues Moreau\nUniversit ´e Gustave Eiffel\nParis, France\nhugues.moreau.pro@gmail.com´Etienne C ˆome\nUniversit ´e Gustave Eiffel\nParis, France\netienne.come@univ-eiffel.frAllou Sam ´e\nUniversit ´e Gustave Eiffel\nParis, France\nallou.same@univ-eiffel.frLatifa Oukhellou\nUniversit ´e Gustave Eiffel\nParis, France\nlatifa.oukhellou@univ-eiffel.fr\nAbstract —The Covid-19 pandemic drastically changed urban\nmobility, both during the height of the pandemic with government\nlockdowns, but also in the longer term with the adoption of\nworking-from-home policies. To understand its effects on rail\npublic transport ridership, we propose a dedicated Regression\nMixture Model able to perform both the clustering of public\ntransport stations and the segmentation of time periods, while\nignoring variations due to additional variables such as the official\nlockdowns or non-working days. Each cluster is thus defined by a\nseries of segments in which the effect of the exogenous variables is\nconstant. As each segment within a cluster has its own regression\ncoefficients to model the impact of the covariates, we analyze how\nthese coefficients evolve to understand the changes in the cluster.\nWe present the regression mixture model and the parameter\nestimation using the EM algorithm, before demonstrating the\nbenefits of the model on both simulated and real data. Thanks\nto a five-year dataset of the ridership in the Paris public transport\nsystem, we analyze the impact of the pandemic, not only in terms\nof the number of travelers but also on the weekly commute. We\nfurther analyze the specific changes that the pandemic caused\ninside each cluster.\nIndex Terms —Clustering, Segmentation, Mixture of Regres-\nsions, Generative models, Expectation-Maximization\nI. I NTRODUCTION\nFinding similarities between groups of samples, a problem\nknown as clustering, has been an active research domain\nin the last decades. However, when the samples are time\nseries, one might want to find similarities between different\nperiods. Segmentation, or change-point detection [19], is the\nproblem of dividing the temporal axis into intervals in which\nthe data follow a constant distribution. To combine these two\napproaches, we can use a sequential approach, by computing\nclusters first and finding change points inside each cluster.\nHowever, sequential approaches are noticeably unoptimized.\nConsequently, researchers have developed joint approaches,\nsuch as SegClust by Picard et al. [16], or ClustSeg [18].\nThe introduction of exogenous variables in the mixture\nmodel serves two purposes: firstly, it can help the model\nto remain unaffected by their effects. In the urban mobility\ndomain, it could be interesting to include exogenous variables\nthat represent the three lockdowns in order to enable the\nmodel to explain the changes in the data using regressioncoefficients rather than new segments. Secondly we want our\nmodel to be able to use the influence of these variables to be\nimportant to our model: it is worth considering that during\ntypical pandemic behavior, the disparity between weekends\nand weekdays may be smaller than in the period prior to the\nyear 2020. This is the rationale of the publications which\nperform a joint regression and clustering [7] or regression\nand segmentation [11], [12]. By explaining the variations\nin the data with exogenous variables, adding regression to\na clustering-segmentation model helps the segmentation to\ndetect only change points for meaningful variations, instead of\nreacting to variations that can be considered trivial or easily\nexplainable given the context.\nFor the analysis of public transport ridership, clustering\nand regression have already been used successfully either to\nknow which types of travel need improvement [15], or to\ndetermine how investigate one metro line can help alleviate the\nflow of passengers when another line or station is unavailable\n[6]. However, these studies focus on signals for which the\ndistribution has the same parameters for the whole duration.\nThe goal in the present paper is to go further and include\nthe effects of the pandemic in the analysis. For instance, to\nmodel the difference between weekdays and weekends, we\nmay compute one mean for each type of day [15], or with an\nadditive model [6]. The latter allows the model to combine the\ninfluence of several variables, such as the day of the week and\nthe position in the year. Furthermore, having several segments,\neach with its regression coefficients, could help us understand\nhow the pandemic changed travel behaviors by looking at\nwhich coefficients differ before and after the pandemic crisis.\nThis paper aims at better understanding how the pandemic\naffected the usage of the Paris public transport network. To\ndo so, we used a public transport ridership dataset published\nby the transport organisation authority called ˆIle-de-France\nMobilit ´es (IdFM) which contains records of public transport\nentries on the rail network of Paris city and its suburbs\nbetween 2017 and June 2022. The data cover the start of\nthe pandemic, as well as the three lockdowns. Obviously, the\nofficial lockdowns had a strong impact on public travel but\nwe are also interested in the impact of the pandemic on travelarXiv:2402.12392v1  [stat.AP]  16 Feb 2024behaviors. We would expect that, if some stations are relatively\nunaffected by the pandemic, a clustering-segmentation model\ncould create a cluster for these stations, in which the transition\npoints are located at different dates than the pandemic.\nThe present work focuses on the model that allows this in-\nterpretation to occur, and its contributions can be summarized\nas follows:\n•we propose a regression mixture model which performs\nclustering and segmentation, and allows the data to follow\na regression model where covariates can be of any type.\n•we develop a formulation that models both univariate and\nmultivariate data.\n•we evaluate this model against a complete set of base-\nlines, both on synthetic and real data.\nThe rest of this paper is organized as follows: after a\nliterature review, section III presents both our model and\nthe parameter estimation algorithm. Then, we evaluate our\nproposed model, on synthetic data (section IV-B) and on the\npublic transport ridership dataset in section IV-D. Finally,\nsection V provides some concluding remarks and proposes\noptions for future work. To foster research on this domain, we\npublish our code at https://github.com/HuguesMoreau/GMM\nClustering Segmentation.\nII. R ELATED WORK\nOur work consists of the intersection of three research\ndomains: clustering, segmentation, and regression.\nAs the scope of these domains is very broad, we provide\nonly a concise overview, focusing on their intersections.\nClustering is a field of research aiming to group samples\naccording to their similarity. We will use the formalism of\nGaussian Mixture Models , it is assumed that the cluster each\nsample belongs to is a hidden variable, which is estimated\nusing the Expectation-Maximization (EM) algorithm [8].\nLike clustering, segmentation is also an active domain of\nresearch, that consists in finding intervals of time in which\nthe behavior of the variables is as constant as possible. The\ntemporal ordering allows the formulation of dynamic program-\nming algorithms that guarantee to find a global optimum of the\nobjective function (see Truong et al. [19] for a comprehensive\nreview). Apart from dynamic programming algorithms and\ntheir variants, another type of model used to perform the\nsegmentation of time series are Hidden Markov Models [10].\nHowever, the transitions of Hidden Markov Models cannot al-\nways be controlled as precisely as required without additional\nconstraints [14].\nWhen the segmentation is paired with clustering, using\ndynamic programming requires re-computing a clustering for\neach time step (as in [16]) which might make the whole\nalgorithm too long to compute. To prevent this problem, Sam ´e\net al. [18] developed a model named ClustSeg that performs\nboth clustering and segmentation. To obtain an estimate in a\nreasonable time, the authors also present the rules to estimate\nthe whole partition using a single EM run. This is the approach\nwe expand by adding the contribution of exogenous variables.\nNote that the ClustSeg model already projects the data on abasis of functions that can be understood as exogenous vari-\nables that are independent of the individuals. This is the reason\nwhy Chamroukhi renamed this model Piecewise Regression\n[3], even though the regression is, as in [18], only on temporal\nvariables. Our contribution is to allow the incorporation of any\ntype of variable and not only time covariates.\nTo account for the exogenous variables, we will use a\nlinear model. De Veaux [7] combined linear regression with\nclustering to obtain a mixture of regression , a model where\nthe mean of each cluster is a linear combination of some\nother variables. The author also proposed an adaptation of\nthe EM algorithm to estimate the parameters of the model,\nwhich is still relatively close to the estimation procedure of\nclassical mixture models. Another family of models that falls\nin the intersection between clustering and regression consists\nin using use a model named Latent class clustering-based\nrandom parameter ordered logit model (LCROL, [4]), where\nthe probability for any individual to belong to a cluster follows\na logistic regression, depending on the provided covariates.\nThis idea is similar to the segmentation used by Sam ´eet al.\n[18] and Chamrouki [3], except that the only variable available\nin the logistic regression is time (for further details, see section\nIII-B).\nThe last field of interest is the intersection of segmentation\nand regression, named segmented regression . However, this\nname often designates studies where the breakpoint is fixed\nbefore the start of the analysis. This kind of segmented\nregression has been used in medicine [2], or for the evaluation\nof public policies [13]. Most of the time, only two segments\nare used to know whether the difference between the two is\nsignificant.\nIn our case, we will exhibit the results our model obtains\non a public transport dataset. But first, we need to present\nthe formulation of our model, along with the formulation of\nthe EM algorithm to estimate the parameters of the mixture\nmodel.\nIII. M ODEL DEFINITION AND PARAMETER ESTIMATION\nA. Definitions and notations\nThe data to be analyzed in this article consist of a set of\ntime series observed over the same time grid indexed by t.\nThe data we want to model are a set of individuals , which\nwe want to group into Kclusters . The dataset covers a time\nseries observed over discrete timesteps t(in our case, days).\nFor each individual i, and each day t, the data is a D-\ndimensional vector noted yi,t. Note that the model does not\nassume that all couples of individuals and days are present:\nsome may be missing from the dataset (and will be in the case\nof the Public Transport dataset, see fig. 1). In all cases, we\nwant to obtain, for each cluster k, a series of Ssegments , that\nis, an interval of timestamps with similar behavior. However,\nthere are several causes for change: in the data which we\nwant to exclude from our analysis. This is why we include\nexogenous variables that will help to explain the changes in\nthe observed variables without resorting to new segments. We\nnotexi,t,lthe value of the lthvariable for individual iandtimestep t. We assume that all variables are continuous: if one\nvariable has more than two categories, we break it down into\nseveral dummy (zero-one) variables. Note that in practice, the\nvariables we will use are mostly constant along all timesteps,\nor for all individuals.\nB. Model definition\nLetzibe the cluster assignment of individual i, and wi,tthe\nsegment of individual iat time t. We assume that the cluster\nassignments ziare i.i.d., following a multinomial distribution\nwhose weights are parameters of our model:\nP(zi=k) =πks.t.KX\nk=1πk= 1. (1)\nGiven the cluster kof individual i, we assume that the\nsegment assignment wi,tfollows a multinomial law defined\nby the probabilities:\nP(wi,t=s|zi=k) =κk,s\nt=exp(t.uk,s+vk,s)P\ns′exp(t.uk,s′+vk,s′),(2)\nwhere uk,sandvk,sare model parameters. In other words, the\nonly reason why we see definitive transitions is the fact that the\narguments of the softmax functions are monotonous. This way\nof modelling segments using independent day assignments\nmight seem unusual. We use the formulation in equation 2\nbecause the independence of the segment assignments allows\nus to make use of the EM algorithm (see section III-C).\nThe partition of the time axis resulting from the application\nof the Maximum a Posteriori (MAP) rule may not be a\nseries of contiguous segments. We thus applied the following\nconstraints to make sure that the distributions of the temporal\nclass κk,s\ntvary faster than a given threshold λ:\n∀k,∀s∈ {1. . . S−1}, u k,s+1−uk,s> λ (3)\nNote that λis not a parameter of our model, but a value we\nchoose before any parameter estimation. Numerically, we set\nit in such a way that the temporal classes distributions κk,s\nt\ngo from 0.01to0.99in less than three months1. We found\nthese constraints paramount: without them, the distribution of\ntemporal classes κk,s\ntwould remain nearly constant (equal to\napproximately 1/S, where Sis the number of segments), and\nthe posteriors would not have the contiguity expected from\nsegments.\nObservation model: Similarly to ClustSeg [18], our obser-\nvation model is a Gaussian linear regression model:\nyi,t|(zi=k, wi,t=s)∼ N(µk,s\ni,t,Σk,s) (4)\nHowever, contrary to ClustSeg, in which the statistical mean\nµdepended only on the cluster, segment, and time covariates,\n1To do so, we solve exp(λ.t+c) = 0 .01andexp(λ(t+ ∆t) +c) =\n0.99, yielding λ= (log(0 .99/0.01))/∆t. As the time tvaries between zero\nand one in all our formulas, the corresponding ∆tfor three months is equal\nto 90 days divided by the number of days in the interval T.our formulation introduces exogenous variables that vary for\nboth individuals iand days t. As in mixtures of regressions [7],\nthe mean is a sum of the linear contributions of all exogenous\nvariables:\nµk,s\ni,t=mk,s+LX\nl=1xi,t,l.αl,k,s, (5)\nwhere mk,sis the intercept of the linear regression model, and\ntheαl,k,s are the contributions of the Ltemporal variables.\nFor the covariance matrix of cluster k, segment s,Σk,s,\nwe choose to have a diagonal covariance (meaning that all\ndimensions are independent) but other constraints are possible\n(full covariance, spherical, etc.), similarly to the fact that\none can choose the type of covariance for classical Gaussian\nMixture Models.\nTo summarize this section, the free parameters of the\nmodel, denoted as θ, are the mixture proportions (πk)k, the\nsegmentation parameters (uk,s, vk,s)k,s, the mean mk,sof\ncluster kand segment s, the contributions of the exogenous\nvariables (αl,k,s)l,k,s, and the covariance matrices (Σk,s)k,s.\nThe next section explains how to estimate them via the EM\nalgorithm [8].\nC. Parameter estimation\nThe parameter estimation is done by maximizing the log-\nlikelihood:\nL(θ) = log P(Y|X,θ) (6)\n=X\nilogX\nkY\ntπkX\nsκk,s\ntN(yi,t;µk,s\ni,t,Σk,s)\nThe formulation of our model allows us to use the\nExpectation-Maximization algorithm [8], a powerful algorithm\naiming to estimate the parameters of latent (hidden) variable\nmodels. In our case, the latent variable is the union of\nthe cluster and segment, which we estimate using both the\nobserved variable ( yi,t) and the exogenous variables ( xi,t).\nThe EM algorithm consists in alternating two steps.\nE-step: In this step, we compute the posterior distribution\nof the latent variables, using the values of the parameters at\nthe current iteration. As mentioned in [18], this is done in two\nsteps. First, we need to compute the posterior probability of a\ngiven individual ito belong to cluster k, which we note ρk\ni:\nρk\ni=P(zi=k|yi,xi;θ) (7)\n=πkQ\ntP\nsκk,s\ntN(yi,t;µk,s\ni,t,Σk,s)\nP\nk′πk′Q\ntP\nsκk′,s\ntN(yi,t;µk′,s\ni,t,Σk′,s),(8)\nwhere we note yi= (yi,t)t∈{1,...,T}.\nThen, we can obtain the posterior probability for any day t\nto belong to segment s:rk,s\ni,t=P(wi,t=s, zi=k|yi,xi;θ) (9)\n=ρk\niκk,s\ntN(yi,t;µk,s\ni,t,Σk,s)\nP\ns′κk,s′\ntN(yi,t;µk,s′\ni,t,Σk,s′)(10)\nrk,s\ni,t=P(zi=k, wi,t=s|yi,xi;θ), (11)\nOnce we have computed the responsibilities, we update the\nparameters during the Maximization step.\nM-step: In the maximization step, we use the responsi-\nbilities rk,s\ni,tcomputed earlier, and obtain the parameters that\nmaximize the expectation of the log-likelihood. Conveniently,\nwe can write it as a sum of terms which can be optimized\nseparately:\nQ(Y,x;θ) =X\ni,t,k,srk,s\ni,tlogπk\n+X\ni,t,k,srk,s\ni,tlogexp(t.uk,s+vk,s)P\ns′exp(t.uk,s′+vk,s′)(12)\n+X\ni,t,k,srk,s\ni,tlogN(yi,t;µk,s\ni,t,Σk,s)\nThus, each set of parameter values can be computed in-\ndependently from the others, the only exception being the\nvariance which requires the segment mean and variable con-\ntributions of the current step. In this section, we display only\nthe results of the maximization step.\na) cluster proportions: As with many mixture models,\nthe mixture proportions are the proportion of the responsibil-\nities of the samples. For each cluster k:\nπk= (X\ni,t,srk,s\ni,t)/(X\ni,t,k′,srk′,s\ni,t) (13)\nb) segment borders: The segment parameters u, v are\nfound by computing the maximum of the following logistic\nregression problem:\nX\ni,t,k,srk,s\ni,tlogexp(t.uk,s+vk,s)P\ns′exp(t.uk,s′+vk,s′)(14)\nThis problem is convex (even with the constraints mentioned\nin equation 3), and we solve it using scipy’s optimize\npackage implementation of the quasi-Newton method from [5].\nWe ensure identifiability by settingP\nsuk,s=P\nsvk,s= 0\nfor all k.\nc) mean, regression coefficients, and covariance: We\ncan find the optimal values for the last term of equation 12\nseparately for each segment. For all k, s, the values of mk,s,\nαk,s, andΣk,sminimize:\nX\ni,trk,s\ni,t\u0010\n(yi,t−µk,s\ni,t)Σ−1\nk,s(yi,t−µk,s\ni,t)′+ log det Σk,s\u0011\n,\n(15)where x′denotes the transpose of vector xandµk,s\ni,t=mk,s+P\nlxi,t,l.αl,k,s. This is a common linear regression problem\nwhere the rk,s\ni,tare weights [17].\nAs mentioned earlier, we use these formulas during each\nof the maximization steps to estimate new values of the\nparameters. The complete algorithm can be summarized as\nfollows:\nAlgorithm 1 the EM algorithm\nRequire: observed data (yi,t)i,t, exogenous variables\n(xi,t)i,t, a number of clusters Kand segments S, and an\ninitial parameter θ\nwhile the log-likelihood has not converged do\nE-step : Compute the expectation of the latent variables\nrk,s\ni,tusing the parameters of the model θ\nM-step : Find the parameters of the model θthat best\nexplain the data\nend while\nWe keep on alternating the expectation step and maximiza-\ntion steps until the log-likelihood stops increasing by more\nthan10−4in ten iterations. Only then is parameter estimation\nover and we can start evaluating our model.\nInitialization: To initialize our algorithm, we start by setting\nthe mixture proportions equal to 1/K, where Kis the number\nof clusters. The segment parameters u, v are set in such\na way as to respect the constraints we set (see equation\n3), and the borders between segments delimit segments of\nequal duration. The segment means mk,sare estimated by\nassigning each individual a random cluster and each day its\ncorresponding segment, and computing the mean of values\ninside each segment. Finally, we end the initialization by\nsetting the contribution of exogenous variables to zero.\nIV. E XPERIMENTS\nThis section is devoted to presenting the experimental\nresults that highlight the efficacy of the proposed model. We\nwill compare the performance of our model against a set of\nbaselines using artificial data. Subsequently, we will assess its\nperformance on the public transport ridership dataset through\ncross-validation. Lastly, we will analyze the results obtained by\nthe model to gain insights into the stations that experienced the\nmost significant impacts due to behavioral changes resulting\nfrom the pandemic.\nA. Protocol\nThe model presented above allows us to cover cluster-\ning, segmentation, and regression at once. To justify its use\ndespite the additional complexity, we compare it to several\nalternatives. Most of these methods consist in using ClustSeg\n[18] without exogenous variables, which means that the basis\nof constant functions is used to represent the time series.\nConversely, as the proposed model includes covariates in the\nparameter estimation, the basis of functions we choose to\nrepresent the time series can be found in the variables we\nhave (in our case, splines of degree 2). To demonstrate theinterest of simultaneously performing the segmentation and\nthe clustering, we include in our comparison two methods that\ncompute clusters before segments:\n•Reg→Clust →Seg: This method operates in\nthree sequential steps. Firstly, we estimate regression\ncoefficients ( α, see eq. 5) common to the whole dataset.\nSecondly, we perform a clustering of the residual time\nseries derived from the first step. Thirdly, we perform a\nsegmentation of the clusters obtained in the second step.\nNote that for synthetic data, we generate the ground truth\nregression coefficients with uniform distributions. Hence,\ncomputing regression coefficients on the whole dataset\ndoes not make much sense. In other words, we expect\nthis method to be completely irrelevant for synthetic data.\nHowever, it might prove somewhat useful for real data,\nwhen the contributions of the exogenous variables might\nbe similar across clusters and segments. We include it\nnonetheless for both types of data.\n•(Clust+Reg) →(Seg+Reg) : This baseline in-\nvolves fitting a mixture of regression model, before\nperforming a segmented regression inside each cluster.\nIt is important to note that we estimate new variable\ncontributions during both the clustering and segmentation\nprocesses.\nWe also include three baselines where the clusters are\nestimated at the same time as the segments, meaning that the\nmodel can use the change points in the samples to estimate\nhow close all individuals are to each cluster:\n•Clust+Seg : We simply reuse the method from [18],\nwithout any covariates. As ClustSeg requires a basis\nof (temporal) functions, using it without any covariates\nis equivalent to determining piecewise constant cluster\nprototypes. The rationale for including this method is\nmainly to demonstrate the relevance of including the\nexogenous variables with real data.\n•Reg→(Clust+Seg) : This model is similar to the\none we propose, except that we assume that the linear\nregression parameters are constant across all clusters and\nsegments. Note that to estimate the parameters for this\nmodel, we perform a linear regression on the whole\ndataset (before any affectation onto clusters and seg-\nments), before using ClustSeg (without any exogenous\nvariables) on the residuals. This estimation procedure\nis equivalent to estimating the parameters of the model\nusing the classical EM algorithm based on Gaussian\nmixture models. Similarly to Reg→Clust →Seg,\nwe expect the common regression coefficients to be ill-\nadapted to synthetic data.\n•(Clust+Seg) →Reg: This method involves first us-\ning ClustSeg [18] without any covariables, then perform-\ning a regression on each segment. It has the same set of\nparameters as the one we propose, the only difference\nbetween the two being that the model we present has\naccess to changing regression coefficients during the\nclustering and segmentation. Given that both this methodand the first one ( Clust+Seg ) compute the clusters and\nsegments without any variables, they assign the same\npartition to the data, which is why we do not use it\non synthetic data, where the evaluation criterion is the\nrelevance of the partition.\nUsing different evaluation criteria, we compare the parti-\ntioning obtained with our model to the ones resulting from\nthese methods. For synthetic data (simulations), the creation\nof the dataset allows us to have access to the latent variables\nthat generated the data as ground truth (cluster and segment of\neach individual and day). We consider that two samples belong\nto the same partition if and only if their cluster and segment\nboth match, and use the Adjusted Rand Score to compare to\nthe evaluated models’ estimations.\nFor real-life data, there is no such reference to use. Hence,\nwe look at the log-likelihood on an unseen dataset: we separate\nthe dataset into two subsets for training and validation. We use\nthe training dataset to estimate the parameters of the model.\nThen, we calculate the log-likelihood of the model by applying\nit to the validation dataset. The higher the log-likelihood, the\nbetter the model explains the validation data, and the better\nwe consider our model.\nB. Validation of the model on synthetic data\nParameter generation: Before generating data, we need\nto fix the model parameters. We select arbitrarily K= 4\nclusters and S= 4 segments. The mixture proportions πk\nare drawn from a Dirichlet distribution with parameter 2,\nto make sure all clusters appear for at least one sample.\nSimilarly, for each cluster, we draw segment proportions with\nthis same distribution, where all slopes uare set so that a\nprobability changes from 10% to90% in one-tenth of the\ntime interval. We sample the cluster and segment means mk,s\nwithN(0,1). To imitate the exogenous variables from the real\ndata, we use three exogenous variables: the first is constant\nacross timesteps ( xi,t,1=yi,1,1), the second is constant across\nindividuals ( xi,t,2=x1,t,2), and the third varies for both\nindividuals and timesteps. All are drawn from N(0,1). Then,\nwe sample the contributions corresponding to these exogenous\nvariables ( αcoefficients) from N(0,Σα), where the value of\nΣαchanges between experiments. Finally, we generate the\nobserved variables using the variance Σk,s= 1. The choice\nof a relatively high unexplained variance Σk,sis voluntary, to\nplace the model in a difficult scenario.\nDuring the first set of experiments, we set Σα=I,\nand change the number of individuals and timesteps in\n{50,100,500,1000}. In the second series of tests, the number\nof individuals and timesteps remains constant and equal to\n100 while the parameter that gives the importance of the\ncontributions Σαvaries in {0,0.5,1.0,1.5}. The case Σα= 0,\nin particular, is interesting, for it means that the model will\nsee variables that do not contribute anything to the observed\nvariables. In this case, if its performance is significantly lower\nthan a model that does not have access to the variables, this\nwould mean that the proposed model overfits the exogenous\nvariables. In order to mitigate the randomness, we repeat the2017 2018 2019 2020 2021 2022\ndate2.0\n1.5\n1.0\n0.5\n0.0normalized log10 of the\n number of validationsLA DEFENSE-GRANDE ARCHE\nGARE DE LYON\nCHARLES DE GAULLE AIRPORT 211-09 18-09 25-09Fig. 1: Some examples of series in the dataset we process. The\nzoom-in represents two weeks starting September 11, 2019 (on\na Monday).\nexperiments ten times for each combination of parameters,\ngenerating a whole new set of parameters each time.\nResults: The results are summarized in table I. As\nexpected, the models which do not take into account the\nvariables ( Clust+Seg ) or which try to fit a regression be-\nfore any clustering ( Reg→Clust →Seg andReg→\n(Clust+Seg) ) obtain ARI scores that are noticeably low.\nThis is because the contributions we generated are independent\nof each other, meaning that fitting regression coefficients on\nthe whole dataset always produces statistically insignificant\nparameters.\nThe only case when these models are relevant is the case\nwhereΣα= 0, meaning that the exogenous contributions\nare always zero. For this experiment, the score only depends\non whether clustering and segmentation are simultaneously\ncarried out: Reg→Clust →Seg and(Clust+Reg)\n→(Seg+Reg) are both worse than the other three models,\nwhich perform similarly. As expected, the position of the\nlinear regression in the estimation process has no significant\nimportance for this case.\nC. The Public Transport Ridership Dataset\nTo validate our model on real data, we used a recording of\ndaily entries in the Parisian public transport system.2Gathered\nby the transport organization authority called ˆIle-De-France\nMobilit ´es (IdFM), this dataset covers five and a half years,\nfrom January 1, 2017, to June 30, 2022. We also focus on the\nrail network (underground, public train, and tramway). Note\nthat the records only cover the entries of travelers entering\nthe network, which means that we have the origin of trips but\nnot the destination. Note also that due to the organization of\nthe ticketing system, one person may validate several times\nfor a single trip when changing fare zones (if the journey\nincludes connections between different modes). We do not try\nto compensate for this. Finally, we must emphasize that even\nthough we use the term ”individual” to designate each time\nseries, one series of entries corresponds to a station, and not\nto a traveler.\n2available at https://data.iledefrance-mobilites.\nfr/explore/dataset/histo-validations-reseau-ferre/ and\nhttps://data.iledefrance-mobilites.fr/explore/dataset/\nvalidations-reseau-ferre-nombre-validations-par-jour-1er-semestre/\ninformation/a) Preprocessing: The raw data contain the number of\nentries for each type of ticket or subscription the travelers used.\nAs we do not make use of this information, we simply consider\nthe sum of all entries for each day and each station. However,\nonly the couples ( station, day ) with at least one entry are\npresent. One might be tempted to say that an absence of data\nfor a given day means that no user entered this station (which\nmight have happened during the lockdowns, for instance), but\nwe have no way of knowing which days are missing due to\nactual errors in the data collection process, and which days are\nabsent because of the lack of users. Thus, we leave missing\ndata as-is.\nTo ensure anonymity, when a station recorded less than five\nentries for the same type of ticketing in a day, the dataset only\ncontains the mention ”less than five” for this type of ticket.\nIn such cases, we considered that there were three entries for\nthis day, station, and ticket type. We removed stations with\nmore than 60% missing days or less than 500 entries per day\non average. To handle the many noisy days in the data, we\ncompute a two-week moving average of each station’s series\nof entries. Each value that is below one-tenth of the result is\nremoved.\nFinally, given the strong imbalance between stations, we\nnormalized the number of entries of each station. For each\nstation, we divide the number of entries by the average number\nof entries during the first year, strikes excluded (from January\n1, 2017 to December 31, 2017). We consider this period to\nbe representative of usual behavior. Then, similarly to [6], we\ntook the log in base 10 of the number of entries.\nb) Exogenous variables: We include several dummy\n(binary) variables that relate to the type of day: one variable to\nknow whether the day is a working day (meaning no holidays\nor weekends), seven variables for the days of the week, and\ntwo variables for strikes: as one strike was particularly between\nDecember 2019 and January 2020, we dedicate one variable\nto this period. The other ’strike’ variable is the same for all\nother strikes. Similarly, we create three variables for the three\nlockdowns declared by the government. Finally, to account\nfor tendencies throughout the year, such as a lower ridership\nduring the summer, we use 20 year-periodic splines with\ndegree 2. We also include four variables that are proper to\neach station, denoting the presence of four commercial types\nof train: underground (Paris only), RER (express regional\nnetwork), transilien (a network going further than RERs),\nand commercial train (presence of national line in the same\nstation).\nD. Experiments with real data\nSection IV-B demonstrated that the model is able to find the\ntrue cluster of individuals generated using Gaussian distribu-\ntions. This section shows that even with real data, the model\nwe propose outperforms the baselines defined in section IV-B.\nHowever, for many real-life datasets, we have no way of\nknowing which individuals belong to each cluster, or even\nif there is a ’right’ number of clusters. We first begin to\nselect the number of clusters using the slope heuristic [1].‘Σα=I\nI=T= 50 I=T= 100 I=T= 500 I=T= 1000\nReg→Clust →Seg 0.334±0.152 0.293±0.062 0.286±0.061 0.377±0.088\n(Clust+Reg) →(Seg+Reg) 0.589±0.141 0.643±0.161 0.620±0.081 0.788±0.114\nReg→(Clust+Seg) 0.463±0.099 0.440±0.125 0.414±0.112 0.511±0.123\nClust+Seg 0.478±0.106 0.453±0.120 0.431±0.120 0.561±0.152\nClust+Seg+Reg (Proposed) 0.708±0.143 0.784±0.110 0.865±0.088 0.895±0.072\nI=T= 100\nΣα=0 Σα= (0.52)I Σα=I Σα= (1.52)I\nReg→Clust →Seg 0.568±0.178 0.456±0.125 0.346±0.063 0.323±0.090\n(Clust+Reg) →(Seg+Reg) 0.542±0.177 0.605±0.122 0.661±0.174 0.626±0.145\nReg→(Clust+Seg) 0.645±0.165 0.548±0.094 0.467±0.121 0.442±0.133\nClust+Seg 0.686±0.092 0.586±0.137 0.499±0.179 0.434±0.107\nClust+Seg+Reg (Proposed) 0.689±0.104 0.794±0.075 0.815±0.088 0.804±0.121\nTABLE I: The mean and standard deviation of the Adjusted Rand Score obtained with ten generations of synthetic data. The\nmethod (Clust+Seg) →Reg is absent because it produces the same partition as Clust+Seg . The best result each time\nis in bold.\nnumber of individuals 542 stations\ndays 2,004\nMeasures per day 1\ncovariates\n(number of variables)type of transport available (4),\nday of the week (7), lockdowns (3),\nstrikes (2), holidays (2),\nyear-periodic splines (20), time (1)\nTABLE II: An overview of the real dataset.\nAs we expect the data to exhibit one change point at the\nbeginning of the pandemic, we select S= 2 segments a\npriori . We begin by fitting the parameters for models with one\nto nineteen clusters. Each time, we obtain a (training) log-\nlikelihood, which increases with the number of parameters.\nThe slope heuristic consists in applying a penalty equal to\ntwice the slope of the curve giving the log-likelihood as a\nfunction of the number of parameters in each model [1]. In\nour case, with S= 2 segments, the optimal value is reached\nforK= 5 clusters (results not shown).\nTo evaluate our model on real data, we perform five-\nfold cross-validation, dividing the dataset into five subsets\nof mutually-independent samples. This constraint of mutual\nindependence is not trivial: if one split the dataset into separate\nperiods, for instance, the folds would not be independent. The\nreason is that the cluster assigned to each individual remains\nconstant: if we know the segment a given day belongs to, we\ncan infer the individual’s cluster, which provides information\nabout the cluster the individual belongs to outside of the\ntraining interval. This is why we perform cross-validation by\ndividing the dataset into five groups of individuals. We use\nthe first four to estimate the parameters of the model, keeping\nthe last group to measure the log-likelihood of the unseen\ndata. We repeat the process four additional times, changing the\nvalidation dataset, and display the mean and standard deviation\nin table III.We can draw several conclusions from the results in ta-\nble III: firstly, the two methods that perform clustering and\nsegmentation separately are noticeably worse than the rest\nwhich means that performing clustering and segmentation at\nthe same time is paramount. Secondly, the model that imposes\nthe regression coefficients to be equal across clusters and\nsegments ( Reg→(Clust+Seg) ) explains unseen data as\nwell as a model that does not have access to the exogenous\nvariables ( Clust+Seg ), hinting at overfitting.\nFinally, the best two models are the ones that both per-\nform clustering and segmentation at the same time, while\nstill allowing variable contributions to differ between seg-\nments ( (Clust+Seg) →Reg andClust+Seg+Reg ).\nBetween them, the proposed approach that performs\nclustering/segmentation and regression at the same time\n(Clust+Seg+Reg ) is significantly better than the sequential\nmodel where the estimation of the contributions is done after\nthe estimation of the partition ( p < 0.01using an unpaired\nt-test).\nHowever, this improvement comes at the cost of a higher\ncomputational burden: using a desktop computer3, the imple-\nmentation that estimates all parameters at the same time with\nthe EM algorithm takes fifteen minutes to converge on average\n(15.4±2.25min), which is four times longer to run than its\nsequential counterpart ( 4.0±0.6min for (Clust+Seg) →\nReg).\nUsing cross-validation, we demonstrated that the proposed\nmodel is able to reach a higher log-likelihood than its coun-\nterparts, indicating a better ability to model the data. We will\nnow use this model to understand the effects of the pandemic\non public transport ridership.\n3a computer with an Intel I7 @ 2.80GHz and a 15 Go RAM, along with\nUbuntu 20.04Model validation LLnumber of\nparametersSegments and clusters\nare computed ...Regression\ncoefficients are...\nReg→Clust →Seg −2.02×106±3.0×10373 Sequentially Common\n(Clust+Reg) →(Seg+Reg) −1.96×106±8.2×102424 Sequentially Different\nClust+Seg 2.03×106±5.6×10334 Simultaneously Absent\nReg→(Clust+Seg) 2.05×106±5.9×10373 Simultaneously Common\n(Clust+Seg) →Reg 2.86×106±7.0×103424 Simultaneously Different\nClust+Seg+Reg (Proposed) 3.14×106±8.2×103424 Simultaneously Different\nTABLE III: The mean and standard deviation of the log-likelihood computed on the validation dataset, using five-fold cross-\nvalidation. The best result is in bold.\nE. Analysis of the results on public transport ridership\nAs mentioned at the beginning of the previous section, we\napply the proposed model with five clusters and two segments.\nAs the changes between segments are easier to explain than the\ndifference between clusters, we will begin by explaining how\nthe model provides insights into the public transport ridership\nbefore and after the pandemic, before using this behavior to\nunderstand how clusters differ from each other.\nFigure 2 displays the sum of the mean and contributions\nof the time and the year-periodic splines. As expected, the\nmean of the second segment is noticeably lower than the first\nsegment’s, and the (positive) regression coefficient associated\nwith the time models the recovery of each cluster. The impact\nof the pandemic on ridership is the gap between the recon-\nstructions of the two segments, when their prior probability is\nequal (approximately early 2020, depending on the clusters).\nAs expected, the gap between the extension of the first segment\nand the second segment is at its highest at the start of the\npandemic, after which the number of entries increase to model\nthe recovery of the network.\nGiven the importance of the pandemic on the population’s\ntrips, one could expect the model to pick up the start of the\npandemic (and the first lockdown) as the change point between\nsegments. What happened is slightly different: a major strike\noccurred between December 2019 and January 2020. Even\nthough we created a dummy variable to model the effect of\nthis specific strike, one variable is not enough to model the\ncontinuous recovery of the signal. As a consequence, for some\nclusters, the model underestimates the effect of this strike,\nand assigns the lowest days to the next segment, the segment\ndedicated to the pandemic. This is why the transition between\nsegments is sometimes located at the very beginning of the\nyear 2020.\nThe small number of days assigned this way to the second\nsegment is not, however, enough to prevent the model from\naccounting correctly for the events of the pandemic. In March\n2020, the French government decreed a lockdown to fight\nagainst the pandemic. While people were allowed to travel\nto work, companies and institutions alike were encouraged\nto implement teleworking to avoid their employees taking\npublic transport. After the end of the first lockdown, even\nthough travel for personal reasons became allowed again, the\nteleworking measures remained in place to a certain extent andkept on affecting public transport ridership. This is what we\nwant to measure with our model.\nAnother source of information is the regression coefficients\nfor each day of the week (fig. 3). Both segment one (cor-\nresponding to normal behavior) and segment two (during and\nafter the pandemic) have an increased number of entries during\nthe weekdays and a decreased ridership on weekends. But\nwhen we look at the difference between the coefficients of\ntwo segments, we notice that in several cases, the coefficients\nassociated with weekdays decreased with the pandemic and\nthe weekend’s coefficients increased. This means that the\ndifference between weekdays and weekends is less noticeable\nafter the pandemic. We hypothesize that this effect is due to\nthe use of teleworking policies that reduced the number of\ncommuters using public transport every day. Please note that\nfig. 3 is not affected by modal shift, as the effect of the sum\nof the seven covariates is absorbed by the mean ( mk,sin eq.\n5) and not the variables encoding for the day of the week.\nThe influence of the covariates varies from cluster to cluster,\nwhich will be helpful when understanding what makes clusters\nspecific.\nTo know which features distinguish one cluster from an-\nother, the first method is to look at the regression coefficients\n(fig. 3). We perform a t-test on the difference of regression\ncoefficients across a couple of clusters to know which are the\ncoefficients that change significantly between two clusters. To\ncompare clusters, we focus on the first segment, corresponding\nto a more usual behavior. Although the entries of all clusters\nincrease during weekdays and decrease during weekends, the\namplitude of weekly variations differs between clusters which\nis informative to us.\nThe first cluster has the highest variations between week-\ndays and weekends, indicating a higher rate of commuters\nthan the other clusters. It includes stations from the North,\nWest, and South suburbs, as well as the center and West Paris\nboroughs (results not shown), two regions with moderate to\nhigh rates of employments-to-surface [9]. As we expected,\nthe stations with the highest weekly variations are also those\nwhere this variation dampened the most after the pandemic:\nthe stations with the most commute are also the ones where\nteleworking policies are the most impactful on public trans-\nport.\nThe next three clusters have average weekly variations.2017 2018 2019 2020 2021 2022\ndate0.8\n0.6\n0.4\n0.2\n0.00.2normalized log10 validations\ncluster 1, segment 1\ncluster 1, segment 2cluster 2, segment 1\ncluster 2, segment 2cluster 3, segment 1\ncluster 3, segment 2cluster 4, segment 1\ncluster 4, segment 2cluster 5, segment 1\ncluster 5, segment 2Fig. 2: The splines associated with each cluster, along with the effect of time. Dashed lines denote the continuation of each\nsegment in regions where it is unlikely.\nMonday Tuesday Wednesday Thursday Friday Saturday Sunday60\n40\n20\n02040relative increase\nSegment 1\nMonday Tuesday Wednesday Thursday Friday Saturday Sunday60\n40\n20\n02040relative increase\nSegment 2\nMonday Tuesday Wednesday Thursday Friday Saturday Sunday7.5\n5.0\n2.5\n0.02.55.07.510.0relative increase\nDifference between segments\n cluster 1 \n cluster 2 \n cluster 3 \n cluster 4 \n cluster 5 \nFig. 3: The effect of the seven days of the week on each of the two segments (left, center) and the difference in regression\ncoefficients for each cluster (all values are translated from the log scale to a linear scale). We deal with the co-linearity by\ncomputing the coefficients such that the effect of the sum of the coefficients is absorbed by each segment’s mean (which does\nnot appear on this figure).\nCluster two is, by all accounts, close to the average, except for\nits variance (see table IV): both its first and second segment\nhave particularly low variances. This means that this cluster\nis dedicated to stations with little noise and average weekly\nvariations.\nConversely, cluster three has a noticeably high variance\nbefore the pandemic. This means that this cluster comprises all\nthe stations where variations in the number of entries cannot\nbe explained using the covariates. Surprisingly, variance in the\nsecond segment is moderate.\nCluster four stands out for its high variance after the\npandemic. It groups all stations for which the number of\nentries cannot be accurately explained using the covariates\nwe provided (such as the official lockdowns, for instance).\nIt should also be noted that the decrease in the number of\nentries in these stations is the largest out of all five clusters,\nalong with cluster three (fig. 2). This is compensated by thehighest recovery rate of all clusters: if we consider the impact\nof time on the logarithm of the number of entries, the number\nof entries in the second segment increases by 74% per year\non average. In comparison, the second segment of cluster\nthree increases by 60% per year, while the other three clusters\nincrease by 37% to42% per year during the period following\nthe pandemic. Thus, cluster four comprising all the stations\nwhose series of entries were affected by the pandemic.\nFinally, cluster five has the smallest differences between\nweekdays and weekends of all the clusters. Like cluster two, it\nhas a small variance, indicating that the stations in this cluster\nhave variations which the model explains correctly using the\nexogenous variables. The decrease in the number of entries\n(fig. 2) is among the lowest of all clusters, along with cluster\ntwo. Geographically, the stations in this cluster are located\nalmost exclusively in inner Paris (table IV), where housings\nare the most common [9].ClusterVariance of\nsegment oneVariance of\nsegment twoAverage distance to\nthe center of Paris ( km)\n1 1.8×10−25.4×10−216.5\n2 5.7×10−32.2×10−27.2\n3 3.5×10−25.1×10−216.0\n4 7.9×10−31.7×10−111.8\n5 4.8×10−31.7×10−25.8\nTABLE IV: The variance in each segment of each cluster,\nalong with the average distance between the stations in the\ncluster and the Center of Paris.\nTo sum up, the main element that distinguishes the clusters\nis the degree of variation between weekdays and weekends:\nclusters one and five have the largest and smallest variations,\nrespectively. Among the middle three clusters, one of them\n(cluster two) had a small variance around an average behavior,\nwhile the remaining two had a high variance either before of\nafter the pandemic. Surprisingly, no cluster had particularly\nhigh variance for both its segments. Additionally, one could\nhave expected the consequences of the pandemic to be more\ndiverse on the stations with high a commute, but the model\ngroups all these stations into a single cluster.\nV. C ONCLUSION\nClustering, segmentation, and regression are three well-\nexplored research problems, to the point the intersection\nbetween any two of them has been covered in the literature.\nWe extended the clustering-segmentation model from [18],\nand completed it by adding the contribution of exogenous\nvariables. We developed the estimation of parameters using the\nExpectation-Maximization algorithm, and experimented with\nit on both synthetic and real data, demonstrating the interest\nin a joint estimation of parameters as opposed to a sequential\none.\nUsing this model on a dataset of entries in the Paris public\ntransport network allowed us to understand how the Covid-19\npandemic affected the public transport ridership. As expected,\nall stations saw the number of entries decrease sharply at the\nstart of the first lockdown, with a slow recovery during the two\nyears that followed. In addition, most clusters saw the number\nof entries further decrease during weekdays and increase\nduring weekends, compared to their respective pre-pandemic\nlevels. We attribute this effect to teleworking policies all\nthe more so because the cluster which is defined by the\nhighest proportion of commuters saw the strongest decrease in\nridership during weekdays and the strongest increase during\nweekends. Whenever the variables we provided are not enough\nfor the model, the model assigns a high variance to the\nconcerned segment, as this was the case for another of the five\nclusters the model uncovered: the second segment of cluster\nfour had a high variance, indicating that the evolution of the\ntime series assigned to it is more irregular than the simple\neffects we included in with the covariates.\nThe next step would be to find a way to reduce the compu-\ntational load of the proposed model. One could, for instance,\nthink about initializing the EM algorithm with coefficientsfound from a simpler version of the model. Another possibility\nfor improvement would be to use model selection criteria\nto choose the number of segments automatically, such as\npenalized criteria (BIC, AIC, etc.) or dedicated heuristics [1].\nACKNOWLEDGMENTS\nThis work was supported by Vinci through the funding of\nthe Eco Conception Chair in ´Ecole des Ponts ParisTech.\nREFERENCES\n[1] J.-P. Baudry, C. 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Machine learning approach\nfor study on subway passenger flow. Scientific Reports , 12(1):2754, Feb.\n2022. Number: 1 Publisher: Nature Publishing Group.\n[16] F. Picard, S. Robin, E. Lebarbier, and J.-J. Daudin. A Segmenta-\ntion/Clustering Model for the Analysis of Array CGH Data. Biometrics ,\n63(3):758–766, 2007.\n[17] C. E. Rasmussen. Gaussian Processes in Machine Learning. In\nO. Bousquet, U. von Luxburg, and G. R ¨atsch, editors, Advanced Lectures\non Machine Learning: ML Summer Schools 2003, Canberra, Australia,\nFebruary 2 - 14, 2003, T ¨ubingen, Germany, August 4 - 16, 2003, Revised\nLectures , Lecture Notes in Computer Science, pages 63–71. Springer,\nBerlin, Heidelberg, 2004.\n[18] A. Sam ´e, F. Chamroukhi, G. Govaert, and P. Aknin. Model-based\nclustering and segmentation of time series with changes in regime.\nAdvances in Data Analysis and Classification , 5(4):301–321, Dec. 2011.\n[19] C. Truong, L. Oudre, and N. Vayatis. Selective review of offline change\npoint detection methods. Signal Processing , 167:107299, Feb. 2020."    },    {        "title": "2402.12428v1.Pre_perihelion_Monitoring_of_Interstellar_Comet_2I_Borisov.pdf",        "content": "MNRAS 000, 1–16 (2024) Preprint 21 February 2024 Compiled using MNRAS L ATEX style file v3.0\nPre-perihelion Monitoring of Interstellar Comet 2I/Borisov\nGeorge P. Prodan,1★Marcel Popescu,1,2Javier Licandro,3,4Mohammad Akhlaghi,5Julia de León,3,4\nEri Tatsumi,6Bogdan Adrian Pastrav,7Jacob M. Hibbert,8Ovidiu Văduvescu,8,3,2\nNicolae Gabriel Simion,1Enric Pallé,3,4Norio Narita,9,10,3Akihiko Fukui,9,3Felipe Murgas3,4\n1Astronomical Institute of the Romanian Academy, 5 Cut ,itul de Argint, 040557 Bucharest, Romania\n2Faculty of Sciences, University of Craiova, Craiova, Romania\n3Instituto de Astrofísica de Canarias (IAC), C/Vía Láctea s/n, E-38205, La Laguna, Spain\n4Departamento de Astrofísica (ULL), E-38205, La Laguna, Spain\n5Centro de Estudios de Física del Cosmos de Aragón (CEFCA), Plaza San Juan 1, 44001 Teruel, Spain\n6Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency (JAXA), Kanagawa, Japan\n7Institute of Space Science, Atomis ,tilor 409, 077125, Bucharest-Măgurele, Romania\n8Isaac Newton Group of Telescopes (ING), Apto. 321, E-38700 Santa Cruz de la Palma, Canary Islands, Spain\n9Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan\n10Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan\nAccepted 2024 February 12. Received 2024 February 10; in original form 2024 January 15\nABSTRACT\nThe discovery of interstellar comet 2I/Borisov offered the unique opportunity to obtain a detailed analysis of an object coming\nfrom another planetary system, and leaving behind material in our interplanetary space. We continuously observed 2I/Borisov\nbetween October 3 and December 13, 2019 using the 1.52-m Telescopio Carlos Sánchez equipped with MuSCAT2 instrument,\nand the 2.54-m Isaac Newton Telescope with Wide Field Camera. We characterize its morphology and spectro-photometric\nfeatures using the data gathered during this extended campaign. Simultaneous imaging in four bands ( 𝑔,𝑟,𝑖, and𝑧𝑠) reveals a\nhomogeneous composition and a reddish hue, resembling Solar System comets, and as well a diffuse profile exhibiting familiar\ncometary traits. We discern a stationary trend fluctuating around a constant activity level throughout October and November\n2019. Subsequently, a reduction in activity is observed in December. Dust production and mass loss calculations indicate\napproximatelyanaverageof4kg/sbeforeperihelion,whileafterperihelionthenetmasslossisabout0.6kg/s.Oursimulations\nindicate the most probable size of coma dust particles should be in the range 200-250 nm, and the terminal speed around 300\nm/s. The spectrum acquired with the 4.2-m William Herschel Telescope shows the presence of a strong CN line for which we\nfindagasproductionrateof 1.2×1024𝑠−1.WealsodetectedNH 2andOIbands.TheratiobetweenNH 2andCNproductionsis\nlog(𝑁𝐻 2/𝐶𝑁)=−0.2. Overall, this observing campaign provides a new understanding of 2I/Borisov’s unique characteristics\nand activity patterns.\nKey words: comets:individual:2I/Borisov–methods:observational–techniques:imagingspectroscopy–methods:numerical\n1 INTRODUCTION\nComets are ancient celestial objects that offer insight into the origin\nof the Solar System. The recent discovery of 2I/Borisov, the first\nknown interstellar comet (Borisov & Shustov 2021) and the second\ninterstellar object after 1I/’Oumuamua (Trilling et al. 2018), has\ncaptured the attention of astronomers as a witness of the formation\nofotherplanetarysystems.Theorbitofthiscomethasledtovarious\nproposals regarding its origin, such as being a stardust comet due to\nits water abundance (Eubanks 2019), or originating from the Wolf\n630 dynamical stream or the Ross 573 stellar system (Bailer-Jones\net al. 2020). Also, it was found by simulations that its velocity is\n★E-mail: prodangp9@gmail.comverysimilartothoseofthestarslocatedinthevicinityoftheSun(de\nLeón et al. 2020).\nComet 2I/Borisov was discovered by G. Borisov at the Crimean\nAstrophysicalObservatoryonAugust30,2019(MinorPlanetCenter\n2019).Atthattime,thecometwaslocatedatthegeocentricdistance\nΔ = 2.98au following a hyperbolic orbit with an eccentricity of\n3.35, and an apparent magnitude in the 𝑉-band of roughly 𝑉≈18.\nAccording to the pre-discovery observations (Ye et al. 2020), the\ncomet was already active at 5 – 7 au with the onset around 8 au in\nDecember-November 2018, suggesting the presence of species that\nare more volatile than the water, such as CO or CO 2.\nMany groups investigated the spectro-photometric properties of\n2I/Borisov.Firstresultsfrom2019alreadyhighlightedsomesimilar-\nities of Borisov with the Solar System comets (Jewitt & Luu 2019;\nYang et al. 2020; Ye et al. 2020; Sekanina 2019; Guzik et al. 2019;\n©2024 The AuthorsarXiv:2402.12428v1  [astro-ph.EP]  19 Feb 20242 G. P. Prodan et al.\ndeLeónetal.2020;Bolinetal.2020).Notlongafter,highresolution\nimagesallowedforabetterestimationofthenucleusradius,between\n0.2and0.5km,asreportedbyJewittetal.(2020).Thedatagathered\nwith Hubble Space Telescope (HST) on multiple observation runs\nmadebetweenNovember2019andJanuary2020revealedadusttail\noriented between the anti-solar direction and the negative heliocen-\ntric velocity vector, with a subtle inner coma asymmetry (Kim et al.\n2020). On the other hand, the HST images taken in September and\nOctober 2019 (Jewitt & Luu 2019; Manzini et al. 2020), showed\nthat the dust was scattered mostly towards the anti-solar direction.\nManzinietal.(2020)arguedthatthereleaseddustbecomesdispersed\natarelativelyshortdistancefromthenucleusandquicklyundergoes\ndeflectionduetosolarradiationpressure,eventuallymergingintothe\ntail.Thissuggestsalowejectionspeed,possiblyassociatedwiththe\nprevalence of smaller-sized dust particles. Also, they suggested that\nthemorphologyof2IissimilartothatofcometC/2014B1exhibiting\nsimilaractivationprocessesasthenativecometsoftheSolarSystem.\nThe high-resolution interferometric observations of 2I/Borisov\nmade with the Atacama Large Millimeter/submillimeter Array\n(ALMA) indicate the presence of compact pebbles of 1 mm radii\nYangetal.(2021).Also,Busarevetal.(2021)foundthattheMg-Fe\nconglomerates and organic dust components are predominant in the\n2I coma. Opitom et al. (2021) detected similarities to Solar System\ncomets in several bands, including FeI and NiI, presenting an abun-\ndanceratioof log(Ni/Fe)=0.21±0.18.Thedustproductionrateand\nmasslosshavebeencomputedinseveralstudies.However,different\nresults have been obtained because of the use of various models or\nassumptions. Implementation of Monte Carlo dust tail models (de\nLeón et al. 2020; Kim et al. 2020; Manzini et al. 2020) showed that\nthe size distribution of the tail dust particles is characterized by a\nmean value around 100 𝜇m - 1 mm, whereas smaller particles are\nmore likely to be found closer to the nucleus.\nRegardingthespectroscopy,evenif2I/Borisovhaspropertiessim-\nilartothoseofcometsinourSolarSystem,itexhibitsanunusualhigh\nratio of CO with respect to water and HCN (Bodewits et al. 2020;\nCordineretal.2020).Thisaspectisinterestingas2Iisconsideredthe\nmost pristinecomet everobserved, withpolarimetric propertiesthat\nsuggest that it has not interacted with any object since its formation\n(Bagnulo et al. 2021), maintaining its original characteristics. How-\never, active comets in the Solar System are usually rich in pristine\nmaterial.\nIndeed, no absorption bands of water ice were found at 1.5 and 2\n𝜇m (Yang et al. 2020) in September – October 2019. On the other\nhand,strongCNlinesat388nmweredetectedbyFitzsimmonsetal.\n(2019); de León et al. (2020); Aravind et al. (2021), HCN being\nsuspected as the parent molecule.\nAnotherspectralfeatureistheC 2depletionbeforereachingperihe-\nlion,asprovedbyseveralgroups(Linetal.2020;Opitometal.2019;\nBannisteretal.2020;Karetaetal.2020).Otherspecieswerealsode-\ntected,likeNH 2,forwhichBannisteretal.(2020)foundlargeabun-\ndances compared to those found for Solar System comets. Further-\nmore,highspectralresolutionmeasurementsbyMcKayetal.(2020)\nproved the presence of the forbidden oxygen lines, specifically the\nO(1D)lineat630nm.Thislinecanbeanexcellenttrackerforwater,\nasitreleasesOIintothecomabythephoto-dissociationoftheparent\nmolecule, which is usually H 2O (Feldman et al. 2004). They found\nthegasproductionrateofwater 𝑄(𝐻2𝑂)=(6.3±1.5)×1026𝑠−1on\nNovember 10, 2019. The abundance of the parent species depends\non the flux ratio between the oxygen’s green line at 557 nm and red\ndoublet lines at 630 nm and 636 nm, denoted usually by 𝐺/𝑅. The\nother potential parent species, such as CO or CO 2, can be neglected\nif they are not dominant species in the composition. However, thisis not the case for 2I, as CO was found to be dominant. Opitom\netal.(2021)found 𝐺/𝑅valuesrangingfrom0.3inNovember2019-\nFebruary2020to0.6inFebruary–March2020,showingthatCOis\nmoreprobabletoberesponsiblefortheforbiddenoxygenlinesfound\nin the spectrum of 2I.\nAnother remark on the 2I spectrum of the coma is its similarity\nto the spectra of D-type asteroids, as it was first suggested by de\nLeón et al. (2019), and later confirmed by other authors (Yang et al.\n2020; de León et al. 2020). The above mentioned features of the\ncomet,regardingthecomposition,seemtoagreewiththisidea.The\nD-type asteroids are characterized by featureless spectra and are\nsupposed to contain organics and volatiles. Tagish Lakes is one of\nthe meteorite spectral analogues for these bodies (e.g. Barucci et al.\n2018; Gartrelle et al. 2021).They are considered the most primitve\namong the asteroids population.\nThisworkfocusesontheanalysisofspectro-photometricdatacol-\nlectedwiththe1.52-mCarlosSánchezTelescope(TCS),the2.53-m\nIsaac Newton Telescope (INT), and the spectral data obtained with\nthe 4.2-m William Herschel Telescope (WHT). The observations\nwere performed during a two and a half months observational cam-\npaignbetweenOctober3-December13,2019.Thedatacoveredthe\nperihelion passage of 2I/Borisov on December 8, 2019 at a distance\nof𝑞= 2.0065 au (according to NASA JPL Small-Body Database\nBrowserephemerides1).Thegoalsofthisstudyaretodeterminethe\ncomet’s morphology and spectrophotometry, as well as to calculate\nthedustproductionrateandthenetmasslossofcometarydust.This\narticle is organized as follows. In Section 2 we present the observa-\ntionalcampaignof2I/Borisov.Section3describesthedatareduction\nandtheanalysismethod(themathematicalmodelsusedareshownin\nAppendixAandAppendixB).TheresultsarepresentedinSection4.\nThe discussion and conclusions are provided in Section 5.\n2 OBSERVATIONS\nOurobservingcampaignaimedtomonitortheactivityof2I/Borisov\nusing broad-band filters. The most convenient instrument for this\nwas the MuSCAT2 (Multicolor Simultaneous Camera for study-\ning Atmospheres of Transiting exoplanets), mounted on the 1.52-\nm TCS, which allows simultaneous imaging with four broad band\nfilters (Narita et al. 2019). We complemented our observations with\nspectro-photometricimagesobtainedwiththe2.54-mINTtelescope.\nAlso, we succeeded to acquire a spectrum with the WHT telescope.\nThese instruments are briefly described bellow.\n2.1 The 1.52-m Carlos Sanchez Telescope (TCS)\nWe conducted an extensive observational program using the TCS,\nwhich is located at the Teide Observatory in the Canary Islands,\nSpain. This observatory is well-known for its good seeing (0.8–1.2\narcsec) and the high number of clear nights (about 80%).\nWecollecteddataduring24observationalnights,spreadbetween\n3Octoberand13December,2019.Thistimeintervalcoversmainly\nthe pre-perihelion passage of 2I/Borisov (which occurred on 8 De-\ncember). The observing log is provided in Table A1.\nTheTCSisaCassegrainreflectingtelescopewitha1.52mdiam-\neter primary mirror and a focal length of f/13.8. For the MuSCAT2\ninstrument a system of lenses reduces the focal length to a ratio of\n𝑓/4.4.Thissetuphasfour1024 ×1024pixelcharge-coupleddevice\n1https://ssd.jpl.nasa.gov/horizons/\nMNRAS 000, 1–16 (2024)Interstellar Comet 2I/Borisov 3\nFigure 1. MuSCAT2 transmittance as a function of wavelength for g (blue),\nr (green), i (orange) and z 𝑠(red) bands (Narita et al. 2019).\n(CCD) cameras with a scale of 0.434 arcsec per pixel. Each camera\nis designed to capture a specific band as follows. There are 𝑔(400-\n550 nm),𝑟(550-700 nm), 𝑖(700-820 nm), and 𝑧𝑆(820-920 nm)\ncameras, providing simultaneous imaging in multiple bands using a\ndichroic mirrors system. The transmittance functions of these filters\nareillustratedinFig.1(Naritaetal.2019)andtheyalsoconsiderthe\nquantum efficiencies of the CCDs.\nThe target was observable in the mornings, and the observations\nstarted when the object has risen above 27 degaltitude (because of\nthemountmechanicallimitoftheTCStelescope).Thus,weobserved\n2I/Borisovatanairmassof1.5–2.Weacquiredconsecutiveimages\nwithanexposuretimeof30seconds.Themediannumberofimages\nacquiredduringeachsessionwas100.Allthesewerecombinedinto\natrack and stack frame (theimagesareshiftedtofollowthecomet),\nanda sky frame (theimagesareshiftedaccordingtosideraltacking).\nBecauseofthelowairmassandtelescopetrackingissues,theseeing\nmeasured on the sky frame ranged from 1.9 to 4.7 arcsec with a\nmedian value of 2.5 arcsec.\n2.2 The 2.54-m Isaac Newton Telescope (INT)\nSpectro-photometric observations were also obtained with the 2.54-\nmINTtelescopeduringfivenights(TableA2),threeatthebeginning\nof October 2019 and two during the second half of November. This\nis located at the astronomical observatory Roque de los Muchachos\nObservatory,ontheislandofLaPalmaintheCanaryIslands,Spain.\nThe Wide Field Camera (WFC) was used. This instrument consists\nof four CCDs. To minimize the readout time a subframe of 1821 ×\n1821 pixels from CCD4 (the central one) was set for observations.\nThis covers a field of view of 10.14 ×10.14 arcmin2at a sampling\nrate of 0.33 arcsec/pixel.\nThe images were acquired with the Johnson/Bessel filters 𝐵\nand𝑉, and with the Sloan filters 𝑟,𝑖, and𝑧. The following ac-\nquisition sequence was repeated during the observing time frame\n𝐵−𝑉−𝑅−𝑉−𝑖−𝑉−𝑧−𝑉.Becauseofthestrongfringing\neffect in the𝑖and𝑧filters we discarded most of these observations.\nAdditionally,duringthenightof25September,2019weobtained\nalowresolutionspectrumof2I/BorisovusingtheIntermediateDis-\npersion Spectrograph (IDS) with a configuration that includes the\nR150grismandtheEEV10CCDcamera.Becauseofthelowsignal\nto noise ratio of these data, the spectrum was only used to asses the\noverall spectral trend.2.3 The 4.2-m William Herschel Telescope (WHT)\nSpectroscopic observations of 2I were performed on the night of 26\nNovember, 2019 (at 5.40 UT, 27 November) with the 4.2-m WHT\nlocated at Roque de los Muchachos Observatory, La Palma Canary\nIslands.\nWe used the Intermediate-dispersion Spectrograph and Imaging\nSystem (ISIS) mounted at the f/11 Cassegrain focus2. In order to\ncover the spectral range from the ultraviolet to the near-infrared\nbetween 320 – 900 nm, we used both the redand the bluearms.\nFor the bluearm the setup included the R300B diffraction element\nwith the default detector EEV12, which allows a dispersion of 0.86\nÅ/pixel. In the same manner, for the redarm is set with the R316R\ngrating and the RED+ CCD camera, which provides a dispersion of\n0.98Å/pixel.Aslitwiththewidthof1arcsecwasputforbotharms.\nWith this configuration the resolution element for the bluearm is\n4.10 Å, and for the red arm is 3.80 Å.\nThe throughput of the dichroic (which allows to observe with\nboth arms simultaneously) drops around 5500 ±200 Å. Also, for\nwavelengths longer than 8000 Å the camera is affected by a strong\nfringing effect, while bellow 3500 Å the signal for 2I/Borisov was\nburied in the noise. Consequently, we discarded the data from these\nintervals.\nIn order to remove the spectrum corresponding to Sun light, we\nobserved with the same configuration three well known solar ana-\nlogues, HD 30246, Hyades 64, and SA 98-978. The observations\nwere made before (for the first two stars), and after (for SA 98-978)\nthe observations of 2I, at similar airmasses. Additionally, to obtain\nthewavelength calibrationthe spectraof aCuAr +CuNe lampwere\nobtained before, between, and after the observations of the main\ntarget (three images).\nFour exposures of 1200 seconds each were made for 2I. The data\nwas acquired between 05:00 to 06:15 UT (the beginning of the first\nandlastexposure),27November,2019.Duringthistimetheairmass\nof the comet changed form 1.77 to 1.40.\n3 DATA REDUCTION AND ANALYSIS METHODS\nThe first step for all data reduction (both spectro-photometric and\nspectroscopic) consists in applying the bias and the flat field cor-\nrections. These calibration images were acquired at the beginning\nor at the end of each observing session (or observing night). Then,\nthe spectro-photometric images were reduced using the Photometry\nPipeline(PP)software(Mommert2017),whilethespectraldatawas\nreduced using Image Reduction and Analysis Facility - IRAF (Tody\n1986).\nThe analysis methods include the characterization of comet mor-\nphology, the computation of dust production rates using the model\nproposedbyFink&Rubin(2012),andthegasproductionrateusing\ntheHasermodel(Haser1957).ForthisanalysisweusedGNUAstro\n(Akhlaghi & Ichikawa 2015; Akhlaghi 2019), Python (Van Rossum\n& Drake 2009) with astropy (Astropy Collaboration et al. 2022) for\nephemerides computations and other helpful operations, and PyMi-\neScatt (Sumlin et al. 2018) for computing the scattering efficiencies\natdifferentrefractiveindices.Also,apartoftheprocessingwasdone\nwith GNU Octave (Eaton et al. 2021).\n2https://www.ing.iac.es/astronomy/instruments/isis/\nMNRAS 000, 1–16 (2024)4 G. P. Prodan et al.\n3.1 The spectro-photometric data\nWe used the PP software (Mommert 2017) and several additional\nPython scripts in order to calibrate the images. The PP software is\nan envelope for SExtractor (Source Extractor) and SCAMP (Bertin\n2006). The astrometric and photometric calibration are performed\nusingthelargedatacataloguesgeneratedbyGAIA(GaiaCollabora-\ntion2018)andPanSTARRS(Chambers2016)surveys,respectively.\nFor the data obtained with TCS we removed the camera patterns\nuncorrected by the bias and flat procedures using the astnoisechisel\nroutine(fromGNUAstronomypackage)withthedefaultparameters\n(Akhlaghi & Ichikawa 2015; Akhlaghi 2019). The astnoisechisel is\na noise-based non-parametric technique dedicated to the detection\nof very faint nebulous objects which are buried in noise. This tool\ndetects the sky background as an intermediary step, and allows to\nremove it. We also used the astnoisechisel procedure to search for\nthe faintest structures of the 2I/Borisov coma.\nTheastrometriccalibrationisperformedforeveryimage.Then,for\neach observing session and for each band, the images are combined\nin order to obtain the track and stack frame – by shifting the images\nto follow the apparent motion of 2I/Borisov at that time, and the sky\nframe. For combining the images we used two methods, the median\nand the average. The median has the advantage of removing most\nof the background stars from the track and stack frame, but it may\nintroduceuncertaintiesinthephotometricdata(thiswasresolvedby\ncomparing the results with those obtained by combining the images\nwith the average method). The photometric calibration constant, the\nzero point, was determined using the combined sky frame. The un-\ncertainty of these determinations depends on the number of stars\navailable for each frame and it spans between 0.02 – 0.06 mag.\n3.2 The spectral data\nThe spectroscopic observations were reduced with the help\nof IRAF, specifically the apallroutine, which is included in\nnoao.twodspec.apextract package. The spectrum of the comet was\nextracted using an aperture of 4 arcsec, while the sky background\nwas measured between 10 to 20 arcsec from the central trace, on\neach side of the spectrum. Each spectral image was reduced in the\ninteractive mode of IRAF, which allowed us to chose the optimal\nmethod for tracing the spectrum.\nAfterthespectraextractionfromtheimages,weprocessedthere-\nsultsusingGNUOctave(Eatonetal.2021).Wemadethewavelength\ncalibration using the lines of the CuAr + CuNe lamp. We corrected\nfortheatmosphericextinctionusingthemodeldescribedinthetech-\nnicalnotesprovidedbytheIsaacNewtonGroup3.Then,wedivided\nthe obtained spectrum of 2I with that of the analogue solar Hyades\n64 as shown by Tatsumi et al. (2022), this star being one of the best\nsolar analogues match across the 350 - 800 nm. The other two stars\nwere used for comparison. The flux calibration was performed con-\nsidering the apparent magnitude obtained in the 𝑔filter with TCS\nwhich was observing simultaneously. According to Willmer (2018),\nthe spectral density flux zeropoint of Vega for 𝑔band is 20.64, cor-\nresponding to a value of 5.52×10−9erg s−1cm−2Å−1. Using this\nvalue and integrating the spectral density flux between 400 and 550\nnm,weobtainedthenormalizationcoefficientinordertore-scalethe\nspectrum density values to the absolute ones.\n3https://www.ing.iac.es//Astronomy/observing/manuals/ps/\ntech_notes/tn031.pdf3.3 Analysis methods\nDust model. The dust model, as detailed in Appendix B, follows\nthe approach described in Fink & Rubin (2012) to determine the\ndust production rate of comet 2I/Borisov. The model incorporates\nassumptions of constant outflow velocity for particles of the same\nradius,𝑎,andsphericallysymmetricdustdistribution.Thedustpro-\nduction rate, 𝑄, is derived considering the terminal velocity of dust\nparticles as estimated from the values of the coma length scales in\nthe sunward direction. The model accounts for the size distribution\nof particles using the distribution proposed by Fink & Rubin (2012)\n(UF distribution),\n𝑑𝑄\n𝑑𝑎=𝑔0𝑒−𝑎0\n𝑎\u0010𝑎0\n𝑎\u0011𝛼\n(1)\nIt allows for the computation of total dust grains, mass, and dust\nproduction rate. The parameters in the model include a normaliza-\ntion constant, denoted as 𝑔0, and two free constants, namely 𝑎0and\n𝛼. The size distribution’s peak, indicating the most probable size,\ncorresponds to 𝑎𝑝, which is calculated as the ratio of 𝑎0and𝛼.\nHaser model. For the analysis of the molecular emission from\ncomet 2I/Borisov, the Haser model, outlined in Appendix C, is em-\nployed.Thismodelextrapolatesthetotalnumberofmoleculesinthe\ncoma by considering a radial distribution based on the parent and\ndaughter scale lengths. The fluorescence efficiency, or \"g-factor\", is\nused to compute the total number of molecules in the entire coma.\nThe production rate of molecules is then derived from measured\nfluxes of specific molecular emission bands, ensuring static equilib-\nrium between destruction and production rates.\n4 RESULTS\nInthissectionwepresentboththephotometricandthespectroscopic\nanalysisoftheobservationaldataobtainedwithTCS,INTandWHT.\nThe TCS/MuSCAT2 observations of the comet allows us to charac-\nterize the comet’s morphology and its changes over 3 October - 13\nDecember, 2019 time frame. The results are presented in the first\nsubsection. The dust colors and its production rate are described\nin the second and the third subsections, where we also cross-check\nthe TCS/MuSCAT2 values with those obtained with the INT/WFC\ninstrument. This provides a strong reliability of our determinations\n(both telescopes observed simultaneously during the night of 4 and\n5 October, 2019). The last subsection describes the spectroscopic\nresults obtained with WHT on the morning of 27 November, 2019.\n4.1 Morphology\nThe comet 2I/Borisov exhibits a complex structure as a result of its\ninteraction with the Sun. The radiation pressure is influencing the\nshape of the coma and how the tail is oriented. As a first step, we\nanalyze the morphology using the TCS data in order to describe\nquantitatively how the solar radiation is affecting the coma. Then,\nthe large number of observations allows to make estimations on the\nterminal speed of the coma dust particles. Another key question\nwouldbeifthereareanycolorvariationsalongthebrightnessprofile\nof the comet. Studying the comet’s morphology in four different\nbands can provide answers to this.\nAn𝑟bandstackedimageisshowninFigure2,withcontoursadded\n(using the matplotlib library from Python) to distinguish between\ndifferent intensity levels. The coma appears bright and the particle\ndensity decreases with distance from the nucleus. In addition, the\nanti-velocity and the anti-solar vectors are plotted to illustrate the\nMNRAS 000, 1–16 (2024)Interstellar Comet 2I/Borisov 5\nFigure 2. Composite image in r-band with the comet on November 4, 2019\nobtained with 98 co-added frames. We illustrate the north (N) and east (E)\ndirections of the equatorial coordinate system, together with the image scale\nof 10 arcseconds. The anti-solar ( -⊙) and anti-velocity ( -v) are plotted with\nyellow,andblue,respectively.Thethreecirclesshowthecorrespondingaper-\ntures of radii 5000 km, 10000 km and 15000 km.\norientation of the coma, which is developing on the north side with\nrespect to the equatorial coordinate system, under the influence of\nboth inertial and non-gravitational forces.\nA clear coma resembling that of a comet followed by a dust tail\ncan be observed from the processed images. We retrieved the comet\nprofilebycomputingthesurfacebrightness,definedasthebrightness\nperunitangulararea, Σ𝜆=𝑑𝐵𝜆\n𝑑Ω.Weestimateitbytakingconsecutive\ncircular apertures with a step of 200 km. The flux is interpolated\nusingthe aperture_photometry ofphotutilswiththe subpixelmethod,\ndividing each pixel in 25 subpixels. As one can see in Fig. 3, there\nare no discernible differences in morphology between the surface\nbrightness profiles retrieved from the images taken simultaneously\nin the four spectral bands.\nForeachprofile,weestimatethecomalengthscale, 𝑋𝑅,byspotting\nthe point of coma distortion (the \"knee\"), where the profile trend is\nsubject to an obvious change in steepness (Jewitt 1991). To find this\npoint we apply a linear regression algorithm fitting the profile using\ntwo lines. The slope of each line gives us the surface brightness\ngradientsandwiththe\"knee\"positionwedeterminethecomalength\nscale.\nThe length scale is averaged over values estimated for the four\nbands, and plotted with respect to the square of the heliocentric\ndistance(seeFig.4)expressedinsquaredastronomicalunits, 𝑎𝑢2.As\nweexpectaproportionalrelationship(seeAppendixB),weperform\na linear fit fixing the intercept at the origin. We obtain that 𝑋𝑅/𝑟2=\n2000±300km/au2leading to\n𝑣𝑑(𝑎)=(0.15±0.01)𝑚\n𝑠\u0010𝑎\n1 m\u0011−0.5\n(2)\nThe Eq. 2 describing the dust particles’ terminal speed with respect\ntotheirsizeisinagreementwiththeMonteCarlosimulationsresult\nof0.1423𝑎−0.5(Tenishev et al. 2011), where the magnitude of the\nvelocity is determined in 𝑚/𝑠if we express 𝑎in meters.\nThesurfacebrightnessgradientsmonitoredduringalltheobserv-\ningsessions–theslopesofthetwolinearfits, 𝑚=𝑑logΣ𝜆\n𝑑log𝜃,showus\nthat for the coma we get values between 𝑚=−1.2and𝑚=−1.5,\nhaving an increasing trend near perihelion. This transition from an\nalmost isotropic coma, characterized by 𝑚=−1(Jewitt 1991), is\nFigure 3. Surfacebrightnessprofileof2Iplottedforeachspectralbandon4\nNovember 2019. For each band, the profile is fitted by two grey lines, while\nthesquaremarksthebeginningofthedistortionduetotheradiationpressure\neffects.\nFigure 4. The coma length scale as a function of the squared heliocentric\ndistance. Data is fitted linearly with fixed intercept: 𝑦=2000𝑥+0.\nthe effect of the solar radiation pressure. Beyond the \"knee\", the\nsteepness drops to values around 𝑚=−2.5.\n4.2 Dust colors\nWe calculate the flux collected inside an apparent aperture size of\n15,000 km for both TCS and INT. This aperture corresponds to an\nangularsizerangingfrom8.3to10.3arcsec,dependingontheEarth\n-cometdistanceattheobservingtime.FromthephotometryofTCS\ndata, we find the average color indices over all the observed nights\nand the corresponding standard deviations, (𝑔−𝑟)=0.61±0.08,\n(𝑟−𝑖)=0.16±0.07,and(𝑖−𝑧𝑠)=0.10±0.06.Ontheotherhand,\nwith INT we find(𝑟−𝑖)=0.22±0.04and(𝐵−𝑉)=0.79±0.04.\nThe first two plots of Fig. 5 show the color indices retrieved from\nthe TCS measurements. They seem to clusterize around the mean\nvalueexhibitingseveralvariationswithinthetimeframeoftheobser-\nvationalcampaign.Mostofthesevariationsareprobablytheresultof\ntheobservationalerrors.Thelargeroffsetscorrespondingtothetop-\nleft side of the plots correspond to measurements at different dates,\nnot being correlated to a specific time interval. Overall, the colors\nalign closely with the characteristics of Solar System long-period\nMNRAS 000, 1–16 (2024)6 G. P. Prodan et al.\n(Solontoi et al. 2012) and Trans-Neptunian objects (Ofek 2012) as\nshown in the last plot of Fig. 5.\nAsitcanbeseeninTable1,ourfindingsareinagreementwiththe\nresults reported by other groups. Overall, 2I does not seem to suffer\nobvious compositional changes that may lead to a deviation in color\nindices.\n4.3 Dust production and mass loss\nThe absolute magnitude and the effective scattering cross-section\nallow to characterize the comet activity and subsequently, to cal-\nculate the dust production rate and the mass loss. Eq. 3 provides\nthe formula for 𝐶𝑒– the effective scattering cross-section, where\n𝑟⊕=1.5×108km=1au is the Earth-Sun mean distance, 𝑝𝜈is the\ngeometric albedo taken as 0.1(Zubko et al. 2017) and 𝑚⊙,𝜈is the\napparentmagnitudeoftheSunintheband 𝜈∈{𝑔,𝑟,𝑖,𝑧𝑠},and𝐻𝜈is\nprovidedbytheEq.4.Whenestimatingthefluxandcomputingthese\nquantities,wealwaystakeanapertureradiusof15000km.Thephase\nterm is estimated linearly (Jewitt 1991) as 2.5 log𝜙(𝛼)≈− 0.04𝛼.\nThe photometry allows us to compute the effective scattering\ncross-section in each band,\n𝐶𝑒=𝜋𝑟2\n⊕\n𝑝𝜈10−0.4(𝐻𝜈−𝑚⊙,𝜈)(3)\n𝐻𝜈=𝑚𝜈−5 log(𝑟/𝑟⊕)−5 log(Δ/𝑟⊕)+2.5 log𝜙(𝛼) (4)\nOur results obtained for the effective cross-sections in 𝑔and𝑟\nbandsarecomparedinFig.6withtheresultsfoundbyJewitt&Luu\n(2019) for V-band (aperture radius of 15000 km) and Aravind et\nal. (2021) in R-band (aperture radius of 10000 km). Our data fills\nthe gap in agreement with the measurements reported by the other\nauthors.\nIn Fig. 7, the cross-sections of each band are normalized by di-\nvidingthemtothemedianvalue.Onecannoticeatrendconsistency\nalong the bands. There are small increases at the beginning of each\nmonththatmaysuggestaperiodicalbehaviourduringOctober-De-\ncember2019,ortheyaresimplyfluctuationsandtheperiodicpattern\niscoincidental.Ingeneral,thecross-sectionexperiencesfluctuations\naround an average value ( 155 km2in𝑔,178 km2in𝑟,188 km2in\n𝑖,206 km2in𝑧𝑠) at distances between 2.51 au to 2.01 au, before\nthe perihelion passage. These variations are barely exceeding 10%\nwithrespecttothemedian 𝐶𝑒foreachband.However,theyarecon-\nsistent and large enough ( ≈2𝜎) to not be considered errors. This\nbehaviour persists until December 3, where a noticeable declining\ntrend becomes evident following the last peak.\nApplying the dust model, as detailed in the methods section (Ap-\npendixB),weaimtoestimatethedustproductionrateandtotalmass\nloss.Thenormalizationconstant 𝑔0isdeterminedthroughnumerical\nintegration of Eq. B10, according to the effective scattering cross-\nsections’estimatedvalues.Ouranalysis,illustratedinFig.5,reveals\nan increasing spectrum intensity towards redder colors. This trend\nsignifies an intensification of scattering, resulting in larger values\nfor𝐶𝑒.Theseobservationsalignswithourmeasurements,wherethe\ndependence 𝐶𝑒(𝜆)demonstrates a roughly linear increasing trend.\nThus,itisimperativethatourmodelaccuratelyreproducesthislinear\ntrend, and achieving this requires an appropriate size distribution of\ndust particles having a specific scattering behaviour. As mentioned\nin Appendix B, the scattering coefficients depend on the refractive\nindexofthematerial( 𝑚=𝑛+𝑖𝑘).Itisverymuchlikelythatthedust\ncoma is a mixture of particles made of distinct materials decoupled\nfrom different regions of the surface layer of the comet nucleus.Through multiple simulations involving various materials, such\nas olivine (𝑚=1.80+0.10𝑖), pyroxene ( 𝑚=1.70+0.04𝑖), and\ncarbon (𝑚=2.00+0.50𝑖), we have identified that particles with the\nmost probable size of around 200 – 300 nm emerge as the primary\nscatterers in the coma (15000 km aperture). In Fig. 8 is shown a\nsummary of the resulting curves, which were obtained using the UF\ndistribution with 𝛼=4.7, a value found via grid searching.\nWe can observe that pyroxene with 𝑎𝑝=250nm would be the\nbestchoicewhenreproducingthescatterersbehaviourof2I.Olivine\nwould be compatible as well. However, carbon does not seem to be\npredominant in the composition of 2I, for which we assume that\ncould be a mixture of olivine and pyroxene, or another mixture of\nmaterials with similar scattering properties, i.e. refractive indices.\nLet us consider the density of compact dust as 2650 kg/m3(Yang\netal.2021)andtheparticlesizedistributionfoundforpyroxenewith\n𝑎𝑝=250nm. Thus, the mean value of the effective cross-section\nbeforeperihelionleadstoanestimationofthetotalmassofdustequal\nto(1.9±0.2)×106kg and a dust production of (7.0±0.4)×1014\nparticles/s,equivalentto (4.0±0.2)kg/s.Aftertheperihelionpassage,\nbased on the same considerations, the total mass of dust decreases\nwith(0.6±0.2)kg/sassumingalineartrend.Thedustproductionis\ndecreasing as well after perihelion, reaching a value of 3.3 kg/s on\n13rd December 2019.\n4.4 Spectroscopy\nThe spectrum of 2I/Borisov obtained with WHT is shown in Fig. 9.\nWemergedthedataobtainedwithbotharmsofthespectrograph.We\nshowthecomparisonbetweenthespectraldataandthecorresponding\nvalues of the flux obtained with TCS at the effective wavelengths of\neach band (𝑔,𝑟,𝑖,𝑧𝑠).\nTo calibrate the spectrum we used the apparent magnitudes ob-\ntained with TCS on the morning of November 25 (two days prior to\nthe spectral observations). Thus, the method implies an uncertainty\nof≈0.05 mag caused by the inaccuracy of the photometric deter-\nmination, and by the different observing dates. First, we applied the\ntransferfunctionofthe 𝑔filter(modulatedbythequantumefficiency\nof the camera) to the spectral data. The total response is obtained\nin ADU (analog to digital units). Then, the calibration consists in\ndetermining the spectral flux density values using the zeropoint for\nVega (Willmer 2018) and the corresponding apparent magnitude in\n𝑔-band obtained with TCS.\nTo characterize the dust continuum the spectrum is fitted linearly.\nWe find𝑆′\n𝐵=(18±2)%/1000Å in the range 4000−5000Å and\n𝑆′\n𝑅=(11±1)%/1000Å in the range 5700−7250Å. The value\nobtained in the blue domain is consistent with those reported by\nFitzsimmonsetal.(2019)forSeptember, 19.9 %/1000Å,orLinetal.\n(2020) for the beginning of November, 19.3 %/1000Å. Further, the\nred gradient shows that the spectrum of 2I/Borisov is steeper along\nthe blue domain as suggested by de León et al. (2020) as well.\nTo detect the emission bands, we subtracted the underlying con-\ntinuum caused by sunlight reflected by dust. The comparison of the\nspectrum before and after removing the dust continuum is depicted\ninFig.9,revealingtheprominentCN(0-0)gasemissionattributedto\nsolar fluorescence at 388 nm. The comet spectra showcasing solely\ngas fluorescence emission characteristics is provided in the bottom\nplotofFig.9,wherewespottedalsothereddoubletofoxygenat630\nnm and 636 nm.\nWe integrated numerically the total flux inside the CN band ob-\ntaining(4.8±0.2)×10−14ergs/cm2/s. To estimate the production\nrate, we use the Haser model (see Appendix C). An implementa-\nMNRAS 000, 1–16 (2024)Interstellar Comet 2I/Borisov 7\nFigure 5. Color diagrams of 2I observed with TCS data: (a, b) (𝑟−𝑖)vs.(𝑔−𝑟), and(𝑖−𝑧𝑠)colors, respectively, plotted with errorbars for each night of\nobservations,themeanvalueisshownwithorange;(c) (𝑟−𝑖)vs.(𝑔−𝑟)of2IplottedcomparedwiththecolorsofotheractivecometsfromtheSolarSystem\n(Solontoi et al. 2012) and the trans-Neptunian objects – TNOs (Ofek 2012). Also, the Sun color is plotted to emphasize the reddening.\nFigure 6. Theeffectivecross-sectionsin 𝑔and𝑟forTCS,and𝑉and𝑟forINTarerepresentedwithrespecttotime.Allthevaluesaredeterminedwithinafixed\ncircular aperture radius of 15,000 km. We plot for comparison the values reported by (Jewitt & Luu 2019) in 𝑉band (15,000 km aperture) and (Aravind et al.\n2021) in𝑅band (10,000 km aperture).\nTime Interval 𝑔−𝑟 𝑟−𝑖 𝑖−𝑧𝑠𝐵−𝑉\nTCS Oct 3 - Dec 13 0.61±0.08 0.16±0.07 0.10±0.06 -\nINT Oct 4 - Oct 30 - 0.22±0.04 - 0.79±0.04\nGuzik et al. (2019) Sep 10-13 0.63±0.02 - - -\nde León et al. (2020) Sep 13-26 0.63±0.02 - - -\nHui et al. (2020) Sep 26 - Jan 26 0.68±0.04 0.23±0.03 - -\nBolin et al. (2020) Oct 12 0.63±0.05 0.20±0.05 - -\nSep 12 - - - 0.76±0.12\nOpitom et al. (2019) Sep 30 - Oct 20 - - - 0.82±0.02\nAravind et al. (2021) Nov 30 - Dec 27 - - - 0.80±0.05\nThe Sun (Willmer 2018) 0.44±0.02 0.11±0.02 0.03±0.02 0.64±0.02\nTable 1. Average colors of 2I/Borisov during TCS and INT campaigns and their uncertainties ( ±𝜎). Other values retrieved from the literature and the solar\ncolors are provided for comparison.\ntion of this model is provided by the Lowell Minor Planet Services\nthroughtheirCometsAPI4.Takingintoaccounttheephemeridesof\n2I on 27th November, 𝑟=2.03au and¤𝑟=−5.2km/s, the interpo-\nlatedg-factoris 3.6×10−13ergs/s/molecule(Schleicher2010).The\nparent and daughter molecule lengthscales are those established in\n4https://asteroid.lowell.edu/comet/ghqA’Hearnetal.(1995).Thus,wefind 𝑄(𝐶𝑁)=(1.2±0.1)×1024𝑠−1,\nbeingclosetothevalueof (1.5±0.5)×1024𝑠−1foundbyBannister\netal.(2020)onthesameday,November26,or (1.6±0.5)×1024𝑠−1\nreported by Opitom et al. (2019) on October 20. It seems that the\nproduction rate of CN follows a decreasing trend as Fitzsimmons\net al. (2019) report 𝑄(𝐶𝑁)=(3.7±0.4)×1024𝑠−1on September\n20.Moreover,anothermeasurementinlateSeptember2019leadsto\nMNRAS 000, 1–16 (2024)8 G. P. Prodan et al.\nFigure 7. The effective scattering cross-section with respect to time, and normalized to the median value along each band. Both TCS and INT data are plotted.\nThe dark red line connects the mean values corresponding to each night. The perihelion is marked with a dashed line.\nFigure 8. Average effective scattering cross-section of 2I before perihelion\nwith respect to time, compared to the same cross-section obtained via Mie\nscatteringsimulations.Differentdustparticledistributionsanddifferentmate-\nrialsareemployedtoreplicatetheobservationaldataresults(olivine-yellow\nlines, pyroxene - blue lines, carbon - red lines). The most probable particle\nsize,𝑎𝑝, for each distribution is mentioned in the legend.\n𝑄(𝐶𝑁)=(2.3±0.4)×1024𝑠−1(de León et al. 2020) and Opitom\net al. (2019) find values between (1.8−2.1)×1024𝑠−1for the first\nhalfofthenextmonth.Post-perihelion,Cordineretal.(2020)report\nproductionratesof(0.9−1.2)×1024𝑠−1forthetimeframeDecember\n15-19.\nIn our spectroscopic analysis, we successfully identify the OI red\ndoublet lines at 630 nm and 636 nm. However, we note that the\ngreenlineoftheforbiddenoxygenlines(outatapproximately557.7\nnm) is outside the instrument range, preventing its detection in our\nobservations. Also, the resolution is not sufficient to separate the\nred OI cometary lines from the telluric ones. The two lines are\npartially overlapping each other as it can be seen in Fig. 9. The\nforbiddenoxygenlinesweredetectedbyOpitometal.(2021);McKay\net al. (2020) as well. We observe two lines at 629 nm and 633 nm\ncorresponding to ammonia, NH 2. Assuming the g-factor value of\n0.564·10−3photons/s determined by Kawakita et al. (2001) for the\n˜𝐴(0, 8, 0)Πband of NH 2at 629 nm (Prasad et al. 1988), we derive\n𝑄(𝑁𝐻2)=(0.7±0.2)×1024𝑠−1, meaning that the gas production\nquotient between NH 2and CN is log(𝑁𝐻2/𝐶𝑁)=−0.2±0.2.Thesignal-to-noiseratioforthismeasurementis 𝑆/𝑁≈7.Ifweconsider\nthe lower values of the fluorescence efficiencies for other lines that\nareinthespectralintervalcoveredbyWHT,weexpecttheirsignalto\nbeundetectableatthisnoiselevel.Thisisconfirmedbyouranalysis.\nOn the other hand, the signal detected at 633 nm is actually an\noverlappingoftwounresolvedNH 2bands(Fink&Disanti1990;Fink\nUwe&Hicks1996;Arpignyetal.1987)andfromthetotaldetected\nfluxweobtainanupperlimitforthesebandsof (1.1±0.2)×1024𝑠−1.\nRegarding C 2, we do not detect any lines above the noise level.\n5 DISCUSSION AND CONCLUSIONS\nIn this study, we presented a comprehensive analysis of observa-\ntional data derived from a monitoring campaign focused on the first\ninterstellar comet, 2I/Borisov.\nBy using the observations obtained with TCS, INT and WHT\ntelescopes, we offer key insights into the comet’s morphology and\nspectro-photometry:\n•2I/Borisov exhibited clear signs of activity, featuring a coma\nlength scale extending between 8000 and 13000 km. The observed\nlengthshowedalineartrendwithrespecttothesquaredheliocentric\ndistance.\n•Simultaneous imaging in four bands revealed a uniform com-\nposition across the entire comet surface, indicating homogeneity in\nmorphology. Color indices indicated a reddish hue, akin to active\ncomets within our solar system and Trans-Neptunian objects.\n•Thecomet’sprofileindicatedadiffusenature,aligningwithtyp-\nical characteristics of Solar System comets. The variation in surface\nbrightnessgradientobservedinourstudyrangedfrom 𝑚=−1.2to\n𝑚=−1.5,which,appearedlesssteepthanthegradientreportedby\nJewitt&Luu(2019)forSeptember26( 𝑚=−1.8).Thediscrepancy\ncan be attributed to the changes occurring in the comet’s behavior,\nparticularly around the onset of October, when our analysis identi-\nfied an increasing trend in activity. This minor outburst is followed\nby a stationary trend of the cometary activity exhibiting small fluc-\nMNRAS 000, 1–16 (2024)Interstellar Comet 2I/Borisov 9\nFigure 9. The comet spectrum obtained with WHT on 2019 November 26. The upper spectrum contains the dust continuum, whereas the bottom one is the\nfluorescence spectrum with CN, NH 2, and O I bands highlighted.\ntuations.ThispersistsuntilDecember.Thefirstjumpcorresponding\nto October has been also spotted by Bolin et al. (2020), where it\nwas proposed that the comet passed the water-ice line at 2.5 au as a\npotential cause.\n•Theassumptionofsmallfluctuationsinthecometactivitytrend\nprior to perihelion is based on the presence of two more activity\njumps in early November and December as suggested by the trend\noftheeffectivescatteringcross-section.Regardingthemagnitudeof\nthe cross-section, Kim et al. (2020) found in the same timeframe an\naverage value of roughly 200km2for the effective scattering cross-\nsection in V-band for an aperture of 16000 km, whereas we found\nthe values of(155±9)km2in𝑔band and(178±8)km2in𝑟band\nfor an aperture of 15000 km radius.\n•Employingtheacquireddata,wecomputedthedustproduction\nrate and mass loss by establishing a dust coma of particles made\nfrom pyroxene with the most probable size equal to 250 nm. Using\nthe distribution proposed by Fink & Rubin (2012) we computed a\ndustproductionrateofroughly 7×1014particles/sbeforeperihelion,\nequivalent to 4 kg/s assuming 𝜌=2650kg/m3. For comparison,\nClements(2021)determinedadustproductionbetween11–16kg/s(assuminglargedustparticleswith 𝑎=100𝜇𝑚and𝜌=1000𝑘𝑔/𝑚3,\nand slow ejection speeds between 5 – 9 m/s), exhibiting a slightly\ndecreasing trend after perihelion. Based on the same model and\nassumptions, similar trend and values were obtained by Mazzotta\nEpifani et al. (2021). Depending on the ejection speed value, they\nfind a dust production rate between 3 – 25 kg/s in October, and 2 –\n18 kg/s in December.\nManzini et al. (2020) argued that the emitted dust, once released,\ndisperses over a relatively short distance from the nucleus. Subse-\nquently, it rapidly undergoes deflection under the influence of solar\nradiationpressure,eventuallyamalgamatingintothecomet’stailim-\nplying a low ejection speed. They found particles of 30 – 100 𝜇m\nbeing predominant in the tail, expecting relatively smaller particles\ncloser to the nucleus, in agreement with the simulations performed\nby de León et al. (2020) and Kim et al. (2020). Our analysis was\nfocused on the dust emissions within a 15000 km aperture, i.e. the\ncoma dust. We found that the most probable size of dust particles\nin the coma is in the range 200 – 250 nm, much smaller than in the\ntail.AccordingtoEq.2,theejectionspeedwouldbearound300m/s,\nhigh enough to escape from the comet’s tail.\nMNRAS 000, 1–16 (2024)10 G. P. Prodan et al.\n•The activity began diminishing in December 2019, after the\nperihelion passage. This behaviour was reported by other groups\nas well (Kim et al. 2020; Aravind et al. 2021; Hui et al. 2020)\nand it is quantified by estimating linearly the decrease rate of the\neffective scattering cross-section, 𝑑𝐶𝑒/𝑑𝑡. We found that 𝑑𝐶𝑒/𝑑𝑡=\n(−4.4±1.6)𝑘𝑚2/𝑑duringatimespanbetween2Decemberand12\nDecember 2019, inside a 15000 km aperture radius. Subsequently,\nwedeterminedthenetmasslossofdustinthecomaofapproximately\n–0.6 kg/s during this post-perihelion interval. Considering a longer\nperiod of time after the perihelion passage, Aravind et al. (2021)\nfound(−1.77±0.22)𝑘𝑚2/𝑑foranR-bandapertureradiusof10000\nkm between 30 November and 27 December 2019 leading to a net\nmass loss of –2.7 kg/s assuming a mean size of dust particles equal\nto𝑎=100𝜇𝑚and𝜌=1000𝑘𝑔/𝑚3. On the other hand, Hui et al.\n(2020) reported a decrease rate 𝑑𝐶𝑒/𝑑𝑡=−0.43±0.02𝑘𝑚2/𝑑for\nan r-band aperture of same size, 10000 km, deriving a net mass\nloss of−0.4kg/s, assuming 𝜌=500𝑘𝑔/𝑚3. The discrepancy in the\nvaluesof𝑑𝐶𝑒/𝑑𝑡isattributedtothedifferentobservationtimeframes,\nwith Hui et al. (2020)’s observational campaign extending until late\nJanuary.Inaddition,ourresultisderivedusinga15000kmaperture\nradius, a larger value being expected from this point of view.\nACKNOWLEDGEMENTS\nWe acknowledge financial support from the Agencia Estatal\nde Investigación of the Ministerio de Ciencia e Innovación\nMCIN/AEI/10.13039/501100011033andtheERDF“Awayofmak-\ning Europe” through project PID2021-125627OB-C32, and from\nthe Centre of Excellence “Severo Ochoa” award to the Instituto de\nAstrofisicadeCanarias.ThisworkispartlysupportedbyJSPSKAK-\nENHI Grant Number JP18H05439, JST CREST Grant Number JP-\nMJCR1761.ThispaperisbasedonobservationsmadewiththeMuS-\nCAT2instrument,developedbyABC,atTelescopioCarlosSánchez\noperatedontheislandofTenerifebytheIACintheSpanishObserva-\ntoriodelTeide,andonobservationsmadewiththeWilliamHerschel\nTelescope(WHTproposalSW2019b06)andtheIsaacNewtonTele-\nscope (INT) both operated on the island of La Palma by the Isaac\nNewtonGroupofTelescopesintheSpanishObservatoriodelRoque\nde los Muchachos of the Instituto de Astrofísica de Canarias. The\nwork of M.P. and B.A.P was supported by a grant of the Romanian\nNational Authority for Scientific Research and Innovation, CNCS –\nUEFISCDI,projectnumberPN-III-P1-1.1-TE-2019-1504.Thework\nofM.P.andO.V.wasalsopartiallysupportedbyagrantoftheRoma-\nnianNationalAuthorityforScientificResearch–UEFISCDI,project\nnumber PN-III-P2-2.1-PED-2021-3625.\nDATA AVAILABILITY\nData available on request. The data underlying this article will be\nshared on reasonable request to the corresponding author.\nREFERENCES\nA’Hearn M. F., Millis R. C., Schleicher D. O., Osip D. J., Birch P. V., 1995,\nIcarus, 118, 223\nAkhlaghi M., 2019, arXiv e-prints, p. arXiv:1909.11230\nAkhlaghi M., Ichikawa T., 2015, ApJS, 220, 1\nAravind et al. K. e. a., 2021, Monthly Notices of the Royal Astronomical\nSociety, 502, 3491–3499Arpigny C., Magain P., Manfroid J., Dossin F., Danks A. C., Lambert K. 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The circumstances are\npresented in detail for each filter.\n𝑁𝑖𝑚𝑔Filter UT ZP (mag) 𝜎(mag) FWHM ( ”)𝑟ℎ(au)Δ(au) T (mag) 𝛼(◦)𝜃⊙(◦)𝜃−𝑣(◦)𝛿⊕(◦)\n98 g 2019-10-03 5:41 27.665 0.027 2.4 2.48575 2.97410 17.8 18.5 294.2 329.2 -11.9\n91 r 2019-10-03 5:40 27.678 0.023 2.1 2.48577 2.97413 17.8 18.5 294.2 329.2 -11.9\n94 z𝑠2019-10-03 5:38 26.087 0.035 2.6 2.48578 2.97415 17.8 18.5 294.2 329.2 -11.9\n54 g 2019-10-04 5:36 26.916 0.041 3.3 2.47294 2.95337 17.8 18.7 294.0 329.3 -12.1\n72 r 2019-10-04 5:39 26.821 0.031 2.7 2.47292 2.95333 17.8 18.7 294.0 329.3 -12.1\n58 z𝑠2019-10-04 5:37 25.333 0.017 3.3 2.47293 2.95335 17.8 18.7 294.0 329.3 -12.1\n104 g 2019-10-05 5:43 27.809 0.055 2.3 2.46014 2.93255 17.8 18.9 293.8 329.4 -12.3\n104 r 2019-10-05 5:42 27.698 0.036 2.7 2.46014 2.93256 17.8 18.9 293.8 329.4 -12.3\n103 z𝑠2019-10-05 5:42 26.211 0.031 2.1 2.46015 2.93257 17.8 18.9 293.8 329.4 -12.3\n83 g 2019-10-07 5:42 27.523 0.031 4.4 2.43505 2.89146 17.7 19.3 293.5 329.6 -12.6\n83 r 2019-10-07 5:35 27.345 0.020 4.9 2.43510 2.89155 17.7 19.3 293.5 329.6 -12.6\n81 i 2019-10-07 5:36 26.529 0.022 4.8 2.43509 2.89154 17.7 19.3 293.5 329.6 -12.6\n83 z𝑠2019-10-07 5:39 26.756 0.027 4.0 2.43507 2.89150 17.7 19.3 293.5 329.6 -12.6\n123 g 2019-10-12 5:29 27.794 0.046 2.9 2.37469 2.79086 17.6 20.3 292.7 330.1 -13.6\n109 r 2019-10-12 5:25 27.781 0.031 1.9 2.37472 2.79092 17.6 20.3 292.7 330.1 -13.6\n107 i 2019-10-12 5:24 26.922 0.046 1.9 2.37473 2.79093 17.6 20.3 292.7 330.1 -13.6\n105 z𝑠2019-10-12 5:24 27.096 0.033 2.1 2.37473 2.79093 17.6 20.3 292.7 330.1 -13.6\n97 g 2019-10-13 5:35 27.722 0.056 3.1 2.36296 2.77100 17.6 20.5 292.5 330.1 -13.7\n100 r 2019-10-13 5:36 27.734 0.058 2.7 2.36295 2.77099 17.6 20.5 292.5 330.1 -13.7\n100 i 2019-10-13 5:36 26.871 0.039 3.7 2.36296 2.77099 17.6 20.5 292.5 330.1 -13.7\n101 z𝑠2019-10-13 5:36 27.010 0.036 3.1 2.36296 2.77099 17.6 20.5 292.5 330.1 -13.7\n62 g 2019-10-14 5:53 27.793 0.127 2.6 2.33989 2.73160 17.5 20.9 292.2 330.3 -14.1\n62 r 2019-10-14 5:51 27.767 0.053 2.6 2.33991 2.73163 17.5 20.9 292.2 330.3 -14.1\n62 i 2019-10-14 5:53 26.891 0.056 2.3 2.33989 2.73160 17.5 20.9 292.2 330.3 -14.1\n62 z𝑠2019-10-14 5:53 27.045 0.047 2.2 2.33989 2.73160 17.5 20.9 292.2 330.3 -14.1\n104 g 2019-10-19 5:49 27.806 0.052 2.7 2.29587 2.65511 17.4 21.7 291.7 330.6 -14.8\n85 r 2019-10-19 5:45 27.778 0.054 2.2 2.29590 2.65516 17.4 21.7 291.7 330.6 -14.8\n88 i 2019-10-19 5:47 26.915 0.055 2.2 2.29589 2.65513 17.4 21.7 291.7 330.6 -14.8\n93 z𝑠2019-10-19 5:46 27.018 0.038 2.8 2.29589 2.65514 17.4 21.7 291.7 330.6 -14.8\n51 g 2019-10-20 5:39 27.565 0.106 2.3 2.28532 2.63648 17.4 21.9 291.6 330.6 -14.9\n57 r 2019-10-20 5:41 27.494 0.058 3.3 2.28531 2.63646 17.4 21.9 291.6 330.6 -14.9\n62 i 2019-10-20 5:42 26.608 0.060 3.1 2.28530 2.63645 17.4 21.9 291.6 330.6 -14.9\n62 z𝑠2019-10-20 5:42 26.729 0.052 3.0 2.28530 2.63644 17.4 21.9 291.6 330.6 -14.9\n81 g 2019-11-01 5:50 27.766 0.080 4.7 2.17094 2.42504 17.1 24.1 290.2 331.1 -16.6\n84 r 2019-11-01 5:51 27.716 0.049 2.4 2.17093 2.42504 17.1 24.1 290.2 331.1 -16.6\n84 i 2019-11-01 5:51 26.859 0.044 2.4 2.17094 2.42504 17.1 24.1 290.2 331.1 -16.6\n83 z𝑠2019-11-01 5:51 26.977 0.047 4.4 2.17094 2.42504 17.1 24.1 290.2 331.1 -16.6\n149 g 2019-11-02 5:31 27.699 0.037 2.2 2.16270 2.40894 17.1 24.3 290.1 331.2 -16.7\n137 r 2019-11-02 5:28 27.654 0.018 2.0 2.16272 2.40898 17.1 24.3 290.1 331.2 -16.7\n138 i 2019-11-02 5:28 26.776 0.017 2.1 2.16272 2.40898 17.1 24.3 290.1 331.2 -16.7\n128 z𝑠2019-11-02 5:25 26.923 0.021 1.9 2.16273 2.40901 17.1 24.3 290.1 331.2 -16.7\n106 g 2019-11-04 5:52 27.740 0.037 2.9 2.14646 2.37672 17.1 24.7 289.9 331.2 -16.9\n98 r 2019-11-04 5:47 27.662 0.031 2.7 2.14649 2.37677 17.1 24.7 289.9 331.2 -16.9\n98 i 2019-11-04 5:43 26.763 0.028 2.7 2.14651 2.37681 17.1 24.7 289.9 331.2 -16.9\n92 z𝑠2019-11-04 5:41 26.945 0.033 2.5 2.14653 2.37684 17.1 24.7 289.9 331.2 -16.9\n89 g 2019-11-06 5:51 27.621 0.045 3.1 2.13114 2.34566 17 25 289.8 331.2 -17.1\n91 r 2019-11-06 5:47 27.524 0.027 3.2 2.13116 2.34570 17 25 289.8 331.2 -17.1\n90 i 2019-11-06 5:41 26.737 0.026 2.8 2.13119 2.34576 17 25 289.8 331.2 -17.1\n90 z𝑠2019-11-06 5:39 26.901 0.042 2.7 2.13120 2.34579 17 25 289.8 331.2 -17.1\n89 g 2019-11-07 5:31 27.752 0.059 3.1 2.12388 2.33068 17 25.2 289.7 331.2 -17.1\n83 r 2019-11-07 5:31 27.731 0.033 2.4 2.12388 2.33068 17 25.2 289.7 331.2 -17.1\n83 i 2019-11-07 5:30 26.843 0.028 2.8 2.12389 2.33069 17 25.2 289.7 331.2 -17.1\n83 z𝑠2019-11-07 5:29 27.025 0.028 2.3 2.12389 2.33070 17 25.2 289.7 331.2 -17.1\n64 g 2019-11-10 5:35 27.646 0.029 2.9 2.10301 2.28654 16.9 25.6 289.5 331.1 -17.3\n63 r 2019-11-10 5:34 27.603 0.026 2.6 2.10301 2.28654 16.9 25.6 289.5 331.1 -17.3\n61 i 2019-11-10 5:29 26.765 0.032 2.5 2.10303 2.28659 16.9 25.6 289.5 331.1 -17.3\n58 z𝑠2019-11-10 5:27 26.934 0.037 2.4 2.10304 2.28661 16.9 25.6 289.5 331.1 -17.3\n87 g 2019-11-11 5:31 27.777 0.064 2.4 2.09649 2.27237 16.9 25.8 289.4 331.1 -17.4\nMNRAS 000, 1–16 (2024)Interstellar Comet 2I/Borisov 13\n83 r 2019-11-11 5:30 27.703 0.033 2.2 2.09649 2.27238 16.9 25.8 289.4 331.1 -17.4\n85 i 2019-11-11 5:31 26.848 0.057 2.1 2.09649 2.27238 16.9 25.8 289.4 331.1 -17.4\n83 z𝑠2019-11-11 5:30 26.999 0.027 2.0 2.09649 2.27238 16.9 25.8 289.4 331.1 -17.4\n56 r 2019-11-17 5:40 27.644 0.059 2.3 2.06182 2.19258 16.8 26.7 289.2 330.8 -17.6\n56 i 2019-11-17 5:36 26.786 0.036 2.3 2.06183 2.19262 16.8 26.7 289.2 330.8 -17.6\n52 z𝑠2019-11-17 5:34 26.940 0.040 2.1 2.06184 2.19264 16.8 26.7 289.2 330.8 -17.6\n38 g 2019-11-18 6:05 27.770 0.031 2.6 2.05673 2.18003 16.8 26.8 289.2 330.8 -17.6\n57 r 2019-11-18 6:10 27.705 0.060 1.8 2.05672 2.17999 16.8 26.8 289.2 330.8 -17.6\n56 i 2019-11-18 6:10 26.821 0.043 2.5 2.05672 2.17999 16.8 26.8 289.2 330.8 -17.6\n58 z𝑠2019-11-18 6:09 26.997 0.050 1.8 2.05672 2.18001 16.8 26.8 289.2 330.8 -17.6\n31 g 2019-11-22 6:20 27.718 0.039 2.9 2.03903 2.13344 16.7 27.3 289.1 330.4 -17.5\n31 r 2019-11-22 6:20 27.692 0.031 2.5 2.03903 2.13344 16.7 27.3 289.1 330.4 -17.5\n31 i 2019-11-22 6:20 26.806 0.037 3.1 2.03902 2.13343 16.7 27.3 289.1 330.4 -17.5\n31 z𝑠2019-11-22 6:20 26.992 0.027 2.7 2.03903 2.13344 16.7 27.3 289.1 330.4 -17.5\n97 g 2019-11-25 6:02 27.802 0.046 2.5 2.02827 2.10177 16.7 27.6 289.1 330.1 -17.4\n97 r 2019-11-25 5:57 27.736 0.037 1.9 2.02828 2.10180 16.7 27.6 289.1 330.1 -17.4\n96 i 2019-11-25 5:54 26.867 0.039 2.5 2.02829 2.10182 16.7 27.6 289.1 330.1 -17.4\n96 z𝑠2019-11-25 5:54 27.020 0.033 1.9 2.02829 2.10182 16.7 27.6 289.1 330.1 -17.4\n145 g 2019-12-02 5:59 26.952 0.050 3.6 2.01145 2.03821 16.6 28.2 289.3 329.2 -16.7\n144 r 2019-12-02 5:56 26.782 0.026 4.9 2.01145 2.03822 16.6 28.2 289.3 329.2 -16.7\n143 i 2019-12-02 5:54 26.037 0.031 3.5 2.01146 2.03824 16.6 28.2 289.3 329.2 -16.7\n144 z𝑠2019-12-02 5:51 26.119 0.042 4.3 2.01146 2.03825 16.6 28.2 289.3 329.2 -16.7\n294 g 2019-12-04 5:34 27.271 0.031 2.3 2.00885 2.02299 16.6 28.3 289.4 328.8 -16.5\n294 r 2019-12-04 5:32 27.233 0.021 2.9 2.00885 2.02300 16.6 28.3 289.4 328.8 -16.5\n294 i 2019-12-04 5:27 26.375 0.024 2.5 2.00886 2.02302 16.6 28.3 289.4 328.8 -16.5\n219 z𝑠2019-12-04 5:27 26.575 0.027 5.2 2.00886 2.02302 16.6 28.3 289.4 328.8 -16.5\n52 g 2019-12-08 6:05 27.614 0.051 3.0 2.00655 1.99595 16.6 28.5 289.7 328.1 -15.8\n50 r 2019-12-08 5:59 27.581 0.038 2.5 2.00655 1.99598 16.6 28.5 289.7 328.1 -15.8\n52 i 2019-12-08 6:00 26.738 0.037 2.5 2.00655 1.99597 16.6 28.5 289.7 328.1 -15.8\n49 z𝑠2019-12-08 5:59 26.909 0.039 2.3 2.00655 1.99598 16.6 28.5 289.7 328.1 -15.8\n27 g 2019-12-10 6:23 27.711 0.048 2.4 2.00689 1.98436 16.5 28.6 289.9 327.6 -15.5\n26 r 2019-12-10 6:24 27.669 0.039 2.2 2.00689 1.98436 16.5 28.6 289.9 327.6 -15.5\n29 i 2019-12-10 6:22 26.825 0.038 2.2 2.00689 1.98436 16.5 28.6 289.9 327.6 -15.5\n29 z𝑠2019-12-10 6:23 26.985 0.043 1.9 2.00689 1.98436 16.5 28.6 289.9 327.6 -15.5\n32 g 2019-12-13 5:42 27.686 0.052 2.4 2.00923 1.96962 16.5 28.6 290.3 326.9 -14.9\n25 r 2019-12-13 5:40 27.624 0.022 2.3 2.00923 1.96962 16.5 28.6 290.3 326.9 -14.9\n28 i 2019-12-13 5:41 26.815 0.043 2.5 2.00923 1.96962 16.5 28.6 290.3 326.9 -14.9\n25 z𝑠2019-12-13 5:40 26.982 0.040 2.2 2.00923 1.96962 16.5 28.6 290.3 326.9 -14.9\nTable A2: INT observing campaign log of 2I/Borisov. The circumstances are\npresented in detail for each filter.\n𝑁𝑖𝑚𝑔Filter UT ZP (mag) 𝜎(𝑚𝑎𝑔)FWHM ( ”)𝑟ℎ(au)Δ(au) T (mag) 𝛼(◦)𝜃⊙(◦)𝜃−𝑣(◦)𝛿⊕(◦)\n3 B 2019-10-04 5:11 28.135 0.065 2.3 2.47315 2.95372 17.8 18.7 294.0 329.3 -12.1\n3 i 2019-10-04 5:17 28.237 0.048 2.0 2.47310 2.95364 17.8 18.7 294.0 329.3 -12.1\n3 r 2019-10-04 5:14 28.531 0.048 2.0 2.47313 2.95368 17.8 18.7 294.0 329.3 -12.1\n12 V 2019-10-04 5:14 28.419 0.038 2.2 2.47313 2.95368 17.8 18.7 294.0 329.3 -12.1\n4 B 2019-10-05 4:53 28.286 0.095 2.2 2.46058 2.93328 17.8 18.9 293.8 329.4 -12.3\n4 i 2019-10-05 4:58 28.325 0.058 1.8 2.46052 2.93319 17.8 18.9 293.8 329.4 -12.3\n4 r 2019-10-05 4:55 28.614 0.058 1.8 2.46055 2.93323 17.8 18.9 293.8 329.4 -12.3\n16 V 2019-10-05 4:57 28.487 0.048 2.0 2.46053 2.93321 17.8 18.9 293.8 329.4 -12.3\n3 B 2019-10-06 5:17 28.374 0.069 2.2 2.44775 2.91232 17.8 19.1 293.7 329.5 -12.5\n3 i 2019-10-06 5:22 28.412 0.060 1.9 2.44770 2.91223 17.8 19.1 293.7 329.5 -12.5\n3 r 2019-10-06 5:19 28.688 0.044 2.0 2.44772 2.91227 17.8 19.1 293.7 329.5 -12.5\n12 V 2019-10-06 5:20 28.581 0.042 2.1 2.44772 2.91227 17.8 19.1 293.7 329.5 -12.5\n4 B 2019-10-18 5:07 28.131 0.076 1.9 2.37469 2.79086 17.5 21.5 291.8 330.5 -14.6\n4 i 2019-10-18 5:13 28.005 0.080 1.8 2.37472 2.79092 17.5 21.5 291.8 330.5 -14.6\n4 r 2019-10-18 5:10 28.386 0.012 1.9 2.37473 2.79093 17.5 21.5 291.8 330.5 -14.6\n16 V 2019-10-18 5:13 28.303 0.094 1.8 2.37473 2.79093 17.5 21.5 291.8 330.5 -14.6\n2 B 2019-10-30 5:49 28.808 0.049 2.6 2.18821 2.45834 17.2 23.8 290.4 331.1 -16.3\nMNRAS 000, 1–16 (2024)14 G. P. Prodan et al.\n2 i 2019-10-30 5:57 28.737 0.059 1.7 2.18816 2.45825 17.2 23.8 290.4 331.1 -16.3\n2 r 2019-10-30 5:53 29.074 0.067 2.4 2.18818 2.45829 17.2 23.8 290.4 331.1 -16.3\n8 V 2019-10-30 5:52 28.957 0.055 2.2 2.18819 2.45831 17.2 23.8 290.4 331.1 -16.3\n𝑁𝑖𝑚𝑔-thenumberofimagesusedinstacking,asweremovedapartofTCSimagesbecauseoftrackingerrorsorunfavourablecircumstances\n(i.e. star occultations)\nUT - Universal Time (yyyy-MM-dd HH:mm)\nZP - the zeropoint (calibration via Pan-STARRS catalog), in magnitudes\n𝜎- the zeropoint uncertainty, in magnitudes (by propagation becomes the uncertainty of the estimated magnitude)\nFWHM - The Full Width at Half-Maximum of the Point Spread Function , in arcseconds\n𝑟ℎ- heliocentric distance, in astronomical units\nΔ- distance to the Earth, in astronomical units\n𝑇- estimated total magnitude of the comet, in magnitudes\n𝛼- phase angle, in degrees\n𝜃⊙- position angle of the anti-solar vector, in degrees\n𝜃−𝑣- position angle of the anti-velocity vector, in degrees\n𝛿⊕- orbital plane angle measuring the comet elevation above the ecliptic as seen by a terrestrial observer, in degrees\nTable A3: WHT observing circumstances of 2I/Borisov.\n𝑁𝑖𝑚𝑔 Instrument UT Exposure (s) 𝑟ℎ(au) Δ(au) T (mag) 𝛼(◦)\n13 ISIS red arm 2019-11-27 5:40 4 x 1200 2.02519 2.09191 16.6 27.6\n13 ISIS blue arm 2019-11-27 5:40 4 x 1200 2.02519 2.09191 16.6 27.6\nMNRAS 000, 1–16 (2024)Interstellar Comet 2I/Borisov 15\nAPPENDIX B: DUST MODEL\nThemodeldescribedinthissectionfordeterminingthedustproduc-\ntionratehasbeenproposedinFink&Rubin(2012).Thecontinuum\nspectradetectedbyobserversonEarthfromacometoriginatesfrom\nsunlight that reflects off the cometary dust. When a comet becomes\nactive, gas released by the nucleus carries dust particles until they\nreach their terminal velocity. According to direct simulations using\nthe Monte Carlo approach, the gas-dust decoupling happens around\n10-20 km from a 2 km nucleus (Tenishev et al. (2011)), which is\nrelativelyshortcomparedtothecoma’ssize.Therefore,themodelis\nbased on the following assumptions:\n•the outflow velocity is constant for dust particles of same\nsize,𝑎, denoted as 𝑣𝑑=𝑣𝑑(𝑎). The above mentioned Monte\nCarlo simulations show that the terminal speed is about 𝑣𝑑(𝑎) ≈\n0.1423𝑚\n𝑠\u0010𝑎\n1 m\u0011−0.5\n•the distribution and outflow of cometary dust are spherically\nsymmetric, and there is no dust production or destruction inside the\ncoma.\nThe coma length scale in the sunward direction, as restricted by\nthe solar radiation pressure (Jewitt 1991), can be estimated as\n𝑋𝑅=𝑣2\n𝑑𝑟2\nℎ\n2𝛽𝐺𝑀⊙(B1)\nwhere𝑟ℎis the heliocentric distance, and 𝛽is the radiation pressure\nfactor,whichisroughlyequalto1𝜇𝑚\n𝑎fordielectricspheres(Bohren\n& Huffman 1983). Thus, Eq. B1 leads to 𝑣𝑑(𝑎)=𝑘𝑑𝑎−0.5, and a\nlinear relationship between the coma length scale and the square of\nthe heliocentric distance.\nThe dust production rate ( 𝑄𝑑) represents the total number of par-\nticles generated by the nucleus per unit time. During a time interval\nΔ𝑡,Δ𝑁=𝑄𝑑Δ𝑡particles are created, which will travel through the\nvolumeΔ𝑉=4𝜋𝑟2Δ𝑟at a radial distance 𝑟from the nucleus with a\nconstant outflow velocity of 𝑣𝑑. Therefore,Δ𝑟=𝑣𝑑Δ𝑡and the dust\nvolume density of particles as a function of 𝑟is\n𝑛𝑣𝑜𝑙=Δ𝑁\nΔ𝑉=𝑄𝑑\n4𝜋𝑣𝑑𝑟2(B2)\nThesurfacedensityasadensityofacolumnofdustalongtheline\nof sight at a distance 𝜌from the nucleus and projected in the sky\nplanecanbecalculatedbyintegratingEq.B2alongthelineofsight,\n𝑧. Substituting 𝑟2=𝜌2+𝑧2, and assuming that 𝜌 <<𝑧one gets\n𝑛𝑐𝑜𝑙(𝜌)=∫∞\n−∞𝑄𝑑\n4𝜋𝑣𝑑1\n𝜌2+𝑧2��𝑧=𝑄𝑑\n4𝑣𝑑𝜌(B3)\nThen,thenumberofdustparticlesisderivedbyintegratingoverthe\nconsidered aperture radius 𝑅, such that\n𝑁=∫𝑅\n0𝑛𝑐𝑜𝑙(𝜌)2𝜋𝜌𝑑𝜌=𝑄𝑑𝜋𝑅\n2𝑣𝑑(B4)\nTheterminalvelocityofdustoutflowandthescatteringefficiency\ndepend on the size of the dust particles. Thus, in order to deter-\nmine macroscopic quantities such as mass or dust production rate\none needs to make an assumption about the size distribution of the\nparticles. For this, the size distribution formula proposed in Fink &\nRubin (2012) is\n𝑑𝑄\n𝑑𝑎=𝑔0𝑒−𝑎0\n𝑎\u0010𝑎0\n𝑎\u0011𝛼\n(B5)\nwhere𝑔0is a normalisation constant and, 𝑎0and𝛼are free con-\nstants.Themostprobablesizegivenbythepeakofthesizedistribu-\ntion is𝑎𝑝=𝑎0\n𝛼.The equivalent grain density distribution derives from the equa-\ntions B4 and B5 as\n𝑑𝑁\n𝑑𝑎=𝜋𝑅\n2𝑣𝑑𝑔0𝑒−𝑎0\n𝑎\u0010𝑎0\n𝑎\u0011𝛼\n(B6)\nHence, the total number of dust grains, mass, and dust production\nrate will be\n𝑁=𝜋𝑅\n2∫𝑎𝑚𝑎𝑥\n𝑎𝑚𝑖𝑛𝑔01\n𝑣𝑑𝑒−𝑎0\n𝑎\u0010𝑎0\n𝑎\u0011𝛼\n𝑑𝑎 (B7)\n𝑀=𝜋𝑅\n2∫𝑎𝑚𝑎𝑥\n𝑎𝑚𝑖𝑛𝑔0𝜌(𝑎)4𝜋𝑎3\n3𝑣𝑑𝑒−𝑎0\n𝑎\u0010𝑎0\n𝑎\u0011𝛼\n𝑑𝑎 (B8)\n𝑄𝑑=∫𝑎𝑚𝑎𝑥\n𝑎𝑚𝑖𝑛𝑔0𝑒−𝑎0\n𝑎\u0010𝑎0\n𝑎\u0011𝛼\n𝑑𝑎 (B9)\nFurther,oneneedstoknowthescatteringefficiencyinordertodeter-\nmine the total scattering cross-section of the comet. The scattering\nefficiency𝑞𝑠(𝑎,𝜆)iscomputedbyMiescatteringtheoryMie(1908)\nHansen&Travis(1974)anditdependsonthewavelengthandtheop-\ntical properties of the dust. To compute these scattering coefficients\nwe use PyMieScatt, a Python software developed by Sumlin et al.\n(2018), which applies the Mie solution of the Maxwell’s equation\nto find the scattering coefficient 𝑞𝑠(𝑎,𝜆)for a spherical particle of\nsize𝑎at wavelength 𝜆. We estimate these coefficients for a specific\nrefractive index 𝑚=𝑛+𝑖𝑘considering the effective wavelength of\neach band.\nThetotalscatteringcross-section, 𝐶𝑒,isfoundbyintegratingover\nthe scattering area of each dust particle, 𝜎=𝑞𝑠(𝑎,𝜆)𝜋𝑎2. Namely,\n𝐶𝑒=𝜋𝑅\n2∫𝑎𝑚𝑎𝑥\n𝑎𝑚𝑖𝑛𝑔0𝑞𝑠(𝑎,𝜆)\n𝑣𝑑𝜋𝑎2𝑒−𝑎0\n𝑎\u0010𝑎0\n𝑎\u0011𝛼\n𝑑𝑎 (B10)\nThemaximumsizeofthegrainsdependsonthegravitationalforce\nexerted by the nucleus and the gas drag force entertaining the dust.\nTenishev et al. (2011) One can write the second law of mechanics\nfor a grain having the mass 𝑚=𝜌4𝜋𝑎3\n3as\n𝑚¤®𝑣𝑑=1\n2𝐶𝑑𝜋𝑎2𝜌𝑔(®𝑣𝑔−®𝑣𝑑)|®𝑣𝑔−®𝑣𝑑|−𝐺𝑀𝑛𝑚\n𝑟2®𝑟\n𝑟(B11)\nwhere®𝑣𝑔is the gas velocity, 𝐺is the gravitational constant, 𝑀𝑛=\n𝜌𝑛4𝜋𝑟3\n𝑛\n3isthenucleusmass, 𝜌𝑔isthegasdensityand 𝐶𝑑isthedrag\ncoefficient.Onecanset 𝑎𝑑=0and𝑟=𝑟𝑛(nucleusradius)tofindthe\nsize limit allowed by the gravitational force exerted by the nucleus.\nThus,\n𝑎𝑚𝑎𝑥=3𝐶𝑑𝑟2𝑛𝑣2𝑔𝜌𝑔\n8𝐺𝑀𝑛𝜌(B12)\nA accepted value for the drag coefficient of cometary gas is 𝐶𝑑=2\nGombosi et al. (1986). The outflow gas velocity can be determined\nfrom the gas temperature,\n1\n2𝜇𝑣2\n𝑔=3\n2𝑘𝐵𝑇 (B13)\nwhere𝜇is the molecular mass and 𝑘𝐵the Boltzmann constant.\nHence, equation B12 can be rewritten as\n𝑎𝑚𝑎𝑥=27𝐶𝑑𝑘𝐵𝑇𝜌𝑔\n32𝜋𝐺𝑟𝑛𝜌𝑛𝜌𝜇(B14)\nMNRAS 000, 1–16 (2024)16 G. P. Prodan et al.\nSpecies r (au) 𝑔(erg/s)𝜆𝑃(km)𝜆𝐷(km)\nCN\n(0-0), 388 nm1.0 3.6×10−1313000 210000\n2.0 8.7×10−1453500 865000\nNH2\n(0, 8, 0)1.0 4.44×10−154900 62000\n2.0 1.77×10−1520000 255000\nTable C1. Haser model properties of CN and NH 2species detected in this\nwork\nAPPENDIX C: HASER MODEL\nTo compute the production rate (Q) of molecules in the comet’s\ncoma, we measure the flux 𝐹𝑖=𝐿𝑖/4𝜋Δ2of a specific molecular\nemissionband 𝑖.Then,thenumberofmoleculeswithintheobserving\naperture is computed using the fluorescence efficiency, 𝐿/𝑁, called\nalso \"g-factor\".\nUsing Haser model (Haser 1957; Cochran 1985), we extrapolate\nthetotalnumberofmoleculesintheentirecoma,byconsideringthe\nfollowing density distribution of the molecules,\n𝑛=𝑄\n4𝜋𝑟2𝑣𝜆𝐷𝜆𝑃\n𝜆𝑃−𝜆𝐷(𝑒−𝑟/𝜆𝑃−𝑒−𝑟/𝜆𝐷) (C1)\nOnecannoticethatthisisaradialdistribution,dependingonaradial\nposition,𝑟, outflow velocity 𝑣, and the parent and daughter scale\nlengths,𝜆𝑃,𝐷. The production rate is then derived ensuring static\nequilibrium between destruction and production rates,\n𝑄𝑖=𝑁𝑖\n𝜏𝑖=𝐿𝑖\n𝑔𝑖𝜏𝑖=4𝜋Δ2𝐹𝑖\n𝑔𝑖𝜏𝑖(C2)\nwhere𝜏𝑖is the lifetime of the daughter molecules, and it is equal\nto the ratio between the daughter scale length, 𝜆𝐷, and the outflow\nvelocity, which can be estimated as 𝑣=0.58/√𝑟ℎkm/s (Delsemme\n1982). The values for the g-factors and scale lengths are provided\nin Table C1 for the standard heliocentric distance of 1 au and the\nheliocentric distance at the time we observed the comet at 2 au. The\nscalelengthsat1auforCNareretrievedfromA’Hearnetal.(1995),\nwhereas for NH 2from Fink Uwe & Hicks (1996). Regarding the\nfluorescenceefficiencies,weusethosereportedbySchleicher(2010)\nfor CN and by Kawakita et al. (2001) for NH 2.\nAPPENDIX D: COMPOSITE IMAGES WITH 2I/BORISOV\nIn this appendix we provide the composite images in r-band for the\nnight observed with TCS.\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\nMNRAS 000, 1–16 (2024)Interstellar Comet 2I/Borisov 17\nFigure D1. Composite images of 2I in r-band with the comet during the observational campaign. They are obtained using the TCS data by tracking on the\ncomet’smotionandstacking.Thenumberofco-addedframescanbecheckedintheobservingloginTableA1.North(N)andeast(E)directionsoftheequatorial\ncoordinate system, and the image scale of 10 arcseconds are illustrated. The anti-solar ( -⊙) and anti-velocity ( -v) orientations retrieved with JPL Horizons are\ndrawn with yellow, and blue, respectively.\nMNRAS 000, 1–16 (2024)"    },    {        "title": "2402.12433v1.A_Measurement_of_the_Assembly_of_Milky_Way_Analogues_at_Redshifts__0_5___z___2__with_Resolved_Stellar_Mass_and_Star_Formation_Rate_Profiles.pdf",        "content": "Draft version February 21, 2024\nTypeset using L ATEXtwocolumn style in AASTeX631\nA Measurement of the Assembly of Milky Way Analogues at Redshifts 0.5< z < 2with Resolved\nStellar Mass and Star-Formation Rate Profiles\nVivian Yun Yan Tan\n ,1Adam Muzzin\n ,1Danilo Marchesini\n ,2Visal Sok\n ,1Ghassan T E Sarrouh\n ,1and\nZ. Cemile Marsan\n1\n1Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada\n2Department of Physics and Astronomy, Tufts University, 574 Boston Avenue, MA 02155, USA\n(Received October 16, 2023; Revised February 13, 2024; Accepted February 19, 2024)\nABSTRACT\nThe resolved mass assembly of Milky-Way-mass galaxies has been previously studied in simulations,\nthe local universe, and at higher redshifts using infrared (IR) light profiles. To better characterize the\nmass assembly of Milky Way Analogues (MWAs), as well as their changes in star-formation rate and\ncolor gradients, we construct resolved stellar mass and star-formation rate maps of MWA progenitors\nselected with abundance matching techniques up to z ∼2 using deep, multi-wavelength imaging data\nfrom the Hubble Frontier Fields. Our results using stellar mass profiles agree well with previous studies\nthat utilize IR light profiles, showing that the inner 2 kpc of the galaxies and the regions beyond 2 kpc\nexhibit similar rates of stellar mass growth. This indicates the progenitors of MWAs from z∼2 to the\npresent do not preferentially grow their bulges or their disks. The evolution of the star-formation rate\n(SFR) profiles indicate greater decrease in SFR density in the inner regions versus the outer regions.\nS´ ersic parameters indicate modest growth in the central regions at lower redshifts, perhaps indicating\nslight bulge growth. However, the S´ ersic index does not rise above n∼2 until z <0.5, meaning these\ngalaxies are still disk dominated systems. We find that the half-mass radii of the MWA progenitors\nincrease between 1 .5< z < 2, but remain constant at later epochs ( z <1.5). This implies mild bulge\ngrowth since z∼2 in MWA progenitors, in line with previous MWA mass assembly studies.\nKeywords: Galaxy evolution, Milky Way\n1.INTRODUCTION\nObservations of stars and gas within the Milky Way\nhave long been the primary method of deciphering\nits formation and evolutionary history (Freeman &\nBland-Hawthorn 2002; Freeman et al. 2013; Bovy et al.\n2012; Bovy & Rix 2013; Rix & Bovy 2013, see Bland-\nHawthorn & Gerhard 2016; Helmi 2020 for reviews).\nHowever, the Milky Way is certainly not unique, and\nis a rather typical galaxy in the universe. It is a spiral\ngalaxy with a mass of 5 −6×1010M⊙(Mutch et al.\n2011; Licquia et al. 2015; Bland-Hawthorn & Gerhard\n2016) which exists in a low density environment, and it\ncurrently resides in the green valley, the transition state\nbetween star-forming and quiescent. It has a bulge with\nan extent of R∼2 kpc in the major axis, a bar with\nhalf-length ∼5kpc (Wegg et al. 2015), and a stellar disk\nthat extends out to R∼15kpc (Robin et al. 1992; Ru-\nphy et al. 1996; Minniti et al. 2011). We wish to place\nthe Milky Way in the context of how it formed, espe-cially in relation to Milky Way Analogues in the local\nand the higher redshift universe. One way to do this is\nto utilize observations of Milky Way Analogues (MWAs)\nat higher redshifts to trace the evolution of our galaxy\nover cosmic time. There are many ways to define MWAs;\nhowever, for the purpose of this paper we define them\nonly by stellar mass.\nThe morphology of star-forming galaxies in the lo-\ncal universe suggests that star formation occurred in\ntwo phases, with bulges forming first and disks forming\naround them later, also known as “inside-out” forma-\ntion (van den Bosch 1998; Mu˜ noz-Mateos et al. 2007;\nWang et al. 2011; Goddard et al. 2017), with even the\nMilky Way’s inner and outer disk forming in phases\nas well (Haywood et al. 2013). An alternative picture\nposits that the stellar disks formed first, and then bulges\ngrew later through secular processes (Dekel et al. 2009;\nMartig et al. 2009), which may be the case for lower\nmass galaxies but not higher mass ones, which tend toarXiv:2402.12433v1  [astro-ph.GA]  19 Feb 20242 Tan et al.\nquench inside-out (Cappellari 2013). However, obser-\nvations in the local universe show dwarf galaxies may\nhave had alternating periods of outside-in and inside-out\ngrowth (Ibarra-Medel et al. 2016). There is also evidence\nthat outside-in quenching may be linked to belonging to\ndenser environments such as clusters and large groups\n(Gavazzi et al. 2013; Schaefer et al. 2017). It is still\nan open question whether galaxies like the Milky Way\nformed inside-out or outside-in, or if the color gradient\npresent in the Milky Way and similar spirals developed\nin a different way.\nSpecifically for MWAs, some studies have approached\nunderstanding the formation of MWAs from simula-\ntions (Abadi et al. 2003a,b; O˜ norbe et al. 2015; Torrey\net al. 2017; Buck et al. 2020; Sotillo-Ramos et al. 2022;\nHasheminia et al. 2022). Buck et al. (2020) find MW-\nlike stellar disks form most of their mass in-situ and only\n5% of the mass is from mergers. Sotillo-Ramos et al.\n(2022) find that MWAs can even survive major mergers\nwith their disk intact but somewhat thicker. Hashem-\ninia et al. (2022) found MWAs have no evolution in their\nhalf-mass radii for the last 10 Gyr, which coincides with\nredshift z∼2. While simulations offer one solution to\nuncovering a dynamic picture of the formation history\nof the Milky Way, there are limitations to the utility of\ntheir results. This is related to the particle resolution,\nthe size of the region being simulated, the treatment of\ngas, stars, and dark matter, and other factors such as\ncosmological parameters or hydrodynamics, which have\ndifferent treatments depending on which simulation code\nis used.\nThere are also abundance matching studies that fo-\ncus on the progenitors of massive galaxies in the lo-\ncal universe ( ⋆ > 1011M⊙, such as Ownsworth et al.\n(2014); Mundy et al. (2015); Ownsworth et al. (2016),\nand Hill et al. (2017). Ownsworth et al. (2014) found\nthat major and minor mergers contribute up to 17%\nand 34% respectively of added stellar mass seen at\nz= 0.3 since z∼3, in comparison to the 24% added\nby star-formation, by comparing merger-adjusted num-\nber densities to constant number densities. However,\nmassive galaxies are more likely to be quiescent, and\nare on average less star-forming compared to the Milky\nWay. Mundy et al. (2015) is a simulation-based study\nwhich found that the choice of number density in past\nepochs affects the recovered SFR and SFH of progeni-\ntors. Ownsworth et al. (2016) and Hill et al. (2017) both\nfind the quiescent fraction to be ∼90% at z <1, but\nthe former used a fixed cumulative number density and\nthe latter used an evolving cumulative number density.\nTherefore, at 1 < z < 2, Ownsworth et al. (2016) find\nthe majority of their massive galaxy progenitors to bequiescent, while for the progenitors in Hill et al. (2017),\nthey are still majority star-forming at the same redshift.\nNevertheless, both studies agree that at earlier epochs\n(z≳2), massive galaxies assembled mass primarily via\nstar-formation, while at later epochs, most mass was\nassembled via mergers.\nThree studies that use an observational approach, and\nwhich are seminal to furthering our understanding of the\nevolution of MWAs up to high redshift are van Dokkum\net al. (2013), Patel et al. (2013), and Papovich et al.\n(2015). van Dokkum et al. (2013) (hereafter VD13) used\na constant number density through time to select MWAs\nat redshifts up to z∼3, with stellar mass functions from\nMarchesini et al. (2009). Their results show that the\nbulges and disks grow together in lockstep throughout\nthe probed redshift. Using a constant comoving num-\nber density may not be the most accurate assumption\nto make for high redshift observations, due to galaxy\nmergers changing local density (Behroozi et al. 2013a,b;\nLeja et al. 2013; Lin et al. 2013).\nPatel et al. (2013) (hereafter P13) used a different\napproach for choosing their sample: the star-forming\nmain sequence. They used the relationship between\nspecific star-formation rate (sSFR) to stellar mass, and\nchose their sample of MW progenitors from the sSFR-\nmass plane from Karim et al. (2011). The evolution\nof star-forming galaxies that led to a final mass was at\n1010.5M⊙, as computed by Leitner (2012), was the basis\nof the stellar mass selection of their MWAs at higher\nredshift. On average, P13 find MWA progenitors to be\nmore star-forming and less massive when compared to\nVD13 across the same redshift range. Interestingly, both\nstudies agree in their findings that there are no strong in-\ndications for inside-out or outside-in growth in their pro-\ngenitors, despite the different mass and redshift ranges.\nIn fact, both works show that the inner and outer re-\ngions grow in lockstep with each other from z∼3 to\nthe present epoch.\nPapovich et al. (2015) (hereafter P15) utilized abun-\ndance matching in concordance with number density\nevaluations on the zFOURGE/CANDELS fields to de-\nfine and delineate the different evolutionary phases\nthat progenitors of MW-like and M31-like galaxies go\nthrough since z∼3. For the MW-like progenitors, the\ngalaxies go from blue and star-forming at z >2 to IR-\nbright and dust obscured from 1 < z < 1.8, where the\noverall maximum SFR is measured, and SFR begins to\nsteadily decrease as S´ ersic indices increase at z <1 to\nthe present epoch. P15 also find that M31-like galaxies\nbegin and end each evolutionary stage earlier than MW-\nlike galaxies. All three of these studies listed above rely\nsolely on the observed light profiles to make argumentsMW Analogues 3\nabout morphology and mass assembly. Here we aim to\nextend the study of mass assembly of the MW by us-\ning SED-fitting to create spatially-resolved stellar mass\nprofiles, as well as resolved star-formation rate profiles.\nThis should bring a clearer and more accurate picture of\nthe underlying morphological changes that progenitors\nof Milky Way-like galaxies undergo in the past 10 Gyr.\nIn this paper, we utilize abundance matching of galax-\nies from redshifts 0 .5< z < 2 to select Milky Way ana-\nlogues. We build upon the stellar mass map studies\nin Tan et al. (2022), from which we utilize the same\nmethods to create resolved stellar mass and SFR pro-\nfiles to trace the mass assembly for MW-like progenitors\nover time. Our abundance matching is based on the as-\nsumption that the approximate rank order of galaxies\nremains consistent throughout cosmic time, even as they\ngrow in stellar mass, but also includes merger trees to\naccount for slowly declining number density as galaxy\nhaloes merge together due to gravitational interactions.\nAbundance matching techniques have previously been\nemployed to study both MWAs and massive galaxies\n(Yang et al. 2012; Moster et al. 2013; Marchesini et al.\n2014; Torrey et al. 2017; Kravtsov et al. 2018).\nIn§2, we give further information on the Hubble Fron-\ntier Fields DeepSpace catalogs and which photometric\nfilters were used, as well as the COSMOS/UltraVISTA\nsurvey from which the Schechter functions for comov-\ning number densities is derived from. Our methods for\nconstructing resolved 1-D and 2-D stellar mass and SFR\nprofiles are outlined in detail in §3. In§4, we present the\nresults of our analysis on the mass assembly and change\nin SFR over cosmic time for our sample of MWA progen-\nitors. Our discussions of the implications of these results\nin comparison to previous works that used IR light pro-\nfiles, as well as comparisons to simulations of MW pro-\ngenitors are in §5. For this work we assume a ΛCDM\ncosmological model of the universe with Ω Λ= 0.7 and\nΩM= 0.3, along with a Hubble constant of H0= 70\nkm/s/Mpc. We assume a Chabrier IMF (Chabrier 2003)\nfor our determination of the stellar masses of our sample.\nOur magnitudes are reported in AB magnitudes.\n2.DATA SAMPLE\n2.1. Hubble Frontier Fields\nThe Hubble Frontier Fields (HFF; Lotz et al. 2017)\nare comprised of six lensing clusters at redshifts of\n0.3< z < 0.6 and six flanking fields imaged in the UV,\noptical, and NIR with the Advanced Camera for Surveys\n(ACS) and Wide Field Camera 3 (WFC3) on the Hubble\nSpace Telescope (HST). We use the DeepSpace catalog\nfrom Shipley et al. (2018) to select our sample of MWA\nprogenitors. For this study, we primarily use the par-allel flanking fields of the DeepSpace catalog, but also\ninclude the cluster fields as long as the objects’ redshifts\nare outside the redshift range of the respective cluster’s\nredshift by z±0.1. This is because the Milky Way does\nnot live within a cluster, so the number density of the\nfields is better suited to search for MWAs.\nThe redshift of our MWA sample spans from 0 .3< z <\n2, with only two galaxies at z <0.5 (one at z∼0.31,\nanother at z∼0.42). Based on the WFC3 and ACS\nPSF size, this implies an angular resolution of 0 .25−0.50\nkpc per pixel and 0 .8−1.5. kpc per PSF FWHM. In all\nmodeling we use the images that are PSF-matched and\nconvolved to the WFC3/F160W resolution from Shipley\net al. (2018) to ensure consistent resolution in each filter.\nThe F160W filter has a PSF FWHM of ∼0.177”, which\ntranslates to a pixel size of 2.95 pixels. The PSFs for\neach filter are derived from stacking at least 3 or more\nisolated and unsaturated stars. The total science area\ncovered by HFF DeepSpace is 156.1 arcmin2. Although\nbecause we were not able to make use of Abell1063clu,\nthe total area that is used in this work is only 150.5\narcmin2.\nFor the flanking fields, each of the pointings has 7 fil-\nters (F435W, F606W, F814W, F105W, F125W, F140W,\nand F160W), except for the parallel field of MACS0416,\nwhich was imaged with 9 filters (the same 7 listed before,\nas well as F775W, and F850LP). For a more in-depth\ndiscussion on the DeepSpace catalog, we refer the reader\nto Sections 2 and 3 of Shipley et al. (2018).\n2.2. Stellar Mass functions\nIn order to perform abundance matching to select\nprogenitors of MWAs based on stellar mass, the stel-\nlar mass functions (SMFs) are needed. The stellar\nmass functions used in this paper come from the COS-\nMOS/UltraVISTA Survey (Muzzin et al. 2013b). The\nSMFs were measured from a sample of over 95,000 Ks-\nselected galaxies up to z= 4. They used 30 photo-\nmetric bands to cover a wavelength range of 0 .15µm−\n24µm. In addition to photometry from UltraVISTA\n(McCracken et al. 2012), the catalog also includes\nadditional data from GALEX, Subaru/SuprimeCAM,\nCFHT, and Spitzer.\n3.METHODS\n3.1. Abundance matching\nOur goal is to connect progenitors of MWA over cos-\nmic time to measure their stellar mass growth. To do\nthis, we adopt the semi-empirical approach using abun-\ndance matching presented in Behroozi et al. (2013a).\nThe Behroozi code takes into account merger rates of\ngalaxies at higher redshifts such that the comoving num-4 Tan et al.\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nredshift8.08.59.09.510.010.511.011.512.012.5Mass (Log M)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nredshift3.50\n3.25\n3.00\n2.75\n2.50\n2.25\n2.00\n1.75\n1.50\nNumber density (Mpc3)\nFigure 1. Left panel: The evolution of the median halo mass (light blue) and the median stellar mass (dark blue) of MWAs\nfrom z= 3 to z= 0. Right panel: The median integrated comoving number density for MWAs. The values at each redshift step\nis generated from the code by Behroozi et al. (2013b). The shaded regions represent 1 σfrom the median.\n7 8 9 10 11 12\nStellar Mass log(M/Msun)3.5\n3.0\n2.5\n2.0\n1.5\n1.0\nIntegrated Number Denisty [Mpc3]\n0.2<z<0.5\n0.5<z<1.0\n1.0<z<1.5\n1.5<z<2.0\n2.0<z<2.5\nMWA progenitors\nFigure 2. Integrated number density of galaxies from stel-\nlar mass functions (SMFs) from Muzzin et al. (2013a), and\neach black point is the median stellar mass of MWAs at the\nredshift range represented by each SMF. The number densi-\nties in black are calculated using the Behroozi et al. (2013b)\ncode and are also shown in the right panel of Figure 1.\nber density shifts to a slightly higher number density as\nredshift increases. These merger trees come from sim-\nulations of dark matter halos that the galaxies inhabit.\nTheir code matches the rank order of galaxies accord-\ning to stellar mass or luminosity with dark matter halos\nin decreasing order of peak historical halo mass. For\nour stellar mass range, the comoving number density\nincreases ≲0.1 dex for every 0.5 increase in redshift,\nin comparison to assuming a constant number density\nthroughout cosmic time.\nWe begin by determining the initial comoving num-\nber density for galaxies assuming the stellar mass ofthe Milky Way to be 1010.77M⊙. This stellar mass\nvalue is generally within the accepted range of 10 .5≲\nlog 10( M⋆/M⊙)≲11, albeit on the higher end (de Rossi\net al. 2009; Bland-Hawthorn & Gerhard 2016; Sotillo-\nRamos et al. 2022). This mass agrees with values for\nthe stellar mass and initial comoving number density\ngiven in Marchesini et al. (2009) for a MW-mass galaxy,\nwhich VD13 used in order to determine their MWA pro-\ngenitors. This mass also corresponds with a comoving\nnumber density of log( ρ) =−2.95 log(Mpc−3), which is\nsimilar to the density used in P15 for their MW progen-\nitor sample.\nFigure 1 plots the range of number densities and halo\nmasses at higher redshifts for MWAs, with the solid lines\nindicating the median values, and the shaded blue re-\ngions indicating the 1- σdeviation. This figure demon-\nstrates how the code from Behroozi et al. (2013a) deter-\nmines the possible previous number densities at higher\nredshifts, given an initial number density at z= 0. It is\nalso able to determine the possible previous halo masses\nfrom the initial number density, and vice versa, using a\nrelation between the cumulative number density to halo\nmass. The range of possible number densities and halo\nmasses increases with redshift because the code from\nBehroozi et al. (2013a) takes into account merger trees\nof halos. This means that a halo that exists at a given\ncurrent number density at z= 0 could have formed ei-\nther from one particular halo in the past growing in\nisolation, or from mergers of smaller halos.\nIn order to find the stellar masses of the MWA pro-\ngenitors from the given previous number densities, they\nmust be matched up to stellar mass functions from large\nscale observations. Figure 2 shows the method of obtain-\ning the median stellar mass of the MWA progenitors\nusing the stellar mass functions (SMFs) from the Ul-\ntraVISTA survey from Muzzin et al. (2013a). For eachMW Analogues 5\nof the SMFs that correspond to a range of redshifts, a\nstellar mass is matched to each of the median number\ndensities predicted for that redshift by the code from\nBehroozi et al. (2013a). The minimum and maximum\nstellar mass for each redshift bin is taken to be the me-\ndian redshift ±0.2 dex, which gives a more conservative\nestimate for the stellar mass range than using the cor-\nresponding stellar masses obtained from the SMFs with\nthe number densities ±1σ. Performing the same exercise\nwith mass functions from Davidzon et al. (2017) results\nin the progenitor masses being different by 0 .01−0.14\ndex, making the difference in dex <0.2 dex at all red-\nshifts.\nOur total sample of galaxies is 110 within the redshift\nrange of 0 .3< z < 2, after the removal of 16 galaxies\nfrom the cluster fields for having magnification values\nexceeding 2.5 as computed from the Bradaˇ c et al. (2009)\nlensing models, and a further removal of 18 galaxies from\nboth cluster and flanking fields combined, from inspec-\ntion of faults in their stellar mass maps (i.e. segmen-\ntation map falsely separated one object into two, too\nlow signal to noise for enough spatial bins). There are\n95 galaxies in the parallel fields and 15 galaxies in the\ncluster fields.\n3.2. Stellar mass and SFR map construction\nIn Figures 3 and 4, we display the colour images and\nthe stellar mass maps of all 110 galaxies in our sample.\nFigure 3 contains all 95 MWA progenitors from the par-\nallel fields and Figure 4 contains the 15 progenitors from\nthe cluster fields. The colour images are constructed\nfrom the filters that are the closest match to rest-frame\nU-band , rest-frame V-band, and the F160Wfilter.\nFigure 5 shows the colour image, F160W cutout, stel-\nlar mass map, and SFR map of several galaxies in\nour sample of various compactness. As stated before,\nthe images are all PSF-matched and convolved to the\nWFC3/F160W resolution to ensure consistent resolu-\ntion in each filter. The angular resolution is 0 .25−0.50\nkpc per pixel and 0 .8−1.5. kpc per PSF FWHM, for the\nredshift range 0 .3< z < 2. From the HFF DeepSpace\ncatalogs, the smallest effective radius in our sample is\n3.27 pixels, and the largest effective radius is 18.1 pix-\nels. The size of the FWHM of the PSF in pixels is 2.95,\nso using 2 times the effective radius as the size of the\ngalaxy in pixels, the most compact galaxy is 2.2 times\nthe PSF FWHM, and the most extended galaxy is 12.3\ntimes the PSF FWHM. The effective radii are based\non the detection image, which is a combination of the\nreddest filters from F814W to F160W as measured by\nSExtractor .The first step to generating resolved stellar mass pro-\nfiles as well as resolved SFR profiles is to spatially bin\nthe images containing the photometry of the MWA pro-\ngenitors. We apply the Voronoi tessellation algorithm\nfrom Cappellari & Copin (2003), and SED-fitting with\nFAST++ (Kriek et al. 2009; Schreiber et al. 2015), in\norder to generate the stellar mass maps of the MWA\nprogenitors from their photometry. The reason why we\nchose the Voronoi tessellation method for pixel-binning\nis due to the geometry of the tessellated bins being able\nto best homogenize the SNR throughout the entire im-\nage of the galaxy, and in a way which preserves maxi-\nmum spatial resolution in the geometry of the shape of\nthe bins. For each galaxy, a lower limit for SNR of 10 is\nplaced on each bin. But most galaxies at z <1.5 have\nenough signal to noise that an SNR lower limit of 30 is\nused instead.\nIn the SED-fitting of the photometry of the selected\ngalaxies, we use BC03 stellar populations (Bruzual &\nCharlot 2003), a Chabrier IMF Chabrier (2003), a\nCalzetti dust law (Calzetti et al. 2000), and a delayed\ntau model SFH. The SED-fitting code FAST is used due\nto the accuracy and robustness of the stellar mass out-\nputs. We note that using a Calzetti dust law and a\ndelayed tau model SFH may not be accurate to the star-\nformation history of the Milky Way; delayed tau models\nallow only for a rising SFH at the very beginning, but\nhave a declining SFH for the vast majority of cosmic\ntime. In the redshift range probed by this work how-\never, a declining SFH is a reasonable assumption, since\nthe global trend of cosmic SFR has been declining since\nz∼2.\nOnce a stellar mass is obtained for each spatial bin,\nwe construct the stellar mass map for that object. The\nmaps have the degraded resolution of the Voronoi tessel-\nlation, however we regain the original resolution by an\nadditional scaling applied to each pixel. We distribute\nthe amount of stellar mass within one bin to each of its\nconstituent pixels according to the pixel’s contribution\nto the F160W flux. For more information on the details\nof resolved stellar mass map construction, we refer the\nreader to Tan et al. (2022).\nIn addition we have also created 2-D star-formation\nrate maps. The SFR profiles were notderived from\nFAST SED-fitting, because while assuming a delayed\ntau model SFH results in reasonable stellar mass deriva-\ntions up to z∼3 (see appendix of Muzzin et al. 2013a),\nitdoes not result in the most accurate SFRs for galax-\nies at higher redshifts. Therefore, we instead derive\nSFR from the integrated UV luminosity calculated us-\ning EAZY (Brammer et al. 2008), by placing a UV-filter\n(1400-1600 ˚A) on the rest-frame fitted spectra and ob-6 Tan et al.\nFigure 3. Top: Colour images of MW progenitors in the HFF DeepSpace parallel flanking fields at different redshifts. Bottom:\nResolved stellar mass maps of the same MW progenitors. The min-max scale for the mass maps has the maximum set to\n2×108M⊙. The redshift of the sources is displayed on the bottom left of each thumbnail.MW Analogues 7\nFigure 4. Left: Colour images of MW progenitors in the HFF DeepSpace cluster fields at different redshifts. Every galaxy is\nbeyond the cluster’s redshift, and also has magnification effects of <2.5.Right: Resolved stellar mass maps of the same MW\nprogenitors. The min-max scale for the mass maps has the maximum set to 2 ×108M⊙. The redshift of the sources is displayed\non the bottom left of each thumbnail.\nz = 0.3108\n f160w\n stellar mass\n SFR EAZYabell370par -- ID 4418, redshift = 0.3108\nz = 0.6048\n f160w\n stellar mass\n SFR EAZYmacs0416par -- ID 6511, redshift = 0.6048\nz = 0.8617\n f160w\n stellar mass\n SFR EAZYmacs1149par -- ID 2820, redshift = 0.8617\nz = 1.2202\n f160w\n stellar mass\n SFR EAZYabell370par -- ID 1030, redshift = 1.2202\nz = 1.5007\n f160w\n stellar mass\n SFR EAZYmacs0717par -- ID 2892, redshift = 1.5007\nz = 1.7266\n f160w\n stellar mass\n SFR EAZYabell2744par -- ID 970, redshift = 1.7266\nz = 1.909\n f160w\n stellar mass\n SFR EAZYmacs0717par -- ID 3248, redshift = 1.909\nz = 1.9214\n f160w\n stellar mass\n SFR EAZYabell2744par -- ID 3663, redshift = 1.9214\nFigure 5. Comparison of colour image, F160W , stellar mass, and SFR profiles. The horizontal line in the bottom right\nrepresent 3 ×FWHM of the F160W PSF. The the third object from the top, in the left column(MACS0717PAR ID 2892) is the\nmost compact object in terms of its effective radius respective to the PSF’s FWHM at that redshift.8 Tan et al.\n0.5\n 0.0 0.5 1.0 1.5 2.0\nV - J0.000.250.500.751.001.251.501.752.00U - Vz < 0.5\n0.5 < z < 1\n1.0 < z < 1.5\n1.5 < z < 2\nFigure 6. UVJ Diagram with boundaries from Shipley et al.\n(2018). There are 11 quiescent galaxies at redshift 0 .5< z <\n1, and 4 quiescent galaxies at redshift 1 < z < 1.5.\ntaining the total rest-frame UV luminosity. To lessen the\neffect of parameters on the resulting fits, we still used a\ndelayed tau SFH, a Chabrier IMF, and a Calzetti dust\nlaw. However, the template libraries were from FSPS\n(Conroy & Gunn 2010). We then apply a dust correction\nusing the Calzetti dust law curve with the Avderived\nfrom the HFF DeepSpace catalog (i.e. the Avoutputted\nby FAST while fitting for the integrated stellar mass of\neach object.)\nOnce the stellar mass maps have been constructed, we\nobtain 1D stellar mass density profiles for each galaxy\nby placing down elliptical apertures/annuli centered on\nthe galaxy of increasing radial distance (each are 0.2 kpc\nin width) up to 20 kpc distance in the major axis, and\nthen taking their stellar mass surface density. The same\nprocess is also used on the resolved SFR maps to create\n1D SFR profiles, also up to 20 kpc. The major and mi-\nnor axes, as well as the position angles used to determine\nthe placement of the annuli are from the HFF DeepSpace\nphotometric catalog, extracted from their detection fil-\nters which utilized a combination of F814W,F105W,\nF125W,F140W, and F160Wbands.\nThis makes it a fair comparison with the light profiles\nin the works of P13 and VD13 because the former also\nintegrated the profiles out to 20 kpc and the latter out to\n25 kpc. We also mask out nearby galaxies from the 2D\nprofiles before creating the 1D profile so their mass or\nSFR does not interfere with the surface density calcula-\ntions. We use a UVJ diagram (see Figure 6) to examine\nwhich galaxies are star-forming or quiescent in our sam-\nple. The UVJ boundaries are from Shipley et al. (2018).Out of the 110 total galaxies from all twelve cluster and\nparallel fields, only 15 galaxies lie within the quiescent\nregion of the UVJ diagram. A small number of z <1\ngalaxies are in the upper-right region of the UVJ dia-\ngram, indicating the presence of dust, although this can\nbe an effect of viewing angle. The majority of galax-\nies, especially at z >1 (green and blue points), are not\ndusty and highly star-forming.\nFor each redshift, we stacked and renormalized the 1-\nD mass profiles based on their total stellar masses from\nthe profile, to create the averaged 1-D mass profile at\nthat redshift range. The same was done for the star-\nformation rate maps in order to obtain the normalized\n1-D SFR profiles at each redshift bin. The normalized\nmass profiles are plotted in Figure 7, and the normalized\nSFR profiles are plotted in Figure 8.\nWe expect the MWA progenitors to be highly star-\nforming as redshift increases, and that is reflected in the\nsample. The Milky Way is currently in the green valley,\neither because of its dust content or because it is in the\nprocess of transitioning from star-forming to quiescent.\nWe note that the quiescent fraction is 41% at redshift\n0.5< z < 1, but only 13% at redshift 1 < z < 1.5, and\nall MW progenitors are star-forming at 1 .5< z < 2.\nThe redshift bin z < 0.5 only contains two galaxies,\nand both are star-forming according to the UVJ dia-\ngram in Figure 6. Therefore, the galaxies being major-\nity star-forming disk galaxies at higher redshifts matches\nthe canonical picture of the evolutionary history of the\nMilky Way. (Bland-Hawthorn & Gerhard 2016). Given\nthat the Milky Way is currently in the green valley with\na modest SFR, that ∼60% of its progenitors (by stellar\nmass) at 0.5 < z < 1.0 are star-forming and ∼40% are\nquiescent is also reasonable.\nFor our sample, we used GALFIT (Peng et al. 2002)\nto fit the stellar mass maps in order to obtain the S´ ersic\nindices and half-mass radii in the same procedure as\noutlined in Tan et al. (2022). We fit for S´ ersic index,\neffective radius, axis ratio, and position angle, though\nthe parameters of interest are only the first two. Ini-\ntial guesses for the parameters were SExtractor values\n(Bertin & Arnouts 1996) for effective radius, semimi-\nnor/semimajor axis, and position angle for each object\nfrom the HFF DeepSpace catalog, and n= 2 for the\nS´ ersic index, but with constraints of n= 0 to n= 8. We\nwill compare their redshift evolution with VD13, P13,\nand P15 in §4.5\n4.RESULTS\n4.1. 1-D Stellar Mass Profiles\nIn Figure 7, the stacked and normalized stellar mass\nsurface density profiles for each redshift bin are plot-MW Analogues 9\n1051061071081091010Stellar Mass Surface Density0.5 < z < 1.0 \n(27 galaxies)1.0 < z < 1.5 \n(31 galaxies)1.5 < z < 2.0 \n(50 galaxies)\n0 2 4 6 8 10\nDistance from center (kpc)105106107108109Stellar Mass Surface Densityz < 0.5 \n(2 galaxies)z < 0.5\n0.5 < z < 1\n1 < z < 1.5\n1.5 < z < 2\nsolar \nneighbourhood\n2 4 6 8 10\nDistance from center (kpc)Mass Profiles comparison \nvan Dokkum et al. 2013z=0\nz=0.6\nz=1\nz=1.5\nz=2\nz=2.5\n2 4 6 8 10\nDistance from center (kpc)Mass Profiles comparison \nPatel et al. 2013z=0.05\n0.25 <z < 0.5\n0.5 < z < 0.75\n0.75 < z < 1\n1 < z < 1.2\n1.2 < z < 1.4\nFigure 7. Top Row and Bottom Left panel: Every surface mass density profile created from stacking and normalizing each\n1-D stellar mass profile at each redshift bin. Light grey lines represent the individual stellar mass profiles in each redshift bin.\nShaded regions in the same colour of the profile represent the 1- σdeviation from the stacked profile. The bottom left panel also\ncontains normalized 1-D surface mass profile for each redshift. The grey lines are the two galaxies which are in the redshift bin\n0< z < 0.5, with the shaded grey region representing the 1- σdeviation. The black point is the local solar neighbourhood stellar\nmass density from Bovy et al. (2012). The profiles are integrated up to r= 20kpc. The bottom center and right panels: These\npanels compare our derived profiles at each redshift bin with previous results. Our mass profiles agree well with the profiles\nfrom P13 more than VD13, despite using abundance matching like the latter, instead of the star-formation main sequence like\nthe former. The vertical grey regions in every panel indicates the part of the profile affected by the PSF.\n104\n103\n102\n101\n100SFR from EAZY LUV[M kpc2 yr1]\nz < 0.5\n0.5 < z < 1\n1 < z < 1.5\n1.5 < z < 2\nMilky Way\n2 4 6 8 10\nDistance from center (kpc)\n0.5<z<1\nQuescient Fraction: 0.41\n2 4 6 8 10\nDistance from center (kpc)\n1<z<1.5\nQuescient Fraction: 0.13\n2 4 6 8 10\nDistance from center (kpc)\n1.5<z<2\nQuescient Fraction: 0.0\n0 2 4 6 8 10\nDistance from center (kpc)101\n100101102SFR ratio to MW\nz<0.5\nQuescient Fraction: 0.0\nFigure 8. Top Row: Similar plots to 7, but for stacked and normalized SFR profiles. The same galaxies are in each redshift\nbin. The leftmost plot also has every stacked-normalized SFR profile from each redshift plotted on top for a comparison. In\naddition, the white points in the leftmost panel indicate the Milky Way’s current day SFR density versus R, from Kennicutt &\nEvans (2012). The vertical grey regions in every panel indicates the part of the profile affected by the PSF. Bottom Row: Ratio\nbetween the current Milky Way Σ SFRfrom Kennicutt & Evans (2012), and the Σ SFRat the same Rfor each past redshift bin.\nThe dotted line indicates where the ratio is the same. Although magnitude of change in Σ SFR varies between redshifts, the\noverall shape remains consistent.10 Tan et al.\n0.0 0.5 1.0 1.5 2.0\nredshift10910101011Stellar Mass [M]\nR < 2kpc (inner)\nR > 2kpc (outer)\nTotal Stellar Mass\nMilky Way\n0.0 0.5 1.0 1.5 2.0\nredshift107108109Stellar Mass Density [M]\n0.0 0.5 1.0 1.5 2.0\nredshift101\n100101SFR (<10 kpc) [M/yr]\n0.0 0.5 1.0 1.5 2.0\nredshift104\n103\n102\n101\n100SFR Density [SFR]\nTotal SFR\nR < 2kpc (inner)\nR > 2kpc (outer)\nMilky Way SFR\nFigure 9. Top row: Evolution of the total stellar masses (left) and stellar mass densities (Σ ⋆, right) of MWA progenitors out\ntoz∼2. Σ⋆for each region is calculated by taking the stellar mass and dividing it by the surface area of its respective regions.\nBlack lines represent the galaxy-integrated properties out to 20 kpc, whereas red and blue lines show the inner ( R <2 kpc) and\nouter regions ( R >2 kpc), respectively. The Milky Way Σ ⋆were taken from the measurements of the bulge stellar mass, total\nstellar mass, and B/T ratio from Bland-Hawthorn & Gerhard (2016), but divided by the surface area to give a surface mass\ndensity. This is assuming the bulge is contained within a galactocentric radius of 2 kpc, and that the disk integrates out to 20\nkpc, for consistency with MWA progenitor mass profiles. Bottom row: Similar to plots above, but showing the evolution of total\nSFRs (left) and SFR densities (Σ SFR, right). The Milky Way ΣSFR is again derived from the total SFR in Bland-Hawthorn &\nGerhard (2016) but divided by surface area.MW Analogues 11\nted as a function of galactocentric radius R. Our main\nresult is that these stacked and normalized Σ ⋆profiles\nhave approximately the same overall shape up to 10 kpc.\nIn the bottom left panel where the Σ ⋆profiles for each\nepoch are plotted together, it is clear that the growth\nin Σ ⋆between different epochs do not depend on Rto a\ngreat extent. With the stellar mass density of the solar\nneighbourhood (from Bovy et al. 2012) overlaid on top,\nit is evident that by redshift 0.5, most of the stellar mass\nhas already been built up for this sample of MWA pro-\ngenitors. Even before z= 0, these galaxies have been\nable to reach the stellar mass density of the Milky Way.\nAlthough we note with only two galaxies in the redshift\nbin of 0 < z < 0.5, this result could benefit from a larger\nsample.\nAt each redshift step from z∼2 to z∼0, the in-\nner 2 kpc region appears to grow more at earlier times.\nThe stacked Σ ⋆profile does not change at R≲3kpc be-\ntween z <0.5 (the black curve) and 0 .5< z < 1 (the red\ncurve). The profiles at around 8 kpc, which is the dis-\ntance the solar neighbourhood is from the MW’s galac-\ntic centre, increase in mass density more at earlier times\n(z >1), and less so at later times ( z <1). It increased\nfrom a mass density of log( M⋆/M⊙)kpc−2= 6.42 to\nlog(M⋆/M⊙)kpc−2= 6.81, which is a difference of 0.35\ndex, but from z∼1 toz∼0.5, it only increased in mass\ndensity to 6 .94 log( M⋆/M⊙)kpc−2, a difference of 0.13\ndex. To characterize the mass build-up with more ac-\ncuracy, §4.3 charts the change in total stellar mass and\nstellar mass density Σ ⋆in more detail.\nFor comparison, we have plotted our 1-D stellar mass\nprofiles against the mass profiles of previous works,\nwhich is displayed in the bottom center and bottom\nright panels of Figure 7 against VD13 and P13 respec-\ntively. Our stellar mass profiles are on a similar order\nof magnitude to P13’s results, whereas VD13’s stellar\nmass profiles are at least 1 dex greater than ours across\nall radii. This is slightly counter-intuitive, because both\nVD13 and P15 use a starting mass for the Milky Way\nof 5×1010M⊙or 1010.7M⊙, and P13’s starting mass is\n1010.5M⊙, which is around 3 ×1010M⊙. Our starting\nmass of 1010.77M⊙is actually closer to the former, but\nwe find similar MWA progenitor masses to P13. Our\nmass profile at 0 .5< z < 1 follows closely P13’s density\nprofiles at 0 .7< z < 0.1 from R∼4 kpc to R∼10\nkpc. At R≲4 kpc, the lowest redshift mass profile\nshows that the stellar mass density is slightly ( ∼0.1 to\n0.3 dex) above P13’s at a similar redshift range. Our\nmass profile at 1 < z < 1.5 falls between P13’s profiles\nat similar mass ranges, up until R≳8 kpc, when our\nprofile flattens out above P13’s 1 < z < 1.2 profile. Ingeneral, the shape of our mass profiles are less steep in\nthe centre and level off at a higher mass density.\n4.2. 1-D star formation rate profiles\nThe other important factor to stellar mass growth is\nthe star-formation rate. We make 1-D SFR profiles, rep-\nresenting the SFR versus galactocentric radius in kpc,\nand these are shown in Figure 8 on the top row. In the\ntop-left panel of Figure 8, the star formation rate density\nprofiles show more star formation within the R <2 kpc\nregion, but there are proportionally more stars there.\nOverall the distribution of the SFRs and stellar mass\nshows that the increase in stellar mass due to star for-\nmation in each redshift range would be approximately\nproportional to the stellar mass that is already present.\nFigure 8 shows that the SFRs are overall more than\n1.3 dex higher from 1 < z < 2 as compared to z <1.\nThe sharp drop in overall SFR between these redshifts\nmight be expected as the SFRs of all galaxies start to\ndecline strongly at z <1. Interestingly for the MWA\nprogenitors, the drop in SFR is uniform over the entire\nprofile, implying that the mass growth from star forma-\ntion is not preferentially in the inner or outer regions of\nMWAs. This in conjunction with the stellar mass pro-\nfiles indicate that mass assembly happens in lockstep\nfrom z∼2 onwards, and also slows down in lockstep as\nwell.\nIn addition to the stacked and normalized SFR pro-\nfiles for each redshift bin, in the bottom row of Figure\n8 we also plot the ratio between the Milky Way’s mea-\nsured SFR density against the SFR density at the same\nRfor each of the stack and normalized SFR density\nprofiles. Although the magnitude of the ratio of Σ SFR\nchanges, the overall shape of this ratio of past Σ SFRto\ncurrent day MW Σ SFR does not change qualitatively.\nThis has implications for bulge-growth, as this implies\na decrease in Σ SFR over time occurred at similar rates\nacross the entire galaxy, with no preference for inner or\nouter regions. Therefore it is another example of lock-\nstep growth in both the bulge and the disk regions of\nMWA progenitors.\n4.3. Growth in inner versus outer regions\n4.3.1. Mass assembly in the inner and the outer regions\nTo further investigate the stellar mass assembly in\neach redshift range, we divide the growth into inner\nvs. outer regions. The inner region is defined as within\n2 kpc of the center of each galaxy’s profile, and outer\nregion is defined as beyond 2 kpc. The profiles are in-\ntegrated out to 20 kpc, to remain consistent with the\nlimit used in P13.12 Tan et al.\n107108stellar [M/kpc2]\nTotal mass density evolution comparisonsThis Work\nMilky Way\nvan Dokkum+13\nPatel+13\nPapovich+15\nredshift1071081091010stellar [M/kpc2]\nPatel et al. (2013)Total \nInner \n(<2kpc)\nOuter \n(>2kpc)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nredshift1071081091010stellar [M/kpc2]\nvan Dokkum et al. (2013)Total \nInner \nOuter \nFigure 10. Top Panel: Comparison of the total Σ⋆as a\nfunction of redshift between our densities and P13, VD13,\nand P15. Middle panel: The inner and outer stellar mass\ndensities of our results and P13. Bottom panel: Same re-\ngional stellar mass density comparisons as the middle panel\nbut with VD13’s densities. MW mass density points are\nagain derived from measurements listed in Bland-Hawthorn\n& Gerhard (2016) as in Figure 9.\nIn Figure 9, we plot the stellar mass surface density\nΣ⋆versus redshift in the top panels, and SFR density\nΣSFR versus redshift for the bottom panels. These are\nobtained by integrating over the 1-D density profiles by\nRup to 20 kpc. In addition, we also plot the inner\n(R < 2kpc) and outer (2kpc < R < 20kpc) Σ ⋆and\nΣSFRagainst redshift, which are the red and blue lines\nrespectively. This is not only to demonstrate overall\nmass assembly and change in SFR over time, but also\nwhether the change was more pronounced towards thecentral regions or the outskirts of the galaxies for each\nepoch.\nFigure 9 shows that the stellar masses of the inner re-\ngions ( R <2 kpc) are consistently around 0.5 dex lower\nthan the outer regions at every redshift bin. Compared\nto the total mass and total mass density of the Milky\nWay from local measurements, the Galaxy is not a per-\nfect extrapolation of the trends shown. In the top left\npanel of Figure 9, the outer regions of the Milky Way\n(the blue point at z= 0) have more stellar mass com-\npared to and extrapolation of the trend for the MWA\nprogenitors’ outer regions ( R > 2 kpc) at higher red-\nshifts (the blue line) to z= 0. Also the stellar mass\nof the inner 2 kpc of the Milky Way is similar to the\ntotal stellar mass of the MWA progenitors at z∼0.5.\nThis seems to indicate the inner 2 kpc for the Milky\nWay stopped growing while the outer regions continued\nto grow, however, our result suggests this may not be\nthe case for MWAs in general, if they follow the trends\nseen in the top two panels of Figure 9.\nIt is useful to note that that while our MWA pro-\ngenitors do not indicate a similar type of growth from\nz∼0.5 toz∼0 that would match the current Milky\nWay’s properties, neither do the MWA progenitor sam-\nples of the previous works that they are compared with.\nIf we took the full sample of all local galaxies consid-\nered “Milky Way Analogues” , perhaps the Milky Way’s\nhigher than expected disk growth at low redshifts is\nanomalous among galaxy analogues of similar mass.\n4.3.2. Star-formation at different regions and different\nepochs\nIn Figure 9, the bottom left panel plots overall median\nSFR versus redshift, and the bottom right panel plots\nSFR density versus redshift. Those panels demonstrate\nthe overall trend of star-formation rate decreasing with\ndecreasing redshift. However, the drop in median SFR\nis more drastic from z∼1.5 toz <1 than any other\nepoch. This holds for both the inner and outer regions\nof the MWA progenitor galaxies, which can be seen in\nthe SFR profiles in the bottom row of Figure 9. Since\nz∼2 is what is known as “cosmic noon” when the peak\nof universal star-formation occurred, the decline may\nbe expected. However it is interesting to note that the\ndecrease in median SFR for the MWAs (for all regions)\nis less drastic in the redshift bin 1 .5< z < 2 than the\nredshift bin 1 < z < 1.5.\nThe inner region ( R <2 kpc) and outer regions ( R >2\nkpc) of the the MWA progenitors have similar changes\nin SFR when moving from one redshift bin to another.\nThis indicates lockstep star-formation rates for MWA\nprogenitors since z∼2. This means that when the\nSFR changes, it changes at the same rate over the entireMW Analogues 13\ngalaxy’s surface area , whether it is in the R < 2 kpc\nregion or in the outer region at R > 2 kpc. Generally,\nSFR is decreasing with decreasing redshift, but there is\na small increase in the median SFR when moving from\nthe redshift bin 0 .5< z < 1 to 0 .3< z < 0.5, as seen in\nthe bottom left panel of Figure 9. Looking instead to the\nbottom right panel of Figure 9, where the SFR density\nof the MWA progenitors is plotted versus redshift, there\nis more of an increase in the SFR density of the R < 2\nkpc inner region than the outer region. This might be\nevidence of mild bulge-growth, but because it is minor,\nit does not break the general trend that SFR changes in\nlockstep across all radii.\nMosleh et al. (2017) also finds evidence that star-\nforming galaxies have their central and outer regions\ngrow concurrently. Their range of stellar masses were\nfrom 1010.2M⊙to 1011.2M⊙, so the Milky Way would be\ncomparable to the star-forming galaxies in their highest\nmass bin.\n4.4. Selection of MWA Progenitors and resulting\ndifference in Mass Growth\nIn Figure 10, we plot inner, outer, and total stellar\nmass density vs. redshift, and compare them against\nsimilar plots but from VD13, P13, and P15. The mass\ndensities for the previous studies were obtained by tak-\ning their stellar mass as a function of redshift (inner,\nouter, and total) and dividing it by the area consistent\nwith their defined limits of integration in their respec-\ntive works (i.e. for VD13, the outer limit is 25 kpc, and\nfor P13 and P15 the outer limit was defined to be 20\nkpc.) Although P15 did not construct mass profiles as\nthe other two papers did, their work still included total\nmass assembly for MWA progenitors.\nWe see a clear difference in total stellar mass, as well as\nstellar mass density between P13 versus VD13 and P15.\nAdditionally, P13 selected MWA progenitors using the\nstar-forming main sequence, and VD13 selecting MWA\nprogenitors from a constant comoving density. Despite\nselecting our progenitors via an evolving comoving num-\nber density, our sample is less massive and more star-\nforming compared to VD13 and P15’s analyses, and is\na much closer match to P13. However, there are still\ndifferences such as the rate of stellar mass growth of the\ninner and outer regions. We note that all three previ-\nous studies used IR light profiles as a proxy for stellar\nmass, but we are measuring stellar mass directly from\nthe resolved 2D mass profiles. Thus, this may account\nfor most of the differences in mass assembly seen.\nAll the stellar mass density plots in Figure 10 display\nsigns of lockstep growth at redshifts 1 < z < 2, but when\nthe growth slows down at lower redshifts, it appears thatthe outer regions stop growing earlier than the inner\nregions.\n4.5. Morphological parameters as a function of time\nIn Figure 12, we plot the median S´ ersic index and me-\ndian half-mass radius of each redshift bin. The S´ ersic\nindices and effective radii were obtained by using GAL-\nFIT to fit a one-parameter S´ ersic profile on the objects’\ntwo-dimensional stellar mass maps. The error bars in-\ndicate standard error of the mean. The S´ ersic indices of\nthe progenitor sample show a slowly increasing trend as\nredshift decreases at z <1, but it remains constant be-\ntween 1 < z < 2. The redshift bin of z <0.5 is the only\nepoch when the median S´ ersic index is n > 2. Before\nthat epoch, from 0 .5< z < 2, the S´ ersic index varies\nfrom 1.2 to 1.5, which suggests a disk-like morphology.\nThe left panel of Figure 12 shows that the evolution\nof the S´ ersic indices between this work and P15 are very\nclose at the epoch 1 < z < 2, but diverges drastically\natz <1. P15’s sample in fact, almost reaches a S´ ersic\nindex of n= 4, which describes a de Vaucoulers profile,\nin other words, an entirely bulge-dominated elliptical\ngalaxy.\nIn the right panel of Figure 12 where the effective radii\nare plotted, the median half-mass radius of our sam-\nple of MWA progenitors starts at only 2 .23±0.19 kpc\nat the earliest epoch, but jumps to 3 .56±0.21 kpc at\n1< z < 1.5. The median half-mass radius is also smaller\noverall at 0 .5< z < 1 than at the redshift bin immedi-\nately before and after that epoch. We do note that the\nquiescent fraction at that epoch is higher than at other\nepochs, which could contribute to a smaller median half-\nmass radius, since quiescent galaxies are more compact\nthan star-forming ones. In addition, the median half-\nmass radii at each epoch tends to match both P13 and\nVD13’s results but only at z < 1. At 1 .5< z < 2,\nour results agree very well with P15, but surprisingly at\n1< z < 1.5, it does not match the effective radii of any\nof the previous works. However, due to the magnitude\nof the error bars, the overall trend of the change in half-\nmass radii is consistent with the changes in half-light\nradii of previous MWA studies.\nIn general, we have found that the stellar mass pro-\nfiles and growth histories are reasonably consistent with\nprevious findings which use IR light as a proxy for stel-\nlar mass. Given that galaxies can have color gradients\n(e.g., Figure 3), it is somewhat surprising how consistent\nthese results are. This shows that IR light tracks stellar\nmass quite well for MWA progenitors, even though it is\nnot fully representative of the underlying stellar mass.\nPerhaps one notable contrast is that our result in Fig-\nure 12 shows that the half-mass radius may grow dras-14 Tan et al.\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nredshift104\n103\n102\n101\nSFR Density [M/yr/kpc 2]\nThis work [SFR]\nPatel+13\nPapovich+15\nvan Dokkum+13\nMilky Way SFR\n0.5 1.0 1.5 2.0 2.5 3.0\nredshift\ninner SFR\nouter SFR\nvan Dokkum SFR\nvan Dokkum (< 2 kpc)\nvan Dokkum (> 2 kpc)\nFigure 11. Left panel: Total SFR density as a function of redshift for our results against VD13, P13, and P15. At higher\nredshift, from 1 .5< z < 2, our SFR densities agree well with both P15 and P13, but the SFR density drops faster for our\nsample. Right panel: Total, inner, and outer SFR density as a function of redshift from our results and VD13. Their work also\nused UV luminosity to derive SFRs. This shows the difference that evolving comoving number density has on the SFR of the\ngalaxies selected as progenitors.\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nredshift012345Sersic index\nz < 0.5(mass)\n0.5 < z < 1(mass)\n1 < z < 1.5(mass)\n1.5 < z < 2(mass)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nredshift012345Half mass radius (kpc)\nPatel+13 (light)\nvan Dokkum+13 (light)\nPapovich+15 (light)\nFigure 12. Left panel: S´ ersic index vs redshift, as fit by GALFIT onto the stellar mass maps. Each point represents the median\nS´ ersic index for that redshift bin. Our sample has good agreement with P13’s sample in S´ ersic index at similar redshifts, again\nshowing that selecting MWA by star-formation rate and abundance matching results in similar samples. Right panel: Half-mass\nradius vs redshift. Each point represents the median half-mass radius index for that redshift bin. The average half-mass radius\nat each of our redshift bins are compared against the average half-light radii for P13, VD13, and P15’s galaxies.\ntically at early times, but plateaus in growth at more\nrecent times, perhaps having to do with galaxies turn-\ning quiescent. This is somewhat different than the half-\nlight radii, which have a more steady growth over cosmic\ntime. The fact that the half-mass radius growth is slow,\ncombined with the result that the S´ ersic index increas-\ning as redshift decreases, could imply the progenitors are\ngrowing their bulges. This is further discussed in §5.1.\nThis result could also indicate lack of evolution in the\nhalf-mass radii of MWA progenitors as much as it indi-\ncates very small growth. A simulation paper on MWA\nprogenitors in TNG50 by Hasheminia et al. (2022) seems\nto find similar results (see §5.3 for more details).\n5.DISCUSSION5.1. Mass Maps vs Luminosity Maps\nOur analysis of MWA progenitors differs from previ-\nous work in the usage of resolved stellar mass and SFR\nmaps instead of luminosity maps. IR luminosity is a\nrepresentation of the stellar mass distribution but is not\nitself the mass distribution. But despite that, our re-\nsults agree well with previous light profile studies. We\nagree particularly well with the results in P13 (as seen\nin Figures 10, 11, and 12). And our results come to\nthe same conclusion, which is that MWA progenitors\nassemble their mass in inner and outer regions in lock-\nstep since z∼2, with no preference for inside-out or\noutside-in growth.MW Analogues 15\nEven though P13 relied on the star-forming main se-\nquence to select MWA progenitors, the closeness of our\nmeasurements point to a link between galaxy merger\nrates and the star-formation history as well as mass as-\nsembly of galaxies at a certain stellar mass in a comoving\nvolume of space. This extends even to the closeness of\nthe S´ ersic indices between our two studies. It may seem\ncounter-intuitive that stellar mass profiles are quite sim-\nilar to light profiles, since the Milky Way has a colour\ngradient with older stars near the center and younger\nstars further out in the disk. But looking back at the\ncolour images of Figure 3, there are progenitors that\nexhibit this colour gradient at high zas well as low z.\nOur S´ ersic indices are 0 .2−0.5 greater than the S´ ersic\nindices of P13. Previous studies that examined the dif-\nference in S´ ersic indices between IR light profiles and\nstellar mass profiles include Suess et al. (2019) and\nTan et al. (2022), which also found nmass to be over-\nall greater than nIRby a similar margin. This means\nthe MWA progenitor populations are very similar be-\ntween this work and P13’s work, despite using a different\nstarting mass at z= 0. However, our half-mass radii are\nslightly larger. This is a notable departure from previous\nstudies that typically found half-mass radii are smaller\nthan half-light radii (again, both Suess et al. 2019 and\nTan et al. 2022).\nWe note that differences for the two Repoints in the\nright panel of Figure 12 at redshifts 0 .3< z < 1 are\nminimal between this work and P13; the differences are\n<0.5 kpc for both points. But the Refor our sample\nat redshift 1 < z < 1.5 is much higher, around 1.5 kpc\ngreater than the Remeasured by P13 at the same red-\nshift range. Since the Frontier Fields cover a smaller re-\ngion of the sky than the surveys used in VD13, P13, and\nP15, this could be attributed to sampling bias. In addi-\ntion, P13 used a stellar mass of 1010.5M⊙for MWAs at\nz= 0 compared to our initial stellar mass of 1010.77M⊙\nfor MWAs. Since our half-mass radii are the closest to\ntheir results, the difference in size could be due to the\nphysical sizes of the chosen sample itself. Finally, our\nRemeasurement for the redshifts 1 .5< z < 2 lies on\ntop of the line indicating the Refor the sample from\nP15, which means our Remeasurements are still on the\nwhole consistent with previous results.\nBoth the MWA progenitor sample from VD13 and P15\ncontained more quiescent galaxies than our sample, and\nthis can explain why their S´ ersic indices are larger at\nlater redshifts and their half-light radii are smaller. Both\nof those quantities are associated with quiescent galax-\nies. The quiescent fractions range in P15 is 74% to 41%\nfrom 0 < z < 1.1 and 29% to 12% from 1 .1< z < 2.\nThis may explain why our measurement at 1 .5< z < 2matches the results from P15 at the same redshift range\nin Figure 12. It is because at that redshift there are\nmostly star-forming galaxies. P15 also used the abun-\ndance matching code from Behroozi et al. (2013a) to\nselect their MWA progenitors, so agreement at high red-\nshift between our results and theirs may point to light\nand mass profiles being more similar at higher redshift,\nat least for star-forming galaxies.\n5.2. Mergers and mass assembly\nThe abundance matching code prescribes merger trees\nfor any given z= 0 comoving number density or z= 0\nhalo mass, and then estimates the median cumulative\nnumber density at past epochs for any population of co-\nevolving galaxies (Behroozi et al. 2013a). Due to select-\ning MWA progenitors via abundance matching, it raises\nthe question of how mergers would affect the evolution of\nMWAs. In the case of detecting onging major mergers,\nthere is only one in our sample, at z∼1.9. Most galax-\nies in our sample appear to have star-forming clumps\nthat do not show up in the stellar mass maps (see Fig-\nure 5). Despite the lack of ongoing major mergers in the\nsample, the assumption is still that statistically speak-\ning, mergers still play a role in the mass assembly of\nMWAs on account of choosing progenitors according to\nan evolving co-moving number density.\nIf we assume the gap in total stellar mass density be-\ntween our results and the results from VD13 is domi-\nnated by increase in mergers, then according to 10, 0.1\nto 0.3 dex of the increase in total stellar mass density\n(Σ⋆attributed to star-formation should instead be at-\ntributed to mergers since z∼2, as seen in the topmost\npanel of 10. For the Σ ⋆within R <2kpc, this can be as\nhigh as 0.8 dex of the increase in Σ ⋆, and for the region\nbetween 2 kpc < R < 20 kpc, it is as high as 0.9dex, as\nseen in the bottom panel of 10.\nGiven that we use true stellar mass, and an evolving\nnumber density evolution, our result that the MWA pro-\ngenitor S´ ersic indices are growing at a slower rate than\nthat implied by VD13 or P15 should be a more accurate\npicture about bulge growth. While all results suggests\nbulge strength is increasing in MWAs, ours suggest a\nslower evolution. Assuming that the difference is in part\ndriven by incorporating the merger trees into the abun-\ndance matching, this implies that mergers may affect\nbulge growth. However, since the S´ ersic index growth\nis modest, it is clear that MW-mass galaxies that are\nstill star-forming will not have substantial morphologi-\ncal transformations since z∼2.\nThis is reflected in both the stellar mass densities,\nas well as the star-formation rate densities. Figure 9\nshows very little difference in the stellar mass growth of16 Tan et al.\nthe inner ( R < 2 kpc) and outer ( R > 2 kpc) regions\ncompared to the total mass growth. There is a similar\ntrend in the change in SFR versus redshift – when the\ntotal SFR drops, both inner and outer SFRs also drop\nby similar rates. Since the change in SFR is not de-\npendent on galacto-centric radius this means the newly\nformed stellar mass is added to both inner and outer\nregions, which demonstrates lockstep growth. It may\nseem that this contradicts the increase in S´ ersic index,\nbut the change in S´ ersic index is rather shallow (from\nn= 1.23±0.39 to n= 2.31±0.48), and it may be that\nthe stellar component of these galaxies compactify from\ndynamic friction from the gas.\nIn addition, not all bulge growth is the result of merg-\ners. Secular evolution and clump migration are other\nmethods of bulge growth. Garrison-Kimmel et al. (2018)\nfind that in the FIRE-2 simulations, halo spin (halo\nangular momentum), is the best predictor of morphol-\nogy, which mergers and secular processes (such as bars\nor counter-rotating disks) both greatly affect. How-\never, clump migration in simulations has been shown\nto contribute very little to bulge growth despite seeing\nlarge numbers of bright clumps at high redshift (Man-\ndelker et al. 2017; Garrison-Kimmel et al. 2018), with\nMandelker et al. (2017) finding that clumps only con-\ntribute 0 .1∼3% of a galaxy’s total stellar mass. Star-\nforming clumps occur in the majority of z >1 galaxies\n(Elmegreen et al. 2007), but despite their brightness,\nobservations do not show that clumps are any more\nmassive than the surrounding galaxy in resolved spa-\ntial stellar mass profiles (Wuyts et al. 2012; Sok et al.\n2022). Since the launch of JWST, galaxies with irregu-\nlar features have been found to make up about 40-50%\nof the universe at 3 ≤z≤9 (Jacobs et al. 2023; Kartal-\ntepe et al. 2023) and it is clear that the progenitors of all\nbut the most massive central galaxies in the current uni-\nverse must have had irregular and clumpy morphologies.\nDue to the sheer amount of clumpy galaxies, their effect\non mass assembly should still be investigated. Perhaps\nclumps play a more prominent role in galaxy evolution\nbeyond z= 2.\nThere is a reasonable survival rate (35%) of disks for\nprogenitors of MW and M31 mass galaxies in the TNG50\nsimulation as seen in Sotillo-Ramos et al. (2022), show-\ning that they do not necessarily destroy the stellar disk\nbut only add dynamical heat to all galaxy components\n(i.e. in thickening the stellar disk, another process which\nmay increase the S´ ersic index). This may also explain\nthe lockstep growth in inner and outer regions. Depend-\ning on the properties of the merger itself, it may add stel-\nlar mass to all radii, especially if gas is distributed to the\nstellar disk and induces starburts. Sotillo-Ramos et al.(2022) also find that recent major mergers of their sim-\nulated MW/M31 analogues do not appreciably deplete\nthe gas fraction of those galaxies, allowing for further\nongoing star-formation after the merger event. There is\nreasonable evidence from simulations that mergers play\na role in bulge growth but are not dominant over secu-\nlar processes, nor do they always completely transform\na galaxy’s morphology.\n5.3. Comparisons with MWA progenitors in\nsimulations\nThe formation of MW and M31 progenitors have been\nheavily studied through simulations before, and it is use-\nful to compare the results from observations against the\nresults from simulations. This is especially critical to an-\nswering the question of whether or not the Milky Way\nis an outlier in being a massive, but still disky galaxy\nin the local universe. Abundance matching has been\napplied to the IllustrisTNG simulation in Torrey et al.\n(2017) to search for Milky Way and M31 progenitors in\nthe simulation.\nHasheminia et al. (2022) found in their sample of\nMWA progenitors that the half-mass radius shows lit-\ntle evolution from z∼2 to the present epoch, while the\nS´ ersic index evolves slowly. They do find that for their\ngalaxies, the S´ ersic index reaches upwards of n∼3 to\nalmost n∼4, suggesting that for galaxies of Milky Way\nmass, they should be mostly bulge-dominated in the lo-\ncal universe. But we do not see this reflected in our sam-\nple from observations, where the maximum S´ ersic index\nat the redshift bin 0 < z < 0.5 is only n∼2.3. How-\never, we do reproduce one of their main results which\nis that the half-mass radius is roughly constant for the\npast 10 Gyr, despite growth in stellar mass, and change\nin S´ ersic index. This means there is something funda-\nmental about the way that MWA progenitors assemble\nwhich is reflected in both simulation and observation.\nSotillo-Ramos et al. (2022) examined a population of\nMW and M31 progenitors in the TNG50 simulation,\nfocusing specifically on the result of mergers on the\ndiskiness of the galaxy, as well as the details of the\nchange in morphology due to mergers. They come to\ntwo main conclusions regarding mergers: one related to\nbulge growth, and another related to stellar halo mass\ndistribution. For bulge growth, they find that for their\nsimulated MWA progenitor sample, there is no statisti-\ncally significant difference in the bulge fraction between\ngalaxies with one or more merger events or no merger\nevents. Secondly, they find that for galaxies with a re-\ncent major merger in the last 5 Gyr, their bulge to halo\nstars ratio (with halo stars also being a proxy for disk\nstars) most closely match those of the MW and M31 atMW Analogues 17\nredshift z= 0, but also their bulge fractions are lower\nthan the rest of the sample, indicating a diskier distri-\nbution.\nSince we obtained a lower S´ ersic index of 2 .3 at our\nlowest redshift bin, this implies that accounting for\nmergers should still result in MWA progenitors retain-\ning a mostly disky morphology in the current universe.\nThis is in line with results from various simulations of\nMWAs, which provide reasonable explanations for mass\nassembly, which agrees with our results. Our observa-\ntional results may indicate the Milky Way is not as much\nof an outlier among Milky Way Analogues as was previ-\nously thought.\n6.CONCLUSION\nWe construct resolved stellar mass and SFR maps of\nMWA progenitors up to redshift z∼2 using resolved\nmulti-wavelength photometry from the HFFs and ex-\namine the stellar mass assembly of Milky Way-mass\ngalaxies in a spatially-resolved manner. We used abun-\ndance matching techniques that select progenitors with\nan evolving comoving number density, which is based\non simulated merger histories of MW mass galaxies at\nz= 0. Our main conclusions are as follows:\n•The stellar mass for the Milky Way Analogues\nincreases at the same rate for both inner ( <2\nkpc) and outer regions ( >2 kpc) over our redshift\nrange, i.e. progenitors of MWAs display lockstep\ngrowth in inner and outer regions up to redshift\nz∼2. Our study, which focuses on resolved stellar\nmass profiles of galaxies agrees with previous ob-\nservational results studying the Milky Way’s mass\nassembly through light profile distributions.\n•Generally speaking, morphological parameters de-\nrived from stellar mass profiles follow the trends\nseen in previous MWA progenitor studies. S´ ersic\nindices derived from stellar mass profiles aresmaller than S´ ersic indices derived from light pro-\nfiles from previous works on MWAs, with the\nexception of P13. However, our half-mass radii\nare slightly larger than the half-light radii from\nprevious works. The difference between light-\nderived and mass-derived S´ ersic parameters dis-\nappears around redshift z∼1.5.\n•We also find evidence that accounting for merger\nrates at different evolutionary epochs by using\nabundance matching in concordance with select-\ning by stellar mass and comoving number density\nresults in a higher overall star-formation rate, and\ndiskier morphology in progenitors of Milky Way\nAnalogues. This is shown by our stellar mass den-\nsity and SFR density closely matching the results\nof P13, which used the star-forming main sequence\nto select MWA progenitors at higher redshift.\n•However, according to the SFR profiles, the SFR\nand the SFR density does not decrease uniformly\nin the inner and outer regions, and in fact the in-\nner region decreases more rapidly than the outer\nregions.\n•We find on average slightly more mass assembly\nin the outer 2 kpc region the MWA progenitors,\nespecially in the epoch 0 .3< z < 0.5, which also\nmatches up with slightly elevated median SFR in\nthe outer 2 kpc region at the same epoch. But it\nis not a large enough amount that would make our\nresults inconsistent with lock-step growth found in\nprevious MWA mass assembly studies.\n7.ACKNOWLEDGEMENTS\nThis work is based on data and catalog products from\nHFF-DeepSpace, funded by the National Science Foun-\ndation and Space Telescope Science Institute (operated\nby the Association of Universities for Research in As-\ntronomy, Inc., under NASA contract NAS5-26555).\nREFERENCES\nAbadi, M. G., Navarro, J. F., Steinmetz, M., & Eke, V. R.\n2003a, ApJ, 591, 499, doi: 10.1086/375512\n—. 2003b, ApJ, 597, 21, doi: 10.1086/378316\nBehroozi, P. S., Marchesini, D., Wechsler, R. H., et al.\n2013a, ApJL, 777, L10,\ndoi: 10.1088/2041-8205/777/1/L10\nBehroozi, P. S., Wechsler, R. 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Kamenshchik,1,‡\nFrancisco G. Pedro,1,§Alessandro Tronconi,1,¶and Giovanni Venturi1,∗∗\n1Dipartimento di Fisica e Astronomia, Universit` a di Bologn a, via Irnerio 46, 40126 Bologna, Italy\nI.N.F.N., Sezione di Bologna, viale B. Pichat 6/2, 40127 Bol ogna, Italy\n2Department of Physics and EHU Quantum Center,\nUniversity of the Basque Country UPV/EHU, Barrio Sarriena s /n, 48940 Leioa, Spain\nWe make a conceptual overview of a particular approach to the initial-value problem in canonical\ngauge theories. We stress how the first-class phase-space co nstraints may be relaxed if we interpret\nthem as fixing the values of new degrees of freedom. This idea g oes back to Fock and Stueckelberg,\nleading to restrictions of the gauge symmetry of a theory, an d it corresponds, in certain cases,\nto promoting constants of Nature to physical fields. Recentl y, different versions of this formulation\nhave gained considerable attention in the literature, with several independentiterations, particularly\nin classical and quantum descriptions of gravity, cosmolog y, and electromagnetism. In particular,\nin the case of canonical quantum gravity, the Fock–Stueckel berg approach is relevant to the so-\ncalled problem of time. Our overview recalls the work of Fock and Stueckelberg and its physical\ninterpretation with the aim of conceptually unifying the di fferent iterations of the idea that appear\nin the literature and of motivating further research.\nI. INTRODUCTION\nCanonical gauge theories are the Hamiltonian descrip-\ntionoftheoriesthatpossesslocalsymmetries(alsoknown\nas “gauge symmetries” or “gauge redundancies”). The\nphase-space generators of these local symmetries are as-\nsociated with a set of initial-value constraints that are of\nfirst class, i.e., they Poisson-commuteon the phase-space\nhypersurface defined by all the constraints [1–9]. Yang–\nMills theories and electromagnetism are the well-known\nexamples but also generalrelativity is such a theory, as it\nis invariantunder spacetimediffeomorphisms, which con-\nstitute local symmetries (albeit not of the same kind as\nYang–Mills theories). In particular, diffeomorphism in-\nvarianceimplies the Hamiltonian vanishesas a constraint\n[1, 6, 10, 11],\nH= 0, (1)\nand this means that not all initial values of fields and\ncanonical momenta are allowed. If the theory is quan-\ntized ` a la Dirac [5], the constraint ( 1) leads to a time-\nindependent Schr¨ odinger equation,\nˆH|Ψ/an}b∇acket∇i}ht= 0, (2)\nalso known as the Wheeler-DeWitt equation [11], which\nimplies quantum states are stationary and the usual no-\ntion of time evolution is trivial. This is the so-called\n∗roberto.casadio@bo.infn.it\n†leonardo.chataignier@ehu.eus\n‡kamenshchik@bo.infn.it\n§francisco.soares@unibo.it\n¶tronconi@bo.infn.it\n∗∗giovanni.venturi@bo.infn.itproblem of time in quantum gravity, and it has inspired\nmany proposed solutions [12].\nOne obvious solution is to construct a Hamiltonian\nthat is not constrained. In other words, the Hamilto-\nnian constraint should simply not be satisfied (it should\nbe relaxed). Instead, one considers field configurations\nor quantum states that do not satisfy Eqs. ( 1) or (2), so\nthat the Hamiltonian does not vanish, and thus we have\nthe standard unconstrained evolution (given by the time-\ndependent Schr¨ odinger equation). This would describe\nthe “Schr¨ odinger evolution of the Universe” in the sim-\nple setting of a “minisuperspace” toy model (i.e., a me-\nchanical model obtained by considering a homogeneous\nmodel universe). This trivial solution to the problem\nof time in quantum cosmology was recently discussed in\nthe article [13], and it has also been the subject of many\nprevious works (see, e.g., [14–27]). To the best of our\nknowledge, the idea has its roots in the works of Fock\nin 1937 [28] and Stueckelberg in 1941 and 1942 [29–31],\nalthough these articles are seldom cited.\nFock and, more explicitly, Stueckelberg proposed a\nrelaxation of the relativistic particle’s mass-shell con-\nstraint, which is the particle’s version of Eq. ( 1)\n(the mass-shell constraint follows from diffeomorphism\ninvariance on the worldline, also referred to as time\nreparametrization invariance [32]). The lack of a mass-\nshell constraint simply implies that, instead of fixing\nthe particle’s (rest) mass to be a certain constant (and\nthereby restricting the particle’s motion to a timelike\nworldline), its mass should rather be determined by the\ninitial values of the particle’s four-momentum, so that\ndifferent trajectories could yield different masses. More-\nover, the trajectories need no longer be timelike, a phe-\nnomenonthatStueckelbergused todescribepaircreation\nand annhiliation by identifying antiparticles with parti-\ncles going backwards in Minkowski time.2\nThis idea of Fock and Stueckelberg may also be seen\nas a passage from the Jacobi action principle (see, e.g.,\n[32–34]), which yields trajectories in configuration space\nin reparametrization-invariant way (without a preferred\ntime variable), to the usual Hamilton action principle of\nmechanics, in which there is a preferred time parame-\nter. In the context of general canonical gauge theories,\nthe Fock–Stueckelberg approach is obtained by fixing a\ngauge condition in the action functional before varying\nthe action, i.e., by imposing the principle of stationary\naction for a set of gauge-fixed variables. This automat-\nically leads to the “loss” of a set of equations of motion\n(the constraints) which are responsible for the original\ngauge invariance. In other words, the constraints are re-\nlaxed. Notice that this has nothing to do with quantum\ntheory, but rather with defining two different classical\ntheories, as fixing the gauge at the level of the action or\nthe equations of motion generally yields different results.\nStueckelberg’s work has inspired an approach to rela-\ntivistic particle dynamics developed by Horwitz, Piron\nand others [35–37], which is also predicated on a re-\nlaxation of the Hamiltonian constraint. As shown by\nStueckelberg, the corresponding quantum theory fea-\ntures a Schr¨ odinger equation instead of the usual Klein–\nGordon constraint, which is only obtained for energy\n(mass) eigenstates. Fock and Stueckelberg already noted\nthat this approach corresponds to a modification of the\ntraditional special-relativistic dynamics of the particle.\nIndeed, the formalism can be understood to be equiva-\nlent to promoting the mass to a new degree of freedom\n(rather than a constant), which is conjugate to the par-\nticle’s proper time, also elevated to a new degree of free-\ndom. The unconstrained dynamics can then be recast\nas a new type of constrained (diffeomorphism-invariant)\ndynamics, where the constraint fixes not the particle’s\nfour-momentum, but the initial values of the mass field\n[32].\nThe ideas of the recent article [13] were subsequently\ndeveloped in the sequel papers [38, 39], where the au-\nthors discuss the relaxation of constraints in field theory,\nspecifically in general relativity and electromagnetism.\nThey argued such a relaxation leads to the development\nof a well-defined quantization of canonical gauge theo-\nries, and they subsequently analyzed the meaning of this\nrelaxation in the classical limit. Hence, their title “the\nclassical equations of motion of quantized gauge theo-\nries.”\nIn [38], it was noted that, going beyond mechanical\n(“minisuperspace”) models, the relaxation of the con-\nstraints of general relativity leads to solutions to the full\nset of Einstein equations in the presence of some addi-\ntionaltypesofmatter(darkmatterordarkenergy). This\nis simply another case of including new degrees of free-\ndom ` a la Fock–Stueckelberg, and, indeed, similar ideas\nhave been repeatedly reported in the literature, where\none considered the promotion of constants of Nature to\nnew fields that would be constrained by the value of the\nHamiltonian in general relativity (e.g., [25–27, 40–42]),thereby freeing one from the problem of actually solving\nthe Hamiltonian constraint for the original metric and\nmatter variables. A particular case is the elevation of\nthe cosmological constant to a new field, which is not\nfixeda prioribut determined by the initial values of the\nmatter and geometric fields, in analogy to the relativistic\nparticle’s rest mass. This is effectively what one obtains\nin different approaches to unimodular gravity, which was\nput forward already by Einstein as early as in 1919 [43],\nand later developed in many different ways [14–17].\nUnimodular gravity theories have attracted a consid-\nerable amount of attention in recent years. As is well-\nknown, an approach to unimodular gravity can be ob-\ntained by the requirement that the theory be invari-\nant with respect to a restricted class of diffeomorphisms,\ntheso-calledvolume-preserving,area-preservingordiver-\ngenceless diffeomorphisms [44–52]; i.e., by the diffeomor-\nphisms generated by the divergenceless transverse vector\nfields. In such a formalism, the Hamiltonian constraint\n(1) is not satisfied at all points, but its value is related to\nan integration constant that plays the role of the cosmo-\nlogical constant. As in the case of the Fock–Stueckelberg\nformalism for the relativistic particle, the approaches to\nunimodular gravity can be seen as particular modifica-\ntions of general relativity, whereby one adds new degrees\noffreedomand modifies the localsymmetry group, which\nleads to a relaxation of the original constraints.\nEvidently, this type of modification is not restricted\nto gravitational theories. In principle, one can apply\nthe Fock–Stueckelbergreasoning to any first-class initial-\nvalue constraint, i.e., to any canonical gauge theory.\nWe will variously refer to such treatments of canoni-\ncal gauge theories as the Fock–Stueckelberg theory, ap-\nproach, mechanism, or method. One simple example can\nbe done for electromagnetism, as discussed by the au-\nthors of [13] and [38] in the article [39], where they re-\nlaxedthestandardvacuumGauss-lawconstraint,div /vectorE=\n0, replacing it by\ndiv/vectorE=ρ , (3)\nwhere/vectorEistheelectricfieldand ρencodesthenowrelaxed\n(possiblynonzero)valueoftheconstraint. Itcanbeinter-\npretedasanelectricchargedensity. With this relaxation,\nthey observedthe “emergenceofchargewithout matter.”\nThis is, again, an iteration of the Fock–Stueckelberg ap-\nproach and, as a matter of fact, the theory of electro-\ndynamics without the Gauss-law constraint was already\nconsidered by Dirac in 1951 [53]. Dirac showed in [53]\nthat, by discarding the Gauss-law constraint, one ac-\nquires an additional degree of freedom, which can be\nidentified with the electron. The ideas of Dirac’s new\nelectrodynamicswerefurther developed by manyauthors\n(see, e.g., [54–58]).\nAlthough the formalism proposed by the authors of\n[13, 38, 39] may certainly lead to a well-defined quan-\ntum theory of interest, it is important to emphasize\nthat the relaxation of constraints is by no means a\nmandatory step in the quantization of gauge theories.3\nThis is evidenced by the success of quantum electrody-\nnamics, not to mention the standard model of particle\nphysics, and by the powerful formalism of quantization ` a\nla Batalin–Vilkovisky (BV), Batalin–Fradkin–Vilkovisky\n(BFV), and Becchi–Rouet–Stora–Tyutin (BRST) (see,\ne.g., [9]). In this way, although it may be easier to quan-\ntize a theorywithout (some ofthe) constraints, the relax-\nationofconstraintsis not aprerequisitefora well-defined\nquantum theory (at least given our current knowledge\nof quantum foundations). Rather, as was already men-\ntioned, the relaxation of constraints defines a different\nclassical theory. Both theories, constrained and uncon-\nstrained,maybequantizedandeachmayleadtodifferent\nphenomenologydue to the presenceorabsenceofnew de-\ngrees of freedom (e.g., presence or absence of a mass field\nin the traditional and Fock–Stueckelberg treatments of\nthe relativistic particle).\nFurthermore, it is true that there is a distinction be-\ntweenYang–Millssymmetriesanddiffeomorphisminvari-\nance but this need not imply that the standard treat-\nment of the constraints in general relativity will lead to\na pathological quantum theory. The Yang–Mills sym-\nmetries are internal, as they do not concern a change of\nspacetime point (the argument of fields), whereas diffeo-\nmorphisms are external as they “movepoints.” However,\nby taking an active view of diffeomorphisms, in which\nthe original and transformed fields may differ in func-\ntional form but not in argument, this distinction seems\nless crucial. In fact, both types of local symmetries can\nbe treated in formally the same way when analyzing the\ninvariance of the action functional, provided one adds\nsuitable boundary terms [59]. It thus seems premature\ntoa priori dismiss the standard quantization of general\nrelativity’s constraints. The task of finding such a non-\nperturbative(andpossiblyultravioletcomplete)theoryis\nsurely daunting but not necessarily inconsistent. In this\nway, the Fock–Stueckelberg approach is not inevitable in\nquantum gravity and cosmology.\nEvidently, however, the necessity of going beyond the\nstandard models of particle physics and cosmology may\nmotivate us to consider the elegant method of Fock and\nStueckelberg as a means of accounting for new physics.\nAlthough this was indeed the spirit of the articles [13, 38,\n39], and of the many other preceding articles cited above,\nan overview with an emphasis on the physical meaning\nof the Fock–Stueckelberg theory is long overdue, due to\nthe frequent appearance of versions of this approach in\nthe literature. In the present article, we propose such an\noverview.\nThe article is divided into five sections. In Sec. II,\nwe quickly review the fundamentals of canonical gauge\ntheories, in order to subsequently present the Fock–\nStueckelberg approach to general gauge theories in Sec.\nIII. This section is then subdivided into several partic-\nular cases and examples of interest. In Sec. IIIB, we\ndiscuss how promoting constants of Nature to spacetime\nfields can be seen as an equivalent approach to the Fock–\nStueckelberg theory under certain conditions. The con-straint relaxation for “partially Abelianized” systems is\nthen presented in Sec. IIIC, and subsequently we turn\nour attention to mechanical examples in Secs. IIIDand\nIIIE, where we examine the case of time reparametriza-\ntion invariance and relativistic particles. The Fock–\nStueckelberg approach to electromagnetism is presented\nin Sec.IIIF,whereasthe applicationtogeneralrelativity\nis featured in Sec. IIIG. Section IVdiscusses the impli-\ncations of the Fock–Stueckelberg method to the problem\nof time in quantum gravity, and we finish with our con-\nclusions in Sec. V.\nII. CANONICAL GAUGE THEORIES IN A\nNUTSHELL\nWe begin with a brief review of canonical gauge the-\nories before we discuss a general presentation of what\nwe call the Fock–Stueckelberg approach. Their original\nwork on the relativistic particle will be discussed as a\nparticular example of the general framework.\nA. Constraint algebra and gauge transformations\nLet us consider a set of fields zµ(τ,x) (µ= 1,...,Z)\nind+1 spacetime dimensions. We omit the spatial de-\npendence by adopting DeWitt’s compact notation [60]\nin which the indices also include a spatial dependence,\nzµ(τ,x)≡zµ(τ)≡zµ.1Repeated indices are taken to\nimply a tacit summation over their discrete values to-\ngether with a spatial integration. In the case d= 0 (me-\nchanics), there is of course no spatial dependence and\nno spatial integrations. We assume that the fields zµare\nbosonic for simplicity. The field-space spanned by zµwill\nbe referred to as the phase space.\nA canonical gauge theory can be defined by the action\nfunctional2\nS[z,λ] =/integraldisplayτ1\nτ0dτ{Aµ(z)˙zµ−H0(τ;z]−λaφa(τ;z)},\n(4)\nwhere a dot indicates a derivative with respect to time\nτ,Aµis the symplectic potential, and λaare Lagrange\n1We use the sign ≡to denote equivalent notations. It is also used\nwhen two functions are identically equal (i.e., equal at eve ry\npoint). We will use the sign := to denote the definition of a\nquantity.\n2We use square brackets [ ·] to denote that a quantity is a func-\ntional of its arguments, and round brackets ( ·) to denote an ordi-\nnary function of the arguments. A combination of brackets ( ·;·]\ndenotes a quantity that is a function of some of its arguments and\na functional of the others. For example, if d >0, we may have\nH0(τ;z] =/integraltext\nddxH0(τ;z(τ,x)) with some smooth function H,\nwhereas a function f(z(τ;x)) =/integraltext\nddyf(z(τ;y))δ(y,x). When\nno confusion is possible, we may omit the arguments.4\nmultipliers ( a= 1,...,N≤Z). The field equations of\nzµare3\nΩµν˙zν=∂H\n∂zµ, (5)\nwhere\nΩµν:=∂Aν\n∂zµ−∂Aµ\n∂zν,\nH:=H0+λaφa(6)\nare the symplectic form and the total Hamiltonian, re-\nspectively. We assume that Ω µνis invertible, so that\nZmust be even (otherwise Ω µνwould be singular) and\nΩµνΩνρ=δµ\nρ.4Then, Eq. ( 5) can be rewritten as\n˙zµ= Ωµν∂H\n∂zν. (7)\nThis motivates the definition of the Poisson bracket of\ntwo function(al)s fandgas\n{f,g}:=∂f\n∂zµΩµν∂g\n∂zν. (8)\nIn this way, the field equations associated with zµcan be\nwritten compactly as ˙ zµ={z,H}. On the other hand,\nthe field equations associated with the multipliers λaare\nconstraints on the fields:\nφa(τ;z) = 0. (9)\nWecannowmakeaseriesofsimplifyingassumptionsthat\nwill be sufficient for our purposes. We assume that the\nconstraint functions do not depend explicitly on time,\n∂φa\n∂τ= 0, (10)\nandthatthe φafunctionsareallfunctionallyindependent\nandnotidenticallyzero, sothereisnoconstraintthatcan\nbe reduced to another (e.g., φ2=φ2\n1). The constraint\nequations ( 9) are also assumed to define a hypersurface\nΣ on the space of the zfields. Furthermore, in open\nregions covering Σ, the derivatives ∂φa/∂zµ|Σmust not\ndiverge and must define a matrix with a rank equal to\nthe number Nof constraints. This regularity condition\n3The derivatives with respect to zµare taken in the func-\ntional sense if d >0. We recall that, for a local func-\ntionalFof the field z(τ,x), which is written as F(τ,x;z] =/integraltextd3yF(τ,x,y;z(τ,y),Dz(τ,y),...,Dnz(τ,y)) with Fbeing a\nscalar density in yand Dnbeing the n-th spatial covariant\nderivative, the functional derivative δF/δz(τ,x) ofFrelative\nto a variation z(τ,x)/mapsto→z(τ,x) +εη(τ,x) is obtained from\nδF=εd\ndε/vextendsingle/vextendsingle/vextendsingle\nε=0F(τ,x;z+εη] =:ε/integraltext\nd3xδF\nδz(τ,x)η(τ,x). This\nleads to δz(τ,x) =εη(��,x), as it should.\n4Ifd >0, the delta symbol corresponds to a multiplication of\na Kronecker delta for the discrete variables and a Dirac delt a\ndistribution for the continuous spatial coordinates.[9] implies that the constraint functions can be regarded\nas coordinates in a neighborhood of Σ; i.e., we can define\nlocal phase-space coordinates z′(z) = (˜z(z),φ(z)). Then,\niff′(z′):=f(z(z′)) is such that f′(˜z,φ= 0) = 0, we\nobtain5\nf′(˜z,φ) =/integraldisplay1\n0dℓ∂\n∂ℓf′(˜z,ℓφ)\n=φa/integraldisplay1\n0dℓ1\nℓ∂\n∂φaf′(˜z,ℓφ)\n=:va(˜z,φ)φa.(11)\nThus, if a phase-space function fvanishes on Σ, it can\nbe written as f(z) =va(z′(z))φa(z) for some functions\nva, which are locally defined by Eq. ( 11).\nWe use Dirac’s ‘weak equality’ sign ≈to denote iden-\ntities that hold on Σ [4].6In particular, φa≈0. The\nconstraint equations ( 9) must hold at every instant of\ntime, so that ˙φa≈0, otherwise the theory is not con-\nsistent. The requirement that ˙φa≈0 may lead to new\nconstraints Cathat are functionally independent from\nφa. In the standard Rosenfeld–Bergmann–Dirac analysis\n[1, 2, 5–7, 9], the φaare called ‘primary’ constraints, as\nthey follow from the structure of the action ( 4), whereas\nCaare called ‘secondary’ as they follow from the appli-\ncation of the field equations.7For our purposes, it is suf-\nficient to consider that only the primaries φaare present;\ni.e., the conditions ˙φa≈0 are satisfied on Σ without\nleading to new (secondary) constraints. In this case, if\nwe demand that ˙φa≈0 is satisfied for every arbitrary\nvalue of the multipliers λa, thenφaandH0must obey\na first-class algebra (one that Poisson-commutes on Σ).\nDue to Eq. ( 11), this implies\n{φa,φb}=Fc\nabφc≈0,\n{φa,H0}=Vb\naφb≈0,(12)\nwith structure function(al)s Fc\nab[z]≡ Fc\nabandVb\na[z]≡\nVb\na.8With this, φa≈0 can be seen as restrictions on the\nallowed initial values of the zfields, and the Lagrange\nmultipliers λaare not determined by any field equation,\nbe it Eqs. ( 7) or Eqs. ( 9). On the other hand, the\nEqs. (7) depend on λa. In this way, solutions to the field\nequations will depend on arbitrary functions λa. This is\na defining feature of a gauge theory.\n5See Theorem 1.1 and Appendix 1.A in [9].\n6Identities that hold when (all) the field equations are satis fied\ncan be called ‘on-shell identities’. Since every solution t o the\nfield equations must satisfy Eqs. (9), then on-shell identit ies are\nalso weak equalities. In a slight abuse of terminology, we us e\nboth weak equalities and on-shell identities as synonyms.\n7Repeated application of the field equations might lead to ‘te r-\ntiary’ constraints and so forth, but all these constraints c an be\ncollectively called secondary.\n8For brevity, we refer to Fc\nabandVb\naas structure functions, even\nthough they may be functionals in general.5\nWhat is the significance of these arbitrary functions?\nTheycanbe thoughtofasdetermininga“generalizedref-\nerence frame” or gauge [9, 32]. The fact that the theory\ndoes not determine λaimplies that no frame is preferred.\nA change of frame λa→λa+δλais a gauge transforma-\ntion, which also changes the solution zµ→zµ+δzµdue\nto the dependence of Eqs. ( 7) onλa.\nIs any such transformation allowed? No. If no pre-\nferred frame is to be selected, then the field equations\nmust have the same functional form in all frames. This\nmeans that zµ→zµ+δzµ,λa→λa+δλashould be\na symmetry. Then, Noether’s theorem implies that the\nNoether charge Gmust satisfy [61]\n/parenleftbigg\nΩµν˙zν−∂H\n∂zµ/parenrightbigg\nδzµ−φaδλa+dG\ndτ= 0.(13)\nThis is an ‘off-shell identity’; i.e., it holds regardless of\nwhetherthefield equationsaresatisfied. Ifthefieldequa-\ntions hold, we obtain the ‘on-shell identity’ d G/dτ= 0,\nwhich is the conservation of the Noether charge. As Eq.\n(13) holds for arbitrary off-shell paths, we can take zµ\nand ˙zµ(as well as λa,˙λa) to be independent quantities,\nand we can set their coefficients to zero separately. The\ncoefficients of ˙ zµand˙λalead to\nδzµ={z,G},\n∂G\n∂λa= 0,(14)\nand this implies that the Noether charge generates a\ncanonical transformation that is independent of the (ar-\nbitrary) multipliers. The remaining condition from Eq.\n(13) is\nφaδλa=∂G\n∂τ+{G,H}, (15)\nwhich, due to the arbitrariness of δλaand the fact that\nφa/ne}ationslash= 0 off shell, implies that Gmust depend on Nar-\nbitrary functions of time εa. We can thus set G=εaGa\nand use the independence of εaand their time deriva-\ntives to find that Gamust be a linear combination of the\nprimary constraints. Without loss of generality, we can\nsetGa=φa, such that Eq. ( 15) together with Eqs. ( 12)\nleads to the transformation law for the multipliers [9, 61]\nδλa= ˙εa+εbVa\nb+εbλcFa\nbc. (16)\nNotice that, on shell, the quantities εamay also depend\non the fields zµ, so that we can identify ˙ εa=∂τεa+\n{εa,H}. Thus, a gauge transformation is a (possibly\nfield-dependent, on-shell)canonicaltransformationofthe\nzfields generated by the constraints φaand accompanied\nby the transformation ( 16) of the multipliers (change of\ngeneralizedreferenceframe), describedbyarbitraryfunc-\ntionsεa. It is a local (i.e., spacetime-dependent) symme-\ntry transformation, with Noether charge G=εaφa.The on-shell (with possibly field-dependent εa) gauge\ntransformation of a general quantity Ireads:\nδI={I,G}+∂I\n∂λaδλa\n≈εa/parenleftbigg\n{I,φa}+Vb\na∂I\n∂λb+Fb\nacλc∂I\n∂λb/parenrightbigg\n+∂I\n∂λa˙εa.\n(17)\nIfIis gaugeinvariant, it does not changeunder arbitrary\ngauge transformations, i.e., δI ≈0 for any choice of εa.\nUsing the independence of εaand its derivatives implied\nby its arbitrariness, we see that Eq. ( 17) together with\nδI ≈0 implies the gauge invariance conditions:\n∂I\n∂λa≈0,{I,φa} ≈0. (18)\nA gauge-invariantquantity can be seen as one that is the\nsame in every generalized reference frame. In particular,\nthe constraint functions φa(z) are gauge invariant, δφa≈\n0, and thus their functional form does not change under\na gauge transformation.\nB. Gauge fixing\nA choice of generalized reference frame consists in a\nfixation of λa, which corresponds to a choice of gauge or\ngauge fixing. This can be done by imposing some ad hoc\nconstraints (“gauge conditions”) [8, 9],\nχa(τ;z,λ]≈0, (19)\nwhich must hold at all times. Admissible choices must\nbe accessible; i.e., for every arbitrary value of λa, there\nmust exist a gauge transformation with some function εa\nthat takes the arbitrary initial frame to chosen one. If\nthere exists a gauge transformation with some functions\nεa≡εa\n⋆that preservesEq. ( 19), then this transformation\nis called a ‘residual gauge transformation’ (generated by\nG⋆=εa\n⋆φa), and the gauge-fixing conditions are said\nto be ‘partial’ or to ‘partially fix the gauge’. If there\nare no residual gauge transformations, the gauge-fixing\nconditions are said to be ‘complete’ or to ‘fix the gauge\ncompletely’.\nThe imposition of the extra constraints ( 19) occurs at\nlevel of the field equations (i.e., on shell). Indeed, let\nus consider a complete gauge fixing for simplicity, and\nlet the fixated λathat results from Eq. ( 19) beλa≈\nΛa(τ;z]. Then the gauge-fixed field equations are\n˙zµ≈ {zµ,H0}+Λa(τ;z]{zµ,φa},\nφa(z)≈0, χa(τ;z,λ]≈0,(20)\nwhich can then be subsequently solved for zµ.\nThere are different ways in which one can consistently\nimplement the gauge fixing at the level of the action.9A\n9A powerful method isrelated to the BRST cohomology [9], whic h\nwe do not cover here.6\nsimple approachis to introduce new Lagrangemultipliers\nwato modify the action ( 4) in order to introduce the new\ngauge-fixing constraints:10\nSgf:=/integraldisplayτ1\nτ0dτ/bracketleftbig\nAµ˙zµ−H0−λaφa−wb(χb)p/bracketrightbig\n,(21)\nwherep≥1 is a natural number. The new gauge-fixed\naction (21) leads to the field equations:\n˙zµ={zµ,H0}+{zµ,λaφa}+{zµ,wb(χb)p},\n0≈φa+wb∂\n∂λa(χb)p=:φa+ ˜wa,\n0≈(χb)p.(22)\nOurgoalistoshowthattheseequationsformaconsistent\nsystem that can be made equivalent to Eqs. ( 20). The\nlast two sets of equations in ( 22) are the new constraints,\nwhich must be valid at all times if the dynamics derived\nfrom the action ( 21) is consistent. In this context, we\nuse the sign ≈to indicate equalities on the hypersurface\ndefined by the new constraints. First, notice that the\nconstraints ( χb)p≈0 are equivalent to χb≈0. Further-\nmore, in order to recover the original gauge-fixed field\nequations, we must show that the wmultipliers are spu-\nrious in the sense that they disappear from Eqs. ( 22).\nEvidently, this can be achieved if we can fix wb≡0 on\nshell. For wb/ne}ationslash= 0, if the last term in the field equations\nforzµis to vanish when χb≈0, then either p≥2 or,\nifp= 1,χbmust Poisson-commute with all the zfields\nand, in particular, with φa. Furthermore, in order to\nrecover the primary constraints φa≈0, then the equa-\ntions in the second line of Eqs. ( 22) should not depend\nonwb/ne}ationslash= 0, which implies that p≥2 or, ifp= 1,χb\nmust not depend on the multipliers. We thus see that\nthe case p= 1 with wb/ne}ationslash= 0 is inconsistent: it implies\nthatχbmust be gauge invariant [cf. Eqs. ( 18)], and thus\nthey are not well-defined gauge-fixing conditions, as ev-\nery gaugetransformationpreservesthem. In this way, we\nmust have p≥2 ifwb/ne}ationslash= 0 on shell, which leads us back\nto Eqs. ( 20) (withλa≈Λa(τ;z] following from χa≈0,\nas before).\nA useful particular case is that in which the gauge-\nfixing conditions are only functions of the zfields,\nχa(τ;z,λ]≡χa(τ;z(τ)). These are the so-called ‘canon-\nical gauge-fixing conditions’ or simply ‘canonical gauges’\n[8, 9]. Then, taking p= 2 without loss of generality, the\nconservation equation\n0≈∂χa\n∂τ+{χa,H0}+λb{χa,φb}(23)\nleads to a fixation of the λfields if the system of con-\nstraints ( φa,χb) is not of first class (it is said to be of\n10The term ( χb)pis thep-th power of the gauge condition for\neach value of b(no summation over bimplied). In the term\nwb(χb)p, there is one tacit summation over btogether with a\nspatial integration if d >0.second class). In particular, we must have the Faddeev–\nPopov determinant condition [9, 62]\ndet{χa,φb} /ne}ationslash= 0. (24)\nTypically, this condition holds only locally in field space\n(and thus no canonical gauge fixing condition is valid\nglobally). With this, one can form the inverse matrix\n(∆−1)a\nb, such that (∆−1)a\nb{χb,φc}=δa\nc. The solution\nfor theλfields obtained from Eq. ( 23) then reads\nλa≈ −/parenleftbig\n∆−1/parenrightbiga\nb/parenleftbigg∂χb\n∂τ+{χb,H0}/parenrightbigg\n≡Λa(τ;z),(25)\nand the field equations ( 22) again coincide with Eqs.\n(20).\nIII. TO GAUGE OR NOT TO GAUGE? THE\nAPPROACH OF FOCK AND STUECKELBERG\nA. General formalism\nWe have seen that the gauge-fixed action ( 21) with\np= 1 andwb/ne}ationslash= 0 on shell is inconsistent in the sense that\nit does not yield the gauge-fixed equations of motion ( 20)\nfor well-defined gauge-fixing conditions, and therefore it\nis not compatible with the dynamics of the gaugesystem.\nHowever, it is of course legitimate to consider the p= 1\naction\nSFS:=/integraldisplayτ1\nτ0dτ/bracketleftbig\nAµ˙zµ−H0−λaφa−wbχb/bracketrightbig\n(26)\nas defining a new theory, which we call the Fock–\nStueckelberg theory. Thus, the Fock–Stueckelberg ap-\nproach consists in defining a new theory from a “parent\ngauge theory.”\nWhat are the properties of the Fock–Stueckelberg the-\nory? The fields to be varied in the action ( 26) arezµ,λa\nandwb. Thus, we notice that we have new degrees of\nfreedom with respect to the parent theory: the wbfields.\nIn the action given in Eq. ( 21) withp≥2, these de-\ngrees of freedom are spurious as they do not affect the\nfield equations. This is not necessarily so in the Fock–\nStueckelberg theory, where the field equations are given\nby Eqs. ( 22) withp= 1.\nAs we have seen in Sec. IIB, ifχaare well-defined\ngauge-fixing conditions in the parent theory, then they\ncannotbe independent of λaand simultaneouslyPoisson-\ncommute with the zfields, as they cannot be gauge in-\nvariant. Let us now analyze the case in which χaare\n(not) canonical gauges of the parent theory.\n1. Canonical gauges of the parent theory\nIf∂χb/∂λavanishes identically, then wehavecanonical\ngaugeconditions. In this case, ˜ wa= 0 [cf. Eqs. ( 22) with7\np= 1], and the constraints of the Fock–Stueckelberg the-\nory coincide with the first-class constraints φaand the\ngauge conditions χbof the parent theory. These con-\nstraints do not fix the wmultipliers. However, the con-\nservationof φa≈0in timewith ˜ wa= 0andp= 1implies\nthat\n0≈˙φa≈wb{φa,χb}; (27)\ni.e.,wamust be zero on shell [as the canonical gauges χb\ncannot Poisson-commute with all the φconstraints, cf.\nEq. (24)].11In this way, the Fock–Stueckelberg theory\ncoincides with the parent theory in canonical gauges, and\nthus it does not lead to new physical effects.\n2. Noncanonical gauges and the relaxation of the parent\ntheory’s constraints\nIf∂χb/∂λa/ne}ationslash= 0, then the constraint φa+ ˜wa≈0 [cf.\nEqs. (22) withp= 1] can be seen as a fixation of ˜ wa\nrather than a restriction on the possible initial values of\nthezfields. Theoriginalfirst-classconstraintsofthe par-\nent theory are thus relaxed: we can now allow arbitrary\ninitial values of the zfields, as this will only change the\ncorresponding ˜ wafield. Thevaluesof zµforwhich φa≈0\narerecoveredonlyasaparticularcase, inwhich ˜ waiscor-\nrespondingly fixed to be zero on shell.12Therefore, the\ninclusion of the new degrees of freedom waeffectively\nremoves the first-class constraints (and thus the gauge\nsymmetry) of the parent theory. This is the key feature\nof the Fock–Stueckelberg approach. Evidently, the fixa-\ntion of ˜wa(and, correspondingly, of wa) viaφa+ ˜wa≈0\nandofthe λmultipliersvia χa≈0musthold atalltimes.\nThis will be illustrated in the examples below.\nFor example, a case of interest in applications of the\nFock–Stueckelberg approach is given by the simple non-\ncanonical gauge-fixing conditions\nχa(τ;z,λ]:= (λau−Λau(τ;z])δa\nau,\nau= 1,...,N u≤N ,(28)\n11This is true for a complete canonical gauge fixing. For a par-\ntial canonical gauge fixing, we can introduce Nu< Ncanoni-\ncal gauges χau(au= 1,...,N u), which Poisson-commute with\nφa>Nubut must obey det {χau,φbu} /ne}ationslash= 0 (au,bu= 1,...,N u) in\norder to fix Nuof theλmultipliers via the immediate analogue of\nEq. (23). Thus, in the Fock–Stueckelberg theory, we introdu ce\nNumultipliers wau. Then, ˙φa>Nu≈0 is an on-shell identity,\nwhereas ˙φau≈wbu{φau,χbu}again implies that wbu≈0.\n12If∂χb/∂λa/ne}ationslash= 0 are the coefficients of an invertible matrix, then\nwe see from Eqs. (22) with p= 1 that ˜ wa≈0 implies wa≈\n0, in which case the Fock–Stueckelberg theory becomes again\nequivalent to the parenttheory. Onthe other hand, if ∂χb/∂λa/ne}ationslash=\n0 form a noninvertible matrix, then ˜ wa≈0 only implies that\nwbis a possibly nontrivial null eigenvector, for which the Foc k–\nStueckelberg theory need not coincide with its parent.which fix the first Nu≤Ngauge multipliers (thus, it\nmay be a partial gauge fixing). In this case, the Fock–\nStueckelberg constraints read\nχau≈0,\nφau+wau≈0,\nφa>Nu≈0,(29)\nand the field equations for zµ[cf. Eqs. ( 22)] can be\nrewritten as\n˙zµ={zµ,H0+λaφa+wauχau}\n≈ {zµ,H0}+Λau{zµ,φau+wau}\n+{zµ,λa>Nuφa>Nu}−{zµ,wauΛau}\n≈ {zµ,H0+Λauφau+λa>Nuφa>Nu},(30)\nwhere we used the constraints given in Eqs. ( 29). No-\ntice that wauhavedisappearedfrom these equations, and\ntheir only effect is to relax the constraints φau, as dis-\ncussed above. Thus, the field equations ( 30) together\nwith the constraints given in Eqs. ( 29) are equivalent on\nshell to the equations derived from the action functional\n˜SFS:=/integraldisplayτ1\nτ0dτ/bracketleftbig\nAµ˙zµ−H0−Λauφau−λa>Nuφa>Nu/bracketrightbig\n,\n(31)\nwhich is simply obtained from the action ( 4) by impos-\ningλau= Λau(τ;z] before variation. From the point of\nview of the parent theory, this is an erroneous gauge-\nfixing procedure, whereas it is a mechanism to relax the\nconstraints and introduce new degrees of freedom in the\nFock–Stueckelberg theory.13\nLet us now consider a series of illustrating examples\nand particular cases of the Fock–Stueckelberg theory.\nB. Constants of Nature as spacetime fields\nBefore we analyze more concrete examples, it is in-\nteresting to note that the Fock–Stueckelberg approach\nis equivalent, under certain conditions, to another ap-\nproach that has been the topic of many articles: that of\npromoting constants of Nature to spacetime fields (see,\ne.g., [25–27, 40–42] and the related paper [63], as well\nas the discussions in [18–24]). This will be useful in the\nexamples that follow.\n13As in the parent gauge theory, the Rosenfeld–Bergmann–Dira c\nalgorithm [1, 2, 5–7, 9] that ensures the theory’s consisten cy by\nverifying that the constraints hold at all times should also be ap-\nplied to the Fock–Stueckelberg theory. This means that we mu st\nrequire that the time derivatives of the constraints derive d, for\nexample, from the action given in Eq. (31) vanish on shell. Th is\nmay lead to a fixation of multipliers λ, to redudant constraints\nor to new constraints. In case new constraints appear, we mus t\nagain require they hold at all times.8\nConstants of Nature are quantities that enter as fixed\nparametersinthe definition ofatheory(i.e., inthe action\nfunctional), and they are independent of space and time.\nIf a parent gauge theory described by the action in Eq.\n(4) has constants of Nature Pi(i= 1,...,P), one can\ntrivially define a new theory with action functional\nSnew=/integraldisplayτ1\nτ0dτ/braceleftBig\nAµ(z,P)˙zµ+Pi˙Qi\n−H0(τ;z,P]−λaφa(z,P)/bracerightBig\n,(32)\nwhich is identical to Eq. ( 4) except for the fact that the\nconstants of Nature Piare promoted to fields (degrees\nof freedom), which are interpreted as the canonical mo-\nmenta conjugate to cyclic degrees of freedom Qi. This\nthen implies that the Pfields do not change over time,\nand any initial condition will be frozen. If one assumes\nan initial value for the Pfields that is constant across\nspace (if d >0), then due to Pconservation, the P\nsolution to the field equations will be spatially constant\nat all times. On the other hand, an initial condition that\nchangesacrossspace would lead to “constants”of Nature\nwith a static spatial variation. In any case, the phase\nspace is extended to include the canonical pairs ( Q,P),\nand the new theory described by the action ( 32) does not\nexhibit constants of Nature (theory-defining parameters)\nbut rather only fields, some of which are conserved in\ntime. Of course, we also assume that the constraints φa\nobeytheregularityconditionwithrespecttotheenlarged\nset of fields ( z,P,Q).\n1. Parametrization\nIt is more convenient to express the action in Eq. ( 32)\nas a completely constrained system, for which H0≡0.\nIn caseH0is not identically zero, we can replace it with a\nconstraint by a procedure called “parametrization” [11].\nFirst, if d >0, we assume that the unconstrained\nHamiltonian can be written as (cf. Footnote 2)\nH0(τ;z,P] =/integraldisplay\nddxH0(τ,x;z(τ,x),P),(33)\nand that the function H0satisfies the first-class relation\n{φa(z(τ,x),P),H0(τ,x′;z(τ,x′),P)}\n=N/summationdisplay\nb=1/integraldisplay\nddyVb\na(τ,x,x′,y;z,P]φb(z(τ,y),P),(34)\nwhere we have made the summation over band the ac-\ncompanying spatial integration explicit for clarity. With\nVb\na(τ,x,y;z,P]:=/integraldisplay\nddx′Vb\na(τ,x,x′,y;z,P],(35)then Eqs. ( 33) and (34) yield\n{φa(z(τ,x),P),H0(τ;z,P]}\n=N/summationdisplay\nb=1/integraldisplay\nddyVb\na(τ,x,y;z,P]φb(z(τ,y),P),(36)\nwhich coincides with the second of Eqs. ( 12) after the\nsummation and integration are again ommitted. To con-\ncatenate the notation, we formally define H0to coincide\nwithH0in the case d= 0.\nThe procedure of parametrization then consists in in-\ntroducing the new canonical pair ( T,pT) and the global\ngauge constraint\nφ0=pT+H0(T;x;z,P), (37)\ntogether with the global gauge-fixing condition\nχ0=T(τ,x)−τ , (38)\nso as to reinterpret the action in Eq. ( 32) as the restric-\ntion of14\nSpar\nnew=/integraldisplayτ1\nτ0dτ/bracketleftBigg\nAµ˙zµ+Pi˙Qi\n+/integraldisplay\nddx/parenleftBig\npT˙T−λ0φ0/parenrightBig\n−λaφa/bracketrightBigg(39)\nto the hypersurface defined by φ0≈0,χ0≈0. In this\nway, the field Tcorresponds to a global clock that keeps\ntrack of the evolution.\nIf we include the ( T,pT) pair in the definition of the\nPoisson bracket in the usual way, it is straightforward to\nsee that φ0is a first-class constraint due to the original\nfirst-class algebra given by Eqs. ( 12) and (34), and due\nto the fact that the φaconstraints are assumed to have\nno explicit dependence on time (or T). We can then re-\ndefine the multipliers λnew:= (λ0,λold) and constraints\nφnew= (φ0,φold), so as to obtain a new first-class alge-\nbra with no unconstrained Hamiltonian. As a matter of\nnotation, we can also add pTto the list of Pfields, and,\ncorrespondingly Tto the list of Qfields.15In this way,\nthe parametrized action in Eq. ( 39) is equivalent to\nSpar\nnew=/integraldisplayτ1\nτ0dτ/bracketleftBig\nAµ(z)˙zµ+Pi˙Qi−λaφa(z,P)/bracketrightBig\n,\n(40)\nwhere the indices now run over the appropriate values\nfor the enlarged set of variables together with implicit\nspatial integrations if d >0. We also assume that the\n14Evidently, the integral over xis absent if d= 0.\n15This is just for convenience because, in general, Twill not be\ncyclic and pTwillnot be constant in time. If H0orH0depend on\nTexplicitly, we have ˙ pT(τ,x)≈ −/integraltext\nddyλ0(τ,y)δH0|y/δT(τ,x)\nifd >0 or ˙pT(τ)≈ −λ0(τ)∂H0/∂Tifd= 0.9\nenlarged set of constraints obeys the regularity condition\nwith respect to the ( z,P,Q) fields.\nMore generally, the procedure of parametrization of a\ntheoryconsistsinreinterpretingtheactionastheon-shell\naction of a larger gauge system by introducing new de-\ngrees of freedom and a system of first-class constraints\n[cf. Eq. ( 37)] together with gauge-fixing conditions [cf.\nEq. (38)]. For convenience, we thus consider the ac-\ntion in Eq. ( 40) (withNfirst-class constraints φathat\ncan be numbered a= 1,...,Nwithout loss of general-\nity andH0≡0) to be the starting point for a theory in\nwhich constants of Nature have been promoted to phys-\nical fields.16\n2. Solving the constraints in terms of constants of Nature\nThe new action in Eqs. ( 32) or (40) offers the possi-\nbility of further extending the theory to include interac-\ntions of the ( Q,P) fields, thereby introducing notions of\n“varying constants of Nature.”17However, even without\nintroducing interactions, the promotion of the constants\nof Nature to degrees of freedom instead of mere theory-\ndefining parameters has the advantage that one can now\nattempt to solve the constraints φain terms of the P\nfields rather than the original zfields. This may lead to\na more straightforward solution, as it may be simpler to\nsolveφa≈0 in terms of the Pfields.\nLet us assume that the number Pof constants of mo-\ntion that have been promoted to fields is less than or\nequal to the number of constraints, P≤N. Then, with-\nout loss of generality, we can also assume that the first\nPconstraints are solved in terms of the Pdegrees of\nfreedom, which can then be written as\nPa≈ −Ha(z,Pi>a),(a= 1,...,P),(41)\nin a region of phase space, where the constraint hyper-\n16Eliminating the unconstrained Hamiltonian via parametriz ation\nis a matter of formalism. However, it is a physically relevan t\nquestion whether a global clock field Texists independently of\nthe original zdegrees of freedom. If not, then the parametriza-\ntion which brings the action (32) into (39) would seem to have no\nphysical relevance. This question is also raised by the inte rpreta-\ntion of the physical meaning of the Fock–Stueckelberg appro ach,\nas discussed in Sec. V.\n17It would be interesting to consider these interactions expl icitly\nforsome cosmological toy models. Of course, the idea of “var ying\nconstants of Nature” has been explored in several articles w ith\ndifferent formalisms (see, e.g., [64–68]).surface Σ can be defined by the new set of constraints18\n¯φa:=Pa+Ha(z,Pi>a)≈0,(a= 1,...,P),\n¯φb:=φb≈0,(b=P+1,...,N).(42)\n3. General constraint redefinitions\nWhat can we conclude about the constraint transfor-\nmation in Eqs. ( 42)? If we assume that this transforma-\ntion is invertible in a region of phase space in a neigh-\nborhood of Σ, then it is a particular case of a general\ntransformation φa/ma√sto→¯φa(φ) of constraints that is well de-\nfined and invertible in a neighborhood of Σ and satisfies\n¯φa= 0 if, and only if, φa= 0 (for every a= 1,...,N).\nAny such transformation must satisfy ¯φa=Mb\naφbdue\nto Eq. (11), withMb\nabeing the coefficients of a matrix\nthat must be non-divergent and invertible at least in a\nneighborhood of Σ.\nThe new constraints ¯φaare thus independent, and, by\ndenotingallthefieldsby ξµ= (zν,Qi,Pj), weobtainthe\ngradient ∂¯φa/∂ξµ≈Mb\na∂φb/∂ξµ, which must have the\nsame rank as the matrix given by ∂φb/∂ξµbecauseMb\na\ncorresponds to a full-rank matrix. In this way, the reg-\nularity condition of the constraints φais preserved. The\nnew constraints ¯φa≈0 determine the same hypersurface\nΣ as the original φafunctions, but they will generally\nhave different structure functions.\nIn the general case, the constraint transformation\ninduces a transformation of the multipliers, ¯λa=\nλb/parenleftbig\nM−1/parenrightbiga\nb, so that we may write the Hamiltonian as\nH=λaφa=¯λa¯φa. This redefinition of φaandλais not\na gauge transformation because the functional form of\nthe constraints will generally change, and the transfor-\nmation of λwill generally not involve a time derivative\nof an arbitrary function in the form of Eq. ( 16).\nAs a particular case, we assume that the new con-\nstraints in Eqs. ( 42) are functionally independent, and\nthat they obey the regularity condition because the gra-\ndients of ¯φaon Σ are all linearly independent where Eq.\n(41) is valid. To verify the relation ¯φa=Mb\naφb, one\nmay expand φa(z,P) (a= 1,...,P) in a Taylor series\nin powers of Pto find, in particular,\nφa=φa(−H)+∂φa\n∂Pb¯φb+1\n2∂2φa\n∂PbPc¯φb¯φc+O(¯φ3),\n(43)\n18The parametrization constraint in Eq. (37) is not necessari ly a\nparticular case of Eq. (42) because pTneed not be constant in\ntime, i.e., a constant of Nature (cf. Footnote 15). Furtherm ore,\nnotice thatwhilethe constraints thatcome fromthe paramet riza-\ntion procedure, such as the one in Eq. (37), may be globally\nvalid in phase space, the solutions in Eqs. (42) may hold only in\na certain region. Thus, in general, a parametrized theory wi th\nconstraints of the form given in Eq. (42) may not be globally\nequivalent to gauge theory with constraints φa, with the equiva-\nlence holdinglocallyincertain cases. We alsodisregardap ossible\nexplicit spatial dependence of Hain Eqs. (41) and (42).10\nwherea,b= 1,...,P,¯φaare given by Eq. ( 42), and we\nsuppressedthe argumentsof φaotherthan Pforbrevity.\nThen, since φa(−H) = 0, we have\nφa=¯φb/bracketleftbigg∂φa\n∂Pb+1\n2∂2φa\n∂PbPc¯φc+O(¯φ2)/bracketrightbigg\n=:/parenleftbig\nM−1/parenrightbigb\na¯φb,(a,b= 1,...,P).(44)\nAs we assume that φa≈0 (a= 1,...,P) if, and only\nif, the constraints in Eq. ( 41) hold, then the matrix/parenleftbig\nM−1/parenrightbigb\namust be invertible on Σ. In a neighborhood of\nΣ, fora,b,c= 1,...,P, we can define1\n2∂2φa/∂Pb∂Pc\nto be the element of row aand column bof a matrix Hc\nfor each value of c, so that det M−1= detM−1|¯φ=0(1+\n¯φcTr(MHc)|¯φ=0+O(¯φ2)), which is nonvanishing for ¯φ\nwith sufficiently small but nonzero absolute values. We\ncan then trivially extend ( M−1)b\natoa,b= 1,...,Nbe-\ncause the constraints φawitha=P+ 1,...,Nare not\nchanged in this particular case. The theory can thus be\ndefined from the action\nSdepar\nnew=/integraldisplayτ1\nτ0dτ/bracketleftBig\nAµ(z)˙zµ+Pi˙Qi−¯λa¯φa(z,P)/bracketrightBig\n,\n(45)\nwith the new constraints and multipliers, which is lo-\ncally equivalent to Eq. ( 40). In Sec. IIIB6, we will\nuse this new theory as the parent gauge theory in the\nFock–Stueckelberg approach.\n4. Deparametrization\nA transformation of constraints that involves the so-\nlution of the original constraints in terms of a momen-\ntum variable is called “deparametrization.”19The de-\nparametrization in Eqs. ( 42) motivates a gauge fixing\ngiven by the conditions\n¯χa:=Qa−fa(τ)≈0, (46)\nwherefaare functions solely of τ. These gauge condi-\ntions fix the first Pmultipliers to be [cf. Eq. ( 25)]:\n¯λa≈˙fa≡¯Λa. (47)\nIf we replaceEqs. ( 41) and (46) in the action in Eq. ( 45),\nwe obtain\nSpar\nnew=/integraldisplayτ1\nτ0dτ[Aµ(z)˙zµ−Hred],\nHred:=P/summationdisplay\na=1Ha˙fa+N/summationdisplay\nb=P+1λbφb.(48)\n19This is conceptually the inverse procedure of parametrizat ion,\nhowever, the two procedures may lead to theories that are not\nglobally equivalent. See Footnote 18.This corresponds to the action functional of a system\nwith Hamiltonian Hred, understood as a function solely\noftime and z.20Thus, the solutionofthe constraintsand\nthe gauge fixing in Eqs. ( 41) and (46) effectively yield a\nsystem with a reduced set of degrees of freedom [the z\nfields instead of the set ( z,Q,P)]. For this reason, the\nHamiltonian Hredissometimesreferredtoasthe‘reduced\nHamiltonian�� [11]. Likewise, the gauge freedom related\nto the constraints φa(a= 1,...,P) and its associated\n“gauge parameters” that change with gauge transforma-\ntions have been removed: that is why one says that the\nsystem has been (partially) “deparametrized,” and Eqs.\n(42) are the deparametrized form of the constraints φa\n(a= 1,...,P). If we can set fa0(τ) =τ, then the cyclic\nfieldQa0plays the role of a physical clock. We will see in\nSec.IIIB6how this is related to the Fock–Stueckelberg\napproach,andwediscusssuchadeparametrizationin the\ncontext of general relativity in Secs. IIIG4andIV.\n5. Physical meaning of promoting constants of Nature to\nspacetime fields\nIf the quantities Pare constants of Nature (i.e., fixed\ntheory-defining parameters), then the possibility to write\nthe constraints in the form given in Eq. ( 42) makes it\nclear that the zfields cannot take all possible initial val-\nues but only those for which the quantities Hatake the\nfixed (theory-defining) values −Pa. This is the tradi-\ntional interpretation of first-class constraints in a gauge\ntheory (cf. Sec. II): they restrict the initial-value prob-\nlem of the theory’s degrees of freedom (the zfields).\nWith the promotion of Pto degrees of freedom on\na par with the zfields, the problem is turned upside\ndown: ratherthanarestrictiononthe initial zvalues, the\nconstraints in Eq. ( 42) should be viewed as restrictions\non the otherwise unspecified values of P. In this way,\nthezfields can now take all possible initial values, as\ndifferent choices (different solutions) simply amount to\nassigning different Pvalues. In this sense, the original\ntheory’s constraints on the Ha(seen as functions of z)\nare relaxed. This is exactly the same result of the Fock–\nStueckelberg approach, to which we now turn.\n6. Equivalence with the Fock–Stueckelberg theory\nLet us take the action in Eq. ( 45) as the definition\nof the parent gauge theory, and Eq. ( 47) is taken as\nthe definition of the noncanonical gauge condition (cf.\n20Recall that all constants of Nature that were promoted to fiel ds\nhave been replaced by the corresponding −Ha, and their conju-\ngate cyclic variableswere replaced by fa(τ). Ofcourse, Hredmay\nstill depend on other constant parameters that are not writt en\nexplicitly.11\nSec.IIIA2). Then, the Fock–Stueckelberg action with\nH0≡0 given in Eq. ( 31) becomes\n˜SFS:=/integraldisplayτ1\nτ0dτ/bracketleftBigg\nAµ˙zµ+Pi˙Qi−P/summationdisplay\na=1˙fa¯φa−N/summationdisplay\nb=P+1λbφb/bracketrightBigg\n.\n(49)\nThez-field equations then read [cf. Eqs. ( 30)]\n˙zµ=/braceleftBigg\nzµ,P/summationdisplay\na=1˙faHa+N/summationdisplay\nb=P+1λbφb/bracerightBigg\n={zµ,Hred}(50)\nbecause fa(τ) are functions solely of τ, andzPoisson-\ncommutes with Pby definition. The equations ( 50) are\nexactly those derived from the action in Eq. ( 48). In\nthis way, as far as the zfields are concerned, both the\nFock–Stueckelberg and constants of Nature approaches\nareequivalent. This is ofcourseexpected because the de-\nparametrizationvia constants of Nature induced a choice\nof gauge [cf. Eqs. ( 46) and (47)], which can then be used\nto seed the Fock–Stueckelberg approach. However, it is\nimportant to emphasize that this equivalence is estab-\nlished starting from the action in Eq. ( 45), in which the\nconstraints are already in deparametrized form. As this\nform holds only locally in general, the Fock–Stueckelberg\ntheory defined from a more general parent theory, with\naction given in Eq. ( 40) for example, may generally\nexhibit global differences. Whereas the physical mean-\ning of the constants of Nature approach becomes most\nclear with deparametrization,21the effects of the Fock–\nStueckelberg approach on the relaxation of constraints\nare evident without deparametrization, as they only re-\nquire a well-defined noncanonical gauge condition.\nC. First-class subalgebras\nLet us consider a parent gauge theory with the fol-\nlowing property: we can divide the constraints into\nφ= (u,K) such that ua=φa(a= 1,...,N u) and\nKa=φNu+a(a= 1,...,N−Nu), and we assume that\nthestructurefunctionshavetheproperty Vc\nb= 0,Fc\nab= 0\nifc= 1,...,N u. If this is not the case, it may be pos-\nsible to bring the structure functions into this form by\na redefinition of the constraints (cf. Sec. IIIB3),22and\n21In the constants of Nature approach, the effects of the relaxa tion\nof constraints, as well as the resulting physical equations of mo-\ntion derived from the action in Eq. (48), are most clearly see n if\nthe deparametrization can be explicitly performed.\n22In local regions of field space, itmay be possible to choose Mb\nain\n¯φa=Mb\naφb(cf. Sec. IIIB3)so that the new structure functions\nvanish, and the new constraints locally satisfy an Abelian a lgebra\n{¯φa,¯φb}= 0. In general, however, it is not possible to achieve\nthis ‘local Abelianization’ in a global manner [9].theuconstraints Poisson-commute if the Kconstraints\nare satisfied:23\n{φa,φb}=FNu+c\nabKc,\n{φa,H0}=VNu+b\naKb.(51)\nTheKfunctions lead to a first-class subalgebra.24In\nparticular, the relations ( 51) also lead to the field equa-\ntions\n˙φa=VNu+b\naKb+λbFNu+c\nabKc+{φa,λb}φb.(52)\nIf we can locally define phase-space functions ξa(a=\n1,...,N u), which satisfy {ξb,ua} ≈δb\nafora,b=\n1,...,N uand{ξb,H0} ≈ {ξb,Ka} ≈0, then we can\nconsider the partial gauge fixing:\nχa:=ξa−fa(τ),(a= 1,...,N u),(53)\nwhich is preserved in time if we have\n0χa=0= ˙χa=∂χa\n∂τ+{χa,H0+λbub+λNu+bKb}\n=∂χa\n∂τ+λa+{χa,λb}ub+{χa,λNu+b}Kb(54)\non the hypersurface defined by χa≈0. If we also\nenforceKb≈0, then Eq. ( 54) can be solved for λa\n(a= 1,...,N u) similarly to Eq. ( 25) by fixing the first\nNumultipliers to be solely functions of τ:\nλa=˙fa(τ),(a= 1,...,N u). (55)\nNoticethatweonlyused χa,Kb≈0inthisresult, andwe\ndid not need to enforce ua≈0. Using this partial gauge\ncondition in Eq. ( 52), we find that the uaconstraints\nevolve as\n˙ua=VNu+b\naKb+λbFNu+c\nabKc+{ua,λNu+b}Kb,(56)\nand they are conserved if Ka≈0 is enforced, regardless\nof whether ua≈0 is imposed.\nThe corresponding Fock–Stueckelberg theory with the\nnoncanonical gauge conditions in Eq. ( 55) leads to the\nconstraints λa≈˙fa(τ),ua+wa≈0,Kb≈0 [cf. Eqs.\n(29)]. Due to the conservation of uaon the hypersurface\ndefined by Kb≈0, the effect of the Fock–Stueckelberg\nconstraint ua+wa≈0 is simply to assign a constant\nvalue to the wamultiplier, and the constraints uaare\nrelaxed: instead of being forced to take the value 0, they\ncan now take any constant value. This is useful because\nmany models of interest, such as (bosonic) relativistic\nparticles (which have H0≡0 and an Abelian algebra\nof constraints), can be seen as particular cases of this\nexample.\n23Indices contracted with the index of φaare summed from 1 to\nN. However, indices contracted with the index of uaare to be\nsummed from 1 to Nuonly, whereas indices contracted with the\nindex of Kaare to be summed from 1 to N−Nu.\n24The structure functions may in principle depend on all degre es\nof freedom and may have some functional dependence also on\ntheufunctions.12\nD. A simple mechanical example\nNow consider a mechanical ( d= 0) theory in\nwhich the zdegrees of freedom correspond to canon-\nical pairs ( x,px,y,py) with symplectic potential A=\n(px/2,−x/2,py/2,−y/2) and an action that can be writ-\nten as (up to a boundary term)\nS=/integraldisplayτ1\nτ0dτ/parenleftBigg\npx˙x+py˙y−p2\nx\n2−p2\ny\n2−λpx/parenrightBigg\n.(57)\nOne can verify that the single constraint is of first class\nand Abelian, and we thus have a particular case of the\nexamplediscussedinSec. IIIC.Thesolutiontotheequa-\ntions of motion is readily found to be\nx(τ) =x0+/integraldisplayτ\ndτ′λ(τ′), px(τ) = 0,\ny(τ) =y0+kτ , p y(τ) =k ,(58)\nwithx0,y0andkbeing constants. As xdepends on the\narbitrary function λ(τ), it is “pure gauge;” i.e., it only\nencodes information concerning the generalized reference\nframe, and it does not have any gauge invariant content.\nIndeed, its gauge transformation is δx=ε(τ), which can\nbe integrated to x(τ;s) =x(τ) +sε(τ), where sis the\nparameter along the ‘gauge orbit’, which is the integral\ncurve of the gauge generator. Thus, the gauge condition\nχ=x(τ)≈0 can be used to eliminate x(τ), as it is both\naccessible and complete because, with ε:=−x(τ) being\nindependent of s, the initial arbitrary generalized refer-\nence frame can be mapped into the frame with vanishing\nx(τ) ats= 1,25and only the trivial gauge transforma-\ntion (with ε≡0) preserves the gauge condition. In this\nway, the only degrees of freedom of this model are y0and\nk.\nThecanonicalgauge x(τ)≈0impliesthe noncanonical\ngaugeλ(τ)≈0.26In this case, the Fock–Stueckelberg\naction (31) becomes\n˜SFS=/integraldisplayτ1\nτ0dτ/bracketleftBigg\npx˙x+py˙y−(p2\nx+p2\ny)\n2/bracketrightBigg\n,(59)\nwhich readily leads to the solutions\nx(τ) =x0+kxτ , px(τ) =kx,\ny(τ) =y0+kτ , p y(τ) =k ,(60)\n25The value of sat which the gauge condition is reached of course\ndepends on ε. Here, it is set to 1 for convenience.\n26This follows from Eq. (58) or from the equation of motion\n˙x=px+λ≈λ. Notice also that Eq. (16) leads to the gauge\ntransformation of the multiplier δλ= ˙εor, in integrated form,\nλ(τ;s) =λ(τ)+s˙ε(τ). Starting froman arbitrarygeneralized ref-\nerence frame, the gauge transformationwith ε:=−x(τ)(which is\nindependent of s) leads to λ(τ;s= 1)≈0, as it should. The con-\nverse, however, is not true: the noncanonical gauge λ≈0 does\nnot imply the canonical gauge x(τ)≈0 because λ≈0 is not\ncomplete. It is preserved by the residual gauge transformat ions\nwithε=const., and thus λ≈0 only implies that x(τ)≈const..This theory clearly has more degrees of freedom (and less\ngauge invariance), as the x0,y0,kx,kconstants are freely\nspecifiable. The parentgaugetheoryis onlyrecoveredfor\nthe set of trajectories which obey kx= 0, and thus only\na particular class of solutions of the Fock–Stueckelberg\ntheory corresponds to the solutions of the parent theory\nin the noncanonical gauge λ≈0.\nE. To gauge or not to gauge time?\nMechanical theories ( d= 0) without local symmetries\ncan be defined from the action [cf. ( 4)]\nSmech:=/integraldisplayτ1\nτ0dτ[Aµ(z)˙zµ−H(z)],(61)\nwhichformsthestartingpointoftheusualHamiltonvari-\nationalprinciple. Theparticle’sdegreesoffreedom, given\nby thezµ(µ= 1,...,Z) variables, are scalar fields on its\nworldline, and τis an ‘absolute’ or ‘external’ time pa-\nrameter that cannot be defined from z. Since the Hamil-\ntonian is assumed to have no explicit time dependence, it\nis conserved, and time translations form a global (rigid)\nsymmetry. They can be ‘gauged’ (i.e., turned into a lo-\ncal symmetry) if we introduce a worldline scalar density\nλ(τ) as an independent auxiliary field, H(z)→λH(z),\nso as to make both terms in the Lagrangian (the sym-\nplectic potential term and the Hamiltonian) transform as\nworldline scalar densities. The new action\n˜Smech:=/integraldisplayτ1\nτ0dτ[Aµ(z)˙zµ−λH(z)](62)\nis that of a canonical gauge system, since the equation of\nmotion of λis the Hamiltonian constraint, H≈0, which\nis clearly of first class and Abelian. The result is a gauge\ntheory of time translations: the parameter τis no longer\nabsolute, as it may be redefined [ τ→τ+ε(τ)] without\nchanging the form of the equations of motion, which are\nsaid to be time-reparametrization invariant. This is a\nparticular case of the example considered in Sec. IIIC.\nThe arbitrariness of τis of course inherited from the\narbitrariness of λ, and the fixation of this ‘generalized\nreference frame’ is a gauge choice. In particular, with\ncanonicalgaugeconditions,onecandefinetimeinternally\nby fixing τto be a function of z, the particle’s degrees of\nfreedom [8, 9, 32]. This, together with the Hamiltonian\nconstraint H≈0, means that 2 out of the Zdegrees of\nfreedom of the particle are not independent of the others.\nIn this way, by gauging the time translations, we exclude\nindependent degrees of freedom.\nWe can restore the eliminated degrees of freedom with\nthe Fock–Stueckelberg mechanism. With the noncanon-\nical gauge condition χ=λ−Λ [cf. Eqs. ( 28)], the\nFock–Stueckelberg action ( 26) reads\nSFS:=/integraldisplayτ1\nτ0dτ[Aµ(z)˙zµ−λH(z)−w(λ−Λ)],(63)13\nand it leads to the constraints H+w≈0,λ≈Λ, which\nare satisfied at all times. Since wis an extra degree of\nfreedom, the Hamiltonian constraint is relaxed, and the\nequations of motion are equivalent to those derived from\nthe action given in Eq. ( 31), which reads\n˜SFS:=/integraldisplayτ1\nτ0dτ[Aµ(z)˙zµ−ΛH(z)].(64)\nThis coincides with the original Hamilton-principle ac-\ntion given in Eq. ( 61) if Λ is chosen to be a constant\nequal to 1. Due to the relaxation of the Hamiltonian\nconstraint and the loss of the symmetry under local time\nreparametrizations, the 2 eliminated degrees of freedom\nare restored. Moreover, to return to the parent theory,\none can promote Λ to an arbitrarymultiplier in Eq. ( 64),\nor setw≡0 in Eq. ( 63).\n1. The relativistic particle: from Jacobi to Hamilton and\nback\nAparticularcaseofthis mechanicaltheorycanbe used\nto describe a relativistic particle. If z= (q,p) are canon-\nical pairs with symplectic potential A= (p/2,−q/2) and\nHamiltonian27\nH(q,p) =1\n2gab(q)[pa+fa(q)][pb+fb(q)]\n+m(V(q)−E),(65)\nthen we have the time-reparametrization invariant equa-\ntion of motion ˙ qa=λgab(pb+fb), and so the phase-\nspace action given in Eq. ( 62) (which is defined as a\nfunctional of paths on the cotangent bundle of config-\nuration space) is equivalent to the configuration-space\ntangent-bundle action\nSparticle=/integraldisplayτ1\nτ0dτ/bracketleftbigggab˙qa˙qb\n2λ−fa˙qa−λm(V−E)/bracketrightbigg\n(66)\nafter a Legendre transform. In these variables, the equa-\ntion of motion of λ(the constraint) becomes\n−gab˙qa˙qb\n2λ2−mV+mE= 0. (67)\nBy solving it for λand replacing the solution in Eq. ( 66),\nwe obtain\nSJacobi=/integraldisplayτ1\nτ0dτ/bracketleftbigg\n−σ/radicalBig\n2m(E −V)gab˙qa˙qb−fa˙qa/bracketrightbigg\n,\n(68)\n27The Latin indices for the relativistic particle variables a re not\nto be confused with the multiplier indices used in the previo us\nsections. In particular, we have q0=ct, withcbeing the speed\nof light and tbeing the time coordinate of the spacetime back-\nground on which the particle moves. The background spacetim e\nmetricgabis taken with a mostly plus signature.whereσ=±1 is the sign of λ, which plays the role of\nthe ‘einbein’ on the particle’s worldline. If the function\nfa(q)≡0, and with Ebeing a constant, this is precisely\nthe usual Jacobi action (up to an overall sign) that can\nbe used to define time-repametrization invariant trajec-\ntories in classical mechanics via its extremization (the\nJacobi action principle)[33].28Furthermore, if σ= 1,\nfa=−eAais constant, V(q)≡0 andE ≡ −mc2/2 is\nconstant, the action in Eq. ( 68) reduces to the usual ac-\ntion for a relativistic particle with charge ecoupled to a\nconstant external electromagnetic field. In this case, the\ncorresponding Hamilton–Jacobi equation reads\ngab/bracketleftbigg∂S\n∂qa−eAa/bracketrightbigg/bracketleftbigg∂S\n∂qb−eAb/bracketrightbigg\n+m2c2= 0,(69)\nwhich will be useful in Sec. IIIF. In any case, the time-\nreparametrization invariance is manifest in the action\ngiven in Eq. ( 68).\nWith the Fock–Stueckelberg mechanism [cf. Eq. ( 63)],\nwe are led from the parent theory defined by Eq. ( 66)\n[which is equivalent to Eq. ( 68)] to a theory defined by\nSHamilton=/integraldisplayτ1\nτ0dτ/parenleftBigm\n2gab˙qa˙qb−fa˙qa−V+E/parenrightBig\n,(70)\nwhere we have taken the noncanonical gauge λ= 1/m\nfor simplicity.29The action in Eq. ( 70) is equivalent, via\na Legendre transform, to an action of the form of Eq.\n(64) with Λ ≡1/m, and it is the action used in the or-\ndinary Hamilton variational principle for unconstrained\nmechanics. Thus, we see that the definition of the Fock–\nStueckelbergtheory for a relativistic particle corresponds\nto a transition from the variational principle of Jacobi to\nthat of Hamilton, whereupon time translations are no\nlonger gauged.30\nThe Hamiltonian derived from Eq. ( 70) is identical to\nthe one in Eq. ( 65), but it is not constrained to van-\nish. Rather, due to the global time-translation invari-\nance of Eq. ( 70), it is simply conserved H(q(τ),p(τ)) =\n28Starting from the Arnowitt–Deser–Misner (ADM) formulatio n of\ngeneral relativity [discussed in [11, 69] (and references t herein)\nand briefly in Sec. IIIG2], a similar procedure leads to a squa re-\nroot action for general relativity, known as the Baierlein– Sharp–\nWheeler action [11, 70]. Analogously, starting from the Pol yakov\naction for a relativistic string, one can obtain a square–ro ot ac-\ntion known as the Nambu–Goto action [71–74].\n29For the ordinary relativistic free particle, for which σ= 1,V≡0,\nE=−mc2/2 andfa≡0, we find from Eq. (67) that the gauge\nλ= 1/mcorresponds to fixing τto be the particle’s proper time\n(on a solution to the equations of motion), while the constra int\nin Eq. (67) enforces that the particle travels on a timelike t ra-\njectory. For a massless particle ( m= 0), we cannot define the\nJacobi action [cf. (68)]. Nevertheless, the action in Eq. (6 6) is\nwell defined, and we can transition to an action akin to the one\nin Eq. (70) with the noncanonical gauge λ= Λ =const., where\nthe constant Λ has units of inverse mass. This corresponds to\nfixingτto be an affine parameter for the null trajectory of the\nmassless particle.\n30As discussed above, from the point of view of the parent theor y,\nthe Fock–Stueckelberg theory is obtained from an “erroneou s”\ngauge fixing that restores degrees of freedom.14\nH(q(0),p(0)). To transition back from the Hamilton ac-\ntion principle defined by Eq. ( 70) to the Jacobi principle\nderived from Eq. ( 68), it suffices to restrict oneself to\ninitial values for which the Hamiltonian vanishes,\nH(q(τ),p(τ)) =H(q(0),p(0))→0,(71)\nwhich is equivalent to\nm\n2gab˙qa˙qb+V−E= const. →0,(72)\nas this recovers the Hamiltonian constraint [cf. Eq.\n(67)] as a particular initial-value condition on the Fock–\nStueckelberg theory.31This condition is also responsible\nfor the ensuing time-reparametrization invariance of the\ntrajectories that exhibit such initial values.\nIt is also worthwhile to notice that the Fock–\nStueckelberg trajectories of a relativistic particle (with\nV≡0) need not be timelike or null throughout con-\nfiguration space. Only those trajectories satisfying the\ninitial-value Hamiltonian constraint will conform to such\na causal restriction. In this way, the transition from the\nparentto the Fock–Stueckelbergtheoryallowsmoresolu-\ntions to the equations of motion: whereas the parent the-\nory constrains initial-values, these are free in the Fock–\nStueckelberg theory. This was one of the main motiva-\ntions of Stueckelberg, as the relaxation of this constraint\nmeantthatonecouldconsidertrajectoriesgoingforwards\nand backwardsin Minkowski time, and this could be use-\nful in the description of pair creation and annihiliation\n[29–31].\n2. Time as the canonical conjugate to constants of Nature\nand the reappearance of time in the quantum theory\nFor the Fock–Stueckelbergtrajectoriesthat do not sat-\nisfytheinitial-valueHamiltonianconstraint,wefindfrom\nEq. (72) that the quantity\nm\n2gab˙qa˙qb+V= const. (73)\nis conserved because Eis assumed to be a constant pa-\nrameter (e.g., E=−mc2/2). The conserved value of Eq.\n(73) is, however, free, as it depends on a choice of initial\nconditions.\nOn the other hand, for the trajectories that respect\nthe initial-value Hamiltonian constraint, we obtain from\nEqs. (71) and (72) the equivalent conditions\nm\n2gab˙qa˙qb+V=E,\ngab\n2m(pa+fa)(pb+fb)+V=E.(74)\n31This is equivalent to setting w≡0 in Eq. (63), as wis con-\nstrained to be the value of Hon shell.The parameter Ecan be regarded as a “constant of Na-\nture” ofthis toy theory. If we regardit as a new degreeof\nfreedomthat isconserved, we cantriviallyenlargethe set\nofvariablesfrom( q,p) to(q,p;η,E), where( η,E) isanew\ncanonical pair (cf. Sec. IIIB). We can add −E˙ηto the\naction, so that the new time-reparametrization invariant\naction can be written as\nSE=/integraldisplayτ1\nτ0dτ[−E˙η+pa˙qa−λH(q,p;E)].(75)\nAsηevolveslinearlyinthe“propertime” m/integraltextτdτ′λ(τ′),32\nit can be taken to be a physical clock or a physical def-\ninition of time. In this sense, time can be seen as a\nphysical variable canonically conjugate to a constant of\nNature. For the time-reparametrization invariant parent\ntheory, we can thus consider the canonical gauge condi-\ntionχ=η(τ)−τ, which fixes the particle’s worldline\ntime parameter to be η. Concomitantly, we can solve the\nHamiltonian initial-value constraint not in terms of the\noriginalpairs( q,p) but ratherin terms of E–the solution\nis trivial. The on-shell, gauge-fixed action becomes\nSE=/integraldisplayτ1\nτ0dτ/braceleftbigg\npa˙qa−/bracketleftbigggab\n2m(pa+fa)(pb+fb)+V/bracketrightbigg/bracerightbigg\n.\n(76)\nThis is an unconstrained action of the type of the usual\nHamilton principle. Its global time-translation invari-\nance leads exactly to the conservation law expressed in\nEq. (73), which is what one obtains with the Fock–\nStueckelberg theory. Thus, the Fock–Stueckelberg ap-\nproach leads to a relaxation of first-class constraints by\nadding new degrees of freedom, which generally can be\nseen asthe wmultipliers [e.g., asin Eq. ( 63)], but, in cer-\ntain cases, these new variablescan be identified with con-\nstants of motion (promoted to physical conserved fields),\nas it was discussed in Sec. IIIB, and as we see here in\nthe case of the relativistic particle, where Eis related to\nthe particle’s rest energy.\n3. The work of Fock and Stueckelberg\nThe above formalism is a direct generalization of the\noriginal work of Fock and Stueckelberg. In [28], Fock\nconsidered a charged relativistic particle and the use of\nproper time as an independent variable in the particle’s\naction principle. In that work, it was noted that, in the\nusualformulationofthe relativisticparticle’sactionprin-\nciple [which can be generalized to the action in Eqs. ( 66)\nand (68)], proper time is not an independent variable\n32For the free massive relativistic particle, with fa≡V≡0 and\nE ≡ −mc2/2, one can verify that this quantity indeed coincides\nwith the usual definition of the particle’s proper time (up to\nan overall sign associated with the sign of the chosen λ; e.g.,\nλ= 1/mhas a positive sign, as m >0).15\n(as it depends on the path of the particle) but, with the\nintroduction of a new action [which can be generalized\nto the action in Eq. ( 70)], one could use an independent\nquantity that coincided with proper time on a solution to\nthe equations of motion. The connection with the usual\ntreatment of the relativistic particle proceeded by fixing\na constant of motion that would recover the usual time-\nreparametrization invariant formalism. This is precisely\nwhat is done in Eq. ( 72), which is a generalization of\nthe equation ˙ x2+ ˙y2+ ˙z2−c2˙t2= const. → −c2that\nis discussed in Fock’s article.33Fock also discussed the\nimplications of the use of an independent proper time for\nthe Dirac equation.\nSimilarly, Stueckelberg proposed in [29–31] a “new rel-\nativisticmechanics”bypositinganewactionprinciplefor\nthe relativistic particle. The proposed action was a par-\nticular case of Eq. ( 70), and Stueckelberg stressed that\nthe new formulation could be straightforwardly quan-\ntized ` a la Schr¨ odinger, and it could also be used to de-\nscribe the creation and annihilation of particles not only\nin the quantum theory but also in the classical theory\n[due to the fact that the trajectories that can be derived\nfrom Eq. ( 70) need not be always timelike or null if the\nparticle is interacting with an external field].34In par-\nticular, the particle’s mass was to be understood as an\nintegrationconstantassociatedto eachworldline(an idea\nwhich would later make a comeback concerning the cos-\nmological constant in unimodular gravity, and possibly\nany constant of Nature in quantum gravity [25–27, 40–\n42, 63], cf. Secs. IIIBandIV).\nThe work of Fock and Stueckelberg was developed in\nthe context of the early days of relativistic quantum the-\nory (and ultimately quantum field theory). It has in-\nspired many subsequent applications, including the rela-\ntivistic quantum mechanics which was further developed\nby Horwitz, Piron and others [35–37]. In the context\nof quantum gravity and gauge theories, their work is\nseldom acknowledged. Although Fock and Stueckelberg\nonly considered a particular gauge theory (the relativis-\ntic particle with its local worldline reparametrizations)\nand they did not employ the modern tools of canoni-\ncal gauge theory and first-class constraints discussed in\nthe present article,35the modern readercan clearlygrasp\nthat the essence of their approach is gauge-theoretical, in\n33To obtain ˙ x2+ ˙y2+ ˙z2−c2˙t2= const. → −c2from Eq. (72), we\nsetgabto be the Minkowski metric with mostly plus signature,\nV≡0, andE=−mc2/2.\n34The works [29–31] of Stueckelberg have a conceptual similar ity\nwith but are distinct from the other famous Stueckelberg met hod\n[75, 76]: that of introducing a scalar field to preserve gauge in-\nvariance in a massive Abelian gauge theory of the Yang–Mills\ntype.\n35This is historically understandable. When their works were pub-\nlished, a general theory of first-class constrained Hamilto nian\nsystems did not seem to be generally known, despite early wor ks\nof Rosenfeld [1, 2, 7], and it would only rise to prominence wi th\nthe articles of Bergmann and Dirac [3–6, 10].the sense of producing a new theory from a parent gauge\ntheory, as we have explained in the previous sections.\nF. Electrodynamics and “charge without charge”\nAnother interesting application of the Fock–\nStueckelberg mechanism is in the archetypical gauge\ntheory: electromagnetism. Let us consider a four-\ndimensional Minkowski spacetime with metric signature\n(−,+,+,+), and let us now use Latin indices to indicate\nthe components of spacetime tensors, not the multiplier\nfields as in previous sections. We use units in which\nthe speed of light is c= 1 for simplicity. We also no\nlonger adopt DeWitt’s compact notation for clarity\n(i.e., repeated indices are tacitly summed over their\ndiscrete values but no accompanying spatial integration\nis implicit). The Maxwell Lagrangian reads\nL=−1\n4FabFab,\nFab=∂aAb−∂bAa,(77)\nwhere the zfields are to be understood as the canonical\npairs of spatial components of Ai(i= 1,2,3) and their\nconjugate momenta [obtained after a Legendretransform\nof the variables in Eq. ( 77)], whereas A0plays the role\nof a multiplier that enforces the Gauss law (see, e.g.,\n[8, 11, 77] for the Hamiltonian formulation of Maxwell’s\ntheory).\nLet us consider the gauge condition\nχ=1\n2(AaAa+k2)≈0, (78)\nwithkbeing a constant. This is a noncanonical gauge,\nas it is meant to fix the multiplier A0. We obtain\nA0=±/radicalbig\nA2\n1+A2\n2+A2\n3+k2, which is well defined (in\nthe sense of yielding a real A0and not introducing any\nnew restrictions on the other components of the four-\nvector potential) if k2≥0 (corresponding to a timelike\nor null four-vector potential) [78]. This gauge condition\nis interesting because, given the gauge transformation\nAa=˜Aa+∂aε, where ˜Aais the four-vector potential in\nsome other arbitrary gauge, we find\ngab/parenleftbigg∂ε\n∂xa+˜Aa/parenrightbigg/parenleftbigg∂ε\n∂xb+˜Ab/parenrightbigg\n≈ −k2,(79)\nwhich is equivalent to the Hamilton–Jacobi equation for\na charged relativistic particle [cf. Eq. ( 69)] if we identify\nε≡ −S/e,k=mc/e(withc= 1 in the adopted units),\nand˜Aa↔Aa[53, 78]. Thus, this gauge choice is accessi-\nble only if the Hamilton–Jacobi equation for the charged\nparticle has a solution.\nThis equivalence with the dynamics of charged parti-\ncles motivated Dirac [53] to consider that the Maxwell\ntheory could be the “wrong” classical theory to be quan-\ntized, given the lack of details about the structure of the16\nelectron and the ensuing complications that researchers\nfaced in order to describe the electron’s self-energy in\npoint-charge models36or in models in which the charge\nwas not localized, which proved to be intractable at the\ntime. Rather, Dirac posited, one ought to seek a theory\nthat is simple in the sense that it makes no assumptions\nabout the structure of electrons, so as to lead to a sim-\npler quantization. This is similar to Stueckelberg’s moti-\nvation to propose a “new relativistic mechanics” with a\nmore direct quantization procedure.\nAs the gauge invariance of electromagnetism leads to\nextra variables that do not have physical significance,37\none should break this invariance, according to Dirac,\nand use the extra variables to describe the sought-after\ncharged particles. This led to Dirac’s proposal of a new\nclassical Lagrangian for electrodynamics:38\nL=−1\n4FabFab−w\n2(AaAa+k2), (80)\nwhich is readily understood to be a particular case of the\ngeneral Fock–Stueckelberg approach [cf. Eq. ( 26)]. The\nFock–Stueckelberg Lagrangian in Eq. ( 80) leads to the\nfield equations\n∂aFab=wAb=:−Jb, (81)\nwith∂aJa= 0 due to the antisymmetry of Fab. The\nappearance of the four-current Jasignals the presence of\nnew degreesoffreedom with respect to the parenttheory.\nIn particular, we obtain a modification of the vacuum\nGauss law:\ndiv/vectorE=−wA0=:J0=:ρ . (82)\nAs discussed in the previous sections, this constraint\nrelaxation is a characteristic feature of the Fock–\nStueckelberg theory. For the case of electromagnetism,\nwe obtain “electrons without electrons” or “charge with-\nout charge.” Furthermore, Dirac emphasized that his\nnew electrodynamics is equivalent to Maxwell’s theory\nif thewmultiplier vanishes (absence of charges). As\ndiscussed above, this is a general feature of the Fock–\nStueckelberg approach: if wis zero on shell, then we\nrecover the parent theory, which, in this case, is vacuum\nelectromagnetism.\nDirac’s electrodynamics inspired many further works\n(e.g., [54–58]),39including an extension to non-Abelian\n36This was eventually addressed by renormalization, which Di rac\nconsidered unappealing [53].\n37They are inaccessible “generalized reference frames.”\n38There were of course previous works by other authors that mod -\nified the Maxwell Lagrangian in order to break gauge invarian ce.\nSee, for instance, [79, 80].\n39It is worthwhile to mention that Nambu [78] verified that the\ncorresponding quantum field theory and S-matrix of Dirac’s\nnew electrodynamics with a timelike four-vector potential (with\nk >0) agree with the usual results of quantum electrody-gauge theories in [56], and the idea was recently rein-\ncarnated in [39], which is part of the series of works\n[13, 38, 39], where the argument that the modified the-\nory leads to a more straightforward quantization is again\nrecorded. However, both Stueckelberg and Dirac make\nit clear that one is producing a different classical theory\nwhich is to be quantized, rather than arguing that the\nmodified theory follows from a consistent quantization or\nconcerns the “classical equations of motion of quantized\ngauge theories” (as in [38, 39]). The Fock–Stueckelberg\napproach does not concern the consistency of quantiza-\ntion methods of gauge theories,40but it rather offers an\ningenious way to classically add new degrees of freedom\nby breaking the original gauge symmetry.41\nDirac’s treatment can be generalized by considering\nthe Lagrangian\nL=−1\n4FabFab+Aaja−wχ(A), (83)\nwhereχ(A) is a generic gauge condition on the four-\nvector potential, and jais a “real” four-current (associ-\nated with local degrees of freedom already present in the\nparent theory). The corresponding Fock–Stueckelberg\nfield equations are\n∂aFab=−jb+w∂χ\n∂Ab=:−(jb+Jb),(84)\nwhich, in particular, imply a modification of the parent-\ntheory Gauss-law constraint:\ndiv/vectorE=j0−w∂χ\n∂A0=:j0+J0=:ρ .(85)\nAn example that wasconsidered in [39] is the Weyl gauge\nχ(A) =A0. In this case, we have Jb= (−w,/vector0), so that\nthere is a violation of the parent-theory Gauss law but\nnot of the Amp` ere law. The physical interpretation of\nthe Fock–Stueckelbergeffective currentin this caseis less\nclear than in Dirac’s original treatment connected with\nthe charged particle’s Hamilton–Jacobi equation.\nnamics at least up to the lowest orders in perturbation theor y.\nMore precisely, Nambu considered the Lagrangian −1\n4FabFab−\njaAa, where jais a conserved current, and the gauge condi-\ntion is tacitly already satisfied; i.e., A0is defined as A0:=\n±/radicalBig\nA2\n1+A2\n2+A2\n3+k2. This corresponds to Dirac’s new elec-\ntrodynamics in the presence of some four-current jaand with\nthe gauge condition already imposed at the level of the actio n\n[similarly to Eq. (31)].\n40One such method is the powerful BRST cohomology [9], which\ndoes not require that a gauge be fixed prior to quantization in\norder to achieve a nonambiguous Hamiltonian.\n41Of course, Nature is presumably “quantum first,” and classic al\nbehavior is relegated to an approximate limit. Even so, this does\nnot imply that, in order to be well defined, the fundamental\nquantum theory must break gauge symmetries, which would the n\nonly emerge in particular cases. We will further discuss thi s in\nSec. V.17\nG. Generalized unimodular gravity and “matter\nwithout matter”\nThe Fock–Stueckelberg approach can also be applied\nto general relativity. In what follows, as in the relativis-\ntic particle and electromagnetic cases, repeated indices\nare to be summed over their discrete values but there\nis no implicit spatial integration. As before, we con-\nsider a spacetime with a mostly positive metric signature\nbut now spacetime indices are denoted by Greek letters,\nwhereas spatial indices are denoted by Latin letters.42\nWe also adopt units in which the speed of light cand\nNewton’s constant Gare both equal to one, c=G= 1.\n1. Canonical general relativity\nGiven a spacelike foliation of spacetime, the space-\ntime metric with components gµνand determinant g:=\ndetgµνcan be decomposed as [11, 69]43\ngµν=/parenleftbigg\nNaNa−N2Nb\nNbhab/parenrightbigg\n,\ngµν=/parenleftbigg−1\nN2Na\nN2\nNa\nN2hab−NaNb\nN2/parenrightbigg (86)\nwhere the spatial indices are lowered and raised by the\nspatial metric hab, which has determinant h:= dethab=\n|g|/N2and inverse hab. The quantities N,Naare respec-\ntively the lapse function (taken to be always positive)\nand the shift vector. These are the variables used in\nthe Arnowitt–Deser–Misner (ADM) formulation of gen-\neral relativity [69].\nIn this case, the zfields of the formalism presented in\nSecs.IIandIIIAcorrespond to matter fields and the\ncomponents hijof the spatial metric, whereas N,Naact\nas multipliers.44The first-class constraints associated\nwith these multipliers are, respectively,45\nφ⊥= 16πGabcdpabpcd−√\nh\n16π(3)R+√\nhρ= 0,\nφa=−2Dbpb\na+√\nhJa= 0,(87)\n42This notation is not to be confused with the µandaindices used\nin Secs. II and IIIA.\n43Obviously, there is an abuse of notation in Eq. (86), as we use\nthe components of matrices and vectors to denote the matrice s\nand vectors themselves.\n44This implies that the constraints given in Eqs. (87) are trea ted\nas primary constraints in accordance with the formalism of S ecs.\nII and IIIA. However, we can evidently include the momenta\nconjugate to N,Nainthe theory, so asto treat the lapse and shift\nas ordinary non-multiplier fields. In this case, their momen ta\nmust vanish as primary constraints, whereas the constraint s in\nEqs. (87) become secondary [11, 61].\n45We disregard the presence of a cosmological constant inEqs. (87)\nbecause, as we shall see, it can be recovered in the correspon ding\nFock–Stueckelberg theory.wherepabis the canonical momentum conjugate to the\nspatial metric (and it is a spatial tensor density of weight\none);(3)Rand D aare, respectively, the Ricci scalar and\nthe covariant derivative on a leaf of the foliation; and\nρandJarespectively correspond to the matter energy\ndensity and “Poynting vector” [8, 11, 69, 81, 82]. The\ninverse metric on the gravitational configuration space\nreads\nGabcd=1\n2√\nh(hachbd+hadhbc−habhcd),(88)\nwhereas the gravitational configuration-space metric\nGabcdsatisfiesGabcdGcdef= (δa\neδb\nf+δa\nfδb\ne)/2. Moreover,\nthe Hamiltonian field equations can be used to write the\nmomentum pabas\npab=1\n32πNGabcd/parenleftBig\n˙hcd−DcNd−DdNc/parenrightBig\n.(89)\nThe first-classalgebraof generalrelativityis given by the\nPoisson brackets (suppressing the temporal dependence\nfor brevity)\n{φ⊥(x),φ⊥(y)}=δ,a(x,y)/bracketleftbig\nhab(x)φb(x)+hab(y)φb(y)/bracketrightbig\n,\n{φa(x),φ⊥(y)}=φ⊥(x)δ,a(x,y), (90)\n{φa(x),φb(y)}=φb(x)δ,a(x,y)+φa(y)δ,b(x,y),\nwhereδ,a(x,y) denotes a partial derivative of the Dirac\ndelta distribution with respect to xa.\n2. Fock–Stueckelberg approach to general relativity\nAn example of a partial non-canonical gauge fixing of\nthe kind given in Eq. ( 28) is given by the condition [83]\nχ:=N−Λ(h)≈0, (91)\nwhich fixes the lapse to be a function of the determinant\nof the spatial metric, N≈Λ(h). We assume that Λ( h)\nis always positive. In this case, the constraints of the\nFock–Stueckelberg theory are χ,φ⊥+w,andφa[cf. Eqs.\n(29)], and thus the first-class Hamiltonian constraint φ⊥\nof general relativity, which results from the variation of\nthe action with respect to N, is relaxed. Its value can\nnow be arbitrary, as it simply fixes the value of the new\nmultiplier field w, which is to be interpreted as an effec-\ntive kind of matter.46\nFrom the variation of the Fock–Stueckelberg action\n(26), we obtain the Hilbert energy-momentum tensor of\n46This is analogous to what is accomplished by Synge’s G-metho d\n[84]. See the comments in [85].18\nthe effective matter,47\nTµν\neff:=2/radicalbig\n|g|δ\nδgµν/integraldisplay\nd4x(−wχ),(92)\nwhich can be computed by using the relation\nδgµν=−gµρgσνδgρσ (93)\nto find\nδN=δ/bracketleftBig\n(−g00)−1\n2/bracketrightBig\n=−N3\n2g0µgν0δgµν.(94)\nThe relation δh=hhijδhij=hhijδµ\niδν\njδgµνisalsouseful.\nWith this, the on-shell result is seen to be the energy-\nmomentum tensor of a perfect fluid with four-velocity\nnµ= (1/N,−Ni/N) =−g0µNwithN≈Λ, which is\nthe normal vector to the spatial hypersurfaces, and with\nenergy density\nE≈w√\nh, (95)\nand pressure\nP≈2dlogΛ\ndloghE=:W E. (96)\nWhat can we conclude about the function Λ( h) that de-\nfines the gauge fixing? And what about the new multi-\nplier field wthat characterizes the effective fluid? First,\nnotice that the constraints χ≈0,φ⊥+w≈0,φa≈0 are\nthe ‘primary’ constraints in the Fock–Stueckelberg the-\nory, as they follow from the Fock–Stueckelberg action it-\nself. Theconstraints χ≈0andw≈ −φ⊥fixthemultipli-\nersNandw, respectively, at all times, and their preser-\nvation in time fixes the corresponding derivatives of the\nmultipliers. Ontheotherhand,wemustproceedwiththe\nanalysis of the constraint conservation (` a la Rosenfeld–\nBergmann–Dirac [1, 2, 5–7, 9]), especifically with respect\nto the (“unaltered”) spatial diffeomorphism constraints\nφato determine whether new (‘secondary’) constraints\nappear in the theory and whether the Fock–Stueckelberg\ntheory defined from the parent-theory gauge choice given\nin Eq. (91) is consistent. We can use the field equations\n(30) together with the first-class algebra given in Eqs.\n(90) to find\n0≈˙φa(τ,x)≈/integraldisplay\nd3y{φa(τ,x),Λ(h(τ,y))φ⊥(τ,y)}\n≈ −2∂\n∂xa/parenleftbigg\nw∂Λ\n∂logh/parenrightbigg\n=−∂\n∂xa(wΛW). (97)\n47In the functional variation δF=εd\ndε/vextendsingle/vextendsingle/vextendsingle\nε=0F(t,x;gµν+εηµν] =:\nε/integraltext\nd3xδF\nδgµνηµν(t,x) (cf. Footnote 3), as the metric and\nits (arbitrary) variation ηµνare symmetric tensors, we must\nsymmetrize the functional derivative δF/δgµνwith respect to\ntheµ,νindices. For example, from Eq. (93), we find\nδgµν(τ,x)/δgρσ(τ,y) =−δ(x,y)(gµρgσν+gµσgρν)(τ,x)/2.This is a new (secondary) constraint that is not redun-\ndant with any of the primaries. We must then impose\nit at all times. For this, it is useful to compute [cf. Eq.\n(89)]\n−˙w(τ,x)≈˙φ⊥(τ,x)\n≈/integraldisplay\nd3y{φ⊥(τ,x),Λ(h(τ,y))φ⊥(τ,y)}\n+/integraldisplay\nd3y{φ⊥(τ,x),Na(τ,y)φa(τ,y)}\n≈/bracketleftBigg\nw˙Λ\nΛ−wW∂Na\n∂xa(98)\n−w\nΛNa∂Λ\n∂xa−∂\n∂xa(Naw)/bracketrightbigg\n(τ,x),\nwhich is equivalent to\nd\ndτ(wΛ)−∂\n∂xa[NawΛ(1+ W)]≈0.(99)\nWith Eq. ( 99), we can compute\n0≈ −d\ndτ∂\n∂xa(wΛW)\n=−∂\n∂xa/braceleftbigg\n˙WwΛ+W∂\n∂xb/bracketleftbig\nNbwΛ(1+ W)/bracketrightbig/bracerightbigg\n,(100)\nwhich is a new (‘tertiary’) constraint that can be used to\nfixtheshiftvector Naifitisnotredundantwiththeother\nconstraints.48In general, the solution to the secondary\nconstraint in Eq. ( 97) is\nwΛW≈F(τ), (101)\nwhere F(τ) is a function solely of time. In the cases in\nwhich F(τ)/ne}ationslash= 0 at all times, then the tertiary constraint\nbecomes\n∂\n∂xa/braceleftbigg\n(1+W)∂Nb\n∂xb+/parenleftbiggd\ndτ−Nb∂\n∂xb/parenrightbigg\nlogW/bracerightbigg\n≈0,\n(102)\nwhich is consistent with\n(1+W)∂Nb\n∂xb+/parenleftbiggd\ndτ−Nb∂\n∂xb/parenrightbigg\nlogW≈dlogF\ndτ.\n(103)\nThis generally fixes the shift vector Na(it cannot be left\narbitrary), thereby also breaking the three-dimensional\ndiffeomorphism symmetry on the spatial hypersurfaces.\n48It is important to notice that constraints are relations on p hase\nspace that can be used to restrict either the initial values o f\nthe fields and their momenta or the multipliers, and they are\nenforced for every solution of the theory. Thus, a constrain t of\nthe form given in Eq. (100) can in principle be used to fix Na\nfor every admissible choice of initial values of the fields an d their\nmomenta.19\nThe exceptions are the cases in which Eq. ( 100) is in-\ndependent of Na. These are: (1) wis everywhere zero\non shell (in which case the Fock–Stueckelberg theory is\nequivalent to general relativity); (2) if W≡0, which cor-\nresponds to an effective dust matter [cf. Eq. ( 96)]; (3) if\nW≡ −1, which corresponds to an effective cosmological\nconstant [cf. Eq. ( 96)] (this is the case of unimodu-\nlar gravity). In all three cases, the tertiary constraint is\ntrivialized (i.e., it does not lead to any new restrictions\non the fields and their momenta nor the multipliers). In\ncases (1) and (2), we also have F(τ)≈0 at all times.\nFor case (3), we find that Eqs. ( 99) and (101) determine\nthatFis a spacetime constant. In this case, we obtain\n[cf. Eqs. ( 95), (96) and (101)]\nΛ =Λ0√\nh, w≈F\nΛ0√\nh ,\nE≈F\nΛ0= const. =: E0,(104)\nwhere Λ 0andFare spacetime constants, so as to yield\nthe energy density Ethat corresponds to the constant\nenergy density of a cosmological constant. Furthermore,\nthe gauge condition N≈Λ = Λ 0/√\nhis equivalent to the\ndeterminantcondition |detgµν|= Λ2\n0. BysettingΛ 0= 1,\nwerecovertheusualunimodulargravitycondition. Thus,\nunimodulargravitycanbe seenasaparticularcaseofthe\nFock–Stueckelberg approach to general relativity.\nBy keeping a more general Λ( h), we obtain a the-\nory that can be called “generalized unimodular grav-\nity,” which was discussed in [83] and further developed\nwith different applications in [86–88].49Depending on\nthe choice of Λ( h) [i.e., the choice of partial gauge fix-\ning in Eq. ( 91)] and the consistency of the constraint\npreservation equations ( 97) and (100), different residual\ngauge symmetries might remain (cf. Sec. IIB). Whereas\nthe residual gaugesymmetries ofunimodular gravitycor-\nrespond to the group of volume/area-preserving diffeo-\nmorphisms [and the shift vector Nais not fixed by Eq.\n(100)], the residual gauge symmetries of the generalized\nunimodular theory may be more involved (see [83, 86]).\nOne might also complement the condition in Eq. ( 91)\nwith an extra gauge condition on the shift vector [e.g.,\nNa≈Λa(h)] and subsequently analyze the consistency\nof the Fock–Stueckelberg constraints.\nIn this way, the application of the Fock–St¨ uckelberg\napproach to general relativity naturally leads to new de-\ngrees of freedom [coming from the wmultiplier(s)] that\ncanbeeffectivelyidentifiedwithdifferentkindsofmatter,\nwhose form depends on the chosen gauge condition(s).\nThis effective matter appears in addition to the matter\ncontent that was already present in the parent theory. If\nthe parent theory is purely gravitational,the correspond-\ning Fock–Stueckelberg theory will feature a gravitational\n49In particular, the article [86] discussed the constraint al gebra in\nsome detail.field and an effective fluid. Similarly to Dirac’s electro-\ndynamics, we thus have “matter without matter” in the\nFock–Stueckelberg theory.\nIn [38], this emergence of effective matter from a relax-\nationofthefirst-classconstraintsingeneralrelativitywas\nrediscovered. Itisalsoimportanttomentiondifferentap-\nproaches across the literature, which yield similar results\nor employ a similar reasoning to the Fock–Stueckelberg\napproach. The treatment of constrained Hamiltonian\nsystems via the “unfree gauge symmetry” formalism was\ndiscussed in the series of papers [89–92]. The essential\nidea is to consider gauge transformations where ε(cf.\nSec.IIA)arerestrictedbyadditionalconditions,namely,\nthey must satisfy a set of partial differential equations.\nThis restriction of the gauge transformations leads to a\nnew theory, just as in the case of the Fock–Stueckelberg\nmethod. Indeed, the simplest example is again unimodu-\nlargravity,where the correspondingequationsaresimply\nthe conditions of the transverseness, ∇µεµ= 0 (with µa\nspacetime index). Moreover, unimodular gravity was an-\nalyzed from the point of view of restricted gauge theories\nin [93]. Finally, in the so-called TDiff theories (see, e.g.,\n[94]), the matter part of the action is invariant only with\nrespect to the transverse diffeomorphisms, which leads\nto a weaker version of unimodular gravity. Some models\nof similar theories were considered in [95], where the de-\ntails of the breaking of diffeomorphism invariance were\ndiscussed. The reader is also referred to [96] for a rather\nbroad analysis of gravity theories with different sets of\nconstraints or different structures of gauge algebras. All\nin all, the Fock–Stueckelbergapproachcanbe seen asone\nof the oldest and simplest of these approaches where the\ngauge symmetry is modified to yield a new theory from\na parent.\n3. Homogeneous cosmology\nOne of the most useful applications of generalized\nunimodular gravity is in cosmology. This corresponds\nto an application of the Fock–Stueckelberg approach\nto a toy model of general relativity, such as the\nFriedmann–Lamaˆ ıtre–Robertson–Walker(FLRW) model\nwith a scalar field ϕ. For a spatially flat model, the\nspacetime metric is given by the line element\nds2=−N2(τ)dτ2+a2(τ)dx2,(105)\nwhereN(τ) is the lapse function, the shift vector Na\nand spatial diffeomorphism constraint are absent due\nto the homogeneity assumption, a(τ) is the scale factor\nof the universe, and d x2is the line element of three-\ndimensional Euclidean space. The Fock–Stueckelberg20\nLagrangian reads50\nL=−3a˙a2\nN+a3˙ϕ2\n2N−Na3V(ϕ)−wχ\n=pa˙a+pϕ˙ϕ−Nφ⊥−wχ .(106)\nIn the absence of the spatial diffeomorphism constraints\nφa, the only Fock–Stueckelberg constraints are χ=N−\nΛ≈0 andφ⊥+w≈0, both of which fix the multipliers\nNandwat all times. The Hamiltonian constraint [11,\n81, 82],\nφ⊥=−p2\na\n12a+p2\nϕ\n2a3+a3V(φ), (107)\nisthenrelaxed, andtheanalysisthatledtothesecondary\nand tertiary constraints in Eqs. ( 97) and (100) need be\nperformed no longer. The model is thus mechanical and,\nin fact, it correspondsto arelativisticparticle in acurved\nbackground, so that we can use the formalism of Secs.\nIIIEandIIIE1. The analogue of Eq. ( 99) leads us to\nd\ndτ(wΛ)≈0, (108)\nwhich implies w≈w0/Λ, where w0is a spacetime con-\nstant.\nFor the purposes of cosmology, it suffices to choose a\ngauge condition for which the effective fluid equation-of-\nstateparameter Wisaspacetimeconstant, whichimplies\n[cf. Eqs. ( 95), (96) and (108)]\nΛ = Λ 0a3W, w≈w0\nΛ0a−3W,\nE≈w0\nΛ0a−3(W+1)=:E0a−3(W+1),(109)\nwith Λ 0being a spacetime constant. The quantities in\nEq. (109) reduced to those in Eq. ( 104) forW=−1, as\nthey should.\nFrom the Fock–Stueckelberg Lagrangian in Eq. ( 106),\nwe obtain the equations\n0 = 3/parenleftbigg˙a\nNa/parenrightbigg2\n−1\n2/parenleftbigg˙ϕ\nN/parenrightbigg2\n−V −w0\nΛ0a−3(W+1),\n0 =2\nN2¨a\na+1\nN2/parenleftbigg˙a\na/parenrightbigg2\n−2˙N˙a\nN3a\n+˙ϕ2\n2N2−V −w0W\nΛ0a3(W+1),\n0 = ¨ϕ+3˙a˙ϕ\na−˙N˙ϕ\nN+N2∂V\n∂ϕ,(110)\n50In this homogeneous setting, the spatial integration in the action\nleads to an overall factor of L3, which is a fiducial spatial volume,\nmultiplying the Lagrangian, d τL3L. The arbitrary length scale L\ncan be removed by the redefinitions a→a/L,N→N/L,t→Lt\n[97], which lead to the Lagrangian in Eq. (106). The gauge\ncondition χis defined with respect to these rescaled variables.\nWe have also changed units to set G= 1/(8π).whereN≈Λ. Notice that the terms proportional to\nw0constitute a modification of the standard Friedmann\nequations, which can indeed be interpreted as a con-\nstant equation-of-state parameter Wof an effective fluid.\nThus, in particular, by assuming w0/Λ0>0, we have:\neffective dust (dark matter) for W= 0, for which we\nrecover the gauge N≈Λ0;51an effecitve cosmological\nconstant (dark energy) for W=−1 (which corresponds\nto the unimodular gauge condition in the parent theory);\nan effective curvature contribution for W=−1/3; and\neffective radiation contribution for W= 1/3.\n4. Time from constants of Nature\nAs in Secs. IIIBandIIIE2, there are certain cases\nin which time can be seen as the canonical conjugate\nto a constant of Nature also in the Fock–Stueckelberg\napproach to general relativity (generalized unimodular\ngravity). For example, in the case of homogeneous cos-\nmologywithaconstanteffectiveequation-of-stateparam-\neter, the Fock–Stueckelberg equations of motion are [cf.\nEq. (30)]\n˙f={f,Λ0a3Wφ⊥} (111)\nfor any phase-space function f≡f(a,ϕ), and the Fock–\nStueckelberg Hamiltonian is conserved but not necessar-\nily zero (the constraint is relaxed),\nΛ0a3Wφ⊥≈ −w0, (112)\ndue to the Fock–Stueckelberg constraint φ⊥+w≈0 [cf.\nEqs. (107), (109) and (110)] and the fact that w0is a\nspacetime constant that need not vanish (if it vanishes,\nthen the theory is identical to the ordinary homogeneous\ncosmology derived from general relativity).\nAre the results in Eqs. ( 111) and (112) also obtained\nby promoting a constant of Nature to a physical field\nconjugate to time? Instead of obtaining an effective fluid\nvia the Fock–Stueckelberg approach to general relativ-\nity with a scalar field, let us consider ordinary general\nrelativity coupled to a scalar field and a fluid with en-\nergy density given by E=E0a−3(W+1)(i.e., the fluid is\nnow already present in the parent theory but it is treated\nas a “potential term” dependent on a). The equations of\nmotion of the corresponding homogeneous cosmology are\n˙f={f,Nφ⊥+Na3E},\n0≈a3Wφ⊥+E0,(113)\nwithf≡f(a,ϕ) being a phase-space function. As E0\nis a conserved quantity (constant of Nature) in this toy\n51A generalized unimodular gravity with effective dust matter and\nN= 1 gauge was also considered in the book [98] and the series\nof articles cited therein.21\nmodel, we are thus in a similar situation to the rela-\ntivistic particle case discussed in Sec. IIIE2, with E0\nbeing analogous to the particle’s rest energy. In this\nway, we can promote pη:= Λ0E0to a new, conserved de-\ngree of freedom, and we enlarge the set of variables from\n(a,pa,ϕ,pϕ) to include the new canonical pair ( η,pη) by\nadding to the action the new term pη˙η. As in the parti-\ncle case, ηis a cyclic variable that serves a physical clock\nthatdefinestimeasacanonicalconjugatetoaconstantof\nNature. By imposing the time parametrization η(τ) =τ,\nwe fix the lapse function to be N= Λ0a3W, and we can\nsolve the constraint a3Wφ⊥+E0≈0 forE0to obtain the\non-shell gauge-fixed equations of motion\n˙f={f,Λ0a3Wφ⊥}, (114)\nwith Λ 0a3Wφ⊥being a conserved quantity. This is iden-\ntical to the Fock–Stueckelberg results in Eqs. ( 111)\nand (112), which shows the equivalence of the Fock–\nStueckelberg approach with the definition of time as a\ncanonicalconjugatetoaconstantofNatureinthiscase.52\nThe unimodular gravity case with W=−1 is a partic-\nular case of the equivalence of the Fock–Stueckelbergand\nconstantsofNature approachesnot only in the contextof\nhomogeneous cosmology. In the full field-theoretic case,\na similaranalysiscanbe carriedout for unimodular grav-\nity. We find from Sec. IIIG2that the Fock–Stueckelberg\nfield equations with W=−1 read [cf. Eqs. ( 30) and\n(104) with explicit spatial integrals]\n˙f=/braceleftBigg\nf,/integraldisplay\nd3y/parenleftbiggΛ0√\nhφ⊥+Naφa/parenrightbigg\n(τ,y)/bracerightBigg\n,\nφa≈0,(115)\n52It is worthwhile to mention that promoting pη:= Λ0E0to a new\ndegree of freedom with respect to which the Hamiltonian con-\nstraint can be solved corresponds to acknowledging that the en-\nergy contribution from homogeneous perfect fluid can be trea ted\nnot as a mere “potential term” dependent on a, but rather as a\nnew genuine variable. This is in line with the Schutz formali sm\nfor perfect fluids [99], which is frequently used in the conte xt of\ncanonical cosmology of homogeneous models in order to simpl ify\nthe solution to the Hamiltonian constraint (see, e.g., [100 –103]\nand references therein fordetails and applications). In a n utshell,\nthis formalism allows us to decompose the fluid’s four-veloc ity\nin terms of velocity potentials, Uµ= 1/h(∂µǫ+ζ∂µβ+θ∂µs),\nwhich are related to the description of the fluid. For example ,h\nis the specific enthalpy, sis the specific entropy and ζandβare\nconnected with circulation (vortex motion) and they vanish in\nisotropic models. With this decomposition and thermodynam -\nical relations, one can establish a series of canonical tran sfor-\nmations to establish that the Hamiltonian of the homogeneou s\nperfect fluid can be written as NpTa−3W, where pTis a mo-\nmentum canonically conjugate to a fluid variable related to i ts\nspecific entropy and other potentials. This clearly agrees w ith\nthe Fock–Stueckelberg and constants of Nature approaches i f we\nidentify pTwithpηand take Λ 0= 1. Thus, one can say that,\nin this cosmological setting, the Fock–Stueckelberg, cons tants of\nNature and Schutz formalisms are equivalent when it comes to\nthe relaxation of the Hamiltonian constraint.withfbeingaphase-spacefunction, andΛ 0φ⊥/√\nhbeing\na conserved quantity due to Eq. ( 99) with W≡ −1. The\nsame results can be obtained by starting with the par-\nent theory being general relativity with units in which\nG= 1 and with a cosmological constant E(which can\nbe regarded as a constant of Nature, insofar as it is\na spacetime constant and one does not consider more\ncomplicated dark energy models), and subsequently pro-\nmoting Λ 0E/8πto a conserved field canonically conju-\ngate to a field η(τ,x). As before, we define time via\nthe condition η(τ,x) =τ, which fixes the lapse to be\nN= Λ0/√\nh. We can solve the Hamiltonian constraint\nφ⊥+E√\nh/8π≈0 forE, which leads to on-shell partially\ngauge-fixedfield equationsthatcoincidewith thoseofthe\nFock–Stueckelberg approach given in Eq. ( 115).\nIV. THE REAPPEARANCE OF TIME IN\nQUANTUM GRAVITY\nThe promotion of constants of Nature to spacetime\nfieldsandtheassociateddeparametrizationofconstraints\ndiscussed in Sec. IIIB, which yields equivalent results to\nthe Fock–Stueckelbergapproachat least in regionswhere\nthe deparametrization is valid (cf. Sec. IIIB6), leads to\na strategy to solve the “problem of time” in quantum\ngravity. This problem is a consequence of spacetime dif-\nfeomorphism invariance (which is a local symmetry or\na canonical gauge symmetry), and it concerns the lack\nof a preferred time parameter in general relativity with\nrespect to which the evolution of the wave functional of\ngravitational and matter fields can be described.\nThe Dirac quantization of general relativity’s con-\nstraints leads to timeless equations (cf. Sec. IIIG1)53\nˆφ⊥|Ψ/an}b∇acket∇i}ht= 0,ˆφa|Ψ/an}b∇acket∇i}ht= 0, (116)\nwhich imply a timeless constraint ˆH|Ψ/an}b∇acket∇i}ht=λa(τ)ˆφa|Ψ/an}b∇acket∇i}ht=\n0 (spatial integration and summation over discrete in-\ndices implied). Due to the structure of φ⊥(cf. Sec.\nIIIG1), thetimelessquantumconstraintsleadtoagener-\nalizationoftheKlein–Gordonequationfortherelativistic\nparticle, and thus the problem of time in quantum grav-\nityis inevitablylinkedto difficulties with thedefinition of\nthe Hilbert space.54In fact, the Hamiltonian constraint\nof canonical quantum cosmology of homogeneous mod-\nels coincides with that obtained from mechanical models\ndiscussed in Sec. IIIE, which is nothing but a Klein–\nGordon equation for a relativistic particle with a general\n53The quantum constraints aretimelessinthe sensethatno exp licit\ntime parameter appears, although of course one may attempt t o\ninterpret one of the physical degrees of freedom as an “inter nal”\ntime [11].\n54See, however, [104, 105] for way a promising way to define a\npositive-definite inner product at least for symmetry-redu ced toy\nmodels.22\npotential anda generalbackground(configuration-space)\nmetric.\nIn contrast, the relaxation of the (Hamiltonian) con-\nstraint obtained with the Fock–Stueckelberg or con-\nstants of Nature approaches leads to the ordinary time-\ndependent (functional) Schr¨ odinger equation\ni/planckover2pi1∂\n∂τ|Ψ/an}b∇acket∇i}ht=ˆH|Ψ/an}b∇acket∇i}ht, (117)\ndue to the fact that the time translations are no longer\ngauged. One can thus speak of the “reappearance\nof time,” which is nothing but the breaking of space-\ntime diffeomorphism invariance engendered by the Fock–\nStueckelberg construction. It is important to emphasize\nagain that this construction is not related to the consis-\ntency of the quantization procedure itself, but rather to\nthe definition of a different classical theory to be subse-\nquently quantized. It makes the quantization more con-\nceptually and technically straightforward at the cost of\nthe theory’s local symmetry, and it was one of the moti-\nvations for Stueckelberg to develop his approach for the\nrelativistic particle (by breaking the worldline diffeomor-\nphism invariance), as well as for modern researchers. In-\ndeed, fromthe“original”unimodulargravitytoitsgener-\nalizations[14–17, 38, 83, 86–88], and with severalpropos-\nals to define time as the canonical conjugate to different\nconstants of Nature [13, 18–27, 40–42, 63], this strategy\nhasledtoaseriesofarticlesclaimingtosolvetheproblem\nof time in quantum gravity.\nIt is worthwhile to note, however, that this solution\nto the problem of time entails the introduction of new\nphysical fields, one of which plays the role of a (possibly\nglobal) physical clock. Whether such a field indepen-\ndently exists in Nature is an open question, and other-\nwise the formalism risks being a mere effective treatment\nof different physical clocks, each of which is valid only in\ncertain regions, in which case the simplification brought\nby the deparametrization relative to constants of Nature\nseemsvacuous–the samedifficulties appearwith the tra-\nditional gauge fixing of the zfields; i.e. the traditional\nproblem of multiple choices of clocks. If no such inde-\npendent, new physical clock field can be measured, this\nsolution to the problem of time is at risk of being no so-\nlution at all. Instead of achieving a consistent and phys-\nically meaningful implementation of time reparametriza-\ntions in the quantum theory of general models, the Fock–\nStueckelberg and constants of Nature approaches simply\nbreakthesymmetry55thatcausestheprobleminthefirst\nplace,andthustheymaybeawayofavoidingratherthan\nsolving the problem.\n55The symmetry is broken in the sense that the Fock–Stueckelbe rg\naction in Eq. (31) or the action in Eq. (48) that appears in the\nconstants of Nature approach display a reduced (or residual ) set\nof symmetries with respect to the original action in Eq. (4).V. CONCLUSIONS\nGenerally, one can say that in a theory with gauge\nsymmetries or in a theory with first-class constraints,\nthe relaxation of some of these constraints implies the\nappearance of new physical degrees of freedom. This is\nthe essence of the Fock–Stueckelberg approach that we\nhave discussed in this article. The idea is rather simple,\nand our goal was to present a comprehensive overview\nof this method, which is often independently rediscov-\nered in the literature, and many sources do not include\na discussion of the origins and physical meaning of the\nformalism.\nThe physical meaning of the Fock–Stueckelberg theory\nwas best captured by Dirac in his own proposal of a new\nelectrodynamics [53]: if we do not know the structure\nanddynamicsofthe “sources”ofgaugefields(suchasthe\nelectrons in electromagnetism or certain kinds of matter-\nenergy in gravity), we can consider breaking the gauge\nsymmetry, and promoting some of the “spurious” gauge\ndegrees of freedom to physical fields that describe or de-\nfine the sources themselves. So are Dirac’s “electrons\nwithout electrons,” and the “matter without matter” of\ngeneralized unimodular gravity, which can lead to mod-\nels of dark matter and dark energy without details about\ntheir internal structure. So is the independent mass\n“field”, which leads to an independent “proper time”\nvariable, in Fock’s and Stueckelberg’s relativistic parti-\ncles, which do not suffer from the traditional problem of\ntime of Klein–Gordon-like Hamiltonian constraints, and\nthus they inspire avoidances of this problem in quantum\ngravity.\nAlthough other approaches can consider a breaking of\ngauge symmetries, the Fock–Stueckelberg theory does so\nin an economical way: it promotes a gauge-fixing condi-\ntion to a physical equation on a par with the first-class\nconstraints in the action. This breaks the symmetry by\nrelaxing the original constraints φa, but if we interpret\nthe newdegreesoffreedom wasanew formofmatter, we\ncan consider that new constraints arise, φa+w≈0 [cf.\nEq. (29)]. The consistency of the new Fock–Stueckelberg\nconstraints has to be analyzed. In certain cases, the new\ndegreesoffreedomcanbeidentified withconstantsofNa-\nture promoted to physical fields (cf. Sec. IIIB), and the\nnew constraints φa+w≈0 might still generate the orig-\ninal symmetry, although they are to be deparametrized\nwith respect to the wfields or the constants of Nature,\nthereby freeing the restriction on the values of φa.\nIt is clear that the Fock–Stueckelberg approach gener-\nates a classical theory, possibly different from its parent\ntheory. The question of quantization is secondary (the\nFock–Stueckelberg theory can be subsequently quantized\nwith the usual methods, from Dirac to BRST quantiza-\ntion [8, 9]). This goesagainst somerecent claims made in\n[13, 38, 39] that fixing the gauge ` a la Fock and Stueckel-\nberg(i.e., fixingthegaugeattheleveloftheactionbefore\nvariation)is necessaryin orderto make the quantum the-\nory or, more precisely, the quantum Hamiltonian and the23\npath integral of gauge theories well defined. Those can\nbe well defined, presumably, via the usual methods, pro-\nvided field-theoretic divergences (which come from func-\ntional derivatives, for example) can be regularized, an is-\nsueaboutwhichtheFock–Stueckelbergapproachremains\nsilent. Thus, the physicalmeaning ofthis formalismneed\nnotbetiedtotheconsistencyofthequantizationofgauge\ntheories, and the fundamental quantum theory need not\nbreak gauge invariance. However, it is noteworthy that,\nsince the Fock–Stueckelberg mechanism often leads to a\nmore straightforward quantization, it is often motivated\nwith that end in mind since the original works of Fock\nand Stueckelberg [28–31].\nSo what is the value of the Fock–Stueckelberg formal-\nism? Its value lies in its ability to simplify a local sym-\nmetry to a residual gauge invariance by defining a kind\nof “shadow” matter (such as Dirac’s electrons) from the\noriginal “spurious” gauge degrees of freedom. It frees\nthe theorist, as it did Dirac, to explore new phenomenol-\nogy without introducing many new complications, and,\nas discussed above, it may even lead to a solution to\nthe problem of time that is somewhat trivial. Although\nadvantageous, this economy comes at the cost of postu-\nlating new fields, and eventually it might become neces-\nsary to ascertain whether such degrees of freedom reallyexist and what their fundamental structure is. 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Mavromatos ,1, 2,‡\nTheodoros Nakas ,1,§and Eleftherios Papantonopoulos1,¶\n1Physics Division, School of Applied Mathematical and Physical Sciences,\nNational Technical University of Athens, Zografou Campus, Athens 15780, Greece\n2Theoretical Particle Physics and Cosmology group, Department of Physics, King’s College London, London WC2R 2LS, UK.\nWe consider higher-order derivative gauge field corrections that arise in the fundamental context of dimen-\nsional reduction of String Theory and Lovelock-inspired gravities and obtain an exact and asymptotically flat\nblack-hole solution, in the presence of non-trivial dilaton configurations. Specifically, by considering the gravi-\ntational theory of Euler-Heisenberg non-linear electrodynamics coupled to a dilaton field with specific coupling\nfunctions, we perform an extensive analysis of the characteristics of the black hole, including its geodesics for\nmassive particles, the energy conditions, thermodynamical and stability analysis. The inclusion of a dilaton\nscalar potential in the action can also give rise to asymptotically (A)dS spacetimes and an effective cosmolog-\nical constant. Moreover, we find that the black hole can be thermodynamically favored when compared to the\nGibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) black hole for those parameters of the model that\nlead to a larger black-hole horizon for the same mass. Finally, it is observed that the energy conditions of the\nobtained black hole are indeed satisfied, further validating the robustness of the solution within the theoretical\nframework, but also implying that this self-gravitating dilaton-non-linear-electrodynamics system constitutes\nanother explicit example of bypassing modern versions of the no-hair theorem without any violation of the\nenergy conditions.\nCONTENTS\nI. Introduction 2\nII. String inspired non-linear electrodynamics 3\nIII. Theoretical framework and black-hole solutions 6\nIV . Geodesics and energy conditions 9\nA. Geodesics 9\nB. Energy Conditions 11\nV . Thermodynamic analysis 12\nA. GMGHS black hole 12\nB. Black hole with non-linear electrodynamics 13\nVI. Linear Perturbations 17\nA. Radial Stability 17\nB. Scalar Quasi-Normal Modes 18\nVII. Other Solutions 20\nA. Asymptotically (A)dS spacetimes 20\nB. Solutions for general γ 20\nVIII. Conclusions 21\nAcknowledgements 22\nReferences 22\n∗a.bakop@uoi.gr\n†thanasiskarakasis@mail.ntua.gr\n‡mavroman@mail.ntua.gr\n§theodoros.nakas@gmail.com\n¶lpapa@central.ntua.grarXiv:2402.12459v1  [hep-th]  19 Feb 20242\nI. INTRODUCTION\nIn the pursuit of a comprehensive understanding of gravitational phenomena in the cosmos as well as gravity itself, the theo-\nretical examination of black holes stands as an essential frontier. The General theory of Relativity (GR), while highly successful\nin describing the macroscopic behavior of these celestial entities, becomes subject to scrutiny under extreme conditions. This\ninvestigation prompts the exploration of modified gravitational theories and theories with extra dimensions. Among the theories\nattempting to unify fundamental interactions, String Theory stands as the leading contender. In particular, the heterotic string\ntheory stands as an essential branch within the broader scope of String Theory, distinguished by its capability to unify gravita-\ntional interactions with other fundamental forces. Notably, it excels in synthesizing these interactions into a cohesive framework.\nA focal point of interest lies in the derivation of an effective four-dimensional theory, offering insights into quantum corrections\nthat modify Einstein’s theory of gravity. These corrections can potentially incorporate terms ranging from the Gauss-Bonnet,\nquadratic-curvature term [1–4] to non-linear electromagnetic corrections (see e.g. [5–8] and references within).\nDrawing therefore inspiration only from the aforementioned corrections introduced by string/brane theory, in this article, we\naim to elucidate the implications of departing from the conventional electromagnetic framework and embracing the intricacies\nof non-linear electrodynamics (NED) and scalar fields within the context of black hole solutions. In addition to the above, it is\nimportant to mention that four-dimensional scalar-vector-tensor theories can be also obtained via an appropriate reduction from\na higher-dimensional Lovelock theory [9]. Under this perspective, scalar-tensor-vector theories can be understood as natural\nextensions of the scalar-tensor theories that have been extensively studied in the last decades [4, 10–51]. Such scalar-vector-\ntensor theories offer a very fruitful framework for finding novel compact-object solutions, eluding the constraints imposed by\nthe “no-hair” theorems [52–81]. This departure from traditional limitations is attributed to the dilaton field, which, notably,\nintroduces no additional independent free parameter into the resulting solution [82]. Instead, these solutions manifest a distinct\nfeature known as “secondary hair”, intricately determined by the compact object’s mass, charge, and angular momentum.\nDespite the motivation coming from higher-dimensional theories, there are also additional reasons that lead us to explore non-\nlinear electrodynamics. First and foremost, in regions with strong gravitational fields, such as those near black holes, traditional\nlinear theories may break down. Non-linear electrodynamics becomes important in these strong field regimes, where the intensity\nof electromagnetic fields can become comparable to the strength of gravitational fields. Studying how non-linearities affect the\nbehavior of electromagnetic fields in these regimes is crucial for understanding the physics of objects like black holes, neutron\nstars, and other astrophysical phenomena. Moreover, non-linear electrodynamics is expected to lead to phenomena that are\nabsent in linear theories. In the early universe for example, when energy densities were extremely high, the interplay between\ngravitational and electromagnetic fields was significant. NED can be crucial in modeling the behavior of these fields during\ncosmological evolution. Consequently, NED allows us to investigate how electromagnetic interactions influenced the dynamics\nof the early universe and whether non-linear effects played a role in the formation of cosmic structures. Understanding these\ncosmological implications helps build a more complete picture of the evolution of the universe. For a review on non-linear\nelectrodynamics and its applications, see [83] and references therein.\nIn this paper, we embark on a comprehensive exploration of a gravitational theory extending the classical Euler-Heisenberg\n(EH) electrodynamics coupled to a non-trivial dilaton field. Our motivation for this study stems from the rich theoretical\nlandscape it promises, building upon the established framework of self-gravitating dilaton-linear-electrodynamics. This exten-\nsion allows us to delve into intriguing phenomena, notably exemplified by the Gibbons-Maeda-Garfinkle-Horowitz-Strominger\n(GMGHS) black hole [84, 85], a significant exact solution within this domain. In our investigation, we examine the intricacies\nof our proposed model and unravel its associated black-hole solution in detail. One of our key insights lies in the strategic\nassumption of a specific profile governing the dilaton coupling to the Euler-Heisenberg terms. This choice results in an exact\nanalytic black-hole solution, facilitating a straightforward examination of its physical characteristics.\nHaving the solution at hand, we then commence a rigorous analysis encompassing various facets of our model’s implications.\nThis includes a thorough examination of the geodesics of massive test particles within the black-hole spacetime, followed by\na meticulous scrutiny of the energy conditions. Subsequently, we delve into the thermodynamic aspects of the black hole,\ncomputing the relevant thermodynamic quantities, such as the temperature, the entropy, and the magnetic potential ( Φm), to\ndemonstrate the validity of the first law of thermodynamics. Moreover, within the parameter space of solutions, we unveil the\nexistence of pairs consisting of two distinct black holes characterized by different ratios Qm/M, both more compact than the\nSchwarzschild solution yet sharing identical horizon radii. Intriguingly, despite their geometric similarity, a thermodynamic\nanalysis reveals clear distinctions, with one black hole exhibiting thermodynamic stability while its doppelg ¨anger proves to be\nthermodynamically unstable. Additionally, we explore the radial stability of the black-hole solution under linear perturbations\nand also its scalar quasi-normal modes, shedding light on its potential as an astrophysical entity. Furthermore, we extend\nour discussions to encompass other solutions and extensions of our model theory, including asymptotically (Anti-)de Sitter\n(AdS) spacetimes and more general dilaton couplings, providing a comprehensive overview of the theoretical landscape. In\nconclusion, our work offers a thorough investigation into the gravitational theory of non-linear EH electrodynamics coupled to a\nnon-trivial dilaton field, unraveling a plethora of intriguing phenomena and paving the way for further exploration and theoretical\nadvancements in this domain.\nThe structure of the current article is the following: In the next section II we motivate our study by discussing a gravita-3\ntional theory of non-linear Euler-Heisenberg (EH) electrodynamics coupled to a non-trivial dilaton field. This generalises the\ncorresponding self-gravitating dilaton-linear-electrodynamics case, known to admit the GMGHS black hole [84, 85] as an exact\nsolution. In the subsequent section III, we discuss our model and its associated black hole solution. By assuming a specific\nprofile for the dilaton coupling to the EH terms, in such a way that, additionally to non-trivial dilaton couplings, one has also\ndilaton-independent EH terms, we demonstrate the possibility of studying analytically the corresponding black-hole solution. In\nsection IV we first discuss the geodesics of test particles in such black hole spacetimes, and then demonstrate the satisfaction of\nthe energy conditions for appropriate sets of the parameters of the solution. In section V, we study the thermodynamics of the\nblack hole, and show explicitly, by computing the relevant thermodynamical quantities, that the first law of thermodynamics is\nsatisfied in a coordinate-independent way, as should have been expected. In the parameter space of solutions, it is possible to\nobtain two distinct black holes with different ratios Qm/Mthat are more compact than the Schwarzschild solution and share\nthe same horizon radius. However, these black holes even though they have the same horizon radius, from a thermodynamic\npoint of view, are quite distinguishable, since the solution with a greater value for the ratio Qm/Mis thermodynamically stable,\nwhile its doppelg ¨anger with a lower value for the ratio Qm/Mis thermodynamically unstable. In section VI, we demonstrate\nthe radial stability of the black-hole solution under linear perturbations, and study its scalar quasi-normal models, which provide\ninsights into its properties as a potential astrophysical object. Other solutions of (extensions of) our model theory (3.1), including\nasymptotically (Anti-)de Sitter (AdS) spacetimes, as well as solutions corresponding to more general couplings exp(−2γϕ), of\nthe dilaton to the Maxwell term in the action, rather than the γ= 1 in closed strings, are discussed in section VII. Finally,\nconclusions and outlook are given in section VIII.\nII. STRING INSPIRED NON-LINEAR ELECTRODYNAMICS\nOne of the particular aspects of string/brane-induced non-linear electrodynamics effects is that the higher order in the Maxwell\ntensor can be combined into an all-order expression, the so-called Born-Infeld (BI) Lagrangian [86–91], as a result of re-\nsummation of open string excitations (attached to, e.g., 3-brane worlds in the D-brane extension of string theory, in which case\nthe world-volume of ( d= 3)-brane leads to the DiracBI (DBI) action (see [92–94] and references therein). In such models,\nthe BI electrodynamics in four spacetime dimensions originates from the higher ( d= 10 )-dimensional superstring action upon\neither compactification or appropriate restriction on a 3d-brane volume. It is important to note that in all such string-inspired\nmodels the BI Lagrangian couples to the inverse of the open-string coupling gs=eϕ, where ϕis the (dimensionless) dilaton\nfield, so the corresponding four-dimensional action in a curved four-dimensional background metric (in the Jordan or σ-model\nframe), gJ\nµν, reads\nSJ\nBI=−T2\n4Z\nd4x e−ϕr\nDet\u0010\n−gJµν+T−1\n4Fµν\u0011\n, (2.1)\nwhere Fµνis the Maxwell tensor Fµν=∂µAν−∂νAµ, andT4=1\n2πα′=M2\ns\n2π,is the (open) string tension, with α′=M−2\ns\nthe Regge slope ( Msthe string mass scale, which in general is different from the four-dimensional Planck scale). One may go\naway from string/brane theory and define the BI action as a starting point of an effective modified electrodynamics field theory.\nIn such a case the string tension T4may be considered as an arbitrary phenomenological dimensionful parameter which is not\nrelated to the Regge slope α′. We term such a parameter the BI parameter .\nWe next remark that, in four spacetime dimensions, the determinant in the argument of the square root in the BI action (2.1)\ncan be expanded exactly to yield:\nSBI=−T2\n4Z\nd4xp\n−gJe−ϕs\n1 +1\n2T2\n4FµνFµν−1\n16T4\n4\u0010\nFµνeFµν\u00112\n(2.2)\nwhereeFµν=1\n2εµνρσFρσis the dual of the Maxwell tensor, with εµνρσ the Levi-Civita fully antisymmetric symbol in curved\nspacetime with metric gJ\nµν. Expanding the (square root in the) four-dimensional BI action (2.2) in inverse powers of the BI\nparameter T4, leads to effective dimension 8(and higher) operators in the effective field theory, which make contact with the\ngeneric Euler-Heisenberg (EH) non-linear electrodynamics (NED) [91, 95, 96]:\nSBI=Z\nd4xp\n−gJe−ϕh\n− T2\n4I2− T4\n4I4\u0010\n1 +O(F2)\u0011i\n,\nI2=1\n4T2\n4FµνFµν, I 4=−1\n8T4\n4FµνFνρFρλFλµ+1\n32T4\n4\u0010\nFµνFµν\u00112\n. (2.3)\nHence, ignoring for the moment the dilaton, to fourth order in the field strength Fµνone obtains (up to the dilaton factors) a4\nspecial case of the generic EH NED with dimension 8operators, with Lagrangian:\nLEH=c1\u0010\nFµνFµν\u00112\n+c2FµνFνρFρλFλµ, (2.4)\nwhere the BI Lagrangian corresponds to [95, 96]\nc1=−1\n32T2\n4, c 2=1\n8T2\n4. (2.5)\nThe reader should notice that the ratio of c2/c1=−4exactly, which is a characteristic prediction of the BI theory.\nPhenomenologically, assuming a constant dilaton and flat Minkowski spacetime, the BI parameter T2\n4can be constrained in\ncollider physics, via light-by-light scattering, for which there is clear experimental evidence these days at LHC experiments (see\n[97–99]). Such light-by-light scattering studies [95] can place a lower bound on the BI parameter T4≳100 GeV . In the case\nof string theory, this would lead to a (weak) lower bound of the string mass scale Ms≳0.25 TeV . Notably, extra dimension\ncollider (LHC) searches place currently this bound to Ms≳O(10) TeV . Forecasts for much larger values of the lower bounds\nof the BI parameter in future colliders, in particular FCC, have been given in [96]. Embedding the BI (or more generally Euler-\nHeisenberg) theory into curved spacetime, and fully incorporating the dilaton effects, leads to a whole new area of tests of NED\nby employing the entire machinery of modern gravitational experiments technology.\nThe BI action SBI(2.1) and (2.2) in curved background metrics can be augmented, at an effective field theory level, by\nincluding the dynamics of the gravitational ( gJ\nµν) and dilaton ( ϕ) fields. In this respect, we recall that the D-brane action\nis by construction in the so-called Jordan (or σ-model) frame. Passing into the Einstein frame in four dimensions, via the\ntransformation of the metric: gJ\nµν→gµν=e−2ϕgJ\nµν, we write for the pertinent gravitational action (in geometrized units\nc=G= 1, in which we work from now on):\nS=1\n16πZ\nd4x√−gh\nR −2∇µϕ∇µϕi\n−Z\nd4x√−g e−ϕh\nT2\n4IE\n2+T4\n4e−4ϕIE\n4i\n+. . . , (2.6)\nwhere the quantities IE\ni, i= 2,4are given by the corresponding ones in (2.3), but the indices contraction is made by the\nEinstein-frame metric gµν.\nDeparting from the case of the brane DBI action (2.1), one may consider higher-order (in derivatives, that is in a Regge slope\nα′expression) electromagnetic terms in effective low energy field theories stemming only from closed strings, e.g. the heterotic\nstring [1]. In such theories, unlike the DBI brane or open-string case, there is no resummation in closed form of the gauge terms.\nNonetheless, some authors have generalised the BI effective action in a curved (3 + 1) -dimensional spacetime, by considering\nthe following form of dilaton couplings to the electromagnetic fields in a BI NED setting [100–102]:\nS=1\n16πZ\nd4x√−gh\nR −2∇µϕ∇µϕ+LBIi\n,\nLBI= 4βBIe2γϕ\u0010\n1−s\n1 +e−4γϕ\n2βBIF2−e−8γϕ\n16β2\nBI(FeF)2\u0011\n(2.7)\nwhere the notation has been defined above, γdefines the dilaton coupling, and now βBIplays the role of the generalised BI\nparameter, with mass dimensions +2(which is identified with T2\n4in the case of strings, in which case, to match with the\ncorresponding O(α′)Maxwell terms of the heterotic-string effective action [1], e−2ϕF2, one should fix γ= 1).\nThe above considerations deal with tree-level in string loops, that is first quantised actions on world-sheet with trivial topology\n(2dsphere for closed string sectors, and disc for open one). In general, string loop effective actions are not known in closed\nform. In simplified phenomenological scenarios such effective actions can be expressed in the generic form, e.g. in the closed\nstring sector in the string (or σ-model frame with metric bgµνin (3 + 1 )-dimensions, after string compactification) [103]:\nS=Z\nd4xp\n−bg\u00101\nα′Bg(Φ)bR+1\nα′BΦ(Φ)h\nb2Φ−4b∇µΦb∇µΦi\n−BF(Φ)\n4bFµνbFµν−Bψ(Φ)bψ/bDbψ+. . .\u0011\n, (2.8)\nwhere thed(...)symbol above a tensorial quantity implies contraction of the world-indices with the string-frame metric bgµν,Fµν\ndenotes the field strength of the gauge field, Dµis the gauge covariant derivative, ψare fermionic matter fields and the . . .denote\nother matter fields as well as (an infinity of) higher-derivative (higher order in α′) terms, The quantities Bi(Φ), i=g,Φ, F, ψ are\nnon-derivative functions of the dilaton which arise from summing over (closed) world-sheet topologies, that is these functions\ninvolve powers of the string coupling gs= exp(Φ) of the form g−χ\ns, where χ= 2−2Nwhere Ndenbotes the number of\nhandles, is the genus of the world-sheet surface (sphere has N= 0, torus (one string loop) = 0etc. Thus,\nBi(Φ) = e−2Φ+c(i)\n0+c(i)\n1e2Φ+···+c(i)\n2ne2nΦ+. . . , (2.9)5\nwhere the constant quantities cipertain to effects of string loops, so that the expressions (2.9) involve a power series in the square\nof the string coupling g2\ns= exp(2Φ) . The first term on the right-hand side of (2.9) leads to the standard closed string expression\nfor the gauge field Maxwell terms in the low-energy effective action, e−2ϕF2for standard dilaton kinetic term normalization in\nthe Einstein frame [1–3]).1\nPassing to the Einstein frame, via appropriate redefinitions of [103]: the metric bgµν→gµν=C Bg(Φ)bgµν, where Care\nnumerical normalization constants, the dilaton Φ→ϕ=R\ndΦr\n3\n4\u0010B′g\nBg\u00112\n+ 2(B′\nΦ\nBΦ+BΦ\nBg), where the prime denotes d/dΦ, and\nthe fermionic matter fields, bψ→ψ=C−3/4B−3/4\ngB1/2\nψbψ, leaving the gauge fields as they are, yields the effective action:\nS=Z\nd4x√−g1\n16πGR−1\n8πG+Smatter ,\nSmatter =Z\nd4x√−gh\n−ψ/Dψ−1\n4BF(Φ)FµνFµν+. . .i\n, (2.10)\nwhere the reader should notice the potential existence (depending on the specific type of string theory considered) of constant\n(dilaton-independent) terms involving F2terms ( cf.thec(F)\n0terms on the right-hand side of (2.9)).\nIn case we consider more general theories involving a combination of closed and open strings (the latter attached, e.g. to\nbrane universe)s, for which one obtains effective actions in the Einstein frame that include both closed- and open-string sectors\n(the latter leading to DBI terms of the form appearing in the second integral on the right-hand side of (2.6)), then, the inclusion\nof string loops can lead, following similar arguments to the closed-string case (2.10), to generalised situations, in which the\n(string-loop corrected) effective action acquires the form in the Einstein frame [104]:2\nS=1\n16πZ\nd4x√−gh\nR −2∇µϕ∇µϕi\n−Z\nd4x√−g BF2(ϕ)h\nT2\n4IE\n2i\n−Z\nd4x√−gT4\n4BF4(ϕ)IE\n4i\n+. . . , (2.12)\nwhere the functions BFi(ϕ), i= 2,4admit a power series expansion in the string coupling, summing up terms of the generic\nform\nBFi(ϕ) =X\nχg−χ\nsc(Fi)\nχ, i = 1,2, g s= exp( ϕ), (2.13)\nwhere χ= 2−2N−NH, with NHthe number of holes (or boundaries) (eg disc has genus χ= 1, since N= 0,NH= 1etc),\nwhere finite parts of dilaton tadpoles contribute to the coefficients c(Fi)\nχ.\nIn heterotic strings, which do not involve branes, the higher derivative EH electrodynamics terms do not appear in a closed\nBDI form. In that case a more general action, involving EH terms, after summation over string loops, might then be considered,\nin the Einstein frame:\nS=1\n16πZ\nd4x√−gh\nR −2∇µϕ∇µϕi\n−Z\nd4x√−g BF2(ϕ)h\nT2\n4IE\n2i\n−Z\nd4x√−gT4\n4BF4(ϕ)LEH+. . . , (2.14)\nwhere the ellipsis ( . . .) includes possible string-loop generated dilaton-potential terms, whose precise form is not known at\npresent, as this is a highly string-model-dependent issue, and the functions BF2(ϕ),BF4(ϕ)in this case are given by power\nseries expansions of even powers of the string coupling, of the form (2.9), as only closed world-sheet surfaces are involved. The\nEuler-Heisenberg Lagrangian LEHis given by (2.4), but the coefficients ci, i= 1,2no longer satisfy (2.5), given that the DBI\naction no longer describes the electromagnetic self-interactions in closed form. In this work we shall use the framework (2.14)\nto discuss black hole solutions in a phenomenological manner, keeping the coefficients ciof the EH Lagrangian as arbitrary. As\nalready mentioned, in such a framework, for some sufficiently high string-loop order one can conjecture that there would exist a\ndilaton independent term in the function BF4(ϕ). As we shall discuss in the next section, such a constant term plays a crucial\nrˆole in our phenomenological analysis in this paper in yielding analytic black hole solutions of a dilaton-EH theory that could be\nthe low energy limit of an appropriate underlying string theory. However in our work, we shall be more general, and our analysis\nwill be presented independent of strings.\n1We note for completion that a similar factor accompanies the quadratic gravitational curvature (Gauss-Bonnet (GB)) terms in the action at string-loop tree\nlevel. This is a remnant of the corresponding situation of the ten-dimensional target-spacetrime heterotic-string effective field theory action, which in the extra\n(compact) dumensional sector leads to the celebrated anomaly cancellation by equating the extra-dimensional (non-Abelian) gauge with the corresponding\nquadratic-curvature gravitational GB termsR\nd6xp\n−G(6)e−2Φ\u0010\nTrF2−R2\nGB\u0011\n→0(with the Tr being a group-index trace), which leads to the Heterotic\nstring selecting the E8×E8gauge group as the unique target-space group before compactification to (3+1)-dimensions [2, 3].\n2Indeed, if only Abelian gauge fields are considered then only open world-sheet surfaces are taken into account, in order to evaluate the pertinent contribution\nto the effective action, as discussed explicitly in [104] where it was shown that the loop corrected effective action acquires the form in the sigma-model frame\n(ignoring antisymmetric tensor fields contributions, which are of no interest in the present discussion):\nS=Z\nd4xp\n−bg α′−2e−2ϕh\n−3\n2α′(bR+ 4(∂ϕ)2+. . .] +d1e−ϕr\ndet\u0010\nbgµν+ 2πα′Fµν\u0011\n+d2+d3+d4eϕ+. . . , (2.11)\nwhere di,i= 1,2, . . . , denote finite parts of the dilaton tadpoles, and the dots denote contributions from higher derivative corrections, as well as higher\nstring loops (that is higher powers of the string coupling gs= exp( ϕ). Passing onto the appropriate Einstein frame leads to actions of the form (2.12).6\nIII. THEORETICAL FRAMEWORK AND BLACK-HOLE SOLUTIONS\nIn the geometrised unit system ( c=G= 1), the Einstein-frame action functional that we will occupy us in this article is a\nsimplified version of (2.14) and reads\nS=1\n16πZ\nd4x√−gh\nR −2∇µϕ∇µϕ−e−2ϕF2−f(ϕ)\u0000\n2αFα\nβFβ\nγFγ\nδFδ\nα−βF4\u0001i\n. (3.1)\nSuch a field theoretic gravitational actions also arises as part of a non-diagonal reduction of the Gauss-Bonnet action [9] and\nadmits the GMGHS black hole [84, 85] as an exact solution when f(ϕ) = 0 . In (3.1), Ris the Ricci scalar, F2≡ F µνFµν∼\nE2−B2is the usual Faraday scalar, and F4≡ F µνFµνFαβFαβ, where Fµνstands for the usual field strength Fµν=\n∂µAν−∂νAµandα, β are coupling constants of the theory, with dimensions (length)2, which in our discussion are treated\nphenomenologically. The scalar field ϕand the associated scalar function f(ϕ)are both dimensionless.3For the moment we\ndo not consider a potential for the dilaton, but only its non-linear interactions with the EH terms. The addition of a pure dilaton\npotential V(ϕ)can lead to interesting alternative solutions, including a cosmological constant, which we discuss in section VII.\nThe field equations emanating from (3.1) are of the following form\nGµν= 2∂µϕ∂νϕ−gµν∂αϕ∂αϕ+ 2e−2ϕ\u0012\nFα\nµFνα−1\n4gµνF2\u0013\n+\nf(ϕ)\u001a\n8αFα\nµFβ\nνFη\nαFβη−αgµνFα\nβFβ\nγFγ\nδFδ\nα−4βFξ\nµFνξF2+1\n2gµνβF4\u001b\n, (3.2)\n4□ϕ=−2e−2ϕF2+d f(ϕ)\ndϕ\u0000\n2αFα\nβFβ\nγFγ\nδFδ\nα−βF4\u0001\n, (3.3)\n∂µn√−gh\n4Fµν\u0000\n2βf(ϕ)F2−e−2ϕ\u0001\n−16αFµ\nκFκ\nλFνλio\n= 0. (3.4)\nBy taking into account the higher-order electromagnetic invariants F4andFα\nβFβ\nγFγ\nδFδ\nα, we are interested in extending the\nGMGHS solution [84, 85]. To do so, we introduce the most general spherically symmetric metric ansatz in the form\nds2=−B(r) dt2+dr2\nB(r)+ [R(r)]2dΩ2, (3.5)\nwhere B(r), R(r)are two unknown functions to be determined from the field equations, while dΩ2= dθ2+ sin2θdφ2.4\nMoreover, we consider both electric and magnetic charges, via the following four-vector, which is compatible with spherical\nsymmetry,\nAµ= (V(r),0,0, Qmcosθ), (3.6)\nwhere Qmstands for the magnetic charge carried by the black hole. This ansatz for the electromagnetic field solves by construc-\ntion the φcomponent of the Maxwell equations iff one considers that the scalar field inherits the spacetime symmetries, namely\nϕ≡ϕ(r). Interestingly, one can see that the combination\n2αFα\nβFβ\nγFγ\nδFδ\nα−βF4=4(α−β)Q4\nm\n[R(r)]8+8βQ2\nm[V′(r)]2\n[R(r)]4+ 4(α−β)[V′(r)]4, (3.7)\nwill vanish if one does not consider both electric and magnetic configurations in the case of α=β. In the above, prime denotes\nderivation with respect to r. Maxwell’s equation is very difficult to be integrated for the dyonic case and as a result we will\nconsider pure magnetic fields, that is V(r) = 0 . Consequently, both these non-linear electrodynamics terms will contribute iff\nα̸=β. We will begin our analysis for the scalar free scenario ϕ= 0, f(ϕ= 0) = 1 , for which the solution reads\nB(r) = 1−2M\nr+Q2\nm\nr2+2(α−β)Q4\nm\n5r6, (3.8)\n3The reader should be reminded at this stage that in the special case of (open)string/brane-inspired BI theory at tree-level in string loops, the function f(ϕ)∼\ne−5ϕ(cf.(2.6)), however in such a case the Maxwell term F2in (3.1) should be accompanied by the inverse of the open string coupling, ie.e−ϕ, instead\nofe−2ϕthat appears in (3.1). On the other hand, in the heterotic-string-inspired model (3.1) can be mapped to the model (2.7), upon choosing γ= 1, and\nFeF= 0, that is concentrating on magnetically charged black holes only (in which case, the function f(ϕ)∼e−6ϕ). However, as we have already stressed,\nand we shall argue below, it is crucial for an analytic treatment of the black-hole solution to have a dilaton-independent term in f(ϕ), which, as we have\nargued in the previous section, can be induced by considering higher-order string loop corrections in the underlying string-theory model.\n4Note that throughout this article, φwill always denote the azimuthal coordinate, while ϕwill always denote the scalar field.7\nandR(r) =r. This solution resembles the Einstein-Euler-Heisenberg black hole [105]. The interesting thing to notice in\n(3.8) is that the non-linear electromagnetic terms Fα\nβFβ\nγFγ\nδFδ\nαandF4affect the spacetime geometry in a similar way. It\nis solely the values of the coupling constants αandβthat determine whether this contribution survives or not. Note that in\nthe case of α=βthe higher-order electromagnetic term does not contribute at all. However, in the case where α̸=β, we\nnotice that depending on the signs of the parameters αandβ, the non-linear electromagnetic terms can act either attractively or\nrepulsively. Black holes with a scalar hair in the Euler-Heisenberg theory have been discussed in [64], and it was found that the\nscalar hair results in a more compact black hole (having a smaller radius for the event horizon) when compared to the non-hairy\nEinstein-Euler-Heisenberg black hole.\nLet us now assume a non-trivial profile for the coupling function f(ϕ). In particular, we consider\nf(ϕ) =−[3 cosh(2 ϕ) + 2] ≡ −1\n2\u0000\n3e−2ϕ+ 3e2ϕ+ 4\u0001\n. (3.9)\nNotice here that the coupling function f(ϕ)contains the dilatonic coupling e2ξϕwithξ=±1as well as a constant (dilaton\nindependent) term. At this point the reader is invited, for completion, to compare such couplings with the string-loop corrected\ncoupling functions BF4(ϕ), in the framework of string-inspired models (2.14), discussed in the previous section II. In such\na stringy context, the exponential dilaton terms in the coupling function (3.9) can be written as f(ϕ) =−3\n2(g−2\ns+g2\ns)−2,\nwhere gs= exp( ϕ)is the string coupling. As discussed in section II, the g−2\nsis the standard tree-level dilaton-Maxwell term\ncoupling [1–3], while the g2\nsindicates two-string-loop corrections (genus- χ= 2 world-sheet surfaces). The crucial, for our\nsubsequent discussion, dilaton-independent term in f(ϕ)might be the result of appropriate combinations of higher-string-loop\ncorrections in the Einstein-frame effective action.\nIt is now straightforward to solve the field equations of (3.1), with (3.9), in order to determine the geometry of the spacetime\nand the functional expression for the scalar field. By doing so, one obtains a simple exact, magnetically charged black-hole\nsolution, for which it holds that\nB(r) = 1−2M\nr−2(α−β)Q4\nm\nr3\u0010\nr−Q2m\nM\u00113,[R(r)]2=r\u0012\nr−Q2\nm\nM\u0013\n, (3.10)\nϕ(r) =−1\n2ln\u0012\n1−Q2\nm\nMr\u0013\n,Aµ= (0,0,0, Qmcosθ). (3.11)\nWe observe that in this case, for α=βwe obtain the GHS solution [85], while the radial coordinate r∈(Q2\nm/M,+∞)in order\nto have R∈(0,+∞). In this case, it is also intriguing to observe that the sign of the combination α−βamong the coupling\nconstants determines whether the higher-order electromagnetic terms in the theory will contribute attractively or not.\nTo obtain a better understanding of the spacetime geometry, one may express the line element (3.5) in terms of the physical\ncoordinate system with Rplaying the role of the radial coordinate. By doing so, one finds that\nds2=−B(R)dt2+[W(R)]2dR2\nB(R)+R2dΩ2, (3.12)\nwith functions B(R),W(R), and ϕ(R)being given by\nB(R) = 1−4M2\nQ2m+p\nQ4m+ 4M2R2−2(α−β)Q4\nm\nR6, (3.13)\n[W(R)]2=4M2R2\nQ4m+ 4M2R2, (3.14)\nϕ(R) =−1\n2ln p\nQ4m+ 4M2R2−Q2\nmp\nQ4m+ 4M2R2+Q2m!\n. (3.15)\nIn the physical coordinate system (t, R, θ, φ ), one can verify that the curvature invariant quantities R,RµνRµν, andRαβγδRαβγδ\npossess a single spacetime singularity residing at R= 0, while the function B(R)satisfies the following expansions\nB(R→+∞) = 1−2M\nR+Q2\nm\nR2−Q4\nm\n4MR3+Q8\nm\n64M3R5−2(α−β)Q4\nm\nR6+O(1/R7), (3.16)\nB(R→0) =−2(α−β)Q4\nm\nR6+\u0012\n1−2M2\nQ2m\u0013\n+O(R2). (3.17)8\n0.5 1 5 10 50 100-0.50.00.51.0\n(a)\n0.1 0.5 1 5 10 50 100-2-1012345 (b)\nFIG. 1: The metric function B(R)in terms of R/(2M)for various values of the parameter Qm/Mand (a) (α−β)/M2= 1,\n(b)(α−β)/M2=−1. All parameters are dimensionless, and the horizontal axis in both figures is logarithmic.\nFrom (3.16), it becomes apparent that the spacetime (3.12) is practically indistinguishable from that of a magnetically charged\nReissner-Nordstr ¨om black hole for an observer at infinity, with the parameter Mcorresponding to the ADM mass of the solution.\nHowever, an observer much closer to the black hole (3.12) would perceive a completely different picture. Indeed these quantum-\ngravity corrections are important near the singularity, since the geometry there is determined by their behavior.\nThe radial null-trajectories for the spacetime (3.12), lead to the relation\ndt\ndR=±2MRp\nQ4m+ 4M2R21\n|B(R)|, (3.18)\nwhich by its turn means that the roots of the function B(Rh)correspond to black-hole horizons. In Fig. 1, one can observe the\nbehavior of the metric function B(R)in terms of the dimensionless quantity R/(2M). We see that the solution (3.12) describes\na black hole with a single horizon when (α−β)/M2= 1, while for (α−β)/M2=−1the black-hole horizons can range\nfrom two to none. It is essential to note that the previous assertion holds in general for (α−β)/M2being either greater or\nlower than zero. Analysis of Fig. 1a reveals that a positive value for the combination (α−β)/M2results in black-hole solutions\nfeaturing a single horizon. To facilitate comparison, we have also included the Schwarzschild solution which can be obtained\nby simply setting Qm= 0. One can readily observe that within our theory’s solution spectrum, black holes can exhibit either\ngreater compactness or sparsity relative to the Schwarzschild solution. In astrophysical scenarios where Qmis relatively small\ncompared to the mass, our solution appears more compact. Conversely, when the fraction (α−β)/M2takes a negative value,\nthe solutions range from black holes with two horizons to naked singularities. The transition from one class of solutions to the\nother occurs continuously as the magnetic charge Qmincreases, as depicted in Fig. 1b. Consequently, in this scenario, there\nalways exists a specific value for the ratio Qm/Mthat renders the black hole extremal, meaning the inner and outer horizons\ncoincide.\nIt is crucial to highlight here the intriguing behavior observed in the realm of single-horizon black-hole solutions, for which\nα−β >0. Specifically, there exists a minimum value for the ratio Rh/(2M), which is below unity, resulting in more compact\nblack holes compared to the Schwarzschild solution. Starting from Qm= 0(Schwarzschild) and increasing the magnetic charge,\nthe resulting black holes become progressively more compact until reaching the point where Rh/(2M)attains its minimum\nvalue. Beyond this point, further increase in the ratio Qm/Mcauses Rh/(2M)to rise again, eventually reaching Rh/(2M) = 1 ,\nalbeit now with Qm̸= 0. Subsequently, any additional increase in the ratio Qm/Myields a solution more sparse than the\nSchwarzschild counterpart. This particular behavior is elucidated by analyzing Fig. 2, where the relationship between the ratio\nof the black hole horizon ( Rh) to twice the black hole mass ( 2M) and the ratio Qm/Mis depicted for various values of the\ndimensionless parameters (α−β)/M2. Conversely, it is observed that when α−β <0, the outer horizon radius of the resulting\nblack holes is consistently smaller than that of the corresponding Schwarzschild black hole with the same mass. Furthermore, it\nis important to note that in this scenario, the graph reaches a termination point. This occurs because, beyond a certain threshold\nof the ratio Qm/M(which is less than unity), there is a significant transition in the nature of the compact object. Specifically,\nthe object transitions from being an extremal black hole to a naked singularity. Consequently, for this particular choice of\nparameters, there is no horizon to be depicted. These observations are further corroborated by the findings depicted in Fig. 1b.9\n0.2 0.5 1 20.81.01.21.4\nFIG. 2: The ratio Rh/(2M)in terms of the ratio Qm/Mfor various values of the dimensionless parameter (α−β)/M2. Both\naxes are logarithmic.\nReturning now to the case α−β > 0, the discovery of black-hole solutions sharing identical horizon radii yet varying in the\nratios Qm/Munveils a realm of doppelg ¨anger black holes within the framework of theory (3.1). While it is typical to find black\nholes stemming from different theoretical paradigms with shared horizon radii but differing physical attributes such as mass,\nelectric charge, or secondary scalar hair, such occurrences are notably rare when considering black holes that arise from the\nsame theory. Even more remarkable is the fact that these two doppelg ¨anger black holes, despite having identical horizon radii,\nexhibit distinguishable thermodynamic behaviors. One is thermodynamically stable while the other is unstable. This distinctive\nfeature is thoroughly explored in Section V.\nIV . GEODESICS AND ENERGY CONDITIONS\nA. Geodesics\nIn this subsection, we will examine the geodesic curves of massive particles and the effective gravitational potential generated\nby the spacetime geometry given by eqs. (3.5) and (3.10). We choose to work with the (t, r, θ, φ )coordinate system, as it\nfacilitates a straightforward derivation of the effective gravitational potential Veffthrough a well-established procedure. This will\nhelp us to better comprehend the geometry of the aforementioned black hole solutions. To do so, we introduce the effective\nLagrangian\n2Leff=gµνdxµ\ndτdxν\ndτ=−B(r)˙t2+˙r2\nB(r)+ [R(r)]2\u0010\n˙θ2+ sin2θ˙φ2\u0011\n, (4.1)\nthe Euler-Lagrange equations of which yield the geodesic equations. In the above, τis an affine parameter of motion which can\nbe identified with the proper time of a particle, dot denotes derivation with respect to τ, while 2Leff=−1corresponds to massive\nparticles which follow a timelike path. Note that massless particles will not follow the geodesics induced by the geometry gµν,\ninstead they will follow the geodesics induced by an effective geometry that accounts for photon-photon interactions, introduced\nby the non-linear electromagnetic terms in our action. Upon inspecting the Lagrangian (4.1), it becomes evident that there is\nno explicit dependence on the coordinates (t, φ). As a result, the Euler-Lagrange equations for tandφyield two conserved\nquantities: the energy Eand the angular momentum Jof the particle under consideration, respectively. Hence, we have\nE=B(r)˙t , (4.2)\nJ= [R(r)]2sin2θ˙φ . (4.3)\nThe equation of motion for θreads\n[R(r)]2¨θ+ 2R(r)R′(r) ˙r˙θ−Jcosθ\nsinθ= 0, (4.4)10\n0.5 1 5 10 50-8-6-4-202\n21.950 21.955 21.960 21.965 21.970 21.975 21.9800.4782940.4782940.4782940.4782940.4782940.4782940.4782940.478294\nFIG. 3: The effective potential Veffin terms of the quantity r/M forQm/M= 0.5,J/M = 5, and (α−β)/M2={−2,0,2}.\nAll parameters are dimensionless, and the horizontal axis is logarithmic.\nand by choosing θ=π/2(˙θ= 0), the particles stay fixed at the equatorial plane. Now plugging these results back to (4.1) we\nobtain the radial equation of motion\n1\n2˙r2+Veff(r) =1\n2E2, (4.5)\nwith the effective potential induced by the geometry being\nVeff(r) =B(r)\n2\u0012\n1 +J2\n[R(r)]2\u0013\n, (4.6)\nand the functions B(r)andR(r)given by (3.10) . As we have already mentioned in the previous section, the radial coordinate\nrranges from Q2\nm/Mto plus infinity because the physical radial coordinate R∈(0,+∞).\nIn Fig. 3, we depict the behavior of the effective potential Veffin terms of the dimensionless parameter r/M , considering\nthree distinct values for the fraction (α−β)/M2. Upon close examination, it becomes evident that the scenarios where α=β\nand(α−β)/M2= 2 share a strikingly similar pattern in the effective potential. In both cases, the potential curve features\none maximum and one minimum value, corresponding to unstable and stable circular orbits, respectively. On the other hand,\nin the case where (α−β)/M2=−2, an additional minimum emerges, exhibiting local behavior that closely resembles the\nNewtonian potential. To understand the origin of this difference, we have to examine the expansion of the potential Veffin the\nlimitr→Q2\nm/M, where one can verify that\nVeff(r→Q2\nm/M) =−J2M4(α−β)\nQ4m\u0010\nr−Q2m\nM\u00114+M3(α−β)\u0000\n4J2M2−Q4\nm\u0001\nQ6m\u0010\nr−Q2m\nM\u00113+O\"\u0012\nr−Q2\nm\nM\u0013−2#\n. (4.7)\nWe observe that the first term, which dominates in this particular regime, depends explicitly on the sign of the quantity α−β.\nWhen α−β >0, the potential tends toward negative infinity, whereas for α−β <0, the potential tends toward positive infinity.\nThis alignment precisely mirrors our observations in Fig. 3. Finally, from Fig. 3, it is also clear that for r/M > 2, the effective\npotential in all cases exhibits the same profile, independently of the relative values of the coupling constants αandβ. This can\nbe naively understood through the expansion of the potential at infinity, which is of the following form\nVeff(r→+∞) =1\n2−M\nr+J2\n2r2+\u0012Q2\nm\n2M2−1\u0013\u0014J2M\nr3+J2Q2\nm\nr4+J2Q4\nm\nM r4+J2Q6\nm\nM2r4\u0015\n−(α−β)Q4\nm\nr6+O\u00121\nr7\u0013\n.(4.8)\nIt is obvious that in the asymptotic regime, the coupling constants αandβcease to influence the potential profile, as their first\ncontribution comes into play only in the seventh term of the expansion. Consequently, even at medium distances, we anticipate\nthat beyond a certain point, the coupling constants will have negligible impact on the potential’s behavior.11\n1 2 3 4 5-0.03-0.02-0.010.000.010.020.03\n2.0 2.5 3.0 3.5 4.0 4.5 5.0-0.00010-0.000050.000000.000050.000100.000150.00020\n(a)\n1 2 3 4 5-0.10-0.050.000.050.10\n2.0 2.5 3.0 3.5 4.0 4.5 5.0-0.00010-0.000050.000000.000050.000100.000150.00020 (b)\nFIG. 4: The energy conditions for Qm/M= 0.5with (a) a positive and (b) a negative value assigned to the dimensionless\nquantity (α−β)/M2. The vertical lines correspond to the horizon of the black hole determined by these parameters.\nB. Energy Conditions\nWe will now turn our attention to the energy conditions associated with the stress-energy tensor of our theory. In the physical\ncoordinate system (t, R, θ, φ )the stress-energy tensor is described by an anisotropic fluid which in a covariant form can be\nwritten as\nTµν= (ρE+pθ)uµuν+ (pR−pθ)nµnν+pθgµν. (4.9)\nIn the above, ρEis the energy density of the fluid measured by a comoving observer with the fluid, pRis its radial pressure,\npθis its tangential pressure, while uµandnµare its timelike four-velocity and a spacelike unit vector orthogonal to uµand also\nto both angular directions. The four-vectors uµandnµsatisfy the following relations:\nuµ=u(R)δµ\n0, uµuνgµν=−1, (4.10)\nnµ=n(R)δµ\n1, nµnνgµν= 1. (4.11)\nGiven eqs. (3.2), (3.9), (3.12)-(3.15), and (4.9)-(4.11), one can readily compute that\nρE=−Tt\nt=B(R)\n[W(R)]2\u0012dϕ\ndR\u00132\n+Q2\nm\nR4e−2ϕ+2(α−β)Q4\nm\nR8f(ϕ), (4.12)\npR=TR\nR=B(R)\n[W(R)]2\u0012dϕ\ndR\u00132\n−Q2\nm\nR4e−2ϕ−2(α−β)Q4\nm\nR8f(ϕ), (4.13)\npθ=Tθ\nθ=Tφ\nφ=−B(R)\n[W(R)]2\u0012dϕ\ndR\u00132\n+Q2\nm\nR4e−2ϕ+6(α−β)Q4\nm\nR8f(ϕ). (4.14)\nFor the anisotropic fluid of (4.9), the energy conditions take the following expression:\n• Null Energy Conditions (NEC): ρE+pR≥0 & ρE+pθ≥0,\n• Weak Energy Conditions (WEC): NEC &ρE≥0,\n• Strong Energy Conditions (SEC): NEC &ρE+pR+ 2pθ≥0.12\nIn Figs. 4a and 4b, we illustrate the graphs of the quantities ρE,ρE+pR,ρE+pθ, andρE+pR+ 2pθeach plotted against the\ndimensionless parameter R/M . The free parameters of our model and solution have been chosen to be Qm/M= 0.5, while the\ncombination (α−β)/M2takes values of 1and−1, respectively. It is evident from Fig. 4 that all the aforementioned quantities\nmaintain positive values within the causal region of spacetime and as a result, all energy conditions are satisfied.\nThese results imply, therefore, the existence of a dilaton hair in the black hole’s exterior, while the energy conditions are\nsatisfied, thereby leading to a bypass of the pertinent (modern version of the) no-hair theorems [106, 107] in the spirit of [108].\nThe situation can be understood as a consequence of the fact that the stress-energy tensor of our theory (3.1), with (3.9), is such\nthat the tangential component of the pressure ( pθ=Tθ\nθ) dominates over its radial one ( pR=TR\nR)(in the (t, R, θ, ϕ )coordinate\nsystem), outside the horizon. That is, the following quantity is positive in the exterior region of the black hole,\nG − J =Tθ\nθ−TR\nR>0, (4.15)\nwhere G=ρE+Tθ\nθandJ ≡ ρE+TR\nR. Note that the condition (4.15) follows from NEC. As discussed in detail in [108],\nthe quantity 2G/Ris the effective gradient pressure force , and its positivity (i.e. that of G, since R > 0) explains in a physical\nway the existence of scalar hair in the black-hole’s exterior, without any violation of the energy conditions. The validity of the\ncondition (4.15) can also be explicitly checked in our model from Eqs. (4.13) and (4.14). Thus, the exact black hole solution\nof the self-gravitating scalar-EH (non-linear) electrodynamics examined in this paper constitutes another explicit example of the\ngeneral considerations of [108] for bypassing the no-hair theorem without any violation of the energy conditions.\nV . THERMODYNAMIC ANALYSIS\nIn this section, we will discuss the thermodynamics of both the GMGHS and our black-hole solution by considering their\nEuclidean actions. We will consider the Grand Canonical Ensemble and enclose the black hole spacetime in a cavity with a large\nradius rc. In the Grand Canonical Ensemble, the black hole is allowed to exchange energy/mass and charge with its environment,\nso these two quantities are allowed to flow in and out through the boundary keeping the temperature and the magnetostatic\npotential of the boundary fixed. This effectively means that T(rh) =T(rc)andΦm(rh) = Φ m(rc)and the system black\nhole-cavity is in thermodynamic equilibrium. Note that Tis the black-hole temperature and Φmis the magnetostatic potential.\nThe quantum partition function for the system is then given by\nZ=Z\nd[g(L)\nµν, ψ]eiS(g(L)\nµν,ψ)=Z\nd[g(E)\nµν, ψ]e−IE(g(E)\nµν,ψ), (5.1)\nwhere Sis the Lorentzian action, IEis the Euclidean action and ψdenotes all other possible fields included besides the metric\ntensor. The two actions are related via IE=−iS[109]. The quantity g(L)\nµνis the Lorentzian metric with signature (−+ ++) ,\nwhich corresponds to a R3,1spacetime, while g(E)\nµνis the Euclidean metric with signature (+ + ++) , which is obtained from\nthe Lorentzian one by performing a Wick rotation [110] of the time coordinate ( τ=it). In the standard Matsubara formalism of\nfinite-temperature systems, the Euclidean metric corresponds to a space R3×S1\nβτ, where the radius βτof the S1is the inverse\ntemperature T−1in units of the Boltzman factor kB= 1. Hence, the second integral in (5.1) is evaluated over all possible\nfield configurations that have an imaginary time τwith period βτ. From the partition function, using standard thermodynamic\nrelations one can obtain the Free Energy Gof the system as\nG=−1\nβτlnZ. (5.2)\nBy using the saddle point approximation (Laplace’s method) we will consider that the classical action contributes the most and\nas a result we may drop the integral in the partition function Z. Then the Euclidean action IEcan be related to the free energy\nevaluated on shell through the following relation\nIE=Gβτ. (5.3)\nHaving the expression of the free energy for the black hole solution, we will compare it with the free energy of the grand\ncanonical ensemble in order to extract the mass (internal energy), the entropy, and the magnetostatic potential of the black hole.\nFor more information in the discussion that follows, we refer the reader to the original work of Gibbons and Hawking [111].\nA. GMGHS black hole\nWe start our analysis with the thermodynamics of the GMGHS solution. The Euclidean action, including the appropriate\nboundary terms, is given by\nIE=−1\n16πZ\nΣd4x√g\u0000\nR −2∇αϕ∇αϕ−e−2ϕFµνFµν\u0001\n−1\n8πZ\n∂Σ(K−K0)d3x√\nh . (5.4)13\nIn the Euclidean signature, the GMGHS black hole is described by the following metric:\nds2=B(r) dτ2+dr2\nB(r)+ [R(r)]2dΩ2, (5.5)\nwhere B(r) = 1−2M/r , while R(r)has the same form as in eq. (3.10). In this coordinate system, the Euclidean time coordinate\nis periodic and takes values in the range 0≤τ≤βτ. For the derivation of the thermodynamic quantities, we assume that we\nhave enclosed the black hole in a large cavity with radius rc. Therefore, the radial coordinate takes values in rh≤r < r c.\nFinally, the two angular coordinates take their usual values. The boundary term Krepresents the trace of the extrinsic curvature,\nwhich in our case reads\nK=∇αnα=2R′(r)p\nB(r)\nR(r)+B′(r)\n2p\nB(r), (5.6)\nwhere nα=p\n1/B(r)δr\nαis a normalized spacelike vector field. The Kterm in the above hypersurface integral represents the\nGibbons-Hawking-York boundary term, ensuring a well-defined variational principle. The second boundary term K0serves as a\nsubtraction term to render the action finite for flat space (in the absence of the black hole). For flat space, K0equals 2/r, obtained\nby setting B(r) = 1 andR(r) =rin the above relation. Utilizing these relations, one can readily compute the Euclidean action\n(5.4) to be\nIE=βτM\n2−βτQ2\nm\n4M. (5.7)\nIn the above, we have used that the horizon radius is given by rh= 2M. In the Grand Canonical Ensemble, the Euclidean action\nis identified with the free energy of the thermodynamic system as IE=βτG, thus, we can rewrite (5.7) as\nIE=βτM−βτΦmQm−S , (5.8)\nwhere Sis the entropy and Φmis the magnetostatic potential, Φm=Qm/rh. For the derivation of the above equation, we have\nused the fact that βτ= 8πM≡1/TwithTbeing the temperature of the black hole. By combining now eqs. (5.7) and (5.8) we\ncan evaluate the black-hole entropy S, which is given by the following relation\nS= 2πM\u0012\n2M−Q2\nm\nM\u0013\n=π[R(rh)]2=A\n4, (5.9)\nwhere Adenotes the horizon area. It is evident that in this case, the entropy function has the well-known form of the Bekenstein-\nHawking entropy. For validation, the same result may also be obtained using Wald’s formula or even using the Arnowitt-\nDeser-Misner (ADM) formalism [112]. For a comprehensive analysis of the ADM formalism, readers are directed to [113].\nAdditionally, for its explicit application in black-hole solutions, we refer the interested reader to [114]. In the subsequent\nsubsection, we will utilize the ADM formalism for the thermodynamic analysis of our black-hole solution.\nThe inclusion of the Gibbons-Hawking-York boundary term ensures that the Euclidean action attains an extremum within the\nclass of fields considered here, δIE= 0.As a result, it is evident that the first law of thermodynamics in the Grand Canonical\nEnsemble (keeping the temperature and the magnetic potential fixed) takes the form\nδM=TδS+ ΦmδQm, (5.10)\nderived from (5.8), and holds by construction . The first law is also evident by taking the variation of the entropy with respect\nto the primary black-hole charges. The temperature of this black hole is the same as that of the Schwarzschild black hole, as\npointed out in [115], since the Euclidean continuation does not care about the angular part. Consequently, the heat capacity C\nfor constant charge will also be negative, C=−1/(8πT2); hence, these types of black holes cannot reach thermal equilibrium.\nB. Black hole with non-linear electrodynamics\nWe will now focus on our black-hole solution, emanating from the action (3.1) and characterized by the line element (3.12)-\n(3.14). The scalar and the gauge fields are of the form ϕ=ϕ(R)—with Rbeing the physical radial coordinate—and Aµ=\n(0,0,0, A(θ)), respectively. In this case, to determine the thermodynamic quantities associated with the resulting black-hole\nsolution, we will make use of the Euclidean signature and also utilize the ADM formalism [112, 113]. Hence, we consider the\nline element of the form\nds2= [N(R)]2B(R) dτ2+[W(R)]2dR2\nB(R)+R2dΩ2, (5.11)14\nwhere the Euclidean time takes values in the range 0≤τ≤βτ, while the radial coordinate R∈[Rh,+∞). To obtain the\ntemperature, that is the period of the Matsubara frequency τ, in our case, we follow the calculation of [37]. To this end, we\nfirst ignore the angular part of the line element and perform a series expansion near the horizon. Thus, we are left with a two-\ndimensional line element which is compared with the line element of two-dimensional space expressed in polar coordinates\ndS= dˆR2+ˆR2dΘ2. By doing so, we obtain\ndˆR2=W(Rh)2\nB′(Rh)(R−Rh)dR2, (5.12)\nB′(Rh)(R−Rh) dτ2=ˆR2dΘ2. (5.13)\nThe coordinate Θis periodic with a period 2πwhich implies that τis also a periodic coordinate with a period βτgiven by:\nβτ=1\nT=4πW(R)\nN(R)B′(R)\f\f\f\f\f\nRh, (5.14)\nwhere Tis the temperature of the black hole. For completeness, we also remark at this point that we have also checked that, as\nexpected, the temperature will take on the same values at the event horizon regardless of the coordinate system we are using ( r\norR).\nThe Euclidean action is related to the Lorentzian action via IE=−iSand we will consider the following variational problem\nwhich basically consists of the theory (3.1) alongside a boundary term denoted by BEwhich we will consider in order to have a\nwell-defined variational principle δIE= 0. Thus, we have\nIE=2πβτ\n16πZπ\n0dθZ∞\nRhdR\u0002\n−NR2WsinθL(R, θ)\u0003\n+BE. (5.15)\nHereLdenotes the Lagrangian of the theory which is a function of R, θ coordinates. After canceling total derivatives, the\nEuclidean action reads\nIE=βτZπ\n0dθZ∞\nRhdRˆL(Qi, ∂µQi) +BE, (5.16)\nwithQi={N(R), W(R), B(R), ϕ(R), A(θ)}andˆL(Qi, ∂µQi)given by\nˆL(Qi, ∂µQi) =Nsinθ\n4W2R6\u0014\nWR7B′+W3\u0012\n2f(ϕ)(α−β)(∂θA)4\nsin4θ+e−2ϕR4(∂θA)2\nsin2θ−R6\u0013\n+BR6\u0000\nWR2ϕ′2−2RW′+W\u0001\u0015\n. (5.17)\nFollowing the ADM formalism, we have to vary the above Euclidean action with respect to each one of the dynamical fields Qi\nto obtain the field equations. By doing so, we obtain none other than the well-known Euler-Lagrange equations, namely\n∂ˆL\n∂Qi−∂µ \n∂ˆL\n∂(∂µQi)!\n= 0. (5.18)\nLet us now apply the above equation for the dynamical field Q1=N(R). Upon substituting the expression of ˆLfrom (5.17) into\n(5.18), we find that the second term vanishes identically, while ∂ˆL/∂N =ˆL/N. As a result, the equation for N(R)indicates\nthatˆL= 0, which in turn implies IE=BE. This outcome is anticipated in the ADM formalism, where the metric construction\n(5.11) is specifically tailored to yield this result. Additionally, by solving the field equations (5.18) for all dynamical fields Qi,\none can determine the unknown functions and verify that the resulting solution is the one obtained in Sec. III with line element\n(3.12)-(3.14), alongside a constant Nwhich without loss of generality we may set equal to 1. It is important to mention at this\npoint, that during the derivation of the Euler-Lagrange equations, certain boundary terms were omitted. These terms are of the\nfollowing form\nβτ\u0012R\n2WδB+2BR2ϕ′\nWδϕ−BR\nW2δW\u0013\f\f\f\f\f∞\nRh, (5.19)15\nand\nβτZ∞\nRhdR\u0012We−2ϕ(∂θA)\n2R2sinθ+2(α−β)Wf(ϕ)(∂θA)3\nR6sin3θ\u0013\nδA\f\f\f\f\fθ=π\nθ=0. (5.20)\nThe variation of the boundary term δBEwill account for the neglected boundary terms, ensuring the attainment of a well-\ndefined variational principle δIE= 0. Utilizing the fact that the variation of Ayields δA= (δQm) cosθ, and substituting the\nexpressions for the functions in (5.20), one can integrate and derive the following expression:\nβτQmnp\n4M2R2\nh+Q4m\u0002\nR4\nh−4(α−β)Q2\nm\u0003\n−R4\nhQ2\nmo\n2MR6\nhδQm. (5.21)\nNow, the variation of the dynamical fields at infinity yield\nδB=−2δM\nR+O\u00121\nR2\u0013\n, (5.22)\nδϕ=Qm\nMRδQm−Q2\nm\n2M2RδM+O\u00121\nR3\u0013\n, (5.23)\nδW=Q4\nm\n4M3R2δM−Q3\nm\n2M2R2δQm+O\u00121\nR3\u0013\n, (5.24)\nwhile at the horizon we have that\nδB|Rh=−B′(Rh)δRh, (5.25)\nδϕ|Rh=δϕ(Rh)−ϕ′(Rh)δ(Rh), (5.26)\nδW|Rh=δW(Rh)−W′(Rh)δ(Rh). (5.27)\nNote that the parameters α, β are fixed by the theory and thus not allowed to vary, while MandQmare pure integration constants\nallowed in the variation.\nAs previously mentioned, to ensure a well-defined variational procedure, it is desirable to have δIE= 0. For clarity and\nconvenience, we will partition the variation of the boundary term δBEinto two components: one at infinity and another at the\nevent horizon, expressed as:\nδBE=δBE(∞) +δBE(Rh). (5.28)\nEvaluating now (5.19) at infinity and considering the variation of the boundary term at infinity we find that a zeroth order\ncontribution survives, which according to the variations of the fields leads to\nβτ\n2R(δB−2δW) +δBE(∞) = 0⇒ (5.29)\nδBE(∞) =βτδM . (5.30)\nOn the other hand, eq. (5.19) at the horizon, alongside the variation of the boundary term at the horizon and (5.21) results in\n2πRhδRh+βτQmnp\n4M2R2\nh+Q4m\u0002\nR4\nh−4(α−β)Q2\nm\u0003\n−R4\nhQ2\nmo\n2MR6\nhδQm+δBE(Rh) = 0 , (5.31)\nwhich might be written equivalently as\nδA\n4+βτΦmδQm+δBE(Rh) = 0 , (5.32)\nwhere we have used the fact that the area of the black hole is given by A= 4πR2\nhand we have defined the magnetic potential as\nΦm=Qmnp\n4M2R2\nh+Q4m\u0002\nR4\nh−4(α−β)Q2\nm\u0003\n−R4\nhQ2\nmo\n2MR6\nh. (5.33)16\n0.8 0.9 1.0 1.1 1.2 1.3125\n(a)\n0.75 0.80 0.85 0.90 0.95 1.00 1.050.40.60.81.0 (b)\nFIG. 5: The black-hole temperature for (a) attractive and (b) repulsive higher-order electromagnetic contributions, with varying\nvalues of the magnetic charge ( Qm), while keeping the mass ( M) the same. The axes in both figures are logarithmic.\nConsidering now that we are dealing with the Grand Canonical Ensemble, we keep the temperature and the magnetic potential\nof the system fixed and as a result we can drop the variations to obtain\nBE(∞) =βτM , (5.34)\nBE(Rh) =−A\n4−βτΦmQm. (5.35)\nTherefore, the value of the Euclidean action is given by\nIE=βτM−A\n4−βτΦmQm, (5.36)\nand since the Euclidean action is related to the free energy Gof the system via IE=βτG=βτM − S−βτΦmQmwe can\nidentify, by comparison the conserved black hole mass and the entropy of the black hole as\nM=M , (5.37)\nS=A/4. (5.38)\nFinally, the First Law of Thermodynamics (5.10) holds by construction as in the GMGHS black hole.\nWith the confirmation that the black-hole thermodynamic quantities in our case adhere to the standard relations, we can\nnow proceed to analyze the black hole’s temperature. In Fig. 5, we depict the black-hole temperature as a function of the\ndimensionless quantity Rh/(2M). Notice that the temperature is scaled by the temperature of the Schwarzschild black-hole\nto form a dimensionless quantity, ensuring its independence from the chosen unit system. In Figs. 5a and 5b, we explore\nthe effects of the higher-order electromagnetic contributions on black-hole temperature, considering fixed (yet distinct) values\nfor the coupling constants αandβ, along with varying magnetic charge ( Qm) values, but maintaining the same value for the\nblack-hole mass. Both Figs. 5a and 5b were generated using the following procedure: For each value (α−β)/M2and the\nratioQm/M, we numerically evaluate the value of the ratio Rh/(2M)using eq. (3.13). Subsequently, employing equation\n(5.14), we calculate the temperature of the black hole for each parameter pair. Finally, for each (α−β)/M2we plot the points\nfrom the list {Rh/(2M), T(Rh)/Tsch}. Note that we use the same mass parameter Mfor the temperature calculation of the\nSchwarzschild black hole Tsch. In both figures, we have also incorporated a distinctive dot symbolizing a constant value for the\nquantity T(Rh)/Tsch, irrespective of the ratio Rh/(2M). Apparently, this is not coincidental, as it mirrors the characteristics of\nthe Schwarzschild black hole, where the horizon radius precisely equals 2Mand its temperature is determined by the established\nformula T= 1/(8πM).\nFocusing now on the thermodynamical characteristics of our solution, we observe that regardless of the value (α−β)/M2,\nforQm= 0, our solution reduces to the Schwarzschild black hole and therefore all graphs in Fig. 5 have as a starting point\nthe Schwarzschild point. However, when we depart from this limit, we notice that for α−β >0, as illustrated in Fig. 5a, the\ntemperature of the resulting black holes consistently surpasses that of the Schwarzschild black hole, whereas for α−β <0(Fig.17\n5b), the opposite effect occurs. Furthermore, this temperature increase, in the α−β >0scenario, is independent of whether the\nblack hole under examination possesses a smaller or larger horizon radius compared to the corresponding Schwarzschild black\nhole. As previously observed in Fig. 2 and discussed in Sec. III, it becomes evident in Fig. 5a that there are consistently pairs\nof black-hole solutions, more compact than the Schwarzschild solution, that share the same horizon radius Rhbut with different\nratios Qm/M. However, now we see that although these solutions possess the same horizon radius, their temperatures differ\nsignificantly. This can be understood through the relation (5.14) where it is evident that the formula determining the temperature\nof a black hole depends on the first derivative of the function B(R). This means that black-hole solutions which for different\nratios Qm/Mresult in the same horizon radius Rhthrough the equation B(Rh) = 0 , their temperatures are not necessarily the\nsame since B′(Rh)could differ in these two cases.\nMoreover, we can deduce the thermodynamic stability of these black holes by examining how the temperature changes with\na change in the mass. In Fig. 5a it is evident that there are two distinct branches of black-hole solutions. In the first branch\nwe have black-hole solutions that get colder as the mass is decreasing, while in the second branch we have black holes that are\ngetting hotter as the mass is getting smaller. This implies that the heat capacity C≡dM/dT for the first branch is positive since\nbothdM, dT are negative and the black holes are thermally stable, while the second branch, for which the temperature rises\nwith the decrease of mass, exhibits negative heat capacity and are thermally unstable. Notice also the fact that the Schwarzchild\nblack hole lies in the second (unstable) branch which is a well-known result. Furthermore, the parameter space of these black\nholes exhibits a point where dT= 0indicating the divergence of the heat capacity and as a result a phase transition from hot to\ncold black holes. In Fig. 5b, we can see that as the black holes lose mass they get colder which implies that they are thermally\nstable since they possess positive heat capacity. These results are in agreement with the studies in [116], where the black holes\nare viewed as defects in the thermodynamical spacetime [117].\nIn the next section, we proceed to study the stability of the black holes from a linear-perturbation point of view, which, in\ngeneral, is distinct from the thermodynamic stability. Indeed, as we shall demonstrate explicitly below, such a linearised stability\nanalysis does not necessarily imply thermodynamical stability, in the sense that the thermodynamically unstable branches found\nabove exhibit stability under linear perturbations.\nVI. LINEAR PERTURBATIONS\nA. Radial Stability\nIn this section, we investigate the stability of our solution under radial perturbations. For simplicity, we focus on linear and\nradial perturbations. Therefore, we use the following ansatz:\nds2=−P(R, t)dt2+Q(R, t)dR2+R2dΩ2, A =a0(R, t)dt+Qmcosθ dφ, ϕ =ϕ(R, t), (6.1)\nwhere\nP(R, t) =B(R)\u0002\n1−ϵ e−iωth1(R)\u0003\n, Q (R, t) =1\nH(R)+ϵ e−iωth2(R), (6.2)\nϕ(R, t) =ϕ(R) +ϵ e−iωtϕ1(R), a 0(R, t) =ϵ e−iωta0(R). (6.3)\nFor the stability analysis, it is more convenient to work within the physical coordinate system; hence, in the above equations\nB(R),ϕ(R)andH(R) =B(R)/W2(R)are given in eqs. (3.13–3.15), and correspond to the background/unperturbed space-\ntime. Note that the radial perturbations are associated with the L= 0 perturbation5in the even sector of the gravitational\nperturbations. Therefore, in the electromagnetic part, only the electric-type perturbations contribute, as the magnetic-type corre-\nsponds to the odd sector [121, 122]. The dimensionless constant ϵdetermines the order of the perturbation. Finally, ωspecifies\nthe decomposition of the modes with fixed energy.\nA direct calculation reveals that both the spacetime and the matter field perturbations are determined from the function ϕ1.\nConsequently, the investigation of the system is simplified to a single equation for the perturbation of the scalar field. This\nspecific equation can be expressed in the conventional Schr ¨odinger form:\nd2Ψ(r∗)\ndr∗2+\u0002\nω2− V(R)\u0003\nΨ(r∗) = 0 , (6.4)\n5Lis the “angular momentum” index in the spherical harmonics function YML\nL(θ, φ). For more information about the decomposition of the perturbations in\nspherical harmonics, see [118–121].18\n-10 0 10 200.000.010.020.030.04\n(a)\n-10 0 10 200.0000.0050.0100.0150.0200.0250.030 (b)\nFIG. 6: The graph of the effective potential Vin terms of r∗for various values of the parameter Qm/Mand (a)\n(α−β)/M2= 1, (b)(α−β)/M2=−1.\nwhere we have defined a new perturbation function as Ψ≡Rϕ1and the tortoise coordinate is dr∗≡dR√\nB(R)H(R). The potential\nof the Schr ¨odinger equation has the following form\nV(R) =HB′+ 2BRϕ′\u0002\nϕ′\u0000\nR\u0000\nH′+HRϕ′2\u0001\n+ 4H\u0001\n+ 3HRϕ′′\u0003\n+BH′\n2R\n+B(α−β)Q4\nm\u0010\nR˙fϕ′+¨f\u0011\nR8+Bγe−2ϕQ2\nm(2−Rϕ′)\nR4. (6.5)\nConsidering our emphasis on the stability of black-hole solutions, there is no need to solve eq. (6.4). The time evolution factor,\nexp (−iωt), simplifies the task, requiring us only to ascertain whether the frequency, ω, is purely imaginary or not. In the\nscenario where the frequency ωis purely imaginary, the mode undergoes exponential growth due to the presence of the term\nexp (−iωt)making the solution unstable. Therefore, a negative eigenvalue, ω2<0, that signifies an unstable mode, corresponds\nto a bound state in the Schr ¨odinger equation (6.4). A general result in quantum physics is that for a potential that vanishes in\nboth asymptotic regions, has a barrier form, and is positive definite. Therefore, eq. (6.4) does not exhibit bound states. In Fig. 6,\nwe depict the potential of the Schr ¨odinger equation for two families of solutions. The first one corresponds to (α−β)/M2= 1,\nwhile the second one corresponds to (α−β)/M2=−1. By carefully examining the parametric space of the solutions, we\ndeduce that the potential always takes a form similar to the potentials in Fig. 6. Therefore, we conclude that our solutions are\nstable under radial perturbations.\nAlthough radial stability is a strong indication regarding the stability of a particular solution, a more careful and general\nperturbation analysis has to be performed to extract a stronger result, which, however, lies beyond the scope of this work.\nMoreover, as shown in the previous section V, linear stability does not necessarily imply thermodynamical stability for the black\nhole, in the sense that the latter, although linearly stable, nonetheless it possesses thermodynamically unstable branches.\nB. Scalar Quasi-Normal Modes\nQuasi-normal modes (QNMs) play a crucial role in the study of black holes and other astrophysical objects [123–125]. These\nmodes represent the characteristic vibrations or oscillations of a black hole after a perturbation, such as a gravitational wave or\na scattering event. QNMs are characterized by complex frequencies, i.e. eigenvalues of the Schr ¨odinger equation, consisting of\na real part and an imaginary part. The real part corresponds to the oscillation frequency, while the imaginary part reflects the\ndamping or decay of the mode. The study of QNMs provides valuable insights into the nature and properties of black holes,\noffering a unique window into their internal dynamics.\nFor simplicity, we will consider the propagation of a test scalar field Φin the background of our black hole and extract its\nQNMs. We begin our analysis from the following action functional for the scalar field,\nS=Z\nd4x√−g\u0002\n∇µΦ∇µΦ +m2Φ2\u0003\n, (6.6)19\n(α−β)/M2= 0.01 (α−β)/M2=−1 (α−β)/M2= 1\nQm/M = 0.010.329438 −0.096255 i0.329438 −0.096255 i0.329438 −0.096255 i\nQm/M = 0.30.333767 −0.096712 i0.333767 −0.096728 i0.333767 −0.096696 i\nQm/M = 0.60.348235 −0.098171 i0.348226 −0.098595 i0.348243 −0.097727 i\nTABLE I: The dimensionless L= 1,n= 0quasinormal modes ( Mω) form= 0.\n(α−β)/M2= 0.01 (α−β)/M2=−1 (α−β)/M2= 1\nQm/M = 0.010.401632 −0.050195 i0.401632 −0.050195 i0.401632 −0.050195 i\nQm/M = 0.30.404334 −0.052296 i0.404335 −0.052294 i0.404332 −0.052298 i\nQm/M = 0.60.413700 −0.058602 i0.413733 −0.058522 i0.413667 −0.058679 i\nTABLE II: The L= 1,n= 0dimensionless quasinormal modes ( Mω) form/M = 0.4.\nwhere mis the mass of the test scalar field. The variation of the above action with respect to the scalar field yields the Klein-\nGordon equation in the black hole background\n\u0000\n□−m2\u0001\nΦ = 0 . (6.7)\nNote that the test scalar field Φ, as a perturbation field, does not back-react on the spacetime metric and is a function of all\nspacetime coordinates Φ = Φ( t, R, θ, φ ). For clarity, we choose to work in the physical coordinate system. Therefore, the\nbackground metric is given by eq. (3.12). We can apply the separation of variables as follows\nΦ(t, R, θ, φ ) =e−iωtYML\nL(θ, φ)Ψ(R)\nR, (6.8)\nwhere YML\nL(θ, φ)represents the spherical harmonics function. By using the tortoise coordinate, dr∗=dR√\nB(R)H(R), one can\nrewrite the perturbation equation in a Schr ¨odinger form as:\nd2Ψ(r∗)\ndr∗2+\u0002\nω2−V(R)\u0003\nΨ(r∗) = 0 , (6.9)\nwhere V(R)is the effective potential of the Schr ¨odinger equation and is given by\nV=HB′+BH′\n2R+B\nR2L(L+ 1) + m2B. (6.10)\nThe background metric functions BandH=B/W2are given in eqs. (3.13–3.14).\nIn the pursuit of calculating the QNMs, the WKB (Wentzel-Kramers-Brillouin) approximation stands as a valuable method.\nParticularly useful in the context of wave-like phenomena, the WKB approximation provides an efficient and semi-classical\napproach to estimating the complex frequencies associated with QNMs. By treating the Schr ¨odinger-like equation governing\nthe perturbations as a semiclassical wave equation, the WKB method allows for the determination of QNM frequencies without\nthe need for an exact solution. The WKB method was initially developed for quantum mechanical problems; however, Schutz\nand Will were the first to apply this method to the problem of scattering around black holes [126]. Later, Iyer and Will extended\nthis approach to the third WKB order beyond the eikonal approximation [127], and Konoplya further advanced it to the sixth\norder [125, 128]. Interestingly, the sixth order yields a relative error of approximately two orders of magnitude lower than that\nof the third WKB order [125, 128]. However, for simplicity, in this work, we will employ the first-order WKB approximation,\nin which the QNM frequencies are obtained from the solution of the following equation\nn+1\n2=−i\"\nω2−V(r∗)p\n−2V′′(r∗)#\nr∗=r∗max, n ∈Z≥. (6.11)\nThe expression in the right-hand part of the above equation is evaluated at the maximum of the potential r∗\nmaxwhile nis the\novertone number of the QNMs.\nIn Tables I and II, we present the dimensionless QNM frequencies, denoted by (Mω), for two distinct scenarios: when\nm= 0 andm/M = 0.4, respectively. Notably, as Qmapproaches 0, our solution converges to the Schwarzschild black hole,20\nirrespective of the (α−β)/M2parameter. This convergence is evident in the first row of both tables, where Qm/M= 0.01,\nas the QNM values remain constant across varying (α−β)/M2. Furthermore, as (α−β)/M2tends toward 0, our solution\nadopts the characteristics of the GMGHS black hole. Consequently, the QNMs in the first row of both tables, specifically when\n(α−β)/M2= 0.01, align with those of the GMGHS black hole. The subsequent rows in the tables provide additional insights\ninto the characteristics of our solution. For instance, in the second and third rows, where Qm/M= 0.3and0.6, respectively,\nwe observe a systematic variation in the QNM values with changes in both Qm/Mand(α−β)/M2. This behavior highlights\nthe sensitivity of the QNM frequencies to the parameters characterizing the black hole solution. Furthermore, by comparing\nthese results to the Schwarzschild and GMGHS cases, we discern how our solution deviates from these benchmark scenarios.\nAdditionally, the tables reveal intriguing patterns in the imaginary parts of the QNMs. For varying Qm/Mand(α−β)/M2,\nthe imaginary parts exhibit non-trivial changes, reflecting the impact of the black hole’s charge and the parameter (α−β)/M2\non the damping behavior of the perturbations.\nVII. OTHER SOLUTIONS\nA. Asymptotically (A)dS spacetimes\nLet us now, briefly discuss asymptotically (A)dS spacetimes. Following [129], introducing a scalar potential V(ϕ)in the\naction and considering\nS=Z\nd4x√−g\u0010\nR −2∇µϕ∇µϕ−e−2ϕF2+f(ϕ)\u0000\n−2αFα\nβFβ\nγFγ\nδFδ\nα+βF4\u0001\n−V(ϕ)\u0011\n, (7.1)\nwith aV(ϕ)of the form\nV(ϕ) =1\n3Λe−2ϕ+1\n3Λe2ϕ+4Λ\n3=2\n3Λ(cosh(2 ϕ) + 2) , (7.2)\nwe can obtain B(r)as\nB(r) = 1−2M\nr−2(α−β)Q4\nm\nr3\u0010\nr−Q2m\nM\u0011\n3−1\n3Λr\u0012\nr−Q2\nm\nM\u0013\n, (7.3)\nwhile ϕ(r), R(r)will remain the same. Note here that the potentials V(ϕ)andf(ϕ)are almost identical, and they are both\nLiouville-type potentials [70].6\nB. Solutions for general γ\nAssuming that the coupling term between the dilaton and the Maxwell term is of the form e−2γϕwe can obtain the same\ngeometry with the γ= 1case with the coupling function f(ϕ)now being given by\nf(ϕ) =−3 cosh(2 ϕ)−2−e2ϕ\u0010\u0000\ne−2ϕ\u0001γ−1−1\u0011\nQ6\nm\n2M4(e2ϕ−1)4(α−β). (7.4)\nIn this case, the charge-to-mass ratio is fixed by the theory. As a result, such black holes are described by a constrained phase\nspace of free parameters, since this situation reduces the number of primary black hole hairs from two to one. A more physical\nresult would be to let the form of the dilaton field to be affected by the change of the coupling function with the Maxwell term,\nhowever, we were not able to derive exact results in this case, so one has to employ numerical techniques. Such endeavors may\nbe undertaken in subsequent works.\n6Nonetheless, we should notice that the presence of a positive cosmological constant term in the gravitational effective actions is problematic within the context\nof string/brane-inspired quantum gravity theories, due to the so-called swampland conjecture [130–132]. This issue is open at present, and its study goes\nbeyond the purpose of the current article.21\nVIII. CONCLUSIONS\nIn the quest to comprehend gravitational phenomena and the nature of gravity itself, the theoretical exploration of black holes\nstands as a pivotal frontier. The predictions of General Relativity (GR) are in good agreement with current observations related\nto black holes. This is attributed to the large mass of the observed objects and therefore their large horizon radius and small\nhorizon curvature. Additionally, a plethora of cosmological observations indicates instances where GR exhibits limitations,\nwith the most notable challenges being the Dark Energy Problem and GR’s inability to account for the inflationary epoch in\nour universe. Therefore, the validity of General Relativity is expected to come under scrutiny in extreme conditions. General\nRelativity is commonly acknowledged as an effective theory applicable only within the realm of low energies. Consequently,\nsuch observations motivate us to explore modified gravitational theories, especially in extreme conditions where GR’s validity\nmay be compromised. Among these theoretical frameworks, modifications originating from String Theory, particularly the\nheterotic string theory, emerge as leading contenders. Notably, String Theory offers insights into high-order corrections, ranging\nfrom the Gauss-Bonnet term to non-linear electromagnetic effects, and provides a rich avenue for exploring the behavior of black\nholes under diverse conditions.\nOne intriguing aspect of string/brane-induced non-linear electrodynamics is the emergence of the Born-Infeld (BI) Lagrangian,\nwhich encapsulates higher-order corrections to Maxwell’s theory. This Lagrangian arises from the resummation of open string\nexcitations, particularly in the context of D-brane worlds in string theory. The coupling of the BI Lagrangian to the dilaton field in\ncurved spacetime leads to an effective four-dimensional action, offering a novel perspective on electromagnetic interactions in the\npresence of gravity. Furthermore, considerations of higher-order electromagnetic terms, originating from closed string sectors,\nbroaden the theoretical landscape. The inclusion of string loops leads to generalized effective actions, incorporating both closed\nand open string contributions, and potentially revealing novel phenomena beyond conventional electromagnetic frameworks.\nDeparting from traditional electromagnetic theories, the exploration of non-linear electrodynamics within the context of black\nhole solutions offers a rich avenue for understanding strong-field regimes and cosmological implications. Non-linear effects\nbecome crucial in regions with intense gravitational fields, such as those near black holes, shedding light on phenomena absent\nin linear theories. Moreover, non-linear electrodynamics holds relevance for early universe cosmology, where the interplay\nbetween gravitational and electromagnetic fields played a significant role.\nIn this work, we considered a string-inspired theory that involves a scalar field ϕcoupled to the electromagnetic field via a\nnon-linear function f(ϕ). The action encompasses higher-order electromagnetic invariants, contributing to the field equations\nand leading to novel black hole solutions. Furthermore, we investigated the impact of a non-trivial coupling function f(ϕ), con-\nsidering a specific functional form motivated by string-inspired models. The resulting exact, magnetically charged black hole\nsolution revealed significant departures from the classical General Relativity predictions, with the scalar field and the electro-\nmagnetic field configurations exhibiting non-trivial behavior. We explored the implications of different coupling constants αand\nβon the spacetime geometry and electromagnetic field configurations. The solutions obtained exhibit intriguing features, includ-\ning dependence on the sign of α−βwhich determines whether the higher-order electromagnetic terms contribute attractively or\nrepulsively to the spacetime geometry. Additionally, we examined the horizon structure of the black hole solutions, observing\ntransitions from single to multiple horizons and even to naked singularities as the parameters varied. Notably, the compactness\nof the black holes relative to the Schwarzschild solution depended on the the magnetic charge to mass ratio. Our findings suggest\na rich interplay between the scalar field, electromagnetic field, and spacetime geometry, highlighting the potential implications\nof such theories in astrophysical contexts, and the search for potential signatures of string theory in black hole physics, at least\nthose signatures that can be manifested through effective string-inspired field theory models. It goes without saying, however,\nthat the present work does not deal with a detailed experimental sensitivity analysis of such objects, which still remains to be\ndone.\nThe examination of geodesics and energy conditions delves into the intricate dynamics of particles moving within the space-\ntime geometry described by the black hole solutions under investigation. By analyzing the geodesic equations we unveil the\nbehavior of massive particles in the vicinity of these black holes, elucidating the role of the effective gravitational potential. No-\ntably, the effective potential exhibits distinct features depending on the relative values of the coupling constants, offering insights\ninto the stability and nature of orbits around the black holes. Additionally, the examination of energy conditions associated with\nthe effective stress-energy tensor reveals intriguing properties of the spacetime, indicating the existence of a dilaton hair in the\nblack hole’s exterior while satisfying all energy conditions. This observation challenges the traditional no-hair theorems, under-\nscoring the nuanced interplay between gravitational theories, non-linear electrodynamics, and scalar fields in modified theories.\nOur thermodynamic analysis provided valuable insights into the properties of black holes in both the GMGHS solution and our\nblack-hole solution with non-linear electrodynamics. Our analysis allows for the extraction of important thermodynamic quanti-\nties such as mass, entropy, magnetostatic potential, and the extraction of the first Law of Thermodynamics. Notably, the entropy\nof both black hole is consistent with the Bekenstein-Hawking entropy formula. By examining the behavior of the temperature,\nwe concluded that when the non-linear electrodynamics terms act attractively, there exist two distinct branches of black holes,\none that is getting colder as the mass is decreasing and therefore is thermally stable, and another one that is getting hotter as the\nmass is decreasing which is thermally unstable. On the other hand, when the non-linear electrodynamics terms have a repulsive\neffect, the black holes are getting colder as the mass is decreasing and as a result are thermally stable.22\nFinally, our analysis of linear perturbations and scalar quasi-normal modes (QNMs) provides valuable insights into the sta-\nbility and dynamic behavior of black hole solutions with non-linear electrodynamics. Through a rigorous investigation of radial\nstability, we demonstrated that our black hole solutions remain stable under linear and radial perturbations. This finding under-\nscores the robustness of our black hole solutions against radial perturbations, supporting their viability as physically meaningful\nconfigurations within the framework of non-linear electrodynamics. Furthermore, our examination of scalar QNMs yielded in-\ntriguing results regarding the characteristic vibrations and oscillations of the black hole spacetime. Moreover, the analysis of\nscalar QNMs revealed the intricate interplay between the black hole parameters, such as charge and (α−β)/M2, and the fre-\nquency and damping behavior of perturbations. By systematically varying these parameters, we observed distinct patterns in the\nQNM frequencies, indicating the sensitivity of the black hole’s dynamic properties to its intrinsic characteristics. Notably, our\nresults exhibited convergence to the Schwarzschild black hole in the limit of vanishing charge and alignment with the GMGHS\nblack hole in specific parameter regimes. 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We find an explicit\nexpression for det∆ for the so-called self-adjoint Szeg¨ o extens ion through the Bergman tau-\nfunction on the moduli space of Abelian differentials and the theta-c onstants (corresponding\nto the spinor bundle). This expression can be considered as a versio n of the well-known spin-\n1/2 bosonization formula of Bost-Nelson for the case of flat conform al metrics with conical\nsingularities and a higher genus generalization of the Ray-Singer for mula for flat elliptic curves.\nWe establish comparison formulas for the determinants of two differ ent extensions (e. g., the\nSzeg¨ o extension and the Friedrichs one). The paper answers a qu estion raised by D’Hoker and\nPhong [3]morethanthirtyyearsago. Wealsoreconsider theresult sfrom[3] ontheregularization\nof diverging determinant ratio for Mandelstam metrics (for any spin ) proposing (and computing)\na new regularization of this ratio.\n1 Introduction\nLetωbe a holomorphic one-form with 2 g−2 simple zeros P1,...,P 2g−2on a compact Riemann\nsurfaceXof genusg>1. Using the one-form ωone can construct\n•a hermitian metric hin then-th power ( n=±1,±2,...),Kn, of the canonical bundle K\noverX:\nh=ω−n¯ω−n\nand, in particular, a conformal Riemannian metric\n|ω|2:=ρ−2(z,¯z)|dz|2\nonX(or, equivalently, a hermitian metric h=ω¯ωin the holomorphic line bundle K−1).\n∗alexey.kokotov@concordia.ca\n†dmitrii.v.korikov@gmail.com\n1•a hermitian metric hin the (n+ 1/2)-spinor bundle C⊗Kn, whereCis one of the 4g\nholomorphic line bundles with property C⊗C=K:\nh=|ω|−1ω−n¯ω−n.\nAll these metrics have singularities at the zeroes of ω.\nForL=Knwithn≥2 (orL=Kn⊗Cwithn≥0) introduce the Dolbeault Laplacian\n∆L=−ρ2h−1∂(h¯∂) (1.1)\nRemark 1.1. Notice that for m=n(orm=n+1/2)the operator ∆−\nmfrom(2.3)of[3]coincides\nwith∆Lup to a numerical factor.\nIn [3] the authors, being motivated by the need to compute various partition functions ap-\npearing in the string theory, stated the problem of finding a reason able regularization of the\nformal expression\ndet′∆L\ndet<ei,ej>Ldet<e∗\nk,e∗\nl>L−1⊗K, (1.2)\nwhere det′stands for (modified, i. e. with excluded zero mode) determinant an d{ej},{e∗\nk}are\nthe bases of the spaces of holomorphic sections of LandL−1⊗Krespectfully.\nNotice that the quantities entering ”the Quillen metric” (1.2) are mea ningless due to the\nfollowing reasons:\n•it is unclear how to naturally associate to singular differential expres sion (1.1) an operator\nin Hilbert space, so neither of definitions of the determinant throug h the spectrum is\nimmediately applicable.\n•while the second Gram determinant in the denominator is absent in all t he cases except\nL=C(bundles of negative degree do not have holomorphic sections), th e first Gram\ndeterminant is infinite (again, in all the cases except L=C).\nIn [3] a regularization of (1.2) was proposed (see also ([21]) for a simila r result and the\ndiscussion in the Appendix to [11]), and question of the spectral th eoretic interpretation of (1.2)\nin caseL=Cwas raised.\nThe main idea of D’Hoker-Phong’s regularization (as well as Sonoda’s) is to apply Polyakov’s\nformula (that compares the determinants of Laplacians in two conf ormally equivalent smooth\nmetrics) to the pair (singular metric, Arakelov metric) and then to m ake use of the known\nresults on the Arakelov determinant. Of course, in case of singular metric the Polyakov’s formula\ncanbeappliedonlyformally(logically, thisapplicationconstitutes apar tofdefinition), moreover,\nit now contains diverging integral which should be cleverly regularized .\nThe present paper consists of two parts. In the short first part (Section 2) we propose a\nnew regularization of (1.2) and compute it explicitly (as in [3] no spectr al properties of (1.1) are\nneeded for that). It turns out that this part is relatively easy: th e key idea of the regularization\n(going back to [8]) was already used in a similar context in [12], all the res ults lying in the\nbackground were elaborated in [9], [10], [14] and [13]. In the much longe r second part (Sections\n3, 4, 5, 6) we address the D’Hoker-Phong question concerning the caseK=C. The main results\nof the second part are described in Section 3.\n2Acknowledgements. The research of the first author was supported by NSERC. The re search\nof the second author was supported by Fonds de recherche du Qu ´ ebec under the programme\n“Science en exil - chercheuses et chercheurs”.\nThe first author acknowledges numerous important discussions wit h Luc Hillairet during our\nstay at Max Planck Institute in Bonn in 2018, the first preliminary plan of the present work\nappeared as a result of these discussions. Several important idea s (in particular, the introduction\nof the holomorphic extension and the observation on the almost isos pectrality of the holomorphic\nand the Friedrichs extensions) used below go back to L. Hillairet’s pap ers on the scalar conical\nLaplacians.\n2 Regularization of Quillen metric\nIn this section we closely follow §3.2 from [12], where the case of L=K2and the metric given\nby|Q|, whereQis a holomorphic quadratic differential with simple zeros was considere d. The\nmethod of regularization for (1.2) used in [12] was taken from [8] (wh ere it was applied to scalar\nconical Laplacians) and remains the same in our case. Explicit comput ation of the resulting\nquantity is based on the same ingredients that were used in [12]: the r eference to Theorem 5.8\nfrom [4] and the statement on the essential coincidence of the two regularizations of the scalar\nconical Laplacians (Proposition 6 from [12]).\nLetL=Kn(withn≥2) orL=Kn⊗C(withn≥1). Following [17], introduce the moduli\nspaceHg(1,...,1) of pairs ( X,ω), whereXis a compact Riemann surface of genus gandωis\na holomorphic differential on Xwith 2g−2 simple zeroes. In [3] D’Hoker and Phong put into\ncorrespondence to any pair ( X,ω)∈Hg(1,...,1) (with a canonical basis of cycles on Xchosen) a\nbasis of holomorphic sections {e1,...eN},N= dimH0(L) of the line bundle LoverX; the latter\nbasis can be constructed explicitly (through ω, theta-functions, prime-form, etc). Say, for the\nsimplest case K2the basis of N= 3g−3 holomorphic quadratic differentials on Xcorresponding\nto a pair (X,ω)∈Hg(1,...,1) is given by\nωvk;k= 1,2,...,g;ωΩP2g−2,Pk, k= 1,...,2g−3,\nwhere{vk}g\nk=1is the basis of normalized holomorphic differentials, P1,...,P 2g−2are the zeros of\nωand Ω a−bis the Abelian differential of the third kind with zero a-periods and simple poles at\na,bwith residues 1 and −1. We refer the reader to p. 641 of [3] for the construction of the basis\nin the general case KnorKn⊗C. We will call this basis the D’Hoker-Phong basis .\nIn a vicinity of a simple zero Pkof a holomorphic differential ωthe metric|ω|2coincides with\nmetric of the standard round cone of the angle 4 π\n|ω|2=|xk|2||dxk|2\nin the so-called distinguished holomorphic coordinate\nxk(Q) =/parenleftbigg\n2/integraldisplayQ\nPkω/parenrightbigg1/2\n. (2.1)\nSmoothing all these standard cones in ǫ-vicinities of their tips (the zeros of differential ω) one\ngets a smooth conformal metric ρ−2\n(ǫ)|dz|2=|ω|2\n(ǫ)onX. One can perform this smoothing without\nchanging the area, thus, we assume that/integraltext\nX|ω|2=/integraltext\nX|ω|2\n(ǫ). Simultaneously, one gets a smooth\nhermitian metric h(ǫ)inL.\n3Denote byG({ek};X,|ω|2\n(ǫ),h(ǫ)) the Gram determinant of the hermitian products of the\nelements of the D’Hoker-Phong basis w. r. t. the metrics ( |ω|2\n(ǫ),h(ǫ)).\nNotice, that, since the metrics |ω|2\n(ǫ)andh(ǫ)are smooth, one can introduce the ζ-regularized\nmodified(i. e. withzeromodesexcluded)determinantoftheDolbeau ltLaplaciandet′∆|ω|2\n(ǫ)(X,ω)\ninLcorresponding to these metrics. Moreover, the Gram determinan t for basic holomorphic sec-\ntionsekof the bundle Lequipped with metric h(ǫ)is finite.\nChoose a compact Riemann surface X0of genusgand a holomorphic differential ω0with\n2g−2 simple zeros on X0(in other words we choose an element ( X0,ω0) of the the moduli space\nHg(1,...,1)).\nNow for any ( X,ω)∈Hg(1,...,1) introduce the ratio\nR(X,ω;ǫ) =det′∆|ω|2\n(ǫ)(X,ω)/parenleftBig\nG({ek};X,|ω|2\n(ǫ),h(ǫ))/parenrightBig−1\ndet′∆|ω0|2\n(ǫ)(X0,ω0)/parenleftBig\nG({ek};X0,|ω0|2\n(ǫ),h0\n(ǫ))/parenrightBig−1.\nThe following key observation (in case of scalar conical Laplacians) w as first made in [8]. The\nproofrepeatstheproofofLemma2from[12]verbatim. Essentially , itreducestoastraightforward\napplication of Proposition3.8 from[4] (a version ofPolyakov’s formula forDolbeault Laplacians).\nLemma 2.1. For sufficiently small ǫthe quantity R(X,ω;ǫ)isǫ-independent.\nDefinition 2.2. Let(X,ω)be an element of Hg(1,...,1). LetL=Knwithn≥2orK=Kn⊗C\nwithn≥1and let the metrics on Xand inLare constructed via the holomorphic differntial ω.\nAssign to the formal expression from (1.2)the following value:\ndet′∆L\ndet<ei,ej>L:= lim\nǫ→0R(X,ω;ǫ) (2.2)\nThus, quantity (1.2) is correctly defined up to moduli (i. e. coordina tes in the space\nHg(1,...,1)) independent overall constant (ruled by the choice of basic elem ent (X0,ω0)).\nIt turnsout thatone canexplicitly computethis quantity. To this en dlet us specify thechoice\nof the spinor bundle C. Let∆be the line bundle of degree g−1 (simultaneously, the linear\nequivalence classofdivisors onX)with AP0(∆) =−KP0. HereP0isabasicpoint,AP0istheAbel\nmap,KP0is the vector of Riemann constants. Let p,q∈{0,1\n2}gand letχp,qbe the line bundle\nof degree zero (simultaneously the linear equivalence class of divisor s) withAP0(χp,q) =Bp+q,\nwhereBis the matrix of b-periods of X. Let\nC=∆⊗χp,q.\nAlso, let det′∆0(|ω|2\n(ǫ),X) be theζ-regularized determinant of the scalar Laplacian in the smooth\nmetric|ω2|2\n(ǫ)and let\nc0(X,|ω|2\n(ǫ))−2=det′∆0(|ω|2\n(ǫ),X)\ndetℑBArea(X,|ω|2\n(ǫ))\n(we remind the reader that Area( X,|ω|2\n(ǫ)) = Area(X,|ω|2)).\nLetm= 2n+1 ifL=Kn⊗Candm= 2nifL=Kn. Then\nR(X,ω,ǫ) =/bracketleftbigg\ndet′∆|ω|2\n(ǫ)(X,ω)/parenleftBig\nG({ek};X,|ω|2\n(ǫ),h(ǫ))/parenrightBig−1\nc(3m2−6m+2)\n0 (X,|ω|2\n(ǫ))/bracketrightbigg\ndet′∆|ω0|2\n(ǫ)(X0,ω0)/parenleftBig\nG({ek};X0,|ω0|2\n(ǫ),h0\n(ǫ))/parenrightBig−1×\n×c−(3m2−6m+2)\n0 (X,|ω|2\n(ǫ)).(2.3)\n4According to Theorem 5.8 from [4], the expression in the square brac kets in (2.3), coincides (up\nto a moduli independent constant) for L=C⊗Kn,m= 2n+1 with\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleθ/bracketleftbigp\nq/bracketrightbig/parenleftBig\n(m−1)KP0+/summationtextN\ni=1AP0(x1+···+xN)/parenrightBig/producttextN\ni<jE(xi,xj)\ndet||ei(xj)||/producttextN\nk=1c(xi)N−1\ng−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,\nwherex1,...,x Nare arbitrary points of X,θ/bracketleftbigp\nq/bracketrightbig\nis theta function defined in (1.8) in [4], E(x,y)\nis the prime form, and c(·) is defined in (1.17) in [4]. In case L=Kn,m= 2nthe expression in\nthe square brackets in (2.3) coincides with |Mg,n|2, where\nMg,n=θ((2n−1)KP0)\nW[e1,e2,...,e N](P0)c(P0)(2n−1)2\nwithN= (2n−1)(g−1) andWbeing the Wronskian for the D’Hoker-Phong basis (cf. Theorems\n5.8 and 1. 6 from [4]).\nOntheother hand, according to Proposition6 from[12], thedeterm inant det′∆0(|ω|2\n(ǫ),X)co-\nincides upto moduli independent constant with the determinant det′∆0(|ω|2,X)of the Friedrichs\nextension of the symmetric scalar Laplacian in the conical metric |ω|2, and, therefore, due to the\nresults of [13] one has the relation\nc0(X,|ω|2\n(ǫ))−2= const|τB(X,ω)|2,\nwhereτBis the Bergman tau-function on the space Hg(1,...,1) (see (3.1), [13] for an explicit\nexpression for τB). Since the expression in the denominator of (2.3) is moduli independ ent, all\nthis leads to an explicit expression for (2.2) through ωand holomorphic invariants of the surface\nXup to moduli independent constant.\nCase of the spinor bundle C=∆⊗χp,q\nThe regularization (2.2) with no changes is applicable in the case of the spinor bundle L=C.\nHowever, in this case Theorem 5.8 is no longer relevant, and to compu te the regularized quantity\n(2.2) one instead uses the bosonization formula of Bost and Nelson ( [2]) (cf. [4], Theorem 4.9).\nFor simplicity, we restrict ourselves to the case of even character istics (i. e. even 4 <p,q>). In\nthis caseN=h0(∆⊗χp,q) = 0 generically.\nAccording to the Bost-Nelson result one has\n′\ndet∆(|ω|2\n(ǫ),X) = const/parenleftBigg\ndet′∆0(|ω|2\n(ǫ),X)\nArea(X,|ω|2\n(ǫ))detℑB/parenrightBigg−1/2\n|θ[p\nq](0)|2\nand formula (2.3) (with no Gram determinants!) together with Propo sition 6 from [12] and\nresults from [13] give\nR(X,ω,ǫ) = lim\nǫ→0R(X,ω,ǫ) = const|θ/bracketleftbigp\nq/bracketrightbig\n(0)|2|τB(X,ω)|−1(2.4)\nIn the remaining part of the paper we clarify to what extent the reg ularization (2.4) of the\nspinor conical determinant iscompatiblewithpossible spectral regu larizationsofthedeterminant\nof the Laplacian acting in ∆⊗χp,qwith conical metric.\n53 Expressions for determinants\nAs emphasized after (1.2), the problem arising with the spectral re gularization of Laplacians on\nmanifolds with conical points is that there are many different ways to associate to differential\nexpression (1.1) a self-adjoint operator.\nNotation. We will use the following notation:\n•L2(X;L)is the space of square integrable sections of Lwith scalar product\n(a,b)L2(X;L):=/integraldisplay\nXahbdS, (3.0)\nwherehis a metric on LanddS:=ρ−2dzdz/2iis the area form.\n•˙Xis the surface obtained by deleting all conical points from XandC∞\nc(˙X;L)is the set of\nsmooth sections of Lwith supports contained in ˙X.\n•c...\n...(...)are constants depending only on the indices and the paramete rs in brackets.\n•z/ma√sto→v(z)as a rule means the section vof a line bundle L(which can be identified from the\ncontext)overXwritten in the local parameter z. Also, we denote v(Pk) :=v(xk)|xk=0.\nIn what follows, we deal with the spinor bundle L=Cand denote by ∆ the Laplacian ∆ C\ngiven by (1.1). Westart with the minimal closed extension ∆ ·inL2(X,C) of Laplacian ∆defined\nonC∞\nc(˙X;C) and denote by ∆∗\n·its adjoint in L2(X;C). In view of the Green formula\n(∆Lf,f′)L2(U;L)−(f,∆Lf′)L2(U;L)=1\n2i/integraldisplay\n∂U/parenleftbig\nfh∂f′dz+f′h∂fdz/parenrightbig\n(3.1)\nwithU=XandL=C, we have ∆ ·⊂∆∗\n·.\nNext, we describe all the self-adjoint extensions of ∆ ·. To this end, we consider the quotient\nspace Dom∆∗\n·/Dom∆ ·endowed with the symplectic bilinear form\nG(π(u),π(v)) := (∆∗\n·u,v)L2(X;C)−(u,∆∗\n·v)L2(X;C)(u,v∈Dom∆∗\n·), (3.2)\nwhereπis the canonical projection from Dom∆∗\n·onto Dom∆∗\n·/Dom∆ ·. The operator ∆ ⊂∆∗\n·is\nself-adjointifandonlyif π(Dom∆)isalagrangiansubspaceofDom∆∗\n·/Dom∆ ·. Theselagrangian\nsubspaces are easily described by the use of the following propositio n (proved inparagraph4.1.3).\nProposition 3.1. The Darboux basis on Dom∆∗\n·/Dom∆ ·can be chosen in the form\n{π(χkfk,m,+)}2g−2,2\nk=1,m=−1,{π(χkfk,m,−)}2g−2,2\nk=1,m=−1, (3.3)\nwhere each χkis a smooth cut-off function equal to one near Pk, the support of χkis sufficiently\nsmall, and fk,m,±are local sections of Cgiven by\nfk,m,+(xk) :=xm\nk, f k,m,−(xk) :=xk1\n2−mx1\n2\nk\nπ(m−1\n2)(3.4)\nin local coordinates (2.1).\n6We restrict ourselves to considering the following three self-adjoin t extensions of ∆ ·.\n1) Friedrichs extension ∆F. The following proposition is proved in paragraph 4.2.1.\nProposition 3.2. The domain of ∆Fconsists of all sections u∈Dom∆∗\n·admitting the asymp-\ntotics\nu(xk) = (−2/π)ck,0,−|xk|+ck,1,+xk+ck,2,+x2\nk+(−2/3π)ck,−1,−xk|xk|+ ˜u(xk) (3.5)\nnear eachPk, where˜u∈Dom∆ ·.\nIn Subsection 4.3, we derive the short-time asymptotics for the he at kernel H(·,·,t) corre-\nsponding to the equation ( ∂t+∆F)u= 0. As a corollary, we prove the following statement.\nProposition 3.3. The zeta function s/ma√sto→ζ(s|∆F)of∆Fadmits analytic continuation on the\nwhole complex plane except the pole s= 1.\n2) Szeg¨ o extension ∆S. Recall that, for smooth metrics ρ−2andh, the Szeg¨ o kernel Sis\ndefined by\nS(z,z′) :=−πh(z′)∂z′G(z,z′), (3.6)\nwhereGis the Green function for the (self-adjoint) Laplacian ∆ = −ρ2h−1∂h∂(see p.25, [4]).\nAlso, the inversion formula\nG(z,z′′) =−1\nπ2/integraldisplay\nXS(z,z′)h(z′)S(z′,z′′)dS(z′) (3.7)\nholds. At the same time, for the case of spinor bundle Cobeyingh0(C) = 0, the Szeg¨ o kernel\nis independent of the choice of metrics ρ−2andhand is given by the explicit formula (see p.29,\n[4])\nS(z,z′) =θ/bracketleftbigp\nq/bracketrightbig/parenleftbig\nAP0(z′−z)/parenrightbig\nθ[p\nq](0)E(z,z′). (3.8)\nIn the non-smooth case, we define the Szeg¨ o extension ∆ Sas a self-adjoint extension of ∆ ·\nwhose Green function G=GSis related to Szeg¨ o kernel (3.8) via formulas (3.6) and (3.7). It\nis easy to see that such an extension is unique. Indeed, formula (3.7 ) uniquely determines the\nGreen functionG=GSand, therefore, the kernel Ker∆ Sand the inverse operator ∆−1\nSon the\northogonal complement Ker∆ SinL2(X;C). Thus, ∆ Sis uniquely determined by Svia (3.7).\nTheexistence oftheSzeg¨ o extension andthedescription ofitsdo mainfollowfromProposition\n3.4 proved in paragraph 4.2.2.\nProposition 3.4. The Szeg¨ o extension ∆Sexists and its domain consists of all u∈Dom∆∗\n·\nadmitting the asymptotics\nu(xk) =ck,0,++ck,1,+xk+ck,2,+x2\nk+(−2/3π)ck,−1,−xk|xk|+ ˜u(xk) (3.9)\nnear eachPk, where˜u∈Dom∆ ·.\n3) Holomorphic extension ∆h. The domain of ∆ hconsists of all u∈Dom∆∗\n·admitting\nthe asymptotics\nu(xk) =ck,−1,+x−1\nk+ck,0,++ck,1,+xk+ck,2,+x2\nk+ ˜u(xk) (3.10)\n7near eachPk, where ˜u∈Dom∆ ·. Note that the principal part of the above expansion contains\nonly holomorphic local sections (3.4).\nThepeculiarityoftheholomorphicextension∆ histhatitcoincides, uptounitaryequivalence,\nwith the self-adjoint operator which can serve as a natural spect ral theoretic counterpart of the\nD’Hoker-Phong’s Laplacian, ∆(3\n2), acting on spin-3\n2fields [3].\nBriefly, this can be explained as follows. The multiplication by ωis a unitary operator acting\nfromL2(X;C) ontoL2(X;C⊗K). Thus, the minimal operators (with domains consisting of\nsmooth sections with support in ˙X) corresponding to ∆(3\n2)and ∆ are unitary equivalent. The\nself-adjoint extension ∆(3\n2),hof the minimal operator ∆(3\n2)with domain containing the D’Hoker-\nPhong basis of holomorphic sections (we consider this requirement a s the most natural) turns\nout to be unitary equivalent to the holomorphic extension ∆ h,\n∆(3\n2),h=ω∆hω−1.\nThus, one can define\ndet′∆(3\n2):= det′∆(3\n2),h= det′∆h. (3.11)\nIn paragraph 4.2.2, we prove the following statement.\nProposition 3.5. The operators ∆Fand∆hare almost isospectral. As a corollary, we have\ndet′∆h= det∆ F. (3.12)\nNext, we derive the comparison formulas for the ζ-regularized determinants of ∆ F, ∆h, ∆S.\nFirst, wehavedet′∆h= det∆ FduetoProposition3.5. InSection5, wecomparethedeterminants\nof ∆Fand ∆ S. To this end, we apply the method developed in [9] and based on the st udy of the\nbehavior of the S-matrixS(λ) associated with ∆ F−λand ∆ S−λ. As a result, we obtain the\nfollowing theorem.\nTheorem 3.6. The zeta function s/ma√sto→ζ(s|∆S)of∆Sadmits analytic continuation on the whole\ncomplex plane except the pole s= 1. The determinants of ∆Sand∆Fare related via\ndet∆F= Γ(3/4)4(g−1)detT(0)det∆ S, (3.13)\nwhereT(0)is the(2g−2)×(2g−2)-matrix with entries\nTij(0) =−1\nπ2|θ[p\nq](0)|2/integraldisplay\nXθ/bracketleftbigp\nq/bracketrightbig\n(AP0(Pi−z))θ[p\nq](AP0(z−Pj))\nE(Pi,z)E(z,Pj)dSz\n|ω(z)|.(3.14)\nLet us compare formula (3.13) with the results of D’Hoker-Phong’s r egularization [3]. Fol-\nlowing [3], introduce the ratios (1.2) for Laplacians acting on spin-1\n2and spin-3\n2fields as follows\nW1/2= det∆ S, W 3/2=det′∆(3\n2)\ndet{(φ(3/2)\ni,φ(3/2)\nj)L2(X;S3)}ij,\nwhereφ(3/2)\nj=ωS(Pj,·) are zero modes of ∆(3\n2)(compare with formula (5.10), [3]) while the\nkernel of ∆ Sis trivial. Since the multiplication by ωis a unitary operator acting from L2(X;C)\nontoL2(X;C⊗K), we have\nW3/2=det′∆(3\n2)\ndetS,\n8whereSis the (2g−2)×(2g−2)-matrix with entries\nSkj= (S(Pk,·),S(Pj,·))L2(X;C). (3.15)\nwhich is related to T(0) via\nS=π2T(0) (3.16)\n(see formula (5.12) below). In view of Theorem 3.6, the ratio\nW3/2\nW1/2=det′∆(3\n2)\ndet∆S1\ndetS=/bracketleftbig\n(3.11),(3.12)/bracketrightbig\n=det∆F\ndet∆S1\ndetS=/bracketleftbig\n(3.13),(3.16)/bracketrightbig\n=/parenleftbiggΓ(3/4)\nπ/parenrightbigg4(g−1)\n(3.17)\nis moduli independent, which is, in a sense, compatible with D’Hoker-Ph ong formula (6.2), [3].\nNamely, in [3] the left hand side of (3.17) is defined and computed as a t ensor-like object depend-\ning on the choice of local parameters at the conical points. Naively c hoosing these parameters\nto be distinguished (see (2.1)), one gets a moduli independent cons tant (which, unfortunately,\ndiffers from the right hand side of (3.17)).\nFinally, we find the explicit expressions for det∆ Sthrough the Bergman tau-function on\nthe moduli space of Abelian differentials and the theta-constants ( corresponding to the spinor\nbundle). To this end, we apply the variational technique developed in [13]. First, we introduce\nthe family of the surfaces corresponding to the variation of the co ordinatesνin moduli space\nHg(1,...,1). Next, we study the dependence of the resolvent kernels Rof the (Szeg¨ o) Laplacians\nonνand define and calculate the derivatives Rwith respect to ν. Based on this information, we\ncalculate the derivative of the Laplacian eigenvalues and zeta funct ions with respect to ν. Due\nto this, we compute the det∆ Sup to a multiplicative moduli independent constant c.\nTheorem 3.7. We have\ndet∆S(|ω|,X) =c/vextendsingle/vextendsingleθ/bracketleftbigp\nq/bracketrightbig\n(0)/vextendsingle/vextendsingle2|τB(X,ω)|−1. (3.18)\nThe dependence of det∆ F= det′∆hon moduli and spin structure is more sophisticated and\nis provided by the comparison of formulas (3.13) and (3.18).\nComparing (3.18) with (2.4), we conclude that the method of the reg ularization of det∆\ndescribed in Section 2 is compatible with the spectral regularization. Also, comparing (3.18)\nwith formula (1.10) from [13], we obtain the formula\ndet∆S(|ω|,X) =c/parenleftbiggdet′∆0(|ω|2,X)\nArea(X,|ω|2)detℑB/parenrightbigg−1/2\n|θ[p\nq](0)|2\nwhich extends the Bost-Nelson bosonization result to the case of ( non-smooth) metrics |ω|2on\nX andh=|ω|onC.\n4 Preliminaries\nIn this section, we present auxiliary facts which are used in the proo fs of Theorems 3.6 and 3.7.\nIn particular, we prove Propositions 3.1–3.5.\n94.1 Local properties of solutions to (∆−λ)u=f\nLetu,f∈L2(X;C). We say that uis a generalized solution to the equation (∆ −λ)u=fif\n(f,v)L2(X;C)= (u,(∆−λ)v)L2(X;C)for anyv∈C∞\nc(X;C). Note that, in local coordinates\nz(Q) :=/integraldisplayQ\nQ0ω, (4.1)\nwe haveω=dz,g=h= 1, and\n∆ =−1\n4(∂2\nℜz+∂2\nℑz). (4.2)\nIn this subsection, we describe the well-known properties (such as smoothness and asymptotics\nnear conical points) of the solution uarising from the ellipticity (in local coordinates (4.1)) of\nthe Laplacian ∆. The most used result is the following statement.\nProposition 4.1. Letu∈L2(X;C)be a generalized solution to the equation (∆−λ)u=f\nwheref∈L2(X;C),f∈Cl(˙X), and\n|∂p\nxk∂q\nxkf(xk)|≤cf|xk|T−p−q−4(p+q≤l, T >5/2) (4.3)\nnearPk, wherexkis the distinguished coordinate (2.1). Thenu∈Cl(˙X)and it admits the\nasymptotics\nu(xk) =/summationdisplay\n(m,±)cλ\nk,m,±(u)fk,m,±(xk)/summationdisplay\nn≥0d(n,±iµm)/parenleftbiggλ|xk|4\n4/parenrightbiggn\n+ ˜u(xk) (4.4)\nnearPkwith the remainder ˜usatisfying\n|∂p\nxk∂q\nxk˜u(xk)|≤cλ,ǫ/parenleftbig\ncf+/bardblu/bardblL2(X;C)/parenrightbig\n|xk|T−p−q+ǫ(p+q≤l) (4.5)\nfor anyǫ>0. Herefk,m,±are the local sections of Cgiven by (3.4), the coefficients cλ\nk,m,±(u)are\nlinear functionals of ugiven by\ncλ\nk,m,±(u) = (f,χkfk,m,∓)L2(X;C)−(u,(∆−λ)[χkfk,m,∓])L2(X;C), (4.6)\nwhere the cut-off function χkis defined before (3.4)and(·,·)L2(X;C)denotes the extension of the\nscalar product (3). Also,µm=i(1−2m)/4, and the numbers d(n,q)are defined by the chain of\nequations\nd(0,q) = 1,−n(q+n)d(n,q) =d(n−1,q) (n>0). (4.7)\nThe right hand side of (4.4)includes those terms that are quadratically integrable (in metrics\nρ−2,h)near the vertex Pkand decrease slower than O(|xk|N)asxk→0. The constant cλin(4.5)\nhas at most polynomial growth as |λ|→∞.\nProposition 4.1 is analogous to the statements describing propertie s of solutions to elliptic\nboundaryvalueproblems inpiecewise smoothdomains presented in[1 5, 19]. Unfortunately, there\nis no appropriate reference for the case ofclosed manifolds, and, for theconvenience ofthe reader,\nwe give the full proof, using the key elements of the reasoning from [15, 19] in a simplified and\nself-contained form. Next, applying the same reasoning to describ e the asymptotics of solutions\n∆∗\n·u=f, we prove Proposition 3.1. Finally, we prove that the ellipticity of ∆ implie s the\ndiscreteness of spectra of self-adjoint expansions of ∆ ·, a well-known fact in the smooth case, but\nnevertheless needed to be verified in the non-smooth one.\n10Notation. In the sequel, the neighborhoods Uk, the coordinates zk, and the cut-off functions χk,\n˜χk(k= 1,...,K)satisfy the following properties\n•Uk-s constitute the finite open cover of XandPk∈Ujif and only if k=j.\n•zkis given by (4.1), where the integration path lies in UkandQ0=Pkfork≤2g−2.\n•the mapzk:Uk→Cis injective for k>2g−2.\n•the mapxk:Uk→Cis injective for k≤2g−2, wherexkis the distinguished coordinate\n(2.1). Note that z=x2\nk/2.\n•χk,˜χkare smooth cut-off functions on Xobeying\nsupp˜χk⊂Uk,˜χkχk=χk,K/summationdisplay\nk=1χk= 1.\n•theχk(Q),˜χk(Q)withk≤2g−2depend only on the distance between QandPk.\nFor any section vany line bundle L, we denote by v(k)its representatives in coordinates zk.\nThe representatives v(k)(k≤2g−2)near conical points Pkare considered as functions of the\npolar coordinates\nrk=|xk|2/2, ϕk= 2argxk\nor coordinates sk:= logrk,ϕk. Note that zk=rkeiϕk=esk+iϕk.\nIn addition, we use the cut-off functions χk,ǫonXwith supports shrinking to Pkasǫ→0.\nEachχk,ǫvanish outside Ukand is given by\nχk,ǫ(rk,ϕk) =χ(ε−1rk) (4.8)\ninUk, whereχis a smooth cut-off function on Requal to one near zero.\n4.1.1 Increasing smoothness outside conical points.\nLetk>2g−2i.e.Ukisseparatedfromconicalpoints. Sinceoperator(4.2)iselliptic, the inclusion\nof ˜χkf(k)into the Sobolev space Hl≡Hl(C) (l= 0,1,...) implies the inclusion χku(k)∈Hl+2\nand the estimate\n/bardblχku(k)/bardblHl+2≤cl(/bardbl˜χkf(k)/bardblHl+/bardbl˜χku/bardblL2(X;C)). (4.9)\nAlso, due to Morrey’s inequality we have u(zk)∈Cl≡Cl(C) and\n/bardblχku(k)/bardblCl≤cl/bardbl˜χku(k)/bardblHl+2. (4.10)\nCombining the two last estimates, we obtain\n/bardblχku(k)/bardblCl≤cl(/bardbl˜χkf(k)/bardblCl+/bardbl˜χku/bardblL2(X;C)). (4.11)\nIn addition, if fis smooth outside conical points, then so is uand the equation (∆ −λ)u=f\nholds pointwise on ˙X.\n114.1.2 Asymptotics near conical points.\nReduction to a model problem. Letk≤2g−2 i.e.Ukis a neighborhoodof Pk. Throughout\nthe paragraph, the subscript kis usually omitted in the notation of coordinates xk,zk,rk,ϕkand\ncut-off functions χk, ˜χk. Also, we assume that λ= 0 (the general case will be considered later).\nPut\ny(r,ϕ) :=χu(z), w(r,ϕ) :=−4r2/parenleftbig\nχf(z)+[∆,χ]u(z)/parenrightbig\n. (4.12)\nIn view of (4.2), the equation ∆ u=fimplies\n[∂2\ns+∂2\nϕ]y= [(r∂r)2+∂2\nϕ]y=w. (4.13)\nAlso, since the transition function of the spinor bundle C=√\nKcorresponding to the change of\nvariablesx/ma√sto→zis equal to\n±/radicalbig\n∂x/∂z=±2−1\n4r−1\n4e−iϕ\n4,\nthe representatives of u,fin coordinate zare 4π-antiperiodic in ϕ, i.e.,\nv(r,ϕ+α) =−v(r,ϕ) (v=y,w, α = 4π). (4.14)\nApplying the complex Fourier transform\nˆv(µ,ϕ) :=1√\n2π/integraldisplay+∞\n−∞e−iµsv(s,ϕ)ds,\none can rewrite (4.13), (4.14) as\n[∂2\nϕ−µ2]ˆy(µ,ϕ) = ˆw(µ,ϕ),ˆy(µ,ϕ+α) =−ˆy(µ,ϕ). (4.15)\nThe resolvent kernel of problem (4.15) is given by\nRα(ϕ,ϕ′,µ2) :=sh(µ(|ϕ−ϕ′|−α/2))\n2µch(αµ/2)(ϕ,ϕ′∈[0,α)). (4.16)\nIt is meromorphic function of µwith simple poles\nµm=iπα−1(1−2m) =i(1−2m)/4 (m∈Z). (4.17)\nThe estimate\n/bardblRα(·,·,µ2)/bardblL2([0,α]2)=O(|µ|−2),/bardbl∂ϕRα(·,·,µ2)/bardblL2([0,α]2)=O(|µ|−1),(4.18)\nis valid for large|ℜµ|.\nWeighted spaces. Denote by Hl\nβthe space of α-antiperiodic (with respect to ϕ) functions\nwith finite norms\n/bardblv/bardblHlν=\nsup\nφ0/summationdisplay\np+q≤l+∞/integraldisplay\n0rdrφ0+α/integraldisplay\nφ0dϕ|∂q\nϕ(r∂r)pv(r,ϕ)|2r2(ν−1)\n1/2\n=\n=\nsup\nφ0/summationdisplay\np+q≤l+∞/integraldisplay\n−∞dsφ0+α/integraldisplay\nφ0dϕ|∂q\nϕ∂p\nsv(s,ϕ)|2e2νs\n1/2\n.(4.19)\n12In view of the Parseval identity, we have the following equivalence of the norms\n/bardblv/bardblHlν≍\nsup\nφ0/summationdisplay\np+q≤lνi+∞/integraldisplay\nνi−∞|µ|2p/bardbl∂q\nϕˆv(µ,ϕ)/bardbl2\nL2([φ0,φ0+α])dµ\n1/2\n. (4.20)\nAs it easy to see, the convergence\n/bardbl(1−χǫ)χ1/ǫv−v/bardblHlν→0, ǫ→0\nis valid for any v∈Hl\nν, where the cut-off function χǫis given by (4.8) with omitted k. Hence,\nwe obtain the following fact.\nRemark 4.2. The set of smooth α-antiperiodic (with respect to ϕ)functions vanishing for suf-\nficiently small and large ris dense in Hl\nν.\nLet us provide the analogues of inequality (4.9), (4.10) for weighted spacesHl\nν. Letκ,κbe\nsmooth cut-off functions on Rsuch thatκκ=κ,κ(s) = 1 fors∈[−3/2,3/2] andκ(s) = 0 for\ns∈[−2,2]. Forj∈Z, denoteκj(s) :=κ(s−j) andκj(s) :=κ(s−j). Since the operator ∂2\ns+∂2\nϕ\nis elliptic, the estimate\n/bardblκjv/bardblHl+2(Π)≤c(/bardblκjv/bardblL2(Π)+/bardblκj(∂2\ns+∂2\nϕ)v/bardblHl(Π))\nis valid, where Π = R×[φ0,φ0+α]. Multiplying both parts by e2νjand making the summation\nover integer j, we arrive at the estimate\n/bardblv/bardblHl+2\nν≤c(/bardblv/bardblH0ν+/bardbl(∂2\ns+∂2\nϕ)v/bardblHlν). (4.21)\nAlso, due to the Morrey’s inequality we have\n|∂q\nϕ(r∂r)pv(r,ϕ)|=|∂q\nϕ∂p\nsv(r,ϕ)|≤c/bardblκjv/bardblHl+2(Π)≤\n≤c/bardblv/bardblHl+2\nνe−νk≤c/bardblv/bardblHl+2\nνr−ν,(p+q≤l)(4.22)\n(herejis chosen in such a way that r∈[j−3/2,j+3/2]).\nProblem (4.13), (4.14) in the scale of weighted spaces. With problem (4.13), (4.14), we\nassociate the continuous operator Aν,l:y/ma√sto→(∂2\ns+∂2\nϕ)yacting from Hl+2\nνtoHl\nν.\nSuppose that the line ℑµ=νdoes not intersect the poles µngiven by (4.17). If y∈Hl+2\nν,\nthenw=Aν,ly∈Hl\nνand the complex Fourier transforms ˆ y, ˆware defined and obey (4.15) for\nℑµ=νand almost allℜµ∈R. Expressing ˆ yin terms of ˆ wand the resolvent kernel Rα(given\nby (4.16)) and then applying the inverse Fourier transform, we obt ain\ny(s,ϕ) =1√\n2πνi+∞/integraldisplay\nνi−∞dµα/integraldisplay\n0dϕ′eiµsRα(ϕ,ϕ′,µ2)ˆw(µ,ϕ′). (4.23)\nIf it is known only that w∈Hl\nν, then formula (4.23) provides the (unique) solution y=A−1\nν,lwto\n(4.13), (4.14) which belongs to y∈Hl+2\nν. Indeed, if follows from (4.18) and the Cauchy–Schwarz\ninequality that\n|µ|2/bardblˆy(µ,·)/bardblL2([0,α])+|µ|/bardbl∂ϕˆy(µ,·)/bardblL2([0,α])≤c/bardblˆw(µ,·)/bardblL2([0,α]).\n13Since equation (4.15) implies ∂2(k+1)\nϕˆy=∂2k\nϕˆw+µ2∂2k\nϕˆy, the iteration of the last formula yields\n/bardbl∂2(k+1)+q\nϕˆy(µ,·)/bardblL2([0,α])≤|µ|2/bardbl∂2k+q\nϕˆy(µ,·)/bardblL2([0,α])+/bardbl∂2k+q\nϕˆw(µ,·)/bardblL2([0,α])≤\n≤2k/summationdisplay\nm=0|µ|2m+q/bardbl∂2(k−m)+q\nϕˆw(µ,·)/bardblL2([0,α])(q= 0,1,2(k+1)+q≤l+2).\nSincew∈Hl\nνthe right hand side of (4.20) with w,ˆwinstead ofv,ˆvis finite. Thus, due to above\nestimates, the right hand side of (4.20) with y,ˆy,l+2 instead of v,ˆv,lobeys\n/bardbly/bardblHl+2\nν≤cν/bardblw/bardblHlν, (4.24)\ni.e.,A−1\nν,lis continuous if νdoes not coincide with any ℑµm.\nNow, suppose that ν,ν′/\\e}atio\\slash=ℑµnfor anyn,ν′< ν, andy∈Hl+2\nνandy′∈Hl+2\nν′are two\nsolutions to (4.13), (4.14) with the joint right hand side w=Aν,ly=Aν′,ly′. Thenw∈Hl\nν′′for\nanyν′′∈[ν′,ν] andµ/ma√sto→ˆw(µ,·) is holomorphic in the strip ℑµ∈[ν′,ν]. Then formulas (4.23),\n(4.16), (4.17), (4.18) and the residue theorem imply\ny−y′=1√\n2π/contintegraldisplay\nCdµ4π/integraldisplay\n0dϕ′eiµsR4π(ϕ,ϕ′,µ2)ˆw(µ,ϕ′) =\n=√\n2πi/summationdisplay\nℑµm∈[ν′,ν]4π/integraldisplay\n0dϕ′Res\nµ=µm/parenleftbig\neiµsR4π(ϕ,ϕ′,µ2)ˆw(µ,ϕ′)/parenrightbig\n=\n=/summationdisplay\nℑµm∈[ν′,ν]/summationdisplay\n±c±,m(w)r2m−1\n4e±i2m−1\n4ϕ,(4.25)\nwhereCisthe(positively oriented) boundaryofthestrip ℑµ∈[ν′,ν], andthecoefficients c±,m(w)\nare given by\nc±,m(w) =−1\n4π(1\n2−m)4π/integraldisplay\n0dϕ′+∞/integraldisplay\n−∞dsw(s,ϕ)r−2m−1\n4e±i2m−1\n4ϕ′. (4.26)\nNow, we are ready to prove Proposition 4.1 using the facts above.\nAsymptotics of unearPk: the case λ= 0.Condition (4.3) can be rewritten as\n|∂q\nϕ(r∂r)p(r2f(k)(r,ϕ))|≤cfr−ν0(p+q≤l),\nwhere\nν0=1−2T\n4<−1.\nAlso, the support [∆ ,χk]uis separated from Pk. Due to these facts and estimate (4.9), the right\nhand side of (4.13) (given by (4.12)) satisfies\n/bardblw/bardblHl\nν′≤c(cf+/bardblu/bardblL2(X;C)) (ν′>ν0). (4.27)\nIn view of(4.21), theinclusion u∈L2(X;C)means that y=χku(k)∈Hl+2\n1and, henceA1,ly=w.\nFor arbitrary ν′∈(ν0,ν= 1), puty′=A−1\nν′,lw. Then formula (4.25) provides the asymptotics\n14ofu(z) asz→0. Rewriting (4.25) in local coordinate x=xkand taking into account that the\nformulas\nf(k)\nk,m,+(rk,ϕk) = (±)(2rk)2m−1\n4eiϕk(2m−1)\n4,\nf(k)\nk,m,−(rk,ϕk) = (±)π−1/parenleftBig1\n2−m/parenrightBig−1\n(2rk)1−2m\n4eiϕk(2m−1)\n4(4.28)\nare valid for the representatives of fk,m,+given by (3.4), we arrive at expansion (4.4) with λ= 0,\nwhere the remainder ˜ uobeys ˜u(k)=y′for smallz=zk. Due to (4.24), (4.22), the equality\ny′=A−1\nν′,lwimplies\nrν′|∂q\nϕ(r∂r)p˜u(k)(r,ϕ)|≤c/bardbly′/bardblHl+2\nν′≤cν/bardblw/bardblHl\nν′(p+q≤l).\nRewriting the last condition in the local coordinate x=xkand taking into account (4.27), we\nobtain estimate (4.5).\nAsymptotics of unearPk: the general case. The general case is reduced to the case λ= 0\nby rewriting the equation (∆ −λ)u=fin the form ∆ u=f+λu. However, replacing fwith\nf+λuin the above reasoning yields that the right hand side wof (4.13) contains the additional\nterm−4r2λu(k), and, hence, estimate (4.27) and formula (4.25) are valid only for ν′≥−1.\nTherefore, the repeating of the reasoning above provides only th e expansion\nu(x) =/summationdisplay\n(m,±)cλ\nk,m,±(u)fk,m,±(x)+ ˜u(x) (4.29)\nwhere the sumation is taken over only terms decaying no faster tha nO(x2) while the remainder\nobeys\n|∂p\nx∂q\nx˜u(x)|≤cǫ/parenleftbig\ncf+(1+λ)/bardblu/bardblL2(X;C)/parenrightbig\n|x|3−p−q+ǫ(p+q≤l, ǫ>0).(4.30)\nTo overcome this obstacle and obtain the next terms of the asympt otics ofunearPk, we use\nthe following trick. For each fk,m,±presented in the sum in (4.29), we construct the formal series\nfk,m,±(1+d(1,±iµm)λ|z|2+d(1,±iµm)λ2|z|4+...) (4.31)\nobeying the equation (∆ −λ)v= 0 nearPk; then the coefficients d(n,±iµm) in (4.31) satisfy\n(4.7). Now, consider the section\nu1=u−χk/summationdisplay\n(m,±)cλ\nk,m,±(u)fλ,N\nk,m,±\ninstead ofu, wherefλ,N\nk,m,±are truncated series (4.31) with sufficiently large number, N, of terms.\nNote that\n(∆−λ)fλ,N\nk,m,±=−λN+1fk,m,±d(1,±iµm)|z|4N.\nAlso,\n|cλ\nk,m,±(u)|≤cf(1+|λ|/bardblu/bardblL2(X;C))\ndue to (4.26). Thus, (∆ −λ)u1satisfies (4.3) with cfreplaced by c(cf+|λ|N+2/bardblu/bardblL2(X;C)). Now,\nwe repeat the above reasoning for u1,f1= ∆u1instead ofu,f. Since (4.30) and (4.3) imply\n/bardblw/bardblHl\nν′≤c(cf+(1+|λ|N+2)/bardblu/bardblL2(X;C)) (ν′>max{−13/4,ν0}),\n15we arrive at the asymptotics\nu1(x) =/summationdisplay\n(m,±)cλ\nk,m,±(u1)fk,m,±(x)+ ˜u1(x)\nwhere the summation is taken over the terms decaying faster than O(x2) but slower than\nO(xmin{T,7}), while the remainder obeys\n|∂p\nx∂q\nx˜u1(x)|≤cλ,ǫ/parenleftbig\ncf+(1+λ)/bardblu/bardblL2(X;C)/parenrightbig\n|x|min{T,7}−p−q+ǫ(p+q≤l, ǫ>0),\nwherecλ,ǫis a polynomial in |λ|.\nIfT >7, then we can repeat the above trick to find the next terms of the asymptotics of u\nnearPkand so on. As a result, we prove expansion (4.4) and estimate (4.5). Now, denote by\nXǫ(p1,...,p n)\nthe domain obtained by deleting ǫ-neighborhoods (in the metric ρ−2) of pointsp1,...,p nfromX.\nSubstituteU=Xǫ(P1,...,P 2g−2),f=u, andf′=χkfk,m,∓in the Green formula (3.1) and pass\nto the limit ǫ→0. Calculating the limit of the right hand side of (3.1) by the use of asym ptotics\n(4.4), we arrive at (4.6). Proposition 4.1 is proved.\n4.1.3 Description of Dom∆∗\n·\nInthisparagraph, we proveProposition3.1describing thestructu reofthespaceDom∆∗\n·/Dom∆ ·.\nIntroduce the space Hl\nβof sections of Cwith finite norms\n/bardblv/bardblHlν=/parenleftBigg/summationdisplay\nk≤2g−2/bardblχkv(k)/bardbl2\nHlν+/summationdisplay\nk>2g−2/bardblχkv(k)/bardbl2\nHl/parenrightBigg1/2\n. (4.32)\nNote that the embedding Hl′\nν′⊂Hl\nνis continuous for l′≥landν′≤ν. Also, we have the\nfollowing equivalence of the norms\n/bardbl·/bardblL2(X;C)≍/bardbl·/bardblH0\n1.\nNote thatHl\nνis complete. Moreover, as a corollary of Remark 4.2, the set C∞\nc(˙X;C) is dense\ninHl\nβfor eachl,β. In view of definitions (4.32), (4.19), the convergence vn→vinH2\n−1implies\nthatvn→vand ∆vn→∆vinL2(X;C). Therefore, we have H2\n−1⊂Dom∆ ·.\nConversely, let vn∈C∞\nc(˙X;C) andvn→vand ∆vn→finL2(X;C). Thenvnis a Cauchy\nsequence inH2\n−1due to inequality (4.9) and estimate (4.24), where y=χkv(k)\nn′−χkv(k)\nnandwis\ngiven by (4.13). Since H2\n−1is complete, we have vn→v′inH2\n−1. Hence,vn→v′and ∆vn→∆v′\ninL2(X;C) andv=v′∈H2\n−1,f= ∆v′. Thus, Dom∆ ·⊂H2\n−1. So, we have proved that\nDom∆ ·=H2\n−1. (4.33)\nLet ∆∗\n·u=f; thenu,f∈L2(X;C) anduis a generalized solution to ∆ u=f. So, estimates\n(4.9) hold for k >2g−2 i.e.u∈H2(Ω) for any domain Ω in Xwhose closure does not contain\nconical points. Let us describe the asymptotics of unear conical point Pk. In view of (4.9),\nχku(s)(considered as a function of local coordinate (4.1)) Let us describ e the asymptotics of u\n16near each conical point Pk. Lety,wbe functions given by (4.12) with z=zk,r=rk,ϕ=ϕk.\nThenw∈H0\n−1,y∈H0\n1, and, moreover, y∈H2\n1due to (4.21). So, we have\nDom∆∗\n·⊂H2\n1. (4.34)\nNow formula (4.25) with l= 2,ν= 1,ν′= 0 yields the expansion\nχku(k)(rk,ϕk) =2/summationdisplay\nm=−1/summationdisplay\n±ck,m,±(u)χkf(k)\nk,m,±(rk,ϕk)+y′(rk,ϕk)\nwith remainder obeying y′∈H2\n−1. In other words, we have\nu=2g−2/summationdisplay\nk=12/summationdisplay\nm=−1/summationdisplay\n±ck,m,±(u)χkfk,m,±+ ˜u, (4.35)\nwhere ˜u∈ H2\n−1⊂Dom∆ ·due to (4.33). Thus, the elements (3.3) constitute a basis in\nDom∆∗\n·/Dom∆ ·.\nNow letf,f′∈Dom∆∗; then the expansions (4.35) are valid for them. Then from the Gree n\nformula (3.1) with U=Xǫ(O1,...,O 2g−2),ǫ→0, it follows that\n(∆∗f,f′)L2(X;C)−(f,∆∗f′)L2(X;C)=2g−2/summationdisplay\nk=12/summationdisplay\nm=−1/parenleftbig\nck,m,+(f)ck,m,−(f′)−ck,m,−(f)ck,m,+(f′)/parenrightbig\n.(4.36)\nComparing (4.36) with (3.2), we obtain the equalities\nG(π(χkfk,m,s),π(χkfk′,m′,−s′)) =δk,k′δm,m′δs,s′\nfrom which it follows that (3.3) is a Darbouxbasis on Dom∆∗\n·/Dom∆ ·. Proposition 3.1 is proved.\n4.1.4 Discreteness of spectra\nIn this paragraph, we prove the following statement.\nProposition 4.3. The spectrum of any self-adjoint extension of of ∆·is discrete.\nThe scheme of the proof is similar to that used in Chapter 4, [19]. Let ∆ 1be a self-adjoint\nextension of of ∆ ·; then (∆ 1−i)−1is a continuous operator in L2(X;C)≡H0\n1. Letν0∈(3/4,1).\nWe first prove that (∆ 1−i)−1acts continuously from H0\n1ontoH2\nν0. Second, we prove that the\nembeddingH2\nν0⊂H0\n1≡L2(X;C) is compact. Due to these facts, (∆ 1−i)−1is a compact\noperator in L2(X;C) and, hence, its spectrum is discrete. Thus, the spectrum of ∆ 1is also\ndiscrete.\nContinuity of (∆1−i)−1in weighted spaces. Letf∈L2(X;C) andu= (∆1−i)−1f. Since\n∆1is self-adjoint, we have\n/bardblu/bardblL2(X;C)≤/bardblf/bardblL2(X;C).\nIn view of (4.9), we also have\n/bardblχku(k)/bardblH2≤c(/bardblu/bardblL2(X;C)+/bardblf/bardblL2(X;C)),(k>2g−2).\n17Sinceu∈Dom∆ 1⊂Dom∆∗\n·, formula (4.34) implies y=χku(k)∈H2\n1fork≤2g−2. Also,\nA1,2y=w, where\nw(rk,ϕk) =−4r2(χkf(zk)+iu(z)+[∆,χk]u(z))\nandw∈H0\n−1. Applying formula (4.25) with ν= 1 andν′=ν0∈(3/4,1) and taking into account\nthat the stripℑµ∈[ν0,1] contains no poles (4.17), we obtain y=y′=A−1\nν0,2w∈H2\nν0and\n/bardbly/bardblH2ν0≤c/bardblw/bardblH0\n2≤c(/bardblu/bardblL2(X;C)+/bardblf/bardblL2(X;C))\ndue to (4.24). Combining the above estimates, we obtain the inequalit y\n/bardblu/bardblH2ν0≤c/bardblf/bardblL2(X;C),\nwhich means that (∆ 1−i)−1acts continuously from H0\n1=L2(X;C) ontoH2\nν0.\nCompactness of embeddings of weighted spaces. Letl′> landν′< ν. Now, we prove\nthat the embedding Hl′\nν′⊂Hl\nνis compact. Let Bl′\nν′is the unit ball in Hl′\nν′. To prove the claim,\nit is sufficient to show that, for any ε>0, there is a finite ε-net forBl′\nν′inHl\nν.\nLetε>0. First, each v∈Bl′\nν′can be represented as\nv=K/summationdisplay\nk=2g−2χkv+2g−2/summationdisplay\nk=1/parenleftbig\nχk(1−χǫ)v+χkχǫv/parenrightbig\n.\nIn view of definition (4.19), we have\n/bardblχkχk,ǫv(zk)/bardblHlν≤cǫν−ν′/bardblχkχk,ǫv(zk)/bardblHl′\nν′\nso one can choose ǫ >0 sufficiently small that /bardbl/summationtext2g−2\nk=1χkχk,ǫv/bardblHlν≤ε/3. In what follows, we\nassume that ǫis fixed in such a way.\nIn view of definition (4.32), for any ν′′,l′′, the norm/bardblu/bardblHl′′\nν′′of any section uwhose support\nis contained in Ukand separated by a fixed distance from each conical point, is equiva lent to\n/bardblu(xk)/bardblHl′′(B), whereBis a sufficiently large fixed ball in C. Note that u=χkvoru=χk(1−χǫ)v\nwithv∈Bl′\nν′are examples of such sections, and there is a constant Rsuch that/bardblu(k)/bardblHl′(B)≤\n/bardblv/bardblHl′\nν′=Rfor them.\nLetBl′be the ball of radius RinHl′(B). Due to the Rellich–Kondrachov theorem, the\nembedding Hl′(B)⊂Hl(B) is compact. So, for arbitrary δ >0, one can choose the finite δ-net\na1,...,a nforBl′inHl(B). Then for any u=χkvoru=χk(1−χǫ)vwithv∈Bl′\nν′, there exists\nassuch that\n/bardblu(xk)−as/bardblHl(B)<δ. (4.37)\nIntroduce the section as,ksupported in Ukby the rule as,k(xk) = ˜χkas(xk) ifk >2g−2 and by\nthe ruleas,k(xk) = ˜χk(1−χǫ/C)as(xk) (Cis sufficiently large) if k≤2g−2. Letu=χkvor\nu=χk(1−χǫ)vwithv∈Bl′\nν′; then ˜χku=uor ˜χk(1−χǫ/C)u=u. Now, estimate (4.37) implies\n/bardblu−as,k/bardblHlν≤c/bardblu(xk)−as/bardblHl(B)≤cδ.\nChooseδ=ε/(13cg). As a corollary of the facts above, the sums/summationtextK\nk=1ask,kwithask= 1,...,n\nprovide the finite ε-net forBl′\nν′inHl\nν. By this, we have proved the compactness of the embedding\nHl′\nν′⊂Hl\nν.\n184.2 Self-adjoint extensions.\nIn this subsection, basic properties of the self-adjoint extension s ∆F, ∆S, ∆hof ∆·are described.\n4.2.1 Friedrichs extension.\nIn this paragraph, we prove Proposition 3.2 describing the domain of the Friedrichs extension\n∆Fof ∆·. Also, we prove that ∆ Fis positive defined and admits the representation\n∆F=D∗\nωDω, (4.38)\nwhereDωis the closure (in L2(X;C)) of the operator\nDωu=ω−1∂u(u∈C∞\nc(˙X;C)). (4.39)\nQuadratic form. First, let us recall the construction of the Friedrichs extension. I ntroduce\nthe quadratic form aµ,·[f] := ((∆ ·−µ)f,f) with the domain Dom aµ,·= Dom∆ ·. In view of the\nGreen formula\n(∆f,f)L2(U;C)=/bardbl∂f/bardbl2\nL2(U;C⊗K)+1\n2i/integraldisplay\n∂Ufh∂fdz (4.40)\nwithU=Xǫ(P1,...,P 2g−2),ǫ→0, we have\naµ,·[f] =/bardbl∂f/bardbl2\nL2(X;C⊗K)−µ/bardblf/bardbl2\nL2(X;C),\nandaµ,·is positive definite for µ<0. Since the multiplication operator ω−1:L2(X;C⊗K)/ma√sto→\nL2(X;C) is unitary, one can represent aµ,·as\naµ,·[f] =/bardblDωf/bardbl2\nL2(X;C)−µ/bardblf/bardbl2\nL2(X;C).\nThe formaµ,·admits closure denoted by aµ,F. The Friedrichs extension ∆ Fof ∆·is defined as\n∆F=A+µ, whereAis the (unique) self-adjoint operator corresponding to the form aµ,·(i.e.,\naµ,F[f] = (Af,f)L2(U;C)on DomA⊂Domaµ,F). Note that ∆ Fis independent of µ <0 and the\nexact lower bounds of the operators ∆ ·,∆Fcoincide.\nDomain of ∆F.Let us prove Proposition 3.2. Denote by ∆ F′the self-adjoint extension of ∆ ·\ndefined on the functions with asymptotics (3.5). To prove that ∆ F′= ∆F, it is sufficient to show\nthat Dom∆ F′⊂Domaµ,Fandaµ,F[f] = ((∆ F′−µ)f,f)L2(U;C)on Dom∆ F′.\nRecall that each u∈Dom∆ F′is the sumu=u0+ ˜u, whereu0is a linear combination of the\nsections\nχkfk,m,s(k= 1,...,2g−2,(m,s) = (−1,−),(0,−),(1,+),(2,+)) (4.41)\nwhile ˜u∈H2\n−1due to (4.33). By definitions of χkfk,m,s, we have\n∂p\nϕk(rk∂rk)qu(k)\n0=O(r1/4−p−q\nk),∆u0= 0 (4.42)\nnear eachPk. Hence,u0,u∈H2\n0in view of definitions (4.32), (4.19). Since C∞\nc(˙X;C) is dense in\nH2\n−1, there is a sequence {˜un}⊂C∞\nc(˙X;C) converging to ˜ uinH2\n−1. Denoteun=u0+ ˜un. Then\n19un→uinH2\n0andL2(X;C). Also, since ∆ u0= 0 near conical points, we have ∆ un→∆F′uin\nL2(X;C).\nNext, let us substitute U=Xǫ(P1,...,P 2g−2),ǫ→0 andf=uninto Green formula (4.40).\nIn view of (4.42), we obtain (∆ un,un)L2(X;C)=/bardblDωun/bardbl2\nL2(X;C). Hence,\n(∆u,u)L2(X;C)=/bardblDωu/bardbl2\nL2(X;C). (4.43)\nis valid due to the above convergences.\nFinally, due to definitions (4.32), (4.19), the convergence un→uinH2\n0implies thatDωun→\nDωuinL2(X;C) andaµ,F[un]→/bardblDωu/bardbl2\nL2(X;C)−µ/bardblu/bardbl2\nL2(X;C). Hence,\nu∈Domaµ,F, aµ,F[u] =/bardblDωu/bardbl2\nL2(X;C⊗K)−µ/bardblu/bardbl2\nL2(X;C)= ((∆ F′−µ)u,u)L2(X;C)(4.44)\nfor anyu∈Dom∆ F′andµ∈C. Hence, ∆ F′= ∆Fand Proposition 3.2 is proved.\nIn addition, formula (4.44) implies that ∆ Fis non-negative and each u∈Ker∆Fis holomor-\nphic outside conical points. Also, each conical point Pkis a zero of udue to (3.5). Hence, uhas\nat least 2g−2 zeroes on X. Since the degree of the divisor class of Cis equal tog−1≥1, the\nlatter is possible only if u= 0 onX. So, Ker∆ F= 0 and ∆ Fis positive definite. Note that this\nfact is independent on the assumption h0(C) = 0.\nConnection between ∆FandDω,D∗\nω.Since\n(Dωu,v)L2(X,C)=/integraldisplay\nXω−1∂u·hvρ−2dzdz=/integraldisplay\nX∂(ω−1uhvρ−2)dzdz−/integraldisplay\nXu∂(ω−1hvρ−2)dzdz=\n= 0+/integraldisplay\nXuh/bracketleftbig\n−(ρ−2h)−1∂(ρ−2hω−1v)/bracketrightbig\ndS(u,v∈C∞\nc(˙X;C)),\nthe operatorD∗\nωsatisfies\nD∗\nωu=−(ρ−2h)−1∂/parenleftbig\nρ−2hω−1u/parenrightbig\n=−|ω|−1∂/parenleftbig\n|ω|ω−1u/parenrightbig\n(u∈C∞\nc(˙X;C)).(4.45)\nIn particular,D∗\nωis densely defined and operator (4.39) admits closure (denoted by Dω) while\nthe operatorsD∗\nωDωandDωD∗\nωare self-adjoint. In addition,\nD∗\nωDωu=−|ω|−1∂/parenleftbig\n|ω|ω−1ω−1∂u/parenrightbig\n= ∆·uforu∈C∞\nc(˙X;C).\nTaking the closure of both parts yields ∆ ·⊂D∗\nωDω.\nNext, letu∈Dom∆ F. As shown after (4.41), there us a sequence {un} ⊂C∞\nc(˙X;C)\nconverging to uinH2\n0and inL2(X;C) and such thatD∗\nωDωun= ∆un→∆FuinL2(X;C).\nHence,u∈DomD∗\nωDωandD∗\nωDωu= ∆Fu. Thus, ∆ F⊂D∗\nωDωand (since both operators are\nself-adjoint)D∗\nωDω= ∆F. So, formula (4.38) is proved.\n4.2.2 Szeg¨ o extension.\nIn this paragraph, we prove Proposition 3.4. First, recall that, un der the assumption h0(C) = 0,\nthe Szeg¨ o kernel ( z,z′)/ma√sto→S(z,z′) (the integral kernel of ( −1\nπ∂)−1) is a section of Cin both\nvariables, holomorphic outside the diagonal z=z′, antisymmetricS(z,z′) =−S(z′,z), and\nadmitting the asymptotics\nS(z,z′) =1\nz−z′+O(1)\n20near the diagonal z=z′(see p.29, [4]). Here z,z′are understood as different values of the same\n(arbitrary) holomorphic local coordinate on X. In what follows, we assume that z,z′are given\nby (4.1).\nDenoteby ∆ Sthe self-adjoint extension of ∆ ·defined on the functions with asymptotics (3.9).\nLetGSbe the Green function of ∆ S. Recall that ( z,z′)/ma√sto→GS(z,z′) is a section of C(C) inz\n(z′) obeying ∆GS(·,z′) = 0 outside the diagonal z=z′and conical points. Also, it is symmetric\nGS(z,z′) =GS(z′,z) and obeys\nGS(z,z′) =−2\nπlog(|z′−z|)+O(1) (4.46)\noutside conical points. In view of Proposition 4.1 and formula (3.9), w e have\nG(xk,z′) =ck,0,+(z′)+ck,1,+(z′)xk+ck,2,+(z′)x2\nk+(−2/3π)ck,−1,−(z′)xk|xk|+O(|xk|3) (4.47)\nnear eachPk. Both asymptotics (4.46), (4.47) admit differentiation.\nDenote\n˜S(z,z′) =−πh(z′)∂z′GS(z,z′).\nThen\n−1\nπ∂z′˜S(z,z′) =∂z′h(z′)∂z′GS(z′,z) =−ρ−2(z′)h(z′)∆z′GS(z′,z) = 0,\ni.e.,z′/ma√sto→˜S(z,z′) is holomorphic in z′outside the diagonal z′=zand conical points. Next,\nformula (4.46) implies\n˜S(z,z′) =−π∂z′(−2\nπlog(|z′−z|)+O(1)) =1\nz′−z+O(1).\nFinally, due to the symmetry GS(z,z′) =GS(z′,z), formula (4.47) can be rewritten as\nGS(z′,xk) =c0\nk,0,+(z′)+c0\nk,1,+(z′)xk+c0\nk,2,+(z′)x2\nk+(−2/3π)c0\nk,−1,−(z′)xk|xk|+O(|xk|3).\nDifferentiation of both sides yields\n˜S(z′,xk) = (2/3)c0\nk,−1,−(z′)|xk|−1∂xk(|xk|xk)+O(|xk|) =c0\nk,−1,−(z′)+O(|xk|).\nTherefore ˜S(z′,·) is holomorphic at each conical point. In view of the facts above, [ ˜S−S](z′,·)\nis a holomorphic section of Cfor eachz′. Sinceh0(C) = 0, we have ˜S=Sand formula (3.6) is\nvalid.\nSinceSis the integral kernel of ( −1\nπ∂)−1, the formula\nf(z) =/integraldisplay\nXS(z,z′)/bracketleftbig\n−1\nπ∂f/bracketrightbig\n(z′)dz′dz′\n2i(4.48)\nis valid on smooth sections fofC. Moreover, equality (4.48) admits extension to sections fwith\nlogarithmic singularities. Thus, substituting f=GS(·,z′′) into (4.48) and taking into account\nthatρ=h, we obtain\nGS(z,z′′) =/integraldisplay\nXS(z,z′)/bracketleftbig\n−1\nπ∂z′GS(z′′,z′)/bracketrightbigdz′dz′\n2i=−1\nπ2/integraldisplay\nXS(z,z′)h(z′)S(z′,z′′)dS(z′)\nTherefore, formula (3.7) is valid. Thus, ∆ Sis the Szeg¨ o extension of ∆ ·. Proposition 3.4 is\nproved.\n21Positivity of ∆S.In view of (3.9), each u∈Dom∆ Sadmits representation u=u0+ ˜u, where\n∂xku0(xk) =−πck,−1,−|xk|nearPkwhile ˜u∈H2\n−1due to (4.33). Then formula (4.40) yields (4.43)\nwith ∆ = ∆ S. In particular, ∆ Sis non-negative, and each u∈Ker∆Sis holomorphic outside\nconical points. Due to (3.9), u(xk) is bounded whence uis holomorphic at conical points. Since\nh0(C) = 0, we obtain u= 0. So, the Szeg¨ o extension ∆ Sis positive.\n4.2.3 Holomorphic extension.\nInthisparagraph,weproveProposition3.5onalmost-isospectralit yofholomorphicandFriedrichs\nextensions of ∆ ·. In addition, we describe the Bergman kernel of ∆ h.\nAlmost-isospectrality of ∆Fand∆h.Let us prove Proposition 3.5. To this end, we consider\nthe self-adjoint operator DωD∗\nω(Dωis defined after (4.38)). In view of (4.39) and (4.45), we have\nDωD∗\nω=−ω−1∂/parenleftbig\n|ω|−1∂/parenleftbig\n|ω|ω−1.../parenrightbig/parenrightbig\n=U−1\nωι−1∆0ιUωonC∞\nc(˙X;C), (4.49)\nwhere\nUω:u/ma√sto→|ω|ω−1u\nis the unitary operator acting from L2(X;C) ontoL2(X;C) and\nι:u/ma√sto→u\nisanti-linearisometryfrom L2(X;C)ontoL2(X;C). SinceDωD∗\nωisself-adjoint, soistheoperator\n∆⊙=UωDωD∗\nωU−1\nωinL2(X;C). Sinceιis anti-linear, theformulas ∆ ⊙:=ι∆⊙ι−1and dom∆ ⊙=\nιUωdom(DωD∗\nω) define the linear unbounded operator ∆ ⊙inL2(X;C). The equality\n(u,∆∗\n⊙v)L2(X;C)= (∆⊙u,v)L2(X;C)=/integraldisplay\nX∆⊙uhvdS=(∆⊙u,v)L2(X;C)=\n=(u,∆⊙v)L2(X;C)=/integraldisplay\nXuh∆⊙vdS= (u,∆⊙v)L2(X;C)\nshows that ∆ ⊙is self-adjoint. Due to (4.49), ∆ ⊙is a self-adjoint extension of ∆ ·.\nAccording to Proposition 4.3, the spectrum of ∆ ⊙is discrete. Since\n∆Fu=λu⇒DωD∗\nω(Dωu) =Dω∆Fu=λ(Dωu)⇔∆⊙(ιUωDωu) =λ(ιUωDωu),\n∆⊙v=λv⇔DωD∗\nω(U−1\nωι−1v) =λ(U−1\nωι−1v)⇒∆F(D∗\nωU−1\nωι−1v) =λ(D∗\nωU−1\nωι−1v),\nthe operators ∆ Fand ∆ ⊙are almost isospectral and the anti-linear map\nu/ma√sto→ιUωDωu\nprovides the correspondence between eigensections of ∆ Fand eigensections of ∆ ⊙with non-zero\neigenvalues.\nTo prove Proposition 3.5 it remains to show that ∆ ⊙= ∆h. Let (λ,u) be any eigenpair (i.e.,\na pair of eigenvalue and eigensection) of ∆ Fobeyingλ/\\e}atio\\slash= 0. Then uis smooth outside of the\nconical points. In view of Proposition 4.1 and formula (3.5), the asym ptotics\nu(xk) = (−2/π)ck,0,−|xk|+ck,1,+xk+ck,2,+x2\nk+(−2/3π)ck,−1,−xk|xk|+ck,3,+x3\nk+\n+(−2/5π)ck,−2,−xk2|xk|+ck,4,+x4\nk+(−2/7π)ck,−3,−xk3|xk|+O(|xk|5), x k→0\n22holds and admits differentiation. Hence,\n−π[Dωu](xk) =−πxk−1∂xku(xk) =ck,0,−xk−2|xk|+ck,−1,−xk−1|xk|+\n+ck,−2,−|xk|+ck,−3,−xk|xk|+O(|xk|3), x k→0\nand\n−π[ιUωDωu](xk) =x−1\nk|xk|/parenleftbig\n−πxk−1∂xku(xk)) =\n=ck,0,−x−1\nk+ck,−1,−+ck,−2,−xk+ck,−3,−x2\nk+O(|xk|3), x k→0.\nThus,ιUωDωu∈Dom∆ hdue to (3.10). Therefore, each eigenpair ( ιUωDωλ/\\e}atio\\slash= 0,u) of ∆ ⊙is an\neigenpair of ∆ h. In other words, we have ∆ ⊙(I−P) = ∆ h(I−P), wherePis a projection on\nthe (finite-dimensional) subspace Ker∆ ⊙. At the same time, ∆ ⊙= ∆hon setC∞\nc(˙X;C). Since\nC∞\nc(˙X;C) is dense in L2(X;C), we obtain ∆ ⊙≡∆h. Proposition 3.5 is proved.\nBergman kernel of ∆h.In view of (3.10), each u∈Dom∆ hadmits representation u=u0+˜u,\nwhere∂xku0(xk) = 0 near Pkwhile ˜u∈H2\n−1due to (4.33). Then formula (4.40) yields (4.43)\nwith ∆ = ∆ h. In particular, ∆ his non-negative and each u∈Ker∆his holomorphic outside\nconical points. In view of (3.10), u∈Ker∆hmay have simple poles at conical points. Due to the\nassumption h0(C) = 0, the Szeg¨ o kernels S(Pj,·) (j= 1,...,2g−2) provide a basis in Ker∆ h;\nin particular, dimKer∆ h= 2g−2. Then the Bergman kernel of ∆ his given by\nBh(z,z′) =−/summationdisplay\nkjS(z,Pk)S−1\nkjS(Pj,z′),\nwhereSis given by (3.15).\n4.3 Heat kernel\nIn this subsection, we construct the parametrix H0(·,·,t) for the fundamental solution H(·,·,t)\nto heat equation ( ∂t+ ∆F)u= 0. Next, we derive the short-time asymptotics of H(·,·,t). In\nparticular, we obtain the short-time asymptotics of the heat trac et/ma√sto→K(t|∆F) and prove that\nthe zeta function s/ma√sto→ζ(s|∆F) admits analytic continuation into the neighborhood of s= 0. Due\nto this fact, the determinant of ∆ Fis well-defined.\n4.3.1 Parametrix for heat kernel\nLocal solution outside conical points. Outside conical points, coordinates (4.1) are well-\ndefined and ∆ is of the form (4.2). Hence, the local fundamental so lution to the heat equation\nis given by\nHC(z,z′,t) =1\nπte−|z−z′|2/t,\nwherez,z′are values of coordinate (4.1).\nLocal solution near conical points. NearPk, the heat equation ( ∂t+ ∆F)u= 0 takes the\nform/parenleftbig\n4∂t−r−2((r∂r)2+∂2\nϕ)/parenrightbig\nv(r,ϕ) = 0, (4.50)\n23wherer=rk,ϕ=ϕk. In addition, the representative v=u(k)of the section uofCin coordinate\nzkobeys (4.14), i.e., it is 4 π-antiperiodic in ϕ=ϕk.\nThe fundamental solution to equation (4.50), which is α−periodic in ϕ,ϕ′, is given by\nH/anticlockwise(r,ϕ,r′,ϕ′,t|α) =1\n2παit/integraldisplay\nC∪(−C)exp/parenleftBig\n−r2−2rr′cosϑ+r′2\nt/parenrightBig\ncotθdϑ. (4.51)\n(see [6], see also [7, 13]). Here θ:=πα−1(ϑ+ϕ−ϕ′) andCis the contour in the semi-strip\nℜϑ∈(−π,π),ℑα′>0 running from ϑ=i∞+πtoϑ=i∞−π. We take the function\nH/archleftdown(r,ϕ,r′,ϕ′,t|α) =H/anticlockwise(r,ϕ,r′,ϕ′,t|2α)−H/anticlockwise(r,ϕ+α,r′,ϕ′,t|2α)\nas anα-anti-periodic fundamental solution to (4.50).\nLet us find the asymptotics of H/archleftdown(r,ϕ,r′,ϕ′,t|α) asr→0. From (4.51), it follows that\nH/archleftdown(r,ϕ,r′,ϕ′,t|α) =1\n4παit/integraldisplay\nC∪(−C)exp/parenleftBig\n−r2−2rr′cosϑ+r′2\nt/parenrightBig\n(cotθ\n2+tanθ\n2)dϑ\n=1\nπαt/integraldisplay\nC∪(−C)exp/parenleftBig\n−r2−2rr′cosϑ+r′2\nt/parenrightBiga\na2−1dϑ,(4.52)\nwherea= exp(iθ). Since\nϑ∈C ⇒ | a|<1,a\na2−1=−/summationdisplay\nk≥0a2k+1,\nϑ∈(−C)⇒ |a|>1,a\na2−1=/summationdisplay\nk≤0a2k−1,\nwe have\n−παte(r2+r′2)/tH/archleftdown(r,ϕ,r′,ϕ′,t|α) =/summationdisplay\n±(±)/summationdisplay\n±k≥0/integraldisplay\n±Ce2rr′cos(ϑ)\nta2k±1dϑ=\n=/summationdisplay\n±/summationdisplay\n±k≥0e±iπ(ϕ−ϕ′)(2(±k)+1)\nα/integraldisplay\n±ϑ∈Ce2rr′cos(±ϑ)\nteiπ(±ϑ)(2(±k)+1)\nαd(±ϑ) =\n=2/summationdisplay\nk≥0cos/parenleftBigπ(ϕ−ϕ′)(2k+1)\nα/parenrightBig/integraldisplay\nCe2rr′cos(ϑ)\nteiπϑ(2k+1)\nαdϑ= [ϑ=iγ] =\n=2i/summationdisplay\nk≥0cos/parenleftBigπ(ϕ−ϕ′)(2k+1)\nα/parenrightBig/integraldisplay\n−iCexp/parenleftBig2rr′\ntcoshγ−π(2k+1)\nαγ/parenrightBig\ndγ.\nTaking into account the expression\n1\n2πi/integraldisplay\n−iCexp/parenleftbig\nzcoshs−νs/parenrightbig\nds=Iν(z),\n24for the modified Bessel functions of the first kind, we finally obtain\nH/archleftdown(r,ϕ,r′,ϕ′,t|α) =\n=1\nαtexp/parenleftbig\n−r2+r′2\nt/parenrightbig/summationdisplay\nk≥0cos/parenleftBigπ(ϕ−ϕ′)(2k+1)\nα/parenrightBig\nIπ(2k+1)/α(2rr′\nt) =O(rπ\nα), r→0.(4.53)\nLet us show that H/archleftdown(r,ϕ,r′,ϕ′,t|α) and all its derivatives decay exponentially as t→+0 if\n(r,ϕmodα) is separated from ( r′,ϕ′modα). First, note that one can replace the integration\ncontourC∪(−C) in (4.52) with the union of two lines ±l:={ϑ=±(π−i´ϑ)}´ϑ∈Rand small\nanti-clockwise circles ◦(ϑ∗) centred at the zeroes ϑ∗of functions a±1. The exponent in the right\nhand side of (4.52) decays as t→+0 uniformly in ϑ∈(±l) since cosϑ=−cosh´ϑ <−1 on±l.\nIn addition,|a/(a2−1)|=O(exp(−πα−1|´ϑ|)) holds on the lines ±l. Thus, the integrals over\n±lin the right hand side of (4.52) converge and decay exponentially as t→+0. Similarly, the\nexponent in the right hand side of (4.52) decays uniformly in ϑ∈◦(ϑ∗), whereϑ∗/\\e}atio\\slash= 0. Finally,\nthe integral over◦(0) is present in the right hand side of (4.52) if and only if ϕ=ϕ′(modα). In\nthe last case, this integral is equal to e−(r−r′)2/4t/πtdue to the residue theorem. This implies the\nneeded decay of H/archleftdown(r,ϕ,r′,ϕ′,t|α) ast→+0 for (r,ϕmodα) separated from ( r′,ϕ′modα).\nIfr=r′,ϕ=ϕ′, then (4.52) takes the form\nH/archleftdown(r,ϕ,r,ϕ,t|α) =1\n2πiαt/integraldisplay\nlexp/parenleftbig\n−4r2sin2(ϑ/2)\nt/parenrightbigdϑ\nsin(πϑ/α),\nwere the integration is over the union lof±land◦(ϑ∗)-s. Therefore,\nR/integraldisplay\n0rdrα/integraldisplay\n0dϕH/archleftdown(r,ϕ,r,ϕ,t|α) =1\n16πi/integraldisplay\nldϑ\nsin2(ϑ/2)sin(πϑ/α)exp/parenleftbig\n−4r2sin2(ϑ/2)\nt/parenrightbig/vextendsingle/vextendsingle/vextendsingle0\nr=R.(4.54)\nHere the term with r= 0 equals zero while the term with r=Rcoincides with\n−1\n16πi/integraldisplay\n◦(0)dϑ\nsin2(ϑ/2)sin(πϑ/α)exp/parenleftbig\n−4R2sin2(ϑ/2)\nt/parenrightbig\n=\n=−1\n8Res\nϑ=0/parenleftBigexp/parenleftbig\n−4R2sin2(ϑ/2)\nt/parenrightbig\nsin2(ϑ/2)sin(πϑ/α)/parenrightBig\n=−1\n8/parenleftBigα\n3π+2π\n3α−α\nπ4R2\nt/parenrightBig\nup to a remainder that exponentially decays as t→+0,R≥const>0. Thus, (4.54) implies\n/integraldisplayR\n0χ(r)rdr/integraldisplayα\n0dϕH/archleftdown(r,ϕ,r,ϕ,t ) =\n=1\nπt/integraldisplayR\n0χ(r)rdr/integraldisplayα\n0dϕ−1\n8/parenleftBigα\n3π+2π\n3α/parenrightBig\n+O(e−ǫ/t),(4.55)\nwhereχis a smooth cut-off function on Requal to 1 near zero and ǫis some positive number.\nConstruction of parametrix. Fort>0, let (z,z′)/ma√sto→H0,k(z,z′,t) be the section of C(C) in\nz(z′) which vanishes outside Uk×Ukand is given by\nH0,k(zk,z′\nk,t) =/braceleftbigg˜χk(zk)χk(z′\nk)HC(zk,z′\nk,t) ( k>2g−2),\n˜χk(zk)χk(z′\nk)H/archleftdown(rk,ϕk,r′\nk,ϕ′\nk,t|4π) (k≤2g−2)(4.56)\n25inUk×Uk(herezk,z′\nkdenote different values of the same coordinate). As a parametrix f or the\nfundamental solution to ( ∂t+∆F)u= 0 we take the sum\nH0=/summationdisplay\nkH0,k.\nFort>0 define\n˜f(·,z′,t) := (∂t+∆z)H0(·,z′,t).\nSince each H0,kin (4.56) is a fundamental solution to the heat equation in Uk×Uk, one has\n˜f=/summationtext\nk˜fk, where˜fk(·,·,t) vanishes outside Uk×Uk, and is given by\n˜fk(zk,z′\nk,t) =/braceleftbigg\n−χk(z′\nk)[∆,˜χk](zk)HC(zk,z′\nk,t) ( k>2g−2),\n−χk(z′\nk)[∆,˜χk](zk)H/archleftdown(rk,ϕk,r′\nk,ϕ′\nk,t|4π) (k≤2g−2)\ninUk×Uk. Since ˜χkχk=χk, the coordinate zkis separated from z′\nkon the support of\nχk(z′\nk)[∆,˜χk](zk), and, hence, ˜fk(zk,z′\nk,t) and all its derivatives decay exponentially as t→+0.\nTherefore, ˜f(z,z′,t) and all its derivatives (considered as functions of coordinates zk,z′\nk) decay\nexponentially as t→+0.\nNext, since each H0,kin (4.56) is a fundamental solution to the heat equation in Uk×Uk\nwhile the cut-off functions obey ˜ χkχk=χk,/summationtext\nkχk= 1, one has\nlim\nt→+0H0(z,z′,t) =δ(z−z′) (4.57)\n(here the limit is understood in the sense of distributions).\nFinally, in view of formula (4.53) with α= 4π, the asymptotics of H0(·,z′,t) near conical Pk\nis of the form (3.5). Therefore,\nH0(·,z′,t)∈Dom∆ F(t>0, z′∈˙X). (4.58)\n4.3.2 Short-time asymptotics of heat kernel\nSince the operator ∆ Fis positive, the C0-semigroup of contractions t/ma√sto→exp/parenleftbig\n−t∆F/parenrightbig\nis well-\ndefined. For each z′∈˙X, the (unique) solution H1(·,z′,·)∈C1(R;L2(X;C))∩C(R;Dom∆ F)\n(where Dom∆ Fis endowed with the graph norm) to the Cauchy problem\n(∂t+∆F)H1(·,z′,t) =−˜f(·,z′,t′), t>0,\nH1(·,z′,0) = 0(4.59)\nis given by\nH1(·,z′,t) =−/integraldisplayt\n0exp/parenleftbig\n(t′−t)∆F/parenrightbig˜f(·,z′,t′)dt′.\nSimilarly, the solution H1,l(·,z′,·)∈C(R;Dom∆ F)∩C1(R;L2(X;C)) to\n(∂t+∆F)H1,l(·,z′,t) =−∂l\nt˜f(·,z′,t), t>0,\nH1,l(·,z′,0) = 0\nis given by\nH1,l(z,z′,t) :=/integraldisplayt\n0exp/parenleftbig\n(t′−t)∆F/parenrightbig\n[−∂l\nt′˜f(·,z′,t′)]dt′.\n26The solutions H1andH1,lwithl>0 are related via the equality\nH1(z,z′,t) =t/integraldisplay\n0dtl−1tl−1/integraldisplay\n0dtl−2···t1/integraldisplay\n0dt1H1,l(z,z′,t1). (4.60)\nIndeed, since H1,l(·,z′,·)∈C(R;Dom∆ F)∩C1(R;L2(X;C)), each integration in the right hand\nside of (4.60) commutes with the action of ∆ F. So, the right hand side of (4.60) is also a solution\nto (4.59); hence, it coincides with H(1)(·,z′,t).\nIn view of (4.60), one has H1(·,z′,·)∈C∞(R;Dom∆ F)∩C∞(R;L2(X;C)) and\n(−∆F)lH1(·,z′,t) =H1,l(·,z′,t)+l/summationdisplay\ns=1(−∆F)l−s∂s−1\nt˜f(·,z′,t). (4.61)\nRecall that the L2(X;C)-norms of ∆k\nF∂s\nt˜f(·,z′,t) (wherek,s= 0,1,...) decay exponentially as\nt→+0 and uniformly with respect to z′. Since exp/parenleftbig\n(t′−t)∆F/parenrightbig\nis a contraction in L2(X;C) for\nt′<t, thesameistrueforthe L2(X;C)-normoftherighthandsideof(4.61). Thus, the L2(X;C)-\nnorms of ∆l\nFH1(·,z′,t) decay exponentially and uniformly with respect to z′ast→+0. Hence,\ndue to estimates (4.11), H1(zk,z′,t) (k>2g−2) decays exponentially as t→+0 uniformly with\nrespect tozk,z′. Moreover, due to Proposition 4.1, the functions H1(xk,z′,t) (k≤2g−2) decay\nexponentially as t→+0 uniformly with respect to xk,z′.\nPut\nH:=H0+H1. (4.62)\nIn view of (4.58) and (4.59), we have ( ∂t+∆F)H(·,z′,t) = 0 fort>0 andz∈˙X. Due to (4.57)\nand the exponential decrease of H1ast→+0, we have lim t→+0H(z,z′,t) =δ(z−z′) in the\nsense of distributions.\nLet 0< λF\n1≤λF\n2≤...be the eigenvalues of ∆ Fcounted with their multiplicities and\n{uF\nk}be the orthonormal basis of the corresponding eigensections. Fo rz′∈˙Xandt >0, put\nck(t,z′) := (H(·,z′,t),uF\nk)L2(X;C). Then\nλF\nkck(t,z′) = (H(·,z′,t),∆FuF\nk)L2(X;C)=(∆FH(·,z′,t),uF\nk)L2(X;C)=\n=−(∂tH(·,z′,t),uF\nk)L2(X;C)=−∂tck(t,z′).\nSince lim t→+0H(z,z′,t) =δ(z−z′), we have\nlim\nt→+0ck(t,z′) =uF\nk(z′).\nTherefore,ck(t,z′) = exp(−λF\nkt)uF\nk(z′) and\nH(z,z′,t) =∞/summationdisplay\nk=1e−λF\nktuF\nk(z)uF\nk(z′). (4.63)\nThus, His the integral kernel of the operator exponent e−t∆F(the heat kernel corresponding\nto the Friedrich extension of Laplacian). Due to (4.62) and the expo nential decrease of H1as\nt→+0, the main term of asymptotics of H(z,z′,t) ast→+0 is given by (4.63), (4.56). Hence,\nformula (4.55) yields\n/integraldisplay\nXH(z,z,t)h(z)dS(z) =Area(X,g)\nπt−3(g−1)\n8+O(e−ǫ/t), t→+0, (4.64)\nwhereǫ>0.\n274.3.3 Zeta function of ∆F\nIn view of (4.63), the left hand side of (4.64) coincides with K(t|∆F) := Tre−t∆F=/summationtext\nke−λF\nkt.\nHence,\nK(t|∆F) =Area(X,g)\nπt−3(g−1)\n8+˜K(t), (4.65)\nwhere˜K(t) =O(e−ǫ/t) ast→+0. Also, K(t|∆F) =O(e−λF\n1t) ast→+∞. Since the zeta-\nfunctionζ(s|A) = TrA−sof the self-adjoint operator Ais related to K(t|A) := Tre−tAvia\nζ(s|A) =1\nΓ(s)/integraldisplay+∞\n0ts−1K(t|A)dt, (4.66)\nformula (4.65) yields\nζ(s|∆F) =Area(X,g)\nπΓ(s)/integraldisplay1\n0ts−2dt−3(g−1)\n8Γ(s)/integraldisplay1\n0ts−1dt+\n+1\nΓ(s)/parenleftBig/integraldisplay1\n0ts−1˜K(t|∆F)dt+/integraldisplay+∞\n1ts−1K(t|∆F)dt/parenrightBig\n=Area(X,g)\nπ(s−1)Γ(s)+˜ζ(s),\nwhere˜ζis an entire function. Thus, s/ma√sto→ζ(s|∆F) is continued to a meromorphic function with\ntheunique(simple) poleat s= 1. Asacorollary, thedeterminant det∆ F= exp/parenleftbig\n−∂sζ(s|∆F)/vextendsingle/vextendsingle\ns=0/parenrightbig\nof ∆Fis well-defined.\n5 Comparison formulas\nIn this section, we prove comparison formula (3.13) for determinan ts of ∆ Fand ∆ S.\n5.1 Comparing resolvents\nIn this subsection, we introduce the scattering matrix T(λ) associated with ∆ Fand ∆ Sand\ndescribe its properties. Then, we derive the formula for the differe nce of resolvent kernels RF\nλ\nandRS\nλof ∆Fand ∆ Sin terms of T(λ).\n5.1.1 Scattering matrix\nIntroduce the solutions\nVλ\nk,0,−=χkfk,0,−−(∆S−λ)−1˜Wλ\nk,0,− (5.1)\nto the equation (∆ −λ)V= 0, where λis not an eigenvalue of ∆ Sand\n˜Wλ\nk,0,−= (∆−λ)/parenleftbig\nχkfk,0,−/parenrightbig\n= [∆,χk]fk,0,−−λχkfk,0,−.\nIn view of (3.9) and (3.5), asymptotic expansion (4.4) for u= (∆S−λ)−1˜Wλ\nk,0,−provided by\nProposition 4.1 yields\nVλ\nk,0,−−/parenleftBig\nχkfk,0,−+2g−2/summationdisplay\nj=1Tjk(λ)χjfj,0,+/parenrightBig\n∈Dom∆ S∩Dom∆ F, (5.2)\nwhereTjk(λ) are some coefficients. The (2 g−2)×(2g−2)-matrixT(λ) with entries Tjk(λ) is\ncalled the scattering matrix . Below, we describe the properties of T(λ) .\n28Expression for T(λ)in terms of RS\nλ.Ifz′is not a conical point, then the resolvent kernel\nR⋆\nλof ∆⋆(⋆=F,S) can be represented as\nRS\nλ(·,z′) =ψz′R0\nλ(·,z′)+/bracketleftbig\n(∆S−λ)−1(∆−λ)(ψz′R0\nλ)/bracketrightbig\n(·,z′), (5.3)\nwhereψz′is a smooth cut-off function on Xequal to one near z′, the support of ψz′is sufficiently\nsmall, and R0\nλis a local fundamental solution to (∆ −λ)v= 0 given by\nR0\nλ(z,z′) :=/braceleftbigg\n(2/π)K0(−2i|z−z′|√\nλ) (λ/\\e}atio\\slash= 0)\n(−2/π)log(|z−z′|) (λ= 0),(5.4)\nin coordinates (4.1), where K0is the Macdonald function. Note that the first term in the right\nhand side of (5.3) vanishes near conical points, while the second ter m is smooth in ˙Xand belongs\ntoDom∆ ⋆. Hence, Proposition4.1providesasymptoticexpansion(4.4)of u=RS\nλ(·,z′)neareach\nPk, whereck,0,+=RS\nλ(Pk,z′). Due tothese facts, the substitution U=Uǫ=Xǫ(P1,...,P 2g−2,z),\nL=C, andf=χkfk,0,−,f′=RS\nλ(·,z) into Green formula (3.1) yields\n(˜Wλ\nk,0,−,RS\nλ(·,z))L2(Uǫ;C)−(χkfk,0,−,(∆−λ)RS\nλ(·,z))L2(Uǫ;C)→\nǫ→0[χkfk,0,−](z)−RS\nλ(Pk,z).\nHence, taking into account the symmetry RS\nλ(z,z′) =RS\nλ(z′,z) of the resolvent kernel, we arrive\nat\n(∆S−λ)−1˜Wλ\nk,0,−(z) =/integraldisplay\nX˜Wλ\nk,0,−(z′)RS\nλ(z,z′)h(z′)dS(z′) = [χkfk,0,−](z)−RS\nλ(z,Pk).\nComparing the last equality with (5.1), we obtain\nVλ\nk,0,−(z) =RS\nλ(z,Pk). (5.5)\nIn view of (5.2), the substitution z=xk→0 into (5.5) yields\nTjk(λ) :=RS\nλ(Pj,Pk) =Tkj(λ). (5.6)\nDerivative of scattering matrix. Since RS\nλis meromorphic with respect to λand has poles\nat the eigenvalues of ∆ S, the scattering matrix λ/ma√sto→T(λ) is also meromorphic and each pole of\nTis an eigenvalue of ∆ S.\nLet uscalculate ∂λT(λ). Inview of Proposition4.1, onecandifferentiate the(∆ −λ)Vλ\nk,0,−= 0\nand asymptotics (5.2) with respect to λ. As a result, one obtains (∆ −λ)∂λVλ\nk,0,−=Vλ\nk,0,−and\n∂λVλ\nk,0,−−2g−2/summationdisplay\nj=1∂λTjk(λ)χjfj,0,+∈Dom∆ S∩Dom∆ F.\nIn view of Green formula (4.36), these relations imply\n(Vλ\nk,0,−,Vλ\nj,0,−)L2(X;C)=\n= ((∆−λ)∂λVλ\nk,0,−,Vλ\nj,0,−)L2(X;C)−(Vλ\nk,0,−,(∆−λ)Vλ\nj,0,−)L2(X;C)=∂λTjk(λ)\nand\n∂λTjk(λ) = (RS\nλ(·,Pk),RS\nλ(·,Pj))L2(X;C). (5.7)\n29Asymptotics of scatterring matrix at infinity. Let us represent Vλ\nk,0,−in the form\nVλ\nk,0,−=χkfλ\nk,0,−−(∆S−λ)−1[∆,χk]fλ\nk,0,−, (5.8)\nwherefλ\nk,0,−(xk) =Dλ(|xk|) is a local solution fλ\nk,0,−to (∆−λ)u= 0 nearPk. Since\nλDλ(|xk|) = ∆fλ\nk,0,−=−|xk|−1∂\n∂xk|xk|−1∂\n∂xkDλ(|xk|) =−1\n4|xk|2D′′\nλ(|xk|),\nwe have\nD′′\nλ(ρ)+4λρ2Dλ(ρ) = 0, ρ=|xk|.\nThe solution exponentially decaying as |λ|r4→+∞,|argλ|>ǫ>0 is given by\nDλ(r) =c(λ)D−1/2(2r(−λ)1/4),\nwhereD−1/2is the parabolic cylinder function. In view of the Taylor expansion\nD−1\n2(z) =π3/22−1/4\nΓ(3/4)−π3/221/4\nΓ(1/4)z+O(z4), z→0,\ncondition (5.2) implies c(λ) = (−2/λ)1/4π−3/2Γ(3/4)−1. Therefore,\nfλ\nk,0,−(xk) =fk,0,−(xk)+Γ(3/4)−2(−λ)−1/4+O(|xk|4). (5.9)\nNotethat[∆ ,χk]fλ\nk,0,−andallitsderivativeswithrespectto xkdecayexponentiallyas ℜλ→−∞.\nIn view of Proposition 4.4, the coefficients in asymptotics (3.9) of u= (∆S−λ)−1[∆,χk]fλ\nk,0,−\ndecay exponentially as ℜλ→−∞. Thus, formulas (5.2), (5.8), and (5.9) imply\nTjk(λ) =δjkΓ(3/4)−2(−λ)−1/4+O(e−ǫ|λ|),\nwhereǫ>0. As a corollary, we have\ndetT(λ) =t∞(−λ)p∞+O(e−ǫ|λ|),ℜλ→−∞, (5.10)\nwhere\nt∞:= Γ(3/4)4−4g, p ∞:=1−g\n2. (5.11)\nExpression for T(0).Since expression (3.7) is valid for the Green function GSof the Szeg¨ o\nextension ∆ S, formula (5.6) implies\nTjk(0) =GS(Pj,Pk) =π−2Sjk, (5.12)\nwhereSis given by (3.15). Hence, in view of (3.8), we obtain (3.14).\n305.1.2 Comparing resolvents of ∆Fand∆S\nLetyS:= (∆ S−λ)−1f, wheref∈C∞\nc(˙X;C) andλis not an eigenvalue of ∆ S. In view of\nProposition 4.1 and formula (3.9), the expansion\nyS−/summationdisplay\njcj,0,+(f)χjfj,0,+∈Dom∆ S∩Dom∆ F (5.13)\nis valid, where the coefficients can be found from Green formula (4.36 ) withf=Vλ\nj,0,−,f′=yS;\ncj,0,+(f) = (f,Vλ\nj,0,−)L2(X;C)= [(5.5)] =/integraldisplay\nXf(z′)RS\nλ(Pj,z′)h(z′)dS(z′).(5.14)\nWe search for a solution yFto (∆F−λ)yF=fin the form\nyF=yS−/summationdisplay\nkckVλ\nk,0,−. (5.15)\nIn view of (5.13) and (5.2), the right hand side belongs to Dom∆ Fif and only if\ncj,0,+(f) =/summationdisplay\nkTjk(λ)ck. (5.16)\nCombining formulas (5.15), (5.16), (5.14), and (5.5), we obtain\nyS(z)−yF(z) =/integraldisplay\nXf(z′)/bracketleftBig/summationdisplay\nk,jRS\nλ(z,Pk)T−1\nkj(λ)RS\nλ(Pj,z′)/bracketrightBig\nh(z′)dS(z′).\nSince thef∈C∞\nc(˙X;C) is arbitrary, we have\nRS\nλ(z,z′)−RF\nλ(z,z′) =/summationdisplay\nk,jRS\nλ(z,Pk)T−1\nkj(λ)RS\nλ(Pj,z′).\nUsing the bra–ket notation, we rewrite the last formula as\nD(λ) := (∆ S−λ)−1−(∆F−λ)−1=/summationdisplay\nk,jT−1\nkj(λ)|RS\nλ(·,Pk)/a\\}bracketri}ht/a\\}bracketle{tRS\nλ(·,Pj)|,(5.17)\nwhere|f/a\\}bracketri}htdenotes an element of L2(X;C) while/a\\}bracketle{tf|is the linear bounded functional on L2(X;C)\ngiven by/a\\}bracketle{tf|f′/a\\}bracketri}ht:= (f′,f)L2(X;C). Formula (5.17) shows that the zeroes of det T(λ) are the\neigenvalues of ∆ F. In view of (5.7) and the equality\nTr|a/a\\}bracketri}ht/a\\}bracketle{tb|=/a\\}bracketle{tb|a/a\\}bracketri}ht,\nformula (5.17) implies\nTrD(λ) =/summationdisplay\nk,jT−1\nkj(λ)(RS\nλ(·,Pk),RS\nλ(·,Pj))L2(X;C)=/summationdisplay\nk,jT−1\nkj(λ)∂λTjk(λ) =\n= Tr/parenleftbig\nT−1(λ)∂λT(λ)/parenrightbig\n=∂λTrlogT(λ) =∂λlogdetT(λ).(5.18)\n315.2 Comparing zeta functions\nIn this subsection, we derive formula (3.13). To this end, we expres s the difference of zeta\nfunctions of ∆ Fand ∆ Sin terms of the trace of the difference of their resolvents, and the n apply\nformula (5.18).\n5.2.1 Some operator relations\nFormula for powers of operator. IfAis a positive definite operator with discrete spectrum,\nthen\nA−s=1\n2πi/integraldisplay\nγξ−s(A−ξ)−1dξ(ℜs>0), (5.19)\nwhereγis the union of the paths\nγ±(t) =|t|e±i(π+0)(±t∈(ǫ,+∞)), γ ◦(t) =ǫeiϕ(ϕ∈(−iπ,+iπ)),(5.20)\nand the number ǫ >0 is sufficiently small. Indeed, since ( ζ−λ)−1=O(|λ|−1) asℜλ→−∞\nuniformly with respect to ζ∈(0,∞)⊃Sp(A), one has/bardbl(A−λ)−1/bardbl=O(|λ|−1) asℜλ→−∞.\nThen the norm of the integrand is O(|ξ|−1−ℜs) asγ∋ξ→∞and the integral in the right hand\nside converges (in the operator norm). In particular, the right ha nd side is a bounded operator,\nwhile the left hand side is a bounded operator since |ξ−s|is bounded on the spectrum of A. So,\nit remains to check (5.19) on elements of some orthonormal basis {uk}k. Let us choose this basis\nin such a way that ukis an eigenfunction of Acorresponding to the eigenvalue λk. Applying the\nboth sides of (5.19) to uk, and taking into account that ℜs>0, one obtains\n(1\n2πi/integraldisplay\nγξ−s(λk−ξ)−1dξ)uk=−Resξ=λk(ξ−s(λk−ξ)−1) =λs\nkuk=A−suk.\nHence, formula (5.19) is proved.\nConnection between zeta functions of self-adjoint extensi ons of symmetric operator.\nLetA′andAbe positive definite operators ( A,A′≥ǫI >0) with discrete spectra. Also, we\nsuppose that A′andAare self-adjoint extensions of the same symmetric operator A0. Then, for\nanyλ/\\e}atio\\slash∈[ǫ>0,+∞), the difference of their resolvents\nD(λ) := (A′−λ)−1−(A−λ)−1\nis an operator of finite rank not exceeding the deficiency index of A0. According to (5.19), we\nhave\nA′−s−A−s=1\n2πi/integraldisplay\nγξ−sD(ξ)dξ(ℜs>0),\nwhere the integral in the right hand side converges in the operator norm as well as in the trace-\nnorm/bardbl·/bardbl1. Therefore, one can apply the trace to both sides and interchang e the trace and the\nintegration in the right hand side. As a result, one obtains\nζ(s|A′)−ζ(s|A) = Tr(A′−s−A−s) =1\n2πi/integraldisplay\nγξ−sTrD(ξ)dξ, (5.21)\nwhere the right hand side is well-defined at least for ℜs>0.\n325.2.2 Comparing ζ(·|∆S)andζ(·|∆F).\nLet us substitute A′= ∆S,A= ∆Finto (5.21) and apply (5.18). As a result, we obtain\nζ(s|∆S)−ζ(s|∆F) =1\n2πi/integraldisplay\nγξ−st′(ξ)\nt(ξ)dξ,(ℜs>0,s/\\e}atio\\slash= 1),\nwheret:= detT. Denote the right hand side by J(s); thenJ=J1+J2, where\nJ1(s) : =1\n2πi/integraldisplay\nγ◦ξ−sdlogt(ξ),\nJ2(s) : =1\n2πi/integraldisplay\nγ++γ−ξ−st′(ξ)\nt(ξ)dξ=sin(πs)\nπ/integraldisplay−ǫ\n−∞|ξ|−st′(ξ)\nt(ξ)dξ\nandγ◦,γ±are given by (5.20). Note that s/ma√sto→J1(s) is holomorphic for any s∈C. Recall that\nthe function λ/ma√sto→t(λ) = detT(λ) is holomorphic and non-zero outside spectra of ∆ Sand ∆ F.\nDue to this, one can choose a branch of log twhich is holomorphic outside [2 ǫ,+∞). Integrating\nby parts, we obtain\nJ1(s) =−sin(πs)\nπǫ−slogt(−ǫ)+sJ3(s),\nwhere\nJ3(s) =1\n2πi/integraldisplay\nγ◦ξ−(s+1)logt(ξ)dξ,\nIn view of the Cauchy formula, we have J3(0) = logt(0). Formula (5.10) can be rewritten as\nlogt=p∞log(−λ)+logt∞+˜q, where ˜g(λ) and its derivatives decay exponentially as ℜλ→−∞.\nHence,J2(s) =J4(s)+J5(s), where\nJ4(s) :=p∞sin(πs)\nπ/integraldisplay+∞\nǫξ−s−1dξ, J 5(s) :=sin(πs)\nπ/integraldisplay−ǫ\n−∞|ξ|−s∂ξ˜q(ξ)dξ.\nNote that J4(s) =−p∞(πs)−1sin(πs)ǫ−sforℜs >0; the same equality is valid for analitic\ncontinuation of J4on the whole C. Since∂ξ˜q(ξ) decays exponentially as ℜλ→−∞, the integral\nJ5is analytic on C. In addition, integration by parts yields\nJ5(s) =sin(πs)\nπ/parenleftBig\nǫ−s˜q(−ǫ)−sJ6(s)/parenrightBig\nwhere\nJ6(s) =/integraldisplay−ǫ\n−∞˜q(ξ)|ξ|−s−1dξ.\nSumming up the above equalities, we obtain\nJ(s) =sJ3(s)−p∞(πs)−1sin(πs)ǫ−s−sin(πs)\nπ/parenleftBig\nǫ−s(logt(−ǫ)−˜q(−ǫ))+sJ6(s)/parenrightBig\n=\n=sJ3(s)−(πs)−1sin(πs)/parenleftBig\nǫ−s/bracketleftBig\np∞+s(p∞log(ǫ)+logt∞)/bracketrightBig\n−s2J6(s)/parenrightBig\n.\nwhere the right hand side is an entire function of s. In particular,\nJ(0) =p∞, J′(0) =J3(0)−logt∞= log(t(0)/t∞).\n33Hence,\ndet∆F\ndet∆S=exp/parenleftbig\n−∂sζ(s|∆F)/vextendsingle/vextendsingle\ns=0/parenrightbig\nexp/parenleftbig\n−∂sζ(s|∆S)/vextendsingle/vextendsingle\ns=0/parenrightbig=eJ′(0)=detT(0)\nt∞= [(5.11)] = Γ(3 /4)4(g−1)detT(0).\nTheorem 3.6 is proved.\n6 Dependence of determinants on moduli\nIn this section, we prove formula (3.18) which describes the depend ence of det∆ Son moduli.\n6.1 Surface families corresponding to variation of moduli\nThe moduli space Hg(1,...,1) is a complex orbifold of dimension 4 g−3 (see [17]). Let ( X,ω)∈\nHg(1,...,1). Choose paths ai,bi(i= 1,...,g) representing a canonical basis in the homol-\nogy group H1(X,Z). Letlk(k= 2,...,2g−3) be a path which connects P1toPkand\ndoes not intersect any of ai,bi. Then{ai,bi,lk}provides a basis in a relative homology group\nH1(X,{P1,...,P 2g−2},Z) (a “marking”). One can continuously transport this basis to near by\n(X′,ω′) inHg(1,...,1) and thereby endow them with markings a′\ni,b′\ni,l′\nk(see [5], some explicit\nway to do this is described below).\nGiven a marking ai,bi(i= 1,...,g), the spin structure ConXis determined by specifying\nits characteristic ( p,q) (wherep,q∈{0,1\n2}2g) in the equation\nAz0(C) =−Kz0+Bp+q, (6.1)\nwhereAz0,Kz0,Bare the Abel transform, the vector of Riemann constants, and t heb-period\nmatrix associated with X, respectively. This spin structure is extended to nearby ( X′,ω′) in\nsuch a way that\nA′\nz′\n0(C′) =−K′\nz′\n0+B′p+q, (6.2)\nwhereC′is a spinor bundle over XwhileA′\nz′\n0,K′\nz′\n0,B′are the Abel transform, the vector of\nRiemann constants, and the period matrix associated with X′, respectively. It is worth noting\nthat the marking ai,bi,lkand the spin structure Cmay be changed after continuation along the\nclosed loop in Hg(1,...,1) (see [1]).\nWitheachtriple( X′,ω′,C′), weassociate theSzeg¨ o self-adjoint extension ∆′\nSoftheDolbeault\nLaplacian on C′. From now on, we deal only with Szeg¨ o extensions and omit the symb olSin\nthe notation of operators, resolvent kernels, Green functions e .t.c.. In addition, primed symbols\ndenote the objects related to the pair ( X′,ω′), while unprimed ones are associated with ( X,ω).\nThe local coordinates (from now on, moduli) of (X,ω) are given by\nAi=/integraldisplay\najω, Bi=/integraldisplay\nbjω, Ck=/integraldisplay\nlkω(6.3)\n(see [16], p.5). For generic points of Hg(1,...,1) (for all the points if g= 2, see [18], Theorem\n1.7), coordinates (6.3) can be visualized as follows (see [13], pp.5,6). C onsider complex tori\n34Tj=C/Λj, where Λ j=ZAj+ZBj. Endow these tori with the system of cuts\n[C1,C2](modΛ 1) on T1\n[C1,C2](modΛ 2),[C3,C4](modΛ 2) on T2\n[C3,C4](modΛ 3), [C5,C6](modΛ 3) on T3\n... ...\n[C2g−5,C2g−4](modΛ g−1) [C2g−3,C2g−2](modΛ g−1) onTg−1\n[C2g−3,C2g−2](modΛ g) on Tg(6.4)\nMaking cross-gluing along the edges of the cuts, we obtain the comp act Riemann surface Xof\ngenusgwhile the differential dzon a common universal cover Cof allTjgives rise to the Abelian\ndifferential ωonX. The holomorphic local coordinates near the ends Ck≡Pkof the cuts are\ngiven by (2.1). Thus, each Pkis a zero of ω. The basis cycles in H1(X,Z) are given by\naj= [ej,Aj+ej](modΛ j), bj= [ej,Bj+ej](modΛ j) onTj, (6.5)\nwhereejis an arbitrary constant such that aj,bjdo not intersect the cuts on Tj. In the con-\nstruction of the perturbed surface X′, we will also use the “shifted” cycles\nˇaj= [cj,Aj+cj](modΛ j),ˇbj= [cj,Bj+cj](modΛ j) onTj, (6.6)\nwherecjis an arbitrary constant such that cj/\\e}atio\\slash=ej(modΛ j) and ˇaj,ˇbjdo not intersect the cuts\nonTj.\nLetǫ0>0 be sufficiently small and denote by Dktheǫ0-neighborhood of Pk(in the metric\n|ω|2). Let¯Tjbe the domain obtained from the torus Tjby making the cuts (6.4) and (6.5) and\nremoving all Dk. Continuous sections fof the bundle ConXcan be identified with collections\n{fi;˜fk}ofcontinuous(uptotheboundaries)functions fion¯Tiand˜fkonDkobeyingthefollowing\nconditions (cf. the case of a single torus explained in [20], pp.275, 27 6):\na) theboundaryvaluesof fjonthedifferentsidesofthecuts aj,bjarerelatedbymultiplication\nby the automorphy factors σ(aj|C) =±1,σ(bj|C) =±1;\nb)fj=±fj+1onLj\n±, whereLj\n±are sides of the cut [ C2j−1,C2j](modΛ j) inTj(identified with\nthe corresponding sides of the cut [ C2j−1,C2j](modΛ j+1) inTj+1).\nc) If∂Dk∩∂¯Tiis nonempty, then ˜fk=fj√xkon it, where xkis given by (2.1).\nLemma 6.1. Any spinor bundle ConXcan be constructed in the way described above. Two\nspinor bundles C1andC2are isomorphic (as holomorphic bundles )if and only if σ(aj|C1) =\nσ(aj|C2)andσ(bj|C1) =σ(bj|C2)for allj.\nProof.Let{fi;˜fk}obey a)-c), then{fi;˜fk}defines a section fof some holomorphic line bundle\nC. In view of a),b), there is a continuous function Fon˙Xsuch thatf2\nj=F|¯Tjfor anyj. If\n∂Dk∩∂¯Tiis nonempty, then ˜fk2=f2\njxk= (dzk/dxk)F. Then the equations\nf2=f2\njdzon¯Tj, f2=˜fk2dxkonDk\ndefine a continuous section f2of the canonical bundle KonX. Therefore, C2=K, i.e.,Cis a\nspinor bundle.\n35Next, letC1andC2be two spinor bundles and f(1)≡{f(1)\ni,˜fk}andf(2)≡{f(2)\ni;˜f(2)\nk}be\ntheir sections. Then\nh≡{hi=f(2)\ni/f(1)\ni;˜hk=˜f(2)\nk/˜f(1)\nk}≡f(2)/f(1)\nis a section of the bundle B:=C2C−1\n1. The bundles C1andC2are isomorphic if and only if B\nis trivial, i.e., if and only if there is a holomorphic section hwhich has no zeroes on X.\nIn view of a), the boundary values of hjon the different sides of the cuts aj,bjare related\nvia multiplication by the automorphy factors\nσ(aj|B) =σ(aj|C2)/σ(aj|C1), σ(bj|B) =σ(bj|C2)/σ(bj|C1).\nOn the both sides of the the cut [ C2j−1,C2j](modΛ j) inTj(identified with the corresponding\nsides of the cut [ C2j−1,C2j](modΛ j+1)), we have\nhj=sjhj+1(sj=±1).\nHeresj= +1 if the same choice of the “plus” side Lj\n+of the cut [C2j−1,C2j](modΛ j) was used in\nthe construction of both C1,C2; otherwise, we have sj=−1. Similarly, if ∂Dk∩∂¯Tiis nonempty,\nthen\n˜hk=skjhj(skj=±1)\non it. Here skj= +1 if the same choice of the branch of the square root√xkwas used in the\nconstruction of both C1,C2; otherwise, we have skj=−1. Multiplying all hiwithi≤jby−1\n(or multiplying arbitrary ˜hkby−1), one obtains a section of a bundle isomorphic to B. Thus,\none can assume that all sjandskjare equal to 1.\nA sectionhofBcan be identified with the continuous (up to the boundary) function Hon\nX\\/uniontextg\ni=1(ai∪bi) such that hj=H|¯Tjand˜hk=H|Dkfor anyj,k. Then the boundary values of H\non the different sides of the cuts aj,bjare related by multiplication by the automorphy factors\nσ(aj|B) =±1,σ(bj|B) =±1. The bundle Bis trivial if and only if there is such Hwhich is\nholomorphic and has no zeroes on X. In the last case, H2can be continued to a holomorphic\nfunction on the whole X. Thus,H2andHare a non-zero constants on X. The latter is possible\nonly if allσ(aj|B) andσ(bj|B) are equal to 1.\nTherefore, thebundle Cisdetermined(uptoisomorphism)by2 gautomorphyfactors σ(aj|C) =\n±1,σ(bj|C) =±1. Note that different choices of the “plus” sides Lj\n+in condition c) and the\nbranches of the square roots in condition c) lead to isomorphic bund les. To complete the proof,\nit remains to note that the number of all possible choices of {σ(aj|C),σ(bj|C)}g\nj=1is equal to 22g\nand coincides with the number of all non-equivalent spinor bundles.\n6.1.1 Varying Aj,Bj\nIn order to vary Ajwhile keeping the other coordinates (6.3) unchanged, we implement t he\nfollowing deformation of ( X,ω). Introduce a small variation δAj=αAj+βBjof coordinate Aj,\nwhereα,β∈R. Forα≤0, we remove the tubular neighborhood\nV={[tBj−sδAj+cj](modΛ j)|t,s∈[0,1]} (6.7)\nof ˇcjfrom the torus TjofX. Then we identify the points\n[tBj+δAj0](modΛ j)←→[tBj−δAj(1+0)](modΛ j). (6.8)\n36As a result, we obtain the surface X′. The restriction of the ωto the domain ´X=X\\Vis\nextended by continuity to the differential ω′onX′. All paths ai,bi,lk, exceptaj, belong to ´X\nand thus can be chosen as a′\ni,b′\ni,l′\nk, respectively. At the same time, the path a′\njis obtained by\nidentification of the ends of the straight line segment [ αAj+cj,Aj+βBj+cj](modΛ j) inTj∩´X.\nForα>0, we cutXalongˇbj. Then we glue the resulting surface ´Xand the cylinder\nV={[tBj+sδAj+cj](modBjZ)|t,s∈[0,1]}\nalong their boundaries according to the following identification rules\n∂´X∋[tBj−δAj0](modΛ j)←→[tBj](modBjZ)∈∂V,\n∂´X∋[tBj+δAj0](modΛ j)←→[tBj+δAj](modBjZ)∈∂V.\nThe differential ω′onX′is given byω′=ωon´Xandω=dzonV. All paths a′\ni,b′\ni,l′\nk, exceptaj,\ncoincide with ai,bi,lkwhilea′\njis a union of straight line segments [ cj,Aj+β(1+α)−1Bj](modΛ j)\nand [Aj+β(1+α)−1Bj,δAj](modBjZ) of´XandV, respectively.\nThe spinor bundle C′overX′is defined by conditions a)-c) with the automorphy factors\nσ(a′\nj|C′) =σ(aj|C), σ(b′\nj|C′) =σ(bj|C),(j= 1,...,g). (6.9)\nNote that, in both cases α≤0,α >0, one can consider ´Xas a common part of XandX′.\nMoreover, the restrictions of the bundles CandC′on´Xare isomorphic,\nC|´X≡C′\n´X. (6.10)\nIn other words, in the domain ´X, one can add sections of C′and sections of C. Note that a small\nshifts of the paths ˇ aj,ˇbj, i.e., a small change of cjin (6.6), do not affect ( X′,ω′,C′).\nLemma 6.2. The bundle C′defined by (6.9)obeys(6.2)for sufficiently small δAj.\nProof.In view of the above procedure for constructing ( X′,ω′,C′), for small δAjthere exists\na near-isometric diffeomorphism q: (X,|ω|2)/ma√sto→(X′,|ω′|2) obeyingq◦ai=a′\ni,q◦bi=b′\ni\n(i= 1,...,g) and such that qis identity on the common tori Ti(i/\\e}atio\\slash=j) ofXandX′and near\neachPk. For simplicity, one can consider X′as a surface Xendowed with another complex\nstructure provided by the operator ∂′. Thenβis identity on X0and∂=∂′on some domain\nU⊂X0. Moreover, in each chart one has\n∂′= (1+a(z))∂+˜a(z)∂,where/bardbla/bardblCl+/bardbl˜a/bardblCl−→\nδAj→00. (6.11)\nSectionsfofbothCandC′canbeconsidered ascollections {fi}g\ni=1,{˜fk}2g−2\nk=1offunctions fion¯Ti\nand˜fkonDkobeying conditions a),b) given after (6.5). For l= 1,..., denote byHlthe subspace\ninHl(¯T1)×···×Hl(¯Tg)×Hl(D1)×···×Hl(D2g−1) consisting of collections obeying a)-c). Since\n∂is an elliptic operator and Cadmits no holomorphic sections, its inverse ∂−1:Hl→Hl+1is\nwell defined and continuous. In view of (6.11), the same is true for t he (elliptic) operator ∂′−1,\nand\n/bardbl∂′−1−∂−1/bardblHl→Hl+1−→\nδAj→00 (l= 1,...). (6.12)\nLetzbe a holomorphic (with respect to both complex structures ∂and∂′) coordinate on U.\nLetz0∈Uandχbe a cut-off function equal to one near z0and vanishing outside U. Let\n37w(z) = (z−z0)−1onU. The Szego kernels of the bundles CandC′admit the representations\n(see 4.2.2)\nS(·,z0) =χ w−∂−1/parenleftBig\n[∂,χ]w/parenrightBig\n,S′(·,z0) =χ w−∂′−1/parenleftBig\n[∂′,χ]w/parenrightBig\n.\nIn view of (6.11) and (6.12), we have\n/bardblS′(·,z0)−S(·,z0)/bardblHl−→\nδAj→00 (l= 1,...).\nTherefore, in view of the Morrey’s inequality, the zeroes o′\n1,...,o′\ngofS′(·,z0) tend to the corre-\nsponding zeroes o1,...,o gofS(·,z0),\no′\ni−→\nδAj→0oi. (6.13)\nSimilarly, one can prove the convergences\n/bardblω′−ω/bardblCl(X)+/bardblA′\nz0−Az0/bardblC(X)−→\nδAj→00,B′−→\nδAj→0B, K′\nz0−→\nδAj→0Kz0,(6.14)\nwhere{υ′\ns}g\ns=1be a basis of Abelian differentials on X≡(X0,∂) (X′≡(X0,∂′)) dual to the ho-\nmology basis{ai,bi}g\ni=1,B′be theb-period matrix of X′,K′\nz0be the vector of Riemann constants,\nandA′\nz0be the Abel map of X′.\nNote thatD=o1+···+og−z0andD′=o′\n1+···+o′\ng−z0are divisors of the bundles C\nandC′, respectively. Since C′is a spinor bundle, we have\nA′\nz0(D′) =A′\nz0(C′) =−K′\nz0+B′p′+q′\nwithp′,q′∈{0,1\n2}2g. In view of convergences (6.13) and (6.14), the comparison of the last\nequation with (6.1) yields p′=p,q′=qfor sufficiently small δAj. Formula (6.2) is proved.\nVariation of Bjis performed similarly (one needs only to interchange ˇ aj,Ajandˇbj,Bjin the\nabove construction).\n6.1.2 Varying Ck\nIn order to vary Ckwhile keeping the other coordinates (6.3) unchanged, we implement t he fol-\nlowing deformation of ( X,ω). Recall that the coordinates zkandxk=√2zknearPkare given by\n(2.1) and(4.1), respectively. One candeform lk(preserving its class in H1(X,{P1,...,P 2g−2},Z))\nin such a way that it coincides with the ray xk/√2δCk≥0 nearPk. Remove from Xthe neigh-\nborhood Dk={zk∈X||zk|<ǫ0}(withǫ0>|δCk|), then attach the disk D′\nk:={x′\nk∈C||x′\nk|<√2ǫ1}(withǫ1>ǫ0) by identifying the points xkandzk∈X\\Dkandx′\nk∈D′\nksuch that\nx′\nk=/radicalbig\n2(zk−δCk). (6.15)\nAs a result, we obtain the surface X′. Then´X:=X\\Dkcan be considered as a common part of\nXandX′whileD′\nkis considered as a domain in X′. The differential ω′onX′is given by ω′=ω\non´X′and byω′=dz′\nkonD′\nk, wherezk=x2\nk/2. Then the zero P′\nkofω′is given by z′\nk= 0 while\nthe other zeroes P′\ns=Ps(s/\\e}atio\\slash=k) belong to ´X. The paths a′\ni:=ai,b′\ni:=bi,l′\ns:=ls(s/\\e}atio\\slash=k) are\ncontained in ´Xwhile the path l′\nicoincides with lion´Xand with the ray x′\nk/√2δCkonD′\nk.\nThe spinor bundle C′onX′is defined (up to isomorphism) by condition (6.10). By repeating\narguments used in the prof of Lemma 6.2, one can verify the equality (6.2). Note that ( X′,ω′,C′)\ndoes not depend on the choice of ǫ0,ǫ1in the above procedure.\n386.2 Variation of resolvent kernels\nLetRλ=Rλ(·,·|X,ω,C) and R′\nλ=Rλ(·,·|X′,ω′,C′) be the resolvent kernels of ∆′= ∆′\nSand\n∆ = ∆ S, respectively. In view of (6.10), the difference\nuλ(z,z′) =Rλ(z,z′|X′,ω′,C′)−Rλ(z,z′|X,ω,C)\nis well defined on the joint domain ´XofXandX′. In addition, the complements of ´XinXand\nX′shrink to conical points Pkor curves ˇaj,ˇbjasδAj,δBj,δCk→0. Moreover, by changing the\nconstantcjin (6.6), the curves ˇ aj,ˇbjcan be slightly shifted along the torus Tjwithout affecting\nthe resulting surface ( X′,ω′,C′). Due to these facts, one can define the derivatives\n∂Rλ(z,z′)\n∂ν,∂Rλ(z,z′)\n∂ν(z,z′∈˙X, ν=Aj,Bj,Ck).\nIn this subsection, we prove the following fact.\nProposition 6.3. Theresolventkernel Rλisdifferentiablewith respectto ℜν,ℑν(ν=Ai,Bi,Ck),\nand\n2i∂Rλ(z,z′)\n∂ν=/integraldisplay\nν†ω(ς)−1/parenleftBig\nλRλ(z,ς)Rλ(ς,z′)ρ−2(ς)h(ς)dς−∂ςRλ(z,ς)∂ς(h(ς)Rλ(ς,z′))dς/parenrightBig\n.\n(6.16)\nHereA†\ni:=−ˇbi,B†\ni:= ˇai, andC†\nk=ckis homologous to the small (counterclockwise )circle\n|ξk|=ǫin˙X.\nBefore proving Proposition 6.3, let us note that the integrand in the right hand side of (6.16)\nis a closed 1-form and, thus, the right hand side of (6.16) depends o nly on the homology class of\nν†inH(˙X,Z).\n6.2.1 Boundedness of uλ(z,z′)\nFor simplicity, we consider the case in which only the coordinate ν=Ajvaries (the other cases\nare considered similarly). Suppose that λ∈Cis not an eigenvalue of ∆ = ∆ S. First, let us show\nthatλis separated from the spectra of ∆′for sufficiently small variations. Suppose the contrary;\nthen there exists a sequence of the triples ( X′\n(n),ω′\n(n),C′\n(n)) constructed in paragraph 6.1.1 and\nrepresenting the variations δA(n)\nj→0 ofAj, and a sequence of eiqenpairs ( λ′\nn,u′\nn) of the (Szeg¨ o)\nLaplacians ∆′\n(n)on (X′\n(n),ω′\n(n),C′\n(n)) such that λ′\nn→λand/bardblu′\nn/bardblL2(X′\n(n),C′\n(n))= 1.\nLetǫ>0 be sufficiently small and let\nX1\nǫ=/braceleftBig\n[(t+is)Bj+cj](modBjZ)/vextendsingle/vextendsinglet∈[0,1], s∈/parenleftBig\n−ǫ\n|Bj|,ǫ\n|Bj|/parenrightBig/bracerightBig\nbe theǫ0-neighborhood of the path ˇbjin (X,|ω|2). Let also X0\nǫ=X\\X1\nǫ/2. For small δA(n)\nj, one\ncan consider X0⊂´Xandˇbj⊂´Xas a domain and a curve in X′\n(n). The (closed) ǫ0-neighborhood\nof the path ˇbjin (X′\n(n),|ω′\n(n)|2) is isometric to the cylinder X1\nǫ. ThenX′\n(n)can be represented as\nthe union of X0\nǫandX1\nǫwhere a common point of X0\nǫandX1\nǫis identified with points\n[(t±is)Bj+cj](modΛ j)←→[(t±is)Bj∓δA(n)\nj/2+cj](modBjZ) (6.17)\n39ofTj∩X0\nǫandX1\nǫ, respectively.\nNote thatC′\n(n)|Xmǫis isomorphic to C|Xmǫdue to (6.9). Let Hl\nν(Xm\nǫ) (m= 0,1) be the space\nof sections of C|Xmǫwith finite norms (4.32), where Xis replaced by Xm\nǫ. The convergence\n/bardbl(∆′\n(n))qu′\nn/bardblL2(X′\n(n);C′\n(n))=λq\nn→λq, q= 0,1,...\nand local estimates (4.9), (4.21) imply that the norms /bardblu′\nn|Xmǫ/bardblHl′\nν′(Xmǫ)are bounded for some ν′\nand (arbitrarily large) l′. The compactness of the embedding Hl′\nν′(Xm\nǫ)⊂Hl\nν(Xm\nǫ) (withν >ν′\nandl′< l) can be proved in the same way as in 4.1.4. As a corollary, there is a sub -sequence\n{un(s)}∞\ns=1obeying\nun(s)|Xmǫ→yminHl\nν(Xm\nǫ) (m= 0,1). (6.18)\nIn view of (6.17) and (6.9), y0,y1are restrictions on X0\nǫ,X1\nǫof some section yofC. Then (6.18)\nand the equations ∆′\n(n)un=λ′\nnunimply ∆y=λyon˙X. In addition, un(s)admit asymptotics\n(3.9) near each Ok∈´X. The same is true for ydue to (6.18) and Proposition 4.1 (with X\nreplaced by Xm\nǫ). Thus, ∆ Sy=λy, which gives a contradiction. Therefore, λis separated from\nthe spectra of ∆′for sufficiently small δAj.\nNext, letδAjbe sufficiently small and let ˙X0\nǫbe a domain in Xobtained by removing the\nǫ-neighborhoods (in the metric |ω|2) ofˇbjand conical points. Let us show that\n|uλ(z,z′)|+|∂zuλ(z,z′)|≤C(z∈´X, z′∈˙X0\nǫ) (6.19)\nuniformly in ( X′,ω′), where the coordinates z,z′are given by (4.1). Let χbe a cut-off function\nequal to one on X\\X1\nǫ/2and equal to zero outside of X\\X1\nǫ/4. Represent uλas the sum u1,λ+u2,λ,\nwhere\nu1,λ(·,z′) = (χ−1)Rλ(·,z′), u 2,λ(·,z′) =R′\nλ(·,z′)−χRλ(·,z′).\nSince Rλ(z,z′) and R′\nλ(z,z′) are smooth outside conical points and have the same logarithmic\nsingularity (given by (5.4)) at the diagonal z=z′, the section u2,λ(·,z′) is smooth for z′∈˙X0\nǫ.\nNote that (∆−λ)u2,λ(·,z′) =−[∆,χ]Rλ(·,z′) does not depend on ( X′,ω′). Since the spectra of\n∆′are separated from λ, the sum/summationtextl\ns=1/bardbl∆su2,λ(·,z′)/bardblL2(X′;C)is bounded uniformly in z′∈˙X0\nǫ\nand (X′,ω′). Thus, (6.19) follows from (4.9) and (4.10).\n6.2.2 Varying Aj,Bj\nFor simplicity, we consider the case in which only the coordinate ν=Ajvaries, and δAj=\nαAj+βBj,α≤0 (the other cases are considered similarly). In this case X′is obtained by\nidentification (6.8) of the boundary points of ´X=X\\V, whereVis given by (6.7).\nRecall that the resolvent kernels Rλ(z,z′),R′\nλ(z,z′) are smooth outside conical points and\nhave the same logarithmic singularity (given by (5.4)) at the diagonal z=z′. So, their difference\nuλ(z,z′) is smooth and obeys (∆ −λ)uλ(·,z′) = 0 on ´X∩˙X. Since Rλ(z,z′),R′\nλ(z,z′) correspond\nto the Szeg¨ o extensions of the Laplacians on ( X,ω,C) and (X′,ω′,C′), their difference uλ(z,z′)\nadmits the asymptotics of the form (3.9) near conical points. Thus , Green formula (3.1) with\nf=uλ(·,z′),f′=Rλ(·,z) =Rλ(z,·), andU=´Xyields\n0 = ((∆−λ)uλ(·,z′),Rλ(·,z))L2(˘X;S)−(uλ(·,z′),(∆−λ)Rλ(·,z))L2(˘X;S)=\n=uλ(z,z′)+/integraldisplay\n∂´Xh(ς)\n2i/parenleftbig\nuλ(ς,z′)∂ςRλ(z,ς)dς+∂ςuλ(ς,z′)Rλ(z,ς)dς/parenrightbig\n.(6.20)\n40Here∂´Xis a union of ˇbj= [cj,Bj+cj](modΛ j) and the curve [ Bj−δAj+cj,−δAj+cj](modΛ j)\nobtained by shifting ˇbj(along the torus Tj) and reversing its orientaiton. Thus,(6.20) can be\nrewritten as\nuλ(z,z′) =−/integraldisplay\nσ∈ˇbjh(ς)\n2i/parenleftbig\nuλ(ς,z′)∂ςRλ(z,ς)dς+∂ςuλ(ς,z′)Rλ(z,ς)dς/parenrightbig/vextendsingle/vextendsingle/vextendsingleς=σ\nς=σ−δAj,(6.21)\nwhereσ,ζare given by (4.1) while h(ς) = 1. Since the bundle C′obeys (6.10) while its section\nz/ma√sto→R′\nλ(z,z′) is smooth outside z=z′and conical points, we have\nR′\nλ(ς,z′)/vextendsingle/vextendsingle/vextendsingleζ=σ\nς=σ−δAj=∂ζR′\nλ(ς,z′)/vextendsingle/vextendsingle/vextendsingleζ=σ\nς=σ−δAj= 0\nand\nuλ(ς,z′)/vextendsingle/vextendsingle/vextendsingleς=σ\nς=σ−δAj=−Rλ(ς,z′)/vextendsingle/vextendsingle/vextendsingleς=σ\nς=σ−δAj, ∂ ςuλ(ς,z′)/vextendsingle/vextendsingle/vextendsingleς=σ\nς=σ−δAj=−∂ςRλ(ς,z′)/vextendsingle/vextendsingle/vextendsingleς=σ\nς=σ−δAj.\nSince Rλ(·,z′) is smooth outside z′=zon˙X, the equalities\nRλ(ς,z′)/vextendsingle/vextendsingle/vextendsingleς=σ\nς=σ−δAj=1\nω(σ)/bracketleftbig\nh−1(σ)∂σh(σ)/bracketrightbig\nRλ(σ,z′)δAj+1\nω(σ)∂σRλ(σ,z′)δAj,\nRλ(z,ς)/vextendsingle/vextendsingle/vextendsingleς=σ\nς=σ−δAj=1\nω(σ)∂σRλ(z,σ)δAj+1\nω(σ)/bracketleftbig\nh−1(σ)∂σh(σ)/bracketrightbig\nRλ(z,σ)δAj\nand\n∂ςRλ(ς,z′)/vextendsingle/vextendsingle/vextendsingleς=σ\nς=σ−δAj=1\nω(σ)/bracketleftbig\nh−1(σ)∂σh(σ)∂σ/bracketrightbig\nRλ(σ,z′)δAj+/bracketleftbig\n∂σ1\nω(σ)∂σ/bracketrightbig\nRλ(σ,z′)δAj\n=/bracketleftBig\nh−1(σ)∂σh(σ)∂σ=−ρ−2(σ)∆σ/bracketrightBig\n=−λρ−2(σ)\nω(σ)Rλ(σ,z′)δAj+/bracketleftbig\n∂σ1\nω(σ)∂σ/bracketrightbig\nRλ(σ,z′)δAj\n∂ςRλ(z,ς)/vextendsingle/vextendsingle/vextendsingleς=σ\nς=σ−δAj=/bracketleftbig\n∂σ1\nω(σ)∂σ/bracketrightbig\nRλ(z,σ)δAj+1\nω(σ)/bracketleftbig\nh−1(σ)∂σh(σ)∂σ/bracketrightbig\nRλ(z,σ)δAj\nhold uptoo(|δAj|)-terms uniformly in z′∈´Xseparated from ˇbj. Substituting the above formulas\ninto (6.21) and taking into account the identity\n(ab)/vextendsingle/vextendsingle+\n−=a(+)(b/vextendsingle/vextendsingle+\n−)+(a/vextendsingle/vextendsingle+\n−)b(−) =a(+)(b/vextendsingle/vextendsingle+\n−)+(a/vextendsingle/vextendsingle+\n−)b(+)−(a/vextendsingle/vextendsingle+\n−)(b/vextendsingle/vextendsingle+\n−),\nwe obtain\nuλ(z,z′) =O/parenleftbig\n|δAj|(C(uλ)+|δAj|)/parenrightbig\n+\n+δAj\n2i/integraldisplay\nσ∈−ˇbjh(σ)\nω(σ)/bracketleftBig\nλρ−2(σ)Rλ(z,σ)Rλ(σ,z′)dσ−∂ςRλ(z,σ)/bracketleftbig\nh−1(σ)∂σh(σ)/bracketrightbig\nRλ(σ,z′)dσ/bracketrightBig\n+δAj\n2i/integraldisplay\nσ∈−ˇbjh(σ)\nω(σ)/bracketleftBig\n∂σRλ(σ,z′)∂ςRλ(z,σ)dσ−/bracketleftbig\nω(σ)∂σ1\nω(σ)∂σ/bracketrightbig\nRλ(σ,z′))Rλ(z,σ)dσ/bracketrightBig\n,\n(6.22)\n41where\nC(uλ) := max\nς∈ˇbj(|u(ς,z′)|+|∂ςu(ς,z′)|).\nIn view of (6.19), we have C(uλ)≤Cuniformly in ( X′,ω′). Then, formula (6.22) implies\nuλ(z,z′) =O(|δAj|) uniformly in z,z′∈´Xseparated from ˇbjand conical points. The same\nestimate for the derivatives of uλ(z,z′) with respect to z,zfollows from the equation ∆ uλ(·,z′) =\nλuλ(·,z′) in´Xand local estimates (4.9), (4.10). As mentioned after (6.10), a sligh t shift of the\npathˇbj(by small change of cjin (6.6)) in the construction procedure for ( X′,ω′,C′) does not\naffect (X′,ω′,C′). Therefore, in the above estimates, one can omit the condition th atz,z′are\nseparated from ˇbj. As a corollary, one has C(uλ) =O(|δAj|). Now, formula (6.22) implies that\nRλ(z,z′|(X,ω)) is differentiable with respect to AjandAj, and equality (6.16) is true with\nν=Aj.\n6.2.3 Varying Ck\nIn this case, the surface X′is obtained by identification (6.15) of points of ´X=X\\Dkand\nD′\nk, whereDk={zk∈X||zk|< ǫ0}andD′\nk:={x′\nk∈C||x′\nk|<√2ǫ1}. The choice of\n(sufficiently small) ǫ0,ǫ1does not affect ( X′,ω′,C′) untilǫ1> ǫ0>|δCk|. In what follows, we\nchooseǫ0= 2|δCk|p0, wherep0∈(0,1].\nDenotez′\nk=x′2\nk/2. Letχ′\nkbe a cut-off function in D′\nkwhich is equal one for |z′\nk|≤ǫ1/3 and\nequal to zero for|z′\nk|≥2ǫ1/3. Put\n˜f′\nλ(·,z′) := (∆−λ)/parenleftbig\nχ′\nkR′\nλ(·,z′)) = [∆,χ′\nk]R′\nλ(·,z′). (6.23)\nLet the coordinates z,z′on´Xbe given by (4.1). Suppose that zandz′are separated from each\nother and the conical points P1,...,P 2g−2. In what follows, all estimates and asymptotics are\nassumed to be uniform in z,z′and (X′,ω′). By repeating the reasoning of 6.2.1, one can show\nthat the resolvent kernel R′\nλ(z,z′) =Rλ(z,z′) +uλ(z,z′) and its derivatives with respect to z\nare bounded uniformly in ( X′,ω′). In particular, ˜f′\nλ(z′\nk,z′) and its derivatives with respect to z′\nk\nare bounded uniformly in z′and (X′,ω′). In addition, we have |z′\nk|∈[ǫ0,2ǫ0] on the support of\n˜f′\nλ(·,z′). Now, applying Proposition 4.1 (with Xreplaced by D′\nk) yields the asymptotics\nR′\nλ(z′\nk,z′) =/summationdisplay\n(m,±)c′λ\nk,m,±(z′)f(k)\nk,m,±(r′\nk,ϕ′\nk)/parenleftBig/summationdisplay\nn≥0d(n,±iµm)(λ|z′\nk|2)n/parenrightBig\n+O(r′N\nk).(6.24)\nHere (r′\nk,ϕ′\nk) are polar coordinates near P′\nk(i.e.,z′\nk=r′\nkeiϕ′\nk) whilef(k)\nk,m,±are given by (4.28).\nAsymptotics (6.24) admits differentiation with respect to z′\nk. The coefficients in (6.24) are given\nby\nc′λ\nk,m,±(z′) = (˜f′\nλ(·,z′),χ′\nkfk,m,∓)L2(D′\nk;C)−(R′\nλ(·′\nk,z′),(∆−λ)[χ′\nkfk,m,∓])L2(D′\nk;C).(6.25)\nThe number Nin (6.24) can be taken arbitrarily large while the sum in the right hand sid e\ncontains only the terms decreasing slower than O(r′N\nk). Since the resolvent kernel R′\nλcorresponds\nto the Szeg¨ o extension of the Laplacian on ( X′,ω′,C′), the first four terms in (6.24) are given by\nR′\nλ(z′\nk,z′) =c′λ\nk,0,+f(k)\nk,0,++c′λ\nk,1,+f(k)\nk,1,++c′λ\nk,2,+f(k)\nk,2,++c′λ\nk,−1,−f(k)\nk,−1,−+O(|z′\nk|5/4).\nThe same formulas with omitted primes are valid for the resolvent ker nelRλ(·,z) of the Szeg¨ o\nLaplacian on ( X,ω,C).\n42Green formula (3.1) with f=uλ(·,z′),f′=Rλ(·,z) =Rλ(z,·), andU=´Xyields\nuλ(z,z′) =/integraldisplay\n∂´Xh(ς)\n2i/parenleftbig\nuλ(zk,z′)∂zkRλ(z,zk)dzk+∂zkuλ(zk,z′)Rλ(z,zk)dzk/parenrightbig\n. (6.26)\nHere∂´Xis the (counterclockwise oriented) circle |zk|= 2|δCk|p0. In view of (6.24) and (6.15),\nthe integrand in the right hand side of (6.26) is O(|zk|−1/2). Then formula (6.26) with p0= 1,\nthe equality ∆ uλ(·,z′) =λuλ(·,z′), and local estimates (4.9), (4.10) imply\n|∂l1\nz∂l2\nzuλ(z,z′)|=O(|δCk|ρ), (6.27)\nwherel1,l2= 0,1,...andρ= 1/2. In view of (6.27), (6.15), we have\n|˜f′\nλ(z′\nk,z′)−˜fλ(zk,z′)|=O(|δCk|ρ)\nwhere˜f′\nλis given by (6.23) and ˜f′\nλ(zk,z′) = [∆,χ′\nk(zk)]Rλ(zk,z′). Thus, (6.25) yields\n|c′λ\nk,m,±(z′)−cλ\nk,m,±(z′)|=O(|δCk|ρ), (6.28)\nfor the coefficients in asymptotics (6.24) for R′\nλ(z′\nk,z′) and Rλ(zk,z′). Since\nz′α\nk−zα\nk=zα\nk/parenleftbig\n(1−δCk/zk)α−1/parenrightbig\n=−αδCkzα−1\nk+O(|δCk|2|zk|α−2)/parenleftbig\n|δCk|/rk→0/parenrightbig\n,\nwe have\nf(k)\nk,m,±(r′\nk,ϕ′\nk)−f(k)\nk,m,±(rk,ϕk) =\n=/parenleftbig\n1∓(1/2+m)/parenrightbig\n(ℜδCk±iℑδCk)f(k)\nk,m∓2,±(rk,ϕk)+O(|δCk|2r−2±(2m−1)/4\nk ).(6.29)\nCombining (6.24), (6.28), (6.29), we obtain\nuλ(z′\nk,z′) =/summationdisplay\n(m,±)/parenleftbig\n1∓(1/2+m)/parenrightbig\ncλ\nk,m,±(z′)f(k)\nk,m∓2,±(zk)(ℜδCk±iℑδCk)+\n+O(|δCk|2r−9\n4\nk)+O(rN\nk)+O(|δCk|ρr−1\n4\nk),\n∂z′\nkuλ(z′\nk,z′) =/summationdisplay\nm(3/2+m)cλ\nk,m,−(z′)∂z′\nkf(k)\nk,m+2,±(zk)(ℜδCk−iℑδCk)+\n+O(|δCk|2|r−9\n4\nk|)+O(rN−1\nk)+O(|δCk|ρr−1\n4\nk).\nMoreover, expansion (6.24) with omitted primes implies ∂zkRλ(zk,z′) =O(r−1/4\nk). In view of the\nabove asymptotics and the equality\n1\n2i/integraldisplay\nrk=c/parenleftbig\nfk,m,sh∂fk,m′,s′dz+fk,m′,s′h∂fk,m,sdz/parenrightbig\n=δm,m′δs,−s′,\nformula (6.20) implies\nuλ(z,z′) =1\n2/parenleftBig\ncλ\nk,1,+(z′)cλ\nk,−1,−(z)−cλ\nk,0,+(z′)cλ\nk,−2,−(z)/parenrightBig\nδCk+\n+1\n2/parenleftBig\ncλ\nk,−2,−(z′)cλ\nk,0,+(z)−cλ\nk,−1,−(z′)cλ\nk,1,+(z)/parenrightBig\nδCk+O(|δCk|˜ρ),(6.30)\n43where ˜ρ:= min{2−3p0/2,1+ 3p0/2,5p0/2,ρ+p0/2}. The choice p0= 2/3 in (6.30) provides\nestimate (6.27) with ρ= 5/6. Due to this, one can choose ρ= 5/6 andp0= 1/2 in the above\nformulas; then ˜ ρ= 13/12 and the remainder in (6.30) is o(|δCk|). Thus, formula (6.30) provides\ndifferentiability of Rλ(z,z′|(X,ω)) with respect to ℜCk,ℑCkas well as the equality\n2i∂Rλ(z,z′)\n∂Ck=i/parenleftBig\ncλ\nk,1,+(z′)cλ\nk,−1,−(z)−cλ\nk,0,+(z′)cλ\nk,−2,−(z)/parenrightBig\n. (6.31)\nAt the same time, the substitution of asymptotics (6.24) into the rig ht hand side of (6.16) with\nν=Ckandν†={ζk(t) =ǫeit}t∈[0,4π),ǫ→0 yields the same value as in the right hand side of\n(6.31). Thus, formula (6.16) is proved for ν=Ck.\nSo, Proposition 6.3 is proved. Moreover, we have proved that the a symptotics\nRλ(z,z′|X′,ω′,C′)−Rλ(z,z′|X,ω,C) =∂Rλ(z,z′)\n∂νδν+∂Rλ(z,z′)\n∂νδν+o(|δν|) (ν=Ai,B0,Ck)\n(6.32)\n(with∂Rλ(z,z′)/∂νgiven by (6.16)) is uniform in ( X′,ω′) andz,z′∈´Xseparated by a fixed\ndistance (in the metric |ω|2) from the conical points of X.\n6.3 Variation of eigenvalues\nLetλ1≤λ2≤...be the sequence of eigenvalues of the Szeg¨ o extension ∆ = ∆ Sof the spinor\nLaplacian on ( X,ω,C) counted with multiplicities. Let unbe the corresponding normalized\neigensections. The eigenvalues and eigensections of the (Szeg¨ o) Laplacian ∆′on (X′,ω′,C′) are\ndenoted by λ′\nn,u′\nn, respectively. For a bounded domain U⊂C, denote\nΣU= ΣU(Aj,Bj,Ck) :=/summationdisplay\nλn∈Uλn,Σ′\nU:=/summationdisplay\nλ′n∈Uλ′\nn.\nProposition 6.4. If∂Udo not intersect the spectrum of ∆, thenΣUis differentiable with respect\ntoℜνandℑν(ν=Ai,Bi,Ck), and\n2i∂ΣU\n∂ν=/summationdisplay\nλn∈U/integraldisplay\nν†ω(ς)−1/parenleftBig\n∂ς/parenleftbig\nh(ς)uk(ς)/parenrightbig\n∂ςuk(ς)dς−|uk(ς)|2ρ2(ς)h(ς)λndς/parenrightBig\n.(6.33)\nIn the remaining part of this subsection, we prove Proposition 6.4.\nNon-concentration of eigensections near conical points. Forǫ>|δν|, denote by ˘X(ǫ,δν)\nthe domain in Xobtained from ´Xby removing ǫ-neighborhoods (in the metric |ω|2) of all conical\npoints. Then ˘X(ǫ,δν) can be also considered as a domain in X′. Denote ��X(ǫ,δν) =X\\˘X(ǫ,δν)\nand˜X′(ǫ,δν) =X′\\˘X(ǫ,δν). Let us prove the estimate\n/bardblu′\nn/bardbl2\nL2(˜X′(ǫ,δν);C′)≤cUǫ(ǫ>|δν|,λn∈U). (6.34)\nDenote by ˜X′\nk(ǫ) theǫ-neighborhood (in the metric |ω′|2) ofP′\nk. Letǫ0>0 be a sufficiently small\nnumber. Since (∆′)lu′\nn= (λ′\nn)lu′\nnand/bardblu′\nn/bardblL2(X′;C′)= 1, Proposition 4.1 (with Xreplaced by\n˜X′\nk(ǫ0)) provides the asymptotics\nu′\nn(x′\nk) =c′\nn,k,0,++c′\nn,k,1,+x′\nk+c′\nn,k,2,+x′2\nk+(−2/3π)c′\nn,k,−1,−x′\nk|x′\nk|+O(x′3\nk)\n44which is uniform in ( X′,ω′). Here, the coordinate x′\nk∈˜X′\nk(ǫ0) is given by (2.1) with Pk,ω\nreplaced by P′\nk,ω′, respectively. Note that the above asymptotics contains only the terms that\nallowu′\nnto be an element of Dom∆′= Dom∆′\nS(cf. with (3.9)). In particular, u′\nn(x′\nk) =O(1) on\n˜X′\nk(ǫ0) uniformly in ( X′,ω′). Sinceρ′(z′\nk) =h′(z′\nk) =|z′\nk|−1in coordinates x′\nk, we have\n/bardblu′\nn/bardbl2\nL2(˜X′\nk(ǫ))≤cUǫ3.\nIfν=Ck, then˜X′(ǫ,δν) is contained in the union of ˜X′\ns(ǫ) (s= 1,...,2g−2) and the last\ninequality implies (6.34).\nTo prove (6.34) for the case ν=Aj(orν=Bj), it remains to estimate /bardblu′\nn/bardbl2\nL2(X′\\´X;C′). Note\nthatX′\\´Xis contained in ǫ-neighborhood (in the metric |ω′|2) of the path ˇbj⊂´X(or ˇaj⊂´X).\nIn view of (4.9) and (4.10), the equations (∆′)lu′\nn= (λ′\nn)lu′\nnimply that|u′\nn(z′)|is bounded on\nX′\\´Xuniformly in ( X′,ω′). Here the coordinate z′is given by (4.1) with ω′instead ofω. Since\nthe area of X′\\´X(in the metric|ω′|2) isO(|δν|) =O(ǫ), we have/bardblu′\nn/bardbl2\nL2(X′\\´X;C′)≤cUǫ. Thus,\n(6.34) is proved for the case ν=Aj,Bj.\nNote that formula (6.34) remains valid with omitted primes.\nConnection between eigenvalues and resolvent kernel. Since\nR′\nλ(z,z′) :=/summationdisplay\nnu′\nn(z)u′n(z′)\nλ′\nn−λ, (6.35)\nwe have\n−1\n2πi/integraldisplay\n∂UR′\nλ(z,z′)Q(λ)dλ=/summationdisplay\nλ′n∈Uu′\nn(z)u′n(z′)Q(λ′\nn) (6.36)\nfor any function Qholomorphic in the bounded domain U⊂Cand continuous up to ∂U. The\nsame formulas with omitted primes are valid for Rλ.\nContinuity of ν/ma√sto→λn(ν).DenoteC=♯{λk∈U}andC′=♯{λ′\nk∈U}. In formula (6.36), put\nz′=zandQ= 1, then integrate over ˘X(ǫ,δν). In view of (6.34), we obtain\n−1\n2πi/integraldisplay\n˘X(ǫ,δν)/parenleftBig/integraldisplay\n∂UR′\nλ(z,z′)dλ/parenrightBig/vextendsingle/vextendsingle/vextendsingle\nz′=zh(z)dS(z) =/summationdisplay\nλ′n∈U/bardblu′\nn/bardbl2\nL2(˘X(ǫ,δν);C)=C′(1+O(ǫ)).\nThis formula remains valid with omitted primes. Therefore,\nC′−C=−1\n2πi/integraldisplay\nλ∈∂U/integraldisplay\n˘X(ǫ,δν)uλ(z,z)h(z)dS(z)dλ+O/parenleftbig\n(C′+C)ǫ/parenrightbig\n=O(|δν|)ǫ+O/parenleftbig\n(C′+C)ǫ/parenrightbig\n,\nwhere the index ǫinO(|δν|)ǫmeans that the estimate is not uniform in ǫ(but still uniform in\n(X′,ω′)). Since the left-hand side is integer, we have C′=Cfor sufficiently small |δν|. SinceU\nis arbitrary, the eigenvalues λn=λn(ν) are continuous functions of ν=Aj,Bj,Ck.\n45Differentiability of ΣUwith respect to ν.Without loss of generality, one can assume\nthatUcontains eigenvalues λn=···=λn+m−1of ∆. Letvn,...,v n+m−1be the corresponding\neigenfunctions orthonormalized with respect to scalar product in L2(˘X(ǫ,δν);C′) (for small ǫ,\nthe orthonormalization is possible due to (6.34)). Introduce the m×m-matricesM′andMwith\nentries\nM′\npq= (u′\nn+p,vn+q)L2(˘X(ǫ,δν);C),Mpq= (un+p,vn+q)L2(˘X(ǫ,δν);C).\nDenote ˆQ= diag{Q(λn),...,Q(λn+m−1)}andˆQ′= diag{Q(λ′\nn),...,Q(λ′\nn+m−1)}. Multiplying\nboth parts of (6.36) by vn+p(z′)vn+q(z) and integrating, we obtain\n/integraldisplay\n˘X(ǫ,δν)/integraldisplay\n˘X(ǫ,δν)/parenleftBig−1\n2πi/integraldisplay\n∂UR′\nλ(z,z′)Q(λ)dλ/parenrightBig\nvn+p(z′)vn+q(z)h(z)dS(z)h(z′)dS(z′) = (M′∗ˆQ′M′)pq.\nThis equality remains valid with omitted primes. Then asymptotics (6.32 ) implies\n(M′∗ˆQ′M′)pq−(M∗ˆQM)pq=\n=−1\n2πi/integraldisplay\nλ∈∂U/integraldisplay\n˘X(ǫ,δν)/integraldisplay\n˘X(ǫ,δν)uλ(z,z′)Q(λ)vn+p(z′)vn+q(z)h(z)dS(z)h(z′)dS(z′)dλ=\n=D(ν|Q,ǫ)pqδν+D(ν|Q,ǫ)pqδν+o(|δν|)ǫ,(6.37)\nwhereD(·|Q,ǫ) is them×m-matrix with entries\nD(·|Q,ǫ)pq:=−1\n2πi/integraldisplay\nλ∈∂U/integraldisplay\n˘X(ǫ,δν)/integraldisplay\n˘X(ǫ,δν)∂Rλ(z,z′)\n∂·Q(λ)vn+q(z)vn+p(z′)h(z)dS(z)h(z′)dS(z′)dλ.\nLetM′=U′A′be a polar decomposition of M′, whereU′andA′are unitary and hermitian\nmatrices, respectively. Let also M=UAbe a polar decomposition of M. Then (6.37) with\nQ= 1 implies\nA′2−A2=M′∗M′−M∗M=D(ν|1,ǫ)δν+D(ν|1,ǫ)δν+o(|δν|)ǫ.\nThus,\nA′−1−A−1=−1\n2/parenleftbig\nD(ν|1,ǫ)δν+D(ν|1,ǫ)δν/parenrightbig\n+o(|δν|)ǫ=O(|δν|)ǫ.\nLetQ(λ) =λ−λn, thenˆQ= 0 and ˆQ′= Σ′\nU−ΣU. In addition, (6.37) yields\nU′∗ˆQ′U′=A′−1M′∗ˆQ′M′A′−1=A′−1/parenleftBig\nD(ν|Q,ǫ)δν+D(ν|Q,ǫ)δν+o(|δν|)ǫ/parenrightBig\nA′−1=\n=A−1/parenleftBig\nD(ν|Q,ǫ)δν+D(ν|Q,ǫ)δν/parenrightBig\nA−1+o(δν)ǫ.\nTaking the trace from both sides, we arrive at\nΣ′\nU−ΣU= Tr/bracketleftbig\nA−1D(ν|Q,ǫ)A−1/bracketrightbig\nδν+Tr/bracketleftbig\nA−1D(ν|Q,ǫ)A−1/bracketrightbig\nδν+o(δCk)ǫ.\nAs a corollary, Σ Uis differentiable with respect to ℜν,ℑν, and\n∂ΣU/∂ν= Tr/bracketleftbig\nA−1D(ν|Q,ǫ)A−1/bracketrightbig\n. (6.38)\nIn particular, the right hand side is independent of ǫ.\n46Calculation of ∂ΣU/∂ν.In view of (6.34), the eigensections u′\nn,...,u′\nn+m−1can be chosen in\nsuch a way thatM−I=O(ǫ1/2\n0). Then passing to the limit as ǫ→0 in (6.38) yields\n∂ΣU\n∂ν=/summationdisplay\npD(ν|Q,0)pp=\n=−/summationdisplay\np/integraldisplay\nX/integraldisplay\nXRes\nλ=λn/bracketleftBig\n(λ−λn)∂Rλ(z,z′)\n∂ν/bracketrightBig\nun+p(z)un+p(z′)h(z)dS(z)h(z′)dS(z′) =\n=−/integraldisplay\nXRes\nλ=λn/bracketleftBig\n(λ−λn)∂Rλ(z,z′)\n∂ν/bracketrightBig/vextendsingle/vextendsingle/vextendsingle\nz′=zh(z)dS(z).(6.39)\nIn view of (6.35), the equality\n(6.40)\n/integraldisplay\nXRes\nλ=λn/parenleftBig\n(λn−λ)a(λ)PςRλ(ς,z)·QςRλ(z,ς)/parenrightBig\nh(z)dS(z) =−/summationdisplay\nλk=λnPςuk(ς)·Qςuk(ς)a(λk)\nis valid for any differential operators P,Qand functions λ/ma√sto→a(λ) holomorphic near λn. Com-\nbining (6.39), (6.16), and (6.40), we arrive at (6.33). Proposition 6.4 is proved.\n6.4 Variation of zeta function\nIn this section, we prove Theorem 3.7.\n6.4.1 Variation of ζ(2|∆−µ).\nLetz,z′be coordinates (4.1). Formally differentiating the series for ζ(s|∆) =ζ(s|∆s(X,ω)) with\nrespect toν=Aj,Bj,Ckand taking into account formula (6.33) and the equalities\nρ2(z)∂2\nλ/parenleftbig\nλh(z)Rλ(z,z′)/parenrightbig\n=/summationdisplay\nn2λnun(z)un(z′)ρ2(z)h(z)\n(λn−λ)3\n∂2\nλ∂z∂z′/parenleftbig\nh(z)Rλ(z,z′)/parenrightbig\n=/summationdisplay\nn2∂z(h(z)un(z))∂z′un(z′)\n(λn−λ)3,\nwe obtain\n2i∂νζ(2|∆−µ) =−2/summationdisplay\nn(λn−µ)−3/parenleftBig\n2i∂λn\n∂ν/parenrightBig\n=\n=/integraldisplay\nν†ω(z)−1/bracketleftBig/bracketleftbig\nρ2(z)∂2\nµ/parenleftbig\nµh(z)Rµ(z,z′)/parenrightbig/bracketrightbig/vextendsingle/vextendsingle/vextendsingle\nz′=zdz−/bracketleftbig\n∂2\nµ∂z∂z′/parenleftbig\nh(z)Rµ(z,z′)/parenrightbig/bracketrightbig/vextendsingle/vextendsingle/vextendsingle\nz′=zdz/bracketrightBig\n.(6.41)\nTo justify the above calculations, we should show that the derivativ es of the resolvent kernel in\nthe right hand side of (6.41) are well defined on the diagonal z′=z.\n47Near-diagonal asymptotics of resolvent kernel. In view of (4.2), the resolvent kernel Rλ\nof ∆ = ∆ Sadmits representation\nRλ(z,z′) =Rq,λ(·,z′)+[(∆ S−λ)−1(∆−λ)Rq,λ(·,z′)](z,z′) (6.42)\noutside conical points, where\nR0,λ(z,z′) =−2\nπlog|z′−z|·χz′(z),\nR1,λ(z,z′) =−2\nπlog|z′−z|·(1−λ|z′−z|2)χz′(z),\n...\nR∞,λ(z,z′) =2\nπK0(−2i|z′−z|√\nλ)χz′(z).\nHere,χz′is a cut-off function equal to one near z′and to zero, the support of χz′is sufficiently\nsmall,K0is the Macdonald function. In view of (4.11), the second term in the r ight hand side\nof (6.42) is (1+2 q)-differentiable with respect to z,z′at the diagonal z=z′. Then the functions\nΦλ(z) : =λ/bracketleftbig\nRλ(z,z′)−R0,λ(z,z′)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle\nz′=z,\nΨλ(z) : =∂z∂z′/bracketleftbig\nRλ(z,z′)−R1,λ(z,z′)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle\nz′=z.(6.43)\narewell-definedandholomorphicwithrespectto λoutsidethespectrumof∆. Since ∂2\nλ/parenleftbig\nλR0,λ(z,z′)) =\n0 and∂2\nλ∂z∂z′R1,λ(z,z′) = 0, andh(z) =ρ(z) = 1, one can rewrite (6.41) as\n2i∂νζ(2|∆−µ) =∂2\nµ/integraldisplay\nz∈ν†/parenleftbig\nΦµ(z)dz−Ψµ(z)dz/parenrightbig\n. (6.44)\nAsymptotics of ΦλandΨλasℜλ→−∞.In view of (4.2), we have\n(∆−λ)R∞,λ(·,z′) = [∆,χ′\nz]2\nπK0(−2i|z′−z|√\nλ).\nSince the support of z/ma√sto→χ′\nz(z) is separated from the diagonal z=z′, andK0(κ) decays exponen-\ntially asℜκ→+∞, the right hand side and all its derivatives decay exponentially as ℜλ→−∞.\nHence, in view of (4.11), the function\nz,z′/ma√sto→[(∆S−λ)−1(∆−λ)R∞,λ(·,z′)](z,z′) =Rλ(z,z′)−R∞,λ(z,z′)\nand all its derivatives decay exponentially as ℜλ→−∞uniformly with respect to z,z′separated\nfrom comical points. As a corollary, we have\nΨλ(z) =∂z∂z′/bracketleftbig\nRλ(z,z′)−R∞,λ(z,z′)+R∞,λ(z,z′)−R1,λ(z,z′)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle\nz′=z=O(eǫℜλ)+\n+∂z∂z′/bracketleftBig2\nπ/parenleftBig\nK0(−2i|z′−z|√\nλ)+(1−λ|z′−z|2)log|z′−z|/bracketrightBig/vextendsingle/vextendsingle/vextendsingle\nz′=z=O(eǫℜλ)+0(6.45)\nuniformly with respect to zseparated from conical points, where ǫ>0. Similarly, we have\nΦλ(z) :=λ/bracketleftbig\nRλ(z,z′)−R∞,λ(z,z′)+R∞,λ(z,z′)−R0,λ(z,z′)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle\nz′=z=O(eǫℜλ)+\n+2λ\nπ/parenleftBig\nK0(−2i|z′−z|√\nλ)+log|z′−z|/parenrightBig/vextendsingle/vextendsingle/vextendsingle\nz′=z=λ\nπlogλ+λ/parenleftBig\ni−2γ\nπ/parenrightBig\n+O(eǫℜλ).(6.46)\n48Calculation of Φλ(0)andΨλ(0).Sinceλ= 0 is not an eigenvalue of ∆ = ∆ S, formulas (6.42)\nand (6.43) yield\nΦ0= 0. (6.47)\nNext, since ∆ = ∆ S, formula (3.6) is valid, and\nΨ0(z) =∂z∂z′/parenleftBig\nG(z,z′)+2\nπlog|z′−z|/parenrightBig/vextendsingle/vextendsingle/vextendsingle\nz′=z=−1\nπ∂z/parenleftBig\nS(z,z′)−1\nz′−z/parenrightBig/vextendsingle/vextendsingle/vextendsingle\nz′=z.(6.48)\nThe asymptotics\nS(z,z′)−1\nz′−z=g/summationdisplay\ni=1∂ξilogθ/bracketleftbigp\nq/bracketrightbig\n(ξ= 0)υi(z)+/bracketleftBigg\nS0(z)\n6+g/summationdisplay\ni=1∂ξilogθ/bracketleftbigp\nq/bracketrightbig\n(ξ= 0)∂zυi(z)+\n+g/summationdisplay\ni,j=1∂2\nξiξjθ/bracketleftbigp\nq/bracketrightbig\n(ξ= 0)\nθ[p\nq](0)vi(z)vj(z)/bracketrightBigg\nz′−z\n2+O/parenleftbig\n(z′−z)2/parenrightbig\n, z′→z.\n(6.49)\nis valid (see p.29, [4]), where υ1,...,υ gis the basis in the space of Abelian differentials on X\ndual to the canonical homology basis. The Bergman projective con nectionS0transforms as\nS0(x′) =S0(x)/parenleftBig∂x\n∂x′/parenrightBig2\n+{x,x′}\nunder a holomorphic change of variables x/ma√sto→x′. Due to the chain rule\n{f,x′}={f,x}/parenleftBig∂x\n∂x′/parenrightBig2\n+{x,x′}\nfor Schwarzian derivative, the formula\nW(x) =S0(x)−/braceleftBig/integraldisplayx0\nxω,x/bracerightBig\n(6.50)\ndefines a quadratic differential such that W(z) =S0(z). In view of (6.48), (6.49), (6.50), and\n(6.63), we have\nΨ0=W\n12πω+1\n2π/bracketleftBigg/summationdisplay\ni=1∂ξilogθ/bracketleftbigp\nq/bracketrightbig\n(ξ)∂/parenleftBigυi\nω/parenrightBig\n+g/summationdisplay\ni,j=1∂2\nξiξjθ/bracketleftbigp\nq/bracketrightbig\n(ξ)\nθ[p\nq](ξ)vivj\nω/bracketrightBig/vextendsingle/vextendsingle/vextendsingle\nξ=0.(6.51)\n6.4.2 Variation of ζ(s|∆)\nConnection between ζ(s|∆S)andζ(2|∆S−µ).From (4.66), the formula K(t|A−µ) =\neµtK(t|A), and the inversion formula for the Laplace transform it follows tha t\nK(t|A) :=Γ(s)\n2πiµ0+i∞/integraldisplay\nµ0−i∞e−µtt1−sζ(s|A−µ)dµ. (6.52)\n49This formula is valid for A= ∆Fifℜs >1 andµ0∈(0,λF\n1). In view of Proposition 3.6, it is\nalso valid for A= ∆Sifℜs >1 andµ0∈(0,λS\n1). Puts= 2 and substitute (6.52) into (4.66).\nAs a result, we obtain\n(s−1)ζ(s|A−λ) =s−1\n2πiΓ(s)µ0+i∞/integraldisplay\nµ0−i∞/parenleftBig+∞/integraldisplay\n0e(λ−µ)tts−2dt/parenrightBig\nζ(2|A−µ)dµ=\n=1\n2πi/integraldisplay\nγ(µ−λ)1−sζ(2|A−µ)dµ,(6.53)\nwhereA= ∆F,∆S,ℜs>1 andγis the union of the paths (5.20). For ℜs≤1, formula (6.53) is\nalso valid but its left hand side and right hand side are understood as a nalytic continuations of\nthem from the half-plane ℜs>1.\nLet us make the analytic continuation of (6.53) more explicit by exclud ing the singular terms.\nSuppose that A= ∆F. Rewrite asymptotics (4.65) as\nK(t|A) =e−µ0t/bracketleftBig\nArea(X,|ω|2)1+µ0t\nπt+3g−1\n8/bracketrightBig\n+˜K(t|A) (6.54)\nwhere˜K(t|A) =O(t) ast→+0 and ˜K(t|A) =O(e−µ0t/2) ast→−∞. The substitution of\n(6.54) into (4.66) yields\nζ(s|A−λ) =ζ0(s,λ)+˜ζ(s|A−λ), (6.55)\nHere\nζ0(s,λ) :=Area(X,|ω|2)\nπ(µ0−λ)s−1(s−1)/parenleftBig\n1+µ0(s−1)\nµ0−λ/parenrightBig\n+3g−1\n8(µ0−λ)s=\n=1\n2πi(s−1)/integraldisplay\nγ(µ−λ)1−sζ0(2,λ)dµ.(6.56)\nThe remainder\n˜ζ(s|A−λ) =1\nΓ(s)/integraldisplay+∞\n0ts−1eλt˜K(t|A)dt=1\n2πi(s−1)/integraldisplay\nγ(µ−λ)1−s˜ζ(2|A−µ)dµ(6.57)\nis well-defined and holomorphic in s,λforℜs >−1,ℜλ < µ 0. In view of Proposition 3.6,\nformulas (6.55), (6.57) are also valid for A= ∆S. Note that one can differentiate the right hand\nside of (6.55) with respect to ν=Ai,Bi,Ckforℜs>−1. From this fact it follows that one can\ndifferentiate both sides of (6.53) with respect to νforℜs>−1.\nVariation of det∆S.Letℜs>1. Differentiating (6.53) with respect to ν, taking into account\n(6.44), and changing the order of integrations, we obtain\n2i(s−1)∂νζ(s|∆S) =/integraldisplay\nz∈ν†/parenleftbig\nJΦ\ns(z)dz−JΨ\ns(z)dz/parenrightbig\n,\nwhere\nJΦ\ns(z) :=1\n2πi/integraldisplay\nγµ1−s∂2\nµΦµ(z)dµ,JΨ\ns(z) :=1\n2πi/integraldisplay\nγµ1−s∂2\nµΨµ(z)dµ.\n50In view of (6.45), the function JΨ\ns(z) is analytic in s���Cand the representation\nJΨ\ns=π−1e−iπssin(πs)JΨ,−,ǫ\ns+JΨ,◦,ǫ\ns,\nholds, where\nJΨ,−,ǫ\ns(z) :=/integraldisplay−ǫ\n−∞µ1−s∂2\nµΨµ(z)dµ,JΨ,◦,ǫ\ns(z) :=1\n2πi/integraldisplay\nγ◦µ1−s∂2\nµΨµ(z)dµ\nare analytic in s∈C, and\nJΨ,−,0\n0=/parenleftbig\nµ∂µΨµ−Ψµ/parenrightbig/vextendsingle/vextendsingle/vextendsingleµ=0\nµ=−∞=−Ψ0JΨ,◦,0\n0=∂sJΨ,◦,0\ns=0= 0\ndue to (6.45) and (6.51). Hence,\nJΨ\n0= 0,−∂sJΨ\ns=0= Ψ0. (6.58)\nWe have\nJΦ\ns=π−1e−iπssin(πs)JΦ,−,ǫ\ns+JΦ,◦,ǫ\ns,\nwhere\nJΦ,−,ǫ\ns(z) :=/integraldisplay−ǫ\n−∞µ1−s∂2\nµΦµ(z)dµ,JΦ,◦,ǫ\ns(z) :=1\n2πi/integraldisplay\nγ◦µ1−s∂2\nµΦµ(z)dµ.\nSince the integration contour γ◦is compact, the function JΦ,◦,ǫ\nsis holomorphic in s∈C. Also,\nsince∂2\nµΦµis holomorphic in µoutside the spectrum of ∆ S(hence, near zero), we have JΦ,◦,ǫ\n0(z) =\n0. Similarly, since ∂2\nµΦµ,∂µΦµ,Φµare holomorphic in µnear zero, and Φ 0= 0 due to (6.47),\nintegration by parts yields\n−∂sJΦ,◦,ǫ\ns=0=1\n2πi/integraldisplay\nγ◦µlogµ∂2\nµΦµdµ=\n=1\n2πi/bracketleftbig\nµlogµ∂µΦµ−Φµ−logµΦµ/bracketrightbig/vextendsingle/vextendsingle/vextendsingleǫe+i(π−0)\nǫe−i(π−0)+/integraldisplay\nγ◦µ−1Φµdµ/bracketrightBig\n=−ǫ∂µΦµ=−ǫ−Φ−ǫ.\nLet us represent JΦ,−,ǫ\nsas\nJΦ,−,ǫ\ns(z) =1\nπ/integraldisplay−ǫ\n−∞µ−sdµ+J˜Φ,−,ǫ\ns(z), (6.59)\nwhere\nJ˜Φ,−,ǫ\ns(z) :=/integraldisplay−ǫ\n−∞µ1−s∂2\nµ˜Φµ(z)dµ,˜Φµ(ζ) := Φ µ(ζ)−π−1µ/parenleftbig\nlogµ+πi−2γ/parenrightbig\n.\nThe integral in the right hand side of (6.59) is equal to ( −ǫ)1−s/π(1−s) forℜs>1, and admits\nthe analytic continuation to the whole complex plane except the pole a ts= 1. In view of (6.46),\nthe integral J˜Φ,−,ǫ\ns(z) is analytic in s∈C. Integrating by parts and taking into account (6.46),\nwe obtain\nJ˜Φ,−,ǫ\n0(z) :=/parenleftBig\nµ∂µ˜Φµ−˜Φµ/parenrightBig/vextendsingle/vextendsingle/vextendsingleµ=−ǫ\nµ=−∞=−ǫ∂µ˜Φµ=−ǫ−˜Φ−ǫ.\n51Due to the above formulas, we have\nJΦ\n0= JΦ,◦,ǫ\n0= 0,\n∂sJΦ\ns=0= JΦ,−,ǫ\ns=0+ǫ∂µΦµ=−ǫ+Φ−ǫ=ǫ∂µΦµ=−ǫ+Φ−ǫ−ǫ\nπ−ǫ∂µ˜Φµ=−ǫ−˜Φ−ǫ= 0.(6.60)\nIn view of (6.58) and (6.60), formula (6.53) with s= 0 yields\n∂νζ(0|∆S) = 0.\nNow, differentiate (6.53) with respect to s, puts= 0, and take into account (6.58) and (6.60).\nAs a result, we obtain\n2i∂νlog det∆ S=−∂ν∂sζ(s|∆S)|s=0=/integraldisplay\nζ∈ν†/parenleftbig\n∂sJΦ\ns=0(ζ)dζ−∂sJΨ\ns=0(ζ)dζ/parenrightbig\n=/integraldisplay\nζ∈ν†Ψ0(ζ)dζ.(6.61)\nNote that one can interchange the analytic continuation from half- planeℜs >1 to the neigh-\nbourhood of s= 0 and the differentiation with respect to ν=Ai,Bi,Ckin the above calculations\ndue to explicit (admitting differentiation with respect to ν) formulas (6.55), (6.56), (6.57) for the\nanalytic continuation of the left hand side and right hand side of (6.53 ) withA= ∆S.\nCompleting the proof of (3.18). Substituting (6.51) into (6.61) and taking into account\nthat the integral of ∂(υi/ω) =d(υi/ω) over any closed cycle equals zero, one gets\n∂νlog det∆ S=1\n2πi/integraldisplay\nν†/bracketleftBigW\n12ω+g/summationdisplay\ni,j=1∂2\nξiξjθ/bracketleftbigp\nq/bracketrightbig\n(ξ)\nθ[p\nq](ξ)/vextendsingle/vextendsingle/vextendsingle\nξ=0vivj\n2ω/bracketrightBig\n. (6.62)\nAccording to the definition of the Bergman tau function τonHg(1,...,1) (see (3.1), [13]), we\nhave∂logτ\n∂ν=−1\n12πi/contintegraldisplay\nν†W\nω,∂τ\n∂ν= 0. (6.63)\nRecall that the theta-function satisfies the heat equation\n∂θ/bracketleftbigp\nq/bracketrightbig\n(ξ|B)\n∂Bij=1\n4πi∂2θ/bracketleftbigp\nq/bracketrightbig\n(ξ|B)\n∂ξi∂ξj. (6.64)\nIn addition, the dependence of Bon coordinates (6.3) is described by the following formula (see\n(2,28) and the remark after (2.81), [13])\n∂Bij\n∂ν=/contintegraldisplay\nν†υiυj\nω,∂Bij\n∂ν= 0. (6.65)\nIn view of (6.63), (6.64), and (6.65), equality (6.62) takes the form\n∂ν/bracketleftBig\nlog det∆ S+1\n2logτ]/bracketrightBig\n=1\n4πiθ[p\nq](0|B)g/summationdisplay\ni,j=1∂2\nξiξjθ/bracketleftbigp\nq/bracketrightbig\n(0|B)∂Bij\n∂ν=\n=1\nθ[p\nq](0|B)∂θ/bracketleftbigp\nq/bracketrightbig\n(0|B)\n∂ν=∂logθ/bracketleftbigp\nq/bracketrightbig\n(0|B)\n∂ν=1\n2∂νlog|θ/bracketleftbigp\nq/bracketrightbig\n(0|B)|2.\nThe last formula can be rewritten as\n∂ν/parenleftbig\nleft hand side of (3.18) −right hand side of (3.18)/parenrightbig\n= 0.\nSince both sides of (3.18) are real-valued, one can replace ∂νwith∂νand, hence, with din the\nformula above. By this, Theorem 3.7 is proved.\n52References\n[1] Michael F. Atiyah. Riemann surfaces and spin structures. An-\nnales scientifiques de l’ ´Ecole Normale Sup´ erieure 4, 4(1) (1971): 47–62.\nhttp://www.numdam.org/articles/10.24033/asens.1205/\n[2] J. B. Bost and P. Nelson. Spin-(1/2) Bosonization on Compact Su rfaces.Phys. Rev. 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DOI:\nhttps://doi.org/10.1016/0550-3213(87)90578-5\n54"    },    {        "title": "2402.12592v1.Global_existence_for_non_homogeneous_incompressible_inviscid_fluids_in_presence_of_Ekman_pumping.pdf",        "content": "arXiv:2402.12592v1  [math.AP]  19 Feb 2024Global existence for non-homogeneous incompressible\ninviscid fluids in presence of Ekman pumping\nMarco Bravin1and Francesco Fanelli2a,b,c\n1Universidad de Cantabria, E.T.S. de Ingenieros Industrial es y de Telecomunicacíon\nDepartmento de Matemática Aplicada y Ciencias de la Computa cíon\nAvd. Los Castros 44, 39005 Santander, SPAIN\n2aBCAM – Basque Center for Applied Mathematics\nAlameda de Mazarredo 14, E-48009 Bilbao, Basque Country, SP AIN\n2bIkerbasque – Basque Foundation for Science\nPlaza Euskadi 5, E-48009 Bilbao, Basque Country, SPAIN\n2cUniversité Claude Bernard Lyon 1 ,Institut Camille Jordan – UMR 5208\n43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, FR ANCE\nEmail addresses:1marco.bravin@unican.es ,2ffanelli@bcamath.org\nFebruary 21, 2024\nAbstract\nIn this paper, we study the global solvability of the density -dependent incompressible Euler equations, supplemented\nwith a damping term of the form Dγ\nα(ρ,u) =αργu, whereα >0andγ∈ {0,1}. To some extent, this system can\nbe seen as a simplified model describing the mean dynamics in t he ocean; from this perspective, the damping term\ncan be interpreted as a term encoding the effects of the celebr ated Ekman pumping in the system.\nOn the one hand, in the general case of space dimension d/greaterorequalslant2, we establish global well-posedness in the Besov\nspaces framework, under a non-linear smallness condition i nvolving the size of the initial velocity field u0, of the\ninitial non-homogeneity ρ0−1and of the damping coefficient α. On the other hand, in the specific situation of\nplanar motions and damping term with γ= 1, we exhibit a second smallness condition implying global ex istence,\nwhich in particular yields global well-posedness for arbit rarily large initial velocity fields, provided the initial d ensity\nvariations ρ0−1are small enough. The formulated smallness conditions rely only on the endpoint Besov norm\nB1\n∞,1of the initial datum, whereas, as a byproduct of our analysis , we derive exponential decay of the velocity\nfield and of the pressure gradient in the high regularity norm sBs\np,r.\n2020 Mathematics Subject Classification: 35Q35 (primary); 35A01, 35B65, 76B03 (secondary).\nKeywords: incompressible Euler equations; density variations; damp ing term; global in time well-posedness.\n1 Introduction\nOcean dynamics is an intricate phenomenon, determined by se veral complementary, and some-\ntimes conflicting, factors [14, 23]. Because of the large scal es involved in the process, the two\npredominant ones are the action of the Coriolis force, which tends to drive the flow to a purely\nplanar (2-D, horizontal) configuration, and the action of gravity, wh ich instead tends to stratify\nthe density on the vertical ( i.e., orthogonal to the previously mentioned effect) direction. Thus,\nif we can neglect temperature and salinity fluctuations in a fi rst approximation, we cannot do so\nfor the variations of the density.\n1Another important feature of the motion of water currents in oceans is the so-called Ekman\npumping effect. This is a mechanism of dissipation of kinetic energy, which originates from a\nboundary layer (the celebrated Ekman layer) close to the oce anic bottom, but involves in fact a\nglobal dynamics, affecting also the mean flow in the bulk. On a r igorous ground, the mathematical\nanalysis of Ekman boundary layers was initiated by the pione ering works [20] (for the well-prepared\ndata case) and [22] (for ill-prepared initial data). In thos e works, it is proven that, in the fast\nrotation regime, an Ekman pumping term appears in the target momentum equation. This Ekman\npumping term takes the form of a linear damping term αu, whereα >0. We refer to the book\n[9] for a complete overview of the subject of (homogeneous) fl uids in fast rotation, as well as for\na study of Ekman layers. We also refer to [3] for a description of Ekman boundary layers for\ncompressible flows in presence of stratification, and to [7] f or a new approach to the derivation of\nthe Ekman pumping effect (without a description of the Ekman l ayers, though), implemented in\nthe context of non-homogeneous incompressible fluids.\nIn this work, we are interested in the (global) well-posedne ss issue for a model which may\nbe seen as a rough approximation of ocean dynamics in the regi me of fast rotation. Notice that\nthe fluid can be assumed to be incompressible, at least at a fai rly good level of approximation.\nIn addition, at large scales, the Reynolds number of the flow i s typically very high (see e.g.the\ndiscussion in Section 4.4 of [14]), so viscous effects may be d iscarded.\nMotivated by the previous considerations, and restricting our attention to the (mean) motion\nin the bulk, it is thus natural to consider the dynamics of an i ncompressible inviscid fluid presenting\ndensity variations and energy dissipation through a dampin g term.\n1.1 The equations under study\nAt the mathematical level, the system we are interested in is described by the so-called non-\nhomogeneous incompressible Euler equations . If we denote by ρ=ρ(t,x)/greaterorequalslant0the density of the\nfluid, byu=u(t,x)∈Rd, withd/greaterorequalslant2, its velocity field and by Π = Π(t,x)∈Rits pressure field,\nthose equations can be written in the following way:\n(1)\n\n∂tρ+u·∇ρ= 0\nρ∂tu+ρu·∇u+∇Π +αργu= 0\ndivu= 0.\nHere,α>0is a positive parameter and γ∈ {0,1}. The term\n(2) Dγ\nα(ρ,u) :=αργu\nis thus a damping term, whose role is, precisely, to encode th e Ekman pumping effect of ocean\ndynamics. On the other hand, for the sake of simplicity, in th e system above we discard remainders\ncoming from to the (fast) Earth rotation.\nNotice that a somewhat similar version of the previous syste m, but with a static density profile\nρ=ρ(x), was derived in [7] from a singular limit problem for non-hom ogeneous incompressible\nviscous fluids in thin domains, in the fast rotation regime. A lthough the target system coming\nfrom fast rotating fluids should be only two-dimensional, fo r the sake of generality in this work\nwe set system (1) in\nR+×Rd, withd/greaterorequalslant2.\nWe are interested in the initial value problem related to equ ations (1), thus we supplement them\nwith an initial datum\n(3)/parenleftbig\nρ,u/parenrightbig\n|t=0=/parenleftbig\nρ0,u0/parenrightbig\nsuch that divu0= 0.\nThroughout this work, we assume absence of vacuum . This means that there exist two positive\nconstants 0<ρ∗/lessorequalslantρ∗such that\n(4) 0<ρ∗/lessorequalslantρ0/lessorequalslantρ∗.\n21.2 Previous results\nWhenα= 0, we observe that the previous system reduces to the non-homo geneous incompressible\nEuler equations, namely\n(ddE)\n\n∂tρ+u·∇ρ= 0\nρ∂tu+ρu·∇u+∇Π = 0\ndivu= 0.\nThis system can be seen as a generalisation of the classical E uler system (obtained when ρ≡1)\nto the case of fluids presenting density variations.\nOn the mathematical side, we notice that, despite the presen ce of the (non-local) pressure term,\nequations (ddE) retain the structure of a transport system, at least under the assumption (4) of\nabsence of vacuum. Thus, their well-posedness can be addres sed in any reasonable functional\nspaces continuously embedded in the set of globally Lipschi tz functions W1,∞. For instance,\npapers [15, 16] established local well-posedness in the cla ss of Besov spaces Bs\np,r=Bs\np,r(Rd)\nverifyingBs\np,r֒→W1,∞, up to the endpoint case p= +∞ands=r= 1. Recall that the previous\nembedding holds true if and only if the triplet of indices (s,p,r)∈R×[1,+∞]×[1,+∞]satisfies\nthe condition\n(5) s >1 +d\npor s= 1 +d\npandr= 1.\nPaper [17] proved instead well-posedness under tangential (also dubbed “striated” and “conormal”)\nregularity assumptions à la Chemin . We refer to the introduction of those papers for a more\ncomplete overview of the literature concerning well-posed ness of equations (ddE).\nIt is worth to point out that all those results hold true only locally in time , even in the two-\ndimensional case. This is a purely heterogeneity effect, ina smuch as variations of the density are\nresponsible for vorticity production, thus causing the bre aking of all the techniques available from\nthe classical case ρ≡1to prove global well-posedness. As a matter of fact, global e xistence of\nregular solutions for equations (ddE) still remains as an op en problem in general (however, see\nthe recent preprint [10] for a study about global asymptotic stability around the Couette flow).\nThe study of the case α >0, namely of system (1), seems instead new in the literature,\neven though some previous results are in fact available for s ystems arising from fluid dynamics in\npresence of a damping term.\nFor instance, the compressible counterpart of equations (1 ), where one drops the condition\ndivu= 0and lets the pressure term Π = Π(ρ)depend on the density ρ, was first investigated\nin [24]. There, the authors proved that, for small initial da ta, the presence of the damping term\nremoves the appearing of finite time singularities and enabl es to prove global existence results.\nFine results (global well-posedness and long time dynamics ) about the damped compressible\nEuler system have recently been obtained in the series of pap ers [11, 12, 13], as particular cases\nof a general theory dealing with partially dissipative hype rbolic systems in a critical functional\nframework, a theory originating from the work [5] of Beauchar d and Zuazua.\nIt is worth mentioning that, in all the works quoted above, th e damping term was of the same\nform asDγ\nαin (2), with γ= 1. Roughly speaking, in this case and for small density pertur bations\naround a constant state, one can then divide the momentum equ ation byρand obtain a damped\nequation for u, for which it is then possible to derive an exponential decay ine.g.L2. Observe\nthat, despite this simple heuristics, the fact that this dec ay implies global well-posedness is not\nstraightforward a priori , and is indeed a remarkable property due to the structure of t he hyperbolic\nsystem: the dissipation does not act on all the components of the solution, but only on some of\nthem. We refer to [5] for a more in-depth discussion about thi s point.\n3From this point of view, our study about the incompressible c ase (1) will suffer of the same\ndifficulty. Nonetheless, the introduction of non-local effec ts due to the pressure function deter-\nmines the loss of the hyperbolic structure which was underly ing the compressible equations. In\nother words, the coupling between mass and momentum conserv ation equations becomes weaker\nin the incompressible setting, thus making the approach of t he above mentioned [11, 12] out of\nuse here and, at the same time, implying the appearance of new difficulties in the analysis.\nOn the other hand, previous results about fluid systems with d amping in the velocity equation\nexist also in the incompressible case, albeit (to the best of our knowledge) only in the homogeneous\nor quasi-homogeneous situation. By this, we mean that the sys tems under study were obtained\nunder the so-called Boussinesq approximation; as a conseque nce, the equation for the velocity\nfieldudid not involve any heterogeneity in the convective term ∂tu+u·∇u. For instance, this\nis the framework considered in [6], in the context of the Invi scid Porus Medium equation, and\nin [8] in the context of the 2-D Boussinesq system. Both those works address the question of\nthe asymptotic stability of stably stratified solutions. We remark that, as a consiquence of the\nBoussinesq approximation, at principal order the velocity fi eldusatisfies an Euler-type equation\nwith pressure ∇Π. Correspondingly, the damping term considered in those wor ks is of the form\nDγ\nαwithγ= 0, namely it reduces to αuas in the case of homogeneous fluid flows. It follows\nthat an application of the curl operator immediately erases the presence of the pressure term and\ngives a damping effect on the vorticity function.\nBecause of the presence of density variations in equations (1 ), our case will be very different\nfrom the ones discussed in [6, 8], even for the choice γ= 1. In particular, the pressure gradient\ncannot be eliminated from the equations; at the same time, it is not clear, in our context, that one\ncan define a “damped mode” as in [6] (see also works [5, 11, 12] f or the compressible Euler equa-\ntions) and give a precise description of the asymptotic beha viour of the solutions when t−→+∞.\nWe refer to the next subsection for more details about this is sue.\n1.3 Main results and strategy of the proof: an overview\nIn the present paper, we address the question of the global so lvability for the non-homogeneous\nincompressible Euler equations with damping, namely of sys tem (1)-(2) with α>0, in both cases\nγ= 0andγ= 1. Under suitable smallness assumptions (see more comments b elow), we are able\nto establish global in time well-posedness in the framework of non-homogeneous Besov spaces\nBs\np,rsatisfying either one of the two conditions appearing in (5) . Recall that each one of those\nconditions guarantees the embedding property of Bs\np,rin the space of globally Lipschitz functions.\nHere, we want to give an overview of our main results and of the strategy of the proof. We\nrefer to Section 2 for the precise statements.\nThe results. As already said, we prove global existence and uniqueness of solutions under\nsuitable smallness assumptions. As a byproduct of our proof , we also establish exponential decay\nof the velocity field uand of the pressure term ∇Πin the high regularity norms. For the sake of\nclarity, let us divide our discussion into two cases, depend ing on the type of results we obtain.\nThe first class of results concerns the case of general space d imensiond/greaterorequalslant2; the precise\nstatements are Theorem 2.1 (for the case γ= 1) and Theorem 2.5 (for the case γ= 0). In those\nresults, we formulate a non-linear smallness assumption guaranteeing global existence, wher e the\nterm “non-linear” refers to the dependence of the condition on the norms of the initial datum/parenleftbig\nρ0,u0/parenrightbig\nand on the damping coefficient α. Roughly speaking, the meaning of such a condition\nis to require either the norm of the initial datum to be small, or the damping coefficient to be\nsufficiently large, in order to deduce global in time results. Interestingly, the norms entering\ninto play in the smallness conditions are only of lower order , resting on the B1\n∞,1norm of the\nheterogeneity ρ0−1andu(actually, when γ= 0one must require a control on the full L2∩B1\n∞,1\n4norm ofu). Notice that, in the case γ= 0, the smallness assumption reads in a slightly more\ncomplicated way, as the analysis of the system becomes more i nvolved than in the case γ= 1.\nThe second class of results, instead, concerns the case d= 2; however, we are forced to consider\nγ= 1. Under these assumptions, in Theorem 2.6 we establish globa l existence under a different\nsmallness condition, resting on the size of the non-homogeneity ρ0−1only. In particular, fixed the\nsize of the damping coefficient α>0and of the initial velocity field u0, which can be arbitrarily\nlarge, then there exists a unique global solution, provided ρ0−1is small enough in a suitable\nnorm. Unfortunately, when d= 2andγ= 0our method of proof breaks down and it is not clear\nwhether or not a similar result can be obtained also in this ca se.\nApproach. The strategy of the proof of our main results is based on three main steps: first,\nproving a local existence result (this is a quite simple adap tation of the theory of [15, 16] for\nthe caseα= 0); second, proving a continuation criterion of Beale-Kato-M ajda type, resting only\non theL∞norm of ∇u; finally, controlling the low regularity norms globally in t ime. On the\none hand, this is the key in order to get a smallness assumptio n which depends only on the low\nregularity norms of the initial datum. On the other hand, we p oint out that this approach yields\nexponential decay of the solution (more precisely, of uand∇Π) only in the low regularity norms,\nand in principle not in the high regularity ones, as one would expect. Actually, the sought decay\nfor the higher regularity norms is obtained a posteriori , using the decay of low regularity norms\nand fine estimates, which are similar in spirit to the ones whi ch are necessary to establish the\ncontinuation criterion.\nAs already remarked above, it turns out that the case γ= 1is slightly easier than the case\nγ= 0. The reason for this is that, when γ= 1and in absence of vacuum (4), one can divide\nthe momentum equation by ρand find an equation of Euler type for the rescaled velocity fie ld\n/tildewideu:=eαtu: after setting /tildewideΠ :=eαtΠ, one has\n∂t/tildewideu+e−αt/tildewideu·∇/tildewideu=−eαt1\nρ∇/tildewideΠ.\nNotice that the exponential growth of the right-hand side is only a fake: taking the divergence of\nthe previous equation, one discovers that (as it is the case i n the incompressible Euler equations)\nthe pressure gradient ∇/tildewideΠis a quadratic term in /tildewideu. As a consequence, the right-hand side of the\nprevious equation actually decays exponentially in time. I n the special case d= 2andγ= 1, one\nhas to complement the previous observations with improved t ransport estimates in Besov spaces\nof regularity index s= 0, originally due to Vishik [25] (see also [21]). Controlling theB0\n∞,1norm\nof the vorticity linearly in the Lipschitz norm of the velocity field is then the key (tog ether with\nthe absence of the stretching term) to get a smallness condit ion depending upon the size of ρ0−1\nonly.\nWhenγ= 0, instead, the equation for the velocity field ubecomes\n∂tu+u·∇u+1\nρ∇Π +α\nρu= 0,\nfrom which we can deduce an exponential decay with sharp rate −α∗tonly inL2, where we have\ndefinedα∗:=α/ρ∗,ρ∗being the maximum value of the density, recall (4). When pass ing to\nthe estimates of high regularity norms, instead, the presen ce of a variable coefficient 1/ρin the\ndamping term is a source of difficulty, because it creates rema inders, which must be treated as\nforcing term though they do not possess any decay in time. The basic idea to overcome this\ndifficulty is then to split the damping term as follows, for any 0<β/lessorequalslantα∗:\nα\nρu=βu+/parenleftbiggα\nρ−β/parenrightbigg\nu.\nFrom the first term on the right, we can deduce exponential dec ay similarly to the previous case\nγ= 1, whereas for the second term on the right we use the positivit y of its coefficient to keep it\n5into the left-hand side of the estimates and, finally, discar d it. In doing so, however, we cannot\nforget about commutator terms arising from the estimates fo r the Besov norms (either endpoint\nnorms inB1\n∞,1, or higher regularity norms in Bs\np,r): the basic idea is to implement a systematic use\nof sharp commutator estimates and interpolation in order to make always a (lower order) norm\nof the velocity appear, which in turn can be controlled by the L2norm, for which one disposes of\nthe sharp decay.\nTo conlude this part, let us mention a second major difficulty a rising in the case γ= 0, namely\nthe presence of a linear term in the pressure equation. Because of this term, when γ= 0it is\nno more true that the pressure gradient is quadratic in the ve locity field, but only linear, thus\npreventing us from using the argument of the case γ= 1and making the analysis much more\ndelicate. In fact, it turns out that the same idea depicted ab ove (based on fine commutator\nestimates and interpolation, to make an L2norm ofuappear) allows us to handle this second\nobstacle and get the desired global in time result also in the instanceγ= 0.\nOrganisation of the paper\nThe paper is organised in the following way. In the next secti on, we state our main results.\nIn Section 3, we collect some tools from harmonic analysis, i n particular the Littlewood-Paley\ntheory, paradifferential calculus and transport estimates in Besov spaces. In Section 4 we study\nlocal well-posedness for system (1) and establish a continu ation criterion in terms of the Lipschitz\nnorm of the velocity field only. The analysis of this section i s carried out for both cases γ= 1\nandγ= 0in parallel, as no essential differences arise. Section 5 is d evoted to the proof of the\nglobal well-posedness statements related to the case γ= 1, namely Theorem 2.1 (in general space\ndimension) and Theorem 2.6 (related to the case d= 2). Finally, in Section 6 we tackle the proof\nof Theorem 2.5, namely of global well-posedness for the case γ= 0.\nNotation\nBefore presenting the main results of the work, let us fix some n otations. For an interval I⊂R\nand a Banach space X, we denote by C/parenleftbig\nI;X/parenrightbig\nthe space of continuous bounded functions on I\nwith values in X. For anyp∈[1,+∞], the symbol Lp/parenleftbig\nI;X/parenrightbig\nstands for the space of measurable\nfunctions on Isuch that the map t/ma√sto→ /ba∇dblf(t)/ba∇dblXbelongs to Lp(I). We also define Cb/parenleftbig\nI;X/parenrightbig\n=\nC/parenleftbig\nI;X/parenrightbig\n∩L∞/parenleftbig\nI;X/parenrightbig\n. Similarly, if Q⊂R1+d=Rt×Rd\nx, we setCb(Q) =C(Q)∩L∞(Q).\nIn addition, let Ybe the predual of Xand denote by /an}b∇acketle{t·,·/an}b∇acket∇i}htX×Ythe duality pairing between\nYandX=Y∗. Then we set Cw/parenleftbig\nI;X/parenrightbig\nthe space of functions ffromItoXwhich are weakly\ncontinuous, namely such that, for any φ∈Y, the mapt/ma√sto→ /an}b∇acketle{tf(t),φ/an}b∇acket∇i}htX×Yis continuous over I. We\nuse the same notation for scalar-valued and vector-valued f unctions.\nAcknowledgements\nThe work of the second author has been partially supported by the project CRISIS (ANR-20-CE40-0020-\n01), operated by the French National Research Agency (ANR).\n2 Main results\nIn this section, we give the precise statement of our main res ults. We start by discussing the case\nγ= 1, which looks physically more pertinent and, to some extent, simpler. Then, we consider\nthe caseγ= 0, which is technically more involved. Finally, we focus on th e special case of space\ndimensiond= 2and choice γ= 1, for which we are able to give an improvement of the previous\nresults.\n62.1 The case γ= 1whend/greaterorequalslant2\nOur first result states global well-posedness under a non-li near smallness condition linking the\nnorm of the initial datum and the damping term, and holds true in any space dimension.\nTheorem 2.1. Letd/greaterorequalslant2and fixγ= 1in system (1). Let us fix indices (s,p,r)∈R×[1,+∞]×\n[1,+∞]such thatp/greaterorequalslant2and condition (5)is satisfied. Consider two constants 0<ρ∗<ρ∗.\nThere exists an absolute constant K >0, only depending on ρ∗andρ∗, and an exponent\nη=η(d)>0, only depending on the space dimension d/greaterorequalslant2, such that the following property holds\ntrue.\nTake any initial datum/parenleftbig\nρ0,u0/parenrightbig\nverifying (3)-(4)with exactly the previous constants ρ∗and\nρ∗, such that u0∈L2∩Bs\np,rand∇ρ0∈Bs−1\np,r. Takeα>0in equations (1). Assume that\n(6)1\nα/ba∇dblu0/ba∇dblB1\n∞,1exp/parenleftbigg/parenleftbigg\n1 +/ba∇dblρ0−1/ba∇dblη\nB1\n∞,1/parenrightbigg\neK/parenleftbigg1\nα/ba∇dblu0/ba∇dblL2+ 1/parenrightbigg/parenrightbigg\n<2\nThen, there exists a unique global in time solution/parenleftbig\nρ,u,∇Π/parenrightbig\nto system (1), related to the\ninitial datum/parenleftbig\nρ0,u0/parenrightbig\n, such that:\n(i)ρ∈ Cb/parenleftbig\nR+×R2/parenrightbig\n, withρ∗/lessorequalslantρ(t)/lessorequalslantρ∗for anyt∈R+, with in addition ∇ρ∈ Cw/parenleftbig\nR+;Bs−1\np,r/parenrightbig\nin the case p<+∞,ρ∈ Cw/parenleftbig\nR+;Bs\n∞,r/parenrightbig\nifp= +∞;\n(ii)u∈ C1/parenleftbig\nR+;L2/parenrightbig\n∩Cw/parenleftbig\nR+;Bs\np,r/parenrightbig\n;\n(iii)∇Π∈ C/parenleftbig\nR+;L2/parenrightbig\n∩Cw/parenleftbig\nR+;Bs\np,r/parenrightbig\n.\nMoreover the map\n(7) t/ma√sto→eαt/ba∇dblu(t)/ba∇dblL2∩Bsp,r+e2αt/ba∇dbl∇Π(t)/ba∇dblL2∩Bs∞,ris bounded over R+.\nWhenr <+∞, the continuity in time with values in Besov spaces holds wit h respect to the\nstrong topology.\nObserve that the precise value 2on the right-hand side of (6) is taken just for convenience,\nin order to give a quantitative version of the smallness cond ition which is required to deduce\nglobal well-posedness. However, one can take any other cons tant, up to modifying the value of\nthe constant K.\nNext, let us formulate a remark about the previous smallness conditions (6).\nRemark 2.2. Conditions (6) rest only on α, on theB1\n∞,1norm ofρ−1and on the L2∩B1\n∞,1\nnorm ofu0. In particular, the initial data may have large norms in high er regularity spaces.\nFrom Theorem 2.1, we immediately deduce the following state ment, which tells us that, for\nany initial datum/parenleftbig\nρ0,u0/parenrightbig\nas above, then global existence holds, provided the damping coefficient\nαis large enough.\nCorollary 2.3. Letd/greaterorequalslant2and fixγ= 1in system (1). Fix indices (s,p,r)∈R×[1,+∞]×[1,+∞]\nsuch thatp/greaterorequalslant2and condition (5)is satisfied. Consider two constants 0< ρ∗< ρ∗and take an\ninitial datum/parenleftbig\nρ0,u0/parenrightbig\nverifying conditions (3)and(4), withu0∈L2∩Bs\np,rand∇ρ0∈Bs−1\np,r.\nThen, there exists α0=α0/parenleftBig\nρ∗,ρ∗,d,/ba∇dblρ0−1/ba∇dblB1\n∞,1,/ba∇dblu0/ba∇dblL2∩B1\n∞,1/parenrightBig\n>0, depending only on\nthe quantities in the brackets, such that, for any α > α 0, there exists a unique global in time\nsolution/parenleftbig\nρ,u,∇Π/parenrightbig\nto system (1), related to the initial datum/parenleftbig\nρ0,u0/parenrightbig\nand satisfying the regularity\nproperties (i)-(ii)-(iii) stated in Theorem 2.1.\nOf course, also the “dual” statement holds true, in which one fixes the value of αand requires\nthe smallness of the norm of the initial datum. The interesti ng point of condition (6) is that it\nimposes smallness only on the B1\n∞,1norm of the initial velocity, once one fixes the size of the\nheterogeneity and of the initial kinetic energy.\n7Corollary 2.4. Letd/greaterorequalslant2and takeγ= 1andα >0in system (1). Fix indices (s,p,r)∈\nR×[1,+∞]×[1,+∞]such thatp/greaterorequalslant2and condition (5)is satisfied. For 0<ρ∗<ρ∗, consider\nan initial datum/parenleftbig\nρ0,u0/parenrightbig\nverifying the assumptions of Corollary 2.3.\nThen, there exists c0=c0/parenleftBig\nρ∗,ρ∗,d,α,/ba∇dblρ0−1/ba∇dblB1\n∞,1,/ba∇dblu0/ba∇dblL2/parenrightBig\n>0, depending only on the\nquantities in the brackets, such that, if /ba∇dblu0/ba∇dblB1\n∞,1< c0, there exists a unique global in time solu-\ntion/parenleftbig\nρ,u,∇Π/parenrightbig\nto system (1), related to the initial datum/parenleftbig\nρ0,u0/parenrightbig\nand satisfying the regularity\nproperties (i)-(ii)-(iii) of Theorem 2.1, together with th e decay estimate (7).\n2.2 The case γ= 0ford/greaterorequalslant2\nIn this subsection, we still focus on the case of general dime nsiond/greaterorequalslant2, but for the choice γ= 0\nin the damping coefficient. We immediately notice that this ca se is much more involved than the\nprevious one γ= 1. As a matter of fact, a linear term survives in the analysis of the pressure\ngradient, which then is no more expected to be quadratic in u, thus to possess a better decay\n(compare with the decay property (7) above).\nNonetheless, by using positivity properties of the damping term and careful decay estimates,\none can prove a global well-posedness result also in this cas e, provided some smallness conditions,\nmore involved than (6), are satisfied. The precise statement reads as follows.\nTheorem 2.5. Letd/greaterorequalslant2and fixγ= 0in system (1). Let us fix indices (s,p,r)∈R×[1,+∞]×\n[1,+∞]such thatp/greaterorequalslant2and condition (5)is satisfied. Consider two constants 0<ρ∗<ρ∗.\nThere exists an absolute constant K >0, only depending on ρ∗andρ∗and on the space\ndimensiond/greaterorequalslant2, and an exponent η=η(d)only depending on d, such that the following property\nholds true.\nTake any initial datum/parenleftbig\nρ0,u0/parenrightbig\nverifying (3)-(4)with exactly the previous constants ρ∗and\nρ∗, such that u0∈L2∩Bs\np,rand∇ρ0∈Bs−1\np,r. Takeα>0in equations (1). Define\nR:= 1+/ba∇dblρ0−1/ba∇dblη\nB1\n∞,1\nand assume that\nKReKR/ba∇dblu0/ba∇dblL2∩B1\n∞,1\nα<2 andKR3eKR/ba∇dblu0/ba∇dbl2\nL2∩B1\n∞,1\nα<4.\nThen, there exists a unique global in time solution/parenleftbig\nρ,u,∇Π/parenrightbig\nto system (1), related to the initial\ndatum/parenleftbig\nρ0,u0/parenrightbig\n, such that the regularity properties (i)-(ii)-(iii) state d in Theorem 2.1 still hold\ntrue. Moreover, after defining θ0∈]0,1[as the interpolation exponent such that /ba∇dblf/ba∇dblBs−1\np,r/lessorequalslant\nC/ba∇dblf/ba∇dblθ0\nL2/ba∇dblf/ba∇dbl1−θ0\nBsp,rand setting\nβ0:=θ0\n1+θ0α\nρ∗,\none has that the map\n(8) t/ma√sto→eβ0t/ba∇dblu(t)/ba∇dblL2∩Bsp,r+eβ0t/ba∇dbl∇Π(t)/ba∇dblL2∩Bs∞,ris bounded over R+.\nOf course, the same comments formulated after Theorem 2.1, t ogether with Remark 2.2 and\nCorollary 2.3, apply also to the previous theorem, obtained in the case γ= 0. However, for the\nsake of conciseness, we avoid to formulate them explicitly a lso here.\n2.3 The case d= 2andγ= 1\nNow, we focus on the special case of planar flows, namely d= 2. Observe that, when ρ→1,\nequations (1) formally converge to the classical (homogene ous) incompressible Euler equations,\n8which are globally well-posed when d= 2. Then, it is natural to wonder if, in this case, Theorem\n2.1 can be improved by imposing a smallness condition on the s ize of the non-homgeneity ρ0−1\nonly.\nIn this section, we give a positive answer to the previous que stion, but only in the case γ= 1.\nThe key ingredient is the use of improved transport estimate s [25, 21] in Besov spaces of regularity\nindexs= 0to control the growth of the vorticity, together with the abs ence of the stretching term\nin its transport equation. In the case γ= 0, the problem is that it is not clear how to reproduce\nthe fundamental cancellation mentioned at the beginning of Subsection 2.2 when estimating the\nnorm of the vorticity in the critical spaceB0\n∞,1.\nForγ= 1, the precise result is contained in the following statement .\nTheorem 2.6. Letd= 2and fixγ= 1in system (1). Let us fix indices (s,p,r)∈R×[1,+∞]×\n[1,+∞]such thatp/greaterorequalslant2and condition (5)is satisfied. Fix two constants 0<ρ∗<ρ∗.\nThere exists an absolute constant K >0, only depending on ρ∗andρ∗, such that the following\nproperty holds true.\nTake any initial datum/parenleftbig\nρ0,u0/parenrightbig\nverifying (3)-(4)with exactly the given constants 0<ρ∗<ρ∗,\nsuch thatu0∈L2∩Bs\np,rand∇ρ0∈Bs−1\np,r. In addition, define the function ΦKby\n∀z/greaterorequalslant0, ΦK(z) :=e2K\nαzeKexp/parenleftbig\nK\nαz/parenrightbig\nand assume that\n(9) /ba∇dblρ0−1/ba∇dblB1\n∞,1/parenleftbigg\n1 +/ba∇dblρ0−1/ba∇dblη\nB1\n∞,1/parenrightbigg\n/ba∇dblu0/ba∇dblL2∩B1\n∞,1ΦK/parenleftBig\n/ba∇dblu0/ba∇dblL2∩B1\n∞,1/parenrightBig\n<4,\nwhere the exponent η>5can be chosen1as close to 5as one wants.\nThen, there exists a unique global in time solution/parenleftbig\nρ,u,∇Π/parenrightbig\nto system (1), related to the\ninitial datum/parenleftbig\nρ0,u0/parenrightbig\nand satisfying the regularity properties (i)-(ii)-(iii) s tated in Theorem 2.1,\ntogether with the decay estimate (7)therein.\nOnce again, we remark that there is no specific reason to take t he critical value 4in the\ninequality defining the smallness condition of the previous statement. Next, we formulate a\nremark, similar in spirit to Remark 2.2.\nRemark 2.7. Also in this case, only the low regularity norm /ba∇dblρ0−1/ba∇dblB1\n∞,1is involved in the\nsmallness condition.\nThe smallness condition (9) has to be interpreted as a smalln ess condition over the non-\nhomogeneity ρ0−1: even in the case when the velocity field is large and the dampi ng coefficient\nα>0is small, if the non-homogeneity is small enough, then the so lution is globally defined. More\nprecisely, as an immediate consequence of Theorem 2.6, we ob tain the next result, in the same\nvein of Corollary 2.3 above.\nCorollary 2.8. Letd= 2and fixγ= 1and someα>0in system (1). Take indices (s,p,r)∈\nR×[1,+∞]×[1,+∞]such thatp/greaterorequalslant2and condition (5)is satisfied. Consider two constants\n0<ρ∗<ρ∗.\nFor any initial velocity field u0∈L2∩Bs\np,r, withdivu0= 0, there exists a positive constant\nC0=C0/parenleftBig\nρ∗,ρ∗,α,/ba∇dblu0/ba∇dblL2∩B1\n∞,1/parenrightBig\n>0, depending only on the quantities in the brackets, such that\nthe following fact holds true.\nFor any initial density ρ0verifying condition (4)and∇ρ0∈Bs−1\np,r, and such that\n/ba∇dblρ0−1/ba∇dblB1\n∞,1/lessorequalslantC0,\n1The exponent ηcomes from an interpolation argument, see inequality (49) b elow. It only depends on the space\ndimension, which is fixed here d= 2.\n9there exists a unique global in time solution/parenleftbig\nρ,u,∇Π/parenrightbig\nto system (1), related to the initial datum/parenleftbig\nρ0,u0/parenrightbig\nand satisfying the regularity properties (i)-(ii)-(iii) s tated in Theorem 2.1, together with\nthe decay property (7).\nThe rest of the work is devoted to the proof of the previous sta tements. Before continuing,\nlet us mention that, although the vorticity of the fluid does n ot appear in the formulation of the\nmain results, this quantity plays an important role in our an alysis. Therefore, let us spend some\nlines to introduce it.\nGiven an incompressible velocity field u, we define the vorticity matrix Ωofuas twice the\nskew-symmetric part of its Jacobian matrix, namely Ω :=Du− ∇u, whereDuis the Jacobian\nmatrix ofuand∇u=tDuis its transpose matrix. We adopt the convention of identify ingΩwith\nthe vector field ω:=∇×uwhend= 3and, whend= 2, with the scalar field ω:=∂1u2−∂2u1.\nThis having been recalled, we can move on and collecting, in t he next section, some preliminary\nmaterial which is needed in our study.\n3 Toolbox\nThe present section is intended to be a functional toolbox, w hich we stock with all the material\nwhich is needed in the course of the proof of the main results. We start by recalling some basic\nfacts of Littlewood-Paley theory in Rd, withd/greaterorequalslant1, the definition of non-homogeneous Besov\nspaces and some interpolation inequalities of Gagliardo-N irenberg type. Then, in Subsection 3.2\nwe introduce some elements of paradifferential calculus, na mely paraproduct decomposition and\nestimates for composition of Besov functions. Finally, in Su bsection 3.4 we recall some well-known\nestimates for smooth solutions to transport and elliptic eq uations.\n3.1 Littlewood-Paley theory and Besov spaces\nWe recall here the main ideas of Littlewood-Paley theory in Rd. These are classical results, for\nwhich, if not otherwise specified, we refer to Chapter 2 of [2] for details.\nTo begin with, we fix a smooth radial function χsupported in the ball B(0,2), equal to 1in a\nneighborhood of B(0,1)and such that r/ma√sto→χ(re)is nonincreasing over R+for all unitary vectors\ne∈Rd. Setϕ(ξ) =χ(ξ)−χ(2ξ)andϕj(ξ) :=ϕ(2−jξ)for allj/greaterorequalslant0. The dyadic blocks (∆j)j∈Z\nare defined by2\n∆j:= 0 ifj/lessorequalslant−2,∆−1:=χ(D) and ∆j:=ϕ(2−jD)ifj/greaterorequalslant0.\nWe also introduce the following low frequency cut-off operat or:\n(10) Sju:=χ(2−jD) =/summationdisplay\nk/lessorequalslantj−1∆k forj/greaterorequalslant0.\nNote thatSjand∆jare convolution operators by L1kernels, whose norms are independent of\nthe indexj. Thus, they act continuously from Lpinto itself, for any p∈[1,+∞].\nRemark that the function χcan be chosen so that, for all ξ∈Rd, one has the equality\nχ(ξ) +/summationtext\nj/greaterorequalslant0ϕj(ξ) = 1 . Based on this partition of unity on the Fourier space, we obta in the\nso-called non-homogeneous “Littlewood-Paley decomposit ion” of tempered distributions:\n∀u∈ S′, u =/summationdisplay\nj/greaterorequalslant−1∆ju in the sense of S′.\nLet us also mention the so-called Bernstein inequalities , which explain the way derivatives act\non spectrally localised functions. They play a key role in th e whole theory.\n2Throughout, we agree that f(D)stands for the pseudo-differential operator u/mapsto→ F−1[f(ξ)/hatwideu(ξ)].\n10Lemma 3.1. Let0<r<R . A constant C >0exists so that, for any integer k/greaterorequalslant0, any couple\n(p,q)in[1,+∞]2, withp/lessorequalslantq, and any function u∈Lp, for allλ>0we have:\nSupp/hatwideu⊂B(0,λR) =/braceleftbig\nξ∈Rd/vextendsingle/vextendsingle|ξ|/lessorequalslantλR/bracerightbig\n=⇒ /ba∇dbl∇ku/ba∇dblLq/lessorequalslantCk+1λk+d/parenleftBig\n1\np−1\nq/parenrightBig\n/ba∇dblu/ba∇dblLp\nSupp/hatwideu⊂/braceleftbig\nξ∈Rd/vextendsingle/vextendsinglerλ/lessorequalslant|ξ|/lessorequalslantRλ/bracerightbig\n=⇒C−k−1λk/ba∇dblu/ba∇dblLp/lessorequalslant/ba∇dbl∇ku/ba∇dblLp/lessorequalslantCk+1λk/ba∇dblu/ba∇dblLp.\nBy use of Littlewood-Paley decomposition, we can define the cl ass of Besov spaces.\nDefinition 3.2. Lets∈Rand1/lessorequalslantp,r/lessorequalslant+∞. The non-homogeneous Besov space Bs\np,r=\nBs\np,r(Rd)is defined as the subset of tempered distributions ufor which\n/ba∇dblu/ba∇dblBsp,r:=/vextenddouble/vextenddouble/vextenddouble/parenleftbig\n2js/ba∇dbl∆ju/ba∇dblLp/parenrightbig\nj/greaterorequalslant−1/vextenddouble/vextenddouble/vextenddouble\nℓr<+∞.\nBesov spaces are interpolation spaces between the more class ical Sobolev spaces. In fact, for\nanyk∈Nandp∈[1,+∞], we have the following chain of continuous embeddings:\nBk\np,1֒→Wk,p֒→Bk\np,∞,\nwhereWk,pstands for the classical Sobolev space of Lpfunctions with all the derivatives up to\nthe orderkinLp. When1<p<+∞, we can refine the previous result (see Theorems 2.40 and\n2.41 in [2]): we have Bk\np,min(p,2)֒→Wk,p֒→Bk\np,max(p,2). In particular, for all s∈R, we deduce the\nequivalence Bs\n2,2≡Hs, with equivalence of norms. In addition, for any k∈Nand anyε∈]0,1[,\none has the equivalence Bk+ε\n∞,∞≡ Ck,εwith the classical Hölder classes, with equivalence of norm s.\nAs an immediate consequence of the first Bernstein inequality , one gets the following embed-\nding result.\nProposition 3.3. The spaceBs1p1,r1is continuously embedded in the space Bs2p2,r2for all indices\nsatisfyingp1/lessorequalslantp2and\ns2< s1−d/parenleftbigg1\np1−1\np2/parenrightbigg\nor s2=s1−d/parenleftbigg1\np1−1\np2/parenrightbigg\nandr1/lessorequalslantr2.\nIn particular, we get the following chain of continuous embe ddings:\nBs\np,r֒→Bs−d\np∞,r֒→B0\n∞,1֒→L∞,\nprovided that the triplet (s,p,r)∈R×[1,+∞]2satisfies the condition\n(11) s>d\npor s=d\npandr= 1.\nSimilarly, as announced at the beginning of the manuscript, for(s,p,r)verifying (5), we get\nBs\np,r֒→W1,∞.\nTo conclude this part, we recall some useful inequalities by means of Littlewood-Paley de-\ncomposition. They are all consequences of a unique simple te chnique, which consists in cutting\na tempered distribution into low and high frequencies and th en optimising the size Nat which\nrealising the cut.\nLemma 3.4. Letdbe the space dimension and let p0>d. For any function f∈ S, the following\nGagliardo-Nirenberg type inequalities hold true:\n/ba∇dblf/ba∇dblL∞/lessorsimilar/ba∇dblf/ba∇dblα1\nL2/ba∇dbl∇f/ba∇dbl1−α1\nL∞ and /ba∇dbl∇f/ba∇dblL∞/lessorsimilar/ba∇dbl∇f/ba∇dblα2\nL2/ba∇dbl∆f/ba∇dbl1−α2\nLp0,\nfor two suitable exponent α1,2∈]0,1[. The above inequalities extend also to vector-valued func-\ntions. In addition, for any vector field u∈ Ssuch that div(u) = 0, one has\n/ba∇dblu/ba∇dblL∞/lessorsimilar/ba∇dblu/ba∇dblα3\nL2/ba∇dblΩ/ba∇dbl1−α3\nLp0\nwhereΩ :=Du−t∇udenotes the vortixity matrix of uandα3∈]0,1[is a suitable exponent.\n113.2 Paradifferential calculus\nWe now apply Littlewood-Paley decomposition to state some u seful results from paradifferential\ncalculus. Again, we refer to Chapter 2 of [2] for details and f urther results.\nTo begin with, let us introduce the paraproduct operator (after J.-M. Bony [4]). Constructing\nthe paraproduct operator relies on the observation that, fo rmally, any product of two tempered\ndistributions uandvmay be decomposed into the sum\n(12) uv=Tuv+Tvu+R(u,v),\nwhere we have defined\nTuv:=/summationdisplay\njSj−1u∆jv, and R(u,v) :=/summationdisplay\nj/summationdisplay\n|k−j|/lessorequalslant1∆ju∆kv.\nThe above operator Tis called “paraproduct” whereas Ris called “remainder”. The paraprod-\nuct and remainder operators have many nice continuity prope rties over Besov spaces, which are\ncollected in the next statement.\nProposition 3.5. For any (s,p,r)∈R×[1,+∞]2andt>0, the paraproduct operator Tmaps\ncontinuously L∞×Bs\np,rinBs\np,randB−t\n∞,∞×Bs\np,rinBs−t\np,r. Moreover, the following estimates hold:\n/ba∇dblTuv/ba∇dblBsp,r/lessorsimilar/ba∇dblu/ba∇dblL∞/ba∇dbl∇v/ba∇dblBs−1\np,rand /ba∇dblTuv/ba∇dblBs−t\np,r/lessorsimilar/ba∇dblu/ba∇dblB−t\n∞,∞/ba∇dbl∇v/ba∇dblBs−1\np,r.\nFor any(s1,p1,r1)and(s2,p2,r2)inR×[1,+∞]2such thats1+s2>0,1/p:= 1/p1+1/p2/lessorequalslant\n1and1/r:= 1/r1+ 1/r2/lessorequalslant1, the remainder operator Rmaps continuously Bs1p1,r1×Bs2p2,r2\nintoBs1+s2p,r. In the case s1+s2= 0, if in addition r= 1, the operator Ris continuous from\nBs1p1,r1×Bs2p2,r2with values in B0\np,∞.\nThe consequence of this proposition is that, when s >0, the spaces Bs\np,r∩L∞are Banach\nalgebras; in particular, Bs\np,ris a Banach algebra as long as condition (11) holds with s >0.\nMoreover, in that case, we have the so-called tame estimates .\nCorollary 3.6. Let(s,p,r)∈R×[1,+∞]2be such that that s >0. Then, for any fandg\nbelonging to L∞∩Bs\np,r, the product fgalso belongs to that space and we have\n(13) /ba∇dblfg/ba∇dblBsp,r/lessorsimilar/ba∇dblf/ba∇dblL∞/ba∇dblg/ba∇dblBsp,r+/ba∇dblf/ba∇dblBsp,r/ba∇dblg/ba∇dblL∞.\nWe remark that the spaces B0\np,rarenotBanach algebras, as the continuity of the remainder\noperator R(f,g)with values in B0\np,rfails in this case. However, following the analysis in [16], we\nremark some information can still be obtained in some specia l cases, which we now highlight. We\nfocus on the case B0\n∞,1, in the end the only relevant one for our scopes.\nFirst of all, by requiring some better regularity for g, we can replace estimate (13) by e.g.\n(14) /ba∇dblfg/ba∇dblB0∞,r/lessorsimilar/ba∇dblf/ba∇dblB0\n∞,1/ba∇dblg/ba∇dblB1/2\n∞,1.\nOn the other hand, we can use some special structure in presen ce of differentiation. More precisely,\nfor a function f∈B1\n∞,1and a vector field G∈B0\n∞,1such that div(G) = 0, we can use the Bony\ndecomposition (12) and write\n∇f·G= div(fG) =T∇fG+TG∇f+ div/parenleftbig\nR(f,G)/parenrightbig\n.\nThus, by using\n/vextenddouble/vextenddoublediv/parenleftbig\nR(f,G)/parenrightbig/vextenddouble/vextenddouble\nB0\n∞,1/lessorsimilar/ba∇dblR(f,G)/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblf/ba∇dblB1\n∞,1/ba∇dblG/ba∇dblB0\n∞,1,\n12it is easy to deduce that\n(15) /ba∇dbl∇f·G/ba∇dblB0\n∞,1/lessorsimilar/ba∇dblf/ba∇dblB1\n∞,1/ba∇dblG/ba∇dblB0\n∞,1.\nInequalities (14) and (15) will be needed in the computation s of Subsection 5.1.\nNext, we recall the following result, in the same spirit of th e classical Meyer’s paralinearisation\ntheorem. It will be important when comparing the Sobolev nor ms of the density ρwith the ones\nof its inverse 1/ρand oflogρ. The proof can be found in [15].\nProposition 3.7. LetIbe an open interval of Rand letF:I→Rbe a smooth function.\nThen for all compact subset J⊂I, alls>0and all(p,r)∈[1,+∞], there exists a constant\nC >0such that, for any function avalued inJand with gradient in Bs−1\np,r, we have ∇(F◦a)∈Bs−1\np,r\ntogether with the estimate\n/ba∇dbl∇(F◦a)/ba∇dblBs−1\np,r/lessorequalslantC/ba∇dbl∇a/ba∇dblBs−1\np,r.\nFinally, we need a few commutator estimates. Recall that, gi ven two operators AandB,\nwe denote by [A,B] :=AB−BAtheir commutator operator. The first commutator estimate\ncorresponds to Lemma 2.100 of [2] (see also Remark 2.101 ther ein).\nLemma 3.8. Lets>0and(s,p,r)∈R×[1,+∞]2such that condition (5)is satisfied. Let vbe\na vector field on Rd, belonging to Bs\np,r. Then, for any f∈Bs\np,r, one has\n2js/vextenddouble/vextenddouble/bracketleftbig\nv·∇,∆j/bracketrightbig\nf/vextenddouble/vextenddouble\nLp/lessorsimilarcj/parenleftBig\n/ba∇dbl∇v/ba∇dblL∞/ba∇dblf/ba∇dblBsp,r+/ba∇dbl∇v/ba∇dblBs−1\np,r/ba∇dbl∇f/ba∇dblL∞/parenrightBig\n,\nfor some sequence/parenleftbig\ncj/parenrightbig\nj/greaterorequalslant−1belonging to the unit ball of ℓr.\nThe second commutator estimate we need involves the commuta tor of a paraproduct operator\nwith a Fourier multiplier. It is contained in Lemma 2.99 of [2 ], but we report it here for the reader\nconvenience.\nLemma 3.9. Letψbe a smooth function on Rd, which is homogeneous of degree maway from a\nneighbourhood of 0. Then, for any vector field vsuch that ∇v∈L∞and for any s∈R, one has:\n∀f∈Bs\np,r,/vextenddouble/vextenddouble/bracketleftbig\nTv,ψ(D)/bracketrightbig\nf/vextenddouble/vextenddouble\nBs−m+1\np,r/lessorsimilar/ba∇dbl∇v/ba∇dblL∞/ba∇dblf/ba∇dblBsp,r.\nThe third commutator estimate, contained in the next statem ent, is used to control the damp-\ning term. Although this estimate seems classical, we were no t able to find a reference for it; so\nwe briefly explain its proof in the Appendix.\nLemma 3.10. Let(s,p,r)∈R×[1,+∞]2satisfying (5).\nThen, for any vector field vonRdsuch thatv∈Bs−1\np,rand anyf∈L∞(Rd)such that\n∇f∈Bs−1\np,r, one has the estimate\n2js/vextenddouble/vextenddouble/bracketleftbig\nf,∆j/bracketrightbig\nv/vextenddouble/vextenddouble\nLp/lessorsimilarcj/parenleftBig\n/ba∇dblf/ba∇dblB1\n∞,1/ba∇dblv/ba∇dblBs−1\np,r+/ba∇dbl∇f/ba∇dblBs−1\np,r/ba∇dblv/ba∇dblL∞/parenrightBig\n,\nfor a suitable sequence/parenleftbig\ncj/parenrightbig\nj/greaterorequalslant−1belonging to the unit ball of ℓr. The (implicit) multiplicative\nconstant intervening in the previous inequality is “univer sal”, namely it does not depend on the\nvector field v, nor on the function f.\n133.3 Elliptic equations with variable coefficients\nHere, we turn our attention to the study of the following elli ptic equation:\n(16) −div/parenleftbigg1\nρ∇Π/parenrightbigg\n= div(F) inRd,\nwhereρ=ρ(x)is a given regular bounded function satisfying\na∗:= inf\nx∈R21\nρ>0.\nWe shall use the following result, based on Lax-Milgram’s th eorem (this is Lemma 2 in [15]).\nLemma 3.11. For all vector field Fwith coefficients in L2, there exists a tempered distribution\nΠ, unique up to constant functions, such that ∇Π∈L2and equation (16)is satisfied. In addition,\nwe have\n/ba∇dbl∇Π/ba∇dblL2/lessorequalslant1\na∗/ba∇dblF/ba∇dblL2.\nNext, we discuss how to derive higher regularity estimates f or equation (16) in Besov spaces.\nPropagation of higher regularity for the pressure for the no n-homogeneous incompressible Euler\nsystem (1) (actually, in the case α= 0) has been obtained in [15, 16, 17]. More precisely, one can\nprove that\n(17) /ba∇dbl∇Π/ba∇dblBsp,r/lessorsimilar/parenleftBig\n1+/ba∇dbl∇ρ/ba∇dblη\nBs−1\np,r/parenrightBig\n/ba∇dblF/ba∇dblL2+/ba∇dblρdiv(F)/ba∇dblBs−1\np,r.\nHowever, in view of applications to the continuation criter ion, which will play a key role in our\nanalysis, we have to use more precise estimates, in the same s pirit of the bounds exhibited in\n[18, 19] for a different (yet related) non-homogeneous fluid s ystem.\nLet us collect the needed estimates in the next statement, wh ose proof is postponed to Ap-\npendix A. It goes without saying that those estimates, toget her with an interpolation argument,\nimply (17) above.\nLemma 3.12. Fix indices (s,p,r)∈R×[1,+∞]×[1,+∞]such thatp/greaterorequalslant2and condition (5)is\nsatisfied. Then, there exists an exponent η >0, only depending on the triplet (s,p,r), such that\nthe solution to equation (16)satisfies the following estimate:\n/ba∇dbl∇Π/ba∇dblBsp,r/lessorsimilar/parenleftbig\n1+/ba∇dbl∇ρ/ba∇dblη\nL∞/parenrightbig\n/ba∇dblF/ba∇dblL2+/ba∇dbl∇ρ/ba∇dblBs−1\np,r/ba∇dbl∇Π/ba∇dblL∞+/ba∇dblρdiv(F)/ba∇dblBs−1\np,r,\nwhere the (implicit) multiplicative constant only depends ona∗anda∗:= supx∈R21/ρ(x).\n3.4 Transport estimates in Besov spaces\nSystem (1) has essentially the structure of a coupling of tra nsport equations, at least under\ncondition (4). As a matter of fact, in our study we will make a g reat use of a priori estimates in\nBesov spaces for smooth solutions to linear transport equati ons, which we collect in this subsection.\nMost of the material presented here can be found in Chapter 3 o f [2].\nLet us consider the initial value problem\n(18)/braceleftBigg∂tf+v·∇f=g\nf|t=0=f0.\nThroughout this part, the velocity field v=v(t,x)will always be assumed to be divergence-free,\ni.e.divv= 0, and Lipschitz continuous with respect to the space variabl e.\nThe following statement contains the conclusions of Theore ms 3.14 and 3.19 of [2], in the case\nwhen the transport field vis Lipschitz-continuous.\n14Theorem 3.13. Let(s,p,r)∈R×[1,+∞]2satisfy the Lipschitz condition (5)andT >0be\nfixed. Assume that vis a divergence-free vector field belonging to L1/parenleftbig\n[0,T];Bs\np,r/parenrightbig\nsuch that, for\nsomeq>1andM >0,v∈Lq/parenleftbig\n[0,T];B−M\n∞,∞/parenrightbig\n. Finally, let σ∈Rbe such that\nσ/greaterorequalslant−dmin/braceleftbigg1\np,1\np′/bracerightbigg\nor, if divv= 0, σ/greaterorequalslant−1−dmin/braceleftbigg1\np,1\np′/bracerightbigg\n.\nThen, for any external force g∈L1/parenleftbig\n[0,T];Bσ\np,r/parenrightbig\nand any initial datum f0∈Bσ\np,r, the transport\nequation (18)has a unique solution fin the space:\n•C/parenleftbig\n[0,T];Bσ\np,r/parenrightbig\nifr<+∞;\n•/parenleftBig/intersectiontext\nσ′<σC/parenleftbig\n[0,T];Bσ′\np,∞/parenrightbig/parenrightBig\n∩Cw/parenleftbig\n[0,T];Bσ\np,∞/parenrightbig\n, in the case r= +∞.\nMoreover, after defining the function V(t)as\nV(t) :=/integraldisplayt\n0/ba∇dbl∇v(τ)/ba∇dblBs−1\np,rdτ,\nthe unique solution fsatisfies the following estimate, for a suitable universal c onstantC >0:\n∀t∈[0,T],/ba∇dblf(t)/ba∇dblBσp,r/lessorequalslanteC V(t)/parenleftbigg\n/ba∇dblf0/ba∇dblBσp,r+/integraldisplayt\n0e−C V(τ)/ba∇dblg(τ)/ba∇dblBσp,rdτ/parenrightbigg\n.\nIn the case when v=f, the previous estimate holds true with V′(t) =/ba∇dbl∇f(t)/ba∇dblL∞.\nNext, let us recall improved transport estimates for Besov sp aces whose index of regularity\nisσ= 0. Those estimates, discovered by Vishik in [25] and later pro ved again viaa different\nargument by Hmidi and Keraani in [21], establish that the B0\np,rnorm of the solution grows\nlinearly , and not exponentially as in Theorem 3.13, with the Lipschit z norm of the transport field\nv. We refer to Theorem 3.18 of [2] for the statement presented h ere.\nTheorem 3.14. Assume that the velocity field vsatisfies ∇v∈L1\nT(L∞)anddivv= 0. Let\n(p,r)∈[1,+∞]2.\nThen there exists a constant C=C(d)>0, only depending on the space dimension d/greaterorequalslant2,\nsuch that, for any solution fto problem (18)inC/parenleftbig\n[0,T];B0\np,r/parenrightbig\n, or inCw/parenleftbig\n[0,T];B0\np,∞/parenrightbig\nifr= +∞,\nwe have\n/ba∇dblf/ba∇dblL∞\nT(B0p,r)/lessorequalslantC/parenleftBig\n/ba∇dblf0/ba∇dblB0p,r+/ba∇dblg/ba∇dblL1\nT(B0p,r)/parenrightBig/parenleftbigg\n1+/integraldisplayT\n0/ba∇dbl∇v(τ)/ba∇dblL∞dτ/parenrightbigg\n.\nTheorem 3.14 plays a key role in the proof of Theorem 2.6, name ly global existence when\nγ= 1andd= 2, with a smallness condition on the Besov norm of the non-homog eneity only.\n4 Local well-posedness\nIn this section, as a first step towards the proof of the global existence results, we investigate local\nwell-posedness of the Cauchy problem (1)-(3), together wit h the validity of a suitable continuation\ncriterion.\nTo begin with, we state the following local existence and uni queness result.\nProposition 4.1. Letα >0andγ∈ {0,1}. Fix indices (s,p,r)∈R×[1,+∞]×[1,+∞]such\nthatp>1and one of the two conditions in (5)is verified. Take an initial datum/parenleftbig\nρ0,u0/parenrightbig\nsuch that\nconditions (3)and(4)are satisfied, for two suitable constants 0<ρ∗/lessorequalslantρ∗. Assume in addition\nthat∇ρ0∈Bs−1\np,rand thatu0∈L2∩Bs\np,r.\nThen, there exist a positive time T >0and a unique solution/parenleftbig\nρ,u,∇Π/parenrightbig\nto equations (1)-(3)-\n(4)on[0,T]×Rdsuch that:\n15(i)ρ∈ Cb/parenleftbig\n[0,T]×Rd/parenrightbig\n, withρ∗/lessorequalslantρ(t)/lessorequalslantρ∗for anyt∈[0,T]and∇ρ∈ Cw/parenleftbig\n[0,T];Bs−1\np,r/parenrightbig\n;\n(ii)u∈ C1/parenleftbig\n[0,T];L2/parenrightbig\n∩ Cw/parenleftbig\n[0,T];Bs\np,r/parenrightbig\n;\n(iii)∇Π∈ C/parenleftbig\n[0,T];L2/parenrightbig\n∩ Cw/parenleftbig\n[0,T];Bs\np,r/parenrightbig\n.\nThe time continuity with values in Besov spaces holds with re spect to the strong topology if r<+∞.\nThe previous well-posedness statement easily follows from the results of [15, 16], for instance\nby looking at the damping term Dγ\nα(ρ,u) =αργuas a forcing term in the equations, and\nimplementing an iterative approximation procedure.\nThe goal of the present paper is to show that the previous solu tions are global, namely that\nT= +∞in Proposition 4.1. In this respect, we notice that a global L2bound foruand global\nL∞bounds for ρ±1are easily obtained a priori by simple energy and transport estimates for,\nrespectively, the momentum equation and the mass conservat ion equation. Indeed, we have\n(19) ∀t∈[0,T], ρ ∗/lessorequalslantρ(t)/lessorequalslantρ∗,\ntogether with the kinetic energy balance\n(20)1\n2d\ndt/integraldisplay\nRdρ|u|2dx+/integraldisplay\nRdDγ\nα(ρ,u)·udx= 0.\nAsDγ\nα(ρ,u)·u/greaterorequalslant0, this latter relation in particular implies, together with (19), that\n(21) ∀t∈[0,T], /ba∇dblu(t)/ba∇dblL2/lessorsimilar/ba∇dblu0/ba∇dblL2.\nIt remains to show that also the high regularity norms remain globally bounded. For this, the\nkey point will be the following continuation criterion in te rms of the velocity field only. Such a\ncriterion improves the ones of [15, 16], which involved in ad dition a low regularity norm of the\npressure gradient, and extends the one of [1], which conside red only the case of high Sobolev\nregularity, to the class of Besov spaces with critical regula rity.\nLemma 4.2. Let the assumptions of Proposition 4.1 be in force. In additi on, assume that p/greaterorequalslant2.\nLet/parenleftbig\nρ,u,∇Π/parenrightbig\nbe a local solution to system (1)-(3)-(4)related to the initial datum/parenleftbig\nρ0,u0/parenrightbig\n, defined\non[0,T∗[×Rd(for some time T∗>0) and such that the regularity properties (i)-(ii)-(iii) st ated\nin Proposition 4.1 hold true for any 0<T <T∗.\nIfT∗<+∞and\n(22)/integraldisplayT∗\n0/ba∇dbl∇u(t)/ba∇dblL∞dt <+∞,\nthen/parenleftbig\nρ,u,∇Π/parenrightbig\ncan be continued beyond the time T∗into a solution of (1)-(3)-(4)with the same\nregularity.\nProof. Without loss of generality, we can fix the value α= 1in the argument below. Moreover,\nlet us restrict our attention to the case γ= 0, which is, to some extent, the most difficult one, as\nit involves the presence of an additional term in the pressur e equation.\nThe method of the proof is classical. It consists in showing t hat, under the integral condition\n(22), one has\n(23) sup\nt∈[0,T∗[/parenleftBig\n/ba∇dbl∇ρ(t)/ba∇dblBs−1\np,r+/ba∇dblu(t)/ba∇dblBsp,r+/ba∇dbl∇Π(t)/ba∇dblL2∩Bsp,r/parenrightBig\n<+∞.\nWe recall that global in time bounds for uinL2and for the L∞norms ofρ±1are available in,\nrespectively, (21) and (19).\n16First of all, we differentiate the mass equation with respect toxj, forj= 1,2, to get\n(24) ∂t∂jρ+u·∇∂jρ=−∂ju·∇ρ.\nA classical argument (see the details e.g.in Section 4 of [15]) allows one to obtain, for all\nt∈[0,T∗[, the inequality\n(25) /ba∇dbl∇ρ/ba∇dblBs−1\np,r/lessorsimilar/ba∇dbl∇ρ0/ba∇dblBs−1\np,r+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∇u/ba∇dblL∞/ba∇dbl∇ρ/ba∇dblBs−1\np,r+/ba∇dbl∇ρ/ba∇dblL∞/ba∇dbl∇u/ba∇dblBs−1\np,r/parenrightBig\ndt,\nat least in the case p<+∞, orp= +∞ands>1. In the endpoint case p= +∞ands= 1(and\nthenr= 1as well), we rather work with ρ(see Section 3 of [16], this time) to get\n(26) /ba∇dblρ/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblρ0/ba∇dblB1\n∞,1+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∇u/ba∇dblL∞/ba∇dblρ/ba∇dblB1\n∞,1+/ba∇dbl∇ρ/ba∇dblL∞/ba∇dblu/ba∇dblB1\n∞,1/parenrightBig\ndt.\nNext, we consider the velocity field. Owing to the absence of v acuum, see (19), we can divide\nthe momentum equation by ρand recast it in the following form (recall that we have fixed α= 1\nandγ= 0at the beginning of the proof):\n(27) ∂tu+u·∇u+1\nρ∇Π +1\nρu= 0.\nBy transport estimates, together with commutator and tame es timates, we can bound, for any\nt∈[0,T∗[, the norm of the velocity field as follows:\n/ba∇dblu(t)/ba∇dblBsp,r/lessorsimilar/ba∇dblu0/ba∇dblBsp,r+/integraldisplayt\n0/parenleftBigg\n/ba∇dbl∇u/ba∇dblL∞/ba∇dblu/ba∇dblBsp,r+1\nρ∗/ba∇dblu/ba∇dblBsp,r+1\nρ∗/ba∇dbl∇Π/ba∇dblBsp,r(28)\n+/ba∇dblu/ba∇dblL∞/ba∇dbl∇ρ/ba∇dblBs−1\np,r+/ba∇dbl∇Π/ba∇dblL∞/ba∇dbl∇ρ/ba∇dblBs−1\np,r/parenrightBigg\ndt.\nFor notational convenience, let us set\nN(t) :=/ba∇dbl∇ρ(t)/ba∇dblBs−1\np,r+/ba∇dblu(t)/ba∇dblBsp,r.\nIn addition, we notice that the Gagliardo-Nirenberg type in equalities of Lemma 3.4 imply\n(29) /ba∇dblu/ba∇dblL∞/lessorsimilar/ba∇dblu/ba∇dblα1\nL2/ba∇dbl∇u/ba∇dbl1−α1\nL∞ and /ba∇dbl∇Π/ba∇dblL∞/lessorsimilar/ba∇dbl∇Π/ba∇dblα2\nL2/ba∇dbl∆Π/ba∇dbl1−α2\nLp0,\nwhich hold true for two suitable exponents α1,2∈]0,1[, as soon as p0> d. If the index pfrom\nthe statement verifies p > d , we simply take p0=p, if not we choose a different p0∈]d,+∞[.\nAfter an application of the Young inequality and summing up t he resulting expression with (25)\n(or with (26) in case p= +∞), we find, for any t∈[0,T∗[, the estimate\nN(t)/lessorsimilarN(0)+/integraldisplayt\n0/parenleftBig/parenleftbig\n1+/ba∇dbl∇ρ/ba∇dblL∞+/ba∇dbl∇u/ba∇dblL∞+/ba∇dbl∇Π/ba∇dblL2+/ba∇dbl∆Π/ba∇dblLp0/parenrightbig\nN(τ)+/ba∇dbl∇Π/ba∇dblBsp,r/parenrightBig\ndτ,\nfor an implicit multiplicative constant which depends also onρ∗,ρ∗and/ba∇dblu0/ba∇dblL2. Observe that,\nunder our assumptions, one always has Bs−1\np,r֒→Lp∩L∞. Then, from equation (24) and classical\ntransport estimates, one deduces that, for all q∈[p,+∞]and allt∈[0,T∗[, one has\n/ba∇dbl∇ρ(t)/ba∇dblLq/lessorsimilar/ba∇dbl∇ρ0/ba∇dblLqexp/parenleftbigg/integraldisplayt\n0/ba∇dbl∇u(τ)/ba∇dblL∞dτ/parenrightbigg\n,\nwhich is of course finite if condition (22) is satisfied:\n(30) sup\nt∈[0,T∗[/ba∇dbl∇ρ(t)/ba∇dblLp∩L∞/lessorequalslantC.\n17In addition, by computing the divergence of the momentum equ ation (27), we discover that\nthe pressure term Πsatisfies the elliptic equation\n(31) −div/parenleftbigg1\nρ∇Π/parenrightbigg\n= div(F), withF:=u·∇u+1\nρu.\nAn application of Lemma 3.11 thus yields\n(32)1\nρ∗/ba∇dbl∇Π/ba∇dblL2/lessorequalslant/vextenddouble/vextenddouble/vextenddouble/vextenddoubleu·∇u+1\nρu/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilar/ba∇dblu0/ba∇dblL2/ba∇dbl∇u/ba∇dblL∞+1\nρ∗/ba∇dblu0/ba∇dblL2,\nwhere we have also used the inequalities in (19) and (21).\nInserting (30) and (32) into the previous estimate for N(t), we infer that, if condition (22)\nholds, for any t∈[0,T∗[one has\n(33) N(t)/lessorsimilarN(0) +/integraldisplayt\n0/parenleftBig/parenleftbig\n1+/ba∇dbl∇u/ba∇dblL∞+/ba∇dbl∆Π/ba∇dblLp0/parenrightbig\nN(τ) +/ba∇dbl∇Π/ba∇dblBsp,r/parenrightBig\ndτ,\nfor a new multiplicative constant, also depending on /ba∇dbl∇ρ0/ba∇dblL∞.\nNext, let us bound the Lp0norm of ∆Π. For this, we develop the derivatives in (31) to find\nan equation for ∆Π: we get\n−∆Π =−∇logρ·∇Π +ρ∇u:∇u−u·∇logρ.\nThus, we deduce\n/ba∇dbl∆Π/ba∇dblLp0/lessorequalslant1\nρ∗/ba∇dbl∇ρ/ba∇dblLp0/ba∇dbl∇Π/ba∇dblL∞+ρ∗/ba∇dbl∇u/ba∇dblLp0/ba∇dbl∇u/ba∇dblL∞+/ba∇dblu/ba∇dblL∞/ba∇dbl∇ρ/ba∇dblLp0.\nRecall that, with our choice of p0, we always have p/lessorequalslantp0/lessorequalslant+∞. Thus, we can use (30) to see\nthat, under assumption (22), the Lp0norm of ∇ρis finite over [0,T∗[. In addition, we can use\nthe interpolation inequalities (29) again together with th e Young inequality to get, pointwise in\nthe interval [0,T∗[, the bound\n/ba∇dbl∆Π/ba∇dblLp0/lessorsimilar1 +/ba∇dbl∇u/ba∇dblL∞+/ba∇dbl∇u/ba∇dblLp0/ba∇dbl∇u/ba∇dblL∞.\nAt this point, by cutting ∇uinto low and high frequencies, we see that\n/ba∇dbl∇u/ba∇dblLp0/lessorsimilar/ba∇dblu/ba∇dblL2+/ba∇dblΩ/ba∇dblLp0/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblΩ/ba∇dblLp0,\nwhereΩis the vorticity matrix and of course, the L2norm ofumay be dispensed of in the case\np/lessorequalslantp0<+∞(by Calderón-Zygmund theory). The previous bound implies t hat\n(34) /ba∇dbl∆Π/ba∇dblLp0/lessorsimilar1 +/ba∇dbl∇u/ba∇dblL∞+/ba∇dbl∇u/ba∇dblL∞/ba∇dblΩ/ba∇dblLp0.\nIn turn, we have moved the problem of bounding /ba∇dbl∆Π/ba∇dblLp0to the one of bounding Ωin the\nsame norm. The advantage is that, from (27), we can compute an equation for the vorticity Ω,\nwhich reads\n∂tΩ +u·∇Ω + Ω·Du+∇u·Ω +∇/parenleftbigg1\nρ/parenrightbigg\n∧∇Π +1\nρΩ +∇/parenleftbigg1\nρ/parenrightbigg\n∧u= 0,\nwhere, given two vectors v,w∈Rd, we have defined v∧wto be the skew-symmetric matrix with\ncomponents/parenleftbig\nv∧w/parenrightbig\njk=vkwj−vjwk.\n18Notice that, in the specific cases d= 2andd= 3, the equation for Ωactually simplifies; however,\nfor the sake of generality, let us work in any space dimension d/greaterorequalslant2. Applying transport estimates\ninLp0to the previous equation, we find, for any time t∈[0,T∗[, the inequality\n/ba∇dblΩ(t)/ba∇dblLp0/lessorsimilar/ba∇dblΩ0/ba∇dblLp0+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∇u/ba∇dblL∞/ba∇dblΩ/ba∇dblLp0+/ba∇dbl∇ρ/ba∇dblLp0/ba∇dbl∇Π/ba∇dblL∞+/ba∇dbl∇ρ/ba∇dblLp0/ba∇dblu/ba∇dblL∞/parenrightBig\ndτ\n/lessorsimilar/ba∇dblΩ0/ba∇dblLp0+/integraldisplayt\n0/parenleftBig/parenleftbig\n1+/ba∇dbl∇ρ/ba∇dblLp0/parenrightbig\n/ba∇dbl∇u/ba∇dblL∞/ba∇dblΩ/ba∇dblLp0+/ba∇dbl∇ρ/ba∇dblLp0/parenleftbig\n1+/ba∇dbl∇u/ba∇dblL∞/parenrightbig/parenrightBig\ndτ,\nwhere, for passing from the first inequality to the second one , we have used the interpolation\ninequalities (29) combined with the Young inequality, and e stimates (32) and (34) for the pressure.\nThe implicit multiplicative constant depends on ρ∗,ρ∗and/ba∇dblu0/ba∇dblL2, of course. Remark again that,\nowing to our assumptions, one has Bs−1\np,r֒→Lp∩L∞andp/lessorequalslantp0/lessorequalslant+∞, so theLp0norm of\nthe initial datum is finite. We also notice that, in the previo us estimate, we could neglect the\ncontribution coming from the term1\nρΩ, because it has the right sign.\nAt this point, we remark that, under condition (22), estimat e (30) holds true, hence the Lp0\nnorm of∇ρremains bounded on [0,T∗[. Thus, an application of the Grönwall inequality yields\n∀t∈[0,T∗[,/ba∇dblΩ(t)/ba∇dblLp0/lessorsimilar/parenleftbigg\n/ba∇dblΩ0/ba∇dblLp0+/integraldisplayt\n0/parenleftbig\n1+/ba∇dbl∇u/ba∇dblL∞/parenrightbig\ndτ/parenrightbigg\nexp/parenleftbigg\nC/integraldisplayt\n0/ba∇dbl∇u/ba∇dblL∞dτ/parenrightbigg\n,\nfor a suitable positive constant C >0. Therefore, under condition (22), we find\n(35) sup\nt∈[0,T∗[/ba∇dblΩ(t)/ba∇dblLp0/lessorequalslantC.\nNow, thanks to (34), from (22) and (35), we deduce that\n(36)/integraldisplayT∗\n0/ba∇dbl∆Π(t)/ba∇dblLp0dt <+∞.\nAs a last step of the argument, we need to bound the pressure te rm∇Πin the Besov norm\nBs\np,r, which correspond to the last term appearing in (33). The sta rting point is the estimate of\nLemma 3.12, which reads\n/ba∇dbl∇Π/ba∇dblBsp,r/lessorsimilar/parenleftbig\n1+/ba∇dbl∇ρ/ba∇dblη\nL∞/parenrightbig\n/ba∇dblF/ba∇dblL2+/ba∇dbl∇ρ/ba∇dblBs−1\np,r/ba∇dbl∇Π/ba∇dblL∞+/ba∇dblρdiv(F)/ba∇dblBs−1\np,r,\nwhereFis defined in (31). By making use of (30), (32) and the interpola tion inequality (29), we\nobtain the bound\n(37)/ba∇dbl∇Π/ba∇dblBsp,r/lessorsimilar1 +/ba∇dbl∇u/ba∇dblL∞+ (1+/ba∇dbl∇u/ba∇dblL∞+/ba∇dbl∆Π/ba∇dblLp0)N+/ba∇dblρdiv(F)/ba∇dblBs−1\np,r.\nWe have now to find a control on the last term on the right-hand s ide of the previous inequality.\nIn order to avoid the appearing of a dangerous /ba∇dbl∇u/ba∇dbl2\nL∞term, let us write\nρdiv(F) =ρ∇u:∇u+ρ∇/parenleftbigg1\nρ/parenrightbigg\n��u=∇/parenleftbig\nρu/parenrightbig\n:∇u−/parenleftbig\nu·∇u/parenrightbig\n·∇ρ−u·∇logρ.\nHence, by a repeated use of the tame estimates, when s>1we gather\n/ba∇dblρdiv(F)/ba∇dblBs−1\np,r/lessorsimilar/ba∇dbl∇u/ba∇dblL∞/vextenddouble/vextenddouble∇/parenleftbig\nρu/parenrightbig/vextenddouble/vextenddouble\nBs−1\np,r+/ba∇dbl∇u/ba∇dblBs−1\np,r/vextenddouble/vextenddouble∇/parenleftbig\nρu/parenrightbig/vextenddouble/vextenddouble\nL∞\n+/ba∇dblu·∇u/ba∇dblBs−1\np,r/ba∇dbl∇ρ/ba∇dblL∞+/ba∇dblu·∇u/ba∇dblL∞/ba∇dbl∇ρ/ba∇dblBs−1\np,r\n+/ba∇dblu/ba∇dblL∞/ba∇dbl∇logρ/ba∇dblBs−1\np,r+/ba∇dblu/ba∇dblBs−1\np,r/ba∇dbl∇logρ/ba∇dblL∞.\n19Observe that, by using Leibniz rule and Bony’s paraproduct de composition, we can bound\n/vextenddouble/vextenddouble∇/parenleftbig\nρu/parenrightbig/vextenddouble/vextenddouble\nBs−1\np,r/lessorequalslant/ba∇dblu∇ρ/ba∇dblBs−1\np,r+/ba∇dblρ∇u/ba∇dblBs−1\np,r\n/lessorsimilar/ba∇dbl∇ρ/ba∇dblBs−1\np,r/ba∇dblu/ba∇dblL∞+/ba∇dblu/ba∇dblBs−1\np,r/ba∇dbl∇ρ/ba∇dblL∞+/ba∇dbl∇u/ba∇dblBs−1\np,r+/ba∇dblT∇u(ρ−1)/ba∇dblBs−1\np,r,\nwhere, owing to Proposition 3.5, we can bound\n/ba∇dblT∇u(ρ−1)/ba∇dblBs−1\np,r/lessorsimilar/ba∇dblu/ba∇dblL∞/ba∇dbl∇ρ/ba∇dblBs−1\np,r.\nIn particular, this argument shows that we have indeed avoid ed the appearing of the term /ba∇dbl∇u/ba∇dbl2\nL∞\nin the continuation criterion.\nIfs= 1(and thenr= 1as well), instead, the argument for bounding ρdiv(F)is slightly more\ninvolved, as B0\n∞,1isnotan algebra. However, we can use the fact that ρanduactually belong to\nB1\n∞,1and thatB1/2\n∞,1is still an interpolation space between B1\n∞,1andL2. Thus, arguing as done\nfor arriving to (14) and (15), we can bound\n/vextenddouble/vextenddouble∇/parenleftbig\nρu/parenrightbig\n:∇u/vextenddouble/vextenddouble\nB0\n∞,1/lessorsimilar/ba∇dblρu/ba∇dblB1\n∞,1/ba∇dbl∇u/ba∇dblL∞+/vextenddouble/vextenddouble∇/parenleftbig\nρu/parenrightbig/vextenddouble/vextenddouble\nL∞/ba∇dbl∇u/ba∇dblB0\n∞,1\n/ba∇dblu·∇logρ/ba∇dblB0\n∞,1/lessorsimilar/ba∇dblu/ba∇dblB1/2\n∞,1/ba∇dbl∇logρ/ba∇dblL∞+/ba∇dblu/ba∇dblL∞/ba∇dbl∇logρ/ba∇dblB0\n∞,1,\nwhere we have exploited the fact that ∇/parenleftbig\nρu/parenrightbig\n:∇u= div/parenleftbig\nρu· ∇u/parenrightbig\nand that ∇logρ·u=\ndiv (logρu). As for the last term, we have\n/parenleftbig\nu·∇u/parenrightbig\n·∇ρ=/summationdisplay\nj,kuj∂juk∂kρ,\nso we can argue as above to get\n/vextenddouble/vextenddouble/parenleftbig\nu·∇u/parenrightbig\n·∇ρ/vextenddouble/vextenddouble\nB0\n∞,1/lessorsimilar/ba∇dblu/ba∇dblB0\n∞,1/ba∇dbl∇u·∇ρ/ba∇dblL∞+/ba∇dblu/ba∇dblL∞/ba∇dbl∇u·∇ρ/ba∇dblB0\n∞,1,\nwhere, owing to the fact that, for any jfixed,/summationtext\nk∂juk∂kρ= div/parenleftbig\nρ∂ju/parenrightbig\n, one has\n/ba∇dbl∇u·∇ρ/ba∇dblB0\n∞,1/lessorsimilar/ba∇dbl∇u/ba∇dblB0\n∞,1/ba∇dbl∇ρ/ba∇dblL∞+/ba∇dbl∇u/ba∇dblL∞/ba∇dblρ/ba∇dblB1\n∞,1.\nIn any case, one arrive at the following bound for the Besov nor m ofρdiv(F):\n/ba∇dblρdiv(F)/ba∇dblBs−1\np,r/lessorsimilar/parenleftBig\n1+/ba∇dblu/ba∇dblL∞+/ba∇dbl∇ρ/ba∇dblL∞/parenrightBig\n/ba∇dbl∇u/ba∇dblL∞N\n+/ba∇dblu/ba∇dblL∞/ba∇dbl∇ρ/ba∇dblL∞N+/parenleftBig\n/ba∇dblu/ba∇dblL∞+/ba∇dbl∇ρ/ba∇dblL∞/parenrightBig\nN,\nwhere we have also used the paralinearisation result of Prop osition 3.7. In order to control the\n/ba∇dblu/ba∇dblL∞, we can apply the second interpolation inequality of Lemma 3 .4 to deduce the existence\nof aα3∈]0,1[such that\n/ba∇dblu/ba∇dblL∞/lessorsimilar/ba∇dblu/ba∇dblα3\nL2/ba∇dblΩ/ba∇dbl1−α3\nLp0,\nwherep0is the same index appearing above (in particular, p0=pifp>d, whilep<p0<+∞\notherwise). In particular, by estimates (21) and (35), we de duce that, under condition (22), we\nhavesup[0,T∗[/ba∇dblu/ba∇dblL∞<+∞. Combining this property with (30), from (37) we infer the in equality\n(38) /ba∇dbl∇Π/ba∇dblBsp,r/lessorsimilar1 +/ba∇dbl∇u/ba∇dblL∞+ (1+/ba∇dbl∇u/ba∇dblL∞+/ba∇dbl∆Π/ba∇dblLp0)N.\nAt this point, we can plug the previous bound (38) into (33) an d get, under condition (22),\nthe following inequality:\n∀t∈[0,T∗[, N(t)/lessorsimilarN(0) +/integraldisplayt\n0/parenleftBig\n1+/ba∇dbl∇u/ba∇dblL∞+/ba∇dbl∆Π/ba∇dblLp0/parenrightBig\nN(τ)dτ.\n20An application of Grönwall inequality and the use of (22) and (36) then imply that\nsup\nt∈[0,T∗[N(t)<+∞.\nThanks to the embedding Bs\np,r֒→W1,∞, in turn we deduce that also /ba∇dbl∇u/ba∇dblL∞is uniformly\nbounded in [0,T∗[, thus also /ba∇dbl∇Π/ba∇dblL2, owing to (32), and /ba∇dbl∆Π/ba∇dblLp0, thanks to (34). Coming back\nto (38), we finally gather that sup[0,T∗[/ba∇dbl∇Π/ba∇dblBsp,r<+��. Hence, we infer that (23) holds true,\nas sought.\nFrom (23), one can use a standard argument (which we omit here ) to complete the proof of\nthe continuation criterion.\n5 Global well-posedness in the case γ= 1\nIn this section, we carry out the proof of Theorems 2.1 and 2.6 in the case γ= 1. This case is\nsomehow simpler, inasmuch as, owing to assumption (4) of abs ence of vacuum, we can divide the\nmomentum equation by ρand write it in the following form:\n∂tu+u·∇u+1\nρ∇Π +αu= 0.\nThus, if we define the new velocity field /tildewideuand the new pressure function /tildewideΠas\n(39) /tildewideu:=eαtu and /tildewideΠ :=eαtΠ,\nthen system (1) can be recasted in the following form:\n(40)\n\n∂tρ+e−αt/tildewideu·∇ρ= 0\nρ∂t/tildewideu+ρe−αt/tildewideu·∇/tildewideu+∇/tildewideΠ = 0\ndiv/tildewideu= 0.\nOf course, systems (1) and (40) are equivalent in our functio nal framework.\nThroughout this section, we consider a solution/parenleftbig\nρ,u,∇Π/parenrightbig\nto system (1), related to an initial\ndatum/parenleftbig\nρ0,u0/parenrightbig\nsatisfying the assumptions fixed in Theorems 2.1 and 2.6 abov e. Let us call\nT=T(ρ0,u0)>0the lifespan of that solution, so that/parenleftbig\nρ,u,∇Π/parenrightbig\nis defined on [0,T[×Rd. The\ntransformation (39) produces a solution/parenleftbig\nρ,/tildewideu,∇/tildewideΠ/parenrightbig\nto system (40), related to the same initial\ndatum, defined on the same time interval.\nOur goal is to prove that T= +∞. Arguing by contradiction, in what follows we are going to\nassume that T <+∞instead.\nRecall that, by the mass equation, we immediately get the bou nd (19), namely\n(41) ∀t∈[0,T[, ρ ∗/lessorequalslantρ(t)/lessorequalslantρ∗.\nOn the other hand, an energy estimate performed on the “momen tum equation” of system (40)\nimplies that\n1\n2d\ndt/vextenddouble/vextenddouble√ρ/tildewideu/vextenddouble/vextenddouble2\nL2= 0,\nwhich implies, together with (4) and (41), that\n(42) ∀t∈[0,T[, /ba∇dbl/tildewideu(t)/ba∇dblL2/lessorsimilar/ba∇dblu0/ba∇dblL2 (i.e./ba∇dblu(t)/ba∇dblL2/lessorsimilare−αt/ba∇dblu0/ba∇dblL2).\nThus, we need to prove a global uniform (in time) bound also fo r the higher order regularity\nnorms of the solution/parenleftbig\nρ,/tildewideu,∇/tildewideΠ/parenrightbig\n. For this, we will exploit the continuation criterion of Lem ma\n214.2 in a fundamental way: we know that the Sobolev norms of/parenleftbig\nρ,/tildewideu,∇/tildewideΠ/parenrightbig\nremain bounded on\n[0,T[if and only if the integral condition (22) is verified.\nNow, cutting into low and high frequencies, it is standard to bound\n/ba∇dbl∇u/ba∇dblL∞/lessorsimilar/ba∇dblu/ba∇dblL2+/ba∇dblΩ/ba∇dblB0\n∞,1,\nwhere we recall that Ω :=Du− ∇udenotes the vorticity of the fluid. Recall that we have\nassumed, by contradiction, that the lifespan TverifiesT+∞. Hence, owing to (42), the failure\nof condition (22) implies that\n(43)/integraldisplayT\n0/ba∇dblΩ(t)/ba∇dblB0\n∞,1dt=/integraldisplayT\n0e−αt/vextenddouble/vextenddouble/vextenddouble/tildewideΩ(t)/vextenddouble/vextenddouble/vextenddouble\nB0\n∞,1dt= +∞,\nwhere we have denoted by /tildewideΩ :=D/tildewideu− ∇/tildewideuthe vorticity of the divergence-free velocity field /tildewideu.\nFrom now on, for the sake of clarity, we split our argument int o two parts: in Subsection\n5.1 we focus on the 2-D situation and prove global existence under a smallness as sumption on\nthe non-homogeneity only (namely, we prove Theorem 2.6), wh ile in Subsection 5.2 we treat the\ngeneral case of space dimension d/greaterorequalslant2and we prove Theorem 2.1.\n5.1 The case of planar flows: proof of Theorem 2.6\nTo begin with, we consider the case of two-dimensional flows d= 2. We aim at proving Theorem\n2.6, namely global well-posedness under a smallness condit ion relying only on the size of the initial\nnon-homogeneity ρ0−1.\nRecall that, in the case d= 2, the vorticities Ωand/tildewideΩ, related respectively to the divergence-\nfree velocity fields uand/tildewideu, can be identified with the scalar functions\nω:= curl(u) =∂1u2−∂2u1and /tildewideω:= curl(/tildewideu) =∂1/tildewideu2−∂2/tildewideu1.\nNow, we implement the following strategy: we perform B0\n∞,1estimates for the vorticity /tildewideωand\nshow that, under the assumptions of Theorem 2.6, relation (4 3) cannot hold, thus reaching the\nsought contradiction.\nThe advantage of the two-dimensional situation is that, as i s well-known, the equation for the\nvorticity simplifies, due to special cancellations which im ply the vanishing of the stretching term.\nMore precisely, if we start from the momentum equation in (40 ) and we compute an equation for\n/tildewideω, asd= 2we find\n(44) ∂t/tildewideω+e−αt/tildewideu·∇/tildewideω=−∇⊥/parenleftbigg1\nρ/parenrightbigg\n·∇/tildewideΠ,\nwhere, for any vector field v∈R2,v=/parenleftbig\nv1,v2/parenrightbig\n, we have denoted by v⊥:=/parenleftbig\n−v2,v1/parenrightbig\nits rotation\nof angleπ/2.\nAt this point, in order to bound /tildewideωinB0\n∞,1, we can apply the improved transport estimates\nfrom Theorem 3.14 to equation (44): we get, for any t∈[0,T[, the inequality\n(45)/ba∇dbl/tildewideω(t)/ba∇dblB0\n∞,1/lessorsimilar/parenleftBigg\n/ba∇dblω0/ba∇dblB0\n∞,1+/integraldisplayt\n0/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇⊥/parenleftbigg1\nρ/parenrightbigg\n·∇/tildewideΠ/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nB0\n∞,1dτ/parenrightBigg/parenleftbigg\n1 +/integraldisplayt\n0e−ατ/ba∇dbl∇/tildewideu/ba∇dblL∞dτ/parenrightbigg\n,\nwhere we have set ω0= curl(u0).\nThe bound of the non-linear term appearing in the right-hand side of the previous inequality\nmay appear complicated, as the space B0\n∞,1isnotalgebra. However, we can rewrite\n∇⊥/parenleftbigg1\nρ/parenrightbigg\n·∇/tildewideΠ =∇⊥/parenleftbigg1\nρ−1/parenrightbigg\n·∇/tildewideΠ = curl/parenleftbigg/parenleftbigg1\nρ−1/parenrightbigg\n∇/tildewideΠ/parenrightbigg\n22and argue as for (15). Using a careful paraproduct decomposi tion (see also Section 4 of [16]),\ntogether with paralinearisation estimate, one can actuall y prove that\n(46)/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇⊥/parenleftbigg1\nρ/parenrightbigg\n·∇/tildewideΠ/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nB0\n∞,1/lessorsimilar/ba∇dblρ−1/ba∇dblB1\n∞,1/vextenddouble/vextenddouble/vextenddouble∇/tildewideΠ/vextenddouble/vextenddouble/vextenddouble\nB0\n∞,1.\nOn the other hand, applying classical Besov estimates to the t ransport equation for ρin (40),\nwhich of course holds true also for the quantity ρ−1, we find\n(47) /ba∇dblρ(t)−1/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblρ0−1/ba∇dblB1\n∞,1exp/parenleftbigg\nC/integraldisplayt\n0e−ατ/ba∇dbl/tildewideu/ba∇dblB1\n∞,1dτ/parenrightbigg\n.\nIt remains us to bound the B0\n∞,1norm of∇/tildewideΠ. Actually, we are going to bound its B1\n∞,1norm,\nfor it is not clear to us how to take really advantage of this lo wer order requirement. Computing\nan equation for ∇/tildewideΠfrom the second equation in (40) yields\n(48) −div/parenleftbigg1\nρ∇/tildewideΠ/parenrightbigg\n= div/parenleftbig\ne−αt/tildewideu·∇/tildewideu/parenrightbig\n.\nBefore applying inequality (17) with F=e−αt/tildewideu·∇/tildewideu, let us notice that\n/ba∇dble−αt/tildewideu·∇/tildewideu/ba∇dblL2/lessorequalslante−αt/ba∇dbl/tildewideu/ba∇dblL2/ba∇dbl∇/tildewideu/ba∇dblL∞\n/ba∇dblρdiv(e−αt/tildewideu·∇/tildewideu)/ba∇dblB0\n∞,1/lessorsimilare−αt/ba∇dblρ/ba∇dblB1\n∞,1/ba∇dbl/tildewideu/ba∇dbl2\nB1\n∞,1,\nwhere we used div/parenleftbig\ne−αt/tildewideu·∇/tildewideu/parenrightbig\n=e−αt∇/tildewideu:∇/tildewideu, together with (14) and (15). These estimates\ntogether with (17) imply the bound\n/vextenddouble/vextenddouble/vextenddouble∇/tildewideΠ/vextenddouble/vextenddouble/vextenddouble\nB1\n∞,1/lessorsimilare−αt/parenleftbigg\n1 +/ba∇dblρ−1/ba∇dblη\nB1\n∞,1/parenrightbigg/parenleftBig\n/ba∇dbl/tildewideu/ba∇dblL2/ba∇dbl∇/tildewideu/ba∇dblL∞+/ba∇dbl/tildewideu/ba∇dbl2\nB1\n∞,1/parenrightBig\n. (49)\nFor notational convenience, let us set\n/tildewideU(t) :=/ba∇dbl/tildewideu(t)/ba∇dblL2+/ba∇dbl/tildewideω(t)/ba∇dblB0\n∞,1and /tildewideU0:=/ba∇dblu0/ba∇dblL2+/ba∇dblω0/ba∇dblB0\n∞,1.\nNotice that, for any t∈[0,T[, we have that /tildewideU(t)is equivalent to /ba∇dbl/tildewideu(t)/ba∇dblL2∩B1\n∞,1. Then, we can\nplug inequalities (47) and (49) into (46) and use the resulti ng expression in (45): thanks to the\nenergy inequality (42), we get, for any t∈[0,T[, the bound\n/tildewideU(t)/lessorsimilar/parenleftbigg\n/tildewideU0+R0/integraldisplayt\n0eC0/integraltextτ\n0e−ασ/tildewideU(σ)dσe−ατ/parenleftBig\n/tildewideU(τ)/parenrightBig2\ndτ/parenrightbigg/parenleftbigg\n1 +/integraldisplayt\n0e−ατ/tildewideU(τ)dτ/parenrightbigg\n, (50)\nwhereC0>0in the exponential term is a suitable constant and where we ha ve defined\nR0:=/ba∇dblρ0−1/ba∇dblB1\n∞,1/parenleftbigg\n1 +/ba∇dblρ0−1/ba∇dblη\nB1\n∞,1/parenrightbigg\n.\nAt this point, we notice that using a classical procedure (se e for instance the definition of the\ntimeT1in the next subsection) would lead to a smallness requiremen tbothon the size of the\ninitial non-homogeneity and on /tildewideU0/α. In order to avoid the appearing of the latter condition, a\nmore subtle argument is required, which we are going to prese nt now.\nLet us define the time T0>0as\nT0:= sup/braceleftBigg\nt∈[0,T[/vextendsingle/vextendsingle/vextendsingleR0/integraldisplayt\n0eC0/integraltextτ\n0e−ασ/tildewideU(σ)dσe−ατ/parenleftBig\n/tildewideU(τ)/parenrightBig2\ndτ/lessorequalslant4/tildewideU0/bracerightBigg\n.\n23Then, from the previous estimate (50), we get that\n(51) ∀t∈[0,T0],/tildewideU(t)/lessorequalslantC1/tildewideU0/parenleftbigg\n1 +/integraldisplayt\n0e−ατ/tildewideU(τ)dτ/parenrightbigg\n,\nwhere the constant C1only depends on ρ∗andρ∗. Thus, if we define the function\nf(t) := 1 +/integraldisplayt\n0e−ατ/tildewideU(τ)dτ,\nin turn we deduce that\n∀t∈[0,T0], f′(t)/lessorequalslantC1/tildewideU0e−αtf(t).\nAn application of the Grönwall inequality and estimate (51) then imply\n(52) ∀t∈[0,T0], f (t)/lessorequalslanteC1\nα/tildewideU0and/tildewideU(t)/lessorequalslantC1/tildewideU0eC1\nα/tildewideU0.\nUsing now the definition of the time T0, we see that\n4/tildewideU0=R0/integraldisplayT0\n0eC0/integraltextt\n0e−ατ/tildewideU(τ)dτe−αt/parenleftBig\n/tildewideU(t)/parenrightBig2\ndt=R0/integraldisplayT0\n0eC0/parenleftbig\nf(t)−1/parenrightbig\nf′(t)/tildewideU(t) dt.\nMaking use of the bounds from (52), we get\n4/tildewideU0/lessorequalslantR0eC0exp/parenleftbigC1\nα/tildewideU0/parenrightbig\nC1/tildewideU0eC1\nα/tildewideU0/parenleftbig\nf(T0)−1/parenrightbig\n/lessorequalslantR0C1/tildewideU0e2C1\nα/tildewideU0eC0exp/parenleftbigC1\nα/tildewideU0/parenrightbig\n.\nThus, after defining Cmax:= max/braceleftbig\nC0,C1/bracerightbig\n, ifR0andU0are such that the inequality\nR0Cmax/tildewideU0e2Cmax\nα/tildewideU0eCmaxexp/parenleftbig\nCmax\nα/tildewideU0/parenrightbig\n<4\nis satisfied, we see that one must have T0≡T, which in turn implies, thanks to (52), that\nsup\nt∈[0,T[/tildewideU(t)≈sup\nt∈[0,T[/ba∇dbl/tildewideu/ba∇dblL2∩B1\n∞,1<+∞.\nThis bound obviously contradicts (43), thus implying that T= +∞,i.e.the constructed solution\nis global.\nObserve that, as a byproduct of our argument, we get also that\n∀t∈R+, /ba∇dblu(t)/ba∇dblL2∩B1\n∞,1/lessorsimilare−αt/ba∇dblu0/ba∇dblL2∩B1\n∞,1eC1\nα/ba∇dblu0/ba∇dblL2∩B1\n∞,1,\nthanks to inequalities (42) and (52), and that\n∀t∈R+, ρ ∗/lessorequalslantρ(t)/lessorequalslantρ∗and /ba∇dblρ(t)−1/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblρ0−1/ba∇dblB1\n∞,1,\nowing to (47). Going back to (49), we also see that\n∀t∈R+, /ba∇dbl∇Π(t)/ba∇dblL2∩B1\n∞,1/lessorsimilare−2αt.\nWith those bounds at our disposal, by working on system (40), we can propagate the Besov\nregularity of ρ,/tildewideuand∇/tildewideΠ, globally in time, and deduce the sought regularity propert ies.\nThis observation concludes the proof of Theorem 2.6, provid ed we show similar decay estimates\nalso for the higher order Besov norms of the solution. We postp one the proof of this latter fact\nto Subsection 5.3.\n245.2 The general case: proof of Theorem 2.1\nLet us consider now the case of any space dimension d/greaterorequalslant2and prove global well-posedness under\na smallness condition on the norm of the initial datum. Now, i n order to avoid the appearing\nof theL2norm ofu0in the smallness condition, it is more convenient to work wit h the velocity\nformulation of the equations, rather than with the vorticit y.\nSimiarly to what done in the previous section, by arguing by c ontradiction, we are going to\nshow that, under the smallness assumption of Theorem 2.1, th e blow-up (43) cannot happen. To\nsee this, we perform transport estimates directly on the equ ation for the velocity field, namely\n∂t/tildewideu+e−αt/tildewideu·∇/tildewideu+1\nρ∇/tildewideΠ = 0.\nBy mimicking the proof of Theorem 3.13, from the previous equa tion it is plain to get the following\nbound, for any t∈[0,T[:\n/ba∇dbl/tildewideu(t)/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblu0/ba∇dblB1\n∞,1+/integraldisplayt\n0e−ατ/parenleftbigg\n/ba∇dblρ/ba∇dblB1\n∞,1/vextenddouble/vextenddouble/vextenddouble∇/tildewideΠ/vextenddouble/vextenddouble/vextenddouble\nB1\n∞,1+/ba∇dbl/tildewideu/ba∇dbl2\nB1\n∞,1/parenrightbigg\ndτ,\nfor an implicit multiplicative constant also depending on ρ∗andρ∗appearing in (4).\nIn order to estimate the pressure term, we can still use (49), up to take a higher η=η(d),\ndepending only on the space dimension d. After setting R1:= 1 +/ba∇dblρ0−1/ba∇dblη\nB1\n∞,1, we combine\nestimate (49) with (47) to get\n/vextenddouble/vextenddouble/vextenddouble∇/tildewideΠ/vextenddouble/vextenddouble/vextenddouble\nB1\n∞,1/lessorsimilare−ατR1/parenleftBig\n/ba∇dbl/tildewideu/ba∇dblL2+/ba∇dbl/tildewideu/ba∇dblB1\n∞,1/parenrightBig\nexp/parenleftbigg\nηC/integraldisplayτ\n0e−ασ/ba∇dbl/tildewideu/ba∇dblB1\n∞,1dσ/parenrightbigg\n,\nUp to change ηintoη+1in the definition of R1, making use of (42) we then obtain\n/ba∇dbl/tildewideu(t)/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblu0/ba∇dblB1\n∞,1+R1/integraldisplayt\n0e−ατeC/integraltextτ\n0e−ασ/ba∇dbl/tildewideu/ba∇dblB1\n∞,1dσ/parenleftBig\n/ba∇dblu0/ba∇dblL2+/ba∇dbl/tildewideu/ba∇dblB1\n∞,1/parenrightBig\n/ba∇dbl/tildewideu/ba∇dblB1\n∞,1dτ,\nand an application of the Grönwall lemma finally yields, for a nyt∈[0,T[, the bound\n/ba∇dbl/tildewideu(t)/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblu0/ba∇dblB1\n∞,1exp/parenleftbigg\nR1/integraldisplayt\n0e−ατeC/integraltextτ\n0e−ασ/ba∇dbl/tildewideu/ba∇dblB1\n∞,1dσ/parenleftBig\n/ba∇dblu0/ba∇dblL2+/ba∇dbl/tildewideu/ba∇dblB1\n∞,1/parenrightBig\ndτ/parenrightbigg\n. (53)\nWith this estimate at hand, the rest of the argument follows t he main lines of the one employed\nin Subsection 5.1 above. Let us define the time T1as\nT1:= sup/braceleftBigg\nt∈[0,T[/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0e−ατ/ba∇dbl/tildewideu(t)/ba∇dblB1\n∞,1dτ/lessorequalslant2/bracerightBigg\n.\nThen, from (53) we deduce, for t∈[0,T1], the inequality\n/ba∇dbl/tildewideu(t)/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblu0/ba∇dblB1\n∞,1exp/parenleftbigg\nR1e2C/parenleftbigg1\nα/ba∇dblu0/ba∇dblL2+ 1/parenrightbigg/parenrightbigg\n. (54)\nThis in particular implies that, for all t∈[0,T1], it holds\n/integraldisplayt\n0e−ατ/ba∇dbl/tildewideu(τ)/ba∇dblB1\n∞,1dτ/lessorsimilar1\nα/ba∇dblu0/ba∇dblB1\n∞,1exp/parenleftbigg\nR1e2C/parenleftbigg1\nα/ba∇dblu0/ba∇dblL2+ 1/parenrightbigg/parenrightbigg\n.\nNow, by definition of T1, we must have\n2/lessorequalslant1\nα/ba∇dblu0/ba∇dblB1\n∞,1exp/parenleftbigg\nR1e2C/parenleftbigg1\nα/ba∇dblu0/ba∇dblL2+ 1/parenrightbigg/parenrightbigg\n,\n25which, however, cannot be verified if the left-hand side is sm all enough. This implies that T1=T,\nhence\nsup\nt∈[0,T[/ba∇dbl/tildewideu(t)/ba∇dblB1\n∞,1<+∞,\nowing to (54), but this property is clearly in contradiction with (43). Therefore, we infer that the\nconstructed solution must be globally defined in time.\nIn addition, analogously to what done in Subsection 5.1, we n otice that our argument also\nimplies the following bounds: first of all,\n∀t∈R+, /ba∇dblu(t)/ba∇dblL2∩B1\n∞,1/lessorsimilare−αt/ba∇dblu0/ba∇dblL2∩B1\n∞,1,\nowing to inequalities (42) and (54), and secondly\n∀t∈R+, ρ ∗/lessorequalslantρ(t)/lessorequalslantρ∗and /ba∇dblρ(t)−1/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblρ0−1/ba∇dblB1\n∞,1,\nowing to (47). Similarly, we obtain the time-decay of the Beso v norm of ∇Π. With those bounds\nat hand, by working on system (40), we can propagate the (high er order, in the case s>1) Besov\nnorms ofρ,/tildewideuand∇/tildewideΠ, globally in time, getting the sought regularity propertie s.\nRemark that, in order to complete the proof to Theorem 2.1, it remains us to prove the time\ndecays of the higher order Besov norms of the solution. As it wa s the case in Subsection 5.1, this\nfact will be shown in Subsection 5.3 below.\n5.3 Decay estimates for higher order Besov norms\nIn this subsection, we prove that not only the B1\n∞,1norm ofuand∇Πdecay exponentially in\ntime, but also the higher order Besov norms Bs\np,r. This will conclude the proof of both Theorems\n2.1 and 2.6.\nLemma 5.1. Letd/greaterorequalslant2,γ= 1andα>0fixed. Take indices (s,p,r)∈R×[1,+∞]×[1,+∞]such\nthatp/greaterorequalslant2and on of the two conditions in (5)is verified. Take an initial datum/parenleftbig\nρ0,u0/parenrightbig\nsuch that\nconditions (3)-(4)are satisfied, for two suitable constants 0<ρ∗/lessorequalslantρ∗. Assume that u0∈L2∩Bs\np,r\nand∇ρ0∈Bs−1\np,rand that, in addition, there exists a unique global in time so lution/parenleftbig\nρ,u,∇Π/parenrightbig\nto system (1)associated to that initial datum, verifying the regularity properties (i)-(ii)-(iii) of\nTheorem 2.1. Finally, assume that there exists a constant C >0such that\n(55) ∀t/greaterorequalslant0, /ba∇dbl∇ρ(t)/ba∇dblB0\n∞,1+eαt/ba∇dblu(t)/ba∇dblL2∩B1\n∞,1+e2αt/ba∇dbl∇Π(t)/ba∇dblL2∩B1\n∞,1/lessorequalslantC.\nThen, for a possibly different constant /tildewideC >0, one has\n∀t/greaterorequalslant0, /ba∇dbl∇ρ(t)/ba∇dblBs−1\np,r+eαt/ba∇dblu(t)/ba∇dblL2∩Bsp,r+e2αt/ba∇dbl∇Π(t)/ba∇dblL2∩Bsp,r/lessorequalslant/tildewideC.\nProof. Without loss of generality, we can assume to have either 2/lessorequalslantp <+∞, orp= +∞and\ns>1(otherwise, the decay estimates have already been proved in the previous Sections 5.2 and\n5.1). In order to proof the previous statement, we are going t o repeat the main steps of the\nproof of the continuation criterion, trying to making each t ime a lower order term appear, which\npossesses the right decay property by assumption.\nWe start by estimating the density gradient: by using the not ation introduced in (39), from\ninequality (25) we gather\n/ba∇dbl∇ρ(t)/ba∇dblBs−1\np,r/lessorsimilar/ba∇dbl∇ρ0/ba∇dblBs−1\np,r+/integraldisplayt\n0e−ατ/parenleftBig\n/ba∇dbl∇/tildewideu/ba∇dblL∞/ba∇dbl∇ρ/ba∇dblBs−1\np,r+/ba∇dbl∇ρ/ba∇dblL∞/ba∇dbl∇/tildewideu/ba∇dblBs−1\np,r/parenrightBig\ndt\nfor any time t/greaterorequalslant0. Notice that, as s>1always, the space Bs−1\np,ris an algebra.\n26By transport estimates, together with commutator and tame es timates, we can bound the\nnorm of the velocity field, for any time t/greaterorequalslant0, as follows:\n/ba∇dbl/tildewideu(t)/ba∇dblBsp,r/lessorsimilar/ba∇dbl/tildewideu0/ba∇dblBsp,r+/integraldisplayt\n0e−ατ/parenleftBig\n/ba∇dbl∇/tildewideu/ba∇dblL∞/ba∇dbl/tildewideu/ba∇dblBsp,r\n+eατ1\nρ∗/vextenddouble/vextenddouble/vextenddouble∇/tildewideΠ/vextenddouble/vextenddouble/vextenddouble\nBsp,r+eατ/vextenddouble/vextenddouble/vextenddouble∇/tildewideΠ/vextenddouble/vextenddouble/vextenddouble\nL∞/ba∇dbl∇ρ/ba∇dblBs−1\np,r/parenrightBig\ndt.\nAt this point, we make use of assumption (55) in the previous e stimates: taking advantage\nalso of the inequality of Lemma 3.12 applied to the pressure e quation (48), we obtain\n/ba∇dbl∇ρ(t)/ba∇dblBs−1\np,r/lessorsimilar/ba∇dbl∇ρ0/ba∇dblBs−1\np,r+/integraldisplayt\n0e−ατ/parenleftBig\n/ba∇dbl∇ρ/ba∇dblBs−1\np,r+/ba∇dbl/tildewideu/ba∇dblBsp,r/parenrightBig\ndt\n/ba∇dbl/tildewideu(t)/ba∇dblBsp,r/lessorsimilar/ba∇dbl/tildewideu0/ba∇dblBsp,r+/integraldisplayt\n0e−ατ/parenleftBig\n/ba∇dbl/tildewideu/ba∇dblBsp,r+/ba∇dbl∇ρ/ba∇dblBs−1\np,r/parenrightBig\ndt.\nThen, the Grönwall lemma implies the desired estimate: for a llt/greaterorequalslant0, one has\n/ba∇dbl∇ρ(t)/ba∇dblBs−1\np,r+/ba∇dbl/tildewideu(t)/ba∇dblBsp,r/lessorsimilar/ba∇dbl∇ρ0/ba∇dblBs−1\np,r+/ba∇dbl/tildewideu0/ba∇dblBsp,r.\nFinally, estimate (17) applied to (48) implies the exponent ial decay also for the pressure term.\nThe lemma is thus completely proved.\n6 The case γ= 0: proof of Theorem 2.5\nThe proof of the global existence in the case γ= 0is more delicate. As a matter of fact, when\nγ= 0, the momentum equation reads\n(56) ρ∂tu+ρu·∇u+∇Π +αu= 0.\nThus, even though the term D0\nα(ρ,u) =αuis still a damping term, it creates complications when\nlooking for higher order estimates for the solution. As a mat ter of fact, for obtaining higher order\nestimates on u, we divide the momentum equation by ρ, but, when doing so, D0\nα(ρ,u)presents a\nvariable coefficient1\nρin front, keep in mind equation (27). As a consequence, it is n ot clear which\nis the optimal decay rate for the solution of (56).\nRecall that, given an initial datum/parenleftbig\nρ0,u0/parenrightbig\nverifying the assumptions of Theorem 2.5, we are\nable to construct a local solution/parenleftbig\nρ,u,∇Π/parenrightbig\ndefined on some maximal time interval [0,T[. As\ndone in Section 5, our goal is to prove that T= +∞by resorting to the continuation criterion of\nLemma 4.2. By contradiction, in what follows we assume that T <+∞instead.\n6.1 Preliminaries and reformulation\nTo begin with, we observe that the bound (19) still holds true forρ, owing to the transport\nstructure of the mass conservation equation:\n(57) ∀t∈[0,T[, ρ ∗/lessorequalslantρ(t)/lessorequalslantρ∗.\nNext, using that D0\nα(ρ,u) =αuin (20), we deduce\n1\n2d\ndt/integraldisplay\nRdρ|u|2dx+α/integraldisplay\nRd|u|2dx= 0.\nThus, after defining\nα∗:=α\nρ∗,\n27we easily find\n1\n2d\ndt/ba∇dbl√ρu/ba∇dbl2\nL2+α∗/ba∇dbl√ρu/ba∇dbl2\nL2/lessorequalslant0,\nwhich in turn implies, thanks also to the use of (57), the ineq uality\n(58) ∀t∈[0,T[, /ba∇dblu(t)/ba∇dblL2/lessorsimilare−α∗t/ba∇dblu0/ba∇dblL2.\nNext, we turn our attention to the higher order estimates. Th e two key ideas in order to bypass\nthe difficulty mentioned above are the following ones. First o f all, we decompose the damping\ntermD0\nα(ρ,u)into two parts: one of them will give an exponential decay (ev en though with a\nnon-optimal rate), while the other one will be a positive ter m, yielding a contribution with the\nright sign in the estimates. Once implemented in the Besov fun ctional framework, this procedure\ncreates remainder terms, which are however of lower order: t he second main ingredient of the\nproof is then to use an interpolation argument to make a L2norm ofuappear, for which one can\nuse the (optimal) decay estimate (58).\nLet us make explicit what we have just announced. Given any\n0< β/lessorequalslantα∗:=α\nρ∗,\nwe can write the momentum equation (56) as\n∂tu+u·∇u+1\nρ∇Π +/parenleftbiggα\nρ−β/parenrightbigg\nu+βu= 0.\nIn other terms, if we define\nuβ:=eβtu and Πβ:=eβtΠ,\nwe get the equation\n(59) ∂tuβ+e−βtuβ·∇uβ+1\nρ∇Πβ+/parenleftbiggα\nρ−β/parenrightbigg\nuβ= 0.\nAnalogously, the mass conservation equation can be recaste d as\n(60) ∂tρ+e−βtuβ·∇ρ= 0.\nNow, similarly to what done in Section 5, we observe that the f ailure of condition (22) of\nLemma 4.2 implies that\n(61)/integraldisplayT\n0/ba∇dblu(t)/ba∇dblB1\n∞,1dt=/integraldisplayT\n0e−βt/ba∇dbluβ(t)/ba∇dblB1\n∞,1dt= +∞.\nThus, we are going to bound the Besov norm of uβand show that, under the smallness condition\nof Theorem 2.5, this cannot happen.\n6.2 Estimates for the Besov norm of the velocity field\nIn this part, we are going to bound the Besov norm of uβ, where we recall that the value of\nβ∈]0,α∗]has not been fixed yet. We have to proceed carefully, in order t o exploit (as anticipated\nabove) the positive sign of the last term appearing in the lef t-hand side of equation (59).\nApplying the operator ∆jof a Littlewood-Paley decomposition (see Section 3) to rela tion (59),\nwe find\n(62) ∂t∆juβ+e−βtuβ·∇∆juβ+/parenleftbiggα\nρ−β/parenrightbigg\n∆juβ=−∆jPβ+C1\nj+C2\nj,\n28where we have defined the pressure-density term\nPβ:=1\nρ∇Πβ\nand the two commutator terms\nC1\nj:=e−βt[uβ·∇,∆j]uβ and C2\nj:=α/bracketleftbigg1\nρ,∆j/bracketrightbigg\nuβ.\nNow, we perform a Lqestimate, for 2/lessorequalslantq<+∞, on the transport-type equation (62): we get\n/ba∇dbl∆juβ(t)/ba∇dblLq+/integraldisplayt\n0/integraldisplay\nRd/parenleftbiggα\nρ−β/parenrightbigg\n|∆juβ|qdxdτ\n/lessorsimilar/ba∇dbl∆juβ(0)/ba∇dblLq+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆jPβ/ba∇dblLq+/vextenddouble/vextenddoubleC1\nj/vextenddouble/vextenddouble\nLq+/vextenddouble/vextenddoubleC2\nj/vextenddouble/vextenddouble\nLq/parenrightBig\ndτ.\nfor anyt∈[0,T[. Using the positivity propertyα\nρ−β/greaterorequalslant0and sending q→+∞yield\n/ba∇dbl∆juβ(t)/ba∇dblL∞/lessorsimilar/ba∇dbl∆juβ(0)/ba∇dblL∞+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆jPβ/ba∇dblL∞+/vextenddouble/vextenddoubleC1\nj/vextenddouble/vextenddouble\nL∞+/vextenddouble/vextenddoubleC2\nj/vextenddouble/vextenddouble\nL∞/parenrightBig\ndτ.\nFrom the previous bound, by multiplying both sides by 2jand summing up over j/greaterorequalslant−1, in turn\nwe deduce an estimate for the B1\n∞,1norm ofuβ: for anyt∈[0,T[, we obtain\n/ba∇dbluβ(t)/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblu0/ba∇dblB1\n∞,1+/integraldisplayt\n0\n/ba∇dblPβ/ba∇dblB1\n∞,1++∞/summationdisplay\nj=−12j/vextenddouble/vextenddoubleC1\nj/vextenddouble/vextenddouble\nL∞++∞/summationdisplay\nj=−12j/vextenddouble/vextenddoubleC2\nj/vextenddouble/vextenddouble\nL∞\ndτ. (63)\nLet us now bound one by one the terms appearing under the integ ral on the right-hand side\nof the previous inequality.\nThe commutator terms\nThe estimate of the first commutator term is classical: by usi ng Lemma 3.8, we get\n+∞/summationdisplay\nj=−12j/vextenddouble/vextenddoubleC1\nj/vextenddouble/vextenddouble\nL∞/lessorsimilare−βt/ba∇dbl∇uβ/ba∇dblL∞/ba∇dbluβ/ba∇dblB1\n∞,1.\nNext, let us focus on the second commutator term appearing in (63). By Lemma 3.10 and the\nparaliearisation result of Proposition 3.7, we can bound it as follows:\n+∞/summationdisplay\nj=−12j/vextenddouble/vextenddoubleC2\nj/vextenddouble/vextenddouble\nL∞/lessorsimilar/ba∇dblρ−1/ba∇dblB1\n∞,1/ba∇dbluβ/ba∇dblB0\n∞,1\n/lessorsimilar/ba∇dblρ−1/ba∇dblB1\n∞,1/ba∇dbluβ/ba∇dblθ\nL2/ba∇dbluβ/ba∇dbl1−θ\nB1\n∞,1/lessorsimilar/ba∇dblρ−1/ba∇dblB1\n∞,1e−θ(α∗−β)t/ba∇dblu0/ba∇dblθ\nL2/ba∇dbluβ/ba∇dbl1−θ\nB1\n∞,1,\nfor a suitable exponent θ=θ(d)∈]0,1[only depending on the space dimension d/greaterorequalslant2. Indeed,\none has the following chain of continuous embeddings: B1\n∞,1֒→B0\n∞,1֒→B−d/2−δ\n∞,1, whereδ>0is\narbitrarily small. Hence, noticing that L2֒→B−d/2\n∞,2֒→B−d/2−δ\n∞,1, we can apply an interpolation\nargument and deduce the existence of such an exponent θ. In addition, in passing from the\npenultimate inequality to the last one, we have used the expo nential decay given by (58).\nIn order to simplify the notation in the argument below, from now on we fix the value of\nβ=β∗∈]0,α∗[such that\n(64) β∗=θ/parenleftbig\nα∗−β∗/parenrightbig\n=⇒ β∗=θ\n1+θα∗.\n29This choice implies that the two commutator terms have the sa me exponential decay. Then,\nplugging the previous bounds into (63), we infer the inequal ity\n/ba∇dbluβ∗(t)/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblu0/ba∇dblB1\n∞,1+/integraldisplayt\n0/parenleftBigg\n/ba∇dblρ/ba∇dblB1\n∞,1/ba∇dbl∇Πβ∗/ba∇dblB1\n∞,1+e−β∗τ/ba∇dbl∇uβ∗/ba∇dblL∞/ba∇dbluβ∗/ba∇dblB1\n∞,1(65)\n+e−β∗τ/ba∇dblρ−1/ba∇dblB1\n∞,1/ba∇dblu0/ba∇dblθ\nL2/ba∇dbluβ∗/ba∇dbl1−θ\nB1\n∞,1/parenrightBigg\ndτ\nfor allt∈[0,T[, where we have also used that\n/ba∇dblPβ/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblρ/ba∇dblB1\n∞,1/ba∇dbl∇Πβ/ba∇dblB1\n∞,1,\nas a consequence of the fact that B1\n∞,1is an algebra and of Proposition 3.7.\nBounds for the density and the pressure\nWe still have to bound the density and the pressure gradient a ppearing in estimate (65). The\nanalysis is similar to the one performed in the previous sect ion, so we will only sketch it.\nFirst of all, we observe that, from the mass equation (60) and classical transport estimates in\nBesov spaces, we have, for any t∈[0,T[, the inequality\n(66) /ba∇dblρ(t)−1/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblρ0−1/ba∇dblB1\n∞,1exp/parenleftbigg\nC/integraldisplayt\n0e−β∗τ/ba∇dbluβ∗(τ)/ba∇dblB1\n∞,1dτ/parenrightbigg\n.\nAs for the pressure, we start by writing an equation for Πβ∗: by applying the divergence\noperator to both sides of relation (59), we find\n−div/parenleftbigg1\nρ∇Πβ∗/parenrightbigg\n= div/parenleftbigg\ne−β∗tuβ∗·∇uβ∗+α\nρuβ∗/parenrightbigg\n,\nowing to the fact that divuβ∗= 0. Before applying estimate (17) with F=e−β∗tuβ∗·∇uβ∗+\nα\nρuβ∗, let us notice that\n/vextenddouble/vextenddouble/vextenddouble/vextenddoublee−β∗tuβ∗·∇uβ∗+α\nρuβ∗/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilare−β∗t/ba∇dbluβ∗/ba∇dblL2/ba∇dbl∇uβ∗/ba∇dblL∞+α/ba∇dbluβ∗/ba∇dblL2 (67)\n/lessorsimilare−β∗t/ba∇dbluβ∗/ba∇dblL2/ba∇dbl∇uβ∗/ba∇dblL∞+αe−β∗t/ba∇dblu0/ba∇dblL2,\nwhere in the last inequality we have used (58) and the fact tha tβ∗=θ(α∗−β∗)/lessorequalslant(α∗−β∗).\nMoreover, by arguing similarly to Section 5.1, see below equ ation (48), one can bound\n(68)/vextenddouble/vextenddouble/vextenddoubleρdiv(e−β∗tuβ∗·∇uβ∗)/vextenddouble/vextenddouble/vextenddouble\nB0\n∞,1/lessorsimilare−β∗t/ba∇dblρ/ba∇dblB1\n∞,1/ba∇dbluβ∗/ba∇dbl2\nB1\n∞,1.\nFinally, we notice that the following equalities hold true:\nρdiv/parenleftbiggα\nρuβ∗/parenrightbigg\n=−α∇logρ·uβ∗=−αdiv/parenleftbig\nlogρuβ∗/parenrightbig\n.\nUsing (15) and the paralinearization theorem, we deduce tha t\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleρdiv/parenleftbiggα\nρuβ∗/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nB0\n∞,1/lessorsimilarα/ba∇dblρ/ba∇dblB1\n∞,1/ba∇dbluβ∗/ba∇dblB0\n∞,1/lessorsimilarαe−β∗t/ba∇dblρ/ba∇dblB1\n∞,1/ba∇dblu0/ba∇dblθ\nL2/ba∇dbluβ∗/ba∇dbl1−θ\nB1\n∞,1, (69)\nwhereθ∈]0,1[is the same exponent appearing above. Estimates (67), (68) a nd (69) together\nwith (17) imply the bound\n/ba∇dbl∇Πβ∗/ba∇dblB1\n∞,1/lessorsimilare−β∗t/parenleftbigg\n1 +/ba∇dblρ−1/ba∇dblη\nB1\n∞,1/parenrightbigg\n(70)\n×/parenleftBig\nα/ba∇dblu0/ba∇dblL2+/ba∇dbluβ∗/ba∇dbl2\nL2∩B1\n∞,1+α/ba∇dblu0/ba∇dblθ\nL2/ba∇dbluβ∗/ba∇dbl1−θ\nB1\n∞,1/parenrightBig\n.\n306.3 End of the argument\nWe can now plug estimates (66) and (70) into (65). After defini ng\nR2:= 1 +/ba∇dblρ0−1/ba∇dblη+1\nB1\n∞,1,\nwe get that the following inequalities hold true, for any t∈[0,T[:\n/ba∇dbluβ∗(t)/ba∇dblB1\n∞,1/lessorsimilar/ba∇dblu0/ba∇dblB1\n∞,1+/integraldisplayt\n0e−β∗τ/parenleftbigg\n1 +/ba∇dblρ−1/ba∇dblη+1\nB1\n∞,1/parenrightbigg\n×/parenleftBig\nα/ba∇dblu0/ba∇dblL2+/ba∇dbluβ∗/ba∇dbl2\nL2∩B1\n∞,1+α/ba∇dblu0/ba∇dblθ\nL2/ba∇dbluβ∗/ba∇dbl1−θ\nB1\n∞,1/parenrightBig\ndτ\n/lessorsimilar/ba∇dblu0/ba∇dblB1\n∞,1+R2/integraldisplayt\n0e−β∗τexp/parenleftbigg\nC(η+1)/integraldisplayτ\n0e−β∗σ/ba∇dbluβ∗/ba∇dblB1\n∞,1dσ/parenrightbigg\n×/parenleftBig\nα/ba∇dblu0/ba∇dblL2+/ba∇dbluβ∗/ba∇dbl2\nL2∩B1\n∞,1+α/ba∇dblu0/ba∇dblθ\nL2/ba∇dbluβ∗/ba∇dbl1−θ\nB1\n∞,1/parenrightBig\ndτ\n/lessorsimilar/ba∇dblu0/ba∇dblB1\n∞,1+α/ba∇dblu0/ba∇dblL2R2/integraldisplayt\n0e−β∗τexp/parenleftbigg\nC(η+1)/integraldisplayτ\n0e−β∗σ/ba∇dbluβ∗/ba∇dblB1\n∞,1dσ/parenrightbigg\n+R2/integraldisplayt\n0e−β∗τexp/parenleftbigg\nC(η+1)/integraldisplayτ\n0e−β∗σ/ba∇dbluβ∗/ba∇dblB1\n∞,1dσ/parenrightbigg\n×/parenleftBig\n/ba∇dbluβ∗/ba∇dbl2\nL2∩B1\n∞,1+α/ba∇dbluβ∗/ba∇dblB1\n∞,1/parenrightBig\ndτ,\nwhere, for passing from the penultimate bound to the last one , we have used the Young inequality.\nAt this point, the argument can be closed in a very similar fas hion as done in Subsections 5.1\nand 5.2 above. Let us introduce the notation\nUβ∗(t) :=/ba∇dbluβ∗(t)/ba∇dblL2+/ba∇dbluβ∗(t)/ba∇dblB1\n∞,1,\nwith obvious changes when t= 0. Then, from the previous estimate for the Besov norm of uβ∗\nand an application of the Grönwall inequality, for any t∈[0,T[we infer that\nUβ∗(t)/lessorsimilareαR2h(t)/parenleftBigg\nUβ∗(0)/parenleftbig\n1 +αR2h(t)/parenrightbig\n(71)\n+R2/integraldisplayt\n0e−β∗τeC/integraltextτ\n0e−β∗σUβ∗(σ)dσ/parenleftbig\nUβ∗(τ)/parenrightbig2dτ/parenrightBigg\n,\nfor a new constant C >0, where we have defined\nh(t) :=/integraldisplayt\n0e−β∗τeC/integraltextτ\n0e−β∗σUβ∗(σ)dσdτ\nNow, let us define the time T2>0as\nT2:= sup/braceleftBigg\nt∈[0,T[/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0e−β∗τUβ∗(τ)dτ/lessorequalslant2\nandR2/integraldisplayt\n0e−β∗τeC/integraltextτ\n0e−β∗σUβ∗(σ)dσ/parenleftbig\nUβ∗(τ)/parenrightbig2dτ/lessorequalslant4Uβ∗(0)/bracerightBigg\n.\nFirst of all, we notice that, for any t∈[0,T2], one has\nαR2h(t)/lessorequalslantR2α\nβ∗e2C=R21+θ\nθρ∗e2C=R2C,\n31where we have used the definition of β∗,α∗andα. Observe that the new constant Cdepends\nonly on the fixed quantities ρ∗andρ∗appearing in (4), but not on α,β∗andu0.\nBy the previous inequality and inequality (71), we gather, fo r anyt∈[0,T2], the following\nbound:\n(72) Uβ∗(t)/lessorsimilar(5 +R2C)eR2CUβ∗(0).\nIn particular, this implies on the one hand that, for all t∈[0,T2], we have\n/integraldisplayt\n0e−β∗τUβ∗(τ)dτ/lessorsimilar(5 +R2C)eR2CUβ∗(0)\nβ∗\nand, on the other hand, that\nR2/integraldisplayt\n0e−β∗τeC/integraltextτ\n0e−β∗σUβ∗(σ)dσ/parenleftbig\nUβ∗(τ)/parenrightbig2dτ/lessorsimilarR2e2C(5 +R2C)2e2R2C(Uβ∗(0))2\nβ∗.\nTherefore, by definition of the time T2, we see that one must have\neither 2/lessorsimilar(5+R2C)eR2CUβ∗(0)\nβ∗or4/lessorsimilarR2e2C(5+R2C)2e2R2CUβ∗(0)\nβ∗.\nNone of those inequalities can be satisfied, if the quantity Uβ∗(0)/β∗is small enough, according\nto the conditions imposed in Theorem 2.5, where we take R=R2. This implies that one must\nhaveT2=T. Hence, by (72), we see that Uβ∗remains bounded in [0,T[, which then contradicts\nthe blow-up property stated in (61).\nWe thus deduce that T= +∞, as sought. In addition, arguing as done at the end of Subsect ion\n5.2, we recover the time decay of the critical Besov norms of th e solution: it follows from (72)\nand (70) that\n∀t/greaterorequalslant0, /ba∇dblu(t)/ba∇dblL2∩B1\n∞,1+/ba∇dbl∇Π(t)/ba∇dblL2∩B1\n∞,1/lessorequalslantCe−β∗t,\nfor an absolute constant C >0.\nIn order to complete the proof of Theorem 2.5, it remains us to show the time decay of the\nhigher order Besov norms of the solution: this will be done in t he next subsection.\n6.4 Decay of higher regularity norms when γ= 0\nOur next goal is to prove the decay of the high regularity norm s of the solution, recall estimate\n(8). This will follow from the counterpart of Lemma 5.1 in the caseγ= 0: the precise statement\nis the following one.\nLemma 6.1. Letd/greaterorequalslant2,γ= 0andα >0fixed. Take indices (s,p,r)∈R×[1,+∞]×[1,+∞]\nsuch thatp/greaterorequalslant2and on of the two conditions in (5)is verified. Take an initial datum/parenleftbig\nρ0,u0/parenrightbig\nsuch that conditions (3)-(4)are satisfied, for two suitable constants 0< ρ∗/lessorequalslantρ∗. Assume that\nu0∈L2∩Bs\np,rand∇ρ0∈Bs−1\np,rand that, in addition, there exists a unique global in time so lution/parenleftbig\nρ,u,∇Π/parenrightbig\nto system (1)associated to that initial datum, verifying the regularity properties (i)-\n(ii)-(iii) of Theorem 2.1. Finally, assume that there exist an exponent β∗>0and a constant\nC >0such that\n(73) ∀t/greaterorequalslant0, /ba∇dbl∇ρ(t)/ba∇dblB0\n∞,1+eβ∗t/ba∇dblu(t)/ba∇dblL2∩B1\n∞,1+eβ∗t/ba∇dbl∇Π(t)/ba∇dblL2∩B1\n∞,1/lessorequalslantC.\nThen, for a possibly different constant /tildewideC >0and for a new exponent 0<β0<β∗, defined in\nrelation (74)below, one has\n∀t/greaterorequalslant0, /ba∇dbl∇ρ(t)/ba∇dblBs−1\np,r+eβ0t/ba∇dblu(t)/ba∇dblL2∩Bsp,r+eβ0t/ba∇dbl∇Π(t)/ba∇dblL2∩Bsp,r/lessorequalslant/tildewideC.\n32Proof. Notice that we can safely assume that p <+∞, orp= +∞ands >1here. As for the\nproof of Lemma 5.1 above, the strategy is to reproduce fine est imates (in the same spirit of the\ncontinuation criterion) on the Besov norm of the solution, an d take care of making lower order\nBesov norms (for which the decay is guaranteed by assumption) appear.\nBy going along the lines of the computations of Subsection 6.2 , one easily sees that it is natural\nto define\n(74) β0:=θ0\n1+θ0α∗, i.e.β0=θ0/parenleftbig\nα∗−β0/parenrightbig\n,\nwhereα∗:=α/ρ∗as above and θ0is the exponent related to the interpolation\n/ba∇dblf/ba∇dblBs−1\np,r/lessorsimilar/ba∇dblf/ba∇dblθ0\nL2/ba∇dblf/ba∇dbl1−θ0\nBsp,r.\nObserve that, if we fix the index σ=d/2+δ(whereδ>0can be taken arbitrarily small) in order\nto have the embeddings Bs\np,r֒→Bs−1\np,r֒→B0\n∞,1֒→B−σ\n∞,1, we find that\nθ=1\n1+σand θ0=1\ns+σ,\nwhereθis the exponent appearing in (64). Then, as easy computation shows that β∗/β0>1,\nnamely\n0< β0< β∗.\nThis having been pointed out, our next goal is to perform Bs\np,restimates for uβ0. The starting\npoint is the analogous of (63) in the Bs\np,rframework, namely\n/ba∇dbluβ0(t)/ba∇dblBsp,r/lessorsimilar/ba∇dblu0/ba∇dblBsp,r+/integraldisplayt\n0/parenleftbigg\n/ba∇dblPβ0/ba∇dblBsp,r+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig\n2js/vextenddouble/vextenddoubleC1\nj/vextenddouble/vextenddouble\nLp/parenrightBig\nj/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nℓr+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig\n2js/vextenddouble/vextenddoubleC2\nj/vextenddouble/vextenddouble\nLp/parenrightBig\nj/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nℓr/parenrightbigg\ndτ.\nTo begin with, we notice that, similarly to inequalities (68 ) and (69), one has the estimates\n/vextenddouble/vextenddouble/vextenddoubleρdiv(e−β0tuβ0·∇uβ0)/vextenddouble/vextenddouble/vextenddouble\nBs−1\np,r/lessorsimilare−β0t/parenleftBig\n/ba∇dbl∇ρ/ba∇dblBs−1\np,r/ba∇dbluβ0/ba∇dblL∞/ba∇dbl∇uβ0/ba∇dblL∞\n+ρ∗/ba∇dbl∇uβ0/ba∇dblL∞/ba∇dbluβ0/ba∇dblBsp,r/parenrightBig\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleρdiv/parenleftbiggα\nρuβ0/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nBs−1\np,r/lessorsimilar/ba∇dblρ/ba∇dblBs−1\np,r/ba∇dbluβ0/ba∇dblL∞+/ba∇dbl∇ρ/ba∇dblL∞/ba∇dbluβ0/ba∇dblBs−1\np,r\n/lessorsimilar/ba∇dblρ/ba∇dblBs−1\np,r/ba∇dbluβ0/ba∇dblL∞+/ba∇dbl∇ρ/ba∇dblL∞/ba∇dbluβ0/ba∇dblθ0\nL2/ba∇dbluβ0/ba∇dbl1−θ0\nBsp,r\n/lessorsimilar/ba∇dblρ/ba∇dblBs−1\np,r/ba∇dbluβ0/ba∇dblL∞+e−(α∗−β0)θ0t/ba∇dbl∇ρ/ba∇dblL∞/ba∇dbluβ0/ba∇dbl1−θ0\nBsp,r.\nThen, using also Lemma 3.12 and the definition of β0in (74), we deduce that\n/ba∇dbl∇Πβ0/ba∇dblBsp,r/lessorsimilar/parenleftbig\n1+/ba∇dbl∇ρ/ba∇dblη\nL∞/parenrightbig/parenleftBig\ne−tβ0/ba∇dbluβ0/ba∇dblL2/ba∇dbl∇uβ0/ba∇dblL∞+/ba∇dbluβ0/ba∇dblL2/parenrightBig\n+/ba∇dbl∇ρ/ba∇dblBs−1\np,r/ba∇dbl∇Πβ0/ba∇dblL∞\n+e−β0t/parenleftBig\n/ba∇dbl∇ρ/ba∇dblBs−1\np,r/ba∇dbluβ0/ba∇dblL∞/ba∇dbl∇uβ0/ba∇dblL∞+/ba∇dbl∇uβ0/ba∇dblL∞/ba∇dbluβ0/ba∇dblBsp,r/parenrightBig\n+e−β0t/ba∇dbl∇ρ/ba∇dblL∞/ba∇dbluβ0/ba∇dbl1−θ0\nBsp,r+/parenleftBig\n1 +/ba∇dbl∇ρ/ba∇dblBs−1\np,r/parenrightBig\n/ba∇dbluβ0/ba∇dblL∞.\nUsing assumption (73) and recalling that β0<β∗, in turn we find\n/ba∇dbl∇Πβ0/ba∇dblBsp,r/lessorsimilare−tβ0/parenleftBig\n/ba∇dbluβ0/ba∇dblBsp,r+/ba∇dbl∇ρ/ba∇dblBs−1\np,r+ 1/parenrightBig\n(75)\n+e−(α∗−β0)θ0t/parenleftBig\n1 +/ba∇dbluβ0/ba∇dblBsp,r/parenrightBig\n+e−(β∗−β0)t/parenleftBig\n1 +/ba∇dbl∇ρ/ba∇dblBs−1\np,r/parenrightBig\n,\n33where the implicit multiplicative constant depends also on /ba∇dblu0/ba∇dblL2. Then, making use of assump-\ntion (73) again, we infer the bound\n/ba∇dblPβ0/ba∇dblBsp,r/lessorsimilar/ba∇dbl∇Πβ0/ba∇dblBsp,r+/ba∇dbl∇ρ/ba∇dblBs−1\np,r/ba∇dbl∇Πβ0/ba∇dblL∞/lessorsimilare−tλ/parenleftBig\n/ba∇dbluβ0/ba∇dblBsp,r+/ba∇dbl∇ρ/ba∇dblBs−1\np,r+ 1/parenrightBig\n,\nwhere we have denoted by λthe minimum value among β0,(α∗−β0)θ0andβ∗−β0.\nLet us now bound the commutator terms. On the one hand, an imme diate application of\nLemma 3.8 yields\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig\n2js/vextenddouble/vextenddoubleC1\nj/vextenddouble/vextenddouble\nLp/parenrightBig\nj/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nℓr/lessorsimilare−β0t/ba∇dbl∇uβ0/ba∇dblL∞/ba∇dbluβ0/ba∇dblBsp,r/lessorsimilare−β0t/ba∇dbluβ0/ba∇dblBsp,r.\nOn the other hand, Lemma 3.10 implies that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig\n2js/vextenddouble/vextenddoubleC2\nj/vextenddouble/vextenddouble\nLp/parenrightBig\nj/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nℓr/lessorsimilar/ba∇dblρ/ba∇dblB1\n∞,1/ba∇dbluβ0/ba∇dblBs−1\np,r+/ba∇dbl∇ρ/ba∇dblBs−1\np,r/ba∇dbluβ0/ba∇dblB1\n∞,1.\nPutting all those estimates together, we finally arrive at th e following inequality:\n/ba∇dbluβ0(t)/ba∇dblBsp,r/lessorsimilar/ba∇dblu0/ba∇dblBsp,r+/integraldisplayt\n0e−λτ/parenleftBig\n/ba∇dbluβ0/ba∇dblBsp,r+/ba∇dbl∇ρ/ba∇dblBs−1\np,r+ 1/parenrightBig\ndτ\n+/integraldisplayt\n0/parenleftbigg\n/ba∇dblρ/ba∇dblB1\n∞,1/ba∇dbluβ0/ba∇dblBs−1\np,r+/ba∇dbl∇ρ/ba∇dblBs−1\np,r/ba∇dbluβ0/ba∇dblB1\n∞,1/parenrightbigg\ndt.\nAt this point, we use the following two facts,\n/ba∇dbluβ0/ba∇dblB1\n∞,1=e−t(β∗−β0)/ba∇dbluβ∗/ba∇dblB1\n∞,1/lessorsimilare−t(β∗−β0)\n/ba∇dbluβ0/ba∇dblBs−1\np,r/lessorsimilar/ba∇dbluβ0/ba∇dblθ0\nL2/ba∇dbluβ0/ba∇dbl1−θ0\nBsp,r/lessorsimilare−(β∗−β0)θ0t/parenleftBig\n1 +/ba∇dbluβ0/ba∇dblBsp,r/parenrightBig\n,\nwhich hold true owing to (73) and (58). Then, up to taking a sma llerλ >0if necessary, we\nobtain the bound\n/ba∇dbluβ0(t)/ba∇dblBsp,r/lessorsimilar/ba∇dblu0/ba∇dblBsp,r+/integraldisplayt\n0e−λτ/parenleftBig\n/ba∇dbluβ0/ba∇dblBsp,r+/ba∇dbl∇ρ/ba∇dblBs−1\np,r+ 1/parenrightBig\ndτ.\nOn the other hand, it follows from inequality (25) and the fac t that, under our assumptions,\none hass>1always, that\n/ba∇dbl∇ρ/ba∇dblBs−1\np,r/lessorsimilar/ba∇dbl∇ρ0/ba∇dblBs−1\np,r+/integraldisplayt\n0e−β0τ/parenleftBig\n/ba∇dbl∇uβ0/ba∇dblL∞/ba∇dbl∇ρ/ba∇dblBs−1\np,r+/ba∇dbl∇ρ/ba∇dblL∞/ba∇dbluβ0/ba∇dblBsp,r/parenrightBig\ndt.\nTherefore, defining\nUβ0(t) :=/ba∇dbluβ0(t)/ba∇dblBsp,r+/ba∇dbl∇ρ/ba∇dblBs−1\np,r\nand using (73) again, we easily find\nUβ0(t)/lessorsimilarUβ0(0) +/integraldisplayt\n0e−λτ/parenleftBig\nUβ0(τ) + 1/parenrightBig\ndτ.\nWe then conclude, by an application of the Grönwall lemma, th at\n∀t/greaterorequalslant0, /ba∇dbl∇ρ(t)/ba∇dblBs−1\np,r+/ba∇dbluβ0(t)/ba∇dblBsp,r<+∞.\nInserting this bound into (75), also the sought decay for the pressure gradient follows. The proof\nof the lemma is thus completed.\n34A Appendix: proof of some technical lemmas\nIn this appendix, we show the proof of some technical lemmas f rom Section 3, which have been\nneeded in the course of our proof.\nWe start by presenting the proof of Lemma 3.10.\nProof of Lemma 3.10. The beginning of the proof follows the main lines of the proof of classical\ncommutator estimates (see Chapter 2 of [2] for details). Mor e precisely, we use Bony’s paraproduct\ndecomposition to write\n[f,∆j]v= [Tf,∆j]v+T∆jvf−∆jTvf+R/parenleftbig\nf,∆jv/parenrightbig\n−∆jR(f,v) =5/summationdisplay\nn=1A(j)\nn\nWe are going to estimate the terms A(j)\nnseparately.\nThe bound for A(j)\n1= [Tf,∆j]vis analogous to the one stated in Lemma 3.9 (in fact, by\nspectral localisation, it would be easy to reduce the proof t o that lemma). Indeed, by using\nLemma 2.97 of [2], we can write\n2js/vextenddouble/vextenddouble/vextenddoubleA(j)\n1/vextenddouble/vextenddouble/vextenddouble\nLp/lessorsimilar/summationdisplay\n|i−j|/lessorequalslant52js2−j/ba∇dbl∇Si−1f/ba∇dblL∞/ba∇dbl∆iv/ba∇dblLp/lessorsimilar2j(s−1)/ba∇dbl∇f/ba∇dblL∞/summationdisplay\n|i−j|/lessorequalslant5/ba∇dbl∆iv/ba∇dblLp\n/lessorsimilarc1\nj/ba∇dbl∇f/ba∇dblL∞/ba∇dblv/ba∇dblBs−1\np,r,\nfor a suitable sequence/parenleftbig\nc1\nj/parenrightbig\nj∈ℓrof unitary norm.\nWe now focus on the second term\nA(j)\n2=T∆jvf=/summationdisplay\ni/greaterorequalslantj−3∆jSi−1v∆if.\nAs only the high frequencies of fappear in the previous expression, we can make use of the seco nd\nBernstein inequality to estimate\n2js/vextenddouble/vextenddouble/vextenddoubleA(j)\n2/vextenddouble/vextenddouble/vextenddouble\nLp/lessorsimilar2js/summationdisplay\ni/greaterorequalslantj+12−i2i/ba∇dbl∆if/ba∇dblL∞/ba∇dbl∆jSi−1v/ba∇dblLp\n/lessorsimilar2j(s−1)/ba∇dbl∇f/ba∇dblB0\n∞,1/ba∇dbl∆jv/ba∇dblLp/lessorsimilarc2\nj/ba∇dbl∇f/ba∇dblB0\n∞,1/ba∇dblv/ba∇dblBs−1\np,r,\nwith the sequence/parenleftbig\nc2\nj/parenrightbig\njbelonging to the unit ball of ℓr.\nNext, for the third term A(j)\n3=−∆jTvf, we notice that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig\n2js/vextenddouble/vextenddouble/vextenddoubleA(j)\n3/vextenddouble/vextenddouble/vextenddouble\nLp/parenrightBig\nj/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nℓr≈ /ba∇dblTvf/ba∇dblBsp,r.\nHence, we can use the continuity properties of the paraprodu ct operator, see Proposition 3.5,\ntogether with Remark 2.83 oc [2], to bound\n2js/vextenddouble/vextenddouble/vextenddoubleA(j)\n3/vextenddouble/vextenddouble/vextenddouble\nLp/lessorsimilarc3\nj/ba∇dbl∇f/ba∇dblBs−1\np,r/ba∇dblv/ba∇dblL∞,\nwith/parenleftbig\nc3\nj/parenrightbig\nj∈ℓrof unitary norm.\nAs forA(j)\n4, we use spectral localisation properties to rewrite\nA(j)\n4=/summationdisplay\n|l−j|/lessorequalslant1/summationdisplay\n|l−k|/lessorequalslant1∆kf∆j∆lv=/summationdisplay\n|k−j|/lessorequalslant2/summationdisplay\n|l−k|/lessorequalslant1∆kf∆j∆lv.\n35Then, we can bound\n2js/vextenddouble/vextenddouble/vextenddoubleA(j)\n4/vextenddouble/vextenddouble/vextenddouble\nLp/lessorsimilar2js/summationdisplay\n|k−j|/lessorequalslant2/summationdisplay\n|l−k|/lessorequalslant1/ba∇dbl∆kf∆j∆lv/ba∇dblLp/lessorsimilarc4\nj/parenleftBig\n/ba∇dbl∇f/ba∇dblBs−1\np,r/ba∇dblv/ba∇dblL∞+/ba∇dblf/ba∇dblL∞/ba∇dblv/ba∇dblL∞/parenrightBig\n,\nfor a suitable sequence/parenleftbig\nc4\nj/parenrightbig\nj∈ℓrof unitary norm. Notice that the presence of the term\n/ba∇dblf/ba∇dblL∞/ba∇dblv/ba∇dblL∞is needed in order to control the terms for which k=l= 0, so that no high\nfrequencies are involved (nor of f, nor ofv, so one cannot make a gradient appear in the esti-\nmates).\nFinally, we consider the last term A(j)\n5=−∆jR(f,v). For controlling it, we can directly\napply Proposition 3.5 and get\n2js/vextenddouble/vextenddouble/vextenddoubleA(j)\n5/vextenddouble/vextenddouble/vextenddouble\nLp/lessorsimilarc5\nj/ba∇dblR(f,v)/ba∇dblBsp,r/lessorsimilar/ba∇dblf/ba∇dblB1∞,∞/ba∇dblv/ba∇dblBs−1\np,r,\nwhere, again, the sequence/parenleftbig\nc5\nj/parenrightbig\njbelongs toℓrand has unitary norm.\nIn the end, after adding all the estimates for the terms A(j)\nn, we deduce the proof of the lemma\nby settingcj:=/summationtext5\nn=1cn\nj∈ℓr.\nNow, we present the proof of Lemma 3.12, concerning fine estim ates for the pressure gradient\nin Besov spaces. The guidelines of the proof being similar to [ 16, 17], we only give a sketch of the\nargument here, indicating only the main modifications in ord er to get the estimates used in our\nwork.\nProof of Lemma 3.12. We are going to argue in a similar way as in Section 5 of [16], so we will be\na bit sketchy here and refer to that paper for details. In addi tion, for simplicity we will consider\nhere only the case (s,p,r) = (1,∞,1), beacause is the one that we will use; besides, this case is\nthe most involved one, as we will need to estimates non-linea r term in the spae B0\n∞,1which, as\nalready remarked several times, is not a Banach algebra.\nFirst of all notice that we can rewrite equation (16) as follo ws:\n−∆Π =−∇log(ρ)·∇Π +ρdivF.\nHence, by cutting ∇Πinto low and high frequency and using that p/greaterorequalslant2, we have\n/ba∇dbl∇Π/ba∇dblB1\n∞,1/lessorsimilar/ba∇dbl∇Π/ba∇dblL2+/ba∇dbl∆Π/ba∇dblB0\n∞,1\n/lessorsimilar/ba∇dbl∇Π/ba∇dblL2+/ba∇dbl∇logρ·∇Π/ba∇dblB0\n∞,1+/ba∇dblρdivF/ba∇dblB0\n∞,1.\nUsing inequality (14) and Proposition 3.7, we get\n/ba∇dbl∇logρ·∇Π/ba∇dblB0\n∞,1/lessorsimilar/ba∇dbl∇(ρ−1)/ba∇dblB0\n∞,1/ba∇dbl∇Π/ba∇dblB1/2\n∞,1.\nHowever, this estimate makes a non-linear dependence appea r on the Besov norm of ρ−1and\nthis is not suitable for our scopes. Therefore, we rather use a Bony’s paraproduct decomposition,\ntogether with Proposition 3.7 again, and bound\n/ba∇dbl∇logρ·∇Π/ba∇dblB0\n∞,1/lessorsimilar/ba∇dbl∇(ρ−1)/ba∇dblB0\n∞,1/ba∇dbl∇Π/ba∇dblL∞+/ba∇dbl∇(ρ−1)/ba∇dblL∞/ba∇dbl∇Π/ba∇dblB1/2\n∞,1\n/lessorsimilar/ba∇dbl∇(ρ−1)/ba∇dblB0\n∞,1/ba∇dbl∇Π/ba∇dblL∞+/ba∇dbl∇(ρ−1)/ba∇dblL∞/ba∇dbl∇Π/ba∇dblδ\nL2/ba∇dbl∇Π/ba∇dbl1−δ\nB1\n∞,1,\nfor a suitable δ∈]0,1[coming from interpolation. After an application of the Youn g inequality,\nthat estimate in turn yields the bound\n/ba∇dbl∇Π/ba∇dblB1\n∞,1/lessorsimilar/parenleftbig\n1 +/ba∇dbl∇ρ/ba∇dblη\nL∞/parenrightbig\n/ba∇dbl∇Π/ba∇dblL2+/ba∇dbl∇(ρ−1)/ba∇dblB0\n∞,1/ba∇dbl∇Π/ba∇dblL∞+/ba∇dblρdiv(F)/ba∇dblB1\n∞,1,\nwhere we have set η= 1+1/δ. Finally, Lemma 3.11 implies the sought inequality (17).\n36References\n[1] H. Bae, W. Lee, J. Shin: A blow-up criterion for the inhomogeneous incompressible Eul er equations .\nNonlinear Anal., 196(2020), 111774, 9 pp.\n[2] H. Bahouri, J.-Y. Chemin, R. Danchin: “Fourier analysis and nonlinear partial differential equa-\ntions” . Grundlehren der mathematischen Wissenschaften [Fundame ntal Principles of Mathematical\nSciences], 343. Springer, Heidelberg, 2011.\n[3] E. Bocchi, F. Fanelli, C. Prange: Anisotropy and stratification effects in the dynamics of fast rotating\ncompressible fluids . Ann. Inst. H. Poincaré C Anal. Non Linéaire, 39(2022), n. 3, 647-704.\n[4] J.-M. Bony: Calcul symbolique et propagation des singularités pour les é quations aux dérivées par-\ntielles non linéaires . Ann. Sci. École Norm. Sup., 14(1981), n. 2, 209-246.\n[5] K. Beauchard, E. Zuazua: Large time asymptotics for partially dissipative hyperbolic systems . Arch.\nRation. Mech. Anal., 199(2011), n. 1, 177-227.\n[6] R. Bianchini, T. Crin-Barat, M. Paicu: Relaxation approximation and asymptotic stability of stra t-\nified solutions to the IPM equation . Arch. Ration. Mech. Anal., 248(2024), n. 1, Paper No. 25, 35\npp.\n[7] M. Bravin, F. 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Fanelli: Conservation of geometric structures for non-homogeneous inv iscid incompressible fluids .\nComm. Partial Differential Equations, 37(2012), n. 9, 1553-1595.\n[18] F. Fanelli, R. Granero-Belinchón, S. Scrobogna: Well-posedness theory for non-homogeneous incom-\npressible fluids with odd viscosity . Submitted (2022), arxiv preprint 2211.15768 .\n[19] F. Fanelli, A. F. Vasseur: Effective velocity and L∞-based well-posedness for incompressible fluids\nwith odd viscosity . Submitted (2024), arxiv preprint 2401.17085 .\n[20] E. Grenier, N. Masmoudi: Ekman layers of rotating fluids, the case of well prepared init ial data .\nComm. Partial Differential Equations, 22(1997), n. 5-6, p. 953-975.\n[21] T. Hmidi, S. Keraani: Incompressible viscous flows in borderline Besov spaces . Arch. Ration. Mech.\nAnal., 189(2008), n. 2, 283-300.\n37[22] N. Masmoudi: Ekman layers of rotating fluids: the case of general initial da ta. Comm. Pure Appl.\nMath., 53(2000), n. 4, p. 432-483.\n[23] J. Pedlosky: “Geophysical fluid dynamics” . Springer-Verlag, New York, 1987.\n[24] T. Sideris, B. Thomases, D. Wang: Long time behaviour of solutions to the 3D compressible Eule r\nequations with damping . Comm. Partial Differential Equations, 28(2003), n. 3-4, 795-816.\n[25] M. Vishik: Hydrodynamics in Besov spaces . Arch. Ration. Mech. Anal., 145(1998), n. 3, 197-214.\n38"    },    {        "title": "2402.12600v1.Gravitational_wave_asteroseismology_of_dark_matter_hadronic_stars.pdf",        "content": "Gravitational wave asteroseismology of dark matter hadronic stars\nC´ esar V. Flores∗\nUniversidade Estadual da Regi˜ ao Tocantina do Maranh˜ ao, UEMASUL,\nCentro de Ciˆ encias Exatas, Naturais e Tecnol´ ogicas, Imperatriz, CEP 65901-480, MA, Brazil and\nUniversidade Federal do Maranh˜ ao, UFMA, Departamento de F´ ısica-CCET,\nCampus Universit´ ario do Bacanga, S˜ ao Lu´ ıs, MA, CEP 65080-805, Brazil\nC. H. Lenzi,†M. Dutra,‡and O. Louren¸ co§\nDepartamento de F´ ısica, Instituto Tecnol´ ogico de Aeron´ autica,\nDCTA, 12228-900, S˜ ao Jos´ e dos Campos, SP, Brazil\nJos´ e D. V. Arba˜ nil¶\nDepartamento de Ciencias, Universidad Privada del Norte,\nAvenida el Sol 461 San Juan de Lurigancho, 15434 Lima, Peru and\nFacultad de Ciencias F´ ısicas, Universidad Nacional Mayor de San Marcos,\nAvenida Venezuela s/n Cercado de Lima, 15081 Lima, Peru\nThe influence of the dark matter mass ( Mχ) and the Fermi momentum ( kDM\nF) on the f0-mode\noscillation frequency, damping time parameter, and tidal deformability of hadronic stars are studied\nby employing a numerical integration of hydrostatic equilibrium, nonradial oscillation, and tidal\ndeformability equations. The matter inside the hadronic stars follows the NL3* equation of state.\nWe obtain that the influence of MχandkDM\nFis observed in the f0-mode, damping tome parameter,\nand tidal deformability. Finally, the correlation between the tidal deformability of the GW170817\nevent with MχandkDM\nFare also investigated.\nI. INTRODUCTION\nIn astrophysics, it is estimated that dark matter (DM)\ncorresponds to approximately 27% of the matter in the\nuniverse today. The name coined for this matter refers\nto the fact that it appears not to interact with light or\nelectromagnetic field, hence the difficulty in its direct de-\ntection. However, this matter would interact gravitation-\nally, thus leaving unequivocal observational signals of its\nexistence [1].\nThe study of dark matter began in the 1930s when\nFritz Zwicky observed that the velocities dispersion of\ngalaxies within the Coma Cluster exceeded what could be\naccounted for by the visible matter alone [2] (review also\n[3]). He suggested that an additional, unseen mass—who\ncoined the name dark matter— must be exerting gravita-\ntional influence to accelerate these orbital motions. Sub-\nsequent investigations developed by Ford and Rubin in\nthe 1970s [4], would show a consistent presence of unex-\nplained high-speed orbits in every galaxy became appar-\nent, solidifying the consensus around the existence of an\nexotic matter.\nIn the search for the exact nature of dark matter, to\ndate, a large number of experiments have been carried\nout using a variety of theoretical models. Among the the-\noretical proposed candidates, we find weakly (WIMPs)\n∗cesar.vasquez@uemasul.edu.br\n†chlenzi@ita.br\n‡marianad@ita.br\n§odilon.ita@gmail.com\n¶jose.arbanil@upn.pe[5] and feebly (FIMP) [6, 7] interacting massive parti-\ncles, the neutralino [8–10], Axions [11], among others;\nhowever, no concrete conclusion has been reached about\nthe nature of dark matter. Within this set of possibil-\nities, in particular, the weakly interacting massive par-\nticles (WIMPs) are those more promising; review, for\nexample, [12, 13].\nMotivated by the detections of gravitational waves\ncaused by the merger of a binary system recorded by\nthe LIGO-Virgo collaboration (LVC) [14–25], in recent\nyears, numerous investigations have explored the inte-\ngration of dark matter into hadronic matter [10, 26–39],\nas well as the effects of dark matter in relation to quarks\nand hybrid stars [40, 41] and on some properties of neu-\ntron stars, namely, how this type of exotic matter would\ninfluence some parameters macroscopic observations such\nas in the equilibrium configuration [26–32], radial stabil-\nity [24, 25], tidal deformability [10, 27, 31, 33–38], and\nf0-mode frequency of oscillation into Cowling approxi-\nmation [39].\nIn the present work, we study the effect of dark matter\non the stellar structure configuration, f0-mode frequency\nof oscillation from complete general relativity, and tidal\ndeformability. For the fluid inside the compact stars, we\nemploy the relativistic mean field model with the NL3*\nparameterization [42]. The contribution of dark matter\nis included through a kinetic term as done in [33, 36],\nwhere we vary the Fermi momentum by taking different\nvalues of the neutralino mass, as detailed throughout the\narticle. Once the equation of state (EOS) is defined, the\nmass, radius, tidal deformability, oscillation frequency,\nand damping time are analyzed for both hadronic stars\nand for hadronic stars in a binary system. These resultsarXiv:2402.12600v1  [hep-ph]  19 Feb 20242\nare contrasted with observational data reported by LVC\nin the articles aforementioned. In addition, the f0-mode\nderived from the complete general relativity is compared\nwith the ones derived from the nonradial oscillation equa-\ntions through the Cowling approximation.\nThe next sections are distributed as follows: in Sec. II\nthe structure of the hadronic model with dark matter\ncontent is described and the main equations of state of\nthe system are obtained. In Sec. III the equilibrium\nequation, nonradial oscillation equation, and tidal de-\nformability equation are presented. In Sec. IV we dis-\ncuss the results and in Sec. V we conclude the paper.\nFinally, throughout the text, we adopt the metric sig-\nnature ( −,+,+,+) and we employ in geometrized units\nc= 1 = G.\nII. HADRONIC MODEL WITH DARK MATTER\nCONTENT\nThe impact of dark matter on the properties related to\nnonradial oscillations in neutron stars is performed here\nby considering the system, composed of hadronic matter\nadmixed with DM, described by the following Lagrangian\ndensity\nL=LHAD+LDM, (1)\nin which\nLHAD=¯ψ(iγµ∂µ−Mnuc)ψ+gσσ¯ψψ−gω¯ψγµωµψ\n−gρ\n2¯ψγµ⃗bµ⃗ τψ+1\n2(∂µσ∂µσ−m2\nσσ2)−A\n3σ3\n−B\n4σ4−1\n4FµνFµν+1\n2m2\nωωµωµ−1\n4⃗Bµν⃗Bµν\n+1\n2m2\nρ⃗bµ⃗bµ, (2)\nwith nucleon and mesons fields denoted by ψ,σ,ωµ,\nand⃗bµ, respectively (masses given by Mnuc,mσ,mω,\nand mρ). The tensors Fµνand ⃗Bµνare defined as\nFµν=∂µων−∂νωµand⃗Bµν=∂µ⃗bν−∂ν⃗bµ. More details\nof this kind of relativistic mean-field (RMF) model ap-\nplied to symmetric and asymmetric nuclear matter can\nbe found, for instance, in Refs. [43, 44]. The free pa-\nrameters of the model are the couplings gσ,gω,gρ,A,B,\nand the particular set used here for these constants is the\none given by the NL3* model [42]. Such parametrization\nwas chosen due to its capability of reproducing ground\nstate binding energies, charge radii, and giant monopole\nresonances of a set of spherical nuclei, namely,16O,34Si,\n40Ca,48Ca,52Ca,54Ca,48Ni,56Ni,78Ni,90Zr,100Sn,\n132Sn, and208Pb, as well as macroscopic properties of\nneutron stars. The complete study performed with more\nthan 400 other parametrizations of the RMF model is\nfound in Ref. [45]. The dark sector of the model is givenby [10, 26, 27, 30–36, 39, 46, 47]\nLDM= ¯χ(iγµ∂µ−Mχ)χ+ξh¯χχ+1\n2(∂µh∂µh−m2\nhh2)\n+fMnuc\nvh¯ψψ, (3)\nwith the Dirac field χrepresenting the dark fermion of\nmass Mχ. The scalar field hdenotes the Higgs boson\nwith mass mh= 125 GeV, and the strength of the Higgs-\nnucleon interaction is controlled by fM nuc/v, with v=\n246 GeV being the Higgs vacuum expectation value. The\nconstant ξregulates the Higgs-dark particle coupling.\nThe field equations of the system are calculated by\nusing the mean-field approximation [43, 44], namely,\nm2\nσσ=gσρs−Aσ2−Bσ3, (4)\nm2\nωω0=gωρ, (5)\nm2\nρb0(3)=gρ\n2ρ3, (6)\n[γµ(i∂µ−gωω0−gρb0(3)τ3/2)−M∗]ψ= 0,(7)\nm2\nhh=ξρDM\ns+fMnuc\nvρs, (8)\n(γµi∂µ−M∗\nχ)χ= 0, (9)\nwith τ3= 1 for protons and −1 for neutrons, and effective\ndark particle and nucleon masses written as\nM∗\nχ=Mχ−ξh, (10)\nand\nM∗=Mnuc−gσσ−fMnuc\nvh, (11)\nrespectively. The densities related to hadronic and dark\nsectors are\nρs=ρsp+ρsn, (12)\nρ=ρp+ρn, (13)\nρ3=\n¯ψγ0τ3ψ\u000b\n=ρp−ρn= (2yp−1)ρ, (14)\nρsp,n=\n¯ψp,nψp,n\u000b\n=M∗\nπ2ZkFp,n\n0k2dk\n(k2+M∗2)1/2,(15)\nρDM\ns=⟨¯χχ⟩=M∗\nχ\nπ2ZkDM\nF\n0k2dk\n(k2+M∗2χ)1/2. (16)\nThe proton fraction is yp=ρp/ρ, and the vector densi-\nties are related to the respective Fermi momenta through\nρp,n= 2k3\nFp,n/(3π2), and ρχ= 2kDM\nF3/(3π2). In our\nstudy, we keep this last quantity as a free parameter, as\nwell as the dark fermion mass, for which the values are\ntaken from the range 50 GeV ⩽Mχ⩽500 GeV. Such\ninterval for Mχ, along with ξ= 0.01, ensures that the\nspin-independent scattering cross-section is compatible\nwith data provided by PandaX-II [48], LUX [49], and\nDarkSide [50] collaborations.\nThe energy density and pressure of the system is calcu-\nlated from the energy-momentum tensor, Tµν, obtained3\nthrough Eq. (1). Such expressions are given, respectively,\nby\nε(σ, ω0, b0(3), ρ, y p, h, ρ χ) =m2\nσσ2\n2+Aσ3\n3+Bσ4\n4\n−m2\nωω2\n0\n2−m2\nρb2\n0(3)\n2+gωω0ρ+gρ\n2b0(3)ρ3+m2\nhh2\n2\n+εp\nkin+εn\nkin+εDM\nkin, (17)\nand\nP(σ, ω0, b0(3), ρ, y p, h, ρ χ) =−m2\nσσ2\n2−Aσ3\n3−Bσ4\n4\n+m2\nωω2\n0\n2+m2\nρb2\n0(3)\n2−m2\nhh2\n2+Pp\nkin+Pn\nkin+PDM\nkin,\n(18)\nwith the kinetic terms written as\nεDM\nkin=1\nπ2ZkDM\nF\n0k2(k2+M∗2\nχ)1/2dk, (19)\nPDM\nkin=1\n3π2ZkDM\nF\n0k4dk\n(k2+M∗2χ)1/2, (20)\nϵp,n\nkin=1\nπ2ZkFp,n\n0k2(k2+M∗2)1/2dk, (21)\nPp,n\nkin=1\n3π2ZkF n,p\n0k4dk\n(k2+M∗2)1/2. (22)\nIn these expressions, σ,ω0,b0(3), and hare the mean-field\nvalues of the respective mesonic fields of the model.\nWe also include electrons and muons in the system\nin order to correctly describe the core of neutron stars.\nTherefore, energy density and pressure become\nϵcore=ε(σ, ω0, b0(3), ρ, y p, h, ρ χ) +µ4\ne\n4π2\n+1\nπ2Z√\nµ2µ−m2µ\n0dk k2(k2+m2\nµ)1/2, (23)\nand\npcore=P(σ, ω0, b0(3), ρ, y p, h, ρ χ) +µ4\ne\n12π2\n+1\n3π2Z√\nµ2µ−m2µ\n0dk k4\n(k2+m2µ)1/2. (24)\nThe last terms of the above equations represent the ther-\nmodynamical quantities of massless electrons, and muons\nwith mass mµ= 105 .7 MeV. The chemical potentials\nof these leptons, denoted by µeandµµ, are related to\ntheir respective densities through ρe=µ3\ne/(3π2), and\nρµ= [(µ2\nµ−m2\nµ)3/2]/(3π2). For a beta-equilibrated sys-\ntem submitted to charge neutrality, the following condi-\ntions apply, namely, ρp−ρe=ρµ, and µn−µp=µe=µµ.\nFinally, we consider two different regions for the descrip-\ntion of the neutron star crust: the outer (OC) and theinner crust (IC). For the former we use the equations pro-\nposed by Baym, Pethick and Sutherland (BPS) [51] in a\ndensity region of 6 .3×10−12fm−3⩽ρ⩽2.5×10−4fm−3.\nFor the inner crust we use a polytropic relation be-\ntween energy density and pressure, namely, pIC(ϵIC) =\nA+Bϵ4/3\nIC. We match this form to the BPS, and to the\ncore equations. The latter is connected at the the core-\ncrust transition pressure and energy density, calculated\nthrough the thermodynamical method [52–54]. Total en-\nergy density and total pressure of the stellar matter is\nthen given by\nϵ=ϵcore+ϵOC+ϵIC (25)\nand\np=pcore+pOC+pIC, (26)\nrespectively.\nIII. GENERAL RELATIVISTIC FORMALISM\nA. The stellar equilibrium equations\nThe static equilibrium configurations of a compact star\ncomposed of a perfect fluid are obtained through the\nTolman-Oppenheimer-Volkoff equations\ndp\ndr=−ϵm\nr2\u0012\n1 +p\nϵ\u0013\u00121 +4πpr3\nm\n1−2m\nr\u0013\n, (27)\ndν\ndr=−2\nϵdp\ndr\u0012\n1 +p\nϵ\u0013−1\n, (28)\ndm\ndr= 4πr2ϵ. (29)\nThe variables m(r) and ν(r) are respectively the gravi-\ntational mass inside the radius rand a metric potential.\nThe pressure pand the mass-energy density ϵare con-\nnected by the equations of state. To integrate the system\nof Eqs. (27)-(29) from the center ( r= 0) to the star’s\nsurface ( r=R), some conditions are required. At the\ncenter of the star ( r= 0), we have\nϵ(0) = ϵc, p(0) = pc,and m(0) = 0 . (30)\nThe surface of the star is found when p(R) = 0. At this\npoint\nν(R) = ln\u0012\n1−2M\nR\u0013\n, (31)\nwith Mrepresenting the total stellar mass.\nB. The nonradial oscillation equations\nTo investigate the non-radial oscillation, both space-\ntime and fluid variables are perturbed. The perturba-\ntions are replaced in the Einstein field equation, in the4\nenergy-momentum tensor, and in the baryon number\nconservation maintaining only the first-order variables.\nFollowing Ref. [55, 56], we adopt the perturbed line\nelement of the form\nds2=−eν(1 +rℓH0Yℓ\nmeiωt)dt2−2iωrℓ+1H1Yℓ\nmeiωtdtdr\n+eλ(1−rℓH0Yℓ\nmeiωt)dr2\n+r2(1−rℓKYℓ\nmeiωt)(dθ2+ sin2θdϕ2), (32)\nwith H0=H0(r),H1=H1(r), and K=K(r) being\nfunctions of the radial coordinate ronly, ωdepicting the\neigenfrequency of oscillation, and Yℓ\nm=Yℓ\nm(θ, ϕ) rep-\nresenting the even-parity spherical harmonic functions.\nSince the small perturbations are set by the Lagrangian\nfluid displacement ξβ, we consider:\nξr=rℓ−1e−λ/2WYℓ\nmeiωt, (33)\nξθ=−rℓ−2V ∂θYℓ\nmeiωt, (34)\nξϕ=−rℓ(rsinθ)−2V ∂ϕYℓ\nmeiωt, (35)\nwhere ξ0= 0.\nWith this, non-radial oscillations are described by the\nfollowing set of first-order linear differential equations\n[56]:\nH′\n1=−r−1\u0014\nℓ+ 1 +2Meλ\nr+ 4πr2eλ(p−ϵ)\u0015\nH1\n+eλr−1[H0+K−16π(ϵ+p)V], (36)\nK′=r−1H0+ℓ(ℓ+ 1)\n2rH1−\u0014(ℓ+ 1)\nr−ν′\n2\u0015\nK\n−8π(ϵ+p)eλ/2r−1W , (37)\nW′=−(ℓ+ 1)r−1W+reλ/2h\ne−ν/2γ−1p−1X\n−ℓ(ℓ+ 1)r−2V+1\n2H0+K\u0015\n, (38)\nX′=−ℓr−1X+(ϵ+p)eν/2\n2\"\u00121\nr+ν′\n2\u0013\nH0\n+\u0012\nrω2e−ν+ℓ(ℓ+ 1)\n2r\u0013\nH1+\u00123\n2ν′−1\nr\u0013\nK\n−ℓ(ℓ+ 1)r−2ν′V−2r−1 \n4π(ϵ+p)eλ/2\n+ω2eλ/2−ν−r2\n2(e−λ/2r−2ν′)′!\nW#\n, (39)\nwhere the prime stands a derivative with respect to rand\nγdepicts the adiabatic index. The function Xis given\nby\nX=ω2(ϵ+p)e−ν/2V−p′\nre(ν−λ)/2W\n+1\n2(ϵ+p)eν/2H0, (40)\nwith H0fulfilling the algebraic relation\na1H0=a2X−a3H1+a4K, (41)where\na1= 3M+1\n2(l+ 2)( l−1)r+ 4πr3p, (42)\na2= 8πr3e−ν/2, (43)\na3=1\n2l(l+ 1)( M+ 4πr3p)−ω2r3e−(λ+ν),(44)\na4=1\n2(l+ 2)( l−1)r−ω2r3e−ν\n−r−1eλ(M+ 4πr3p)(3M−r+ 4πr3p).(45)\nOutside the stellar structure configuration, the pertur-\nbation functions that describe the motion of the fluid W\nandVvanish, and the system of differential equations\nreduces to the Zerilli equation:\nd2Z\ndr∗2= [VZ(r∗)−ω2]Z, (46)\nwhere Z(r∗) and dZ(r∗)/dr∗are related to the functions\nH0(r) and K(r) through the transformations given in\nRefs. [55, 56]. Z(r∗) depends on the “tortoise” coordi-\nnate which is given by\nr∗=r+ 2Mln\u0010r\n2M−1\u0011\n, (47)\nand the effective potential VZ(r∗) yields\nVZ(r∗) =(1−2M/r)\nr3(nr+ 3M)2[2n2(n+ 1)r3+ 6n2Mr2\n+18nM2r+ 18M3],(48)\nwith n= (l−1)(l+ 2)/2.\nFor given values of landω, the set of Eqs. (36) −(39)\nhas four linearly independent solutions. The physical so-\nlution obtained must verify boundary conditions: (a) The\nperturbation functions have to be regular throughout the\nentire star; thus implying that in r= 0 must also be fi-\nnite, since at this point non-radial oscillation equations\nare singular. To insert such condition, the solution near\nr= 0 must be expanded using a power series (for more\ndetail about the process, see Ref. [55]. (b) Since the La-\ngrangian perturbation of pressure vanishes at the star’s\nsurface r=R. This boundary condition is equivalent to\nX(R) = 0. For a set of values of landω, there is a so-\nlution that fulfills the above boundary conditions inside\nthe star.\nIn general, in the exterior of the star, the perturbed\nredline element depicts a mixture of outgoing and ingoing\ngravitational waves. Due to we focus on the purely outgo-\ning gravitational radiation at r=∞, the Zerilli equation\nis used. The eigenfrequency of oscillation that satisfies\nthis requirement represents the quasinormal modes of the\nstellar model. process to solve the above equations is de-\ntailed in [55, 56].\nDifferent oscillation modes can be investigated using\nthe aforementioned equations. The purely gravitational\nmodes ( w-modes) do not induce fluid motion and are5\n6 8 10 12 14 1611.522.53\n  R [km]  M [Solar Masses]kf DM = 0 GeV\nkf DM = 0.02 GeV\nkf DM = 0.04 GeV\nkf DM = 0.06 GeV\nFIG. 1. Mass, normalized in Sun’s masses, versus radius for\nsome values of MχandkDM\nF. The curve in magenta repre-\nsents the case without dark matter. The curves of the same\ncolor represent the results for a fixed value of kDM\nFand the\ncurves of the same type depict results for a fixed value of Mχ.\nThe solid curves correspond to Mχ= 50 MeV, dashed curves\ncorrespond to Mχ= 200 MeV, and solid curves with circles\ncorrespond to the case Mχ= 500 MeV. Red and blue re-\ngions represent the masses and radii intervals (95%) from PSR\nJ0030+0451 and PSR J0740+6620 measured by NICER [57–\n60]. The magenta horizontal line includes all observed neu-\ntron stars over 2 solar masses, which include the pulsars PSR\nJ1614 −2230, PSR J0348 + 0432, and PSR J0740 + 6620 [61–\n63]\n.\nhighly damped. The fluid pulsation modes ( f0,g, and\np-modes) are usually grouped according to the origin of\nthe restoring force which prevails in bringing the per-\nturbed element of fluid back to the equilibrium position;\ne.g., the buoyancy in the case of g-modes or a gradient\nof pressure for p-modes. It is widely known that the fre-\nquencies of the g-modes are lower than those of p-modes,\nand these two sets are separated by the frequency of the\nf0-mode. The f0-mode frequency is proportional to the\nsquare root of the mean density of the star and tends to\nbe independent of the details of the stellar structure. In\nthis work, we analyze only on the f0-mode because it is\nexpected to be the most excited in astrophysical events\nand therefore the main contribution to the GW emission\nof the star should be expected at this frequency.\nC. The tidal deformability equations\nThe analysis of tidal deformability is commonly inves-\ntigated in binary compact object systems. The grav-\nitational effects caused by one star can bring out the\ndeformation of its partner. This deformation was inves-\ntigated by Damour and Nagar, Binnington and Poisson\n[64, 65]. They found that the tidal deformation of a neu-tron star is determined by the gravito-electric Kel\n2and\ngravito-magnetic Kmag\n2Love numbers, which are respec-\ntively connected with the mass quadrupole and with the\ncurrent quadrupole induced by the companion star. Ad-\nditional research carried out by Flanagan and Hindeler\nfound that a single detection should be sufficient to im-\npose upper limits on Kel\n2at 90% confidence level [66].\nSince then, intensive research has been carried out on the\ncalculation of the Love numbers of neutron stars [67–71].\nIn a binary stellar system, the induced quadrupole mo-\nment Qijin one neutron star because of the external tidal\nfieldEijinduced by a partner compact object can be ex-\npressed as [71]\nQij=−λEij, (49)\nwhere λis the tidal deformability parameter, which can\nbe written in terms of the quadrupole ( l= 2) tidal Love\nnumber k2as\nλ=2\n3k2R5. (50)\nTo determine k2we need to solve the first-order differen-\ntial equation\nrdy\ndr+y2+yF(r) +r2Q(r) = 0 , (51)\nwith the functions F(r) and Q(r) given by\nF(r) = [1 −4πr2(ε−p)]/E (52)\nQ(r) = 4 π\u0014\n5ε+ 9p+ (ε+p)\u0012∂p\n∂ε\u0013\n−6\n4πr2\u0015\n/E\n−4\u0014m+ 4πr3p\nr2E\u00152\n, (53)\nwith E= 1−2m/r. Therefore the Love number k2is\ncomputed to be\nk2=8C5\n5(1−2C)2[2 +C(yR−1)−yR]×\n×n\n2C[6−3yR+ 3C(5yR−8)]\n+ 4C3[13−11yR+C(3yR−2) + 2 C2(1 +yR)]\n+ 3(1−2C2)[2−yR+ 2C(yR−1)]ln(1 −2C)o−1\n,\n(54)\nwhere the function yR=y(r=R) and C=M/R is the\ncompactness parameter. Equation (51) has to be solved\ncoupled with the TOV equations.\nThe dimensionless tidal deformability Λ (i.e., the di-\nmensionless version of λ) is connected with the Love num-\nber through the relation\nΛ =2k2\n3C5. (55)6\n1 1.5 2 2.5101102103\n  M [Solar Masses]  Λkf DM = 0 GeV\nkf DM = 0.02 GeV\nkf DM = 0.04 GeV\nkf DM = 0.06 GeV\nFIG. 2. Tidal deformability versus mass different values of Mχ\nandkDM\nF. The curve in magenta stands for the case without\ndark matter. The curves with the same type of color represent\nresults obtained with the same value of kDM\nFand the curves\nof the same type depict the results found with the same value\nMχ. The solid curves correspond to Mχ= 50 MeV, dashed\ncurves correspond to Mχ= 200 MeV, and solid curves with\ncircles correspond to the case Mχ= 500 MeV. The vertical\nsolid line depicts Λ 1.4= 190+390\n−120published in Ref. [23].\nkDM\nFMχMmax R f0 τ\n(GeV) (MeV) (M⊙)(km) (kHz) (s)\n0.00 - 2.760 13.39 1.930 1.940\n50 2.750 13.33 1.920 1.920\n0.02 200 2.720 3.110 1.980 1.940\n500 2.670 12.72 2.000 1.870\n50 2.680 12.85 1.990 1.890\n0.04 200 2.690 11.61 2.150 1.780\n500 2.220 10.08 3.280 1.230\n50 2.520 11.83 2.120 1.780\n0.06 200 2.100 9.450 2.580 1.520\n500 1.660 7.330 3.280 1.230\nTABLE I. Maximum mass values Mmax, and their correspond-\ning radius R, fundamental mode frequency f0, and damping\ntime of the fundamental mode τ, calculated for different val-\nues of fermi moment of dark matter kDM\nFand dark matter\nmass Mχ.\nIV. RESULTS\nA. General remarks\nThe full general relativistic equations for the oscilla-\ntions of neutron stars (when the fluid and spacetime vari-\nables are perturbed), where the energy-momentum ten-\nsor is coupled with gravitational radiation (for l= 2, i.e.\nquadrupole oscillations), are treated in the following sub-\nsections. In addition, there is a well-known framework\ncalled the Cowling approximation, where it is considered\nthat the fluid perturbations are weakly coupled with the\nspacetime vibrations, so in this case we can consider only\na set of differential equations for the fluid variables.\nIn the case of results obtained by using the Cowlingapproximation, we employ a code that implements the\nshooting method. In this procedure, there exist bound-\nary conditions to be considered in the center and on the\nsurface of the star. In the first stage, in the center of the\nstar, we consider the initial values of the functions W\nandV(which satisfy regularity conditions), and we also\nintroduce a test value for the f0-mode frequency, which\nis typically around 2 .5 kHz. Of course, this proof value\nis suitable for a standard neutron star. After, we pro-\nceed with the numerical integration which starts in the\ncenter and ends on the surface of the star; this numer-\nical interaction is developed through the Runge Kutta\nmethod. After finishing the integration process, we ar-\nrive at the surface of the star. At this point, we use\nthe Newton-Raphson method to see if the estimated fre-\nquency value (the value of the eigenfunction) satisfies the\nboundary condition. If we do not find a good precision,\nthe test value for the frequency is corrected and all the\nprocess is initialized again, until reach the desired preci-\nsion for the oscillation mode, which will be the desired\nfrequency. Finally, we have to mention that the numer-\nical estimation considering the Cowling approximation\nhas two main objectives; the first one is the direct cal-\nculation of the frequency of the f0-mode and the second\none is to use this value as the best proof value for the\nnumerical integration of the full version of the general\nrelativistic oscillation equations. This numerical code al-\nlows us to obtain the f0-mode reported in literature, e.g.,\nit reproduces the results of Ref. [72].\nIn a more rigorous treatment of neutron star seismol-\nogy, we have to consider the coupling between the fluid\nmovement and the metric perturbations. In this case, the\nnumerical procedure is a very elaborate and long process\nof shooting method. The first step is to use as our best\ntest value for the gravitational wave frequency the one ob-\ntained in the framework of the Cowling approximation,\ni.e.,ωCow. With such proof value, we begin the integra-\ntion inside the star, from the center towards the star’s\nsurface. At first, it is selected three linearly independent\nsolutions compatible with the regularity conditions in the\ncenter of the star, and then we begin with the numeri-\ncal integration from r= 0 to the point R/2 inside the\nstar. After that, a second integration is realized, where\nwe select two linearly independent solutions compatible\nwith the boundary condition on the star’s surface, this\nprocess is made from the surface Rto the point R/2. To\ncomplete the procedure inside the star, the five solutions\nare combined to obtain compatibility with the boundary\nconditions at the center and on the surface of the star.\nThereafter, we continue with the integration outside the\nstar. The boundary values for the Zerilli function and\nits derivative at the surface of the star can always be ob-\ntained from the values of H0(R) and K(R) obtained from\nthe integration inside the compact star. Then, Z(r∗) can\nbe determined outside the star by integrating the Zer-\nilli equation. At an asymptotically large radius (which\nrepresents the infinity), the two linearly independent so-\nlutions of the Zerilli equation may be expressed as power7\n0 0.5 1 1.5 2 2.5 30.511.522.533.5\n  M [Solar Masses]  f0 [kHz]\n1 1.5 2 2.5100200300400500600700800\n  M [Solar Masses]  τ[ms]kf DM = 0 GeV\nkf DM = 0.02 GeV\nkf DM = 0.04 GeV\nkf DM = 0.06 GeV\n0 200 400 600 800 1000 1200 14001.522.533.5\n  Λ  f0 [kHz]\n0 200 400 600 800 1000 1200 1400100200300400\n  Λ  τ[ms]\nFIG. 3. Fundamental mode frequency f0and damping time of fundamental mode τagainst the total mass M/M ⊙are shown\non the top panel and versus the tidal deformability Λ on the bottom panel. In all cases, different values of MχandkDM\nF\nare employed. The curves in magenta depict the case without dark matter. The curves of the same color represent the\nresults for a fixed value of kDM\nFand the curves of the same type stand results for a fixed value of Mχ. The solid curves\ncorrespond to Mχ= 50 MeV, dashed curves correspond to Mχ= 200 MeV, and solid curves with circles correspond to the\ncaseMχ= 500 MeV. In the bottom panels, the dashed vertical lines represent the tidal deformability obtained from the event\nGW170817, i.e., Λ 1.4= 190+390\n−120, reported Ref. [23].\nseries\nZ−(r∗) =e−iωr∗∞X\nj=0βjr−j(56)\nand\nZ+(r∗) =eiωr∗∞X\nj=0¯βj (57)\nwhere Z−depicts the radiated waves and Z+the ingoing\nwaves. The general asymptotic solution is given by the\nlinear combination\nZ(r∗) =A(ω)Z−(r∗) +B(ω)Z+(r∗), (58)\nwith A(ω) and B(ω) being complex numbers. The inte-\ngration of the Zerilli differential equation from the surfaceof the star to the infinity ( r∞≈55ω−1) allows finding Z\nanddZ/dr . Then, by using Z(r∗) atr∗=r∗\n∞, we can ob-\ntainB(ω) = 0 against to Z(r∗\n∞) and dZ/dr (r∗\n∞) for each\nvalue of ω. The boundary condition at infinity requires\nthat we only have outgoing gravitational radiation, i.e.,\nto find the frequencies of quasi-normal modes we must\nobtain the roots of B(ω) = 0. This is achieved by deter-\nmining B(ω) for three different close trial values of the\neigenfrequency ω. Then, we fit a quadratic polynomial\nB(ω) =γ0+γ1ω+γ2ω2to the computed values of ωto\nobtain an approximate root of B(ω) = 0. Finally, we iter-\nate this procedure using the real part of the approximate\nroot as an input for the next integration of the oscillation\nequations. This interaction is repeated until the real part\nvalue of the quasi-normal mode changes from one step to\nthe next by less than one part in 108. The imaginary8\npart of ωis connected to the gravitational damping time\nof the oscillation τ. The numerical code implemented in\nthis case allows us to reproduce the results shown in the\nreference [73].\nIn this work, we analyze the effects of dark matter\ninside neutron stars. Since dark matter would interact\nvery weakly with normal matter, its possible effects just\ncould be felt only in a gravitational field.\nFor our purposes, the dark matter mass Mχand the\nFermi momentum of dark matter particles kDM\nFvalues\nwere selected to satisfy the restrictions obtained of both\nstellar mass and the tidal deformability of neutron stars\nobserved; namely, we use Mχ= 50 ,200, and 500 MeV\nandkDM\nF= 0,0.02,0.04, and 0 .06 GeV. As a consequence,\nby using the NL3* EoS, we reproduce the mass, radius,\nf0-mode frequency, damping time of the fundamental\nmode, and tidal deformability.\nB. Tidal deformability, frequency of oscillations,\nand damping time of fundamental mode for a\nhadronic star\nFrom Fig. 1, the mass, normalized in solar masses, as\na function of the total radius is plotted for some values of\nMχandkDM\nF. The observation data belong to the NICER\nrestrictions obtained from the pulsars PSR J0030 + 0451\n[57, 60] and PSR J0740 + 6620 [58, 59]. There are also\nshown the bands of the pulsars PSR J0740 + 6620 [63],\nPSR J0348 + 0432 [62], and PSR J1614 + 2230 [61]. In\nthe figure, we observe that there is a strong dependence\nof the stellar mass on the parameters MχandkDM\nF. For\na fixed Mχ, an increase of kDM\nFbring as a consequence\na decrease of the maximum mass. It means that the in-\ncrease in the fermi momentum of dark matter produces\na softness in the EOS, therefore there is a lower pressure,\nthus obtaining an equilibrium configuration with lower\nmasses. In the same figure, we also observe that an in-\ncrease in Mχinduces a decrease in the stellar mass; i.e.,\nwe have a softer EOS when that parameter is increased.\nFrom these results, we observe that the change of Mχand\nkDM\nFallows us to find some results more precise and closer\nto empirical evidence of neutron stars PSR J0030+0451,\nPSR J0740+6620, PSR J0740+6620, PSR J0348+0432,\nand PSR J1614 + 2230.\nThe deformation of a compact star within a binary sys-\ntem is measured by the tidal deformability Λ. In this way,\nFig. 2 shows the deformability parameter as a function\nof the mass. These results are compared with the case\nΛ1.4= 190+390\n−120reported by LVC in [23]. In all curves, we\nfound a monotonic decay of the deformability with the\nincrement of the mass. In addition, we observe the effects\nof the quantities MχandkDM\nFon the tidal deformability.\nFor an interval of total masses, we note that lower values\nΛ are obtained when larger values of both MχandkDM\nF\nare used. Note that there are curves that are into the\ninterval Λ 1.4described by LVC in [23].\nSince our objective is to determine a relationship be-\n1 1.5 2 2.5020406080100\n  M [Solar Masses]  diff [%]FIG. 4. The relative difference between the frequency of the\nfundamental mode in the full linearized equations and the rel-\nativistic Cowling approximation, diff = 100 ×(f0−fcow\n0)/f0,\nversus the total mass for all MχandkDM\nFconsidered in Fig.\n4.\ntween the mass stellar configuration, tidal deformability,\nand frequency of the fundamental mode we plot the f0-\nmode frequency and its respective damping time as a\nfunction of the total mass on the top panel of Fig. 3\nand against the tidal deformability on the bottom panel\nof Fig. 3, where are employed for different values of\nMχandkDM\nF. The results on the bottom panels are\ncompared with the tidal deformability derived from the\nevent GW170817, Λ 1.4= 190+390\n−120, communicated in [23].\nThese plots have justification in the physics of the bi-\nnary system because when both stars are orbiting each\nother, there could exist resonance and, in principle, the\nfundamental mode of both stars could be excited. In this\nway, in this figure, we see the change in MχandkDM\nFpro-\nduces a systematic shifting in the curves of f0(M),τ(M),\nf0(Λ), and τ(Λ); review also Table I to see the change of\nthe maximum total masses with their respective param-\neters R,f0, and τwith MχandkDM\nFemployed in Fig. 3.\nMoreover, in the range placed by the observation, we see\nthat the f0-mode frequency has a nearly linear behavior.\nIn addition, the frequency of the f0mode obtained in\nthe full linearized equations of general relativity is com-\npared with the relativistic Cowling approximation in Fig.\n4. We found that the f0-mode with the approximation\ndiffers by less than about 40%, in low-mass stars, and\ndiffers by less than about 20%, in high-mass stars.\nThef0-mode frequency of oscillations, damping time\nmode, and tidal deformability as a function of the fermi\nmomentum of dark matter are shown in Fig. 5 for three\nvalues of dark matter mass Mχ= 50,200, and 500 MeV.\nThe results presented correspond to the equilibrium con-\nfiguration with total mass M= 1.4M⊙. The curves\nΛ×kDM\nFare constrasted with the case Λ 1.4= 190+390\n−120re-\nported by LVC in Ref. [23]. From the figure, we see that\nequilibrium solutions with 1 .4 solar masses with larger f0-9\n0 0.01 0.02 0.03 0.04 0.05 0.061.522.53\n  kfDM [GeV]  f0[kHz]Mχ* = 50MeV\nMχ* = 200MeV\nMχ* = 50MeV\n0 0.01 0.02 0.03 0.04 0.05 0.060.10.150.20.250.30.35\n  kfDM [GeV]  τ[s]\n0 0.01 0.02 0.03 0.04 0.05 0.06020040060080010001200\n  kfDM [GeV]  Λ\nFIG. 5. The f0-mode oscillation frequency, damping time parameter, and tidal deformability against the fermi momentum of\ndark matter are shown respectively on the left, middle, and right panels. The results connected on straight dashed lines on red,\nblue, and black correspond to Mχ= 50,200, and 500 MeV. All values presented concern the equilibrium configurations with\ntotal masses M= 1.4M⊙. The horizontal straight line represent �� 1.4= 190+390\n−120obtained by LVC in Ref. [23].\n0 200 400 600 800 1000 1200 14000500100015002000\n  Λ1  Λ2\n  90% \n  50% kf DM = 0.04GeV; Mχ*  = 200MeV\nkf DM = 0.04GeV; Mχ*  = 500MeV\nkf DM = 0.06GeV; Mχ*  = 50MeV\nkf DM = 0.06GeV; Mχ*  = 200MeV\nkf DM = 0.06GeV; Mχ*  = 500MeV\n2 2.5 322.53\n  f0(1)[kHz]  f0(2)[kHz]\n100 150 200 250 300 350100150200250300350\n  τ1[ms]  τ2[ms]\nFIG. 6. In the Left, middle, and bottom panels are respectively plotted Λ 2×Λ1,f0(2)×f0(1), and τ2×τ1for the components\nof the event GW170817 for different values of kDM\nFandMχ. The wine lines on the left panel depict the LIGO-Virgo confidence\ncurves of 50% and 90% levels [22] and the solid diagonal appearing on the left, middle, and right panels correspond respectively\nto Λ 1= Λ 2,f0(1)=f0(2), and τ1=τ2.\nmode frequency and lower tidal deformability are found\nwhen higher values of kDM\nFandMχare considered. Com-\nparing the results reported here with those published in\n[23], we can put limits on the values of kDM\nF, and Mχ\nused.\nC. Tidal deformability, frequency of oscillations,\nand damping time of fundamental mode for a binary\nhadronic star system\nIn Ref. [22], by using the data reported by LVC, au-\nthors implement some constraints on the dimensionless\ntidal deformability of a binary system Λ 1and Λ 2, with\nΛ1representing a star’s dimensionless tidal deformability\nin the system and Λ 2depicting the same parameter but\nof its companion. In this way, in Fig. 6, we present the\nrelation Λ 2×Λ1, where the numerical results are obtained\nby choosing a value of M1and finding M2through the\nchirp mass M= 1.188M⊙[22] which is determined by\nM=(M1M2)3/5\n(M1 +M2)1/5. (59)In addition, the value of M1andM2are respectively in\nthe interval 1 .36M⊙≤M1≤1.60M⊙and 1 .17M⊙≤\nM2≤1.36M⊙. In the figure are also placed the credi-\nbility levels linked to the GW170817 event instituted by\nLVC in the low-spin prior scenario. We note that the\nlarger values of kDM\nFand lower values of Mχ, and vice\nversa, allow us to obtain values within the confidence\nlines extracted from Ref. [22]. In Fig. 6 also appears\nthe curves f0(2)×f0(1) and τ2×τ1for the same binary\nsystem. In these two relations, for all values of kDM\nFand\nMχconsidered, we see the dependence of the frequency\nand the damping time of one star concerning its partner.\nIn the first one, we note that f0(2)decays with the in-\ncrement of f0(1)and in the second one τ2grows with the\ndecline of τ1.\nV. CONCLUSIONS\nIn the present article we study the effects of the dark\nmatter mass Mχand the Fermi momentum of dark mat-\nter particles kDM\nFin the fluid pulsation mode, damp-\ning time parameter, and tidal deformability of hadronic\nstars. These studies are developed by the numerical in-10\ntegration of the hydrostatic equilibrium, nonradial os-\ncillation, and tidal deformability equations. The matter\nwithin the stars is considered the NL3* equation of state.\nAbout the total mass, it is observed that it changes no-\ntoriously with MχandkDM\nF. From an appropriate com-\nbination of MχandkDM\nF, we can determine that some\nsolutions that are close to the empirical evidence of neu-\ntron stars PSR J0030 + 0451, PSR J0740 + 6620, PSR\nJ0740+6620, PSR J0348+0432, and PSR J1614 +2230.\nConcerning the fluid pulsation mode, damping time\nparameter, and tidal deformability, they are considerably\naffected by MχandkDM\nF. We found that the f0-mode\ngrows with MχandkDM\nFand the damping time and tidal\ndeformability decay with these parameters. We also con-\ntrast these results with the observational reported by the\nLVC from the GW170817 event. In this scheme, the nu-\nmerical results reported are in the range of the observa-\ntional data aforementioned.\nWe also calculated the frequencies of the fundamental\nmode by using the Cowling approximation, as done in\nthe reference [39]; where the authors showed that dark\nmatter affects the f-mode frequency in the RMF model\nwith and without hyperons. In our model, it was de-\nrived the same effect in the presence of dark matter in\nhadronic stars. We showed that for larger mass values,\nthe Cowling method provides a good approximation com-\npared to the solution of the complete equation; the dif-\nference between these two models is below 20% for starswith masses above 2 M⊙(see Fig. 4).\nThe influence of MχandkDM\nFon the tidal deformabil-\nity,f-mode frequency, and damping time parameter for\na binary hadronic star system with equal chirp mass as\nthe GW170817 event have been also studied. We found\nthe dependence of f0(1)-mode, τ1, and Λ 1of one star with\nf0(2)-mode, τ2, and Λ 2of its partner and how these re-\nlations change with the dark matter mass and the fermi\nmomentum of dark matter particles.\nACKNOWLEDGMENTS\nThis work is a part of the project INCT-FNA proc.\nNo. 464898 /2014−5. It is also supported by Conselho\nNacional de Desenvolvimento Cient´ ıfico e Tecnol´ ogico\n(CNPq) under Grants No. 401565 /2023−8 (C.H.L., O.L,\nM.D.), No. 305327 /2023−2 (C.H.L.), No. 312410 /2020−\n4 (O. L.), and No. 308528 /2021−2 (M. D.). J. D. V.\nA. thanks Universidad Privada del Norte and Universi-\ndad Nacional Mayor de San Marcos for the financial sup-\nport - RR N º005753-2021-R /UNMSM under the project\nnumber B21131781. C. V. F. also makes his acknowl-\nedgements for the financial support of the productivity\nprogram of the Conselho Nacional de Desenvolvimento\nCient´ ıfico e Tecnol´ ogico (CNPq), with project number\n304569 /2022−4.\n[1] Planck Collaboration, Aghanim, N., Akrami, Y., Ash-\ndown, M., Aumont, J., Baccigalupi, C., Ballardini, M.,\nBanday, A. J., Barreiro, R. B., Bartolo, N., Basak, S.,\nBattye, R., Benabed, K., etal., A&A 641, A6 (2020).\n[2] F. Zwicky, Helvetica Physica Acta 6, 110 (1933).\n[3] H. Andernach and F. 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In\nthis paper, we provide a theoretical analysis\nof using the sigmoid loss in contrastive learn-\ning, in the perspective of the geometric struc-\nture of learned embeddings. First, we pro-\npose the double-Constant Embedding Model\n(CCEM), a framework for parameterizing var-\nious well-known embedding structures by a\nsingle variable. Interestingly, the proposed\nCCEMis proven to contain the optimal em-\nbedding with respect to the sigmoid loss. Sec-\nond, we mathematically analyze the optimal\nembedding minimizing the sigmoid loss for\ncontrastive learning. The optimal embedding\nranges from simplex equiangular-tight-frame\nto antipodal structure, depending on the tem-\nperature parameter used in the sigmoid loss.\nThird, our experimental results on synthetic\ndatasets coincide with the theoretical results\non the optimal embedding structures.\n1 Introduction\nContrastive learning (CL), which learns the embedding\nspace based on the comparison between training data,\nProceedings of the 27thInternational Conference on Artifi-\ncial Intelligence and Statistics (AISTATS) 2024, Valencia,\nSpain. PMLR: Volume 238. Copyright 2024 by the au-\nthor(s).is recently considered as one of the popular method for\nlearning good representations (Liu et al., 2021; Jaiswal\net al., 2020; Chen et al., 2020a; Tian et al., 2020). The\nbasic idea is, to train an encoder such that similar train-\ning data (called positive pairs) are mapped to closer\nembedding vectors, while the embedding vectors for\ndissimilar training data (called negative pairs) are far\napart. For example, in Chen et al. (2020a), different\naugmentations of the same image are labeled as a posi-\ntive pair, while augmentations of different images are\nlabeled as a negative pair. In Radford et al. (2021),\nthe caption text and the image extracted from a sin-\ngle captioned image are considered as a positive pair,\nwhile those extracted from different captioned images\nare considered as a negative pair. In general, different\nviews(augmentations or modalities) on a single entity\nform a positive pair, while views on different entities\nform a negative pair.\nInterestingly, self-supervised learning trained by such\ncomparison-based criterion alone, without any labeled\ndata, is sufficient to learn meaningful features (He\net al., 2020; Chen et al., 2020a; Radford et al., 2021;\nJia et al., 2021; Girdhar et al., 2023), recently outper-\nforming the features learned by supervised learning.\nVarious ways of defining contrastive loss have been con-\nsidered in the past years (Schroff et al., 2015; Oh Song\net al., 2016; Sohn, 2016; Gutmann and Hyvärinen, 2010,\n2012; Hyvarinen et al., 2019), see Appendix A for more\ndetails. Especially, one of the pioneering works on\nmulti-modal CL called CLIP (Radford et al., 2021)\nuses the InfoNCE loss (Oord et al., 2018), which is\ntheoretically analyzed in recent papers (Saunshi et al.,\n2019, 2022; Wang and Isola, 2020; Wang and Liu, 2021;\nChen et al., 2021; Arora et al., 2019). Above all, Lu and\nSteinerberger (2022) shows that the optimal solution\nof the variational problem of minimizing cross-entropy\nloss, which coincides with the InfoNCE loss, is struc-\ntured by the simplex equiangular-tight-frame (ETF) if\nfeature vectors are in a sufficiently high-dimensional\nspace.\nRecently, a variant of CLIP called SigLIP is pro-\nposed (Zhai et al., 2023), which uses the sigmoid loss\ninstead of the InfoNCE loss, motivated by the fact thatarXiv:2402.12613v1  [cs.LG]  20 Feb 2024Analysis of Using Sigmoid Loss for Contrastive Learning\ncomputing with the sigmoid loss is much more efficient\nthan using the InfoNCE loss. More precisely, CL us-\ning the InfoNCE loss reveals remarkable performance\nonly if the negative sample size, which is limited by\nits mini-batch size, is sufficiently large (Chen et al.,\n2020a; Zhang et al., 2022a). However, the necessity for\na larger batch size imposes the computation burdens\non CL using the InfoNCE loss. On the other hand,\nlearning CL with SigLIP is independent of batch size,\nenabling comparable performance with significantly re-\nduced computational demands. Unfortunately, there is\nno theoretical discussions on the behavior of SigLIP.\nOur Contributions In this paper, we have taken the\nfirst step towards understanding SigLIP, or in general,\nusing the sigmoid loss for contrastive learning. Specifi-\ncally, we have the following contributions, in terms of\nanalyzing the embeddings learned by the sigmoid loss.\n•We define the double-Constant Embedding Model\n(CCEM), a framework that models various struc-\ntures formed by multiple embedding vectors. This\nmodel contains well-known structures including\nthesimplexETFstructureandtheantipodalstruc-\nture.\n•We show that optimal embedding which minimizes\nthe sigmoid loss can be found in the proposed\nCCEM. In other words, it is sufficient to find the\noptimal solution within the structures that can be\nrepresented in the CCEMframework.\n•The optimal embedding structure that minimizes\nsigmoid loss is specified. The optimal embedding\nforms simplex ETF when the temperature parame-\nter is sufficiently large, while it forms an antipodal\nstructure when the temperature is smaller than a\nthreshold.\n•Our experimental results on synthetic datasets\nsupport our theoretical findings on the behavior\nof optimal embeddings.\n2 Related Works\nContrastive Learning Self-supervised learning has\ngained significant attention as it enables learning from\nlarge-scale data without the need for laborious super-\nvision labeling (Goyal et al., 2019; Chen et al., 2020a;\nHe et al., 2020; Chen et al., 2020b). Among various\ntechniques, CL has been widely utilized in computer\nvision (Caron et al., 2020; Chen and He, 2021; Grill\net al., 2020) and natural language processing (Giorgi\net al., 2020; Wu et al., 2020) domains due to the ex-\ncellent performance of learned embeddings. CL is a\nsimilarity-based training method, aiming to position\nsimilar data close to each other in the embedding spacewhile pushing dissimilar data far apart. Moreover, it is\nknown that CL can be applied in multi-modal scenar-\nios(Radfordetal.,2021;Jiaetal.,2021;Maetal.,2020;\nElizalde et al., 2023; Li et al., 2021; Mu et al., 2022;\nYao et al., 2021; Goel et al., 2022; Zhai et al., 2023),\nwhich leverage data with multiple views from different\nmodalities such as images paired with text captions.\nIn this paper, we provide analysis on the CL, in the\nperspective of the learned embedding structure, which\ncan be applied for both uni-modal and multi-modal\nscenarios.\nAnalysis on Geometry of Embeddings Several\nresearchers focused on the geometric structure of opti-\nmal embeddings learned by CL such as antipodal and\nsimplex ETF (Lu and Steinerberger, 2022; Awasthi\net al., 2022; Sreenivasan et al., 2023; Graf et al., 2021).\nEspecially, Lu and Steinerberger (2022) shows that the\noptimal feature vector solution of a variational prob-\nlem related to cross-entropy loss forms a simplex ETF,\nbased on previous results on the ETFs (Sustik et al.,\n2007; Fickus and Mixon, 2015; Strohmer and Heath Jr,\n2003). On this basis, Sreenivasan et al. (2023) demon-\nstrated that optimal representations for minimizing the\ncontrastive loss form a simplex ETF when d≥N−1.\nOur paper is similar to this work in terms of finding\noptimal embeddings for minimizing contrastive losses.\nWhile existing works solve the optimization problem for\nsoftmax-based losses, our paper solves the problem for\nthe sigmoid loss. In addition, we define a novel embed-\nding structure model which includes the simplex ETF\nand antipodal structure in a single framework, which\nhelps find the embeddings minimizing the sigmoid loss.\nAnalysis on Temperature Parameter of CL Var-\nious theoretical studies have explored on the temper-\nature parameter of CL (Zhang et al., 2021, 2022a,b).\nEspecially, Wang andLiu (2021) shows that thetemper-\natureparameterofCLcontrolsthestrengthofpenalties\non the hard negative sample by using gradient analysis.\nKukleva et al. (2023) introduces a schedule for the tem-\nperature parameter of CL to improve the representa-\ntion quality. However, previous research focused on the\nrole of the temperature parameter during the learning\nprocess without considering the optimal embeddings\nfrom different temperature parameters. Particularly,\nwe demonstrated that the use of the sigmoid loss re-\nsults in embeddings that are highly dependent on the\ntemperature parameter.\n3 Problem Setup\nHere we first provide backgrounds on CL problem we\nare focusing on, and introduce several notations neces-\nsary for defining our problem setup. Then, we provideChungpa Lee, Joonhwan Chang, Jy-yong Sohn\nthe formal description of our target problem of optimiz-\ning the embedding with the sigmoid loss, and describe\nthe skeleton of our paper.\n3.1 Backgrounds\nCL uses Npaired instances {(xi,yi)}N\ni=1, where xi\nandyifor the same index iare two different views or\nmodalities of the same object. The goal of CL is to\ntrain encoders fxandfyin a way that the output of\nthe encoders for different views ( xiandyi) of the same\nobject are mapped into similar embedding vectors, i.e.,\nfx(xi)≈fy(yi).\nDenote the collections of embedding vectors by U=\n[u1,u2,···,uN]andV= [v1,v2,···,vN]where ui=\nfx(xi)∈Rdandvi=fy(yi)∈Rdfori∈[N]. For\nmathematical simplicity, we use the normalized embed-\nding space where u⊤\niui=v⊤\nivi= 1for all i∈[N],\nwhichiscommonlyusedinrelatedworks(Awasthietal.,\n2022; Sreenivasan et al., 2023; Huang et al., 2023).\nHere we focus on two popular contrastive losses: the\nInfoNCE loss\nLInfoNCE(U,V) :=−1\nNNX\ni=1logexp(u⊤\nivi/t)PN\nj=1exp(u⊤\nivj/t)\n−1\nNNX\ni=1logexp(u⊤\nivi/t)PN\nj=1exp(u⊤\njvi/t)\n(1)\nconsidered in Oord et al. (2018); Radford et al. (2021),\nand the sigmoid loss\nLsig(U,V) :=−1\nNNX\ni=1log1\n1 + exp( −tu⊤\nivi+b)(2)\n−1\nNNX\ni=1X\nj∈[N]\\{i}log1\n1 + exp( tu⊤\nivj−b)\nproposed by Zhai et al. (2023) as an alternative to\nthe InfoNCE loss LInfoNCEin Eq. 1 for the purpose\nof efficient computation. Here, t >0andb≥0are\ntemperature and bias, respectively.\nUnder this setting, one important question is about\nfinding the optimal embedding vectors\n(U⋆,V⋆) := arg min\nU,VL(U,V)\nlearned by a pre-defined contrastive loss L. A recent\nwork (Lu and Steinerberger, 2022) shows that the op-\ntimal embedding vectors for the InfoNCE loss form a\nsimplex ETF defined below:Definition 1. Suppose d≥N−1. A set of Nvectors\n{wi}N\ni=1inRdis called (N−1)-simplex ETF, if\nw⊤\niwi= 1,∀i∈[N],\nw⊤\niwj=−1\nN−1,∀i, j∈[N]withi̸=j.\nCorollary 1 (Theorem 1 of Lu and Steinerberger\n(2022)).Let(U⋆,V⋆) =arg minU,VLInfoNCE(U,V).\nSuppose d≥N−1, where dis the dimension of embed-\nding vectors u⋆\ni,v⋆\ni. Then, {u⋆\ni}N\ni=1forms a (N−1)-\nsimplex ETF and u⋆\ni=v⋆\niholds for all i∈[N].\nNote that Corollary 1 slightly differs from Theorem 1 of\nLu and Steinerberger (2022), which demonstrates that\nthe optimal embedding vectors for the InfoNCE loss\nLInfoNCEwithout any temperature parameter tform a\nsimplex ETF. However, even with the inclusion of a\ntemperature parmeter tin the InfoNCE loss LInfoNCE,\nthe result remains unchanged. The proof is on Ap-\npendix B.3.\nWe define the antipodal structure as follows.\nDefinition2. A set of 2Nvectors {ui}N\ni=1and{vi}N\ni=1\ninRdis called antipodal, if\nui=−vj,∀i, j∈[N].\nNote that if vectors {ui,vi}N\ni=1form the antipodal\nstructure, then the positive pairs are heading to the\nopposite direction, while the negative pairs are aligned,\nwhich is achieving the opposite of what contrastive\nlearning is aiming at.\n3.2 Skeleton of This Paper\nIn this paper, we analyze the embedding U⋆,V⋆opti-\nmized by the sigmoid loss Lsigin Eq. 2, denoted by\n(U⋆,V⋆) = arg min\nU,VLsig(U,V).\nBefore solving this problem, in Sec. 4, we first pro-\npose the double-Constant Embedding Model (CCEM),\nwhich models the evolution of different embedding vec-\ntors using a single parameter δ≥0. Note that this\nmodel contains various embedding structures including\nsimplex ETF. Interestingly, under some mild assump-\ntion (which holds in practice) on the contrastive loss,\nour proposed model is shown to contain the optimal\nembedding (U⋆,V⋆).\nBased on this observation, in Sec. 5, we use the pro-\nposed CCEMto explore the optimal embedding for\nsigmoid loss. Our mathematical results specify the op-\ntimal embedding for various hyperparameter settings.\nFurthermore, our theoretical findings are supported by\nexperimental results on toy datasets.Analysis of Using Sigmoid Loss for Contrastive Learning\nRemark 1. Note that our analysis focuses on the case\nofd≥N, similar to related works (Lu and Steiner-\nberger, 2022; Sreenivasan et al., 2023). This is because\nwhen d < N −1, the ETF solution, which is con-\nnected with the optimal solution for our target CL, does\nnot have a standard format for all N, dvalues (Sustik\net al., 2007; Fickus et al., 2012; Fickus and Mixon,\n2015; Azarija and Marc, 2018).\n4 Double-Constant Embedding Model\nIn this section, we propose a framework that models\nvarious types of geometric structures formed by em-\nbedding vectors {(ui,vi)}N\ni=1. Since it is not trivial to\nmodel arbitrary structures of 2Nembedding vectors,\nhere we restrict ourselves to model a set of vectors\nsatisfying\nu⊤\nivi=c1for all i∈[N], (3)\nu⊤\nivj=c2for all i, j∈[N]with i̸=j\nfor some c1, c2∈[−1,1],i.e.,the matrix Mcontain-\ning elements Mij=u⊤\nivjisdouble-constant (O’Neill,\n2021), where all diagonal elements are c1and all off-\ndiagonal elements are c2. Interestingly, as shown later\nin Corollary 2, the optimal embedding (U⋆,V⋆)min-\nimizing the sigmoid loss satisfies the double-constant\npropertyinEq.3, whichshowsthatwecansafelysearch\nthe embedding vectors represented by our framework,\nwithout losing the optimality. With this in mind, we\nnow introduce the double-Constant Embedding Model\n(CCEM).\nDefinition 3 (Double-Constant Embedding Model) .\nSuppose d≥N. Let{wi}N\ni=1be the (N−1)-simplex\nETF in Rd−1. The double-Constant Embedding Model\n(CCEM) constructs 2Nvectors ({uδ\ni,vδ\ni}N\ni=1)as\nuδ\ni:= 1√\n1+δ2wi\n1√\n1+δ2δ!\n∈Rd,vδ\ni:= 1√\n1+δ2wi\n−1√\n1+δ2δ!\n∈Rd,\nwhich are parameterized by δ≥0. The corresponding\nembedding matrix is denoted by\nU(δ) = [uδ\n1,···,uδ\nN],V(δ) = [vδ\n1,···,vδ\nN].\n4.1 Properties and Examples of CCEM\nOur proposed model has properties summarized below:\nProposition 1. Let{uδ\ni,vδ\ni}N\ni=1be2Nembedding vec-\ntors formed by the CCEMin Def. 3, for a given δ≥0.\nThe inner products between embedding vectors are\n(uδ\ni)⊤vδ\ni=1−δ2\n1 +δ2∀i∈[N], (4)\n(uδ\ni)⊤vδ\nj=−1\nN−1+δ2\n1 +δ2∀i, j∈[N]withi̸=j,\n(a)δ= 0 (b)δ= 1(c)δ=∞\nFigure 1: The proposed double-constant embedding\nmodel in Def. 3 for different δvalues, when we have\nN= 3embedding vector pairs {(uδ\ni,vδ\ni)}N\ni=1ind= 3\ndimensional space. (1) δ= 0(simplex ETF), the angle\nbetween positive pair ( u0\niandv0\ni) is 0. (2) δ= 1,\nthe angle between u1\niandv1\niisπ\n2. (3) δ=∞, the\nangle between u∞\niandv∞\niisπ, forming ‘antipodal’\nembedding.\ni.e., the double-constant property in Eq. 3 is satisfied.\nWhen δ= 0, the embedding vectors satisfy\n{u0\ni}N\ni=1form (N−1)-simplex ETF,\nu0\ni=v0\ni∀i∈[N].\nWhen δ=∞, the embedding vectors satisfy\nu∞\ni=−v∞\nj∀i, j∈[N]\nwhich is ‘antipodal’ embedding defined in Def. 2.\nBelow we provide an example of embeddings formed\nby our model, when d= 3andN= 3.\nExample 1. Fig. 1 illustrates an example of the CCEM\nproposed in Def. 3, where three embedding vector pairs\n(N= 3) are in three dimensional space ( d= 3). When\nδ= 0, the embedding structure model follows precisely\nthe simplex ETF, which coincides with the result of\nProp. 1. As δincreases, the angle between positive\npairs (ui,vi)increases for all i∈[N], and the distance\nbetween negative pairs (ui,vj)also increases for all i̸=\nj. This means that increasing δis pushing both positive\npairs and negative pairs apart. In the extreme case of\nδ=∞, we have ui=−vjfor all i, j, meaning that\npairs with positive labels ( uiandvi) are ignored and\npairs with negative labels ( uiandvj∀i̸=j) dominate\nthe behavior. Thus, we call this case as the ‘antipodal’\nstructure.\n4.2 Optimal Embeddings are in CCEM\nNow we provide theoretical results supporting that the\nCCEMproposedinDef.3issufficienttofindtheoptimal\nembedding that minimizes the sigmoid loss. We start\nwith a theorem stating that the optimal embedding of\nCL can be found in the CCEM, under a mild condition\non the loss function.Chungpa Lee, Joonhwan Chang, Jy-yong Sohn\nTheorem 1. Consider an arbitrary contrastive loss\nthat follows the form of\nL(U,V) =NX\ni=1lp(u⊤\nivi) +NX\ni=1X\nj∈[N]\\{i}ln(u⊤\nivj),(5)\nwhere lpis an arbitrary convex decreasing function for\npositive pairs, and lnis an arbitrary convex increasing\nfunction for negative pairs.\nThen, the optimal embeddings (U⋆,V⋆) =\narg minU,VL(U,V)can be uniquely found in\ntheCCEM, i.e.,\n∃!δ⋆≥0such that (U(δ⋆),V(δ⋆)) = (U⋆,V⋆).\nNote that in the above theorem, we assume the loss\nfunction Lsatisfies Eq. 5. This is quite mild assump-\ntion, since various loss functions used in practice satisfy\nthe condition. For example, the sigmoid loss Lsigin\nEq. 2 is in the form of Eq. 5 (as shown below), thus\nthe above theorem implies the following.\nCorollary 2. Consider the sigmoid loss Lsig(U,V)\ndefined in Eq. 2. Then,\nmin\nU,VLsig(U,V) = min\nδ≥0Lsig(U(δ),V(δ)).\nIn other words, the optimal embedding vectors for the\nsigmoid loss can be found in the CCEM.\nProof.The sigmoid loss Lsigin Eq. 2 can be reformu-\nlated as Eq. 5, when lp(x) =log(1 +exp(−tx+b))and\nln(x) =log(1 + exp(tx−b))for a given temperature\nt >0and bias b≥0. Combining this with Theorem 1\ncompletes the proof.\nNow we provide a proof of Theorem 1. In order to\nestablish Theorem 1, we introduce the mathematical\nresult, which demonstrates that the CCEMdefined in\nDef. 3 is a structure that minimizes the sum of the\nnegative pairs’ inner product, for a given sum of the\npositive pairs’ inner product:\nLemma 1. Suppose d≥N. Let {ui,vi}N\ni=1be2N\ndistinct vectors in Rd, satisfying u⊤\niui=v⊤\nivi= 1for\nalli∈[N]. Then, for every c∈[−1,1], there exists\nunique δ≥0such that\n(U(δ),V(δ)) = arg min\n(U,V)∈S(c)NX\ni=1X\nj∈[N]\\{i}u⊤\nivj.(6)\nwhere S(c)=n\n(U,V) :1\nNPN\ni=1u⊤\nivi=co\nProof.See Appendix B.1.It should be pointed out that there could be other\nembedding vectors that satisfy the minimization of\nEq. 6. However, the unique existence of δin Lemma 1\nimpliesthatthereisaunique CCEMstructuresatisfying\nthe minimization property of Eq. 6. This is crucial as\nwe require the double-constant properties of the CCEM\nin Eq. 4 to prove Theorem 1 as below.\nProof of Theorem 1\nProof.Given two embedding matrices UandV, define\nMwith Mij=u⊤\nivjas the similarity matrix of two\nembeddings. Define the function hofc∈[−1,1]as\nh(c) := min\n{(U,V):1\nNPN\ni=1u⊤\nivi=c}NX\ni=1X\nj∈[N]\\{i}u⊤\nivj,\nwhich is the minimum summation of the off-diagonal\nterms of similarity matrix M, among embedding ma-\ntrices U,Vsatisfying that the sum of diagonal terms\nisNc.\nTo simplify the notation, define\nl(x, y) :=lp\u0010x\nN\u0011\n+ (N−1)ln\u0012y\nN(N−1)\u0013\n.\nBy using Jensen’s inequality on the loss Lin Eq. 5,\n1\nNL ≥lp \n1\nNNX\ni=1u⊤\nivi!\n+1\nNNX\ni=1X\nj∈[N]\\{i}ln(u⊤\nivj)\n≥lp \n1\nNNX\ni=1u⊤\nivi!\n+ (N−1)·ln\n1\nN(N−1)NX\ni=1X\nj∈[N]\\{i}u⊤\nivj\n\n=l\nNX\ni=1u⊤\nivi,NX\ni=1X\nj∈[N]\\{i}u⊤\nivj\n\n≥l NX\ni=1u⊤\nivi, h \n1\nNNX\ni=1u⊤\nivi!!\n(7)\nwhere the equality conditions of Jensen’s inequalities\n(1st and 2nd inequalities in the above equation) are\nu⊤\nivi=c1∀i, (8)\nu⊤\nivj=c2∀i, j∈[N]with i̸=j.\nThe last inequality holds since l(x, y)is an increasing\nfunction with respect to y. For every c∈[−1,1], by\nLemma 1, there is a unique δ=δ(c)≥0such that\n(U(δ),V(δ)) = arg min\n(U,V)∈S(c)NX\ni=1X\nj∈[N]\\{i}u⊤\nivjAnalysis of Using Sigmoid Loss for Contrastive Learning\nwhere S(c)=n\n(U,V) :1\nNPN\ni=1u⊤\nivi=co\n. It implies\nNX\ni=1X\nj∈[N]\\{i}\u0000\nuδ\ni\u0001⊤vδ\nj=h \n1\nNNX\ni=1\u0000\nuδ\ni\u0001⊤vδ\ni!\n.(9)\nMoreover, every CCEMfulfills the equality conditions\ngiven in Eq. 8 , due to its double-constant properties\nin Eq. 4. Thus, combining these facts with Eq. 9, the\ninequalities of Eq. 7 reduces to equality as\nL(U(δ),V(δ)) (10)\n=N·l NX\ni=1\u0000\nuδ\ni\u0001⊤vδ\ni, h \n1\nNNX\ni=1\u0000\nuδ\ni\u0001⊤vδ\ni!!\n.\nSince δ=δ(c)is a function of c=1\nNPN\ni=1u⊤\nivi, from\nEq. 7, we have\nL ≥N·l NX\ni=1u⊤\nivi, h \n1\nNNX\ni=1u⊤\nivi!!\n≥N min\n1\nNPN\ni=1u⊤\nivil NX\ni=1u⊤\nivi, h \n1\nNNX\ni=1u⊤\nivi!!\n=Nmin\nδ≥0l NX\ni=1\u0000\nuδ\ni\u0001⊤vδ\ni, h \n1\nNNX\ni=1\u0000\nuδ\ni\u0001⊤vδ\ni!!\n= min\nδ≥0L(U(δ),V(δ))\nwhere the last equality holds from Eq. 10.\nTherefore, the optimal embeddings ( U(δ⋆),V(δ⋆))\nfound from the CCEMminimize the contrastive loss\nover all possible embeddings.\nRemark 2. The proposed CCEMcan be effectively\nutilized not only for the loss in Eq. 5 format, but also\nfor many other contrastive losses, e.g., InfoNCE loss.\nSimilar to Corollary 2, it can be shown that the optimal\nembedding vectors for InfoNCE loss can be found in\nCCEM, as formally stated below:\nCorollary 3. Consider InfoNCE loss LInfoNCE(U,V)\ndefined in Eq. 1. Then,\nmin\nU,VLInfoNCE(U,V) = min\nδ≥0LInfoNCE(U(δ),V(δ)).\nProof.See Appendix B.2.\n5Embedding Learned by Sigmoid Loss\nIn this section, we explore the optimal embedding\nlearned by the sigmoid loss using the suggested CCEM.\nIn Sec. 5.1, we provide theoretical results explaininghow the structure of optimal embedding varies depend-\ning on the temperature and the bias term used in the\nsigmoid loss, assuming d≥N. The behavior proven in\nSec. 5.1 is empirically confirmed in Sec. 5.2, not only\nford≥Ncase but also for d < Ncase.\n5.1 Theoretical Results\nRecall that Corollary 2 implies that the optimal em-\nbedding vectors U⋆,V⋆can be found by solving\nmin\n(U,V)∈SLsig(U,V),\nwhich is equivalent to finding the optimal parameter\nδ⋆:= arg min\nδ≥0Lsig(U(δ),V(δ)).\nThe below theorem provides the behavior of δ⋆:\nTheorem 2 (Optimal Embedding for sigmoid loss) .\nConsider the case when t=b, i.e., the temperature and\nthe bias are identical.\nWhen N= 3, we have δ⋆= 0, i.e., the optimal embed-\nding for sigmoid loss forms a simplex ETF, regardless\noftvalue.\nOn the other hand, when N≥4, the optimal δ⋆is a\nmonotonic decreasing function of t, satisfying\nδ⋆=(\n0,if and only if t >N−1\nNlog (N−3)\n∞,if and only if t <1\n2logN−2\n2.\nProof.See Appendix B.4.\nRemark3. The above theorem implies that the optimal\nembeddings form a simplex ETF when the temperature\nis sufficiently large, while they reduce to the ‘antipo-\ndal’ embedding when the temperature is smaller than a\nthreshold. Note that the threshold temperature value is\nin the order of Θ(log N).\nRemark 4. According to the empirical results reported\nin Zhai et al. (2023), SigLIP enjoys high performance\nonly when the temperature parameter is set to a suffi-\nciently large number. This coincides with our result in\nTheorem 2 stating that sufficiently large tis necessary\nto let the trained embeddings form a simplex ETF with\nδ⋆= 0, instead of the antipodal structure with δ⋆=∞.\nNote that Theorem 2 implies that for a given N, the\noptimal δ⋆(representing the distance between positive\npairs) decreases as the temperature parameter t=b\nincreases, leading to a tighter alignment of the positive\npairs,i.e.,u⊤\niviapproaches to 1. Now we provide a\ncorollary relating the optimal embeddings obtained by\nInfoNCE loss and sigmoid loss.Chungpa Lee, Joonhwan Chang, Jy-yong Sohn\nFigure 2: The normalized similarity sof positive pairs\nmeasured for the embeddings trained by the sigmoid\nlossLsig, for various Nandt. We test for two cases:\nd=N(upper) and d=N\n2(below). The blue dashed\nline satisfies t=N−1\nNlog(N−3), and the red dashed\nline satisfies t=1\n2logN−2\n2, which are the threshold\nvalues obtained in Theorem 2. This plot shows that our\ntheoretical results coincide with empirical observations.\nCorollary 4. Consider the case when t=b. When\nN= 3, the optimal solution of minimizing Lsigis equal\nto that of minimizing LInfoNCE, for any temperature\nparameter t. When N≥4, both solutions are identical,\nif and only if t >N−1\nNlog (N−3).\nProof.Recall that the optimal embedding for InfoNCE\nlossLInfoNCEalways follows the simplex ETF irre-\nspective of the temperature parameter t, as in Corol-\nlary 1. Combining this with Theorem 2 completes the\nproof.\nTherefore, by using an appropriate temperature param-\netert >N−1\nNlog (N−3), the simplex ETF solution\nis attainable by training with the sigmoid loss (Zhai\net al., 2023), which is shown to be more efficient than\ntraining with InfoNCE loss.\n5.2 Experimental results\nNow we provide empirical observations supporting our\ntheoretical results. Specifically, we test the behavior\nof the optimal embedding trained by the sigmoid loss\nfor various settings, and check whether it coincides\nwith our result in Theorem 2. Although our theoretical\nresults hold only when d≥N, we test whether the\nbehavior of optimal embeddings coincides with the\nresult of Theorem 2 for two cases: d=Nandd=N\n2.\nSynthetic data without Encoder We train em-\nbedding vectors {ui,vi}N\ni=1, each of which is initial-\nized by sampling from a standard normal distribution\nFigure 3: The normalized similarity s=1\n2(1 +\n1\nNPN\ni=1u⊤\nivi)of positive pairs measured for the em-\nbeddings trained by sigmoid loss Lsig, for various N\nandtwhen d=N. Unlike the setting for Fig 2, we\ntrain a encoder (two-layer fully-connected ReLU net-\nwork) which outputs embeddings, rather than directly\noptimizing the embedding vectors.\nN(0,Id), followed by projecting it to a unit sphere.\nUnlike real-world scenarios, we do not use encoder\nstructures; instead, we directly optimize the embed-\nding vectors {ui,vi}N\ni=1by training them for 50,000\nsteps with the sigmoid loss, using learning rate γ= 0.5.\nDuring the training process, the updated embeddings\nare projected back to the unit sphere, to make sure\nthatu⊤\niui=v⊤\niui= 1for all i∈[N]. Here, we test\nvarious temperature parameters t=b, ranging from\n0.1 to 5.\nGiven the trained embeddings {ui,vi}N\ni=1, we measure\nthe normalized similarity of positive pairs, denoted by\ns=1\n2(1 +1\nNPN\ni=1u⊤\nivi). Note that s= 1holds when\nall positive pairs are aligned, while s= 0holds when\nall positive pairs are antipodal.\nFig. 2 shows the normalized similarity sfor various N\nandt. The above plot is the result for d=N, while the\nbelow plot is for d=⌈N/2⌉. We have two dashed lines\nvisualizing the theoretical threshold values obtained in\nTheorem 2. The blue line satisfies t=N−1\nNlog(N−3),\nthe minimum value allowing the optimal embedding\nto be the simplex ETF having s= 1. The red line\nsatisfies t=1\n2logN−2\n2, the maximum value allowing\nthe optimal embedding to be the antipodal solution\nhaving s= 0. Thus, our theoretical results claim that\nthe set of (N, t)points above the blue line have s= 1,\nwhile the points below the red line have s= 0, which\ncoincide with the empirical results in Fig. 2.\nSynthetic data with Encoder Furthermore, we\nextend our experiments to more realistic setting where\nwe have an encoder that generates the embedding vec-\ntors. In contrast to the previous synthetic experiments\nwhere the encoder output is directly optimized, weAnalysis of Using Sigmoid Loss for Contrastive Learning\nemploy a two-layer fully-connected ReLU network de-\nnoted as f. Especially, Npair of instances {xi,yi}N\ni=1\nare generated from a standard normal distribution\nN(0,I2d), which passes through the encoder fthat\ntransforms each 2d-dimensional vector ( xioryi) into a\nd-dimensional embedding vector ( f(xi)orf(yi)) on a\nunit sphere. Then, the encoder is trained to minimize\nthe sigmoid loss Lsig. Fig. 3 shows the similarity of\npaired embeddings as a function of the temperature\nparameter tused in the sigmoid loss, when d=N.\nThis result exhibits a tendency similar to Fig 2, imply-\ning that the empirical results in a realistic setting also\nalign with the theoretical findings.\nRemark 5. Our theoretical results are based on the\nassumption of d≥N, while our empirical results have\nsimilar behaviors for both d≥Nandd < Ncases.\nThis empirical finding can be interpreted as follows. Re-\ncall that when dis sufficiently large (i.e., d≥N), the\nlearned embeddings satisfy both uniformity and align-\nment (Wang and Isola, 2020), while the uniformity\nis not attainable for smaller d(especially, d < N).\nOur empirical results, which measure the alignment of\npositive pairs, imply that smaller ddoes not harm the\nalignment property, although the uniformity is broken.\n6 Conclusion\nThis paper investigated the embedding structure\nlearned by contrastive learning with the sigmoid loss,\nmotivated by the recent success of SigLIP, a variant of\nCLIP that uses sigmoid loss instead of InfoNCE loss.\nDuring the route towards answering the question of\nfinding the optimal embedding for sigmoid loss, we first\ndefined the double-Constant Embedding Model (CCEM)\nframework that models various embedding structures.\nBased on our observation that the optimal embeddings\nare in the CCEM, we used the CCEMto specify the\noptimal embedding structure learned by the sigmoid\nloss. When the temperature parameter is sufficiently\nlarge, the optimal embedding forms a simplex ETF\nstructure, while the optimal embedding reduces to the\nantipodal structure when the temperature parameter\nis set to be a small number, which coincides with the\nempirical observations made by the authors of SigLIP.\nWe also conducted experiments on synthetic datasets,\nthe results of which coincide with our theoretical results\non the behavior of optimal embeddings.\nAcknowledgement\nThis research was supported by the MSIT (Ministry\nof Science and ICT), Korea, under the ICAN (ICT\nChallenge and Advanced Network of HRD) support\nprogram (RS-2023-00259934) supervised by the IITP(Institute for Information & Communications Technol-\nogy Planning & Evaluation).Chungpa Lee, Joonhwan Chang, Jy-yong Sohn\nReferences\nArora, S., Khandeparkar, H., Khodak, M., Plevrakis,\nO., and Saunshi, N. (2019). 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How does simsiam\navoid collapse without negative samples? a unified\nunderstanding with self-supervised contrastive learn-\ning.arXiv preprint arXiv:2203.16262 .\nZhang, O., Wu, M., Bayrooti, J., and Goodman, N.\n(2021). Temperature as uncertainty in contrastive\nlearning. arXiv preprint arXiv:2110.04403 .Analysis of Using Sigmoid Loss for Contrastive Learning\nA Comparison of Various Contrastive Loss Functions\nAs metioned in Sec. 1, there have been various approaches of defining contrastive loss function in the past year.\nIn this section, we compare and highlight the distinctive points of the three loss functions relevant to our paper;\nNCE loss LNCE(Gutmann and Hyvärinen, 2010, 2012), InfoNCE loss LInfoNCE(Oord et al., 2018; Radford et al.,\n2021), and Sigmoid loss Lsig(Zhai et al., 2023).\nTo simplify the notation, we omit the instance pair index i, the temperature parameter, and the bias term. For\na given embedding u, we consider the the positive embedding v+and the negative embeddings {v(j)\n−}N\nj=1with\nrespect to u. If there is only one negative embedding, we remove the index (j)and denote it as v−. Then, the\nthree loss functions are defined as follows.\nLNCE(u,V) =−log\u0012exp(u⊤v+)\nexp(u⊤v+) + exp( u⊤v−)\u0013\n=−log\u00121\n1 + exp ( −u⊤v++u⊤v−)\u0013\nLInfoNCE(u,V) =−log\nexp(u⊤v+)\nexp(u⊤v+) +P\njexp\u0010\nu⊤v(j)\n−\u0011\n\n=−log\n1\n1 +P\njexp\u0010\n−u⊤v++u⊤v(j)\n−\u0011\n\nLsig(u,V) =−log\u00121\n1 + exp ( −u⊤v+)\u0013\n−X\njlog\n1\n1 + exp\u0010\nu⊤v(j)\n−\u0011\n\nThere are two points to note. First, LNCEhas only one negative pair, whereas LInfoNCE, an extension of LNCE,\nincludes multiple negative pairs, similar to Lsig. Second, the NCE-based loss functions, LNCEandLInfoNCE, are\nderived from the softmax function, while Lsigis formulated based on the sigmoid function. As a result, Lsigcan\nseparate the term for the positive pair and negative pairs through summation, whereas the NCE-based losses\ncannot.\nB Proofs\nNote that the (N−1)-regular-simplex is a set of Nvectors which makes the geometry of simplex with equal\nedges. Detailed concepts and properties are on El-Gebeily and Fiagbedzi (2004); Gale (1953); Fickus et al. (2018);\nSustik et al. (2007).\nB.1 Proof of Lemma 1\nLemma. Suppose d≥N. Let {ui,vi}N\ni=1be2Ndistinct vectors in Rd, satisfying u⊤\niui=v⊤\nivi= 1for all\ni∈[N]. Then, for every c∈[−1,1], there exists unique δ≥0such that\n(U(δ),V(δ)) = arg min\n{(U,V):1\nNPN\ni=1u⊤\nivi=c}NX\ni=1X\nj∈[N]\\{i}u⊤\nivj. (11)Chungpa Lee, Joonhwan Chang, Jy-yong Sohn\nProof.By using Jensen’s inequality,\nNX\ni=1∥ui−vi∥2\n2≥N\r\r\r\r\r1\nNNX\ni=1(ui−vi)\r\r\r\r\r2\n2\n=N\r\r\r\r\r1\nNNX\ni=1(ui+vi)\r\r\r\r\r2\n2−4N \n1\nNNX\ni=1ui!⊤ \n1\nNNX\ni=1vi!\n≥ −4N \n1\nNNX\ni=1ui!⊤ \n1\nNNX\ni=1vi!\nwhere the equality conditions are\n(\nui−vi=c′for all i∈[N],for some constant vector c′\n1\nNPN\ni=1ui+1\nNPN\ni=1vi= 0 .(12)\nThis implies that \n1\nNNX\ni=1ui!⊤ \n1\nNNX\ni=1vi!\n≥ −1\n4NNX\ni=1∥ui−vi∥2\n2. (13)\nNote that the inner product of centroids is\n \n1\nNNX\ni=1ui!⊤\n1\nNNX\nj=1vj\n=1\nN2NX\ni=1u⊤\nivi+1\nN2NX\ni=1X\nj∈[N]\\{i}u⊤\nivj.\nSince∥ui−vi∥2\n2= 2−2u⊤\nivifor all i∈[N], we have\n1\nN2NX\ni=1X\nj∈[N]\\{i}u⊤\nivj= \n1\nNNX\ni=1ui!⊤\n1\nNNX\nj=1vj\n−1\nN2NX\ni=1u⊤\nivi\n= \n1\nNNX\ni=1ui!⊤\n1\nNNX\nj=1vj\n+1\n2N2NX\ni=1∥ui−vi∥2\n2−1\nN\n≥ −1\n4NNX\ni=1∥ui−vi∥2\n2+1\n2N2NX\ni=1∥ui−vi∥2\n2−1\nN(14)\n=N−2\n2N2NX\ni=1u⊤\nivi−1\n2(15)\nwhere the inequality is from Eq. 13.\nRecall that our target problem in Eq. 11 is about finding the optimal embedding matrices (U,V)that minimizesPN\ni=1P\nj∈[N]\\{i}u⊤\nivj, under the constraint of1\nNPN\ni=1u⊤\nivi=c. One can confirm that the necessary and\nsufficient condition for achieving this optimal (U,V)is nothing but the equality condition for Eq. 14, which is in\nEq. 12.\nNote that the construction of the CCEMin Def. 3 fulfills these equality conditions of Eq. 12, which also attains\nthe minimum of Eq. 15, since\n1\nN2NX\ni=1X\nj∈[N]\\{i}\u0000\nuδ\ni\u0001⊤vδ\nj=−1\nN+N−1\nNδ2\n1 +δ2\n=N−2\n2N2NX\ni=1\u0000\nuδ\ni\u0001⊤vδ\ni−1\n2.Analysis of Using Sigmoid Loss for Contrastive Learning\nholds from Prop. 1.\nThus, we can conclude that there exists unique δsuch that (U(δ),V(δ))is the optimal embedding. Actually, we\ncan specify such δ: from Eq. 4, δ=q\n1−c\n1+cholds when1\nNPN\ni=1u⊤\nivi=c. The uniqueness of δarises from the\nfact thatq\n1−c\n1+cis a strictly decreasing function with respect to c∈[−1,1].\nB.2 Proof of Corollary 3\nCorollary. Consider InfoNCE loss LInfoNCE(U,V)defined in Eq. 1. Then,\nmin\nU,VLInfoNCE(U,V) = min\nδ≥0LInfoNCE(U(δ),V(δ)).\nProof.Define the function hofc∈[−1,1]as\nh(c) := min\n{(U,V):1\nNPN\ni=1u⊤\nivi=c}NX\ni=1X\nj∈[N]\\{i}u⊤\nivj.\nTo simplify the notation, define the convex function\nl(x) = log\u0012\n1 + (N−1) exp\u00121\ntN(N−1)x\u0013\u0013\nfor any N≥3andt >0.\nBy using Jensen’s inequality on the loss LInfoNCEin Eq. 1,\nLInfoNCE=1\nNNX\ni=1log\n1 +X\nj∈[N]\\{i}exp\u0000\nu⊤\ni(vj−vi)/t\u0001\n+1\nNNX\ni=1log\n1 +X\nj∈[N]\\{i}exp\u0000\n(uj−ui)⊤vi/t\u0001\n\n≥1\nNNX\ni=1log\n1 + (N−1) exp\n1\nN−1X\nj∈[N]\\{i}u⊤\ni(vj−vi)/t\n\n\n+1\nNNX\ni=1log\n1 + (N−1) exp\n1\nN−1X\nj∈[N]\\{i}(uj−ui)⊤vi/t\n\n\n≥log\n1 + (N−1) exp\n1\nN(N−1)NX\ni=1X\nj∈[N]\\{i}u⊤\ni(vj−vi)/t\n\n\n+ log\n1 + (N−1) exp\n1\nN(N−1)NX\ni=1X\nj∈[N]\\{i}(uj−ui)⊤vi/t\n\n\n= 2l\nNX\ni=1X\nj∈[N]\\{i}u⊤\nivj−(N−1)NX\ni=1u⊤\nivi\n\n≥2l \nh \n1\nNNX\ni=1u⊤\nivi!\n−(N−1)NX\ni=1u⊤\nivi!\nwhere the equality conditions of Jensen’s inequalities are u⊤\ni(vj−vi) =c1and(uj−ui)⊤vi=c2for all i, j∈[N]\nwith i̸=j. The last inequality holds since lis an increasing function.Chungpa Lee, Joonhwan Chang, Jy-yong Sohn\nFor every c∈[−1,1], by Lemma 1, there is a unique δ(c)≥0such that\n(U(δ),V(δ)) := arg min\n{(U,V):1\nNPN\ni=1u⊤\nivi=c}NX\ni=1X\nj∈[N]\\{i}u⊤\nivj\nwhich implies\nNX\ni=1X\nj∈[N]\\{i}\u0000\nuδ\ni\u0001⊤vδ\nj=h \n1\nNNX\ni=1\u0000\nuδ\ni\u0001⊤vδ\ni!\nMoreover, every CCEMfulfills the equality conditions of Jensen’s inequalities, due to its double-constant properties\nin Eq. 4. Thus,\nLInfoNCE(U(δ),V(δ)) = 2 l \nh \n1\nNNX\ni=1\u0000\nuδ\ni\u0001⊤vδ\ni!\n−(N−1)NX\ni=1\u0000\nuδ\ni\u0001⊤vδ\ni!\nSince δis function of1\nNPN\ni=1u⊤\nivi, we have\nLInfoNCE≥2l \nh \n1\nNNX\ni=1u⊤\nivi!\n−(N−1)NX\ni=1u⊤\nivi!\n≥2 min\n1\nNPN\ni=1u⊤\nivil \nh \n1\nNNX\ni=1u⊤\nivi!\n−(N−1)NX\ni=1u⊤\nivi!\n= 2 min\nδ≥0l \nh \n1\nNNX\ni=1\u0000\nuδ\ni\u0001⊤vδ\ni!\n−(N−1)NX\ni=1\u0000\nuδ\ni\u0001⊤vδ\ni!\n= min\nδ≥0LInfoNCE(U(δ),V(δ)).\nThus, the optimal embeddings ( U(δ⋆),V(δ⋆)) found from CCEMminimize the contrastive loss over all possible\nembeddings.\nB.3 Proof of Corollary 1, using the proposed CCEM\nHere we show that Corollary 1 proven by Lu and Steinerberger (2022), can be also proved using the CCEM\n(Double-constant Embedding Model) framework proposed by us in Def. 3.\nCorollary (Equivalent to Theorem 1 of Lu and Steinerberger (2022)) .LetU,Vbe the optimal embedding\nminimizing the InfoNCE loss LInfoNCE. Then U=Vholds and Uforms a simplex ETF, regardless of the\ntemperature parameter t >0and the number of paired instances N≥3.\nProof.It is sufficient to use the CCEMin Def. 3 to find the minimum by Corollary 3. Then, by introducing the\nCCEMto the InfoNCE loss LInfoNCEof Eq. 1,\nLInfoNCE(U(δ),V(δ)) =−1\nNNX\ni=1logexp\u0010\u0000\nuδ\ni\u0001⊤vδ\ni/t\u0011\nPN\nj=1exp\u0010\u0000\nuδ\ni\u0001⊤vδ\nj/t\u0011−1\nNNX\ni=1logexp\u0010\u0000\nuδ\ni\u0001⊤vδ\ni/t\u0011\nPN\nj=1exp\u0010\u0000\nuδ\nj\u0001⊤vδ\ni/t\u0011\n= 2 log \n1 + (N−1) exp \n−N\nN−1\n1 +δ2/t!!\nwhich is strictly monotonic increasing function with respect to δfor all t >0andN≥3.\nTherefore, LInfoNCEminimize at δ= 0for all t >0andN≥3.Analysis of Using Sigmoid Loss for Contrastive Learning\nB.4 Proof of Theorem 2\nTheorem (Optimal Embedding for sigmoid loss) .Consider the case when t=b, i.e., the temperature and the\nbias are identical. When N= 3, we have δ⋆= 0, i.e., the optimal embedding for sigmoid loss forms a simplex\nETF, regardless of tvalue. On the other hand, when N≥4, the optimal δ⋆is a monotonic decreasing function of\nt, satisfying\nδ⋆=(\n0,if and only if t >N−1\nNlog (N−3)\n∞,if and only if t <1\n2logN−2\n2.\nProof.It is sufficient to use the CCEMin Def. 3 to find the minimum by Corollary 2. Then, by introducing the\nCCEMof Def. 3 to the sigmoid loss Lsigof Eq. 2,\nLsig(U(δ),V(δ)) =1\nNNX\ni=1log\u0010\n1 + exp\u0010\n−t\u0000\nuδ\ni\u0001⊤vδ\ni+b\u0011\u0011\n+1\nNNX\ni=1X\nj∈[N]\\{i}log\u0010\n1 + exp\u0010\nt\u0000\nuδ\ni\u0001⊤vδ\nj−b\u0011\u0011\n= log\u0012\n1 + exp\u0012\n−t1−δ2\n1 +δ2+b\u0013\u0013\n+ (N−1) log \n1 + exp \n−t1\nN−1+δ2\n1 +δ2−b!!\n= log\u0012\n1 + exp\u0012\n−t−2δ2\n1 +δ2\u0013\u0013\n+ (N−1) log \n1 + exp \n−tN\nN−1+ 2δ2\n1 +δ2!!\n.\nThen,\n∂\n∂δLsig=exp\u0010\n−t−2δ2\n1+δ2\u0011\n1 + exp\u0010\n−t−2δ2\n1+δ2\u0011\u0012\n−t−4δ\n(1 +δ2)2\u0013\n+ (N−1)exp\u0012\n−tN\nN−1+2δ2\n1+δ2\u0013\n1 + exp\u0010\n−tN\nN−1+2δ2\n1+δ2\u0011 \n−t2(N−2)\nN−1δ\n(1 +δ2)2!\n=2tδ\n(1 +δ2)2\n2\n1 + exp\u0010\nt−2δ2\n1+δ2\u0011−(N−2)\n1 + exp\u0010\ntN\nN−1+2δ2\n1+δ2\u0011\n\n=2tδ\n(1 +δ2)2g(N, t, δ )\u0010\n1 + exp\u0010\nt−2δ2\n1+δ2\u0011\u0011\u0010\n1 + exp\u0010\ntN\nN−1+2δ2\n1+δ2\u0011\u0011\nwhere we define g(N, t, δ ) = (4 −N) + 2 exp\u0012\ntN\nN−1+2δ2\n1+δ2\u0013\n−(N−2) exp\u0010\nt−2δ2\n1+δ2\u0011\n.\nNote that g(N, t, δ )is strictly monotone increasing with respect to δ >0for all t >0andN≥3, since\n∂\n∂δg(N, t, δ ) =4t(N−2)δ\n(1 +δ2)2 \n1\nN−1exp \ntN\nN−1+ 2δ2\n1 +δ2!\n+ exp\u0012\nt−2δ2\n1 +δ2\u0013!\n>0.\nAlso, note that∂\n∂δLsig\f\f\nδ=0= 0,∂\n∂δLsig\f\f\nδ→∞= 0, and∂2\n∂δ2Lsig\f\f\f\nδ=0= 2t(3−N)+exp (tN\nN−1)\n1+exp (tN\nN−1).\nTo find out the condition tofNthat minimize Lsig, we split the cases for N= 3andN≥4.\n[N= 3case ] First, consider N= 3. Then, for all t >0,\n∂2\n∂δ2Lsig\f\f\f\f\nδ=0,N=3= 2texp\u0000\nt3\n2\u0001\n1 + exp\u0000\nt3\n2\u0001\n>0Chungpa Lee, Joonhwan Chang, Jy-yong Sohn\nFurthermore,\ng(3, t, δ) = 1 + 2 exp\u0012\nt3\n2+ 2δ2\n1 +δ2\u0013\n−exp\u0012\nt−2δ2\n1 +δ2\u0013\n= 1 + 2 exp\u0012\nt\u0012\n2−1\n2\n1 +δ2\u0013\u0013\n−exp\u0012\nt\u0012\n−2 +2\n1 +δ2\u0013\u0013\n≥2 exp\u00123\n2t\u0013\n>0\nwhere the inequality holds by plugging δ= 0, because it is monotone increasing with respect to δ2. Since\ngis non-negative for all t, δ > 0, it is also hold that∂\n∂δLsig\f\f\nδ=0,N=3>0for all δ > 0. Therefore, 0 =\narg minδLsig(U(δ),V(δ))forN= 3.\n[N≥4case ] Second, consider the case of N≥4.\nFor any t >N−1\nNlog (N−3),∂2\n∂δ2Lsig\f\f\f\nδ=0>0holds and\ng(N, t, δ )≥g(N, t,0)\n= 2(3−N) + 2 exp\u0012\ntN\nN−1\u0013\n>0\nwhere the first inequality comes from the fact that gis strictly monotone increasing with respect to δ≥0. Since\ngis non-negative for above case, it is also hold that∂\n∂δLsig>0for all δ >0. As a result, Lsigminimize at δ= 0.\nOn the other hand, for 0< t <1\n2logN−2\n2,\ng(N, t, δ )≤g(N, t,∞)\n= (4−N) + 2 exp (2 t)−(N−2) exp ( −2t)\n<0\nholds. Therefore,∂\n∂δLsig<0for all δ >0, which implies Lsigminimize at δ=∞.\nFinally, for1\n2logN−2\n2< t <N−1\nNlog (N−3), by applying the inequality opposite to the above, we can easily\ngetg(N, t,0)<0andg(N, t,∞)>0. Since gis strictly increasing continuous function with respect to δ, there\nis unique δ⋆∈(0,∞)such that g(N, t, δ⋆) = 0by the intermediate value theorem. Therefore, there is unique\nδ⋆∈(0,∞)such that minimize Lsig.\nFurthermore, since∂\n∂δg(N, t, δ )>0and∂\n∂tg(N, t, δ )>0holds for all t, δ > 0andN≥3, by taking partial\nderivative of g(N, t, δ⋆) = 0for fixed N,\n0 =\u0012∂\n∂tg(N, t, δ⋆)\u0013\n+\u0012∂δ⋆\n∂t\u0013\u0012∂\n∂δ⋆g(N, t, δ⋆)\u0013\nimplies∂δ⋆\n∂t<0for all t, δ > 0andN≥3. Therefore, the optimal δ⋆is a monotonic decreasing function of tfor\nfixed N."    },    {        "title": "2402.12630v1.FAST__An_Optimization_Framework_for_Fast_Additive_Segmentation_in_Transparent_ML.pdf",        "content": "FAST: An Optimization Framework for Fast Additive\nSegmentation in Transparent ML\nBrian Liu\nMassachusetts Institute of Technology\nCambridge, Massachusetts, USA\nbriliu@mit.eduRahul Mazumder\nMassachusetts Institute of Technology\nCambridge, Massachusetts, USA\nrahulmaz@mit.edu\nABSTRACT\nWe present FAST, an optimization framework for fast additive seg-\nmentation. FAST segments piecewise constant shape functions for\neach feature in a dataset to produce transparent additive models.\nThe framework leverages a novel optimization procedure to fit these\nmodels∼2 orders of magnitude faster than existing state-of-the-art\nmethods, such as explainable boosting machines [ 20]. We also de-\nvelop new feature selection algorithms in the FAST framework to\nfit parsimonious models that perform well. Through experiments\nand case studies, we show that FAST improves the computational\nefficiency and interpretability of additive models.\n1 INTRODUCTION\nAdditive models are popular in machine learning for balancing a\nhigh degree of explainability with good predictive performance\n[2,3]. These models, when fit on a dataset with 𝑝features, take the\nformÍ𝑝\n𝑗=1𝑠𝑗(𝑥𝑗). Each additive component 𝑠𝑗is the shape func-\ntion of feature 𝑥𝑗, and since the contribution of each feature can be\nreadily observed from its shape function, additive models are said\nto be inherently transparent. One such additive model, explainable\nboosting machines (EBMs), combines this inherent transparency\nwith the powerful predictive performance of tree ensembles [ 20].\nEBMs use single-feature decision trees, fit via a cyclic boosting\nheuristic, to construct shape functions. As such, the shape func-\ntions built are piecewise constant, a departure from classical and\npopular smooth components such as those based on polynomials\nor splines [ 7]. Using piecewise constant shape functions, EBMs can\ncapture discontinuities in the underlying data, patterns that are\nunobserved by smooth additive models and which often have real-\nworld significance [ 2,14]. EBMs have also been shown to match\nthe predictive performance of black box methods in various ap-\nplications while preserving model transparency [ 3]. Due to these\nadvantages, EBMs are rapidly becoming ubiquitous in high-stakes\napplications of ML, such as criminal justice [ 3] and healthcare [ 2],\nwhere model explainability is critical.\nInspired by the success of EBMs, and stemming from a reinter-\npretation of the method, we propose an alternative, FAST. FAST is a\nformal optimization-based procedure to fit piecewise constant addi-\ntive models (PCAMs). Both methods construct piecewise constant\nshape functions, but FAST does so by minimizing a regularized\noptimization objective while EBMs use a cyclic boosting heuristic.\nMoreover, the main goal of FAST is to address the limitations\nof EBMs that result from this cyclic boosting heuristic. Starting\nfrom the null model, EBMs are fit by cycling round-robin over the\nfeatures and building single-feature decision trees on the boosted\nresiduals, which are dampened by a learning rate. To ensure that\nthe ordering of the features is irrelevant, this learning rate must bekept small. As a result, many cyclic boosting iterations and trees\nare required to fit an EBM that performs well. This increases the\ncomplexity and computational cost of the algorithm and conse-\nquently, EBMs struggle to scale for larger datasets. As a motivating\nexample, consider the UK Black Smoke dataset (9 million rows and\n14 columns) used by [ 30] to test the computational feasibility of\nsplines. It takes the InterpretML package [ 20] nearly 4 hours to fit\nan EBM using the default hyperparameters, which are optimized for\ncomputation time. FAST, on the other hand, leverages a specialized\ngreedy optimization algorithm to fit a PCAM that performs the\nsame in terms of accuracy in under 1 minute . The cyclic heuristic\nused to fit EBMs also produces feature-dense models by design. This\nmay harm interpretability since an EBM fit on a high dimensional\ndataset (𝑝>50features) will contain too many shape functions for\na practitioner to explain. FAST introduces two novel feature selec-\ntion algorithms to remedy this, and these new methods outperform\nexisting feature-sparse PCAM algorithms by up to a 30% reduction\nin test error. We summarize the contributions of our paper below.\nMain Contributions\n•We introduce FAST, an efficient optimization framework to fit\nPCAMs that supports feature sparsity.\n•FAST uses a novel procedure to improve computational efficiency.\nTo solve optimization problems in FAST, we apply a computation-\nally cheap greedy block selection rule to an implicit reformula-\ntion of our original problem in order to guide a block coordinate\ndescent algorithm. This procedure can fit PCAMs 2 orders of\nmagnitude faster than existing SOTA methods.\n•We introduce 2 new feature selection algorithms to build sparse\nPCAMs, an iterative algorithm that relies on our greedy block\nselection rule and a group ℓ0-regularized optimization algorithm.\n•We investigate how correlated features impact feature selec-\ntion and shape functions in PCAMs and discuss implications for\nmodel trustworthiness.\nWe first discuss the advantages of PCAMs over smooth additive\nmodels and overview existing algorithms to build PCAMs. Fol-\nlowing these preliminaries, we introduce the FAST optimization\nframework (§2) and present its novelties: the greedy optimization\nprocedure used to accelerate computation (§3) and the feature se-\nlection algorithms used to support feature sparsity (§4).\n1.1 Why PCAMs?\nCompared to smooth additive models such as splines, PCAMs have\nthe advantage that they are able to capture discontinuities in the\nSupplementary Material: github.com/brianliu12437/FAST_segmentationarXiv:2402.12630v1  [stat.ML]  20 Feb 2024Brian Liu & Rahul Mazumder\nshape functions. These discontinuities can reveal interesting in-\nsights about the underlying data. Consider the example shown in\nFigure 1. The scatterplot shows the daily number of car accidents\nin New York City over a 12-year period and there is a large jump\ndiscontinuity in early 2020 due to the COVID-19 pandemic [ 23].\nThis discontinuity is captured by the shape function from a PCAM\n(in blue) but is interpolated and obscured by the smoothing spline\n(in orange).\n2012 2014 2016 2018 2020 2022 2024\nDate200400600800Daily Number of Car AccidentsNew York City Car Accidents\nSmooth\nPCAM\nFigure 1: PCAM shape\nfunctions can be used to\nuncover discontinuities\nin the underlying data.\nThe discontinuities observed in PCAM shape functions have\nbeen used to uncover hidden patterns in mortality risk models for\nICU patients [ 14] and patients with pneumonia [ 2]. These patterns\nwould have been difficult to detect with smooth additive models\nor black-box methods. PCAMs also have the advantage that fitted\npiecewise constant shape functions can be represented by a set\nof breakpoints. As a result, fitted PCAMs are straightforward to\nproductionize and can be hard-coded into any language with condi-\ntional statements (e.g. SQL). Finally, PCAM predictions only require\nlookups and addition so PCAMs are extremely fast at inference [ 20].\n1.2 Existing PCAM Algorithms:\nAs mentioned earlier, EBMs use single-feature decision trees, fit via\na cyclic boosting heuristic, to build PCAMs [ 18]. EBMs are inter-\npretable and perform well, but are slow to train and feature-dense\nby design [ 20]. Besides EBMs, various methods have been used to\nconstruct PCAMs. Additive isotonic models use isotonic regression\nwith backfitting to build PCAMs with monotonic shape functions\n[1]. Spline-based frameworks can also fit PCAMs using zero-degree\nsplines [ 27]. More recently, the fused LASSO has been used to fit\nPCAMs via ADMM [ 4] or cyclic block coordinate descent [ 25]. The\nlatter approach is better known as the fused LASSO additive model\n(FLAM) and is considered a SOTA algorithm for building PCAMs.\nAs such, we primarily compare FAST against FLAMs and EBMs for\nfitting feature-dense PCAMs.\n2 FAST OPTIMIZATION FRAMEWORK\nWe introduce FAST and outline the optimization algorithm used to\nsolve problems in our framework. More importantly, we motivate\nwhy our greedy optimization procedure (§3) improves efficiency.\nGiven data matrix 𝑋∈R𝑛×𝑝and target vector 𝑦∈R𝑛, our goal\nis to fit additive modelÍ𝑝\n𝑗=1𝑠𝑗(𝑥𝑗), where each shape function 𝑠𝑗\nis piecewise constant. To accomplish this, we introduce a decision\nvariable for each entry in 𝑋. These decision variables are grouped\ninto decision vectors 𝛽𝑗∈R𝑛for𝑗∈[𝑝], where each decision\nvector𝛽𝑗represents the block of decision variables that correspondto feature𝑥𝑗. The decision variables in 𝛽𝑗are ordered with respect\nto the sorted values of 𝑥𝑗and the sum of decision vectors gives\nthe prediction of our model. We fit this prediction to 𝑦and recover\nshape functions 𝑠𝑗from the fitted decision vectors 𝛽∗\n𝑗.\n2.1 Optimization Problem\nLet𝛽denote the set of decision vectors {𝛽1...𝛽𝑝}. FAST minimizes\nthe objective 𝐿(𝑦,𝛽)+𝑆(𝛽)to fit PCAMs, where 𝐿is a loss function\nthat captures data fidelity and 𝑆is a segmentation penalty that\nencourages piecewise constant segmentation in the fitted decision\nvectors. The optimization problem can be written as:\nmin\n𝛽1,...,𝛽𝑝1\n2∥𝑦−𝑝∑︁\n𝑗=1𝑄⊺\n𝑗𝛽𝑗∥2\n2+𝜆𝑓𝑝∑︁\n𝑗=1∥𝐷𝛽𝑗∥1. (1)\nThe first term in the objective is quadratic loss, where 𝑄𝑗∈{0,1}𝑛×𝑛\nis the square sorting matrix for feature 𝑥𝑗. In other words, 𝑄𝑗𝑥𝑗re-\nturns the elements of 𝑥𝑗sorted in ascending order and 𝑄⊺\n𝑗(𝑄𝑗𝑥𝑗)=\n𝑥𝑗. Since each decision vector 𝛽𝑗is ordered with respect to the\nsorted values of 𝑥𝑗,Í𝑝\n𝑗=1𝑄⊺\n𝑗𝛽𝑗gives the prediction of our model.\nThe second term in the objective is the fused LASSO segmentation\npenalty, where 𝜆𝑓is the parameter that controls the number of\npiecewise constant segments in the shape functions. Higher values\nof𝜆𝑓result in less flexible shape functions with fewer segments.\nMatrix𝐷∈{− 1,0,1}(𝑛−1)×𝑛is the differencing matrix, where 𝐷𝛽𝑗\nreturns a vector of the successive differences of 𝛽𝑗.\nProblem 1 fits feature-dense PCAMs. An optional group sparsity\nconstraint can applied over the blocks 𝛽𝑗to select features and we\ndiscuss this further in §4.\n2.2 Optimization Algorithm\nProblem 1 is convex and separable over blocks 𝛽𝑗; we develop a\nblock coordinate descent (BCD) algorithm to solve this problem to\noptimality. Our algorithm has two components: block selections\nand block updates, and starting with all blocks 𝛽𝑗=0we alternate\nbetween the two until convergence.\nBlock Updates: It is critical to note that block updates in FAST\nare expensive. For a selected block 𝑘, let𝛿={1...𝑝}\\𝑘and define\nresidual vector 𝑟=𝑦−Í\n𝑗∈𝛿𝑄⊺\n𝑗𝛽𝑗. Each block update solves:\nmin\n𝛽𝑘1\n2∥𝑟−𝑄⊺\n𝑘𝛽𝑘∥2\n2+𝜆𝑓∥𝐷𝛽𝑘∥1, (2)\nwhich is equivalent to a fused LASSO signal approximation (FLSA)\nproblem on 𝑄𝑘𝑟. These FLSA problems are solved using dynamic\nprogramming [12] which is computationally expensive.\nBlock Selections: Since block updates are expensive, improving\nthe efficiency of our BCD algorithm relies on reducing the number\nof block updates that we conduct. To do so, we try to select the block\nthat makes the most progress towards the optimal solution in each\nBCD iteration. Other selection rules, such as cyclic or randomized\nselection [22], bottleneck BCD with unnecessary updates.\nWe also must select blocks cheaply since block selection would\nbe ineffective if the cost of selecting the best block to update is\nsimilar to the cost of updating all blocks. One novelty in FAST is\nthat we develop a greedy optimization procedure to select blocks\nextremely efficiently. We present this procedure below.FAST: An Optimization Framework for Fast Additive Segmentation in Transparent ML\n3 GREEDY OPTIMIZATION PROCEDURE\nOur greedy optimization procedure hinges on the fact that we can\ntransform Problem 1 into an equivalent LASSO problem with 𝑛\nrows and(𝑛−1)𝑝variables. While many LASSO algorithms exist\n[6], it is infeasible to solve this problem directly since there are\ntoo many variables when 𝑛is large and the variables are heavily\ncorrelated by design [ 26]. Rather, we use this LASSO reformulated\nproblem to guide block selection when we apply BCD to Problem 1.\nImportantly, we exploit the structure of the design matrix in\nour LASSO reformulation to derive an extremely efficient block\nselection rule. In fact, our block selection rule only requires an\nimplicit LASSO reformulation of the original problem (Problem 1),\nwhere the design matrix is not explicitly constructed. This is crucial\nsince constructing the design matrix requires a space complexity of\n𝑂(𝑛2𝑝), which is infeasible for large data. For example, the design\nmatrix for the UK Black Smoke problem (9 million rows and 14\ncolumns) mentioned in the introduction would take over 109TB of\nmemory if explicitly constructed.\n3.1 Implicit LASSO Reformulation\nWe define a new set of decision vectors 𝜃𝑗∈R𝑛−1for𝑗∈ [𝑝],\nwhere each vector 𝜃𝑗contains the successive differences of vector\n𝛽𝑗. Let𝐴∈{0,1}𝑛×(𝑛−1)be a padded lower triangular matrix with\nzeros in the first row. We first reformulate Problem 1 as:\nmin\n𝜃1,...,𝜃𝑝1\n2∥𝑦−𝑝∑︁\n𝑗=1𝑄⊺\n𝑗𝐴𝜃𝑗∥2\n2+𝜆𝑓𝑝∑︁\n𝑗=1∥𝜃𝑗∥1. (3)\nLet𝜃∈R(𝑛−1)𝑝represent the decision vectors {𝜃1...𝜃𝑝}ver-\ntically stacked. Let 𝐴′∈ {0,1}𝑛𝑝×(𝑛−1)𝑝be the matrix formed\nby stacking 𝐴submatrices 𝑝times along the main diagonal. Let\n𝑄⊺∈{0,1}𝑛𝑝×𝑛𝑝be the matrix formed by stacking {𝑄⊺\n1...𝑄⊺\n𝑝}\nalong the main diagonal. Finally, let 𝑀∈{0,1}𝑛×𝑛𝑝be the matrix\nformed by stacking 𝑝identity matrices of dimension 𝑛×𝑛horizon-\ntally. We show a visualization of these matrices in the appendix\n(suppl. A). Problem 3 and Problem 1 are equivalent to:\nmin\n𝜃1\n2∥𝑦−𝑀𝑄⊺𝐴′𝜃∥2\n2+𝜆𝑓∥𝜃∥1, (4)\nwhich is a block-separable LASSO problem with design matrix\n𝑀𝑄⊺𝐴′∈R𝑛×(𝑛−1)𝑝. We show in the next section that we do not\nneed to construct this matrix for our greedy selection rule.\n3.2 Block Selection Rule (BGS rule)\nSince Problem 4 is an equivalent LASSO reformulation of Problem\n1, we use this reformulation to select which blocks to update when\nperforming BCD on Problem 1. For each BCD iteration, we apply\na block Gauss Southwell (BGS) greedy selection rule to Problem 4\nto select the next block to update. BGS selection has been shown\nin theory and in practice to make more progress per iteration than\ncyclic or random selection [ 5,21], however, on many problems,\nBGS selection is prohibitively expensive [ 22]. One critical aspect\nof our procedure is that we exploit problem structure to develop a\nBGS steepest direction (BGS-s) rule that is cheap to compute.\nLet𝑓(𝜃)=1\n2∥𝑦−𝑀𝑄⊺𝐴′𝜃∥2\n2. For BGS-s selection, we first\ncompute vector 𝑑∈R𝑛𝑝which stores the magnitude of the mostnegative directional derivative for each coordinate. This vector is\ndefined coordinate-wise by\n𝑑𝑖=(\n|𝑆𝜆𝑓(∇𝑖𝑓(𝜃)| if𝜃𝑖=0\n|∇𝑖𝑓(𝜃)+sign(𝜃𝑖)𝜆𝑓|if𝜃𝑖≠0,(5)\nwhere𝑆𝜆𝑓is the soft-thresholding operator. Let 𝑑𝑘∈R𝑛represent\nthe elements in vector 𝑑associated with block 𝑘. We select the best\nblock𝑘∗to update via:\n𝑘∗=arg max\n𝑘∈[𝑝]∥𝑑𝑘∥2\n2.(6)\nEquations 5 and 6 form our BGS selection rule, which is compu-\ntationally bottlenecked by the cost of computing the full gradient\n∇𝑓(𝜃). The LASSO design matrix 𝑀𝑄⊺𝐴′is also only used to com-\npute this gradient. Below, we show how to efficiently compute\ngradient∇𝑓(𝜃)without forming the LASSO design matrix.\nFast Gradient Procedure: We have that∇𝑓(𝜃)=−𝐴′⊺𝑄𝑀⊺𝑟′,\nwhere𝑟′=𝑦−𝑀𝑄⊺𝐴′𝜃. Since our algorithm is zero-initialized, we\ncan store𝑟′and update the residual vector at each BCD iteration\nto avoid multiplying the design matrix with 𝜃. Matrix𝑀⊺consists\nof𝑝identity matrices stacked vertically which makes the gradient\nexpression block-separable. For a fixed block 𝑘∈[𝑝], we have that\n∇𝑘𝑓(𝜃)=−𝐴⊺𝑄𝑘𝑟′, where𝑄𝑘is the sorting matrix for feature 𝑥𝑗.\nThe matrix𝐴⊺is a padded upper triangular matrix, so computing\nthe gradient for block 𝑘simply involves ordering 𝑟′with respect to\nthe sorted values of 𝑥𝑗and taking a rolling sum down the ordered\nvector, which is extremely efficient. Computing the full gradient\ncan be embarrassingly parallelized across blocks.\nWith this procedure, our BGS-s selection rule is efficient, par-\nallelizable, and can be computed without constructing the LASSO\ndesign matrix. Below, we formalize our greedy block coordinate\ndescent (GBCD) algorithm and analyze its convergence properties.\n3.3 BGS-GBCD Algorithm\nTo solve Problem 1, we use the following GBCD algorithm. Start\nwith𝛽𝑗=0for all blocks 𝑗∈[𝑝]and repeat until convergence:\napply our BGS selection rule to Problem 4 (LASSO reformulation) to\nselect a block to update and solve Problem 2 (original block update\nproblem) with dynamic programming to update the block. This\nreturns a sequence of solutions 𝛽𝑡that correspond to a sequence\nof decreasing objective values.\n3.3.1 Convergence Analysis. The sequence of solutions 𝛽𝑡returned\nby BGS-GBCD converges to the minimizer for Problems 1 and 4.\nMore generally, we show that BGS-GBCD converges to optimality\nwhen applied to block-separable LASSO problems. We prove the\nnext proposition in the appendix (suppl. B.1).\nProposition 1. Given composite problem\nmin\n𝜃𝐹(𝜃)=𝑓(𝜃)+𝜆∥𝜃∥1,\nwhere𝑓is convex and coordinate-wise L-smooth and 𝜃is both block\nand coordinate separable, every limit point of BGS-GBCD coincides\nwith a minimizer for 𝐹(𝜃). Any sequence of solutions 𝜃𝑡generated\nby BGS-GBCD converges to a limit/minimum point.\nWe prove for the first time that greedy block coordinate descent\nusing block Gauss-Southwell-s selection converges to the minimumBrian Liu & Rahul Mazumder\npoint when applied to ℓ1-composite problems. We also show that\nunder certain conditions, BGS-GBCD updates make provably good\nprogress towards the minimum. Proposition 2 states a property that\nwe exploit in §4 when developing feature selection algorithms.\nProposition 2. If block𝜃𝑡\n𝑘=0is selected via the BGS rule, the\nprogress after one GBCD update is bound by:\n𝐹(𝜃𝑡+1)−𝐹(𝜃𝑡)≤ min\n𝛾∈R𝑛∇𝑓(𝜃𝑡)⊺𝛾+\nL\n2snorm(𝛾)+𝜆∥𝜃𝑡+𝛾∥1−𝜆∥𝜃𝑡∥1,\nwhere snorm( 𝛾) sums theℓ2-norm of each block in 𝛾.\nEach block in our optimization problem corresponds to a fea-\nture;𝛽𝑗gives the contribution of feature 𝑥𝑗to the additive model.\nProposition 2 states that when the BGS rule is used to select a fea-\nture (block) to enter the support, the corresponding block update\nmakes substantial progress towards the minimum. The proof for\nthis proposition is also in the appendix (suppl. B.2).\n3.4 Discussion\nIn most LASSO problems, greedy selection offers little advantage\nover cyclic selection since the computational cost of selecting the\nblock with the steepest directional derivatives is similar to the\ncost of updating all of the blocks [ 31]. Greedy BCD is effective in\nFAST, however, since block selections, which involve embarrass-\ningly parallel summations are much cheaper than block updates,\nwhich require expensive dynamic programming calls.\nWe observe that BGS-GBCD greatly reduces the number of dy-\nnamic programming block updates (Problem 2) required to solve\nProblem 1, compared to cyclic block selection. For example, in Fig-\n0 1 2 3 4\n# Dynamic Programming Block Updates (log10)0.10.20.30.40.5Training ErrorCyclic vs. Greedy Block Selection\nCyclic Block Selection\nGreedy Block Selection\nFigure 2: Greedy selec-\ntion reduces the num-\nber of dynamic pro-\ngramming block up-\ndates by 2 orders of\nmagnitude.\nure 2, we use greedy and cyclic BCD to fit FAST on the Elevators\ndataset [ 29]. The horizontal axis shows the number of dynamic\nprogramming block updates (log10) and the vertical axis shows\ntraining loss. We observe that BGS-GBCD requires nearly 100 ×\nfewer updates to converge. This corresponds to substantial compu-\ntational speedups, which we show in our experiments in §5.1.\n3.5 Binning\nFAST can also incorporate binning, a popular heuristic used by\nEBMs [ 20] and LightGBMs [ 13], to reduce computation time for a\nnominal cost in model expressiveness. FAST performs binning using\na novel equivalent optimization formulation while existing methods,\nsuch as EBMs, pre-process the data. Given a set of bins for eachfeature𝑥𝑗, we add the constraints for all entries (𝑖1,𝑖2)∈[𝑛]that\nif entries(𝑥𝑗)𝑖1and(𝑥𝑗)𝑖2fall in the same bin, then (𝛽𝑗)𝑖1=(𝛽𝑗)𝑖2.\nWe show in the appendix (suppl. C) that we can reformulate these\nconstraints into a weighted smooth loss function in the objective\nand efficiently solve this unconstrained problem with BGS-GBCD.\nBinning directly reduces the number of decision variables in FAST\nby a factor of # bins over # rows and combining BGS-GBCD with\nbinning further reduces computation time.\n4 FEATURE-SPARSE FAST\nOur FAST framework is quite flexible; here we discuss an extension\nof the framework to explicitly account for variable selection. We add\nthis group sparsity constraint to Problem 1:Í𝑝\n𝑗=11(𝛽𝑗≠0)≤𝐾,\nwhere𝐾is the maximum number of features to select. Problem 1\nwith this constraint is NP-hard and difficult to solve to optimality\ndue to the large number of variables; we have a variable for each\nentry of𝑋. As such, we develop two approximate algorithms to\nfind good solutions. These algorithms have different strengths in\nterms of solution quality and runtime, but both algorithms rely\non the BGS rule presented in §3.2 and the fact that BGS selection\nmakes provably good progress when selecting features to enter the\nsupport (Prop. 2).\n4.1 Approximate Greedy Iterative Selection\nFor Approximate Greedy Iterative Selection (AGIS), we partition\nthe blocks into the support 𝑆={𝑗∈[𝑝]|𝛽𝑗≠0}and complement\n𝑆𝑐and start with all blocks equal to 0. We use the BGS rule to select\nthe best block 𝑘∈𝑆𝑐to update and we perform a block update by\nsolving Problem 2 to add 𝑘into𝑆. If|𝑆|>1, we iterative through\nthe blocks in 𝑆and conduct block updates until convergences. We\nrepeat this procedure, interlacing BGS selection with sweeps on\nthe support 𝑆until the condition |𝑆|=𝐾is reached. AGIS returns a\nsequence of PCAMs with every feature sparsity level from 1...𝐾 .\nTo improve solution quality across all sparsity levels we apply this\nlocal search heuristic.\n4.1.1 BGS Local Search: After each sweep of 𝑆converges, use the\nBGS rule to select the best block to update in 𝑆𝑐and denote that\nblock𝛽∗\n𝑗. This is the block that we will swap into the support. To find\nthe best block to swap out of the support, iterate over 𝛽𝑗∈𝑆. For\neach block, set 𝛽𝑗=0and conduct a block update on 𝛽∗\n𝑗, and select\nthe block in 𝑆that when swapped improves the objective the most.\nAfter this swap, conduct another sweep over 𝑆until convergence\nto obtain the final solution. We present our full AGIS algorithm,\nwith local search, in Algorithm 1.\n4.2 Group ℓ0-FAST\nIn addition to AGIS, we can use a group ℓ0-sparsity penalty to select\nfeatures in FAST. This approach often obtains better solutions at\nthe cost of increased computation time, which we discuss in §5.2.FAST: An Optimization Framework for Fast Additive Segmentation in Transparent ML\nAlgorithm 1: AGIS\nInput:𝐾,𝜆𝑓,𝐷,𝑄𝑗∀𝑗∈[𝑝]\n1Initialize𝛽𝑗=0∀𝑗∈[𝑝],𝑆=∅,𝑆𝑎𝑙𝑙=∅\n2repeat\n3 Use BGS rule to select 𝑘∈𝑆𝑐.\n4 Update block 𝑘(Problem 2).\n5𝑆=𝑆∪𝛽𝑘, 𝑆𝑐=𝑆𝑐\\𝛽𝑘\n6 repeat\n7 Sweep through 𝑆and update blocks (Problem 2).\n8 until converged\n9 BGS local search.\n10𝑆𝑎𝑙𝑙=𝑆𝑎𝑙𝑙∪𝑆\n11until|𝑆|=𝐾\nOutput: Sequence of models 𝑆𝑎𝑙𝑙\nWe use this Lagrangian formulation:\nmin\n𝛽1\n2∥𝑦−𝑝∑︁\n𝑗=1𝑄⊺\n𝑗𝛽𝑗∥2\n2+\n𝜆𝑓𝑝∑︁\n𝑗=1∥𝐷𝛽𝑗∥1+𝜆𝑠𝑝∑︁\n𝑗=11(𝛽𝑗≠0),(7)\nwhere𝜆𝑠is the sparsity hyperparameter.The group sparsity penalty\nis block-separable over 𝛽𝑗so we can apply BCD methods to find\ngood solutions to this problem. Given fixed block 𝑘and residual\nvector𝑟, we can write each block update problem as:\nmin\n𝛽𝑘1\n2∥𝑄𝑘𝑟−𝛽𝑘∥2\n2+𝜆𝑓∥𝐷𝛽𝑘∥1+𝜆𝑠 1(𝛽𝑘≠0).\nThis problem can be solved by first setting 𝜆𝑠=0and solving the\nFLSA for𝛽∗\n𝑘. We then check the thresholding condition:\n1\n2∥𝑟∥2\n2−1\n2∥𝑄𝑘𝑟−𝛽∗\n𝑘∥2\n2−𝜆𝑓∥𝐷𝛽∗\n𝑘∥1≤𝜆𝑠\nand set𝛽∗\n𝑘=0if the condition is satisfied. We show the derivation\nfor this in the appendix (suppl. D).\nSince the group sparsity penalty is not continuous, it is not clear\nif BGS-GBCD can be extended here. To find high-quality solutions\nto Problem 7, we use cyclic block coordinate descent and apply\nour BGS local search heuristic (§4.1.1) when CBCD converges. We\ninterlace CBCD sweeps with local search steps until the objective\nno longer improves.\n4.3 Discussion\nWe show an example of the impact of local search on solution\nquality and discuss the strengths and weaknesses of both group\nℓ0-FAST and AGIS.\n4.3.1 Local Search Performance. We observe empirically that our\nBGS local search heuristic improves the out-of-sample performance\nof both feature selection algorithms. For example, in Figure 3, we\nuse groupℓ0-FAST and AGIS to build feature-sparse PCAMs on the\nElevators dataset (16500 rows and 16 columns) [ 29]. We vary𝐾, the\nsparsity budget in the model from 1 to 10, and compare the test\nperformance of the model measured via MSE. For both methods,\nthe local search heuristic improves performance.\n4.3.2 Group ℓ0-FAST vs. AGIS. In our experiments in §5.2, we ob-\nserve that group ℓ0-FAST generally outperforms AGIS at build-\ning sparse PCAMs. AGIS, however, is computationally faster since\n2 4 6 8 100.20.30.40.50.6Test MSEAGIS no LS\nAGIS w/ LS\n2 4 6 8 10Group/lscript0no LS\nGroup/lscript0w/ LS\nNumber of FeaturesEffect of Local SearchFigure 3: BGS local search improves the solution quality for\nboth of our feature-sparse PCAM algorithms.\nthe algorithm can leverage greedy block selection. Fitting group\nℓ0-FAST requires CBCD updates due to the non-convexity of the\ngroup sparsity penalty. In addition, AGIS is easier to use since the\nalgorithm by design outputs a sequence of PCAMs with with every\nsupport size from 1to𝐾. The sparsity hyperparameter 𝜆𝑠in group\nℓ0-FAST must be tuned and the algorithm may skip certain support\nsizes due to non-convexity [9].\n5 EXPERIMENTS\nWe evaluate the computation time of FAST against existing algo-\nrithms and assess how well the framework performs at building\nfeature-sparse PCAMs.\n5.1 Computation Time Experiment\nWe compare the computation time of FAST against existing SOTA\nalgorithms for building feature-dense PCAMs: EBMs and FLAMs.\n5.1.1 Experimental Procedure. On 10 large regression benchmark\ndatasets from OpenML [ 29], we use FAST, EBM, and FLAM to\nfit PCAMs. For the competing methods, we use the InterpretML\npackage [ 20] to fit EBMs in Python and the FLAM package in R\n[24]. We use the default hyperparameters for InterpretML EBMs,\nwhich are optimized for fast runtime. For FLAM, we match the\nfusion hyperparameter with the value used in FAST. The test errors\nof the models fit using the 3 methods, under these configurations,\nare comparable (as intended). We conduct this experiment on a M2\nMacbook Pro with 10 cores and match the number of cores used\nin the methods that support multiprocessing (FAST and EBMs).\nAdditional details can be found in the appendix (suppl. E).\n5.1.2 Results. Table 1 shows the results of our experiment. The\nleftmost column shows dataset names and dimensions: (𝑛,𝑝). In\neach cell in the other columns, the top entry shows the computation\ntime of the method averaged over runs along with the standard\ndeviation. The bottom entry shows the test MSE of the model.\nThe top 5 rows of this table show timing results on large 𝑛\ndatasets with more than 500000 rows. On these datasets, we are\nunable to apply FLAM due to problem scale so we compare FAST\nagainst EBMs. We observe that FAST fits PCAMs up to 2 orders of\nmagnitude faster than EBMs. For example on an augmented version\nof the UK Black Smoke dataset, with 9 million rows and 41 columns,\nit takes over 15 hours to fit an EBM. FAST on the other hand can\nfit a PCAM that performs the same in around 5 minutes.Brian Liu & Rahul Mazumder\nDataset / Method FAST EBM FLAM\nBlack Smoke +\n(9214951, 41)329.6s (1.2)\n0.3715h 49m 31s\n0.37__\nBlack Smoke\n(9214951, 14)43s (2.8)\n0.383h 57m 9s\n0.38__\nPhysiochemical\n(5023496, 9)33.4s (0.3)\n0.5243m 31s (31.9)\n0.53__\nAuto Horsepower\n(900000, 17)1.63s (0.01)\n0.4285s (3.9)\n0.43__\nAilerons BNG\n(669994, 38)2.57s (0.08)\n0.4485s (2.0)\n0.44__\nSlice Localization\n(35845, 351)6.7s (0.05)\n0.1858.7s (2.2)\n0.2014m 50s (30.4)\n0.18\nSuperconduct\n(21263, 79)0.45s (0.01)\n0.207.6s (0.05)\n0.2113.0s (0.03)\n0.20\nScm1d\n(8828, 280)0.7s (0.01)\n0.107.9s (0.47)\n0.12190s (5.1)\n0.10\nRf2\n(8212, 448)1.34s (.01)\n0.0278.5s (4.3)\n0.02180s (3.0)\n0.03\nIsolet\n(7017, 613)2.50s (0.02)\n0.328.67s (0.3)\n0.32360s (8.0)\n0.34\nTable 1: Timing experiment results. FAST achieves 2 orders\nof magnitude speedups for large problems.\nThe bottom 5 rows of Table 1 show results on large 𝑝datasets\nwith more than 50 columns, but less than 50,000 rows. We observe\nhere that FAST fits PCAMs 2 orders of magnitude faster than FLAM\nand around 1 order of magnitude faster than EBMs. For example on\nthe Slice Localization dataset, with over 300 columns, it takes nearly\n15 minutes to fit a FLAM. FAST can fit a PCAM that performs the\nsame in under 10 seconds.\nIn all, we find that FAST substantially outperforms EBMs and\nFLAMs in terms of computation time across various large problems.\n0 10 20 30 400.40.5Test MSE1\n2\n3\n4\n5Black Smoke n≈107\n0.0 0.5 1.0 1.50.50.61\n2\n3\n45Auto Horsepower n≈106\nTraining Time (seconds)\nFigure 4: FAST can fit low optimization tolerance models that\nperform well extremely quickly.\n5.1.3 Low Optimization Tolerance Models. As an aside, we note\nthat we can leverage our greedy BCD algorithm to fit FAST with low\noptimization tolerances, in order to quickly produce a PCAM that\nstill performs well out-of-sample. In Figure 4, we show the test error\nof FAST (vertical axes) plotted against the training time in seconds\n(horizontal axes) for the UK Black Smoke and Auto Horsepower\n[29] datasets. We vary the training time of FAST by early-stopping\nthe optimization algorithm after a fixed number of iterations, the\nfirst 5 GBCD iterations are plotted in red. In both examples, thefirst 5 iterations greatly reduce the testerror of the model. For the\nUK Black Smoke dataset, FAST can fit a low optimization tolerance\nmodel that performs well in less than 10 seconds.\n5.2 Feature Selection Experiment\nHere we evaluate how well FAST performs at building feature-\nsparse PCAMs.\n5.2.1 Experimental Setup. We repeat this procedure on 20 regres-\nsion datasets from OpenML and use a 10-fold CV on each dataset.\nThe full list of datasets can be found in the appendix (suppl. E). On\nthe training folds, we use group ℓ0-FAST and AGIS to fit feature-\nsparse PCAMs by varying the sparsity budget 𝐾∈{2,4,6,8,10}.\nWe evaluate the MSE of each sparse model on the test fold.\nWe compare the performance of these models against the fol-\nlowing SOTA algorithms to construct feature-sparse PCAMs.\n•FLAM-GL (2016): In FLAM group LASSO [ 25], we fit a FLAM\nwith a group LASSO penalty over the features. We tune the spar-\nsity hyperparameter such that at most 𝐾features are selected.\n•EBM-RS (2019): In EBM rank and select [ 20], we first fit an EBM\non the training data and rank the features by importance scores;\nthe contribution of each feature averaged over the training ob-\nservations. We select the top 𝐾features and refit an EBM. This\nmethod is computationally expensive since it fits two PCAMs.\n•ControlBurn (2021): ControlBurn [ 15] is a flexible framework\nfor building feature-sparse nonlinear models. The feature selec-\ntion algorithm in the framework first constructs a specialized\ntree ensemble that is diverse, where each tree in the ensemble\nuses a different subset of features. Then, the weighted LASSO is\nused to select feature-sparse subsets of trees that perform well.\nWe refit the final model, in this case, an EBM, on the 𝐾selected\nfeatures. ControlBurn with an EBM has been used to construct\nhigh-performing, feature-sparse PCAMs for heart failure predic-\ntion in clinical machine learning [28].\n•FastSparseGAM (2022): FastSparseGAM [ 16] is a package for\nsparse regression built on top of the L0Learn framework [ 9,10].\nThe package can be adapted to construct extremely sparse PCAMs\nby one-hot encoding the features and selecting a small subset of\nthe resulting components [17].\nWe also compare feature-sparse FAST against two traditional al-\ngorithms that produce non-piecewise constant additive models,\nSparse Additive Models ( SAM ), which uses the group LASSO to\nsparsify splines, and the linear LASSO . Additional details on our\nexperimental procedure can be found in the appendix (suppl. E).\n5.2.2 Results. For each run of our experiment, we compute the\npercent decrease in test MSE between feature-sparse FAST and each\ncompeting algorithm, given by:\n% decrease MSE =MSE Competing Alg. −MSE FAST\nMSE FAST,\nfor each sparsity budget. A positive percent decrease in test er-\nror indicates that feature-sparse FAST performs better than the\ncompeting algorithm for that sparsity budget.\nIn Table 2 we report the average percent decrease in test error\nbetween group ℓ0-FAST and the competing algorithms ( topvalues)\nand AGIS and the competing algorithms ( bottom values) across all\nsparsity budgets. These averages are taken across all datasets andFAST: An Optimization Framework for Fast Additive Segmentation in Transparent ML\n2 4 6 8 100%100%200%300%400%500%600%Group/lscript0-FAST % Decrease in Test MSE\n2 4 6 8 10AGIS % Decrease in Test MSE\nFLAM-GL\nEBM-RS\nControlBurn\nFastSparseGAM\nSAM\nLASSO\nSparsity\nFigure 5: Distribution of results from our feature selection experiment (averages shown in Table 2). The distributions are mostly\nentirely positive, which indicates that feature-sparse FAST outperforms our competing algorithms.\nCompeting Alg. /\nSparsity2 4 6 8 10\nFLAM-GL84.1%\n81.5%102.0%\n81.6%77.9%\n62.3%50.8%\n48.6%45.1%\n41.4%\nEBM-RS48.9%\n45.1%24.3%\n18.3%29.7%\n22.7%23.7%\n22.0%22.8%\n19.8%\nControlBurn28.4%\n27.7%57.5%\n47.5%71.3%\n60.1%68.1%\n65.4%70.4%\n65.6%\nFastSparseGAM131.5%\n126.5%85.6%\n73.4%72.4%\n60.7%50.9%\n48.6%48.2%\n44.2%\nSAM103.5%\n102.4%64.0%\n51.7%52.8%\n40.4%30.0%\n28.4%23.5%\n20.5%\nLASSO89.6%\n89.5%71.9%\n65.8%56.1%\n49.1%46.0%\n44.1%41.2%\n38.2%\nTable 2: Average % decrease in test error between feature-\nsparse FAST and our competing algorithms across sparsity\nbudgets (distributions shown in Figure 5). Positive values\nindicate that feature-sparse FAST outperforms the competing\nalgorithm. In each cell, the top value shows group ℓ0-FAST\nand the bottom value shows AGIS.\nfolds in our experiment. In Figure 5, we show the full distributions\nof our results. In each plot, the horizontal axis shows the sparsity\nbudget and the vertical axis shows the percent decrease in test error\nbetween feature-sparse FAST and the competing algorithm; the\nleft plot shows group ℓ0-FAST and the right plot shows AGIS. The\ngrouped violin plots show the distribution of the results for each\nsparsity budget and the averages of each distribution are marked\nby horizontal lines, which correspond to the averages in Table 2.\nFrom Figure 5, we see that group ℓ0-FAST and AGIS largely\noutperform all of our competing algorithms. The distributions of\nthe percent decrease in test error between these two methods and\nour competing algorithms are nearly entirely positive across all\nsparsity budgets. We also observe that group ℓ0-FAST consistently\nperforms slightly better than AGIS. In Table 2, the average percent\ndecrease in test error for group ℓ0- FAST (top value) is always higher\nthan that for AGIS (bottom value). However, as discussed in §4.3.2,\nAGIS also has several advantages in terms of speed and ease of use.For sparsity budget 𝐾=2, groupℓ0-FAST attains a 28% decrease\nin test error compared to the best competing algorithm, Control-\nBurn. Interestingly, the performance of ControlBurn degrades as 𝐾\nincreases. This is because the framework selects features indepen-\ndently of the final EBM that is refitted [ 15]. While ControlBurn is\nuseful for selecting a few important features, our results show that\nthe framework fails at building sparse PCAMs for larger values of\n𝐾. For𝐾∈{4,6,8,10}, groupℓ0-FAST attains up to a 30% decrease\nin test error compared to the best competing algorithm, EBM-RS.\nIn all, we observe that group ℓ0-FAST consistently outperforms the\nbest competing algorithm across all sparsities.\nIn addition, feature-sparse FAST substantially outperforms FLAM-\nGL and FastSparseGAM, by over a 100% decrease in test error for\nsome sparsities. FLAM-GL is affected by over-shrinkage from the\ngroup LASSO, which is especially pronounced since the FLAM for-\nmulation uses a large number of variables. The ℓ0-based penalties\nand constraints in feature-sparse FAST are shrinkage-free and ro-\nbust to this effect. FastSparseGAM indirectly selects features by\nsparsifying piecewise segments in the additive model [ 17]. Feature-\nsparse FAST, on the other hand, directly accounts for feature spar-\nsity in the optimization framework and outperforms this competing\nmethod. We also note that feature-sparse FAST substantially out-\nperforms our competing algorithms that do not fit PCAMs: SAM,\nwhich also uses the group LASSO, and the linear LASSO.\nFinally, we observe that many distributions in Figure 5 have\nheavy positive tails, notably for the EBM-RS, FLAM-GL, and Con-\ntrolBurn competing algorithms. These heavy tails typically contain\nthe results from datasets with correlated features; we show correla-\ntion matrices and the distribution of errors in the appendix (suppl.\nF). In §6.1, we show through a semi-synthetic experiment that cor-\nrelated features degrade the performance of EBM-RS, FLAM-GL,\nand ControlBurn. Group ℓ0-FAST and AGIS, on the other hand, can\neffectively build sparse PCAMs regardless of feature correlations.Brian Liu & Rahul Mazumder\n6 PCAMS AND FEATURE CORRELATIONS\nWe conclude by investigating how correlated features impact the\ninterpretability of PCAMs.\n6.1 Correlated Feature Selection\nWe present here a semi-synthetic example to investigate how corre-\nlated features affect our feature selection experiment (§5.2). We start\nwith the Houses dataset [ 29] and build feature-sparse PCAMs using\nFAST, EBM-RS, FLAM-GL, and ControlBurn by varying sparsity\nbudget𝐾. The top left plot in Figure 6 shows the test performance\nof these sparse models.\n0.40.60.8Original Data\nAGIS\nGroup/lscript0\nEBM-RS\nFLAM-GL\nControlBurn\n3 Correlated Features Added\n2 4 6 80.40.60.86 Correlated Features Added\n2 4 6 89 Correlated Features Added\nSparsityTest MSE\nFigure 6: Group ℓ0-FAST and AGIS perform well even after\nadding correlated features.\nWe then add 3, 6, and 9 correlated features to the data. As the\nother plots in Figure 6 show, the performances of EBM-RS, Control-\nBurn, and FLAM-GL degrade significantly with added correlations\nbut the performances of group ℓ0-FAST and AGIS remain unaffected.\nEBM feature importance scores capture the contribution of each\nfeature to the prediction of the model, averaged over all training\nobservations. Given a pair of highly correlated features, the cyclic\nround-robin algorithm used to fit EBMs will split the contribution\nof the features evenly between the pair. As such, the average feature\nimportance score/ranking of a group of correlated features will be\nsuppressed, which degrades the performance of EBM-RS. This effect\nis analogous to the so-called correlation bias observed in random\nforest feature rankings by the ControlBurn paper [15].\nControlBurn attempts to address correlation bias by using the\nweighted LASSO to select features and we indeed observe in Figure\n6 that the algorithm is more robust than EBM-RS to added cor-\nrelations. However, the LASSO penalty used in ControlBurn still\nimparts shrinkage which biases sparse selection in the presence of\nmulticollinearity [ 8,11,19]. Additionally, the group LASSO penalty\nin FLAM-GL is known to suffer from over-shrinkage and performs\neven worse at selecting sparse subsets of correlated groups [ 11].\nThe penalties and constraints used to select features in FAST are\nshrinkage-free and, as a result, our algorithms are unaffected by\nthe added correlated features.6.2 Identifying Discontinuities\nOne attractive property of PCAMs is their ability to capture dis-\ncontinuous patterns in the underlying data. Here, we present a\ncase study to demonstrate how correlated features can degrade the\nability of EBMs to identify discontinuities.\nWe use the Houses [ 29] dataset to build PCAMs to predict house\nprices using demographic features. Unsurprisingly, there is a nearly\nlinear relationship between the median income of a district and\nthe price of homes in that district. We add the following artificial\ndiscontinuity to the data: for all districts with a median income\nabove $40,000 a year, we drop the price of homes by $20,000.\n-1-0.50FAST original\nMedian Income\nCorrelated FeaturesFAST w/ correlations\n0 4 8-1-0.50EBM original\n0 4 8EBM w/ correlations\nFeature ValueContribution\nFigure 7: Correlated features can mask discontinuities in\nPCAM shape functions.\nWe then fit feature-dense PCAMs using FAST and EBMs. Both\nmethods capture the discontinuity in the shape function for median\nincome, as shown in the left two plots in Figure 7. Next, we add 10\nsynthetic features that are correlated with median income and refit.\nThe right two plots in Figure 7 show the new shape functions.\nThe red lines show median income and the grey lines show the syn-\nthetic correlated features. We observe that for FAST (top right), the\nshape function of median income is preserved and that the discon-\ntinuous pattern can still be easily identified. The shape functions\nof the synthetic correlated features are reduced in magnitude and\nseveral are implicitly regularized to zero. The EBM shape functions\n(bottom right) tell a different story. Due to the cyclic algorithm used\nto fit EBMs, the contribution of median income is evenly distributed\namong the noisy correlated features. All the shape functions in this\ngroup are compressed and the discontinuity is difficult to detect.\nIf a dataset contains a group of correlated features, and only one\nfeature in that group contains an interesting discontinuity, FAST\nmay be more likely to capture this pattern compared to EBMs. Also,\nconsider the case where a dataset contains a sensitive attribute\nwith a discontinuous pattern. A potential adversarial attack would\nbe to add features correlated with this sensitive attribute to mask\nthis discontinuous pattern from EBMs. Due to greedy model fitting,\nFAST again may be more robust to this attack.\nWe emphasize that both FAST and EBMs produce transparent\nPCAMs, however, the interpretations of the models change depend-\ning on whether the PCAMs were fit greedily or cyclically. ModelFAST: An Optimization Framework for Fast Additive Segmentation in Transparent ML\ntransparency does not guarantee trustworthiness and practitioners\nshould still interpret transparent models cautiously.\nConclusion: FAST is an optimization-based framework that\nleverages a novel greedy optimization procedure to fit PCAMs up\nto 2 orders of magnitude faster than SOTA methods. The frame-\nwork also introduces two feature selection algorithms that signif-\nicantly outperform existing methods at building sparse PCAMs.\nUsing FAST, we investigate how correlated features impact the\ninterpretability of PCAMs in terms of selecting important features\nand interpreting shape functions. These phenomena should be con-\nsidered when evaluating the trustworthiness of additive models.\nREFERENCES\n[1]Peter Bacchetti. Additive isotonic models. Journal of the American Statistical\nAssociation , 84(405):289–294, 1989.\n[2] Rich Caruana, Yin Lou, Johannes Gehrke, Paul Koch, Marc Sturm, and Noemie\nElhadad. Intelligible models for healthcare: Predicting pneumonia risk and\nhospital 30-day readmission. In Proceedings of the 21th ACM SIGKDD international\nconference on knowledge discovery and data mining , pages 1721–1730, 2015.\n[3] Chun-Hao Chang, Sarah Tan, Ben Lengerich, Anna Goldenberg, and Rich Caru-\nana. How interpretable and trustworthy are gams? 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Coordinate descent algorithms for lasso\npenalized regression. 2008."    },    {        "title": "2402.12645v1.Optimal_PSPACE_hardness_of_Approximating_Set_Cover_Reconfiguration.pdf",        "content": "Optimal PSPACE-hardness of\nApproximating Set Cover Reconfiguration\nShuichi Hirahara* Naoto Ohsaka†\nFebruary 21, 2024\nAbstract\nIn the M INMAX SETCOVER RECONFIGURATION problem, given a set system Fover\na universe and its two covers CstartandCgoalof size k, we wish to transform Cstart\nintoCgoalby repeatedly adding or removing a single set of Fwhile covering the uni-\nverse in any intermediate state. Then, the objective is to minimize the maximize size\nof any intermediate cover during transformation. We prove that M INMAX SETCOVER\nRECONFIGURATION and M INMAX DOMINATING SETRECONFIGURATION arePSPACE -\nhard to approximate within a factor of 2 −1\npolyloglog N, where Nis the size of the universe\nand the number of vertices in a graph, respectively, improving upon Ohsaka (SODA\n2024) [Ohs24] and Karthik C. S. and Manurangsi (2023) [KM23]. This is the first re-\nsult that exhibits a sharp threshold for the approximation factor of any reconfigura-\ntion problem because both problems admit a 2-factor approximation algorithm as per\nIto, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno (Theor. Comput. Sci.,\n2011) [IDHPSUU11]. Our proof is based on a reconfiguration analogue of the FGLSS\nreduction [FGLSS96] from Probabilistically Checkable Reconfiguration Proofs of Hira-\nhara and Ohsaka (2024) [HO24]. We also prove that for any constant ε∈(0,1), M INMAX\nHYPERGRAPH VERTEX COVER RECONFIGURATION on poly( ε−1)-uniform hypergraphs\nisPSPACE -hard to approximate within a factor of 2 −ε.\n*National Institute of Informatics, Japan. s_hirahara@nii.ac.jp\n†CyberAgent, Inc., Tokyo, Japan. ohsaka_naoto@cyberagent.co.jp ;naoto.ohsaka@gmail.comarXiv:2402.12645v1  [cs.CC]  20 Feb 2024Contents\n1 Introduction 3\n1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.2 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.3 Proof Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n1.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n2 Preliminaries 7\n2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n2.2 Reconfiguration Problems on Constraint Graphs . . . . . . . . . . . . . . . . . . 8\n2.3 Probabilistically Checkable Reconfiguration Proof Systems . . . . . . . . . . . 10\n3 Subconstant Error PCRP Systems and FGLSS Reduction 11\n3.1 Bounded-degree PCRP Systems with Subconstant Error . . . . . . . . . . . . . 11\n3.2 FGLSS Reduction and PSPACE -hardness of Approximation for P ARTIAL 2CSP\nRECONFIGURATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n3.3 Reducing P ARTIAL 2CSP R ECONFIGURATION to L ABEL COVER RECONFIGU -\nRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n4 Applications 21\n4.1 PSPACE -hardness of Approximation for S ETCOVER RECONFIGURATION . . . 21\n4.2 PSPACE -hardness of Approximation for H YPERGRAPH VERTEX COVER RE-\nCONFIGURATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\n21 Introduction\n1.1 Background\nIn the field of reconfiguration, we study the reachability and connectivity over the space of\nfeasible solutions under an adjacency relation. Given a source problem that asks the exis-\ntence of a feasible solution, its reconfiguration problem requires to decide if there exists a\nreconfiguration sequence , namely, a step-by-step transformation between a pair of feasible\nsolutions while always preserving the feasibility of any intermediate solution. One of the\nreconfiguration problems we study in this paper is S ETCOVER RECONFIGURATION [IDH-\nPSUU11], whose source problem is S ETCOVER . In the S ETCOVER RECONFIGURATION\nproblem, for a set system Fover a universe Uand its two covers CstartandCgoalof size\nk, we seek a reconfiguration sequence from CstarttoCgoalconsisting only of covers of size\nat most k+1, each of which is obtained from the previous one by adding or removing a sin-\ngle set of F. Countless reconfiguration problems have been defined from a variety of source\nproblems, including Boolean satisfiability, constraint satisfaction problems, and graph prob-\nlems. Studying reconfiguration problems may help elucidate the structure of the solution\nspace of combinatorial problems [GKMP09].\nThe computational complexity of reconfiguration problems has the following trend: a re-\nconfiguration problem is likely to be PSPACE -complete if its source problem is intractable\n(say, NP-complete); e.g., S ETCOVER [IDHPSUU11], 3SAT [GKMP09], and I NDEPENDENT\nSET[HD05, HD09]; a source problem in Pfrequently induces a reconfiguration problem in\nP; e.g., S PANNING TREE [IDHPSUU11] and 2SAT [GKMP09]. Some exception are how-\never known; e.g., 3C OLORING [CvJ11] and S HORTEST PATH [Bon13]. We refer the readers\nto the surveys by Nishimura [Nis18] and van den Heuvel [van13] and the Combinatorial\nReconfiguration wiki [Hoa23] for more algorithmic and hardness results of reconfiguration\nproblems.\nTo overcome the computational hardness of a reconfiguration problem, we consider its\noptimization version , which affords to relax the feasibility of intermediate solutions. For ex-\nample, M INMAX SETCOVER RECONFIGURATION [IDHPSUU11] is an optimization version\nof S ETCOVER RECONFIGURATION , where we are allowed to use any cover of size greater\nthan k+1, but required to minimize the maximum size of any covers in the reconfiguration\nsequence (see Section 4.1 for the formal definition). Solving this problem approximately, we\nmay be able to find a “reasonable” reconfiguration sequence for S ETCOVER RECONFIGURA -\nTION which consists of covers of size at most, say, 1% larger than k+1. Unlike S ETCOVER ,\nwhich is NP-hard to approximate within a factor smaller than ln n[DS14, Fei98, LY94],\nMINMAX SETCOVER RECONFIGURATION admits a 2-factor approximation algorithm due to\nIto, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno [IDHPSUU11, Theorem 6].\nAn immediate question is: Is this the best possible?\nHere, we summarize known hardness-of-approximation results on M INMAX SETCOVER\nRECONFIGURATION . Ohsaka [Ohs24] showed that M INMAX SETCOVER RECONFIGURA -\n3TION isPSPACE -hard to approximate within a factor of 1 .0029 assuming the Reconfigu-\nration Inapproximability Hypothesis [Ohs23], which was recently proved [HO24, KM23].\nKarthik C. S. and Manurangsi [KM23] proved NP-hardness of the (2 −ε)-factor approxima-\ntion for any constant ε∈(0,1). Both results are not optimal: Ohsaka’s factor 1 .0029 is far\nsmaller than 2, while Karthik C. S. and Manurangsi’s result is not PSPACE -hardness. This\nleaves a tantalizing possibility that there may exist a polynomial-length reconfiguration\nsequence that achieves a 1 .0030-factor approximation for M INMAX SETCOVER RECONFIG -\nURATION , and hence the approximation problem may be in NP. Note that the PSPACE -\nhardness result of Ohsaka disproves the existence of a polynomial-length witness (in par-\nticular, a polynomial-length reconfiguration sequence) for the 1 .0029-factor approximation\nunder the assumption that NP̸=PSPACE .\n1.2 Our Results\nWe present optimal results of PSPACE -hardness of approximation for three reconfiguration\nproblems. Our first result is that M INMAX SETCOVER RECONFIGURATION isPSPACE -\nhard to approximate within a factor smaller than 2, improving upon Ohsaka [Ohs24, Corol-\nlary 4.2] and Karthik C. S. and Manurangsi [KM23, Theorem 4]. This is the first result\nthat exhibits a sharp threshold for the approximation factor of any reconfiguration problem:\napproximating within any factor below 2 is PSPACE -complete and within a 2-factor is in P\n[IDHPSUU11].\nTheorem 1.1 (informal; see Theorem 4.1) .For a set system Fof universe size Nand its two\ncoversCstartandCgoalof size k, it is PSPACE -complete to distinguish between the following\ncases :\n•(Completeness )There exists a reconfiguration sequence from CstarttoCgoalconsisting\nonly of covers of size at most k+1.\n•(Soundness )Every reconfiguration sequence contains a cover of size greater than (2−\nε(N))(k+1), where ε(N):=(polyloglog N)−1.\nIn particular, MINMAX SETCOVER RECONFIGURATION isPSPACE -hard to approximate\nwithin a factor of 2−1\npolyloglog N.\nAs a corollary of Theorem 4.1 along with [Ohs24], the following PSPACE -hardness of ap-\nproximation is established for D OMINATING SETRECONFIGURATION , which also admits a\n2-factor approximation [IDHPSUU11] (please refer to [Ohs24] for the problem definition).\nCorollary 1.2 (from Theorem 4.1 and [Ohs24, Corollary 4.3]) .MINMAX DOMINATING SET\nRECONFIGURATION isPSPACE -hard to approximate within a factor of 2−1\npolyloglog N, where\nNis the number of vertices in a graph.\n4Our third result is a similar inapproximability result for H YPERGRAPH VERTEX COVER\nRECONFIGURATION , which is defined analogously to S ETCOVER RECONFIGURATION (see\nSection 4.2 for the formal definition). M INMAX HYPERGRAPH VERTEX COVER RECONFIG -\nURATION is easily shown to be 2-factor approximable [IDHPSUU11]; we prove that this is\noptimal.\nTheorem 1.3 (informal; see Theorem 4.3) .For any constant ε∈(0,1), apoly(ε−1)-uniform\nhypergraph, and its two vertex covers CstartandCgoalof size k, it is PSPACE -complete to\ndistinguish between the following cases :\n•(Completeness )There exists a reconfiguration sequence from CstarttoCgoalconsisting\nonly of vertex covers of size at most k+1.\n•(Soundness )Every reconfiguration sequence contains a vertex cover of size greater than\n(2−ε)(k+1).\nIn particular, MINMAX HYPERGRAPH VERTEX COVER RECONFIGURATION onpoly(ε−1)-\nuniform hypergraphs is PSPACE -hard to approximate within a factor of 2−ε.\nWe highlight here that the size of hyperedges in a H YPERGRAPH VERTEX COVER RE-\nCONFIGURATION instance of Theorem 4.3 depends (polynomially) only on the value of ε−1,\nwhereas the size of subsets in a S ETCOVER RECONFIGURATION instance of Theorem 4.1\nmay depend on the universe size N.\n1.3 Proof Overview\nAt a high level, our proofs of Theorems 1.1 and 1.3 are given by combining the ideas devel-\noped in [HO24, KM23, Ohs23, Ohs24]. Karthik C. S. and Manurangsi [KM23] proved NP-\nhardness of the (2 −ε)-factor approximation of M INMAX SETCOVER RECONFIGURATION as\nfollows.\n1. Starting from the PCP theorem for NP[ALMSS98, AS98], they applied the FGLSS\nreduction [FGLSS96] to prove NP-hardness of the O(ε−1)-factor approximation of an\nintermediate problem, which we call M AXPARTIAL 2CSP.\n2. The O(ε−1)-factor approximation of M AXPARTIAL 2CSP is reduced to the (2 −ε)-factor\napproximation of a reconfiguration problem, which we call L ABEL COVER RECONFIG -\nURATION (Problem 2.3).\n3. L ABEL COVER RECONFIGURATION can be reduced to M INMAX SETCOVER RECON -\nFIGURATION via approximation-preserving reductions of Lund and Yannakakis [LY94]\nand Ohsaka [Ohs24].\n5Here, M AXPARTIAL 2CSP is defined as follows. The input consists of a graph G=(V,E), a\nfinite alphabet Σ, and constraints ψe:Σ2→{0,1}for each edge e∈E. Apartial assignment\nis a function f:V→Σ∪{⊥}, where the symbol ⊥indicates “unassigned.” The task is to\nmaximize the fraction of assigned vertices in a partial assignment fthat satisfies ψefor\nevery e=(v,w)∈E; i.e., ψe(f(v),f(w))=1 iff(v)̸= ⊥and f(w)̸= ⊥.\nTo improve this NP-hardness result to PSPACE -hardness, we replace the starting point\nwith the PCRP (Probabilistically Checkable Reconfiguration Proof) system of Hirahara and\nOhsaka [HO24], which is a reconfiguration analogue of the PCP theorem. We also replace\nMAXPARTIAL 2CSP with its reconfiguration analogue, which we call P ARTIAL 2CSP R E-\nCONFIGURATION (Problem 2.2). The proof of PSPACE -hardness is outlined as follows.\n1. Starting from the PCRP theorem forPSPACE [HO24], we apply the FGLSS reduction\n[FGLSS96] to prove PSPACE -hardness of P ARTIAL 2CSP R ECONFIGURATION (Sec-\ntions 3.1 and 3.2).\n2. We reduce P ARTIAL 2CSP R ECONFIGURATION to L ABEL COVER RECONFIGURATION\n(Section 3.3).\n3. We reduce L ABEL COVER RECONFIGURATION to M INMAX SETCOVER RECONFIGU -\nRATION by the reductions of [LY94, Ohs24] (Section 4.1).\nThe second and third steps are similar to the previous work [KM23]. Our main technical\ncontribution lies in the first step, which we explain below.\nRoughly speaking, the PCRP theorem [HO24] shows that any PSPACE computation on\ninputs of length ncan be encoded into a sequence π(1),···,π(T)∈{0,1}poly( n)of exponentially\nmany proofs such that any adjacent pair of proofs π(t)andπ(t+1)differs in at most one bit, and\neach proof π(t)can be probabilistically checked by reading q(n) bits of the proof and using\nr(n) random bits, where q(n)=O(1) and r(n)=O(logn). The FGLSS reduction [FGLSS96]\ntransforms such a proof system into a graph G=(V,E), an alphabet Σ, and constraints\n(ψe)e∈Esuch that each vertex v∈V:={0,1}r(n)corresponds to a coin flip sequence of a veri-\nfier, each value α∈Σ={0,1}q(n)corresponds to a local view of the verifier, and the constraints\nψecheck the consistency of two local views of the verifier. This reduction works in the case\nof the PCP theorem and proves NP-hardness of M AXPARTIAL 2CSP [KM23]. However,\nthe reduction does not work in the case of the PCRP theorem: We need to ensure that the\nreconfiguration sequence of proofs π(1),···,π(T)is transformed into a sequence of partial as-\nsignments f(1),···,f(T), each adjacent pair of which differs in at most one vertex. The issue\nis that changing one bit in the original proof π(t)may result in changing the assignments of\nmany vertices in a partial assignment f(t):V→Σ∪{⊥}.\nTo address this issue, we employ the ideas developed in [Ohs23, Ohs24], called the al-\nphabet squaring trick , and modify the FGLSS reduction as follows. Given a verifier that\nreads q(n) bits of a proof, we define the alphabet as Σ={0,1,01}q(n). Intuitively, the symbol\n6“01” means that we are taking 0 and 1 simultaneously. This enables us to construct a re-\nconfiguration sequence of partial assignments f(1),···,f(T)from a reconfiguration sequence\nof proofs π(1),···,π(T). Details can be found in Section 3.2.\n1.4 Related Work\nIto, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno [IDHPSUU11] showed that\noptimization versions of SAT R ECONFIGURATION and C LIQUE RECONFIGURATION areNP-\nhard to approximate, relying on NP-hardness of approximating M AXSAT [Hås01] and M AX\nCLIQUE [Hås99], respectively. Note that their NP-hardness results are not optimal since\nSAT R ECONFIGURATION and C LIQUE RECONFIGURATION arePSPACE -complete. Toward\nPSPACE -hardness of approximation for reconfiguration problems, Ohsaka [Ohs23] proposed\ntheReconfiguration Inapproximability Hypothesis (RIH), which postulates that a reconfigu-\nration analogue of C ONSTRAINT SATISFACTION PROBLEM isPSPACE -hard to approximate,\nand demonstrated PSPACE -hardness of approximation for many popular reconfiguration\nproblems, including those of 3SAT, I NDEPENDENT SET, VERTEX COVER , CLIQUE , DOMI-\nNATING SET, and S ETCOVER . Ohsaka [Ohs24] adapted Dinur’s gap amplification [Din07]\nto demonstrate that under RIH, optimization versions of 2CSP R ECONFIGURATION and\nSETCOVER RECONFIGURATION arePSPACE -hard to approximate within a factor of 0 .9942\nand 1 .0029, respectively.\nVery recently, Hirahara and Ohsaka [HO24] and Karthik C. S. and Manurangsi [KM23]\nannounced the proof of RIH independently, implying that the above PSPACE -hardness re-\nsults hold unconditionally . Karthik C. S. and Manurangsi [KM23] further proved that (op-\ntimization versions of) 2CSP R ECONFIGURATION and S ETCOVER RECONFIGURATION are\nNP-hard to approximate within a factor smaller than 2, which is quantitatively tight be-\ncause both problems are (nearly) 2-factor approximable. Our result partially resolves an\nopen question of [KM23, Section 6]: “ Can we prove tight PSPACE -hardness of approximation\nresults for GapMaxMin-2- CSP qand Set Cover Reconfiguration? ”\nOther reconfiguration problems whose approximability was investigated include those\nof S ETCOVER [IDHPSUU11], S UBSET SUM[ID14], and S UBMODULAR MAXIMIZATION\n[OM22]. We note that optimization variants of reconfiguration problems frequently re-\nfer to those of the shortest reconfiguration sequence [BHIKMMSW20, IKKKO22, KMM11,\nMNPR17], which are orthogonal to this study.\n2 Preliminaries\n2.1 Notations\nFor a nonnegative integer n∈N, let [ n]:={1,2,...,n}. Unless otherwise specified, the base\nof logarithms is 2. A sequence Sof a finite number of objects S(1),...,S(T)is denoted by\n7(S(1),...,S(T)), and we write S(t)∈Sto indicate that S(t)appears in S. LetΣbe a finite set\ncalled alphabet . For a length- nstring π∈Σnand a finite sequence of indices I⊆[n]∗, we use\nπ|I:=(πi)i∈Ito denote the restriction of πtoI. The Hamming distance between two strings\nf,g∈Σn, denoted by ∆(f,g), is defined as the number of positions on which fand gdiffer;\nnamely,\n∆(f,g):=¯¯¯n\ni∈[n]¯¯¯fi̸=gio¯¯¯. (2.1)\n2.2 Reconfiguration Problems on Constraint Graphs\nConstraint Graphs. In this section, we formulate reconfiguration problems on constraint\ngraphs. The notion of constraint graph is defined as follows.\nDefinition 2.1. Aq-ary constraint graph is defined as a tuple G=(V,E,Σ,Ψ) such that\n•(V,E) is a q-uniform1hypergraph called the underlying graph ,\n•Σis a finite set called the alphabet , and\n•Ψ=(ψe)e∈Eis a collection of q-ary constraints , where each ψe:Σe→{0,1}is a circuit.\nA binary constraint graph is simply referred to as a constraint graph .\nFor an assignment f:V→Σ, we say that fsatisfies a hyperedge e={v1,...,vq}∈E(or a\nconstraint ψe) ifψe(f(e))=1, where f(e):=(f(v1),...,f(vq)), and fsatisfies Gif it satisfies all\nthe hyperedges of G. In the qCSP R ECONFIGURATION problem, for a q-ary constraint graph\nGand its two satisfying assignments fstartandfgoal, we are required to decide if there exists\na reconfiguration sequence from fstarttofgoalconsisting only of satisfying assignments for\nG, each adjacent pair of which differs in at most one vertex. qCSP R ECONFIGURATION is\nPSPACE -complete in general [GKMP09, IDHPSUU11]; thus, we formulate its two optimiza-\ntion versions.\nPARTIAL 2CSP R ECONFIGURATION .For a constraint graph G=(V,E,Σ,Ψ=(ψe)e∈E),\napartial assignment is defined as a function f:V→Σ∪{⊥}, where the symbol ⊥indicates\n“unassigned.” We say that a partial assignment f:V→Σ∪{⊥}satisfies an edge e=(v,w)∈E\nifψe(f(v),f(w))=1 whenever f(v)̸= ⊥and f(w)̸= ⊥. The sizeoff, denoted by ∥f∥, is defined\nas the number of vertices whose value is assigned; namely,\n∥f∥:=¯¯¯n\nv∈V¯¯¯f(v)̸= ⊥o¯¯¯. (2.2)\nFor two satisfying partial assignments fstartand fgoalforG, areconfiguration partial as-\nsignment sequence from fstarttofgoalis a sequence F=(f(1),...f(T)) of satisfying partial\n1A hypergraph is said to be q-uniform if each of its hyperedges has size exactly q.\n8assignments such that f(1)=fstart,f(T)=fgoal, and ∆(f(t),f(t+1))É1 (i.e., f(t)and f(t+1)dif-\nfer in at most one vertex) for all t. For any reconfiguration partial assignment sequence\nF=(f(1),...,f(T)), we define ∥F∥minas\n∥F∥min:=min\n1ÉtÉT∥f(t)∥. (2.3)\nPARTIAL 2CSP R ECONFIGURATION is formally defined as follows:\nProblem 2.2 (PARTIAL 2CSP R ECONFIGURATION ).For a constraint graph G=(V,E,Σ,Ψ=\n(ψe)e∈E) and its two satisfying partial assignments fstart,fgoal:V→Σ∪{⊥}, we are required\nto find a reconfiguration partial assignment sequence Ffrom fstarttofgoalsuch that ∥F∥min\nis maximized.\nLetMaxPar G(fstart↭fgoal) denote the maximum value of∥F∥min\n|V|over all possible reconfigu-\nration sequences Ffrom fstarttofgoal; namely,\nMaxPar G(fstart↭fgoal):= max\nF=(fstart,...,fgoal)∥F∥min\n|V|. (2.4)\nNote that 0 ÉMaxPar G(fstart↭fgoal)É1. For every numbers 0 ÉsÉcÉ1, G APc,sPAR-\nTIAL 2CSP R ECONFIGURATION requests to determine for a constraint graph Gand its\ntwo satisfying partial assignments fstartand fgoal, whether MaxPar G(fstart↭fgoal)Êcor\nMaxPar G(fstart↭fgoal)<s. Note that we can assume ∥fstart∥ = ∥ fgoal∥ = |V|when c=1.\nLABEL COVER RECONFIGURATION .For a constraint graph G=(V,E,Σ,Ψ), amulti-\nassignment is defined as a function f:V→2Σ. We say that a multi-assignment fsatisfies\nedge e=(v,w)∈Eif there exists a pair ( α,β)∈f(v)×f(w) such that ψe(α,β)=1. The size of\nf, denoted by ∥f∥, is defined as the sum of |f(v)|over all v∈V; namely,\n∥f∥:=X\nv∈V|f(v)|. (2.5)\nFor two satisfying multi-assignments fstartandfgoalforG, areconfiguration multi-assignment\nsequence from fstarttofgoalis a sequence F=(f(1),...,f(T)) of satisfying multi-assignments\nsuch that f(1)=fstart,f(T)=fgoal, and\nX\nv∈V¯¯¯f(t)(v)△f(t+1)(v)¯¯¯É1 for all t. (2.6)\nFor any reconfiguration multi-assignment sequence F=(f(1),...,f(T)), we define ∥F∥maxas\n∥F∥max:=max\n1ÉtÉT∥f(t)∥. (2.7)\nLABEL COVER RECONFIGURATION is formally defined as follows.2\n2This problem can be thought of as a reconfiguration analogue of M INREP[CHK11].\n9Problem 2.3 (LABEL COVER RECONFIGURATION ).For a constraint graph G=(V,E,Σ,Ψ)\nand its two satisfying multi-assignments fstart,fgoal:V→2Σ, we are required to find a\nreconfiguration multi-assignment sequence Ffrom fstarttofgoalsuch that ∥F∥maxis mini-\nmized.\nLetMinLab G(fstart↭fgoal) denote the minimum value of∥F∥max\n|V|+1over all possible reconfigu-\nration multi-assignment sequences Ffrom fstarttofgoal; namely,\nMinLab G(fstart↭fgoal):= min\nF=(fstart,...,fgoal)∥F∥max\n|V|+1. (2.8)\nNote that MinLab G(fstart↭fgoal)Ê|V|\n|V|+1. For every numbers 1 ÉcÉs, GAPc,sLABEL\nCOVER RECONFIGURATION requests to determine whether MinLab G(fstart↭fgoal)Écor\nMinLab G(fstart↭fgoal)>sfor a constraint graph Gand its two satisfying multi-assignments\nfstartand fgoal. Note that we can assume∥fstart∥\n|V|+1=∥fgoal∥\n|V|+1É1 when c=1.\n2.3 Probabilistically Checkable Reconfiguration Proof Systems\nFirst, we formally define the notion of verifier .\nDefinition 2.4. Averifier with randomness complexity r:N→Nand query complexity\nq:N→Nis a probabilistic polynomial-time algorithm Vthat given an input x∈{0,1}∗,\ntosses r=r(|x|) random bits Rand uses Rto generate a sequence of q=q(|x|) queries\nI=(i1,...,iq) and a circuit D:{0,1}q→{0,1}. We write ( I,D)∼V(x) to denote the random\nvariable for a pair of the query sequence and circuit generated by Von input x∈{0,1}∗.\nDenote by Vπ(x):=D(π|I) the output of Von input xgiven oracle access to a proof π∈{0,1}∗.\nWe say that V(x)accepts a proof πifVπ(x)=1; i.e., D(π|I)=1 for ( I,D)∼V(x).\nWe proceed to the definition of Probabilistically Checkable Reconfiguration Proofs (PCRPs)\ndue to Hirahara and Ohsaka [HO24], which offer a PCP-type characterization of PSPACE .\nFor any pair of proofs πstart,πgoal∈{0,1}ℓ, areconfiguration sequence from πstarttoπgoalis\na sequence ( π(1),...,π(T))∈({0,1}ℓ)∗such that π(1)=πstart,π(T)=πgoal, and ∆(π(t),π(t+1))É1\n(i.e.,π(t)andπ(t+1)differ in at most one bit) for all t.\nTheorem 2.5 (PCRP theorem of Hirahara and Ohsaka [HO24]) .For any language Lin\nPSPACE , there exists a verifier Vwith randomness complexity r(n)=O(logn)and query com-\nplexity q(n)=O(1), coupled with polynomial-time computable functions πstart,πgoal:{0,1}∗→\n{0,1}∗, such that the following hold for any input x∈{0,1}∗:\n•(Completeness )Ifx∈L, there exists a reconfiguration sequence Π=(π(1),...,π(T))from\nπstart(x)toπgoal(x)over {0,1}poly(|x|)such that V(x)accepts every proof with probability\n1;namely,\n∀t∈[T],Ph\nV(x)accepts π(t)i\n=1. (2.9)\n10•(Soundness )Ifx∉L, every reconfiguration sequence Π=(π(1),...,π(T))from πstart(x)to\nπgoal(x)over {0,1}poly(|x|)includes a proof that is rejected by V(x)with probability more\nthan1\n2;namely,\n∃t∈[T],Ph\nV(x)accepts π(t)i\n<1\n2. (2.10)\nWe further introduce the notion of regular verifier. We say that a verifier is regular if\neach position in its proof is equally likely to be queried.3\nDefinition 2.6. For a verifier Vand an input x∈{0,1}∗, the degree of a position iof a proof\nis defined as the number of times iis queried by V(x) over r(|x|) random bits; namely,\n¯¯¯n\nR∈{0,1}r(|x|)¯¯¯i∈IRo¯¯¯=P\n(I,D)∼V(x)h\ni∈Ii\n·2r(|x|), (2.11)\nwhere ris the randomness complexity of VandIRis the query sequence generated by V(x)\non the randomness R. A verifier Vis said to be ∆-regular if the degree of every position is\nexactly equal to ∆.\n3 Subconstant Error PCRP Systems and FGLSS Reduc-\ntion\nIn this section, we construct a bounded-degree PCRP verifier with subconstant error us-\ning Theorem 2.5 in Section 3.1, and prove PSPACE -hardness of approximation for P ARTIAL\n2CSP R ECONFIGURATION and L ABEL COVER RECONFIGURATION by the FGLSS reduction\n[FGLSS96] in Sections 3.2 and 3.3, respectively.\n3.1 Bounded-degree PCRP Systems with Subconstant Error\nStarting from Theorem 2.5, we first obtain a regular PCRP verifier for any PSPACE lan-\nguage, whose proof uses the degree reduction technique due to Ohsaka [Ohs23].\nProposition 3.1. For any language LinPSPACE , there exists a ∆-regular PCRP verifier V\nwith randomness complexity r(n)=O(logn), query complexity q(n)=O(1), perfect complete-\nness, and soundness 1−ε, for some constant ∆∈Nandε∈(0,1).\nProof. Here, the suffix “ W” will designate the restricted case of qCSP R ECONFIGURATION\nwhose alphabet size |Σ|isW. By Theorem 2.5, for some integer q∈N, there is a PCRP\nverifier VforLwith randomness complexity O(logn), query complexity q, perfect complete-\nness, and soundness1\n2. For an input x∈{0,1}∗, the verifier V(x) can be transformed into an\n3Note that regular verifiers are sometimes called smooth verifiers, e.g., [Par21]. Since the term “regularity”\nis compatible with that of (hyper) graphs, we do not use the term “smoothness” but “regularity.”\n11problem/verifier alph. size soundness notes reference\nq-query verifier 21\n2— [HO24, Theorem 5.1]\n2CSP R ECONF 3 1 −ε(εde-\npends on q)— [Ohs23, Lemmas 3.2 and 3.6]\n2CSP R ECONF 6 1 −ε(εde-\npends on ε)max. degree ∆\ndepends on ε[Ohs23, Lemma 3.7]\n3SAT R ECONF 2 1 −Ω(ε) each variable\nappears O(∆)\ntimes[Ohs23, Lemma 3.2]\n3-query verifier 2 1 −Ω³\nε\n∆´\nO(∆)-regular —\nTable 1: Sequence of reductions used in Proposition 3.1.\ninstance of G AP1,1\n2qCSP 2RECONFIGURATION in a canonical manner, e.g., [HO24, Proposi-\ntion 4.9]. By [Ohs23, Lemmas 3.2 and 3.6], we then obtain an instance of G AP1,1−ε2CSP 3\nRECONFIGURATION , where ε���(0,1) depends only on q. Further applying the degree reduc-\ntion step of [Ohs23, Lemma 3.7], we get an instance of G AP1,1−ε2CSP 6RECONFIGURATION ,\nwhose underlying graph has maximum degree bounded by ∆, where ε∈(0,1) and ∆∈Nde-\npend only on ε. Using [Ohs23, Lemma 3.2], an instance of G AP1,1−Ω(ε)3SAT R ECONFIGURA -\nTION is produced, where each Boolean variable appears in at most O(∆) clauses. By padding\nwith trivial constraints, this instance is transformed into an instance of G AP1,1−Ω(ε\n∆)3CSP 2\nRECONFIGURATION , whose underlying graph is O(∆)-regular. That is to say, there exists\na 3-query O(∆)-regular PCRP verifier eVforLwith randomness complexity O(logn), query\ncomplexity O(1), perfect completeness, and soundness 1 −ε′, where ε′=Ω³\nε\n∆´\n, as desired.\nSee Table 1 for a sequence of reductions used to obtain eV.\nSubsequently, using a randomness-efficient sampler over expander graphs (e.g., [HLW06,\nSection 3]), we construct a bounded-degree PCRP verifier with subconstant error .\nProposition 3.2. For any language LinPSPACE and any function δ:N→Rwith δ(n)=\nΩ(n−1), there exists a bounded-degree PCRP verifier Vwith randomness complexity r(n)=\nO(logδ(n)−1+logn), query complexity q(n)=O(logδ(n)−1), perfect completeness, and sound-\nnessδ(n). Moreover, for any input x∈{0,1}∗, the degree of any position is poly(δ(|x|)−1).\nVerifier Description. Our PCRP verifier is described as follows. By Proposition 3.1, let\nVbe a∆-regular PCRP verifier for a PSPACE -complete language Lwith randomness com-\nplexity r(n)=O(logn), query complexity q(n)=q∈N, perfect completeness, and soundness\n1−ε, where ∆∈Nandε∈(0,1). The proof length, denoted by ℓ(n), is polynomially bounded\nsince ℓ(n)Éq(n)2r(n)=poly( n). Hereafter, for any r(n) random bit sequence R, let IRand\n12DRrespectively denote the query sequence and circuit generated by V(x) on the randomness\nR. Given a function δ:N→Rwith δ(n)=Ω(n−1), we construct the following verifier eV:\nBounded-degree verifier eVwith subconstant error.\nInput: a∆-regular verifier Vwith soundness 1 −ε, a function δ:N→R, and an input\nx∈{0,1}n.\nOracle access: a proof π∈{0,1}ℓ(n).\n1:construct a ( d,λ)-expander graph Xover vertex set {0,1}r(n)withλ\nd<ε\n4.\n2:letρ:=§2\nεlnδ(n)−1¨\n=O(logδ(n)−1).\n3:uniformly sample a ( ρ−1)-length random walk R=(R1,...,Rρ) over Xusing r(n)+\nρ·logdrandom bits.\n4:for each 1ÉkÉρdo\n5: execute V(x) onRkto generate a query sequence IRk=(i1,...,iq) and a circuit\nDRk:{0,1}q→{0,1}.\n6: ifDRk(π|IRk)=0then\n7: declare reject .\n8:declare accept .\nCorrectness. The perfect completeness and soundness for a fixed proof π∈{0,1}ℓ(n)are\nshown below, whose proof relies on the property about random walks over expander graphs\ndue to Alon, Feige, Wigderson, and Zuckerman [AFWZ95].\nClaim 3.3. IfV(x)accepts πwith probability 1, theneV(x)accepts πwith probability 1. If\nV(x)accepts πwith probability less than 1−ε, theneV(x)accepts πwith probability less than\nδ(n).\nTo prove Claim 3.3, we refer to the following property about random walks over expander\ngraphs.\nLemma 3.4 (Alon, Feige, Wigderson, and Zuckerman [AFWZ95]) .LetXbe a(d,λ)-expander\ngraph, Sbe any vertex set of X, and R:=(R1,...,Rρ)be aρ-tuple of random variables de-\nnoting the vertices of a uniformly chosen (ρ−1)-length random walk over X. Then, it holds\nthat\nµ|S|\n|V(X)|−2λ\nd¶ρ\nÉP\nRh\n∀k∈[ρ],Rk∈Si\nɵ|S|\n|V(X)|+2λ\nd¶ρ\n. (3.1)\nProof of Claim 3.3. Suppose first V(x) accepts πwith probability 1; then, it holds that\nPh\neV(x) accepts πi\n=P\nRh\n∀k∈[ρ],DRk(π|IRk)=1i\n=1. (3.2)\nSuppose then V(x) accepts πwith probability less than 1 −ε. Define\nS:=n\nR∈{0,1}r(n)¯¯¯DR(π|IR)=1o\n. (3.3)\n13Note that|S|\n|V(X)|<1−ε. Then, it holds that\nPh\neV(x) accepts πi\n=P\nRh\n∀k∈[ρ],DRk(π|IRk)=1i\n=P\nRh\n∀k∈[ρ],Rk∈Si\n. (3.4)\nApplying Lemma 3.4, we derive\nP\nRh\n∀k∈[ρ],Rk∈Si\nɵ|S|\n|V(X)|+2λ\nd¶ρ\n<³\n1−ε\n2´ρ\n(byλ\nd<ε\n4)\nÉexp³\n−ε\n2ρ´\nÉδ(n), (byρ=»2\nεlnδ(n)−1¼\n)(3.5)\ncompleting the proof.\nWe are now ready to prove Proposition 3.2.\nProof of Proposition 3.2. We first show the perfect completeness and soundness. Suppose\nx∈L, then there exists a reconfiguration sequence Π=(π(1),...,π(T)) from πstart(x) toπgoal(x)\nsuch that P[V(x) accepts π(t)]=1 for all t. By Claim 3.3, we have P[eV(x) accepts π(t)]=1\nfor all t. Suppose x∉L, then for every reconfiguration sequence Π=(π(1),...,π(T)) from\nπstart(x) toπgoal(x), it holds that P[V(x) accepts π(t)]<1−εfor some t. By Claim 3.3, we have\nP[eV(x) accepts π(t)]<δ(n) for such t.\nSince ρ=O(logδ(n)−1), the randomness complexity of eVis equal to er(n)=r(n)+ρ·logd=\nO(logδ(n)−1+logn), and the query complexity is eq(n)=q(n)·ρ=O(logδ(n)−1). Note that\ndandλmay depend only on ε, and a ( d,λ)-expander graph Xover {0,1}r(n)can be con-\nstructed in polynomial time in 2r(n)=poly( n), e.g., by using an explicit construction of near-\nRamanujan graphs [Alo21, MOP21].\nObserve finally that eVqueries each position i∈[ℓ(n)] of a proof with probability equal to\nP\nR\"\n_\n1ÉkÉρ³\ni∈IRk´#\n. (3.6)\nSince Vis∆-regular, it holds that\nP\nR∼{0,1}r(n)h\ni∈IRi\n=∆\n2r(n). (3.7)\nUsing the fact that each Rkis uniformly distributed over {0,1}r(n), we bound Eq. (3.6) as\nfollows:\nP\nR\"\n_\n1ÉkÉρ³\ni∈IRk´#\nÉ|{z}\nunion boundX\nk∈[ρ]P\nRh\ni∈IRki\n=ρ·∆\n2r(n)=Oµlogδ(n)−1\n2r(n)¶\n. (3.8)\n14Consequently, the degree of each position iwith respect to eVis at most\nP\nR\"\n_\n1ÉkÉρ³\ni∈IRk´#\n·2er(n)=Oµlogδ(n)−1\n2r(n)¶\n·2r(n)+ρ·logd\n=O(logδ(n)−1)·2O(logδ(n)−1)\n=poly(δ(n)−1),(3.9)\nwhich completes the proof.\n3.2 FGLSS Reduction and PSPACE -hardness of Approximation for\nPARTIAL 2CSP R ECONFIGURATION\nWe now establish the FGLSS reduction from Proposition 3.2 and show that P ARTIAL 2CSP\nRECONFIGURATION isPSPACE -hard to approximate within a factor arbitrarily close to 0.\nTheorem 3.5. For any function ε:N→Rwith ε(n)=Ω³\n1\npolylog n´\n,GAP1,ε(N)PARTIAL 2CSP\nRECONFIGURATION with alphabet size poly(ε(N)−1)isPSPACE -complete, where Nis the\nnumber of vertices.\nReduction. We describe a reduction from a bounded-degree PCRP verifier to P ARTIAL\n2CSP R ECONFIGURATION . Define δ(n):=ε(poly( n))\n2, whose precise expression is given later.\nFor any PSPACE -complete language L, let Vbe a bounded-degree PCRP verifier of Propo-\nsition 3.2 with randomness complexity r(n)=O(logδ(n)−1+logn), query complexity q(n)=\nO(logδ(n)−1), perfect completeness, and soundness δ(n). The proof length ℓ(n) is polynomi-\nally bounded. Suppose we are given an input x∈{0,1}n. Let πstart,πgoal∈{0,1}ℓ(n)be the two\nproofs associated with V(x). Because the degree of Vis bounded by poly( δ(n)−1), for some\nconstant κ∈N, we have\nP\n(I,D)∼V(x)h\ni∈Ii\nÉδ(n)−κ\n2r(n)for any i∈[ℓ(n)]. (3.10)\nHereafter, for any r(n) random bit sequence R, let IRand DRdenote the query sequence\nand the circuit generated by V(x) on the randomness R, respectively. Construct a constraint\ngraph G=(V,E,Σ,Ψ) as follows:\nV:={0,1}r(n), (3.11)\nE:=n\n(R1,R2)∈V×V¯¯¯IR1∩IR2̸= ;o\n, (3.12)\nΣ:=n\n{0},{1},{0,1}oq(n)\n, (3.13)\nΨ:={ψe}e∈E, (3.14)\n15where we define ψR1,R2:Σ×Σ→{0,1}for each edge ( R1,R2)∈Eso that ψR1,R2(f(R1),f(R2))=\n1 for an assignment f:V→Σif and only if the following three conditions are satisfied:\n∀α∈Y\ni∈IRf(R1)i,DR1(α)=1, (3.15)\n∀β∈Y\ni∈IRf(R2)i,DR2(β)=1, (3.16)\n∀i∈IR1∩IR2,f(R1)i⊆f(R2)iorf(R1)i⊇f(R2)i. (3.17)\nHere, for the sake of simple notation, we consider f(R) as if it were indexed by IR(rather\nthan [ q(n)]); namely, f(R)∈{{0},{1},{0,1}}IR. Thus, f(R) for each R∈Vcorresponds the local\nview of V(x) onR.\nFor any proof π∈{0,1}ℓ(n), we associate it with an assignment fπ:V→Σsuch that\nfπ(R):=³\n{πi}´\ni∈IRfor all R∈V. (3.18)\nNote that fπ(R)∈{{0},{1}}IR. Constructing two assignments fstartfrom πstartand fgoalfrom\nπgoalby Eq. (3.18), we obtain an instance ( G,fstart,fgoal) of P ARTIAL 2CSP R ECONFIGU -\nRATION . Observe that fstartand fgoalsatisfy Gand∥fstart∥ = ∥ fgoal∥ = |V|. Note that\nN:= |V| Éncfor some constant c∈N. Letting δ(n):=ε(nc)\n2=Ω³\n1\npolylog n´\nensures that the\nalphabet size is |Σ| =O(3q(n))=poly(ε(N)−1). This completes the description of the reduc-\ntion.\nCorrectness. We first prove the completeness.\nLemma 3.6 (Completeness) .Ifx∈L, then MaxPar G(fstart↭fgoal)=1.\nProof of Lemma 3.6. It is sufficient to consider the case that πstartandπgoaldiffer in exactly\none position, say, i⋆∈[ℓ(n)]; namely, πstart\ni⋆̸=πgoal\ni⋆andπstart\ni=πgoal\nifor all i̸=i⋆. Note\nthat fstartand fgoalmay differ in two or more vertices. Consider a reconfiguration partial\nassignment sequence Ffrom fstarttofgoalobtained by the following procedure:\nReconfiguration sequence Ffrom fstarttofgoal.\n1:for each R∈Vsuch that i⋆∈IRdo\n2: change the i⋆thentry of R’s current value from fstart(R)i⋆={πstart\ni⋆}to{0,1}.\n3:for each R∈Vsuch that i⋆∈IRdo\n4: change the i⋆thentry of R’s current value from {0,1}tofgoal(R)i⋆={πgoal\ni⋆}.\nObserve that any partial assignment f◦ofFsatisfies Gfor the following reasons:\n•Since f◦(R)i⋆⊆{0,1}={πstart\ni⋆,πgoal\ni⋆}=fstart(R)i⋆∪fgoal(R)i⋆when i⋆∈IR,f◦satisfies\nEqs. (3.15) and (3.16).\n16•Letting K:={f◦(R)i⋆|i⋆∈IR}, we find Kto be either {{0}},{{1}},{{0,1}},{{0},{0,1}}, or\n{{1},{0,1}}by construction; i.e., f◦satisfies Eq. (3.17).\nSince ∥f◦∥ = |V|, it holds that MaxPar G(fstart↭fgoal)Ê∥F∥max\n|V|=1, completing the proof.\nLemma 3.7 (Soundness) .Ifx∉L, then\nMaxPar G(fstart↭fgoal)<δ(n)+q(n)·δ(n)−κ\n2r(n). (3.19)\nThe proof of Theorem 3.5 follows from Lemmas 3.6 and 3.7 because for any sufficiently\nlarge nsuch thatq(n)·δ(n)−κ\n2r(n)Éδ(n) (note that δ(n)=Ω³\n1\npolylog n´\n), the following hold:\n•(Perfect completeness) If x∈L, then MaxPar G(fstart↭fgoal)=1;\n•(Soundness) If x∉L, then MaxPar G(fstart↭fgoal)<2δ(n)=ε(N).\nProof of Lemma 3.7. We prove the contrapositive. Suppose MaxPar G(fstart↭fgoal)ÊΓfor\nsome Γ∈(0,1), and there is a reconfiguration partial assignment sequence F=(f(1),...,f(T))\nfrom fstarttofgoalsuch that ∥F∥min=MaxPar G(fstart↭fgoal). Define then a (not neces-\nsarily reconfiguration) sequence Π=(π(1),...,π(T)) over {0,1}ℓ(n)such that each proof π(t)is\ndetermined based on the plurality vote over f(t); namely,\nπ(t)\ni:=argmax\nb∈{0,1}¯¯¯n\nR∈V¯¯¯i∈IRandb∈f(t)(R)io¯¯¯for all i∈[ℓ(n)], (3.20)\nwhere ties are broken so that 0 is chosen. In particular, π(1)=πstartandπ(T)=πgoal. Observe\nthe following:\nObservation 3.8. For any t∈[T]and R∈V, it holds that\nf(t)(R)̸= ⊥ =⇒ DR(π(t)|IR)=1. (3.21)\nSincePR∼V[f(t)(R)̸= ⊥]= ∥f(t)∥ ÊΓ, by Observation 3.8, we have that for all t,\nPh\nV(x) accepts π(t)i\n=P\nR∼{0,1}r(n)h\nDR(π(t)|IR)=1i\nÊP\nR∼Vh\nf(t)(R)̸= ⊥i\nÊΓ. (3.22)\nUnfortunately, Πisnota reconfiguration sequence because π(t)andπ(t+1)may differ in two\nor more positions. Since f(t)and f(t+1)differ in a single vertex R∈V, we have π(t)\ni̸=π(t+1)\ni\nonly if i∈IR, implying ∆(π(t),π(t+1))É |IR| =q(n). Using this fact, we interpolate between\nπ(t)andπ(t+1)to find a valid reconfiguration sequence Π(t)such that V(x) accepts every proof\nofΠ(t)with probability Γ−o(1).\nClaim 3.9. There exists a reconfiguration sequence Π(t)from π(t)toπ(t+1)such that for every\nproof π◦ofΠ(t),\nPh\nV(x)accepts π◦i\nÊΓ−q(n)·δ(n)−κ\n2r(n). (3.23)\n17Concatenating Π(t)’s of Claim 3.9 for all t, we obtain a valid reconfiguration sequence Π\nfrom πstarttoπgoalsuch that\nmin\n1ÉtÉTPh\nV(x) accepts π(t)i\nÊΓ−q(n)·δ(n)−κ\n2r(n). (3.24)\nSubstituting δ(n)+q(n)·δ(n)−κ\n2r(n)forΓ, we have that if MaxPar G(fstart↭fgoal)Êδ(n)+q(n)·δ(n)−κ\n2r(n),\nthen V(x) accepts every proof π(t)ofΠwith probability at least δ(n); i.e., x∈L. This com-\npletes the proof of Lemma 3.7.\nWhat remains to be done is to prove Observation 3.8 and Claim 3.9.\nProof of Observation 3.8. Suppose f(t)(R)̸= ⊥for some t∈[T] and R∈V. We will show that\nπ(t)\ni∈f(t)(R)ifor every i∈IR. Define\nK:=n\nf(t)(R′)i¯¯¯∃R′∈Vs.t.i∈IR′and f(t)(R′)̸= ⊥o\n. (3.25)\nThen, any pair α,β∈Kmust satisfy that α⊆βorα⊇βbecause otherwise, f(t)would violate\nEq. (3.17) at edge ( R1,R2) such that i∈R1∩R2,f(t)(R1)i=α, and f(t)(R2)i=β, which is a\ncontradiction. For each possible case of K, the result of the plurality vote π(t)\niis shown below,\nimplying that π(t)\ni∈f(t)(R)i.\nK {} {{ 0}} {{ 1}} {{ 0,1}} {{ 0},{0,1}} {{ 1},{0,1}}\nπ(t)\ni0 0 1 0 0 1\nSince f(t)(R) must satisfy a self-loop ( R,R)∈E, by the definition of ψR,R, we have\n∀α∈Y\ni∈IRf(t)(R)i,DR(α)=1, (3.26)\nOn the other hand, it holds that\nπ(t)|IR∈Y\ni∈IRf(t)(R)i, (3.27)\nimplying DR(π(t)|IR)=1, as desired.\nProof of Claim 3.9. Recall that π(t)andπ(t+1)may differ in at most q(n) positions. Consider\nany trivial reconfiguration sequence Π(t)from π(t)toπ(t+1)by simply changing at most q(n)\npositions on which π(t)andπ(t+1)differ. By construction, any proof π◦ofΠ(t)differs from π(t)\n18in at most q(n) positions, say, I◦∈¡ℓ(n)\nÉq(n)¢\n. Then, we derive the following:\nPh\nV(x) accepts π◦i\n=P\n(I,D)∼V(x)h\nD(π◦|I)=1i\nÊP\n(I,D)∼V(x)h\nD(π◦|I)=1 and I∩I◦= ;i\n=P\n(I,D)∼V(x)h\nD(π(t)|I)=1 and I∩I◦= ;i\n=P\n(I,D)∼V(x)h\nD(π(t)|I)=1i\n| {z }\n=P[V(x) accepts π(t)]ÊΓ−P\n(I,D)∼V(x)h\nD(π(t)|I)=1 and I∩I◦̸= ;i\nÊΓ−P\n(I,D)∼V(x)h\nI∩I◦̸= ;i\n.\nRecall that P(I,D)∼V(x)[i∈I]Éδ(n)−κ\n2r(n)for any i∈[ℓ(n)] by assumption. Since |I◦| Éq(n), taking\na union bound, we have\nP\n(I,D)∼V(x)h\nI∩I◦̸= ;i\nÉX\ni∈I◦P\n(I,D)∼V(x)h\ni∈Ii\nÉq(n)·δ(n)−κ\n2r(n), (3.28)\nimplying that\nPh\nV(x) accepts π◦i\nÊΓ−q(n)·δ(n)−κ\n2r(n). (3.29)\nThis completes the proof.\n3.3 Reducing P ARTIAL 2CSP R ECONFIGURATION to L ABEL COVER\nRECONFIGURATION\nSubsequently, we show PSPACE -hardness of approximation for L ABEL COVER RECONFIG -\nURATION by reducing from P ARTIAL 2CSP R ECONFIGURATION , whose proof is similar to\n[KM23]. Note that L ABEL COVER RECONFIGURATION admits a 2-factor approximation,\nsimilarly to M INMAX SETCOVER RECONFIGURATION (see Section 4.1).\nTheorem 3.10. For any function ε:N→Rwithε(n)=Ω³\n1\npolylog n´\n,GAP1,2−ε(N)LABEL COVER\nRECONFIGURATION with alphabet size poly(ε(N)−1)isPSPACE -complete, where Nis the\nnumber of vertices. In particular,\n•for any constant ε∈(0,1),GAP1,2−εLABEL COVER RECONFIGURATION with constant\nalphabet size is PSPACE -complete, and\n•GAP1,2−1\npolyloglog NLABEL COVER RECONFIGURATION with alphabet size polyloglog Nis\nPSPACE -complete.\n19Proof. We present a gap-preserving reduction from G AP1,ε(N)\n2PARTIAL 2CSP R ECONFIGU -\nRATION to G AP1,2−ε(N)LABEL COVER RECONFIGURATION . Let ( G=(V,E,Σ,Ψ),fstart,fgoal)\nbe an instance of P ARTIAL 2CSP R ECONFIGURATION with Nvariables and alphabet size\npoly(ε(N)−1), where ∥fstart∥ = ∥ fgoal∥ = |V|. Without loss of generality, we can safely as-\nsume that Nis sufficiently large so that NÊ4\nε(N)because ε(N)=Ω³\n1\npolylog N´\n. Construct\nthen multi-assignments f′start,f′goal:V→2Σsuch that f′start(v):={fstart(v)}and f′goal(v):=\n{fgoal(v)}for all v∈V. Observe that f′startand f′goalsatisfy G; thus, ( G,f′start,f′goal) is an\ninstance of L ABEL COVER RECONFIGURATION , completing the reduction.\nWe first show the perfect completeness; namely,\nMaxPar G(fstart↭fgoal)Ê1=⇒MinLab G(f′start↭f′goal)É1. (3.30)\nSuppose there is a reconfiguration partial assignment sequence F=(f(1),...,f(T)) from fstart\ntofgoalsuch that ∥F∥min= |V|. Construct then a sequence F′=(f′(1),f′(1.5),...,f′(T−0.5),f′(T))\nof multi-assignments such that f′(t)(v)={f(t)(v)}for all t∈[T] and v∈V, and f′(t+0.5)for each\nt∈[T−1] is defined as follows: Given that f(t)and f(t+1)differ only in v⋆, we let\nf′(t+0.5)(v):=(©\nf(t)(v⋆),f(t+1)(v⋆)ª\nifv=v⋆,\n©\nf(t)(v)ª\notherwise ,for all v∈V. (3.31)\nIn particular, it holds that f′(1)=f′startand f′(T)=f′goal. Observe that f′(t+0.5)is a satisfying\nmulti-assignment with ∥f′(t+0.5)∥ =N+1 for all t, and thatP\nv∈V|f′(t)△f′(t+0.5)| =1; i.e., F′is\na reconfiguration multi-assignment sequence from f′starttof′goalsuch that ∥F′∥max=N+1;\ni.e.,MinLab G(f′start↭f′goal)É∥F′∥max\nN+1=1.\nWe then prove the soundness; i.e.,\nMaxPar G(fstart↭fgoal)<ε(N)\n2=⇒MinLab G(f′start↭f′goal)>2−ε(N). (3.32)\nSuppose we are given a reconfiguration multi-assignment sequence F′=(f′(1),...,f′(T)) from\nf′starttof′goalsuch that∥F′∥max\nN+1=MinLab G(f′start↭f′goal). Define\nV(t)\n1:=n\nv∈V¯¯¯|f′(t)(v)| =1o\n. (3.33)\nConstruct then a sequence F=(f(1),...,f(T)) of partial assignments such that each f(t):V→\nΣ∪{⊥}is defined as follows:\nf(t)(v):=(\nunique α∈f′(t)(v) if v∈V(t)\n1,\n⊥ otherwise ,for all v∈V. (3.34)\nIn particular, it holds that f(1)=fstartand f(T)=fgoal. Observe easily that f(t)is a satisfying\npartial assignment, and f(t)and f(t+1)differ in at most one vertex; i.e., Fis a reconfigura-\ntion partial assignment sequence from fstarttofgoal. By assumption, there exists a partial\n20assignment f(t)inFsuch that |V(t)\n1| = ∥f(t)∥ <ε(N)\n2·N. Simple calculation derives that\n∥f′(t)∥ Ê1·|V(t)\n1|+2·|V\\V(t)\n1|\n=2N−|V(t)\n1|\n>µ\n2−ε(N)\n2¶\n·N\nÊ|{z}\nNÊ4\nε(N)(2−ε(N))·(N+1),(3.35)\nimplying that MinLab G(f′start↭f′goal)=∥F′∥max\nN+1Ê∥f′(t)∥\nN+1>2−ε(N), which completes the\nproof.\n4 Applications\nIn this section, we apply Theorem 3.10 to show optimal PSPACE -hardness of approximation\nresults for M INMAX SETCOVER RECONFIGURATION (Theorem 4.1) and M INMAX HYPER -\nGRAPH VERTEX COVER RECONFIGURATION (Theorem 4.3).\n4.1 PSPACE -hardness of Approximation for S ETCOVER RECONFIG -\nURATION\nWe first prove that M INMAX SETCOVER RECONFIGURATION isPSPACE -hard to approx-\nimate within a factor smaller than 2. Let Ube a finite set called the universe andF=\n{S1,...,Sm}be a family of msubsets of U. Acover for a set system ( U,F) is a subfamily\nofFwhose union is equal to U. For any pair of covers CstartandCgoalfor (U,F), arecon-\nfiguration sequence from CstarttoCgoalis a sequence C=(C(1),...,C(T)) of covers such that\nC(1)=Cstart,C(T)=Cgoal, and |C(t)△C(t+1)| É1 (i.e., C(t+1)is obtained from C(t)by adding or\nremoving a single set of F) for all t. In S ETCOVER RECONFIGURATION [IDHPSUU11], for\na set system ( U,F) and its two covers CstartandCgoalof size k, we are asked to decide if\nthere is a reconfiguration sequence from CstarttoCgoalconsisting only of covers of size at\nmost k+1. Next, we formulate its optimization version. Denote by opt(F) the size of the\nminimum cover of ( U,F). For any reconfiguration sequence C=(C(1),...,C(T)), its cost is\ndefined as the maximum value of|C(t)|\nopt(F)+1over all C(t)’s inC; namely,4\ncostF(C):=max\nC(t)∈C|C(t)|\nopt(F)+1, (4.1)\n4Here, division by opt(F)+1 is derived from the nature that we must first add at least one set whenever\n|Cstart| = |Cgoal| =opt(F) andCstart̸=Cgoal.\n21In M INMAX SETCOVER RECONFIGURATION , we wish to minimize costF(C) subject to C=\n(Cstart,...,Cgoal). For a pair of covers CstartandCgoalfor (U,F), let costF(Cstart↭Cgoal)\ndenote the minimum value of costF(C) over all possible reconfiguration sequences Cfrom\nCstarttoCgoal; namely,\ncostF(Cstart↭Cgoal):= min\nC=(Cstart,...,Cgoal)costF(C). (4.2)\nFor every 1 ÉcÉs, GAPc,sSETCOVER RECONFIGURATION requests to distinguish whether\ncostF(Cstart↭Cgoal)ÉcorcostF(Cstart↭Cgoal)>s.\nFor the sake of completeness, we here present a 2-factor approximation algorithm for\nMINMAX SETCOVER RECONFIGURATION of [IDHPSUU11]:5\n2-factor approximation for M INMAX SETCOVER RECONFIGURATION .\n1:▷start from Cstart. ◁\n2:insert each set of Cgoal\\Cstartinto the current cover in any order.\n3:discard each set of Cstart\\Cgoalfrom the current cover in any order.\n4:▷end with Cgoal. ◁\nOur main result is stated below, whose proof uses a gap-preserving reduction from\nLABEL COVER RECONFIGURATION to M INMAX SETCOVER RECONFIGURATION [LY94,\nOhs24].\nTheorem 4.1. GAP1,2−1\npolyloglog NSETCOVER RECONFIGURATION isPSPACE -complete, where\nNis the universe size. In particular, MINMAX SETCOVER RECONFIGURATION isPSPACE -\nhard to approximate within a factor of 2−1\npolyloglog N.\nTheorem 4.1 along with [Ohs24] implies that M INMAX DOMINATING SETRECONFIG -\nURATION isPSPACE -hard to approximate within a factor of 2 −1\npolyloglog N, where Nis the\nnumber of vertices (see Corollary 1.2).\nProof of Theorem 4.1. The reduction from L ABEL COVER RECONFIGURATION to M INMAX\nSETCOVER RECONFIGURATION is almost the same as that due to Lund and Yannakakis\n[LY94] and Ohsaka [Ohs24]. Let ( G=(V,E,Σ,Ψ),fstart,fgoal) be an instance of L ABEL\nCOVER RECONFIGURATION with Nvertices and alphabet size |Σ| =polyloglog N, where\n∥fstart∥ = ∥ fgoal∥ = |V|. Define B:={0,1}Σ. For each α∈Σand S⊆Σ, we construct Qα⊂B\nandQS⊂Baccording to [Ohs24] in 2O(|Σ|)time. Let ≺be an arbitrary order over V. Create\nan instance of M INMAX SETCOVER RECONFIGURATION as follows. For each vertex v∈V\nand each value α∈Σ, we define Sv,α⊂E×Bas\nSv,α:=Ã\n[\ne=(v,w)∈E:v≺w{e}×Qα!\n∪Ã\n[\ne=(v,w)∈E:v≻w{e}×Qπe(α)!\n, (4.3)\n5Similarly, a 2-factor approximation algorithm can be obtained for M INMAX DOMINATING SETRECONFIG -\nURATION and M INMAX HYPERGRAPH VERTEX COVER RECONFIGURATION .\n22where πe(α):={β∈Σ|ψe(α,β)=1}. Then, a set system ( U,F) is defined as\nU:=E×BandF:=n\nSv,α¯¯¯v∈V,α∈Σo\n. (4.4)\nFor a satisfying multi-assignment f:V→2ΣforGwith∥f∥ = |V|,6we associate it with a\nsubfamily Cf⊂Fsuch that\nCf:=n\nSv,α¯¯¯v∈V,α∈f(v)o\n, (4.5)\nwhich is a minimum cover for ( U,F) [Ohs24]; i.e., |Cf| = |V| =opt(F). Constructing min-\nimum covers Cstartfrom fstartandCgoalfrom fgoalby Eq. (4.5), we obtain an instance\n((U,F),Cstart,Cgoal) of M INMAX SETCOVER RECONFIGURATION . This completes the de-\nscription of the reduction.\nHere, we will show that\nMinLab G(fstart↭fgoal)=costF(Cstart↭Cgoal), (4.6)\nwhich implies the completeness and soundness; for this, we use the following lemma [Ohs24].\nLemma 4.2 ([Ohs24, Observation 4.4, Claim 4.7]) .Letf:V→2Σbe a multi-assignment\nandC⊆Fbe a subfamily such that for any v∈Vandα∈Σ,α∈f(v)if and only if Sv,α∈C.\nThen, fsatisfies an edge e=(v,w)∈Eif and only if Ccovers {e}×B. In particular, fsatisfies\nGif and only if CcoversE×B. Moreover, it holds that ∥f∥ = |C|.\nWe first show that MinLab G(fstart↭fgoal)ÊcostF(Cstart↭Cgoal). For any reconfigura-\ntion multi-assignment sequence F=(f(1),...,f(T)) from fstarttofgoalsuch that ∥F∥max=\nMinLab G(fstart↭fgoal), we can construct a reconfiguration sequence C=(Cf(1),...,Cf(T))\nfromCstarttoCgoalby Eq. (4.5). By Lemma 4.2, each Cf(t)covers U; thus,Cis a valid re-\nconfiguration sequence from CstarttoCgoal. Moreover, costF(Cstart↭Cgoal)ÉcostF(C)=\n∥F∥max=MinLab G(fstart↭fgoal), as desired. We then show that MinLab G(fstart↭fgoal)É\ncostF(Cstart↭Cgoal). For any reconfiguration sequence C=(C(1),...,C(T)) from Cstartto\nCgoalsuch that costF(C)=costF(Cstart↭Cgoal), we can construct a sequence F=(f(1),...,f(t))\nof multi-assignments such that f(t):V→2Σis defined as follows:\nf(t)(v):=n\nα∈Σ¯¯¯Sv,α∈C(t)o\nfor all v∈V. (4.7)\nBy Lemma 4.2, each f(t)satisfies G; thus, Fis a valid reconfiguration multi-assignment\nsequence from fstarttofgoal. Moreover, MinLab G(fstart↭fgoal)É ∥F∥max=costF(C)=\ncostF(Cstart↭Cgoal), which completes the proof of Eq. (4.6).\nSince |Σ| =polyloglog N, the reduction takes polynomial time in N, and it holds that\n|U| = |E×B| =O(N2·2polyloglog N)=O(N3); i.e., N=Ω(|U|1\n3). By Theorem 3.10, G AP1,2−1\npolyloglog N\n6In other words, each f(v) is a singleton.\n23LABEL COVER RECONFIGURATION with alphabet size polyloglog NisPSPACE -complete;\nthus, G AP1,2−1\npolyloglog |U|SETCOVER RECONFIGURATION isPSPACE -complete as well, accom-\nplishing the proof.\n4.2 PSPACE -hardness of Approximation for H YPERGRAPH VERTEX\nCOVER RECONFIGURATION\nWe conclude this section with a similar inapproximability result for M INMAX HYPERGRAPH\nVERTEX COVER RECONFIGURATION onO(1)-uniform hypergraphs. A vertex cover of a hy-\npergraph H=(V,E) is a vertex set C⊆Vthat intersects every hyperedge in E; i.e., e∩C̸= ;\nfor every e∈E. For any pair of vertex covers CstartandCgoalofH, areconfiguration sequence\nfromCstarttoCgoalis a sequence C=(C(1),...,C(T)) of vertex covers such that C(1)=Cstart,\nC(T)=Cgoal, and |C(t)△C(t+1)| É1 for all t. Denote by β(H) the size of the minimum ver-\ntex cover of H. For a reconfiguration sequence C=(C(1),...C(T)), its cost is defined as the\nmaximum value of|C(t)|\nβ(H)+1over all C(t)’s inC; namely,\ncost H(C):=max\nC(t)∈C|C(t)|\nβ(H)+1. (4.8)\nIn the M INMAX HYPERGRAPH VERTEX COVER RECONFIGURATION problem, for a hyper-\ngraph Hand its two vertex covers CstartandCgoal, we wish to minimize cost H(C) subject to\nC=(Cstart,...,Cgoal). Let cost H(Cstart↭Cgoal) denote the minimum value of cost H(C) over\nall possible reconfiguration sequences CfromCstarttoCgoal; namely,\ncost H(Cstart↭Cgoal):= min\nC=(Cstart,...,Cgoal)cost H(C). (4.9)\nFor every 1 ÉcÉs, GAPc,sHYPERGRAPH VERTEX COVER RECONFIGURATION requires to\ndistinguish whether cost H(Cstart↭Cgoal)Écorcost H(Cstart↭Cgoal)>s. Our inapprox-\nimability result is shown below, whose proof reuses the reduction of Theorem 4.1.\nTheorem 4.3. For any constant ε∈(0,1),GAP1,2−εHYPERGRAPH VERTEX COVER RECON -\nFIGURATION onpoly(ε−1)-uniform hypergraphs is PSPACE -complete. In particular, MIN-\nMAX HYPERGRAPH VERTEX COVER RECONFIGURATION onpoly(ε−1)-uniform hypergraphs\nisPSPACE -hard to approximate within a factor of 2−ε.\nProof. We build an “inverted index” of G AP1,2−εSETCOVER RECONFIGURATION of The-\norem 4.1. Let ( G=(V,E,Σ,Ψ),fstart,fgoal) be an instance of L ABEL COVER RECONFIGU -\nRATION with Nvariables and alphabet size |Σ| =poly(ε−1), where ∥fstart∥ = ∥ fgoal∥ = |V|.\nDefine B:={0,1}Σand Sv,α’s by Eq. (4.3). Construct then a hypergraph H=(W,F) as\n24follows:\nW:=V×Σ, (4.10)\nF:=n\nTe,q⊂W¯¯¯(e,q)∈E×Bo\n,where (4.11)\nTe,q:=n\n(v,α)∈W¯¯¯(e,q)∈Sv,αo\nfor all ( e,q)∈E×B. (4.12)\nNote that the size of hyperedge Te,qis\n|Te,q| =¯¯¯n\n(v,α)∈W¯¯¯(e,q)∈Sv,αo¯¯¯\nɯ¯¯n\n(v,α)∈V×Σ¯¯¯v∈eo¯¯¯\nÉ2|Σ| =poly(ε−1).(4.13)\nTo make Hinto 2 |Σ|-uniform, we simply augment each hyperedge Te,qwith a set of fresh\n2|Σ|−|Te,q|vertices. For a satisfying multi-assignment f:V→2ΣforGwith∥f∥ = |V|, we\nassociate with it a vertex set Cf⊂Wsuch that\nCf:=n\n(v,α)∈V×Σ¯¯¯v∈V,α∈f(v)o\n, (4.14)\nwhich is a minimum vertex cover of H(i.e.,|Cf| = |V| =β(H)), shown similarly to the proof of\nTheorem 4.1. Constructing minimum vertex covers Cstartfrom fstartandCgoalfrom fgoalby\nEq. (4.14), we obtain an instance ( H,Cstart,Cgoal) of M INMAX HYPERGRAPH VERTEX COVER\nRECONFIGURATION on a poly( ε−1)-uniform hypergraph. This completes the description of\nthe reduction.\nSimilarly to the proof Theorem 4.1, we can show that\nMinLab G(fstart↭fgoal)=cost H(Cstart↭Cgoal), (4.15)\nimplying the completeness and soundness. By Theorem 3.10, G AP1,2−εLABEL COVER RE-\nCONFIGURATION with alphabet size poly( ε−1) isPSPACE -complete; thus, G AP1,2−εHYPER -\nGRAPH VERTEX COVER RECONFIGURATION on poly( ε−1)-uniform hypergraphs is PSPACE -\ncomplete as well, which completes the proof.\nReferences\n[AFWZ95] Noga Alon, Uriel Feige, Avi Wigderson, and David Zuckerman. “De-\nrandomized graph products”. In: Comput. Complex. 5 (1995), pp. 60–75\n(↰p. 13).\n[ALMSS98] Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario\nSzegedy. “Proof Verification and the Hardness of Approximation Prob-\nlems”. In: J. ACM 45.3 (1998), pp. 501–555 ( ↰p. 5).\n25[Alo21] Noga Alon. “Explicit Expanders of Every Degree and Size”. In: Comb.\n41.4 (2021), pp. 447–463 ( ↰p. 14).\n[AS98] Sanjeev Arora and Shmuel Safra. “Probabilistic Checking of Proofs: A\nNew Characterization of NP”. In: J. 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In: WSDM . 2022, pp. 764–774 ( ↰p. 7).\n[Par21] Orr Paradise. “Smooth and Strong PCPs”. In: Comput. Complex. 30.1\n(2021), p. 1 ( ↰p. 11).\n[van13] Jan van den Heuvel. “The Complexity of Change”. In: Surveys in Combi-\nnatorics 2013 . Vol. 409. Cambridge University Press, 2013, pp. 127–160\n(↰p. 3).\n28"    },    {        "title": "2402.12651v1.Null_controllability_for_stochastic_semi_discrete_parabolic_equations.pdf",        "content": "arXiv:2402.12651v1  [math.OC]  20 Feb 2024Null controllability for stochastic\nsemi-discrete parabolic equations∗\nQingmei Zhao†\nAbstract\nIn this paper, we present a null controllability result for a class of stochastic semi-\ndiscrete parabolic equations. For this purpose, an observa bility estimate is established\nforbackwardstochasticsemi-discreteparabolicequation s, withanexplicitobservability\nconstant that depends on the discretization parameter. Thi s estimate is obtained by a\nnew Carleman estimate for backward stochastic semi-discre te parabolic operators.\n2010 Mathematics Subject Classification . Primary 93B05; Secondary 93B07, 93C20\nKey Words . Stochastic semi-discrete parabolic equations, controllability, obs ervability,\nglobal Carleman estimate.\n1 Introduction and main result\nLetT >0 and (Ω,F,{Ft}t≥0,P) be a complete filtered probability space on which a\none-dimensional standard Brownian motion {B(t)}t≥0is defined such that {Ft}t≥0is the\nnatural filtration generated by B(·), augmented by all the P-null sets in F. Write F\nfor the progressive σ-field with respect to F. LetHbe a Banach space. Denote by\nL2\nFτ(Ω;H) the space of all Fτ-measurable random variables ξsuch that lE |ξ|2\nH<∞; by\nL2\nF(0,T;H) the Banach space consisting of all H-valued{Ft}t≥0-adapted processes X(·)\nsuch that E(|X(·)|2\nL2(0,T;H))<∞, with the canonical norm; by L∞\nF(0,T;H) the Banach\nspace consisting of all H-valued{Ft}t≥0-adapted essentially bounded processes; and by\nL2\nF(Ω;C([0,T];H)) the Banach space consisting of all H-valued{Ft}t≥0-adapted continu-\nous processes X(·) such that E(|X(·)|2\nC([0,T];H))<∞.\nLetωbe a given nonempty open subset of (0 ,1), and denote by χωthe characteristic\nfunction of the set ω. We recall the following controlled stochastic parabolic equation with in\na continuous framework:\n\ndy−yxxdt= (a1y+χωu)dt+(a2y+v)dB(t) in (0,1)×(0,T),\ny(0,t) = 0, y(1,t) = 0 on (0 ,T),\ny(x,0) =y0(x) in (0 ,1),(1.1)\n∗This work is partially supported by the NSF of China under grants 119 71333and 11931011. The author\ngratefully acknowledges Professor Xiaoyu Fu for her guidance and suggestions.\n†School of Mathematics Sciences, Sichuan Normal University, Chen gdu 610066, China. E-mail address:\nqmmath@163.com.\n1with suitable coefficients a1(·),a2(·)∈L∞\nF(0,T;L∞(0,1)). In (1.1), the control variable\nconsists of the pair ( u,v),yis the state, and y0∈L2(Ω,F0,P;L2(0,1)) is the initial datum.\nThe system (1.1) is said to be null controllable at time Tif for any given y0, there exist\ncontrols (u,v)∈L2\nF(0,T;L2(ω))×L2\nF(0,T;L2(0,1)) such that the corresponding solution\nsatisfiesy(x,T) = 0,P-a.s.\nIn the literature, significant progress has been achieved in investig ating null controllabil-\nity for both deterministic and stochastic parabolic equations within t he continuous frame-\nwork(see [9, 10, 11, 14, 17] andthe references therein). Morer ecently, important progress has\nbeen made in addressing control problems for semi-discrete appro ximations of deterministic\nparabolic equations (see [1, 5, 7, 8, 15] and the references there in). However, to the best of\nour knowledge, within the stochastic framework, there is a noticea ble gap in addressing the\ncorresponding controllability issues for stochastic semi-discrete p arabolic equations.\nFor givenN∈lN, we set the space discretization parameter h:= 1/(N+1). We consider\nthe pairs (xi,t) witht∈(0,T), andxi=ih,i= 0,...,N+ 1. Applying the centered finite\ndifference method to the spatial variable for the system (1.1), we o btain the following semi-\ndiscrete system:\n\n\ndyi−1\nh2(yi+1−2yi+yi−1)dt= (ai\n1yi+χi\nωui)dt+(ai\n2yi+vi)dB(t),\nt∈(0,T), i= 1,...,N,\ny0(t) = 0, yN+1(t) = 0, t∈(0,T),\nyi(0) =yi\n0, i= 1,...,N,(1.2)\nwhereai\n1=a1(xi),ai\n2=a2(xi),χi\nω=χω(xi),ui=u(xi), andvi=v(xi).\nTo present our main result, we first introduce some notations. Firs t, we define a regular\npartition of the interval [0 ,1] as follows:\nK:={xi|i= 0,1,...,N,N +1}.\nConsidering any set of points W ⊂ K, we define the dual meshes W′andW∗, respectively,\nby\nW′:=τ+(W)∩τ−(W),W∗:=τ+(W)∪τ−(W), (1.3)\nwhere\nτ±(W) :={x±h\n2|x∈ W}.\nWe denote W:= (W∗)∗and˚W:= (W′)′. Note that if˚W=W, then for two consecutive\npointsxi,xi+1∈ W, we havexi+1−xi=h. Thus, any subset W ⊂ Kthat verifies˚W=Wis\ncalled regular mesh. Finally, we define the boundary of a regular mesh Was∂W:=W\\W.\nWe introduce, using (1.3), the semi-discrete sets. We define the se mi-discrete set G:=\nW ×(0,T). Moreover, we say that the semi-discrete set is regular if the spa ce variable is a\nregular mesh. Additionally, we define the dual semi-discrete sets by G′:=W′×(0,T) and\nG∗:=W∗×(0,T). Similarly, the semi-discrete boundary is given by ∂G=∂W ×(0,T).\n2Next, we define the average operator Ahand the difference operator DhonGby\nAh(u)(x,t) :=τ+u(x,t)+τ−u(x,t)\n2,\nDh(u)(x,t) :=τ+u(x,t)−τ−u(x,t)\nh,\nwhereτ±u(x,t) =u(x±h\n2,t).\nHere and in what follows, denote by C(W) the set of real-valued functions defined in W,\nbyL2\nh(W) the Hilbert space with inner product\n/a\\}bracketle{tu,v/a\\}bracketri}htW:=/integraldisplay\nWuv:=h/summationdisplay\nx∈Wu(x)v(x).\nFurther, we define the outward normal for ( x,t)∈∂Gas\nν(x,t) :=\n\n1, if(τ−(x),t)∈G∗and(τ+(x),t)/∈G∗,\n−1, if(τ−(x),t)/∈G∗and(τ+(x),t)∈G∗,\n0, otherwise.\nMeanwhile, we introduce the trace operator on ∂Gas\n∀(x,t)∈∂G, tr(u) :=\n\nτ−u(x,t), if ν(x,t) = 1,\nτ+u(x,t), if ν(x,t) =−1,\n0, if ν (x,t) = 0.\nFinally, we introduce the discrete integration on the boundary on ∂Was\n/integraldisplay\n∂Wu:=/summationdisplay\nx∈∂Wu(x).\nThus, defining Q:=M ×(0,T) whereM:=˚K, the controlled semi-discrete system (1.2)\ncan be written as\n\n\ndy−D2\nhydt= (a1y+χωv)dt+(a2y+v)dB(t),(x,t)∈Q,\ny(0,t) = 0, y(1,t) = 0, t∈(0,T),\ny(x,0) =y0, x∈ M.(1.4)\nThe main purpose of this paper is to study the null controllability for t he semi-discrete\nstochastic parabolic equation (1.4). Similar to the continuous case in (1.1), for any given\ninitial value y0∈L2(Ω,F0,P;L2\nh(M)), we wonder to find controls ( u,v)∈L2\nF(0,T;L2\nh(ω∩\nM))×L2\nF(0,T;L2\nh(M)) such the solution to system (1.4) satisfies yi(T) = 0,P-a.s., for\nalli= 1,...,N. If the control exists we would say that the system (1.4) is uniform ly null-\ncontrollable.\n3It is well known that the uniform null controllability property is very h ard to address\neven for deterministic case. The underlying difficulty lies in demonstra ting some uniform\nbehavior with respect to the discretization parameter. For the un iform null controllability of\ndeterministic semi-discrete parabolic equations, there are some re sults in some special cases\n(see [12, 13, 18, 19]). However, as pointed out in [18], these method s are not applicable\nto semi-discrete parabolic equations with variable coefficients. Unifo rm null controllability\nproperty may not even hold in the two-dimensional case, as demons trated by the counterex-\nample provided in [19]. Consequently, alternative definitions have bee n proposed to relax\nthe requirement for uniform null controllability. One such relaxed de finition isφ-null con-\ntrollability, i.e. |y(T)|L2\nh(M)≤φ(h), whereφis a real-valued function that tend to zero when\nspace discretization parameter htends to zero. This definition proves to be highly useful for\nobtaining well-characterized controls.\nOn the other hand, in practice, due to the interference of random factors, stochastic\nprocesses give a natural replacement for deterministic functions in mathematical descrip-\ntions. Compared with deterministic case, some substantially difficultie s arise in the study of\nthe stochastic partial differential equations. For example, the so lution to a stochastic par-\ntial differential equation is non-differentiable with respect to noise v ariable, and the usual\ncompactness embedding result is not valid for solution spaces of the stochastic evolution\nequations. Indeed, many tools and methods, which are effective in t he deterministic case,\ndo not work anymore in the stochastic setting.\nDespite these challenges, to the best of our knowledge, no resear ch has been conducted\non the null controllability of stochastic semi-discrete parabolic equa tions. In this paper,\nwe address this gap by investigating the controllability problems of ge neral stochastic semi-\ndiscrete parabolicequations. Toachieve thisobjective, weestablis h anobservability estimate\nfor backward stochastic semi-discrete parabolic equations, with a n explicit observability con-\nstant that depends on the discretization parameter. This estimat e is derived through a novel\nCarleman estimate tailored for backward stochastic semi-discrete parabolic operators. It is\nwell known that Carleman estimate is a powerful tool in the study of control problems and\ninverse problems for partial differential equations. Here, we refe r to [9] for the Carleman\nestimate of deterministic parabolic operator in continuous framewo rk, [17] for the Carleman\nestimate of continuous stochastic parabolicoperator, [2, 3, 4, 6, 16] forCarleman estimates of\ndeterministic semi-discrete elliptic equations and parabolic equations . Unlike the references\nmentioned above, establishing Carleman estimates for stochastic s emi-discrete parabolic op-\nerators poses new challenges. To address this, we introduce a new auxiliary function to\ntackle the difficulties arising from the diffusion term.\nOur main result of this paper can be stated as follows.\nTheorem 1.1 There exist Candh0, for allh≤h0, there exist (u,v)∈L2\nF(0,T;L2\nh(ω∩\nM))×L2\nF(0,T;L2\nh(M))such that the solution to (1.4)satisfies\nlE/integraldisplay\nQ|v|2dt+lE/integraldisplayT\n0/integraldisplay\nω∩M|u|2dt≤ClE/integraldisplay\nM|y0|2,\nand\nlE/integraldisplay\nM|y(T)|2≤Ce−C\nhlE/integraldisplay\nM|y0|2.\n4The rest of this paper is organized as follows. In section 2, we give so me needed pre-\nliminaries. In section 3, we prove a global Carleman estimate for a bac kward stochastic\nsemi-discrete parabolic operator. In section 4, we prove the obse rvability result for the back-\nward stochastic semi-discrete parabolic equation. Finally, in section 5, we prove the main\ncontrollability result.\n2 Preliminaries\nIn this section, we give some preliminaries we needed.\nLemma 2.1 ([6]) For any u,v∈C(W), we have for the difference operator\nDh(uv) =DhuAhv+AhuDhv, onW∗. (2.1)\nSimilarly, the average of the product gives\nAh(uv) =AhuAhv+h2\n4DhuDhv, onW∗. (2.2)\nFinally, on ˚Wwe have\nu=A2\nhu−h2\n4D2\nhu. (2.3)\nNext, we recall the following integration by parts formulas for the d iscrete difference and\naverage operators.\nLemma 2.2 ([6]) Let Wbe a semi-discrete regular mesh. For u∈C(W) andv∈C(W∗),\nwe have/integraldisplay\nWuDhv=−/integraldisplay\nW∗Dhuv+/integraldisplay\n∂Wutr(v)ν (2.4)\nand/integraldisplay\nWuAhv=/integraldisplay\nW∗Ahuv−h\n2/integraldisplay\n∂Wutr(v). (2.5)\nFurther, we have the following Lemma, which gives the relationship be tween disctete\noperators and derivative operators.\nLemma 2.3 ([6])Letfbea(n+4)-timesdifferentiable functiondefinedonlRand m,n∈lN.\nThen\nAm\nhDn\nhf=f(n)+RDn\nh(f)+RAm\nh(f)+RAm\nhDn\nh(f),\nwhere\nRDn\nh(f) :=h2n/summationdisplay\nk=0/parenleftbiggn\nk/parenrightbigg\n(−1)k/parenleftBign−2k\n2/parenrightBign+2/integraldisplay1\n0(1−σ)n+1\n(n+1)!f(n+2)/parenleftBig\n·+(n−2k)h\n2σ/parenrightBig\ndσ,\nRAm\nh(f) :=h2\n2m+2n/summationdisplay\nk=0/parenleftbiggm\nk/parenrightbigg\n(m−2k)2/integraldisplay1\n0(1−σ)f(2)/parenleftBig\n·+(m−2k)h\n2σ/parenrightBig\ndσ,\nandRAm\nhDn\nh=RAm\nh◦RDn\nh.\n5In what follows, we will present some results related to discrete ope rations performed on\nthe Carleman weight functions. To begin with, we introduce the weigh t functions. By [9],\nwe know that there exists a real-valued function ψ∈Cp(/tildewideG),psufficiently large, so that\nψ(x)>0 in/tildewideG;|∂xψ(x)|>0 in/tildewideG\\ω0; and (∂xψ)(0)>0,(∂xψ)(1)<0,(2.6)\nwhere/tildewideGisasmoothopenandconnected neighborhoodof[0 ,1]andω0isanygivennonempty\nopen subset of ωsatisfyingω0⊆ω.\nFor any positive parameters λ>1,µ>1, write\nφ(x) =eµψ(x)−e2µ|ψ|∞, ϕ(x) =eµψ(x), r(t,x) =es(t)φ(x), ρ(t,x) =r−1,\ns(t) =λθ(t), θ(t) =1\n(t+δT)(T+δT−t),(2.7)\nfor 0<δ <1\n2.\nIn the sequel, for any n∈lN, we denote by O(sn) a function of order sn, for sufficiently\nlarges, and by Oµ(sn) a function of order nfor fixedµand sufficiently large s.\nBy (2.7) and Lemma 2.3, we have the following estimation of coefficients involved in the\nenergy terms of Carleman estimate.\nLemma 2.4 Forλh(δT2)−1≤1, we have\n\n\nrD2\nhρ=Ah(rD2\nhρ) =Oµ(s2), ∂t(rD2\nhρ) =∂tAh(rD2\nhρ) =s2TθOµ(1),\nD2\nh(rD2\nhρ) =Oµ(s2), ∂ tD2\nh(rD2\nhρ) =s2TθOµ(1),\nAhDh(rD2\nhρ) =Oµ(s2),(2.8)\nand\n/braceleftBigg\nr2D2\nhρAhDhρ=Oµ(s3), Dh(r2D2\nhρAhDhρ) =Oµ(s3),\nAhDh(r2D2\nhρAhDhρ) =−3s3ϕ3µ4(∂xψ)4+s3ϕ3O(µ3)+Oµ(s2)+s3O((sh)2).(2.9)\nAnd moreover,\n/braceleftBigg\nr2A2\nhρAhDhρ=−sϕµ∂xψ+sOµ((sh)2),\nDh(r2A2\nhρAhDhρ) =−sϕµ2(∂xψ)2+sϕO(µ)+sOµ((sh)2).(2.10)\nProof. By (2.7) and Lemma 2.3, one can obtain the estimations of (2.8)–(2.10 ) di-\nrectly. For the reader’s convenience, here we only give the estimat ions ofrD2\nhρ,∂tAh(rD2\nhρ),\nAhDh(r2D2\nhρAhDhρ) andDh(r2A2\nhρAhDhρ).\nBy (2.7), it is easy to see that\n/braceleftBigg\n|∂tθ| ≤CTθ2,|∂ttθ| ≤CTθ3,\n∂xρ=−sµϕρ∂xψ, ∂2\nxρ=s2µ2ϕ2ρ(∂xψ)2+sϕρO(µ2).(2.11)\nThen, by Lemma 2.3, we have\nrD2\nhρ=r∂2\nxρ+rRD2\nh(ρ) =s2ϕ2O(µ2)+s2Oµ((sh)2) =Oµ(s2),\n6which is used the fact that sh≤1 andr(x)∂4\nxρ(x+σh) =Oµ(s4) for anyσ∈[0,1]. By\nLemma 2.3, it follows that\nAh(rD2\nhρ) =Ah(r∂2\nxρ)+Ah[rRD2\nh(ρ)] =r∂2\nxρ+RAh(r∂2\nxρ)+rRD2\nh(ρ)+RAh[rRD2\nh(ρ)].\nForσ∈[0,1],∂t∂2\nx(r∂2\nxρ)(x+σh,t) =TθOµ(s2)by(2.11),then ∂t[RAh(r∂2\nxρ)] =TθOµ((sh)2).\nAnd by∂t[r(x,t)∂4\nxρ(x+σh,t)] =TθOµ(s4), we have ∂t[rRD2\nh(ρ)] =s2TθOµ((sh)2). Simi-\nlarly,∂t[RAh[rRD2\nh(ρ)] =TθOµ((sh)4). Then, for sh≤1, it holds that\n∂tAh(rD2\nhρ) =TθOµ(s2)+s2TθOµ((sh)2)+TθOµ((sh)4) =TθOµ(s2).\nNext, by Lemma 2.3, we have\nD2\nhρ=∂2\nxρ+RD2\nh(ρ), AhDhρ=∂xρ+RAh(ρ)+RDh(ρ)+RDhAh(ρ).\nThus, combining the the above estimate, Lemma 2.3 and (2.11), we co nclude that for sh≤1,\nAhDh(r2D2\nhρAhDhρ) =∂x(r2∂2\nxρ∂xρ)+s3Oµ((sh)2)\n=∂x[−s3µ3ϕ3(∂xψ)3]+s2ϕ2O(µ4)+s3O((sh)2)\n=−3s3ϕ3µ4(∂xψ)4+s3ϕ3O(µ3)+s2ϕ2O(µ4)+s3O((sh)2).\nSimilarly, by Lemma 2.3 and (2.11), for sh≤1, we can get that\nDh(r2A2\nhρAhDhρ) =∂x(r2ρ∂xρ)+sOµ((sh)2)\n=∂x(−sµϕ∂xψ)+sOµ((sh)2)\n=−sϕµ2(∂xψ)2+sϕO(µ)+sOµ((sh)2).\nThis completes the proof of Lemma 2.4.\nRemark 2.1 In fact, what is needed in the proof of Lemma 2.4issh≤1. Notice that\ns=λθandθ(t)≤θ(0) =θ(T)<(δT2)−1, for allt∈[0,T]. Therefore, the condition we\npresent in Lemma 2.4isλh(δT2)−1≤1.\nFinally, for any v∈L2\nF(Ω;C([0,T];L2\nh(W))), we use the following notations:\nX1:= lE/integraldisplay\nQ/bracketleftBig\ns3ϕ3O(µ3)+Oµ(s2)+θOµ(s2)+s3Oµ((sh)2)/bracketrightBig\n|v|2dt\n+lE/integraldisplay\nQ∗/bracketleftBig\nsϕO(µ)+Oµ(1)+θOµ(1)+sOµ((sh)2))/bracketrightBig\n|Dhv|2dt,(2.12)\nX2:= lE/integraldisplay\nQOµ(s2)|dv|2+lE/integraldisplay\nQ∗Oµ((sh)2)|Dh(dv)|2, (2.13)\nX3:= lE/integraldisplay\n∂Q/bracketleftBig\nsOµ((sh)2)+Oµ(1)/bracketrightBig\ntr(|Dhv|2)dt, (2.14)\n7and\nX4:= lE/integraldisplay\nMOµ(s2)v2/vextendsingle/vextendsingle/vextendsingle\nt=0+lE/integraldisplay\nM∗/bracketleftBig\nOµ(1)+Oµ((sh)2)/bracketrightBig\n|Dhv|2/vextendsingle/vextendsingle/vextendsingle\nt=0\n+lE/integraldisplay\nMOµ(s2)v2/vextendsingle/vextendsingle/vextendsingle\nt=T+lE/integraldisplay\nM∗/bracketleftBig\nOµ(1)+Oµ((sh)2)/bracketrightBig\n|Dhv|2/vextendsingle/vextendsingle/vextendsingle\nt=T.(2.15)\nIn fact, we can see that X1is the lower order term about energy, X2is the estimate about\ndrift term, X3is the lower order term about space boundary, and X4is the time boundary\nterm.\n3 Semi-discreteCarleman estimate for uniform meshes\nIn the section, we give the semi-discrete Carleman estimate for the backward stochastic\nsemi-discrete parabolic operator. Then, we have the following Carle man estimate.\nTheorem 3.1 There exists a positive constant µ0such that for any µ > µ 0, one can find\ntwo positive constants ε0>0andh0>0, with0< ε0≤1and a sufficiently large λ0≥1,\nsuch that for λ≥λ0, andh≤h0andλh(δT2)−1≤ε0, it holds that\nlE/integraldisplay\nQs3e2sφw2dt+lE/integraldisplay\nQ∗se2sφ|Dhw|2dt\n≤C/parenleftBig\nlE/integraldisplayT\n0/integraldisplay\nω∩Ms3e2sφw2dt+lE/integraldisplay\nQs2e2sφ|g|2+lE/integraldisplay\nQe2sφ|f|2dt\n+h−2lE/integraldisplay\nMe2sφ|w|2/vextendsingle/vextendsingle/vextendsingle\nt=0+h−2lE/integraldisplay\nMe2sφ|w|2/vextendsingle/vextendsingle/vextendsingle\nt=T/parenrightBig\n,\nfor allw∈L2\nF(Ω;C([0,T];L2\nh(W)))satisfyingdw+D2\nhwdt=fdt+gdB(t)withw= 0on\n∂M.\nProof. The whole proof is divided into six steps.\nStep 1. At first, let\nLhw:= dw+D2\nhwdt=fdt+gdB(t). (3.1)\nSettingv=rwand noting that ρ=r−1, we have\nrLh(w) =rLh(ρv) =I1dt−Φ(v)dt+I2, (3.2)\nwhere\n\n\nI1=rD2\nhρA2\nhv+rA2\nhρD2\nhv−λφ∂tθv+Φ(v),\nI2=dv+2rAhDhρAhDhvdt,\nΦ(v) =h2\n2Dh/bracketleftBig\nDh(rD2\nhρ)Ahv/bracketrightBig\n.(3.3)\n8Therefore, by (3.2), it holds that\n2rLh(w)I1= 2|I1|2dt−2I1Φ(v)dt+2I1I2. (3.4)\nHence, combining (3.4) with (3.1), it follows that\nlE/integraldisplay\nQ2rfI1dt= 2lE/integraldisplay\nQ|I1|2dt−2lE/integraldisplay\nQI1Φ(v)dt+2lE/integraldisplay\nQI1I2.\nBy using the triangle inequality, it is easy to see that\nlE/integraldisplayT\n0|rf|2dt≥2lE/integraldisplay\nQI1I2−lE/integraldisplay\nQ|Φ(v)|2dt. (3.5)\nThe next step is to provide an estimate for the right-hand side of (3 .5). Denote the terms\nin the right-hand side of I1andI2in (3.3) by Ii\n1(i= 1,2,3,4) andIj\n2(j= 1,2), respectively.\nThen\n2lE/integraldisplay\nQI1I2=4/summationdisplay\ni=12/summationdisplay\nj=1Jij, Jij:= 2lE/integraldisplay\nQIi\n1Ij\n2. (3.6)\nStep 2. Let us compute “ J11+J21+J31+J41”. By (2.3) and (3.3), noting that rρ= 1,\nwe have\nJ11+J21+J31+J41\n= 2lE/integraldisplay\nQ/parenleftBig\nrD2\nhρA2\nhv+rA2\nhρD2\nhv−λφ∂tθv+Φ(v)/parenrightBig\ndv\n= 2lE/integraldisplay\nQ/parenleftBig\nrD2\nhρv+h2\n4rD2\nhρD2\nhv+rρD2\nhv+h2\n4rD2\nhρD2\nhv−λφ∂tθv+Φ(v)/parenrightBig\ndv\n= 2lE/integraldisplay\nQ/parenleftBig\nq11v−λφ∂tθv+D2\nhv+h2\n2q11D2\nhv+Φ(v)/parenrightBig\ndv,(3.7)\nwhere\nq11:=rD2\nhρ. (3.8)\nForλh(δT2)−1≤1, by (2.8), (2.11), (3.8) and Itˆ o’s formula, we can get that\n2lE/integraldisplay\nQ/parenleftBig\nq11v−λφ∂tθv/parenrightBig\ndv\n= lE/integraldisplay\nM(q11−λφ∂tθ)|v|2/vextendsingle/vextendsingle/vextendsingleT\n0−lE/integraldisplay\nQ(∂tq11−λφ∂ttθ)|v|2dt\n−lE/integraldisplay\nQ(q11−λφ∂tθ)|dv|2≥ −X1−X2−X4.(3.9)\nNext, by Lemma 2.2 and Itˆ o’s formula, noting that v= 0 on∂Q, we have\n2lE/integraldisplay\nQD2\nhvdv=−2lE/integraldisplay\nQ∗DhvDh(dv)+2/integraldisplay\n∂Qdvtr(Dhv)ν\n= lE/integraldisplay\nQ∗|Dh(dv)|2−lE/integraldisplay\nM∗|Dhv|2/vextendsingle/vextendsingle/vextendsingleT\n0≥lE/integraldisplay\nQ∗|Dh(dv)|2−X4.(3.10)\n9Further, by Lemmas 2.1 and 2.2, noting that v= 0 on∂Q, it yields that\nh2lE/integraldisplay\nQq11D2\nhvdv\n=−h2lE/integraldisplay\nQ∗Ahq11Dh(dv)Dhv−h2lE/integraldisplay\nQ∗Dhq11Ah(dv)Dhv.(3.11)\nBy 2Ah(dv)Dhv=d(Dh(v2))−2Ah(dv)Dh(dv)−2AhvDh(dv), recalling (3.3) for the\ndefinition of Φ( v), andv= 0 on∂Qandtr(v) = 0 on∂Q∗, we have\n−h2lE/integraldisplay\nQ∗Dhq11DhvAh(dv)+2lE/integraldisplay\nQΦ(v)dv\n=−h2\n2lE/integraldisplay\nQ∗Dhq11d(Dh(v2))+h2lE/integraldisplay\nQ∗Dhq11Ah(dv)Dh(dv)\n+h2lE/integraldisplay\nQ∗Dhq11AhvDh(dv)+2lE/integraldisplay\nQΦ(v)dv\n=−h2\n2lE/integraldisplay\nM∗Dhq11Dh(v2)/vextendsingle/vextendsingle/vextendsingleT\n0+h2\n2lE/integraldisplay\nQ∗∂t(Dhq11)Dh(v2)dt\n+h2\n2lE/integraldisplay\nQ∗Dhq11Dh[(dv)2].(3.12)\nCombining (3.11)-(3.12), by Lemma 2.2 and Itˆ o’s formula, noting tha tv= 0 on∂Qand\ntr(v) = 0 on∂Q∗, we have\nh2lE/integraldisplay\nQq11D2\nhvdv+2lE/integraldisplay\nQΦ(v)dv\n=−h2\n2/bracketleftBig\nlE/integraldisplay\nM∗Ahq11|Dhv|2/vextendsingle/vextendsingle/vextendsingleT\n0−lE/integraldisplay\nQ∗∂t(Ahq11)|Dhv|2dt−lE/integraldisplay\nQ∗Ahq11|Dh(dv)|2\n−lE/integraldisplay\nMD2\nhq11v2/vextendsingle/vextendsingle/vextendsingleT\n0+lE/integraldisplay\nQ∂t(D2\nhq11)v2dt+lE/integraldisplay\nQD2\nhq11|dv|2/bracketrightBig\n.(3.13)\nForλh(δT2)−1≤1, by (2.8), (3.7)–(3.10) and (3.13), we end up with\nJ11+J21+J31+J41≥lE/integraldisplay\nQ∗|Dh(dv)|2−X1−X2−X4. (3.14)\nStep 3. Let us compute “ J12+J22+J32+J42”.\nRecalling (3.3) for the definitions of I1,I2and Φ(v), for simplicity, we set\nq12=r2D2\nhρAhDhρ, q22=r2A2\nhρAhDhρ, q32=λφ∂tθrAhDhρ, q42=rAhDhρ.(3.15)\nBy Lemma 2.1, it is easy to see that\nA2\nhv=v+h2\n4D2\nhv,2D2\nhvAhDhv=Dh(|Dhv|2). (3.16)\n10Then, by (3.3), (3.15) and (3.16), we know that\nJ12+J22+J32+J42\n= 2lE/integraldisplay\nQ/bracketleftBig\n2q12A2\nhv+2q22D2\nhv−2q32v+2q42Φ(v)/bracketrightBig\nAhDhvdt\n= 4lE/integraldisplay\nQq12vAhDhvdt+lE/integraldisplay\nQ/parenleftBigh2\n2q12+2q22/parenrightBig\nDh(|Dhv|2)dt\n−4lE/integraldisplay\nQq32AhDhvvdt+4lE/integraldisplay\nQq42AhDhvΦ(v)dt.(3.17)\nNoting that ( Ahv)2=Ah(v2)−h2\n4(Dhv)2,tr(v) = 0 on∂Q∗andv= 0 on∂Q, we have\n4lE/integraldisplay\nQq12vAhDhvdt\n=−4lE/integraldisplay\nQ∗Ahq12DhvAhvdt−4lE/integraldisplay\nQ∗Dhq12|Ahv|2dt\n=−2lE/integraldisplay\nQ∗Ahq12Dh(v2)dt−4lE/integraldisplay\nQ∗Dhq12Ah(v2)dt+h2lE/integraldisplay\nQ∗Dhq12|Dhv|2dt\n=−2lE/integraldisplay\nQDhAhq12v2dt+h2lE/integraldisplay\nQ∗Dhq12|Dhv|2dt.(3.18)\nNext, by Lemma 2.2, we have\nlE/integraldisplay\nQ/parenleftBigh2\n2q12+2q22/parenrightBig\nDh(|Dhv|2)dt\n=−lE/integraldisplay\nQ∗/parenleftBigh2\n2Dhq12+2Dhq22/parenrightBig\n|Dhv|2dt+lE/integraldisplay\n∂Q/parenleftBigh2\n2q12+2q22/parenrightBig\ntr(|Dhv|2)νdt.(3.19)\nCombining (3.15), (3.17)–(3.19) with (2.9)–(2.10), by (2.12) and (2.1 4), we have\nJ12+J22+J32+J42≥6lE/integraldisplay\nQs3ϕ3µ4(∂xψ)4v2dt+2lE/integraldisplay\nQ∗sϕµ2(∂xψ)2|Dhv|2dt\n−2lE/integraldisplay\n∂Qsϕµ∂xψtr(|Dhv|2)νdt−X1−X3\n−4lE/integraldisplay\nQq32AhDhvvdt+4lE/integraldisplay\nQq42AhDhvΦ(v)dt.(3.20)\nBy Lemmas 2.1 and 2.2, and noting that v= 0 on∂Q,tr(v) = 0 on∂Q∗, we have\n−4lE/integraldisplay\nQq32AhDhvvdt= 4lE/integraldisplay\nQ∗Dhq32|Ahv|2dt+2lE/integraldisplay\nQ∗Ahq32Dh(v2)dt\n= 4lE/integraldisplay\nQ∗Dhq32|Ahv|2dt−2lE/integraldisplay\nQDhAhq32v2dt\n≥ −lE/integraldisplay\nQ∗θOµ(s2)|Ahv|2dt−2lE/integraldisplay\nQθOµ(s2)v2dt,(3.21)\n11where we have used the following facts:\nDh(q32) =Dh(λφ∂tθrAhDhρ) =θOµ(s2), AhDh(q32) =θOµ(s2).\nBy|Ahv|2≤Ah(v2), Lemma 2.2, and tr(v) = 0 on∂Q∗, it yields that\n−lE/integraldisplay\nQ∗θOµ(s2)|Ahv|2dt≥ −lE/integraldisplay\nQ∗θOµ(s2)Ah(v2)dt=−lE/integraldisplay\nQ∗θOµ(s2)|v|2dt.(3.22)\nNext, by (3.3), we get\n4lE/integraldisplay\nQq42AhDhvΦ(v)dt\n= 2h2lE/integraldisplay\nQq42AhDhvDh/bracketleftBig\nDh(rD2\nhρ)Ahv/bracketrightBig\ndt\n= 2h2lE/integraldisplay\nQq42D2\nh(rD2\nhρ)A2\nhvAhDhvdt+2h2lE/integraldisplay\nQq42AhDh(rD2\nhρ)|AhDhv|2dt\n≥ −lE/integraldisplay\nQsOµ((sh)2)|A2\nhv|2dt−lE/integraldisplay\nQsOµ((sh)2)|AhDhv|2dt,(3.23)\nwhere we have used the following facts:\nrD2\nh(rD2\nhρ)AhDhρ=∂2\nx(r∂2\nxρ)r∂xρ+s3Oµ((sh)2) =Oµ(s3),\nand\nrAhDh(rD2\nhρ)AhDhρ=∂x(r∂2\nxρ)r∂xρ+s3Oµ((sh)2) =Oµ(s3).\nBy|A2\nhv|2≤Ah(|Ahv|2) and Lemma 2.2, we have\n−lE/integraldisplay\nQsOµ((sh)2)|A2\nhv|2dt≥ −lE/integraldisplay\nQsOµ((sh)2)Ah(|Ahv|2)dt\n=−lE/integraldisplay\nQ∗sOµ((sh)2)|Ahv|2dt+h\n2lE/integraldisplay\n∂QsOµ((sh)2)tr(|Ahv|2)dt\n≥ −lE/integraldisplay\nQ∗sOµ((sh)2)|Ahv|2dt.(3.24)\nIn the same manner, we can obtain that\n−lE/integraldisplay\nQ∗sOµ((sh)2)|Ahv|2dt≥ −lE/integraldisplay\nQsOµ((sh)2)|v|2dt, (3.25)\nand\n−lE/integraldisplay\nQsOµ((sh)2)|AhDhv|2dt≥ −lE/integraldisplay\nQsOµ((sh)2)|Dhv|2dt. (3.26)\n12Then, combining (3.21)–(3.26), we have\n−4lE/integraldisplay\nQq32AhDhvvdt+4lE/integraldisplay\nQq42AhDhvΦ(v)dt≥ −X1. (3.27)\nBy (3.20) and (3.27), we end up with\nJ12+J22+J32+J42≥6lE/integraldisplay\nQs3ϕ3µ4(∂xψ)4v2dt+2lE/integraldisplay\nQ∗sϕµ2(∂xψ)2|Dhv|2dt\n−2lE/integraldisplay\n∂Qsϕµ∂xψtr(|Dhv|2)νdt−X1−X3.(3.28)\nStep 4. Combining (3.14), (3.28) with (3.6), for λ>1 andλh(δT2)−1≤1, it holds that\n2lE/integraldisplay\nQI1I2\n≥6lE/integraldisplay\nQs3ϕ3µ4(∂xψ)4v2dt+2lE/integraldisplay\nQ∗sϕµ2(∂xϕ)2|Dhv|2dt+lE/integraldisplay\nQ∗|Dh(dv)|2\n−2/integraldisplay\n∂Qsϕµ∂xψtr(|Dhv|2)νdt−X1−X2−X3−X4.(3.29)\nNext, recall (3.3) for the definition of Φ( v), by (2.8), we have\n−lE/integraldisplay\nQ|Φ(v)|2dt≥ −lE/integraldisplay\nQOµ((sh)2)|A2\nhv|2dt−lE/integraldisplay\nQOµ((sh)2)|AhDhv|2dt.(3.30)\nSimilar to the derivation of (3.24), it follows that\n−lE/integraldisplay\nQOµ((sh)2)|A2\nhv|2dt≥ −lE/integraldisplay\nQOµ((sh)2)|v|2dt, (3.31)\nand\n−lE/integraldisplay\nQOµ((sh)2)|AhDhv|2≥ −lE/integraldisplay\nQOµ((sh)2)|Dhv|2dt. (3.32)\nCombining (3.31)-(3.32) with (3.30), we have\n−lE/integraldisplay\nQ|Φ(v)|2dt≥ −X1. (3.33)\nCombining (3.29), (3.33) with (3.5),\n6lE/integraldisplay\nQs3ϕ3µ4(∂xψ)4v2dt+2lE/integraldisplay\nQ∗sϕµ2(∂xϕ)2|Dhv|2dt+lE/integraldisplay\nQ∗|Dh(dv)|2\n≤2/integraldisplay\n∂Qsϕµ∂xψtr(|Dhv|2)νdt+lE/integraldisplay\nQ|rf|2dt+X1+X2+X3+X4.\n13Remember that ψsatisfying (2.6), thenby elementary calculation, weconclude thatt here\nexists a positive constant µ0such that for any µ≥µ0, one can find two positive constants\nε1>0 andh1>0, with 0<ε1≤1 and a sufficiently large λ1≥1, such that for λ≥λ1, and\nh≤h1andλh(δT2)−1≤ε1, there exist a constant C >0 such that\nlE/integraldisplay\nQs3ϕ3µ4v2dt+lE/integraldisplay\nQ∗sϕµ2|Dhv|2dt\n≤lE/integraldisplayT\n0/integraldisplay\nω0∩Ms3ϕ3µ4v2dt+lE/integraldisplayT\n0/integraldisplay\nω0∩M∗sϕµ2|Dhv|2dt\n+C/parenleftBig\nlE/integraldisplay\nQ|rf|2dt+lE/integraldisplay\nQs2|dv|2+h−2lE/integraldisplay\nM|v|2/vextendsingle/vextendsingle/vextendsingle\nt=0+h−2lE/integraldisplay\nM|v|2/vextendsingle/vextendsingle/vextendsingle\nt=T/parenrightBig\n,(3.34)\nwhere we have used that\n|Dhv|2≤Ch−2(|τ−v|2+|τ+v|2).\nStep 5. We now return vtow. Recalling w=ρv, we haveDhw=DhρAhv+AhρDhv.\nTherefore, by Lemma 2.3, it holds that\nlE/integraldisplay\nQ∗sϕµ2|rDhw|2dt\n≤C/parenleftBig\nlE/integraldisplay\nQ∗s3ϕ3µ4|Ahv|2dt+lE/integraldisplay\nQ∗s3Oµ((sh)2)|Ahv|2dt\n+lE/integraldisplay\nQ∗sϕµ2|Dhv|2dt+lE/integraldisplay\nQ∗sOµ((sh)2)|Dhv|2dt/parenrightBig\n.(3.35)\nSimilar to the derivation of (3.22), and by Lemma 2.3, we have\nlE/integraldisplay\nQ∗s3ϕ3µ4|Ahv|2dt≤lE/integraldisplay\nQ∗s3ϕ3µ4Ah(v2)dt= lE/integraldisplay\nQs3µ4Ah(ϕ3)|v|2dt\n= lE/integraldisplay\nQs3µ4ϕ3|v|2dt+lE/integraldisplay\nQsOµ((sh)2)|v|2dt.(3.36)\nAnd we have\nlE/integraldisplay\nQ∗s3Oµ((sh)2)|Ahv|2dt≤lE/integraldisplay\nQs3Oµ((sh)2)|v|2dt. (3.37)\nThen, combining (3.35)–(3.37), it follows that\nlE/integraldisplay\nQ∗sϕµ2|rDhw|2dt\n≤C/parenleftBig\nlE/integraldisplay\nQs3µ4ϕ3|v|2dt+lE/integraldisplay\nQ∗s3Oµ((sh)2)v2dt+lE/integraldisplay\nQ∗sϕµ2|Dhv|2dt\n+lE/integraldisplay\nQ∗sOµ((sh)2)|Dhv|2dt/parenrightBig\n.(3.38)\n14Notice that rAhρDhv=rDhw−rDhρAhv, then by Lemma 2.3, we have Dhv=rDhw−\nrDhρAhv−Oµ((sh)2)Dhv. Then\nlE/integraldisplayT\n0/integraldisplay\nω0∩M∗sϕµ2|Dhv|2dt\n≤C/parenleftBig\nlE/integraldisplayT\n0/integraldisplay\nω0∩W∗sϕµ2r2|Dhw|2dt+lE/integraldisplayT\n0/integraldisplay\nω0∩M∗s3ϕ3µ4|Ahv|2dt\n+lE/integraldisplayT\n0/integraldisplay\nω0∩M∗s3Oµ((sh)2)|Ahv|2dt+lE/integraldisplayT\n0/integraldisplay\nω0∩M∗sOµ((sh)2)|Dhv|2dt/parenrightBig\n.(3.39)\nSimilar to (3.24), and by Lemma 2.3, we can obtain that\nlE/integraldisplayT\n0/integraldisplay\nω0∩M∗s3ϕ3µ4|Ahv|2dt+lE/integraldisplayT\n0/integraldisplay\nω0∩M∗s3Oµ((sh)2)|Ahv|2dt\n≤lE/integraldisplay\nω0∩Ms3µ4ϕ3|v|2dt+lE/integraldisplay\nω0∩Ms3Oµ((sh)2)|v|2dt.(3.40)\nBy (3.39)–(3.40), we have\nlE/integraldisplayT\n0/integraldisplay\nω0∩M∗sϕµ2|Dhv|2dt\n≤C/parenleftBig\nlE/integraldisplayT\n0/integraldisplay\nω0∩M∗sϕµ2r2|Dhw|2dt+lE/integraldisplayT\n0/integraldisplay\nω0∩Ms3ϕ3µ4v2dt\n+lE/integraldisplayT\n0/integraldisplay\nω0∩Ms3Oµ(sh)2)v2dt+lE/integraldisplayT\n0/integraldisplay\nω0∩M∗sOµ(sh)2)|Dhv|2dt/parenrightBig\n.(3.41)\nCombining (3.38), (3.41) with (3.34) and noting\nlE/integraldisplay\nQs2|dv|2= lE/integraldisplay\nQs2|rdw|2= lE/integraldisplay\nQs2|rg|2dt,\nwe conclude that, for any µ≥µ0, one can find two positive constants ε2>0 andh2>0,\nwith 0< ε2≤1 and a sufficiently large λ2≥1, such that for λ≥λ2, andh≤h2and\nλh(δT2)−1≤ε2, there exist a constant C >0 such that\nlE/integraldisplay\nQs3ϕ3µ4e2sφw2dt+lE/integraldisplay\nQ∗sϕµ2e2sφ|Dhw|2dt\n≤C/parenleftBig\nlE/integraldisplayT\n0/integraldisplay\nω0∩Ws3ϕ3µ4e2sφw2dt+lE/integraldisplayT\n0/integraldisplay\nω0∩W∗sϕµ2e2sφ|Dhw|2dt\n+lE/integraldisplay\nQs2e2sφ|g|2dt+lE/integraldisplay\nQe2sφ|f|2dt+h−2lE/integraldisplay\nMe2sφ|w|2/vextendsingle/vextendsingle/vextendsingle\nt=0+h−2lE/integraldisplay\nMe2sφ|w|2/vextendsingle/vextendsingle/vextendsingle\nt=T/parenrightBig\n.(3.42)\nStep 6. Next, we choose a real-valued function ξ∈C∞\n0(w;[0,1]) such that ξ= 1 inw0.\n15By Itˆ o’s formula and (3.1), then it is easy to show that\nlE/integraldisplay\nω∩Msϕξ2e2sφ|w|2/vextendsingle/vextendsingle/vextendsingle\nt=T−lE/integraldisplay\nω∩Msϕξ2e2sφ|w|2/vextendsingle/vextendsingle/vextendsingle\nt=0\n= lE/integraldisplayT\n0/integraldisplay\nω∩Mξ2/bracketleftBig\nϕ∂t(se2sφ)|w|2dt+2sϕe2sφwdw+sϕe2sφ(dw)2/bracketrightBig\n= lE/integraldisplayT\n0/integraldisplay\nω∩Mξ2/bracketleftBig\nϕ∂t(se2sφ)|w|2+2sϕe2sφw(−D2\nhw+f)+sϕe2sφg2/bracketrightBig\ndt\n= lE/integraldisplayT\n0/integraldisplay\nω∩M/bracketleftBig\nξ2ϕ∂t(se2sφ)|w|2+2sξ2ϕe2sφwf+sξ2ϕe2sφg2/bracketrightBig\ndt\n+lE/integraldisplayT\n0/integraldisplay\nω∩M∗/bracketleftBig\n2sAh(ξ2ϕe2sφ)|Dhw|2+2sDh(ξ2ϕe2sφ)AhwDhw/bracketrightBig\ndt.\nTherefore, we conclude that\n2lE/integraldisplayT\n0/integraldisplay\nω∩M∗sAh(ξ2ϕe2sφ)|Dhw|2dt+lE/integraldisplayT\n0/integraldisplay\nω∩Msξ2ϕe2sφg2dt\n=−lE/integraldisplayT\n0/integraldisplay\nω∩M/bracketleftBig\nξ2ϕ∂t(se2sφ)|w|2+2sξ2ϕe2sφwf/bracketrightBig\ndt+lE/integraldisplay\nω∩Msϕξ2e2sφ|w|2/vextendsingle/vextendsingle/vextendsingle\nt=T\n−lE/integraldisplay\nω∩Msϕξ2e2sφ|w|2/vextendsingle/vextendsingle/vextendsingle\nt=0−2lE/integraldisplayT\n0/integraldisplay\nω∩W∗sDh(ξ2ϕe2sφ)AhwDhwdt.(3.43)\nBy Lemma 2.3, for λh(δT2)−1≤1, it follows that Ah(ξ2ϕe2sφ) =ξ2ϕe2sφ+Oµ((sh)2)e2sφ.\nThen the first term in the the left-hand of (3.43) can be estimated b y\n2lE/integraldisplayT\n0/integraldisplay\nω∩M∗sAh(ξ2ϕe2sφ)|Dhw|2dt\n≥2lE/integraldisplayT\n0/integraldisplay\nω∩M∗sξ2ϕe2sφ|Dhw|2dt−lE/integraldisplayT\n0/integraldisplay\nω∩M∗sOµ((sh)2)e2sφ|Dhw|2dt.(3.44)\nFor the first term in the right-hand of (3.43), we have\n−lE/integraldisplayT\n0/integraldisplay\nω∩Mξ2ϕ∂t(se2sφ)|w|2dt≤lE/integraldisplayT\n0/integraldisplay\nω∩MθOµ(s2)e2sφ|w|2dt. (3.45)\nMoreover, it is easy to check that\n−lE/integraldisplayT\n0/integraldisplay\nω∩M2sξ2ϕe2sφwfdt\n≤ClE/integraldisplayT\n0/integraldisplay\nω∩Ms2µ2ϕ2e2sφ|w|2dt+ClE/integraldisplayT\n0/integraldisplay\nMµ−2e2sφ|f|2dt.(3.46)\nNext, we estimate the last term in the the right-hand of (3.43). By L emma 2.3, we have\n16Dh(ξ2ϕe2sφ) =∂x(ξ2ϕe2sφ)+sOµ((sh)2)e2sφ=sO(µ)ξϕ2e2sφ+sOµ((sh)2)e2sφ, then\n−lE/integraldisplayT\n0/integraldisplay\nω∩M∗2sDh(ξ2ϕe2sφ)AhwDhwdt\n= lE/integraldisplayT\n0/integraldisplay\nω∩M∗/bracketleftBig\ns2O(µ)ξϕ2e2sφ+sOµ((sh)2)e2sφ/bracketrightBig\nAhwDhwdt\n≤/integraldisplayT\n0/integraldisplay\nω∩M∗ξ2sϕe2sφ|Dhw|2dt+ClE/integraldisplayT\n0/integraldisplay\nω∩M∗s3µ2ϕ3e2sφ|Ahw|2dt\n+/integraldisplayT\n0/integraldisplay\nω∩M∗sOµ((sh)2)e2sφ|Ahw|2dt+/integraldisplayT\n0/integraldisplay\nω∩M∗sOµ((sh)2)e2sφ|Dhw|2dt.(3.47)\nSimilarto(3.24),andnotingthat Ah(ϕ3e2sφ) =ϕ3e2sφ+Oµ((sh)2)e2sφ,Ah(e2sφ) =Oµ(1)e2sφ,\nwe have\nlE/integraldisplayT\n0/integraldisplay\nω∩M∗s3µ2ϕ3e2sφ|Ahw|2dt+/integraldisplayT\n0/integraldisplay\nω∩M∗sOµ((sh)2)e2sφ|Ahw|2dt\n≤lE/integraldisplayT\n0/integraldisplay\nω∩Ms3µ2ϕ3e2sφ|w|2dt+/integraldisplayT\n0/integraldisplay\nω∩Ms3Oµ((sh)2)e2sφ|w|2dt.(3.48)\nHence, combining (3.47) and (3.48), the last term in the the left-han d of (3.43) can be\nestimated by\n−lE/integraldisplayT\n0/integraldisplay\nω∩M∗2sDh(ξ2ϕe2sφ)AhwDhwdt\n≤/integraldisplayT\n0/integraldisplay\nω∩M∗ξ2sϕe2sφ|Dhw|2dt+ClE/integraldisplayT\n0/integraldisplay\nω∩M∗s3µ2ϕ3e2sφ|w|2dt\n+/integraldisplayT\n0/integraldisplay\nω∩M∗s3Oµ((sh)2)e2sφ|w|2dt+/integraldisplayT\n0/integraldisplay\nω∩M∗sOµ((sh)2)e2sφ|Dhw|2dt.(3.49)\nTherefore, combining (3.43)-(3.46), (3.49), we conclude that for anyµ≥µ0, one can find\ntwo positive constants ε3>0 andh3>0, with 0< ε3≤1 and a sufficiently large λ3≥1,\nsuch that for λ≥λ3, andh≤h3andλh(δT2)−1≤ε3, it holds that\nlE/integraldisplayT\n0/integraldisplay\nω0∩M∗sϕe2sφ|Dhw|2dt≤lE/integraldisplayT\n0/integraldisplay\nω∩M∗sξ2ϕe2sφ|Dhw|2dt\n≤C/parenleftBig\nlE/integraldisplayT\n0/integraldisplay\nω∩Ms3µ2ϕ3e2sφ|w|2dt+lE/integraldisplayT\n0/integraldisplay\nω∩Mµ−2e2sφ|f|2dt\n+h−1lE/integraldisplay\nω∩Me2sφ|w|2/vextendsingle/vextendsingle/vextendsingle\nt=T/parenrightBig\n.(3.50)\nBy (3.42) and (3.50), then we complete the proof.\n174 The observability for the backward stochastic semi-\ndiscrete parabolic equations\nThe controllability result for (1.2) can be established by proving an ob servability estimate\nfor its adjoint system. In this section, we first consider the followin g backward stochastic\nsemi-discrete parabolic equation:\n\n\ndz+D2\nhzdt=−(a1z+azZ)dt+ZdB(t),(x,t)∈Q,\nz(0,t) = 0, z(1,t) = 0, t∈(0,T),\nz(x,T) =zT, x∈ M.(4.1)\nThen the following observability inequality holds.\nTheorem 4.1 There exist h0>0andCsuch that for all h≤h0, the following observability\ninequality holds\nE/integraldisplay\nM|z(0)|2≤C/parenleftBig\nlE/integraldisplay\nQ|Z|2dt+lE/integraldisplayT\n0/integraldisplay\nω∩M|z|2dt+e−C\nhlE/integraldisplay\nM|z|2/vextendsingle/vextendsingle/vextendsingle\nt=T/parenrightBig\n.\nfor any solution to system (4.1).\nProof.Let\nA=|a1|L∞\nlF(0,T;L∞\nh(M))+|a2|L∞\nlF(0,T;L∞\nh(M)).\nBy Theorem 3.1, it deduce that\nlE/integraldisplay\nQs3e2sφz2dt+lE/integraldisplay\nQ∗se2sφ|Dhz|2dt\n≤C/bracketleftBig\nlE/integraldisplay\nω∩Ms3e2sφz2dt+lE/integraldisplay\nQs2e2sψ|Z|2dt+A2lE/integraldisplay\nQe2sφ(|z|2+|Z|2)dt\n+h−2lE/integraldisplay\nMe2sφ|z|2/vextendsingle/vextendsingle/vextendsingle\nt=0+h−2lE/integraldisplay\nMe2sφ|z|2/vextendsingle/vextendsingle/vextendsingle\nt=T/bracketrightBig\n.\nAfter choosing a sufficiently large λ, we fix parameters λ. It follows that\nlE/integraldisplay\nQθ3e2sφz2dt+lE/integraldisplay\nQ∗θe2sφ|Dhz|2dt\n≤C/bracketleftBig\nlE/integraldisplay\nω∩Mθ3e2sφz2dt+lE/integraldisplay\nQθ2e2sφ|Z|2dt+h−2lE/integraldisplay\nMe2sφ|z|2/vextendsingle/vextendsingle/vextendsingle\nt=0\n+h−2lE/integraldisplay\nMe−2sφ|z|2/vextendsingle/vextendsingle/vextendsingle\nt=T/bracketrightBig\n.\nRecallingthedefinitionof θ,wehaveθ(t)≥θ(T\n2)≥T−2,since0<δ<1\n2,andθ≤θ(T\n4)≤16\n3T2\nfort∈[T\n4,3T\n4]. Then\nlE/integraldisplay\nQθ3e2sφz2dt≥lE/integraldisplay3T\n4\nT\n4/integraldisplay\nMθ3e2sφz2dt≥Ce−CT−2lE/integraldisplay3T\n4\nT\n4/integraldisplay\nMz2dt. (4.2)\n18On the other hand, notice that d(|z|2) = 2zdz+|dz|2. Hence, for any 0 ≤t1≤t2≤T,\nE/integraldisplay\nM|z(t2)|2−E/integraldisplay\nM|z(t1)|2\n=E/integraldisplayt2\nt1/integraldisplay\nM/braceleftBig\n2z/bracketleftBig\n−D2\nhz−(a1z+azZ)/bracketrightBig\n+|Z|2/bracerightBig\ndt\n=E/integraldisplayt2\nt1/integraldisplay\nM/bracketleftBig\n2|Dhz|2−2a1|z|2−2a2zZ+|Z|2/bracketrightBig\ndt\n≥ −E/integraldisplayt2\nt1/integraldisplay\nM/bracketleftBig\nC√\nA|z|2+CA|z|2/bracketrightBig\ndxdt\n≥ −C(1+A)E/integraldisplayt2\nt1/integraldisplay\nM|z|2dt.\nBy the Gronwall inequality, it holds that\nE/integraldisplay\nM|z(t1)|2≤CE/integraldisplay\nM|z(t2)|2,0≤t1≤t2≤T. (4.3)\nNoting that θ(T) =θ(0)≥2\n3(δT2)−1, then\nh−2lE/integraldisplay\nMe2sφ|z|2/vextendsingle/vextendsingle/vextendsingle\nt=T+h−2lE/integraldisplay\nMe2sφ|z|2/vextendsingle/vextendsingle/vextendsingle\nt=T\n≤Ch−2e−C\nδT2/bracketleftBig\nlE/integraldisplay\nM|z|2/vextendsingle/vextendsingle/vextendsingle\nt=0+lE/integraldisplay\nM|z|2/vextendsingle/vextendsingle/vextendsingle\nt=T/bracketrightBig\n≤Ch−2e−C\nδT2lE/integraldisplay\nM|z|2/vextendsingle/vextendsingle/vextendsingle\nt=T.(4.4)\nMoreover,θ(t)≥θ(T\n2)≥T−2, we conclude that\nlE/integraldisplay\nω∩Mθ3e2sφz2dt+lE/integraldisplay\nQθ2e2sφ|Z|2dt\n≤e−C\nT2lE/integraldisplay\nω∩Mz2dt+e−C\nT2lE/integraldisplay\nQ|Z|2dt.(4.5)\nCombining (4.2)–(4.5), we have\nE/integraldisplay\nM|z|2(0)≤CTE/integraldisplay3T\n4\nT\n4/integraldisplay\nM|z|2(x,t)dt≤CeCT−2lE/integraldisplay\nQθ3e−2sϕz2dt\n≤C/parenleftBig\nlE/integraldisplay\nω∩Mz2dt+lE/integraldisplay\nQ|Z|2dt/parenrightBig\n+Ch−2e−(C\nδT2−C\nT2)lE/integraldisplay\nM|z|2/vextendsingle/vextendsingle/vextendsingle\nt=T\n≤C/parenleftBig\nlE/integraldisplay\nω∩Mz2dt+lE/integraldisplay\nQ|Z|2dt/parenrightBig\n+Ch−2e−C\nδT2lE/integraldisplay\nM|z|2/vextendsingle/vextendsingle/vextendsingle\nt=T,\nforδ≤δ0, whereδ0is sufficiently small. Notice that we need h < h 0andλh(δT2)−1≤ǫ0,\nwhereλis fixed, then let h1=ǫ0δ0T2\nλ. For every h≤min{h0,h1}, we choose δ=h\nh1δ0≤δ0,\n19thenλh(δT2)−1=ǫ0. Therefore,\nE/integraldisplay\nM|z|2(0)≤C/parenleftBig\nlE/integraldisplay\nω∩Mz2dt+lE/integraldisplay\nQ|Z|2dt/parenrightBig\n+Ch−2e−C\nhlE/integraldisplay\nM|z|2/vextendsingle/vextendsingle/vextendsingle\nt=T/parenrightBig\n.\nThis completes the proof of Theorem 4.1.\n5 The controllability for the stochastic semi-discrete\nparabolic equation\nThis section is devoted to the proof of our controllability result, The orem 1.1. The proof is\nalmost standard. However, we give the details for the sake of comp leteness.\nProof of Theorem 1.1. For anyy0∈L2\nh(M), define a functional J(·) onL2\nFT(Ω,L2\nh(M))\nas follows:\nJ(zT) =1\n2lE/integraldisplay\nQ|Z|2dt+1\n2lE/integraldisplayT\n0/integraldisplay\nω∩M|z|2dt+ǫ\n2lE/bardblzT/bardbl2\nL2\nh(M)−lE/a\\}bracketle{ty0,z(0)/a\\}bracketri}htL2\nh(M),\nwhereǫ=e−C\nh, and (z,Z) is the solution to (4.1) with final value zT. It is a simple matter\nto see that the functional J(·) is continuous and convex. Now we prove that it is coercive.\nBy Young’s inequality and Theorem 4.1, we have\nJ(zT)≥1\n2lE/integraldisplay\nQ|Z|2dt+1\n2lE/integraldisplayT\n0/integraldisplay\nω∩M|z|2dt+ǫ\n2/bardblzT/bardbl2\nL2\nh(M)\n−ε/bardblz(0)/bardbl2\nL2\nh(M)−1\nε/bardbly0/bardbl2\nL2\nh(M)\n≥1\n4lE/integraldisplay\nQ|Z|2dt+1\n4lE/integraldisplayT\n0/integraldisplay\nω∩M|z|2dt+ǫ\n4/bardblzT/bardbl2\nL2\nh(M)−C/bardbly0/bardbl2\nL2\nh(M).\nThen,J(·) is coercive. Therefore, J(·) admits a unique minimizer z∗\nT. Denote (z∗,Z∗) the\nsolution to (4.1) with final value z∗\nT. Then we have\nlE/integraldisplay\nQZZ∗dt+lE/integraldisplayT\n0/integraldisplay\nω∩Mzz∗dt+ǫlE/a\\}bracketle{tzT,z∗\nT/a\\}bracketri}htL2\nh(M)−lE/a\\}bracketle{ty0,z(0)/a\\}bracketri}htL2\nh(M)= 0,\nforanyzT, withthe associated solution( z,Z) to (4.1). We set u∗=−z∗χω∩Mandv∗=−Z∗.\nWe claim that (4.1) with u=u∗andv=v∗satisfy our needs. By the duality of (4.1) with\n(1.4), we have\nlE/integraldisplay\nMy(T)zT−lE/integraldisplay\nMy0z(0) = lE/integraldisplay\nQvZdt+lE/integraldisplayT\n0/integraldisplay\nω∩Mzudt.\nTheny(T) =ǫz∗\nT. On the other hand,\nlE/integraldisplay\nQ|Z∗|2dt+lE/integraldisplayT\n0/integraldisplay\nω∩M|z∗|2dt+ǫlE/bardblz∗\nT/bardbl2\nL2\nh(M)= lE/a\\}bracketle{ty0,z∗(0)/a\\}bracketri}htL2\nh(M)\n≤1\n4εlE/integraldisplay\nM|y0|2dt+εlE/integraldisplay\nM|z∗(0)|2dt.\n20Notice that\nlE/integraldisplay\nM|z∗(0)|2dt≤C/parenleftBig\nlE/integraldisplay\nQ|Z∗|2dt+lE/integraldisplayT\n0/integraldisplay\nω∩M|z∗|2dt+ǫlE/integraldisplay\nM|z∗|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=T/parenrightBig\n.\nTherefore\nlE/integraldisplay\nQ|Z∗|2dt+lE/integraldisplayT\n0/integraldisplay\nω∩M|z∗|2dt≤ClE/integraldisplay\nM|y0|2dt,\nand\nǫlE/bardblz∗\nT/bardbl2\nL2\nh(M)≤ClE/integraldisplay\nM|y0|2dt.\nHence\nlE/integraldisplay\nQ|v|2dt+lE/integraldisplayT\n0/integraldisplay\nω∩M|u|2dt≤ClE/integraldisplay\nM|y0|2dt,\nand\nlE/bardbly(T)/bardbl2\nL2\nh(M)≤CǫlE/integraldisplay\nM|y0|2dt.\nThen, we complete the proof.\nReferences\n[1] D. Allonsius, F. Boyer, and M. Morancey, Spectral analysis of discrete elliptic operators\nand applications in control theory. Numer. Math., 140(4) (2018), 857–911.\n[2] F. Boyer, F. Hubert, and J. Le Rousseau, Discrete Carleman estimates for elliptic\noperators and uniform controllability of semi-discretize d parabolic equations. J. Math.\nPures Appl., 93(2010), 240–276.\n[3] F. Boyer, F. Hubert, and J. Le Rousseau, Discrete Carleman estimates for elliptic\noperators in arbitrary dimension and applications, SIAM J. Control Optim., 48(8),\n(2010), 5357–5397.\n[4] F. Boyer and J. Le Rousseau, Carleman estimates for semi-discrete parabolic operators\nand application to the controllability of semi-linear semi -discrete parabolic equations.\nAnn. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 31(2014), 1035–1078.\n[5] C. Carthel, R. Glowinski, and J.-L. Lions, On exact and approximate boundary control-\nlabilities for the heat equation: a numerical approach. J. Optim. Theory Appl., 82(3),\n(1994), 429–484.\n[6] E. Cerpa, R. Lecaros, T. N. T. Nguyen, and A. P´ erez, Carleman estimates and control-\nlability for a semi-discrete fourth-order parabolic equat ion.J. Math. Pures Appl., 164\n(2022), 93–130.\n21[7] E. Fern´ andez-Cara and A. M¨ unch, Numerical null controllability of semi-linear 1-D heat\nequations: fixed point, least squares and Newton methods. Math. Control Relat. Fields,\n2(3), (2012), 217–246.\n[8] E. Fern´ andez-Cara and A. M¨ unch, Numerical exact controllability of the 1D heat equa-\ntion: duality and Carleman weights. J. Optim. Theory Appl., 163(1), (2014), 253–285.\n[9] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture\nNotes Series 34, Seoul National University, Seoul, 1996.\n[10] X. Fu andX. Liu, Controllability and observability of some stochastic comp lex Ginzburg-\nLandau equations, SIAM J. Control Optim., 55(2017), 1102–1127.\n[11] X. Fu, Q. L¨ u and X. Zhang, Carleman Estimates for Second Order Partial Differential\nOperators and Applications, Springer, 2019.\n[12] S. Labb´ e and E. Tr´ elat, Uniform controllability of semidiscrete approximations o f\nparabolic control systems, Syst. Control Lett., 55(7) (2006), 597–609.\n[13] A. Lopez and E. Zuazua, Some new results related to the null controllability of the 1 -d\nheat equation, In S´ eminaire sur les ´Equations aux D´ eriv´ ees Partielles, 1997–1998, pages\nExp. No. VIII, 22. ´Ecole Polytech., Palaiseau, 1998.\n[14] Q. L¨ uand X. Zhang, Mathematical control theory for stochastic partial differe ntial equa-\ntions,Probab. Theory Stoch. Model., Springer, Cham, 2021.\n[15] A. M¨ unch and E. Zuazua, Numerical approximation of null controls for the heat equa-\ntion: ill-posedness and remedies. Inverse Problems, 26(8), (2010), 085018.\n[16] T. Nguyen, Carleman estimates for semi-discrete parabolic operators with a discontinu-\nous diffusion coefficient and applications to controllabilit y,Math. Control Relat. Fields,\n4(2) (2014), 203–259.\n[17] S. Tang and X. Zhang, Null controllability for forward and backward stochastic p arabolic\nequations ,SIAM J. Control Optim., 48(2009), 2191–2216.\n[18] E. Zuazua, Propagation, observation, and control of waves approximat ed by finite dif-\nference methods , SIAM Review, 47(2) (2005), 197–243.\n[19] E. Zuazua, Control and numerical approximation of the wave and heat equ ations, In\nInternational Congress of Mathematicians. Vol. III, pages 1389 –1417. Eur. Math. Soc.,\nZ¨ urich, 2006.\n22"    },    {        "title": "2402.12661v2.Dynamics_of_Symmetry_Protected_Topological_Matter_on_a_Quantum_Computer.pdf",        "content": "Dynamics of Symmetry-Protected Topological Matter on a Quantum Computer\nMiguel Mercado,1,∗Kyle Chen,2Parth Hemant Darekar,3, 4Aiichiro Nakano,5Rosa Di Felice,1, 6and Stephan Haas1\n1Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484\n2Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL 60637\n3Department of Electrical and Computer Engineering,\nUniversity of Southern California, Los Angeles, CA 90089-0484\n4Department of Electrical and Computer Engineering,\nUniversity of Maryland, College Park, Maryland 20742\n5Collaboratory for Advanced Computing and Simulations,\nUniversity of Southern California, Los Angeles, CA 90089-0484\n6CNR Institute of Nanoscience, Modena, Italy\n(Dated: March 15, 2024)\nControl of topological edge modes is desirable for encoding quantum information resiliently against\nexternal noise. Their implementation on quantum hardware, however, remains a long-standing prob-\nlem due to current limitations of circuit depth and noise, which grows with the number of time steps.\nBy utilizing recently developed constant-depth quantum circuits in which the circuit depth is in-\ndependent of time, we demonstrate successful long-time dynamics simulation of bulk and surface\nmodes in topological insulators on noisy intermediate-scale quantum (NISQ) processors, which ex-\nhibits robust signatures of localized topological modes. We further identify a class of one-dimensional\ntopological Hamiltonians that can be readily simulated with NISQ hardware. Our results provide\na pathway towards stable long-time implementation of interacting many-body systems on present\nday quantum processors.\nIntroduction. —Understanding topological properties\nof matter is a current frontier of physics [1–7]. Chiral\nphenomena, in particular, have been ubiquitously sought\nafter in recent years for their novel emergent behaviors\nwhich have been observed, for example, in acoustic and\nmechanical systems [8–14], photonics [15–19], and mag-\nnetic materials [20–25]. Symmetry-protected topologi-\ncal (SPT) phases, the paradigmatic models for realizing\nsuch chiral states, are characterized by their protection\nvia global symmetries which have become well-studied in\nthe space-domain [26–31]. A consequence of topological\nsymmetry protection is the predicted robustness of SPT\nphases against local noise and perturbations, opening\navenues for foundational research and numerous device\napplications in quantum science [32–36], such as novel\nhigh-performance transistors [37–39], quantum sensors\n[40, 41], and protected room-temperature transport de-\nvices [42–45].\nDespite this enticing context, thorough investigation\nof chiral topological modes within the time-domain curi-\nously remains ambiguous, especially within the context\nof open system time evolution [46, 47]. Understanding\nthe role of topology in the dynamics of quantum sys-\ntems therefore, is a timely endeavor, which - however -\nhas been exceptionally challenging to realize and control\ndue to the inherent difficulty of accessing sensitive many-\nbody quantum states [48–50].\nAs intrinsically quantum platforms, a pioneering appli-\ncation of quantum computers is the physical simulation\nof many-body systems [51–55], where in special-use cases\npractical quantum advantage has recently been shown\n∗momercad@usc.eduto be achievable [56]. Such is the potential of quantum\nsimulation that topological physics is currently being ex-\nplored using photonic simulators [57–61], ultracold atoms\n[9, 62–66], hybrid-quantum simulation [67], and periodi-\ncally driven quantum simulators [68–70].\nWhile these provide a promising route to investigate\ncondensed matter systems, many-body interactions that\nwould otherwise be intractable for analog quantum sim-\nulators are well suited for digital quantum simulation,\nwhich provides universal capability to realize any finite-\ndimensional local Hamiltonian using sequences of quan-\ntum circuits [55, 71, 72]. Although topological states\nhave been prepared and observed using digital quantum\nsimulation [73–79], the analysis of transient behavior of\ntopological modes in the time-domain presents a partic-\nular set of challenges. A primary reason is the effect of\nread-out noise and gate-error rates, which have sharply\nlimited reliable quantum simulation of interacting many-\nbody systems [80].\nThe computational power and effectiveness of current\nnoisy intermediate-scale quantum (NISQ) computers is\nrestricted by the number and quality of qubits. In fact,\nNISQ-era qubits have short coherence times and high\ngate-error rates [81], which demand the requirement for\nnoise mitigation and error correction schemes. Moreover,\nquantum simulations for generic Hamiltonians require\nthat the circuit depth scales linearly at minimum with\nthe number of simulation time steps, according to the\nNo-Fast-Forwarding Theorem [82–84]. These restraints\nmake prolonged time evolution of many-body systems\nexceedingly challenging for conventional approaches.\nTo date, a reliable method to realize the long-time dy-\nnamics of topological phases of matter using digital quan-\ntum simulation is not available due to the aforementionedarXiv:2402.12661v2  [quant-ph]  14 Mar 20242\nlimitations. Prior simulations are largely restricted in\nthe number and width of time-step intervals, making the\nanalysis of transient dynamics, which is non-negligible\nwhen interaction with the environment is present, un-\nreachable.\nIn this Letter , we introduce a method to realize the\nlong-time quantum dynamics of topological matter us-\ning contemporary quantum hardware, positioning NISQ\ncomputation as a favorable environment to probe novel\nphases of matter. In comparison to previous methods\n[73–79], which have been limited in the total time dura-\ntion or restricted entirely to the space-domain, our proto-\ncol enables the quantum simulation of topological states\nin one spatial dimension nominally up to arbitrarily long\ntimes. Our one-dimensional protocol can be extended to\nhigher dimensions with larger system implementations\nand mathematical developments—we call to attention a\nsubset of Hamiltonians with novel topological properties\nthat are now tractable with this result. Its ability to pre-\ndict physical results does not rely on employing readout\nerror mitigation or post-processing techniques, though\nthe output precision can be complemented by such tech-\nniques. By engineering interactions between qubits, we\nimplement and analyze three different topologically non-\ntrivial states. Specifically, we construct the desired topo-\nlogical phases via staggered coupling, using a topolog-\nical mirror, and introducing a topological defect. We\nutilize the time-step compression of recently developed\nconstant-depth quantum circuits [85], which possess the\nkey property that the depth required to execute a quan-\ntum simulation is constant with respect to the number\nof time steps compared to linear or exponential. This in-\nherent feature of matchgates enables the construction of\ntopological Hamiltonians as quantum circuits to be sim-\nulated for arbitrary time. This is achieved because the\nconstituents of constant-depth circuits, matchgates, have\nthe symmetric property that the product of two match-\ngates is itself a matchgate, allowing for the decomposi-\ntion into native gates with only two CNOT gates, while\nconventionally three CNOT gates are needed. This re-\nduction enables the Trotter error to be made negligible\nby breaking the simulation down into small time steps.\nWe subsequently analyze the dynamics of bulk and edge\nmodes, and observe that the topological modes are co-\nherently stabilized in each case using our method, where\ntransient data is clearly visible.\nDynamics of topological insulators as constant-depth\ncircuits. —We simulate the time evolution of coupled SPT\ninsulator systems using quantum circuits. In this ap-\nproach, the non-trivial topology is introduced via locally\nvarying nearest-neighbor couplings between lattice sites\nthat host magnetic dipoles.\nWe implement the topological states using a 1D Quan-\ntum Spin Hamiltonian, given by\nH=−N−1/summationdisplay\ni=1Jz,iZiZi+1−hxN/summationdisplay\ni=1Xi, (1)\nFIG. 1. Schematic illustration of topological insulator mod-\nels with alternating couplings JandJ′(> J), realized in this\nwork on quantum processors with five qubits and extendable\nto larger odd numbers of qubits. (a) Spin-1/2 chain with a\ntopological surface mode observed on the first qubit. (b) Spin-\n1/2 chain with topological surface modes on the first and fifth\nqubits. (c) Spin-1/2 chain with a topological defect mode on\nthe central qubit.\nwhere Zi,Xidenote the spin-1/2 Pauli zandxma-\ntrices acting on qubit i,Jz,idenotes the z-direction cou-\npling between qubits iandi+ 1, and hxis an external\nmagnetic field in the xdirection. By particular choices of\nJz,i, the signature bulk and edge modes of topological in-\nsulator systems emerge and are tunable; we extensively\nexamine multiple cases. We note that while Eq. (1)\npresents a quantum Hamiltonian with couplings between\nqubits along the zdirection and external field in the x\ndirection, the model generalizes to couplings and fields\nalong any direction, where nearest neighbor interactions\nare confined to two or less directions, and the external\nfield acts along one perpendicular direction.\nA schematic of the topological configurations is given\nin Fig. 1. In general, when the coupling J′is greater than\nJ, the system enters a topologically nontrivial regime,\nand exhibits protected topological modes that exist on\nthe weakly-coupled edges of the system [28, 86–93]. Fig.\n1(a) shows a topological edge mode, which can be cre-\nated and protected via alternating weak couplings Jand\nstrong couplings J′in a chain. Fig. 1(b) shows a mirror\nconfiguration, which can be created by using weak cou-\nplings Jconnecting the qubits on the edge, and strong\ncouplings J′connecting the “bulk” qubits, thus creating\nedge modes at the two ends of the chain. Fig. 1(c) illus-\ntrates a topological defect, which can be created in the\ncenter of the chain by choosing strong couplings J′at the3\nFIG. 2. Building constant depth circuits for a 5-qubit sys-\ntem. (a) Matchgate symmetry is enforced on a set of match-\ngates outlined in bold. (b) Pairs of matchgates are mapped to\nsingles using the SU(2) downfolding property of matchgates.\nProcess is repeated for remaining matchgates in each set as\ndenoted by the dotted line. (c) New circuit with number of\nmatchgate columns reduced by one is produced.\nedges and weak couplings Jconnecting the bulk qubits.\nAs a special class of quantum circuits, constant-depth\ncircuits have a constant scaling rate in circuit depth with\ntime step, as compared to conventional circuits or low-\ndepth circuits which scale linearly or exponentially. This\nclass of Hamiltonians thus embodies a nontrivial excep-\ntion to the No-Fast-Forwarding Theorem [84, 94]. The\ntime-step compression of constant-depth circuits follows\ndirectly from their intrinsic matchgate property, which\nutilizes symmetry to downfold the total amount of cir-\ncuits needed to construct quantum algorithims on native\ngates.\nAccording to this property, the product of two match-\ngates is also a matchgate, constituting an SU(2) Lie Al-\ngebra [85]. This allows the decomposition of a match-\ngate into native gates, which conventionally requires\nthree CNOT gates, to be executed with only two CNOT\ngates with constant-depth circuits. Therefore, N(N−1)\nCNOT gates are required to simulate a system with N\nspins. This compression in resource requirement allows\nthe Trotter error, which is proportional to the size of the\ntime step, to be negligible, by breaking down the simu-\nlation into miniature time steps which can be repeated\nmany times. We elaborate on this method in Sec. II of\nthe Supplementary Material.\nIn Fig. 2, we illustrate the construction of constant-\ndepth circuits for 5-qubit spin systems. We map SPTinsulators into constant-depth circuits, splitting the total\nevolution into discretized time steps ∆ t= 0.1 in units of\nthe inverse applied magnetic field ℏ/hxfollowing the con-\nvention ℏ= 1. In principle, one can continually reduce\nthe Trotter error by breaking down the total simulation\ntime into increasingly smaller time-step intervals.\nIn order to generate the circuits for these systems, we\nemploy numerical optimization to identify optimal circuit\nparameters. This is accomplished by first computing the\ntime evolution operator associated with each topological\nspin system, which defines a target matrix. After con-\nstructing the constant-depth circuit structure, which con-\ntains Nmatchgate columns for a system with Nqubits,\nwe compute its corresponding circuit matrix. Then, we\ngenerate optimized constant-depth circuits by minimiz-\ning the distance between the target and circuit matrix.\nThis procedure is detailed in Sec. III of Supplementary\nMaterial. We point out that generation of constant-depth\ncircuits can be achieved independently of the optimiza-\ntion method used.\nRealization on quantum hardware. —We implement the\ntime-evolution of SPT insulator models using an IBM\ntransmon-based quantum computer [95], demonstrating\nthat topological physics can be coherently observed with\nibmq manila , a quantum processor with five qubits. Our\nfindings highlight that large system size is not required to\nobserve topological phenomena. On the other hand, the\nsystem size can be expanded to explore open quantum\nlattice systems. We compare our results with classical\nsimulations, using Exact Diagonalization to numerically\nsolve the closed-system dynamics as ground truth (Sec.\nIV of the Supplementary Material).\nThe IBM hardware results for the topological mirror\ncase in Fig. 1(b) are shown in Fig. 3, where the z-\ndirection magnetization is reported for each qubit as a\nfunction of time, over the total open system evolution.\nPresented in the left panel of Fig. 3 is the real-space\ndata obtained from the quantum hardware, while the\nright panel of Fig. 3 shows the corresponding power\nspectrum in frequency space generated by performing a\nFourier Transform.\nFor this configuration, we expect strongly localized\ntopological surface modes on both open ends of the quan-\ntum spin chain. Furthermore, since there is spatial mir-\nror symmetry in the system parameters, qubits 1 and 5\nshould be equivalent, as well as qubits 2 and 4.\nIndeed, the result obtained from the quantum hard-\nware accurately corroborates these expectations. A clear\nsignature of topological surface modes is evidenced by\nthe strong temporal oscillations of the magnetization in\nqubits 1 and 5, translating into sharp low energy modes\nin the power spectrum, indicated by arrows in Fig. 3.\nTheir frequency depends on the choice of Hamiltonian\nparameters J, J′, hx, whereas their lifetime is charac-\nterized by the damping rate γas in a fitting formula,\nm(t)∝exp(−γt). These topological mode signatures are\nthe salient feature preserved amidst noise in the hardware\nresults. Constant-depth circuits have stabilized the SPT4\nFIG. 3. Local dynamics of a 5-qubit Hamiltonian in a topologically non-trivial regime, illustrated in Fig. 1(b), obtained by\nNISQ processors. J= 2,J′= 4,hx= 1. Left: real-space magnetization in the z-direction for each qubit. Transient dynamics\nis highlighted with orange lines. Black stars indicate the noisy finite system equilibriation times for each qubit (defined in Sec.\nIIID of the Supplementary Material). Time is reported in units of the inverse magnetic field 1 /hx. Right: Fourier-transformed\npower spectrum in frequency space. Sharp surface modes are clearly visible on the first and fifth qubits. Frequency is reported\nin units of the magnetic field hx.\nmodes, producing longer-lived states at the edge of the\nsystem, which has not been observable with conventional\nquantum circuits (see Supplementary Fig. S13).\nIn contrast, the“bulk” qubits 2, 3, and 4 do not display\nany oscillations, in analogy with the equivalent closed\nquantum system dynamics [48] discussed in Sec. IV of\nthe Supplementary Material. Furthermore, we observe\nsite dependent transient dynamics in the time window\nt∈[0,4], with the odd site qubits settling into their\nsteady state markedly earlier than the even sites, as in-\ndicated by the black dots in the left panel of Fig. 3, which\nwere obtained using the criteria for equilibriation times of\nnoisy finite systems outlined in Ref. [96], namely, when\nthe expectation value of an observable approaches its av-\nerageAand begins fluctuating about it with fluctuations\nδA=/radicalig\nA2−A2. The orange lines are obtained by fit-\nting the exponential damping function to the short-time\ndata, indicating with the black star where this transient\nfunctional dependence does not apply anymore. Finally,\nthe observed large DC (zero energy) feature in the power\nspectrum is due to the polarization of the qubit spins\nalong the z-direction.\nMoving to the other topological configurations shown\nin Figs. 1(a) and 1(c), we also find agreement between\nreal-time evolution on NISQ hardware and the equivalent\nclosed system quantum dynamics. Specifically, in the\nstaggered coupling model of Fig. 1(a), a sharp amplitude\nin Fourier space is detected on the first qubit, which is\nweakly coupled with the bulk, indicating a long lived\ntopological surface mode. In the topological defect case\nof Fig. 1(c), a strongly localized state with temporallow-frequency oscillation is observed on the center qubit,\n(Supplementary Material Sec. III).\nWe emphasize that the presented results are mea-\nsured data on NISQ devices, without any noise mit-\nigation or error correction techniques. Despite this,\nthe constant-depth method measurably outperforms con-\nventional Trotterization, in which each qubit undergoes\nrapid decay where noise scrambles discernable signatures\nin the data. Our method demonstrates that coherent dy-\nnamics of novel phases of matter can be realized at the\ncircuit-level using minimum resource requirements, over-\ncoming the known challenges of noise and limited system\nsize in NISQ simulation protocols.\nFast-forwardable topological models. —The results re-\nported here are general and can be utilized to realize\nthe long-time dynamics for a host of topological sys-\ntems. We illustrate how to apply our method to probe\ntopological phenomena in open systems and highlight\na notable subset of models, pointing out experimen-\ntally accessible procedures to control non-trivial states.\nFirst, one can straightforwardly realize protected modes\nby manufacturing corresponding symmetries of a de-\nsired Hamiltonian Hthrough artificial design of cou-\nplings J, demonstrated by models depicted in Fig 1.\nParticularly, the presence or absence of chiral symme-\ntryCdetermines the existence of topological modes in\nSPT phases [28, 93, 97–99], which can be further catego-\nrized into chiral-symmetric Hamiltonians that maintain\nor break combinations of time-reversal Tsymmetry and\nparticle-hole (charge-conjugation) Psymmetry. It has\nbeen proven that quadratic Hamiltonians can be fast-5\nforwarded [84], meaning that a system with Nspins re-\nquires only N(N−1) CNOT gates to simulate. This fol-\nlows from the fact that matchgates for quadratic Hamil-\ntonians can be decomposed into native circuits requiring\ntwo CNOT gates. For further technical details, see Sup-\nplementary Material Sec. II.\nTherefore, one can completely tune the presence and\ncharacter of topological modes through our method by\nmodulating interactions Jon two or less axes to real-\nize discrete symmetries in Hof particular interest, such\nas chiral symmetry C, which is then decomposed into\nmatchgates. This can be written generally as H=c†Hc,\nwhere\nH=/parenleftbigg\nA B\n−B∗−A∗/parenrightbigg\n, (2)\nwith arbitrary matrices A,Bsatisfying A=A†and\nBT=−B, and c†\njandcjdenoting the fermionic cre-\nation and annihilation operators which define the col-\numn vector operator c= (c1. . . cnc†\n1. . . c†\nn)T. obeying\nU−1\nCHUC=−H, where UCis defined as the chiral sym-\nmetry operator, C=P · T .\nWe argue that this opens many interesting directions\nin one dimension, particularly in studying open system\ndynamics of models in the AIII, BDI, CI, and DIII uni-\nversality classes [98]. In higher dimensions, it has been\nshown that n-band topological insulator Hamiltonians\ncan be mapped to 1D hardcore boson chains by rep-\nresenting their single-particle hopping as an n-particle\nhopping living on distinct sublattices of a 1D chain as\nshown in Ref. [77], which can be applied to generalize\nour method through application of matchgate symme-\ntries. We point out that it has recently been shown that\ncertain number-conserving bosonic Hamiltonians can also\nbe fast-forwarded [94]. Another procedure to probe topo-\nlogical modes is to induce spatial localization within the\nlattice by modulating J. This is exemplified by the model\nin Fig. 1(c), which realizes a topological defect which\ncan be interpreted as a domain wall in the center of the\nsystem between two configurations [100, 101], we further\nelaborate on this process in Sec. IC of the Supplementary\nMaterial.\nDiscussion. —We introduced a method that greatly im-\nproves dynamical simulations of quantum systems us-\ning contemporary quantum processors. By utilizing the\ntime-step compression of constant-depth circuits, thetransient dynamics of novel many-body systems is achiev-\nable up to arbitrary long times, which to date has re-\nmained one of the major challenges of the NISQ era.\nUsing this approach, we showed how open system\nquantum dynamics can be stabilized by exploiting topo-\nlogical properties of matter. While conventional qubits\nsuffer from rapid decoherence, one can create much\nlonger lived qubits through utilizing the topologically sta-\nbilized surface states as shown in our study. These long\nlived states present a host of potential applications. For\nexample, one may use such topologically stabilized states\nto create robust and highly local quantum sensing devices\nthrough tuning the magnetic field hxand couplings Jand\nJ′closer relative to each other to induce a critical state\n[102]. In this state, the qubits have higher sensitivity,\nwhich combined with the advantage of being longer-lived\ncan potentially be applied as a quantum sensor. Alter-\nnatively, one can tune hxfarther than JandJ′to maxi-\nmize the contrast between JandJ′. In principle, tuning\nthe system as such will increase the effectiveness of using\ntopological modes as qubits with longer-lived coherence\ntimes.\nThe constant-depth quantum circuits method provides\na clear pathway towards executing long-time dynami-\ncal simulations of exotic interacting many-body systems.\nWhile in this study we demonstrate the versatility of\nthis approach by executing dynamics of simple paradig-\nmatic topological models with relatively small resource\nrequirements, one can also apply the method to study\nlong-time dynamics of more complex cases, such as in\nAubry-Andr´ e-Harper models [103–108], generalized Su-\nSchrieffer-Heeger models [109–111], and Kitaev chains\n[32, 112]. Another promising future direction is dy-\nnamical studies of Higher Order Topological Insulators\n(HOTI)s [113–115], which are now tractable using this\nconstant-depth circuit implementation.\nAcknowledgements We thank Lindsay Bassman Of-\ntelie of CNR-NANO, Pisa, Italy for valuable discussion\nand insight. AN was supported by an NSF grant OAC-\n2118061. RDF was partially supported by the Defense\nAdvanced Research Projects Agency (DARPA), Contract\nNo. HR001122C0063. We acknowledge the use of IBM\nQuantum services for this work and to advanced services\nprovided by the IBM Quantum Researchers Program.\nThe views expressed are those of the authors, and do\nnot reflect the official policy or position of IBM or the\nIBM Quantum team.\n[1] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,\nRev. Mod. Phys. 88, 035005 (2016).\n[2] T. Ozawa, H. M. Price, A. Amo, N. Goldman,\nM. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Si-\nmon, O. Zilberberg, and I. Carusotto, Rev. Mod. 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Blatt, Science 345, 302 (2014).Supplementary Material: Dynamics of Symmetry-Protected Topological Matter on a\nQuantum Computer\nMiguel Mercado,1,∗Kyle Chen,2Parth Hemant Darekar,3, 4Aiichiro Nakano,5Rosa Di Felice,1, 6and Stephan Haas1\n1Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484\n2Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL 60637\n3Department of Electrical and Computer Engineering,\nUniversity of Southern California, Los Angeles, CA 90089-0484\n4Department of Electrical and Computer Engineering,\nUniversity of Maryland, College Park, Maryland 20742\n5Collaboratory for Advanced Computing and Simulations,\nUniversity of Southern California, Los Angeles, CA 90089-0484\n6CNR Institute of Nanoscience, Modena, Italy\n(Dated: March 15, 2024)\nI. TOPOLOGICAL LATTICE MODELS\nHere, we discuss the salient features of the models depicted in Figs. 1(a)-(c) of the main text and consider\nstraightforward methods to realize non-trivial topological degeneracy in interacting many-body systems.\nA. Staggered Coupling\nFirst, we design an N-site spin-1/2 molecule in analogy to the fermionic Su-Schrieffer Heeger model (SSH) [1], as\ndepicted in Fig. 1(a) of the main text. Since their inception, symmetry-protected topological (SPT) states of matter\n[2–7] have become emblematic due to their topological surface modes [8] which emerge in the presence of one or more\nglobal symmetries of the Hamiltonian. Physically, the soliton states of SPT phases can be interpreted as boundary\nstates between regimes with different ground states whose eigenenergies are within a gap between two bulk energy\nbands [9, 10]. In the 1d SSH model, the global chiral symmetry Cof the Hamiltonian plays this role, resulting in\nprotected surface states that reside at zero energy within an energy gap [4].\nThe 1d SSH model provides an intuitive picture for realizing such chiral states via dimerization, i.e., staggered\ncouplings between two different sublattices of the system. We emulate the physics of the SSH model in an N-site\nspin-1/2 molecule by introducing intra-cell (weak) Jand inter-cell (strong) J′couplings between unit cells, which is\nimplemented via alternating interactions between the spin-1/2 sites,\nH=−N−1/summationdisplay\ni=1Jµ\niσµ\niσµ\ni+1−hµN/summationdisplay\ni=1σµ\ni, (S1)\nwhere µrepresents axes which are specified along {x, y, z},σµ\niare the µPauli matrices acting on site i,Jµ\niis the\ninteraction between neighboring Pauli matrices along the axis µ, and hµis an external field acting along the µ\ndirection. One recovers the familiar Transverse-Field Ising (TFI) model when Jµ\niis chosen to be uniform along the\nentire lattice with an applied magnetic field hµacting in its transverse direction. We remark that Eq. S1 is not the\ndescription for a fully general spin chain that can be can implemented using the method presented in the main text.\nOne may, for example, select interactions Jµ\niacting on two different axes. We discuss this in further detail in Sec. II\nand Sec. III. In our study, we select interactions along the z-axis{Jz\ni, σz\ni}with an external magnetic field along the\nx-axis{hx, σx\ni}.\nTo create the model featured in Fig. 1(a) of the main text, we induce topological soliton states analogous to those\nof the SSH model by enforcing two distinct sublattices via dimerization in an open chain: Jz\n1= 2, Jz\n1= 4, Jz\n3= 2,\nJz\n4= 4,Jz\n5= 0, with hx= 1, noting that the last term J5\n1= 0 is due to the open boundary condition (OBC) necessary\nto host chiral edge states in 1d. To make our notation more intuitive, we define\n∗momercad@usc.eduarXiv:2402.12661v2  [quant-ph]  14 Mar 20242\n/braceleftigg\nJ=Jz\niwhen Jz\ni= 2\nJ′=Jz\niwhen Jz\ni= 4\nto recover the familiar weak couplings Jand strong couplings J′depicted in Fig. 1 of the main text. Spatially\nlocalized surface states are sharpened by increasing the contrast between JandJ′.\nThis spin model has two distinct sublattices which can be viewed as an analogous version of the fermionic intra-\ncell and inter-cell hopping amplitudes between each unit cell. When the inter-cell hopping amplitude between two\nunit cells overcomes the intra-cell hopping amplitudes between two spins in one unit cell, our model gives rise to a\nbound soliton spin state at zero energy along the weakly-coupled edge of the spin lattice which is supported by the\nclosed dynamics result discussed in Sec. IV. This state is localized within the gap between bulk energy eigenstates\nand is therefore protected against local perturbations such as the observed environmental noise affecting the bulk\nqubits; in principle the topological degeneracy of the system would be affected if environmental noise affects the\nhopping amplitudes in such a manner that the intra-cell hopping Jovercomes the inter-cell J′, which would close the\nenergy gap and hybridize the surface and bulk states—which according to Fig. 3 of the main text, is not observed.\nInterestingly, our result demonstrates that the effect of spatial localization induced by sublattice dimerization is visible\nwithin the time-evolution of a system with minimal lattice length, which was previously not well-studied.\nWe now briefly discuss the transverse magnetic field dependence. The ferromagnetic phase of the spin chain J > h x\npreserves the long-order (two-fold degeneracy) of the ground state, and thus the topological soliton zero state of the\nstaggered coupling is coherently observed in this regime, hence why we choose hx= 1. We note that increasing the\nmagnetic field J << h xlifts this degeneracy, and thus soliton states that originally result as a consequence of the\ntopological degeneracy of the model hence become indistinguishable during simulation, consistent with our control\nhardware tests.\nFinally, it is useful to analyze the topological modes from the viewpoint of symmetry as discussed in the main text.\nTopological protection of SPT phases arises from enforcing global symmetries of the Hamiltonian. The SSH-inspired\nmolecule in this case has chiral (sublattice) symmetry U−1\nCHUC=−H, which is responsible for the large degree of\nobserved spatial localization of the topological mode on qubit 1 of the staggered coupling model limit of Eq. S1.\nHowever, one may for example, design other global symmetries such as combinations of time-reversal and charge-\nconjugation symmetries, or crystalline symmetries [11]. Spin-1/2 models such as the one presented in the main text\ncan be related to fermionic/hardcore bosonic models through the non-local Jordan-Wigner transformation,\nσx\ni=i−1/productdisplay\nj(1−2nj)(c†\ni+ci) σy\ni=i−1/productdisplay\nj(1−2nj)(c†\ni−ci) σz\ni= (1−2ni) (S2)\nwhere c†\ni,cidenote the usual fermionic creation and annhilation operators and ni=c†\nicithe fermionic number\noperator acting on site i. We point out that the SSH model and Kitaev chain [12] obtained from applying the Jordan-\nWigner transformation to the TFI model fall within the same universality class [13], which is another way of viewing\nhow the presented model exhibits robust edge modes on weakly-coupled boundaries in the topologically non-trivial\nphase.\nB. Mirror Model\nBuilding from the model described in Sec. I A, we investigate topologies with more complex eigenvalue degeneracy\nand symmetry-breaking patterns. The topological mirror model featured in the main text is depicted by Fig. 1(b).\nWe define the mirror model HMas a topologically non-trivial limit of Eq. S1 with specified couplings Jz\n1= 2,Jz\n2= 4,\nJz\n3= 4,Jz\n4= 2,Jz\n5= 0, with hx= 1. We discuss the spectrum of this model below.\nConsider HMprepared in its initial state, before the quantum quench is performed, where hx= 0. Let us examine\nthe degeneracy of the initial phase of HMwhen ( t <0) to characterize the effect of enforcing the above couplings.\nSinceHMis a quadratic Hamiltonian, it is convenient to employ the fermionic Bogoliubov-de Gennes (BdG) approach\nto diagonalize HM[14–16].\nApplying the Jordan-Wigner transformation S2 to Hgives the fermionic Hamiltonian with OBC,\nHf=−N−1/summationdisplay\ni=1(Jz\nic†\nici+1+c†\nic†\ni+1+h.c.) +hxN/summationdisplay\ni=1(2c†\nici−1). (S3)3\nApplying the same transformation S2 to HMgives the analogous fermionic Hamiltonian with OBC in the absence of\nmagnetic field ( hx= 0),\nHf=−N−1/summationdisplay\ni=1(Jz\nic†\nici+1+c†\nic†\ni+1+h.c.). (S4)\nLet Ψ = ( c1. . . cnc†\n1. . . c†\nn)T, Ψ†= (c†\n1. . . c†\nnc1. . . cn) be the Nambu spinors which obey the usual fermionic anti-\ncommutation relations. We can rewrite our problem into the standard BdG form as\n⟨Hf\nM⟩= Ψ†HΨ = Ψ†/parenleftbigg\nA B\n−B∗−A∗/parenrightbigg\nΨ (S5)\nwhere His the 2 N×2NBdG block matrix containing matrices AandBsatisfying A=A†andBT=−B. It can be\nshown that in the case of the Ising model, AandBmust be real and symmetric matrices whose elements yield the\nfollowing structure for OBC [16]:\nAi,i=hxBi,i= 0 Ai,i+1=Ai+1,i=−Jz\ni\n2Bi,i+1=Bi+1,i=−Jz\ni\n2\nRecalling that we examine the initial state where hx= 0, we obtain the following matrix\nH=1\n2\n−J\n−J0−J′\n−J′0−J′\n−J′0−J\n−J0−J\nJ0−J′\nJ′0−J′\nJ′0−J\nJ 0\nJ\n−J0J′\n−J′0J′\n−J′0J\n−J0J\nJ0J′\nJ′0J′\nJ′0J\nJ0\n(S6)\nThe eigenvectors corresponding to the solution of Eq. S5 can be organized into a unitary matrix form\nUM=/parenleftbigg\nU V∗\nV U∗/parenrightbigg\nwhere the eigenvectors constitute the columns of the block matrices. To analyze the eigenspectrum obtained by\nsolving Eq. S5, define new Nambu fermion operators Φ = UM†Ψ, such that\nΦ =/parenleftbigg\nU†V†\nVTUT/parenrightbigg/parenleftbig\nc1. . . c nc†\n1. . . c†\nn/parenrightbigT\nThe key feature of the UMrepresentation is to extract the Nambu fermion operator pairs that emerge from writing\nthe linear combinations of the eigenvectors associated with each energy eigenvalue. Written in terms of Nambu fermion\noperators Φ, we obtain the following spectrum of our mirror model:4\nE1= 8 : Φ 1=−1√\n2c3−1\n2√\n2(c†\n2−c2) +1\n2√\n2(c†\n4+c4)\nE2= 8 : Φ 2=1\n2(c†\n2−c2) +1\n2(c†\n3+c3)\nE3= 4 : Φ 3=1\n2(c†\n3−c3) +1\n2(c†\n4+c4)\nE4= 4 : Φ 4=1\n2(c†\n4−c4) +1\n2(c†\n5+c5)\nE5= 0 : Φ 5=1√\n2(c†\n1+c1)\nE6=−8 : Φ 6=1√\n2c3−1\n2√\n2(c†\n2−c2) +1\n2√\n2(c†\n4+c4)\nE7=−8 : Φ 7=−1\n2(c†\n2−c2) +1\n2(c†\n3+c3)\nE8=−4 : Φ 8=−1\n2(c†\n3−c3) +1\n2(c†\n4+c4)\nE9=−4 : Φ 9=−1\n2(c†\n4−c4) +1\n2(c†\n5+c5)\nE10= 0 : Φ 10=1√\n2(c†\n5−c5)\nWe analyze the emergence of Nambu fermion operator pairs in the mirror model. The energies occupying the bulk\nmodes form pairs which are Hermitian conjugate to one another: Φ 2and Φ 7, Φ3and Φ 8, Φ4and Φ 9. However, we\nobserve the presence of unpaired operators: Φ 5and Φ 10which directly correspond to the known unpaired Majorana\nmodes localized at zero energies which live at the edges of the spin chain [12], and Φ 1and Φ 6, which we identify as\nforming a topological mirror point localized on the third qubit. One can explicitly check that the Hermitian conjugate\nof Φ 1or Φ 6differs from each other by the fermionic creation/annihilation operator c†\n3/c3occupying site 3 of the\nlattice.\nThe structure of the above spectrum is directly visible in the dynamics of the model. From our analysis, we find\nthree localized states: two at zero energy, and conversely, one far from zero energy occupying E = |8|which can be\nviewed as a topological mirror point. As expected, the two unpaired zero energy states which live on weakly linked\nedges are highly robust during open system time evolution due to protection via high degree of localization within\nthe energy gap induced by staggered mirror couplings. We remark that when the quench is performed, these localized\nstates will be shifted to a finite value instead of exactly zero. Conversely, the topological mirror point, which exists\nat energies far from zero, behaves as a bulk state experiencing minimal protection with weak temporal oscillations of\nthe magnetization. These predictions are consistent with the closed system result for the mirror model obtained in\nFig. S17.\nC. Topological Defect Model\nWe design a topological defect model HD, depicted by Fig. 1(c). Similarly, we define the defect model HDas a\ntopologically non-trivial limit of Eq. S1 with specified couplings Jz\n1= 4,Jz\n2= 2,Jz\n3= 2,Jz\n4= 4,Jz\n5= 0, with hx= 1.\nWe exactly follow the same procedure previously outlined in Sec. I B to analyze the corresponding spectrum of HD.\nWe solve the following BdG Hamiltonian obtained after performing the Jordan-Wigner transformation S2 of HDwith\nOBC before the quench is performed ( t >0),\n⟨Hf\nD⟩= Ψ†HΨ = Ψ†/parenleftbigg\nA B\n−B∗−A∗/parenrightbigg\nΨ (S7)5\nwhere His the 2 N×2NBdG block matrix containing matrices AandBsatisfying A=A†andBT=−B. We\nobtain the corresponding matrix Hfor the defect model\nH=1\n2\n−J′\n−J′0−J\n−J0−J\n−J0−J′\n−J′0−J′\nJ′0−J\nJ 0−J\nJ0−J′\nJ′0\nJ′\n−J′0J\n−J0J\n−J0J′\n−J′0J′\nJ′0J\nJ0J\nJ0J′\nJ′0\n(S8)\nFollowing the earlier procedure, the eigenvectors corresponding to the solution of Eq. S7 can be organized into the\nunitary block diagonal matrix form UDand can be analyzed by defining new Nambu fermion operators Φ = UDΨ\nwith the exact same structure as above. We obtain the following spectrum written in terms of the Nambu fermion\noperators:\nE1= 8 : Φ 4=1\n2(c†\n1−c1) +1\n2(c†\n2+c2)\nE2= 8 : Φ 2=1\n2(c†\n4−c4) +1\n2(c†\n5+c5)\nE3= 4 : Φ 3=1\n2(c†\n2−c2) +1\n2(c†\n3+c3)\nE4= 4 : Φ 1=−1√\n2c3−1\n2√\n2(c†\n2−c2) +1\n2√\n2(c†\n4+c4)\nE5= 0 : Φ 5=1√\n2(c†\n1+c1)\nE6=−8 : Φ 9=−1\n2(c†\n1−c1) +1\n2(c†\n2+c2)\nE7=−8 : Φ 7=−1\n2(c†\n4−c4) +1\n2(c†\n5+c5)\nE8=−4 : Φ 8=−1\n2(c†\n2−c2) +1\n2(c†\n3+c3)\nE9=−4 : Φ 6=1√\n2c3−1\n2√\n2(c†\n2−c2) +1\n2√\n2(c†\n4+c4)\nE10= 0 : Φ 10=1√\n2(c†\n5−c5)\nWe identify key features of this spectrum which underlie the observed dynamics. First, in agreement with Sec. I B,\nwe find the emergence of Nambu fermion pairs which classify the bulk modes Φ 1and Φ 6, Φ2and Φ 7, Φ3and Φ 8which\nare Hermitian conjugates of themselves. We find unpaired Nambu fermion operators Φ 5and Φ 10which correspond to\nthe unpaired Majorana modes found in the above models of Sec. I A and Sec. I B. Additionally, we find the Nambu\nfermion pairing Φ 4and Φ 9which are not Hermitian conjugate due to the creation/annihilation operator c†\n3/c3acting\non site three, which we identify in this case to correspond to the topological defect on qubit 3 in the center of the\nchain. Therefore, we find the existence three topological modes in this model: two on the ends of the chain, and one\ndefect mode in the center.\nNotably, one can characterize the difference in dynamics produced by the topological modes of HDwhen compared\nto the other two models discussed in Sec. I A and I B. In our case, the topological defect model can be viewed as a\ndomain wall between two topological dimer configurations which introduces a pair of localized states at both ends\nof the chain, which in the 5-qubit case results in the producing of three distinct surface modes [9, 10]. In contrast\nto the other two models, whose topological edge modes live on the weakly-coupled links, the two edge modes of the\ndefect model live on strongly coupled links and hence experience a smaller degree of localization which is reflected\nin their open system dynamics. For the topological defect mode created by the union of two weakly-coupled links at\nthe center of the model, we expect to observe signatures of topological localization due to their occupancy at energies\nnearer to zero, in contrast to the topological mirror point discussed in I B which exists farther from zero energy and\ntherefore no observable Fourier signature. This is accurately predicted by the closed system result in Fig. S18.6\nII. CONSTANT-DEPTH CIRCUITS CONSTRUCTION\nWe construct constant depth circuits [17] in the following manner. Since the simulated Hamiltonians are quadratic\ncontaining nearest neighbor interactions, they can be implemented in the form of two qubit gates acting on adjacent\nqubits. Using these gates, we construct the circuit for the time evolution of one time-step. As an example, for a\n5-qubit system the circuit is shown in Fig. S1.\nFIG. S1: Parameterized circuit structure for a 5-qubit system for 1 time-step.\nHere, each G(Θi) is a two qubit gate parameterized by a set of parameters Θ i. Note that this parameter set is\ndifferent for each gate G. Below, we will drop Θ iin the labelling of the gates for convenience.\nFor every subsequent time-step, the circuit in Fig. S1 needs to be repeated. It is straightforward to observe that\nfor a time evolution of ntime-steps, there are 2 n‘columns’ of the Ggates. To downfold the circuit, i.e. to reduce\nthe number of columns of the Ggates, thereby reducing the circuit depth, we can exploit special properties of these\ngates. The Ggates belong to a special group of two qubit gates known as matchgates [18], which we discuss below.\nWe report here only salient features that are needed for this work.\nDefinition II.1 (Matchgate) .Consider two matrices A, B inSU(2):\nA=/bracketleftbigg\na b\nc d/bracketrightbigg\n, B=/bracketleftbigg\ne f\ng h/bracketrightbigg\n, (S9)\nwhere det( A)=det( B). Then the two qubit matchgate G(A, B) is defined as\nG=\na b\ne f\ng h\nc d\n (S10)\nMatchgates have two properties that allow for the downfolding of circuits. First, the product of two matchgates is\na matchgate. The proof for this is obtained via matrix multiplication. The second property is an empirical conjecture\nthat is found to be satisfied using numerical optimization for quadratic Hamiltonians. Namely, for distinct matchgates\nG1,G2, and G3there exist matchgates G4,G5, and G6of the same type such that ( G1⊗I)(I⊗G2)(G3⊗I) =\n(I⊗G4)(G4⊗I)(I⊗G6) [17]. This property is visualized with the circuit configuration shown in Fig. S2.\nFIG. S2: Illustration of the empirical conjecture ( G1⊗I)(I⊗G2)(G3⊗I) = (I⊗G4)(G4⊗I)(I⊗G6) [17].\nUsing this conjecture we can construct mirror identities for matchgates. An example is presented in Fig. S3 for a\n5-qubit system, which is implemented to generate matchgates for the topological models depicted in Sec. I.\nTo construct the constant depth circuits, we first create the Nqubit parameterized circuit for simulating ntime-\nsteps that will contain 2 ncolumns of the Ggates. Next we apply the mirror identity to the last Ncolumns of the\nGgates. This allows the combination of two matchgates into one, and by extension allows the combination of two\ncolumns of matchgates into one. This process is repeated until Ncolumns of matchgates remain. This process for a7\nFIG. S3: Matchgate mirror identity for 5 qubits.\n(a) The 5-qubit parameterized circuit for simulating n\ntime-steps. Note that it will contain 2 ncolumns of matchgates.\n(b) The circuit after applying the mirror identity to the last 5\ncolumns of the circuit in (a). The two sets of highlighted\nmatchgates can be combined into one.\n(c) Final circuit after repeated application of the process in (a)\nand (b) until only 5 columns of matchgates remain.\nFIG. S4: Construction of constant depth circuits using matchgate properties for a 5-qubit system. The Hamiltonian\nis simulated for ntime-steps.\n5-qubit system is depicted in Fig. S4. Note that irrespective of n, the final circuit will have Ncolumns of matchgates,\nleading to a circuit depth scaling that is constant with respect to the number of time steps.\nIn practice, we directly construct the parameterized circuit similar to the one in Fig. S4c. As mentioned previ-\nously, there are Ncolumns of matchgates for the Nqubit system. Next, we compute the equivalent matrix of this\nparameterized circuit which is then compared with the desired unitary evolution operator to find the optimized circuit\nparameters. The desired unitary evolution operator is obtained by exact diagonalization. Note that every time-step\nhas a separate constant depth circuit. For example, if a system has to be simulated according to a particular Hamil-\ntonian for 100 time-steps, then there are 100 constant depth circuits. These circuits are independent, and hence their\noptimization can be executed in parallel.\nThe matchgates themselves are decomposed into native gate circuits. For certain Hamiltonians [19], these match-8\ngates require only two CNOT gates, unlike a general two qubit gate that requires at most three CNOT gates. For\ntheNqubit system, the constant depth circuit contains N(N−1) CNOT gates. Decompositions of matchgates into\nthe native gates for different Hamiltonians are provided in [17]. Fig. S5 shows the decomposition for the matchgates\ncorresponding to the Hamiltonian the we have studied in this work. Each θiset is determined after optimizing the\nconstant depth circuits for the Hamiltonian of interest. Rx(θ), Rz(θ) are rotation gates that rotate the qubit about\nthex, zaxis respectively by angle θ. Further details on the construction of the optimized circuits are given in Sec.\nIII.\nIt is observed that adding an additional column of matchgates leads to better results. Hence our results with the\nconstant depth circuits have N+ 1 columns of matchgates for a Nqubit system instead of N. An example of an\nFIG. S5: Decomposition of the matchgate corresponding to the Hamiltonian simulated in our model.\noptimized circuit is shown in Fig. S6. Here, the circuit is for the first time-step for a 5-qubit system. Each Gis of the\nform in Fig. S5 with 4 optimized parameters. To provide clarity, we show a few optimized parameters as an example.\nWe point out that this optimization can be performed independent of the algorithim employed.\nFIG. S6: Optimized constant depth circuit for a 5 qubit system for the simulation upto the first time-step. Each G\ngate is of the form in Fig. S5. Note that these optimized parameters correspond to a subset of the constructed total\ngates.\nIII. QUANTUM HARDWARE IMPLEMENTATION\nA. Methodology\nWe implement the 1D Quantum Spin Hamiltonian given in Eq. 1 in the main text for a 5-qubit system for different\nvalues of J, J′. We consider a total number of 100 time-steps. Each time-step has a value of 0 .1 units of inverse\nmagnetic field ℏ/hxfollowing the convention ℏ= 1. Note that each time-step has a separate parameterized constant\ndepth circuit leading to a total of 100 parameterized constant depth circuits that are optimized as described in Sec.\nII.\nWe begin by creating 100 target matrices for the 100 constant depth circuits using exact diagonalization. Then, we\ncreate 100 parameterized constant depth circuits. The optimal parameters for each of these 100 circuits are calculated\nby maximizing the ‘overlap’ (i.e. minimizing a distance function) between the target matrix and the equivalent\nmatrix of the constant depth circuit. This distance function, motivated by the definition of the Hilbert-Schmidt inner\nproduct, can be defined as follows. Letting dUbe the dimension of the (unitary) target matrix UTand parameterized\nconstant-depth circuit matrix U, their distance is given by\nD(U, U T) = 1−Tr(U†UT)\ndU(S11)9\nThe closer U†UTis to the identity IdU,U†UT≈IdU, the more indistinguishable the parameterized matrix Uis from\nthe target matrix UTis. When U†UT=IdU,D(U, U T) = 0 as expected, and hence the parameterization is exact. For\ntechnical details the reader is referred elsewhere [20]. Once the optimal parameters are calculated, the final optimal\nconstant depth circuit for each of the time-step is constructed. We point out that all of these tasks along with the\nexecution of the circuits on IBM Quantum hardware can be performed directly using Python and the IBM Qiskit\nsoftware package [21]. To summarize our methodology, we identify five steps in the computational protocol to follow\nthe Hamiltonian dynamics of our chosen spin chain on the IBM quantum hardware, Fig. S7.\nImplementing Constant-Depth Circuits on Quantum Hardware\n1. Construct nconstant-depth circuits for ndesired time steps of the Hevolution.\n2. Create a target matrix UTfor each constant-depth circuit.\n3. Generate parameterized constant-depth circuits by minimizing a distance function (Eq. S11) between the target matrix UT\nand constant-depth circuit matrix U.\n4. Transpile parameterized circuits for a given backend.\n5. Execute circuits on backend.\nFIG. S7: Flow of our computational protocol for implementing constant-depth circuits on quantum hardware.\nWe now turn to specific details regarding hardware implementation of our digital quantum simulation protocol.\nIBM quantum processors are based on transmon superconducting qubit architectures. Gate operations are executed\nby performing sequences of microwave pulses to control states of system qubits which are physically realized via\nnonlinear LC circuits with Josephson junctions [22].\nWe execute these circuits implementing our methodology by utilizing the IBM Quantum Processor ibmq manila ,\nwhich is accessible online to run experiments via the IBM Quantum Network [23]. The reported average energy\nrelaxation time of a qubit T1 for ibmq manila is 100 .56µs, dephasing time T2 is 101 .29µs; the reported average\nreadout fidelity is 0 .9739 and CNOT fidelity is 0 .99 [24]. The qubit layout for the device is shown in Fig. S8. We\npoint out that while as of 2023 Sep 26th, ibmq manila is retired and is no longer accessible via the IBM cloud, our\nresults have been tested and shown to be consistent between different quantum processors and simulators provided\nby IBM.\nFIG. S8: Qubit geometry of ibmq manila [21], the quantum processor used in this work. The color code complies\nwith IBM settings [21].\nFollowing convention, we define |0⟩=/bracketleftbigg\n1\n0/bracketrightbigg\nand|1⟩=/bracketleftbigg\n0\n1/bracketrightbigg\n. In each of our simulations, for simplicity, we choose the\ninitial state for the circuits to be |00000⟩. Operationally, Qiskit follows a flipped notation for qubits represented by\nstate kets, i.e., the state of the first qubit is denoted by the right-most position in the ket and is read right to left.\nAs an example, for the initial state consisting of three qubits, |001⟩, the qubit 1 is in state |1⟩, qubit 2 in state |0⟩,\nand qubit 3 in state |0⟩.\nWe execute a simulation known as a quantum quench. We initiate the ground state of our Hamiltonian with the\napplied magnetic field turned off: H(t <0) =−/summationtextN−1\ni=1Jz,iZiZi+1. Then, at t= 0, the field is instantaneously turned\non, and the evolution of the final Hamiltonian H(t≥0) =−/summationtextN−1\ni=1Jz,iZiZi+1−hx/summationtextN\ni=1Xiis executed. During\nevolution, series of projective measurements are performed on each time-step for the observable σz\ni, for each qubit\ni={1,2,3,4,5}. To ensure that measurement outcomes are sampled with statistical sufficiency, repetitions of each\ncircuit, known as shots, are executed along with their corresponding measurements. We then define our observable\nof interest, the local magnetization in the z-direction at each site, given by the corresponding expectation value\n⟨σz\ni⟩=pi(0)−pi(1)\nwhere pi(0) and pi(1) are the averaged probabilities of obtaining the result 0 and 1 respectively from performing a\nprojective measurement on qubit iover all shots. In our simulations, a total of 8192 shots are executed to obtain the\naverage the magnetization for each time step.10\nFig. 3 of the main text reports the results obtained with this approach for the dynamics of the spin model illustrated\nin Fig. 1(b) of the main text and described in Sec. IB of this document. Figs. S9 and S10 display the results obtained\nfor the dynamics of the spin models illustrated in Figs. 1(a) and 1(c) of the main text and described in Sec. IA\nand Sec. IC of this document, respectively. In Fig. S11, we show results for a quantum spin chain with uniform\ncoupling. As in Fig. 1 of the main text, in Figs. S9, S10 and S11 the plots in the left column show the observed\non-site magnetization as a function of time. Applicable equilibration time methods are discussed in Sec. III D. The\nplots in the right column show the corresponding Fourier-transformed power spectra of the on-site magnetization.\nB. Additional Results\nImplementing the procedure discussed above, we comment on the dynamics of open system evolution for the models\ndepicted by Fig. 1(a) and Fig. 1(c) (Sec. I A and Sec. I C respectively) of the main text.\nThe dynamics for the staggered coupling model are shown in Fig. S9. In analogy with the mirror model shown as\nFig. 3 in the main text, strong temporal oscillations of the magnetization of qubit 1 produce a strong peak in the\nfrequency spectrum, denoting the presence of a topological mode localized to qubit 1. This is further supported by\nthe observation that this peak occurs at comparable value of power and frequency as the two peaks observed in Fig.\n3. We infer that constant-depth circuits have capability of stabilizing the topological mode, resulting in a long-time\nsimulation where the robustness of qubit 1 is visible through transient dynamics.\nFIG. S9: Local dynamics of a 5-qubit Hamiltonian in a topologically non-trivial regime, illustrated in Fig. 1(a) of\nthe main text, obtained by NISQ processors. J= 2,J′= 4,hx= 1. Left: real-space magnetization in the\nz-direction for each qubit. Time is reported in units of the inverse magnetic field 1 /hx. Right: Fourier-transformed\npower spectrum in frequency space. A sharp surface mode signature is clearly visible on the first qubit, as expected\nbased on the theory presented in Sec. I A. Frequency is reported in units of the magnetic field hx.\n.\nThe dynamics for the topological defect model are shown in Fig. S10. The signature of a topological defect mode\nis visible on the center qubit indicated by the small peak lying in the interval of frequency (1 ,2), which is consistent\nwith its corresponding closed system dynamics simulation result in Fig. S18. We also identify a slight peak in the\npower spectrum of qubit 1 lying in a comparable frequency range. We observe that open system dynamics affects\ntopologically non-trivial states hierarchically. As discussed in Sec. I C, the topological ’trimer’ defect [9, 10], which\ncan be viewed as a domain wall between two topologically distinct configurations, introduces a pair of localized states\nat both ends of the chain. While traces of the three localized states of qubits 1 and 3 are intelligible, noise limits\nany sharp values from being visible in the corresponding power spectrum, particularly on qubit 5. The degree of\ntopological protection against noise is heavily limited in comparison to the models depicted in Fig. 1(a) and Fig. 1(b)11\nof the main text, while remnants of protection are visible in comparison to the uniform case with no such features.\nMoreover, the bulk and boundary of the system hybridize as a result of open system dynamics. The finite size of the\nsystem has limited the degree of separation between the bulk and boundary of the system. This is corroborated by\nthe findings in Fig. S21, where the localized states of the edge have a larger degree of separation from the center of\nthe system.\nFIG. S10: Local dynamics of a 5-qubit Hamiltonian in a topologically non-trivial regime, illustrated in Fig. 1(c) of\nthe main text, obtained by NISQ processors. J= 2,J′= 4,hx= 1. Left: real-space magnetization in the\nz-direction for each qubit. Time is reported in units of the inverse magnetic field 1 /hx. Right: Fourier-transformed\npower spectrum in frequency space. A topological signature imprinted by the defect mode is visible on the center\nqubit at frequency ≈1.2 units of magnetic field hx. The bulk and boundary states delocalize due to strong outer\ncouplings in conjunction with finite-size effects (compare to Figs. S18 and S21).\nFinally, we present as a benchmark in Fig. S11 the on-site magnetization in the time and frequency domains\nobtained from the dynamics simulation of a uniform 5-site spin chain. In this configuration, J=J′, and hence the\nabsence of dimerization means that there are no topological features. As expected, the power spectra show that each\nqubit is indistinguishable from the rest of the system. We conclude by noting that, in each case, the observed large\nDC (zero energy) feature in the power spectrum is due to the polarization of the qubit spins along the z-direction, as\npointed out in the main text.12\nFIG. S11: Local dynamics obtained from quantum hardware for a uniform coupling model, J=J′= 2,h= 1. This\nis equivalent to the non-topological regime of Hfor each model discussed in the main text. Time is reported in units\nof the inverse magnetic field 1 /hx, frequency is reported in units of the magnetic field hx. As expected, the absence\nof topological modes is evidenced by the noisy power spectrum.\nC. Trotterization\nIn this section, we present the results for the time-dependent on-site magnetization obtained with Hamiltonian\ndynamics simulations utilizing traditional Trotterization [25]. This method exploits the Suzuki-Trotter decomposition\nto break the total circuit implementation for the time evolution operator into discrete Trotter steps [26] (Eq. S12).\ne−iH∆t=/productdisplay\nne−iHn∆t+O(1\nM2) (S12)\nThe time evolution of His broken up into steps e−iHn∆tover which the instantaneous Hamiltonian His conbstant.\nEach step is implemented directly into the circuit, resulting in an error, known as Trotter error, that scales with the\nnumber of time steps MasO(1\nM). The corresponding dynamics obtained using Trotterization for each model in Fig.\n1 of the main text and of the uniform quantum spin chain S11 are shown in Figs. S12, S13, S14, and S15, respectively.\nThe individual time-step value used in the Trotterization simulations is chosen to be 0 .1 in units of inverse magnetic\nfield ℏ/hx, likewise following the convention ℏ= 1.\nIn all cases, the circuits with time-dependent depth generate dynamical simulations which are inconsistent with the\nground truth of each model that is presented in Sec. IV. Using the downfolding property of constant-depth circuits,\nhowever, eliminates this issue and yields theory-consistent magnetization data. We can rephrase this evidence in terms\nof reduction of the Trotter error. In fact, the matchgates allow us to afford reasonably long simulations times even\nwith tiny discretized time intervals. Compared to existing methods, our method produces viable results without the\nrequirement of error mitigation [27]. In other words, our method makes long-time dynamical simulations achievable\nintrinsically at the circuit-level.13\nFIG. S12: Local dynamics obtained from quantum hardware for the staggered spin model of Fig. 1(a), obtained with\nTrotterization. Time is reported in units of the inverse magnetic field 1 /hx, frequency is reported in units of the\nmagnetic field hx. No clear frequency peak occurs in the power spectrum, at odds with the corresponding Fig. S9.\nFIG. S13: Local dynamics obtained from quantum hardware for the topological mirror spin model of Fig. 1(b),\nobtained with Trotterization. Time is reported in units of the inverse magnetic field 1 /hx, frequency is reported in\nunits of the magnetic field hx. No clear frequency peak occurs in the power spectrum, at odds with the\ncorresponding Fig. 3 of the main text.14\nFIG. S14: Local dynamics obtained from quantum hardware for the topological defect spin model of Fig. 1(c),\nobtained with Trotterization. Time is reported in units of the inverse magnetic field 1 /hx, frequency is reported in\nunits of the magnetic field hx. No clear frequency peak occurs in the power spectrum, at odds with the\ncorresponding Fig. S10.\nFIG. S15: Local dynamics obtained from quantum hardware for for a uniform coupling model, obtained with\nTrotterization. Time is reported in units of the inverse magnetic field 1 /hx, frequency is reported in units of the\nmagnetic field hx. No clear frequency peak occurs in the power spectrum, in line with Fig. S11.15\nD. Equilibration Time\nThe equilibration time Teqis the time required for the expectation value of an observable to approach its average,\nA, and fluctuate about it with fluctuations δA=/radicalig\nA2−A2, which are due to finite system size [28]. Operationally,\nwe calculate Teqby obtaining the first time when A(t) =A. This choice is justified as a simple, intuitive measure\nfor equilibriation times in quantum systems comparable to ours, where constant-depth circuits generate results where\nthe oscillatory dynamics is stable over a long period, and does not immediately decay according to an exponential\nfunction. This definition is suitable for finite-size systems, noting that Teqscales with the system size N. We remark,\nhowever, that the equilibriation time is a measure with a degree of arbitrariness, whose usefulness depends directly\non the system itself.\nIV. CLOSED SYSTEM DYNAMICS\nIn this section we report results obtained by exact diagonalization, which pertain to closed systems that represent\nour models in an ideal setting. These results serve as benchmarks for the dynamics obtained on quantum hardware,\nwhich embody open quantum systems. Additionally, exact diagonalization sheds insight into finite-size scaling effects\nof the systems. We argue that, while constant-depth circuit simulations are constrained by the number of qubits\navailable on the quantum hardware, classical exact diagonalization is constrained by the size of the Hilbert space of\nthe simulated system and available classical computational resources.\nFigs. S16, S17, S10 display the closed system dynamics of the models showcased in Fig. 1(a), Fig. 1(b), and Fig.\n1(c) of the main text, respectively. Fig. S19 displays the closed system dynamics of the uniform coupling model.\nFIG. S16: Closed system dynamics of the 5-qubit staggered coupling model (Fig. 1(a) of the main text). We\nreplicate the same initial conditions as the simulation depicted in Fig. S9. J= 2,J′= 4,hx= 1. Left:\ntime-dependent real-space magnetization in the z-direction for each qubit. Time is reported in units of the inverse\nmagnetic field 1 /hx. Right: Fourier transformed power spectrum in the frequency space for each qubit. Frequency is\nreported in units of the magnetic field hx.\nThe closed system analogues of each model exhibit similar behavior and supplement the open system result. Inter-\nestingly, we observe noticeably different dynamics in the non-topological 5-qubit uniform coupling model, where the\nbulk and boundary are indistinguishable, and the Fourier transformed power spectrum exhibits more pronounced and16\nFIG. S17: Closed system dynamics of the 5-qubit topological mirror model (Fig. 1(b) of the main text). We\nreplicate the same initial conditions as the simulation depicted in Fig. 3 of the main text. J= 2,J′= 4,hx= 1.\nLeft: time-dependent real-space magnetization in the z-direction for each qubit. Time is reported in units of the\ninverse magnetic field 1 /hx. Right: Fourier transformed power spectrum in the frequency space for each qubit.\nFrequency is reported in units of the magnetic field hx.\nvaried peaks before reaching frequency = 2 in units of the applied magnetic field, when compared to the topological\nmodels, which notably exhibit power spectrums with patterned features.\nAdditionally, we provide the closed system dynamics for three topological 7-qubit systems and a uniform 7-qubit\nchain. We accomplish this by employing the same Hamiltonian, H=−/summationtextN−1\ni=1Jz,iZiZi+1−hx/summationtextN\ni=1Xi, with N= 7\nspins. Fig. S20 pertains to the staggered coupling model extended to 7 sites, inspired by the SSH model [1]. Next,\nwe present two systems which analogue the topological defect model, created by imprinting a topological defect in\nthe center (qubit 4) between two topological configurations. In Fig. S21, we weakly link two spin chains consisting of\nweakly coupled outer qubits and strongly coupled inner qubits. In Fig. S22, we implement the opposite by strongly\nlinking two spin chains consisting of strongly coupled outer qubits and weakly coupled inner qubits.\nBulk-boundary separation affects the obtained results, as discussed in Sec. III B. The weakly linked 7-qubit topo-\nlogical defect model of Fig. S21 analogues the 5-qubit ’trimer’ defect model discussed in Fig. S10. For the 7-qubit\nchain, larger separation between the system bulk and boundary creates dimer pairs with higher localization at the\nedges of the lattice.17\nFIG. S18: Closed system dynamics of the 5-qubit topological defect model (Fig. 1(c) of the main text). We replicate\nthe same initial conditions as the simulation depicted in Fig. S10. J= 2,J′= 4,hx= 1. Left: time-dependent\nreal-space magnetization in the z-direction for each qubit. Time is reported in units of the inverse magnetic field\n1/hx. Right: Fourier transformed power spectrum in the frequency space for each qubit. Frequency is reported in\nunits of the magnetic field hx.18\nFIG. S19: Closed system dynamics of the 5-qubit uniform coupling model, J=J′= 2. Time is reported in units of\nthe inverse magnetic field 1 /hx, frequency is reported in units of the magnetic field hx. This should be compared to\nthe data in Fig. S11.\nFIG. S20: Closed system dynamics of a 7-qubit staggered coupling chain with alternating couplings JandJ′.\nJ= 2,J′= 4,hx= 1. Time is reported in units of the inverse magnetic field 1 /hx, frequency is reported in units of\nthe magnetic field hx. Since the system has an odd number of sites, one topological mode is visible on the\nweakly-coupled edge qubit 1.19\nFIG. S21: Closed system dynamics of a 7-qubit topological defect system with weak coupling in the center, specified\nbyJ1= 2,J2= 4,J3= 2,J4= 2,J5= 4,J6= 2,hx= 1. Time is reported in units of the inverse magnetic field\n1/hx, frequency is reported in units of the magnetic field hx.\nFIG. S22: Closed system dynamics of a 7-qubit topological defect system with strong coupling in the center,\nspecified by J1= 4,J2= 2,J3= 4,J4= 4,J5= 2,J6= 4,h= 1. Time is reported in units of the inverse magnetic\nfield 1 /hx, frequency is reported in units of the magnetic field hx.20\nFIG. S23: Closed system dynamics of a 7-qubit uniform chain, J=J′= 2,h= 1. Time is reported in units of the\ninverse magnetic field 1 /hx, frequency is reported in units of the magnetic field hx.21\n[1] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).\n[2] X. G. Wen, Int. J. Mod. Phys. B 4, 239 (1990).\n[3] X.-L. Qi and S.-C. Zhang, Reviews of Modern Physics 83, 1057–1110 (2011).\n[4] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, New Journal of Physics 12, 065010 (2010).\n[5] T. Senthil, Annual Review of Condensed Matter Physics 6, 299–324 (2015).\n[6] J. Gooth, S. Galeski, and T. Meng, Reports on Progress in Physics 86, 044501 (2023).\n[7] L. Li, S. Mu, C. H. Lee, and J. Gong, Nature Communications 12, 10.1038/s41467-021-25626-z (2021).\n[8] R. Jackiw and J. Schrieffer, Nuclear Physics B 190, 253 (1981).\n[9] M. P. Estarellas, I. D’Amico, and T. P. Spiller, Scientific Reports 7, 10.1038/srep42904 (2017).\n[10] P. Delplace, D. Ullmo, and G. Montambaux, Phys. Rev. B 84, 195452 (2011).\n[11] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).\n[12] A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001).\n[13] Y. Lin, W. Hao, M. Wang, J. Qian, and H. Guo, Scientific Reports 7(2017).\n[14] N. N. Bogolyubov, V. V. Tolmachev, and D. V. Shirkov, Fortsch. Phys. 6, 605 (1958).\n[15] J. G. Valatin, Nuovo Cim. 7, 843 (1958).\n[16] P. Pfeuty, Annals of Physics 57, 79 (1970).\n[17] L. B. Oftelie, R. V. Beeumen, E. Younis, E. Smith, C. Iancu, and W. A. de Jong, Materials Theory 6, 10.1186/s41313-\n022-00043-x (2022).\n[18] L. G. Valiant, SIAM Journal on Computing 31, 1229 (2002), https://doi.org/10.1137/S0097539700377025.\n[19] L. Bassman, M. Urbanek, M. Metcalf, J. Carter, A. F. Kemper, and W. A. de Jong, Quantum Science and Technology 6,\n043002 (2021).\n[20] M. G. Davis, E. Smith, A. Tudor, K. Sen, I. Siddiqi, and C. Iancu, 2020 IEEE International Conference on Quantum\nComputing and Engineering (QCE) , 223 (2020).\n[21] Qiskit contributors, Qiskit: An open-source framework for quantum computing (2023).\n[22] T. E. Roth, R. Ma, and W. C. Chew, An introduction to the transmon qubit for electromagnetic engineers (2021),\narXiv:2106.11352 [quant-ph].\n[23] G. Garc´ ıa-P´ erez, M. A. C. Rossi, and S. Maniscalco, npj Quantum Information 6, 10.1038/s41534-019-0235-y (2020).\n[24] J. Wang, G. Guo, and Z. Shan, Entropy 24(2022).\n[25] I. M. Georgescu, S. Ashhab, and F. Nori, Rev. Mod. Phys. 86, 153 (2014).\n[26] A. Smith, M. S. Kim, F. Pollmann, and J. Knolle, npj Quantum Information 5, 10.1038/s41534-019-0217-0 (2019).\n[27] R. Takagi, S. Endo, S. Minagawa, and M. Gu, npj Quantum Information 8, 10.1038/s41534-022-00618-z (2022).\n[28] L. Campos Venuti, S. Yeshwanth, and S. Haas, Phys. Rev. A 87, 032108 (2013)."    },    {        "title": "2402.12665v1.Antifragile_Perimeter_Control__Anticipating_and_Gaining_from_Disruptions_with_Reinforcement_Learning.pdf",        "content": "Antifragile Perimeter Control: Anticipating and Gaining from\nDisruptions with Reinforcement Learning\nLinghang Suna,∗, Michail A. Makridisa,∗, Alexander Gensera, Cristian Axenieb, Margherita\nGrossic, Anastasios Kouvelasa\naInstitute for Transport Planning and Systems, ETH Zurich, Zurich, 8093, Switzerland\nbComputer Science Department and Center for Artificial Intelligence, Technische Hochschule\nN¨ urnberg, N¨ urnberg, 90489, Germany\ncIntelligent Cloud Technologies Lab, Huawei Munich Research Center, Munich, 80992, Germany\nAbstract\nThe optimal operation of transportation networks is often susceptible to unexpected disrup-\ntions, such as traffic incidents and social events. Many established control strategies rely on\nmathematical models that struggle to cope with real-world uncertainties, leading to a signifi-\ncant decline in effectiveness when faced with substantial disruptions. While previous research\nworks have dedicated efforts to improving the robustness or resilience of transportation systems\nagainst disruptions, this paper applies the cutting-edge concept of antifragility to better design\na traffic control strategy for urban road networks. Antifragility sets itself apart from robustness\nand resilience as it represents a system’s ability to not only withstand stressors, shocks, and\nvolatility but also thrive and enhance performance in the presence of such adversarial events.\nHence, modern transportation systems call for solutions that are antifragile. In this work, we\npropose a model-free deep Reinforcement Learning (RL) scheme to control a two-region urban\ntraffic perimeter network. The system exploits the learning capability of RL under disruptions\nto achieve antifragility. By monitoring the change rate andcurvature of the traffic state with\nthe RL framework, the proposed algorithm anticipates imminent disruptions. An additional\nterm is also integrated into the RL algorithm as redundancy to improve the performance under\ndisruption scenarios. When compared to a state-of-the-art model predictive control approach\nand a state-of-the-art RL algorithm, our proposed method demonstrates two antifragility-related\nproperties: (a) gradual performance improvement under disruptions of constant magnitude; and\n(b) increasingly superior performance under growing disruptions.\nKeywords: Antifragility, Reinforcement Learning (RL), Perimeter control, Traffic disruptions,\nMacroscopic Fundamental Diagram (MFD)\n1. Introduction\nTransportation networks serve as vital channels for the movement of people and goods. The\noptimization of transportation systems has become a focal point for researchers, resulting in a\nmultitude of research endeavors and practical implementations in the field of Intelligent Trans-\nportation Systems (ITS), as in Figueiredo et al. (2001); Haque et al. (2013). Given that various\n∗Corresponding author.\nEmail addresses: lisun@ethz.ch (Linghang Sun), mmakridis@ethz.ch (Michail A. Makridis)\nPreprint submitted to Transportation Research Interdisciplinary Perspectives February 21, 2024arXiv:2402.12665v1  [eess.SY]  20 Feb 2024sorts of unexpected events, such as traffic accidents, social events, unfavorable weather con-\nditions, etc., often occur unexpectedly in real-world networks, examining the robustness and\nresilience of the transportation system is crucial in the research of ITS (Ganin et al., 2019).\nWith the ever-growing population in the cities and urbanization, traffic systems have gained\nboth volume and complexity. According to the Federal Statistical Office of Switzerland, private\nmotorized road traffic experienced a steady increase of 54% between 1980 and 2019 (Federal Sta-\ntistical Office, 2020). The rise in traffic volume can lead to an escalation of intensified congestion\nand more traffic incidents (Chang and Xiang, 2003). Therefore, urban road networks need to\nsecure a decent level of service even when confronted by various disruptions of unforeseen mag-\nnitudes. This also raises the question of whether the current levels of robustness and resilience\nin transportation systems are sufficient to handle such challenges.\nTo address such issues, antifragility has shed light on a feasible solution and provides a\npossible new evaluation criterium. First introduced in the bestseller Antifragile: Things that\nGain from Disorder by Nassim Nicholas Taleb in 2012, it has then been mathematically explained\nin Taleb (2013); Taleb and Douady (2013), providing insights on designing systems that can\nbenefit from disruptions and perform better under growing volatility and randomness. Since\nthen, antifragility has gained significant interest from both the public and academia, particularly\nin the field of risk engineering (Aven, 2015). The potential of antifragility can also be leveraged\nwithin transportation systems to tackle the increasingly severe traffic problems and non-linear\ndynamics that modern cities face today.\nHowever, how to design antifragile transportation systems remains unexplored. One promis-\ning approach to induce antifragility is by using learning-based algorithms. With the rapid ad-\nvancement of big data and sensor techniques, using Machine Learning (ML), particularly Rein-\nforcement Learning (RL)-based methods has also become a trending practice in designing traffic\nmanagement and control strategies (Zhu et al., 2019; Haydari and Yılmaz, 2022). Through in-\nteracting with a given environment, an RL agent enhances its decision-making ability over time\n(Sutton and Barto, 2018). One advantage of RL over established controllers is that it allows\nfor more flexibility and competence in dealing with multivariate nonlinearities in complex envi-\nronments (Li, 2018; Mysore et al., 2021). RL agents can gradually adjust their decision-making\nwhen deployed to an environment subject to variations, whereas established controllers, such\nas PID controllers, may need intensive manual tuning of parameters. Also, additional informa-\ntion can be fed to the RL agent with ease as a representation of the environment, regardless of\nknowledge of the model dynamics. In contrast, structured model dynamics of the system are\noften required when designing a controller (Bemporad, 2006). This feature of RL can also be\nexploited to explore hidden information from the vast amount of data collected through various\nsensors. As a result, modern traffic control systems have the potential to acquire knowledge from\ntraffic disruptions, respond preemptively with merely early signs, and exhibit increasingly better\nperformance as disruptions escalate.\nThe main goal of this paper is to design an RL algorithm to empower traffic control strategies,\ni.e., perimeter control in this study, to be antifragile against volatility and randomness, which\nare instantiated through disruptions with various magnitudes and onsets. First, we distinguish\nthe concept of antifragility as opposed to other more commonly used terms in the field of trans-\nportation, namely robustness and resilience. Then we introduce how antifragility can be induced\nby integrating derivatives (i.e., change rate and curvature of traffic state) and redundancy into\nthe state space and reward function of an RL algorithm. We simulate perimeter control in a\ncordon-shaped urban road network, subject to demand and model disruptions. Based on the\n2simulation results, we propose a quantitative approach to evaluate and compare the antifragility\nproperty among different methods.\nThe key contributions of this work can be summarized as follows: This work is the first\nto embrace the concept of antifragility in the context of the daily operation of transportation\nsystems. This study demonstrates the effectiveness of incorporating derivatives and redundancy\nto handle system disruptions. Also, the proposed antifragile model-free RL-based algorithm\nachieves better performance compared to both a model-based control strategy that requires a\npriori knowledge and another state-of-the-art model-free RL-based method.\n2. Literature review\nThis section provides a review of the relevant literature with three topics covered in this work.\nFirst, a macroscopic traffic model and the control strategy applied in this paper are introduced,\nwhich serve as the basis of model dynamics for the simulation environment. Following that,\nsince antifragility itself is a novel research topic and the link between RL and antifragility is yet\nabsent, we present state-of-the-art research on leveraging RL algorithms to induce robustness\nand resilience of traffic control strategies. And eventually, based on the literature, we introduce\nantifragility conceptually and how an antifragile system can be designed.\nAlleviating urban network congestion can be realized through various traffic control strategies.\nSince Daganzo (2007) proved the existence of the Macroscopic Fundamental Diagram (MFD)\ntheoretically and Geroliminis and Daganzo (2008) demonstrated the presence of the MFD with\nempirical data, the relationship between traffic flow and density has been established through the\naggregation of individual microscopic data points. This relationship has paved the way for the\ndevelopment of control strategies on a macroscopic level, enabling more computationally feasible\nreal-time control strategies for large-scale networks (Knoop et al., 2012), such as perimeter control\n(Keyvan-Ekbatani et al., 2012; Geroliminis et al., 2013; Kouvelas et al., 2017; Yang et al., 2017),\npricing (Zheng et al., 2012; Zheng and Geroliminis, 2016; Genser and Kouvelas, 2022) and route\nguidance (Yildirimoglu et al., 2015; Fu et al., 2022).\nPerimeter control is among the strategies that have attracted immense attention and research.\nReal-world implementation as shown in Amb¨ uhl et al. (2018) also demonstrated its applicability\nas an effective approach to regulating urban traffic. By refraining the incoming vehicles from\nadjacent regions into a protected zone, the traffic density in the protected area remains below\nthe critical density, so a satisfactory level of service can be upheld (Keyvan-Ekbatani et al.,\n2012). Geroliminis et al. (2013) proposed an optimal perimeter control method using Model\nPredictive Control (MPC) and proved its effectiveness compared to a greedy controller in a\ncordon-shaped network. One major issue with the previous works is the MFD heterogeneity.\nTo tackle this challenge, a substantial amount of effort has been made in investigating the\npartitioning algorithms so that a well-defined MFD can be referred to for a certain sub-network\n(Amb¨ uhl et al., 2019; Saedi et al., 2020). However, MFDs in the real world can hardly be well-\ndefined, as demonstrated in Amb¨ uhl et al. (2021) with loop detector data over a year. Wang\net al. (2015) and Ji et al. (2015) also showed that adverse weather conditions and traffic incidents\ncan alter the shape of the MFDs and even a recovery from the peak-hour congestion may lead\nto a hysteresis (Gayah and Daganzo, 2011). These phenomena could potentially violate the\nmathematical model that serves as the foundation for the established model-based perimeter\ncontrollers.\nTo tackle the parameter uncertainties in the models caused by real-world disruptions, re-\ncent years have also witnessed a growing trend towards utilizing non-parametric learning-based\n3approaches in traffic control (Nguyen et al., 2018). Among different ML algorithms, RL has\nbeen researched extensively in transportation operations, e.g., traffic light control (Wei et al.,\n2019; Chen et al., 2020), dynamic pricing (Wang et al., 2022), delay management (Zhu et al.,\n2021). Particularly for perimeter control, recent works from Ni and Cassidy (2019) and Zhou\nand Gayah (2021) have illustrated the capability of RL algorithms to achieve similar or even\nsuperior performance compared to the established control methods.\nBefore introducing antifragility, two associated terms robustness andresilience commonly\nused in evaluating traffic control strategies and how they can be induced with RL are introduced.\nAs Tang et al. (2020) mentioned, these terms are sometimes used interchangeably in transporta-\ntion research works. Therefore, we follow the principles proposed by Zhou et al. (2019), that\nrobustness is concerned with assessing a system’s capacity to preserve its initial state and re-\nsist performance deterioration in the presence of uncertainty and disturbances, while resilience\nstresses the ability and speed of a system to recover from major disruptions to the original state.\nWhile researchers have endeavored to produce review articles in either the applications of RL\nin transportation (Haydari and Yılmaz, 2022; Haghighat et al., 2020) or robustness/resilience\nin transportation systems (Zhou et al., 2019; Tamvakis and Xenidis, 2012), a summarization\nof using RL algorithms in transportation to achieve robust or resilient urban networks remains\nabsent. Here we briefly reviewed the existing studies that apply RL to regulate urban traffic that\nhave demonstrated the robustness or resilience of their proposed methods. By transferring the\nknowledge of robust or resilient design, we can induce antifragility in our traffic control design\nthrough similar approaches. Table 1 shows a selection of papers around perimeter control and\ntraffic signal control regulated with RL algorithms.\nAs can be seen, only two out of the ten papers are focused on the resilience of the RL-based\ntraffic control strategies and the studied scenarios are traffic demand uncertainties and MFD\nerrors (Zhou and Gayah, 2023) as well as lane closures (Korecki et al., 2023). When examining the\nsystem’s robustness, the types of scenarios under consideration become more diverse, including\nincidents (Aslani et al., 2018; Rodrigues and Azevedo, 2019), sensor failures (Wu et al., 2020).\nTable 1 also summarizes the RL setups to demonstrate the property of being either robust\nor resilient of each traffic control strategy. In these papers, some authors proved the superior ro-\nbustness or resilience of their proposed methods directly through testing against the established\nmethods as benchmarks, whereas others induced such properties by adding specific terms (ital-\nicized in Table 1) in the state space of the RL-algorithm. As a result, the algorithms are given\nadditional information, potentially empowering the algorithms to anticipate ongoing disruptions.\nFor instance, Rodrigues and Azevedo (2019) induced robustness by adding the elapsed time since\nthe last green signal for each phase, Tan et al. (2020) experimented with speed or pressure (resid-\nual queue) as an additional state representation in the state space of the RL algorithm, Chu et al.\n(2020) supplemented the control policies of neighboring intersections as additional information\nto the agents, and Zhou and Gayah (2023) used an extra binary congestion indicator in the state\nspace. The analysis of the state-of-the-art RL studies considers reversals or sudden changes in\nthe state-action-reward dynamics, which evokes unanticipated uncertainty. The problem in these\ncontexts is often to respond to unexpected results appropriately since they might indicate a shift\nin the environment. In this case, exploration refers to the process of looking for new information\nto improve the RL agent’s understanding of the traffic dynamics under disruptions, which would\nthen be used to identify better courses of action.\nEver since the concept of antifragility was proposed, it has become an increasingly popular\nconcept in many disciplines, such as economy (Manso et al., 2020), biology (Kim et al., 2020),\n4Table 1: Traffic control strategies using RL with demonstrated robustness or resilience\nLiterature Category Objective Scenarios Robust Resilient Benchmark RL method State Action Reward\nZhou and\nGayah\n(2021)Perimeter\ncontrolMaximize trip\ncompletion under\nuncertaintiesDemand and\nMFD\nuncertaintiesTestedNo-control,\nMPCDDPGNo. of vehicles,\ntraffic demandPerimeter\ncontrol\nvariablesTrip\ncompletion\nChen et al.\n(2022)Perimeter\ncontrolMinimize total time\nspent under uncer-\ntainties/disturbancesDemand\nuncertainties,\ndemand surgeTested MPC AC-IRL No. of vehiclesPerimeter\ncontrol\nvariablesDivergence\nfrom critical\naccumulation\nSu et al.\n(2023)Perimeter\ncontrolMaximize trip\ncompletion under\nuncertaintiesDemand\nuncertaintiesTested FT/Webster/\nMP + PIDQNFlow, density,\nspeedPerimeter\ncontrol\nvariablesRatio of flow\nover max flow\nZhou and\nGayah\n(2023)Perimeter\ncontrolMaximize trip\ncompletion under\nuncertainties and\nerrorsDemand and\nMFD\nuncertaintiesInduced1No-control,\nMPCMADDPGNo. of vehicles,\ntraffic demand,\ncongestion\nindicatorPerimeter\ncontrol\nvariablesTrip\ncompletion\nAslani et al.\n(2018)Traffic\nsignal\ncontrolMinimize average\ntravel time under\ndisturbancesIncidents,\nsensor noiseTestedCross\ncomparisonDQN, AC,\nSARSAPhase, pressureGreen time\ndurationQueue length\nChu et al.\n(2020)Traffic\nsignal\ncontrolMinimize queue\nlength and delay,\nmaximize vehicle\nspeed and trip\ncompletion under\ndemand uncertaintiesDemand\nuncertainties,\ndemand surgeInducedIA2C,\nIQL-LR,\nIQL-DNNMAA2CTotal delay, no.\nof vehicles,\nadjacent infoSignal con-\nfigurationQueue length,\ntotal delay\nRodrigues\nand\nAzevedo\n(2019)Traffic\nsignal\ncontrolMinimize total travel\ntime, total delay, and\ntotal stop timeDemand surge,\nincidents,\nsensor failuresInducedSelf-\ncomparisonDDQNQueue length,\nelapsed time in\neach phaseTime\nextension,\nphase\nselectionNumerical\nderivative of\nqueue length\nTan et al.\n(2020)Traffic\nsignal\ncontrolMinimize queue\nlength at the\nintersection under\nuncertaintiesDemand\nuncertaintiesInducedSelf-\ncomparisonDDQNphase info,\nqueue length,\nspeed, pressurePhase con-\nfigurationQueue length,\nduration over\ndetection\nWu et al.\n(2020)Traffic\nsignal\ncontrolMinimize average\nvehicle delay under\ndifferent scenariosDemand\nuncertainties,\nsensor noiseTestedFixed-time,\nactuatedDDQNNo. of vehicles,\nphase, elapsed\ntime, neighbor\nphasesNo-change/\nextend/\nterminateQueue length\nKorecki\net al. (2023)Traffic\nsignal\ncontrolMinimize average\ntravel time under\ndisruptionsLane closure TestedRandom,\ncyclical,\ndemand,\nanalytical+DQN,\nDQN +\nheuristicVehicle\noccupanciesGreen time\ndurationNegative\npressure\nTested/Induced: Tested means the robustness or resilience is demonstrated through direct testing against other benchmark methods, without identifying which component in the\nRL algorithm results in such properties. Induced means the author identified that adding certain terms in the RL algorithm improves the robustness or resilience.\nItalicized terms represent the additional information the authors applied as effective to design a robust/resilient system.\n5medicine (Axenie et al., 2022), and robotics (Axenie and Saveriano, 2023). However, current\nstudies on antifragility in the field of engineering mostly pertain to post-disaster reconstruc-\ntion efforts (Fang and Sansavini, 2017; Priyadarshini et al., 2022). Leveraging the potential of\nantifragility for the daily operations and optimization of transportation systems is a new and\nunexplored notion.\nIn Taleb (2012), in which the concept of antifragility was first introduced, two different\nlevels of antifragility have been indicated: proto-antifragility and antifragility. To differentiate\nthese two terms more clearly, we rename the latter as progressive antifragility. In contrast to\nrobustness or resilience, proto-antifragility describes systems that can improve performance from\nexperiencing disruptions, within a certain magnitude. A very illustrated example can be the\nbiological process of hormesis. In this sense, proto-antifragility resembles adaptiveness, research\nworks such as Fang and Sansavini (2017) are related to proto-antifragility of infrastructure under\nhazardous events. The higher level of antifragility, i.e., the progressive antifragility, lays emphasis\non the concave response of the system regarding an increasing magnitude of disruptions. Its\nantonym, fragility, refers to systems that suffer from exponentially growing loss when faced with\nlinearly increasing disruptions (Taleb and Douady, 2013), which is characterized by convexity and\ncan be mathematically formulated with Jensen’s inequality E[g(X)]≥g(E[X]). On the contrary,\nthe nonlinear relationship between external stressors and responses for antifragile systems is\nconcave with E[g(X)]≤g(E[X]). A system exhibiting proto-antifragility is not necessarily\nprogressive antifragile at the same time.\nResearchers have also proposed methods to incentive antifragile property of a system by em-\nphasizing the derivatives to capture the temporal evolution patterns of the system dynamics,\ni.e., how fast the system state deviates towards a possible black swan event, and the curvature of\nthis deviation (Taleb and Douady, 2013; Taleb and West, 2023; Axenie et al., 2022). With this\nadditional information, the system can anticipate ongoing disruptions and be more responsive to\ndrastic changes. Similar to the function of redundancy in resilience (Tan et al., 2019; Kamalah-\nmadi et al., 2022), redundancy can also be added in the system to induce antifragility de Bruijn\net al. (2020); Johnson and Gheorghe (2013); Munoz et al. (2022). Other feasible approaches also\ninclude time-scale separation and attractor dynamics (Axenie et al., 2022; Axenie, 2022).\n3. Problem Formulation\nThis paper studies the problem of perimeter control between two homogeneous regions. A\ncordon-shaped urban network is investigated as in Geroliminis et al. (2013); Zhou and Gayah\n(2021), with the inner region representing a city center, as shown in Figure 1(a). Traffic demand\nfor an Origin-Destination (OD) pair from region ito region jat time tis denoted as qij(t). The\ninner and outer regions have different MFDs due to the difference in capacity to accommodate\nvehicles in the road networks within the city center and the surrounding region, defined as\nGi(ni(t)) as illustrated in Figure 1(b). Given the total number of vehicles with presence in region\niat time t, denoted as ni(t), the total trip completion rate for this region i, denoted as Mi(t),\ncan be determined using the corresponding MFD, which comprises both the intraregional trip\ncompletion, i.e., Mii(t) and the interregional transfer flow, i.e., Mij(t) (i̸=j) with i, j∈ {1,2}.\nIn order to protect both regions from being overflown by possible high traffic demand, the\npercentage of the transfer flow allowed to go across the region perimeter at time tis regulated\nby two perimeter controllers denoted as uij(t) (i̸=j). A list of all notations used in this\npaper, including the notations used in defining the RL algorithm and the antifragile terms, is\nsummarized in Table 2.\n6(a) The cordon-shaped urban network\n (b) MFDs for the inner and outer regions.\nFigure 1: The network structure and the related MFDs.\nEq. 1 describes the change rate of the intraregional vehicle accumulation of region i. It\nis the sum of intraregional traffic demand in this region, denoted as qii(t), together with the\nperimeter control regulated transfer flow from region jto region i, defined as uji(t)·Mji(t), then\ndeducted by the trip completion within region i, denoted as Mii(t). Likewise, the change rate of\ninterregional traffic accumulation, as in Eq. 2 shows, is the difference between the interregional\ntraffic demand, denoted as qij(t), and the regulated transfer flow uij(t)·Mij(t):\ndnii(t)\ndt=qii(t) +uji(t)·Mji(t)−Mii(t) (1)\ndnij(t)\ndt=qij(t)−uij(t)·Mij(t),(i̸=j) (2)\nThe total trip completion, i.e., Mi(t) for region iat time tis calculated based on the trip\naccumulation and the related MFD, defined as Gi(ni(t)), and is the sum of the intraregional trip\ncompletion, i.e., Mii(t), in Eq. 3 and the interregional transfer flow, i.e., Mij(t) (i̸=j), in Eq. 4:\nMii(t) =nii(t)\nni(t)·Gi(ni(t)) (3)\nMij(t) =nij(t)\nni(t)·Gi(ni(t)),(i̸=j) (4)\nni(t) =X\nj=1,2nij(t) (5)\nThe objective function is to maximize the throughput of this cordon-shaped network, which\nis the sum of the intraregional trip completion in both regions.\nJ= max\nuij(t)Ztend\n0X\ni=1,2Mii(t)dt (6)\nsubject to the following boundary conditions:\n7Table 2: List of notations\nSymbol Meaning\n1. General notations in problem formulation\nt Time\n∆t Time step\ntend Total simulation time\nnij(t) Vehicle accumulation with OD from region itojat time t\nni(t) Vehicle accumulation in region iat time t\nuij(t) Perimeter control variables regulating flow from region itojat time t\nqij(t) Traffic demand with OD pair iandjat time t\nGi(ni(t)) Sum of trip completion and transfer flow in region iat time t\nMij(t) Trip completion with OD from region itoj(i̸=j) at time t\nni,cap(t) Maximal number of vehicles (jam accumulation) in region iat time t\nni,crit(t) Vehicle accumulation with highest completion rate in region iat time t\nJ Objective function\n2. Notations in reinforcement learning\nS State space, the whole set of states the RL agent can transition to\nst st∈ S, the observable state in simulation at time t\nA Action space, the whole set of actions the RL agent can act out\nat at∈ A, the action taken in simulation at time t\nR The reward function for the RL agent\nrt rt=R(st, at), the received reward with state stand action atat time t\nγ Discount factor to favor rewards in the near future\nQ(s(t), a(t)) Expected long-term return for taking action a(t) in state s(t) at time t\nθµWeight parameter of the deep neural network for the actor network\nθQWeight parameter of the deep neural network for the critic network\nyi Expected long-term return calculated with the target critic network\nL The loss of the critic network\nρβAll possible trajectories of st\nI The objective function for the actor-network\n3. Notations for the additional antifragile terms applied in reinforcement learning\nϵ(t) Additional reward term in RL based on derivatives and redundancy\nωh The weight of first derivative in the additional reward term ϵ(t)\nω∆h The weight of second derivative in the additional reward term ϵ(t)\nαi(t) Binary variable determining the term to be reward/penalty\nhi(t) The first derivative of traffic state at time t\n∆hi(t) The second derivative of traffic state at time t\n8nij(t)≥0 (7)\nni(t)≤ni,cap (8)\numin≤uij(t)≤umax (9)\nIntraregional and interregional vehicle accumulation, i.e., nii(t) and nij(t) are non-negative\nvalues, and ni,capis the maximal possible number of vehicles accumulated in the region i. At this\nvehicle accumulation, a gridlock will occur in the network. uminandumaxrepresent the lower and\nupper limit for the perimeter control variable uij(t) for both directions and such applications are\nin line with Geroliminis et al. (2013); Zhou and Gayah (2021). These bounds are due to the fact\nthat perimeter control is normally implemented through signalization. While umaxaccounts for\nthe lost time caused by the interchange between the red and green phases, uminis necessitated\nsince an indefinite long red light is rare in real-world cases.\nIn contrast to the control-based strategies, for the RL-based algorithms, following the idea\nof redundancy, an additional term ϵ(t) is added into the objective function JRL, referred to as\nreward rt∈ R in the context of RL, leading to:\nJRL= max\nuij(t)Ztend\n0[X\ni=1,2Mii(t) +ϵ(t)]dt (10)\nThe term ϵ(t) aims to build up a proper redundancy so that the proposed RL algorithm\ndoes not reward the agent for targeting the exact critical accumulation point. A comprehensive\nexplanation of the term ϵ(t) for the reward in RL will be provided in the following Section 4.3.\n4. Methodology\nThis section gives a brief explanation of how MPC and RL are applied in this paper, and how\nwe design an antifragility perimeter control strategy based on the derivatives and redundancy in\ndetail.\n4.1. MPC with average history disruption\nMPC is an established control method with wide applications in engineering to regulate\ndynamic systems (Qin and Badgwell, 2003), and is used as one of the benchmark methods in\nthis work. First applied in Geroliminis et al. (2013), MPC has been proven to be an effective and\nrobust method to regulate perimeter control in comparison to a greedy controller. For details\nof the implementation of MPC in general practice or under the context of perimeter control,\nwe refer the readers to Darby and Nikolaou (2012) and Geroliminis et al. (2013). The applied\nMPC toolkit in this paper is introduced in Lucia et al. (2017), which uses the CasADi framework\n(Andersson et al., 2019) and the NLP solver IPOPT (W¨ achter and Biegler, 2006).\nThe drawback associated with MPC lies in its presumption of ample a priori knowledge,\nwhich includes traffic demand at each time step. However, such data is rarely accessible in\npractical real-world scenarios. Although traffic forecasting Vlahogianni et al. (2014) may serve\nas a substitute for traffic demand, the prediction accuracy is not always guaranteed with all sorts\nof disruptions in the real world. To make the MPC less sensitive to disruptions, in our study,\nthe benchmark MPC algorithm averages all the historical profiles, including both normal and\ndisruption profiles, and uses this aggregated profile as part of the simulation environment.\n94.2. RL algorithm\nIn RL algorithms, an agent or multiple agents interact with a preset environment and improve\nthe performance of decision-making, defined as action atin an action space A, based on the\nobservable state stin the state space Sand the reward, defined as rt=R(st, at), where Ris\nthe reward function. The improvement of decision-making is commonly realized through a deep\nneural network as a function approximator. The RL algorithm applied in this work is Deep\nDeterministic Policy Gradient (DDPG) as proposed in Lillicrap et al. (2015). By applying an\nactor-critic scheme, DDPG can manage a continuous action space instead of only choosing from\na limited set of discrete values as in the Deep Q-Network (DQN) algorithm (Mnih et al., 2013),\nwhich are commonly applied as Table 1 shows. Also, Zhou and Gayah (2021) has demonstrated\nthat an RL algorithm with continuous action space can achieve better performance compared to\ndiscrete action space. The DDPG algorithm can be divided into two main components, namely\nthe actor and the critic, which are updated at each step through policy gradient and Q-value,\nrespectively. The scheme of the DDPG algorithm applied in this paper is schematically illustrated\nin Figure 2.\nThe state st∈ Sis defined distinctively according to different methods applied in this work.\nOur proposed method consists of three terms, the vehicle accumulation regarding the OD pair\nnij(t), the change rate of vehicle accumulation at each time step dnij(t) (first derivative) as well\nas the second derivative d2nij,t. In Zhou and Gayah (2021), a state stdefined as [ nij(t), qij(t)] is\nadopted. However, since traffic demand in the real world is hardly measurable, qij(t) would be an\nunobservable state for the agent. The action at∈ Ais defined the same as the control variables\nuij(t). For the reward rt, while Zhou and Gayah (2021) uses merely the completion rate, in our\nproposed method, the reward is defined with an additional ϵ(t) term, as Eq. 10 shows.\nThe actor-network is represented by µ(·) and it determines the best action at, which is the\npercentage of vehicles that are allowed to travel across the periphery, based on the current state\nstand the weight parameters at a certain time step t:\nThe nature of the best action atwas also explored in the previous work of Axenie (2022),\nwhere in the framework of variable structure control, the authors demonstrate the need for a\ndiscontinuous signal. The critic network, denoted by Q(·), takes the responsibility to evaluate\nwhether a specific state-action pair at a certain time step yields the maximal possible discounted\nfuture reward Q(st, at). A common technique used in DDPG is to create a target actor network\nµ′(·) and a target critic network Q′(·), which are a copy of the original actor and critic network but\nupdated posteriorly to stabilize the training process and prevent overfitting (Zhang et al., 2021).\nThe target maximal discounted future reward for the target critic network can be calculated as\nin Eq. 11.\nyi=ri+γQ′(si+1, µ′(si+1|θµ′)|θQ′) (11)\nSimilar to DQN, the critic network can be updated by calculating the temporal difference\nbetween the predicted reward and the target reward and minimizing the loss for a mini-batch N\nsampled from the replay buffer:\nL=1\nNX\ni(yi−Q(si, ai|θQ))2(12)\nAfterward, the actor network can be updated with sampled deterministic policy gradient\n(Silver et al., 2014):\n10Figure 2: DDPG scheme\nI=Est∼ρβ[r(s, µ(s|θµ))|s=st] (13)\n∇θµI=Est∼ρβ[∇aQ(s, a|θQ)|s=st,a=µ(st)∇θµµ(s|θµ)|s=st] (14)\n∇θµI≈1\nNX\nt∇aQ(s, a|θQ)|s=st,a=µ(st)∇θµµ(s|θµ)|st (15)\nWhen training the agent, we add disruptions into the simulation environment starting from a\ncertain training episode, in the form of surging traffic demand or MFD disruption. These various\nforms of volatility ought to optimally elicit various learning and decision-making processes. The\nRL agent would include the results of their prior choices to create and update their weight\nparameters in a situation with high outcome volatility, and being capable of generating and\nupdating expectations after sensing a change in a high-volatility environment.\n4.3. Antifragility and the antifragile terms in RL\nIn our work, following the same idea of modifying the state space Sas in the research works in\nTabel 1, we add additional terms based on derivatives (Taleb and Douady, 2013) and redundancy\n(de Bruijn et al., 2020), in both the state space Sand the reward function Rof the RL algorithm.\nFor the state space S, we replace qijusing the first and second derivatives of the vehicle\naccumulation dnij(t) and d2nij(t). With this additional information, the RL agent is aware of a\npossible demand surge of MFD disruption and d2nij(t) can reflect the curvature of such changes.\n11For the reward function R, the trip completion at each time step acts as the main component\nof our proposed method. The term ϵ(t) in the objective function JRLin Equation (10) acts\nas an additional term to build up redundancy in the system. Similar to the creation of the\nadditional term in the state space, we create redundancy also through the calculation of the\nderivates, but instead of the derivatives of the vehicle accumulation, we calculate the derivatives\nof the traffic state. This creates a one-to-one correspondence between the derivatives of the\nvehicle accumulation and the derivatives of the traffic state. To explain this antifragile term\nin the reward, we summarize ϵ(t) as the sum of two terms, with H(t) being an overall term\nrepresenting the first derivative of the traffic state and ∆ H(t) representing the second derivative:\nϵ(t) =H(t) + ∆ H(t) (16)\nHere, H(t) and ∆ H(t) can be expanded as:\nH(t) =X\ni=1,2Hi(t) =ωhX\ni=1,2f(ni(t), ni,crit, ni,cap)·αi(t)·hi(t) (17)\n∆H(t) =X\ni=1,2∆Hi(t) =ω∆hX\ni=1,2f(ni(t), ni,crit, ni,cap)·∆hi(t) (18)\nhi(t) and ∆ hi(t) are the first and second numerical derivatives of the traffic states on the\nMFD, hi(t) is defined as the difference of trip completion over vehicle accumulation at the end\nof a time step versus at the beginning of the same time step, as in Eq. 19 shows, and the second\nderivative ∆ hi(t) is calculated as the difference between the first derivatives of two consecutive\ntime steps, as in Eq. 20 shows:\nhi(t) =Mi(t)−Mi(t−1)\nni(t)−ni(t−1)(19)\n∆hi(t) =hi(t)−hi(t−1) (20)\nSince in the RL algorithms implemented in this paper, all variables involved in the deep\nneural network should be normalized to facilitate the training process, meaning the exact values\nof the derivatives are not of importance, ωhandω∆hare introduced as the weight constants for\nthe first and second derivatives to regulate their impact on the reward side R.\nThe binary variable αi(t) was designed in the first derivative to reward the agent when moving\ntowards the desired direction on the MFD. For instance, the derivative of any data point in the\ncongested zone of the MFD is negative. In this case, when the vehicle accumulation is still\ngetting larger, a penalty will be applied. However, if the vehicle accumulation is decreasing\nthrough perimeter control, this binary αi(t) variable will turn it into a reward. For the second\nderivative, an additional binary variable is not necessary since the two consecutive first derivatives\nare able to determine whether ∆ hi(t) is either positive or negative.\nαi(t) =(\n1,ifni(t)≥ni(t−1),\n−1,otherwise.(21)\nThe term f(ni(t), ni,crit, ni,cap) is a reduction factor to constrain the impact of the ϵ(t) term\nwhen the accumulation is either on a very lower level (empty network) or on a very higher\nlevel (gridlock). The area near the critical accumulation is where the ϵ(t) term should have the\n12greatest impact. Here we use a modified trigonometric function to realize this purpose. It should\nbe noted that other functions, such as normal distribution, are also valid for achieving the same\npurpose.\nf(ni(t), ni,crit, ni,cap) =\n\n1 + cos\u0010\n−π·ni,crit−ni(t)\nni,crit\u0011\n2,ifni(t)≥ni,crit,\n1 + cos\u0010\n−π·ni(t)−ni,crit\nni,cap−ni,crit\u0011\n2,otherwise.(22)\nAfter considering all the modifiers above, we show H(t) and ∆ H(t) using a single MFD as\nan example in Figure 3 and 4. The first derivative H(t), in Figure 3(a), rewards the agent\nmore when it’s moving towards the critical accumulation to maximize its trip completion rate.\nHowever, when the number of vehicles approaches the critical accumulation, this term drops\nsignificantly and becomes a penalty when it exceeds the critical point. Since the first derivative\nH(t) is a complementary term in addition to the trip completion in the reward function, we\nshowcase the influence of this term on the MFD after normalization in Figure 3. With increasing\nweight coefficient ωh, the critical accumulation of the modified MFD becomes marginally smaller\ncompared to the original MFD, and the reward that the RL agent can receive also decreases faster\nafter the accumulation exceeds the critical accumulation. Although the trip completion still\nfollows the original MFD in the simulation environment, The agent learns to get more rewards\nfollowing the modified MFD. In this way, redundant overcompensation has been established\nto prevent accumulation from exceeding the critical accumulation when disruption takes place\nunexpectedly.\nAn interesting note is that estimation uncertainty, also known as second-order uncertainty,\nis another factor that affects disruptions that take place unexpectedly. This is in fact the im-\nprecision of the learner’s current beliefs about the environment, and what the antifragile terms\ncapture. This amount reduces with sampling if beliefs are acquired by learning as opposed to\ninstruction (e.g. anticipation through redundant overcompensation). When estimating uncer-\ntainty is substantial, unlikely samples could partly be attributed to the agent’s false assumptions\nabout the environment’s structure rather than a change in that structure (e.g. around critical\naccumulation).\nThe second derivative ∆ H(t) is shown in Figure 4. The x-axis is the vehicle accumulation,\nsame as in Figure 3(a), while the y-axis represents how fast the traffic state is changing on the\nMFD. The faster it increases to reach the critical accumulation, the greater the penalty will be\napplied to the RL agent. This observation is consistent with the redundant overcompensation and\ntime-scale separation principles formalized in Taleb and Douady (2013) and practically applied\nin Axenie (2022) and Axenie and Saveriano (2023). On the contrary, if the vehicle accumulation\ndecelerates, a reward will be applied. Similar to H(t), this complementary term ∆ H(t) is also\ndependent on the normalization factor ω∆h.\nWith H(t) and ∆ H(t), the agent learns to be conservative when regulating the perimeter\ncontrol variables when the accumulation is about to reach critical, in case disruptions take\nplace. Therefore, as can be concluded, although H(t) and ∆ H(t) apply the same concept of\nthe derivative as the dnij(t) and d2nijin the state space S, the purpose of H(t) and ∆ H(t) is\npreserving redundancy in the system instead of feeding additional information to the agent. This\nbehavior is consistent with the locally discontinuous shape of the action signals uij(t) applied to\n13(a)Hfor the first derivative hi(t).\n (b) ∆ Hfor the second derivative ∆ hi(t).\nFigure 3: Illustration of the term H(t) and its effect on the MFD\nFigure 4: Illustration of the term ∆ H(t)\nthe cordon network, as suggested by the control theoretic study of Axenie (2022).\n5. Experiment application\nThis section introduces the environment setup of the simulation and the evaluation procedure.\nWe investigate a total number of three perimeter control strategies in addition to a no-control\nscenario, based on 2 different scenarios, i.e., under constant or incremental disruptions for testing\nproto-antifragility and progressive antifragility respectively:\n•No control\n14•MPC modified from (Geroliminis et al., 2013), averaging all history profiles including dis-\nruption as a new demand or MFD profile.\n•State-of-the-art RL-based method proposed by (Zhou and Gayah, 2021) as baseline:\nState st: vehicle accumulation and traffic demand [ nij(t), qij(t)]\nReward rt: trip completion in both regions [P\ni=1,2Mii(t)]\n•The proposed RL-based antifragile method:\nState st: vehicle accumulation and its derivatives [ nij(t), dnij(t), d2nij(t)]\nReward rt: trip completion and the ϵ(t) redundancy term [P\ni=1,2Mii(t) +ϵ(t)]\n5.1. Simulation environment parametrization\nAs introduced in Section 3, we simulate a cordon-shaped urban network with inner and outer\nregions with different MFDs. Some minor modifications have been adapted in the congested zone\nof the MFD to ensure the derivative of the MFD is continuous. The MFD of the outer region is\nlargely in line with the observations in Yokohama (Geroliminis and Daganzo, 2008). Other critical\nindicators, e.g., critical accumulation and maximal trip completion, remain unchanged. The non-\nidentical MFDs reflect the fact that the area of the periphery is much larger than the city center,\nand thus the maximal accumulation would be significantly higher. The traffic demand under no\ndisruption is approximated based on Geroliminis and Daganzo (2008). Instead of a trapezoidal\ndemand profile, the chosen profile is in the shape of a normal distribution in accordance with\nMazloumi et al. (2010) to describe the peak hour traffic flow. The total simulation duration\ntendis 2 hours with each time step ∆ tas 60 seconds, and the second hour has significantly\nfewer vehicles so that the network would be able to clear the vehicles accumulated in the first\nhour. Other notable constraints and initializations include the lower and upper bounds for the\nperimeter control variable uij∈[0.1,0.9] (Geroliminis et al., 2013) and an initial accumulation\nin the network nij,0= [600 ,1300,300,2400].\nWe simulate two different disruption scenarios in our work, namely surging traffic demand\nand MFD disruption due to adversarial events. Also, as introduced in Section 2, antifragility can\nbe defined in two different levels, i.e., proto-antifragility and progressive antifragility. Therefore,\nwe consider two different disruption profiles regarding each level of antifragility, as shown in\nFigure 5.\nFirst, to validate proto-antifragility under surging demand, after the agent has been trained\nfor the first 50 episodes with a base demand profile, as shown in Figure 6(a), we introduce a\nnew demand profile with disruption, which characterizes a surging demand within a short period\nfrom the outer region into the inner region, as shown in Figure 6(b). The new environment\nlasts for another 50 episodes with a constant magnitude of 2,500 vehicles following normal dis-\ntribution. On the other side, to test progressive antifragility, instead of a constant magnitude\nof disruption, an incremental disruption is applied for another 50 episodes following the training\nwith 50 episodes of the based demand profile. The magnitude of disruption grows linearly from\n0 to 5,000 vehicles, which is twice the number of vehicles in the constant disruption. Both the\nRL-based methods are trained 15 times and the average is calculated as their learning curves.\nSome adversarial events will exert a negative effect on the MFD, leading to a worse roadside\nperformance. However, since these events may take place in different forms, weather, accidents,\nblockage, etc., little research effort has been made to unveil the exact correlation between each\ntype of event and the change of the MFD. Hence, we simulate three possible MFD disruption pro-\nfiles and assume these profiles can account for the majority of the possible negative consequences.\n15Figure 5: Illustration of simulation scenarios\n(a) Base demand profile\n (b) Demand profile under disruption\nFigure 6: Demand profile with or without surging demand\nThe first profile is a decrease of the free flow speed without critical and maximal accumulation\ndropping, as shown in Figure 7(a) representing the MFD of the inner region. This can be caused\nby unfavorable weather, such as slippery and icy roads or low visibility. The second profile,\nFigure 7(b), is a decrease of critical density without a reduction of the free flow speed. For\nexample, road maintenance work on an arterial road can significantly reduce the serviceability\nof this region. The last profile is based on Amb¨ uhl et al. (2020), in which a parameter λis de-\nfined as the infrastructure potential to quantify the decrease of flow values due to infrastructure,\nbetween-vehicle interactions, etc. Networks in the real world can have a λaround 0 .03−0.07\nand a greater λindicates the infrastructure is less efficiently utilized. Since this method is based\non the method of cuts (Daganzo and Geroliminis, 2008) instead of a polynomial function, the\nshape of the MFD for this profile is marginally different from the other two MFDs.\nFor the first two profiles, while proto-antifragility is validated with a 12.5% drop of the\nmaximal flow, progressive antifragility is tested against an increasing value of drop from 0% to\n25%. For the third MFD profile, a constant λof 0.095 is applied to test the proto-antifragility,\nand the λincreases from 0 .07 to 0 .12 to represent the growing magnitude of disruption as the\nsimulation environment for exhibiting progressive antifragility. Interestingly, these three profiles\n16also represent three different patterns of how the critical accumulation evolves with the drop of\nflow values, i.e., constant, decreasing, and increasing critical vehicle accumulation.\n(a) Decrease of free flow speed\n (b) Decrease of maximal traffic flow\n(c) Decrease of infrastructure potential\nFigure 7: MFD change due to adversarial events\n5.2. Performance evaluation\nAlthough the reward defined in the RL algorithm is based on trip completion, to better\nshowcase antifragility, the main performance indicator being evaluated here is the Total Time\nSpent (TTS), which is calculated by adding up the number of vehicles within the network at\neach second of the simulation. The main reason for choosing TTS over completion is that\ncompletion is lower bounded with increasing magnitude of disruption by the number of vehicles\nthat finished their trips before the disruption, meaning we may observe a concave trip completion\nwith increasing disruptions at first but then it becomes approximately constant. This makes the\nmanifestation of antifragility difficult since the performance curve could be partially concave\nand partially convex. However, by using TTS as the performance indicator, we circumvent this\nproblem since TTS still grows linearly with increasing disruptions even under extreme cases\ninstead of being lower-bounded. Therefore, the full convexity or concavity of the performance\ncurve can be maintained and we can roughly approximate the simulation results with a second-\ndegree polynomial.\n17As urban road networks are always subject to capacity constraints, fully antifragile traffic\ncontrol strategies may be impossible to design. Therefore, we normalize all the other perimeter\ncontrol strategies over the RL baseline method to study the relative antifragility. To quantify the\nprogressive antifragility of different methods, we calculate the skewness of the TTS distribution.\nA negative value of skewness indicates the distribution has a longer or fatter left tail and thus a\nhigher degree of concavity in the function.\n6. Results\nThis section presents the simulation results of the four methods, i.e., no-control, MPC, base-\nline RL-based method, and our proposed antifragile RL-based method under the two scenarios\nto show the property of proto-antifragility and progressive antifragility, respectively.\n6.1. Proto-antifragility\nSystems with the property of proto-antifragility can learn from past adverse events and an-\nticipate possible ongoing disruptions of similar magnitudes to enhance future performance. To\nvalidate this property, we apply disruptions with constant magnitude after training the agent\nwith the base demand profile for a certain number of episodes.\n6.1.1. Surging traffic demand\nUnlike the no-control and MPC approaches, RL-based methods are subject to performance\nvariations over simulation episodes. Therefore, in Figure 8, besides the performance curves of\nthe no-control and MPC approaches, we also show the learning curves of the baseline RL method\nand our proposed antifragile RL method. The curves are averaged from 15 simulations, while\nthe shadowed area indicates the standard deviation of the simulation results. After conducting\n50 episodes of training together with testing under disruptions, the performance of different\nalgorithms under no disruptions is then again validated. Hence, an addition of 2 testing phases\nwas implemented under and under no disruption, using the weight parameters of the neural\nnetworks obtained from the last training episode.\nIt can be noticed that two significant phenomena occurred during the initial 50 episodes with\nthe base demand. First, the proposed antifragile method can achieve better performance with less\nTTS when there is no disruption in the network. While the baseline method has almost converged\nafter 50 episodes, the proposed antifragile method seems to be capable of further improving the\nperformance with more training episodes. The reason for not continuing the training process\nuntil the convergence of both methods is to avoid overtraining the non-disruption-related weight\nparameters as well as the underfitting of the disruption-related parameters. In addition, the\nshadowed area of our proposed method represents a less significant performance variation of\nabout 6.0% less compared to the baseline method, based on the averaged performance between\nepisodes 30 and 50. This indicates that the proposed antifragile method exhibits comparably\nhigher training stability.\nAfter disruption with constant magnitude is introduced into the network. The performance\nof the two RL-based methods has first changed drastically. Then, while the proposed method\nhas an obvious tendency to reduce TTS over the following 50 episodes, the baseline method\nshows little sign of improving performance. Also, the training stability of the proposed method\nbecomes increasingly stable over episodes as the width of the shadowed area gets smaller.\nWhile the results of testing under disruption follow the same pattern as the last episode of the\ntraining process, however, when the simulation environment reverts to the previous no-disruption\n18Figure 8: Performance curves under constant disruptions\ncondition, the performance of different methods behaves differently. Since our modified MPC\naverages demand history, after experiencing episodes with disruptions, performance deterioration\nof 1.3% can be observed when there is no disruption. A similar pattern can be observed for the\nbaseline RL method. Not only is the TTS 1.5% higher in this testing episode compared to the\ntraining episode 50, but the performance variation is also significantly larger. On the contrary,\nour proposed antifragile RL method can both maintain excellent performance and achieve smaller\nvariation.\nWe normalize all the other methods over the result of the baseline RL-based method averaged\nfrom the 15 simulations, as shown in Figure 9. The performance improvement of both MPC and\nthe proposed antifragile methods can be observed. Since we average all the past experiences of the\ndemand profile, including disruptions for MPC, it is capable of performing better under the same\ndisrupted scenario, and even marginally better than the proposed antifragile method. Therefore,\nMPC can, to some extent, demonstrate the property of proto-antifragility. While maintaining\nroughly similar performance compared to the baseline method at the beginning of the scenario\nswitch, and thus better performance than MPC, the proposed method gains capability in dealing\nwith disruptions when experiencing more and more disruptions of the same magnitude, and\nreaches a reduction of 4.9% of TTS in the final episode compared to the baseline RL method.\nThis also demonstrates the property of proto-antifragility.\n6.1.2. MFD disruption\nHere we showcase the performance of different methods under the influence of three MFD\ndisruption scenarios, as illustrated in Figure 5, namely decrease of free flow speed, decrease of\nmax accumulation, and decrease of infrastructure potential, and these three scenarios correspond\nto the MFD profiles in Figure 7. After 50 episodes of training without disruption, we introduce\na constant magnitude of disruption into the simulation environment, in the form of an MFD\ncapacity drop, for another 50 episodes. Figure 10 shows the simulation results of the three MFD\ndisruptions. The comparison between these three MFD disruption settings first shows that, for\nthe baseline RL method, the difficulty in adjusting to new MFD profiles is different. While Figure\n19Figure 9: Performance difference normalized over the baseline method under constant disruptions\n10(b) with decreasing maximal traffic flow demonstrates that the baseline RL method gradually\nadapts to the new profile, Figure 10(c) shows little sign of the baseline method learning from dis-\nruptions. Figure 10(a) exhibits the adaptation to a certain degree between the other disruption\nprofiles. It is speculated that since the shape of MFD remains unchanged in Figure 10(b), this\ncreates a less deviated scenario for the RL agent to learn, whereas the change of the MFD profile\nis the greatest in Figure 10(c) as the critical accumulation also increases during such disruptions.\nThat being said, the learning curves from the proposed antifragile method demonstrate notice-\nable performance improvement in all three MFD disruption scenarios, indicating its property of\nproto-antifragility. Also, the standard deviation indicated by the shadowed area of the proposed\nantifragile method is also significantly smaller compared to the baseline method in Figure 10(a)\nand 10(c).\n6.2. Progressive antifragility\nProgressive antifragility describes a system that exhibits a nonlinear response to a linearly\nincreasing magnitude of volatility. In terms of TTS, since an increase in the traffic density or\nvehicle accumulation within the network will result in a decrease in the vehicle speed or trip\ncompletion rate, and inevitably increase the TTS, a concave response between TTS and vehicle\naccumulation can indicate a progressive antifragile system.\n6.2.1. Surging traffic demand\nThe four studied methods are tested in an environment with linearly increasing traffic demand\nfor 50 episodes after being trained with the same base demand as in the previous scenario for\n50 episodes. Furthermore, another testing phase of 20 episodes is carried out following the same\ndemand increment and the demand magnitude of the last training episode. Figure 11 shows the\nTTS of different methods under incremental disruptions in both training and testing phases. It\n20(a) Decrease of free flow speed\n (b) Decrease of maximal traffic flow\n(c) Decrease of infrastructure potential\nFigure 10: MFD change due to adversarial events\nshould be noted that Figure 11(a) and Figure 11(b) do not share the same y-axis and the result\nof the no-control approach is not presented in 11(b) due to its exceedingly high value.\nUnder this scenario, the TTS of the no-control approach grows sharply, doubling the difference\nbetween the last and the first episode under the incremental disruption of the other methods. The\nperformance deviation of the proposed method is relatively low compared to the baseline method\nunder the disruptions scenario. However, the performance deviation of both methods grows with\nthe increase in the magnitude of disruption. The deviation of the baseline method is significantly\nlarger compared to the proposed antifragile method in the last simulation episodes. Although\nthe performance of MPC under the constant disruptions scenario is marginally better than the\nproposed antifragile method, the performance falls behind the other 2 RL-based methods, as the\nmagnitude of the disruption remains low and grows linearly after the first 50 episodes. Therefore,\nwhen averaging the past demand experiences with increasing disruptions, MPC cannot gain\nsufficient knowledge of the upcoming magnitude of the disruptions. In the testing phase, it can\nbe observed that the performance deviation of the baseline RL method is significantly larger\n21(a) Training for 50 episodes\n (b) Testing for 20 episodes\nFigure 11: Performance curves under incremental disruptions\ncompared to the proposed antifragile RL method. Since in the testing phase, the agents’ weight\nparameters are no longer subject to updates as typically observed during the training phase in\nRL, the performance curves of both RL-based methods appear to be linear.\nSince the performance differences between the baseline Zhou and Gayah (2021) and the other\nmethods are hard to observe in this figure, we need to normalize the performance of all the other\nmethods over the baseline methods. Before the normalization process, to better showcase the\nnonlinear response and compare the concavity or convexity of different methods, the performance\ndeviation in the learning curves of the RL-based methods needs to be minimized. As part of the\nprocess, we utilize second-degree polynomials to approximate the performance curves of different\nmethods. After obtaining the polynomial coefficients through fitting, we reconstruct the data\npoints accordingly, with which the normalization process can be implemented, and the results\nare shown in Figure 12.\nOne advantage of using a polar coordinate system instead of Cartesian coordinates is its ability\nto accentuate performance differences, particularly when there is a high level of disruption. While\nboth the no-control approach and MPC exhibit a positive increase in performance compared to\nthe baseline method, the proposed antifragile approach stands out as the only one maintaining\na negative difference, with its absolute value becoming even larger once the disruption exceeds\napproximately 4,000 vehicles. It is important to note that the curvature of the plot in polar\ncoordinates does not necessarily imply that the data points for a specific method follow a concave\nor convex distribution due to the inherent characteristics of polar coordinates. However, it is\nevident that the curve representing the proposed method initially approaches the baseline curve,\ndemonstrating, for instance, a performance difference of -0.5% when the disruption shows roughly\n3200 additional vehicles. Subsequently, it diverges and reaches -0.5% again with around 5000\nvehicles. This behavior indicates that the proposed method exhibits relative concavity when\ncompared to the baseline method, highlighting its progressive antifragility.\nFurthermore, for a quantitative comparison of progressive antifragility across various methods,\nwe compute the skewness of each distribution and present the results in Table 3. All three\nmethods exhibit negative skewness values, implying a form of relative progressive antifragility\ncompared to the baseline method. Among them, the no-control approach displays the least\n22Figure 12: Performance difference normalized over the baseline method under incremental disruptions\ndegree of relative antifragility, while our proposed antifragile method has demonstrated the most\nsubstantial antifragility property, as evidenced by the highest absolute skewness value in its\ndistribution.\nTable 3: Normalized skewness over baseline method under incremental traffic demand\nMethods Skewness\nNo control -0.326\nMPC -0.753\nAntifragile RL-based method -1.106\n6.2.2. MFD disruption\nFigure 13 shows the learning curves of the four methods and the performance difference\nnormalized over the baseline method under different MFD disruptions. In general, the perfor-\nmance quantified by TTS shows that our proposed method has a superior performance. More\nsignificantly, the skewness of the performance difference curves under all three types of MFD\ndisruptions, as in Table 4 summarized, are negative, showcasing the property of our proposed\nmethod being progressively antifragile. In the scenario with the decrease of the critical accumu-\nlation, the skewness has reached a value of −0.991, demonstrating the most relative antifragile\nproperty of our proposed method compared to the baseline method.\n7. Conclusion\nThis research work introduces the concept of antifragility by comparing it with two other\nterms robustness and resilience, which are commonly used in transportation and traffic control\n23(a) Under decreasing free flow speed\n (b) Performance difference\n(c) Under decreasing critical flow\n (d) Performance difference\n(e) Under infrastructure potential loss\n (f) Performance difference\nFigure 13: Performance curves under incremental disruptions\n24Table 4: Normalized skewness over baseline method under growing MFD disruptions\nMFD disruption Skewness\nDecrease of the free flow speed -0.795\nDecrease of the critical accumulation -0.991\nDecrease of the infrastructure capacity -0.392\nstrategies. Through a literature review on using RL algorithms in traffic control to realize a\nrobust or resilient design, whether and how such properties can be induced in RL is investigated.\nFollowing the same idea, we manage to induce antifragility in perimeter control by modifying the\nstate space and reward function based on a state-of-the-art RL-based perimeter control method.\nWe incorporate the change rate and curvature of the traffic states in the state space to leverage the\npotential of such derivatives to feed more information to the RL algorithm. Also, a redundant\novercompensation term in the reward function has been carefully composed to empower the\nsystem to be more antifragile to disruptions.\nWe conducted a comprehensive comparison between our proposed antifragile RL-based perime-\nter control approach and three other methods: no-control, MPC, and the state-of-the-art RL-\nbased method as a baseline. This comparison was carried out on two different levels, disruptions\nwith constant and incremental magnitude, for validating proto-antifragility and progressive an-\ntifragility, respectively. Proto-antifragility describes a system that learns from past experience\nand increases performance while progressive antifragility depicts a nonlinear response to linearly\ngrowing disruption. We investigated two different types of disruptions, namely surging traffic\ndemand and MFD disruptions. The results from the study clearly demonstrated the effectiveness\nand antifragility of our proposed method, for both proto-antifragility and progressive antifragility.\nFurthermore, we put forward a novel method for quantification by comparing the skewness of the\ndistribution. Our proposed antifragile method exhibits the greatest negative skewness among the\nmethods examined, indicating its relative progressive antifragility property against disruptions.\nIn conclusion, this study is the first of its kind to pioneer the application of the antifragility\nconcept in enhancing the daily operation of engineering systems to improve performance during\nunforeseen disruptions using a learning-based algorithm. It introduces a new possibility for\nevaluating system operation under disruptive conditions. Moreover, the concept can be extended\nnot only to other traffic control strategies like traffic signal control and pricing but also to various\nengineering disciplines and industries, broadening its potential impact.\n8. CRediT authorship contribution statement\nLinghang Sun: Conceptualization, Investigation, Methodology, Visualization, Writing – orig-\ninal draft. Michail A. Makridis: Conceptualization, Methodology, Supervision, Writing - review\n& editing. Alexander Genser: Methodology, Visualization, Writing - review & editing. Cris-\ntian Axenie: Project administration, Resources, Writing - review & editing. Margherita Grossi:\nProject administration, Resources, Writing - review & editing. Anastasios Kouvelas: Supervi-\nsion, Writing - review & editing.\n259. Declaration of Competing Interest\nThis research was kindly funded by the Huawei Munich Research Center under the frame-\nwork of the Antigones project, with one of our co-authors being employed at the said company.\nOtherwise, the authors declare that they have no known competing financial interests or personal\nrelationships that could have appeared to influence the work reported in this paper.\nReferences\nAmb¨ uhl, L., Loder, A., Bliemer, M.C.J., Menendez, M., Axhausen, K.W., 2020. A functional\nform with a physical meaning for the macroscopic fundamental diagram. 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Dodin1, 2\n1Department of Astrophysical Sciences, Princeton Universi ty, Princeton, New Jersey 08543,\nUSA\n2Princeton Plasma Physics Laboratory, Princeton, New Jerse y 08540, USA\n(Received xx; revised xx; accepted xx)\nThis paper explores structure formation in two-dimensiona l magnetohydrodynamic\n(MHD) turbulence as a modulational instability (MI) of turb ulent fluctuations. We focus\non the early stages of structure formation and consider simp le backgrounds that allow for\na tractable model of the MI while retaining the full chain of m odulational harmonics. This\napproach allows us to systematically examine the validity o f popular closures such as the\nquasilinear approximation and other low-order truncation s. We find that, although such\nsimple closures can provide quantitatively accurate appro ximations of the MI growth\nrates in some regimes, they can fail to capture the modulatio nal dynamics in adjacent\nregimes even qualitatively, falsely predicting MI when the system is actually stable. We\nfind that this discrepancy is due to the excitation of propaga ting spectral waves (PSWs)\nwhich can ballistically transport energy along the modulat ional spectrum, unimpeded\nuntil dissipative scales, thereby breaking the feedback lo ops that would otherwise sustain\nMIs. PSWs can be self-maintained as global modes with real fr equencies and drain energy\nfrom the primary structure at a constant rate until the prima ry structure is depleted.\nTo describe these waves within a reduced model, we propose an approximate spectral\nclosure that captures them and MIs on the same footing. We als o find that introducing\ncorrections to ideal MHD, conservative or dissipative, can suppress PSWs and reinstate\nthe accuracy of the quasilinear approximation. In this sens e, ideal MHD is a ‘singular’\nsystem that is particularly sensitive to the accuracy of the closure within mean-field\nmodels.\n1. Introduction\n1.1.Motivation\nCoherent-structure formation from turbulence is ubiquito us in nature, intrinsically\ncompelling, and one of the precious few aspects of turbulenc e that are relatively yielding\nto analytical efforts ( Hussain 1986 ;Karimabadi et al. 2013;Smolyakov et al. 2000;\nKrashennikov et al. 2008). For magnetohydrodynamic (MHD) turbulence ( Biskamp\n2003;Beresnyak 2019 ;Schekochihin 2022 ) and the turbulent-dynamo problem\n(Brandenburg et al.2012;Tobias 2021 ;Rincon et al.2016), this has spawned a vast body\nof work known as mean-field electrodynamics, or mean-field dy namo theory ( Rädler 2007 ;\nKrause & Rädler 1980 ;Moffatt 1978 ;Brandenburg 2018 ;Gruzinov & Diamond 1994 ). In\nsuch approaches, the velocity and magnetic fields are split i nto fluctuations and mean, or\naverage, fields (the definition of average depends on the prob lem); then, various closures\nare implemented to obtain the mean-field evolution in respon se to the fluctuations\n†Email address for correspondence: sjin@pppl.gov2 S. Jin and I. Y. Dodin\n(Blackman & Field 2002 ;Brandenburg & Subramanian 2005 b;Pouquet et al. 1976;\nKraichnan 1977 ;Nicklaus & Stix 1988 ).\nThese closures ultimately rely on a truncation of a formally infinite chain of correlation\nfunctions of increasingly high order. †This is a common approach to reduced modeling\nof nonlinear systems, and one can expect it to work when the flu ctuations are small\ncompared to the mean field. However, in turbulent structure f ormation, the usual roles\nof perturbation and background are switched, that is, mean fi elds emerge as small\nmodulations of order-one correlation functions that chara cterize turbulent fluctuations.\nHence, the question of the validity of low-order closures be comes nontrivial.\nIn this paper, we are concerned with the popular closure call ed the first-order smooth-\ning (Krause & Rädler 1980 ) or second-order correlation approximation ( Rädler 1982 ).\nIt implies that the mean-field evolution is determined only b y second-order correlations\nof the fluctuations, which, in a broader context, is also know n as the quasilinear ap-\nproximation ( Dodin 2022 ). On the one hand, one can expect its validity domain to\nbe extremely limited. It can be rigorously justified only for small magnetic Reynolds\nnumbers or short correlation times ( Brandenburg & Subramanian 2005 a;Rincon 2019 ),\nand comparisons with direct numerical simulations often co rroborate this expectation\n(Pipin & Proctor 2008 ;Schrinner et al. 2005). On the other hand, the quasilinear ap-\nproximation endures in its popularity as a workhorse that is too convenient to cast\naside in analytical calculations ( Squire & Bhattacharjee 2015 ;Lingam & Bhattacharjee\n2016;Masada & Sano 2014 ;Gopalakrishnan & Singh 2023 ). Furthermore, in some cases,\nquasilinear calculations have been found to produce good ag reement with direct numerical\nsimulations (DNS) even outside of their formal validity dom ain (Käpylä et al.2006).‡It is\ntherefore important to understand when, and how exactly, th e quasilinear approximation\nfails. Answering this question requires exploring a mean-fi eld model that retains high-\norder correlations. That is what the present paper is about.\n1.2.Outline\nHere, we explore structure formation in MHD turbulence as a m odulational instability\n(Zakharov & Ostrovsky 2009 ) (MI) of the flow-velocity and magnetic-field fluctuations\ncomprising the background turbulence. (For a recent overvi ew of this approach in appli-\ncation to drift-wave and hydrodynamic turbulence, see Zhu & Dodin (2021);Tsiolis et al.\n(2020).) For simplicity, we restrict our consideration to two-di mensional (2-D) turbulence.\nAlthough bounded planar flows cannot sustain long-lasting m agnetic fields in two dimen-\nsions (an anti-dynamo theorem by Zeldovich (1957)), 2-D turbulence nevertheless hosts\nrich physics and has long served as popular testing grounds f or various aspects of MHD\nturbulence theory due to its (relative) tractability ( Pouquet 1978 ;Biskamp & Schwarz\n2001;Nazarenko 2007 ;Giorgio et al.2017;Biskamp et al.1996). The goal of this paper is\nto complement those works with a detailed study of MIs using b oth analytical calculations\nand DNS.\nSpecifically, we study how energy is transferred from primar y fluctuations to mean\nmagnetic field and mean velocity during the early stages of st ructure formation, when\nthese mean fields are weak and the primary structure can be tre ated as fixed. To analyze\n†Here, we use the term ‘truncation’ to refer to low-order clos ures in general, as opposed to\nprecisely the closure choice of setting all correlations to zero after some cutoff. The latter, which\nwe will refer to as a simple truncation, is done in the quasili near approximation discussed below\nbut not, for example, in the various τapproximations ( Brandenburg & Subramanian 2005 b;\nBlackman & Field 2002 ).\n‡Confusingly, various analytical closures and DNS can confli ct and agree in various\ncombinations ( Zhou & Blackman 2021 ). But this issue is beyond the scope of the present paper.Modulational dynamics in MHD 3\nthis process in detail, we assume a maximally reduced model o f the turbulent fluctuations,\nnamely, a spatially monochromatic primary structure. The s implicity of this model has\nbeen detracting attention from it in the literature in favor of more realistic turbulence\nsettings, but there is more to it that meets the eye. Even with a monochromatic primary\nstructure, the modulational dynamics remains remarkably i ntricate and, depending on\nthe system parameters, can change drastically (figure 1) in ways that are not captured by\nthe common closures or obvious from the standard eigenmode a nalysis ( Friedberg 1987 ).\nHowever, the simplicity of our model helps unravel this myst ery.\nWe find that, when the primary structure trulyexperiences a MI, this MI can be de-\nscribed well with typical truncated models that neglect hig h-order correlations. However,\nalready in adjacent regimes, such truncated models can fail spectacularly and produce\nfalse positives for instability. To study this process in de tail, we propose an ‘extended’\nquasilinear theory (XQLT) that treats the primary structur e as fixed but includes the\nentire spectrum of modulational harmonics (as opposed to ju st the low-order harmonics,\nas usual). From XQLT, also corroborated by DNS, we find that th e difference between\nsaid regimes is due to a fundamental difference in the structu re of the modulational\nspectrum. For unstable modes, the spectrum is exponentiall y localised at low harmonic\nnumbers, so truncated models are justified. But this localiz ation does not always occur. At\nother parameters, modulational modes turn into constant-a mplitude waves propagating\ndown the spectrum, unimpeded until dissipative scales. The se spectral waves are self-\nmaintained as global modes with real frequencies and cause b allistic energy transport\nalong the spectrum, breaking the feedback loops that could o therwise sustain MI. The\nballistic transport drains energy from the primary structu re at a constant rate until the\nprimary structure is depleted. Notably, these effects are ov erlooked if one assumes from\nthe start that the modulation spectrum is confined and that, a ccordingly, dissipation\nentirely vanishes in the limit of zero viscosity and resisti vity.\nA comment is due here regarding how the established truncate d MHD closures fit\ninto this picture. Given the indispensability of such simpl e closures, a higher-resolution\nunderstanding of their limitations can help guide and inter pret their inevitable applica-\ntion. For example, we show that both conservative and dissip ative corrections to ideal\nMHD tend to restore the validity of the quasilinear approxim ation. In this sense, ideal\nincompressible MHD is a ‘singular’ theory that is particula rly sensitive to the accuracy\nof the closure within mean-field models. That said, it is an im portant conclusion of our\nwork that, unless deviations from ideal incompressible MHD are substantial, changing the\nturbulence parameters even slightly can produce qualitati ve inaccuracies in an otherwise\nworkable reduced model. Therefore, efforts to extrapolate c losures or infer them from\ngeneral considerations should be supplemented with first-p rinciple calculations based on\nthe underlying modulational dynamics. As a step in this dire ction, we propose a spectral\nclosure that captures both the propagating spectral waves a nd MIs within our model.\n(But, of course, more work remains to be done for generic MHD t urbulence.)\nThe paper is organised as follows. In section 2, we present the basic equations and\nderive the XQLT. In section 3, we examine the ability of truncated models of low-order\nto predict the MI rate. In section 4, we introduce spectral waves and describe their\nproperties. In section 5, we propose a closure that can simultaneously capture MIs an d\nspectral waves. In section 6, we demonstrate the impact of corrections to ideal MHD\non the modulational dynamics. In section 7, we summarise our main results. Auxiliary\ncalculations are presented in appendices.4 S. Jin and I. Y. Dodin\nFigure 1. DNS of modulational dynamics seeded at τ= 0with random noise. (This figure is\ndiscussed in detail in later sections. For the notation, see sections 2.1and2.2.) Figures (a), (c),\nand (e) correspond to a modulationally unstable primary mod e (2.17) withθ±=±π/6, and (b),\n(d), and (f) correspond to a modulationally stable primary m ode withθ±=±π/3. Figures (a)\nand (b) show z+\nx(t,y), (c) and (d) show z+\ny(t,x), and (e) and (f) show the corresponding\nenergy breakdown. Specifically, Evis the normalised kinetic energy ( 2.21c),Eb=Eb/Etotis\nthe normalised magnetic energy ( 2.21d),Emodis the normalised modulational-mode energy\n(2.21b), andEpriis the normalised primary-mode energy ( 2.21a). The colour bar shows the\nfield amplitudes normalised to A/p. The spatial coordinates are normalised to 1/p.\n2. Extended quasilinear theory\n2.1.Base model\n2.1.1. Incompressible resistive MHD\nAs a base model, we assume incompressible resistive MHD with homogeneous mass\ndensityρ=const. The corresponding governing equations are\n∂tv+(v·∇)v= (b·∇)b−∇P+ν∇2v, (2.1a)\n∂tb=∇×(v×b)+η∇2b. (2.1b)\nHerevis the fluid velocity; b.=B/√4πρis the local Alfvén velocity (the symbol.=\ndenotes definitions), i.e. rescaled magnetic field B; andP.= (Pkin+B2/8π)/ρis the\nnormalised total pressure, with Pkinbeing the kinetic pressure. Also, νis the viscosity\nandηis the resistivity, both of which are assumed either constan t or negligible (except\nat small enough scales). Due to incompressibility and magne tic Gauss’s law, one also has\n∇·v= 0,∇·b= 0. (2.1c)Modulational dynamics in MHD 5\nIn order to put ( 2.1) in a more symmetric form, let us rewrite them in terms of the\ntwo Elsässer fields z±(Elsasser 1950 ), which are also solenoidal:\nz±.=v±b,∇·z±= 0. (2.2)\nThis leads to two coupled equations for z±:\n∂tz±=−(z∓·∇)z±−∇P+ν+∇2z±+ν−∇2z∓, (2.3)\nwhereν+.= (ν+η)/2andν−.= (ν−η)/2. The normalised total pressure Pcan be found\nas follows. By taking the divergence of ( 2.1a) and using ( 2.2), one obtains\n∇2P=−∇·[(z∓·∇)z±]. (2.4)\nLet us introduce the wavevector operator ˆk.=−i∇, soˆk2.=ˆk2=−∇2. Let us also assume\nsome appropriate (say, periodic) boundary conditions. The n, (2.4) yields\nP=−ˆk−2ˆk·[(z∓·ˆk)z±]+const, (2.5)\nwhence∇Pin (2.3) can be expressed through z±. Alternatively, ∇Pcan be eliminated\nfrom ( 2.3) by taking the curl of this equation and rewriting the result in terms of the\nElsässer vorticities\nw±.=∇×z±. (2.6)\nIndeed, due to ( 2.2), one can express z±using a vector potential a±such that\nz±=∇×a±. Let us assume the gauge such that ∇·a±= 0. Then,\nw±=∇×(∇×a±) =−∇2a±≡ˆk2a±, (2.7)\nwhence\nz±= iˆk−2(ˆk×w±). (2.8)\n(Here, we assume that z±have zero spatial average, so ˆk2is invertible.) Then, ( 2.3) can\nbe expressed through w±alone:\n∂tw±=−ˆk×{[(ˆk׈k−2w∓)·ˆk](ˆk׈k−2w±)}−ˆk2(ν+w±+ν−w∓). (2.9)\n2.1.2. 2-D collisionless model\nFor simplicity, below we adopt a 2-D model, meaning that z±lie in the (x,y)plane\nand∂z= 0. In this case, the only potentially nonzero component of w±is thez-\ncomponent, w±.= (∇×z±)z. We also forgo an explicit treatment of the viscosities\nν±below. (However small, though, these viscosities remain no nnegligible in that they\ndetermine absorbing boundary conditions at infinity in the s pectral space; see section 4.)\nThen, the vector equation ( 2.9) can be replaced with a scalar equation for w±:\n∂tw±=ǫlmz/bracketleftBigg/parenleftBiggˆkm\nˆk2w∓/parenrightBigg/parenleftBig\nˆklw±/parenrightBig\n−/parenleftBiggˆknˆkl\nˆk2w∓/parenrightBigg/parenleftBiggˆknˆkm\nˆk2w±/parenrightBigg/bracketrightBigg\n, (2.10)\nwhereǫlmzis the Levi–Civita symbol with the third index fixed to z. Note that the right-\nhand side of ( 2.10) vanishes whenever w+orw−is zero. This means that any stationary\nw+is a solution as long as w−is zero and vice versa.\n2.1.3. Fourier representation\nAssuming the Fourier representation\nw±=/summationdisplay\nkw±\nk(t)eik·x, (2.11)6 S. Jin and I. Y. Dodin\nwherex.= (x,y), one arrives at the following equation for the Fourier coeffic ientsw±\nk:\n∂tw±\nk=/summationdisplay\nk1,k2T(k1,k2)w∓\nk1w±\nk2δk,k2+k1, (2.12)\nwhere the dot is the time derivative, δis the Kronecker delta, T(k1,k2)are the coupling\ncoefficients given by\nT(k1,k2) = [ez·(k2×k1)](k2+k1)·k2\nk2\n1k2\n2, (2.13)\nandejis the unit vector along the jth axis. Accordingly, the kinetic and magnetic energy\nare given by\nEv=ρ\n8/summationdisplay\nk1\nk2|w+\nk+w−\nk|2≡/summationdisplay\nkEv,k, (2.14a)\nEb=ρ\n8/summationdisplay\nk1\nk2|w+\nk−w−\nk|2≡/summationdisplay\nkEb,k, (2.14b)\nwhereEv,k, andEb,kare the corresponding spectral energy densities. Then, the total\nenergy density is\nE=ρ\n4/summationdisplay\nk1\nk2(|w+\nk|2+|w−\nk|2)≡/summationdisplay\nkEk, (2.15)\nwhereEkis the spectral total-energy density.\n2.2.Harmonic coupling mediated by a primary structure\n2.2.1. Equation for the modulational spectrum\nLet us explore modulational stability of a Fourier harmonic with a given wavevector\nk=pand the corresponding order-one Fourier amplitudes w±\np. We will call it the primary\nharmonic or, if multiple ps are present, primary structure. Let us choose axes such tha t\np=peyand assume a perturbation with a wavevector q. As can be seen from ( 2.13),\nnontrivial coupling requires a component of qperpendicular to p, and the component of\nqparallel topdoes not qualitatively affect the modulational dynamics. †For simplicity\nthen, let us assume q=qex. Let us also assume w±\np=O(1),w±\nq+np=O(ǫ), and same for\nany otherw±\nk/negationslash=p, whereǫ≪1is a small parameter. (Remember that O(ǫ)means, loosely\nspeaking, ‘scales as ǫor smaller’ but not necessarily ‘of order ǫ’.) Then, by linearizing\n(2.12) inǫ, one obtains the following chain of equations:\n∂tw±\nk≈T(p,k−)w∓\npw±\nk−+T(−p,k+)w∓\n−pw±\nk+\n+T(k−,p)w∓\nk−w±\np+T(k+,−p)w∓\nk+w±\n−p, (2.16)\nwherek±.=k±pandk=q+npwith integer n. For all other k, one obtains ∂tw±\nk≈0.\nIn particular, this means that w±\npcan be considered fixed; i.e. A±.=w±\np≈const. In terms\nof the Elsässer fields, this corresponds to a primary structu re\nz±=−2\npA±sin(py+θ±)ex, (2.17)\nwhereA±.=|A±|andθ±.= argA±.\n†As shown below, the modulational dynamics is determined mos tly by the asymptotic form\nof the coefficients α±\nnandβ±\nn(introduced in section 2.2.2) at largen, and this form is insensitive\nto the component of qparallel to p.Modulational dynamics in MHD 7\nFor the modulation energy density and the primary-wave ener gy density,\nEmod.=/summationdisplay\nk=q+npEk=/summationdisplay\nnρ\n4/summationdisplay\nσ=±1|wσ\nq+np|2\nq2+n2p2, (2.18a)\nEpri.=Ep=ρ\n4/summationdisplay\nσ=±1|wσ\np|2\np2, (2.18b)\none has\ndEmod\ndt=/summationdisplay\nnρ\n4/summationdisplay\nσ=±1wσ∗\nq+np∂twσ\nq+np\nq2+n2p2+c.c., (2.19a)\ndEpri\ndt=ρ\n4/summationdisplay\nσ=±1wσ∗\np˙wσ\np\np2+c.c., (2.19b)\nwhere ‘c.c.’ stands for ‘complex conjugate’. Using\n∂twσ\np=/summationdisplay\nk=q+npT(k,p−k)w−σ\nkwσ\np−k, (2.20)\nfork=q+np, one finds that our approximate equations conserve the total energy\ndensityEtot=Epri+Emodexactly within this approximation. (For brevity, from now o n\nwe will refer to the energy densities simply as energies.) We will assume that the initial\nmodulational energy is negligible, so Etot=ρA2/4p2, whereA2.= (A+)2+(A−)2, and\nwe will also use the following dimensionless energies:\nEpri=Epri/Etot, (2.21a)\nEmod=Emod/Etot, (2.21b)\nEv=Ev/Etot, (2.21c)\nEv=Eb/Etot. (2.21d)\n2.2.2. Dimensionless equations\nIn order to reduce the number of free parameters, let us perfo rm a variable transfor-\nmationw±\nq+np∝ma√sto→ψ±\nn:\nw±\nq+np=/radicalBigg\nA3(q2+n2p2)\nA∓p2exp/parenleftbigg\ninθ++θ−\n2/parenrightbigg\nψ±\nn, (2.22)\nwhereψ±\nn=O(ǫ). Then, ( 2.16) becomes\n∂tψ±\nn=A∓e∓iθα−\nnψ±\nn−1+√\nA+A−e±iθβ−\nnψ∓\nn−1\n+A∓e±iθα+\nnψ±\nn+1+√\nA+A−e∓iθβ+\nnψ∓\nn+1, (2.23)\nwhereθ.= (θ+−θ−)/2andα±\nnandβ±\nnare given by\nα±\nn.=∓rr2+n(n±1)/radicalbig\n(r2+n2)(r2+(n±1)2), (2.24a)\nβ±\nn.=−rn/radicalbig\n(r2+n2)(r2+(n±1)2). (2.24b)\nNote that these coefficients depend only on the ratio r.=q/prather than on qandp\nseparately. From ( 2.23), one can also see that the individual phases θ+andθ−per se\nare, expectedly, unimportant and the dynamics is affected by these phases only in the\ncombination θ+−θ−.8 S. Jin and I. Y. Dodin\nIt is also convenient for our purposes to introduce\nτ.=At (2.25)\nas the new dimensionless time, as well as φ.= arctan( A−/A+)∈[0,π/2], so that\nA+=Acosφ,A−=Asinφ. (2.26)\nIn particular, φ= 0andφ=π/2correspond to structures consisting only of w+\npandw−\np,\nrespectively, and φ=π/4corresponds to a ‘balanced’ background with A+=A−. For\nsimplicity,φ=π/4will be assumed from now on throughout the paper (except wher e\nspecified otherwise). In this case, one has\nEpri,b\nEpri,v=|w+\np−w−\np|2\n|w+p+w−p|2= tan2θ, (2.27)\nwhereEpri,vandEpri,bare the kinetic and magnetic energy densities in the primary\nmode respectively. Hence, θcan be understood as the parameter that determines the\nrelative weights of the kinetic and magnetic-field energy in the primary-mode energy.\nIn particular, θ= 0corresponds to a primary structure that involves only veloc ity\nperturbations, while θ=π/2corresponds to a primary structure that involves only\nmagnetic-field perturbations.\nUsing this notation, one can express ( 2.23) in the following compact form:\n˙ψn=/summationdisplay\nσ=±1Gσ\nnψn+σ (2.28)\n(the dot denotes ∂τ), or even more compactly as\n˙ψ=Mψ. (2.29)\nHere,ψn.= (ψ−\nn,ψ+\nn)⊺is a two-component column vector (the symbol⊺denotes trans-\nposition),ψ.= (...,ψ−1,ψ0,ψ1...)⊺is an infinite-dimensional block vector consisting\nofψn, the matrices Gσ\nnare given by\nGσ\nn.=/parenleftbigg\nασ\nncosφe−iσθβσ\nn√cosφsinφeiσθ\nβσ\nn√cosφsinφe−iσθασ\nnsinφeiσθ/parenrightbigg\n, (2.30)\nandMis the following block matrix:\nM.=\n...... ...\n0G+\n−20 0 0\nG−\n−10G+\n−10 0\n...0G−\n00G+\n00...\n0 0 G−\n10G+\n1\n0 0 0 G−\n20\n... ......\n.\nIt is readily seen then that the system’s dynamics is control led only by the dimensionless\nparameters r,θ, andφ, whileAandpdetermine only its characteristic temporal and\nspatial scales, respectively.\nEquations ( 2.28) and ( 2.29) are similar to the equations that describe a one-dimension al\nchain of coupled linear oscillators. The nth elementary cell of the chain consists of two\ndifferent oscillators characterised by ψ+\nnandψ−\nn, respectively. Similar equations emerge\nin studies of phase-space turbulence for the coupling betwe en Hermite moments of the\nparticle phase-space probability distribution ( Hammett et al. 1993;Nastac et al. 2023;Modulational dynamics in MHD 9\n10-5100|0+|(a)\nXQL\nNL\n10-5100|1+|(c)\n0 25 5010-6100|2+|(e)-202Im  0+10-5\n(b)\n-202Im  25+10-5\n(d)\n0 100 200-202Im  50+10-5\n(f)\nFigure 2. Comparison of the time evolution of various ψ+\nnas predicted by DNS of the nonlinear\nequation ( 2.3) (red dots) and XQLT ( 2.28) (black curves) for two representative cases of the MI\n(θ=π/6) [(a), (c), (e)] and stable modulational dynamics ( θ=π/3) [(b), (d), (f)]. Nonlinear\nDNS are seeded with random noise, resulting in a broadband mo dulational dynamics, but the\nresults shown are for a modulation corresponding to r= 0.5, and the XQLT simulations are\ninitialised accordingly. For the θ=π/6case, the harmonics shown are: (a) n= 0, (b)n= 1,\nand (c)n= 2. In this case, XQLT adequately approximates the nonlinear d ynamics until about\nτ∼30. For theθ=π/3case, the harmonics shown are: (d) n= 0, (e)n= 25, and (f)n= 50.\nIn this case, XQLT adequately approximates the nonlinear dy namics indefinitely.\nAdkins & Schekochihin 2018 ;Parker 2016 ;Kanekar et al. 2015). Throughout this work,\nwe use ( 2.28) and ( 2.29) to study the nature of collective oscillations in the chain of\nmodulational harmonics, which can be understood as Floquet modes of the linearised\nsystem. Unstable oscillations of this system (i.e. MIs) lea d to structure formation on top\nof the primary structure.\nOne can also consider ( 2.29) as a Schr��dinger equation with a Hamiltonian iM. This\nHamiltonian is not Hermitian, because modulations are para metrically coupled with\nthe primary mode (through which energy can be either gained o r lost) and dissipate\natn→ ∞. At the same time, this Hamiltonian is invariant under the ti me-reversal\ntransformation ( i→ −i) and the parity transformation in the spectral space ( n→ −n,\nσ→ −σ). This makes the system ( 2.28)PT-symmetric ( Bender 2005 ;Bender et al.\n2019). Depending on the balance of sources and sinks, such system s can support modes\nwith entirely real frequencies (unbroken PTsymmetry) and pairs of modes whose fre-\nquencies are mutually complex-conjugate (broken PTsymmetry). This will be discussed\nfurther in section 4.\nAs long as the underlying ordering assumption [ |w±\nq+np|/|w±\np| ∼ O(ǫ)] holds, XQLT10 S. Jin and I. Y. Dodin\n(2.28) provides an excellent approximation of the nonlinear modu lational dynamics of\n(2.3) (figure 2). However, practical applications of XQLT require truncat ions of this\nsystem. In the remainder of this work, we discuss to what exte nt such truncations can\nbe adequate and compare our analytical results to DNS of ( 2.28), in lieu of the full ( 2.3).\n3. Truncated models\n3.1.Basic equations\nA common way to truncate a problem like ( 2.29) is to postulate that all ψnwith\n|n|>Nare negligible. One can also understand this as imposing refl ective boundary\nconditions in the oscillator chain {ψσ\nn}:\nψσ\nN+1=ψσ\n−(N+1)= 0. (3.1)\nRetained in this case are harmonics with wavevectors pandq+np, withn=\n0,±1,...,±N. This means that the resulting system has Q= 1+(1+2 N)degrees\nof freedom. We call this procedure Q-mode truncation ( QMT). The corresponding\ntruncation of ψwill be denoted as ψQMT, and the corresponding truncation of Mwill\nbe denoted as MQMT, so (2.29) becomes\n˙ψQMT=MQMTψQMT. (3.2)\nBecause of ( 3.1), eigenmodes of this system can be understood as standing wa ves in\nthe oscillator chain {ψσ\nn}and have the form ψQMT=Yexp(−iωτ), whereωare global\nfrequencies and Yare constant ‘polarization vectors’. According to ( 2.29), these vectors\nsatisfyDY= 0, whereD= iω1+MQMTand1is a unitQ×Qmatrix. Then, ωcan\nbe found from\ndetD= 0 (3.3)\nand MIs correspond to modes with Imω>0.\nThe success of this approach hinges on the assumption that hi gher harmonics truly\nremain negligible. In other words, a QMT should be able to adequately describe the\nevolution of a modulational mode if its eigenvector ψQMTis heavily weighted towards\nthe low-order harmonics. Below, we explore to what extent th is assumption is satisfied\nin various regimes.\n3.2.Growth rates\nThe lowest-order QMT is 4MT, which corresponds to N= 1. Within this model, the\nprimary harmonic with k=pis assumed to interact with the modulational mode at\nk=qonly through two sidebands k=q±p; then,ψ4MT= (ψ−1,ψ0,ψ1)⊺and\nM4MT=\n0G+\n−10\nG−\n00G+\n0\n0G−\n10\n. (3.4)\n(This model can be understood as quasilinear, because, alth ough the modulational\ndynamics remains nonlinear, the second- and higher-order h armonics of the perturbation\nare neglected.) The 4MT equations yield\nω2=r4\n1+r2/parenleftbigg\n1±/radicalBig\n1−(1−r−4cos22θ)sin22φ/parenrightbigg\n. (3.5)\n[Note that these are normalised frequencies corresponding to the normalised time τ. To\nobtain the actual physical frequencies, one must multiply t he right-hand side of ( 3.5) byModulational dynamics in MHD 11\nA.] Assuming φ=π/4, one finds that two out of four branches of this dispersion rel ation\nare unstable at |r|<1. The corresponding growth rate Γ.= Imωis maximised at θ= 0\nandθ=π/2at the value\nΓ=r/radicalbigg\n1−r2\n1+r2(3.6)\n(from now on, we assume r>0for clarity of notation), and the corresponding polarizati on\nvectors are as follows:\nY0=/braceleftBigg\n(1,+1)⊺, θ= 0,\n(1,−1)⊺, θ=π/2.(3.7)\nThe caseθ= 0can be recognised as a purely hydrodynamic Kelvin–Helmholt z instability\n(KHI), i.e. one that does not involve magnetic field. The case 0<θ<π/ 4can be\nunderstood as the MHD generalization of the KHI.\nPotentially more accurate is a model with N= 2, or 6MT, which accounts for har-\nmonics withk=q±2p; then,ψ4MT= (ψ−2,ψ−1,ψ0,ψ1,ψ2)⊺and\nM6MT=\n0G+\n−20 0 0\nG−\n−10G+\n−10 0\n0G−\n00G+\n00\n0 0 G−\n10G+\n1\n0 0 0 G−\n20\n. (3.8)\nOne can derive an analytical expression for ωfrom here just like we derived ( 3.5) from\n(3.4). However, this expression is cumbersome and not particula rly instructive, so it is\nnot presented explicitly. Truncations with larger Q, albeit also explored by us, will not\nbe presented either for the same reason.\nFigure 3shows a comparison of the 4MT and 6MT predictions and the MI gr owth\nrates inferred numerically from nonlinear DNS of ( 2.9). Note that the truncated models\nquantitatively agree with each other and with the DNS at some parameters but drastically\ndisagree at other parameters. In particular, the 4MT predic ts thatΓis an even function\nofπ/2−θ, while the 6MT predicts that this is not the case. Furthermor e, the DNS\npredicts that, for example, at r= 0.5, the system is stable at all θ>π/2[figure 3(b)].\nLet us discuss this in detail.\nAs expected from the analytical formula ( 3.6) based on the 4MT, and as we have\nalso found numerically, MI is typically suppressed at r/greaterorsimilar1; hence, below we focus on\nthe regime r∈(0,1). A comparison of the growth rate predicted by nonlinear DNS a nd\ntruncated models in this range is presented figure 5. The corresponding primary modes\nwithθ/lessorsimilarπ/4are modulationally unstable, and the growth rates predicte d by 4MT and\n6MT are in good agreement with DNS. Such primary structures a re dominated by the\nvelocity field, Epri,v>Epri,b[see (2.27)], and thus will be called v-dominated primary\nmodes (VDPMs). At θ/greaterorsimilarπ/4, primary structures are dominated by the magnetic field,\nEpri,b>Epri,v, and thus will be called b-dominated primary modes (BDPMs). As seen\nin figure 5, BDPMs are modulationally stable, but truncated models pre dict otherwise.\nFigures 4(a) and (b) show the discrepancy between the predictions of n onlinear DNS,\n4MT, and 6MT for representative parameters, specifically, θ∈(0,π/2)andφ=π/4.\n(Stability is primarily determined by θandr. Deviations of φfromπ/4result only\nin quantitative adjustments, unless φis close to 0orπ/2.) In other words, truncated\nmodes tend to overestimate the growth rate and systematical ly produce false positives\nfor instability. Figure 5shows that, although the magnitude of disagreement decreas es\nwithN, the systematic overprediction of instability persists.12 S. Jin and I. Y. Dodin\nFigure 3. (a) The growth rate Γ(of the most unstable mode) versus the normalised wavenumbe r\nratθ= 0: 4MT (blue), 6MT (red), and nonlinear DNS of ( 2.9) (black markers). (b) Same for\nΓversusφatr= 0.5. The 4MT predicts identical growth rates at φ= 0andφ=π, while the\n6MT and nonlinear DNS predict that the system is unstable at φ= 0and stable at φ=π. The\ngrowth rates are calculated with nonlinear DNS by adding ran dom noise with ky= 0noise to\nthe primary mode, allowing this modulational noise to evolv e, and then picking out the growth\nrate of the most unstable mode. The noise amplitude is taken s mall enough such that it does\nnot affect the results.\nFigure 4. The difference between the growth rate predicted by a QMT and that inferred\nfrom XQLT DNS, ΓQMT−ΓDNS, versus (θ,r)forφ=π/4: (a) 4MT, (b) 6MT. The dashed\ncurves mark the stability boundary as determined by a parame ter scan with nonlinear DNS.\nThe preponderance of the red colour, especially outside the instability domain, indicates that\ntruncated models tend to produce false positives for instab ility.\nThe reason for this overprediction can be understood by anal yzing the eigenmode\nstructure. (See also appendix Afor an alternative explanation.) As to be discussed\nin section 4, unstable modes are evanescent spectral waves , whose |Yn|exponentially\ndecrease with |n|[figure 6(a)]:\n|Yn| ∝exp(−|n|/l). (3.9)\n(We assume φ=π/4for simplicity, so |Y+\nn|=|Y−\nn|and the upper index in |Yσ\nn|can be\nomitted.) That is why truncated models correctly predict MI for modes that are actually\nunstable. The spectral scale ldepends on the system parameters, and the smaller l\nthe more accurate QMT is. At some parameters, though, particularly when θ→π/4,\nlbecomes large or even infinite, such that |Yn|asymptotes to a nonzero constant atModulational dynamics in MHD 13\nFigure 5. The growth rates Γobtained through a QMT forθ= 0(blue markers) and θ=π/2\n(orange markers) versus the number of modes retained, N. The solid coloured lines indicate the\ncorresponding DNS growth rates, to which the rates predicte d by the truncated models converge\nat largeN.\n|n| → ∞ [figure 6(b)]. In this case, QMT is bound to fail. The corresponding modes are\npropagating spectral waves (PSWs). While they also receive energy from the primary\nstructure at low n, PSWs transport this energy along the spectrum to |n| → ∞ . There,\nthe energy is unavoidably dissipated by viscosity or resist ivity, however small those are.\nThis dissipation makes PSWs more stable than any QMT would predict, because QMTs\nassume that the mode energy forever resides at small n, where dissipation is small or\nzero. Below, we discuss these effects in more detail.\n4. Spectral waves\n4.1.Dynamics at large |n|\nFirst, let us consider spectral waves at |n| ≫1. In this limit, one has α±\nn→ ∓rand\nβ±\nn→0, so (2.29) yields two decoupled wave equations for ψ+\nnandψ−\nn:\n˙ψ−\nn=rcosφ/parenleftbig\neiθψ−\nn−1−e−iθψ−\nn+1/parenrightbig\n, (4.1a)\n˙ψ+\nn=rsinφ/parenleftbig\ne−iθψ+\nn−1−eiθψ+\nn+1/parenrightbig\n. (4.1b)\nThese equations allow solutions in the form of monochromati c waves:\nψσ\nn=Ψσexp(−iωτ+iKσn), (4.2)\nwith constant amplitude Ψσ, frequency ω, and ‘wavenumber’ Kσ. The quotation marks\nare added as a reminder that the corresponding waves propaga te in the spectral space\nrather than in the physical space. Also note that K±are defined unambiguously only\nwithin the first Brillouin zone, K±∈(−π,π).\nFrom ( 4.1), one readily finds that spectral waves obey the following di spersion relations:\nω=ωσ\n0sinκσ, κ±.=K±±θ, (4.3)\nwhereω+\n0= 2rsinφandω−\n0= 2rcosφ. Accordingly,spectral waves in the nspace along\ntheψσchannel have the group velocity\nvσ\ng=ωσ\n0cosκσ(4.4)\nand can propagate either up or down the spectrum (figure 7). The dependence of ωand\nv±\ngonθcan be removed by a gauge transformation. Specifically, κ±becomes the true\nwavenumber if one adopts the variables ¯ψ±\nn.=ψ±\nne∓inθinstead ofψ±\nn. Also notably, to the14 S. Jin and I. Y. Dodin\nFigure 6. Typical structures of eigenmodes, specifically, |Yn|versusn, atr= 0.5andφ=π/4\nfor various θ: (a) unstable modes supported by VDPMs; (b) oscillatory sol utions supported by\nBDPMs. (The upper index in |Yσ\nn|is omitted because |Y+\nn|=|Y−\nn|for the assumed parameters.)\n|Yn|decreases exponentially with nfor the former (notice the logarithmic scale) but asymptote\nto nonzero constants for the latter. The asymptotes are indi cated by the dotted lines. These\nresults are obtained through DNS of ( 2.29) using the initial conditions of the form ( 4.6) and\nnormalised such that |Y0|= 1.\nextent that the (large) index ncan be approximately considered a continuous variable,\n(4.1) can be written as usual nondispersive-wave equations for ζ±(τ,n).=¯ψ±\nn(τ):\n∂τζ±=−ω±\n0∂nζ±. (4.5)\nThis model corresponds to the small- κσlimit of ( 4.3), that is,ω≈ωσ\n0κσ.\nThe above equations can be used to describe PSW packets local ised at large |n|but not\nthe global eigenmodes (because the latter involve dynamics at small |n|). In particular,\n(4.3) are not enough to find the global-mode frequencies. Yet they allow one to determine\nthe mode structure at |n| ≫1if one knows the mode frequency from other considerations\n(or, vice versa, to find the mode frequency if κ±are known). In particular, unstable modes\nhave complex ω, so, according to ( 4.3), their asymptotic wavenumber cannot be real.\nThis explains why unstable modes are evanescent. In contras t, for realω, (4.3) allows\nfor real wavenumbers (provided that |ω|/lessorequalslantω±\n0), i.e. predicts PSWs. These results are\ncorroborated by DNS of ( 2.29). For example, figure 6shows the mode structures inferred\nfrom the asymptotic dynamics of ψσ\nn(τ)at large enough τfor the initial conditions\nψσ\nn(τ= 0) =/braceleftBigg\nǫexp(iξσ\nn), n=±1,\n0, n ∝ne}ationslash=±1.(4.6)\nHere,ǫis a constant small amplitude (within the linear approximat ion, one can always\nrescaleψσ\nnsuch thatǫ= 1), andξnare parameters that change the polarization of the\ninitial conditions while keeping the initial modulation en ergy fixed. (The specific values\nofξσ\nnare given in the captions of figures in which initial conditio n dependent results of\nDNS are presented.) This form of the initial conditions is ch osen to emulate a simple\nmodulation of the primary mode.\nThe fact that the properties of spectral waves are determine d only by the asymptotic\nproperties of α±\nnandβ±\nnmakes these waves particularly robust. In particular, note that\nthe asymptotic form of α±\nnandβ±\nnat|n| → ∞ remains the same even when pandq\nare not orthogonal, so the above results readily extend to ge neralq. Likewise, a similar\npicture holds also for three-dimensional interactions, al though the mode polarization can\nbe more complicated in this case. Notably, PSWs provide ball istic rather than diffusive (or\nsuper-diffusive) energy transport along the spectrum. This distinguishes them from theModulational dynamics in MHD 15\nFigure 7. Numerical simulations showing PSW packets propagating: (a ) down the spectrum\n(|n|of the packet center increases with time) and (b) up the spect rum (|n|of the packet\ncenter decreases with time). In both cases, r= 0.5andθ=π/3. Both packets are initialised\nusingψ+\nn(τ= 0) = exp[ −(n−n0)2/ς+iK+n]withn0= 200 ,ς= 150 , and: (a) K+=−π/6,\n(b)K+=π/2.\ncascades that are commonly discussed in turbulence theorie s and result from eddy–eddy\ninteractions, or wave–wave collisions ( Galtier et al. 2000).\n4.2.Global modes in the form of PSWs\nIn realistic settings, spectral waves are excited at finite |n|and propagate down the\nspectrum. Evanescent waves reach |n| ∼l[see (3.9)] in finite time τ0, get reflected, and\neventually (asymptotically) settle as stationary standin g waves on time scales of the\norder of a few τ0. Such waves can be adequately described by truncated models . In\ncontrast, PSWs continue to propagate toward infinity indefin itely (until they dissipate\nat scales where viscosity or resistivity is no longer neglig ible). Thus, they never develop\ncomponents propagating up the spectrum at large |n|and so they never become quite\nlike the standing-wave eigenmodes predicted by truncated m odels (section 3.1). This\nmeans that actual PSWs cannot be adequately described by ana lyzingQMT in principle.\nInstead, they should be considered in the context of the init ial-value problem, much like\nLangmuir waves appear in plasma kinetic theory within Landa u’s initial-value approach\nas opposed to the Case–van Kampen true-eigenmode approach ( Stix 1992 ).\nConsequently, even though ( 2.29) has infinitely many degrees of freedom, the number\nofrelevant global modes, i.e. those that propagate towards infinity, is finite. We find\nthat there are only two of them, which can be attributed to the fact that dimψn= 2.\n(Likewise, at a given wavenumber, only one, Langmuir, branc h of relevant electrostatic\nwaves in electron plasma remains relevant after transients are gone.) For example, for\nφ=π/4, we find from DNS that\nω=±√\n2rcosθ. (4.7)\nThis dependence and the corresponding mode structures are i llustrated in figure 8. Each\nof the two modes has two wavenumbers associated with it: one d eterminesψσ\nnatn→+∞,\nand the other one determines ψ−σ\nnatn→ −∞ (figure 9). Thus, there are four κσ\ndoverall,\nwhereσdenotes the corresponding component of ψn, andddenotes the direction of\npropagation ( d=±1is the sign of the group velocity at |n| → ∞ ). They can be found16 S. Jin and I. Y. Dodin\nFigure 8. (a) The dependence of the global-mode frequency ω>0onθfor various r. The\ndashed lines are the inferred solutions ( 4.7). (b)|Y+\nn|(blue) and |Y−\nn|(orange) for the mode\nwithω>0. The eigenmode amplitudes for the ω<0mode are identical with the roles of |Y+\nn|\nand|Y−\nn|switched.\nby combining ( 4.7) with ( 4.3) and applying sgnvg=d, i.e.ωσ\n0dcosκσ\nd>0. This yields\nω>0 :κ+\n+=π\n2−θ, κ−\n−=−3π\n2+θ, (4.8a)\nω<0 :κ+\n−=3π\n2−θ, κ−\n+=−π\n2+θ, (4.8b)\nwhereκσ\nd=Kσ\nd+σθand the values of Kσ\ndare defined modulo the Brillouin zone (i.e.\nK−\n−=−3π/2should be understood as π/2and so on). Remember, though, that this\napplies only to global eigenmodes. Transient waves propaga ting in the region |n| ≫1can\nhave anyκσ(figure 7).\nThe fact that the frequencies of these modes are real indicat es that the modes are self-\nsustained notwithstanding the dissipation that they unavo idably experience at |n| → ∞ .\nThe corresponding evolution of the wave spectrum is illustr ated in figure 9(a) and the\ncorresponding evolution in the physical space is shown in fig ure10. This remarkable\ndynamics is understood from the fact that the energy drain vi a this dissipation is balanced\nby the energy injection from the primary mode at small |n|. As seen from ( 2.19) and\n(2.16), the dimensionless modulational energy ( 2.21b), or\nEmod.=/summationdisplay\nσA\nA−σ|ψσ\nn|2≡/summationdisplay\nnEn, (4.9)\nis governed by\n˙En=/summationdisplay\nσ(Iσ\nn+Fσ\nn)+c.c., (4.10)\nwhereInandFnare given by\nIσ\nn=A√\nAσ\n(A−σ)3/2(β−\nneiσθψ−σ\nn−1+β+\nne−iσθψ−σ\nn+1)ψσ∗\nn, (4.11)\nFσ\nn=A\nA−σ(α−\nne−iσθψσ\nn−1+α+\nneiσθψσ\nn+1)ψσ∗\nn. (4.12)\nBecause/summationtext\nnFσ\nn= 0, the terms Fσ\nnrepresent the energy flux that is carried along the\nmodulation spectrum and is conserved within each sub-chann elσ. In contrast, Incan be\nunderstood as injection terms in that they also appear, with the opposite signs, in theModulational dynamics in MHD 17\nFigure 9. DNS of ( 2.29) showing global-mode PSWs for r= 0.5,θ=π/3, andφ=π/4:\n(a)Reψ+\nn/ǫfor a PSW seeded by the initial conditions ( 4.6) withξσ\nd=dexp(σiπ/4)(colour\nbar). The dashed lines indicate the fronts propagating at: ( i) the maximum spectral speed √\n2rand (ii) the actual group velocity of the mode. The field betwe en these dashed lines\nconsists of transients, which are negligible behind the sec ond front. (b) The total normalised\nenergy of the modulation, Emod/ǫ2, versusτ(black), along with its kinetic (red) and magnetic\n(blue) components. (c) The profiles of the spectral energy de nsity atτ= 100,150,200, clearly\nexhibiting left- and right-propagating fronts. The horizo ntal dashed line indicates the average\nspectral energy density En, where the average is taken over the PSW period and over nbetween\nthe energy fronts.\nequation for the primary wave,\n˙Epri=−/summationdisplay\nn/summationdisplay\nσ=±1Iσ\nn+c.c., (4.13)\nwhich follows from ( 2.20).\nThe linear spectral waves discussed so far correspond to the regime when the change\nofEpriis negligible. Eventually, though, as higher harmonics are populated and thus\nabsorb more energy, the primary wave will be depleted. This m eans, in particular, that\nno primary wave can be truly stationary. Even an arbitrarily small perturbation will\ngenerally launch a PSW, and this PSW will eventually deplete the primary wave.18 S. Jin and I. Y. Dodin\nFigure 10. The global-mode PSW mode with ω>0,r= 0.5, andθ=π/3: same as figure 9, but\nin the real space as opposed to the spectral space. Specifical ly shown are: (a) z+\nyas a function\nof(τ,x)aty= 0, (b)z+\nxas a function of (τ,y)atx= 0.\n4.3.Anomalous dissipation\nSince a PSW mode carries energy towards |n| → ∞ , eventually, a large enough |n|\nis reached where the energy is dissipated regardless of how s mall the viscosity and\nresistivity are. Thus, a PSW exhibits an effective, or ‘anoma lous’, dissipation rate γthat\nis independent of νandηin the limit ν,η→0. This effect is different from the anomalous\ntransport caused by eddy–eddy collisions in turbulence (fo r example, see Donzis et al.\n(2005)) in that the energy transport caused by a PSW is ballistic. A s a result, γis\nstraightforward to calculate, which is done as follows.\nAs discussed in section 4.2, a global PSW at large |n|has a flat mode structure, i.e.\na structure with Enindependent of n. This structure establishes itself as an expanding\n‘shelf’ whose edges (wave fronts) propagate across the modu lational spectrum to n→ ±∞\nat the group velocity vg(figure 11). Since the height of the shelf, En(the overbar here\ndenotes averaging over the PSW temporal period and over all nwithin the shelf), remains\nconstant, this process drains energy from the primary mode l inearly in time:\n˙Epri=−2|vg|En=const, (4.14)\nwhere we consider the dynamics average over the PSW temporal period. Hence,\nγ.=−˙Epri\nEpri= 2|vg|En\nEpri∼ω±\n0ǫ2. (4.15)\n(The rate relative to the physical time t, as opposed to τ, isAtimes thisγ.)\nLet us again assume the initial conditions ( 4.6) andφ=π/4. Through DNS\nof (2.29), we find that the global PSW-mode amplitude is maximised whe n\n|arg(ξσ\n±1/ξ−σ\n±1)|=|arg(ξσ\n±1/ξσ\n∓1)|=π, irrespective of θ. (Any particular choice of ξσ\nn\nthat satisfies this condition affects only phases of the modul ational dynamics and does\nnot impact averaged quantities that determine γ.) This corresponds to the case when\nall modulational-seed energy is in the magnetic field; i.e. Eb=EandEv= 0. Figure 11\nshows the corresponding anomalous dissipation rate normal ised to the seed energy, along\nwith its determining factors – the system-dependent PSW gro up velocity vgand the\naverage spectral energy density En, which is determined by the initial conditions. It can\nbe seen that, for the MI unstable VDPMs ( θ<π/4), spectral waves are relatively slowModulational dynamics in MHD 19\nFigure 11. (a) The magnitude of the group velocity |vg|of the global PSW mode (i.e. the speed\nat which the energy front propagates along the spectrum) ver susθfor various r. (b) Same for\nthe average spectral energy density En/ǫ2. (c) Same for the resulting average drain rate γ/ǫ2,\nwhereγis defined in ( 4.15). The initial conditions used throughout these figures are g iven by\n(4.6), withξσ\n1=−ξσ\n−1= exp(σiπ/2).\nand have a small amplitude. The transition to modulational s tability occurs at |vg| ∼Γ.\nThis condition can be understood as a threshold beyond which (i.e. at|vg|/greaterorsimilarΓ) PSWs\nprovide a sufficiently fast escape route for the energy inject ed at small |n|, such that the\npositive feedback loops [see ( 4.11)] supporting MIs can no longer be sustained. In the\ncontext of PTsymmetry (section 2.2.2), the spectral group velocity can be understood\nas the effective coupling between the links in the oscillator chain, connecting the energy\ninjection from the primary mode at small nto the energy sink at n→ ∞. As|vg|\nincreases with θ, the system transitions from broken to unbroken PTsymmetry.\nAlso, notice the following. If modulations at multiple qs are present, they launch inde-\npendent cascades along k=q+npand thus the drain on the shared primary mode will\nbe additive. This regime is illustrated by figures 1(d)-(f), which show the modulational\ndynamics for a BDPM ( θ=π/3) seeded with noise. The rather nondescript modulational\ndynamics seen in figures 1(d) and (e) can be understood as the superposition of the many\nstructures like that in figure 10for variousq. Also, the linear-in-time depletion of the\nprimary mode seen in figure 1(f) can be understood as the sum of the PSW-driven energy\nfluxes along the constituent modulational channels.\n5. Unified closure\nAs discussed in section 3.1, naiveQMTs assume the perfectly reflecting boundary con-\nditions ( 3.1) for spectral waves, thus precluding PSWs and ignoring pote ntial dissipation\nat large |n|. One can expect that, instead of ( 3.1), it would be more accurate to adopt\nthe following closure in anticipation of spectral waves:\nψσ\nN+1= eiKσψσ\nN (5.1)\n(and similarly for ψσ\n−(N+1)). The value of Kσis found, for given ω, from ( 4.3):\neiKσ= e−iσθ/bracketleftBig\niΩσ+/radicalbig\n1−(Ωσ)2/bracketrightBig\n, (5.2)\nwhereΩσ.=ω/ωσ\n0, and we restrict our attention to the range |ReΩ|/lessorequalslant1to avoid\ndiscontinuities, a choice which is consistent with the rang e of observed solutions. The\nsign of the square root is chosen to enforce outward propagat ion atImω= 0, which\nalso enforces that unstable solutions exponentially decay in the direction of propagation20 S. Jin and I. Y. Dodin\nFigure 12. Theθ-dependence of the global-mode frequencies derived from tr uncated models\nwith the closure ( 5.3) for various N:Reω(left column) and Imω(right column). (a) and (b):\nN= 2. (c) and (d): N= 3. (e) and (f): N= 4. The red and blue curves indicate the branches\nthat are the best match to the, respectively, unstable and st able solutions obtained through\nDNS (black dashed lines). The gray curves show the remaining spurious solutions due to finite\nN. Note that MIs exist only for θ∈(0,π/4), while PSWs exist in the entire range θ∈(0,π/2).\n|exp(iKσ\n+)|<1.†(Conversely, damped solutions grow in the direction of prop agation.)\nAs seen from figure 12, the resulting closure\nψσ\nN+1= e−iσθ/bracketleftBig\niΩσ+/radicalbig\n1−(Ωσ)2/bracketrightBig\nψσ\nN (5.3)\nleads to reasonably accurate results already N= 2, both for MIs and PSWs.\nBecause the closure becomes exact only in the limit N→ ∞, it also yields spurious\nsolutions (gray curves in figure 12) at finiteN. One can understand this from the analogy\nwith the Case–van Kampen modes mentioned in section 4.2. These solutions become\nincreasingly stable, and thus less of an issue, as Nincreases. Also note that the closure\nadequately describes true MIs. This is explained by the fact thatψσ\nnat large |n|are\nexponentially small and do not matter for unstable modes any way.\n6. Quasilinear approximation beyond ideal MHD\nAs discussed in the previous sections, the existence of PSWs undermines the stan-\ndard quasilinear approximation in application to ideal inc ompressible MHD. Given the\nubiquity of quasilinear modeling in the literature, it may s eem concerning that the\nquasilinear approximation can fail so spectacularly. Inte restingly, though, the quasilinear\n†A different choice of signs would also yield formally valid so lutions, but would not have the\nproperties that correspond to PSWs of physical interest.Modulational dynamics in MHD 21\napproximation is somewhat more robust beyond the ideal-MHD limit, both due to\nconservative corrections and dissipation.\nLet us discuss the former first. Without attempting to descri be any particular physical\nsystem, let us consider the following modified version of ( 2.3):\n∂tz±=−(z∓·∇)z±−∇P+λ∂x∇2z±, (6.1)\nwhereλis a constant parameter. The last term is intended as a simple ad hoc correc-\ntion that, while causing deviation from ideal MHD, leaves th e primary-mode evolution\nunaffected and preserves MHD’s key invariants, specifically , the energy and cross helicity\n(appendix B). Perhaps notably, this term introduces rotational asymme try in the (x,y)\nplane. Similar terms can appear due to a background magnetic field or differential rotation\n(Heinonen et al. 2023).\nTakingφ=π/4for simplicity, one arrives at the following corresponding modification\nof the linear ( 2.23):\n√\n2(∂τ+iδn)ψ±\nn= e∓iθα−\nnψ±\nn−1+e±iθβ−\nnψ∓\nn−1+e±iθα+\nnψ±\nn+1+e∓iθβ+\nnψ∓\nn+1,(6.2)\nwhereδ±\nn.=Λr(r2+n2),Λ.=λ/Ap3, andα±\nn,β±\nnare as in ( 2.24). In the large- |n|limit,\none hasα±\nn∼1,β±\nn→0,δn∼Λn2, so one obtains the following scaling:\n|ψ±\nn+1| ∼|ψ±\nn|\nΛn2. (6.3)\nThis shows that the harmonic magnitude |ψn|decreases rapidly (super-exponentially)\nwith|n|, and thus low-order truncations may be justified. Note that a lthough the opposite\nscaling,|ψ±\nn−1| ∼ |ψ±\nn|/Λn2is also formally possible, such modes cannot be excited, as i s\nthe case with inward propagating PSWs. Figure 13(a) shows that, indeed, the agreement\nbetween analytical QL growth rates and nonlinear DNS improv es as the parameter Λ\nincreases. The agreement becomes nearly perfect for Λ/greaterorsimilar0.5. Figure 13(b) shows that\nthe same effect can be achieved if, instead of the λterm in ( 6.1), one introduces sufficiently\nstrong viscosity. In this case, one also has a modified ( 2.23) with the exact form of ( 6.2),\nbut withδ±\nn.=µ(r2+n2), whereµ.=ν+/Ap2(withν−= 0for simplicity). Again, the\nagreement becomes nearly exact for µ/greaterorsimilar0.5.\n7. Summary\nIn this paper, we explore structure formation in two-dimens ional MHD turbulence as\na modulational instability (MI) of turbulent fluctuations. We focus on the early stages of\nstructure formation and consider simple backgrounds that a llow for a tractable model of\nthe MI while retaining the full chain of modulational harmon ics. This approach allows for\na systematic examination of the importance of high-order co rrelations that are typically\nignored in mean-field theories.\nWe find that, when the primary structure truly experiences a M I, this MI can be de-\nscribed well with typical truncated models that neglect hig h-order correlations. However,\nalready in adjacent regimes, such truncated models can fail spectacularly and produce\nfalse positives for instability. To study this process in de tail, we propose an ‘extended’\nquasilinear theory (XQLT) that treats the primary structur e as fixed but includes the\nentire spectrum of modulational harmonics (as opposed to ju st the low-order harmonics,\nas usual). From XQLT, also corroborated by DNS, we find that th e difference between\nsaid regimes is due to a fundamental difference in the structu res of the modulational\nspectra. For unstable modes, the spectrum is exponentially localised at low harmonic\nnumbers, so truncated models are justified. But this localiz ation does not always occur. At22 S. Jin and I. Y. Dodin\nFigure 13. The growth rate Γ, atν−= 0, versus: (a) Λ.=λ/Ap3and (b)µ.=ν+/Ap2. The\ncorresponding ‘center of energy\" of the unstable eigenmode ,n.=/radicalbig/summationtext\nnEnn2/E, is shown in (c)\nand (d), respectively. In all figures, the colour markers ind icate the results obtained through\nDNS of ( 2.29), while the black solid curves indicate solutions obtained from the 4MT truncation\nof (6.2). The results are presented for the representative cases θ= 0andθ=π/2, both atr= 0.5\nandφ=π/4.\nother parameters, modulational modes turn into constant-a mplitude waves propagating\ndown the spectrum, unimpeded until dissipative scales. The se spectral waves are self-\nmaintained as global modes with real frequencies and cause b allistic energy transport\nalong the spectrum, breaking the feedback loops that could o therwise sustain MI.\nThe ballistic transport by PSWs drains energy from the prima ry structure at a constant\nrate until the primary structure is depleted. Because globa l PSWs exist at almost\nall parameters, this means that almost any primary structur e in ideal incompressible\nMHD will eventually be depleted. This means, in particular, that sustainability of MHD\nstructures is not entirely limited to the issue of exponenti ally growing linear instabilities,\nas PSWs must also be taken into consideration. To describe th em within a reduced\nmodel, we propose a closure that successfully captures both the propagating spectral\nwaves and MIs for the assumed primary structure.\nFinally, we find that departures from ideal MHD constrains th e form of modulational\neigenmodes, in turn suppressing the amplitude and impact of high harmonics. This allows\nus to end on an informed yet optimistic note regarding the app licability of the quasilinear\napproximation for structure formation in dispersive forms of MHD. That said, it is an\nimportant conclusion of our work that, unless deviations fr om ideal incompressible MHD\nare substantial, changing the turbulence parameters even s lightly can utterly destroy the\napplicability of an otherwise workable reduced model. An un derstanding of the complex\nmodulational dynamics supported by (nearly) incompressib le MHD provides important\ncontext in the interpretation of existing simple closures a nd potentially opens the path\nto building more reliable alternatives.Modulational dynamics in MHD 23\nThis research was supported by the U.S. Department of Energy through contract No.\nDE-AC02-09CH11466.\nAppendix A. Quasilinear time scale\nThe quasilinear approximation holds when the quasilinear M I growth rate exceeds the\nrate at which energy escapes to higher harmonics. In the main text, we take the latter\nto be the group velocity of the PSW global mode. However, one m ight object that the\nproperties of PSWs are well defined only at large n, while the departure from quasilinear\noccurs at the modest n= 2. It may be instructive then to develop an alternative argume nt\nthat would not assume large n. Here, we propose such an argument by considering the\ninitial-value problem.\nFor simplicity, let us assume the initial conditions such th at onlyψ0(τ= 0) is\nnonzero, with all other ψn/negationslash=0(τ= 0) are zero. We also adopt (φ,θ) = (π/4,0)and\n(φ,θ) = (π/4,π/2)as representative cases for VDPMs and BDPMs respectively. F or\nthese parameters, ( 2.28) can be further reduced due to the parity of the initial condi tions.\nAlso, the coupling matrices Gσ\nnbecome particularly simple:\nGσ\nn=\n\n1√\n2/parenleftBigg\nασ\nnβσ\nn\nβσ\nnασ\nn/parenrightBigg\n, θ= 0,\niσ√\n2/parenleftBigg\n−ασ\nnβσ\nn\n−βσ\nnασ\nn/parenrightBigg\n, θ=π\n2.(A1)\nFor the case θ= 0, we assume assume ψ+\n0=ψ−\n0atτ= 0. Then,˙ψ+\nn(τ= 0) =˙ψ−\nn(τ= 0)\nas well, and thus ψ+\nn=ψ−\nnat all times, i.e. b≡0. We will refer to this case as v-only\ninitial conditions (VIC). The corresponding dynamics can b e described by a single\nfunctionψn.=ψ+\nn=ψ−\nn, which satisfies the equation\n˙ψn=α−\nn+β−\nn√\n2ψn−1+α+\nn+β+\nn√\n2ψn+1. (A2)\nFor the case θ=π/2, we assume ψ+\n0=−ψ−\n0atτ= 0. Similarly to the VIC case, this\nimplies that v≡0. We will refer to this case as b-only initial conditions (BIC). The\ncorresponding dynamics can be described by a single functio nψn.=−inψ+\nn= (−i)nψ−\nn,\nwhich satisfies the equation\n˙ψn=α−\nn−(−1)nβ−\nn√\n2ψn−1+α+\nn−(−1)nβ+\nn√\n2ψn+1. (A3)\nTo estimate the nonlinear time scale τNLon whichψ2might become comparable with\nψ0, let us consider ( A2) and ( A3) forn= 2at early times, when ψ3remains negligible:\n¨ψ2=\n\nα−\n2+β−\n2√\n2/parenleftBig\nα−\n1+β−\n1√\n2ψ0+α+\n1+β+\n1√\n2ψ2/parenrightBig\nfor VIC,\nα−\n2−β−\n2√\n2/parenleftBig\nα−\n1+β−\n1√\n2ψ0+α+\n1+β+\n1√\n2ψ2/parenrightBig\nfor BIC.(A4)\nInitially, |ψ0| ≫ |ψ2|, so the second term in the brackets can be neglected in both ca ses\nand the nonlinear time scale can be readily estimated:\nτ−2\nNL=1\n2×/braceleftBigg\n(α−\n2+β−\n2)(α−\n1+β−\n1)for VIC,\n(α−\n2−β−\n2)(α−\n1+β−\n1)for BIC.(A5)\nIn contrast, the quasilinear time scale τQL, which follows from the 4MT equations ( 3.4),24 S. Jin and I. Y. Dodin\nis identical in both cases:\nτ−2\nQL=α−\n0(α−\n1+β−\n1). (A6)\nThe ratio of time scales is then\n/parenleftbiggτQL\nτNL/parenrightbigg2\n=/braceleftBigg\n2√\nr2+4 for VIC,\n2r√\nr2+4\nr+4for BIC.(A7)\nFor the range of interest ( |r|<1), this always yields\nτQL\nτNL,BIC<1<τQL\nτNL,VIC. (A8)\n(For example, at r= 0.5, one hasτQL/τNL,VIC≈0.5, andτQL/τNL,BIC≈1.5.) Thus,\nquasilinear is sufficient for VIC but not for BIC, in agreement with our numerical results.\nAppendix B. Properties of ( 6.1)\nThe operator introduced in ( 6.1) conserves energy (here it is normalised to the constant\nmass density):\nE.=1\n4/integraldisplay\ndx(|z+|2+|z−|2), (B 1)\nand cross helicity:\nHC.=1\n4/integraldisplay\ndx(|z+|2−|z−|2). (B 2)\nThese can be expressed in terms of the Elsasser energies\nE±.=/integraldisplay\ndx|z±|2/4, (B 3)\nas\nE=E++E−, H C=E+−E−. 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Starting with a minimal five-band tight-binding\nmodel, we sort out all the topological states into five groups, which are determined by the interlayer and\nintralayer hopping parameters. Combined with the model, we design an algorithm to obtain a series of\nexperimentally synthesized topological semimetals with kagome and honeycomb layers, i.e., IAMX family\n(IA = Alkali metal element, M = Rare earth metal element, X = Carbon group element), in the inorganic crystal\nstructure database. A follow-up high-throughput calculation shows that IAMX family materials are all nodal-\nline semimetals and they will be Weyl semimetals after taking spin-orbit coupling into consideration. To have\nfurther insights into the topology of the IAMX family, a detailed chemical rule analysis is carried out on the\nhigh-throughput calculations, including the lattice constants of the structure, intralayer and interlayer couplings,\nbond strengths, electronegativity, and so on, which are consistent with our tight-binding model. Our study\nprovides a way to discover and modulate topological properties in stacked kagome and honeycomb crystals and\noffers candidates for studying topology-related properties like topological superconductors and axion insulators.\nI. INTRODUCTION\nThe simplest model with topological phases arises from\nthe honeycomb lattice, with the most famous material being\ngraphene, in which the band structure features a Dirac cone\nat K and a van Hove singularity (vHs) at M1–7, as shown\nin Figs. 1 (a-b). Recently, the kagome lattice, a two-\ndimensional (2D) grid structure composed of corner-sharing\ntriangles, has garnered significant attention in condensed\nmatter physics due to its intriguing and diverse array of\nexotic phenomena8–13, including charge density waves14–17,\ntopological states18–23, and quantum spin liquids24–27. The\nkagome lattice shares similar properties in the band structure\nwith the honeycomb lattice, including the Dirac cone and the\nvHs known as saddle point. Nonetheless, it also manifests\ndistinct characteristics of flat band28, as depicted in Figs. 1\n(c-d). Recently discovered kagome superconductors, such as\nCsV 3Sb5, have also sparked curiosity regarding the potential\nof unconventional superconductivity and intertwined phase in\nkagome lattice29–38.\nIn the past, topological materials with kagome lattice and\nhoneycomb lattice were studied separately, including nodal-\nline semimetals and Weyl semimetals39–43, among which\nstudies on kagome lattices were mainly focused on magnetic\nmaterials, such as magnetic Weyl semimetals Co 3Sn2S219,20,44\nand Fe 3Sn245–47. Thus, the interplay between the kagome\nlattice and honeycomb lattice is still in the early stage and\nneeds to be explored on both the analytical model level and\nmaterials level.\nIn this work, we first construct a minimal five-band\ntight-binding (TB) model to study the topology of stacked\nhoneycomb and distorted kagome lattice. Then, based on\nthe results of the TB model, we sweep through the inorganic\ncrystal structure database (ICSD)48and discover 298\nexperimentally synthesized new ideal topological materials,\ni.e., IAMX family with the stacked distorted kagome and\nhoneycomb lattice (IA = Alkali metal element, M = Rare\nearth metal element, X = Carbon group element)49–53. The\ncrystal structure of IAMX family is characterized by the\n(a) (b)\nΓ M K Γ\nΓ M K Γ(d) (c)\n(e) (f)\nΓ M K ΓEnergy Energy EnergyFIG. 1. Schematic diagrams of three lattices and their band\nstructures. (a-b) are for the honeycomb lattice, (c-d) are for the\ndistorted kagome lattice, and (e-f) are for the system with stacked\nkagome and honeycomb lattice.\nhexagonal lattice with Fe 2P-type, with the space group of\nP-62m(No. 189)54,55, and the interlayer interactions of\nthe IAMX family are chemical bonds rather than the van\nder Waals forces comparing to the previous studied kagome\nmaterials. We perform a high-throughput calculation for\nIAMX compounds and discover four distinct topological\nnode-ring configurations in the absence of spin-orbital\ncoupling (SOC), which is closely associated with different\nchemical conditions, including the lattice constants of\nthe structure, interlayer and intralayer interactions, bond\nstrengths, electronegativity, and so on. Our numerical results\nmatch our TB model very well. Further calculation shows that\nLiNdGe is not a superconductor with Tc<10−4K under thearXiv:2402.12700v1  [cond-mat.mes-hall]  20 Feb 20242\nNoYes\nHoneycomblattice:7705Kagomelattice:2275IAMX:298YesYesYesWyckoffpositionwith 3?Wyckoffpositionwith 2?Wyckoffpositionwith 2and3?MaterialsfromPymatgen\nStackedKagomeandhoneycomblattice:1798Hexagonal?\nFIG. 2. The flowchart for screening compounds with kagome lattice,\nhoneycomb lattice, and their stacking lattice. First, the lattice should\nbe a hexagonal one. Then, the crystal is classified based on their\nWyckoff positions. If the Wyckoff position has 2? (? stands for\ndifferent letters to distinguish different Wyckoff positions with the\nsame site symmetry), the compound has an isolated honeycomb\nlattice; If the Wyckoff position has 3?, the compound has an isolated\nkagome lattice; If the Wyckoff positions have both 2? and 3?,\nthe compound has the stacked structure of kagome and honeycomb\nlattice. We obtain 298 IAMX compounds with stacked lattices and\nideal topological properties, which will be further proceeded with\nhigh-throughput calculations based on DFT.\nBCS scenario, which implies IAMX family materials may be\ncandidates for unconventional (topological) superconductors\nwhen the superconductivity is achieved by experiments. Our\ntheory and findings offer a method for the stable realization\nof topological states, particularly Weyl semimetals, in the\nstacked kagome and honeycomb lattices. Combining the\ntopological states with the charge instability arising from vHs\nin kagome material14–17, we provide an effective pathway for\nachieving charge density wave (CDW) in the Weyl semimetal\nstate56–60. Our findings present an efficient approach for\nexploring the axion insulator state in kagome materials.\nII. METHOD\nThe minimal five-band TB model for stacked kagome and\nhoneycomb lattice without SOC is shown in Eq. (1), with pz\norbital at Wyckoff position 1a, forming a honeycomb lattice,\nanddz2orbital at Wyckoff position 3f, forming distorted\nkagome lattice. The model belongs to space group P-\n62m(No.189) and with the next-next-neighbor hopping, as\nshown in Fig. 3 (a)61. Fig. 3 (c-h) are the node-ring\nconfigurations obtained in the tight-binding model without\nSOC with different parameters, as marked by the blue lines.\nAfter considering SOC, these nodal rings will evolve into\nWeyl points marked by red dots.The material screening process follows the flowchart shown\nin Fig. 2. We first pick out all compounds with hexagonal\nlattices, then diagnose whether the compounds are stacked\nkagome and honeycomb lattices by the atomic Wyckoff\npositions. If atoms are located at Wyckoff positions of 3? or\n2?, there will be a kagome and honeycomb sublattice. Here,\n3? means there will be 3 symmetry-related atoms in the unit\ncell, and ? represents different letters for different Wyckoff\npositions with the same site symmetry. In this work, we\ndiscover 400 IAMX family candidates for DFT calculation,\nof which 298 are ideal topological materials for further study.\nHigh-throughput DFT calculations for the IAMX family are\nperformed by Vienna ab initio simulation package (V ASP)62.\nThe exchange-correlation functional is described by the\ngeneralized gradient approximation (GGA) with the Perdew-\nBurke-Ernzerhof (PBE) functional type63,64. The cutoff\nenergy for plane-wave basis is set to 520 eV and the BZ\nis sampled by 7 ×7×11Γ-centered kmesh. The structural\noptimization is performed until forces on each atom are\nless than 0.001 eV/ ˚A. To further obtain the topological\ninformation, we constructed the tight-binding model with d\norbitals of M atoms and porbitals of X atoms based on the\nmaximally-localized Wannier functions (MLWF) method65.\nThe nodal-ring configurations are obtained by WannierTools\npackage66.\nIII. RESULTS AND DISCUSSIONS\nA. TB models\nA minimal five-band TB model is constructed for the\nstacked honeycomb-distorted kagome lattice, with pzorbital\nat Wyckoff position 1a (atomic coordinate of (1/3,2/3,1/2),\nforming the honeycomb lattice), d2\nzorbital at Wyckoff\nposition 3f (atomic coordinate of (x,0,0), forming the\ndistorted kagome lattice), in the non-SOC case:\nHnSOC =X\ni∈α,βϵic†\nici+X\n<i,j> ∈αt1c†\nicj+X\n<i,j> ∈βt2c†\nicj\n+X\ni∈αtz1c†\nici+X\ni∈βtz2c†\nici+X\ni∈α,j∈βtzc†\nicj+h.c.\n(1)\nwhere αandβare the labels for kagome lattice and\nhoneycomb lattice. t1andt2are the hopping between nearest\nsites in two sublattices, i.e., the intralayer hoppings in each\nsublattice, as illustrated in Fig. 3 (a). tzis the interlayer\nhopping between two sublattices. tz1andtz2are the hopping\nfor the same sublattices in different unit cells along the\nstacking direction, respectively. The Hamiltonian can be\nexpressed in the matrix form below:\nHnSOC =\u0012HhHhk\nH†\nhkHk\u0013\n(2)3\nt1 t2\n G      M      K     G      A      L      H      A|L      M|H     K    -3-2-10123Energy(eV)EFEF\n(∆e0+2∆tz+2t1)>t2>\n(∆e0+2∆tz+4t1)/3\nt2>(∆e0+2∆tz+2t1)-(∆e0+2∆tz+4t1)/3>t2\n>-(∆e0+2∆tz+2t1)-(∆e0+2∆tz+2t1)>t2>\n-(∆e0-2∆tz+4t1)/3-(∆e0-2∆tz+4t1)/3>t2>\n-(∆e0-2∆tz+2t1)t2<-(∆e0-2∆tz+2t1)(a) (b) (c)\n(d) (e) (f) (g) (h)non-SOC\nSOC\nii\ni iv\nFIG. 3. (a) Lattice model, (b) band structure, and (c-h) topological configuration for the stacked lattice tight-binding model. The blue lines\nrepresent the nodal ring in the non-SOC case, while the red dots represent the Weyl points in the SOC case. The shape of the nodal ring\nchanges with various t2, with ∆e0=e1−e2,∆tz=tz1−tz2as depicted in the figure. The results of tight-binding model with parameters\ne1= 1, e2=−1, t1=−0.2, tz= 0.2, tz1=−0.5, tz2= 0.05, andt2=0.7 (d), 0.3 (b)-(c), -0.3 (e), -0.7 (f), -0.9 (g), -4 (h).\nHh=\u0012\ne2+ 2tz2coskz t2e−1/3i(2kx+ky)(1 +eikx+ei(kx+ky))\nt2e−1/3i(kx+2ky)(1 +eiky+ei(kx+ky)) e2+ 2tz2coskz\u0013\n(3)\nHhk=\u0012\ntz(−1 +eikz)ei(d1·k)tz(−1 +eikz)ei(d2·k)tz(−1 +eikz)ei(d3·k)\ntz(−1 +eikz)ei(d4·k)tz(−1 +eikz)ei(d5·k)tz(−1 +eikz)ei(d6·k)\u0013\n(4)\nHk=\ne1+ 2tz1coskzt1e−iA(1 +ei(kx+ky))t1e−iB(1 +eiky)\n∗ e1+ 2tz1coskz t1e−iC(1 +eikx)\n∗ ∗ e1+ 2tz1coskz\n (5)\nwhered1-d6are the vectors connecting the nearest neighbor\nkagome lattice and honeycomb lattice sites ( d1= (x−\n1/3,1/3,−1/2),d2= (−1/3, x−2/3,−1/2),d3= (2/3−\nx,1/3−x,−1/2),d4= (x−2/3,−1/3,−1/2),d5=\n(1/3, x−1/3,−1/2),d6= (1/3−x,2/3−x,−1/2)) andA\n-Care the vectors connecting the nearest kagome lattice sites\n(A= (x,1−x,0),B= (2x−1, x,0),C= (x,2x−1,0)).\nBy fine-tuning the parameters of the TB model, nodal-ring\nsemimetals with different configurations can be obtained, as\nthe five different configurations shown in Figs. 3 (d-h). The\nnode-ring topology is protected by mirror symmetry, thus\nensuring their stable presence on the mirror planes50,67–70.\nAfter considering SOC, these nodal-ring band crossings willopen energy gaps, giving rise to the emergence of Weyl points\nat generic momenta, as shown by the red dots in Fig. 3.\nWe note that we only tune t1andt2, i.e., the intralayer\nhopping strengths within two sublattices for simplicity, and\nkeep the others fixed. Fig. 3 (b) shows the typical band\nstructure without SOC depicted by solid blue lines and with\nSOC depicted by dashed red lines. The nodal ring pinned\non the kz= 0 plane shown Fig. 3 (c) is the band crossing\nwith a filling number of 2, which will evolve into 6 pairs of\nWeyl points after SOC is taken into consideration. Figs. 3\n(c-h) show the evolution of nodal-ring configuration as the\nparameter t2increases. The single nodal ring centered at Γ\nwill cross the BZ boundary and become several nodal rings4\n(a) (b)IA\nX M\nkc\nML\nKH A\nkakbΓ M001M210M120 M110-\n(e) (d)the IA atoms\nthe M atoms\nthe X atoms\nNd Ge\n5d 4pa1’\ne’\ne’’dz2\ndxy dx2\ndxz dyzpz\npx py(c)\nNo. 36\nNo. 39No. 43\nNo. 44 No. 45\nNd: dz2Ge: pz\nΓ M HL A K A Γ5\n01234\n-3-2-1Energy (eV)\n36394344, 45LiNdGe\nGeNd\nFIG. 4. (a) Crystal structure of the IAMX family. The green, yellow, and purple balls represent IA, M, and X atoms, respectively. (b) BZ\nfor the IAMX family, where the light green, light yellow, light blue, and dark gray planes are the M1−10,M120andM210andM001planes,\nrespectively. (c) The energy splitting in the bonding process for the orbitals of Nd and Ge atoms in the crystal around the Fermi level. (d) The\nband structure of LiNdGe in the non-SOC case, with the projections of Ge pzorbital and Nd dz2orbital labeled in blue and red, respectively.\n(e) Wavefunctions for the energy bands at Γaround the Fermi level, showing the compositions of Ge pzorbitals and Nd dz2orbitals.\ncentered at K as t2increases, and it gradually transforms into a\nnodal ring centered at the Γpoint in Fig. 3 (c), further evolving\ninto multiple nodal-ring patterns depicted in Fig. 3 (e-h) as t2\nincreases. Such an evolution process indicates the intralayer\nhopping strength in the sublattice will play an important role\nin the topological configuration.\nB. Scanning Materials with Stacked Kagome and Honeycomb\nLattice\nBy following the flowchart shown in Fig. 2 and the minimal\nfive-band TB model, we discover that the IAMX family\n(IA = Alkali metal element, M = Rare earth metal element,\nX = Carbon group element) are ideal topological materials\nwith stacked kagome and honeycomb lattice by scanning\nthe ICSD. They belong to the space group P-62m(No.\n189), which has three equivalent mirror symmetries M1−10,\nM120and M210, an inequivalent mirror symmetry M001\nperpendicular to the z-axis, a threefold rotation symmetry\nC3zalong z-axis and a twofold rotation symmetry C2(110) .IA atoms occupy the Wyckoff position 3g, and M atoms are\nlocated at position 3f, forming a distorted kagome lattice, as\ndepicted in Fig. 1 (c). X atoms occupy two nonequivalent\nWyckoff positions 1a and 2d, and the latter ones form the\nhoneycomb lattice, as shown in Fig. 1 (a). Thus, the\ncrystal of the IAMX family can be regarded as composed\nof distorted kagome lattice and honeycomb lattice stacking\nalong zdirection, as depicted in Fig. 4 (a). Fig. 4 (b) is the\nfirst BZ of the IAMX family, with M1−10,M120andM210\nrepresented by light green, light yellow, and light blue planes,\nrespectively, and the dark gray plane represents the non-\nequivalent (001) mirror plane. It is worth highlighting that\nthe interlayer interactions of the IAMX family are chemical\nbonds rather than the van der Waals forces, which is distinct\nfrom previously discovered two-dimensional or quasi-two-\ndimensional materials featuring the kagome lattice. Thus, the\ninterplay between the kagome lattice and honeycomb lattice\nmay bring new physical properties for their 3D nature.5\nC. Orbital Analysis for the IAMX family\nSince all the IAMX family materials share similar\nelemental components, band structures, and topology, we will\ntake LiNdGe as an example for further analysis. In LiNdGe,\nporbital from Ge forms a honeycomb lattice, and dorbital\nfrom Nd forms a distorted kagome lattice, and they will\nsplit into different bands under the crystal field, as shown in\nFig. 4 (c). After the orbital hybridization between dz2and\npz, bonding and antibonding bands can be obtained around\nthe Fermi level, forming the band structure depicted in Fig. 4\n(d). Fig. 4 (e) shows the orbital charge density distributions\nof different bands forming the kagome and honeycomb bands\nnear the Fermi level, both the band structure and the orbital\ncomponents can be captured by the aforementioned five-band\nTB model.\nD. Topological Analysis for the IAMX family\nTo study the topological information of the IAMX family\ncompounds, we calculate band structures, surface states,\nnodal-ring configuration, etc., through a high-throughput\ncalculation, with SOC taken into account or not, respectively.\n298 ideal topological materials in the IAMX family are\nproceeded for further analysis. Within this dataset, we\ncategorize the configurations of nodal rings into four main\ngroups, as shown in Fig. 5: (i) multiple nodal rings centered\nat K point on kz= 0 plane; (ii) a closed nodal ring located\non the kz= 0 or πplane centered at Γor A; (iii) a nodal-\nchain composed by six nodal rings on the mirror planes and\nan additional nodal ring on kz= 0 plane centered at Γ; (iv)\nthree nested nodal rings on the mirror planes, accompanied\nby a nodal line at the intersection of the Γ-Z path. These\nfour types of nodal-ring configuration highly depend on the\nchemical rule of the IAMX family, which will be discussed in\ndetail below.\nE. Chemical Rules for The Topology\n1. Lattice Constant and Bonding strength\nTo study the influence of the overall bonding strength on\nthe topological configuration, we categorize the distributions\nof 298 IAMX family materials based on their lattice constants\naandcof the compounds in Fig. 5, with type (i)-(iv) node-\nring configurations fully separated in four regions, denoted by\nblue, black, red, and brown dots, respectively. We notice that\ndifferent nodal-ring configurations can be separated by three\nlinear dashed lines, showing a strong correlation between the\ntopological configuration and the lattice constants.\nLattice constant acorresponds to the parameters t1andt2\nin our TB model since it exerts influence on intralayer hopping\nin kagome or honeycomb lattice, while lattice constant c\ncorresponds to the parameter tzsince it specifically impacts\nthe interlayer coupling. If cis fixed and adecreases in\nIAMX family materials, the increase of intralayer coupling\nNaYbPbNaEuPbLiYbSnLiYbSiLiEuSnLiEuPbLiLuSnRbDyGeKTmSiKGdSnKCePb\nRbNdCRbHoCRbEuPbRbEuGe\nRbDyCNaPrCNaGdGe\nNaEuGeNaDyCLiTmSi\nLiNdCLiErCKYbCKPrPb\nKLuCKEuSiKEuGe\n6.57.07.58.08.59.04.04.55.05.5\nLatticeconstanta(Å)Latticeconstantc(Å)\n4.04.55.05.5\n6.57.07.58.08.59.0iviiiiiiFIG. 5. The distribution of four nodal-ring configurations under\ndifferent lattice constants aandcfor the IAMX family. Different\nnodal-ring configurations are labeled in different colors, separated\nby linear slashes.\nwill lead to a change of the nodal-ring configuration from\nthe right to the left of the region in Fig. 5. As adecreases,\ntype-(i) configuration, i.e., nodal rings centered at K points\non the kz= 0 plane, shrink gradually and transform into\ntype-(ii) nodal ring, i.e., a closed nodal ring centered at\nΓ. When the closed nodal ring on the kz= 0 plane\ngradually shrinks and disappears, six nodal rings emerge\non the vertical mirror planes, forming type-(iii) node-ring\nconfiguration, progressively approaching the kzaxis, and\neventually converging into type-(iv) node-ring configuration,\ni.e., three nested nodal-rings. As discussed in the TB model\npart, different node-line configurations can be obtained by\ntunning the parameter t1, which is consistent with the node-\nring configuration transition from (i) to (iv) by decreasing the\nlattice constant a, in which process the bond strength in each\nsublattice will be enhanced.\n2. Intralayer and Interlayer bond strength\nTo explore the relationship between the topological\nconfiguration and the intralayer and interlayer chemical\nbonds, we study the node-ring configuration distribution of the\nIAMX family with different intralayer atomic distances in the\nhoneycomb lattice ( dh), kagome lattice ( dk) and the interlayer\nlattice ( dhk), as shown in Fig. 6.\nFigure 6 (a) illustrates the distribution relationship of all\nnode-ring configurations based on the distance between X\natoms forming the honeycomb lattice ( dh) and the distance\nbetween M atoms constituting the kagome lattice ( dk). All\nthe data points are predominantly aligned along a straight\nline with the same gradient, which can be attributed to the\nlinear dependence of both dhanddkon the lattice constant\nafor different types of configurations. The intercepts of\nthe lines, delineated by points representing distinct node-\nring configurations in the diagram, exhibit correlation with\nthe Wyckoff coordinates of the distorted kagome layer.6\n3.03.54.04.55.05.56.02.53.03.54.04.55.05.5Distancebetweenkagomeatoms(Å)Distancebetweenkagomeatoms(Å)\n2.62.83.03.23.43.63.83.54.04.5dkdh(a)\n2.53.03.54.04.55.05.5\nDistancebetweenhoneycombatoms(Å)3.03.54.04.55.05.56.0\n3.03.54.04.55.0\n2.62.83.03.23.43.63.8(b)\ndhk\nDistancebetweenkagomeandhoneycombatoms(Å)\nFIG. 6. (a) The distribution of all IAMX family compounds with\ndifferent intralayer atomic distances in both the honeycomb lattice\n(dh) and kagome lattice ( dk). (b) The distribution of all IAMX\nfamily compounds with different interlayer atomic distances between\nthe honeycomb and kagome lattice ( dhk). Dots with different colors\nrepresent different node-ring configurations, as shown in Fig. 5. All\nthe data is distributed in a linear way, showing a strong relation\nbetween the node-ring configuration and the atomic distances in the\nIAMX family.\nThus, different types of configurations can be distinguished\nthrough different magnitudes of intercepts, which highlights\nthe significance of intralayer hopping as a key contributing\nfactor influencing the formation of nodal-ring configurations.\nFurthermore, we also analyze the distribution of topological\nconfigurations based on the distance between the atoms\ncomprising the kagome and honeycomb lattice dhkand the\ndistance between the atoms forming the kagome lattice dk, as\ndepicted in Fig. 6 (b). The interlayer hopping leads to more\neffective discrimination of various node-ring configurations,\nand simultaneously, a continuous transition links the type-\n(i) and type-(ii) node-ring configurations labeled by blue and\nblack points, which is in alignment with Fig. 5.\n3. The Role of Chemical Bond Strength for Nodal-line\nConfiguration\nType-(i) configuration corresponds to a single closed nodal\nring centered at Γ. If the radius becomes larger, the nodal\nLatticeconstanta(Å)Radiusofnodal-rings(Å)\n6.57.07.58.08.59.00.020.040.060.080.101/(3#)−0.011\n0.040.080.10\n0.06\n6.57.07.58.08.59.00.02iiiFIG. 7. The distribution of IAMX family materials with the type-(i)\nconfiguration (blue points) and type-(ii) configuration (black points)\nwith respect to the lattice constant aand nodal-ring radius, which\nare distributed in the light blue and grey regions. These two kinds\nof configurations are divided by a dashed line with a slope of1√\n3a,\nwhere ais the lattice constant, showing that these two configurations\nare similar in nature.\nLiYPb\nLiTmPbLiSmPbLiPrPbLiNdPb\nLiLuPbLiLaPbLiGdPbLiErPbLiDyPbLiCePb\nLiYSn\nLiTmSnLiSmSnLiPmSnLiNdSn\nLiLuSnLiLaSnLiHoSnLiGdSn\nLiErSnLiDySnLiCeSn\n7.27.37.47.57.67. 77.80.0000.0050.0100.0150.020\n2*Radiusofnodal-ringsRadiusofnodal-rings(Å)\nLatticeconstanta(Å)0.0000.0050.0100.0150.020\n7.27.37.47.57.67.77.8iii\nFIG. 8. The distribution for the radius of type-(iii) configuration\nmaterials (red points in Fig. 5) with respect to lattice constant a.\nring will cross the BZ boundary and become six nodal\nrings centered at K, which will transform into the type-(ii)\nconfiguration. In order to have a further study on the effect\nrole of chemical bond strength on node-ring configurations,\nwe analyze the relationship between the configurations of\ntype-(i) and type-(ii) nodal rings, i.e., the nodal-ring radius\nwith respect to lattice constants a, as illustrated in Fig. 7.\nThese two configurations are well separated by a dashed line\nwith a slope of 1/(√\n3a), and they show a continuity at the\nboundary of these two regions. The increasing tendency\nfor the radius of the nodal ring as the lattice constant a\nincreases on a whole shows that there is a strong correction\nbetween the topological configuration and the inplane lattice\nconstant. Since the chemical bond strength in the kagome\nand honeycomb sublattices is also strongly corrected to the\nintralayer lattice constant, we conclude that the intralayer\nchemical bond strength plays an important role in modulating7\nthe topological configuration. However, we note that Fig. 7\nshows a downward trend in each local part, which will be\nfurther explored in Part. III E 4.\nSimilarly, we also analyze the type-(iii) materials (red\npoints in Fig. 5) on the relationship between the node-ring\nradius and the lattice constant a, as depicted in Fig. 8.\nOverall, their distribution exhibits an increasing tendency as\naincreases, showing a strong correction between the node-\nring radius and the intralayer bond strength, which matches\nthe results shown in Fig. 7.\n4. The Influence of Electronegativity on Topology\nAs we discussed above, even within the same node-ring\nconfiguration, the relationship between the node-ring radius\nand the lattice constant amay locally show an opposite\ntendency compared to the global tendency, as shown in Fig. 7.\nTo have a deep insight into such phenomena, we take materials\nin type-(ii) configuration (black points) as an example, and\nfurther divide them into six groups, as shown in Fig. 9, with\nX taking the carbon element and IA taking Li, Na, K, and\nRb, respectively. Overall, the radius of the nodal ring and\nthe lattice constant aare linearly related, but with different\nslopes in these six groups. The slope of the linear dispersion\ndecreases and even becomes zero when the IA element varies\nfrom Li to Rb. If we fix the IA element to Na, then the slope\nof the distribution gradually becomes negative, as shown in\nFig. 9 (e-f).\nTo gain further insight, we randomly selected one material\nfrom each group for further calculation. Our results reveal that\neach of the three elements in the IAMX family shows different\nvalence states in each group. When the valence state of IV\ngroup element X is greater than -2, the slope of the distribution\ncurve is positive. By contrast, when the valence of the X\natom is less than -2, the slope is negative. While the valence\nstate of X approaches -2, the slope of the curve tends to be\naround zero. These results suggest that the electronegativity\nor electron affinity of the X element significantly influences\nthe bonding strength, leading to variations in the node-ring\nradius and the topological configurations.\nF. Topological properties of LiNdGe with SOC\nSince materials within the IAMX family share similar\ncrystal structures, band structures, and topological properties,\nwe choose LiNdGe as an example to show the electronic\nstructures and topological properties after taking SOC into\nconsideration.\nThe band structure of LiNdGe without SOC is depicted\nin Fig. 4 (d), where a nodal ring is located on the kz=\n0plane, protected by the M001symmetry. When SOC is\ntaken into account, all band crossings on the nodal ring are\nfully gapped, generating six pairs of Weyl points around the\nmirror plane, making LiNdGe a Weyl semimetal with energy\nbands contributed from the distorted kagome and honeycomb\nlattice. The distribution of these Weyl points is depicted inFig. 10 (a), where blue/red dots represent Weyl points with\nnegative/positive monopole charge, and they are constrained\nby the M001andC3zsymmetries. Figure 10 (b) shows the\nband structure crossing two time-reversal-symmetry-related\nWeyl points along P−Γ−P′, and the energy of Weyl\npoints is just 1.4 meV above the Fermi level, which means that\nIAMX family materials are ideal platforms to study physical\nproperties induced by the Weyl fermions.\nTo have a better understanding of the topological\ninformation, we perform the mirror Chern numbers (MCN)\ncalculations on the two non-equivalent M001andM1−10\nplanes based on the Hamiltonian obtained from Wannier90.\nFig. 10 (c-d) indicate the MCN of M001is 2, while that of\nM1−10is 1. According to the previous studies on TaAs71,72,\nif two non-parallel mirror planes have different MCNs, each\nof the four regions that these two mirror planes divide will\ncontain an odd number of Weyl points. In this case, we find\nthat there are three Weyl points in each of the four regions\ndivided by M001andM1−10, and 12 Weyl points in the whole\nBZ. Conversely, if the MCNs of the two non-parallel mirrors\nare the same, there will be 0 or an even number of Weyl points.\nThus, in the regions divided by any two mirror planes among\nM1−10,M120, and M210, there are 2 or 4 Weyl points in each\nregion, thus there will also be 12 Weyl points in total.\nFermi arc is one of the distinctive hallmarks of Weyl\nsemimetals, and it connects the projected Weyl fermions\nwith opposite chirality on the surface73,74. Thus, we\nperform a surface states calculation based on semi-infinite\nGreen’s function method for LiNbGe along (001) direction\nto demonstrate the existence of the Weyl fermions. 12 Weyl\npoints in LiNbGe project onto 6 momenta (marked by red\ndots) on the surface BZ and each of the projected points\nis composed of two Weyl points with opposite chirality.\nFigure 10 (e) shows that there are six Fermi arcs connecting\ntwo Weyl points that are the second nearest neighbor to each\nother.\nHere, we offer a perspective to understand the connection\nof Fermi arcs in terms of MCN, since the number of Fermi\narcs is constrained by the mirror Chern numbers. The MCN\nofM001is 2, so there are two Fermi arcs cross through half of\nM001on the projection surface that is perpendicular to M001.\nLikewise, the MCNs of M1−10,M120, andM210are all equal\nto 1, so there is one Fermi arc crossing through the half of\nthese three mirror planes on the (001) surface, forming the\nFermi arcs connection shown in Fig. 10 (e).\nIV . SUMMARY\nWe study the topological properties and their chemical rules\nfor systems with stacked kagome and honeycomb lattice, in\nboth analytical ways based on the TB model and numerical\nways based on the DFT calculation. The minimal five-band\nTB model shows that distinct node-ring semimetals and Weyl\nsemimetals can be obtained by tuning the parameters, in the\nnon-SOC case and SOC case, respectively. Combined with\nthe TB model, we propose an algorithm for fast-scanning\nthe materials with stacked kagome and honeycomb lattices8\nRbTmCRbTbCRbSmCRbPrCRbPmCRbNdCRbLaCRbGdCRbDyCRbCeC\n7.387.407.427.447.467.480.0460.0480.0500.0520.0540.056LiYCLiTmCLiTbCLiSmCLiPrCLiPmCLiNdC\nLiLaCLiHoCLiGdC\nLiErCLiDyCLiCeC\n6.26.36.46.56.60.0250.0300.0350.0400.045Radiusofnodal-rings(Å)\n0.0440.0500.0560.0250.0350.045\nNaYCNaTmCNaTbCNaSmCNaPrCNaPmCNaNdC\nNaLaCNaHoCNaGdCNaErCNaDyCNaCeC\n6.66.76.86.90.0360.0380.0400.042KYCKTbCKSmCKPrCKPmCKNdCKLuC\nKLaCKHoCKGdCKDyCKCeC\n7.057 .107 .157.207.250.0420.0440.0460.0480.0500.0520.054\nNaYSiNaYGeNaTmSiNaTbSiNaTbGeNaSmSiNaSmGeNaPrSiNaPrGeNaPmSiNaNdSiNaNdGeNaLuSi\nNaLaSiNaLaGeNaGdGeNaErSiNaDySiNaDyGeNaCeSiNaCeGe\n7.37.47.57.67. 77.80.0400.0450.0500.0550.060NaYPbNaTmPbNaTbSnNaTbPbNaSmSnNaSmPbNaPrSnNaPrPbNaPmSnNaPmPbNaNdSnNaNdPbNaLuSn\nNaLaSnNaLaPbNaGdPbNaErSnNaErPbNaDyPbNaCeSnNaCePb7.87.98.08.18.20.0400.0450.0500.0550.0340.0380.0420.0420.0480.054\n0.0400.0500.0600.0400.0500.0550.0456.26.36.46.56.6\n7.387.407.427.447.467.48Latticeconstanta(Å)7.37.47.57.67.77.8Latticeconstanta(Å)6.66.76.86.97.057.107.157.257.20\n7.87.98.08.28.1Latticeconstanta(Å)(a)(b)(c)\n(d)(e)(f)\nNd\nC+1.3-2.3\nNd\nC+1.28-2\nNd\nC+1.27-1.84\nNd\nC+1.26-1.77\nNd\nSi+1.23-1.93\nNd\nGe+1.2-1.9\nNd\nSn+1.18-1.87\nNd\nPb+1.14-1.84\nFIG. 9. Further analysis of type-(ii) nodal-ring configuration in Fig. 5, which corresponds to a single closed nodal-ring centered at Γ. The\ntendencies are different for the radius of the nodal ring and the lattice constant afor different IA elements.\nP ī P’0.4\n-0.4-0.20.00.2Energy (eV)(b)\n(c)\n(d)-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0  0.2  0.4  0.6  0.8  1MCN\nkM001\n-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0  0.2  0.4  0.6  0.8  1MCNM110-\n12\n246810\n-20(e)(a)\nP’\nP\nFIG. 10. (a) Nodal-ring configuration (green line, without SOC)\nand the distribution of Weyl points (with SOC) in the BZ, where\nthe blue and red dots denote the Weyl points with opposite chirality.\n(b) The band structure with SOC along the path P−Γ−P′. (c-d)\nThe average position of the Wannier centers for occupied bands with\nmirror eigenvalue +iforM001(c) and M1−10(d) planes, showing\nmirror Chern number of −2and +1, respectively. (e) The Fermi\nsurface including Weyl Fermi arcs along (001) direction for LiNdGe.in the ICSD and carry out a high-throughput calculation on\ntheir physical properties. 298 ideal topological semimetals\nof the IAMX family are proposed, and their topological\nfeatures are fully captured by the five-band TB model. Further\nanalyses reveal that different topological configurations in\nIAMX family materials are determined by lattice constants,\nintralayer and interlayer bond strength, electronegativity, and\nso on. Our comprehensive study on the stacked kagome and\nhoneycomb crystals paves the way for future studies on both\ntopology-related physical properties in such systems and their\ninterplay with CDW, superconductivity, axion insulators and\nother physical properties in kagome and honeycomb lattices.\nV . ACKNOWLEDGEMENT\nWe acknowledge the support from the National Natural\nScience Foundation of China (Grant Nos. 12047503 and\n12374165).\nAppendix A: The DFT calculation results of some red dots\nIn this section, we show the DFT results of two materials\nfrom red dots (type-iii) as shown in Fig. 11.\nAppendix B: The superconductivity of LiNdGe\nWe calculate the superconducting transition temperature\n(Tc) based on the BCS theorem75. We perform the phonon\nand the electron-phonon coupling matrix using the Quantum\nEspresso package76–78. The PBE functional64with PAW9\nLiLuSn LiTmPb\nΓ      M      K     Γ      A      L      H      A|L      M|H      K    -3-2-10123Energy (eV)EF\nΓ      M      K     Γ      A      L      H      A|L      M|H      K    -3-2-10123Energy (eV)EF\n(a) (b)\n(c) (d)\nFIG. 11. (a, b) The band structures of LiLuSn (a) and LiTmPb (b)\nwithout SOC. (c, d) The corresponding nodal-line configurations of\nLiLuSn (c) and LiTmPb (d).\n01020304050\nΓ       M   K         Γ             A        L   H         AFreq. (meV) phdos\nα2F\nλ\n0                  1                  2FIG. 12. The phonon spectra, density of states, integrated electron-\nphonon coupling strength λ, and the α2Fspectrum of LiNdGe.\nmethod63is used with the f-shell electrons of Nd were\ntreated as the semi-core electrons. The cutoff energy for\nwavefunctions is set as 60 Ry and a 7×7×11k-mesh is set to\nget converged total energy and charge density. We calculate\nthe dynamic matrix at a 2×2×4q-point mesh. 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For an old problem of Euler’s elastica we prove the novel glob al\nproperty that every planar elastica with non-constant mono tone curvature is\nuniquely minimal subject to the clamped boundary condition . We also partly\nextend this unique minimality to the length-penalised case ; this result is new\neven in view of local minimality. As an application we prove u niqueness of\nglobal minimisers in the straightening problem for generic boundary angles.\n1.Introduction\nEuler’s elastica is the oldest variational model of elastic rods initiated by D.\nBernoulli and L. Euler in the 18th century. The problem is to study cr itical points\nof the bending energy\nB[γ] =/integraldisplay\nγk2ds\namong immersed planar curves γunder the fixed-length constraint. Here we use k\nto denote the signed curvature of γ, and usesanddsfor the arc-length parameter\nand measure, respectively. By the multiplier method, such a critical point satisfies\nfor someλ∈Rthe Euler–Lagrange equation of the modified bending energy\nEλ[γ] :=B[γ]+λL[γ] =/integraldisplay\nγ(k2+λ)ds,\nwhereLdenotes the length functional, which takes the form\n(1.1) 2 kss+k3−λk= 0.\nAn arc-length parametrised curve γ: [0,L]→R2is called an elastica (orλ-\nelasticato explicitly specify the multiplier λ) if the curvature satisfies (1.1). As\none may imagine due to their classicality, elasticae have been the subj ect of a great\nnumber of works. However, compared to the level of critical point s as well as\nlocal variational properties (stability), global variational proper ties of those critical\npoints have retained their mystery and are relatively poorly unders tood.\nIn this paper we address global uniqueness and minimality issues subj ect to\nthe classical clamped boundary condition , which means that the endpoints are pre-\nscribed up to first order, exactly as posed by Bernoulli. For closed c urves it follows\nby Cauchy–Schwarz and Fenchel inequalities that minimisers are circ les (the area-\nconstrained counterpart is however much more nontrivial, see rec ent works [3,5]).\nInthe general(non-closed)case, globalminimisers alwaysexist, b ut theirproperties\nare largely unknown, including their uniqueness.\nDate: February 21, 2024.\n2020Mathematics Subject Classification. 49Q10 and 53A04.\nKey words and phrases. calculus of variations, elastica, boundary value problem, uniqueness.\n12 TATSUYA MIURA AND GLEN WHEELER\nHere we adopt the natural energy space of Sobolev class W2,2. ForL >0 we\ndefine the set of arc-length parametrised curves of length Lby\nW2,2\narc(0,L;R2) :=/braceleftBig\nγ∈W2,2(0,L;R2) :|γ′| ≡1/bracerightBig\n.\nWhen we say that an arc-length parametrised curve γ: [0,L]→R2is an elastica,\nwe mean that γ∈W2,2\narc(0,L;R2) and the curvature scalar kofγsatisfies (1.1)\nin the distributional sense (which automatically implies smoothness). In addition,\ngivenγ∈W2,2\narc(0,L;R2) and˜L>0, letA(γ;˜L) denote the set of curves that have\nlength˜Land agree with γat the endpoints up to first order, that is,\nA(γ;˜L) :=/braceleftBig\n˜γ∈W2,2\narc(0,˜L;R2) : ˜γsatisfies (1.2)/bracerightBig\n,\nwhere (1.2) are the following clamped boundary conditions:\n(1.2) ˜ γ(0) =γ(0), ˜γ(˜L) =γ(L), ˜γs(0) =γs(0), ˜γs(˜L) =γs(L).\nIn particular, if ˜L=L, we simply write\nA(γ) :=A(γ;L).\nOur first main result is the following sufficient condition on unique minimalit y.\nTheorem 1.1. Letγ: [0,L]→R2be an arc-length parametrised elastica with\nnon-constant and monotone signed curvature. Then γis the unique minimiser of\nthe bending energy Bin the set A(γ).\nRecall that in general an elastica may not be minimal, and even if minimal, may\nnot be unique; see e.g. [9, Figure 5] for a typical scenario. The case of constant\ncurvature is easier but needs to be addressed independently. A st raight segment\nis always uniquely minimal. A circular arc is uniquely minimal if and only if it is\nless than one fold; see [9, Appendix B] for minimality and [13, Theorem 2.1] for\nnon-minimality. In particular, the exactly one-fold circle is minimal but not unique\nby reflection; Theorem 1.1 also rules out such symmetry-induced no n-uniqueness.\nFew results are previously known about sufficient conditions for glob al minimal-\nity. One instance is the following fact claimed by Sachkov in [18]: for an y elasticaγ\nthere is a short subarc γ|[0,s0]which is minimal. Our theorem gives an entirely new\nsufficient condition, without any implicit smallness assumption. Not only that, our\nresult (combined with the constant-curvature case) can be rega rded as an improve-\nment of Sachkov’s claim since the subarc always has monotone curva turek[0,s0]for\nsmalls0. We also mention that Sachkov–Sachkova obtained a general rest riction\non the number of global minimisers [19]. Concerning necessary cond itions, see [13]\nand the references therein.\nAs for stability (local minimality) there are several known results. A mong them,\nby Born’s pioneering result [2] it is classically known that if an elastica is lo cally\nstrictly convex, then it is stable. More recently, Sachkov [17,18] developed a precise\nstability theory by an optimal control approach. In particular, Sa chkov’s results\nimplythat ifanelasticahasmonotonecurvature, then itisalsostable . Ourtheorem\nreveals, unexpectedly to the authors, that Sachkov’slocal princ iple directly extends\nto its global counterpart (up to the case of constant curvature ). Note that such a\nglobal extension does not hold for Born’s principle since long orbitlike e lasticae are\nalways non-minimal. See also [6,8] for related stability analysis of plana r or spatial\nelasticae subject to various boundary conditions.UNIQUENESS AND MINIMALITY OF EULER’S ELASTICA 3\nOur next result partly extends Theorem 1.1 from the length-fixed t o the length-\npenalised problem. For γ∈W2,2\narc(0,L;R2), we define the set of all curves which\nagree with γat the endpoints up to first order but may have different length by\n/tildewideA(γ) :=/uniondisplay\n˜L>0A(γ;˜L).\nTheorem 1.2. Letλ∈Randγ: [0,L]→R2be an arc-length parametrised λ-\nelastica with non-constant and monotone signed curvature. Ifγis not a subcritical\nwavelike elastica, then γis a unique minimiser of the modified bending energy Eλ\nin the set /tildewideA(γ).\nRemark1.3.Wavelikeelastica, eachdetermined byanelliptic parameter m∈(0,1),\ncan be either supercritical ,subcritical , orcritical. The latter case is that of the\nfigure-eight elastica with m=m0, whereas the remaining two have mlarger or\nsmaller than that of the figure-eight elastica. See Definition 2.6 for d etails.\nTo the authors’ knowledge this is the first result about global minima lity in the\nlength-penalised problem. Note that length-penalised minimality does not follow\nby length-fixed minimality, while the converse does follow. Indeed, Th eorem 1.2\nrequires the additional “non-subcriticality” assumption. Although its optimality is\nnot clear, if we remove it then there are counterexamples (see Rem ark 2.8).\nIn addition, Theorem 1.2 would be new even in view of stability. The auth ors\nare only aware of the fact that Born’s argument in [2] directly works even in the\nlength-penalised case (but only for smooth perturbations), see a lso [9, Section 5.3].\nMost of the other known stability results are based on the fixed-len gth constraint.\nFinally, we apply Theorem 1.2 to directly solve Euler’s elastica problem. M ore\nprecisely, for the classical clamped boundary value problem, we aim a t detecting\nglobal minimisers onlyfrom given boundary data. Given ℓ∈[0,L) andθ0,θ1∈\n(−π,π], we define the (nonempty) admissible set\n(1.3) AL\nℓ,θ0,θ1:=/braceleftBig\nγ∈W2,2\narc(0,L;R2) :γsatisfies (1.4)/bracerightBig\n,\nwhere the clamped boundary condition is given by\n(1.4)γ(0) =/parenleftbigg\n0\n0/parenrightbigg\n, γ(L) =/parenleftbigg\nℓ\n0/parenrightbigg\n, γs(0) =/parenleftbigg\ncosθ0\nsinθ0/parenrightbigg\n, γs(L) =/parenleftbigg\ncosθ1\nsinθ1/parenrightbigg\n.\nThis is similar to the set A(γ) considered earlier. Note that this formulation does\nnot lose generality thanks to geometric invariances. As mentioned b efore, there\nalways exists a minimiser of the bending energy BinAL\nθ0,θ1,ℓ, and also every min-\nimiser (or indeed critical point) is a smooth elastica. However, the min imisers are\nnot necessarily unique, and their shape is in general hard to rigorou sly detect just\nfrom the given boundary data. For results with some special bound ary data, see\ne.g. [4,11].\nIn [9] the first authorgavea detailed analysisof globalminimisers in th estraight-\nening problem , whereℓis close toL, for generic boundary angles θ0,θ1∈(−π,π)\\\n{0}. However, uniqueness was obtained only in the convex case θ0θ1<0 for a\ntechnical reason. Armed with Theorem 1.2, now we can extend it to t he remaining\nnon-convex case θ0θ1>0 almost directly. As a result, we obtain the following\nTheorem 1.4. LetL>0andθ0,θ1∈(−π,π)\\{0}. Then there exists ℓ∗∈(0,L)\nsuch that for any ℓ∈(ℓ∗,L)the bending energy Bhas a unique minimiser γℓin the\nsetAL\nℓ,θ0,θ1. In addition:4 TATSUYA MIURA AND GLEN WHEELER\n(a)Ifθ0θ1<0, then the above γℓhas strictly monotone tangential angle from\nθ0toθ1; in particular, γℓhas no inflection point.\n(b)Ifθ0θ1>0, then the above γℓhas strictly monotone curvature with a\nunique zero; in particular, γℓhas exactly one inflection point.\nFinally, in the limit ℓ→L, the rescaled curve defined by\nˆγℓ(s) :=1\nεℓγℓ(εℓs), εℓ:=L−ℓ\n4√\n2(sin2θ0\n4+sin2θ1\n4),\nlocally smoothly converges to the borderline elastica γθ0\nBdefined below. Similar con-\nvergence behaviour holds at the other endpoint by symmetry.\nHere the borderline elastica with initial angle θ0∈(−π,π)\\ {0}is explicitly\ndefined by\n(1.5)γθ0\nB(s) :=/integraldisplays\n0/parenleftbiggcosϕ(u+u0)\n(signθ0)sinϕ(u+u0)/parenrightbigg\ndu, ϕ(u) := 4arctan/parenleftBig\ne−u√\n2/parenrightBig\n,\nwhereu0:=−√\n2log(tan|θ0|\n4), which is the unique positive number such that\nϕ(u0) =|θ0|. This curvecan be characterisedas the unique arc-lengthparame trised\nelasticaγ: [0,∞)→R2withλ= 1 andγ(0) = (0,0)⊤,γ′(0) = (cosθ0,sinθ0)⊤,\nlims→∞γ′(s) = (1,0)⊤. See [9] for details.\nWe emphasise that we will apply the length-variable result in Theorem 1 .2 to\nprove the fixed-length result in Theorem 1.4. This is because the sing ular limit\napproach developed in [9] is heavily based on the analysis of the length -penalised\nproblem, which is then applied to analyse the original fixed-length pro blem.\nNow we discuss the idea of our proof, focusing on Theorem 1.1. The m ain tool\nis the classically known fact that planar elasticae are classified in term s of Jaco-\nbian elliptic integrals and functions (Theorem 2.4). Although we alread y have this\ncomplete classification of critical points, it is almost infeasible to find a ll solutions\nto the general clamped boundary value problem by solving the assoc iated system\nof transcendental equations (see e.g. [7, Theorem 2.2]) and also to detect global\nminimisers by computing their energy. This point is one of the common d ifficulties\nin elastica theory, in spite of the availability of the classification.\nOur main idea provides a new way to circumvent this common difficulty, p artly\ninspiredbypreviouswork[15]. Moreprecisely, weformulateand solve atailor-made\nfamily of free boundary problems such that every elastica with non- constant mono-\ntone curvature appears (or part thereof) as the unique minimiser of one of those\nproblems. Our choice of free boundary conditions is given in terms of right-angle\nconditions on fixed parallel lines. Then any critical point satisfies the additional\nso-called ‘no flux’ boundary condition that the endpoints must be ve rtices (Lemma\n2.1). This gives a very effective reduction of possible minimisers since in this case\nall the critical points are “computable”, in the sense that they invo lve only com-\nplete elliptic integrals. In this way we obtain unique explicit minimisers; alo ng the\nway we also employ the recently-developed “cut-and-paste” trick [13]. By varying\nthe free-boundary parameters we can exhaust all the elasticae u nder consideration,\nwith the only exception being the borderline case.\nBorderline elasticae require special treatment. They may be regar ded as the\ndegenerate case where the horizontal intercept of one of the pa rallel lines diverges\nto infinity. Then we encounter several additional issues. First, it is unclear how\nto appropriately incorporate the length constraint since in this cas e admissibleUNIQUENESS AND MINIMALITY OF EULER’S ELASTICA 5\ncurves must have infinite length. A simple idea would be to instead cons ider a\nlength-penalised problem for Eλbut then the energy diverges since the length is in-\nfinite. An additional difficulty is that the divergence of the free-bou ndary line loses\none boundary condition. To overcome those issues simultaneously, we introduce\na suitably adapted energy ˜Eλ(see Theorem 3.7). This energy admits appropriate\ninfinite-length curves as competitors, and also reflects the desire d boundary con-\ndition (at infinity) in terms of its integrability. In particular, half of a b orderline\nelastica appears as a unique minimiser of the adapted energy ˜Eλ. Not only that,\nthe adapted energy is chosen so that any finite-length subarc of t he minimiser of\n˜Eλis also minimal for the original energy Eλ. These are the crucial points of the\nproof.\nWe finally remark that minimality and stability of spatial elasticae are sig nifi-\ncantly different from the planar case and less-understood, see e.g . [13, Section 7]\nand references therein.\nAcknowledgements. The first author is supported by JSPS KAKENHI Grant\nNumbers 20K14341, 21H00990, and 23H00085. The second autho r was partially\nsupported by AEGiS Advance grant number PR6064 from the Univer sity of Wol-\nlongong. Much of this work was completed while the second author wa s visiting\nthe first at the Tokyo Institute of Technology. He is grateful for their support and\nkind hospitality.\n2.Preliminaries\nLetℓ∈R. We say that a curve γhasfree boundary (with horizontal extent ℓ)if\nγ∈ F+\nℓorγ∈ F−\nℓwhere\nF±\nℓ:=/braceleftBigg\nγ∈/uniondisplay\nL>0W2,2\narc(0,L;R2) :(γ(0),e1) = 0,(γ(L),e1) =ℓ\nγ′(0) =e1, γ′(L) =±e1/bracerightBigg\n.\nHeree1= (1,0)⊤ande2= (0,1)⊤denote the canonical basis of R2. The space F±\nℓ\ndoes not restrict the length of its members. It requires γ(0) to lie on the vertical\nlinet/ma√sto→te2andγ(L) to lie on the vertical line t/ma√sto→ℓe1+te2; and it requires γto\nsatisfy a perpendicularity condition on these lines. The lines support the boundary\nofγ∈ F±\nℓ, so we call them support lines . Note that ℓmay be negative; this simply\nmeans that the line t/ma√sto→ℓe1+te2is to the left of the starting line t/ma√sto→te2.\nNow we observe that the well-known first variation formulae yield an e mergent\nboundary condition for critical points with free boundary. For the moment, let us\nconsider a general (smooth) planar curve γ:¯I→R2, whereI= (a,b), which is\nimmersed ( |∂xγ(x)|>0) but not necessarily in the arc-length parametrisation. The\nlength ofγin its given parametrisation is\nL[γ] :=/integraldisplay\nI|∂xγ(x)|dx.\nThe arc-length derivative ∂sand measure dsare defined by ∂s=|∂xγ|−1∂xand\nds:=|∂sγ|dx, respectively. The unit tangent vector is given by T:=∂sγ, and the\nunit normal by its counterclockwiseπ\n2-rotationN:= rotT. The curvature scalar k\nis then defined so that ∂2\nsγ=kN. Accordingly the bending energy is given by\nB[γ] =/integraldisplay\nI|∂2\nsγ|2ds.6 TATSUYA MIURA AND GLEN WHEELER\nRecall also the Frenet–Serret formula ∂sN=−kT. Hereafter we essentially only\nwork in the arc-length parametrisation, so we use′to denote the (arc-length) de-\nrivative.\nFor an arc-length parametrised curve γ∈W2,2\narc(0,L;R2), the following formula\nholds in any direction η∈W2,2(0,L;R2) (see e.g. [12, Lemma A.1]):\n(2.1)d\ndε/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nε=0Eλ[γ+εη] =/integraldisplayL\n0/parenleftBig\n−3k2(T,η′)+2k(N,η′′)+λ(T,η′)/parenrightBig\nds.\nIfγis sufficiently smooth, then we can further perform integration by p arts to\ndeduce that\nd\ndε/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nε=0Eλ[γ+εη] =/integraldisplayL\n0(2k′′+k3−λk)(N,η)ds\n+/bracketleftbig\n(2kN,η′)−(2k′N+(k2+λ)T,η)/bracketrightbigL\n0.(2.2)\nLemma 2.1 (No flux condition) .Letλ∈Randγ∈ F±\nℓbe a critical point of Eλin\nF±\nℓin the sense that the first variation (2.1)vanishes for any admissible direction\nη∈W2,2(0,L;R2)such that (η(0),e1) = (η(L),e1) = (η′(0),e2) = (η′(L),e2) = 0.\nThenγis aλ-elastica (and hence smooth). In addition,\nk′(0) =k′(L) = 0.\nProof.The fact that γis a smooth elastica follows by a known bootstrap argument,\nwhere we only need to require (2.1) to vanish for all η∈C∞\nc(0,L;R2) (see e.g. [12]).\nThis means that the integral term always vanishes in (2.2). Hence by assumption,\nfor any admissible η,\n[(2kN,η′)−(2k′N+(k2+λ)T,η)]L\n0= 0.\nIn particular, if we take η=φN, whereφis a smooth cutoff function satisfying\nφ(0) = 1,φ′(0) = 0, and φ(s) = 0 fors>L\n2, then\n0 = [2kφ′−2k′φ]L\n0= 2k′(0),\nas required. By symmetry we also have k′(L) = 0. /square\nOne difficult aspect of Euler’s elastica problem is that the bending ener gy has\na significant degeneracy. That is particularly relevant to us here, a s our main\ngoal is to establish uniqueness of elastica in various senses. To be pr ecise, any\nisometryM:R2→R2preserves the bending energy and the length, as does any\nreparametrisation φ:J/ma√sto→I; we have B[M◦γ◦φ] =B[γ] andL[M◦γ◦φ] =L[γ].\nWe work in the parametrisation throughout, and we aim to prove uniq ueness in\nthis parametrisation. This almost removes the degeneracy in param etrisation (only\nshifts and reflections remain), and so it does not represent a signifi cant obstacle in\nour analysis. However for isometries the situation is more complicate d. While both\nbending and length are invariant under the action of an isometry M, the image\nγ(I) is not invariant; that is, ( M◦γ)(I)/\\e}atio\\slash=γ(I). This means that many isometries\nwill not be admissible for our free boundary problems. To be more precise, we have\nthe following result.\nLemma 2.2. The admissible subgroup MofF±\nℓ-invariant isometries consists of\nthe vertical reflection (x1,x2)⊤/ma√sto→(x1,−x2)⊤, the vertical translation (x1,x2)⊤/ma√sto→\n(x1,x2+α)⊤with arbitrary α∈R, and their compositions.UNIQUENESS AND MINIMALITY OF EULER’S ELASTICA 7\nThe remaining action we wish to discuss is that of scaling. Scaling is not a n\nisometry, but weshallshowthat it neverthelessmapsfromanelast icatoanelastica,\nand so it is important for us to consider. Given a curve γ, we define its rescaling\nby a factor Λ >0 to be\nγΛ(s) := Λγ(s/Λ).\nThe division by Λ is to ensure that γΛremains parametrised by arc-length. Then\nstraightforward calculation yields dsΛ= Λds,κΛ= Λ−1κ, and hence\n(2.3) B[γΛ] = Λ−1B[γ] andL[γΛ] = ΛL[γ].\nFor the energy Eλ, this means that scaling introduces a competition between the\nbending and the length. This implies the following.\nLemma 2.3. Supposeγ: [0,L]→R2is aλ-elastica. Let Λ>0. Then the rescaled\ncurveγΛis a(λΛ−2)-elastica.\nProof.Observe that 2 k′′\nΛ+k3\nΛ= Λ−3(2k′′+k3) = Λ−3λk= (Λ−2λ)kΛ. /square\nOne of the most remarkable achievements of classical investigation into pla-\nnar elasticae was their complete classification, through the use (an d in the first\ninstance, invention) of elliptic functions. Very roughly speaking, th is follows by\nintegrating the Euler–Lagrange equation once after multiplying by ks, using the\nPicard–Lindel¨ of theorem to obtain existence, and then explicit exp ressions through\nproperties of the elliptic functions and consideration of many cases . For more de-\ntailed arguments, see e.g. [16, Appendix B] or [12].\nTo state the classification we briefly recall Jacobi elliptic integrals an d functions.\nFor more details see classical textbooks, e.g. [20, Chapter XXII] (and also [1]), or\nsee [11, Appendix A] which collects all the necessary properties we w ill use here\nwith the same conventionsand notation. The incomplete elliptic integral of the first\nkindF(x,m) andof the second kind E(x,m) withparameterm∈(0,1) (squared\nelliptic modulus) are defined by\nF(x,m) :=/integraldisplayx\n0dθ/radicalbig\n1−msin2θ, E(x,m) :=/integraldisplayx\n0/radicalbig\n1−msin2θdθ,\nrespectively. The complete elliptic integral of first kind K(m) andof second kind\nE(m) are then defined by\nK(m) :=F(π/2,m), E(m) :=E(π/2,m),\nrespectively. The (Jacobi) amplitude function is defined by\nam(·,m) :=F−1(·,m) onR.\nThe(Jacobi) elliptic functions are then given by\ncn(x,m) := cos(am( x,m)),sn(x,m) := sin(am( x,m)),\ndn(x,m) :=/radicalBig\n1−msin2(am(x,m)).\nTheorem 2.4 (Explicit parametrisations of planar elasticae) .Letγ: [0,L]→R2\nbe an elastica in R2. Then, up to the action of the similarity group on R2and\nreparametrisation,\nγ(s) =γ∗(s+s0)\nwiths0∈R, whereγ∗is one of the following five parametrisations:8 TATSUYA MIURA AND GLEN WHEELER\n(I) (Linear elastica)\nγℓ(s) =/parenleftbiggs\n0/parenrightbigg\n.\nIn this case, θℓ(s) = 0andkℓ(s) = 0.\n(II) (Wavelike elastica) There exists m∈(0,1)such that\nγw(s) =γw(s,m) =/parenleftbigg\n2E(am(s,m),m)−s\n−2√mcn(s,m)/parenrightbigg\n.\nIn this case, θw(s) = 2arcsin(√msn(s,m))andkw(s) = 2√mcn(s,m).\n(III) (Borderline elastica)\nγb(s) =/parenleftbigg2tanhs−s\n−2sechs/parenrightbigg\n.\nIn this case, θb(s) = 2arcsin(tanh s)andkb(s) = 2sechs.\n(IV) (Orbitlike elastica) There exists m∈(0,1)such that\nγo(s) =γo(s,m) =1\nm/parenleftbigg2E(am(s,m),m)+(m−2)s\n−2dn(s,m)/parenrightbigg\n.\nIn this case, θo(s) = 2am(s,m)andko(s) = 2dn(s,m).\n(V) (Circular elastica)\nγc(s) =/parenleftbiggcoss\nsins/parenrightbigg\n.\nIn this case, θc(s) =s+π/2andkc(s) = 1.\nHere the tangential angle θ∗and the curvature k∗satisfy the relations ∂sγ∗=\n(cosθ∗,sinθ∗)⊤and∂sθ∗=k∗.\nRemark 2.5.The tangential angle of the borderline elastica is twice the Guderman -\nnian function, and thus has several equivalent expressions: θb(s) = 2am(s,1) =\n2arcsin(tanh s) = 2arctan(sinh s) = 4arctan( es)−π. In particular, up to isometry\nand reparametrisation, the curve γθ0\nBdefined in the introduction (1.5) is given by√\n2γb(s/√\n2).\nDefinition 2.6. A wavelike elastica is supercritical (resp.critical,subcritical ) if\nthe corresponding parameter m∈(0,1) satisfiesm>m 0(resp.m=m0,m<m 0),\nwherem0∈(0,1)is the uniquerootofthe decreasingfunction m/ma√sto→2E(m)−K(m).\nRemark 2.7.The wavelike elastica with m0≃0.8261 produces the so-called figure-\neight elastica , which is the only non-circular closed planar elastica.\nRemark 2.8.We can directly compute the multipliers of the base curves:\n(I)γℓis aλ-elastica for any λ∈R.\n(II)γw=γw(·,m) is aλ-elastica for λ= 2(2m−1).\n(III)γbis aλ-elastica for λ= 2.\n(IV)γo=γo(·,m) is aλ-elastica for λ= 2(2−m).\n(V)γcis aλ-elastica for λ= 1.\nMoreover,thesignofthemultiplierispreservedundersimilarityactio ns. Therefore,\nthe multiplier can be negative if and only if γis either a linear elastica or a wavelike\nelastica with parameter m∈(0,1\n2). In addition, any 0-elastica (also called free\nelastica) is linear or wavelike with m=1\n2. Sincem0>1\n2(see [10] for a proof),\nthe class of subcritical wavelike elasticae contains all elasticae with n on-positiveUNIQUENESS AND MINIMALITY OF EULER’S ELASTICA 9\nmultipliers λ≤0. Therefore, the assumption in Theorem 1.2 automatically implies\nthatλ>0. On the other hand, for any λ-elasticaγwith negative multiplier λ<0,\nwe can always construct a large-length competitor in /tildewideA(γ) such that Eλ→ −∞,\nand henceγdoes not have the minimality as in Theorem 1.2.\n3.Free boundary problems with fixed length\nIn this section we prove Theorem 1.1. Non-constant monotone cur vature elasti-\ncae come in three flavours: an arc of a wavelike elastica (type (II)) , of a borderline\nelastica (type (III)), or of an orbitlike elastica (type (IV)). For a ll wavelike and\norbitlike elasticae, we present distinct free boundary minimisation pr oblems that\neach type uniquely solves. The borderline elastica may be regarded a s a degenerate\ncase with infinite horizontal extent.\nLet us begin with the wavelike elastica.\nTheorem 3.1. Letℓ,L∈Rbe fixed parameters satisfying −L<ℓ<L . Consider\nthe setALof admissible curves given by\nAL=/braceleftBig\nγ∈ F+\nℓ:L[γ] =L/bracerightBig\n.\nThen there exists a smooth minimiser γℓ,Lof the bending energy B:AL→(0,∞).\nIn addition, there is a unique parameter m∈(0,1), depending only on the ratio\nℓ/L, such that any minimiser is parametrised by\n(3.1) γℓ,L(s) =L\n2K(m)Mγw(2K(m)\nLs,m)\nfor someM∈ M(see Lemma 2.2), where γwis the wavelike elastica in Theorem\n2.4. In particular, the minimiser is unique up to vertical tr anslations and reflection.\nMoreover, the map φ:ℓ/L/ma√sto→mis given by the inverse of\nm/ma√sto→2E(m)\nK(m)−1.\nThe mapφis a strictly increasing bijective function from (−1,1)to(0,1)such that\nφ(0) =m0. In particular, if ℓ>0, thenm∈(m0,1); ifℓ<0, thenm∈(0,m0); if\nℓ= 0, thenm=m0and henceγ0,Lis (half of) the figure-eight elastica.\nProof.The direct method in the calculus of variations implies the existence of a\nminimiser; for example follow the proof of [14, Proposition 4.1] with the additional\nargument that, without loss of generality, we can take a minimising se quence{γj}\nsatisfyingγj(0) = (0,0)⊤up to vertical translations. Then in view of the standard\nmultiplier method, the minimiser must be a critical point Eλfor someλ∈Rin the\nsense of Lemma 2.1. (See [14, Appendix A] for basic facts, and also [ 15, Appendix\nA] for a similar boundary condition.) Hence the minimiser is a smooth elas tica.\nIn what follows we prove uniqueness. Any minimiser γℓ,Lmust belong to the\nclassification given by Theorem 2.4. This means that it has parametris ation\n(3.2) γℓ,L(s) =MΛγ∗/parenleftbig\n(s+s0)Λ−1/parenrightbig\n,\nwhere Λ>0,M:R2→R2is an isometry, and γ∗is one ofγℓ,γw,γb,γo, orγc\ncorrespondingto cases(I)–(V) in Theorem2.4. Note that the act ionofreparametri-\nsation is already taken into consideration, since the speed of the pa rameter is fixed\nby arc-length parametrisation, and the orientation can be incorpo rated into the\nchoices ofMands0thanks to periodicity and symmetry. This formula therefore\ntakes into account sll of the degeneracy mentioned earlier.10 TATSUYA MIURA AND GLEN WHEELER\nThe goal for the remainder of the proof is to systematically reduce and simplify\nthe freedom and form of (3.2).\nRuling out Case (I) . Linear elasticae (I) satisfying the boundary condition have\nL=ℓand so they are not members of ALsince hereℓ<L.\nRuling out Case (III) . The borderline case γ∗=γbcan also not be possible since\nγℓ,Lneeds to satisfy the no flux condition in Lemma 2.1 at both the endpoin ts, but\nthe curvature kbofγbhas only one vertex.\nRuling out Case (IV). We prove by contradiction. Suppose that γ∗=γo. By\nLemma 2.1 a candidate minimiser γℓ,Lmust have vertices on its boundary. By\nperiodicity of ko, we may assume that s0∈ {0,ΛK(m)}, andL=jΛK(m) for\nsome positive integer j. In addition, since θo(iK(m)) =iπholds for any integer\ni, the boundary condition γ′\nℓ,L(0) =γ′\nℓ,L(L) =e1implies that not only jis even\nbut alsoγ′\nℓ,L(L\n2) =−e1. This means that, if we reflect vertically the arc γℓ,L|[L\n2,L]\nwhile keeping the former half γℓ,L|[0,L\n2]as it is, we can construct a new (admissible)\nminimiser. However this flipped curve has a curvature jump at s=L\n2. This con-\ntradicts the smoothness of minimisers. (This cut-and-paste argu ment is a variant\nof the method developed in [13].)\nRuling out Case (V). For circular elasticae, the boundary condition immediately\nimplies that ℓ= 0 andγℓ,Lto be given by a whole circle, possibly multiply covered.\nThen the same reflection argument as in case (IV) again implies a cont radiction.\nCase (II). The only remaining possibility is the wavelike case γ∗=γw. By\nLemma 2.1 and periodicity, we deduce that s0∈ {0,2K(m)Λ}andL= 2jΛK(m)\nfor some positive integer j. Up to a vertical reflection (which does not lose the\nadmissibility) we may assume that s0= 0. Recall that in general the isometry M\nis of the form M=A+bfor a linear isometry Aand a translation vector b. By\nthe boundary condition γ′\nℓ,L(0) =e1, we haveAγ′\nw(0) =e1, i.e.,Ais either the\nidentity or the vertical reflection. (At this point γ′\nℓ,L(L) =e1holds automatically.)\nThen, since ( γw(0),e1) = 0, the boundary condition ( γℓ,L(0),e1) = 0 implies that\n(b,e1) = 0, i.e.,bis a vertical translation. In summary, we deduce that\nγℓ,L(s) = ΛMγw(s/Λ,m)\nfor someM∈ Mand somem∈(0,1), with the scaling factor Λ =L\n2jK(m).\nMoreover, if j≥2, then by periodicity the dilated subarc ˜ γ(s) :=jγℓ,L|[0,L\nj](s/j)\nis still admissible and has less energy than γℓ,Lby the scaling property of the\nbending energy B, but this contradicts the minimality of γℓ,L. Hencej= 1, i.e.,\nΛ =L/(2K(m)).\nFinally, we detect a unique parameter m∈(0,1). To this end we use the\nremaining boundary condition ( γℓ,L(L),e1) =ℓ. This condition, combined with all\nthe obtained properties (as well as some basic properties like am( iK(m),m) =iπ/2\nandE(iπ/2,m) =iE(m) fori∈Z), implies that 2Λ(2 E(m)−K(m)) =ℓ.Thus\n(3.3)2E(m)\nK(m)−1 =ℓ\nL.\nIt iswell known(and not difficult toshow, seee.g. [10,16,21]) that th e mapf:m/ma√sto→\n2E(m)/K(m)−1 is strictly monotone and bijective from (0 ,1) to (−1,1), and has\nthe unique root m=m0. This implies the uniqueness of m∈(0,1), and also that\nthe inverse φ=f−1satisfies all the desired properties. The proof is complete. /squareUNIQUENESS AND MINIMALITY OF EULER’S ELASTICA 11\nIn particular, this implies unique minimality in the wavelike case.\nCorollary 3.2. Letm∈(0,1)andγbe an elastica given by a subarc of a similarity\nimage of the half-period wavelike elastica γw(·,m)|[0,2K(m)]. Thenγis the unique\nminimiser of BinA(γ).\nProof.In Theorem 3.1, if we fix the left endpoint of minimisers, then the vert ical\ntranslation is uniquely determined. Moreover, since\n(γw(0,m),e2) =−2√m/\\e}atio\\slash= 2√m= (γw(2K(m),m),e2),\nif we additionally fix the right endpoint, then the vertical reflection is also uniquely\ndetermined. Hence the minimiser must be unique. Therefore, consid eringL:=\n2K(m) andℓ:= 2K(m)φ−1(m) in Theorem 3.1, we deduce that the curve ˜ γ:=\nγw(·,m)|[0,2K(m)]is the unique minimiser in A(˜γ). This also implies unique mini-\nmality of any subarc of ˜ γ, since otherwise we could replace the subarc with a new\ncurve to construct an admissible curve ξ∈ A(˜γ) such that ξ/\\e}atio\\slash= ˜γandB[ξ]≤B[˜γ],\nwhich contradicts unique minimality of ˜ γ. As the property of unique minimality\nis invariant with respect to similarity actions, the same is true for any similarity\nimage of ˜γ. /square\nRemark3.3.Note that, assimilarity sends unique minimisers to unique minimisers,\nwe may without loss of generality fix e.g. L= 1 for these uniqueness results. Since\nthis does not simplify the arguments significantly however, we have c hosen not to\npursue this normalisation.\nNext we discuss the orbitlike case.\nTheorem 3.4. Letℓ,L∈Rbe fixed parameters satisfying −L<ℓ< 0. Consider\nthe setBLof admissible curves given by\nBL=/braceleftBig\nγ∈ F−\nℓ:L[γ] =L/bracerightBig\n.\nThen there exists a smooth minimiser γℓ,Lof the bending energy B:BL→(0,∞).\nIn addition, there is a unique parameter m∈(0,1), depending only on the ratio\nℓ/L, such that any minimiser is parametrised by\n(3.4) γℓ,L(s) =L\nK(m)Mγo(K(m)\nLs,m)\nfor someM∈ M(see Lemma 2.2), where γois the orbitlike elastica in Theorem\n2.4. In particular, the minimiser is unique up to vertical tr anslations and reflection.\nMoreover, the map ψ:ℓ/L/ma√sto→mis given by the inverse of\nm/ma√sto→2E(m)\nmK(m)+1−2\nm.\nThe mapψis a strictly decreasing bijective function from (−1,0)to(0,1).\nProof.As in the proof of Theorem 3.1 the existence of a minimiser follows by us ing\nthe direct method, and any minimiser must be given by\n(3.5) γℓ,L(s) =MΛγ∗/parenleftbig\n(s+s0)Λ−1/parenrightbig\n,\nwhere Λ>0,M:R2→R2is an isometry, and γ∗is one ofγℓ,γw,γb,γo, orγcin\nTheorem 2.4. We now work to reduce the space of possible minimisers.\nRuling out Cases (I), (III), (V) . Linear elasticae do not change angle, and so can\nnot be members of BL. Circular elasticae do not satisfy the boundary condition12 TATSUYA MIURA AND GLEN WHEELER\nunlessℓ= 0, so are not members of BLsince hereℓ <0. Borderline elasticae are\nalso ruled out for the same reason as in the proof of Theorem 3.1.\nRuling out Case (II). For wavelike elasticae, arguing as in the proof of Theorem\n3.1, Lemma 2.1 implies that the vertices of MΛγwmust appear at the endpoints.\nHowever, at all vertices the tangent vector of MΛγwis in a same direction, so for\nthe boundary conditions of BL, the wavelike case can not occur.\nCase (IV). As in the proof of Theorem 3.1 for Case (II), Lemma 2.1 with peri-\nodicity and symmetry implies that\nγℓ,L(s) =MΛγo((s+s0)Λ−1)\nfor somes0∈ {0,K(m)Λ}, some isometry M:R2→R2such that\n•ifs0= 0, thenM∈ M;\n•ifs0=K(m)Λ, thenM=M′NwhereM′∈ MandN: (x1,x2)⊤/ma√sto→\n(−x1,−x2)⊤+Λγo(K(m)),\nandL=jK(m)Λ for some integer j≥1. The scaling argument as before implies\nj= 1, that is,\nΛ =L/K(m).\nNow, assuming that s0= 0, we prove the uniqueness of m∈(0,1). By the\nboundary condition ( γℓ,L(L),e1) =ℓ, we computeΛ\nm(2E(m)+(m−2)K(m)) =ℓ,\nand hence\n(3.6)2E(m)\nmK(m)+1−2\nm=ℓ\nL.\nWe now show that the map\nf:m/ma√sto→2E(m)\nmK(m)+1−2\nm\nis strictly decreasing. Since (let us omit the argument mfor brevity)\nK′=E−(1−m)K\n2m(1−m), E′=E−K\n2m,\nwe have\nf′=2\nm2K2(mE′K−EK−mEK′+K2) =2\nm2K2(1−m)K2−E2\n2(1−m).\nIt now suffices to check the negativity of g(m) := (1−m)K(m)2−E(m)2. Note\nthatg(0) = 0 asK(0) =E(0) =π\n2. In addition,\ng′=−K2+(1−m)2KK′−2EE′\n=1\nm(−mK2+K(E−(1−m)K)−E(E−K)) =−1\nm(K−E)2,\nwhich is negative for m∈(0,1), and hence gis negative for m∈(0,1). Therefore\nfis strictly decreasing on (0 ,1), and in particular there is a unique m∈(0,1)\nsatisfying (3.6).\nNow let us show that s0=K(m)Λ can not occur. If this were the case, then\n(γℓ,L(L),e1)−(γℓ,L(0),e1) =/parenleftbig\nNΛγo(2K(m)),e1/parenrightbig\n−/parenleftbig\nNΛγo(K(m)),e1/parenrightbig\n= Λ/parenleftbig\n−γo(2K(m))+γo(K(m)),e1/parenrightbig\n−0\n=−Λ\nm(2E(m)+(m−2)K(m)),UNIQUENESS AND MINIMALITY OF EULER’S ELASTICA 13\nwhich is strictly positive by the previous paragraph. This contradict s the boundary\ncondition of BLwithℓ<0. Therefore s0= 0 is the only possible case.\nFinally, by using Rolle’s theorem for the above f, we compute f(0+) = 0, and\nalso by recalling that E(1) = 1 and K(1−) =∞, we havef(1−) =−1. Hencefis\nbijective from (0 ,1) to (−1,0). The inverse ψ:=f−1gives the desired map. /square\nTheorem 3.4 allows us to establish unique minimality in the orbitlike case.\nCorollary 3.5. Letm∈(0,1)andγbe an elastica given by a subarc of a similarity\nimage of the half-period orbitlike elastica γo(·,m)|[0,K(m)]. Thenγis the unique\nminimiser of BinA(γ).\nThe proof of Corollary 3.5 follows along the same lines as the argument for\nCorollary 3.2.\nRemark 3.6.The case 0 < ℓ < L can also be covered by Theorem 3.4 up to\nsymmetry. The case ℓ= 0 is slightly different but it is easy to deduce by the\nCauchy–Schwarzinequalitythattheonlyminimiserisahalf-circleupto invariances.\nFinally we address the borderline case.\nTheorem 3.7. Let\nCN=/braceleftbigg\nγ∈W2,2\narc(0,N;R2) : (γ(0),e1) = 0, γ′(0) =e1/bracerightbigg\n,\nand consider Cthe set of admissible curves given by\nC=/braceleftbigg\nγ: [0,∞)→R2:∀N >0, γ|[0,N]∈ CN/bracerightbigg\n.\nLetλ>0. Consider the adapted energy\n˜Eλ[γ] :=B[γ]+λ/integraldisplay∞\n0/parenleftbig\n1+(γs,e1)/parenrightbig\nds.\nThen there exists a smooth minimiser ˜γof the adapted energy ˜Eλ:C →(0,∞).\nIn addition, any minimiser is parametrised by\n˜γ(s) =Mγb(s)\nfor someM∈ M(see Lemma 2.2), where γbis the borderline elastica in Theorem\n2.4. In particular, the minimiser is unique up to vertical tr anslations and reflection.\nProof.Tobegin, observethat ahalf-period ˜ γb=γb|[0,∞)ofthe borderlineelasticais\nan element of Cand has finite energy ˜Eλ[˜γb]<∞. This can be explicitly calculated\nusingθb(s) = 2arcsin(tanh s) andkb(s) = 2sech(s). Note for later that k′\nb(0) = 0\nand this is the only vertex of γb.\nNow we proceed with a direct method. Take a minimising sequence {γj−γj(0)}\nfor˜Eλon the (non-empty) admissible set C. We prove that there exists a sub-\nsequence that is W2,2-weakly convergent to a limit ˜ γ∈ Con every bounded in-\ntervalK⊂⊂[0,∞). In order to see this, we use a diagonal-type argument. Let\n˜γj:=γj−γj(0), and first consider the restricted sequence {˜γj|[0,1]}. This is uni-\nformlybounded in W2,2(0,1;R2) bythe energybound /bardbl˜γ′′\nj|[0,1]/bardbl2\nL2=B[˜γj]≤˜Eλ[˜γj]\nandbyarc-lengthparametrisation. So it hasaweaklyconvergents ubsequence(that\nwe use the same index for, abusing notation). Suppose we found a s ubsequence\n{˜γj}weakly converging on [0 ,N] for someN∈N. Consider now the restriction14 TATSUYA MIURA AND GLEN WHEELER\n{˜γj|[0,N+1]}. This is uniformly bounded in W2,2(0,N+ 1;R2) and so has a (fur-\nther) subsequence that converges weakly. Taking the diagonal s equence, we find\nthe desired limit ˜ γ∈ C. Lower semicontinuity of ˜Eλimplies that for any N, the\nrestriction\n˜γN:= ˜γ|[0,N]\nsatisfies\n˜Eλ[˜γN]≤liminf\nj→∞˜Eλ[˜γj|[0,N]]≤liminf\nj→∞˜Eλ[˜γj] = inf\nC˜Eλ.\nSince lim N→∞˜Eλ[˜γN] =˜Eλ[˜γ] by the monotone convergence theorem, we find that\n˜γis a minimiser of ˜EλinC.\nLet ˜γbe any minimiser. Then the first variation (2.1) vanishes at least for a ll\nsmooth perturbations η: [0,∞)→R2with compact support such that ( η(0),e1) =\n(η′(0),e2) = 0. Taking an appropriate family of functions η∈C∞\nc(0,∞;R2) with\nbounded support shows that for every [0 ,N]⊂⊂[0,∞) the restriction ˜ γNis critical\nfor˜Eλ. Calculating, for any ξ: [0,N]→R2,\n˜Eλ[ξ] =B[ξ]+λL[ξ]+λ/integraldisplayN\n0∂s(ξ,e1)ds\n=B[ξ]+λL[ξ]+λ(ξ(N)−ξ(0),e1),\nwhichmeansthatthe firstvariationfor ˜Eλwith respecttoperturbationsofbounded\nsupport is equal to that for B+λL. Thus ˜γNis also critical for B+λL. SinceN\nis arbitrary, the (infinite-length) curve ˜ γmust be aλ-elastica, and it must fit into\nthe classification in Theorem 2.4.\nFirst, we note that a similarity image of case (III) is a possible candida te for ˜γ.\nAll the other cases can be ruled out by the uniform energy bounded ness˜Eλ[˜γ]<∞.\nIndeed, case (I) is clearly excluded as the second term of ˜Eλis unbounded for any\nstraight line in C. All the remaining cases (II), (IV), (V) have nonzero periodic\ncurvature and thus have unbounded bending energy as well.\nThus ˜γmust be a similarity image of the borderline elastica γb. Sinceγbis a\n2-elastica (see Remark 2.8) the scaling factor Λ >0 needs to satisfy λ= 2Λ−2(see\nLemma 2.3) and hence for some isometry M,\n˜γ(s) =/radicalBig\n2\nλMγb/parenleftBig/radicalBig\nλ\n2(s+s0)/parenrightBig\n.\nBy using the boundary condition of Cats= 0 and finiteness of the second term of\n˜Eλ[˜γ] (as a boundary condition at s=∞) we deduce that M∈ Mands0= 0./square\nThis settles the borderline case not only for the length-fixed but als o the length-\nvariable problem.\nCorollary 3.8. Letλ>0andγbe anyλ-elastica given by a (finite length) subarc\nof a similarity image of the half-period borderline elastic aγb|[0,∞). Thenγis the\nunique minimiser of Eλin/tildewideA(γ).\nIn particular, γis also the unique minimiser of BinA(γ).\nProof.By almost the same argument for Corollary 3.2 we deduce that the su barc\nγis uniquely minimal for ˜Eλin the length-variable admissible set /tildewideA(γ). This\nminimality is equivalent to that for Eλ=B+λEas already observed in the proof\nof Theorem 3.7. Hence the first assertion follows, and also the last a ssertion follows\ndirectly since λ>0. /squareUNIQUENESS AND MINIMALITY OF EULER’S ELASTICA 15\nRemark 3.9.Almost the same energy as our adapted energy was used in previous\nwork[9] for arelatedbut different purpose. In that workit wasrep resentedin terms\nof the tangential angle θ, namely in the form of a periodic potential perturbed by\nthe Dirichlet energy ε2/integraltext\n|θs|2ds+/integraltext\n(1−cosθ)ds. This expression highlights the\nfact that the borderline elastica can be regarded as a kind of “tran sition layer”.\nWe are now in a position to complete the proof of our main theorem.\nProof of Theorem 1.1. By Theorem 2.4, any elastica with non-constant monotone\ncurvature satisfies the assumption of one of Corollaries 3.2, 3.5, an d 3.8. Hence the\ndesired unique minimality follows. /square\n4.Free boundary problems with variable length\nIn this section we extend some of previous results to the length-pe nalised prob-\nlem. We first address the wavelike case. Here a discontinuous trans ition of min-\nimisers occurs at ℓ= 0, and nontrivial minimisers only appear for ℓ≤0.\nTheorem 4.1. Letλ >0andℓ∈Rbe fixed parameters. Then there exists a\nsmooth minimiser γλ,ℓof the modified bending energy Eλ:F+\nℓ→(0,∞).\nIfℓ>0, then any minimiser is a horizontal line segment.\nIfℓ≤0, then there is a unique parameter m∈[m0,1)(see Definition 2.6),\ndepending only on ℓ√\nλ, such that any minimiser is parametrised by\n(4.1) γλ,ℓ(s) =/radicalBig\n4m−2\nλMγw/parenleftBig/radicalBig\nλ\n4m−2s,m/parenrightBig\n, s∈/bracketleftbig\n0,2K(m)/radicalBig\n4m−2\nλ/bracketrightbig\n,\nfor someM∈ M(see Lemma 2.2), where γwis the wavelike elastica in Theorem\n2.4. Moreover, the map ˜φ:ℓ√\nλ/ma√sto→mis given by the inverse of\nm/ma√sto→2/parenleftbig\n2E(m)−K(m)/parenrightbig√\n4m−2.\nThe map ˜φis a strictly decreasing bijective function from (−∞,0]to[m0,1).\nProof.The caseℓ>0 is trivial so hereafter we only consider the case ℓ≤0.\nThe existence of a minimiser follows by using the direct method as in the fixed-\nlength case and so we do not repeat it. The only difference is that her e we use\nthe a-priori bound on the energy Eλto uniformly control the length Lalong a\nminimising sequence, with the assumption λ>0, instead of having it fixed a-priori.\nLetγλ,ℓbe any minimiser of EλinF+\nℓ. Thenγλ,ℓis also a minimiser of Bin the\nfixed-length subclass ALwith\nL:=L[γλ,ℓ].\nBy Theorem 3.1 this minimiser must be given by formula (3.1) with a param eter\nm∈(0,1) such that (3.3) holds. On the other hand, γλ,ℓis clearly a λ-elastica,\nwhile by Lemma 2.3 and Remark 2.8 we can compute the multiplier of the RH S of\n(3.1) to deduce that λ= 2(2m−1)(L\n2K(m))−2.Sinceλ >0, we have m >1\n2and\nhence\nL= 2K(m)/radicalBig\n4m−2\nλ.\nSubstituting this Lto (3.1) and (3.3), respectively, we obtain (4.1) and\n2/parenleftbig\n2E(m)−K(m)/parenrightbig√\n4m−2 =ℓ√\nλ.\nAs the RHS is non-positive, so is the LHS, and hence m∈[m0,1) (see Definition\n2.6). Form∈[m0,1) the map m/ma√sto→2E(m)−K(m) is strictly decreasing and16 TATSUYA MIURA AND GLEN WHEELER\nhas range ( −∞,0]. Sincem0>1\n2, the mapm/ma√sto→2(2E(m)−K(m))√4m−2 also\nhas the same property. So the inverse gives the desired map ˜φ, and in particular\nm∈[m0,1) is uniquely determined. /square\nAs in the length-fixed problem we obtain the following unique minimality, w hich\nonly covers critical and supercritical wavelike elasticae as so does T heorem 4.1.\nCorollary 4.2. Letλ>0,m∈[m0,1), andγbe aλ-elastica given by a subarc of\na similarity image of the half-period wavelike elastica γw(·,m)|[0,2K(m)]. Thenγis\nthe unique minimiser of Eλin/tildewideA(γ).\nBefore moving on to the orbitlike problem we mention the following\nRemark 4.3.In the proof of Theorem 4.1 we only used the monotonicity of the\nfunctionf:m/ma√sto→2(2E(m)−K(m))√4m−2on[m0,1), butthebehavioron(1\n2,m0]\nis also useful for other problems. Calculating, f′′<0 on (1\n2,1) andf(1\n2) =f(m0) =\n0 softakes its maximum M∗:= maxf≃0.837 atm=m∗≃0.628∈(1\n2,m0).\nWe can apply this fact, for example, to classify all critical points of t he modified\nbending energy Eλwithλ>0 subject to the so-called pinned boundary condition.\nMore precisely, consider the class of curves such that one endpoin t is fixed at the\norigin and the other is at ℓe1whereℓ≥0. Then as in the fixed-length case, the\ncurvature of any critical point must vanish at the endpoints. This im plies that\nany critical point is either a trivial line segment, or a wavelike elastica g iven by an\nisometric image of\nγ(s) :=N/radicalBig\n4m−2\nλγw/parenleftBig\nN−1/radicalBig\nλ\n4m−2(s−K(m)),m/parenrightBig\n, s∈[0,2NK(m)],\nwherem∈(1\n2,1) andN∈Nsuch that\n/vextendsingle/vextendsingle/vextendsingle2(2E(m)−K(m))√\n4m−2/vextendsingle/vextendsingle/vextendsingle=ℓ√\nλ/N.\nFinding all pairs of ( m,N) is reduced to understanding the behavior of the above\nfunctionf. More precisely,\n•forN <ℓ√\nλ/M∗there is a unique m∈(m0,1);\n•forN=ℓ√\nλ/M∗there are two solutions, m=m∗andm∈(m0,1);\n•forN > ℓ√\nλ/M∗there are three solutions, m∈(1\n2,m∗),m∈(m∗,m0),\nandm∈(m0,1).\nInparticular,ifwefocusonafixed“ N-mode”,thenas ℓ→+0thelasttwosolutions\n“converge” to a figure-eight elastica ( m→m0), while the first one “shrinks” but\nafterrescalingconvergestoafree elastica( m→1\n2). Asimilarobservationis already\ngiven in [7, Section 2.3] in terms of the limit λ→+0.\nNow we turn to the orbitlike case.\nTheorem 4.4. Letλ >0andℓ <0be fixed parameters. Then there exists a\nsmooth minimiser γλ,ℓof the modified bending energy Eλ:F−\nℓ→(0,∞).\nIn addition, there is a unique parameter m∈(0,1), depending only on ℓ√\nλ, such\nthat any minimiser is parametrised by\n(4.2)γλ,ℓ(s) =/radicalBig\n4−2m\nλMγo/parenleftBig/radicalBig\nλ\n4−2ms,m/parenrightBig\n, s∈/bracketleftbig\n0,K(m)/radicalBig\n4−2m\nλ/bracketrightbig\n,UNIQUENESS AND MINIMALITY OF EULER’S ELASTICA 17\nfor someM∈ M(see Lemma 2.2), where γois the orbitlike elastica in Theorem\n2.4. Moreover, the map ˜ψ:ℓ√\nλ/ma√sto→mis given by the inverse of\nm/ma√sto→/parenleftbig\n2E(m)+(m−2)K(m)/parenrightbig√4−2m\nm.\nThe map ˜ψis a strictly decreasing bijective function from (−∞,0)to(0,1).\nProof.The existence of a minimiser again follows by using the direct method, s o\nwe only argue about uniqueness.\nLetγλ,ℓbe any minimiser of EλinF−\nℓ. Thenγλ,ℓis also a minimiser of Bin the\nfixed-length subclass BLwithL:=L[γλ,ℓ]. By Theorem 3.1 this minimiser must be\ngiven by formula (3.4) with a parameter m∈(0,1) such that (3.6) holds. On the\nother hand, γλ,ℓis clearly a λ-elastica, while by Lemma 2.3 and Remark 2.8 we can\ncompute the multiplier of the RHS of (3.4) to deduce that λ= 2(2−m)(L\nK(m))−2.\nHence\nL=K(m)/radicalBig\n4−2m\nλ.\nSubstituting this Lto (3.4) and (3.6), respectively, we obtain (4.2) and\n/parenleftbig\n2E(m)+(m−2)K(m)/parenrightbig√4−2m\nm=ℓ√\nλ.\nLetfdenote the function on the LHS of the above equation. We calculate\nf′=−(m2−8m+8)E−(4m2−12m+8)K\n(1−m)m2√4−2m. (4.3)\nSinceE−√1−mK≥0 (to see this note that it is zero at m= 0 and the derivative\nis positive) we find\n(m2−8m+8)E−(4m2−12m+8)K\n≥/parenleftBig\n(m2−8m+8)√\n1−m−(4m2−12m+8)/parenrightBig\nK.(4.4)\nFurthermore, since\n(m2−8m+8)2(1−m)−(4m2−12m+8)2=m4(1−m)>0,\nwe find (m2−8m+ 8)√1−m >(4m2−12m+8). Using this in (4.3) and (4.4)\nyieldsf′<0. Thus noting f′(0+) = 0 we conclude that mis unique and the inverse\noffgives the desired ˜ψ. /square\nRemark 4.5.The caseℓ≥0 can also be treated as in Remark 3.6.\nCorollary 4.6. Letλ>0,m∈(0,1), andγbe aλ-elastica given by a subarc of a\nsimilarity image of the half-period orbitlike elastica γo(·,m)|[0,K(m)]. Thenγis the\nunique minimiser of Eλin/tildewideA(γ).\nProof of Theorem 1.2. The proof is almost same as that of Theorem 1.1. Here we\nuse Corollaries 4.2, 4.6, and 3.8. /square18 TATSUYA MIURA AND GLEN WHEELER\n5.Straightening problem\nIn this final section we apply Theorem 1.2 to prove Theorem 1.4. Our a rgument\nfollowsthegenerallineof[9], thusgoingthroughthecorresponding length-penalised\nproblem. For this application it is important to first check the following lemma,\nwhich is almost a direct consequence of the singular limit analysis in [9]. We\nintroduce the variable-length admissible set\nAθ0,θ1,ℓ:=/uniondisplay\nL>ℓAL\nθ0,θ1,ℓ\nby using the fixed-length admissible set AL\nθ0,θ1,ℓdefined in (1.3). Also, just for\ncompatibility with [9] we define the notation\nEε:=ε2B+L.\nClearly, minimization of Eεis equivalent for that of Eλwithλ= 1/ε2.\nLemma 5.1. Letℓ >0andθ0,θ1∈(−π,π)withθ0θ1>0. Then there exists\nε∗>0with the following property: Let ε∈(0,ε∗)andγbe any minimiser of\nthe energy Eεin the set Aθ0,θ1,ℓ. Thenγis a supercritical wavelike elastica with\nmonotone curvature.\nProof.By [9, Theorem 2.4] there is a (small) ε∗>0 such that for any ε∈(0,ε∗),\nany minimiser γis an elastica with exactly one sign change in its curvature. By\nclassification (Theorem 2.4), up to reparametrisationand similarity, the curve must\nbe of the form γ=γw(·,m)|[a,b]for some subinterval [ a,b]⊂(−K(m),3K(m)) and\nsome parameter m∈(0,1). Sinceγhas multiplier 1 /ε2>0, we have m∈(1\n2,1).\nThanks to the boundary layer analysis [9, Theorem 2.2 (1)] which ens ures a\nrescaled convergence to a suitable borderline elastica as ε→0, taking smaller ε∗\nif necessary, we may also assume that the absolute curvature |k|ofγis decreasing\n(resp. increasing) in a neighborhood of the endpoint s= 0 (resp.s=L[γ]). This\nimplies that the above subinterval must be contained in [ a,b]⊂[0,2K(m)] since\nγwhas absolute curvature 2√m|cn(s,m)|, which is increasing on ( −K(m),0) and\n(K(m),2K(m)) as well as decreasing on (0 ,K(m)) and (2K(m),3K(m)). There-\nfore,γis a wavelike elastica with monotone curvature.\nFinally we provethat γis supercritical in the sense ofDefinition 2.6. Note that γ\nhas curvature of the form 2√mΛ−1cn(Λ−1s+s0,m), and as in the proof of formula\n(4.2) we can determine the scaling factor Λ = ε√4m−2. This function has period\n4K(m)Λ = 4εK(m)√4m−2. Sinceγis contained in a half period, meaning that\n2εK(m)√4m−2≥L[γ], and since the length has the bound L[γ]>ℓ, we have\n2εK(m)√\n4m−2>ℓ>0.\nTherefore, if εis sufficiently small, K(m) needs to be so large that m>m 0./square\nNow the key uniqueness is easy to verify.\nCorollary 5.2. Letℓ >0andθ0,θ1∈(−π,π)withθ0θ1/\\e}atio\\slash= 0. Then there exists\nε∗>0such that for any ε∈(0,ε∗)the energy Eεhas a unique minimiser γεin the\nsetAθ0,θ1,ℓ.\nProof.Ifθ0θ1>0, then this directly followsby Lemma 5.1 combined with Theorem\n1.2, while the case of θ0θ1<0 is already treated in [9, Theorem 2.6]. /squareUNIQUENESS AND MINIMALITY OF EULER’S ELASTICA 19\nArmed with this uniqueness, we can prove Theorem 1.4 in the case of θ0θ1>0\nin the completely same way as the proof in [9, Section 6] for the case ofθ0θ1<0.\nHere we give a refined argument for the control of the length of min imisers of Eε,\nand then apply it to complete the proof of Theorem 1.4.\nLemma 5.3. Letℓ≥0andθ0,θ1∈(−π,π]such that Aθ0,θ1,ℓdoes not admit a\nstraight segment. Let 0<ε0<ε1. Forj= 0,1, letγjbe any minimiser of Eεjin\nAθ0,θ1,ℓ. ThenL[γ0]<L[γ1].\nProof.We first recall that L[γ0]≤L[γ1] (as in [9, Proposition 6.1]), which follows\nby combining the two inequalities due to minimality:\n(5.1)ε2\n0B[γ0]+L[γ0]≤ε2\n0B[γ1]+L[γ1], ε2\n1B[γ1]+L[γ1]≤ε2\n1B[γ0]+L[γ0].\nNow we prove that L[γ0]/\\e}atio\\slash=L[γ1] by contradiction. If L[γ0] =L[γ1], then by (5.1) we\nfurther have B[γ0] =B[γ1], which implies that γ0(resp.γ1) is also a minimiser of\nEε1(resp.Eε0) inAθ0,θ1,ℓ. Then each of γ0andγ1is both a (1 /ε2\n0)-elastica and a\n(1/ε2\n1)-elastica, but this is impossible since the curvature cannot vanish id entically\ndue to the boundary condition. We thus obtain a contradiction. /square\nCorollary 5.4. Under the situation of Corollary 5.2, the well-defined funct ion\nε/ma√sto→L[γε]on(0,ε∗)is strictly increasing and continuous. In addition,\nlim\nε→0L[γε]−ℓ\nε= 4√\n2/parenleftbigg\nsin2θ0\n4+sin2θ1\n4/parenrightbigg\n.\nProof.Uniqueness in Corollary 5.2 implies well-definedness directly, implies stric t\nmonotonicity combined with Lemma 5.3, implies continuity combined with [9 ,\nProposition 6.3], and also implies the last equation on the limit combined wit h\n[9, Proposition 6.4]. /square\nProof of Theorem 1.4. By Corollary 5.4, taking the inverse map of ε/ma√sto→L[γε], we\ncandefineacontinuous,strictlyincreasing,bijectivefunction ˜ ε: (ℓ,L[γε∗])→(0,ε∗)\nwith\nlim\nL→ℓL−ℓ\n˜ε(L)= 4√\n2/parenleftbigg\nsin2θ0\n4+sin2θ1\n4/parenrightbigg\n.\nThen for any L∈(ℓ,L[γε∗]), a curveγis a minimiser of BinAL\nθ0,θ1,ℓif and only if γ\nis a minimiser of E˜ε(L)inAθ0,θ1,ℓ. This directly gives a one-to-one correspondence\nbetween the fixed-length and length-penalised problems in the stra ightening regime\nunder consideration. Therefore, up to a simple dilation argument as in [9, Section\n6.3], we can directly apply Corollary 5.2 to deduce the uniqueness in The orem 1.4,\nand also apply [9, Theorems 2.2 and 2.4] to deduce property (b) and t he rescaled\nconvergence in Theorem 1.4. /square\nReferences\n[1] M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions with formulas,\ngraphs, and mathematical tables , Dover Publications, Inc., New York, 1992. Reprint of the\n1972 edition. MR1225604\n[2] M. Born, Untersuchungen ¨ uber die stabilit¨ at der elastischen lini e in ebene und raum: Unter\nverschiedenen grenzbedingungen , Ph.D. Thesis, 1906.\n[3] D. Bucur and A. Henrot, A new isoperimetric inequality for elasticae , J. Eur. Math. Soc.\n(JEMS) 19(2017), no. 11, 3355–3376. MR3713042\n[4] A. Dall’Acqua and K. Deckelnick, Elastic graphs with clamped boundary and length con-\nstraints, to appear in Ann. Mat. Pura Appl. (2023).20 TATSUYA MIURA AND GLEN WHEELER\n[5] V. Ferone, B. Kawohl, and C. Nitsch, The elastica problem under area constraint , Math. Ann.\n365(2016), no. 3-4, 987–1015. MR3521079\n[6] J. Langer and D. A. Singer, Curve straightening and a minimax argument for closed elast ic\ncurves, Topology 24(1985), no. 1, 75–88. MR790677\n[7] A. Linn´ er, Curve-straightening and the Palais-Smale condition , Trans. Amer. Math. Soc. 350\n(1998), no. 9, 3743–3765. MR1432203\n[8] J. H. Maddocks, Stability of nonlinearly elastic rods , Arch. Rational Mech. Anal. 85(1984),\nno. 4, 311–354. MR746170\n[9] T. Miura, Elastic curves and phase transitions , Math. Ann. 376(2020), no. 3-4, 1629–1674.\n[10] T.Miura, Li-Yau type inequality for curves in any codimension , Calc. Var.PartialDifferential\nEquations 62(2023), no. 8, Paper No. 216, 28. MR4631455\n[11] T. Miura, M. M¨ uller, and F. Rupp, Optimal thresholds for preserving embeddedness of elastic\nflows, to appear in Amer. J. Math., arXiv:2106.09549.\n[12] T. Miura and K. Yoshizawa, Complete classification of planar p-elasticae , arXiv:2203.08535.\n[13] T. Miura and K. Yoshizawa, General rigidity principles for stable and minimal elastic curves,\narXiv:2301.08384.\n[14] T. Miura and K. Yoshizawa, Pinned planar p-elasticae , arXiv:2209.05721, to appear in Indi-\nana Univ. Math. J.\n[15] T. Miura and K. Yoshizawa, Variational stabilization of degenerate p-elasticae ,\narXiv:2310.07451.\n[16] M. M¨ uller and F. Rupp, A Li-Yau inequality for the 1-dimensional Willmore energy , Adv.\nCalc. Var. 16(2023), no. 2, 337–362. MR4565935\n[17] Y. L Sachkov, Conjugate points in the Euler elastic problem , J. Dyn. Control Syst. 14(2008),\nno. 3, 409–439.\n[18] Y. L Sachkov, Maxwell strata in the Euler elastic problem , J. Dyn. Control Syst. 14(2008),\nno. 2, 169–234.\n[19] Yu. L. Sachkov and E. F. Sachkova, Exponential mapping in Euler’s elastic problem , J. Dyn.\nControl Syst. 20(2014), no. 4, 443–464. MR3255084\n[20] E. T. Whittaker and G. N. Watson, A course of modern analysis. An introduction to the\ngeneral theory of infinite processes and of analytic functio ns: with an account of the principal\ntranscendental functions , Fourth edition. Reprinted, Cambridge University Press, N ew York,\n1962. MR0178117\n[21] K. Yoshizawa, The critical points of the elastic energy among curves pinne d at endpoints ,\nDiscrete Contin. Dyn. Syst. 42(2022), no. 1, 403–423. MR4349792\n(T.Miura) Department of Mathematics, Tokyo Institute of Technology, 2-12-1Ookayama,\nMeguro-ku, Tokyo 152-8551, Japan\nEmail address :miura@math.titech.ac.jp\n(G. Wheeler) School of Mathematics and Applied Statistics, University of W ollon-\ngong, Wollongong NSW 2522 Australia\nEmail address :glenw@uow.edu.au"    },    {        "title": "2402.12791v1.Dual_polarization_huge_photonic_spin_Hall_shift_and_deep_subwavelength_sensing_based_on_topological_singularities_in_one_dimensional_photonic_crystals.pdf",        "content": "Dual-polarization huge photonic spin Hall shift and deep-subwavelength sensing based\non topological singularities in one-dimensional photonic crystals\nYufu Liu1,3, Xianjun Wang1,3, Yunlin Li1,3, Haoran Zhang1,3, Langlang\nXiong2,3, Xingchao Qi2,3, Zhen Lai1,3, Xuezhi Wang1,3, and Xunya Jiang1,2,3∗\n1Department of Illuminating Engineering and Light Sources,\nSchool of Information Science and Engineering, Fudan University, Shanghai, 200433, China\n2Institute of Future Lighting, Academy for engineering and technology,\nFudan University, Shanghai, 200433, China and\n3Engineering Research Center of Advanced Lighting Technology,\nFudan University, Ministry of Education, Shanghai, 200433, China\n(Dated: February 21, 2024)\nAlthough several efforts have been taken to enhance the photonic spin Hall shift in deep-\nsubwavelength region, according to effective medium theory, the fundamental confliction between\nnear-zero reflection coefficient and near-zero incident angle still hinders the further application.\nHere, we reveal a fundamental breakdown of effective medium theory due to the existing of topo-\nlogical singularity in deep-subwavelength region in one-dimensional photonic crystals. We find that\nnear the topological singularity, huge photonic spin Hall shift can be achieved for s-polarization\nand p-polarization. At the topological singularity, the reflected filed is split as dipole-like distribu-\ntion with zero photonic spin Hall shift for both-polarizations, which is resulted from the interfere\nof the spin-maintained normal light and spin-flipped abnormal light. Based on the theoretical\nresearch, dual-polarizations thickness and dielectric constant sensing devices can be designed in\ndeep-subwavelength region. Further more, by applying more complicated layered structure, multi-\nchannels dual-polarizations detection and broadband dual-polarizations huge spin Hall shift platform\ncan be designed. This work paves the way to exploring the topological properties and polarization\ncontrol of photonic crystals and provides a prospective method for the design of multi-channels\nsensitive detection spin optical devices.\nKeywords : topological singularity, photonic spin Hall\nshift, effective medium theory\nI. INTRODUCTION\nThe spin-orbital interaction of light is of great impor-\ntance in modern photonics, as it generates mutual con-\nversions between spin angular momentum and orbital an-\ngular momentum [1–5]. One types of spin-orbital inter-\naction, a conversion from spin angular momentum to ex-\ntrinsic orbital angular momentum, resulting in photonic\nspin-Hall effect [6–10]. The photonic spin-Hall effect has\nbeen widely investigated both theoretically [11–13] and\nexperimentally [14–16] in recent years, in which a spa-\ntially confined light beam (such as Gaussian beam) splits\ninto two spin components (i.e., left-handed and right-\nhanded) in the direction perpendicular to the plane of\nincidence when reflected or refracted from an interface,\nwhich is so-called as the photonic spin-Hall shift (PSHS).\nTake p-polarization incidence as an example, the trans-\nverse PSHS for reflected light can be approximately writ-\nten as [17, 18]:\nδ=−λ\n2π[1 +|rs|\n|rp|cos(ϕs−ϕp)] cot θi (1)\n∗jiangxunya@fudan.edu.cnwhere, ϕs/pis the reflected phase of s/p-polarization and\nθiis the incident angle. Eq.(1) remains applicable to\ns-polarization as long as the roles of sandpare inter-\nchanged. Eq.(1) indicates that, to get an PSHS as large\nas possible, the following two features should be satisfied:\n(i)|rp| →0; (ii) θiis near-normally incident.\nGenerally, the tiny spin-dependent displacements in\nphotonic spin-Hall effect are too small to detect di-\nrectly. Several efforts have been taken to satisfy these\nconditions to enhance the photonic spin-Hall displace-\nments and efficiency in PSHE [19–24]. Unfortunately,\nthese two requirements in are conflicting with each\nother in deep-subwavelength region, where the wave-\nlength is much larger than the scale of structure. In the\ndeep-subwavelength region, the effective medium theory\n(EMT) is widely regarded as an approximation method\nused to replacing complex inhomogeneous structure as a\nsimple homogeneous material that provides the same re-\nsponse [25–29]. According to EMT, condition (i) |rp| →\n0 requires that the incident angle θiis large enough,\nwhich contradicts with condition (ii). This confliction\nis also applied for transmitted light. As a result, it is\nfundamentally difficult to observe large PSHS in deep-\nsubwavelength region experimentally.\nA few years ago, in one-dimensional (1D) photonic\ncrystals (PhCs) with inversion symmetry, the concept\nand evolution of topological singularities, which can be\ndirectly positioned by the zero-scattering condition, has\nbeen extensively researched [30–35]. The topological sin-\ngularity is regarded as a topological charge, where therearXiv:2402.12791v1  [physics.optics]  20 Feb 20242\nis a phase jump near the singularity [30], so that it in-\ndicates nontrivial topology of a band. In inversion sym-\nmetry gauge, the topological singularity can be directly\npositioned by the zero-scattering condition, which means\nzero-reflection of period structure. Very recently, it was\nfound that, the topological singularity can be achieved\nin deep-subwavelength region when normally incident,\nwhich breaks the EMT [33, 34]. This counterintuitive\nphenomenon means that zero reflection (i.e., |rs,p| →0)\nwith near-zero incident angle θican be achieved in the\ndeep-subwavelength region, which will provide a perfect\nplatform to research photonic spin-Hall effect.\nIn this work, we reveal a fundamental breakdown of\nEMT due to the existing of topological singularity in\ndeep-subwavelength region in 1D PhCs with inversion\nsymmetry. We find that near the topological singular-\nity in the deep-subwavelength region, huge PSHS can\nbe achieved for s-polarization and p-polarization, which\nbreaks the EMT. Besides, at the topological singularity,\nthe reflected filed is split as dipole-like distribution with\nzero PSHS for both polarizations, which results from the\ninterfere of the spin-maintained normal light and spin-\nflipped abnormal light. Based on this exotic property,\ndual-polarizations thickness and dielectric constant sens-\ning device can be designed, which can be applied into\na wide range of fields, such as the sub-nanometer de-\ntection, non-linear power detection and environmental\nsensors. Finally, by applying more complicated layers,\nmulti-channels dual-polarizations detections and dual-\npolarizations broadband huge-PSHS platform can be de-\nsigned. Theoretically, this work builds the bridge be-\ntween topology study of 1D PhCs and light-matter in-\nteraction in deep-subwavelength region. In practice, our\nmodel provides an unprecedented platform to research\nphotonic spin-Hall effect, which can been applied in\nmany applications, such as optical metrology, sensing and\nsuper-resolution imaging.\nII. DEEP-SUBWAVELENGTH\nDUAL-POLARIZATION PHOTONIC SPIN HALL\nSHIFT\nFig.(1a) shows the principal illustration of our model\nwith a linearly polarized (LP) Gaussian beam reflects\nfrom an 1D finite PhCs with cell number N. The re-\nflected beams are composed of left-handed circular po-\nlarized (CP) Gaussian beam (marked by |−⟩) with PSHS\n∆y−and right-handed CP Gaussian beam (marked by\n|+⟩) with PSHS ∆ y+. The PhCs is supported by a dielec-\ntric substrate with a relative permittivity of εsuband cov-\nered by a dielectric superstrate with a relative permittiv-\nity of εsuper. Below and after, we set εsub=εsuper = 2.1,\nwhich is exact the relative permittivity of silica in in-\nfrared region. The structure of 1D PhCs is shown in\nFig.(1b), where the PhCs is consist of five layers in\nthe unit cell with relative permittivity as εA= 2.1,\nεB= 3.24,εC= 1.44,εB= 3.24,εA= 2.1 and thewidths of five layers are set as dA/2,dB/2,dC,dB/2,\ndA/2, respectively. Obviously, the PhCs exhibits inver-\nsion symmetry at the central point of layer-C. This struc-\nture is called ”ABCBA-model”. In this work, the relative\npermeability of all dielectric materials is supposed to be\nµr= 1. In Fig.(1c), in the deep sub-wavelength region\nwith λ >> a (where a=dA+dB+dCis the length\nof unit cell and is set as a= 1µm), the ABCBA-model\ncan be modelled by the same effective uniaxial medium\nwith a relative permittivity of ¯¯ε= [ε∥, ε∥, ε⊥]. Accord-\ning to the EMT [28, 36], we have ε∥=εAdA+εBdB+εCdC\na\nandε⊥=a\ndA/εA+dB/εB+dC/εC. Transfer matrix method\n(TMM) is a strict and useful method when considering\nmulti-layer structure, so that we will use TMM to calcu-\nlate the properties of our model below and after.\nFirst, we discuss the evolution of topological singu-\nlarity in deep-subwavelength region. In this work, the\ndeep-subwavelength is defined as λ≥25a. We also set\nλ > n effNa/2, where neffis the effective index of our\nmodel, so that there is no Bragger reflections resonant\nmode [37] in this region. In Fig.(2a), we depict the re-\nflection coefficient (in log) in {dB, λ}space when nor-\nmally incident, where dA= 0.2a,dC=a−dA−dB\nandN= 8. The deep red region indicates the first gap,\nmarked by white dashed lines. The deep blue lines, in-\ndicating perfect transmission and zero reflection, is the\ntrajectory of topological singularity in {dB, λ}space. To\nverify the correctness of our results from TMM, we use\nthe finite element method (FEM) as the numerical exper-\niments to obtain the trajectory of topological singularity\n(marked by the black squares), which agrees well with\nour results from TMM. We find that, with the decrease\nofdB, the singularity will move towards lower frequency\n(higher wavelength) region and it reaches zero frequency\n(infinite wavelength) at dB0= 0.2944a, which satisfies\n[34]:\nεsuper =εAdA+εBdB+εCdC\na(2)\nFurther decreasing dB, the topological singularity will\nevolve into the pure imaginary frequency region, details\nare shown in the supplementary section S1. When the\nlight is obliquely incident with angle θi, the reflection\nproperties of p-polarization and s-polarization are differ-\nent. In Fig.(2b) and Fig.(2c), we depict the reflection co-\nefficient of p-polarization and s-polarization in deep sub-\nwavelength region ( a/λ≤0.04) with dB= 0.295a. The\nred dashed lines and black squares are the trajectory of\ntopological singularity for both polarizations calculated\nby TMM and FEM, which agree well with each other. we\nshow that the singularity trajectories of s-polarization are\nmore flat than those of p-polarization because the effec-\ntive dielectric constant ε∥= 2.1 is larger than ε⊥= 1.44\n[35]. Therefore, the singularities for p-polarization and s-\npolarization will gradually separate as the incident angle\nincreases. The zero-reflection of topological singularities\nin deep-subwavelength region provides a perfect platform3\n(a)\n𝒚𝒚𝒙𝒙\n𝒛𝒛\n+−\n𝚫𝚫𝒚𝒚+𝚫𝚫𝒚𝒚−\nSubstrate��𝜺𝜺=diag𝜺𝜺∥,𝜺𝜺∥,𝜺𝜺⊥xz𝜽𝜽𝒊𝒊Superstrate\nSubstrateABCBA…\n…\n…xz𝜽𝜽𝒊𝒊Superstrate\nEMT(b)\n𝑳𝑳𝑷𝑷(c)\nFIG. 1. (a) The schematic of a p-polarized Gaussian beam reflects from 1D PhCs. The reflected beams are composed of\nleft-handed ( |+⟩) and right-handed ( |−⟩) circular polarization with PSHS ∆ y+and ∆ y−. (b) The schematic of PhCs in x-z\nview with periodic all-dielectric multi-layer constituents composed of three dielectrics A, B and C. The lattice constant of\nPhCs is aand cell number is fixed as N= 8. (c) The schematic of homogenized effective medium with relative permittivity\n¯¯ε= [ε∥, ε∥, ε⊥]. Below and after, we set wavelength λ >> a andλ > Na/ 2. The waist radius of Gaussian beam is set as:\nw0= 100 λ.\n1st gap\n0.04deep subwavelength(a) (b) (c)\nFEM\n FEM\nFIG. 2. (a) The reflection coefficient of s- (p-) polarization when normally incident. The deep red region is the first gap and\nthe deep blue line is the trajectory of topological singularity, which is also verified by the black squares calculated by FDTD.\nThe topological singularity reaches zero frequency when dB0= 0.2944a, marked by ret dots.Deep sub-wavelength is defined\nasλ≥25a, as the black dashed line in (a). (b) and (c) are the reflection coefficient of p- and s-polarization in {θ, λ}space\nwith dB= 0.295a. The pink dashed lines and black squares are the trajectory of topological singularity for both polarizations\ncalculated by TMM and FDTD, respectively.\nto research photonic spin-Hall effect.\nNext, we will investigate the PSHS of topological sin-\ngularity in deep sub-wavelength region and compare with\nthe effective medium model. According to the three-\ndimensional rotation matrix, near the topological singu-\nlarity with rs,p→0, the approximation of Eq.(1) fails,\nand the transverse spin-shift ∆ y±\ns,pof reflected Gaussian\nbeam with finite waist radius w0can be approximately\nrewritten as [18]:\n∆y±\ns,p=±δs,p\n1 +|∆s,p|2+|δs,p|2\nw2\n0(3)where,\n∆s,p=−2π\nλ∂ln(rp,s)\n∂θ(4)\nδs,p=−2π\nλ(1 +rp,s\nrs,p) (5)\nDetailed derivations are shown in the supplementary sec-\ntion S2. According to Eq.(3), it is obvious that the PSHS\nof p-polarization and s-polarization are determined by\nthe relative valus of |∆p,s|and|δp,s|.\nIn Fig.(3a) and Fig.(3d), the reflection coefficient |r|\nwith incident angle θi= 5◦for p-polarization and s-4\n(a)\n(e)\nD𝑠𝑠𝜆𝜆(f)\nD𝑝𝑝𝜆𝜆\n(d)(b) (c)Δ𝑦𝑦𝑝𝑝+𝜆𝜆 Δ𝑦𝑦𝑠𝑠+𝜆𝜆𝑟𝑟𝑝𝑝𝑟𝑟𝑠𝑠\nFIG. 3. The spin-Hall shift for both-polarizations and the breakdown of effective medium theory in deep-subwavelength region.\n(a)-(c) for p-polarization and (d)-(f) for s-polarization. (a) and (d) are the reflection coefficient |r|calculated by TMM (blue\nsolid lines), EMT (black dashed lines) and FEM (green squares) for p-polarization and s-polarization. (b) and (e) are the spin\nHall shift ∆ y+\ns,pof reflected right-handed CP Gaussian beam |+⟩(units of λ) calculated by Eq.(3) and EMT for p-polarization\nand s-polarization. (c) and (f) are the difference of spin Hall shift between Eq.(3) and EMT for p-polarization and s-polarization.\nThe position of topological singularity is marked by red and green pentagram for p-polarization and s-polarization, respectively.\npolarization are calculated by TMM (solid lines) and ver-\nified by FEM (squares), respectively. The results from\nTMM agree well with FEM. We show that there is a\ntopological singularity at a/λ= 0.26 for p-polarization\n(marked by red pentagram) and a topological singularity\nata/λ= 0.15 for s-polarization (marked by green penta-\ngram). Besides, the reflection coefficients |rp,s|calculated\nby EMT are also calculated in Fig.(3a) and Fig.(3d),\nwe show that there is no zero-reflection for EMT, so\nthat the breakdown of EMT is occured in the deep-\nsubwavelength region. We further calculate the PSHS of\nreflected right-handed CP Gaussian beam |+⟩for both\npolarizations in Fig.(3b) and Fig.(3e). The solid lines\nare calculated by Eq.(3) and dashed lines are calculated\nby EMT. We show that, near singularity, huge PSHS\nis achieved for both-polarizations, which reaches to 24 λ\nfor p-polarization and 50 λfor s-polarization with waist\nradius w0= 100 λ. Besides, the PSHS for both polariza-\ntion is near zero at the singularity due to the fact that\nδs,p>> w 0andδs,p>> w 0at singularity. While for\nEMT, it remains tiny due to the absence of singularity,\nwhich is accordance with Eq.(1). In Fig.(3c) and Fig.(3f),\nwe further calculate the difference of PSHS calculated\nbetween Eq.(3) and EMT (i.e., Ds,p= ∆Eq(3)\ny,s/p−∆EMT\ny,s/p)\nfor p-polarization and s-polarization, respectively. We\nshow a dramatic breakdown of EMT approximation is\noccurred near the topological singularity in the deep-\nsubwavelength, as the gery shadows shown in the pic-\ntures. Away from the topological singularity, the EMTstill holds in deep-subwavelength region.\nIn this section, we show that, due to the existing\nof topological singularity in deep-subwavelength, huge\nPSHS for dual-polarizations can be achieved and the\nEMT is breakdown in this case. We note that when the\ntopological singularity is away from sub-wavelength re-\ngion (such as, moving into the higher frequency region or\nmoving into the pure imaginary frequency domain), the\nEMT will be applicable again. The details can be seen\nin the supplementary section S1.\nIII. DEEP-SUBWAVELENGTH DETECTION\nBASED ON THE TOPOLOGICAL SINGULARITY\nIn the previous section, we have shown that the PSHS\nreaches zero at singularity and huge PSHS for both po-\nlarizations can be achieved near the singularity. In this\nsection, we will show that the topological singularity is\nsensitive to the structure and material parameters and\ndramatically changes of PSHS near the topological singu-\nlarity will be beneficial to design dual-polarizations spin-\nsensitive thickness and refractive index sensing devices.\nFor p-polarization, the PSHS is shown in Fig.(4a). The\nminimum values, zero and maximum values of PSHS are\nmarked by A, B and C, where B is the position of topo-\nlogical singularity with λ= 38.2a. The filed distribution\nof reflected CP Gaussian beam can be written as:5\nABC(a)\n(d)\nsingularity𝑫𝑫𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑\n𝑫𝑫𝒉𝒉𝒑𝒑𝒉𝒉𝒉𝒉(c)\n𝒓𝒓++=−𝒓𝒓+−𝑪𝑪\n𝑬𝑬++𝑬𝑬+−(e)\n(b)\nB A C\n𝑥𝑥/𝜆𝜆0 -100 100\n𝑥𝑥/𝜆𝜆0 -100 100\n𝑥𝑥/𝜆𝜆0 -100 100𝑦𝑦/𝜆𝜆\n-1000100\n0Max𝑦𝑦/𝜆𝜆\n-1000100\n𝑦𝑦/𝜆𝜆\n-1000100\n0Max\n0Max\n𝑥𝑥 𝑥𝑥 𝑥𝑥\n𝐸𝐸+2𝐸𝐸+2𝐸𝐸+2p-polarizationΔ𝑦𝑦𝑝𝑝+𝜆𝜆\nD\nE\nFs-polarization\n𝑦𝑦 𝑦𝑦 𝑦𝑦𝐸𝐸+2 𝐸𝐸+2𝐸𝐸+2𝑥𝑥/𝜆𝜆0 -100 100\n𝑥𝑥/𝜆𝜆0 -100 100\n𝑥𝑥/𝜆𝜆0 -100 100𝑦𝑦/𝜆𝜆\n-1000100\n𝑦𝑦/𝜆𝜆\n-1000100\n𝑦𝑦/𝜆𝜆\n-1000100\n0Max\n0Max\n0Max\n𝑯𝑯𝟏𝟏\n𝑯𝑯𝟐𝟐D F E (f) (g)\nFIG. 4. (a) Photonic spin Hall shift for p-polarization. (b) The 2D and 1D filed distribution of the reflected right-handed\nCP|+⟩, marked by A, B and C in (a). The waist radius of Gaussian beam is set as w0= 100 λ. (c) The normalized filed\ndistribution of B-point in (b) among x-direction with y= 0. (d) The resolution ratio Rpfor p-polarization. Green shadow\nindicates Rp>1. (e) Fresnel coefficients r++andr+−for the CP plane waves. r++=−r+−at the topological singularity.\nInset: the illustration of interfere between abnormal and normal light.(f) Photonic spin Hall shift for s-polarization. (b) The\n2D and 1D filed distribution of the reflected |+⟩CP field, marked by D, E and F in (f).\n|E(x, y)|2= exp( −2x2+y2\nw0)×|rs,p|2\n2{[(2∆s,p\nw0)2x2+\n(2δs,p\nw0)2y2]|+⟩+ [(2∆s,p\nw0)2x2−(2δs,p\nw2\n0)2y2]|−⟩}\n(6)\nDetailed derivations are shown in the supplementary sec-\ntion S3. According to Eq.(6), near the topological sin-\ngularity, the reflected field distributions of p-polarization\nand s-polarization are determined by relative values be-\ntween |∆s,p|and|δs,p|(also see in the supplementary\nsection S3). The two-dimensional (2D) and 1D field dis-\ntribution of the right-handed CP Gaussian beam |+⟩\nmarked by A, B and C are shown in Fig.(4b). For A(C), the field is arc-like distribution with negative (pos-\nitive) transverse spin Hall shift. Interesting, for B, the\nfield is split as dipole-like distribution in xdirection with\nnear-zero transverse spin Hall shift due to the fact that\n|∆s,p|>|δs,p|. This phenomenon is also discovered for\nthe Brewster angle in the single interface system [38]\nand perfect spin-conversion points in multi-layered sys-\ntem [35]. Quantitatively, in Fig.(4c), we depict the nor-\nmalized filed distribution of B-point among x-direction\nwith y= 0. There are two peaks at x=w0/√\n2 and zero\nvalue at x= 0, which is consistent with Fig.(4b). We\ndefine that Dpeakis the distance between two peaks and\nDhalfis the half of one of peak. To better distinguish the\ncase of B and A/C, we further define the ratio between6\nDpeakandDhalfas:\nRp=Dpeak\nDhalf(7)\nwhere, Rpis the resolution factor for p-polarization.\nWhen Rp>1, the field can be regarded as dipole-like\ndistribution. When Rp<1, the field can be regarded as\narc-like distribution. We mention that although reflected\nCP Gaussian beam is rather weak, it can be measured\nby signal enhancement technique experimentally [14–16].\nIn Fig.(4d), the ratio Rpis plotted in the deep sun-\nwavelength region, where green shadow indicates Rp>1.\nWe show that a narrow frequency-region near the topo-\nlogical singularity with Rp>1 can be regarded as dipole-\nlike distribution.\nThe similar phenomenon is also shown for s-\npolarization. In Fig.(4f), we also depict the PSHS for\ns-polarization in deep-sub-wavelength region, where the\nmaximum values, zero and minimum values of PSHS are\nmarked by D, E and F, in which E is the position of topo-\nlogical singularity with λ= 67.2a. The 2D and 1D field\ndistribution of the reflected right-handed CP Gaussian\nbeam |+⟩marked by D, E and F are shown in Fig.(4g).\nFor D (F), the field is circular-like distribution with pos-\nitive (negative) transverse spin Hall shift. Interesting,\nwe show that a split of reflected field at singularity (E)\nis occurred among y-direction, where two peaks H1and\nH2are located at y=±w0/√\n2. This phenomenon is\ndue to the fact that |∆s,p|<|δs,p|in this case (details\nare shown in the supplementary materials). The quanti-\ntatively distinguish the reflected field of E and D/F, we\nfurther define the resolution factor for s-polarization as:\nRs=min(H1, H2)\nmax(H1, H2)(8)\nWhen Rs>0.5, the field can be regarded as dipole-like\ndistribution. When Rs<0.5, the field can be regarded\nas circular-like distribution.\nNext, we will give the illustration of the split field dis-\ntribution at topological singularity physically. The p-\npolarization can be written in the CP basis:\n|p⟩=1\n2(|+⟩+|−⟩) (9)\nUnder paraxial approximation, the reflected CP Gaus-\nsian beam can be obtained as:\n|+⟩r=Er\n+++Er\n+−≈r++|+⟩i+r+−|−⟩i\n|−⟩r=Er\n−−+Er\n−+≈r−−|−⟩i+r−+|+⟩i(10)\nwhere r++=rp+rs\n2andr+−=rp−rs\n2are the Fresnel\ncoefficients for the CP plane waves. Er\n++(Er\n−−) is the\nreflected spin-maintained normal light with zero spin Hall\nshift and Er\n+−(Er\n−+) is reflected the spin-flipped abnor-\nmal light with spin Hall shift ∆ y±=∓2 cot( θi)\nk0≈5λ\nin our case. The spin Hall shift is rather tiny compar-\ning with the waist w0= 100 λ. At the singularity forp-polarization with rp= 0, we obtain that r+−=−r++.\nTherefore, according to Eq.(10), the reflected normal\nlight E++and abnormal light E+−are destructive inter-\nference near the origin, as the inset shown in Fig.(4e).\nThe destructive interference between spin-maintained\nnormal light and spin-flipped abnormal light results the\nsplit field distribution.\nIn above, we show that near the topological singularity,\nthe field will undergo a dramatic variation from arc-like\nto dipole-like to arc-like mathematically and physically.\nPractically, the position of the singularity are closely re-\nlated to the structure properties (i.e., thickness) and ma-\nterial properties (i.e., dielectric constant), so it has broad\nprospects in the field of precious detections.\nThe first device is the sensitive sub-nanometer thick-\nness detection. In Fig.(5a) and Fig.(5d), the trajec-\ntory of topological singularity for p-polarization and s-\npolarization with the thickness of layer-B are calculated\nby TMM (solid lines) and verified by FEM (squares),\nwhich agrees well with each other. When dBis in-\ncreased from 0 .295ato 0.296a, the singularity will be\nfrom λ= 38.2atoλ= 30.4afor p-polarization and from\nλ= 67.2atoλ= 40.2afor s-polarization. The sensitivity\nof the thickness sensor can be described by S= ∆λ/∆d,\nwhere ∆ λis the wavelength shift value of the singular-\nity caused by the thickness change ∆ d. The Svalues\nof p-polarization and s-polarization reach 7800 RIU−1\nand 27000 RIU−1, respectively. Therefore, the topo-\nlogical singularity is very sensitive to the dBfor both-\npolarizations. Additionally, in Fig.(5b) and Fig.(5e), we\nfurther depict the resolution ratio RpandRswith the\nchange of dB, in which the grey shadow indicates Rp≥1\nandRs≥1. The 2D and 1D field distributions are\nshown in Fig.(5c) and Fig.(5f). For p-polarization, we\nshow that, in the region with Rp>1, the field is well de-\nfined as spilt dipole-like distribution (shown in Fig.(5c)),\nwhich can be easily and clearly distinguished from the\ncase cin Fig.(5b). So a sensitive thickness detection\ncan be achieved by the reflection field shape of Gaus-\nsian beam. Therefore, in Fig.(5b), we show that the\nwell defined dipole-like distribution region with Rp>1\nis very sensitive to the thickness of dB, where we can\ndistinguish 0 .0275% change of dBby the reflected field\ndistribution. For the structure with the lattice constant\na= 1µmthat we set before, we can detect 0 .081nm\nchange of the thickness of layer-B, which can be applied\ninto sub-nanometer thickness error sensor. The sensitive\nthickness detection can be also applied for s-polarization.\nAs shown in Fig.(5e), we can even distinguish 0 .00373%\n(∼0.011nm) change of dBby the shape of reflected field\ndistribution.\nThe second device is the sensitive dielectric constant ε\nsensor. In Fig.(6a) and Fig.(6d), the trajectory of topo-\nlogical singularity for p-polarization and s-polarization\nwith the dielectric constant of layer-B ( εB) are calculated\nby TMM (solid lines) and verified by FEM (squares).\nWe show that the topological singularity is also sensi-\ntive to εB. When εBis increased from 3 .24 to 3 .241,7\n𝒂𝒂\n𝒄𝒄𝒂𝒂\n𝒃𝒃\n𝒄𝒄\n(a) (b) (c)\n𝐸𝐸2𝐸𝐸2𝐸𝐸2𝑥𝑥\n𝑥𝑥\n𝑥𝑥0.000081 0.0275%𝒃𝒃\n𝒅𝒅 𝒇𝒇 (d)\n𝒇𝒇(e) 𝒆𝒆\n𝐸𝐸2𝐸𝐸2𝐸𝐸2\n𝑦𝑦 𝑦𝑦 𝑦𝑦(f) 𝒆𝒆\n𝒅𝒅\n1.1𝑒𝑒−50.00373%p-polarization\ns-polarization\nFIG. 5. Thickness detection for p-polarization ((a)-(c)) and s-polarization ((d)-(f)). (a) and (d) are the trajectory of topological\nsingularity with different dBcalculated by TMM (blue lines) and FEM (black squares). (b) and (e) are resolution ratio Rp\n(with incident wavelength λ= 38.2a) and Rs(with incident wavelength λ= 67.2a) with different dB, where the grey shadow\nindicates Rp≥1 and Rs≥0.5. (c) and (f) are the 1D and 2D filed distribution of the reflected right-handed CP Gaussian\nbeam |+⟩, marked by a-f in (b) and (e).\nthe singularity will be from λ= 38.2atoλ= 35.6afor\np-polarization and from λ= 67 .2atoλ= 58 .9afor s-\npolarization. The sensitivity of the dielectric constant\nsensor can be described by S= ∆λ/∆ε, where ∆ λis\nthe wavelength shift value of the singularity caused by\nthe dielectric constant change ∆ ε. The Svalues of p-\npolarization and s-polarization reach 2600 µm/ RIU−1\nand 8300 µm/ RIU−1, respectively. Further more, in\nFig.(6b) and Fig.(6e), we further depict the resolution\nratio RpandRswith the change of εB, in which the grey\nshadow indicates Rp≥1 and Rs≥1. The 2D and 1D\nfield distributions are shown in Fig.(6c) and Fig.(6f). We\nshown that a sensitive dielectric constant sensor can be\nachieved by the reflection field shape of Gaussian beam.\nIn Fig.(6b) for p-polarization, we can distinguish 0 .0154%\n(∼0.0005) change of εBby the reflected field distribu-\ntion. In Fig.(6e) for s-polarization, we can even distin-\nguish 0 .0022% ( ∼0.00007) change of εBby the reflected\nfield distribution. We mention that the sensitive dielec-\ntric constant sensor can also be applied to the temper-\nature detection and power detection if the material is\nthe nonlinear material. We also note that the topolog-\nical singularity is sensitive to the dielectric constant of\nthe substrate and superstrate, which can further applied\ninto medical diagnosis[39], environmental detection[40],\netc. Details are shown in the supplementary section S4.\nIn this section, we illustrate that the reflected CPfiled is split at singularity numerically and physically.\nMeanwhile, the topological singularity is rather sen-\nsitive to the thickness and dielectric constant in the\ndeep-subwavelength, which can be applied to the single-\nchannel sensitive detection of the thickness and dielectric\nconstant for both-polarizations. Practically, this design\ncan be applied into the manufacturing error detection,\ntemperature detection and nonlinear power detection.\nIV. DUAL-POLARIZATIONS\nMULTI-CHANNELS DETECTION IN\nSUBWAVELENGTH REGION\nIn the above sections, we have shown that for\n”ABCBA-model”, there is a singularity in deep-\nsubwavelength for both-polarizations so that single-\nchannel dual-polarizations sensitive detection can be de-\nsigned utilizing the topological singularity. In this sec-\ntion, we will show that for more complicated layered\nstructure, more topological singularities can be achieved\nin the subwavelength region for both polarizations so\nthat multi-channels dual-polarizations detection can be\ndesigned.\nThe more complicated layered structure is shown in\nFig.(7a), where the PhCs is consist of seven layers in\nthe unit cell with relative permittivity as εA= 5.999,8\n(d)\n𝒋𝒋\n𝒍𝒍(e)𝒋𝒋 𝒌𝒌 𝒍𝒍\n𝐸𝐸2𝐸𝐸2𝐸𝐸2\n𝑦𝑦 𝑦𝑦 𝑦𝑦(f)\n7𝑒𝑒−50.0022%𝒌𝒌\ns-polarization𝒊𝒊𝒈𝒈\n𝒉𝒉\n𝒊𝒊\n(a) (b)\n𝐸𝐸2𝐸𝐸2𝐸𝐸2(c)\n𝑥𝑥\n𝑥𝑥\n𝑥𝑥𝒉𝒉\n𝒈𝒈\n0.0005 0.0154%\np-polarization\nFIG. 6. Dielectric constant detection for p-polarization ((a)-(c)) and s-polarization ((d)-(f)). (a) and (d) are the trajectory of\ntopological singularity with different εBcalculated by TMM (blue lines) and FEM (black squares). (b) and (e) are resolution\nratio Rp(with incident wavelength λ= 38.2a) and Rs(with incident wavelength λ= 67.2a) with different εB, where the grey\nshadow indicates Rp≥1 and Rs≥0.5. (c) and (f) are the 1D and 2D filed distribution of the reflected right-handed CP\nGaussian beam |+⟩, marked by g-l in (b) and (e).\nεB= 4, εC= 9, εD= 4.8,εC= 9, εB= 4,\nεA= 5.999 and the thickness of seven layers are set as\ndA/2 = 0 .041a,dB/2 = 0 .1a,dC/2 = 0 .15a,dD= 0.418,\ndC/2 = 0 .15a,dB/2 = 0 .1a,dA/2 = 0 .041a, where\na=dA+dB+dC+dDis the length of unit cell. The\nstructure is called ”ABCDCBA-model” [34]. In Fig.(7b)\nand Fig.(7c), the log of reflection coefficients are depicted\nin{θ, λ}space for s-polarization and p-polarization. We\nfind that there are two singularities in the subwavelength\nregion with λ≥15a(i.e., a/λ < = 0.06). The trajecto-\nries of topological singularities are labeled by blue and\nblack dashed-solid lines. For s-polarization in Fig.(7b),\nthe trajectories of two singularities are nearly fixed with\nthe increase of θ. While for p-polarization in Fig.(7c), two\nsingularities will gradually separate with each other with\nthe increase of θ. In Fig.(6d), the reflection coefficient of\nfinite PhCs with cell number N= 2 and incident angle\nθ= 1◦for p-polarization (blue line) and s-polarization\n(orange line) are calculated. The positions of topological\nsingularities are marked by red pentagrams (i.e., λ= 71a\nandλ= 19 a) for p-polarization and green pentagrams\n(i.e., λ= 40 aandλ= 21 a) for s-polarization, respec-\ntively. In Fig.(6e), we further calculate the PSHS for\np-polarization (blue line) and s-polarization (red line)\nin subwavelength region. We show that, two chan-\nnels dual-polairzations huge PSHS can be achieved for\n”ABCDCBA-model”, in which PSHS reaches 50 afor p-polarization and 100 afor s-polarization with waist radius\nw0= 200 a. Meanwhile, near the four topological singu-\nlarities (two singularities for s-polarization and two sin-\ngularities for p-polarization), the PSHS changes dramat-\nically, which means that there are two sub-wavelength\ndetection channels for s-polarization (marked by orange\nshadows) and two subwavelength detection channels p-\npolarization (marked by blue shadows), respectively. Ad-\nditionally, more singularities can be achieved in more\ncomplicated layered structure or periodic modulation\nmodel [34] so that more detection channels can be de-\nsigned applied to multi-frequency multiplexing applica-\ntions. These dual-polarizations multi-channels can be\nalso applied into the thickness, dielectric constant and\nbackground sensor, ect.\nIn the above devices, huge PSHS can be achieved\nnear the topological singularity, which is very sensitive\nto the incident frequency (wavelength) and less robust.\nHere, except for these applications, a broadband platform\nwith huge-PSHS can be realized for both-polarizations\nin subwavelength region in the ”ABCDCBA-model”. In\nFig.(8a), the log of reflection coefficient with θi= 1◦are\ndepicted in {dD, λ}space for p-polarization, in which the\ntrajectories of singularities are marked by black dashed\nlines. We show that with the increasing of dD, two singu-\nlarities will gradually approach each other and they will\nannihilate each other in real frequency domain at dD=9\nSubstrate…\n…\n…𝜽𝜽𝒊𝒊Superstrate\nABCBA\nC\nDABCBA\nC\nDxz(a)\n0.5 1 1.50.020.040.06𝑎𝑎/𝜆𝜆log10𝑟𝑟𝑠𝑠\n0\n𝜃𝜃(degree)(b)\n-4\n-6\n-8\n0.5 1 1.5\n𝜃𝜃(degree)0log10𝑟𝑟𝑝𝑝\n0.020.040.06𝑎𝑎/𝜆𝜆-4\n-6\n-8\n0 0.02 0.04 0.06-100-50050100\ndata\ndata\n0.02 0.04 0.06\n𝑎𝑎/𝜆𝜆-100-50050100Shift 𝜆𝜆(c)\n(e)\n0.02 0.04 0.06\n𝑎𝑎/𝜆𝜆×10−5\n|𝑟𝑟𝑝𝑝|\n024𝑟𝑟(d)0-10\n0Δ𝑦𝑦𝑝𝑝+\nΔ𝑦𝑦𝑠𝑠+ |𝑟𝑟𝑠𝑠|\nFIG. 7. (a) The multi-layers ABCDCBA-model. (b) and (c) are the log of reflection coefficient in {θ, λ}space for s-polarization\nand p-polarization. The trajectories of topological singularities are marked by solid-dashed lines. (d) The reflection coefficient\nof finite PhCs for p-polarization (blue line) and s-polarization (orange line) with cell number N= 2 and incident angle θ= 1◦.\nTopological singularities are marked by red pentagrams for p-polarization and blue pentagrams for s-polarization. (e) Photonic\nspin Hall shift for p-polarization (blue line) and s-polarization (red line). The waist radius is set as w0= 200 λ. There are two\ndetection channels for both polarizations, marked by blue and orange shadows.\n𝒅𝒅𝑫𝑫=𝟎𝟎.𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒 𝒅𝒅𝑫𝑫=𝟎𝟎.𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒 𝒅𝒅𝑫𝑫=𝟎𝟎.𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒\n0.02 0.04 0.06\n𝑎𝑎/𝜆𝜆0.02 0.04 0.06\n𝑎𝑎/𝜆𝜆0.02 0.04 0.06\n𝑎𝑎/𝜆𝜆\n0.02 0.04 0.06\n𝑎𝑎/𝜆𝜆0.02 0.04 0.06\n𝑎𝑎/𝜆𝜆0.02 0.04 0.06\n𝑎𝑎/𝜆𝜆05×10−5\n040\n-40 𝑟𝑟𝑝𝑝Δ𝑦𝑦𝑝𝑝+𝜆𝜆\n040\n-40Δ𝑦𝑦𝑝𝑝+𝜆𝜆\n040\n-40Δ𝑦𝑦𝑝𝑝+𝜆𝜆00.51\n00.51𝑟𝑟𝑝𝑝\n𝑟𝑟𝑝𝑝×10−4×10−4 (a) (b) (c) (d)\nFIG. 8. (a) The log of reflection coefficient in {dD, λ}space for p-polarization with θ= 1◦. The trajectories of topological\nsingularities are marked by black solid-dashed lines. Two singularities will annihilate at dD= 0.43428 a. (b)-(d) The reflection\ncoefficient (upper) and PSHS (downer) for p-polarization with dD= 0.424a,dD= 0.42425 aanddD= 0.4243a, respectively.\nBlue shadow in (c) indicates huge-PSHS platform for p-polarization.\n0.45428 aand then evolve into the complex frequency\ndomain [34]. In Fig.(8b)-Fig.(8d), The reflection coef-\nficients (upper) and PSHSs (downer) for p-polarization\nare depicted with dD= 0.424a,dD= 0.42425 aand\ndD= 0.4243a, respectively, labeled by dashed lines in\nFig.(8a). The topological singularities are marked by\ngreen pentagrams. The upper panels in Fig.(8b)-Fig.(8d)\nshow the singularities approach-annihilation process with\nthe increase of dD. In the downer panels in Fig.(8b)-\nFig.(8d), we shown that huge PSHS ( ∼50λ) can be\nachieved near two topological singularities. More inter-estingly, we find that when singularities approach each\nother, the PSHS between two singularities will become\nflatter. When dD= 0.42425 a, there is a wide wavelength-\nrange λ= 25a∼28aplatform with huge-PSHS reaches\nto ∆+\np≥37λ, as the blue shadow shown in the downer\npanel of Fig.(8c), which provides a perfect platform to\nresearch PSHS and apply into the detection devices. Fur-\nther increasing dD, two singularities will annihilate each\nother and the huge-PSHS will disappear due to the disap-\npearing of topological singularities in the real frequency\ndomain. The huge PSHS platform with wide frequency10\nrange can also be realized for s-polarizations in subwave-\nlength region, details are shown in the supplementary\nsection S5.\nV. CONCLUSION\nIn this work, we study the PSHS based on the topo-\nlogical singularity in deep-subwavelength in 1D multi-\nlayer PhCs with inversion symmetry. First, huge PSHS\nnear the topological singularity can be achieved for s-\npolarization ( ∼50λ) and p-polarization ( ∼25λ) with\nwaist radius w0= 100 ain deep-subwavelength, which\nbreaks the EMT. Second, the split reflected field dis-\ntribution with zero PSHS at the topological singular-\nity is demonstrated numerically and physically, which is\nresulted from the interfere of the spin-maintained nor-\nmal light and spin-flipped abnormal light. Third, based\non this exotic property, single-channel dual-polarizations\nthickness and dielectric constant sensor can be designed.\nFinally, by applying more complex ”ABCDCBA-model”,\ntwo singularities can be achieved in the subwavelength\nso that multi-channels detection can be designed. Fur-\nther more, a huge PSHS platform with wide frequency\nrange can be realized for both-polarizations in subwave-\nlength region. Theoretically, this work builds the bridge\nbetween topology study of 1D PhCs and light-matter\ninteraction in deep-subwavelength region. In practice,our model provides an unprecedented platform to re-\nsearch photonic spin-Hall effect, which can been applied\nin many applications, such as optical metrology, sensing\nand super-resolution imaging.\nMETHODS\nNumerical simulation . Finite element method\nverifications were calculated using commercial software\n(COMSOL Multiphysics). For reflection coefficient, a\nunit cell with Bloch boundary conditions along the x−\nandy−axes and ports along the z-axis was simulated.\nACKNOWLEDGEMENT\nThis work is supported by National Natural Science\nFoundation of China (Grant No. 12174073).\nDATA AVAILABILITY\nAll data that support the findings of this study are\nincluded within the article (and any supplementary files).\nCONFLICT OF INTEREST\nThe authors declare that they have no conflict of in-\nterest.\n[1] D. O’connor, P. Ginzburg, F. J. Rodr´ ıguez-Fortu˜ no,\nG. A. Wurtz, and A. V. Zayats, Nature communications\n5, 5327 (2014).\n[2] K. Y. Bliokh, F. J. Rodr´ ıguez-Fortu˜ no, F. 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Plasma Phys. 1\nThe Fundamental Parameters of\nAstrophysical Plasma Turbulence and its\nDissipation: Nonrelativistic Limit\nGregory G. Howes\nDepartment of Physics and Astronomy, University of Iowa, Io wa City IA 54224, USA\n(Received ?; revised ?; accepted ?. - To be entered by editori al office)\nA specific set of dimensionless plasma and turbulence parameters is in troduced to char-\nacterize the nature of turbulence and its dissipation in weakly collision al space and as-\ntrophysical plasmas. Key considerations are discussed for the de velopment of predictive\nmodels of the turbulent plasma heating that characterize the part itioning of dissipated\nturbulent energy between the ion and electron species and betwee n the perpendicular\nand parallel degrees of freedom for each species. Identifying the kinetic physical mech-\nanisms that govern the damping of the turbulent fluctuations is a cr itical first step in\nconstructing such turbulent heating models. A set of ten general plasma and turbulence\nparameters are defined, and reasonable approximations along with the exploitation of\nexisting scaling theories for magnetohydrodynamic turbulence are used to reduce this\ngeneral set of ten parameters to just three parameters in the is otropic temperature case.\nAcriticalstep forwardin this study isto identify the dependence of allofthe proposedki-\nnetic mechanisms for turbulent damping in terms of the same set of f undamental plasma\nand turbulence parameters.Analytical estimations ofthe scaling o feach damping mecha-\nnism on these fundamental parameters are presented, and this in formation is synthesized\nto produce the first phase diagram for the turbulent damping mech anisms as a function\nof driving scale and ion plasma beta.\n1. Introduction\nTurbulence is a fundamental, yet incompletely understood, proces s in space and as-\ntrophysical plasmas that mediates the transfer of the energy of chaotic plasma flows and\nelectromagneticfields into the energyof the plasmaparticles, eithe r asheat ofthe plasma\nspecies or as acceleration of a small population of particles (Howes 2 017). The 2013 NRC\nHeliophysics Decadal Survey (National Research Council 2013) ide ntifies plasma turbu-\nlence as a ubiquitous phenomenon occurring both within the heliosphe re and throughout\nthe Universe. Predicting the heating or acceleration of the differen t plasma species by\nturbulence, based on the observable turbulence and plasma param eters at large scales,\nconstitutes one of the grand challenge problems in heliophysics and a strophysics.\nTurbulent plasma heating plays a key role in governing the flow of ener gy in the he-\nliosphere, impacting the mesoscale and macroscale evolution of key h eliospheric environ-\nments comprising the coupled solar-terrestrial system and conne cting the solar corona,\nsolar wind, and Earth’s magnetosphere. The accuracy of ongoing e fforts (Miki´ c et al.\n1999; Lionello et al.2009; van der Holst et al.2014; Adhikari et al.2020) to model glob-\nally the flow of energy from the Sun through the interplanetary med ium to the Earth,\nto the other planets, and on to the boundary of the heliosphere wit h the surrounding\ninterstellar medium (Opher et al.2020) relies on the availability of prescriptive models\nof the turbulent plasma heating (Howes 2010; Chandran et al.2011; Rowan et al.2017).2\nSuchturbulentheatingprescriptionsarealsocriticalfortheinter pretationofremoteas-\ntronomical observations of emissions from black hole accretion disk s, such as the ground-\nbreaking observations by the Event Horizon Telescope of the supe rmassive black holes at\nthe centerofM87(Event Horizon Telescope Collaboration et al.2019) andat Sagittarius\nA∗intheMilkyWay(Event Horizon Telescope Collaboration et al.2022),Inthesecases,\nalternative turbulent heating prescriptions (Howes 2010; Rowan et al.2017) have been\nshown to yield drastically different predictions for the emitted radiat ion (Chael et al.\n2019).\n1.1.How Does Turbulence Mediate Energy Transport and Plasma Hea ting?\nThe transport of energy mediated by plasma turbulence is depicted in Fig. 1, where (a) a\nplot of the turbulent magnetic energy wavenumber spectrum show s that energy injected\ninto the turbulent cascade at the outer scale ( k0ρi= 10−4) is transferred nonlinearly\nwithout loss through the inertial range to the transition into the dis sipation range at\nion scales ( kρi∼1). At the ion scales and smaller ( kρi/greaterorsimilar1), poorly understood mech-\nanisms can remove energy from the turbulence, transferring ene rgy to the ions (blue\narrows) or electrons (magenta arrows). In Fig. 1(b), modern pla sma turbulence theory\n[(Goldreich & Sridhar 1995, hereafter GS95) or (Boldyrev 2006, hereafter B06)] predicts\nthat the turbulent cascade generates a scale-dependent anisot ropy in wavevector space\n(blue shaded region); isotropy at the driving scale ( k⊥0ρi=k/bardbl0ρi= 10−4) develops into\na significant anisotropy k⊥≫k/bardblas the cascade transfers energy to smaller scales †. In ad-\ndition, kinetic instabilities can generate unstable turbulent fluctuat ions with wavevectors\nkρi∼1(redovalsinboth panels),whichcanmediate thenonlocaltransfe rofenergyfrom\nthe large-scale motions directly to the ion kinetic scales. I explicitly de fine “turbulence”\nhere as the collection of physical mechanisms involved in the convers ion of the energy of\nlarge-scaleplasma flows and magnetic fields into plasma heat or the en ergy of accelerated\nparticles, including the nonlinear scale-to-scale energy transfer o f the turbulent cascade,\nthe mechanisms damping the turbulence at small scales, and any pos sible instabilities\nthat may generate turbulent fluctuations or govern nonlocal ene rgy transfer.\n1.2.What Do We Want to Predict?\nA long-term goal of the heliophysics community is to characterize th e partitioning of\nturbulent energy in terms of observable plasma and turbulence par ameters, producing a\npredictiveheatingmodelthatcanbeusedinnumericalmodelingofth eglobalevolutionof\nthe heliosphere, thereby advancing our capability for predictive mo deling of the coupled\nsolar-terrestrial system. Understanding how the multiscale phys ics of plasma turbulence\ncouples the large-scale plasma conditions and evolution to the microp hysical heating and\nparticle energization is essential to creating predictive heating mod els. Unfortunately,\ncurrent numerical codes cannot simulate the vast dynamic range o f scales inherent to\nthe multiscale nature of heliospheric plasma turbulence while simultane ously capturing\nthe kinetic physics governing the conversion of turbulent energy t o some form of particle\nenergy. Developing turbulent heating models that capture this cou pling is one promising\navenue for formulating predictive models at a computationally feasib le cost. To develop\nsuch models, it is essential to establish a thorough understanding o f the microphysical\nkinetic processes that govern the energization of particles.\nIn a statistically steady-state turbulent cascade in a weakly collision al astrophysi-\ncal plasma, the energy injected into the turbulence at the outer s cale is transferred\nlocally in scale down to increasingly smaller scales until reaching the kine tic length\n†Parallel andperpendicular here are defined relative to the local mean magnetic field dire c-\ntion.Fundamental Parameters of Turbulence 3ρik =1\nk−2.8α\nkρe=1 \nαk−5/3~10 −4 log(E  ) B\nk d i~1 \n0 −4 −3 −2 −1 1Damping\n2Nonlinear Energy Transfer \nElectron\nDampingDissipation Range Inertial Range\nInjection by\nKineticInstabilities\nlog(k     )ρi(a)\nρi Outer Scale, kIon\ni log(k     )kρe=1 ρik =1\nρi=1 k\nkkα1/3 ρi log(k     ) \nkkα2/3 \nρikρik ~10−4 ~Isotropic\nOuter Scale\n0 −4 −3 −2 −1 1 2 Inertial Range Dissipation Range\n−4 −3 −2 −1 01\nCritically Balanced\nAlfvenic Cascade(b)\nInjection by\nKinetic\nInstabilities\nEnergy Transfer \nρNonlocal \nFigure 1. (a) Schematic diagram of the turbulent magnetic-energy spe ctrum in the solar wind,\ndepicting the local transfer of energy from large scales (sm all wavenumbers k) to small scales\n(largek) through a turbulent cascade (black arrows). Instabilitie s may alter this energy flow by\nnonlocally transporting energy directly to small, kinetic scales (red arrow). (b) In wavevector\nspace, the local transfer of the Alfv´ enic turbulent cascad e follows the anisotropic, critically\nbalanced path, here plotted quantitatively for the case of t heGS95scaling theory.4\nQuantity Symbol Bounded\nTurbulent energy density cascade rate ε\nPlasma energy density heating rate Q\nIon-to-Electron heating ratio Qi/QeQi/Q\nAnisotropic ion energy density heating ratio Q⊥,i/Q/bardbl,iQ⊥,i/Qi\nAnisotropic electron energy density heating ratio Q⊥,e/Q/bardbl,eQ⊥,e/Qe\nTable 1. The key input quantities (top section) and deliverables (bo ttom section) for predictive\nturbulent heating models. Bounded quantities are normaliz ed such that their values vary only\nover the interval [0 ,1].\nscales at which damping mechanisms can collisonlessly remove energy f rom the tur-\nbulence and energize the different particle species, as depicted in Fig . 1(a). This local\ncascade of energy is mediated by nonlinear interactions among the t urbulent fluctua-\ntions, such as between counterpropagating Alfv´ en wave packet s, often denoted “Alfv´ en\nwave collisions” (Kraichnan 1965; Sridhar & Goldreich 1994; Goldreich & Sridhar 1995;\nNg & Bhattacharjee1996;Galtier et al.2000;Howes et al.2012b;Howes & Nielson2013;\nNielsonet al.2013; Howes et al.2013; Drake et al.2013; Howes 2016; Verniero et al.\n2018; Verniero & Howes 2018; TenBarge et al.2021; Ripperda et al.2021). Nonlocal en-\nergytransfermechanisms—suchaskinetic instabilitiesdrivenbythe large-scaledynamics\nthat can cause the plasma conditions to deviate from local thermod ynamic equilibrium—\ncan also directly generate turbulent fluctuations at the small kinet ic length scales, as\nillustrated in Fig. 1(b). Together, these mechanisms lead to a turbu lent energy den-\nsity cascade rate εfrom the large scales down to the small scales of the turbulence.\nThis nonlinear transfer of the turbulent energy density is ultimately damped at small\nscales by kinetic plasma physics mechanisms, converting the energy density of the small-\nscale turbulent plasma flows and magnetic fields into particle energy w ith a statistically\nsteady-state total plasma energy density heating rate Q∼ε.\nFor a single-ion-species plasma, such as a fully ionized hydrogenic plas ma of protons\nand electrons, the primary deliverable of a turbulent heating model is the determination\nof the partitioning of the energy by the kinetic turbulent damping me chanisms between\nthe ions and electrons, Qi/Qe. Unlike in the case of strongly collisional plasmas, in which\nthe dissipated turbulent energy would be rapidly equilibrated betwee n ion and electron\nspecies by collisions, under the weakly collisional plasma conditions typ ical of space\nand astrophysical plasmas, the differential heating of the plasma s pecies can impact the\nthermodynamic evolution of the plasma on mesoscales and macrosca les.\nFurthermore, energy transferred to a given plasma species may le ad to energization\nof the parallel degree of freedom or the perpendicular degrees of freedom relative to the\ndirection of the local magnetic field. Thus, secondary deliverables o f a turbulent heating\nmodel are measures of the anisotropic ion energy density heating r ateQ⊥,i/Q/bardbl,iand\nthe anisotropic electron energy density heating rate Q⊥,e/Q/bardbl,e. In a strongly collisional\nplasma,theanisotropicheatingforeachspecieswouldbe rapidlyisot ropizedbycollisions,Fundamental Parameters of Turbulence 5\nbut weakly collisional plasmas allow the parallel and perpendicular †degrees of freedom\nto be energized independently (Kawazura et al.2019).\nDifferent kinetic particle energization mechanisms may energize differ ent degrees of\nfreedom— e.g., Landau damping energizes particles parallel to the magnetic field, b ut cy-\nclotron damping energizes particles perpendicular to the magnetic fi eld—so determining\nQ⊥,i/Q/bardbl,iandQ⊥,e/Q/bardbl,efirst requires one to identify which kinetic physical mecha-\nnisms govern the damping of the turbulent fluctuations as a functio n of the plasma\nand turbulence parameters. Therefore, identifying the contribu ting turbulent damping\nmechanisms as a function of the plasma and turbulence parameters is an additional de-\nliverable of a predictive turbulent heating model. Such information ab out the dominant\nturbulent damping mechanisms, as a function of the key dimensionles s parameters de-\nscribing the turbulent plasma, can be concisely presented in a phase diagram for the\ndissipation of kinetic plasma turbulence, analogous to that develope d for the behavior of\nmagnetic reconnection in astrophysical plasmas (Ji & Daughton 20 11). A first attempt\nat a phase diagram for the turbulent damping mechanisms in weakly co llisional space\nand astrophysical plasmas is presented here in Fig. 11.\nThe key input quantities for a turbulent heating model ( ε,Q) and the deliverables\n(Qi/Qe,Q⊥,i/Q/bardbl,i,Q⊥,e/Q/bardbl,e) for a turbulent heating model are summarized in Table 1.\nNote that for ion cyclotron damping, which energizes only the perpe ndicular degrees\nof freedom in the ion velocity distribution function, the anisotropic io n heating ratio\nQ⊥,i/Q/bardbl,iwould be formally infinite. To avoid these inelegant infinities, we may cho ose\nto use alternative bounded versions of the heating ratios in the lower part of Table 1\nthat vary only over the range [0 ,1]. These bounded quantities are related to the original\nquantities by Qi/Q= (Qi/Qe)/[1+ (Qi/Qe)] for the species partition and Q⊥,s/Qs=\n(Q⊥,s/Q/bardbl,s)/[1+(Q⊥,s/Q/bardbl,s)] for the anisotropic heating ratio for each species.\nIt is worthwhile noting that the determination of the partitioning of t he turbulent\nenergy between ions and electrons Qi/Qecan be simplified by the properties of the\nturbulent cascade that are evident in Fig. 1(a). The inertial range is defined as the\nrange of scales over which the turbulent energy density is transfe rred to smaller scales\nwithout loss, leading to a constant energy density cascade rate εthroughout this range ‡.\nTherefore, the turbulent cascade rate εat the onset of damping at ion kinetic scales at\nkρi∼1 is the same as that determined at the outer scale of the turbulenc e atk0ρi≪1.\nThecollisionlesswave-particleinteractionsthatservetoremoveen ergyfromtheturbulent\nfluctuations and transfer that energy to the ionsgenerally play a significant role only\nwithin the range of length scales similar to the ion kinetic length scales, kρi∼1: (i)\nat larger scales kρi≪1, the collisionless damping rate with the ions γiis typically\nnegligible in comparison to the turbulent cascade rate γi≪ωnl; and (ii) at smaller\nscaleskρi≫1, the gyromotion of the ions averages out the effect of the small-s cale\nelectromagnetic fluctuations, so the ions decouple from the turbu lent fields and do not\nexchange significant energy. Whatever turbulent energy manage s to cascade beyond the\nion scales to smaller scales at kρi≫1 will ultimately be deposited with the electrons.\nThus, the task of determining Qi/Qeboils down to determining what fraction of the\nturbulent energy is damped on the ions as the cascade progresses through the ion kinetic\nscales at kρi∼1.\n†Note that, under typical astrophysical plasma conditions i n which the turbulent fluctuation\nfrequencies are not much faster than the species cyclotron f requencies, ω/lessorsimilarΩs, the two perpen-\ndicular degrees of freedom are rapidly equilibrated by the g yromotion of the particles about the\nmagnetic field, leading to gyrotropic plasma conditions wit hT⊥,1∼T⊥,2.\n‡This standard definition of the inertial range from fluid turb ulence theory neglects the\npossibility of a nonlocal transfer of energy that may arise i n weakly collisional plasmas.6\nUltimately, the kinetic mechanisms that serve to damp the turbulent fluctuations at\nkinetic length scales depend on the nature of the turbulent fluctua tions at those scales.\nTherefore, a viable strategy for developing a useful phase diagra m for plasma turbulence\nis to use turbulence scaling theories ( e.g.,GS95orB06) to predict the properties of the\nturbulent fluctuations upon reaching the ion and electron kinetic sc ales from the plasma\nand turbulence parameters at large scales. The primary aim of this in vestigation is to\nestablish a theoretical framework upon which to base such predict ions of the dominant\nturbulent damping mechanisms as a function of the plasma and turbu lence parameters.\n1.3.What is the Approach for Developing a Predictive Turbulent H eating Model?\nThe Buckingham Pi Theorem (Buckingham 1914) uses dimensional an alysis (Barenblatt\n1996) to determine the minimum number of dimensionless parameters upon which the\nphysical behavior of a system depends. This approach is the key pr inciple underlying the\ndevelopmentofpredictivemodelsofturbulentplasmaheatinginterm softhefundamental\ndimensionless parameters of kinetic plasma turbulence. For sufficien tly simple systems,\nin which there are not multiple physical quantities of importance that have the same\ndimensions, one can define a unique set of fundamental dimensionles s parameters. But\nfor more complicated systems, in which there may be more than one r elevant physical\nquantity with the same dimensions, such as multiple characteristic len gth scales, the\ndetermination of the fundamental dimensionless parameters is gen erally non-unique. The\ncase of kinetic turbulence in a weakly collisional plasma certainly falls int o this more\ncomplicated category. In that situation, it is essential to use phys ical insight into the\nsystem’s properties and behavior to define a particular set of dimen sionless parameters\nthat is most useful in characterizing the behavior of the system.\nThe goal of this paper is to define a particular set of fundamental d imensionless pa-\nrameters that can be used to characterizethe properties of kine tic plasma turbulence and\nits damping mechanisms. Modern scaling theories for anisotropic plas ma turbulence can\nthen be expressed in terms of these fundamental parameters to determine the properties\nof the turbulent fluctuations at the kinetic length scales at which kin etic physical mecha-\nnisms can act to remove energy from the turbulent fluctuations an d energize the plasma\nparticles. How those kinetic damping mechanisms depend on the plasm a parameters and\nproperties of the turbulent fluctuations can then be used to pred ict the dominant mech-\nanisms for the damping of turbulence as a function of the dimensionle ss parameters.\nAll of this information can then be synthesized to generate improve d turbulent heating\nprescriptions for comparison to direct numerical simulations of tur bulent dissipation and\nfor use in global modeling efforts for space and astrophysical plasm a systems, such as the\nheliosphere or supermassive black hole accretion disks.\n2. The Fundamental Dimensionless Parameters of Kinetic Plasma\nTurbulence\nThe first step in developing a predictive model for plasma turbulence in weakly col-\nlisional space and astrophysical plasmas is to identify the key dimens ionless parameters\nupon which the turbulent energy cascade and its dissipation depend . In this paper, the\nfocus is to characterize kinetic plasma turbulence in the non-relativ istic limit. There are\ntwo categories of governing parameters: Plasma Parameters andTurbulence Parameters .\nModern scaling theories for anisotropic plasma turbulence can be em ployed to reduce\nthe number of parameters needed to characterize the turbulenc e, as described in §3.1.\nThe reduced set of dimensionless parameters can then be used to d evelop useful phase\ndiagrams that identify the dominant physical mechanisms of turbule nt dissipation andFundamental Parameters of Turbulence 7\nwhich of these key parameters has the strongest impact on how pa rticles are energized\nby the turbulence.\n2.1.Plasma Parameters\nThePlasma Parameters play a critical role in determining the nature of the turbulent\nfluctuations and their interaction with the underlying particle velocit y distributions. For\nsimplicity of notation, we assume here a fully ionized, hydrogenic plasm a of protons and\nelectrons with bi-Maxwellian equilibrium velocity distributions characte rizedby separate\nparallel and perpendicular temperatures †for each species, T/bardbl,sandT⊥,s. In this idealized\ncase, the equilibrium ion and electron number densities are equal, ni=ne.\nThe first and most important plasma parameter is the parallel ion plas ma beta, β/bardbl,i=\n8πniT/bardbli/B2, which characterizes the ratio of parallel thermal pressure to th e magnetic\npressure in the plasma and dominantly controls the phase speeds of different linear wave\nmodes in a magnetized plasma. Note that with definitions for the para llel ion thermal\nvelocity v2\nt/bardbl,i= 2T/bardbl,i/miand the Alfv´ en velocity v2\nA=B2/(4πnimi), the parallel ion\nplasma beta can be alternatively expressed as β/bardbl,i=v2\nt/bardbl,i/v2\nA. The second parameter\nis the ion-to-electron parallel temperature ratio τ/bardbl=T/bardbl,i/T/bardbl,e. The third and forth\nparameters describe the ion and electron temperature anisotrop ies,Ai=T⊥,i/T/bardbl,iand\nAe=T⊥,e/T/bardbl,e. The fifth parameter characterizes the collisionality of the plasma b y\ncomparing the mean free path scale for electron-electron collisions relative to the parallel\ndriving scale k/bardbl0λmfp,e.\nAlthough for a hydrogenic plasma of protons and electrons the ion- to-electron mass\nratioµ=mi/mehas a fixed physical value of 1836, this ratio is included as an additiona l\nindependent parameter in the calculations below since a reduced mas s ratio is often used\nin numerical simulations of plasma turbulence for reasons of comput ational efficiency.\nNote also that, once µandτare specified, the single dimensionless parameter k/bardbl0λmfp,e\nis sufficient to specify all possible charged particle collision rates, νee,νei,νii, andνie.\nTherefore, ( β/bardbl,i,τ/bardbl,Ai,Ae,k/bardbl0λmfp,e) are five key dimensionless plasma parameters\nthat govern the properties of plasma turbulence, along with the fix ed mass ratio µ.\nThese Plasma Parameters are summarized in Table 2.\nNotethatiftheplasmacontainsmultipleionspecies,possiblyeachwith multiplecharge\nstates, the number of parameters increases dramatically, requir ing a minimum of five\nnew dimensionless parameters for each new species s: (i) mass ratio ms/mi, (ii) number\ndensity ratio ns/ne, (iii) charge ratio qs/qe, (iv) parallel temperature ratio T/bardbl,s/T/bardbl,i, and\n(v) temperature anisotropy T⊥,s/T/bardbl,s. Partially ionized plasmas will similarly require\nadditional dimensionless parameters to be characterized. Since th e intention here is not\nto treat kinetic plasma turbulence in full generality, but rather to d escribe how one\ncan use the Pi Theorem and turbulence scaling theories to minimize th e number of\nrelevant dimensionless parameters, the remainder of this paper will focus on the case of a\nfully ionized, hydrogenic plasma of protons and electrons with bi-Max wellian equilibrium\nvelocity distributions. Refinements of turbulent heating models to in clude minor ions,\nsuch as α-particles that comprise a small fraction of ions in space and astrop hysical\nplasmas, can be constructed following the same principles presente d here.\n2.2.Turbulence Parameters\nTheTurbulence Parameters characterize the scale, amplitude, and properties of the tur-\nbulent driving though five additional parameters. For simplicity, the focus here is re-\n†Note that we absorb the Boltzmann constant into the temperat ure throughout this paper,\ngiving temperature in units of energy.8\nPlasma Parameters\nParameter Symbol and Definition\nParallel Ion Plasma Beta β/bardbl,i= 8πniT/bardbli/B2\nIon-to-Electron Parallel Temperature Ratio τ/bardbl=T/bardbl,i/T/bardbl,e\nIon Temperature Anisotropy Ai=T⊥,i/T/bardbl,i\nElectron Temperature Anisotropy Ae=T⊥,e/T/bardbl,e\nPlasma Collisionality k/bardbl0λmfp,e\nIon-to-Electron Mass Ratio µ=mi/me\nTurbulence Parameters\nParameter Symbol and Definition\nPerpendicular Driving Scale k⊥0ρi\nDriving Scale Anisotropy k/bardbl0/k⊥0\nNonlinear Parameter χ0=k⊥0δB⊥0θ0/(k/bardbl0B0)\nImbalance Parameter Z+\n0/Z−\n0\nDriving Compressibility Ecomp/Einc\nTable 2. For the general case of homogeneous turbulence in a fully ion ized, hydrogenic plasma\nwith bi-Maxwellian equilibrium velocity distributions, t hese are the fundamental dimensionless\nparameters of kinetic plasma turbulence, including Plasma Parameters and Turbulence Param-\neters. Note that the Boltzmann constant is absorbed to give t emperature in units of energy.\nstricted to only the case of turbulence in a spatially homogeneous sy stem in which the\nscalelengthofgradientsin the equilibriumquantitiesaremuchlargert han theouterscale\nof the system. Dimensionless ratios of length scales are notationally simplified by charac-\nterizing the length-scalesofthe turbulent fluctuations in terms of their perpendicular and\nparallel wavenumbers. Below quantities at the outer scale of the tu rbulence, commonly\ndenoted the driving scale or energy injection scale, are denoted by the subscript “0”.\nThe driving of the turbulence can be characterized by three dimens ionless parameters.\nThe first parameter is the perpendicular wavenumber of the driving -scale fluctuations\nnormalized by the ion Larmor radius, k⊥0ρi. The second parameter is the wavevec-\ntor anisotropy of the fluctuations at the driving scale, k/bardbl0/k⊥0. The third parame-\nter is the the nonlinearity parameter that characterizes the amplit ude of the driving,\nχ0= (k⊥0δB⊥0θ0)/(k/bardbl0B0), where δB⊥0is the amplitude of the perpendicular magnetic\nfield fluctuations, B0is the equilibrium magnetic field magnitude, and θ0is the angle\nof dynamic alignment at the driving scale in radians (Boldyrev 2006). I n the next sec-\ntion, we demonstrate how modern scaling theories for anisotropic M HD turbulence can\nbe used to combine these three dimensionless parameters into a sing le parameter, the\nisotropic driving wavenumber ,k0ρi. Note also that, in principle, the dynamic alignment\nangleθ0atdrivingscaleisanadditionaldimensionlessparameter,but dynamic alignment\nis typically understood to be a phenomenon that develops self-cons istently due to the\nturbulence as it cascadesto smaller scales, so we neglect this param eter at the outerscale,\ngenerally taking θ0∼1.\nThe nature of the turbulent fluctuations at the driving scale also pla ys an impor-Fundamental Parameters of Turbulence 9\ntant role in determining the ultimate fate of the turbulent energy. T he fourth turbu-\nlence parameter is the imbalance parameter, measuring the ratio of the amplitudes in\nupward-to-downward Elsasser fields, Z+\n0/Z−\n0, where the Elsasser fields at the outer scale\nare defined by Z±\n0=δU⊥,0±δB⊥,0/√4πnimi(Elsasser 1950), and where δU⊥,0and\nδB⊥,0are the perpendicular perturbed plasma flow velocity and perpendic ular perturbed\nmagnetic field of the turblent fluctuations at the driving scale. Phys ically, Alfv´ en wave\npackets propagating up the mean magnetic field have Z+= 0 and Z−∝negationslash= 0, and those\npropagating down the field have Z+∝negationslash= 0 and Z−= 0. The Elsasser fields effectively\ncharacterize the wave energy flux in each direction along the magne tic field†of the typ-\nically dominant Alfv´ enic fluctuations in heliospheric plasma turbulence (Tu & Marsch\n1995; Schekochihin et al.2009; Bruno & Carbone 2013). Since the nonlinear interactions\nthat mediate the turbulent cascade in Alfv´ enic turbulence at MHD s cales arise only\nbetween counterpropagating and perpendicularly polarized Alfv´ e n waves (Kraichnan\n1965; Sridhar & Goldreich 1994; Goldreich & Sridhar 1995; Ng & Bhatt acharjee 1996;\nGaltieret al.2000; Howes & Nielson 2013; ?), the imbalance plays an important role\nin characterizing the turbulence. The importance of Z+\n0/Z−\n0under sufficiently imbal-\nanced conditions, such as occurs in the inner heliosphere where wav e energy fluxes are\ndominantly anti-sunward, has been demonstrated clearly with the r ecent discovery of\nnew phenomenon known as the helicity barrier in plasma turbulence (Squire et al.2022,\n2023).\nA fifth turbulence parameter is the ratio of the energy in compress ible fluctuations\nto that in incompressible fluctuations, Ecomp/Einc. Since a magnetized plasma supports\ntwo distinct compressible wave modes, the fast and slow magnetoso nic waves (differing\nby whether the thermal and magnetic pressure perturbations as sociated with the wave\nact in conjunction or in opposition), one could further characteriz e the compressible\ncomponent of the turbulent fluctuations by the additional ratio of the energy in the fast\nmagnetosonic fluctuations to the total energy of compressible flu ctuations EF/Ecomp,\nwhere the total compressible energy is the sum of the contrbution s from fast-mode and\nslow-mode fluctuations, Ecomp=EF+Es. The ratio EF/Ecompis not included here as\none of the fundamental parameters based on a study of the inert ial range in solar wind\nturbulence at 1 AU that showed negligible energy in fast-mode mode fl uctuations, or\nEF/Ecomp≪1 (Howes et al.2012a). In significant energy exists in fast-mode fluctua-\ntions in other turbulent astrophysical plasmas, EF/Ecompmay need to be included.\nTherefore, ( k⊥0ρi,k/bardbl0/k⊥0,χ0,Z+\n0/Z−\n0,Ecomp/Einc) are five key dimensionless turbu-\nlence parameters that govern the properties of plasma turbulenc e. These Turbulence\nParameters are summarized in Table 2.\nIf the restriction to spatially homogeneousturbulent plasmas is rela xed, the large-scale\ncontext ofthe turbulent plasmacan alsoimpact the nonlineardynam ics ofthe turbulence\nand the evolution of the turbulent fluctuations. In the solar wind, t he spherical expan-\nsion of the solar wind governs the evolution of the plasma equilibrium te mperatures\nwith heliocentric radius, possibly triggering kinetic temperature anis otropy instabilities\nthat can mediate nonlocal transfer of energy to small scales, as d epicted in Fig. 1(b).\nIn rotating plasmas, such as found in the solar convection zone, th e turbulence may be\ndriven with helical motions that can initiate the growth of magnetic fie ld energy through\na dynamo effect (Parker 1955; Glatzmaier 1984, 1985; Parker 199 3; Ossendrijver 2003).\n†Note that the imbalance of counterpropagating wave energy fl uxes in plasma turbulence has\nalternatively beencharacterized intheliterature usingt hecross helicity(Dobrowolny et al.1980;\nMarsch & Tu 1990; Matthaeus et al.1982; Perez & Boldyrev 2009, 2010) or the normalized\nPoynting flux (Hnat et al.2003; Di Mare & Howes 2024).10\nDifferential rotationin astrophysicalaccretiondisks,which can tr iggerplasmaturbulence\nthrough the magnetorotational instability (Balbus & Hawley 1991; H awley & Balbus\n1991; Balbus & Hawley 1998) has been recently shown to generate t urbulence with a\ndominant fraction of its energy in compressible fluctuations (Kawaz uraet al.2022). Fi-\nnally, kinetic instabilities arising in the foot and ramp regions of collisionle ss shocks\n(Brownet al.2023) can generate turbulent fluctuations that are swept into th e down-\nstream region. In all of these astrophysically relevant cases, add itional dimensionless tur-\nbulence parameters will be required to characterize properly the t urbulent fluctuations\nand their evolution.\n3. Scaling Theories for Anisotropic Plasma Turbulence\nThe key to predicting the evolution of the turbulent fluctuations as energy is cas-\ncaded to ever smaller scales, and therefore also to developing a use ful phase diagram\nfor plasma turbulence, is to use turbulence scaling theories to pred ict the properties of\nthe turbulence upon reaching the ion kinetic scales from the charac teristics of the tur-\nbulence at large scales. In particular, we will include the two prominen t theories for the\nanisotropic cascade of Alfv´ enic turbulence in MHD plasmas, the Gold reich-Sridhar 1995\n(GS95) formulation for weak and strong MHD turbulence (Sridhar & Goldre ich 1994;\nGoldreich & Sridhar 1995), and the modification of this theory by Bold yrev (B06) that\naccountsforaweakeningofthe nonlinearitybydynamicalignment(B oldyrev2006).Note\nthat since these are MHD theories of plasma turbulence—where str ong collisionality is\nassumed in the standard MHD approximation (Kulsrud 1983; Gurnet t & Bhattacharjee\n2017)—the equilibrium species temperatures are assumed to be anis otropic and equal,\ncorresponding to τ/bardbl=Ai=Ae= 1. For the case of weakly collisional plasma turbulence\nthat is relevant to most space and astrophysicalplasma environme nts, these theories may\nneed to be modified appropriately, but such development of refined turbulence theories\nis beyond the scope of the treatment here.\nThe particular scalings of the parallel wavenumber, intermediate wa venumber, align-\nment angle, and magnetic field perturbation as a function of the per pendicular wavenum-\nber are given by\nk/bardbl∝k2/(3+α)\n⊥(3.1)\nki∝k3/(3+α)\n⊥(3.2)\nθ∝k−α/(3+α)\n⊥(3.3)\nδB⊥∝k−1/(3+α)\n⊥(3.4)\nwhereα= 0 corresponds to the GS95theory and α= 1 corresponds to the B06theory.\nHere the turbulent fluctuations as a function of scale are charact erized by the three\nspatial components of the wave vector relative to the local magne tic field direction:\n(i) the parallel wavenumber k/bardbl; and (ii) the intermediate wavenumber kiand (iii) the\nperpendicular wavenumber k⊥, which togetherdescribe the anisotropyofthe fluctuations\nin the plane perpendicular to the local magnetic field, ki≪k⊥, arising from the dynamic\nalignment of the fluctuations (Boldyrev 2006; Mason et al.2006; Boldyrev et al.2009;\nPerezet al.2012) and predicting the development of current sheets at small s cales. These\nscaling theories predict the scaling of k/bardbl,ki, the angle of dynamic alignment θ, and the\namplitude of the perpendicular magnetic field fluctuations δB⊥as a function of the\nperpendicular wavenumber k⊥.Fundamental Parameters of Turbulence 11\nThese theories yield specific predictions for the perpendicular magn etic energy spec-\ntrum ofturbulence inthe MHDregime, k⊥ρi≫1,andforthe scale-dependentanisotropy\nof the turbulence, a qualitative feature that is well-established in MH D turbulence\n(Cho & Vishniac 2000; Maron & Goldreich 2001; Cho et al.2002; Cho & Lazarian 2003;\nMasonet al.2006;Boldyrev et al.2009;Narita et al.2010;Sahraoui et al.2010;Perez et al.\n2012; Roberts et al.2015). The perpendicular magnetic energy spectrum for the two t he-\nories is given by\nEB⊥(k⊥)∝(δB⊥)2\nk⊥∝k−(5+α)/(3+α)\n⊥=/braceleftBigg\nk−5/3\n⊥α= 0\nk−3/2\n⊥α= 1. (3.5)\nThe parallel-to-perpendicular wavevector anisotropy is given by\nk/bardbl\nk⊥∝k−(1+α)/(3+α)\n⊥=/braceleftBigg\nk−1/3\n⊥α= 0\nk−1/2\n⊥α= 1, (3.6)\ndemonstrating that both theories predict that the turbulent fluc tuations develop a scale-\ndependent anisotropy, whereby the fluctuations become progre ssively more elongated\nalongthemagneticfieldastheycascadetosmallerscale,eventuallyy ieldingk/bardbl/k⊥≪1at\nsufficiently smallscales.This k/bardbl/k⊥anisotropyfollowsfromthe imposing ofacriticalbal-\nance between the linear and nonlinear timescales of the turbulence ( Goldreich & Sridhar\n1995; Mallet et al.2015). The intermediate-to-perpendicular wavevector anisotro py is\ngiven by\nki\nk⊥∝k−α/(3+α)\n⊥=/braceleftbiggk0\n⊥α= 0\nk−1/4\n⊥α= 1. (3.7)\nHere, the GS95theory predicts no development of anisotropyin the perpendicular plane,\nat odds with the findings of current sheets at small perpendicular s cales in numerical\nsimulations (Matthaeus & Montgomery 1980; Meneguzzi et al.1981; Uritsky et al.2010;\nZhdankin et al.2012, 2013) and spacecraft observations (Borovsky 2008; Osm anet al.\n2011).The B06theorypredictsthedevelopmentofanisotropyintheperpendicula rplane,\ncorresponding to the development of current sheets with ki/k⊥≪1 at sufficiently small\nscales,potentiallytriggeringtearinginstabilities(Furth et al.1963;Zocco & Schekochihin\n2011)thatenablemagneticreconnectiontodisrupttheturbulent cascade(Boldyrev & Loureiro\n2017; Loureiro & Boldyrev 2017 b,a; Malletet al.2017b,a; Walker et al.2018).\nIt is worthwhile emphasizing that the idea of critical balance in the tur bulent cascade\ndoesnotimplythatalloftheturbulentenergyisconcentratedattheparalle lwavenumber\nkcb\n/bardbldetermined by critical balance. Rather, kcb\n/bardblis the upper limit of parallel wavenumber,\nand thus generally also implies an upper limit on the linear wave frequenc y (sinceω∝k/bardbl\nforω≪Ωi). Rather, the turbulent fluctuation powerextends overthe ran ge0/lessorequalslantk/bardbl/lessorequalslantkcb\n/bardbl\n(Goldreich & Sridhar 1995; Maron & Goldreich 2001). Accounting for how the turbulent\nenergy is distributed over this k/bardblrange would lead to order unity or less quantitative\nchanges in the heating rates. Since the goal of predictive turbulen t heating models is\ngenerally to calculate the lowest-order, order-of-magnitude qua ntities for species heating\nrates, any effects due to the distribution of turbulent energy ove r 0/lessorequalslantk/bardbl/lessorequalslantkcb\n/bardblare likely\nto yield only higher-order corrections to these predictions; thus, the effect of this range\nof turbulent energy in k/bardblwill be neglected in the first generation of turbulent heating\nmodels addressed here.\nIn closing, note that these theories for MHD turbulence are largely focused on the\ntransport of energy in Alfv´ enic fluctuations, so if significant ener gy is transferred non-\nlinearly into other (non-Alfv´ enic ) wave modes, this may significantly alter the energy12\ntransport from the predictions shown here, with potentially qualita tively different pre-\ndictions for the damping mechanisms of these non-Alfv´ enic turbule nt fluctuations.\n3.1.Exploiting Turbulent Scaling Theories to Reduce the Number of Fundamental\nDimensionless Parameters\nGiventhelargenumberoffundamentaldimensionlessparameterst hatcharacterizeweakly\ncollisional plasma turbulence, shown in Table 2, it is imperative to reduc e the number\nof important parameters to make progress in developing predictive models of turbulent\nheating. Adopting idealized approximations, such as maintaining isotr opic equilbrium\nvelocity distributions for the plasma, enables one to eliminate some of the parameters in\nTable 2. Another approach is to exploit the MHD turbulent scaling the ories described\nin§3 to combine several parameters into a new single parameter, an ap proach presented\nhere.\nIn general, the driving of the plasma turbulence depends on at least three independent\nparameters: (i) the perpendicular wavenumber at which the turbu lence is driven k⊥0ρi,\n(ii) the anisotropyofthe driven fluctuationswith respect tothe loc almagneticfield direc-\ntionk/bardbl0/k⊥0, and (iii) the nonlinearity parameter which characterizesthe stren gth of the\ndriving,χ0≡k⊥0δB⊥0θ0/(k/bardbl0B0). Recall that quantities evaluated at the driving scale\nare given the subscript “0”. Fortunately, for turbulence driven a t large scales (relative to\nthe ion kinetic scales), as is typical in most space or astrophysical p lasma environments,\nMHD scaling theory can be used to predict the properties of the tur bulence upon reach-\ning the kinetic scales, where kinetic plasma processes act to dissipat e the turbulence. In\nfact, as detailed here, it is possible to combine these three dimension less parameters into\na new, single dimensionless parameter: the isotropic driving wavenumber ,k0ρi.\nMHD turbulent scaling theories can be used, for a given set of driving parameters\n(k⊥0ρi,k/bardbl0/k⊥0,χ0),topredictthepropertiesoftheturbulentfluctuations( e.g.,anisotropy,\namplitude) upon reaching the ion kinetic scales, here taken to be the perpendicular\nwavenumber k⊥ρi= 1. The isotropic driving wavenumber k0ρiis the unique wavenum-\nber that produces the same conditions at k⊥ρi= 1 as if the turbulence were driven\nisotropically at that wavenumber with k⊥0=k/bardbl0≡k0and in a condition of critical\nbalanceχ0= 1.\nThe key procedure for determining the isotropic driving wavenumbe rk0ρiis illus-\ntrated in Fig. 2. Panel (a) shows the perpendicular wavenumber ma gnetic energy spec-\ntrumEB⊥(k⊥) driven weakly with χ0= 0.1 at perpendicular wavenumber k⊥0ρi=\n10−5, where a weak turbulent cascade leads to a steep weak MHD turbule nce spec-\ntrum with EB⊥∝k−2\n⊥(Sridhar & Goldreich 1994; Galtier et al.2000; Boldyrev & Perez\n2009), followed by a flatter strong MHD turbulence spectrum with EB⊥∝k−(5+α)/(3+α)\n⊥\n(Goldreich & Sridhar 1995; Boldyrev 2006). The cascade reaches t he perpendicular ion\nkinetic scale k⊥ρi= 1,andathigherwavenumbersthe cascadecontinuesasstrongk inetic\nAlfv´ en wave (KAW) turbulence with a significantly steeper magnetic energy spectrum\nwithEB⊥∝k−2.8\n⊥(Sahraoui et al.2009; Kiyani et al.2009; Alexandrova et al.2009;\nChenet al.2010; Howes et al.2011; Sahraoui et al.2013). When the collisionless damp-\ningoftheKAWturbulentcascadebeginstodiminishtherateofnonline arenergycascade,\nthe spectrum begins to fall off exponentially in the weakly dissipating K AW turbulence\n(WDKT) regime (Howes et al.2011), ultimately terminating the turbulent cascade at\nscalesk⊥ρe/greaterorsimilar1.\nIn Fig. 2(b) is presented the characteristic path of the anisotrop ic cascade of energy\nthrough ( k⊥,k/bardbl) wavevector space for the different regimes of the turbulent cas cade.\nThere is no parallel cascade in the weak MHD turbulence regime (Sridh ar & GoldreichFundamental Parameters of Turbulence 13\nFigure 2. Diagram illustrating how to determine the isotropic drivin g wavenumber k0ρiin\nterms of the perpendicular driving wavenumber k⊥0ρi, the driving anisotropy k/bardbl0/k⊥0, and the\nnonlinearity parameter of the driving, χ0=k⊥0δB⊥θ0/(k/bardbl0B0).\n1994;Galtier et al.2000),followedbyascale-dependentanisotropygivenby k/bardbl∝k2/(3+α)\n⊥\nin the strong MHD turbulence regime (Goldreich & Sridhar 1995; Boldy rev 2006). The\nturbulence transitions to the strong KAW regime at k⊥ρi∼1, with the anisotropy be-\ncoming stronger at k⊥ρi>1, scaling as k/bardbl∝k1/3\n⊥(Cho & Lazarian 2004; Howes et al.14\n2008a, 2011). As the turbulence weakens due to collisionlessdamping of tu rbulent fluctu-\nations in the WDKT regime, the parallel cascade ceases once again (H oweset al.2011).\nThe diagrams in Fig. 2 provide guidance to understand how MHD turbu lence scaling\ntheories can be used to combine ( k⊥0ρi,k/bardbl0/k⊥0,χ0) into a single new parameter, the\nisotropicdrivingwavenumber, k0ρi.Iftheturbulenceisdrivenatcriticalbalance( χ0= 1)\nand isotropically ( k/bardbl0/k⊥0= 1), then the isotropic driving wavenumber is simply the\nsameastheperpendiculardrivingwavenumber, k0ρi=k⊥0ρi.Therefore,itisnecessaryto\nuse MHDturbulencescalingtheoryto determine k0ρionlyinthe casethatthe turbulence\nis driven weakly χ0<1 or anisotropically k/bardbl0/k⊥0∝negationslash= 1.\nFor weakly driven turbulence with χ0<1, MHD turbulence theory predicts that the\nparallel cascade is inhibited and that the perpendicular wavenumber magnetic energy\nspectrum of the turbulence scales as EB⊥∝(δB⊥)2/k⊥∝k−2\n⊥. Therefore, in terms of\nthe driving amplitude δB⊥0and perpendicular driving wavenumber k⊥0, the amplitude\nof the turbulence as a function of k⊥is given by δB⊥(k⊥) =δB⊥0(k⊥0/k⊥)1/2. Given\nthe lack of parallel cascade in weak MHD turbulence, we fix k/bardbl=k/bardbl0, and obtain the\nfollowing scaling †for the nonlinearity parameter as a function of k⊥,\nχ(k⊥) =χ0/parenleftbiggk⊥ρi\nk⊥0ρi/parenrightbigg1/2\n. (3.8)\nMHD turbulence theory predicts that the nonlinearity parameter w ill increase through\nthe weak turbulent cascade until it reaches a value of unity, χ∼1, at which point the\nturbulence has reached a state of critical balance in the strong tu rbulence regime. From\nthat point, theory predicts that an anisotropic cascade will ensue (transferring energy\nto smaller scales more rapidly in the perpendicular than the parallel dir ection), with the\nparallel and perpendicular cascades balanced to maintain a nonlinear ity parameter of\nunity,χ∼1. Therefore, the key point in the cascade is perpendicular wavenu mber at\nwhich the turbulence first reaches χ= 1, given by\nk⊥ρi|χ=1=1\nχ2\n0k⊥0ρi (3.9)\nAn example of this weakly turbulent, strictly perpendicular cascade in (k⊥,k/bardbl) space\nis depicted in Fig. 2(b), where turbulence driven weakly with χ0= 0.1, cascades to\nk⊥ρi= 100k⊥0ρi, at which point the turbulence becomes strong, with χ= 1.\nTo obtain the isotropic driving wavenumber k0ρifrom this point in wavevector space\n(k⊥ρi|χ=1,k/bardbl0), we need to extend the cascade backward along the critical balan ce line\n(solid black and red dashed lines) until it intersects with isotropy k/bardbl=k⊥(dotted line).\nFrom the MHD scaling theories in §3, for a critically balanced, strong turbulent cascade\nthat is driven strongly and isotropically at k0ρi, the anisotropy as a function of k⊥scales\nas\nk/bardbl\nk⊥=/parenleftbiggk0ρi\nk⊥ρi/parenrightbigg1+α\n3+α\n. (3.10)\nThe anisotropy at the point ( k⊥ρi|χ=1,k/bardbl0) is given by k/bardbl/k⊥= (k/bardbl0/k⊥0)χ2\n0. Substitut-\ning this expression for k/bardbl/k⊥on the left-hand side of (3.10) and substituting (3.9) into\nthe denominator on the right-hand side of (3.10), we can solve for t he isotropic driving\n†Here we have assumed that the dynamic alignment angle θ0also does not evolve as a\nfunction of k⊥in weak MHD turbulence, as there is no prediction in the liter ature for the\nscaling of dynamic alignment in weak turbulence.Fundamental Parameters of Turbulence 15\nwavenumber k0ρi, obtaining the result\nk0ρi=k⊥0ρi/parenleftbiggk/bardbl0\nk⊥0/parenrightbigg3+α\n1+α\nχ4\n1+α\n0 (3.11)\nIn Fig. 2(b), the isotropic driving wavenumber k0ρi(red circle) is the shown where the\ncriticalbalanceline(solid blackandreddashedlines) intersectsthe is otropyline, k/bardbl=k⊥\n(dotted line). Note that this reduction of the three driving parame ters tok0ρiis valid\nonly in the limit that the turbulent cascade becomes strong before it reaches the ion\nkinetic scales at k⊥ρi∼1.\nThe relation (3.11) providesthe means to combine the three turbule nce driving param-\neters (k⊥0ρi,k/bardbl0/k⊥0,χ0) into the single new effective parameter, the isotropic driving\nwavenumber k0ρi. The properties of the turbulent fluctuations at perpendicular sc ales\nsmaller than the point where the turbulent cascade becomes stron g, ork⊥ρi> k⊥ρi|χ=1,\nwill be the same for the original case with driving parameterized by ( k⊥0ρi,k/bardbl0/k⊥0,χ0)\nas for turbulence driven stronglyand isotropicallyat k0ρi. In the development ofa turbu-\nlent heating model, it is the properties of the turbulent fluctuations at the small, kinetic\nlength scales that will determine the kinetic damping mechanism and th e resulting par-\nticle energization, so reducing three turbulence driving parameter s to a single parameter\nsubstantially reduces the dimensionality of the parameter space to be modeled.\nIt is worthwhile noting that the evolution of the dynamic alignment ang le with per-\npendicular wavenumber θ(k⊥) is not well modeled by k0ρi. The development of current\nsheets at very small scales due to dynamic alignment, and the possib ility for the inter-\nruption of the turbulent cascade by magnetic reconnection (Boldy rev & Loureiro 2017;\nLoureiro & Boldyrev 2017 b,a; Malletet al.2017b,a; Walker et al.2018), will depend on\nthis scaling of θ(k⊥). But, from a practical point of view, this issue in modeling θ(k⊥)\nwhile reducing ( k⊥0ρi,k/bardbl0/k⊥0,χ0) tok0ρiis likely to be a problem only in rare cases.\nFirst, the scaling of dynamic alignment angle θ∝k−1/4\n⊥inB06is rather weak, scaling\nas the−1/4 power, so only when the MHD inertial range spans many orders of m ag-\nnitude in the perpendicular length scale will the development of curre nt sheets likely\nhave a significant impact the turbulent dynamics. Second, dynamic a lignment appears\nto arise only in strong MHD turbulence, so it seems reasonable to tak eθ0∼1 radian\n(roughly equivalent to isotropy in the perpendicular plane) at the pe rpendicular scale\nk⊥ρi|χ=1where the turbulence becomes strong. One may then estimate the evolution of\nthat dynamic alignment angle θ(k⊥) from that point for the cases in which it impacts the\nevolution. Finally, although the development of current sheets in pla sma turbulence is\nwell established (Matthaeus & Montgomery 1980; Meneguzzi et al.1981; Borovsky2008;\nUritskyet al.2010; Osman et al.2011; Zhdankin et al.2012, 2013), it is not clear that\nthe scaling theory in B06indeed properly describes this development; an alternative\nexplanation of the development of current sheets due to from Alfv ´ en wave collisions ex-\nists (Howes & Nielson 2013; Howes 2015, 2016; Verniero et al.2018), although a specific\nscaling for this alternative mechanism has not yet been proposed.\n3.2.Efficiently Modeling the Small-Scale End of the Turbulent Ine rtial Range\nThe multiscale physics of astrophysicalplasma turbulence present s a particular challenge\nfor numerical modeling that covers the entire dynamic range of the turbulent cascade\nwhilesimultaneouslycapturingthekineticmechanismsthegovernthe dampingofthetur-\nbulence under weakly collisional plasma conditions, in particular since p hysics of plasma\nturbulence is a inherently three-dimensional (Howes 2015). For th e development of tur-\nbulent heating models, it is critical to investigate the physical mecha nisms that remove16\nenergy from the turbulence and energize the plasma particles. For tunately, since the iner-\ntial range of turbulence is defined as the range of scales over which the turbulent energy\ncascades to smaller scales but the dissipation is negligible, as depicted in Fig. 1(a), it\nis not strictly necessary to model the the entire inertial range in or der to capture the\nphysics of the turbulent dissipation.\nOne less computationally costly approach to capture numerically the mechanisms of\nturbulent damping is to choose to model directly only the small-scale e nd of the inertial\nrange, preserving as much of the resolved dynamic range of the sim ulation as possible\nto capture the dissipation mechanisms. To do so, one takes advant age of the scaling\nrelations for turbulence in the MHD regime k⊥ρi≪1, summarized in §3, to “drive” the\nturbulence at the domain scale of the simulation with turbulent fluctu ations that have\nthe properties dictated by the scaling theories.\nAs an example of how to set up simulations in this way, consider turbule nce physically\ndriven strongly and isotropically at an isotropic driving wavenumber o fk0ρi= 10−4,\nas depicted in Fig. 3. For a hydrogenic plasma of protons and electro ns with a realistic\nmass ratio mi/me= 1836, the turbulent cascade extends at least down to the perpe n-\ndicular scales of the electron Larmor radius k⊥ρe∼1 (Sahraoui et al.2010; Howes et al.\n2011; TenBarge et al.2013; Kiyani et al.2015). Thus, a dynamic range of Lo/ρe=\nk⊥max/k0= (k⊥maxρe)/(k0ρi)(ρi/ρe)∝(k0ρi)−1(Ti/Te)1/2(mi/me)1/2∼4.3×105\nin each of the three spatial dimensions is necessary when Ti/Te∼1, well beyond the\ncapabilities of any kinetic simulation code. If the simulation code can re solve a dynamic\nrange of 103in each spatial dimension, we can choose a perpendicular domain scale in\nthe middle of the physical inertial range at k⊥Dρi= 10−2and use a reduced mass ratio\nmi/me= 100 so that k⊥ρe= 1 corresponds to k⊥ρi= 10 for Ti/Te= 1. Thus, the\nresolved perpendicular length scales of the simulation span 10−2/lessorequalslantk⊥ρi/lessorequalslant10, corre-\nsponding to 10−3/lessorequalslantk⊥ρe/lessorequalslant1 in terms of the electron Larmor radius ρe, as shown in\nFig. 3.\nTo drive the turbulence in the middle of the turbulent inertial range a tk⊥Dρi= 10−2,\nwe use the scaling relations (3.1)–(3.4) to determine the appropriat e turbulent scale-\ndependent anisotropyand amplitude ofthe fluctuations atthe num ericaldomainscalefor\naphysical inertial range driven strongly ( χ0= 1) and isotropically ( k⊥0=k/bardbl0=ki0≡k0\nandθ0= 1 rad)at k0ρi= 10−4.Forthis example,wewill adoptthe B06scaling,choosing\nα= 1. The wavevector anisotropy and amplitude of the fluctuations a t the domain scale\ncan be expressed using scaling relations (3.1)–(3.4) to yield\nk/bardblDρi=k0ρi/parenleftbiggk⊥Dρi\nk0ρi/parenrightbigg1/2\n(3.12)\nkiDρi=k0ρi/parenleftbiggk⊥Dρi\nk0ρi/parenrightbigg3/4\n(3.13)\nθD=θ0/parenleftbiggk⊥Dρi\nk0ρi/parenrightbigg−1/4\n(3.14)\nδB⊥D\nB0=/parenleftbiggk⊥Dρi\nk0ρi/parenrightbigg−1/4\n(3.15)\nThe properties of the domain scale fluctuations at k⊥Dρi= 10−2determined by these\nscaling relations are illustrated in Fig. 3 for a turbulent cascade para meterized by di-\nmensionless parameters k0ρi= 10−4andθ0= 1 rad. The normalized value of energy\nspectrum at k⊥Dρi= 10−2is (k0/B0)2EB⊥(k⊥) = 10−3, consistent with the solutionFundamental Parameters of Turbulence 17\nFigure 3. Modeling the small-scale end of the turbulent inertial rang e using MHD scaling\nrelations, choosing B06scaling with α= 1,mi/me= 100,k0ρi= 10−4,k⊥Dρi= 10−2.\nδB⊥D/B0= 0.32 of (3.15) at the domain scale, given by the red circle in Fig. 3(a). Th e\nthree-dimensional anisotropy of the wavevector of the turbulen t fluctuations at the per-\npendicular domain scale k⊥Dρi= 10−2is characterized by: (i) k/bardblDρi= 10−3from (3.12),\ngiven by the blue circle in Fig. 3(b); and (ii) kiDρi= 3.2×10−3from (3.13), given by\nthe green circle in Fig. 3(b). The dynamic alignment of the velocity and magnetic field18\nfluctuations at the domain scale is given by θD= 0.32 rad from (3.14). Note that for\nthese parameters, the scalings lead to a constant nonlinearity par ameter as a function of\nperpendicular wavenumber χ(k⊥) =χ0= 1 at all scales k⊥ρi<1, asdictated by the con-\njecture of critical balance in strong turbulence (Goldreich & Sridha r 1995; Mallet et al.\n2015). These scaling calculations determine the nature of the turb ulent fluctuations at\nthe numerical domain scale, in the middle of the physical inertial rang e being modeled.\nSince the large-scale of the resolved dynamic range of the simulation atk⊥Dρiis con-\nstantlybeing fed turbulent cascadeenergymediated bynonlinearin teractionsamongtur-\nbulent fluctuationsatscalesslightlylargerthanthe domainscalewith k⊥< k⊥D,it isim-\nperative that these turbulence simulations are driven, rather tha n simply simulating the\ndecay of the energy turbulent fluctuations initialized at the domain s cale. The nonlinear\ninteractionsamongperpendicularlypolarized,counterpropagatin gAlfv´ enwavesdrivethe\ncascade of energy in Alfv´ enic plasma turbulence (Kraichnan 1965; Sridhar & Goldreich\n1994;Goldreich & Sridhar1995;Ng & Bhattacharjee1996;Galtier et al.2000;Howes et al.\n2012b; Howes & Nielson 2013; Nielson et al.2013; Howes et al.2013; Drake et al.2013;\nHowes 2016; Verniero et al.2018; Verniero & Howes 2018). Exploiting this physical in-\nsight, driving counterpropagatingAlfv´ en waves, polarized in both dimensions perpendic-\nular to the equilibrium magnetic field, has been shown to generate suc cessfully a steady-\nstate plasma turbulent cascade (Howes et al.2008b; Howeset al.2011; TenBarge et al.\n2013; Told et al.2015; Verniero et al.2018; Verniero & Howes 2018). A finite-time corre-\nlated driving, such as an antenna driven by a Langevin equation (Ten Bargeet al.2014),\nmimics the unsteady nature of the turbulent cascade from larger s cales expected of tur-\nbulent driving at a chosen scale within the turbulent inertial range.\nTo set up the simulation to describe optimally the domain scale fluctuat ions, it has\nbecome common practice to utilize a numerical domain that is elongate d along the direc-\ntion of the equilibrium magnetic field, consistent with the anisotropics caling of turbulent\nfluctuations at small scales, k/bardbl/k⊥≪1. For the example shown in Fig. 3, individual\nAlfv´ enic fluctuations are predicted to have a perpendicular plane a nisotropy given by\nkiD/k⊥D= 0.32. But since these Alfv´ enic fluctuations can be polarized in either o f the\ndirections in the perpendicular plane, it is necessary for both perpe ndicular directions\nto be large enough to resolve the intermediate scale (the larger of t he two perpendicular\nscales since kiD/k⊥D<1). Thus, the simulation domain should be defined by L/bardbl×L2\ni,\nwhereL/bardbl= 2πρi(k/bardblDρi)−1andLi= 2πρi(kiDρi)−1. Within this domain, counterpropa-\ngating Alfv´ enic fluctuations should be driven, polarized in both perp endicular directions,\nwith wavevectors characterized by ( k⊥D,kiD,k/bardblD), with dynamic alignment character-\nized byθD, and with amplitudes δB⊥D. This approach enables the efficient simulation of\nthe small-scale end of the inertial range consistent with the scaling p roperties of MHD\nturbulence, enabling kinetic simulations to focus on the physical mec hanisms damping\nthe turbulent fluctuations.\n3.3.A Reduced Parameter Space for Kinetic Plasma Turbulence\nAlthough the physics of plasma turbulence, its dissipation, and the r esulting heating of\nthe plasma species formally depends on all ten of the general plasma and turbulence\nparameters ( β/bardbl,i,τ/bardbl,Ai,Ae,k/bardbl0λmfp,e;k⊥0ρi,k/bardbl0/k⊥0,χ0,Z+\n0/Z−\n0,Ecomp/Einc) listed in\nTable 2, focusing initially on the development of completely general tu rbulent heating\nmodels on a ten-dimensional parameter space is unlikely to be succes sful. Instead, first\ndeveloping predictive turbulent heating models over a reduced para meter space is likely\na better strategy.\nNote that kinetic temperature anisotropy instabilities generally bou nd the tempera-\nturesinthevicintyofisotropy,althoughiftheseinstabilitiesaretrig gered,at βi≫1,theyFundamental Parameters of Turbulence 19\nPlasma Parameters\nParameter Symbol and Definition\nIon Plasma Beta βi= 8πniTi/B2\nIon-to-Electron Temperature Ratio τ=Ti/Te\nIon-to-Electron Mass Ratio µ=mi/me\nTurbulence Parameters\nParameter Symbol and Definition\nIsotropic Driving Wavenumber k0ρi\nTable 3. Isotropic Temperature Case : Reduced set of fundamental dimensionless parameters\nfor isotropic Maxwellian equilibrium velocity distributi ons with Ai=Ae= 1, collisionless con-\nditionsk/bardbl0λmfp,e≫1, strong turbulence assumed at the ion kinetic scales, bala nced turbulence\nZ+\n0/Z−\n0∼1, and negligible energy in compressible fluctuations Ecomp/Einc≪1. This results in\na reduced set of plasma and turbulence parameters ( βi,τ;k0ρi).\ncan impact both the linear wave physics at play in the turbulent dynam ics of critically\nbalancedturbulence(Squire et al.2016,2017 b,a)andmediatethenonlocaltransferofen-\nergy directly from the large driving scales to the kinetic length scales (Arzamasskiy et al.\n2023). As an initial exploration of Alfv´ enic turbulence and its kinetic damping mecha-\nnisms, however, we choose to focus on the simplified limit of isotropic t emperatures,\nmodifying our approach as needed for cases that violate these con ditions.\nFor many of the calculations in this study, a reduced isotropic tempe rature case is\nadopted, assuming that the equilibrium velocity distribution for each species remains\nwell described as an isotropic Maxwellian distribution. Note that kinet ic temperature\nanisotropyinstabilities generallybound the species temperature an isotropiesin the vicin-\nity of isotropy As∼1 (Kasper et al.2002; Hellinger et al.2006;ˇStver´ aket al.2008;\nBaleet al.2009;ˇStver´ aket al.2009), although if these instabilities are triggered, they\ncan impact both the turbulent dynamics and energy transfer to pa rticles (Squire et al.\n2016, 2017 b,a; Arzamasskiy et al.2023). In this isotropic temperature case, the temper-\nature anisotropies for ions and electrons are both set to unity, Ai= 1 and Ae= 1.\nIn addition, the ion plasma beta and ion-to-electron temperature r atio are redefined in\nterms of these isotropic temperatures, yielding the isotropic para metersβi= 8πniTi/B2\nandτ=Ti/Te. In addition, for this reduced parameter space, we will also assume that\nthe turbulent inertial range is sufficiently large, or the driving is suffic iently strong, that\nthe dynamics at the perpendicular ion kinetic length scales k⊥ρi∼1 are in a state of\nstrong plasma turbulence with a nonlinearity parameter χ∼1. Under these conditions,\nwe can replace the three turbulent driving parameters ( k⊥0ρi,k/bardbl0/k⊥0,χ0) by the single\nisotropic driving wavenumber k0ρi, as detailed in §3.1.\nWe will also make two other simplfying assumptions for the isotropic te mperature\ncase: We will assume that the turbulence is balanced, with equal wav e energy fluxes up\nand down the local magnetic field Z+\n0/Z−\n0∼1, which is a necessary condition to ap-\nply the MHD turbulence scaling relations given by (3.1)–(3.4). Addition ally, based on20\ndirect spacecraft measurements of turbulence in the heliosphere which find that incom-\npressible Alfv´ enic fluctuations appear to dominate turbulent dyna mics in space plasmas\n(Belcher & Davis 1971; Tu & Marsch 1995; Alexandrova et al.2008; Bruno & Carbone\n2013), as well as theoretical considerations that suggest that t he dynamics of Alfv´ en\nwaves dominate the turbulent cascade (Lithwick & Goldreich 2001; M aron & Goldreich\n2001; Cho & Lazarian 2003; Schekochihin et al.2009; Howes 2015), we will also assume\na negligible amount of energy in compressible turbulent fluctuations, Ecomp/Einc≪1.\nTherefore, we propose the isotropic temperature case , a reduced parameter space for\nweakly collisional plasma turbulence in the limit of strong, balanced, inc ompressible tur-\nbulence, with a governing set of three dimensionless parameters, ( βi,τ;k0ρi), as shown\nin Table 3. Although the isotropic temperature case eliminates a numb er of important\nphysical effects such as temperature anisotropy instabilities, imba lanced turbulent wave\nenergy fluxes, and compressible fluctuations, it nonetheless repr esents an excellent start-\ning point for the development of refined turbulent heating models. S uch models can\nproduce theoretical predictions that can be compared to kinetic n umerical simulations\nand spacecraft observations of weakly collisional plasma turbulenc e, exemplified by the\nphase diagram for turbulent damping mechanisms shown in Fig. 11.\nIntermsofthefundamentalparametersoftheisotropictemper aturecase,theturbulent\nscaling relations (3.1)–(3.4) simplify to the following easy-to-use for ms\nk/bardblρi= (k0ρi)(1+α)/(3+α)(k⊥ρi)2/(3+α)(3.16)\nkiρi= (k0ρi)α/(3+α)(k⊥ρi)3/(3+α)(3.17)\nθ=θ0/parenleftbiggk0ρi\nk⊥ρi/parenrightbiggα/(3+α)\n(3.18)\nδB⊥\nB0=/parenleftbiggk0ρi\nk⊥ρi/parenrightbigg1/(3+α)\n(3.19)\n4. Review of Previous Turbulent Heating Models\nSeveral previous studies have constructed turbulent heating mo dels that attempt to\nquantify how the energy removed from the turbulent cascade is pa rtitioned between\nprotons and electrons and between parallel and perpendicular deg rees of freedom. Here\nwe briefly summarize these existing turbulent heating models, includin g the dissipation\nmechanisms considered, the assumptions of how that energy is rem oved from the turbu-\nlence, and that stated limitations of each model.\nOne ofthe earliestefforts to estimate the proton and electronene rgizationratesby tur-\nbulenceduetocollisionlessdampingmechanismswascarriedoutbyQua taert & Gruzinov\n(1999) (hereafter QG99), following earlier results by each of these authors to determine\nthe collisionless damping rates under astrophysically relevant condit ions (Quataert 1998;\nGruzinov 1998). This study adopted the GS95scaling for dominantly Alfv´ enic turbu-\nlence at perpendicular scales restricted to the MHD limit, k⊥ρi/lessorsimilar1. The turbulent\nfluctuations upon reaching the end of the MHD limit at the perpendicu lar scale of the\nion Larmor radius k⊥ρi∼1 are predicted to be significantly anisotropic with k⊥≫k/bardbl;\nsince the MHD Alfv´ en wave frequency ω=k/bardblvAis proportional to k/bardbl, this yields turbu-\nlent fluctuations at this perpendicular ion scale with low frequencies r elative to the ion\ncyclotron frequency, ω≪Ωi. Thus, all collisionless damping is provided by the n= 0\nLandau resonance, specifically Landau damping and transit-time da mping. In addition,\nthis model assumed that all turbulent energy at a given perpendicu lar wavenumber k⊥ρiFundamental Parameters of Turbulence 21\nwas concentrated at the parallel wavenumber given by the conjec ture of critical bal-\nance (Goldreich & Sridhar 1995; Quataert & Gruzinov 1999). They c onstructed a simple\ncascade model that balanced the collisionless damping by ions and elec trons given by\nthe Vlasov-Maxwell linear dispersion relation (Stix 1992) over the MH D range of scales,\nk⊥ρi/lessorsimilar1. Beyond these scales at k⊥ρi/greaterorsimilar1 in the kinetic regime, uncertainty about\nthe nature of the turbulence at these sub-ion length scales led the m to assume simply\nthat any energy reaching k⊥ρi/greaterorsimilar1 ultimately heated the electrons. They calculated\nthe fraction of the turbulent energy that heats the electrons, Qe/Q, as function of total\nplasma beta β=βi+βe=βi(1+Ti/Te), yielding a numerical result for Qe/Q(β). Their\nresulting energy partition depended strongly on a constant assum ed in the model that\ncharacterized the balance of the nonlinear turbulent cascade rat e to the linear collision-\nless damping rate; uncertainty in the value of this constant (by a fa ctor from 1 /4 to 4)\nled the resulting prediction of Qe/Q(β) to vary by more than one order-of-magnitude\nover the range of 3 /lessorsimilarβ/lessorsimilar100.\nThe Howes2010(hereafter H10) turbulent heating model (Howes2010) wasbasedon a\nturbulent cascademodel (Howes et al.2008a) that extended the GS95turbulence scaling\nfromMHDscalesat k⊥ρi/lessorsimilar1intothe kinetic regimeat k⊥ρi/greaterorsimilar1,generalizingthe conjec-\ntureofcriticalbalancetoaccountforthedispersivenatureofkin eticAlfv´ enwavesatthese\nsub-ion length scales (Biskmap et al.1999; Cho & Lazarian 2004; Krishan & Mahajan\n2004; Shaikh & Zank 2005; Schekochihin et al.2009). The H10model was based on\nthree primary assumptions: (i) the Kolmogorov hypothesis that th e energy cascade is\ndetermined by local interactions (Kolmogorov 1941); (ii) the conje cture that turbulence\nmaintains a state of critical balance at all scales (Goldreich & Sridhar 1995); and (iii)\nthe speculation that linear collisionless damping rates are applicable ev en in the presence\nof the strong nonlinear interactions that mediate the turbulent ca scade to small scales.\nThe casade model used by H10balanced at each perpendicular wavenumber k⊥ρithe\nnonlinear energy cascade rate given by the extended turbulent sc aling theories with the\nlinear collisionless damping rates at that value of k⊥ρi. Similar to QG99, it assumed all\nturbulent energy at a given value of k⊥ρiresided at the parallel wavenumber governed\nby the extended critical balance. The H10model assumed turbulence under nonrelativis-\ntic conditions vte/c≪1 with isotropic ion and electron temperatures, strictly Alfv´ enic\nturbulence with Ecomp/Einc= 0, balance turbulence with Z+\n0/Z−\n0= 1, and strongly and\nisotropically driven turbulence with χ0= 1 and k/bardbl0/k⊥0= 1 (the driving was there-\nfore characterized by the isotropic driving wavenumber, k0ρi). Furthermore, the same as\nQG99, the model assumed that the inertial range of the turbulence is su fficiently large\nthat the anisotropic fluctuations predicted by the turbulent scalin g relations remain at a\nlow enough frequency relative to the ion cyclotron frequency ω≪Ωithat the cyclotron\nresonance plays a negligible role; the H10model calculated an estimate of the maximum\nvalue ofk0ρiasa function of( βi,Ti/Te) for this assumption to be satisfied.Thus, the H10\nturbulent heating model predicted the damping of the turbulent ca scade due to Landau\ndamping and transit-time damping overthe full range of the turbule nt cascade, including\nMHD scales at k⊥ρi/lessorsimilar1 and kinetic scales at k⊥ρi/greaterorsimilar1. TheH10model predicted the\npartitioning of energy between ions and electrons as a function of t he ion plasma beta\nβiand the ion-to-electron temperature ratio Ti/Te, providing a simple analytical fit to\nthe numerically calculated predictions, Qi/Qe(βi,Ti/Te). In short, the model found that\nQi/Qeis a montonically increasing function of βiwith weak dependence on Ti/Te, where\nthe transition from Qi/Qe<1 toQi/Qe>1 happens at approximately βi∼1.\nThe Chandran 2011 (hereafter C11) turbulent heating model (Chandran et al.2011)\nlooked to incorporate the recently quantified damping rate due to n onlinear ion stochas-\ntic heating (Chandran et al.2010; Chandran 2010) into a model that also accounted for22\nion and electron Landau damping and transit-time damping. Similar to QG99andH10,\nit assumed a sufficiently large inertial range that no significant cyclot ron damping would\noccursince the anisotropiccascadewould leadto turbulent fluctua tions with lowfrequen-\ncies,ω≪Ωi. Focusing also on Alfv´ enic turbulence that transitions from an MHD Alfv´ en\nwave cascade at k⊥ρi/lessorsimilar1 and to a kinetic Alfv´ en wave cascade at k⊥ρi/greaterorsimilar1, theC11\nmodel assumed that dissipation occured dominantly in two distinct wa venumber ranges:\n(i) atk⊥ρi∼1, where collisionlessdamping via the Landau ( n= 0) resonancecan lead to\nparallel energizationof the ions and electronsand alsowhere nonline arstochastic heating\ncan lead to perpendicular energizationof the ions; and (ii) at k⊥ρi≫1, whereall remain-\ning energy in the turbulent cascade is ultimately deposited with the ele ctrons. The model\nused fitting forms for the Landau-resonant collisionless damping ra tes for the ions and\nelectrons at k⊥ρi= 1 over 10−3< βi<10 and 1 < Ti/Te<5. It also included an empir-\nically derived formula for the ion stochastic heating rate at k⊥ρi= 1 (Chandran et al.\n2010). This empirical prescription for the ion stochastic heating de pended on two di-\nmensionless constants: one governed the scaling of the stochast ic heating rate relative\nto the turbulent frequencies; and the other, which appeared as a factor in the argument\nof the exponential function, effectively established a threshold am plitude below which\nion stochastic heating was negligible. Thus, the C11model calculated how the turbulent\ncascade rate at k⊥ρi= 1 is partitioned among ion Landau and transit-time damping,\nelectron Landau and transit-time damping, ion stochastic heating, and nonlinear trans-\nfer to smaller scales with k⊥ρi>1. The model provided a partitioning of the total\nturbulent heating rate into parallel ion heating by ion Landau and tra nsit-time damping\nQ/bardbl,i/Q, perpendicular ion stochastic heating Q⊥,i/Q, and electron heating Qe/Q, each\nas a function of ion plasma beta βi, the ion-to-electron temperature ratio Ti/Te, and the\namplitude of the turbulent fluctuations at the ion Larmor radius sca le determined using\ntheB06scaling for the amplitude in terms of the driving scale, as given here by (3.19),\nindicating that the amplitude dependence can be characterized by k0ρi.\nMotivated by the desire to understand the partitioning of dissipate d turbulent energy\nbetween ions and electrons in hot collisionless accretion flows, the Ro wan 2017 (hereafter\nR17) heating model (Rowan et al.2017) took a fundamentally different approach from\nthat adoptedin the QG99,H10, andC11turbulent heating models byfocusing instead in\nthe heating arising from transrelativistic magnetic reconnection. I n this case, the protons\nare nonrelativistic, but the electrons can be ultrarelativistic. Utilizin g a large suite of 2D\nparticle-in-cell (PIC) simulations of anti-parallel reconnection, th eR17model studied\nthe irreversible particle energization over a range of ion plasma beta 10−4/lessorsimilarβi/lessorequalslant2 and\nmagnetization 0 .1/lessorequalslantσw/lessorequalslant10, where σwis defined as the ratio of the magnetic energy\ndensity to the enthalpy density. Starting with a Harris current she et (Harris 1962) of the\nantiparallel magnetic field (given by a tanh profile) with width a∼40de, they carefully\nseparated the irreversible heating associated with an increase in th e entropy from the\nreversible heating associated with adiabatic compression. From the ir synthesis of the\nresults of a large suite of simulations varying different parameters, they fit a formula to\ntheirnumericalresultsthatprovidedtoelectron-to-totalheat ingratio Qe/Qasafunction\nof the ion plasma beta βiand magnetization σw.\nThe numerical study by Kawazura et al.(2019) (hereafter K19) used a large suite\nof hybrid gyrokinetic simulations to determine directly the ion-to-ele ctron heating ra-\ntioQi/Qeover a broad range of ion plasma beta 0 .1/lessorequalslantβi/lessorequalslant100 and ion-to-electron\ntemperature ratio 0 .05/lessorequalslantTi/Te/lessorequalslant100. Using a hybrid code (Kawazura & Barnes 2018)\nthat evolves the ions gyrokinetically and the electrons as an isother mal fluid, this study\nadopted many of the same assumptions as the H10model: (i) the anisotropic ( k⊥≫k/bardbl)Fundamental Parameters of Turbulence 23\nAlfv´ enic turbulence was driven in a statistically balanced manner with Z+\n0/Z−\n0= 1; (ii)\nthe equilibrium ion and electron temperatures were assumed to be iso tropic; (iii) the gy-\nrokinetic approximation eliminated the physics of the ion cyclotron re sonances, leaving\nthe Landau resonances and collisionless magnetic reconnection in th e large-guide-field\nlimit as potential damping mechanisms; and (iv) the strategy explaine d in§3.2 was ex-\nploited to model numerically only the small-scale end of the inertial ran ge, where the\nisotropicdrivingwavenumber k0ρi,characterizingthephysicaldrivingscale,wasassumed\nto be sufficiently small that the amplitudes ofthe turbulent fluctuat ions atk⊥ρi∼1 were\nsmall enough that ion stochastic heating was inhibited (Chandran et al.2010, 2011). The\nK19model provides a numerically validated prescription for Qi/Qeas a function of βi\nandTi/Te, presenting a simple analytical formula that fit the numerical result s well. The\nnumerical results largely validated the qualitative predictions of the H10turbulent heat-\ning model that Qi/Qeis a monotonic function of βiwith little dependence on Ti/Te, with\na couple of minor quantitative differences: the K19model found a ceiling of Qi/Qe≃30\natβi≫1, cutting off at lower values than the H10model; and the numerical results\ndid not find the drop in the heating rate Qi/Qe≪0.1 predicted by the H10model at\nβi≪1.\nSubsequently, using the same hybrid gyrokineticcode, Kawazura et al.(2020) explored\nthe effect of compressible driving Ecomp/Einc>0 on the ion-to-electron heating ratio\nQi/Qe(hereafter K20). The turbulence was driven at MHD scales k⊥ρi≪1 with an\nadjustable mixture of incompressible Alfv´ enic fluctuations and com pressible slow mag-\nnetosonic fluctuations (fast magnetosonic wave modes are order ed out of gyrokinetic\nsimulations by the perpendicular pressure balance that is maintained under the gy-\nrokinetic approximation (Howes et al.2006)). At βi≪1, they found that Qi/Qe=\nEcomp/Einc, confirminga previoustheoreticalprediction that compressiveen ergystrictly\nheats ions at low ion plasma beta βi≪1 (Schekochihin et al.2019). Surprisingly, the\nK20model also found that Qi/Qe=Ecomp/Eincwhen the compressible driving dom-\ninates,Ecomp/Einc≫1. A simple fitting formula was constructed that simply added\nthe term Ecomp/Eincto the previous prescription for Qi/Qedue to Alfv´ enic turbu-\nlence in the K19model. Thus, the K20model provided a simple analytical formula for\nQi/Qe(βi,Ti/Te,Ecomp/Einc).\nThe application of the K20model to understand the ion-to-electron heating ratio in\naccretion disks where the magnetorotational instability (MRI) (Ba lbus & Hawley 1991;\nHawley & Balbus 1991; Balbus & Hawley 1998) drives the turbulence re quires a deter-\nmination of the ratio of the compressible-to-incompressible driving Ecomp/Einc. The\napplication of a rotating reduced MHD (RRMHD) model to simulate the collisional MRI\nturbulence threaded by a nearly aximuthal magnetic field in a 3D pseu do-spectral, shear-\ning reduced MHD code (Kawazura 2022) found a ratio of compressib le-to-incompressible\nturbulent energy of 2 /lessorsimilarEcomp/Einc/lessorsimilar2.5 (Kawazura et al.2022), providing a solid\nfoundation for the application of the K20model to astrophysical accretion disks.\n5. Dependence of Dissipation Mechanisms on Fundamental\nParameters\nThe development of turbulent heating models for weakly collisional sp ace and astro-\nphysical plasmas requires two fundamental steps: (i) identifying t he mechanisms that\nplay a role in the damping of the turbulence; and (ii) determining upon w hich of the\nfundamental dimensionless parameters of turbulence each mecha nism depends and ap-\nproximating a functional form for those dependencies.\nIt is worthwhile noting an important subtlety regarding irreversible p lasma heating in24\nthe thermodynamic picture of turbulent dissipation in weakly collisiona l plasmas (Howes\n2017). In thestronglycollisionallimit offluid plasmaturbulence, thed issipationofturbu-\nlence isfundamentallycollisionalonamicroscopiclevel.Forexample,in a nMHDplasma,\nthe physical mechanisms of viscosity and resistivity arise in the limit of a non-negligible\ncollisional mean free path of the plasma particles. These two dissipat ive mechanisms may\nbe derived rigorouslybeginning with kinetic theory through a hierarc hy of moment equa-\ntions derived in the limit of small mean free path by the Chapman-Ensk og procedure\n(Chapman & Cowling 1970) for neutral gases, or an analogous proc edure for plasma sys-\ntems (Spitzer 1962; Grad 1963; Braginskii 1965). Because viscos ity and resistivity are\nultimately collisional in nature, the resulting energy transfer from t he turbulent fluctua-\ntions to the particles is irreversible, leading to an increase in the entr opy of the system.\nUnder the weakly collisional conditions typical of many space and ast rophysical plas-\nmas, the collisionless removal of energy from the turbulent fluctua tions and the ultimate\nconversion of that removed energy into plasma heat is a two-step p rocess (Howes et al.\n2006;Howes2008;Schekochihin et al.2009;Navarro et al.2016;Howes2017;Howes et al.\n2018). First, collisionless damping mechanisms remove energy from t he turbulent fluc-\ntuations, transferring that energy into nonthermal energy con tained in non-Maxwellian\nfluctuations in the particle velocity distributions, an energy transf er that is reversible in\nprinciple. That free energy in the fluctuations of the velocity distrib ution about the equi-\nlibrium may be transferred to smaller scales in velocity space— e.g., via linear phase mix-\ning by the ballistic term of the Boltzmann equation (Landau 1946; Ham mettet al.1992;\nSnyderet al.1997; Schekochihin et al.2009) or via nonlinear phase mixing, a process re-\ncently denoted the entropycascade(Dorland & Hammett 1993; Sc hekochihin et al.2009;\nTatsuno et al.2009; Plunk et al.2010; Plunk & Tatsuno 2011; Kawamori 2013)—until\nit reaches sufficiently small velocity-space scales that an arbitrarily weak collisionality is\nsufficient to smooth out those fluctuations, providing an irreversib le conversion of that\nenergy to plasma heat and thereby increasing the entropy of the s ystem (Howes 2008;\nSchekochihin et al.2009).\nThe subtleties of energy conversion and plasma heating in weakly collis ional plasmas\nthat are not in a state of local thermodynamic equilibrium remains a to pic of vigor-\nous investigation, with many recent developments that will not be re viewed in detail\nhere(Schekochihin et al.2016;Parker et al.2016;Yang et al.2016, 2017 a,b;Howeset al.\n2018;Kawazura et al.2019;Meyrand et al.2019,2021;Barbhuiya & Cassak2022;Cassak & Barbhuiya\n2022; Cassak et al.2022; Squire et al.2022; Cassak et al.2023). In this paper, we use\nthe term turbulent dissipation to mean the entire process of removing energy from the\nturbulent fluctuations and subsequently thermalizing that energy via collisions to re-\nalize irreversible heating of the plasma and an increase in the entropy of the system.\nWhen we refer to only the first step of the process, the collisionless removal of energy\nfrom the turbulent fluctuations, we denote those specific proces ses asturbulent damping\nmechanisms .\nA primary aim of this study is to provide a unified framework for chara cterizing the\ndependenceoftheturbulenceanditsdissipationonacommonsetof dimensionlessparam-\neters. The first step pursued hereis to identify the fundamental dimensionless parameters\nupon which each proposed turbulent damping mechanism primarily dep ends, where the\nset of dimensionless parameters for the general case is presente d in Table 2 and for the\nisotropic temperature case is in Table 3. The key parameter depend encies found here are\nsummarized for each proposed damping mechanism in Table 4. Althoug h, for any single\nparticular mechanism, different choicesof dimensionless parameter smay be more natural\n(e.g., takingβeinstead of βiin the case of magnetic reconnection), the particular sets of\ndimensionless parameters in Tables 2 and 3 are proposed here as a us eful framework forFundamental Parameters of Turbulence 25\nCategory Abbrev Mechanism Key Parameters\nResonant iLD Ion Landau Damping βi\neLD Electron Landau Damping βi,Ti/Te, (µ)\niTTD Ion Transit-Time Damping βi\neTTD Electron Transit-Time Damping βi,Ti/Te, (µ)\niCD Ion Cyclotron Damping k0ρi,βi\nNon- iSH Ion Stochastic Heating k0ρi,βi\nResonant iMP Ion Magnetic Pumping Ecomp/Einc,βi,Ti/Te,k/bardbl0λmfp,e,\nk/bardbl0/k⊥0,χ0\neMP Electron Magnetic Pumping βi,Ti/Te,k/bardbl0λmfp,e,k/bardbl0/k⊥0,χ0,\nEcomp/Einc, (µ)\niVH Ion Kinetic Viscous Heating βi,k⊥0/k/bardbl0,χ0\nCoherent iRXN Ion Magnetic Reconnection k0ρi,βi,Ti/Te, (µ)\nStructures eRXN Electron Magnetic Reconnection k0ρi,βi,Ti/Te, (µ)\nTable 4. Proposed turbulent damping mechanisms and their key dimens ionless parameter\ndependencies.\ndescribing allof the proposed turbulent damping mechanisms, thereby simplifying the\ntask of detemining how different mechanisms compete with each othe r throughout the\nfull parameter space.\n5.1.Proposed Damping Mechanisms for Kinetic Turbulence\nIn weakly collisional space and astrophysical plasmas, the propose d kinetic damping\nmechanisms that remove energy from the turbulent fluctuations a nd transfer it to the\nparticlespeciesfallintothreebroadcategories:(i)resonantwav e-particleinteractions,(ii)\nnon-resonant wave-particle interactions, and (iii) damping occurr ing in coherent struc-\ntures.\nResonant mechanisms for the damping of the turbulent fluctuation s in a magnetized\nplasma transfer energy to particles in regions of the velocity distrib ution that satisfy the\nresonant condition\nω−k/bardblv/bardbl−nΩs= 0, (5.1)\nwhere Ω sis the cyclotron frequency for species sand|n|>1 indicates the harmonics\nof the cyclotron frequency. The n= 0 resonance is often denoted the Landau resonance\nand includes two damping mechanisms: (i) Landau damping is mediated b y the parallel\ncomponent of the electric field E/bardbldoing work on the charged particles (Landau 1946;\nLeamon et al.1998b,a; Quataert 1998; Leamon et al.1999; Quataert & Gruzinov 1999;\nLeamon et al.2000; Howes et al.2008a; Schekochihin et al.2009; TenBarge & Howes\n2013; Howes 2015; Li et al.2016; Chen et al.2019; Afshari et al.2021); and (ii) transit-\ntime damping, also known as Barnes damping, is mediated by the magne tic mirror force\nassociated with magnetic field magnitude fluctuations doing work on t he magnetic mo-\nmentofthechargedparticlegyromotion(Barnes1966;Quataert 1998;Quataert & Gruzinov\n1999; Howes et al.2008a; Huang et al.2024). Both of these Landau-resonant collision-\nless damping mechanisms energize particles in the degree of freedom parallel to the local\nmagnetic field. The n∝negationslash= 0 resonances yield cyclotron damping of the turbulent flucuta-26\ntions by E⊥, energizing particles in the two perpendicular degrees of freedom ( Coleman\n1968; Denskat et al.1983; Isenberg & Hollweg 1983; Goldstein et al.1994; Leamon et al.\n1998a; Gary 1999; Isenberg et al.2001; Hollweg & Isenberg 2002; Isenberg & Vasquez\n2019; Afshari et al.2023).\nNon-resonant damping mechanisms can remove energy from turbu lent fluctuations\nthrough wave-particle interactions that do not require satisfying a particular resonance\ncondition. Stochastic heating can arise when sufficiently large amplitu de turbulent elec-\ntromagnetic fluctuations can effectively scatter the otherwise or ganized gyromotion of\nparticles, leading to a heating of the particles in the perpendicular de grees of freedom\n(Johnson & Cheng 2001;Chen et al.2001; White et al.2002; Voitenko & Goossens2004;\nBourouaine et al.2008; Chandran et al.2010; Chandran 2010; Chandran et al.2011;\nBourouaine & Chandran2013;Chandran et al.2013;Klein & Chandran2016;Arzamasskiy et al.\n2019; Martinovi´ c et al.2020; Cerri et al.2021). Magnetic pumping energizes particles\nthrough a combination of two effects: (i) particles undergo the rev ersible, double adi-\nabatic evolution of their perpendicular and parallel velocities in a magn etic field with\na time varying magnitude; and (ii) those particles are scattered by c ollisions during\nthis evolution, introducing irreversibility and leading to a small net tra nsfer of energy\nto the particles over a full oscillation cycle (Spitzer & Witten 1951; Be rgeret al.1958;\nLichkoet al.2017; Lichko & Egedal 2020; Montag & Howes 2022). One final prop osed\nnon-resonant damping mechanism is the recently identified kinetic “v iscous” heating me-\ndiated by the effective collisionality associated with temperature anis otropy instabilities\ndriven by large-scale compressible fluctuations in high beta plasmas ( Arzamasskiy et al.\n2023).\nThe final category involves the removal of turbulent fluctuation e nergy in coherent\nstructures, such as particle energization arising from collisionless m agnetic reconnection\nincurrentsheetsthatarefoundtoarisenaturallyinplasmaturbule nce(Ambrosiano et al.\n1988;Dmitruk et al.2004;Markovskii & Vasquez2011;Matthaeus & Velli2011;Osman et al.\n2011;Servidio et al.2011;Osman et al.2012b,a;Wanet al.2012;Karimabadi et al.2013;\nZhdankin et al.2013; Dalena et al.2014; Osman et al.2014a,b; Zhdankin et al.2015b,a;\nMalletet al.2017b,a; Loureiro & Boldyrev 2017 b).\nEach of the kinetic damping mechanisms in these three categories (w ith helpful abbre-\nviations) are summarized in Table 4. Also included in Table 4 are the prima ry dimen-\nsionless parameters upon which each damping mechanism depends, a s explored in more\ndetail in the following subsections.\n5.2.A Competition between Turbulent Cascade Rates and Damping R ates\nOneofthemostchallengingaspectsofmodelingthe dampingofplasma turbulenceisthat\nthe net energy transfer to each species arises from the competit ion between the turbulent\ncascaderateandthekineticdampingratesasafunctionofscale,a sdepictedbythearrows\nin Fig. 1(a). Nonlinear interactions among turbulent fluctuations—d enoted Alfv´ en wave\ncollisions (Howes & Nielson 2013) in the case of Alfv´ enic turbulence—m ediate the trans-\nferofenergyfromlargetosmallscalesthatisdominantlylocalinsca lespace(Howes et al.\n2011; Told et al.2015), where energy is transferred locally from a wavenumber kto a\nwavenumber 2 k, then on to 4 k, and so on. This is indicated by the black curved arrowsin\nthe figure. Because the physics of the cascade is dominantly local in nature (Howes et al.\n2011; Told et al.2015), scaling theory implies that the energy transfer rate at a giv en\nwavenumber kis determined by the conditions at that scale. Note that, when analy zed in\nFourier space, the energy transfer to small scales in the turbulen t cascade is the sum of\nthe nonlinearinteractions amongmany triads ofplane-wavemodes, with a wide spreadof\nenergytransferratesaboutzero;the net energytransfergiv enbythis sum atagivenscaleFundamental Parameters of Turbulence 27\n+-\n*iQ (k )e*(k )ε\nε(k <k )*\n**+-(k )\nQ (k )k\nk\nk*-(k <k )\nFigure 4. Diagram ofthebalanceoftheenergytransfer rates inastead y-stateturbulentcascade\nat a local wavenumber k∗, where the following terms sum to zero: (i) the turbulent non linear\nenergy transfer rate from lower wavenumbers ε(k−)(k−<k∗); (ii) the turbulent nonlinear energy\ntransfer rate to higher wavenumbers ε(k∗)(k∗<k+); and (iii) the local (in scale) kinetic damping\nrates of the turbulent energy, due to the sum of ion energizat ion rates Qi(k∗) (blue) and electron\nenergization rates Qe(k∗) (magenta).\nis much smaller in amplitude than the spread of individual three-wave in teraction rates,\nand the sum typically indicates a forwardcascade ofenergyto small scales(Coburn et al.\n2014, 2015). The turbulent cascade rate εused here specifically refers to this net energy\nrate obtained by integrating over all contributing nonlinear interac tions involving that\nscale.\nCompeting with that turbulent cascade rate at each scale is the dam ping rate due to\ncollisionless interactions between the electromagnetic fields and the plasma particles, as\nshownbytheblue arrowsforthe transferofenergytoionsandth e magentaarrowsforthe\ntransfer of energy to electrons in Fig. 1(a). The net damping rate of the entire turbulent\ncascade, resulting from the sum of the energization rates for eac h particle species s, can\nbe computed by integration over all scales of the turbulent cascad e. In a steady-state,\nat each wavenumber k∗, the sum of three terms must balance to zero, as illustrated in\nFig. 4: (i) the energy transfer from lower wavenumbers k−< k∗(larger scales), denoted\nbyε(k−)(k−<k∗); (ii) the energytransfer to higher wavenumbers k+> k∗(smaller scales),\ndenoted by ε(k∗)(k∗<k+); and (iii) the kinetic damping of the fluctations at wavenumber\nk∗, given by Q(k∗), which is the sum of the local (in scale) ion damping Qi(k∗) (blue) and\nelectron damping Qe(k∗) (magenta). Complicating the determination of the partitioning\nof damped turbulent energy between ion and electrons Qi/Qeis the fact that the kinetic\ndamping rates due to ions and electrons often overlap as a function of scale (Howes et al.\n2011; Howes et al.2011; Told et al.2015).\nDetermining this balance of cascade and damping rates as a function of scale, and ulti-28\nFigure 5. Weakened cascade model results for a turbulent cascade driv en weakly at\nk⊥0ρi=k/bardblρi= 10−5with nonlinearity parameter χ(k⊥0) = 0.1 in a plasma with isotropic\nequilibrium Maxwellian velocity distributions with Ti/Te= 9 and βi= 1: (a) One-dimensional\nperpendicular magnetic energy spectrum EB⊥(k⊥), (b) anisotropic cascade through wave vector\nspace given by , k/bardbl(k⊥), (c) energy transfer rates to the ions Qi(k⊥) (blue) and electrons Qe(k⊥)\n(magenta), and (d) integrated energy damping rates by ions/integraltextk⊥\n0dk′\n⊥Qi(k′\n⊥) (blue) and by elec-\ntrons/integraltextk⊥\n0dk′\n⊥Qe(k′\n⊥) (magenta), all plotted as a function of the perpendicular w avenumber\nk⊥ρi.\nmatelyusingthattodeterminethepartitioningofturbulentenergy amongspeciesandde-\ngreesoffreedom—parameterizedby Qi/Qe,Q⊥,i/Qi, andQ⊥,e/Qe—has been attempted\ninempiricalturbulentcascademodels(Pao1965;Howes et al.2008a;Podesta et al.2010;\nHoweset al.2011; Zhao et al.2013; Schreiner & Saur 2017). An example of the modeling\nof the turbulent energy cascade and its kinetic dissipation is presen ted in Fig. 5 from\nthe results of the weakened cascade model by Howes et al.(2011). In addition to bal-\nancing the physics of the local cascade rate ε(k⊥) with the kinetic damping rates due\nto ionsQi(k⊥) and electrons Qe(k⊥) via the Landau resonance, this cascade model alsoFundamental Parameters of Turbulence 29\nincludes the contributions to the local cascade rate from nonlocal interactions by large-\nscale shearing and small-scale diffusion. The model results shown her e in Fig. 5 are from\nthe same calculation as presented in Fig. 8 of Howes et al.(2011), where we reproduce\n(a) the one-dimensional perpendicular magnetic energy spectrum EB⊥(k⊥) and (b) the\nanisotropic cascade through wave vector space characterized b y the characteristic paral-\nlel wavenumber as a function of the perpendicular wavenumber, k/bardbl(k⊥). This model for\na plasma with βi= 1 and isotropic equilibrium Maxwellian velocity distributions with\nTi/Te= 9 is driven weakly with χ(k⊥0) = 0.1 atk⊥0ρi=k/bardblρi= 10−5, taking the model\nKolmogorov constants to be C1= 1.4 andC2= 1.0.\nIn Fig. 5(c) is plotted the rate of energy transfer to the ions Qi(k⊥) (blue) and to\nthe electrons Qe(k⊥) (magenta) as a function of k⊥ρi, both normalized by the turbulent\ncascaderateatthe drivingscale ε0. Thisshowsthat the ionkinetic damping occursat the\nionkineticscales k⊥ρi∼1,whereastheelectrondampingrisesmonotonicallyfromtheion\nscales, peaking around k⊥ρi∼30 for this case. To obtain the net partitioning of energy\nbetween ions and electrons Qi/Qe, it is necessaryto integratethese energydamping rates\nover the entire cascade, as shown in Fig. 5(d): the turbulent ener gy cascade rate ε(k⊥)\n(black) is ultimately terminated at k⊥ρi≫1 by the integrated energy damping rates by\nions/integraltextk⊥\n0dk′\n⊥Qi(k′\n⊥) (blue) and by electrons/integraltextk⊥\n0dk′\n⊥Qe(k′\n⊥) (magenta). Note that this\nturbulent cascade model determines the solutions presented in Fig . 5 by computing the\nbalance of the turbulent cascade rates with the local kinetic dampin g rates, as depicted\nin Fig. 4. Integration of those damping rates over the entire casca de yields the predicted\npartitioning of dissipated turbulent energy between ions and electr onsQi/Qe.\nThe energy transfer rates Qi(k⊥) andQe(k⊥) that remove energy from the turbulent\nfluctuations and transfer it to the plasma particles in turbulent cas cademodels require as\ninput the kinetic dampingratesdue toions γi(k⊥) andelectrons γe(k⊥). In thecaseofthe\nweakened cascade model shown in Fig. 5, the damping rates used ar e collisionless damp-\ning rates via the Landau ( n= 0) resonance for the Alfv´ en wave mode (and its extension\nas a kinetic Alfv´ en wave at scales k⊥ρi/greaterorsimilar1) from the Vlasov-Maxwell linear dispersion\nrelation (Howes et al.2011). Previous studies have called into question whether resonan t\ncollisionless wave-particle interactions, such as Landau damping, ca n effectively damp\nturbulent fluctuations in the presence of the strong nonlinear inte ractions that mediate\nthe turbulent cascade (Plunk 2013; Schekochihin et al.2016). Using the field-particle\ncorrelation technique (Klein & Howes 2016; Howes et al.2017; Klein et al.2017), anal-\nysis of kinetic simulations (Klein et al.2017; Howes et al.2018; Klein et al.2020) and\nspacecraft observations (Chen et al.2019; Afshari et al.2021, 2023) of plasma turbu-\nlence find clear velocity-space signatures of Landau damping. Thes e findings indicate\nindeed that resonant wave-particle interactions do play a role, and possibly a dominant\nrole (Afshari et al.2021), in the damping of space and astrophysical plasma turbulenc e,\nsettling the question of whether Landau damping can effectively dam p strong plasma\nturbulence.\nIn quantifying how a particular damping mechanism competes with the nonlinear\nturbulent cascade, the key dimensionless quantity to evaluate is th e ratio of linear damp-\ning rate due to nonlinear cascade rate γ/ωnl(Howeset al.2008a, 2011). Adopting the\nconjecture of critical balance between linear and nonlinear timesca les in strong plasma\nturbulence ω∼ωnl(Goldreich & Sridhar 1995; Howes et al.2008a, 2011; Mallet et al.\n2015), the key dimensionless measure of the importance of that pa rticular mechanism is\nthe ratio γ/ω, which can be estimated using the linear theory for each of the differ ent\ndamping mechanisms.30\n5.3.The Relevance of Linear Physics to Strong Plasma Turbulence\nTheconceptofcriticalbalanceinplasmaturbulence(Goldreich & Srid har1995;Mallet et al.\n2015)—which suggests that the nonlinear timescales of the turbule nt cascade remain in\napproximate balance with the timescales of the linear wave physics, ωnl∼ω—implies\nthat the physics of the linear plasma response remains valid in strong plasma turbu-\nlence. But, perhaps in analogy with the case of incompressible hydro dynamic turbu-\nlence in which no linear response exists (Howes 2015), the validity of lin ear physics has\nbeen called into question in the presence of strong plasma turbulenc e (Matthaeus et al.\n2014). Although there exist numerous counterexamples in which line ar physics properties\nhave been shown to be relevant to strong plasma turbulence (Maro n & Goldreich 2001;\nCho & Lazarian 2003; Alexandrova et al.2008; Howes et al.2008b,a; Svidzinski et al.\n2009; Sahraoui et al.2010; Hunana et al.2011; Howes et al.2011; TenBarge et al.2012;\nChenet al.2013), the question has persisted. A more recent careful study of kinetic\nnumerical simulations and spacecraft observations has shown clea rly that both small-\namplitude and large-amplitude fluctuations, including variations inter preted to be co-\nherent structures, preserve the linear properties of Alfv´ enic fl uctuations (Groˇ selj et al.\n2019), hopefully providing sufficient evidence to establish that relev ance of linear physics\nto strong plasma turbulence.\nThe key to predicting qualitatively which kinetic damping mechanisms ar e likely to\ndominate for a given instance of turbulence is to determine how the v arious collision-\nless damping mechanisms in §5.1 depend on the fundamental dimensionless plasma and\nturbulence parameters. For a fully ionized proton and electron plas ma with anisotropic\n(bi-Maxwellian) equilibrium velocity distribution functions (Stix 1992; Q uataert 1998;\nQuataert & Gruzinov 1999; Swanson 2003; Klein & Howes 2015), the complex eigenfre-\nquencies determined by the linear Vlasov-Maxwell dispersion relation depend on the di-\nmensionlessparametersthatcharacterizethenatureofthe tur bulentfluctuationsthrough\nthe typical wave vector components in a plane-wave decomposition of the turbulent fluc-\ntuations k⊥ρiandk/bardblρiand the plasma parameters through the parallel ion plasma\nbetaβ/bardbl,i, the parallel ion-to-electron temperature ratio τ/bardbl=T/bardbl,i/T/bardbl,e, the temperature\nanisotropies of each species Ai=T⊥,i/T/bardbl,iandAe=T⊥,e/T/bardbl,e, and the ratio vt/bardbl,i/c.\nThus, these dependencies can be expressed through the genera l relation\nω=ωaniso\nVM(k⊥ρi,k/bardblρi,β/bardbl,i,τ/bardbl,Ai,Ae,vti/c). (5.2)\nIn the limit of isotropic equilibrium velocity distributions for the ions and electrons—\nAi= 1,Ae= 1, and τ=Ti/Te—this set of dependencies is reduced to\nω=ωVM(k⊥ρi,k/bardblρi,βi,τ,vti/c). (5.3)\nFor both the isotropic and anisotropic equilibirum temperatures, in t he non-relativistic\nlimitvti/c≪1 typical of many space and astrophysical plasmas of interest, th e linear\nVlasov-Maxwell dispersion relation is practically independent of vti/c, so this parameter\nmay effectively be dropped for the case of non-relativistic plasma tu rbulence.\nIn the limit of a sufficiently large inertial range of turbulence, the MHD turbulence\nscaling parameters given by (3.1)–(3.4) predict that turbulent fluc uations at the small-\nscale end of the inertial range, where the kinetic damping mechanism s are believed to\nbecome significant, will be anisotropic with k/bardbl≪k⊥and small amplitude relative to\nthe equilibrium magnetic field |δB| ≪B0(Howeset al.2006, 2008 a; Schekochihin et al.\n2009). In this limit, the turbulent frequencies of the incompressible Alfv´ enic fluctua-\ntions directly observed to dominate turbulence in space plasmas (Be lcher & Davis 1971;\nTu & Marsch1995;Alexandrova et al.2008;Howes et al.2012a;Bruno & Carbone2013)Fundamental Parameters of Turbulence 31\nhave frequencies much smaller than the ion cyclotron frequency, ω/Ωi=k/bardbldi≪1, where\ndi=vA/Ωi=ρi/√βiis the ion inertial length. In this limit, the turbulent fluctuations\nat the small-scale end of the inertial range satisfy the conditions fo r the applicabilty\nof gyrokinetic theory (Rutherford & Frieman 1968; Taylor & Hastie 1968; Catto 1978;\nAntonsen & Lane1980;Catto et al.1981;Frieman & Chen1982;Dubin et al.1983;Hahm et al.\n1988; Brizard 1992; Howes et al.2006; Brizard & Hahm 2007; Schekochihin et al.2009).\nGyrokinetic theory can be rigorouslyderived from plasma kinetic the ory under the limits\nof low frequency ( ω/Ωi≪1) and anisotropic k/bardbl≪k⊥fluctuations in the non-relativisitc\nlimitvti/c≪1. In the gyrokinetic approximation, the physics of the the cyclotr on reso-\nnances and of the fast magnetosonic fluctuations—and their kinet ic extension to whistler\nfluctuations (Howes et al.2014)—are ordered out of the system. Gyrokinetics retains the\nphysics of the E×Bnonlinearity that mediates the turbulent cascade to small scales,\nfull finite-Larmor-radius effects, and the collisionless damping asso ciated with the n= 0\nLandau resonance. In this reduced limit, the complex eigenfrequen cies of the linear gy-\nrokinetic dispersion relation for isotropic equilibrium velocity distribut ions are linearly\ndependent on the parallel wavenumber, and have the significantly r educed dimensionless\nparameter dependency given by (Howes et al.2006)\nω≡ω\nk/bardblvA=ωGK(k⊥ρi,βi,τ). (5.4)\nAlthough the ion-to-electron mass ratio µ=mi/meis not a parameter that varies\nin the physical world, many numerical studies of plasma turbulence c hoose to employ\na reduced mass ratio for computational efficiency. Therefore, we choose to include the\nµdependence of different proposed turbulent damping mechanisms in our calculations\nbelow to determine how the use of a reduced mass ratio may impact th e partitioning of\nturbulent energy removed by different damping mechanisms. Note a lso that it is impor-\ntant to employ a sufficiently large mass ratio to achieve a separation b etween the ion and\nelectron scales, with a previous study suggesting mi/me/greaterorequalslant32 is necessary to model the\ndistinct ion and electron responses to the electromagnetic fluctua tions due to this scale\nseparation (Howes et al.2018).\n5.4.Landau Damping and Transit-Time Damping\nTo explore the two Landau-resonant collisionless damping mechansim s—Landau damp-\ning and transit-time damping—we take n= 0 in the resonance condition in (5.1), obtain-\ning the condition that particleswith parallelvelocities equalto the pa rallelphase velocity\nofthewaves, v/bardbl=ω/k/bardbl,mayresonantlyexchangeenergywith theelectromagneticwaves .\nFor simplicity in this analysis, we will assume the isotropic temperature case for a\nproton-electron plasma with linear wave properties given by the isot ropic linear Vlasov-\nMaxwell dispersion relation (5.3); extending these calculations to allo w for anisotropic\ntemperatures can be done simply following the same logical procedur e and using the lin-\near Vlasov-Maxwell dispersion relation for anisotropic temperatur es given by (5.2). Fur-\nthermore, justified by observations in solar wind turbulence that in compessible Alfv´ enic\nfluctuations energetically dominate the turbulence (Belcher & Davis 1971; Tu & Marsch\n1995; Alexandrova et al.2008; Bruno & Carbone 2013), we will focus on the linear col-\nlisionless damping rates of the Alfv´ enic fluctautions, including both A lfv´ en waves in the\nMHD limit at k⊥ρi≪1, and kinetic Alfv´ en waves at k⊥ρi/greaterorsimilar1. Consideration of the\ndamping of compressible turbulent fluctuations, specifically fast an d slow magnetosonic\nfluctuations at k⊥ρi≪1 (Howes et al.2012a), is easily performed using the same pro-\ncedure with the linear dispersion relations for these alternative wav e modes†.\n†Although, it is important to keep in mind that the recent iden tified nonlinear effect of32\nFigure 6. Diagram demonstrating how Landau resonant damping qualita tively depends on\nwhere the parallel phase velocity ω/k/bardblof the wave falls within the velocity distribution of\nparticle species s, with weak damping for ω/k/bardbl≪vtsorω/k/bardbl≫vts, and strong damping for\nω/k/bardbl∼vts.\nNormalizing the resonant parallel velocity by the thermal velocity fo r both species\nin turn and converting the result into our dimensionless parameters for the isotropic\ntemperature case, we obtain the key relations\nv/bardbl\nvti=ω\nk/bardblvti=ωβ−1/2\ni,v/bardbl\nvte=ω\nk/bardblvte=ω/parenleftbiggτ\nβiµ/parenrightbigg1/2\n, (5.5)\nwhereω≡ω/k/bardblvA. For Alfv´ en waves, this normalized frequency ωis well approximated\nby (Howes et al.2014)\nω(k⊥ρi,βi,τ)≡ω\nk/bardblvA=/radicalBigg\n1+(k⊥ρi)2\nβi+2/(1+1/τ), (5.6)\nan expressionthat is valid in the limits that the ion plasma beta is not ver ylowβi≫µ−1\nand that the wave frequency remains smaller than the ion cyclotron frequency ω/Ωi≪1,\nequivalent to the condition k/bardbldi≪1, ork/bardblρi≪β1/2\ni.\nWhether damping by the Landau-resonantmechanisms is weak or st rong primarily de-\npends on where the parallel wave phase velocity falls within the equilibr ium particle ve-\nlocity distribution, as quantified by the expressions in (5.5). The app lication of the weak-\ngrowth-rateapproximationtothekinetictheoryforLandaudamp ing(Krall & Trivelpiece\n1973) indicates that the linear damping rate is proportional to the s lope of the distribu-\ntion at the resonant velocity, as depicted in Fig. 6. For phase velocit ies that fall deep in\nthe core of the velocity distribution ω/k/bardbl≪vts(blue dotted) where the slope of fs(v/bardbl)\nis flat, the damping is weak because nearly equal numbers of particle s withv/bardbl< ω/k /bardbl\nare accelerated by the wave as particles with v/bardbl> ω/k /bardblare decelerated by the wave,\nanti-phase-mixing in turbulent plasmas can suppress the co llisionless damping predicted by\nlinear kinetic theory for the compressible wave modes in the inertial range of scales at k⊥ρi≪1\n(Schekochihin et al.2016; Parker et al.2016).Fundamental Parameters of Turbulence 33\nleading to little net energy exchange. Similarly, for phase velocities fa r out in the tail of\nthe velocity distribution ω/k/bardbl≫vts(black dashed), there are few particles to resonate\nwith the waves and the slope of fs(v/bardbl) is also flat, so the damping is weak. Only if the\nparallel phase velocity falls in the steep region of the velocity distribu tion atω/k/bardbl∼vts\n(red solid) is there a significant net energy transfer to the particle s: more particles with\nv/bardbl< ω/k /bardblare accelerated by the wave than those particles with v/bardbl> ω/k /bardblare deceler-\nated by the wave, so that particles gain net energy at the expense of the wave.\nThe ultimate goalis to characterizethe turbulent plasma heating— s pecifically Qi/Qe,\nQ⊥,i/Qi,andQ⊥,e/Qe—due to Landau damping and transit-time damping as a function\nof the fundamental dimensionless parameters of turbulence. To a ccomplish this goal, it\nis necessary to integrate the damping rates over the full range of the turbulent cascade,\nas discussed in §5.2 and illustrated in Figures 4 and 5. This integration over all scales\neffectively eliminates the dependence of the damping mechanisms on t he components of\nthe wavevector k⊥ρiandk/bardblρi, leaving only the desired dependence on the plasma and\nturbulence parameters enumerated in Tables 2 and 3.\nWith the scale dependence on k⊥ρieliminated, the ion Landau damping (iLD) and\nion transit-time damping (iTTD) rates depend primarily on βi, and the electron Landau\ndamping (eLD) and electron transit-time damping (eTTD) rates dep end onβi,τ, and\nµ, as shown by (5.5). Note that the dependence of eLD and eTTD on τandµprimarily\narises from those parameters controlling the scaling of the ion to ele ctron Larmor radius\nscales,ρi/ρe∼(τµ)1/2.\nIt is worthwhile pointing out that the physics of collisionless damping via the Landau\nresonance is fully captured in the gyrokinetic approximation, with th e linear wave fre-\nquencies and damping rates both linearly proportional to k/bardbl, as shown in (5.4). Thus,\nthe ratio of the damping-to-cascade rate †,γ/ωnl≃γ/ω, is formally independent of k/bardbl,\nleaving the expected parameter dependence on the plasma parame tersβiandτ, along\nwith the mass ratio µ.\nNote that dependence of the linear gyrokinetic dispersion relation o n (k⊥ρi,βi,τ) in\n(5.4) is consistent with the results of our analysis of the parameter dependence of the\nLandau-resonantdampingmechanisms.Thelineargyrokineticdispe rsionrelationalsode-\nterminestheamplitudesandphasesofthekeyelectromagneticfield sE/bardblandδB/bardblthat me-\ndiate the energizationby Landau damping and transit-time damping. As done in the H10\nturbulent heating model, the linear dispersion relation can simply be us ed to calculatedi-\nrectly the linear collisionless damping rates as a function of ( k⊥ρi,βi,τ), but these linear\ndamping rate predictions may be modified under strongly turbulent p lasma conditions.\nHowever, field-particle correlation analyses of numerical simulation s (Klein et al.2017,\n2020; Huang et al.2024) and spacecraft observations (Chen et al.2019; Afshari et al.\n2021) of plasma turbulence have clearly shown that Landau damping and transit-time\ndamping do indeed play a role in the damping of strong plasma turbulenc e.\nOne final step is to estimate how the ion-to-electron heating ratio Qi/Qedepends\non the remaining key parameters βi,τ, andµ. As illustrated in Fig. 5(c) and Fig. 7,\nthe ion damping rate γipeaks at scales k⊥ρi∼1, and this Landau-resonant damping\nontothe ionsisessentiallyindependent oftheratioion-to-electron temperatureratio.The\nelectrondampingrate γe,ontheotherhand,increasesmonotonicallyastheperpendicular\nwavevector increases to the electron Larmor radius scale k⊥ρe→1, eventually yielding a\nnormalizedcollisionlessdampingratethatissufficientlystrongtoterm inatetheturbulent\ncascade with γe/ω→1, as shown in Fig. 7. Thus, any energy in the turbulent cascade\n†Herewe haveadoptedthecondition ofcriticial balance for s trongplasma turbulence ωnl≃ω\n(Goldreich & Sridhar 1995; Mallet et al.2015).34\nthat passes beyond the ion kinetic scales at k⊥ρi∼1 will ultimately be transferred to\nthe electrons. Since the ion damping rate depends dominantly on βiwith very weak\ndependence on τ, yielding Qi(βi) to lowest order, this means that the ratio Qi/Qeis\neffectively determined by the ion plasma beta alone given a total turb ulent energization\nratethateventuallymustbalancethetotalcascaderate Q∼ε0,sinceitcanbeshownthat\nQe/Qi=ε0/Qi(βi)−1. This dominant dependence of the Landau-resonant collisionless\ndamping mechanisms on βiis consistent with existing turbulent heating models (Howes\n2010; Kawazura et al.2019).\nOne final unexpected physical behavior of Landau damping and tra nsit-time damping\nis also illustrated in Fig. 7. For isotropicMaxwellian velocity distributions of the ions and\nelectrons, the imaginary component of the complex frequency, ωc≡ω−iγ, determined\nby the linear dispersion relation is always negative (solid black curve), corresponding\nto a damping of the wave with γ >0. However, for this particular plasma parameter\nchoice of βi= 3,Ti/Te= 1, and mi/me= 1836, the individual energy transfer rates to\nthe ions through Landau damping mediated by E/bardblare actually negative over the range\nk⊥ρi/lessorsimilar2 (red dotted), indicating a net transfer of energy from the ions t o the wave fields\nwhen averaged over the full period (or, alternatively, over 2 πin phase) of the Alfv´ en\nwave. But the contribution from transit-time damping mediated by E⊥(Huanget al.\n2024; Howes et al.2024) is positive and much larger (green long dashed), leading to\na net damping of the wave ( γi>0, thin blue) by the sum of the Landau-resonant\ninteractions of ions with the Alfv´ en wave. Whether the wave-perio d-averaged energy\ntransfer to ions is positive or negative depends on the relative phas es ofE/bardblandE⊥to\nthe self-consistently determined ion current density ji. For the case of βi= 3 pictured\nin Fig. 7, at scales k⊥ρi/lessorsimilar2 (red dotted), this relative phase δφbetween E/bardblandj/bardbl,i\nmust fall in the range π/2/lessorequalslantδφ/lessorequalslant3π/2 (corresponding to wave growth at the expense\nof ion energy), while the relative phase between E⊥andj⊥,imust fall in the range\n−π/2/lessorequalslantδφ/lessorequalslantπ/2 (corresponding to wave damping and ion energy gain). Because ea ch\nindividual mechanism can possibly lead to negative energy transfer w hile the sum always\nleads to damping, it suggests that perhaps a separation of the par ticle energization rates\nby Landaudampingand transit-timedampingis not particularlyimport antin unraveling\nthe turbulent heatingofthe plasma(especially since both ofthese m echansimsultimately\nlead to energization of the parallel degree of freedom of the partic les), so that only the\nsummed effect of the Landau-resonant collisionless damping rates is important.\nSummarizing our results for Landau-resonant collisionless damping o f the turbulent\ncascade, the resulting ion-to-electron heating is predicted to be d ominantly a function of\nthe ion plasma beta, Qi/Qe(βi), with a weak dependence on the ion-to-electron tempera-\ntureratio τandmassratio µ.BothLandaudamping,mediated bythe parallelcomponent\nof the electric field E/bardbl, and transit-time damping, mediated by the perpendicular com-\nponent of the electric field E⊥, lead to energization of the degrees of freedom parallel to\nthe magnetic field, so Q⊥,i/Qi= 0 and Q⊥,e/Qe= 0.\n5.5.Cyclotron Damping\nFor the fundamental ( n= 1) cyclotronresonance,the resonancecondition in (5.1) simpli-\nfies to (ω−k/bardblv/bardbl)/ωs= 1. The numerator involves the fluctuation frequency modified by\nthe Doppler shift of the particles motion through spatial variations parallel to the mag-\nnetic field. When that Doppler-shiftedfrequency matchesthe ionc yclotronfrequency,the\nperpendicular compnent of the electric field E⊥can resonantly interact with the particles\nto energize the particles in the directions perpendicular to the magn etic field, yielding\nQ⊥,s/Qs= 1. To estimate the conditions required for cyclotron resonant da mping, it is\noften sufficient to require the condition ω/ωs→1.Fundamental Parameters of Turbulence 35\nFigure 7. Plot of the normalized damping rate γ/ω(black dashed) for the Alfv´ en and kinetic\nAlfv´ en wave over the perpendicular wavenumber range 10−2/lessorequalslantk⊥ρi/lessorequalslant102for a plasma with\nβi= 3,Ti/Te= 1, and mi/me= 1836, where the complexeigenfrequency is givenby ωc=ω−iγ,\nso thatγ >0 corresponds to wave damping and γ <0 to wave growth. The contributions from\nthe ion damping rate γi(thin blue) and electron damping rate γe(thin magenta) are separately\nplotted, with this decomposition failing at strong damping rates with γ/ω/greaterorsimilar0.5, occurring\nfork⊥ρi/greaterorsimilar40. Ion damping due to Landau damping γi,LD>0 (red dashed) and transit-time\ndamping γi,TTD>0 (green long dashed) are separated, showing a region of net e nergy transfer\nfromionstothe wave for ion Landau damping with γi,LD>0 (red dotted) at k⊥ρi/lessorequalslant2.\nIn typical space and astrophysical plasmas, under the conditions derived below the\nturbulent cascade can reach the ion cyclotron frequency ω/ωi→1, but since the electron\ncyclotron frequency is higher by the mass ratio, Ω e/Ωi=mi/me, there is typically\nlittle energy remaining in the cascade at the very high frequencies ne eded to transfer\nis significant amount of energy to the electron through their cyclot ron resonance. Thus,\nhere we consider only the physics of the ion cyclotron resonance as a possible physical\nmechanism for the dissioation of plasma turbulence.\nEvaluating the conditions needed to obtain ω/ωi→1 in terms of the dimensionless\nparameters of turbulence in the isotropic temperature case, we fi nd\nω\nΩi=ωk/bardbldi=ωk/bardblρi\nβ1/2\ni, (5.7)\nwhereωis defined by (5.6). Note that the middle expression indicates that th e most\nnatural unit to normalize the length scale parallel to the magnetic fie ld is the ion inertial\nlengthdi=c/ωpi=vA/Ωi, but to maintain consistency with the length normalizations\nin Tables 2 and 3, we substitute di=ρiβ−1/2\ni. Utilizing the generalscaling ofthe parallel36\nwavenumber in terms of the isotropic driving wavenumber,\nk/bardblρi= (k0ρi)(1+α)/(3+α)(k⊥ρi)2/(3+α), (5.8)\nto determine the frequency achieved when the cascade reaches k⊥ρi= 1, we obtain the\ncondition\nω\nΩi=ω\nβ1/2\ni(k0ρi)(1+α)/(3+α). (5.9)\nThis indicates clearly that ion cyclotron damping is more likely to occur f or small ion\nplasma beta βi≪1 or a smaller turbulent inertial range with a value of k0ρithis is\nnot too small. For the GS95 scaling with α= 0, we obtain a condition for the onset of\nion cyclotron damping of k0ρi=β3/2\ni/ω3; for the B06 scaling with α= 1, we obtain\nk0ρi=β/2\ni/ω2. For simplicity, note that at the small-scale end of the inertial range with\nk⊥ρi∼1, the normalized wave frequency ω∼1.\nIn summary, ion cyclotron damping is predicted to play a role in the dam ping of\nplasma turbulence for conditions with a smaller inertial range and a low plasma beta\nvaluesβi≪1. The nature of the resulting plasma heating is strictly ion energizat ion\nQi/Q= 1 in the perpendicular degrees of freedom Q⊥,i/Qi= 1.\nNote that recent theoretical and numerical study of turbulence with a sufficiently large\namount of imbalance Z+\n0/Z−\n0≫1, such as may be encountered in the inner heliosphere,\nhas uncovered a new phenomenon denoted the helicity barrier , which can lead to the\ngeneration of sufficiently high-frequency fluctuations, specifically ion cyclotron waves, to\nlead to iCD energization of the plasma particles (Squire et al.2022, 2023).\n5.6.Stochastic Heating\nThe stochastic heating of ions has long been proposed as a means to energize ions in tur-\nbulentplasmas(Johnson & Cheng2001;Chen et al.2001;White et al.2002;Voitenko & Goossens\n2004; Bourouaine et al.2008), with more recent work providing significant theoretical\nand numerical evidence for ion stochastic heating as a significant me chanism for turbu-\nlent damping in space and astrophysical plasmas (Chandran et al.2010; Chandran 2010;\nChandran et al.2011;Bourouaine & Chandran2013;Chandran et al.2013;Klein & Chandran\n2016; Arzamasskiy et al.2019; Martinovi´ c et al.2020; Cerri et al.2021).\nChandran et al.(2010) used test particle simulations of randomly phased Alfv´ en an d\nkinetic Alfv´ en wavesto developan empirical formulafor the rateof ion stochastic heating\nby turbulent fluctuations with frequencies much lower than the ion c yclotron frequency\nω/Ωi≪1. Following this, Chandran et al.(2011) developed a refined implementation to\npredict the partitioning of turbulent energy among parallel ion heat ing, perpendicular\nion heating, and electron heating, with a formula for the damping rat e by ion stochastic\nheating given by\nγSH=c1/parenleftBigg\nδv(ρi)\n⊥\nvti/parenrightBigg\nΩie−c2vti/δv(ρi)\n⊥ (5.10)\nwhereδv(ρi)\n⊥is the root-mean-square amplitude of the perpendicular velocity flu ctua-\ntions at the perpendicular scale of the ion Larmor radius k⊥ρi= 1. Here c1andc2are\nfitting constants with typical values c1= 0.18 and 0 .15/lessorsimilarc2/lessorsimilar0.34 (Chandran et al.\n2011). The key dimensionless parameter controlling the onset of st ochastic heating in\nthis formulation is the ratio of the perpendicular velocity fluctuation to the thermal\nvelocity, ǫ≡δv(ρi)\n⊥/vti; these studies found a critical threshold ǫcrit≃0.19 that deter-\nmines the minimum fluctuation amplitude required for the onset of sto chastic heating\n(Chandran et al.2010).Fundamental Parameters of Turbulence 37\nUsing the relation for Alfv´ en waves to connect the perpendicular fl uid velocity fluctua-\ntionstotheperpendicularmagneticfieldfluctuations δv⊥/vA=±ωδB⊥/B0(Howeset al.\n2008a) and the MHD scaling of the turbulent perpendicular magnetic field flu ctuation\namplitude given by (3.4) with the B06 scaling with α= 1, we obtain the expression\nδv⊥\nvti=ωβ−1/2\ni/parenleftbiggk0ρi\nk⊥ρi/parenrightbigg1/4\n(5.11)\nAtk⊥ρi∼1, we obtain the normalized frequency ω∼1 and find the key relation for the\namplitude in terms of our fundamental turbulence parameters\nǫ≡δv(ρi)\n⊥\nvti=β−1/2\ni(k0ρi)1/4(5.12)\nTo estimate the importance of damping by ion stochastic heating rela tive to the tur-\nbulent cascade rate, we can calculate γSH/ωnl∼γSH/ωto obtain the final result\nγSH\nω=c1(k0ρi)−1/4e−c2β1/2\ni(k0ρi)−1/4(5.13)\nwhere we have used the subsitution Ω i/ω=β1/2\ni/k/bardblρi.\nIn summary, the proposed turbulent damping mechanism of ion stoc hastic heating\ndepends primarily on the fundamental parameters βiandk0ρi, where lower ion plasma\nbeta and larger isotropic driving wavenumber (which corresponds t o a smaller inertial\nrange) both conspire to yield larger turbulent amplitude at the ion La rmor radius scale\nk⊥ρi∼1, enhancing stochastic heating of the ions. The exponential effec tively provides\na threshold turbulent amplitude at the ion scales below which stochas tic ion heating is\nineffective. Note that the exponential dependence on the consta ntc2makes its accurate\ndetermination critical to assess the importance of ion stochastic h eating in any particular\ncase; its value is likely to depend on the detailed nature of the turbule nt fluctuations at\nthe perpendicular scale of the ion Larmor radius (Chandran et al.2011). This heating\nmechanism is predicted to lead to strictly ion energization Qi/Q= 1 in the perpendicular\ndegrees of freedom Q⊥,i/Qi= 1.\nNote that, because the amplitude of the turbulent fluctuations is t ypically found to\ndecrease monotonically as the turbulent cascade progress to sma ller scales, it is unlikely\nthat electrons will experience significant stochastic heating, so we do not explore electron\nstochastic heating here. Further research will be necessry to to evaluate thoroughly the\npotential for electron stochastic heating to damp turbulence at t he perpendicular scale\nof the electron Larmor radius, k⊥ρe∼1.\n5.7.Magnetic Pumping\nMagneticpumping(Spitzer & Witten1951;Berger et al.1958;Lichko et al.2017;Lichko & Egedal\n2020; Montag & Howes 2022) is a particle energization mechanism in wh ich variations\nof magnetic field magnitude at low frequency ωlead to an oscillating transfer of energy\nfrom parallelto perpendicular degreesof freedom and back, obey ing the double-adiabatic\n(Chew-Goldberger-Low, CGL) equations of state (Chew et al.1956). In the absence of\nany collisions, the net change of particle energy over a full oscillation is zero. But in the\npresence small but finite collisionality †there can be a non-zero transfer of energy from\nthe oscillating electromagnetic fields to the particles over the cours e of a full oscillation.\n†Note that the collisionality can be due to Coulomb collision s or to an effective collisionality\ndue to pitch-angle scattering through collisionless wave- particle interactions with small-scale\nelectromagnetic fluctuations, such as those driven bykinet ic temperature anisotropy instabilities\n(Kunzet al.2018; Arzamasskiy et al.2023).38\nAsaproposedmechanismforthedampingofturbulence,magneticp umpingnecessarily\nrequires turbulent fluctuations that involve changes of the magne tic field magnitude from\nthe equilibrium field strength B0, orδB≡ |B|−B0∝negationslash= 0. For Alfv´ enic turbulence, which\nis dominantly incompressible ‡in the MHD regime at k⊥ρi≪1, significant fluctuations\nof the magnetic field magnitude arise only within the kinetic Alfv´ en wav e regime at\nk⊥ρi/greaterorsimilar1 under conditions of ion plasma beta βi>1. Alternatively, if the turbulence has\na signficant component of compressible magnetosonic fluctuations withEcomp/Einc>\n0, because the turbulent amplitudes typically decrease monotonica lly with decreasing\nscale lengths, the largest scale compressible fluctuations at the dr iving scale may lead to\nmagnetic pumping, directly energizing particles from these large-sc ale fluctuations.\nWe determine first the parameters controlling magnetic pumping in Alf v´ enic turbu-\nlence, following a similar procedure later to analyze the magnetic pump ing arising from\ncompressible turbulence with Ecomp/Einc>0. The foundation of magnetic pumping is\nthe double-adiabatic evolution of the weakly collisional plasma which co nserves the par-\nticle magnetic moment µm≡mv2\n⊥/2Band the action integral of the parallel bounce\nmotionJ/bardbl≡/contintegraltext\nv/bardbldl(Lichkoet al.2017; Montag 2018). Conservation of these adiabatic\ninvariants requires a scale separation of the perpendicular wavenu mber relative to the\nparticle species Larmor radius k⊥ρs≪1 and of the wave parallel phase velocity to the\nparticle thermal velocity ω/k/bardblvts≪1.\nKinetic Alfv´ en waves give rise to non-negligible magnetic field magnitud e variations\nover the perpendicular wavenumber range 1 /lessorsimilark⊥ρi/lessorsimilar(τµ)1/2for values of βi/greaterorsimilar1.\nThis range of δBfluctuations is incompatible with the necessary condition k⊥ρi≪1\nfor the ions, so we expect that magnetic pumping by kinetic Alfv´ en waves cannot ener-\ngize the ions . For the electrons, on the other hand, the requisite condition k⊥ρe≪1\ncan be rewritten as k⊥ρi≪(τµ)1/2, so we expect that kinetic Alfv´ en waves may be\nable to damp turbulent fluctuations and energize the electrons und er ion plasma beta\nconditions with βi/greaterorsimilar1. The phase velocity condition for electrons can be expressed as\nω≪(βiµ/τ)1/2. Under the ion plasma beta condition βi/greaterorsimilar1, (5.6) can be approximated\nbyω≃k⊥ρi/β1/2\ni, so this requirement can be converted to a condition on the perpen -\ndicular wavenumber given by k⊥ρi≪βi(µ/τ)1/2, which can certainly be satisfied for the\nperpendicular wavenumber range of kinetic Alfv´ en waves with signifi cant magnetic field\nmagnitude fluctuations. Therefore, the necessary conditions fo r the magnetic pumping\nof electrons to play a role in the dissipation of Alfv´ enic plasma turbule nce with βi/greaterorsimilar1\nappear to be satisfied.\nPrevious studies have shown that the ratio of the damping rate due to magnetic pump-\ningγMPto the pump wave frequency ωis given by (Lichko et al.2017; Montag & Howes\n2022)\nγMP\nω=χc,s\n1+χ2c,s/parenleftbiggδB\nB0/parenrightbigg2\n(5.14)\nwhereχc,s≡6νs/ωis a dimensionless measure of the collision frequency νsfor species s\nandδBis the amplitude of the magnetic field magnitude fluctuations.\nTo determine the dependence of the magnetic pumping of electrons on the plasma and\nturbulence parameters in Tables 2 and 3, we assume isotropic driving k/bardbl0/k⊥0= 1 for\nsimplicity, write the electron-electron collision frequency as νe=vte/λmfp,e, and use the\n‡Note, however, that the magnetic field magnitude variations arising from strong turbulent\ndriving at the outer scale with |δB| ∼B0can lead to the “interruption” of linearly polar-\nized Alfv´ en waves by kinetic temperature anisotropy insta bilities at sufficient high βi≫1\n(Squireet al.2016, 2017 b,a)Fundamental Parameters of Turbulence 39\ngeneral parallel wavenumber scaling for arbitrary αgiven by (3.1) to obtain\nχc,e=6\nω(k/bardbl0λmfp,e)/parenleftbiggβiµ\nτ/parenrightbigg1/2/parenleftbiggk0ρi\nk⊥ρi/parenrightbigg2/(3+α)\n. (5.15)\nFor a sufficiently large turbulent inertial range with k0ρi≪1, kinetic Alfv´ en waves\nover the perpendicular wavenumber range 1 /lessorsimilark⊥ρi/lessorsimilar(τµ)1/2will have relatively small\nturbulent amplitudes δB⊥/B0≪1, so the magnetic field magnitude fluctuations can\nbe expressed to lowest order as variations of the parallel compone nt of the perturbed\nmagnetic field δB≃δB/bardbl(Huanget al.2024). The turbulent magnetic field magnitude\nvariations may then be expressed as\nδB\nB0=/parenleftbiggδB/bardbl\nδB⊥/parenrightbigg/parenleftbiggk0ρi\nk⊥ρi/parenrightbigg1/(3+α)\n, (5.16)\nwhere the ratio δB/bardbl/δB⊥for kinetic Alfv´ en waves over the relevant range from the linear\ndispersion relation is strictly a function of βi, orδB/bardbl/δB⊥=f(βi). Therefore, electron\nmagnetic pumping by Alfv´ enic turbulent fluctuations at scales k⊥ρi/greaterorsimilar1 depends on the\nparameters βi,τ,µ,k0ρi, andk/bardbl0λmfp,e.\nNextweconsiderthatcasethattheturbulenceisdrivencompress iblywith Ecomp/Einc>\n0. In this situation, the magnetic field magnitude fluctuations at the driving scale, which\ntypically have the largest amplitude of δB/B0, may contribute to magnetic pumping of\nboth the ions and electrons. In this case of turbulent damping at th e driving scale, it\nis necessary to use the three driving parameters ( k⊥0ρi,k/bardbl0/k⊥0,χ0) in Table 2, rather\nthan the simplified parameterization through the isotropic driving wa venumber k0ρi, to\ndetermine the parameter dependence of the turbulent damping via magnetic pumping.\nFirst, the normalized fast and slow magnetosonic wave phase velocit ies can be ex-\npressed by the linear dispersion relation (Klein et al.2012)\n/tildewideω4−/tildewideω2(1+β)+βcos2Θ0= 0, (5.17)\nwhich has four solutions given by\n/tildewideω2=1\n2/bracketleftBig\n1+β±/radicalbig\n(1+β)2−4βcos2Θ0/bracketrightBig\n, (5.18)\nwhere the plus (minus) sign correspondsto the two fast (slow)mag netosonicwavemodes.\nHere we define the normalized wave frequency /tildewideω≡ω/kvA, the total (MHD) plasma beta\nβ=βi(1+1/τ), and the wavevectorcomponents ( k/bardbl,k⊥) are alternatively parameterized\nby (k,Θ), where k/bardbl=kcosΘ,k⊥=ksinΘ,k= (k2\n/bardbl+k2\n⊥)1/2, and Θ = tan−1(k⊥/k/bardbl).\nThe dimensionless form of the solutions in (5.18) makes clear that the fast and slow mag-\nnetosonic wave phase velocities depend only on two dimensionless par ameters,/tildewideω(β,Θ).\nWe may determine the parameterdependence of the normalized ion c ollision frequency\nbyusingtherelationfortheion-ioncollisionrateintermsoftheelectr on-electroncollision\nrateνi=µ−1/2τ−3/2νeand writing the pumping wave frequency at the driving scale as\nω=/tildewideω0k0vAto obtain\nχc,i=6\n/tildewideω0(k/bardbl0λmfp,e)τ2β1/2\nicosΘ0. (5.19)\nNote that the ion collisional mean free path can be expressed in term s of the electron\ncollisional mean free path by the relation λmfp,i=λmfp,eτ2, so the denominator is\nactually independent of the ion-to-electron temperature ratio τ, depending rather only\non the normalized ion mean free path, k/bardbl0λmfp,i.\nTo evaluate the turbulent magnetic field magnitude variations for ion magnetic pump-40\ning by compressible fluctuations at the driving scale, we first note th at the phenomenon\nof dynamic alignment, parameterizied by the alignment angle θ0, will not play a role: dy-\nnamic alignment develops through the nonlinear interactions as the t urbulence cascades\nto smaller scales, but does not come into play at the driving scale, so w e setθ0= 1,\nthus reducing the nonlinearity parameter to χ0=k⊥0δB⊥0/(k/bardbl0B0). If the magnetic\nperturbations at the driving scale have |δB|/B0∼1, the linear theory for waves breaks\ndown, and the nonlinearity must be taken into account in the lowest- order description\nof the wave dynamics; in this case, one may directly take the measur ed magnetic field\nmagnitude fluctuations δB/B0as the parameter to estimate the damping rate by mag-\nnetic pumping in (5.14). However, if the magnetic perturbations at t he driving scale have\nǫ≡ |δB|/B0<1, the binomial expansion may be used to obtain the expression\nδB/B0≃δB/bardbl/B0+(δB/bardbl/B0)2/2+(δB⊥/B0)2/2, (5.20)\nwhere the first term is of order ǫand the next two terms are of order ǫ2. Thus, to lowest\norder, we may estimate δB≃δB/bardbl, where the error in this estimation scales as ǫ2/2,\nyielding a fractional error of 25% for |δB|/B0= 1/2 and of 6% for |δB|/B0= 1/4.\nUsing this approximation, we can derive an expression the normalized fluctuations of\nthe magnetic field magnitude by δB/B0≃δB/bardbl/B0= (δB⊥/B0)(δB/bardbl/δB⊥). If we esti-\nmateδB/bardbl/δB⊥∼(Ecomp/Einc)1/2, we can then obtain the expression for the normalized\namplitude of the magnetic field magnitude fluctuations\nδB\nB0=k/bardbl0\nk⊥0χ0/parenleftbiggEcomp\nEinc/parenrightbigg1/2\n. (5.21)\nCombining the dependencies of χc,iin (5.19) and of δB/B0in (5.21), ion magnetic\npumping by compressible turbulent fluctuations at the driving scale d epends on the\nparameters βi,k/bardbl0λmfp,i= (k/bardbl0λmfp,e)τ2,k/bardbl0/k⊥0,χ0, andEcomp/Einc.\nFor magnetic pumping of electrons by compressible turbulent fluctu ations at the driv-\ningscale,wecanfollowananalogousproceduretoobtainthenormaliz edelectroncollision\nfrequency\nχc,e=6\n/tildewideω0(k/bardbl0λmfp,e)/parenleftbiggβiµ\nτ/parenrightbigg1/2\ncosΘ0. (5.22)\nand the same expression in (5.21) for the the normalized fluctuation s of the magnetic\nfield magnitude by δB/B0, which is independent of the species properties. Thus, electron\nmagnetic pumping by compressible turbulent fluctuations at the driv ing scale depends\non the parameters βi,τ,µ,k/bardbl0λmfp,e,k/bardbl0/k⊥0,χ0, andEcomp/Einc.\nIn summary, Alfv´ enic turbulent fluctuations can damping turbulen ce and energize\nelectrons through turbulent fluctuations at perpendicular scales 1/lessorsimilark⊥ρi/lessorsimilar(τµ)1/2for\nplasma conditions with βi/greaterorsimilar1, but ions are not expected to be energized by Alfv´ enic\nfluctuations. The resulting magnetic pumping of electrons by Alfv´ e nic turbulent fluctua-\ntions depends on the parameters βi,τ,µ,k0ρi, andk/bardbl0λmfp,e. If the turbulence is driven\ncompressibly with Ecomp/Einc>0, we expect that the turbulent fluctuations at the\ndriving scales, which have the largest amplitude of δB/B0, may lead to damping of the\nturbulence through the magnetic pumping of both the ions and elect rons. The resulting\nmagneticpumpingofionsbycompressibleturbulenceatthe drivingsc alesdepends onthe\nparameters βi,k/bardbl0λmfp,i= (k/bardbl0λmfp,e)τ2,k/bardbl0/k⊥0,χ0, andEcomp/Einc. Analogously,\nthe magnetic pumping of electrons by compressible turbulence depe nds on the param-\netersβi,τ,µ,k/bardbl0λmfp,e,k/bardbl0/k⊥0,χ0, andEcomp/Einc. These parameter dependencies\nfor magnetic pumping of ions and electrons are summarized in Table 4.Fundamental Parameters of Turbulence 41\n5.8.Kinetic Viscous Heating\nAs discussed above for the case of magnetic pumping, in weakly collisio nal plasmas, low-\nfrequency turbulent fluctuations that generate variationsin the magnetic field magnitude\nwill lead to a double-adiabatic (CGL) (Chew et al.1956) evolution of the plasma that\npreserves to lowest order the particle magnetic moment µmand the action integral of the\nparallel bounce motion J/bardbl(Lichkoet al.2017; Montag 2018; Arzamasskiy et al.2023).\nThis dynamics leads to the development of anisotropies in the velocity distribution of the\nplasma particles exhibiting temperature and pressure anisotropies withp/bardbl∝negationslash=p⊥. These\ndeviations from local thermodynamic equilibrium can trigger rapidly gr owing kinetic\ninstabilities that can disrupt the linear wave dynamics (Squire et al.2016, 2017 b,a) and\nmediate nonlocal transfer of turbulent energy directly from the la rge driving scales to\nkinetic length scales (Kunz et al.2018; Arzamasskiy et al.2023), as illustrated by the\nred arrow in Fig. 1(b). The effect of these kinetic instabilities on the t urbulent dynamics\nhas recently been modeled as an effective viscosity (Arzamasskiy et al.2023).\nTo properly describe the kinetic physics of turbulence in temperatu re anisotropic plas-\nmas,it is necessaryto employthe full setofplasmaandturbulence p arametersin Table2.\nThe relevant kinetic instabilities driven by anisotropies in the ion velocit y distribution\nare the four ion temperature anisotropy instabilities: (i) the paralle l (or whistler) firehose\ninstability (Kennel 1966; Gary et al.1976) and (ii) the Alfv´ en (or oblique) firehose in-\nstability (Hellinger & Matsumoto 2000), which both are relevant to pla smas with an ion\ntemperature anisotropysufficiently less than unity Ai=T⊥,i/T/bardbl,i<1 and with a parallel\nion plasma beta βi>1; and (iii) the mirror instability (Vedenov & Sagdeev 1958; Tajiri\n1967; Southwood & Kivelson 1993) and (iv) the ion cyclotron instabilit y (Gary et al.\n1976), which both are relevant to plasmaswith an ion temperature a nisotropysufficiently\ngreater than unity Ai=T⊥,i/T/bardbl,i>1 for any value of the parallel ion plasma beta βi.\nWhen the plasma exceedsa thresholdvalue ofthe proton temperat ureanisotropy( ????),\nthese instabilities can tap the free energy associated with the aniso tropic proton tem-\nperature, driving electromagnetic fluctuations and ultimately redu cing the temperature\nanisotropy, thereby moving the plasma back toward a state of mar ginal stability.\nRecent theoretical and numerical studies have highlighted the fac t that, at sufficiently\nhighparallelionplasmabeta βi≫1,the marginalstabilityboundaryforthesekineticion\ntemperatureanisotropyinstabilitiesasymptotestowardsisotrop y,sothe double-adiabatic\nevolution of turbulent fluctuations can easily exceed these bounda ries and trigger these\nrapidly growing instabilities. Large-scale Alfv´ en waves in the MHD limit a tk⊥ρi≪1\nyield only perpendicular magnetic field perturbations δB⊥∝negationslash= 0 with δB/bardbl= 0, so the\nvariation in the magnetic field magnitude δBis negligible in the limit of small amplitude\nwavesδB⊥/B0≪1. However, at sufficiently large amplitudes δB⊥/B0→1, these fluctu-\nations can lead to significant variation in the magnetic field magnitude, as made clear by\n(5.20). Thus, at sufficiently high values of parallel ion plasma beta βi≫1, the physics of\nlinearly polarized †Alfv´ en waves can be “interrupted” by these kinetic ion temperatu re\nanisotropy instabilities (Squire et al.2016, 2017 b,a).\nRecent theoretical and numerical work has extended this concep t to consider the effect\nof kinetic instabilities on the turbulent cascade under conditions of h igh parallel ion\nplasmabeta βi≫1,findingthattheseinstabilitiescannonlocallytransferenergydire ctly\nto kinetic length scales (Arzamasskiy et al.2023), as illustrated by the red arrow in\nFig. 1(b). This new channel of energy transfer can compete with t he traditional local\ntransfer of energy to smaller scales ofthe turbulent cascade tha t is mediated by nonlinear\n†Notethat circularly polarized Alfv´ enic fluctuations, whi chhave nomagnetic fieldmagnitude\nvariation, do not suffer this kinetic-instability-driven i nterruption (Squire et al.2016).42\ninteractions, as illustrated by the black arrows in Fig. 1(a). The effe ct of these kinetic\ninstabilities on the large-scale turbulent dynamics has been quantifie d as an effective\nviscosity by Arzamasskiy et al.(2023). Here we denote this kinetic-instability-mediated\nviscosity as the kinetic viscosity ‡to distinguish it from the usual definition of viscosity\nthat arises through Coulomb collisions among the plasma particles.\nThese ion kinetic instabilities act to limit the ion pressure anisotropy in t he plasma to\nvalues corresponding to marginal stability, given by ∆ pi/pi/lessorsimilar1/βi, where ∆ pi≡p⊥i−\np/bardbliandpi≡(2p⊥i+p/bardbli)/3 (Kunz et al.2014; Melville et al.2016; Arzamasskiy et al.\n2023). Using this marginalstability scalingwith βi, one can estimate an effective collision\nfrequency νeff∼βiω(δB⊥/B0)2(Arzamasskiy et al.2023), where ωis the linear wave\nfrequency at the scale where the dynamics generate that largest pressure anisotropy, and\nthushavethemaximumgrowthrate.Sincetheturbulentfluctuatio namplitudesgenerally\ndecreasemonotonicallywith scale,the kinetic temperature anisotr opyinstabilities will be\ndriven most strongly by turbulent fluctuations at the driving scale. As with the case for\nmagnetic pumping by compressible fluctuations that is dominated by t he driving scale\ndyanmics, here we must use the three driving parameters ( k⊥0ρi,k/bardbl0/k⊥0,χ0) in Table 2,\nrather than k0ρi, to estimate the turbulent damping rate by the kinetic viscosity.\nUsing the effective collision frequency νeffabove, we can estimate the ratio of the\ndamping rate due to kinetic viscous heating to the wave frequency a t the driving scales,\nobtaining\nγV H\nω∼βi/parenleftbigg\nχ0k/bardbl0\nk⊥0/parenrightbigg2\n, (5.23)\nwhere the definition of the nonlinearity parameter χ0= (k⊥0/k/bardbl0)(δB⊥0/B0)θ0with\nθ0= 1 at the driving scale is used to replace the amplitude of the driving-s cale turbulent\nfluctuations δB⊥0/B0.\nIn summary, the kinetic viscous heating due to ion temperature anis otropyinstabilities\narising from the nonthermal velocity distributions driven by the larg e-scale turbulent\nmotions occurs in weakly collisional turbulent plasmas with parallel ion p lasma beta\nβi≫1. The kinetic viscous damping rate depends on the parallel ion plasma betaβi,\nthe nonlinearity parameter χ0, and the anisotropy of the turbulent driving k⊥0/k/bardbl0.\nNote that the scaling used to derive the form of the effective collision frequency νeff\ndepends on the scaling of the marginal stability boundaries that are generally calculated\nin quiescient plasmas in the absence of turbulence. The scaling of mar ginal stability\nboundaries in the presence of strong plasma turbulence may differ, although evidence\nfrom spacecraft observations (Hellinger et al.2006; Bale et al.2009; Maruca et al.2011)\nand from hybrid kinetic ion and fluid electron simulations of temperatu re anisotropic\nplasma turbulence (Kunz et al.2014; Bott et al.2021; Arzamasskiy et al.2023) appear\nto suggest that any modifications of these boundaries due to the p resence of turbulence\nare relatively small.\n5.9.Collisionless Magnetic Reconnection\nNumerical simulations of plasma turbulence demonstrate the ubiquit ous development of\ncurrent sheets at small scales (Matthaeus & Montgomery 1980; M eneguzzi et al.1981;\nBiskamp & Welter1989;Spangler1998,1999;Biskamp & M¨ uller2000; Maron & Goldreich\n2001; Merrifield et al.2005; Greco et al.2008), and it has been found that the dissipa-\ntion of the turbulence is largely concentrated in the vicinity of these current sheets\n(Uritsky et al.2010; Wan et al.2012; Karimabadi et al.2013; TenBarge & Howes 2013;\n‡Thiskinetic viscosity terminology for kinetic-instability-mediated viscosity should not be\nconfused with the kinematic viscosity defined in hydrodynamic fluids.Fundamental Parameters of Turbulence 43\nFigure 8. Diagram of the current sheet geometry generated self-consi stently in plasma turbu-\nlence in the MHD regime at k⊥ρi/lessorsimilar1, with the scalngs of ( k⊥,ki,k/bardbl) given by the B06scaling.\nThe current sheet has thickness a∼k⊥and width w∼1/kiin the perpendicular plane and\nlengthl∼1/k/bardblalong the equilibirium magnetic field B0. The wavelength of the mode unstable\nto the collisionless tearing instability is given by λ∼1/k(red), and must fall within the bounds\ndetermined by the current sheet scaling, yielding an orderi ngki/lessorsimilark≪k⊥.\nWuet al.2013; Zhdankin et al.2013), giving rise to the proposal that magnetic recon-\nnection may play a role in the damping of plasma turbulence. Motivated by these find-\nings, statistical analyses of observations of turbulence in the sola r wind have sought evi-\ndence for such spatially localized heating (Osman et al.2011; Borovsky & Denton 2011;\nOsmanet al.2012b,a; Perriet al.2012; Wang et al.2013; Wu et al.2013; Osman et al.\n2014b). Yet the physical mechanisms that lead to the development of the se current sheets\nin plasma turbulence is yet to be fully understood.\nUnder the B06theory for the scaling of MHD turbulence, the phenomenon of dyna mic\nalignment (Boldyrev 2006) is predicted to lead to the development of current sheets with\na widthw∼1/kito thickness a∼1/k⊥ratio in the plane perpendicular to the magnetic\nfield given by w/a=k⊥/ki= [(k⊥ρi)/(k0ρi)]1/4, according to the scaling in (3.2) for\nα= 1. The geometry of the current sheets expected to develop in pla sma turbulence\nis illustrated in Fig. 8, where for sufficiently small perpendicular scales relative to the\ndriving scale, k⊥ρi≫k0ρi, the current sheets have a parallel extent along the mean\nmagnetic field B0, given by l∼1/k/bardbl, and a scale separation of thickness to width to\nlength obeying a≪w≪l; the corresponding three components of the wavenumber\nhave the complementary scaling k/bardbl≪ki≪k⊥. In Fig. 8, the wavelength of a mode\nthat is unstable to the collisionless tearing instability is illustrated by λ∼1/k(red).\nNote that the minimum value of this tearing unstable wavenumber kis equal to the\nintermediate wavenumber kicorresponding to the width of the current sheet, w∼1/ki.\nThe maximum value of kassumed in calculations of the linear collisionless tearing in-\nstability (Loureiro & Boldyrev 2017 a; Mallet et al.2017b) must be much smaller than\nthe perpendicular wavenumber k⊥characterizingthe current sheet thickness ( a∼1/k⊥),\nthereby yielding k/k⊥=ka≪1. Thus, the range of kis given by\nkiρi= (k0ρi)1/4(k⊥ρi)3/4/lessorsimilarkρi≪k⊥ρi/lessorsimilar1, (5.24)\nwhere the final constraint k⊥ρi/lessorsimilar1 is needed for the tearing instability to arise for a cur-\nrent sheet thicknesswithin the MHD regime,otherwisedifferent sca lingsforelectron-only\nreconnection (Phan et al.2018) in the regime k⊥ρi/greaterorsimilar1 would need to be incorporated.\nNote that, in turbulence obeying the B06scaling, the self-consistent generation of cur-\nrent sheets arises only at sufficiently small perpendicular scales, k⊥ρi≫k0ρi. Therefore,\nthe ratio of the perpendicular magnetic field perturbation to the me an magnetic field\nwill be small, δB⊥/B0= (k0ρi/k⊥ρi)1/4≪1, so the reconnection that arises in plasma44\nturbulence necessarily falls in the limit of strong-guide-field magnetic reconnection, as\nillustrated in Fig. 8 (green). An implication of this fact is that, if numer ical simulations\nare employed to assess the importance of magnetic reconnection in turbulence, it is criti-\ncal that those simulations be modeled in three spatial dimensions (3D ). Previous studies\nof 2D vs. 3D plasma turbulence have shown that 3D simulations exhibit a significantly\nfaster turbulent cascade rate (Li et al.2016), so enabling full 3D evolution is likely to\nbe essential to determine the competition between the turbulent c ascade and magnetic\nreconnection.\nThe tearinginstability growthrate typicallyincreaseswith anincreas ingperpendicular\naspect ratio w/aratio of the current sheets. If this ratio reaches a sufficiently larg e value,\nit was recently recognized that the growth rate of the tearing inst ability could exceed\nthe turbulent cascade rate at that scale, γ/ωnl/greaterorsimilar1, triggering a reconnection flow that\nwould disrupt the turbulent dynamics, deviating from the scaling rela tions summarized\nin§3 and potentially affecting the damping of turbulent cascade (Boldyr ev & Loureiro\n2017; Loureiro & Boldyrev 2017 b,a; Malletet al.2017b,a; Walker et al.2018).\nIt is worth noting here that an alternative explanation for the deve lopment of current\nsheetsinplasmaturbulenceduetothenonlinearinteractionsbetwe encounterpropagating\nAlfv´ en waves—a phenomenon often denoted Alfv´ en wave collisions (Howes & Nielson\n2013)—was proposed by Howes (2016), but no scaling for the resu lting aspect ratio of\nthe current sheets was provided. If the scaling of this alternative mechanism differs from\nthat of the B06theory, than the scaling predictions for the onset of reconnectio n would\nalso change.\nHere we adopt the B06scaling for the development of current sheets to determine\nthe dependence of magnetic reconnection as a turbulent damping m echanism on the\nfundamental plasma and turbulence parameters for the isotropic temperature case in\nTable 3. This calculation follows previous investigations that assess t he conditions under\nwhich collisionless magnetic reconnection may disrupt the turbulent c ascade and thereby\nimpact how the turbulent energy is channeled into particle energy (L oureiro & Boldyrev\n2017a; Mallet et al.2017b). Our goal here is to express the results from these previous\nstudies in the dimensionless parameters for the isotropic temperat ure case proposed in\nTable 3.\nThe growth rate of the collisionless tearing instabilty, which is the inst ability that\ninitiates the process of magnetic reconnection under the weakly co llisional conditions\ntypical of space and astrophysical plasmas, has been derived in th e limit of very low\nelectron plasma beta βe∼µ−1(Zocco & Schekochihin 2011), where we convert βe=\nβi/τ. The physics of the collisionless tearing instabilty depends on three le ngth scales: (i)\nthe thickness of the current sheet a∼1/k⊥, assumed here to occur on MHD scales with\nk⊥ρi≪1; (ii) the ion sound Larmor radius ρs≡ρi/(2τ)1/2where two fluid effects begin\nto lead to differences between the ion and electrondynamics; and (iii) the inner boundary\nlayer scale δinwhere the magnetic flux can be unfrozen from the electron fluid flow . We\nwill restrict our analysis to βe≫µ−1, such that the flux unfreezing arises from electron\ninertia (Zocco & Schekochihin 2011; Loureiro & Boldyrev 2017 a; Mallet et al.2017b),\noccuring approximately at the electron inertial length scale de≡c/ωpe=ρi/(βiµ)1/2.\nThese scales are ordered by δin∼de≪ρs≪a. Together, these restrictions limit us to\nmoderately small plasma beta,\nµ−1≪βi/τ≪1. (5.25)\nThe tearing instabilty depends on the tearing mode instability parame ter ∆′, deter-\nmined by the outer-region solution (at MHD scales k⊥ρi≪1) (Zocco & Schekochihin\n2011). This parametercan be expressedovertwo limiting regimes in a dimensionless formFundamental Parameters of Turbulence 45\n(Loureiro & Boldyrev 2017 a) as\n∆′a= (ka)−nfor 1 < n/lessorequalslant2, (5.26)\nwherekis the unstable wavenumber along the width of the current layer gov erning\nthe reconnection flow, as illustrated in Fig. 8 by the red sinusoidal cu rve. Here n= 1\ncorresponds to a Harris current sheet (a hyperbolic tangent pro file of the reconnecting\nmagnetic field across the current sheet thickness) (Harris 1962) , whilen= 2 corresponds\nto a sinusoidal variation of the reconnection magnetic field (Loureir o & Boldyrev 2017 a),\nwhich maybe the morelikelycasefortheonsetofreconnectionatcu rrentsheetsthat self-\nconsistently arisein aturbulent plasma. The tearinginstability growt hrates aresolvedin\ntwo limits of ∆′(Zocco & Schekochihin 2011; Loureiro & Boldyrev 2017 a; Mallet et al.\n2017b): (i) ∆′δin≪1,\nγ∼kvA⊥deρs∆′\nawith δin∼deρ1/2\ns∆′1/2, (5.27)\nand (ii) ∆′δin/greaterorsimilar1,\nγ∼kvA⊥d1/3\neρ2/3\ns\nawith δin∼d2/3\neρ1/3\ns, (5.28)\nwherevA⊥≡δB⊥/(4πnimi)1/2is the Alfv´ en speed based on the magnitude of onlythe\nreconnecting component of the magnetic field δB⊥. Since the growth rates above are\ncalculated in the limit k⊥ρi≪1, we will focus our assessment on whether magnetic\nreconnection will disrupt the turbulent cascade at MHD scales k⊥ρi/lessorsimilar1, which is the\nsame approach as taken in previous studies (Loureiro & Boldyrev 20 17a; Mallet et al.\n2017b).\nUsing (5.26) and either (5.27) or (5.28) for δin, the normalized tearing mode instability\nparameter ∆′δincan be expressed in terms of the dimensionless quantities in Table 3 by\n∆′δin=k⊥ρi\n(2τβ2\niµ2)1/6/parenleftbiggk⊥ρi\nkρi/parenrightbiggn\n. (5.29)\nThis equation shows that ∆′δinincreases as the perpendicular wavenumber k⊥ρiin-\ncreases, corresponding to a thinner current sheet, or as the un stable wavenumber kρi\ndecreases, requiring to a wider current sheet. Since the width w∼1/kiof the cur-\nrent sheets in B06scaling theory governed by (3.2) bounds the unstable wavenumber\nkfrom below by the intermediate wavenumber, k > ki, the maximum value of instabil-\nity parameter ∆′δinincreases with the perpendicular aspect ratio of the current shee ts\nw/a=k⊥/ki≫1. Note also that ∆′δin∝β−1/3\ni, so that lower values of ion plasma beta\nlead to a larger instability parameter.\nNote that the value of the unstable wavenumber kρiat the transition between the\nlimiting regimes of the tearing mode instability parameter, where ∆′δin= 1, is given by\nkρi=ktρi≡(k⊥ρi)(n+1)/n\n(2τβ2\niµ2)1/6n, (5.30)\nwhich provides a bound on the value of kρifor the validity of the tearing growth rate\nformulas in (5.27) and (5.28).\nTo assess whether the collisionless tearing instabilty grows fast eno ugh to enable\nthe onset of magnetic reconnection to disrupt the turbulent casc ade, we will assess\nγRXN/ωnl∼γRXN/ωassuming critical balance of linear and nonlinear timescales ωnl∼\nω(Goldreich & Sridhar 1995; Mallet et al.2015). Taking the linear wave frequency for\nMHD Alfv´ en waves ω=k/bardblvA= (k/bardblρi)1/2(k⊥ρi)1/2vA/ρiusing (3.1) with α= 1 for the46\nB06scaling of k/bardbl, and substituting vA⊥=vA(δB⊥/B0) =vA[(k0ρi)/(k⊥ρi)]1/4using\n(3.4), we find that the growth rate in the ∆′δin≪1 limit, which corresponds to the large\nunstable wavenumber klimit by inspection of (5.29), can be expressed as\nγRXN\nω∼(k⊥ρi)(5+4n)/4\n(kρi)n−1(k0ρi)1/4(2τβiµ)1/2, (5.31)\nwhere this expression is formally valid in the MHD limit with k⊥ρi≪1. Combining\nthe limitations on the possible unstable wavenumber from (5.24) and t he requirement\n∆′δin/lessorsimilar1using (5.30),theformulain(5.31)isvalidovertherangeofnormalize dunstable\nwavenumbers\nmax/bracketleftbigg\n(k0ρi)1/4(k⊥ρi)3/4,(k⊥ρi)(n+1)/n\n(2τβ2\niµ2)1/6n/bracketrightbigg\n/lessorsimilarkρi≪k⊥ρi/lessorsimilar1. (5.32)\nNote that this valid unstable wavenumber range can be used to show that no range with\n∆′δin/lessorsimilar1 exists if βi/lessorsimilar(2τµ2)−1/2. For typical values of τ∼1, however, the very low\nvalues of βirequired to violate this condition would already have violated the assu med\nmoderate beta limit given by (5.25), so this limitation is of no practical c oncern.\nThetearinggrowthratein(5.31)demonstratesthat,inthelarge klimit,thenormalized\ngrowth rate is independent of kwithγRXN/ω∝k0for Harris-type current sheets with\nn= 1, and is inversely proportional to kwithγRXN/ω∝k−1for then= 2 case of\nsinusoidalvariationsoftheperpendicularmagneticfieldcomponent thatmaybeexpected\nto occur in plasma turbulence.\nThe maximum growth rate γRXN/ωfrom (5.31) increases with k⊥ρi, but also re-\nquires the limit k⊥ρi≪1 for the validity of this linear tearing growth rate calculation.\nTherefore, to estimate quantitatively the maximum normalized tear ing growth rate, we\ntakek⊥ρi→ F, whereF ≪1 is the maximum value of the normalized perpendicular\nwavenumber for which (5.31) remains valid; numerical simulations of c ollisionless mag-\nnetic reconnection will be invaluable to provide estimates of the maxim um value of F.\nThe maximum growth rate γRXN/ωalso occurs for the minimum unstable wavenumber\nkρiallowed by the range in (5.32). Thus, taking these two limits and requir inga collision-\nless tearing instability growth sufficient to disrupt the turbulent cas cade,γRXN/ω/greaterorsimilar1,\nyields the constraint\nk0ρi/lessorsimilarF(4+5n)/n\n(2τ)(4n+2)/3n(βiµ)(2n+4)/3n(5.33)\nif the minimum kρiis constrained by the condition ∆′δin/lessorsimilar1. Alternatively, if the\nminimum kρiis constrained by the width of the current sheet given by (5.24), th en the\ncondition γRXN/ω/greaterorsimilar1 yields the constraint\nk0ρi/lessorsimilarF(5+4n)/n\n(2τβiµ)2/n. (5.34)\nThe normalized growth rates in the ∆′δin/lessorsimilar1 limit vs. normalized unstable wavenum-\nberkρiare illustrated in Fig. 9(a) for n= 1 (thin blue) and n= 2 (thick blue) for\nturbulent parameters βi= 0.01,τ= 1,µ= 1836, and k0ρi= 10−4and for three values\nof the maximum of k⊥ρigiven by F= 1.0 (solid), 0 .3 (dashed), and 0 .1 (dotted). The\ncorresponding values of ∆′δinvs.kρiare presented in Fig. 9(b) for each of the nand\nk⊥ρicases. For the turbulence parametersspecified in this example, we findγRXN/ω/greaterorsimilar1\nonly for the k⊥ρi= 1.0 case over the unstable wavenumber range 0 .3/lessorsimilarkρi/lessorsimilar1 for the\nn= 1 case (thin solid blue) and over 0 .6/lessorsimilarkρi/lessorsimilar1 for the n= 2 case (thick solid blue).\nNext we assess the initiation of collisionless tearing in the ∆′δin/greaterorsimilar1 limit with theFundamental Parameters of Turbulence 47\nFigure 9. (a) Scaling the normalized collisionless tearing instabil ity growth rate γRXN/ωfor a\nturblent plasma with βi= 0.01,τ= 1,µ= 1836, and k0ρi= 10−4as a function of the normal-\nized unstable wavenumber kρi. Colors indicate the solutions in the low klimit (corresponding\nto ∆′δin/greaterorsimilar1) (red) and in the high klimit (corresponding to ∆′δin≪1) (blue). Line thickness\nindicates the either a sinusoidal current profile with n= 2 (thick) or a Harris-like hyperbolic\ntangent current profile with n= 1 (thin). Line styles indicate different values of maximum p er-\npendicular wavenumber F= 1.0 (solid), 0 .3 (dashed), and 0 .1 (dotted). (b) The corresponding\nnormalized tearing mode instability parameter ∆′δinvs.kρifor each case.\ngrowth rate given by (5.28). Using similar substitutions for ωandvA⊥(as detailed above\nfor the ∆′δin≪1 limit), we find that the growth rate in the ∆′δin/greaterorsimilar1 limit, which is\nthe small unstable wavenumber klimit as shown by (5.29), can be expressed as\nγRXN\nω∼(kρi)(k⊥ρi)1/4\n(k0ρi)1/4(4τ2βiµ)1/6, (5.35)\nwhere this expression is formally valid in the MHD limit with k⊥ρi≪1. Combining\nthe limitations on the possible unstable wavenumber from (5.24) and t he requirement\n∆′δin/greaterorsimilar1using (5.30),theformulain(5.35)isvalidovertherangeofnormalize dunstable\nwavenumbers\n(k0ρi)1/4(k⊥ρi)3/4/lessorsimilarkρi≪max/bracketleftbigg\nk⊥ρi,(k⊥ρi)(n+1)/n\n(2τβ2\niµ2)1/6n/bracketrightbigg\n/lessorsimilar1. (5.36)\nThis valid unstable wavenumber range can be used to show that no ra nge with ∆′δin/greaterorsimilar1\nexists if\nk⊥ρi/lessorsimilar(k0ρi)n/(n+4)(2τβ2\niµ2)2/3(n+4). (5.37)\nThis finding is consistent with the fact the that value of ∆′δindecreases with decreasing48\nk⊥ρiin (5.29), so by decreasing k⊥ρienough, one will always reach a point where there\nis no range of kρiwith ∆′δin/greaterorsimilar1.\nThe tearing growth rate in (5.35) shows that, in the small klimit, the normalized\ngrowth rate increases linearly with the unstable wavenumber kand increasesweakly with\nthe perpendicular wavenumber k⊥ρi, independent of whether the turbulently generated\ncurrent sheets are Harris-like or sinusoidal (thus, independent o fn).\nOnce again the maximum growth rate γRXN/ωfrom (5.35) increases with k⊥ρi(al-\nthoughly weakly with the 1 /4 power), so once again we take the limit k⊥ρi→ F, where\nF ≪1 is the maximum value of the normalized perpendicular wavenumber fo r which\n(5.35) remains valid. The maximum growth rate will also occur for the m aximum un-\nstable wavenumber kρigiven by (5.36). Thus, taking these two limits and requiring a\ncollisionless tearing instability growth rate sufficient to distrupt the t urbulent cascade,\nγRXN/ω/greaterorsimilar1, yields the constraint\nk0ρi/lessorsimilarF(4+5n)/n\n(2τ)(4n+2)/3n(βiµ)(2n+4)/3n. (5.38)\nif the maximum kρiis constrained by the condition ∆′δin/greaterorsimilar1. Note that this condition\nis the same as (5.33), as it must be since this maximum growth rate occ urs at ∆′δin= 1,\nwhere the growth rates for the ∆′δin/lessorsimilar1 and ∆′δin/greaterorsimilar1 limits cross. Alternatively,\nif the maximum kρiis constrained by the maximum k⊥ρi→ F, then the condition\nγRXN/ω/greaterorsimilar1 yields the constraint\nk0ρi/lessorsimilarF(5+4n)/n\n(4τ2βiµ)2/3(5.39)\nThe normalized growth rates in the ∆′δin/greaterorsimilar1 limit vs. normalized unstable wavenum-\nberkρiare illustrated in Fig. 9(a) (thick red) for turbulent parameters βi= 0.01,τ= 1,\nµ= 1836,and k0ρi= 10−4andfor threevalues ofthe maximumof k⊥ρigivenby F= 1.0\n(solid), 0 .3 (dashed), and 0 .1 (dotted). The corresponding values of ∆′δinvs.kρiare pre-\nsented in Fig. 9(b) for each of the nandk⊥ρicases. For the turbulence parameters\nspecified in this example, we find γRXN/ω/greaterorsimilar1 only for the k⊥ρi= 1.0 case over the\nunstable wavenumber range 0 .2/lessorsimilarkρi/lessorsimilar0.3 for the n= 1 case (thick solid red) and over\n0.2/lessorsimilarkρi/lessorsimilar0.6 for the n= 2 case (thick solid red).\nIt is necessaryto highlight here a few important caveats in using the linear collisionless\ntearing mode growth rates and MHD turbulence scaling theory to pr edict when magnetic\nreconnection may disrupt the dynamics of the turbulent cascade. First, the linear tearing\nmode growth rates used here are rigorously derived in asymptotic lim its, but we employ\nthese same rates up to the boundary of those limits. For example, t he linear growth rates\nformally require k⊥ρi≪1; also, the ∆′δin≪1 limit is used to estimate the tearing mode\ngrowth rates as ∆′δin→1. But such rates derived in asymptotic limits may not remain\nquantitatively accurate as we extend beyond their regime of forma l validity. We take the\napproach that the growth rates from the limits are at least a reaso nable lowest-order\nestimate of the rate beyond their formally limited range. Furhermor e, the growth rates\nfor the collisionless tearing mode are determined under smooth cond itions, whereas to\ndetermine whetherreconnectionarisesin the currentsheetstha t naturallyariseinplasma\nturbulence one needs to determine the collisionless tearing growth r ates in the presence\nof turbulence; once again, the estimated tearing mode growth rat es used here we take\nsimply asthe lowest-orderestimation. Computing the tearingmode g rowthratesin a tur-\nbulent plasma, and determining the competition of the tearing mode w ith the nonlinear\ncascade of energy due to turbulence, are both necessary for mo re accurate calculations,Fundamental Parameters of Turbulence 49\nwhich likely will require detailed kinetic numerical simulations. Finally, alte rnative pro-\nposed mechanisms for the development of current sheets in plasma turbulence—such as\nthe nonlinear interaction of counterpropagating Alfv´ en wave collis ions (Howes 2016), as\nopposed to dynamic alignment (Boldyrev 2006)—may yield a different s caling for the\nwidth and thickness of current sheets; the resulting predictions f or the onset of magnetic\nreconnection in a turbulent plasma would thereby also likely change.\nIn summary, we have followed previous investigations (Loureiro & Bo ldyrev 2017 a;\nMalletet al.2017b) to estimate how the collisionless tearing mode may arise in the\ncurrent sheets that develop self-consistently in plasma turbulenc e, potentially disrupting\nthe nonlinear cascade of turbulent energy to small scales. We rest rict ourselves to the\nmoderately low electron plasma beta limit µ−1≪βe≪1, which can be expressed in\nterms of the turbulence parameters in Table 3 as the limit 1 ≪βiµ/τ≪µ. In this limit,\nmagnetic flux conservation is broken due to electron inertia at elect ron inertial length\nscalesde(Zocco & Schekochihin 2011; Loureiro & Boldyrev 2017 a; Malletet al.2017b).\nWeconsideronlythecasewhencollisionlessmagneticreconnectionar isesincurrentsheets\nwith thicknesses a∼1/k⊥in the MHD regime k⊥ρi/lessorsimilar1. We find that the growth rate of\nreconnection relativeto the turbulent fluctuation frequencies γRXN/ωis a function ofthe\nisotropic driving wavenumber k0ρi, ion plasma beta βi, and ion-to-electron temperature\nratioτ, as well as the ion-to-electron mass ratio µ.\nIn general, the normalized collisionless tearing instabilty growth rate sγRXN/ω—given\nby (5.31)inthe∆′δin≪1limitandby (5.35)inthe∆′δin/greaterorsimilar1limit—increaseas k0ρiand\nβidecrease. For specified values of the plasma parameters βiandτ, we obtain different\nconditions on the maxmimum value of k0ρineeded for reconnection to arise, given by\n(5.33) and (5.34) in the ∆′δin≪1 limit and by (5.38) and (5.39) in the ∆′δin/greaterorsimilar1\nlimit. These scalings mean that reconnection is more likely to arise in tur bulence with a\nvery large driving scale relative to the ion Larmor radius scale k0ρi≪1 and for low ion\nplasma beta condtions βi≪1.\nIt is worthwhile emphasizing that these estimates on the importance of magnetic re-\nconnection in plasma turbulence are calculated only if for current sh eets with a thickness\na∼1/k⊥in the MHD regime k⊥ρi≪1 so that the MHD scaling predictions in §3 ap-\nply. It is also possible that electron-only magnetic reconnection (Ph anet al.2018) could\noccur with the range of scales in the kinetic dissipation range k⊥ρi≪1, but to estimate\nwhether this will occur would require the use of modified turbulence s calings appropriate\nfor the dissipation range (Howes et al.2008a, 2011).\nFinally, how the incorporation of magnetic reconnection into the tur bulent cascade im-\npacts the damping of the turbulence and the resulting energization of the plasma remains\nan open question. Collisionless magnetic reconnection effectively con verts magnetic en-\nergy into other forms, specifically the kinetic energy of the bulk plas ma outflows and\npossibly nonthermal energy in the particle velocity distribution func tions (Drake et al.\n2005;Egedal et al.2008,2012;Drake & Swisdak2012;Loureiro et al.2013;Jiansen et al.\n2018; Pucci et al.2018; Mu˜ noz & B¨ uchner 2018; Dahlin 2020; McCubbin et al.2022).\nBut, the linear terms in the equations of evolution for the turbulent plasma mediate the\nongoing oscillation of energy from magnetic energy to the kinetic ene rgy of the turbulent\nbulk plasma motions and back (Howes 2015). For example, the physic s of undamped\nAlfv´ en waves involve the transformation of the bulk kinetic energy of the perpendicular\nwave motions to magnetic energy as the frozen-in magnetic field is st retched; magnetic\ntension serves as the restoring force for the wave, decelerating the perpendicular wave\nmotions and ultimately re-accelerating them back towards the stat e where the magnetic\nfield is not bent, thereby converting magnetic energy back to kinet ic energy. Thus, col-\nlisionless magnetic reconnection does not necessarily lead directly to damping of the50\n10 −2 10 −1 10 2\n10 −4 \n10 −5 iβ\n10 −4 10 −3 10 −2 10 −1 10 210 310 410 610 5\nλmfp,ek||0 λmfp,ek||T110 \n1 10 CoronaSolar\nLAPD\nExptHeliosphereInnerFluid\nRegimeKinetic Regime\nIntraclusterMedium\nGalacticCenterOuter Heliosphere\nJET FusionExptNear−EarthSolar WindMagneto−\nsheath\nExtent of Inertial Range from                to\nInertial Range \nEnergy CascadeEnergy\nInjection10 \nFor each box−3 \nFigure 10. Phase Diagram of the extent of the turbulent inertial range f rom driving scales at\nk/bardbl0λmfp,eto the transition to the dissipation range at k/bardblTλmfp,evs. the ion plasma beta βi.\nThe plot indicates that, at the small scale end of the inertia l range where damping mechanisms\ncan remove energy from the turbulence, for many plasma syste ms of interest, the governing\ndynamics are weakly collisional with t k/bardblTλmfp,e≫1, so kinetic plasma theory is necessary to\ncapture the physics of turbulent dissipation.\nturbulent motions, but rather is simply a channel, in additon to magne tic tension, for\nconverting magnetic to kinetic energy. The triggering of magnetic r econnection, however,\nwill impact the scaling of the three-dimensional wave vector of the t urbulent motions\nthat result from the reconnection flow (Boldyrev & Loureiro 2017; Loureiro & Boldyrev\n2017b,a;Malletet al.2017b,a;Walker et al.2018), andthesedifferentscalingsmaychan-\nnel energy into different kinetic physical mechanisms that transfe r turbulent energy into\nparticle energy.\n6. Phase Diagrams for Plasmas Turbulence\nThe scalings of the different proposed kinetic dissipation mechanisms listed in §5.1\nenable us to map the importance of the different mechanisms for the damping of turbu-\nlence as a function of the fundamental plasma parameters by crea tingphase diagrams for\nplasma turbulence , analogous to a similar effort to create a phase diagram for magnetic\nreconnection (Ji & Daughton 2011). Here we present two such ph ase diagrams based on\nthe plasma and turbulence parameters estimated for different spa ce and astrophysical\nplasmas.Fundamental Parameters of Turbulence 51\n6.1.Collisionality of the Dissipation Range of Plasma Turbulen ce\nWe first create a phase diagram indicating that, for many plasmas of interest, even if\nthe turbulence is driven at fluid scales with k/bardbl,0λmfp,e≪1, the end of the inertial\nrange generally occurs at scales that are weakly collisional. Therefo re, the kinetic damp-\ning mechanisms analyzed here are responsible for removing energy f rom the turbulent\ncascade, rather than viscosity and resistivity, which are based on microscopic Coulomb\ncollisions between particles (Braginskii 1965; Chapman & Cowling 1970 ). Since the mean\nfree path must be compared to the parallel length scales of the tur bulent fluctuations\ncharacterized by k/bardbl, we must first determine the parallel length scales associated with\nthe small-scale end of the inertial range at the transition to the diss ipation range, cor-\nresponding to a perpendicular scale k⊥ρi∼1. Specifying the GS95scaling with α= 0\nfor this example, the parallel wavenumber at the transiton to the d issipation range k/bardblT,\ndetermined by (3.1), is given by k/bardblTρi= (k0ρi)1/3(k⊥ρi)2/3= (k0ρi)1/3, here taking\nk⊥ρi∼1 as the end of the inertial range. For simplicity, we assume here tha t the tur-\nbulence is driven strongly and isotropically with χ0= 1 and k/bardbl0/k⊥0= 1. Normalizing\nthis parallel scale in terms of the electron collisional mean free path λmfp,e, we obtain\nk/bardblTλmfp,e= (k0ρi)1/3(λmfp,e/ρi).\nThe range of plasma and turbulence parameters for a number of pla sma systems of\ninterest are denoted by the colored boxes in Fig. 10. Plotted on the horizontal axis of\nFig. 10 is the range of parallel scales bounding the turbulent inertial range normalized\nto the electron collisional mean free path λmfp,e: (i) the normalized parallel driving scale\nk/bardbl0λmfp,eof the turbulence is represented by the left end of each box; (ii) th e turbulent\nenergy cascades to smaller scales to the right within each box (wavy red arrow below\nplot); and (iii) the parallel scale correspondingto the small-scale end of the inertial range\natk/bardblTλmfp,eisrepresentedbytherightendofeachbox.Theverticalextento fthecolored\nboxes indicates the range of ion plasma beta βifor each of the chosen systems. The fluid\nregime (yellow region) indicates conditions under which the dynamics o f the turbulent\nfluctuations occur on parallel scales that are collisional with k/bardbl0λmfp,e≪1; the kinetic\nregime (light blue region)indicates conditions under which the dynamic s of the turbulent\nfluctuations occur on parallel scales that are weakly collisional with k/bardbl0λmfp,e≫1.\nIn Fig. 10 are shown these parameter ranges for the near Earth s olar wind (red), the\nEarth’s magnetosheath and the solar corona(green), the inner h elisophere at heliocentric\nradiiR≪1 AU and the outer helisophere at R≫1 AU (blue), the Galactic Center and\nintracluster medium (black), and terrestrial laboratory experime nts including the Large\nPlasma Device (LAPD)at UCLA (Gekelman et al.1991, 2016) and the Joint European\nTorus (JET) Fusion experiment (JET Team 1992). The main message of Fig. 10 is that,\nfor most turbulent plasma systems of interest, the parallel length scales associated with\nthe end of the inertial range correspond to weakly collisional plasma conditions with\nk/bardblTλmfp,e≫1. Therefore, to explore the physical damping mechanisms respon sible for\nremoving energy from the turbulent fluctuations and consequent ly energizing the plasma\nparticles, plasma kinetic theory is essential.\n6.2.Turbulent Damping Mechanisms on the (βi,k0ρi)Plane\nA single phase diagram that demonstrates the power of using the dim ensionless plasma\nand turbulence parameters for the isotropic temperature case p resented in Table 3 is a\nplotofthepredictedregionsonthe( βi,k0ρi)planewheredifferentkineticmechanismsare\nexpeected to play a role the damping of weakly collisional plasma turbu lence, presented\nhere in Fig. 11.\nFor this phase diagram, we specify the ion-to-electron temperatu re ratioτ= 1 and52\nFigure 11. Phase diagram for the kinetic damping mechanisms of weakly c ollisional plasmas\nturbulence as a function of isotropic driving wavenumber k0ρiand ion plasma beta βi, showing\nregions ofthis parameter space where differentmechanisms a re likelytocontributetotheheating\nof the plasma species: ion Landau damping (iLD) and ion trans it time damping (iTTD) (blue);\nelectron Landau damping (eLD) and electron transit time dam ping (eTTD) (blue); ion cycltron\ndamping (iCD) (black); ion stochastic heating (iSH) (green ); ion kinetic viscous heating (iVH)\n(magenta); and collisionless magnetic reconnection (RXN) (red) for intermittent current sheets\n(n= 1, red dashed) and sinusoidal current sheets ( n= 2, red solid). The extent of the turbulent\nnear-Earth solar wind on the ( βi,k0ρi) plane is indicated (gray shading).\nthe ion-to-electron mass ratio µ= 1836. For the isotropic temperature case, the species\ntemperature anisotropies Ai=Ae= 1, and we take the turbulence to be driven strongly\n(χ0= 0) and isotropically ( k/bardbl0/k⊥0= 1) so that the turbulent driving can be character-\nized by the isotropic driving wavenumber k0ρi. Furthermore, we assume the turbulence\ndriving to be balanced ( Z+\n0/Z−\n0= 1) and incompressible ( Ecomp/Einc= 0). In addi-\ntion, the turbulence is assumed to be weakly collisional with k/bardbl0λmfp,e≪1, eliminating\nmagnetic pumping as a potential turbulent damping mechanism. Finally , we specifically\nchoose to use the B06scaling for MHD turbulence with α= 1 to assess the contributions\nof the different proposed damping mechanisms.\nWe use the dependencies discussed in §5 of the proposed turbulent damping mech-\nanisms on the dimensionless parameters in Table 3 to assess the regio ns of the two-\ndimensional ( βi,k0ρi) parameter space over which each mechanism is expected to con-\ntribute.Fundamental Parameters of Turbulence 53\nFor the Landau-resonant( n= 0) collisionless damping mechanisms—Landau damping\nand transit-time damping—the primary parameter dependencies su mmarized in §5.4 are\nβifor ion Landau damping (iLD) and ion transit-time damping (iTTD) and ( βi,τ,µ) for\nelectron Landau damping (eLD) and electron transit-time damping ( eTTD). Since the\nsum of LD and TTD for each species is always leads to damping for isotr opic equilibrium\ntemperatures, but seperately the net energy transfer by eithe r mechanism to that species\nover a full wave period can be positive or negative—as discussed in §5.4 and illustrated\nin Fig. 7— we do not separate the contributions LD and TTD for a single species.\nFurthermore, since any turbulent cascade energy not deposited onto the ions by iLD\nand iTTD at the ion scales k⊥ρi∼1 will cascade to k⊥ρi≫1 and ultimately be\ntransferred to the electrons, the partitioning of energy betwee n ions and electrons by\nthese Landau-resonant mechanisms is simply a function of βi. Solutions of the linear\nVlasov-Maxwell dispersion relation for the normalized ion damping rat e by iLD and\niTTD, (γiLD+γiLD)/ωindicate that ion damping is non-neglible for βi/greaterorsimilar0.5, so iLD\nand iTTD will contribute in this region (blue line with right arrow), indepe ndent of the\ndrivign scale k0ρi. Forβi/lessorsimilar0.5, eLD and eTTD will dominate the Landau-resonant\ndamping (blue line with left arrow), but eLD and eTTD will still contribut e to terminate\nthe remaining turbulent cascade energy not removed by ions (blue r ight arrow).\nAs discussed in §5.5, ion cyclotron damping (iCD) will contribute to the damping of\nturbulence if the turbulent frequencies approach the ion cyclotro n frequency ω/Ωi→1,\nwith the scaling of this ratio given by (5.9). We can solve this equation f ork0ρiin terms\nofβiwithα= 1 for the B06scaling, yielding\nk0ρi=/parenleftbiggω\nΩi1\nω/parenrightbigg2\nβi. (6.1)\nTaking the factor ( ω/Ωi)/ω= 1 gives an estimate for when ω/Ωi= 1 (black dotted line),\nbut iCD often becomes significant at lower values of the frequency, ω/Ωi/lessorsimilar1. Thus,\nwe estimate the onset of iCD by taking ( ω/Ωi)/ω= 1/2π(black solid line and upward\narrow).\nTheimpactofionstochasticheating(iSH)onturbulentfluctuations fortheB06scaling\nis given by (5.13) in §5.6. The exponential dependence in this equation for γSH/ωyields\na strong inhibition of ion stochastic heating unless c2β1/2\ni(k0ρi)−1/4/lessorsimilar1. Thus, we obtain\na condition for k0ρiin terms of βifor the onset of iSH, given by\nk0ρi/greaterorsimilarc4\n2β2\ni. (6.2)\nNote that the dependence of this condition on c4\n2means that an accurate determination\nof this constant is critical to assess the role of iSH on the damping of plasma turbulence.\nTaking the value of c2= 0.34 from Chandran et al.(2010), we plot (6.2) for the onset of\nion stochastic heating (green line and upward arrow) on Fig. 11.\nKinetic viscous heating of ions (iVH) mediated by temperature anisot ropy instabilities\na high values of ion plasma beta βi≫1 depends on the βiand the amplitude of the\nturbulent fluctuations at the driving scale, as shown by (5.23) in §5.8. Because this\nmechanism only contributes when βiis sufficiently high that the temperature anisotropy\nexceeds the threshold values for the proton temperature anisot ropy instabilities, we take\nthis kinetic damping mechanism as dependent only on the value of the io n plasma beta,\nwith an estimated onset at βi/greaterorsimilar10 (magenta line and right arrow).\nFinally, if the driving scale k0ρi≪1 and the ion plasma beta βi≪1 of the turbulence\nare sufficiently low, current sheets with a large width-to-thickness ratio in the plane\nperpendicular to the equilibrium magnetic field can be unstable to a rap id collisionless54\ntearing instability in the large-guide-field limit, as detailed in §5.9. If growth rate of the\ntearing instability is faster than the rate of the nonlinear cascade o f energy to small\nscales by the turbulence γRXN/ωnl/greaterorsimilar1, collisionless magnetic reconnection can disrupt\nthe turbulent cascade. The condition that the normalized collisionles s tearing growth\nrates satisfy γRXN/ω/greaterorsimilar1 occurs for the maximum growth rate of the tearing instability\nisgivenbyeither(5.33) or(5.38)(both expressionsyieldthat samem aximumvalueatthe\ntransitional value of ∆′δin= 1). In these expressions, n= 1 corresponds to a Harris-like\ncurrent sheet with a hyperbolic tangent profile, and n= 2 corresponds to a sinusoidal\nvariation of the reconnection magnetic field. The determination of t he tearing growth\nrate demands k⊥ρi≪1 for the validity of this linear tearing growth rate calculation, so\nwe takek⊥ρi→ F, whereF ≪1 is the maximum value of the normalized perpendicular\nwavenumber for which the growth rate derviations remain valid. Her e, we specify a value\nF= 0.3 for our estimation of the onset of collisionless magnetic reconnect ion, plotting\nthe boundaries for reconnection in intermittent current sheets ( n= 1) (dashed red line\nand downward arrow) and for reconnection in sinusoidal current s heets (n= 2) (solid red\nline and downward arrow). Note that the dependence of k0ρionFin both (5.33) and\n(5.38) is very strong, scaling as F9forn= 1 and as F7forn= 2, so the determination of\nthe maximal value of k⊥ρi≡ Ffor the tearing instability growth has a strong influence\non whether reconnection will arise.\nThe phase diagram for the turbulent damping mechanisms on the ( βi,k0ρi) plane in\nFig. 11 can be used in the following manner to predict which damping mec hanisms will\ncontribute to the dissipation of turbulence in a specific space or ast rophysical plasma\nsystem. In the near-Earth solar wind, the observed ion plasma bet a values span the\nrange 0.05/lessorsimilarβi/lessorsimilar10 (Wilson et al.2018), taking a 5% and 95% limit on the distribution\nofβivalues. Furthermore, the isotropic driving wavenumberin the near -Earthsolar wind\nspans the typical values 10−4/lessorsimilark0ρi/lessorsimilar10−3(Tu & Marsch 1990, 1995; Matthaeus et al.\n2005; Howes et al.2008a; Kiyani et al.2015). This rangeof parametersfor the solarwind\nat 1 AU is plotted as the gray shaded region on Fig. 11. Reading the ph ase diagram,\nthe prediction for the damping mechanisms in the near-Earth solar w ind is that Landau-\nresonantdampingforions(iLDandiTTD) andelectrons(eLDandeTT D)willcontribute\nto the damping overthe full rangeofsolarwind turbulence paramet ers.At lowervaluesof\nβi/lessorsimilar0.3 and higher values of k0ρi, stochastic heating is predicted to play a role. Indeed,\nin the Earth’s magnetosheath (which has a similar footprint on the ( βi,k0ρi) plane as\nthe near-Earth solar wind), there is direct evidence from Magnetospheric MultiScale\nspacecraft observations of electron Landau damping playing a sign ificant role in the\ndamping ofthe turbulent fluctuations (Chen et al.2019; Afshari et al.2021). In addition,\nin the solar wind in the inner heliosphere, where the values of βitend to be lower\nthan in the near-Earth solar wind, analysis of observations from th eHeliosspacecraft\n(Martinovi´ c et al.2019) and Parker Solar Probe spacecraft (Martinovi´ c et al.2020) find\nevidence for ion stochastic heating in the turbulent plasma. Thus, p hase diagrams like\nFig. 11 can play a valuable role in predicting the mechanisms that will con tribute to the\ndamping of weakly collisional plasma turbulence in space and astrophy sical plasmas.\n7. Conclusions\nTurbulence is a ubiquitous phenomenon that arises in space and astr ophysical plasmas\nand mediates the transfer of the energy of large-scale plasma flow s and electromagnetic\nfields to sufficiently smallscalesthat dissipationmechanisms can conv ertthat energyinto\nthe microscopicenergyofthe plasma particles.A grandchallengepr oblem in heliophysics\nand astrophysics is to predict the heating or acceleration of the diff erent plasma speciesFundamental Parameters of Turbulence 55\nby the turbulence in terms of the dimensionless parameters that ch aracterize the plasma\nand the nature of the turbulence. In particular, a long-term goal is to develop predictive\nmodels of the turbulent plasma heating that determine the partition ing of energy be-\ntween the ion and electron species, Qi/Qe, and between the perpendicular and parallel\ndegrees of freedom for each species, Q⊥,i/Q/bardbl,iandQ⊥,e/Q/bardbl,e. An essential step in the\ndevelopment of such predictive turbulent heating models is to identif y the microphysical\nkinetic processes that govern the transfer of the turbulent ene rgy to the particles as a\nfunction of the dimensionless parameters,information that can be summarized in a phase\ndiagram for the dissipation of plasma turbulence, as shown in Fig. 11.\nThe first step in this long term effort is to specify a set of the key dime nsionless\nparameters upon which the turbulent energy cascade and its dissip ation depend. In this\npaper, we propose a specific set of ten dimensionless parameters t hat characterize the\nstate of the plasma and the nature of the turbulence,\n(βi,τ,Ai,Ae,k/bardbl0λmfp,e;k⊥0ρi,k/bardbl0/k⊥0,χ0,Z+\n0/Z−\n0,Ecomp/Einc),\nsummarized in Table 2. Although this set of ten parameters is sufficien t to characterize a\nwide variety ofturbulent space and astrophysicalplasmas,the de velopmentof completely\ngeneral turbulent heating models on a ten-dimensional parameter space is unlikely to be\nsuccessful. Fortunately, we can capture many of the dominant de pendencies of the turbu-\nlence and its damping mechanisms by adopting of reduced set of para meters in the limits\nthat (i) the equilibrium velocity distribution for each species is an isotr opic Maxwellian\nvelocity and (ii) the turbulence is driven at sufficiently large scale or su fficiently strongly\nthattheturbulentfluctuationsatthe small-scaleendoftheinertia lrangek⊥ρi∼1satisfy\na state of strong turbulence, χ∼1. With these idealizations to the isotropic temperature\ncase, there remain just three dimensionless parameters\n(βi,τ;k0ρi),\nsuumarized in Table 3.\nA number of kinetic mechanisms have been proposed to govern the d amping of weakly\ncollisional plasma turbulence, as enumerated in §5.1. The critical advance presented in\nthis paper is to express the dependence of each of these mechanis ms on the same set\nof fundamental plasma and turbulence parameters . The results of this analysis of the\ndependencies for each of the turbulent damping mechanisms is summ arized in Table 4.\nFinally, the scalings of each of the turbulent damping mechanisms with the isotropic\ndriving wavenumber k0ρiand the ion plasma beta βican be synthesized to generate\nthe first phase diagram for turbulent damping mechanisms as a func tion of (βi,k0ρi),\npresented in Fig. 11. For a chosen space or astrophysical plasma s ystem, such as the\nnear-Earth solar wind, the range of dimensionless plasma and turbu lence parameters\ncan be identified on plotted on the phase diagram, as illustrated by th e gray shaded\nregion in the figure. One may then read off which mechnisms are predic ted to contribute\nto the damping of the turbulence in that system. Kinetic numerical s imulations and\nspacecraft observations of weakly collisional turbulence in space o r astrophysicalplasmas\ncan be used to update and refine this phase diagram, or to extend it s applicability as\nadditional dimensionless parameters are varied, such as the ion-to -electron temperature\nratioτ=Ti/Te.\nUlimately,thegoalistoproducemoreaccurateandcompletepredic tiveturbulentheat-\ning models than those reviewed in §4, facilitating improved global modeling of important\nturbulent space and astrophysical plasmas, such as the mesosca le and macroscale en-\nergy transport through the heliosphere or the interpretation of Event Horizon Telescope\nobservations of accretion disks around supermassive black holes.56\nAcknowledgements\nMorethan fifteen yearsofturbulent modelingefforts aresynthes izedhere, havingbene-\nfitted from interactions overmany yearswith Steve Cowley, Bill Dor land, Eliot Quataert,\nAlex Schekochihin, Greg Hammett, Tomo Tatsuno, Ryusuke Numata , Ben Chandran,\nStuart Bale, Kevin Nielson, JasonTenBarge,Jan Drake, Fred Skiff, CraigKletzing, Steve\nVincena, TroyCarter,ChadiSalem, KristopherKlein, ChrisChen, J im Schroeder,Daniel\nTold, Frank Jenko, Foaud Sahraoui, Tak Chu Li, Suranga Ruhunusir i, Jasper Halekas,\nSofiane Bourouaine, Jaye Verniero, Matt Kunz, Andrew McCubbin, Francesco Valentini,\nRobert Wicks, Daniel Verscharen, Lloyd Woodham, Sarah Conley, A rya Afshari, Peter\nMontag, Daniel McGinnis, Rui Huang, and Alberto Felix.\nFunding\nSignificant funding supporting this long-term was provided by NSF gr ants PHY-\n10033446, CAREER AGS-1054061 and AGS-1842561, DOE grant DE -SC0014599, and\nNASA grants NNX10AC91G, 80NSSC18K0643,80NSSC18K1217,an d 80NSSC18K1371.\nREFERENCES\nAdhikari, L., Zank, G. 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J.\n771, 124.\nZocco, A. & Schekochihin, A. A. 2011 Reduced fluid-kinetic equations for low-frequency\ndynamics, magnetic reconnection, and electron heating in l ow-beta plasmas. Phys. Plasmas\n18(10), 102309."    },    {        "title": "2402.12832v5.Nearly_Optimal_Fault_Tolerant_Distance_Oracle.pdf",        "content": "arXiv:2402.12832v5  [cs.DS]  3 Apr 2024Nearly Optimal Fault Tolerant Distance Oracle\nDipan Dey\nIIT Gandhinagar\nGandhinagar\nIndia\ndey_dipan@iitgn.ac.inManoj Gupta\nIIT Gandhinagar\nGandhinagar\nIndia\ngmanoj@iitgn.ac.in\nAbstract\nWe present an f-fault tolerant distance oracle for an undirected weighted graph where each edge\nhas an integral weight from [1...W]. Given a set Foffedges, as well as a source node sand a\ndestination node t, our oracle returns the shortest path from stotavoiding FinO((c flog(nW))O(f2))\ntime, where c>1 is a constant. The space complexity of our oracle is O(f4n2log2(nW)). For a\nconstant f, our oracle is nearly optimal both in terms of space and time ( barring some logarithmic\nfactor).\n1 Introduction\nIn real-life networks, which can be represented as graphs, d etermining the distance between any two\nvertices is often necessary . This problem is commonly known as the “all-pair shortest path” problem\nin the literature. In the data structure version of this prob lem, the goal is to preprocess the graph to\nanswer distance queries between any two vertices efficientl y .\nQUERY(s,t): Find the weight of the shortest path from stot.\nFor unweighted graphs, the shortest path between all pairs o f vertices can be found using the\nbreadth-first search (BFS) algorithm, and the information c an be stored in O(n2)space. The query\ntime is O(1). In the case of weighted graphs, there is extensive research on all-pair shortest path\nalgorithms[Flo67 ,War62 ,Fre76 ,Dob90 ,Tak92 ,Han04 ,Tak04 ,Zwi04 ,Tak05 ,Cha05 ,Han06 ,Cha07 ,\nHan08 ,Cha08 ,Wil14]. Using the state-of-the-art algorithm, we can construct a d ata structure of size\nO(n2)with O(1)query time.\nHowever, real-life networks are prone to failures. Thus, we need to find the distance between\nvertices, avoiding certain edges or vertices, known as faul ts in the literature. This paper focuses on\nedge faults. Let’s formally de��ne the problem. We are given a n undirected weighted graph G. We can\npreprocess Gand construct some suitable data structures. These data str uctures are used to answer\nqueries of the form:\nQUERY(s,t,F): Find the shortest path (and its weight) from stotavoiding the edge set F.\nThe algorithm that handles the aforementioned query is know n as the query algorithm. It utilises\nthe prepared data structures to respond to the query . This co mbination of data structures and the query\nalgorithm is referred to as a distance oracle in the literature. Since we are dealing with faults, our orac le\nis known as f-fault-tolerant distance oracle when|F|≤f. A fault-tolerant distance oracle is evaluated\nbased on its size, query time (the time required to answer a qu ery), and the number of faults it can\nhandle, i.e., the maximum size of F. In a naive approach, we could store all distances, avoiding allFaults Space Query time Remarks Reference\n1 ˜O(n2) O(1) ˜Ohides polylog nfactor. [DTCR08]\n2 ˜O(n2) O(logn) - [DP09]\nf ˜O(n3−α) ˜O(n2−(1−α)/f)α∈[0,1]when the preprocess-\ning time is O(W n3.376−α)and edge\nweights are integral and in the range\n[−W...W].[WY13]\nf ˜O(W n2+α) O(W n2−αf2+W nfω)α∈[0,1], edge weights are integral\nand in the range [−W...W]andω\nis the matrix multiplication exponent\n[CW87 ,Sto10 ,Wil12 ,Gal14 ,AW21][vdBS19]\nf O(n4) O(fO(f)) - [DR22]\nf O(n2) ˜O(nf) Edge weights are in the range\n[1 ...W][KS23]\nTable 1: Relevant results for fault-tolerant distance orac le\npossible faults and return the distance upon query . However , this would consume significant space,\nalbeit with a small query time. Hence, the primary research c hallenge in this field is to reduce the\nspace while not significantly increasing the query time.\nWe summarise relevant results in this area in Table 1 . When f=1, the oracle presented in\n[DTCR08]is optimal. For f=2, Duan and Pettie [DP09]designed a nearly optimal distance ora-\ncle (barring some polylog nfactor). However, for f>2, our understanding of this problem is lim-\nited. The results can be categorised into two groups: those w ith high query time but low space\n[WY13 ,vdBS19 ,KS23], and those with low query time but high space [DR22]. An important ques-\ntion in this field is to develop an oracle that minimises both s pace (as in[KS23]) and query time (as in\n[DR22]). In fact, this question is explicitly raised in [DR22], and we quote it here (with minor notation\nchanges).\n“Problem 1: Fix a large constant f . Is there an f -failure oracle for handl ing exact dis-\ntance queries in undirected graphs with query time poly( logn,logW) and a reasonable size\nbound?”[DR22]\nWe answer this question in the affirmative. The main result of our paper is:\nTheorem 1.1. There is an f -fault-tolerant distance oracle that takes O (f4n2log(nW))space and has a\nquery time of O(c(f+1)2f8(f+1)2log2(f+1)2(nW)), where c>1is a constant. Moreover, the oracle returns a\npath in(f+1)-decomposable form.\nIn the above theorem, we have used the term “(f+1)-decomposable form” . We will describe this\nterm now without delving into technical details. Most of the existing fault-tolerant distance oracles\nreturn the weight of the shortest path, not the path itself. I f they do return the shortest path, their\nquery time becomes Ω(n). However, our algorithm returns the shortest path while avo iding Fino(n)\ntime. One might wonder if the output size could be as large as O(n), how can we claim a query time\nindependent of n. We will demonstrate that the shortest path from stot, avoiding F, can be represented\nin a suitable form of size O(f). This form is termed as (f+1)-decomposable form. We will formally\ndefine this term in Section 2 .\nFor a fixed f, compared to[DR22], our oracle has better space and compared to [WY13 ,vdBS19 ,\nKS23], it has a better query time. In fact, for a fixed f, our oracle is nearly optimal both in terms of space\nand time (barring some polylog (nW)factor). Also note that the results in [vdBS19 ,KS23]have a linear\n2dependency on Win terms of space and /or query time, while our result has a polylog dependency on\nW. One drawback of our result compared to [WY13 ,vdBS19]is that it works for graphs with weights\n[1 ...W]as opposed to[−W...W]. Another drawback is that the preprocessing time (approxim ately\nΩ(nf)) of our oracle is huge.\n1.1 Related works\nSingle fault tolerant distance oracle, f=1:Demetrescu et al. [DTCR08]designed a single fault toler-\nant distance oracle in O(mn2+n3logn)construction/preprocessing time (the time required to construct\nthe oracle before taking any query) with ˜O(n2)space and O(1)query time. Bernstein and Karger [BK08]\nimproved the preprocessing time to ˜O(n2/radi⌋allowm)and later to ˜O(mn)time in[BK09a]. The preprocessing\ntimes of the above results are Ω(n3). Grandoni and Williams [GW19]designed the first single fault\ntolerant distance oracle with subcubic preprocessing time and sublinear query time. Chechik and Co-\nhen[CC20]kept the preprocessing time as subcubic but improved the que ry time to ˜O(1). After that,\nGu and Ren[GR21]constructed a single fault tolerant distance oracle with O(n2.6146M)preprocessing\ntime and constant query time where Mis the maximum edge weight.\nFault tolerant approximate distance oracle: Another direction of work is approximate fault tol-\nerant distance oracle. Here, instead of reporting the exact distance, the oracle returns the approxi-\nmate distance between two vertices in the presence of faults . Baswana and Khanna [BK13]designed\n(2k−1)(1+ε)-approximation distance oracle with O(kn1+1/klog3n/ε4)(where k>1). For multiple\nfaults, Chechik et al. [CCFK17]designed a(1+ε)-approximate distance oracle which handled upto\nf=o(logn\nlog log n)many faults. However, their algorithm had a super quadratic space requirement. Later,\nBilò et al.[BCC+23]designed a(3+ε)-approximate distance oracle in sub-quadratic space and su blinear\nquery time.\n2 Preliminaries\nWe are given an undirected weighted graph Gwhere the weight of an edge is integral and is in the range\n[1 ...W]. For an edge e,w(e)denotes the weight of e. The notation strepresents the shortest path from\nvertex sto vertex t, and|st|denotes its weight. For an edge e=(u,v),|se|is defined as the minimum\nof|su|and|sv|. Let Pbe an arbitrary path from stot. We write P[s,t]to indicate that the path starts\nfrom sand ends at t.P[t,s]is defined symmetrically . When considering an arbitrary pat hP, in which\nvertex uappears before vertex v,P[u,v]refers to the subpath from utov, and|P[u,v]|represents its\nweight. Given two paths P[u,v]andQ[v,w],P⊙Qis the concatenation of Pwith Q. The notation\nst⋄Fdenotes the shortest path from stot, avoiding a set of edges F, and|st⋄F|corresponds to its\nweight. For an edge e=(u,v),se⋄F:=arg min{|su⋄F|,|sv⋄F|}andes⋄F:=arg min{|us⋄F|,|vs⋄F|}.\nV(F)denotes the set of vertices with endpoints in F. For any path P,(P)2:=2⌊log|P|⌋.\nWe will assume that the shortest path between any two vertice s in the graph is unique, even under\nedge failures. Uniqueness can be achieved by adding suitabl e small weights to each edge (e.g., see\n[PP13]) – we can perturb the edge weights such that any two paths betw een a pair of vertices have\ndifferent weights. This approach has also been used in [BK09b ,HS01 ,GS18 ,DR22].\nFor any vertex s,Tsdenotes the shortest path tree from s.Ts(x)represents the subtree rooted at\nnode xinTs. For each s, we will store Ts. This takes O(n2)space for all the vertices in the graph. After\nconstructing Ts, we can preprocess it such that it takes O(1)time to find the shortest path from sto any\nother vertex in the graph.\nFor each 0≤i≤log(nW), we will find the first vertex on the stpath that is at a distance ≥2ifrom\ns. We can pre-process Tsto store this information for every t∈G. This additional data structure takes\nO(n2log(nW))space over all shortest path trees. Thus, we will assume that we can find the first vertex\nthat is at a distance ≥2ion the stpath in O(1)time.\nWe can extend the above approach to find the 2i-th vertex not only on a single path but on a\ncollection of shortest paths. To this end, we define the follo wing:\n3Definition 2.1. A path is termed f -decomposable if it is a concatenation of at most f shortest paths\ninterleaved with at most f −1edges.\nThe shortest paths in the graph are 1-decomposable. If a path Pisf-decomposable, then it can be\nrepresented as L1⊙e1⊙L2⊙e2⊙...⊙ek−1⊙Lkwhere each Liis a shortest path and k≤f. Also,\nthe interleaving edges may or may not exist. Thus, each eiis either an edge or may be empty . It takes\nonly O(f)space to represent Pin the above form, making it easy to process. In our algorithm s, we will\nalways output paths in the above form. Let us define one more te rm to formalise this notion:\nDefinition 2.2. If a path P is represented as L1⊙e1⊙L2⊙e2⊙...⊙ek−1⊙Lk, where each Li(where\n1≤i≤k) is a shortest path, each eiis either empty or is an edge and f ≥k, then we say that P is in an\nf -decomposable form .\nIfPis in f-decomposable form, it becomes easy to process it, as can be s een from the next lemma:\nLemma 2.3. Let P[s,t]be a path in f -decomposable form. Then, the first vertex at a di stance≥2ifrom\ns on P can be found in O (f+log(nW))time, where 0≤i≤lognW.\nThis is an easy algorithmic problem, and we prove it in Appendix A .\n3 Overview\nWe begin with a fundamental result in this field.\nTheorem 3.1. ([ABK+02, Theorem 2]) Let G be an undirected weighted graph. For any F ∈Ef, st⋄F is\n(f+1)-decomposable.\nConsequently , st⋄Fcontains at most f+1segments , each of which is a shortest path.\nDefinition 3.2. (Intermediate vertex) A vertex x ∈st⋄F is called an intermediate vertex if it does not lie\non the first and last segment of st ⋄F.\nA straightforward strategy for finding st⋄Fis to first locate an intermediate vertex xon it and then\nrecursively search for sx⋄Fandx t⋄F. Both sx⋄F,x t⋄Faref-decomposable, reducing the problem\nto a smaller subproblem. The key challenge is finding an inter mediate vertex. But how to find xif you\ndon’t even know st⋄F? To solve this, we will try to find a “hitting set” Hthat will satisfy two properties:\n(I) The size of His polynomial in f.\n(II) There exists a vertex in Hthat lies on st⋄Fand serves as an intermediate vertex.\nThere is a simple hitting set that satisfies (I)but only partially satisfies (II). Consider a set of edges\nF∗of size fwhose removal maximises the distance between sand t. We can show that either (i)\nst⋄F∗=st⋄For (ii) an edge of F∗lies on the path st⋄F. In case (i), the answer is essentially the\nsame as the output of the maximiser. However, in case (ii), we find a small set V(F∗)of size O(f)that\nintersects st⋄F. Unfortunately , this set may include vertices from the first or last segments of st⋄F,\nmaking this maximiser less useful.\nIn[DR22], Ren and Duan developed an algorithm that uses different max imisers. Using these\nmaximisers, they identify a “hitting set” Hof size O(f3)containing an intermediate vertex. Although\nwe won’t delve into their approach in detail, their maximise r has size O(n4), and it is not clear how to\nreduce this space.\n3.1 A new approach\nIn this section, we describe our approach for handling a sing le fault in an unweighted graph. Let\nR=st⋄e, where e=(u,v). We assume vertex uis closer to salong the stpath. The path Rdeviates\nfrom the stpath at pand merges back at q(refer to Figure 1 ). The detour of Ris the path between p\nandqonR, orR[p,q].\nWe ask the following question: can we find vertices x,y∈Rsuch that\n4stpathst⋄e\ns p q t u v e x y\nFigure 1: Xis the set of green vertices and Yis the set of blue vertices. xsatisfies (a)and ysatisfies\n(b).\n(a) If uis the nearest vertex of V(F)from x, then|xp|<(xu)2, and\n(b) If vis the nearest vertex of V(F)from y, then|yq|<(yv)2\nIntuitively , xis closer to pthan the nearest faulty edge. Similarly , yis closer to qthan the nearest\nfaulty edge. We will see later that we can design maximisers u sing xandy, such that the output of the\nmaximiser not only hits Rbut also the detour of R. This increases our chances of finding an intermediate\nvertex. However, the challenge is finding xandywhen we don’t know R,p, and q.\nFor one fault, we will construct a small sets XandYthat contains xand y, respectively . Let\nX={x1=s,x2,... , xk=u}be vertices on stpath, where the distance between xi−1andxiis(xi−1u)2.\nFor instance, if|su|=2ℓ−1,x1=s,x2is at a distance 2ℓ−1from x1,x3is at a distance 2ℓ−2from x2\nand so on. Similarly , we create sequence Y={y1=t,y2,...}on the tspath, with distances between\nyi−1andyiequal to(yi−1v)2. Both sequences are of length O(logn).\nConsider a vertex x∈Xjust before pon the stpath. If|xp|≥(xu)2, then there exists another\nvertex of Xbetween xandponstpath, a contradiction. So |xp|<(xu)2and we satisfy (a). Similarly ,\nlety∈Ybe the vertex just before qon the tspath. Again,|yq|<(yv)2. Also, Rpasses through both\nxandy. Thus, we satisfy (a)and(b)andR=sx⊙x y⋄e⊙y t. InSection 6 , we demonstrate that it’s\nstraightforward to find x y⋄eusing the following maximiser:\n/Dbbb(x,y,d1=(xu)2,d2=(yv)2)=arg max\ne∗∈E/braceleftbig5\n|x y⋄e∗|such that|xe∗|≥d1and|ye∗|≥d2/bracerightbig5\nAs we don’t know xand yin advance, we try all possible pairs in X×Y. Consequently , the time\ncomplexity of our algorithm is O(log2n). Also, the size of our maximiser is O(n2log2n).\nNow, let’s generalise this approach to handle two faults. Su ppose R=st⋄{e1,e2}, with e1∈stand\ne2∈st⋄e1. Let pbe the vertex on Rfrom s, after which Rdeviates from st∪st⋄e. Symmetrically , define\nq. Let us assume that plies on the path st⋄e1and q lies on st(See Figure 2 ). Consider a sequence\nX={x1=s,x2,...}on the st⋄e1path, where distance between any xi−1andxiis minu∈V(F){(xi−1u)2}.\nSimilarly , we find a sequence Yontspath. Interestingly , even for two faults, we show that the si ze of\nXandYremains O(logn), which is a significant generalisation detailed in Section 5 .\nAs before, there exists vertices x∈Xand y∈Ysatisfying (a)and(b)respectively . Also, R=\nsx⊙x y⋄F⊙y t. We will demonstrate how to find x y⋄Fby appropriately utilising multiple maximisers.\nHowever, when multiple faults are present, finding x y⋄Fis a bit more involved and leverages ideas\nfrom[DR22]. Specifically , in Section 8 , we will show that we can identify a hitting set of size O(f3)\nthat contains an intermediate vertex of x y⋄F. This completes the overview.\n4 Organisation of the paper\nInSection 5 , we introduce a new tool “ Jump Sequence” which formalises th e new approach taken in\nthis paper. In Section 6 , as a warm-up, we apply our new approach to design a single fau lt tolerant\ndistance oracle in undirected weighted graph. In Section 7 andSection 8 , we extend our results to\nf-faults.\n5sp\nq te1e2x\nystpathst⋄e1st⋄{e1,e2}\nFigure 2: Xis the set of green vertices and Yis the set of blue vertices. xsatisfies (a)and ysatisfies\n(b).\n5 A new tool: Jump Sequence\nLetR=st⋄F′where F′⊆FandFis a set of edges of size f. Let ube a vertex in V(F). Consider the\nsequence of vertices x1=s,x2,x3,... , xkwhere, for 1≤i≤k−1,xiis the first vertex on path Rfrom\nxi−1satisfying\n|R[xi−1,xi]|≥max{1,(xi−1u⋄F)2} (1)\nImagine sitting at xi−1attached to a rope which starts at u. The rope’s length is (xi−1u⋄F)2,\nrepresenting how far we can jump on path R. The action of moving from xi−1toxiis called a jump\nfrom xi−1using u . Observe that xiis at a distance at least (xi−1u⋄F)2from xi−1along R. Ifxi−1u⋄F=0,\nthen we move to the vertex adjacent to xi−1onR.\n(xi−1u⋄F)2\nsxi−1 xi\ntu\nFigure 3: A jump from xi−1using u.\nNote that we can start the sequence from either sort, yielding different sequences. We denote\nJUMP(R[s,t])as the sequence starting from sand J UMP(R[t,s])as the other sequence. In this section,\nwe focus on J UMP(R[s,t]), and all our results apply to R[t,s]by symmetry .\nIn JUMP(R[s,t]), we advance along R[s,t]in steps of(x1u⋄F)2,(x2u⋄F)2, and so on. Note that if\nuis on the path R, we simply transition from uto its adjacent vertex on R. For a clearer presentation\nin the subsequent section, we’ll assume uis not on R, although our arguments remain valid even if uis\nonR.\nConsider the sequence {(x1u⋄F)2,(x2u⋄F)2,... ,(xk−1u⋄F)2}. We claim that this sequence roughly\nbehaves like a bitonic sequence. That is, we can divide it at a vertex, say xj, such that:\n(A1) In the sequence {(x1u⋄F)2,(x2u⋄F)2,... ,(xju⋄F)2}, the numbers decrease by at least a factor\nof two after at most four jumps. Here, the jump from x1tox2in JUMP(R[s,t])implies jumping\nfrom(x1u⋄F)2to(x2u⋄F)2in the above sequence.\n(A2) In the sequence {(xju⋄F)2,(xj+1u⋄F)2,... ,(xku⋄F)2}, the numbers increase by a factor of two\nafter every four jumps.\nThe above assertion implies the following lemma:\n6Lemma 5.1. The length of JUMP(R[s,t])is≤8 log(nW)\nWe now prove (A1). Let(x1u⋄F)2=2ℓ(for someℓ≥0). Consider the next four vertices after\nx1. If, for any i∈{2,3,4,5},(xiu⋄F)2≤2ℓ−1, then after at most 4 jumps, we find a vertex xiwhere\n(xiu⋄F)2decreases by atleast half. We repeat this process until we re ach a vertex, say xj, for which\nthis condition no longer holds. This proves (A1).\nNow, let’s prove (A2). Assume(xju⋄F)2=2ℓ. By construction, for all i∈{j+1,j+2,j+3,j+4},\n(xiu⋄F)2≥2ℓ. To prove (A2), we first need to demonstrate that (xj+4u⋄F)2≥2ℓ+1. To do so, we\nestablish the following stronger statement:\nLemma 5.2. Let(xju⋄F)2=2ℓand for all i∈{j+1,j+2,j+3,j+4},(xiu⋄F)2≥2ℓ. Then, for all\ni≥j+4p,(xiu⋄F)2≥2ℓ+pwhere p≥1.\nProof. We prove this lemma by induction on p.\nBase Case ( p=1):Fori≥j+4, assume for contradiction that (xiu⋄F)2≤2ℓ. By construction,\nR[xj,xi]≥4×2ℓbecause xj,xj+1,xj+2,xj+3,xj+4are consecutively at a distance ≥2ℓonR[s,t]. How-\never, there is an alternate path from xjtoxiavoiding F, which is xju⋄F⊙xiu⋄F. The sum of their\nweights is<2ℓ+1+2ℓ+1=4×2ℓ, contradicting the fact that R[xj,xi]is the shortest path from xjtoxi\navoiding F′⊆F.\nInductive Step ( p>1):Assume that for all i≥j+4(p−1),(xiu⋄F)2≥2ℓ+p−1. We will now show\nthat for any i≥j+4p,(xiu⋄F)2≥2ℓ+p. For contradiction, assume that (xiu⋄F)2≤2ℓ+p−1.\nConsider the path R[xj,xi]. As in the base case, the subpath R[xj,xj+4]ofR[xj,xi]has weight\n≥4×2ℓ. Similarly , using the induction hypothesis, the weight of t he path from the vertex xj+4(p−1)to\nxj+4pis≥4×2ℓ+p−1. Thus, the weight of the path R[xj,xi]is≥4×2ℓ+4×2ℓ+p−1>2ℓ+1+2ℓ+p.\nHowever, there is an alternate path from xjtoxithat avoids F. This path is xju⋄F⊙xiu⋄F. The sum\nof the weights of these two paths is <2ℓ+1+2ℓ+p. This is a contradiction as R[xj,xi]is the shortest\npath from xjtoxiavoiding F′⊆F. This completes the proof of the lemma.\nWe now extend our result to all vertices in V(F).\n5.1 Jump Sequence using the fault set F\nLet’s create another jump sequence J UMP(R[s,t])that utilizes each vertex of V(F). The first vertex in\nJUMP(R[s,t])isx1:=s. The i-th vertex in J UMP(R[s,t])is defined as follows: we first find the vertex\nu=argminv∈V(F){(xi−1v)2}. Then, we jump from xi−1using uon the path R. In other words, xiis the\nfirst vertex after xi−1on the path Rthat satisfies:\n|R[xi−1,xi]|≥max{1,(xi−1u⋄F)2} (2)\nNotice that by definition, xi−1u⋄F=xi−1u. This is because uis the closest vertex of V(F)toxi−1.\nTherefore, instead of writing xi−1u⋄F(as in the previous section), we simply write xi−1u. Assuming\nthe setting in above paragraph, we claim the following lemma :\nLemma 5.3. The length of JUMP(R[s,t])is≤16flog(nW).\nProof. Letu∈V(F). Let{y1,y2,... , yℓ}be the subsequence of J UMP(R[s,t])where we jump using\nu. Consider the sequence {(y1u)2,(y2u)2,... ,(yℓ−1u)2}. We claim that this sequence also behaves like\na bitonic sequence satisfying (A1) and(A2). The proof remains exactly the same as in the previous\nsection. Thus, the length of this subsequence is ≤8 log(nW). Since there are at most 2 fvertices in\nV(F), the length of J UMP(R[s,t])is≤16flog(nW).\n76 Single fault tolerant distance oracle\nIn our new single fault tolerant distance oracle, we will cru cially use the jump sequence with a suitable\nmaximiser. Before describing our maximiser, we note its cru cial property :\nLemma 6.1. ([DR22 , Claim 4.2]) Let/Dbbbbe a maximiser returning an edge set F∗maximising|st⋄F∗|. If\nF satisfies the maximiser’s condition, then either (1) st ⋄F=st⋄F∗or (2) an edge in F∗lies on the path\nst⋄F.\nProof. Let us assume that no edge of F∗lies on the path st⋄F. So, st⋄Favoids F∗also. Hence,\n|st⋄F|≥|st⋄F∗|/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n(α)asst⋄F∗is the shortest path avoiding F∗.\nNow, if Fsatisfies the maximiser’s conditions and F∗is the output of the maximiser, then |st⋄F∗|≥|st⋄F|/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n(β).\n(α)and(β)implies|st⋄F|=|st⋄F∗|. Also, since we preprocessed the graph ensuring unique shor test\npath, st⋄F=st⋄F∗. This completes the proof.\nFor all our maximisers, Fwill satisfy its condition. Case (1) in Lemma 6.1 is good for us, as it allows\nus to return st⋄F∗. In most parts of this paper, we focus on case (2). Now, let’s e xamine the maximiser\nfor our single fault-tolerant distance oracle.\n6.1 Maximiser\n/Dbbb(x,y,d1,d2)=arg max\ne∗∈E/braceleftbig5\n|x y⋄e∗|such that|xe∗|≥d1and|ye∗|≥d2\nwhere both d1andd2are a power of 2./bracerightbig5\n(3)\nIn addition to returning e∗, our maximiser will also return x y⋄e∗. In our algorithm, we will write\n(e∗,x y⋄e∗)←/Dbbb(x,y,d1,d2)to indicate this fact. By Theorem 3.1 ,x y⋄e∗is 2-decomposable and we\ncan represent it in 2-decomposable form. Thus, the space req uired to store(e∗,x y⋄e∗)isO(1). For a\nfixed pair x,y, there are O(log(nW))possible values of d1andd2. Thus, there are O(log2(nW))pos-\nsible maximisers each taking O(1)space. We construct maximisers for all pair of vertices in th e graph.\nThus, the size of our data-structure is O(n2log2(nW)).\n6.2 Query Algorithm\nLet us now describe our algorithm Q UERY(s,t,e=(u,v))(See Algorithm 1 ). Let us assume that elies\non the stpath such that uis closer to sthan v. We first find J UMP(st)and J UMP(ts)using e. This is\ndone by calling the function F INDJUMP().\nWe store our final answer in P ATH, which is initialised to /emptysetstress. We define|PATH|=∞, if P ATH=/emptysetstress. For\neach pair(x,y)in JUMP(st)and J UMP(ts), we call the maximiser /Dbbb(x,y,(xu)2,(yv)2). Note that by\nconstruction, esatisfies the condition of the maximiser. The maximiser retu rns(e∗,x y⋄e∗). Then, we\nsetP=sx⊙x y⋄e∗⊙y t. We then check if Pcontains eor not. If Pdoes not contain e, then we update\nour P ATH. This completes the description of our algorithm.\n8Algorithm 1: QUERY(s,t,e= (u,v)): re-\nturns st⋄e\n1JUMP(st)←FINDJUMP(st,{e});\n2JUMP(ts)←FINDJUMP(ts,{e});\n3PATH←/emptysetstress;\n4foreach(x,y)∈JUMP(st)×JUMP(ts)do\n5(e∗,x y⋄e∗)=/Dbbb(x,y,(xu)2,(yv)2);\n6 P←sx⊙x y⋄e∗⊙y t\n7 ife/∈Pthen\n8 PATH←argmin{|P|,|PATH|};\n9return P ATH;Algorithm 2: FINDJUMP(P[s,t],F)\n1X←/emptysetstress;\n2x←s;\n3do\n4 X←X∪{x};\n5 u=arg minv∈V(F){(xv)2};\n6 Letybe the first vertex on\nP[x,t]path at distance\n≥max{1,(xu)2}from x;\n7 ify exists then\n8 x←y;\n9while y exists ;\n10return X\n6.2.1 Correctness\nIn our algorithm, we always update P ATHwith Pthat does not contain e. Thus,|PATH|≥|st⋄e|. We\nwill now show that there is a pair, say (x,y)∈JUMP(st)×JUMP(ts)for which both Pand P ATHwill be\nset to st⋄e. This will imply that our algorithm is correct.\nLetR=st⋄e. Let us also assume that the detour of Rstarts at the vertex pon the supath and ends\nat the vertex qon the vtpath (See Figure 1 ). Consider the jump sequence J UMP(st). Let xbe the last\nvertex before pon J UMP(st)on the stpath. Thus, we have, xp≤(xu)2. Since the detour of Rstarts at\nthe vertex p,R[x,p]=xp. Thus, we have:\n|R[x,p]|≤(xu)2 (4)\nSimilarly , in the jump sequence J UMP(ts), let ybe the last vertex such that t y≤tq. Consider the\nmaximiser on the pair (x,y):/Dbbb(x,y,(xu)2,(yv)2). Let e∗be the edge returned by this maximiser.\nNote that e∗must lie on the x ypath, as the weight of any other path from xtoyis strictly greater than\nthe weight of the x ypath. Thus, we have:\ne∗lies on the shortest x ypath which is a subpath of st (5)\nThere are two cases:\n1.e∗does not lie on the path x y⋄e.\nSince ealso satisfies the conditions of the maximiser, by Lemma 6.1 ,x y⋄e∗=x y⋄e. Thus, we\ncorrectly set Pto:\nP=sx⊙x y⋄e∗⊙y t=sx⊙x y⋄e⊙y t=R\nNow, we show that we can represent Pin 2-decomposable form in our algorithm. By construction,\nx y⋄e∗is in 2-decomposable form, say L1⊙e′⊙L2. But even Ris 2-decomposable and x y⋄e∗is\na subpath of R. Thus, sx⊙L1must be a shortest path. Similarly , L2⊙y tis also a shortest path.\nThus, the path Pis in 2-decomposable form.\n2.e∗lies on the path x y⋄eorR[x,y]\nWe will see this case cannot arise. By the condition of the max imiser,|xe∗|≥(xu)2orR[x,e∗]≥\n(xu)2. Using the previous inequality and Equation (4) ,e∗lies after ponR. But the detour of R\nstarts at p. Thus, e∗cannot lie on xupath. Similarly , we can show that e∗does not lie on yv.\nThis implies that e∗does not lie on x ypath. But this violates Equation (5) . So this case cannot\narise.\n96.2.2 Running Time\nLet us first find the time taken by F INDPATH(st). By Lemma 5.3 , there are O(log(nW))vertices in\nJUMP(st). Thus, in F INDPATH(st), the while loop runs for O(log((nW))time. In the while loop, in\nLine 6 , we find a vertex which is at a distance 2i(for some i) on some shortest path. This can be done\ninO(1)time. Apart from this, all other operations in the while loop can also be performed in O(1)time.\nThus, the running time of F INDPATH(st)(and similarly F INDPATH(ts)) isO(log(nW)).\nNext, we process every pair of vertices in J UMP(st)and J UMP(ts). We will show that the time taken\nto process each pair is O(1)implying that the running time of our query algorithm is (log2(nW)).\nWhile processing a pair in J UMP(st)×JUMP(ts), the dominating step is to check if Pcontains e(in\nLine 7 ). To check if a shortest path, say ab, contains e, we can check if|ae|+w(e)+|eb|=|ab|. Since\nPis in 2-decomposable form, we can check if Pcontains eor not in O(1)time.\nThus, we have proven the following:\nLemma 6.2. There is a single fault tolerant distance oracle that takes O (n2log2(nW))space and answers\neach query in O(log2(nW))time. Moreover, it returns a path in 2-decomposable form.\n7 From single fault to ffaults\nIn the single-fault algorithm, we know that the fault emust lie on the stpath. Also, the detour of st⋄e\nstarts and ends on the stpath. But what happens with multiple faults? How do we define t he start and\nend of the detour? To address this, we introduce some new nota tions.\nWe label the shortest path from stotas path P1. Let e1∈Fbe an edge on P1. Then, st⋄e1is\nreferred to as path P2. Let us assume that e2∈Fis an edge on P2. Then P3isst⋄{e1,e2}. We now\ngeneralise this recursive definition.\nDefinition 7.1. (Path Pi’s)\nFor the base case, let P1=st. Let e1,e2,... , ei−1be the distinct edges of F on paths P1,P2,... , Pi−1\nrespectively where 1≤i≤f . Then, Piis st⋄{e1,e2,... , ei−1}.\nFor notational convenience, we define P0=/emptysetstress. We will now define the notion of the detour of a\npath, its start and end.\nDefinition 7.2. (Prefix and suffix of Pi) Let e be the first faulty edge on Pifrom s. Then PREFIX(Pi)=\nPi[s,e]. Similarly, if e is the first faulty edge on Pifrom t, then SUFFIX(Pi)=Pi[t,e].\nDefinition 7.3. (Detour of st⋄F, its start and end)\nLet poutbe the first vertex in st ⋄F from s that does not lie in ∪f\nj=1PREFIX(Pj)and p be the vertex\nbefore poutin st⋄F. If p is in PREFIX(Pi)but not in PREFIX(P0)∪PREFIX(P1)∪···∪ PREFIX(Pi−1)(where\n1≤i≤f ), then the detour of st ⋄F starts on path Piat p. We define the end of a detour symmetrically\n(using SUFFIX instead of PREFIX ). Let q be the vertex where the detour of st ⋄F ends. The path between p\nand q on st⋄F is called the detour of st ⋄F.\nFor example, in Figure 2 , the detour of st⋄Fstarts on the path P2atpasp∈PREFIX(P2)but not\nin P REFIX(P0)∪PREFIX(P1). Similarly , the detour of st⋄Fends on P1atqasq∈SUFFIX(P1)but not in\nSUFFIX(P0).\nAssume wlog that the detour of st⋄Fstarts on p∈Piand ends on q∈Pj. Similar to the single fault\ntolerant distance oracle, we find J UMP(Pi[s,t])and J UMP(Pj[t,s]). In J UMP(Pi[s,t]), let xbe the last\nvertex before pinPi[s,t](see Figure 2 ). Similarly , let y∈JUMP(Pj[t,s])be the last vertex before qin\nPj[t,s]. For the pair(x,y), we use an appropriate maximiser, (F∗,x y⋄F∗)←/Dbbb(x,y,·,·)to find x y⋄F\n(we will define the maximiser later). In the single fault tole rant distance oracle, we proved no edge in\nF∗lies on x y⋄F. Unfortunately , here, a vertex in V(F∗)may lie on x y⋄F. Let’s consider the easier\ncase. If a vertex u∈V(F∗)is an intermediate vertex on x y⋄F, we can recurse on su⋄Fandut⋄F.\nBut, if ulies on the first or last segment of x y⋄F, we follow[DR22]. In this case, we will see that u\nwill be a \" clean \" vertex. We now define this notion.\n10Definition 7.4. (Intact from failure F)\nA path is intact from failure F if no edge of F lies on the path. A tree T is intact from failure F if none\nof the vertices of V (F)are in T.\nDefinition 7.5. (x-clean and y-clean from failure F)\nLet R=x y⋄F. A vertex u∈R is called x-clean from failure F if xu and Tx(u)is intact from failure\nF. We call u an x-clean vertex, dropping the term “from the fai lure F” where it is clear from the context.\nSimilarly, define a y-clean vertex.\nAnx-clean vertex ulies on the first segment of R, asx∈Randxuis intact from failure F. Addition-\nally ,Tx(u)is also intact from failure F. As in[DR22], once we find an x-clean vertex u, we can build a\nsuitable maximiser that finds a fault set F′that:\nmaximises|x y⋄F′|such that uisx-clean from failure F′.\nAssume that a vertex v∈V(F′)lies on x y⋄F. We will later show that vcannot lie on the first\nsegment of x y⋄F, which fixes the problem for the first segment. Similarly , we c an address the issue\nfor the last segment by finding a y-clean vertex. Assuming we have found both an x-clean vertex uand\nay-clean vertex v, we can construct a suitable maximiser that finds a fault set F′′that:\nmaximises x y⋄F′′, ensuring that uisx-clean from failure F′′andvisy-clean from failure F′′.\nWe will show that a vertex in V(F′′)cannot lie on the first or last segments of x y⋄F; if it lies on x y⋄F,\nit must be an intermediate vertex. Though we are following an approach similar to [DR22], the space\nrequired for all our maximisers (which are different from th eir maximisers) will be O(f4n2log2(nW)).\nThis completes the overview of our algorithm.\n8f- fault Tolerant Distance Oracle\nIn this section, we will prove Theorem 1.1 using induction on f. For the base case, i.e., f=1, we can\nuse the single fault tolerant distance oracle in Section 6 . Let us assume using the induction hypothesis\nthat we have a fault tolerant distance oracle for 1,2,... , f−1 faults. In the rest of the paper, we design\nanf-fault tolerant distance oracle. We will assume that we want to find the shortest path from stot\navoiding F.\nTo find st⋄F, we will call the function Q UERY(s,t,r=f+1). For brevity , we drop the fault set\nFfrom the arguments given to the query algorithm. The first arg ument of Q UERY is the source and\nthe second argument is the destination. The third argument rsuggests to us that the path st⋄Fis\nr-decomposable. Initially , we call Q UERY with r=f+1. Let us now describe the Q UERY algorithm (See\nAlgorithm 3 ).\nBy induction hypothesis, using 1-fault, 2-fault,... , f−1-fault tolerant distance oracles, we find paths\nP2,... , Pfrespectively . This is done in the function F INDPATH. InSection 10 , we will prove:\nLemma 8.1. FINDPATHreturns paths P1,P2,... , Pfin≤10cf2f8f2+1log2f2(nW)time.\nFor each path Pi, we then find J UMP(Pi[s,t])and J UMP(Pi[t,s]). Our query algorithm will return\nthe answer stored in the variable P ATHwhich is initialised to /emptysetstress.\nThe detour of st⋄Fmay start and end on PiandPj, respectively . Since we do not know iand\njbeforehand, we will run our algorithm for every possible i,jpair. We consider each pair (x,y)in\nJUMP(Pi[s,t])×JUMP(Pj[t,s]). Ifxorysatisfies the condition of an intermediate vertex, then we ad d\nit to the set of I NTERMEDIATE vertices. Otherwise, we call F IND-INTERMEDIATE1(x,y)which returns a\ntuple(P,I). In the tuple, Pis a path and Iis a candidate set of intermediate vertices. In Section 9 , we\nwill show that F IND-INTERMEDIATE1(x,y)finds an intermediate vertex in x y⋄Fif(x,y)satisfies the\nfollowing setting:\n11Algorithm 3: QUERY(s,t,r)\n1ifr=0then\n2 return∞;\n3P1,P2,... , Pf←FINDPATH(s,t,F);\n4foreach i=1to fdo\n5 JUMP(Pi[s,t])←FINDJUMP(Pi[s,t],F);\n6 JUMP(Pi[t,s])←FINDJUMP(Pi[t,s],F);\n7PATH=φ;\n8for(i,j)∈[1 ... f]×[1 ...f]do\n9 INTERMEDIATE←/emptysetstress;\n10 foreach(x,y)∈JUMP(Pi[s,t])×JUMP(Pj[t,s])do\n//xsatisfies the condition of an intermediate vertex\n11 ifsx and x t contain an edge from F then\n12 INTERMEDIATE←INTERMEDIATE∪{x};\n//ysatisfies the condition of an intermediate vertex\n13 else if s y and yt contain an edge from F then\n14 INTERMEDIATE←INTERMEDIATE∪{y};\n15 else\n16(P,I)←FIND-INTERMEDIATE1(x,y);\n17 P′←sx⊙P⊙yt;\n18 PATH=argmin{|PATH|,|P′|};\n19 INTERMEDIATE←INTERMEDIATE∪I;\n20 foreach u∈INTERMEDIATE do\n21 PATH←argmin{|QUERY(s,u,r−1)⊙QUERY(u,t,r−1)|,|PATH|};\n22return P ATH;\n12(S1) Let st⋄Fbe a r-decomposable path and the detour of st⋄Fstarts on Piat the vertex pand ends\nonPjat the vertex q. Let xbe the last vertex of J UMP(Pi[s,t])before ponPi[s,t]. Similarly , ybe\nthe last vertex of J UMP(Pj[t,s])before qonPj[t,s]. Also, xandyare not intermediate vertices\nonst⋄F(See Figure 2 for an example).\nLemma 8.2. Let(P,I)←FIND-INTERMEDIATE1(x,y). The function FIND-INTERMEDIATE1comes with the\nfollowing guarantees: P is a path that avoids F, I is a set of si ze≤8f3and the running time of procedure\nFIND-INTERMEDIATE1(x,y)is O(f4).\nMoreover, if(x,y)satisfies (S1), then either\n1. P=x y⋄F or\n2. I contains an intermediate vertex on the path x y ⋄F.\nMoreover, P is in r-decomposable form.\nSome remarks are in order. The function F IND-INTERMEDIATE1(x,y)has no guarantees, except for\nrunning time, fault free path Pand the size of set I, if(x,y)does not satisfy (S1). If(x,y)satisfies\n(S1), then in the first case, we get x y⋄Fitself. Else, at least one vertex in Iis an intermediate vertex\nonx y⋄F.\nIf(x,y)satisfies (S1), an intermediate vertex in x y⋄Fis also an intermediate vertex in st⋄F.\nSince we do not know the intermediate vertex beforehand, we a dd all vertices of Ito INTERMEDIATE\nand recurse our algorithm on each vertex in I NTERMEDIATE . This completes the description of our\nalgorithm. We now show that our algorithm is correct.\n8.1 Correctness\nWe will first show that our algorithm always returns a P ATHthat avoids F. In Q UERY(x,y,r), PATH\nis set at Line 18 andLine 21 . InLine 18 ,sxandy tare intact from failure Fas both xandyare not\nintermediate vertices. Also, using Lemma 8.2 ,Pavoids F. Thus, P′and in turn P ATHavoids F. Similarly ,\nwe can prove using induction that the P ATHset in Line 21 also avoids F. Thus,|PATH|≥|st⋄F|. We\nwill now show that there is a pair (x,y)for which P ATHis set to st⋄Fin our algorithm. To this end,\nwe prove the following lemma.\nLemma 8.3. If st⋄F is r-decomposable, then QUERY(s,t,r′)will return it in r-decomposable form where\nr′≥r.\nProof. We prove this lemma using induction on r.\nBase Case ( r=2):Let us prove the base case when r′=2. The proof remains the same even if\nr′>2.\n1. Q UERY(s,t,2)returns the correct path.\nAsst⋄Fis 2-decomposable, no pair (x,y)processed by our algorithm can be intermediate vertices.\nLet us assume that we process a pair (x,y)that satisfies (S1). In that case, we call (P,I)←\nFIND-INTERMEDIATE1(x,y).\nSince st⋄Fis 2-decomposable, so is x y⋄F. Thus, Iis empty . By Lemma 8.2 ,P=x y⋄F, and\nwe have found the correct path, that is, sx⊙P⊙y t.\n2. The path returned by Q UERY(s,t,2)is in 2-decomposable form.\nByLemma 8.2 ,Pis in 2-decomposable form, say L1⊙e′⊙L2. But even st⋄Fis 2-decomposable\nandPis a subpath of st⋄F. Thus, sx⊙L1must be a shortest path. Similarly , L2⊙y tis also a\nshortest path. Thus, the path sx⊙P⊙y tcan be represented in 2-decomposable form.\n13Using the induction hypothesis, let us assume that Q UERY returns the correct answer for paths that are\nfdecomposable.\nInductive step ( r=f+1):Again, let us prove the lemma when r′=f+1.\n1. Q UERY(s,t,f+1)returns the correct path.\nLet us assume that we process a pair (x,y)in Q UERY(s,t,f+1). Ifxis an intermediate vertex,\nthen we can recurse using Q UERY(s,x,f)⊙QUERY(x,t,f). By induction hypothesis, both these\nfunctions return the correct path, and we are done. Similarl y , ifyis an intermediate vertex, we\nare done. So, let us assume that neither xnoryis an intermediate vertex. Specifically , look at\nthe pair(x,y)that satisfies (S1). In that case, we call (P,I)←FIND-INTERMEDIATE1(x,y). If\nP=x y⋄F, then we have found the correct path, that is sx⊙P⊙y t.\nOtherwise, using Lemma 8.2 , there is an intermediate vertex, say u∈Ion the path x y⋄F. Since\nx y⋄Fis a subpath of st⋄F,uis an intermediate vertex even for st⋄F. We then recurse using\nQUERY(s,u,f)⊙QUERY(u,t,f). By induction hypothesis, both these functions return the c orrect\npath, and we are done.\n2. The path returned by Q UERY(s,t,f+1)is in f+1-decomposable form.\nThere are two places where we set P ATH. They are:\n(a)Line 18\nInLine 18 , the algorithm sets P ATHtosx⊙P⊙y t. ByLemma 8.2 ,Pis in f+1-decomposable\nform. Let the first and the last segment of Pbexaandb y(for some vertex aandb). But\neven st⋄Fisf+1-decomposable. Also, since xandyare not intermediate vertices, xlies\non the first segment of st⋄Fandylies on its last segment. Thus, sx⊙xais the first segment\nofst⋄F. Similarly , b y⊙y tis the last segment of st⋄F. Thus, we also return a path in f+1\ndecomposable form in Line 18 .\n(b)Line 21\nLetube an intermediate vertex in st⋄F. Since st⋄Fisf+1 decomposable, we can\nrepresent is as st⋄F=L1⊙e1⊙L2⊙e2⊙...⊙ef⊙Lf+1where each Li(where 2≤i≤f)\nis a shortest path and each eiis an interleaving edge. Wlog, let us assume that uis in Li. By\ninduction hypothesis, Q UERY(s,u,f)will return a path in idecomposable form as su⋄Fis\nani-decomposable path. Similarly , Q UERY(u,t,f)will return the path ut⋄Fin(f−i+2)\ndecomposable form. As the last shortest path of su⋄Fand the first shortest path of ut⋄F\nhas no interleaving edges, we can merge them and get a path in (f+1)decomposable form.\nThis completes our proof.\n8.2 Running Time\nWe now calculate the running time of Q UERY(s,t,r).\nLemma 8.4. The running time of QUERY(s,t,r)is≤cf2+rf8(f2+r)log2(f2+r)(nW)where c>1is a con-\nstant.\nIf we put r=f+1 in the above equation, we get the running time of Q UERY(s,t,f+1)to be\n≤c(f+1)2f8(f+1)2log2(f+1)2(nW)as required.\nProof. We will prove using induction on r. Let T(r)be the running time of Q UERY(s,t,r). For the base\ncase, r=0,T(0)=1. Using induction hypothesis, let us assume that T(1),T(2),... , T(r−1)satisfies\nthe statement of the lemma. We want to find T(r). Let us first find the running time of Q UERY(s,t,r)\n14excluding recursive calls. In Q UERY(s,t,r), we find the paths P1,P2,... , Pfusing F INDPATH(). Using\nLemma 8.1 , FINDPATH()takes≤10cf2f8f2+1log2f2(nW)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nαtime.\nThen, we find the jump sequence for each path Pi. Using induction hypothesis, the path Piis in\ni-decomposable form or ≤f-decomposable form. Thus, using Lemma 2.3 , each jump vertex on Pican\nbe found in O(f+log(nW))time. Using Lemma 5.3 , since there are ≤16flog(nW)jump vertices\non path Pi, the total time taken to find J UMP(Pi[s,t])and J UMP(Pi[t,s])is≤c f2log2(nW)(for some\nsuitable constant c>1). Thus, the time taken to find all the jump sequences is ≤c f3log2(nW)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nη.\nThen, we go over each possible pair (i,j)∈[1 ...f]×[1 ...f]. There are f2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nβsuch pairs. For\neach pair(x,y)∈JUMP(Pi[s,t])×JUMP(Pj[t,s]), we may call the function F IND-INTERMEDIATE1(x,y).\nUsing Lemma 5.3 , there are≤256f2log2(nW)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nγpossible(x,y)pairs. Using Lemma 8.2 , the running\ntime of F IND-INTERMEDIATE1(x,y)isO(f4)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nδ. Thus, the total time Ytaken by Q UERY(s,t,r)(excluding\nrecursive calls) is:\nY≤10cf2f8f2+1log2f2+1(nW)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nα+c f3log2(nW)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nη+c f2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nβ×256f2log2(nW)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nγ×f4\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nδ\nThe first term on the RHS dominates the rest if we set cto a suitable high constant. Thus, Y≤\n12cf2f8f2+1log2f2(nW). Let us now find the total number of recursive calls in Q UERY(s,t,r). The\nnumber of intermediate vertices returned by F IND-INTERMEDIATE1(x,y)is 8f3\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nθ. For each intermediate\nvertex, we call Q UERY(·,·,r−1)two times. Thus, the total number of recursive calls Xcan be bounded\nas:\nX≤f2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nβ×256f2log2(nW)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nγ×8f3\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nθ×2=5096 f7log2(nW)≤5096 f8log2(nW)\nHence, we get the following recurrence relation, T(r)≤X T(r−1)+Yor\nT(r)≤5096 f8log2(nW)T(r−1)+12cf2f8f2+1log2f2(nW)\nUsing induction hypothesis, we put T(r−1)≤cf2+r−1f8(f2+r−1)log2(f2+r−1)(nW)in the above equa-\ntion. Thus, we get:\nT(r)≤5096 cf2+r−1f8(f2+r−1)+8log2(f2+r−1)+2(nW)+12cf2f8f2+1log2f2(nW)\n≤cf2+rf8(f2+r)log2(f2+r)(nW) (ifc≥10192)\nTo complete the proof of Theorem 1.1 , we prove Lemma 8.2 and Lemma 8.1 inSection 9 and\nSection 10 respectively . In Section 11 , we bound the space taken by our oracle.\n9 Proof of Lemma 8.2\nBefore looking at the procedure F IND-INTERMEDIATE1(x,y), let’s first examine the maximisers we will\nemploy within it.\n159.1 Maximisers\nWe will use three new maximisers.\nFirst Maximiser: Our first maximiser is as follows:\n/Dbbb1(x,y,d1,d2)=arg max\nF∗∈Ef/braceleftbig5\n|x y⋄F∗|such that the closest vertex of V(F∗)is at a distance\nat least d1from xandd2from ywhere both d1andd2are power of 2/bracerightbig5\n(6)\nAs in Section 6 , we assume the maximiser returns F∗and the path x y⋄F∗is in f+1-decomposable\nform. We denote this as (F∗,x y⋄F∗)←/Dbbb1(x,y,d1,d2). The sizes of F∗andx y⋄F∗are both O(f),\nresulting in a space usage of O(f)per maximiser. For fixed xand y, the total number of maximis-\ners is O(log2(nW)). Thus, the total space taken by all of /Dbbb1maximisers is/summationtext\nx,yO(flog2(nW))=\nO(f n2log2(nW)).\nSecond Maximiser: For each vertex v∈V(F∗)returned by maximiser /Dbbb1(x,y,d1,d2), we make two\nmaximisers: (1)/Dbbb2(x,y,d1,v)and (2)/Dbbb2(x,y,v,d2). We define the first maximiser. The definition of\nthe second maximiser is symmetric.\n/Dbbb2(x,y,d1,v)=arg max\nF∗∈Ef/braceleftbig5\n|x y⋄F∗|such that the closest vertex of V(F∗)is at a distance\nat least d1from xwhere d1is a power of 2 and\nvisy-clean from the failure F∗/bracerightbig5\n(7)\nFor/Dbbb2(x,y,v,d2), in the conditions, visx-clean from the failures F∗and the closest vertex of V(F∗)\nis at a distance d2from y. For each maximiser/Dbbb1, we create O(f)/Dbbb2maximisers, each requiring O(f)\nspace. Therefore, the total space used by all /Dbbb2maximisers is O(f2n2log(nW)).\nThird Maximiser: For each vertex u∈V(F∗)returned by maximiser /Dbbb2(x,y,d1,v), we make a max-\nimiser/Dbbb3(x,y,u,v).\n/Dbbb3(x,y,u,v)=arg max\nF∗∈Ef/braceleftbig5\n|x y⋄F∗|such that uisx-clean and visy-clean from failure F∗/bracerightbig5\n(8)\nSimilar is the case for the other variant of /Dbbb2maximiser. Thus, for each maximiser /Dbbb2, we create O(f)\n/Dbbb3maximisers. The total size of /Dbbb3maximisers can be bounded to O(f3n2log(nW)).\nProperties of our maximisers: Whenever we will use our maximisers, we will ensure that Fsatisfies\ntheir conditions. We will later see all our maximisers retur n a set F∗with one or more of the following\nguarantees for a pair (x,y)that satisfies (S1):\n(G1) x y⋄F=x y⋄F∗.\n(G2) V(F∗)contains a vertex that is x-clean from failure F.\n(G3) V(F∗)contains a vertex that is y-clean from failure F.\n16(G4) V(F∗)contains an intermediate vertex of x y⋄F.\nLet us now go over the cases when these maximisers will be used .\n1. Maximiser/Dbbb1is used when we have neither found an x-clean or y-clean vertex.\nWe will use/Dbbb1in our main procedure F IND-INTERMEDIATE1(See Section 9.4 ). The maximiser\ncomes with the following guarantees: either (G1) or(G2) or(G3) or(G4) is true. Thus, in the\nworst case, we will either get an x-clean or a y-clean vertex from this maximiser.\n2. Maximiser/Dbbb2is used when we either found an x-clean or a y-clean vertex.\nLet us assume that we have found a y-clean vertex. In procedure F IND-INTERMEDIATE2(See\nSection 9.3 ), we will use/Dbbb2. It comes with the following guarantees: either (G1) or(G2) or(G4)\nis true. Thus, in the worst case, we will get an x-clean vertex from this maximiser.\n3. Maximiser/Dbbb3is used when we have found both an x-clean and y-clean vertex.\n/Dbbb3will be used in F IND-INTERMEDIATE3(See Section 9.2 ). It comes with the following guarantee:\neither (G1) or(G4) is true. Thus, in the worst case, we get an intermediate verte x from this\nmaximiser.\nBefore we move ahead, we would like the reader to again refres h setting (S1) as its assumptions will\nbe used in our arguments henceforth. In the setting (S1), we have assumed that st⋄Fisr-decomposable.\nAlso, xandylie on the first and the last segment of st⋄F. Thus,\nx y⋄Fis also r-decomposable. (9)\nLet us now go over the above three cases in reverse order.\n9.2 Found an x-clean and a y-clean vertex\nLet us assume that uisx-clean and visy-clean from failure F. In that case, we call the proce-\ndure F IND-INTERMEDIATE3(x,y,u,v)(See Algorithm 4 ). In this procedure, we will use the maximiser\n/Dbbb3(x,y,u,v). Note that by definition of a clean vertex, Fsatisfies the conditions of /Dbbb3. Let F∗be the\noutput of/Dbbb3(x,y,u,v). Ifx y⋄F∗does not contain any edge from F, we update P ATH.\nAlgorithm 4: FIND-INTERMEDIATE 3(x,y,u,v)\n1(F∗,x y⋄F∗)=/Dbbb3(x,y,u,v);\n2INTERMEDIATE←/emptysetstress;\n3ifx y⋄F∗does not contain any edges of F then\n4 PATH←x y⋄F∗;\n5forz∈V(F∗)do\n//zsatisfies the definition of an intermediate vertex\n6 ifboth xz and z y contain an edge from F then\n7 INTERMEDIATE←INTERMEDIATE∪{z};\n8return(PATH,INTERMEDIATE);\nThen, we go over all z∈V(F∗). Ifzsatisfies the definition of an intermediate vertex, then we ad d\nit to the I NTERMEDIATE set. Else, we discard it.\n9.2.1 Correctness\nWe will now show that either (G1) or(G4) is true for F IND-INTERMEDIATE3(x,y,u,v). IfF∗does not hit\nx y⋄F, then by Lemma 6.1 ,(G1) is true. Also, since x y⋄Fisr-decomposable (using Equation (9) ),\nwe also return it in r-decomposable form. So, let us move to the interesting case w hen an edge of F∗\nlies in the path x y⋄F. In this case, we will show that (G4) is true.\n17Lemma 9.1. If an edge of F∗, say e=(z,z′), lies in the path x y ⋄F, then at least one endpoint of e, say\nz, is an intermediate vertex.\nProof. To show that zis an intermediate vertex, we have to show it lies neither on t he first segment nor\nthe last segment of x y⋄F. Let us show that zcannot lie on the first segment of x y⋄F.\nSince uisx-clean from failure F,ulies on the first segment of x y⋄F. By the condition of the\nmaximiser, uisx-clean from failure F∗too. Thus, (1) no edge in F∗can lie on xupath and (2) no\nvertex in V(F∗)can lie on Tx(u).\nBy (1), elies after uonx y⋄F. Or, zlies after uonx y⋄F. By (2), zcannot lie on Tx(u). Let us\nassume that zlies on the first segment of x y⋄F. By (1), it lies after u(asealso lies after u). But then\nz∈Tx(u), which is a contradiction. Thus, zcannot lie on the first segment of x y⋄F. Similarly , we can\nshow that zcannot lie on the last segment of x y⋄F. Hence, zis an intermediate vertex.\n9.2.2 Running time and the size of I NTERMEDIATE set\nThe maximiser/Dbbb3returns the path x y⋄F∗in≤f+1 decomposable form, taking O(f)space. Consider\nany shortest path, say ab, in the decomposition. In O(f)time, we can find if any edge of Flies on\nab. So it takes O(f2)time to check if x y⋄F∗contains any edges from F. To check if a vertex zis an\nintermediate vertex, we need to check if xzandyzcontains any edge from F. Again, this takes O(f)\ntime. Thus, the running time of F IND-INTERMEDIATE3(x,y,u,v)isO(f2). Also, we can add at most 2 f\nvertices in I NTERMEDIATE . Thus, we claim the following lemma:\nLemma 9.2. The running time of FIND-INTERMEDIATE3(x,y,u,v)is O(f2)and the size of INTERMEDIATE\nset returned by it is ≤2f .\n9.3 Found a y-clean vertex\nLet us assume that we have found a vertex vthat is y-clean from the failure F. In that case, we call the\nfunction F IND-INTERMEDIATE2(x,y,φ,v). The third parameter of this function indicates that we have\nnot found any x-clean vertex, and the fourth parameter is the y-clean vertex.\nWe now describe function F IND-INTERMEDIATE2(x,y,φ,v)(See Algorithm 5 ). In this function, we’ll\ncalculate d1, which represents the largest power of 2 that is less than or e qual to the distance from point\nxto the nearest vertex of V(F). Then, we use the maximiser /Dbbb2(x,y,d1,v). The reader can check that\nFsatisfies the conditions of maximiser. Let F∗be the edges returned by /Dbbb2(x,y,d1,v). Ifx y⋄F∗does\nnot contain any edge from F, we update P ATH. Then, we go over all vertices u∈V(F∗). Ifusatisfies the\ncondition of being an x-clean vertex, then we call function F IND-INTERMEDIATE3(x,y,u,v). Ifusatisfies\nthe definition of an intermediate vertex, then we add it to the INTERMEDIATE set. Else, we discard u.\nBefore moving forward, let us state an important observatio n which implies that in F IND-INTERMEDIATE3\nwe are using maximisers that are created in Section 9.1 .\nRemark 9.3. InFIND-INTERMEDIATE3(x,y,u,v), we will use maximiser /Dbbb3(x,y,u,v)which is one of the\nO(f)maximisers created from /Dbbb2(x,y,d1,v).\n9.3.1 Correctness\nWe will now show that either (G1) or(G2) or(G4) is true for F IND-INTERMEDIATE2(x,y,φ,v). Let\n(F∗,x y⋄F∗)be the output of the maximiser /Dbbb2(x,y,d1,v). IfF∗does not hit x y⋄F, then by Lemma 6.1 ,\n(G1) is true. Also, since x y⋄Fisr-decomposable (using Equation (9) ), we also return it in r-decomposable\nform. Let us now move to the interesting case when an edge of F∗, say e, lies on x y⋄F. Or let ube the\nfarthest endpoint of efrom xthat lies on the path x y⋄F. There are three cases:\n(Case I) uis an intermediate vertex.\nIfuis an intermediate vertex, then (G4) is true.\n18Algorithm 5: FIND-INTERMEDIATE 2(x,y,φ,v)\n1d1=mina∈V(F){(xa)2};\n2(F∗,x y⋄F∗)=/Dbbb2(x,y,d1,v);\n3INTERMEDIATE←/emptysetstress;\n4ifx y⋄F∗does not contain any edges of F then\n5 PATH←x y⋄F∗;\n6foru∈V(F∗)do\n//usatisfies the definition of an x-clean vertex\n7 ifxu and Tu(x)is intact from failure F then\n8(P,I)←FIND-INTERMEDIATE3(x,y,u,v);\n9 PATH←argmin{|P|,|PATH|};\n10 INTERMEDIATE←INTERMEDIATE∪I;\n//usatisfies the definition of an intermediate vertex\n11 ifboth xu and yu contain a failure from F then\n12 INTERMEDIATE←INTERMEDIATE∪{u};\n13return(PATH,INTERMEDIATE);\n(Case II) ulies on the last segment of x y⋄F.\nWe will show that this case cannot arise. Our argument is simi lar to the one given in the proof of\nLemma 9.1 . The vertex visy-clean from failure Fso,vlies on the last segment of x y⋄F. Since\nF∗also satisfies the condition of the maximiser, visy-clean from failure F∗too. Thus, (1) no\nedge in F∗can lie on the yvpath and (2) no vertex in V(F∗)can lie on Ty(v). By (1), ecannot\nlie on the last segment of x y⋄F. By (2), ucannot lie in Ty(v). As in the proof of Lemma 9.1 ,\nifuis in the last segment of x y⋄F, then by (1), it lies after v. But then u∈Ty(v), which is a\ncontradiction. This implies that ucannot lie on the last segment of x y⋄F.\n(Case III) ulies on the first segment of x y⋄F.\nWe will show that umust be an x-clean vertex. Thus, (G2) is true. To prove uisx-clean, we have\nto show that:\n(P1) xuis intact from failure Fand\n(P2) Tx(u)is intact from failure F.\nSince ulie on the first segment of x y⋄F,(P1) is true. We now prove (P2). Let us look back at\nthe setting which led us to call F IND-INTERMEDIATE2(x,y,φ,v). To this end, we encourage the\nreader to see Setting (S1) again as we will be using it in the ensuing argument.\nLetR=st⋄F. In F IND-INTERMEDIATE2(x,y,φ,v), we set d1=mina∈V(F){(xa)2}. Since, the\ndetour of Rstarts at the vertex pon the path Pi, and xis the last vertex of J UMP(Pi[s,t])before\npon the path Pi, we have:\nPi[x,p]≤d1\nAgain, since the detour of Rstarts at the vertex pon the path Piandxlies before pon path Pi,\nwe can write the above equation as:\nR[x,p]≤d1 (10)\n19But according to the condition in /Dbbb2(x,y,d1,v),emust be at a distance ≥d1from x. Since e\n(and thus u) lies on the first segment of x y⋄F(by the assumption in this case) and xlies on\nthe first segment of R(by assumption in (S1)), we have R[x,e]≥d1. But, by definition, uis the\nfarthest endpoint of ein the path x y⋄F. Thus, we have:\nR[x,u]>d1 (11)\nThus, using Equation (10) andEquation (11) ,ulies after ponR(See Figure 4 for an illustration).\nBut the detour of Rstarts at ponPi(by assumption in (S1)). This implies the following:\nulies on path Rbut not on Pi[s,t] (12)\ns tR\nPip xu\nFigure 4: ucan lie only on the violet subpath of R. The horizontal black path is Pi.\nSome remarks are in order. The above statement is one of the mo st important deductions in our\npaper. We are able to prove that udoes not lie on Pibut lies on the detour of R. This will help us\nin proving that usatisfies (P2).\nWe will now show that Tx(u)contains no endpoints from V(F). For contradiction, let ek=(ak,bk)\nbe the faulty edge whose endpoint is nearest to uinTx(u). Remember that we have found fpaths\nP1,P2,... , Pfusing F INDPATH. Let us assume that ekis the first edge on Pk[s,t]andakis closer to\nsonPk. There are two cases depending on whether Pk=Pior not.\n(a)Pk=Piandek=ei\nWe will show that neither ainorbilies in Tx(u).\ni.aicannot lie in Tx(u)\nSince ulies on the first segment of x y⋄F(by assumption in this case), and xlies on the\nfirst segment of R(by assumption in (S1)),saiavoids Fand the shortest path from s\ntoeipasses through u. Or, Pi[s,t]passes through u. But this contradicts Equation (12)\n(See Figure 5 for an illustration).\nii.bicannot lie in Tx(u)\nIfbiis nearest to uinTx(u), then the path sbiavoids F. But then there is a shorter path\nfrom stobithat avoids ei. This is su⊙ubi. Thus, Pi[s,bi]does not pass through ei, a\ncontradiction.\n(b)Pk/ne}ationslash=Pi\nWe will show that neither aknorbklies in Tx(u).\ni.akcannot lie in Tx(u)\nIfakis the nearest vertex to uinTx(u), then sakavoids F. So, sek=su⊙uek. Since, the\nshortest sekpath contains u,Pkalso passes through u. Now, by Equation (12) ,udoes\nnot lie on Pi[s,t]butRpasses through u. But then the detour of Rmust start on that\npath Pkafter the vertex u(See Figure 6 for an illustration). This is a contradiction as\nwe assumed that the detour of Rstarts on path Piin Setting (S1).\n20u\nai\nbis\ntx\nPiR\nFigure 5: If ai∈Tx(u), then Pi[s,t]contains uand this contradicts Equation (12) .\nii.bkcannot lie in Tx(u)\nIfbkis nearest to uinTx(u), then our arguments are the same as in Case (a)(ii).\nThus, usatisfies (P2) and it is an x-clean vertex.\nu\nak\nbks\ntx\nPi\nPkR\nFigure 6: Here the blue path is Pi.suis intact from failure Fand is in the first segment of R. So,\nthe detour of Rstarts somewhere in the dashed path from utoak. Or the detour starts on Pk. This\ncontradicts our assumption in (S1) that the detour starts on Pi.\n9.3.2 Running time and the size of I NTERMEDIATE set\nThe running time is dominated by Line 8 of FIND-INTERMEDIATE2(x,y,φ,v). In this line, the procedure\nFIND-INTERMEDIATE3(x,y,u,v)is invoked. But this invocation can happen only O(f)times. Using\nLemma 9.2 , the running time of F IND-INTERMEDIATE3(x y,u,v)isO(f2). Thus, the running time of\n21FIND-INTERMEDIATE2(x,y,φ,v)isO(f3). Moreover, F IND-INTERMEDIATE3(x y,u,v)returns 2 fcandi-\ndate intermediate vertices. Thus, in the worst case, we can a dd at most 4 f2vertices in I NTERMEDIATE .\nThus, we claim the following lemma:\nLemma 9.4. The running time of FIND-INTERMEDIATE2is O(f3), and the size of INTERMEDIATE set re-\nturned by it is≤4f2.\n9.4 Neither found an x-clean or a y-clean vertex\nAlgorithm 6: FIND-INTERMEDIATE1(x,y)\n1d1=mina∈V(F){(xa)2};\n2d2=mina∈V(F){(a y)2};\n3(F∗,x y⋄F∗)=/Dbbb1(x,y,d1,d2);\n4PATH←/emptysetstress;\n5INTERMEDIATE←/emptysetstress;\n6ifx y⋄F∗does not contain any edge of F then\n7 PATH←x y⋄F∗;\n8foru∈V(F∗)do\n//usatisfies the conditions of an x-clean vertex\n9 ifxu and Tu(x)is intact from failure F then\n10(P1,I1)←FIND-INTERMEDIATE2(x,y,u,φ);\n11 PATH←argmin{|P1|,|PATH|};\n12 INTERMEDIATE←INTERMEDIATE∪I1;\n//usatisfies the conditions of a y-clean vertex\n13 ifyu and Tu(y)is intact from failure F then\n14(P2,I2)←FIND-INTERMEDIATE2(x,y,φ,u);\n15 PATH←argmin{|P2|,|PATH|};\n16 INTERMEDIATE←INTERMEDIATE∪I2;\n//usatisfies the conditions of an intermediate vertex\n17 ifxu and Tu(x)is intact from failure F then\n18 INTERMEDIATE←INTERMEDIATE∪{u};\n19return(PATH,INTERMEDIATE);\nThis is the most general case in which we call F IND-INTERMEDIATE1(x,y). Let us describe this\nfunction (See Algorithm 6 ). We first find the distance of the closest vertex of V(F)from xandy. That is,\nwe set d1=mina∈V(F){(xa)2}andd2=mina∈V(F){(ya)2}. Then, we use the maximiser /Dbbb1(x,y,d1,d2).\nFirst, note that Fsatisfies the condition of the maximiser.\nLetF∗be the edges returned by /Dbbb2(x,y,d1,d2). Ifx y⋄F∗does not contain any edge from F, we\nupdate P ATH. Then, we go over all vertices u∈V(F∗). Ifusatisfies the conditions of an x-clean vertex,\nthen we call the function F IND-INTERMEDIATE2(x,y,u,φ). Similar is the case when uis a potential y-\nclean vertex. Else, if usatisfies the conditions of an intermediate vertex, then we a dd it to I NTERMEDIATE\nset. Else, we discard u.\nAs in the previous section, we claim the maximisers /Dbbb2used by us in F IND-INTERMEDIATE2(x,y,u,φ)\nare created from/Dbbb1maximisers.\nRemark 9.5. InFIND-INTERMEDIATE2(x,y,u,φ), we will use maximiser /Dbbb2(x,y,u,d2)which is one of\nthe O(f)maximisers created from /Dbbb1(x,y,d1,d2). Similarly, in FIND-INTERMEDIATE2(x,y,φ,u), we will\nuse maximiser/Dbbb2(x,y,d1,u)which is one of the O (f)maximisers created from /Dbbb1(x,y,d1,d2).\n229.4.1 Correctness\nWe will now show that at least one of the guarantees from (G1) to(G4) is true. Let(F∗,x y⋄F∗)be\nthe tuple returned by the maximiser /Dbbb1in F IND-INTERMEDIATE1(x,y). IfF∗does not hit x y⋄F, then\nbyLemma 6.1 ,(G1) is true. Let us now move to the interesting case when an edge of F∗, say e, lies on\nx y⋄F. Or an endpoint of e, say u, lies on the path x y⋄F. There are three cases:\n1.uis an intermediate vertex.\nIn this case, (G4) is true.\n2.ulies on the first segment of x y⋄F.\nIn this case, we can show that uis an x-clean vertex or (G2) is true. Our arguments are identical\nto the one used in (Case III) .\n3.ulies on the last segment of x y⋄F.\nBy symmetry , we claim that uis ay-clean vertex or (G3) holds.\n9.4.2 Running time and the size of I NTERMEDIATE set\nThe running time of function F IND-INTERMEDIATE1(x,y)is dominated by the number of times we\ncallLine 10 orLine 14 . Note that either Line 10 orLine 14 is executed in each iteration of the\nfor loop. Wlog, let us assume that Line 14 is executed in each iteration. In Line 14 , the procedure\nFIND-INTERMEDIATE2(x,y,φ,u)is invoked. But this can happen only O(f)time. Using Lemma 9.4 , run-\nning time of F IND-INTERMEDIATE2(x,y,φ,u)isO(f3). Thus, running time of F IND-INTERMEDIATE1(x,y)\nisO(f4). Moreover, F IND-INTERMEDIATE2(x,y,φ,u)returns 4 f2candidate intermediate vertices. Thus,\nin the worst case, we can add at most 8 f3vertices in I NTERMEDIATE . Thus, we claim the following\nlemma:\nLemma 9.6. The running time of FIND-INTERMEDIATE1(x,y)is O(f4)and the size of INTERMEDIATE set\nreturned by it is≤8f3.\n9.5 Summary of the the proof of Lemma 8.2\nIf the pair(x,y)satisfies (S1), then we can use our discussion in the above three sections. I n\nFIND-INTERMEDIATE1(x,y), in the best case, we find x y⋄Fand return it. Or, we find an intermediate\nvertex. But in the worst case, we either find an x-clean or a y-clean vertex. Let us assume that we find\nay-clean vertex v. Again, in F IND-INTERMEDIATE2(x,y,φ,v), in the worst case, we find an x-clean\nvertex u. And lastly , in F IND-INTERMEDIATE3(x,y,u,v), we are sure to find an intermediate vertex in\nthe worst case. Thus, we either return the path x y⋄F(in the best case) or a set I NTERMEDIATE which\ncontains an intermediate vertex of x y⋄F(in the worst case). Moreover, in the best case we return\nx y⋄Finr-decomposable form. Thus, we have satisfied all the requirem ent of Lemma 8.2 when the\npair(x,y)satisfy setting (S1).\nLet us now discuss the other guarantees in Lemma 8.2 which donot assume Setting (S1). In al-\ngorithm F IND-INTERMEDIATE1,FIND-INTERMEDIATE2and F IND-INTERMEDIATE3, before we update P ATH,\nwe always check if the path contains For not. Thus, the P ATHreturned by F IND-INTERMEDIATE1(x,y)\navoids F. Using Lemma 9.6 , the running time of F IND-INTERMEDIATE1(x,y)isO(f4)and the size of\nINTERMEDIATE set is≤8f3. Thus, we have proven all the requirements of Lemma 8.2 .\n10 Proof of Lemma 8.1\nWe use Algorithm 7 to find P1,... , Pf.P1is the stpath. For notational convenience, we assume that a\n“0-fault tolerant” distance oracle finds the path P1. To find Pi(i≥2), using induction hypothesis, we\nuse the(i−1)-fault tolerant distance oracle, which returns it in i-decomposable form. If Pidoes not\ncontain any faulty edge, then we can return Pias the answer and exit the query algorithm.\n23To check if Picontains an edge e∈For not, we just need four more invocation of (i−1)-distance\noracle, two times in Line 6 andLine 7 . Thus, we invoke the i−1-fault tolerant distance oracle at most\n5ftimes in the i-th iteration of the for loop. Thus, the time taken in the i-th iteration of the for loop\ncan be bounded to <5f ci2i8i2log2i2(nW). Summing it over all i, the running time of F INDPATHS is/summationtextf\ni=15f ci2i8i2log2i2(nW)≤10cf2f8f2+1log2f2(nW)(if we set c≥2). This proves Lemma 8.1 .\nAlgorithm 7: FINDPATHS(s,t,F)\n1F′←/emptysetstress;\n2fori=1to fdo\n3 Pi←st⋄F′found using an i−1-fault distance oracle;\n4 foreach e∈F−F′do\n5 f lag←0;\n6 R1←se⋄F′found using i−1-fault tolerant distance oracle;\n7 R2←et⋄F′found using i−1-fault tolerant distance oracle;\n//Check if elies on the path Pi\n8 if|R1|+w(e)+|R2|=|st⋄F′|then\n9 F′←F′∪{e};\n10 f lag←1;\n11 break;\n12 iff lag=0then\n13 Piis our answer;\n14 exit;\n15return{P1,P2,... , Pf}\n11 Space taken by our oracle\nInSection 9.1 , we defined maximisers /Dbbb1,/Dbbb2, and/Dbbb3. The combined size of these maximisers is\nO(f3n2log2(nW)). In F INDPATH, we employ an i-fault tolerant distance oracle, which also utilizes\nmaximisers/Dbbb1,/Dbbb2, and/Dbbb3. However, the size of these maximisers is O(i3n2log2(nW)) as here the\nfault set is of size i. For each 1≤i≤f, these three maximisers are constructed only once. 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In\nProceedings of the 44th Symposium on Theory of Computing Con ference, STOC 2012, New\nYork, NY, USA, May 19 - 22, 2012 , pages 887–898, 2012. 2\n[Wil14]Ryan Williams. Faster all-pairs shortest paths via circuit complexity . In Proceedings of the\nforty-sixth annual ACM symposium on Theory of computing , pages 664–673, 2014. 1\n[WY13]Oren Weimann and Raphael Yuster. Replacement paths and dist ance sensitivity oracles via\nfast matrix multiplication. ACM Trans. Algorithms , 9(2), March 2013. 2,3\n[Zwi04]Uri Zwick. A slightly improved sub-cubic algorithm for the a ll pairs shortest paths problem\nwith real edge lengths. In International Symposium on Algorithms and Computation , pages\n921–932. Springer, 2004. 1\nA Proof of Lemma 2.3\nLetxbe the first vertex at a distance ≥2ifrom sonP[s,t]. Since P[s,t]is in f-decomposable form, let\nus assume that it is represented as L1⊙e1⊙L2⊙...⊙ef−1⊙Lf, where each Liis a shortest path. Since\nwe know the weights of these paths and interleaving edges, we can efficiently identify the segment\nLicontaining x. Let Lirepresent the shortest path pq. Suppose we’ve determined that xis at least\ndistance dfrom p.\nIfdis a power of 2, we can use information stored at vertex qin the tree Tpto find x. Else, letℓbe\nsuch that 2ℓ≤d<2ℓ+1. Then we locate the vertex at a distance ≥2ℓ-th on the pqpath, denoting it as\np1. We then recursively examine the path p1q.\nIn a maximum of log (nW)recursive steps, we can identify x. Initially , it takes O(f)time to find the\nshortest path Li, and then an additional log (nW)time to locate xonLi. Therefore, the overall time\ncomplexity is O(f+log(nW)).\n27"    },    {        "title": "2402.12839v1.Critical_thresholds_in_pressureless_Euler__Poisson_equations_with_background_states.pdf",        "content": "CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON\nEQUATIONS WITH BACKGROUND STATES\nYOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\nAbstract. We investigate the critical threshold phenomena in a large class of one dimensional\npressureless Euler–Poisson (EP) equations, with non-vanishing background states. First, we es-\ntablish local-in-time well-posedness in proper regularity spaces, which are adapted for a certain\nneutrality condition to hold. The neutrality condition is shown to be necessary: we construct\nsmooth solutions that exhibit instantaneous failure of the neutrality condition, which in turn yields\nnon-existence of solutions, even locally in time, in the classical Sobolev spaces Hs(R),s⩾2. Next,\nwe study the critical threshold phenomena in the neutrality-condition-satisfying pressureless EP\nsystems, where we distinguish between two cases. We prove that in the case of attractive forcing,\nthe neutrality condition can further restrict the sub-critical region into its borderline, namely —\nthe sub-critical region is reduced to a single line in the phase plane. We then turn to provide a\nrather definitive answer for the critical thresholds in the case of repulsive EP systems with variable\nbackgrounds. As an application, we analyze the critical thresholds for the damped EP system\nfor cold plasma ion dynamics, where the density of electrons is given by the Maxwell–Boltzmann\nrelation .\nContents\n1. Introduction and statement of main results 1\n2. Local well-posedness and the neutrality condition 12\n3. Attractive forcing and the restricted borderline threshold 16\n4. Repulsive forcing and threshold with variable background 17\nAppendix A. A priori estimates in the proof of Theorem 1.1 23\nReferences 24\n1.Introduction and statement of main results\n1.1.Systems and notions. In this paper, we are interested in the analysis of critical threshold\nphenomena in a large class of pressureless Euler–Poisson (EP) equations in one dimension. More\nprecisely, the main purpose of our work is to propose a new method based on Lyapunov functions\nto investigate the super-critical region with finite-time breakdown and the sub-critical region with\nDate : February 21, 2024.\n2020 Mathematics Subject Classification. Primary, 35Q35; Secondary, 35B30, 76N10.\nKey words and phrases. Pressureless Euler–Poisson system, non-vanishing background states, neutrality condition,\ncritical thresholds, well-posedness, non-existence.\nAcknowledgment. The authors thank Junsik Bae, Bongsuk Kwon, and Changhui Tan for helpful discussion\non the Euler–Poisson system for cold ion dynamics and Euler–Poisson–alignment system. The research of YPC was\nsupported by NRF grant 2022R1A2C1002820. The research of ET was supported in part by ONR grant N00014-\n2112773.\n1arXiv:2402.12839v1  [math.AP]  20 Feb 20242 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\nglobal regularity of C1solutions to the following equations:\n\n∂tρ+∂x(ρu) = 0 ,\n∂tu+u∂xu=−νu−k∂xϕ,\n−∂xxϕ=ρ−c(1.1)\ninR×(0, T) subject to initial data\n(ρ, u)(0, x) = (ρ0, u0)(x), x∈R. (1.2)\nHere, ρ=ρ(t, x),u=u(t, x) and ϕ=ϕ(t, x) are respectively, the density, velocity, and potential of\nthe flow. The parameter ν⩾0 indicates the strength of damping. The parameter kon the right of\n(1.1) 2is a physical constant, signifying the underlying forcing of the system — either an attractive\norrepulsive forcing depending on whether k <0 or, respectively, k >0; for simplicity, we use the\nre-scaled parameter k=±1 throughout the paper.\nEquation (1.1) 3involves a positive background state, c=c(t, x), which is assumed to be uniformly\nbounded away from vacuum,\n0< c−⩽c(t, x)⩽c+,∀(t, x)∈R+×R,\nto vary smoothly in a manner specified in (1.8) below, and to preserve its mass fluctuation in the\nsense that Z\nR(c(t, x)−c0(x)) dx= 0,∀t∈[0, T). (1.3)\nFinally, we augment (1.1) with the following neutrality condition\nρ(t,·)−c(t,·)∈L1(R) andZ\nR(ρ(t, x)−c(t, x)) dx= 0 ∀t∈[0, T). (1.4)\nThis neutrality condition plays a central role in our discussion. Accordingly, we define a classi-\ncal solution to be a pair, ( ρ, u), satisfying both — the system (1.1) pointwise, and the neutrality\ncondition (1.4). It is essential to observe that the neutrality condition is independent of the main\nsystem (1.1). Indeed, in Theorem 1.4 below we construct so-called anomalous solutions , in the\nsense of solving the main system (1.1) with neutrality-satisfying initial data, yet these solutions fail\n(1.4) instantaneously after the initial time. Observe that since c(t, x)⩾c−>0, then total massZ\nRρ(t, x)dxis inifinte; we impose (1.3) as a compatiblity condition with the neutrality (1.4), so that\nthe total mass fluctuation ,Z\nR(ρ(t, x)−ρ0(x)) dx, is conserved in time.\n1.2.History and review. The system (1.1) is one of the fundamental fluid models describing the\ndynamics of many important physical flows, including charge transport, plasma waves, and cold\nions ( k >0) as well as collapse of stars due to self-gravitation ( k <0), see [2, 3, 8, 12, 13, 15, 23]\nand references therein.\nA mathematical study of finite-time loss of regularity (or singularity formation as shocks) and\nglobal-in-time regularity of solutions to Euler equations with nonlocal interaction forces are by now\nwell established. Since the literature-related results are too vast to be mentioned here, we focus on\nresults for the EP systems. The critical thresholds in the 1D pressureless EP system (1.1) with,\nν⩾0, and c= ¯c⩾0 are studied in [13]. The damped EP system or EP system with the quadratic\nconfinement is also taken into account in [4, 9]. In the multi-dimensional case, the critical thresholds\nare investigated for the restricted EP system in [20, 21] and EP system with radial symmetry in\n[6, 26, 27].\nThe case with pressure i.e., the pressure term p(ρ) =ργwith γ⩾1 is added to the EP system,\nbut with c= 0 and ν= 0, is also studied in [25]. The critical thresholds for the EP system with\npressure and nonzero background state are a challenging open problem. We refer to [14] for theCRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 3\nglobal-in-time existence of solutions with small amplitude to the 1D EP system for electrons. The\ncritical thresholds for the 1D Euler-alignment or Euler–Poisson-alignment systems are analyzed in\n[7, 10, 24]. The finite-time singularity formation for the 1D EP system for cold ions is also discussed\nin [1, 19].\nBeyond those developments, there is little literature on critical thresholds in EP systems with\nvariable background. In this context, we mention the recent work [5] in which critical thresholds\nconditions in the system (1.1) with the attractive forces are investigated. In order to handle new\ndifficulties arising from the consideration of the variable background, a combination of phase plane\nanalysis and comparison principle is used in [5]. However, as mentioned in [5], that strategy would\nnot be worked in the presence of repulsive forces.\n1.3.Main results. In this section, we present our main results on Euler–Poisson equation with\nbackground states. The purpose of this work is four-fold:\n(i) we present a new framework for the well-posedness theory for the EP system with the back-\nground, which includes a rigorous treatment of the neutrality condition (1.4), while justifying this\nframework by presenting concrete examples;\n(ii) we point out that in the attractive case, the set of relevant configurations that yield global\nsmooth solutions — with variable and even with constant background, is, in fact, restricted to the\nborderline case due to the neutrality condition;\n(iii) we present an in-depth threshold analysis for global solutions of the repulsive EP system with\nvariable background; and\n(iv) as an application of repulsive case, we study the case of cold plasma corresponding to expo-\nnential background c=eϕ.\nBelow, we present the main results in more detail.\n1.3.1. Local well-posedness and neutrality condition .We first develop a local-in-time well-\nposedness theory for the EP system with a non-vanishing background state (1.1). The EP system\nwith zero-background was studied by many authors, and in particular we refer to the Hs(R) well-\nposedness in [17, Appendix]. However, we cannot specify an appropriate reference for the local-\nin-time well-posedness theory for the EP system with non-vanishing background, (1.1). Here, the\nneutrality condition, (1.4), implies the non-integrability ρ(t,·)/∈L1(R), in order to match the\ninfinite mass that ‘enters’ the system from infinity,Z\nRc(t,·)dx=∞. Consequently, the local well-\nposedness of (1.1) requires a careful study of proper regularity space, beyond Hs(R). This is the\ncontent of our first theorem. We fix s⩾2 throughout the paper.\nTheorem 1.1 (Local well-posedness). Suppose that the initial data (ρ0, u0)satisfy\n(ρ0−c0, u0)∈Hs∩˙H−1(R)×Hs+1(R). (1.5)\nThen there exists a positive constant T > 0such that the system (1.1)–(1.2) admits a unique\nsolution (ρ, u)satisfying\n(ρ−c, u)∈L∞(0, T;Hs∩˙H−1(R))×L∞(0, T;Hs+1(R)). (1.6)\nMoreover, if the solutions ρand∂xuare bounded over the time interval [0, T], i.e.\n|∥{ρ, u}|∥[0,T]:= sup\n0⩽t⩽T(∥ρ(t)∥L∞+∥∂xu(t)∥L∞)<∞,\nthen there is propagation of higher order regularity of (ρ, u)expressed in terms of\n|∥{ρ(t,·), u(t,·)}|∥s:=∥(ρ−c)(t)∥Hs∩˙H−1+∥u(t)∥Hs+1,\nsup\n0⩽t⩽T|∥{ρ(t,·), u(t,·)}|∥s⩽C|∥{ρ0(·), u0(·)}|∥seηT, (1.7)\nwhere η∼ |∥{ ρ, u}|∥[0,T]+∥∂tc∥L∞(0,T;Hs∩˙H−1)+∥∂xc∥L∞(0,T;Hs)+ 1.4 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\nWe emphasize that the local well-posedness stated in Theorem 1.1 does not necessarily require\nthe neutrality condition (1.4) to hold. On the other hand, the following result shows that it is\nimpossible to construct the solution map without ˙H−1in (1.6). Thus, Theorems 1.1 and 1.2 tell us\nthat it is essential to equip ρ0−c0with additional ˙H−1regularity to secure the local well-posedness.\nTheorem 1.2 (Non-existence). For the EP system (1.1)–(1.2) with c= ¯c > 0, there exists\n(ρ0, u0)satisfying\n(ρ0−¯c, u0)∈Hs(R)×Hs+1(R)\nsuch that there is no solution to (1.1)–(1.2) with initial data (ρ0, u0), which fulfills\n(ρ−¯c, u)∈L∞\u0000\n[0, δ];Hs(R)×Hs+1(R)\u0001\nfor any δ >0.\nIn particular, Theorem 1.2 implies that the EP system (1.1)–(1.2) is ill-posed inHs×Hs+1(R).\nOur next theorem identifies a proper subset of initial configurations (1.5), which secure the\npropagation of neutrality, a question that seems to be mostly ignored in the literature. To provide\na rigorous treatment of the extended system (1.1),(1.4), we introduce proper regularity conditions\nfor the background state c(t, x), given by\n(\n∂tc∈L∞(R+;Hs∩˙H−1∩L1(R)),\n∂xc∈L∞(R+;Hs(R)).(1.8)\nIn particular, the regularity imposed in (1.8) 1implies that the background function csatisfies the\ncompatibility condition (1.3). Indeed, any function fin˙H−1∩L1(R), has a continuous Fourier\ntransform bf(·) such that |ξ|−1bf(ξ)∈L2(R),and in particular, it delivers that bf(0) = 0, i.e.,Z\nRf(x) dx= 0. Hence, we deduceZ\nR∂tc(t, x)dx= 0 a.e. t∈R+, and by Fubini’s theorem, we\nobtain the compatibility (1.3),\nZ\nR(c(t, x)−c0(x))dx=Z\nRZt\n0∂tc(t, x)dtdx=Zt\n0Z\nR∂tc(t, x)dxdt= 0,∀t >0.\nTheorem 1.3 (Well-posedness with neutrality). Suppose that the initial data (ρ0, u0)satisfy\n(ρ0−c0, u0)∈Hs∩˙H−1∩L1(R)×Hs+1∩BV(R). (1.9)\nThen the system (1.1)-(1.2) and(1.4) admits a classical solution (ρ(t,·), u(t,·))such that\nsup\n0⩽t<T(∥(ρ−c)(t)∥L1+∥(∂xu)(t)∥L1)<∞ (1.10)\nfor some T >0.\nIn view of the ˙H−1regularity of ρ0−c0secured in Theorem 1.1 and its integrability in Theorem\n1.3, there follows the neutrality condition (1.4),Z\nR(ρ(t, x)−c(t, x)) dx= 0,∀t∈[0, T). Thus,\nTheorem 1.3 identifies the set of initial data (1.9), which give rise to classical solution (ρ, u) of the\nextended system (1.1), (1.4). Again, Theorem 1.3 emphasizes that the density ρ(t,·) cannot be\nintegrable; only ( ρ−c)(t,·) is.\nIn our next Theorem 1.4, we construct an anomalous solution , exhibiting the non-propagation of\nthe neutrality in time even with the neutralized initial data. To circumvent this anomaly, additional\nregularity must be imposed, for instance, u0∈BV(R). Consult Lemma 2.1 for the discussion on\nthe constant background case, where the necessary and sufficient condition for the propagation of\nneutrality is presented.CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 5\nTheorem 1.4 (Non-propagation of neutrality: Anomalous solution). Consider the EP\nsystem (1.1) withc= ¯c >0. There exists a initial data (ρ0, u0)satisfying\n(ρ0−¯c, u0)∈Hs∩˙H−1∩L1(R)×Hs+1(R),\nsuch that any solution (ρ, u)of class C11to(1.1)–(1.2) with initial data (ρ0, u0)does not fulfill the\nneutrality condition (1.4).\n1.3.2. The case of attractive forcing .In this part, we discuss how the neutrality condition\naffects the critical threshold phenomenon of the attractive EP system, which is considered to be\nirrelevant. The threshold conditions for the attractive EP system were derived earlier in [13, 9, 5].\nThe sub-critical condition for the existence of a global classical solution of the attractive EP system\nis given by \n−ν+p\nν2+ 4c−\n2!\nu′\n0(x)⩾(ρ0(x)−c−)∀x∈R, (1.11)\nwhere c−:=c >0 for the constant [13, Theorem 3.2], [5, Theorem 2.5] and c−:= min c(·,·)>0\nfor the variable background case [5, Theorem 2.2]. In particular, this threshold is sharp for the\nconstant background case; if there exists x∈Rwhich fails to satisfy the inequality in (1.11), then\nthe solution forms singularity in a finite time. In the following theorem, however, we find that\nthe sub-critical region should be further restricted to its borderline zone — initial configurations\nsatisfying equality in (1.11), in order for the neutrality condition to be fulfilled.\nTheorem 1.5 (Reduction to the borderline). Suppose that (ρ, u)is a global-in-time classical\nsolution of the attractive EP system (1.1)–(1.2)–(1.4), subject to initial data satisfying (1.11) . Then,\nthe set of admissible c(·,·)’s is necessarily restricted to the constant background, namely — c(·,·) =\nc−and(1.11) is reduced the borderline case\n \n−ν+p\nν2+ 4c−\n2!\nu′\n0(x) = (ρ0(x)−c−)∀x∈R. (1.12)\nRemark 1.6.Theorem 1.5 shows that in order to entertain the neutrality condition, the sub-critical\ncriteria for variable backgrounds, (1.11), is too restrictive in the sense that it will rule out all but\nthe borderline cases. In particular, for the constant background case c=c−, the solution ( ρ, u) of\nneurality-satisfying attractive EP system has global-in-time regularity if and only if its initial data\n(ρ0, u0) satisfies (1.12).\nThis reduction to the borderline is a main theme in the attractive case. We demonstrate this in\nthe context of variable background, c=c(t, x). Now assume that |u0(x)||x|→∞−→ 0. We integrate\n(1.11) to find\n \n−ν+p\nν2+ 4c−\n2!\nu0(x)\f\f\fR\n−R⩾RZ\n−R\u0000\nρ0(x)−c0(x)\u0001\ndx+RZ\n−R\u0000\nc0(x)−c−\u0001\ndx. (1.13)\nBy assumption, u0(x)\f\f\fR\n−Rvanishes as R↑ ∞ (in fact, equals zero on the periodic case) and the\nneutrality condition yieldsZ\nR\u0000\nc0(x)−c−\u0001\ndx= 0, which implies c0(x) =c−and we get equality at\n1In this paper, we say fis of class C1iff∈ C1([0, T)×R) i.e., |f(t, x)|+|∂tf(t, x)|+|∂xf(t, x)|is uniformly\nbounded in [0 , T′]×Rfor any T′< T. We note that any solution constructed in Theorem 1.1 is of class C1by means\nof Morrey’s embedding.6 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\n(1.13) R→∞. By assumption, both sides tend to zero with R↑ ∞, hence the equality (1.11) takes\nplace almost everywhere. Indeed, for initial data satisfying (1.11), we have\n \n−ν+p\nν2+ 4c−\n2!\n∂xu(t, x)⩾(ρ(t, x)−c−)∀x∈R,\nso that we can repeat the previous argument for each t >0; in particular, we get c(·,·) =c−. In\nSection 3, we recover the critical threshold results for attractive EP systems with constant/variable\nbackgrounds via the Lyapunov function method. It captures the solution trajectories in a geometric\nway so that it can be more intuitively understood than the previous approaches. More importantly,\na notable feature of the Lyapunov-based approach is that it goes beyond the attractive forcing case;\nit can be employed in obtaining the critical thresholds for the repulsive EP systems, in particular,\nwith variable backgrounds.\n1.3.3. The case of repulsive forcing .We now turn our attention to the study of critical thresh-\nolds for the repulsive EP systems. Unlike the attractive interaction case, there is no literature on\nthe sub-critical criteria for the repulsive EP system in the variable backgrounds. We provide the\nfirst sub-critical results in this case. In order to present our results more precisely, we must intro-\nduce the method of phase plane analysis. To this end, we consider the auxiliary variables ( s, w)\n(introduced by H. Liu, S. Engelberg and the last author [13] in their study of critical thresholds of\nEP system with constant backgrounds),\ns:=1\nρ,and w:=∂xu\nρ.\nLet□′:=˙□(t, x(t)) denote differentiation along particle path x(t) =x(t;α),\n˙x(t;α) =u(t, x(t;α)), x(0;α) =α. (1.14)\nWe find that (abusing notations) w(t) =w(t, x(t)) and s(t) =s(t, x(t)) satisfy\n(\nw′=−νw+k(1−cs)\ns′=w.(1.15)\nWe observe that the question of global-in-time regularity vs. finite-time breakdown is reduced\nto whether s(t) attains zero — ρ(t,·) blows up — in a finite time. There are two classes of initial\nconfigurations ( w0, s0) of our interest. We say Σ♭is a sub-critical region of (1.15) if it is an invariant\nset, i.e., ( w0, s0)∈Σ♭implies ( w(t), s(t))∈Σ♭for all t >0. On the other hand, we say Σ♯is a\nsuper-critical region of (1.15) if for each ( w0, s0)∈Σ♯, there exists T∗>0 such that s(T∗) = 0.\nHence, the set of sub/super-critical region gives rise to the criteria of initial data ( ρ0, u0) for which\nthe local-in-time classical solution of (1.1)–(1.2) becomes global or forms singularity in a finite time.\nLastly, we say the critical threshold is sharp if\nR×R+= Σ♭∪Σ♯.\nFor the constant backgrounds, where the solutions to (1.15) can be obtained explicitly, the sharp\ncritical threshold was obtained in [13] for the undamped case, precisely given as:\nΣ♭=n\n−p\ns(2−¯cs)< w <p\ns(2−¯cs)o\n,Σ♯=R×R+\\Σ♭, (1.16)\nwhere c= ¯c >0 denotes the background. Recently, Bhatnagar and Liu obtained the sharp critical\nthresholds for the damped case [4].\nFor the variable backgrounds, however, we note that c=c(t, x(t, α)) in (1.15) 2varies along the\ncharacteristic and the system (1.15) is not closed. We overcome this non-locality by exploiting the\nuniform bounds of c. We compare the system (1.15) to the one associated with its extremes c=c±,\nand we establish the critical thresholds for the variable background by the solutions originatingCRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 7\nfrom the constant backgrounds. This idea — a Lyapunov-based approach — is widely used for\nattractive and repulsive force in Sections 3 and 4. By nature of this comparison method, all our\nresults in the sequel are sharp in the sense that we can recover the sharp critical threshold results\nfor the constant backgrounds once we set c−=c+.\nThroughout this paper, we use\nL(w, s) =w+f(s)\nas a general form of the Lyapunov function. fwill be chosen so as to make the signofLdetermine\nthe critical threshold regions. In particular, L(w, s) = 0 designates the boundary of sub/super-\ncritical regions — often, we will call this by threshold line.\nTo obtain the threshold lines of the (damped) repulsive EP system, we make use of the following\nLyapunov functions,\nLP(w, s) :=w+p\n2P(s),LN(w, s) :=w−p\n2N(s), (1.17)\nwhere PandNare the maximal solutions of the following ODEs\ndP\nds=νp\n2P(s) + 1−c1s, P (0) = 0 , (1.18)\ndN\nds=−νp\n2N(s) + 1−c2s, N (s∗) = 0 , (1.19)\nwith constants c1, c2>0,ν⩾0 and s∗>1\nc2. The choice of c1andc2will depend on the context,\neither ¯ cfor the constant background or c±for the variable background case.\nTo exploit these Lyapunov functions, we characterize the domains ofp\n2P(·) andp\n2N(·). By\nabusing notation, we denote the domain ofp\n2P(·) by Dom( P). We appeal the parallel statement\ntoN. Indeed, the domains are characterized as\nDom( P) =(\n[0,∞) ν⩾2√c1\n[0,˜s] 0 ⩽ν <2√c1, Dom( N) =(\n(−∞, s∗] ν⩾2√c2\n[s∗∗, s∗] 0 ⩽ν <2√c2,(1.20)\nwhere\n˜s=1 +eγ1\nc1, s∗∗=1\nc2−\u0012\ns∗−1\nc2\u0013\neγ2, γ i:=πν√\n4ci−ν2, i∈ {1,2}. (1.21)\nIn the context of constructing the sub/super-critical regions, we always choose s∗:= ˜sso as to\nmake these regions as large as possible. Here, we mention that the formula of s∗∗plays a vital role\nin establishing the critical regions.\nThe main novelty of this approach originated with M. Bhatnagar & H. Liu in [4], with the purpose\nof replacing an explicit algebraic description of the critical threshold, by a differential description\nexpressed in terms of a maximal solution of the ODE in w–splane:\ndQ(s)\nds=ν+1−¯cs\nQ(s). (1.22)\nIndeed, while√\n2Pand−√\n2Ncoincide with the maximal solution Qof [4] in the case of c1=c2= ¯c,\nhere we make a distinction between√\n2Pand−√\n2N, as we trace the the signofQ, corresponding\nto the positive and negative solution. In Section 4, we provide precise characterization of domains\n(1.20) – (1.21), replacing the graphical representation associated with (1.22) in [4].\nComparison Principle. We now introduce our comparison principles, deploying the Lyapunov\nfunctions (1.17), to establish the critical thresholds. For this, we denote P±by the solution of\n(1.18) with c1=c±andN±by that of (1.19) with c2=c±. The weak comparison principle is given\nas follows.\nLP−(w(t), s(t))⩽0 if LP−(w0, s0)⩽0,\nLN+(w(t), s(t))⩾0 if LN+(w0, s0)⩾0(1.23)8 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\nprovided that\n∀τ∈[0, t], s(τ)∈Dom( P−) (resp. s(τ)⩾0 and s(τ)∈Dom( N+)).\nThis gives rise to the construction of super-critical regions.\nΣ♯:=R×R+\\n\n−p\n2P−(s)< wo\nforν⩾2√c−,and\nΣ♯:=R×R+\\n\n−p\n2P−(s)< w <p\n2N+(s)o\nfor 0⩽ν <2√c−. (1.24)\nOn the other hand, we have strong comparison principle.\nLP+(w(t), s(t))>0 if LP+(w0, s0)>0,\nLN−(w(t), s(t))<0 if LN−(w0, s0)<0(1.25)\nas long as\n∀τ∈[0, t], s(τ)∈Dom( P+) (resp. s(τ)⩾0 and s(τ)∈Dom( N−)).\nIt leads us to define\nΣ♭:=n\n−p\n2P+(s)< wo\nforν⩾2√c+,and\nΣ♭:=n\n−p\n2P+(s)< w <p\n2N−(s)o\nfor 0⩽ν <2√c+. (1.26)\nIt is not always the case that the set Σ♭(1.26) is sub-critical; some extra conditions should be met\nto secure the sub-criticality.\nClosing Condition. In order to prove that Σ♭(1.26) and Σ♯(1.24) are indeed sub/super-critical\nregions, we require\ns∗∗⩽0 (1.27)\nto hold. We call (1.27) by closing condition . The term closing is motivated by the geometric\ndepiction that the Σ♭andR×R+\\Σ♯are enclosed by its boundaries as in Figures 1 and 3.\nThe closing condition plays a key role in constructing a sub/super-critical region. We illustrate\nthis by demonstrating that Σ♯in (1.24) is a super-critical set. We readily check that the closing\ncondition is validated for Σ♯. Indeed, by assigning ( c1, c2) = (c−, c+) to (1.21), we obtain\ns∗⩾2\nc−>1\nc+,and s∗∗⩽1\nc+−\u00122\nc−−1\nc+\u0013\n⩽0.\nIn particular, it implies that LN+(w, s) = 0 line crosses w-axis atp\n2N+(0)⩾0 so that the\nboundary of Σ♯can be decomposed as:\n{LP−(w, s) = 0} ∪ {L N+(w, s) = 0 , s⩾0} ∪ {w⩾p\n2N+(0), s= 0} ∪ {w⩽0, s= 0}.\nBy weak comparison principle (1.23), we deduce that if ( w0, s0)∈Σ♯, then ( w(t), s(t)) can\nescape Σ♯only through s= 0 line (with w⩽0 because s(t)′=w(t)⩾p\n2N+(0)>0 on the\nright part of horizontal boundary.) In fact, for each ( w0, s0)∈Σ♯, there exists t∗<+∞such that\ns(t∗) = 0. Consult Proposition 4.4, 4.5. It proves that Σ♯is a super-critical set. Hence, we obtain\na super-critical condition for the undamped repulsive EP with variable backgrounds.\nTheorem 1.7 (Finite-time breakdown). Consider the repulsive EP system (1.1)–(1.2). For a\ngiven initial data (ρ0, u0), if there exists x∈R, which does not satisfy:\n• − ρ0(x)s\n2P−\u00121\nρ0(x)\u0013\n< u′\n0(x) when ν⩾2√c−;\n• − ρ0(x)s\n2P−\u00121\nρ0(x)\u0013\n< u′\n0(x)< ρ0(x)s\n2N+\u00121\nρ0(x)\u0013\nwhen 0 ⩽ν <2√c−,CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 9\nthen the solution (ρ, u)will lose C1regularity in a finite time.\nRemark 1.8.For the undamped ν= 0 case, we have\nΣ♯=R×R+\\(\n−p\ns(2−c−s)< w <s\ns(2−c+s) +4\nc−\u0012c+\nc−−1\u0013)\n. (1.28)\nNote that Σ♯coincides with (1.16) if we set c−=c+= ¯c. Moreover, it improves previously known\nsuper-critical results for repulsive EP systems with variable backgrounds [1, 19]. In the phase plane\ncoordinates, these previous super-criteria can be expressed as s⩾2\nc−andw+√\n2s⩽0, respectively.\nIn fact, (1.28) contains these two regions. We display this in Figure 1 for the reader’s convenience.\n(a)The super-critical set in (1.28)\n (b)Violet: s⩾2/c−, Blue: w+√\n2s⩽0\nFigure 1. Ilustration of the super-critical regions\nWe now turn our attention to the sub-critical case. The closing condition can also be exploited\nto show that Σ♭in (1.26) is a sub-critical set. Once we assume the closing condition hold for Σ♭,\nthen the boundary of Σ♭can be decomposed as\n\b\nLP+(w, s) = 0\t\n∪\b\nLN−(w, s) = 0 , s > 0\t\n∪n\n0< w⩽p\n2N−(0), s= 0o\n. (1.29)\nNote that Σ♭is an open set, and we can prove that any trajectory issued inside this set cannot hit\nthe boundary by strong comparison principle. Consult Proposition 4.7 for the details.\nUnlike the super-critical set, the closing condition is not satisfied for Σ♭in (1.26) in general. For\ninstance, if ν= 0 with c−< c+, then the closing condition is violated because\ns∗∗=2\nc−−2\nc+>0.\nIn fact, the closing condition is not merely an artifact of technique but reflects the difficulties\nin obtaining sub-critical regions for the repulsive EP system. To exhibit this, we introduce the\nfollowing system of ODEs.\n(\nw′= 1−(1 +ϵsin(t))s,\ns′=w,(1.30)\nwith ( w0, s0) = (0 ,1). Although the deviation of background from 1 is assumed to be small as\n|(1 + ϵsin(t))−1|⩽ϵ, the oscillation of ( w(t), s(t)) and c(t) can be aligned so that it makes10 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\naresonance . In particular, s(t) attains zero in a finite time. We implemented the numerical\nexperiment of (1.30) with ϵ= 0.05 using the built-in ode45 Matlab command. See Figure 2 for\nits depiction. This example convinces us that the closing condition is necessary to construct a\nsub-critical region.\n0 5 10 15 20 25 30 35 40 45 50-0.500.511.522.5\n-1.5 -1 -0.5 0 0.5 1 1.500.511.52\nFigure 2. The evolution of s(left) and the phase plane plot (right) of the solution\ntrajectory of (1.30).\nIn the presence of damping ( ν >0), the closing condition always holds for ν⩾2√c+case. For\n0< ν < 2√c+case, the closing condition can be entertained by imposing suitable conditions on\nc+,c−, and ν. The heuristic is that the ratio of c+overc−is close enough to 1, and the damping\neffect is large enough to control the oscillation so that the system is similar to that of a constant\nbackground case.\nTheorem 1.9 (Global-in-time regularity). Consider the repulsive EP system (1.1)–(1.2). It\nadmits a global classical solution, depending on the relative size of νand√c±. Specifically — global\nclassical solution exists if the initial data (ρ0, u0)falls into one of the following cases.\n•Case #1.ν⩾2√c+and−ρ0(x)s\n2P+\u00121\nρ0(x)\u0013\n< u′\n0(x);\n•Case #2.0< ν < 2√c+and\n−ρ0(x)s\n2P+\u00121\nρ0(x)\u0013\n< u′\n0(x)< ρ0(x)s\n2N−\u00121\nρ0(x)\u0013\n, s +:=1 +eγ+\nc+, γ±:=πνp\n4c±−ν2,\nfor all x∈Rand one of the following two sub-cases is fulfilled:\n(\n•Case #2.1 : either ν⩾2√c−and s+c−>1,\n•Case #2.2 : or 0 < ν < 2√c−and eγ−(s+c−−1)⩾1.(1.31)\nRemark 1.10.In fact, the condition (1.31) is equivalent to the closing condition (1.27). We arrive\nat (1.31) is by putting ( c1, c2) = ( c+, c−) in (1.21). If we assume c−=c+, the closing condition\nis satisfied without any extra condition, and we recover the sharp critical threshold result for the\ndamped EP system in [4].\nWe display the critical threshold results for the repulsive interaction case in Figure 3.CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 11\n(a)ν >2√c+case\n (b)2√c−⩽ν <2√c+case\n (c)0< ν < 2√c−case\nFigure 3. Illustration of the sub/super-critical regions for the repulsive EP system\n1.3.4. Application to the EP system for cold plasma ion dynamics .Our final result is\nfinite-time singularity formation and the global regularity of the damped EP system for cold plasma\nion dynamics. Consider a plasma consisting of electrons and ions on R, where both electrons and\nions have constant temperature. We assume that the temperatures of electrons and ions equal one\nand zero, respectively. In this case, our main system is given by\n\n\n∂tρ+∂x(ρu) = 0 ,\n∂tu+u∂xu=−νu−∂xϕ,\n−∂xxϕ=ρ−eϕ,(1.32)\nwhere the density of electrons ρeis given by Maxwell–Boltzmann relation [11], i.e., ρe=eϕ, where\nϕ=ϕ(t, x) is the electric potential.\nNote that the system (1.32) corresponds to (1.1) with ( k, c) = (1 , eϕ). We consider the system\n(1.32) around a constant state (1 ,0), i.e.\n(ρ, u)(t, x)→(1,0) as |x| → ∞ . (1.33)\nWe notice from the above that ϕ(t, x)→0 as|x| → ∞ . For the system (1.32)–(1.33), the local-in-\ntime existence and uniqueness of smooth solutions are studied in [16].\nIn order to apply the framework developed in Section 1.3.3, we need to secure lower and upper\nbounds of eϕ. To this end, we introduce the total energy function H=H(t) for (1.1)–(1.2) by\nH=H(t) :=Z\nR\u00121\n2ρu2+1\n2|∂xϕ|2+U(ϕ)\u0013\ndx, (1.34)\nwhere U(r) := ( r−1)er+ 1, which is nonnegative for all r∈R.\nThen, as long as there exist classical solutions, one can readily check that the total energy is not\nincreasing in time, i.e., H(t)⩽H0:=H(0) for all t⩾0, and conserved in time when ν= 0. It\nwas investigated in [1] that a uniform bound of the potential function, ϕ, can be obtained by H0.\nHence, we can apply the results in the previous section to this case.\nTheorem 1.11 (Finite-time breakdown). Consider the repulsive EP system (1.32) –(1.33) with\nν= 0. For a given initial data (ρ0, u0), if there exists x∈Rwhich does not satisfy\n−p\n2ρ0(x)−c−< u′\n0(x)<s\n2ρ0(x)−c++4\nc−\u0012c+\nc−−1\u0013\nρ0(x)2,\nthen the classical solution (ρ, u)will lose C1regularity in a finite time. Here, c−:= exp\u0000\nV−1\n−(H0)\u0001\nandc+:= exp\u0000\nV−1\n+(H0)\u0001\ndenote positive constants, where V−1\n±are inverses of V±defined in (4.7).12 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\nFor the damped Euler–Poisson system for a cold plasma, the global existence and uniqueness\nofHssolutions are obtained in [18] under the smallness assumptions on the initial data ( ρ0, u0)\nnear the constant equilibrium state (1 ,0) in Hsnorm. However, the following theorem shows that\nthe global-in-time existence of classical solutions to the damped Euler–Poisson system for a cold\nplasma can be obtained onlyunder the smallness assumption on the initial energy H0compared to\nthe strength of damping.\nTheorem 1.12 (Global-in-time regularity). For a class of initial data (ρ0, u0), whose initial\nenergy H0(1.34) is small enough, we assume that\nν⩾Cp\nH0, (1.35)\nfor some absolute constant C >0and\n∥ρ0−1∥L∞+∥u′\n0∥L∞⩽Λ0, (1.36)\nwhere Λ0>0is a small absolute constant. Then the damped cold-ion system (1.32) –(1.33) admits\nglobal classical solution with initial data (ρ0, u0).\nRemark 1.13.We note that (1.36) restricts ( ρ0, u0) to fall into the sub-critical regions in Theorem\n1.9. We can take Λ 0=1\n2, for instance. Thus, we are only concerned with the smallness of H0and\nthe requirement on the strength of damping (1.35).\n1.4.Outline of the paper. This paper is organized as follows. In Section 2, we establish the\nlocal-in-time well-posedness of the EP system without imposing the neutrality condition. We then\nprove the non-existence result of Theorem 1.2. Moreover, we study the well-posedness of the EP\nsystem equipped with the neutrality condition and construct the anomalous solutions in Theorem\n1.4. In Section 3, we present our Lyapunov-based approach to establish the critical thresholds in\nthe attractive case. In Section 4, we discuss the repulsive interaction case, completing the proofs\nof Theorems 1.7 and 1.9. As a consequence, we prove Theorem 1.12, which concerns the global\nregularity of damped cold plasma ion dynamics.\n2.Local well-posedness and the neutrality condition\nIn this section, we present a rigorous mathematical treatment of the local well-posedness and the\nneutrality condition of the EP system introduced in Section 1.3.1. We begin by proving the local\nwell-posedness theory, which is independent of the neutrality condition.\n2.1.Local well-posedness and Non-existence.\nProof of Theorem 1.1. We only provide a priori estimates of solutions in the desired regularity\nspaces. The local well-posedness can be obtained by the standard arguments developed for the\ntypes of conservation laws; see [4, 5, 22] for instances. Indeed, we obtain\nd\ndt\u0000\n∥ρ−c∥Hs∩˙H−1+∥u∥Hs+1\u0001\n⩽C(∥∂xu∥L∞+∥ρ∥L∞+∥c∥L∞+∥∂xc∥Hs+ 1)\u0000\n∥ρ−c∥Hs∩˙H−1+∥u∥Hs+1\u0001\n+∥∂tc∥Hs∩˙H−1, (2.1)\nwhere C > 0 is an absolute constant, see Appendix A for the details of (2.1). We apply the\nGr¨ onwall’s lemma to obtain (1.7), thereby concluding the proof. □\nIn fact, the proofs of Theorems 1.2 and 1.4 are the main themes in this section. They restrict\nour attention to the case of a constant background state. For the sake of simplicity in presenting\nour results, we only focus on the undamped case. The damped case can be treated in a similar\nway. Our analysis relies on the characteristic method and, especially, the explicit formulation of\nJacobian of the characteristic flow. We briefly explain the main tools and the strategy.CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 13\nFor aC1solution ( ρ, u) to EP system (1.1)–(1.2) with c= ¯c, we consider the characteristic flow\nx(t, α) defined in (1.14). Write\nΓ(t, α) :=∂αx(t, α),\nsignifying the Jacobian of the map x(t,·) :R→R. We note that Γ(0 , α) = 1 and x(t,·) is a\ndiffeomorphism as long as Γ( t,·)>0. In particular, we have\nlim\nα→±∞x(t, α) =±∞. (2.2)\nTaking ∂αto (1.14), we obtain\n∂xu(t, x(t, α)) =∂tΓ(t, α)\nΓ(t, α). (2.3)\nIn view of the mass conservation ρ′=−ρ∂xu, we find that\nρ(t, x(t, α)) =ρ0(α)\nΓ(t, α). (2.4)\nFinally, we recall the explicit formulas of Γ( t, α) for the undamped ( ν= 0) case [13]. They will lay\nthe foundation of the proofs throughout this section.\n•Repulsive case:\nΓ(t, α) = 1 +\u0012ρ0(α)\n¯c−1\u0013\n(1−cos(√\n¯ct)) +u′\n0(α)sin(√¯ct)√¯c, (2.5)\n•Attractive case:\nΓ(t, α) = 1 +\u0012ρ0(α)\n¯c−1\u0013\n(1−cosh(√\n¯ct)) +u′\n0(α)sinh(√¯ct)√¯c. (2.6)\nWe now prove Theorem 1.2 based on a contradiction argument. Assuming the existence of a\nsolution satisfying (2.7), subject to “non-neutral” initial data, i.e.,\nZ\nR(ρ0(x)−¯c) dx̸= 0,\nthe non-neutral effect prevents ufrom decaying at the far field that u(t,·)/∈Hs+1as soon as t >0.\nProof of Theorem 1.2. We set initial data ( ρ0, u0) by assigning u0= 0 and ρ0= ¯c+g(x) where\ng(x)∈L1∩Hs(R) andZ\nRg(x) dx̸= 0.\nWe stress that g(x)/∈˙H−1(R); otherwise, it would violate the non-vanishing integral condition\nabove. Indeed, there are infinitely many such functions; for instance, we choose g(x) = (1 + x2)−1\nin this context.\nSuppose, by way of deriving a contradiction, that there exists a solution to EP system (1.1)–(1.2)\nsatisfying\n(ρ−¯c, u)∈L∞\u0000\n[0, δ];Hs(R)×Hs+1(R)\u0001\n(2.7)\nfor some δ >0. We note that this, in particular, requires uto decay at the far field,\nlim\nα→±∞u(t, α) = 0 ,∀t∈[0, δ]. (2.8)\nTo investigate the solution behavior at infinity, we observe that ∥∂xu∥L∞is uniformly bounded\nint∈[0, δ]. Hence, we can employ the forward characteristic x(t,·) defined in (1.14). For each14 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\nt∈[0, δ] and R >0, we infer from (2.3) that\nu(t, x(t, R))−u(t, x(t,−R)) =Zx(t,R)\nx(t,−R)(∂xu)(t, x) dx\n=ZR\nR(∂xu)(t, x(t, α))dx(t, α)\n=ZR\n−R∂tΓ(t, α) dα.\nHence, an application of the explicit formula (2.5), for the case of repulsive forcing, supplies that\nu(t, x(t, R))−u(t, x(t,−R)) =ZR\n−Rg(α) dα·sin(√¯ct)√¯c.\nThen, by taking R→ ∞ , we find that, for all small t >0,\nlim\nR→+∞(u(t, x(t, R))−u(t, x(t,−R)))̸= 0,\nyielding a contradiction to (2.8) by means of (2.2). The attractive forcing case can be tackled in\nthe same manner by recalling (2.6). The conclusion of the theorem is now immediate. □\n2.2.Neutrality condition and Anomalous solution. We next discuss the local well-posedness\nencompassing the neutrality condition. We prove that the propagation of neutrality is secured in\nthe regularity space recorded in Theorem 1.3.\nProof of Theorem 1.3. We estimate the evolution of ∥(ρ−c)(t)∥L1and∥(∂xu)(t)∥L1. Straightfor-\nward computations give\nd\ndt∥ρ−c∥L1⩽∥∂x((ρ−c)u)∥L1+∥∂tc∥L1+∥∂x(cu)∥L1\n⩽∥ρ−c∥H1∥u∥H1+∥∂tc∥L1+∥c∥L∞∥∂xu∥L1+∥∂xc∥L2∥u∥L2\n⩽|∥{ρ(t,·), u(t,·)}|∥2\ns+∥∂tc∥L1+∥c∥L∞∥∂xu∥L1+∥∂xc∥2\nL2+|∥{ρ(t,·), u(t,·)}|∥2\ns\nand\nd\ndt∥∂xu∥L1⩽∥∂xu∥2\nL2+∥u∥L2∥∂2\nxu∥L2+∥ρ−c∥L1⩽2|∥{ρ(t,·), u(t,·)}|∥2\ns+∥ρ−c∥L1.\nBy adding these two inequalities, we obtain\nd\ndt(∥ρ−c∥L1+∥∂xu∥L1)\n⩽(1 +∥c∥L∞) (∥ρ−c∥L1+∥∂xu∥L1) + 4|∥{ρ(t,·), u(t,·)}|∥2\ns+∥∂tc∥L1+∥∂xc∥2\nL2.\nApplying the Gr¨ onwall’s lemma to the above yields the desired result (1.10), and the theorem\nfollows. □\nFor the case of constant background, we identify the necessary and sufficient condition of the\nneutrality condition. It plays a central role in constructing the anomalous solutions.\nLemma 2.1 (Neutrality condition: constant background). Consider the EP system (1.1)\nwith c= ¯c >0. Let (ρ, u)be aC1solution of (1.1)–(1.2) in[0, T)×Rwith neutrality satisfying\ninitial data, i.e,\nρ0−¯c∈L1(R),withZ\nR(ρ0(x)−¯c) dx= 0. (2.9)\nThen, the propagation of neutrality condition (1.4) holds if and only if\nu0∈BV(R),withZ\nRu′\n0(x) dx= 0.CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 15\nProof of Lemma 2.1. Letx(t, α) be the forward characteristic flow satisfying (1.14). Note that\nΓ(t,·)>0 is well-defined for each t∈[0, T). We claim that\nρ(t,·)−¯c∈L1(R) if and only if Γ( t,·)−1∈L1(R).\nIn this context, we apply (2.4) to see that\nZx(t,R)\nx(t,−R)|ρ(t, x)−¯c|dx=ZR\n−R|ρ(t, x(t, α))−¯c|dx(t, α)\n=ZR\n−R\f\f\f\fρ0(α)\nΓ(t, α)−¯c\f\f\f\fΓ(t, α) dα\n=ZR\n−R|(ρ0(α)−¯c)−¯c(Γ(t, α)−1)|dα.\nOn applying the triangle’s inequality with (2.2), the claim follows. Hence, we deduce from the\nexplicit formulas of Γ( t,·)−1 recorded in (2.5) and (2.6) that u′\n0∈L1(R) if and only if Γ( t,·)−1∈\nL1(R), which is equivalent to ρ(t,·)−¯c∈L1(R).\nWe now assume that u′\n0∈L1(R). Similarly, in view of (2.4), one sees that\nZ\nR(ρ(t, x)−¯c) dx=Z\nR(ρ0(α)−¯c)−¯c(Γ(t, α)−1) dα=−¯cZ\nR(Γ(t, α)−1) dα.\nBy inserting (2.5), (2.6) into Γ( t,·)−1 and integrating over R, we discern that\nZ\nR(ρ(t, x)−¯c) dx= 0 if and only ifZ\nRu′\n0(α) dα= 0.\nThis confirms the conclusion of the lemma. □\nRemark 2.2.It is worth noting that if u0/∈BV(R), we infer that the local C1solution instanta-\nneously fails the neutrality condition by taking T > 0 arbitrarily small. Hence, this lemma leads\nus to find anomalous solutions in Theorem 1.4.\nProof of Theorem 1.4. We fix ρ0= ¯c >0. It satisfies (2.9), evidently. In view of Theorem 1.1 and\nLemma 2.1, we aim to choose u0satisfying\nu0∈Hs+1\\BV(R).\nConsider\nu0(x) :=sinx\n(1 +x2)3\n8.\nSince u0∈Hs+1(R), we obtain a local smooth solution ( ρ, u) via Theorem 1.1. In particular, we\nfind by means of Morrey’s embedding that this solution is of class C1. However, upon noting\nu′\n0(x) =cosx\n(1 +x2)3\n8−3\n4xsinx\n(1 +x2)11\n8/∈L1(R),\nwe find that ( ρ, u) fails the neutrality condition instantaneously by Lemma 2.1. This completes the\nproof. □\nWe, therefore, conclude from Theorems 1.3 and 1.4 that it is necessary to assume additional\nregularity u0∈BV(R) to secure the propagation of the neutrality condition.16 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\n3.Attractive forcing and the restricted borderline threshold\nIn this section, we focus on the critical threshold phenomenon of the attractive EP system (1.1)–\n(1.2). To apply the method of phase plane analysis introduced in Section 1.3.3, we consider the\nfollowing ODE system:\n(\nw′=−νw−(1−c(t)s),\ns′=w,(3.1)\nsubject to ( w(0), s(0)) = ( w0, s0), where c(t) is smooth function satisfying 0 < c−⩽c(t)⩽c+.\n3.1.Constant background case. To give a motivation for the Lyapunov function, we initiate\nthe discussion for the constant background case — c(t) = ¯c. In this case, (3.1) can be rewritten in\nthe matrix form as,\n w\ns−1\n¯c!′\n=\u0012\n−ν¯c\n1 0\u0013 w\ns−1\n¯c!\n. (3.2)\nThe dynamics of solution trajectories are determined by the eigenvalues of its coefficient matrix,\nwhose eigenvalues are given by\nλs=−ν−√\nν2+ 4¯c\n2<0 and λu=−ν+√\nν2+ 4¯c\n2>0,\nwith its corresponding eigenvectors Xs=\u0012\nλs\n1\u0013\nandXu=\u0012\nλu\n1\u0013\n. In particular, the equilibrium\npoint (0 ,1/¯c) of (3.2) is unstable; the solution curves are attracted to (0 ,1/¯c) along Xsdirection\nwhile repulsed from (0 ,1/¯c) along Xudirection.\nThis motivates us to define the following Lyapunov functions,\nLs(w, s) :=w−λs\u0012\ns−1\n¯c\u0013\nandLu(w, s) :=w−λu\u0012\ns−1\n¯c\u0013\n.\nIn particular, the solution ( w(t), s(t)) of (3.2) satisfies\nd\ndtLs(w(t), s(t)) =λuLs(w(t), s(t)) andd\ndtLu(w(t), s(t)) =λsLu(w(t), s(t)),\nobtaining\nLs(w(t), s(t)) =Ls(w0, s0)eλutandLu(w(t), s(t)) =Lu(w0, s0)eλst. (3.3)\nNote that Ls(w0, s0)⩾0 implies Ls(w(t), s(t))⩾0 for all t⩾0. This, combined with s′=wand\nthat w-intercept of the line Ls(w, s) = 0 is positive, yields that {Ls(w, s)⩾0, s > 0}isinvariant .\nOn the other hand, if Ls(w0, s0)<0, then (3.3) implies that\nLs(w(t), s(t))→ −∞ ,Lu(w(t), s(t))→0 as t→ ∞ .\nBy noticing\nLs(w, s)− L u(w, s) = (−λs+λu)\u0012\ns−1\n¯c\u0013\n,\nwe conclude that s(t) attains zero in a finite time. This argument recovers the sharp critical\nthreshold for the attractive EP with constant background [13, Theorem 3.2], [5, Theorem 2.5].CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 17\n3.2.Variable background case. To study the critical threshold for the variable background case,\nwe consider the following Lyapunov functions:\nL±\ns(w, s) :=w−λ±\ns\u0012\ns−1\nc±\u0013\nandL−\nu(w, s) :=w−λ−\nu\u0012\ns−1\nc−\u0013\n,\nwhere\nλ±\ns=−ν−p\nν2+ 4c±\n2<0 and λ±\nu=−ν+p\nν2+ 4c±\n2>0.\nWe readily check that\nd\ndtL−\ns(w(t), s(t))⩾λ−\nuL−\ns(w(t), s(t)), s(t)⩾0.\nSimilarly, this implies that {L−\ns(w, s)⩾0, s > 0}is a sub-critical region. On the other hand, we\nhave\nd\ndtL+\ns(w(t), s(t))⩽λ+\nuL+\ns(w(t), s(t)),d\ndtL−\nu(w(t), s(t))⩾λ−\nsL−\nu(w(t), s(t)), s(t)⩾0.\nBy Gr¨ onwall’s lemma, we observe that\n\u0000\n−λ+\ns+λ−\nu\u0001\ns(t) =L+\ns(w(t), s(t))− L−\nu(w(t), s(t)) +1\nλ+u−1\nλ−s⩽L+\ns(w0, s0)eλ+\nut+O(1)\nas long as s(t) is non-negative; in particular, if L+\ns(w0, s0)<0, then s(·) attains zero in a finite\ntime. Hence, we obtain the sub/super-critical thresholds for attractive EP systems with variable\nbackgrounds. They coincide with [5, Theorems 2.2 and 2.3].\nFigure 4. Illustration of the sub-critical {L−\ns(w, s)⩾0}and super-critical\n{L+\ns(w, s)<0}criteria for the attractive case\n4.Repulsive forcing and threshold with variable background\nIn this section, we aim to establish the critical thresholds for the repulsive EP system (1.1)–\n(1.2). To this end, we conduct the phase plane analysis as introduced in Section 1.3.3; for repulsive\ninteraction, we recall the transformed ODE system (1.15):\n(\nw′=−νw+ 1−c(t)s\ns′=w,(4.1)\nsubject to initial point ( w0, s0). Here we assume that 0 < c−⩽c(t)⩽c+.18 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\nWe first confirm the comparison principles. Note that\nd\ndtLP(w(t), s(t)) =−νw(t) + 1−c(t)s(t) + \nνp\n2P(s(t)) + 1 −c1s(t)p\n2P(s(t))!\nw(t)\n= \n1−c1s(t)p\n2P(s(t))!\nLP(w(t), s(t)) + ( c1−c(t))s(t).(4.2)\nSimilarly, we observe that\nd\ndtLN(w(t), s(t)) = \nc2s(t)−1p\n2N(s(t))!\nLN(w(t), s(t)) + ( c2−c(t))s(t). (4.3)\nIn particular, putting ( c1, c2) = (c−, c+) yields\nLP−(w(t), s(t))⩽LP−(w0, s0) exp Zt\n01−c−s(τ)p\n2P−(s(τ))dτ!\n,\nLN+(w(t), s(t))⩾LN+(w0, s0) exp Zt\n0c+s(τ)−1p\n2N+(s(τ))dτ!\n,\nprovided that s(τ) satisfies P−(s(τ))⩾0 and (resp. N+(s(τ))⩾0) for all τ∈[0, t], and the the\nweak comparison principle (1.23) follows. Similarly, by taking ( c1, c2) = ( c+, c−), we obtain the\nstrong comparison principle (1.25).\nThe Lyapunov-based approach explained in Section 1.3.3 — essentially for the closing condition\n(1.27) — makes use of the domains of√\n2Pand√\n2N. By witnessing the defining equation of P\nandNin (1.18), (1.19), we observe that the domains ofp\n2P(·),p\n2N(·) correspond to the set of\ns∈Rfor which P(s)⩾0,N(s)⩾0, respectively. It implies that the endpoints of the domains\nare precisely determined by zeros of PandN. The domains (1.20) can be obtained by the explicit\nsolutions for (4.1).\nWe first investigate the exact formula of ˜ sands∗∗in (1.21). We take index i∈ {1,2}. For\n0⩽ν <2√cicase, the solution ( w(t), s(t)) of (4.1) with c(t) =ci>0, (w0, s0) = (0 , a) is explicitly\ngiven as:\nw(t) =−1\nµi(¯ca−1) sin( ��it)e−νt\n2, µ i=r\nci−ν2\n4,\ns(t) =1\n¯c+\u0012\na−1\n¯c\u0013\u0012\ncosµit+ν\n2µisinµit\u0013\ne−νt\n2.(4.4)\nTo unveil the formula of ˜ s(1.21), we take a= ˜sand index i= 1 so that\nLP(w0, s0) = 0 +p\n2P(˜s) = 0 ,\nwhere Psatisfies (1.18). By (4.2), we observe that\nLP(w(t), s(t)) =w(t) +p\n2P(s(t)) = 0 , t∈\u0014\n0,π\nµ1\u0015\n.\nNote that the range of tcan be obtained by the criteria P(s(t))⩾0 and (4.4). In particular, we\ndeduce that\ns\u0012π\nµi\u0013\n= 0⇔˜s=s0=1 +eγ1\nc1.\nSimilarly, by taking a=s∗andi= 2, we obtain\np\n2N(s(t)) =w(t)⩾0 for t∈\u0014\n−π\nµ2,0\u0015\n.CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 19\nBy noting s0=s∗, we infer that\ns∗∗=s\u0012\n−π\nµ2\u0013\n=1\nc2−\u0012\ns∗−1\nc2\u0013\neγ2.\nThis confirms (1.21).\n4.1.Finite-time breakdown. In this subsection, we prove that finite breakdown occurs for the\nsuper-critical initial data. To this end, we present the following lemma, which is crucially utilized\nin obtaining an upper bound of the maximal time for which a local smooth solution persists.\nLemma 4.1. For the solution (w(t), s(t))of(4.1), the following identity holds:\nt2−t1⩽Zs(t1)\ns(t2)dsp\n2P−(s),\nprovided that\nw0+p\n2P−(s0)⩽0\nands(τ)∈Dom( P−)for all τ∈[t1, t2]. Indeed, we have equality in the integral if c(t) =c−and\nw0+p\n2P−(s0) = 0 .\nProof. By change of coordinates s=s(t), we have\nZs(t1)\ns(t2)dsp\n2P(s)=Zt1\nt2w(t)p\n2P(s(t))dt⩾Zt1\nt2(−1) dt=t2−t1,\nwhere the inequality comes from the weak comparison principle (1.23). Under the latter assumption,\nwe note that (4.2) turns the inequality into equality. This completes the proof. □\nSeveral remarks are in order. Suppose that Psatisfies (1.18) with c1= ¯c.\nRemark 4.2.For 0⩽ν <2√¯ccase, we obtain an explict integral formula:\nZ\nDom( P)dsp\n2P(s)=π\nµ¯c,with µ¯c:=r\n¯c−1\n4ν2.\nThe initial condition in (1.18) can be switched to P(a) = 0 for any a̸= 1/¯c. In particular, for ν= 0\ncase, we obtain the so-called elliptic integral :Za\n2\n¯c−adsp\n(a−s)(¯c(a+s)−2)=π√¯c.\nRemark 4.3.Forν⩾2√¯ccase, we have\nZs0\n0dsp\n2P(s)⩽log (λs0)∨2\nλ,with λ:=ν−√\nν2−4¯c\n2>0,\nfor any s0>0. In particular, it implies that Dom( P) = [0 ,+∞).\nProposition 4.4. Letν⩾2√c−. For any (w0, s0)∈Σ♯={(w, s) :w⩽−p\n2P−(s), s > 0}, the\nsolution to (4.1) with initial data (w0, s0)satisfies s(t∗) = 0 for some finite t∗>0.\nProof. We claim that s(t∗) = 0 with t∗⩽T0where\nT0:=Zs0\n0dsp\n2P−(s).\nSuppose not, i.e., there exists ( w0, s0)∈Σ♯such that s(t)>0 for all 0 ⩽t⩽t∗satisfying t∗> T 0.\nThen, by weak comparison principle (1.23), we have\nw(t)⩽−p\n2P−(s(t)),0⩽t⩽t∗.20 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\nHence, by Lemma 4.1, we note in passing\nt∗⩽Zs0\ns(t∗)dsp\n2P−(s)⩽T0< t∗,\nwhich leads to a contradiction. Hence, the conclusion of the proposition follows by recalling that\nT0<+∞via Remark 4.3. □\nProposition 4.5. Suppose that 0⩽ν <2√c−. For any\n(w0, s0)∈Σ♯={(w, s) :s >0} \\n\n(w, s) :−p\n2P−(s)< w <p\n2N+(s), s > 0o\n,\nthe solution to (4.1) with initial data (w0, s0)satisfies s(t∗) = 0 for some finite t∗>0.\nProof. We decompose Σ♯into three subsets:\nΣ♯\n1:=n\nw⩽−p\n2P−(s),0< s⩽s∗o\n∪ {w⩽0, s > s ∗},\nΣ♯\n2:={w >0, s⩾s∗},and Σ♯\n3:=n\nw⩾p\n2N+(s),0< s < s ∗o\n.\n•For ( w0, s0)∈Σ♯\n1, we put a:= max {s0, s∗}.Consider P∗(·) satisfying\ndP∗\nds=νp\n2P∗(s) + 1−c−s, P∗(a) = 0 .\nThen, we have\nLP∗(w0, s0) =w0+p\n2P∗(s0)⩽0.\nIndeed, for s0> s∗, we have w0+p\n2P∗(s0) =w0+p\n2P∗(a) =w0⩽0, and for 0 < s0⩽s∗,\nwe notice that w0+p\n2P∗(s0) =w0+p\n2P−(s0)⩽0. The weak comparison principle (1.23) also\nholds for Pas well. Hence, following the argument in the proof of Proposition 4.4 leads us to find\nthats(t1) = 0 with\nt1⩽Za\n0dsp\n2P(s)⩽π\nµ−, µ−:=r\nc−−ν2\n4,\nwhere the last inequality follows from Remark 4.2.\n•For ( w0, s0)∈Σ♯\n2, we prove that w(t2) = 0, s(t2)⩾s∗with t2⩽π/2√c−. To this end, we\ntemporarily introduce the following Lyapunov function, defined by\nθ−(w, s) := arctan\n√c−\u0010\ns−1\nc−\u0011\nw\n.\nWe claim that\nd\ndtθ−((w(t), s(t)))⩾√c−\nas long as s(t)⩾1\nc−andw(t)>0. It can be shown by the following straightforward computation:\nd\ndtθ−(w, s)√c−=1\n1 +c−\u0012s−1\nc−\nw\u00132\nw2−\u0010\ns−1\nc−\u0011\n(−νw+ 1−c(t)s)\nw2\n\n=1\n1 +c−\u0012s−1\nc−\nw\u00132\n1 +c− \ns−1\nc−\nw!2\n+ν \ns−1\nc−\nw!\n+\u0010\ns−1\nc−\u0011\n(c(t)−c−)s\nw2\n⩾1.CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 21\nHence, we find that θ−(w(t2), s(t2)) =π\n2with\nt2⩽π/2−θ−(w0, s0)√c−⩽π/2√c−.\nIn particular, we obtain w(t2) = 0.\n•For ( w0, s0)∈Σ♯\n3, we argue as in the proof of Proposition 4.4 that s(t3) =s∗,w(t3)⩾0 with\nt3⩽Zs∗\ns0dsp\n2N+(s)⩽π\nµ+, µ +:=r\nc+−ν2\n4.\nHenceforth, we deduce that, for any ( w0, s0)∈Σ♯,s(t∗) = 0 with\nt∗⩽π\nµ−+π/2√c−+π\nµ+.\nThis delivers the finite upper bound of t∗and completes the proof of the proposition. □\nProof of Theorem 1.7. The proof is based on a combination of the weak comparison principle (1.23)\nwith Propositions 4.4 and 4.5. For any regular initial data ( ρ0, u0) in the sense of Theorem 1.1, if\nthere exists x∈Rsuch that \u0012u′\n0(x)\nρ0(x),1\nρ0(x)\u0013\n∈Σ♯,\nthen ρ(·, t) blows up in a finite time, and the conclusion of the theorem follows. □\n4.2.Global-in-time regularity. To establish the global regularity of the initial data in sub-\ncritical regime, we prove the strong comparison principle (1.25) in the sequel. Recall from (4.2),\n(4.3) that if s(τ) satisfies P+(s(τ))⩾0 and (resp. N−(s(τ))⩾0) for all τ∈[0, t], then we get\nLP+(w(t), s(t))⩾LP+(w0, s0) exp Zt\n01−c+s(τ)p\n2P+(s(τ))dτ!\n,\nLN−(w(t), s(t))⩽LN−(w0, s0) exp Zt\n0c−s(τ)−1p\n2N−(s(τ))dτ!\n.(4.5)\nThe proof for the strict inequalities in the strong comparison principle (1.25) requires us to control\nthe integral in the exponential term in (4.5). We demonstrate this in detail in the following\npropositions.\nProposition 4.6. Letν⩾2√c+. Then,\nΣ♭=n\n(w, s) :−p\n2P+(s)< w, s > 0o\nis an invariant set for the system (4.1).\nProof. We prove that for any ( w0, s0)∈Σ♭, the solution to (4.1) satisfies ( w(t), s(t))∈Σ♭for all\nt⩾0. Suppose not, i.e., there exists ( w0, s0)∈Σ♭such that\nT∗:= inf{t >0 : (w(t), s(t))/∈Σ♭}<+∞.\nIn particular, we have ( w(T∗), s(T∗))∈∂Σ♭. We decompose the boundary of Σ♭into two subsets:\n∂Σ♭=∂Σ♭\n1∪∂Σ♭\n2:=n\nw+p\n2P+(s) = 0 , s⩾0o\n∪ {w >0, s= 0}.\nWe shall demonstrate that T∗<+∞leads to a contradiction. Indeed, the nature of proof is the\nsame for 0 < ν < 2√c+case, which is slightly more complicated. Hence, we defer the details of the\nproof to the next proposition. □22 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\nProposition 4.7. Let0< ν < 2√c+. Then, the following set\nΣ♭=n\n(w, s) :−p\n2P+(s)< w <p\n2N−(s)o\nis an invariant set for the system (4.1) provided that (1.31) holds.\nProof. We prove that for any ( w0, s0)∈Σ♭, the solution to (4.1) satisfies ( w(t), s(t))∈Σ♭for all\nt⩾0. Suppose, on the contrary, that there exists ( w0, s0)∈Σ♭such that\nT1:= inf{t >0 : (w(t), s(t))∈∂Σ♭}<+∞.\nWe decompose the boundary of Σ♭into three subsets in the sense of (1.29) that\n∂Σ♭\n1∪∂Σ♭\n2∪∂Σ♭\n3\n:=\b\nLP+(w, s) = 0 ,0⩽s < s ∗\t\n∪\b\nLN−(w, s) = 0 ,0< s⩽s∗\t\n∪n\n0< w⩽p\n2N−(0), s= 0o\n.\nWe now consider the following sub-three cases.\n•(w(T1), s(T1))∈∂Σ♭\n1. We claim that there exists a small ϵ >0 and Cϵ>0 such that\nZT1\nT1−ϵ1−c+s(τ)p\n2P+(s(τ))dτ⩾−Cϵ. (4.6)\nIfs(T1)<1\nc+, then we can take small ϵ >0 such that the integrand becomes non-negative for all\nτ∈[T1−ϵ, T1]. On the other hand, if s(T1)⩾1\nc+, then we have s(T1)< s∗by the assumption. Sincep\n2P+(s∗) = 0, we can take small ϵ >0 such that P+(τ) is away from zero for all τ∈[T1−ϵ, T1].\nHence, this supplies the estimate (4.6). By combining this lower bound of integral (4.6) with the\ninequality (4.5), we find that\nw(T1) +p\n2P+(s(T1))⩾\u0010\nw(T1−ϵ) +p\n2P+(s(T1−ϵ))\u0011\nexp(−Cϵ)>0,\nleading to a contradiction.\n•(w(T1), s(T1))∈∂Σ♭\n2. Similarly, we can prove that there exist a small ϵ >0 and Cϵ>0 such\nthatZT1\nT1−ϵc−s(τ)−1p\n2N−(s(τ))dτ >−Cϵ.\nWe find by means of the previous case with (4.5) that\nw(T1) +p\n2N−(s(T1))⩽\u0010\nw(T1−ϵ) +p\n2N−(s(T1−ϵ))\u0011\nexp(−Cϵ)<0,\nwhich is a contradiction\n•(w(T1), s(T1))∈∂Σ♭\n3. Noting that s(T1) = 0 and w(T1)>0 conveys us to the following\n0< w(T1) =s′(T1) = lim\nt→T1−s(T1)−s(t)\nT1−t⩽lim sup\nt→T1−−s(t)\nT1−t⩽0,\nthat we arrive at the contradiction. This completes the proof. □\nProof of Theorem 1.9. In view of the conclusion that Σ♭isinvariant for (4.1) from Proposition 4.6\nand 4.7, for any regular initial data ( ρ0, u0) satisfying\n\u0012u′\n0(x)\nρ0(x),1\nρ0(x)\u0013\n∈Σ♭,∀x∈R,\nwe find that the classical solution ( ρ, u) is indeed global by invoking a priori estimate (1.7). Hence,\nwe arrive at the conclusion of the theorem. □CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 23\nAs a direct consequence of this result, we present the proof of Theorem 1.12.\nProof of Theorem 1.12. We begin by discussing the smallness assumption on H0. In [1], the lower\nand upper bounds on the electric potential for a classical solution to the system (1.32)–(1.33) are\ninvestigated as follows:\n0< c−⩽eϕ(t,x)⩽c+, c−:= exp\u0000\nV−1\n−(H0)\u0001\nand c+:= exp\u0000\nV−1\n+(H0)\u0001\n,\nwhere V−1\n±are inverses of V±which is defined by\nV(z) :=\n\nV+(z) :=Zz\n0p\n2U(r) drforz⩾0,\nV−(z) :=Z0\nzp\n2U(r) drforz⩽0.(4.7)\nNote that V±(z) have the inverse functions and V∈ C2(R), see [1, Section 2.1].\nIndeed, careful estimates via Taylor’s expansion suggest that\nV±(z) =1\n2z2+1\n9z3+O(z4),\nso that\nV−1\n±(x) =±√\n2x+O(x), 0< x≪1.\nHence, for small enough H0≪1, we have\nc+\nc−= 1 + 2p\n2H0+O(H0).\nWe are now ready to investigate what secures the closing condition to hold; we go through as:\ns∗∗⩽0⇐⇒\u0012\n(1 +eγ+)c−\nc+−1\u0013\neγ−⩾1\n⇐= 1 + eγ+⩾2c+\nc−\n⇐=γ+⩾2\u0012c+\nc−−1\u0013\n= 4p\n2H0+O(H0)\n⇐=νπ⩾p\n4c+−ν2\u0010\n4p\n2H0+O(H0)\u0011\n.\nSincep\n4c+−ν2=O(1) for small ν >0, we find that (1.35) guarantees the closing condition as\ndesired. □\nAppendix A.A priori estimates in the proof of Theorem 1.1\nIn this appendix, we provide the details of the a priori estimate (2.1).\n•Estimates of ∥ρ−c∥Hs∩˙H−1: We find by means of the continuity equation (1.1) 1and the dual\nformulation of ˙H−1that\nd\ndt∥ρ−c∥˙H−1⩽∥ρu∥L2+∥∂tc∥˙H−1⩽∥ρ∥L∞∥u∥L2+∥∂tc∥˙H−1. (A.1)\nFor the higher-order estimates of ρ−c, we see that the continuity equation can be read as\n∂t(ρ−c) +∂x((ρ−c)u) =−∂tc−∂x(cu).24 YOUNG-PIL CHOI, DONG-HA KIM, DOWAN KOO, AND EITAN TADMOR\nDifferentiating it by ∂l\nxfor each l= 0,1, . . . , s , we test ∂l\nx(ρ−c) to find that\n1\n2d\ndtZ\nR|∂l\nx(ρ−c)|2dx=−Z\nR∂l\nx(ρ−c)u∂l+1\nx(ρ−c) dx−Z\nR∂l\nx(ρ−c)[∂l+1\nx, u](ρ−c) dx\n−Z\nR∂l\nx(ρ−c)∂l\nx∂tcdx−Z\nR∂l\nx(ρ−c)∂l+1\nx(cu) dx\n=:I+II+III+IV,\nwhere [ ·,·] stands for the commutator operator, i.e. [ A, B] =AB−BA. Then, standard commutator\nestimates and Sobolev inequalities yield\nI⩽∥∂xu∥L∞∥∂l\nx(ρ−c)∥2\nL2,\nII⩽∥∂l\nx(ρ−c)∥L2∥[∂l+1\nx, u](ρ−c)∥L2\n⩽C∥∂l\nx(ρ−c)∥L2\u0010\n∥∂xu∥L∞∥∂l\nx(ρ−c)∥L2+∥∂l+1\nxu∥L2∥ρ−c∥L∞\u0011\nIII⩽∥∂l\nx∂tc∥L2∥∂l\nx(ρ−c)∥L2,and\nIV⩽∥∂l\nx(ρ−c)∥L2∥∂l+1\nx(cu)∥L2\n⩽C∥∂l\nx(ρ−c)∥L2\u0010\n∥∂l+1\nxc∥L2∥u∥L∞+∥∂l+1\nxu∥L2∥c∥L∞\u0011\n⩽C∥∂l\nx(ρ−c)∥L2\u0010\n∥∂l+1\nxc∥L2∥u∥H1+∥∂l+1\nxu∥L2∥c∥L∞\u0011\n.\nThus, we have\nd\ndt∥∂l\nx(ρ−c)∥L2⩽C(∥ρ−c∥L∞+∥c∥L∞+∥∂xu∥L∞)\u0010\n∥∂l\nx(ρ−c)∥L2+∥∂l+1\nxu∥L2\u0011\n+C∥∂l+1\nxc∥L2∥u∥H1+∥∂l\nx∂tc∥L2. (A.2)\n•Estimates of ∥u∥Hs+1: We first estimate\n1\n2d\ndtZ\nRu2dx⩽∥∂xu∥L∞∥u∥2\nL2+∥∂xϕ∥L2∥u∥L2=∥u∥L2\u0000\n∥∂xu∥L∞∥u∥L2+∥ρ−c∥˙H−1\u0001\n.(A.3)\nFor the higher-order estimates of u, where l= 0,1, . . . , s , we deduce\n1\n2d\ndtZ\nR|∂l+1\nxu|2dx⩽−Z\nR(∂l+1\nxu)(∂l+2\nxu)udx−Z\nR∂l+1\nxu[∂l+1\nx, u]∂xudx+kZ\nR∂l+1\nxu∂l\nx(ρ−c) dx\n⩽C∥∂xu∥L∞∥∂l+1\nxu∥2\nL2+∥∂l+1\nxu∥L2∥∂l\nx(ρ−c)∥L2.\n(A.4)\nCombining (A.1), (A.2), (A.3), and (A.4), we conclude the estimate (2.1).\nReferences\n[1] J. Bae, J. Choi, and B. Kwon, Formation of singularities in plasma ion dynamics, preprint, arXiv:2012.09657.\n[2] J. Bae and B. Kwon, Small amplitude limit of solitary waves for the Euler–Poisson system, J. Differential\nEquations, 266, (2019), 3450–3478.\n[3] J. Bae and B. Kwon, Linear stability of solitary waves for the isothermal Euler–Poisson system, Arch. Ration.\nMech. Anal., 243, (2022), 257–327.\n[4] M. Bhatnagar and H. Liu, Critical thresholds in one-dimensional damped Euler–Poisson systems, Math. Models\nMethods Appl. Sci., 30, (2020), 891–916.\n[5] M. Bhatnagar and H. Liu. Critical thresholds in 1D pressureless Euler–Poisson systems with variable background,\nPhysica D, 414, (2020) 132728.\n[6] M. Bhatnagar and H. Liu, A complete characterization of sharp thresholds to spherically symmetric multidi-\nmensional pressureless Euler–Poisson systems, preprint, arXiv:2302.04428.\n[7] M. Bhatnagar, H. Liu, and C. Tan, Critical thresholds in the Euler–Poisson–alignment system, J. Differential\nEquations, 375, (2023), 82–119.CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS 25\n[8] M. P. Brenner and T. P. Witelski, On spherically symmetric gravitational collapse, J. Stat. Phys., 93, (1998),\n863–899.\n[9] J. A. Carrillo, Y.-P. Choi, and E. Zatorska, On the pressureless damped Euler–Poisson equations with quadratic\nconfinement: critical thresholds and large-time behavior, Math. Models Methods Appl. Sci., 26, (2016), 2311–\n2340.\n[10] J. A. Carrillo, Y.-P. Choi, E. Tadmor, and C. Tan, Critical thresholds in 1D Euler equations with nonlocal forces,\nMath. Models Methods Appl. Sci., 26, (2016), 185–206.\n[11] F. Chen, Introduction to Plasma Physics and Controlled Fusion (Plenum Press, New York, 1984), Vol. 1.\n[12] Y. Deng, T.-P. Liu, T. Yang, and Z. Yao, Solutions of Euler–Poisson equations for gaseous stars, Arch. Ration.\nMech. Anal., 164, (2002), 261–285.\n[13] S. Engelberg, H. Liu and E. Tadmor, Critical threshold in Euler–Poisson equations, Indiana Univ. Math. J., 50,\n(2001), 109–157.\n[14] Y. Guo, L. Han, and J. Zhang, Absence of shocks for one dimensional Euler–Poisson system, Arch. Rational\nMech. Anal., 223, (2017), 1057–1121.\n[15] D. Holm, S. F. Johnson, and K. E. Lonngren, Expansion of a cold ion cloud, Appl. Phys. Lett., 38, (1981),\n519–521.\n[16] D. Lannes, F. Linares, and J. C. Saut, The Cauchy problem for the Euler–Poisson system and derivation of the\nZakharov–Kuznetsov equation, in: M. Cicognani, F. Colombini, D. Del Santo (Eds.), Studies in Phase Space\nAnalysis with Applications to PDEs, in: Progr. Nonlinear Differential Equations Appl., vol. 84, Birkh¨ auser, pp.\n183–215, (2013).\n[17] F. Lin and P. Zhang, On the hydrodynamic limit of Ginzburg–Landau wave vortices. Comm. Pure and Applied\nMath., 55(7) (2002), 831–856.\n[18] C. Liu and Y Peng, Global convergence of the Euler–Poisson system for ion dynamics, Math. Mech. Appl. Sci.,\n42, (2019), 1236–1248.\n[19] H. Liu, Wave breaking in a class of nonlocal dispersive wave equations, J. Nonlinear Math. Phys., 13, (2006),\n441–466.\n[20] H. Liu and E. Tadmor, Spectral dynamics of velocity gradient field in restricted flows, Comm. Math. Phys., 228,\n(2002), 435–466.\n[21] H. Liu and E. Tadmor, Critical thresholds in 2-D restricted Euler-Poisson equations, SIAM J. Appl. Math., 63,\n(2003), 1889–1910.\n[22] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lect. Notes Math. 9,\nCourant Institute of Mathematical, Sciences, New York, 2003.\n[23] P. A. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Berlin, Heidelberg,\nNew York, 1990.\n[24] E. Tadmor and C. Tan, Critical thresholds in flocking hydrodynamics with nonlocal alignment, Proc. Royal Soc.\nA, 372, (2014), 20130401.\n[25] E. Tadmor and D. Wei, On the global regularity of sub-critical Euler–Poisson equations with pressure, J. Eur.\nMath. Soc., 10, (2008), 757–769.\n[26] C. Tan. Eulerian dynamics in multi-dimensions with radial symmetry, SIAM J. Math. Anal., 53, (2021), 3040–\n3071.\n[27] D. Wei, E. Tadmor, H. Bae. Critical Thresholds in multi-dimensional Euler–Poisson equations with radial sym-\nmetry, Commun. Math. Sci., 10, (2012), 75–86.\nDepartment of Mathematics, Yonsei University, Seoul 03722, Republic of Korea\nEmail address :ypchoi@yonsei.ac.kr\nDepartment of Mathematics, Yonsei University, Seoul 03722, Republic of Korea\nEmail address :skyblue898@yonsei.ac.kr\nDepartment of Mathematics, Yonsei University, Seoul 03722, Republic of Korea\nEmail address :dowan.koo@yonsei.ac.kr\nDepartment of Mathematics and Institute for Physical Science and Technology, University of\nMaryland, College Park, MD 20742, USA\nEmail address :tadmor@umd.edu"    },    {        "title": "2402.12875v1.Chain_of_Thought_Empowers_Transformers_to_Solve_Inherently_Serial_Problems.pdf",        "content": "Chain of Thought Empowers Transformers to Solve Inherently\nSerial Problems\nZhiyuan Li1,2, Hong Liu1, Denny Zhou3, and Tengyu Ma1\n1Stanford University,2Toyota Technological Institute at Chicago,3Google\nAbstract\nInstructingthemodeltogenerateasequenceofintermediatesteps, a.k.a.,achainofthought\n(CoT), is a highly effective method to improve the accuracy of large language models (LLMs)\non arithmetics and symbolic reasoning tasks. However, the mechanism behind CoT remains\nunclear. This work provides a theoretical understanding of the power of CoT for decoder-only\ntransformers through the lens of expressiveness. Conceptually, CoT empowers the model with\ntheabilitytoperforminherentlyserialcomputation,whichisotherwiselackingintransformers,\nespecially when depth is low. Given input length n, previous works have shown that constant-\ndepth transformers with finite precision poly(n)embedding size can only solve problems in\nTC0withoutCoT.Wefirstshowaneventighter expressivenessupperboundforconstant-depth\ntransformerswithconstant-bitprecision,whichcanonlysolveproblemsin AC0,apropersubset\nofTC0. However,with TstepsofCoT,constant-depthtransformersusingconstant-bitprecision\nandO(logn)embedding size can solve any problem solvable by boolean circuits of size T.\nEmpirically,enablingCoTdramaticallyimprovestheaccuracyfortasksthatarehardforparallel\ncomputation, including the composition of permutation groups, iterated squaring, and circuit\nvalue problems, especially for low-depth transformers.\n1 Introduction\nChain-of-Thought (CoT) reasoning, the technique of including intermediate steps besides the final\nanswers, through prompting (Wei et al., 2022; Chowdhery et al., 2022; Lampinen et al., 2022;\nKojima et al., 2022; Zhou et al., 2023) or finetuning (Cobbe et al., 2021; Nye et al., 2021; Chung\net al., 2022; Huang et al., 2022), dramatically enhances the performance of large language models\n(LLMs)onvariousreasoningbenchmarks(Wangetal.,2023;Lewkowyczetal.,2022;Suzgunetal.,\n2022; Shi et al., 2023; Yao et al., 2022; Drozdov et al., 2022; OpenAI, 2023; Google, 2023). The\nsuccess of CoT has even been extended to multi-modal QA (Zhang et al., 2023; Yang et al., 2023)\nand code generation (Chen et al., 2022; Gao et al., 2022). However, despite the evident empirical\nsuccess, the fundamental mechanism of CoT remains largely elusive (Wang et al., 2022a).\nA natural potential explanation is the intermediate steps provide extra information to the LLMs\nabout the tasks and the most efficient approach to solving the tasks, so that LLMs can imitate those\nsteps and solve the tasks more effectively. However, intriguingly, the efficacy of generating thought\nsteps extends to zero-shot CoT prompting (Kojima et al., 2022), where the LLMs are only simply\ninstructed with the prompt “let’s think step by step”, and to even using incorrect reasoning steps in\nthe few-shot examples (Wang et al., 2022a; Madaan & Yazdanbakhsh, 2022).\nThese observations suggest that in the original few-shot CoT settings, the form of CoT is as\nimportantas(ifnotmoreimportantthan)thecontentsinthem,becausemerelyinstructingtheLLMs\nto generate the intermediate steps helps.\n1arXiv:2402.12875v1  [cs.LG]  20 Feb 2024ThispaperaimstostudywhytheformofCoTimprovesthereasoningcapabilityofLLMs. Our\nhypothesis is that CoT allows for performing more serial computations that a vanilla transformer\ncannotdowithoutCoT.Weformulateandanalyzethishypothesisthroughthelensofexpressiveness\nwith and without CoT. We adopt the language of circuit complexity to discuss the capability of\ntransformers. Previousworks(Liuetal.,2022b;Merrill&Sabharwal,2023b)haveshownstandard\ndecoder-only transformers (that output answers directly) are efficient parallel computers and can\nonly express functions computable in an O(1)-parallel run-time with threshold circuits, TC0, a\ncomputationalmodelthatallowsthe AND,OR,NOTandMAJORITY functionwithmultipleinputs\nto be computed efficiently in parallel. We first show a tighter upper bound (Theorem 3.1) for\nexpressivenessofconstant-precisiontransformer–itcanonlyexpressapropersubsetclassof TC0,\nAC0, where MAJORITY gates are not allowed. Our upper bound is also more realistic because\nit handles the rounding issue or iterative addition of floating point numbers, while most previous\nresults essentially only work for fixed-point number addition.\nWethenshowthattransformersequippedwithCoT—allowingthetransformertoauto-regressively\ngenerateasequenceofintermediatetokensbeforeansweringthequestions—cansolvecomplexprob-\nlems that inherently require serial computations (assuming well-known conjectures in complexity\ntheory). Intuitively, without CoT, the number of serial computations conducted by the transformer\nis bounded by the depth (which is considered as a fixed constant for this work), whereas with T\nintermediate steps, the number of serial computations possible is boosted to T. Note that Tcan\neasily increase as the sequence length increases where the depth is a fixed number that depends on\nthe architecture.\nConcretely,weprovethataconstant-precisiontransformerwith Tintermediatestepsandembed-\ndingdimensionlogarithmicinthesequencelengthcanexpressanyfunctionscomputablebyacircuit\nofsize TinTheorem3.3. Taking Ttobepolynomialinthesequencelength,theresultsuggeststhat\ntransformerswithpolynomiallymanyintermediatestepsarecapableofcomputingallcircuitsinwith\npolynomialsize, P/poly,asuperclassofP.Theorem3.3alsoimpliesthattransformerswithlinearly\nmany intermediate steps can compute all regular languages, including composition of non-solvable\ngroups, like permutation group over five elements, S5, which does not belong to AC0and is also\nwidely conjectured to be out of TC0. As such, CoT makes transformers with bounded depth and\nprecision strictly more powerful.\nTo corroborate our theoretical analysis, we empirically evaluate the capability of transformers\nin solving four core problems: modular addition, permutation composition, iterated squaring, and\ncircuit value problem. We learn transformers to solve these tasks with a large amount of synthetic\ndata, with and without CoT, or with additional hint but not CoT. The modular addition belongs to\nTC0, meaning it can be easily solved in parallel. Liu et al. (2022a) shows it is solvable by constant-\ndepth transformers with log-precision and, indeed empirically depth 1 is sufficient for the parity\nproblem (Modulo 2 addition). The other three tasks are all conjectured to require inherently serial\ncomputations. Asexpected,thevanillatransformereitherrequiresahugedepthtosolvethesetasks\n(because the depth is the upper bound on the number of serial computation by transformers), or\ncannotsolvethetasksatall. Ontheotherhand,CoTcansolvethesetasksaslongasthedepthexceeds\na small threshold. These experiments demonstrate CoT can provide more serial computations to\nsolve complex reasoning tasks.\n2 Notations and Preliminaries\nWeuse NandRtodenotethesetofnaturalnumbersandrealnumbersrespectively. Forany n∈N+,\nwe define [n]≜{1,2, . . . , n }. We define relu(x)≜max( x,0). For vector x, we use xa:bto denote\n2the vector containing coordinates of xfrom position ato position b. For matrix M, we define\nMa1:b1,a2:b2to denote the submatrix by selecting rows from a1tob1, columns from a2tob2. We\nalso use a1:to denote the subset of indices from a1to the end, :b1to denote the subset of indices\nfrom the beginning (1) to b1and:to denote all indices. Given two non-negative functions f, g,\nwe say f(n) =O(g(n))(resp. f(n) = Ω( g(n))) iff there exists C > 0, such that for all n≥0,\nf(n)≤Cg(n)(resp. f(n)≥Cg(n)). We use poly(n)≜{T:N→N| ∃k >0, T(n) =O(nk)}\nto denote the set of functions with at most polynomial growth rate.\nWe use ϕ(x) =P|x|\ni=12|x|−ixito denote the value of binary number represented by binary\nstring x. We use bink(x)to denote the usual binary encoding of natural number xusing kbinary\nbits in the sense that ϕ(bink(x)) = xandsbink(x)to denote the signed binary encoding, which\nis2bink(x)−(1, . . . , 1). For any n∈N+, we define softmax : Rn→Rnas(softmax( x))i=\nexp(xi)/Pn\ni=1exp(xi)for any x∈Rnandi∈[n]. We use ⊙to denote the element-wise product\nof two vectors. We use a∥bor(a, b)to denote the concatenation of two vectors aandb.\n2.1 Decoder-only Transformers\nGiven a vocabulary V, adecoder-only transformer with parameter θand maximal input length\nnmaxmapsasequenceofinputtokens (x1, . . . , x n)∈ Vntoaprobabilitydistributionover Vforall\nn≤nmax, denoted by pθ(· |x1, . . . , x n). We also define function TFθ(x)by the token in Vthat\nmaximizes pθ(· |x1, . . . , x n), that is, TFθ(x)≜arg maxx∈Vpθ(x|x1, . . . , x n).\nNext-token Generator: Given a vocabulary V, a next-token generator with parameter θand\nmaximal input length nmaxis a mapping from ∪nmax\nn=1VntoV. The main next-token generator we\nare interested in this work is decoder-only transformers, TFθ(x1, . . . , x n)where xi∈ Vfor all i∈\n[n]. We also recursively define TFi\nθ(x1, . . . , x n)≜TFi−1\nθ(x1, . . . , x n,TFθ(x1, . . . , x n)), for every\npositiveinteger iandnsatisfyingthat i+n≤nmax−1withthebasecasethat TF1\nθ(x1, . . . , x n)≜\nTFθ(x1, . . . , x n). In other words, for all 0≤i≤nmax−n−1, the output with isteps of CoT is\nxn+i+1=TFi+1\nθ(x1, . . . , x n) =TFθ(x1, . . . , x n, xn+1, . . . , x n+i).\nTransformerArchitectureOverview: Thedecoder-onlytransformermodelweconsiderinthis\npaper is very similar to GPT style architectures (Radford et al., 2019) and consists of four parts: a\ntokenembeddinglayer( TE),apositionencodinglayer( PE),anoutputlinearlayer( OUTPUT ),anda\nstackof Lidenticallayersservingasthe“decoder”where Lisalsocalledthedepthofthemodel. Each\ndecoderlayerhastwosub-layers: amulti-headself-attentionlayer( ATTN)andaposition-wisefully-\nconnectedfeed-forwardnetwork( FF). Eachlayermentionedabovehasitsowntrainableparameters\nand is indexed by the layer name and the depth for attention and feedforward layers. 1That is we\ncan split the model parameter θin the following way: θ= (θPE, θTE, θOUTPUT ,{θ(l)\nATTN, θ(l)\nFF}L−1\nl=0),\nwhich are all trainable. (See formal definition in Algorithm 2). Throughout this paper, we use dto\ndenote the embedding size of a transformer.\nSelf-Attention Mechanism: Given attention parameter θATTN = (WQ, WK, WV, WO)∈\nRd×d×Rd×d×Rd×d×Rd×d, we define the Attention layer with mask for decoder-only trans-\nformer in Algorithm 3. Note allowing multi-head attention will not change the class of problems\nsolvablebyconstantlayerdecoder-onlytransformersaswecansimulate1multi-headattentionlayer\nwith any constantly many heads with multiple single-head attention layers. Thus for simplicity of\npresentation, we do not include multi-head attention in the definition below.\nFeed-Forward Network: Given the parameter of fully-connected feedforward network layer\nθFF= (W1, b1, W2, b2)∈Rd×d×Rd×Rd×d×Rd,wedefinethefully-connectedfeedforwardlayer\n1We ignore the LayerNorm (Ba et al., 2016) in the usual transformer architecture for simplicity. Our expressiveness\nanalysis can extend to the transformers with LayerNorm with more careful treatment. See Appendix F.1 for discussion.\n3Algorithm 1 Causal Self-Attention, ATTN\nRequire: Parameter θATTN = (WQ, WK, WV, WO), Input embedding h= (h1, . . . , h n)∈Rnd.\nEnsure: Output embedding h′= (h′\n1, . . . , h′\nn)≜ATTN θATTN(h1, . . . , h n).\n1:qi≜WQhi, ki≜WKhi, vi≜WVhi,∀i∈[n]\n2:si≜softmax( ⟨qi, k1⟩, . . . ,⟨qi, ki⟩)∥(0, . . . , 0).\n3:h′\ni≜WOPn\nj=1(si)jvj.\nFFθFF:Rd→RdasFFθFF(h)≜W2relu(W1h+b1) +b2.\nToken Embedding: Given the parameter of token embedding layer θTE∈Rd×|V|, we define\nthe token embedding layer by viewing θTEas a mapping from VtoRd, that is, for all x∈ V, the\ntoken embedding is θTE(x).\nPosition Encoding: Given the parameter of position encoding layer θPE∈Rd×nmax, we define\nthetokenembeddinglayerbyviewing θPEasamappingfrom [nmax]toRdthatis,forall n∈[nmax],\nthe position embedding is as θPE(n).\nOutput Layer: Given the parameter of output layer θOUTPUT ∈R|V|×d, we define the output\nlayer OUTPUT θOUTPUT :Rd→ VasOUTPUT θOUTPUT (h)≜softmax( θOUTPUT h)for all h∈Rd.\nAlgorithm 2 Decoder-only Transformer, TFθandpθ\nInput:Transformer parameter θ= (θPE, θTE, θOUTPUT ,{θ(l)\nATTN, θ(l)\nFF}L−1\nl=0)and input tokens x=\n(x1, . . . , x n)∈ Vn.\nOutput: Output distribution pθ(· |x1, . . . , x i)for all i∈[n]and output token TFθ(x).\n1:h(0)\ni←θTE(xi) +θPE(i),∀i∈[n]\n2:forl= 0, . . . , L −1do\n3: (h(l+0.5)\n1 , . . . , h(l+0.5)\nn )←(h(l)\n1, . . . , h(l)\nn) +ATTNθ(l)\nATTN(h(l)\n1, . . . , h(l)\nn)\n4: h(l+1)\ni←h(l+0.5)\ni +FFθ(l)\nFF(h(l+0.5)\ni ),∀i∈[n]\n5:end for\n6:pθ(· |x1, . . . , x i)←OUTPUT θOUTPUT (h(L)\ni),∀i∈[n]\n7:TFθ(x)←arg maxxpθ(x|x1, . . . , x n).\n2.2 Circuit Complexity\nProblem. In this paper we consider the following notion of problems: given a sequence of input\ntokens, output a token as the answer. Mathematically, given a vocabulary V, we call a mapping\nL:∪k∈N+Vk→ Vaproblem. Ifthe correctanswer isalways 0or1, wecall La decisionproblem.\nIn circuit complexity, such Lis also called a language.\nThough the standard definition of circuit complexity only deals with binary strings, given any\nfinite vocabulary V, we can always replace each token in Vby its binary representation, and the\nlengthoftheinputonlyblowsupbyaconstantfactor. Thereforewecanextendexistingcomplexity\nclasses listed to arbitrary finite vocabulary naturally.\nP.The class Pcontains all problems solvable by a deterministic Turing machine in polynomial\ntime.\nBooleanCircuit. ABooleancircuitover nvariablesisadirectedacyclicgraphwherenodesare\nAND,OR, orNOTgates. The gates with in-degree 0 are the inputs, which are assigned one of the\nnboolean variables. Given the inputs, the circuit computes the value of each non-input gate based\n4on the value of the incoming gates and outputs a number at the output gate.\nSIZE[T(n)].Givenanyfunction T,SIZE[T(n)]denotestheclassofproblemsthatcanbesolved\nby boolean circuits with O(T(n))gates when the input length is n. Formally, a problem Lis in\nSIZE[T(n)]if and only if there exists a sequence of circuits {Cn}such that each circuit Cnhasn\ninputsand1output,thesizeofeachcircuit Cnisatmost O(T(n)),andforallstrings x,xisinLif\nand only if C|x|(x) = 1.\nP/poly.We define the class P/polyas the set of problems that can be solved by a family of\npolynomial-size circuits, that is, P/poly≜∪k∈N+SIZE[nk]. Since any Turing Machine with time\nbound T(n)can be simulated by a circuit of size T(n) logT(n)(Pippenger & Fischer, 1979), we\nknow that P⊆P/poly.\nNC,AC, and TC.The class NCcontains all problems that can be solved in a small parallel\nruntime—polylogarithmicininputlength—andwithapolynomialnumberofprocessors. Formally,\nfor a positive integer k, a problem Lis in NCkif and only if there exists a polynomial p(n)\nand a family of circuits {Cn}such that each circuit Cnhasninputs and 1 output, the fan-in\nof the gates is at most 2, the size of each circuit Cnis at most p(n), the depth of each circuit\nCnisO((log n)k), and for all strings x,xis in if and only if C|x|(x) = 1. Finally we define\nNC=∪k∈NNCk. The class ACkis defined almost the same as NCkfor each k∈N+, except the\nANDandORgates in ACkallow unbounded fan-in. The class TCkallows a more powerful type of\ngate, MAJORITY , compared to ACk.MAJORITY gate can have unbounded fan-in and is defined\nasMAJORITY (x1, . . . , x n) =⌊1\n2+(Pn\ni=1xi)−1/2\nn⌋.\nIt holds that NCi⊆ACi⊆TCi⊆NCi+1for all natural number i. Therefore NC=AC=TC,\nwhich all stands for the problem class that can be solved in polylogarithmic time with polynomial\nparallel processors.\n3 ExpressivenessTheoryforTransformerswithChainofThought(CoT)\nInthissection,westudytheexpressivenessoftransformerswithCoTfromatheoreticalperspective.\n3.1 Finite Precision Modeling\nIn practice, training and inference of transformers are typically done with 16- or 32-bit floating\npointnumbers. Thusinthispaper,wemainlyfocusonthecomputationmodelof constant-precision\ntransformers, where the output of each arithmetic operation is rounded to the closest floating point\nnumber representable by a fixed number of digits following IEEE 754 standard (Definition 3.2),\nthus avoiding the unrealistic infinite precision assumption made by prior works (Pérez et al., 2019;\nDehghani et al., 2018).\nBelow we give a formal definition of the floating-point number androunding operation. Recall\nϕ(a) =Pk\ni=12k−iaidenotethevalueofbinarynumberrepresentedby a∈ {0,1}kforany k∈N+.\nDefinition 3.1 (Floating-point Representation) .Letebe the number of bits for exponents and sbe\nthe number of bits for significand. A (e+ 2s+ 1)-bit binary string a= (a1, a2, . . . a e+2s+1)∈\n{0,1}e+2s+1is afloating-point binary representation of number ϕe,s(a)≜sign(a)·2exponent( a)·\nsignificand( a)withe-bitexponentand 2s-precision,wherethesignis sign(a)≜2a1−1,thesignif-\nicand is significand( a)≜2−sϕ(a2:2s+1), and the exponent is exponent( a)≜ϕ(a2s+2:2s+e+1)−\n2max(0 ,e−1). We further use Fe,sto denote all the floating numbers representable using e-bit\nexponent and 2s-bit precision (significand), that is, Fe,s≜{S·2−s+E| −22s+ 1≤S≤\n22s−1,−2max(0 ,e−1)≤E≤2e−1−2max(0 ,e−1), E, S∈N}. We define Be,s≜maxFe,s.\n5We also use ψe,s:Fe,s→ {0,1}e+2s+1to denote the inverse of ϕe,s. We note that when the\nnumber of exponent bits is larger than 0, there are multiple ways to represent a number in Fe,sby a\nbinarystringandweassign ψe,s(x)asthestring a∈ {0,1}e+2s+1withthesmallest |exponent( a)|,\nwhich is unique for all non-zero numbers. For 0we additionally set sign(ψe,s(0)) = 1.\nDefinition 3.2 (Correct Rounding) .For any x∈Rand any closed subset of Rcontaining 0,F, we\ndefinecorrectrounding round (x,F)astheclosestnumberto xinF. Webreakthetiebypickingthe\none with a smaller absolute value.\nInparticular,wedenotetheroundingoperationwith e-bitexponent, 2s-bitprecisionby round e,s(·)≜\nround (·,Fe,s), which is also denoted by [·]e,sfor convenience. We extend the definition of round\nandround e,sto vector inputs by rounding coordinate-wisely.\nOurnotionoffloating-pointnumbersimplifiestheIEEE754StandardforFloating-pointArith-\nmetic (IEEE, 2008) by removing ∞and−∞. When overflow happens, we always round the\noutput to the (negative) largest representable number in Fe,s. For unary functions like exp(·)and\nbinaryfunctionsincludingaddition,subtraction,multiplication,anddivision,wesimplydefinetheir\nroundedversionbyroundingtheiroutputs. Wheneverdivisionby 0happens,wetreatitasthemodel\noutputs the wrong result.\nNext,wedefinefinite-precisionsummationovermoretwonumbersbydecomposingitasachain\nof rounded binary addition in a fixed order. 2\nDefinition 3.3 (Summation with Iterative Rounding) .For any s, n∈N+and vector x∈Rn, we\ndefinesummation with iterative rounding to 2s-bit precision assume,s:∪n∈N+(Fe,s)n→Fe,s,\nwhereforany n∈N+andx∈Rn,sume,s(x)≜\"\u0014h\n[x1+x2]e,s+x3i\ne,s+···xn−1\u0015\ne,s+xn#\ne,s.\nWe further define the following operations:\n•Finite-precision inner product: ⟨x, y⟩e,s≜sume,s(x⊙y);\n•Finite-precision matrix product: (A×e,sB)i,j≜\n(Ai,:)⊤, B:,j\u000b\ne,s;\n•Finite-precision softmax: softmax e,s(x)≜h\n[exp( x)]e,s/sume,s([exp( x)]e,s)i\ne,s.\nFinally, a finite-precision transformer can be defined by replacing all the infinite-precision\noperations by their finite-precision counterparts listed above. (See details in Algorithm 4). We\npostponethedetailsofthefinite-precisionversionofindividualtransformerlayersintoAppendixB.\n3.2 CoT: Complexity Class for Constant-depth Transformers with CoT\nIn this subsection, we define the complexity class consisting of all the problems that can be solved\nby some decoder-only transformers with CoTwith finite precision.\nDefinition 3.4 (CoT).Given a finite vocabulary Vand four functions T(n), d(n), s(n), e(n), in-\nformally, CoT[T(n), d(n), s(n), e(n)]is the family of problems solvable by a transformer with a\nconstant depth, s(n)bits of precision, e(n)bits of exponent, embedding size d(n)andT(n)steps\nof CoT. Formally, we say a problem L:∪n∈N+Vn→ {0,1}is in CoT[T(n), d(n), s(n), e(n)]iff\nthere is an integer Land three functions T′(n) =O(T(n)), d′(n) =O(d(n)), s′(n) =O(s(n)),\ne′(n) =O(e(n)),suchthatforeverypositiveinteger n,thereisa L-layerdecoder-onlytransformer,\ndenoted by TFθnwith embedding size d′(n),2s′(n)bits of precision, and e′(n)bits of exponent,\n2Technically speaking, instead of a chain, the summation could also proceed like a tree. This is a more complicated\ncase and we leave it for future work.\n6thatcanoutput L(x)givenanyinput xinVn,using T′(n)stepsofchainofthought. Mathematically,\nit means\nTF1+T′(n)\nθn(x) =L(x),∀x∈ Vn. (1)\nWealsoextendthedefinitionof CoTtoaclassoffunctioninsteadofasinglefunction. Forexample,\nCoT[T(n),poly(n), s(n), e(n)]≜∪k∈N+CoT[T(n), nk, s(n), e(n)].\nDefinition 3.5 (T).We define T[d(n), s(n)]≜CoT[0, d(n), s(n)]as the problems that a constant-\ndepth,constant-precisiondecoder-onlytransformercansolvewith O(s(n))bitsofprecision, O(d(n))\nembedding size and without CoT (or with only 0step of CoT).\nBy definition, CoT[T(n), d(n), s(n), e(n)]is monotone in all T(n), d(n), s(n), e(n),e.g.,\nCoT[T′(n), d(n), s(n), e(n)]⊆CoT[T(n), d(n), s(n), e(n)]ifT′(n)≤T(n)for all n∈N. In\nparticular, we have T[d(n), s(n)]≜CoT[0, d(n), s(n)]⊆CoT[T(n), d(n), s(n)].\nNotetheabove-definedcomplexityclass CoTisnon-uniform,thatis,itallowsadifferentprogram\nforeveryinputsize. Thisisincontrasttopreviousworks(Pérezetal.,2019,2021;Yaoetal.,2021;\nWeissetal.,2021;Chiangetal.,2023;Haoetal.,2022;Merrill&Sabharwal,2023a;Merrilletal.,\n2022)whichfocusontheuniformtransformerclasses. PleaserefertoAppendixGforadiscussion.\n3.3 Tighter Upper Bounds on Transformer Expressiveness\nExisting works (Merrill & Sabharwal, 2023b; Liu et al., 2022a) have shown that constant depth,\npolynomial width, and log precision transformers can be simulated in a small parallel time, i.e.,\nusing TC0circuits. These results are built on the fact that multiplication and division of n-bits\nbinary numbers (Hesse, 2001), as well as the iterated addition over ndifferent n-bit binary integers\nare in TC0.\nHowever, such TC0expressiveness upper bounds may be unrealistic for transformers operating\nwith floating point numbers. (Merrill & Sabharwal, 2023b; Liu et al., 2022a) implicitly assumes\nwhen adding more than one floating-point number, the algorithm first computes the exact answer\nwithoutroundingusingarbitrarilymoreprecisionandonlyperformsroundingintheend. However,in\npracticeroundinghappensaftereachadditionbetweentwonumbersanditisopenifsuch TC0upper\nboundsstillholds. Immediateroundingmakesiteratedadditionoverfloatingpointnumbersnolonger\nassociative (Goldberg, 1991), for example, round (a+round (b+c))̸=round (round (a+b) +c)).\nThe associativity of integer addition plays a crucial role in the fact that the iterated addition over n\ndifferent n-bit binary integers are in TC0.\nInthissection,wepresenttwonovelexpressivenessupperboundsfortransformerswhichrounds\nthe immediate result after step of the arithmetic operation. First, we show a strictly tighter upper\nboundthan TC0,whichis AC0,forconstant-depthtransformerswithbothconstantbitsofprecision\nandexponents. (Theorem3.1)Thissuggestswheninputlengthissufficientlylong,constant-precision\ntransformers cannot count eventually, even in the sense of modular. For example, it is well known\nthat no AC0circuits can decide the parity of a binary string.\nTheorem 3.1. T[poly(n),1,1]⊆CoT[1,poly(n),1,1]⊆AC0.\nOursecondresult,Theorem3.2,showsthatwhenthenumberofbitsforexponentis 0(i.e.fixed-\npointnumbers), TC0upperboundsforexpressivenessofconstant-depth,log-precisiontransformers\nstill holds, even with the correct rounding defined in Definition 3.2.\nTheorem 3.2. T[poly(n),log(n),0]⊆CoT[1,poly(n),log(n),0]⊆TC0.\n7Hint Label: Input: (23145)(23415)(51432)CoTLabel: Input: (23145)(12435)(54321)=AB(231…4321)=(2313…1432)BA(2313…1432)A\nFigure 1: Permutation Composition ( S5). The label is the composition of all the permutations, where given\ntwo permutation σ= (σ1, . . . , σ 5),π= (π1, . . . , π 5), we define σ◦π≜(σπ1, . . . , σ π5). The chain of\nthoughts and hints are the partial compositions. Only CoT can solve this task well, as predicted by our\nTheorem 3.5. Note for the most time the accuracy without CoT is ∼20%, which is no better than randomly\nguessing a number between 1and5.\nThe main technical difficulties in above two results are showing sume,s: (Fe,s)n→Fe,shas\nAC0(resp. TC0) circuits when e, sare both constants (resp. e= 0,s=O(log(n))). We view\niterated addition with rounding over Fe,sas an automaton with both state space and vocabulary\nbeingFe,s. The first result are due to a novel application of classical Krhon-Rhodes decomposition\ntheorem for automata (Theorem C.2), where we use the property of rounded addition that for all\nx, x′∈Fe,s, y∈Fe,s,x≥x′=⇒[x+y]e,s≥[x′+y]e,s. We formalize this property in\nDefinition D.2 as orderedautomata and show all ordered automata are counter-free Theorem D.3\nand thus can be simulated by AC0circuits (McNaughton & Papert, 1971).\nThe proof technique for Theorem 3.1 does not generalize to Theorem 3.2 because the depth of\nAC0circuits constructed before depends on the number of the states of the automaton and thus is\nnot constant. Our proof for Theorem 3.2 is motivated by Algorithm 1 in Liu et al. (2022a) for the\nautomaton named ‘GridWorld’.\nHowever, it remains open whether constant-depth, log-precision transformers with log bits for\nexponents T[poly(n),log(n),log(n)]orevenconstantbitsforexponents T[poly(n),log(n),1]have\nTC0circuits.\n3.4 CoT Makes Transformers More Expressive\nNowwearereadytopresentourmaintheoreticalresults(Theorem3.3)whichcharacterizetheexpres-\nsivenessofconstant-depth,constant-precisiontransformerswithCoTand O(log(n))embeddingsize.\nlog(n)embeddingsizesisnecessarytoensurethatthepositionembeddingsfor ninputsaredifferent.\nAllthelowerboundsfortransformerexpressiveness(withorwithoutCoT)areprovedforfixed-point\nnumbers, i.e.,withoutusinganyexponentbits. Allowingexponentbitswillonlymaketransformers\nmore expressive. For convenience, we define CoT[T(n), d(n), s(n)]≜CoT[T(n), d(n), s(n),0].\nThe omitted proofs in this section can be found in Appendix E.\n8Theorem 3.3. For any polynomial function T:N+→N+,SIZE[T(n)]⊆CoT[T(n),logn,1]. In\nparticular, P/poly=CoT[poly(n),logn,1].\nCompared to Theorems 3.1 and 3.2, Theorem 3.3 shows that allowing polynomial steps of CoT\nstrictlymakesconstant-depth,constant-precision,decoder-onlytransformermoreexpressiveandlog-\nprecisiontransformersmoreexpressiveunderastandardhardnessassumptionthat TC0⊊P/poly.3\nProof sketch of Theorem 3.3. The high-level proof idea is that we use each step in CoT to simulate\nonegateoperationinthetargetcircuitandwritethegateoutputasnextinput. Todothat,weuseone\nposition encoding to store the information for each gate, which contains four parts: the current gate\ntype{AND,OR,NOT,TRUE ,FALSE }, the two input gates id and the current gate id. Since there\nare total poly(n)gates, d(n) = Θ(log n)embedding size suffices to store the above information.\nAndtheCoThereisconstructedtobethevaluesofeachgateintheincreasingorderofid. Therefore,\nineachstep,wecanuseattentiontopullthevalue(eithercomputedalreadyoritisinput)ofthetwo\ninput gates and use a feedforward network to compute the value of the current gate.\nAs we can see from proof sketch, a crucial step for CoT to simulate any depth circuit is to write\nthe output token back to the next input position. This action resets the “depth” of the intermediate\noutputinthecircuitto 0. OurtheoryexplainstheablationexperimentinWeietal.(2022)thatwhen\nthemodelispromptedtooutputaonlysequenceofdots(. . .) equaltothenumberoftokensneeded\nto solve the problem, the performance is no better than directly outputting the answer.\nBecause every regular languages can be recognized by a finite state automaton (Definition C.1)\nand finite state automata can clearly be simulated by linear size circuits. The following holds as a\ndirect corollary of Theorem 3.3\nCorollary 3.4. Every regular language belongs to CoT[n,logn,1].\nBelowwegiveaconcreteregularlanguagethatconstant-depth,poly-embedding-sizetransform-\ners can solve only with CoT, the wording problem of permutation group over 5elements, S5in\nTheorem 3.5, under a standard hardness assumption that TC0⊊NC1(Yao, 1989).\nDefinition3.6 (Wordingproblemofgroup G).Given nelementsfrom G,(g1, . . . , g n),weuse LG\nto denote the decision problem of whether g1◦g2◦ ··· ◦ gnis equal to the identity of G.\nFor convenience, in this paper we extend the domain of LGto the sequence of groups encoded\nby binary strings. The proof of Theorem 3.5 is a direct consequence of Theorems 3.2, 3.3 and 3.6.\nTheorem 3.5. Assuming TC0⊊NC1, the wording problem of S5,LS5is in CoT[n,logn,1]but\nnotT[poly(n),logn].\nTheorem 3.6 (Barrington (1986)) .The wording problem of S5isNC1-complete under AC0reduc-\ntions. That is, for any decision problem LinNC1, there is a family of AC0circuits {Cn}∞\nn=1\n(constantdepth, poly(n)fan-outs),suchthatforany n∈N+andx∈ {0,1}n,L(x) =LS5(Cn(x)).\nProof of Theorem 3.5. FirstLS5isaregularlanguage,thusbelongingto CoT[n,logn,1]byCorol-\nlary3.4. Since LS5isNC1-completebyTheorem3.6,assuming TC0⊊NC1,LS5doesnotbelongto\nTC0. ThisproofiscompletedbyapplyingTheorem3.2,whichsays T[poly(n),log(n)]⊆TC0.\n3Indeedsuchseparationcanbeshownfor anypolynomialstepsofCoTbypaddingpolynomiallymanytokenstoinput.\n9Results for poly(n)embedding size: So far we have been focusing on the expressiveness of\ntransformer with O(logn)embedding size, so it is natural to ask whether transformers can also\nbenefit from having a larger embedding size, say poly(n)? Our Theorem 3.7 answers this question\npositivelybyshowingthatlog-precision(resp. constant-precision)constant-depthpoly-embedding-\nsizedecoder-onlytransformerswith T(n)stepsofCoTcansimulateany T(n)-sizecircuitwithsome\nTC0(resp. AC0) oracle gates with poly(n)input.\nFormally, given a decision problem L:∪∞\nn=1{0,1}n→ {0,1}, we use Lnto denote the\nrestriction of Lon{0,1}n, which can also be viewed as an single gate with nfan-ins. We define\nproblems that can be solved by circuits with certain sizes of gates (including oracle gates) by\nDefinition 3.7. 4\nDefinition3.7 (SIZEL).Foranydecisionproblem LandT(n)⊆O(poly(n)),wedefine SIZEL(T(n))\nasthesetofdecisionproblems L′suchthatthereexists p(n)∈poly(n)andcircuits {Cn}∞\nn=1where\nCncontainsatmost O(T(n))AND,OR,NOT,andLp(n)gates. Foracomplexityclass C,wedefine\nSIZEC(T(n))≜∪L∈CSIZEL(T(n)).\nTheorem3.7. Forany T(n)∈poly(n),itholdsthat SIZETC0[1+T(n)] = CoT[T(n),poly(n),logn].\nSpecifically,for T(n) = 0,wehave TC0=SIZETC0[1] = CoT[0,poly(n),logn] =T[poly(n),logn].\nTheorem3.8. Forany T(n)∈poly(n),itholdsthat SIZEAC0[1 +T(n)] = CoT[T(n),poly(n),1].\nSpecifically, for T(n) = 0, we have AC0=SIZEAC0[1] = CoT[0,poly(n),1] = T[poly(n),1].\nTheorem3.8showsthatfor T(n) =poly(n)stepsofCoT,using poly(n)embeddingsizedoesnot\nimproveexpressivenessoverusing log(n)embeddingsize(Theorem3.3),because SIZETC0[poly(n)] =\nSIZEAC0[poly(n)] = SIZE[poly(n)]. However, Theorem 3.9 shows that for any specific polynomial\nT(n) =nkstepsofCoT,increasingembeddingwidthfrom O(log(n))topoly(n)maketransformers\nstrictly more powerful.\nTheorem 3.9. For any s(n) =O(logn),T[logn, s(n)]⊊T[poly(n), s(n)]and for all k∈N,\nCoT[nk,logn, s(n)]⊊CoT[nk,poly(n), s(n)].\n4 CoTEmpiricallyImprovesExpressivenessofLow-DepthTransform-\ners on Inherently Serial Problems\nThis section is an empirical study of the expressiveness of decoder-only transformers with CoT\non four different arithmetic problems: modular addition, permutation composition ( S5), iterated\nsquaring, and circuit value problem. The first problem is parallelizable and can be solved by\nconstant-depth transformers with log-precision while the latter three are inherently serial under\nsomestandardhardnessassumptionsincomputationalcomplexityorcryptography. Asaprediction\nof our theory, we expect to see a huge improvement in accuracy when CoT is turned on.\nGeneral Setup. To examine the expressiveness of decode-only transformers with and without\nCoT on these four types of problems, we train the transformer using Adam(Kingma & Ba, 2014)\nfrom random initialization in the online supervised setting for each problem and each different\nsequencelength n. Ateachstep,wesampleabatchoftrainingdatafromadistribution pn(x)where\nx= (x1, . . . , x n)is training data and y=f∗(x1, . . . , x n)is the label. We always set xnto be ’ =’.\nWe consider three different settings, base,cot, andhint:\n4Ourdefinitionofcomplexityclasssolvablebycircuitswithoracleisslightlydifferentfromthatinliterature(Wilson,\n1985), where the size of the oracle circuits refers to the number of wires, whereas ours refers to the number of gates.\n10Hint Label: Input: 1363100123456=CoTLabel: Input: 1363100123456=1363100\nFigure 2: Modular Addition( C7). The label is the sum of the inputs modulo a positive integer, which is 7in\nthiscase. Thechainofthoughtsandhintsarethepartialmodularsum. Low-depthtransformerswith hintcan\nsolve this task well for a reasonable input sequence length, but with cotthe performance is much better,\nespecially with a long input sequence, as predicted by our Theorem 3.3. See experiments for C2in Figure 5.\n•base: The optimization objective is simply ℓbase(θ)≜Ex∼pℓce(pθ(x), f∗(x)).\n•cot: Wemanuallydesignachainofthoughtforeachinstance x,whichisastringin Vandwe\ndenote by c(x). We ensure the last token of c(x)is always equal to the answer, f∗(x). With\n˜x≜(x, c(x)),theconcatenationof xandc(x),andmbethelengthof c(x),theoptimization\nobjective is ℓcot(θ)≜1\nmEx∼pPn+m−1\ni=n−lnpθ(˜xi+1|˜x1, . . . , ˜xi)).\n•hint: Evenifthetransformerhasbetterperformancein cotsettingthan basesetting,onemay\nargue that besides the difference expressiveness, cotsetting also has a statistical advantage\noverbase,ascotprovidesmorelabelsandthusmoreinformationaboutthegroundtruth f∗to\nthemodel. Thismotivatesustodesignthefollowinglosswhichprovidesthechainofthought\nc(x)asthelabels. Hereforsimplicity,weassumethelengthof c(x)isequalto n.5Formally\nwe define ℓhint(θ)≜1\nnEx∼pPn\ni=1−lnpθ(ci(x)|x1, . . . , x i)).\nPerformanceEvaluation. Sincewetraintransformersusingfreshsampledsyntheticdataeach\nstep,thetrainingaccuracy/lossisjustthesameasvalidationaccuracy/loss. For baseandhintsetting,\nwe evaluate the accuracy of the final answer directly. For cotsetting, directly evaluating the final\nanswer is too easy because it only measures the ability of the transformer to correctly compute the\nlaststepsinceCoTisgivenasinputs. Ideally,weshouldmeasuretheansweroutputbytransformers\nafter auto-regressively generating |c(x)|tokens. But for computational efficiency, we measure the\nprobabilitythattransformerscanpredictalltokensinthegivenCoTcorrectly. Notethisprobability\nisalowerboundoftheidealmetricbecausethereisasmallpossibilitythattransformerscananswer\ncorrectly with a wrong CoT. Nevertheless, even with this slightly more difficult evaluation metric,\ntransformers in cotsetting still optimize much faster than without CoT.\nDue to space limit, we defer the details of the training and each setting to Appendix A. Our\nexperimental results are presented in Figures 1 to 4.\nOur Findings: Unsurprisingly, the accuracy in hintsetting is always higher than basesetting.\nDuetothespacelimit,wepostponeallresultsfor basesettingsintoAppendixA.Fortheproblems\nhard for parallel computation, i.e., permutation composition, iterated squaring, and circuit value\nproblem,wefindthat cotisalwaysbetterthan hintandbase,andtheimprovementishugeespecially\n5NotesuchalignmentisingeneralimpossiblebecausetheCoT c(x)canbeevenlongerthan xitself. However,inour\nfour settings, there exist meaningful ways to set CoT c(x)as hints for earlier tokens in x.\n11Hint Label: Input: CoTLabel: Input: 9972^2^2^2^2=4162567317319972^2^2^2^2=416256731416256731\nFigure3: IteratedSquaring(IS).Thevocabulary V≜{0,1, . . . , T −1,=,ˆ 2}withT= 1000. Werandomly\ngenerate input of format (r, p,ˆ 2, . . . ,ˆ 2,=)with1≤r, p≤T−1,pbeing a prime and random number\nof ˆ2 tokens (at most m). The label is fr,p(n)≡(r2n) mod p. CoT and hints are (fr,p(i))n\ni=1. Though\nour construction does not exactly satisfy the technical conditions of the hardness assumption, this problem is\ndifficult for transformers without CoT to learn, but can be perfectly expressed with CoT even with depth 1.\nwhen depth is small. Our experiments suggest that turning on CoT drastically improves the expres-\nsiveness of low-depth transformers on problems which are hard to be parallel computed, i.e., those\ninherently serial problems.\n5 Related Works\nDespitethenumerousempiricalachievements,unansweredquestionsconcerningtheinnerworkings\nof neural networks capable of algorithmic reasoning. The ability of self-attention to create low-\ncomplexitycircuitshasbeenrecognized(Edelmanetal.,2022;Hahn,2020;Merrilletal.,2021),as\nwellasitscapacitytoformdeclarativeprograms(Weissetal.,2021),andTuringmachines(Dehghani\net al., 2018; Giannou et al., 2023; Pérez et al., 2021). Moreover, it has been demonstrated that\ninterpretable symbolic computations can be drawn from trained models (Clark et al., 2019; Tenney\net al., 2019; Vig, 2019; Wang et al., 2022b).\nLiuetal.(2022a)isacloselyrelatedworktoours,whichstudiestheexpressivenessoflow-depth\ntransformers for semi-automata. Their setting corresponds to using only 1 step of CoT and our\ncontribution is to show that allowing more steps of CoT enables the transformers to solve more\ndifficult problems than semi-automata, especially those inherently serial problems, like the circuit\nvalue problem, which is P-complete.\nConstantprecisionversuslogarithmicprecision: Wenotethatmostpreviousliteratureonthe\nexpressivenessoftransformersfocusesonthesettingoflogarithmicprecision,including(Merrill&\nSabharwal, 2023b; Merrill et al., 2022, 2021; Liu et al., 2022a), etc. One main reason as argued by\nMerrill & Sabharwal (2023a) is that log precision allows the transformer to use uniform attention\nover the rest tokens. However, recent advancements in LLMs showed that uniform attention might\nnot be necessary towards good performance, at least for natural language tasks. For example, one\nof the most successful open-sourced LLM, LLAMA2 (Touvron et al., 2023) takes the input of\n12Hint Label: Input: CoTInput: TNANA1FNANA2AND123NOT3NA4=TNANA1FNANA2ANDTFFNOTFNATTABTNANA…3NA4=TNANA1…FNATTTNANA1…FNATTAAB\nFigure4: CircuitValueProblem(CVP).Givenarandomlygeneratedcircuitwith mgates(sortedbytopolog-\nical order), the vocabulary V= [m]∪ {TRUE ,FALSE ,AND,OR,NOT,NA,=}. Each gate is represented\nby four consecutive tokens, which are gate type, two input gate ids, and the current gate id. The output is the\nvalueofthelastgate m. CoTandhintsalsocontain4tokensforeachgate,whicharegatetype,twoinputgate\nvalues, and the current gate value.\na sequence of 4096tokens and uses BF16precision, which has 1 sign bit, 8 exponent bits and 7\nmantissabits(plusoneextraleadingbit). Asaconsequence,forexample, BF16cannotexpressany\nfloating-point number between 28= 256and28+ 2 = 258 , which makes LLAMA2 impossible to\ncompute uniform attention over 257elements.\nA concurrent work Feng et al. (2023) also studies the benefit of CoT via the perspective of\nexpressiveness, where they show with CoT, transformers can solve some specific P-complete prob-\nlem. Our result is stronger in the sense that we give a simple and clean construction for each\nproblem in P/poly. We also note the slight difference in the settings, while we mainly focus on\nconstant-precision transformers with O(logn)embedding size, they focus on O(log(n))precision\ntransformers with bounded embedding size.\n6 Conclusion\nWestudythecapabilityofCoTfordecoder-onlytransformersthroughthelensofexpressiveness. We\nadoptthelanguageofcircuitcomplexityanddefineanewcomplexityclass CoT[T(n), d(n), s(n), e(n)]\nwhich corresponds to a problem class solvable by constant-depth, constant-precision decoder-only\ntransformerswith O(T(n))stepsofCoT, O(d(n))embeddingsizeandfloating-pointnumberswith\nO(e(n))bits of exponents and O(s(n))bits of significand. Our theory suggests that increasing\nthe length of CoT can drastically make transformers more expressive. We also empirically verify\nour theory in four arithmetic problems. We find that for those three inherently serial problems,\ntransformers can only express the groundtruth function by using CoT.\n13Acknowledgement\nThe authors would like to thank the support from NSF IIS 2045685. 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International Conference on Learning Representations (ICLR) , 2023.\n18Contents\n1 Introduction 1\n2 Notations and Preliminaries 2\n2.1 Decoder-only Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n2.2 Circuit Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n3 Expressiveness Theory for Transformers with Chain of Thought(CoT) 5\n3.1 Finite Precision Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n3.2 CoT: Complexity Class for Constant-depth Transformers with CoT . . . . . . . . . 6\n3.3 Tighter Upper Bounds on Transformer Expressiveness . . . . . . . . . . . . . . . 7\n3.4 CoT Makes Transformers More Expressive . . . . . . . . . . . . . . . . . . . . . . 8\n4 CoT Empirically Improves Expressiveness of Low-Depth Transformers on Inherently\nSerial Problems 10\n5 Related Works 12\n6 Conclusion 13\nA Additional Experimental Results 20\nB Details on Finite-Precision Layers 22\nC Preliminary of Automata and Krohn-Rhodes Decomposition Theorem 23\nC.1 The Krohn-Rhodes Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . 24\nC.2 Counter-free Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\nD Proofs for Expressiveness Upper Bounds (Section 3.3) 25\nD.1 Proofs for Theorem D.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\nD.2 Proofs for Theorem D.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27\nE Proofs for Expressiveness Lower Bounds (Section 3.4) 28\nE.1 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\nE.2 Proof of Theorems 3.7 and 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31\nE.3 Proof of Theorem 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34\nE.4 Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34\nF Discussion on Variants in Transformer Architecture 36\nF.1 Extension to Transformers with LayerNorm . . . . . . . . . . . . . . . . . . . . . 36\nF.2 Extension to Transformers with Multihead Attention . . . . . . . . . . . . . . . . 36\nG Discussion on Non-uniformity 36\n19A Additional Experimental Results\nIn this section present the experimental results for basesetting which is omitted in the main paper\nand the details of training and each task. We use the nanogpt 6codebase for language modeling.\nTraining Details. For all settings we use Adamwith 10−5learning rate, 0weight decay,\nβ1= 0.9,β2= 0.95, and gradient clipping with threshold equal to 1.0. The total training budget\nis106steps and we use a linear warmup in the first 2000steps starting from 10−6. For each step,\nwe use a fresh sampled batch of size 64from population distribution. We turn off dropout and use\nfloat16. We vary the depth of the transformer for different settings while the embedding size and\nthe number of attention heads are fixed to be 512and8respectively.\nBelow we present the setting and the experimental results of each problem respectively.\nModular Addition ( Cp).Given any positive integer p, the vocabulary of modular addition\nproblem is {0,1, . . . , p −1,=}. We generate x= (x1, . . . , x n)in the following way: for each\ni∈[n−1], we independently sample xifrom{0,1, . . . , p −1}and set xn= ‘=’. The label is\nf∗(x) =Pn−1\ni=1ximod pand CoT c(x)is(Pk\ni=1ximod p)n−1\nk=1. Unsurprisingly, this task is an\neasy task for transformers because attention can easily express the average function across different\npositions, and so is the sum function. Then the feedforward layers can compute the modulus of the\nsumand m. WenotethatthehightrainingaccuracyhereisnotcontradictorywithourTheorem3.1,\nbecause our sequence length is not long enough and float16is like log-precision. This intuitive\nargument is elegantly extended to all solvable groups by leveraging Khron-Rhodes decomposition\ntheorem by Liu et al. (2022a).\nPermutationComposition( Sp).Givenany p∈N+,thevocabularyofpermutationcomposition\nproblem is {1, . . . , p, (,),=}. We pick n= (p+ 2)m+ 1and generate x= (x1, . . . , x n)in the\nfollowingway: foreach i∈[m],weset x(p+2)(i−1)+1as’(’, x(p+2)ias’)’andindependentlysample\na random permutation over [p],σi= (x(p+2)(i−1)+2, . . . , x (p+2)(i−1)+p+1). We set xnto be ’=’.\nDifferent from other settings which only have the label at one position, we have plabels for this\nsetting,whichisthecompositionof σ1◦. . .◦σn. TheCoT c(x)isthepartialcompositionfrom σ1\ntoσn.\nAsmentionedinSection3,unless TC0=NC1,compositionof Spcannotbecomputedby TC0\nfor any p≥5, since composition of Spimplies the wording problem of Sp, which is NC1-complete\nunder AC0reductions. Sinceallconstant-depthpoly-embedding-sizetransformerscanbesimulated\nbyTC0circuits (Theorem 3.2), shallow transformers are not able to solve the composition problem\nofSpforp≥5. Our experimental results in Figure 1 matches this theoretic prediction very well.\nIteratedSquaring(IS). Iteratedsquaringreferstothefollowingproblem: givenintegers r, n, p,\nwe define the iterated squaring function fr,p(n)≜r2nmod p. It is often used as hardness\nassumptions in cryptography (Rivest et al., 1996; Lombardi & Vaikuntanathan, 2020) that iterated\nsquaring cannot be solved in n−o(n)time even with polynomial parallel processors under certain\ntechnical conditions ( e.g.,pis the product of two primes of a certain magnitude and there is some\nrequirement on the order of ras an element of the multiplicative group of integers modulo p). In\notherwords,peopleconjecturethereisnofasterparallelalgorithmthandoingsquaringfor ntimes.\nCircuitValueProblem(CVP). Circuitvalueproblemisthecomputationalproblemofcomput-\ningtheoutputofagivenBooleancircuitonagiveninput. Itiscompletefor Punder AC0-reductions.\nThis means if one can solve CVP with constant-depth transformers (or any TC0circuits), then any\nproblem in Pbecomes solvable by TC0, which is widely believed to be impossible.\n6https://github.com/karpathy/nanoGPT\n20Hint Label: Input: 1001100110101=CoTLabel: Input: 1001100110101=1001100\nFigure 5: Results of Modular Addition C2.\n369121518212427303336\nLength129654321Depth100 21 20 21 21 20 20 20 20 20 20 20\n100 21 20 20 20 20 21 20 20 20 20 20\n100 21 21 22 20 21 20 21 20 20 20 20\n100 20 20 20 20 20 20 20 20 21 23 20\n100 20 20 21 21 20 21 20 20 20 20 20\n100 21 21 20 20 20 20 20 20 20 20 20\n100 20 21 20 20 21 20 21 20 21 20 21\n72 20 20 20 21 21 20 21 20 21 20 20S5, Base, accuracy %\n6 14 30 62 126\nLength1296321Depth52 25 17 15 11\n57 27 16 13 11\n52 27 17 13 13\n40 19 16 11 11\n30 20 14 11 12\n19 14 12 10 12IS1000, Base, accuracy %\n7 15 31 63\nLength1296321Depth100 100 80 66\n100 95 64 78\n100 100 59 73\n100 94 67 73\n100 77 72 64\n100 66 75 62CVP, Base, accuracy %\nFigure 6: Results of baseon Permutation Composition, Iterated Squaring, and Circuit Value Problem.\n7 15 31 63 127\nLength3 2 1Depth100 50 51 51 49\n100 50 51 51 48\n51 50 50 50 49C2, Base, accuracy %\n7 15 31 63 127\nLength3 2 1Depth15 16 14 14 14\n13 14 17 14 14\n15 13 15 16 15C7, Base, accuracy %\nFigure 7: Results of Modular Addition baseonC2andC7.\nWe can observe that the accuracy of basesetting is also lower than that of hintsetting.\n21B Details on Finite-Precision Layers\nIn this section, we give the definition of the finite-precision version of different transformer layers.\nRecall that given s∈N+, the numbers representable using 2s-bit significand and e-bit exponent is\nFe,s≜{S·2−s+E| −22s+1≤S≤22s−1,−2max(0 ,e−1)≤E≤2e−1−2max(0 ,e−1), E, S∈N}.\nSelf-Attention Mechanism: Given attention parameter θATTN = (WQ, WK, WV, WO)∈\nFd×d\ne,s×Fd×d\ne,s×Fd×d\ne,s×Fd×d\ne,s, we define the self-attention layer with causal mask for decoder-\nonly transformer in Algorithm 3.\nAlgorithm 3 Finite-Precision Causal Self-Attention, ATTN\nInput:Integer s∈N+,e∈N, Parameter θATTN = (WQ, WK, WV, WO), Input embedding\nh= (h1, . . . , h n)∈Fnd\ne,s.\nOutput: Output embedding h′= (h′\n1, . . . , h′\nn)≜ATTN θATTN(h1, . . . , h n).\n1:qi≜WQ×e,shi, ki≜WK×e,shi, vi≜WV×e,shi,∀i∈[n]\n2:si≜softmax e,s(⟨qi, k1⟩e,s, . . . ,⟨qi, ki⟩e,s)∥(0, . . . , 0). ▷ n−i’s 0; Mask for Causal\nAttention;\n3:h′\ni≜WO×e,ssume,s([v1, . . . , v n]×e,ssi).\nFeed-Forward Network: Given s∈N+,e∈N, and the parameter of fully-connected feed-\nforward network layer θFF= (W1, b1, W2, b2)∈Fd×d\ne,s×Fd\ne,s×Fd×d\ne,s×Fd\ne,s, we define the fully-\nconnectedfeedforwardlayer FFθFF:Fd\ne,s→Fd\ne,sasFFθFF(h)≜h\nW2×e,srelu([W1×e,sh+b1]e,s) +b2i\ne,s.\nToken Embedding: Given s∈N+,e∈N, and the parameter of token embedding layer\nθTE∈Fd×|V|\ne,s,wedefinethetokenembeddinglayerbyviewing θTEasamappingfrom VtoRd,that\nis, for all x∈ V, the token embedding is θTE(x).\nPosition Encoding: Given s∈N+,e∈N, and the parameter of position encoding layer\nθPE∈Fd×nmaxe,s, we define the token embedding layer by viewing θPEas a mapping from [nmax]to\nRdthat is, for all n∈[nmax], the position embedding is as θPE(n).\nOutput Layer: Given s∈N+,e∈N, and the parameter of output layer θOUTPUT ∈\nF|V|×d\ne,s, we define the output layer OUTPUT θOUTPUT :Fd\ne,s→ VasOUTPUT θOUTPUT (h)≜\nsoftmax e,s(θOUTPUT ×e,sh)for all h∈Fd\ne,s.\nFinally, we define finite-precision decoder-only transformers below.\nAlgorithm 4 Finite-precision Decoder-only Transformer, TFθandpθ\nInput:Integer s∈N+,e∈N. Transformerparameter θ= (θPE, θTE, θOUTPUT ,{θ(l)\nATTN, θ(l)\nFF}L−1\nl=0)\nwith2s-bit precision and e-bit exponent. Input tokens x= (x1, . . . , x n)∈ Vn.\nOutput: Output distribution pθ(· |x1, . . . , x i)for all i∈[n]and output token TFθ(x).\n1:h(0)\ni≜[TE(xi) +PE(i)]e,s,∀i∈[n]\n2:forl= 0, . . . , L −1do\n3: (h(l+0.5)\n1 , . . . , h(l+0.5)\nn )≜h\n(h(l)\n1, . . . , h(l)\nn) +ATTNθ(l)\nATTN(h(l)\n1, . . . , h(l)\nn)i\ne,s\n4: h(l+1)\ni≜h\nh(l+0.5)\ni +FFθ(l)\nFF(h(l+0.5)\ni )i\ne,s,∀i∈[n]\n5:end for\n6:pθ(· |x1, . . . , x i)≜h\nOUTPUT θOUTPUT (h(L)\ni)i\ne,s,∀i∈[n]\n7:TFθ(x)≜arg maxxpθ(x|x1, . . . , x n).\n22C PreliminaryofAutomataandKrohn-RhodesDecompositionTheo-\nrem\nIn this section we recap the basic notations and definitions for automata theory and Krohn-Rhodes\nDecompositionTheorem(Krohn&Rhodes,1965),followingthenotationandpresentationofMaler\n(2010).\nDefinitionC.1 (Automaton) .Adeterministicautomatonistriple A= (Σ, Q, δ )where Σisafinite\nsetofsymbolscalledtheinputalphabet, Qisafinitesetofstatesand δ:Q×Σ→Qisthetransition\nfunction.\nThe transition function can be lifted naturally to input sequences, by letting δ(q, wσ )≜\nδ(δ(q, w), σ)for all w∈Σ∗recursively.\nAnautomatoncanbemadeanacceptorbychoosinganinitialstate q0∈Qandasetofaccepting\nstates F⊆Q. As such it accepts/recognizes a set of sequences, also known as a language, defined\nasL(A, q0, F) ={w∈Σ∗:δ(q0, w)∈F}. Kleene’s Theorem states that the class of languages\nrecognizable by finite automata coincides with the regular languages.\nDefinitionC.2 (AutomatonHomomorphism) .Asurjection ϕ:Q→Q0isanautomatonhomomor-\nphismfrom A= (Σ, Q, δ )toA0= (Σ, Q0, δ0)ifforevery q∈Q, σ∈Σ, ϕ(δ(q, σ)) =δ0(ϕ(q), σ).\nInsuchacasewesaythat A0ishomomorphicto Aanddenoteitby A0≤ϕA. When ϕisabijection,\nAandA0are said to be isomorphic.\nThe conceptual significance of Automaton Homomorphism is that, if we can simulate any A\nandA0≤ϕA, we can ‘almost’ simulate A0as well, in the sense of following lemma:\nLemma C.1. For any two automata A= (Σ , Q, δ ),A0= (Σ , Q0, δ0)satisfying that A0≤ϕ\nAfor some function ϕ, for any F0⊆Q,q0∈Q,ϕ(q) =q0, it holds that L(A0, q0, F0) =\nL(A, q, ϕ−1(F0)).\nProof of Lemma C.1. We claim for any w∈Σ∗, it holds that ϕ(δ(q, w)) =δ(ϕ(q), w). This claim\nholds by definition of automaton homomorphism for all |w| ≤1. suppose the claim already holds\nfor all wno longer than nfor some n, for any w′=wσwith|w|=nandσ∈Σ, it holds that\nϕ(δ(q, w′)) = ϕ(δ(δ(q, w), σ)) = δ(ϕ(δ(q, w)), σ) =δ(δ(ϕ(q), w), σ) =δ(ϕ(q), w′). Therefore\nδ0(q0, w)∈F0⇐⇒ δ0(ϕ(q), w)∈F0⇐⇒ ϕ(δ(q, w))∈F0⇐⇒ δ(q, w)∈ϕ−1(F0). Thus\nwe conclude that L(A0, q0, F0) ={w∈Σ∗|δ(q0, w)∈F0}={w∈Σ∗|δ(q, w)∈ϕ−1(F0)}=\nL(A, q, ϕ−1(F0)).\nDefinition C.3 (Semigroups, Monoids and Groups) .A Semigroup is a pair (S,·)where Sis a set\nand·is a binary associative operation (“multiplication”) from S×StoS. A Monoid (S,·,1)is a\nsemigroupadmittinganidentityelement 1suchthat s·1 = 1···=sforevery s∈S. Agroupisa\nmonoidsuchthatforevery s∈Sthereexistsanelement s−1∈S(aninverse)suchthat s·s−1= 1.\nDefinition C.4 (Semigroup Homomorphisms) .A surjective function ϕ:S→S0is a semigroup\nhomomorphism from (S,·)to(S0,∗)if for every s1, s2∈S, ϕ(s1·s2) =ϕ(s1)∗ϕ(s2). In such a\ncase we say that S0is homomorphic to Sand denote it by S0≤ϕS. Two mutually homomorphic\nsemigroups are said to be isomorphic.\nDefinition C.5 (Transformation Semigroup) .The transformation semigroup of an automata A=\n(Σ, Q, δ )is the semigroup generated by {δ(·, σ) :Q→Q|σ∈Σ}.\n23C.1 The Krohn-Rhodes Decomposition Theorem\nBelow we give the definition of the cascade product of two automata, which is a central concept\nused in Krohn-Rhodes Decomposition Theorem for automata.\nDefinition C.6 (Cascade Product) .LetB1= (Σ , Q1, δ1)andB2= (Q1×Σ, Q2, δ2)be two\nautomata. The cascade product B1◦ B2is the automaton C= (Σ, Q1×Q2,δ)where\nδ((q1, q2), σ)≜(δ1(q1, σ), δ2(q2,(q1, σ))).\nThecascadeproductofmorethantwoautomataisdefinedas B1◦ B2◦ ··· ◦ B k= (···((B1◦ B2)◦\nB3···)◦ Bk.\nDefinition C.7 (Permutation-Reset Automata) .A automaton A= (Σ , Q, δ )is a permutation-reset\nautomaton if for every letter σ∈Σ,σis either a permutation or reset. If the only permutations are\nidentities, we call it a reset automaton.\nTheorem C.2 (Krohn-Rhodes; cf. Maler (2010)) .For every automaton A there exists a cascade\nC=B1◦ B2◦ · · · ◦ B ksuch that:\n1. Each Biis a permutation-reset automaton;\n2. There is a homomorphism ϕfromCtoA;\n3. Any permutation group in some Biis homomorphic to a subgroup of the transformation\nsemigroup of A.\nThe pair (C, ϕ)is called a cascaded decomposition of A.\nC.2 Counter-free Automata\nNext we introduce a key concept used in the proof of Theorem D.1 (and thus Theorem 3.1) –\nCounter-free Automaton.\nDefinitionC.8 (Counter-freeAutomaton,(McNaughton&Papert,1971)) .Anautomatoniscounter-\nfree if no word w∈Σ∗induces a permutation other than identity on any subset of Q.\nA subclass of the regular languages is the class of star-free sets defined as:\nDefinition C.9 (Star-Free regular languages) .The class of star-free regular languages over Σis the\nsmallestclasscontaining Σ∗andthesetsoftheform {σ}where σ∈σ∪ {ϵ},whichisclosedunder\nfinitely many applications of concatenation and Boolean operations including union, intersection,\nand complementation.\nItiswell-knownthatlanguagesrecognizedbycounter-freeautomatahavethefollowingequiva-\nlent characterizations.\nTheorem C.3 (McNaughton & Papert (1971)) .Suppose Lis a regular language not containing the\nempty string. Then the following are equivalent:\n1.Lis star-free;\n2.Lis accepted by a counter-free automata.\n243.Lis non-counting, i.e., there is an n∈Nso that for all x,y, and zand all m≥n,\nxymz∈L⇐⇒ xym+1z∈L.\nCounter-freepropertyofanautomatoncanalsobecharacterizedviaitstransformationsemigroup\nby Lemma C.4, whose proof is straightforward and skipped.\nLemma C.4. An automaton is counter-free if and only if the transformation semigroup of the\nautomaton is group-free, i.e., it has no non-trivial subgroups. A semigroup (S,·)is group-free if\nand only if it is aperiodic ,i.e., for all s∈S, there exists k∈N,sk=sk+1.\nThus Theorem C.5 holds as a corollary of Theorem C.2.\nTheorem C.5 (Corollary of Theorem C.2) .For every counter-free automaton Athere exists a\ncascade C=B1◦ B2◦ · · · ◦ B ksuchthateach Biisaresetautomatonandthereisahomomorphism\nϕfromCtoA.\nUsingTheoremC.5thefollowingtheoremconnectsthecounter-freeautomatatoconstant-depth\npoly-size circuits with unbounded fan-in. The high-level proof idea is that any reset automaton can\nbe simulated using constantly many depth and any counter-free automaton can be decomposed into\nthe cascade product of a finite number of reset automaton.\nTheorem C.6. [Theorem 2.6, Chandra et al. (1983)] Suppose A= (Σ , Q, δ )is an counter-free\nautomaton. Then there is a circuit of size O(n3)with unbounded fan-in and constant depth that\nsimulates δ(q, w)for any q∈Qandw∈Σ∗satisfying |w|=n, where O(·)hides constants\ndepending on the automaton.\nD Proofs for Expressiveness Upper Bounds (Section 3.3)\nThe main technical theorems we will prove in this section are Theorems D.1 and D.2. Their proofs\ncan be found in Appendices D.1 and D.2 respectively.\nRecall ψe,s:Fe,s→ {0,1}e+2s+1is the binary representation of floating point with e-bit\nexponent and 2s-bit precision.\nTheorem D.1. For any fixed e∈N, s∈N+,sume,s: (Fe,s)n→Fe,shasAC0circuits.\nIn detail, there is a family of AC0circuits {Cn}such that for all x1, . . . , x n∈Fe,s, it holds that\nCn(ψe,s(x1)∥. . .∥ψe,s(xn)) =ψe,s(sume,s(x1, . . . , x n)) (2)\nTheorem D.2. Fors(n) =O(poly(n)),sum 0,s(n): (F0,s(n))n→F0,s(n)hasTC0circuits.\nIn detail, there is a family of TC0circuits {Cn}such that for all x1, . . . , x n∈F0,s(n), it holds\nthat\nCn(ψ0,s(n)(x1)∥. . .∥ψ0,s(n)(xn)) =ψ0,s(n)(sume,s(x1, . . . , x n)) (3)\nWithTheoremsD.1andD.2ready,Theorems3.1and3.2arestandard(e.g.,seeproofofTheorem\n4 in Liu et al. (2022a)) and thus are omitted.\nD.1 Proofs for Theorem D.1\nDefinitionD.1 (TotalOrder) .Atotalorder ≤onsomeset Xisabinaryrelationshipsatisfyingthat\nfor all a, b, c∈X:\n1.a≤a(reflexive)\n252.a≤b,b≤c=⇒a≤c(transitive)\n3.a≤b,b≤a=⇒a=b(antisymmetric)\n4.a≤borb≤a. (total)\nDefinitionD.2 (OrderedAutomaton) .Wesayanautomaton A= (Σ, Q, δ )isorderedifandonlyif\nthere exists a total order ≤onQand for all σ∈Σ,δ(·, σ)preserves the order, that is,\n∀q, q′∈Q, q ≥q′=⇒δ(q, σ)≥δ(q′, σ).\nTheorem D.3. All ordered automata are counter-free. Languages recognizable by any ordered\nautomata belong to AC0.\nProof of Theorem D.3. To show an ordered automaton A= (Σ , Q, δ )is counter-free, it suffices to\nits transformation semigroup is group-free, or aperiodic. We first recall the definition of aperiodic\nsemigroupsLemmaC.4. Let πw:Q→Q, π w(q)≜δ(q, w)bethetransformationinducedbyword\nw∈Σ∗. Transformation semigroup of Ais aperiodic iff for any w∈Σ∗, there exists k∈N, such\nthat(πw)k= (πw)k+1.\nNow We claim for any q∈Q, there is k∈N, such that (πw)k(q) = ( πw)k+1(q). Since Qis\nfinite, this implies that there exists k∈N, such that (πw)k= (πw)k+1and thus the transformation\nsemigroup of Ais aperiodic. First, note that Ais ordered, we know πσis order-preserving for all\nσ∈Σ. Let w=w1···wnwhere |w|=n, we have πw=πw1◦ ··· ◦ πwnis also order-preserving\nand thus for all q≥q′∈Q,πw(q)≥πw(q′). Then we proceed by three cases for each q∈Q:\n1.πw(q) =q. In this case, it suffices to take k= 0;\n2.πw(q)≥q. Since πwis order-preserving, we know for any k∈N,(πw)k+1(q)≥(πw)k(q).\nSince Qis finite, there must exist some k∈Nsuch that (πw)k+1(q) = (πw)k.\n3.πw(q)≤q. Same as the case of πw(q)≥q.\nSince≥is a total order, at least one of the three cases happens. This concludes the proof.\nThe second claim follows directly from Theorem C.3.\nFor any e, s∈N, iterated addition on floating point numbers with e-bit exponent and s-bit\nsignificand Fe,scan be viewed Ae,s= (Fe,s,Fe,s, δ+).\nTheorem D.4. Automaton Ae,s= (Fe,s,Fe,s, δ+)is ordered, where δ+(x, y)≜[x+y]e,sfor any\nx, y∈Fe,s.\nProof of Theorem D.4. The total order we use for Fe,sas the state space of automaton Acoincides\nwiththeusualorder ≤onR. Recalltheroundingoperationisdefinedas [x]e,s∈arg minx′∈Fe,s|x−x′|,\nwhich means rounding operation is order preserving, that is, for any x≥x′∈Fe,s,[x]e,s≥[x′]e,s.\nThus for any x, x′, y∈Fe,swithx≤x′, it holds that δ+(x, y) = [ x+y]e,s≥[x′+y]e,s=\nδ+(x′, y). Thus Ae,sis ordered.\nThe following theorem Theorem D.1 is a direct consequence of Theorem D.4.\n26D.2 Proofs for Theorem D.2\nWefirstclaimthatthefollowingalgorithmAlgorithm5correctlycomputes sum 0,s(n)overnnumbers\ninF0,s(n).\nLemma D.5. Algorithm 5 outputs sum 0,s(n)(x1, . . . , x n)for all n∈N+andx1, . . . , x n∈F0,s(n).\nProof of Lemma D.5. Notethat y−2= 0,[y−1+y0]0,s(n)= sign( x1)·Bs(n),and[sign( x1)·Bs(n)+\ny1]0,s(n)=x1, thus we conclude sum 0,s(n)(x1, . . . , x n) = sum 0,s(n)(y−2, y−1, y0, y1, . . . , y n).\nWithout loss of generality, we can assume that x1>0. Therefore S−2= 0, S−1=Bs(n), and\nS0= 2Bs(n), which further implies that L−2≤0andU−2≥2Bs(n). This ensures i∗is always\nwell-defined. Forconvenienceweuse Hitodenote sum 0,s(n)(y−2, y−1, y0, y1, . . . , y i−1)intherest\nof this proof.\nNow we claim either Si∗=Li∗orSi∗=Ui∗. By definition of i∗, if neither of these two\nequalities happen, we have that i∗≤n−1,Ui∗=Ui∗+1, and Li∗=Li∗+1, which contradicts with\nthe maximality of i∗since Ui∗+1−Li∗+1=Ui∗−Li∗≥2Bs(n). Without loss of generality, we\nassume Si∗=Li∗and the analysis for the other case is almost the same. Now we claim that for all\ni > i∗, no negative overflow happens at position i, that is, Hi+yi≥ −Bs(n).\nWe will prove this claim for two cases respectively depending on whether there exists some\ni∗< j < isuch that sum 0,s(n)(y−2, y−1, y0, y1, . . . , y j) =Bs(n). The first case is such jdoes not\nexist. Then neither positive or negative overflow happens through i∗toi, and thus\nHi−1+yi=Hi∗+ (Si−Si∗)≥Hi∗≥ −Bs(n). (4)\nIf such jexists, we let j∗to be the maximum of such j. Then neither positive or negative\noverflow happens through j∗toi. Due to the optimality of i∗, we know that for all i, j≥i∗,\n|Si−Sj|<2Bs(n)<. Thus\nHi−1+yi=Hj∗+ (Si−Sj∗)≥Bs(n)−2Bs(n)≥ −Bs(n). (5)\nNowweclaim Hk∗=Bs(n). Becausethereisnonegativeoverflowbetween i∗andk∗,wehave\nthatHk∗−Hi∗≥Sk∗−Si∗≥2Bs(n)and the fist inequality is only strict when positive overflow\nhappens at some i∗≤j≤k∗. If there is no such j, then Bs(n)≥Hk∗≥Hi∗+ 2Bs(n)≥Bs(n)\nand thus Hk∗=Bs(n). Otherwise such jexists and j∗be the maximum of such j. Then\nHk∗≥Hj∗+ (Sk∗−Sj∗)≥Bs(n)+ (Sk∗−Sj∗)≥Bs(n), where the last inequality is due\nto the optimality of k∗. Thus in both cases we conclude that Hk∗=Bs(n).\nFinally we will show there is neither negative or positive overflow from k∗+ 1tonand thus\nHn=Hk∗+Sn−Sk∗, which would justify the correctness of the algorithm. We have already\nshown there is no negative overflow. Suppose there is a positive overflow at some j > k∗in\nthe sense that Hj−1+yj≥Bs(n)and we let j∗be the first positive overflow after k∗. By\ndefinition of j∗, there is neither positive and negative overflow between k∗+ 1andj∗and thus\nHj∗−1+yj∗=Hk∗+ (Sj∗−Sk∗)≤Hk∗=Bs(n), which is contradictory to the assumption that\nthere is a positive overflow at j∗. This concludes the proof.\nProof of Theorem 3.2. It suffices to show that Algorithm 5 can be implemented by a family of AC0\ncircuitssinceLemmaD.5guaranteesthecorrectnessofAlgorithm5. Wecantreatallthefixed-point\nfloatingnumbersintheAlgorithm5asintegerswithasuitablerescaling,whichis 2s(n). Sinceboth\nsortingandadding nbinaryintegerswithpolynomialbitshave AC0circuits,eachlineinAlgorithm5\ncan be implemented by an AC0circuits (for all indexes isimultaneously if there is any).\n27Algorithm 5 AC0algorithm for iterative addition for poly-precision floating point numbers\nInput:Integer n∈N+,s(n) =O(poly(n)). Floating numbers x1, . . . , x n∈F0,s(n).\nOutput: ans=sum 0,s(n)(x1, . . . , x n)∈F0,s(n).\n1:y−2←0, y−1, y0←sign(x1)·Bs(n),y1←x1−sign(x1)·Bs(n),yi←xi,∀i∈ {2, . . . , n };\n2:Si←Pi\nj=0yj,∀i∈ {− 2, . . . , n };\n3:Ui←max i≤j≤nSj, Li←mini≤j≤nSj,∀i∈ {− 2, . . . , n };\n4:i∗←max{−2≤i≤n|Ui−Li≥2Bs(n)};\n5:ifSi∗=Ui∗then\n6: k∗←max{i∗≤k≤n|Sk=Li∗};\n7: O← −Bs(n);\n8:else\n9: k∗←max{i∗≤k≤n|Sk=Ui∗};\n10: O←Bs(n);\n11:end if\n12:ans←O+Sn−Sk∗.\nE Proofs for Expressiveness Lower Bounds (Section 3.4)\nWe first introduce some notations. Since in the construction of the lower bounds we are only using\nfixed-point numbers, we will use the shorthand Fs≜F0,s={c·k·2−s|c∈ {− 1,1},0≤\nk≤22s−1, k∈N}and rounding operation [·]s≜[·]0,s. We use 1sto denote all-one vectors of\nlength s. Similarly we define ⟨·,·⟩s,×s, and softmax s. We recall that for any s∈N+and integer\n0≤x≤2s−1, we use bins(x)∈ {0,1}sto denote the usual binary encoding of integer xusing s\nbinary bits in the sense that x=Ps\ni=12i(bins(x))iandsbins(x)∈ {− 1,1}sto denote the signed\nbinary encoding, which is 2bins(x)−(1, . . . , 1).\nRecall Bs= max Fs= 2s−2−s.\nLemma E.1. For any s∈N+, it holds that [exp(−Bs)]s= 0.\nProof of Lemma E.1. By the definition of rounding operation for 2s-bit precision (Definition 3.2),\nitsufficestoshowthat exp(−Bs)≤2−s−1,thatis, Bs≥ln 2·(s+ 1). Notethat 2s≥s+ 1forall\ns≥1, we have Bs/(s+ 1)≥Bs2−s= 1−2−2s≥3/4≥ln 2.\nUsing the same argument above, we also have Lemma E.2.\nLemma E.2. For any s∈N+, it holds that [exp( Bs)]s=Bs.\nE.1 Proof of Theorem 3.3\nGiven two vectors x, yof the same length s, we use x⌢yto denote their interleaving, that is,\n(x⌢y)2i−1=xi,(x⌢y)2i=yifor all i∈[e].\nLemma E.3. For any s∈N+, letqi=sbins(i)⌢1sandki=Bs·(sbins(i)⌢(−1s))for all\ni∈[2s−1], it holds that\u0002\nexp(⟨qi, kj⟩s)\u0003\ns=1[i=j]for all i, j∈[2s−1].\nProof of Lemma E.3. It suffices to prove that ⟨qi, kj⟩s=−Bsifi̸=jand⟨qi, kj⟩s= 0ifi=j.\nThe rest is done by Lemma E.1.\nGiven any i, j∈[2s−1], by definition of finite-precision inner product, we know that for any\nl∈[2s−1], it holds that al=\u0002\nal−1+ [(qi)l(kj)l]s\u0003\nswhere a0≜0andal≜⟨(qi):l,(kj):l⟩sfor\nl∈[2s].\n28Forall l∈[e],wehavethat [(qi)2l(kj)2l]s=−Bsand[(qi)2l−1(kj)2l−1]s=Bs·1[(sbins(i))l=\n(sbins(j))l]. Ifi=j, it is straightforward that a2l−1=−Bsanda2l= 0for all l∈[e]. Ifi̸=j,\nthen there exists l∈[s−1]such that (sbin(i)))l̸= (sbin(j)))l. Thus [(qi)2l−1(kj)2l−1]s=\n[(qi)2l(kj)2l]s=−Bs, which implies a2l=−Bsregardless of the value of a2l−2. Again use\ninduction we can conclude that a2l′=−Bsfor all l≤l′≤e.\nProof of Theorem 3.3. For any L ∈ SIZE[T(n)], by definition there is a family of boolean circuit\n{Cn}which compute Lfor all inputs of length nusing O(T(n))many NOTandANDgates.\nWithout loss of generality, let us assume the number of non-input gates in CnbeT(n). We will\nshow that for each n, there is a 2-layer decoder-only transformer TFθnthat computes Cn(x)using\nT(n)stepsofCoTforall x∈ {0,1}n. Moreprecisely,wewillconstructatransformerthatsimulates\none boolean gate in Cn, following the topological order of the circuit, in each step of its chain of\nthought.\nWefirstindexthegates(includinginputgates)from 1ton+T(n)accordingtothetopological\norder. For each gate i∈[n+ 1, n+T(n)], we use a(i)andb(i)to denote its two input gates.\nSince NOTonly has one input, we set a(i)as its input and b(i)as0. We let c(i) = 0ifith gate\nisNOTandc(i) = 1ifith gate is AND. For any input gate 1≤i≤n,a(i), b(i), and the gate\ntype are not meaningful and their choice will not affect the output and thus can be set arbitrarily.\nFor convenience, we will set a(i) =b(i) =c(i) = 0. We use gi(x)to denote the output of\nnon-input gate i(n+ 1≤i≤n+T(n)) on the circuit input x∈ {0,1}n, which is equal to\n(1−c(i))(1−xa(i)) +c(i)(relu(xa(i)+xb(i)−1)).\nNow we describe the construction of the vocabulary V, the token embedding θTE, and position\nencoding θPE. We set precision sas any positive integer larger than 1,V={0,1},k≜k(n) =\n⌈log2(T(n) +n)⌉=O(logn)since T(n)is a polynomial, d′(n) = 3 k+ 6,θTE(0) = 0 ·e1,\nθTE(1) = 1 ·e1, and\n(θPE)⊤\ni−1= [0,0,0,0, c(i),sbink(i),sbink(a(i)),sbink(b(i)),1],∀2≤i≤n+T(n).(6)\nWeuse h0\ni,h0.5\ni,h1\ni,h1.5\ni,andh2\nitodenotetheintermediateembeddingsatposition ianddifferent\ndepths. Here, depth 0.5and1.5refer to the output of the Attention layer inside each transformer\nlayer.\n1. For the first attention layer, denoting embedding at the ith position h0\nibyh, we set the query as\nqi≜(hk+6:2k+5)⌢(h3k+6·1s), the key as ki≜(Bsh6:k+5)⌢(−h3k+6·1s), and the value as\nvi≜h1·e2.7\n2. For the first fully-connected layer, we skip it by setting its weights to be 0.\n3. For the second attention layer, denoting embedding at the ith position h1\nibyh, we set the query\nasqi≜(h2k+6:3k+5)⌢(h3k+6·1s), the key as ki≜(Bsh6:k+5)⌢(−h3k+6·1s), and the value\nasvi≜h1·e3.\n4. Forthesecondfully-connectedlayer,wedefine F(a, b, c )≜relu(1−a−c)+relu(a+b+c−2).\nDenote the embedding at position i,hi\n1.5byh. The output of the second fully-connected layer\nis defined as F(h2, h3, h4)·e4. Note this expression is valid because it can be expressed by\ntwo-layer ReLU nets with constant bits of precision and a constant number of neurons.\n5. The final output at position iis(h2\ni)4.\n7Note here the dimension of kiandqiare the same but less than d′, which does not strictly satisfy our definition of\ntransformerinAlgorithm3. Thisisfornotationalconvenienceandiswithoutlossofgeneralitybecausewecanpadextra\nzeros.\n29Below we first describe the value of the internal variables of the transformer and then show\nthere exist parameters making such computation realizable. Let (x1, . . . , x n)be the input tokens,\nwedefine xn+i≜TFi\nθn(x1, . . . , x n),∀1≤i≤T(n). Weclaimthereexiststransformersparameter\nθnsuch that TFT(n)\nθn(x1, . . . , x n) =gT(n)(x)(i.e.Cn(x)). More specifically, we claim that our\nconstructions will ensure the following inductively for all n+ 1≤i≤n+T(n),\n1.h0\ni−1= [xi−1,0,0,0, c(i),(sbink(i))⊤,(sbink(a(i)))⊤,(sbink(b(i)))⊤,1]⊤;\n2.h0.5\ni−1= [xi−1, xa(i),0,0, c(i),(sbink(i))⊤,(sbink(a(i)))⊤,(sbink(b(i)))⊤,1]⊤;\n3.h1\ni−1= [xi−1, xa(i),0,0, c(i),(sbink(i))⊤,(sbink(a(i)))⊤,(sbink(b(i)))⊤,1]⊤;\n4.h1.5\ni−1=[xi−1, xa(i), xb(i),0, c(i),(sbink(i))⊤,(sbink(a(i)))⊤,(sbink(b(i)))⊤,1]⊤;\n5.h2\ni−1=[xi−1, xa(i), xb(i), gi(x), c(i),(sbink(i))⊤,(sbink(a(i)))⊤,(sbink(b(i)))⊤,1]⊤;\n6.xi≜TFθn(x1, . . . , x n, . . . , x i−1) =gi(x).\nNowweexplainwhytheaboveconditionsholdforanyposition iusinginduction, i.e.,assuming\nitistrueforall n+1≤i′≤i. Wefirstnoticethatbyourconstruction,forall 1≤i′≤n−1,itholds\nthat(hk\ni′)1=xi′and(hk\ni′)6:k+5for all k∈ {0,0.5,1,1.5,2}. Note these are the only information\nthat will be used in the later attention layers.\n1. This is simply by the construction of θTEandθPE.\n2. In the first attention layer, at the jth position, we have qj≜sbink(a(j))⌢1sas the query,\nkj≜Bs·sbink(j)⌢(−1s)as the key and vj≜xjas the value for all j∈[n+T(n)].\nNote here we reduce the sizes of hidden embeddings for simplicity of demonstration. This\nis valid because we can fill the extra coordinates by 0. This is valid because we can al-\nways set the extra coordinates to be 0. By Lemma E.3, we know that\u0002\nexp(⟨qi, kj⟩s)\u0003\ns=\n1[a(i) = j]for all j∈[n+T(n)]. Recall that the attention scores are defined as\nsi≜softmax( ⟨qi, k1⟩s, . . . ,⟨qi, ki⟩s)∥(0, . . . , 0), we know that si=ea(i).\n3. Wesettheparametersinthefirstfully-connectedfeedforwardlayertobeall 0andlettheskip\nconnection pass the intermediate values.\n4. The second attention layer attains xb(i)and places it in the third coordinate in the same way\nas step 2.\n5. In the fully-connected feedforward layer we compute F(xa(i), xb(i), c(i)) = relu(1−xa(i)−\nc(i)) + relu(xa(i)+xb(i)+c(i)−2)for all xa(i), xb(i), c(i)∈ {0,1}. We can verify that\nF(xa(i), xb(i), c(i)) = (1 −c(i))(1−xa(i))+c(i)(relu(xa(i)+xb(i)−1)) = gi(x),whichisthe\ndesirrdoutputofthegate i. Thisisbecausewhen c(i) = 0,theoutputis 1−a(i) =NOTa(i)\nand when c(i) = 0, the output is relu(a(i) +b(i)−1) = a(i)ANDb(i).\n6. The output layer uses the fourth coordinate of h2\ni, which is gi(x)according to induction, as\nthe output.\nThis completes the proof of Theorem 3.3.\n30E.2 Proof of Theorems 3.7 and 3.8\nIn this subsection, we prove Theorems 3.7 and 3.8. We first prove a useful lemma that gives an\nequivalent characterization of SIZEAC0andSIZETC0.\nLemma E.4. For any T(n)∈poly(n)satisfying T(n)≥1,∀n∈N+, a decision problem\nL:∪k∈N+{0,1}k→ {0,1}belongs to SIZEAC0[T(n)](resp. SIZETC0[T(n)]) if and only if\nthere exist a polynomial S(n), a function T′(n) =O(T(n)), and a depth L∈N+such that for\nevery n∈N+there exist a sequence of sizes- S(n), depth- Lcircuits, {Ci\nn}T′(n)\ni=1, with unlimited-\nfanin AND,ORandNOTgates (with additionally MAJORITY gates for SIZETC0) and that for all\nx∈ {0,1}n,\nL(x) =xn+T′(n),where xn+i≜Ci\nn(x1, . . . , x n+i−1),∀i∈[T′(n)]. (7)\nProof of Lemma E.4. Wewillprovefor SIZETC0onlyandtheprooffor SIZEAC0isalmostthesame.\nThe“ =⇒”directionisstraightforward. Bydefinitionof SIZETC0[T(n)](Definition3.7),forany\nL ∈SIZETC0[T(n)], there is a function p(n)∈poly(n)and a family of TC0circuits {C′\ni}∞\ni=1such\nthatforevery n∈Nandx∈ {0,1}n,Ln(x1, . . . , x n)canbecomputedbyasize- O(T(n))threshold\ncircuitswithoraclegate Cp(n). Nowwesortallthenodesinthecircuitswithoraclegatesalongthe\ntopological order as x1, . . . , x n+T′(n)where x1, . . . , x nare the inputs and T′(n) =O(T(n))is the\nnumber of the gates, then clearly xn+iis a function of x1, . . . , x n+i−1for all i∈[T′(n)]and this\nfunction can be implemented by a different threshold circuit of constant depth and poly(n)size for\neachi. This completes the proof of “ =⇒” direction.\nNow we prove the other direction “ ⇐=”. We first show that given T′(n)sizes- S(n), depth- L\ncircuits, {Ci\nn}T′(n)\ni=1, there is a depth- (L+ 1), size O(T′(n)S(n))circuit C′\nn, such that\nC′\nn(x1, . . . , x n+T′(n)−1, ej) =Cj\nn(x1, . . . , x n+j−1),∀j∈[T′(n)], x∈ {0,1}n+T′(n)−1,(8)\nwhere 1j∈ {0,1}T′(n)istheone-hotvectorwithits jthcoordinatebeing 1. Indeed,itsufficestoset\nC′\nn(1, . . . , x n+T′(n)−1, y1, . . . , y T′(n)) =∨T′(n)\nj=1\u0000\nyj∧Cj\nn(x1, . . . , x n+j−1)\u0001\n.(9)\nOnce we have such oracle gate C′\nn, given input x1, . . . , x n, we can recursively define\nxn+i≜C′\nn(x1, . . . , x n+i−1,0T′(n)−n, ej). (10)\nThuswecancompute L(x) =xn+T′(n)using T′(n)oraclegate C′\nn. Wecangetconstantgate 0and\n1by using x1∧ ¬x1andx1∨ ¬x1. respectively. This completes the proof.\nNowwearereadytoproveTheorems3.7and3.8. WewillproveTheorem3.7firstandtheproof\nof Theorem 3.8 is very similar to Theorem 3.7.\nProof of Theorem 3.7. We first show that SIZETC0[T(n) + 1] ⊇CoT[T(n),poly(n),logn]. For\nthe case that the vocabulary of transformer V={0,1}, by Theorem 3.2, we know for any θn,\nTFθn(x1, . . . , x i)can be expressed by a TC0circuit whose depth is uniformly upper bounded by\nsome constant for all n≤i≤n+O(T(n)). This completes the proof when V={0,1}. When\nV ̸={0,1}, we can use the binary encoding of elements in Vas the inputs of those TC0gates\nconstructed for the later layers of the transformer.\nNowweturntotheprooffortheotherdirection: SIZETC0[T(n)+1]⊆CoT[T(n),poly(n),logn].\nIn high-level speaking, the proof contains two steps:\n311. We show that TC0⊆T[poly(n),logn]⊆CoT[1,poly(n),logn]. The first step has two key\nconstructions: (a). using attention to copy all the weights to the same position; (b). we can\nuse polysize two-layer FC net with ReLU activation to simulate MAJORITY ,AND,ORgate\nwith unbounded fan-in (Lemma E.5);\n2. We can do the first step for all positions i=n+ 1, . . . , n +O(T(n) + 1)simultaneously.\nBy Lemma E.4, we know that for any problem L ∈SIZETC0[T(n) + 1], there exist constant L,\npolynomial S(n),and T′(n) =O(T(n) + 1),suchthatforevery n∈N+,thereexistasequenceof\nthreshold circuits, {Ci\nn}T′(n)\ni=1, whose sizes are uniformly bounded by S(n)and depth are uniformly\nbounded by L, and that for all x∈ {0,1}n,\nL(x) =xn+T′(n),where xn+i≜Ci\nn(x1, . . . , x n+i−1),∀i∈[T′(n)].(11)\nTo simplify the notation of the proof, without loss of generality, we assume for each i∈[T′(n)],\ncircuit Ci\nnhas the exactly same size S(n)and depth L.\nNow we present the construction of the constant-depth, constant-precision decoder-only trans-\nformer, TFθnwhich computes problem Lwhen input length is n. Without loss of generality we\nonly consider the case where T′(n) =T(n) + 1. We set vocabulary V={0,1}, embedding width\nd(n) = 1 + 3( T(n) +n) + 2 S(n)T(n) =O(poly(n)), depth equal to L+ 2, CoT length T(n)\nand precision s(n) =⌈log2S(n)⌉so the precision is high enough for simulating all the poly(n)\nsizeMAJORITY gates used in Ci\nn(Lemma E.5). We set (θTE)0= 0·e1,(θTE)1= 1·e1, and\n(θPE)i=ei+1for all i∈[n+T(n)], where we use ei∈ {0,1}d(n)to denote the one-hot vector\nwhose ith coordinate is 1fori∈[d(n)]andei∈ {0,1}n+T(n)to denote one-hot vector whose ith\ncoordinate is 1fori∈[n+T(n)].\nBelow we first describe the value the internal variables of the transformer and then show there\nexist parameters making such computation realizable. To make our claims more interpretable, we\nonly write the non-zero part of the embedding and omit the remaining 0’s. the Let (x1, . . . , x n)be\nthe input tokens and ∆≜3n+ 3T(n) + 1, our constructions will ensure that\n1.xn+i=TFi\nθn(x1, . . . , x n),∀i∈[T(n)].\n2.h0\ni=xie1+ei+1= (xi,ei)∀i∈[n+T(n)];\n3.h0.5\ni=xie1+ei+1= (xi,ei),∀i∈[n+T(n)];\n4.h1\ni=xie1+ei+1+xien+T(n)+i+1= (xi,ei, xiei),∀i∈[n+T(n)];\n5.h1.5\ni= (xi,ei, xiei, x1, . . . , x i),∀i∈[n+T(n)]\n6.(h1+l\nn+i−1)∆+(i−1)S(n)+1:∆+ iS(n)∈ {0,1}S(n)storestheintermediateresultofcircuit Ci\nn(x1, . . . , x n+i−1)\nat layer l,∀i∈[T(n)]andl∈[L];\n7.(hL+2\nn+i−1)∆+(i−1)S(n)+1:∆+ iS(n)is the intermediate result of circuit Ci\nn(x1, . . . , x n+i−1)at\nlayerl,∀i∈[T(n)],butrepresentedusing {−1,1}. Thatis, (hL+2\nn+i−1)∆+(i−1)S(n)+1:∆+ iS(n)=\n2·(hL+2\nn+i−1)∆+(i−1)S(n)+1:∆+ iS(n)−1. Meanwhile, (hL+2\nn+i−1)∆+(j−1)S(n)+1:∆+ jS(n)= 0,\n∀i, j∈[T(n)], j̸=i\n8.(θOUTPUT hL+2\nn+i−1)0= 0,(θOUTPUT hL+2\nn+i−1)1=PT(n)\nj=1(hL+2\nn+i−1)∆+jS(n)= 2Ci\nn(x1, . . . , x n−i+1)−\n1, for all i∈[T(n)].\n32Nowweexplainthepurposeofeachlayerandhowtosettheparameterssuchthattherequirements\nabove are met.\n1.xn+i=TFi\nθn(x1, . . . , x n),∀i∈[T(n)]is the goal of the construction;\n2. This is by our construction of θPEandθTE;\n3. The first attention layer does nothing by setting all weights to 0;\n4. By Lemma E.5, ANDcan be simulated by 2-layer ReLU networks using 2hidden neurons.\nThusweusethefirstfeedforward-layertocomputethefunction (h1\ni)n+T(n)+1+ j= (h0.5\ni)1+j∧\n(h0.5\ni)1forall i, j∈[n+T(n)]withtotally 2(n+T(n))hiddenneurons. Thereforeif j̸=i,\nthen(h0.5\ni)1+j= 0, which implies (h1\ni)n+T(n)+1+ j= 0; ifj=i, then (h0.5\ni)1+j= 1, thus\n(h1\ni)n+T(n)+1+ j(h0.5\ni)1=xi.\n5. Thisstepexactlyrequires (ATTNθ(1)\nATTN(h(1)\n1, . . . , h(1)\nn))i=Pi\nj=1e∆+jxj. Itsufficestosetthe\nattentionscoreofthesecondlayerat ithposition si= (1, . . . , 1,0. . . ,0) = (1 i,0n+T(n)−i)for\nalli∈[n+T(n)−1]. Thiscanbedonebysetting q1.5\ni=WQh1.5\ni= (Bs(n),0n+T(n)−1), k1.5\ni=\nWKh1.5\ni= (1,0n+T(n)−1). ByLemmaE.2,wehave [exp([⟨qi, kj⟩]s(n))]s(n)= [exp( Bs(n))]s(n)=\nBs(n). Since rounded sum of any number of Bs(n)is still Bs(n)and[Bs(n)/Bs(n)]s(n)= 1,\nwe know that\nsi= softmax s(n)(Bs(n)1i)∥(0, . . . , 0) = (1 , . . . , 1,0. . . ,0) = (1 i,0n+T(n)−i)\nfor all i∈[n+T(n)−1]. Note in this step we use our specific rounding rule to copy all the\nprevious xiwith a sum of attention score larger than 1. We can just also use approximately\nuniformattentionscoreswithanadditionalcoefficientbefore xisincewehave lognprecision.\nFinally we set v1.5\ni=WVh1.5\ni=e∆+ixiandWO=Id(n).\n6. The second attention layer is the only attention layer which has non-zero weights. Using the\nfeedforwardReLUnetworksfromlayer 2toL+1,wecansimulatethecircuits Ci\nninparallelfor\nalli∈[T(n)]byLemmaE.5. Indetail,LemmaE.5ensuresthatwecanuseatwo-layerfully-\nconnected ReLU network with weights to simulate a layer of the TC0circuits Ci\nn. Moreover,\nthere is enough space in the embedding to reserve S(n)’s1needed by Lemma E.5. And we\nalways append the value of intermediate gates after the gate values which have already been\ncomputedateachlayerusingtheindices ∆ + ( i−1)S(n) + 1 : ∆ + S(n)forhl\nn+i−1foreach\ni∈[T(n)]. Noteforeach i,onlythecomputationintherange ∆+( i−1)S(n)+1 : ∆+ S(n)\nis meaningful and the computation for other indices will not be used later.\n7. Thisissimilartostep3. Weskiptheattentionlayerandsimplyset (hL+2\nn+i−1)∆+(i−1)S(n)+k=\n(hL+1.5\nn+i−1)n+i∧(hL+1.5\nn+i−1)∆+(i−1)S(n)+kforall k∈[S(n)]usingthefeedforwardfully-connected\nnetwork.\n8. This step holds directly due to the property guaranteed in step 8. We note that with\nthe property claimed in step 9, we have that (θOUTPUT hL+2\nn+i−1)1−(θOUTPUT hL+2\nn+i−1)0=\n2Ci\nn(x1, . . . , x n−i+1))−1. Thus if Ci\nn(x1, . . . , x n−i+1) = 1, then (θOUTPUT hL+2\nn+i−1)1−\n(θOUTPUT hL+2\nn+i−1)0= 1,whichimplies TFθn(x1, . . . , x n+i−1) = 1,otherwiseif Ci\nn(x1, . . . , x n−i+1) =\n0, then TFθn(x1, . . . , x n+i−1) = 0. In both cases, we have that\nCi\nn(x1, . . . , x n−i+1) =TFθn(x1, . . . , x n+i−1) (12)\n33So far we have finished the proof for the general T(n). Specifically, when T(n) =T′(n) = 0, our\nproofshowsthattheconstant-depthtransformercanstillsimulateanyconstant-depthcircuit,which\nmeans TC0⊆T[poly(n)]⊆CoT[1,poly(n)] = SIZETC0(1) = TC0. Thus all the inclusions are\nequivalence, that is TC0=T[poly(n)] = CoT[1,poly(n)] = SIZETC0(1).\nProof of Theorem 3.8. We first show that SIZEAC0[T(n) + 1] ⊇CoT[T(n),poly(n),1]. For the\ncase that the vocabulary of transformer V={0,1}, by Theorem 3.1, we know for any θn,\nTFθn(x1, . . . , x i)can be expressed by a TC0circuit whose depth is uniformly upper bounded\nby some constant for all n≤i≤n+O(T(n)). This completes the proof when V={0,1}. When\nV ̸={0,1}, we can use the binary encoding of elements in Vas the inputs of those TC0gates\nconstructed for the later layers of the transformer.\nThe other direction is almost the same as that of Theorem 3.7, except that we now only need\nconstant bits of precision because we do not need to simulate MAJORITY gates (Lemma E.5).\nE.3 Proof of Theorem 3.9\nProof of Theorem 3.9. By Lemma E.7, it holds that for all k∈N,AC0⊊SIZE[nk]. By Theo-\nrem 3.7, we know that AC0⊆CoT[1,poly(n),1]⊆CoT[nk,poly(n),1]for any k∈N. Thus\nCoT[nk,poly(n),1]⊊SIZE[nk]for all k, k′∈N. Also, note that the attention layer and fully-\nconnected layer can be computed using poly-size circuits. Thus for any k∈N,CoT[nk,log(n)]⊆\nSIZE[nk′]forsomeinteger k′≥k. Combiningtheseweconcludethatforany k∈N,CoT[nk,log(n)]⊊\nCoT[nk,poly(n)].\nE.4 Auxiliary Lemmas\nIn this subsection, we prove a few auxiliary lemmas that are used in the proofs in Section 3.4.\nLemma E.5. Unlimited-fanin AND,OR(resp. MAJORITY ) :{0,1}n→ {0,1}can be simulated\nby some 2-layer feedforward ReLU network with constant (resp. logn) bits of precision constant\nhidden dimension and additional nconstant inputs of value 1.\nMathematically, let FF[s(n)]be the set of functions C:{0,1}n→ {0,1}which can be a two-\nlayerfeedforwardReLUnetworkwithatmost s(n)bitsofprecisionandconstanthiddendimension\nFFθ:{0,1}2n→ {0,1},FFθ(x′) =W2×srelu([W1×sx′+b1]s), where θ= (W2, W1, b1), such\nthat for any x∈ {0,1}n,\nFFθ(x1,1, x2,1, . . . , x n,1) = C(x). (13)\nWe have unlimited-fanin AND,OR∈FF[1]andMAJORITY ∈FF[logn].\nThe proof of Lemma E.5 is based on the following straightforward lemma (Lemma E.6).\nLemmaE.6. Forany e∈N+anda∈Z∩Fs,relu([a]s)−relu([a−1]s) =1[a >0]. Inparticular,\nfor any a∈Z,relu(a)−relu(a−1) =1[a >0].\nProof of Lemma E.5. Recall that x⌢ydenotes (x1, y1, x2, y2, . . . , x n, yn)for any x, y∈ {0,1}n.\nWe have that sums(x⌢(−1n))≤0for all e≥2andx∈ {0,1}n. Moreover, sums(x⌢(−1n)) =\n0⇐⇒ ∀ i∈[n], xi= 1. Similarly, we have that sums(x)≥0andsums(x) = 0 ⇐⇒ ∀ i∈\n[n], xi= 0. In other words, we have\n•AND(x) =1[sums(x⌢(−1n)]≥0) =1[⟨x⌢1n,1n⌢(−1n)⟩s+ 1>0];\n34•OR(x) =1[sums(x′\ni)>0] =1[⟨x⌢1n,1n⌢(0n)⟩s>0].\nThereforefor AND,wecanset θAND≜(WAND\n1, WAND\n2, bAND\n1)withWAND\n1≜\u00141n⌢(−1n)\n1n⌢(−1n)\u0015\n, b1=\n\u00141\n0\u0015\n, WAND\n2= [1,−1], and we have that\nFFθAND(x⌢1n) = [relu(⟨x⌢1n,1n⌢(−1n)⟩s+ 1)−relu(⟨x⌢1n,1n⌢(−1n)⟩s)]s\n=1[⟨x⌢1n,1n⌢(−1n)⟩s+ 1>0] (by Lemma E.6)\n=AND(x)\nSimilarly for OR, we can set θOR≜(WOR\n1, WOR\n2, bOR\n1)with WOR\n1≜\u00141n⌢0n\n1n⌢0n\u0015\n, b1=\n\u00140\n−1\u0015\n, WOR\n2= [1,−1], and we have that\nFFθOR(x⌢1n) = [relu(⟨x⌢1n,1n⌢0n⟩s)−relu(⟨x⌢1n,1n⌢0n⟩s−1)]s\n=1[⟨x⌢1n,1n⌢0n⟩s>0] (by Lemma E.6)\n=OR(x)\nThe proofs for ANDandORare thus completed.\nNextwedealwith MAJORITY . Notethatfor s(n)≥log2n+1,wehavethatPn\ni=1(2xi−1) =\n⟨x⌢1n,2n⌢(−1n)⟩sfor all x∈ {0,1}n.\nMAJORITY (x) =1[nX\ni=1(2xi−1)>0] =1[⟨x⌢1n,2n⌢(−1n)⟩s>0]\n= [relu(⟨x⌢1n,2n⌢(−1n)⟩s)−relu(⟨x⌢1n,2n⌢(−1n)⟩s−1)]s\n=FFθMAJORITY (x⌢1n), (14)\nwhere θMAJORITY≜(WMAJORITY\n1 , WMAJORITY\n2 , bMAJORITY\n1 )withWMAJORITY\n1 ≜\u00142n⌢−1n\n2n⌢−1n\u0015\n, b1=\n\u00140\n−1\u0015\n, WMAJORITY\n2 = [1,−1].\nLemma E.7. For all k∈N,AC0̸⊆SIZE[nk].\nProof of Lemma E.7. We first define SIZE[T(n)]as the problems solvable by circuits with T(n)\nstandard AND,OR,NOTgates exactly. Thus SIZE[nk] =∪C∈N+SIZE(Cnk). Now we claim that\nfor each C∈N, there is a N∈N+, such that for all n≥N, it holds that there is a conjunction\nnormal form (CNF) with at most nk+1clauses over {x1, . . . , x n}that cannot be expressed by\nany circuit of size Cnk. This claim holds because of a simple counting argument. There are at\nleast 2nk+1different such CNFs. On the other hand, it is well known that one can represent a\nT(n)-size circuit only allowing standard AND,NOT,ORgates with 3T(n) logT(n)bits (we need\nlogT(n)bits to encode the id of a gate). Thus the total number of different circuits of size at most\nCnkis at most 23Cnk(klogn+C), which is smaller than 2nk+1for sufficiently large n. We denote\nsuchnfor each CbyNC. Now we define the following language LCNF: if the input length of\nxisNCfor some C, use the nk+1-clause CNF’s output which cannot be expressed by size- Cnk\ncircuits as the output; otherwise rejects (output 0). Then clearly LCNF/∈SIZE(Cnk)for all C,\nthusLCNF/∈ ∪C∈N+SIZE(Cnk) =SIZE[nk]. By construction, LCNF∈AC0. This completes the\nproof.\n35F Discussion on Variants in Transformer Architecture\nF.1 Extension to Transformers with LayerNorm\nAllowingLayerNormchangesthefunctionclassthatatransformercanexpressandthepositionofthe\nlayer norm also matters (Xiong et al., 2020). However, the expressiveness results mentioned in this\nwork still hold for the two most popular transformer architecture variants with LayerNorm — Post\nLayerNorm and Pre LayerNorm. The upper bounds on transformer expressiveness Theorems 3.1\nand3.2clearlydon’tgetaffectedbyaddingLayerNorm,whichcanbecomputedinpolynomialtime\nfor each token.\nBelow we focus on the upper bound of the expressiveness of decoder-only transformers with or\nwithout CoT. In detail, we will explain why Theorems 3.3 and 3.7 still holds even with LayerNorm.\nHerethekeyobservationisthat,ifeachcoordinateof h∈Rdrangesfrom {−1,1}and−1,1appear\ninpairs,then LayerNorm (h) =h. Thusitsufficestoshowthatwecanslightlytwisttheconstruction\nof transformers in Theorems 3.3 and 3.7 that for all i∈[n+T(n)], l∈ {0,0.5,1, . . . , L },hl\ni\nis composed of −1and1and they appear in pairs so the sum is always 0. Note that in the\ncurrent construction, each hl\nionly contains 0,−1,1. It suffices to replace each dimension with four\ndimensions, in the sense 0→(1,−1,1,−1),1→(1,1,−1,−1)and−1→(−1,−1,1,1). This\ncanbedonebychangingtheweightsofthetokenembedding,positionencoding,andtheweightsof\nthesecondlayerofeachfully-connectedlayer. Fortheoutgoinglayer,wejustusetheaverageofthe\nnew representation, which is exactly the same as the original value in all three cases.\nF.2 Extension to Transformers with Multihead Attention\nInthispaper,forsimplicity,weonlyfocusonthecasewherethereisonlyoneattentionheadineach\nlayer. Themainresultsinthispaperstillapplyifweallowconstantlymanyattentionheads,because\nwe can simulate an attention layer with kheads with kattention layers with one head. Allowing an\narbitrarynumberofattentionheadswhilefixingtotalembeddingsizemightmaketheconstant-depth\ntransformers strictly more expressive in certain settings and we leave it for future works.\nG Discussion on Non-uniformity\nNon-uniform computation models allow a different program for each different input length, like\nboolean circuits. However, the complexity class defined by circuits can also be uniform, if we add\nadditional assumption on the correlation between circuits of different input lengths, e.g., one can\nrequirethecircuitsforinputlength ncanbegeneratedbyaTuringMachinetaken nasinputinusing\na certain amount of time and space.\nThe complexity class CoTintroduced in this paper can also be made uniform by enforcing an\nadditional assumption, that the parameters of the transformer can be generalized by some Turing\nMachinegiventheinputsequencelength n. Itiswell-knownthatonecansimulatetheexecutionof\nthe Turing Machine for any Tsteps by a family of uniform boolean circuits of size O(T2). Thus if\nwe enforce the parameters of transformers in CoTto be uniform, our main theorem would imply\nthat constant-depth transformers with uniform parameters and polynomially many steps of chain of\nthoughts can solve all problems in P. Also note that the inference of transformers can also be done\nin polynomial time, we conclude it is exactly equal to P.\nOne natural question about non-uniformity is that whether having a different transformer for\neachinputsequencelengthispractical, giventhatasignificantportionofprevioustheoreticalworks\non transformer expressiveness focuses on the uniform setting. This problem is kind of ill-defined\n36becausewehaven’tbeenabletoscaleuptheinputlengthtoarbitrarylengthinpractice,andthusitis\nnotclearifitisnecessarytokeepscalingupthesizeofLLMsforlongerinputsequencelength. But\nat least for the LLMs that have been seen in practice, it seems quite common to scale up the model\nsize when dealing with longer input sequence length. Also taking the GPT architecture (Radford\netal.,2019)thatwefocusoninthispaper,havingmoretrainableparametersisnecessaryforlonger\ninput sequence length, due to the trainable absolute position encoding.\nStill, one needs to note that there is a difference between natural language tasks and complexity\nclass,wheretheformerhasalotofmemorizationanddoesnotrequireastrongabilitytosolvemath\nproblems of any sequence length. In contrast, to learn this complexity class like the composition of\npermutationofanylength,transformersneedtohavetheabilityof lengthgeneralization ,whichdoes\nseemimpossibleforcertainnon-uniformmodels, e.g.,likeGPTarchitectureswithtrainableabsolute\nposition encoding, because there is no way to learn the position encoding at an unseen position in\nthe training dataset. Of course, length generalization would still be possible if GPT architecture\nlearned the ground truth without using the trainable position encoding at all.\n37"    },    {        "title": "2402.12896v1.Evidence_of_apsidal_motion_and_a_possible_co_moving_companion_star_detected_in_the_WASP_19_system.pdf",        "content": "Astronomy &Astrophysics manuscript no. main ©ESO 2024\nFebruary 21, 2024\nEvidence of apsidal motion and a possible co-moving companion\nstar detected in the WASP-19 system\nL. M. Bernabò1\n, Sz. Csizmadia1,2\n, A. M. S. Smith1\n, H. Rauer1,3,4\n, A. Hatzes5\n, M. Esposito5\n, D.\nGandolfi6\n, and J. Cabrera1\n1Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin\n2ELKH-SZTE Stellar Astrophysics Research Group, H-6500 Baja, Szegedi út Kt. 766, Hungary\n3Zentrum für Astronomie und Astrophysik, Technische Universität Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany\n4Institut für Geologische Wissenschaften, Freie Universität Berlin, 12249 Berlin, Germany\n5Thüringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany\n6Dipartimento di Fisica, Università degli Studi di Torino, Via Pietro Giuria, 1, 10125 Torino, Italy\ne-mail: lia.bernabo@dlr.de\nSubmitted 9 May 2023 ; accepted 8 January 2024\nABSTRACT\nContext. Love numbers measure the reaction of a celestial body to perturbing forces, such as the centrifugal force caused by rotation,\nor tidal forces resulting from the interaction with a companion body. These parameters are related to the interior density profile. The\nnon-point mass nature of the host star and a planet orbiting around each other contributes to the periastron precession. The rate of this\nprecession is characterized mainly by the second-order Love number, which o ffers an opportunity to determine its value. When it is\nknown, the planetary interior structure can be studied with one additional constraint beyond the mass, radius, and orbital parameters.\nAims. We aim to re-determine the orbital period, eccentricity, and argument of the periastron for WASP-19Ab, along with a study of\nits periastron precession rate. We calculated the planetary Love number from the observed periastron precession rate, based on the\nassumption of the stellar Love number from stellar evolutionary models.\nMethods. We collected all available radial velocity (RV) data, along with the transit and occultation times from the previous investiga-\ntions of the system. We supplemented the data set with 19 new RV data points of the host star WASP-19A obtained by HARPS. Here,\nwe summarize the technique for modeling the RV observations and the photometric transit timing variations (TTVs) to determine the\nrate of periastron precession in this system for the first time.\nResults. We excluded the presence of a second possible planet up to a period of „4200 d and with a radial velocity amplitude\nbigger than»1 m s´1. We show that a constant period is not able to reproduce the observed radial velocities. We also investigated\nand excluded the possibility of tidal decay and long-term acceleration in the system. However, the inclusion of a small periastron\nprecession term did indeed improve the quality of the fit. We measured the periastron precession rate to be 233`25\n´352d´1. By assuming\nsynchronous rotation for the planet, it indicates a k2Love number of 0.20`0.02\n´0.03for WASP-19Ab.\nConclusions. The derived k2,pvalue of the planet has the same order of magnitude as the estimated fluid Love number of other\nJupiter-sized exoplanets (WASP-18Ab, WASP-103b, and WASP-121b). A low value of k2,pindicates a higher concentration of mass\ntoward the planetary nucleus.\n1. Introduction\nThe understanding of planetary interiors constitutes one of the\nfour major challenges of contemporary exoplanetary science\n(Schneider 2018). It is crucial for assessing their potential hab-\nitability, formation, and evolution. Interior processes play an es-\nsential role in creating and maintaining the physical conditions\nthat are required to support life (see, e.g., Van Hoolst et al. 2019).\nThey are useful for determining which planetary formation pro-\ncess is most plausible; namely, whether it is core accretion (Pol-\nlack et al. 1996 and references therein) or gravitational instabil-\nity (Boss 1997 and references therein). According to the former,\nthe accretion of planetesimals would result in the formation of\na core comprised of heavy elements. On the other hand, planets\nforming through disc instability can rapidly accrete gas. These\ntwo scenarios lead to di fferent core sizes. The size of the core is\nrelated to the second order fluid k2,pLove number (Love 1911),\nas explained in Becker & Batygin (2013). Knowledge of the in-\nterior of a giant planet can help us distinguish between these twoscenarios, in addition to its Love number, as we describe in this\nwork.\nLove numbers can also be used to break the known mass-\nradius-composition degeneracy of exoplanets (Baumeister et al.\n2020). When an exoplanet is characterizable both by transit and\nradial velocity techniques, we can derive its radius and mass –\nand, thus, the planetary mean density. However, mass, radius and\nmean density are degenerate when it comes to determining the\ninterior structure of bodies, such as the radial density, temper-\nature, pressure, and composition profiles. We can find multiple\nsolutions for the planetary interior producing the same planetary\ntotal mass and radius (see, e.g., Valencia et al. 2007; Wagner\net al. 2011; Damasso et al. 2018). This kind of degeneracy can\nbe decreased based on the strong assumption that the planet and\nits host star share the same metallicity (Dorn et al. 2015).\nFor Solar System planets, the degeneracy between mean den-\nsity and interior structure can be reduced by in situ measure-\nments of the gravitational field, as well as by seismic mea-\nsurements, such as InSight (Banerdt et al. 2020), and by ob-\nArticle number, page 1 of 22arXiv:2402.12896v1  [astro-ph.EP]  20 Feb 2024A&A proofs: manuscript no. main\nservations of moon motions. However, for exoplanets, gravita-\ntional moments cannot be measured. Hence, Love numbers were\nproposed as further observables in exoplanetary interior studies\n(see, e.g., Batygin et al. 2009; Ragozzine & Wolf 2009 and ref-\nerences therein), as has already been applied to eclipsing binary\nstars systems (see Russell 1928 and references therein). These\nauthors measured the deformations and mass re-distribution in-\nside the planet due to the tidal interaction with the host star. As\nshown by Baumeister et al. (2020), thanks to the knowledge of\nthe second-order fluid Love number of the planet, we can better\ninfer the distribution of possible thickness of each interior layer.\nIn this work, we employ a technique used by Csizmadia\net al. (2019), which we further refined to determine k2,pfrom\nperiastron precession. This precession is derived from radial ve-\nlocity (RV) measurements and transit and occultation mid-times.\nWe apply it to the system WASP-19 based on an approach that\nrequires the calculation of the secular evolution of the orbital\nelements, which is related to the mass distribution of the star\nand the planet through their second-degree Love numbers.\nThis paper is organized as follows. Section 2 and Appendix A\npresent the theory of apsidal motion. Section 3 describes the\nsystem WASP-19 and provides evidence of a possible compan-\nion star in the WASP-19 system. In Sect. 4, we describe archival\ntransit and occultation timing observations as well as the archival\nradial velocity investigations of the system which are supple-\nmented by our recent RV study of the system. Section 5 contains\nour data analysis along with the search for a second planetary\ncompanion. Our conclusions are given in Sect. 6.\n2. Apsidal motion\nApsidal motion, namely, the secular perturbation in the argument\nof periastron, can be detected either in RV or in TTV datasets (or\ntheir combination) to constrain the internal structure of planets.\nThis type of analysis on RVs was introduced by Kopal (1959,\n1978). Here, we make use of such equations, however, we have\nrewritten them in a way that allows them to be directly applied\nto the observed data sets. We describe their derivations more in\ndetail in Appendix A.\nThe total rate of apsidal advance is the result of the three\ncontributions:\n9ω“dω\ndt“dωGR\ndt`dωtidal\ndt`dωrot\ndt. (1)\nThe general relativistic term dωGR{dt(Einstein 1915) is:\ndωGR\ndt“6πGM‹\nac2p1´e2q1\nPa, (2)\nwhere Gis the gravitational constant, M‹the mass of the star,\naandethe semi-major axis and eccentricity of the orbit, cthe\nspeed of light in vacuum, and Pathe anomalistic period.\nThe secular perturbations of the tidal and rotational compo-\nnents dωtidal{dtanddωrot{dtare derived in Appendix A and can\nbe expressed more explicitly if approximated to the second or-\nder, as in Csizmadia et al. (2019):\ndωtidal\ndt»15\n2n\np1´e2q5«\nmp\nM‹k2,‹ˆR‹\na˙5\n`\n`M‹\nmpk2,pˆRp\na˙5ffˆ\n1`3\n2e2`1\n8e4˙\n, (3)k2,p“0.01 k2,p“1.5\n9ωGR 2.852d´12.852d´1\n9ωtid,‹ 0.472d´10.472d´1\n9ωrot,‹ 0.152d´10.152d´1\n9ωtid,p 10.532d´11580.162d´1\n9ωrot,p 0.702d´1105.462d´1\nTotal 9ω 14.702d´11689.102d´1\nPeriod 241.25 yrs 2.10 yrs\nTable 1. Expected gravitational, rotational and tidal contribution to the\nperiastron precession in the extreme Love number cases (mass-point\nand homogeneous planet) and the corresponding apsidal motion period,\nassuming synchronous rotation for the planet.\nand\ndωrot\ndt»1\n2n\np1´e2q2«ˆPorb\nProt,‹˙2\nk2,‹ˆR‹\na˙5ˆ\n1`mp\nM‹˙\n`\n`ˆPorb\nProt,p˙2\nk2,pˆRp\na˙5ˆ\n1`M‹\nmp˙ff\n. (4)\nIn these equations, nis the mean motion, mpandRpthe mass\nand radius of the planet, R‹the radius of the star, and Prot,pand\nProt,‹are the rotational periods of the two components. Also, k2,p\nandk2,‹are their second-order fluid Love number correspond-\ning to the double the apsidal motion constant ((Csizmadia et al.\n2019)):\nk2,i“2k2,aps,i, (5)\nboth for the star ( i“ ‹) and the planet ( i“p). These two\nequations were obtained by Sterne (1939). Just as in Eq. (A.8),\nit is assumed that the rotational axis of both components are\nperpendicular to the orbital plane.\nRegardless of whether the stellar and planetary rotation\nis direct or retrograde, the corresponding terms in Eq. (4) are\npositive, therefore they cause the apsidal line to advance.\nAssuming the values of the orbital parameters of WASP-19Ab\nfrom literature and the Love number of the star from Claret\n(2004), we calculated the amplitude of the general relativity term\nand the stellar tidal and rotational contributions, as reported in\nTable 1. Moreover, a first estimate for the planet can be calcu-\nlated assuming synchronous rotation for the planet and k2,pbe-\ntween the values 0.01-1.5 (corresponding to the extreme cases\nof a mass point and a homogeneous body, respectively). The to-\ntal contribution is between about 14.71 - 16892d´1»0.0041 -\n0.479˝d´1, respectively when assuming k2,p=0.01 and k2,p=1.5.\nThe apsidal motion period U=2π{9ωis therefore between »2.10\nyrs and»241.25 yrs.\nThe equations in Appendix A show that the angle of\nperiastronωoscillates over one orbital period but also shows\na gradual long-term change that can be retrieved in the RV\nobservations. Due to the finite length of the typical datasets\nof observations (12 years in the case of WASP-19Ab), in\nRV datasets, the change in the angle of periastron ωcan be\nretrieved in the radial velocity of the star (see, e.g., Jackson\net al. 2008 for more details on the timescales). In long baselines\nof observations, all orbital elements must be considered along\nArticle number, page 2 of 22Bernabò et al.: Periastron precession of WASP-19Ab\nwith periastron precession. As described in Kopal (1978), they\ninclude the change in semi-major axis (tidal decay), eccentricity\n(circularization) and time of periastron passage. The semi-major\naxis and its eccentricity would diminish in value and the time of\nperiastron passage is shifted in time.\n3. WASP-19Ab\n3.1. The system\nThe planet WASP-19Ab, hosted by the bright, active G-dwarf\nstar ( V=12.3, Zacharias et al. 2012 and Teff,‹»5568 K, Tor-\nres et al. 2012), was discovered through the transit method by\nthe Wide Angle Search for Planets (WASP)-South observatory\n(Hebb et al. 2010). It is a hot Jupiter with mass mp=1.139 ME,\ninflated radius rp=1.410 RE(Mancini et al. 2013) and orbital\nperiod Porb»0.79 d.\nSo far, the system is among the best studied from the point of\nview of orbital and planetary parameters (e.g., Lendl et al. 2013;\nMancini et al. 2013; Sedaghati et al. 2015, 2017) and atmosphere\ncharacterization (very pronounced water absorption, Huitson\net al. 2013; Sedaghati et al. 2017, and titanium oxide detected\nin the transmission spectrum, Sedaghati et al. 2017). The prox-\nimity of WASP-19Ab to its host star results in significant tidal\ninteraction that may govern the evolution of the planetary orbit,\nwhich makes the system a good candidate for our study.\nBased on transit observations, certain studies have suggested\nthe possibility of variations in the mid-transit time. Mancini et al.\n(2013) suggested a non-linear ephemeris. However, Petrucci\net al. (2020) discarded orbital decay with 74 complete transit\nlight curves spanning over ten years of observations. Moreover,\nas described more in detail in Sect. 5.1, Cortés-Zuleta et al.\n(2020) used transit timing variations (TTVs) to infer an upper\nmass limit for an additional candidate planetary companion in\nthe system.\n3.2. Companion star: WASP-19B\nWe searched the catalogue of the third Gaia data release (DR3)\n(Gaia Collaboration et al. 2016, 2022) for possible co-moving\ncompanions to WASP-19A. We searched all objects within 41of\nthe target, which corresponds to a physical separation of around\n0.3 pc at the distance of WASP-19A, for those with compat-\nible parallax ( θ) and proper motions (PM). Only one object\nwas found to have parallax and proper motion values in both\nthe RA (α) and Dec ( δ) directions that are compatible with\nthose of the target to within 5 σ. This object ( Gaia DR3 ID =\n5411725145916602496), which is 8.8 magnitudes fainter in the\nG-band, has a parallax that di ffers from that of WASP-19A by\n1.6σand PMαand PMδvalues di ffering by just 0 .1σand 0.6σ\nrespectively. This star is found in the southernmost part of the\nsouth-east region of WASP-19A, at a separation of 682. The\nmagnitude and BP-RP colour of the companion suggests that this\nstar is a mid M-dwarf, with a best matching spectral type for the\nmagnitude of M6V and for the colour M2V , using the updated\ntable1of Pecaut & Mamajek (2013).\nTo estimate the probability of this putative companion aris-\ning from a chance matching of WASP-19 in parallax and proper\nmotions, we searched a much larger area of sky in Gaia DR3,\nstill centered on WASP-19, but with a radius of 2˝. This pro-\n1https://www.pas.rochester.edu/~emamajek/EEM_dwarf_\nUBVIJHK_colors_Teff.txtParameter WASP-19A WASP-19B\nGaia DR3 ID 5411736896952029568 5411725145916602496\nApparent separation (2) 0.0 68 .0347˘0.0009\nGaia G-magnitude 12 .1012 20 .8859\nGaia BP´RPcolour 0 .953˘0.001 2 .213˘0.12\npmRA (mas yr´1) ´35.4571˘0.0089´35.33˘1.18\npmDec (mas yr´1) 17 .378˘0.009 18 .22˘1.30\nParallax (mas) 3 .7516˘0.0090 1 .92˘1.16\nDistance (pc)˚ 266.54˘0.64 487`564\n´184\nProjected physical separation (AU) 0.0 19976 ˘48\nSpectral type:G8V M2V - M6V\nTable 2. WASP-19A and its stellar companion\n* Assuming no extinction.\n:Assuming that the companion is also a star on the main-sequence.\nduced 77 objects (including the companion; hereafter: ‘WASP-\n19B’) that match WASP-19 within 5 σin all ofθ, PMα, and PM δ.\nThis implies that there is an 8.6% chance of finding a match-\ning object within our initial search radius of 41, and just a 0.7%\nchance of finding an object as close as WASP-19B. Relaxing the\n5σlimit to 3σchanges these probabilities to 2% and 0.16%,\nrespectively. Furthermore, none of these additional matches, all\nof which are more than ten times further from WASP-19A than\nWASP-19B, demonstrate as good a match as WASP-19B. Math-\nematically, this is expressed as WASP-19B having the lowest\nvalue ofσtot“σθ`σPMα`σPMδ, where for each parame-\nter,X(where Xstands forθ, PMα, PMδ)σX“|X1´X2|b\nσ2\nX1`σ2\nX2, where\nX1˘σX1andX2˘σX2are the values and uncertainties of param-\neterXfor WASP-19 and its companion, respectively. We recog-\nnise that the uncertainty on the parallax of WASP-19B is rela-\ntively large. Even if WASP-19B is likely to be a true, gravita-\ntionally bound, companion to WASP-19A, some doubt remains\nas to whether it truly is a bound companion to WASP-19A. We\nadopted the planet name WASP-19Ab because it orbits around\nthe brighter primary star. Details of WASP-19B are given in Ta-\nble 2. With Eq. (9) of Csizmadia et al. (2019), we also deter-\nmined that the e ffect of the companion star on the radial velocity\nof star A is negligible. Star B may be the cause of the presence of\nan eccentricity di fferent from zero in the orbit on the transiting\nplanet. While tidal interaction leads to circularization, the exci-\ntation of the orbit due to a distant companion might prevent it, as\ndiscussed in detail in Appendix B.\n4. Observations\n4.1. Transits and occultations\nMid-transit and occultation times used in this paper are listed\nin Tables C.1 - C.4 in Appendix C. We collected publically\navailable data, including some amateur observations listed in\ntheTRansiting ExoplanetS and CAndidates (TRESCA)2project\ndatabase, part of the Exoplanet Transit Database (Poddaný et al.\n2010). In particular, we included amateur data that were selected\nby Mancini et al. (2013) and other high-quality data showing a\nclear light curve without gaps. As detailed in the tables, we also\nre-analyzed some transits with the Transit and Light Curve Mod-\neller (TLCM, Csizmadia 2020).\nThe time unit of each mid-transit and mid-occultation time\nwas carefully checked and eventually converted into BJD TDB\n2http://var2.astro.cz/EN/tresca/index.php\nArticle number, page 3 of 22A&A proofs: manuscript no. main\nID Nobs Instrument Reference\n(out of transit)\n1 34 (30) Coralie Hebb et al. (2010)\n2 3 (2) Coralie Hellier et al. (2011)\n3 36 (21) HARPS Hellier et al. (2011)\n4 20 (6) PFS Albrecht et al. (2012)\n5 3 (3) HIRES Knutson et al. (2014)\n6, 7 88 (16, 6) ESPRESSO Sedaghati et al. (2021)\n8 19 (15) HARPS This work\nTable 3. Radial velocity datasets used in this work.\nRV datasets are described with ID number, number of points, instrument\nand reference for the publication of the corresponding data and results.\nIn brackets, one can find the number of out-of-transit as those are the\nonly RV datapoints that are used in the fit.\n(barycentric Julian date, in the barycentric dynamical time sys-\ntem). The di fference between BJD UTCand BJD TDBdepends on\nthe number of leap seconds (Eastman et al. 2010). For our\ndataset, it is around one minute. If the time unit used in the\nreference paper was not clear, the main author was contacted\nand the issue was clarified; otherwise, the data were discarded\nand not reported here. We used the estimated mid-times for each\nprimary transit and occultation to evaluate whether they exhibit\ndeviations from a Keplerian orbit over a long baseline of obser-\nvations (see Sect. 5.2 for more details) as well as to constrain the\nsidereal period of the planet.\n4.2. RVs\nWe collected all available radial velocity measurements from\n2008 onward and we present new RV observations (see ID 8\nbelow). Information on the RV observations are listed in Table 3\nand the complete datasets are shown in Appendix D.\nHere, we list the main details on the datasets, with the labels\ncorresponding to the discussion in Appendix D:\nID 1 Hebb et al. (2010) (discovery paper) reported 34 RV\nmeasurements obtained with the CORALIE spectrograph on\nthe 1.2 m Euler telescope between 2008 and 2009. To the\nbest of our knowledge and after contacting the authors of\nthe paper, the dataset reported in the paper is in BJD UTC. We\npresent the data transformed to BJD TDB;\nID 2 Hellier et al. (2011) reported 3 RV points from the\nCORALIE spectrograph on the Swiss Euler 1.2 m telescope\nin 2009. The data were reported in BJD UTCin Hellier et al.\n(2011), so they are transformed here into BJD TDB;\nID 3 Hellier et al. (2011) reported 36 points during and around\nmore than one transit in 2010 using the HARPS spectrograph\non the ESO 3.6 m at La Silla, taken in order to measure the\nRossiter-Mclaughlin e ffect. As well as for ID 2, the data\nwere reported in BJD UTCin the original paper, so they have\nbeen transformed here into BJD TDB;\nID 4 Albrecht et al. (2012) reported 20 data points taken in 2010\nas measurements of the Rossiter-McLaughlin e ffect with the\nMagellan II (Clay) 6.5 m telescope and the Planet Finder\nSpectrograph (PFS);\nID 5 Knutson et al. (2014) reported 3 RV measurements obtained\nwith the High Resolution Echelle Spectrometer (HIRES) on\nthe 10 m Keck I telescope between 2008 and 2013;\n5000 6000 7000 8000 9000\ntime - 2 450 000 [BJDTDB]0.3\n0.2\n0.1\n0.00.10.20.3RV [km/s]\nID 1\nID 2\nID 3\nID 4\nID 5\nID 6\nID 7\nID 8Fig. 1. Radial velocity observations of WASP-19Ab, time vs radial ve-\nlocities, with corresponding observational errorbars. The di fferent data\nsets (see Sect. 4) are represented by di fferent colours. Between the\ndatasets, there are significant o ffsets between di fferent instruments and\nstudies. They have been removed accordingly to the results in Table 4.\nIDs 6,7 Sedaghati et al. (2021) reported 88 RV points taken\nwith VLT /ESPRESSO between 2019 and 2020, used to\nderive the Rossiter-McLaughlin e ffect and characterize the\natmosphere of the planet. The two di fferent IDs are given\nbecause of some setup changes in ESPRESSO, as described\nby Sedaghati et al. (2021), which lead to the introduction\nof a small o ffset. Moreover, as clarified with the first author\nof that paper, ID 6 was reported in BJD UTCin the original\npaper, while ID 7 in BJD TDB, therefore the first one is\ntransformed;\nID 8 reported 19 RV points taken by us using the High Accu-\nracy Radial velocity Planet Searcher (HARPS, Mayor et al.\n2003) at the ESO La Silla 3.6m telescope under the pro-\ngramme 0104.C-0849 (PI: Sz. Csizmadia). The observations\nwere performed between 6 and 21 February 2020. We tried\nto obtain 1-3 points per night and we planned the obser-\nvations to have a good phase coverage. The exposure time\nwas 1200 seconds while the spectral resolution as 115,000.\nTheobjAB observing mode of HARPS was used, covering\nthe 378-691 nm wavelength range. The measured signal-to-\nnoise ratio (S /N) varied between 10.2 and 21.6 depending on\nweather and seeing conditions. The data were reduced by the\nstandard HARPS pipeline available at La Silla Observatory\nand the results with their corresponding uncertainties are re-\nported in Tables D.3 and D.4.\nAll RV observations are shown in Fig. 1. As described in\nSect. 5.2, the datasets have di fferent o ffsets {which are sub-\ntracted in the figure. They are fitted in our model by including\nthe term Vi,instrin Eq. (7). As for the transit and occultation mid-\ntimes, the time unit of the observation was transformed and uni-\nformized to BJD TDBwhen necessary.\n5. Data analysis\n5.1. Long-term effects in RVs\nIn this section, we study the long-term e ffects a ffecting the RV\ncurve. If present, they must be identified and subtracted before\nmodeling and estimating the rate of periastron precession in\nArticle number, page 4 of 22Bernabò et al.: Periastron precession of WASP-19Ab\nSect. 5.2.\nFirst, we applied the l1periodogram (Hara et al. 2017) to the\narchive and newly acquired RV data (ID 8, see Fig. 2) to search\nfor further companions in the system, in addition to the con-\nfirmed planet in the system. This method was developed specif-\nically to find companions in RV datasets and can be used in a\nsimilar way to a Lomb–Scargle periodogram. To verify the sig-\nnificance of the peaks, we followed the re-sampling approach\nsuggested by Hara et al. (2017): we randomly removed 10–20%\nof the data and re-computed the periodogram. Some peaks did\nnot appear after the re-sampling, therefore, we did not consider\nthem real. After the re-sampling, we identified only one peak at\n0.78 d, as shown in the top plot of Fig. 2. It was retrieved in cor-\nrespondence with the orbital period of WASP-19Ab, with a false\nalarm probability (FAP) of log 10pFAPq=-30.9 (FAP»10´31).\nThe analysis shows no evidence of any additional periodic sig-\nnal in the RV data.\nWe also re-computed the periodogram on the RV residuals\nafter the subtraction of the signal of the first planet (see bottom\nplot in Fig. 2). After a re-sampling of the data, only a few small\npeaks are left. However, their FAP is ě0.2, meaning that any\npeaks are due to artefacts.\n100101102103104\nPeriod (days)0255075100125150175200coefficient amplitude\n0.78\n1 periodogram\nPeak periods (d)WASP_19A_outoftransit 1 periodogram\n100101102103104\nPeriod (days)0.00.51.01.52.02.53.0coefficient amplitude\n0.71\n10.9\n0.671 periodogram\nPeak periods (d)WASP-19A 1 periodogram\nFig. 2. l1periodogram of all RV data outside the transit (top). There\nis only one significant peak with false alarm probability »10´31. The\nperiodogram of the residuals after the subtraction of the contribution of\nplanet b (bottom). All residual peaks have false alarm probabilities of\n>20%.\nWe estimated the sensitivity of our search for perturbing bodies\nin the data by performing a sensitivity study based on the transit\ntiming variations. We calculated the maximum mass of a per-\nturber in the system as a function of the maximum amplitude of\nthe TTVs (see Eq. (2) in Holman & Murray 2005). Our analysis\nshows that the maximum allowed mass of the perturber, Mc, as\na function of its orbital period, Pc, assuming di fferent values of\nthe orbital eccentricity (e =0.0, 0.2, 0.4,0.6, and 0.8). The anal-\nysis shows that the region above 4000 MEand period below 20\ndays is excluded by the TTV observations.\nWe also performed a sensitivity study based on the RVs. The\nmaximum amplitude of the RV residuals after the subtraction of\nthe signal of the orbit of planet b (with Sedaghati et al. 2021’s\nparameters) is around 52 m s´1. Through Kepler’s third law, weestimated the maximum mass of the putative perturber. With an\norbital period equal to 12 yrs (the total length of the RV dataset),\nwe obtained Mc»3.9MEand an orbital distance of »5 AU. We\nfitted the data with an additional Keplerian signal (beyond the\nsignal of planet b) mimicking a hypothetical candidate for planet\nc. We injected a candidate and repeated the experiment two bil-\nlion times with di fferent random eccentric orbital parameters and\nfitted the new HARPS RV data. We varied the candidate parame-\nters as follows:?e¨sinω0and?e¨cosω0between -1.0 and 1.0,\nthe RV semi-amplitude Kcbetween 0.1 and 100 m s´1(to widely\ninvestigate the amplitude of the RV residuals with errorbars) and\nthe orbital period between 0.1 and the length of the dataset ( »19\nd for the first attempt with the only HARPS dataset and »12 yr\nfor the second attempt with the whole dataset). Moreover, from\nKepler’s third law, we set the following condition to ensure that\nthe orbit would not collide with the star:\nPcě2πR3{2\n‹?GM‹, (6)\nwhere Gis the gravitational constant, while R‹andM‹are\nthe radius and mass of the host star, respectively. With these\nconstraints and varying the orbital parameters in the aforemen-\ntioned ranges, the candidate mass range is found to be between\n»1.4 and»400 M C.\nThe results of our simulation (see Fig. 3) suggest scenarios\nthat satisfy the condition χ2ď500.The top plot shows the χ2\nof the model when including the proposed perturber, corre-\nsponding to five more free parameters: transit epoch, period,\nRV semi-amplitude,?e¨sinω0and?e¨cosω0(plotted as\na function of the orbital period). Many of these candidates\nare discarded because the corresponding χ2is too high or the\namplitude of the perturbing body on the RV curve would be\nover a dozen m s´1, so they would definitely be visible in the\nRV data and in the periodogram. The red lines underline the\nperiods corresponding with the stable orbital regions present in\nthe stability map by Cortés-Zuleta et al. (2020). Their analysis\nallowed for the presence of a second planet either in 1:2, 5:3,\n2:1, 5:2, or 3:1 resonance with planet b, corresponding to orbital\nperiods of 0.39 d, 1.31 d, 1.58 d, 1.97 d, or 2.37, respectively.\nSome of these periods correspond with regions in the figure\nthat are densely populated by our best candidates. The bottom\nplot shows the mass of the candidate plotted against its orbital\nperiod. Again, we highlight the stable orbit by Cortés-Zuleta\net al. (2020) with red dashed lines.\nThe expected χ2would be around the number of degrees\nof freedom (19 total datapoints for dataset 8, of which 15 were\nfitted as out-of-transit points and 11 free parameters), while the\nlowestχ2we obtained during the simulations is »134 (clearly\nvisible in the top plot of Fig. 3 at »10 d orbital period). We\nfit the HARPS data with a one-planet eccentric orbit model to\ncompare it to the scenarios with a perturbed. Without adding\nany RV jitter, the χ2of this model is »135. Since the two-planet\nmodel has four more free parameters, by applying the Bayesian\ninformation criterion (BIC), we found no evidence of the\npresence of a perturber in the system with a period within the\nlength of the dataset with this approach. For weak (respectively,\nstrong) evidence (see Kass & Raftery 1995 for more details)\nthat the two-planet model is preferred, a χ2value below 115\n(respectively, 98) is needed.\nWe also injected a candidate in the whole RV dataset (IDs\nfrom 1 to 8), but we obtained an even higher χ2, resulting in an\neven more unlikely presence of a perturbed. Our results exclude\nit up to an RV amplitude of »1 m s´1.\nArticle number, page 5 of 22A&A proofs: manuscript no. main\n0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0\nPeriod Pc [d]01002003004005002\nCandidates\n0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0\nPeriod Pc [d]050100150200250Mass  mc [ME]\nFig. 3. Properties of third bodies excluded by RV data. The plot shows\nthe results for the simulated scenarios including a putative companion\nplanet c of di fferent masses at di fferent periods. Each black dot repre-\nsents one of the around 20 000 selected scenarios (see the text for more\ndetails).\nTop plot :χ2value vs the orbital period of the perturber. The red lines\ncorrespond to the period of the stable orbits predicted by Cortés-Zuleta\net al. (2020). The two horizontal dashed lines represent the χ2required\nto have weak (upper line) or strong (lower line) evidence of a second\nplanet, as discussed in the text.\nBottom plot : Mass and orbital period of the selected candidates.\nThis simulation confirms the conclusion of Knutson et al.\n(2014) when looking for the acceleration of the system due\nto possible long-period companions. The cited paper does not\nfind an acceleration that di ffers from zero by more than 3 σ. In\nthis way, we extended the orbital period range of investigation\nin Knutson et al. (2014) from between 0 and 1702 d up to\n»4200 d.Since we did not find any additional companion and there is no\nevidence for long-term trends in the data, we proceeded with our\nanalysis by introducing periastron precession in the fit.\n5.2. Rate of apsidal motion\nIn this section, we describe the code used for the data analysis\n(by allowing for non-zero periastron precession) with the fitted\nparameters and the results.\nWe used the approach and idlcode developed by Csizmadia\net al. (2019) to analyze the RV data in parallel with mid-transit\nand occultation times. Since then, the code has been modified\nand updated. Now the Genetic Algorithm minimization is\nfollowed by a di fferential-evolution Markov chain Monte Carlo\n(DE-MCMC, Ter Braak 2006) algorithm to explore the param-\neter space. The values of the fitted parameters (see Table 4 for\nthe list of such parameters) obtained by the Genetic Algorithm\nare used as a starting point for the DE-MCMC analysis to derive\nthe posterior probability distributions. The Genetic Algorithm\nis initialized with a population set made of 200 individuals.\nIn the DE-MCMC we run 90 chains. The first 6 ¨104steps of\neach chain are discarded as part of the burn-in phase. They are\nfollowed by 104more DE-MCMC steps, which are used for\nthe posterior distributions. The convergence is checked by the\nGelman - Rubin test applied on the steps that follow the burn-in\nphase. At the end of the run, all parameters resulted in having\nR<1.2 (Gelman & Rubin 1992).\nThe priors that are used on the fitted parameters are set from\nthe results in the literature (mainly from Hellier et al. 2011)\nand noted in Table 4 along with the results. The prior on the\neccentricity is set as the weighted mean and standard deviation\nof the results in the literature (Hebb et al. 2010; Anderson et al.\n2010, 2013; Hellier et al. 2011; Mandell et al. 2013; Lendl et al.\n2013; Zhou et al. 2014; Knutson et al. 2014). From this, the\npriors on?e¨sinω0and?e¨cosω0are calculated as a combi-\nnation of the two parameters. The contribution on the χ2of the\npriors was calculated the same way as in Csizmadia et al. (2019).\nWe modeled the radial velocity of the host star with a Keplerian\nmotion but with a time-variable ω:\nVrad,i,j“Vγ`Vi,instr`K¨pe¨cosωptjq`cospvj`ωptjqqq`δVptjq,\n(7)\nwith a secular advance of the longitude of periastron, 9ω=dω{dt,\ncaused by the tidal and rotational potential (as expressed in\nEq. (A.8)):\nωptjq“ω0` 9ωptj´t0q, (8)\nwhere iis the index of the instrument used and jis the index\nof the time when the observation was taken; Vγis the system-\natic velocity of the barycenter of WASP-19Ab system relative to\nthe barycenter of the Solar System. Di fferent datasets may have\ndifferent instrumental zero-point o ffsets, Vi,instr. Finally, Kis the\nradial velocity half-amplitude:\nK“2πa‹sini\nPa?\n1´e2. (9)\nIn this expression, a‹is the semi-major axis of the star around\nthe common center of mass and iis the inclination between the\norbital plane and the plane of the sky ( perpendicular to the line\nof sight).\nArticle number, page 6 of 22Bernabò et al.: Periastron precession of WASP-19Ab\nFig. 4. Star-planet system geometry when tidal interactions occur. The\nspherical shapes that star and the planet would have in the absence of\ntidal interactions are drawn in red and dark brown. The tidal bulges\nare drawn in orange and light brown, respectively. They arise on the\nstellar and planetary surfaces (respectively in orange and light brown)\nand rotate with the star itself and are elongated in the direction of the\nother body. The tidal lags are also drawn. The image is not to scale.\nHere,δVis the non-orbital apparent contribution that arises from\nthe rotation of the star around its own axis. The stellar and plane-\ntary shapes are distorted ellipsoids with the longest axis in the di-\nrection of the radius vector connecting them, as shown in Fig. 4.\nTheir apparent area varies continuously during an orbital revo-\nlution, therefore the emitted light changes (i.e., the ellipticity ef-\nfect). Moreover, in close binary systems, part of the radiation of\neach of the two components falls on the surface of the compan-\nion. It is absorbed and re-emitted (or scattered) in all directions,\nvarying during the orbital phase (i.e., the reflection e ffect).\nThese two photometric e ffects in the RV data due to distor-\ntion are described by Kopal (1959) as a non-orbital contribution\nto the RV curve:\nδV“´3\n2Ka\n1´e2M‹`mp\nM‹Porb\nProt,‹ˆR‹\na˙4\n¨\n¨ˆ\n1`k2,‹\n2˙\nsini¨sinp2pv`ωqqf2, (10)\nwhere the Love number of the star is assumed from theoretical\ncalculations (we take it from Claret 2004). Then, Prot,‹is the\nrotational period of the star around its axis and f2takes into\naccount the limb darkening. For more details on f2see Kopal\n(1959) and Csizmadia et al. (2019).\nArras et al. (2012) already includes such calculations for\nprecise RV analysis of WASP-18A, but neglects the e ffect of\nstellar rotation. In other words, a synchronous rotation of the\nstar is assumed. However, for WASP-19A this is not the case.\nIt is known to be active, showing a rotational modulation with\nProt,‹=10.5 d (Hebb et al. 2010). Therefore, the term Porb/Prot,‹\nis not expected to be 1, but »0.79 d /10.5 d»0.08, introducing a\nfactor of about 12 di fference in the amplitude of such e ffect cal-\nculated by Arras et al. (2012) and Kopal (1959). The amplitude\nof this contribution is around 0.19 m s´1, so it is negligible for\nWASP-19A if compared to the typical observational errorbars.\nThe period of this e ffect is linked to the sine term of the true\nanomaly in Eq. (10) and, therefore, it is half of the orbital period.\nMoreover, we added jitter terms for each spectrograph to the RV\nmodel (see, e.g., Baluev 2009 for a discussion of instrumental\njitters). The jitter is composed of an instrumental and a stellar\ncontribution. It is not possible to separate them. As a starting\npoint, if present, we used the values proposed by the correspond-ing literature. In particular, 13 m s´1for Hebb et al. (2010)’s data\n(ID 1, as in Anderson et al. 2010), 14.1 m s´1and 6.9 m s´1for\nCoralie (ID 2) and HARPS (ID 3) Hellier et al. (2011)’s data, re-\nspectively, 20 m s´1for Albrecht et al. (2012)’s (ID 4) and 17.8\nm s´1for Knutson et al. (2014)’s (ID 5) RVs. Then, we refined\nall the jitter values by minimizing the χ2as:\nχ2\njit,instr“ndataÿ\ni“1pOi´Ciq2\nσ2\ni`σ2\njitinstr“ndata´1, (11)\nwhere OiandCiare the observed points (see Tables D.1 and\nD.2) and the (calculated) expected values from the model,\nrespectively; σirepresents the observed errors corresponding\ntoOiandndatais the number of datapoints in the subset. This\nprocedure was iteratively repeated until the condition of Eq. (11)\nwas fulfilled. Due to the highly inhomogeneous set of data, this\nprocedure allows for every RV dataset to have a weight in the fit\nproportional to the number of its data points.\nWe also added a jitter value for transits and one for oc-\ncultations to account for inhomogeneities in the dataset and\nfor a perturber with an orbital period longer than 12 years, as\ndescribed in Secion 5.1. The fitted values for the jitter terms\ncan be found in Table 4, along with all the fitted and derived\nparameters in our model.\nThe orbital eccentricity is a key issue for Love number studies:\nin circular orbits, apsidal motion is not observable. In addition,\nthe argument of periastron is hard to measure if eccentricity\nis not well known. When the eccentricity is small, then its\nuncertainty range is often large and the investigators find that\nthe orbit is compatible with a circular one. However, such results\nin the literature do not necessarily imply that the orbit is circular,\ndespite many investigators setting a zero eccentricity (and not\npropagating the error bars of eccentricity to the corresponding\nresult). It only means that an upper limit exists for eccentricity.\nMoreover, a certain dataset may indicate the presence of an\neccentric orbit with an accurate result. However, the sample size\nis not big enough or not aptly distributed to get the uncertainty\nranges accurate enough. The results are imprecise and the real\nvalue of eccentricity is vague.\nAlthough we collected more RV and TTV data than any\nprevious study of the WASP-19Ab system, the datasets come\nfrom various instruments with di fferent precisions and, thus, are\nnon-homogeneous. The value of eccentricity may be a ffected\nby the di fferent data accuracy, as some of the data sets are\ndiluted by less precise measurement points. Because of these\nreasons, any previous estimate of eccentricity can be accurate\nbut imprecise.\nApplying a prior on the eccentricity in the fitting procedure\ncan be a valid approach. The average and standard deviation of\nthe previously reported eccentricities can better represent reality\nthan any single determination from a limited dataset. Therefore,\nwe calculated the weighted average of the previous eccentricity\nmeasurement and their weighted standard error using their\nuncertainty range as weights. Therefore, we performed a fit\nusing this mean eccentricity as a prior with their weighted\nstandard error as a Gaussian width.\nIn addition, in the calculation of the χ2of the fit, priors also\non the stellar temperature, Teff,‹, and Love number, k2,‹, the\nmass ratio, q, the orbital inclination, i, the orbital over rotational\nperiod ratio, F ‹, and the ratio, rp{R‹, were added as described\nin Csizmadia et al. (2019).\nArticle number, page 7 of 22A&A proofs: manuscript no. main\nParameter Symbol Fit Value Priors\nAssumed parameters\nStellar rotation Vsinpiq 4.63 km s´1:1\nStellar mass M‹ 0.970 Md\nPlanet-to-star radius ratio rp/R‹ 0.14248\nSemi-major axis a 0.01655 AU\nFirst transit epoch T0 2 454 775.3372 BJD TDB\nAdjusted parameters\nJitters ID 1 jit 1 RVs 13.5 m s´1\nID 2 jit 2 RVs 35.0 m s´1\nID 3 jit 3 RVs 9.8 m s´1\nID 4 jit 4 RVs 11.5 m s´1\nID 5 jit 5 RVs 30.4 m s´1\nID 6 jit 6 RVs 13.1 m s´1\nID 7 jit 7 RVs 9.8 m s´1\nID 8 jit 8 RVs 22.3 m s´1\nTransits jit tr Tr 0.00005 d\nOccultations jit occ Occ 0.0001 d\nFitted parameters\nOffset velocity of the system V γ RVs (20.7791 ˘0.0060) km s´1U[-8,+8]\nRV semi-amplitude K RVs (253.48 ˘3.87 ) m s´1U[-8,+8]\nOffset instrumental velocity\nID 2 V instr,2 RVs (0.0258 ˘0.0071)km s´1U[-8,+8]\nID 3 V instr,3 RVs (0.0221 ˘0.0074 ) km s´1U[-8,+8]\nID 4 V instr,4 RVs (-20.827 ˘0.016 ) km s´1U[-8,+8]\nID 5 V instr,5 RVs (-20.712 ˘0.014 ) km s´1U[-8,+8]\nID 6 V instr,6 RVs (-0.0919 ˘0.0092 ) km s´1U[-8,+8]\nID 7 V instr,7 RVs (-0.1149 ˘0.0139 ) km s´1U[-8,+8]\nID 8 V instr,8 RVs (0.0750 ˘0.0090) km s´1U[-8,+8]\nAnomalistic period Pa RVs and Tr /Occ (0.788951`0.000012\n´0.000017) d U[-8,+8]?e¨sinω0 RVs and Tr /Occ 0.03920`0.00458\n´0.00682U[-1,+1]?e¨cosω0 RVs and Tr /Occ -0.00056`0.01385\n´0.01717U[-1,+1]\nPeriastron precession rate dω{dt RVs and Tr /Occ (0.065`0.007\n´0.010)˝d´1=\n(233`25\n´35)2d´1U[-8,+8]\nStellar Love number k2,‹ RVs 0.0200 ˘0.0069 N(0.02;0.004)\nStellar temperature Teff,‹ RVs (5498.65 ˘172.86)K N(5500 K;100 K)\nPlanet-to-star mass ratio q=mp/M‹ RVs 0.001150 ˘0.000057 N(0.00115;3.3¨10´5)\nStellar radius over semi-major axis R‹/a RVs 0.2785 ˘0.0070 N(0.2784;0.0040)\nOrbital-to-rotational stellar period F‹=Porb/Prot,‹RVs 0.0729 ˘0.0074 N(0.0729;0.0043)\nOrbital inclination i RVs and Tr /Occ (79.40 ˘0.69)˝N(79.4;0.4)\nDerived parameters\nEccentricity e 0.00172`0.00035\n´0.00033\nPeriastron angle ω0 (89.71`24.68\n´19.12)˝\nSidereal period Ps (0.788839267`0.000000018\n´0.000000019) d\nPlanetary Love number k2,p 0.20`0.02\n´0.03\nTable 4. Parameters of the WASP-19Ab system. Assumed, adjusted – according to Eq. (11) – fitted parameters and their priors, and derived\nparameters for our model which fits RVs and transit /occultation mid-times in parallel. The third column indicates if the parameter was fitted to\nmodel radial velocity (\"RV\") and /or mid-transit and mid-occultation times (\"Tr /Occ\"). The last column shows the results of our analysis. The\noffset related to ID 1 ( Vinstr,1) was not included because the first dataset was used to derive the relative velocity between the observer and the\ncenter of mass of the system ( Vgamma ).\nThe solutions are calculated only from those steps of the DE-MCMC that fall within the distribution around the highest peak of the posterior\ndistribution of 9ω(see Fig. (6), solution B).\nErrorbars are given with 1 σconfidence level.\nThe reported values and errorbars of the sidereal period Ps,ω0andeare not calculated - as Ps=Pa¨p1´Pa¨9ω{nq, arctanp?e¨sinω0{?e¨cosω0q\nandp?e¨sinω0q2+p?e¨cosω0q2respectively - from the reported values of Pa,9ω,?e¨sinω0and?e¨cosω0but from the median of the\ndistribution resulting from the combination of the posterior distributions of?e¨sinω0and?e¨cosω0.\n:1Hellier et al. 2011\nArticle number, page 8 of 22Bernabò et al.: Periastron precession of WASP-19Ab\n0.2\n 0.1\n 0.0 0.1 0.2 0.3 0.4\n [° d1]\n4104204304404504604702\nFig. 5.χ2values of the fit when varying 9ωin the model. The horizontal\ndashed line separates the statistically worse models (over the line) from\nthe best models (below the dashed lines). The models below the dashed\nlines are statistically equivalent within 1 σ(68% confidence level for\nthe red line) and 2 σ(95% confidence level for the green line). Note: the\nsignificantly larger χ2at 9ω=0. The high amount of local minima is an\nindication of the complexity of the problem.\nTogether with the RVs, we modeled mid-transit and mid-\noccultation times listed in Tables C.1 - C.4 mainly to refine\nthe determination of the first transit epoch and the period. To\ninclude periastron precession in the fit of predicted mid-times,\nwe followed the approach of Gimenez & Garcia-Pelayo (1983)\nand Csizmadia et al. (2019).\n5.3. Presence of apsidal motion\nBecause of the large parameter correlations between the mean\nmotion and periastron precession rate (see Csizmadia et al.\n2019), as a first analysis, we calculated 80 di fferent models with\nfixed values of 9ω, the so-called semi-grid analysis. This proce-\ndure also helps to get an initial range for the fitted value of 9ω. The\nlimiting values were 9ω“´ 0.25 and`0.45˝d´1. We also fixed\nthe epoch to 2 454 775.3372 BJD TDB, while all other parameters\nwere free in the DE-MCMC analysis. The result is presented in\nFig. (5), showing the assumed value of the periastron preces-\nsion rate and the corresponding χ2of the fit. The figure clearly\nshows that the joint fit of the RV and TTV data is significantly\nworse if 9ω“0, namely, no periastron precession is present. A\nsmall amount of periastron precession improves the quality of\nthe fit significantly. The figure shows that small negative perias-\ntron precession rates also improve the fit but still result in a statis-\ntically worse fit than the small positive precession values. From\nthat figure and the statistical analysis, we conclude that small\npositive periastron precession rates are needed to have a good fit\nof the joint RV and TTV data of WASP-19Ab, and we are able\nto give a lower and an upper limit of 0 .06À 9ωÀ0.075˝d´1.\nThe result is refined in the next section.Solution 9ω[0d´1]χ2\nA -0.064 416\nB +0.065 408\nC +0.163 423\nD +0.261 424\nE +0.479 430\nTable 5. Median values of the five posterior distributions shown in\nFig. 6 and corresponding χ2(calculated before introducing the contri-\nbution of the jitters to the uncertainty, as in Eq. (11)).\n6. Discussion of the results\n6.1. Apsidal motion rate\nIn this section, we show and discuss the results of the analysis\nperformed with the model described in Sect. 5. Contrarily to the\nsemi-grid method (Sect. 5.3), here we fit 9ωalong with the other\nparameters.\nDuring the analysis, we noticed that di fferent DE-MCMC chains\nconverged to di fferent values of 9ω. In particular, as it is shown\nin Fig. 6 and in Table 5, during the run multiple chains explored\nvarious positive and negative values of 9ω. Figure 6 clearly shows\nalso the correlation between the anomalistic period and the peri-\nastron precession rate, as discussed in Csizmadia et al. (2019). It\nis due to the formula that links the anomalistic period Paand the\nsidereal period Ps:\nPs“Pap1´ 9ω{nq. (12)\nThis formula that links the two quantities gives a between pe-\nriod and periastron precession rate, but no unique solution. The\nadded value of including transit and occultation mid-times lies\nin the fact that they constrain the sidereal period, and therefore a\nrange of values for Paand 9ω(as shown in Fig. 6), while avoiding\nsome other ranges.\nA different chain convergence shows why it was important to\nrun various chains. Only one chain would have converged to one\nof the possible values, excluding the other results or could have\nbeen trapped in a local minimum (the same local minima found\nby the semi-grid method approach; see Fig. 5).\nThe highest peak of 9ω(solution B) in the posterior distribu-\ntion results around „+0.065˝d´1(summarized in Table 5). The\npositive solutions C, D, and E can be discarded because of their\nhighχ2values. Moreover, solution E would lead to a planetary\nLove number higher than 1.5, which is unphysical.\nThe negative periastron precession rate (solution A) would\ncorrespond either to a retrograde orbit (of the planet with respect\nto the rotation of the star) or to an unseen perturber. The ret-\nrograde orbit can be excluded from Rossiter-McLaughlin e ffect\nstudies and TESS light curves (Wong et al. 2020). They confirm\nthat the spin axis of the star is almost perpendicular to the orbital\nplane.\nWe investigated in detail the presence of a perturber in\nSect. 5.1 and no evidence is found. Assuming a negative pe-\nriastron precession amplitude as observed of „-0.064˝d´1=\n-229.682d´1, through Eqs. (12) and (16) in Borkovits et al.\n(2011), we can estimate the period of the body producing such a\nprecession rate. We subtracted the general relativity contribution\n(calculated in Sect. 5.2 as 2.852d´1) from these solutions and\nderived the orbital period of the perturber producing such an 9ω.\nFor a circular orbit, a body with a mass in the range 0.1 M Cand\n20 M Ewould have an orbital period below 10 d, as shown in\nFig. 7. Such a planet would be retrievable in the RV data, while\nArticle number, page 9 of 22A&A proofs: manuscript no. main\nFig. 6. Posterior distributions and correlation between the anomalistic\nperiod and periastron precession rate where the chains converged. The\nmiddle plot shows the strong correlation between these two free param-\neters, as in Eq. (12). On the sides, are the corresponding histograms of\nthe posterior distributions. As can be seen from the histograms on the\nsides, many chains converged to various positive and negative peaks, as\npredicted by the semi-grid method and reported in Table 5.\nno long-term (over a 10-day orbital period) companion in this\nmass range can produce a negative periastron precession. There-\nfore the solutions yielding negative 9ωvalues can be discarded.\nThe results for the fitted parameters of the 9ω-positive solution\n(solution B) with lowest χ2are shown in Table 4, along with\ntheir errorbars. With 99 RV points (outside the transit, 203 in to-\ntal), 162 mid-transit and 11 mid-occultation points, and 19 free\nparameters, the reduced χ2is 1.02. The observed value of the\ntotal contribution 9ωfrom the DE-MCMC analysis of the data is\n(233`25\n´35)2d´1(see Table 4), corresponding to an apsidal motion\nperiod of U=2π{9ω»15 years.\nFigure 8 shows the residuals of the transit and occultation mid-\ntimes with such a fit. Figure 9 shows the phase-folded RV curve\nand the residuals of our periastron precession fit.\nWe subtract the General Relativity term and the tidal and ro-\ntational components relative to the star from the observed 9ω.\nWhat is left in Eq. (1) are the tidal and rotational components\ndue to the planet. They are related to the interior of the planet and\nshow clearly the dependence on Love number (see Eqs. (3) and\n(4)). Then, k2,pcan then be estimated by inverting these equa-\ntions.\nThe only unknown parameter is the rotational period Prot,pof\nthe planet itself. Since there is no measurement of the rotational\nperiod of the planet, we can only assume a value for Prot,p. The\ntwo extreme cases are presented when:\n- tidal forces lead to the disruption of the planet ( Prot,p=0.14\nd), then we obtain k2,p=0.07;\n0 2 4 6 8 10\nPeriod P2 [d]0.02.55.07.510.012.515.017.520.0Mass M2 [MJup]Fig. 7. Mass and orbital period of a second body that would cause a neg-\native periastron precession rate of „-0.0638˝d´1(corresponding to the\nnegative solution of the DE-MCMC analysis) on planet b, as deduced\nfrom Borkovits et al. (2011). See text for more details.\n- the planet is non-rotating ( Porb{Prot,pÑ0), then we obtain\nk2,p=0.22.\nThe measured Love number of the planet must lie in between\nthese two values. We can suppose that tidal forces synchronize\nthe orbital and rotational periods quite fast. This means that Porb\n=Prot,p. We can then derive k2,pfrom Eqs. (A.2) and (A.7):\nk2,p=0.20`0.02\n´0.03, with a precision of 15%.\nFor comparison, the most up-to-date values of the tidal\nLove number for planets and satellites in the Solar System and\nexoplanets are listed in Table 6. The value obtained for the fluid\nLove number of WASP-19Ab has the same order of magnitude\nas the other values derived for exoplanets.\nThe lower the value, the more matter is condensed to-\nward the center of the body. The low Love number obtained,\nk2,p=0.20, is an indication of a high-mass core. This has to\nbe tested and verified with simulations of planetary interior\nstructure through the equations of state, however, this is beyond\nthe scope of this paper.\nMoreover, if hydrostatic equilibrium is assumed (whereby the\nsurface of the body can be described as an equipotential surface\nof the tidal and rotational potentials), the second-order fluid\nLove number h2pcan also be calculated. This assumption is\nclose to reality if the body is a giant gaseous planet. In this case,\nh2,p=k2,p+1 (Munk & MacDonald 1960), therefore h2,p=1.20 for\nsynchronous rotation. This causes a small-shaped deformation\nthat can be hard to observe photometrically (Hellard et al. 2019).\nDuring the analysis, we noticed that the main contribution to the\nuncertainty of the fit is related to the transit fit. It is due to under-\nestimated errorbars on the mid-transit times, which compromise\nthe reliability of the results, increasing the χ2of the fit. This\nwas already suggested by Mancini et al. (2013) regarding the 54\ntransits they analysed. Therefore, we increased the errorbars to\nArticle number, page 10 of 22Bernabò et al.: Periastron precession of WASP-19Ab\n0 1000 2000 3000 4000 5000 6000\nCycle400\n200\n0200400Residuals [s]\nTransits\nOccultations\nFig. 8. Residuals of the periastron precession fit for transits and\noccultation mid-times, calculated with the sidereal period. Transit\nmid-times are plotted as blue circles, occultation mid-times as red\ncircles. The errorbars include the jitters, as in Table 4.\n0.0 0.2 0.4 0.6 0.8 1.00.3\n0.1\n0.2\n0.00.10.20.3RV [km/s]\n-model\n0.0 0.2 0.4 0.6 0.8 1.0\nPhase100\n50\n050100Residuals [m/s]\nFig. 9. Data with errorbars, phase-folded with our fit including perias-\ntron precession, shown in the upper plot. The colours represent di fferent\ndatasets, as in Fig. (1) (see also Sect. 4.2 for more details).\nResiduals of the periastron precession rate fit are shown in the lower\nplot. The errorbars include the jitters, as in Table 4.\nIn both plots, the in-transit points are not shown as they were not used\nin the fit.\n0.0005 d (approximately 43 s) if the reported uncertainties were\nlower than this threshold.\nWe also underline the importance of acquiring new light\ncurves. Precise transit and, mostly, occultation mid-times are\nneeded to break the degeneracy between periastron precession\nand the change of orbital period due to for instance companions\nor orbital decay.\n6.2. Alternative explanations\nWe investigated alternative scenarios to the presence of perias-\ntron precession. In particular, we searched for distant massive\ncompanions with the approach of Knutson et al. (2014). The\nlong-term acceleration of the system was fitted by adding a γ\nacceleration to the RV Eq. (7) as:\nVaccptjq“ 9Vγ¨ptj´t0q. (13)\nWe ran the DE-MCMC algorithm in the case of a circular and\nan eccentric orbit. The posterior distribution of 9Vγhas medianof (7.37˘5.02)¨10´7m s´2for the circular case and (7.43\n˘4.89)¨10´7m s´2for the eccentric one. Both results are\ncompatible with zero and reproduce the results of Knutson et al.\n(2014) of (7.52 ˘3.94)¨10´7m s´2. Our results increase the\nconfidence level of a non-detection of long-term acceleration\nfrom Knutson et al. (2014)’s 1.9 σto 1.5σ.\nWe also investigated the tidal decay scenario, as it has been\nanalysed and confirmed in the system WASP-12b (see respec-\ntively Patra et al. 2017; Yee et al. 2020; Turner et al. 2021; Wong\net al. 2022). The observed TTVs can be due to tidal decay or pe-\nriastron precession. We fitted the RV semi-amplitude Kin Eq. (9)\nas a time-varying parameter:\nKptjq“K0` 9Kptj´t0q. (14)\nDeriving Kepler’s laws with respect to time, we can see that a\nchange in the RV semi-amplitude results in an opposite change\nin the semi-major axis.\nFor the circular orbit case, we obtained 9K=(-2.77˘\n2.80)¨10´8m s´2while for the eccentric case (-2.60 ˘2.73)\n¨10´8m s´2. As Rosário et al. (2022), we did not find signif-\nicant evidence of orbital decay. Both in the case of a long-term\nvelocity and of tidal decay, the results are compatible with zero\nand therefore considered as constant.\n7. Conclusions\nIn this study, we review the theory on tidal interaction, as\ndescribed by Kopal (1978) and reported with modern notation.\nUsing assumptions on the Love number of the star, based\non well-established theoretical models of its evolution and\ninterior structure, the apsidal motion rate allows us to constrain\nthe Love number k2,pof the planet. We applied the results\nto a case study, namely, the system WASP-19Ab, by using\narchival mid-transit times, occultations, and radial velocity data,\nincluding previously unpublished RV observations made by\nHARPS. This dataset covers twelve years in total. First, we\nruled out previous claims of additional planetary companions in\nthe system as possible perturbers in the radial velocity curve that\ncould hide the small contribution of the periastron precession\nmotion. Moreover, we investigated the presence of a long-term\nacceleration in the RV dataset and of tidal decay. Both these\nscenarios have been rejected with respect to being compatible\nwith a zero trend. Then, we used a model that accounts for an\neccentric orbit and apsidal motion in order to fit the data. We\ndetected the presence of apsidal motion in the planetary orbit\nand linked it to the Love number of the planet. By assuming\nsynchronous rotation for the planet, we derived k2,p=0.20`0.02\n´0.03.\nThe planetary fluid Love number is constrained in this\nsystem for the first time. There are few systems for which\nit has been measured. We take the opportunity to compare\nWASP-19Ab to them. Table 6 shows the fluid and tidal Love\nnumbers of Solar system planets and bodies and exoplanets\nwhich are reported in the literature. The derived value of the\nfluid Love number k2,pof WASP-19Ab is on the same order of\nmagnitude as the one of other Jupiter-like planets (WASP-4b,\nWASP-18Ab, WASP-103b and WASP-121b). A low value\nsuggests the presence of a dense core in the planet, similar to the\ncase of Saturn, Neptune, and WASP-121b. Note that the Love\nnumbers of WASP-12b and WASP-4b (derived by Campo et al.\n2011 and Bouma et al. 2019, respectively) are intentionally left\nout of the table. In the first case, the TTVs might be explained\nArticle number, page 11 of 22A&A proofs: manuscript no. main\nBody Fluid Love number Method Reference\nk2,p\nWASP-19Ab 0.20`0.02\n´0.03RVs and TTVs: periastron precession This work\nWASP-18Ab 0.62`0.55\n´0.19RVs and TTVs: periastron precession Csizmadia et al. (2019)\nWASP-103b:0.59`0.23\n´0.27LCs: shape deformation Barros et al. (2022)\nWASP-121b:0.39`0.71\n´0.81LCs: shape deformation Hellard et al. (2020)\nJupiter 0.565 ˘0.006 Juno: Doppler shift measurements Durante et al. (2020)\nSaturn 0.390 ˘0.024 Cassini: gravitational perturbations on Enceladus, Tethys, Dione and Rhea Lainey et al. (2017)\nEnceladus 0.9896 ˘0.0103 Cassini: quadrupole gravity field and its hemispherical asymmetry Taubner et al. (2016)\nNeptune 0.392 Orbital perturbations on Triton and Nereid Jacobson (2009)\nBody Tidal Love number Method Reference\nHAT-P-13b 0.31`0.08\n´0.05RVs and TTVs: 9eand 9ω Buhler et al. (2016)\n\" \" 0.81 ˘0.10 RVs and TTVs: 9eand 9ω Hardy et al. (2017)\nMercury 0.53 ˘0.03 MESSENGER: Topography Konopliv et al. (2020)\nVenus 0.295 ˘0.066 Magellan and Pioneer Venus Orbiter: Doppler tracking Konopliv & Yoder (1996)\nMoon 0.02416 ˘0.00022 GRAIL and LLR: mapping of the gravitational field Williams et al. (2014)\nMars 0.183 ˘0.009 Mars Express: Doppler tracking and characterization of the orbits of Phobos and Deimos Jacobson & Lainey (2014)\nTitan 0.637 ˘0.112 Cassini: response to Saturn’s variable tidal field Iess et al. (2012)\nTable 6. Fluid and tidal Love numbers measured for Solar System planets and minor bodies and exoplanets. In the first half of the table, we show\nfluid Love numbers, which are comparable with our result. In the second half, we report tidal Love numbers. They refer to di fferent quantities, so\nthey are not comparable, except at longer periods when elastic stresses relax.\n:Calculated from h2assuming hydrostatic equilibrium\nby tidal decay (Yee et al. (2020) and reference therein). In the\nsecond case, the presence of a second, long-period planet in\nthe system (Turner et al. 2022) further complicates the TTV\nanalysis. The results of Bouma et al. (2020) were discussed\nfurther by Harre & Smith (2023) in the light of this second\nplanet, making the determination of k2,puncertain.\nNo general conclusion on planetary physics can be drawn\nfrom this set of measured Love numbers because the sample\nis too small to be representative of the existing exoplanet\npopulation. We need observations of the Love number of more\nexoplanets.\nWe also characterize, for the first time, the stellar multiplicity\nof the system. We find a stellar companion with an apparent\nseparation of 682, corresponding to a physical distance of\nalmost 20 000 AU, from the system under analysis. However, its\nexpected e ffect on the RV of WASP-19A is negligible.\nAcknowledgments We acknowledge the support of DFG Re-\nsearch Unit 2440: \"Matter Under Planetary Interior Conditions:\nHigh Pressure, Planetary, and Plasma Physics\" and of DFG\ngrants RA 714 /14-1 within the DFG Schwerpunkt SPP 1992,\nExploring the Diversity of Extrasolar Planets. L. M. B. would\nalso like to thank Carl Knight for providing us with two transits\nof WASP-19Ab observed by the Ngileah Observatory, New\nZealand, in May 2022. L. M. B. would also like to acknowledge\nH. Hußmann and F. Sohl for the useful discussion on the\ndifferent kinds of Love numbers.\nThe work is based on observations collected at the European\nSouthern Observatory under ESO programme 0104.C-0849.\nThis research has made use of the NASA Exoplanet Archive,\nwhich is operated by the California Institute of Technology,\nunder contract with the National Aeronautics and Space Admin-\nistration under the Exoplanet Exploration Program.\nThis work presents results from the European Space Agency\n(ESA) space mission Gaia (https: //www.cosmos.esa.int /gaia).\nGaia data are being processed by the Gaia Data Pro-\ncessing and Analysis Consortium (DPAC) (DPAC,\nhttps: //www.cosmos.esa.int /web/gaia/dpac/consortium). Fund-\ning for the DPAC is provided by national institutions, in\nparticular the institutions participating in the Gaia MultiLateralAgreement (MLA).\nReferences\nAbe, L., Gonçalves, I., Agabi, A., et al. 2013, A&A, 553, A49\nAlbrecht, S., Winn, J. N., Johnson, J. 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In this configuration, θequals 0 for transits and\nπfor occultations.\nAppendix A: Equations of apsidal motion\nIn this Appendix, we rewrite the equations derived by Kopal\n(1959, 1978) with modern notation.\nFirst, we define the stellar and planetary disturbing acceler-\nations RiandSi, where the subscript irepresents either the star\n(i“‹) or the planet ( i“p).Riacts in the direction of the radius\nvector and Siin the plane of the orbit in the direction perpendic-\nular to the radius vector and positive in the direction of motion.\nThe radial perturbation accelerations can be expressed as:\nR‹“´3\n4GpM‹`mpqmp\nM‹k2,‹R5\n‹\nr4r3ϵ‹p3 cos2ϵ‹´1q, (A.1)\nand\nRp“´3\n4GpM‹`mpqM‹\nmpk2,pr5\np\nr4r3ϵpp3 cos2ϵp´1q, (A.2)\nwhere Gis the Gravitational constant, M‹andmpare the masses\nof the star and the planet, respectively, R‹andrptheir radii and\nk2,‹andk2,ptheir second-order fluid Love number.\nHere, ris the radius vector:\nr“ap1´e2q\n1`e¨cosv, (A.3)\nwhere ais the semi-major axis of the orbit, eis the orbital eccen-\ntricity and vis the true anomaly of the planetary orbit, as shown\nin Fig. A.1.\nA tidal lag in latitude arises if the spin axis of the two bodies\nis not perpendicular to the orbital plane and in general is small\nor even zero. On the other hand, a lag in longitude arises from\nasynchronism between spin and orbital rotation. The angles ϵ‹andϵprepresent the tidal lag of the star or the planet in astrocen-\ntric longitude (Efroimsky & Makarov 2013) as:\nϵi“1\n2arcsinˆ\n´1\nQisignrPorb{Prot,i´1s˙\n, (A.4)\nwhere Porbis the orbital period and Prot,iis the rotational period\nof the star or the planet. The function sign takes the value of +1,\n0 or -1, depending on the sign of the expression in the square\nbrackets. Qiis the tidal quality factor. A high Qfactor corre-\nsponds to a weak dissipation and vice-versa (see Kaula (1964)\nand reference therein).\nThe radius vector, rϵ, is defined as:\nrϵ,i“ap1´e2q\n1`e¨cospv´ϵiq. (A.5)\nThe di fference between randrϵiis due to a finite speed\nof progression of the radial tidal waves in viscous matter. The\namplitude of tidal distortion is not maximum when the bodies\nare closest to each other, but at a later time. We consider a fluid\ncase, which corresponds to ϵi=0.\nIn addition, we define the tangential perturbation accelerations:\nS‹“3\n4GpM‹`mpqmp\nM‹k2,‹R‹5\nr4r3ϵ‹sinp2ϵ‹q, (A.6)\nfor the star and\nSp“3\n4GpM‹`mpqM‹\nmpk2,pr5\np\nr4r3ϵpsinp2ϵpq, (A.7)\nfor the planet. Finally, R=R‹+RpandS=S‹+Sp.\nThere is a third perturbing acceleration component beyond Sand\nR. It is normal to the plane of the orbit and positive toward the\nnorth pole. However, its contribution is zero as it is proportional\nto the latitude component of the angle of tidal lag. It can arise\nonly if the orbital plane is inclined with respect to the equatorial\nplane of the body (star or planet). We neglect this term because\nthe Rossiter-McLaughlin measurements of WASP-19A show an\nequatorial orbit (see, e.g., Sedaghati et al. 2021).\nWith the introduction of the terms, SandR, the perturbation\nequations can be easily written following Kopal (1959, 1978).\nThe temporal change of the argument of periastron measured\nfrom the ascending node is:\ndω\ndt“?\n1´e2\nnae„\n´Rcosv`Sˆ\n1`r\nap1´e2q˙\nsinvȷ\n,\n(A.8)\nwhere nis the mean motion that can be defined through Kepler’s\nlaw or through the orbital period Porb.\nFigure A.2 shows the e ffect of the periodical and secular per-\nturbations on ω, as in Eq. (A.8). The plots were made setting the\nphysical and orbital parameters to those of the exoplanetary sys-\ntem WASP-19Ab (see Sect. 3) with the Love number of WASP-\n18Ab (Csizmadia et al. 2019) as a first guess. We present how it\nevolves over 1, 10, 50, 100, and 500 orbits.\nArticle number, page 14 of 22Bernabò et al.: Periastron precession of WASP-19Ab\n0.0 0.2 0.4 0.6 0.8174.725174.750174.775[°]\n0 2 4 6 8174.75174.80[°]\n0 10 20 30 40174.8175.0[°]\n0 20 40 60 80174.8175.0175.2[°]\n0 100 200 300 400\nTime [days]175176[°]\nFig. A.2. Temporal change of the osculating orbital element ωptqdue to\ntidal and rotational potentials, as described by Eq. (A.8), assuming the\norbital properties of the system WASP-19 (see Sect. 3.1) and the Love\nnumber of WASP-18Ab Csizmadia et al. (2019).\nAppendix B: Planetary eccentricity driven by\nWASP-19B\nHere, we show that a companion star on a very eccentric orbit\nis able to cause an eccentric orbit of the planet in the WASP-19\nsystem.\nBorkovits et al. (2011) gave analytic expressions for the vari-\nations of the orbital elements up to the sixth order of the inner\neccentricity for any arbitrary orientation of the orbits. However,\nwe do not know the spatial orientation of the orbit of the com-\npanion star B nor the longitude of the node of the planet b. There-\nfore, we assume co-planar orbits for the planet and star B, and\nwe also assume that their semi-major axes are aligned. In this\nspecific case, the integration of Eq. (9) of Borkovits et al. (2011)\ngives the simple expression for the eccentricity of the planet.\neb“d\neAL{2¨fpvBq\n1`eAL{2¨fpvBq, (B.1)\nwhere fpvBqis\nfpvBq“1\n2cos 2 vB`4eB\n3cos3vB, (B.2)\nand,AL,\nAL“15\n8Pb\nPBMB\nMA`MB`Mb1\np1´eBq3{2. (B.3)\nHere vBis the true anomaly of star B and it holds the time-\ndependence, PandMare the orbital periods around star A and\nthe masses of the bodies and eis the eccentricity. Indices Aand\nBstand for stars A and B respectively, while index bdenotes the\nplanet.\nThe lower period limit of star B around star A can be esti-\nmated assuming that PBis minimum if star B is in apoastron.\nIn this case, the observed separation between stars A and B cor-\nresponds to a distance equal to aBp1`eBq. For di fferent valuesofeB, we can determine the semi-major axis that yields the or-\nbital period, PB, via Kepler’s third law. In case of eB“0.995\naB«104AU and PB«1 Myrs (note: the periastron point is\nstill located at 50 AU from star A).\nSince we do not know the exact orbit of star B, we just es-\ntimate the planetary eccentricity. Taking the Taylor series of the\nexponential function in Eq. (B.1) and neglecting the time depen-\ndence, we obtain\neb«d\nAL{2ˆˆ1\n2`4eB\n3˙\n. (B.4)\nUsing the aforementioned numerical values and assuming MB„\n0.4Md, we get eb«0.0011. This calculation leads to an order\nof magnitude estimation of whether any (or a specific) combi-\nnation of orbital elements can be responsible for the observed\neccentricity. A deviation from this estimate can be due to the\nunknown spatial orientation of the di fferent orbits.\nArticle number, page 15 of 22A&A proofs: manuscript no. main\nAppendix C: Tables of transits and occultations\nArticle number, page 16 of 22Bernabò et al.: Periastron precession of WASP-19Ab\nCycle number T mid[days] Uncertainty [days] Observations Reference\nN BJD TDB- 2 450 000\n0 4775.33720 0.00150 WASP-South, 2.0m - Faulkes Telescope South (FTS) Hebb et al. (2010)\n2 4776.91566 0.00019 HAWK-I (VLT) Anderson et al. (2010)\n53 4817.14633 0.00021 2.0m - Faulkes Telescope South (FTS) Lendl et al. (2013), from Hebb et al. (2010)\n499 5168.96879 0.00009 EFOSC on NTT, WASP-South and Hebb et al. (2010)’s LCs Hellier et al. (2011)\n517 5183.167890 0.000068 IRAC on Spitzer Anderson et al. (2013)\n538 5199.73343 0.000134 0.25m - Observatorio Astronómico Beta Orionis:1F. Tifner (TRESCA)\n604 5251.79657 0.00014 EFOSC2 on NTT Tregloan-Reed et al. (2013)\n605 5252.58544 0.00010 EFOSC2 on NTT Tregloan-Reed et al. (2013)\n609 5255.74077 0.00012 EFOSC2 on NTT Tregloan-Reed et al. (2013)\n609 5255.741050 0.000145 EFOSC on NTT Dragomir et al. (2011), from Hellier et al. (2011)\n614 5259.68459 0.000359 0.40m - Observatorio Cerro Armazones:2Colque (TRESCA) re-analysed by Petrucci et al. (2020)\n632 5273.88282 0.00062 0.25m - Rarotonga Observatory:3P. Evans (TRESCA)\n664 5299.12768 0.00055 0.24m - YSVP Observatory:4I. Curtis (TRESCA)\n687 5317.27076 0.00006 ASTEP 400 Telescope Abe et al. (2013)\n709 5334.62540 0.00021 DFOSC on 1.54m - Danish Telescope Mancini et al. (2013)\n714 5338.56929 0.00023 TRAPPIST Lendl et al. (2013)\n733 5353.55659 0.00024 DFOSC on 1.54m Danish Telescope Mancini et al. (2013)\n746 5363.81131 0.00041 0.30m - MAGS Observatory:5G. Milne (TRESCA)\n752 5368.54285 0.00212 DFOSC on 1.54m - Danish Telescope Mancini et al. (2013)\n969 5539.72329 0.00030 EulerCam on Euler-Swiss Lendl et al. (2013)\n1007 5569.69827 0.00036 TRAPPIST Lendl et al. (2013)\n1021 5580.74124 0.00058 1.0m - CTIO:8Dragomir et al. (2011)\n1024 5583.10979 0.00089 0.24m - YSVP Observatory:4I. Curtis (TRESCA)\n1026 5584.68693 0.00024 TRAPPIST Lendl et al. (2013)\n1026 5584.68685 0.00019 EulerCam on Euler-Swiss Lendl et al. (2013)\n1038 5594.15188 0.00168 0.30m - PEST:6TG Tan re-analysed by Mancini et al. (2013)\n1047 5601.25164 0.00071 D0.30m - PEST:6TG Tan (TRESCA) re-analysed by Mancini et al. (2013)\n1049 5602.83138 0.00046 TRAPPIST Lendl et al. (2013)\n1052 5605.19414 0.00180 0.30m - PEST:6TG Tan (TRESCA) re-analysed by Mancini et al. (2013)\n1054 5606.77464 0.00022 EulerCam on Euler-Swiss Lendl et al. (2013)\n1055 5607.56241 0.00033 TRAPPIST Lendl et al. (2013)\n1074 5622.55057 0.00026 TRAPPIST Lendl et al. (2013)\n1076 5624.12787 0.00142 DFOSC on 1.54m Danish Telescope Mancini et al. (2013)\n1087 5632.80612 0.00025 EulerCam on Euler-Swiss Lendl et al. (2013)\n1116 5655.68222 0.00045 TRAPPIST Lendl et al. (2013)\n1135 5670.66976 0.00064 TRAPPIST Lendl et al. (2013)\n1144 5677.77038 0.00195 0.30m - PEST:6TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)\n1158 5688.81201 0.00333 0.30m - PEST:6TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)\n1159 5689.60276 0.00030 DFOSC on 1.54m Danish Telescope Mancini et al. (2013)\n1163 5692.75674 0.00255 0.30m - PEST:6TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)\n1164 5693.54639 0.00013 DFOSC on 1.54m Danish Telescope Mancini et al. (2013)\n1177 5703.79933 0.00411 0.30m - PEST:6TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)\n1178 5704.59078 0.00034 DFOSC on 1.54m Danish Telescope Mancini et al. (2013)\n1183 5708.53495 0.00015 DFOSC on 1.54m Danish Telescope Mancini et al. (2013)\n1228 5744.032190 0.000040 WFC3 on HST Mandell et al. (2013)\n1254 5764.54014 0.00125 CTIO 1m:8Cortés-Zuleta et al. (2020), analyzed by TLCM\n1270 5777.16366 0.00022 IRAC on Spitzer Wong et al. (2016)\n1283 5787.41863 0.00023 IRAC on Spitzer Wong et al. (2016)\n1409 5886.81234 0.00208 0.30m - PEST:6TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)\n1421 5896.27611 0.00210 0.30m - PEST:6TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)\n1445 5915.20980 0.00065 0.30m - PEST:6TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)\n1450 5919.15485 0.00103 0.30m - PEST:6TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)\n1454 5922.30966 0.00555 0.30m - PEST:6TG Tan (TRESCA) re-analyzed by Mancini et al. (2013)\n1552 5999.616301 0.00007 MMIRS on the Magellan II Telescope Bean et al. (2013)\n1580 6021.703740 0.000085 MMIRS on the Magellan II Telescope Bean et al. (2013)\n1595 6033.53645 0.00014 GROND:7Mancini et al. (2013)\n1595 6033.53651 0.00007 GROND:7Mancini et al. (2013)\n1595 6033.53643 0.00007 GROND:7Mancini et al. (2013)\n1595 6033.53652 0.00009 GROND:7Mancini et al. (2013)\n1590 6029.59251 0.00035 EulerCam on Euler-Swiss Lendl et al. (2013)\n1633 6063.51175 0.00030 TRAPPIST Lendl et al. (2013)\n1977 6334.87208 0.00053 0.25m - Rarotonga Observatory:3P. Evans (TRESCA)\n1996 6349.86037 0.00081 0.25m - Rarotonga Observatory:3P. Evans (TRESCA)\n2016 6365.63794 0.00010 DFOSC on 1.54m Danish Telescope Cortés-Zuleta et al. (2020), analyzed by TLCM\n2048 6390.87994 0.00043 0.25m - Rarotonga Observatory:3P. Evans (TRESCA)\n2064 6403.50164 0.00022 DFOSC on 1.54m Danish Telescope Cortés-Zuleta et al. (2020), analyzed by TLCM\n2465 6719.82642 0.00050 0.25m - Rarotonga Observatory:3P. Evans (TRESCA)\n2490 6739.551364 0.000046 IMACS on Magellan I Telescope Espinoza et al. (2019)\n2490 6739.54756 0.00029 0.30m - FRAM:9M. Mašek, K. Ho ˇnková, J. Juryšek (TRESCA)\nTable C.1. Transits found in the literature: cycle number, referred to in Hebb et al. (2010) - mid-transit point in BJD TDB, its error, details on\nthe observations, and reference. The time unit of each mid-transit time was checked and uniformized to BJD TDB. Some digits are scientifically\ninsignificant, but are left the same as in the reference papers.\n:1Beta Orionis Observatory, Argentina\n:2Cerro Armazones Observatory, Argentina\n:3Rarotonga Observatory, Cook Islands, New Zealand\n:4YSVP Observatory, South Australia, Australia\n:5Mount Albert Grammar School Observatory, New Zealand\n:6Perth Exoplanet Survey Telescope, West Australia, Australia\n:7Gamma Ray Burst Optical and Near-Infrared Detector on MPG /ESO, La Silla, Chile\n:8Cerro Tololo Inter-American Observatory, Chile\n:9F/Photometric Robotic Atmospheric Monitor telescope, Pierre Auger Observatory, Argentina, Ebr et al. (2014)\nArticle number, page 17 of 22A&A proofs: manuscript no. main\nCycle number T mid[days] Uncertainty [days] Observations Reference\nN BJD TDB- 2 450 000\n2532 6772.67789 0.00052 TRAPPIST-South Patra et al. (2020)\n2537 6776.625805 0.000066 IMACS on Magellan I Telescope Espinoza et al. (2019)\n2792 6977.77653 0.00019 FORS2 on VLT Sedaghati et al. (2015)\n2930 7086.63573 0.00020 DFOSC on 1.54m Danish Telescope Cortés-Zuleta et al. (2020), analyzed by TLCM\n3044 7176.565524 0.000094 IMACS on Magellan I Telescope Espinoza et al. (2019)\n3351 7418.736819 0.000234 FORS2 on VLT Sedaghati et al. (2017) re-analyzed by Petrucci et al. (2020)\n3389 7448.712917 0.0000766 FORS2 on VLT Sedaghati et al. (2017) re-analyzed by Petrucci et al. (2020)\n3446 7493.67653 0.00024 DFOSC on 1.54m - Danish Telescope Cortés-Zuleta et al. (2020), analyzed by TLCM\n3477 7518.131796 0.000380 0.30m - PEST:6Ivshina & Winn (2022) from Petrucci et al. (2020), TG Tan (TRESCA)\n3513 7546.53025 0.00038 TRAPPIST-South Patra et al. (2020)\n3513 7546.52975 0.0002759 1.54m - EABA:10Ivshina & Winn (2022) from Petrucci et al. (2020)\n3518 7550.472701 0.0004477 1.54m - EABA:10Ivshina & Winn (2022) from Petrucci et al. (2020)\n3551 7576.504698 0.0007447 0.30m - FRAM:9M. Mašek, K. Ho ˇnková, J. Juryšek (TRESCA),\nre-analyzed by Ivshina & Winn (2022) from Petrucci et al. (2020)\n3739 7724.807825 0.000828 2.15m - CASLEO:11E. Fernández-Lajús, R. P. Di Sisto (TRESCA), re-analysed by Petrucci et al. (2020)\n3758 7739.794151 0.000537 1.54m - EABA:10Petrucci et al. (2020)\n3806 7777.657426 0.001404 2.15m - CASLEO:11Petrucci et al. (2020)\n3815 7784.758917 0.000176 1.54m - EABA:10Petrucci et al. (2020)\n3825 7792.647526 0.000759 0.30m -FRAM:9M. Mašek, K. Ho ˇnková, J. Juryšek (TRESCA), re-analysed by Petrucci et al. (2020)\n3830 7796.591396 0.000316 0.30m - FRAM:9M. Mašek, K. Ho ˇnková, J. Juryšek (TRESCA), re-analysed by Petrucci et al. (2020)\n3830 7796.592241 0.000061 IMACS on Magellan I Telescope Espinoza et al. (2019)\n3834 7799.74616 0.00013 DFOSC on 1.54m - Danish Telescope Cortés-Zuleta et al. (2020), analysed by TLCM\n3896 7848.655592 0.000068 IMACS on Magellan I Telescope Espinoza et al. (2019)\n3901 7852.59809 0.00028 DFOSC on 1.54m - Danish Telescope Cortés-Zuleta et al. (2020), analysed by TLCM\n3906 7856.543843 0.000111 IMACS on Magellan I Telescope Espinoza et al. (2019)\n3906 7856.542283 0.000524 0.30m - FRAM:9M. Mašek, K. Ho ˇnková, J. Juryšek (TRESCA), re-analysed by Petrucci et al. (2020)\n4127 8030.87615 0.00213 DFOSC on 1.54m - Danish Telescope Cortés-Zuleta et al. (2020), analysed by TLCM\n4189 8079.785673 0.0002142 1.54m - EABA:10Petrucci et al. (2020)\n4227 8109.763837 0.001527 2.15m - CASLEO:11Petrucci et al. (2020)\n4359 8213.887473 0.000243 0.30m - Ngileah Observatory:12Petrucci et al. (2020) from C. Knight (TRESCA)\n4757 8527.845759 0.000453 2.15m - CASLEO:11Petrucci et al. (2020)\n4778 8544.41145 0.0004479 TESS Ivshina & Winn (2022)\n4779 8545.19907 0.0003870 TESS Ivshina & Winn (2022)\n4780 8545.98902 0.0004050 TESS Ivshina & Winn (2022)\n4781 8546.77731 0.0004406 TESS Ivshina & Winn (2022)\n4782 8547.56698 0.0004338 TESS Ivshina & Winn (2022)\n4783 8548.35489 0.0004466 TESS Ivshina & Winn (2022)\n4784 8549.14427 0.0005019 TESS Ivshina & Winn (2022)\n4784 8549.144717 0.000534 0.32m - Bathurst Observatory:13Petrucci et al. (2020) from A. Wünsche (TRESCA)\n4785 8549.93223 0.0004417 TESS Ivshina & Winn (2022)\n4786 8550.72120 0.0004841 TESS Ivshina & Winn (2022)\n4787 8551.50999 0.0004451 TESS Ivshina & Winn (2022)\n4788 8552.29907 0.0004410 TESS Ivshina & Winn (2022)\n4789 8553.08782 0.0005204 TESS Ivshina & Winn (2022)\n4790 8553.87694 0.0004185 TESS Ivshina & Winn (2022)\n4791 8554.66562 0.0004163 TESS Ivshina & Winn (2022)\n4792 8555.45470 0.00011 TESS Wong et al. (2020)\n4795 8557.82154 0.0005118 TESS Ivshina & Winn (2022)\n4796 8558.60951 0.0005083 TESS Ivshina & Winn (2022)\n4797 8559.39917 0.0004697 TESS Ivshina & Winn (2022)\n4798 8560.18675 0.0004762 TESS Ivshina & Winn (2022)\n4799 8560.97736 0.0004474 TESS Ivshina & Winn (2022)\n4800 8561.76555 0.0004190 TESS Ivshina & Winn (2022)\n4801 8562.55365 0.0004069 TESS Ivshina & Winn (2022)\n4802 8563.34279 0.0004283 TESS Ivshina & Winn (2022)\n4803 8564.13199 0.0004019 TESS Ivshina & Winn (2022)\n4804 8564.92114 0.0005022 TESS Ivshina & Winn (2022)\n4805 8565.70943 0.0004585 TESS Ivshina & Winn (2022)\n4806 8566.49870 0.0003981 TESS Ivshina & Winn (2022)\n4807 8567.28776 0.0004885 TESS Ivshina & Winn (2022)\n4808 8568.07637 0.0004006 TESS Ivshina & Winn (2022)\n5714 9282.76417 0.0003798 TESS Ivshina & Winn (2022)\n5715 9283.55339 0.0004562 TESS Ivshina & Winn (2022)\n5716 9284.34253 0.0005261 TESS Ivshina & Winn (2022)\n5717 9285.13026 0.0004526 TESS Ivshina & Winn (2022)\n5718 9285.91981 0.0004843 TESS Ivshina & Winn (2022)\n5719 9286.70858 0.0004234 TESS Ivshina & Winn (2022)\n5720 9287.49712 0.0004700 TESS Ivshina & Winn (2022)\n5721 9288.28588 0.0004688 TESS Ivshina & Winn (2022)\n5722 9289.07500 0.0004201 TESS Ivshina & Winn (2022)\n5723 9289.86356 0.0003895 TESS Ivshina & Winn (2022)\n5724 9290.65264 0.0004974 TESS Ivshina & Winn (2022)\n5725 9291.44097 0.0004855 TESS Ivshina & Winn (2022)\n5726 9292.23008 0.0004141 TESS Ivshina & Winn (2022)\nTable C.2. Continuation of Table C.1.\n:9F/Photometric Robotic Atmospheric Monitor telescope, Pierre Auger Observatory, Argentina\n:10Estación Astrofísica de Bosque Alegre, Argentina\n:11Complejo Astronómico El Leoncito, Argentina\n:12Ngileah Observatory, New Zealand\n:13Bathurst Observatory, New South Wales, Australia\nArticle number, page 18 of 22Bernabò et al.: Periastron precession of WASP-19Ab\nCycle number T mid[days] Uncertainty [days] Observations Reference\nN BJD TDB- 2.450.000\n4824 8580.69724 0.00040 TRAPPIST-South Patra et al. (2020)\n4829 8584.641667 0.000268 1.54m - EABA:9Petrucci et al. (2020)\n5730 9295.38506316 0.0005686 TESS Rosário et al. (2022)\n5731 9296.17539 0.0004310 TESS Ivshina & Winn (2022)\n5732 9296.96307 0.0004743 TESS Ivshina & Winn (2022)\n5733 9297.75141 0.0005115 TESS Ivshina & Winn (2022)\n5734 9298.54007 0.0004491 TESS Ivshina & Winn (2022)\n5735 9299.32928 0.0004109 TESS Ivshina & Winn (2022)\n5736 9300.11829 0.0004992 TESS Ivshina & Winn (2022)\n5737 9300.90761 0.0004934 TESS Ivshina & Winn (2022)\n5738 9301.69654 0.0004158 TESS Ivshina & Winn (2022)\n5739 9302.48453 0.0004775 TESS Ivshina & Winn (2022)\n5740 9303.27451 0.0004810 TESS Ivshina & Winn (2022)\n5741 9304.06354 0.0004313 TESS Ivshina & Winn (2022)\n5742 9304.85186 0.0004499 TESS Ivshina & Winn (2022)\n6258 9711.89359 0.00034 0.30m - Ngileah Observatory:12C. Knight (TRESCA), analysed by TLCM\n6272 9722.93627 0.00084 0.30m - Ngileah Observatory:12C. Knight (TRESCA), analysed by TLCM\n6505 9906.734941 0.00058 0.43m - Deep Sky, Chile Y . Jongen (TRESCA)\n6141 9619.60147 0.00039 0.43m - Deep Sky, Chile J.-P. Vignes (TRESCA)\n6145 9622.756750 0.00042 0.43m - Deep Sky, Chile J.-P. Vignes (TRESCA)\n6557 9947.75209 0.00114 0.15m - Observatorio Vuelta por el Universo:14M. Anzola (TRESCA)\nTable C.3. Continuation of Table C.1.\n:10Estación Astrofísica de Bosque Alegre, Argentina\n:12Ngileah Observatory, New Zealand\n:14Observatorio Vuelta por el Universo, Argentina\nCycle number T mid[days] Uncertainty [days] Observations Reference\nN BJD TDB- 2 450 000\n2 4777.313572`0.00013\n´0.00014HAWK-I (VLT) Anderson et al. (2010)\n185 4921.66849 0.00190 HAWK-I (VLT) Gibson et al. (2010)\n1018 5578.7684 0.0039 ULTRACAM on NTT Burton et al. (2012)\n1199 5721.5508`0.0028\n´0.0031DFOSC on 1.54m - Danish Telescope Mancini et al. (2013)\n1269 5776.77020 0.00083 IRAC on Spitzer Wong et al. (2016)\n1270 5777.55733 0.00092 IRAC on Spitzer Wong et al. (2016)\n1282 5787.02383 0.00077 IRAC on Spitzer Wong et al. (2016)\n1283 5787.81228 0.00076 IRAC on Spitzer Wong et al. (2016)\n1549 5997.6440 0.001 MMIRS on Magellan II telescope Bean et al. (2013)\n1582 6023.6763 0.001 MMIRS on Magellan II telescope Bean et al. (2013)\nTable C.4. Occultations found in the literature. The cycle number is referred to the previous transit (half phase before the occultation). Light travel\ntime across the system (around 16.3 s) is subtracted from all mid-occultation times.\nArticle number, page 19 of 22A&A proofs: manuscript no. main\nAppendix D: Tables of RVs Instr. ID BJD TDB RVobs Uncertainty\n´2 450 000.0 [km s´1] [km s´1]\n1 4616.46326 20.965 0.0218\n1 4623.46713 20.712 0.0289\n1 4624.46191 21.019 0.0220\n1 4652.46587 20.512 0.0210\n1 4653.46543 20.770 0.0231\n1 4654.46530 21.007 0.0219\n1 4656.47564 20.511 0.0193\n1 4657.46805 20.902 0.0386\n1 4658.46376 21.008 0.0302\n1 4660.46652 20.604 0.0400\n1 4661.46434 20.974 0.0217\n1 4662.46549 20.922 0.0191\n1 4663.46607 20.547 0.0188\n1 4664.46573 20.645 0.0362\n1 4665.46709 21.077 0.0305\n1 4827.74752 20.683 0.0188\n1 4832.74362 21.050 0.0242\n1 4833.66844 20.860 0.0212\n1 4834.67693 20.548 0.0191\n1 4837.67024 20.726 0.0227\n1 4838.68496 20.562 0.0244\n1 4839.70064 20.936 0.0185\n1 4890.61491 20.637 0.0187\n1 4894.68743 20.518 0.0290\n1 4895.69238 20.894 0.0179\n1 4896.66180 21.041 0.0145\n1 4897.65715 20.674 0.0175\n1 4898.66015 20.544 0.0197\n1 4939.53373 20.578 0.0164\n1 4940.52747 20.642 0.0154\n1 4941.53928 20.996 0.0156\n1 4942.52421 20.899 0.0169\n1 4943.53594 20.559 0.0191\n1 4944.52955 20.798 0.0155\n2 4972.4990 20.8874 0.0176\n2 4973.4656 20.6042 0.0182\n2 4999.4865 20.5571 0.0341\n3 5242.6892 20.7469 0.0036\n3 5242.8465 21.0203 0.0050\n3 5243.6598 21.0391 0.0042\n3 5243.8250 20.9547 0.0043\n3 5244.7229 20.7355 0.0031\n3 5247.7159 21.0254 0.0088\n3 5272.5453 20.5741 0.0044\n3 5272.7784 20.9778 0.0052\n3 5274.5336 21.0391 0.0073\n3 5274.6155 20.9318 0.0124\n3 5274.6232 20.9129 0.0112\n3 5274.6307 20.8819 0.0094\n3 5274.6383 20.8842 0.0088\n3 5274.6459 20.8711 0.0084\n3 5274.6533 20.8914 0.0086\n3 5274.6610 20.8473 0.0094\n3 5274.6687 20.8214 0.0100\n3 5274.6762 20.7819 0.0108\n3 5274.6838 20.7568 0.0118\n3 5274.6915 20.7315 0.0108\n3 5274.6991 20.7138 0.0103\nTable D.1. Radial velocity of WASP-19Ab system. The indices of the\nobservations are the same as in Fig. 1 and Sect. 4.\nArticle number, page 20 of 22Bernabò et al.: Periastron precession of WASP-19Ab\nInstr. ID BJD TDB RVobs Uncertainty\n´2 450 000.0 [km s´1] [km s´1]\n3 5274.7066 20.7262 0.0106\n3 5274.7142 20.7093 0.0104\n3 5274.7218 20.7151 0.0118\n3 5274.7296 20.7056 0.0120\n3 5274.7371 20.6582 0.0116\n3 5274.7446 20.6687 0.0104\n3 5274.7522 20.6387 0.0119\n3 5274.7599 20.6452 0.0113\n3 5274.7675 20.6136 0.0105\n3 5274.7748 20.6079 0.0106\n3 5274.7826 20.5771 0.0123\n3 5274.8224 20.5707 0.0069\n3 5275.5229 20.6841 0.0052\n3 5275.7903 20.6841 0.0077\n3 5276.5154 20.5850 0.0062\n4 5338.47873 0.13439 0.01522\n4 5338.48498 0.11220 0.01214\n4 5338.49280 0.09687 0.01226\n4 5338.50068 0.06777 0.01130\n4 5338.50859 0.05913 0.01108\n4 5338.51623 0.04541 0.01068\n4 5338.52418 0.00975 0.01071\n4 5338.53215 0.00564 0.01140\n4 5338.53993 0.00808 0.01054\n4 5338.54787 0.06322 0.01216\n4 5338.55585 -0.00877 0.01121\n4 5338.56351 -0.01221 0.01281\n4 5338.57167 -0.05589 0.01214\n5 6018.8117983 -0.168663 0.003077\n5 6290.1405875 -0.137606 0.003032\n5 6318.9215917 0.306274 0.002318\n6 8498.59068874 20.82646 0.00818\n6 8498.60180095 20.7993 0.00716\n6 8498.61279395 20.7785 0.00819\n6 8498.62459247 20.76231 0.00651\n6 8498.63454273 20.76321 0.0145\n6 8498.63972473 20.77791 0.01274\n6 8498.64464221 20.79349 0.01338\n6 8498.64956116 20.73967 0.01424\n6 8498.65436899 20.69906 0.01462\n6 8498.65942065 20.69273 0.01402\n6 8498.66430660 20.67341 0.01475\n6 8498.66926300 20.63671 0.01324\n6 8498.67428582 20.61285 0.01011\n6 8498.67907061 20.60608 0.00899\n6 8498.68407176 20.61570 0.00897\n6 8498.68899962 20.61782 0.00781\n6 8498.69387378 20.62464 0.00745\n6 8498.70087685 20.61316 0.00457\n6 8498.71044096 20.60040 0.00456\n6 8498.71956719 20.59894 0.00435\n6 8498.72920549 20.56458 0.00439\n6 8498.73862103 20.55429 0.00399\n6 8546.70109917 20.81404 0.00542\n6 8546.71810796 20.77644 0.00864\n6 8546.73102276 20.74880 0.00911\n6 8546.74715133 20.74357 0.00749\n6 8546.75677225 20.74175 0.01771\n6 8546.76244128 20.70465 0.00183\n6 8546.76773966 20.74182 0.02089\n6 8546.77334046 20.66661 0.01599\n6 8546.77859110 20.65585 0.01502\nTable D.2. Continuation of Table D.1.Instr. ID BJD TDB RVobs Uncertainty\n´2 450 000.0 [km s´1] [km s´1]\n6 8546.78395751 20.66390 0.01514\n6 8546.78909247 20.57827 0.02002\n6 8546.79475258 20.59739 0.01890\n6 8546.80036398 20.55623 0.02008\n6 8546.80553459 20.63033 0.01792\n6 8546.81107313 20.62332 0.01776\n6 8546.81638670 20.59751 0.01665\n6 8546.82327218 20.61843 0.01500\n6 8546.83186421 20.50721 0.01337\n6 8546.84018927 20.57332 0.01128\n6 8546.84855560 20.55209 0.01556\n6 8546.85730223 20.51568 0.01358\n6 8565.63006930 20.85050 0.00419\n6 8565.64050392 20.83263 0.00415\n6 8565.65068188 20.81295 0.00392\n6 8565.66089612 20.79194 0.00431\n6 8565.67121537 20.76472 0.00420\n6 8565.68117929 20.77662 0.00364\n6 8565.68892203 20.78292 0.00850\n6 8565.69398195 20.76429 0.00868\n6 8565.69899674 20.74261 0.00836\n6 8565.70399522 20.72989 0.00813\n6 8565.70895259 20.68067 0.00844\n6 8565.71404306 20.66203 0.00896\n6 8565.71903842 20.64335 0.00830\n6 8565.72397488 20.61127 0.00858\n6 8565.72902717 20.61737 0.00865\n6 8565.73403920 20.61288 0.00769\n6 8565.73901729 20.62037 0.00740\n6 8565.74406099 20.61360 0.00733\n6 8565.75100246 20.60935 0.00391\n6 8565.75992839 20.59612 0.00389\n6 8565.76886384 20.57339 0.00399\n6 8565.77776409 20.56269 0.00403\n6 8565.78359563 20.54291 0.01537\n7 8860.65868965 20.80633 0.00247\n7 8860.66953784 20.78622 0.00219\n7 8860.68086427 20.77083 0.00224\n7 8860.69232959 20.75252 0.00205\n7 8860.70329540 20.73853 0.00201\n7 8860.71235224 20.76021 0.00306\n7 8860.71764990 20.75988 0.00329\n7 8860.72296232 20.74915 0.00298\n7 8860.72842374 20.72062 0.00296\n7 8860.73374936 20.68917 0.00305\n7 8860.73902683 20.65960 0.00289\n7 8860.74434671 20.62669 0.00289\n7 8860.74960873 20.60702 0.00300\n7 8860.75500031 20.59131 0.00293\n7 8860.76033814 20.59485 0.00285\n7 8860.76571960 20.60629 0.00280\n7 8860.77096557 20.60589 0.00288\n7 8860.77847463 20.59486 0.00195\n7 8860.78808536 20.57681 0.00200\n7 8860.79770303 20.56175 0.00208\n7 8860.80730381 20.54011 0.00213\n7 8860.81678339 20.52403 0.00211\n8 8885.81831 21.05010 0.00590\n8 8886.63492 21.04724 0.00588\n8 8886.74637 20.91119 0.00495\nTable D.3. Continuation of Table D.2.\nArticle number, page 21 of 22A&A proofs: manuscript no. main\nInstr. ID BJD TDB RVobs Uncertainty\n´2 450 000.0 [km s´1] [km s´1]\n8 8887.70434 20.62644 0.01004\n8 8887.78381 20.61007 0.00596\n8 8888.61415 20.65507 0.00593\n8 8888.65498 20.69769 0.00576\n8 8889.67038 21.11750 0.00694\n8 8889.82412 21.04723 0.00663\n8 8893.66960 21.10599 0.01208\n8 8893.76426 21.04400 0.00549\n8 8893.83745 20.91112 0.00615\n8 8894.63550 20.90340 0.00821\n8 8894.69107 20.75424 0.00781\n8 8894.77058 20.62550 0.00732\n8 8897.68716 21.06761 0.00712\n8 8897.75751 20.97271 0.00668\n8 8899.85008 21.01457 0.00592\nTable D.4. Continuation of Table D.3.\nArticle number, page 22 of 22"    },    {        "title": "2402.12903v1.Inverse_problems_for_semilinear_Schrödinger_equations_at_large_frequency_via_polynomial_resolvent_estimates_on_manifolds.pdf",        "content": "arXiv:2402.12903v1  [math.AP]  20 Feb 2024INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER\nEQUATIONS AT LARGE FREQUENCY VIA POLYNOMIAL\nRESOLVENT ESTIMATES ON MANIFOLDS\nKATYA KRUPCHYK, SHIQI MA, SUMAN KUMAR SAHOO, MIKKO SALO,\nAND SIMON ST-AMANT\nAbstract. We study inverse boundary problems for semilinear Schr¨ odinger\nequations on smooth compact Riemannian manifolds of dimensions ≥2 with\nsmooth boundary, at a large fixed frequency. We show that certa in classes of\ncubic nonlinearities are determined uniquely from the knowledge of th e non-\nlinear Dirichlet–to–Neumann map at a large fixed frequency on quite g eneral\nRiemannian manifolds. In particular, in contrast to the previous res ults avail-\nable, here the manifolds need not satisfy any product structure, may have\ntrapped geodesics, and the geodesic ray transform need not be in jective. Only\na mild assumption about the geometry of intersecting geodesics is re quired.\nWe also establish a polynomial resolvent estimate for the Laplacian on an arbi-\ntrary smooth compact Riemannian manifold without boundary, valid f or most\nfrequencies. This estimate, along with the invariant construction o f Gaussian\nbeam quasimodes with uniform bounds for underlying constants and a station-\nary phase lemma with explicit control over all involved constants, co nstitutes\nthe key elements in proving the uniqueness results for the consider ed inverse\nproblems.\n1.Introduction and statement of results\nThe anisotropic Calder´ on problem seeks to determine the electrica l conductivity\nmatrix of a medium by performing electrical measurements along its b oundary,\nsee [5, 46]. To state the geometric version of this problem, let ( M,g) be a smooth\noriented compact Riemannian manifold of dimension n≥2 with smooth bound-\nary. Associated to the metric g, we have the Laplace operator −∆ggiven in local\ncoordinates by\n−∆g=−n/summationdisplay\nj,k=1|g|−1/2∂xj(|g|1/2gjk∂xk),\nwhere (gjk) = (gjk)−1and|g|= det(gjk). Let us introduce the Cauchy data set\non∂Mof harmonic functions on M,\nCg={(u|∂M,∂νu|∂M) :u∈C∞(M) such that −∆gu= 0 inMint},\n12 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nwhereνistheunitouternormaltotheboundaryof MandMint=M\\∂Mstands\nfor the interior of M. In the geometric version of the anisotropic Calder´ on prob-\nlem, one wishes to recover the Riemannian manifold ( M,g) from the knowledge\nof the Cauchy data set Cgon∂M. BecauseCψ∗g=Cgfor any smooth diffeomor-\nphismψ:M→Msuch thatψ|∂M= Id, see [29], one can only hope to recover\na Riemannian manifold from Cgup to this isometry. In dimension n= 2, this\nproblem, with an additional obstruction arising from the conformal invariance of\nthe Laplacian, is solved in [34] for the case of Riemannian metrics on bo unded\ndomains in the plane, and in [27] for the case of Riemannian surfaces. In dimen-\nsionsn≥3, the anisotropic Calder´ on problem is widely open. It is solved for\nreal analytic manifolds in [29, 27, 26], see also [16], and in the C∞case in [8, 10],\nfor metrics in a fixed conformal class on a conformally transversally anisotropic\n(CTA) manifold, assuming that the geodesic X-ray transform on th e transversal\nmanifold is injective. In the latter case, the anisotropic Calder´ on in verse problem\ncan be reduced to an inverse boundary problem of recovery of the potential in the\nSchr¨ odinger equation from the knowledge of the Cauchy data of it s solutions on\ntheboundary. Theapproachoftheworks [8, 10] relies onaconstr uction ofspecial\nsolutions to the Schr¨ odinger equation, called complex geometric op tics solutions.\nThe requirement that the manifold should be CTA guarantees the ex istence of\nso-called limiting Carleman weights on such manifolds, which are importa nt to\nconstruct such solutions using the technique of Carleman estimate s. However, a\ngeneric manifold of dimension n≥3 does not admit limiting Carleman weights,\nsee [30, 1]. Also, there are examples of manifolds for which the geode sic X-ray\ntransform is non-injective, see [20, Example 1.7]. Thus, it is of great significance\nto try to remove the assumption of the injectivity of the geodesic X -ray trans-\nform on the transversal manifold in the case of CTA manifolds, or ev en to try\nto remove the assumption that the manifold is CTA altogether when s olving the\nanisotropic Calder´ on problem.\nIn this direction, the works [25, 14, 13, 21] demonstrated that wh en solving\ninverse boundary problems for semilinear Schr¨ odinger equations, no assumption\non the transversal manifold is required, while still keeping the assum ption that\nthe manifold should enjoy a CTA structure, see also [23, 32, 6, 11].\nThe recent work [47] demonstrated that the recovery of certain classes of po-\ntentials in the Schr¨ odinger equation at a large fixed frequency did n ot require\nthe manifold to be CTA. The work [31] extended the result of [47] and showed\nthat the manifold only needed to be non-trapping, and the geodesic X-ray trans-\nform needed to be stably invertible and continuous. We refer to the recent work\n[42], where inverse problems of the recovery of connections from t he Dirichlet–\nto–Neumann maps at a large fixed frequency were studied.\nMotivated by [47, 31], the purpose of this paper is to show that an inv erse bound-\nary problem for a semilinear Schr¨ odinger equation for a certain clas s of cubicINVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 3\nnonlinearities, at a large fixed frequency, can be solved in quite gene ral geome-\ntries. Specifically, we do not require our manifold to be non-trapping and we do\nnot need any assumption related to the geodesic X-ray transform on it. Only a\nmild assumption about the geometry of geodesics is needed.\nTo state the inverse problem, let λ≥0 be a frequency and let us consider the\nDirichlet problem for the semilinear Schr¨ odinger operator,/braceleftBigg\n−∆gu−λ2u+qu3= 0 inMint,\nu=f on∂M,(1.1)\nwhereq∈C0,α(M) with some 0 < α <1. Here and in what follows Ck,α(M),\nk∈N, stands for the H¨ older space on M. Assume that\n(A)λ2is not a Dirichlet eigenvalue of −∆gonM.\nItfollowsfrom[21, Theorem B.1], seealso[14, Proposition1]and[25, P roposition\n2.1], that under the assumption (A), there is δ >0 andC >0 such that when\nf∈Bδ(∂M) :={f∈C2,α(∂M) :/ba∇dblf/ba∇dblC2,α(∂M)< δ}, the problem (1.1) has a\nunique solution u=uf∈C2,α(M) satisfying /ba∇dblu/ba∇dblC2,α(M)<Cδ. Associated to the\nproblem (1.1), we define the Dirichlet–to–Neumann map at the frequ encyλas\nfollows\nΛλ\nqf=∂νuf|∂M,\nwheref∈Bδ(∂M). The inverse problem that we areinterested in isto determine\nqfrom the knowledge of the Dirichlet–to–Neumann map Λλ\nqfor a large but fixed\nfrequencyλ.\nFirst, motivated by [9, Definition 1.2] and [20, Definition 2.1], we impose t he\nfollowing geometric condition on the manifold ( M,g):\n(H) for every point x0∈Mint, there are two non-tangential unit speed geode-\nsics, passing through x0, such that they do not have self-intersections at\nx0andx0is their only point of intersection.\nOur first main result is as follows, where |·|denotes the Lebesgue measure on R.\nTheorem 1.1. Let(M,g)be a smooth compact oriented Riemannian manifold\nof dimension n≥2with smooth boundary, satisfying the condition (H). Let\n0< α <1and letq1,q2∈C0,α(M). Then for any δ >0, there exists a set\nJ⊂[1,∞),J=J(M,g,δ), satisfying |J| ≤δ, andλ0=λ0(M,g,δ,q 1−q2)>0\nsuch that if for some λ≥λ0,λ /∈J, andλsatisfying the assumption (A), we\nhaveΛλ\nq1= Λλ\nq2thenq1=q2inM.\nRemark 1.2. In contrast to most prior results concerning elliptic parti al differ-\nential equations in dimensions n≥3, the manifolds considered in Theorem 1.1\nare not limited to having any specific product structure. The y may contain trapped\ngeodesics, and the geodesic X-ray transform is not required to be injective.4 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nLet us proceed to make some comments regarding the geometric as sumption (H).\nExample 1.3. If the manifold (M,g)is simple, i.e. a compact simply connected\nRiemannian manifold with strictly convex boundary so that n o geodesic has con-\njugate points, then the geometric assumption (H) holds. Thi s follows directly\nfrom the fact that the exponential map expxis a diffeomorphism onto Mfor any\nx∈M[35], and hence any two geodesics expx(tv)andexpx(tw)wherev/\\e}atio\\slash=w,\n|v|=|w|= 1, will satisfy (H).\nExample 1.4. If the manifold (M,g)satisfies the strict Stefanov–Uhlmann reg-\nularity condition at every point (x0,ξ0)∈S∗Mint, see[20, Definition 1.8] , then\nthe assumption (H) holds, see [9, Lemma 3.1] . As an example of such manifolds,\nconsiderMto be the closure of a neighborhood of a geodesic arc from the n orth\npole to the south pole of the unit sphere S3⊂R4, see[9]and[20, Example 1.10] .\nNote that the manifold Mcontains conjugate points so it is not simple.\nExample 1.5. LetM=S1×[0,a],a>0, be a cylinder with its usual flat metric\ng. The geodesics on Mare straight lines, circular cross sections, and helices th at\nwind around the cylinder. It is shown in [20, Appendix A] that(M,g)satisfies\nthe assumption (H). Furthermore, the geodesic X-ray transf orm is not invertible\nonM, since the kernel contains functions of the form f(eit,s) =h(s)where\nh∈C∞\n0((0,a))integrates to zero over [0,a], and(M,g)has trapping.\nWe shall next propose a geometric condition on the manifold ( M,g) which guar-\nantees that the assumption (H) holds for almost every point in Mint, and which\nmight be of independent interest. To motivate the proposed geome tric condition,\nwe first note that the fact that all the geodesics on spheres inter sect twice can be\nattributed to the high order of conjugacy of the sphere Sn. To state our result, we\nshall introduce some definitions, following [28, Section 10], [7, Chapt er 5, Section\n3]. Given a geodesic γ:I→M, and two points p=γ(a),q=γ(b),a,b∈I, we\nsay thatpandqare conjugate along γif there is a Jacobi field along γvanishing\natt=aandt=bbut not identically zero. The order of conjugacy of prelative\ntoqalongγdenoted conjγ(p,q), is the dimension of the space of Jacobi fields\nalongγthat vanish at aandb. Note that conjγ(p,q)≤n−1, and that equality\nis achieved on the sphere Snby takingpandqto be antipodal points, and γany\ngreat circle connecting them, see [28, Section 10, page 299]. We defi ne the order\nof conjugacy of p∈Mas the maximal value of conjγ(p,q) over all other points\nq∈Mand geodesics γconnecting ptoq. Similarly, the order of conjugacy of\nMis the maximal order of conjugacy of all points in M. We have the following\nresult.\nTheorem 1.6. Let(M,g)be a smooth compact Riemannian manifold of dimen-\nsionn≥3with smooth boundary. Suppose that the order of conjugacy of Mis\nat mostn−3. Then the assumption (H) holds for almost all x∈Mint.INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 5\nExample 1.7. Let(Mj,gj)be a smooth compact Riemannian manifold of di-\nmensionnj≥1without boundary, j= 1,2,3. IfM1has order of conjugacy k1\nandM2has order of conjugacy k2, thenM1×M2has order of conjugacy k1+k2.\nIndeed, ifJis a Jacobi field vanishing at p= (p1,p2)∈M1×M2, then\nJ(t) =d(exp(p1,p2))(tv1,tv2)(tw1,tw2) =d(expp1)tv1(tw1)⊕d(expp2)tv2(tw2)\ndecomposes as the direct sum of Jacobi fields J1andJ2onM1andM2vanishing\natp1andp2respectively, see [28, Proposition 10.10] . Hence,Jvanishes at q=\n(q1,q2)if and only if J1andJ2vanish atq1andq2respectively. Since the manifold\nMjhas order of conjugacy at most nj−1, the product of three manifolds M1×\nM2×M3has order of conjugacy at most n1+n2+n3−3, wheren1+n2+n3is the\ndimension of M1×M2×M3. Hence, any smooth compact Riemannian manifold\nwith smooth boundary that can be isometrically embedded as a codimension 0\nsubmanifold (also known as a regular domain) of a product of t hree manifolds\nM1×M2×M3satisfies the assumption of Theorem 1.6.\nNext, we let 0 <α<1 be fixed and following [31], for any nonzero p∈C0,α(M),\nwe introduce the frequency function N(p) by\nN(p) =/ba∇dblp/ba∇dblC0,α(M)\n/ba∇dblp/ba∇dblL∞(M),\nand we setN(p) = 0 ifp= 0. When B >0, we define the set A(B) of admissible\nperturbations by\nA(B) ={p∈C0,α(M) :N(p)≤B}.\nNote that for any q1,q2∈C0,α(M), there exists a constant B >0 such that\nq1−q2∈ A(B). Recall that in Theorem 1.1, the frequency λ0depends on the\ndifferenceq1−q2. Specifically, it follows from the proof of Theorem 1.1 that the\nvalue ofλ0is contingent on the following factors:\n•the constant B >0 such that q1−q2∈ A(B),\n•the specific point x0∈Mwhere supx∈M|q1(x)−q2(x)|is reached,\n•thechoiceoftwonon-tangentialgeodesicspassingthrough x0andfulfilling\nthe assumption (H), including their lengths and the angle between th em.\nOur next result aims to improve Theorem 1.1 by eliminating the depende ncy\nof the frequency λ0on a specific point x0and the particular choice of two non-\ntangentialgeodesicspassingthrough x0andsatisfyingassumption(H).Toachieve\nthis, we will impose the following geometric condition on the manifold ( M,g),\nwhich will replace the condition (H). To state this condition, we let ∂+SMbe a\npart of the boundary of the unit sphere bundle SM, consisting of inward vectors,\nsee Section 4.1 below. For ( x,w)∈∂+SM, we letγ=γx,w: [0,τ(x,w)]→Mbe\nthe unit speed geodesic such that γ(0) =x, and ˙γ(0) =w.6 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\n(H1) There are constants T >0, 0< θ0< π/2, 0< r <Inj(M)/2, and\nc0>0, such that for almost every point x0∈Mint, there exist two\nnon-tangential unit-speed geodesics γ=γx1,w1: [0,τ(x1,w1)]→Mand\nη=ηx2,w2: [0,τ(x2,w2)]→M, (x1,w1),(x2,w2)∈∂+SMsuch that\n(i)γ(t0) =η(τ0) =x0,t0∈(0,τ(x1,w1)),τ0∈(0,τ(x2,w2)),\n(ii) the length of γandηdoes not exceed T,\n(iii)γandηform an angle θatx0belonging to the interval [ θ0,π/2],\n(iv) ifd(γ(t),η(τ))< r, then|t−t0| ≤c0d(γ(t),η(τ)) and|τ−τ0| ≤\nc0d(γ(t),η(τ)).\nHere, Inj(M) denotes the injectivity radius of M, defined as follows. Let ( S,g)\nbe a closed extension of ( M,g), see [35, Lemma 3.1.8], and we assume that ( S,g)\nis fixed once for all. We write Inj( M) := Inj(S), and we have Inj( M)>0 asS\nis compact, see [28, Lemma 6.16]. We denote the Riemannian distance a sd(·,·).\nThe angle between γandηat the point x0is denoted by θ, where 0 ≤θ≤π,\nand cosθ=g(˙γ(t0),˙η(τ0)). Our next main result is as follows.\nTheorem 1.8. Let(M,g)be a smooth compact oriented Riemannian manifold\nof dimension n≥2with smooth boundary. Assume that (M,g)satisfies the\ncondition (H1) with some constants T >0,0<θ0<π/2,0<r<Inj(M)/2, and\nc0>0. LetB >0. Assume that q1,q2∈C0,α(M)are such that q1−q2∈ A(B).\nThen for any δ >0, there exists a set J⊂[1,∞),J=J(M,g,δ), satisfying\n|J| ≤δ, andλ0=λ0(M,g,δ,B,T,θ 0,r,c0)>0such that if for some λ≥λ0,\nλ /∈J, andλsatisfying the assumption (A), we have Λλ\nq1= Λλ\nq2thenq1=q2\ninM.\nLet us make some comments regarding the geometric assumption (H 1). The\nconditionr <Inj(M)/2 ensures that the geodesics γandηin assumption (H1)\ndonothaveself-intersections for |t−t0|<rand|τ−τ0|<r, respectively. Thefirst\nthree conditions are straightforward. As for condition (iv), it is a q uantitative\nway to state that the geodesics γandηintersect only at x0and they do not have\nself-intersections at x0. Proposition D.1 in Appendix D shows that condition (iv)\nlocally follows from condition (iii).\nExample 1.9. If the manifold (M,g)is simple then the geometric assumption\n(H1) holds, see Appendix D for the details.\nExample 1.10. The condition (H1) holds for the cylinder M=S1×[0,a],a>0,\nwith its usual flat metric, see Appendix D for the details.\nLet us next make some comments about the class of potentials in the cubic\nnonlinearity considered inTheorem 1.8. The assumption that p∈ A(B) issimilar\nto the assumption that the perturbation is angularly controlled in [37 , Theorem\n2] or horizontally controlled in [36]. In particular, the assumption p∈ A(B) is\nalways satisfied for some Bifplies in a finite-dimensional space. We refer toINVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 7\nAppendix C for an example of an infinite set of linearly independent adm issible\nperturbations.\nLetusproceedtodiscuss themainideasintheproofofTheorem1.1a ndTheorem\n1.8. The crucial ingredient in both proofs is the polynomial resolvent estimate\nfor the Laplacian on a smooth compact Riemannian manifold without bo undary,\nvalid for most frequencies, established in Theorem 2.1 below. This res olvent\nestimate could also be of independent interest. Its proof is quite sim ple and uses\nonly the self-adjoint resolvent bound, see (2.2), and a rough vers ion of the Weyl\nlaw, see (2.4). We refer to [24], see also [44, 43], for polynomial resolv ent bounds\nfor most frequencies in the black box scattering.\nOnce the polynomial resolvent estimates have been established, th e proof of The-\norem 1.1 proceeds as follows. First, using the third order linearizatio n of the\nproblem (1.1), we derive the integral identity (3.8) valid for four solu tions of the\nHelmholtz equation ( −∆g−λ2)v= 0 inMint. The construction of such solutions\nis based on Gaussian beam quasimodes to the Helmholtz equation. To c onvert\nthese approximate solutions into exact solutions, we rely on the solv ability re-\nsult which is a consequence of the polynomial resolvent estimate of T heorem 2.1.\nSubstituting the constructed solutions into the integral identity ( 3.8) and using\na rough stationary phase argument combined with the boundary de termination\nof the potential, allows us to complete the proof of Theorem 1.1.\nThe proof of Theorem 1.8 follows a similar approach to that of Theore m 1.1.\nHowever, we introduce a crucial new element that allows us to remov e the de-\npendence of the frequency λ0on a specific point x0and the specific choice of\ntwo non-tangential geodesics passing through this point. This cru cial element\ninvolves the invariant construction of Gaussian beam quasimodes, w ith uniform\nbounds for the underlying constants, as recently established in [31 , Theorem 6.2],\nsee also [42]. Additionally, we conduct a more thorough analysis of all c onstants\ninvolved during the proof.\nThe paper is organized as follows. Section 2 contains the proof of po lynomial\nresolvent estimates forthe Laplacianona smoothcompact Riemann ian manifold.\nThe proof of Theorem 1.1 is given in Section 3. Section 4 is devoted to t he proof\nof Theorem 1.8. The proof of Theorem 1.6 is presented in Section 5. A ppendix\nA discusses a standard rough version of the stationary phase lemm a required for\nthe proofs of Theorem 1.1 and Theorem 1.8. A boundary determinat ion of a\npotential from the integral identity (3.8), used in the proofs of Th eorem 1.1 and\nTheorem 1.8, is presented in Appendix B. Appendix C contains an exam ple of an\ninfinite set of linearly independent admissible perturbations. Finally, A ppendix\nD provides examples of manifolds that satisfy the assumption (H1).8 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\n2.Polynomial resolvent estimates for the Laplacian on a smoot h\ncompact Riemannian manifold\nLet(N,g)beasmoothcompactRiemannianmanifoldofdimension n≥2without\nboundary. Let −∆gbe the positive Laplace operator on N, associated with the\nmetricg. It is a self-adjoint operator on L2(N) with the domain H2(N), the\nstandardSobolevspaceon N, andithasadiscretespectrumSpec( −∆g)⊂[0,∞).\nWhenλ>0,λ2/∈Spec(−∆g), the resolvent ( −∆g−λ2)−1:L2(N)→L2(N) is\na bounded operator. Motivated by [24, Theorem 1.1, Theorem 3.3], we have the\nfollowing result establishing a polynomial resolvent estimate for the L aplacian on\nN, valid for most frequencies. In what follows we let |·|stand for the Lebesgue\nmeasure on R.\nTheorem 2.1. Givenδ >0andε >0, there exist C=C(N,g,δ,ε)>0and a\nsetJ=J(N,g,δ,ε)⊂[1,∞)with|J| ≤δsuch that\n/ba∇dbl(−∆g−λ2)−1/ba∇dblL(L2(N),L2(N))≤Cλn+ε, (2.1)\nfor allλ∈[1,∞)\\J.\nProof.We shall first consider the resolvent of the semiclassical Laplacian −h2∆g,\n0<h≤1. Whenz /∈Spec(−h2∆g), by the spectral theorem we have\n/ba∇dbl(−h2∆g−z)−1/ba∇dblL(L2(N),L2(N))=1\ndist(z,Spec(−h2∆g)).(2.2)\nWe shall first work on the frequency interval [1 ,2]. Letr(h)>0 be an arbitrary\nfunction of h∈(0,1]. Then it follows from (2.2) that\n/ba∇dbl(−h2∆g−z)−1/ba∇dblL(L2(N),L2(N))≤1\nr(h),\nfor allz∈[1,2]\\J(h), where\nJ(h) ={z∈[1,2] : dist(z,Spec(−h2∆g))<r(h)}.\nTo bound the measure of the set J(h), we observe that\nJ(h)⊂/uniondisplay\nzj(h)∈Spec(−h2∆g)∩[1,2](zj(h)−r(h),zj(h)+r(h)), (2.3)\nwherezj(h),j= 1,2,..., stand for the eigenvalues of −h2∆g. It follows from\nthe Weyl law that the number of the eigenvalues of −h2∆gon the interval [1 ,2],\ncounting with multiplicities, is given by\n#(Spec(−h2∆g)∩[1,2]) =O(h−n), (2.4)\nfor all 0<h≤1, see [48, Theorem 14.11]. Thus, it follows from (2.3), (2.4) that\n|J(h)| ≤/summationdisplay\nzj(h)∈Spec(−h2∆g)∩[1,2]2r(h)≤ O(h−n)r(h),INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 9\nfor all 0<h≤1. Lettingδ′>0 andε′>0 be fixed to be chosen later, we set\nr(h) =δ′hn+ε′\nso that\n|J(h)| ≤δ′O(hε′)→0, (2.5)\nash→0. Summarizing the discussion so far, we have shown that for all 0 <h≤\n1, there exists a subset J(h)⊂[1,2] satisfying (2.5) such that\n/ba∇dbl(−h2∆g−z)−1/ba∇dblL(L2(N),L2(N))≤(δ′)−1h−n−ε′, (2.6)\nfor allz∈[1,2]\\J(h).\nNext, we shall obtain the bound (2.1) for the nonsemiclassical reso lvent\n(−∆g−λ2)−1=h2(−h2∆g−h2λ2)−1,\nforλ∈[1,∞), outside some set of small measure, containing the spectrum of/radicalbig\n−∆g. In doing so, we shall follow the proof of [24, Theorem 3.3], and [41, p age\n767]. We write\n[1,∞) =∞/uniondisplay\nl=0[2l,2l+1).\nNow ifλ∈[1,∞) thenλ2∈[2l,2l+1) for some unique l= 0,1,2,..., and there-\nfore, 2−lλ2∈[1,2)⊂[1,2]. Letting h= 2−l/2∈(0,1], and using (2.6), we obtain\nthat\n/ba∇dbl(−∆g−λ2)−1/ba∇dblL(L2(N),L2(N))=h2/ba∇dbl(−h2∆g−h2λ2)−1/ba∇dblL(L2(N),L2(N))\n≤(δ′)−1h2−n−ε′≤(δ′)−12l/2(n+ε′−2)≤(δ′)−1λn−2+ε′,(2.7)\nfor allλ2∈[2l,2l+1)\\/tildewideJl. Here we let /tildewideJl:= 2lJ(2−l/2)∩[2l,2l+1). We set\n/tildewideJ:=∞/uniondisplay\nl=0/tildewideJl\nso that the bound (2.7) holds for all λ2∈[1,∞)\\/tildewideJ. Lettingε>0 be arbitrary\nfixed, choosing\nε′= 2+ε, (2.8)\nand using (2.5), we get\n|/tildewideJ| ≤∞/summationdisplay\nl=02l|J(2−l/2)| ≤ O(δ′)∞/summationdisplay\nl=0/parenleftbigg1\n2ε′/2−1/parenrightbiggl\n=O(δ′)∞/summationdisplay\nl=0/parenleftbigg1\n2ε/2/parenrightbiggl\n≤ Oε(δ′).\n(2.9)\nLettingδ >0 be arbitrary fixed, and choosing δ′=δ′(δ,ε)>0 so that Oε(δ′)≤\n2δ, we get from (2.9) that\n|/tildewideJ| ≤2δ. (2.10)\nIt follows from (2.7), (2.8), and (2.10) that\n/ba∇dbl(−∆g−λ2)−1/ba∇dblL2(N)→L2(N)≤Cλn+ε,10 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nfor allλ2∈[1,∞)\\/tildewideJ, whereC=C(δ,ε)>0 and/tildewideJsatisfies (2.10). We define\nthe setJ⊂[1,∞) so thatλ∈[1,∞)\\Jif and only if λ2∈[1,∞)\\/tildewideJ. Letting\nχJandχ/tildewideJbe the characteristic functions of Jand/tildewideJ, respectively, we get\n|/tildewideJ|=/integraldisplay∞\n1χ/tildewideJ(y)dy=/integraldisplay∞\n1χ/tildewideJ(x2)2xdx=/integraldisplay∞\n1χJ(x)2xdx≥2|J|,\nand therefore, |J| ≤ |/tildewideJ|/2≤δ. /square\nRemark 2.2. A result similar to Theorem 2.1 is valid when −∆gis replaced by\nthe Schr¨ odinger operator −∆g+qwithq∈L∞(N;R). We refer to [33, Theorem\n10.1, Remark 10.2] for the Weyl law for −∆g+q.\nLetHs(N),s∈R, be the standard Sobolev space on N, equipped with the norm\n/ba∇dblu/ba∇dblHs(N)=/ba∇dbl(1−∆g)s/2u/ba∇dblL2(N),\nwhere the Bessel potential (1 −∆g)s/2is defined by the self-adjoint functional\ncalculus. Note that\n/ba∇dblu/ba∇dbl2\nH1(N)= ((1−∆g)u,u)L2(N)=/ba∇dbl∇gu/ba∇dbl2\nL2(N)+/ba∇dblu/ba∇dbl2\nL2(N). (2.11)\nWe have the following consequence of Theorem 2.1.\nCorollary 2.3. Givenδ >0andε>0, there exists C=C(N,g,δ,ε)>0and a\nsetJ=J(N,g,δ,ε)⊂[1,∞)with|J| ≤δsuch that\n/ba∇dbl(−∆g−λ2)−1/ba∇dblL2(N)→H1(N)≤Cλ1+n+ε,\nfor allλ∈[1,∞)\\J.\nProof.The result follows by multiplying the equation −∆gu−λ2u=fbyu,\nintegrating by parts, using (2.11), and Theorem 2.1. /square\nLet (M,g) be a smooth compact Riemannian manifold of dimension n≥2 with\nsmooth boundary and let λ≥1. In the proofs of Theorem 1.1 and Theorem 1.8,\nwe shall construct complex geometric optics solutions to the Helmho ltz equation\n−∆gu−λ2u= 0 inMint. In doing so, we shall need the following consequence\nof the resolvent estimates of Theorem 2.1. To state the result, fo r convenience,\nwe use the semiclassical notation and write h=λ−1. Assume, as we may, that\n(M,g) is embedded in a compact smooth Riemannian manifold ( N,g) without\nboundary of the same dimension. Let U⊂Nbe open such that M⊂U. Let\nHs(N),s∈R, be the standard Sobolev space on N, equipped with the natural\nsemiclassical norm,\n/ba∇dblu/ba∇dblHs\nscl(N)=/ba∇dblJsu/ba∇dblL2(N), (2.12)\nwhere the Bessel potential Js= (1−h2∆g)s/2is defined by the self-adjoint func-\ntional calculus on L2(N).INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 11\nProposition 2.4. Lets∈R. Givenδ >0andε >0, there exists C=\nC(N,g,δ,ε)>0and a setJ=J(N,g,δ,ε)⊂[1,∞)with|J| ≤δsuch that\n/ba∇dblu/ba∇dblHs\nscl(N)≤Ch−2−n−ε/ba∇dbl(−h2∆g−1)u/ba∇dblHs\nscl(N), u∈C∞\n0(Mint),(2.13)\nfor allh>0small enough such that h−1∈[1,∞)\\J.\nProof.We shall follow [8, Lemma 4.3], see also [22, Proposition 2.2], [31, Lemma\n3.2]. First it follows from Theorem 2.1 that\n/ba∇dblu/ba∇dblL2(N)≤Ch−2−n−ε/ba∇dbl(−h2∆g−1)u/ba∇dblL2(N), u∈C∞\n0(U), (2.14)\nfor allh>0 such that h−1∈[1,∞)\\J.\nNow note that Jsis a semiclassical pseudodifferential operator of order sonN,\ni.e.Js∈Oph(Ss(T∗N)), see [40, Proposition 16.1.1], and see [22, Section 2]\nfor the definition of the symbol class Ss(T∗N) and the standard h-quantization\nOph(Ss(T∗N)). Therefore, we have the following pseudolocal estimates: if ψ,χ∈\nC∞(N) withχ= 1 near supp( ψ) and ifα,β∈Rthen\n(1−χ)Jsψ=O(h∞) :Hα\nscl(N)→Hβ\nscl(N). (2.15)\nall 0< h≤1. Letχ∈C∞\n0(U) be such that χ= 1 nearM. Then using\n(2.14), (2.12), and (2.15), we get for for all h>0 such that h−1∈[1,∞)\\Jand\nu∈C∞\n0(Mint) that\n/ba∇dblu/ba∇dblHs\nscl(N)≤ /ba∇dblχJsu/ba∇dblL2(N)+/ba∇dbl(1−χ)Jsu/ba∇dblL2(N)\n≤Ch−2−n−ε/ba∇dbl(−h2∆g−1)(χJsu)/ba∇dblL2(N)+O(h∞)/ba∇dblu/ba∇dblHs\nscl(N)\n≤Ch−2−n−ε/parenleftbig\n/ba∇dblχ(−h2∆g−1)(Jsu)/ba∇dblL2(N)+/ba∇dbl[−h2∆g,χ](Jsu)/ba∇dblL2(N)/parenrightbig\n+O(h∞)/ba∇dblu/ba∇dblHs\nscl(N)\n≤Ch−2−n−ε/ba∇dbl(−h2∆g−1)(Jsu)/ba∇dblL2(N)+O(h∞)/ba∇dblu/ba∇dblHs\nscl(N)\n≤Ch−2−n−ε/ba∇dblJs(−h2∆g−1)u/ba∇dblL2(N)+O(h∞)/ba∇dblu/ba∇dblHs\nscl(N)\n≤Ch−2−n−ε/ba∇dbl(−h2∆g−1)u/ba∇dblHs\nscl(N)+O(h∞)/ba∇dblu/ba∇dblHs\nscl(N).\n(2.16)\nHere we have used that\n/ba∇dbl[−h2∆g,χ](Jsu)/ba∇dblL2(N)≤ O(h∞)/ba∇dblu/ba∇dblHs\nscl(N),\nwhich is a consequence of (2.15), and the fact that [ −h2∆g,Js] = 0. Absorbing\nthe error term O(h∞)/ba∇dblu/ba∇dblHs\nscl(N)in theleft handside of (2.16), we get the estimate\n(2.13) forhsmall enough such that h−1∈[1,∞)\\J. /square\nUsing a standard argument, see [8], we convert the a priori estimat e (2.13) into\nthe following solvability result.12 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nProposition 2.5. Lets∈R,δ >0andε >0. Then ifh >0is small enough\nsuch thath−1∈[1,∞)\\J,|J| ≤δ, then for any v∈Hs(Mint), there is a solution\nu∈Hs(Mint)of the equation\n(−h2∆g−1)u=vinMint,\nwhich satisfies\n/ba∇dblu/ba∇dblHs\nscl(Mint)≤Ch−2−n−ε/ba∇dblv/ba∇dblHs\nscl(Mint), C=C(M,g,δ,ε)>0.\nHere\nHs(Mint) ={V|Mint:V∈Hs(N)}, s∈R,\nwith the norm\n/ba∇dblv/ba∇dblHs\nscl(Mint)= inf\nV∈Hs(N),v=V|Mint/ba∇dblV/ba∇dblHs\nscl(N).\n3.Proof of Theorem 1.1\n3.1.Gaussian beam quasimodes and solutions to Helmholtz equati ons.\nLet (M,g) be a smooth compact oriented Riemannian manifold of dimension\nn≥2 with smooth boundary. Let γ: [0,T]→M, 0<T <∞, be a unit speed\nnon-tangential geodesic on M. Here following [10], we say that a unit speed\ngeodesicγ: [0,T]→M, 0< T <∞, is non-tangential if γ(0),γ(T)∈∂M,\nγ(t)∈Mintfor all 0<t<T, and ˙γ(0),˙γ(T) are non-tangential vectors on ∂M.\nWe shall need the following well-known result concerning the constru ction of\nGaussian beam quasimodes on M, localized to the non-tangential geodesic γ, see\n[10, 14, 25]. As in [25], it will be convenient to normalize our quasimodes in\nL4(M), as later we have to work with products of four such quasimodes.\nProposition 3.1. For anyk∈N∪{0}andR≥1, there exist N∈Nand a family\nof Gaussian beam quasimodes v(·;λ)∈C∞(M),λ≥1, such that supp(v(·;λ))is\nconfined to a small neighborhood of γ([0,T])and\n/ba∇dbl(−∆g−λ2)v(·;λ)/ba∇dblHk(Mint)=O(λ−R), (3.1)\n/ba∇dblv(·;λ)/ba∇dblL4(M)=O(1),/ba∇dblv(·;λ)/ba∇dblL∞(M)=O(λn−1\n8),\nasλ→ ∞. The local structure of the family v(·;λ)is as follows: let p∈γ([0,T])\nand lett1<···< tNpbe the times in [0,T]whenγ(tl) =p,l= 1,...,Np. In a\nsufficiently small neighborhood Uof a pointp, we have\nv|U=v(1)+···+v(Np),\nwhere each v(l)has the form\nv(l)(x;λ) =λn−1\n8eiλϕ(l)(x)a(l)(x;λ).\nHereϕ=ϕ(l)∈C∞(U;C)satisfies for tneartl,\nϕ(γ(t)) =t,∇ϕ(γ(t)) = ˙γ(t),Im(∇2ϕ(γ(t)))≥0,Im(∇2ϕ)|˙γ(t)⊥>0,INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 13\nanda(l)∈C∞(U)are of the form,\na(l)(·;λ) =/parenleftbiggN/summationdisplay\nj=0λ−ja(l)\nj/parenrightbigg\nχ/parenleftbiggy\nδ′/parenrightbigg\n,\nwherea(l)\n0(t,y) =a(l)\n00(t) +O(|y|)witha(l)\n00(t)/\\e}atio\\slash= 0for allt. Here(t,y)are the\nFermi coordinates for γwhentneartl,χ∈C∞\n0(Rn−1)is such that 0≤χ≤1,\nχ= 1for|y| ≤1/4andχ= 0for|y| ≥1/2, andδ′>0is a fixed number that\ncan be taken arbitrarily small.\nLetλ≥1. Next, we shall construct complex geometric optics solutions to t he\nHelmholtz equation\n−∆gu−λ2u= 0 inMint, (3.2)\nbased on the Gaussian beam quasimodes of Proposition 3.1. Specifica lly, we look\nfor solutions to (3.2) in the form,\nu(·;λ) =v(·;λ)+r(·;λ),\nwherev(·;λ)istheGaussianbeamquasimode giveninProposition3.1, and r(·;λ)\nistheremainder term. Hence, usolves (3.2)provided that rsatisfies theequation,\n(−∆g−λ2)r=−(−∆g−λ2)v,inMint. (3.3)\nLetk∈NandR≥1 be large. Letting h=λ−1, it follows from Proposition\n2.5 that for all h >0 small enough such that h−1∈[1,∞)\\J,|J| ≤δ, there is\nr∈Hk(Mint) satisfying (3.3) such that\n/ba∇dblr/ba∇dblHk\nscl(Mint)≤ O(h−2−n−ε)/ba∇dbl(−h2∆g−1)v/ba∇dblHk\nscl(Mint)≤ O(hR−n−ε).(3.4)\nHere we have used (3.1). Thus, for any Kandk, there isRlarge enough such\nthat (3.4) gives that\n/ba∇dblr/ba∇dblHk(Mint)≤h−k/ba∇dblr/ba∇dblHk\nscl(Mint)=O(hK).\nWe summarize the discussion above in the following proposition.\nProposition 3.2. Letk∈N,K≥1, andδ >0. Then for λ≥1large enough\nsuch thatλ∈[1,∞)\\J,|J| ≤δ, there isu=u(·;λ)∈Hk(Mint)solving\n(−∆g−λ2)u= 0inMintand having the form\nu(·;λ) =v(·;λ)+r(·;λ),\nwherev(·;λ)∈C∞(M)is the Gaussian beam quasimode given in Proposition 3.1\nandr(·;λ)∈Hk(Mint)such that /ba∇dblr/ba∇dblHk(Mint)=O(λ−K), asλ→ ∞.\nRemark 3.3. Takingk>n/2+3and using the Sobolev embedding Hk(Mint)⊂\nC3(M), we conclude that u∈C3(M)with\n/ba∇dblr/ba∇dblC3(M)=O(λ−K),\nasλ→ ∞.14 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\n3.2.Main part of the proof of Theorem 1.1. Following [14, 25, 21], we start\nbyperformingthethirdorderlinearizationoftheproblem(1.1)andt heDirichlet–\nto–Neumann map. To that end letting ε= (ε1,ε2,ε3)∈C3andfk∈C2,α(∂M),\nk= 1,2,3, we consider the Dirichlet problem\n/braceleftBigg\n−∆guj−λ2uj+qj(x)u3\nj= 0 inMint,\nuj=ε1f1+ε2f2+ε3f3on∂M,(3.5)\nj= 1,2. It follows from [21, Theorem B.1], see also [14, Proposition 1] and [25 ,\nProposition 2.1], that for all |ε|sufficiently small, the problem (3.5) has a unique\nsmall solution uj=uj(x,ε), which depends holomorphically on ε∈neigh(0,C3).\nDifferentiating (3.5) with respect to εl,l= 1,2,3, and using that uj(x,0) = 0,\nwe get/braceleftBigg\n−∆gv(l)\nj−λ2v(l)\nj= 0 inMint,\nv(l)\nj=fl∂M,(3.6)\nwherev(l)\nj=∂εuj|ε=0. By the uniqueness and elliptic regularity for (3.6), we\nhavev(l):=v(l)\n1=v(2)\n2∈C2,α(M),l= 1,2,3, see [15, Theorem 6.15]. Applying\n∂ε1∂ε2∂ε3|ε=0to (3.5), we get the third order linearization,\n/braceleftBigg\n−∆gwj−λ2wj+6qj(x)v(1)v(2)v(3)= 0 inMint,\nwj= 0 on ∂M,(3.7)\nwherewj:=∂ε1∂ε2∂ε3uj|ε=0.\nThe fact that for some λ∈[1,∞)\\Jsufficiently large, we have Λλ\nq1(ε1f1+ε2f2+\nε3f3) = Λλ\nq2(ε1f1+ε2f2+ε3f3) for all small εandfj∈C2,α(∂M) yields that\n∂νu1|∂M=∂νu2|∂M. Therefore, ∂νw1|∂M=∂νw2|∂M. Multiplying the difference\nof two equations in (3.7) by v(4)∈C2,α(M) such that −∆gv(4)−λ2v(4)= 0 in\nMint, and using Green’s formula, we get\n/integraldisplay\nMpv(1)v(2)v(3)v(4)dVg= 0. (3.8)\nHerep=q1−q2. Note that (3.8) holds for all v(l)∈C2,α(M) such that −∆gv(l)−\nλ2v(l)= 0 inMint,l= 1,...,4.\nWe shall next show that p= 0. To that end, let x0∈Mbe such that\nsup\nx∈M|p(x)|=|p(x0)|. (3.9)\nAssume first that x0∈Mint. By the condition (H), there are two non-tangential\nunit speed geodesics γ: [0,T]→Mandη: [0,S]→M, 0< T,S < ∞, such\nthatγandηdo not have self-intersections at x0andx0is the only point of their\nintersections.INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 15\nFurthermore, there is a constant B >0 such that p∈ A(B). Throughout the\nfollowing discussion, it is important to note that all implicit constants m ight rely\nonseveral factors,suchasthemanifold( M,g),theconstant Bensuringp∈ A(B),\nthe pointx0, and the specific geodesics γandη, including their lengths and the\nangle between them. Since in Theorem 1.1, we simply assert that the f requency\nλ0is dependent on ( M,g) andp, without specifying the precise nature of this\ndependence, to avoid clutter, we will not explicitly state the depend ence on these\nfactors in the implicit constants mentioned below.\nUsing Proposition 3.2 and Remark 3.3, for λ∈[1,∞)\\Jlarge enough, we have\nv(l)∈C3(M) solving −∆gv(l)−λ2v(l)= 0 inMint,l= 1,2, of the form\nv(1)=v+r1, v(2)=w+r2, (3.10)\nwherev=v(·;λ),w=w(·;λ)∈C∞(M) are Gaussian beam quasimodes concen-\ntrating near the geodesics γandη, respectively, given in Proposition 3.1, and\n/ba∇dblrl/ba∇dblL∞(M)=O(λ−K), (3.11)\nasλ→ ∞,K≫1. We set\nv(3)=v(1), v(4)=v(2). (3.12)\nIt follows from Proposition 3.1 that in a sufficiently small neighborhood of the\npointx0, we have\nv(x;λ) =λn−1\n8eiλϕ(x)a(x;λ), w(x;λ) =λn−1\n8eiλψ(x)b(x;λ),(3.13)\nwhere\nϕ(γ(t)) =t,∇ϕ(γ(t)) = ˙γ(t),Im(∇2ϕ(γ(t)))≥0,Im(∇2ϕ)|˙γ(t)⊥>0,\nψ(η(τ)) =τ,∇ψ(η(τ)) = ˙η(τ),Im(∇2ψ(η(τ)))≥0,Im(∇2ψ)|˙η(τ)⊥>0,\n(3.14)\nand\na(t,y;λ) =N/summationdisplay\nj=0λ−j/tildewideaj(t,y),/tildewideaj(t,y) =aj(t,y)χ/parenleftbiggy\nδ′/parenrightbigg\n,\nb(τ,z;λ) =N/summationdisplay\nj=0λ−j/tildewidebj(τ,z),/tildewidebj(τ,z) =bj(τ,z)χ/parenleftbiggz\nδ′/parenrightbigg\n.(3.15)\nHere\na0(t,y) =a00(t)+O(|y|), a00(t)/\\e}atio\\slash= 0,∀t,\nb0(τ,z) =b00(τ)+O(|z|), b00(τ)/\\e}atio\\slash= 0,∀τ,\n(t,y) and (τ,z) are the Fermi coordinates for γandη,χ∈C∞\n0(Rn−1) is such that\n0≤χ≤1,χ= 1 for|y| ≤1/4 andχ= 0 for|y| ≥1/2, andδ′>0 is a fixed16 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nnumber that can be taken arbitrarily small. We also have\n/ba∇dblv/ba∇dblL4(M)=O(1),/ba∇dblw/ba∇dblL4(M)=O(1),\n/ba∇dblv/ba∇dblL∞(M)=O(λn−1\n8),/ba∇dblw/ba∇dblL∞(M)=O(λn−1\n8),(3.16)\nasλ→ ∞. Now thanks to the bounds (3.11), (3.16), for any L>0, we can find\nKlarge so that we have\nv(1)v(2)v(3)v(4)=|v|2|w|2+R, (3.17)\nwhere\n/ba∇dblR/ba∇dblL∞(M)=O(λ−L). (3.18)\nSubstituting (3.17) into the integral identity (3.8), and using (3.18) , we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nMp|v|2|w|2dVg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ O(λ−L)/ba∇dblp/ba∇dblL∞(M). (3.19)\nNow recall that x0is the only intersection point of the geodesics γandη. Let\n/tildewideδ >0. Takingδ′>0 in (3.15) to be sufficiently small but fixed, we could achieve\nthat the support of the product |v|2|w|2is contained in an open geodesic ball\nB/tildewideδ(x0) inMof radius/tildewideδcentered at x0. InB/tildewideδ(x0), using (3.13) and (3.15), we\nget\n|v|2|w|2=λn−1\n2e−2λ(Imϕ+Imψ)/parenleftbig\n|/tildewidea0|2|/tildewideb0|2+OL∞(B/tildewideδ(x0))/parenleftbig\nλ−1/parenrightbig/parenrightbig\n.(3.20)\nLetting\nΨ = 2(Imϕ+Imψ)≥0, (3.21)\nwe have from (3.21) and (3.14) that\nΨ(x0) = 0, dΨ(x0) = 0,∇2Ψ(x0)>0. (3.22)\nThe latter inequality follows from the fact that the Hessians of Im ϕand Imψat\nx0are positive semidefinite and positive definite in the directions orthog onal to\nγandη, respectively. If z= (z1,...,zn) are normal coordinates centered at x0,\nwe have\nΨ(z) =1\n2Ψ′′(0)z·z+O(|z|3),\nwhere the Hessian Ψ′′(0)>0. Thus, by choosing /tildewideδsmall enough, we get\nΨ(z)≥c|z|2inB/tildewideδ(x0), (3.23)\nfor somec>0.INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 17\nSubstituting (3.20) into (3.19), we obtain that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleλn−1\n2/integraldisplay\nB/tildewideδ(x0)pe−λΨ|/tildewidea0|2|/tildewideb0|2dVg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤O(λ−L)/ba∇dblp/ba∇dblL∞(M)\n+O(λn−1\n2−1)/ba∇dblp/ba∇dblL∞(M)/integraldisplay\nB/tildewideδ(x0)e−λΨdVg.\n(3.24)\nWriting the integral in the normal coordinates, using dVg(z) =|g(z)|1/2dzand\n(3.23), we get\n/integraldisplay\nB/tildewideδ(x0)e−λΨdVg≤ O(1)/integraldisplay\nRne−λc|z|2dz=O(λ−n/2). (3.25)\nThus, fixing\nL= 3/2, (3.26)\nwe obtain from (3.24) and (3.25) that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleλn\n2/integraldisplay\nB/tildewideδ(x0)pe−λΨ|/tildewidea0|2|/tildewideb0|2dVg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ O(λ−1)/ba∇dblp/ba∇dblL∞(M). (3.27)\nNext, weshallusetheroughstationaryphaseLemmaA.1intheinteg ralin(3.27).\nTo that end, we shall write the integral in the normal coordinates zcentered at\nx0, and redefine /tildewideδ >0 to be smaller if necessary. We get\nλn\n2/integraldisplay\nB/tildewideδ(x0)pe−λΨ|/tildewidea0|2|/tildewideb0|2dVg=λn\n2/integraldisplay\nB/tildewideδ(0)p(z)e−λΨ(z)|/tildewidea0(z)|2|/tildewideb0(z)|2|g(z)|1/2dz\n=(2π)n\n2\n(det∇2Ψ(x0))1/2|p(x0)||a0(x0)|2|b0(x0)|2+O(1)λ−α\n2/ba∇dblp/ba∇dblC0,α(M).\n(3.28)\nHere we have also used (A.1) and the fact that |g(0)|= 1 in the normal coordi-\nnates. Combining (3.27) and (3.28), we obtain that\n1\n(det∇2Ψ(x0))1/2|p(x0)||a0(x0)|2|b0(x0)|2≤ O(λ−α/2)/ba∇dblp/ba∇dblC0,α(M).(3.29)\nTherefore, using that a0(x0)/\\e}atio\\slash= 0, andb0(x0)/\\e}atio\\slash= 0, we get from (3.29) that\n|p(x0)| ≤ O(λ−α/2)/ba∇dblp/ba∇dblC0,α(M), (3.30)\nforλ∈[1,∞)\\Jlarge enough, satisfying the assumption (A).\nIt follows from (3.9) and (3.30) that\n/ba∇dblp/ba∇dblL∞(M)≤ O(λ−α/2)/ba∇dblp/ba∇dblC0,α(M). (3.31)\nSincep∈ A(B), get from (3.31) that\n/ba∇dblp/ba∇dblL∞(M)≤BO(λ−α/2)/ba∇dblp/ba∇dblL∞(M). (3.32)18 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nNow choosing λ∈[1,∞)\\Jlarge enough, satisfying the assumption (A) and\ndepending on B, we conclude from (3.32) that p= 0 inM.\nIfx0∈∂Mthen (3.8) and Proposition B.1 imply that p(x0) = 0, and therefore,\n(3.9) yields that p= 0 inM. This completes the proof of Theorem 1.1.\n4.Proof of Theorem 1.8\n4.1.Gaussian beam quasimodes and solutions to Helmholtz equati ons\nwith uniform constants. Let (M,g) be a smooth compact oriented Riemann-\nian manifold of dimension n≥2 with smooth boundary, and let TxMbe the\ntangent space at the point x∈M. We write /a\\}b∇acketle{t·,·/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htg=g(·,·) for theg–inner\nproduct for tangent vectors and | · |=| · |g=/radicalbig\n/a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htgfor the norm of tangent\nvectors. The unit sphere bundle SMofMis defined as\nSM=∪x∈MSxM, SxM={(x,w)∈TxM:|w|g= 1}.\nThe boundary of SMis given by ∂(SM) ={(x,w)∈SM:x∈∂M}. It is the\nunion of the sets of inward and outward pointing vectors,\n∂±SM={(x,w)∈∂(SM) :±/a\\}b∇acketle{tw,ν/a\\}b∇acket∇i}ht ≤0},\nwhereνis the outward unit normal to the boundary of M.\nLet (x,w)∈∂+SM, and letγ=γx,w(t) : [0,τ(x,w)]→Mbe the maximally\nextended unit speed geodesic such that γ(0) =xand ˙γ(0) =w.\nLetT >0 be fixed. Following [31, Theorem 6.2], we allow the manifold ( M,g)\nto have trapped geodesics, i.e. τ(x,w) maybe + ∞for some (x,w), but we shall\nonly work with ( x,w)∈ GTwhere\nGT={(x,w)∈∂+SM:τ(x,w)≤T}.\nNote thatτ(x,w) is the length of the geodesic γx,w.\nWe will utilize a recently established result, see [31, Theorem 6.2], con cerning the\nconstruction of Gaussian beam quasimodes on M, localized to γx,w, with uniform\nbounds over ( x,w)∈ GT.\nTheorem 4.1. LetT >0,k∈N∪{0}, andR≥0be given. There are constants\nC=C(M,g,T,k,R )>0,/tildewideC=/tildewideC(M,g,T,k,R )>0, andN=N(M,g,T,k,R )∈\nNsuch that for any (x,w)∈ GT, there exists a family of Gaussian beam quasi-\nmodesv=v(·;λ) =vx,w(·;λ)∈C∞(M),λ≥1, associated to γ=γx,w, and\nsatisfying\n/ba∇dbl(−∆g−λ2)v(·;λ)/ba∇dblHk(Mint)≤Cλ−R, (4.1)\n/ba∇dblv(·;λ)/ba∇dblL4(M)≤C,/ba∇dblv(·;λ)/ba∇dblL∞(M)≤Cλn−1\n8,INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 19\nasλ→ ∞. There is also a symmetric complex (1,1)-tensorH(t) =Hx,w(t)on\nTγ(t)M, depending smoothly on t∈[0,τ(x,w)]and satisfying\nIm(H(t)♭)≥0,Im(H(t)♭)|˙γ(t)⊥≥/tildewideC−1g.\nThe local structure of the family v(·;λ)is as follows: if p∈γ([0,τ(x,w)])and\nt1<···<tNpare the times in [0,τ(x,w)]whenγ(tl) =p,l= 1,...,Np, then in\na small neighborhood Uofp, we have\nv|U=v(1)+···+v(Np),\nwhere each v(l)has the form\nv(l)(x;λ) =λn−1\n8eiλϕ(l)(x)a(l)(x;λ).\nHereϕ=ϕ(l)∈C∞(U;C)satisfies for tneartl,\nϕ(γ(t)) =t,∇ϕ(γ(t)) = ˙γ(t),∇2ϕ(γ(t)) =H(t)♭,/ba∇dblϕ/ba∇dblCk(U)≤/tildewideC,(4.2)\nanda(l)∈C∞(U)are of the form,\na(l)(·;λ) =/parenleftbiggN/summationdisplay\nj=0λ−ja(l)\nj/parenrightbigg\nρ, (4.3)\nwhereρis a smooth cutoff function supported near γwhentis neartl, and\na(l)\n0(γ(t)) = exp/bracketleftbigg\n−1\n2/integraldisplayt\n0trg(H(s))ds/bracketrightbigg\n. (4.4)\nOne also has /ba∇dbla(l)\nj/ba∇dblCk(U)≤/tildewideC,l= 1,...,Np, j= 0,...,N.\nRemark 4.2. According to the proof presented in [31, Theorem 6.2] , the cut-\noff function ρ=ρ(l)used in the amplitude a(l)in(4.3)is selected as ρ(t,y) =\nχ(|y|/δ1), whereδ1=δ1(M,g,T,k,R )>0is small, see [31, formula (6.6) and\ndiscussion before it] . Here,(t,y)are the Fermi coordinates for γwhentnear\ntl, and the function χis fixed and belongs to the space C∞\n0(R), satisfying the\nproperties: 0≤χ≤1,χ(t) = 1for|t| ≤1/2, andχ(t) = 0for|t| ≥2/3.\nRemark 4.3. While the constant Cin Theorem 4.1 depends on δ1, the constants\n/tildewideCandNare independent of δ1. The independence of /tildewideCofδ1will be crucial in\nthe proof of Theorem 1.8 below.\nRemark 4.4. If(M,g)is non-trapping, i.e. τ(x,w)<∞for all(x,w)∈∂+SM,\nthenGT=∂+SMfor sufficiently large T, see[31, Remark 6.1] .\nLetK≥1. Through a similar line of reasoning as in establishing Proposition\n3.2 and Remark 3.3, fixing εin Proposition 2.5 and k > n/2 + 3, and letting\nR=K+k+n+ε, we arrive at the following result regarding the construction\nof complex geometric optics solutions for the Helmholtz equation with uniform\nconstants based on the Gaussian beam quasimodes of Theorem 4.1.20 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nProposition 4.5. LetK≥1andδ >0. Then for λ∈[1,∞)\\Jlarge enough,\n|J| ≤δ, there isu=u(·;λ)∈C3(M)solving(−∆g−λ2)u= 0inMintand\nhaving the form\nu(·;λ) =v(·;λ)+r(·;λ),\nwherev(·;λ)∈C∞(M)is the Gaussian beam quasimode constructed in The-\norem 4.1 with the constants C=C(M,g,T,K,δ 1),/tildewideC=/tildewideC(M,g,T,K ),N=\nN(M,g,T,K ), andr(·;λ)∈C3(M)such that /ba∇dblr/ba∇dblC3(M)≤C1λ−K, asλ→ ∞\nwithC1=C1(M,g,δ,T,K,δ 1)>0.\n4.2.Main part of the proof of Theorem 1.8. LetT >0, 0< θ0< π/2,\n0<r<Inj(M)\n2andc0>0 be constants given in the condition (H1). Let ET,θ0,r,c0\nbe the set of points x∈Mintsatisfying the condition (H1).\nBased on the proof of Theorem 1.1, we begin with the integral identit y (3.8). We\nshall show that p= 0. If supx∈M|p(x)|is attained at some point on the boundary\nofMthen by Proposition B.1 we conclude from (3.8) that p= 0 at this point,\nand we are done. Thus, we only need to consider the case when supx∈M|p(x)|is\nnot attained at any boundary point, and therefore, supx∈M|p(x)|is achieved at\nsome point in Mint. In this case, supx∈M|p(x)|= supx∈Mint|p(x)|=/ba∇dblp/ba∇dblL∞(M).\nLetε >0. The fact that ET,θ0,r,c0is dense in Mintand continuity of pimplies\nthat there is x0∈ET,θ0,r,c0such that\n|p(x0)|>/ba∇dblp/ba∇dblL∞(M)−ε. (4.5)\nAsx0∈ET,θ0,r,c0, there are two non-tangential unit-speed geodesics γ=γx1,w1:\n[0,τ(x1,w1)]→Mandη=ηx2,w2: [0,τ(x2,w2)]→M, (x1,w1),(x2,w2)∈ GT,\nsuch thatγ(t0) =η(τ0) =x0,t0∈(0,τ(x1,w1)),τ0∈(0,τ(x2,w2)). Furthermore,\nbothγandηadhere to all other requirements outlined in (H1).\nLetδ >0 andK≥1. By Proposition 4.5, for λ∈[1,∞)\\Jlarge enough,\n|J| ≤δ, we havev(1),v(2)∈C3(M) solving ( −∆g−λ2)v(l)= 0 inMint,l= 1,2,\nof the form\nv(1)=v+r1, v(2)=w+r2, (4.6)\nwherev=v(·;λ),w=w(·;λ)∈C∞(M) are the Gaussian beam quasimodes\nconcentrating near the geodesics γandη, respectively, given in Theorem 4.1, and\nrl,l= 1,2, satisfies\n/ba∇dblrl/ba∇dblL∞(M)≤C1λ−K, (4.7)\nasλ→ ∞, withC1=C1(M,g,δ,T,K,δ 1)>0. It follows from Theorem 4.1 that\n/ba∇dblv/ba∇dblL4(M)≤C,/ba∇dblw/ba∇dblL4(M)≤C,/ba∇dblv/ba∇dblL∞(M)≤Cλn−1\n8,/ba∇dblw/ba∇dblL∞(M)≤Cλn−1\n8,\n(4.8)\nasλ→ ∞, whereC=C(M,g,T,K,δ 1)>0. We let\nv(3)=v(1), v(4)=v(2). (4.9)INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 21\nUtilizing (4.9), (4.7), and (4.8), we get\nv(1)v(2)v(3)v(4)=|v|2|w|2+R, (4.10)\nwhere/ba∇dblR/ba∇dblL∞(M)≤Cλ−K+3(n−1)\n8and the dependence of the implicit constant is\nas follows: C=C(M,g,δ,T,K,δ 1), cf. (3.17) and (3.18).\nMotivated by (3.26), we fix K=3(n−1)\n8+3\n2so that/ba∇dblR/ba∇dblL∞(M)≤Cλ−3\n2with\nC=C(M,g,δ,T,δ 1). Therefore, by substituting (4.10) into the integral identity\n(3.8), we arrive at\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nMp|v|2|w|2dVg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cλ−3\n2/ba∇dblp/ba∇dblL∞(M), (4.11)\nwherein the implicit constant C=C(M,g,δ,T,δ 1)>0, cf. (3.19) and (3.26).\nLet us mention that, since Kis fixed, the only dependence in the constants C,\n/tildewideC, andNin Theorem 4.1 is as follows: C=C(M,g,T,δ 1),/tildewideC=/tildewideC(M,g,T), and\nN=N(M,g,T). Furthermore, the only dependence in δ1in the cutoff function\nρused in the amplitude of the Gaussian beam quasimodes vandwis as follows:\nδ1=δ1(M,g,T), see Remark 4.2.\nWehavethatthesupportsof vandwarecontainedwithintubularneighborhoods\nwith radius δ1ofγandη, that is,\nsupp(v(·;λ))⊂ {y∈M:d(y,γ([0,τ(x1,w1)]))<δ1},\nsupp(w(·;λ))⊂ {y∈M:d(y,η([0,τ(x2,w2)]))<δ1},(4.12)\nfor allλ≥1.\nNext, we will demonstrate that if δ1<min{r/2,Inj(M)/(1+2c0)}, the condition\n(iv) in assumption (H1) implies that for all λ≥1,\nsupp(v(·;λ)w(·;λ))⊂B(1+2c0)δ1(x0)∩M. (4.13)\nHere,B(1+2c0)δ1(x0) is an open geodesic ball in Scentered at x0with a radius of\n(1+2c0)δ1, where (S,g) is the closed extension of ( M,g).\nIndeed, ify∈supp(v(·;λ)w(·;λ))⊂supp(v(·;λ))∩supp(w(·;λ)) then it follows\nfrom (4.12) that d(y,γ(t′))< δ1andd(y,η(τ′))< δ1for somet′∈[0,τ(x1,w1)]\nandτ′∈[0,τ(x2,w2)]. By the triangle inequality, we obtain d(γ(t′),η(τ′))<\n2δ1<r, and therefore, condition (iv) in assumption (H1) implies that\n|t′−t0| ≤2c0δ1. (4.14)\nHence, since γis a unit-speed geodesic, we have\nd(y,x0)≤d(y,γ(t′))+d(γ(t′),x0)<δ1+2c0δ1. (4.15)\nTherefore,y∈B(1+2c0)δ1(x0) and this establishes the claim (4.13).22 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nIf we takeτ=τ0in condition (iv) of assumption (H1), we see that if d(γ(t),x0)<\nr, then|t−t0| ≤c0d(γ(t),x0). Let\nρ= min/parenleftbigg\nr,Inj(M)\nc0/parenrightbigg\n. (4.16)\nIfd(γ(t),x0)< ρthen|t−t0|<Inj(M), and therefore, d(γ(t),γ(t0)) =|t−t0|.\nA similar statement also holds for η.\nWe now claim that if δ1<ρ\n2, the Gaussian beams vandwtake the simple form\ngiven by (4.21) within Bρ\n2(x0)∩M, whereBρ\n2(x0) is an open geodesic ball in S\ncentered at x0with a radius ofρ\n2. Indeed, first, let us show that the set\n{y∈M:d(y,γ(t))<δ1for somet∈[0,τ(x1,w1)],|t−t0| ≥ρ}∩Bρ\n2(x0) (4.17)\nis empty. To prove this, we assume the contrary, that there exist s a pointyin\nthe intersection of the sets in (4.17). Then, we can find a value t′in the interval\n[0,τ(x1,w1)] with|t′−t0| ≥ρsuch thatd(y,γ(t′))<δ1andd(y,x0)<ρ\n2. Using\nthe triangle inequality and the fact that δ1<ρ\n2, we obtain d(γ(t′),x0)<ρ. Given\nthe equation (4.16), as reasoned in the preceding paragraph, we c an conclude\nthatd(γ(t′),x0) =|t′−t0| ≥ρ. This contradiction proves that the set in (4.17)\nis empty.\nMoreover, using the same logic, we can show that\nγ([0,τ(x1,w1)])∩Bρ(x0)⊂ {γ(t) :t∈[0,τ(x1,w1)],|t−t0|<ρ}.(4.18)\nSinceρ≤r <Inj(M)\n2, the geodesic γ(t) does not exhibit self-intersections when\n|t−t0|<ρ. This, in combination with (4.17) and (4.18), establishes the claim for\nthe Gaussian beam quasimode v. Similarly, we can demonstrate the same claim\nfor the Gaussian beam quasimode w.\nHence, we can redefine δ1asδ1=δ1(M,g,T,r,c 0) to ensure that it is sufficiently\nsmall, such that\nδ1<min/parenleftbiggρ\n2,ρ\n2(1+2c0)/parenrightbigg\n. (4.19)\nThis condition guarantees that\nsupp(vw)⊂B(1+2c0)δ1(x0)∩M⊂Bρ\n2(x0)∩M. (4.20)\nAccordingtoTheorem4.1,seealsotheproofof[31, Theorem6.2],wit hinBρ\n2(x0)∩\nM, the Gaussian beam quasimodes vandwtake on a simple form, i.e.\nv(x;λ) =λn−1\n8eiλϕ(x)a(x;λ), w(x;λ) =λn−1\n8eiλψ(x)b(x;λ),(4.21)INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 23\nwhere\nϕ(γ(t)) =t,∇ϕ(γ(t)) = ˙γ(t),∇2ϕ(γ(t)) =Hx1,w1(t)♭,/ba∇dblϕ/ba∇dblC3(Bρ\n2(x0)∩M)≤/tildewideC,\nψ(η(τ)) =τ,∇ψ(η(τ)) = ˙η(τ),∇2ψ(η(τ)) =Hx2,w2(τ)♭,/ba∇dblψ/ba∇dblC3(Bρ\n2(x0)∩M)≤/tildewideC.\n(4.22)\nHereHx1,w1(t) andHx2,w2(τ) are symmetric complex (1 ,1)-tensors on Tγ(t)M\nandTη(τ)M, depending smoothly on t∈[0,τ(x1,w1)] andτ∈[0,τ(x2,w2)] and\nsatisfying\nIm(Hx1,w1(t)♭)≥0,Im(Hx1,w1(t)♭)|˙γ(t)⊥≥/tildewideC−1g,\nIm(Hx2,w2(τ)♭)≥0,Im(Hx2,w2(τ)♭)|˙η(τ)⊥≥/tildewideC−1g,(4.23)\nrespectively. We also have in Bρ\n2(x0)∩M,\na(·;λ) =N/summationdisplay\nj=0λ−j/tildewideaj,/tildewideaj=ajρ, b(·;λ) =N/summationdisplay\nj=0λ−j/tildewidebj,/tildewidebj=ajρ,(4.24)\nand\n/ba∇dblaj/ba∇dblC3(Bρ\n2(x0)∩M)≤/tildewideC,/ba∇dblbj/ba∇dblC3(Bρ\n2(x0)∩M)≤/tildewideC, j= 0,...,N. (4.25)\nFurthermore,\na0(γ(t)) = exp/bracketleftbigg\n−1\n2/integraldisplayt\n0trg(Hx1,w1(s))ds/bracketrightbigg\n,\nb0(η(τ)) = exp/bracketleftbigg\n−1\n2/integraldisplayτ\n0trg(Hx2,w2(s))ds/bracketrightbigg\n.(4.26)\nThe implicit constants in (4.22), (4.23), (4.24), and (4.25) are expre ssed as/tildewideC=\n/tildewideC(M,g,T)>0 andN=N(M,g,T)∈N.\nNow inBρ\n2(x0)∩M, in view of (4.21), we have\n|v|2|w|2=λn−1\n2e−2λ(Imϕ+Imψ)|a|2|b|2.\nLetting\nΨ = 2(Imϕ+Imψ)≥0,\nwe deduce from equations (4.22) and (4.23) the following properties :\nΨ(x0) = 0, dΨ(x0) = 0,∇2Ψ(x0)≥cg, (4.27)\nwherec=c(M,g,T,θ 0)>0. To show the last inequality in (4.27), first in Fermi\ncoordinates along γnearx0, we have\nIm(∇2ϕ(γ(t)))♯=/parenleftbigg\n0 0\n0 Im(Hx1,w1(t))|˙γ(t)⊥/parenrightbigg\n(4.28)\nwhere the block decomposition of the matrix is understood with resp ect to the\ndecomposition Tγ(t)M=R˙γ(t)⊕(˙γ(t))⊥.24 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nWe have a similar decomposition for Im( ∇2ψ(η(τ)))♯in Fermi coordinates along\nη. Lett0andτ0be the unique times such that γ(t0) =η(τ0) =x0, and let\nW= (˙γ(t0))⊥∩(˙η(τ0))⊥⊂Tx0Mint. We can write any v∈Tx0Mintuniquely as\nv=vγ˙γ(t0) +vη˙η(τ0) +wwithvγ,vη∈R, andw∈W. By using that ˙ γ(t0) is\nin the kernel of Im( ∇2ϕ(x0))♯and ˙η(τ0) is in the kernel of Im( ∇2ψ(x0))♯, we can\nexpand\n(∇2Ψ(x0)♯v,v) = (2Im( ∇2ϕ(x0))♯(vη˙η(τ0)+w),vη˙η(τ0)+w)+\n+(2Im(∇2ψ(x0))♯(vγ˙γ(t0)+w),vγ˙γ(t0)+w).(4.29)\nHere and in what follows ( ·,·) = (·,·)g. We can decompose vη˙η(τ0) +wwith\nrespect to R˙γ(t0)⊕(˙γ(t0))⊥as\nvη˙η(τ0)+w= (vηcosθ)˙γ(t0)+/tildewidew (4.30)\nfor some unique /tildewidew∈(˙γ(t0))⊥with cosθ= (˙γ(t0),˙η(τ0)). Taking norms on both\nsides of (4.30), we see that |/tildewidew|2=v2\nηsin2θ+|w|2sincewis orthogonal to ˙ η(τ0).\nMoreover, as Im( ∇2ϕ(x0))♯˙γ(t0) = 0, we get\n(Im(∇2ϕ(x0))♯(vη˙η(τ0)+w),vη˙η(τ0)+w) = (Im(∇2ϕ(x0))♯/tildewidew,/tildewidew)≥/tildewideC−1|/tildewidew|2\nby (4.23), since /tildewidew∈(˙γ(t0))⊥. By working analogously with the second term in\n(4.29) and combining, we get\n(∇2Ψ(x0)♯v,v)≥2/tildewideC−1((v2\nγ+v2\nη)sin2θ+2|w|2).\nAs|v|2=v2\nγ+v2\nη+ 2vγvηcosθ+|w|2andv2\nγ+v2\nη≥1\n2(v2\nγ+v2\nη+2vγvηcosθ), it\nfollows that ∇2Ψ(x0)≥(/tildewideC−1sin2θ)g≥(/tildewideC−1sin2θ0)g, sinceθ∈[θ0,π/2]. This\nshows the last inequality in (4.27).\nNow ifz= (z1,...,zn) are normal coordinates centered at x0∈Mint, we have\nΨ(z) =1\n2Ψ′′(0)z·z+Ψ3(z),\nwhere Ψ′′(0)≥cI,c=c(M,g,T,θ 0)>0, in view of the last inequality in (4.27).\nThanks to (4.22), we get |Ψ3(z)| ≤/tildewideC|z|3forz∈exp−1\nx0(Bρ\n2(x0)∩M)⊂Bρ\n2(0)\nwith/tildewideC=/tildewideC(M,g,T)>0. Here the exponential map expx0:Bρ\n2(0)→Bρ\n2(x0) is\na diffeomorphism.\nNow, if 0</tildewideδ<ρ\n2is small enough to satisfy the condition\n/tildewideC/tildewideδ≤c/4, (4.31)\nthen we have\nΨ(z)≥1\n2c|z|2−/tildewideC|z|3≥(c/2−/tildewideC/tildewideδ)|z|2≥c\n4|z|2, z∈exp−1\nx0(B/tildewideδ(x0)∩M)⊂B/tildewideδ(0).\n(4.32)\nNote that/tildewideδ=/tildewideδ(/tildewideC,c,ρ) =/tildewideδ(M,g,T,θ 0,r,c0).INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 25\nIn light of (4.31) and (4.19), we can redefine δ1=δ1(M,g,T,θ 0,r,c0)>0 to be\nsufficiently small, ensuring that\nδ1≤min/parenleftBigg\nρ\n2,ρ\n2(1+2c0),c\n4/tildewideC(1+2c0)/parenrightBigg\n. (4.33)\nBy letting/tildewideδ:= (1+2c0)δ1, we can see from (4.33) and (4.20) that\nsupp(vw)⊂B/tildewideδ(x0)∩M⊂Bρ\n2(x0)∩M,\nand (4.32) holds. It is essential to emphasize that the independenc e of the con-\nstant/tildewideCfromδ1in Theorem 4.1 is crucial for (4.33).\nThischoiceof δ1willremainfixedinthesubsequent discussion. Asaconsequence,\nthe constants, previously dependent on δ1, will now depend on M,g,T,θ0,r,\nandc0.\nWe shall proceed to thoroughly examine the rest of the proof of Th eorem 1.1,\nwith a specific focus on controlling the dependencies of all the const ants involved.\nThroughout this examination, we will represent all constants as C, and it is\nimportant to note that these constants might change from line to lin e.\nUsing the fact that the support of the product |v|2|w|2is entirely confined within\nB/tildewideδ(x0)∩Mand (4.25), we obtain (3.20), where the implicit constant is denoted\nasC=C(M,g,T)>0.\nNow, (4.11) and (3.20) imply that the bound (3.24) holds with L= 3/2, and\nthe implicit constant is denoted as C=C(M,g,δ,T,θ 0,r,c0)>0. Moreover, the\nintegrals in (3.24) are taken over B/tildewideδ(x0)∩M. By using (4.32), we obtain (3.25),\nwhere the implicit constant is expressed as C=C(M,g,T,θ 0,r,c0)>0. Thus,\nwe get (3.27) with an implicit constant of C=C(M,g,δ,T,θ 0,r,c0)>0, and\nwith the integral taken over B/tildewideδ(x0)∩M.\nAssume first that B/tildewideδ(x0)⊂Mint. Thanks to (4.31), we can now apply the\nrough stationary phase Lemma A.1 to the integral in (3.27). This allow s us to\nobtain (3.28), where the implicit constant is C=C(M,g,T,θ 0,r,c0)>0. If\nB/tildewideδ(x0)∩∂M/\\e}atio\\slash=∅, then first, by Proposition B.1, we conclude from (3.8) that\np|∂M= 0. Extending /tildewidep:=p|/tildewidea0|2|/tildewideb0|2by zero to B/tildewideδ(x0)\\(B/tildewideδ(x0)∩M) and\ndenoting this extension by /tildewidepagain, we see that /tildewidep∈C0,α(B/tildewideδ(x0)), and supp(/tildewidep)⊂\nB/tildewideδ(x0) is compact, see Lemma A.3. In view of Remark A.2, applying Lemma\nA.1 to the integral in (3.27), we obtain (3.28), where the implicit const ant is\nC=C(M,g,T,θ 0,r,c0)>0. Consequently, in both cases, we arrive at (3.29),\nwith the implicit constant expressed as C=C(M,g,δ,T,θ 0,r,c0)>0.\nNext, it follows from (4.26) that\na0(x0) = exp/bracketleftbigg\n−1\n2/integraldisplayt0\n0trg(Hx1,w1(s))ds/bracketrightbigg\n,26 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nwhere 0< t0< τ(x1,w1)≤T. Working with the open cover of γconsisting\nof/tildewideN=/tildewideN(M,g,T) elements, and using (4.2) in each element of the cover, we\nconclude that |a0(x0)| ≥C, whereC=C(M,g,T)>0. We refer to the proof of\n[31, Theorem 6.2] for the existence of such a cover. Similarly, we hav e|b0(x0)| ≥\nC. Furthermore, |det∇2Ψ(x0)| ≤C, whereC=C(M,g,T)>0, due to (4.22).\nHence, from (3.29) and the fact that p∈ A(B), we deduce that\n|p(x0)| ≤CBλ−α/2/ba∇dblp/ba∇dblL∞(M), (4.34)\nforλ∈[1,∞)\\Jlarge enough, satisfying the assumption (A). Here, C=\nC(M,g,δ,T,θ 0,r,c0)>0.\nBy combining (4.34) with (4.5), we obtain the inequality\n/ba∇dblp/ba∇dblL∞(M)(1−CBλ−α/2)<ε,\nfor allε>0. Now, by letting ε→0 and choosing λ=λ(M,g,δ,T,θ 0,B,r,c 0)∈\n[1,∞)\\Jlarge enough, satisfying the assumption (A), we conclude that p= 0.\nThis completes the proof of Theorem 1.8.\n5.Proof of Theorem 1.6\n5.1.Existence of non-tangential geodesics between boundary po ints.\nWe start by showing that almost all points in Mintlie on a non-tangential geo-\ndesic. We will follow the approach used in [38, Lemma 3.1], but modify it a s we\ndo not want to require strict convexity of the boundary.\nLemma 5.1. Let(M,g)be a smooth compact Riemannian manifold of dimension\nn≥2with smooth boundary. Then almost every point of Mintlies on some non-\ntangential geodesic.\nBefore giving the proof of Lemma 5.1, let us do some preparations. L et (M,g)\nbe embedded in a closed manifold ( N,g). LetSMintandSNdenote the unit\nsphere bundle of MintandNrespectively and let ϕtdenote the geodesic flow on\nSN. For (x,v)∈SMint, we define the future and past exit times respectively as\nl+(x,v) = sup{T >0 :ϕt(x,v)∈SMintfor all 0≤t<T},\nl−(x,v) = inf{T <0 :ϕt(x,v)∈SMintfor allT <t≤0}.INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 27\nConsider the following sets of good and bad directions,\nG={(x,v)∈SMint:l−(x,v) andl+(x,v) are finite with transversal\nintersections },\nB1={(x,v)∈SMint: exactly one of l±(x,v) is finite },\nB2={(x,v)∈SMint:l−(x,v) andl+(x,v) are infinite },\nB3={(x,v)∈SMint:l−(x,v) andl+(x,v) are finite with a tangential\nintersection }.\nIn the definition of G, by transversal intersections, we mean that the unit speed\ngeodesic going through ( x,v) at time 0 intersects ∂Mtransversally at times\nl−(x,v) andl+(x,v). On the opposite, in the definition of B3, by a tangen-\ntial intersection, we mean that the unit speed geodesic going throu gh (x,v) at\ntime 0 intersects ∂Mtangentially at least once at times l−(x,v) orl+(x,v).\nLetmdenote the Riemannian volume measure on ( M,g) and letµdenote the\nLiouville measure on SN, see [35, Section 3.6.2].\nLemma 5.2. The setsB1andB3have Liouville measure 0.\nProof.It follows directly from [38, Lemma 3.2] that µ(B1) = 0. ForB3, let\nE={ϕt(x,v) : (x,v)∈∂0SMand 0≤t<1},\nwhere∂0SM:=S∂M. We have µ(E) = 0 as dim( ∂0SM) = 2n−3 and\ndim(SN) = 2n−1. By invariance of the Liouville measure, µ(ϕt(E)) = 0\nfor allt∈R. If (x,v)∈B3, the unit speed geodesic going through ( x,v) at\ntime 0 intersects ∂Mtangentially at least once at times l−(x,v) orl+(x,v),\nthat is,ϕl−(x,v)(x,v)∈∂0SMorϕl+(x,v)(x,v)∈∂0SM. It follows that B3⊂/uniontext∞\nk=−∞ϕk(E) and soµ(B3) = 0. /square\nProof of Lemma 5.1. Consider the set\nA={x∈Mint: for anyv∈SxMint,eitherl−(x,v) =−∞,orl+(x,v) =∞,\norl−(x,v) andl+(x,v) are finite with a tangential intersection }.\nHere, asbefore, byatangentialintersection, wemeanthattheu nitspeedgeodesic\ngoing through ( x,v) at time 0 intersects ∂Mtangentially at least once at times\nl−(x,v) orl+(x,v). The claim will be proved if we can show m(A) = 0.\nSuppose that Ahas positive measure. By the Lebesgue density theorem, we can\nfind a point x0∈Asuch that\nlim\nε→0m(A∩Bε(x0))\nm(Bε(x0))= 1.\nBy [19, Lemma 2.10], there exists v0∈Sx0Mintsuch that the geodesic γx0,v0\nminimises the distance between x0and∂M. As∂Mis closed, such a v0always28 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nexists, although it might not be unique. By minimality, as ∂Mis smooth, this\ngeodesicexits Mnormally. Therefore, thereis v0∈Sx0Mintsuchthatl+(x0,v0)<\n∞and the geodesic through ( x0,v0) exitsMnormally.\nThe implicit function theorem guarantees the existence of a neighbo rhoodUof\n(x0,v0) inSMintfor whichl+(x,v)<∞and the geodesic through ( x,v) exitsM\ntransversally for all ( x,v)∈U. Following the notation in [38, Section 3], we can\nfindε0and setsSBε(x0),W⊂U,ε<ε0, for which\n(x,v)∈SBε(x0),W∩SA=⇒(x,v)∈(B1∪B3)∩SA.\nSince by Lemma 5.2 µ(B1∪B3) = 0, it follows by the same reasoning as in [38,\nthe proof of Lemma 3.1] that m(A∩Bε(x0)) = 0,ε<ε0, contradicting the fact\nthatx0was a point of density one in A. /square\n5.2.Perturbation of pairs of geodesics. For (x,v)∈SMint, we denote by\nγx,v: [−T1,T2]→M, 0< T1,T2<∞, the unique unit-speed geodesic such\nthatγx,v(0) =xand ˙γx,v(0) =v. We say that the smooth family of geodesics\n(γs)s∈(−ε,ε), is a variation through geodesics of γx,vwith variation field Jifγ0=\nγx,vandJ(t) =d\nds/vextendsingle/vextendsingle\ns=0γs(t).\nLemma 5.3. Letγx,u: [−Tu,Tu]→Mandγx,v: [−Tv,Tv]→Mbe geodesic\nsegments intersecting only at xand letγsbe a variation through geodesics of γx,v\nwith variation field J. Suppose there are nonzero sequences sj,tj→0such that\nγsjintersectsγx,uat the point γsj(tj)for allj∈N. Then,\nJ(0)∈span({u,v}) = span( {˙γx,u(0),˙γx,v(0)}).\nProof.Let (τ,y) be Fermi coordinates about xadapted to the geodesic γx,u. The\ncoordinates induce the frame∂\n∂τ,∂\n∂y1,...,∂\n∂yn−1. In these coordinates, γx,uis given\nby{y= 0}and henceu= ˙γx,u(0) =∂\n∂τ. We can express γs(t) as (τs(t),ys(t)).\nLet/tildewideJ(t) and/tildewide˙γx,v(t) be the respective components of J(t) and ˙γx,v(t) in they\ncoordinates. We have\nys(t) =y0(t)+s/tildewideJ(t)+O(s2).\nSinceγsjintersectsγx,uat thepoint γsj(tj), we haveysj(tj) = 0 =y0(0). Plugging\nthis in the above expression and rearranging gives\ny0(0)−y0(tj)\ntjtj\nsj=/tildewideJ(tj)+O(sj). (5.1)\nThe right-hand side converges to /tildewideJ(0) asj→ ∞. As for the left-hand side, the\nfirst fraction converges to −/tildewide˙γx,v(0). It follows that tj/sjhas to converge since\n/tildewide˙γx,v(0) is nonzero, and so /tildewideJ(0) is a multiple of /tildewide˙γx,v(0). Asu=∂\n∂τgenerates the\nfirst components of both J(0) andv, the result follows. /squareINVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 29\nLemma 5.4. Let(M,g)be a smooth compact Riemannian manifold of dimension\nn≥3with smooth boundary. Let x∈Mintbe such that there exist distinct\ndirectionsu,v∈SxMintsuch that the geodesics γx,uandγx,vare non-tangential.\nSuppose that the order of conjugacy of xis at most n−3. Then, there is a\nsequence (vn)n∈N⊂SxMint,vn→v, such that for all n∈Nsufficiently large,\nγx,vndoes not intersect γx,u(except atx) andγx,vndoes not self-intersect at x.\nRemark 5.5. The non-intersection result of Lemma 5.4 is global, while th at of\nLemma 5.3 is only local.\nThe main idea of the proof of Lemma 5.4 is to find a direction towards wh ich\nwe can perturb γx,vaway from γx,u. We wish to use Lemma 5.3. Say that\nboth geodesics intersect at q∈M, that is, we have tandrsuch thatγx,v(t) =\nγx,u(r) =q. If we can find a variation through geodesics γsofγx,vwith variation\nfieldJsuch thatJ(t) is transversal to span( {˙γx,v(t),˙γx,u(r)}), then Lemma 5.3\nguarantees that γsandγx,udo not intersect for all nonzero ssmall enough. The\ncondition on the order of conjugacy allows us to do this around ever y point of\nintersection between γx,vandγx,u. The sequence vnis then obtained from the\ndifferent values of ˙ γs(0).\nProof.To simplify notation, we write γx,v=γvas all the geodesics will start at\nx. Letγv: [−T1,T2]→Mbe maximal and intersect γuat the points qjat times\n(tj,rj) with\n−T1≤t−k−≤ ··· ≤t−1<0<t1≤ ··· ≤tk+≤T2\nandrj< rj+1whenevertj=tj+1, that is,γv(tj) =γu(rj) =qj. Without loss of\ngenerality, we can assume that the geodesics do not intersect at t he boundary.\nOtherwise, we can make Mslightly larger and extend the geodesics by adding\na small open neighborhood of ∂MinN. We denote vj=tjv∈TxMintso that\nexpx(vj) =γv(tj). The behaviour of the differential of the exponential map at\nvj∈TxMint,d(expx)vj:TvjTxMint→TqjMintis closely related to the order of\nconjugacy of Matx. Indeed, denoting the restriction of γvto [0,tj] asγj, we\nhave\ndimkerd(expx)vj= conjγj(x,qj),\nsee [7, Proposition 3.5, page 117].\nConsider the linear spaces\nEj={ξ∈TvjTxMint:d(expx)vj(ξ) is parallel to ˙ γu(rj)}.\nNote that the kernel of d(expx)vjis a subspace of Ejand it is the whole space\nunless ˙γu(rj) is in the range of d(expx)vj, in which case the dimension of Ejis\nconjγj(x,qj) + 1. Through the identification of TvjTxMintwithTxMint, we can\nidentifyEjwith the affine plane\nPj={vj+ξ:ξ∈Ej} ⊂TxMint.30 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nBecause the order of conjugacy at xis at mostn−3, the dimension of the affine\nplanePjis at mostn−2. Moreover, the planes Pjare transversal to vj.\nLetπ:TxMint→SxMintbe the natural projection from the tangent space to\nthe unit sphere bundle at x. Forj≥1, we have π(vj) =vwhile forj≤ −1,\nπ(vj) =−v. In any case, near π(vj), the image of Pjthroughπis an immersed\nsubmanifold/tildewidePj⊂SxMintof codimension at least 1. Hence, Tv/tildewidePjis a proper\nsubspace of TvSxMintand since a vector space over an infinite field cannot be\nrealized as the finite union of proper subspaces, there is α∈TvSxMintsuch that\nα/\\e}atio\\slash∈ ∪k+\nj=1Tv/tildewidePj. Similarly, we may also require that −α/\\e}atio\\slash∈ ∪k−\nj=1T−v/tildewideP−j, where\n−α∈T−vSxMintis obtained through the identification with TxMint.\nWe claim that\nvn= expS\nv(α/n)\nis the desired sequence, where expS\nvis the exponential map on TvSxMinttaking\nvalues in the sphere SxMint.\nAsngoes to∞, it is clear that vnconverges to v. It remains to show that γvn\ndoes not intersect γufor allnsufficiently large. Consider the variation through\ngeodesics of γv,\nγs(t) = expx(texpS\nv(sα)).\nNote thatγsis non-tangential for sufficiently small values of sas it is a variation\nof the non-tangential geodesic γv. Since expS\nv(sα) is equal to v+sαto first order,\nwe have\nJ(t) =d\nds/vextendsingle/vextendsingle/vextendsingle\ns=0γs(t) =d\nds/vextendsingle/vextendsingle/vextendsingle\ns=0expx(t(v+sα)) =d(expx)tv(tα)\nwhich is a normal Jacobi field along γv.\nForδ >0, consider the pairs of geodesic segments γv|[tj−δ,tj+δ]andγu|[rj−δ,rj+δ].\nChoosingδsmall enough, we can guarantee that all such pairs intersect exac tly\nonce atqj. By compactness, there is δ >0 such that for all small enough\nvariationss, the only possible intersections between γsandγuhave to occur\nbetween the segments γs|[tj−δ,tj+δ]andγu|[rj−δ,rj+δ]. We can see γs|[tj−δ,tj+δ]as a\nvariation through geodesics of γv|[tj−δ,tj+δ]with variation field J|[tj−δ,tj+δ]. At the\npointst=tjwhereγvintersectsγu, we have\nJ(tj) =d(expx)vj(tjα).\nBy the choice of α, we know that tjαis not inEj. Hence,J(tj) is nonzero and\ntransversal to span( {˙γv(tj),˙γu(rj)}).\nWe claim that there is some εj>0 such that γs|[tj−δ,tj+δ]does not intersect\nγu|[rj−δ,rj+δ]for alls∈(−εj,εj)\\{0}. If such aεjdid not exist, there would be a\nsequencesl→0 and times tj,l∈[tj−δ,tj+δ] such that γslintersectsγu|[rj−δ,rj+δ]\nat the point γsl(tj,l). By compactness, tj,lwould admit a convergent subsequence\nwhich must converge to tjby continuity and because it is the only time whenINVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 31\nγv|[tj−δ,tj+δ]intersectsγu|[rj−δ,rj+δ]. But then, by Lemma 5.3, this would imply\nJ(tj)∈span({˙γv(tj),˙γu(rj)}) which contradicts the above choice of α.\nAs we accounted for all the pairs of segments that could intersect and there are\nfinitely many such pairs, we know that γsdoes not intersect γu(except atxat\ntime 0) for all nonzero ssmall enough. Moreover, for such s,γsdoes not self-\nintersect at xas it would then intersect γu. The claim follows since taking ssmall\ncorresponds to taking nlarge enough. /square\n5.3.Proof of Theorem 1.6. By Lemma 5.1, almost every point of Mintlies on\nsome non-tangential geodesic. Let x∈Mintbe such a point and let γx,vbe a non-\ntangential geodesic passing through x. Asγx,vis non-tangential, we can perturb\nvin some open neighborhood to get another non-tangential geodes icγx,upassing\nthroughx. By Lemma 5.4, we can perturb γx,vto obtainγ1that intersects γx,u\nonly atxand does not self-intersect at x. Ifγx,udoes not self-intersect at x, then\nwe setγ2=γx,uand we are done. Otherwise, we can again apply Lemma 5.4\nto get a geodesic γ2that only intersects γ1atxand does not self-intersect at x.\nThe proof of Theorem 1.6 is complete.\nAppendix A.A rough stationary phase argument\nLetU⊂Rnbe an open set and let 0 < α <1. We define the space C0,α(U) of\nthe H¨ older continuous functions to be the subspace of C(U) consisting of those\nfunctionsufor which supx,y∈U,x/negationslash=y|u(x)−u(y)|\n|x−y|αis finite. The space C0,α(U) is a\nBanach space with norm given by\n/ba∇dblu/ba∇dblC0,α(U)= sup\nx,y∈U,x/negationslash=y|u(x)−u(y)|\n|x−y|α+/ba∇dblu/ba∇dblL∞(U).\nWe note that C0,α(U) is an algebra under pointwise multiplication, and\n/ba∇dbluv/ba∇dblC0,α(U)≤C/parenleftbig\n/ba∇dblu/ba∇dblC0,α(U)/ba∇dblv/ba∇dblL∞(U)+/ba∇dblu/ba∇dblL∞(U)/ba∇dblv/ba∇dblC0,α(U)/parenrightbig\n, u,v∈C0,α(U),\n(A.1)\nsee [18, Theorem A.7].\nIn the proofs of Theorem 1.1 and Theorem 1.8, we need a version of t he sta-\ntionary phase lemma with quite explicit control of all the constants in volved.\nFurthermore, note that in our applications, amplitudes have compa ct supports\nin a geodesic ball. To state the required result, let Br={x∈Rn:|x|< r}be\nan open ball centered at 0 of radius r >0, and leta∈C0,α(Br) be such that\n0∈supp(a) anda|∂Br= 0. Let Ψ ∈C∞(Br;R) be such that\nΨ(0) = 0,Ψ′(0) = 0,Ψ′′(0)>0. (A.2)\nThus, we have\nΨ′′(0)≥cI, (A.3)32 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nwith somec>0. Taylor expanding the phase function Ψ and using (A.2), we get\nΨ(x) =1\n2Ψ′′(0)x·x+Ψ3(x), (A.4)\nwhere\n|Ψ3(x)| ≤C|x|3, x∈Br, (A.5)\nwith someC >0.\nLemma A.1. Letr>0be such that\nCr≤c\n4. (A.6)\nThen forλ≥1, we have\nλn\n2/integraldisplay\nBre−λΨ(x)a(x)dx=(2π)n\n2\n(detΨ′′(0))1/2a(0)+OC,c(1)λ−α\n2/ba∇dbla/ba∇dblC0,α(Br).(A.7)\nHere the implicit constant depends on Candconly and is independent of a.\nProof.First, let us extend aby zero to Rn\\Brand still denote this extension by\na. By Lemma A.3, as a|∂Br= 0, we conclude that a∈C0,α(Rn) and\n/ba∇dbla/ba∇dblC0,α(Rn)=/ba∇dbla/ba∇dblC0,α(Br). (A.8)\nUsing (A.4), we write/integraldisplay\nBre−λΨ(x)a(x)dx=J1+J2+J3,\nwhere\nJ1:=/integraldisplay\nRne−λ\n2Ψ′′(0)x·xa(0)dx, J 2:=/integraldisplay\nRne−λ\n2Ψ′′(0)x·x(a(x)−a(0))dx,\nJ3:=/integraldisplay\nBre−λ\n2Ψ′′(0)x·x/parenleftbig\ne−λΨ3(x)−1/parenrightbig\na(x)dx.\nMaking the change of variables x/mapsto→λ1\n2xin the integral J1, we obtain that\nJ1=a(0)λ−n\n2/integraldisplay\nRne−1\n2Ψ′′(0)x·xdx=(2π)n\n2\n(detΨ′′(0))1/2a(0)λ−n\n2.(A.9)\nTo bound |J2|, first using that a∈C0,α(Rn) and (A.8), we get\n|a(x)−a(0)| ≤ /ba∇dbla/ba∇dblC0,α(Br)|x|α, x∈Rn. (A.10)\nThenusing (A.10), (A.3), andmaking thechangeofvariables x/mapsto→λ1\n2x, weobtain\nthat\n|J2| ≤ /ba∇dbla/ba∇dblC0,α(Br)/integraldisplay\nRne−λc\n2|x|2|x|αdx=/ba∇dbla/ba∇dblC0,α(Br)λ−n\n2−α\n2/integraldisplay\nRne−c\n2|x|2|x|αdx\n=Oc(1)λ−n\n2−α\n2/ba∇dbla/ba∇dblC0,α(Br).\n(A.11)INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 33\nTo bound |J3|, in view of (A.5), we first observe that\n|e−λΨ3(x)−1|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0d\ndte−λΨ3(x)tdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤λ|Ψ3(x)|eλ|Ψ3(x)|≤Cλ|x|3eCλ|x|3,(A.12)\nforx∈Br. Then using (A.3), (A.12), (A.6), and making the change of variables\nx/mapsto→λ1\n2x, we get\n|J3| ≤/integraldisplay\nBre−λc\n2|x|2Cλ|x|3eCλ|x|3|a(x)|dx≤ /ba∇dbla/ba∇dblC0,α(Br)/integraldisplay\nBreλ(−c\n2+Cr)|x|2Cλ|x|3dx\n≤ /ba∇dbla/ba∇dblC0,α(Br)/integraldisplay\nRne−λc\n4|x|2Cλ|x|3dx=/ba∇dbla/ba∇dblC0,α(Br)Cλ−n\n2−1\n2/integraldisplay\nRne−c\n4|x|2|x|3dx\n=OC,c(1)λ−n\n2−1\n2/ba∇dbla/ba∇dblC0,α(Br)≤ OC,c(1)λ−n\n2−α\n2/ba∇dbla/ba∇dblC0,α(Br),\n(A.13)\nforλ≥1. Combining (A.9), (A.11), and (A.13), we obtain (A.7). /square\nRemark A.2. Lemma A.1 also holds under the following assumptions: a∈\nC0,α(Br)is such that supp(a)⊂Bris compact, 0∈supp(a),Ψ∈C∞(supp(a);R)\nsatisfies (A.2), and there exists a constant C >0such that (A.5)holds for all\nx∈supp(a).\nThe following result is standard and is presented here for the comple teness and\nconvenience of the reader only.\nLemma A.3. Leta∈C0,α(Br)be such that a|∂Br= 0. The extension of ato\nRn\\Brby zero, denoted also as a, belongs to the class C0,α(Rn), and we have\n/ba∇dbla/ba∇dblC0,α(Rn)=/ba∇dbla/ba∇dblC0,α(Br).\nProof.LetL= supx,y∈Br,x/negationslash=y|a(x)−a(y)|\n|x−y|α. Sincea∈C(Br), we have\n|a(x)−a(y)| ≤L|x−y|α, (A.14)\nfor allx,y∈Br. Letx,y∈Rn, and let us show (A.14). First if both x,y∈Br\nor bothx,y∈Rn\\Br, (A.14) clearly holds. The only case we have to consider\nis whenx∈Brandy∈Rn\\Br. In this case, we consider the straight-line\nsegment connecting xandy. It intersects ∂Brat some point z∈∂Brso that\n|x−z| ≤ |x−y|. Using (A.14), we get\n|a(x)−a(y)|=|a(x)−a(z)| ≤L|x−z|α≤L|x−y|α.\nThe result follows. /square\nAppendix B.Boundary determination\nIn the proofs of Theorem 1.1 and Theorem 1.8 we need the following bo undary\ndetermination result, which is essentially known, see [3], [4], [17, Appen dix], [12,34 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nAppendix A], [21] and references therein, and presented here for the completeness\nand convenience of the reader.\nProposition B.1. Let(M,g)be a compact smooth Riemannian manifold of di-\nmensionn≥2with smooth boundary. Let q∈C(M),λ >0, and assume that\nλ2is not a Dirichlet eigenvalue of −∆ginM. If\n/integraldisplay\nMqu1u2u3u4dVg= 0, (B.1)\nfor alluj∈C2,α(M)such that\n(−∆g−λ2)uj= 0inMint, (B.2)\nj= 1,...,4, thenq|∂M= 0.\nProof.We shall follow [3], [4], constructing an explicit family of functions vµ,\nwhose boundary values have a highly oscillatory behavior as µ→0, while be-\ncoming increasingly concentrated near a given point on the boundar y ofM, and\nwe shall follow [12, Appendix A], correcting vµinto an exact solution to (B.2).\nLetx0∈∂Mand let (x1,...,xn) be the boundary normal coordinates centered\natx0so that in these coordinates, x0= 0, the boundary ∂Misgiven by {xn= 0},\nandMintisgiven by {xn>0}. Inthelocalcoordinates, Tx0∂M=Rn−1, equipped\nwith the Euclidean metric. The unit tangent vector τis then given by τ= (τ′,0)\nwhereτ′∈Rn−1,|τ′|= 1. Letη∈C∞\n0(Rn;R) be such that supp( η) is in a small\nneighborhood of 0, η(0)/\\e}atio\\slash= 0, and\n/integraldisplay\nRn−1η(x′,0)2dx′= 1.\nLet1\n3≤α≤1\n2, and following [4], we let\nv(x;µ) =µ−α(n−1)\n2−1\n2η/parenleftbiggx\nµα/parenrightbigg\nei\nµ(τ′·x′+ixn),0<µ≪1, (B.3)\nin the boundary normal coordinates, so that v∈C∞(M), with supp( v) inO(µα)\nneighborhood of x0= 0. Hereτ′is viewed as a covector. A direct computation\ngives that\n/ba∇dblv(·;µ)/ba∇dblL2(M)=O(1), (B.4)\nasµ→0. We look for a solution u1to (B.2) in the form\nu1(·;µ) =v(·;µ)+r(·;µ), (B.5)\nwhereris the solution to the Dirichlet problem,\n/braceleftBigg\n(−∆g−λ2)r= (−∆g−λ2)vinMint,\nr|∂M= 0.(B.6)INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 35\nByboundaryellipticregularity, wehave r∈H1\n0(Mint)∩C∞(M). By[12, Theorem\nA.3], we get\n/ba∇dblr/ba∇dblL2(M)≤ /ba∇dblr/ba∇dblH1/2+ε(Mint)≤C/ba∇dbl(−∆g−λ2)v/ba∇dblH−3/2+ε(Mint)\n≤C/ba∇dbl∆gv/ba∇dblH−3/2+ε(Mint)+Cλ2/ba∇dblv/ba∇dblH−1(Mint),(B.7)\nfor 0<ε<1/2. It is shown in [12] that with α= 1/3 andε= 1/12,\n/ba∇dbl∆gv/ba∇dblH−3/2+ε(Mint)=O(µ1/12), (B.8)\nsee formula (A.14) in [12], and that\n/ba∇dblv/ba∇dblH−1(Mint)=O(µ1−α) =O(µ2/3), (B.9)\nsee formula (A.19) in [12] and discussion after it. Combining (B.7), (B.8 ), and\n(B.9), we get\n/ba∇dblr/ba∇dblL2(M)=O(µ1/12). (B.10)\nSubstituting u1given by (B.5) and u2=u1into the integral identity (B.1), and\nusing (B.4), (B.10), we obtain that/integraldisplay\nM(qu3u4)|v|2dVg=−/integraldisplay\nM(qu3u4)(vr+rv+|r|2)dVg=o(1),(B.11)\nasµ→0. It is computed in [12], see formula (A.24), that\nlim\nµ→0/integraldisplay\nM(qu3u4)|v|2dVg=1\n2q(0)u3(0)u4(0). (B.12)\nIt follows from (B.11) and (B.12) that\nq(0)u3(0)u4(0) = 0, (B.13)\nwhereu3,u4∈C2,α(M) are any solutions of (B.2). Letting u3(·;h) =v(·;h) +\nr(·;h), 0< h≪1, wherev(·;h) is given by (B.3) with µreplaced by h,r(·;h)\nis the solution to (B.6) with v=v(·;h), and letting u4=u3, we get from (B.13)\nthatq(0) = 0. Here we have used that η(0)/\\e}atio\\slash= 0. /square\nAppendix C.Example of an infinite set of linearly independent\nadmissible perturbations\nLetMbe a smooth compact Riemannian manifold of dimension n≥2 with\nboundary. We write C0,α(M), 0< α <1, for the space of H¨ older continuous\nfunctions on M, equipped with the norm,\n/ba∇dblϕ/ba∇dblC0,α(M)=/ba∇dblϕ/ba∇dblL∞(M)+ sup\nx,y∈M,x/negationslash=y|ϕ(x)−ϕ(y)|\nd(x,y)α,\nwhered(x,y) is the Riemannian distance between xandy, see [2, Chapter 2, §3,\n§9], [45, Section 13.8]. We define the C1-norm onMby\n/ba∇dblϕ/ba∇dblC1(M)=/ba∇dblϕ/ba∇dblL∞(M)+/ba∇dbl∇gϕ(x)/ba∇dblL∞(M),36 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nwhere∇gis the gradient with respect to the Riemannian metric g. We have the\ncontinuous embedding C1(M)⊂C0,α(M), 0< α <1, i.e. there exists B >0\nsuch that\n/ba∇dblϕ/ba∇dblC0,α(M)≤B/ba∇dblϕ/ba∇dblC1(M), ϕ∈C1(M). (C.1)\nIn what follows we refer to the definition and properties of simple man ifolds to\n[35, Section 3.8]. We have the following result.\nProposition C.1. Let(M,g)be simple. The set of admissible perturbations\nA(2B), withBgiven in (C.1), contains an infinite set of linearly independent\nadmissible perturbations.\nProof.First we have ( M,g)⊂⊂(U,g) where (U,g) is simple. Let ω∈U\\M\nand let (r,θ),r>0 andθ∈Sn−1, be the polar normal coordinates in ( U,g) with\ncenterω. In these coordinates the metric ghas the form,\ng(r,θ) =/parenleftbigg\n1 0\n0g1(r,θ)/parenrightbigg\n, (C.2)\nwhereg1(r,θ) is some (n−1)×(n−1) positive definite matrix. Consider the\npolynomial\np(r,θ) =N/summationdisplay\nk=0akrk∈C∞(M),\nwhereak≥0,k= 0,1,...N,\nkak≤ak−1, k= 1,...N, (C.3)\nandN∈N. It follows from (C.2) that ∇gp= (∂rp,0,...,0). Thus, using (C.2)\nand (C.3), we get\n|∇gp|g=|∂rp|=N/summationdisplay\nk=1akkrk−1≤N/summationdisplay\nk=1ak−1rk−1≤p.\nHence,/ba∇dblp/ba∇dblC1(M)≤2/ba∇dblp/ba∇dblL∞(M). Combining this with (C.1), we get /ba∇dblp/ba∇dblC0,α(M)≤\n2B/ba∇dblp/ba∇dblL∞(M), showing that p∈ A(2B). Thus, in particular, we have {/summationtextN\nk=0rk\nk!:\nN= 1,2,...} ⊂ A(2B).\n/square\nAppendix D.Discussion of the geometric assumption (H1) in\nTheorem 1.8\nD.1.Local result. Thefollowing result demonstrates thatlocally, condition(iv)\nin assumption (H1) can be derived from condition (iii).INVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 37\nProposition D.1. Let(N,g)be a smooth compact Riemannian manifold without\nboundary whose sectional curvatures are bounded from above byκ >0. Letρ <\nmin(Conv(N),π√κ), whereConv(N)>0is the convexity radius of N. There is\na constant C=C(N,g,ρ)such that for all x0∈Nand all pairs of geodesics\nγ,η: (−ρ,ρ)→Nwithγ(0) =η(0) =x0,\nd(γ(t),η(τ))≥C|t|sinθ, d(γ(t),η(τ))≥C|τ|sinθ, t,τ∈(−ρ,ρ),\nwhereθis the angle of intersection between γandη.\nProof.Let (xi) denote normal coordinates around x0with respect to the metric\ngonU=Bρ(x0). In these coordinates, the geodesics γandηare simply given\nby lines through the origin. Let α: [0,T]→Ube the shortest path between γ(t)\nandη(τ), which is unique and entirely contained in Usinceρ <Conv(N). The\ndistance between γ(t) andη(τ) is then\ndg(γ(t),η(τ)) =/integraldisplayT\n0|˙α(t)|gdt\nWe parametrise αby arclength with respect to the coordinates ( xi) equipped\nwith the standard flat metric g0ofRn, so that |˙α(t)|g0= 1. With respect to g0,\nthe shortest path from γ(t) toηintersectsηat a right angle. Therefore,\nT=dg0(γ(t),η(τ))≥dg0(γ(t),η) =|t|sinθ\nfor allt∈(−ρ,ρ). Letgκbe a metric of constant sectional curvature κonU\nwhich exists since ρ <π√κ. The manifold ( U,gκ) is isometric to a spherical cap,\nand hencegκcan be extended to the closure of U. By compactness, there is a\nconstantC=C(κ)>0 such that gκ≥Cg0as bilinear forms. By [28, Theorem\n11.10], we have g≥gκonUsinceρ<π√κ. Therefore,\ndg(γ(t),η(τ))≥/integraldisplayT\n0|˙α(t)|gκdt≥C/integraldisplayT\n0|˙α(t)|g0dt≥C|t|sinθ.\n/square\nD.2.Discussion related to Example 1.9. Let (M,g) be a simple Riemannian\nmanifold. We shall demonstrate that it satisfies the geometric assu mption (H1).\nIndeed, consider the bundle\nSM×πSM={((x,v),(y,w))∈SM×SM:x=y}.\nAsMisasimplemanifold, byapplyingthestoppedgeodesicflowinbackwarda nd\nforwardtime, wecansee SM×πSMasaparameterspaceforpairsofgeodesicson\nMthat intersect at a single point. Indeed, to each pair (( x,v),(x,w)) = (x,v,w),\nwe associate the two non-tangential geodesics γx,vandγx,w.38 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nFor 0< θ0< π/2, letEθ0⊂SM×πSMbe the set of pairs ( x,v,w) such that\n|gx(v,w)| ≥cosθ0. The setEθ0is compact andconsists of geodesics that intersect\nat an angle θ∈[θ0,π/2].\nFor eachx0∈Mint, choose a pair ( x0,v,w)∈Eθ0and letγ=γx0,vandη=γx0,w.\nLet us show that such pairs of geodesics satisfy the assumption (H 1).\nFirst,γ(0) =η(0) =x0and so (i) holds (with shifted times so that t0=τ0= 0).\nAsMis simple, it is non-trapping and there exists a T >0 such that all geodesics\nhave length at most Tso (ii) holds. By our choice of pairs within Eθ0, (iii) holds\nwithθ0as above.\nItremainstoshow(iv). Embed( M,g)inasmoothcompactRiemannianmanifold\nwithout boundary ( N,g)whose sectional curvatures are bounded above by κ>0.\nLetρbe as in Proposition D.1 and consider the function F:Eθ0→[0,∞] given\nby\nF(x,v,w) = dist(γx,v(Ix,v\\(−ρ,ρ)),γx,w(Ix,w))\nwhereIx,vis the closed interval on which γx,vis defined in M,F(x,v,w) =∞\nifIx,v⊂(−ρ,ρ), and dist( A,B) = inf{dg(a,b) :a∈A,b∈B}for any pair\nof subsetsAandBofM. By continuous dependence of the geodesics γx,von\n(x,v)∈SM, the set\nVρ={(x,v,w)∈Eθ0:Ix,v⊂(−ρ,ρ)}\nis open and Fis continuous on Eθ0\\Vρ. It follows that the sets F−1((y,∞]) are\nopen for all y∈R, that is,Fis lower semicontinuous.\nThe function ( x,v,w)/mapsto→min(F(x,v,w),F(x,w,v)) is then also lower semicon-\ntinuous. Therefore, by compactness of Eθ0and the fact that geodesics in M\nintersect at most once, there is c=c(M,g,ρ)>0 such that\nd(γx,v(t),γx,w(τ))≥c\nwhenever |t| ≥ρor|τ| ≥ρforall(x,v,w)∈Eθ0. Thisholdsbecause F(x,v,w)≥\ncif and only if d(γx,v(t),γx,w(τ))≥cfor all|t| ≥ρ. Hence, by choosing r <\nmin(c,Inj(M)/2), we see that if d(γ(t),η(τ))≤r, then|t|< ρand|τ|< ρ.\nProposition D.1 then guarantees there is C=C(M,g,ρ)>0 such that\n|t| ≤d(γ(t),η(τ))\nCsinθ0,|τ| ≤d(γ(t),η(τ))\nCsinθ0.\nHere we have also used that θ∈[θ0,π/2]. Condition (iv) follows from choosing r\nas above and c0= (Csinθ0)−1. Sinceρonly depends on ( M,g), the constants r,\nc0, andθ0can be chosen uniformly.\nD.3.Discussion related to Example 1.10. LetM=S1×[0,a],a >0, be\nthe cylinder with the usual flat metric. We shall show it satisfies the g eometricINVERSE PROBLEMS FOR SEMILINEAR SCHR ¨ODINGER EQUATIONS 39\nassumption (H1). Indeed, let x0= (eit0,s0)∈Mintand forθ∈(0,π/2), consider\nthe geodesics\nγx0(t) = (eit0,t), t∈[0,a],\nηθ\nx0(τ) = (ei(t0−s0tanθ+τsinθ),τcosθ), τ∈/bracketleftBig\n0,a\ncosθ/bracketrightBig\n.\nThe geodesics γx0andηθ\nx0intersect at the point x0=γx0(s0) =ηθ\nx0(s0/cosθ)\nwith an intersection angle of θ. If|−s0tanθ+τsinθ|<2πfor allτ∈/bracketleftbig\n0,a\ncosθ/bracketrightbig\n,\nthe geodesics γx0andηθ\nx0do not intersect at any other points. This condition is\nguaranteed if we require\n|−s0tanθ+τsinθ| ≤2atanθ<2π,\nwhich implies that 0 <θ<arctan(π\na).\nLetθ0=1\n4arctan(π\na). Ifθ∈[θ0,2θ0), thenηθ\nx0rotates at most half a turn\naround the cylinder. Unwrapping that half of the cylinder into [0 ,π]×[0,a],\nwe see that the distance between γx0(t) andηθ\nx0(τ) is the same as the Euclidean\ndistance. The shortest path from γx0(t) to the curve ηθ\nx0intersectsηθ\nx0at a right\nangle and therefore d(γx0(t),ηθ\nx0) =|t−s0|sinθsinceγx0(s0) =x0. Similarly,\nd(γ,ηθ\nx0(τ)) =|τ−s0/cosθ|sinθsinceηθ\nx0(s0/cosθ) =x0. Therefore,\nd(γx0(t),ηθ\nx0(τ))≥sinθmax(|t−s0|,|τ−s0/cosθ|),\nfor allt∈[0,a] andτ∈/bracketleftbig\n0,a\ncosθ/bracketrightbig\n.\nTherefore, if we choose a pair of geodesics γ=γx0andη=ηθ\nx0withθ∈[θ0,2θ0)\nfor every point x0∈Mint, we get a family of pairs of geodesics that satisfies\nassumption (H1) with θ0as above,T=a\ncos(2θ0), 0<r<π/ 2, andc0= (sinθ0)−1.\nAcknowledgements\nThe research of K.K. is partially supported by the National Science F oundation\n(DMS 2109199). The research of S.M. is partially supported by the N SF of China\nunder the grant No. 12301540. S.K.S is partially supported by the Ac ademy of\nFinland (Centre of Excellence in Inverse Modelling and Imaging, grant 284715)\nand by the European Research Council under Horizon 2020 (ERC Co G 770924).\nThe research of M.S. is supported by Research Council Finland (CoE in Inverse\nModelling and Imaging and FAME flagship, grants 353091 and 359208) and by\nthe European Research Council under Horizon 2020 (ERC CoG 7709 24). The\nresearch ofS.S.-A. is partiallysupported bythe NaturalSciences a ndEngineering\nResearch Council of Canada.40 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\nReferences\n[1] Angulo-Ardoy, P., On the set of metrics without local limiting Carleman weight s, Inverse\nProbl. Imaging 11(2017), no. 1, 47–64.\n[2] Aubin, T., Nonlinear analysis on manifolds. 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Soc. 4(1991), no. 4, 729–769.\n[42] St-Amant, S., Gaussian beams and Calder´ on problem for connections at lar ge frequency ,\npreprint 2024.42 KRUPCHYK, MA, KUMAR SAHOO, SALO, AND ST-AMANT\n[43] Stefanov, P., Resonance expansions and Rayleigh waves , Math. Res. Lett. 8(2001), no.\n1–2, 107–124.\n[44] Tang, S.-H., Zworski, M., From quasimodes to resonances , Math. Res. Lett. 5(1998), no.\n3, 261–272.\n[45] Taylor, M., Partial differential equations III. Nonlinear equations , Applied Mathematical\nSciences, 117. Springer, New York, 2011.\n[46] Uhlmann, G., Inverse problems: seeing the unseen , Bull. Math. Sci. 4(2014), no. 2, 209–\n279.\n[47] Uhlmann, G., Wang, Y., The anisotropic Calder´ on problem for high fixed frequency , SIAM\nJ. Math. Anal., to appear.\n[48] Zworski, M., Semiclassical analysis , Graduate Studies in Mathematics, 138. American\nMathematical Society, Providence, RI, 2012.\nK. Krupchyk, Department of Mathematics, University of Cali fornia, Irvine, CA\n92697-3875, USA\nEmail address :katya.krupchyk@uci.edu\nS. Ma, School of Mathematics, Jilin University, Changchun, China\nEmail address :mashiqi@jlu.edu.cn\nS. K. Sahoo, Seminar for Applied Mathematics, Department of Mathematics,\nETH Z¨urich, Switzerland\nEmail address :susahoo@ethz.ch\nM. Salo, Department of Mathematics and Statistics, Univers ity of Jyv ¨askyl¨a,\nJyv¨askyl¨a, FI-40014, Finland\nEmail address :mikko.j.salo@jyu.fi\nS. St-Amant, Department of Pure Mathematics and Mathematic al Statistics,\nUniversity of Cambridge, Cambridge CB3 0WB, UK\nEmail address :sas242@cam.ac.uk"    },    {        "title": "2402.12954v1.Conditional_Logical_Message_Passing_Transformer_for_Complex_Query_Answering.pdf",        "content": "Conditional Logical Message Passing Transformer for Complex\nQuery Answering\nChongzhi Zhang\ncschongzhizhang@mail.scut.edu.cn\nSouth China University of Technology\nGuangzhou, ChinaZhiping Peng\nGuangdong University of Petrochemical Technology\nMaoming, China\nJiangmen Polytechnic\nJiangmen, China\nJunhao Zheng\njunhaozheng47@outlook.com\nSouth China University of Technology\nGuangzhou, ChinaQianli Ma∗\nqianlima@scut.edu.cn\nSouth China University of Technology\nGuangzhou, China\nABSTRACT\nComplex Query Answering (CQA) over Knowledge Graphs (KGs)\nis a challenging task. Given that KGs are usually incomplete, neural\nmodels are proposed to solve CQA by performing multi-hop logical\nreasoning. However, most of them cannot perform well on both one-\nhop and multi-hop queries simultaneously. Recent work proposes\na logical message passing mechanism based on the pre-trained\nneural link predictors. While effective on both one-hop and multi-\nhop queries, it ignores the difference between the constant and\nvariable nodes in a query graph. In addition, during the node em-\nbedding update stage, this mechanism cannot dynamically measure\nthe importance of different messages, and whether it can capture\nthe implicit logical dependencies related to a node and received\nmessages remains unclear. In this paper, we propose Conditional\nLogical Message Passing Transformer (CLMPT), which considers\nthe difference between constants and variables in the case of using\npre-trained neural link predictors and performs message passing\nconditionally on the node type. We empirically verified that this\napproach can reduce computational costs without affecting per-\nformance. Furthermore, CLMPT uses the transformer to aggregate\nreceived messages and update the corresponding node embedding.\nThrough the self-attention mechanism, CLMPT can assign adaptive\nweights to elements in an input set consisting of received messages\nand the corresponding node and explicitly model logical depen-\ndencies between various elements. Experimental results show that\nCLMPT is a new state-of-the-art neural CQA model.\n∗Corresponding author\nPermission to make digital or hard copies of all or part of this work for personal or\nclassroom use is granted without fee provided that copies are not made or distributed\nfor profit or commercial advantage and that copies bear this notice and the full citation\non the first page. Copyrights for components of this work owned by others than ACM\nmust be honored. Abstracting with credit is permitted. To copy otherwise, or republish,\nto post on servers or to redistribute to lists, requires prior specific permission and/or a\nfee. Request permissions from permissions@acm.org.\nConference acronym ’XX, June 03–05, 2018, Woodstock, NY\n©2018 ACM.\nACM ISBN 978-1-4503-XXXX-X/18/06\nhttps://doi.org/XXXXXXX.XXXXXXXCCS CONCEPTS\n•Computing methodologies →Reasoning about belief and\nknowledge ;Knowledge representation and reasoning ;•In-\nformation systems →Data mining .\nKEYWORDS\nKnowledge Graph, Logical Reasoning, Complex Query Answering,\nGraph Neural Network\nACM Reference Format:\nChongzhi Zhang, Zhiping Peng, Junhao Zheng, and Qianli Ma. 2018. Condi-\ntional Logical Message Passing Transformer for Complex Query Answering.\nInProceedings of Make sure to enter the correct conference title from your\nrights confirmation emai (Conference acronym ’XX). ACM, New York, NY,\nUSA, 13 pages. https://doi.org/XXXXXXX.XXXXXXX\n1 INTRODUCTION\nKnowledge graphs (KGs) store factual knowledge in the form of\ntriples that can be utilized to support a variety of downstream tasks\n[42,54,58]. However, given that modern KGs are usually auto-\ngenerated [ 46] or built through crowd-sourcing [ 50], so real-world\nKGs [ 7,9,43] are often considered noisy and incomplete, which is\nalso known as the Open World Assumption [ 22,26]. Answering\nqueries on such incomplete KGs is a fundamental yet challenging\ntask. To alleviate the incompleteness of KGs, knowledge graph rep-\nresentation methods [ 8,15,44,47], which can be viewed as neural\nlink predictors [ 3], have been developed. They learn representa-\ntions based on the available triples and generalize them to unseen\ntriples. Such neural link predictors can answer one-hop atomic\nqueries on incomplete KGs. But for multi-hop complex queries,\nthe query-answering models need to perform multi-hop logical\nreasoning over incomplete KGs. That is, not only to utilize avail-\nable knowledge to predict the unseen one but also execute logical\noperators, such as conjunction ( ∧), disjunction (∨), and negation\n(¬) [39, 57].\nRecently, neural models [ 2,39,67] have been proposed to solve\nComplex Query Answering (CQA) over incomplete KGs. The com-\nplex queries that these models aim to address belong to an important\nsubset of the first-order queries, specifically Existentially quantified\nFirst Order queries with a single free variable (EFO-1) [ 56,57]. Such\nEFO-1 queries contain logical operators, including conjunction,arXiv:2402.12954v1  [cs.LG]  20 Feb 2024Conference acronym ’XX, June 03–05, 2018, Woodstock, NY Chongzhi Zhang, Zhiping Peng, Junhao Zheng, and Qianli Ma\nFigure 1: EFO-1 query and its query graph for the question\n“Who starred the films that were directed by Mizoguchi but\nnever won a Venice Film Festival Award?”.\ndisjunction, and negation, as well as the existential quantifier ( ∃)\n[39]. As shown in Figure 1, one can get the corresponding EFO-1\nformula and query graph given a question. The query graph is a\ngraphical representation of the EFO-1 query, where each edge is an\natomic formula that contains a predicate with a (possible) negation\noperator, and each node represents an input constant entity or a\nvariable. Most neural CQA models convert the query graph into\nthe computation graph, where logical operators are replaced with\ncorresponding set operators. These models embed the entity set\ninto specific vector space [ 10,39,63,67] and execute set operations\nparameterized by the neural networks according to the computa-\ntion graph. However, these models cannot perform well on both\none-hop and multi-hop queries simultaneously. They tend to be\nless effective than classical neural link predictors [47] on one-hop\natomic queries [3, 56].\nTo this end, Logical Message Passing Neural Network (LMPNN)\n[56] proposes a message passing framework based on pre-trained\nneural link predictors. Specifically, for each edge in the query graph,\nLMPNN uses the pre-trained neural link predictor to infer an in-\ntermediate embedding for a node given neighborhood information.\nThe intermediate embedding can be interpreted as a logical mes-\nsage passed by the neighbor on the edge. Figure 1 demonstrates\nthe logical message passing with red/blue arrows. Then, LMPNN\naggregates messages and updates node embeddings in a manner\nsimilar to Graph Isomorphism Network (GIN) [ 60]. While effec-\ntive on both one-hop and multi-hop queries, LMPNN ignores the\ndifference between constant and variable nodes. Logical message\npassing can be viewed as learning an embedding for the variable\nby utilizing information of constant entities. It makes this embed-\nding similar or close to the embedding of the solution entity in the\nembedding space of the pre-trained neural link predictor. However,\nas shown by the red arrows in Figure 1, LMPNN also passes mes-\nsages to constant entities with pre-trained embeddings and updates\ntheir embeddings, which we believe is unnecessary and experi-\nmentally demonstrated. In addition, for node embedding updating,\ndespite the expressive power of GIN in graph structure learning\n[60], it cannot dynamically measure the importance of different\nlogical messages. Furthermore, it is unclear whether the GIN-like\napproach can capture the implicit complex logical dependencies be-\ntween messages received by a node and between messages and that\nnode. Explicitly modeling the above implicit logical dependencieswhile being able to measure the importance of different messages\ndynamically is an open challenge we take on in this paper.\nWe propose Conditional Logical Message Passing Transformer\n(CLMPT), a special Graph Neural Network (GNN) for CQA. Each\nCLMPT layer has two stages: (1) passing the logical message of\neach node to all of its neighbors that are not constant entity nodes;\n(2) updating the embedding of each variable node based on the\nlogical messages received by the variable node. This means that\nduring forward message passing, CLMPT does not use the pre-\ntrained neural link predictor to infer the intermediate embeddings\n(i.e. logical messages) for constant entity nodes with pre-trained\nembeddings and does not update constant node embeddings. We\ncall such a mechanism Conditional Logical Message Passing, where\nthe condition depends on whether the node type is constant or\nvariable. We empirically verified that it can reduce computational\ncosts without affecting performance. Furthermore, CLMPT uses\nthe transformer [ 49] to aggregate messages and update the corre-\nsponding node embedding. The self-attention mechanism in the\ntransformer can dynamically measure the importance of different\nelements in the input and capture the interactions between any two\nelements. It allows us to explicitly model the logical dependencies\nbetween messages received by a node and between messages and\nthat node. We conducted experiments on three popular KGs: FB15k\n[8], FB15k-237 [ 46], and NELL995 [ 59]. The experimental results\nshow that CLMPT is a new state-of-the-art neural CQA model.\n2 RELATED WORK\nNeural Link Predictors. Reasoning over KGs with missing knowl-\nedge is one of the fundamental problems in Artificial Intelligence\nand has been widely studied. Traditional KG reasoning tasks such\nas link prediction [ 17,47] are essentially one-hop atomic query\nproblems. Representative methods for link prediction are knowl-\nedge graph representation methods [ 8,15,35,44,47,62], which can\nbe regarded as neural link predictors [ 3]. Specifically, they embed\nentities and relations into continuous vector spaces and predict un-\nseen triples by scoring triples with a well-defined scoring function.\nSuch latent feature models can effectively answer one-hop atomic\nqueries over incomplete KGs. Other methods for link prediction in-\nclude rule learning [ 40,65], text representation learning [ 41,51,53],\nand GNNs [48, 66, 68].\nNeural Complex Query Answering. Complex queries over KGs can\nbe regarded as one-hop atomic queries combined with existential\nfirst-order logic operators. The scope of complex queries that exist-\ning works can answer is expanded from conjunctive queries [ 20,24],\nto Existential Positive First-Order (EPFO) queries [ 3,38], and more\nrecently to the existential first-order [ 39], or more specifically to\nEFO-1 queries [ 56,57]. Recently, neural query embedding (QE) mod-\nels have been proposed to answer complex queries. Most QE models\nembed entity sets into specific continuous vector spaces with var-\nious forms, including geometric shapes [ 2,12,20,27,34,38,67],\nprobabilistic distributions [ 11,39,63], fuzzy logic [ 10], and bounded\nhistograms on uniform grids [ 55]. They represent the complex\nquery in the form of the computation graph and perform neural\nset operations in the corresponding embedding spaces according\nto the computation graph. Such neural set operations are usually\nmodeled by complex neural networks [ 45,49]. In spite of gettingConditional Logical Message Passing Transformer for Complex Query Answering Conference acronym ’XX, June 03–05, 2018, Woodstock, NY\nquery embedding with neural set operations, other QE models rep-\nresent the queries as graphs to embed queries with GNNs [ 1,14]\nor Transformers [ 24,28,52]. However, these graph representation\nlearning models only focus on EPFO queries, and how they handle\nthe logical negation operator is unclear. Moreover, the aforemen-\ntioned neural models are still less effective than a simple neural\nlink predictor [47] on one-hop atomic queries.\nThere is also a class of neural CQA models based on pre-trained\nneural link predictors. CQD-CO [ 3] proposes a continuous opti-\nmization strategy that uses pre-trained neural link predictors [ 47]\nand logical t-norms [ 23] to solve complex queries. However, accord-\ning to [ 56], CQD-CO performs badly on negative queries. LMPNN\n[56], which is most related to our work, proposes a logical message\npassing framework based on pre-trained neural link predictors.\nThrough one-hop inference on atomic formulas, LMPNN performs\nbidirectional logical message passing on each edge in the query\ngraph. However, such a mechanism ignores the difference between\nthe constant node with a pre-trained embedding and the variable\nnode whose embedding needs update. Our experimental results\nshow that passing messages to the constant node and updating its\nembedding is unnecessary. In addition, at the node level, LMPNN\naggregates the received logical messages and updates the node em-\nbeddings in a manner similar to GIN [ 60]. Despite the expressive\npower of GIN in graph structure learning, it cannot dynamically\nmeasure the importance of different messages. Whether such a GIN-\nlike approach can capture the implicit logical dependencies between\nmessages received by a node and between messages and that node\nremains unclear. By contrast, our work only performs conditional\nmessage passing on the query graph, which avoids unnecessary\ncomputation costs. Furthermore, by using the transformer [ 49] to\naggregate the message and update the corresponding node embed-\nding, our work can assign adaptive weights to elements in an input\nset consisting of received messages and the corresponding node and\nexplicitly model logical dependencies between various elements.\nSymbolic Integration Models. In addition to neural CQA models,\nsome recent studies integrate symbolic information into neural mod-\nels, which can be called symbolic integration models, also known\nas neural-symbolic models [ 3,4,6,61,69]. These models estimate\nthe probability of whether each entity is the answer at each inter-\nmediate step in the process of multi-hop logical reasoning. This\nmakes the size of the intermediate states for symbolic reasoning\nin symbolic integration models scale linearly with the size of the\nKG. Therefore, compared with neural CQA models where the inter-\nmediate embeddings are of fixed dimension, symbolic integration\nmodels require more computing resources and are more likely to\nsuffer from scalability problems on large-scale KGs [37, 56].\n3 BACKGROUND\nModel-theoretic Concepts for Knowledge Graphs. A Knowledge\nGraphKG consists of a set of entities Vand a set of relations\nR. It can be defined as a set of triples E=(𝑒ℎ𝑖,𝑟𝑖,𝑒𝑡𝑖) ⊆ V×\nR×V , namelyKG=(V,E,R). A first-order language Lcan\nbe defined as(F,R,C), whereF,R, andCare sets of symbols\nfor functions, relations, and constants, respectively [ 56]. Under\nlanguageLKG, theKG can be represented as a first-order logic\nknowledge base [ 3]. In this case, the relation symbols in Rdenotebinary relations, the constant symbols in Crepresent the entities,\nand the function symbol set satisfies F=∅. In other words,KG\nis anLKG-structure, where each entity 𝑒∈V is also a constant\n𝑐∈ C=Vand each relation 𝑟∈ R is a set𝑟⊆ V×V . We\nhave𝑟(𝑡1,𝑡2)=𝑇𝑟𝑢𝑒 when(𝑡1,𝑡2)∈𝑟. An atomic formula is either\n𝑟(𝑡1,𝑡2)or¬𝑟(𝑡1,𝑡2), where𝑡𝑖is a term that can be a constant or a\nvariable, and 𝑟is a relation that can be viewed as a binary predicate\n[3]. A variable is a bound variable when associated with a quantifier.\nOtherwise, it is a free variable. By adding connectives (conjunction\n∧, disjunction∨, and negation¬) to such atomic formulas and\nadding quantifiers (existential ∃and universal∀) to variables, we\ncan inductively define the first order formula [33, 56].\nEFO-1 Queries. In this paper, we consider Existential First Order\nqueries with a single free variable (EFO-1) [ 56,57]. Such EFO-1\nqueries are an important subset of first-order queries, using exis-\ntential quantifier, conjunction, disjunction, and atomic negation.\nFollowing the previous studies [ 38,39,56], we define the EFO-1\nquery as the first order formula in the following Disjunctive Normal\nForm (DNF) [13]:\n𝑞[𝑦,𝑥1,...,𝑥𝑘]=∃𝑥1,...,∃𝑥𝑘(𝑎1\n1∧...∧𝑎1\n𝑛1)\n∨...∨(𝑎𝑑\n1∧...∧𝑎𝑑\n𝑛𝑑),(1)\nwhere𝑦is the only free variable, 𝑥1,...,𝑥𝑘are existential variables.\n𝑎𝑖\n𝑗are atomic formulas that can be either negated or not:\n𝑎𝑖\n𝑗=\u001a𝑟(𝑒𝑐,𝑣)𝑜𝑟 𝑟(𝑣,𝑣′)\n¬𝑟(𝑒𝑐,𝑣)𝑜𝑟¬𝑟(𝑣,𝑣′), (2)\nwhere𝑒𝑐∈V is an input constant entity, 𝑣,𝑣′∈{𝑦,𝑥1,...,𝑥𝑘}are\nvariables,𝑣≠𝑣′. To address the EFO-1 query 𝑞, the corresponding\nanswer entity set⟦𝑞⟧⊆V needs to be determined, where ⟦𝑞⟧is\na set of entities such that 𝑒∈⟦𝑞⟧iff𝑞[𝑦=𝑒,𝑥1,...,𝑥𝑘]=𝑇𝑟𝑢𝑒 . In\nparticular, since we use DNF to represent the first order formula, the\nEquation 1 can also be expressed as the disjunction of conjunctive\nqueries:\n𝑞[𝑦,𝑥1,...,𝑥𝑘]=𝐶𝑄1(𝑦,𝑥1,...,𝑥𝑘)∨...∨𝐶𝑄𝑑(𝑦,𝑥1,...,𝑥𝑘),(3)\nwhere𝐶𝑄𝑖=∃𝑥1,...,∃𝑥𝑘𝑎𝑖\n1∧...∧𝑎𝑖𝑛𝑖is a conjunctive query.\nAccording to [ 38,56], we can get the answer set ⟦𝑞⟧by taking\nthe union of the answer sets of each conjunctive query, namely\n⟦𝑞⟧=Ð𝑑\n𝑖=1⟦𝐶𝑄𝑖⟧. This means that solving all conjunctive queries\n𝐶𝑄𝑖,1≤𝑖≤𝑑yields the answer set ⟦𝑞⟧for the EFO-1 query 𝑞.\nSuch DNF-based processing can solve complex queries involving\ndisjunction operators in a scalable way [36–38].\nQuery Graph. Since disjunctive queries can be solved in a scalable\nmanner by transforming queries into the disjunctive normal form,\nit is only necessary to define query graphs for conjunctive queries.\nWe follow previous works [ 38,56] and represent the conjunctive\nquery as the query graph where the terms are represented as nodes\nconnected by the atomic formulas. That is, each edge in the query\ngraph is an atomic formula. According to the definition in Equation\n2, an atomic formula contains relations, negation information, and\nterms. Therefore, each node in the query graph is either a constant\nentity or a free or existential variable, as illustrated in Figure 1.Conference acronym ’XX, June 03–05, 2018, Woodstock, NY Chongzhi Zhang, Zhiping Peng, Junhao Zheng, and Qianli Ma\nNeural Link Predictors. A neural link predictor has a correspond-\ning scoring function that is used to learn the embeddings of entities\nand relations in KG. For a triple, the embeddings of the head entity,\nrelation, and tail entity are ℎ,𝑟, and𝑡, respectively. The scoring\nfunction𝜙(ℎ,𝑟,𝑡)can calculate the likelihood score of whether the\ntriple exists. By using the scoring function 𝜙(ℎ,𝑟,𝑡)and the sig-\nmoid function 𝜎, neural link predictors can produce a continuous\ntruth value𝜓(ℎ,𝑟,𝑡)∈[ 0,1]for a triple. For example, the scoring\nfunction of ComplEx [47] is as follows:\n𝜙(ℎ,𝑟,𝑡)=𝑅𝑒\u0000\nℎ⊗𝑟,𝑡\u000b\u0001, (4)\nwhere⊗represents the element-wise complex number multiplica-\ntion,⟨·,·⟩is the complex inner product, and 𝑅𝑒means taking the\nreal part of a complex number. Based on 𝜙(ℎ,𝑟,𝑡), we can obtain\nthe corresponding truth value:\n𝜓(ℎ,𝑟,𝑡)=𝜎(𝜙(ℎ,𝑟,𝑡)). (5)\nIn this work, the neural link predictor handles not only specific\nentity embeddings but also variable embeddings. That is, the em-\nbeddingsℎ,𝑡correspond to embedding of terms, which can be the\nembeddings of constant entities or the embeddings of variables.\nFollowing previous works [ 3,56], in our work, we use ComplEx-N3\n[25, 47] as the neural link predictor of CLMPT.\nOne-hop Inference on Atomic Formulas. As shown in Figure 1,\neach edge in a query graph is an atomic formula containing the\ninformation of relation, logical negation, and terms. Passing a logi-\ncal message on an edge is essentially an operation of utilizing this\ninformation to perform one-hop inference on an atomic formula.\nOn each edge, when a node is at the head position, its neighbor is\nat the tail position, and vice versa. A logical message passed from a\nneighbor to a node is essentially an intermediate embedding of that\nnode. Prior work [ 56] proposes to obtain the intermediate embed-\ndings of nodes by one-hop inference that maximizes the continuous\ntruth value of the (negated) atomic formulas. Specifically, a logical\nmessage encoding function 𝜌is proposed to perform one-hop infer-\nence. Such function has four input parameters, including neighbor\nembedding, relation embedding, direction information ( ℎ→𝑡or\n𝑡→ℎ), and logical negation information ( 0for no negation and 1\nfor with negation), and it has four cases depending on the input\nparameters. Given the tail embedding 𝑡and relation embedding 𝑟\non a non-negated atomic formula, 𝜌is formulated in the form of\ncontinuous truth value maximization to infer the head embedding\nˆℎ:\nˆℎ=𝜌(𝑡,𝑟,𝑡→ℎ,0):=𝑎𝑟𝑔𝑚𝑎𝑥\n𝑥∈D𝜓(𝑥,𝑟,𝑡),(6)\nwhereDis the search domain for the embedding 𝑥. Similarly, the\ntail embedding ˆ𝑡on a non-negated atomic formula can be inferred\ngivenℎand𝑟:\nˆ𝑡=𝜌(ℎ,𝑟,ℎ→𝑡,0):=𝑎𝑟𝑔𝑚𝑎𝑥\n𝑥∈D𝜓(ℎ,𝑟,𝑥).(7)\nAccording to the fuzzy logic negator [ 19], one has𝜓(ℎ,¬𝑟,𝑡)=\n1−𝜓(ℎ,𝑟,𝑡). Therefore, the estimation of head and tail embeddings\non negated atomic formulas can be defined as follows:\nˆℎ=𝜌(𝑡,𝑟,𝑡→ℎ,1):=𝑎𝑟𝑔𝑚𝑎𝑥\n𝑥∈D[1−𝜓(𝑥,𝑟,𝑡)],(8)ˆ𝑡=𝜌(ℎ,𝑟,ℎ→𝑡,1):=𝑎𝑟𝑔𝑚𝑎𝑥\n𝑥∈D[1−𝜓(ℎ,𝑟,𝑥)].(9)\nFor the situation where ComplEx-N3 [ 25,47] is selected as the\nneural link predictor, according to the propositions and proofs in\n[56], the logical message encoding function 𝜌has the following\nclosed-form with respect to the complex embedding 𝑟and𝑡when\ninferring the head embedding ˆℎon a non-negated atomic formula:\n𝜌(𝑡,𝑟,𝑡→ℎ,0)=𝑎𝑟𝑔𝑚𝑎𝑥\n𝑥∈C𝑑\b\n𝑅𝑒(⟨𝑟⊗𝑡,𝑥⟩)−𝜆∥𝑥∥3\t\n=𝑟⊗𝑡√︁\n3𝜆∥𝑟⊗𝑡∥,(10)\nSimilarly, for the other three cases, the encoding functions 𝜌are as\nfollows:\n𝜌(ℎ,𝑟,ℎ→𝑡,0)=𝑟⊗ℎ√︁\n3𝜆∥𝑟⊗ℎ∥, (11)\n𝜌(𝑡,𝑟,𝑡→ℎ,1)=−𝑟⊗𝑡√︁\n3𝜆∥𝑟⊗𝑡∥, (12)\n𝜌(ℎ,𝑟,ℎ→𝑡,1)=−𝑟⊗ℎ√︁\n3𝜆∥𝑟⊗ℎ∥. (13)\nFor Equations 10–13, the 𝜆is a hyperparameter. In our work, we\nfollow prior work [ 56] and use the above logical message encoding\nfunction𝜌to compute messages passed to the variable nodes.\n4 PROPOSED METHOD\nIn this section, we propose Conditional Logical Message Passing\nTransformer (CLMPT) to answer complex queries. As a special\nmessage passing neural network [ 18], each CLMPT layer has two\nstages: (1) passing the logical message of each node to all of its\nneighbors that are not constant entity nodes; (2) updating the em-\nbedding of each variable node based on the messages received by\nthe variable node. Figure 2 illustrates how these two stages work\nusing the query graph in Figure 1. Then, we can use the final layer\nembedding of the free variable node to predict the answer entities\nto the corresponding query. In the following subsections, we first\ndescribe how each layer of CLMPT performs conditional logical\nmessage passing and updates the variable node embeddings. After\nthat, we represent the training objective of our method. Finally, we\nintroduce how to answer complex queries with CLMPT.\n4.1 Conditional Logical Message Passing\nA query graph contains two types of nodes: constant entity nodes\nand variable nodes. We decide whether to pass logical messages to\na node based on its type. Specifically, for any neighbor of a node,\nwhen the neighbor is a variable node, we use the logical message\nencoding function 𝜌to calculate the corresponding message and\npass it to the neighbor; when the neighbor is a constant entity node,\nwe do not pass the logical message to it. Figure 2 (a) demonstrates\nthe conditional logical message passing with blue arrows. Every\nnode in the query graph except the constant entity nodes receives\nthe logical message from all of its neighbors.Conditional Logical Message Passing Transformer for Complex Query Answering Conference acronym ’XX, June 03–05, 2018, Woodstock, NY\nFigure 2: An illustration of the two-stage procedures of CLMPT. (a) Passing the logical message to the variable node in the\nneighborhood. (b) Updating the existential variable node embedding with the received messages and a transformer encoder.\nSimilarly, such a node embedding update scheme is applied to other variable nodes, such as the free variable node 𝑦.\n4.2 Node Embedding Conditional Update\nScheme\nFor a constant entity node, we denote its embedding at the 𝑙-th\nlayer as𝑧(𝑙)\n𝑒and let𝑧(𝑙)\n𝑣be the embedding of a variable node at\nthe𝑙-th layer. Next, we discuss how to calculate the 𝑧(𝑙)\n𝑒and𝑧(𝑙)\n𝑣\nfrom the input layer 𝑙=0to latent layers 𝑙>0. For a constant\nentity node, 𝑧(0)\n𝑒is the corresponding entity embedding in the pre-\ntrained neural link predictor, which can be frozen or optimized\ncontinuously. For 𝑧(0)\n𝑣, we follow previous works [ 56] and assign\ntwo learnable embeddings 𝑣𝑥,𝑣𝑦for the existential variable node 𝑥\nand the free variable node 𝑦respectively, and set that all existential\nvariable nodes 𝑥𝑖share one𝑣𝑥, namely𝑧(0)\n𝑥𝑖=𝑣𝑥and𝑧(0)\n𝑦=𝑣𝑦.\nSimilar to conditional logical message passing, for updating the\nnode embeddings, we also do not consider constant entity nodes.\nThat is, we only update node embeddings for variable nodes in the\nquery graph. Thus, the embeddings of the constant entity nodes are\nthe same at each layer, namely 𝑧(𝑙)\n𝑒=𝑧(0)\n𝑒. While for𝑧(𝑙)\n𝑣, we use\nthe corresponding information from the (𝑙−1)-th layer to update it.\nSpecifically, for a variable node 𝑣, we represent its neighbor set in\nthe query graph asN(𝑣). For each neighbor node 𝑛∈N(𝑣), one can\nobtain information about the edge (i.e., the atomic formula) between\n𝑛and𝑣, which contains the neighbor embedding 𝑧(𝑙−1)\n𝑛∈D, the\nrelation𝑟𝑛𝑣∈R, the direction 𝐷𝑛𝑣∈{ℎ→𝑡,𝑡→ℎ}, and the\nnegation indicator 𝑁𝑒𝑔𝑛𝑣∈{0,1}. Based on the edge information,\nwe can use the logical message encoding function 𝜌to compute the\nlogical message 𝑚(𝑙)that𝑛passes to𝑣:\n𝑚(𝑙)=𝜌(𝑧(𝑙−1)\n𝑛,𝑟𝑛𝑣,𝐷𝑛𝑣,𝑁𝑒𝑔𝑛𝑣). (14)\nLet𝑘𝑣be the number of neighbor nodes in N(𝑣), where𝑘𝑣≥1.\nFor the variable node 𝑣, it receives 𝑘𝑣logical messages from 𝑘𝑣\nneighbors:𝑚(𝑙)\n1,...,𝑚(𝑙)\n𝑘𝑣. We use the transformer [ 49] to encode the\ninput set consisting of these logical messages and the variable node\nembedding𝑧(𝑙−1)\n𝑣 to obtain the updated embedding of node 𝑣:\n𝑧(𝑙)\n𝑣=𝑀𝑒𝑎𝑛(𝑇𝐸(𝑙)(𝑚(𝑙)\n1,...,𝑚(𝑙)\n𝑘𝑣,𝑧(𝑙−1)\n𝑣)), (15)\nwhere𝑀𝑒𝑎𝑛 is a mean pooling layer, 𝑇𝐸is a standard transformer\nencoder. Through the self-attention mechanism in the transformer,\nwe can explicitly model the complex logical dependencies between\nvarious elements in the input set and dynamically measure the\nimportance of different elements. Figure 2 (b) shows how to obtainthe updated embedding of the existential variable node 𝑥through\nthe encoding process defined by Equation 15.\n4.3 Training Objective\nWe use the Noisy Contrastive Estimation (NCE) loss proposed in\n[32] to train CLMPT. Specifically, we denote the positive data as\n{(𝑎𝑖,𝑞𝑖)}𝑁\n𝑖=1, where𝑎𝑖∈⟦𝑞𝑖⟧. Our optimization involves 𝐾uni-\nformly sampled noisy answers from the entity set. The training\nobjective is as follows:\n𝐿𝑁𝐶𝐸=−1\n𝑁𝑁∑︁\n𝑖=1𝑙𝑜𝑔\"\n𝐹(𝑧𝑎𝑖,𝑧(𝑞𝑖),𝑇)\n𝐹(𝑧𝑎𝑖,𝑧(𝑞𝑖),𝑇)+Í𝐾\n𝑘=1𝐹(𝑧𝑛𝑖𝑘,𝑧(𝑞𝑖),𝑇)#\n,\n(16)\n𝐹(𝑥,𝑦,𝑧)=𝑒𝑥𝑝[𝑐𝑜𝑠(𝑥,𝑦)/𝑧], (17)\nwhere𝑐𝑜𝑠(·,·)is the cosine similarity, 𝑇is a hyperparameter, 𝑧𝑎𝑖\nis the embedding of positive answer 𝑎𝑖,𝑧𝑛𝑖𝑘is the embedding of\nthe𝑘-th noisy samples, and 𝑧(𝑞𝑖)is the embedding of the predicted\nanswer to the query 𝑞𝑖, corresponding to the embedding of the free\nvariable of the query 𝑞𝑖at the final layer of CLMPT.\n4.4 Answering Complex Queries with CLMPT\nFor a complex query defined in Equation 3, i.e., a DNF query, we\nfollow the previous works [ 38,39] and use DNF-based processing\nto get answer entities. Specifically, we estimate the predicted an-\nswer embeddings for each sub-conjunctive query of the complex\nquery. Then, the answer entities are ranked by the maximal cosine\nsimilarity against these predicted answer embeddings. Therefore,\nCLMPT only needs to consider the query graph of the conjunctive\nquery.\nFor a given conjunctive query 𝑞, let the depth of CLMPT be 𝐿, and\nwe apply the CLMPT layers 𝐿times to the query graph of 𝑞. Then,\nwe can obtain the free variable embedding 𝑧(𝐿)\n𝑦at the final layer. We\nuse𝑧(𝐿)\n���as the embedding of the predicted answer entity, namely\n𝑧(𝑞)=𝑧(𝐿)\n𝑦. Based on the cosine similarity between 𝑧(𝑞)and the\nentity embeddings in the pre-trained neural link predictor, we can\nrank the entities to retrieve answers. For the depth 𝐿, according\nto the analyses in [ 56],𝐿should be the largest distance between\nthe constant entity nodes and the free variable node to ensure the\nfree variable node successfully receives all logical messages from\nthe constant entity nodes. This means 𝐿is not determined, and 𝐿\nshould change dynamically with different conjunctive query types.Conference acronym ’XX, June 03–05, 2018, Woodstock, NY Chongzhi Zhang, Zhiping Peng, Junhao Zheng, and Qianli Ma\nFigure 3: All the query types considered in our experiments,\nwhere𝑝,𝑖,𝑢, and𝑛represent projection, intersection, union,\nand negation, respectively. The naming of each query type\nreflects how they were generated in the BetaE paper [39].\nThus, we follow prior work [ 56] and assume all 𝐿CLMPT layers\nshare the same transformer encoder.\nIn this case, the parameters in CLMPT include two embeddings\nfor existential and free variables, a transformer encoder, and the\nparameters in the pre-trained neural link predictor. For the pre-\ntrained neural link predictor, it can be frozen or not. In our work, we\ncontinue to optimize pre-trained neural link predictors by default,\nthat is, we do not freeze pre-trained embeddings of entities and\nrelations.\n5 EXPERIMENTS\n5.1 Experimental Settings\n5.1.1 Datasets and Queries. We evaluate our method on three\ncommonly-used standard knowledge graphs: FB15k [ 8], FB15k-237\n[46], and NELL995 [ 59]. For a fair comparison with previous works,\nwe use the datasets of complex queries proposed by BetaE [ 39],\nwhich consist of five conjunctive query types ( 1𝑝/2𝑝/3𝑝/2𝑖/3𝑖),\nfive query types with atomic negations ( 2𝑖𝑛/3𝑖𝑛/𝑖𝑛𝑝/𝑝𝑖𝑛/𝑝𝑛𝑖), and\nfour zero-shot query types ( 𝑝𝑖/𝑖𝑝/2𝑢/𝑢𝑝), as illustrated in Figure\n3. The datasets provided by BetaE introduce complex queries with\nhard answers that cannot be obtained by traversing the KG directly.\nThat is, the evaluation focuses on discovering such hard answers to\ncomplex queries. We follow previous works [ 39,56,67] and train\nour model with five conjunctive query types and five query types\nwith atomic negations. When evaluating the model, we consider all\nquery types including the four zero-shot query types. The statistics\nfor each dataset are presented in Table 1. More details about the\ndatasets can be found in Appendix A.\n5.1.2 Evaluation Protocol. The evaluation scheme follows the pre-\nvious works [ 39,56], which divides the answers to each complex\nquery into easy and hard sets. For test and validation splits, we de-\nfine hard answers as those that cannot be obtained by direct graph\ntraversal on KG. In order to get these hard answers, the model is re-\nquired to impute at least one missing edge, which means the modelTable 1: Statistics of different query types used in the bench-\nmark datasets.\nSplit Query Types FB15k FB15k-237 NELL995\nTrain1p, 2p, 3p, 2i, 3i 273,710 149,689 107,982\n2in, 3in, inp, pin, pni 27,371 14,968 10,798\nValid1p 59,078 20,094 16,910\nOthers 8,000 5,000 4,000\nTest1p 66,990 22,804 17,021\nOthers 8,000 5,000 4,000\nneeds to complete non-trivial reasoning [ 38,39]. Specifically, for\neach hard answer of a query, we rank it against non-answer entities\nbased on their cosine similarity with the free variable embedding\nof the query and calculate the Mean Reciprocal Rank (MRR).\n5.1.3 Baselines. We consider the state-of-the-art neural complex\nquery answering models for EFO-1 queries in recent years as our\nbaselines to compare: BetaE [ 39], ConE [ 67], Q2P [ 5], MLP [ 2],\nGammaE [ 63], CylE [ 34], CQD-CO [ 3], and LMPNN [ 56], where\nCQD-CO and LMPNN use pre-trained neural link predictor, while\nother models do not. In addition, some other neural CQA models\ncannot handle logical negation operators, such as BIQE [ 24], PREM\n[11], kgTransformer [ 28], etc., which use the EPFO queries gener-\nated by Q2B [ 38] for training and evaluation. We compare these\nworks on the Q2B datasets in Appendix C. We also compare our\nwork with symbolic integration CQA models in Appendix E.\n5.1.4 Model Details. Following LMPNN [ 56] and CQD-CO [ 3], we\nchoose ComplEx-N3 [ 25,47] as the neural link predictor in our\nwork and use the ComplEx-N3 checkpoints released by CQD-CO\nto conduct experiments. The rank of ComplEx-N3 is 1,000, and the\nepoch for the checkpoints is 100. For all datasets, we use a two-\nlayer transformer encoder with eight self-attention heads, where\nthe dimension of the hidden layer of the feed-forward network\n(FFN) in the transformer is 8,192. For the training objective, the\nnegative sample size 𝐾is 128, and𝑇is chosen as 0.05 for FB15k-\n237 and FB15k and 0.1 for NELL995. For more details about the\nimplementation and experiments, please refer to Appendix B.\n5.2 Main Results\nTable 2 shows the MRR results of CLMPT and neural CQA base-\nlines about answering EFO-1 queries over three KGs. It is found\nthat CLMPT reaches the best performance on average for EPFO\nqueries across all three datasets. Compared with LMPNN, which is\nmost relevant to our work, CLMPT achieves average performance\nimprovements of 8.9%, 7.5%, and 2.0% for EPFO queries on FB15k,\nFB15K-237, and NELL995, respectively. Since conditional logical\nmessage passing has little impact on the model performance (it\nwill be later discussed in Section 5.3.1), these performance improve-\nments are mainly due to the transformer-based node embedding\nupdate scheme. This shows that this approach, which explicitly\nmodels dependencies between input elements by assigning adap-\ntive weights to the messages and the corresponding node through\nthe self-attention mechanism, is more suitable for the logical query\ngraphs of EPFO queries than the GIN-like approach. For the neg-\native queries, CLMPT achieves the best average performance onConditional Logical Message Passing Transformer for Complex Query Answering Conference acronym ’XX, June 03–05, 2018, Woodstock, NY\nTable 2: The MRR results of baselines and our model over three KGs. Avg prepresents the average score of EPFO queries and\nAvg ndenote the average score of queries with negation. The boldface indicates the best results, and the second-best ones are\nmarked with underlines .\nDataset Model 1p 2p 3p 2i 3i pi ip 2u up 2in 3in inp pin pni Avg pAvg n\nFB15kBetaE 65.1 25.7 24.7 55.8 66.5 43.9 28.1 40.1 25.2 14.3 14.7 11.5 6.5 12.4 41.6 11.8\nConE 73.3 33.8 29.2 64.4 73.7 50.9 35.7 55.7 31.4 17.9 18.7 12.5 9.8 15.1 49.8 14.8\nQ2P 82.6 30.8 25.5 65.1 74.7 49.5 34.9 32.1 26.2 21.9 20.8 12.5 8.9 17.1 46.8 16.4\nMLP 67.1 31.2 27.2 57.1 66.9 45.7 33.9 38.0 28.0 17.2 17.8 13.5 9.1 15.2 43.9 14.5\nGammaE 76.5 36.9 31.4 65.4 75.1 53.9 39.7 53.5 30.9 20.1 20.5 13.5 11.8 17.1 51.3 16.6\nCylE 78.8 37.0 30.9 66.9 75.7 53.8 40.8 59.4 33.5 15.7 16.3 13.7 7.8 13.9 53.0 13.5\n(Using pre-trained neural link predictor)\nCQD-CO 89.4 27.6 15.1 63.0 65.5 46.0 35.2 42.9 23.2 0.2 0.2 4.0 0.1 18.4 45.3 4.6\nLMPNN 85.0 39.9 28.6 68.2 76.5 46.7 43.0 36.7 31.4 29.1 29.4 14.9 10.2 16.4 50.6 20.0\nCLMPT 86.1 43.3 33.9 69.1 78.2 55.1 46.6 46.1 37.3 21.2 22.9 17.0 13.0 15.3 55.1 17.9\nFB15k-237BetaE 39.0 10.9 10.0 28.8 42.5 22.4 12.6 12.4 9.7 5.1 7.9 7.4 3.6 3.4 20.9 5.4\nConE 41.8 12.8 11.0 32.6 47.3 25.5 14.0 14.5 10.8 5.4 8.6 7.8 4.0 3.6 23.4 5.9\nQ2P 39.1 11.4 10.1 32.3 47.7 24.0 14.3 8.7 9.1 4.4 9.7 7.5 4.6 3.8 21.9 6.0\nMLP 42.7 12.4 10.6 31.7 43.9 24.2 14.9 13.7 9.7 6.6 10.7 8.1 4.7 4.4 22.6 6.9\nGammaE 43.2 13.2 11.0 33.5 47.9 27.2 15.9 13.9 10.3 6.7 9.4 8.6 4.8 4.4 24.0 6.8\nCylE 42.9 13.3 11.3 35.0 49.0 27.0 15.7 15.3 11.2 4.9 8.3 8.2 3.7 3.4 24.5 5.7\n(Using pre-trained neural link predictor)\nCQD-CO 46.7 10.3 6.5 23.1 29.8 22.1 16.3 14.2 8.9 0.2 0.2 2.1 0.1 6.1 19.8 1.7\nLMPNN 45.9 13.1 10.3 34.8 48.9 22.7 17.6 13.5 10.3 8.7 12.9 7.7 4.6 5.2 24.1 7.8\nCLMPT 45.7 13.7 11.3 37.4 52.0 28.2 19.0 14.3 11.1 7.7 13.7 8.0 5.0 5.1 25.9 7.9\nNELL995BetaE 53.0 13.0 11.4 37.6 47.5 24.1 14.3 12.2 8.5 5.1 7.8 10.0 3.1 3.5 24.6 5.9\nConE 53.1 16.1 13.9 40.0 50.8 26.3 17.5 15.3 11.3 5.7 8.1 10.8 3.5 3.9 27.2 6.4\nQ2P 56.5 15.2 12.5 35.8 48.7 22.6 16.1 11.1 10.4 5.1 7.4 10.2 3.3 3.4 25.5 6.0\nMLP 55.2 16.8 14.9 36.4 48.0 22.7 18.2 14.7 11.3 5.1 8.0 10.0 3.6 3.6 26.5 6.1\nGammaE 55.1 17.3 14.2 41.9 51.1 26.9 18.3 15.1 11.2 6.3 8.7 11.4 4.0 4.5 27.9 7.0\nCylE 56.5 17.5 15.6 41.4 51.2 27.2 19.6 15.7 12.3 5.6 7.5 11.2 3.4 3.7 28.5 6.3\n(Using pre-trained neural link predictor)\nCQD-CO 60.8 18.3 13.2 36.5 43.0 30.0 22.5 17.6 13.7 0.1 0.1 4.0 0.0 5.2 28.4 1.9\nLMPNN 60.6 22.1 17.5 40.1 50.3 28.4 24.9 17.2 15.7 8.5 10.8 12.2 3.9 4.8 30.7 8.0\nCLMPT 58.9 22.1 18.4 41.8 51.9 28.8 24.4 18.6 16.2 6.6 8.1 11.8 3.8 4.5 31.3 7.0\nFB15k-237 and sub-optimal performance on the remaining two KGs.\nOne potential explanation for these sub-optimal results is that the\ntransformer architecture lacks inductive biases and relies more on\nthe training data. As shown in the statistics in Table 1, the number\nof negative queries in the training set is an order of magnitude less\nthan the EPFO queries, which may lead to the tendency of the trans-\nformer encoder to learn the patterns relevant to answering EPFO\nqueries, resulting in insufficient modeling of relevant patterns that\nanswer negative queries.\n5.3 ABLATION STUDY\n5.3.1 Conditional Logical Meassage Passing. The logical message\npassing mechanism can be viewed as learning an embedding for\nthe variable by utilizing information of constant entities. It makes\nthis embedding similar to the embedding of the solution entity\nin the embedding space of the pre-trained neural link predictor.\nAs in the previous analyses, we argue that for a constant node\nwith the corresponding entity embedding, it is unnecessary to\nupdate its embedding in the forward passing of the model. To\nverify this, we conduct experiments on whether to pass messages\nto the constant node and update its embedding, and the results areTable 3: The average MRR results of models with or with-\nout conditional logical message passing on answering EFO-1\nqueries.↓Memory and↓Time represent the GPU memory\nusage reduction and time savings with conditional logical\nmessage passing, respectively. The boldface indicates the best\nresults.\nDataset Model Avg pAvg n↓Memory↓Time\nFB15k-237LMPNN 24.1 7.8\n↓1.2%↓5.0%LMPNN w/ C 24.3 7.8\nCLMPT w/o C 25.8 7.7\n↓9.6%↓12.3%CLMPT 25.9 7.9\nNELL995LMPNN 30.7 8.0\n↓0.6%↓8.3%LMPNN w/ C 31.0 8.2\nCLMPT w/o C 31.4 7.0\n↓8.0%↓11.0%CLMPT 31.3 7.0\nshown in Table 3, where \"w/ C\" indicates \"with conditional logical\nmessage passing\" and \"w/o C\" indicates \"without conditional logical\nmessage passing\". From the experimental results, the performanceConference acronym ’XX, June 03–05, 2018, Woodstock, NY Chongzhi Zhang, Zhiping Peng, Junhao Zheng, and Qianli Ma\nTable 4: The MRR results of different hyperparameter settings and different pooling approaches. The best results are bold.\nModel 1p 2p 3p 2i 3i pi ip 2u up 2in 3in inp pin pni Avg pAvg n\n𝑇𝐸layer = 1 45.8 13.2 10.9 37.3 51.9 27.7 18.2 14.6 10.9 7.7 13.3 8.1 5.3 4.5 25.6 7.8\n𝑇𝐸layer = 3 45.8 13.9 11.4 37.3 51.8 29.0 19.2 14.4 11.2 7.6 13.1 8.1 5.0 4.9 26.0 7.7\n𝑑FFN= 4096 45.8 13.5 11.1 37.3 51.7 27.9 18.6 14.3 11.2 7.7 13.2 8.1 5.1 4.9 25.7 7.8\nself-attention head = 4 45.9 13.6 11.3 37.4 52.0 28.1 18.7 14.6 11.3 7.4 13.1 8.1 5.0 4.8 25.9 7.7\nself-attention head = 16 45.0 12.8 10.3 36.4 50.4 26.8 18.0 14.5 10.6 7.5 12.9 7.7 5.2 4.3 25.0 7.5\n𝑇= 0.01 45.8 12.9 10.3 35.1 48.9 26.4 16.8 14.1 10.3 7.1 10.8 7.4 5.6 4.0 24.5 7.0\n𝑇= 0.1 44.9 13.5 10.7 34.4 47.2 26.5 18.7 15.6 11.1 7.6 10.2 8.1 4.2 5.8 24.7 7.2\nMax Pooling 46.1 13.8 11.1 36.8 50.5 26.0 17.5 14.4 11.1 5.1 9.5 7.3 3.7 3.4 25.2 5.8\nSum Pooling 45.9 13.8 11.0 37.5 51.9 28.5 18.7 14.3 11.3 7.6 13.6 8.2 5.0 4.9 25.9 7.9\nDefault Setting 45.7 13.7 11.3 37.4 52.0 28.2 19.0 14.3 11.1 7.7 13.7 8.0 5.0 5.1 25.9 7.9\nof LMPNN with conditional logical message passing on the two\nKGs is not reduced but improved. CLMPT achieves better results\non FB15k-237 than the variant that does not use conditional logical\nmessage passing, only slightly underperforming the variant on\nthe EPFO queries of NELL995. In addition, we evaluate how much\nunnecessary computational cost the conditional logical message\npassing mechanism can avoid on an RTX 3090 GPU. Specifically, we\ncalculate the relative percentage of the reduction in GPU memory\nusage and the relative percentage of training time saved:\n↓Memory =(𝑢𝑠𝑎𝑔𝑒𝑤𝑜𝐶−𝑢𝑠𝑎𝑔𝑒𝐶)/𝑢𝑠𝑎𝑔𝑒𝑤𝑜𝐶, (18)\n↓Time =(𝑡𝑤𝑜𝐶−𝑡𝐶)/𝑡𝑤𝑜𝐶, (19)\nwhere𝑢𝑠𝑎𝑔𝑒𝐶and𝑡𝐶respectively refer to the GPU memory usage\nand the required training time when the model uses conditional log-\nical message passing, while 𝑢𝑠𝑎𝑔𝑒𝑤𝑜𝐶 and𝑡𝑤𝑜𝐶 correspond to the\nsituation without conditional logical message passing. As shown\nin Table 3, conditional logical message passing reduces compu-\ntational costs without significantly negatively impacting model\nperformance. In particular, the computational cost reduction is\nparticularly significant for CLMPT, which uses the more complex\ntransformer architecture. The above results confirm our view that\nthe model does not need to consider the constant entity nodes in\nthe query graph during the forward message passing.\n5.3.2 Hyperparamaters and Pooling Approaches. We evaluate the\nperformance of CLMPT with different hyperparameter settings and\npooling approaches on FB15k-237. The default setting we adopted\nis described in Section 5.1.4. As is shown in Table 4, regardless of\nreducing the number of transformer encoder layers or halving the\ndimensions of the hidden layer of the FFN, the model still achieves\nthe best performance than the baselines on FB15k-237, which re-\nflects the effectiveness of our proposed method. When the number\nof transformer encoder layers increases, the performance changes\nlittle, meaning the two-layer transformer encoder is sufficient. In\naddition, we can find that 𝑇in NCE loss is very important to the\nperformance. When 𝑇is 0.01 or 0.1, the performance of the model\nis still competitive on EPFO queries, but there is a large gap with\nthe average performance under the default settings. For the pool-\ning approaches, we evaluate max pooling and sum pooling. The\nexperimental results show max pooling does not perform as well\nas sum pooling and mean pooling.5.3.3 Frozen or Trainable Neural Link Predictors. For the pre-\ntrained neural link predictor, it can be frozen or trainable. As men-\ntioned above (see Section 4.4), in our work, we optimize all param-\neters in the pre-trained neural link predictor by default. It is worth\nnoting that the pre-trained neural link predictor is frozen in LMPNN.\nIn order to explore the impact of freezing the pre-trained neural\nlink predictor on model performance, we conduct experiments on\nthree KGs, and the results are shown in Table 5.\nTable 5: The average MRR results for experiments on whether\nto freeze the Neural Link Predictor (NLP). The boldface indi-\ncates the best results.\nDataset ModelFrozen NLP Trainable NLP\nAvg pAvg nAvg p Avg n\nFB15kLMPNN 50.6 20.0 51.5 18.9\nCLMPT 54.2 17.3 55.1 17.9\nFB15k-237LMPNN 24.1 7.8 24.4 7.5\nCLMPT 25.4 7.7 25.9 7.9\nNELL995LMPNN 30.7 8.0 30.1 7.9\nCLMPT 32.0 6.8 31.3 7.0\nFor LMPNN, using frozen neural link predictors performs better\nthan using trainable ones, except for EPFO queries on FB15k and\nFB15k-237. Therefore, in the main results, we only compare the\nMRR results of LMPNN using frozen neural link predictors, i.e. the\nresults reported in the original paper [ 56]. For CLMPT, with the\nexception of EPFO queries on NELL995, CLMPT using the trainable\nneural link predictor performs relatively better. Thus, we continue\nto optimize the pre-trained neural link predictor on complex queries\nby default. For these results of CLMPT, according to the progress\nof pre-training and fine-tuning paradigms [ 16,29], one possible\nreason is that the neural link predictor is pre-trained on one-hop\nqueries, and tuning its parameters on complex queries can make the\nembeddings of entities and relations more suitable for the patterns\nand characteristics of CQA. The better performance of LMPNN\nusing trainable neural link predictors on EPFO queries of FB15k and\nFB15K-237 can also support our argument to some extent because\nthe number of EPFO queries in the datasets is much more than\nnegative ones; in this case, the model is more inclined to learn\npatterns that answer EPFO queries.Conditional Logical Message Passing Transformer for Complex Query Answering Conference acronym ’XX, June 03–05, 2018, Woodstock, NY\nAlthough CLMPT, which uses the trainable neural link predictor,\nhas more training parameters, it still significantly outperforms\nLMPNN on EPFO queries across all datasets when the neural link\npredictor is frozen. This reflects the effectiveness of CLMPT. For\nmore experimental analyses of the model parameters, please refer\nto Appendix D.\n6 CONCLUSION\nIn this paper, we propose CLMPT, a special message passing neural\nnetwork, to answer complex queries over KGs. Based on one-hop\ninference by pre-trained neural link predictor on atomic formu-\nlas, CLMPT performs logical message passing conditionally on the\nnode type. In the ablation study, we verify that this conditional mes-\nsage passing mechanism can effectively reduce the computational\ncosts and even improve the performance of the model to some\nextent. Furthermore, CLMPT uses the transformer to aggregate\nmessages and update the corresponding node embedding. Through\nthe self-attention mechanism, CLMPT can explicitly model logical\ndependencies between messages received by a node and between\nmessages and that node by assigning adaptive weights to the mes-\nsages and the node. The experimental results show that CLMPT\nachieves a strong performance through this transformer-based node\nembedding update scheme. 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Advances in\nNeural Information Processing Systems 34 (2021), 19172–19183.\n[68] Zhanqiu Zhang, Jie Wang, Jieping Ye, and Feng Wu. 2022. Rethinking graph\nconvolutional networks in knowledge graph completion. In Proceedings of the\nACM Web Conference 2022 . 798–807.\n[69] Zhaocheng Zhu, Mikhail Galkin, Zuobai Zhang, and Jian Tang. 2022. Neural-\nsymbolic models for logical queries on knowledge graphs. In International Con-\nference on Machine Learning . PMLR, 27454–27478.\n[70] Zhaocheng Zhu, Zuobai Zhang, Louis-Pascal Xhonneux, and Jian Tang. 2021.\nNeural bellman-ford networks: A general graph neural network framework for\nlink prediction. Advances in Neural Information Processing Systems 34 (2021),\n29476–29490.\nA MORE DETAILS ABOUT THE DATASETS\nThe statistics of the three knowledge graphs used in our experiment\nare shown in Table 6. As we described in section 5.1.1, we conduct\nexperiments on the complex query datasets proposed in the BetaE\npaper [ 39]. The statistics for the complex queries contained in these\ndatasets are shown in Table 1. The statistics for the answers for\neach query type are shown in Table 7.\nAs mentioned in Section 5.1.3, there are some other neural CQA\nmodels [ 11,24,28] that are trained and evaluated on the EPFO query\ndatasets proposed in the Q2B paper [ 38]. To compare these models,\nwe also conduct experiments (see Appendix C) on the FB15k-237\nand NELL995 configured by the Q2B setting. We denote these two\ndatasets as FB237-Q2B and NELL-Q2B. The statistics for the EPFO\nqueries and corresponding answers for Q2B datasets are shown in\nTable 8 and Table 9.\nOne difference between the BetaE datasets and the Q2B datasets\nis the limit on the number of answers to the queries, since the Q2B\ndatasets have queries with more than 5000 answers, almost 1/3of\nsome datasets. Furthermore, the Q2B datasets do not contain queries\nwith negation. These differences mean that BetaE [ 39] datasets\nare more challenging, so most CQA models in recent years chose\nto evaluate their performance on the BetaE datasets. To obtain\nconvincing comparison results, we mainly conduct experiments\non the BetaE datasets to compare existing competitive neural CQA\nmodels.Conditional Logical Message Passing Transformer for Complex Query Answering Conference acronym ’XX, June 03–05, 2018, Woodstock, NY\nTable 6: Datasets statistics, as well as training, validation and test edge splits.\nDataset Entities Relations Training Edges Validation Edges Test Edges Total Edges\nFB15k 14,951 1,345 483,142 50,000 59,071 592,213\nFB15k-237 14,505 237 272,115 17,526 20,438 310,079\nNELL995 63,361 200 114,213 14,324 14,267 142,804\nTable 7: The average number of answers divided by query structure.\nDataset 1p 2p 3p 2i 3i ip pi 2u up 2in 3in inp pin pni\nFB15k 1.7 19.6 24.4 8.0 5.2 18.3 12.5 18.9 23.8 15.9 14.6 19.8 21.6 16.9\nFB15k-237 1.7 17.3 24.3 6.9 4.5 17.7 10.4 19.6 24.3 16.3 13.4 19.5 21.7 18.2\nNELL995 1.6 14.9 17.5 5.7 6.0 17.4 11.9 14.9 19.0 12.9 11.1 12.9 16.0 13.0\nTable 8: The number of training, validation, and test queries generated for different query structures of the Q2B datasets [ 38].\nQueries Training Validation Test\nDataset 1p others 1p Others 1p Others\nFB237-Q2B 149,689 149,689 20,101 5,000 22,812 5,000\nNELL-Q2B 107,982 107,982 16,927 4,000 17,034 4,000\nTable 9: The average number of answers divided by query structure for Q2B datasets [38].\nDataset 1p 2p 3p 2i 3i ip pi 2u up\nFB237-Q2B 13.3 131.4 215.3 69.0 48.9 593.8 257.7 35.6 127.7\nNELL-Q2B 8.5 56.6 65.3 30.3 15.9 310.0 144.9 14.4 62.5\nB MORE EXPERIMENTAL DETAILS\nOur code is implemented using PyTorch. We use NVIDIA Tesla\nA10 GPU (24GB) and NVIDIA GeForce RTX 3090 GPU (24GB) to\nconduct all of our experiments. We use the AdamW [ 30] optimizer,\nwhose weight decay is 1e-4, to tune the parameters. The learning\nrate is 5e-5 for NELL995 and FB15k-237 and 1e-4 for FB15k. The\nbatch size is 512 for NELL995 and FB15k-237 and 1,024 for FB15k.\nThe training epoch for all datasets is 100. Following CQD-CO [ 3]\nand LMPNN [ 56], we choose ComplEx-N3 [ 25,47] as the neural\nlink predictor for fair comparison. In this case, we consider the\nlogical message encoding function 𝜌corresponding to Equations\n10–13. For these equations, the 𝜆is a hyperparameter that needs\nto be determined. In LMPNN application, for simplicity, LMPNN\njust lets 3𝜆∥·∥=1and then all denominators in these closed-form\nexpressions are 1, namely:\n𝜌(𝑡,𝑟,𝑡→ℎ,0):=𝑟⊗𝑡, (20)\n𝜌(ℎ,𝑟,ℎ→𝑡,0):=𝑟⊗ℎ, (21)\n𝜌(𝑡,𝑟,𝑡→ℎ,1):=−𝑟⊗𝑡, (22)\n𝜌(ℎ,𝑟,ℎ→𝑡,1):=−𝑟⊗ℎ. (23)To make a fair comparison with LMPNN, in CLMPT application,\nwe also use the logical message encoding function 𝜌defined in\nEquations 20–23 to compute messages passed to the variable nodes.\nFor the experiments (see Section 5.3.1) to evaluate how much unnec-\nessary computational costs can be avoided by conditional logical\nmessage passing mechanism, we repeat the experiments three times\non an RTX 3090 GPU and then average the calculated ↓Memory\n(see Equation 18) and ↓Time (see Equation 19) metrics.\nC COMPARISON WITH MORE NEURAL CQA\nMODELS ON Q2B DATASETS\nTo further evaluate the performance of CLMPT, we also consider\ncomparing neural CQA models that cannot handle logical negation\nand are trained and evaluated on Q2B datasets, including GQE [ 20],\nQ2B [ 38], BIQE [ 24], PERM [ 11], kgTransformer [ 28], and SILR [ 52].\nAmong these models, BIQE, kgTransformer, and SILR all use the\ntransformer [ 49] to encode complex queries directly. BIQE and SILR\nserialize the query graph and treat the CQA task as the sequence\nlearning task to solve. kgTransformer uses the pre-training and\nfine-tuning strategies based on Heterogeneous Graph Transformer\n(HGT) [ 21] to answer complex queries. In essence, kgTransformer is\nalso a transformer-based graph neural network. We use the resultsConference acronym ’XX, June 03–05, 2018, Woodstock, NY Chongzhi Zhang, Zhiping Peng, Junhao Zheng, and Qianli Ma\nTable 10: The Hits@3 results of other neural CQA baselines on answering EPFO queries generated by Q2B [ 38].Avg prepresents\nthe average results of these EPFO query types. The boldface indicates the best results.\nDataset Model 1p 2p 3p 2i 3i pi ip 2u up Avg p\nFB237-Q2BGQE 40.5 21.3 15.3 29.8 41.1 18.2 8.5 16.7 16.0 23.0\nQ2B 46.7 24.0 18.6 32.4 45.3 20.5 10.8 23.9 19.3 26.8\nBIQE 47.2 29.3 24.5 34.5 47.3 21.1 9.6 7.5 14.5 26.2\nPERM 52.0 28.6 21.6 36.1 49.0 21.2 12.8 30.5 23.9 30.6\nkgTransformer 45.9 31.2 27.6 39.8 52.8 28.6 18.9 26.3 21.4 32.5\nSILR 47.1 30.2 24.9 35.8 48.4 22.2 11.3 28.3 18.1 29.6\nCLMPT 50.5 31.3 26.7 39.0 52.1 27.7 16.3 28.6 24.1 32.9\nNELL-Q2BGQE 41.7 23.1 20.3 31.8 45.4 18.8 8.1 20.0 13.9 24.8\nQ2B 55.5 26.6 23.3 34.3 48.0 21.2 13.2 36.9 16.3 30.6\nBIQE 63.2 31.0 33.2 37.0 52.5 16.4 9.1 17.3 18.4 30.9\nPERM 58.1 28.6 24.3 35.2 50.8 19.5 14.3 46.0 20.0 32.8\nkgTransformer 62.5 40.1 36.7 40.5 54.6 30.6 20.3 46.9 27.0 39.9\nSILR 64.1 32.9 33.7 37.6 53.2 17.7 5.5 45.0 18.0 34.2\nCLMPT 64.3 39.9 36.5 42.1 54.1 28.1 22.1 48.9 33.4 41.1\nreported in these papers [ 28,52] for comparison. Since they report\nthe Hits@3 results, for a fair comparison, we also use Hits@3 as\nthe evaluation metric, which calculates the proportion of correct\nanswer entities ranked among the top 3.\nAs shown in the results in Table 10, CLMPT reaches the best per-\nformance on average across all datasets. For those models that use\nthe transformer to encode the entire query graph directly, namely\nBIQE, SILR, and kgTransformer, we speculate that their insufficient\ngraph inductive biases may negatively affect their performance. To\nencode the entire query graph at once using the transformer, BIQE\ndesigns a special positional encoding scheme, and SILR designs a\nparticular graph serialization method and structural prompt. These\ndesigns can be viewed as introducing specific graph inductive biases\ninto the transformer to encode the query graph. For kgTransformer,\nit relies primarily on the graph inductive biases of the HGT ar-\nchitecture. According to the progress of the graph transformers\n[31,64], graph inductive biases are critical to using the transformer\nto encode graph data. In particular, structural encoding is important\nfor the graph transformers without message passing. Therefore, we\nspeculate that the graph inductive biases introduced by these CQA\nmodels may not be sufficient to encode the query graph that defines\ncomplex logical dependencies. On the other hand, it is not clear how\nto introduce appropriate graph inductive biases for query graphs\nthat define first-order logical dependencies. By contrast, CLMPT,\nbased on logical message passing, uses the transformer to aggregate\nmessages and update node embeddings rather than to encode the\nentire query graph. This difference makes CLMPT less dependent\non graph inductive biases than the transformer-based CQA models\nmentioned above. However, it also suggests that appropriate graph\ninductive biases may enhance the performance of CLMPT. This\nextension can be explored in the future.\nD ANALYSES ON MODEL PARAMETERS\nSince CLMPT introduces the transformer, it has more parameters\nthan LMPNN. To further evaluate the effectiveness of CLMPT, we\ncompare LMPNN using a smaller model whose parameters are\nclose to those of LMPNN. Specifically, we set 𝑇𝐸layer =1and𝑑FFN=4096 , and we represent such a smaller model as CLMPT-\nsmall. Since the parameters of the neural link predictor in LMPNN\nare frozen, we consider using both frozen and trainable neural link\npredictors for fair comparison. We represent the models in these\ntwo cases as CLMPT-small(F) and CLMPT-small(T), respectively.\nThe experimental results are shown in Table 11. When their model\nparameters are close, CLMPT still achieves the best average per-\nformance on EPFO queries. In the case of freezing the neural link\npredictor, CLMPT-small(F) has fewer trainable parameters than\nLMPNN on NELL995, but it still significantly outperforms LMPNN\non average in anwsering EPFO queries. These experimental results\nreflect the effectiveness of CLMPT.\nE COMPARISON WITH SYMBOLIC\nINTEGRATION MODELS\nWe discuss the differences between symbolic integration CQA mod-\nels and neural CQA models in Section 2. The model proposed in\nthis paper is a neural CQA model, so our comparisons and related\ndiscussions are mainly carried out within the scope of neural CQA\nmodels. Since symbolic information can enhance the performance\nof neural CQA models [ 56], in this section, we consider two repre-\nsentative symbolic integration models, GNN-QE [ 69] and ENeSy\n[61], for comparison with CLMPT to show the potential. We list\nthe MRR results reported in their original papers. The results are\nshown in Table 12. It is found that CLMPT is also competitive even\nwith the symbolic integration models on answering EPFO queries.\nSpecifically, CLMPT achieves a better average performance than\nthese symbolic integration models on EPFO queries for NELL995.\nFor FB15k-237, CLMPT still has a gap in performance with GNN-\nQE, but it can outperform ENeSy on EPFO queries. For the negative\nqueries, there are still gaps between the neural CQA models and\nthese symbolic integration models because the fuzzy sets equipped\nwith symbolic information used in the reasoning process of these\nmodels can effectively deal with logical negation through proba-\nbilistic values. These results suggest that neural models can be po-\ntentially improved with symbolic integration. However, it is worthConditional Logical Message Passing Transformer for Complex Query Answering Conference acronym ’XX, June 03–05, 2018, Woodstock, NY\nTable 11: The MRR results of CLMPT-small using frozen neural link predictor and using trainable neural link predictor. Param T\nandParam Frepresent trainable and frozen parameters of the model, respectively. The F in parentheses indicates that the model\nuses a frozen neural link predictor, and T indicates that it uses a trainable one.\nDataset Model 1p 2p 3p 2i 3i pi ip 2u up 2in 3in inp pin pni Avg pAvg nParam TParam F\nFB15k-237LMPNN 45.9 13.1 10.3 34.8 48.9 22.7 17.6 13.5 10.3 8.7 12.9 7.7 4.6 5.2 24.1 7.8 16M 30M\nCLMPT-small(F) 46.1 12.6 10.0 36.1 50.4 26.7 17.5 14.9 10.3 7.7 11.6 7.5 4.4 5.0 25.0 7.3 32M 30M\nCLMPT-small(T) 45.7 13.2 11.0 37.2 51.6 27.4 18.2 14.3 11.1 7.4 13.1 8.1 5.1 4.7 25.5 7.7 62M 0M\nNELL995LMPNN 60.6 22.1 17.5 40.1 50.3 28.4 24.9 17.2 15.7 8.5 10.8 12.2 3.9 4.8 30.7 8.0 33M 128M\nCLMPT-small(F) 60.6 21.7 17.7 42.2 51.7 30.7 24.5 19.4 15.6 6.4 7.9 11.0 3.7 4.5 31.6 6.7 32M 128M\nCLMPT-small(T) 59.1 21.3 17.5 41.8 51.8 29.0 23.5 19.0 15.6 6.4 7.9 11.2 3.8 4.5 31.0 6.8 160M 0M\nTable 12: The MRR results of symbolic integration models and CLMPT. The boldface indicates the best results.\nDataset Model 1p 2p 3p 2i 3i pi ip 2u up 2in 3in inp pin pni Avg pAvg n\nFB15k-237GNN-QE 42.8 14.7 11.8 38.3 54.1 31.1 18.9 16.2 13.4 10.0 16.8 9.3 7.2 7.8 26.8 10.2\nENeSy 44.7 11.7 8.6 34.8 50.4 27.6 19.7 14.2 8.4 10.1 10.4 7.6 6.1 8.1 24.5 8.5\nCLMPT-small(F) 46.1 12.6 10.0 36.1 50.4 26.7 17.5 14.9 10.3 7.7 11.6 7.5 4.4 5.0 25.0 7.3\nCLMPT 45.7 13.7 11.3 37.4 52.0 28.2 19.0 14.3 11.1 7.7 13.7 8.0 5.0 5.1 25.9 7.9\nNELL995GNN-QE 53.3 18.9 14.9 42.4 52.5 30.8 18.9 15.9 12.6 9.9 14.6 11.4 6.3 6.3 28.9 9.7\nENeSy 59.0 18.0 14.0 39.6 49.8 29.8 24.8 16.4 13.1 11.3 8.5 11.6 8.6 8.8 29.4 9.8\nCLMPT-small(F) 60.6 21.7 17.7 42.2 51.7 30.7 24.5 19.4 15.6 6.4 7.9 11.0 3.7 4.5 31.6 6.7\nCLMPT 58.9 22.1 18.4 41.8 51.9 28.8 24.4 18.6 16.2 6.6 8.1 11.8 3.8 4.5 31.3 7.0\nnoting that introducing additional symbolic information requires\nlarger computational costs.\nAccording to [ 37,56], for larger knowledge graphs, neural CQA\nmodels and symbolic integration models have different scalabil-\nities. As we discussed in Section 2, symbolic integration models\nrequire more computing resources and are more likely to suffer\nfrom scalability issues. For example, GNN-QE employs NBFNet [ 70]\nto compute message passing on the whole KG, resulting in complex-\nity that is linear to (|E|+|V|)𝑑, where|E|is the number of edges\nin KG,|V|is the number of nodes in KG, and 𝑑is the embedding\ndimension. For CLMPT, which only performs message passing on\nthe query graph, the complexity is just 𝑂(𝑑). Besides, GNN-QE es-\ntimates the probability of whether each entity is the answer at eachintermediate step, making the size of its fuzzy sets scale linearly\nwith|V|. As a result, GNN-QE requires much more computational\ncosts, requiring 128GB GPU memory to run a batch size of 32. For\nCLMPT-small(F), which has only 32M trainable parameters, it only\nrequires less than 12GB GPU memory to run a batch size of 1024.\nEven in this case, CLMPT-small(F) can still achieve performance\ncompetitive with GNN-QE on NELL995.\nWe suspect that integrating symbolic information into CLMPT\ncan improve the performance, especially on negative queries. How-\never, exploring how to integrate symbolic information into CLMPT\nis beyond the scope of this paper. This extension is left for future\nwork."    },    {        "title": "2402.12970v2.See_Further_Than_CFAR__a_Data_Driven_Radar_Detector_Trained_by_Lidar.pdf",        "content": "See Further Than CFAR: a Data-Driven Radar\nDetector Trained by Lidar\nIgnacio Roldan1, Andras Palffy1-2, Julian F. P. Kooij2, Dariu M. Gavrila2, Francesco Fioranelli1, Alexander Yarovoy1\n1Microwave Sensing, Signals and Systems (MS3) Group, Department of Microelectronics\n2Intelligent Vehicles (IV) Group, Department of Cognitive Robotics\nDelft University of Technology, Delft, The Netherlands\n{i.roldanmontero, a.palffy, j.f.p.kooij, d.m.gavrila, f.fioranelli, a.yarovoy }@tudelft.nl\nAbstract —In this paper, we address the limitations of tra-\nditional constant false alarm rate (CFAR) target detectors in\nautomotive radars, particularly in complex urban environments\nwith multiple objects that appear as extended targets.\nWe propose a data-driven radar target detector exploiting a\nhighly efficient 2D CNN backbone inspired by the computer\nvision domain. Our approach is distinguished by a unique cross-\nsensor supervision pipeline, enabling it to learn exclusively from\nunlabeled synchronized radar and lidar data, thus eliminating\nthe need for costly manual object annotations.\nUsing a novel large-scale, real-life multi-sensor dataset\nrecorded in various driving scenarios, we demonstrate that\nthe proposed detector generates dense, lidar-like point clouds,\nachieving a lower Chamfer distance to the reference lidar point\nclouds than CFAR detectors. Overall, it significantly outperforms\nCFAR baselines detection accuracy.\nIndex Terms —Automotive radar, radar target detection, deep\nlearning, point cloud generation.\nI. I NTRODUCTION\nIn recent years, the landscape of automotive technology\nhas witnessed a transformative shift, with the integration of\nadvanced sensor systems playing a pivotal role in enhancing\nvehicle safety and autonomy. Among these sensors, radar has\nemerged as a cornerstone technology, contributing significantly\nto the evolution of intelligent transportation systems. Once\nmostly confined to defence applications, radar is routinely\nused for autonomous driving in collision avoidance, adaptive\ncruise control, and overall vehicular awareness [1]. Unlike\nsensing modalities such as cameras and lidar, radar can operate\neffectively in adverse environmental and weather conditions\nwith low visibility. This adaptability positions radar as a robust\nand reliable solution for real-time perception in the complex\nand dynamic context of urban and highway driving [2], [3].\nHowever, despite its merits, integrating radar into automo-\ntive systems is challenging. One such challenge arises from\nthe use of Constant False Alarm Rate (CFAR) detectors to\ngenerate the radar point cloud from the dense radar cube.\nCFAR detectors are optimal in specific scenarios [4] but have\nlimitations in the context of automotive radar [5]. CFAR\nalgorithms, designed to maintain a constant false alarm rate in\nthe presence of varying clutter conditions, may not be well-\nsuited for the dynamic and rapidly changing environments\nencountered on roadways. Issues such as non-homogeneous\nclutter, target masking, and shadowing can compromise theefficacy of CFAR in automotive radar applications. Moreover,\nCFAR detectors have the inherent problem of a fixed, prede-\nfined, expected target size, given how the guard and training\ncells are defined. In the automotive context, the targets of\ninterest have very different sizes, from small targets such as\npedestrians, to large trucks. On top of that, the perceived size\nin the angular dimension depends on the range, and objects\nthat occupy many cells in the near field may be point-like\ntargets in the farther area.\nTo overcome the limitations of CFAR detectors, a novel\ndata-driven detection approach based on deep learning has\nbeen designed and tested in this work using real automo-\ntive radar data. Data-driven detectors have been previously\nexplored in the open literature [6]–[8]; however, this work\nintroduces some new concepts that, to the best of the authors’\nknowledge, are investigated for the first time, namely:\n•The use of the full 4D range-azimuth-elevation-Doppler\nspace as input, ensuring that no weak targets are omitted\ndue to an initial coarse thresholding.\n•The usage of pre-processed lidar data as ground truth to\nenhance radar detection.\n•The comparison against conventional CFAR detectors us-\ning grid-based probabilities of detection and false alarm,\nas well as point-cloud level metrics that capture the\nspatial similarities between lidar and radar data.\n•The verification with a large-scale multi-sensor dataset,\nincluding realistic automotive scenarios. This dataset,\ncontaining lidar, radar, GPS, and camera data will be\npublished for the research community upon publication.\nThe rest of the paper is organized as follows. Section II de-\nscribes the proposed data-driven detector. Section III presents\nthe data collected for the validation of the proposed approach,\nwith results presented in Section IV. Finally, Section V con-\ncludes the paper.\nII. P ROPOSED METHOD\nThe radar detection problem can be formulated as a binary\ndecision task per cell in the radar cube. Before the proposed\nmethod is introduced, some definitions are needed to clarify\nthe terminology used in the rest of the paper:\n•Radar cube refers to the spherical, voxelized represen-\ntation of the radar data, meaning the range, azimuth,arXiv:2402.12970v2  [eess.SP]  27 Feb 2024T raining Inference\nradarlidar\nTimesync\nWindowing +\n2D FFTDoppler\ndisambiguity\n& Doppler\nCorrectionZero filling +\n2D FFTADC DataRDC\nCubeCorrected\nRDC\nCubeRAED\nCubeRoI crop and\nsingle peak\nfinding in\nelevationRAD\npower\ncube\nRAD\nelevation\ncubeTransformation\nto radar framelidar\npoint clouds\nTimestampSyncronized\nlidar Ground\nPlane\nremovalCalibrated\nlidarMapping to\nSpherical formatNon-ground\nobjectsVoxelization and\nthresholding\nProposed\nnetworklidar in spherical\ncoordinates\n3D occupancy grid\n(Ground truth)\n3D occupancy\ngrid\n(Output)(Inputs)Cube parametersFig. 1. Overview of the proposed method. The steps to generate the 3D lidar occupancy grid are on the top row, which will be used as ground truth\nfor training the neural network. The radar signal processing pipeline is shown at the bottom, and is needed to generate the input data. RDC stands for\nrange-Doppler-channel (no angle estimation), and RAED stands for range-azimuth-elevation-Doppler.\nelevation, and Doppler estimation have already been per-\nformed. Each voxel in the radar cube contains information\nabout the reflected power in that cell.\n•3D occupancy grid refers to a binary cube, also in\nspherical coordinates, which contains ones in voxels\nwhere there are targets and zeros otherwise. This can be\ngenerated with lidar data serving as ground truth or with\nradar data being the main task of this work.\n•Point cloud refers to the Cartesian coordinates set of\npoints that results from selecting only those cells con-\ntaining ones in the 3D occupancy grid and converting\nthem to Cartesian coordinates.\nThe proposed method uses a neural network to produce the\n3D occupancy grid only with radar data, using the lidar data\nas ground truth, as shown in the block diagram in Fig. 1. As\ncan be seen, the data collected with both sensors must be pre-\nprocessed before it can be used for the task. For the radar,\nsignal processing has to be applied to estimate the radar cube,\nwhile for the lidar, the 3D occupancy grid has to be generated.\nThe next subsections explain in detail both processes.\nA. Radar Signal Processing\nFrequency-modulated continuous-wave (FMCW) radar with\nmultiple-input multiple-output (MIMO) arrays has been estab-\nlished as the standard for automotive radars [2]. While several\nstrategies to ensure orthogonality between transmitted waves\nare possible [10], the most extended one is time division\nmultiple access (TDMA) due to its ease of implementation.\nHowever, this introduces two undesirable effects that need to\nbe compensated. First, the PRI (Pulse Repetition Interval) is\nenlarged by a factor of the number of transmitters; therefore,\nthe maximum unambiguous Doppler is reduced, as can be seen\nin equation (1),\nvmax=c\n4fcPRI, (1)\nwhere cis the speed of light and fcis the carrier frequency.\nThis effect is especially problematic in the automotive\ncontext, where targets can have high relative speeds. Moreover,\nthe phase difference between signals received from differenttransmitters will depend on both the angle of arrival of the\nsignal and the velocity of the targets (due to the target’s move-\nment between transmission times) [11]. This phase migration\nterm is shown in equation (2),\nϕmig=4π\nλv∆t, (2)\nwhere λis the wavelength, vis the relative speed of the\ntarget and ∆tis the time difference between transmitters. This\nterm must be compensated before performing angle estimation\nto avoid significant estimation distortions. Both undesirable\neffects of TDMA are solved using the overlapped virtual\nantennas present in the radar system with the algorithms\nprovided in [12].\nAn overview of the radar signal processing pipeline is\nshown in the bottom row of Fig. 1. As it can be seen,\napart from the TDMA compensation step, it is a standard\nprocessing pipeline, with the peculiarity that only a single\nvalue in elevation is saved per range-azimuth-Doppler cell.\nThis is done for two reasons. First, due to the sparse antenna\nsystem in the vertical direction, there are grating lobes in the\nelevation plane. Thus, the elevation dimension is cropped in\nthe±20 degree region to avoid the grating lobes, and the\nhighest peak is selected as a single estimated elevation value.\nSecond, the output data size is significantly reduced this way,\nfrom a full range-azimuth-elevation-Doppler (RAED) cube to\na range-azimuth-Doppler (RAD) cube with two channels, one\nwith the power and one with the elevation. This eases data\nhandling for later network training. This processing pipeline\ncan be summarized as follows:\n1) 2D FFT with a Hamming window in the range and\nDoppler dimensions.\n2) Maximum unambiguous Doppler extensions and phase\nmigration compensation due to TDMA.\n3) Zero filling for missing virtual elements, and 2D FFT in\nazimuth and elevation.\n4) Region of interest (RoI) cropping in elevation at ±20◦\nand single peak selection.(a)\n (b)\n (c)\nFig. 2. In (a) the original lidar point cloud as provided by the sensor. In (b) the lidar point cloud after the RoI cropping to mimic the radar field of view\n(i.e.,±70◦in azimuth and ±20◦in elevation). In (c), the point cloud after the road surface removal using PatchWork++ [9].\n5) The output consists then of two range-azimuth-Doppler\n3D cubes, one with the power values of each cell, and\none with the elevation estimation in each cell.\nB. Lidar Processing\nIn order to use the lidar data as ground truth, it has to\nbe converted into a 3D occupancy grid, as shown in Fig. 1.\nThe first step is to time synchronize the data using the\nradar timestamps and perform an extrinsic calibration using\nprecomputed parameters obtained measuring reference targets.\nThen, the lidar data is cropped to the same field of view\nof the radar, i.e., ±70◦in azimuth and ±20◦in elevation,\nwith a maximum range of 50m. An example of this cropped\nRoI is shown in Fig. 2b compared to the original in Fig. 2a.\nMoreover, removing all the returns from the road surface is\nessential since the road surface is hardly visible to the radars.\nThus, these points would provide noisy supervision for training\nthe network. The Patchwork++ algorithm is used to this end\n[9]. The resulting lidar point cloud after removing the road\nsurface points can be seen in Fig. 2c. Finally, the lidar point\ncloud has to be voxelized into a cube. The voxelization process\ncan be understood as generating a 3D occupancy grid, where\neach voxel contains ‘one’ if at least one lidar point is inside\nand ‘zero’ otherwise. However, it is important to note that the\nradar cube grid is not uniform due to the Fourier Transform\nprocessing and its relationship with the sinof the estimated\nangle. This effect, which makes the cells thinner at boresight\nand broader at the edge of the field of view, must be considered\nto generate the same non-uniform lidar 3D occupancy grid.\nC. Neural Network\nThe network’s inputs are two range-azimuth-Doppler radar\ncubes with dimensions R×A×D, where in practice R=\n500, A= 240 andD= 128 . These values are higher than\nthe number of fast-time samples, the number of chirps, and\nthe number antenna elements due to zero padding before the\nFFT processing. One radar cube contains information about\nthe power in each voxel while the other encodes the selected\nelevation bin, as described in Subsection II-A. However, the\nlidar ground truth does not include Doppler information, andtherefore, the Doppler dimension should be processed before\nestimating the 3D occupancy grid. It is known that there is\na dependency between the Doppler and the angle for moving\ntargets (or moving platforms) [13], [14], and therefore, the\nDoppler information can be used to enhance the angular\nresolution. To this end, the first part of the network is designed\nto extract all the Doppler information in each range-azimuth\ncell and encode it into the channel dimension. This is achieved\nby using two 3D convolutional layers followed by a 3D max\npool layer, transforming the 2×R×A×Dinput tensor\ninto a 64×R×Atensor, where the 64 channel dimension\ncontains the encoded information of Doppler and elevation.\nThen, an off-the-shelf 2D CNN backbone is applied to estimate\ntheR×A×E(500×240×44) 3D occupancy grid. The\nsignificant advantage of using such 2D CNN backbones is\ntheir compatibility with hardware accelerators (e.g., GPUs and\nTPUs) and major machine learning frameworks (e.g., Tensor-\nFlow, PyTorch), leading to enhanced computational efficiency.\nWhile our current implementation employs ResNet18 [15], our\nmodular design allows for different backbones, enabling the\nsystem to be tailored to the specific memory and computa-\ntional requirements of the intended platform. Fig. 3 shows a\nschematic representation of the proposed network. Since the\ndata is highly imbalanced (mostly empty), the focal loss [16]\nbetween the predicted 3D occupancy grid and the ground truth\nlidar 3D occupancy grid is used, instead of using the standard\nbinary cross-entropy, setting the sparsity regularizer to 0.95.\nIII. D ATA COLLECTION\nTo support our experiments, a new synchronized and cali-\nbrated multi-sensor dataset was recorded, similar to the View-\nof-Delft dataset [17] and using the same demonstrator vehicle.\nSpecifically for this paper, a Texas Instrument MMWCAS-\nRF-EVM [18] radar board has been installed on the roof next\nto a RoboSense Ruby Plus Lidar (128 layer rotating lidar1).\nThe radar board is a Multiple-Input Multiple-Output (MIMO)\nradar with four cascaded chips resulting in 16 receivers and 12\n1https://www.robosense.ai/en/rslidar/RS-Ruby Plus3D Conv Layer (5x3x3)RAD Power\nRadar Cube2x128x500x240\n64x500x240\n3D occupancy\ngrid\n44x500x2403D Conv Layer (5x3x3)\n3D Max Pool\nResNet 18 (18 Conv layers with\nskip connections)Doppler\nEncoderSegmentation\nBackboneRAD Elevation\nRadar CubeFig. 3. Block diagram of the proposed neural network. The first part is a\nDoppler encoder network to reduce the Doppler dimension. Then, a Resnet18\nsegmentation backbone is used to estimate the 3D occupancy grid.\nTABLE I\nRADAR WAVEFORM PARAMETERS USED IN THE DATA COLLECTION\nParameter Value\nStart Frequency (GHz) 76\nEffective Bandwidth (MHz) 750\nChirp Slope (MHz/ µs) 35\nChirps Length ( µs) 28\nNumber ADC Samples per Chirp 256\nNumber of Chirps per Frame 128\nSampling Frequency (Msps) 12\ntransmitters. The resulting virtual array is an 86-element half-\nwavelength spaced array in the x-direction (used for azimuth\nestimation) and a 4 Minimum Redundancy Array in the z-\ndirection (elevation estimation).\nThe data collection took place in various real-life scenar-\nios with different object characteristics, such as suburban,\ncampus, and Delft old-town locations. The recordings have\nbeen organized into seven independent scenes of 5 minutes,\nyielding around 3000 radar frames per scene (with a frame rate\nof 10Hz) with their associated lidar point cloud with a total\nof 21000 frames. Both sensors have been time-synchronized\nusing the provided timestamps and spatial calibrated using\nreference targets. The dataset has been split into training,\nvalidation, and test sets by assigning five scenes to the training\nand validation and two to the test. For the train validation\nsplit within the five scenes, 10% of the frames are used for\nvalidation.\nIV. R ESULTS\nAfter training, the proposed neural network can be used to\nestimate the 3D occupancy grid for each frame. The results\npresented in this section have been obtained using only the\ntest set, which contains unseen data for the proposed method.The results are compared against different types of CFAR\ndetectors with different parameters, trying always to tune\nthem to achieve the best performance. However, optimizing\nCFAR parameters in every realistic driving situation is hardly\npossible due to the wide variety of conditions and targets.\nTheoretically, the order statistics (OS)-CFAR yields better\nresults in multi-target situations [19]. However, the OSCA-\nCFAR algorithm proposed in [20] has also been tested since\nthe computational complexity is lower while keeping good\nperformance. Moreover, to avoid expensive 3D detectors, in all\ncases, a 2D CFAR in either range-Doppler or range-azimuth\nmatrices has been applied, followed by a 1D CFAR or a peak\ndetector in the remaining dimension of the radar data. For\neach CFAR implementation, different combinations of their\nhyperparameters have been tested for a fair evaluation, but\nonly the top five best are reported in this paper.\nQualitative results of the ground truth and generated point\ncloud of the baseline and our proposed method are shown in\nFig. 4. All the point clouds have been projected into the camera\nimage to give a sense of the 3D scene (top), but the bird’s\neye view projection is also shown (bottom). For simplicity,\nthe point clouds have been cropped to a maximum range of\n30 meters. Moreover, as a visual aid in the top view, cyclists\nare highlighted with a blue hexagon, and cars are highlighted\nwith a red ellipsoid. In Fig. 4a, the original lidar point cloud is\npresented, where many details of the scene can be appreciated.\nFig. 4b shows the detections generated using the proposed\nneural network, and as can be seen, most of the details are\npreserved. Objects are slightly overestimated in size, but the\noverall scene is clear. Finally, Fig. 4c shows the output of\nthe quantitatively best CFAR (i.e., a 2D OS-CFAR in range-\nazimuth followed by a 1D OS-CFAR in Doppler). In the top-\ndown view, it is possible to appreciate how points for the main\ntargets are detected, but the overall structure of the point cloud\nis much sparser and noisier. For further evaluation, a different\nframe is shown in Fig. 5 with the same approach, highlighting\nagain the cars and cyclists.\nFor quantitative evaluation, two different metrics are pre-\nsented. First, an analysis of the probability of detection and\nprobability of false alarms has been performed, where the lidar\n3D occupancy grid is used as the ground truth. However, these\nmetrics do not capture any spatial relationship, i.e., a slight\nshift due to sensor misalignment in one of the dimensions\nwould make the values of these two probabilities rather poor,\neven if the overall spatial estimation may be reasonable. For\nthis reason, a point-cloud level metric has been included\nusing the Chamfer distance between the lidar point cloud and\nthe generated radar point clouds. The Chamfer distance is a\ncommon metric used to evaluate how similar two point clouds\nare [21]–[23]. For each point in each set (the sets do not need\nto have the same number of points), the nearest neighbor is\nfound, and the distances are squared and summed up. The\nformal definition can be seen in (3),\nd(S1, S2) =X\nx∈S1min\ny∈S2||x−y||2\n2+X\ny∈S2min\nx∈S1||x−y||2\n2.(3)(a)\n (b)\n (c)\nFig. 4. Data frame example #1. In (a), the original lidar point cloud projected into the camera as well as a top-down view. In (b), the radar point cloud\ngenerated with the proposed method. In (c) the radar point cloud generated with the best implemented CFAR (2D OS-CFAR in range-azimuth followed by\nan OS-CFAR in Doppler).\n(a)\n (b)\n (c)\nFig. 5. Data frame example #2. In (a), the original lidar point cloud projected into the camera as well as a top-down view. In (b), the radar point cloud\ngenerated with the proposed method. In (c) the radar point cloud generated with the best implemented CFAR (2D OS-CFAR in range-azimuth followed by\nan OS-CFAR in Doppler).\nTable II shows the results for the top 5 best CFAR detectors\nand the proposed method. The CFAR guard cells in each\ndimension have been set to 10 in range and 16 in Doppler\nand azimuth. For the OS-CFAR, the rank has been set to 0.75\ntimes the number of training cells and no guard cells have\nbeen used as recommended in [19]. Only those peaks with\na minimum height of 10dB below the maximum have beenselected for peak detection. An important remark is needed\nregarding the results in Table II. As mentioned before, the\ncomputation of the PfaandPdis highly dependent on a good\nalignment of the radar cubes. A shift on a single resolution\ncell may be enough to reduce the Pddrastically. Therefore,\nit is possible that a calibration misalignment exists between\nthe lidar and the radar that the neural network is learningTABLE II\nPERFORMANCE RESULTS . RAD STANDS FOR RANGE ,AZIMUTH ,AND\nDOPPLER ,MEANING IT IS THE DIMENSION WHERE EACH DETECTOR IS\nAPPLIED .\nMethod Pd(%) Pfa(%) Chamfer distance ( m2)\nProposed Method 65.32 1.99 1.60\n2D OS(RA) +\n1D OS(D)1.72 0.09 4.98\n2D OS(RD) +\n1D OS(A)1.51 0.08 5.24\n2D CAOS(RA) +\n1D OS(D)0.22 0.07 6.98\n2D CAOS(RA) +\nPeak Detector(D)0.15 0.03 6.40\n2D CAOS(RD) +\nPeak Detector(A)0.63 0.02 4.85\nto resolve, significantly boosting the Pdcompared to the\nconventional CFAR approaches. Moreover, CFAR results are\nmuch more sparse than lidar data; thus, the Pdis dropping.\nEven with this in mind, the proposed method has an excellent\nperformance of ∼65%Pdcompared with lidar. On the other\nhand, the relatively high Pfa(∼2%) can be explained by\nthe overestimation of most targets to the neighboring cells,\nwhich, while not desirable, may not be critical for most\nautomotive situations. In terms of the Chamfer distance, where\nthe possible misalignment does not play a significant role, the\nproposed method still outperforms all the CFAR approaches.\nV. C ONCLUSIONS\nIn this paper, a novel approach to radar target detection\nleveraging the capabilities of neural networks is presented. A\ncomprehensive dataset comprising over 30 minutes of real-\nworld driving scenarios has been collected using a vehicle\nequipped with both lidar and radar sensors, resulting in 21000\nradar frames with their corresponding lidar ground truth. The\ndataset contains a variety of scenes such as suburban and\ncrowded city center environments.\nThe work thoroughly compares the proposed neural\nnetwork-based radar detector against various Constant False\nAlarm Rate (CFAR) methods. The results reveal that the\nproposed method consistently outperforms conventional CFAR\ntechniques across various challenging scenarios. Two different\ntypes of metrics have been used: the classical probability\nof detection and probability of false alarm and the Chamfer\ndistance, a point cloud level metric used to capture the spatial\nrelationships and similarity between point clouds. The pro-\nposed neural network-based detector achieves enhanced per-\nformance in both metrics, especially in dynamic and cluttered\nenvironments.\nThe success of this approach opens avenues for future\nresearch, encouraging the exploration of innovative machine-\nlearning techniques for radar signal processing; therefore, after\nsuitable organization, the dataset used here will be publicly\nreleased in the near future, containing the raw radar data.REFERENCES\n[1] I. Bilik, O. Longman, S. Villeval, and J. Tabrikian, “The rise of radar\nfor autonomous vehicles: Signal processing solutions and future research\ndirections,” IEEE Signal Processing Magazine , vol. 36, no. 5, pp. 20–31,\n2019.\n[2] S. Sun, A. P. Petropulu, and H. V . Poor, “MIMO radar for advanced\ndriver-assistance systems and autonomous driving: Advantages and\nchallenges,” IEEE Signal Processing Magazine , vol. 37, no. 4, pp. 98–\n117, jul 2020.\n[3] A. Palffy, J. Dong, J. F. Kooij, and D. M. Gavrila, “CNN based road\nuser detection using the 3d radar cube,” IEEE Robotics and Automation\nLetters , vol. 5, no. 2, pp. 1263–1270, 2020.\n[4] M. A. Richards, J. A. Scheer, and W. A. Holm, Principles of Modern\nRadar: Basic Principles . Scitech Publishing Inc, 2010, vol. 1.\n[5] J. Yoon, S. Lee, S. Lim, and S.-C. Kim, “High-density clutter recognition\nand suppression for automotive radar systems,” IEEE Access , vol. 7, pp.\n58 368–58 380, 2019.\n[6] D. Brodeski, I. Bilik, and R. Giryes, “Deep radar detector,” in 2019\nIEEE Radar Conference (RadarConf) , 2019, pp. 1–6.\n[7] Y . Cheng, J. Su, M. Jiang, and Y . Liu, “A novel radar point cloud gen-\neration method for robot environment perception,” IEEE Transactions\non Robotics , vol. 38, no. 6, pp. 3754–3773, 2022.\n[8] Y . Lin, X. Wei, Z. Zou, and W. Yi, “Deep learning based target detection\nmethod for the range-azimuth-doppler cube of automotive radar,” in\n2023 IEEE 26th International Conference on Intelligent Transportation\nSystems (ITSC) , 2023, pp. 2868–2873.\n[9] S. Lee, H. Lim, and H. Myung, “Patchwork++: Fast and robust ground\nsegmentation solving partial under-segmentation using 3d point cloud,”\nin2022 IEEE/RSJ International Conference on Intelligent Robots and\nSystems (IROS) , 2022, pp. 13 276–13 283.\n[10] H. Sun, F. Brigui, and M. Lesturgie, “Analysis and comparison of MIMO\nradar waveforms,” in 2014 International Radar Conference , 2014, pp.\n1–6.\n[11] D. Zoeke and A. Ziroff, “Phase migration effects in moving target\nlocalization using switched mimo arrays,” in 2015 European Radar\nConference (EuRAD) , 2015, pp. 85–88.\n[12] C. M. Schmid, R. Feger, C. Pfeffer, and A. Stelzer, “Motion compensa-\ntion and efficient array design for tdma fmcw mimo radar systems,” in\n2012 6th European Conference on Antennas and Propagation (EUCAP) ,\n2012, pp. 1746–1750.\n[13] S. Yuan, F. Fioranelli, and A. Yarovoy, “Vehicular motion-based doa\nestimation with a limited amount of snapshots for automotive MIMO\nradar,” IEEE Transactions on Aerospace and Electronic Systems , pp.\n1–15, 2023.\n[14] W. Zhang, P. Wang, N. He, and Z. He, “Super resolution doa based\non relative motion for fmcw automotive radar,” IEEE Transactions on\nVehicular Technology , vol. 69, no. 8, pp. 8698–8709, 2020.\n[15] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image\nrecognition,” 2015.\n[16] T.-Y . Lin, P. Goyal, R. Girshick, K. He, and P. Doll ´ar, “Focal loss for\ndense object detection,” 2018.\n[17] A. Palffy, E. Pool, S. Baratam, J. F. P. Kooij, and D. M. Gavrila, “Multi-\nclass road user detection with 3+1d radar in the view-of-delft dataset,”\nIEEE Robotics and Automation Letters , vol. 7, no. 2, pp. 4961–4968,\n2022.\n[18] T. I. Inc. (2019) Design guide: Tidep-01012—imaging radar using\ncascaded mmwave sensor reference design (rev. a). [Online]. Available:\nhttps://www.ti.com/lit/ug/tiduen5a/tiduen5a.pdf\n[19] H. Rohling, “Radar CFAR thresholding in clutter and multiple target\nsituations,” IEEE Transactions on Aerospace and Electronic Systems ,\nvol. AES-19, no. 4, pp. 608–621, 1983.\n[20] M. Kronauge and H. Rohling, “Fast two-dimensional CFAR procedure,”\nIEEE Transactions on Aerospace and Electronic Systems , vol. 49, no. 3,\npp. 1817–1823, 2013.\n[21] C. Duan, S. Chen, and J. Kovacevic, “3d point cloud denoising via deep\nneural network based local surface estimation,” 2019.\n[22] P. Hermosilla, T. Ritschel, and T. Ropinski, “Total denoising: Unsuper-\nvised learning of 3d point cloud cleaning,” 2019.\n[23] H. Fan, H. Su, and L. Guibas, “A point set generation network for 3d\nobject reconstruction from a single image,” 2016."    },    {        "title": "2402.12992v1.On_gravito_inertial_surface_waves.pdf",        "content": "Submitted to Contemporary Mathematics\nOn gravito-inertial surface waves\nYves Colin de Verdi` ere and J´ er´ emie Vidal\nThis paper is dedicated to the memory of Steve.\nAbstract. In geophysical environments, wave motions that are shaped by\nthe action of gravity and global rotation bear the name of gravito-inertial\nwaves. We present a geometrical description of gravito-inertial surface waves,\nwhich are low-frequency waves existing in the presence of a solid boundary.\nWe consider an idealized fluid model for an incompressible fluid enclosed in\na smooth compact three-dimensional domain, subject to a constant rotation\nvector. The fluid is also stratified in density under a constant Brunt-V¨ ais¨ al¨ a\nfrequency. The spectral problem is formulated in terms of the pressure, which\nsatisfies a Poincar´ e equation within the domain, and a Kelvin equation on the\nboundary. The Poincar´ e equation is elliptic when the wave frequency is small\nenough, such that we can use the Dirichlet-to-Neumann operator to reduce the\nKelvin equation to a pseudo-differential equation on the boundary. We find\nthat the wave energy is concentrated on the boundary for large covectors, and\ncan exhibit surface wave attractors for generic domains. In an ellipsoid, we\nshow that these waves are square-integrable and reduce to spherical harmonics\non the boundary.\n1. Introduction\nGlobal rotation and buoyancy are ubiquitous ingredients of many geophysical\nsystems (e.g. the Earth’s oceans, or the liquid cores of planets). When the density\nstratification is stable (i.e. when a light fluid lies above a denser one), the Coriolis\nand buoyancy forces can sustain gravito-inertial waves. Since these waves are often\nbelieved to be key to understand the dynamics of geophysical flows, they have\nreceived considerable interest. From a mathematical viewpoint, these waves are\ngoverned by a mixed hyperbolic-elliptic equation [ FS82a ], which is referred to as\nthe Poincar´ e equation below. Indeed, the latter reduces to the so-called Poincar´ e\nequation (denoted by Cartan [ Car22 ] after the seminal work of poincar´ e [ Poi85 ])\nfor pure inertial waves without density stratification [ CdVV23 ]. Let us consider\nthe simplest case (known as the f−plane approximation in geophysical modeling),\nwhere the fluid is subject to a constant global rotation− →Ω = ( f/2)− →e3and stratified\n2020 Mathematics Subject Classification. Primary 53Z05, 35Q35; Secondary 76U60, 76B70.\nKey words and phrases. Spectral theory, microlocal analysis, gravito-inertial waves, surface\nwaves.\nJV received funding from European Research Council ( erc) under the European Union’s\nHorizon 2020 research and innovation programme (grant agreement no. 847433, theia project).\n1arXiv:2402.12992v1  [math.AP]  20 Feb 20242 YVES COLIN DE VERDI `ERE AND J ´ER´EMIE VIDAL\nin density with a constant Brunt-V¨ ais¨ al¨ a frequency Nunder the constant gravity− →g=−g− →e3. In unbounded fluids, the angular frequency ω∈R\\0 of gravito-inertial\nwaves is then given by\n(1.1) ω||ξ||2=N2(ξ2\n1+ξ2)2+f2ξ2\n3,ξ= (ξ1, ξ2, ξ3),\nwhich is obtained when the principal symbol of the Poincar´ e equation vanishes. Dis-\npersion relation (1.1) shows that gravito-inertial waves only exist when min( N, f)≤\n|ω| ≤max( N, f). Hence, there is a frequency gap without waves when the fluid\nis unbounded. However, low-frequency gravito-inertial waves can exist in this fre-\nquency gap when the fluid domain is bounded [ FS82a ]. Motivated by geophys-\nical applications, we recently revisited this canonical problem and (surprisingly)\nfound these low-frequency waves are polynomials when the boundary is an ellipsoid\n[VCdV24 ].\nThe goal of this study is to present a geometrical description of these gravito-\ninertial surface waves, which are denoted below by Kelvin waves. Indeed, they\nshare similar properties with surface waves studied by Lord Kelvin in oceanogra-\nphy [Kel80 ]. This is based on a reduction to the boundary of the wave equa-\ntions, following our previous works on inertial [ CdVV23 ] and gravito-inertial\nwaves [ VCdV24 ]. We consider an incompressible fluid in a smooth compact three-\ndimensional (3D) domain D, which is subject to a constant global rotation. The\nfluid is also supposed to be stably stratified in density with a constant Brunt-V¨ ais¨ al¨ a\nfrequency N. The motion of the fluid can be recovered from the pressure ϕ, which\nsatisfies a Poincar´ e equation inside D, and a Kelvin equation on the boundary ∂D.\nIf the frequency ωis small enough, the Poincar´ e equation is elliptic. Thus, we\ncan use the Dirichlet-to-Neumann operator (see Appendix B) to reduce the Kelvin\nequation to a pseudo-differential equation on the boundary. In this case, the eigen-\nsolutions concentrate on the boundary for large covectors (i.e. large wave numbers\nin the physical terminology).\nThe manuscript is organized as follows. We first present general properties of\nthe Poincar´ e and Kelvin equations in §2 and §3. Then, a detailed derivation of the\nKelvin equation is given in §4. Next, a microlocal analysis of the Kelvin equation\nis presented in §5 and §6. Finally, the case where ∂Dis an ellipsoid is considered in\n§7. Notably, we show that the spectrum is pure point with polynomial eigenvectors,\nand that the pressure field reduces to a spherical harmonic on ∂D.\n2. The Poincar´ e operator\nWe consider a fluid in a compact domain D⊂R3with a smooth boundary ∂D,\nand we equip R3with the canonical (orthogonal) basis vectors (− →e1,− →e2,− →e3). We\nassume that the fluid domain is submitted to a constant global rotation− →Ω (with\nrespect to an inertial frame), and that the fluid is stably stratified in density with\na Brunt-V¨ ais¨ al¨ a frequency N > 0 under the imposed gravity− →g=−g− →e3. In this\npaper, we always assume that− →gandNare constant. We seek small-amplitude\nperiodic motions, with angular frequency ω, with respect to a hydrostatic reference\nconfiguration. We refer the interested reader to [ VCdV24 ] for further details about\nthe physical model. In the linear theory, the equations in the rotating frame are\n(2.1) iω− →u+ 2− →Ω∧− →u−ρ− →g=−∇ϕ,div− →u= 0, iωρ =N2u3/g,ON GRAVITO-INERTIAL SURFACE WAVES 3\nwhere the velocity vector field− →uis tangent to the boundary ∂D, where ϕis the\npressure, and ρis the density perturbation. We can rewrite these equations in a\nsymmetric form as\n(2.2) iω− →u+ 2− →Ω∧− →u+Nρ1− →e3=−∇ϕ,div− →u= 0, iωρ 1=Nu3,\nwhere ρ1:=ρg/N is a rescaled density of the fluid.\nUsing the Leray projector denoted by Π (see Appendix A) and defining Π 0the\noperator acting on a pair (− →u , ρ) by\nΠ0(− →u , ρ 1) := (Π− →u , ρ 1),\nwe can reformulate the spectral problem as the spectral properties of the Poincar´ e\noperator Pdefined by\n(2.3) P:=iΠ0AΠ0, A =\n0−2Ω32Ω20\n2Ω3 0−2Ω10\n−2Ω22Ω1 0 N\n0 0 −N 0\n,\nwhere iAis self-adjoint on L2(D,C4). The eigenvalues of iAare±ω±with\n(2.4) ω±=s\n1\n2\u0012\nN2+f2±q\n(N2+f2)2−16N2Ω2\n3\u0013\n,\nwhere we have introduced f:= 2∥− →Ω∥(which is called the Coriolis parameter in\nthe geophysical literature). Then, the spectrum of P(i.e. the set of values of ω\nfor which P −ωIis not invertible, where Iis the identity operator) is given by the\nfollowing theorem.\nTheorem 2.1.The Poincar´ e operator Pis a bounded self-adjoint operator in\nL2(D,C4), and its spectrum σ(P)is given by\nσ(P) = [−ω+, ω+].\nProof. We have ∥Π0∥= 1, while the eigenvalues of the Hermitian matrix iA\nare±ω±. This proves that ∥P∥ ≤ ω+. Moreover, Pis not elliptic when ω−≤\n|ω| ≤ω+, and we will see in Section 5.3 that the Kelvin equation is not elliptic for\n0<|ω|< ω−. The theorem follows from these observations and Appendix C (see\nalso Theorem 2.1. of [ CdV20 ]). □\nOur main interest in this paper will be the study of σ(P)∩]−ω−, ω−[. The\nlow-frequency waves in this interval are refereed as class-II solutions in [ FS82a ].\nThis part of the spectrum is controlled by an equation Kωϕ= 0 satisfied by the\npressure on the boundary, which we call the Kelvin equation. The solutions of the\nKelvin equation for large covectors are localized on the boundary. This shows that\nthese solutions are associated with surface waves of the Poincar´ e operator.\n3. Equations for the boundary value of the pressure\nWe want to recast the equations for the velocity− →uand the density ρ1as\na boundary-value problem for the scalar pressure ϕ. Indeed, this will ease the\nmicrolocal analysis of the problem. The pressure satisfies an equation in D, called\nthe Poincar´ e equation, and boundary conditions given by the Kelvin equation. We\nexplain below how to obtain these two equations in the general case.4 YVES COLIN DE VERDI `ERE AND J ´ER´EMIE VIDAL\n3.1. Poincar´ e equation. When ω̸=±ω±, we can solve the equation\n(3.1)− →u=Mω(i∇ϕ)\nwith Mω=j⋆(ω−iA)−1jwhere j:R3→R4is the injection− →u→(− ���u ,0)⊤. Then,\nwe make use of the divergenceless condition div(− →u) = 0 to obtain the Poincar´ e\nequation Pωϕ= 0 for ϕ. This is a partial differential equation of degree two with\nconstant coefficients, which is given by\n(3.2) Pωϕ=ω\n(ω2−N2)∆ + N2∂2\n∂x2\n3−4 3X\ni=1Ωi∂\n∂xi!2\nϕ.\nNote that, if we do that directly, the equation that we obtain has a simple zero at\nω= 0. Equation (3.2) is also defined using a determinant as\nPωϕ=−\f\f\f\f\f\f\f\f\f\fω−iA\n∂1\n∂2\n∂3\n0\n\n\u0000∂1∂2∂30\u0001\n0\f\f\f\f\f\f\f\f\f\fϕ\nThe symbol of Pωis\nσω(ξ) =ω\u0012\n(N2−ω2)∥ξ∥2−N2ξ2\n3+ 4\u0010\nξ.− →Ω\u00112\u0013\n,\nwhere ξ= (ξ1, ξ2, ξ3) is the covector of norm ||ξ||= (ξ2\n1+ξ2\n2+ξ2\n3)1/2. We also\ndefine the real-valued matrix Σ ω, associated with the quadratic form σω, as\nσω(ξ) :=⟨Σωξ|ξ⟩\nwhere ⟨.|.⟩is the canonical duality product. If 0 < ω < ω −, Σωis a real-valued,\nsymmetric, and strictly positive matrix. The Poincar´ e operator is elliptic if and\nonly if 0 <|ω|< ω−or|ω|> ω +.\n3.2. Kelvin equation. Using again the equation− →u=Mω(i∇ϕ), we write\nthat− →uis tangent to ∂D. This allows us to obtain the Kelvin equation. Since\nthis equation is quite complicated in general, we will compute it in some special\ncases (see §4). In the general case, the Kelvin equation can be written as a pseudo-\ndifferential equation Kωϕ= 0 on the boundary ∂D, which satisfies Theorem 3.1.\nTheorem 3.1.The Poincar´ e equation is elliptic when 0<|ω|< ω−, and the\nKelvin equation reads\n(3.3) Kωϕ=DtN ωϕ+i− →Wωϕ,\nwhere the Dirichlet to Neumann operator DtN ω(see Appendix B) is defined using\nthe Euclidian metric gωassociated to the symbol σωofPω, that is gω= Σ−1\nω,\nand− →Wωis a real-valued vector field tangent to the boundary. Moreover,− →Wωis\ndivergenceless with respect to the area vωassociated to the restriction gω,∂ofgωto\n∂D. In particular, Kωis a self-adjoint pseudo-differential operator of degree 1on\nL2(∂D, v ω)whose principal symbol is\n(3.4) kω(x,ξ) =q\ng⋆\nω,∂(x,ξ)− ⟨ξ|− →Wω(x)⟩,\nwhere (x,ξ)∈T⋆(∂D)andg⋆\nω,∂is the dual of gω,∂. The Kelvin equation Kωϕ= 0\nis elliptic at x∈∂Dif and only if ∥− →Wω(x)∥gω,∂<1.ON GRAVITO-INERTIAL SURFACE WAVES 5\nProof. For any two smooth functions ϕ1andϕ2onD, let us consider the\nHermitian bracket\n((ϕ1|ϕ2)) =⟨i∇ϕ1|Mω(i∇ϕ2)⟩L2\nwhere ⟨u1|u2⟩L2=R\nDu1·u2|dx|where uis the complex conjugate of u. Integrating\nby part, we get\n((ϕ1|ϕ2)) =−Z\nDϕ1div (Mω∇ϕ2)|dx|+Z\n∂Dϕ1Lωϕ2vω\nwhere Lωis a complex-valued vector field. Let us assume now that Pωϕ2= 0, and\nwe get\n((ϕ1|ϕ2)) =Z\n∂Dϕ1Lωϕ2vω\nThis vanishes if and only if Mω(∇ϕ2) is orthogonal to all gradient fields and, hence,\nis tangent to the boundary. In other words, the equation Lωϕ= 0 is the boundary\ncondition for ϕ, that is the Kelvin equation. Assuming also that Pωϕ1= 0, we see\nthatLωis self-adjoint. We can always decompose the vector field Lωas Ψ− →nω+i− →W1\nwhere Ψ is a function,− →nωis the unit outgoing normal to ∂Dfor the metric gω,\nand− →Wis a complex vector field tangent to ∂D. Then, we can rewrite Lωas\nLω= ΨDtN ω+i− →W1. The self-adjointness of the operator Lωimplies that the\nprincipal symbol is real-valued. Ψ is thus real-valued, and− →W1is a real-valued\nvector field. Taking the adjoint, we get\nLω=L⋆\nω=DtN ωΨ +i− →W1+idivω(− →W1).\nThis implies, by separation of the real and imaginary parts, that div ω(− →W1) = 0 and\nΨDtN ω−DtN ωΨ = 0. The last equation implies that Ψ : ∂D→Ris invariant by\nthe geodesic flow of gω,∂and, hence, constant. Removing this real constant, we get\nthe final result. □\n4. Computing the Kelvin equation\nLet us fix a point x∈∂D, and we assume that the tangent plane to ∂Dis\ngiven by a1x1+a2x2+a3x3= 0 where a= (a1, a2, a3) is a non zero outward\nnormal covector of D. The condition that− →uis tangent on ∂Dcan be written as a\nvanishing condition for the determinant given by\n(4.1) Bω(∇ϕ,a) =\f\f\f\f\f\f\f\f\f\fω−iA\n∂1ϕ\n∂2ϕ\n∂3ϕ\n0\n\n\u0000a1a2a30\u0001\n0\f\f\f\f\f\f\f\f\f\f.\nThe Hermitian form Bωcan be written as Bω= Σ ω+iΞω, where Ξ is real and\nantisymmetric. We can rewrite Bω(∇ϕ,a) as a complex-valued vector field acting\nonϕ, which gives\nBω(∇ϕ,− →a) =\u0010− →N+i− →W1\u0011\nϕ\nwith− →N= Σω(a). Using the fact that gω= Σ−1\nω, we get\ngω(− →N ,− →e) =⟨a|− →e⟩.6 YVES COLIN DE VERDI `ERE AND J ´ER´EMIE VIDAL\n  \nFigure 1. Non-ellipticity of the Kelvin equation for different\ncases. Left: Aligned case with− →Ω∝− →g. The Kelvin equation is\nnon-elliptic near the equator when cos ϕ≤ω/N (see §5.1). Right :\nThe Kelvin equation is elliptic when the tangent plane (gray) is\nparallel to gravity (see §5.2), whereas it is non-elliptic when the\ntangent plane is orthogonal to gravity (see §5.3).\nThis confirms that− →Nisgω-orthogonal to ∂D. We see that the vector− →Nis outward\nbecause, if− →nis an outgoing vector, gω(− →N ,− →n) =⟨a|− →n⟩is positive. Moreover, we\nhave\ngω(− →N ,− →N) =⟨a|− →N⟩=σω(a).\nHence, we have− →N=p\nσω(a)− →nω. Finally, we get the Kelvin equation\np\nσω(a)DtN ωϕ+i− →W1ϕ= 0,\nwhich is of the form (3.3) with− →W=− →W1/p\nσω(a).\nThe Kelvin equation is non-elliptic at points where the tangent plane is given\nbyaif and only if\n∥− →W1∥2\nω> σω(a).\nAt the points x∈∂Dsuch that− →W1(x)̸= 0 for ω= 0, this holds when ωis small\nenough. Indeed, there is a coefficient 1 /ωin front of gωand a coefficient ωin\nfront of σ(ω). Note also that DtN ωand− →W1are even functions of ωsuch that, if\nKωϕ= 0, we have K−ω¯ϕ= 0. This allows us to obtain, as expected, real-valued\neigenvectors.\n5. Explicit computations\nAs illustrated in figure 1, we seek below whether the Kelvin equation is elliptic\nor not for different cases. To check the ellipticity of Kω, we need to compute the\nHermitian form Bωand the metric gω. Without loss of generality, we can assume\nthat− →Ω = (Ω 1,0,Ω3)⊤with f2= 4(Ω2\n1+ Ω2\n3). The symbol of the Poincar´ e equation\nis written as\nσω(ξ) =ω\u0002\n(N2+ 4Ω2\n1−ω2)ξ2\n1+ (N2−ω2)ξ2\n2+ (4Ω2\n3−ω2)ξ2\n3+ 8Ω 1Ω3ξ1ξ3\u0003\nwithξ= (ξ1, ξ2, ξ3), and the associated metric is\ngω=1\nδω\u0000\n(4Ω2\n3−ω2)dx2\n1+ (N2+ 4Ω2\n1−ω2)dx2\n3−8Ω1Ω3dx1dx3\u0001\n+dx2\n2\nω(N2−ω2).ON GRAVITO-INERTIAL SURFACE WAVES 7\nwith δ= (ω2−ω2\n−)(ω2−ω2\n+). Moreover, we have\n− →W1= 2Ω 3(N2−ω2)(a1∂2−a2∂1) + 2Ω 1ω2(a2∂3−a3∂2),\nwhich is obtained from equation (4.1). We see that− →W1is everywhere transverse to\nthe meridians. This implies the wave propagation in the prograde direction (e.g. is\neastward when Ω 3>0). When ωis small, we also have\n− →W1= 2Ω 3N2(a1∂2−a2∂1) +O(ω2)\nand− →W=− →W1/p\nσω(a). We can now evaluate these formulas for different cases.\n5.1. Vertical stratification and rotation. We describe here the Kelvin\nequation when the rotation vector− →Ω is parallel to gravity (known as the f−plane\napproximation in geophysical literature). The corresponding Kelvin equation was\nderived in [ FS82a ], and revisited in [ VCdV24 ] using microlocal arguments when\n∂Dis an ellipsoid. We have Ω 1= 0 and define the Coriolis parameter f:= 2Ω 3\nsuch that\nω−= min( N, f), ω += max( N, f),\nandδ= (ω2−N2)(ω2−f2). The problem is then invariant by rotation around the\nx3-axis. Hence, we can assume that a= (cos ϕ,0,sinϕ) with |ϕ| ≤π/2, where ϕ\ncan be called the latitude. Then, we get\nσω(a) =ω\u0002\n(N2−ω2) cos2ϕ+ (f2−ω2) sin2ϕ\u0003\nand\n− →W1=f(N2−ω2) cosϕ∂2,∥− →W1∥2\nω=f2(N2−ω2)2cos2ϕ\nω(N2−ω2).\nWe can check that the Kelvin equation is elliptic at large latitudes, that is when\n|cosϕ|< ω/N . In particular, the operator is never elliptic at the equator ϕ= 0.\nWe can also compute the symbol of kωofKω. It is given by\nkω(x,ξ) =ω+Ncosϕ(x)ξsp\nξ2s+ξ2u,\nwhere ξ= (ξs, ξu) are the canonical coordinates of ξin an orthonormal basis of\nTx∂Dwith xsa length parametrisation of the “parallel” through xoriented to the\n“east” (i.e. in the prograde direction when f >0). In particular, we see that\n•When 0 < N≤f,T⋆∂D\\0 is foliated by the cones H−1(ω) with |ω| ≤\nω−=N.\n•When 0 < f < N , we have\n∪|ω|≤fH−1(ω) =(\f\f\f\f\fcosϕξsp\nξ2s+ξ2u\f\f\f\f\f≤f/N)\n,\nwhich is not the full cotangent space. This will have some implications on\nthe Weyl asymptotics for the eigenvalues when ∂Dis an ellipsoid.8 YVES COLIN DE VERDI `ERE AND J ´ER´EMIE VIDAL\n5.2. Stratification parallel to the boundary. We consider the case where\nthe tangent plane is horizontal {x3≤0}, such that a= (0,0,1) and the stratifica-\ntion with N > 0 is parallel to the tangent plane. Then, we have\n− →W1=−2Ω1ω2∂2,∥− →W1∥2\nω=4Ω2\n1ω4\nω(N2−ω2), σ ω(a) =ω(4Ω2\n3−ω2).\nThe ellipticity condition is given by\n∥− →W1∥2\nω< ω(4Ω2\n3−ω2).\nAfter some calculations, this gives\n(ω2−ω2\n−)(ω2−ω2\n+)>0\nwhich is satisfied when 0 <|ω|< ω−. Hence, there are no surface waves in this\ncase.\n5.3. Stratification orthogonal to the boundary. We assume that a=\n(1,0,0) and Ω 3̸= 0. Then, we have\n− →W1= 2(N2−ω2)Ω3∂2,∥− →W1∥2\nω=4(N2−ω2)Ω2\n3\nω, σ ω(a) =ω(N2+ 4Ω2\n1−ω2).\nAfter some calculations, this gives the ellipticity condition\n(ω2−ω2\n−)(ω2−ω2\n+)<0.\nWe see that the Kelvin equation is non-elliptic when 0 <|ω|< ω−. Hence, there\nare always surface waves located near the equator. This completes the fact that\nthe spectrum is always [ −ω+, ω+], as stated in Theorem 2.1.\n6. Dynamics of waves\nWe can study a more general problem. Let ( X, g) be a closed Riemannian\nsurface, and− →Wbe a divergenceless vector field on X. We consider a self-adjoint\noperator Kdefined by\nK=E−1\ni− →W,\nwhere Eis an elliptic self-adjoint pseudo-differential operator of principal symbol\neof degree 1. The principal symbol kofKis given by\nk(x,ξ) =e(x,ξ)− ⟨ξ|− →W(x)⟩.\nIf the characteristic cone Σ = k−1(0) is non-degenerate, Kis called of principal\ntype. The projection on Xof Σ is S=πM(Σ ={∥− →W∥g≥1}. For each x∈∂D\nsuch that ∥− →W(x)∥g>1, there is exactly two covectors directions in Σ. The base of\nthe cone Σ consists of two copies of Sglued along the boundaries {∥− →W∥g= 1}. If\nthe Hamiltonian vector field is non-radial on Σ, this basis is the union of 2D-tori\n(because it supports a non-vanishing vector field: the unscaled Hamiltonian field\nof the symbol).\nIfkis allowed to smoothly vary as a function of a spectral parameter ω(e.g.\nkωin the previous sections), the projection of the classical (the group velocity)\ndynamics (the group velocity) onto Xis given by\n\u0012∂kω\n∂ω\u0013−1\u0010− →G+− →W\u0011\n,ON GRAVITO-INERTIAL SURFACE WAVES 9\n0 5 10 15 20\nDegreen010203040Number of eigenvalues2n+ 3\nN/f = 0.5\nN/f = 1.5\nN/f = 5\nFigure 2. Number of eigenvalues in the interval 0 <|ω|< ω−\nfor polynomial eigenvectors of degree less than n. Aligned rotation\nand gravity (as considered in §5.1) when ∂Dis a sphere. For every\ndegree n, the number of eigenvalues is bounded by the number\n2n+3 of spherical harmonics of degree n+1. Numerical calculations\nfollowing the method presented in [ VCdV24 ].\nwhere− →Gis the projection of the geodesic field for the corresponding values of ξ.\nThe main direction is then given by ±− →Wwith±=sign∂kω\n∂ω. This gives the motion\nto the east in our case, at least for small values of ω.\nWe can apply the study of attractors initiated in [ CdVSR20, CdV20 ]. We\nassume that the Hamiltonian flow of ksatisfies the Morse-Smale assumptions of\n[CdVSR20 ]. Then, the solutions of the Kelvin equation are Lagrangian distri-\nbutions located over the attractors and the repulsors. In the case of the Kelvin\nequation, we thus predict the existence of some attractors for the surface waves\n(which remain to be observed in physics).\n7. The case of ellipsoids\nNow, we assume that ∂Dis an ellipsoid. Ellipsoidal models have proven useful\nfor geophysical applications (e.g. to model some peculiar vortices in the Earth’s\noceans [ VCdV24 ]). In an ellipsoid, the spectrum of the Poincar´ e operator is pure-\npoint and dense in [ −ω+, ω+], with polynomial eigenvectors. In particular, gravito-\ninertial surface modes were found in [ VCdV24 ] when 0 <|ω|< ω−. Numerical\ncalculations (figure 2) show that the number of eigenvalues associated with each\npolynomial vector space of degree nis bounded by 2 n+ 3, which is the number of\nspherical harmonics of degree n+ 1. Indeed, if the velocity− →uis of degree n≥1,\nthen the pressure ϕis of degree n+ 1 according to equation (3.1). This strongly\nsuggests a connection, when ∂Dis an ellipsoid, between the gravito-inertial surface\nmodes and the theory of spherical harmonics. This is the subject of the subsection\n7.1. Moreover, the numerical results also confirm the microlocal predictions detailed\nin§5.1, which show that the surface waves do not span the full cotangent space\nwhen N≥f. Another natural question is thus to estimate, when the polynomial10 YVES COLIN DE VERDI `ERE AND J ´ER´EMIE VIDAL\ndegree ntends to infinity, the asymptotic number of eigenvalues of the Poincar´ e\noperator in the interval ]0 , ω−[. A conjecture for this question is given in Section 8.\n7.1. Spherical harmonics. We consider the case of an ellipsoid E:=L(B),\nwhere Bis the unit Euclidean ball and Lis linear. The pull-back of the Poincar´ e\noperator to Bis then of the form PE= Π ECEΠEwhere CEis a 4 ×4 matrix\nand Π Eis the Leray projector for the pull-back metric L⋆(g0). It turns out that\nPEcommutes with the Legendre operator as used in [ CdVV23 ]: recall that the\nLegendre operator defined on functions with values in Cdis defined by\nL⊕d:= (−∆−L⋆L)⊗IdCd\nwith L=P3\ni=1xi∂i, and that L⊕4commutes with the Poincar´ e operator.\nLemma 7.1.L⊕3and∇satisfy a commutator formula given by\nL⊕3∇ − ∇L⊕1=−2∇(1 +L).\nProof. We have [∆ ,∇] = 0, [ L,∇] =−∇and [ L⋆,∇] =∇. Then, the proof\neasily follows by using L⋆=−L−3. □\nLemma 7.2.For any smooth function ϕonB, we have\n\u0000\nL⊕3∇ϕ\u0001\n|S2=∇(∆S2−2)ϕ|S2.\nProof. From the previous lemma, we have\nL⊕3∇ϕ=∇(−∆−L⋆L−2(1 + L))ϕ.\nUsing polar coordinates ( r, θ) with θ∈S2, we get\n−∆−L⋆L−2(1 + L) =−d2\ndr2−2\nrd\ndr+1\nr2∆S2−L⋆L−2L−2.\nPutting r= 1 in the previous equation, and using L⋆L=−r2d2\ndr2−4L, we get\n((−∆−L⋆L−2(1 + L))ϕ)|S2= (∆ S2−2)\u0000\nϕ|S2\u0001\n. □\nHence, the Laplace-Beltrami operator on S2will play the role of L⊕3for the\nwaves in the bulk. As a consequence, the gravito-inertial surface waves satisfy:\nTheorem 7.3.The restriction to ∂Eof the pressure associated with every\neigenvector of the Poincar´ e operator is a spherical harmonic. More precisely, if the\nellipsoid E=L(B)where Bis the unit Euclidean ball and Lis linear, the pull-back\nof the pressures on the boundary of EareS2-spherical harmonics.\nProof. It follows then from the equation ∇ϕ= (ω−iA)(− →u , ρ 1)⊤that, if− →u\nandρ1are of degree nand have all components in ker ( L −n(n+ 3)), L⊕3(∇ϕ) =\nn(n+ 3)∇ϕ. Then, we use the commutation relation given in Lemma 7.2 and get\n∇\u0000\n∆S2ϕ|S2−(n+ 1)( n+ 2)ϕ|S2\u0001\n= 0.\nThe function ϕ|S2is a spherical harmonic of degree n+ 1 up to a constant (which\ncan be assumed to vanish). Note that this holds for any eigenvector, without any\nrestriction on the value of ω. □ON GRAVITO-INERTIAL SURFACE WAVES 11\n7.2. Liouville integrability. Let us fix ωwith 0 < ω < ω −and assume that\nthe characteristic cone Cω:=k−1\nω(0) is a smooth 3D sub-cone of T⋆∂D\\0 (i.e. dkω\ndoes not vanish on Cω). Then, we consider Tω=Cω∩U⋆S2, where U⋆S2is the\nunit conormal bundle of S2. We have the following propositions.\nTheorem 7.4.The set Tωis a smooth Lagrangian torus invariant by the\ngeodesics flow of S2and the Hamiltonian flow of kω.\nProof. We consider a sequence of eigenpairs ( ϕj,n, ωj,n) with\n∆ϕj,n= (n+ 1)( n+ 2)ϕj,n, K ωj,nϕj,n= 0,\n∥ϕj,n∥= 1, and lim n→∞ωj,n=ω. Such a sequence exists because the eigenvalues\nωj,nare dense in ]0 , ω−[. We consider h= 1/nas a semi-classical parameter. We\nlook then at the h-wavefront set of this sequence. We already know that it is\ncontained in Tω, and is invariant by the geodesic flow. Moreover, it follows from\nLemma 7.5 that it is invariant by the Hamiltonian dynamics of kω. Hence, the\nwavefront set is Tω(which is Lagrangian). □\nLemma 7.5.Assume that Kωj,nϕj,n= 0, then the wavefront set of the sequence\nϕj,nis invariant by the Hamiltonian dynamics of kω.\nProof. This is a local property. Hence, we can build a smooth family of\nFourier integral operators Uω′forω′close to ωsuch that\nU⋆\nω′KωUω′=Kω′,\nUω= Id, and the associated canonical transform χω′converges to the Identity.\nHence, we have KωUωj,nϕj,n= 0 and the wavefront set Zj,nofUωj,nϕj,nis invariant\nby the flow of kω. Moreover, we have WF(Uωj,nϕj,n) =χωj,nWF(ϕj,n), which\nconverges to the wavefront set of the sequence ϕj,n. □\nIt follows that, in the case of ellipsoids, the dynamics of waves is quasi-periodic.\nUnfortunately, we were not able to find a direct proof of this fact, that is to prove\nthat the Poisson brackets {kω, g⋆\ncan}vanish. This would directly imply that the\nsurface dynamics is Liouville integrable.\n8. Perspectives for future work\nSeveral questions have remained unanswered in the present study, and will be\nconsidered in future work. When ∂Dis an ellipsoid, we are interested in a more\nprecise description of the low-frequency spectrum in the interval ]0 , ω−[. It would\nbe worth explicitly obtaining the number of eigenvalues with eigenvectors of degree\nn. Then, the asymptotic distribution of the eigenvalues in ]0 , ω−[ could be obtained\nwhen n→ ∞ . This would be complementary to the Weyl asymptotic formula in the\ninterval [ ω−, ω+], which has been obtained in [ VCdV24 ]. For ( x,ξ)∈T⋆(∂D)\\0,\nlet us denote by n(x,ξ)∈Nthe number of values of ωsuch that kω(x,ξ) = 0 Then,\nwe conjecture the following asymptotics:\nConjecture 8.1.Letµj,nbe the eigenvalues of the Poincar´ e operator with\npolynomial eigenvectors of degree nand, for an ellipsoid E,HEbe the Hamiltonian\nof the Kelvin equation. Then, for 0< a, b < ω −, we have\n#{µj,n∈[a, b]} ∼2n+ 3\n8π2Z\nU⋆S2∩{(x,ξ)|kω(x,ξ)=0, ω∈[a,b]}ndL\nwhen n→+∞, where Lis the Liouville measure on U⋆S2.12 YVES COLIN DE VERDI `ERE AND J ´ER´EMIE VIDAL\n  \nFigure 3. Discrete and essential spectra defined in Theorem C.1.\nProof. This two-dimensional Weyl formula could be proved using Bohr - Som-\nmerfeld rules (following [ CdV80 ]). □\nAnother interesting extension would be to consider an unstable stratification\n(i.e. when N2<0). In this case, the Poincar´ e operator is no longer self-adjoint.\nThen, it would be worth describing the essential spectrum (i.e. to determine the\nset of values of ωfor which the boundary conditions are elliptic). This is a rather\nclassical problem, which is for instance described in Chapter XX of [ H¨ o85 ]. Finally,\nanother interesting application would be the case where− →gis not a constant vector.\nFor instance, this situation occurs inside planets or stars (where gravity is central).\nThen, for a given ω, the Poincar´ e operator can be elliptic in some regions of Dand\nhyperbolic in others [ FS82b ]. The calculus of the symbol of the Kelvin operator at\nx∈∂Ddepends only on the value of− →gat these points, such that the microlocal\nanalysis presented in this work could be re-employed to investigate this problem.\nAppendix A. The Leray projector\nRecall the Helmoltz-Leray decomposition. The space of L2vector fields in Dis\nan orthogonal sum of the closures of ∇C∞(D,C3) and the space of divergenceless\nvector fields tangent to ∂D. The Leray projector Π is the orthogonal projection on\nthe second space. It turns out that Π is a pseudo-differential operator of degree 0,\nwhose symbol π(x,ξ) is the orthogonal projector on ker ξ.\nAppendix B. The Dirichlet-to-Neumann map\nLet us consider a Laplace-Beltrami operator ∆ ginD. The Dirichlet-to-Neumann\noperator DtN gacts on functions on ∂Das follows. If f:∂D→C, we extend fas\nF:D→Csuch that ∆ gF= 0 and F∂D=f. Then, we have DtN gf=∂F/∂n g,\nwhere ngis the outgoing unit normal along the boundary. The operator DtN g\nis a self-ajoint pseudo-differential operator of degree 1 on L2(∂D, v g), where vgis\nthe Riemannian volume on ∂Dwith respect to the restriction g∂ofgto∂D. The\nprincipal symbol of DtN gispg⋆\n∂, where g⋆\n∂is the dual metric of g∂. Further details\nare given in Chapter 7 of [ Tay13 ], in Part IV of [ Gru08 ], or in [ GKLP22 ].\nAppendix C. Ellipticity and spectra\nWe denote by Pthe analytic family of pseudo-differential equations Pω(with\nω∈C) on a closed manifold Xwith degree( Pω)≡0 (we can always assume the lastON GRAVITO-INERTIAL SURFACE WAVES 13\ncondition by using a left product with an invertible elliptic operator). The operators\nPωare continuous and linear from L2(X) into itself. The spectrum of this family\nis defined as the set of ω∈Cso that Pωis not invertible. If Pωis invertible, the\ninverse is also continuous. We also remind the reader that a pseudo-differential\noperator Qis elliptic at ( x,ξ)∈T⋆X\\0 if and only if the principal symbol p(x,ξ)\nis invertible. Qis elliptic if this holds for all ( x,ξ)∈T⋆X\\0. We will need the\nfollowing theorem defining the different spectra (as illustrated in figure 3).\nTheorem C.1.LetPbe as defined above, and U⊂Cbe the open set of the\nvalues of ωwhere Pωis elliptic. Then, there exists a discrete subset σd(P)ofU\nsuch that Pωis not invertible. If ω∈σd(P), then dim(ker( Pω))is finite. More\nprecisely, for all ω∈U,Pωis Fredholm. If ω∈σe(P) =C\\U, then Pωis not\ninvertible.\nIfPωis elliptic, we can define a right parametrix as\nPωRω= Id + Aω\nwith Aωsmoothing and, similarly, a left parametrix Lω. Hence, Pωis Fred-\nholm and we can apply the Fredholm analytic theorem. Conversely, if Pωis non-\nelliptic, it is non-invertible. We can use test functions of the form a(x)eiλxξ0with\npω(a(x0),ξ0) = 0, where pωis the principal symbol of Pω, in order to show that\nthe operator does not admit a continuous inverse (see [ CdV20 ], section 2, for more\ndetails). This result also applies to the boundary calculus of pseudo-differential\noperator’s [ BdM71, Gru08 ].\nIt follows from this that the essential spectrum of the Poincar´ e operator is the\nsame, in the interval 0 < ω < ω −, as the set of values of ωfor which the Kelvin equa-\ntion is invertible. Both properties are equivalent to say that the boundary symbol is\nnon-elliptic. This depends on the symbolic calculus of boundary pseudo-differential\noperator’s, which is a rather technical issue. The identification of discrete spectra is\nmuch simpler. Indeed, we can transfer any eigenmodes of the Poincar´ e operator to\naH1solution of the Kelvin equation (and conversely). More details will be given in\nfuture work, where we consider also the case of unstable stratification (i.e. N2<0)\nfor which a greater part of that calculation is needed.\nReferences\n[BdM71] Louis Boutet de Monvel. Boundary problems for pseudo-differential operators. Acta\nMath. , 126:11–51, 1971.\n[Car22] ´Elie Cartan. Sur les petites oscillations d’une masse fluide. Bull. Sci. Math. , 46:317–\n369, 1922.\n[CdV80] Yves Colin de Verdi` ere. Spectre conjoint d’op´ erateurs pseudo-diff´ erentiels qui com-\nmutent. II. Le cas int´ egrable. Math. Z. , 171:51–73, 1980.\n[CdV20] Yves Colin de Verdi` ere. Spectral theory of pseudodifferential operators of degree 0\nand an application to forced linear waves. Anal. PDE , 13(5):1521–1537, 2020.\n[CdVSR20] Yves Colin de Verdi` ere and Laure Saint-Raymond. Attractors for two-dimensional\nwaves with homogeneous Hamiltonians of degree 0. Commun. Pure Appl. Math. ,\n73(2):421–462, 2020.\n[CdVV23] Yves Colin de Verdi` ere and J´ er´ emie Vidal. The spectrum of the poincar´ e operator in\nan ellipsoid. arxiv:2305.01369, 2023.\n[FS82a] Susan Friedlander and William L. Siegmann. Internal waves in a contained rotating\nstratified fluid. J. Fluid Mech. , 114:123–156, 1982.\n[FS82b] Susan Friedlander and William L. Siegmann. Internal waves in a rotating stratified\nfluid in an arbitrary gravitational field. Geophys. Astrophys. Fluid Dyn. , 19(3-4):267–\n291, 1982.14 YVES COLIN DE VERDI `ERE AND J ´ER´EMIE VIDAL\n[GKLP22] Alexandre Girouard, Mikhail Karpukhin, Michael Levitin, and Iosif Polterovich. The\nDirichlet-to-Neumann map, the boundary Laplacian, and H¨ ormander’s rediscovered\nmanuscript. J. Spectr. Theory , 12(1):195–225, 2022.\n[Gru08] Gerd Grubb. Distributions and operators . Graduate Texts in Mathematics. Springer,\n2008.\n[H¨ o85] Lars H¨ ormander. The Analysis of Linear Partial Differential Operators III: Pseudo-\ndifferential operators . Springer-Verlag, 1985.\n[Kel80] Lord Kelvin. On gravitational oscillations of rotating water. Proc. R. Soc. Edinburgh ,\n10:92–100, 1880.\n[Poi85] Henri Poincar´ e. Sur l’´ equilibre d’une masse fluide anim´ ee d’un mouvement de rotation.\nActa Math. , 7:259–380, 1885.\n[Tay13] Michael Taylor. Partial differential equations II: Qualitative studies of linear equa-\ntions . Springer, 2013.\n[VCdV24] J´ er´ emie Vidal and Yves Colin de Verdi` ere. Inertia-gravity waves in geophysical vor-\ntices. P. R. Soc. A , In press, 2024. https://doi.org/10.48550/arXiv.2402.10749.\nUniversit ´e Grenoble-Alpes, CNRS, Institut Fourier, 38000 Grenoble (France)\nEmail address :yves.colin-de-verdiere@univ-grenoble-alpes.fr\nUniversit ´e Grenoble-Alpes, CNRS, ISTerre, 38000 Grenoble (France)\nEmail address :jeremie.vidal@univ-grenoble-alpes.fr"    },    {        "title": "2402.13003v1.The_influence_of_silicon_on_the_formation_and_transformation_of_corrosion_products.pdf",        "content": "The influence of silicon on the formation and\ntransformation of corrosion products\nFabio E. Furcas†, Shishir Mundra†, Barbara Lothenbach§, Camelia N. Borca‡, Thomas\nHuthwelker‡, Ueli M. Angst†∗\n†Institute for Building Materials, ETH Zürich, 8093, Zürich, Switzerland\n§Empa Concrete & Asphalt Labortory, 8600 Dübendorf, Switzerland\n‡Swiss Light Source, Paul Scherrer Institut, 5232 Villigen, Switzerland\n∗Corresponding author: uangst@ethz.ch\nAbstract\nAccurate model predictions of corrosion-driven damage in reinforced concrete structures necessitate\na comprehensive understanding of the rate of corrosion product formation. Here, we investigate the\ninfluence of dissolved Si characteristic of cementitious systems on the rate of corrosion product transfor-\nmation at alkaline pH. Compared to systems aged in the absence of Si, small amounts of Si retard the\nformation rate of the thermodynamically stable corrosion product goethite by a factor of 10. The esti-\nmated first order rate constant of transformation kdecreases exponentially as a function of the dissolved\nSi concentration and follows the progression log10k=log10k0−14.65×[Si]0.28. Findings further suggest\nthat the observed retardation is primarily due to the formation of a mobile aqueous Fe-Si complex. The\nconcentration of Si in cementitious systems has a crucial influence, and additional research is required\nto fully incorporate this factor into reactive transport models, ultimately essential for accurate service\nlife predictions.arXiv:2402.13003v1  [cond-mat.mtrl-sci]  20 Feb 20241 Introduction\nCorrosion of steel in concrete is widely recognised as one of the main causes for the structural degrada-\ntion of reinforced concrete structures(1). Apart from the corrosion process itself and the reduction in\nthe reinforcement cross-sectional area, the dissolution of iron at the steel-concrete interface may lead to\nthe precipitation of corrosion products in the concrete pore network. Over time, the formation of these\niron (hydr)oxide phases can generate internal stresses leading to cracks, spalling of the concrete cover,\ningress of chlorides, carbonates and moisture(2), thus rendering the degradation process self-sustaining.\nWithin this vicious cycle, both the concrete pore solution chemistry as well as the type of corrosion\nproducts formed play a crucial role.\nOver the course of the last decades, the formation of iron (hydr)oxide phases in aqueous electrolytes\nhas been researched extensively in the fields of soil(3,4)and environmental science(5,6). Whilst the\nmajority of studies published focused on their stability at mildly acidic to mildly alkaline pH, selected\npublications investigated the transformation of corrosion products in mill raffinates(7), air pollution\ncontrol residues(8)and other highly alkaline systems(9). In comparison, comparatively little research\nhas been devoted to the formation of corrosion products in highly alkaline environments and in the\npresence of other solutes commonly encountered in cementitious systems. In an attempt to forecast the\ncorrosionperformanceoftraditional, highlyalkalinebindersincludingPortlandcementandfacilitatethe\ndevelopment and usage of alternative, environmentally benign cementitious materials, a more thorough\nscientific understanding about the formation of corrosion products in aqueous electrolytes characteristic\nto these systems must be established(10).\n1.1 Some corrosion products are more expansive than others\nDepending on the crystal structure and the bulk chemical composition, some iron (hydr)oxides have\nsignificantly larger specific molar volumes than others. Polymorphs of the FeOOH type generally feature\na small specific molar volume, ranging from 20.9 cm3mol−1for goethite ( α−FeOOH(s)) to approx-\nimately 25.5 cm3mol−1for akaganéite ( β−FeOOH(s))(11,12). Amorphous iron hydroxides including\n2- and 6-line ferrihydrite ( 2,6l−Fe(OH)3(s)) feature a V0of up to 34.0 cm3mol−1(13)and are thus\nalmost 5 times as voluminous as crystalline Fe0(V0= 7.10 cm3mol−1)(14). From a concrete durability\nviewpoint, the formation of either of these comparatively small FeOOH polymorphs is favourable, as\nthe magnitude of internal stresses generated is expected to be lower.\n11.2 The influence of the pore solution chemistry\nThe rate and mechanism of corrosion product formation and transformation is heavily dependent on\nboth the pH and the presence of pore solution constituents other than aqueous Fe. 2-line ferrihydrite,\nthe most soluble iron hydroxide often preceding the formation of more crystalline iron phases(15), has\nbeen observed to exclusively transform to goethite ( α−FeOOH(s)) at pH > 13(16), i.e. characteris-\ntic to the pore solution of the majority of uncarbonated Portland cement based binders(17,18,19). At\nlower pH values (pH = 9 to 11), as typical for the pore solution of carbonated concrete(20), MgO-based\ncements(21)and MSH(22), 2-line ferrihydrite transforms into mixtures of hematite ( α−Fe2O3(s)) and\ngoethite(5). As the overall process is rate-limited by the dissolution of 2-line ferrihydrite, transformation\nrates are greatly accelerated the higher the degree of alkalinity(23).\nOther ionic species including sulphates, carbonates and silicon have been reported to retard phase\ntransformation or completely alter the type(s) of iron (hydr)oxide(s) stabilised. Amongst all ionic con-\nstituents commonly encountered in the pore solution of cementitious systems, silicon arguably plays the\nlargest role. Small amounts of Si in the orders of 1×10−4M are characteristic for the pore solution\nof a diverse landscape of cementitious systems including CEM I(17,18), CEM I with additions of cal-\ncium sulfoaluminates(19), fly-ash(24,25,26), blast-furnace slag(26), silica fume(26,27,28), metakaolin(28)or\nCEMIII/B with nanosilica(29). At pH ∼13, Si retards the dissolution of 2-line ferrihydrite and readily\nadsorbs onto goethite(15,30). Schulz et al.(31)show that the Fe(II)-catalysed transformation of 2-line\nferrihydrite and lepidocrocite ( γ−FeOOH(s)) to goethite and magnetite ( α−Fe3O4(s)) is inhibited\nin the presence of silicate at neutral pH. Other studies evidence the formation of aqueous Fe-Si com-\nplexes(32), that could influence total aqueous iron concentration.\nThis paper investigates the effect of dissolved silicon on the formation of corrosion products at\nalkaline pH ≥13. A combination of X-ray diffraction (XRD) and X-ray absorption spectroscopy (XAS)\nmeasurements is used to determine the type and formation rate of iron (hydr)oxide phases aged in\nthe presence of 0.1 and 0.5 mM Na2SiO3. Aqueous Fe and Si concentration measurements determined\nvia inductively coupled plasma optical emission spectroscopy (ICP-OES) provide further insights into\nthe likely mechanism of corrosion product transformation, the effect of silicon on the dissolution and\nre-precipitation of solid phases involved and the formation of aqueous Fe-Si complexes. In combination\nwith the results of a previous study investigating the rate of corrosion product transformation in the\nabsence of silicon, we derive a semi-empirical relationship that quantifies the estimated first-order rate\n2constant of transformation kas a function of the aqueous Si concentration. Findings can be used to\ncalculatetherateofiron(hydr)oxideformationacrossabroadrangeofcementitioussystems. Thisstudy\nis intended to serve as a base-case for future investigations into the formation of corrosion products in\nthe presence of sulfates, carbonates and other prominent cementitious pore solution constituents.\n2 Methods\n2.1 Preparation of supersaturated stock solutions\nThe sample preparation has been described in detail in Furcas et al.(16). Briefly, 1 M of FeCl3·6H2O(cr)\nhas been dissolved in 2 wt. % HNO3. 5mLof the resultant acidic stock solution was mixed with 245\nmLof 4 basic stock solutions containing 0.104 M (pH = 13) and 1.024 M (pH = 14) NaOH as well\nas 0.100 and 0.500 mM of Na2SiO3, respectively. Stock solutions were stirred rigorously and aged in\nfresh polyethylene containers at ambient temperature for up to 30 days. Solid phases were extracted by\ncentrifugation at 10000 rpm for 15 min and subsequent freeze drying and stored as dry powders. Liquid\nphasealiquotsweretakenbyextractingapprox. 1 gfromtheagedstocksolutionwithasyringe. Aliquots\nwere filtered using 0.20 µmnylon filters (Semadeni AG, Ostermundigen, Switzerland) and mixed with\n2 wt. % HNO3(EMSURE) purchased from Merck Group (Merck KGaA, Darmstadt, Germany), at a\nratio of 1:10 for early and 1:8 for late equilibration times to prevent further precipitation.\n2.2 Iron hydroxide reference standards\nA total of 3 reference standards are considered in this study, 2-line ferrihydrite ( 2l−Fe(OH)3(s)),\ngoethite ( α−FeOOH(s)) and lepidocrocite ( γ−FeOOH(s)). They were selected based on a previous\ninvestigation into the formation of corrosion products at alkaline pH(16). Both 2-line ferrihydrite and\nlepidocrocite were synthesised following the experimental procedure of Schwertmann and Cornell(15),\ngoethitewaspurchasedfromThermoFisherScientific, Waltham, MA,USA.Theidentityofallstandards\nwas confirmed by XRD(16). Their extended X-ray absorption fine structure (EXAFS) spectra were\nrecorded in a previous study(16).\n2.3 Inductively coupled plasma optical emission spectroscopy\nThe composition of the solution was measured using the Agilent 5110 ICP-OES instrument (Agilent\nTechnologies Inc., Santa Clara, CA, USA). Iron and silica concentrations were determined via linear\ninterpolation on a set of 8 discrete reference concentrations, ranging from 0.01 to 50 ppm for Fe and\n30.01 to 50 ppm for Si. The limit of quantitation (LOQ) was determined for each analyte emission\nline, Fe 234.350, 238.204, 239.563, 259.940 nmand Si 250.690, 251.611, 288.158 nm, according to the\nreccomendations of Caruso et al.(33)(Supporting information, Tab. S1). At 0.163 and 1.105 µmol L−1,\nFe 259.940 nmand Si 250.690 nmfeature the highest LOQ of all analyte spectral lines recorded and\nwill thus be considered the global limits of quantitation. The detailed account of all reference standard\ncompositions and calibration lines is presented in Supporting Information, Tab. S2, S3. and Fig. S4,\nS5 and S6. For more details, the reader is referred to Furcas et al.(16).\n2.4 X-ray diffraction\nPowder X-ray diffraction patterns were recorded from 4◦to80◦in steps of 0.02 2θ, using the Bruker\nD8 Advance. The diffractometer uses a coupled θ−2θconfiguration, Co K αratiation ( λ= 1.7902Å)\nand a LynxEye XE-T detector. Diffraction patterns of reference components and solids extracted from\nsupersaturated stock solution were analysed with the open source X-ray diffraction and Rietveld and\nrefinement programme Profex(34). The identity of the reference standards was confirmed by comparison\nto the reference powder diffraction spectra of 2-line ferrihydrite(35), goethite(36,37)and lepidocrocite\n(PDF entry 00-044-1415), as detailed in(16).\n2.5 X-ray absorption spectroscopy\nAll synchrotron-based investigations were carried out at the PHOENIX (Photons for the Exploration of\nNature by Imaging and XAFS) beamline at the Swiss Light Source (SLS), Paul Scherrer Institute (PSI),\nVilligen, Switzerland. The beamline specifications have been described elsewhere(38). Iron hydroxide\npowders were uniformly applied to conductive carbon tape on a roughened-up copper sample holder\nand measured at ambient temperature under vacuum at approx. 10−6bar. The Total Fluorescent Yield\n(TFY) of the sample was measured using a four-element vortex detector. The Total Electron Yield\n(TEY) of the sample was measured as the incident photon flux on a Ni-coated polyester foil, located\nin a different vacuum chamber at approx. 5·10−8bar. Iron hydroxide powder samples were measured\n3−5times, the reference standards 8 times.\nFe K-edge spectra including the X-ray absorption near edge structure (XANES) and the extended X-\nray absorption fine structure (EXAFS) were recorded from 7041 to 7858 eV, i.e. from 70 eV below\nto approximately 750 eV above the absorption edge. The recorded data was analysed and normalised\nusing the IFFEFIT (ATHENA) software package(39,40). The attenuated TFY signal was corrected for\nself-absorption by adjusting it to the TEY using the software in-built Fluo- µ(E)algorithm, providing\n4the molecular formula of various samples analysed. Radial distribution functions were generated from\nk3-weighted EXAFS spectra by performing a Fourier transform over 1.5 to 10.0 Å−1, using the Kaiser-\nBessel window function. The relative amount of different iron hydroxides in the sample was determined\nby linear combination fitting (LCF) in the XANES region extending from 20 eV below to 40 eV above\nthe Fe K-edge. The weights of all reference standards, 2-line ferrihydrite, lepidocrocite and goethite,\nwere constrained to lie between the values of 0 and 1 and forced to add up to 1.\n3 Results and discussion\n3.1 Aqueous concentration measurements\nThe aqueous iron and silicon concentrations measurements in equilibrium with various solid iron hy-\ndroxide phases suggest the transformation of highly soluble 2-line ferrihydrite towards a more crystalline\nFe-bearing phase. Fig. 1 show the decrease of [Fe]and[Si]as a function equilibration time. At pH =\n14 (Fig. 1a and Fig. 1c), the initial reduction in the aqueous iron concentration observed in silicon-free\nsolutions prevails across all silicon concentrations tested. Levelling off at approximately 5×10−5M\nafter 1 hour, the aqueous iron concentration in the presence of 0.5 mM Na2SiO3is 5 times higher than\nthat measured in the absence of silicon. The iron concentration in solutions containing 0.1 mM Na2SiO3\ndecreases faster than that measured in the presence of 0.5 mM Na2SiO3and slower compared to the\nsilicon-free samples. After 3 hours, the aqueous iron concentration in the presence of both 0 mM and\n0.1 mM Na2SiO3plateaus at ∼1×10−5M, i.e. approximately 100 times higher than the calculated\nsolubility of goethite at pH 14. All measured [Fe], within 1 minute of equilibration time and there-\nafter, were found to be below the solubility limit of 2-line ferrihydrite and approach the aqueous iron\nconcentration in equilibrium with goethite at rates inversely correlated to the initial concentration of\nNa2SiO3. The aqueous silicon concentration on the other hand rapidly drops to 40 % of its initial value,\nirrespective of the dissolved amounts of Na2SiO3. At pH = 13 (Fig. 1b and Fig. 1d), the progression\nof[Si]rapidly reduces to ∼6×10−5M, i.e. approximately 10 % of the initial concentration. At the\nsame time, aqueous iron concentrations are significantly lower than those measured at pH = 14. Even\nthough the solubility limit of 2-line ferrihydrite decreases by one order of magnitude from pH 14 to\n13 (compare Figure 1a and 1b), the same initial reduction in the aqueous iron concentration within 1\nminute of equilibration time is observed. No clear correlation can be established between the decrease\nof[Fe]and the dissolved aqueous silicon concentration. Within the standard deviation of various [Fe]in\nthe presence of 0.1 mM and 0.5 mM Na2SiO3, it appears that the aqueous iron concentration initially\n50 60 120 180 240 300 36010−710−610−510−410−310−210−1\nSolubility limit of 2-line ferrihydrite\nLimit of quantitation\nSolubility limit of goethite[Fe],mol L−1[Si] = 0 .5mM\n[Si] = 0 .1mM\n[Si] = 0 mMa pH = 14\n1\n0 60 120 180 240 300 36010−910−810−710−610−510−410−310−210−1\nSolubility limit of 2-line ferrihydrite\nLimit of quantitation\nSolubility limit of goethite[Si] = 0 .5mM\n[Si] = 0 .1mM\n[Si] = 0 mMb pH = 13\n1\n0 60 120 180 240 300 36010−510−410−3\nLimit of quantitation\nTime t, min[Si],mol L−1[Si] = 0 .5mM\n[Si] = 0 .1mMc pH = 14\n1\n0 60 120 180 240 300 36010−510−410−3\nLimit of quantitation\nTime t, min[Si] = 0 .5mM\n[Si] = 0 .1mMd pH = 13\n1Figure 1: Total aqueous concentration of iron (Fig. 1a and Fig. 1b) and silicon (Fig. 1c and Fig. 1d)\nat pH = 14 (Fig. 1a and Fig. 1c) and pH = 13 (Fig. 1b and Fig. 1d) over time. Concentrations are\nmeasured by ICP-OES at the emission lines Fe 259.940 nmand Si 250.690 nm. The error bars displayed\nrepresent the standard deviation of the determined elemental concentrations in three independent ex-\nperimental runs. Note that for the measurements of Si-free solutions, the level of quantitation amounts\nto1×10−6mol L−1due to the increased dilution with 2 wt. % nitric acid(16).\ndecreases below the measured concentrations in the absence of silicon, but then increases above it at\nlonger equilibration times, striving towards a consistent solubility of 2×10−6after 2 hours. The [Fe]\nin the presence of Na2SiO3is in the order of 3×10−7M within the first 2 hours of equilibration, just\nabove the LOQ of the ICP device, and thus features significantly higher standard errors than the mea-\nsurements at pH = 14. Note that the aqueous silicon concentration measured from solutions containing\n0.1 mM Na2SiO3at pH = 13 are below the respective LOQ across all analyte emission lines.\n63.2 Mineral phase identification\nTheobtainedX-raydiffractionpatternsandabsorptionspectraindependentlycorroboratetheformation\nof goethite from 2-line ferrihydrite at the investigated physiochemical conditions. As illustrated in Fig.\n2, dried solids extracted from supersaturated iron stock solution at pH = 14 are completely amorphous\nto X-ray diffraction at early equilibration times. Both diffraction patterns at 20 minutes feature two\nbroad peaks at ∼40◦and∼74◦2θ, characteristic to amorphous 2-line ferrihydrite as well as negligible\namounts of halite (NaCl(s)). In the presence of 0.1 mM Na2SiO3(Fig. 2a), all major peaks of goethite\nα−FeOOH(s) emerge from the initial diffraction pattern within one day, similar to the observations in\nthe absence of Si(16). At elevated concentrations of 0.5 mM Na2SiO3(Fig. 2b), only the most prominent\nlattice plane reflections are discernible from the amorphous diffraction pattern of 2-line ferrihydrite after\n1 day. Peaks after 30 days are generally sharper than those measured in the 0.1 mM Na2SiO3sample\nor in the absence of silicon (Supporting information, Fig. S1) at the same point in time. The observed\nincrease in peak sharpness can be attributed to the adsorption of Si onto the terminal goethite crystal\nfaces, leading to preferential growth on the other crystal faces(15).\n75 10 15 20 25 30 35 40 45 50 55 60 65 70 75 8020 min6 h18 h24 h3 d5 d30 dα\nh\n2l2l h(020)\n(110)\n(120)\n(130)\n(021)\n(111)\n(121)\n(140)\n(041)\n(211)\n(221)\n(151)\n(002)\n(061)\n(112)\n2θCo kαIntensity, a.u.a\n1\n5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 8020 min6 h18 h24 h3 d5 d30 dα\nh\n2l2l h(020)\n(110)\n(120)\n(130)\n(021)\n(111)\n(121)\n(140)\n(041)\n(211)\n(221)\n(151)\n(002)\n(061)\n(112)\n2��Co kαIntensity, a.u.b\n1Figure 2: XRD patterns of iron hydroxide phases extracted from stock solutions containing 20.0 mM\nFeCl3·6H2O(cr) and 0.1 mM Na2SiO3(Fig. 2a) and 0.5 mM Na2SiO3(Fig. 2b) at pH = 14. Main\npeak positions of goethite ( α−FeOOH(s)), 2-line ferrihydrite ( 2l−Fe(OH)3(s)) and halite (NaCl(s))\nare indicated with a red vertical bar ( |), 2l and an h.\n8Experimentally determined Fe K-edge XANES spectra (Fig. 3) exhibit a pre-edge feature at ap-\nproximately 7113.5±0.5eV, resulting from quadrupolar 1s →3d electronic transitions in the absorber\niron ion, characteristic to Fe(III) compounds(41,42). Over the course of the experiment, the integrated\npeak intensity of the pre-edge feature reduces, whilst the white line intensity as well as the magnitude\nof the first destructive (A) and constructive (B) interference after the rising edge increases, indicating a\ntransition towards a more centrosymetric, octrahedral coordination environment(43,44). Here, the white\n7,100 7,120 7,140 7,160 7,180 7,2002l−Fe(OH) 3(s)20 min6 h18 h24 h3 d5 d30 dα−FeOOH(s) AB\nEnergy E, eVNormalised absorbance, a.u.20 mM Fe + 0.1 mM Si\n20 mM Fe + 0.5 mM Sia\n1\n7,110 7,112 7,114 7,116 7,118 7,1202l−Fe(OH) 3(s)20 min6 h18 h24 h3 d5 d30 dα−FeOOH(s)\nEnergy E, eVb\n1\nFigure 3: Normalised Fe K-edge XANES spectra of iron hydroxide powders extracted from supersat-\nurated stock solutions at pH = 14 containing 20.0 mM FeCl3·6H2O(cr) and 0.1 mM Na2SiO3(black\ncontinuous lines) and 0.5 mM Na2SiO3(red dashed lines), together with the reference standards spectra\nof 2-line ferrihydrite ( 2l−Fe(OH)3(s)) and goethite ( α−FeOOH(s)). Fig. 3a highlights the white line\nintensity as well as the first destructive (A) and constructive (B) interference after the absorption edge,\nFig. 3b shows a zoomed-in section of the pre-edge feature.\nline and pre-edge feature intensities of spectra corresponding to an initial silicon concentration of 0.1\nmM are in between those aged in the presence of 0.5 mM Na2SiO3and in the absence of silicon (Fig. 4).\nXANES spectra of iron hydroxide powders extracted from supersaturated stock solution after 30 days\nclosely resemble the spectrum of the goethite reference standard. The observed transformation is also\nevidenced from the radial distribution functions (RDF) of iron hydroxide powders extracted at different\nequilibration times. As illustrated in Fig. 5, the peaks in the Fourier transformed k3-weighted EXAFS\n97,100 7,120 7,140 7,160 7,180 7,20020 min1 d30 d\nEnergy E, eVNormalised absorbance, a.u.20 mM Fe, Furcas et al.(16)\n20 mM Fe + 0.1 mM Si\n20 mM Fe + 0.5 mM Sia\n1\n7,110 7,112 7,114 7,116 7,118 7,12020 min1 d30 d\nEnergy E, eVb\n1Figure 4: Normalised Fe K-edge XANES spectra of iron hydroxide powders extracted from solutions\ncontaining 20.0 mM FeCl3·6H2O(cr) and no additional silicon(16), 0.1 mM Na2SiO3(black continuous\nlines) and 0.5 mM Na2SiO3(red dashed lines).\nspectra constitute 3 distinct contributions due to the presence of Fe −O bonds at distances of ∼2.0\nÅ(45), Fe−Fe edges at ∼2.4Å to∼2.9Å and Fe −Fe double corners at ∼3.5Å, respectively(32,46).\nAt low equilibration times, the coordination environment of Fe resembles that of 2-line ferrihydrite,\nfeaturing little to no contribution of the octahedra-linking Fe −Fe corners. At successively higher equi-\nlibration times, both Fe −Fe pairs in the two first iron coordination shells become significantly more\npronounced. The evolution clearly indicates a change in the local coordination environment of iron and\nthe formation of a corner-sharing Fe −(O,OH,H2O)6octahedral structure. Moreover, the ratio of the\nsecond (Fe −Fe edge) to the third (Fe −Fe double corner) peak intensity decreases as a function of\nequilibration time. At 30 days, the relative peak intensities of both contributions fall short of those\nmeasured in the synthetic goethite reference standard. This is presumably due to a lower degree of crys-\ntallinity of the goethite phase stabilised at alkaline pH and in the presence of silicon when compared\nto its synthetic counterpart. The RDF of synthetic lepidocrocite only exhibits an intense Fe −Fe edge\npeak and a comparatively weak corner contribution. This feature is unique amongst all common iron\n(hydr)oxides and arises from the topology of lepidocrocite chains.\n100 1 2 3 4 5 62l−Fe(OH) 3(s)0 h6 h18 h24 h3 d5 d30 dα−FeOOH(s)Fe−(O,OH,H2O)6\nFe−Fe\nedgesFe−Fe\ndouble\ncorners\nRadial distance R, Å|χ(R)|, Å−420 mM Fe + 0.1 mM Si\n20 mM Fe + 0.5 mM Sia\n1\n0 1 2 3 4 5 6 7 8 9 10 11 122l−Fe(OH) 3(s)0 h6 h18 h24 h3 d5 d30 dα−FeOOH(s)\nWavenumber k, Å−1k3×χ(k), Å−320 mM Fe + 0.1 mM Si\n20 mM Fe + 0.5 mM Sib\n1Figure 5: Fourier transforms of k3-weighted Fe K-edge EXAFS spectra of iron hydroxide powders\nextracted from supersaturated stock solutions at pH = 14 (Fig. 5a), together with the k3-weighted\nEXAFS spectra (Fig. 5b). Solid black and dashed red lines correspond to solutions containing 20.0 mM\nFeCl3·6H2O(cr) and 0.1 mM Na2SiO3and 0.5 mM Na2SiO3, respectively. The Fourier transform was\nperformed using the Kaiser-Bessel window function over a k-space ranging from 1.5 to 10.0 Å−1. 11As schematically illustrated in Fig. 6, one octahedron of lepidocrocite ( γ−FeOOH(s)) shares a total\nof 6 edges and consequentially only 2 corners with the adjacent ones(46). The respective Fe-Fe double\nV0= 34.0 cm3mol−1V0= 22.4 cm3mol−1V0= 20.9 cm3mol−12l−Fe(OH) 3(s) γ−FeOOH(s) α−FeOOH(s)\n1\nFigure 6: The structural models of iron hydroxides lepidocrocite ( γ−FeOOH(s)), 2-line ferrihydrite\n(2l−Fe(OH)3(s)) and goethite ( α−FeOOH(s)), together with their specific molar volumes in cm3mol−1.\nThe central iron atoms are depicted in orange, oxygen in red and hydrogen atoms in white. The\nstructural models are generated from the .cif files of their respective reference standards, plotted in\nVesta.\ncorner contributions in 2-line ferrihydrite ( 2l−Fe(OH)3(s)) and goethite ( α−FeOOH(s)), as displayed\nin Fig. 7 are significantly higher. It is thus likely that the measured iron hydroxide powders do not\ncontain lepidocrocite in quantities sufficiently large to be detected by EXAFS. Comparing the obtained\nRDF at the same equilibration times, it can be recognised that both the first coordination shell Fe −O\nas well as various Fe −Fe edge and double corner contributions are suppressed in the presence of 0.5 mM,\ncompared to 0.1 mM Na2SiO3. Analogous to the hindered polymerisation of octahedrally coordinated\nFe(III) in Si containing solution, as described by Pokrovski et al.(32), the observed peak intensity\nattenuation strongly suggests the presence of Si ligands within the second coordination shell around\nthe ferric ion. The observed retardation is also in agreement with a number of studies investigating\nthe transformation of solid primary to secondary iron (hydr)oxides and is commonly attributed to the\nadsorption or incorporation of silicon into the crystal or nucleus of the primary phase(4,47,30,48).\n3.3 The kinetics of corrosion product phase transformation\nThe rate of phase transformation was calculated by linear combination fitting (LCF) of various XAS\nspectra of iron hydroxide powders aged in the presence of 0.1 and 0.5 mM Si in the near-edge region from\n20eVbelowto40eVabovetheabsorptionedge(Supportinginformation, Fig. S2). Fromwithinthepool\nof reference standards employed, 2-line ferihydrite, goethite and lepidocrocite, weighted combinations\nof 2-line ferrihydrite and goethite best reproduce the measured XANES spectra of aged iron hydroxide\n120 1 2 3 4 5 62l−Fe(OH) 3(s)α−FeOOH(s)γ−FeOOH(s)\nRadial distance R, Å|χ(R)|, Å−4\n1Figure 7: Fe K-edge k3-weighted RDF of solid iron hydroxide reference standards 2-line ferrihydrite\n(2l−Fe(OH)3(s)),lepidocrocite( γ−FeOOH(s))andgoethite( α−FeOOH(s)),togetherwiththegraphical\nrepresentation of their unit-cell crystal structure. Octahedra making up the unit cell of lepidocrocite\nare predominantly joined at their faces and consequentially feature a small Fe −Fe double corner\ncontribution. The RDF of goethite entails a close to equal contribution of Fe −Fe edges and double\ncorners, as evident from the topology of its unit cell(46). The ferric cation in 2-line ferrihydrite on the\nother hand is coordinated both tetrahedrally and octahedrally(35). The corresponding RDF features\ntwo discernable, low-intensity contributions from Fe −Fe edges as well as double corners.\npowders. Including the third standard, lepidocrocite, yielded a less robust fit with comparatively high\nstandard deviations in the predicted solid molar fractions (Supporting information, Fig. S3). The\ncomputed molar fractions of 2-line ferrihydrite as a function of time are displayed in Fig. 8. It is clear\nthat there is a positive correlation between the dissolved silicon concentration and the observed solid\nfraction of 2-line ferrihydrite. In the absence of silicon, the formation of goethite is rapid, consuming\ncloseto75%of2-lineferrihydritewithin24hours, asreportedinFurcasetal.(16). Inthesametimespan,\nthe conversion extent amounts to approximately 35 % and 8 % in the presence of 0.1 mM and 0.5 mM\nSi, respectively. After 30 days, all progressions approach full conversion. Assuming the time-dependent\namount of 2-line ferrihydrite (Fe (t)) is proportional to its rate of transformation,\ndFe(t)\ndt=−kFe(t), (1)\n130 1 2 3 4 5 6 300.00.10.20.30.40.50.60.70.80.91.01.1\n0 mM Si0.1 mM Si0.5 mM Si\nTime t, daysSoild mol fraction Fe (t)/Fe0, mol/mol\n1Figure 8: Time dependent molar fractions of 2-line ferrihydrite in the presence of 0, 0.1 and 0.5 mM\nSi at pH = 14. The data for 0 mM Si (circular empty markers) is taken from Furcas et al.(16). Molar\nfractions at 0.1 mM and 0.5 mM Si are obtained from LCF of the XANES spectra displayed in Fig. S2,\nusing the reference standards of 2-line ferrihydrite and goethite.\ncan be computed by fitting its mol fraction with respect to the initial iron concentration Fe (t)/Fe0to\nthe integrated first order rate equation Fe (t) =Fe0×exp(−kt). Fig. 9 displays the resultant estimated\nrate constant of phase transformation kins−1as a function of the aqueous Si concentration. It is\nshown that the presence of 1×10−4M of Si, corresponding to an initial iron to silicon ratio of 1:200,\nis sufficient to reduce the rate of corrosion product transformation by one order of magnitude. Further\nreduction is exponential with respect to the silicon concentration and follows\nlog10k=log10k0−14.65×[Si]0.28, (2)\nwhere k0= (4.04±1.53)×10−5s−1is the rate of 2-line ferrihydrite transformation at pH = 14 in the\nabsence of silicon.\n140 1×10−42×10−43×10−44×10−45×10−46×10−410−610−510−4\nlog10k=log10k0−14.65×[Si]0.28\n[Si],mol L−1First order rate constant k,s−1Furcas et al.(6)\nThis work\n1Figure 9: The estimated first order rate constants of corrosion product transformation kins−1as a\nfunction of the aqueously dissolved silicon concentration [Si]inmol L−1at pH = 14. Various kwere\ncomputed by fitting the time dependent molar fraction of 2-line ferrihydrite to the integrated first order\nrate equation Fe (t) =Fe0×exp(−kt). The dataset corresponding to the rate of transformation in the\nabsence of silicon was taken from Furcas et al.(16).\n153.4 AsuggestedmechanismoftheinfluenceofSionthetransformationofcorrosion\nproducts\nXRD (Fig. 2) and EXAFS (Fig. 5) measurements demonstrate a delayed formation of the crystalline\ncorrosionproductgoethitefromamorphous2-lineferrihydriteinthepresenceofsilicon. TheRDFandX-\nraydiffractogramsofironhydroxidephasesagedinthepresenceof0.5mMNa2SiO3areequivalenttothe\nmeasurements of solid phases stabilised in the presence of 0.1 mM Na2SiO3at earlier equilibration times.\nChangesintheaqueousphasecompositionoverthecourseofthetransformationfurtherindicatethatthe\nconcentrationsofironareincreasedinthepresenceofsilicon(compareFigure1). Viceversa, thefraction\nofsiliconthatremainsintheaqueousphasewithrespecttotheamountinitiallydissolvedincreasesw.r.t.\nthe aqueous iron concentration. In either case, the balance, i.e. the difference between the initial and\naqueously dissolved number of moles of Fe and Si, partitions into one of the two solid phases, 2-line\nferrihydrite and goethite. These trends are summarised in Fig. 10. From the aqueous Si concentration\n10−110010110210310410−310−210−1100\n[Si]/[Si]0\n[Fe]/[Fe]0[Si]0= 0.5mM [Si]0= 0.1mM\n[Si]0= 0.5mM\n[Si]0= 0.1mMpH = 14\npH = 13\nTime t, minAqueous mol fraction xi, mol/mol\n1\nFigure 10: The aqueous mol fraction of iron and silicon with respect to their initial concentrations [Fe]0\nand[Si]0as a function of time, the pH and the initial silicon concentration. The fraction of aqueous\nSi is approximately constant over time and increases as a function of the aqueous iron concentration\nand thus, the pH. The progressions of [Fe]/[Fe]0decrease as a function of time, where values measured\nin the presence of [Si]0= 0.5mM (empty markers) exceed those corresponding to an initial silicon\nconcentration of 0.1 mM (filled markers). Note that the aqueous silicon concentration at pH = 13 and\n[Si]0=0.1 mM is below the LOQ.\n16measurements presented in Figure 1, it is apparent that the initial reduction in the dissolved silicon\nconcentration is concluded within the first few minutes of equilibration time. As evident from the\nextended X-ray absorption fine structures presented in Figure 5, the solids stabilised across the same\ntimespan consist exclusively of 2-line ferrihydrite. Consequentially, the difference between the initial\nand final aqueous silicon concentration must have predominantly associated to 2-line ferrihydrite, rather\nthan goethite. These observations are in agreement with Schulz et al.(31), demonstrating that the vast\nmajority of dissolved Si is associated with 2-line ferrihydrite during its Fe(II)-catalysed transformation\nto other iron (hydr)oxides at neutral pH. The remaining amount of Si that is not adsorbed or taken-up\nby 2-line ferrihydrite within the first hour of equilibration time appears to influence the morphology of\ngoethite crystals stabilised and the aqueous iron concentration in equilibrium with both solid phases.\nCornell and Giovanoli(30)show that high levels of Si (9.6 mol % relative to the total number of moles\nincluding water) significantly enhances the development of the terminal (021), (111) and (121) crystal\nfaces. Although the aqueous Si concentration of 0.1 and 0.5 mM used in this paper is significantly lower,\nthe onset of the (110) and (111) face development over the (002) face is evidenced from an increase in\ntheir crystallinity at 0.5 mM (compare Supplementary information, Figure S1).\nProvided that the adsorption of Si onto 2-line ferrihydrite slows down its dissolution(15,31), the aque-\nous iron concentration resulting of the dissolution of 2-line ferrihydrite is expected to be lower in the\npresence of Si. Given additionally that the dissolution of 2-line ferrihydrite limits the rate of goethite\nprecipitation(23), while the precipitation of goethite is not occurring instantly as is evidenced by the fact\nthat aqueous iron concentrations plateau above the goethite solubility limit (irrespective of the presence\nof Si, see Figure 1), the overall aqueous iron concentration as measured by ICP-OES resultant of the\ncombined dissolution of 2-line ferrihydrite and precipitation of goethite is also expected to be lower in\nthe presence of Si. As the measured [Fe]are instead higher, it is likely that some of the aqueous iron\nthat would otherwise be coordinated as Fe(OH)–\n4forms an aqueous Fe-Si complex(32). In comparison to\nthe transformation of 2-line ferrihydrite in the absence of silicon, the observed retardation would thus\nbe due to the competitive formation of goethite and the hitherto unconsidered Fe-Si complex, effectively\nreducing the amount of Fe(OH)–\n4at disposal to precipitate. The formation of such a complex could\nexplain the strongly increased aqueous iron concentration in cementitious systems containing dissolved\nSi compared to systems aged in the absence of silicon(13,49,50). Figure 11 illustrates these findings in the\nform of a schematic reaction diagram. Further deconvolution of the influence of silicon on the transfor-\nmation mechanism would require a more detailed investigation of (i) the nature of Fe −Si complexes\nformed as well as (ii) the rate of silicon adsorption onto both solid phases and (iii) the rate of 2-line\n17Adsorption/uptake \nof the majority of \nsilica onto Fe(OH)3 \nsurface \nPrecipitation of \namorphous \nFe(OH)3Redissolution of \nFe(OH)3, complex -\nation of silica and \naqueous Fe(III)Crystallisation of \ngoethite from \nsolution, adsorp -\ntion of small \namounts of silicaFull conversion\nseconds minutes days/monthsApproximated first order \ntransformation \nFe(t) = Fe0 /gid00051 exp(- kt)\nRate constant \nlog10k = log10k/gid00131 - 14.65 /gid00051/gid00001[Si]0.28Silicon\nIronFigure 11: Suggested mechanism of the influence of silicon on the transformation of 2-line ferrihydrite\nto goethite at alkaline pH. Initially, large quantities of silicon are adsorbed onto 2-line ferrihydrite,\nthe remainder forms aqueous Fe −Si complexes or adsorbs onto goethite during a later stage in the\ntransformation process. The estimated rate constants k, as computed by fitting the molar fractions\nof 2-line ferrihydrite and goethite to the integrated first order rate equation Fe (t) =Fe0×exp(−kt),\ndecreaseasafunctionoftheinitialaqueoussiliconconcentrationaccordingtothesemi-empiricalrelation\nlog10k=log10k0−14.65×[Si]0.28, where k0= (4.04±1.53)×10−5s−1is the standard rate constant of\ntransformation at pH = 14 in the absence of silicon.\nferrihydrite dissolution as a function of the amount of silicon adsorbed. Moreover, it cannot be excluded\nthat small quantities of Si are incorporated into or alter the structure of corrosion products formed.\nTo this effect, the utilised approach to quantify the transformation of 2-line ferrihydrite to goethite by\nmeans of XAS LCF may not be applicable at higher silicon concentrations. Instead, the structure and\nstoichiometryofSi-bearingironphasesrequiresrigorousmodellingofthelocalcoordinationenvironment\nof both ligands.\n3.5 Implications for the durability of reinforced concrete structures\nWith regard to the durability of reinforced concrete structures, these findings imply that the pore\nsolution chemistry, i.e. the high degree of alkalinity and the presence of Si, maintains an aqueous Fe(III)\nconcentrationwellaboveitsthermodynamicsolubilitylimit. IrrespectiveofwhethertheseFe(III)species\nare coordinated as one of the Fe(III) hydrolysis products or in the form of an aqueous Fe-Si complex,\n18they remain mobile in the aqueous phase. Cementitious systems dissolving high amounts of silicon\ncould thus facilitate iron transport and therefore the precipitation of corrosion products distant from\nthe steel-concrete interface. Moreover, the kinetically hindered transformation of 2-line ferrihydrite to\ngoethite could lead to a faster accumulation of internal stresses, as the high-volume corrosion product 2-\nline ferrihydrite dissolves comparatively slower. As these competing reactive mass-transport phenomena\nare highly dependent on the local solute concentrations as well as the pore shape in which the corrosion\nproducts precipitate, further experimental and computational studies are needed to assess whether the\ndelayed transformation of corrosion products contributes to the structural degradation on a macroscopic\nlevel.\n4 Conclusions\nThe findings presented in this study and other research on the precipitation of corrosion products at\nalkaline pH substantiate the following major conclusions:\n•The stabilisation of the thermodynamically favourable corrosion product goethite from amorphous\n2-line ferrihydrite is delayed in the presence of silicon.\n•At the aqueous silicon concentration characteristic to a broad range of cementitious systems, the\nestimated rate constant of transformation decreases exponentially with respect to [Si].\n•The observed retardation of goethite formation is likely a consequence of 3 distinct reactive phe-\nnomena, (i)thehindereddissolutionof2-lineferrihydrite, (ii)theformationofasolubleiron-silicon\ncomplex and (iii) changes to the morphology of the goethite crystals.\n•Whilst the influence of Si on the dissolution of 2-line ferrihydrite is known(15,31), the structure and\nformation mechanism of the hypothesized Fe-Si complex as well as the precipitation of goethite\nin the presence of Si must be further researched.\n•With regard to the durability of reinforced concrete structures, the observed retardation prospec-\ntivelygeneratesmoreinternalstressesduetotheprecipitationofcorrosionproducts, asthespecific\nmolarvolumeof2-lineferrihydriteissignificantlyhigherthanthatofgoethite. Moreover, theaque-\nous iron concentration in equilibrium with both phases is higher compared to systems containing\nno silicon, facilitating the transport of iron complexes across the concrete pore network.\n19Acknowledgements\nAll XAS measurements were performed at the PHOENIX beamline at the Swiss Light Source, Paul\nScherrer Institut, Villigen, Switzerland. The authors thank Dr. Michael Plötze from the ETH Zürich\nClayLab at the Institute for Geotechnical Engineering for access to the XRD equipment, help with\nthe measurement and discussion. The authors also thank Raphael Kuhn and Dr. Alexandru Pirvan\nfrom the Empa Concrete & Asphalt Laboratory for their help with the ICP-OES measurements. The\nauthors are further grateful for the financial support provided for Fabio Enrico Furcas by the European\nResearch Council (ERC) under the European Union’s Horizon 2020 research and innovation program\n(grant agreement no. 848794).\nAuthor contributions\nFabio E. Furcas: Conceptualization, Investigation, Formal analysis, Visualization, Writing - Original\nDraft\nShishir Mundra: Conceptualization, Investigation, Writing - Review & Editing, Supervision\nBarbara Lothenbach: Conceptualization, Formal analysis, Writing - Review & Editing, Supervision\nCamelia N. Borca: Investigation, Writing - Review & Editing\nThomas Huthwelker: Investigation, Writing - Review & Editing\nUeli M. Angst: Conceptualization, Writing - Review & Editing, Supervision, Funding acquisition\nData availability\nThe experimental data generated in this study is openly available in the ETH Research Collection,\nhttps://doi.org/10.3929/ethz-b-000658759.\n20Supplementary information\n23 24 25 26\n(112)Intensity, a.u.a\n110\n1\n41 42 43 44b\n111\n1\n46 47 48 49\n2θCo kαIntensity, a.u.c\n140\n121\n0 mM Si\n0.5 mM Si0.1 mM Si\n1\n71 72 73 74\n2θCo kαd\n002\n1\nFigure S1: X-ray diffractogram 110 (Fig. S1a), 111 (Fig. S1b), 121, 140 (Fig. S1c) and 002 (Fig. S1d)\npeaks of the iron hydroxide powders extracted from supersaturated stock solution after 30 days the\npresence of no silica (continuous line), 0.1 mM Na2SiO3(circular marker) and 0.5 mM Na2SiO3(cross\nmarker).\n217,100 7,110 7,120 7,130 7,140 7,150 7,160 7,170 7,180 7,190 7,20020 min6 h18 h24 h3 d5 d30 d\nEnergy E, eVNormalised absorbance, a.u.a\n1\n7,100 7,110 7,120 7,130 7,140 7,150 7,160 7,170 7,180 7,190 7,20020 min6 h18 h24 h3 d5 d30 d\nEnergy E, eVNormalised absorbance, a.u.b\n1FigureS2: NormalisedFeK-edgeXANESspectraofironhydroxidepowders(continuouslines)extracted\nfrom supersaturated stock solutions at pH = 14 containing 20.0 mM FeCl3·6H2O(cr) and 0.1 mM\nNa2SiO3(Fig. S2a) and 0.5 mM Na2SiO3(Fig. S2b), together with their linear combination fits\n(circular markers). Spectra were fitted using the reference standards 2-line ferrihydrite and goethite in\nthe XANES region from 20 eV below to 40 eV above the absorption edge.220 1·1052·1053·1054·1055·1052.59·1060.00.10.20.30.40.50.60.70.80.91.01.1\nγ−FeOOH(s)α−FeOOH(s)2l−Fe(OH) 3(s)\nTime t, sSoild mol fraction xi, mol/mola\n1\n0 1·1052·1053·1054·1055·1052.59·1060.00.10.20.30.40.50.60.70.80.91.01.1\nγ−FeOOH(s)α−FeOOH(s)2l−Fe(OH) 3(s)\nTime t, sSoild mol fraction xi, mol/molb\n1Figure S3: Time dependent solid molar fractions of the reference standards 2-line ferrihydrite, lepi-\ndocrocite and goethite in the presence of 0.1 mM Na2SiO3(Fig. S3a) and 0.5 mM Na2SiO3(Fig. S3b)\nat pH = 14. The data is obtained from LCF of the XANES spectra displayed in Fig. S2.\n23Table S1: LOD and LOQ in µg L−1andµmol L−1for the low-concentration element Fe. Concentrations\nare obtained by means of a linear 8-point interpolation containing 0.01, 0.10, 0.50, 1.00, 5.00, 10.00,\n25.00 and 50.00 ppm of the measured elements.\nElement Spectral line, nm SD LOD LOQ\n(ppm) ( µg L−1) ( µmol L−1) ( µg L−1) ( µmol L−1)\nFe 234.350 2.961·10−40.888 0.016 2.961 0.053\n238.204 4.726·10−41.418 0.025 4.726 0.085\n239.563 4.348·10−41.304 0.023 4.348 0.078\n259.940 9.081·10−42.724 0.049 9.081 0.163\nSi 212.412 2.739·10−38.215 0.293 27.385 0.975\n250.690 3.102·10−39.307 0.331 31.023 1.150\n251.611 2.224·10−36.672 0.238 22.204 0.792\n288.158 2.639·10−37.918 0.282 26.394 0.940\n240 10 20 30 40 50 600×1001×1052×1053×1054×1055×1056×1057×1058×1059×105\nConcentration [Fe], ppmIntensity, a.u.234.350 nm\n238.204 nm\n239.563 nm\n259.940 nma\n1\n0 0.1 0.2 0.3 0.4 0.5 0.60×1001×1032×1033×1034×1035×1036×1037×1038×1039×1031×104\nConcentration [Fe], ppmIntensity, a.u.234.350 nm\n238.204 nm\n239.563 nm\n259.940 nmb\n1Figure S4: Inductively coupled plasma optical emission spectroscopy calibration for the element iron,\ncorresponding to the solution composition displayed in Table S2. Figure S4a displays the full calibration\nrange (0.01, 0.05, 0.10, 0.50, 1.00, 2.50, and 5.00 ppm) and Figure S4b shows a zoomed-in section of the\nthree lowest standards above the blank (0.01, 0.05 and 0.10 ppm). All linear fits feature a coefficient of\ndetermination R2≥0.999.\n250 1 2 3 4 5 60×1001×1042×1043×1044×1045×1046×1047×1048×1049×1041×105\nConcentration [Fe], ppmIntensity, a.u.234.350 nm\n238.204 nm\n239.563 nm\n259.940 nma\n1\n0.00 0.02 0.04 0.06 0.08 0.10 0.120×1001×1032×1033×103\nConcentration [Fe], ppmIntensity, a.u.234.350 nm\n238.204 nm\n239.563 nm\n259.940 nmb\n1Figure S5: Inductively coupled plasma optical emission spectroscopy calibration for the element iron,\ncorresponding to the solution composition displayed in Table S3. Figure S5a displays the full calibration\nrange (0.01, 0.05, 0.10, 0.50, 1.00, 2.50, and 5.00 ppm) and Figure S5b shows a zoomed-in section of the\nthree lowest standards above the blank (0.01, 0.05 and 0.10 ppm). All linear fits feature a coefficient of\ndetermination R2≥0.999.\n260 2 4 6 8 10 12 140×1001×1042×1043×1044×1045×1046×1047×1048×1049×1041×1051.1×1051.2×1051.3×105\nConcentration [Si], ppmIntensity, a.u.212.412 nm\n250.690 nm\n251.611 nm\n288.158 nma\n1\n0 0.05 0.1 0.15 0.2 0.25 0.30×1001×1032×1033×1034×103\nConcentration [Si], ppmIntensity, a.u.212.412 nm\n250.690 nm\n251.611 nm\n288.158 nmb\n1Figure S6: Inductively coupled plasma optical emission spectroscopy calibration for the element silicon,\ncorresponding to the solution composition displayed in Table S3. Figure S6a displays the full calibration\nrange (0.02, 0.12, 0.25, 1.25, 2.50, 6.50, and 12.50 ppm) and Figure S6b shows a zoomed-in section of\nthe three lowest standards above the blank (0.02, 0.12 and 0.25 ppm). All linear fits feature a coefficient\nof determination R2≥0.999.\n27Table S2: Elemental composition of the ICP calibration solutions used to determine the total amount\ndissolved aqueous iron at pH = 13 and 14 in the absence of silica. For all runs, the blank solution\nconsisted of 2 wt.% HNO3in UPW.\nStandard Element, ppm\nNumber Na K Ca Mg Al Fe S Si P\n8 50.5506 50.2203 50.3531 50.4780 20.2052 50.3020 50.2344 50.3595 50.3269\n7 25.3110 25.1456 25.2121 25.2747 10.1169 25.1865 25.1527 25.2153 25.1990\n6 10.1346 10.0684 10.0950 10.1201 4.0508 10.0848 10.0712 10.0963 10.0898\n5 5.0766 5.0435 5.0568 5.0693 2.0291 5.0517 5.0449 5.0574 5.0542\n4 1.0239 1.0172 1.0199 1.0224 0.4093 1.0188 1.0175 1.0200 1.0194\n3 0.5096 0.5063 0.5076 0.5089 0.2037 0.5071 0.5064 0.5077 0.5640\n2 0.1017 0.1010 0.1013 0.1016 0.0407 0.1012 0.1011 0.1013 0.1013\n1 0.0101 0.0101 0.0101 0.0102 0.0041 0.0101 0.0101 0.0101 0.0101\nTable S3: Elemental composition of the ICP calibration solutions used to determine the total amount\ndissolved aqueous iron and silica at pH = 13 and 14 for [Si] = 1·10−4M and [Si] = 5·10−4M. For all\nruns, the blank solution consisted of 2 wt.% HNO3in UPW.\nStandard Element, ppm\nNumber Na K Ca Mg Al Fe S Si P\n7 50.4817 50.1734 49.6668 20.2355 20.2262 5.0717 50.3728 12.6314 -\n6 26.1642 26.0044 25.7418 10.4878 10.4830 2.6286 26.1077 6.5467 -\n5 10.4703 10.4064 10.3013 4.1970 4.1951 1.0519 10.4477 2.6198 -\n4 5.2400 5.2080 5.1555 2.1005 2.0995 0.5264 5.2287 1.3111 -\n3 1.0496 1.0432 1.0327 0.4207 0.4206 0.1055 1.0474 0.2626 -\n2 0.5293 0.5261 0.5208 0.2122 0.2121 0.0532 0.5282 0.1324 -\n1 0.1057 0.1051 0.1040 0.0424 0.0424 0.0106 0.1055 0.0265 -\n28References\n[1] Luca Bertolini, Bernhard Elsener, Pietro Pedeferri, Elena Redaelli, and Rob Polder. Corrosion of\nSteel in Concrete , volume 392. Wiley-Vch, Weinheim, Germany, 2013.\n[2] HS Wong, YX Zhao, AR Karimi, NR Buenfeld, and WL Jin. On the penetration of corrosion\nproducts from reinforcing steel into concrete due to chloride-induced corrosion. Corrosion Science ,\n52(7):2469–2480, 2010. 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Natural “amorphous” ferric hydroxide. Geoderma , 10(3):237–247,\n1973. ISSN 0016-7061.\n[49] Belay Zeleke Dilnesa, Barbara Lothenbach, Guillaume Renaudin, Adrian Wichser, and Dmitrii\nKulik. Synthesis and characterization of hydrogarnet Ca3(AlxFe1–x)2(SiO4)y(OH)4(3−y).Cement\nand Concrete Research , 59:96–111, 2014.\n[50] Fabio E Furcas, Barbara Lothenbach, O Burkan Isgor, Shishir Mundra, Zhidong Zhang, and Ueli M\nAngst. Solubility and speciation of iron in cementitious systems. Cement and Concrete Research ,\n151:106620, 2022. ISSN 0008-8846.\n33"    },    {        "title": "2402.13021v1.Dirichlet_Problems_in_Perforated_Domains.pdf",        "content": "arXiv:2402.13021v1  [math.AP]  20 Feb 2024Dirichlet Problems in Perforated Domains\nRobert Righi∗Zhongwei Shen†\nAbstract\nIn this paper we establish W1,pestimates for solutions uεto Laplace’s equation with the\nDirichlet condition in a bounded and perforated, not necessarily per iodically,C1domain Ω ε,η\ninRd. The bounding constants depend explicitly on two small parameters εandη, whereε\nrepresents the scale of the minimal distance between holes, and ηdenotes the ratio between the\nsize of the holes and ε. The proof relies on a large-scale Lpestimate for ∇uε, whose proof is\ndividedintotwoparts. Inthefirstpart,weshowthatas ε,ηapproachzero,harmonicfunctionsin\nΩε,ηmay be approximated by solutions of an intermediate problem for a Sc hr¨ odinger operator\nin Ω. In the second part, a real-variable method is employed to estab lish the large-scale Lp\nestimate for ∇uεby using the approximation at scales above ε. The results are sharp except in\nthe cased≥3 andp=dord′.\nKeywords : Uniform Estimates; Dirichlet Problem; Perforated Domain; Homoge nization.\nMR (2020) Subject Classification : 35Q35; 35B27; 76D07.\n1 Introduction\nIn this paper we continue the study of the Dirichlet problem f or Laplace’s equation,\n/braceleftBigg\n−∆uε=F+div(f) in Ω ε,η,\nuε= 0 on ∂Ωε,η,(1.1)\nwhere Ω ε,ηis a domain perforated, not necessarily periodically, with a large number of tiny holes.\nGivenF∈Lp(Ωε,η) andf∈Lp(Ωε,η;Rd), it iswell known that theDirichlet problem(1.1) possesse s\na unique solution uε=uε,ηinW1,p\n0(Ωε,η), if 1<p<∞and Ωε,ηis a bounded C1domain in Rd,\nd≥2. LetAp(Ωε,η) andBp(Ωε,η) denote the smallest constants for which the W1,pestimate,\n/ba∇dbl∇uε/ba∇dblLp(Ωε,η)≤Ap(Ωε,η)/ba∇dblf/ba∇dblLp(Ωε,η)+Bp(Ωε,η)/ba∇dblF/ba∇dblLp(Ωε,η), (1.2)\nholds. We are interested in bounds of Ap(Ωε,η) andBp(Ωε,η) that exhibit explicit and sharp\ndependence on the sizes of the holes as well as on the distance s between them.\nLetQ(x,r) denote the open cube centered at xwith side length r. To describe the perforated\ndomain Ω ε,η, let{Ys\nz:z∈Zd}be a family of bounded domains with connected and uniform C1\nboundaries such that\nB(0,c0)⊂Ys\nz⊂B(0,1/8) (1.3)\nfor somec0>0. Let{xz:z∈Zd}be a sequence of points in Q(0,1/2) and\nTz,η=z+xz+ηYsz, (1.4)\n∗Supported in part by NSF grants DMS-1856235 and DMS-2153585 .\n†Supported in part by NSF grants DMS-1856235 and DMS-2153585 .\n1whereη∈(0,1/2). For a domain Ω in Rdand 0<ε≤1, define\nΩε,η= Ω\\/uniondisplay\nzεTz,η, (1.5)\nwhere the union is taken over those z’s inZdfor whichε(z+Y)⊂Ω, whereY=Q(0,1). Thus,\nthe perforated domain Ω ε,ηis obtained from Ω by removing a hole εTz,η, centered at ε(z+xz) and\nof sizeεη, from each cell ε(z+Y) of sizeεand contained in Ω. Roughly speaking, the parameter ε\nrepresents the scale of the minimal distances between holes , while the parameter ηrepresents the\nscale of the ratios between the sizes of the holes and ε. We point out that the holes need not to be\nidentical, nor they are placed periodically, unless the seq uences{xz}, and{Ys\nz}are independent of\nz.\nTostatethemainresultsfor Ap(Ωε,η), wenotethat A2(Ωε,η) = 1, andthat Ap(Ωε,η) =Ap′(Ωε,η),\nwherep′=p\np−1, by duality. As a result, it suffices to consider the case 2 <p<∞. Let\nσε=σε,η=/braceleftBigg\nεη1−d\n2ifd≥3,\nε|lnη|1\n2ifd= 2.(1.6)\nThe asymptotic behaviors of Ap(Ωε,η) andBp(Ωε,η), asε,η→0, depend on σε. Our first theorem\ntreats the case of relatively large holes, where σε≤1, while our second theorem handles the case\nof relatively small holes, where σε≥1.\nTheorem 1.1. Suppose that 0< σε≤1and2< p <∞. LetΩbe a bounded C1domain in Rd\nandΩε,ηbe given by (1.5). Then\nAp(Ωε,η)≤\n\nCη−d|1\n2−1\np|ifd≥3,\nCη−2|1\n2−1\np||lnη|−1\n2ifd= 2,(1.7)\nwhereCdepends only on d,p,Ω, and{Ys\nz}.\nTheorem 1.2. Suppose that σε≥1and2< p <∞. LetΩbe a bounded C1domain in Rdand\nΩε,ηbe given by (1.5). Then\nAp(Ωε,η)≤\n\nC(1+ε−1ηd\np−1) ifd≥3and2<p<d,\nC(ε−1+|lnη|1−1\nd)ifd≥3andp=d,\nCε−1ηd\np−1ifd≥3andd<p<∞,\nCε−1η2\np−1|lnη|−1ifd= 2and2<p<∞,(1.8)\nwhereCdepends only on d,p,Ω, and{Ys\nz}.\nWe point out that the upper bounds for Ap(Ωε,η) in (1.7) are sharp. The upper bounds of\nAp(Ωε,η) in (1.8) are also sharp for d= 2 as well as for d≥3 andp/\\e}atio\\slash=d. Whether the upper bound\nis sharp for the remaining case, where d≥3 andp=d(ord′), is not known. Indeed, if Ω ε,ηis a\nperiodically perforated domain, given by (1.5) with the seq uences{xz}and{Ys\nz}independent of\nz, we will show that\nAp(Ωε,η)≥\n\ncη−d|1\n2−1\np|ifd≥3,\ncη−2|1\n2−1\np||lnη|−1\n2ifd= 2,(1.9)\n2in the case σε≤1, and that\nAp(Ωε,η)≥\n\nc(1+ε−1ηd\np−1) ifd≥3 and 2<p<d,\ncε−1ηd\np−1ifd≥3 andd≤p<∞,\ncε−1η2\np−1|lnη|−1ifd= 2 and 2<p<∞,(1.10)\nin the case σε≥1. The constants c >0 in (1.9)-(1.10) are independent of εandη. See Theorem\n9.6. Note that in the case d≥3 andp=d, our upper and lower bounds differ by a term C|lnη|1−1\nd.\nThe main results for Bp(Ωε,η) are given in Theorems 3.1, 3.2 and 3.3. The question of their\nsharpness is addressed in Section 9.\nMotivated by the homogenization theory of boundary value pr oblems for elliptic equations in\nperforated domains [8, 5, 1, 2, 7, 4], the study of the uniform W1,pestimate (1.2) in Ω ε,ηwas\ninitiated by N. Masmoudi [9] in the case η= 1. The general case 0 < η <1 was first studied by\nthe second author of the present paper in [13]. Both the bound ed domain Ω ε,ηand the unbounded\nperforated domain,\nωε,η=Rd\\/uniondisplay\nz∈Zdε(z+ηYsz), (1.11)\nwere considered in [13], where Ω ε,ηandωε,ηneed not to be periodically perforated. The upper\nbounds obtained in [13] for Apare not sharp, although they are only off by an arbitrary small\npower ofη. In [14] J. Wallace and the second author studied the case of t he unbounded domain\nωε,η. Under the additional assumption that the holes {Ys\nz}are identical and thus ωε,ηis periodic,\nthey were able to obtain the estimate (1.2) with sharp consta ntsApandBp. The argument in [14]\nuses a large-scale Lipschitz estimate for harmonic functio nsuεin perforated domains, whose proof\nrelies on the observation that the difference uε(x+εej)−uε(x) is also harmonic. It is not clear\nhow to extend this proof to the non-periodic case and to the ca se of bounded domains.\nInthispaperweintroduceanewapproach to thestudyof W1,pestimates inperforateddomains,\nwhich allows us to handle the case of non-periodic holes as we ll as boundary estimates for bounded\nperforated domains. To show Theorems 1.1 and 1.2, we start wi th a localization argument as in\n[14] and reduce the Lpestimates of ∇uεto anLpestimate of the operator Sεdefined by\nSε(F,f) =/parenleftBigg \nQ(x,2ε)|∇uε|2/parenrightBigg1/2\n, (1.12)\nwhereuεis the solution of (1.1), extended to Rdby zero. It is not hard to see that\n/ba∇dblSε(F,f)/ba∇dblL2(Rd)=/ba∇dbl∇uε/ba∇dblL2(Ωε,η).\nThus, by energy estimates,\n/ba∇dblSε(F,f)/ba∇dblL2(Rd)≤ /ba∇dblf/ba∇dblL2(Ωε,η)+Cmin(σε,1)/ba∇dblF/ba∇dblL2(Ωε,η). (1.13)\nTheorem 1.3. LetΩbe a bounded C1domain in RdandΩε,ηbe given by (1.5). Then\n/ba∇dblSε(F,f)/ba∇dblLp(Rd)≤C/braceleftbig\n/ba∇dblf/ba∇dblLp(Ωε,η)+min(σε,1)/ba∇dblF/ba∇dblLp(Ωε,η)/bracerightbig\n(1.14)\nfor2<p<∞, whereCdepends only on d,p,{Ys\nz}andΩ.\n3By a localization argument as well as Lpestimates for uεin [13], Theorems 1.1 and 1.2 follow\nfrom Theorem 1.3. Since Sεis defined by averaging ∇uεover a cell of size 2 ε, the estimate (1.14)\nshould be regarded as a large-scale W1,pestimate for uε. Theorem 1.3 shows that such average of\n∇uεbehaves much better than ∇uεinLpspaces forp/\\e}atio\\slash= 2. Indeed, as ε,η→0,Ap(Ωε,η)→ ∞for\np/\\e}atio\\slash= 2 andσε≤1, while the operator norm /ba∇dblSε(0,·)/ba∇dblLp→Lpremains bounded.\nTo prove Theorem 1.3, we use a real-variable method from [10] , which reduces the problem\nto a reverse H¨ older inequality for harmonic functions uεin perforated domains. The proof of\nthe reverse H¨ older inequality again relies on the same real -variable argument. As such, for each\nsubdomain Dof size greater than ε, one needs to approximate ∇uεby a function that behaves well\nin theLpnorm. To this end, for each hole in the domain, we introduce an associated nonnegative\npotential, supported in a neighborhood of the hole. Let Vεdenote the sum of these potentials. A\nkey observation of this paper is that a harmonic function uεin a perforated domain Dε,ηis well\napproximated by χε,ηvε, whereχε,ηis a corrector for Dε,ηwithχε,η= 1 on∂D, andvεis the\nsolution of an intermediate equation,\n(−∆+σ−2\nεVε)vε= 0 (1.15)\nin the non-perforated domain Dwith boundary data vε=uεon∂D. We note that even though\nthe holes may not be periodically placed, the potential Vεbehaves like a constant. As a result, ∇vε\nsatisfies the reverse H¨ older inequality for any p>2.\nThe rest of the paper is organized as follows. In Section 2 we g ive the proof of Theorems 1.1\nand 1.2, using Theorem 1.3. The estimates of bounding consta ntsBp(Ωε,η) in (1.2) are given in\nSection 3. Sections 4 - 8 are devoted to the proof of Theorem 1. 3. Specifically, the construction\nand estimates of the correctors χε,ηare given in Sections 4 and 5. The estimates for solutions of\nthe intermediate problem (1.15) are given in Section 7. A con vergence rate for ∇(uε−χε,ηvε) inL2\nis established in Section 6, and the proof of Theorem 1.3 is gi ven in Section 8. Finally, we address\nthe question of sharpness of Theorems 1.1 and 1.2 in Section 9 .\n2 Reduction to the large-scale estimates\nRecallQ(x,r) denotes the open cube centered at xwith size length r. Let{Ys\nz:z∈Zd}be\na family of bounded domains with connected and uniform C1boundaries such that B(0,c0)⊂\nYs\nz⊂B(0,1/8). Let{xz:z∈Zd}be a sequence of points in Q(0,1/2). Forz∈Zd, we define\nQz=z+Q(0,1) andTz,η=z+xz+ηYsz, whereη∈(0,1/2). Note that B(z+xz,c0η)⊂Tz,η⊂Qz\nand dist(Tz,η,∂Qz)≥c>0.\nThe following lemma will be useful to us. Its proof for p= 2 is well known [1]. The proof for\np/\\e}atio\\slash= 2 is similar and can be found in [13, Lemma 2.1].\nLemma 2.1. Letd≥2and1≤p<∞. Suppose that u∈W1,p(Q(0,ε))andu= 0onB(x0,εη)\nfor somex0∈Q(0,ε/2)and0<η<1/4. Then\nˆ\nQ(0,ε)|u|pdx≤Cˆ\nQ(0,ε)|∇u|pdx·\n\nεpηp−dif1≤p<d,\nεp|lnη|d−1ifp=d,\nεpifd<p<∞,(2.1)\nwhereCdepends on dandp.\nLemma 2.2. LetΩbe a bounded Lipschitz domain in RdandΩε,ηbe given by (1.5). Then\n/ba∇dblu/ba∇dblL2(Ωε,η)≤Cmin(σε,1)/ba∇dbl∇u/ba∇dblL2(Ωε,η) (2.2)\n4for anyu∈H1\n0(Ωε,η), whereσεis given by (1.6)andCis independent of εandη.\nProof.It follows from Lemma 2.1 that if u∈H1(Ωε,η) andu= 0 on∂Ωε,η\\∂Ω, then\n/ba∇dblu/ba∇dblL2(Ωε,η)≤Cσε/ba∇dbl∇u/ba∇dblL2(Ωε,η). (2.3)\nThis, together with the Poincar´ e inequality /ba∇dblu/ba∇dblL2(Ω)≤C/ba∇dbl∇u/ba∇dblL2(Ω)foru∈H1\n0(Ω), yields (2.2) for\nu∈H1\n0(Ωε,η).\nForF∈L2(Ω) andf∈L2(Ω;Rd), letuε∈H1\n0(Ωε,η) be the solution of the Dirichlet problem\n(1.1). Using (2.2), it is not hard to show that\n/ba∇dbl∇uε/ba∇dblL2(Ω)≤ /ba∇dblf/ba∇dblL2(Ω)+Cmin(σε,1)/ba∇dblF/ba∇dblL2(Ω), (2.4)\nwhere we have extended uεto Ω by zero. The following theorem gives the Lpestimate of uεfor\np>2.\nTheorem 2.3. LetF∈Lp(Ω)andf∈Lp(Ω;Rd)for somep≥2. Letuεbe the solution of (1.1),\nwhereΩis a bounded Lipschitz domain. Then\n/ba∇dbluε/ba∇dblLp(Ω)≤Cmin(σε,1)/ba∇dblf/ba∇dblLp(Ω)+Cmin(σ2\nε,1)/ba∇dblF/ba∇dblLp(Ω), (2.5)\nwhereCdepends only on d,p,c0andΩ.\nProof.The casep= 2 follows readily from the energy estimate (2.4) and (2.2). The casep>2 was\nproved in [13].\nTo establish the Lpestimates for ∇uε, we consider the local W1,pestimates for solutions of\n/braceleftBigg\n−∆uε=F+div(f) inε(/tildewideQz\\Tz,η),\nuε= 0 in εTz,η,(2.6)\nwherez∈Zdand/tildewideQz=Q(z,17/16). Let\nΦp(R) =\n\n1 if d≥3 and 2<p<d,\n(lnR)1−1\ndifd≥3 andp=d,\nR1−d\np ifd≥3 andd<p<∞,\nR1−2\np(lnR)−1ifd= 2 and 2<p<∞,(2.7)\nforR>2.\nLemma 2.4. Let2<p<∞. Letuεbe a solution of (2.6)withF∈Lp(ε/tildewideQz)andf∈Lp(ε/tildewideQz;Rd).\nThen ford≥3,\n/parenleftbigg \nεQz|∇uε|p/parenrightbigg1/p\n≤C|α|ε−1ηd\np−1+CΦp(η−1)/parenleftbigg \nε/tildewideQz(|εF|+|f|)p/parenrightbigg1/p\n+Cε−1Φp(η−1)/parenleftBigg \nε/tildewideQz\\Q(yz,ε/3)|uε−α|2dx/parenrightBigg1/2\n,(2.8)\n5and ford= 2\n/parenleftbigg \nεQz|∇uε|p/parenrightbigg1/p\n≤C|α|ε−1Φp(η−1)+CΦp(η−1)/parenleftbigg \nε/tildewideQz(|εF|+|f|)p/parenrightbigg1/p\n+Cε−1Φp(η−1)/parenleftBigg \nε/tildewideQz\\Q(yz,ε/3)|uε−α|2dx/parenrightBigg1/2\n,(2.9)\nwhereα∈R,yz=z+xzandCdepends only on d,pand{Ys\nz}.\nProof.The case where ε= 1 andxz= 0 was proved in [14, Theorem 6.1], using a result in [3].\nThe same argument, together with the correctors constructe d in this paper, also gives (2.8)-(2.9)\nforε= 1 andxz∈Q(0,1/2) with constants independent of xz. We point out that there is a minor\nflaw in the proof of Theorem 6.1 in [14]. In the case d= 2, the corrector defined in [14, (6.6)] does\nnot satisfy the condition in [14, (6.11)]. The mistake can be fixed easily by using the corrector in\nSection 5.\nThe general case 0 < ε <1 follows readily from the case ε= 1 by dilation. Note that the\nparameterηremains invariant under dilation.\nRemark 2.5. Letuεbe the solution of (1.1) with F∈Lp(Ω) andf∈Lp(Ω;Rd) for somep>2.\nLet\nΩ′\nε,η=/uniondisplay\nzε(Qz\\Tz,η), (2.10)\nwhere the union is taken over those z’s inZdfor whichεQz⊂Ω. It follows from (2.8)-(2.9) with\nα= 0 and a covering argument that\nˆ\nΩ′′ε,η|∇uε|p≤C|Φp(η−1)|pˆ\nΩ(ε|F|+|f|)p+Cε−p|Φp(η−1)|pˆ\nΩ|uε|p, (2.11)\nwhere Ω′′\nε,η={x∈Ω′\nε,η: dist(x,∂Ω)≥cε}. For the region near ∂Ω and away from the holes, we\nuse the boundary W1,pestimates for Laplace’s equation in C1domains [6] to obtain\nˆ\nB(x,cε)|∇uε|p≤Cε−pˆ\nB(x,2cε)|uε|p+Cˆ\nB(x,2cε)(ε|F|+|f|)p, (2.12)\nwherex∈Ω andB(x,2cε)∩Ω =B(x,2cε)∩Ωε,η. By a covering argument we obtain\nˆ\nΩε,η\\Ω′′ε,η|∇uε|p≤Cˆ\nΩ(ε|F|+|f|)p+Cε−pˆ\nΩε,η|uε|p. (2.13)\nThis, together with (2.11), gives\nˆ\nΩ|∇uε|p≤C|Φp(η−1)|pˆ\nΩ(ε|F|+|f|)p+Cε−p|Φp(η−1)|pˆ\nΩ|uε|p\n≤C|Φp(η−1)|p/parenleftbig\n1+ε−1min(σε,1)/parenrightbigpˆ\nΩ|f|p\n+C|Φp(η−1)|p/parenleftbig\n1+ε−1min(σε,1)/parenrightbig2pˆ\nΩ|εF|p,\nwhere we have used (2.5) for the last inequality. As a result, we have proved that\n/ba∇dbl∇uε/ba∇dblLp(Ω)≤CΦp(η−1)(1+ε−1min(σε,1))/ba∇dblf/ba∇dblLp(Ω)\n+CΦp(η−1)/parenleftbig\n1+ε−1min(σε,1)/parenrightbig2/ba∇dblεF/ba∇dblLp(Ω)(2.14)\n6forp >2 andd≥2. Suppose F= 0. In the case 0 < σε≤1,d≥3 andp > d, it follows from\n(2.14) that\n/ba∇dbl∇uε/ba∇dblLp(Ω)≤Cηd\np−d\n2/ba∇dblf/ba∇dblLp(Ω). (2.15)\nSince (2.15) also holds for p= 2, by Riesz Thorin Interpolation Theorem, (2.15) holds for 2<p<\n∞. This gives the proof of Theorem 3.1 for the case d≥3.\nRemark 2.6. Supposeσε≥1. It follows from (2.14) that if F= 0,\n/ba∇dbl∇uε/ba∇dblLp(Ω)≤Cε−1Φp(η−1)/ba∇dblf/ba∇dblLp(Ω). (2.16)\nIt follows that\n/ba∇dbl∇uε/ba∇dblLp(Ω)≤\n\nCε−1ηd\np−1/ba∇dblf/ba∇dblLp(Ω) ifd≥3 andp>d,\nCε−1η2\np−1|lnη|−1/ba∇dblf/ba∇dblLp(Ω)ifd= 2 and 2<p<∞.(2.17)\nThis gives the estimates in Theorem 1.2 for the case d≥2 andd<p<∞.\nTo deal with the remaining cases in Theorem 1.1 and 1.2, we let\nα= \nε/tildewideQz\\Q(yz,ε/3)uε\nin (2.8)-(2.9). By Poincar´ e’s inequality we obtain\n/parenleftbigg \nεQz|∇uε|p/parenrightbigg1/p\n≤Cε−1ηd\np−1 \nε/tildewideQz|uε|+CΦp(η−1)/parenleftbigg \nε/tildewideQz(|εF|+|f|)p/parenrightbigg1/p\n+CΦp(η−1)/parenleftbigg \nε/tildewideQz|∇uε|2/parenrightbigg1/2(2.18)\nford≥3, and\n/parenleftbigg \nεQz|∇uε|p/parenrightbigg1/p\n≤Cε−1Φp(η−1) \nε/tildewideQz|uε|+CΦp(η−1)/parenleftbigg \nε/tildewideQz(|εF|+|f|)p/parenrightbigg1/p\n+CΦp(η−1)/parenleftbigg \nε/tildewideQz|∇uε|2/parenrightbigg1/2(2.19)\nford= 2. Define\nSε(F,f)(x) =/parenleftBigg \nQ(x,2ε)|∇uε|2/parenrightBigg1/2\n.\nIt follows from (2.18)-(2.19) by a covering argument that\nˆ\nΩ′′ε,η|∇uε|p≤Cε−pηd−pˆ\nΩ|uε|p+C|Φp(η−1)|pˆ\nΩ(|εF|+|f|)p\n+C|Φp(η−1)|pˆ\nΩ|Sε(F,f)|p(2.20)\nford≥3, and that\nˆ\nΩ′′ε,η|∇uε|p≤Cε−p|Φp(η−1)|pˆ\nΩ|uε|p+C|Φp(η−1)|pˆ\nΩ(|εF|+|f|)p\n+CΦp(η−1)|pˆ\nΩ|Sε(F,f)|p(2.21)\n7ford= 2, where Ω′′\nε,ηis the same as in (2.11). For the region near ∂Ω and away from the holes, we\nuse the local W1,pestimate in C1domains,\n/parenleftBigg \nQ(x,cε)|∇uε|p/parenrightBigg1/p\n≤C/parenleftBigg \nQ(x,2ε)|∇uε|2/parenrightBigg1/2\n+C/parenleftBigg \nQ(x,2ε)(|εF|+|f|)p/parenrightBigg1p\n,(2.22)\nwherex∈Ω, dist(x,∂Ω)≤CεandB(x,2cε)∩Ω =B(x,2cε)∩Ωε,η. By a covering argument, this\nleads to ˆ\nΩε,η\\Ω′′ε,η|∇uε|p≤Cˆ\nΩ|Sε(F,f)|p+Cˆ\nΩ(|εF|+|f|)p. (2.23)\nLemma 2.7. LetΩbe a bounded C1domain in Rd. Letuε∈H1\n0(Ωε,η)be the solution of (1.1)\nwithF∈Lp(Ω)andf∈Lp(Ω;Rd)for somep>2. Then\n/ba∇dbl∇uε/ba∇dblLp(Ω)≤Cε−1ηd\np−1/ba∇dbluε/ba∇dblLp(Ω)\n+CΦp(η−1)/braceleftbig\n/ba∇dblε|F|+|f|/ba∇dblLp(Ω)+/ba∇dblSε(F,f)/ba∇dblLp(Ω)/bracerightbig (2.24)\nford≥3, and\n/ba∇dbl∇uε/ba∇dblLp(Ω)≤Cε−1Φp(η−1)/ba∇dbluε/ba∇dblLp(Ω)\n+CΦp(η−1)/braceleftbig\n/ba∇dblε|F|+|f|/ba∇dblLp(Ω)+/ba∇dblSε(F,f)/ba∇dblLp(Ω)/bracerightbig (2.25)\nford= 2, whereCdepends on d,p,{Ys\nz}andΩ.\nProof.This follows from (2.20)- (2.21) and (2.23).\nUsing Theorem 1.3, we now give the proof of Theorems 1.1 and 1. 2.\nProof of Theorem 1.1. Suppose 0 < σε≤1 and 2< p <∞. The case d≥3 is treated already\nin Remark 2.5. Assume d= 2. Then σε=ε|lnη|1/2and Φp(η−1) =η2\np−1|lnη|−1. Letuεbe the\nsolution of (1.1) with F= 0. By (2.25) and Theorem 1.3,\n/ba∇dbl∇uε/ba∇dblLp(Ω)≤Cε−1Φp(η−1)/ba∇dbluε/ba∇dblLp(Ω)+CΦp(η−1)/ba∇dblf/ba∇dblLp(Ω)\n≤CΦp(η−1)ε−1σε/ba∇dblf/ba∇dblLp(Ω)\n=Cη2\np−1|lnη|−1\n2/ba∇dblf/ba∇dblLp(Ω),(2.26)\nwhere we have used Theorem 2.3 for the second inequality.\nProof of Theorem 1.2. Supposeσε≥1 and 2< p <∞. The case for d≥2 andd < p < ∞is\ntreated in Remark 2.6. Assume d≥3 and 2<p≤d. Letuεbe the solution of (1.1) with F= 0.\nIt follows from (2.24) and Theorem 1.3 that\n/ba∇dbl∇uε/ba∇dblLp(Ω)≤Cε−1ηd\np−1/ba∇dbluε/ba∇dblLp(Ω)+CΦp(η−1)/ba∇dblf/ba∇dblLp(Ω)\n≤C/parenleftBig\nε−1ηd\np−1+Φp(η−1)/parenrightBig\n/ba∇dblf/ba∇dblLp(Ω),(2.27)\nwhere we have used Theorem 2.3 for the last inequality. In vie w of (2.7) we obtain\n/ba∇dbl∇uε/ba∇dblLp(Ω)≤/parenleftBig\nε−1ηd\np−1+1/parenrightBig\n/ba∇dblf/ba∇dblLp(Ω),\nfor 2<p<d, and\n/ba∇dbl∇uε/ba∇dblLp(Ω)≤/parenleftBig\nε−1+|lnη|1−1\nd/parenrightBig\n/ba∇dblf/ba∇dblLp(Ω)\nforp=d. This gives the estimates in Theorem 1.2 for the case 2 <p≤d.\n83 Estimates of Bp(Ωε,η)\nIn this section we establish upper bounds for Bp(Ωε,η) in (1.2). We begin with the case 1 <p≤2.\nThe estimate (3.1) below is sharp. See Theorem 9.4.\nTheorem 3.1. Suppose 1<p≤2. LetΩbe a bounded Lipschitz domain in Rd,d≥2, andΩε,η\nbe given by (1.5). Then\nBp(Ωε,η)≤Cmin(σε,1), (3.1)\nwhereCdepends only on d,p,Ω, and{Ys\nz}.\nProof.This was proved in [13], using (2.5) and a duality argument.\nNext, we consider the case 2 < p<∞and 0<σε≤1. The estimate (3.2) below is sharp for\nd≥3 and 2<p<∞. See Theorem 9.5.\nTheorem 3.2. Suppose that 0< σε≤1and2< p <∞. LetΩbe a bounded C1domain in Rd\nandΩεbe given by (1.5). Then\nBp(Ωε,η)≤Cεη1−d+d\np, (3.2)\nwhereCdepends only on d,p,Ω, and{Ys\nz}.\nProof.It follows from (2.14) that\nBp(Ωε,η)≤CΦp(η−1)/braceleftbig\n1+ε−1min(σε,1)/bracerightbig2ε. (3.3)\nUsing 0<σε≤1, we obtain\nBp(Ωε,η)≤CΦp(η−1)σ2\nεε−1. (3.4)\nIn the case d= 2, since Φ p(η−1) =η2\np−1|lnη|−1andσε=ε|lnη|1/2, this gives (3.2).\nIn the case d≥3 andp>d, we haveσε=εη1−d\n2and Φp(η−1) =ηd\np−1. It follows from (3.4) that\n(3.2) holds. Since (3.2) also holds for p= 2, by Riesz-Thorin Interpolation Theorem, the estimate\n(3.2) holds for all 2 <p<∞.\nFinally, we treat the case 2 <p<∞andσε≥1. The estimate (3.5) below is sharp in the case\nd≥3 and 2<p<d. See Theorem 9.5.\nTheorem 3.3. Suppose that σε≥1and2< p <∞. LetΩbe a bounded C1domain in Rdand\nΩε,ηbe given by (1.5). Then\nBp(Ωε,η)≤\n\nC(1+ε−1ηd\np−1) ifd≥3and2<p<d,\nC(ε−1+|lnη|1−1\nd)ifd≥3andp=d,\nCε−1ηd\np−1ifd≥3andd<p<∞,\nCε−1η2\np−1|lnη|−1ifd= 2and2<p<∞,(3.5)\nwhereCdepends only on d,p,Ω, and{Ys\nz}.\nProof.Usingσε≥1, we obtain from (3.3) that\nBp(Ωε,η)≤CΦp(η−1)ε−1. (3.6)\n9In view of (2.7), this gives (3.5) if d≥2 andp>d. For the remaining case d≥3 and 2<p≤d,\nwe letuε∈W1,p\n0(Ωε,η) be a solution of −∆uε=Fin Ωε,η, whereF∈Lp(Ω). We extend uεtoRd\nby zero. It follows from (2.24) and Theorem 1.3 that\n/ba∇dbl∇uε/ba∇dblLp(Ω)≤Cε−1ηd\np−1/ba∇dbluε/ba∇dblLp(Ω)+CΦp(η−1)/ba∇dblF/ba∇dblLp(Ω)\n≤C/parenleftBig\nε−1ηd\np−1+Φp(η−1)/parenrightBig\n/ba∇dblF/ba∇dblLp(Ω),(3.7)\nwhere we have used Theorem 2.3 for the last inequality. In vie w of (2.7) we obtain (3.5) for d≥3\nand 2<p≤d.\n4 Correctors: the case d≥3\nThis section is dedicated to constructing and establishing estimates for the corrector χε,ηin the\ncased≥3. The two-dimensional case will be handled separately in th e next section.\nLet Ωε,ηbe given by (1.5), where Tz,ηbe given by (1.4). The corrector χε,ηfor Ωε,ηis defined\nin a piecewise manner, following an idea in [1]. For z∈Zd, letyz=z+xzandQz=z+Q(0,1).\nIfεQz⊂Ω, we define\nχε,η(x) =\n\n1 in εQz\\εB(yz,1/5),\nφz\n∗(x−εyz\nεη) inεB(yz,1/6)\\εTz,η,\n0 in εTz,η,(4.1)\nwhereφz\n∗is a solution to the exterior problem,\n\n\n−∆φz\n∗= 0 in Rd\\Ysz,\nφz\n∗= 0 on ∂Ys\nz,\nφz\n∗→1 as |x| → ∞.(4.2)\nIn the region εB(yz,1/5)\\εB(yz,1/6) we letχε,ηsolve the following Dirichlet problem,\n\n\n−∆χε,η= 0 in εB(yz,1/5)\\εB(yz,1/6),\nχε,η=φz\n∗/parenleftBigx−εyz\nεη/parenrightBig\non∂B(εyz,ε/6),\nχε,η= 1 on ∂B(εyz,ε/5).(4.3)\nFinally, ifεQzdoes not lie entirely in Ω, we define χε,η= 1 inεQz. As a result, the corrector χε,ηis\ndefined in Rd, taking value 1 in Ωcand 0 in the holes εTz,ηinside Ω. The definition in (4.3) bridges\nthe gap between the boundary data of the exterior problem and 1. Thus, we see that χε,η∈H1(Ω),\nχε,η= 1 on∂Ω andχε,η= 0 on∂Ωε,η\\∂Ω.\nIt is known that for each φz\n∗that satisfies (4.2),\n\n\nφz\n∗(x) = 1−cz\n∗|x|2−d+O(|x|1−d),\n∇φz\n∗(x) =−cz\n∗∇(|x|2−d)+O(|x|−d),\n∇2φz\n∗(x) =−cz\n∗∇2(|x|2−d)+O(|x|−d−1),(4.4)\nas|x| → ∞, wherecz\n∗=Cdµz\n∗is the Newtonian capacity of Ys\nzwith\nµz\n∗=ˆ\n∂Yszn·∇φz\n∗dσ (4.5)\n10andCd=1\n(d−2)|∂B(0,1)|. See [3, 15]. The condition (1.3) on Ys\nzensures that there exist µ0,µ1>0\nsuch thatµ0≤µz\n∗≤µ1for anyz∈Zd.\nLemma 4.1. Letχε,ηbe defined as above. Then\n/ba∇dblχε,η−1/ba∇dblLp(Ωε,η)≤\n\nCηd−2for1≤p<d\nd−2,\nCηd−2|lnη|d−2forp=d\nd−2,\nCηd\np forp>d\nd−2,(4.6)\nwhereCdoes not depend on εorη.\nProof.SupposeεQz⊂Ω. Notethat χε,η−1 = 0inεQz\\B(εyz,ε/5) andisharmonicin B(εyz,ε/5)\\\nB(εyz,ε/6). Hence, by the maximum principle,\n/ba∇dblχε,η−1/ba∇dblL∞(B(εyz,ε/5)\\B(εyz,ε/6))≤ /ba∇dblχε,η−1/ba∇dblL∞(∂B(εyz,ε/6)).\nSinceχε,η(x) =φz\n∗((x−εyz)/(εη)) on∂B(εyz,ε/6), it follows from (4.4) that\n/ba∇dblχε,η−1/ba∇dblL∞(B(εyz,ε/5)\\B(εyz,ε/6))≤Cηd−2.\nNext, we note that\nˆ\nB(εyz,ε\n6)\\εTz,η|χε,η−1|pdx=ˆ\nB(εyz,ε\n6)\\εTz,η|φz\n∗/parenleftBigx−εyz\nεη/parenrightBig\n−1|pdx\n=εdηdˆ\nB(0,1\n6η)\\Ysz|φz\n∗(y)−1|pdy.\nIn view of (4.4) we obtain\nˆ\nB(εyz,ε\n6)\\εTz,η|χε,η−1|pdx≤Cεdηdˆ\nB(0,1\n6η)\\Yszdy\n|y|(d−2)p\n≤Cεdηdˆ1\n6η\ncrd−1−dp+2pdr.\nThus,\n/ba∇dblχε,η−1/ba∇dblp\nLp(εQz\\εTz,η)≤\n\nCεdηp(d−2)for 1≤p<d\nd−2,\nCεdηd|lnη|forp=d\nd−2,\nCεdηdforp>d\nd−2.(4.7)\nSumming over all cubes εQzin Ω gives the desired result, as the number of such cubes is bo unded\nbyCε−d.\nRemark 4.2. SupposeεQz⊂Ω. Consider ξ(x) =χε,η(εx+εyz)−1. The function ξis harmonic\ninB(0,1/5)\\B(0,1/6) withξ= 0 on∂B(0,1/5) andξ=φz\n∗(η−1x)−1 on∂B(0,1/6). It follows\nthat inB(0,1/5)\\B(0,1/6),\n|∇ξ| ≤C/ba∇dblφz\n∗(η−1x)−1/ba∇dblC1,1(∂B(0,1/6))≤Cηd−2,\nwhere we have used (4.4) for the last inequality. This implie s that\n|∇χε,η| ≤Cε−1ηd−2\ninB(εyz,ε/3)\\B(εyz,ε/4).\n11Lemma 4.3. SupposeεQz⊂Ω. Then\n/parenleftbigg \nεQz|∇χε,η|p/parenrightbigg1/p\n≤\n\nCε−1ηd−p\np ifp>d′,\nCε−1ηd−2|lnη|1\npifp=d′,\nCε−1ηd−2ifp<d′,(4.8)\nwhered′=d\nd−1and the constant Cdoes not depend on εorη.\nProof.We start by decomposing the integral along the sub-regions w ithin the cube to obtain\nˆ\nεQz|∇χε,η|p=ˆ\nB(εyz,ε\n5)\\B(εyz,ε\n6)|∇χε,η|p+ˆ\nB(εyz,ε\n6)\\εTz,η|∇χε,η|p. (4.9)\nBy Remark 4.2, the first term in the right-hand side of (4.9) is bounded by Cεd(ε−1ηd−2)p. For\nthe second term, we have\nˆ\nB(εyz,ε\n6)\\εTz,η|∇χε,η|p=ˆ\nB(εyz,ε\n6)\\εTz,η|∇(φz\n∗/parenleftBigx−εyz\nεη/parenrightBig\n)|pdx\n= (εη)−pˆ\nB(εyz,ε\n6)\\εTz,η|∇φz\n∗(x−εyz\nǫη)|pdx\n= (εη)d−pˆ\nB(0,1\n6η)\\Ysz|∇φz\n∗(y)|pdy\n≤C(εη)d−pˆ1\n6η\ncr(1−d)p+d−1dr,(4.10)\nwhere we have used (4.4) as well as the fact that ∇φz\n∗isLpintegrable near ∂Ys\nzunder the condition\nthatYs\nzis uniformly C1. It follows that\n/parenleftBigg \nεB(yz,1/6)|∇χε,η|p/parenrightBigg1/p\n≤\n\nCε−1ηd−p\np ifp>d′,\nCε−1ηd−p\np|lnη|1\npifp=d′,\nCε−1ηd−2ifp<d′.(4.11)\nCombining (4.11) with the estimate for B(εyz,ε/5)\\B(εyz,ε/6) gives (4.8).\nRecall that σε=εη−d−2\n2ford≥3 andµz\n∗is given by (4.5).\nLemma 4.4. SupposeεQz⊂Ω. Then\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nεQz∇χε,η·∇φ−ˆ\nεQzσ−2\nεµz\n∗φ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cε−1ηd−2ˆ\nεQz|∇φ|, (4.12)\nwhereφ∈H1(εQz)andφ= 0inεTz,η.\nProof.Note that\nˆ\nεQz∇χε,η·∇φ−ˆ\nεQzσ−2\nεµz\n∗φ\n=ˆ\nεQz\\B(εyz,ε/6)∇χε,η·∇φ+ˆ\nB(εyz,ε/6)\\εTz,η∇χε,η·∇φ−ˆ\nεQzσ−2\nεµz\n∗φ.(4.13)\n12By Remark 4.2, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nεQz\\B(εyz,ε/6)∇χε,η·∇φ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cε−1ηd−2ˆ\nεQz|∇φ|. (4.14)\nFor the remaining two terms in the right-hand side of (4.13), we use integration by parts to get\nˆ\n∂B(εyz,ε/6)/parenleftBigg\n∂χε,η\n∂n− \n∂B(εyz,ε/6)∂χε,η\n∂n/parenrightBigg\n·(φ−α)+ \n∂B(εyz,ε/6)∂χε,η\n∂nˆ\n∂B(εyz,ε/6)φ\n−ˆ\nεQzσ−2\nεµz\n∗φ,(4.15)\nwherendenotes the outward unit normal and αis a constant to be determined. Here we also use\nthe fact that χε,ηis harmonic in B(εyz,ε/6)\\εTz,η. Using (4.4), we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\n∂B(εyz,ε/6)/parenleftBigg\n∂χε,η\n∂n− \n∂B(εyz,ε/6)∂χε,η\n∂n/parenrightBigg\n·(φ−α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤Cε−1ηd−2ˆ\n∂B(εyz,ε/6)|φ−α|\n≤Cε−1ηd−2/parenleftBigg\n1\nεˆ\nB(εyz,ε/6)|φ−α|+ˆ\nB(εyz,ε/6)|∇φ|/parenrightBigg\n,(4.16)\nwhere we have used a trace inequality for the last step (see Re mark 4.5). Picking α=ffl\nB(εyz,ǫ/6)φ\nand applying the Poincar´ e inequality gives the desired bou nd.\nTo bound /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle \n∂B(εyz,ε/6)∂χε,η\n∂nˆ\n∂B(εyz,ε/6)φ−ˆ\nεQzσ−2\nεµz\n∗φ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (4.17)\nwe move the average onto the integral of φand note that\nˆ\n∂B(εyz,ε/6)∂χε,η\n∂n= (εη)−1ˆ\n∂B(εyz,ε/6)∂φz\n∗\n∂n/parenleftBigx−εyz\nεη/parenrightBig\ndσ(x)\n= (εη)d−2ˆ\n∂B(0,1\n6η)∂φz\n∗\n∂ndσ(y)\n= (εη)d−2ˆ\n∂Ysz∂φz\n∗\n∂ndσ(y)\n=µz\n∗(εη)d−2.\nAs a result, (4.17) becomes\nµz\n∗(εη)d−2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle \n∂B(εyz,ε/4)φ− \nεQzφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nwhich, in view of (4.21), is bounded by\nCε−1ηd−2ˆ\nεQz|∇φ|.\nThis completes the proof.\n13Remark 4.5. In (4.16) we have used the trace inequality,\nˆ\n∂B(x0,r)|φ| ≤d\nrˆ\nB(x0,r)|φ|+ˆ\nB(x0,r)|∇φ|, (4.18)\nforφ∈H1(B(x0,r)). This follows by writing\nˆ\n∂B(x0,r)|φ|=1\nrˆ\n∂B(x0,r)|φ|((x−x0)·n)dσ(x)\nand applying the divergence theorem. Replacing φin (4.18) by φ−α, whereα=ffl\nεQzφ, we obtain\nˆ\n∂B(εyz,cε)|φ− \nεQzφ| ≤Cˆ\nεQz|∇φ|, (4.19)\nwhere we have used a Poincar´ e inequality,\nˆ\nεQz|φ− \nεQzφ| ≤Cεˆ\nεQz|∇φ|. (4.20)\nIt follows from (4.19) that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle \n∂B(εyz,cε)φ− \nεQzφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cε \nεQz|∇φ|. (4.21)\n5 Correctors: the two-dimensional case\nIn this section we introduce a corrector χε,ηfor the case d= 2 and establish its estimates analogous\nto those in Lemmas 4.1, 4.3 and 4.4.\nForz∈Z2, letφz\n∗be a solution to the exterior problem,\n\n\n−∆φz\n∗= 0 in R2\\Ysz,\nφz\n∗= 0 on ∂Ys\nz,\nφz\n∗(x)−ln|x|=O(1) as |x| → ∞.(5.1)\nIt is known that \n\n∇φz\n∗(x) =x\n|x|2+O(|x|−2),\n∇2φz\n∗(x) =O(|x|−2),(5.2)\nas|x| → ∞. Recall that yz=z+xzandQz=z+Q(0,1). IfεQz⊂Ω, we let\nχε,η(x) =\n\n1 in εQz\\εB(yz,1/5),\nφz\n∗(x−εyz\nεη)/|lnη|inεB(yz,1/6)\\εTz,η,\n0 in εTz,η.(5.3)\nIn the region εB(yz,1/5)\\εB(yz,1/6), we letχε,ηsolve the following Dirichlet problem,\n\n\n−∆χε,η= 0 in εB(yz,1/5)\\εB(yz,1/6),\nχε,η=φz\n∗/parenleftBigx−εyz\nεη/parenrightBig\n/|lnη|on∂B(εyz,ε/6),\nχε,η= 1 on ∂B(εyz,ε/5).(5.4)\n14Finally, ifεQzis not contained in Ω, let χε,η= 1 inεQz, as in the case of d≥3. By construction,\nwe haveχε,η∈H1(Ω). Moreover, χε,η= 1 in Ωcandχε,η= 0 in Ω \\Ωε,η. Furthermore, in the\nregionεB(yz,1/5)\\B(yz,1/6), we have L∞estimates,\n|χε,η−1| ≤C|lnη|−1and|∇χε,η| ≤Cε−1|lnη|−1. (5.5)\nAs in the case d≥3, the estimates in (5.5) follow from the maximum principle a nd the Lipschitz\nestimates for harmonic functions.\nLemma 5.1. Letχε,ηbe defined as above. Then for 1≤p<∞,\n/ba∇dblχε,η−1/ba∇dblLp(Ωε,η)≤C|lnη|−1, (5.6)\nwhereCdoes not depend on εorη.\nProof.Note that if εQz⊂Ω,\nˆ\nεQz\\εTz,η|χε,η−1|p≤Cε2|lnη|−p+ˆ\nB(εyz,ε/6)\\εTz,η/vextendsingle/vextendsingle/vextendsingle/vextendsingleln|(x−εyz)/ε|\nlnη/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\ndx\n≤Cε2\n|lnη|p/braceleftBiggˆ\nB(0,1)|ln|x||pdx+1/bracerightBigg\n≤Cε2\n|lnη|p,\nwhere we have used (5.5) as well as the fact that φz\n∗(x) = ln|x|+O(1) as|x| → ∞. Summing over\nall such cubes yields\n/ba∇dblχε,η−1/ba∇dblp\nLp(Ωε,η)≤C|lnη|−p,\nwhich yields (5.6).\nRecall that σε=ε|lnη|1/2ford= 2.\nLemma 5.2. SupposeεQz⊂Ω. Then\n/parenleftbigg \nεQz|∇χε,η|p/parenrightbigg1/p\n≤\n\nCσ−1\nε|lnη|−1/2if1<p<2,\nCσ−1\nε ifp= 2,\nCσ−1\nεη2−p\np|lnη|−1/2if2<p<∞,(5.7)\nwhere the constant Cdoes not depend on εorη.\nProof.Note that\nˆ\nεQz|∇χε,η|p≤Cε2−p|lnη|−p+ˆ\nB(εyz,ε/6)\\εTz,η|∇χε,η|pdx\n=Cε2−p|lnη|−p+ˆ\nB(εyz,ε/6)\\εTz,η/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggC\n|lnη||x−εyz|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\ndx\n=Cε2−p|lnη|−p/braceleftbigg\n1+ˆ1\ncηr1−pdr/bracerightbigg\n,\n15where we have used (5.5) and (5.2). It follows that\n/parenleftbigg \nεQz|∇χε,η|p/parenrightbigg1/p\n≤\n\nCε−1|lnη|−1if 1<p<2,\nCε−1|lnη|−1\n2ifp= 2,\nCε−1η2−p\np|lnη|−1if 2<p<∞,\nwhich gives (5.7)\nLemma 5.3. SupposeεQz⊂Ω. Then\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nεQz∇χε,η·∇φ−2πσ−2\nεˆ\nεQzφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cε−1|lnη|−1ˆ\nεQz|∇φ|, (5.8)\nwhereφ∈H1(εQz)andφ= 0inεTz,η.\nProof.The proof is similar to that of Lemma 4.4. Note that\nˆ\nεQz∇χε,η·∇φ−2πσ−2\nεˆ\nεQzφ\n=ˆ\nB(εyz,ε/5)\\B(εyz,ε/6)∇χε,η·∇φ+ˆ\nB(εyz,ε/6)\\εTz,η∇χε,η·∇φ−2π\n|lnη| \nεQzφ\n=ˆ\nB(εyz,ε/5)\\B(εyz,ε/6)∇χε,η·∇φ+/braceleftBiggˆ\n∂B(εyz,ε/6)∂χε,η\n∂nφ−2π\n|lnη| \nεQzφ/bracerightBigg\n=I1+I2,(5.9)\nwhere we have used integration by parts and the fact that χε,ηis harmonic in B(εyz,ε/6)\\εTz,η.\nBy (5.5), it is easy to see that |I1|is bounded by the right-hand side of (5.8).\nTo boundI2, we write it as\nI2=ˆ\n∂B(εyz,ε/6)/parenleftBigg\n∂χε,η\n∂n− \n∂B(εyz,ε/6)∂χε,η\n∂n/parenrightBigg\n·(φ−α)\n+/braceleftBiggˆ\n∂B(εyz,ε/6)∂χε,η\n∂n \n∂B(εyz,ε/6)φ−2π\n|lnη| \nεQzφ/bracerightBigg\n=I21+I22,(5.10)\nwhereαis a constant to be determined. Using (5.2), one may show that\n|I21| ≤C\nε|lnη|ˆ\n∂B(εyz,ε/6)|φ−α|\n≤C\nε|lnη|ˆ\nεQz|∇φ|,(5.11)\nwhere we have choose αto be the average of φoverεQz, as in the proof of Lemma 4.4.\nFinally, to handle I22, we note that by (5.2),\nˆ\n∂B(εyz,ε/6)∂χε,η\n∂n=1\n|lnη|{2π+O(η)}.\n16This, together with (4.21), shows that\n|I22| ≤C\nε|lnη|ˆ\nεQz|∇φ|+Cη\nε|lnη|ˆ\nεQz|φ|\n≤CC\nε|lnη|ˆ\nεQz|∇φ|,\nwhere we have used Lemma 2.1 with p= 1 for the last inequality. This completes the proof.\n6 Convergence Rates\nIn this section we study the Dirichlet problem with nonhomog eneous boundary conditions,\n\n\n−∆uε= 0 in Ω ε,η,\nuε=hon∂Ω,\nuε= 0 on ∂Ωε,η\\∂Ω,(6.1)\nwhere Ω ε,ηis given in (1.5) and h∈H1/2(∂Ω). LetVε(x) =V(x/ε), where\nV(y) =/summationdisplay\nz∈Zdµz\n∗χQ(z,1)(y), (6.2)\nandµz\n∗is given by (4.5) for d≥3 andµz\n∗= 2πford= 2. In (6.2) we have used χQ(z,1)to denote\nthe characteristic function of the cell Q(z,1). Note that c0≤V(y)≤c1for somec0,c1>0. Let\nχε,ηbe the corrector constructed in the last two sections for the domain Ω ε,η. We will show that\nthe solution uεof (6.1) is well approximated by χε,ηu0,ε, whereu0,εis the solution of a boundary\nvalue problem in Ω,/braceleftBigg\n−∆u0,ε+σ−2\nεVε(x)u0,ε= 0 in Ω ,\nu0,ε=hon∂Ω,(6.3)\nwhichwewill call theintermediate problem, for aSchr¨ odin geroperator −∆+σ−2\nεVε. Theparameter\nσε, which depends on εandη, is given by (1.6). The rest of this section will be divided in to two\nparts. The first part deals with the case d≥3, while the second treats the case d= 2.\n6.1 Dimension d≥3\nThe goal of this subsection is to prove the following.\nTheorem 6.1. Letuεbe a solution of (6.1)andu0,εa solution of (6.3). Let\nrε=uε−χε,ηu0,ε, (6.4)\nwhereχε,ηis the corrector constructed in Section 4. Assume that u0,ε∈W1,p(Ω)for some 2<p<\n∞. Then for d≥3,\n/ba∇dbl∇rε/ba∇dblL2(Ωε,η)≤/braceleftBigg\nCηd−2\n2/parenleftbig\nσ−1\nε/ba∇dblu0,ε/ba∇dblLp(Ω)+/ba∇dbl∇u0,ε/ba∇dblLp(Ω)/parenrightbig\nifp≥d,\nCηd/q/parenleftbig\nσ−1\nε/ba∇dblu0,ε/ba∇dblLp(Ω)+/ba∇dbl∇u0,ε/ba∇dblLp(Ω)/parenrightbig\nif2<p<d,(6.5)\nwhere1\n2=1\np+1\nqandCdoes not depend on εorη.\n17Note thatχε,η∈W1,p(Ω) for any p>2. Moreover, χε,η= 1 on∂Ω andχε,η= 0 on∂Ωε,η\\∂Ω.\nIt follows that rε∈H1\n0(Ωε,η). The proof of Theorem 6.1 will rely heavily on the estimates forχε,η\nin Section 4.\nLemma 6.2. Letφ∈H1\n0(Ωε,η). Then\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩε,η∇χε,η·∇φ−ˆ\nΩε,ησ−2\nεVεφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cε−1ηd−2ˆ\nΩε,η|∇φ|, (6.6)\nwhereVε=V(x/ε)andV(y)is defined in (6.2).\nProof.We extend φtoRdby zero. Note that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩε,η∇χε,η·∇φ−ˆ\nΩε,ησ−2\nεVεφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/summationdisplay\nz∈Ω′ε/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nεQz∇χε,η·∇φ−ˆ\nεQzσ−2\nεVεφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nFσ−2\nεVεφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle,(6.7)\nwhere Ω′\nεis the union of cubes εQzlying entirely inside of Ω and Fis the union of cubes with\nnon-empty intersection with the boundary of Ω. By Lemma 4.4, the first term in the right-hand\nside of (6.7) is bounded by the right-hand side of (6.6). For t he second term we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nFσ−2\nεVεφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cσ−2\nεˆ\ndist(x,∂Ω)≤Cε|φ|\n≤Cεσ−2\nεˆ\ndist(x,∂Ω)≤Cε|∇φ| ≤Cε−1ηd−2ˆ\nΩε,η|∇φ|,\nwhere we have used a Poincar´ e-Sobolev inequality for the se cond inequality.\nProof of Theorem 6.1. Letuεbe a solution of (6.1) and u0,εa solution of (6.3). Assume further\nthatu0,ε∈W1,p(Ω) for some p>2. Letrε=uε−χε,ηu0,ε. Note that for any φ∈H1\n0(Ωε,η),\nˆ\nΩε,η∇rε·∇φ=−ˆ\nΩε,η(∇χε,η·∇φ)u0,ε−ˆ\nΩε,ηχε,η(∇u0,ε·∇φ)\n=−ˆ\nΩε,η∇χε,η·∇(u0,εφ)+ˆ\nΩε,η[∇(χε,η−1)·∇u0,ε]φ\n−ˆ\nΩε,η(χε,η−1)(∇u0,ε·∇φ)−ˆ\nΩε,η(∇u0,ε·∇φ).(6.8)\nBy using integration by parts, the right-hand side of (6.8) b ecomes\n−ˆ\nΩε,η∇χε,η·∇(u0,εφ)−2ˆ\nΩε,η(χε,η−1)(∇u0,ε·∇φ)\n−ˆ\nΩε,η(χε,η−1)∆u0,ε·φ+ˆ\nΩε,η(∆u0,ε·φ),\nwhich, in view of (6.3), equals to\n−ˆ\nΩε,η∇χε,η·∇(u0,εφ)+σ−2\nεˆ\nΩε,ηVεu0,εφ−2ˆ\nΩε,η(χε,η−1)(∇u0,ε·∇φ)\n−σ−2\nεˆ\nΩε,η(χε,η−1)Vεu0,ε·φ.\n18Note thatVε(x)≤c2. We now apply Lemma 6.2 to obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩε,η∇rε·∇φ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cε−1ηd−2ˆ\nΩε,η|∇(u0,εφ)|+2ˆ\nΩε,η|χε,η−1||∇u0,ε||∇φ|\n+Cσ−2\nεˆ\nΩε,η|χε,η−1||u0,ε||φ|\n≤Cε−1ηd−2ˆ\nΩε,η|∇u0,ε||φ|+Cε−1ηd−2ˆ\nΩε,η|∇φ||u0,ε|\n+2ˆ\nΩε,η|χε,η−1||∇u0,ε||∇φ|+Cσ−2\nεˆ\nΩε,η|χε,η−1||u0,ε||φ|.\nApplying H¨ older’s inequality yields\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩε,η∇rε·∇φ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cσ−1\nεηd−2\n2/ba∇dbl∇φ/ba∇dblL2(Ωε,η)||u0,ε/ba∇dblL2(Ωε,η)\n+Cσ−1\nεηd−2\n2/ba∇dblφ/ba∇dblL2(Ωε,η)/ba∇dbl∇u0,ǫ/ba∇dblL2(Ωε,η)\n+C/ba∇dblχε,η−1/ba∇dblLq(Ωε,η)/ba∇dbl∇φ/ba∇dblL2(Ωε,η)/ba∇dbl∇u0,ε/ba∇dblLp(Ωε,η)\n+Cσ−2\nε/ba∇dblχε,η−1/ba∇dblLq(Ωε,η)/ba∇dblφ/ba∇dblL2(Ωε,η)/ba∇dblu0,ε/ba∇dblLp(Ωε,η),(6.9)\nwhere1\n2=1\nq+1\np. By applying the Poincar´ e inequality /ba∇dblφ/ba∇dblL2(Ωε,η)≤Cσε/ba∇dbl∇φ/ba∇dblL2(Ωε,η)in (6.9) we\nhave\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩε,η∇rε·∇φ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cηd−2\n2/ba∇dbl∇φ/ba∇dblL2(Ωε,η)/parenleftbig\nσ−1\nε/ba∇dblu0,ε/ba∇dblL2(Ω)+/ba∇dbl∇u0,ε/ba∇dblL2(Ω)/parenrightbig\n+C/ba∇dblχε,η−1/ba∇dblLq(Ωε,η)||∇φ/ba∇dblL2(Ωε,η)/parenleftbig\n/ba∇dbl∇u0,ε/ba∇dblLp(Ω)+σ−1\nε/ba∇dblu0,ε/ba∇dblLp(Ω)/parenrightbig\n.\nChoosingφ=rε, we obtain\n/ba∇dbl∇rε/ba∇dblL2(Ωε,η)≤C/parenleftBig\nηd−2\n2+/ba∇dblχε,η−1/ba∇dblLq(Ωε,η)/parenrightBig/parenleftbig\n/ba∇dbl∇u0,ε||Lp(Ω)+σ−1\nε/ba∇dblu0,ε/ba∇dblLp(Ω)/parenrightbig\n.\nApplying Lemma 4.1 gives (6.5).\n6.2 Dimension d= 2\nWe now consider the case d= 2.\nTheorem 6.3. Letuεbe a solution of (6.1)amdu0,εa solution of (6.3)Assume that u0,ε∈\nW1,p(Ω)for some 2<p<∞. Letrε=uε−χε,ηu0,ε. Then\n/ba∇dbl∇rε/ba∇dblL2(Ωε,η)≤C|lnη|−1/2/parenleftbig\nσ−1\nε/ba∇dblu0,ε/ba∇dblLp(Ω)+/ba∇dbl∇u0,ε/ba∇dblLp(Ω)/parenrightbig\n, (6.10)\nwhereCdoes not depend on εorη.\nThe following lemma plays the role of Lemma 6.2.\nLemma 6.4. Letχε,ηbe the corrector constructed in Section 5. Then, for any φ∈H1\n0(Ωε,η),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩε,η∇χε,η·∇φ−2πσ−2\nεˆ\nΩε,ηφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C\nε|lnη|ˆ\nΩε,η|∇φ|. (6.11)\n19Proof.The proof is similar to that of Lemma 6.2. Let φ∈H1\n0(Ωε,η) and extend it to R2by zero.\nThe left-hand side of (6.11) is bounded by\n/summationdisplay\nz/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nεQz∇χε,η·∇φ−2πσ−2\nεˆ\nεQzφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nF2πσ−2\nεφ/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (6.12)\nwhere the sum is taken over those z’s for which εQz⊂Ω andFis the union of cubes εQzwith\nnon-empty intersections with ∂Ω. As in the proof of Lemma 6.2, the second term in (6.12) can be\nhandled readily by a Poincar´ e inequality, since φ= 0 on∂Ω. For the first term we use Lemma 5.3\nto bound it by the right-hand side of (6.11).\nProof of Theorem 6.3. Letuεbe a solution of (6.1) and uε,0a solution of (6.3). Assume further\nthatu0,ε∈W1,p(Ω). Letrε=uε−χε,ηu0,ε. As in the proof of Theorem 6.1, for φ∈H1\n0(Ωε,η), we\nhaveˆ\nΩε,η∇rε·∇φ=−ˆ\nΩε,η∇χε,η·∇(u0,εφ)+2πσ−2\nεˆ\nΩε,ηu0,εφ\n−2ˆ\nΩε,η(χε,η−1)(∇u0,ε·∇φ)−2πσ−2\nεˆ\nΩε,η(χε,η−1)u0,εφ.\nBy applying Lemma 6.4 we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩε,η∇rε·∇φ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C\nε|lnη|ˆ\nΩε,η|∇(u0,εφ)|+2ˆ\nΩε,η|χε,η−1||∇u0,ε||∇φ|\n+2πσ−2\nεˆ\nΩε,η|χε,η−1||u0,ε||φ|\n=I.\nNext, by H¨ older’s inequality,\nI≤Cσ−1\nε|lnη|−1\n2/parenleftbig\n/ba∇dbl∇φ/ba∇dblL2(Ωε,η)/ba∇dblu0,ε/ba∇dblL2(Ωε,η)+/ba∇dblφ/ba∇dblL2(Ωε,η)/ba∇dbl∇u0,ε/ba∇dblL2(Ωε,η)/parenrightbig\n+C/ba∇dblχε,η−1/ba∇dblLq(Ωε,η)/parenleftbig\n/ba∇dbl∇φ/ba∇dblL2(Ωε,η)/ba∇dbl∇u0,ε/ba∇dblLp(Ωε,η)+σ−2\nε/ba∇dblφ/ba∇dblL2(Ωε,η)/ba∇dblu0,ε/ba∇dblLp(Ωε,η)/parenrightbig\n,\nwhere1\n2=1\np+1\nq. By applying the Poincar´ e inequality /ba∇dblφ/ba∇dblL2(Ωε,η)≤Cσε/ba∇dbl∇φ/ba∇dblL2(Ωε,η)and Lemma\n5.1, we obtain\nI≤C|lnη|−1\n2/ba∇dbl∇φ/ba∇dblL2(Ωε,η)/parenleftbig\n/ba∇dbl∇u0,ε/ba∇dblLp(Ω)+σ−1\nε/ba∇dblu0,ε/ba∇dblLp(Ω)/parenrightbig\n.\nBy choosing φ=rε, we obtain (6.10).\n7 An Intermediate Problem\nIn this section we consider the boundary value problem for a S chr¨ odinger operator,\n/braceleftBigg\n−∆u+λ2V(x)u=Fin Ω,\nu=gon∂Ω,(7.1)\nwhereλ>0 andV=V(x) is a potential satisfying the condition 0 <µ0≤V≤µ1.\n20Lemma 7.1. LetΩbe abounded Lipschitz domain in Rdandλ>0. Suppose that u∈H1(B(x0,2r)∩\nΩ)and/braceleftBigg\n−∆u+λ2V(x)u= 0 inB(x0,2r)∩Ω,\nu= 0 onB(x0,2r)∩∂Ω,(7.2)\nwherex0∈∂Ωand0<r<r 0. Then\nsup\nB(x0,r)∩Ω|u| ≤C/parenleftBigg \nB(x0,2r)∩Ω|u|2/parenrightBigg1/2\n, (7.3)\nwhereCdepends on d,Ωand(µ0,µ1). Moreover,\n/parenleftBigg \nB(x0,r)∩Ω|∇u|p/parenrightBigg1/p\n≤C/parenleftBigg \nB(x0,2r)∩Ω|∇u|2/parenrightBigg1/2\n, (7.4)\nwhere2< p <3 +δford≥3,2< p <4 +δford= 2, andδ >0depends on dandΩ. The\nconstantCin(7.4)depends on d,p,Ωand(µ0,µ1). IfΩis a bounded C1domain, the estimate\n(7.4)holds for any 2<p<∞.\nProof.Letφ∈C∞\n0(B(x0,2r)). Then\nˆ\nB(x0,2r)∩Ω∇u·∇(uφ2)+ˆ\nB(x0,2r)∩Ωλ2Vu·uφ2= 0.\nIt follows by Cauchy’s inequality that\nˆ\nB(x0,2r)∩Ω|∇u|2φ2+ˆ\nB(x0,2r)∩Ωλ2Vu2φ2≤Cˆ\nB(x0,2r)∩Ω|u|2|∇φ|2.\nBy choosing a suitable cut-off function φ, we see that\nλ2ˆ\nB(x0,tr)∩Ωu2≤C\nr2(s−t)2ˆ\nB(x0,sr)∩Ωu2\nfor 1≤t<s≤2. This implies that\nˆ\nB(x0,tr)∩Ωu2≤C\n1+(s−t)2λ2r2ˆ\nB(x0,sr)∩Ωu2. (7.5)\nBy iterating (7.5) ktimes we obtain\n/parenleftBigg \nB(x0,tr)∩Ωu2/parenrightBigg1/2\n≤Ck,t,s\n(1+λr)2k/parenleftBiggˆ\nB(x0,sr)∩Ωu2/parenrightBigg1/2\n(7.6)\nfor anyk≥1, where 1 ≤t<s≤2.\nNow, since\n∆u=λ2VuinB(x0,2r)∩Ω and u= 0 onB(x0,2r)∩∂Ω,\nthe boundary L∞estimates for Laplace’s equation in Lipschitz domains give\n/parenleftBigg \nB(x0,tr)∩Ω|u|p/parenrightBigg1/p\n≤Ct,s/parenleftBigg \nB(x0,sr)∩Ω|u|2/parenrightBigg1/2\n+Ct,sr2λ2/parenleftBigg \nB(x0,sr)∩Ω|u|q/parenrightBigg1/q\n,\n21where 0<1\nq−1\np<2\nd, and\nsup\nB(x0,tr)∩Ω|u| ≤Ct,s/parenleftBigg \nB(x0,sr)∩Ω|u|2/parenrightBigg1/2\n+Ct,sr2λ2/parenleftBigg \nB(x0,sr)∩Ω|u|q/parenrightBigg1/q\n,\nwhereq>(d/2). Thesetwo estimates, together with(7.6), yield(7.3)by abootstrappingargument.\nNote that the W1,pestimate for Laplace’s equation in C1domains [6] gives\n/parenleftBigg \nB(x0,ts)∩Ω|∇u|p/parenrightBigg1/p\n≤Ct,s/parenleftBigg \nB(x0,sr)∩Ω|∇u|2/parenrightBigg1/2\n+Ct,srλ2/parenleftBigg \nB(x0,sr)∩Ω|u|q/parenrightBigg1/q\n,\nwhere 1< t < s < 2 and 0<1\nq−1\np<1\nd. If Ω is a Lipschitz domain, we need to impose the\nadditional conditions that 2 < p <3 +δford≥3, and 2< p <4 +δford= 2, where δ >0\ndepends on dand Ω. Thus, by (7.6) and (7.3),\n/parenleftBigg \nB(x0,r)∩Ω|∇u|p/parenrightBigg1/p\n≤C/parenleftBigg \nB(x0,2r)∩Ω|∇u|2/parenrightBigg1/2\n+Cr−1/parenleftBigg \nB(x0,2r)∩Ω|u|2/parenrightBigg1/2\n≤C/parenleftBigg \nB(x0,2r)∩Ω|∇u|2/parenrightBigg1/2\n,(7.7)\nwhere we have used a Poincar´ e inequality and the fact that u= 0 onB(x0,2r)∩∂Ω for the last\nstep.\nRemark 7.2. Suppose that u∈H1(B(x0,2r)) and−∆u+λ2Vu= 0 inB(x0,2r). Then\nsup\nB(x0,r)|u| ≤C/parenleftBigg \nB(x0,2r)|u|2/parenrightBigg1/2\n, (7.8)\nsup\nB(x0,r)|∇u| ≤C/parenleftBigg \nB(x0,2r)(|∇u|2+λ2|u|2)/parenrightBigg1/2\n, (7.9)\nwhereCdepends on dand (µ0,µ1). The proof is similar to that of Lemma 7.1.\nAn operator Tis called sublinear if there exists K≥1 such that\n|T(f+g)| ≤K(|T(f)|+|T(g)|). (7.10)\nThe following theorem was proved in [10].\nTheorem 7.3. LetΩbe a bounded Lipschitz domain in Rd. LetTbe a bounded sublinear operator\nonL2(Ω)with\n/ba∇dblT/ba∇dblL2→L2≤C0. (7.11)\nLetq>2. Suppose that\n/parenleftBigg \nΩ∩B(x0,r)|T(f)|q/parenrightBigg1/q\n≤N/parenleftBigg \nΩ∩B(x0,2r)|T(f)|2/parenrightBigg1/2\n(7.12)\n22for any ball B(x0,r)with the property that 0< r < r 0and eitherB(x0,4r)⊂Ωorx0∈∂Ωand\nfor anyf∈C∞\n0(Ω)with supp (f)⊂Ω\\B(x0,4r). Then for any F∈Lp(Ω),\n/ba∇dblT(F)/ba∇dblLp(Ω)≤C/ba∇dblF/ba∇dblLp(Ω),\nwhere2<p<qandCdepends at most on d,p,q,C 0,N,r0,ΩandK.\nTheorem 7.4. LetΩbe a bounded Lipschitz domain in Rd. Letu∈W1,2\n0(Ω)be a solution of\n/braceleftBigg\n−∆u+λ2V(x)u=FinΩ,\nu= 0 on∂Ω,(7.13)\nwhereF∈L2(Ω). SupposeF∈Lp(Ω), where2<p<3+δford≥3, and2<p<4+δford= 2.\nThen\nλ/ba∇dbl∇u/ba∇dblLp(Ω)+λ2/ba∇dblu/ba∇dblLp(Ω)≤C/ba∇dblF/ba∇dblLp(Ω), (7.14)\nwhereCdepends on d,p,Ωand(µ0,µ1). IfΩis a bounded C1domain, the estimate (7.14)holds\nfor any2<p<∞.\nProof.To prove (7.14), we consider the sublinear operator,\nTλ(F) =λ|∇u|+λ2|u|,\nwhereuis a solution of (7.13), and apply Theorem 7.3. Note that by th e energy estimate,\n/ba∇dblTλ/ba∇dblL2(Ω)→L2(Ω)≤C, whereCdepends on d, Ω and (µ0,µ1). To verify the condition (7.12), let\nf∈C∞\n0(Ω) such that f= 0 inB(x0,2r), where 0<r <r 0and eitherx0∈∂Ω orB(x0,4r)⊂Ω.\nLetwbe the solution of (7.13) with fin the place of F. Then−∆w+λ2Vw= 0 inB(x0,4r)∩Ω\nandw= 0 onB(x0,4r)∩∂Ω (ifx0∈∂Ω). If Ω is a bounded C1domain, it follows from Lemma\n7.1 and Remark 7.2 that\n/parenleftBigg \nB(x0,r)∩Ω(λ|∇w|+λ2|w|)q/parenrightBigg1/q\n≤C/parenleftBigg \nB(x0,2r)∩Ω(λ|∇w|+λ2|w|)2/parenrightBigg1/2\n(7.15)\nfor any 2< q <∞, whereCdepends on d,q, Ω, and (µ0,µ1). As a result, by Theorem 7.3, we\nobtain/ba∇dblTλ(F)/ba∇dblLp(Ω)≤C/ba∇dblF/ba∇dblLp(Ω)for any 2< p <∞. If Ω is a bounded Lipschitz domain, the\nestimate (7.15) holds for 2 < q <3+δifd≥3, and for 2 < q <4+δifd= 2. As a result, we\nobtain (7.14) for 2 <p<3+δifd≥3, and for 2 <p<4+δifd= 2, where δ >0 depends on d\nand Ω.\nTheorem 7.5. LetΩbe a bounded Lipschitz domain in Rdwith connected boundary. Let u∈\nW1,2(Ω)be a solution of/braceleftBigg\n−∆u+λ2V(x)u= 0 inΩ,\nu=gon∂Ω,\nwhereg∈W1,2(∂Ω). Then\n/braceleftBigg\n/ba∇dblu/ba∇dblLp(Ω)≤C/ba∇dblg/ba∇dblL2(∂Ω),\n/ba∇dbl∇u/ba∇dblLp(Ω)≤C/braceleftbig\n/ba∇dbl∇tg/ba∇dblL2(∂Ω)+λ/ba∇dblg/ba∇dblL2(∂Ω)/bracerightbig\n,(7.16)\nwherep=2d\nd−1,∇tgdenotes the tangential gradient of gon∂Ω, andCdepends on d,Ωand(µ0,µ1).\n23Proof.We start by solving the Dirichlet problem,\n/braceleftBigg\n−∆G= 0 in Ω ,\nG=gon∂Ω.\nBy the well known nontangential-maximal-function estimat es for harmonic functions in Lipschitz\ndomains (see e.g. [15]),\n/ba∇dbl(∇G)∗||L2(∂Ω)≤C/ba∇dbl∇tg/ba∇dblL2(∂Ω)and/ba∇dbl(G)∗/ba∇dblL2(Ω)≤ /ba∇dblg/ba∇dblL2(∂Ω),\nwhere∇tgdenotes the tangential gradient of gon∂Ω. It follows that\n/ba∇dbl∇G/ba∇dblLp(Ω)≤C/ba∇dbl(∇G)∗/ba∇dblL2(∂Ω)≤C/ba∇dbl∇tg/ba∇dblL2(∂Ω),\n/ba∇dblG/ba∇dblLp(Ω)≤C/ba∇dbl(G)∗/ba∇dblL2(∂Ω)≤C/ba∇dblg/ba∇dblL2(∂Ω),\nwherep=2d\nd−1and we have used the inequality /ba∇dblw/ba∇dblLp(Ω)≤C/ba∇dbl(w)∗/ba∇dblL2(∂Ω)for functions win Ω.\nNext, letv=u−G. Thenvsatisfies the following equation,\n/braceleftBigg\n−∆v+λ2V(x)v=Fin Ω,\nv= 0 on ∂Ω,\nwhereF=−λ2V(x)G. Letp=2d\nd−1. Note that p= 4 ford= 2,p= 3 ford= 3, andp <3 for\nd≥4. In view of Theorem 7.4 we have\nλ2/ba∇dblu−G/ba∇dblLp(Ω)≤C/ba∇dblF/ba∇dblLp(Ω)≤Cλ2/ba∇dblG/ba∇dblLp(Ω).\nThis implies\n/ba∇dblu/ba∇dblLp(Ω)≤C/ba∇dblG/ba∇dblLp(Ω)≤C/ba∇dblg/ba∇dblL2(∂Ω) (7.17)\nforp=2d\nd−1. Moreover, by Theorem 7.4,\nλ/ba∇dbl∇(u−G)/ba∇dblLp(Ω)≤C/ba∇dblλ2VG/ba∇dblLp(Ω).\nIt follows that\n/ba∇dbl∇u/ba∇dblLp(Ω)≤ /ba∇dbl∇G/ba∇dblLp(Ω)+Cλ/ba∇dblG/ba∇dblLp(Ω)\n≤C/ba∇dbl∇tg/ba∇dblL2(∂Ω)+Cλ/ba∇dblg/ba∇dblL2(∂Ω)(7.18)\nforp=2d\nd−1.\n8 Large-scale W1,pestimates\nLetuεbe a solution of /braceleftBigg\n−∆uε=F+div(f) in Ω ε,η,\nuε= 0 on ∂Ωε,η,(8.1)\nwhere Ω ε,ηis given by (1.5). Let\nSε(F,f) =/parenleftBigg \nQ(x,2ε)|∇uε|2/parenrightBigg1/2\n,\n24where we have extended uεtoRdby zero. Note that\n/ba∇dblSε(F,f)/ba∇dblL2(Rd)=/ba∇dbl∇uε/ba∇dblL2(Ωε,η).\nIt follows that\n/ba∇dblSε(F,f)/ba∇dblL2(Rd)≤ /ba∇dblf/ba∇dblL2(Ωε,η)+Cmin(σε,1)/ba∇dblF/ba∇dblL2(Ωε,η). (8.2)\nThis section is dedicated to the Lpestimates for Sε(F,f) in Theorem 1.3.\nTheorem 8.1. LetΩbe a bounded C1domain in RdandΩε,ηbe given by (1.5). Then,\n/ba∇dblSε(0,f)/ba∇dblLp(Ω)≤C/ba∇dblf/ba∇dblLp(Ω) (8.3)\nfor2<p<∞, whereCdepends only on d,p,Ω, and{Ys\nz}.\nIn the following we will assume 0 <η<η 0, whereη0>0 is sufficiently small. The case η≥η0\nwith any fixed η0>0 is the same as the case η= 1, which is contained in [9]. To prove Theorem\n8.1, we use Theorem 7.3. The L2boundedness condition (7.11) is given by (8.2). To verify th e\nreverse H¨ older condition (7.12), it suffices to show that if ∆ uε= 0 inQ(x0,4R)∩Ωε,ηanduε= 0\nonQ(x0,4R)∩∂Ωε,η, where 0<R<c 0and eitherQ(x0,4R)⊂Ω orx0∈∂Ω, then\n\n \nQ(x0,R)∩Ω/parenleftBigg \nQ(x,2ε)|∇uε|2/parenrightBiggp/2\ndx\n1/p\n≤C/parenleftBigg \nQ(x0,2R)∩Ω/parenleftBigg \nQ(x,2ε)|∇uε|2/parenrightBigg\ndx/parenrightBigg1/2\n(8.4)\nforp >2. It is not hard to see that (8.4) holds for 0 < R < Cε . The proof of (8.4) for the\nlarge-scale case R≥Cεuses a real-variable argument and relies on the estimates fr om the previous\ntwo sections.\nWe begin with the interior case Q(x0,4R)⊂Ω. The following theorem was proved in [11,\nTheorem 3.2].\nTheorem 8.2. LetF∈L2(Q(x0,2R))and2< p < q. Suppose that for each cube Q=Q(y,r)\nwithy∈Q(x0,R)and0<r<c 0R, there exists two functions RQandFQsuch that\n|F| ≤ |RQ|+|FQ|in2Q, (8.5)\n/parenleftbigg \n2Q|RQ|q/parenrightbigg1/q\n≤N/parenleftbigg \n8Q|F|2/parenrightbigg1/2\n, (8.6)\n/parenleftbigg \n2Q|FQ|2/parenrightbigg1/2\n≤δ/parenleftbigg \n8Q|F|2/parenrightbigg1/2\n, (8.7)\nwhereN >1and0< c0<1. Then there exists δ0>0, depending only on d,N,c0,p,q, with the\nproperty that if 0≤δ<δ0, thenF∈Lp(Q(x0,R))and\n/parenleftBigg \nQ(x0,R)|F|p/parenrightBigg1/p\n≤C/parenleftBigg \nQ(x0,2R)|F|2/parenrightBigg1/2\n, (8.8)\nwhereCdepends at most on d,c0,p,q,N.\nWeuseQε,η(x0,r)todenoteΩ ε,η, whereΩ = Q(x0,r). Wepointoutthatingeneral, Qε,η(x0,r)/\\e}atio\\slash=\nQ(x0,r)∩Ωε,η, as there is no hole near the boundary of Q(x0,r) inQε,η(x0,r). However, under\nthe assumption r≥Cε, it is possible to find Q(y0,t) such that Q(x0,r)⊂Q(y0,t)⊂Q(x0,2r) and\nQε,η(y0,t) = Ωε,η∩Q(y0,t). This observation is used in the proof of the following lemm a.\n25Lemma 8.3. Letu∈H1(Q(x0,4r)), wherer≥8εandQ(x0,8r)⊂Ω. Suppose that ∆u= 0in\nQ(x0,4r)∩Ωε,ηandu= 0onQ(x0,4r)\\Ωε,η. Then there exists v∈H1(Q(x0,2r))such that\n/parenleftBigg \nQ(x0,r)|∇(u−v)|2/parenrightBigg1/2\n≤Cψ(η)/parenleftBigg \nQ(x0,3r)|∇u|2/parenrightBigg1/2\n, (8.9)\nmax\nx∈Q(x0,r)/parenleftBigg \nQ(x,2ε)|∇v|2/parenrightBigg1/2\n≤C/parenleftBigg \nQ(x0,3r)|∇u|2/parenrightBigg1/2\n, (8.10)\nwhereψ(η) =η1/2ford≥3andψ(η) =|lnη|−1/2ford= 2. The constant Cdepends only on d\nand{Ys\nz}.\nProof.Sincer≥8ε, without loss of generality, we may assume that x0= 0 andr= 2jεfor some\nj≥2. By dilation we may also assume r= 1. Note that the parameter ηremains invariant under\ndilation. Since u= 0 onQ(0,3)\\Ωε,η, it follows by Lemma 2.1 that\nσ−2\nεˆ\nQ(0,3)|u|2≤Cˆ\nQ(0,3)|∇u|2. (8.11)\nWe claim that there exists t∈[2,3] such that Qε,η(0,t) =Q(0,t)∩Ωε,ηand\nσ−2\nεˆ\n∂Q(0,t)|u|2+ˆ\n∂Q(0,t)|∇u|2≤Cˆ\nQ(0,3)|∇u|2. (8.12)\nTo see (8.12), we consider the set E={t∈[2,3] :|t−kε| ≤c0εfor some 1 ≤k≤2j+2}. Note that\n|E| ≥c>0 andQε,η(0,t) =Q(0,t)∩Ωε,ηfort∈E. If (8.12) fails for all t∈E, we may integrate\nthe left-hand side of (8.12) with respect to toverEand use (8.11) to obtain a contradiction.\nNext, we let wbe the solution of\n/braceleftBigg\n−∆w+σ−2\nεV(x/ε)w= 0 in Q(0,t),\nw=uon∂Q(0,t),(8.13)\nwhereV(x) is given by (6.2). It follows by Theorems 6.1 and 6.3 that\n/ba∇dbl∇(u−χε,ηw)/ba∇dblL2(Q(0,t))≤Cψ(η)/braceleftbig\nσ−1\nε/ba∇dblw/ba∇dblLp(Q(0,t))+/ba∇dbl∇w/ba∇dblLp(Q(0,t))/bracerightbig\n,(8.14)\nwherep=2d\nd−1andχε,ηis the corrector for the domain Qε,η(0,t). This, together with Theorem\n7.5, yields\n/ba∇dbl∇(u−χε,ηw)/ba∇dblL2(Q(0,t))≤Cψ(η)/braceleftbig\nσ−1\nε/ba∇dblu/ba∇dblL2(∂Q(0,t))+/ba∇dbl∇u/ba∇dblL2(∂Q(0,t))/bracerightbig\n≤Cψ(η)/ba∇dbl∇u/ba∇dblL2(Q(0,3)),(8.15)\nwhere we have used (8.12) for the last step.\nFinally, we let v=χε,ηw. Note that the estimate (8.9) is given by (8.15). To see (8.10 ), we\nnote that\n/parenleftBigg \nQ(x,2ε)|∇v|2/parenrightBigg1/2\n≤max\nQ(x,2ε)|w|/parenleftBigg \nQ(x,2ε)|∇χε,η|2/parenrightBigg1/2\n+ max\nQ(x,2ε)|χε,η||∇w|\n≤Cmax\nQ(x,2ε)(σ−1\nε|w|+|∇w|),\n26where we have used Lemmas 4.3 and 5.2. This, together with Rem ark 7.2, gives\n/parenleftBigg \nQ(x,2ε)|∇v|2/parenrightBigg1/2\n≤Cmax\nQ(0,3/2)(σ−1\nε|w|+|∇w|)\n≤C/braceleftbig\nσ−1\nε/ba∇dblw/ba∇dblL2(Q(0,2))+/ba∇dbl∇w/ba∇dblL2(Q(0,2))/bracerightbig\n≤C/ba∇dbl∇u/ba∇dblL2(Q(0,3))\nfor anyx∈Q(0,1).\nLemma 8.4. Letu∈H1(B(x0,4R)), whereR≥8εandB(x0,8R)⊂Ω. Suppose that ∆u= 0in\nQ(x0,4R)∩Ωε,ηandu= 0inQ(x0,4R)\\Ωε,η. Let\nv(x) =/parenleftBigg \nQ(x,2ε)|∇u|2/parenrightBigg1/2\n. (8.16)\nThen for 2<p<∞,/parenleftBigg \nQ(x0,R)|v|p/parenrightBigg1/p\n≤C/parenleftBigg \nQ(x0,2R)|v|2/parenrightBigg1/2\n, (8.17)\nwhereCdepends only on d,pand{Ys\nz}.\nProof.Without loss of generality we may assume x0= 0. By dilation we may assume R= 1.\nTo prove (8.17), we apply Theorem 8.2 with F=v. LetQ=Q(y0,r), wherey0∈Q(0,1) and\n0<r<(1/8). If 0<r<8ε, we letRQ=vandFQ= 0. Note that\nmax\n2QRQ≤C/parenleftBigg \nQ(y0,2r+2ε)|∇u|2/parenrightBigg1/2\n≤C/parenleftbigg \n4Q|v|2/parenrightbigg1/2\n.\nIfr≥8ε, we letRQ=|∇v|andFQ=|∇(u−v)|, wherevis given by Lemma 8.3. In view of (8.9)\nand (8.10), we obtain (8.6) and (8.7) with δ=Cψ(η). It follows by Theorem 8.2 that the estimate\n(8.17) holds if η<η0, whereη0>0 depends only on d,pand{Ys\nz}.\nNext, we treat the boundary case where x0∈∂Ω. Since Ω is Lipschitz, there exists r0>0 such\nthatB(x0,r0)∩Ω =B(x0,r0)∩DandB(x0,r0)∩∂Ω =B(x0,r0)∩∂D, where, after a rotation of\nthe coordinate system, Dis given by\nD=/braceleftBig\n(x′,xd)∈Rd:x′∈Rd−1andxd>ϕ(x′)/bracerightBig\n, (8.18)\nfor some Lipschitz function ϕ:Rd−1→R.\nTheorem 8.5. Let2<p<qandDbe given by (8.18). Letx0∈∂DandF∈L2(Q(x0,2R)∩D).\nSuppose that for each cube Q=Q(y,r)with the property that 0< r < c 0Rand either y∈\nQ(x0,R)∩∂Dor4Q⊂Q(x0,2R)∩D, there exist two measurable functions FQandRQin2Q∩D\nsuch that\n|F| ≤ |RQ|+|FQ|in2Q∩D, (8.19)\n/parenleftbigg \n2Q∩D|RQ|q/parenrightbigg1/q\n≤N/parenleftbigg \n4Q∩D|F|2/parenrightbigg1/2\n, (8.20)\n27/parenleftbigg \n2Q∩D|FQ|2/parenrightbigg1/2\n≤η/parenleftbigg \n4Q∩D|F|2/parenrightbigg1/2\n, (8.21)\nwhereN >1and0< c0<1. Then there exists η0>0, depending only on d,N,c0,,p,qand\n/ba∇dbl∇ϕ/ba∇dbl∞, with the property that if 0≤η<η0, thenF∈Lp(Q(x0,R)∩D)and\n/parenleftBigg \nQ(x0,R)∩D|F|p/parenrightBigg1/p\n≤C/parenleftBigg \nQ(x0,2R)∩D|F|2/parenrightBigg1/2\n, (8.22)\nwhereCdepends at most on d, N,c0,p,qand/ba∇dbl∇ϕ/ba∇dbl∞.\nThe proof of the above theorem can be found in [11] or [12, pp. 7 9-82].\nLemma 8.6. LetΩbe a bounded C1domain. Let u∈H1(Q(x0,4r)∩Ω), wherer≥8εand\nx0∈∂Ω. Suppose that ∆u= 0inQ(x0,4r)∩Ωε,ηandu= 0onQ(x0,4r)∩∂Ωε,η. Then there\nexistsv∈H1(Q(x0,2r)∩Ω)such that\n/parenleftBigg \nQ(x0,r)∩Ω|∇(u−v)|2/parenrightBigg1/2\n≤Cψ(η)/parenleftBigg \nQ(x0,3r)∩Ω|∇u|2/parenrightBigg1/2\n, (8.23)\n\n \nQ(x0,r)∩Ω/parenleftBigg \nQ(x,2ε)∩��|∇v|2/parenrightBiggp/2\n1/p\n≤C/parenleftBigg \nQ(x0,3r)∩Ω|∇u|2/parenrightBigg1/2\n, (8.24)\nfor anyp >2, whereψ(η) =η1/2ford≥3andψ(η) =|lnη|−1/2ford= 2. The constant C\ndepends only on d,p,{Ys\nz}andΩ.\nProof.Theproofis similar to that of Lemma8.3. Without loss of gene rality we may assume x0= 0.\nBy dilation we may also assume r= 1. Choose t∈[2,3] such that ( Q(0,t)∩Ω)ε,η=Q(0,t)∩Ωε,η\nand\nσ−2\nεˆ\nΩ∩∂Q(0,t)|u|2+ˆ\nΩ∩∂Q(0,t)|∇u|2≤Cˆ\nΩ∩Q(0,3)|∇u|2. (8.25)\nLetwbe the solution of\n/braceleftBigg\n−∆w+σ−2\nεV(x/ε)w= 0 in Q(0,t)∩Ω,\nw=uon∂(Q(0,t)∩Ω).(8.26)\nAs in the proof of Theorem 8.3, we have\n/ba∇dbl∇(u−χε,ηw)/ba∇dblL2(Q(0,t)∩Ω)≤Cψ(η)/ba∇dbl∇u/ba∇dblL2(Q(0,3)∩Ω), (8.27)\nwhereχε,ηdenotes the corrector for the domain Qε,η(0,t) =Q(0,t)∩Ωε,η. Letv=χε,ηw. Then\n/parenleftBigg \nQ(x,2ε)∩Ω|∇v|2/parenrightBigg1/2\n≤Cσ−1\nεmax\nQ(x,2ε)∩Ω|w|+C/parenleftBigg \nQ(x,2ε)∩Ω|∇w|2/parenrightBigg1/2\n.(8.28)\nSincew=u= 0 onQ(0,t)∩∂Ω, it follows from Lemma 7.1 that\nσ−1\nεmax\nQ(0,3/2)∩Ω|w| ≤C/parenleftBigg \nQ(0,2)∩Ω|∇w|2/parenrightBigg1/2\n,\n/parenleftBigg \nQ(0,3/2)∩Ω|∇w|p/parenrightBigg1/p\n≤C/parenleftBigg \nQ(0,2)∩Ω|∇w|2/parenrightBigg1/2(8.29)\n28for 2<p<∞. As a result, we obtain\nˆ\nQ(0,1)∩Ω/parenleftBigg \nQ(x,2ε)∩Ω|∇v|2/parenrightBiggp/2\n≤C/parenleftBigg \nQ(0,2)∩Ω|∇w|2/parenrightBiggp/2\n≤C/braceleftbig\n/ba∇dbl∇u/ba∇dblL2(∂(Q(0,t)∩Ω))+σ−1\nε/ba∇dblu/ba∇dblL2(∂Q(0,t)∩Ω)/bracerightbigp,\nThis, together with (8.25), gives (8.24).\nProof of Theorem 8.1. As we discussed earlier, it suffices to prove the reverse H¨ old er estimate (8.4),\nwhere 0<R<c 0and eitherQ(x0,4R)⊂Ω orx0∈∂Ω. Note that the interior case Q(x0,4R)⊂Ω\nis given by Lemma 8.4. For the boundary case where x0∈∂Ω, we apply Theorem 8.5. To this end,\nfor each cube Q=Q(y,r) with the property that 0 <r <c 0Rand either 4 Q⊂Q(x0,2R)∩Ω or\ny∈Q(x0,2R)∩∂Ω, we need to construct two functions FQandRQsuch that (8.19)-(8.21) hold.\nAgain, the case 4 Q⊂Q(x0,2R)∩Ω is given by Lemma 8.4. For the case y∈∂Ω and 0<r<8ε,\nwe letFQ=FandRQ= 0. Finally, for r≥8ε, we use Lemma 8.6.\nTheorem 8.7. LetΩbe a bounded C1domain and Ωε,ηbe given by (1.5). Then,\n/ba∇dblSε(F,0)/ba∇dblLp(Ω)≤Cmin(σε,1)/ba∇dblF/ba∇dblLp(Ω) (8.30)\nfor any2<p<∞, whereCdepends only on d,p,Ω, and{Ys\nz}.\nProof.To prove (8.30), we consider the operator\nTε(F) = (min(σε,1))−1Sε(F,0)\nand apply Theorem 7.3. Note that /ba∇dblTε/ba∇dblL2→L2≤C. Moreover, the reverse H¨ older condition (7.12)\nis reduced to the estimate (8.4), as in the case of Sε(0,f). As a result, by Theorem 7.3, we obtain\n/ba∇dblTε/ba∇dblLp→Lp≤Cfor anyp>2. This gives the estimate (8.30) for p>2.\nProof of Theorem 1.3. Since\nSε(F,f)≤Sε(F,0)+Sε(0,f),\nTheorem 1.3 follows readily from Theorems 8.1 and 8.7.\n9 Lower bounds for Ap(Ωε,η)andBp(Ωε,η)\nInthis section we discussthesharpnessof the estimates in T heorems 1.1 and 1.2 for Ap(Ωε,η) as well\nas those in Section 3 for Bp(Ωε,η). Our approach is similar to that used in [13] for an unbounde d\nand periodically perforated domain,\nωǫ,η=Rd\\/uniondisplay\nz∈Zdε(z+ηYs), (9.1)\nwhereYsis a bounded domain with connected C1boundary such that B(0,c0)⊂Ys⊂B(0,1/4).\nFollowing an idea of [7], we consider an ε-periodic corrector defined by\n−∆ψε,η=ε−2ηd−2inωε,ηandψε,η= 0 on Rd\\ωε,η. (9.2)\n29Lemma 9.1. Letψε,ηbe theε-periodic corrector defined in (9.2). Then ifd≥3,\n \nQ(0,ε)ψε,η≈1and/parenleftBigg \nQ(0,ε)|∇ψε,η|2/parenrightBigg1/2\n≈ε−1ηd−2\n2. (9.3)\nIfd= 2, we have\n \nQ(0,ε)ψε,η≈ |lnη|and/parenleftBigg \nQ(0,ε)|∇ψε,η|2/parenrightBigg1/2\n≈ε−1|lnη|1/2. (9.4)\nProof.The caseε= 1 was proved in [13, Lemma 4.4]. The case 0 <ε<1 follows by rescaling.\nLemma 9.2. Letψε,ηbe the same as in Lemma 9.1. Suppose 2<p<∞. Then ifd≥3,\n/parenleftBigg \nQ(0,ε)|ψε,η|p/parenrightBigg1/p\n≤Cand/parenleftBigg \nQ(0,ε)|∇ψε,η|p/parenrightBigg1/p\n≥Cε−1ηd\np−1. (9.5)\nIfd= 2, we have\n/parenleftBigg \nQ(0,ε)|ψε,η|p/parenrightBigg1/p\n≤C|lnη|and/parenleftBigg \nQ(0,ε)|∇ψε,η|p/parenrightBigg1/p\n≥Cε−1η2\np−1.(9.6)\nProof.Thecaseε= 1wasprovedin[13, Lemma5.3], whilethegeneralcasefollo wsbyrescaling.\nFor a bounded Lipschitz domain Ω, define\nΩε,η= Ω\\/uniondisplay\nzε(z+ηYs), (9.7)\nwhere the union is taken over those z’s inZdfor whichεQ(z,1)⊂Ω; i.e., Ω ε,ηis given by (1.5)\nwithxz= 0 andYs\nz=Ysfor allz∈Zd. Recall that Ap(Ωε,η) andBp(Ωε,η) denote the smallest\nbounding constants for which (1.2) holds, where uεis the solution of the Dirichlet problem (1.1) in\nΩε,η. Similarly, we use Cp(Ωε,η) andDp(Ωε,η) to denote the smallest bounding constants for which\ntheLpestimate,\n/ba∇dbluε/ba∇dblLp(Ωε,η)≤Cp(Ωε,η)/ba∇dblf/ba∇dblLp(Ωε,η)+Dp(Ωε,η)/ba∇dblF/ba∇dblLp(Ωε,η), (9.8)\nholds. By a duality argument we have\nAp(Ωε,η) =Ap′(Ωε,η), Dp(Ωε,η) =Dp′(Ωε,η) andBp(Ωε,η) =Cp′(Ωε,η),(9.9)\nwhere 1<p<∞andp′=p\np−1.\nWe begin with the sharpness of Dp(Ωε,η).\nTheorem 9.3. LetΩbe a bounded Lipschitz domain and Ωε,ηbe given by (9.7). Then, for d≥2\nand1<p<∞,\nDp(Ωε,η)≈min(σ2\nε,1). (9.10)\n30Proof.The upper bound Dp(Ωε,η)≤Cmin(σ2\nε,1) is contained in Theorem 2.3. To show the lower\nboundDp(Ωε,η)≥cmin(σ2\nε,1), by (9.9), we may assume 1 <p≤2. Without loss of generality we\nmay also assume Q(0,4r0)⊂Ω.\nSupposeεis sufficiently small and Rε≤r0for someR∈N. Letφ∈C∞\n0(Q(0,2Rε)) be a cut-off\nfunction such that 0 ≤φ≤1,φ= 1 inQ(0,Rε),|∇φ| ≤C/(Rε) and|∇2φ| ≤C/(Rε)2. Note that\n−∆(ψε,ηφ) =ε−2ηd−2φ−2∇ψε,η·∇φ−ψε,η∆φ (9.11)\nin Ωε,ηandψε,ηφ= 0 on∂Ωε,η. It follows that\ncRd/p/ba∇dblψε,η/ba∇dblLp(Q(0,ε))≤ /ba∇dblψε,ηφ/ba∇dblLp(Ωε,η)\n≤CDp(Ωε,η)/braceleftBig\nε−2+d\npηd−2Rd\np+ε−1Rd\np−1/ba∇dbl∇ψε,η/ba∇dblLp(Q(0,ε))+ε−2Rd\np−2/ba∇dblψε,η/ba∇dblLp(Q(0,ε))/bracerightBig\n,\nwhere we have used the periodicity of ψǫ,η. This gives\nεd\np \nQ(0,ε)|ψε,η|\n≤CDp(Ωε,η)\n\nεd\np−2ηd−2+R−1εd\np−1/parenleftBigg \nQ(0,ε)|∇ψε,η|2/parenrightBigg1/2\n+(εR)−2/ba∇dblψε,η/ba∇dblLp(Q(0,ε))\n\n,(9.12)\nwhere we have used the fact 1 <p≤2.\nCase 1:d≥3and0<σε≤1.\nBy applying Lemmas 9.1 and 9.2 to (9.12), we obtain\nc≤Dp(Ωε,η)/braceleftBig\nε−2ηd−2+R−1ε−2ηd−2\n2+(εR)−2/bracerightBig\n. (9.13)\nSinceσε=εη1−d\n2≤1, we may choose R∈Nsuch thatR≈η1−d\n2in (9.13) to get\nc≤Dp(Ωε,η)·ε−2ηd−2.\nHence,Dp(Ωε,η)≥cσ2\nε=cmin(σ2\nε,1).\nCase 2:d≥3andσε≥1.\nIn this case we choose R∈Nsuch thatR≈ε−1in (9.13). It follows that\nc≤Dp(Ωε,η)/braceleftBig\nε−2ηd−2+ε−1ηd−2\n2+1/bracerightBig\n.\nSinceσε≥1, we obtain Dp(Ωε,η)≥c=cmin(σ2\nε,1).\nCase 3:d= 2andσε≤1.\nApplying Lemmas 9.1 and 9.2 to (9.12) now yields\nc|lnη| ≤Dp(Ωε,η)/braceleftBig\nε−2+R−1ε−2|lnη|1\n2+(εR)−2|lnη|/bracerightBig\n. (9.14)\nPickingR∈Nsuch thatR≈ |lnη|1\n2in (9.14) gives\nDp(Ωε,η)≥cε2|lnη|=cmin(σ2\nε,1).\n31Case 4:d= 2andσε≥1.\nChoosingR∈Nsuch thatR≈ε−1in (9.14) yields\nc|lnη| ≤Dp(Ωε,η)/braceleftBig\nε−2+ε−1|lnη|1\n2+|lnη|/bracerightBig\n≤CDp(Ωε,η)|lnη|.\nIt follows that Dp(Ωǫ)≥c=cmin(σ2\nε,1).\nNext, we consider the case Bp(Ωε,η) for 1<p≤2.\nTheorem 9.4. LetΩbe a bounded Lipschitz domain and Ωε,ηbe given by (9.7). Then\nBp(Ωε,η)≈min(σε,1) (9.15)\nford≥2and1<p≤2,\nProof.It follows from Theorem 2.3 that\nCp(Ωε,η)≤Cmin(σε,1)\nfor 2≤p<∞. SinceBp(Ωε,η) =Cp′(Ωε,η), we obtain the upper bound\nBp(Ωε,η)≤Cmin(σε,1)\nfor 1<p≤2. To establish the lower bounds, we first note that D2(Ωε,η)≤Cmin(σε,1)B2(Ωε,η),\nwhich follows (2.2). As a result, by Theorem 9.3, B2(Ωε,η)≥cmin(σε,1).\nFor the case 1 <p<2, we proceed as in the proof of Theorem 9.3. It follows from (9 .11) that\n/ba∇dblψε,ηφ/ba∇dblLq(Ωε,η)≤C/ba∇dbl∇(ψε,ηφ)/ba∇dblLp(Ωε,η)\n≤CBp(Ωε,η)/braceleftBig\nεd\np−2ηd−2Rd\np+ε−1Rd\np−1/ba∇dbl∇ψε,η/ba∇dblLp(Q(0,ε))+ε−2Rd\np−2/ba∇dblψε,η/ba∇dblLp(Q(0,ε))/bracerightBig\n,\nwhere1\nq=1\np−1\nd. This implies that\n \nQ(0,ε)|ψε,η| ≤CBp(Ωε,η)\n\nε−1ηd−2R+/parenleftBigg \nQ(0,ε)|∇ψε,η|2/parenrightBigg1/2\n+(εR)−1/parenleftBigg \nQ(0,ε)|ψε,η|p/parenrightBigg1/p\n\n.\n(9.16)\nCase 1:d≥3andσε≤1.\nWe apply Lemmas 9.1 and 9.2 to (9.16) to obtain\nc≤Bp(Ωε,η)/braceleftBig\nε−1ηd−2R+ε−1ηd−2\n2+ε−1R−1/bracerightBig\n. (9.17)\nLettingR≈η−d−2\n2in (9.17) yields\nc≤Bp(Ωǫ)·ε−1ηd−2\n2.\nThus,Bp(Ωε,η)≥cσε=cmin(σε,1).\nCase 2:d≥3andσε≥1.\n32LetR≈ε−1in (9.17). Then\nc≤Bp(Ωε,η)/braceleftBig\nε−2+ε−1ηd−2\n2+C/bracerightBig\n.\nThus,Bp(Ωε,η)≥c=cmin(σε,1).\nCase 3:d= 2andσε≤1.\nApplying Lemmas 9.1 and 9.2 to (9.16) now yields\nc|lnη| ≤Bp(Ωε,η)/braceleftBig\nε−1R+ε−1|lnη|1\n2+(εR)−1|lnη|/bracerightBig\n. (9.18)\nPickingR≈ |lnη|1\n2in (9.18) gives\nc|lnη| ≤Bp(Ωε,η)·ε−1|lnη|1\n2.\nThus,Bp(Ωε,η)≥cε|lnη|1/2=cmin(σε,1).\nCase 4:d= 2andσε≥1.\nTakingR≈ε−1in (9.18) leads to\nc|lnη| ≤Bp(Ωε,η)/braceleftBig\nε−2+ε−1|lnη|1\n2+|lnη|/bracerightBig\n≤CBp(Ωε,η)|lnη|.\nThis implies that Bp(Ωε,η)≥c=cmin(σε,1).\nWe now consider Bp(Ωε,η) for 2<p<∞.\nTheorem 9.5. LetΩε,ηbe given by (9.7). Assumed≥3. Then if 0<σε≤1and2<p<∞,\nBp(Ωε,η)≈εη1−d+d\np. (9.19)\nIfσε≥1and2<p<d, we have\nBp(Ωε,η)≈1+ε−1ηd\np−1. (9.20)\nProof.Suppose 0<σε≤1. The upper bound Bp(Ωε,η)≤Cεη1−d+d\npis contained in Theorem 3.2.\nTo prove the lower bound, we use the Poincar´ e inequality,\n/ba∇dblu/ba∇dblLp(Ωε,η)≤Cmin(εη1−d\np,1)/ba∇dbl∇u/ba∇dblLp(Ωε,η) (9.21)\nforu∈W1,p\n0(Ωε,η), where 2<p<d. The inequality (9.21) follows readily from Lemma 2.1. As a\nresult, we see that\nDp(Ωε,η)≤CBp(Ωε,η)min(εη1−d\np,1). (9.22)\nBy Theorem 9.3, we have Dp(Ωε,η)≥cσ2\nε=cε2η2−d. This, together with (9.22), gives (9.19) for\nthe case 2<p<d.\nNext, we use a convexity argument to handle the case 0 < σε≤1 andd≤p <∞. Choose\n2<q<dandt∈(0,1) so that\n1\nq=1−t\n2+t\np.\n33By Riesz-Thorin Theorem we have\nBq(Ωε,η)≤[B2(Ωε,η)]1−t[Bp(Ωε,η)]t.\nIt follows that\nBp(Ωε,η)≥[Bq(Ωε,η)]1\nt[B2(Ωε,η)]1−1\nt.\nThe desired estimate follows by using B2(Ωε,η)≤CσεandBq(Ωε,η)≥cεη1−d+d\nq.\nFinally, we consider the case σε≥1 and 2<p<d. The upper bound is contained in Theorem\n3.3. SinceDp(Ωε,η)≥c, by (9.22), we obtain\nBp(Ωε,η)≥cε−1ηd\np−1\nHowever, (9.22) also yields Bp(Ωε,η)≥cDp(Ωε,η)≥c. As a result, we obtain Bp(Ωε,η)≥c(1 +\nε−1ηd\np−1).\nFinally, we establish lower bounds for Ap(Ωε,η).\nTheorem 9.6. Let2<p<∞. LetΩε,ηbe given by (9.7). Then, if 0<σε≤1,\nAp(Ωε,η)≥/braceleftBigg\ncη−d|1\n2−1\np|ifd≥3,\ncη−2|1\n2−1\np||lnη|−1\n2ifd= 2.(9.23)\nIfσǫ≥1we have\nAp(Ωε,η)≥\n\nc(1+ε−1ηd\np−1)ifd≥3and2<p<d,\ncε−1ηd\np−1ifd≥3andd≤p<∞,\ncε−1η2\np−1|lnη|−1ifd= 2and2<p<∞.(9.24)\nThe constants c>0only depend on d,p,YsandΩ.\nProof.Letuεbe a solution to −∆uε= div(f) in Ω ε,ηwithuε= 0 on∂Ωε,η. By a Sobolev\nimbedding, we have\n/ba∇dbluε/ba∇dblLq′(Ωε,η)≤C/ba∇dbl∇uε/ba∇dblLp′(Ωε,η)≤CAp(Ωε,η)/ba∇dblf/ba∇dblLp′(Ωε,η),\nwhere1\nq′=1\np′−1\ndand 1<p′<d.By duality this implies that if −∆vε=Gin Ωε,ηandvε= 0 on\n∂Ωε,η,then\n/ba∇dbl∇vε/ba∇dblLp(Ωε,η)≤CAp(Ωε,η)/ba∇dblG/ba∇dblLq(Ωε,η).\nThus, if−∆uε=F+div(f) in Ωε,ηwithuε= 0 on∂Ωε,η, then\n/ba∇dbl∇uε/ba∇dblLp(Ωε,η)≤CAp(Ωε,η)/braceleftbig\n/ba∇dblF/ba∇dblLq(Ωε,η)+/ba∇dblf/ba∇dblLp(Ωε,η)/bracerightbig\n, (9.25)\nwhered′<p<∞and1\nq=1\np+1\nd.\nLetφandψε,ηbe the same as in the proof of Theorem 9.3. Note that\n−∆(ψε,ηφ) =ε−2ηd−2φ−2div(ψε,η∇φ)+ψε,η∆φ\n34in Ωε,ηandψε,ηφ= 0 on∂Ωε,η. It follows from (9.25) that\nRd\np/ba∇dbl∇ψε,η/ba∇dblLp(Q(0,ε))≤C/ba∇dbl∇(ψε,ηφ)/ba∇dblLp(Ωε,η)\n≤CAp(Ωε,η)/braceleftBig\nεd\nq−2ηd−2Rd\nq+ε−1Rd\np−1/ba∇dblψε,η/ba∇dblLp(Q(0,ε))+ε−2Rd\nq−2/ba∇dblψε,η/ba∇dblLq(Q(0,ε))/bracerightBig\n,\nwhere we have used the periodicity of ψε,η. Hence,\n/ba∇dbl∇ψε,η/ba∇dblLp(Q(0,ε))\n≤CAp(Ωε,η)/braceleftBig\nεd\np−1ηd−2R+ε−1R−1/ba∇dblψε,η/ba∇dblLp(Q(0,ε))+ε−2R−1/ba∇dblψε,η/ba∇dblLq(Q(0,ε))/bracerightBig\n.(9.26)\nCase 1:d≥3and0<σε≤1.\nApplying Lemma 9.2 to (9.26) yields\nηd\np−1≤CAp(Ωε,η)/braceleftBig\nηd−2R+R−1/bracerightBig\n. (9.27)\nSinceσε≤1, we may pick R≈η−d−2\n2in (9.27) and obtain\nAp(Ωε,η)≥Cηd\np−d\n2=Cη−d|1\np−1\n2|.\nCase 2:d= 2and0<σε≤1.\nApplying Lemma 9.2 to (9.26) now yields\nη2\np−1≤CAp(Ωε,η)/braceleftbig\nR+|lnη|R−1/bracerightbig\n. (9.28)\nPickingR≈ |lnη|1\n2in (9.28) gives\nη2\np−1≤CAp(Ωε,η)|lnη|1\n2,\nwhich leads to\nAp(Ωε,η)≥cη−2|1\n2−1\np||lnη|−1\n2.\nCase 3:d≥3andσε≥1.\nChoosingR≈ε−1in (9.27) yields\nηd\np−1≤CAp(Ωε,η)/braceleftBig\nηd−2ε−1+ε/bracerightBig\n≤CεAp(Ωε,η).\nThus,\nAp(Ωε,η)≥cε−1ηd\np−1.\nThis gives the lower bound in (9.24) for the case d≤p <∞. Note that if 2 < p < d , by the\nPoincar´ e inequality /ba∇dbluε/ba∇dblLp(Ωε,η)≤C/ba∇dbl∇uε/ba∇dblLp(Ωε,η), we have\nAp(Ωε,η)≥cCp(Ωε,η) =cBp′(Ωε,η)≥c.\nHence,\nAp(Ωε,η)≥c(1+ε−1ηd\np−1).\n35Case 4:d= 2andσε≥1.\nNow we take R≈ε−1in (9.28). This gives\nη2\np−1≤CAp(Ωε,η)/braceleftbig\nε−1+ε|lnη|/bracerightbig\n≤CεAp(Ωε,η)|lnη|,\nwhich leads to\nAp(Ωε,η)≥Cε−1η2\np−1|lnη|−1.\nThis completes the proof.\nReferences\n[1] Gr´ egoire Allaire. Homogenization of the Navier-Stoke s equations in open sets perforated with\ntiny holes. I. Abstract framework, a volume distribution of holes.Arch. Rational Mech. Anal. ,\n113(3):209–259, 1990.\n[2] Gr´ egoire Allaire. Homogenization of the Navier-Stoke s equations in open sets perforated with\ntiny holes. II. Noncritical sizes of the holes for a volume di stribution and a surface distribution\nof holes. Arch. Rational Mech. Anal. , 113(3):261–298, 1990.\n[3] C. Amrouche, V. Girault, and J. Giroire. Dirichlet and Ne umann exterior problems for the\nn-dimensional Laplace operator: an approach in weighted Sob olev spaces. J. Math. Pures\nAppl. (9) , 76(1):55–81, 1997.\n[4] Xavier Blanc and Sylvain Wolf. Homogenization of the Poi sson equation in a non-periodically\nperforated domain. Asymptot. Anal. , 126(1-2):129–155, 2022.\n[5] Do¨ ına Cioranescu and Fran¸ cois Murat. Un terme ´ etrang e venu d’ailleurs. II. In Nonlinear\npartial differential equations and their applications. Coll ` ege de France Seminar, Vol. III (Paris,\n1980/1981) , volume 70 of Res. Notes in Math. , pages 154–178, 425–426. Pitman, Boston,\nMass.-London, 1982.\n[6] David Jerison and Carlos E. Kenig. The inhomogeneous Dir ichlet problem in Lipschitz do-\nmains.J. Funct. Anal. , 130(1):161–219, 1995.\n[7] Wenjia Jing. A unified homogenization approach for the Di richlet problem in perforated\ndomains. SIAM J. Math. Anal. , 52(2):1192–1220, 2020.\n[8] J.-L. Lions. Asymptotic expansions in perforated media with a periodic structure. Rocky\nMountain J. Math. , 10(1):125–140, 1980.\n[9] Nader Masmoudi. Some uniform elliptic estimates in a por ous medium. C. R. Math. Acad.\nSci. Paris , 339(12):849–854, 2004.\n[10] Zhongwei Shen. Bounds of Riesz transforms on Lpspaces for second order elliptic operators.\nAnn. Inst. Fourier (Grenoble) , 55(1):173–197, 2005.\n[11] Zhongwei Shen. The Lpboundary value problems on Lipschitz domains. Adv. Math. ,\n216(1):212–254, 2007.\n36[12] Zhongwei Shen. Periodic homogenization of elliptic systems , volume 269 of Operator Theory:\nAdvances and Applications . Birkh¨ auser/Springer, Cham, 2018. Advancesin Partial Di fferential\nEquations (Basel).\n[13] Zhongwei Shen. Uniform estimates for Dirichlet proble ms in perforated domains. Vietnam J.\nMath., 51(4):845–867, 2023.\n[14] Zhongwei Shen and Jamison Wallace. Uniform W1,pestimates and large-scale regularity for\nDirichlet problems in perforated domains. J. Funct. Anal. , 285(10):Paper No. 110118, 33,\n2023.\n[15] Gregory Verchota. Layer potentials and regularity for the Dirichlet problem for Laplace’s\nequation in Lipschitz domains. J. Funct. Anal. , 59(3):572–611, 1984.\nRobert Righi, Department of Mathematics, University of Ken tucky, Lexington, Kentucky 40506,\nUSA.\nE-mail: robert.righi@uky.edu\nZhongwei Shen, Department of Mathematics, University of Ke ntucky, Lexington, Kentucky 40506,\nUSA.\nE-mail: zshen2@uky.edu\n37"    },    {        "title": "2402.13032v1.Observed_epochal_variations_in_X_ray_lines_from_the_O_Supergiant__ζ__Puppis_do_not_require_substantial_changes_in_the_wind_mass_flux.pdf",        "content": "MNRAS 000, 1–10 (2023) Preprint 21 February 2024 Compiled using MNRAS L ATEX style file v3.0\nObserved epochal variations in X-ray lines from the O Supergiant 𝜁\nPuppis do not require substantial changes in the wind mass flux\nSean J. Gunderson1,2★, Kenneth G. Gayley2, David P. Huenemoerder1, Pragati Pradhan3, Nathan A. Miller4\n1Massachusetts Institute of Technology, Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Ave., Cambridge, MA 02139, USA\n2Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA\n3Embry Riddle Aeronautical University, Department of Physics Prescott Campus, 3700 Willow Creek Road, Prescott, AZ 86301\n4Department of Physics and Astronomy, University of Wisconsin–Eau Claire, Eau Claire, WI 54701, USA\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nWe fit the high resolution Chandra X-ray spectra of the O supergiant 𝜁Puppis using the variable boundary condition (VBC)\nline model to test the stability of its mass-loss rate between two epochs of observation: 2000 March and 2018 July – 2019\nAugust. At issue is whether the observed variations are induced by global changes in the cool (unshocked) wind itself or are\nisolated to the local pockets of hot gas (i.e., changes in the frequency and location of the shocks). Evidence in the literature\nfavored the possibility of a 40 per cent increase in the mass flux of the entire stellar wind, based on X-ray reabsorption from a\nline-deshadowing-instability-inspiredparameterization,whereasourfitparametersareconsistentwithaconstantmassfluxwith\na change in the velocity variations that determine the locations where shocks form. Our results suggest the shocks in the more\nrecent data are formed at somewhat larger radii, mimicking the enhanced blueshifts and increased line fluxes interpreted in the\nprevious analysis as being due to increases in both the X-ray generation and reabsorption from an overall stronger wind.\nKey words: stars: mass-loss – stars: massive – X-rays: stars – stars: winds, outflows – line: profiles\n1 INTRODUCTION\nAtonly332pcaway(Howarth&vanLeeuwen2019), 𝜁Puppis(HD\n66811)isoneofthebrightestOstarsinthenightsky.Thishasmade\nit the the canonical O supergiant and one of the most well studied\nsingle massive stars across all wave bands. Its well studied nature\nhas also made it an ideal calibration target for some of our most\nsensitive X-ray instruments (e.g., XMM-Newton’s RGS and EPIC1).\nAs a massive star, 𝜁Pup’s stellar wind is relatively well understood\nto be driven by radiative forces acting on hypersonically Doppler\nshifted UV line opacity from metal ions in its atmosphere, As such,\nit is expected to have a wind that, despite being clumpy (Martínez-\nNúñezetal.2017),shouldonthewholebeassteadyasitsluminosity.\nThisistrueeveninthecaseofthestarbeingarapidrotator,asisthe\ncase for𝜁Pup. The source of the high rotation rate, along with its\nrunaway status, is thought to be evidence of a prior companion in 𝜁\nPup’sevolutionaryhistorywhosesupernovaresultedin 𝜁Pup’shigh\nspeed ejection (Woermann et al. 2001).\nWhile many of 𝜁Pup’s stellar properties have been well con-\nstrained, a number of studies have also revealed that our picture\n★Contact e-mail: seang97@mit.edu\n1https://xmm-tools.cosmos.esa.int/external/xmm_user_\nsupport/documentation/uhb/routinecal.htmlis perhaps incomplete. Observations in both the optical (Ramiara-\nmanantsoa et al. (2018) using BRITE) and X-ray (Nazé et al. (2018)\nusingXMM-Newton andSwiftand recounting earlier documented\nvariabilities,andNicholsetal.(2021)using Chandra)regimeshave\nhighlighted apparent variabilities in the star. The main periodicity\ndetected in the optical band has a period of 1.78 days (originally\ndetected by Howarth & Stevens (2014)) and been argued to arise\nfrom bright spots on 𝜁Pup’s surface and its rapid rotation (Rami-\naramanantsoa et al. 2018; Nichols et al. 2021). There is also an\napparent2.56-dayperiodicity(Marchenkoetal.1998)thatHowarth\n& van Leeuwen (2019) corroborated in the Hipparcos data, which\ninterestinglydoesnotshowevidenceoftheaforementioned1.78-day\nsignal.\nAddinganotherlayerofcomplexity,theperiodicitiesdonotappear\nto be constant. Nazé et al. (2018) noted that the X-ray periodicity\nappear to change in phase and period across many XMM-Newton\nobservations, at times correlating with the BRITEsignal and other\ntimes not. The strength of the X-ray signal also appears to wax and\nwane:itwasstrongerduringthe2007-2011thanbeforeorafter.The\noptical1.78-daysignalalsoappearstobevaryingasHowarth&van\nLeeuwen (2019) posited that its amplitude has grown with time to\nexplainwhy Hipparcos doesnotshowit.Olderobservationsreported\nlonger period variations than those today. These include periods of\n5.21 days (Balona 1992) and 5.075 days (Moffat & Michaud 1981)\n©2023 The AuthorsarXiv:2402.13032v1  [astro-ph.HE]  20 Feb 20242S. J. Gunderson et al.\nusing optical data, and 5.1 days (Howarth et al. 1995) in the UV.\nHow these longer periods fit into the picture is less clear. They were\noriginallyproposedtobeconnectedtotherotationof 𝜁Pup,butthe\nrotation period is faster (<3 days; Howarth & van Leeuwen 2019)\nthan these signal periods. Thus there appears to be multiple layers\nofperiodicitywhicharethemselvesvaryinginfrequency,amplitude,\nand phase relationship between various wavebands.\nThere is an important distinction to be made between X-ray vari-\nability and variability in other wavebands. As a massive star, 𝜁Pup\ngenerates X-rays not at its static photosphere but farther out in its\nhypersonic wind through embedded shocks (Lucy & White 1980;\nFeldmeier et al. 1997) as clumps experience differential accelera-\ntion. This X-ray generating mechanism is dependent on the number,\nsize, speed, and many other factors of the wind clumps, so it has an\ninherent stochasticity. This randomness is associated with the flow\ntime of the wind 𝑡flow=𝑅∗/𝑣∞, which is on the order of a cou-\nple of hours, a much shorter timescale than those discussed above.\nAn example of a possible short-period signal is the 0.694-day pe-\nriod Berghoefer et al. (1996, reported as 1.44 cycles/day) found in\nsimultaneous X-ray and H 𝛼observations. X-ray observations are\ntypically long enough compared with this timescale to average out\nany stochastic effects (see Table 2). The noted X-ray periodicities\naremorecoherentsignalsthatpersistthroughthetimeaveraging,so\nchangesintheX-raysignalmaycorrespondtoglobalchangesinthe\ntime-averaged wind.\nIn summary, many recent investigations of the relationship be-\ntween optical and X-ray variations of this star on short time scales\nhaverevealedinterestinglinksbetweenthetwowavebands.Henceit\nis timely to ask if there exists longer term changes, either in pockets\nof X-ray generating shocks, or more globally over the cool wind.\nCapitalizingonthedecadesofhighresolutionX-rayprofileinforma-\ntionthatexistsfor 𝜁Puppis,ourfocusisexploringlongtermchanges\nin the wind and its X-ray heating\nThe first piece of evidence for changes in 𝜁Pup as an X-ray\nsource is its brightening in the last few years as detected with both\nChandra (Cohenetal.2020)and XMM-Newton .Thelatterisshown\nin Figure 1, where the plotted flux values were measured by XMM-\nNewton’s EPIC detector and reported in 4XMM-DR13 (Webb et al.\n2020) Cohen et al. (2020) proposed that the increase in X-ray flux\nis due to the wind mass-flux increasing (more on this immediately\nbelow), but the exact origin is not entirely clear due to the the many\nfactors which influence overall X-ray output from embedded wind\nshocks. More mass within the wind would mean more clumps and\nsubsequently more shocks, but the increase in flux could also be\ncaused by stronger shocks producing more light or a wind that is\nmore porous to X-rays due to changes in clump size or density.\nThe second is changes in the wind as inferred from a line shape\nanalysis,somethingthatcanonlybemeasuredusingthehigherspec-\ntral resolution of Chandra (though the time coverage is much less\nfrequent). Constraints on the physical effects listed above can be\nextracted from the X-ray spectral line shapes via some appropriate\nparameterization of its features. Understanding what the increased\nflux and changes to the line shapes allude to requires an appropriate\nlinemodeltoparameterizequantitiesofinterest.Theinferenceofthe\nphysical changes to the wind using a line profile parameterization is\na goal of this paper, for which the chosen parameterization will be\ndiscussed in § 2.\nAn important parameterization comes from Owocki & Cohen\n(2001), which Cohen et al. (2010, 2020) used to calculate the mass-\nloss rate of𝜁Pup from X-ray lines measured by Chandra. In this\npaperwewillbereferringtothismodelassimplythe“Cohenmodel\"\ntoaccountforthenumberofpapersthathavemadeuseofit(Owocki\n1.601.651.701.751.801.851.90Flux  (10−11\n erg cm−2\n s−1\n)\n2000 2005 2010 2015 2020\nY ear of Observation2.55.0Error (10−14\n \nerg cm−2\n s−1\n)Figure1. Year-to-yearX-rayfluxof 𝜁Pupmeasuredby XMM-Newton’s EPIC\ndetector. Vertical lines denote the Cycle 1 Chandra observation and the start\noftheCycle19campaign(seeTable2fordates).Valueswereretrievedfrom\n4XMM-DR13 (Webb et al. 2020). The reported errors in the catalogue are\ntoo small to visualize as vertical bars so are plotted in the sub-plot.\n& Cohen 2001; Cohen et al. 2010, 2014, 2020). The most recent of\nthesepapers,Cohenetal.(2020)cametotheremarkableconclusion\nthat the mass-loss rate of 𝜁Pup has increased by 40 per cent in a\nroughly 20-year time frame, based on the changes to the line shapes\n(i.e., centroid and width) in Chandra High Resolution Transmission\nGratingSystem(HETGS)datafromCycles1and19(seeTable2for\nobservationdates).Suchalargechangeinthemass-lossrateofastar\nbrings to light the need to independently verify that observed pro-\nfilechangesarerobustlyinterpretableaschangesintheglobalwind,\nrather than in just its hot component, given that the hot component\nisasmallfractionofthetotalwind(Cohenetal.1997;Owockietal.\n2013; Gayley 2016) and hence might be more susceptible to larger\nvariations.\n1.1 The Challenge of Inferring Mass-loss Rates\nTheproblemofdeterminingmass-lossratesofmassivestarsisnota\nsimpleone.InTable1,welistanumberofcalculatedmass-lossrates\nfor𝜁Pupintheliterature.ThetheoreticalvalueexpectedfromVink\netal.(2000)isunsurprisinglyhighastheirtheorydoesnotaccountfor\nthe clumps that make up the wind. Puls et al. (2006) made detailed\ncomputations of the clumping-included mass-loss rates using H 𝛼,\nIR, mm, and radio, finding values that bracket the predicted value\nfromVinketal.(2000).Thewidedifferencebetweenthevaluesisa\nreflectionofdifferingassumptionsofwhereHerecombinationstarts\nin the wind. These four wave bands provide precise values for the\nmass-loss rate, but the specific values determined are still heavily\ndependent on the assumed clumping factor of the wind.\nTheX-rayvaluesfromCohenetal.(2010)andCohenetal.(2020)\narefreeofthisassumptionsincesmallscaleclumpingdoesnotaffect\nthe reabsorption of X-rays by the cool wind. This allows measure-\nments of X-ray line profiles to directly parameterize the mass-loss\nrate (see Section 2). Additionally, if the X-ray lines are fit individu-\nallyinisolatedregions,the(relativelyflat)localcontinuumrequires\nno correction for interstellar absorption, so there is no dependence\non the distance to 𝜁Pup. Thus there is no need for distance cor-\nrections like those made by Howarth & van Leeuwen (2019) to the\nflux-dependentcalculationsdonebyPulsetal.(2006).However,the\nuse of X-rays for inferring mass-loss rates is not without its own\nMNRAS 000, 1–10 (2023)X-ray variations in 𝜁Pup do not require mass flux changes 3\nTable 1.Archival values of 𝜁Pup’s mass-loss rate\nReference ¤𝑀(10−6M⊙yr−1)\nVink et al. (2000)a6.4\nPuls et al. (2006)b4.2, 8.5\nCohen et al. (2014)c1.76+0.13\n−0.12\nHowarth & van Leeuwen (2019)d2.6\nCohen et al. (2020)e2.47+0.09\n−0.09\nNotes–aTheoretical predicted value.\nbDerived from simultaneous H 𝛼, IR, and radio modelling.\nThe two values correspond to different assumed radii of He\nrecombination.\ncDerived from X-ray line profile modelling Cycle 1 Chandra\ndata.\ndThe published mass loss rate have been scaled to reflect the\nauthor’s newly constrained distance measurement, Puls et al.\n(2006,¤𝑀∝𝑑3/2).\neDerivedfromX-raylineprofilemodellingCycle19 Chandra\ndata.\nproblems.Gundersonetal.(2022,hereafterG2022)determinedthat\ntheseestimatesaredependentontheassumptionsgoingintothewind\nprofile model used to fit the data.\n1.2 Expected Stability of the Global Wind Mass Flux\nIn classical theories of massive stars (Castor et al. 1975; Owocki\n2004), the mass-loss rate is regarded as a stable quantity due to the\nnature of the line-driving of the wind. Therefore, assuming one’s\ncalculated value can be trusted, a large change in the mass-loss rate\nmay be indcative of an incomplete understanding of the stability of\nthese stars over longer periods. Variations on long timescales is not\nunheard of for stars; for example the Sun undergoes a 22 year cycle\n(Russell et al. 2016). The disks of Be stars also exhibit similar long\ntimescales (Kee 2020).\nTo know if Cohen et al. (2020)’s reported changes are part of a\nlong-period variations, we need much more data to see more than\none period to determine if it is indeed even periodic, or stochastic\nchange over a longer time scale. As a first step, however, we need to\nknow if the secular changes that have been highlighted in our above\ndiscussioncorrespond torealchanges intheglobal windproperties.\nThus is the goal of this paper. We will investigate the X-ray line\nshapes of𝜁Pup for the Cycle 1 and 19 Chandra observations to\ninvestigatethelargechangeinmass-lossratereportedbyCohenetal.\n(2020). To do so, we will use the VBC model from G2022. While\nthespecificvaluesofmass-lossratesderivedthroughX-raylinesare\ndependentonthemodelused,anylargechangeinaglobalparameter\nsuch as the mass-loss rate should be relativity model independent.\nToexplorethis,wewillbecomparingmanyofourmodellingresults\nwiththoseofCohenetal.(2010)andCohenetal.(2020),sowewill\nhereafter refer to these works as C2010 and C2020 respectively.\nThis paper is organized into the following sections. In § 2, we\ngive a short summary of the VBC model while including some\nnew insights. In § 3, we provide details on the data reduction and\nmodelling.Finally,in§4and5,wegiveourresultsandconclusions.2 VARIABLE BOUNDARY CONDITION MODEL\nForamoredetailedderivationoftheVBCmodel,readersarereferred\ntoG2022.Herewegiveasummaryofthemodelanditsparameters,\nwhichstartswiththewind’svelocity.Aswithmostmassive-starwind\nmodels, the velocity field is assumed to follow the usual 𝛽-velocity\nlaw\n𝑣(𝑟)=𝑣∞\u0012\n1−𝑅∗\n𝑟\u0013𝛽\n. (1)\nWechoose𝛽=1toallowforanalyticresultsinsubsequentequations.\nFor example, with 𝛽=1the optical depth takes the form\n𝜏(𝜇,𝑟;𝜏∗)=𝜏∗𝑅∗\n𝑧𝑡(\n𝑡−(𝜇,𝑟)𝜇≥0\n𝑡+(|𝜇|,𝑟)+𝜋 𝜇< 0, (2)\nwhere𝜇=cos𝜃and\n𝑡±=arctan\u0012𝑅∗\n𝑧𝑡\u0013\n±arctan\u0012𝛾\n𝜇\u0013\n, (3)\n𝑧𝑡=√︃\u00001−𝜇2\u0001𝑟2−𝑅2∗ (4)\n𝛾=𝑅∗−𝑟(1−𝜇2)\n𝑧𝑡. (5)\nThe variable\n𝜏∗≡𝜅(𝜆)\n4𝜋𝑅∗𝑣∞¤𝑀 (6)\nis a fiducial optical depth describing the amount of absorption a\nphoton produced at 𝑟=𝑅∗experiences in a constant velocity wind\n(𝛽=0) with the same mass-loss rate as an accelerating wind. For\nthe case of 𝛽=1, it is instead the optical depth at a radius of\n𝑟/𝑅∗=e/(e−1)≈1.58for a photon on the same central ray.2For\ninformation on how 𝜏∗can be used to calculatea mass-loss rate, see\nC2010. Additionally, details on the role of complex numbers in the\nderivations of Equations (2)–(5) are given in Appendix A.\nThe main distinguishing feature of the VBC model is its direct\nparameterization of the heating of the gas. This is in contrast to the\nuse of a hot-gas filling factor, a number which implicitly conflates\ntheeffectsoftheheatingandcoolingrates.Thisisachievedthrough\ntracking pockets of gas that are assumed to follow a slightly faster\nvelocity field with terminal velocities of 𝑣∞,𝑓than the ambient,\nslower pockets. Note that this velocity is not used in the definition\nof𝜏∗because it is the ambient, unshocked gas described by 𝑣∞\nthat absorbs the photons generated in shocks. The faster clumps are\nassumed to have a differential probability\nd𝑃\nd𝑟=1\nℓ0(7)\nof shocking with a slower clump. The above equation contains the\nfinal model parameter ℓ0which parameterizes the mean free path\nbetween shocks. Thus for a given line can be described by how\nmuch optical depth 𝜏∗the photons traveled through, a characteristic\nformation radius 𝑅∗+ℓ0, and the terminal speed 𝑣∞,𝑓of the faster\nclump that caused the shocked.\nThe line shape described by these parameters is modelled using\nthe relative luminosity per frequency bin\n𝐿(𝜉)=∫∞\n𝑟𝑚(𝜉)d𝜇\nd𝜉e−(𝑟−𝑅∗)/ℓ0e−𝜏(𝜉,𝑟;𝜏∗)d𝑃\nd𝑟d𝑟, (8)\n2This radius interestingly corresponds to the average value of 𝑅0found by\nC2010 and C2020\nMNRAS 000, 1–10 (2023)4S. J. Gunderson et al.\nwhere𝜉≡−𝜇𝑣(𝑟)/𝑣∞,𝑓is the frequency shift from line center in\nterminal speed units and\nd𝜇\nd𝜉=1\n2𝑣∞,𝑓\n𝑣(𝑟). (9)\nisthelocalmappingfromsolidangletofrequencyspace.Thelower\nbound of the integral is the minimum radius capable of producing\nenough Doppler shift to reach the 𝜉in question. It is a complicated\nfunction dependent on whether the emission is toward the forward\n(𝜇 > 0,𝜉 < 0) or backward ( 𝜇 < 0,𝜉 > 0) hemisphere of the star.\nReaders are encouraged to see G2022 for details on this minimum\nradius.InAppendixB,wegivedetailsonapproximatingEquation(8)\nwith a Gauss-Laguerre quadrature.\n2.1 Terminal Velocity Parameter\nIn G2022, a constant, frozen fast-gas terminal velocity of 𝑣∞,𝑓=\n2500km s−1for each line was assumed. This was done to simplify\nthe model fitting. We use the same process in this work but provide\na more mathematical backing for this choice.\nLetusassumethatthefastgasterminalvelocityisasimplescaling\nconstantwithrespecttotheslowgas, 𝑣∞,𝑓=𝑎𝑣∞.Then,foragiven\npocketoffastgasthatshockswithinamean-freepath,itsvelocityis\n𝑣𝑓=𝑣∞,𝑓\u0012\n1−𝑅∗\n𝑅∗+ℓ0\u0013\n=𝑎𝑣∞ℓ0\n𝑅∗+ℓ0∼𝑣∞𝑎ℓ0\n𝑅∗(10)\nThus the true parameter to investigate is 𝑎ℓ0. This is difficult to\ninterpret and compare with existing work however, as other line\nprofile models only parameterize the slow gas terminal velocity. It\nis for this reason that we choose to fix the fast-gas terminal velocity,\ni.e. assume a fixed 𝑎, and varyℓ0to better compare against spatial\nparameters in other models.\n2.2 Source Function Description\nIn this section we will discuss how the VBC model compares with\npreviouslypublishedmodelswhenputintothestandardlanguage.In\ngeneral, a line profile function should have the form\n𝐿(𝜆)=∫∞\n𝑉min𝑆(𝜆,𝑉)e−𝜏(𝜆,𝑉)d𝑉, (11)\nwhere𝑆(𝜆,𝑉)istheemissivityfunctionofthelightproducedinthe\nvolume𝑉of the wind. An example of this 𝑆function is the emis-\nsivity𝜂𝜆used in the Cohen model. If we change the integration of\nEquation(8)tothatofvolume,itfollowsthatourmodel’sequivalent\nemissivity function (when converted back to radial form) is\n𝑆VBC=d𝑃\nd𝑟e−(𝑟−𝑅∗)/ℓ0. (12)\nExample curves for this emissivity function are plotted for various\nℓ0values in Figure 2.\nFor comparison, in the bottom half of that figure we have plotted\nthe emissivity function of the Cohen model. The full emissivity\nfunction is given in Owocki & Cohen (2001), but for our purposes,\nfor a photon emitted at line center, it scales as\n𝑆Cohen =𝜂𝜆∝𝜌2(𝑟)𝑓(𝑟), (13)\nwhere𝑓isanX-rayemissionfillingfactor.Theauthorsofthismodel\nassume, based on the work of Ignace (2001), that this filling factor\nis a power law 𝑓(𝑟)∝𝑟−𝑞. Calculations of 𝑆Cohenas a function of\nradiusareshowninthebottomplotofFigure2forarangeofvalues\n0.00.20.40.60.81.0S\nVBC (Normalized emissivity)/uni2113\n0= 0.5 R\n*\n/uni2113\n0= 1 R\n*\n/uni2113\n0= 1.5 R\n*\n/uni2113\n0= 2 R\n*\n1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0\nr ( R\n*)0.00.20.40.60.81.0S\nCohℓn (Normalized emissivity)q = 0\nq = 1\nq = 2\nq = 3Figure 2. Plot of the line emissivities for the VBC model (top panel) and\nCohen model (bottom panel; Owocki & Cohen 2001), both starting at a\nradiusof 1.5𝑅∗.NotethatfortheVBCmodel,theequivalentoftheemissivity\nfunction Equation (12) is describing emission over a distance ℓ0, not a local\nemissivity. Both 𝑆VBCand𝑆Cohenare normalized at 𝑟=1.5𝑅∗.\nof𝑞. The velocity function 𝑣(𝑟)which appears in 𝜌=¤𝑀/4𝜋𝑣(𝑟)𝑟2\nis calculated using a 𝛽-law with the value 𝛽=1.\nFigure 2 highlights an important difference between these model\nparameterizations:theextentoftheX-rayemission.Usingaconstant\nor slowly varying filling factor causes the emission to stem from a\nrather narrow radius interval where the cool wind emission mea-\nsure is high. But since the VBC approach parameterizes the shock\nheating and assumes efficient radiative cooling, the cool wind emis-\nsion measure is of no consequence, and the X-ray generation can be\nmoreextendediftheshockdistributionismoreextended.Ultimately,\nonly observations and more detailed simulations can clarify which\napproach better characterizes the spatial distribution of the heating.\nTherefore our current intent is to understand how robust the conclu-\nMNRAS 000, 1–10 (2023)X-ray variations in 𝜁Pup do not require mass flux changes 5\nTable 2.Chandra HETGS Observations of 𝜁Pup.\nObs ID Exposure Time (ks) Date\n640a67.74 2000 Mar 28\n21113b17.72 2018 Jul 1\n21112b29.70 2018 Jul 2\n20156b15.51 2018 Jul 3\n21114b19.69 2018 Jul 5\n21111b26.86 2018 Jul 6\n21115b18.09 2018 Jul 7\n21116b43.39 2018 Jul 8\n20158b18.41 2018 Jul 30\n21661b96.88 2018 Aug 3\n20157b76.43 2018 Aug 8\n21659b86.35 2018 Aug 22\n21673b14.95 2018 Aug 24\n20154b46.97 2019 Jan 25\n22049b27.69 2019 Feb 1\n20155b19.69 2019 Jul 15\n22278b30.51 2019 Jul 16\n22279b26.05 2019 Jul 17\n22280b25.53 2019 Jul 20\n22281b41.74 2019 Jul 21\n22076b75.12 2019 Aug 1\n21898b55.70 2019 Aug 17\nNotes–aCXO Proposal Cycle 1 observation.\nbCXO Proposal Cycle 19 observation.\nsionsinferredfromlineprofileshapesarewhenworkingwithinthese\ntwo schemes.\n3 DATA AND MODELLING\nThe observations of 𝜁Pup included all archival HETGS data avail-\nable in the Chandra Archive. The list of observations with their\nObservationID(ObsID)aregiveninTable2.Eachobservationused\ntheChandraHETGSinstrumentwhichsimultaneouslyprovidesdata\nfromtwogratingarrays:themediumenergygrating(MEG)andhigh\nenergy grating (HEG). These gratings have resolutions of 0.023 Å\nand 0.012 Å respectively (Canizares et al. 2005).\nEach observation was retrieved from the Chandra archive and re-\nprocessedusingthestandardpipelinein ciaoversion4.13(Fruscione\net al. 2006). This process produced two first-order spectra for both\nthe HEG and MEG arrays, corresponding to the positive and nega-\ntivediffractionorders.Eachofthepositiveandnegativeorderswere\nsubsequently co-added to produce a single first order data set for\ntheirrespectivegratingarrayandObsID.FortheCycle19datasets,\nwe additionally summed the 42 datasets (21 for the HEG and MEG\nrespectively) to produce a single total observation for each of indi-\nvidual grating array. Thus in the end we had 1 HEG and 1 MEG\nspectra for Cycle 1 and 1 HEG and 1 MEG spectra for Cycle 19.\nThese final datasets were then rebinned by a constant factor of 3 formodelfitting.Thiscorrespondstobinssizesof0.015ÅfortheMEG\nand 0.0075 Å for the HEG.\nWhen comparing data taken by Chandra separated by almost two\ndecades,weneededtoaccountforchangesinthedetectorproperties\nduring that time. The most important change is a result of contam-\nination build up on Chandra’s ACIS-S detector used for HETGS\nobservations (Marshall et al. 2004; O’Dell et al. 2017). This con-\ntamination decreases the detector’s effective area, particular in the\nlong-wavelength region of HETGS spectra, reducing overall count\nrates. However, this change in the detector should not have an im-\nportant affect on the analysis carried out here. The contamination\nis calibrated in the standard data products (i.e., date-dependent re-\nsponsefiles),andwehaveusefuldiagnosticsoutto17Åasindicated\nin Figure 3. In addition, it is worthwhile to note that the Cycle 19\nobservations have a much greater exposure time, so even with the\ngreater contamination effects there are many more total counts in\nmost of the X-ray lines compared to the Cycle 1 observation.\nUsing xspecversion 12.12 (Arnaud 1996), through the python\nwrap around pyxspec, we fit the same list of lines as in G2022 in\nisolated regions. The total source model used consisted of the VBC\nmodel3discussed above plus a constant continuum component. For\nthelinefeaturesSi xiiiHe𝛼,4He-likeLyman 𝛼isshortenedtoSi xiii\nHe𝛼.Mg xiHe𝛼,Fexviiat15.01Å,andFe xviiat16.78Å,atotal\nof three line model components were used to model the three lines\nthatoccurinclosespacingatthesewavelengths.The 𝜏∗andℓ0were\nthesameforeachlineofthesemulti-componentfits,beingtiedtothe\nline with the shortest wavelength of the feature’s fitted region. The\nshortestwavelengthemissionlinewaschosenforbeingthestrongest\nsingle line in these features. Each line component in a feature had a\nfreelyfitnormalization 𝑁.WeassumedPoissonstatisticsanduseda\nCash (maximum likelihood) statistic (Cash 1979).\n3.1 Model Fitting Behavior\nThe fitting algorithm used for our analysis was the Markov chain\nMonte Carlo (MCMC) function provided in pyxspec. For each fit,\ntheresultingchainwasanalyzedforallstepsafteraconvergencewas\nachieved, giving parameter distributions like those in Figure C1 in\nAppendixCfortheMg xiiH𝛼line.Thislineiswithinanisolatedre-\ngionwithlittlecontinuum,soitprovidesawellconstrainedexample\nof the model behavior,\nThere are two interesting trends that can be seen in the parameter\npairs. First is the inverse relationship between 𝜏∗andℓ0. This is\nsimilar to the discussion in G2022 when comparing observable-\nparameterpaircontours.Amoreinterestingeffectisbetween ℓ0and\nthe normalization, which show a slight positive correlation. This is\nlikely due to the factor of 1/ℓ0that applies to the entire integral in\nEquation (8) through the relationship defined in Equation (7). Since\nsuch a constant factor would normally be considered a part of the\nnormalization, it is not surprising that these would be linked in this\nway.\n4 RESULTS\nThe results of our modelling are summarized in Table 3 where we\nreport the best-fit values of free parameters in our model. Note that\n3Sourcecodeforimplementingthemodellocallycanbefoundhere: https:\n//github.com/sjgunderson/VBC_Line_Model\n4In this paper, we shorten the H- and He-like Lyman transition names for\nconciseness. For example, Si xiii\nMNRAS 000, 1–10 (2023)6S. J. Gunderson et al.\nforlinefeaturesfitwithmultiplecomponentswereportthetotalflux\nin the feature. These best-fit values were extracted from the MCMC\nchains after they stabilized along with their 68 per cent confidence\nintervals.TheactualmodelfitsareplottedinAppendixD.Figures3\nand 4 show these parameter results visually.\n4.1 Determination of Mass-loss Rate\nFocusing first on Figure 3, we can draw two important conclusions.\nFirst, it appears that mass-loss rates calculated from X-ray line pro-\nfilesisindependentoftheassumedheatingmodel.Thebest-fitvalues\nfor our Cycle 1 analysis (blue dots) and C2010 (green up-triangles)\nshowanover-allconsistencywhenlookingat68percentconfidence\nintervals. This is further bolstered by the values from our Cycle 19\nanalysis (orange squares) as the consistency is maintained. For all\nthree fit groups, we would find the same mass-loss rate, since 𝜏∗is\na simple rescaling of ¤𝑀based on Equation (6), save for deviations\naccounted for by errors in a trend-line fitting process.\nGiven this overall consistency, the second conclusion we draw\nfrom Figure 3 is that 𝜁Pup’s mass-loss rate has notchanged. If\nsuch a change occurred, particularly one suggested to be so large,\nour independent analysis on the same datasets would have found a\nsimilar result. However, we do confirm that there is a real change\nin the lines. The flux in all but one of our fitted line features has\nincreased, just as was found by C2020. Combined with the XMM-\nNewtonflux in Figure 1, there is high confidence in more X-rays\nbeing produced and/or escaping the wind.\n4.2 Issues with Epochal Analysis\nThe result that mass-loss rate determinations from X-ray lines is\nmodelindependentisincontrasttoG2022thatconcludedtheoppo-\nsite by analyzing only Cycle 19 data. The choice in only looking at\nCycle19datamayhavebeenthesourceoftheerroneousconclusion\nas G2022 was comparing against the results of C2020, which is the\nonly set to break from the noted consistency. Why this is the case is\nnot immediately clear. For lines below 10 Å, all four groups are in\nagreementexceptforSi xiiiHe𝛼,thoughonlymarginallyso.Forthe\nlonger wavelengths, there are large differences in the values found\nfor the Cohen model. It is possible that the limited fitting region of\ntheNe xH𝛼andFe xviiat15ÅlinesusedbyC2020isresponsible\nforsomeoftheinflatedvaluesof 𝜏∗.Thefitspresentedherecontrast\nthis by fitting the entire red-half of the profile of Ne xH𝛼and all\nthree lines in the Fe xviiat 15 Å feature (Figures D3 and D6). This\nis simply speculation as to what differences may exist between the\ntwomodellingapproachesthatwouldresultinthedifferent 𝜏∗values\nfound. Whatever the source is, there may be anomalous behavior in\ntheC2020fittinganalysisthatisalsoresponsiblefortheirclaimofa\n40 per cent increase in 𝜁Pup’s mass-loss rate.\nFrom their values in Table 3, we argue that this difference stems\nfromtheℓ0parameter,whichisalsothesourceofthelineshapedif-\nferencesthatwerenotedbyC2020.Putanotherway:thedistribution\nof the embedded wind shocks has changed. Figure 4 more clearly\nshows that the mean-free path between shocks in 𝜁Pup’s wind has\nincreased by a significant amount. Note that in this figure we plot\n𝑅∗+ℓ0so as to compare directly with the turn-on radius 𝑅0of the\nCohen model. What this means is that the shocks are occurring far-\nther out in the wind, allowing for more X-ray emission to escape,\nthereby raising the amount of flux that we see.\nThis presents two competing interpretations of the same data:\neither the mass-loss rate has changed or the shock distribution. Incomparing these, it is worth considering the implications each has\nover the entire wind. Given C2020’s increased mass-loss rate, the\nentirewindisrequiredtobechangedtoaccountforthedifferenceswe\nseeinthelineprofileshapes.Withoutevidenceof 𝜁Puphavinglarge\nscale variabilities, i.e. those that persist after time-averaging during\nanobservation,thatalterthestellarparameters,suchalargeincrease\ninthemass-lossratewouldimplyeitherthat 𝜁Puphasundiscovered\nproperties or our understanding of massive stars, particularly O-\ntypes,ismissinganimportantpiece.Giventheimportanceof 𝜁Pup\nas a prototype, guide star, and calibration target, it has many further\nimplicationsineveryotherwaveband.TheconclusionfromtheVBC\nmodelanalysisthatonlytheshockdistributionhaschanged,andnot\nthe global mass-loss rate, has less drastic implications involving a\nmuch smaller fraction of the wind.\n4.3 Shock-Temperature Spatial Distribution\nTheprimarymotivationforusingtheVBCanalysisistounderstand\nhowthegasisheated,nottodeterminethecoolwindmasslossrate.\nSo now we turn to the issue of using the VBC parameterization to\ninterpret the shock distribution, and contrast that with the picture\nfrom Cohen’s parametrization. For example, part of the motivation\nof using a turn-on radius 𝑅0in the Cohen model is the idea that the\nline deshadowing instability requires a certain stand-off distance in\nordertohaveroomtotakeeffect,whereastheVBCpictureholdsthat\nfastgasisdestinedtoovertakeslowgasoverthelengthscale ℓ0.Inthe\nlatter picture, it is natural that more significant boundary variations\nshould yield more violent encounters between fast and slow wind,\nallatlowerℓ0,whereasintheformerpicture,therecouldbemoreof\na tendency for 𝑅0to be independent of shock strength, or even rise\nwith shock strength (Driessen et al. 2021).\nG2022visuallyidentifiedatrendin ℓ0vs.thetemperatureofmaxi-\nmumemissivity 𝑇max.Thattrendcanbedemonstratedquantitatively\nusing the Pearson correlation coefficients\n𝜌𝑥𝑦=𝑛Í����𝑖𝑦𝑖−Í𝑥𝑖Í𝑦𝑖√︃\n𝑛Í𝑥2\n𝑖−(Í𝑥𝑖)2√︃\n𝑛Í𝑦2\n𝑖−(Í𝑦𝑖)2(14)\nfor the plotted points in Figure 4. The values of the coefficients\n𝜌ℓ0𝑇maxare given in Table 4. Note that this correlation coefficient is\ninvariant under linear transformations, so the calculated values are\nthesameforboth ℓ0and𝑅∗+ℓ0.Additionally,wecalculated 𝜌ℓ0𝑇max\nusing log(𝑇max), but for readability of the coefficient subscripts, we\nomit the logarithm.\nWeused arandomsampling methodofthe ℓ0parameterdistribu-\ntionstocalculatethereportedvalues.Thecentralvaluesarethemost\nfrequent of the resulting 𝜌ℓ0𝑇maxdistribution while the uncertainties\nare then taken as the 68 per cent levels.\nWe are interested in investigating trends related to the shock\nstrength, for which the temperature of maximum emissivity 𝑇max\nis a useful proxy. From the values in Table 4, a negative correlation\ndoesexistbetweenthe ℓ0andtheshockstrengthforbothCycle1and\nCycle19.Thetrendisnotexactlylinear,whichwouldcorrespondto\n𝜌ℓ0𝑇max=−1, but it does imply an inverse trend nonetheless.\nOf note is how the Cycle 1 results were handled. Two of the lines\nfit with our model, Si xivH𝛼and Ne ixHe𝛽, show significantly\nmoreuncertaintyintheirbestfit ℓ0comparedtotherestofthelines.\nWhen these points are included, the correlation is consistent with 0,\nmeaningnolinearrelationshipexists.However,therestoftheCycle1\nresultsshowatightertrendwith log(𝑇max),sowealsocalculatedthe\ncorrelation coefficients with these two lines removed. The restricted\ncoefficient is constrained to only negative values and is statistically\nMNRAS 000, 1–10 (2023)X-ray variations in 𝜁Pup do not require mass flux changes 7\nTable 3.VBC model fitting results for Cycles 1 and 19.\nLine 𝜆p(Å)alog(𝑇max)(log(K))b𝜏∗ ℓ0(𝑅∗) Flux ( 10−5ph cm−2s−1)\nCycle 1 Cycle 19 Cycle 1 Cycle 19 Cycle 1 Cycle 19\nSixiiiHe𝛽c5.681 7.07 – 0.68+0.32\n−0.20– 1.18+0.42\n−0.16– 0.90+0.07\n−0.07\nSixivH𝛼 6.180 7.40 0.17+1.37\n−0.0040.26+0.10\n−0.061.03+4.34\n−0.150.85+0.14\n−0.100.75+0.19\n−0.151.05+0.05\n−0.05\nSixiiiHe𝛼d6.648, 6.688, 6.740 7.03 0.34+0.13\n−0.100.51+0.06\n−0.050.96+0.17\n−0.091.35+0.07\n−0.0512.63+0.45\n−0.4015.18+0.16\n−0.14\nMgxiiH𝛼 8.419 7.19 0.75+0.35\n−0.230.60+0.07\n−0.060.82+0.29\n−0.151.18+0.08\n−0.073.00+0.31\n−0.283.84+0.10\n−0.09\nMgxiHe𝛼d9.169, 9.231, 9.314 6.84 0.75+0.18\n−0.180.60+0.07\n−0.071.15+0.20\n−0.111.69+0.12\n−0.0919.67+0.70\n−0.6022.96+0.28\n−0.31\nNexH𝛽 10.238 6.97 1.63+0.76\n−0.311.84+0.24\n−0.220.82+0.47\n−0.120.88+0.16\n−0.093.62+0.58\n−0.393.59+0.23\n−0.22\nNeixHe𝛽 11.544 6.64 0.54+0.48\n−0.181.34+0.23\n−0.162.41+1.58\n−0.431.46+0.23\n−0.046.07+0.76\n−0.827.07+0.40\n−0.37\nNexH𝛼 12.132 6.94 1.41+0.19\n−0.151.71+0.12\n−0.111.03+0.15\n−0.101.32+0.11\n−0.0726.92+1.36\n−1.4527.24+0.80\n−0.81\nFexviid15.014, 15.176, 15.261 6.73 1.48+0.22\n−0.181.57+0.16\n−0.151.27+0.17\n−0.101.75+0.15\n−0.1079.13+3.81\n−3.0192.54+2.55\n−2.62\nFexviid16.780, 17.051, 17.096 6.70 2.75+0.67\n−0.373.14+0.80\n−0.491.28+0.19\n−0.142.06+0.43\n−0.2762.67+2.95\n−2.9182.40+4.23\n−3.73\nNotes–aPredicted line center wavelength as reported by AtomDB (Foster et al. 2012; Smith et al. 2001).\nbTemperature of maximum emissivity as reported by AtomDB (Foster et al. 2012; Smith et al. 2001).\ncThis line is not confidently detected in the Cycle 1 data.\ndThe reported flux for these line features is the total flux in the three components used.\n6 8 10 12 14 16\nWavelength (Å)01234τ\n*This work Cycle 1\nThis work Cycle 19\nC2010 Cycle 1\nC2020 Cycle 19\nFigure 3. Plot of best-fit 𝜏∗values for the four analyses discussed. Error bars represent 68 per cent uncertainty. Note that we have applied constant offsets to\nthe different result groups to make the error bars more visible. The plotted wavelengths correspond to 𝜆pfor the blue dots, 𝜆p+0.1Å for the orange squares,\n𝜆p+0.2Å for the green up triangles, and 𝜆p+0.3Å for the red down triangles. Data points that do not have a corresponding group of four are due to that line\nfeature not being fit in some of the works considered here.\nMNRAS 000, 1–10 (2023)8S. J. Gunderson et al.\n6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4\nlog( T\nmax) log (K )1.01.52.02.53.03.51 + /uni2113\n0, ℓ\n0 ( ℓ\n*)This work Cycle 1\nThis work Cycle 19\nC2010 Cycle 1\nC2020 Cycle 19\nFigure 4. Plot of best-fit spatial parameters values for the four analyses\ndiscussed against the temperatures of maximum emissivity taken from the\nAPED. The VBC model’s parameter is plotted as 𝑅∗+ℓ0in units of stellar\nradii.Errorbarsrepresent68percentuncertainty.Pointswithlargeerrorsare\nnot plotted in entirety for readability and are represented by error bars with\narrows.\nTable 4.Pearson correlation coefficients.\nDataset 𝜌ℓ0𝑇max\nCycle 1−0.56+1.02\n−0.06a\n−0.82+0.42\n−0.06b\nCycle 19−0.75+0.11\n−0.07Notes–aCorrelationcoefficientwhenoutlier\nvalues from Si xivH𝛼and Ne ixHe𝛽are\nincluded.\nbCorrelation coefficient without outlier val-\nues from Si xivH𝛼and Ne ixHe𝛽.\nconsistent with the computed value with all lines considered. If the\nuncertainties in the outlier lines can be better constrained, the true\nvalue is likely to be within the overlap of these two values. These\ncoefficients confirm the prediction that under a VBC initiation for a\nmassive star’s wind, the stronger shocks will occur at lower radii.\nThere are interesting ramifications for how the shock heating dis-\ntribution appears to have changed with time. While 𝑅0parameter\nin the Cohen model is broadly unchanged, the ℓ0parameter of the\nVBC model has increased significantly. Such a difference between\nthe temporal changes in the shock distributions is likely explained\nfrom the fact that in the VBC model it is only the hot gas that has\nchanged. This requires a more significant adjustment of the heating\ndistributioninsteadoftheentirewind.Thelatterofwhichisrequired\nby the Cohen model based on the wind mass flux increase (C2020).\nThus we can conclude that the hot gas has moved to farther radii in\nrecent years while the cool, unshocked gas has not changed, though\nsome changes can not be ruled out.\n5 CONCLUSIONS\nWepresentnewandbetterconstrainedresultsfortheVBClinemodel\nfit to both epochs of X-ray line emission data on 𝜁Pup as observed\nbyChandra.Basedonthebest-fittingvaluesofthe 𝜏∗parameter,we\nfind that𝜁Pup’s mass-loss rate has not undergone any large-scale\nchangeswithintheneartwodecadebetweenofthetwoobservations.\nAdditionally, the consistency between our Cycle 1 and 19 𝜏∗values\nand those of C2010 implies that mass-loss rates inferred using X-raylineemissionareindependentofmodelparameterizingandthus\nrobust as a technique.\nItisconcerningthenthata40percentchangeinthemass-lossrate\nshowsupinoneparameterizationandnotintheother.However,rather\nthan reflecting an overall incompatibility between the two models,\nthe difference is possibly from the specifics of the line analysis. For\nexample,thereappearstobeadependenceonwhetheranentireline\nfeature, such as the three resolved lines of the Fe xviiat 15 Å, is fit\nversus a single line of the feature. This is based on our modelling\naccounting for the three lines in this feature for both Cycle 1 and 19\nand finding a smaller 𝜏∗value than C2010 and C2020, the former\nstillbeingwithinerror,whichfitonlytheFe xviilineat15.014Å.It\nremainsunclearwhatsystematicuncertaintiesarepresentinthemass\nfluxcalculationsduetothecomplexityofthemodellingchoicesand\nad-hoc nature of the heating distributions used in the two models\ndiscussed. Further work should be done on two fronts to confront\ntheseproblems.First,thesourceofuncertaintiesinmodellingshould\nbe investigated as the potential source for the discrepancies noted in\nthiswork.Secondly,modelswithmoreself-consistentshockphysics\nshould be developed to better approach this problem.\nIf indeed𝜁Pup’s mass-loss rate has not changed, we must then\nexplain why there are observed increases in the line fluxes that are\nnot due to changes in Chandra’s detectors. Our spatial parameter\nℓ0provides such an explanation: the embedded wind shocks are\noccurring at farther radii now than before. The further out a shock\noccurs, the more photons can escape since they travel through less\nmaterial. The increase in ℓ0can be traced back to the surface where\nthelagbetweentheboundaryvariationsisgettinglonger,givingthe\nslower gas a longer head start before the fast gas is launched. So if\nthe shocks are seeded by boundary fluctuations, this would suggest\nthe timescale of the fluctuations has gotten longer. That any such\nchange is possible on a 20-year timescale is already interesting, and\na challenge to interior models, but some sort of stellar cycle seems\npotentiallyrelevant.Futuremonitoringof 𝜁PupintheX-raybandis\nneeded to test the apparent long-term variations.\nACKNOWLEDGEMENTS\nThe scientific results in this article are based on data retrieved from\ntheChandra Data Archive, software provided by the Chandra X-\nray Center (CXC) in the application packages ciao, and software\nprovidedbyNASA’sHighEnergyAstrophysicsScienceArchiveRe-\nsearch Center (HEASARC) in the application xspec. This research\nwas supported in part through high performance computational re-\nsources provided by The University of Iowa, Iowa City, Iowa. This\nresearch has made use of data obtained from the 4XMM XMM-\nNewtonSerendipitous Source Catalog compiled by the 10 institutes\nof theXMM-Newton Survey Science Centre selected by ESA.\nSupport for SJG was provided by the University of Iowa’s CLAS\nDissertationWritingFellowshipandaChandraX-rayCenter(CXC)\nResearchVisitorAward.SupportforDPHandSJGwasprovidedby\nNASA through the Smithsonian Astrophysical Observatory (SAO)\ncontract SV3-73016 to MIT for Support of the CXC and Science\nInstruments. CXC is operated by SAO for and on behalf of NASA\nunder contract NAS8-03060.\nThis work was original published in SJG’s doctoral thesis in par-\ntial fulfillment of the University of Iowa’s Graduate College Ph.D.\nrequirements. (Gunderson 2023)\nWe also thank R. Ignace and C. Erba for their discussions on\nputting the VBC model into context of the usual modelling ap-\nproaches and interpreting our results.\nMNRAS 000, 1–10 (2023)X-ray variations in 𝜁Pup do not require mass flux changes 9\nDATA AVAILABILITY\nTheX-rayspectraldatausedinthisarticleareavailableinthe Chan-\ndraData Archive at https://cxc.cfa.harvard.edu/cda/ . Ob-\nservations are uniquely identified by an observation identifier (Obs\nID) given in Table 2.\nREFERENCES\nArnaud K. A., 1996, in Jacoby G. H., Barnes J., eds, Astronomical Society\nof the Pacific Conference Series Vol. 101, Astronomical Data Analysis\nSoftware and Systems V. p. 17\nBalona L. A., 1992, MNRAS, 254, 404\nBerghoefer T. W., Baade D., Schmitt J. H. M. M., Kudritzki R. P., Puls J.,\nHillier D. J., Pauldrach A. W. A., 1996, A&A, 306, 899\nCanizares C. R., et al., 2005, PASP, 117, 1144\nCash W., 1979, ApJ, 228, 939\nCastor J. I., Abbott D. C., Klein R. I., 1975, ApJ, 195, 157\nCohen D. H., Cassinelli J. P., MacFarlane J. 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J., Otrupcek R., 2001, MNRAS, 325, 1213\nAPPENDIX A: OPTICAL DEPTH EXPRESSION: REAL VS.\nCOMPLEX\nAswasnotedinthediscussionoftheminimumradiusfortheintegral\nof Equation (8) in G2022, complex numbers tend to show up due to\ntherootsthatarebeingtakenintheformulasintheVBCmodel.One\nmore source of complex numbers that has not been mention is that\nof the optical depth in Equation (2).\nTo be clear, as a variable corresponding to a physical quantity,\n𝜏(𝜇,𝑟;𝜏∗)isalwaysreal.Theproblemofcomplexnumbersarisesin\nthesub-equationsthatmakeitup.Specifically, 𝑧𝑡fromEquation(4)\nbecomesimaginarywhen 𝑟 < 𝑅∗/√︁\n1−𝜇2.Ifonewishesorneedsto\navoidthisintheirownimplementationoftheVBCmodelequations,\nthe following expanded form of Equation (3) can be used\n𝑡±= \narctan\u0010𝑅∗\n����𝑡\u0011\n±arctan\u0010𝛾\n𝜇\u0011\n𝑟≥𝑅∗√\n1−𝜇2\n−\u0010\narctanh\u0010𝑅∗\n𝑧𝑡\u0011\n±arctanh\u0010𝛾\n𝜇\u0011\u0011\n𝑟 <𝑅∗√\n1−𝜇2,(A1)\nwhich then requires that\n𝑧𝑡= \n√︃\u00001−𝜇2\u0001𝑟2−𝑅2∗𝑟≥𝑅∗√\n1−𝜇2√︃\n𝑅2∗−\u00001−𝜇2\u0001𝑟2𝑟 <𝑅∗√\n1−𝜇2.(A2)\nAPPENDIX B: GAUSS-LAGUERRE QUADRATURE\nAPPROXIMATION\nTheuseofaGaussianquadraturealgorithmforcalculatingnumerical\nintegrals is a well-known method of fast and precise approximation,\nparticularlyforsituationsofinfiniteintegrationbounds.FortheVBC\nmodel, the Gauss-Laguerre quadrature is the appropriate option to\nuse when re-written in an applicable form. To do this, we will make\nthe substitution 𝑦=(𝑟−𝑟𝑚(𝜉))/ℓ0. This turns Equation 8 into the\nform\n𝐿(𝜉)=e−(𝑟𝑚(𝜉)−𝑅∗)/ℓ0∫∞\n0e−𝑦ℎ(𝑦,𝜉)d𝑦 (B1)\nwhere\nℎ(𝑦,𝜉)=1\n2𝑦ℓ0+𝑟𝑚(𝜉)\n𝑦ℓ0+𝑟𝑚(𝜉)−𝑅∗e−𝜏(𝑦,𝜉)(B2)\nWe can then write a semi-analytic expression for the line profile\n𝐿(𝜉)≈e−(𝑟𝑚(𝜉)−𝑅∗)/ℓ0𝑛∑︁\n𝑖=1𝑤𝑖ℎ(𝑦𝑖,𝜉). (B3)\nHere𝑦𝑖are the𝑖-th roots of the classical orthogonal Laguerre poly-\nnomialL𝑛and the weights are\n𝑤𝑖=𝑦𝑖\n(𝑛+1)2L𝑛+1(𝑦𝑖)2(B4)\nTo show that this semi-analytic form does not introduce uncer-\ntainties greater than the instrumental sources, we have plotted the\nMNRAS 000, 1–10 (2023)10S. J. Gunderson et al.\n−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00\nξ0.0000.0020.0040.0060.0080.0100.0120.0140.016|L\nQ− L\nGL|/(( L\nQ+ L\nGL)/ 2)\nFigure B1. Relative difference between the general purpose quadrature 𝐿Q\nand Gauss-Laguerre quadrature 𝐿GLintegration methods available in the\nVBC model codes.\nrelativedifferencebetweenusing scipy’sgeneralpurposequadrature\nalgorithm𝐿Qand the Gauss-Laguerre algorithm 𝐿GLfor𝑛=20in\nFigureB1.Atthegreatest,thedifferencebetweenthesetwomethods\nare of order 1.7 per cent, far below any uncertainties from statistical\nsources.Bothofthesemethodsareincludedintheavailablefilesfor\nimplementingtheVBCmodelonpersonaldistributionsof pyxspec.\nAPPENDIX C: VBC PARAMETER DISTRIBUTION\nEXAMPLE\nAPPENDIX D: FIT PLOTS\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\nMNRAS 000, 1–10 (2023)X-ray variations in 𝜁Pup do not require mass flux changes 11\n0250005000075000100000125000150000175000200000\n0.91.01.11.21.31.41.51.6ℓ\n0 ( R\n*)\n050000100000150000200000\n0.4 0.6 0.8\nτ\n*3.43.63.84.04.2N (10−5\n ph cm−2\n s−1\n)\n0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6\nℓ\n0 ( R\n*)3.4 3.6 3.8 4.0 4.2\nN (10−5\n ph cm−2\n s−1\n)05000010000015000020000068 percent\n90 percent\n95 percent\nFigureC1. Exampleparameter-pairconfidencecontoursfortheVBCmodelfittoMg xiiH𝛼with𝑣∞,𝑓frozen.Dashedverticallinesinthemarginaldistributions\nare the 68 per cent levels for that parameter. Parameter-pair contours are representative of model behavior for most lines fit.\nMNRAS 000, 1–10 (2023)12S. J. Gunderson et al.\nSi XIV H αSi XIV H α\nSi XIII He αSi XII He α\nMg XII H αMgXII H α\nFigure D1. Cycle 1 best fit plots of Si xivH𝛼, SixiiHe𝛽, and Mg xiH𝛼. Left column is the HEG data while the right is the MEG. Fits that used more than\none component also include the individual components. All plots are shown binned by a factor 3, as was used for the fitting process.\nMNRAS 000, 1–10 (2023)X-ray variations in 𝜁Pup do not require mass flux changes 13\nMgXI He αMgXI He α\nNeXH βNeXH β\nNeIXHe βNe IX He β\nFigure D2. Cycle 1 best fit plots of Mg xiHe𝛼, NexH𝛽, and Ne ixHe𝛽. Left column is the HEG data while the right is the MEG. Fits that used more than\none component also include the individual components. All plots are shown binned by a factor 3, as was used for the fitting process.\nMNRAS 000, 1–10 (2023)14S. J. Gunderson et al.\nNe X H αNe X H α\nFe XVIIFe XVII\nFe XVIIFe XVII\nFigureD3. Cycle1bestfitplotsofNe xH𝛼,Fexviiat15.01Å,andFe xviiat16.68Å.LeftcolumnistheHEGdatawhiletherightistheMEG.Fitsthatused\nmore than one component also include the individual components. All plots are shown binned by a factor 3, as was used for the fitting process.\nMNRAS 000, 1–10 (2023)X-ray variations in 𝜁Pup do not require mass flux changes 15\nSi XIII He βSi XIII He β\nSi XIV H αSi XIV H α\nSi XIII He αSi XII He α\nFigure D4. Cycle 19 best fit plots of Si xiiiHe𝛽, SixivH𝛼, and Si xiiiHe𝛼. Left column is the HEG data while the right is the MEG. Fits that used more\nthan one component also include the individual components. All plots are shown binned by a factor 3, as was used for the fitting process.\nMNRAS 000, 1–10 (2023)16S. J. Gunderson et al.\nMg XII H αMgXII H α\nMgXI He αMgXIHe α\nNe X H βNe X H β\nFigureD5. Cycle19bestfitplotsofMg xiiH𝛼,Mg xiHe𝛼,andNe xH𝛽.LeftcolumnistheHEGdatawhiletherightistheMEG.Fitsthatusedmorethan\none component also include the individual components. All plots are shown binned by a factor 3, as was used for the fitting process.\nMNRAS 000, 1–10 (2023)X-ray variations in 𝜁Pup do not require mass flux changes 17\nNe IX He βNeIXHeb\nNeXH αNeX H α\nFe XVIIFe XVII\nFigureD6. Cycle19bestfitplotsofNe ixHe𝛽,NexH𝛼,andFe xviiat15.01Å.LeftcolumnistheHEGdatawhiletherightistheMEG.Fitsthatusedmore\nthan one component also include the individual components. All plots are shown binned by a factor 3, as was used for the fitting process.\nMNRAS 000, 1–10 (2023)18S. J. Gunderson et al.\nFe XVIIFe XVII\nFigure D7. Cycle 19 best fit plots of Fe xviiat 16.78 Å. Left column is the HEG data while the right is the MEG. Fits that used more than one component also\ninclude the individual components. All plots are shown binned by a factor 3, as was used for the fitting process.\nMNRAS 000, 1–10 (2023)"    },    {        "title": "2402.13107v2.An_Improved_Lower_Bound_on_the_Number_of_Pseudoline_Arrangements.pdf",        "content": "An Improved Lower Bound on the Number of\nPseudoline Arrangements∗\nFernando Cortés Kühnast /envel⌢pe/orcid\nInstitute of Mathematics, Technische Universität Berlin, Germany\nJustin Dallant /envel⌢pe/orcid\nAlgorithms Research Group, Université libre de Bruxelles, Belgium\nStefan Felsner /envel⌢pe/orcid\nInstitute of Mathematics, Technische Universität Berlin, Germany\nManfred Scheucher /envel⌢pe/orcid\nInstitute of Mathematics, Technische Universität Berlin, Germany\nAbstract\nArrangements of pseudolines are classic objects in discrete and computational geometry. They\nhave been studied with increasing intensity since their introduction almost 100 years ago. The\nstudy of the number Bnof non-isomorphic simple arrangements of npseudolines goes back to\nGoodman and Pollack, Knuth, and others. It is known that Bnis in the order of 2Θ(n2)and finding\nasymptotic bounds on bn=log2(Bn)\nn2remains a challenging task. In 2011, Felsner and Valtr showed\nthat 0.1887≤bn≤0.6571for sufficiently large n. The upper bound remains untouched but in 2020\nDumitrescu and Mandal improved the lower bound constant to 0.2083. Their approach utilizes the\nknown values of Bnfor up ton= 12.\nWe tackle the lower bound by utilizing dynamic programming and the Lindström–Gessel–Viennot\nlemma. Our new bound is bn≥0.2721for sufficiently large n. The result is based on a delicate\ninterplay of theoretical ideas and computer assistance.\n2012 ACM Subject Classification Mathematics of computing →Combinatorics; Theory of compu-\ntation→Computational geometry\nKeywords and phrases counting, pseudoline arrangement, recursive construction, bipermutation,\ndivide and conquer, dynamic programming, computer-assisted proof\nDigital Object Identifier 10.4230/LIPIcs.SoCG.2024.XX\nSupplementary Material Our supplemental data is available on github [1]\nFunding S. Felsner was partially supported by DFG Grant FE 340/13-1. F. Cortés Kühnast and\nM. Scheucher were supported by DFG Grant SCHE 2214/1-1. J. Dallant was supported by the\nFrench Community of Belgium via the funding of a FRIA grant.\n1 Introduction\nLevi [22] introduced arrangements of pseudolines as a natural generalization of line arrange-\nments in 1926. Ringel studied them in the 1950’s and Grünbaum [ 16] popularized them in his\n1972 monograph Arrangements and Spreads . In the 1980’s Goodman and Pollack initiated a\nthorough study and related them to many other objects of interest in discrete geometry. For\n∗This article is a merge of the following two independent SoCG’24 submissions:\n1) Justin Dallant – Improved Lower Bound on the Number of Pseudoline Arrangements [7]\n2) Fernando Cortés Kühnast, Stefan Felsner, and Manfred Scheucher – An Improved Lower Bound on\nthe Number of Pseudoline Arrangements [ 21]. First steps were made in the Bachelor’s thesis of the first\nauthor [6].\nWhile in 1) and 2) the estimates bn≥0.2604andbn≥0.2526were shown, we here establish\nbn≥0.2721by combining our two approaches.\n©Fernando Cortés Kühnast, Justin Dallant, Stefan Felsner, and Manfred Scheucher;\nlicensed under Creative Commons License CC-BY 4.0\n40th International Symposium on Computational Geometry (SoCG 2024).\nEditors: Wolfgang Mulzer and Jeff M. Phillips; Article No.XX; pp.XX:1–XX:19\nLeibniz International Proceedings in Informatics\nSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, GermanyarXiv:2402.13107v2  [math.CO]  21 Mar 2024XX:2 An Improved Lower Bound on the Number of Pseudoline Arrangements\na more detailed account to the history and the relevant references we refer the interested\nreader to the handbook article [11].\nAnarrangement of pseudolines in the Euclidean plane R2is a finite family of simple\ncurves, called pseudolines, such that each curve approaches infinity in both directions and\nevery pair intersects in exactly one point where the two curves cross. More generally, we\ncall a collection of pseudolines partial arrangement if every pair intersects in at most one\ncrossing-point. Pseudolines which do not intersect are said to be parallel. Note that, while\nfor partial arrangements of properlines the relation ’parallel’ is transitive, this is no longer\ntrue in partial pseudoline arrangements.\nIn this article, the focus will be on simplearrangements, that is, no three or more\npseudolines intersect in a common point (called multicrossing ). Moreover, we consider all\narrangements to be marked, that is, they have a unique marked unbounded cell, which is\ncallednorth-cell . Two arrangements are isomorphic if one can be mapped to the other by an\norientation preserving homeomorphism of the plane that also preserves the north-cell.\nWhile it is known that the number Bnof non-isomorphic arrangements of npseudolines\ngrows as 2Θ(n2), it remains a challenging problem to bound the multiplicative factor of\nthe leading term of log2Bn= Θ(n2). Determining precise values for small values of nis\na challenging task as well, see Table 1. Our focus will be on finding better estimates on\nthe lower bound constant c−:=lim infn→∞log2Bn\nn2. One can analogously define the upper\nbound constant c+:=lim supn→∞log2Bn\nn2. It seems to be open whether c+andc−coincide,\ni.e., whether the limit exists.\nn B n\n3 2\n4 8\n5 62\n6 908\n7 24698\n8 1232944\n9 112018190 [19, Section 9]\n10 18410581880 [10]\n11 5449192389984 [28]\n12 2894710651370536 [26]\n13 2752596959306389652 [18]\n14 4675651520558571537540Yuma Tanaka 201315 14163808995580022218786390\n16 76413073725772593230461936736 Günter Rote 2021 [17]\nTable 1 The known values of Bn. Sequence A6245 in the OEIS [27].\nA lot of work has been done on finding good estimates for c−andc+. In the 1980’s Good-\nman and Pollak [ 15] investigated pseudopoint configurations, which are dual to pseudoline\narrangements, and established the lower bound c−≥1\n8. An alternative and slightly easier\nconstruction for c−≥1\n12was given by Matoušek in his Lectures on Discrete Geometry [24,\nChapter 6]. As pointed out in [ 8] one also obtains c−≥1\n8from Matoušek’s construction\nvia recursion. Concerning the upper bound, Edelsbrunner, O’Rourke and Seidel [ 9] showed\nc+<∞via an algorithm that constructs an arrangement of npseudolines in O(n2)time. It\nis worth noting that an upper bound can also be obtained via planar graphs: The dual graphF. Cortés Kühnast, J. Dallant, S. Felsner, and M. Scheucher XX:3\nof arrangement of npseudolines is planar quadrangulation on m=/parenleftbign\n2/parenrightbig\n+n+ 1vertices. Since\nthe number of m-vertex planar graphs is at most 2O(m)[3, 4], it follows that Bn≤2O(n2).\nIn the 1990’s Knuth [ 19, Section 9] improved the bounds to c−≥1\n6andc+<0.7925, and\nconjectured that c+≤0.5. The upper bound was lowered to c+<0.6974by Felsner [ 10].\nIn 2011, Felsner and Valtr [ 13] further narrowed the gap by showing c−>0.1887and\nc+<0.6571, and, in 2020 Dumitrescu and Mandal [ 8] proved the currently best lower\nboundc−>0.2083.\nIn this article, we make a substantial step on the lower bound by proving c−>0.2721.\n▶Theorem 1. The number Bnof non-isomorphic simple arrangements of npseudolines\nsatisfies the inequality Bn≥2cn2−O(nlogn)withc>0.2721.\n2 Outline\nOur approach is in the spirit of several previous bounds. We consider a specific partial\narrangementLofnlines consisting of kbundlesL1,...,Lkof equally-spaced parallel lines.\nWe then define a class of local perturbations to Land consider the number of arrangements\nthat can be obtained by these perturbations. This number is a lower bound on Bn, and\nit can be improved by recursively applying the same construction to each of the parallel\nclassesLi.\nThe main difference between the approaches lies in the number of bundles kand the\nchoice of locality. Matoušek and also Felsner and Valtr used three bundles but the locality\nwas increased from considering just a triple intersection with its two simple resolutions to the\nfull intersection pattern of three bundles. Dumitrescu and Mandal [ 8] increased the number k\nof bundles to 12 but restricted the locality to the possible resolutions of the multicrossings.\nOur approach combines higher values of k(we discuss cases with kequal to 4, 6, and 12\nin this paper) with an increased locality for the perturbations, i.e., we allow reroutings of sets\nof pseudolines in designated regions, which we call patches. Figure 1 gives an illustration.\nWhen rerouting a partial arrangement within a patch P, the order of the crossings along\nthe pseudolines may change. In Section 5.1 we show that the boundary information of P\ndetermines which pairs of pseudolines cross within P, but the order of crossings along the\npseudolines is not determined in general. Outside of P, the arrangement remains unaffected,\nwhich allows us to count the number of reroutings for each patch independently. The\ntotal number of perturbations is obtained as the product of the numbers computed for the\nindividual patches. Details on how we computed the number of possibilities within a patch\nare given in Section 5 .\nTo eventually use computer assistance, we choose patches of high regularity and reasonably\nsmall complexity. In fact, since the initial arrangement is extremely regular, it is sufficient\nto determine the rerouting possibilities only for a small number of patch-types. Only a\nnegligible fraction of patches along the boundaries are different. As we only want to find an\nasymptotic lower bound on Bn, the small number of irregular patches along the boundary of\nthe regions will be ignored for the counting.\nTo eventually prove Theorem 1, we perform the following two steps:\nIn the first step (Section 3) we specify the parameters of the construction: For k= 4\n(Section 3.1), k= 6(Section 3.2), and k= 12(Section 3.3) we construct the initial\narrangement with kbundles of/floorleftbign\nk/floorrightbig\nparallel lines and cover the multicrossing points by\npatches. By resolving the multicrossing points within the patches, and taking the product\nSoCG 2024XX:4 An Improved Lower Bound on the Number of Pseudoline Arrangements\nL1\nL2\nL3\nFigure 1 Left: An arrangement of k= 3bundles of parallel lines and a collection of interior-\ndisjoint patches (highlighted red) such that each multicrossing point is covered by a patch. Right: A\npartial pseudoline arrangement with the same parallel bundles obtained by rerouting within the\npatches.\nover all patches we obtain a lower bound on the number Fk(n)of partial arrangements\nwithkbundles of/floorleftbign\nk/floorrightbig\nparallel pseudolines.\nInthesecondstep(Section4), werecursivelyaccountforcrossingsinbundlesofpseudolines\nwhich had been parallel before. The product of the so-computed possibilities yields the\nimproved lower bound on the number Bnof simple arrangements on npseudolines.\n3 Step 1: bundles of parallel lines, patches, and perturbations\nFor the start we fix an integer kand construct an arrangement Lofkbundles of/floorleftbign\nk/floorrightbig\nparallel\nlines as in [ 8]. Ifnis not an odd multiple of k, the remaining lines are discarded, or not used\nin the counting. We then cover all multicrossing points by a family of disjoint patches, and\nreroute the line segments within the patches so that all multicrossing points are resolved\nand the arrangement is simple. Since we use computers it is convenient to construct patches\nwith a high regularity. Moreover, for simplicity in the computational part (cf. Section 5) we\nensure that no crossings of Lare located on the boundary of a patch and no patch has a\ntouching with a line.\nExtremum 1: If we use one tiny patch for each of the multicrossings, the counting\nwill give the same results as in [ 8], where each multicrossing was rerouted locally in all\npossible ways for various configurations with kup to 12.\nExtremum 2: If we choose one gigantic patch containing all crossings of L, then all\npartialarrangements of npseudolines with the same parallel bundles as in Lwill be\ncounted, that is Fk(n). For the case of k= 3bundles, Felsner and Valtr [ 13] determined\nthatF3(n) = 2cn2+o(n2)wherec=log2(3)\n2−2\n3≈0.1258.\n3.1 Construction with 4 bundles\nFirst we construct a partial arrangement Lofn= 4mlines, consisting of 4bundlesL1,...,L4\nofmparallel lines. See Figure 2 for an illustration.F. Cortés Kühnast, J. Dallant, S. Felsner, and M. Scheucher XX:5\nL2\nL4\nL3\nL1\nR3\nR4\nR3\nR3\nR3\nL2\nL4\nL3\nL1\nFigure 2 Construction with 4 bundles as in [8].\nThe construction comes with crossings of order 2, 3, and 4. We restrict our attention\nto regions with multicrossings since regions with 2-crossings do not allow reroutings. As\nillustrated on the right-hand side figure, there are two types of regions with multicrossings:\nR3contains multicrossings of degree 3;\nR4contains multicrossings of degree 4.\nWe use different patches in the regions R3andR4but in both cases rectangular patches\ncan be used to tile the full region apart from a negligible area at its boundary. See Figure 3\nfor the patch P4we used to cover R4. Note that each copy of P4intersects withLin an\nequivalent way. This will be true implicitly of all patches we consider from now on. Since\nR3has a special pattern of intersection, we were able to compute the number of reroutings\nusing a more efficient method (see Section 5.2) than our more general dynamic programming\napproach (see Section 5.1). This enabled us to use a square of side length 1000as our patch\nP3forR3, which is much larger geometrically compared to P4.\nFigure 3 An illustration of the tiling for region R4in the 4-slope construction. The patch P4is\na square of side length 8√\n2\n2and area 32.\nSoCG 2024XX:6 An Improved Lower Bound on the Number of Pseudoline Arrangements\nNext we determine the numbers µiof patches of type Pi. Since the number of crossings\nin our construction is asymptotically quadratic in nand each patch contains only a constant\nnumber of crossings, the number of patches of type Piis also quadratic. It is important to\nnote that the patches along the boundary of Ribehave differently. However, since there are\nonly linearly many of these irregular patches, they only affect the lower order error term.\nHence we can omit them in the calculations.\nWe use a simple area calculation to obtain the µi’s. First we fix area(R3∪R4) = (m−1)2\nwhich implies area(R3) =area(R4) =(m−1)2\n2. Next we determine the areas of the patches.\nSince at this point all of our patches are rectangular, this amounts to determining their side\nlengths. The distance between adjacent lines in L2orL4is1and between those in L1orL3\nit is√\n2\n2. Thus the areas of the patches are\narea(P3) = 10002\narea(P4) = (8·√\n2\n2)2= 32.\nNow we can easily write down the µ′\nis:\nµ3(P3,n) =area(R3)\narea(P3)−O(n) =m2/2\n10002−O(n) =n2\n32000000−O(n)\nµ4(P4,n) =area(R4)\narea(P4)−O(n) =m2/2\n32−O(n) =n2\n1024−O(n).\nTo compute the number F(P4)of all possible reroutings within the patches of type P4,\nwe ran our dynamic program; see Section 5.1 for details. To compute the number F(P3)of\nall possible reroutings within the patches of type P3, we made use of the Lindström-Gessel-\nViennot lemma; see Section 5.2. We obtained:\nF(P3)>21397192\nF(P4) = 10233480626615962155895931163981261674 .\nCombining the possibilities from the two types of patches according to the estimate Fk(n)≥/producttextk\ni=3F(Pi)µi(Pi,n)yields\n▶Proposition 2. F4(n)≥2cn2−O(n)withc>0.1637.\nBy writing ci:=limn→∞µi(n)\nn2·log2F(Pi), we can see the contributions of the patches P3\nandP4to the leading constant c=c3+c4from Proposition 2; see Table 2.\nregion log2(# of reroutings )# of patches contribution computing time\nR3* 1397192.00 1/32000000 0.04366 -\nR4 122.94 1/1024 0.12006 20199.30s/summationtext\n- - 0.16373 20199.30s\nTable 2 Summary of our computational results for each of the patches in the k= 4slope\nconstruction. The first entry (marked with a star) is computed using the LGV lemma; see Table 5\nfor details. The first column shows the name of the region; the second column shows the number\nof reroutings ( log2(F)); the third column shows the number of patches ( limµ\nn2); the forth column\nshows the contribution of the region to the leading constant, i.e., the product of the second and\nthird column; the last column shows the computing time in CPU seconds.\n3.2 Construction with 6 bundles\nInthissectionweconsiderapartialarrangement Lofnlinesconsistingof 6bundlesL1,...,L6\nofmparallel lines. See Figure 4 for an illustration.\nThe construction comes with four types of regions with multicrossings:\nRifori∈{3,4,5}only contains multicrossings of order iandF. Cortés Kühnast, J. Dallant, S. Felsner, and M. Scheucher XX:7\nR6contains multicrossings of order 3 and 6.\nNote that in contrast to the k= 4bundle construction from Section 3.1, multicrossings of\norder 3 now occur in R3andR6. For each of the four regions Riwe will use a different\nR3\nR3\nR3\nR3\nR3\nR4\nR4\nR4\nR4\nR5\nR5\nR5\nR5\nR5\nR6\nR5\nR4\nL3\nL5\nL1\nL6\nL4\nL2\nR3\nR4\nFigure 4 Construction with 6 bundles as in [8].\ntype of patch Pithat is based on a regular tiling of the plane to ensure regularity. Figure 5\nillustrates the tiling for R6,R5andR4. ForR3we use a rhombic tiling such that each patch\nP3contains exactly 1000·1000crossings of order 3; see Section 5.2. Pause to note that P3\nandP4are affine transformations of the patches used in the 4 slopes setting in Section 3.1.\nTherefore they allow the same number of reroutings F(P3)andF(P4).\nWe have to determine the number µiof patches of type i. Towards this end we perform a\nsimilar area calculation as we did for k= 4bundles. This time both our patches and regions\nare a little more complex and numerous. To avoid human error, we automated the calculation\nof both area types in different ways. For the patches, we decided to simply count the number\nof highest degree crossing within the patch, because that can be done automatically without\nrounding. There is a one-to-one correspondence between such crossings and the fundamental\nparallelogram σof the lattice of crossings between lines from the bundles L2,L4andL6.\nSince every patch of the same type contains the same number of such crossings and the\npatches are part of a potentially infinite tiling, we obtain the area of Piby simply multiplying\nthe two numbers: area(Pi) = #{crossings of highest degree }·area(σ). If we take the area of\nσto be 1then the count corresponds exactly to the desired area. We obtain\narea(P3) = 10002, area(P4) = 32, area(P5) = 12, area(P6) = 7.\nSoCG 2024XX:8 An Improved Lower Bound on the Number of Pseudoline Arrangements\n(a)\n (b)\n (c)\nFigure 5 Three of the four types of patches for our construction on k= 6bundles: (a) For R6\nwe use a hexagonal tiling such that each patch P6contains exactly 7 crossings of order 6 and\n14 crossings of order 3. (b) For R5we use a hexagonal tiling such that each patch P5contains\nexactly 12 crossings of order 5. (c) For R4we use a rectangular tiling such that each patch P4\ncontains exactly 8·4 = 32crossings of order 4.\nSince each region is made up of convex polygons whose supporting lines are known, their\nareas are easy to compute using computer algebra software. In our supplemental data [ 1], we\nprovide a Sage script which carries out this task, and can be used for different constructions\nas well. The areas are\narea(R3) =3(m−1)2\n4,area(R4) =(m−1)2\n4,area(R5) =(m−1)2\n4,area(R6) =(m−1)2\n4.\nUsingn= 6mwe can calculate the µi’s as we did previously:\nµ3(P3,n) =area(R3)\narea(P3)−O(n) =3n2\n144·10002−O(n)\nµ4(P4,n) =area(R4)\narea(P4)−O(n) =n2\n144·32−O(n)\nµ3(P5,n) =area(R5)\narea(P5)−O(n) =n2\n144·12−O(n)\nµ3(P6,n) =area(R6)\narea(P6)−O(n) =n2\n144·7−O(n).\nTo compute the numbers F(Pi)of all possible perturbations within the patch type Pi\nfori= 4,5,6, we used our dynamic programming approach; see Section 5.1 for more details.\nFor region F(P3)we were able to use the same number already computed for Section 3.1.\nThe computation was based on the LGV lemma; see Section 5.2. Altogether we computed\nF(P3)>21397192\nF(P4) = 10233480626615962155895931163981261674\nF(P5) = 32207077855497546508132740267\nF(P6) = 5489259325039519956333073658900 .\nFromFk(n)≥/producttextk\ni=3F(Pi)µi(n), we can now derive:\n▶Proposition 3. F6(n)≥2cn2−O(n)withc>0.2118.\nBy writing ci:=limn→∞µi(n)\nn2·log2F(Pi), we can see the contributions of the patches P3,\nP4,P5andP6to the leading constant c=c3+c4+c5+c6from Proposition 3; see Table 3.F. Cortés Kühnast, J. Dallant, S. Felsner, and M. Scheucher XX:9\nregion log2(# of reroutings )# of patches contribution computing time\nR3* 1397192.00 1/48000000 0.02911 -\nR4 122.94 1/4608 0.02668 18787.90s\nR5 94.70 1/1728 0.05480 4219.49s\nR6 102.11 1/1008 0.10130 868737.00s/summationtext\n- - 0.21190 891744.39s\nTable 3 Summary of our computational results for each of the patches in the k= 6slope\nconstruction. See Table 2 for description of columns.\n3.3 Rectangular construction with 12 bundles\nFinally we consider a partial arrangement Lofnlines consisting of 12bundlesL1,...,L12\nofmparallel lines. See Figure 6 for an illustration.\nUp to symmetries of L, the construction produces 19regions that contain intersections of\nat least three bundles, see Figure 7. Since structurally different regions may meet the same\nnumbers of bundles, we differentiate them by adding lowercase Latin letters to their label\narbitrarily. Similar as above, we cover each region (except for a small area at its boundary)\nusing a regular tiling and count the numbers of reroutings within each tile using a computer.\nFor region R12even a 1×1square is too complex for our dynamic program to handle\nin reasonable time/memory. Therefore, we used two sub-patches as illustrated in Figure 8.\nMultiplying the obtained numbers of reroutings in the two sub-patches yields a lower bound\nfor the 1×1square. Note that the smaller patch contains only one crossing of degree 12.\nSince all pairs of pseudolines cross, the number of reroutings is exactly the number B12of\narrangements of 12pseudolines.\nFor the other regions we used specific patches which are available in our supplemental\ndata [1]. To obtain the numbers µixof copies of patch Pix, we simply divide the area of the\nwhole region Rixby the area of the patch:\nµix=area(Rix)\narea(Pix)−O(n),\nsee Table 4. To compute the areas of the patches we count the number of highest degree\ncrossings they contain again. In this construction, those correspond to 1×1squares, so the\nresulting count will be equal to the patch’s area. The areas of the regions can be computed\nautomatically from the specification of the bundles using a Sage script that we provide with\nour supplemental data [1].\nNow we compute the numbers F(Pix)of possible reroutings within each patch, see\nTable 4. To cope with the large patches in the two regions R3aandR3b, we again used\nthe Lindström-Gessel-Viennot lemma, see Section 5.2. For all the other patches we used a\ndynamic programming approach, see Section 5.1. Using the estimate Fk(n)≥/producttextF(Pix)µix(n)\nwe derive:\n▶Proposition 4. F12(n)≥2cn2−O(n)withc>0.2494.\nBy writing cix:=limn→∞µix(n)\nn2·log2F(Pix), we can see the contributions of the patches\nto the leading constant c=/summationtextcixfrom Proposition 4. All the data discussed above are\nsummarised in Table 4.\nSoCG 2024XX:10 An Improved Lower Bound on the Number of Pseudoline Arrangements\nFigure 6 The construction with 12 bundles as in [8].\n(a)\n (b)\nFigure 7 Regions with multi-crossings in the rectangular 12-slope construction.\n(a) shows the nine outer regions R3a,R3b,R4a,R4b,R5a,R5b,R6a,R6b,R7a.\n(b) shows the ten inner regions R7b,R8a,R8b,R8c,R9a,R9b,R10a,R10b,R11,R12.F. Cortés Kühnast, J. Dallant, S. Felsner, and M. Scheucher XX:11\nFigure 8 To partition region R12in thek= 12bundle construction, we use a square-shaped\npatchP12which is further partitioned into two sub-patches.\nregion log2(# of reroutings )# of patches contribution computing time\nR3a* 1397192.00 1/144000000 0.00970 -\nR3b* 1397192.00 1/432000000 0.00323 -\nR4a 122.94 1/6912 0.01779 18345.70s\nR4b 122.94 1/69120 0.00178 18345.70s\nR5a 94.70 1/6480 0.01461 4463.95s\nR5b 94.70 1/17280 0.00548 4463.95s\nR6a 111.79 1/17280 0.00647 98184.90s\nR6b 113.74 1/17280 0.00658 38082.70s\nR7a 93.74 1/7560 0.01240 1204.09s\nR7b 91.12 1/20160 0.00452 8392.94s\nR8a 148.77 1/24192 0.00615 148199.00s\nR8b 109.24 1/15120 0.00722 57856.50s\nR8c 105.70 1/6048 0.01748 142085.00s\nR9a 107.55 1/30240 0.00356 8577.60s\nR9b 102.98 1/10080 0.01022 316.13s\nR10a 144.70 1/17280 0.00837 68991.30s\nR10b 70.78 1/4320 0.01639 1430.85s\nR11 99.03 1/4320 0.02292 2059.67s\nR12 128.89 1/1728 0.07459 47.15s/summationtext\n- - 0.24946 621047.12s\nTable 4 Summary of our computational results for each of the patches in the k= 12slope\nconstruction. See Table 2 for description of columns.\nSoCG 2024XX:12 An Improved Lower Bound on the Number of Pseudoline Arrangements\n4 Step 2: resolving parallel bundles\nWith the second and final step, we obtain a simple arrangement of pairwise intersecting\npseudolines from a partial arrangement of kbundles ofm=/floorleftbign\nk/floorrightbig\nparallel pseudolines. To do\nso, we use a recursive scheme as in [ 13,8] to make each pair of parallel pseudolines cross:\nFor eachi= 1,...,k, we consider a disk Disuch that\n(1)Diintersects all parallel pseudolines of the bundle Liand no other pseudolines, and\n(2)no two disks overlap.\nWithin each disk Diwe can place any of the Bmarrangements of mpseudolines. This makes\nall the pseudolines of a bundle cross. Figure 9 gives an illustration for the case k= 3.\nL1\nL2\nL3\nD3\nD2\nD1\nFigure 9 Left: A partial arrangement of 3bundles of parallel pseudolines and a collection of\ninterior-disjoint disks (highlighted blue) such that each bundle is covered by one disk.\nRight: A proper pseudoline arrangement obtained by rerouting within the disks.\nSince allD′\nisare independent and there are Bmpossibilities to reroute within each Di,\nwe obtain the estimate\nBn≥Fk(n)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nStep 1·(Bm)k\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nStep 2,\nwherem=/floorleftbign\nk/floorrightbig\n. With the following lemma we can derive c−≥k\nk−1cwherecis the constant\nobtained in Section 3. The construction with k= 4bundles yields c−>0.2183, which is\nalready an improvement to the previous best bound by Dumitrescu and Mandal [ 8]. The\nconstruction with k= 6bundles allows a bigger step: It gives the lower bound c−>0.2542.\nThe biggest improvement comes from the construction with k= 12bundles, which yields\nc−>0.2721and therefore completes the proof of Theorem 1.\n▶Lemma 5. IfFk(n)≥2cn2−O(n)for somec>0thenBn≥2k\nk−1cn2−O(nlogn).\nThe proof of Lemma 5 is defered to Appendix A.\n5 Counting all possible reroutings in a patch\nIn the following we present two approaches how to compute the number F(P)of all possible\nperturbations within a patch P. In Section 5.1 we discuss a general approach based on\ndynamic programming which can deal with any patch. In Section 5.2 we discuss the special\ncase of 3 slopes, where the reroutings can be derived more efficiently with the Lindström-\nGessel-Viennot lemma.F. Cortés Kühnast, J. Dallant, S. Felsner, and M. Scheucher XX:13\n5.1 Counting reroutings via dynamic programming\nConsider a patch Pwith boundary curve C. SinceChas no touching with any line, the\nintersection of Pwith any line L∈Lyields proper line segments. And because Cpasses\nno crossings all the end points of segments are distinct. We can label the end points on C\naccording to the index of the resulting line segments and since every index occurs twice, we\ncall the cyclic sequence of end points along Cabipermutation . Note that in the case of a\nnon-convex boundary curve C, the intersection of a line with Pconsist of several segments,\nthey have to be distingushed by distinct labels. Figure 10 gives an illustration.\nP\n1\n1\n2\n2\n3\n3\n4\n4\n5\n5\n6\n6\n7\n7\nP2\n1\n1\n2\n3\n4\n6\n6\n7\nP1\nP2\n1\n1\n2\n2\n3\n4\n6\n6\n7\nP1\n3\n4\n5\n7\n4\n3\n3\n4\n4\n7\n7\n3\n5\n2\n3\n7\n7\n4\n3\n7\n5\n4\n5\nP2\n1\n1\n2\n2\n3\n4\n6\n6\n7\nP1\n3\n4\n5\n7\n4\n3\n3\n7\n7\n4\n5\nFigure 10 An illustration of how to recurse on a patch P. When cutting along segment 1,\nhighlighted purple, there are intersections with the segments 3,4, and 7. As the segments 3and\n7do no cross within P, there are only three possibilities for placing the three crossings along the\nsegment 1, namely 4–3–7(right top), 3–4–7(right center) and 3–7–4(right bottom).\nThe number of possible reroutings within a patch Pis computed recursively: Choose a\nsegmentz, it splitsPinto two parts. For every segment a̸=zwe can determine whether a\ncrosseszby looking at the occurrences of aandzin the bipermutation. If up to cyclic shifts\nthe pattern is azazthey cross in P, if the pattern is aazzthey are parallel in P.\nNow for every legal order πof the crossing segments on zwe combine the two arcs of the\nbipermutation defined by the two occurrences of zand combine them with πto obtain the\nbipermutations of the two smaller patches P1andP2. Figure 10 gives an example.\nThe legal orderings of crossings on zcan be found as follows: Consider a pair of segments\na,bthatbothcross z. Eitheraandbareparallelin P(i.e., wesee abzbazinthebipermutation)\nor they form a crossing in Pwhich we denote by a×b(i.e., we see abzabz). In Figure 10\nthe segments 3and7are parallel, and segments 3and4cross, and the crossing point 3×4\ncan lie on either side of the segment 1. In the case where aandbare parallel, the order\nof the two crossings a×zandb×zalongzis uniquely determined. In the case where a\nandbcross, the crossing point a×bcan lie on either side of z, and the choice of the part\nfurther determines the order of the two crossings a×zandb×zalongz. Altogether, the\nSoCG 2024XX:14 An Improved Lower Bound on the Number of Pseudoline Arrangements\ne1\ne2 e3s1 s2\ns3s'1\ns'2\ns'3e'1\ne'2\ne'3s1 s2\ns3 e1\ne2 e3\nFigure 11 Illustration of the correspondence between arrangements of the chord diagram and\ndisjoint paths in the directed acyclic graph.\nbipermutation of Pdetermines a partial order ≺of all crossings on zsuch that the legal\norderings correspond to linear extensions of this partial ordering.\nTo count all possible reroutings within P, we iterate over all linear extensions ≺⋆of≺\nand continue recursively. Note that linear extensions can be enumerated efficiently via back-\ntracking: pick a maximal element and recurse on the remaining elements. Each extension ≺⋆\nuniquely determines the order of crossings along zand therefore the two sides P1(z,≺⋆)and\nP1(z,≺⋆)can be explicitely given via their bipermutations. By cumulatively summing up\nthe obtained numbers, we get\nF(P) =/summationdisplay\n≺⋆linear extension of ≺F(P1(z,≺⋆))·F(P2(z,≺⋆)).\nWe provide a computer-assisted framework [ 1] that allows to fully automatically compute\nF(P)for a given patch P. The input is given as IPE-file1. The program reads the collection\nof lines and the polygonal boundary of P, computes the bipermutation, and then performs\na dynamic program to determine the number of reroutings within P. More specifically,\nwe compute for each bipermutation the lexicographic minimal among all relabelings of the\nelements, and reuse previously computed values whenever possible.\n5.2 Counting reroutings via the Lindström-Gessel-Viennot lemma\nWe illustrate our approach on a small patch defined by a square of side length l= 2(see\nFigure 11). We can count the number of reroutings in this patch by starting with only the\nblack grid, and counting the number of combinatorially distinct ways to insert three curves\nstarting at s1,s2ands3respectively, and ending at e1,e2ande3respectively, such that\nthese three curves do not intersect, and each curve crosses every black segment at most once.\nWe can in turn translate this into counting the number of ways to have three paths in the\ndual grid graph (with edges oriented left to right and top to bottom), starting and ending\nat prescribed vertices, such that these paths do not cross (middle illustration in Figure 11).\nNote that here by “not crossing” we do not mean vertex- or edge-disjoint, but that if a path\nstarts above/right of another, it can at no point go below/left of it. Given three such paths,\nordered from top right to bottom left, we can shift the second path one unit down and left\nalong the grid and the second path two units down and left along the grid. This results in\nthree vertex-disjoint paths (right illustration in Figure 11). The reverse is also true: given\n1IPE [5] is a drawing editor for creating vector graphics in XML or PDF format. Besides the supplemental\ninput files, also all figures in this article were created with it.F. Cortés Kühnast, J. Dallant, S. Felsner, and M. Scheucher XX:15\nFigure 12 The20reroutings of the patch.\nthree vertex-disjoint paths on the directed acyclic graph starting at s′\n1,s′\n2,s′\n3and ending\nate′\n1,e′\n2,e′\n3, reversing the shifts produces three non-crossing paths starting at s1,s2,s3and\nending ate1,e2,e3.\nThus, our question reduces to counting the number of ways to have three such vertex-\ndisjoint paths, whose set of starting vertices is s′\n1,s′\n2,s′\n3and set of ending vertices is e′\n1,e′\n2,e′\n3.\nNote that the vertex-disjointness ensures that in such a collection of paths, the path starting\nats′\niwill end at e′\nifor all 1≤i≤3. The Lindström–Gessel–Viennot lemma gives us an\nefficient method to do so:\n▶Lemma 6 (Lindström [ 23], Gessel & Viennot [ 14]).LetGbe a finite directed acyclic graph.\nConsider starting vertices S={s1,...,sk}and destination vertices E={e1,...,ek}. For\nany two vertices uandv, letp(u,v)be the number of paths from utov. Assume that for\nany tuple of kvertex-disjoint paths starting in Sand ending in E, the path starting at si\nnecessarily ends at ei, for all 1≤i≤k. Then the number of distinct such tuples is the\ndeterminant of the matrix\nM=\np(s1,e1)p(s1,e2)... p (s1,ek)\np(s2,e1)p(s2,e2)... p (s2,ek)\n............\np(sk,e1)p(sk,e2)... p (sk,ek)\n.\nIn our specific case, the entries of this matrix are the number of paths going only down\nor right between two specified vertices in a grid, which can be easily expressed as a binomial\ncoefficient. We have\nM=\n/parenleftbig2\n1/parenrightbig /parenleftbig3\n0/parenrightbig\n0/parenleftbig3\n3/parenrightbig /parenleftbig4\n2/parenrightbig /parenleftbig3\n0/parenrightbig\n0/parenleftbig3\n3/parenrightbig /parenleftbig2\n1/parenrightbig\n=\n2 1 0\n1 6 1\n0 1 2\n.\nThe determinant of this matrix is det(M) = 20, corresponding to the 20reroutings of the\npatch illustrated in Figure 12. In general, for a square patch of size l×l, the corresponding\nSoCG 2024XX:16 An Improved Lower Bound on the Number of Pseudoline Arrangements\nmatrix isM= (mij)1≤i,j≤2l−1, with\nmij=/parenleftbigg2l−|l−i|−|l−j|\n2l−|l−i|−|l−j|+3|i−j|\n2/parenrightbigg\n.\nFor our actual bounds, we use a square patch of side length l= 1000. The number of\nreroutings of this patch is too big to write explicitly here. In base 10, it is 420 597digits long.\nHowever, it is enough for us to know that log2(# of reroutings )>1397192. Table 5 shows\nthe numbers and computation times for other side lengths. The computations suggest that\nthe ratiolog2(# of reroutings )\nl2 is converging as lincreases. Hence, we expect that increasing l\nonly gives negligible improvements.\nllog2(# reroutings )ratio computing time\n10 130.523 1.3052 0.02s\n20 539.561 1.3489 0.09s\n50 3444.189 1.3777 0.62s\n100 13877.972 1.3878 3.44s\n200 55719.146 1.3930 37.96s\n500 349033.080 1.3961 2473.19s\n1000 1397192.106 1.3972 66467.25s\nTable 5 Summary on the LGV computations for l×lsquare patches for different sizes l.\n6 Discussion\nAll results presented in Tables 2–5 were computed on cluster nodes of TU Berlin with up to\n1TB of RAM. The programs can also be run with fewer resources but the dynamic program\ncomes with a memory–time trade-off. Our programs and further details are available in our\nsupplemental data [ 1]. We also provide simpler patches for which the program only needs few\nCPU seconds and low RAM. The simpler patches have also been verified by an independent\nimplementation [7]. They, however, give a slightly worse bound.\nWe performed quite some experiments to optimize the set of parameters. For the 4-\nslope construction in Section 3.1, we so far manged to obtain c−>0.2183and for the\n6-slope construction in Section 3.2, c−>0.2542. For the rectangular 12-slope construction\nin Section 3.3, we obtained c−>0.2721, which is the currently best lower bound. It is\ninteresting that already the 4-slope construction gives a significant improvement to the\nprevious best bound by Dumitrescu and Mandal [ 8]. In the future we plan to investigate\nalso the construction with 16 slopes, from which we expect an improvement to 0.28.\nWhile the results from [ 8] suggest that larger values of kgive better bounds, the com-\nputations get more and more complex. In fact, as the number kincreases, the number of\ndifferent shape types and the complexity of the patches increases. Since our program can\nonly deal with patches containing about 30 to 40 segments in reasonable time, depending on\nthe structure of crossings within it, there is a trade-off between the number of crossings and\nthe number of bundles involved in a patch.\nAs long as one fixes k, the counting approach is implicitly limited by Fk(n), which is\nmuch smaller than Bn. SinceF3(n) = 2cn2+o(n2)withc=log2(3)\n2−2\n3≈0.1258is known [ 13],\nit would be interesting to determine limn→∞log2Fi(n)\nn2fori= 4,..., 12. We wonder how far\nfrom the truth the constant in Propositions 2 and 3 is. However, since our tiling approachF. Cortés Kühnast, J. Dallant, S. Felsner, and M. Scheucher XX:17\nasserts a particular global structure on the arrangement, it is impossible to approximate Fk–\neven under the assumption that we can process arbitrarily large (but finite) patches.\nThere is a lot of freedom for choosing the patches for each region, and we indeed\nexperimented with various shapes. The ones presented along Section 3 led to the best results.\nAs mentioned along Section 5.1, when recursively computing F(B), the choice of zcan have\na significant impact on the computing time. Experiments showed that the best practical\nperformance is obtained by choosing a cutting-segment zsuch that\nthe complexity of the larger part P2is as small as possible.\nIn practice this strategy tends to lead to\nbalanced cuts, that is, P1andP2are of similar sizes;\nshortcuts, that is, the number of intersections on zis relatively small; and\nrelatively small numbers of is legal orders.\nIn general each of those criteria may lead to a different cutting strategy. However, a properly\nbalanced cut may come with larger parts and hence might be less efficient. Also the shortest\ncut will often be very unbalanced, resulting in one half that is only slightly smaller than the\nwhole path and thus more recursion steps might be required to deal with the patch.\nA good splitting strategy should also take the number of legal orders into account because\na large number of legal orders will also negatively affect the computation time. The main\nadvantage of our strategy is that it minimizes the complexity of both P1andP2. This allows\nthe algorithm to reach the bottom of the recursive search tree as fast as possible. In practice\nit appeared to be more important to reach a low level in the search than to minimize the\nnumber of legal orders because the number of cache-hits (pre-computed values) increases fast\nas the level decreases. To obtain the best possible run-times we performed benchmarks and\nmade statistics for the computing time used in different layers in the recursive search tree.\nHowever, in general, it is hard to tell which cutting strategy is the best because we do not\nhave any a priori estimates for the computation time or the number of reroutings for a patch.\nLast but not least, since local rerouting is a well-known and frequently used technique in\ncombinatorial geometry, our technique might also be adapted to various other combinatorial\nstructures to derive improved lower bounds. We see for example great potential to improve\nthe lower bounds for arrangements of pseudocircles [ 12], simple drawings of the complete\ngraph [25,20], or higher dimensional pseudohyperplane arrangements [ 2, Corollary 7.4.3],\nwhich are all natural generalizations of pseudoline arrangements.\nReferences\n1Supplemental data. https://github.com/fcorteskuehnast/counting-arrangements .\n2Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler.\nOriented Matroids , volume 46 of Encyclopedia of Mathematics and its Applications . Cambridge\nUniversity Press, 2 edition, 1999. doi:10.1017/CBO9780511586507 .\n3Nicolas Bonichon, Cyril Gavoille, and Nicolas Hanusse. An information-theoretic upper bound\nof planar graphs using triangulation. In Annual Symposium on Theoretical Aspects of Computer\nScience (STACS 2003) , pages 499–510. Springer, 2003. doi:10.1007/3-540-36494-3_44 .\n4Nicolas Bonichon, Cyril Gavoille, and Nicolas Hanusse. An Information-Theoretic Upper\nBound on Planar Graphs Using Well-Orderly Maps , pages 17–46. Birkhäuser, 2011. doi:\n10.1007/978-0-8176-4904-3_2 .\n5Otfried Cheong. The Ipe extensible drawing editor. http://ipe.otfried.org/ .\n6Fernando Cortés Kühnast. On the number of arrangements of pseudolines. Bachelor’s thesis,\nTechnische Universität Berlin, Germany, 2023. https://fcorteskuehnast.github.io/files/\nbachelor_thesis.pdf .\nSoCG 2024XX:18 An Improved Lower Bound on the Number of Pseudoline Arrangements\n7Justin Dallant. Improved Lower Bound on the Number of Pseudoline Arrangements.\narXiv:2402.13923, 2024.\n8Adrian Dumitrescu and Ritankar Mandal. New lower bounds for the number of pseudoline\narrangements. Journal of Computational Geometry , 11:60–92, 2020. doi:10.20382/jocg.\nv11i1a3.\n9Herbert Edelsbrunner, Joseph O’Rourke, and Raimund Seidel. Constructing arrangements of\nlines and hyperplanes with applications. SIAM Journal on Computing , 15(2):341–363, 1986.\ndoi:10.1137/0215024 .\n10Stefan Felsner. On the Number of Arrangements of Pseudolines. Discrete & Computational\nGeometry , 18(3):257–267, 1997. doi:10.1007/PL00009318 .\n11Stefan Felsner and Jacob E. Goodman. Pseudoline Arrangements. In C.D. Toth, J. O’Rourke,\nand J.E. Goodman, editors, Handbook of Discrete and Computational Geometry (3rd ed.) .\nCRC Press, 2018. doi:10.1201/9781315119601 .\n12Stefan Felsner and Manfred Scheucher. Arrangements of Pseudocircles: On Circularizability.\nDiscrete & Computational Geometry, Ricky Pollack Memorial Issue , 64:776–813, 2020. doi:\n10.1007/s00454-019-00077-y .\n13Stefan Felsner and Pavel Valtr. Coding and Counting Arrangements of Pseudolines. Discrete\n& Computational Geometry , 46(3), 2011. doi:10.1007/s00454-011-9366-4 .\n14Ira Gessel and Gérard Viennot. Binomial determinants, paths, and hook length formulae.\nAdvances in Mathematics , 58(3):300–321, 1985. doi:10.1016/0001-8708(85)90121-5 .\n15Jacob E. Goodman and Richard Pollack. Multidimensional Sorting. SIAM Journal on\nComputing , 12(3):484–507, 1983. doi:10.1137/0212032 .\n16Branko Grünbaum. Arrangements and Spreads , volume 10 of CBMS Regional Conference\nSeries in Mathematics . AMS, 1972 (reprinted 1980). doi:10/knkd .\n17Günter Rote. NumPSLA – An experimental research tool for pseudoline arrange-\nments and order types, 2021. https://github.com/guenterrote/NumPSLA/blob/main/\nNumPSLA-paper-2024-03-06.pdf .\n18Jun Kawahara, Toshiki Saitoh, Ryo Yoshinaka, and Shin-ichi Minato. Counting Primitive\nSorting Networks by πDDs. 2011. URL: https://api.semanticscholar.org/CorpusID:\n9525480.\n19Donald E. Knuth. Axioms and Hulls , volume 606 of LNCS. Springer, 1992. doi:10/bwfnz9 .\n20Jan Kynčl. Enumeration of simple complete topological graphs. European Journal of Combin-\natorics, 30(7):1676–1685, 2009. doi:10.1016/j.ejc.2009.03.005 .\n21Fernando Cortés Kühnast, Stefan Felsner, and Manfred Scheucher. An Improved Lower Bound\non the Number of Pseudoline Arrangements. arXiv:2402.13107v1, 2024.\n22Friedrich Levi. Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade.\nBerichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig,\nMathematisch-Physische Klasse , 78:256–267, 1926.\n23Bernt Lindström. On the vector representations of induced matroids. Bulletin of the London\nMathematical Society , 5(1):85–90, 1973. doi:10.1112/blms/5.1.85 .\n24Jiří Matoušek. Lectures on Discrete Geometry . Springer, 2002. doi:10.1007/\n978-1-4613-0039-7 .\n25János Pach and Géza Tóth. How many ways can one draw a graph? Combinatorica ,\n26(5):559–576, 2006. doi:10.1007/s00493-006-0032-z .\n26Matthew J. Samuel. Word posets, complexity, and Coxeter groups. arXiv:1101.4655, 2011.\n27Neil J. A. Sloane. The On-Line Encyclopedia of Integer Sequences.\nhttp://oeis.org .\n28Katsuhisa Yamanaka, Shin-ichi Nakano, Yasuko Matsui, Ryuhei Uehara, and Kento Nakada.\nEfficient enumeration of all ladder lotteries and its application. Theoretical Computer Science ,\n411(16):1714–1722, 2010. doi:10.1016/j.tcs.2010.01.002 .F. Cortés Kühnast, J. Dallant, S. Felsner, and M. Scheucher XX:19\nA Proof of Lemma 5\nChooseL>0as a sufficiently large constant such that Fk(n)≥2cn2−Lnholds for all n≥1.\nWe show by induction that Bn≥2˜cn2−G(n)whereG(n) := ˜Lnlogk(˜Ln),˜L:=L+ 2˜c, and\n˜c:=k\nk−1·c. The base case n≤kis clearly satisfied if Lis chosen sufficiently large. For the\ninduction step, we have\nlog2Bn≥log2/parenleftbig\nFk(n)·(Bm)k/parenrightbig\n= log2Fk(n) +klog2Bm\nI.H.\n≥(cn2−Ln) +k/parenleftbig\n˜cm2−G(m)/parenrightbig\n≥cn2−Ln+˜c\nk(n−k)2−kG(m)\n= (c+˜c\nk)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=˜cn2−(L+ 2˜c)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=˜Ln−kG(m) +ck/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≥0,\nwherem=/floorleftbign\nk/floorrightbig\n. SincekG(m) +˜Ln≤G(n)holds by definition, this completes the proof of\nLemma 5.\nSoCG 2024"    },    {        "title": "2402.13127v2.Explicit_upper_bounds_on_the_average_of_Euler_Kronecker_constants_of_narrow_ray_class_fields.pdf",        "content": "arXiv:2402.13127v2  [math.NT]  23 Feb 2024EXPLICIT UPPER BOUNDS ON THE A VERAGE OF\nEULER-KRONECKER CONSTANTS OF NARROW RAY CLASS FIELDS\nNEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nABSTRACT . For a number field K, the Euler-Kronecker constant γKassociated to Kis an arith-\nmetic invariant the size and nature of which is linked to some of the deepest questions in number\ntheory. This theme was given impetus by Ihara who obtained bo unds, both unconditional as\nwell as under GRH for Dedekind zeta functions. In this note, w e study the analogous constants\nassociated to the narrow ray class fields of an imaginary quad ratic field. Our goal for studying\nsuch families is twofold. First to show that for such familie s, the conditional bounds obtained by\nIhara can be improved on the average, again under GRH for Dede kind zeta functions. Further,\nour family of number fields are non-abelian while such averag e bounds have earlier been studied\nfor cyclotomic fields. The technical part of our work is to mak e the dependence of these upper\nbounds on the ambient number field explicit. Such explicit de pendence is essential to further our\nobjective.\n1.INTRODUCTION\nThe Euler-Mascheroni constant γ, introduced by Euler [ 10] is given by following limit\nγ= lim\nx→∞\n/summationdisplay\nn≤x1\nn−logx\n.\nIt is not known if γis rational or irrational. It is perhaps prudent not to study γin isolation,\nbut as a member of some family whose behaviour can be studied e n masse. One such family\nwas envisaged by Ihara [ 20] in 2005, the members of which are referred to as Euler-Kronec ker\nconstants. To motivate these constants, let us note that γis the constant term in the Laurent\nseries expansion of the Riemann zeta function around s= 1. Following this line of thought,\none can define the Euler-Kronecker constant of a number field Kas follows.\nThe Dedekind zeta function of a number field Kis given by\nζK(s) =/summationdisplay\na⊆OK\na/negationslash=(0)1\nNas,ℜ(s)>1.\n2020 Mathematics Subject Classification. 11R18, 11R29, 11R42, 11Y60.\nKey words and phrases. Euler-Kronecker constants, Generalized Riemann Hypothes is, Dedekind zeta functions,\nNarrow ray class fields, Generalized Dirichlet characters, Hecke L-functions.\n12 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nIt is known to have a meromorphic continuation to the entire c omplex plane with only a simple\npole at the point s= 1. Thus we have a Laurent series expansion for ζKarounds= 1. This is\ngiven by\nζK(s) =ρK\ns−1+cK+O(s−1). (1)\nFollowing Ihara [ 20], we now define the Euler-Kronecker constant of Kas\nγK:=cK/ρK.\nSinceγKis evidently real, it is natural to ask its sign. Ihara [ 21] conjectured that γKis positive\nwhenKis a cyclotomic field. However it has been shown by Ford, Luca an d Moree [ 11] that\nγQ(ζp)is negative for p= 964477901 . Further, Ihara [ 21] showed that γKis negative infinitely\noften under the Generalised Riemann Hypothesis (GRH) for ζK(s).\nAnother possible line of investigation is to obtain bounds f orγKin terms of other invariants\nof the ambient number field K. This was initiated by Ihara in his papers [ 20], [21].\nFor a number field Kof degree at least 3, he showed under GRH that\n−2(nK−1)DK−nK+1\nDK+nK−1/parenleftbigg\nlogDK\nnK−1+1/parenrightbigg\n−1≤γK≤/parenleftbiggDK+1\nDK−1/parenrightbigg\n(2logDK+1),(2)\nwhereDK= log/radicalbig\n|dK|andnKanddKdenote the degree and discriminant of Krespectively.\nRecently, Dixit [ 6] showed the following unconditional bounds for |γK|for certain families\nof number fields. We say that a number field Kis almost normal if there is a tower of number\nfields\nK=Kn⊃Kn−1⊃ ··· ⊃K1=Q,\nsuch that Ki+1/Kiis Galois for all 1≤i≤n.\nTheorem 1.1. (Dixit, [ 6])LetKbe an almost normal number field, not containing any quadratic\nsubfields. Then\nγK≪(log|dK|)4n3\nK.\nOn the other hand if Kis a number field with solvable normal closure, not containing an y quadratic\nsubfields. Then\nγK≪(log|dK|)cloglog|dK|\nfor some absolute positive constant c.\nFor an arbitrary number field, Dixit and Ram Murty [ 7] relatedγKto the hypothetical Siegel\nzero ofζK(s). They showed the following.\nTheorem 1.2. (Dixit and Ram Murty, [ 7,8]For any number field K, we have\nγK≪log|dK|EULER-KRONECKER CONSTANTS 3\nifζK(s)has no Siegel zero. If it has a Siegel zero β0, then the bound is\nγK=1\nβ0(1−β0)+O(log|dK|).\nIn the case of cyclotomic fields, Ihara suggested the followin g:\nConjecture 1. (Ihara, [ 20])There are positive constants 0<a0,a1≤2such that for any msufficiently\nlarge and any ǫ>0, we have\n(a0−ǫ)logm<γQ(ζm)<(a1+ǫ)logm,\nwhereQ(ζm)denotes them-th cyclotomic field.\nThe results of Ford, Luca and Moree [ 11] indicate that γQ(ζm)can be negative, thus the above\nlower bound suggested by Ihara does not hold.\nAssuming GRH for Dedekind zeta functions of cyclotomic fields , Ihara, Kumar Murty and\nShimura [ 22] proved that\nγQ(ζm)≪(logm)2.\nIn 2010, Badzyan [ 3] improved this result and showed that under GRH for Dedekind zeta\nfunctions of cyclotomic fields,\nγQ(ζm)≪(logm)(loglogm).\nIn 2011, Kumar Murty [ 24] proved that the upper bound of |γQ(ζp)|coming from Ihara’s\nconjecture is true on average when restricted to primes. More precisely,\nTheorem 1.3. (Kumar Murty, [ 24])We have\n1\nπ∗(Q)/summationdisplay\n1\n2Q<p≤Q|γQ(ζp)| ≪logQ,\nwhere the sum is over prime numbers pin the interval (1\n2Q,Q]andπ∗(Q)denotes the number of primes\nin this interval.\nFouvry in [ 12] studied the case of Q(ζm)wheremis not necessarily prime and showed the\nfollowing.\nTheorem 1.4. (Fouvry, [ 12])Uniformly for Q≥3, we have\n1\nQ/summationdisplay\n1\n2Q<m≤QγQ(ζm)= logQ+O(loglog Q).\nThe aforementioned result of Fouvry has recently been refined by Hong, Ono and Zhang\n[19] under the following conjecture.4 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nConjecture 2. [Elliott and Halberstam conjecture [ 9]]For every real number θ <1and for every\npositive integer A>0, one has\n/summationdisplay\nq≤xθmax\ny≤xmax\n(a,q)=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nn≤y\nn≡amodqΛ(n)−y\nϕ(q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪A,θx\nlogAx\nfor all real numbers x>2. HereΛis the Von Mangoldt function.\nThe precise result of Hong, Ono and Zhang is as follows.\nTheorem 1.5. (Hong, Ono and Zhang, [ 19])Assume the Elliott-Halberstam conjecture, for Q→ ∞ ,\nwe have\n1\nQ/summationdisplay\nQ<m≤2Q|γQ(ζm)−logm|=o(logQ),\nwhere the sum is over integers m.\nWe observe here that in a recently published work, Dixit and R am Murty [ 7] have given\nan alternative proof of the Theorem 1.3using certain new observations for arbitrary number\nfields. Further they use the same method to show the following r esult.\nTheorem 1.6. (Dixit and Ram Murty, [ 7])Assuming the Elliott-Halberstam conjecture, we have\n/summationdisplay\nQ<p≤2Q|γQ(ζp)−logp|=o(Q).\nSince the above results focus on cyclotomic fields, the raison d’etre of our work is to carry\nout such investigations over families of Galois number fields which are non-abelian. One such\nfamily originates from narrow ray class fields of imaginary qu adratic fields and it is the study\nof this family which we undertake. More precisely, we prove th e following theorem.\nTheorem 1.7. LetKbe an imaginary quadratic field with class number hK. For a non-zero principal\nprime ideal qofOK, letK(q)be the narrow ray class field modulo q. Assume GRH for the Dedekind\nzeta functions of K(q)for all non-zero principal prime ideals q. Then forQ≥8exp(8·1045|dK|), we\nhave\n1\nπ∗(Q)/summationdisplay′\n1\n2Q<Nq≤Q|γK(q)|<|γK|+(6000h2\nK+1017hK+11)logQ, (3)\nwhere′over the sum indicates that the sum is over principal prime idea ls ofOKandπ∗(Q)denotes the\nnumber of principal prime ideals qofOKwith norm in the interval (1\n2Q,Q].\nAt this juncture, it is perhaps worthwhile to highlight a few points about this result.EULER-KRONECKER CONSTANTS 5\n•The explicit dependence of the upper bound in ( 3) onhKis what allows us to improve\nthe lower bound in ( 2) for infinitely many ray class fields of K. To the best of our\nknowledge, this is the first instance of a non-abelian family o verQfor which the bound\nhas been improved.\n•Even though our final result is conditional under GRH, at vario us places, we have cho-\nsen unconditional bounds and not those under GRH unless esse ntial.\n•We have not striven to optimise the numerical constants in th e above theorem.\nThis paper is organised as follows. In the next section we out line the strategy of our proof. In\nsection 3, we list some results needed for our proofs while in section 4 we derive some lemmas\nand propositions needed to prove our theorem. The penultima te section gives a complete proof\nof Theorem 1.7. Finally in the last section we show that there are infinitely m any non-abelian\nfields in the family of narrow ray class fields considered by us.\n2. S TRATEGY OF PROOF\nIn this section, we quickly outline the strategy of the proof of Theorem 1.7. In [ 22], Ihara,\nKumar Murty and Shimura define, for a number field Kand Hecke character χmoduloqofK,\n(cf. section 3)\nΦK,χ(x) =1\nx−1/integraldisplayx\n1/parenleftig/summationdisplay\na⊂OK\n1≤Na≤tΛ(a)\nNaχ(a)/parenrightig\ndt,(x>1)\nand show that for χ/\\e}atio\\slash=χ0, (χ0is the principal Hecke character)\n−L′\nL(1,χ) = lim\nx→∞ΦK,χ(x),\nwhereL(s,χ)is the Hecke L function corresponding to χ. We know that\nγK(q)= lim\ns→1/parenleftiggζ′\nK(q)\nζK(q)(s)+1\ns−1/parenrightigg\nandζK(q)(s) =ζK(s)/productdisplay\nχ/negationslash=χ0L(s,χ∗)\nwhereχ∗is the primitive character inducing the Hecke character χmodulo q. It now follows\nthat\n|γK(q)| ≤ |γK|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nχ/negationslash=χ0ΦK,χ(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nχ/negationslash=χ0/parenleftig\nΦK,χ(x)−ΦK,χ∗(x)/parenrightig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/summationdisplay\nχ/negationslash=χ0/vextendsingle/vextendsingle/vextendsingle/vextendsingleL′\nL(1,χ∗)+ΦK,χ∗(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nThe crux of the proof lies in estimating each of these sums on a verage as we vary norm of q\nbetweenQ/2andQwhereQis a large real parameter. In Theorem 5.2, we estimate the average\nof the second term on the right by breaking the average into tw o parts. For small primes,\nwe use an analogue of the Brun-Titchmarsh theorem (see Theor em4.7) and some elementary\nestimates. For large primes, we use the explicit Chebotarev density theorem under GRH due to\nGrenie and Molteni (see Theorem 3.12). The third term is relatively harmless and we estimate6 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nit directly (estimated after proof of Theorem 5.2). To estimate the last term we use an explicit\nformula for\nL′\nL(1,χ∗)+Φχ∗(x)\nproved by Ihara, Kumar Murty and Shimura in [ 22] and deduce bounds under the assumption\nof GRH. Theorem 1.7is obtained by combining all these estimates.\n3.PRELIMINARIES\n3.1. Notation. Throughout this article, Kwill denote an algebraic number field of degree nK,\ncontained in C. LetOKbe its ring of integers and dKits discriminant with respect to Q. Further,\nlethKdenote the class number of OKandµKthe set of roots of unity in OK. For any finite set\nS, we use |S|to denote its cardinality.\nFor an ideal a⊆ OK, letNadenote the order of the finite group OK/a. We use PKto denote\nthe set of all prime elements of OK. Further, pandqshall denote prime ideals in OK. We define\nthe generalized Euler-phi function as follows: For any non- zero ideal a⊆ OK,\nϕK(a) =Na/productdisplay\np|a/parenleftbigg\n1−1\nNp/parenrightbigg\n.\nLetρKdenote the residue of the Dedekind zeta function of Kats= 1. For an imaginary\nquadratic field K, the class number formula gives us\nρK=2πhK\n|µK|/radicalbig\n|dK|. (4)\nThe following result gives bounds for ρKin terms of dK.\nLemma 3.1. (Deshouillers, Gun, Ramaré and Sivaraman [ 5])We have\n9\n25/radicalbig\n|dK|≤ρK≤6/parenleftbigg2π2\n5/parenrightbiggnK\n|dK|1/4.\nWe note that when Kis imaginary quadratic, the above can be refined to obtain\nπ\n3/radicalbig\n|dK|≤ρK≤6/parenleftbigg2π2\n5/parenrightbigg2\n|dK|1/4.\nFor number fields K⊂L, we recall that the relative discriminant ∆L/Kis defined as the OK-\nideal generated by the discriminants of all bases of LoverKcontained in OL. In this context,\nwe record the following result.\nLemma 3.2. (Neukirch, page 202, [ 28])For a tower of fields Q⊂K⊂L, let[L:K]denote the\ndegree of L/Kand∆L/Kdenote the relative discriminant of LoverK. Then we have the following\nrelation\n|dL|=|dK|[L:K]N∆L/K.EULER-KRONECKER CONSTANTS 7\nNow we state the Conductor-Discriminant formula which will be used to bound the dis-\ncriminant in Theorem 5.2.\nTheorem 3.3. (Neukirch, page 534 (Conductor-Discriminant formula), [ 28])LetL/Kbe a Galois\nextension of number fields and ∆L/Kdenote its relative discriminant. Then we can express it by the\nfollowing decomposition\n∆L/K=/productdisplay\nχirred.f(χ)χ(1),\nwhereχvaries over the irreducible characters of the Galois group of L/Kandf(χ)denotes its Artin\nconductor (see page 527 of [28]).\nRecall that when K=Q,ζK(s)is the Riemann zeta function ζ(s).LetΓ′\nΓ(s)andζ′\nζ(s)denote\nthe logarithmic derivatives of the gamma function and Riema nn zeta function evaluated at s\nrespectively. Now we record the bounds on these functions.\nLemma 3.4. (Hall and Tenenbaum, page 146, [ 18])For realσ>1, we have\n−ζ′\nζ(σ)<1\nσ−1.\nLemma 3.5. (Ahn and Kwon, [ 1])Assume that ℜ(s)>1\n2. We have\nℜ/parenleftbiggΓ′\nΓ(s)/parenrightbigg\n≤1.08log(|s|+2).\nFor an arithmetic function fand a positive arithmetic function g,f(z) = O∗(g(z))means\nthat|f(z)| ≤g(z).\n3.2. Narrow ray class groups and Hecke L-functions. Letcbe a non-zero integral ideal of K.\nFor any element α∈K, we say that α≡1 mod∗cifαsatisfies the following two conditions:\n(1) The element αis of the form a/bfora,b∈ OKwhere the ideals (a)and(b)are coprime\ntocanda≡bmodc.\n(2) For any real embedding σofK,σ(α)>0.\nDefinition 1. LetI(c)denote the set of non-zero fractional ideals relatively prim e tocandPcthe group\nof non-zero principal fractional ideals (α)⊂ OKsuch thatα≡1 mod∗c. Then the group I(c)/Pcis\ncalled the narrow ray class group of Kmoduloc, denoted by Hc(K).\nDefinition 2. A character χ, ofHc(K)will be called a generalized Dirichlet character modulo c. We\nshall useχ0to denote the principal generalized Dirichlet character.\nThere is a natural restriction from Hc(K)toHc′(K)for allc′|c.\nDefinition 3. We say that a character of Hc(K)is primitive if it does not factor through Hc′(K)for\nevery proper divisor c′ofc. Further, the conductor of a generalized Dirichlet character χmodulocis the\nsmallest divisor c′ofcsuch thatχfactors through a character modulo c′.8 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nFurther for a generalized Dirichlet character χmoduloc, we define the associated Hecke L-\nfunction as\nL(s,χ,K) =/summationdisplay\n0/negationslash=a⊆OK\n(a,c)=OKχ([a])\nNas,ℜ(s)>1,\nwhere[a]denotes the class of the ideal ainHc(K). We will henceforth use L(s,χ)to denote\nL(s,χ,K). It is known that when χis not the principal character, L(s,χ)has an analytic con-\ntinuation to the entire complex plane. If χis principal then L(s,χ)has a meromorphic contin-\nuation to the entire complex plane with only a simple pole at s= 1. The Generalized Riemann\nHypothesis (GRH) for Dedekind zeta functions states that th e zeros of a Dedekind zeta func-\ntion in the strip 0<ℜ(s)<1lie on the line ℜ(s) = 1/2. Now we state the following lemma\nof Lagarias and Odlyzko.\nLemma 3.6. (Lagarias and Odlyzko, [ 25])Letχbe a generalized Dirichlet character modulo integral\nidealcand letL(s,χ)be the corresponding Hecke L-function. Then we have\nL′\nL(s,χ)+L′\nL(s,χ) =/summationdisplay\nρ/parenleftbigg1\ns−ρ+1\ns−ρ/parenrightbigg\n−log(|dK|Nc′)−1χ/parenleftbigg2\ns+2\ns−1/parenrightbigg\n−2Γ′\nχ\nΓχ(s),\nwhere the sum is over non-trivial zeros of L(s,χ),1χis1ifχis a principal character otherwise it is 0,\nc′is the conductor of χand for some fixed integer 0≤aχ≤nK,\nΓχ(s) :=/bracketleftig\nπ−s+1\n2Γ/parenleftbiggs+1\n2/parenrightbigg/bracketrightigaχ/bracketleftig\nπ−s\n2Γ/parenleftigs\n2/parenrightig/bracketrightignK−aχ. (5)\nLetχ/\\e}atio\\slash=χ0be a generalized Dirichlet character on an imaginary quadra tic field Kand\nL(s,χ)be the corresponding Hecke L- function. Now we consider the following function\nΦχ(x) :=1\nx−1/integraldisplayx\n1/parenleftig/summationdisplay\na⊂OK\n1≤Na≤tΛ(a)\nNaχ([a])/parenrightig\ndt, (6)\nas defined in [ 22]. Then we have the following result proved by Ihara, Kumar Murt y and\nShimura.\nTheorem 3.7. (Ihara, Kumar Murty and Shimura, [ 22])Forx >1and any primitive generalized\nDirichlet character χ, we have\nL′\nL(1,χ)+Φχ(x) =1\nx−1/summationdisplay\nρxρ−1\nρ(1−ρ)+logx\nx−1+1\nx−1logx,\nwhere the sum is over all non-trivial zeros of L(s,χ)counted with multiplicities, and\n/summationdisplay\nρxρ−1\nρ(1−ρ)= lim\nT→∞/summationdisplay\n|Im(ρ)|<Txρ−1\nρ(1−ρ).EULER-KRONECKER CONSTANTS 9\n3.3. Counting integral ideals. We now state a result on counting the number of integral ideal s\nofOKfor number field Kdue to Gun, Ramaré and Sivaraman. For any embedding σofK, the\nMinkowski embedding θofKtoR2mapsxto(ℜ(σ(x)),ℑ(σ(x))). In this setup, we have the\nfollowing counting theorem.\nTheorem 3.8. (Gun, Ramaré and Sivaraman, [ 16])Leta,qbe co-prime ideals of OK,Cbe the ideal\nclass ofaqin the class group of OKandΛ(aq)be the lattice θ(aq)inR2, whereθis as defined above.\nAlso let\nSβ/parenleftbig\na,q,t2/parenrightbig\n={α∈a:|θ(α)|2≤t2,α��βmodq}\nfor some fix β∈ OK. Then for any real number t≥1, we have\n/vextendsingle/vextendsingleSβ/parenleftbig\na,q,t2/parenrightbig/vextendsingle/vextendsingle=2π/radicalbig\n|dK|N(aq)t2+ O∗/parenleftigg\n1013.66N(C−1)\n|N(aq)|1\n2t+1/parenrightigg\n, (7)\nwhere\nN(C−1) =maxb∈C−11\n|N(b)|1\n2.\nOne can ignore 1in the error term when q=OK.\nAs a corollary the authors of [ 16,17] deduce the following two theorems.\nTheorem 3.9. (Gun, Ramaré and Sivaraman, [ 16])LetKbe an imaginary quadratic field. Let qbe\nan integral ideal of Kand[b]be an element of Hq(K). For any real number x≥1, we have\n/summationdisplay\na⊂OK\n[a]=[b]\nNa≤x1 =ρKϕ(q)\n|Hq(K)|x\nNq+O∗/parenleftigg\n1021/parenleftbiggx\nNq/parenrightbigg1/2\n+4·105/parenrightigg\n.\nTheorem 3.10. (Gun, Ramaré and Sivaraman, [ 17])LetKbe an imaginary quadratic field. For any\nreal number x≥1,\n/summationdisplay\na⊆OK\nNa≤x1 =ρKx+O∗/parenleftig\n1015(hKlog(3hK))1/2x1/2/parenrightig\n.\nWe now state a result of Garcia and Lee [ 13] which will be used in the proof of Theorem 1.7.\nTheorem 3.11. LetKbe an imaginary quadratic field and x≥2. We have\n/summationdisplay\nNp≤xlogNp\nNp= logx+O∗/parenleftigg\n3+e75|dK|1/3(log|dK|)2\nρK/parenrightigg\n.\nLetL/Kbe an abelian extension with Galois group Gand letnLandnKbe the respective\ndegrees over Q.10 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nDefinition 4. Letpbe a non-zero prime ideal of OK, unramified in Landqbe a prime ideal above it in\nOL. The unique Galois element σin the Galois group of L/Ksuch that\nσ(a)≡aN(p)modqfor alla∈ OL\nis called the Artin symbol corresponding to pand the extension L/K. The Artin symbol is extended\nmultiplicatively to the set of all non-zero integral ideals o fOKwhich are supported on the set of prime\nideals which do not ramify in L/K. For such an integral ideal a, we use(L/K,a)to denote the Artin\nsymbol.\nUnder the above notation, we have the following theorem from [15].\nTheorem 3.12. (Grenié and Molteni, [ 15])Letσbe an element of the group Gand let1σdenote the\ncharacteristic function of σ. We define\nψ(x,L/K,σ) =/summationdisplay\na⊂OK\npunramified ∀p|a\n1≤Na≤x1σ((a,L/K))Λ(a).\nUnder GRH, for all x≥1,\n/vextendsingle/vextendsingle/vextendsingle|G|ψ(x,L/K,σ)−x/vextendsingle/vextendsingle/vextendsingle≤√x/parenleftbigg/parenleftbigglogx\n2π+2/parenrightbigg\nlog|dL|+/parenleftbigglog2x\n8π+2/parenrightbigg\nnL/parenrightbigg\n.\nThe above result is an extension of their previous result for number fields [ 14] which we\nshall now state in a weaker form as required by our proof.\nTheorem 3.13. (Grenié and Molteni, [ 14])LetKbe a number field. We define ψ(x) =/summationdisplay\na⊂OK,\n1≤Na≤xΛ(a).\nUnder GRH, for all x≥3,\n|ψ(x)−x| ≤7log|dK|√xlogx+√xlog2x+19√x.\nBefore we proceed further, we would like to introduce the not ion of the narrow ray class\nfield modulo a non-zero integral ideal cofOK. To do so, we first define the Artin map in the\nfollowing manner : Given an abelian extension L/Kand a non-zero integral ideal csuch that\nevery prime of Kthat ramifies in Ldividesc, we define ΨL/K,c:I(c)→Gal(L/K)as the map\n/productdisplay\np∤c\npprime ideal of OKpmp/ma√sto→/productdisplay\np∤c\npprime ideal of OK(L/K,p)mp, m p∈N\nwhere the integer mpis0for all but finitely many primes p. It is known from class field theory\nthat this map is surjective.\nDefinition 5. For a non-zero ideal c, the unique extension LofKsuch that\n(1)every prime ideal of Kthat ramifies in Ldividescand\n(2)the kernel of ΨL/K,cisP(c)EULER-KRONECKER CONSTANTS 11\nis called the narrow ray class field of Kwith respect to c. We shall denote this field by K(c).\nIt is known that every finite abelian extension Lof an imaginary quadratic field Kis con-\ntained in a narrow ray class field K(c)ofKfor some integral ideal c. The greatest common\ndivisor of all such integral ideals is called the conductor o fLwith respect to K.\nCorollary 1.2 of [ 15] restates the above bounds in terms of the prime counting fun ction\nπ(x,L/K,σ), where\nπ(x,L/K,σ) =/summationdisplay\n(0)/negationslash=p⊂OK,\npunramified\n1≤Np≤x1σ((p,L/K)).\nIn this setup, we now deduce a corollary about principal prim e ideals. We first recall that\nthe Hilbert class field of an imaginary quadratic field Kis the narrow ray class field of K\ncorresponding to the ideal OK. By class field theory it is also the unique abelian extension o f\nKwhere all the non-zero principal prime ideals of OKsplit completely. For a number field K,\nwe useK(OK)to denote its Hilbert class field.\nCorollary 3.14. LetKbe an imaginary quadratic field and K(OK)the Hilbert class field of K. Letσ0\nbe the trivial element of the Galois group of K(OK)/K. Under GRH, ∀x≥2,\nπ(x,K(OK)/K,σ0) =/vextendsingle/vextendsingle/vextendsingle{p⊆ OK: 1≤Np≤x,psplits completely in K(OK)}/vextendsingle/vextendsingle/vextendsingle\n=1\nhK/integraldisplayx\n21\nlogtdt+O∗/parenleftbigg\n5√xlog|dK|+2√x/parenleftbigglogx\n8π+9/parenrightbigg/parenrightbigg\n.\n4.REQUISITE PROPOSITIONS AND LEMMAS\nIn this section, we state and prove some lemmas and propositi ons required in the proofs of\nour theorems. The first lemma gives us a bound on the cardinalit y ofHq(K)for an imaginary\nquadratic field K.\nLemma 4.1. LetKbe an imaginary quadratic field. For any non-zero prime ideal qinOK,\n1\n12hKNq≤ |Hq(K)|<hKNq.\nProof. See Theorem 1 of Chapter VI [ 26] (page 127). /square\nWe now state and prove some lemmas to compute the sum of the inv erse of the function ϕK\nintroduced earlier. For a number field Kand a non-zero ideal a⊆ OK, we define\nσ(a) :=/summationdisplay\nb⊆OK\nb|aNb.12 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nLemma 4.2. LetKbe an imaginary quadratic field. Then for x≥1, we have\n/summationdisplay\na⊆OK\n1≤Na≤xσ(a)\nNa≤ζK(2)ρKx+1015ζK(3/2)(hKlog(3hK))1/2x1/2.\nProof. We have\n/summationdisplay\na⊆OK\n1≤Na≤xσ(a)\nNa=/summationdisplay\na⊆OK\n1≤Na≤x1\nNa/summationdisplay\nb⊆OK\nb|aNb=/summationdisplay\na⊆OK\n1≤Na≤x/summationdisplay\nb⊆OK\nb|a1\nNb=/summationdisplay\nb⊆OK\n1≤Nb≤x1\nNb/summationdisplay\nc⊆OK\n1≤Nc≤x\nNb1\n≤/summationdisplay\nb⊆OK\n1≤Nb≤x1\nNb/parenleftbigg\nρKx\nNb+1015(hKlog(3hK))1/2/parenleftigx\nNb/parenrightig1/2/parenrightbigg\n≤ζK(2)ρKx+1015ζK(3/2)(hKlog(3hK))1/2x1/2.\nHere we have used Theorem 3.10 in the penultimate step. Using equation ( 4), we get the desired\nresult.\n/square\nLemma 4.3. For a number field Kanda⊂ OK, we have\nNa\nϕK(a)≤ζK(2)σ(a)\nNa.\nProof. Forp|a, letαpbe such that pαp/bardbla. We now consider\nNa\nϕK(a)=/productdisplay\np|a/parenleftbigg\n1−1\nNp/parenrightbigg−1\n=/productdisplay\np|a1+1\nNp\n1−1\nNp2=/productdisplay\np|aNpαp+Npαp−1\nNpαp/parenleftig\n1−1\nNp2/parenrightig≤ζK(2)σ(a)\nNa.\n/square\nLemma 4.4. LetKbe an imaginary quadratic field. We now have for x≥3,\n/summationdisplay\na⊂OK\n1≤Na≤x1\nϕK(a)≤ζK(2)2ρKlogex+2·1015ζK(3/2)ζK(2)(hKlog(3hK))1/2.\nProof. Consider\n/summationdisplay\na⊆OK\n1≤Na≤x1\nϕK(a)=/summationdisplay\n1≤t≤x/summationdisplay\na⊆OK\nNa=t1\nϕK(a)=/summationdisplay\n1≤t≤x1\nt/summationdisplay\na⊆OK\nNa=tNa\nϕK(a).\nLetat=/summationdisplay\na⊆OK\nNa=tNa\nϕK(a)andA(x) =/summationdisplay\n1≤t≤xat. We now apply partial summation formula to get\n/summationdisplay\na⊆OK\n1≤Na≤x1\nϕK(a)=/summationdisplay\n1≤t≤xat\nt=A(x)\nx+/integraldisplayx\n1A(u)\nu2du. (8)EULER-KRONECKER CONSTANTS 13\nBy Lemma 4.2and Lemma 4.3,\nA(x) =/summationdisplay\n1≤t≤xat=/summationdisplay\n1≤t≤x/summationdisplay\na⊆OK\nNa=tNa\nϕK(a)=/summationdisplay\na⊆OK\n1≤Na≤xNa\nϕK(a)\n≤ζK(2)2ρKx+1015ζK(3/2)ζK(2)(hKlog(3hK))1/2x1/2.\nUsing the above bound in ( 8), we get the desired result.\n/square\nNow we compute the upper bound of the absolute discriminant o f the narrow ray class field\nK(q)corresponding to modulus qof number field K.\nLemma 4.5. LetKbe an imaginary quadratic field. Let qbe a non-zero prime ideal in OKanddK(q)be\nthe discriminant of K(q). Then,\nlog|dK(q)| ≤ρK/radicalbig\n|dK|Nqlog(|dK|Nq).\nProof. Using Lemma 3.2, we have\n|dK(q)|=|dK||Hq(K)|N∆K(q)/K.\nSinceGal(K(q)/K)is abelian, every irreducible character of Gal(K(q)/K)is of degree 1. Ifχ\nis a Hecke character of Gal(K(q)/K),f(χ)is also the conductor of a subfield of K(q)which\ncontains K(see page 535, Prop 11.10 of [ 28]). Therefore, f(χ)dividesq. By Theorem 3.3, we\nhave\n∆K(q)/K=/productdisplay\nχirred.f(χ).\nTaking norm on both sides, we get\nN∆K(q)/K=/productdisplay\nχirred.Nf(χ)≤/productdisplay\nχirred.Nq=Nq|Hq(K)|.\nUsing Lemma 4.1, we get\nlog|dK(q)|=|Hq(K)|log|dK|+logN∆K(q)/K≤ρK/radicalbig\n|dK|Nqlog(|dK|Nq).\n/square\n4.1. Selberg’s sieve. Let(t)be a non-zero principal integral ideal of OK. For a non-zero inte-\ngral ideal b, letb(t)denote the maximal divisor of bco-prime to (t), that is\nb(t)=/productdisplay\n(p,(t))=1pνp(b),14 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nwhereνp(b)denotes the valuation of batp. Also letd(b)denote the number of divisors of band\nΩK(b)denote the number of prime ideals dividing bcounted with multiplicity. For a non-zero\nintegral ideal b, we define\nρ(b) = 2ΩK(b(t)), f(t)(b) =Nb\n2ΩK(b(t))andf1,(t)(b) =/summationdisplay\na|bµ(a)f(t)/parenleftbiggb\na/parenrightbigg\n.\nEvidently all these functions are multiplicative. Let\nP(z) =/productdisplay\nNp≤z\n(p,(2310))=1p.\nFurther, for an ideal eofOKcoprime to (2), we define\nSe(z) =/summationdisplay\na⊆OK\n1≤N(a)≤z\n(a,(2310)e)=OKµ2(a)\nf1,(t)(a),andλe=µ(e)f(t)(e)Se(z\nN(e))\nf1,(t)(e)SOK(z).\nWe observe here that 2310 is the product of all the primes upto 12. Working with primes greater\nthan12allows us to get explicit lower bounds for certain sums which appear in the Selberg\nsieve (see equation ( 9) below). For non-zero integral ideals b1andb2, we use [b1,b2]to denote\nthe least common multiple of the ideals b1andb2. In this setup, we have the following result.\nLemma 4.6. We have\n/summationdisplay\nb1,b2|P(z),\nNbi≤z|λb1λb2|ρ([b1,b2])/radicalbig\nN([b1,b2])≤ζK/parenleftbigg3\n2/parenrightbigg16\nz.\nProof. The proof of this lemma follows along the same lines as Lemma 1 1 of [ 16]. We consider\nthe sum\n/summationdisplay\nb1,b2|P(z),\nNbi≤z|λb1λb2|ρ([b1,b2])/radicalbig\nN([b1,b2])=/summationdisplay\nd|P(z),\nNd≤z√\nNd\nρ(d)/summationdisplay\nbi|P(z),\nd=(b1,b2)\nNbi≤z|λb1λb2|ρ(b1)ρ(b2)/radicalbig\nN(b1b2).\nFrom the expression of λband withy=z/Nd, we get\n/summationdisplay\nNc≤y,\n(c,d(2310))= OK|λdc|ρ(c)√\nNc≤√yNd\nρ(d)f1,(t)(d)/productdisplay\nNp>11/parenleftbigg\n1+ρ(p)√Np(Np−ρ(p))/parenrightbigg\n.\nWe thus get\n/summationdisplay\nb1,b2|P(z),\nNbi≤z|λb1λb2|ρ([b1,b2])/radicalbig\nN([b1,b2])≤z/productdisplay\nNp>11/parenleftbigg\n1+ρ(p)√Np(Np−ρ(p))/parenrightbigg2/parenleftbigg\n1+ρ(p)√Np\n(Np−ρ(p))2/parenrightbigg\n.EULER-KRONECKER CONSTANTS 15\nNote that\n/productdisplay\nNp>11/parenleftbigg\n1+ρ(p)√Np(Np−ρ(p))/parenrightbigg\n≤ζK/parenleftbigg3\n2/parenrightbigg4\nand/productdisplay\nNp>11/parenleftbigg\n1+ρ(p)√Np\n(Np−ρ(p))2/parenrightbigg\n≤ζK/parenleftbigg3\n2/parenrightbigg8\n.\nThis completes the proof of the lemma. /square\nProposition 4.7. LetKbe an imaginary quadratic field and tbe a fixed non-zero element in OK, which\nis not a unit in OK. We have for u≥2exp/parenleftbig\n8·1045·|dK|/parenrightbig\n/summationdisplay\n1≤N(α)≤u\nα,tα+1∈ PK1≤/parenleftigg\n2·1010\n/radicalbig\n|dK|ρ2\nK/parenrightigg\nN(t)u\nϕ((t))log2u1/4\n2+1030u3/4.\nProof. Letzbe a real number such that√z >12. We would like to estimate the sum\n/summationdisplay\n1≤N(α)≤u\nα,tα+1∈PK1≤/summationdisplay\n1≤N(α)≤z\nα,tα+1∈PK1+/summationdisplay\nz<N(α)≤u\n1≤N(tα+1)≤z\nα,tα+1∈PK1+/summationdisplay\nz<N(α)≤u\n((α(tα+1)),P(z))=11\n≤/summationdisplay\n1≤N(α)≤u\n((α(tα+1)),P(z))=11+4|µK|z.\nLet us consider the first sum\n/summationdisplay\n1≤N(α)≤u\n((α(tα+1)),P(z))=11 =/summationdisplay\n1≤N(α)≤u\n/summationdisplay\nb|((α(tα+1)),P(z))µ(b)\n≤/summationdisplay\n1≤N(α)≤u\n/summationdisplay\nb|((α(tα+1)),P(z))λb\n2\n.\nRearranging the sums we get\n/summationdisplay\n1≤N(α)≤u\n/summationdisplay\nb|((α(tα+1)),P(z))λb\n2\n=/summationdisplay\nb1,b2|P(z)\nNbi≤zλb1λb2/summationdisplay\n1≤N(α)≤u\nbi|(α(tα+1))1.\nNote that b= [b1,b2]is squarefree. If b(t)is the maximal divisor of bco-prime to (t), we need\nto countαin2ΩK(b(t))classes in OK/b. Using Theorem 3.8fora=OK,q=b, we get for z≤u1\n2\n/summationdisplay\nb1,b2|P(z)\nNbi≤zλb1λb2/summationdisplay\n1≤N(α)≤u\n[b1,b2]|(α(tα+1))1 =/summationdisplay\nb1,b2|P(z)\nNbi≤zλb1λb2/parenleftigg\ncK2ΩK([b1,b2](t))u\nN[b1,b2]+O∗/parenleftbigg\n10142ΩK([b1,b2](t))/radicalbiggu\nN[b1,b2]/parenrightbigg/parenrightigg\n,\nwherecKis the constant given by2π√\n|dK|. We first deal with the error term. By Lemma 4.6, we\nget\n/summationdisplay\nb1,b2|P(z)\nNbi≤z|λb1λb2|2ΩK([b1,b2](t))\n/radicalbig\nN[b1,b2]≤332·z.16 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nThe main term now becomes/summationdisplay\nb1,b2|P(z)λb1λb2\nf(t)([b1,b2])which in turn is\n/summationdisplay\nb1,b2|P(z)λb1λb2f(t)((b1,b2))\nf(t)(b1)f(t)(b2)=/summationdisplay\nb1,b2|P(z)λb1λb2\nf(t)(b1)f(t)(b2)/summationdisplay\na|(b1,b2)f1,(t)(a) =/summationdisplay\na|P(z)f1,(t)(a)\n/summationdisplay\nc|P(z)\na|cλc\nf(t)(c)\n2\n.\nFurther, we observe that\n/summationdisplay\nc|P(z)\na|cλc\nf(t)(c)=SOK(z)−1/summationdisplay\nc|P(z)\na|cµ(c)\nf1,(t)(c)/summationdisplay\n1≤N(g)≤z\nN(c)\n(g,(2310)c)=OKµ2(g)\nf1,(t)(g).\nWritingc=hawith(h,a) =OK, we get\nµ(a)\nf1,(t)(a)SOK(z)−1/summationdisplay\n1≤N(h)≤z\nN(a)\nh|P(z)\n(h,a)=OKµ(h)\nf1,(t)(h)/summationdisplay\n1≤N(g)≤z\nN(ha)\n(g,(2310)ha)=OKµ2(g)\nf1,(t)(g)\n=µ(a)\nf1,(t)(a)SOK(z)−1/summationdisplay\n1≤N(h)≤z\nN(a)\nh|P(z)\n(h,a)=OK/summationdisplay\n1≤N(g)≤z\nN(ha)\n(g,(2310)ha)=OKµ(h)µ2(gh)\nf1,(t)(gh).\nSettinga1=ghgives\n/summationdisplay\nc|P(z)\na|cλc\nf(t)(c)=µ(a)\nf1,(t)(a)SOK(z)−1/summationdisplay\n1≤N(a1)≤z\nN(a)\n(a1,(2310)a)=OK\na1|P(z)µ2(a1)\nf1,(t)(a1)/summationdisplay\nh|a1µ(h) =SOK(z)−1µ(a)\nf1,(t)(a).\nCombining the above, the main term is cKuSOK(z)−1.However,\nSOK(z) =/summationdisplay\n1≤N(a)≤z\n(a,(2310))= OKµ2(a)\nf1,(t)(a)=/summationdisplay\n1≤N(a)≤z\n(a,(2310))= OKµ2(a)\nf(t)(a)·f(t)(a)\nf1,(t)(a)\n=/summationdisplay\n1≤N(a)≤z\n(a,(2310))= OKµ2(a)\nf(t)(a)·/productdisplay\np|af(t)(p)\nµ(p)+f(t)(p)\n≥/summationdisplay\n1≤N(a)≤z\n(a,(2310))= OK1\nf(t)(a)−/summationdisplay\n1≤N(a)≤z\nanon-sq-free\n(a,(2310))= OK1\nf(t)(a).\nConsider\n/summationdisplay\n1≤N(a)≤z\nanon-sq-free\n(a,(2310))= OK1\nf(t)(a)≤/summationdisplay\n1≤Nb≤z\n(b,(2310))= OK/summationdisplay\nNp≤√z\n(p,(2310))= OK1\nf(t)(b)f(t)(p)2=/summationdisplay\n1≤Nb≤z\n(b,(2310))= OK1\nf(t)(b)\n/summationdisplay\nNp≤√z\n(p,(2310))= OK1\nf(t)(p)2\n.EULER-KRONECKER CONSTANTS 17\nLooking at the sum\n/summationdisplay\nNp≤√z\n(p,(2310))= OK1\nf(t)(p)2≤/summationdisplay\nNp≤√z\n(p,(2310))= OK4\nN(p)2≤8\n11.\nTherefore,\nSOK(z)≥3\n11/summationdisplay\n1≤N(a)≤z\n(a,(2310))= OK1\nf(t)(a). (9)\nLet us consider\n/summationdisplay\n1≤Na≤z\n(a,(2310))= OK1\nf(t)(a)=/summationdisplay\n1≤Na≤z\n(a,(2310))= OK2ΩK(a(t))\nNa≥/summationdisplay\n1≤Na≤z\n(a,(2310))= OKd(t)(a)\nNa.\nHered(t)(a)is the number of ideals dividing aand co-prime to (t). We now look at\n/productdisplay\np|(t)/parenleftbigg\n1−1\nNp/parenrightbigg−1/summationdisplay\n1≤Na≤z\n(a,(2310))= OKd(t)(a)\nNa=/summationdisplay\n1≤Na≤z\n(a,(2310))= OKd(t)(a)\nNa/summationdisplay\np|b⇒p|(t)1\nNb\n=/summationdisplay\n(0)/negationslash=b⊂OK1\nNb/summationdisplay\n1≤Na≤z,a|b\n(a,(2310))= OK\np|b\na⇒p|(t)d(t)(a)\n≥/summationdisplay\nb⊂OK\n1≤Nb≤z\n(b,(2310))= OKd(b)\nNb.\nIn order to compute the above sum, we consider the following\n/summationdisplay\nb⊂OK\n1≤Nb≤z\n(b,(2310))= OKd(b) =/summationdisplay\nb⊂OK\n1≤Nb≤z\n(b,(2310))= OK/summationdisplay\ne|b1 =/summationdisplay\nNe≤z\n(e,(2310))=1/summationdisplay\nb⊂OK\n1≤Nb≤z\ne|b,(b,(2310))= OK1 =/summationdisplay\nNe≤z\n(e,(2310))=1/summationdisplay\nb⊂OK\n1≤Nb≤z\nNe\n(b,(2310))= OK1.\nWe have by Theorem 3.9\n/summationdisplay\nb⊂OK\n1≤Nb≤z\nNe\n(b,(2310))= OK1 =ρKϕ((2310))z\n23102Ne+O∗/parenleftbigg\n3·1024hK/parenleftigz\nNe/parenrightig1\n2+3·1012hK/parenrightbigg\n.\nTherefore, we have\n/summationdisplay\nb⊂OK\n1≤Nb≤z\nNe\n(b,(2310))= OK1≥ρKϕ((2310))z\n23102Ne−3·1024hK/parenleftigz\nNe/parenrightig1\n2−3·1012hK.18 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nBy partial summation formula,\n/summationdisplay\ne⊂OK\nNe≤z\n(e,(2310))= OK1\nNe≥/parenleftbiggρKϕ((2310))\n23102/parenrightbigg\nlogez−7·1024hK.\nSimilarly, again by partial summation, we get\n/summationdisplay\ne⊂OK\nNe≤z\n(e,(2310))= OK1√\nNe≤2ρKϕ((2310))\n23102√z+6·1024hKlogz.\nCombining the above inequalities,\n/summationdisplay\nb⊂OK\n1≤Nb≤z\n(b,(2310))= OKd(b)≥/summationdisplay\nNe≤z\n(e,(2310))= OK/parenleftbiggρKϕ((2310))z\n23102Ne−4·1024hK/parenleftigz\nNe/parenrightig1\n2/parenrightbigg\n≥/parenleftbiggρKϕ((2310))\n23102/parenrightbigg2\nzlogez−4.81·1049ρKhK/radicalbig\n|dK|z\n≥1\n2/parenleftbiggρKϕ((2310))\n23102/parenrightbigg2\nzlogez,\nforlogez≥23102·1050hK√\n|dK|\nρKϕ((2310))2. Now, using the fact that hK≥ρK/radicalbig\n|dK|/(2π)and thatϕ((2310)) ≥4802,\nwe getlogez≥2·1045|dK|. We now apply partial summation once again to get\n/summationdisplay\nb⊂OK\n1≤Nb≤z\n(b,(2310))= OKd(b)\nNb≥(6ρK)2\n2·1010log2zforlogez≥2·1045|dK|.\nCombining the above for logez≥2·1045|dK|,\nSOK(z)−1≤/productdisplay\np|(t)/parenleftbigg\n1−1\nNp/parenrightbigg−122·1010\n3(6ρK)2log2z.\nIf we suppose further that z≤√u, we get\n/summationdisplay\n1≤N(α)≤u\nα,tα+1∈ PK1≤4|µK|z+2π/radicalbig\n|dK|/productdisplay\np|(t)/parenleftbigg\n1−1\nNp/parenrightbigg−122·1010u\n3(6ρK)2log2z+1014·332·z√u\n≤2π/radicalbig\n|dK|·8·1010N(t)u\nϕ((t))(6ρK)2log2z+1030z√u.EULER-KRONECKER CONSTANTS 19\nFurther, since we have logz≤2√z, we get for z=u1/4\n2,\n/summationdisplay\n1≤N(α)≤u\nα,tα+1∈ PK1≤/parenleftigg\n2·1010\nρ2\nK/radicalbig\n|dK|/parenrightigg\nN(t)u\nϕ((t))log2u1/4\n2+1030u3/4.\n/square\nLemma 4.8. ForQ≥exp(1045|dK|), we have\nhKπ∗(Q)≥2Q\n25logQ.\nProof. We first observe that for y≥8, we have\n/integraldisplayy\n21\nlogtdt≥y\nlogy.\nFory≥2005, we have 4/log2≤y1/5/logyand this implies\n/integraldisplayy\n21\nlogtdt≤/integraldisplayy4/5\n21\nlog2dt+/integraldisplayy\ny4/51\nlogy4/5dt≤y4/5\nlog2+5y\n4logy≤3y\n2logy.\nWe have\nπ∗(Q) =π(Q,K(OK)/K,σ0)−π(Q/2,K(OK)/K,σ0),\nwhereσ0is the trivial element of the Galois group of K(OK)/K. We now get by Corollary 3.14,\nhKπ∗(Q)≥Q\nlogQ−5hK/radicalbig\nQlog|dK|−2hK/radicalbig\nQ/parenleftbigglogQ\n8π+9/parenrightbigg\n−3Q\n4logQ/2−5hK/radicalbigg\nQ\n2log|dK|−2hK/radicalbigg\nQ\n2/parenleftigg\nlogQ\n2\n8π+9/parenrightigg\n.\nFurther since Q≥26, we get\nhKπ∗(Q)≥Q\nlogQ−5hK/radicalbig\nQlog|dK|−2hK/radicalbig\nQ/parenleftbigglogQ\n8π+9/parenrightbigg\n−18Q\n20logQ−5hK/radicalbigg\nQ\n2log|dK|−2hK/radicalbigg\nQ\n2/parenleftigg\nlogQ\n2\n8π+9/parenrightigg\n.\nSimplifying the above and using Lemma 3.1and Lemma 4.1, we get\nhKπ∗(Q)≥Q\nlogQ/parenleftigg\n1\n10−103|dK|3/4log|dK|logQ√Q−400|dK|3/4logQ√Q/parenleftbigglogQ\n8π+9/parenrightbigg/parenrightigg\n.\nWe now bound each of the negative terms separately using the l ower bound for Qas follows.\nUsing the fact that logx/√xis a decreasing function for x≥8, for the first term, we get\n103|dK|3/4log|dK|logQ√Q≤1048|dK|7/4log|dK|\nexp(1045|dK|\n2)≤1\n100.20 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nSimilarly, for the second term,\n400|dK|3/4logQ√Q/parenleftbigglogQ\n8π+9/parenrightbigg\n≤400·1090|dK|11/4\n8πexp(1045|dK|\n2)+3600·1045|dK|7/4\nexp(1045|dK|\n2)≤1\n100.\nThis gives us the lemma. /square\nNext two lemmas are devoted to the proof of Proposition 4.11. We first compute an upper\nbound of the real part of the functionΓ′\nχ\nΓχ(s)fors∈C(see equation ( 5) for its definition).\nLemma 4.9. Forℜ(s)>1and number field K, we have\nℜ/parenleftbiggΓ′\nχ\nΓχ(s)/parenrightbigg\n<nK/parenleftbigg\nlog/parenleftbigg|s+1|\n2+2/parenrightbigg\n−logπ\n2/parenrightbigg\n.\nProof. Taking the logarithmic derivative on both sides of ( 5), we obtain\nΓ′\nχ\nΓχ(s) =−nK\n2logπ+aχ\n2Γ′\nΓ/parenleftbiggs+1\n2/parenrightbigg\n+(nK−aχ)\n2·Γ′\nΓ/parenleftigs\n2/parenrightig\n.\nUsing Lemma 3.5, forℜ(s)>1, we get\nℜ/parenleftbiggΓ′\nχ\nΓχ(s)/parenrightbigg\n<−nK\n2logπ+aχlog/parenleftbigg|s+1|\n2+2/parenrightbigg\n+(nK−aχ)log/parenleftbigg|s|\n2+2/parenrightbigg\n<nK/parenleftbigg\nlog/parenleftbigg|s+1|\n2+2/parenrightbigg\n−logπ\n2/parenrightbigg\n.\n/square\nThe following lemma gives us an upper bound of the absolute va lue of the logarithmic\nderivative of Hecke L- functions.\nLemma 4.10. LetL(s,χ)be the Hecke L-function corresponding to a generalized Dirichlet character χ\nmodulo integral ideal c. Then for ℜ(s) =σ>1and number field K, we have\n/vextendsingle/vextendsingle/vextendsingleL′\nL(s,χ)/vextendsingle/vextendsingle/vextendsingle<nK\nσ−1.\nProof. Taking logarithmic derivative of Euler product of Hecke L fu nction and Dedekind zeta\nfunction, we obtain\n/vextendsingle/vextendsingle/vextendsingleL′\nL(s,χ)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/summationdisplay\np∤c\np/negationslash=(0)logNp\n(Np)s\nχ([p])−1/vextendsingle/vextendsingle/vextendsingle≤/summationdisplay\np/negationslash=(0)logNp\n(Np)σ−1=−ζ′\nK\nζK(σ).\nSimilarly, we note that\n−ζ′\nK\nζK(σ)≤ −nKζ′\nζ(σ)<nK\nσ−1, σ>1.\nThe last inequality follows from Lemma 3.4. Now combining the above identities, we get\n/vextendsingle/vextendsingle/vextendsingleL′\nL(s,χ)/vextendsingle/vextendsingle/vextendsingle<nK\nσ−1.EULER-KRONECKER CONSTANTS 21\n/square\nNow letL(s,χ)be the Hecke L-function corresponding to generalized Dirichlet charact er\nχmodulo candc′be the conductor of χ. Then following Lagarias and Odlyzko [ 25, Lemma\n5.4], we have the following result which gives us an explicit bound on the number of zeros of\nL(s,χ).\nProposition 4.11. For a real number t, letNt,χdenote the number of zeros (denoted ρ=α+iβ) of\nL(s,χ)with0<α<1and|β−t| ≤1. Then\nNt,χ<5(3(nK+1)+log( |dK|Nc′)+2nKlog(|t|+2)).\nMoreover, when K/\\e}atio\\slash=Qthen\nNt,χ<50nKlog(|dK|Nc′(|t|+2)).\nProof. From Lemma 3.6, we have\nℜ/parenleftigg/summationdisplay\nρ/parenleftbigg1\ns−ρ+1\ns−ρ/parenrightbigg/parenrightigg\n=ℜ/parenleftbiggL′\nL(s,χ)/parenrightbigg\n+ℜ/parenleftbiggL′\nL(s,χ)/parenrightbigg\n+log(|dK|Nc′)\n+1χℜ/parenleftbigg2\ns+2\ns−1/parenrightbigg\n+2ℜ/parenleftbiggΓ′\nχ\nΓχ(s)/parenrightbigg\n.\nEvaluating it at s= 2+itand using Lemma 4.9and Lemma 4.10, we obtain\nℜ/parenleftigg/summationdisplay\nρ/parenleftbigg1\ns−ρ+1\ns−ρ/parenrightbigg/parenrightigg\n<nK(2−logπ)+log(|dK|Nc′)+2ℜ/parenleftbigg1\n2+it+1\n1+it/parenrightbigg\n+2nKlog/parenleftbigg|3+it|\n2+2/parenrightbigg\n<3nK+3+log( |dK|Nc′)+2nKlog(|t|+2). (10)\nClearly, when K/\\e}atio\\slash=Q, the above quantity is bounded above by 10nKlog(|dK|Nc′(|t|+2)).\nAs noticed by Lagarias and Odlyzko [ 25, Lemma 5.4], we have\nℜ/parenleftigg/summationdisplay\nρ/parenleftbigg1\ns−ρ+1\ns−ρ/parenrightbigg/parenrightigg\n≥/summationdisplay\nρ\n|β−t|≤12−α\n(2−α)2+(t−β)2>1\n5Nt,χ, (11)\nsinceαis between 0 and 1. Substituting ( 11) into ( 10), we get our desired results. /square\n5. B OUNDS ON THE AVERAGE VALUE OF THE EULER -KRONECKER CONSTANT\nThroughout this section, we will use the following notation s. LetKbe an imaginary qua-\ndratic number field and qbe a non-zero principal prime ideal of K. Letχbe a generalized22 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nDirichlet character modq. Now after summing over all non-trivial characters in the fo llowing\nfunction (see equation ( 6))\nΦχ(x) =1\nx−1/integraldisplayx\n1/parenleftig/summationdisplay\na⊂OK\n1≤Na≤tΛ(a)\nNaχ([a])/parenrightig\ndt,\nwe get by orthogonality of characters of a finite group,\n/summationdisplay\nχ/negationslash=χ0Φχ(x) =1\nx−1/integraldisplayx\n1/parenleftig\n|Hq(K)|/summationdisplay\na⊂OK\n1≤Na≤t\na∈[1]qΛ(a)\nNa−/summationdisplay\na⊂OK\n1≤Na≤t\n(a,q)=1Λ(a)\nNa/parenrightig\ndt, (12)\nwhere[1]qis the trivial element of the group Hq(K). Let\nψ(x,q,[1]q) =/summationdisplay\na⊂OK\n1≤Na≤x\na∈[1]qΛ(a)andψ(x) =/summationdisplay\na⊂OK\n1≤Na≤xΛ(a).\nSince the Galois group of K(q)/Kis isomorphic to the group Hq(K)under the Artin map,\nwe can identify the trivial element of the Galois group of K(q)/Kwith the trivial element\n[1]q∈Hq(K). Therefore, ψ(x,q,[1]q) =ψ(x,K(q)/K,σo)whereσois the identity element of the\nGalois group of K(q)/K. For a non-zero prime ideal q,\n/summationdisplay\na⊂OK\n1≤Na≤x\n(a,q)=1Λ(a)\nNa≥/summationdisplay\na⊂OK\n1≤Na≤xΛ(a)\nNa−logNq\nNq−1. (13)\nBy Abel’s summation formula, we get\n/summationdisplay\na⊂OK\n1≤Na≤x\na∈[1]qΛ(a)\nNa=ψ(x,q,[1]q)\nx+/integraldisplayx\n1ψ(u,q,[1]q)\nu2du and/summationdisplay\na⊂OK\n1≤Na≤xΛ(a)\nNa=ψ(x)\nx+/integraldisplayx\n1ψ(u)\nu2du.\nIntegrating over ton both sides of ( 13), we get\n/integraldisplayx\n1/summationdisplay\na⊂OK\n1≤Na≤t\n(a,q)=1Λ(a)\nNadt≥/integraldisplayx\n1ψ(t)\ntdt+/integraldisplayx\n1/integraldisplayt\n1ψ(u)\nu2dudt−logNq\nNq−1(x−1)\n≥/integraldisplayx\n1ψ(t)\ntdt+/integraldisplayx\n1/integraldisplayx\nuψ(u)\nu2dtdu−logNq\nNq−1(x−1)\n≥x/integraldisplayx\n1ψ(u)\nu2du−logNq\nNq−1(x−1).\nSimilarly, we get/integraldisplayx\n1/summationdisplay\na⊂OK\n1≤Na≤t\na∈[1]qΛ(a)\nNadt=x/integraldisplayx\n1ψ(u,q,[1]q)\nu2du.EULER-KRONECKER CONSTANTS 23\nHence, ( 12) becomes\n/summationdisplay\nχ/negationslash=χ0Φχ(x)≤x\nx−1/integraldisplayx\n1/parenleftbigg\n|Hq(K)|ψ(u,q,[1]q)\nu2−ψ(u)\nu2/parenrightbigg\ndu+logNq\nNq−1. (14)\nLemma 5.1. We have\n/summationdisplay\n1\n2Q<Nq≤QlogNq\nNq−1≤14+4e75|dK|1/3(log|dK|)2\nρK,\nwhere the sum runs over prime ideals qwithNqin the interval (1\n2Q,Q].\nProof. Using Theorem 3.11, we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\n1≤Nq≤xlogNq\nNq−logx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤3+e75|dK|1/3(log|dK|)2\nρK.\nIt implies that\n/summationdisplay\n1\n2Q<Nq≤QlogNq\nNq−1≤/summationdisplay\n1\n2Q<Nq≤Q2logNq\nNq≤2log2+12+4e75|dK|1/3(log|dK|)2\nρK.\n/square\nRemark 5.1. We note, from the above, that we have\n/summationdisplay\nQ\n2≤Nq≤QlogNq\nNq−1<8·1045|dK|.\nTheorem 5.2. Assuming GRH, for x>2andQ≥8exp(8·1045|dK|), we have\n/summationdisplay′\n1\n2Q<Nq≤Q/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nχ/negationslash=χ0Φχ(x)/vextendsingle/vextendsingle/vextendsingle≤(213·1013hK+6000h2\nK+10)π∗(Q)logQ.\nwhere the outer sum is over principal prime ideals and π∗(Q)denotes the number of principal prime\nideals with norm in the interval (1\n2Q,Q].\nProof. From now onwards, we will drop the prime over summation and al ways assume the\nsum to be over principal prime ideals in the given interval.\nFrom equation ( 14), forx≥2, we get\n/summationdisplay\n1\n2Q<Nq≤Q/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nχ/negationslash=χ0Φχ(x)/vextendsingle/vextendsingle/vextendsingle≤2/integraldisplayx\n1/summationtext\n1\n2Q<Nq≤Q/vextendsingle/vextendsingle/vextendsingle|Hq(K)|ψ(u,q,[1]q)−ψ(u)/vextendsingle/vextendsingle/vextendsingle\nu2du+C1(K), (15)\nwhereC1(K) = 14+4e75|dK|1/3(log|dK|)2\nρK.\nFor any principal ideal a⊂ OK, letTabe the set of all principal prime ideals qsuch that there24 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nexists a generator πaofasuch thatπa≡1modq. Then,\n/summationdisplay\n1\n2Q<Nq≤Q|Hq(K)|ψ(u,q,[1]q) =/summationdisplay\na⊂OK\n1≤N(a)≤u\naprincipalΛ(a)/summationdisplay\nq∈Ta\n1\n2Q<Nq≤Q|Hq(K)|. (16)\nFirst we shall consider the case u≤Q5/2.\nContribution from prime powers pkfork≥2:Consider a principal ideal a⊂ OK,1≤Na≤u\nand letp1,...,pjbe the distinct prime ideals with norm in (Q/2,Q]such that a∈[1]pifor all\n1≤i≤j. Therefore for each 1≤i≤j, we haveπa,pi, generators of a, such thatπa,pi≡1modpi.\nWe observe that since ais a principal ideal, all the πa,piare associates. Then we have\nj/productdisplay\ni=1pi|/productdisplay\nw∈µK(wπa,p1−1).\nLetc=|µK|. Then\nQj\n2j≤N(p1...pj)≤/productdisplay\nw∈µKN(wπa,p1−1)≤(4N(πa,p1))c≤(4u)c≤(4Q5/2)c, (17)\nwhere we have used the fact that N(α−1)≤4N(α)for any non-zero integral element αin an\nimaginary quadratic field. For Q>8, comparing both sides of ( 17), we note that\n26j−15c<Q2j−5c≤22j+4c.\nThis implies that j≤5|µK|. So, the number of terms in the inner sum of ( 16) is bounded above\nby5|µK|forQ >8. Using Lemma 4.1, we see that the contribution of powers of prime ideals\nto the sum ( 16) is\n/summationdisplay\na=pk⊂OK\n1≤Npk≤u\npkprincipal\nk≥2Λ(a)/summationdisplay\nq∈Ta\n1\n2Q<Nq≤Q|Hq(K)| ≤(5|µK|)hKQ/summationdisplay\n1≤Npk≤u\nk≥2logNp≤10hK|µK|Q√ulogu.\nPutting this in ( 15), we get\n2/integraldisplayQ5/2\n1\n/summationdisplay\na=pk⊂OK\n1≤Npk≤u\npkprincipal\nk≥2Λ(a)/summationdisplay\nq∈Ta\n1\n2Q<Nq≤Q|Hq(K)|\ndu\nu2≤20hK|µK|Q/integraldisplayQ5/2\n1logu\nu3/2du≤480hKQ,\nwhere we use ( 4) and the fact that for ℜ(s)>0,\n/integraldisplay∞\n1lognt\nts+1dt=Γ(n+1)\nsn+1. (18)EULER-KRONECKER CONSTANTS 25\nSinceQ>exp(1045|dK|), using Lemma 4.8and the fact that log|dK| ≥log2 , we get\n2/integraldisplayQ5/2\n1\n/summationdisplay\npk⊂OK\n1≤Npk≤u\npkprincipal\nk≥2Λ(a)/summationdisplay\nq∈Ta\n1\n2Q<Nq≤Q|Hq(K)|\ndu\nu2≤6000h2\nKπ∗(Q)logQ. (19)\nContribution from primes p:We now want to compute the contribution of prime ideals to\n(16). To do so, we consider the sum\n/summationdisplay\np⊂OK\n1≤Np≤u\npprincipallogNp/summationdisplay\n1\n2Q<Nq≤Q\np∈[1]q|Hq(K)|.\nTo bound this term for each pin the outer sum, we choose a generator πpand count the number\nof ideals qsuch thatπp≡1 modq. Therefore\n/summationdisplay\np⊂OK\n1≤Np≤u\npprincipallogNp/summationdisplay\np∈[1]q\n1\n2Q<Nq≤Q|Hq(K)| ≤/summationdisplay\nπp∈OK\n1≤N(πp)≤u\nπp∈PKlogN(πp)/summationdisplay\nπp≡1 modq\n1\n2Q<Nq≤Q|Hq(K)|.\nFor each principal prime ideal q, we now fix a generator πq. The condition πp≡1 modqis\nequivalent to the existence of a δ∈ OKwithN(δ)≤8u/Q such thatπp−1 =δπq.Therefore,\nwe have\n/summationdisplay\np⊂OK\n1≤Np≤u\npprincipallogNp/summationdisplay\np∈[1]q\n1\n2Q<Nq≤Q|Hq(K)| ≤/summationdisplay\nπp∈OK\n1≤N(πp)≤u\nπp∈PK/summationdisplay\n1\n2Q<Nq≤Q\nπp−1=πqδ\nfor some δ∈OK|Hq(K)|logNπp\n≤/summationdisplay\nπp∈OK\n1≤N(πp)≤u\nπp∈PK/summationdisplay\n1\n2Q<N(πq)≤Q\nπp−1=πqδ\nfor some δ∈OK|H(πq)(K)|logN(πp)\n≤/summationdisplay\nδ∈OK\n1≤N(δ)≤8u\nQ/summationdisplay\nπp,πq∈PK\nN(πp)≤u\n1\n2Q<N(πq)≤Q\nπp−1=δπq|H(πq)(K)|logN(πp).(20)\nWe split the outer summation as\n/summationdisplay\nδ∈OK\n1≤N(δ)≤8u\nQ=/summationdisplay\nδ∈OK\n1≤N(δ)<u\nQ+/summationdisplay\nδ∈OK\nu\nQ≤N(δ)≤8u\nQ26 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nNow we shall compute the following integrals\nI1=/integraldisplayQ5/2\n1/summationdisplay\nδ∈OK\n1≤N(δ)<u\nQ/summationdisplay\nπp,πq∈PK\nN(πp)≤u\n1\n2Q<N(πq)≤Q\nπp−1=δπq|H(πq)(K)|logN(πp)du\nu2(21)\nI2=/integraldisplayQ5/2\n1/summationdisplay\nδ∈OK\nu\nQ≤N(δ)≤8u\nQ/summationdisplay\nπp,πq∈PK\nN(πp)≤u\n1\n2Q<N(πq)≤Q\nπp−1=δπq|H(πq)(K)|logN(πp)du\nu2. (22)\nEstimatingI1:Forδ∈ OK\\{0}, we note that counting the number of pairs of prime elements\n(πp,πq)such thatπp−1 =δπqin the inner sum in ( 21) is same as counting number of α∈ PK\nsuch thatδα+1also belongs to PK. Using the conditions πp−1 =δπqandQ/2<N(πq)≤Q,\nwe getN(πp)≤4N(πp−1)≤4N(δ)Q. Also,N(α) =N(πq)≤Q. Using Proposition 4.7we\nbound this term by\n/summationdisplay\nπp,πq∈PK\nN(πp)≤u\n1\n2Q<N(πq)≤Q\nπp−1=δπq1≤/summationdisplay\n1≤N(α)≤Q\nα,δα+1∈PK1≤25·1010·N(δ)Q\nρ2\nK/radicalbig\n|dK|ϕ((δ))log2Q\n16+1030Q3/4.\nUsing equation ( 5) and Lemma 4.1, we get\nI1≤hKQ/integraldisplayQ5/2\n1/summationdisplay\nδ∈OK\n1≤N(δ)<u\nQ/parenleftigg\n25·1010·N(δ)Q\nρ2\nK/radicalbig\n|dK|ϕ((δ))log2Q\n16+1030Q3/4/parenrightigg\nlog4QN(δ)du\nu2.\nInterchanging sum and integral, we get\nI1≤hKQ/summationdisplay\nδ∈OK\n1≤N(δ)<Q3/2/parenleftigg\n25·1010·N(δ)Q\nρ2\nK/radicalbig\n|dK|ϕ((δ))log2Q\n16+1030Q3/4/parenrightigg\nlog4QN(δ)/integraldisplay∞\nQN(δ)du\nu2\n≤hKQ/summationdisplay\nδ∈OK\n1≤N(δ)<Q3/2/parenleftigg\n25·1010·N(δ)Q\nρ2\nK/radicalbig\n|dK|ϕ((δ))log2Q\n16+1030Q3/4/parenrightigg\nlog4QN(δ)\nQN(δ).\nLet\nI11=hKQ/summationdisplay\nδ∈OK\n1≤N(δ)<Q3/2/parenleftigg\n25·1010·N(δ)Q\nρ2\nK/radicalbig\n|dK|ϕ((δ))log2Q\n16/parenrightigg\nlog4QN(δ)\nQN(δ)\nand\nI12=hKQ/summationdisplay\nδ∈OK\n1≤N(δ)<Q3/21030Q3/4log4QN(δ)\nQN(δ).EULER-KRONECKER CONSTANTS 27\nComputing the summation in the first term and using Lemma 4.4, we get\nI11≤24·5·1010hKQlog(2Q)\nρ2\nK/radicalbig\n|dK|log2Q\n16/summationdisplay\nδ∈OK\n1≤N(δ)<Q3/21\nϕ((δ))\n≤24·5·1010hKQlog(2Q)\nρ2\nK/radicalbig\n|dK|log2Q\n16/parenleftig\nζK(2)2ρKlog4Q3/2+√\n3·2·1015ζK(3/2)ζK(2)hK/parenrightig\n.\nForQ>29we have log2Q≤2logQ\n16. Further, since hK≤ρK/radicalbig\n|dK|, we get\nI11≤28·3·1011Q+1040hKQ\nρKlogQ\n16.\nSinceQ≥8exp(8·1045·|dK|), we have logQ/16≥1045·|dK|. This gives us\nI11≤28·3·1011Q+Q/radicalbig\n|dK|.\nUsing Lemma 4.8, we get\nI11≤/parenleftbigg\n25·3·1013hK+1\n2/parenrightbigg\nπ∗(Q)logQ. (23)\nWe now consider\nI12≤5\n2·1030hKQ3/4log2Q/summationdisplay\nδ∈OK\n1≤N(δ)<Q3/21\nN(δ).\nBy partial summation formula, we have\n/summationdisplay\nδ∈OK\n1≤N(δ)<Q3/21\nN(δ)≤2ρKlog2Q+2·√\n3·1015hK. (24)\nWe now substitute the bound ( 24) in bound for I12. Using Lemma 4.8, the fact that log2Q≤\n16(2Q)1/16, and the bound Q≥8exp(8·1045·|dK|)we get\nI12≤1035ρKh2\nKπ∗(Q)logQ\nQ1/4+1049h3\nKπ∗(Q)logQ\nQ1/4≤1\n2π∗(Q)logQ. (25)\nTherefore,\nI1≤(25·3·1013hK+1)π∗(Q)logQ. (26)\nEstimatingI2:In the same way as before, for δ∈ OK\\{0}, counting the number of pairs of\nprime elements (πp,πq)such thatπp−1 =δπqin the inner sum in ( 22) is same as counting\nnumber ofα∈ PKsuch thatδα+1also belongs to PK. Using the condition πp−1 =δπqwe get\nN(δα) =N(δπq) =N(πp−1)≤4N(πp)≤4u. Using Proposition 4.7we bound this term by\n/summationdisplay\nπp,πq∈PK\nN(πp)≤u\n1\n2Q<N(πq)≤Q\nπp−1=δπq1≤/summationdisplay\n1≤N(α)≤4u/N(δ)\nα,δα+1∈PK1≤27·1010u\nρ2\nK/radicalbig\n|dK|ϕ((δ))log2u\n4N(δ)+1030/parenleftbigg4u\nN(δ)/parenrightbigg3/4\n. (27)28 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nUsing Lemma 4.1and ( 27), we get\nI2≤27·1010hKQ\nρ2\nK/radicalbig\n|dK|log2(Q\n32)/integraldisplayQ5/2\n1/summationdisplay\nδ∈OK\nu\nQ≤N(δ)≤8u\nQu−1logu\nϕK((δ))du+ 3·1030hKQ/integraldisplayQ5/2\n1/summationdisplay\nδ∈OK\nu\nQ≤N(δ)≤8u\nQlogu\nN(δ)3/4u5/4du.\nWe shall denote henceforth by\nI21=27·1010hKQ\nρ2\nK/radicalbig\n|dK|log2(Q\n32)/integraldisplayQ5/2\n1/summationdisplay\nδ∈OK\nu\nQ≤N(δ)≤8u\nQu−1logu\nϕK((δ))du\nand\nI22= 3·1030hKQ/integraldisplayQ5/2\n1/summationdisplay\nδ∈OK\nu\nQ≤N(δ)≤8u\nQlogu\nu5/4N(δ)3/4du.\nLet us first consider I21, interchanging sum and integral and using the bound hK≤ρK/radicalbig\n|dK|,\nwe get\nI21≤27·1010Q\nρKlog2(Q\n32)/summationdisplay\nδ∈OK\n1≤N(δ)≤8Q3/21\nϕK((δ))/integraldisplayQN(δ)\nQN(δ)\n8logu\nudu.\nIntegrating and using Lemma 4.4and the fact that log(3hK)≤8(3hK)1/8, we get\nI21≤26·1010·log8·Q\nρKlog2(Q\n32)/summationdisplay\nδ∈OK\n1≤N(δ)≤8Q3/21\nϕK((δ))log/parenleftbiggQ2N(δ)2\n8/parenrightbigg\n≤29·1011·log8·Qlog2(2Q)\nlog2(Q\n32)+28·34·1026·Qlog(2Q)\nρKlog2(Q\n32)(hKlog(3hK))1/2\n≤29·1011·log8·Qlog2(2Q)\nlog2(Q\n32)+211·34·1026·Qlog(2Q)\nρKlog2(Q\n32)h9/16\nK.\nUsing the fact that 2log(Q/32)≥log(2Q)forQ≥211, we get\nI21≤213·1011·log8·Q+212·34·1026Q\nρKlog(Q\n32)h9/16\nK.\nSinceQ≥8exp(8·1045|dK|)and bounding hK, we get\nI21≤213·1011·log8·Q+Q\n2·1010ρ7/16\nK|dK|23/32.\nWe now apply Lemma 4.8and we bound ρKusing Lemma 3.1to get\nI21≤210·1013·log8·hKπ∗(Q)logQ+1\n2π∗(Q)logQ. (28)EULER-KRONECKER CONSTANTS 29\nLet us now consider I22. We have\nI22≤6·1030hKQ/summationdisplay\nδ∈OK\n1≤N(δ)≤8Q3/21\nN(δ)Q1/4/integraldisplayQN(δ)\nQN(δ)\n8logu\nudu\n≤3·log8·1030hKQ/summationdisplay\nδ∈OK\n1≤N(δ)≤8Q3/21\nN(δ)Q1/4logQ2N(δ)2\n8\n≤15·log8·1030hKQlog(2Q)\nQ1/4/summationdisplay\nδ∈OK\n1≤N(δ)≤8Q3/21\nN(δ).\nSincelog(2Q)≤20Q1/8, we have\nI22≤3·log8·1032hKQ\nQ1/8/summationdisplay\nδ∈OK\n1≤N(δ)≤8Q3/21\nN(δ).\nNow, using ( 24), we get\nI22≤1033hKQ\nQ1/8(2ρKlog8Q+2·√\n3·1015hK)\n≤1035ρKhKQ\nQ1/16+·1049h2\nKQ\nQ1/8.\nBy Lemma 3.1, Lemma 4.8, the bound on hKand the fact that Q≥8exp(8·1045|dK|), we get\nI22≤1\n2π∗(Q)logQ. (29)\nCombining ( 28) and ( 29), we get\nI2≤(212·1013·hK+1)π∗(Q)logQ. (30)\nFinally from ( 26) and ( 30), we get\n2/integraldisplayQ5/2\n1\n/summationdisplay\np⊂OK\n1≤Np≤u\npprincipalΛ(a)/summationdisplay\nq∈Ta\n1\n2Q<Nq≤Q|Hq(K)|\ndu\nu2≤(213·1013·hK+2)π∗(Q)logQ. (31)\nContribution from ψ(u):Using Theorem 3.13, under the Riemann hypothesis we have for\nu≥3,\nψ(u)≤u+7log|dK|√ulogu+√ulog2u+19√u.\nPutting this in ( 15), we get\n2/integraldisplayQ5/2\n1/summationtext\n1\n2Q<Nq≤Qψ(u)\nu2du≤(5logQ+56log|dK|+109)π∗(Q)≤6·π∗(Q)logQ. (32)30 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nWe will now consider the case u>Q5/2. Since\nψ(u,q,[1]q) =/summationdisplay\na⊂OK\n1≤Na≤u\na∈[1]qΛ(a) =/summationdisplay\na⊂OK\n1≤Na≤u\n(a,q)=OK\n(a,K(q)/K)=σ0Λ(a) =/summationdisplay\na⊂OK\n1≤Na≤u\n(a,q)=OK1σo((a,K(q)/K))Λ(a),\nwhere(a,K(q)/K)denotes the Artin symbol of aandσothe trivial element of Gal(K(q)/K)\nwith1σoas its characteristic function. Using Theorem 3.12, Lemma 4.1, Lemma 4.5and equa-\ntion ( 4), we get\n/vextendsingle/vextendsingle/vextendsingle|Hq(K)|ψ(u,q,[1]q)−u/vextendsingle/vextendsingle/vextendsingle≤√u/parenleftbigg/parenleftbigglogu\n2π+2/parenrightbigg\nlog|dK(q)|+/parenleftbigglog2u\n4π+4/parenrightbigg\n|Hq(K)|/parenrightbigg\n≤2ρK/radicalbig\n|dK|Nq/parenleftig\nlog(Nq|dK|)logu+log2u/parenrightig√u.\nTherefore, for Q≥8exp(8·1045|dK|),\n2/integraldisplayx\nQ5/2/summationtext\n1\n2Q<Nq≤Q/vextendsingle/vextendsingle/vextendsingle|Hq(K)|ψ(u,q,[1]q)−u/vextendsingle/vextendsingle/vextendsingle\nu2du\n≤4ρK/radicalbig\n|dK|Qπ∗(Q)/integraldisplayx\nQ5/2log(Q|dK|)logu+log2u\nu3/2du\n≤1\n2π∗(Q)logQ,\nwhere in the last step we have used Lemma 3.1. By Theorem 3.13, forQ≥3, we get\n|ψ(u)−u| ≤7(log|dK|)√ulogu+√ulog2u+19√u.\nThis implies that for Q≥8exp(8·1045|dK|),\n2/integraldisplayx\nQ5/2/summationtext\n1\n2Q<Nq≤Q|ψ(u)−u|\nu2du≤(56log|dK|+108)π∗(Q)≤1\n2π∗(Q)logQ,\nwhere we have used equation ( 18). Thus, the contribution of Q5/2<u<x to (15) is\n2/integraldisplayx\nQ5/2/summationtext\n1\n2Q<Nq≤Q/vextendsingle/vextendsingle/vextendsingle|Hq(K)|ψ(u,q,[1]q)−ψ(u)/vextendsingle/vextendsingle/vextendsingle\nu2du≤π∗(Q)logQ. (33)\nCombining Lemma 5.1, (19), (31), (32) and ( 33), we get\n/summationdisplay\n1\n2Q<Nq≤Q/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nχ/negationslash=χ0Φχ(x)/vextendsingle/vextendsingle/vextendsingle≤(213·1013hK+6000h2\nK+10)π∗(Q)logQ.\n/squareEULER-KRONECKER CONSTANTS 31\nLetqbe a non-zero principal prime ideal of OK. We note that given a generalized Dirichlet\ncharacterχmodulo qit is either primitive or induced by a character of the class g roup (also\ncalled a generalized Dirichlet character modulo OK). We now consider the sum\n/summationdisplay\nχ/negationslash=χ0(Φχ∗(x)−Φχ(x)) =/summationdisplay\nχmodq\nχ∗modq(Φχ∗(x)−Φχ(x))+/summationdisplay\nχmodq\nχ∗modOK\nχ/negationslash=χ0(Φχ∗(x)−Φχ(x))\n=/summationdisplay\nχmodq\nχ∗modOK\nχ/negationslash=χ0(Φχ∗(x)−Φχ(x))\n=/summationdisplay\nχ∗modOK\nχ/negationslash=χ01\nx−1/integraldisplayx\n1\n/summationdisplay\nNa≤t,\nq|aΛ(a)\nNaχ∗(a)\ndt.\nSinceqis a prime ideal, it follows that the inner sum only consists o f terms for which a=qk.\nFurther since qis principal so is a. Therefore, we get\n/summationdisplay\nχ/negationslash=χ0(Φχ∗(x)−Φχ(x)) =1\nx−1/integraldisplayx\n1\n/summationdisplay\nNa≤t,\nq|aΛ(a)\nNa/summationdisplay\nχ∗modOK\nχ/negationslash=χ0χ∗(a)\ndt\n≤hK\nx−1/integraldisplayx\n1\n/summationdisplay\nk≥1\nNqk≤tlogNq\nNqk\ndt\n≤hKx\nx−1·logNq\nNq−1.\nTherefore for x≥2, using Lemma 5.1, we obtain\n/summationdisplay\nQ\n2<Nq≤Q/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nχ/negationslash=χ0(Φχ(x)−Φχ∗(x))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2hK/parenleftigg\n14+4e75|dK|1/3(log|dK|)2\nρK/parenrightigg\n. (34)\nProposition 5.3. LetQ≥2andx≥Q4. Letχbe a generalized Dirichlet character modulo q, where\nqis a non-zero principal prime ideal in OK. Also letL′\nL(1,χ)denote the logarithmic derivative of the\ncorresponding Hecke L-function evaluated at 1. Then under GRH, we have\n/summationdisplay′\n1\n2Q<Nq≤Q/summationdisplay\nχ/negationslash=χ0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleL′\nL(1,χ∗)+Φχ∗(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<2010ρK/radicalbig\n|dK|π∗(Q)log(5|dK|Q)\nQ,\nwhere the outer sum is over principal prime ideals in the inter val(1\n2Q,Q]andπ∗(Q)denotes the number\nof principal prime ideals with norm in the interval (1\n2Q,Q].32 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nProof. Using Theorem 3.7, we have\nL′\nL(1,χ∗)+Φχ∗(x) =1\nx−1/summationdisplay\nρχ∗xρχ∗−1\nρχ∗(1−ρχ∗)+logx\nx−1+1\nx−1logx, (35)\nwhere the sum is over all non trivial zeros of L(s,χ∗)counted with multiplicities. Using Propo-\nsition 4.11, we get\nNt,χ∗= #{ρ=1\n2+iβ:|β−t| ≤1,L(ρ,χ∗) = 0}<100log(|dK|Nq(|t|+2)).\nFor a non-trivial zero ρχ∗=1\n2+iβofL(s,χ∗), we have |ρχ∗(1−ρχ∗)|=/parenleftbig1\n2/parenrightbig2+β2. For|β| ≤1,\nwe get\n/summationdisplay\nρχ∗=1/2+iβ\n|β|≤11\n|ρχ∗(1−ρχ∗)|≤4N0,χ∗<400log(2 |dK|Nq). (36)\nAlso, for |β|>1, we get\n/summationdisplay\nρχ∗=1/2+iβ\n|β|>11\n|ρχ∗(1−ρχ∗)|≤∞/summationdisplay\nk=1/summationdisplay\nρχ∗=1/2+iβ\nk<|β|≤k+11\nβ2≤∞/summationdisplay\nk=11\nk2(Nk,χ∗+N−k,χ∗)\n<200∞/summationdisplay\nk=1log(|dK|Nq(k+2))\nk2.\nWe have\n∞/summationdisplay\nk=1log(|dK|Nq(k+2))\nk2≤ζ(2)log(|dK|Nq)+∞/summationdisplay\nk=1log(k+2)\nk2≤3log(5|dK|Nq).\nIt implies that\n/summationdisplay\nρχ∗=1/2+iβ\n|β|>11\n|ρχ∗(1−ρχ∗)|≤600log(5 |dK|Nq). (37)\nPlugging ( 36) and ( 37) in ( 35), we get\n/vextendsingle/vextendsingle/vextendsingleL′\nL(1,χ∗)+Φχ∗(x)/vextendsingle/vextendsingle/vextendsingle<√x+1\nx−1/summationdisplay\nρχ∗1\n|ρχ∗(1−ρχ∗)|+logx\nx−1+2logx\nx\n<2000√xlog(5|dK|Nq)+1\nx−1+4√x\n<2010√xlog(5|dK|Nq).EULER-KRONECKER CONSTANTS 33\nSumming over χ/\\e}atio\\slash=χ0moduloqand then over the principal prime ideals qwithNq∈(1\n2Q,Q],\nwe get\n/summationdisplay′\n1\n2Q<Nq≤Q/summationdisplay\nχ/negationslash=χ0/vextendsingle/vextendsingle/vextendsingleL′\nL(1,χ∗)+Φχ∗(x)/vextendsingle/vextendsingle/vextendsingle<2010√xhKQπ∗(Q)log(5|dK|Q)\n≤2010√xρK/radicalbig\n|dK|Qπ∗(Q)log(5|dK|Q),\nwhere we use equation ( 4) in the last step. Choosing x≥Q4, we get our desired result. /square\n5.1. Proof of Theorem 1.7.Letχbe a generalized Dirichlet character modulo q, whereqis a\nnon-zero principal prime ideal in OK. We have (see Chapter VIII, [ 4] for more details)\nζK(q)(s) =/productdisplay\nχmodqL(s,χ∗), (38)\nwhere the product is over all generalized Dirichlet charact ers modulo q. For any number field\nK, an equivalent definition of γKis given by\nγK= lim\ns→1/parenleftbiggζ′\nK\nζK(s)+1\ns−1/parenrightbigg\n.\nTherefore, taking the logarithmic derivative on both sides of (38), we get\nγK(q)=γK+/summationdisplay\nχ/negationslash=χ0L′\nL(1,χ∗).\nIt now follows that\nγK(q)=γK−/summationdisplay\nχ/negationslash=χ0Φχ(x)+/summationdisplay\nχ/negationslash=χ0/parenleftig\nΦχ(x)−Φχ∗(x)/parenrightig\n+/summationdisplay\nχ/negationslash=χ0/parenleftigL′\nL(1,χ∗)+Φχ∗(x)/parenrightig\n.\nTherefore,\n|γK(q)| ≤ |γK|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nχ/negationslash=χ0Φχ(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nχ/negationslash=χ0/parenleftig\nΦχ(x)−Φχ∗(x)/parenrightig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nχ/negationslash=χ0/parenleftigL′\nL(1,χ∗)+Φχ∗(x)/parenrightig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nUsing Theorem 5.2, (34) Proposition 5.3and the fact that Q≥8exp(8·1045|dK|), we get\n/summationdisplay′\n1\n2Q<Nq≤Q|γK(q)|<|γK|π∗(Q)+(6000h2\nK+213·1013hK+10)π∗(Q)logQ\n+ 2hKlogQ+π∗(Q)logQ.\nThis implies that\n1\nπ∗(Q)/summationdisplay′\n1\n2Q<Nq≤Q|γK(q)|<|γK|+(6000h2\nK+1017hK+11)logQ.34 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\n6.CONCLUDING REMARKS\nWe conclude by showing that infinitely many narrow ray class fiel ds corresponding to prin-\ncipal prime moduli of an imaginary quadratic field are non-abe lian. Given any narrow ray\nclass field K(q)ofKmodulo an integral ideal q, the Hilbert class field of Kis a subfield of\nK(q)containing K. By a result of Ankeny and Chowla [ 2], we know that there are infinitely\nmany imaginary quadratic fields whose class number is divisib le by3. By genus theory (see\nchapter 1 of [ 23]) if the Hilbert class field of Kis abelian over Q, the class number must be a\npower of 2. Therefore for the family of fields suggested by the result of A nkeny and Chowla,\nthe Hilbert class field must be non-abelian and therefore all t he narrow ray class fields of such\nan imaginary quadratic are also non-abelian.\nData Availability Statement: Data sharing is not applicable to this article as no datasets were\ngenerated or analysed during the current study.\nConflict of Interest Statement: The authors certify that there is no conflict of interest.\nAcknowledgments. All the authors would like to thank Prof. Sanoli Gun for sugge sting the\nproblem and comments on the manuscript. They would also like to thank Prof. Purusottam\nRath for helpful comments on earlier versions of the draft. T he first author would also like\nto thank Prof. Alessandro Languasco for drawing her attenti on to the reference [ 8]. Part of\nthis work was done during visits of the first author to the Sorbo nne university and the second\nauthor to the Sorbonne and the Paris Cité universities under the support of the Indo-French\nProgram in Mathematics (IFPM). The first and second authors woul d like to thank the IFPM for\npartial financial support. They would also like to thank the DA E number theory plan project.\nAll the authors thank the Institute of Mathematical Sciences for providing academic facilities.\nThe third author would like to thank the Chennai Mathematical Institute and the Indian Sta-\ntistical Institute for providing academic support. Furthe r, the third author would like to thank\nthe National Board for Higher Mathematics and the Department of Science and Technology for\nfinancial support. Finally the authors would like to thank the referee for a careful reading of\nthe paper and a number of relevant suggestions.\nREFERENCES\n[1] J.- H. Ahn and S.- H. Kwon, An explicit upper bound for the least prime ideal in the Chebo tarev density theorem , Ann.\nInst. Fourier (Grenoble) 69(2019), no. 3, 1411–1458.\n[2] N. C. Ankeny and S. Chowla, On the divisibility of the class number of quadratic fields , Pacific J. Math. 5(1955), no.\n3, 321-324.\n[3] A. I. Badzyan, The Euler-Kronecker constant , Mat. Zametki 87(2010), no. 1, 35–47.\n[4] J. W. S. Cassels and A. Frohlich, Algebraic Number Theory ,Thompson book company INC. , Washington D. C.,\n1967.EULER-KRONECKER CONSTANTS 35\n[5] J.- M. Deshouillers, S. Gun, O. Ramaré and J. 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Moree, Values of the Euler φ-function not divisible by a given odd prime, and the distrib ution\nof Euler-Kronecker constants for cyclotomic fields , Math. Comp., 83(2014), no. 287, 1447-1476\n[12] E. Fouvry, Sum of Euler–Kronecker constants over consecutive cycloto mic fields , J. Number Theory, 133(2013), no. 4,\n1346–1361.\n[13] S. R. Garcia and E. S. Lee, Unconditional explicit Mertens’ theorems for number fields and Dedekind zeta residue\nbounds , The Ramanujan Journal, 57(2022), 1169–1191.\n[14] L. Grenié and G. Molteni, Explicit versions of the prime ideal theorem for Dedekind ze ta functions under GRH , Math.\nComp., 85(2016), no. 298, 889–906.\n[15] L. Grenié and G. Molteni, An explicit Chebotarev density theorem under GRH , J. Number Theory, 200(2019), no. ,\n441–485.\n[16] S. Gun, O. Ramaré and J. Sivaraman, Counting ideals in ray classes , J. Number Theory, 243 (2023), 13–37.\n[17] S. Gun, O. Ramaré and J. Sivaraman, An application of counting ideals in ray classes , submitted.\n[18] R. R. Hall and G. Tenenbaum, Divisors, Cambridge University Press , 1988.\n[19] L. Hong, K. Ono and S. Zhang, Euler-Kronecker constants for cyclotomic fields , Bull. Austral. Math. Soc., 107(2023),\n79–84.\n[20] Y. Ihara, The Euler-Kronecker invariants in various families of glob al fields , Arithmetics, geometry and coding theory\n(AGCT 2005), Sémin. Congr., 21(2005), Soc. Math. France, 79–102.\n[21] Y. Ihara, On the Euler-Kronecker constants of global fields and primes with small norms , Algebraic geometry and\nnumber theory, Progr. Math., 253(2006), Birkhäuser Boston, Boston, MA, 407–451.\n[22] Y. Ihara, V . Kumar Murty and M. Shimura, On the logarithmic derivatives of Dirichlet L-functions at s= 1, Acta\nArith., 137(2009), no. 3, 253–276.\n[23] M. Ishida, The genus fields of algebraic number fields, Le cture Notes in Mathematics, 555,Springer-Verlag,\nBerlin-New York (1976).\n[24] V . Kumar Murty, The Euler-Kronecker constant of a cyclotomic field , Ann. Sci. Math. Québec, 35(2011), no. 2,\n239–247.\n[25] J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem , Algebraic number fields\n(Durham, 1975), Academic press, London,(1977), 409–464.\n[26] S. Lang, Algebraic number theory, Springer-Verlag , second edition, New York, 1994.\n[27] H. L. Montgomery, Zeros of L-functions , Invent. Math., 8(1969), pp. 346–354.\n[28] J. Neukirch, Algebraic Number Theory, Springer-Verlag , Berlin, 1999.36 NEELAM KANDHIL, RASHI LUNIA AND JYOTHSNAA SIVARAMAN\nNEELAM KANDHIL ,\nMAX-PLANCK -INSTITUT FÜR MATHEMATIK , VIVATSGASSE 7, D-53111 B ONN , GERMANY .\nRASHI LUNIA ,\nTHEINSTITUTE OF MATHEMATICAL SCIENCES , A CI OFHOMI BHABHA NATIONAL INSTITUTE , CIT C AMPUS ,\nTARAMANI , CHENNAI 600 113, I NDIA .\nJYOTHSNAA SIVARAMAN ,\nSTATISTICS AND MATHEMATICS UNIT, INDIAN STATISTICAL INSTITUTE , 8THMILE, M YSORE RD, RVCE P OST,\nBENGALURU , KARNATAKA 560059, I NDIA .\nEmail address :kandhil@mpim-bonn.mpg.de\nEmail address :rashisl@imsc.res.in\nEmail address :jyothsnaa.s@gmail.com"    },    {        "title": "2402.13134v1.An_architecture_for_two_qubit_encoding_in_neutral_ytterbium_171_atoms.pdf",        "content": "An architecture for two-qubit encoding in neutral ytterbium-171 atoms\nZhubing Jia,1,∗William Huie,1Lintao Li,1Won Kyu Calvin Sun,1Xiye Hu,1Aakash,1\nHealey Kogan,1Abhishek Karve,1Jong Yeon Lee,1, 2,†and Jacob P. Covey1,‡\n1Department of Physics, The University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA\n2Department of Physics, The University of California, Berkeley, CA 94618, USA\nWe present an architecture for encoding two qubits within the optical “clock” transition and\nnuclear spin-1/2 degree of freedom of neutral ytterbium-171 atoms. Inspired by recent high-fidelity\ncontrol of all pairs of states within this four-dimensional ququart space, we present a toolbox for\nintra-ququart (single atom) one- and two-qubit gates, inter-ququart (two atom) Rydberg-based\ntwo- and four-qubit gates, and quantum nondemolition (QND) readout. We then use this toolbox\nto demonstrate the advantages of the ququart encoding for entanglement distillation and quantum\nerror correction which exhibit superior hardware efficiency and better performance in some cases\nsince fewer two-atom (Rydberg-based) operations are required. Finally, leveraging single-state QND\nreadout in our ququart encoding, we present a unique approach to studying interactive circuits\nas well as to realizing a symmetry protected topological phase of a spin-1 chain with a shallow,\nconstant-depth circuit. These applications are all within reach of recent experiments with neutral\nytterbium-171 atom arrays or with several trapped ion species.\nI. INTRODUCTION\nAlthough most quantum computing architectures focus\non two-level quantum systems for encoding qubits, the\nneed for auxiliary quantum systems or extra quantum\nstates is ubiquitous. These extra degrees of freedom are\nused to, e.g., mediate gates between qubits, perform mea-\nsurements or cooling during a computation, or improve\nthe qubit connectivity. For neutral atom and trapped\nion quantum computers, these auxiliary degrees of free-\ndom often take the form of a second atomic species [1–3]\nand/or atoms that are moved during the circuit [1, 4–7].\nRecently, the potential of extra states within the atoms\nto address these needs has gained intense interest. The\nability to use two or more portions of the atomic level\nstructure to perform disparate functions simultaneously\nwithout crosstalk has recently enabled “mid-circuit” op-\nerations such as readout and cooling [1, 8–14]. Addition-\nally, it has been shown how the ability to programmably\nrepurpose an atom for these roles by moving it be-\ntween the different sets of levels opens the door to im-\nproved hardware efficiency and more flexible circuit com-\npiling [9, 11, 15].\nWhen using auxiliary states within an atom to per-\nform ancillary functions for our qubits, a natural ques-\ntion is whether it may be advantageous to include these\nstates in the computational space and encode quan-\ntum information in a higher dimension. For instance,\nwhen using a pair of extra states to provide two dis-\ntinct qubit encodings within one atom, we could alter-\nnatively think of this four-level system as a “ququart”\ncomposed of two qubits. Higher-dimensional encodings\nwith “qudits” have attracted much attention for neutral\n∗zjia11@illinois.edu\n†jongyeon@illinois.edu\n‡jcovey@illinois.edu\n1S0\nmF -1/2 1/23P03P13S1\n(n*㲊\u0012)\n|0 |1|0|1\nNuclear\n(“n”) qubit Optical \n(“o”) qubit \n|00 |01|10 |11(a) (b)FIG. 1. Overview of ququart encoding architecture in\nthe omg level structure of171Yb atoms . (a) The relevant\nenergy levels and operations. We encode a ququart in the1S0\nand3P0manifolds of171Yb atom, each containing two nuclear\nZeeman states. Qubit rotations of the states in the same mani-\nfold are performed with rf magnetic field or stimulated Raman\ntransitions, while1S0and3P0states are connected through\noptical clock transitions. Readouts are performed on1S0sub-\nstates using1S0↔3P1transition. Inter-ququart entangling\ngates are driven by single-photon Rydberg excitation from\n3P0states to high- nRydberg states denoted as3S1. (b) Two-\nqubit encoding in one ququart. The ququart can be treated\nnaturally as the tensor product of an optical (“o”) and a nu-\nclear (“n”) qubit.\natoms [14, 16–20], trapped ions [21–25] and supercon-\nducting circuits [26–29]. Qudit systems offer improved\nhardware efficiency including logical encoding [30] and\npotentially have better performance if the intra-atom\ngates have higher fidelity than the inter-atom gates. How-\never, the required intra-atom operations quickly become\nnumerous and challenging as the dimension grows, and\nthus relatively small internal spaces whose states are al-\nready widely used for myriad qubit operations offer an\nattractive starting point for architectures with higher-\ndimensional encoding.\nHere, we focus on a four-level “ququart” encoding ofarXiv:2402.13134v1  [quant-ph]  20 Feb 20242\nquantum information in neutral ytterbium-171 (171Yb)\nby employing the optical “clock” transition and the nu-\nclear spin-1/2 degree of freedom. We show that this\nququart can equivalently be represented as two qubits\n– the optical (“o”) qubit and the nuclear (“n”) qubit –\nwith the full gate set of intra-atom one- and two-qubit\ngates, inter-atom two- and four-qubit gates, and measure-\nments. We show several example applications of our en-\ncoding such as entanglement distillation, quantum error\ncorrection, and a new approach to interactive quantum\ncircuits that leverages single-state quantum nondemoli-\ntion readout. Our work builds upon the growing litera-\nture of higher-dimensional encoding schemes for neutral\natoms [14, 16, 18, 19], trapped ions [21–24] and super-\nconducting circuits [26–29], and presents a comprehen-\nsive blueprint for qudit encodings that are specifically\nfocused on a direct mapping with qubit circuits.\nII. QUQUART OPERATIONS IN171YB ATOMS\nWe now focus on the level structure and operations\nsurrounding the ququart encoding. Specifically, we fo-\ncus on operations that are easily translated to the more\nwidely used language of qubits because they allow us\nto apply our ququart system to any qubit-based algo-\nrithm. Figure 1(a) shows the relevant energy levels and\nthe transitions related to our ququart operations. The\nququart is encoded in the F= 1/21S0(ground) and\n6s6p3P0(metastable) manifolds, each with two nuclear\nZeeman substates. Transitions between the ground and\nmetastable states are driven by a direct optical “clock”\ncoupling, and transitions between nuclear Zeeman states\nare driven by a stimulated Raman coupling. The map-\nping between our ququart states and two-qubit states\n(|00⟩,|01⟩,|10⟩and|11⟩) is based on treating the ququart\nstates as the tensor product of an optical clock (“o”-)\nqubit and a nuclear Zeeman (“n”-) qubit, as is shown in\nFig. 1(b). We discuss this mapping more formally and\nconsider ququart state tomography in Appendices A and\nB, respectively.\nBefore going into detail on this encoding scheme and\nthe required operations, we briefly summarize the state\nof the art and the primary challenges. The ground and\nmetastable nuclear spin qubits have both been manipu-\nlated with gate fidelities of F ≈ 0.999 and have coher-\nence times of T∗\n2>7 seconds [12, 31–35]. The primary\nchallenge centers around realizing fast, high-fidelity op-\ntical qubit operations with a long coherence time. Long-\nterm phase stability is needed for running long circuits,\nhigh-frequency phase noise must be suppressed when per-\nforming fast gates, and deleterious motional effects from\nfinite temperature and finite trap frequencies must be\nmitigated. Techniques have been devised to filter high-\nfrequency phase noise on ultrastable lasers [11, 36–38],\nand motional ground-state cooling schemes [11, 14, 33]\ncombined with carefully chosen pulse design [11, 14] have\nbeen used to obviate the effects of atomic motion. Fi-\n|00 |01|10|11\n|01|10\n|10 |11\n“o”-qubit rotation “n”-qubit rotation Intra-ququart SWAP Intra-ququart CNOT(a) (b) (c) (d)\nRn(θ)Rn(θ)“o”\n“n”3D1\n3P1RX(π)RX(π)FIG. 2. Intra-ququart gates through clock and Ra-\nman transitions . (a) Single-qubit rotations Rˆn(θ) of angle\nθalong rotation axis ˆ non the “o”-qubit via pairs of clock\ntransitions. To ensure that the state of the other qubit is un-\naffected, the two transitions within a pair should perform the\nsame rotations on each pair of states. (b) Single-qubit rota-\ntions Rˆn(θ) on the “n”-qubit driven by pairs of stimulated\nRaman transitions. (c) Intra-ququart swap gate is achieved\nby applying a π-rotation between |01⟩and|10⟩states. This\noperation swaps the information encoded in “o”- and “n”-\nqubits. (d) Intra-ququart CNOT gate is achieved by applying\naπ-rotation between |10⟩and|11⟩states. When the control\nqubit (“o”-qubit) is in |1⟩state, a bit flip operation is applied\nto the target qubit (“n”-qubit).\nnally, we assume that the atoms are trapped at a “magic”\nwavelength for which the potential experienced for all\nfour states is identical [11, 34, 35, 39]. Optical clock tran-\nsitions have been manipulated with gate fidelities of\nF ≈ 0.998 and have coherence times of T∗\n2≈3 sec-\nonds [11, 14, 40].\nA. Intra-ququart gates\nUsing the two-qubit encoding described above, intra-\nququart one- and two-qubit gates can be performed by\ndriving optical clock or Raman transitions. Figure 2\nshows four typical sets of intra-ququart gates, including\nsingle-qubit rotations, two-qubit swap, and CNOT gates.\nFurther details on the intra-ququart gates can be found\nin Appendix C.\nThe single-qubit gates on the “o”- or “n”-qubit require\none pair of clock or Raman transitions mediated by dif-\nferent states [Fig. 2(a)(b)]. For the “o”-sector rotations,\nwe assume that both transitions have the same resonance\nfrequency. This is a good approximation in a small mag-\nnetic field [11] or under application of a light shift on one\nof the four states when operating at a high field (see Ap-\npendix C). For the “n”-qubit rotations, the Raman tran-\nsitions in the ground and metastable states do not have\ncrosstalk since the wavelengths associated with the in-\ntermediate states are separated by ∼102THz. The Rabi\nfrequencies of each Raman transition can be matched.\nIntra-ququart two-qubit gates are often simpler than\nsingle-qubit gates: Figure 2(c)(d) shows the intra-3\n|11 |10 |113S1\n(n*㲊\u0012)(a) (b)\nInter-ququart CCCZ “o”-qubit CZδ δ\nδ\n“o”\n“n”\n“o”\n“n”\nFIG. 3. Inter-ququart gates based on Rydberg transi-\ntions . It is accessed through a single-photon transition of λ=\n302 nm from3P0states to high- nstates. (a) One stretched\nRydberg transition is driven on two nearby atoms, achiev-\ning inter-atom entanglement via Rydberg blockade. With the\ntwo-qubit encoding, this inter-ququart gate performs a CCCZ\ngate. (b) Inter-ququart CZ gates on the “o”-qubits of two\nququarts. Both stretched Rydberg transitions are driven with\nthe same Rabi frequency, detuning and phase such that both\n3P0states can couple to Rydberg states.\nququart swap and CNOT operations. The swap gate\nis achieved by driving a π-pulse between |01⟩and|10⟩\nstate, effectively swapping the information encoded in\n“o”-qubit and “n”-qubit. The CNOT gate with the con-\ntrol qubit “o” and the target qubit “n” is achieved by\nswapping the population in |10⟩and|11⟩states. Note\nthat operations such as the CNOT gate induce a differen-\ntial light shift in the ‘o’ sector, leading to a phase accumu-\nlation during its execution. This effect can be neutralized\nwith an additional light shift or deliberately engineered\nto produce a trivial 2 πphase shift.\nB. Inter-ququart gates\nFor neutral atoms, entangling two physical qubits\nis commonly achieved through Rydberg block-\nade [41–46]. Specifically, two-qubit entangling gates\nbetween metastable nuclear spin qubits in171Yb [12]\nand optical clock qubits in strontium-88 [14, 47] have\nbeen achieved with single-photon Rydberg excitations.\nTo perform an entangling gate, one commonly-used way\nis to selectively couple one of the two qubit states to\na Rydberg state, such that due to Rydberg blockade\nbetween nearby atoms, the qubit states accumulate\na state-dependent phase, generating a CZ gate under\nproper experimental settings [46].\nWith our ququart encoding, we first consider a four-\nqubit CCCZ gate: the CZ gate that was recently demon-\nstrated for the metastable nuclear spin qubit of171Yb [12]\nbecomes a CCCZ gate directly with the inclusion of thetwo ground states into the computational space. As is\nshown in Fig. 3(a), the phase shift due to Rydberg inter-\nactions is only present in the |11⟩state of each ququart.\nThe gate’s performance is simulated in Appendix E 1,\nwhere we demonstrate its effectiveness at the 99.9% fi-\ndelity level.\nWe also demonstrate an inter-ququart CZ gate between\nthe two “o”-qubits of the ququarts. Figure 3(b) shows\nthe gate scheme of this “o”-sector inter-ququart CZ gate,\nwhich must be agnostic to the “n” sector. To excite both\nthe states in the optical |1⟩state to the Rydberg mani-\nfold, we drive two stretched Rydberg transitions between\n6s6p3P0|mF=±1/2⟩and 6 sns3S1|mF=±3/2⟩states\nwith the same Rabi frequency, detuning, and phase.\nApart from the necessity of these dual-tone Rydberg ex-\ncitation, our approach adheres to the standard Rydberg\nCZ gate protocol [46, 48]. This gate scheme requires Ry-\ndberg states of different mF’s to blockade each other. In\nAppendix E 1, we show that the blockade strengths be-\ntween Rydberg states of different mF’s are expected to\nbe of the same order of magnitude (limited spectroscopic\ndata is available), and we perform a numerical simulation\nof this (2 ×8)-dimensional space for realistic parameters.\nNotably, we can also perform gates between an atom\nwith ququart encoding and an atom functioning only as\na qubit which is encoded in the metastable nuclear spin.\nThere are two approaches: One option is to apply only\none Rydberg drive ( σ+orσ−) to the qubit atom but\napply both ( σ+andσ−) to the ququart atom. This will\nperform a CZ gate between the metastable nuclear spin\nof the qubit atom and the “o”-qubit of the ququart atom.\nA second option is to first convert the metastable nuclear\nspin of the qubit atom into an optical qubit via the swap\ngate in Fig. 2(c), then apply the CZ gate with both Ry-\ndberg tones on each atom (global pulses), and then swap\nthe qubit-atom’s encoding back to the metastable nuclear\nspin.\nC. Readout\nThe readout of higher-dimensional spaces like qudits\ncan be cumbersome. In neutral atoms and trapped ions,\nthe ground state manifold typically exhibits fluorescence\n(’bright’), in contrast to the non-fluorescent (’dark’)\nmetastable manifolds. The usual method for qudit read-\nout in such cases involves ’shelving’ all but one basis\nstate into the metastable manifold, with the chosen state\ntransitioned (mapped) into the ground state for readout.\nFollowing fluorescence readout, this ground state is reset\nand then shelved into the metastable manifold, with the\nprocess sequentially repeated for each state [23]. Hence,\nreadout of a d-level qudit usually involves up to d−1\niterations of this cycle as only one state is probed at a\ntime. This approach is exponentially less efficient than\nmeasuring multiple qubits in parallel.\nInstead, here we perform a 2-round deterministic read-\nout of the ququart states based on the quantum nonde-4\n(a)\n(c) (b)|00 |01|10 |11\n|00|01\n|10|11\n|00 |01|10 |113P1\n|0X\nX“o”\n“n”\nX\nX|01|10\nFIG. 4. Different ququart readout schemes . (a) 2-round\ndeterministic readout of ququart states. Two stretched transi-\ntions1S0↔3P1are simultaneously driven, and the resultant\nfluorescences are detected in an indistinguishable manner, ef-\nfectively measuring the “o”-qubit. Note that the cyclicity of\nthe transitions ensures that the population in1S0substates\nremains in the same state after measurement. Afterward, an\nintra-ququart SWAP is applied, swapping the populations in\n|01⟩and|10⟩states. Finally, the same two stretched tran-\nsitions are used to effectively measure the “n”-qubit before\nthe swap operation. The four combinations of two readout\nresults can thus be mapped to the four ququart states. (b)\nSingle-state QND readout of |00⟩, within the1S0ground state\nmanifold. (c) The two-qubit analog of the single-state QND\nmeasurement via an ancilla qubit and a Toffoli gate.\nmolition (QND) readout of the1S0nuclear spin ground\nstates [34, 35], which reduces the upper limit of the re-\nquired number of measurement operations from 3 to 2.\nFigure 4(a) shows the 2-round ququart readout scheme.\nThis readout protocol requires simultaneously probing on\ntwo1S0↔3P0stretched transitions and collecting the\nscattered photons. Both stretched transitions are closed\n(“cycling”) under a magnetic field of ≳102G, which\nallows us to perform readout of the “o”-qubit without\ntransferring the population between ��n” qubit states.\nWe apply two rounds of the dual-stretched readout with\nan intra-qubit SWAP operation sandwiched in between.\nConceptually, the first readout measures the “o” qubit\nwhile the second readout measures the “n” qubit. The\ntwo bits of information from the readout process de-\ntermine the projective measurement outcome and post-\nmeasurement state. For instance, “bright”-“bright” refers\nto|00⟩while “bright”-“dark” refers to |01⟩, etc.\nNote that readout of only one qubit within the ququart\nwithout affecting the other qubit remains challenging\nwith this readout scheme. One exception is that if the\n“o”-qubit QND readout measures |1⟩, then the “n”-qubit\nis encoded only in the3P0states, which is left unaffected.Therefore, the information encoded in the “n” qubit is\npreserved. As described below, this condition is sufficient\nfor an entanglement distillation protocol in which “o”-\nqubit readout with outcome |1⟩heralds the generation\nof purified entanglement in the “n” sector. Accordingly,\nalthough an “o”-qubit readout outcome of |0⟩would col-\nlapse the “n” qubit, the distillation protocols fail anyway\nand must be re-attempted.\nFinally, we consider a unique opportunity for qudit\nreadout. The typical shelving-probing protocol discussed\nabove that requires d−1 rounds is cumbersome if all\nbasis states must be measured. However, this scenario\npresents a unique opportunity to perform a QND read-\nout of a portion of the computational space. Specifi-\ncally, we propose to perform single-state QND readout\nvia probing with only one stretched transition from the\nground manifold [34, 35] [see Fig. 4(b)]. In this case, |00⟩\nwill be “bright” while all other ququart states will be\n“dark”. A measurement outcome of “bright” will col-\nlapse the ququart to |00⟩, while an outcome of “dark” al-\nlows a re-normalized superposition of |01⟩,|10⟩, and|11⟩\nto survive. Naturally, we must be aware of light shifts\non the |01⟩state due to the probe light as discussed\nin Appendix D 2. The two-qubit analog of this single-\nstate QND readout paradigm requires an ancilla qubit\nas shown in Fig. 4(c). As shown below, this paradigm\nprovides new opportunities for interactive circuits.\nIII. APPLICATIONS\nQudit architectures offer improved hardware effi-\nciency, enabling the encoding of more quantum infor-\nmation—potentially including logical qubits—within a\nsingle atom. Moreover, the fidelity of two-atom gates\noften still lags behind that of one-atom gates in both\ntrapped ion and neutral atom systems. Consequently, the\nproposed ququart architecture promises potential perfor-\nmance enhancements over traditional qubit systems. This\npotential is contingent upon maintaining the quality of\none-atom gates amidst the complexities introduced by\nthe ququart’s expanded gate set and its additional con-\nstraints.\nIn this section, we focus on leveraging this improved\nhardware efficiency and potentially superior perfor-\nmance, as well as on applications that exploit the unique\nreadout properties of qudit systems for interactive cir-\ncuits. In the discussions that follow, we use black lines in\nthe circuit diagrams to represent standard qubits, while\ntwo adjacent lines, colored yellow and orange, denote the\n”o”- and ”n”-qubits of a ququart, respectively.5\nKeep if\nboth in\noptical\n  |1   state\nEntanglement distillation(a)\n“o”\n“n”\n“o”\n“n”\nEntanglement \ngeneration(b)\nFIG. 5. Entanglement distillation with two ququarts .\n(a) Circuit diagram of entanglement distillation, with two\nnearby yellow and red qubit as a ququart. The two\ncurved lines denote entanglement between two qubits in two\nququarts. After the entanglement generation, intra-ququart\ntwo-qubit gates are applied, followed by a measurement on the\n“o”-qubits. We only keep the post-distilled state when both\n“o” qubits measure |1⟩, which indicates that the entanglement\npurification works and the “n”-qubit coherence is unaffected.\n(b) Simulation results of Bell state infidelities before and after\ndistillation assuming depolarization and Pauli-X errors.\nA. Quantum information processing\n1. Entanglement distillation\nEntanglement distillation is a protocol to generate a\nhigh-fidelity entangled two-qubit state from two low-\nfidelity entangled two-qubit states using local operations\nand classical communication [49, 50], which has been\ndemonstrated in several physical platforms with qubit\narchitectures [51–53].\nWithin our proposed ququart architecture, the distilla-\ntion protocol is executed more efficiently, requiring only\ntwo atoms with operations internal to the ququart and\na global readout mechanism. This contrasts with tradi-\ntional qubit systems, where the distillation necessitates\nfour atoms, the application of inter-atomic gates, and\nmid-circuit (local) readout techniques.\nAs illustrated in Figure 5(a), the entanglement distil-\nlation protocol begins with a pair of two-qubit entangled\nstates. In our ququart architecture, we assume that the\npre-distilled state is given as follows:\n|Ψ0⟩=O\ni∈{o,n}1√\n2\u0000\n|01⟩i|02⟩i+|11⟩i|12⟩i\u0001\n. (1)\nHere, the subscript indicates the atom, while the super-\nscript distinguishes whether the qubit is optical (o) or\nnuclear (n). We highlight the presence of entanglement\nbetween the optical qubits across two distinct atoms, as\nwell as between their nuclear qubits. This arrangement,\ncomprising a pair of Bell states, can be efficiently pro-\nduced using inter-ququart gates.\nWe want the final two-qubit state to be encoded in the\nnuclear qubits of two atoms. Hence, we apply SWAP op-\nerations to both ququarts so that we preserve “n” qubits\nrather than “o” qubits. Finally, the “o” qubit in each\nququart is measured in the Pauli Z-basis and the success\nof the distillation protocol is heralded by an outcomeof|1⟩(dark) for both, such that the resulting two-qubit\nstate is indeed encoded in nuclear qubits. We emphasize\nthat this “o”-qubit readout is performed globally as the\noptical qubit state |1⟩can be measured without interfer-\ning with nuclear qubits as discussed in Sec. II C.\nIn Figure 5(b), we present the infidelities between both\npre-distilled and distilled states compared to the ideal\nBell state, aiming to showcase the circuit’s efficacy in\ndistillation. To this end, we consider two distinct error\nmodels impacting the inter-ququart gates during the en-\ntanglement generation step: the depolarization channel\nand Pauli-X noise. For both scenarios, we factor in a\nconsistent measurement error rate of 1% and an intra-\nququart gate infidelity of 0.1%, with both types of errors\nsimulated via the depolarization channel. The yield un-\nder these two types of errors is shown in Fig. S6.\nThe distillation circuit is specifically designed to mit-\nigate Pauli- Xerrors on Bell states. However, it does\nnot address Pauli- Zerrors, leading to its diminished ef-\nfectiveness in combating depolarization errors, as illus-\ntrated by the plot. Given the varied nature of inter-\nququart gate errors, which realistically may fall between\nthese two models, we anticipate an improvement in Bell\nstate fidelity under certain conditions. We note that the\nperformance of our ququart distillation protocol is com-\nparable to that of the standard, qubit-based approach.\nConsequently, the post-distillation states are expected to\nprovide high-fidelity entangled qubit pairs, proving ben-\neficial for a range of applications involving metastable\nqubits.\n2. Flag-based quantum error correction\nFor large-scale quantum computing, quantum error cor-\nrection (QEC) becomes necessary to overcome hardware\nimperfections [54–57]. QEC is achieved by encoding logi-\ncal qubits using multiple physical data qubits. With this\nredundancy, one can measure specific multi-body oper-\nators called stabilizers to identify errors in the physical\nqubits without interfering with logical qubits, thereby de-\ntermining the necessary corrections. When the error rate\nof the physical qubits and their associated operations are\nbelow a specific threshold, logical qubits can achieve an\nerror rate lower than the base error rate of the physical\nqubits. A QEC code that employs nphysical qubits to\nencode klogical qubits with code distance dis denoted\nas a [[ n, k, d ]] code.\nHowever, QEC is effective only if the quantum circuits\nthat identify and correct errors do not proliferate those\nerrors across physical qubits – a requirement called fault-\ntolerance (FT) [58, 59]. Unfortunately, a straightforward\napproach using a single ancilla qubit (syndrome qubit)\nfor stabilizer measurement generally fails to meet this\nFT standard. To address this, various strategies have\nbeen proposed, including the use of a block of ancilla\nqubits [60–62].\nMore recently, a novel approach emerged to minimize6\n|0|+ H(a) (b)\n“o”\n“n”\nFIG. 6. Flag-based QEC with ququarts (a) Circuit to\nfault-tolerantly extract the stabilizer XXXX . Black lines are\ndata qubits and yellow and red lines are syndrome and flag\nqubits encoded in a ququart. The CNOTs between syndrome\nand flag qubits are performed with intra-ququart gates, which\nshould suppress the infidelity to the same level as single-\nqubit gates. (b) Simulated threshold for [[5 ,1,3]] and Steane\n[[7,1,3]] codes. Circles represent normal flag EC and triangles\nrepresent flag EC with ququart by assuming intra-ququart\ngates have perfect fidelity. The intersection of the grey line\n(physical error rate) and the curve representing logical error\nrate of a certain case gives the physical error rate, below which\nthe QEC code outperforms the bare qubit. Flag ququart im-\nproves the threshold, and in practice, the threshold should lie\nin between the normal flag case and the ququart flag case.\nthe large number of ancilla qubits traditionally needed.\nBy introducing one extra flagqubit in stabilizer mea-\nsurements, one can track and contain an error that could\nhave proliferated through the circuit [63–65] with rela-\ntively low qubit overhead (only two auxiliary qubits are\nrequired). This concept has shown promising results in\nsolid-state spin qubits [66].\nWith ququarts, this flag-based QEC can be naturally\nimplemented as illustrated in Fig. 6(a). The “o”-qubit\nserves as the ancilla qubit while the “n”-qubit serves as\nthe flag qubit. Two-qubit gates between a normal qubit\nand the “o”-qubit within a ququart can be performed\ndirectly if the qubit is encoded in the optical transition,\nor by swapping its nuclear spin encoding to the optical\nsector before and after the gate (see section II B). With\nhigh-fidelity intra-ququart two-qubit gates between the\nancilla and flag qubits, an increase in the error correction\nperformance is possible. Additionally, using just a single\nauxiliary atom for fault-tolerant operations reduces the\nhardware overhead required to achieve a specified level\nof qubit connectivity.\nTo demonstrate the advantage of having ququarts in\nflag-based QEC, we calculated logical error rates with\ngates of an error rate pfor two different scenarios in Fig-\nure 6(b): In the first scenario, the error rate between flag\nand syndrome qubits is also p, while in the second sce-\nnario, the error rate is zero. The latter case corresponds\nto the ququart implementation with an almost-perfect\nintra-ququart gate. In practice, the intra-ququart gates\ngenerally have higher fidelity than the gates between\nququart and qubit, so the ququart flag performance\nshould lie in the shaded region in Fig. 6(b). We comparedthe performance using two types of QEC code: the five-\nqubit [[5 ,1,3]] code and Steane seven-qubit [[7 ,1,3]] code.\nDetails about the simulation can be found in Appendix\nF. For both QEC codes, ququart flag increases the error\nthreshold due to the higher-fidelity intra-ququart gates.\n3. Four-qubit code error detection\nAnother useful direction towards fault-tolerant quan-\ntum computation is the quantum error detection code.\nOne example of quantum error detection code is the\nsmallest instance of surface codes, the four-qubit [[4 ,2,2]]\ncode [67]. Unlike QEC codes that can correct errors\non physical qubits, the four-qubit code can detect er-\nrors on a single qubit but is not able to correct the er-\nrors. The four-qubit code can act as a building block\nfor large-scale fault-tolerant quantum computations [62],\nand fault-tolerant operations on the four-qubit code have\nbeen demonstrated in several physical platforms [68–70].\nThe four-qubit code is defined by the stabilizers\nSX=X1X2X3X4,\nSZ=Z1Z2Z3Z4.(2)\nHere XiandZiare Pauli X and Z operators on the i-\nth physical qubit. The logical operators assigned to two\nlogical qubits are\nXL1=X1X3, ZL1=Z1Z2,\nXL2=X1X2, ZL2=Z1Z3.(3)\nThe logical states are\n|00⟩L= (|0000⟩+|1111⟩)/√\n2,\n|01⟩L= (|0011⟩+|1100⟩)/√\n2,\n|10⟩L= (|0101⟩+|1010⟩)/√\n2,\n|11⟩L= (|0110⟩+|1001⟩)/√\n2.(4)\nSpecifically, one can fix the first logical qubit as a gauge\nand the code turns into a [[4 ,1,2]] subsystem code.\nWe encode the four-qubit code using the “o” and “n”\nqubits of two ququarts, as is shown in fig. 7(a). To\navoid correlated two-qubit errors from stabilizer mea-\nsurement, an auxiliary ququart is implemented to func-\ntion as the ququart flag discussed in section III A 2 (see\nfig. 7(b)). Therefore, the fault-tolerant operations on the\nfour-qubit code can be demonstrated using only three\nindividual atoms. We further highlight that the specific\nlogical states,\n|0±⟩= (|00⟩ ± |11⟩)⊗2/2, (5)\ncan be easily prepared with two ququarts by applying\nclock π/2-rotations between |00⟩and|11⟩states.7\n“o” “n”Atom 1\nAtom 21 2\n3 41\n2\n3\n4\n|0|+ H(a) (b)\nFIG. 7. Four-qubit code encoded in two ququarts . (a)\nThe “o” qubit and “n” qubit of two ququarts are used to en-\ncode the four-qubit code. (b) Measurement of the stabilizer\nSXusing an extra ancilla ququart. The extra ququart flag is\nimplemented to fault-tolerantly measure the stabilizer with-\nout introducing logical errors into the state.\nB. Interactive circuits\n1. Realizing topological phases\nOne fascinating aspect of ququarts is that a natural\nmeasurement does not bisect the local Hilbert space un-\nlike in a system made up of qubits1. The quantum chan-\nnel for measuring |00⟩in a ququart is expressed as:\nρ⊗ |0⟩⟨0|E7→ P 0ρP0⊗ |0⟩⟨0|E+P¯0ρP¯0⊗ |¯0⟩⟨¯0|E\nP0=|00⟩⟨00|\nP¯0=|01⟩⟨01|+|10⟩⟨10|+|11⟩⟨11|, (6)\nwhere |0⟩Eand|1⟩Eare the states of a measurement\napparatus. Using any intra-qubit unitary transformation\nUrotating |00⟩into (|10⟩ − |01⟩)/√\n2, through this mea-\nsurement one can perform a measurement to distinguish\nspin-singlet ( s= 0) and spin-triplet ( s= 1) sectors of a\ngiven ququart. This leads to a new protocol to prepare\ninteresting quantum many-body wavefunctions.\nTo illustrate our approach, consider a spin-1\nchain model by Affleck, Kennedy, Lieb, and Tasaki\n(AKLT) [71, 72] defined by the following Hamiltonian:\nHAKLT =X\nih\n(⃗Si·⃗Si+1) +1\n3(⃗Si·⃗Si+1)2i\n,(7)\nwhere ⃗Si= (Sx\ni, Sy\ni, Sz\ni) is the spin-1 operator at site\ni. Its ground state is a canonical example of nontrivial\nsymmetry protected topological order, exhibiting several\ninteresting properties [73–75]: ( i)Non-local order param-\neter: For any arbitrary length |i−j|, the expectation\nvalue of the following string of operators takes a finite\nvalue:\nOα\nij:=Sα\nieiπPj−1\nl=i+1Sα\nlSα\nj, α =x, y, z. (8)\n1For a quantum circuit consisting of qubits, such a measurement\noperation would require introducing an extra ancilla qubit and\napplying a non-Clifford gate.The finite expectation value of a nonlocal operator Oα\nij\nimplies that the state can be utilized as an entanglement\nresource for quantum computation [76–79]. ( ii)Anoma-\nlous boundary modes: In the open boundary condition,\neffective spin-1/2 edge degrees of freedom exist at each\nedge albeit consisting of spin-1 in the microscopic model.\nThere are various protocols suggested to realize this\nstate using qubits and local unitary gates. However, these\nschemes require a gate depth linearly scaling [80–86] with\nthe system size due to a finite correlation length of the\nAKLT state, which requires some exponential tail of uni-\ntary gates. As we will show, this can be circumvented by\nour novel approach.\nA schematic diagram for our protocol is shown in\nFig. 8(a). First, we prepare a wavefunction where an “o”-\nqubit at the ith-site and a “n”-qubit at the ( i+1)th-\nsite form a singlet1√\n2(|0⟩o\ni|1⟩n\ni+1−|1⟩o\ni|0⟩n\ni+1). Second, we\nproject each ququart onto its spin-triplet ( S= 1) mani-\nfold via measurement. If all ququarts are properly pro-\njected, they immediately form the AKLT state, which\nis the ground state of Eq. (7). However, each ququart\nHilbert space decomposes into spin-singlet and triplet\nsectors, and the probability of getting a spin-0 measure-\nment outcome is 25%. Thus, it is natural to ask whether\nwe can get a whole chain of the AKLT state instead of a\nset of small AKLT islands.\nThe solution to this problem is straightforward: by\ndiscarding neutral atoms that are projected onto spin-\nsinglets, we can ensure that the remaining ququarts will\nform a single AKLT state. To elucidate this idea, let us\nexamine the scenario illustrated in Fig. 8(b), where we la-\nbel qubits as 1, 2, 3, and 4 in a cardinal manner. Initially,\nthe qubits (1,2) and (3,4) are in singlet states, resulting\nin a zero total spin. Now, we project the pair (2 ,3) onto\na singlet. Given that the initial total spin is zero and that\nthe conservation of total spin is a requisite in each pro-\njected component, it logically follows that the pair (1,4)\nmust also be in a spin-singlet state. As a result, while\nququarts projected onto spin-singlets become disentan-\ngled from the rest of the system, those projected onto\nspin-triplets contribute to forming a single AKLT state.\nThe size of this resultant AKLT state is approximately\n3/4 of the original system.\nUsing experimental controls discussed in the previous\nsection, the protocol consisting of three parts can be di-\nrectly implemented as shown Fig. 8(c):\n•Inter-ququart Singlet Formation : As our inter-\nququart operations are limited among optical\nqubits, we have to introduce SWAP operations. Us-\ning intra-ququart SWAP and inter-ququart CNOT\noperations in a parallel manner, this step can be\ndone within five steps.\n•Triplet Projection : To perform this projection,\nwe have to conjugate our |00⟩measurement by\nthe basis rotation using single-qubit rotations and\nintra-ququart CNOT gate.8\n(i) Singlet formation|00⟩(ii) Triplet Projection |L⟩|1⟩ |−⟩|1⟩ |−⟩|1⟩ |−⟩|R⟩|00⟩|00⟩|00⟩Rx(π)Ry(π/2)(iii) RearrangementRemove spin-0 atoms (c)\n<latexit sha1_base64=\"FKMyf5tAxCfi95Ag34W7V7E655A=\">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</latexit>=(|01Í≠|10Í)/Ô2\ni th atom<latexit sha1_base64=\"VRZjrmmxbwUWcFkzKOiBQd943Vc=\">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</latexit>PS=0iSi=0S=1S=1S=1S=1(a)\n(b)Spin-1 AKLT state⇓Optical qubitNuclear qubitTriplet ProjectionRx(π)Rx(π)Rx(π)Ry(π/2)Ry(π/2)Ry(π/2)Rx(π)Ry(−π/2)Rx(π)Ry(−π/2)Rx(π)Ry(−π/2)Rx(π)Ry(−π/2)\nFIG. 8. AKLT state preparation protocol (a) Within each ququart, there are two qubits (optical and nuclear). Starting\nfrom a chain of singlets between optical and nuclear qubits across different sites, we project each ququart onto the spin-1\nsubspace. The projected wavefunction is the exact ground state of the AKLT Hamiltonian. (b)Ifi-th ququart is projected\nonto the spin-0 subspace instead, the “o”-qubit of the ( i−1)-th ququart and “n”-qubit of the ( i+ 1)-th ququart automatically\nforms a maximally entangled singlet. (c)The quantum circuit of the actual experimental procedure consists of three steps. ( i)\nInitialize the system into the product state of a particular pattern, which is followed by the intra-ququart SWAP and inter-\nququart CNOT to create a chain of singlets. ( ii) Apply individual qubit rotations and intra-ququart gates to rotate |01⟩ − |10⟩\ninto|00⟩, measure |00⟩, and then apply the reverse operation. This sequence effectively performs singlet/triplet projections.\n(iii) After the measurement, ququarts projected onto the spin-1 subspace automatically form the AKLT state, while spin-0\nququarts completely disentangle. One can rearrange neutral atoms to throw away spin-0 ququarts.\n•Rearrangement : Conditioned on the measure-\nment outcomes, we can remove atoms projected\nonto spin-singlet states either virtually (through\npost-processing) or physically (through the optical\ntweezer technique [7]).\nFurthermore, if we want to create an AKLT state with\na definite size, it can be achieved by the fusion pro-\ncess [87]: Let l1andl2be the length of an AKLT state.\nThen, one can apply gates to transform “o”-qubit in\nthe last atom of the first AKLT chain and “n”-qubit in\nthe first atom of the second AKLT chain into a singlet\nstate, followed by triplet projection of the said atoms.\nThen, the resulting state would become a single AKLT\nchain with length L=l1+l2−m, where mis the num-\nber of atoms projected onto singlet states. Note that\nP(m=0) = 9 /16,P(m=1) = 6 /16, and P(m=2) = 1 /16.\nTherefore, our protocol combines local unitary gates,\nmeasurements, and feedback to create an AKLT state in\na constant depth, offering a novel experimental route to\nrealize exotic topological systems in quantum simulator\nplatforms.\nInterestingly, the illustrated protocol can be consid-\nered as a particular experimental implementation of a\nso-called parton construction, which is heavily utilized in\nthe theoretical study of quantum many-body physics. In\nthis construction, an exotic quantum state is created by\nstarting from a simple wavefunction consisting of frac-\ntional degrees of freedom and then projecting it onto\nphysical degrees of freedom [88–92]. Therefore, our ap-\nproach paves the road to implement this parton construc-tion in experiments.\n2. Adaptive dynamics: absorbing transitions\nThe capacity to perform mid-circuit measurements and\nfeedback operations based on the outcomes in ququarts\noffers experimental access to adaptive dynamics , where\nthe system evolves under random local unitaries, mea-\nsurement, and local feedback operations based on mea-\nsurement outcomes. In recent years, the entanglement\nstructure of such adaptive dynamics has been theoreti-\ncally explored where the measurement probability pof\neach qubit for each cycle of the circuit has been the main\ntuning knob. In particular, two interesting transitions\nhave been theoretically discussed: the volume to area-\nlaw entanglement transition across a critical measure-\nment rate [93, 94], and directed percolation or steering\ntransitions between two distinct area law phases across\nanother critical tuning parameter [95–97].\nOur ququarts encoding, or more generically higher-\ndimensional encodings offer a unique opportunity to ex-\nplore these phenomena. In fact, the qudit single-state\nQND readout discussed above has been utilized in re-\ncent theoretical proposals for studying directed percola-\ntion transitions [95]. Our single-state QND readout of\n|00⟩makes it an “absorbing state”: A ququart that is\nmeasured to be in this state becomes “inactive” and does\nnot participate in subsequent unitary operations. Con-\nversely, if the ququart is measured to be notin|00⟩, then\nit remains “active” and its Hilbert space dimension re-9\nduces from four to three. Accordingly, above a certain\nmeasurement rate, the system transitions into a many-\nbody absorbing state whose universality class is given as\na directed percolation transition.\nWe now outline an exemplary circuit to study\nmeasurement-induced phase transitions (MIPTs) with\nqudits as introduced in Ref. [95]. We perform a |00⟩-state\nmeasurement of all ququarts in each cycle. After the first\nmeasurement layer, the active and inactive sectors remain\nseparated. The knob that we will use to re-couple the\nsectors and introduce phase transitions are “reset” oper-\nations in which select ququarts are re-initialized to |00⟩,\nwhich can be achieved via local optical pumping. Note\nthat single-ququart dissipative operations can be per-\nformed without deleterious effects on the other ququarts\nby trapping the target ququart(s) in tweezers of a dif-\nferent clock-magic wavelength than the other ququarts\nfor which the probe/reset transition is sufficiently light-\nshifted from that of the nominal tweezers. Mid-circuit\nreadout has already been demonstrated via this tech-\nnique [35]. The probability pof performing such a reset\noperation on each ququart in each cycle is the parameter\nthat allows us to traverse the phase diagram.\nThis architecture presents a relatively simple approach\nto studying MIPTs, specifically directed percolation tran-\nsitions. Since it is known to be very challenging to study\nentanglement transitions between volume and area law\nphases in large systems due to the exponential post-\nselection problem [93, 94], the study of percolation tran-\nsitions between area law phases appears to be a better\ntarget for large-scale quantum hardware [95–97]. Hence,\na concrete and feasible route towards this realization con-\nstitutes an important development for interactive quan-\ntum circuits.\nIV. CONCLUDING DISCUSSION\nWe have presented a toolbox for encoding two qubits\nwithin the four-level structure of alkaline earth-like atom\nisotopes with nuclear spin-1/2 like171Yb. We focused\non the application of this ququart encoding to entan-\nglement distillation and quantum error correction. We\nhave also discussed how the higher-dimensional encoding\noffers new opportunities for interactive circuits to pre-\npare topological phases and study measurement-induced\nphase transitions. This ququart architecture is a natural\nextension of the ‘omg’ architecture [11, 15, 98] in which\nall pairs of states within the four-level system are used\nto encode qubits with various functionalities. We believe\nthe use of ququarts does not require further capabilities\nthan the ‘omg’ scheme yet offers a significant hardware\nadvantage as well as new opportunities.\nAlkaline earth-like atoms without nuclear spin may still\nprovide access to our ququart encoding. For instance, we\ncould replace the nuclear spin-1/2 degree of freedom with\ntwo harmonic motional states [14]. Motional ground state\npreparation via Raman sideband cooling has providedaccess to temperatures for which this encoding is real-\nistic [11, 14, 33]. Indeed, there is at least one scenario\nin which the motional encoding may offer an advantage:\nA crucial capability that is proposed in this work is the\ninter-ququart gate that performs a CZ gate between the\noptical qubit (“o”) sector of each ququart without affect-\ning the nuclear qubit (“n”) sector. If we instead encode\nthe second qubit in the motional sector and do not have\nnuclear spin, then this inter-ququart gate between “o”-\nsectors would be given for free with a single-tone Ryd-\nberg excitation. In other words, the standard Rydberg-\nmediated gate is already agnostic to the motional state\nin the low temperature regime.\nAcknowledgments .—We acknowledge Ken Brown, Xiao\nChen, Sagar Vijay, Matteo Ippoliti, Bryan Clark, and\nHannes Bernien for stimulating discussions; and Yichao\nYu for critical reading of the manuscript. J.P.C. acknowl-\nedges funding from the NSF QLCI for Hybrid Quan-\ntum Architectures and Networks (NSF award 2016136);\nthe NSF PHY Division (NSF award 2112663); the NSF\nQuantum Interconnects Challenge for Transformational\nAdvances in Quantum Systems (NSF award 2137642);\nthe ONR Young Investigator Program (ONR award\nN00014-22-1-2311); the AFOSR Young Investigator Pro-\ngram (AFOSR award FA9550-23-1-0059); and the U.S.\nDepartment of Energy, Office of Science, National Quan-\ntum Information Science Research Centers. J.Y.L. ac-\nknowledges funding from the Simons Investigator Award.\nAPPENDIX A: Utilizing a single ququart as two\neffective qubits\nHere we describe one simple approach to control\na single ququart as two effective qubits via resonant\ntones. Consider a qudit with bare Hamiltonian H0=P\nkω0,k|k⟩⟨k|, with an external drive given by Hc(t) =P\nj,kHjk\nc(t), where\nHjk\nc(t) = Ωjk\n0cos(qjkt+ϕjk)(|j⟩⟨k|+hc). (A1)\nWe now assume for simplicity that the presence of\ndrives (Ωjk\n0̸=0) between states j, kdo not introduce par-\nasitic light shifts to the rest of the states. Then, going\ninto the rotating frame given by the bare Hamiltonian\nandTr=e−iHrtwith Hr=P\nkν0,k|k⟩⟨k|, the effective\nHamiltonian becomes\nHeff=Tr(H0+Hc(t))T†\nr+Hr\n=X\nk(ω0,k−ν0,k)|k⟩⟨k|+X\njkΩjk\n0\n2(eiθjk|j⟩⟨k|+ h.c.)\n(A2)\nwhere θjk= (qjk−∆kj)t+ϕjkand ∆ kj=νk−νj. Thus\napplying drive frequencies qjkresonant with the various\ntransition ∆ kj=νk−νj, the Hamiltonian becomes time-\nindependent.10\nFocusing on the ququart ( d= 4) system, we see the\ndrives resonant with various pairs of states |j⟩yield effec-\ntive two-qubit rotations with relabeling j={0,1,2,3} 7→\n{00,01,10,11}. Below we list some pertinent drives\nthat yield generalized CNOT’s (controlled rotations or\n“CRot’s”) and flip-flop/SWAP Hamiltonians ( H12\nc):\nH01\nc≈Ω01\n0\n2|0⟩⟨0|0⊗(σx\n1cosϕ01+σy\n1sinϕ01),\nH23\nc≈Ω23\n0\n2|1⟩⟨1|0⊗(σx\n1cosϕ23+σy\n1sinϕ23),\nH02\nc≈Ω02\n0\n2(σx\n0cosϕ02+σy\n0sinϕ02)⊗ |0⟩⟨0|1,\nH13\nc≈Ω13\n0\n2(σx\n0cosϕ13+σy\n0sinϕ13)⊗ |1⟩⟨1|1,\nH12\nc≈Ω12\n0\n2(σx\n0σx\n1+σy\n0σy\n1)/2,\n(A3)\nwhere σjrefers to Pauli operator on the j-th qubit. Note\nthe combination of CRot’s conditioned on both |0⟩⟨0|j\nand|1⟩⟨1|jof the j-th qubit yields arbitrary single-qubit\nrotation on the other qubit.\nThus given both arbitrary single-qubit rotations as well\nas two-qubit interactions we find the use of resonant tones\nin a ququart, combined with state initialization and mea-\nsurement, allow universal control over the effective two-\nqubit system.\nWe remark that thus far we have assumed there is no\nlight shifts added to the system due the presence of the\napplied drives themselves (Ωjk\n0̸=0), which may result in,\ne.g., undesired effective phase evolution between various\nstates. Such light shifts on states coupled by Hjk\nccould\narise from states not only inside but also outside the d=\n4 computational subspace of interest. Thus in practice for\nevery choice of the d= 4 states as well as applied controls\nHjk\ncit may be necessary to identify and correct for these\nlight shifts, for instance by specific timing or applying\nphysical or virtual Z-gates on appropriate pairs of states\nvia laser pulses or adjusting phases on the software end.\nAPPENDIX B: Quantum state tomography of\nququart states\nHere we discuss two approaches to perform quantum\nstate tomography (QST) of the target d= 4 system,\nclosely following earlier works [99, 100]. Depending on\nthe experimentally available control, one approach may\nbe more favorable than the other.\nFirst, if effective single-qubit controls within the qu-\ndit (of dimension d= 2N) are readily accessible, it may\nbe favorable to perform QST of the qudit as an effec-\ntiveN-qubit system, by describing the quantum state to\nbe reconstructed ρin the Pauli basis with (4N−1) realparameters αij. For a ququart,\nρ=X\nj,k∈{0,x,y,z}αjkσj⊗σk. (B1)\nHere, σ0=Iis the identity and σj∀j∈ {x, y, z}are\nPauli matrices. By the trace property of a density ma-\ntrix Tr ρ= 1, α00= 1/d. Thus, to perform QST it re-\nmains to experimentally find the (4N−1) unknowns.\nWhile there is no unique procedure to achieve QST, we\ndescribe one rather intuitive approach to identify ρvia ar-\nbitrary single-qubit rotations and projective population\nmeasurements (access to projective measurement opera-\ntorsPjk=|jk⟩⟨jk|, with j, k∈ {0,1}).\nFirst, notice that the direct population measurements\nTrρPjk≡ ⟨Pjk⟩=⟨jk|ρ|jk⟩gives a system of unique\nequations which can be solved:\n⟨P00⟩= (1 + ⟨σz\n0⟩+⟨σz\n1⟩+⟨σz\n0σz\n1⟩)/4,\n⟨P01⟩= (1 + ⟨σz\n0⟩ − ⟨σz\n1⟩ − ⟨σz\n0σz\n1⟩)/4,\n⟨P10⟩= (1− ⟨σz\n0⟩+⟨σz\n1⟩ − ⟨σz\n0σz\n1⟩)/4,\n⟨P11⟩= (1− ⟨σz\n0⟩ − ⟨σz\n1⟩+⟨σz\n0σz\n1⟩)/4.\n(B2)\nNote the left-hand side (LHS) are given by experimen-\ntally measured values, while the right-hand side (RHS) is\ngiven by a linear combination of the unknowns αjkthat\ndescribe our state of interest ρ.\nHaving identified ⟨σz\nj⟩and⟨σz\n0σz\n1⟩, we can now ac-\ncess other Pauli strings by transforming the effec-\ntive measurement operators with single-qubit rotations\nRab(θa, θb) =Ra(θa)⊗Rb(θb), with a, b∈ {0, x, y}and\nRa(θa) =e−iθaσa/2. Concretely, by applying Rab(θa, θb)\nbefore measuring population, we will measure Pab\njk=\nRab(θa, θb)†PjkRab(θa, θb).\nFor instance, utilizing only π/2-pulses ( θa=θb=π/2)\nfor simplicity, we find\n⟨Pxy\n00⟩= (1− ⟨σy\n0⟩+⟨σx\n1⟩ − ⟨σy\n0σx\n1⟩)/4,\n⟨Pxy\n01⟩= (1− ⟨σy\n0⟩ − ⟨σx\n1⟩+⟨σy\n0σx\n1⟩)/4,\n⟨Pxy\n10⟩= (1 + ⟨σy\n0⟩+⟨σx\n1⟩+⟨σy\n0σx\n1⟩)/4,\n⟨Pxy\n11⟩= (1 + ⟨σy\n0⟩ − ⟨σx\n1⟩ − ⟨σy\n0σx\n1⟩)/4.\n(B3)\nTherefore, with access to population measurements,\n3N= 9 unique measurement settings suffice to fully iden-\ntify the 4N−1 = 15 unknowns of the (effective) N-qubit\nsystem and hence ρ. For a general qudit, however, all pop-\nulation measurements per single experiment (quantum\ncircuit) may not be readily available either due to the\nphysical quantum system or available controls; in such a\ncase, up to 4N−1 measurements may be required to fully\nsolve for ρ. Finally, due to experimental uncertainty the\nreconstructed ρmay not satisfy all three desired proper-\nties of density matrices; thus to remedy this, a maximum\nlikelihood estimation (MLE) of the ‘true’ density matrix\ncan be reconstructed using ρas an initial estimate as\npioneered by earlier works [99].11\nWhile the above approach may be favorable if effective\nsingle-qubit rotations (in the qudit) are readily accessi-\nble, the QST of the qudit as a general d-state system can\nbe done with access to resonant drives between various\ndstates as pioneered in earlier works [100]. Following\n[100] we now describe the procedure to identify ρof the\nd= 4 system, and for concreteness we assume access to\n“single-quantum” transitions H01\nc, H12\nc, H23\ncas defined\nin Appendix A.\nTo simplify notations in ref. [100], we define ρjj(U) =\nTrPjjUρU†, where ρis the state to be reconstructed.\nSimilar to above, we can start by extracting the state\npopulations ρjj(I),∀j∈ {0,1,2,3}. To access the co-\nherences ρj,k̸=j, we can now apply effective π/2-rotations\n(and their combinations) to ρbefore measurement. For\ninstance, applying a π/2-pulse on the |0⟩ ↔ | 1⟩transition\nalong both xandy, to call X01andY01, one can identify\ntheir coherence:\nρ01=ℜρ01+iℑρ01,\nℜρ01= (ρ00(Y01)−ρ11(Y01))/2,\nℑρ01= (ρ00(X01)−ρ11(X01))/2.\n(B4)\nSimilarly, the remaining coherences ρjkcan be deter-\nmined by applying π/2-pulses on relevant transitions:\nℜρ12=\u0002\nρ11(Y12)−ρ22(Y12)\u0003\n/2,\nℑρ12=\u0002\nρ11(X12)−ρ22(X12)\u0003\n/2,\nℜρ23=\u0002\nρ22(Y23)−ρ33(Y23)\u0003\n/2,\nℑρ23=\u0002\nρ22(X23)−ρ33(X23)\u0003\n/2,\nℜρ02=\u0002\nρ11(X12X01)−ρ22(X12X01)\n−√\n2ℑρ12\u0003\n/√\n2,\nℑρ02=\u0002\nρ22(Y12X01)−ρ11(Y12X01)\n+√\n2ℜρ12\u0003\n/√\n2,\nℜρ13=\u0002\nρ22(X23X12)−ρ33(X23X12)\n−√\n2ℑρ23\u0003\n/√\n2,\nℑρ13=\u0002\nρ33(Y23X12)−ρ22(Y23X12)\n−√\n2ℜρ23\u0003\n/√\n2,\nℜρ03=ρ22(Y23Y12Y01)−ρ33(Y23Y12Y01)\n−√\n2ℜρ23+ℜρ13,\nℑρ03=ρ33(X23X12X01)−ρ22(X23X12X01)\n+√\n2ℑρ23+ℜρ13.\n(B5)\nAs noted above, the experimentally reconstructed ρcould\nbe further improved via MLE [99].\nAPPENDIX C: Intra-ququart gates\nHere we analyze the intra-ququart gates shown in\nFig. 2. We note that there are two operating regimes:\n(a)\n(b)\n(c)FIG. S1. Equalization of nuclear splittings via light\nshift . (a) The detuning to the red of the1S0|mF= +1 /2⟩ ↔\n3P1F= 1/2|mF=−1/2⟩transition required for the total\nenergy differences between the ground states in the1S0and\n3P0manifolds to be equal, as a function of applied laser inten-\nsity in units of the saturation intensity, Isat≈0.14 mW /cm2.\nThe detuning is held fixed to this function in (b) and (c). (b)\nThe effective scattering rate on this transition while light is\napplied. (c) The total probability of scattering a photon for\ntotal illumination time 20 µs, corresponding to a single “o”-\nqubit X rotation.\n1) Low magnetic field ( B≲5 G) where the nuclear\nspin splitting is ∼kHz-level and thus negligible compared\nto the Rabi frequencies used for clock, stimulated Ra-\nman, and Rydberg transitions. This regime was used in\nRefs. [12, 31–33].\n2) High magnetic field ( B≳100 G) where the dif-\nferential nuclear spin splitting between the ground and\nmetastable manifolds is non-negligible. This regime was\nused in Refs. [34, 35] for QND readout of the ground\nnuclear spin. Our proposed ququart readout techniques\nrely on this ability. However, all other operations could\nbe performed in the low-field regime.\n1. ‘o’-qubit X rotations\nA pure “o”-qubit X operation requires that the state\nrotation be completely agnostic to the “n”-sector qubit.\nTo facilitate “o”-qubit X rotations as shown in Fig. 2(a),\nwe propose two approaches. The first is to operate in\nthe low-field regime for which the nuclear spin degree of\nfreedom has been demonstrated to be preserved during\noptical clock transitions [11]. The second approach is de-\nsigned for high-field operation. We propose to apply a\nlight shift such that the total energy difference between\nthe|00⟩and|01⟩states becomes equal to that between\n|10⟩and|11⟩. With either approach, both transitions\nshown in Fig. 2(a) are driven by the same, monochro-\nmatic pulse. Hence, driving an “o”-qubit X rotation is12\ntantamount to driving two identical two-level systems in\nparallel, thus making it additionally compatible with var-\nious other methods to combine high-fidelity population\ntransfer with preservation of the atom’s motional state\nin the trap [11, 101].\nWe now briefly discuss both the single-state light shift\nand the motion-preserving “o” gates.\na. Single-state light shifts\nIn the presence of a 120 G magnetic field, the Zeeman\nsplitting between |10⟩and|11⟩is≈50 kHz larger than\nthat between |00⟩and|01⟩. Perhaps the most convenient\nway to accomplish the desired light shift is by apply-\ning red-detuned light with σ−polarization, far-detuned\nfrom the |01⟩ ↔3P1F= 1/2|mF=−1/2⟩transition.\nThe detuning required for the ground nuclear spin split-\nting to match that of the metastable state is shown in\nFig. S1(a) as a function of the intensity of the applied\nlight. Fig. S1(b) and (c) show the scattering rate and\nprobability of scattering for an assumed application time\nof≈20µs corresponding to a single “o”-qubit gate (see\nbelow). We require an intensity of I/Isat≈2.8×103and\ndetuning ∆ /2π≈240 MHz for a scattering probability of\n≲10−3.\nb. Motional-state preserving “o”-qubit rotations\nUnder a magic trapping condition, one challenge to-\nwards achieving high-fidelity operations on the “o” qubit\nis eliminating the effects from motional states. Even after\nthe atoms are cooled to motional ground state, when the\noptical Rabi frequency is comparable to or larger than\nthe trap frequency (high-Rabi regime), motional side-\nband transitions are off-resonantly excited during clock\noperations, leading to leakage from qubit (ququart) states\nto other motional states. This issue can be solved by op-\nerating at Rabi frequency much smaller than the trap\nfrequency, at the cost of longer gate time. Alternative\nfaster approaches via composite pulse sequences have\nbeen demonstrated for ground-clock shelving, which in\nququart language represents “o”-qubit π-rotations [11].\nHere we present two motional-state preserving “o”-qubit\ngate designs for π-rotations and Hadamard gates, both\nworking at high-Rabi regime.\nInstead of using composite pulse sequences, we apply\na continuous, modulation-free clock pulse on resonance\nto the clock carrier transition, with total gate time ap-\nproximately 3 times the clock π-time, as is shown in\nFig. S2(b). Figure S2(a) shows the theoretical change\nof the average motional quanta after the pulse and gate\ninfidelity over certain range of parameter space, assum-\ning the atoms start from motional ground state. We find\nthat with a certain ratio of gate time τ, Rabi frequency\nΩ and trap frequency ω(Ω≈3ω,τ≈3π/Ω), we can\nachieve a ∼99.9% fidelity π-pulse while preserving themotional ground state, labeled with a star in Fig. S2. Fig-\nure S2(c) shows the evolution of the motional wave packet\nin⟨x⟩-⟨p⟩phase space. The trajectory returns back to the\nstarting point, indicating the motional state remains un-\nchanged together with Fig. S2(a).\nRealizing arbitrary-angle rotations while preserving\nmotional states is however non-trivial. To overcome this\nand to move towards a motional state-preserving univer-\nsal gate set on the “o”-qubit, we instead use a detuned\nclock Rabi oscillation to perform a Hadamard gate on\nthe “o” qubit. Figure S3 shows the change of average\nmotional quanta, gate infidelity, pulse design and evo-\nlution in phase of the Hadamard gate. Similar to the\nπ-rotation, the Hadamard pulse is also modulation-free,\nas is shown in Fig. S3(b). After searching through the\nparameter space, we find a set of optimized parameter\nsetting (detuning δ≈Ω∼3.4ω,τ= 3π/Ω), labeled with\na star in Fig. S3(a). The evolution in phase space also\nreturns to the starting point as is shown in Fig. S3(c).\nThe above gate schemes only consider a two-level sys-\ntem. To act as an “o” qubit gate in a ququart, we first\nuse the methods described in section C 1 a to add single-\nstate light shift to one of the ququart state, such that\nthe energy difference between |00⟩and|10⟩and between\n|01⟩and|11⟩are the same. Then one π-polarized clock\nbeam can be used to drive both pairs of clock rotations\nsimultaneously while being agnostic to the “n” qubit.\n2. “n”-qubit X rotations\nX rotations on the “n” qubit are driven via two stim-\nulated Raman process applied simultaneously to the\nground and clock manifolds. As shown in Fig. 2(b), the\ntwo clock states will use the3D1F= 1/2mF=−1/2\nstate as the intermediate in the process, while the ground\nstates will use the analogous state in the3P1manifold.\nAlthough it is also possible to drive the two appropri-\nate magnetic transitions simultaneously with an oscillat-\ning magnetic field when the ground and clock manifolds\nhave equalized splittings (see the previous section), we\nprefer this scheme due to its access to ∼MHz-scale Rabi\nfrequencies [98].\nMore precisely, we consider for both Raman processes\nillumination of the ququart by two beams: one with σ−\npolarization and one with πpolarization. Using single-\nbeam Rabi frequencies of ∼10-100 MHz with detunings\nof≈800 MHz to the excited states, it is possible to tune\nthe pulse timings such that the accrued phase on the “o”\nqubit during the “n” qubit rotation due to light shifts\nfrom the Raman beams is zero modulo 2 π, while addi-\ntionally maintaining effective Raman Rabi frequencies of\n≈1 MHz and excited state populations of ≲10−3. We\nnote that Raman beam intensity fluctuations will intro-\nduce fluctuations in the phase shift of the “o” qubit, but\nsuch fluctuations will similarly affect the ability to per-\nform high-fidelity Raman rotations. Thus, we believe that\nno additional fidelity constraints are imposed on the Ra-13\n15 16 17 18\n (s)\n0.0800.0850.0900.0950.100/2\n(a)Final n\n0 5 10 15\nTime (s)\n0.00.20.40.60.81.0P(t)(c)\nP(0)\nP(1)\n15 16 17 18\n (s)\n0.0800.0850.0900.0950.100/2\n(b)log10(1-)\n0.5\n 0.0 0.5\nx\n0.50\n0.25\n0.000.250.50p\n(d)\n0.10\n0.05\n0.00\n0\n1\n2\n3\n0.00.51.0Time/\nFIG. S2. Clock motion-insensitive π-rotation. (a)(b)\nTheoretical ∆¯ nand gate infidelity under different gate time\nand Rabi frequency, with trap frequency fixed to 30 kHz. The\n∆¯nand infidelity are calculated by starting from a product of\nqubit state |0⟩and thermal ground state. We see that there\nexists a region with minimal change in ¯ nand high final state\nfidelity. The stars indicate the conditions under which Rabi\noscillations are shown in (c) and (d). (c) Spin evolution under\nthe parameters labeled in (a) (Ω /2π= 88 .5 kHz, gate time\n16.78 µs). At the end of gate time the two spins are flipped.\n(d) Simulation of the phase space trajectory of the motional\nstate during the evolution. The diamond denotes the motional\nstate at the end of the gate.\nman “n”-qubit rotations due to the addition of the “o”-\nqubit in the computational space.\n3. “o”- and “n”-qubit Z rotations\nZ rotations on the “o” and “n” sectors are to be ac-\ncomplished by selectively applying light shifts. Specifi-\ncally, we seek to apply a uniform light shift to the two\nground ququart states for an “o”-qubit Z rotation and a\nuniform light shift to the two mF= +1 /2 ququart states\nfor a “n”-qubit Z rotation. Incorporating the beam de-\nscribed in Appendix C 1, the “n”-qubit light shift can be\napplied using that same beam, as well as another tuned\nanalogously to the3D1F= 1/2 manifold with the same\nσ−polarization. The “o”-qubit light shift could be ap-\nplied with a linearly polarized, far-detuned laser since\nthe goal is to essentially produce a non-‘clock-magic’ op-\ntical potential. Specifically, a wavelength of 532 nm has\na large differential scalar polarizability for the1S0and\n3P0states. This approach was used to light-shift atoms\nout of resonance with the clock laser for mid-circuit op-\nerations [11].\n16 17 18 190.0900.0950.1000.1050.110/2\n(a)Final n\n0.00.51.0P(t)(c)\nP(0)\nP(1)\n0 5 10 15\nTime (s)\n0.00.51.0P(t)P(+i)\nP(-i)\n16 17 18 19\nGate time (s)\n0.0900.0950.1000.1050.110/2\n(b)log10(1-)\n0.25\n 0.00 0.25\nx\n0.2\n0.00.2p\n(d)0.04\n0.02\n0.00\n0\n1\n2\n3\n0.00.51.0Time/\nFIG. S3. Clock motion-insensitive Hadamard gate.\n(a)(b) Theoretical ∆¯ nand gate infidelity under different gate\ntime and Rabi frequency, with trap frequency fixed to 30 kHz\nand starting from motional ground state. The stars indicate\nthe conditions under which Rabi oscillations are shown in (c)\nand (d). (c) Spin evolution under the parameters labeled by\nthe star ( ω/2π= 30 kHz, δ/2π= 100 kHz, Ω /2π= 106 kHz,\ngate time 17.1 µs) starting from |0⟩and|+i⟩states respec-\ntively. At the end of gate time, |0⟩state transforms to |+⟩\nstate and |+⟩state transforms to | −i⟩. (d) Simulation of the\nphase space trajectory of the motional state during the evo-\nlution, starting from |0⟩spin state. The diamond denotes the\nmotional state at the end of the gate.\n4. Intra-ququart CNOT and SWAP, and\ncompensatory Z rotations\nDuring application of both the intra-ququart CNOT\nand SWAP gates [see Fig. 2(c) and 2(d)], two of the\nququart states accrue phases relative to the others due to\nlight shifts. In the CNOT case |10⟩and|11⟩are shifted\ndue to the operation itself, while in the SWAP case |00⟩\nand|11⟩are shifted due to various other transitions close\nto 578 nm, primarily that from the ground states to the\n3P1manifold at 556 nm. Both cases amount to a combi-\nnation of a Z rotation on the “o” qubit with a differential\nZ rotation between the ground and metastable nuclear\nspin qubits, and can hence be compensated via applied\nlight shifts. For instance, the σ−-polarized beam detuned\nfrom the3P1F= 1/2 manifold from Appendix C 1 can\nbe used to create a shift of the ground nuclear spin split-\nting. This can be combined with the far off-resonant non-\n‘clock-magic’ optical trap from Appendix C 3 that creates\na shift between the ground and metastable manifolds.14\nAPPENDIX D: Readout\n1. The effect of imperfections on the two-shot\nququart readout\nThe two-shot ququart state readout described in sec-\ntion D is based on the repetitive QND readout proto-\ncol of nuclear ground state qubit from ref. [34]. Com-\npared with single-shot readout, more sources of imper-\nfections can affect the two-shot readout fidelity: (a) The\nsingle-shot readout fidelity Fm, (b) the atom loss proba-\nbility after first readout Pl, (c) the ground state flip rate\nPf, if the atom scatters photons during the first read-\nout, and (d) the intra-ququart SWAP fidelity Fπ. Fig-\nure S4 shows how these imperfections can cause different\ntwo-shot measurement results for different initial states.\nThe dark/bright result from each readout is indicated\nby the notation D/B. The red arrows indicate the error-\nfree path, while errors occurring at different steps can\nlead to other wrong readout results. For instance, if we\ntake Fm=Fπ= 99 .9%,Pf= 0.1% and Pl= 1%, the\nprobabilities of returning the right result conditioned on\ncertain input states are\nP(BB||00⟩) = 98 .7%,\nP(BD||01⟩) = 99 .6%,\nP(DB||10⟩) = 98 .7%,\nP(DD||11⟩) = 99 .8%.(D1)\n2. Light shift of |01⟩due to |00⟩readout\nDuring single-state readout [see Fig. 4(b)], the |01⟩\nground state incurs a light shift. Referring to previous\nexperimental work [34] for QND probing conditions, we\nset the intensities of the probe light to I≈Isaton all po-\nlarizations and the detuning from the targeted stretched\ntransition to ∆ ≈Γ. In this configuration, the probe\nlight produces a light shift of ≈30 Hz on |01⟩. For a\nτ≈10 ms probe time, this results in the accumulation of\na≈1.9 rad phase, which must be compensated with ei-\nther an additional blue-detuned tone in the probing light\nor an applied light shift using the beam described in Ap-\npendix C 1. It may also be possible to choose other probe\nlight parameters or timing such that the accrued phase\nis exactly 2 π. In any case, it will be important that the\nprobe intensity and detuning are well controlled – beyond\nthe level typically required for readout.\nAPPENDIX E: Two-ququart Rydberg-based gates\nWe propose two flavors of inter-ququart gates using\nthe single-photon Rydberg coupling from the3P0nuclear\nspin states. Both gate protocols extend that detailed in\nRef. [46] to the two-ququart computational space. Oper-\nating on the space of states {|00⟩⊗|00⟩,|00⟩⊗|01⟩,|00⟩⊗|10⟩, . . . ,|11⟩⊗|11⟩}, the first is a CCCZ gate that applies\na phase shift φto all states for which exactly one atom is\nin the |11⟩ququart state, a different phase shift φ′to the\n|11⟩ ⊗ |11⟩state such that 2 φ−π=φ′, and zero phase\nshift to the remaining states. The second gate is a CZ gate\nbetween only the “o” sectors of the two ququarts, apply-\ningφto any states of the forms |1a⟩⊗|0b⟩or|0a⟩⊗|1b⟩,\nφ′to any states of the form |1a⟩ ⊗ |1b⟩, and zero phase\nshift to the remaining states.\nThe CCCZ gate has the simpler of the two implementa-\ntions, requiring a drive of only a single transition from the\n3P0mF= +1 /2 clock state to the n3S1F= 3/2, mF=\n+3/2 (n= 59, for example) Rydberg state [see Fig. 3(a)].\nThe protocol itself has been experimentally realized us-\ning this transition in Ref. [32]; we only analyze its effects\non the total ququart space. The analysis of this gate is\nstraightforward through numerical simulation (later in\nthis appendix).\nOn the other hand, the CZ gate requires the applied\nphase shifts to be fully agnostic to the “n” sectors of\nboth ququarts, for which the pulse sequence must then\nbe duplicated with identical detuning and power on the\ntransition between the3P0mF=−1/2 and n3S1mF=\n−3/2 states [Fig. 3(b)]. This second configuration also\nrequires that atoms in states of different mFin the n3S1\nmanifold experience an energy shift that is similar to that\nfor states of identical mF– a configuration that has not\nyet been explicitly studied to our knowledge.\nIn the sections that follow, we perform numerical sim-\nulations of the proposed ququart CZ and CCCZ proto-\ncols. In our calculations, we neglect effects from laser\nphase noise and Doppler shifts due to finite temperature\nsince we seek to only highlight challenges arising from the\nhigher-dimensional encoding. For the ququart CZ gate,\nwe examine interactions between Rydberg states of dif-\nfering mFvalues and use the results in our simulations.\n1. Rydberg interactions for multiple Zeeman states\nDetermination of the σz-type interaction strength be-\ntween Rydberg states of differing mFvalues amounts to a\ncalculation of the C6van der Waals interaction potential\ncoefficient. We base our calculations on the perturbative\ntreatment of the problem in Ref. [102] for I= 0 bosonic\nspecies, which gives the value of the C6coefficient for dif-\nferent pair states as the eigenvalues of a C6matrix whose\nelements are calculated as\n(C6)ij=−X\np\"\f\fR11(s0, sp)\f\f2\nE0−EpD11(si, sp,ˆρ)D11(sp, sj,ˆρ)#\n.\n(E1)\nThis sum runs over pindexing pair states of the form\nsp= (n, α, m J) where nandmJare the principal and\ntotal electronic angular momentum projection quantum\nnumbers and, for brevity, αis a vector containing all\nother relevant total-spin quantum numbers. Epis the en-\nergy of state sp, and the subscript 0 denotes the initial15\nReadout 1 Readout 2Intra-ququart\nSWAPReadout 1 Readout 2Intra-ququart\nSWAP\nInitialstate\n|00⟩B_\nFm\nD_1–FmBright|00⟩ –Pf–Pl+1\nBright|01⟩PfBright|ø⟩ Pl\nDark|00⟩ –Pf–Pl+1\nDark|01⟩PfDark|ø⟩PlSWAP,\nBright|00⟩ 1\nSWAP,\nBright|10⟩FπSWAP,\nBright|01⟩1–FπSWAP,\nBright|ø⟩1\nSWAP,\nDark|00⟩1\nSWAP,\nDark|10⟩FπSWAP,\nDark|01⟩1–FπSWAP,\nDark|ø⟩1BD1–Fm\nBBFm\nFm\n1–Fm1–Fm\nFmFm\n1–Fm\nDD1–Fm\nDBFm\nFm\n1–Fm1–Fm\nFmFm\n1–FmInitialstate\n|01⟩B_\nFm\nD_1–FmBright|00⟩ Pf\nBright|01⟩–Pf–Pl+1Bright|ø⟩ Pl\nDark|00⟩ Pf\nDark|01⟩–Pf–Pl+1Dark|ø⟩PlSWAP,\nBright|00⟩ 1\nSWAP,\nBright|10⟩FπSWAP,\nBright|01⟩1–FπSWAP,\nBright|ø⟩1\nSWAP,\nDark|00⟩1\nSWAP,\nDark|10⟩FπSWAP,\nDark|01⟩1–FπSWAP,\nDark|ø⟩1BD1–Fm\nBBFm\nFm\n1–Fm1–Fm\nFmFm\n1–Fm\nDD1–Fm\nDBFm\nFm\n1–Fm1–Fm\nFmFm\n1–Fm\nInitialstate\n|10⟩B_ 1–Fm\nD_FmBright|10⟩\n1–Pl\nBright|ø⟩Pl\nDark|10⟩1–Pl\nDark|ø⟩PlSWAP,\nBright|10⟩1–Fπ\nSWAP,\nBright|01⟩Fπ\nSWAP,\nBright|ø⟩1\nSWAP,\nDark|10⟩ 1–Fπ\nSWAP,\nDark|01⟩Fπ\nSWAP,\nDark|ø⟩1BDFm\nBB1–Fm\n1–FmFm\nFm1–Fm\nDDFm\nDB1–Fm\n1–FmFm\nFm1–FmInitialstate\n|11⟩B_ 1–Fm\nD_FmBright|11⟩1–Pl\nBright|ø⟩Pl\nDark|11⟩1–Pl\nDark|ø⟩PlSWAP,\nBright|11⟩1\nSWAP,\nBright|ø⟩1\nSWAP,\nDark|11⟩1\nSWAP,\nDark|ø⟩1BDFm\nBB1–Fm\nFm\n1–Fm\nDDFm\nDB1–Fm\nFm\n1–Fm(a) (b)\n(c)(d)\nFIG. S4. The effects of readout infidelity, atom loss, ground state flipping, and intra-ququart SWAP fidelity on the 2-shot\nreadout result for (a) |00⟩, (b)|01⟩, (c)|10⟩and (d) |01⟩initial states. The red arrows denote the error-free path to the expected\nresult for each initial state.\neigenstate of the unperturbed Hamiltonian. ˆ ρis a unit\nvector ˆ ρ= (θ, φ) describing the angles in spherical coor-\ndinates between the two atoms relative to the quantiza-\ntion axis.\nThe functions R11andD11are the appropriate radial\nand angular parts of the matrix elements of the electricdipole operator,\nRk1,k2(s′, s)\n≈\u0012Z∞\n0dr1Rn′\n1,L′\n1(r1)Rn1,L1(r1)r2+k1\n1\u0013\n×\u0012Z∞\n0dr2Rn′\n2,L′\n2(r2)Rn2,L2(r2)r2+k2\n2\u0013\n(E2)\nDk1,k2(s′, s,ˆρ)\n= (−1)k2\u0014\n4π(2k1+ 2k2)!\n(2k1)!(2k2)!(2L1+ 1)(2 L2+ 1)\n(2k1+ 2k2+ 1)\u00151/2\n×[(2J1+ 1)(2 J2+ 1)]1/2CL′\n10\nL10,k10CL′\n20\nL20,k20\n×\u001a\nJ1k1J′\n1\nL′\n1S L 1\u001b\u001a\nJ2k2J′\n2\nL′\n2S L 2\u001b\n×k1+k2X\nq=−(k1+k2)k1X\nq1=−k1k2X\nq2=−k2h\nC(k1+k2)q\nk1q1,k2q2CJ1mJ1\nJ1mJ1,k1q1CJ2mJ2\nJ2mJ2,k2q2Yk1+k2,q(ˆρ)i\n(E3)16\nwhere Rn,ℓ(r) is a Hydrogen radial wavefunction,\nCJ3,m3\nJ1,m1;J2,m2is the Clebsch-Gordan coefficient\n⟨J1, m1;J2, m2|J3, m3⟩, and Yℓ,mis a spherical har-\nmonic. At the time of writing, the Rydberg states of\n171Yb are not yet well-studied, and we lack sufficient\nquantum defect data for a proper calculation of R11.\nGiven the results of Refs. [12, 32], however, we assume a\nnominal Rydberg interaction following C′\n6= 5 THz µm6\nforn= 74, scaled down to the n= 59 ,3S1F= 3/2\nRydberg states, and focus on only the angular factors inE1. Thus we reduce the calculation to\n(˜C6)ij=−X\nph\nD11(si, sp,ˆρ)D11(sp, sj,ˆρ)i\n(E4)\nin order to show only that the expected variation in the\nsize of the Rydberg interaction within the targeted man-\nifold would not be disruptive to the implementation of\nthe CZ gate.\nWe therefore rewrite Eq. E3 by way of the Wigner-\nEckart theorem as\nDk1,k2→˜Dk1,k2(s′, s,ˆρ)\n= (−1)−F1+k1−m′\nF1−F2+k2−m′\nF2+J′\n1+J′\n2−L1−L2+2I+2S+k1+k2\n×(−1)k2\u00144π\n(2k1+ 2k2+ 1)(2k1+ 2k2)!\n(2k1)!(2k2)!\u00151/2\n×\u0002\n(2F′\n1+ 1)(2 F′\n2+ 1)(2 J′\n1+ 1)(2 J′\n2+ 1)(2 L′\n1+ 1)(2 L′\n2+ 1)\n×(2F1+ 1)(2 F2+ 1)(2 J1+ 1)(2 J2+ 1)(2 L1+ 1)(2 L2+ 1)\u00031/2\n×\u001a\nJ′\n1F′\n1I\nF1J1k1\u001b\u001a\nJ′\n2F′\n2I\nF2J2k2\u001b\u001a\nL′\n1J′\n1S\nJ1L1k1\u001b\u001a\nL′\n2J′\n1S\nJ2L2k2\u001b\u0012\nL′\n1k1L1\n0 0 0\u0013\u0012\nL′\n2k2L2\n0 0 0\u0013\n×k1+k2X\nq=−(k1+k2)k1X\nq1=−k1k2X\nq2=−k2\u0012\nF′\n1k1F1\n−m′\nF1q1mF1\u0013\u0012\nF′\n2k2F2\n−m′\nF2q2mF2\u0013\nC(k1+k2)q\nk1q1,k2q2Y(k1+k2)q(ˆρ)(E5)\nusing the particular choices S1=S2=S′\n1=S′\n2≡S= 1\nandI1=I2=I′\n1=I′\n2≡I= 1/2, which are motivated\nbelow. Following the diagonalization of Eq. E4, we cal-\nculate ˜C6values for all possible pair states in a chosen\nbasis. These values are dimensionless and act only as a\nweighting factor to estimate the strength of the Rydberg\ninteraction between a given pair of states in the n3S1\nmanifold. Specifically, the ˜C6value for the k-th pair state\nis calculated as\n˜C(k)\n6=X\nj⟨k|j⟩⟨j|˜C6|j⟩⟨j|k⟩ (E6)\nwhere |j⟩is the j-th eigenvector of ˜C. The final C6value\nfor the k-th pair state used in numerical simulations (see\nbelow) is then C(k)\n6=\u0000\nC′\n6/max k|˜C(k)\n6|\u0001\nטC(k)\n6.\nAlthough we only consider these angular weighting fac-\ntors for interactions between states within the targeted\nRydberg manifold, the calculation as a whole still de-\npends critically on effects from other, nearby Rydberg\nstates whose contributions to Eq. E1 eventually vanish\nas the energy difference factors in the denominator be-\ncome large and the overlaps between radial wavefunc-\ntions become small. Lacking both in Eq. E4, we are then\nforced to decide independently on a particular basis over\nwhich the sum is carried out. From the analysis in pre-\nvious work [98], we also restrict the basis to states for\nwhich S= 1 and F=I+J. Thus the chosen basiscomprises all hyperfine states associated with the mani-\nfolds{3S1F= 3/2,3P0F= 1/2,3P1F= 3/2,3P2F=\n5/2,3D1F= 3/2, . . .}which we choose to cut off at\n3D1F= 3/2 (for a total of 20 states) as an estimated\nthreshold for where contributions to Eq. E1 would mostly\nvanish in an ideal calculation. Note that Eq. E4 does not\ndepend on the principal quantum number nbecause we\nhave discarded all energy and radial factors. The results\nof the full calculation for the targeted3S1F= 3/2 man-\nifold are shown in Table I.\nTable I. C6/C′\n6weighting factors between3S1F= 3/2\nhyperfine states calculated from Eq. E4, normalized to their\nlargest-magnitude value. The row indexes the mFquantum\nnumber of the first atom while the column indexes that of the\nsecond. Note that the Rydberg interaction is symmetric with\nrespect to the interchange of the two states.\nm(1)\nF m(2)\nF\n−3/2−1/2 +1 /2 +3 /2\n−3/2−0.989−0.980−0.984−1.000\n−1/2−0.980−0.969−0.970−0.984\n+1/2−0.984−0.970−0.969−0.980\n+3/2−1.000−0.984−0.980−0.98917\n2. Numerical simulations\nWe examine both the CCCZ and CZ ququart gate\nprotocols by numerically simulating the ground-clock-\nRydberg state manifold using the methods from previous\nwork [98] under the application of the pulse sequence de-\nscribed in Ref. [46]. The actions of the gate protocols were\nevaluated by simulating the appropriate pulse sequence\nwith the two-ququart system initialized in each compu-\ntational basis state. The simulations were performed for\na typical Rabi frequency Ω Ryd= 2π×3 MHz on the\ntargeted Rydberg transitions (with laser detuning deter-\nmined by the Rabi frequency; ∆ /ΩRyd≈0.375 for both\nprotocols) and nominal interaction strength 2 GHz in the\npresence of a magnetic field B= 120 G without effects\nfrom laser phase noise or finite temperature. For both\nCZ and CCCZ, all excitation pulses were performed for\nhorizontally polarized light with angle of incidence per-\npendicular to the quantization axis.\nNotably, the forms of both gates are defined in terms\nof two phases φandφ′= 2φ−πwhere φcan in general\nbe any real number modulo 2 π, which leaves the “ideal”\nform of the gate (in the case of perfect fidelity) param-\neterized by φ. We therefore calculate the fidelity of our\nsimulated gates according to\nFCCCZ = max\nφ1\n16\f\fTr\u0000\nCCCZ†(φ) CCCZ′\u0001\f\f (E7)\nFCZ= max\nφ1\n16\f\fTr\u0000\nCZ†(φ) CZ′\u0001\f\f (E8)\nwhere CCCZ( φ) and CZ( φ) are the ideal realizations of\nthe gates, CCCZ′and CZ′are the forms of the gates re-\nalized from simulation, and we normalize by the identity\noperator on the total (4 ×4)-state two-ququart basis.\nIt was assumed that the two ground states would not\nhave significant coupling to any other nearby states from\nthe Rydberg excitation pulses, and it was calculated that\nany associated light shifts from dipole-allowed transitions\ncould be neglected at the ∼10−5Hz level. Under the\nconditions identified above, we find FCCCZ≳0.999 and\nFCZ≳0.999. We do not identify additional limitations\ninFCCCZ associated with the ququart encoding; however,\nFCZdepends on the assumed Rydberg interactions, as we\nnow describe.\nAlthough the CCCZ protocol merely extends that\nidentified in Ref. [46] to include effects on addi-\ntional states in the ququart encoding scheme, the CZ\nprotocol requires consideration of the magnitude of\nthe Rydberg interaction between different states in\nthen3S1manifold. Specifically, we consider effects\nfrom non-uniform energy shifts on the |∗⟩m(1)\nF, m(2)\nF∈\n{|−3/2,−3/2⟩,|+3/2,+3/2⟩,|+3/2,−3/2⟩,|−3/2,+3/2⟩}\ntwo-ququart Rydberg states. We parameterize their C6\n(a)\n(b)FIG. S5. Inter-ququart CZ gate performance . (a) Ac-\ncrued phases φfor all two-ququart basis states in units of\nπas a function of ∆ /Ω for η= 0.989, corresponding to the\nweighting values shown in Table I, and overall Rydberg en-\nergy shift 2 GHz. Lines are colored by the number of 1’s in\nthe “o” positions of both atoms’ ququart bitstrings – 0 (pur-\nple), 1 (blue), and 2 (orange), giving distinct accruals φ0,φ1,\nandφ2, respectively. The dot-dash patterns of each of the\nlines correspond to their associated states’ two-ququart bit-\nstrings. The CZ gate protocol from Ref. [46] is implemented\nfor 2φ1−π=φ2(red points). (b) CZ gate infidelities as a func-\ntion of η∈[−1,1], parameterizing the uniformity of the Ryd-\nberg interaction between the various states of the 593S1man-\nifold. The lines of different opacities show the gate infidelity\nfor varied overall energy shift from 30 GHz (most opaque) to\n50 MHz (least opaque), with equal logarithmic spacing. The\ncondition used in (a) is shown in red. Note that the authors\nof Ref. [32] estimate their shift to be of size ≈26 GHz.\nvalues by ηas\nC(−3/2,+3/2)\n6 →ηC′\n6 (E9)\nC(+3/2,−3/2)\n6 →ηC′\n6 (E10)\nC(+3/2,+3/2)\n6 →C′\n6 (E11)\nC(−3/2,−3/2)\n6 →C′\n6 (E12)\nwhere η= 1 corresponds to perfect uniformity, and\nη= 0.989 to the result of the calculations from the previ-\nous section. Here we assume that the size of the shifts for\nthe|±3/2,±3/2⟩states are equal due to symmetry, and\nneglect parameterization of all other pairs of Rydberg\nstates due to selectivity in the applied pulses’ detunings\nand polarizations. We vary ηfrom -1 to 1 (always as-\nsuming all interactions have the same sign) as well as the\noverall size of the Rydberg shift (equal to that for the\n|±3/2,∓3/2⟩states). We show the simulated phase ac-\ncrual of all two-ququart states for η= 0.989 in Fig. S5(a),\nand gate infidelity as a function of ηand overall shift\nmagnitude in Fig. S5(b). In particular, we see that the18\n0.00 0.02 0.04 0.06\nEntanglement generation infidelity0.420.440.460.48YieldImperfect distillation, depol\nImperfect distillation, X error\nFIG. S6. The yield of imperfect distillation with respect to\ndifferent values entanglement generation infidelity for differ-\nent error models. The initial deviation from 0.5 is due to the\nassumed 1% measurement error.\ngate fidelity is more robust to effects from non-uniform\ninteractions for larger shift sizes. For expected interaction\nshifts of ∼GHz, we can tolerate an interaction disparity\nof 1−η≳0.95 between the two pair-states to still obtain\na gate fidelity of F ≈0.999.\nAPPENDIX F: Simulating Entanglement\nDistillation and Flag-based FTQEC\nIn section III, we proposed several possible applications\nrelated to the ququart architecture and simulated how\nthe behavior gets improved when using ququarts. Here we\ndiscuss several details about the simulations, including\nerror models for gates, circuit-level error models for QEC\nsimulations, and the stabilizer lists and logical operators\nfor the QEC code we used to perform simulations.\n1. Entanglement distillation\nFigure 5(a) shows the circuit of entanglement distilla-\ntion, with the entanglement generation presented as two\nwavy lines. We simulate the pre- and post-distillation en-\ntanglement infidelity and yield of the circuit in Fig. 5(a),\nwhose result is shown in Fig. 5(b) and Fig. S6. Here we fix\nthe measurement error rate to 1% and the intra-ququart\ngate infidelity to 0 .1%, and vary the error rate in the\ntwo different error models to change the pre-distillation\nentanglement infidelity.\n2. Flag-based FTQEC\nFigure 6(b) simulates how the threshold for certain\ncodes under flag-based QEC can increase by replacingthe ancilla and flag qubits with one ququart. We per-\nformed a circuit-level simulation, using the Pauli frame\nto track how single-qubit faults propagate through two-\nqubit CNOT gates during the whole circuit. The faults\non physical qubits/ququarts are generated via the de-\npolarization channel: During the state preparation each\nphysical qubit has a probability pto be replaced with\na maximally mixed state, but post-selection by mea-\nsuring the stabilizers ensures that the state preparation\nprocess is fault-tolerant. The measurement has probabil-\nitypto be faulty, and after each inter-qubit (ququart)\ngate the state is replaced with a totally-mixed two-qubit\nstate with probability p. To show the improvement from\nusing ququart and due to the limitations of the Pauli\nframe (only uncorrelated Pauli errors on each qubit can\nbe tracked), we treat intra-ququart CNOT gates as per-\nfect, but existing single-qubit faults can still propagate\nthrough intra-ququart CNOT gates. After one round of\ncircuit-level error correction, we perform another round\nof perfect decoding. The circuit-level decoding is denoted\nas successful if, after the perfect decoding, the logical\nstate gives no logical errors.\nUsing the model described above, we simulate the logi-\ncal error rate of [[5 ,1,3]] and Steane [[7 ,1,3]] codes using\nflag-based error correction scheme under different values\nofp. For convenience, we list out the stabilizer lists and\nlogical operators for [[5 ,1,3]] and [[7 ,1,3]] code respec-\ntively:\n[[5,1,3]]:\nS1=X1Z2Z3X4,\nS2=X2Z3Z4X5,\nS3=X1X3Z4Z5,\nS4=Z1X2X4Z5,\nXL=X1X2X3X4X5,\nZL=Z1Z2Z3Z4Z5.(F1)\n[[7,1,3]]:\nS1=X1X2X3X4,\nS2=X2X3X5X6,\nS3=X3X4X6X7,\nS4=Z1Z2Z3Z4,\nS5=Z2Z3Z5Z6,\nS6=Z3Z4Z6Z7,\nXL=X1X2X3X4X5X6X7,\nZL=Z1Z2Z3Z4Z5Z6Z7.(F2)\nThe ququart flag shows an improved threshold, making\nthe logical qubit more robust against the physical error\nratepas shown in Fig. 6.19\n[1] D. Leibfried, R. Blatt, C. Monroe, and D. 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LOUIS, Long-range interactions22\nin one-and two-electron rydberg atoms , Ph.D. thesis,\nDurham University (2014)."    },    {        "title": "2402.13138v1.Transcendence_of_generalized_Euler_Kronecker_constants.pdf",        "content": "arXiv:2402.13138v1  [math.NT]  20 Feb 2024TRANSCENDENCE OF GENERALIZED EULER-KRONECKER CONSTANTS\nNEELAM KANDHIL AND RASHI LUNIA\nABSTRACT . We introduce some generalizations of the Euler-Kronecker constant of a number field\nand study their arithmetic nature.\n1.INTRODUCTION AND PRELIMINARIES\nIn 1740, Euler introduced the Euler-Mascheroni constant, wh ich is defined as\nγ= lim\nx→∞\n/summationdisplay\nn≤x1\nn−logx\n. (1)\nSince then it has been extensively studied but many question s about its behaviour are yet\nunanswered. For example, it is not known if γis rational or irrational. For a survey on Euler’s\nconstant, please refer [4]. In [1], Diamond and Ford introdu ced a generalization of Euler’s\nconstant as follows. For a non-empty finite set of distinct pri mesΩ, letPΩdenote the product\nof the elements of ΩandδΩ=/productdisplay\np∈Ω(1−1\np). Then the generalized Euler constant is defined as\nγ(Ω) = lim\nx→∞\n/summationdisplay\nn≤x\n(n,PΩ)=11\nn−δΩlogx\n.\nNote that, when Ω =∅, we have PΩ= 1 =δΩandγ(Ω) =γ. In this context, Murty and\nZaytseva [8] proved the following theorem.\nTheorem 1.1. (Murty and Zaytseva) At most one number in the infinite list {γ(Ω)}, asΩvaries over\nall finite subsets of distinct primes, is algebraic.\nWe note that γappears as the constant term in the Laurent series expansion ofζ(s)around\ns= 1. This observation led Ihara [3] to define the Euler-Kronecker c onstant associated to a\nnumber field as follows.\n2020 Mathematics Subject Classification. 11J81, 11J86.\nKey words and phrases. Linear forms in logarithms, generalized Euler-Kronecker c onstants.\n12 NEELAM KANDHIL AND RASHI LUNIA\nLetKbe a number field of degree nand letOKdenote its ring of integers. The Dedekind zeta\nfunction of Kis given by\nζK(s) =/summationdisplay\n(0)/negationslash=a⊆OK1\nNas,ℜ(s)>1.\nIt has a meromorphic continuation to the entire complex plan e with only a simple pole at the\npoints= 1. Its Laurent series expansion around s= 1is given by\nζK(s) =ρK\ns−1+cK+O(s−1), (2)\nwhereρK/\\e}atio\\slash= 0is the residue of ζKats= 1.The ratio\nγK:=cK/ρK, (3)\nwas defined by Ihara [3] as the Euler-Kronecker constant of K. In the next section, an expression\nanalogous to (1) is given for γK.\nThe aim of this article is to study the arithmetic nature of ge neralizations of Euler-Kronecker\nconstants. To do so, we introduce some notations. Let PKdenote the set of non-zero prime\nidealspofOKand letΩbe a non-empty subset of PK(possibly infinite) such that\n/summationdisplay\np∈ΩlogN(p)\nN(p)−1<∞. (4)\nForK=Q, the set of Pjateckii- ˇSapiro primes is an example of such an infinite subset. Let\nNΩ={p∩Z|p∈Ω}. We set\nP(Ω(x)) =/productdisplay\np∈Ω(x)p andδK(Ω(x)) =/productdisplay\np∈Ω(x)/parenleftbigg\n1−1\nN(p)/parenrightbigg\n, (5)\nwhereΩ(x) ={p∈Ω|N(p)≤x}.Then by (4), lim\nx→∞δK(Ω(x))exists and equals\nδK(Ω) =/productdisplay\np∈Ω/parenleftbigg\n1−1\nN(p)/parenrightbigg\n. (6)\nNote that δK(Ω) = 1 forΩ =∅. The generalized Euler-Kronecker constant associated to Ωis\ndenoted by γK(Ω)and is defined as\nlim\nx→∞\n1\nρK/summationdisplay\n0/negationslash=a⊂OK\nN(a)≤x\n(a,P(Ω(x))=11\nN(I)−δK(Ω(x))logx\n.\nIn §3, we will show that this limit exists. We note that γK(Ω) =γKwhenΩ =∅. In this set up,\nwe have the following theorem.GENERALIZED EULER-KRONECKER CONSTANTS 3\nTheorem 1.2. Let{Ωi}i∈Ibe a family of subsets of PKsatisfying equation (4). Further suppose that\nNΩi\\NΩjis nonempty and finite for all i,j∈Iandi/\\e}atio\\slash=j. Then at most one number from the infinite\nlist /braceleftbiggγK(Ωi)\nδK(Ωi)/vextendsingle/vextendsingle/vextendsinglei∈I/bracerightbigg\nis algebraic.\nWe digress here a little to make an interesting observation. ForK=Q, it is known by Merten’s\ntheorem that as x→ ∞ ,\nδQ(Ω(x))∼e−γ\nlogx.\nThis makes one wonder if γKappears as an exponent in the expression for K/\\e}atio\\slash=Qtoo. A result\nof Rosen [9] shows that this is not true in general. More precis ely he showed that as x→ ∞ ,\nδK(Ω(x))∼e−γ\nρKlogx.\n2.PRELIMINARIES AND LEMMAS\nLetKbe a number field of degree n. Throughout this section, pdenotes a non-zero prime\nideal ofOK. We recall the following result on counting the number of int egral ideals of OK.\nLemma 2.1. [7, Ch 11] Letambe the number of integral ideals of OKwith norm m. Then, as xtends\nto infinity,\nx/summationdisplay\nm=1am=ρKx+O(x1−1/n).\nUsing this result we get the following expression for γK, which is analogous to (1).\nLemma 2.2. For any number field K, the limit\nlim\nx→∞\n1\nρK/summationdisplay\n0/negationslash=a⊂OK\nN(a)≤x1\nN(a)−logx\n (7)\nexists and equals γK.\nProof. Applying partial summation formula and Lemma 2.1, the resul t follows. /square\nThe Möbius function µKand the von Mangoldt function ΛKare defined on OKas follows.\nµK(a) =\n\n1 ifa=OK\n(−1)rifais a product of rdistinct prime ideals\n0 otherwise ,4 NEELAM KANDHIL AND RASHI LUNIA\nΛK(a) =\n\nlogNp ifa=pmfor some pand some integer m≥1\n0 otherwise .\nWe record the following identities satisfied by these functio ns which can be derived using\ntechniques similar to [6, Exercise: 1.1.2, 1.1.4, 1.1.6].\n/summationdisplay\nJ|IµK(J)\nN(J)=/productdisplay\np|I/parenleftbigg\n1−1\nN(p)/parenrightbigg\n, (8)\nµK(I)logN(I) =−/summationdisplay\nJ|IΛK(J)µK/parenleftbig\nIJ−1/parenrightbig\n. (9)\nWe end this section by stating the key ingredient in the proof of Theorem 1.2.\nLemma 2.3 (Lindemann [5]) .Ifα/\\e}atio\\slash= 0,1is an algebraic number, then logαis transcendental, where\nlogdenotes any branch of logarithmic function.\n3. G ENERALIZED EULER -KRONECKER CONSTANTS\nLetPKdenote the set of non-zero prime ideals of OK. For any non-empty finite set Ωf⊂ PK,\nwe set\nP(Ωf) =/productdisplay\np∈Ωfp andδK(Ωf) =/productdisplay\np∈Ωf/parenleftbigg\n1−1\nN(p)/parenrightbigg\n, (10)\nwith the convention that P(Ωf) = 1 =δK(Ωf), whenΩf=∅. SinceOKis a Dedekind domain,\nevery integral ideal can be uniquely expressed as a product o f prime ideals. We define the gcd\nof ideals aandbin the following way.\nFor\na=/productdisplay\np∈PKpvp(a),b=/productdisplay\np∈PKpvp(b),\nwhere all but finitely many vp(a),vp(b)are zero, we define\n(a,b) = gcd(a,b) =/productdisplay\np∈PKpmin(vp(a),vp(b)),\nwhere we have denoted p0byOK. Hence if prime factors of aandbare all distinct, (a,b) =OK.\nWe notice that (a,b) =a+basvp(a+b) = min( vp(a),vp(b)).From now on, OKwill be denoted\nby 1.GENERALIZED EULER-KRONECKER CONSTANTS 5\nLemma 3.1. For a number field Kand a finite set Ωf, the limit\nlim\nx→∞\n1\nρK/summationdisplay\n0/negationslash=I⊂OK\nN(I)≤x\n(I,P(Ωf))=11\nN(I)−δK(Ωf)logx\n\nexists and is denoted by γK(Ωf).\nProof. LetΩf⊂ PKandp∈ PKbe a prime ideal not in Ωf. Using\n/summationdisplay\n0/negationslash=I⊆OK\nN(I)≤x\n(I,pP(Ωf))=11\nN(I)=/summationdisplay\n0/negationslash=I⊆OK\nN(I)≤x\n(I,P(Ωf))=11\nN(I)−1\nN(p)/summationdisplay\n0/negationslash=I⊆OK\nN(I)≤x\nN(p)\n(I,P(Ωf))=11\nN(I),\nthe result follows by induction on cardinality of Ωf. /square\nLemma 3.2. LetΩfbe a finite set of non-zero prime ideals. Then\nγK(Ωf) =δK(Ωf)\nγK+/summationdisplay\np∈ΩflogN(p)\nN(p)−1\n.\nProof. We have\n/summationdisplay\n0/negationslash=I⊂OK\nN(I)≤x\n(I,P(Ωf))=11\nN(I)=/summationdisplay\n0/negationslash=I⊂OK\nN(I)≤x1\nN(I)/summationdisplay\nJ|(I,P(Ωf))µ(J)\n=/summationdisplay\nJ|P(Ωf)µ(J)\nN(J)/summationdisplay\n0/negationslash=J0⊂OK\nN(J0)≤x\nN(J)1\nN(J0)\n=/summationdisplay\nJ|P(Ωf)µ(J)\nN(J)/braceleftbig\nρKlogx\nN(J)+ρKγK+o(1)/bracerightbig\n=δK(Ωf)/parenleftbig\nρKlogx+ρKγK+o(1)/parenrightbig\n−ρK/summationdisplay\nJ|P(Ωf)µ(J)\nN(J)logN(J).6 NEELAM KANDHIL AND RASHI LUNIA\nWe now consider the last term.\n−/summationdisplay\nJ|P(Ωf)µ(J)\nN(J)logN(J) =/summationdisplay\nJ|P(Ωf)1\nN(J)/summationdisplay\nJ0|JΛ(J0)µ/parenleftbig\nJJ−1\n0/parenrightbig\n=/summationdisplay\nJ0|P(Ωf)Λ(J0)\nN(J0)/summationdisplay\nJ1|P(Ωf)J−1\n0µ(J1)\nN(J1)\n=/summationdisplay\np′∈ΩfΛ(p′)\nN(p′)/summationdisplay\nJ1|P(Ωf)p′−1µ(J1)\nN(J1)\n=/summationdisplay\np′∈ΩflogN(p′)\nN(p′)/parenleftigg\nδK(Ωf)\n1−1\nN(p′)/parenrightigg\n=δK(Ωf)/summationdisplay\np∈ΩflogN(p)\nN(p)−1.\nThus\nlim\nx→∞\n1\nρK/summationdisplay\n0/negationslash=I⊂OK\nN(I)≤x\n(I,P(Ωf))=11\nN(I)−δK(Ωf)logx\n=δK(Ωf)\nγK+/summationdisplay\np∈ΩflogN(p)\nN(p)−1\n.\n/square\nCorollary 3.3. For a number field Kand any set Ω⊂ PKsatisfying (4), the limit\nlim\nx→∞\n1\nρK/summationdisplay\n0/negationslash=I⊂OK\nN(I)≤x\n(I,P(Ω(x)))=11\nN(I)−δK(Ω(x))logx\n\nexists and equals\nδK(Ω)\nγK+/summationdisplay\np∈ΩlogN(p)\nN(p)−1\n.\nWe denote this limit by γK(Ω).\nProof. Follows from Lemma 3.2 since Ω(x)is a finite set. /square\n4. P ROOF OF THEOREM 1.2\nSuppose there exist i,j∈Isuch that\nγK(Ωi)\nδK(Ωi)andγK(Ωj)\nδK(Ωj)GENERALIZED EULER-KRONECKER CONSTANTS 7\nare algebraic. Using Corollary 3.3,\nγK(Ωi)\nδK(Ωi)−γK(Ωj)\nδK(Ωj)=/summationdisplay\np∈ΩilogN(p)\nN(p)−1−/summationdisplay\np∈ΩjlogN(p)\nN(p)−1, (11)\nwhich is also an algebraic number. Since the sets NΩi\\NΩjandNΩi\\NΩjare non-empty and\nfinite, the sets Ωi\\ΩjandΩj\\Ωiare also finite. Let\nΩi\\Ωj={p1,p2,···,pn},Ωj\\Ωi={q1,q2,···,qm}.\nThen equation (11) implies\n/summationdisplay\np∈ΩilogN(p)\nN(p)−1−/summationdisplay\np∈ΩjlogN(p)\nN(p)−1=n/summationdisplay\ns=1logpfss\npfss−1−m/summationdisplay\nt=1logqgt\nt\nqgt\nt−1= log\n/producttextn\ns=1p/parenleftbigg\nfs\npfss−1/parenrightbigg\ns\n/producttextm\nt=1q/parenleftbigg\ngt\nqgt\nt−1/parenrightbigg\nt\n,(12)\nwhereN(ps) =pfssandN(qt) =qgt\nt. Using Lemma 2.3 and unique prime factorisation of natural\nnumbers, the expression in (12) becomes a transcendental nu mber, which gives a contradiction.\nAcknowledgments. The authors would like to thank Prof. S. Gun for suggesting th e problem\nand IMSc for providing academic facilities. They would also l ike to thank Prof. P . Moree\nand J. Sivaraman for comments on an earlier version of the man uscript, which improved the\nexposition. The authors would like to thank the anonymous re free for careful reading of the\npaper and helpful comments. The first author would like to than k the Max- Planck-Institut\nfür Mathematik for providing a friendly atmosphere. The seco nd author would like to thank\nNumber Theory plan project, Department of Atomic Energy for financial support.\nREFERENCES\n[1] H. Diamond and K. Ford, Generalized Euler constants , Math. Proc. Cambridge Philos. Soc. 145(2008), no. 1,\n27–41.\n[2] L. Euler, De Progressionibus harmonicis observationes , Commentarii academiae scientiarum Petropolitanae, 7\n(1740), 150–161\n[3] Y. Ihara, The Euler-Kronecker invariants in various families of glob al fields , Arithmetics, geometry and coding theory\n(AGCT 2005), 79–102, Sémin. Congr., 21, Soc. Math. France.\n[4] Jeffrey C. Lagarias, Euler’s constant: Euler’s work and modern developments , Bull. Amer. Math. Soc. 50(2013),\n527–628.\n[5] F. Lindemann, Über die Zahl π, Math. Ann. 20(1882), no. 2, 213–225.\n[6] M. Ram Murty, Problems in analytic number theory, Second edition, Springer , New York, 2008.\n[7] M. Ram Murty and J. Esmonde, Problems in Algebraic Number Theory, Springer-Verlag , New York, 2005.\n[8] M. Ram Murty and A. Zaytseva, Transcendence of generalized Euler constants , American Math. Monthly, 120\n(2013), no. 1, 48–54.\n[9] M. Rosen, A generalization of Mertens’ theorem , J. Ramanujan Math. Soc. 14(1999), no. 1, 1–19.8 NEELAM KANDHIL AND RASHI LUNIA\nNEELAM KANDHIL ,\nMAX-PLANCK -INSTITUT FÜR MATHEMATIK , VIVATSGASSE 7, D-53111 B ONN , GERMANY .\nEmail address :kandhil@mpim-bonn.mpg.de\nRASHI LUNIA ,\nTHEINSTITUTE OF MATHEMATICAL SCIENCES , A CI OFHOMI BHABHA NATIONAL INSTITUTE , CIT C AMPUS ,\nTARAMANI , CHENNAI 600 113, I NDIA .\nEmail address :rashisl@imsc.res.in"    },    {        "title": "2402.13139v1.Deterministic_Dynamic_Edge_Colouring.pdf",        "content": "Deterministic Dynamic Edge-Colouring\nAleksander B. G. Christiansen∗\nTechnical University of Denmark\nAbstract\nGiven a dynamic graph Gwith nvertices and medges subject to insertion an deletions of\nedges, we show how to maintain a (1+ ε)∆-edge-colouring of Gwithout the use of randomisation.\nMore specifically, we show a deterministic dynamic algorithm with an amortised update time\nof 2˜Ologε−1(√logn)using (1 + ε)∆ colours. If ε−1∈2O(log0.49n), then our update time is sub-\npolynomial in n.\nWhile there exists randomised algorithms maintaining colourings with the same number\nof colours [Christiansen STOC’23, Duan, He, Zhang SODA’19, Bhattacarya, Costa, Panski,\nSolomon SODA’24] in polylogarithmic and even constant update time, this is the first deter-\nministic algorithm to go below the greedy threshold of 2∆ −1 colours for all input graphs.\nOn the way to our main result, we show how to dynamically maintain a shallow hierarchy of\ndegree-splitters with both recourse and update time in no(1). We believe that this algorithm\nmight be of independent interest.\n∗This work was supported by VILLUM FONDEN grant 37507 “Efficient Recomputations for Changeful Problems”.\n1arXiv:2402.13139v1  [cs.DS]  20 Feb 2024Contents\n1 Introduction & Related Work 3\n1.1 Main technical challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.2 Further Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2 Informal Overview 5\n2.1 The overall approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n2.2 Dynamic t-splitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n2.3 (∆ max+ 1)-edge-colouring of low degree graphs . . . . . . . . . . . . . . . . . . . . . 10\n2.4 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n3 Preliminaries & Notation 11\n3.1 Chains and shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n3.2 Algorithms for constructing and extending Vizing chains . . . . . . . . . . . . . . . . 14\n3.3 Path-stepping processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n4 Dynamic t-splitters 19\n4.1 Analysing the Local Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n4.2 Dynamically Maintaining Invariant 20 . . . . . . . . . . . . . . . . . . . . . . . . . . 23\n4.3 Analysing the Dynamic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\n4.3.1 Analysis of Round-Robin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\n4.3.2 Analysing the Recourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27\n4.3.3 Analysing the Update Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29\n5 Maintaining a hierarchy of t-splitters 30\n5.1 The hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31\n5.2 An algorithm for maintaining a hierarchy . . . . . . . . . . . . . . . . . . . . . . . . 32\n5.3 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\n5.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35\n6 Deterministic Dynamic Edge-Colouring of Low-Degree Graphs 36\n6.1 Structural Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36\n6.2 Contents of the data structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\n6.3 Maintaining the data structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40\n6.4 Analysing the data structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43\n6.5 The proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49\n7 Putting it all together 50\n7.1 The final algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50\n7.2 Dynamic ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51\n8 Acknowledgements 51\n21 Introduction & Related Work\nEdge-colouring is a classic and fundamental problem studied extensively in both graph theory as\nwell as many different models of computation [GNK+85, Ber22, Chr23, DHZ19, BGW21]. Given\na simple graph G= (V, E), ak-edge-colouring of Gis a function c:E(G)→[k] that maps every\nedge to a colour in some colour set [ k] ={1,2, . . . , k }, An edge-colouring is proper ife∩e′̸=∅\nimplies that c(e)̸=c(e′) for any pair of distinct edges e, e′. The chromatic index χ′(G) ofGis\nthen the smallest kfor which a proper k-edge-colouring of Gexists. It is not difficult to check\nthatχ′(G)≥∆ ifGhas maximum degree ∆. A classic result of Vizing [Viz64] shows that in fact\nχ′(G)≤∆ + 1. Deciding whether χ′(G) = ∆ or χ′(G) = ∆ + 1 is, however, not easy, and it was\nshown by Holyer that determining χ′(G) exactly is, in fact, NP-complete [Hol81].\nRecently, the edge-colouring problem has received a lot of attention in the dynamic algorithms\ncommunity. Here one studies dynamic graphs , where edges are continually inserted and deleted, and\nthe challenge is to maintain some proper edge-colouring of the graph as it is subjected to updates.\nThere has been some exciting progress on developing efficient randomised dynamic algorithms\nusing only (1 + ε)∆ colours. This number of colours go significantly below the greedy-threshold of\n2∆−1 colours. Duan, He, and Zhange [DHZ19] showed how to maintain such a (1 + ε)∆-edge-\ncolouring with an update time of O(log7n\nε2) provided that ∆ = Ω(log2n\nε2). Christiansen got rid of\nall requirements on ∆ while maintaining a worst-case update time in O(poly( ε−1,logn)). Very\nrecently Bhattacarya, Costa, Panski, and Solomon [BCPS] pushed the update time all the way\ndown to ˜O(poly( ε−1)) under some additional assumptions on ∆ and ε.\nHowever, all of the above algorithms rely on randomness in several and crucial ways, and the\nlandscape of deterministic and dynamic algorithms is a lot less colourful. Barenboim and Mai-\nmon [BM17] gave a dynamic algorithm with O(√\n∆) worst-case update time using O(∆) colours.\nThis was subsequently improved by Bhattacarya, Chakrabarty, Henzinger and Nanongkai [BCHN18]\nwho gave a fully dynamic algorithm for computing a (2∆ −1)-edge-colouring with worst case\nO(log ∆) update time. Bhattacarya, Costa, Panski, and Solomon [BCPS23a] and independently\nChristiansen, Rotenberg, and Vlieghe [CRV23] showed how to maintain a (∆+ O(α))-edge-colouring\nin amortised O(poly(log n)) update-time, where αis the arboricity ofG. However, if ∆ ∼α, this\nalgorithm will not always use a number of colours below the greedy threshold as αcan be as big as\n∆\n2. In particular, until now no deterministic and dynamic algorithm always using less than 2∆ −1\ncolours with sub-linear update time was known.\nIn general it is an interesting and challenging task to design deterministic algorithms with guar-\nantees similar to the randomised ones. In this paper, we will address this gap between randomised\nand deterministic dynamic algorithm for (1 + ε)∆-edge-colouring by providing a deterministic and\nfully-dynamic algorithm for maintaining a (1 + ε)∆-edge-colouring in no(1)amortised update-time:\nTheorem 1 (Informal) .LetGbe a dynamic graph and ε−1≤2poly(log log n)a given parameter.\nThen, we can maintain a proper ((1 + ε)∆)-edge-colouring of Ginno(1)amortised update time per\noperation.\n1.1 Main technical challenges\nIn this section, we will pinpoint two issues towards designing deterministic dynamic edge-colouring\nalgorithms with few colours. These are barriers we know how to overcome in either the static\nsetting or by using randomisation, but so far, to the best of our knowledge, no current techniques\ndeal with these issues directly.\nMany static algorithms, distributed algorithms, and dynamic randomised algorithms produce\nedge-colourings with few colours via a two-step approach. The first step is a form for dimenstionality\n3reduction ; one attempts to reduce the problem to a case where ∆ is small. The second step is then\nto apply an algorithm that is fast and efficient for graphs with small maximum degree.\nMany static algorithms [GNK+85, BCPS23b, Sin19] and many distributed algorithms [GKMU18,\nGHK+20, GS17] adress the first step by computing a hierarchy of degree-splitters . A degree-splitter\nsplits a graph Ginto two covering subgraphs G1, G2such that the maximum degree of Giis roughly\nhalf of the maximum degree of G. By applying this procedure recursively one can reduce the prob-\nlem to∆\npoly( ε−1,logn)sub-problems each of maximum-degree poly( ε−1,logn). However, these static\nand precise approaches for splitting the degree, seems less compatible with changes to the graph.\nIf the maximum degree is large enough, it is easy to realise something similar in the randomised\nsetting, even dynamically against an oblivious adversary. Indeed, one can place each edge into\none of∆\npoly( ε−1,logn)subgraphs uniformly at random. It then follows from a standard application\nof Chernoff-bounds that the maximum degree of each such subgraph is split nicely. A different\napproach for reducing ∆ is to sample uniformly at random a palette S⊂[t] of colours such that\nthe graph induced by all edges coloured with a colour from S, say G[S], exhibits useful properties.\nIndeed, one can show that with high-probability the maximum degree of G[S] is small and every\nvertex in Ghas a free colour in S. One can then colour a new edge, by viewing it as being inserted\nintoG[S]. This is the approach taken in [Chr23, DHZ19]. It is, however, not clear how to perform\nsuch a reduction in the deterministic dynamic setting; naively building a hierarchy can easily result\nin an insurmountable recourse (see section 2), and sampling colours or edges relies crucially on the\npower of randomisation.\nThe second challenge is to efficiently colour a low-degree graph. In the static setting one\ncould rely on parallelism [GNK+85], and in the dynamic & distributed settings recent progress\nhas shown that the recourse for extending colourings on low-degree graphs is small [Ber22, BD23,\nChr23, DHZ19] and, more importantly, can be efficiently realised using randomisation [Ber22,\nBD23, Chr23, DHZ19]. However, naively searching for such an extending sequence of recolourings\ncan be expensive, and so it is unclear how to locate it without using randomisation.\n1.2 Further Related Work\nGabow, Nishizeki, Kariv, Leven, and Tereda [GNK+85] showed how to compute a (∆ + 1)-edge-\ncolouring in ˜O(m√n) or ˜O(m∆) time. This improved over the O(mn) time algorithm implied by\nthe work of Vizing [Viz64]. Recently this was improved by Sinnamon [Sin19], and in special settings\nby Bernshteyn and Dhawan [BD23] and by Bhattacarya, Costa, Panski, and Solomon [BCPS23b].\nAlon [Alo03] and Cole, Ost and Schirra [COS01] gave a near linear-time algorithm for ∆-edge-\ncolouring bipartite graphs. By applying a reduction due to Karloff and Shmoys [KS87], these\nalgorithms yield 3 ⌈∆\n2⌉-edge-colouring algorithms for general graphs. When ∆ = Ω( ε−1logn),\nDuan, He and Zhang [DHZ19] gave a O(ε−2mlog6n) time algorithm for computing a (1+ ε)∆-edge-\ncolouring. This has recently been improved by Bhattacarya, Costa, Panski, and Solomon [BCPS]\nto˜O(mpoly( ε−1)) time for many values of ∆ and ε.\nThe edge-colouring problem has also been extensively studied in the LOCAL model [Ber22,\nBD23, CHL+18, Chr23, GKMU18, GHK+20, GS17, SV19] (see for instance [CHL+18, GKMU18]\nfor extensive surveys). Some of these algorithms rely on degree-splitting either rely on or directly\nconsider the degree-splitting problem [GKMU18, GHK+20, GS17].\n1.3 Results\nOur main result it the following theorem:\n4Theorem 2. LetGbe a dynamic graph, and 0< ε < 1a given parameter. We can maintain a\nproper (1 +ε)∆-edge-colouring of Gin2˜Ologε−1(√logn)amortised1update time per operation.\nOn the way to this result, we present several algorithms which we believe might be of inde-\npendent interest. First of all, we design what we call a dynamic t-splitter; a dynamic algorithm\nmaintaining a t-edge-colouring such that the graph induced by each colour has low-degree.\nTheorem 3. LetGbe a dynamic graph, and suppose ε >0,t∈N, and an upper bound ∆maxon the\nmaximum degree of Gthrough the entire update sequence is given. Assume that∆max\nt≥104log2n\nε2.\nThen there is a dynamic algorithm that computes a t-splitter of Gwith an amortised update-time\nofO(t3ε−4log3m).\nSecond of all, we show how to build a shallow hierarchy of t-splitters that splits the degree\nall the way down to poly(log n, ε−1). We believe that this might be a useful algorithmic tool for\ndeveloping efficient dynamic algorithms for other problems as well.\nTheorem 4. LetGbe a dynamic graph, and let 1> ε > 0and an upper bound ∆max≥107log5n\nε3\non the maximum degree of Gthrough the entire update sequence be given. Set t1=⌊210√logn⌋and\nt2=⌊logn\nε⌋. Then there exists a dynamic algorithm maintaining parameters h, i, j withh=i+j≤\n4√lognand a set of graphs Ghsatisfying that:\n1.|Gi+j| ≤ti\n1tj\n2\n2.ˆ∆i+j≤(1 +ε\n16)∆max\nti\n1tj\n2\nThe algorithm has amortised recourse and update time both in 2˜Ologε−1(√logn).\nFinally, we show how to efficiently (∆ max+ 1)-edge-colour graphs with low maximum degree:\nTheorem 5. LetGbe a dynamic graph, and let ∆maxbe a known upper-bound on the maximum\ndegree throughout the entire update sequence. Then, we can maintain a proper (∆max+ 1)-edge-\ncolouring of Gin(∆max+ 1)˜O(√logn)worst-case update time per operation.\n2 Informal Overview\nIn this section, we will give a high-level overview of the main ideas and the algorithm. We first\nintroduce the overall approach:\n2.1 The overall approach\nFrom now on, unless otherwise noted, we will assume that we have a known upper bound ∆ max\non the maximum degree ∆. This assumption will later be removed by using a degree-scheduler\nidentical to the one used by Duan, He, and Zhang [DHZ19].\nWe will take a two-step approach that, at a high-level, is inspired by the distributed algorithm\nof Ghaffari, Kuhn, Maus, and Uitto [GKMU18]. The idea is to partition the graph into a set of at\nmost∆\ntsubgraphs Ghsuch that {E(Gi)}Gi∈Ghpartitions E(G) and such that ∆( Gi)≤(1+ε\n2)·tfor\nallGi∈ Gh. Then one can colour each Gi∈ Ghwith a separate palette using some algorithm that is\nefficient in low-degree graphs, and then combine all of these colourings into a (1+ ε)∆-edge-colouring\nof the original graph G.\n1Here ˜Ologε−1hides log log nand log ε−1factors.\n5As a warm-up, we briefly sketch how to boot-strap a simple static algorithm for computing\na (1 + ε)∆-edge-colouring in near-linear time using the algorithms of Gabow et Al. [GNK+85].\nThis algorithm was recently, and independently, obtained by Elkin and Khuzman [EK24], while\nthis manuscript was under preparation, and we therefore refer to their manuscript for a formal\ndescription of the algorithm.\nThe idea is the following: a degree-splitter with discrepancy δis a partition of the edge set into\ntwo sets E1andE2, such that for all vertices v, we have that dG[E1](v), dG[Ew](v)≤∆\n2+δ, where\nG[Ei] is the graph induced in Gby the set Eii.e. the subgraph of Gcontaining all edges of Ei. This\nis a relaxed definition of degree-splitting similar to the one used in [GS17], but other variants have\nalso been studied (see for example [GHK+20]). Given an efficient degree-splitter with small enough\ndiscrepancy, one can easily compute a partition Ghof the desired form by recursively applying the\ndegree splitter to G1andG2to build a hierarchy whose lowest level form Gh.\nGabow et Al. [GNK+85] showed how to compute a degree-splitter with a discrepancy of only\n2 in the static setting. By recursively applying this degree splitter roughly h= log ∆ −loglogn\nε\ntimes (or 0 times if ∆ = O(logn\nε)) one can achieve a split into roughly∆\nlogn\nεsubgraphs, each with\nmaximum degree at mostlogn\nε+O(logn+P∞\ni=02\n2i). By rescaling εa bit, this yields the desired\npartition. Finally, one can colour each graph in Ghwith a distinct palette efficiently using another\nalgorithm of Gabow et Al. [GNK+85].\nAs we have already briefly touched upon in section 1.1, the two main challenges with making\nsuch an approach dynamic are the following: 1) how to maintain the degree splitters, and 2) how\nto efficiently colour the low-degree graphs.\nAs for 1), we note that first of all, as far as we are aware, no deterministic dynamic degree-\nsplitting algorithm is known. While it is easy to devise a randomised such algorithm against an\noblivious adversary for graphs with not too small maximum degree (simply partition the edges u.a.r\ninto 2-edge sets), it is unclear how to do this deterministically. Second of all, if a deterministic\ndegree-splitter has recourse at least 10, then naively maintaining a hierarchy of degree-splitters\nbecomes infeasible as the recourse at depth hof the hierarchy can be as large as 10h. In order\nto split the degree from n0.75ton0.25, we would require h≥1\n2logn, thus resulting in super-linear\nrecourse, and therefore a super-linear update time.\nWe overcome this as follows: instead of designing a dynamic degree-splitter, we consider the\ndynamic t-splitting problem, where we are interested in splitting the graph into t-subgraphs, each\nwith some restrictions on the maximum degree:\nProblem 6. Given a dynamic graph Gsubject to insertions and deletions, maintain a colouring\nc:E(G)→[t]such that for all vertices v∈V(G)and all colours i∈[t], we have that the number\nof edges coloured iincident to v,di(v), is upper bounded by di(v)≤(1 +ε)∆\nt.\nWe show in Section 4 that we can, in fact, maintain a dynamic t-splitter in ˜O(poly( t,log(n), ε))\nupdate time, but crucially with only poly(log( n), ε) recourse. This data-structure is far from trivial,\nand requires many ideas of its own. We will discuss it in Section 2.2.\nSince the t-splitter has low recourse, we can tune tvery aggressively. In turn, we can get a\ndynamic∆\npoly(log n,ε−1)-splitter by maintaining a very shallow hierarchy of correctly tuned t-splitters.\nIndeed, we can set t∼2√lognand split the degree down to ∼2√lognusing a hierarchy of depth\nonly h∼√logn. This allows us to bound the recourse R(h) at depth has:\nR(h)≤poly(log( n), ε)h= 2˜Ologε(√logn)\nThen, we can set t′∼poly(log n, ε−1) to further split the degree down to poly(log n, ε−1), while\nonly increasing the recursive depth by an extra additive factor of√logn. Since each update takes\n6Figure 1: An augmenting Vizing chain is shifted on the left, and a truncated Vizing chain is shifted\non the right. This illustration is from [Chr23]\nonly ˜O(poly( t,log(n), ε)) time, this allows us to split the degree down to poly(log n, ε−1) with both\nrecourse and update time in 2˜Ologε−1(√logn).\nIn order to address 2), we will make use of multi-Step Vizing chains , a term coined by Bern-\nshteyn [Ber22] and a topic that has seen a lot of recent development [Ber22, BD23, GP20, Chr23].\nHere, one attempts to find a sequence of recolourings that creates an uncoloured edge with a free\ncolour. The idea is to construct such a sequence of recolourings by first constructing a 1-step Vizing\nchain, which is a construction that goes back all the way to Vizing’s original paper [Viz64]: a Vizing\nchain consists of two parts: a) a fanFof edges and b) a bichromatic path Pof edges coloured with\ntwo colours κ1andκ2. The fan Fconsists of a center vertex utogether with edges uw1, . . . , uw k\nsuch that the colour of uwi+1is available at wi, meaning that no edges incident to wihas the colour\nc(uwi+1). The bichromatic path Pthen has to begin at wkand consist only of edges coloured κ1\nandκ2, where κ1is available at uandκ2is available at wk. A Vizing chain can be shifted by\nrecolouring edges in Fand subsequently in Pas shown in Figure 1 to the left. Observe that if P\nis not too long, we are already happy with the construction, as we need only recolour few edges.\nHowever, Pcould potentially have length Ω( n). In this case, we can truncate it by uncolouring an\nedge along Pand only shifting the first part of Pas shown in Figure 1 to the right.\nThen we may build a new Vizing chain on top of the new uncoloured edge in the hope that\nthis chain has a shorter length. We may iterate this to construct multi-step Vizing chains (see\nFigure 2), as has been done in similar fashion in [Ber22, BD23, Chr23, GP20]. The goal is to get a\nshort Vizing chain which is augmenting in the sense that after shifting it, the remaining uncoloured\nedge has a free colour. In order to more readily reason about these chains, we will require them to\nbenon-overlapping in the sense that we impose restrictions on how the ithextension can overlap\nwith earlier chains.\nThe main issue is that all known results about Vizing chains are either existential or obtained\nvia randomised constructions. As noted by Bernshteyn & Dhawan [BD23] naively applying the\nargument of Christiansen [Chr23] to search for a Vizing chain can result in linear update time,\nand so it seems hopeless to perform some deterministic search for a short and augmenting Vizing\nchain. However, we will show that one can efficiently maintain data structures contianing enough\n7Figure 2: On the left, we see a non-overlapping multi-step Vizing chain. On the right, we see a\nmulti-step Vizing chain overlapping in a single edge. This illustration is from [Chr23]\ninformation to be able to locate certain classes of short and augmenting Vizing chains efficiently.\nThe main observation is that, under the correct tuning of parameters, one has that while one\nuncoloured edge can be part of many different multi-step Vizing chains, not too many vertices\ncan actually reach the edge via a multi-step Vizing chain. This means that assigning an edge a\nnew colour, does not result in the need of updating too many different data-structures. While\nmaintaining these data structures is intuitively quite simple, it is also quite technical. Therefore,\nwe will give a separate overview of how to maintain and use these data structures in Section 2.3.\n2.2 Dynamic t-splitters\nIn this section, we will give an overview of how to maintain a dynamic t-splitter with low recourse.\nWe begin by sketching an algorithm for the case t= 2. We note already now that this case\nis considerable easier than the case t >2. For the case t= 2 , the algorithm has worst-case\nguarantees, but for t >2 all guarantees are amortised. After sketching the approach, we discuss\nthe technical difficulties with generalising this approach, and sketch how we overcome them.\nWe will colour the graph with 2 colours, colour 1 and colour 2. For a vertex v, we define its\nsurplus of colour ito be si(v) = max {di(v)−∆max\nt,0}. We will maintain a colouring satisfying the\nfollowing local invariant for some parameter η:\nInvariant 7. For all uv∈E(G), we have that c(uv) =iimplies and all i∈[t], we have that\nmax{si(u), si(v)} ≤η+ max {si+1(u), si+1(v)}\nwhere addition is performed modulo 2.\nThis invariant is useful for two reasons: first of all, one can show that if it holds everywhere, then\nthe maximum surplus is never too big, and second of all, whenever an edge uvis inserted one can\nrestore the invariant by only recolouring edges along a path with alternating colours. These paths\nbear some resemblance to the augmenting paths used in a distributed algorithm due to Ghaffari\nand Su [GS17].\n8Indeed, to see that the invariant implies a low maximal surplus, observe that if si(v) is large,\nthen any vertex wneighbouring vvia an edge coloured imust have large surplus si+1(w)≥\nsi(v)−η. Applying this approach recursively yields set V1, V2, . . . , V kwhere u∈Vjimplies that\nsi+j−1 mod 2 (u)≥si(v)−(j−1)η. In particular, one has:\n|E(G[∪k\nh=1Vh])]≥ | ∪k\nh=1Vh| ·(si(v)−(k−1)η)\nIndeed, observe that one double counts no edges as vertices in Vjcannot be connected by an edge\ncoloured i+j−1 mod 2, without violating the local invariant.\nFor a reasonable choice of k, ηandsi(v), we have that si(v)−(k−1)η >(1+ε\n16)∆\n2. In particular,\nsince the maximum subgraph density ρ(G) = max J⊂G{|E(J)|\n|V(J)|} ≤∆\n2by the hand-shaking lemma, it\nfollows by induction that | ∪k\nh=1Vh| ≥(1 +ε\n16)| ∪k−1\nh=1Vh|. Hence, we conclude that k=O(logn\nε),\nwhich naturally bounds how big si(v) can be.\nIn order to see that one need only recolour edges along an alternating path, assume the edge uv\nis inserted and receives colour iminimizing max {si(u), si(v)}. The choice of ireadily implies that\nat most one of si(u) and si(v) can be larger than∆max\n2. Ifsi(w)≤∆max\n2after uvis coloured by\ni, then no edge incident to wwill suddenly violate the invariant. So we need only focus on edges\nincident to the endpoint with the larger surplus in i. WLOG say si(u)>∆max\n2. If so, then only\nedges coloured iincident to ucan become violating. If an edge uxbecomes violating, then it must\nbe that\nmax{si(u), si(x)}> η+ max {si+1(u), si+1(x)}\nHence, recolouring uxto colour i+ 1 mod 2 restores the invariant at all edges incident to u, since\nsi+1(u)<∆max\n2. Now some edges coloured i+ 1 incident to xmight be violating. These can be\nfixed as above. Note that since si+1(x)≤si(u)−η, this process cannot be iterated too many\ntimes before one visits a vertex with less than ηsurplus, and one restores the invariant. In order\nto implement the above efficiently, one can locate violating in-edges by updating information lazily\nvia a round-robin scheme. The use of the round-robin scheme is inspired by a sequence of recent\nwork on dynamic algorithms for maintaining an estimate of the maximum subgraph density ρ,\nsee [SW20, CCH+], where a lazy round robin scheme is used to locate a chain of directed edges to\nre-orient.\nIn order to generalise the above to t >2, we will work with the following local invariant instead:\nInvariant 8. For all uv∈E(G)and all i∈[t], we have that\nL(uv) =sc(uv)(u) +sc(uv)(v)≤η+si(u) +si(v)\nMany of the difficulties that arise when generalising the above approach to t >2 stem from the\nfact that the problem becomes a lot less symmetric. The generalisation of the local invariant is\nchosen carefully to account for this asymmetry. Again we wish to show that 1) the local invariant\nimplies that the colouring is t-splitter, and 2) that we can restore the invariant in some efficient\nmanner.\nAs for 1) the asymmetry is reflected in the fact that one can no longer extract dense subgraphs\nas in the argument above. Indeed, for t >2 it is unclear how to extend a density based argument.\nBefore there was a natural way of assigning a high-surplus edge to the corresponding high-surplus\nendpoint. This allowed us to extract a dense subgraph, which we could use to show that the\nmaximum surplus could not be too large. However, for t >2 it is unclear which endpoint one\nshould assign a high-surplus edge to; indeed assigning it to the high-surplus vertex as before risks\nthat some vertices do not get assigned more than∆max\n2edges. Furthermore, there is the additional\n9caveat that it might not even be well-defined which endpoint has high-surplus, as both endpoints\ncould contribute equally to sc(uv)(u) +sc(uv)(v).\nInstead, we show how in Section 4 that if sc(uv)(u) +sc(uv)(v) is large, then one can extract a\nsequence of bipartite subgraphs {Bj}10 log( n)ε−1\nj=1 , possibly with low density, containing an exponen-\ntially increasing number of vertices. This again allows us to bound sc(uv)(u) +sc(uv)(v), and show\nthat we indeed maintain a t-splitter.\nThe asymmetry is also reflected in 2). Before one could restore the invariant by recolouring\na path, since fixing violating edges coloured idid not cause edges coloured i+ 1 at the same\nvertex to become violating. However, if one has multiple colours to choose from, one might have to\nrecolour multiple edges incident to each vertex. Hence, one has to analyse the recourse completely\ndifferently. We show that each recolouring of a violating edge cause the following potential to drop:\nΦ(G, c) =tX\ni=1X\nv∈V(G)si(v)X\nj=1j\nthus allowing us to bound the recourse of the algorithm in an amortised fashion.\nFinally, we arrive at the dynamic t-splitter by carefully adjusting the lazy round-robin schemes\nand the data structures used to account for the asymmetry and the fact that t >2.\n2.3 (∆max+ 1)-edge-colouring of low degree graphs\nAs a warm-up we will informally sketch how to maintain a (∆ max+1)-edge-colouring with ˜O(poly(∆ max)·\nn2\n3) update-time. Then we sketch how to generalise this approach in order to achieve the faster\nupdate time of (∆ max+ 1)o(1).\nThe starting point is this: suppose that given any proper partial (∆ max+ 1)-edge-colouring\nand an uncoloured edge e, there exists an augmenting, non-overlapping 3-step Vizing chain with\nlength O(poly(∆ max)·n1\n3). The existence of such chains is easy to derive from the work of Chris-\ntiansen [Chr23]. Technically speaking, these derivations used ∆ and not ∆ max, but it is straight-\nforward to check that one can replace ∆ by the less restrictive ∆ max. Since the final algorithm\nworks with more steps, we do not explicitly derive this bound, but it follows from near-identical\narguments to the proof of Corollary 46.\nWhile a naive search for such a chain might take O(poly(∆ max)n) time, one can readily observe\nthat any single edge can only be part of a 3-step Vizing chains beginning at one of O(poly(∆ max))\ndifferent points. Indeed, any coloured edge is part of at most ∆ max+ 1 different bichromatic paths\nand at most 2 different fans. Since any bichromatic path can be a part of at most 2(∆ max+ 1)\ndifferent 1-step Vizing chains, the statement follows for 1-step Vizing chains. To see that the\nstatement holds for 3-step Vizing chains, observe that in order to extend the chain through e, one\nhas to be able to reach an edge incident to an edge sharing an endpoint with e. Since there are\nat most 2(∆ max+ 1) such edges, one can apply the above argument again to see that there are at\nmost O(poly(∆ max)) choices for the starting point of 2-step Vizing chains. Iterating this argument\nthen gives a similar bound for 3-step Vizing chains.\nThe above argument relies crucially on the fact that we require the Vizing chain to be non-\noverlapping (see Definition 11 for a formal definition). Indeed, an issue could be that shifting the\nfirst step of a Vizing chain could create many new bichromatic paths not accounted for by the\nabove argument. But for that to happen, the extension must circle back and overlap orintersect\nan ealier Vizing chain. This in turn means that the Vizing chain becomes overlapping , and thus\nis not considered for the above argument. This type of argument is not new, and has been used\n10for instance by Bernshteyn [Ber22] and Christiansen [Chr23] to analyse the recourse of certain\nmulti-step Vizing chain constructions.\nIn this paper, we use this fact very explicitly in our algorithm. The idea is to explicitly maintain\nenough information about bichromatic paths in the graph to be able to construct an augmenting\nand non-overlapping 3-step Vizing chain fast: if each vertex knows the O(poly(∆ max)n1\n3) shortest\naugmenting 1-step Vizing chains going through it, then one can update information about the short-\nest augmenting and non-overlapping 2-step Vizing chain fast: simply consider all O(poly(∆ max)n1\n3)\ndifferent extension points, locate the shortest augmenting 1-step Vizing chains and extend via the\nshortest such chains. We can then update information about 3-step Vizing chains in a similar\nfashion.\nSince any edge is part of only O(poly(∆ max)) colouring, re-colouring, or un-colouring such\nan edge only affects at most O(poly(∆ max)) data-structures. By determining all data-structures\ncontaining augmenting 1-step Vizing chains and augmenting 2-step Vizing chains, we can first\nupdate all of the 1-step structures and then use them to update the 2-step structures efficiently,\nand so on.\nThe above approach has some glaring issues: first of all, we need the extensions to be non-\noverlapping. This requires us to check whether each extension is overlapping. However, by ex-\ntending in a controlled manner [Ber22, GP20], one can ensure that not too many extension are\noverlapping, and thus one can actually afford to check for each chain whether it is overlapping, in\n˜O(poly(∆ max)n1\n3) time per chain. By carefully choosing how many augmenting chains need to be\nsaved for each number of steps, one can ensure that one actually finds the shortest such chains.\nSecondly, changing the colour of edges might cause some of the fan-constructions to become\nout-dated, and so explicitly storing the Vizing chains might suddenly require information about a\ndifferent type of extension. To remedy this issue, we consider a stepping process , which emulates the\nstructure of consistent Vizing chains: in each step the process is allowed to jump to a neighbouring\nvertex and then follow a new bichromatic path. This allows us to maintain information about\nprocesses for many different kind of extensions, which in turn allows us to quickly update the data\nstructures whenever a fan-construction becomes out-dated.\nThe final algorithm is then similar to above, but instead we will work with step-length of roughly\n2O(√\n( logn))andO(√logn) steps. By carefully designing data-structures, we show that then the\nabove approach can in fact be generalised to yield a dynamic algorithm with (∆ max+ 1)˜O(√\n( logn))\nworst-case update time.\n2.4 Outline of the paper\nIn Section 3, we define notation and discuss some preliminaries. Then in Section 4, we present\nour dynamic t-splitter, before we show how to build a hierarchy of such splitters in Section 5.\nIn Section 6, we present our algorithm for colouring low-degree graphs. Finally, in Section 7 we\ncombine all of the above to get our final algorithm.\n3 Preliminaries & Notation\nThe first parts of this section is very similar to Section 2 of the full-version of [Chr23], as we need\nsimilar terminology. For a positive integer t, we let [ t] ={1,2, . . . , t}be the set of numbers in\nNthat are greater than 0, but at most t. We let G= (V, E) be a graph on nvertices and m\nedges. A subgraph H⊂GofGis a graph such that V(H)⊂V(G) and E(H)⊂E(G). The t-hop\n11neighbourhood ofHinG,Nt(H) consists of all of the vertices in Gof distance at most t(inG) to\na vertex in H. In particular, H⊂N(H). A proper (partial) k-edge-colouring cofGis a function\nc:E(G)7→[k]∪ {¬} such that if c(e)̸=¬andc(e′)̸=¬, then e∩e′implies that c(e)̸=c(e′). An\nedge esuch that c(e) =¬is said to be uncoloured . Two uncoloured edges may share an endpoint\nin a proper partial colouring. If no edges are uncoloured, we say that the colouring is a proper\nk-edge-colouring. For a vertex set U⊂V(G) (resp. edge set U′⊂E(G)), we let G[U] (resp. G[U′])\nbe the induced graph of UinGi.e. the subgraph of Gon vertex set V(G[U]) =Ucontaining the\nmaximum number of edges (resp. the induced graph of U′inGi.e. the subgraph with edge set\nE(G[U′]) =U′and vertex set V(G[U′]) =V(G)). Given a proper (partial) k-edge-colouring cand a\nset of colours S⊂[k], we let G[S] be the graph containing all edges coloured with a colour from S,\nand all of the vertices incident to an edge coloured with a colour from S. Given a proper (partial)\nt-edge-colouring of a graph G, we define the set A(v) ofavailable colours at a vertex vto consist\nof precisely the colours of [ t] that no edge incident to vhas received.\nFor a directed graph G, any edge denoted by ( uv)∈E(G) is oriented from utowards v. We\ndenote by din(v) (resp. dout(v)) the in-degree ofv(resp. out-degree ofv). Similarly, if E′⊂E(G)\nis a subset of edges and any subgraph H, we let Nj\nE′,in(H) denote the j-hop in-neighbourhood of\nHin the graph induced by E′. Similarly, we let EE′,in(H) be the edge set containing all in-edges\nofHalso belonging to E′.\n3.1 Chains and shifts\nIn order to extend a partial edge-colouring, we will have to assign some coloured edge new colours.\nIn order to describe such a process formally, we briefly recall the chain and shift terminology as it\nwas used by for example Bernshteyn [Ber22] (see [Ber22] for a more in-depth treatment).\nLete1, e2∈E(G) be two adjacent edges in a graph G, and let cbe a proper partial colouring\nofG. Then we define a colouring Shift( c, e1, e2) by setting:\nShift( c, e1, e2)(e) =\n\nc(e2) if e=e1\n¬ ife=e2\nc(e) if e /∈ {e1, e2}\nWe say that such a pair of adjacent edges e1, e2arec-shiftable ifc(e1) =¬andc(e2)̸=¬and the\ncolouring Shift( c, e1, e2) defined above is a proper partial colouring. If the colouring cis clear from\nthe context, we will sometimes leave out this argument. We will only consider shiftable chains.\nAchain Cof size kis then a set of edges C= (e1, . . . , e k) such that eiandei+1are adjacent for\nalliandc(ei) =¬if and only if i= 1. For 0 ≤j≤k−1, we can j-shift such a chain by performing\nShift j(c, C) defined as:\nShift 0(c, C) =c\nShift i(c, C) = Shift(Shift i−1(c, C), ei, ei+1)\nWe say that Cisc-shiftable if every pair of edges ei, ei+1is Shift i−1(c, C)-shiftable. It is straightfor-\nward to check that j-shifting such a chain Cyields a proper colouring, where the unique uncoloured\nedge in Cis the edge ej+1. We say that the chain ends at the edge ekand at the vertex vthat is\nshared by ekandek−1. Note that in the context of simple graphs, this vertex is well defined. We\nwill refer to the act of k-shifting a chain of size kas simply shifting the chain. We let cbe the\npre-shift colouring of Cand Shift( c, C) the post-shift colouring of C. We briefly note that from an\nalgorithmic point of view, we can view Shift( c, e1, e2)(e) as first uncolouring e2, and then assigning\n12e1the colour c(e2). This ensures that during all steps of the recolouring, the current colouring of\nthe graph is proper.\nOften we wish to find a chain Csuch that shifting Cyields an uncoloured edge uvsuch that\nA(u)∩A(v)̸=∅so that we may colour uvto extend our partial colouring. If this is the case, we say\nCis an augmenting chain , otherwise we refer to it is a truncated chain . We say that κisavailable\natuvifκ∈A(u)∩A(v).\nThe initial segment of length sof a chain Cis then the chain C|s= (e1, . . . , e s). One can\nconcatenate two chains C1= (e1, . . . , e s1) and C2= (es1, f2, . . . , f s2) to get the chain C1+\nC2= (e1, . . . , e s1, f2, . . . , f s2). Then in order to shift such a chain, we use Shift( c, C1+C2) =\nShift(Shift( c, C1), C2).\nAfanchain with center uis a chain Fof the form F= (uw1, . . . , uw k). Here uis the center\nof the fan, denoted by center( F). Bernshteyn [Ber22] showed that for such fans to be shiftable\nchains, we require c(uwi+1)∈A(wi). We will let the colour of c(uwi+1) be the representative\navailable colour atwi. We will also pick a representative available colour at wkinA(wk). If this\nrepresentative available colour is also either available at uor if it is the representative available\ncolour for some wjwith j < k , then we say that the fan is maximal .\nApath chain is a shiftable chain of the form P= (e1, . . . , e k), such that the edges in the chain\nform a path in the graph. The length of a path chain P= (e1, . . . , e k) isk. Abichromatic path\nchain is then a c-shiftable chain P= (e1, . . . , e k) that forms a path in Gsuch that c(e1) =¬\nand the colour of c(e2i) =κ1for all 1 ≤i≤ ⌊k\n2⌋andc(e2i+1) =κ2for all 1 ≤i≤ ⌊k−1\n2⌋, and\nfurthermore such that e2is either the first or the last edge in some maximal ( κ1, κ2)-bichromatic\npath. In order to specify the colours present in the chain, we will sometimes refer to such a chain\nas a ( κ1, κ2)-bichromatic path chain.\nVizing originally showed how to choose both a fan chain Fand a bichromatic path chain P\nthat together form a bigger chain F+Psuch that Shift( c, F+P) leaves some edge uvwith an\navailable colour. This type of chain has been referred to as a Vizing chain in the literature [Ber22,\nBD23, Chr23, GP20].\nDefinition 9. Given a proper partial colouring cofGand an edge e=uv∈E(G)withc(e) =¬,\naVizing chain onewith center uis a chain F+P, where F= (uw1, . . . , uw k)is a fan chain\nwith center (F) =uandP= (p0p1, . . . , p t−1pt)is a(κ1, κ2)-bichromatic path chain with p0=u,\np1=wk,κ1∈A(u), and κ2∈A(wk).\nWe say that F+Pends at the edge pt−1ptand at the vertex pt−1. We let Shift( c, F+P) be\nthepost-shift colouring. We let |E(P)|be the length of F+P.\nDefinition 10. Given a proper partial colouring cofG, an i-step Vizing chain is a c-shiftable\nchain of the form F1+P1+F2+P2+. . . F i+Piwhere Fj+Pjis a Vizing chain for all j.\nA (multi-step) Vizing chain is augmenting if shifting the chain leaves the final edge of the chain\nwith a colour that is available at both endpoints of the edge. If the chain is not augmenting, it is\ntruncated . The length of a multi-step Vizing chain is the sum of the lengths of the chains Fj+Pj.\nWe will only be considering Vizing chains that are non-overlapping :\nDefinition 11. Ani-step Vizing chain is non-overlapping if every pair of Vizing chains Fj+Pj\non edge ejandFk+Pkon edge ekshare an edge exactly when j=k−1and the shared edge is ek,\nand if, furthermore, for all k≤ino edge in the chain Fk+Pk= (e1, . . . , e s)is repeated i.e. ej=eh\niffj=h.\n133.2 Algorithms for constructing and extending Vizing chains\nIn this subsection, we briefly recall some well-known algorithms for constructing Vizing chains. In\nsome cases, one is allowed to make arbitrary choices in the constructions. In order to perform the\nconstructions in a consistent manner, we will order the colours [ s] according to the strict ordering <\nonN. Now, whenever an arbitrary choice between colours (possibly satisfying some conditions) can\nbe made, we will always choose the smallest colour according to <. We say that a fan is consistent\nif the representative available colour αiof each edge uwiis the smallest colour in A(wi) subject to\nthe requirements that 1) if A(u)∩A(wi)̸=∅, then αi∈A(u)∩A(wi) and 2) else if jis the smallest\ninteger j < i with αj∈A(wi) then αi=αj. Given a consistent fan-chain F= (uw1, . . . , uw k), a\n(κ1, κ2)-bichromatic path chain Pisconsistent with Fif:κ1∈A(u),κ2∈A(wk) are also chosen\nas small as possible according to <.\nThe following algorithm will be used to construct the first fan of every Vizing chain. It is a\nstandard result that we have slightly adjusted to have a consistent output.\nLemma 12 ([Ber22, BD23, Viz64]) .LetGbe a graph with maximum degree ∆satisfying ∆≤∆max.\nLetcbe a proper partial (∆max+ 1)-edge-colouring, and let e=uv∈E(G)be an uncoloured edge.\nThen there exists a unique consistent maximal fan chain F= (uw1, . . . , uw k)with center u.\nFurthermore, Fmay be computed in ˜O(∆3\nmax)time.\nProof. We first describe how to construct the fan: we set w1=v. We pick an available represen-\ntative colour α1inA(v) for vas follows: if A(v)∩A(u)̸=∅, we let α1be the smallest element of\nA(v)∩A(u). In this case, the construction terminates, and we conclude that Fis unique. Oth-\nerwise, we let α1be the smallest element of A(v). We let w2be the unique neighbour of uwith\nc(uw2) =α1. In this case, we note that any consistent maximal fan chain must have ( uw1, uw 2) as\nan initial segment.\nNext assume that we have inductively constructed a fan chain ( uw1, uw 2, . . . , uw j) with available\nrepresentative colours α1, . . . , α j−1, and that any consistent maximal fan chain on emust have\n(uw1, uw 2, . . . , uw j) as an initial segment. We extend the chain as follows: we pick an available\nrepresentative colour αjinA(wj) for wjas follows:\n1. if A(wj)∩A(u)̸=∅, we let αjbe the smallest element of A(wj)∩A(u). In this case, the\nconstruction terminates, and we conclude that Fis unique by the induction hypothesis.\n2. else if there exists a h < i such that αh∈A(wj), we let h′be the smallest such hand set\nαj=αh. In this case, the construction terminates, and we conclude that Fis unique by the\ninduction hypothesis.\n3. Otherwise, we let αjbe the smallest element of A(wj). We let wj+1be the unique neighbour\nofuwith c(uwj+1) =αj. In this case, we note that any consistent maximal fan chain must\nhave ( uw1, uw 2, . . . , uw j+1) as an initial segment.\nThe above construction must terminate. Indeed, when j=d(u)−1 the third case cannot occur,\nasuhas at most d(u)−1 coloured edges incident to it.\nAs for the furthermore part, we can check 1) in ˜O(∆2\nmax) by just checking for each colour in\nA(wj) if it belongs to A(u), we can check 2) in O(∆2\nmax) time, since we can check if αk∈A(wj) in\nO(∆max) time. Finally, we can do 3) in ˜O(∆max) time by simply computing A(wj) and extracting\nthe minimum.\nGiven a consistent maximal fan chain F= (uw1, . . . , uw k) and a parameter t∈Z≥0, we can\nextend Fto a Vizing chain ( F′+P)t, or simply F′+Piftis clear from the context, in a consistent\n14manner (note F′might not be equal to F), as follows: if αk∈A(u), we set P= (uwk) and F′=F.\nOtherwise, we let κ1be the smallest colour in A(u) and κ2=αk. Let P′=p1, p2, . . . , p t′be the\nunique maximal ( κ1, κ2)-bichromatic path with p1=wk. Ift < t′−1, we let F′=F, and we let\nP= (uwk, wkp2, . . . p tpt+1). Otherwise, if t≥t′−1, there are 3 cases:\nCase 1: ifpt′=wiwithi < k , we let F′= (uw1, . . . , uw i) and we let P= (uwi, wipt′−1, pt′−1pt′−2, . . . , p 2p1).\nNote that in this case F′is not a maximal fan.\nCase 2: ifpt′−1=wi+1andpt′=uwith i+ 1< k, we let F′= (uw1, . . . , uw i+1) and we let\nP= (uwi+1, wi+1pt′−2, pt′−2pt′−3, . . . , p 2p1). Note that in this case F′is not a maximal fan.\nCase 3: ifpt′/∈ {w1, . . . , w k}, we let F′=Fand we let P= (uwk, wkp2, p2p3, . . . , p t′−1pt′).\nIt is straight-forward to check that we can construct F′+PinO(∆3\nmax+ ∆ max·t) time, and\nthat the construction is unique. If e=uvis any uncoloured edge and Fis the unique maximal\nfan chain guaranteed by Lemma 12 centered on u, then for any given t, we say that ( F′+P)tis a\nconsistent Vizing chain with center u. Note that ( F′+P)tis in fact unique.\nNext we will see a different algorithm needed to extend any i-step Vizing chain to an ( i+ 1)-\nstep Vizing chain in a consistent way. Again we will first construct a fan chain, and then explain\nhow to compute the accompanying bichromatic path chain. We will use the so-called ’second fan\nlemma’ [Ber22, GP20].\nLemma 13 ([Ber22, GP20]) .LetGbe a graph with maximum degree ∆satisfying ∆≤∆max. Let\ncbe a proper partial (∆max+ 1)-edge-colouring, and let e=uv∈E(G)be an uncoloured edge. Let\nκ1∈[∆ + 1] be an available colour at uand let κ2∈[∆ + 1] be an available colour at v. Then there\nexists a fan chain uw1, . . . , uw kwithw1=vsuch that either\n1. the representative available colour at wkis also available at u.\n2.κ2/∈A(u)and the fan is maximal subject to the constraint that the colours κ1orκ2are\nconsidered unavailable to every vertex.\n3.wk̸=v, the representative available colour at wkisκ2and no edge uwiis coloured κ2.\nIf the representative available colours are chosen as small as possible subject to first 1), then 2),\nand finally 3)above, then Fis unique.\nFurthermore, we may compute Fin˜O(∆3\nmax)time.\nProof. We construct a maximal fan almost exactly as in the proof of Lemma 12. Again we begin\nby setting w1=v. We pick an available representative colour α1inA(v) for vas follows: if\nA(v)∩A(u)̸=∅, we let α1be the smallest element of A(v)∩A(u). In this case, the construction\nterminates, and we conclude that Fis unique. Otherwise, we let α1be the smallest element of\nA(v)\\ {κ2}. Note that since uvis uncoloured, |A(v)| ≥2. We let w2be the unique neighbour\nofuwith c(uw2) =α1. In this case, we note that any consistent maximal fan chain must have\n(uw1, uw 2) as an initial segment.\nNext assume that we have inductively constructed a fan chain ( uw1, uw 2, . . . , uw j) with available\nrepresentative colours α1, . . . , α j−1, and that any consistent maximal fan chain on emust have\n(uw1, uw 2, . . . , uw j) as an initial segment. We extend the chain as follows: we pick an available\nrepresentative colour αjinA(wj) for wjas follows:\n151. if A(wj)∩A(u)̸=∅, we let αjbe the smallest element of A(wj)∩A(u). In this case, the\nconstruction terminates, and we conclude that Fis unique by the induction hypothesis.\n2. else if there exists a h < i such that αh∈A(wj), we let h′be the smallest such hand set\nαj=αh. In this case, the construction terminates, and we conclude that Fis unique by the\ninduction hypothesis.\n3. else if |A(wj)\\ {κ2}| ≥ 1, we let αjbe the smallest element of A(wj)\\ {κ2}. We let wj+1\nbe the unique neighbour of uwith c(uwj+1) =αj. In this case, we note that any consistent\nmaximal fan chain must have ( uw1, uw 2, . . . , uw j+1) as an initial segment.\n4. Otherwise, we set αj=κ2and conclude the construction.\nThe above construction must terminate. Indeed, when j=d(u)−1 the third case cannot occur,\nasuhas at most d(u)−1 coloured edges incident to it.\nSimilarly to Lemma 12, we can construct the fan in ˜O(∆3\nmax) time.\nNow given some i-step Vizing chain F1+P1+F2+P2+. . . F i+Piand some parameter t∈Z≥0,\nwe can extend it in a consistent manner by applying Lemma 13 to the colouring Shift( c, F1+P1+\nF2+P2+. . . F i+Pi) and the final edge pti−1pti=u′v′to get a unique fan F= (u′w′\n1, u′w′\n2, . . . , u′w′\nk).\nLetPibe (κ1, κ2)-bichromatic with c(u′v′) =κ2, then we apply Lemma 13 with κ1andκ2. If we\nare in Case 1 ., we set Fi+1=FandPi+1=u′w′\nk. If we are in Case 2 ., we set Fi+1=Fand let τ1∈\nA(u′)\\ {κ1, κ2}andτ2=αkbe the smallest available colours. Note that |A(u′)| ≥2 in the shifted\ncolouring, and that κ2/∈A(u) by assumption. Then we let P′=p′\n1p′\n2. . . p′\nt′be the maximal ( τ1, τ2)-\nbichromatic path beginning at p′\n1=w′\nk. Ift < t′−1, we set Pi+1= (u′w′\nk, wkp′\n2, . . . p tpt+1), otherwise\nwe set Pi+1= (u′w′\nk, wkp′\n2, . . . p t′−1pt′). Otherwise, we are in Case 3). Again we set Fi+1=F, but\nthis time we let P′=p′\n1p′\n2. . . p′\nt′be the maximal ( κ1, κ2)-bichromatic path beginning at p′\n1=w′\nk. If\nt < t′−1, we set Pi+1= (u′w′\nk, w′\nkp2, . . . p tpt+1), otherwise we set Pi+1= (u′w′\nk, wkp′\n2, . . . p t′−1pt′).\nSimilarly to before, we can perform the above extension in ˜O(∆3\nmax+∆ max·t) time. We will say\nthat we extend the Vizing chain via Lemma 13 in a consistent manner. Note that if every extension\nin a Vizing chain F1+P1+F2+P2+. . . F i+Piis consistent, then Pj+1will never overlap Pjin\nany edges for all j. Indeed, the colours of Pj+1are either disjoint from those of Pjor identical to\nthose of Pj. In the latter case, Pj+1andPjbelong to different components in the shifted colouring.\nIn this paper, we will only be working with Vizing chains satisfying certain restrictions. The\nrestrictions we will impose are as follows:\nDefinition 14. LetV C=F1+P1+···+Fj+Pjbe a non-overlapping multi-step Vizing chain.\nWe say that V Cisconsistent if the following conditions hold:\n•F1+P1is constructed in a consistent manner (Lemma 12 and following discussion).\n•Fori≥2we have that Fi+Piis constructed in a consistent manner (Lemma 13 and following\ndiscussion).\n•center (Fj)is disjoint from the 2-hop neighbourhood of center (F1) +P1+···+center (Fj−2) +\nPj−2+center (Fj−1)for all j.\n•For all j, we have that Pjis disjoint from the 2-hop neighbourhood of center (Fk)for all\nk≤j−1and from Pkfor all k≤j−2.\n•For all j, we have that the maximal bichromatic path containing Pjunder cis different from\nthe maximal bichromatic path containing Pj+1under c.\n163.3 Path-stepping processes\nWe will now consider a looser process than the one used to construct Vizing chains. Beginning\nfrom some vertex w, we let one step be the process of first moving to a neighbour of w, say y, and\nthen following some bichromatic path Phaving yas an endpoint. As with the multi-step Vizing\nchains we are allowed to truncate the bichromatic path at any point to end up at some new vertex\nw′. We may then iteratively repeat this process at w′. If we do so itimes, we say that we run an\ni-step process starting from w1. We denote by w1+y1+ˆP1+w2+y2+ˆP2+. . . w i+yi+ˆPithe\ni-step process beginning at w, that first jumps to the neighbour y1ofw, then follows a truncated\nmaximal bichromatic path P1to end up at w2, and then jumps to the neighbour y2ofw2and\nfollows a truncated maximal bichromatic path P2and so on and so forth. We will say that the\nprocess is non-overlapping if for all k, jwe have that V(Pk)∩V(Pj)̸=∅implies that |k−j| ≤1, if\nE(Pk)∩E(Pj)̸=∅implies k=j, ifV(Pk)∩N2(wj)̸=∅fork≥jimplies that k=j, and, finally,\nif the maximal bichromatic path Qjcontaining Pjis equal to the maximal bichromatic path Qk\ncontaining Pk, then j=k. In other words, we will only allow a bichromatic path to intersect the\n2-hop-neighbourhood of the immediately preceding path, and we will not allow it to intersect the\nedge set of any previous paths. If |Pj|+ 1≤ℓfor all j, we say that the process has step length at\nmost ℓ. IfPiis a maximal bichromatic path, we say that the i-step process is augmenting .\nMaintaining information about this type of process is useful, as it allows us to accurately\nrepresent where consistent Vizing chains might go, while at the same time avoiding the need to\nworry about exactly which extensions are possible at what times. In particular, we observe that any\nconsistent and non-overlapping i-step Vizing chain F1+P1+···+Fi+Piwhere F1is centered at w\nnaturally induces an i-step process on w, by letting wj= center( Fj), letting yjbe the second point\nofPj, and letting ˆPjbe the the bichromatic sub-path of the path Pjexcluding only the first edge\nof the chain Pj. To make this observation more precise, we will define and consider semi-consistent\nprocesses. Here, we will put some further restrictions on the processes so that they model consistent\nVizing chains exactly.\nGiven an i-step process w1+y1+P1+w2+y2+P2+. . . w i+yi+Pi, we say that the extension\nwj+yj+Pjforj≥2 isconsistent with Lemma 13, if the following is the case: consider the\ncolouring c′obtained from cas follows: suppose Pj−1=p1p2. . . p tis (κ1, κ2)-bichromatic with\nc(pt−1pt) =κ1. Suppose the P=p1p2. . . p tpt+1. . . p sis a maximal bichromatic path containing\nPj−1. Note that if s=t, then Pj−1is augmenting, and it cannot be extended consistently.\nWe obtain c′by first un-colouring the edge ptpt+1i.e.c′(ptpt+1) =¬, and then setting c′(pkpk+1) =\nκ2ifc(pkpk+1) =κ1and vice versa for k < t −1 setting c′(pkpk+1) =κ1ifc(pkpk+1) =κ2. It\nis straight-forward to check that if cis a proper (∆ max+ 1)-edge-colouring, then so is c′. Now,\napply Lemma 13 to pt−1ptwith the colouring c′,u=pt−1,v=pt, and the colours κ1andκ2as\nbefore. By fixing a length t′=|Pi|, the lemma together with the discussion that follows it produces\na fan chain Fand a bichromatic-path chain Q= (q0q1, . . . , q ˆtqˆt+1) with ˆt≤t′such that F+Qis\na 1-step Vizing chain. We say the extension wj+yj+Pjforj≥2 isconsistent with Lemma 13\nifwj= center( Fj) =q0, and if yj=q1, and finally Pj=q1, q2, . . . , q t′. Ifˆt=t′−1, then Pjis\naugmenting. If ˆt < t′−1, then the extension is not consistent.\nNext, we define a semi-consistent process:\nDefinition 15. LetC=w1+y1+P1+w2+y2+P2+. . . w i+yi+Pibe a non-overlapping i-step\nprocess beginning at w1. We say that Cissemi-consistent if the following holds:\n•Forj≥2we have that wj, yjandPjis an extension consistent with Lemma 13.\n•We have that wjis disjoint from the 2-hop neighbourhood of w1+y1+P1+. . . w j−2+yj−2+\nPj−2+wj−1for all j.\n17•For all j, we have that Pjis disjoint from the 2-hop neighbourhood of wkfor all k≤j−1\nand from Pkfor all k≤j−2.\nWe observe that for any j < i the length of Pjis at least 3. Indeed, otherwise Item 2 in\nDefinition 15 fails. The following Lemma is immediate:\nLemma 16. LetGbe a graph endowed with a proper (partial) edge-colouring c. LetSCP be the\nset of semi-consistent processes on (G, c), and let NOP be the set of non-overlapping processes on\n(G, c). Then we have:\nSCP ⊂ NOP\nSemi-consistent processes are in correspondence with consistent Vizing chains in the following\nsense:\nLemma 17. There is a mapping γfrom the set of consistent i-step Vizing chains to the set of\nsemi-consistent i-step processes. In particular, if a consistent i-step Vizing chains augmenting, so\nis its image, and similarly if a semi-consistent i-step process is augmenting, so is every Vizing\nchain in its pre-image.\nProof. LetV C=F1+Q1+···+Fi+Qibe any consistent i-step Vizing chain. Then we will let\nV C=F1+Q1+···+Fi+Qicorrespond to the i-step process C=w1+y1+P1+···+wi+yi+Pi,\nwhere wj= center( Fj), ifQj= (q0q1, . . . , q tj+1), then yj=q1andPj=q1, q2. . . , q tj.\nNext we show that Cis semi-consistent. First of all, we need to check that Cis non-overlapping.\nThis follows from the fact that V Cis consistent, which puts precisely the same constraints on the\nPj’s due Definition 14 as is required for Cto be non-overlapping. The first item of Definition 15 is\nsatisfied since the second item of Definition 14 holds. This is the case since no edge outside of Pj−1\nin the two-hop neighbourhood of the center of Fjhas changed colour before Lemma 13 is applied.\nThis means that applying it directly to Pj−1yields the same result as using it to extend the Vizing\nchain. Finally, items 3 and 4 of Definition 14 are equivalent to the last two items of Definition 15.\nFurthermore, observe that solely the length of the maximal path containing Pidetermines\nwhether CandV Care augmenting and so Cis augmenting if and only if V Cis.\nThe other direction is symmetrical.\nIf a semi-consistent process is the image of a consistent i-step Vizing chain, we say that the\nprocess is consistent . Finally, we note that under the right conditions, we can use semi-consistent\nprocesses to build consistent Vizing chains:\nLemma 18. LetGbe a graph with maximum degree ∆≤∆max, and let cbe a proper (∆max+ 1)-\nedge-colouring of G. Let C=w1+y1+P1+···+wi+yi+Pibe an augmenting semi-consistent\ni-step process with step-length ≤ℓ. Let e=w1vbe an un-coloured edge. Suppose that applying\nLemma 12 and the following discussion yields a consistent 1-step Vizing chain (F+Q1)ℓsuch that\n•center (F) =w1\n•y1=xkwithF= (w1x1, w1x2, . . . w 1xk\n•P1=q1q2. . . q ℓwithQ= (q0q1, q1q2, . . . , q ℓqℓ+1)\nThen there is an augmenting consistent i-step Vizing chain on ewith the same step length as C.\n18Proof. We will construct the augmenting consistent Vizing chain on e, sayV C=F1+Q1+F2+Q2+\n···+Fi+Qias follows: we set F1=FandQ1=Q. Then we extend F1+Q1+F2+Q2+···+Fj+Qj\nby applying Lemma 13 and the following discussion making sure that the length of Fj+1+Qjis\nequal to tj+ 2 with tj=|Pj|.\nWe now observe that γ(V C) =C. Indeed, by construction every extension results in Qj=\n(q0,jq1,j, . . . , q tj+1,jqtj+2,j) such that Pj=q1,jq2,j. . . q tj+1,j. This follows from the fact that since\nCis semi-consistent, the extension performed on Pjuses a colouring c′which only differs from\nthe colouring Shift( c, F1+Q1+F2+Q2+···+Fj+Qj) outside of the 2-hop neighbourhood of\nwj. Indeed, c′and Shift( c, F1+Q1+F2+Q2+···+Fj+Qj) agree on the colours along Pjby\nconstruction, and wj+ 1 is disjoint from the two-hop neighbourhood of C=w1+y1+P1+···+\nwj−1+yj−1+Pj−1+wjalso by construction. Since Cis semi-consistent, the extension ensures\nthatQj= (q0,jq1,j, . . . , q tj+1,jqtj+2,j) such that Pj=q1,jq2,j. . . q tj+1,j.\nIt now follows that γ(V C) =C, and therefore Lemma 17 now guarantees that V Cis also\naugmenting.\n4 Dynamic t-splitters\nGiven a graph Gequipped with any (not necessarily proper) edge-colouring c, we let di(u) be\nthe degree of uin the graph induced in Gby all edges coloured with the colour i. We let ˜d=\nmax κmax vdκ(v). Given two disjoint subsets of vertices A, B⊂V(G), we let G[A, B], be the\nbipartite subgraph of Gon vertex set A∪Bcontaining exactly the edges going from a vertex in A\nto a vertex in B. We let si(v) = max {di(v)−∆max\nt,0}be the surplus ofi-coloured edges around\nv. Finally, we let L:E(G)→Zbe a function that maps uv7→sc(uv)(u) +sc(uv)(v), and we set\nmaxL(e) =˜L.\nIn this section, we will show Theorem 3, which we restate below for convenience:\nTheorem 19 (Identical to Theorem 3) .LetGbe a dynamic graph, and suppose ε >0,t∈N, and\nan upper bound ∆maxon the maximum degree of Gthrough the entire update sequence is given.\nAssume that∆max\nt≥104log2n\nε2. Then there is a dynamic algorithm that computes a t-splitter of G\nwith an amortised update-time of O(t3ε−4log3m).\nIn this entire section, we will work with two parameters ∆ max, which we assume is a known\nupper-bound on the maximum degree of some dynamic graph G, as well as a parameter η≥4.\nNote that it is important that η≥1, but we will only apply the theory below, when this is the\ncase. We will analyse a colouring csatisfying the following invariant:\nInvariant 20. For all uv∈E(G)and all i∈[t], we have that\nL(uv) =sc(uv)(u) +sc(uv)(v)≤η+si(u) +si(v)\nWe first show that if 1 ≤η≤ε2\n128 log n∆max\nt, then ˜d≤(1 +ε)∆max\nt. We note already now that\nwe will only apply the above Theorem, when∆max\nt= Ω(poly( ε−1,logn)), so the assumption that\nη≥1 is not dangerous.\n4.1 Analysing the Local Condition\nWe first show the following structural theorem:\nTheorem 21. Let0< ε≤1\n2and let Gbe equipped with a t-colouring csatisfying Invariant 20.\nLet1≤η≤ε2\n128 log n∆max\nt. Then it holds that ˜d≤(1 +ε)∆max\nt.\n19Before we give this proof, we first show some helpful lemmas. The first lemma shows that if an\nedge ehas a large surplus, then any endpoint of ehas a large surplus in some colour.\nLemma 22. Suppose Invariant 20 holds. Suppose furthermore that e=uvhasL(uv) =D. Then\nthere exists a colour κsuch that sκ(u)≥D−η.\nProof. Consider v. Since d(v)≤∆max, it follows by the pigeon-hole principle that there exists some\ncolour κ′for which dκ′(v)≤∆\nt. Since we assumed Invariant 20 holds, we have that\nD≤η+sκ′(u) +sκ′(v) =η+sκ′(u)\nHence, we deduce that D−η≤sκ′(u), and we are done by setting κ=κ′.\nWe let ˜L:= max e∈E(G)L(e). The next lemma shows how to inductively determine a large set\nof bipartite subgraphs containing vertices with large surpluses.\nLemma 23. Suppose Invariant 20 holds, and let µ∈Zbe a parameter. Then there exists two\ncolours κxandκy, as well as two family of sets (Xi)µ\ni=1and(Yi)µ\ni=1such that for all i, we have\nXi, Yi⊂V(G)and that the following two conditions hold:\n∀u∈Xi:sκx(u)−sκy(u)≥˜L−(1 + 2( i−1))η (1)\n∀v∈Yi:sκy(v)−sκx(v)≥˜L−2iη (2)\nProof. We first determine κxandκy. Then we construct the sets inductively. Pick an arbitrary\nedge uvachieving L(uv) = max L(e) =˜L. Now we may apply Lemma 22 to see that there exists a\ncolour κxsuch that dκx(u)≥∆max\nt+˜L−η. By the pigeon-hole principle there is some colour κy\nincident to usuch that dκy(u)≤∆max\nt.\nWith κxandκydefined, we construct the family of sets inductively. Initially, we set X1={u}\nandY1=NG[κx](u)− {u}. We confirm that the required conditions hold: (1) is clear from our\nchoice of κxandκy. We get (2) by observing that by Invariant 20, we find that for any v∈Y1, we\nhave:\nsκy(v) +sκy(u) +η≥sκx(v) +sκx(u)\nHence:\nsκy(v)−sκx(v)≥sκx(u)−sκy(u)−η\n≥˜L−2η\nby above.\nFor the inductive step, we now set X2=NG[κy](Y1)−Y1, and Y2=NG[κx](X2)−X2. Again we\nconfirm the two conditions:\n•(1): Consider u∈Xi. We have ∀v∈Yi−1thatsκy(v)−sκx(v)≥˜L−2(i−1)ηby the induction\nhypothesis. By applying Invariant 20, we find that\nsκx(v) +sκx(u) +η≥sκy(v) +sκy(u)\nsince c(uv) =κy. Hence:\nsκx(u)−sκy(u)≥sκx(v)−sκy(v)−η\n≥˜L−(1 + 2( i−1))η\nas before.\n20•(2) is symmetric: consider any v∈Yi. We have ∀u∈Xithatsκx(u)−sκy(u)≥˜L−(1+ 2( i−\n1))ηby induction. By applying Invariant 20, we find that\nsκy(v) +sκy(u) +η≥sκx(v) +sκx(u)\nsince c(uv) =κx. Hence:\nsκy(v)−sκx(v)≥sκx(u)−sκy(u)−η\n≥˜L−2iη\nas before. Note that the above argument can be repeated ad infinitum, and so in particular\nalsoµtimes.\nWe are now ready to show Theorem 21:\nProof of Theorem 21. Suppose for contradiction that ˜d >(1 + ε)∆\nt. We will show that this im-\nplies the existence of a family of bipartite subgraphs {Bj}10 log( n)ε−1\nj=1 ={(Pj, Qj))}10 log( n)ε−1\nj=1 ofG\nsatisfying that |Qj| ≥(1 +ε\n4)j.\nThis implies that for some j≤10 log( n)ε−1, we have that\n|Qj| ≥(1 +ε\n4)10 log( n)ε−1≥e10\n8logn> n\nwhere we used the standard in-equality 1 −x≤e−xwhich holds for all real x. In particular, it\nfollows that for all x≤1/2, we have that ex≤1\n1−x≤1 +x\n1−x≤1 + 2 x. This contradicts the fact\nthat we also must have Bj⊂Gand so |V(Bj)| ≤ |V(G)|=n.\nIn order to construct {Bj}10 log( n)ε−1\nj=1 , we invoke Lemma 23 with µ=⌈10 log( n)ε−1⌉to get\nsets ( Xi)µ\ni=1and ( Yi)µ\ni=1and colours κxandκysatisfying the condition of the lemma. Then for\nappropriate values of i, we set\nB2i−1= (G[κx])[Xi, Yi]\nwith P2i−1=XiandQ2i−1=Yias well as\nB2i= (G[κy])[Yi, Xi+1]\nwith P2i=YiandQ2i=Xi+1These graphs are clearly bipartite by construction, so all we need to\nshow is that Qimust expand. We show this by induction on i. The base case is is as follows: since\nwe have tacitly assumed that η≥1 it follows that ˜L−ηis large, say at least 2. In particular, we\nhave that |Q1|=|Y1| ≥˜L−η≥2.\nInstead of showing that Q2is also sufficiently large, we proceed to the induction step directly.\nWe will first show that if Q2i−1≥(1 +ε\n4)2i−1, then Q2i≥(1 +ε\n4)2i. Then we show that if\nQ2i≥(1 +ε\n4)i, then Q2i+1≥(1 +ε\n4)2i+1. In this way, we only need to show the base case\n|Q1|=|Y1| ≥˜L−η≥2, as we have already done.\nSo for the inductive step, we first consider B2i. We note that, by construction, we have that\n∀u∈Xi+1\nsκx(u)−sκy(u)≥˜L−(1 + 2 i)η\nsimilarly for all v∈Yiwe have\nsκy(v)−sκx(v)≥˜L−2iη\n21This implies that every vertex v∈Yihassκy(v)≥˜L−2iηand that every vertex u∈Xi+1has\nsκy(u)≤(1 + 2 i)η. Since the graph is bipartite, we have that\nX\nu∈Xi+1dκy(u)≥X\nv∈Yidκy(v)\nApplying the above, we find that\n((1 + 2 i)η+∆\nt)|Xi+1| ≥X\nu∈Xi+1dκy(u)≥X\nv∈Yidκy(v)≥(˜L−2iη+∆\nt)|Yi|\nSince ˜d≤∆\nt+˜Lby definition and since η22 log( n)\nε≤ε∆\n4tby choice of η, we have that\n˜L+∆\nt−η22 log( n)\nε≥˜d−η22 log( n)\nε≥(1 +3ε\n4)∆\nt\nand that\n∆\nt+η22 log( n)\nε≤(1 +ε\n4)∆\nt\nHence we can write\n(1 +ε\n4)∆\nt|Xi+1| ≥(1 +3ε\n4)∆\nt|Yi|\nTherefore, we find that\n|Xi+1| ≥(1 +3ε\n4)\n(1 +ε\n4)|Yi|\n≥(1−ε\n4)(1 +3ε\n4)|Yi|\n≥(1 +3ε\n4−2ε\n4))|Yi|\n≥(1 +ε\n4)|Yi|\nsince Q2i=Xi+1andQ2i−1=Yi, applying the induction hypothesis yields the first case. The\nsecond case is symmetric, but we include it for completeness.\nNow consider B2i+1. We note that, by construction, we have that ∀u∈Xi+1\nsκx(u)−sκy(u)≥˜L−(1 + 2 i)η\nsimilarly for all v∈Yi+1we have\nsκy(v)−sκx(v)≥˜L−2(i+ 1)η\nThis implies that every vertex v∈Yihassκx(v)≤2(i+ 1)ηand that every vertex u∈Xi+1has\nsκx(u)≥˜L−(1 + 2 i)η. Since the graph is bipartite, we have that\nX\nv∈Yi+1dκx(v)≥X\nu∈Xi+1dκx(u)\nApplying the above, we find that\n(2(i+ 1)η+∆\nt)|Yi+1| ≥X\nu∈Xi+1dκy(u)≥X\nv∈Yidκy(v)≥(˜L−(1 + 2 i)η+∆\nt)|Xi+1|\n22Since ˜d≤∆\nt+˜Lby definition and since η22 log( n)\nε≤ε∆\n4tby choice of η, we have similarly to before\nthat\n˜L+∆\nt−η22 log( n)\nε≥˜d−η22 log( n)\nε≥(1 +3ε\n4)∆\nt\nand that\n∆\nt+η22 log( n)\nε≤(1 +ε\n4)∆\nt\nSo we can write\n(1 +ε\n4)∆\nt|Yi+1| ≥(1 +3ε\n4)∆\nt|Xi+1|\nTherefore, we find, as before, that\n|Yi+1| ≥(1 +ε\n4)|Xi+1|\nsince Q2i+1=Yi+1andQ2i=Xi+1, applying the induction hypothesis thus yields the second\ncase.\nNote that Theorem 21 shows that any colouring satisfying the local condition is a t-degree\nsplitting colouring.\n4.2 Dynamically Maintaining Invariant 20\nAgain, we assume we have access to some upper bound ∆ maxon the maximum degree over the\nentire update sequence, and some input parameter t. We further require that∆max\nt≥104log2n\nε2. We\nwill set η=⌊ε2\n128 log n⌋∆max\nt, and note that η≥10 always due to our assumptions.\nWe will maintain the following data structures. At each vertex u, we will store:\n•The surplus for each colour in an index-able array SP(u).\n•For each colour κ, we store a BBST, MSD κ(u), containing all estimated surplus differences\nof the form:\n˜sc(wx)(w) + ˜sc(wx)(x)−(˜sκ(w) + ˜sκ(x))\n•For each colour κ, we store a doubly-linked-list LL(u, κ) containing all edges coloured κ\nincident to u.\n•For each colour κ, we store a counter ι(u, κ)∈[dk] and a pointer to the ι(u, κ)thelement of\nLL(u, κ).\nFor each edge e, we store a BBST over the estimated surpluses for each colour: Ts\ne.\nThe algorithm now works as follows (see below for pseudo-code). Whenever an edge e=uvis\ninserted, we calculate the surplus sκ(u) +sκ(v) for all choices of κ, and assign ean arbitrary colour\nκ′with minimum surplus. We also initialize all of e’s data structures, and update its contributions\ntouandv’s data structures. Then, we insert einto the linked-lists LL(u, κ′) and LL(v, κ′), ensuring\nthateis inserted just before position ι(u, κ′) inLL(u, κ′) resp. ι(v, κ′) inLL(v, κ′). Then we update\ndata structures for other edges faround w∈ {u, v}in a round-robin fashion: starting with the edge\nstored at position ι(w, κ′) inLL(w, κ′), we recompute the contents of the next ⌈500∆max\nη⌉edges in\nLL(w, κ′) and update ι(w, κ′) =ι(w, κ′) +⌈500∆max\nη⌉mod dκ′(w). Then, we check for all colours\nif any violations have arisen. If so, we push the endpoint and the colour with too large a surplus\ndifference, unto a queue Q. After this, we process Qone pair at the time by recolouring with a\n23similar procedure as above, possibly pushing new violating pairs unto Q. Note that to get the\nedge to recolour, we extract the observed minimum and check if its surplus difference is still large\nenough. If yes, we process it by recolouring it, if no, we conclude our processing of this element of\nQ. Deletions are symmetric. Here, we have to remember to remove efrom any doubly-linked-lists\ncontaining it, and to re-adjust the relevant ι(·,·) by decrementing them by 1, if e’s position in\nLL(·,·) was before the element pointed to by ι(·,·).\nNote that these running times can probably be optimised with respect to t, but since any such\noptimisations are likely to only change constant factors hidden by big-Oh notation anyways in the\nfinal edge-colouring algorithm, we have not attempted them here. Such optimisation, however,\nwould yield a better t-splitter.\nWe update Ts\nebe calculating the surplus for each colour of eand inserting it into a search\ntree. We then update (deleting the old, and inserting the new) the surplus difference for einto\nMSD κ(w) for all colours κand all endpoints wofe. These data structures always contain correct\nestimates immediately after being updated. Hereafter they might slowly deteriorate in quality, until\nthey are updated again. We will later show that this deterioration is small enough to be handled\nefficiently. We will denote by ˜ sκ(u) + ˜sκ(v) the estimate of a surplus stored in a data structure\nand by sκ(u) +sκ(v) the actual value of a surplus. In particular, we will denote by ∆˜ sκ(w) the\nmaximum estimated surplus difference stored in MSD κ(w):\n∆˜sκ(w) = max\nx∈N(w)˜sc(wx)(w) + ˜sc(wx)(x)−(˜sκ(w) + ˜sκ(x))\nNote that this value can be found by querying MSD κ(w) and returning the minimum.\nAlgorithm 1: Insert (uv)\nadd(uv) toG\nInitialize-DS (uv)\nQ←ReColour (uv,∅)\nwhile Q̸=∅do\n(w, κ)←pop(Q)\nif∆˜sκ(w)>η\n2then\ne←edge maximising the above\nexpression.\nQ←ReColour (e, Q)Algorithm 2: Delete (− →uv)\nRemove (uv) from G.\nUpdate uv’s influence on relevant LL(·,·),\nι(·,·),MSD ·(·)\nRound-Robin (uv)\nfor all κ∈[t] and w∈ {u, v}do\nif∆˜sκ(w)>η\n2then\nPush((w, κ)) to Q\nwhile Q̸=∅do\n(w, κ)←pop(Q)\nif∆˜s(w)>η\n2then\ne←edge maximising the above\nexpression.\nQ←ReColour (e, Q)\n24Algorithm 3: ReColour (uv, Q )\nifc(uv) =∅then\nc(uv)←ExtractMinColour (Ts\nuv)\nUpdate uv’s influence on relevant LL(·,·)\nandι(·,·)\nUpdate-DS (uv)\nelse\nc(uv)←ExtractMinColour (Ts\nuv)\nUpdate uv’s influence on relevant LL(·,·),\nι(·,·),MSD ·(·)\nUpdate-DS (uv)\nRound-Robin (uv)\nfor all κ∈[t] and w∈ {u, v}do\nif∆˜sκ(w)>η\n2then\nPush((w, κ)) to Q\nReturn QAlgorithm 4: Round-Robin (uv)\nfor all κ∈[t] and w∈ {u, v}do\nfor all j∈[⌈500∆ max\nη⌉]do\ne←LL(w, κ)[ι(w, κ) + (j−1)\nmod dκ(w)]\nUpdate-DS (e)\nι(w, κ)←ι(w, κ) +⌈500∆ max\nη⌉mod dκ(w)\nUpdate pointer associated with ι(w, κ)\nAlgorithm 5: Initialize-DS (uv)\nifd(w) = 1 for w∈ {u, v}then\nInitialize DS for w\nInitialize BBST Ts\nuv.\nfor all κ∈[t]do\ninsert (sκ(u) +sκ(v)) into Ts\nuv\nUpdate uv’s contribution to MSD κ(u) and\nMSD κ(v).Algorithm 6: Update-DS (uv)\nUpdate SP(u) and SP(v)\nfor all κ∈[t] and w∈ {u, v}do\nUpdate (sκ(u) +sκ(v)) in Ts\nuv\nfor all κ∈[t] and w∈ {u, v}do\nUpdate uv’s contribution to MSD κ(w)\n4.3 Analysing the Dynamic Algorithm\nIn this subsection, we will analyse the dynamic algorithm above. We begin by analysing how precise\nthe estimates ˜ sκ(u) + ˜sκ(v) in the relevant data structures are. Then we show that the algorithm\nactually maintains Invariant 20, thus establishing correctness via Theorem 21. Then we analyse\nthe recourse of the algorithm, before we pick explicit implementations of the data structures and\nanalyse the final running time.\n4.3.1 Analysis of Round-Robin\nIn this section, we will analyse the round-robin scheme applied by the algorithm. In particular, we\nwill show that the algorithm maintains Invariant 20. We will do this by showing that even-though\nthe algorithm sometimes works with (slightly) out-dated information, the information is still precise\nenough for the algorithm to detect necessary changes and perform them efficiently.\nTo this end, we first show that the data structures used by the algorithm contain good estimates.\nBefore doing this, we will introduce some notation. For any data structure D, we will let ( D)τbe\nthe state of the data structure after the ( τ−1)thcall to ReColour concluded, and just before the\nτthcall to ReColour commences. Similarly, for any value p(x) we will let (˜ p(x))D,τdenote the\nestimate of this value stored in data-structure Dat time τ, and similarly we will let ( p(x))τdenote\nthe actual value of the parameter at time τ. If the data-structure is clear from the context, we will\nleave out this argument to ease readability.\n25Lemma 24. At all times τ, we have that:\n1. For any edge e=uvand any colour κ, the estimates (˜S)τ:= (˜sκ(u) + ˜sκ(v))τinTs\nuvsatisfy\nthat:\n|(˜S)τ−((sκ(u))τ+ (sκ(v))τ)| ≤η\n16\n2. For any vertex u, any neighbour v, and any colour κ, the estimates\n(˜E(u, uv, κ ))τ= ((˜sc(uv)(u)) + (˜ sc(uv)(v)))−((˜sκ(u)) + (˜ sκ(v)))τ\nstored in MSD κ(u)satisfy that:\n|(E(u, uv, κ ))τ−(˜E(u, uv, κ ))τ| ≤η\n8\nwhere\n(E(u, uv, κ ))τ= ((sc(uv)(u)) + ( sc(uv)(v)))−((sκ(u)) + ( sκ(v)))τ\nProof. We will show item 1) first. Let τ′< τbe the last time before time τ, where ˜Swas updated.\nWe then have that\n(˜S)τ= (˜S)τ′= (˜sκ(u) + ˜sκ(v))τ′= ((sκ(u))τ′+ (sκ(v))τ′\nand so it is sufficient to estimate differences of the form |sκ(w)τ−sκ(w))τ′|. To this end, we note\nthat due to the round-robin scheme being run every time an edge incident to wchanges its colour,\nwe can change sκ(w) by at most\n∆max\n⌈500∆ max\nη⌉≤η\n500\nbefore it is updated again. As a result, we obtain that |(sκ(w))τ−(sκ(w))τ′| ≤η\n32, and so the\nabove observations yield:\n|(˜S)τ−((sκ(u))τ+ (sκ(v))τ)| ≤ |((sκ(u))τ+ (sκ(v))τ)−((sκ(u))τ+ (sκ(v))τ)|\n≤ |(sκ(u))τ−(sκ(u))τ|+|(sκ(v))τ−(sκ(v))τ|\n≤η\n16\nFor the second item, first observe that every time MSD κ(u) is updated, it is updated with a\ndifference of entrances from some BBST Ts\nux. This means that if c(uv) =κ′, then for all κ, we\nactually have that:\n(˜E(u, uv, κ ))τ= (˜sκ′(u) + ˜sκ′(v))τ−(˜sκ(u) + ˜sκ(v))τ\nBy construction, we also have that:\n(E(u, uv, κ ))τ= ((sκ′(u))τ+ (sκ′(y))τ)−((sκ(u))τ+ (sκ(y))τ)\nThus we can apply 1 .to conclude that:\n|(E(u, uv, κ )τ−(˜E(u, uv, κ ))τ| ≤ |(˜sκ′(u) + ˜sκ′(v))τ−((sκ′(u))τ+ (sκ′(v))τ)|\n+|((sκ(u))τ+ (sκ(v))τ)−(˜sκ(u) + ˜sκ(v))τ|\n≤η\n16+η\n16=η\n8\n26Next we will argue that this information is accurate enough to maintain Invariant 20, provided\nthat the algorithm terminates. Afterwards, we show that the algorithm always terminates and in\nfact does so with small amortised recourse.\nLemma 25. LetGbe a dynamic graph, and let cibe the colouring in Gimaintained by the algorithm\nafter iupdates. Assume that the algorithm terminates after each update and that∆max\nt≥104log2n\nε2.\nThen cisatisfies Invariant 20 for all i.\nProof. It suffices to show that after every update, we must have ∆ sκ(w)< η for all vertices w\nand all colours κ. We first note that by Lemma 24, it follows that if ∆˜ sκ(w)≤η\n2, then we have\n∆sκ(w)≤∆˜sκ(w) +η\n8< η.\nHence, if we can show that after the algorithm is done processing any update, we must have\n∆˜sκ(w)≤η\n2for all vertices w∈Vand all colours κ∈[t], then we are done. To see this, we\napply induction on the number of updates processed. The base case is when no updates have been\nprocessed yet in which case what we claim clearly is true. Next assume that the above holds after\niupdates have been processed, and consider the state of the data structures just after the ( i+ 1)th\nupdate concluded.\nSince the round-robin scheme, which updates the values stored in MSD κ(w), is only called on\nvertices weither directly incident to edges that were inserted, deleted or recoloured or vertices\nneighbouring such vertices, the desired statement holds for any other vertex by induction. In\nparticular, MSD κ(w) is only changed after a call to Initialize-DS or a call to Update-DS . The\nlatter only happens during a call to Round-Robin (uv).\nWe first note that we need only worry about vertices directly incident to inserted, deleted or\nrecoloured edges, and not neighbours of such vertices. Indeed, we always update an edge uv’s\ncontribution to MSD κ(w) for w∈ {u, v}for both endpoints to the same value. This means\nthat after a call to Update-DS (uv) or Initialize-DS terminates, we always have ˜E(u, uv, κ ) =\n˜E(v, uv, κ ) i.e. for all τwe have ( ˜E(u, uv, κ ))τ= (˜E(v, uv, κ ))τ. Hence, it is sufficient to verify that\nany endpoint wdirectly incident to an insertion, deletion or recolouring satisfies ∆˜ sκ(w)≤η\n2after\nthe update procedure ends.\nTo this end, we note that after a call to Initialize-DS the degree of wis exactly one, and so\nsince η≥10, this call can never cause ∆˜ sκ(w)>η\n2.\nAfter every Round-Robin (uv) call, any vertex wwith ∆˜ sκ(w)>η\n2is immediately added to a\nqueue Q. The vertex wcan only leave Qin two ways: either because it is verified at some time step\nτthat (∆˜ sκ(w))τ≤η\n2or because some edge incident to wis recoloured. In the first case, wcan\nonly enter Qagain if some edge incident to it is recoloured at some time-step τ′> τin which case\nthe exact same argument can be applied again. If no such future update occurs, we immediately\nhave that ∆˜ sκ(w)≤η\n2after the update procedure ends. In the latter case, it is checked after the\nrecolouring whether ∆˜ sκ(w)>η\n2and if so, then wis added back to Qbut at a later time-step. If\n∆˜sκ(w)≤η\n2, then wimmediately leaves Qand similar arguments to the first case applies.\nNext, we show that the algorithm actually terminates and analyse its recourse.\n4.3.2 Analysing the Recourse\nIn this sub-section, we analyse the recourse of the algorithm and show that the it terminates. We\nfirst show that any edge re-coloured by the algorithm results in a significant lowering of the surplus\nof this edge. This allows us to show that a certain global potential of the graph drops enough to\nensure that the amortised number of recolourings become bounded.\nTo this end, we first argue that any recolouring performed by the algorithm lowers the surplus\nof an edge significantly:\n27Lemma 26. Lete=uvby any edge recoloured by the algorithm during time-step τ. Suppose that\n(c(uv))τ−1=κ1and that (c(uv))τ=κ2i.e. the colour of the edge is changed from κ1toκ2. Then\nat time τ−1, we have that\nδuv\nκ1→κ2:= ((sκ1(u))τ−1+ (sκ1(v))τ−1)−((sκ2(u))τ−1+ (sκ2(v))τ−1)≥η\n4\nProof. Anyκ-coloured edge incident to uorvis only recoloured during time-step τif ∆˜sκ(u)≥η\n2\nor ∆˜sκ(v)≥η\n2. Since ewas chosen as an edge maximising the surplus difference, it must be the\ncase that\n˜δuv\nκ1→κ2:= ((˜sκ1(u) + ˜sκ1(v))−(˜sκ2(u) + ˜sκ2(v)))τ−1\n= (˜sκ1(u) + ˜sκ1(v))τ−1−(˜sκ2(u) + ˜sκ2(v))τ−1\n≥η\n2\nwhere the second equality follows from the fact that MSD κ(u) always is updated by a difference\nof surpluses stored in some BBST Ts(uv). Hence, it follows that:\nδuv\nκ1→κ2+ (˜sκ1(u) + ˜sκ1(v))τ−1−((sκ1(u))τ−1+ (sκ1(v))τ−1)−((˜sκ2(u) + ˜sκ2(v))τ−1\n−((sκ2(u))τ−1+ (sκ2(v))τ−1))≥˜δuv\nκ1→κ2≥η\n2\nConsequently, we can apply Lemma 24 part 1) in order to observe that:\nδuv\nκ1→κ2≥˜δuv\nκ1→κ2− |(˜sκ1(u) + ˜sκ1(v))τ−1−((sκ1(u))τ−1+ (sκ1(v))τ−1)|−\n|(˜sκ2(u) + ˜sκ2(v))τ−1−((sκ2(u))τ−1+ (sκ2(v))τ−1)|\n≥η\n2−2η\n16≥η\n4\nas claimed.\nThe idea is now to consider the following potential:\nΦ(G, c) =tX\ni=1X\nv∈V(G)si(v)X\nj=1j\nAs we will show below, Lemma 26 implies that every recolouring performed by the algorithm will\nlower the potential substantially. In particular, this ensures that the amortised recourse of the\ndynamic algorithm is low. This is shown in the following Lemma:\nLemma 27. Let˜d= max cmax v˜s(v)be the maximum surplus of any colouring satisfying In-\nvariant 20 with 0< ε < 1. Then the amortised recourse of the algorithm is upper bounded by\n16˜d\nη≤210logn\nε2=O(logn\nε2)provided that η=⌊ε2\n128 log n⌋∆max\nt≥16and∆max\nt≥104log2n\nε2.\nIn particular, under these assumptions,the algorithm always terminates.\nProof. We first note that after inserting and colouring any edge e=uv, the potential can increase\nby at most 2( ˜d+ 1), since between insertions the algorithm guarantees that Invariant 20 holds,\nprovided that the algorithm terminates, by Lemma 25. Note that a deletion will always lower the\npotential.\nThe second observation is that each recolouring will lower the potential by Ω( η). Indeed, if e\nis recoloured from having colour κ1to having colour κ2during time-step τ, then it follows from\nLemma 26 that:\nδuv\nκ1→κ2= ((sκ1(u))τ−1+ (sκ1(v))τ−1)−((sκ2(u))τ−1+ (sκ2(v))τ−1)≥η\n4\n28Notice that if the recolouring performed in step τtransforms ( G, c) to ( G, c′), then we have that:\nΦ(G, c)−Φ(G, c′) =δuv\nκ1→κ2−2≥η\n4−2≥η\n8\nIndeed, the above holds since recolouring ereduces sκ1(u) and sκ1(v) by 1, and hence Φ by\n(sκ1(u))τ−1+ (sκ1(v))τ−1. On the other hand, sκ2(u) and sκ2(v) are both increased by one, which\nincreases the potential by ( sκ2(u)))τ−1+ (sκ2(v)))τ−1+ 2.\nSince Φ is always non-negative, since deletions do not lower Φ, and since insertions increase Φ\nby at most 2( ˜d+ 1), we find that we can witness at most:\nr·2(˜d+ 1)\nη\n8= 16r·˜d\nη\nrecolourings over a sequence of updates containing rinsertions. Consequently, the algorithm always\nterminates and it does so with recourse bounded by O(˜d\nη). To recover the last part of the Lemma,\nnote that Theorem 21 gives that ˜d≤(1 +ε)∆max\ntand that η=⌊ε2\n128 log n⌋∆max\nt.\nWe briefly remark that it is quite important that η≥16. This ensures that any recolouring\nactually lowers the potential, and that one does not end up in a situation where the algorithm\ncontinually recolours the same edges. In the next section, we will analyse the update time as a\nfunction of the number of recolourings i.e. the recourse performed by the algorithm. We will then\nuse Lemma 27 to upper bound the total update time.\n4.3.3 Analysing the Update Time\nWe begin by implementing the basic procedures in the following lemma. The proof is straight-\nforward, but we include a proof for completeness.\nLemma 28. For vertex v, we can support the following operations with deterministic and worst-\ncase guarantees as listed below:\n•For all edges uvincident to v, we can perform Initialize-DS (uv)inO(tlogm)time.\n•For all edges uvincident to v, we can perform Update-DS (uv)inO(tlogm)time.\n•For all edges uvincident to v, we can perform Round-Robin (uv)inO(t3ε−2log2m)time.\n•Not counting the recursive call, for all edges e, we can perform one call to ReColour (e)in\nO(t3ε−2log2m)time.\n•We can extract an edge (and the value) maximising ∆˜sκ(v)inO(logm)time.\n•We can perform ExtractMinColour (Ts\nuv)inO(logm)time.\nProof. We will only require our BBST’s to support very basic operations, so any standard BBST\nsuch as red-black trees [CLRS22] will do.\nFor the first item, we note that initializing MSD κ(w) for any κ∈[t] and any w∈V(G) takes\nconstant time since MSD κis initialised as an empty BBST. Initialising the doubly linked-lists over\nneighbouring edges can also be done trivially in O(1) time for each κ. Since, we have to do the\nabove for tchoices of κand up to 2 choices of w, this part takes O(t) time. We can initialize Ts\nuv\ninO(1)-time and insert the tsurplus sums into Ts\nuvinO(tlogt) time. Finally, we can insert the\n29surplus differences into the relevant MSD ·(·)’s in O(t·logm) time, as MSD κ(w) contains O(m)\nelements for all κ, w.\nFor the second item, we note that each update to Ts\nuvtakes O(logt) time, since we can compute\nsκ(u) +sκ(v) in constant time. Each update to MSD κ(v) can also be performed in O(logn) time.\nIndeed, we can compute ( sc(uv)(u) +sc(uv)(v))−(sκ(u) +sκ(v)) in O(1) time, and insert it in\nO(logm) time. Similarly, by storing a pointer to the element at w, we can also remove the old\nentrance in O(logm) time. All of the above has to be performed ttimes, leaving us with the stated\nupdate time.\nFor the third item, we note that we perform 2 t·⌈500∆\nη⌉calls to Update-DS (e), which dominates\nthe update time. Indeed, we can index into the doubly-linked-list in O(1) time, provided that we\nhave stored a pointer to the element, we wish to index, as well as its position in the linked list.\nApplying the second item therefore yields the running time.\nFor the fourth item, we note that we can perform ExtractMinColour (Ts\nuv) in log ttime by\nsimply computing the minimum in the BBST. Hence, if we disregard further recursive call, the\nupdate time is dominated by the call to Round-Robin (uv), so the statement follows from the third\nitem.\nFor the fifth item, we note that by storing a pointer from each entrance in MSD κ(v) to the\nedge responsible for the entrance, we can, given access to an entrance, locate the corresponding\nedge in O(1) time. As such, we only need to compute the maximum entrance in MSD κ(v) which\ncan be done in O(logm) time.\nFor the sixth and final item, we note that extracting the maximum entrance in a BBST can be\nperformed in O(logn) time by a standard search always going to the largest child.\nEverytime we call ReColour on some edge, we perform a recolouring. Hence, by Lemma 27,\nafter iinsertions, we have performed at most 16 ·i·˜d\nηcalls to ReColour . Thus, in an amortised\nsense, the number of calls to ReColour isO(˜d\nη) per update. If we ignore the recursive calls, then\nby Lemma 28 the time spent for each call is dominated by the call to Round-Robin , and so each\ncall to Recolour takes only O(t3ε−2log2m) time. Hence, we have shown the following bound on\nthe amortised running time of insert anddelete .\nLemma 29. The amortised running time of insert anddelete is upper-bounded by O(t3ε−4log3m).\nCombining Lemma 29, Lemma 27, and Lemma 25 now gives Theorem 3.\n5 Maintaining a hierarchy of t-splitters\nIn this section, we will show how to build a hierarchy of dynamic t-splitters in order to split the\ndegree all the way down to O(poly( ε−1,logn)) while retaining an update time of 2˜Ologε−1(√logn).\nThus we will show Theorem 4, which we restate below:\nTheorem 30 (Identical to Theorem 4) .LetGbe a dynamic graph, and let 1> ε > 0and an\nupper bound ∆max≥107log5n\nε3 on the maximum degree of Gthrough the entire update sequence be\ngiven. Set t1=⌊210√logn⌋andt2=⌊logn\nε⌋. Then there exists a dynamic algorithm maintaining\nparameters h, i, j withh=i+j≤4√lognand a set of graphs Ghsatisfying that:\n1.|Gi+j| ≤ti\n1tj\n2\n2.ˆ∆i+j≤(1 +ε\n16)∆max\nti\n1tj\n2\n30The algorithm has amortised recourse and update time both in 2˜Ologε−1(√logn).\nIn this section, we will again assume we are given an upper bound ∆ maxon ∆. We first give a\nformal description of the hierarchy and an algorithm that maintains it.\n5.1 The hierarchy\nWe first describe the hierarchy in words. The hierarchy will consist of two types of t-splitters. The\nfirst type of t-splitter, we will use, has t1=⌊210√logn⌋, and the second type has t2=⌊logn\nε⌋. Given\nany graph G, any ∆ max, and any 0 < ε < 1, we will select some h. The hierarchy then consists of\na sequence of sets of graphs {Gi}h\ni=0and a sequence of values {ˆ∆i}h\ni=0such that for all 0 ≤i≤h\nand all H∈ Gi, we have that max v∈V(H)d(v)≤ˆ∆ii.e. the sequence of ˆ∆iprovides a sequence of\nupper bounds on the max degrees.\nWe note already here that since the algorithm from Theorem 3 is nothistory independent and so\n’running’ it on a static graph is ill-defined. We will ignore this in the following informal description,\nand then provide a proper definition afterwards. We will initialise G0={G}andˆ∆0= ∆ max, and\nproceed as follows: given Giand ˆ∆i, we let Gi+1be the set of graphs obtained in the following\nmanner. If ∆ max≥210i√logn, we will run the algorithm from Theorem 3 with the parameter\nη=ε′2\nlognandt=t1on all H∈ Gi. For each H, every colour-class κinduces a subgraph HκofH.\nFor each choice of Handκ, we include HκinGi+1after removing all degree 0 vertices. Finally,\nby Theorem 3, we can choose ˆ∆i+1= (1 + η)ˆ∆i\nt. Otherwise we set t=t2and repeat the above\nconstruction. We will pick hsuch that the construction terminates with ˆ∆h=O(poly( ε−1,logn)),\nbut still large enough for any application of Theorem 3 to go through.\nThe idea is that the edge sets of the graphs in Ghform a partition of the edges of Gsuch that\neach partition induces a low-degree subgraph of G. Since the t-splitters used are fully-dynamic,\nwe can maintain the above hierarchy dynamically as well (see Algorithms 7 8 9). To do so, we\nsimply build the t-splitters as specified, and whenever an edge is inserted, deleted or recoloured, we\nremove it from the graphs it previously belonged to (if any) and instead insert it into the graphs\nspecified by its new colour (if any). The order of operations does not matter too much, as long\nas we guarantee that any t-splitter algorithm that is called is allowed to finish and that the ˆ∆i’s\nremain valid upper-bounds through-out the update sequence. In Algorithms 7 8 9 we will explicitly\nfix the order of operations.\nNext, we introduce some basic notation in order to make the above description formal. First\nof all, we define a ( G,∆max, ε)-valid-hierarchy. This is any hierarchy obtained by splitting via\nany of t-splitter satisifying Invariant 20, and not necessarily only via the dynamic t-splitter from\nTheorem 3. This deals with the technicality that the data structure from Theorem 3 might not\nbe history-independent, and so we cannot dynamically hope to maintain a very strictly defined\nhierarchy.\nDefinition 31. Given G,∆max, and ε. A (G,∆max, ε)-valid-hierarchy of depth his a sequence\nof sets of graphs {Gi}h\ni=0, a sequence of sets of colourings {Ci}h\ni=0, and two sequences of integers\n{ˆ∆i}h\ni=0and{t(i)}h−1\ni=0such that:\n•G0={G}andˆ∆0= ∆ max.\n•For all i: if∆max≥ ⌊210√logn⌋⌊2√logn⌋i−1then t(i) =t1, else t(i) =t2.\n•For each H∈ Githere is a corresponding t(i)-splitting colouring cH∈ Ci.\n•Gi+1:={ˆH:∃H∈ Gi, κ∈[t′]withH[κ] =ˆHunder cH}.\n31•For all i:ˆ∆i+1≤(1 +ε\n128 log n)ˆ∆i\nt′\n•For all i, allH∈ Gi, and all v∈V(H):dH(v)≤ˆ∆i\nRemark 32. We remark that the choice of the value 2(10+( i−1)) log nis not extremely precise. One\ncould have equally well chosen 2(9+(i−1)) log nor20·2(10+( i−1)) log n. One then need only change the\nconstant in Algorithms 7 9 8, in Definition 31, and in the subsequent proofs mutatis mutandi. In\nparticular, one can always discretize the values and slightly over- or under-estimate all parameters\nlike∆max, εor2(10+( i−1)) log nto ensure that one does not need access to real RAM in order to do\nfast comparisons.\nIn particular, any hierarchy that uses a valid instantiation of Theorem 3 for every t-splitter\nyields a ( G,∆max, ε)-valid-hierarchy; regardless of the insertion order. We will also observe that\nfor all i, the edge sets of the graphs in Giform a valid partition of E(G). We will let a vector χ(e)\ndescribe the hierarchy-profile of an edge:\nDefinition 33. For any edge eand any (G,∆max, ε)-valid-hierarchy, we let the hierarchy-profile\nofebe the vector χ(e)∈Zhsuch that if e∈E(H)forH∈ Gi, then χ(e)i=cH(e).\nFor any i≤hand any vector v∈Zh, we let the ithinitial segment of vbe the vector v(i)∈Zi\nwhose first icoordinates agree with the first icoordinates of v. For any vector v∈Ziand constant\nk∈Z, we also let [ v(i), k] be the vector in Zi+1with ithinitial segment equal to vand [ v, k]i+1=k\nWe can index the graphs in the hierarchy by providing an index set Jcontaining every initial\nsegment of the hierarchy-profile of every edge e∈G. Here, we let G∅=G, and we let Gjfor any\nj∈Jbe the graph containing exactly the edges efor which jis an initial segment of χ(e). We let\n|j|:=dim(j) denote the dimension of the vector j. Also |j|is the level ofGj.\n5.2 An algorithm for maintaining a hierarchy\nThe algorithm for dynamically maintaining the hierarchy is quite straight-forward. First of all,\none initializes values t(i) and ˆ∆iso that they conform to the constraints of Definition 31 (see\nAlgorithm 7). This only has to be done once.\nThe hierarchy is then maintained by repeatedly applying Theorem 3 to further split graphs into\nones with lower degree. First one performs the update on level 0, which then propagates the update\nalong with any extra recolourings to the next level of the hierarchy. One has to be a bit careful\nwith determining the order of deletions and insertions, in order to ensure that the ˆ∆i’s remain valid\nupper-bounds through-out the update sequence. Therefore, the algorithm first processes an update\nin the top level, deletes all edges that were recoloured from the old graphs they were put into, and\nthen inserts any new edges into the further recursive structures. This ensures that ˆ∆i’s remain\nvalid upper-bounds. Finally, we note that the hierarchy consists of many graphs, which can either\nbe constructed in a pre-processing phase or simply allocated online by only inserting vertices into\na graph once the vertex actually has degree at least one.\nNext, we give pseudo-code for the above dynamic algorithm. If an edge ehas just been re-\ncoloured, we let c(e)oldrespectively χ(xy)olddenote the colours of ejust before the recolouring.\nThe hierarchy is maintained via the following set of recursive procedures. The first procedure shows\nhow to initialize the parameters needed for the following procedures. We note already now that to\nactually perform an insertion or a deletion, one must call the respective recursive procedure with\nthe correct parameter setting (see Remark 34):\n32Algorithm 7: Initialize-Parameters (∆max, ε)\ni1= 0\ni2= 0\nˆ∆0= ∆ max\nG0={G}\nη=⌊ε\n128 log n⌋\nwhile ˆ∆i1+i2≥ ⌈107log5n\nε3⌉do\nif∆max\n⌊2√logn⌋i1−1≥ ⌊210√logn⌋then\nt(i1+i2) =t1\ni1=i1+ 1\nˆ∆i1+i2= (1 + η)i1+i2∆max\nti1\n1ti2\n2else\nt(i1+i2) =t2\ni2=i2+ 1\nˆ∆i1+i2= (1 + η)i1+i2∆max\nti1\n1ti2\n2h=i1+i2\nAlgorithm 8: Insert-Hierarchy (uv, j)\ninsert(uv) intoGjusing Theorem 3 with (∆ max, t, ε′) = ( ˆ∆|j|, t(|j|), η)\nfor all recoloured edges xy̸=uvinGjdo\nfork=htoh− |j|do\nDelete-Hierarchy (xy, χ (xy)(k)\nold, χ(xy)old)\nInsert-Hierarchy (xy,[j, cGj(xy)])\nInsert-Hierarchy (uv,[j, cGj(uv)])\nAlgorithm 9: Delete-Hierarchy (uv, j, v )\ndelete(uv) from Gjusing Theorem 3 with (∆ max, t, ε′) = ( ˆ∆|j|, t(|j|), η)\nif|v| ≥ |j|+ 1then\nDelete-Hierarchy (uv, v(|j|+1), v)\nfor all recoloured edges xy̸=uvinGjdo\nfork=htoh− |j|do\nDelete-Hierarchy (xy, χ (xy)(k)\nold, χ(xy)old)\nInsert-Hierarchy (xy,[j, cGj(xy)])\nRemark 34. We remark that to actually perform an insertion, one must call Insert-Hierarchy (uv,∅),\nand to actually perform a deletion, one must call Delete-Hierarchy (uv,∅, χ(uv)).\n5.3 Correctness\nIn this section, we show that the algorithms from above actually maintain a ( G,∆max, ε)-valid-\nhierarchy with parameters that give the guarantees stated in Theorem 4. We first show that the\nalgorithm maintains a ( G,∆max, ε)-valid-hierarchy:\nLemma 35. Algorithms 7 8 9 implement a (G,∆max, ε)-valid-hierarchy.\n33Proof. The proof is a straight-forward verification that the hierarchy returned satisfies the items in\nDefinition 31. To that end, we note that Initialize-Parameters (∆max, ε) initialises in accordance\nwith item 1. This part is never changed, and so through-out all updates Item 1 holds.\nFurthermore, Initialize-Parameters (∆max, ε) ensures that the sequence {t(i)}h−1\ni=0is in ac-\ncordance with the second item of Definition 31.\nThe hierarchy is also in accordance with the third, fifth, and sixth items. Indeed, if at all times\nthe conditions of Theorem 3 is met, then it follows by induction and the theorem itself that the\ncomputed colourings are t(i)-splitting for all i, and furthermore satisfy the fifth and sixth items.\nTo check that this is the case, we check that the upper bounds on the degrees ˆ∆iremain valid at\nall times during an update. This holds since the algorithms immediately delete all further copies of\nan edge upon a deletion or recolouring. Indeed, suppose the graph Gvis a graph with |v|minimum\nthat invalidates the upper-bound ˆ∆|v|. Then it must be because the colouring on G|v|(|v|−1)is not\nat(|v| −1)-splitter, or because Gvat some point is not a subgraph of G|v|(|v|−1)[v|v|]. The first case\ncannot happen, since the t(|v| −1)-splitter on G|v|(|v|−1)always has all requirements satisfied by\nassumption. The second case cannot happen, since any time an edge coloured v|v|is deleted or\nrecoloured in G|v|(|v|−1), it is immediately removed from Gv. Hence, any time that Gvis subjected\nto an insertion, it is a subgraph of G|v|v[v|v|] and thus satisfies the degree-bound.\nSince we construct Gi+1by applying Theorem 3 to every graph in Gi, and since Theorem 3\nreturns a valid t(i)-splitter, we thus also recover Item 4.\nWe will use the following standard estimate repeatedly:\nClaim 36. For0< ε < 1and0< η≤ε\n128 log n, we have that (1 +η)4 logn≤(1 +ε\n16)\nProof. It follows from the well-known fact that 1 + r≤erfor all r∈Rthat (1 +x\nq)q≤exfor all\nreals x, q̸= 0. Indeed, we can write;\n(1 +η)4 logn≤(1 +ε\n128 log n)4 logn≤eε\n32\nFurthermore, for r >−1 it is well-known thatr\n1+r≤log(1 + r) and so we have\n(1 +η)4 logn≤eε\n16(1+ε\n16)≤(1 +ε\n16)\nas claimed.\nNext, we show that:\nLemma 37. For any graph Gwith at least one edge and any upper bound ∆max≤n, we have that\nany(G,∆max, ε)-valid-hierarchy of depth hmust have h≤4√logn.\nProof. Observe that we can perform at most i≤2√lognsplits with t=t1. Indeed, after i\nsplits, we have∆max\nti\n1≤n\nti\n1, so setting i > 2√lognforces∆max\nti−1\n1<⌊210√logn⌋and so ∆ max<\n⌊210√logn⌋ · ⌊2√logn⌋i−1. This means that t(j)̸=t1for all j≥i.\nA similar argument shows that we perform at most 2√lognsplits with t=t2before210√logn\nti−1\n2\ngoes below 1 /2. By Lemma 36, we have (1 + η)4√logn<2, and so ˆ∆4√logn<1, contradicting the\nfact that ˆ∆2√lognmust upper bound the maximum degree of all graphs in G4√logn, one of which\nmust have degree at least one, since G2√lognpartitions G.\n34Finally, we show that any ( G,∆max, ε)-valid-hierarchy with ∆ max≥107log5n\nε3satisfies the con-\nditions of Theorem 4. We note that if ∆ max<107log5n\nε3, we do not need to apply the hierarchy, as\nwe can simply apply the techniques develop in Section 6 directly on the graph.\nLemma 38. Given any (G,∆max, ε)-valid-hierarchy with ∆max≥107log5n\nε3. Let ibe the smallest\ninteger ifor which∆max\nti\n1<⌊210√logn⌋. Similarly, let jbe the smallest integer jfor which∆max\nti\n1tj\n2<\n107log5n\nε3 Then, assuming the hierarchy has depth i+j, we have:\n1.|Gi+j| ≤ti\n1tj\n2\n2.ˆ∆i+j= (1 + η)i+j∆max\nti\n1tj\n2≤(1 +ε\n16)∆max\nti\n1tj\n2\nProof. We show the first item first. Observe that we perform isplits into at most t1graphs first,\nmeaning that |Gi| ≤ti\n1by an easy induction on i, since |G0|= 1. A similar induction on jyields\nthe result, since this time each split is into t2graphs. The proof of the first part of the second item\nis a completely synchronous induction on first iand then jusing item 5 of Definition 31.\nThe last in-equality of item 2 follows by observing that i+j≤4√lognby Lemma 37 and then\napplying Claim 36.\nLemma 35 and Lemma 38 together establish correctness of the Algorithm. In order to prove\nTheorem 4, we need now only analyse the running time and the recourse of the hierarchy. We do\nso in the next sub-section.\n5.4 Analysis\nThe following Lemma analyses the recourse the algorithm for maintaining the hierarchy:\nLemma 39. Starting from an empty graph, a sequence of σinsertion or deletions to graphs at level\nitriggers at most σ\u0000\n212ε−2logn\u0001j−iinsertions or deletions to graphs at level j≥i.\nProof. The proof is by strong induction on j−i. The base case where j−i= 0 clearly holds. We\nproceed to the induction step. Assume that for j−i= 0,j−i= 1, . . . , j−1 =kthe lemma holds.\nNow assume j−i=k+ 1. First of all, we note that any insertion or deletion at a lower level might\ntrigger a similar insertion or deletion at all higher levels. By the induction hypothesis, the number\nof such insertion or deletions, U1, at level jis bounded as follows:\nU1≤j−1X\nℓ=iσ\u0000\n212ε−2logn\u0001ℓ−i≤σ\n2\u0000\n212ε−2logn\u0001j−i\nsincePy\ni=1xi≤xy+1\nx−1forx≥2. Note that these insertions and deletions account not only for direct\ninsertions or deletions into G, but also for insertions and deletions resulting from re-colourings at\nsome level ℓ < j−1. Indeed, these re-colourings trigger insertions and deletions at higher levels,\nwhose contributions are already accounted for in the quantity above.\nThus it only remains to account for the number of insertions or deletions, U2, caused by re-\ncolourings at level j−1. Observe that by Lemma 27 it follows that over any sequence of σ′insertions\nor deletions to a graph at level j−1, at most σ′·210logn\nε2recolourings are performed at level j−1.\nEach recolouring corresponds to one insertion and one deletion on the next level, and so we find\nthat at most\nU2≤2·σ\u0000\n212ε−2logn\u0001j−1−i·210logn\nε2≤σ\n2\u0000\n212ε−2logn\u0001j−i\n35insertions or deletions are performed at level jas a result of re-colourings at level j−1. Summing\nU1andU2finishes the induction.\nThe following lemma is then immediate:\nLemma 40. The amortised recourse of an insertion or deletion to Gis in 2˜Ologε−1(√logn). Fur-\nthermore, the amortised update time is also in 2˜Ologε−1(√logn).\nProof. Starting from any empty graph undergoing a sequence of σupdates Lemma 39 and Lemma 37\ntogether imply that at most σ·2˜Ologε−1(√logn)insertion and deletions are performed at level ℓfor\nanyℓ≤h. Since the hierarchy maintained has depth h≤4√lognby Lemma 37, an insertion or\ndeletion triggers at most\nσ·2˜Ologε−1(√logn)\nre-colourings, insertion or deletions to some level of the hierarchy. Hence, over the sequence of σ\nupdates the recourse is bounded by σ·2˜Ologε−1(√logn).\nTo recover the second part of the lemma, we know by above that, starting from an empty graph,\nany sequence of σoperations trigger at most σ·2˜Ologε−1(√logn)insertion and deletions. Hence, for\nthe usual index set I, we know that if Gvexperience σvinsertion or deletions, then\nσ·2˜Ologε−1(√logn)=X\nv∈Iσv\nBy Theorem 3, we can process a sequence of σvinsertions or deletion to a graph of level |v|in\ncσvlog(m)t3\n1update time for some constant c. As a result, the total update time is\nX\nv∈Icσvlog(m)t3\n1=clog(m)t3\n1σ·2˜Ologε−1(√logn)=σ·2˜Ologε−1(√logn)\nand so the amortised update time is as claimed.\nFinally, we note that Lemma 37 shows that the while -loop of Initialize-Parameters (∆max, ε)\nhas few iterations. Secondly, each iteration of the while -loop can be implemented efficiently pro-\nvided that all values are properly discretised as noted in Remark 34. The time spent on this step is\ntherefore only O(√logn). This step only has to be performed once upon initialization. Combining\nall of the above now yields Theorem 4.\n6 Deterministic Dynamic Edge-Colouring of Low-Degree Graphs\nIn this section, we show how to deterministically maintain a (∆ max+1)-edge-colouring in a dynamic\ngraph with maximum degree ∆ ≤∆max. In Section 7.2, we will show how to adjust the algorithm\nto the case where no upper bound on ∆ is needed, at the cost of the algorithm using (1+ ε)∆ colours\ninstead. We will follow the approach sketched in the informal overview. We begin by showing some\nstructural corollaries of previous work:\n6.1 Structural Corollaries\nAs mentioned earlier, a key observation is that a particular bichromatic path cannot be reached\nfrom too many different starting points. Similar observation has been used in the literature be-\nfore [Ber22, BD23, Chr23] to analyse algorithms. Here, we will apply the fact more explicitly and\n36actually update data structures at every point that may reach a recoloured edge via a short and\nnon-overlapping multi-step process with few steps. We will use the following Lemma due to Chris-\ntiansen [Chr23]. Note that originally the lemma is stated using the maximum degree ∆, but the\nLemma carries over ad-verbatim if one replaces ∆ by an upper bound ∆ maxon ∆:\nLemma 41 (Lemma 26 in full-version of [Chr23]) .LetH⊂Gbe a subgraph of Gand let c, c′be\ntwo (partial) (∆max+ 1)-edge-colourings of Gdiffering only on H. Then, under both candc′, the\nsame at most 2(∆ max+ 1)3· |H|points in G−N1(H)can reach Hvia a 1-step process.\nWe have the following Corollary 42 below:\nCorollary 42. LetGbe a graph with maximum degree ∆≤∆maxendowed with a proper (possibly\npartial) (∆max+ 1) edge-colouring c. Let v∈V(G)be any vertex. Then vcan be reached by at\nmost 3i(∆max+ 1)3inon-overlapping i-step processes.\nFurthermore, if Cis a semi-consistent j-step process with step-length ≤ℓ, then N2(C)can be\nreached by at most 3i(∆max+ 1)3i+2·j·ℓnon-overlapping i-step processes.\nProof. We show both claims via induction on i. We begin with the first claim. To this end, fix\nan arbitrary vertex v. Then vcan be reached by at most 2(∆ max+ 1)3points in G−N1(v) via a\nbichromatic path by Lemma 41. Any point in N1(v) might also reach vby simply hopping to a\nneighbour, and so we conclude that in total at most 2(∆ max+ 1)3+ (∆ max+ 1)≤3(∆ max+ 1)3\npoints are able to reach vby a 1-step process. This concludes the base case. Now let wbe some\npoint that can reach vvia (i−1)-steps of the above process. Then by applying the same argument\nas above, we find that at most 3(∆ max+ 1)3points can be the basis of a 1-step process that can\nreach w. By applying induction we find that there are at most 3(∆ max+ 1)3(i−1)choices for w, and\nso since any i-step process reaching vhas to go through w, we find that there are at most\n3(∆ max+ 1)3·3i−1(∆max+ 1)3(i−1)= 3i(∆max+ 1)3i\nsuch points, which is what we wanted to show.\nThe second claim follows a similar argument to above. We will let H=N2(C) be the one-hop\nneighbourhood of the j-step process, and focus on the first vertex in Hreached by some process.\nThen we can apply Lemma 41 and the same argument as above to find that at most 3(∆ max+1)3|H|\npoints can reach Hvia a 1-step process. The induction step is similar to above, except for the fact\nthat we now have at most 3i−1(∆max+1)3(i−1)|H|choices for w, and so this time we find that there\nat most\n3(∆ max+ 1)3·3i−1(∆max+ 1)3(i−1)|H|= 3i(∆max+ 1)3i|H|= 3i(∆max+ 1)3i+2·j·ℓ\nof the desired points, which is what we wanted to show.\nNext we will need the following notation. Note that the definitions deviate slightly from those\npresented in [Chr23], as we now require the Vizing chains to be consistent according to Definition 14.\nDefinition 43 ([Chr23]) .Given a proper partial colouring and an uncoloured edge e, letv∈Ri(e, ℓ)\ndenote the set of vertices such that there exists a consistent and non-overlapping i-step Vizing chain\nfrom eending at vsuch that every Vizing chain in the i-step Vizing chain has length at most ℓ.\nDefinition 44 ([Chr23]) .Given a proper partial colouring and an uncoloured edge e, let v∈\nNi(e, ℓ)⊂Ri(e, ℓ)if there exists a consistent and non-overlapping i-step chain from eending at v,\nand, furthermore, the multi-step Vizing chain may be extended to a non-overlapping (i+ 1)-Vizing\nchain by concatenating a Vizing chain of length at most ℓatv.\n37In order to make it easier to maintain the data structures used to find short augmenting multi-\nstep Vizing chains, we have imposed some further restrictions on how the chains are allowed to be\nextended. Most of these restrictions are already implicitly applied in for instance Christiansen’s\nanalysis of multi-step Vizing chains [Chr23]. While this analysis allow for Vizing chains not adhering\nto the restrictions, we have imposed, all arguments essentially impose most of these restrictions on\nthe chains anyway, and so the arguments go through with the above restrictions added almost ad\nverbatim. Hence, the following lemma is implicit in [Chr23]:\nLemma 45 (Implicit in Lemma 27 in full-version of [Chr23]) .Suppose we are given a proper\npartial colouring and an uncoloured edge ein a graph of maximum degree ∆≤∆max, and let\nℓ≥80(∆ max+ 1)7. Suppose that for all 1≤j≤i−1it holds that no vertex in Nj(e, ℓ)can be\nextended to an augmenting and non-overlapping (j+ 1)-step Vizing chain via a Vizing chain of\nlength at most ℓ. Suppose furthermore that the conditions below hold for all 1≤j≤i−1:\n1.|Nj(e, ℓ)| ≥ℓ\n8∆3maxj−1P\nk=1|Nk(e, ℓ)|\n2.|Nj(e, ℓ)| ≥ |Rj(e, ℓ)|/2.\nThen condition 1)and2)also hold for j=i.\nThe lemma allows us to deduce the existence of short augmenting and consistent multi-step\nVizing chains taking a sub-log nnumber of steps. This follows from arguments very similar to\nthose from [Chr23], just with a different choice of parameters:\nCorollary 46. LetGbe any simple graph with maximum degree ∆≤∆max. Let cbe any proper\npartial (∆ + 1) -edge-colouring of G. Then for any uncoloured edge e, under the colouring c, there\nis an augmenting, consistent and non-overlapping Vizing chain with at most√lognsteps and total\nlength at most 80(∆ max+ 1)7·2√logn√logn.\nProof. We want to apply Lemma 45 recursively until the vertex set is exhausted, and the assumption\nof the Lemma no longer holds. This will then imply the existence of the object we seek. To this\nend, we will fix ℓ= 80(∆ max+ 1)7·2√lognand fix an arbitrary uncoloured edge e.\nWe first construct an augmenting Vizing chain on eas specified in Section 3, making sure that\nit is consistent with Definition 14. If it has length ≤ℓ, we are done, so we may assume this is not\nthe case. Next we consider the action of truncating the Vizing chain at any of the ℓfirst vertices,\nand extending it via Lemma 13 in a manner consistent with Definition 14.\nAt most 2∆ maxof these extensions will overlap the edges of the path of the first Vizing chain\n– namely if the bichromatic path they extend through coincides with the bichromatic path of\nthe Vizing chain build on e. At most (∆ max+ 1)2bichromatic paths can go through a vertex,\nand therefore at most 2(∆ max+ 1)4of the extensions will intersect the one-hop neighbourhood of\nthe center of the fan situated at e. Hence we find that |R1(e, ℓ)|=ℓby construction and that\n|N1(e, ℓ)| ≥ℓ−2∆max−2(∆ max+ 1)4≥ℓ/2. Hence, we may apply Lemma 45 until we find a short\naugmenting and non-overlapping Vizing chain. In particular, it follows by induction that on jthat\n|Nj| ≥\u0012ℓ\n8∆3max\u0013\nj\nprovided that for every i≤j, we have no augmenting and consistent i-step Vizing chain. In\nparticular, since Nj⊂V(G) for all j, we conclude that\n\u0012ℓ\n8∆3max\u0013\nj≤ |V(G)| ≤n\n38and consequently we have:\ni≤log ℓ\n8∆3maxn≤logn\nlogℓ\n8∆3max≤logn√logn=p\nlogn\nIn particular, this means that there must exist some augmenting and consistent Vizing chain with\nat most√lognsteps and length at most ℓ·√logn= 80(∆ max+ 1)7·2√logn√logn\nAt the same time, it follows from Corollary 46 and Lemma 17, that at most 2˜Ologε(√logn)of\nthei-step consistent and non-overlapping Vizing chains can reach the Vizing chain. The choice of\nℓ= 80(∆ max+ 1)7·2√lognallows us to set i≤√lognwhile ensuring the existence of augmenting,\nconsistent and non-overlapping i-step Vizing chains by Corollary 46. This guarantees the Vizing\nchain has length at most 2˜Ologε(√logn).\nAt the same time, it follows from Corollary 42 and Lemma 17, that at most 2˜Ologε(√logn)of\nnon-overlapping i-step processes can reach the Vizing chain. This allows us to exhaustively locate\nall of these chains if necessary. The combination of these properties is what allows us to build\na dynamic data structure for implicitly maintaining a shortest augmenting and consistent Vizing\nchain for any edge that might be inserted into the dynamic graph. In the next section we will\ndescribe the data structures needed to do so.\n6.2 Contents of the data structures\nWe will describe the data structures in terms of the following graph:\nDefinition 47. LetGbe a graph with maximum degree ∆≤∆max, let cbe a proper (partial)\n(∆max+ 1)-edge-colouring of G, and let ℓbe a parameter. The bichromatic skeleton ofGis the\ndirected graph B(G, c, ℓ )with:\n•Vertex-set V(B(G, c, ℓ )) = X1∪X2, where every maximal bichromatic path PofGunder c\nis represented by a node in X1and every node in V(G)is represented by a node in X2. Note\nthat we allow Pto consist of an isolated vertex.\n•Edge-set E(B(G, c, ℓ )) =Y1∪Y2∪Y3. Here:\n•e= (x2x1)∈Y1with x1∈X1andx2∈X2ifx2is an endpoint of the bichromatic path\nrepresented by x1.\n•e= (xy)∈Y2with x, y∈X2if the nodes in Grepresented by x, yinB(G, c)are neighbours\nvia a coloured edge in G.\n•e= (x1x2)∈Y3with x1∈X1andx2∈X2ifx2belongs to the vertex set of the bichromatic\npath represented by x1, and, furthermore, the distance in the bichromatic path, represented by\nx1, from x2to an endpoint of the bichromatic path is at most ℓ\nIfcandℓis clear from the context, we will often leave the arguments out and simply write\nB(G). Let P(G, c, ℓ ) or simply P, if the arguments are clear from the context, be the set containing\nall bichromatic paths Pthat are contained in some maximal bichromatic path P′ofGunder c\nsuch that Phas length at most ℓand such that P∩P′contains an endpoint of P′. Then there is\na canonical map r:P ∪V(G)→V(B(G)) that maps a vertex in V(G) to its representative in X2,\nand similarly a bichromatic path Pto the representative in X1of the unique maximal bichromatic\npath P′containing P. Below we state some very basic and straight-forward properties of this type\nof graphs:\n39Claim 48. LetG,∆≤∆max,c, and ℓbe as in Definition 47. Then for B(G)the following holds:\n•B(G)contains no parallel edges pointing in the same direction.\n•The maximum in-degree of B(G, c)is upper bounded by 2(∆ max+ 1)2.\n•The maximum out-degree of v∈X2is upper bounded by 2(∆ max+ 1)2.\nProof. It is straight-forward to check that for any u, v∈V(B(G)) at most one edge ( uv) can be\ninserted.\nThe second item follows from the fact that at most (∆ max+ 1)2bichromatic paths can go\nthrough any vertex of G. More specifically, if v∈X1then at most 2 in-edges can be incident to\nv, and if v∈X2, then at most (∆ max+ 1)2in-edge of vcan belong to Y3and at most (∆ max+ 1)\nin-edges of vcan belong to Y2. In this case, no in-edges belong to Y1.\nSimilarly, for the third item, we note that for v∈X2at most (∆ max+ 1) out-edges of vbelongs\ntoY2and at most (∆ max+ 1)2out-edges of vbelongs to Y1, since vcan be the endpoint of at most\n(∆max+ 1)2maximal bichromatic paths. No out-edges of vbelongs to Y3.\nFinally, we note that any non-overlapping i-step process with step-length at most ℓcorresponds\nto a directed path in B(G) in a canonical way: simply map w1+y1+P1+···+wi+yi+Pito the\npath ( r(w1), r(y1), r(P1), . . . , r (wi), r(yi), r(Pi)).\nAgain let G, ∆≤∆max,c, and ℓbe as in Definition 47. We will maintain the data structures\ncontaining the following information:\n•We explicitly maintain the graph B(G).\n•For each edge ( x2, x1)∈Y1, we maintain for all i≤√lognand for all neighbours wof\ny1∈r−1(x2), the following information in Di(e, w): either all or the\np\nlogn(p\nlogn−i)·3√logn(∆max+ 1)3√logn+2·ℓ\nshortest semi-consistent and augmenting i-step processes beginning at wwith y1∈r−1(x2)\nandP1∈r−1(x1).\nIn the next section, we will describe how to efficiently maintain the above information under\ncolouring and un-colouring of edges. This is sufficient to easily construct a dynamic algorithm,\nsince the above data structures readily contain information about which edges to recolour in order\nto extend the maintained colouring.\n6.3 Maintaining the data structures\nIn this section, we will consider two main operations: colour (e, κ) and un-colour (e). Here\ncolour (e, κ) will extend the maintained colouring cby giving the edge ethe colour κ. We re-\nquire that κis free at both endpoints of eforcolour (e, κ) to be called. Oppositely, the operation\nun-colour (e) will reduce the colouring cby assigning ea blank colour ¬. In addition to these op-\nerations, we will also support add(e) which inserts a new edge einitially coloured ¬, and remove (e)\nwhich removes a blankly coloured edge efrom the graph. These last two operations are at this\npoint straight-forward to maintain, as all the work can be pushed to the two main operations. As\nmentioned earlier, we will again assume that at all times ∆( G)≤∆max.\n40Maintaining the first item: In order to explicitly maintain B(G), we will consider the (∆ max+\n1)2graphs induced in Gby any pair of distinct colours κ1, κ2. Since the colouring cis at all times\nproper, these graphs consist only of isolated points, paths, and even cycles. In order to dynamically\nmaintain the representation, we maintain each component in a Top-tree [AHdLT05]. In the case\nof even cycles, we cannot store the component directly in a dynamic tree data structure, but in\nthat case we store the entire component except for one arbitrary edge in a Top-tree. As ancillary\ninformation, we store the number of degree 1 vertices in the component. Note that there can be\nonly 0, 1, or 2 vertices of degree 1. We use this information to explicitly store B(G): whenever a\nmaximal bichromatic path is changed, we remove the corresponding node and all its related edges.\nNext we insert representatives of any un-represented maximal bichromatic paths, and add the edges\nincident to these representatives as specified by Definition 47.\nMaintaining the second item: We first consider the operation un-colour (e). Whenever eis\nun-coloured some of the information stored in the data structures is no longer valid. In particular,\nfor every non-empty maximal bichromatic path Pthat previously contained e, any i-step process\nthat used Pmight no longer be valid. In a similar vein, some extension close to ecould stop being\nconsistent. We can mark every data structure that might be affected by this as follows: any vertex\nthat can reach Pvia a j-step process, could contain faulty information in their data structures for\ni > j + 1. We will later bound this number using Claim 48 as well as the fact that ebelongs to at\nmost (∆ max+ 1)2such P’s.\nNext, we will cleanse the marked data structures and recompute their contents from scratch. We\nwill re-compute the marked data structures using the invariant that all data structures of the form\nDi−1should be recomputed before the first data structure of the form Diis recomputed. Performing\nthe re-computation in the case where i= 1 is straight-forward as there only is one choice for the\n1-step process. To recompute the structures of the form Diassuming that all structures of the form\nDjwith j≤i−1 are correct, now amounts to the following: we perform the first step as prescribed\nin the description of the data structures. Next we use the data-structures of the form Di−1along the\nbichromatic path to find the shortest suggestions of augmenting and consistent ( i−1)-step processes\n(here the number of vertices we need to traverse to get to an extension is factored into the suggested\nlength). Then we attempt each extension in increasing size of their lengths. For each attempted\nextension, we check that it is indeed consistent with our prefix extension by explicitly following the\npath along and checking the conditions as specified in Definition 15 and the following text. If the\nextension is not consistent, we don’t include it and attempt the next extension instead. We continue\nthis process until we have enough extensions. It is crucial here that each i−1 structure contains\nenough suggested path that even if the maximum number of them overlap, the data structure can\nat least suggest the required number of non-overlapping paths. We show that this is indeed the\ncase in the next section.\nPerforming colour (e, κ) is now completely symmetric to the above. Assigning ea colour κ\nalso causes some data structures to, potentially, be out-dated. These can now be fixed similarly to\nbefore. We provide a more in-depth description below:\nMaintaining a bichromatic skeleton: We will let Fκ1,κ2denote the pseudo-forest induced in\nGby all edges coloured κ1orκ2. For some vertex v∈G, we let PTκ1,κ2(v) be the pseudo-tree in\nFκ1,κ2containing v. We store every PTκ1,κ2as a tree Tκ1,κ2stored as a Top-tree [AHdLT05] and\npossibly one extra edge stored in a special slot Sκ1,κ2. We denote by Tκ1,κ2(v) the Top-tree currently\ncontaining v. Note that we will maintain the invariant that Sκ1,κ2=xyonly if x, yalready belong\nto the same Top-tree i.e. if Tκ1,κ2(v) =Tκ1,κ2(u). To perform BCS-colour (uv, κ), we do as follows:\n41•For all ��′̸=κ, we check if Tκ1,κ2(v) =Tκ1,κ2(u). If this is not the case, then we simply call\nlink(u, v) on the Top-tree data-structures. Otherwise, we store uvin the special slot Sκ1,κ2.\n•For each Top-tree we store as ancillary information the number of vertices with degree 1 or\n0 contained in each cluster, not counting the boundary vertices. It is straight-forward to\nmaintain this information under join andsplit .\n•After each modification to the Top-tree, we perform a non-local search for each leaf and check\nif they are incident to the edge stored in Sκ1,κ2(if any such edge). If so, the vertices’ degrees\nare changed accordingly, and they might no longer be classified as vertices with degree 1 or\n0.\n•If the Top-tree has either 1 or 2 such vertices, then it is classified as a bichromatic path.\n•Next, we remove every vertex (and incident edges) representing a bichromatic path whose\nTop-tree, or related special slot, was modified in the above steps.\n•Finally, we add representatives (and incident edges) of all bichromatic paths whose Top-tree,\nor related special slot, was modified in the above steps.\n•To add the incident edges, we access the leaves of the Top-tree to find their neighbours in G\nas well as the first ℓvertices (from either direction) on the leaf-to-leaf path in the Top-tree.\nFinally, we add the edges as prescribed by Definition 47.\n•Finally, we add parallel, oppositely directed edges between r(u) and r(v).\nTo perform BCS-uncolour (uv) we only have to modify the first step. The rest of the steps are\nexactly as before. Instead of calling link(u, v), we call cut(u, v) (or remove uvfrom Sκ1,κ2if it\nstored there). Finally, if Sκ1,κ2=xyis non-empty, we call link(x, y), and set Sκ1,κ2=∅. After\nthis, we do the same clean-up step as we did in BCS-colour , except that insteado of adding parallel,\noppositely directed edges between r(u) and r(v), we remove any such edges.\nMaintaining the Di’s:To perform un-colour (uv) we do as follows:\n•First we un-colour uvinGand update B(G) as described earlier.\n•Next, we mark some of the edges of B(G) with the following scheme: an edge eis marked j\nif the data structures Di(e, w) for j≤i≤√lognneeds to be updated. An edge might be\nmarked multiple times, but we need only keep the lowest mark, it receives.\n•All new edges in B(G) from step 1 are marked 1.\n•We let Rbe the set containing representatives of the 1-neighbourhood of uandvinG, all\nrepresentatives of endpoints of new bichromatic paths of B(G), and all representatives of\nformer endpoints of no longer-existing bichromatic paths.\n•All edges in Y1incident to Rare marked 1.\n•We set M1=R. We will mark all in-edges of vertices belonging to the set NY3,in(NY2,in(M1))\nalso belonging to Y1i.e. the set EY1,in(NY3,in(NY2,in(M1))) with 2. Finally, we set M2=\nNY1,in(NY3,in(NY2,in(M1))).\n42•For all 1 < i≤√logn, we mark all in-edges of vertices belonging to the set NY3,in(NY2,in(Mi))\nbelonging to Y1with i+ 1. Finally, we set Mi+1=NY1,in(NY3,in(NY2,in(Mi))).\n•Having updated all data-structures at level < i, we update Di(e, w) as follows: To update\nDi(x1x2, w) for i= 1, construct every augmenting 1-step process starting at w1=r−1(w)\nwith y1=r−1(x2)andP1∈r−1(x1). If the process is augmenting, we save the vertices in\nthe process. Note that it is crucial that this information can be efficiently computed without\naccess to any other similar data structures.\nFori >1, we ensure that all of the data structures Dj(e, w) have been (re)computed for all\nj < i . To update Di(x2x1, w) we let w1=r−1(w) and y1=r−1(x2). We pick a valid choice\nofP1∈r−1(x1) For each valid choice of P1, there is a unique extension point x=x(P1)\ncorresponding to the last vertex of P1visited, if one traverses P1starting at y1. Note that x\nis among the first ℓvertices of the maximal bichromatic path containing P1. For each choice\nofP1(or equivalently x), we do as follows:\n1. Construct a fan consistent with Lemma 13. We let P2(x) resp. y2(x) be the be bichro-\nmatic path resp. the endpoint of Pconsistent with Lemma 13.\n2. Next we consider Di−1(r(y2(x))r(P2(x), x) and extract the shortest consistent and aug-\nmenting ( i−1)-step process stored here.\n3. Then we check whether this extension is consistent with P1according to Definition 15.\nThis is done by simply following the ( i−1)-step process forward and checking the\nconditions. If not, then we are happy for now. Otherwise, we take the second smallest\nextension and repeat the process until we either exhaust all choices or we find a process\nand concatenate it with r−1(x) +y1+P1to form a consistent i-step process. We let Lx\ndenote the length of the above i-step process.\n4. We repeat the above step for each choice of x, and then store the candidate chains in a\nBBST.\n5. Then we extract the shortest chain, store its length and the vertices in the chain in the\nBBST representing Di(x2x1, w).\n6. If the shortest chain went through x, then we repeat the above process to find the second\nshortest chain that does not overlap with r−1(x)+P1(x) as before. Then L(x) is defined\nin terms of this chain and again inserted into the BBST containing all of the shortest\ncandidate chains.\n7. This is repeated until Di(x2x1, w) contains the desired number of chains. If we at one\npoint run out of chains to check, then we also terminate the above step and conclude\nthatDi(x2x1, w) is updated.\n8. Finally, we un-mark every marked edges.\nIn order to perform colour (uv, κ), we do almost the same as above. The only change is that instead\nof un-colouring uvinG, we colour it in Gwith the colour κ. Then we update B(G) accordingly,\nand then proceed exactly as before.\n6.4 Analysing the data structures\nWe start out by analysing the data structure for maintaining a bichromatic skeleton:\n43Lemma 49. LetGbe a dynamic graph with maximum degree ∆≤∆maxat all times, and let ℓ\nbe a parameter. Let cbe a (∆max+ 1)-edge-colouring of G. Then we can maintain the bichro-\nmatic skeleton B(G, c, ℓ )under the operations add(e),remove (e),un-colour (e), and colour (e, κ)\ninO(∆2\nmax+ ∆ max·ℓ·log(n))worst-case update time per operation.\nProof. We have already described the algorithm in Section 6.3, so what remains is to analyse it\nand prove that it is correct. To show correctness, we have to prove two things. First of all that\nthe maintained bichromatic skeleton has the required vertex set, and second of all that it has\nthe required edge-set. Since X2is cannot be updated, we need only check that X1is correctly\nmaintained. Since add(e) and remove (e) does not alter X1, we need only check that un-colour (e)\nand colour (e, κ) correctly re-establishes X1. We will argue that colour (e, κ) does so, and note\nthat the case for un-colour (e) is completely symmetrical.\nAfter colouring e, the only new vertices of X1that need to be added are representatives of\nbichromatic paths containing e. Any bichromatic component Ccontaining ecan only contain\nedges with colours κandκ′for some κ′̸=κ. Since the colouring maintained is forced to be proper,\nit follows that Cmust be either an even cycle or a path (possibly of length 0). In particular, Ccan\nalways be represented by a pseudo-tree or a tree with one additional special edge. Therefore, the\nalgorithm always correctly identifies the number of of vertices in Cof degree 1 or 0. Hence, the\nalgorithm will always identify if Cis a (possibly empty) bichromatic path, and correctly insert a\nrepresentative of CintoB(G). So X1contains a representative of all current bichromatic paths. The\nalgorithm also deletes any bichromatic path whose Top-tree representation was modified during the\nprocess above. Since eis inserted into the representation of G[κ, κ′] for all κ′, either the Top-tree or\nthe related special slot representing the component of einG[κ, κ′] will be modified. In particular,\neither the Top-tree or the related special slot representing bichromatic paths no longer present will\nbe modified and hence deleted. Therefore, X1is maintained correctly.\nTo see that the edge set is also correctly maintained, we note that the algorithm inserts the\nnew edges in Y2as well as any relevant edges incident to new bichromatic paths. These are the\nonly new edges that need to be added. It also removes any edges going to representatives no longer\npresent.\nTo recover the update times, we note that any operation to the Top-trees can be performed in\nO(logn) time (see [AHdLT05]), and that any non-local search for edges or degree 1 or 0 vertices\ncan be done in O(logn) time [AHdLT05] per edge. It is also standard that the number of such\nvertices can be maintained with only O(1) overhead. Since we update or search only O(∆max)\nTop-trees, the total time spend searching in the Top-trees is in O(∆max·ℓ·logn). Furthermore,\nfor each new (and old) bichromatic path, of which there are at most O(∆max), we have to add (and\ndelete) 2∆ maxedges to (from) Y1and 2 ℓedges to (from) Y3. Each edge is easy to insert (or delete)\nin constant time after performing the searches in the Top-trees already accounted for. Hence this\ntakes additional time at most O(∆2\nmax+ ∆ max·ℓ·logn) and the lemma follows.\nRemark 50. We remark that it is straight-forward to allow insertion and deletion of isolated\nvertices into GinO(1)update time per insertion/deletion.\nNext we analyse the data structure for maintaining the Di’s. We will show the following\nTheorem:\nTheorem 51. LetGbe a dynamic graph, and let ∆maxbe a known upper-bound on the maximum\ndegree throughout the entire update sequence. Then, we can maintain the data structures Di(·,·)in\n(∆max+ 1)˜O(√logn)worst-case update time per operation.\nWe have already described how to update the data structure in Section 6.3. In the following,\nwe show correctness and analyse the update times:\n44Correctness: First we show that any data structure whose contents are no longer valid, will\nindeed be properly marked:\nLemma 52. Assuming that all data-structures Diare initially correct, then after the marking\nprocess of a call to colour (e, κ)oruncolour (e)concludes, we have that: if fis any edge and w\nany vertex with data-structure Di(f, w)no longer correct, then fis marked with j≤i.\nProof. Letf=x2x1,wandibe resp. an edge, a vertex and index such that Di(f, w) is no longer\ncorrect. We will show that then fis marked with j≤i.\nThere are two reasons why Di(f, w) might no longer be correct. The first reason A) is that it\ncontained an i-step process with step-length ≤ℓwhich used a changed bichromatic path and so is no\nlonger valid. Similarly it could also be that a new bichromatic path allows for a shorter consistent\nand augmenting i-step process with step-length ≤ℓ, and this needs to be reflected in Di(f, w).\nThe second reason B) is that changes to the colouring ccauses some invocations of Lemma 13 to\nbe altered and so some of the stored processes are no longer consistent. Of course both reasons\ncould be present at the same time. Observe that it is sufficient to show that if any i-step process\nwith step-length ≤ℓbeginning with w+x2+x1can reach either 1) a changed bichromatic path in\n(i−1)-steps or less or 2) any vertex in the one-hop neighbourhood of ein (i−1)-steps or less, then\nfis marked with j≤i. Indeed, if 1) does not hold, then the reason A) can never occur. Similarly,\nif 2) does not hold, then the reason B) cannot occur.\nThus it is sufficient to show that if 1) or 2) holds then f∈EY1,in(NY3,in(NY2,in(Mj−1))) for some\n2≤j≤iorfis incident to a vertex in R. Indeed, then fis marked with j≤i.\nWe begin by assuming that 1) holds. This means that after the changes to c, there is an i-step\nprocess w1+y1+P1+···+wi+yi+Piwith w1=w,y1=r−1(x2), and P1∈r−1(x1) using a\nchanged bichromatic path Pduring the jthstep i.e. Pjis a sub-path of P. Let j≤ibe the smallest\nnumber for which the above is true.\nThis means that before the change to c,w1+y1+P1+···+wj+yjwas also an initial segment\nof some j-step process. Hence, this initial segment corresponds to a directed path in B(G) of\nthe form ( r(w1), r(y1), r(P1), . . . , r (wj), r(yj) present both before and after the update to c. This\nfollows from the minimality of jand the discussion directly after Claim 48. Observe that then\nyj∈R, (r(wj)r(yj))∈Y2, (r(Pj−1)r(wj))∈Y3. Hence, if j= 1, then f= (x2x1) = ( r(y1)r(P1))\nis incident to R, and so we would be done. More generally note that for all k≤j−1, we have\nthat ( r(yk)r(Pk))∈Y1, (r(wk)r(yk))∈Y2, and if k > 1 that ( r(Pk−1)r(wk))∈Y3. Thus if\nψ:P(V(B(G)))→ P(V(B(G))) maps a set A∈ P(V(B(G))) to the set NY1,in(NY3,in(NY2,in(A))),\nthen yk−1∈ψ({yk}) for 1 < k≤j. In particular, it follows by induction that y1∈ψj−1({yj}). So\nifyj∈R, then y1∈Mjand therefore f= (r(y1)r(P1))∈EY1,in(NY3,in(NY2,in(Mj−1))).\nOn the other hand, assume that 2) holds. Then there is an i-step process w1+y1+P1+···+\nwi+yi+Piwith w1=w,y1=r−1(x2), and P1∈r−1(x1) such that for some jwe have wj=r(z) for\nsome z∈N1(e)∪ein the coloured inclusive neighbourhood of z. Indeed, otherwise the application\nof Lemma 13 cannot be changed at z, as it depends only on the colours of edges incident to vertices\nin the 1-hop neighbourhood of z. Therefore there exist a vertex z′in the coloured neighbourhood\nofzwhose representative r(z′) belongs to R. WLOG we can assume that jis the smallest such\nindex among all choices.\nIfj= 1, then we are now done as r−1(y1) must be one such vertex and therefore fis incident\nto a vertex in R. For j >1 we can conclude, similarly to case 1), that y1∈ψj−1(R) and therfore\nthat f∈EY1,in(NY3,in(NY2,in(Mj−1))) and thusly marked with j. This concludes the proof of the\nlemma.\nNext we show that assuming that all un-marked data structures are maintained correctly, then\n45we will update all marked structures correctly. Together with Lemma 52 above, this shows correct-\nness of the update-algorithm.\nLemma 53. Assume that all data-structures Di(f, z)are correct before a call to colour (e, κ)or\nun-colour (e). Then they are all also correct after.\nProof. It follows from Lemma 52 that if Di(f, z) could be affected by the call to colour (e, κ) or\nun-colour (e), then fis marked with some j≤i. Therefore it is sufficient to instead show the\nfollowing by induction. Suppose all data structures Dk(f, z) are correct for k < i . Then any data\nstructure Di(f′, z′) marked with k≤iis updated correctly.\nThe base case is simple, as we just check all possible choices for semi-consistent and augmenting\n1-step processes with step-length at most ℓ. At most one such process can exist, and if so, we store\nit. Note that this information can be computed without requiring information stored at any other\ndata structures of the form D·(·,·).\nFor the induction step, we assume that all data-structures Dk(f, z) are correct for k < i . We\nfix choices of f′andz′, and show that Di(f′, z′) is correctly computed. We first observe that the\nvalues L(x) correctly lower bound the shortest semi-consistent and augmenting i-step process with\nstep-length at most ��extended through xat the 2ndstep.\nClaim 54. Letf′= (x2x1), z′be as above. Let xbe any extension-point corresponding to some\nP1∈r−1(x1). Then, at all times L(x)is a lower-bound for the shortest semi-consistent and\naugmenting i-step process with step-length at most ℓbeginning with z′+r−1(x2) +P1+xnot\nalready present in Di(f′, z′).\nProof. Any semi-consistent and augmenting i-step process with step-length at most ℓbeginning\nwith z′+x2+P1+xmust be extended by a semi-consistent and augmenting ( i−1)-step process\nwith step-length at most ℓbeginning with x+y2+P2for some sub-path P2ofP, where y2and\nPare chosen consistently with Lemma 13, that furthermore does not overlap z′+x2+P1. This is\nstraight-forward to verify via Definition 15. By the induction hypothesis, we have that whenever\nL(x) is set, it is set to |P1|+1+ the length of the shortest semi-consistent and augmenting ( i−1)-step\nprocess Cwith step-length at most ℓthat satisfies the following:\n1.Cbegins with x+y2+P2for some sub-path P2ofP, where y2andPare chosen consistently\nwith Lemma 13.\n2.Cis shortest among such processes that is noteither already represented in Di(f′, z′) or has\nalready been confirmed to overlap z′+x2+P1in the sense of Definition 15.\nIn particular, it is clear that L(x) therefore lower bounds the length of any sought-after process.\nSecondly, we observe that if we have αsemi-consistent and augmenting ( i−1)-step processes\nwith step-length at most ℓbeginning with x+y2+P2for some P2with the choices of y2andP2\nconsistent with Lemma 13, then Di−1((r(y2)r(P2), r(x)) contains at least min {α,√logn(√logn−\n(i−1))·3√logn(∆max+ 1)3√logn+2·ℓ}augmenting and semi-consistent ( i−1)-step processes which\ncan be added to the 1-step process z+r−1(x2)+P1to form either all or the shortest√logn(√logn−\ni)·3√logn(∆max+ 1)3√logn+2·ℓsemi-consistent and augmenting processes of the form we seek.\nIndeed, we note that by applying Lemma 41 to z′+r−1(x2) +P1, we find that at most i·\n3i(∆max+ 1)3i+2·ℓof all non-overlapping j-step processes with j≤i−1 can overlap with z′+\nr−1(x2) +P1in the sense of Definition 15. Note that this bound also holds for the extensions\nstored in Di−1((r(y2)r(P2), r(x)), as the first path of any extension can only overlap the 2-hop\nneighbourhood of z′, in which case the extension is accounted for above via Lemma 41, or via a\n46later part of the process that loops back and overlaps the two hop-neighbourhood of z′+r−1(x2)+P1.\nBut this later part of the process then forms a j-step process for j < i−1 that starts outside of\nthe two hop-neighbourhood of z′+r−1(x2) +P1and then reaches the two hop-neighbourhood of\nz′+r−1(x2) +P1. Thus it is also accounted for by Lemma 41.\nIn particular, since Di−1((r(y2)r(P2), r(x)) contains the shortest\np\nlogn(p\nlogn−(i−1))·3√logn(∆max+ 1)3√logn+2·ℓ\ncandidate extensions (or fewer, if fewer exists), we either find all extensions that will not overlap with\nz′+r−1(x2)+P1in the sense of Definition 15 or we find√logn(√logn−i)·3√logn(∆max+1)3√logn+2·ℓ\nsuch extensions. In particular, the above together with Claim 54 implies that the algorithm will\nlocate either all or the√logn(√logn−i)·3√logn(∆max+1)3√logn+2·ℓshortest semi-consistent and\naugmenting i-step processes with step-length at most ℓbeginning with z′+r−1(x2) +P′, where\nP′∈r−1(x1). This concludes the induction step.\nLemma 53, Lemma 52, and Lemma 49 together show correctness of the data structure and the\nupdate algorithms for the supported operations.\nImplementation and analysis: We first briefly comment on implementations certain data struc-\ntures. For each BBST, we can use Red-blacks trees [CLRS22] to support basic operations in O(logn)\nworst-case time. For Top-trees we use the version from [AHdLT05]. As for the analysis, we first\nshow that not too many edges are marked:\nLemma 55. In each invocation of the marking scheme, at most 23i(∆max+1)6i|R|edges are marked\nwithi. Hence |Mi| ≤23i(∆max+ 1)6i|R|.\nProof. The proof is by induction on i. For i= 1, we note that by item two and item three of\nClaim 48 at most 4(∆ max+ 1)2· |R|edges are initially marked 1. Also |M1|=|R|. For the\ninduction step, we assume the statement for all j≤i−1. We note that applying item two of\nClaim 48 thrice shows that Mi≤23(∆max+ 1)6|Mi−1|and that the number of edges marked with i\nis also at most 22(∆max+ 1)6|Mi−1|. Applying the induction hypothesis now yields the result.\nNext we show that Ris not too big:\nLemma 56. In each invocation of the marking scheme, we have that |R| ≤2 + 2(∆ max+ 1) +\n6(∆2\nmax+ 1)2≤10(∆2\nmax+ 1)2.\nProof. We note that at most 3(∆2\nmax+1)2bichromatic paths are altered after a removal or insertion\nof a coloured edge: namely 3 for each pair of colours; the bichromatic path that used to contain the\nedge and the two new bichromatic paths that this path was split into (for insertion it is of course\nopposite; one new bichromatic path that is the fusion of two old). There are at most (∆ max+ 1)2\nchoices for such colour pairs. Note that each path has at most 2 endpoints. Finally, there are at\nmost 2(∆ max+ 1) vertices in the one-hop neighbourhood of uandv. Adding these quantities gives\nthe bound.\nThese two lemmas allow us to analyse the time it takes to mark all edges:\nLemma 57. It takes ˜O(23(√logn+4)(∆max+ 1)6(√logn+2))time to mark all edges.\n47Proof. We can mark each edge with constant overhead by performing a backwards BFS search\nalong the relevant edges (assuming we store different types of edges in different BBST’s). Hence,\nthe Lemma follows by combining Lemma 55 with Lemma 56, and summing over all choices of\n1≤i≤√logn:\n10(∆ max+ 1)2√lognX\ni=123i(∆max+ 1)6i≤10(∆ max+ 1)2·23(√logn+1)(∆max+ 1)6(√logn+1)\n≤23(√logn+4)(∆max+ 1)6(√logn+2)\nNext, we bound the time it takes to check whether an extension is overlapping:\nLemma 58. Given a 1-step process w1+y1+P1, with r−1(w2)the second last vertex of P1, and\na semi-consistent and augmenting (i−1)-step process w2+y2+P2+···+wi+yi+Pi, we can\ncheck if C=w1+y1+P1+···+wi+yi+Piis a semi-consistent and augmenting i-step process\ninO((i+ 1)·ℓ·(∆max+ 1)3)time.\nProof. We check whether w2, y2andP2is consistent with Lemma 13 in ˜O(∆3\nmax). Next we can can\ntrace and temporarily mark the two-hop neighbourhood of every vertex in w1+y1+P1, as well\nas the edges in P1. Finally, we can follow along the ( i−1)-step process and check the conditions\nof Definition 15 by using the temporary marks. It takes ˜O((i+ 1)ℓ) time to trace the process and\n˜O(ℓ(∆max+ 1)2) to temporarily mark all of the edges and vertices. We note here that any update\ntime required to correctly locate edges or vertices in memory is captured by the ˜O-notation which\naccounts for look-ups in BBST.\nWe have already seen in the proof of Lemma 53 that not too many of the processes stored in\nneighbouring data structures can be overlapping. This is captured in the following Lemma:\nLemma 59. Suppose we are given a 1-step process w1+y1+P1, with r−1(w2)the second last vertex\nofP1. Let C′be a semi-consistent and augmenting (i−1)-step processes of the form C′=w2+y2+\nP2+···+wi+yi+Pi. Suppose further that the choice of w2, y2andP2is consistent with Lemma 13.\nThen at most i·3i(∆max+ 1)3i+2·ℓchoices of C′can cause C=w1+y2+P2+···+wi+yi+Pi\nto not be a semi-consistent and augmenting i-step process.\nProof. Since the extension is already assumed to be consistent with Lemma 13, we need only check\nthat C′does not overlap w1+y1+P1in the sense of Definition 15. We can however bound the\nnumber of such C′reaching the two-hop neighbourhood of w1by applying Corollary 42 to every\npoint in this neighbourhood. We note that due to the extension being consistent with Lemma 13,\nany other violations has to be because Pjintersects the 2-hop neighbourhood of w1+y1+P1for\nsome j≥3 or because we reuse the maximal bichromatic path containing P1. In the first-case the\nnon-overlapping process C′=w3+y3+P3+···+wi+yi+Pialso intersects the 2-hop neighbourhood\nofw1+y1+P1. But since w3is disjoint from the 2-hop neighbourhood of w1+y1+P1, this process\nis accounted for by Corollary 42. The second case is similarly accounted for by Corollary 42.\nFinally, we analyse the time it takes to update the data structures:\nLemma 60. Letfbe marked with j≤i. Then for any w, we can update Di(f, w)in˜O(3√logn(∆max+\n1)3√logn+4·ℓ3·log2n)time.\n48Proof. Note firstly that the ˜O-notation takes care of any log nfactors arising from inserting, search-\ning or deleting things from BBST’s. Here it is important to note that the BBST’s used to update\nthe data structures never have sizes greater than ˜O(ℓ2·logn·3√logn(∆max+ 1)3√logn+2). In-\ndeed, that is the maximum number of chain extensions consistent with Lemma 13, stored in the\nneighbourhoods of the first ℓvertices along P1.\nSecondly, Lemma 59 implies that at most√logn·3√logn(∆max+1)3√logn+2·ℓof these extensions\ncan be overlapping with the first-step of the process in the sense of Definition 15.\nHence, at most 3√logn(∆max+1)3√logn+2·ℓ·lognsuch chains need to be checked, before Di(f, w)\nis either full or contains all semi-consistent and augmenting i-step processes. Each extension can\nbe checked in ˜O((i+ 1)·ℓ·(∆max+ 1)3) time by Lemma 58 and since it only takes ˜O(ℓ∆3\nmax) to\ncompute and locate the correct choices of Di−1(·,·) to look for extension suggestions in. By noting\nthati≤√logn, the Lemma follows.\nCombining all of the above finalises the proof of Theorem 51.\n6.5 The proof of Theorem 5\nIn this section, we present a proof of Theorem 5 (restated below for convenience).\nTheorem 61 (Identical to Theorem 5) .LetGbe a dynamic graph, and let ∆maxbe a known upper-\nbound on the maximum degree throughout the entire update sequence. Then, we can maintain a\nproper (∆max+ 1)-edge-colouring of Gin(∆max+ 1)˜O(√logn)worst-case update time per operation.\nDescription of the algorithm: Maintaining the data-structures from Section 6.3 represents\nmost of the work required to show Theorem 5. The idea is the following. We will indeed maintain\nthese data-structures and ensure that after every update terminates, the colouring cmaintained\nby the algorithm of Theorem 51 is a proper (∆ max+ 1)-edge-colouring. Given a call to delete (e),\nwe first call un-colour (e) and then subsequently remove (e) in the previous data structures and in\nour representation of G. Given a call to insert (e), we do the following steps:\n•Call add(e).\n•Then we pick an arbitrary vertex u∈e, and construct a consistent fan according to Lemma 12.\nThe fan ends at a vertex y1and outputs a bichromatic path P.\n•If|P| ≤ℓ, we have discovered an augmenting chain, and we can immediately jump to step 5).\nOtherwise, we look in Di(r(y1)r(P), u) for all iin order to find a consistent and augmenting\nj-step process Cfor some j. Note that Corollary 46 and Theorem 51 guarantees that such a\nchain exists.\n•According to Lemma 18, Cguarantees some augmenting and consistent j-step Vizing chain.\nWe can explicitly compute this chain by tracing the process and computing the consistent\nfans according to Lemma 13.\n•We shift the colouring caccording to this chain using the operations un-colour (·) and\ncolour (·,·). Since the colouring is shifted along a chain, we ensure that cis at all times\nproper, and so all of the required assumptions hold.\n•Finally since the chain is augmenting, after shifting it the remaining un-coloured edge fhas\na free colour κ. We colour it using colour (f, κ).\n49Correctness: Corollary 46 guarantees the existence of a consistent and augmenting Vizing chain\nof the form, we week. Lemma 17 and Theorem 51 guarantees that for some ithe data structure\nDi(r(y1)r(P), u) contains at least one process of the claimed form. Next, applying Lemma 18\nensures that this process guarantees a consistent and augmenting Vizing chain. Finally, correctness\nfollows by Theorem 51, as the Theorem guarantees the correctness of un-colour (·) and colour (·,·).\nAnalysis: The running time of delete (e) follows immediately from Theorem 51. As for insert (e),\nwe note that since the shiftable-chain has length O(√logn∆7\nmax·2√logn), we need only call un-colour (·)\nandcolour (·,·)O(√logn∆7\nmax·2√logn) times. This part of the process dominates the update time,\nas we can trace the process in ˜O(√logn·ℓ) time, and compute the at most√lognfans in ˜O(∆3\nmax)\ntime each. This concludes the proof.\n7 Putting it all together\nIn this Section, we combine the results of the previous section with a degree-scheduler from [DHZ19]\nto show Theorem 2\nTheorem 62 (Identical to Theorem 2) .LetGbe a dynamic graph, and 0< ε < 1a given\nparameter. We can maintain a proper ((1 + ε)∆)-edge-colouring of Gin2˜Ologε−1(√logn)amortised\nupdate time per operation.\n7.1 The final algorithm\nFor now, we will still assume that we have access to some upper bound ∆ maxon the maximum\ndegree of Gthroughout the entire update sequence. We will then get rid of this assumption in the\nnext section by using a degree-scheduler due to Duan, He, and Zhang [DHZ19].\nThe algorithm is now the following: If ∆ max≤107log5n\nε3: we simply apply Theorem 5 directly.\nSince ∆ max+ 1≤ ⌈(1 +ε)∆max⌉for any ε >0, we are immediately done by Theorem 5.\nOtherwise, if ∆ max>107log5n\nε3, we first apply Theorem 4 to get a degree-splitting hierarchy of\nheight 0 < h≤2√logn. Then we apply Theorem 5 with the upper bound on the maximum degree\nequal to ∆′\nmax, where ∆′\nmax=ˆ∆h, to each graph of Gh, using a distinct palette to colour each\ngraph. Correctness of the algorithm then follows immediately from Theorem 4 and Theorem 5.\nAs for the update time of the algorithm, we note that by Theorem 4 each sequence of σupdates\ntoGresults in at most σ·2˜Ologε−1(√logn)updates to graphs in Gh. We can handle each such update\nin 2˜Ologε−1(√logn)time by Theorem 5, so in total, we process the σupdates in σ·2˜Ologε−1(√logn)\ntime, thus resulting in an amortised update time of 2˜Ologε−1(√logn)time, as claimed.\nFinally, we analyse the number of colours used. We note that by Theorem 4, there exists\nnon-negative integers i, jwith i+j=hsuch that:\n|Gh| ≤ti\n1tj\n2\nand\n∆′\nmax=ˆ∆h≤(1 +ε\n16)∆max\nti\n1tj\n2\nIn particular, each graph in Ghis coloured using at most (1 +ε\n16)∆max\nti\n1tj\n2+ 1 colours. Therefore, the\ntotal number of colours used is upper bounded by:\n|Gh| ·((1 +ε\n16)∆max\nti\n1tj\n2) + 1) ≤(1 +ε\n2)∆max+ti\n1tj\n2≤(1 +ε)∆max\n50Here we used the fact that Theorem 4 terminates with a hierarchy for which ∆ max≥2·ti\n1tj\n2\nε.\n7.2 Dynamic ∆\nIn this subsection, we finish the proof of Theorem 2. This last part contains no new ideas, as we\ncan use the degree-scheduler of Duan, He and Chang [DHZ19] essentially ad verbatim. For the sake\nof completeness, we briefly sketch their approach below.\nLetµbe some constant not specified until later (we will ultimately set µ= Θ( ε)). Consider the\nexponentially increasing intervals Ii:= [(1 + µ)i−1,(1 +µ)i] for i∈ {0,1, . . . , s }with sminimum\nsubject to (1 + µ)s≥n. For each i, we will maintain a subgraph of G, say Giwith the guarantees\nthat if ∆( G)∈Ii, then Gi=G, and that for all i, we have ∆( Gi)≤(1+µ)i+1always. Furthermore,\nwe will at all times maintain some index jfor which ∆( G)∈Ij.\nGiven the above, one can run any (1 + ε′)∆max-edge-colouring algorithm on all of the Gi’s, and\nthen let the colouring of Gjbe the currently valid edge-colouring. If one sets ∆ max,i= (1 + µ)i+1\nin the instance of the edge-colouring algorithm on Gi, one has the guarantee that at all times the\nalgorithm maintains a (1 + µ)2(1 +ε′)∆-edge-colouring. Indeed, at all times in the copy currently\nindexed by jwill have ∆ ·(1 + µ)2≥∆max,j. Setting ε′, µ≤ε\n3is then sufficient to achieve the\ncorrect number of colours.\nIn order to maintain this information, Duan, He and Chang do as follows: let δ(e) be the\nmaximum degree of an endpoint of e. Ifδ(e)≤(1 +µ)i, we add etoGi. Now estays in Giuntil\nδ(e)>(1 +µ)i+1, at which point in time it is removed, and not added back until δ(e)≤(1 +µ)i\nagain. It is now easy to check that that every time some vertex wcrosses the threshold (1 + µ)k\nfor some k, it performs at most deg(w) insertions into Gkor at most deg(w) deletions from Gk.\nBetween any set of insertions into some Gk,deg(w) must have increased beyond (1 + µ)k+1and\nback down to (1 + µ)k. All of the deletions incident to wthat decreased the degree from (1 + µ)k+1\nto (1 + µ)kcan be charged the cost of performing the deg(w) insertions into Gk. Deletions can\nbe analysed in a symmetric way. In total, the amortised recourse of performing these changes is\nupper bounded by O(µ−1) for each update. 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We introduce two new variants that both insist on a\nlevel-planar drawing of the input graph but relax the requirements on the shape of the clusters.\nIn unrestricted Clustered Level Planarity (uCLP) we only require that they are bounded by\nsimple closed curves that enclose exactly the vertices of the cluster and cross each edge of the graph\nat most once. The problem y-monotone Clustered Level Planarity (y-CLP) requires that\nadditionally it must be possible to augment each cluster with edges that do not cross the cluster\nboundaries so that it becomes connected while the graph remains level-planar, thereby mimicking a\nclassic characterization of clustered planarity in the level-planar setting.\nWe give a polynomial-time algorithm for uCLPif the input graph is biconnected and has a\nsingle source. By contrast, we show that y-CLPis hard under the same restrictions and it remains\nNP-hard even if the number of levels is bounded by a constant and there is only a single non-trivial\ncluster.\n1 Introduction\nAlevel graph (G,γ)is a graphG= (V,E)and a function γ:V→{1,2,...,k}withk∈N\nthat assigns vertices to levels such that no two adjacent vertices are assigned to the same\nlevel. Alevel planar drawing of a level graph (G,γ)is a crossing-free drawing of Gthat maps\neach vertex vto a point on the line y=γ(v)and each edge to a y-monotone curve between\nits endpoints. A level graph is level planar if it has a level planar drawing. Level planarity\ncan be tested in linear time [13].\nLetG= (V,E)be a graph. A clusteringTofGis a rooted tree whose leaves are the\nverticesV. Each inner node µofTrepresents a cluster, which encompasses all leaves Vµ\nof the subtree rooted at µ. The pair (G,T)is called a clustered graph . Aclustered planar\ndrawing of a clustered graph (G,T)is a planar drawing of Gthat also maps every cluster µ\nto a region Rµthat is enclosed by a simple closed curve such that (i) Rµcontains exactly the\nverticesVµ, (ii) no two region boundaries intersect, and (iii) no edge intersects the boundary\nof a cluster region more than once. The combination of (i) and (iii) implies that an edge may\nintersect a cluster boundary if and only if precisely one of its endpoints lies inside the cluster.\nA clustered graph is clustered planar if it has a clustered planar drawing. The problem of\ntesting this property and finding such drawings is called Clustered Planarity . In a\n∗SimonD.FinkwasfundedbytheDeutscheForschungsgemeinschaft(GermanResearchFoundation, DFG)\ngrant RU-1903/3-1 and by the Vienna Science and Technology Fund (WWTF) [10.47379/ICT22029].arXiv:2402.13153v1  [cs.CG]  20 Feb 20242 Clustered Planarity Variants for Level Graphs\n1\n2\n(a)\n1\n2\n3 (b)\nFigure 1 (a)A drawing that is level planar and clustered planar and thus cl-planar, but not\nconvex cl-planar or y-cl-planar. (b)A drawing that is y-cl-planar (with the augmentation edge in\nE′shown dashed in red) and thus also cl-planar, but not convex cl-planar.\nrecent breakthrough, Fulek and Tóth gave the first efficient algorithm for this problem [ 11],\nwhich was soon after improved to a quadratic-time solution [3].\nIn this paper, we seek to explore the combination of the two concepts of level planarity\nand clustered planarity. Namely, our input is a clustered level graph (cl-graph) , which is a\ntuple (G,γ,T )such that (G,γ)is a level graph and (G,T)is a clustered graph. We insist\non a level-planar drawing of G. However, it is not immediately clear which conditions the\ncluster boundaries should fulfill. Forster and Bachmaier [ 10] proposed the problem variant\nConvex Clustered Level Planarity (short cCLP), which requires to draw the clusters\nas convex regions1. They showed that cCLPcan be solved in linear time if the graph is\nproper (i.e., all edges connect vertices on adjacent levels) and the clusters are level-connected\n(i.e., each cluster contains an edge between any pair of adjacent levels it spans). Angelini et\nal. [1] showed that testing cCLPis NP-complete, but can be tested in quadratic time if the\ninput graph is proper, thereby dropping the requirement of level-connectedness.\nIn this paper we consider two new variants that relax the conditions on the drawing\nof the cluster. In unrestricted Clustered Level Planarity (short uCLP) we keep the\nconditions (i)–(iii) as stated above, i.e., the shapes of the clusters are not restricted by the\nlevels. Our second variant y-monotone Clustered Level Planarity (shorty-CLP) is\nbased on the characterization that a planar drawing Gof a graphGis clustered planar w.r.t.\nto a clustering Tif and only if it is possible to insert a set of augmentation edges intoG\nin a planar way such that each cluster becomes connected and no cycle formed by vertices\nof a cluster µencloses a vertex not in µin its interior [ 6]. In analogy to this, we define a\nlevel-planar drawing to be y-cl-planar if it satisfies these conditions but additionally the\naugmentation edges can be added as y-monotone curves. Figure 1 shows that cCLP,uCLP,\ny-CLPare indeed different problems. We are not aware of work that concerns uCLPor\ny-CLP.\nWe show that uCLPcan be solved in polynomial time if the input graph is biconnected\nand has a single source, i.e., a vertex that does not have a neighbor on a lower level. On the\nother hand we show that y-CLPis NP-complete under the same conditions and also if the\nnumber of levels is 5 and there is only a single non-trivial cluster, i.e., that not contains all\nvertices.\n1they only considered this convex setting and used the name CLPinstead of cCLPS.D. Fink, M. Pfretzscher, I. Rutter, M.D. Sieper 3\n2 Single-Source Biconnected (unrestricted) Clustered Level Planarity\nWe show that uCLPcan be solved efficiently if Gis a biconnected graph with a single source.\nTo this end, we combine the polynomial-time solution for Clustered Planarity [3] with a\ncombinatorial description of all level-planar drawings of a biconnected single-source graph [ 5].\nNote that whether a drawing of Gis clustered planar depends only on its combinatorial\nembedding, rather than the precise drawing. Thus, we call an embedding EofGclustered\nplanarif the corresponding drawings are. We call an embedding EofGlevel planar ifG\nadmits a level-planar drawing with embedding E. To solve uCLPfor an instance (G,γ,T ),\nwe need to find an embedding of Gthat is both cluster planar and level planar.\nWe first introduce yet another type of constraints called synchronized fixed-vertex con-\nstraints(sfv-constraints for short). For a graph G= (V,E)ansfv-constraint is a setQof\npairs (v,σv), wherev∈Vandσvis a fixed cyclic order of the edges incident to v, called its\ndefault rotation . An embeddingEofGsatisfies the constraint Qif for each pair (v,σv)∈Q\nthe rotation of vinEis its default rotation or if for each pair (v,σv)∈Qthe rotation of v\ninEis the reverse of its default rotation. Given a set Qof sfv-constraints, we say that an\nembedding of GsatisfiesQif it satisfies each Q∈Q.\nWe use sfv-constraints to bridge from level-planarity to usual planarity. To this end,\nwe introduce a slightly generalized version of Clustered Planarity , where we seek a\nclustered-planar embedding that additionally satisfies a given set Qof sfv-constraints. This\nproblem is called Sync CP . The point is that the algorithm of Bläsius et al. [ 3] reduces\nClustered Planarity to the intermediate problem Synchronized Planarity , which\nincludes the option to directly express sfv-constraints.\n▶Lemma 2.1. Sync CP can be solved in O(n3)time.\nWe now turn to the second ingredient. Let (G,γ)be a biconnected single-source level-\nplanar graph and let Γbe an arbitrary level-planar embedding of G. Brückner and Rutter [ 5]\nshowed that there exists a data structure, called LP-tree, very similar to the famous SPQR-\ntree, that represents precisely the level-planar embeddings of G; see fig. 2 for an example.\nLike the SPQR-tree, the embeddings decisions for the LP-tree are made by (i) arbitrarily\nreordering parallel subgraphs between a pair of vertices and (ii) flipping the embeddings of\nsome disjoint and otherwise rigid structures. Hence, the possible orderings of the edges around\neach vertex vofGin any level-planar embedding can be described by a PQ-tree [ 4,8]Tv,\ncalledlevel PQ-tree that is straightforwardly derived from the LP-tree; it contains one\nP-nodeuv,µfor each parallel structure µin whichvoccurs and one Q-node uv,ρfor each\nrigid structure ρ, in whichvoccurs; see fig. 2a. The level-planar embedding Γis used as\nreference to determine a default rotation σv,ρfor each Q-node uv,ρ.\nIf an embedding EofGis level-planar, the rotation of each vertex vis necessarily\nrepresented by its level PQ-tree Tv. It further holds that all Q-nodes uv,ρwithv∈Vthat\nstem from the same rigid structure either all have their default orientation or all its reversal.\nAn embedding where the last condition holds for each rigid structure is called ρ-consistent .\n▶Lemma 2.2. An embeddingEis level-planar if and only if the rotation of each vertex vis\nrepresented by its level PQ-tree Tvand moreoverEisρ-consistent.\nFor a biconnected single-source level-planar graph Gwith level-planar embedding Γ, we\nderive a new graph G+and a setQof synchronized fixed-vertex constraints. We replace\neach vertex vofGby a tree isomorphic to its level PQ-tree Tv; see Figure 2c. In order to\nenforceρ-consistency, we additionally create for each rigid structure ρof the LP-tree an4 Clustered Planarity Variants for Level Graphs\nw\nv\nTw\nTv\n(a)\nw\nw\nv\nv\nρ (b)\nu\nv\nTw\nTv\n(c)\nFigure 2 (a)A level graph Gwith two level PQ-trees TwandTvderived from (b)its LP-tree.\nP-nodes are represented by black disks, Q-nodes as white double disks. (c)The graph after replacing\nw,vbyTw,Tv; the orange arrow indicates the sfv-constraint due to ρ.S.D. Fink, M. Pfretzscher, I. Rutter, M.D. Sieper 5\nsfv-constraint Qρ={(uv,ρ,σv,ρ)|voccurs in the rigid structure ρ}. LetQdenote the set of\nthese constraints for all rigid structures. Clearly, we can obtain a planar embedding of Gby\ntaking a planar embedding of G+that satisfiesQand contracting each tree Tvback into\na single vertex. The embeddings we can obtain in this way are precisely those where the\nrotation of each vertex vis represented by its level PQ-tree Tvand that are ρ-consistent.\nFinally, it is time to connect clusters and level planarity. To this end, consider a\nclusteringTonG. We naturally obtain a corresponding clustering T+ofG+by placing\neach vertex of G+into the cluster of the vertex of Git replaces.\n▶Lemma 2.3. (G,γ,T )admits a planar embedding that is level-planar and clustered-planar\nif and only if (G+,T+)admits a clustered-planar embedding that satisfies Q.\nAltogether, this reduces the problem uCLPof biconnected single-source graphs to Sync\nCP, which can be solved efficiently by Lemma 2.1.\n▶Theorem 2.4. uCLPcan be solved in O(n3)time for biconnected single-source level\ngraphs.\n3 Hardness of y-monotone Clustered Level Planarity\nIt is easy to see that y-CLPlies in NP by guessing and verifying the augmentation edges and\nan embedding. We show that it is NP-hard even for inputs with very restricted properties.\n▶Theorem 3.1. y-monotone Clustered Level Planarity is NP-complete, even if the\ninput is a biconnected graph with just one source.\nProof Sketch. We reduce from the NP-complete problem Planar Monotone 3-SAT [7],\nwhich asks for the satisfiablity of 3-SATformulas whose incidence graph has a drawing\nwhere all variables lie on a vertical line ℓ, each clause contains only positive or only negative\nliterals, and the positive and negative clauses lie on opposing sides of ℓ; see Figure 3a.\nGiven such a drawing for a formula ϕ, we first reorient the clauses horizontally above\nthe variables as illustrated in Figure 3b. From this drawing, we construct an equivalent\ninstance of y-CLPas illustrated in Figure 3c. The variables and literals are represented by\ntriconnected pillars (a (3×k)-grid for suitable k) that extend vertically towards the clause\ngadgets. The horizontal flip of a pillar represents its truth value. To synchronize pillars of the\nsame variable, we use wedges that enclose a vertex of a disconnected cluster and, due to the\nrequired y-monotonicity for cluster connections, prohibit corresponding wedges of adjacent\npillars to face each other, as such a connection would have to bypass both wedges. Using\ntwo clusters per variable, we can thus ensure consistent flips for all variables; see Figure 3c.\nUsing a similar approach with wedges, we can construct a clause gadget that allows\nall assignments for its literals except when all three literals are false; see Figure 3c for\nthe structure of the gadget. As shown in Figure 4, there exists one configuration for the\nliteral pillars of a clause where no valid embedding is possible, as the cluster of the gadget\ncannot connect with only y-monotone curves. Figure 5 shows valid embeddings for all other\nconfigurations. This way, we can construct in polynomial time an equivalent instance of\ny-CLPwhere the graph is biconnected and only has a single source. ◀\nOur second reduction is from 3-Partition , whose input is a multiset A={a1,...,a 3m}\npositive integers and a bound B∈N+withB/4< ai< B/ 2and/summationtext\na∈Aa=m·B.\nThe question is whether Acan be partitioned into msetsA1,A2,...,Am, such that for\neveryj∈{1,...,m}it is/summationtext\na∈Aja=B.3-Partition is strongly NP-complete, i.e., it\nremains NP-complete even if Bis polynomial in m[12].6 Clustered Planarity Variants for Level Graphs\nA\nB\nC\nD\n(a)\nA\nB\nC\nD (b)\nA\nB\nC\nD\nA\nB\nB\nC\nD\nD\nX\nY\nZ\nX′\npX\npX′\npY\npZ\nA\nB\nC\npD\n(c)\nFigure 3 (a)An instance of Planar Monotone 3-SAT .(b)The modified incidence graph.\n(c)The structure of the corresponding y-CLPinstance. Highlighted are a clause gadget (top right)\nand the gadget for propagating variable assignments (bottom right).S.D. Fink, M. Pfretzscher, I. Rutter, M.D. Sieper 7\nX\nY\nZ\nX′\npZ\n(a)\nX\nY\nZ\nX′\npZ (b)\nFigure 4 Neither flip of X′admits a valid embedding of the clause gadget if all literals are false.\nX\nY\nZ\nX′\nZ\nX′\nX\nX′\nX\nY\nZ\nX′\nX\nY\nZ\nX′\nX\nY\nZ\nX′\nX\nY\nZ\nX′\nY\nX\nY\nZ\nFigure 5 The seven variable assignments for which the clause gadget admits a valid embedding.8 Clustered Planarity Variants for Level Graphs\nr\nt\nv1\nv2\nh\n(a)\nh\ns1\ns2\ns3\nv1 (b)\nwalls\n(c)\nFigure 6 (a)A receiver (bold) with a marked vertex r(box), and a second receiver (non-bold)\nchained to the first one. (b)A 3-plug. (c)A bucket of size 7, filled with two 2-plugs and a 3-plug.\nEvery marked bucket vertex can be connected to a pin with a y-monotone curve.\n▶Theorem 3.2. y-monotone Clustered Level Planarity is NP-complete, even if the\ninput contains only one non-trivial cluster, the number of levels is at most 5, and all vertices\nhave a fixed rotation.\nProof Sketch. Let(m,A,B )be an instance of 3-Partition . We construct an instance of\nuCLPwith a single non-trivial cluster µ. The main idea is to build mbuckets(the structure\nin Figure 6c) by chaining Breceivers (the structure in Figure 6a, each of which contains a\nconnector vertex rthat belongs to µ), and closing the sides of each bucket with two paths\n(also called walls). Note that each of the Bconnector vertices of a bucket must be connected\nto the rest of µdue to the paths of length 2 attached to the vertices marked as tin the\nreceiver; see Figure 6a. For every a∈A, we generate an a-plug; the structure in Figure 6b).\nThe leaves of a plug are called pinsand these are the only vertices that can link the connector\nvertices to the remainder of cluster µ. Given a plug and a bucket, either all of the vertices of\nthe plug are drawn between the two bucket walls, or none of them. Thus, we can model a\nsolutionA1,...,Amwith thembuckets, and a drawing assigns a∈AtoAiif and only if the\ncorresponding a-plug is in the i-th bucket. Since every pin can join at most one connector\nvertex, there are at least Bpins inside a bucket in a valid drawing. Since there are mbuckets\nand a total of m·Bpins, the instance is valid if and only if we can distribute the plugs\nin such a way that there are precisely Bpins per bucket, which corresponds directly to a\nsolution of (m,A,B ). ◀\n4 Conclusion\nWe have introduced the problems unrestricted Clustered Level Planarity andy-mono-\ntone Clustered Level Planarity , gave an polynomial-time algorithm for unrestricted\nClustered Level Planarity if restricted to biconnected single-source graphs, and showed\nthaty-monotone Clustered Level Planarity is NP-complete under very restricted\nconditions.\nWe conclude by providing some open questions. On the one hand, these are inspired by\nthe restrictions imposed by our algorithm. The LP-trees we use in Theorem 2.4 only exist\nfor biconnected single-source instances and it is unlikely that this concept can be extendedS.D. Fink, M. Pfretzscher, I. Rutter, M.D. Sieper 9\nto multiple sources [ 5, Section 5]. Is it possible to extend our algorithm to non-biconnected\ngraphs? More generally, what is the complexity of unrestricted Clustered Level Pla -\nnarity? On the other hand, the NP-hardness results on y-monotone Clustered Level\nPlanarity and (non-proper) Convex Clustered Level Planarity [1] raise the question\nwhether these problems are FPT with respect to natural parameters.\nReferences\n1Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, and Vincenzo\nRoselli. The importance of being proper: (in clustered-level planarity and t-level planarity).\nTheoretical Computer Science , 571:1–9, 2015. doi:10.1016/j.tcs.2014.12.019 .\n2Thomas Bläsius and Ignaz Rutter. A new perspective on clustered planarity as a\ncombinatorial embedding problem. Theoretical Computer Science , 609:306–315, 2016.\narXiv:1506.05673 ,doi:10.1016/j.tcs.2015.10.011 .\n3Thomas Bläsius, Simon D. Fink, and Ignaz Rutter. Synchronized planarity with applications\nto constrained planarity problems. ACM Transactions on Algorithms , 2023. doi:10.1145/\n3607474.\n4Kellogg S. Booth and George S. Lueker. Testing for the consecutive ones property, interval\ngraphs, and graph planarity using PQ-tree algorithms. Journal of Computer and System\nSciences, 13(3):335–379, 1976. doi:10.1016/S0022-0000(76)80045-1 .\n5Guido Brückner and Ignaz Rutter. An SPQR-tree-like embedding representation for level\nplanarity. Journal of Computational Geometry , 14(1):48–77, 2023. doi:10.20382/JOCG.\nV14I1A3.\n6Sabine Cornelsen and Dorothea Wagner. Completely connected clustered graphs. Journal\nof Discrete Algorithms , 4(2):313–323, 2006. doi:10.1016/j.jda.2005.06.002 .\n7Mark de Berg and Amirali Khosravi. Optimal binary space partitions for segments\nin the plane. Int. J. Comput. Geom. Appl. , 22(3):187–206, 2012. doi:10.1142/\nS0218195912500045 .\n8Simon D. Fink, Matthias Pfretzschner, and Ignaz Rutter. Experimental comparison of\nPC-trees and PQ-trees. ACM Journal of Experimental Algorithmics , 28(1):10:1–10:24, 2023.\ndoi:10.1145/3611653 .\n9Simon D. Fink and Ignaz Rutter. Maintaining triconnected components under node\nexpansion. In Marios Mavronicolas, editor, Proceedings of the 13th Conference on Algorithms\nand Complexity (CIAC’23) , volume 13898 of Lecture Notes in Computer Science , pages\n202–216. Springer, 2023. doi:10.1007/978-3-031-30448-4_15 .\n10Michael Forster and Christian Bachmaier. Clustered level planarity. In Peter van Emde Boas,\nJaroslav Pokorný, Mária Bieliková, and Julius Stuller, editors, Proceedings of the 30th\nConference on Current Trends in Theory and Practice of Computer Science (SOFSEM’04) ,\nvolume 2932 of Lecture Notes in Computer Science , pages 218–228. Springer, 2004. doi:\n10.1007/978-3-540-24618-3_18 .\n11Radoslav Fulek and Csaba D. Tóth. Atomic embeddability, clustered planarity, and\nthickenability. Journal of the ACM , 69(2):13:1–13:34, 2022. arXiv:1907.13086v1 ,doi:\n10.1145/3502264 .\n12Michael R. Garey and David S. Johnson. Complexity results for multiprocessor scheduling\nunder resource constraints. SIAM Journal on Computing , 4(4):397–411, 1975. doi:10.\n1137/0204035 .\n13Michael Jünger, Sebastian Leipert, and Petra Mutzel. Level planarity testing in linear\ntime. In Sue Whitesides, editor, Proceedings of the 6th International Symposium on Graph\nDrawing (GD’98) , volume 1547 of Lecture Notes in Computer Science , pages 224–237.\nSpringer, 1998. doi:10.1007/3-540-37623-2_17 .10 Clustered Planarity Variants for Level Graphs\nA Omitted Proofs from Section 2\n▶Lemma 2.1. Sync CP can be solved in O(n3)time.\nProof.We show the stronger statement that an instance (G,T,Q )ofSync CP can be solved\nin timeO(n+d·∆ +/summationtext\nQ∈Q|Q|)time, where dis the total number of crossings between an\nedge and a cluster boundary and ∆is the maximum number of such crossings on a single\ncluster boundary. The planarity of the input graph guarantees that m∈O(n)and thus\nd∈O(n2)and∆∈O(n). Because further/summationtext\nQ∈Q|Q|is inO(n), the statement of the Lemma\nfollows.\nWe will first summarize the linear reduction of Bläsius et al. [ 3] from Clustered Pla -\nnaritytoSynchronized Planarity to later show that it can easily also accomodate the\nsynchronized fixed-vertex constraints. We obtain the skeleton graphGνof the root cluster ν\nfrom the input graph Gby contracting all vertices in any child cluster µofνinto a single\nvertexvµ. We can similarly obtain the skeleton Gµof any non-root cluster µby not only\nindividually contracting all its respective child clusters, but also contracting all vertices\nV\\Vµ(that is, those not in µ) into a vertex vµ. Note that, for each non-root cluster µ, there\nis one vertex vµin its skeleton Gµinto which we contracted V\\Vµand one vertex vµin\nanother skeleton into which we contracted Vµ. A datastructure labeling each cluster in T\nwith its skeleton was introduced under the name CD-tree by Bläsius et al. [2].\nTheSynchronized Planarity instance corresponding to (G,T)is based on the graph\nG′comprising the union of all skeletons. Synchronized Planarity allows us to specify a\nset ofpipesPwhich is a matching on a subset of the vertices of G′. Any valid embedding will\nhave the rotation of one pipe endpoint vertex being the reverse of the rotation of the other\npipe endpoint it is matched with under a bijection given with the pipe. For each non-root\nclusterµ, we add a pipe between vµandvµtoP. The bijection naturally arises from the\nfact that the edges incident to vµandvµstem from the same edges in G, which are those\nbetweenVµandV\\Vµ.\nSynchronized Planarity additionally allows for subsets of the vertices to be assigned\na default rotation. Analogously to synchronized fixed-vertex constraints, in a valid Syn-\nchronized Planarity embedding, all vertices in the same such set are required to either\nall have their default rotation or all have the reverse of their default rotation. As G′includes\nall vertices from V, we can simply formulate our synchronized fixed-vertex constraints Qas\nwhat is called Q-constraints inSynchronized Planarity . The main difference is that in\nSynchronized Planarity , each vertex may be part of only one Q-constraint. This can be\nestablished in O(/summationtext\nQ∈Q|Q|)time by merging sets that constrain a common vertex. (If the\nsame vertex is constrained with default rotations that are neither the same nor the reverse\nof each other, we can immediately conclude that this is a no-instance.)\nAlltogether, we obtain an instance (G′,P,Q )ofSynchronized Planarity ,2for which\nwe will now show equivalence with the input Clustered Planarity with Synchronized\nFixed-Vertex Constraints instance (G,T,Q ).\nLetEbe a valid embedding of (G,T,Q ). We show that there is a valid embedding\nE′of the instance (G′,P,Q ). AsEis a c-planar embedding of (G,T), there is a c-planar\ndrawingGof(G,T)that uses the rotations of E. In this drawing, we can contract the region\nRµrepresenting cluster µ(and thereby also its vertices Vµ) into a single point vµwhile\n2Note that in the original definition of Synchronized Planarity [3], the vertex sets and default\nrotations of Qare given as separate variables. To keep notation consistent, we will not make this\ndistinction here.S.D. Fink, M. Pfretzscher, I. Rutter, M.D. Sieper 11\nmaintaining planarity as well as the rotations of all remaining vertices. Similarly, we can\nalso contract the outside of any region Rµ(and thereby the vertices V\\Vµ) into a single\npointvµ, again maintaining planarity and vertex rotations. In this way, we can obtain a\nplanar drawing and therefore also a planar embedding for each skeleton Gµ. Combining\nthese embeddings yields and embedding E′ofG′. Note that asEis c-planar and thus the\ncyclic orders in which edges cross a cluster region boundary are the same on both sides of\nthe boundary, the rotations of vµandvµare the reverse of each other for every µ. Thus,\nthe constraints arising from the pipes Pare satisfied byE′. Furthermore, note that we\ndid not change the rotation of any vertex of VinE′. As the set of vertices constrained by\nthe synchronized fixed-vertex constraints Qis a subset of V, the Q-constraints Qare also\nsatisfied in the Synchronized Planarity embeddingE′.\nConversely, letE′be a valid embedding of the Synchronized Planarity instance\n(G′,Q,P ). We show there is a valid embedding Eof the corresponding instance (G,T,Q )\nofClustered Planaritywith Synchronized Fixed-Vertex Constraints . LetE′\nν\nbe the planar sub-embedding of E′for the skeleton Gνof the root cluster ν. LetG′\nνbe a\ndrawing ofE′\nνand similarly, letG′\nµbe a drawing of a child cluster µofν. Consider a disk Rµ\naroundvµinG′\nνthat only contains (parts of) the edges incident to vµ. Similarly, consider a\ndiskRµaroundvµinG′\nµ. Thevµ-vµpipe defines a bijection φbetween these incident edges,\nand the cyclic order under which these edges intersect RµandRµare the reverse of each\nother. We can therefore replace the inside of Rµwith the outside of Rµand connect edges\nintersecting the disks according to φwhile mainting planarity. Effectively, this inserts G′\nµ\ninstead ofvµinG′\nνand thereby undoes the contraction of Vµ. Recursively applying this\nprocedure for all non-root clusters undoes all contractions and thus yields a drawing GofG.\nThe drawing is planar as the individual drawings are planar and the orders of the identified\nedges are the reverse of each other. The drawing is also c-planar when using Rµas region\nfor clusterµ. Thus, we obtain a c-planar embedding EofG. It only remains to show the\nsynchronized fixed-vertex contraints Qare satisfied, which again directly follows from the\nrotations of any vertex in Vremaining unchanged by the performed insertions.\nThis concludes the proof that we obtain an equivalent instance of Synchronized Pla -\nnarityfor any instance of Clustered Planaritywith Synchronized Fixed-Vertex\nConstraints . The size of the resulting Synchronized Planarity instance is bounded by\nO(n+d)[3] and it can be solved in O(n+d·∆)time [9], wheredis the total number of\ncrossings between an edge and a cluster boundary and ∆is the maximum number of such\ncrossings on a single cluster boundary. ◀\n▶Lemma 2.2. An embeddingEis level-planar if and only if the rotation of each vertex vis\nrepresented by its level PQ-tree Tvand moreoverEisρ-consistent.\nProof.The necessity is clear, the sufficiency follows from the fact that the conditions ensure\nthat the planar embedding Edetermines either the same embedding of the rigid structure ρ\nas the level-planar embedding Γor its reverse. It therefore follows that an embedding\nthat satisfies these additional conditions is represented by the LP-tree and is therefore\nlevel-planar [5]. ◀\n▶Lemma 2.3. (G,γ,T )admits a planar embedding that is level-planar and clustered-planar\nif and only if (G+,T+)admits a clustered-planar embedding that satisfies Q.\nProof.LetEbe an embedding that is both level-planar and clustered planar. By Lemma 2.2,\nsinceEis level-planar, the rotation around each vertex vis represented by the corresponding\nLP-treeTv. We can therefore perform the replacement of the vertices in the construction12 Clustered Planarity Variants for Level Graphs\nofG+also in the embedding E, which thereby defines a unique planar embedding E+ofG+.\nSince this expansion can be done in a drawing that includes the cluster boundaries without\nintroducing edge–cluster crossings that are not allowed, the resulting embedding E+is\nclustered planar. Again by Lemma 2.2 the fact that Eis level-planar further implies that E\nisρ-consistent and hence that E+satisfiesQ.\nConversely, assume that E+is a clustered-planar embedding of (G+,T+)that satisfiesQ.\nLetEbe the embedding obtained from Gby contracting each of the trees Tvback into a\nsingle vertex v. Since we only contract edges whose endpoints lie in exactly the same clusters,\nit follows thatEis clustered planar. Moreover, since the rotation around each vertex vofG\ninEis obtained by contracting an embedding of TvinE+, it follows that the rotation of v\ninEis represented by its level PQ-tree Tv. Finally, the fact that E+is level-planar further\nimplies thatEisρ-consistent. By Lemma 2.2 it follows that Eis level planar. ◀\n▶Theorem 2.4. uCLPcan be solved in O(n3)time for biconnected single-source level\ngraphs.\nProof.Let(G,T,γ )be a clustered biconnected single-source level graph and let (G+,T+),Q\ndenote the clustered graph and the synchronized fixed-vertex constraints constructed above.\nBy Lemma 2.3, (G,T,γ )is clustered level-planar if and only if (G+,T+)admits a clustered-\nplanar embedding that satisfies Q. The latter is a simple instance of Clustered Planari -\nty with Synchronized Fixed-Vertex Constraints , which can be solved efficiently\nby Lemma 2.1.\nFor the running time, observe that the LP-tree for (G,γ)can be computed in linear\ntime [5]. The total size of all level PQ-trees TvofGisO(n)and they can be computed in\nthe same running time. We can thus compute (G+,T+)and also the constraints Qin linear\ntime. Then the running time follows from Lemma 2.1. ◀\nB Omitted Proofs from Section 3\n▶Theorem 3.1. y-monotone Clustered Level Planarity is NP-complete, even if the\ninput is a biconnected graph with just one source.\nProof.We reduce from the NP-complete problem Planar Monotone 3-SAT [7], which is\ndefined as follows.\nA3-SATformula is monotone if every clause is either positiveornegative, i.e., it contains\nonly positive or only negative literals, respectively. The incidence graph of a3-SATformula\nis a bipartite graph between its clauses and variables, where the edges correspond to the\noccurrences of variables in the clauses. In a monotone rectilinear drawing of the incidence\ngraph, all variables and clauses are represented by rectangles, the variables lie on a vertical\nline and the positive and negative clauses lie on opposing sides of this line. The edges are\nrepresented by horizontal straight-line segments. Given as input a monotone 3-SATformula\nϕtogether with a planar monotone rectilinear drawing Eof its incidence graph, Planar\nMonotone 3-SAT asks whether ϕis satisfiable.\nGiven an instance (ϕ,E)ofPlanar Monotone 3-SAT , we construct an equivalent\ninstance ofy-CLPas follows. Figure 3 shows the overall structure of the resulting instance, as\nwell as the gadgets for clauses and for propagating variable assignments. In our construction,\nwe implicitly allow edges between vertices on the same level, as these can be easily eliminated\nby adding intermediate levels and distributing the adjacent vertices among them.S.D. Fink, M. Pfretzscher, I. Rutter, M.D. Sieper 13\nWe begin by generating a new rectilinear drawing E′out ofEby reorienting the clause\nrectangles horizontally, placing the clauses above the variables; see Figure 3b. This introduces\na bend in every edge. We now construct a clustered level graph out of E′.\nThe variables of ϕare represented by triconnected pillars (a (3×k)-grid for suitable k) that\nare placed on a vertical line, where the horizontal flip of the pillar represents the corresponding\ntruth value. Each variable is associated with a dedicated horizontal area containing pillars\nthat represent the corresponding literals of the variable. The pillars corresponding to positive\nliterals lie to the right of the variable, while the negative literals lie to the left. These pillars\nextend to the top where they join in their respective clause gadgets.\nTo synchronize the rotation of all pillars corresponding to the same variable, we use the\nfollowing technique. For each pillar Dthat requires synchronization, we add a degree-2\nvertexpD, whose level lies above that of its neighbors, to the side of D, forming a wedge\ntowards the top; see Figure 3c. The level of pDis chosen high enough such that the wedge\nmust always lie on the outer face of D. Moreover, we add a cluster Hthat spans both sides of\nthe pillar (red cluster in Figure 3c), such that the end of Hon the side of pDlies within the\nwedge, and the end on the other side lies above pD. The wedge, together with the required\ny-monotonicity, ensures that the vertex of Hit encloses can only connect to vertices that lie\nabovepD. Therefore, if we have two adjacent pillars that contain this structure, the sides\ncontaining the wedges cannot face each other, as otherwise, the vertices of cluster Hcannot\nconnect. Repeating this construction using wedges on the other side of the pillars and a\ndifferent cluster (blue cluster in Figure 3c), we can thus synchronize the rotation of pillars.\nWe use this technique to synchronize all literals corresponding to the same variable using\ntwo clusters.\nFor every clause cofϕ, the corresponding gadget is structured as follows. Let x,yand\nzdenote the three variables contained in c, in the order induced by their appearance in c\ninE. We letX,Y, andZdenote the corresponding pillars. Pillar Yis further subdivided\nsuch that the topmost portion X′can rotate independently to mimic and propagate the\nconfiguration of X. Note that the clause gadget should yield a valid embedding of y-CLPfor\nall variable assignments except one. We ensure this using vertices of a disconnected cluster\nHc(green vertices in Figure 3c) that seek to connect to one another. At the top of the clause\ngadget, there are connected vertices of Hc, allowing vertices of HcinXandZto always\nconnect if they lie on the outer face. Similar to the gadget for synchronizing variables, each\nof the pillars V∈{X,X′,Y,Z}has a degree-2 vertex pVon one side forming a wedge that\nencloses a vertex of Hc. As before, this construction allows us to ensure that the wedges of\nXandX′can never face each other. The wedge pZof pillarZlies below all other wedges\nand, in contrast to the other pillars, the cluster Hcdoes not span to the other side of Z. We\nrefer to Figure 3c for the exact structure of the clause gadget.\nFigure 4 shows the one configuration of truth values for the three variables that does\nnot allow an admissible embedding of the clause gadget. For both possible flips of X′, the\ny-monotonicity we require for the connections between cluster components prohibits a valid\nembedding of the gadget. Either the vertex of cluster Hcthat is enclosed by the wedge\npZinZcannot connect to any other vertex of Hc(Figure 4a), or the cluster component\ncontained in X′cannot connect to the cluster component contained in X(Figure 4b). In\nboth cases, this configuration thus leads to an invalid embedding. For each of the other seven\nconfigurations, a valid embedding is possible; see Figure 5.\nSince the clause gadget enforces valid variable assignments and the literal gadget synchro-\nnizes the flip of corresponding literals, the resulting instance of y-CLPis a yes-instance if\nand only if the original instance of Planar Monotone 3-SAT was a yes-instance. Further,14 Clustered Planarity Variants for Level Graphs\nr\nt\nv1\nv2\nh\n(a)\nh\ns1\ns2\ns3\nv1 (b)\nwalls\n(c)\nFigure 7 (a) A receiver (bold) with a marked vertex r(box), and a second receiver (non-bold)\nchained to the first one. (b) A 3-plug. (c) A bucket of size 7, filled with two 2-plugs and a 3-plug.\nEvery marked bucket vertex can be connected to a pin with a y-monotone curve.\nthe resulting instance is single-source and, after adding an additional path on the outer face\nconnecting the topmost and bottommost variables, the instance is also biconnected. ◀\n▶Theorem 3.2. y-monotone Clustered Level Planarity is NP-complete, even if the\ninput contains only one non-trivial cluster, the number of levels is at most 5, and all vertices\nhave a fixed rotation.\nProof.We describe a reduction from 3-Partition toy-CLPwith 5 levels.\nAreceiver is the structure described in Figure 7(a). Its main component are its two\nendpointsv1,v2on level 1, and a path connecting them that goes from level 1 to a vertex\nton level 4 and back. We mark the unique vertex on the path from v1tot, that lies on\nlevel 3 and call it r. Additionally, there are two paths of length 2, also called blockers\nconnected to t, that both have a vertex on level 3 and then go back to level 4. If the order\nofv1,v2is fixed, Figure 7(a) shows the unique level planar embedding (up to swapping the\nblockers) of a receiver. The two arms assure that the marked vertex rcan only be reached\nwith ay-monotone curve from a vertex on level 2 or lower. Two receivers can be chained\nby identifying v2of the first receiver with v1of the second receiver. A bucketof sizeBis\nobtained by chaining Breceivers in this manner, and then adding a path (also called wall) to\nlevel 5 to each of the two unchained endpoints ( v1of the first receiver, and v2from the last).\nA bucket contains Bmarked vertices r1,r2,...,rB, one for each of the chained receivers. Like\na receiver, the level planar embedding of a bucket is unique up to swapping of blockers and\nmirroring along the y-axis. Thus, every valley formed by two receivers or the first receiver\nwith its wall contains precisely one marked vertex. A plugof sizek(also called a k-plug) are\nkpaths from level 2 to level 4, and a vertex hon level 5 that is connected to all path vertices\non level 4, as shown in Figure 7(b). We call the kvertices on level 2 the pinss1,s2,...,skof\nthek-plug. If a graph contains a plug and a bucket, then in every level-planar drawing of\nthis graph the pin has to lie either completely inside the area bounded by the two walls, or\ncompletely outside of it (that is, either every vertex of the pin is drawn between the two\nwall-vertices of its level, or none of them). Further, if a plug is drawn in a bucket, every pin\nof this plug is drawn in a valley of the chain, and the only marked vertex it can be connected\nto via ay-monotone curve is the (at most one) marked vertex of this valley. Figure 7(c)S.D. Fink, M. Pfretzscher, I. Rutter, M.D. Sieper 15\nshows a bucket of size 7 that contains two 2-plugs and one 3-plug, and in which every marked\nvertex of the bucket can be connected to a pin by a y-monotone curve.\nNow let (m,A,B )be an instance of 3-Partition whereBis polynomial in n, and let\nn= 3m. We construct an instance Gofy-CLP. The graph consists of mbuckets and an\nai-plug for every ai∈A. The clustering is given by tree Twith one inner vertex that has as\nchildren all plug vertices, all marked vertices of the receivers (in total m·B) and the level 4\nand 5 vertices of the bucket-walls. All of the other vertices in Gare children of the root in\nT. In other words, besides of the trivial cluster that contains all vertices, there is only one\nclusterCthat contains the plugs, the marked vertices, and the upper parts of the walls. It is\neasy to verify that the size of this instance is bounded in O(m·B), which is polynomial in\nthe size of the 3-Partition problem. Thus, it can also be generated in polynomial time. It\nis left to show that there exists a solution A1,A2,...,Amfor(m,A,B )if and only if Gis\ny-monotone cl-planar.\nAssumeGisy-monotone cl-planar, and let Gbe ay-monotone cl-planar drawing of G.\nLetPibe a bucket of G. Since the non-trivial cluster Cis connected, every marked vertex\ntinPiis connected to the cluster with a y-monotone curve. It cannot be connected to a\ncluster-vertex of a wall since the arms obstruct this path. So the only cluster-vertices it can\nbe connected to are pins. Since every bucket contains Bmarked vertices there are at least B\npins present in Pi. This holds for all buckets, therefore there are precisely Bpins inPi, and\nif a pin is in Pi, all pins of the same plug are as well. This means there exists a partition\nK1,K2,...,Kmof the plugs, such that the values corresponding to the plugs in every set\nsum up toB. The setsA1,A2,...,Am, where every Aicontains the values corresponding to\nthe numbers of the plugs in Ki, are a valid solution for (m,A,B ).\nNow assume that there exists a valid solution A1,A2,...,Amfor(m,A,B ). We construct\nay-monotone cl-planar drawing GforG, in which every bucket will be filled with plugs as\nin Figure 7(c). In the first step, we draw the buckets P1,...,Pmin this order from left to\nright next to each other, and for each i∈{1,...,m−1}we connect the level-4 vertex of\nthe left wall of Piand the level-5 vertex of the right wall of Pi+1with a straight line (thus\npairwise connecting these cluster components). Now for every i∈{1,...m}we draw the\nplugs corresponding to the set Aiinto the bucket Pifrom left to right, such that every valley\nwith a marked vertex contains precisely one pin. The pins can now be connected with a\nmarked vertex teach, connecting every cluster component of a marked vertex to a plug\ncluster component. At the end, in every bucket we draw connections from the left wall to\nthe first plug, the first to the second plug, the second to the third plug, and the third plug\nto the right wall (always with a straight line from a level-4 vertex to a level-5 vertex). In\nthis manner, Cis connected with y-monotone curves in GandGis cl-level-planar, so Gis a\nvalid solution for G.\nClearly, this result holds even if the rotation system of the graph is fixed. ◀"    },    {        "title": "2402.13159v1.Barking_dogs__A_Fréchet_distance_variant_for_detour_detection.pdf",        "content": "Barking dogs: A Fréchet distance variant for\ndetour detection\nIvor van der Hoog /envel⌢pe\nTechnical University of Denmark, Denmark\nFabian Klute /envel⌢pe\nUniversitat Politècnica de Catalunya, Spain\nIrene Parada /envel⌢pe\nUniversitat Politècnica de Catalunya, Spain\nPatrick Schnider /envel⌢pe\nDepartment of Computer Science, ETH Zürich, Switzerland\nAbstract\nImagine you are a dog behind a fence Qand a hiker is passing by at constant speed along the hiking\npathP. In order to fulfil your duties as a watchdog, you desire to bark as long as possible at the\nhuman. However, your barks can only be heard in a fixed radius ρand, as a dog, you have bounded\nspeeds. Can you optimize your route along the fence Qin order to maximize the barking time with\nradiusρ, assuming you can run backwards and forward at speed at most s?\nWe define the barking distance from a polyline Ponnvertices to a polyline Qonmvertices as\nthe time that the hiker stays in your barking radius if you run optimally along Q. This asymmetric\nsimilarity measure between two curves can be used to detect outliers in Qcompared to Pthat other\nestablished measures like the Fréchet distance and Dynamic Time Warping fail to capture at times.\nWe consider this measure in three different settings. In the discrete setting, the traversals of Pand\nQare both discrete. For this case we show that the barking distance from PtoQcan be computed\ninO(nmlogs)time. In the semi-discrete setting, the traversal of Qis continuous while the one of\nPis again discrete. Here, we show how to compute the barking distance in time O(nmlog(nm)).\nFinally, in the continuous setting in which both traversals are continuous, we show that the problem\ncan be solved in polynomial time. For all the settings we show that, assuming SETH, no truly\nsubquadratic algorithm can exist.\n2012 ACM Subject Classification Theory of computation →Design and analysis of algorithms\nKeywords and phrases Outlier detection, similarity measures, Dynamic Time Warping, Fréchet\ndistance, algorithms, computational geometry\nFunding Ivor van der Hoog : This project has received funding from the European Union’s Horizon\n2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No\n899987.\nFabian Klute : F.K. is supported by a “María Zambrano grant for attracting international talent”.\nIrene Parada : I.P. is a Serra Húnter fellow.\n1 Introduction\nAcurveis any sequence of points in Rdwhere consecutive points are connected by their line\nsegment. Curves may be used to model a variety of real-world input such as trajectories [ 23],\nhandwriting [ 22,30] and even strings [ 8]. Curves in R1may be seen as time series which\nmodel data such as music samples [ 21], the financial market [ 24] and seismologic data [ 29].\nA common way to analyse data that can be modeled as curves is to deploy a curve similarity\nmeasure, which for any pair of curves series (P,Q)reports a real number (where the number\nis lower the more ‘similar’ PandQare). Such similarity measures are a building block\nfor common analysis techniques such as clustering [ 16,27], classification [ 3,18,19] orarXiv:2402.13159v1  [cs.CG]  20 Feb 20242 Barking dogs: A Fréchet distance variant for detour detection\nsimplification [ 5,14,25]. The two most popular similarity measures for curve analysis\nare the Fréchet distance and the Dynamic Time Warping (DTW) distance. The discrete\nFréchet distance for two curves P= (p1,...,pn)andQ= (q1,...,qm)is illustrated as follows.\nImagine a dog walking along Qand its owner walking along P. Both owner and dog start at\nthe beginning of their curves, and in each step the owner may stay in place or jump to the\nnext point along Pand the dog may stay in place or jump to the next vertex along Q, until\nboth of them have reached the end of their curves. Intuitively, the Fréchet distance is the\nminimal length of the leash between the dog and its owner. The DTW distance is defined\nanalogously but sums over all leash lengths instead.\nContinuous distance measures. The downside of discrete distance measures is that they\nare highly dependent on the sample rate of the curve. Both distance measures can be made\ncontinuous by defining a traversal as continuous monotone functions f: [0,1]→Pand\ng: [0,1]→Qwhich start and end at the respective start and end of the curve. Given a\ntraversal (f,g), the cost of a traversal under the Fréchet distance is then the maximum over\nallt∈[0,1]of the distance between the dog and its owner at time t. The DTW distance\nmay also be made continuous, by defining the cost of a traversal as the integral over [0,1]of\nthe distance between the dog and its owner. However, such a direct translation from discrete\nto continuous traversals invites degenerate behavior. To avoid such degeneracies, Buchin [ 13]\nproposed several variants of continuous DTW distances (originally called average Fréchet\ndistance) that each penalise the speed of the dog and its owner.\nComputing Fréchet and dynamic time warping distance. The time complexity of computing\nthe Fréchet and DTW distance of two static curves is well-understood: Given two curves P\nandQof lengthnandm, we can compute their DTW distance and discrete Fréchet distance\nin timeO(nm)by dynamic programming approaches. This running time is almost the best\nknown, up to mild improvements [ 4,17]. For continuous Fréchet distance the best known\nalgorithms run in slightly superquadratic time [ 10]. Bringmann [ 7] showed that, conditioned\non SETH, one cannot compute even a (1 +ε)-approximation of the Fréchet distance between\ncurves in R2under theL1,L2,L∞metric faster than Ω((nm)1−δ)time for any δ>0. This\nlower bound was extended by Bringmann and Mulzer [ 9] to intersecting curves in R1. Buchin,\nOphelders and Speckmann [ 12] show this lower bound for pairwise disjoint planar curves in\nR2and intersecting curves in R1. For the DTW distance similar SETH- or OVH-conditioned\nlower bounds exist [ 1,8] (even for curves in R1). Eor the DTW distanc, even if we relax\nthe goal to a constant-factor approximation no algorithm running in truly subquadratic\ntime is known (see [ 20] for polynomial-factor approximation algorithms and approximation\nalgorithms for restricted input models), though for this approximate setting conditional lower\nbounds are still amiss. Recently, a subquadratic constant-approximation for the continuous\nFréchet distance was presented by van der Horst et al. [26].\nDownsides of existing similarity measures. The existing curve similarity measures each\nhave their corresponding drawback: The Fréchet distance is not robust versus outliers. The\ndiscrete DTW distance is heavily dependent on the sampling rate. The continuous DTW\nvariants are robust to outliers, but they are difficult to compute [11].\nA measure for outliers. We present a new curve similarity measure, specifically designed\nfor computing similarities between curves under outliers, by applying a natural thresholding\ntechnique. Our algorithmic input receives in addition to PandQsome threshold ρ. If there\nexists a traversal of PandQwhere at all times the dog and its owner are within distance ρ\nthen their distance is 0. Otherwise, the distance is the minimal time spent where the dog and\nits owner are more than ρapart. Formally, we define a threshold function θρ(p,q)which is 1I. van der Hoog, F. Klute, I. Parada, and P. Schnider 3\nwheneverd(p,q)>ρand zero otherwise. Given a traversal (f,g), the cost of the traversal is\nthen the integral over [0,1]ofθρ(f(t),g(t)). To avoid the same degenerate behavior as under\nthe continuous DTW distance, we restrict the speeds of the dog and its owner. In particular,\nwe create an asymmetric measure where we set the speed of the human to be constant and\nupper bound the speed of the dog by some input variable s.\n2 Preliminaries\nWe consider Rd, for constant d, where for two points p,q∈Rdwe denote by d(p,q)their\nEuclidean distance. We define a curveas any ordered set of points (vertices) in Rd. When\nsequential vertices are connected by their line segment, this forms a polyline. For any curve\nP,Sis a subcurve of Pwhenever its vertices are a subsequence of those of P. As algorithmic\ninput we receive two curves P= (p1,p2,...,pn)andQ= (q1,q2,...qm). By|P|and|Q|\nwe denote their Euclidean length (the sum of the length of all the edges in the polyline).\nByP[t](Q[t])we denote a continuous constant-speed traversal of P(Q). That is, P[t]is a\ncontinuous, monotone function for t∈[1,n]to the points on the polyline defined by Pwhere\nP[1] =p1,P[n] =pnand for allt,t′∈[1,n]the distance between P[t]andP[t′](alongP) is\nequal to|P|\n|t−t′|.\nDiscrete walks and reparametrizations. Given two curves PandQ, we define discrete walks\nand reparametrizations. First, consider the n×minteger lattice embedded in R2. We can\nconstruct a graph Gnmover this lattice where the vertices are all lattice points and two\nlattice points l1,l2share an edge whenever d(l1,l2)≤√\n2.\n▶Definition 1. For curves PandQ, adiscrete reparametrization Fis any walk in Gnm\nfrom (1,1)to(n,m).Fis a curve in R2and it isx-monotone whenever its embedding is.\nThespeedσ(F)is the size|S|for the largest horizontal or vertical subcurve S⊆F.\n▶Definition 2. For a curve P, a reparametrization of Pis a differentiable (monotone)\nfunctionα: [0,1]→[1,|P|]whereα(0) = 1andα(1) =|P|. For any two curves (P,Q)with\nreparametrizations (α,β)we define the speedas:σ(α,β) := max\nt∈[0,1]max{α′(t),β′(t)}.\n▶Definition 3. For a curve P, a one-sided reparametrization (OSR) from PtoQis a\ndifferentiable (monotone) function γ: [1,|P|]→[1,|Q|]whereγ(1) = 1andγ(|P|) =|Q|. For\nany two curves (P,Q)with reparametrization γwe define the speedasσ(γ) := max\nt∈[1,|P|]|γ′(t)|.\nIntroducing speed restrictions to overcome degeneracies. As discussed above any sensible\ndefinition of continuous DTW distance needs to somehow restrict the speedof the repara-\nmertization. There have been several attempts at this. Buchin [ 13, Chapter 6] proposed\nthree ways of penalising using near-infinite speed by incorporating a multiplicative penalty\nfor the speed in the definition of continuous DTW. Buchin, Nusser and Wong [ 11] present\nthe current best algorithm for computing the first variant in R1, with a running time of\nO(n5). More closely related to our upcoming definition is the third definition in [ 13] which is\none-sided and therefore asymmetric. Alternatively, the speed of the reparametrization may\nsimply be capped by some variable s.\nOutlier detection: speed bounds and thresholds. The Fréchet distance and DTW are both\ndistance measures that are often used to compare sampled (trajectory) data. However, in\npractice sampled trajectory data often contains measurement errors, leading to outliers or\ndetours in the data, which we would like to detect. For this consider two polylines PandQ,4 Barking dogs: A Fréchet distance variant for detour detection\nP\nQ\n\u001a\nd\nFigure 1 An intended trajectory Pand a faulty sample Qof it. The Fréchet distance between P\nandQisdand captures the first detour, but fails to capture the others. Continuous DTW, even\nwith a speed bound, cannot distinguish Qfrom a copy of Ptranslated by ρif the right part is\nsufficiently long. Barking distance with barking radius ρhowever captures all three detours.\nwhere we think of Pas the intended trajectory and Qas a potentially faulty sample. We\nwant to determine whether Qis an accurate sample for Por detect outliers if there are any.\nUnfortunately, as can be seen in Figure 1, both the Fréchet distance as well as (continuous)\nDTW are not adequate measures for this task.\nDefining Barking Distance. The barking distance stems from the following illustration,\nwhich is again dog-based:1assume you are hiking with constant speed along a curve P. A\ndog is running at bounded speed on a curve Q, constantly barking at you. However, the dogs\nbarks can only be heard within radius ρ∈R. The dog tries to optimize its route in order\nto maximize the time you hear it. This maximum time is the barking distance of PtoQ.\nFormally, for ρ∈Rwe define the threshold function as follows:\nθρ(p,q) =/braceleftigg\n1ifd(p,q)>ρ\n0otherwise.\nJust as for Fréchet and DTW distance, we now define our distance measure in the\ncontinuous and discrete variants. Observe that our definition includes a speed constraint s,\nto avoid the aforementioned degeneracy where one avoids detecting outliers by traversing\nthrough them at high speed. We choose a one-sided definition. Like this, our definition stays\nclose to the asymmetrical behavior of our hiker and dog. Additionally, for detecting outliers,\nan asymmetric measure may also be more desirable since the input is inherently asymmetric.\n▶Definition 4. For curvesPandQ, denote by Fthe set of all pairs of discrete x-monotone\nreparametrizations of (P,Q). For anyρ,s∈R, the discrete barking distance is defined as:\nDs\nB(P,Q) = min\nF∈F\nσ(F)≤s/summationdisplay\n(i,j)∈Fθρ(pi,qj).\n1This illustration is inspired by a dog that some of the authors met while on a hike.I. van der Hoog, F. Klute, I. Parada, and P. Schnider 5\n▶Definition 5. For curvesPandQ, denote by Gthe set of all one-sided reparametrizations\nfromPtoQ. For anyρ,s∈R, the continuous barking distance is defined as:\nDs\nB(P,Q) = min\nγ∈G\nσ(γ)≤s/integraldisplay|P|\n0θρ(P[t],Q[γ(t)])dt.\nWe also define a version of the barking distance that is in between the discrete and\ncontinuous barking distance. In this semi-discrete barking distance definition, the person has\na discrete constant traversal of P(spending|(pi,pi+1)|time at vertex pi, before continuing\nto vertexpi+1) whilst the traversal of the dog is continuous. Formally:\n▶Definition 6. For curvesPandQ, denote by Gthe set of all one-sided reparametrizations\nfromPtoQ. Consider the step function α: [0,|P|]whereα[t] :=|P[1,i]|whenevert∈\n[|P[1,i]|,|P[1,i+ 1]|]. For anyρ,s∈R, the semi-discrete barking distance is defined as:\nDs\nB(P,Q) = min\nγ∈G\nσ(γ)≤s/integraldisplay|P|\n0θρ(P[α(t)],Q[γ(t)])dt.\nContribution. Our main contribution are efficient algorithms to compute the barking distance\nin several settings. In Section 3 we show that the discrete barking distance can be computed\nin timeO(nmlogs). In Section 4 we show that the semi-discrete barking distance can be\ncomputed in time O(nmlog(nm)). In Section 5 we show that the continuous barking distance\ncan be computed in time O(n4m3log(nm)). While slower than the best known algorithm for\nDTW for time series [ 11], our algorithm works for curves in Rdfor constant d. In Section 6\nwe give conditional lower bounds showing that no truly subquadratic algorithm may exist to\ncompute any variant, assuming the Strong Exponential Time Hypothesis (SETH). Hence, our\nalgorithms for the discrete and semi-discrete barking distance are near-optimal. A feature of\nour reduction is that the inputs (P,Q)are not only pairwise disjoint but also simple.\n3 The discrete setting\nLetGn,m= (V,E)be a graph defined on top of an n×mlattice in R2wherevi,j∈Vis\nidentified with the lattice point at coordinate (i,j). We find (vi,j,vi′,j′)∈Ewith distinct\nvi,j,vi′,j′∈Vwhenevervi,jandvi′,j′are identified with points of the lattice at distance\n≤√\n2. We say that vi′,j′is thesouthern,south-western ,western,north-western , ornorthern\nneighbor of vi,jifvi′,j′lies in the corresponding cardinal direction in the lattice.\nForvi,j∈Vwe setw(vi,j) =θ(pi,qj), withpithei-th corner of Pandqjthej-th corner\nofQ. Similarly, we set w(π) =/summationtextk\na=0w(via,ja)for a walkπ= (vi1,j1,...,vik,jk)inGn,m. We\nsay thatπismonotone ifja≤ja′whenevera≤a′and we define the lengthofπas|π|,\ni.e., the number of vertices in the walk. A sub-walk of πis said to be horizontal if all its\nvertices correspond to lattice points with the same y-coordinate and verticalif all its vertices\ncorrespond to lattice points with the same x-coordinate. Moreover, we say that πhasspeeds\nif the longest horizontal or vertical sub-walk of πhas length at most s. Let Π(s,ρ)be the set\nof all monotone walks in Gn,mstarting at v1,1and ending at vn,mwith speed sand weight\nfunction depending on the threshold ρ. The next observation now follows with Definitions 1\nand 4.\n▶Observation 7. Given two polygonal curves PandQ, a threshold ρ, and a speed bound s,\nletGn,mbe defined as above, then w(π) =Ds\nB(P,Q)for anyπ∈Π(s,ρ)of minimum weight.6 Barking dogs: A Fréchet distance variant for detour detection\nBy Observation 7 we can restrict our attention to monotone paths from v1,1tovn,mthat\nhave speed at most sand are of minimum weight. Our strategy is to compute for each vertex\nvi,j∈Vthe weight of such a path from v1,1and tovi,j. Our computation will proceed in n\nrounds, where in each round we consider the mvertices of column j. The challenge is to\ncompute the length of a minimum weight monotone path of speed sin timeO(logs).\nLetRi(j1,j2)be the weight of path (vi,j1+1,vi,j1+2,...,vi,j2)andCj(i1,i2)be the weight\nof path (vi1+1,j,vi1+2,j,...,vi2,j). Observe, that these values can be computed in constant\ntime if we have arrays containing at position ithe length of a path from the first element\nin the row or column to the i-th element of the row or column. For each row and column\nand taking either side as the starting vertex. We precompute these arrays for all rows at the\nbeginning and for each column only when we process this column in the computation.\nLetFδ(i,j)withδ∈D={↑,↗,→,↘,↓}be the minimum weight of a monotone path\nof speedsfromv1,1tovi,jwhere the vertex preceding vi,jon the path is the southern,\nsouth-western, western, north-western, or northern neighbor of vi,j, respectively. We set\nFδ(i,j) =∞ifvi,jcannot be reached with any monotone path of speed sfromv1,1.\nWe then compute the minimum weight monotone path of speed sfromv1,1tovi,jas\nF(i,j) =min{Fδ(i,j)|δ∈D}. To compute F(i,j)from left to right along the columns\nwe maintain the relevant minima of paths Fδending at vertices around vi,jfor each row\nand for the current column in separate heaps. Moreover, instead of updating the weights\nof all heap-elements explicitly for each vi,j, we precompute the lengths of paths starting at\nthe beginning or end of a row or column. From this we can in constant time compute the\nnecessary offsets. The runtime of O(nmlogs)then follows as every of the O(nm)elements\ngets only inserted and deleted from some min-heap a constant number of times and at no\npoint any min-heap contains more than selements.\nFor the following proof we rewrite Fd(i,j)as a recurrence taking the speed-bound sinto\naccount for j >1. Recall that w(vi,j)contributes to the values of CjandRi.\nFd(i,j) =\n\nmin{Cj(i−k,i) +Fδ(i−k,j)|δ∈{↗,→,↘}∧k∈[1,s]}ifd=↑\nF(i−1,j−1) +w(vi−1,j−1) ifd=↗\nmin{Ri(j−k,j) +Fδ(i,j−k)|δ∈{↑,↗,↘,↓}∧k∈[1,s]}ifd=→\nF(i+ 1,j+ 1) +w(vi+1,j+1) ifd=↘\nmin{Cj(i+k,i) +Fδ(i+k,j)|δ∈{↗,→,↘}∧k∈[1,s]}ifd=↓\n▶Theorem 8. Given two polygonal curves PandQwithnandmvertices, respectively, the\ndiscrete Barking distance of PtoQcan be computed in time O(nm(logs))wheresis the\nspeed bound or time O(nmlog(nm))ifs>nors>m.\nProof.For the first column, i.e., j= 1, we can compute F(i,j)as follows. Clearly, F(1,1) =\nw(v1,1) =θρ(p1,q1). We setF(i,1) =∞for alli>s. Finally, we find that the remaining\nentriesF(i,1)withi∈[2,s]in a bottom-up traversal as the values C1(1,i). We conclude\nthis step by initializing a min-heap Hifor each row icontaining vertex vi,1as its sole element\nandF(i,1) =F→(i,1)as the key.\nAssume now that we want to compute the entries Fd(i,j)fori∈[1,m]where all entries\nFd(i,j′)withj′<jare already computed and for row iwe have a min-heap Hicontaining\nallF→(i,j−k)fork∈[1,s]ordered by key F→(i,j−k) +Ri(j−k,j−1). From this\ninformation we can for each iimmediately compute F→(i,j)as the minimum of Hi, sayvi,j′\nplusRi(j−j′,j). We then update Hiby deleting all entries for vi,j−sand then inserting\n(vi,j,F↑(i,j)),(v1,j,F↓(i,j)),(v1,j,F↘(i,j)), and (v1,j,F↗(i,j))using as key for comparisonI. van der Hoog, F. Klute, I. Parada, and P. Schnider 7\nF·(i,j) +Ri(j−k,j)in the insertion. Note that since for all elements already present in Hi\ntheir keys change only by w(vi,j)and hence their order remains the same. Moreover, we can\ndirectly compute F↗(i,j)andF↘(i,j)for eachi.\nIt remains to compute F↑(i,j)andF↓(i,j)for eachi∈[1,m]in columnj. We describe\nhow to compute F↑(i,j),F↓(i,j)can be computed symmetrically. We start from v1,j. Clearly,\nF↑(1,j) =∞. We also initialize a min-heap Hand insert (v1,j,F→(1,j)),(v1,j,F↘(1,j)),\nand(v1,j,F↗(1,j) =∞)where the second element is used as key. Assume that we now\nwant to compute F↑(i,j)and that we have a heap Hcontaining for k∈[1,s]the elements\n(vi−k,j,F→(i−k,j)),(vi−k,j,F↘(i−k,j)), and (vi−k,j,F↗(i−k,j))ordered by key F·(i−\nk,j) +Cj(i−k,i−1). This can be done as for the row by just extracting the minimum\nelement from the heap H, say (vi′,j,Fδ(i′,j)), and setting F↑(i,j) =Fδ(i′,j) +Cj(i′,i). We\nupdate the heap as in the case of Hi, with the only difference being that we need to insert\nthe three elements (vi,j,F→(i,j)),(v1,j,F↘(i,j)), and (v1,j,F↗(i,j))\nCorrectness follows since the algorithm computes directly the above recurrence. Moreover,\nsince every element vertex and partial weight combination gets deleted and inserted at\nmost once from some heap over the whole computation and no heap contains more than\n3selements at a time, we obtain the claimed running time of O(nmlogs). Note, that if\ns>mors>nwe obtain a runtime of O(nm(log(m) +log(n)))since again never more than\nO(s)elements are contained in a heap and no more than O(nm)elements can be inserted or\ndeleted. ◀\n4 The semi-discrete setting\nWe show an O(nmlog(nm))time algorithm to compute the semi-discrete barking distance.\nTo this end, we define a few critical variables and functions:\n▶Definition 9. LetPandQbe any two curves with nandmvertices respectively and let\nρ,s∈Rbe fixed. We define for each i∈[n]:\nByΛithe vertical line with x-coordinate xi=|P[1,i]|in[|P|]×[|Q|].\nFor any (λi−1,λi)∈Λi−1×Λiwe define the minimal cost of any path from λi−1toλi.\nFormally let Gidenote all differentiable functions γ: [xi−1,xi]→[0,|Q|]. We define:\nf(λi−1,λi) := min\nγ∈Gi,γ′(t)≤s\nf(xi−1)=λi,f(xi)=λi+1/integraldisplayxi\nxi−1θρ(P[xi−1],Q[f(t)])dt.\nFinally, we define Fi(λi)as the minimum cost of all speed-bounded traversals where the\ndog ends at λi∈Λi. In other words, we have Fi(λi) :=minλi−1(Fi−1(λi−1)+f(λi−1,λi)).\nNote that the barking distance from PtoQis now just the value of Fn(|Q|).\nBarkable Space Diagram. To compute these values, we define the Barkable Space Diagram\n(BSD).2The BSD is a graph over [0,|P|]×[0,|Q|]⊂R2. Thex-axis corresponds to the\nlocation of the hiker in its (fixed) traversal of P, and they-axis corresponds to the location of\nthe dog on Q(parameterized according to Q[t]). In particular, a point (x,y)corresponds to\nthe pair (P[x],Q[y]). For any (x,y)∈[0,|P|]×[0,|Q|]we defineBSD [x,y] :=θρ(P[x],Q[y]).\nWe denote Bρ={(x,y)∈[0,|P|]×[0,|Q|]|BSD [x,y] = 1}and call this set the obstacles .\n2As a geometric object this diagram is the same as the free space diagram for Fréchet distance. We use a\ndifferent name to highlight that the paths through the diagram have different rules.8 Barking dogs: A Fréchet distance variant for detour detection\nA traversal of the dog (i.e. a function γ∈G) is a curve γthrough the BSD. Furthermore,\nthe barking distance of γis the total time spent in obstacles, i.e., the length of the projection\nonto thex-axis of the curve γ∩Bρ. The speed bound for the dog translates into a maximal\nand minimal slope for γ.\nOur BSD naturally partitions into vertical slabsin between lines Λi−1andΛi. For any\ncurveγ∈[0,|P|]×[0,|Q|], we can extract the values λi=min{γ∩Λi}. We determine the\nvaluesλicorresponding to the optimal curve γby reasoning about the ‘shape’ of the BSD:\n▶Lemma 10. For anyi∈[n], the setBρ∩([xi−1,xi]×[0,|Q|])areO(m)axis-parallel\nrectangles whose vertical boundaries are incident to Λi−1andΛi.\nProof.Fix somepi∈P. For anyq∈Q, consider the distance d(pi,q). This partitions\nQinto maximal subcurves where for each subcurve S, allq∈Shave either d(pi,q)≤ρ\nord(pi,q)> ρ. Each maximal subcurve S=Q[y1,y2]where for all q∈S,d(pi,q)> ρ\ncorresponds uniquely to a connected component in Bρ∩(∩)[xi−1,xi]×[0,|Q|])areO(m).\nThis connected component has its vertical boundaries coinciding with ΛiandΛi+1. Its\nvertical boundaries are horizontal segments of height y1andy2.\nNote that each edge of Qcan contain at most one endpoint of a maximal subcurve S\nwhere for all q∈S,d(pi,q)> ρ. This bounds the number of connected components in\nBρ∩([xi−1,xi]×[0,|Q|])to beO(m). ◀\n4.1 Understanding the function f(λi−1,λi)\nThe function f(λi−1,λi)takes values in Λi−1×Λiand returns a cost, which is a value in\nR≥0. In particular, in order to understand the behaviour of the function f, we plot a new\ngraph where we think of λi∈Λito be thex-axis andλi−1∈Λi−1to be they-axis.3\n▶Lemma 11. For alli∈[n], the function f(λi−1,λi)is symmetric, i.e., f(a,b) =f(b,a).\nProof.Consider a (partial) traversal of Qwhich, whilst the human remains stationary at pi,\nstarts atλi−1=aand ends at λi=b. Any such traversal, in the BSD, is an x-monotone\ncurveγcontained within the vertical slab corresponding to pi. As the obstacles are rectangles\nby Lemma 10, mirroring this curve gives an optimal curve from λi−1=btoλi=a.◀\nFix now a slab with boundaries Λi−1andΛi. Consider the obstacles O1,...,Okin the\nslab. We may note for each Ojbybjandtjthe height of the bottom and top boundary of\nOj, respectively. This defines a natural grid structure on the plot of fover Λi−1×Λi.\nWe say that a point λi−1on the vertical line Λi−1, iscovered(with respect to the\nslab [xi−1,xi]×[0,|Q|]) if it lies on the boundary of some obstacle Oj, that is, if we have\nbj≤λi−1≤tjfor somej. Otherwise, we say that it is uncovered (with respect to the current\nslab). Analogously, say that λi∈Λiis covered (with respect to the slab [xi−1,xi]×[0,|Q|])\nifbj≤λi−1≤tjfor somejand uncovered otherwise. In the following, the slab under\nconsideration is always fixed, so we just say that a point is covered or uncovered. This defines\nthree different types of rectangles in the grid structure on the plot of f: a point (λi−1,λi)is\ncompletely covered if bothλiandλi−1are covered, it is half-covered if one of the is covered\nbut the other is uncovered, and it is completely uncovered if bothλiandλi−1are uncovered.\n▶Lemma 12. For all vertices pi∈P, the function f(λi−1,λi)is a piecewise linear function.\nFurther, the pieces can be described in the plot of fas follows:\n3This choice allows us to read off a univariate function that appears later as a function over the x-axis.I. van der Hoog, F. Klute, I. Parada, and P. Schnider 9\na)there are two lines ℓ1andℓ2of slope 1such that for all points (λi−1,λi)belowℓ1or above\nℓ2we havef(λi−1,λi) =∞;\nb)in between the pieces form an (O(m)×O(m))-grid (truncated by ℓ1andℓ2) where the\ncompletely covered cells crossing the diagonal are further divided by segments of slope −1.\nProof.We claim that there is always an optimal traversal (i.e., a traversal of Qwhose cost\ncorresponds to a value of f(λi−1,λi)) whose curve γconsists of at most two line segments of\nmaximal (or minimal) slope and at most one horizontal line segment. Indeed, any segment of\nnon-optimal slope can be replaced by a segment of optimal slope and a horizontal segment.\nAs the borders of the obstacles are horizontal segments by Lemma 10, this can only improve\nthe time spent in an obstacle. Similarly, any two segments of positive (negative) slope with a\nhorizontal segment between them can be replaced by a single segment of positive (negative\nslope) with a horizontal segment afterwards. Finally, as each obstacle is a rectangle, no\noptimal path will zig-zag, thus we may indeed assume that γconsists of at most three\nsegments as claimed above. In particular, note that such an optimal curve γwill only have a\nhorizontal segment in an obstacle if γlies completely in this obstacle. This already implies\nthat the function is piecewise linear: the time an optimal segment stays in an obstacle is\nproportional to the vertical distance of its start or endpoint to the boundary of the obstacle.\nIt remains to prove the claims on the pieces. Claim a) follows immediately from the speed\nbound, as from a point λi−1∈Λi−1we can only reach points λi∈Λiwithλi−1−ℓ≤λi≤\nλi−1+ℓ, where the ℓis a number that depends on the speed bound. For claim b), consider\ntwo points λi−1∈Λi−1andλi∈Λi, where the cost of the optimal path between them by\ndefinition is f(λi−1,λi). We will distinguish the cases whether λi−1andλiare covered or\nuncovered.\nLet us start with the case that both λi−1andλiare uncovered. Then, for any x∈Λi−1\nandy∈Λiin the same uncovered regions we have that if yis reachable from x, thenc(x,y),\nthe cost of the path from xtoy, equalsc(λi−1,λi): as by the above observations any optimal\npath traverses obstacles with maximum speed, this cost is proportional to the width of the\nobstacles between the two uncovered regions.\nLet us now assume that λi−1is covered by some obstacle Oj=Oand thatλiis either\nalso covered by Oor lies in an uncovered region adjacent to the region covered by O. We\ndistinguish three different subcases, depending on the width of the obstacle, starting with the\nlargest and ending with the smallest. Recall that tandbdenote the topmost and bottommost\npoint of the left boundary of the obstacle O. Further denote by t′the lowest point on the\nright boundary that is reachable from tand byb′the highest point reachable from b.\nCase 1: there is a point on the right boundary of the obstacle that can neither be reached\nfromtnor fromb. In particular we have b<b′<t′<t. For an illustration of this case we\nrefer to Figure 3(a). In any cell of the form [b���,t′]×Iand, symmetrically, I×[b′,t′], the\nfunctionfis constant at the highest possible cost c:=xi−xi−1. The cell [b,b′]×[b,b′]is\nsubdivided into two triangles by a segment of slope −1, wherefis constant con the upper\ntriangle and linearly increasing with gradient (1,1)on the lower triangle. Symmetrically, the\ncell[t′,t]×[t′,t]is also subdivided into two triangles, now with the lower one constant and\nthe upper one linear with gradient (−1,−1). Finally the cells of the form [b,b′]×[·,b](and\ntheir symmetric variants) are linear with gradient (1,0)(or the corresponding negative or\nswap of coordinates), whereas the cells [·,b]×[·,b]and[t,·]×[t,·]are constant 0.\nCase 2: every point on the right boundary can be reached either from tor fromb, but\nsome of them only from one of them. We thus have b<t′<b′<t. For the structure of fin\nthis case, we refer to Figure 3(b).\nCase 3: every point on the right boundary can be reached both from tand fromb. We10 Barking dogs: A Fréchet distance variant for detour detection\n\u0015\nt\n\u0015i\n\u0015i+1\ni\ni\u00001\ni+ 1\nFigure 2 Two trajectories PandQ(left), the corresponding BSD (middle) and the plot of\nf(λi−1,λi)(right). In the plot of f(λi−1,λi)there is also the graph of Ψ(λi).\nthus havet′<b<t<b′. For the structure of fin this case, we refer to Figure 3(c).\nThe cases where λi−1is covered and λiis in a different region differ from the above\ncases by the addition of a value that is proportional to the width of the obstacles that are\ncompletely between. Finally, the cases where λi−1is uncovered and λiis covered follow\nfrom symmetry. The size of the grid is thus bounded by the number of obstacles, which by\nLemma 10 is O(m). ◀\nFor an illustration of this, see Figure 2. The lemma can be proven by a careful analysis\nof the possible paths γ. Illustrations of the behaviour of f(λi−1,λi)around differently sized\nobstacles can be found in Figure 3.\nFixing the endpoint of the path γcorresponds to fixing the value λi=y. This defines\na univariate function f(λi−1) =f(λi−1,y), which in the plot of fcorresponds to the\nrestriction to a vertical line. Depending on whether yis covered or uncovered, this line only\npasses through completely covered and half-covered or through half-covered and completely\nuncovered cells, where each cell defines an interval on the line. We call these intervals covered\nifλi−1is covered and uncovered otherwise. The following now follows from Lemma 12.\n▶Lemma 13. For eachλi=y, for the (univariate) function fy(λi−1) =f(λi−1,y)there are\nvaluesp1≤p2≤y≤p3≤p4such that\np1,...,p 4correspond to coordinates of horizontal lines of the grid structure of f(λi−1,λi);\nfy(λi−1) =∞forλi−1<p1andλi−1>p4;\nfy(λi−1)is piecewise linear where all pieces have slope −1on covered intervals and 0on\nuncovered intervals for p1≤λi−1≤p2;\nfy(λi−1)is piecewise linear where all pieces have slope 1on covered intervals and 0on\nuncovered intervals for p3≤λi−1≤p4;\nfy(λi−1)is linear with slope 1and then constant for p2≤λi−1≤y;\nfy(λi−1)is constant and then linear with slope −1fory≤λi−1≤p3.\nFurther, ifλi−1is uncovered, that is, not on the boundary of an obstacle, then we can choose\np2=p3and ignore the last two points.\n4.2 Computing the barking time\nOur algorithm computes Fi(λi)fromFi−1(λi−1)andf(λi−1,λi). For this, we need to\nunderstand for each point λithe pointλi−1∈Λi−1which is intersected by an optimal curveI. van der Hoog, F. Klute, I. Parada, and P. Schnider 11\npi−1\npi\nb\nb′\nt′\nt\nb′\nt′\nb\nb′\nt′\nt\nb\nt\nb\nb′\nt′\nt\n0\n0\n1\n(a)\npi−1\npi\nb\nb′\nt′\nt\nt′\nb′\nb\nb′\nt′\nt\nb\nt\nb\nt′\nb′\nt\n0\n0\n1\n1\n(b)\npi−1\npi\nb\nb′\nt′\nt\nt′\nb′\nb\nb′\nt′\nt\nb\nt\nb\nt′\nb′\nt\nc\n0\n0\n(c)\nFigure 3 The graphs of ffor a large (a) medium (b) and small obstacle (c).12 Barking dogs: A Fréchet distance variant for detour detection\nγ∈Gi. As there might be several optimal curves, we will always consider the highest one, that\nis, the one maximizing λi−1. We define Ψ(λi) :=max{argminλi−1(Fi−1(λi−1)+f(λi−1,λi))}.\n▶Lemma 14. The function Ψ(λi)is monotone.\nProof.Assume for the sake of contradiction that Ψ(λi)is not monotone, that is, there are\nvaluesq1<q2for which Ψ(q1) =:r1>r2:= Ψ(q2). That is, the highest optimal path γ1to\nq1passes through r1and the highest optimal path γ2to the higher q2goes through r2, which\nis lower than r1. In particular, the paths γ1andγ2cross in some point x. Now the cost of γ2\nfrom toxhas to be strictly smaller than the cost of γ1tox, otherwise we would get a path\ntoq2with smaller or equal cost than γ2by following γ1untilxand only following γ2fromx\ntoq2, soγ2would not be the highest optimal path. But then the symmetric construction\ngives us a path to q1with strictly smaller cost than γ1, which is a contradiction. ◀\n▶Lemma 15. The function Fi(λi)is a piecewise linear function with O(im)parts.\nProof.We prove this by induction on i. For the base case we may assume F0(λ0) = 0. By\nLemma 12 f(λi−1,λi)is piecewise linear and its cells form (more or less) a grid structure.\nBy the induction hypothesis F(λi−1)is piecewise linear and as it only depends on λi−1, we\nget that the function Fi−1(λi−1) +f(λi−1,λi)is also piecewise linear and each of its pieces\nis defined by subdividing the pieces of f(λi−1,λi)by horizontal lines corresponding to the\nnon-differentiable points of Fi−1(λi−1). In particular, for each λiwe have that Ψ(λi)lies on\nthe boundary of a piece. This already implies that Fi(λi)is piecewise linear, and it remains\nto bound the number of parts.\nFor this we consider the graph of Ψ(λi). By Lemma 14, Ψ(λi)is monotone, so this\ngraph isxy-monotone, and as it follows the boundaries of the pieces, it must thus consist\nof segments on ℓ1andℓ2and horizontal segments with vertical jumps, i.e., outside of ℓ1\nandℓ2it is the graph of a step function. Further, a new part of Fi(λi)can only occur\nwhenever the graph of Ψ(λi)crosses a vertical boundary of a piece, makes a vertical jump\nor crosses the horizontal boundary of a piece while following ℓ1orℓ2. For the first type of\nchanges we note that all the boundaries that were added with the addition of Fi−1(λi−1)are\nhorizontal, so the number of vertical boundaries is still O(m)as proven in Lemma 12. As\nfor the number of vertical jumps, we note that every vertical jump is either on a vertical\nboundary or crosses a diagonal segment of slope −1, where each such segment can only\nbe crossed once by monotonicity. As by Lemma 12 the number of diagonal segments is\nalsoO(m), these also contribute to O(m)changes. Finally, the number of horizontal lines\nthat can be crossed is O(m)for the grid lines of f(λi−1,λi)andO((i−1)m)for the lines\ncorresponding to non-differentiable points of Fi−1(λi−1). It follows that the total number of\nparts isO(im). ◀\nFrom the above proof it follows that there are two types of non-differentiable points in\nFi(λi). The first type are the ones that were already present in Fi−1(λi−1), i.e., the ones\ncorresponding to points where the graph of Ψ(λi)goes alongℓ1orℓ2and crosses a horizontal\nline induced by Fi−1(λi−1). The second type are the newly introduced ones, that is, the once\ndefined by crossing a grid line of f(λi−1,λi)or a vertical jump. The arguments above show\nthat the number of non-differentiable points of the second type is on O(m). We call these\npointsnew breaking points . We call the horizontal lines induced by the new breaking points\nofFi−1(λi−1)as well as the first horizontal lines before and after them breaking lines .\n▶Lemma 16. Between any two new breaking points the function Fi(λi)is monotonically\nincreasing or decreasing and each piece has integer slope.I. van der Hoog, F. Klute, I. Parada, and P. Schnider 13\nProof.We prove this by induction on i. The base case F0(λ0)is again trivial. We first\nnote that the only case where Fi(λi)is not just a single segment between two new breaking\npoints is if Ψ(λi)followsℓ1orℓ2. Consider such a part of Ψ(λi), without loss of generality\nonℓ1and the horizontal lines it crosses. The crossed horizontal lines correspond to non-\ndifferentiable points a1< a2< ... < a kofFi−1(λi−1)onΛi−1from which to considered\noptimal paths in the current slab are segments of maximal slope. Denote by bjthe endpoint\nof the segment starting at aj. We first note that Fi−1(a1)≤...≤Fi−1(ak), as otherwise\nthe segment of maximal slope would not be an optimal path. As the pieces of Fi−1(λi−1)\nhave integer slopes by the induction hypothesis and f(λi−1,λi) =αx+cfor some constant\ncand anα∈{− 1,0,1}, we thus already get that the slopes of the relevant pieces are in\nthe set{−1,0,1...}. It remains to show that there is no segment of slope −1. Assume\nfor the sake of contradiction that [Fi(bj),Fi(bj+1)]has slope−1. This can only happen\nifFi−1(aj) =Fi−1(aj+1)andf(aj,bj) =f(aj+1,bj+1)−1. In particular, bjandbj+1are\nuncovered. But then the segment from aj+1tobjgives a path to bjwith smaller cost, which\nis a contradiction. ◀\n▶Corollary 17. Leta1anda2be two consecutive non-differentiable points of Fi−1(λi−1)that\ninduce non-differentiable points b1andb2inFi(λi). Thenb1andb2are also consecutive.\n▶Lemma 18. Whenever the graph of Ψ(λi)follows a horizontal line, this line is either a\ngrid line or a breaking line.\nProof.Fix a value λiand assume that the minimum of Fi−1(λi−1) +f(λi−1,λi)does not lie\nonℓ1orℓ2. We want to show that it either lies on or right next to a new breaking point of\nFi−1(λi−1)or on a grid point of f(λi−1,λi). For the sake of contradiction assume otherwise.\nThenλi−1is in a cell of f(λi−1,λi), that is, by Lemma 13 we have that f(λi−1,λi) =αx+c\nfor some constant cand a fixed α∈{− 1,0,1}. Further,λi−1is between two new breaking\npoints ofFi−1(λi−1), thus by Lemma 16 we have that Fi−1(λi−1)is monotonically increasing\nor decreasing and each piece has integer slope. Thus, Fi−1(λi−1)+f(λi−1,λi)is monotonically\nincreasing or decreasing except possibly on the first or last part. But then the minimum is\neither on a new breaking point or right next to it. ◀\n4.3 The algorithm\nFor the algorithm we proceed iteratively over the vertical slabs from left to right. Ideally, we\nwould want to compute the function Fi(λi)iteratively for every vertical slab. The problem\nthough is that by Lemma 15 only guarantees us that there are no more than O(nm)many\nbreakpoints over the whole computation. This would pretty directly lead to a O(nm2logm)\nalgorithm after showing how to compute the argmin in which we maintain and update all\nfunction values at breakpoints explicitly. To improve the running time, we next describe a\ndata structure, which will allow us to query and update the function values Fi−1(λi−1)for\nspecific breakpoints λi−1. Crucially, while this data structure might contain at some point\nO(nm)elements, all its operations can be executed in O(log(nm))time.\nMaintaining the breakpoints. Let(x,g)be akey-function pair wherex∈Randg:R→R\nis a linear function. We assume that the linear function gis represented by the real numbers\nmandbwheremis the slope and bis they-axis value of the function. Hence, when we\nin the following speak of adding two linear functions we just need to add its slope and\ny-axis value. To query Fion the set of up to O(mn)breakpoints we next describe a data\nstructure that stores nkey-function pairs and allows us to efficiently add functions to the14 Barking dogs: A Fréchet distance variant for detour detection\npairs and evaluate them at a point equivalent to some key stored in the data structure.\nOur data structure should support for a pair (x,g)insertion via Insert (x,g), deleting via\nDelete (x,g), evaluating gatxviaEvaluate (x), and the following two operations:\nShiftVertical (a,b,h )Add for each pair (x,g)withx∈(a,b)the function htog.\nShiftHorizontal (a,b,y )Add for each pair (x,g)withx∈(a,b)the valueytoxassuming\nthat the order of elements in the data structure sorted by their key remains the same.\n▶Lemma 19. There is a data structure that maintains a set of nkey-function pairs and\nsupports the above operations in O(logn)time each.\nProof.At the core of our data structure is a modified AVL tree [ 2]Twhose keys are going\nto be the keys of the key-function pairs. We assume that each leaf of the AVL tree has a\npointer At each internal node u∈Twe store a shift-value ξu∈Rand a linear function\nhu:R→Rwhich we call the function at this node.\nTo show correctness we are going to maintain the following invariant for Tafter each\noperation:\n▶Invariant 1. Let(k,g)be a key-function pair stored in the data structure then there exists\na path starting at the root with vertices (ξ1,h1),..., (ξm,hm)inTsuch that\n1.k=/summationtextm\ni=1ξi\n2.g(k) =/summationtextm\ni=1hi(k)\nWe implement Search (k)in our tree such that it returns the node where the key of\nthe left tree is smaller and of the right tree is larger, or the returned node is a leaf. We\nuse the following comparison. Let (ξi,hi)be the current node, then we recurse on the left\nsubtree when k≤ξiand on the right one else using the new key k−ξi. Let (ξ′,g′)be the\nreturned node and (ξ1,h1),..., (ξm,hm)the traversed path from the root, then k=/summationtextm\ni=1ξi\nby Invariant 1 whenever (k,g)is an element of our data structure.\nRotation is implemented in the usual way, but we need to update the functions stored\nat each node. Let u= (ξ,h)be the node around which we execute our rotation. We first\ndescribe a left rotation. First, we update the values, then we rotate as usual. Let ξ′be the\nshift-value and h′the function of the right child urofu. We addξ′to the shift-values of\nthe children of urandh′to the function of the children of ur. We set the shift-value and\nfunction of urto be equal to ξandhrespectively and finally set shift-value and function\nto constant 0foru. Right rotation is executed is implemented symmetrically. Checking all\npossible paths to traverse Tfrom root to any leaf after rotation shows that Invariant 1 is\nfulfilled. Using left and right rotations we can implement the necessary double rotations as\nusual.\nSuppose we want to insert the key-function pair (k,g). IfTis empty we create one\nsingle node (k,g)and return. The invariant then holds by definition. Else, we first search\nfor the location to insert (k,g)using the same comparison as in the search function. Let\n(ξ1,h1),..., (ξm,hm)be the path starting at the root and ending at (ξm,hm). Moreover, let\nξ=/summationtextk\ni=1ξiandh=/summationtextk\ni=1hi. We insert a new node for (k,g)with the values (k−ξ,g−h).\nInvariant 1 holds by definition.\nSupposewewanttodeletethekey-functionpair (k,g). First, locate (k,g)withSearch (k),\nletu= (ξ,h)be the returned node. Again, we only need to describe how to update shift-\nvalues and functions, the other parts are as usual. If uis a leaf we can simply delete it. If u\nhas only one child vwe simply add the shift-value ξand function hto the corresponding\nvalues ofvand replace ubyv. Ifuhas two children, we locate the successor vofuin itsI. van der Hoog, F. Klute, I. Parada, and P. Schnider 15\nright subtree and replace ubyv. In casevwas also the right child of uwe addξtov’s\nshift-value and htov’s function. It remains to handle the case when vwas not the right\nchild ofu. Letξvandhvbev′sshift-value and function. Moreover, let wbe the unique child\nofv. We replace vbywand addξvandhvto the shift-value and function of w. Then, we\nreplaceubyvand set its shift-value and function equal to ξandhrespectively.\nWe argue that Invariant 1 holds after executing the above replacements, but before any\nrotations are executed. Since the invariant holds after a series of rotations it then holds\nafter the whole call to delete has terminated. If uwas a leaf Invariant 1 holds again. Let\n(ξ1,h1),..., (ξm,hm)be a path starting at the root and passing through ubeforeugot\ndeleted. The cases of uhaving had only one child or its successor being a child are easy\nto check since we only need to add the shift-value and function of uto the node replacing\nit without any adjustment to the subtrees. By the same argument, the invariant holds for\nthe subtree previously rooted at the successor of u. Finally, in case the successor was not a\nchild ofuwe simply replaced its shift-value and function by the one of uwhich means the\ninvariant holds.\nIt remains to describe the two shift functions. We begin with ShiftVertical (a,b,h ).\nLetuaandubbe the two nodes of Tfound by calling Search(a) andSearch (b). We now\naddhto the function of the lowest common ancestor rofuaandubinT. In the following\nwe assume that ubis in the right subtree of randuain the left subtree of r. If the sum\nof shift-values on the path from root to uais larger than awe simply subtract hfrom the\nfunction of the left child of ua, else we subtract hfrom the function of uaand add it to\nthe function of its right child. We proceed symmetrically for ubwith left and right children\nreversing their roles.\nLet(k,g)be an element of our data structure. Part one of Invariant 1 holds since the\nshift-values were not altered. Let uk=Search(k) ,ua=Search(a) , andub=Search(b) .\nThere are two cases k∈(a,b)ork̸∈(a,b). Consider the sums sk,sa,sbof shift-values along\npaths from the root to uk,ua, andub, then it holds that sa<sk<sb, meaning that ukis a\nsuccessor or uaand a predecessor of ub. Hence, the lowest common ancestor rofuaandub\nis on the root to ukpath, which means that g(k) =/summationtextm\ni=1(hi(k)) =/summationtextm\ni=1,ui̸=rhi(k) +hr(k)\nwherehris the function stored at r. Note that hrconsists of the sum of the function at\nrbefore the call to ShiftVertical andhas desired. Symmetrically we can prove the\nstatement for k̸∈(a,b).\nNext we describe ShiftHorizontal (a,b,ξ ). Letuaandubbe the two nodes of Tfound\nby calling Search(a) andSearch (b). Again, let rbe the lowest common ancestor of ua\nandubinT. We addξto the shift-value of rand to its function the constant q=−(m·ξ)\nwheremis the slope of the current function at r. In the following we assume that ubis in\nthe right subtree of randuain the left subtree of r. We update the subtree rooted at uaas\nabove with respect to the tree before any changes subtracting ξandqfrom the shift-value\nand function of the left child of uaor subtracting them from uaand adding them to the\nright child of ua. We proceed symmetrically for the shift-value and function of the subtree\nrooted atub.\nLet(k,g)be an element of our data structure after an application of ShiftHorizontal\nSince we assume that the ordering of the key-functions pairs by their keys does not change\nin this operation we can immediately assume that Tis still fulfills all the properties of an\nAVL-tree. It remains to show that Invariant 1 holds. Let uk=Search(k) ,ua=Search(a) ,\nandub=Search(b) . There are two cases k∈(a,b)ork̸∈(a,b). Consider the sums of shift-\nvaluessk,sa, andsbon the respective root-node-paths. Before the call to ShiftHorizontal\nwe know that sa<sk<sb. Hence,ukis a successor of uaand a predecessor of sb. Moreover,16 Barking dogs: A Fréchet distance variant for detour detection\nwe desire that sk=kwhich holds if sk=s′\nk+ξwheres′\nkis the sum on the root-to-node\npath toukbefore the call to ShiftHorizontal . But this is true as either ξis added once\neither atror atrand the right child of uain which case its also subtracted at ua. In case\nk̸∈(a,b)we argue in the same fashion, either ukis not in the subtree rooted at rin which\ncase no change was made to its shift-value or kis in the left subtree of uain which case xis\nfirst added at rand then subtracted at uaor at the left child of ua.\nIt remains to show part two of Invariant 1. After the execution of ShiftHorizontal\nwe desire that g(k) =g′(k+ξ) =mk+mξ+b=g′(k)−mξwhereg′is the function\nbefore the call to ShiftHorizontal . Assumek∈(a,b)and letg(k) =/summationtextm\ni=1hi(k)on a\npath from the root to uk, as above consider hrto be the function at r, then/summationtextm\ni=1hi(k) =/summationtextm\ni=1,ui̸=rhi(k) +hr(k) =/summationtextm\ni=1,ui̸=rh′\ni(k) +h′\nr(k)−mξ=g′(k)−mξ. In casek̸∈(a,b)the\ninvariant holds we simply recover the previous function as desired.\nSince we only need to ever traverse paths of the tree, modify vertices along those paths\nor their direct children, and do constant time algebraic operations the runtime bounds follow\ndirectly from the ones for the AVL-tree. ◀\nPutting things together. From now on we are going to assume that Fi−1is given as a data\nstructure as described in above. Next to inserting and deleting key-function pairs in this\nstructure, we are allowed to evaluate a function at its key, add a linear function on an interval\nof key-function pairs, and add constant shifts to the keys of the elements.\n▶Lemma 20. GivenFi−1(λi−1), we can compute Ψ(λi)in timeO(mlogm).\nProof.We compute Ψ(λi)on the vertical boundary lines. There are O(m)such lines by\nLemma 12, so in order to achieve the claimed running time, we have to be able to do this\nin timeO(logm)per line. For each of these vertical lines, by Lemma 18, Ψcan only be on\none ofO(m)possible points. These are all points corresponding to new breaking points of\nFi−1and we can compute them in O(mlogm)time by querying Fi−1O(m)times. Moreover,\nsince these are all breaking points from the previous iteration, we can assume that the\ncorresponding O(m)coordinates are explicitly stored.\nWe introduce the following four functions:\nℓ+(x) :=x;\nℓ−(x) :=−x;\ns+(x)is continuous piecewise linear where all pieces have slope 1on covered intervals\nand0on uncovered intervals;\ns−(x)is continuous piecewise linear where all pieces have slope −1on covered intervals\nand0on uncovered intervals.\nBy Lemma 13, the function Fi−1(λi−1) +f(λi−1,λi)restricted to any vertical boundary line\ncan be partitioned into four intervals and on each interval it is of the form F(λi) +g+c\nfor a real value cand a function g∈{ℓ+,ℓ−,s+,s−}. In particular, the argmin restricted to\none of these intervals is invariant of the value c. Further, the boundaries p1,...,p 4of the\nintervals can be read off in constant time.\nGiventhevaluesfromabovewecancomputeall Fi−1(λi−1)+g(λi−1)forg∈{ℓ+,ℓ−,s+,s−}.\nWe store the values corresponding to one gin a segment tree [ 15] that supports querying the\nminimum on a given interval of leafs in O(logm)time. Building each such tree takes time at\nmostO(mlogm). For a vertical boundary line we now read off the four intervals and for each\ninterval find the minimum within this interval using the segment tree of the corresponding\nfunctionFi−1(λi−1) +gand return the argument. We then evaluate Fi−1(λi−1) +f(λi−1,λi)I. van der Hoog, F. Klute, I. Parada, and P. Schnider 17\non these four arguments and keep the one minimizing Fi−1(λi−1)+f(λi−1,λi). The computed\nvalues already uniquely determine Ψ(λi)except between two vertical boundaries where Ψ(λi)\njumps over a diagonal segment. In these cases we still need to compute the exact location of\nthe jump. This amounts to evaluating two linear functions and computing where they are\nequal, so it can be done in constant time per diagonal segment, that is, O(m)total time.\n◀\n▶Lemma 21. GivenFi−1(λi−1), we can compute Fi(λi).\nProof.By Lemma 20 we can assume that Ψ(λi)is given and since its description consists of\nonlyO(m)points we may traverse it in a linear fashion. We run the following algorithm to\nperform the necessary updates to Fi−1.\nConsider a vertical segment of Ψbetween Ψ(λ(j)\ni)andΨ(λ(j+1)\ni)we call Delete (Ψ(λa))\non all lines that lie between Ψ(λ(j)\ni)andΨ(λ(j+1)\ni). This can be implemented by retrieving\nall the to-be-deleted nodes of the tree and then calling delete on all of them.\nFor each segment between λ(j)\niandλ(j+1)\nithat lies on ℓ1orℓ2in the plot of fwe first\nexecute ShiftHorizontal (Ψ(λ(j)\ni),Ψ(λ(j+1)\ni),d)if we are on ℓ1and if we are on ℓ2we\nexecute ShiftHorizontal (Ψ(λ(j)\ni),Ψ(λ(j+1)\ni),−d), wheredis the length of the vertical slab\nbetween Λi−1andΛi. Letg(λi) :=f(Ψ(λi),λi)and note that by Corollary 17, between λ(j)\ni\nandλ(j+1)\nithis is a single linear function of the form mx+qwherem∈{− 1,0,1}. Next, we\ncallShiftVertical (λ(j)\ni,λ(j+1)\ni,g(λi)).\nFinally, call Insert (λ(j)\ni,wj)for everyλ(j)\niwherewjis the explicit value for Fi(λ(j+1)\ni).\nTo show correctness we are going to argue that for every key kin our data structure\nit holdsFi(k) =Evaluate (k). Whenever kwas inserted in the last step of the algorithm\nthis follows by correctness of the insert function. For all other elements we only executed\none horizontal and one vertical shift at most. It remains to show that these shift values are\nindeed correct. Which follows with\nFi(λi) =Fi−1(λi±d) +f(Ψ(λi),λi) =Fi−1(λi−1) +f(Ψ(λi),λi). ◀\n▶Theorem 22. Given two polygonal curves PandQwithnandmvertices, respectively,\nthe semi-discrete Barking distance of PtoQcan be computed in time O(nmlog(nm)).\nProof.The semi-discrete Barking distance of PtoQis just the value Fn(|Q|). We can\ncomputeFn(λn)by iteratively computing all Fi(λi). Examining the proof of Lemma 21 we\ncan see that the needed computations can be implemented in O(mlogm). Additionally, every\nquery to the data structure takes only O(log(nm))time by Lemma 19. Finally, by Lemma 15\nthere are only O(nm)breaking points and each one of them is inserted and deleted at most\nonce from our data structure. It follows that the total running time is in O(nmlog(nm)).◀\n5 The continuous setting\nThe underlying ideas are similar to those for the semi-discrete setting. The definitions of the\nfunctionsf(λi−1,λi)andFi(λi) :=minλi−1(Fi−1(λi−1) +f(λi−1,λi))extend. In contrast to\nbefore, these functions are now piecewise quadratic. The BSD has a natural grid structure,\nwhere each rectangular cell corresponds to a choice of a segment PiofPand a choice of a\nsegmentQiofQ. It is a well-known fact (see e.g. Lemma 3 in [ 6]) that for each such rectangle\nRi,j, the clear space of the BSD is the intersection of Ri,jwith an ellipse. We denote this18 Barking dogs: A Fréchet distance variant for detour detection\nclear space by Ci,jand call it cut ellipse . In the following, we restrict our attention to a\nfixed segment PiofP, so we abbreviate the notation to Cj.\nThe main idea for the algorithm is that we can always find an optimal curve in the BSD\nsuch that each segment corresponds to either (i) a segment of maximum or minimum slope or\n(ii) a segment on a special diameter of a Cj. We discretize the problem by describing all the\npotential endpoints of such segments. For each cut ellipse Cjwe consider the longest feasible\ndiameterd(Cj)given by a segment whose supporting line passes through the center of the\n(uncut) ellipse with endpoints coinciding with the intersections of the two vertical tangents\nofCj. Denote by ljandrjthe left and right endpoints of this diameter, respectively.\nWe again denote by Λi−1andΛithe left and right boundary of the slab. For any curve γ\nin the BSD we define its cost as the time γspends outside/uniontextCj. We say that a curve with\nstart point aand endpoint bisoptimalif it minimizes the cost among all such paths.\n▶Lemma 23. For any two points λi−1andλion the boundary of a vertical slab there is an\noptimal polygonal curve from λi−1toλisuch that its segments are either of maximum or\nminimum slope (Type 1) or lies on d(Cj)for some cut ellipse Cj(Type 2).\nProof.Letγbe a polygonal path from λi−1toλithat does not confirm to some of the three\nconditions.\nSubdivideγwith points γ1,...,γkwheneverγintersects the boundary of a cut ellipse.\nLetγjandγj+1be two consecutive points such that the segments on the sub-path γj,j+1are\nnot all of Type 1 and 2.\nFirst, we consider the case that γj,j+1lies in an obstacle. Then, we can replace γj,j+1by\nany polygonal path with segments of Type 1 and 2 that connects the points γjandγj+1. If\nthe new path ever leaves the obstacle, the the original path γwas not optimal.\nThe other case is that γj,j+1lies in cut ellipse C. First, assume that d(C)has absolute\nslope smaller than the maximal absolute slope. We distinguish two cases, either γj,j+1\nintersectsd(C)or it doesn’t. In the first case, we replace γj,j+1by a segment sjof maximal\nabsolute slope with endpoint at γj, a segment sj+1of maximal absolute slope with endpoint\natγj+1, and a segment s′alongd(C)connecting the intersection points of d(C)withsjand\nsj+1. In the latter case, γj,j+1is replaced by the same path or by the two segments sjand\nsj+1solely.\nSecond, assume that d(C)has absolute slope larger than the maximal absolute slope.\nIf the segment with endpoints γjandγj+1has maximal absolute slope we replace γj,j+1\nby this segment. Assume this is not the case, then we replace γj,j+1by three segments of\nmaximum absolute slope. A segment sjof maximal absolute slope with endpoint at γj, a\nsegmentsj+1of maximal absolute slope with endpoint at γj+1, and a segment s′connecting\nthe endpoints of sjandsj+1at maximum absolute slope. Note that sj,s′, andsj+1can\nalways be chosen in this way since the connection of sjandsj+1alongd(C)is too steep, but\nthe segment connecting γjandγj+1is shallower than the maximum absolute slope. Hence,\nthere exists a segment in between lying inside Cand having maximum absolute slope. ◀\nWe now further restrict the optimal curves. For each ellipse Cjcompletely contained in\nthe vertical slab we consider eight lines in total, four with maximum slope and four with\nminimum slope. They are the two tangents t+\nlandt+\nrof maximum and the two tangents\nt−\nlandt−\nrof minimum slope, with the intersection point of t+/−\nlbefore the one of t+/−\nr\ninx-direction. The other four lines are the lines of maximum and minimum slope passing\nthroughljandrj, denoted as s+\nl,s+\nr,s−\nl, ands−\nr. For a cut ellipse Cjthat is cut off, we\nadd the same lines, even if ljorrjare not in the vertical slab. We additionally add lines\nof maximum and minimum slope passing through all intersections of Cjwith grid linesI. van der Hoog, F. Klute, I. Parada, and P. Schnider 19\nof the BSD. Let S+andS−be the resulting set of lines with positive and negative slope,\nrespectively. For a cut ellipse Cj, we call all these up to 24 lines its boundary lines .\nS+(S−) defines a set of parallel stripsthat cover the slab. We define the following lines\nonly forS+, the definition for S−is symmetric. By definition, in each of them any maximum\nslope line intersects the same subset of cut ellipses. Consider a strip τbounded from the left\nby linetl∈S+and from the right by tr∈S+and two cut ellipses Ci,Cjinτ, we denote by\ncτ,Ci,Cj(x)the cost function of going between the boundaries of CiandCjusing a maximum\nslope line at horizontal distance xfromtl. This function is the sum of the costs of traversing\nthe obstacles between consecutive cut ellipse boundaries that lie in between CiandCj. It is\nthus a sum of quadratic functions, and therefore a quadratic function itself. We denote by\nopt+\ni,j,τthe line of maximal slope which minimizes the cost cτ,Ci,Cj(x). Note that if opt+\ni,j,τ\nis not equal to tlortr, then it is unique. We call opt+\ni,j,τthepositive transition line and\nopt−\ni,j,τthenegative transition line ofCiandCjinτ.\n▶Lemma 24. For any two points λi−1andλion the boundary of a vertical slab there is an\noptimal polygonal curve as in Lemma 23 such that all segments of Type 1 in obstacles lie\neither on lines of the type opt+/−\ni,j,τ, a boundary line, or on the four lines of maximum and\nminimum slope at the start and end points of the polygonal curve.\nProof.Letγbe an optimal polygonal path from λi−1toλias in Lemma 23 for which at\nleast one of its segments of Type 1 lies in an obstacle and neither on a transition line nor\non one of the lines of maximal and minimal slope at the start or end vertices of γ. Lets\nbe the first segment seen from the start of γthat lies in an obstacle, but does not lie on\none of our desired lines. We assume without loss of generality that shas maximal slope,\nthe case where shas minimal slope is symmetric. If one endpoint of scoincides with one\nof the endpoints of γ, thenslies on the line of maximal slope through this endpoint. Let\nτbe the strip with bounding lines τlandτrin whichslies. We try to shift stowardsτl.\nNote that cost will not improve as we assumed γto be optimal. If we can shift until the\nsegment before sdegenerates to a single point, we have reduced the number of vertices on\nthe polygonal path γand we continue inductively. If we cannot shift further because the\ncost would increase, there are two cases. If the cost increases in both directions of shifting,\nthenslies on the positive transition line between two cut ellipses. If the cost increases only\ntowardsτl, thenslies on a boundary line. ◀\n▶Observation 25. There areO(m)boundary lines each of which can be computed in time\nO(1). There are O(m2)transition lines and these can be computed as minima of O(m)\nquadratic functions in time O(m3).\nWe are now ready to define our discretization of the problem. On a high level, we define a\nweighted directed graph Giconnecting relevant points on Λi−1with relevant points on Λisuch\nthat optimal curves correspond to shortest paths in Gi. The edges of Giare geometrically\nmotivated by the boundary and transition lines defined above. In order to also include the\ninformation of Fi−1(λi−1), we further do a construction similar to the boundary lines, but for\nthe non-differentiable points of Fi−1. Formally, let x∈Λi−1be a non-differentiable point of\nFi−1(λi−1). We consider two lines through x, one with maximal slope and one with minimal\nslope. Taking these two lines for every non-differentiable point of Fi−1(λi−1), we get a set of\nlines that we call the F-induced lines. We define the graph Gi(and a natural straight-line\ndrawing of it in the BSD). First, we define the following set of vertices and their embedding\nin the BSD.\n(i)For each intersection between two lines which are boundary or F-induced lines lying\noutside or on the boundary of/uniontext\niCi, we add a vertex.20 Barking dogs: A Fréchet distance variant for detour detection\n(ii)For each intersection of a boundary line or an F-induced line with Λi−1orΛi, we add a\nvertex. This includes adding a vertex on Λi−1for each non-differentiable point of Fi−1.\n(iii)For eachCjwe add as vertices the intersection points of the boundary of Cjwith all\nboundary and F-induced lines.\n(iv) For each Cjadd as vertices the intersections of its boundary with its transition lines.\n(v)For each interval between two vertices on Λiwe add one vertex in the interval on Λi. For\neach of those vertices, we now add the lines of maximal and minimal slope to the set of\nF-induced lines and add vertices as in (i), (ii) and (iii).\n(vi) Finally, we add one dummy vertex L, which we think of as lying to the left of ��i−1.\nIt will follow from our arguments that the function Fi(λi)has at most O(im)pieces. Thus\nwe getO(n2m2)vertices in total in the vertical slab. We add four types of edges:\nWe add edges between each vertex of Types (i), (ii), (iii), and (v) and the at most four\nclosest vertices on the lines of maximum or minimum slope that pass through this vertex.\nWe only add such edges if they fully lie in obstacles.\nFor each transition line between cut ellipses CiandCjwe add an edge between the two\nconsecutive vertices of Type (iv) on the line where one belongs to Ciand the other to Cj.\nInside each Ciwe make all possible connections, i.e., we connect two vertices lying on the\nboundary of Ciby an edge if the slope of the segment connecting them does not exceed\nour slope bounds. Note that every Ciis convex and that we might be including edges in\naCiincident to vertices that lie in the shared boundary between CiandCi−1orCi+1.\nWe connect Lwith all vertices lying on Λi−1.\nNote that this defines O(n2m2)edges. We orient all these edges from left to right and assign\nthem weights. Consider an edge enot incident to Land its intersections with cut ellipses.\nThese intersections define sub-segments, some of which fully lie in obstacles. The weight\nofe,w(e), is the sum of the lengths of the projections onto the horizontal line of these\nsub-segments in obstacles. For the edges incident to Lwe define the weight of the edge\n{L,v}asFi−1(v).\n▶Lemma 26. For any fixed λicorresponding to a point on Λiand for each interval\nI= [λi−1,λ′\ni−1]onΛi−1that minimizes f(λi−1,λi)we have that the endpoints of Iare\nvertices ofGi.\nProof.Letλi−1be a point which minimizes f(λi−1,λi)but that is not a vertex of Gi. Let\nγbe a trajectory of optimal cost. By Lemma 24 we may assume that the first segment is of\nmaximal or minimal slope; without loss of generality we assume that it is of maximal slope.\nWe first consider the case in which λi−1lies on the intersection of Λiwith some cut ellipse\nCj. The intersection of ΛiwithCjis in general a segment, and its endpoints are vertices\nofGi. Thus, ifλi−1is not a vertex of Gi, we have that in a small neighborhood of λi−1\nall points lie on the boundary of Cj, implying that λi−1is in the interior of a minimizing\ninterval whose endpoints are vertices of Gi.\nNext we assume that λi−1lies in some obstacle. Let Cjbe the cut ellipse first intersected\nby the first segment of γand letpbe the intersection point. Consider the tangent τtoCjat\npand assume it to be non-vertical having positive slope; the argument for negative slope is\nsymmetric. Since τis of positive slope the leftmost point of Cjlies to the left of p. Hence,\nthere exists a number ε>0such that we can shift the first segment of γbyεunits to the\nbottom and elongate the second segment of γwith a segment lying completely inside Cj.\nLetp′be the intersection point of this shifted segment and Cj, thenp′lies left ofpand we\nfound a path of cost smaller than the one of γ, a contradiction.I. van der Hoog, F. Klute, I. Parada, and P. Schnider 21\nFinally, assume that τis vertical at p. Then,pcoincides with the left-most point on the\nboundary of Cj. But then, we find the starting point of γto lie on the intersection of a\nboundary-line with λi−1for which a vertex was added to Gi. ◀\n▶Lemma 27. Letvbe a vertex of GionΛiand letw(L,v)be the weight of a shortest path\ninGifromLtov. Thenw(L,v) =Fi(v).\nProof.Letπbe a path from LtovinGiwhose weight is w(L,v). By our definition of\nGi, each edge ekof the path defines a sub-trajectory γkwhose cost equals the weight of ek.\nThese sub-trajectories intersect in the vertices of π, soπdefines a trajectory γtovwith cost\nw(L,v). This shows that w(L,v)≥Fi(v).\nLet nowγbe an optimal trajectory to v, which by definition has cost Fi(v). Recall that\nFi(v) =minw{Fi−1(w) +f(w,v)}and that the non-differentiable points of Fi−1are vertices\nofGi. Thus, by Lemma 26, we can assume that the point win whichγintersects Λi−1is a\nvertex ofGi. By Lemma 24, we can further assume that γfulfills the conditions of this Lemma\nand thus, by definition of Gi,γis partitioned into sub-trajectories that are in one-to-one\ncorrespondence with edges in Gi. More precisely, we partition γat its intersections with a\ncut ellipseCjas long as the sub-trajectory does not follow a transition line. These edges\ndefine a path from LtovinGiwhose weight is Fi(v), showing that w(L,v)≤Fi(v).◀\nWith this we can already compute Fi(λi)for anyλithat is a vertex of Gi. In the\nfollowing, we discuss how we can extend the computation to all of Fi(λi). Recall that Fi(λi)\nis a continuous, piecewise quadratic function. In particular, in order to compute Fi(λi), we\nwill also need to compute the non-differentiable points of it. We first observe that Gialready\ngives us many of these. For this, let VΛidenote the vertices of GionΛithat are not of Type\n(v). Note that there is a natural order on VΛigiven by the order on Λi.\nIn order to compute all the non-differentiable points of Fi(λi)we realize that in the\ninterval (v,w)ofΛibetween consecutive vertices of VΛi, the optimal curves come in two\nforms: either the last segment is of maximal or of minimal slope. We define the functions\nF+\ni(λi)andF−\ni(λi)as the cost of an optimal curve to λiwhere we restrict the set of optimal\ncurves to those of Lemma 24 with the last segment having maximal slope and minimal slope,\nrespectively. It then follows from Lemma 24 that Fi(λi) =min(F+\ni(λi),F−\ni(λi)). The next\nlemma tells us that if there is a non-differentiable point of Fi(λi)in the interior of [v,w]it\nmust be at the point in which F+\ni(λi) =F−\ni(λi). We further define the graph G+\ni(G−\ni) as\nthe subgraph of Giin which we restrict the edges incident to vertices on Λito those that\nhave positive slope (resp. negative slope) and observe that we can compute F+\ni(λi)and\nF−\ni(λi)fromG+\niandG−\nijust like we computed Fi(λi)fromGi.\n▶Lemma 28. Letvandwbe two consecutive points in VΛi. ThenF+\ni(λi)andF−\ni(λi),\nrestricted to the interval on Λiwhere they are not infinite, are continuous, piecewise quadratic\nfunctions that do not have any non-differentiable points in the interior of the interval [v,w].\nFurther, for any vertex vinGionΛiletw(L,v)be the weight of a shortest path in G+\ni(G−\ni)\nfromLtov. Thenw(L,v) =F+\ni(v)(w(L,v) =F−\ni(v)).\nProof.We only argue about F+\ni(λi), the arguments for F−\ni(λi)are symmetric. We first argue\nthatF+\ni(λi)is continuous and piecewise quadratic. As by definition we have that Fi(λi) :=\nminλi−1(Fi−1(λi−1) +f(λi−1,λi)), it is enough to show that f(λi−1,λi)is continuous and\npiecewise quadratic. This follows from the definition of fas the sum of distances within\nobstacles of an optimal trajectory, and the fact that the boundaries of the obstacles are\npiecewise quadratic. The fact that all non-differentiable points are on vertices of Giis22 Barking dogs: A Fréchet distance variant for detour detection\nanalogous to the proof of Lemma 26. Finally, the proof that w(L,v) =F+\ni(v)is analogous\nto the proof of Lemma 27. ◀\nWe now describe our algorithm. For each interval [v,w]between two consecutive points\ninVΛiwe have three values of F+\ni(λi)andF−\ni(λi): atv, atw, and at the subdivision vertex\nof Type (v). By Lemma 28, since the functions restricted to [v,w]are quadratic functions,\nwe can obtain them via interpolation and we can compute both F+\ni(λi)andF−\ni(λi)via a\nshortest path computation in the graphs G+\niandG−\ni. As both of them have O(n2m2)many\nvertices edges, each of the O(nm)values ofF+\ni(λi)andF−\ni(λi)can be computed in time\nO(n2m2log(nm))time using Dijkstra’s algorithm. Thus, both F+\ni(λi)andF−\ni(λi)can be\ncomputed in time O(n3m3log(nm)). We further get that both F+\ni(λi)andF−\ni(λi)have\nO(nm)many pieces. As stated before, Fi(λi) =min(F+\ni(λi),F−\ni(λi)), thus we can compute\nFi(λi)fromF+\ni(λi)andF−\ni(λi)in additional time O(nm).\n▶Lemma 29. GivenFi−1(λi−1), we can compute Fi(λi)in timeO(n3m3log(nm)).\nAs before we get an algorithm to compute F(λn), albeit with higher time complexity.\n▶Theorem 30. Given two polygonal curves PandQwithnandmvertices, respectively,\nthe continuous Barking distance of QtoPcan be computed in time O(n4m3log(nm)).\n6 Lower bound\nMany lower bounds on time complexity are based on the Strong Exponential Time Hypothesis\n(SETH). SETH asserts that for every ε>0there is aksuch thatk-SAT cannot be solved in\nO((2−ε)ntime. Given two sets A,B⊆{0,1}dofnvectors each, the Orthogonal Vectors\n(OV) problem asks whether there are two orthogonal vectors u∈Aandv∈B, that is,\nu·v=/summationtextd\ni=1uivi= 0. Assuming d=ω(logn), the Orthogonal Vectors Conjecture (OVC)\nstates that for all ϵ>0, solving OV requires Ω(n2−ϵ)time. SETH implies both OVC [ 28],\nmeaning that the Orthogonal Vectors Conjecture is weaker and thus more likely to be true.\n▶Theorem 31. LetPandQbe two disjoint polygonal curves with nvertices. Assuming\nOVC, solving the barking decision problem where the maximum speed of the dog matches the\nspeed of the human and with constant barking radius ρrequires Ω(n2−ϵ)time for any ϵ>0.\nProof.Consider an instance of the OV problem consisting of sets A,B⊆{0,1}dofnvectors\nandd=ω(logn). We construct an instance of the barking decision problem consisting of\ntwo polylines PandQwithO(n)vertices and rational coordinates. The trajectories for the\nhuman and the dog go along these polylines PandQ, respectively.\nVector gadgets. As it is usually the case, the reduction is based on gadgets corresponding\nto the vectors; refer to Figure 4. To build them, we define a distance u=ρ/25. The polyline\nPhas a gadget for each vector in Aand analogously the polyline Qhas a gadget for each\nvector inB. In these gadgets the horizontal distances are small and thus irrelevant. We\nstart defining a horizontal line κ0′=π0′and, symmetrically on the top and bottom, in that\norder, lines κ0andπ0at distance 4u, linesκ1′andπ1��at distance 16u, and linesκ1andπ1\nat distance 20u.\n▶Observation 32. The vertical distance between any two horizontal lines such that at least\none of them has subindex 0or0′is at most 24u= 24/25ρ < ρ. In contrast, the distance\nbetween two lines with subindices 1or1′is at least 32u= 32ρ/25>ρ.I. van der Hoog, F. Klute, I. Parada, and P. Schnider 23\n1\n10\n0\n00\n00\n0\n10\n1\nlong\nshort\npn\n\u000b\n\u000b0\n\f\n\f0\nFigure 4 Illustration of the reduction. The bottom gadgets show the polylines corresponding to\n00110001000 ∈A(left) and 01001110100 ∈B(right).\nFor vectorv∈A(inB) we create a subpolyline that for each value i∈{0,1}invhas a\nconstant horizontal segment on πi(resp.κi) if the next value is the same, and otherwise\nhas an alternation πi,πi′,πi,πi′,πi(resp.κi,κi′,κi,κi′,κi) of horizontal segments connected\nvertically. The parts corresponding to different consecutive values are connected with vertical\nsegments. Figure 4 (bottom) shows an example.\nPrefixes and suffixes. To force synchronization in the transversal of the vector gadgets, we\nprepend the vector 1...10withd1s at the beginning and we append the reverse vector\n01...1at the end of each vector in B. Analogously, we prepend the vector 0...01withd0s\nat the beginning and we append the reverse vector 10...0at the end of each vector in A.\nOnly in these prefixes and suffixes of Awe have such a long stretch (of length at least 4ud)\nin whichAis within radius ρof vertices on κ1andκ′\n1.\n▶Observation 33. If always staying within barking radius ρ, the dog can only (fully) traverse\na prefix or suffix gadget while the human is traversing one. Moreover, two prefix (two suffix)\ngadgets can only be traversed synchronously. More precisely, when the human is at the last\n(resp. first) vertex of its gadget in A, the dog must be at the part of the gadget corresponding\nto the digit 0.\nSplit gadgets. Between two vector gadgets in P, the polyline briefly goes vertically down\na distance ℓuntil a horizontal line β′. The polyline then goes vertically up until π0. To\nprevent interference with the vector gadgets, this distance ℓbetweenπ0andβ′should be\nlarge enough; ℓ= 2ρ= 50usuffices. This is the P-split gadget.\nBetween two vector gadgets in Q, the polyline does four detours from κ0.24 Barking dogs: A Fréchet distance variant for detour detection\n(i)First, it briefly goes vertically down a distance ℓuntil a horizontal line βand returns\nvertically to κ0.\n(ii)Then, polyline then goes vertically up a distance ℓ+ 10uuntil a horizontal line αand\nreturns vertically.\n(iii)After that, the polyline goes vertically up again, this time a distance ℓuntil reaching the\nhorizontal line α′and then it returns vertically to κ0.\n(iv)The polyline continues down vertically surpassing β′. It then goes horizontally left a\ndistanced; it suffices to also make d= 2ρ= 50u. From there the polyline traces its way\nback as it came, first horizontally and then vertically.\nThis is theQ-split gadget. This gadget is also included at the start of Q, and also at the end\n(there, including just the two first detours suffices).\nGoal.At the end of Pthe polyline has a P-split gadget and then it goes vertically up a\ndistanceℓ+ 30uuntil point pn.\n▶Observation 34. The vertical distance from pnto vertices in Qis only smaller than ρfor\nvertices on α.\nPositioning gadget. The beginning of polyline Pconsists of the polyline that would\ncorrespond to a vector Bwhich was a set of m−1zero vectors with a full Q-split gadget\nat the end. The human traversing this positioning gadget allows the dog to place itself at\nanyQ-split gadget (except the last half one) and wait there until the human finishes the\npositioning gadget.\n▷Claim 35. There is a trajectory for the dog such that when the human finishes the\npositioning gadget, the dog stands at the end of the Q-split gadget before a given vector\ngadget inQand such that the distance between human and dog is less than ρ.\nAt the beginning, the dog moves forward at the same speed as the human, thus staying\nwithin barking range. When in this way the dog reaches the desired Q-split gadget, instead\nof traversing Detour (iv), it returns back to point qbetween Detours (iii) and (iv), unless\nthe desired Q-split gadget is the one before the last vector gadget, in which case the dog\ncontinues forward. After this, when the human is at κ0, so is the dog at point qbetween\nDetours (iii) and (iv) of the Q-split gadget; when the human goes down to β, the dog does\nthe same using Detour (iv); when the human goes up to αso does the dog using Detour\n(iii), returning to q; and when the human goes down and to the left, the dogs does the same,\nreturning again to q. Only when the human does its last detour of the positioning gadget,\nthe dog fully traverses Detour (iv). ◀\nWith the construction description completed, notice that the polylines PandQdo not\nintersect.\nCorrectness. We show that the answer to the OV problem is yesif and only if the answer to\nthe barking decision problem is yes. Assume that there are two orthogonal vectors a,bin\nAandB, respectively. Note that the vectors are still orthogonal after adding the prefixes\nand suffixes. Claim 35 guarantees that the dog can stand at the beginning of the gadget\ncorresponding to awhen the human reaches the beginning of the gadget corresponding to b.\nThen, human and dog traverse those gadgets simultaneously. When the human traverses\nthe nextP-split gadget, the dog fully traverses Detour (i) of the next Q-split gadget. Then\nthe dog stays within barking radius at the point between Detours (i) and (ii) or going down\nalong Detour (i) and returning to the same point while the human traverses the remainingI. van der Hoog, F. Klute, I. Parada, and P. Schnider 25\ngadgets. Finally, when the human goes up to pnthe dog follows Detour (ii) until its top\npoint, thus staying always within barking radius ρ.\nIn the other direction, assume for contradiction that there are no orthogonal vectors in A\nandBand the dog always stays within barking radius ρ. When the human finishes traversing\nthe positioning gadget, which ends with its Detour (iv), the dog must just have traversed\nDetour (iv) of a Q-split gadget Qb, since the (leftmost points of) Detours (iv) (both in P\nandQ) are only within barking radius of themselves (and their close vicinity). Moreover,\nafter the human has finished the positioning gadget, the dog can never again fully traverse\nDetours (iii) or (iv) of a Q-split gadget. Thus, after that, the dog is constrained between\nthe twoQ-split gadgets QbandQb+1. Furthermore, it can only reach Detours (iii) and (iv)\nofQb, Detours (i) and (ii) of Qb+1, and the subcurve in-between. Whenever the human\ntraverses a P-split gadget, the dog must be at a Q-split gadget, more precisely, QborQb+1.\nObservations 32 and 33 imply that the dog can only traverse from QborQb+1ifBhas an\northogonal vector in A. Thus, it cannot reach Detour (ii) of Qb+1. However, Observation 34\nimplies that only points in the Detour (ii) of a Q-split gadget are within barking distance of\nqn. Thus, we reached a contradiction.\nWe remark that in the reduction as presented, the dog does not necessarily end at its\nfinal position. If we would desire that, we can just add a gadget to the human curve at the\nend analogous to the positioning gadget. As before, pncan be reached by a trajectory that\nstays within barking radius if and only if there are two orthogonal vectors in AandB. The\nappended positioning gadget in P(same as the one described minus the two first detours)\nallows the dog to reach the end of Qanalogously to the proof of Claim 35. ◀\n7 Conclusion\nWe introduced the barking distance to detect outliers in curves and have shown algorithms\nto compute this measure as well as a conditional lower bound for the computation time.\nIn the discrete and semi-discrete setting, the runtime of our algorithms match the lower\nbound up to logarithmic factors. For the continuous setting we give an algorithm that is\nlikely not optimal. We believe that using techniques as in Section 4 we can improve the\nruntime to O(nm3logm), but this would still leave a significant gap between upper and\nlower bound. While we conjecture that it is possible to obtain an O(nmlog(nm))algorithm,\nit is likely that new ideas are necessary for this. 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Springer, 2008. doi:10.1007/\n978-3-540-92137-0_51 ."    },    {        "title": "2402.13187v1.Testing_Calibration_in_Subquadratic_Time.pdf",        "content": "Testing Calibration in Subquadratic Time\nLunjia Hu∗Kevin Tian†Chutong Yang‡\nAbstract\nIn the recent literature on machine learning and decision making, calibration has emerged as\na desirable and widely-studied statistical property of the outputs of binary prediction models.\nHowever, the algorithmic aspects of measuring model calibration have remained relatively less\nwell-explored. Motivated by [BGHN23a], which proposed a rigorous framework for measuring\ndistances to calibration, we initiate the algorithmic study of calibration through the lens of\nproperty testing. We define the problem of calibration testing from samples where given ndraws\nfrom a distribution Don(predictions ,binary outcomes ), our goal is to distinguish between the\ncase where Dis perfectly calibrated, and the case where Disε-far from calibration.\nWe design an algorithm based on approximate linear programming, which solves calibration\ntesting information-theoretically optimally (up to constant factors) in time O(n1.5log(n)). This\nimprovesuponstate-of-the-artblack-boxlinearprogramsolversrequiring Ω(nω)time, where ω >\n2istheexponentofmatrixmultiplication. Wealsodevelopalgorithmsfortolerantvariantsofour\ntesting problem, and give sample complexity lower bounds for alternative calibration distances\nto the one considered in this work. Finally, we present preliminary experiments showing that\nthe testing problem we define faithfully captures standard notions of calibration, and that our\nalgorithms scale to accommodate moderate sample sizes.\n∗Stanford University, lunjia@stanford.edu\n†University of Texas at Austin, kjtian@cs.utexas.edu\n‡University of Texas at Austin, cyang98@utexas.edu\n1arXiv:2402.13187v1  [cs.LG]  20 Feb 2024Contents\n1 Introduction 3\n1.1 Our results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.2 Our techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n1.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n2 Preliminaries 8\n2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n2.2 Rounding linear programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n2.3 Box-simplex games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n3 Smooth calibration 13\n3.1 Rounding for empirical smooth calibration . . . . . . . . . . . . . . . . . . . . . . . . 13\n3.2 Sufficiency and insufficiency of Sbase. . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n3.3 Testing via smooth calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n4 Lower distance to calibration 21\n4.1 LDTC preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n4.2 Rounding for empirical U-LDTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\n4.3 Testing via LDTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27\n5 Sample complexity lower bounds for calibration distances 27\n5.1 Lower bound for convolved ECE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29\n5.2 Lower bound for (surrogate) interval CE . . . . . . . . . . . . . . . . . . . . . . . . . 31\n6 Experiments 32\n21 Introduction\nProbabilistic predictions are at the heart of modern data science. In domains as wide-ranging as\nforecasting (e.g. predicting the chance of rain from meteorological data [MW84, Mur98]), medicine\n(e.g. assessing the likelihood of disease [Doi07]), computer vision (e.g. assigning confidence values\nfor categorizing images [VDDP17]), and more (e.g. speech recognition [AAA+16] and recommender\nsystems [RRSK11]), prediction models have by now become essential components of the decision-\nmaking pipeline. Particularly in the context of critical, high-risk use cases, the interpretability of\nprediction models is therefore paramount in downstream applications. That is, how do we assign\nmeaning to the predictions our model gives us, especially when the model is uncertain?\nWe focus on perhaps the most ubiquitous form of prediction modeling: binary predictions, repre-\nsented as tuples (v, y)in[0,1]× {0,1}(where the vcoordinate is our prediction of the likelihood\nof an event, and the ycoordinate is the observed outcome). We model prediction-outcome pairs\nin the binary prediction setting by a joint distribution Dover [0,1]× {0,1}, fixed in the follow-\ning discussion. In this context, calibration of a predictor has emerged as a basic desideratum. A\nprediction-outcome distribution Dis said to be calibrated if\nE(v,y)∼D[y|v=t] =tfor all t∈[0,1]. (1)\nThat is, calibration asks that the outcome is 1exactly 60%of the time, when the model returns a\nprediction v= 0.6. While calibration (or approximate variants thereof) is a relatively weak require-\nment on a meaningful predictor, as it can be achieved by simple models,1it can still be significantly\nviolated in practice. For example, interest in calibration in the machine learning community was\nspurred by [GPSW17], which observed that many modern deep learning models are far from cali-\nbrated. Moreover, variants of calibration have been shown to have strong postprocessing properties\nfor fairness constraints and loss minimization [HKRR18, DKR+21, GKR+22], which has garnered\nrenewed interest in calibration by the theoretical computer science and statistics communities.\nThe question of measuring the calibration of a distribution is subtle; even a calibrated distribution\nincurs measurement error due to sampling. For example, consider the expected calibration error ,\nused in e.g. [NCH15, GPSW17, MDR+21a, RT21b] as a ground-truth notion of calibration distance:\nECE(D) :=E(v,y)∼D\u0002\f\fE(v′,y)∼D\u0002\ny|v′=v\u0003\n−v\f\f\u0003\n.\nUnfortunately, the empirical ECEis typically meaningless; if the marginal density of vis continuous,\nwe will almost surely only observe a single sample with each vvalue. Further, [KF08] observed that\nECEis discontinuous in v. In practice, binned variants of ECEare often used as a proxy, where a\nrange of vis lumped together in the conditioning event. However, hyperparameter choices (e.g. the\nnumber of bins) can significantly affect the quality of binned ECEvariants as a distance measure\n[KLM19, NDZ+19, MDR+21a].2Moreover, as we explore in this paper, binned calibration measures\ninherently suffer from larger sample complexity-to-accuracy tradeoffs, and are less faithful to ground\ntruth calibration notions in experiments than the calibration distance measures we consider.\nRecently, [BGHN23a] undertook a systematic study of various notions of distance to calibration pro-\nposed in the literature. They proposed information-theoretic tractability in the prediction-only ac-\n1The predictor which ignores features and always return the population mean is calibrated, for example.\n2For example, [NDZ+19] observed that, in their words, “dramatic differences in bin sensitivity” can occur “depend-\ning on properties of the (distribution) at hand,” a sentiment echoed by Section 5 of [MDR+21a].\n3cess (PA) model , wherethedistancetocalibrationdefinitioncanonlydependonthejointprediction-\noutcome distribution (rather than the features of training examples),3as a desirable criterion for\ncalibration distances. Correspondingly, [BGHN23a] introduced Definition 1 as a ground-truth no-\ntion of distance to calibration in the PA model, which we also adopt in this work.4\nDefinition 1 (Lower distance to calibration) .LetDbe a distribution over [0,1]×{0,1}. Thelower\ndistance to calibration (LDTC) ofD, denoted dCE(D), is defined by\ndCE(D) := inf\nΠ∈ext(D)E(u,v,y )∼Π|u−v|,\nwhereext(D)is all joint distributions Πover(u, v, y )∈[0,1]×[0,1]×{0,1}satisfying the following.\n•The marginal distribution of (v, y)isD.\n•The marginal distribution (u, y)is perfectly calibrated, i.e. EΠ[y|u] =u.\nDefinition 1 has various beneficial aspects: it is convex in v, computable in the PA model, and (as\nshown by [BGHN23a]) polynomially-related to various other calibration distance notions, including\nsome which require feature access. Roughly, the LDTC of a distribution is the tightest lower bound\non the ℓ1distance between vand any calibrated function of the features which can be made, after\ntaking features into account . The LDTC is the analog of this feature-aware definition of calibration\ndistance when limited to the PA model. We focus on this definition in the remainder of the paper.\n1.1 Our results\nWe initiate the algorithmic study of the calibration testing problem, defined as follows.\nDefinition 2 (Calibration testing) .Letε∈[0,1]. We say algorithm Asolves the ε-calibration\ntestingproblem with nsamples, if given ni.i.d. draws from a distribution Dover [0,1]× {0,1},A\nreturns either “yes” or “no” and satisfies the following with probability ≥2\n3.5\n•Areturns “no” if dCE(D)≥ε.\n•Areturns “yes” if dCE(D) = 0.\nIn this case, we also call Aanε-calibration tester.\nTo our knowledge, we are the first work to formalize the calibration testing problem. Defini-\ntion 2 is natural from the perspective of property testing , an influential paradigm in statistical\nlearning [Ron08, Ron09, Gol17]. In particular, there is an εn= Θ( n−1/2)so that it is information-\ntheoretically impossible to solve the εn-calibration testing problem from nsamples (see Lemma 2),\nso a variant of Definition 2 with an exact distinguishing threshold between “calibrated/uncalibrated”\n3This access model is practically desirable because it abstracts away the feature space, which can lead to significant\nmemory savings when our goal is only to test the calibration of model predictions. Moreover, this matches conventions\nin the machine learning literature: for example, loss functions are typically defined in the PA model.\n4We note that [BGHN23a] introduced an upper distance to calibration , also defined in the PA model, which\nthey showed is quadratically-related to the dCEin Definition 1. However, the upper distance does not satisfy basic\nproperties such as continuity, making it less amenable to estimation and algorithm design.\n5As is standard in property testing problems, the success probability of either a calibration tester or a tolerant\ncalibration tester can be boosted to 1−δfor any δ∈(0,1)at ak=O(log(1\nδ))overhead in the sample complexity.\nThis is because we can independently call kcopies of the tester and output the majority vote, which succeeds with\nprobability ≥1−δby Chernoff bounds, so we focus on δ=1\n3.\n4is not tractable. Hence, Definition 2 only requires distinguishing distributions Dwhich are “clearly\nuncalibrated” (parameterized by a threshold ε) from those which are perfectly calibrated.\nOur first algorithmic contribution is a subquadratic-time algorithm for calibration testing, in the\nregime where the threshold εis information-theoretically optimal up to a constant.\nTheorem 1 (Informal, see Theorem 3, Corollary 3) .Letn∈N, and let ε= Ω( εn), where εn=\nΘ(n−1/2)is minimal such that it is information-theoretically possible to solve the εn-calibration\ntesting problem with nsamples. There is an algorithm which solves the ε-calibration testing problem\nwithnsamples, running in time O(nlog(n)\nε) =O(n1.5log(n)).\nWe prove Theorem 1 by designing an algorithm for estimating smCE (bDn), thesmooth calibration\nerror(Definition 3), an alternative calibration distance measure, of an empirical distribution bDn.\nDefinition3 (Smoothcalibrationerror) .LetWbe the set of Lipschitz functions w: [0,1]→[−1,1].\nThesmooth calibration error of distribution Dover [0,1]× {0,1}, denoted smCE (D), is defined by\nsmCE (D) = sup\nw∈W\f\fE(v,y)∼D[(y−v)w(v)]\f\f.\nIt was shown in [BGHN23a] that smCE (D)is a constant-factor approximation to dCE(D)for all D\non[0,1]× {0,1}(see Lemma 12). Additionally, the empirical smCEadmits a representation as a\nlinear program with an O(n)×O(n)-sized constraint matrix encoding Lipschitz constraints.6Thus,\n[BGHN23a] proposed a simple procedure for estimating smCE (D): draw nsamples from D, and\nsolve the associated linear program on the empirical distribution. While there have been significant\nrecentruntimeadvancesinthelinearprogrammingliterature[LS14,CLS21,vdBLSS20,vdBLL+21],\nall state-of-the-art black-box linear programming algorithms solve linear systems involving the con-\nstraint matrix, which takes Ω(nω)time, where ω > 2.371[WXXZ23] is the current exponent of\nmatrix multiplication. Even under the best-possible assumption that ω= 2, the strategy of exactly\nsolving a linear program represents an Ω(n2)quadratic runtime barrier for calibration testing.\nOur work bypasses this barrier by noting that it suffices to solve the smCElinear program to low\naccuracy, i.e. ε= Ω(n−1/2), making it amenable to approximate (first-order method based) solvers.\nHowever, the smCElinear program (see (11)) is hard-constrained, so standard first-order methods do\nnot readily apply. We develop a novel approximate solver based on a custom combinatorial rounding\nprocedure, used to argue about the error incurred by lifting the smooth calibration constraints into\na penalty term, making it amenable to existing minimax optimization procedures. Our algorithm\nproving the runtime bound in Theorem 1 is given in Section 3.\nWenextdefinea tolerantvariantofDefinition2(seeDefinition4),whereweallowforerrorthresholds\nin both the “yes” and “no” cases; “yes” is the required answer when dCE(D)≤ε2, and “no” is\nrequired when dCE(D)≥ε1. Our algorithm in Theorem 1 continues to serve as an efficient tolerant\ncalibration tester when ε1≥4ε2, with formal guarantees stated in Theorem 3. This constant-factor\nloss comes from a similar loss in the relationship between smCEanddCE, see Lemma 12. We make\ntheobservationthataconstantfactorlossinthetoleranttestingparametersisinherentfollowingthis\nstrategy, via a lower bound in Lemma 13. Thus, even given infinite samples, computing the smooth\n6Formally,thenumberofconstraintsinthe smCElinearprogramis O(n2),butweshowthatinthehard-constrained\nsetting, requiring that “adjacent” constraints are met suffices (see Lemma 7).\n5calibration error cannot solve tolerant calibration testing all the way down to the information-\ntheoreticthreshold ε1≥ε2. Todevelopanimprovedtolerantcalibrationtester,wedirectlyshowhow\nto approximate the LDTC of an empirical distribution, our second main algorithmic contribution.\nTheorem 2 (Informal, see Theorem 4, Corollary 4) .Letn∈N, and let ε1−ε2= Ω( εn), where\nεn= Θ( n−1/2)is minimal such that it is information-theoretically possible to solve the εn-calibration\ntesting problem with nsamples. There is an algorithm which solves the (ε1, ε2)-tolerant calibration\ntesting problem with nsamples, running in time O(nlog(n)\n(ε1−ε2)2) =O(n2log(n)).\nWhile our algorithm in Theorem 2 is slower than that in Theorem 1, it directly approximates the\nLDTC, rather than a related quantity. We mention that state-of-the-art black-box linear program-\nming based solvers, while still applicable to (a discretized variant of) the empirical LDTC, require\nΩ(n2.5)time [vdBLL+21]. This is because the constraint matrix for the ε-approximate empirical\nLDTC linear program has dimensions O(n\nε)×O(n), resulting in a runtime overhead of ≈1\nε=O(√n)\nin both our approach and more direct solvers. We prove Theorem 2 in Section 4.\nIn Section 5, we complement our algorithmic results with lower bounds (Theorems 5, 6) on the\nsample complexity required to solve variants of the testing problem in Definition 2, when dCE\nis replaced with different calibration distances. For several widely-used distances in the machine\nlearning literature, including binned and convolved variants of ECE[NCH15, BN23], we show that\neΩ(ε−2.5)samples are required to the associated ε-calibration testing problem. This demonstrates a\nstatistical advantage of our focus on dCEas our ground-truth notion for calibration testing.\nWecorroborateourtheoreticalfindingswithpreliminaryexperimentalevidenceonrealandsynthetic\ndatainSection6. First, onasimpleBernoulliexample, weshowthat dCEandsmCEtestersaremore\nreliable indicators of calibration distance than a recently-proposed binned ECEvariant. We then\napply our smCEtester to postprocessed neural network predictions to test their calibration levels,\nvalidating against the findings in [GPSW17]. Finally, we implement our iterative method from\nTheorem 1 on our Bernoulli dataset, showing that it scales to moderate dimensions and achieves\nthe expected convergence rate. We note that a current limitation of our work is optimizing the\nimplementation to scale more effectively to high dimensions, an important open direction.7\n1.2 Our techniques\nTheorem 1 and Theorem 2 follow from designing custom first-order methods for approximat-\ning empirical linear programs associated with the smCEand dCEof a sampled dataset bDn:=\n{(vi, yi)}i∈[n]∼i.i.d.D. In both cases, known generalization bounds from [BGHN23a] show it suf-\nfices to approximate the value of the empirical calibration distances to error ε= Ω(n−1/2).\nWe begin by explaining our strategy for estimating smCE (bDn)(Definition 3). By definition, the\nsmooth calibration error of bDncan be formulated as a linear program,\nmin\nx∈[−1,1]n1\nnX\ni∈[n]xi(vi−yi),where |xi−xj| ≤ |vi−vj|for all (i, j)∈[n]×[n]. (2)\nHere, xi∈[−1,1]corresponds to the weight on vi, and there are 2\u0000n\n2\u0001\nconstraints on the decision\n7Ourcodecanbefoundat: https://github.com/chutongyang98/Testing-Calibration-in-Subquadratic-Time.\ngit.\n6variable x, each of which corresponds to a Lipschitz constraint. We can write (2) as an inequality-\nconstrained linear program max x∈[−1,1]n:Ax≤b⟨c, x⟩, for appropriate A, b, c. While first-order meth-\nods (e.g. Frank-Wolfe type algorithms) can sometimes apply to hard-constrained linear programs, it\nis unclear how to implement projection steps onto the Lipschitz constraints Ax≤bin our setting.\nWe instead follow an “augmented Lagrangian” method where we lift the constraints directly into\nthe objective as a soft-constrained penalty term. To prove correctness of this lifting, we follow a line\nof results in combinatorial optimization [She13, JST19]. These works develop a “proof-by-rounding\nalgorithm” framework to show that the hard-constrained and soft-constrained linear programs have\nequal values. This rounding-based framework is summarized in Section 2.2 (see Lemma 3).\nTo use this rounding-based framework, our key technical innovation is showing that if xviolates\neach of a carefully-selected set SofO(n)Lipschitz constraints by at most ∆, we can design a\nprocedure which produces x′with∥x′−x∥∞=O(∆), yet x′is fully feasible for the constraints\nAx′≤b. Our set Sis a multilayer construction, stated formally in (20). Informally, we begin\nby sorting vinto nonincreasing order, and defining a “zeroth-layer” of constraints by enforcing the\nLipschitz condition on all pairs (i, i+ 1)fori∈[n−1]. The triangle inequality shows that exactly\nenforcing these constraints suffices for feasibility of the linear program (Lemma 7). Enforcing these\nzeroth-layer constraints only approximately, however, degrades poorly under iterative rounding.\nAccordingly, we build three dyadic trees of constraints to protect how much our rounding procedure\ncan iteratively degrade constraint violations. After proving correctness of our rounding method for\nthe smooth calibration error linear program, we use it to show that the soft-constrained objective\nmin\nx∈[−1,1]n⟨c, x⟩+∥max (AS:x−bS,0S)∥∞ (3)\nhas the same value as (2), for A, b, cfrom the smCElinear program, and where subscripting by S\nrestricts to only the relevant rows of the matrix A, and the relevant coordinates of b. Finally, we\napply a recent first-order minimax optimization algorithm of [JT23] to solve the problem (3).\nOur high-level approach to proving Theorem 2 is analogous, going through the same rounding-based\naugmented Lagrangian framework as described before. However, the empirical dCElinear program\n(see Lemma 16) is more complicated than that for smCEerror, and imposes two sets of constraints,\nroughly corresponding to marginal satisfaction of (v, y)and calibration of (u, y)(using notation\nfrom Definition 1). We design a two-step rounding procedure, which first fixes the marginals on the\n(v, y)coordinates, and then calibrates the ucoordinates without affecting any (v, y)marginal. One\ninteresting feature of our dCErounding algorithm, compared to prior works [She13, JST19], is that\nit iscost-sensitive : we take into account the metric structure of the linear costs cto argue rounding\ndoes not harm the objective value, rather than naïvely applying Hölder’s inequality.\nFinally, our sample complexity lower bounds in Section 5 for alternative calibration measures fol-\nlow from an information-theoretic analysis using LeCam’s two-point method [LeC73] and Ingster’s\nmethod [IS03]. We construct a perfectly-calibrated distribution as well as a family of miscalibrated\ndistributions with large calibration error ( ≥ε) in the alternative calibration measures (convolved\nECE and interval CE). A testing algorithm for these measures needs to distinguish these two cases.\nWe show that the distinguishing task has high sample complexity ( ≈ε−2.5) by bounding the total\nvariation distance between the two joint distributions of the input examples from the two cases.\nOur proof is inspired by a similar analysis showing a sample complexity lower bound for identity\ntesting in the distribution testing literature (see e.g. Section 3.1 of [Can22]).\n71.3 Related work\nThe calibration performance of deep neural networks has been studied extensively in the literature\n(e.g. [GPSW17, MDR+21b, Rt21a, BGHN23b]). Measuring the calibration error in a meaningful\nway can be challenging, especially when the predictions are not naturally discretized (e.g. in neu-\nral networks). Recently, [BGHN23a] addresses this challenge using the distance to calibration as a\ncentral notion. They consider a calibration measure to be consistent if it is polynomially-related\nto the distance to calibration. Consistent calibration measures include the smooth calibration error\n[KF04], Laplace kernel calibration error [KSJ18], interval calibration error [BGHN23a], and con-\nvolved ECE [BN23].8On the algorithmic front, substantial observations were made by [BGHN23a]\non linear programming characterizations of calibration distances such as the LDTC and smooth cal-\nibration. While there have been significant advances on the runtime frontier of linear programming\nsolvers, current runtimes for handling an n×dlinear program constraint matrix with n≥dremain\nΩ(min( nd+d2.5, nω))[CLS21, vdBLL+21]. Our constraint matrix is roughly-square and highly-\nsparse, so it is plausible that e.g. the recent research on sparse linear system solvers [PV21, Nie22]\ncould apply to the relevant Newton’s method subproblems and improve upon these rates. Moreover,\nwhile efficient estimation algorithms have been proposed by [BGHN23a] for (surrogate) interval\ncalibration error and by [BN23] for convolved ECE, these algorithms require suboptimal sample\ncomplexity for solving our testing task in Definition 2 (see Section 5). To compute their respective\ndistances to error εfrom samples, these algorithms require Ω(ε−5)andΩ(ε−3)time. As compari-\nson, under this parameterization Theorems 1 and 2 require eO(ε−3)andeO(ε−4)time, but can solve\nstronger testing problems with the same sample complexity, experimentally validated in Section 6.\n2 Preliminaries\n2.1 Notation\nThroughout this paper, we use Dto denote a distribution over [0,1]× {0,1}. When Dis clear\nfrom context, we let bDn={(vi, yi)}i∈[n]denote a dataset of nindependent samples from Dand,\nin a slight abuse of notation, the distribution with probability1\nnfor each (vi, yi). We say dis a\ncalibration distance if it takes distributions on [0,1]× {0,1}to the nonnegative reals R≥0, sodCE\n(Definition 1) and smCE(Definition 3) are both calibration distances.\nWhen applied to a vector, we let ∥·∥pdenote the ℓpnorm for p≥1. We denote [n] :={i∈\nN|i≤n}. We let 0dand 1ddenote the all-zeroes and all-ones vectors in dimension d. We let\n∆d:={x∈Rd\n≥0| ∥x∥1= 1}denote the probability simplex in dimension d. The ithcoordinate\nbasis vector is denoted ei. We say ˜x∈Ris an ε-additive approximation of x∈Rif|˜x−x| ≤ε.\nFor a set S⊂R, we say another set T⊂Ris an ε-cover of Sif for all s∈S, there is t∈Twith\n|s−t| ≤ε. We use eOandeΩto hide polylogarithmic factors in the argument.\nWe denote matrices in boldface throughout. For any matrix A∈Rm×n, we refer to its ithrow by\nAi:and its jthcolumn by A:j. Moreover, for a set Sidentified with rows of a matrix A, we let As:\ndenote the row indexed by s∈S, and use similar notation for columns.\n8The calibration distance we call the convolved ECE in our work was originally called the smooth ECE in [BN23].\nWe change the name slightly to reduce overlap with the smooth calibration error (Definition 3), a central object\nthroughout the paper.\n8For any p, q≥1, we define\n∥A∥p→q:= max\nx∈Rn|∥x∥p≤1∥Ax∥q.\nNotice that in particular, ∥A∥1→1is the largest ℓ1norm of any column of A, and∥A∥∞→∞is the\nlargest ℓ1norm of any row. We say that x∈ Xis an ε-approximate minimizer of f:X → Rif\nf(x)−minx′∈Xf(x′)≤ε. We call (x, y)∈ X ×Y anε-approximate saddle point to a convex-concave\nfunction f:X × Y → Rif its duality gap is at most ε, i.e.\nmax\ny′∈Yf(x, y′)−min\nx′∈Xf(x′, y)≤ε.\nThe following simple claim will often be useful.\nLemma 1. Letf:X ×Y → Rbe convex-concave for compact X,Y, and let g(x) := max y∈Yf(x, y)\nforx∈ X. If(x, y)is an ε-approximate saddle point to f,xis an ε-approximate minimizer to g.\nProof.Lety′:=argmaxy∈Yf(x, y)andx′:=argminx′∈Xf(x′, y). The conclusion follows from\nmin\nx⋆∈Xg(x⋆) = min\nx⋆∈Xmax\ny⋆∈Yf(x⋆, y⋆) = max\ny⋆∈Ymin\nx⋆∈Xf(x⋆, y⋆)≥min\nx′∈Xf(x′, y),\nwhere we used strong duality (via Sion’s minimax theorem), so that\ng(x)−min\nx⋆∈Xg(x⋆)≤g(x)−f(x′, y) =f(x, y′)−f(x′, y)≤ε.\nWe next define a tolerant variant of the calibration testing problem in Definition 2.\nDefinition 4 (Tolerant calibration testing) .Let0≤ε2≤ε1≤1. We say algorithm Asolves the\n(ε1, ε2)-tolerantcalibrationtesting problem with nsamples, if given ni.i.d. draws from a distribution\nDover [0,1]×{0,1},Areturns either “yes” or “no” and satisfies the following with probability ≥2\n3.\n•Areturns “no” if dCE(D)≥ε1.\n•Areturns “yes” if dCE(D)≤ε2.\nIn this case, we also call Aan(ε1, ε2)-tolerant calibration tester.\nNote that an algorithm which solves the (ε1, ε2)-tolerant calibration testing problem with nsamples\nalso solves the ε1-calibration testing problem with the same sample complexity. Moreover, we give\na simple impossibility result on parameter ranges for calibration testing.\nLemma 2. Let0≤ε2≤ε1≤1\n2satisfy ε1−ε2=ε. There is a universal constant Ccoinsuch\nthat, given n≤Ccoin\nε2samples from a distribution on [0,1]× {0,1}, it is information-theoretically\nimpossible to solve the (ε1, ε2)-tolerant calibration testing problem.\nProof.Suppose ε≤1\n10, else we can choose Ccoinsmall enough such that n <1. We consider two\ndistributions over [0,1]× {0,1},DandD′, with dCE(D)≥ε1butdCE(D′)≤ε2, so if Asucceeds\nat tolerant calibration testing for both DandD′, we must have dTV(D⊗n,(D′)⊗n)≥1\n3, where we\ndenote the n-fold product of a distribution with ·⊗n. Else, Acannot return different answers from\nnsamples with probability ≥2\n3, as required by Definition 4. Specifically, we define D,D′as follows.\n9•To draw (v, y)∼ D, letv=1\n2+ε1andy∼Bern(1\n2).\n•To draw (v, y)∼ D′, letv=1\n2+ε1andy∼Bern(1\n2+ε).\nWe claim that dCE(D) =ε1. To see this, let Π∈ext(D)and(u, v, y )∼Π, soE(u,v,y )∼Π[u] =1\n2\nbecause uis calibrated. By Jensen’s inequality, we have:\nE(u,v,y )∼Π[|u−v|]≥\f\f\f\fE(u,v,y )∼Π[u]−\u00121\n2+ε1\u0013\f\f\f\f=ε1.\nThe equality case is realized when u=1\n2with probability 1, proving the claim. Similarly, dCE(D′) =\nε2. Finally, let π:=Bern(1\n2),π′:=Bern(1\n2+ε), and π⊗n,(π′)⊗ndenote their n-fold product\ndistributions. Pinsker’s inequality shows that it suffices to show that dKL(π⊗n∥(π′)⊗n)≤1\n5to\ncontradict our earlier claim dTV((D)⊗n,(D′)⊗n)≥1\n3. To this end, we have\ndKL(π⊗n∥(π′)⊗n) =n·dKL\u0000\nπ∥π′\u0001\n=n\n2 \nlog 1\n2\n1\n2+ε!\n+ log 1\n2\n1\n2−ε!!\n=n\n2log\u00121\n1−4ε2\u0013\n≤n\n2·5ε2≤1\n5,\nwhere the first line used tensorization of dKL, and the last chose Ccoinsmall enough.\nWe also generalize Definitions 2 and 4 to apply to an arbitrary calibration distance.\nDefinition5 (dtesting).Letdbe a calibration distance. For ε∈R≥0, we say algorithm Asolves the\nε-d-testingproblem (or, Ais an ε-dtester) with nsamples, if given ni.i.d. draws from a distribution\nDover [0,1]×{0,1},Areturns either “yes” or “no” and satisfies the following with probability ≥2\n3.\n1.Areturns “no” if d(D)≥ε.\n2.Areturns “yes” if d(D) = 0.\nFor0≤ε2≤ε1, we say algorithm Asolves the (ε1, ε2)-tolerant dtesting problem (or, Ais an\n(ε1, ε2)-tolerant dtester) with nsamples, if given ni.i.d. draws from a distribution Dover [0,1]×\n{0,1},Areturns either “yes” or “no” and satisfies the following with probability ≥2\n3.\n1.Areturns “no” if d(D)≥ε1.\n2.Areturns “yes” if d(D)≤ε2.\n2.2 Rounding linear programs\nIn this section, we give a general framework for approximately solving linear programs, following\nsimilar developments in the recent combinatorial optimization literature [She13, JST19]. Roughly\nspeaking, this framework is a technique for losslessly converting a constrained convex program to\nan unconstrained one, provided we can show existence of a rounding procedure compatible with the\nconstrained program in an appropriate sense. We begin with our definition of a rounding procedure.\nDefinition 6 (Rounding procedure) .Consider a convex program defined on the intersection of\nconvex set Xwith linear equality constraints Ax=b:\nmin\nx∈X\nAx=bc⊤x. (4)\n10We say Roundis a(eA,˜b, p)-equality rounding procedure for (A, b, c,X)ifp≥1, and for any x∈ X,\nthere exists x′:=Round (x)∈ Xsuch that Ax′=b,eAx′=˜b, and\nc⊤x′≤c⊤x+\r\r\reAx−˜b\r\r\r\np. (5)\nSimilarly, consider a program on the intersection of a convex set with linear inequality constraints:\nmin\nx∈X\nAx≤bc⊤x. (6)\nWe say Roundis a(eA,˜b, p)-inequality rounding procedure for (A, b, c,X)ifp≥1, and for any\nx∈ X, there exists x′:=Round (x)∈ Xsuch that Ax′≤b,eAx′≤˜b, and\nc⊤x′≤c⊤x+\r\r\rmax\u0010\neAx−˜b,0m\u0011\r\r\r\np, (7)\nwhere mis the number of rows in eA,˜b, and the maxoperation is entrywise.\nIntuitively, rounding procedures replace the hard-constrained problems (4), (6) with their soft-\nconstrained variants, i.e. the soft equality-constrained\nmin\nx∈Xc⊤x+\r\r\reAx−˜b\r\r\r\np, (8)\nand the soft inequality-constrained\nmin\nx∈Xc⊤x+\r\r\rmax\u0010\neAx−˜b,0m\u0011\r\r\r\np, (9)\nrespectively, for some (eA,˜b, p)constructed from the hard-constrained problem instance (parameter-\nized by A, b, c,X). Leveraging the assumptions on our rounding procedure in each case (5), (7), we\nnow show how to relate approximate solutions to these problems, generalizing Lemma 1 of [JST19].\nLemma 3. Letxbe an ε-approximate minimizer to (8), and let Roundbe a(eA,˜b, p)-equality round-\ning procedure for (A, b, c,X). Then x′:=Round (x)is an ε-approximate minimizer to (4). Similarly,\nifxis an ε-approximate minimizer to (9)andRoundis a(eA,˜b, p)-inequality rounding procedure for\n(A, b, c,X),x′:=Round (x)is an ε-approximate minimizer to (6).\nProof.We only discuss the equality-constrained setting (4), (8), as the proof of the inequality-\nconstrained setting is entirely analogous. We first claim that a minimizing solution to (8) satisfies\nthe constraints Ax=b. To see this, given any x∈ X, we can produce x′∈ XwithAx′=band\nsuch that x′has smaller objective value in (8). Indeed, letting x′:=Round (x′), (5) guarantees\nc⊤x′=c⊤x′+\r\r\reAx′−˜b\r\r\r\np≤c⊤x+\r\r\reAx−˜b\r\r\r\np,\nas claimed. Now let x⋆∈ Xsatisfying Ax⋆=bminimize (8). Then, if xis an ε-approximate\nminimizer to (8) and x′=Round (x), we have the desired claim from Ax′=b, and\nc⊤x′=c⊤x′+\r\r\reAx′−˜b\r\r\r\np≤c⊤x+\r\r\reAx−˜b\r\r\r\np≤c⊤x⋆+\r\r\reAx⋆−˜b\r\r\r\np=c⊤x⋆+ε.\n112.3 Box-simplex games\nWe will apply the rounding framework in Section 2.2 to various hard-constrained linear programs,\nin the geometries defined by p= 1,∞. To aid in approximately solving the soft-constrained linear\nprograms arising from our framework, we use the following procedure from [JT23], building upon\nthe recent literature for solving box-simplex games at accelerated rates [She17, JST19, CST21].\nProposition 1 (Theorem 1, [JT23]) .LetA∈Rn×d,b∈Rd,c∈Rn, and ε >0. There is an\nalgorithm which computes an ε-approximate saddle point to the box-simplex game\nmin\nx∈[−1,1]nmax\ny∈∆dx⊤Ay−b⊤y+c⊤x, (10)\nin time\nO\u0012\nnnz(A)·∥A∥1→1logd\nε\u0013\n.\nThe following two corollaries of Proposition 1 will be particularly useful in our development. The\nfirst is immediate using Lemma 1, upon negating the box-simplex game (10), exchanging the names\nof the variables (x, y),(b, c), and explicitly maximizing over y∈[−1,1]n, i.e.\nmin\nx∈∆d⟨c, x⟩+∥Ax−b∥1= min\nx∈∆dmax\ny∈[−1,1]n⟨c, x⟩+y⊤(Ax−b).\nCorollary 1. LetA∈Rn×d,b∈Rn,c∈Rd, and ε >0. There is an algorithm which computes an\nε-approximate minimizer to minx∈∆dc⊤x+∥Ax−b∥1, in time\nO\u0012\nnnz(A)·∥A∥1→1logd\nε\u0013\n.\nWe also have the following analog for a variant of ℓ∞regression.\nCorollary 2. LetA∈Rn×d, b∈Rn, c∈Rd, and ε >0. There is an algorithm which computes\nanε-approximate minimizer to minx∈[−1,1]dc⊤x+∥max (Ax−b,0n)∥∞, where the maxoperation\nis entrywise, in time\nO\u0012\nnnz(A)·∥A∥∞→∞ logn\nε\u0013\n.\nProof.First, observe that we can rewrite the given objective as\nmin\nx∈[−1,1]dmax\ny∈∆n+1c⊤x+y⊤(eAx−˜b),\nwhere eA∈R(n+1)×dappends an extra all-zeroes row to A, and ˜b∈Rn+1appends an extra zero to b.\nNamely, if Ax−b≤0nentrywise, we have y⊤(eAx−˜b) = 0 = ∥max(Ax−b,0n)∥∞, and otherwise\ny⊤(eAx−˜b)chooses the largest entry of Ax−b. The conclusion follows from Proposition 1, using\n∥eA∥∞→∞ =∥A∥∞→∞ =∥A⊤∥1→1.\n123 Smooth calibration\nIn this section, we provide our main result on approximating the smooth calibration of a distribution\non[0,1]×{0,1}. In Section 3.1, we first develop a rounding procedure compatible with the smooth\ncalibration linear program (in the sense of Definition 6), when applied to an empirical distribution.\nWe provide some discussion of a simpler variant of our rounding procedure in Section 3.2. Finally,\nin Section 3.3, we show how to use the solver resulting from our rounding procedure to develop an\nefficient algorithm for solving the calibration testing problems in Definitions 2 and 4.\n3.1 Rounding for empirical smooth calibration\nIn this section, we develop an efficient algorithm for computing the smooth calibration error of an\nempirical distribution. Specifically, throughout the section, we fix a dataset under consideration,\nbDn:={(vi, yi)}i∈[n]⊂[0,1]× {0,1},\nand the corresponding empirical distribution (which, in an abuse of notation, we also denote bDn),\ni.e. we use (v, y)∼bDnto mean that (v, y) = ( vi, yi)with probability1\nnfor each i∈[n]. We also\nassume without loss of generality that the {vi}i∈[n]are in sorted order, so 0≤v1≤. . .≤vn≤1.\nRecalling Definition 3, the associated empirical smooth calibration linear program is\nsmCE (bDn) := max\nx∈X\nAx≤bc⊤x,where X:= [−1,1]n,\nci:=1\nn(yi−vi)for all i∈[n],\nb(i,j,+)=b(i,j,−):=vj−vifor all (i, j)∈[n]×[n]with i < j,\nandA(i,j,+)::=ej−ei,\nA(i,j,−)::=ei−ej,for all (i, j)∈[n]×[n]with i < j.(11)\nHere, xirepresents the value w(vi). Because x=0nis always feasible, the maximum ⟨c, x⟩is non-\nnegative, so we can drop the absolute value in Definition 3. Moreover, Aandbhave rows identified\nwith [2\u0000n\n2\u0001\n], i.e. doubled pairs of sorted indices in [n]×[n], enforcing the Lipschitz constraints\nmax( xj−xi, xi−xj)≤vj−vifor all (i, j)∈[n]×[n]with i < j.\nOur solver for (11) goes through the machinery of Definition 6; we design a (eA,˜b,∞)-inequality\nrounding procedure for (A, b, c,X)in (11). We let eA,˜bbe carefully chosen subsets of the rows of\nA, b, with O(n)rows. Before formally stating our choices of eA,˜b, we give several helper lemmas.\nLemma 4. Leti, j, k∈[n]withi < j < k . Suppose that for ∆≥0andv, x∈Rnwithvi≤vj≤vk,\n|xi−xj| ≤vj−vi+ ∆,|xi−xk| ≤vk−vi,|xj−xk| ≤vk−vj+ ∆.\nThere is y∈Rnwithyℓ=xℓfor all ℓ̸=j,yj∈[min ℓ∈[n]xℓ,max ℓ∈[n]xℓ],|yj−xj| ≤∆, and\n|yi−yj| ≤vj−vi,|yi−yk| ≤vk−vi,|yj−yk| ≤vk−vj.\n13Proof.We split into four cases, depending on which subset of the constraints |xi−xj| ≤vj−viand\n|xj−xk| ≤vk−vjis false. In the first case, neither is false and returning y←xsuffices.\nIn the second case, both are false, i.e.\nvj−vi<|xi−xj| ≤vj−vi+ ∆, vk−vj<|xj−xk| ≤vk−vj+ ∆. (12)\nWe claim that in this case, xj≤min(xi, xk)orxj≥max( xi, xk). To see this, if xi≤xj≤xk, then\n|xj−xi|+|xk−xj|= (xj−xi) + (xk−xj) =xk−xi≤vk−vi= (vj−vi) + (vk−vj),(13)\nwhere we used |xi−xk| ≤vk−vi, so we cannot have (12) hold. Similarly, if xk≤xj≤xi, this again\ncontradicts (12). Now, we claim it suffices to choose y=xcoordinatewise except for yj, defined by\nyj←xj+sign(xi−xj) max (( |xi−xj| −(vj−vi)),(|xk−xj| −(vk−vj)))| {z }\n:=ζ.(14)\nIn other words, we move xjtowards the other two points by the larger of the two violation amounts\nin (12). To prove correctness of (14), suppose xj≤min(xi, xk)without loss of generality, so that\nyj=xj+ζ(the other case is symmetric by negating x). We first observe that\nmax( xi, xk)−xj≥ζ=⇒yj≤max( xi, xk). (15)\nNext, consider the subcase where yj≤min(xi, xk). Then, yi−yj=xi−yj=xi−xj−ζ≤vj−vi\nso the Lipschitz constraint on (yi, yj)is enforced. We symmetrically have yk−yj≤vk−vi.\nThe other subcases are when xi≤yj≤xkorxk≤yj≤xi, i.e. yjis in between xiandxk. By the\ndefinition of ζ, either |yj−xi|=vj−vior|xk−yj|=vk−vjis a tight Lipschitz constraint, and\nin either case (13) shows that the other constraint is also enforced, since |xk−xi| ≤vk−vi.\nIn the third case, we have\nvj−vi<|xi−xj| ≤vj−vi+ ∆,|xj−xk| ≤vk−vj. (16)\nWe claim it suffices to choose\nyj←xj+sign(xi−xj) (|xi−xj| −(vj−vi))| {z }\n:=ζ. (17)\nAgain assume without loss of generality that xj≤xiandyj=xj+ζ. By construction, |yj−xi|=\nvj−vi, so we need to check that |yj−xk| ≤vk−vj. If originally xk≤xj≤xi, then (13) shows\nxj−xk≤(vk−vj)−ζ, so|yj−xk| ≤xk−xj+ζ≤vk−vjas claimed. Otherwise, if xj≤xk, then\n(15) shows yj≤xi. Ifyj∈[xk, xi], then |yj−xi|=vj−viand (13) show that |yj−xk| ≤vk−vj\nas well. Finally, if yj≤xk, then 0≤xk−yj=xk−xj−ζ≤vk−vj, as desired.\nIn the fourth case, we have\nvk−vj<|xk−xj| ≤vk−vj+δ,|xj−xi| ≤vj−vi. (18)\nWe claim it suffices to choose\nyj←xj+sign(xk−xj) (|xk−xj| −(vk−vj))| {z }\n:=ζ. (19)\n14The correctness analysis of (19) proceeds symmetrically to the correctness analysis of (17).\nIt is straightforward to check that in all cases, |yj−xj| ≤∆, and yjstays in the range with endpoints\nmin(xi, xj, xk)andmax( xi, xj, xk), proving the other claimed conditions.\nLemma 4 says that given three Lipschitz constraints induced by three coordinates of v, if the “outer”\nconstraint is satisfied and the other two constraints are violated by ≤∆, we can move the middle\ncoordinate by ≤∆to fix all constraints. Moreover, it guarantees that if x∈ X = [−1,1]n, then\ny∈ X. We next observe that this procedure does not make adjacent constraints worse.\nLemma 5. Leti, j, h∈[n]withi < j < h . Suppose that for ∆≥0andv, x∈Rnwithvi≤vj≤vh,\nvj−vi≤ |xi−xj| ≤vj−vi+ ∆,|xh−xi| ≤vh−vi+ ∆,|xh−xj| ≤vh−vj+ ∆.\nLett=xj−sign(xj−xi)·ζforζ∈[0,|xi−xj| −(vj−vi)]. Then, |xh−t| ≤vh−vj+ ∆.\nProof.The argument is symmetric in xand−x, so assume without loss that xi≤xjand\nt∈[xi+ (vj−vi), xj].\nIfxh≤xj, then since t≤xj, we have |xh−t| ≤max(|xh−xj|,|t−xj|)≤max( vh−vj+ ∆,∆) =\nvh−vj+ ∆. Otherwise, if xh≥xj, since |xh−t|=|xh−xj|+|xj−t|in this case, we have\n|xh−t|=xh−xj+ (xj−xi)−(vj−vi)≤vh−vj+ ∆,\nwhich follows from |xh−xi|=xh−xi≤vh−vi+ ∆.\nWe remark that Lemma 5 applies exactly to moves of the form induced by Lemma 4, regardless of\nthe case of Lemma 4 we are in. We symmetrically have the following claim.\nLemma6. Leth, j, k ∈[n]withh < j < k . Suppose that for ∆≥0andv, x∈Rnwithvh≤vj≤vk,\nvk−vj≤ |xk−xj| ≤vk−vj+ ∆,|xh−xj| ≤vj−vh+ ∆,|xh−xk| ≤vk−vh+ ∆.\nLett=xj−sign(xj−xk)·ζforζ∈[0,|xk−xj| −(vk−vj)]. Then, |xh−t| ≤vj−vh+ ∆.\nOur rounding procedure design (parameterized by eA,˜b) is motivated by the observations in Lem-\nmas 4, 5, and 6. We fix three sets of index pairs in [n]×[n]as follows:\nS:=Sbase∪Sleft∪Sright, L:=⌊log2(n−1)⌋,\nSbase:= ([n]×[n])∩L[\nℓ=0n\u0010\n(i−1)2ℓ+ 1, i2ℓ+ 1\u0011\n|i∈[2L−ℓ]o\n,\nSleft:= ([n]×[n])∩L[\nℓ=1ℓ−1[\nm=0n\u0010\n(i−1)2ℓ−2m+ 1, i2ℓ+ 1\u0011\n|i∈[2L−ℓ]o\n,\nSright:= ([n]×[n])∩L[\nℓ=1ℓ−1[\nm=0n\u0010\n(i−1)2ℓ+ 1, i2ℓ+ 2m+ 1\u0011\n|i∈[2L−ℓ]o\n.(20)\n15We also call the following set of indices in Stheℓthlayer of constraints :\nSℓ:=\u0010\n([n]×[n])∩n\u0010\n(i−1)2ℓ+ 1, i2ℓ+ 1\u0011\n|i∈[2L−ℓ]o\u0011\n. (21)\nIn other words, for each 0≤ℓ≤L, the ℓthlayer of constraints is entirely due to Sbase, and consists\nof index pairs in [n]×[n]spaced 2ℓcoordinates apart. There is a corresponding set of constraints\ndue to Sleftwhich extends the ℓthlayer of constraints, and consists of index pairs in [n]×[n]\nspaced 2ℓ+ 2mcoordinates apart for each 0≤m≤ℓ−1, offset by 2ℓcoordinates each, where\nthe left endpoint is extended 2mbeyond the corresponding constraint in Sbase. We define Sright\nsymmetrically, extending right endpoints of intervals in Sbaseby2minstead.\nOur rounding procedure fixes Lipschitz constraints one layer at a time using Lemma 4. We use\nLemma 5 to make sure that fixing a higher layer does not worsen the violation of any lower layer, by\nusing adjacent constraints in Srightto protect the lower layer. Similarly, we use Lemma 6 to protect\nlower layers using adjacent constraints in Sleft. This also explains our inclusion of the constraints\nSleft,Sright. For completeness, we give a discussion in Section 3.2 on why using only Sbasewould\nresult in losing a logarithmic factor in the rounding guarantee, yielding a similar loss in the runtime.\nFor example, if n= 9, soL= 3,Sdefined in (20) consists of the pairs\nSbase={(i, i+ 1)|i∈[8]} ∪ {(2i−1,2i+ 1)|i∈[4]} ∪ {(4i−3,4i+ 1)|i∈[2]} ∪ {(1,9)},\nSleft={(2i−2,2i+ 1)|2≤i≤4} ∪ {(4,9)} ∪ {(3,9)},\nSright={(2i−1,2i+ 2)|i∈[3]} ∪ {(1,6)} ∪ {(1,7)}.\nWe let eA,˜bbe scaled row subsets of A,bcorresponding to S, i.e. following notation in (11), (20),\neA∈R2|S|×nhaseA(s,+)::=A(s,+):,eA(s,−)::=A(s,−):for all s∈S,\n˜b∈R2|S|has˜b(s,+):=b(s,+),˜b(s,−):=b(s,−)for all s∈S.(22)\nWenextshowhowtorecursivelyapplyLemma4toshowthereexistsanefficient (eA,˜b,∞)-inequality\nrounding procedure for (A, b, c,X), recalling the definition of rounding procedures in Definition 6.\nWe begin by observing that it suffices to enforce the 0thlayer of constraints in Sto show feasibility.\nLemma 7. Ifx, v∈Rn, where vhas monotonically nondecreasing coordinates, and |xi−xi+1| ≤\nvi+1−vifor all i∈[n−1], then |xi−xj| ≤vj−vifor all (i, j)∈[n]×[n]withi < j.\nProof.This follows from the triangle inequality:\n|xi−xj| ≤j−1X\nk=i|xk−xk+1| ≤j−1X\nk=ivk+1−vk=vj−vi.\nWe can now state and analyze our rounding procedure.\nLemma 8. Let(A, b, c,X)be defined as in (11), and let (eA,˜b)be defined as in (20),(22). There\nexists Round, a(eA,˜b,∞)-inequality rounding procedure for (A, b, c,X)running in O(n)time.\n16Proof.For a fixed x∈ X, let\n∆ :=\r\r\rmax\u0010\neAx−˜b,02|S|\u0011\r\r\r\n∞,\nso that ∆is the maximum violation of any Lipschitz constraint in Sat the start of the algorithm.\nWe claim we can produce x′=Round (x)∈ XwithAx′≤b(i.e.x′satisfies all of the Lipschitz\nconstraints in [n]×[n]), such that ∥x−x′∥∞≤∆. This implies Roundis a(eA,˜b,∞)-inequality\nrounding procedure as desired, because (9) then follows from c∈[−1\nn,1\nn]n=⇒ ∥c∥1≤1, and\nc⊤(x′−x)≤ ∥c∥1\r\rx′−x\r\r\n∞≤∆ =\r\r\rmax\u0010\neAx−˜b,02|S|\u0011\r\r\r\n∞.\nNext, we show how to produce x′∈ Xsatisfying Ax′≤band∥x−x′∥∞≤∆. Without loss of\ngenerality, let n= 2L+ 1; otherwise, we create O(n)additional coordinates of xandv, all equal to\nxnandvn. We inductively apply Lemma 4 to each layer of constraints 0≤ℓ≤L, starting from\nℓ←Landx(L+1)←x. Each round of the algorithm corresponds to a single layer ℓ, takes as input\na point x(ℓ+1)∈ X, and modifies it to produce x(ℓ)∈ Xas output. At the end of round ℓ(and\ntherefore the start of ℓ−1), we maintain the following invariants on x(ℓ).\n1. Every Lipschitz constraint sin the ℓthlayer holds, i.e. max(A(s,+):x,A(s,−):x)≤b(s,+).\n2.max(A(s,+):x−b(s,+),A(s,−):x−b(s,+))≤∆for all s∈Sbase.\n3.∥x(ℓ)−x(ℓ+1)∥∞≤∆.\n4. Only coordinates of the form j=i2ℓ+ 1for odd i∈Z≥0have x(ℓ+1)\nj̸=x(ℓ)\nj.\nSuppose for induction that Items 1, 2, 3, and 4 all hold for all rounds in [ℓ, L], and consider the\n(ℓ−1)thround. We apply Lemma 4 to each (ℓ−1)thlayer constraint, where the maximum violation\nof any constraint is ∆due to Item 2 from the previous round. By construction, Lemma 4 preserves\nItems 3 and 4, and enforces Item 1 after completion. We are only left with Item 2. To see that this\nis true, note that for each mthlevel constraint for m < ℓ −1, it can share an endpoint with at most\none(ℓ−1)thlevel constraint; if it shares no endpoints, correctness of Item 2 clearly still holds. If it\nshares one endpoint, depending on whether the endpoint is on the left or the right of the mthlevel\nconstraint, we use Lemma 5 or 6, as well as existence of the relevant constraint in SleftorSright,\nto ensure that Item 2 holds after applying Lemma 4. Finally, at the end of the algorithm, each\ncoordinate has moved at most once, and by at most ∆, via Items 3 and 4. Because all the 0th-layer\nconstraints hold by Item 1, we have feasibility of x(0)via Lemma 7, and can set x′←x(0).\nTo bound the runtime, note that applying Lemma 4 to each set of three coordinates corresponding\nto a consecutive pair of constraints takes O(1)time, and |S|=O(n)because of the following:\n|S| ≤kX\nℓ=02k−ℓ+ 2kX\nℓ=1ℓ−1X\nm=02k−ℓ≤2k+1+ 2kX\nℓ=1ℓ2k−ℓ\n≤2k+1+ 2kX\nℓ=1k2k−ℓ−2kX\nℓ=1(k−ℓ)2k−ℓ\n≤2k+1+ 2k2k−2(k−2)2k≤2n+ 4n=O(n).\n17By combining Lemma 8 with Corollary 2, we have our main result of this section.\nProposition 2. Letε≥0. We can compute x∈ X, anε-approximate minimizer to (11), in time\nO\u0012nlog(n)\nε\u0013\n.\nFurther, the objective value of xin(11)is an ε-additive approximation of smCE (bDn).\nProof.Observe that for eAdefined in (22), we have ∥eA∥∞→∞ = 2and nnz (eA) = 4|S|=O(n).\nTherefore, Corollary 2 shows we can compute an ε-approximate minimizer to\nmin\nx∈[−1,1]nc⊤x+\r\r\rmax\u0010\neAx−˜b,02|S|\u0011\r\r\r\n∞\nwithin the stated runtime. Finally, the definition of Roundfrom Lemma 8 shows that given this\nε-approximate minimizer, we can then produce an ε-approximate minimizer to (11) in O(n)time.\nThe last claim in the lemma statement follows immediately from the definition of (11).\n3.2 Sufficiency and insufficiency of Sbase\nInthissection, wegiveabriefdiscussionofanalternativestrategytothatinSection3.1. Specifically,\nsuppose we let Sin (20) only consist of the pairs in Sbase. We instead define (eA,˜b)as follows:\neA∈R2|Sbase|×nhaseA(s,+)::= (L+ 1)A(s,+):,eA(s,−)::= (L+ 1)A(s,−):for all s∈Sbase,\n˜b∈R2|Sbase|has˜b(s,+):= (L+ 1)b(s,+),˜b(s,−):= (L+ 1)b(s,−)for all s∈Sbase.(23)\nHere, L:=⌊log2(n−1)⌋as in (20). In other words, eA,˜btake a subset of rows of A, band scale\nthem up by L+ 1. We first show that enforcing this simpler set of constraints yields a rounding\nprocedure losing only a logarithmic factor in quality over the more complicated Lemma 8.\nLemma 9. Let(A, b, c,X)be defined as in (11), and let (eA,˜b)be defined as in (20),(23). There\nexists Round, a(eA,˜b,∞)-inequality rounding procedure for (A, b, c,X)running in O(n)time.\nProof.For a fixed x∈ X, let\n∆ :=1\nL+ 1\r\r\rmax\u0010\neAx−˜b,02|S|\u0011\r\r\r\n∞,\nso that ∆is the maximum violation of any Lipschitz constraint in Sat the start of the algorithm,\nsince we have undone the scaling by L+ 1. We claim we can produce x′=Round (x)∈ Xwith\nAx′≤b(i.e.x′satisfies all of the Lipschitz constraints in [n]×[n]), such that ∥x−x′∥∞≤δ(L+1).\nThe proof that Roundis a(eA,˜b,∞)-inequality rounding procedure is then identical to Lemma 8.\nTo produce such x′, we again apply Lemma 4 to one layer of constraints at a time, producing a\nsequence {x(ℓ)}L+1\nℓ=0. Specifically, in place of the invariants in the proof of Lemma 8, we instead\nenforce the following invariants on x(ℓ), for all 0≤ℓ≤L.\n1. Every Lipschitz constraint sin the ℓthlayer of constraints is satisfied, i.e. As:x≤bs.\n182.∥x(ℓ)−x(ℓ+1)∥∞≤∆(L+ 1−ℓ).\n3. Only coordinates of the form j=i2ℓ+ 1for odd i∈Z≥0have x(ℓ+1)\nj̸=x(ℓ)\nj.\nThe proof that Item 2 is inductively maintained replaces the use of Lemmas 5, 6 with the triangle\ninequality, since each constraint is originally violated by at most ∆, and Lemma 4 moves ℓth-layer\nconstraints by at most ∆(L+ 1−ℓ)via Item 2. The rest of the proof is identical to Lemma 8.\nBy plugging in Lemma 9 into Corollary 2 instead of Lemma 8, we lose a log(n)factor in runtime,\nsince∥eA∥∞→∞ = 2(L+1) = O(log(n)). We next show this loss is inherent, if we only enforce Sbase.\nLemma 10. LetL∈N, letn= 2L+ 1, and define A, bas in(11), and Sbaseas in(20). There\nexists x∈[−1,1]nand∆∈[0,1]such that, letting [Ax−b]Sbasedenote the 2|Sbase|coordinates of\nthe vector Ax−bcorresponding to elements of Sbase,\n\r\rmax\u0000\n[Ax−b]Sbase,02|Sbase|\u0001\r\r\n∞= ∆,\nbut∥max([Ax−b]Sbase,02|Sbase|)∥∞>0for any x′∈[−1,1]nwith∥x′−x∥∞≤L∆\n2.\nProof.LetL∈N,n= 2L+ 1, and let L∆≤1. We consider the following hard example:\nxj= (L− ⌈log2(2L−j+ 1)⌉)∆for all j∈[n−1]andxn=L∆,\nvj= 0for all j∈[n−1]andvn=L∆.\nFor example, if L= 3son= 9, we have\nx1=. . .=x4= 0, x5=x6= ∆, x7= 2∆ ,andx8=x9= 3∆ .\nWe first show that none of the constraints is violated by more than ∆in the set Sbasefor our\nconstruction. The Lthlayer is not violated because xn−x1=vn−v1. Now, for the ℓthlayer where\n0≤ℓ < L, we want to show that for any i∈[2L−ℓ], the constraint ((i−1)2ℓ+ 1, i2ℓ+ 1)is violated\nby at most ∆. IfL= 2L−ℓ, this is obvious because\n\f\f\fxn−x(i−1)2ℓ+1\f\f\f≤vn−v(i−1)2ℓ+1=L∆.\nOtherwise, because |⌈log2(2L−(i−1)2ℓ)⌉ − ⌈log2(2L−i2ℓ)⌉| ≤1fori <2L−ℓ,\n\f\f\fxi2ℓ+1−x(i−1)2ℓ+1\f\f\f≤∆ = ∆ + vi2ℓ+1−v(i−1)2ℓ+1.\nFinally, if x′satisfies all Lipschitz constraints, x′\n1=x′\n2L, so one of x1orx2Lmust move byL∆\n2.\n3.3 Testing via smooth calibration\nIn this section, we build upon Proposition 2 and give algorithms that solve the testing problems in\nDefinitions 2 and 4. We begin with a result from [BGHN23a] which bounds how well the smooth\ncalibration of an empirical distribution approximates the smooth calibration of the population.\n19Lemma 11 (Corollary 9.9, [BGHN23a]) .For any ε∈(0,1), there is an n=O(1\nε2)such that if bDn\nis the empirical distribution over ni.i.d. draws from D, with probability ≥2\n3,\n\f\f\fsmCE (D)−smCE (bDn)\f\f\f≤ε.\nFurther, we recall the smooth calibration error is constant-factor related to the LDTC.\nLemma 12 (Theorem 7.3, [BGHN23a]) .For any distribution Dover [0,1]× {0,1}, we have\n1\n2dCE(D)≤smCE (D)≤2dCE(D).\nFor completeness, we make the simple (but to our knowledge, new) observation that, while the\nconstants in Lemma 12 are not necessarily tight, there is a constant gap between dCEandsmCE.\nLemma 13. Suppose for constants B≥A > 0, it is the case that A·dCE(D)≤smCE (D)≤\nB·dCE(D)for all distributions Dover [0,1]× {0,1}. Then,B\nA≥3\n2.\nProof.First, we claim that A≤1. To see this, let (v, y)∼ Dbe distributed where v=1\n2with\nprobability 1, and y∼Bern(1\n2+ε)for some ε∈[0,1\n2]. Clearly, smCE (D) =|1\n2−(1\n2+ε)|=ε.\nMoreover, dCE(D) =ε, which follows from the same Jensen’s inequality argument as in Lemma 2,\nso this shows that A≤1. Next, we claim that B≥3\n2, concluding the proof. Consider the joint\ndistribution over (u, v, y )in Table 3.3, and let Dbe the marginal of (v, y). It is straightforward to\nProbability mass u v y w (v)\n1\n21\n21\n2−ε1 1\n1\n21\n21\n20 1−ε\ncheck (u, y)is calibrated and E|u−v|=ε\n2, sodCE(D)≤ε\n2. Moreover, smCE (v, y)≥3ε\n4, as witnessed\nby the Lipschitz weight function w(v)in Table 3.3, finishing our proof that B≥3\n2:\nsmCE (v, y)≥E[(y−v)w(v)] =1\n2\u0012\u00121\n2+ε\u0013\n·1\u0013\n+1\n2\u0012\u0012\n−1\n2\u0013\n·(1−ε)\u0013\n=3ε\n4.\nUsing these claims, we now give our tolerant calibration tester in the regime ε1>4ε2.\nTheorem 3. Let0≤ε2≤ε1≤1satisfy ε1>4ε2, and let n≥Ctct·1\n(ε1−4ε2)2for a universal\nconstant Ctct. There is an algorithm Awhich solves the (ε1, ε2)-tolerant calibration testing problem\nwithnsamples, which runs in time\nO\u0012nlog(n)\nε1−4ε2\u0013\n.\nProof.Throughout the proof, let α:=ε1\n2−2ε2>0. Consider the following algorithm.\n1. Sample n≥Ctct\n4α2samples to form an empirical distribution bDn, where Ctctis chosen large\nenough so that Lemma 11 guarantees |smCE (D)−smCE (bDn)| ≤α\n4with probability ≥2\n3.\n202. Call Proposition 1 with ε←α\n4to obtain β, anα\n4-additive approximation to |smCE (bDn)|.\n3. Return “yes” if β≤2ε2+α\n2, and return “no” otherwise.\nConditioned on the event that |smCE (D)−smCE (bDn)| ≤α\n4, we show that the algorithm succeeds\nin tolerant calibration testing. First, if dCE(D)≤ε2, then smCE (D)≤2ε2by Lemma 12, and\ntherefore by the guarantee of Proposition 1 and the assumed success of Lemma 11, the algorithm\nwill return “yes.” Second, if dCE(D)≥ε1, then smCE (D)≥ε1\n2by Lemma 12, and similarly the\nalgorithm returns “no” in this case. Finally, the runtime is immediate from Proposition 2 and the\ndefinition of α.\nTheorem 3 has the following implication for (standard) calibration testing.\nCorollary 3. Letn∈Nand let εn∈(0,1)be minimal such that it is information-theoretically\npossible to solve the εn-calibration testing problem with nsamples. For some ε= Θ( εn), there is an\nalgorithm Awhich solves the ε-calibration testing problem with nsamples, which runs in time\nO\u0000\nn1.5log(n)\u0001\n.\nProof.Recall from Lemma 2 that εn= Ω( n−1/2). The conclusion follows by applying Theorem 3\nwith ε1←Θ(εn)andε2←0.\n4 Lower distance to calibration\nIn this section, we provide our main result on approximating the lower distance to calibration of\na distribution on [0,1]× {0,1}. In Section 4.1, we state some preliminary definitions and results\nfrom [BGHN23a] which are used in our algorithm. In Section 4.2, we then develop a rounding\nprocedure compatible with a linear program which closely approximates the empirical lower distance\nto calibration. Finally, in Section 4.3, we use our rounding procedure to design an algorithm for\ncalibration testing, which can solve the testing problem for a larger range of parameters than\nTheorem 4 (i.e. the entire relevant parameter range), at the cost of a slight runtime overhead.\n4.1 LDTC preliminaries\nIn this section, we collect preliminaries for our testing algorithm based on estimating the LDTC.\nFirst, analogouslytoLemma11, werecallaboundfrom[BGHN23a]onthedeviationoftheempirical\nestimate of dCE(D)from the population truth which holds with constant probability.\nLemma 14 (Theorem 9.10, [BGHN23a]) .For any ε∈(0,1), there is an n=O(1\nε2)such that, if\nbDnis the empirical distribution over ni.i.d. draws from D, with probability ≥2\n3,\n\f\f\fdCE(D)−dCE(bDn)\f\f\f≤ε.\nNext, given a set U⊂[0,1], we provide an analog of Definition 1 which is restricted to U.\nDefinition 7 (U-LDTC).LetU⊂[0,1], and let Dbe a distribution over [0,1]× {0,1}. Define\nextU(D)to be all joint distributions Πover(u, v, y )∈U×[0,1]×{0,1}, with the following properties.\n21•The marginal distribution of (v, y)isD.\n•The marginal distribution (u, y)is perfectly calibrated, i.e. EΠ[y|u] =u.\nTheU-lower distance to calibration (U-LDTC) of D, denoted dCEU(D), is defined by\ndCEU(D) := inf\nΠ∈extU(D)E(u,v,y )∼Π|u−v|.\nNote that if we require {0,1} ⊂U, then extU(D)is always nonempty, because we can let u=y\nwith probability 1. We also state a helper claim from [BGHN23a], which relates dCEUtodCE.\nLemma 15 (Lemma 7.11, [BGHN23a]) .LetDbe a distribution over [0,1]× {0,1}, and let Ube a\nfiniteε\n2-covering of [0,1]satisfying {0,1} ⊆U. Then, dCE(D)≤dCEU(D)≤dCE(D) +ε.\nTo this end, in the rest of the section we define, for any ε∈(0,1),\nUε:={0,1} ∪\u001aiε\n2|i∈\u0014\u00162\nε\u0017\u0015\u001b\n, (24)\nwhich is anε\n2-cover of [0,1]satisfying |Uε|=O(1\nε). Finally, we state a linear program whose value\nis equivalent to dCEU(D), when the first marginal of Dis discretely supported.\nLemma 16 (Lemma 7.6, [BGHN23a]) .LetU, V⊂[0,1]be discrete sets, where {0,1} ⊂U, and\nletDbe a distribution over V× {0,1}, where for (v, y)∈V× {0,1}we denote the probability\nof(v, y)∼ DbyD(v, y). The following linear program with 2|U||V|variables Π(u, v, y )for all\n(u, v, y )∈U×V× {0,1}, is feasible, and its optimal value equals dCEU(D):\nmin\nΠ∈R2|U||V|\n≥0X\n(u,v,y )∈U×V×{0,1}|u−v|Π(u, v, y )\nsuch thatX\nu∈UΠ(u, v, y ) =D(v, y),for all (v, y)∈V× {0,1},\nand(1−u)X\nv∈VΠ(u, v,1) = uX\nv∈VΠ(u, v,0),for all u∈U.\n4.2 Rounding for empirical U-LDTC\nAnalogously to Section 3.1, in this section we fix a dataset under consideration,\nbDn:={(vi, yi)}i∈[n]⊂[0,1]× {0,1},\nand the corresponding empirical distribution, also denoted bDn, where (v, y)∼bDnmeans (v, y) =\n(vi, yi)with probability1\nnfor each i∈[n]. We let V:={vi}i∈[n]be identified with [n]in the natural\nway. Moreover, for a fixed parameter ε∈(0,1)throughout, we let U:=Uεdefined in (24). Finally,\nwe denote m:=|U|=O(1\nε), and let U∈[0,1]m×mbe the diagonal matrix whose diagonal entries\ncorrespond to U. We also identify elements of Uwith j∈[m]in an arbitrary but consistent way,\nwriting uj∈[0,1]to mean the jthelement of Uaccording to this identification.\nWe next rewrite the linear program in Lemma 16 into a more convenient reformulation.\n22Lemma 17. The linear program in Lemma 16 can equivalently be written as:\ndCEU(bDn) := min\nx∈X\nMx=1\nn1n\nUB 0x0=(Im−U)B1x1c⊤x,where X:= ∆2mnand we denote x=\u0012x0∈Rmn\nx1∈Rmn\u0013\n,(25)\nwhere we define c∈R2mn,M∈Rn×2mn, andB0,B1∈Rm×mnby\nc(i,j,k):=|uj−vi|for all (i, j, k )∈[n]×[m]× {0,1},\nMi′,(i,j,k):=(\n1yi=k, i′=i\n0elsefor all i′∈[n],(i, j, k )∈[n]×[m]× {0,1},\nand[B0]j′,(i,j,k):=(\n1j=j′, k= 0\n0elsefor all j′∈[m],(i, j, k )∈[n]×[m]× {0,1},\n[B1]j′,(i,j,k):=(\n1j=j′, k= 1\n0elsefor all j′∈[m],(i, j, k )∈[n]×[m]× {0,1}.\nProof.This is clear from observation, but we give a brief explanation of the notation. First, x∈ X\nrepresents the density function of our joint distribution Πover U×V× {0,1}, and has 2mn\ncoordinates identified with elements (i, j, k )∈V×U×{0,1} ≡[n]×[m]×{0,1}. We let the subset\nof coordinates with k= 0be denoted x0∈Rmn, defining x1similarly. Recalling the definition of\nthe linear program in Lemma 16, x(i,j,k)is indeed reweighted by c(i,j,k)=|uj−vi|.\nNext,Mrepresents the marginal constraints in Lemma 16, and enforcing Mx=1\nn1nis equivalent\nto the statement that, for each i′∈[n], the sum of all entries (i, j, k )ofxwith i=i′andk=yiis\n1\nn, since that is the probability density assigned to (vi′, yi′)by the distribution bDn.\nLastly, the jthcalibration constraint in Lemma 16 is enforced by the jthrow of the equation\nUB 0x0= (Im−U)B1x1, which reads uj⟨[B0]j:, x0⟩= (1−uj)⟨[B1]j:, x1⟩. We can check by the\ndefinitions of [B0]j:,[B1]j:that this is consistent with our earlier calibration constraints.\nWe give a convenient way of visualizing the marginal and calibration constraints described in\nLemma 17. For convenience, we identify each x∈∆2mnwith an m×2nmatrix\nmat(x) =X=\u0000\nX0∈Rm×nX1∈Rm×n\u0001\n, (26)\nwhereX0consists of entries of x0arranged in a matrix fashion (with rows corresponding to [m]≡U\nand columns corresponding to [n]≡V), and similarly X1is a rearrangement of x1, recalling (25).\nWhen explaining how we design our rounding procedures to modify Xto satisfy constraints, it will\nbe helpful to view entries of Xas denoting an amount of physical mass which we can move around.\nThereare 2ncolumnsin X, correspondingtopairs (i, k)∈V×{0,1}; amongthese, wesay ncolumns\nare “active,” where column (i, k)is active iff yi=k, and we say the other ncolumns are “inactive.”\nFollowing notation (26), the marginal constraints X=1\nn1nsimply ask that the total amount of\nmass in each active column is1\nn, so there is no mass in any inactive column since x∈∆2mn.\n23Moreover, there are mrows in X, each corresponding to some j∈U. If we let ℓjdenote the amount\nof mass on [X0]j:andrjthe amount of mass on [X1]j:, the jthcalibration constraint simply asks\nthatujℓj= (1−uj)rj, i.e. it enforces balance on the amount of mass in each row’s two halves.\nFinally, for consistency with Definition 6, the linear program in (25) can be concisely written as\nmin\nx∈X\nAx=bc⊤x,whereA:=\u0012M\nB\u0013\n,B:=\u0000\nUB 0−(Im−U)B1\u0001\n, b:=\u00121\nn1n\n0m\u0013\n(27)\nandc,Xare as defined in (25). In the rest of the section, following Definition 6, we develop an\nequality rounding procedure for the equality-constrained linear program in (27) in two steps.\n1. In Lemma 18, we first show how to take x∈ Xwith∥Mx−1\nn1n∥1= ∆, and produce x′∈ X\nsuch that Mx′=1\nn1n(i.e.x′now satisfies the marginal constraints) and ∥x−x′∥1=O(∆).\n2. In Lemma 20, we then consider x∈ Xsuch that, following the notation (25), ∥UB 0x0−(Im−\nU)B1x1∥1= ∆. We show how to produce x′∈ Xsuch that Mx=Mx′(i.e. the marginals of\nx′are unchanged), UB 0x′\n0= (Im−U)B1x′\n1(i.e.x′is calibrated), and ⟨c, x′−x⟩=O(∆).\nOur rounding procedure uses Lemma 18 to satisfy the marginal constraints in (25), and then applies\nLemma 20 to the result to satisfy the calibration constraints in (25) without affecting the marginal\nconstraints. By leveraging the stability guarantees on these steps, we can show this is indeed a valid\nrounding procedure in the sense of (5). We now give our first step for marginal satisfication.\nLemma 18 (Marginal satisfaction) .Following notation in (25), letx∈ Xsatisfy ∥Mx−1\nn1n∥1=\n∆. There is an algorithm which runs in time O(mn), and returns x′with\nMx′=1\nn1n,\r\rx−x′\r\r\n1≤2∆.\nProof.Recall for i∈[n], we say column iofX0is active if yi= 0, and similarly column iofX1is\nactive if yi= 1. We call Ithe set of ninactive columns, and partition A, which we call the set of n\nactive columns, into three sets A>,A=, and A<, where A>are the columns whose sums are >1\nn,\nA<are the columns whose sums are <1\nn, and A=are the remaining columns. Hence, every column\nofXbelongs to I,A>,A=, orA≤. Note that until |A=|=n, we can never have A<=∅, since this\nmeans all column sums in Aare≥1\nn(with at least 1strict inequality), contradicting x∈ X.\nWe first take columns i∈A>one at a time, and pair them with an arbitrary column in i′∈A<,\nmoving mass from column iarbitrarily to column i′until either column ior column i′enters A=.\nWe charge this movement to the marginal constraints corresponding to iandi′, since the constraints\nwere violated by the same amount as the mass being moved. After this process is complete, A>is\nempty, and we only moved mass from columns originally in A>to columns originally in A<.\nNext, we take columns i∈Ione at a time, and pair them with an arbitrary column i′∈A<,\nmoving mass until either column iis0mor column i′enters A=. We can charge half this movement\nto the marginal constraint corresponding to i′, since the sign of the marginal violation stays the\nsame throughout. Hence, the overall movement is ≤2∆. After this is complete, all columns in Iare\n0mand all columns in Aare in A=, so we can return x′∈∆2mncorresponding to the new matrix.\nIt is clear both steps of this marginal satisfaction procedure take O(mn)time, since we can sequen-\ntially process columns in A−until they enter A=, and will never be considered again.\n24We next describe a procedure which takes x∈∆2mn, and modifies it to satisfy the calibration\nconstraints UB 0x= (Im−U)B1xwithout changing the marginals Mx. We first provide a helper\nlemma used in our rounding procedure, which describes how to fix the jthmarginal constraint.\nLemma 19. Letx∈∆2mnandX:=mat(x)as defined in (26). Let j∈[m]correspond to an\nelement uj∈U, letℓj:=∥[X0]j:∥1,rj:=∥[X1]j:∥1, and let ∆j:=|ujℓj−(1−uj)rj|. There exists\nj′∈[m]such that we can move mass from only Xj:toXj′:, resulting in Rm×2n∋X′≡x′∈∆2mn\nsuch that Mx′=Mx,uj∥[X′\n0]j:∥1= (1−uj)∥[X′\n1]j:∥1, and⟨c, x′−x⟩ ≤∆j.\nProof.Without loss of generality, suppose that the row j= 1corresponds to uj= 0, and j=m\ncorresponds to uj= 1. We split the proof into two cases, depending on the sign of ujℓj−(1−uj)rj.\nCase 1: ujℓj>(1−uj)rj.We let j′= 1, i.e. we only move mass from the jthrow to the first row.\nSpecifically, we leave [X1]j:unchanged, and move mass from [X0]j:to[X0]1:, making sure to only\nmove mass in the same column. The total amount of mass we must delete from [X0]j:is\nℓj−1−uj\nuj·rj=ujℓj−(1−uj)rj\nuj=∆j\nuj.\nOur strategy is to arbitrarily move mass within columns until we have deleted∆j\nujtotal mass. If we\ndenote the mass moved in column i∈[n]asδij, and let x′be the result after the move,\n\nc, x′−x\u000b\n≤X\ni∈[n]|c(i,1,0)−c(i,j,0)|δij=X\ni∈[n]||uj−vi| − |u1−vi||δij≤X\ni∈[n]ujδij= ∆ j.\nHere, the first inequality was the triangle inequality, the first equality used the definition of cin (25),\nthe second inequality used u1= 0and the triangle inequality, and the last usedP\ni∈[n]δij=∆j\nuj.\nCase 2: ujℓj<(1−uj)rj.This case is entirely analogous; we move mass arbitrarily from row jto\nrowm, i.e. the last row with um= 1. The amount of mass we must move is\nrj−uj\n1−ujℓj=(1−uj)rj−ujℓj\n1−uj=∆j\n1−uj.\nAgain denoting the amount of mass moved from column i∈[n]asδij, the claim follows:\n\nc, x′−x\u000b\n≤X\ni∈[n]||uj−vi| − |um−vi||δij≤X\ni∈[n](1−uj)δij= ∆ j.\nBy iteratively applying Lemma 19, we have our marginal-preserving calibration procedure.\nLemma 20 (Marginal-preserving calibration) .Following the notation (27), given x∈∆2mnwith\n∥Bx∥1= ∆, we can compute x′withMx′=Mx,Bx′=0m, and⟨c, x′−x⟩ ≤∆inO(mn)time.\nProof.It suffices to apply Lemma 19 to each row i∈[m]. All of the movement in the rows\ni∈[2, m−1]are independent of each other, and do not affect the imbalance in the rows i∈ {1, m}\nwhen we have finished applying Lemma 19, since e.g. u1ℓ1= 0regardless of how much mass is\n25moved to [X0]1:, and a similar property holds for the mthrow. The total change in ⟨c, x′−x⟩is\nthus boundable byP\nj∈[m]∆j≤∆, and applying Lemma 19 to each row takes O(n)time. Finally,\nMx=Mx′follows because we only move mass within the same column, so no marginalchanges.\nBy combining Lemma 18 with Lemma 20, we can complete our rounding procedure.\nLemma 21. Let(A, b, c,X)be defined as in (25),(27), and let (eA,˜b) := (4 A,4b). There exists\nRound, a(eA,˜b,1)-equality rounding procedure for (A, b, c,X), running in O(mn)time.\nProof.Throughout the proof, let ∆M:=\r\rMx−1\nn1n\r\r\n1and∆B:=∥Bx∥1, following the notation\n(27). We also denote the total violation by\n∆ :=\r\r\reAx−˜b\r\r\r\n1= 4∆ M+ 4∆B.\nWe first apply Lemma 18 to xto produce ˜xsatisfying ∥x−˜x∥1≤2∆MandM˜x=1\nn1n, inO(mn)\ntime. Note that, because ∥B∥1→1≤1since all columns of Bare1-sparse, we have\n∥B˜x∥1≤ ∥Bx∥1+∥B∥1→1∥x−˜x∥1≤∆B+ 2∆M.\nNext, we apply Lemma 20 to ˜x, resulting in x′withMx′=1\nn1n,Bx′=0m, and ⟨c, x′−˜x⟩ ≤\n∆B+ 2∆M, inO(mn)time. Recalling the definition (8), we have the conclusion from ∥c∥∞≤1, so\nc⊤(x′−x)≤c⊤(˜x−x) +c⊤(x′−˜x)\n≤ ∥c∥∞∥˜x−x∥1+c⊤(x′−˜x)≤2∆M+ ∆B+ 2∆M≤∆.\nWe conclude by applying the solver from Corollary 1 to our resulting unconstrained linear program.\nProposition 3. Letε≥0. We can compute x∈ X, anε-approximate minimizer to (25), in time\nO\u0012nlog(n)\nε2\u0013\n.\nFurther, the objective value of xin(25)is a2ε-additive approximation of dCE(bDn).\nProof.Observe that for eA= 4A, we have ∥eA∥1→1≤8and nnz (eA) =O(mn), since no column is\nmore than 2-sparse and all entries of Aare in [−1,1]. Further, recalling the definition of Ufrom\n(24), we have m=O(1\nε). So, Corollary 1 shows we can compute an ε-approximate minimizer to\nmin\nx∈∆2mnc⊤x+\r\r\reAx−˜b\r\r\r\n1\nwithin the stated runtime. The rest of the proof follows as in Proposition 2, using Roundfrom\nLemma21, wherewerecall |dCE(bDn)−dCEU(bDn)| ≤εduetoourdefinitionof UandLemma15.\n264.3 Testing via LDTC\nWe now give analogs of Theorems 3 and Corollary 3, using our solver in Proposition 3.\nTheorem 4. Let0≤ε2≤ε1≤1satisfy ε1> ε2, and let n≥Ctct·1\n(ε1−ε2)2for a universal constant\nCtct. There is an algorithm Awhich solves the (ε1, ε2)-tolerant calibration testing problem with n\nsamples, which runs in time\nO\u0012nlog(n)\n(ε1−ε2)2\u0013\n.\nProof.Throughout the proof, let α:=ε1−ε2>0. Consider the following algorithm.\n1. For |U|=m≥6\nα, sample n≥Ctct\nα2samples to form an empirical distribution bDn, where Ctct\nis chosen so Lemma 11 guarantees |dCE(D)−dCE(bDn)| ≤α\n6with probability ≥2\n3.\n2. Call Proposition 1 with ε←α\n6to obtain β, anα\n3-additive approximation to |dCE(bDn)|.\n3. Return “yes” if β≤ε2+α\n2, and return “no” otherwise.\nConditioned on the event that |dCE(D)−dCE(bDn)| ≤α\n6, we show that the algorithm succeeds.\nFirst, if dCE(D)≤ε2, then dCE(bDn)≤ε2+α\n6by assumption, and so β≤ε2+α\n2by Proposition 1,\nso the tester will return “yes.” Second, if dCE(D)≥ε1, then dCE(bDn)≥ε1−α\n6by assumption,\nsoβ≥ε1−α\n2by Proposition 1 and similarly the tester will return “no” in this case. Finally, the\nruntime is immediate from Proposition 3 and the definition of α.\nTheorem 4 has the following implication for (standard) calibration testing.\nCorollary 4. Letn∈Nand let εn∈(0,1)be minimal such that it is information-theoretically\npossible to solve the εn-calibration testing problem with nsamples. For some ε= Θ( εn), there is an\nalgorithm Awhich solves the ε-calibration testing problem with nsamples, which runs in time\nO\u0000\nn2log(n)\u0001\n.\nProof.This claim follows analogously to Corollary 3.\n5 Sample complexity lower bounds for calibration distances\nRecent works [BN23, BGHN23a] have introduced other calibration measures (e.g. the convolved\nECE and interval CE), given efficient estimation algorithms for them, and showed that they are\npolynomially related to the lower distance to calibration dCE. Therefore, an alternative approach to\nthe (non-tolerant) testing problem for dCEis by reducing it to testing problems for these measures.\nThe main result of this section is that this approach leads to suboptimal sample complexity: the\ntesting problems for these measures cannot be solved given only O(ε−2)data points {(vi, yi)}i∈[n].\nTo establish this sample complexity lower bound, we construct a perfectly calibrated distribution\nD0and a family of miscalibrated distributions Dθparameterized by θbelonging to a finite set. We\nuseD⊗n\n0(andD⊗n\nθ) to denote the joint distribution of nindependent examples from D0(andDθ).\nIn Lemma 22, we show that the total variation distance between D⊗n\n0and the mixture E[D⊗n\nθ]\nofD⊗n\nθis small unless nis large, and thus distinguishing them requires large sample complexity.\n27Consequently, the testing problem for a calibration measure has large sample complexity if it assigns\neveryDθa large calibration error. Finally, we show every Dθindeed has large convolved ECE and\ninterval CE, establishing sample complexity lower bounds for these measures in Theorems 5 and 6.\nTo construct D0andDθ, we consider tvalues {ui}i∈[t]∈[1\n3,2\n3]where ui=1\n3+i\n3tfori∈[t]. We will\ndetermine the value of t∈Nlater. We also define the following distribution, a perfectly calibrated\ndistribution which is related to the miscalibrated synthetic dataset used in Section 6.\nDefinition 8. The distribution D0of(v, y)∈[0,1]× {0,1}is defined such that the marginal\ndistribution of vis uniform over {ui}i∈[t]andED0[y|v] =v.\nFixα∈(0,1\n3). For θ∈ {− 1,1}t, we define distribution Dθof(v, y)∈[0,1]× {0,1}such that the\nmarginal distribution of vis uniform over {ui}i∈[t]andEDθ[y|v=ui] =ui+θiα. In other words,\neach conditional distribution given vis miscalibrated by α, but the bias takes a random direction.\nWe now follow a standard approach by [IS03] to bound the total variation between our distributions.\nLemma 22. For any t∈Nandα∈(0,1\n3),\ndTV(D⊗n\n0,Eθ[D⊗n\nθ])≤1\n2s\nexp\u001211α4n2\nt\u0013\n−1.\nHere, to construct the mixture distribution Eθ[D⊗n\nθ], we first draw θ∼unif.{−1,1}t, and then draw\nnindependent examples from Dθ. We denote the distribution of the nexamples by Eθ[D⊗n\nθ].\nProof.By a standard inequality between the total variation distance and the χ2distance, we have\ndTV(D⊗n\n0,Eθ[D⊗n\nθ])≤1\n2q\nχ2(Eθ[D⊗n\nθ]∥D⊗n\n0). (28)\nBy Ingster’s method [IS03] (see also Section 3.1 of [Can22]),\nχ2(Eθ[D⊗n\nθ]∥D⊗n\n0) =Eθ,θ′\n\ntX\ni=1X\nj∈{0,1}Dθ(ui, j)Dθ′(ui, j)\nD0(ui, j)\nn\n−1, (29)\nwhere the expectation is over θ, θ′drawn i.i.d. ∼unif.{−1,1}t. For every i∈[t], we have\nDθ(ui,1)Dθ′(ui,1)\nD0(ui,1)=\u0010\nui+θiα\nt\u0011\u0010ui+θ′\niα\nt\u0011\nui\nt=ui\nt+(θi+θ′\ni)α\nt+θiθ′\niα2\nuit,\nand similarly\nDθ(ui,0)Dθ′(ui,0)\nD0(ui,0)=\u0010\n(1−ui−θiα)\nt\u0011\u0010(1−ui−θ′\niα)\nt\u0011\n1−ui\nt=1−ui\nt−(θi−θ′\ni)\nt+θiθ′\niα2\n(1−ui)t.\nAdding up the two equations, we get\nX\nj∈{0,1}Dθ(ui, j)Dθ′(ui, j)\nD0(ui, j)=1\nt+θiθ′\niα2\nt\u00121\nui+1\n1−ui\u0013\n≤1\nt+9θiθ′\niα2\n2t,\n28where the last inequality uses the fact that ui∈[1\n3,2\n3]. Plugging this into (29), we get\nχ2(Eθ[D⊗n\nθ]∥D⊗n\n0)≤Eθ,θ′\" \n1 +9α2\n2ttX\ni=1θiθ′\ni!n#\n−1\n≤Eθ,θ′\"\nexp \n9α2n\n2ttX\ni=1θiθ′\ni!#\n−1\n=tY\ni=1Eθ,θ′\u0014\nexp\u00129α2n\n2tθiθ′\ni\u0013\u0015\n−1\n≤tY\ni=1Eθ\u0014\nexp\u001281α4n2\n8t2θ2\ni\u0013\u0015\n−1 (by Hoeffding’s lemma)\n= exp\u001281α4n2\n8t\u0013\n−1.\nPlugging this into (28) completes the proof.\n5.1 Lower bound for convolved ECE\nWe now introduce the definition of convolved ECE from [BN23], and show that for every θ∈\n{−1,1}t,DθhasalargeconvolvedECEinLemma24. Thisallowsustoproveoursamplecomplexity\nlower bound for convolved ECE in Theorem 5, by applying Lemma 22.\nDefinition 9 (Convolved ECE [BN23]) .LetπR:R→[0,1]be the periodic function with period 2\nsatisfying πR(v) =vifv∈[0,1], and πR(v) = 2−vifv∈[1,2]. Consider a distribution Dover\n[0,1]×{0,1}. For (v, y)∼ D, define random variable ˆv:=πR(v+η), where ηis drawn independently\nfromN(0, σ2)for a parameter σ≥0. The σ-convolved ECE is defined as follows:\ncECE σ(D) :=E|E[(y−v)|ˆv]|,\nwhere the outer expectation is over the marginal distribution of ˆv, and the inner expectation is over\nthe conditional distribution of (y, v)given ˆv. It has been shown in [BN23] that cECE σ(D)∈[0,1]\nis a nonincreasing function of σ≥0and there exists a unique σ∗≥0satisfying cECE σ∗(D) =σ∗.\nThe convolved ECE cECE (D)is defined to be cECE σ∗(D).\nWe also mention that the following relationship is known between cECEanddCE.\nLemma 23 (Theorem 7, [BN23]) .For any distribution Dover [0,1]× {0,1}, it holds that\n1\n2dCE(D)≤cECE (D)≤2p\ndCE(D).\nWe have the following lower bound on cECE (Dθ):\nLemma 24. For integer t≥3, choose α=1\nt√\nlnt. Then for every θ∈ {− 1,1}t,\ncECE (Dθ)≥1\n100t√\nlnt.\n29Proof.It suffices to show that cECE σ(Dθ)≥1\n100t√\nlntwhenever σ≤1\n100t√\nlnt.\nConsider (v, y)∼ Dandˆv=πR(v+η), where ηis drawn independently from N(0, σ2). By standard\nGaussian tail bounds, we have\nPr\u0014\n|η| ≥1\n6t\u0015\n≤1\nt2. (30)\nNext, consider a function ℓ: [0,1]→[t]such that ℓ(ˆv)∈argmini∈[t]|ui−ˆv|. LetEdenote the event\nthatv=uℓ(ˆv). Let IEandI¬Ebe the indicators of Eand its complement, respectively. We have\nE[(y−v)IE|ˆv, v] =IEE[y−v|ˆv, v] (IEis fully determined by vandˆv)\n=IEE[y−v|v] (yis independent of ˆvgiven v)\n=IEE[y−v|v=uℓ(ˆv)] =IEθℓ(ˆv)α.\nTaking expectation over vconditioned on ˆv, we have\n|E[(y−v)IE|ˆv]|=|Pr[E |ˆv]θℓ(ˆv)α|= Pr[E |ˆv]α.\nWe also have \f\f\fEh\n(y−v)I¬E|ˆvi\f\f\f≤Eh\n|(y−v)I¬E| |ˆvi\n≤Pr[¬E | ˆv].\nTherefore,\n|E[y−v|ˆv]| ≥Pr[E|ˆv]α−Pr[¬E | ˆv].\nTaking expectations over ˆv, we have\ncECE σ(Dθ) =E[|E[y−v|ˆv]|]≥Pr[E]α−Pr[¬E]. (31)\nWhenever Edoes not occur, it must hold that |v−ˆv| ≥1\n6t, which can only hold when |η| ≥1\n6t.\nTherefore, by plugging (30) into (31), we get\ncECE σ(Dθ)≥\u0012\n1−1\nt2\u0013\nα−1\nt2≥1\n100t√\nlnt.\nTheorem 5. IfAis an ε-cECEtester with nsamples (Definition 5), for ε∈(0,1\n3), then\nn= Ω \n1\nε2.5ln0.25(1\nε)!\n.\nProof.Without loss of generality, assume that ε≤10−3. Let t≥3be the largest integer satisfying\nε≤1\n100t√\nlnt. We choose α=1\nt√\nlnt.\nBy Lemma 24, we have cECE (Dθ)≥εfor every θ∈ {− 1,1}t. By the guarantee of the tester, we\nhave\ndTV(D⊗n\n0,Eθ[D⊗n\nθ])≥1\n3.\nCombining this with Lemma 22, we get n= Ω(α−2√\nt) = Ω( t2.5lnt), so the claim holds.\n305.2 Lower bound for (surrogate) interval CE\nTheinterval calibration error was introduced in [BGHN23a] as a modified version of the popular\nbinned ECE to obtain a polynomial relationship to the lower distance to calibration ( dCE). To give\nan efficient estimation algorithm, [BGHN23a] considered a slight variant of the interval calibration\nerror, called the surrogate interval calibration error , which preserves the polynomial relationship.\nBelow we include the definition of the surrogate interval calibration error, and its polynomial rela-\ntionship with dCE. We then establish our sample complexity lower bound (Theorem 6) for surrogate\ninterval CE by showing that every Dθhas a large surrogate interval CE (Lemma 26).\nDefinition 10 ([BGHN23a]) .For a distribution Dover [0,1]×{0,1}and an interval width param-\neterw >0, therandom interval calibration error is defined to be\nRintCE (D, w) :=Er\nX\nj∈Z|E(v,y)∼D[(y−v)I(v∈Iw\nr,j)]|\n, (32)\nwhere the outer expectation is over rdrawn uniformly from [0, w)andIw\nr,jis the interval [r+jε, r+\n(j+ 1)ε). Note that although the summation is over j∈Z, there are only finitely many jthat\ncan contribute to the sum (which are the jthat satisfy Iw\nr,j∩[0,1]̸=∅). Thesurrogate interval\ncalibration error is defined as follows:\nSintCE (D) := inf\nk∈Z≥0\u0010\nRintCE (D,2−k) + 2−k\u0011\n.\nLemma 25 (Theorem 6.11, [BGHN23a]) .For any distribution Dover [0,1]× {0,1}, it holds that\ndCE(D)≤SintCE (D)≤6p\ndCE(D).\nLemma 26. Fort∈N, letα=1\n3t. Then for every θ∈ {− 1,1}t, it holds that SintCE (Dθ)≥1\n3t.\nProof.It suffices to prove that RintCE (Dθ, w)≥1\n3twhenever w <1\n3t, where we recall the definition\n(32). Fix some r∈[0,1]. Every uibelongs to the interval Iw\nr,jifor a unique ji∈Z. Since the interval\nwidth wis smaller than the gap between uiandui′for distinct i, i′, we have ji̸=ji′. Therefore,\nX\nj∈Z|E(v,y)∼D[(y−v)I(v∈Iw\nr,j)]| ≥X\ni∈[t]|E(v,y)∼D[(y−v)I(v∈Iw\nr,ji)]|\n=X\ni∈[t]|E(v,y)∼D[(y−v)I(v=ui)]|\n=X\ni∈[t]Pr[v=ui]E[y−v|v=ui]\n=1\n3t.\nPlugging this into (32), we get RintCE (Dθ, w)≥1\n3t.\nTheorem 6. IfAis an ε-SintCEtester with nsamples (Definition 5), for ε∈(0,1\n3), then\nn= Ω\u00121\nε2.5\u0013\n.\n31Proof.Choose t≥1to be the largest integer satisfying1\n3t≥ε, and choose α=1\n3t. By Lemma 26,\nSintCE (Dθ)≥εfor every θ∈ {− 1,1}t. By the guarantee of the tester, we have\ndTV(D⊗n\n0,Eθ[D⊗n\nθ])≥1\n3.\nCombining this with Lemma 22, we get n= Ω(α−2√\nt) = Ω( ε−2.5).\n6 Experiments\nIn this section, we present experiments on synthetic data and CIFAR-100 supporting our results.\nSynthetic dataset. In our first experiment, we considered the ability of ε-d-testers (Definition 5)\ntodetectthemiscalibrationofasyntheticdataset,forvariouslevelsof ε∈ {0.01,0.03,0.05,0.07,0.1},\nand various choices of d∈ {smCE ,dCE,ConvECE }.9The synthetic dataset we used is nindependent\ndraws from D, where a draw (v, y)∼ Dfirst draws v∼unif.[0,1−ε⋆], and y∼Bern(v+ε⋆), for\nε⋆:= 0.01.10Note that dCE(D) =ε⋆= 0.01, by the proof in Lemma 13. In Table 1, where the\ncolumns index n(the number of samples), for each choice of dwe report the smallest value of εsuch\nthat a majority of 100runs of an ε-d-tester report “yes.” For d=ConvECE , we implemented our\ntester by running code in [BN23] to compute ConvECE and thresholding atε\n2. For d∈ {smCE ,dCE},\nwe used the standard linear program solver from CVXPY [DB16, AVDB18] and again thresholded\natε\n2. We remark that the CVXPY solver, when run on the dCElinear program, fails to produce\nstable results for n >29due to the size of the constraint matrix. As seen from Table 1, both smCE\nanddCEtesters are more reliable estimators of the ground truth calibration error ε⋆than ConvECE .\nn 26+ 1 27+ 1 28+ 1 29+ 1 210+ 1 211+ 1\nsmCE 0.07 0 .05 0 .03 0 .03 0 .01 0 .01\ndCE 0.03 0 .01 0 .01\ncECE 0.1 0 .1 0 .07 0 .07 0 .05 0 .03\nGround Truth 0.01 0 .01 0 .01 0 .01 0 .01 0 .01\nTable 1: Calibration testing thresholds (smallest passed on half of 100runs).\nIn Figure 1, we plot the median error with error bars for each calibration distance, where the xaxis\ndenotes log2(n−1), and results are reported over 100runs.\nPostprocessed neural networks. In [GPSW17], which observed modern deep neural networks\nmay be very miscalibrated, various strategies were proposed for postprocessing network predictions\nto calibrate them. We evaluate two of these strategies using our testing algorithms. We trained\na DenseNet40 model [HLvdMW17] on the CIFAR-100 dataset [Kri09], producing a distribution\nDbase, where a draw (v, y)∼ D baseselects a random example from the test dataset, sets yto be its\nlabel, and vto be the prediction of the neural network. We also learned calibrating postprocessing\nfunctions fisoandftempfrom the training dataset, the former via isotonic regression and the latter\nvia temperature scaling. These induce (ideally, calibrated) distributions Diso,Dtemp, where a draw\n9We implemented ConvECE using code from [BN23], which automatically conducts a parameter search for σ.\n10This is a slight variation on the synthetic dataset used in [BGHN23a].\n32Figure 1: The 25%quantile, median, and 75%quantile (over 100runs) for smCE,dCEandcECE\nrespectively. The x-axis is for dataset with size 2x+ 1.\nD Dbase Diso Dtemp\nEmpirical smCE 0.2269 0 .2150 0 .1542\nTable 2: Empirical smCEon postprocessed DenseNet40 predictions (median over 20runs)\nfromDisosamples (v, y)∼ D baseand returns (fiso(v), y), and Dtempis defined analogously. The\nneural network and postprocessing functions were all trained by adapting code from [GPSW17].\nWe computed the median smooth calibration error of 20runs of the following experiment. In each\nrun, for each D ∈ {D base,Diso,Dtemp}, we drew 4864random examples from D, and computed the\nsmooth calibration error smCEof the empirical dataset. We report our findings in Table 2. Qual-\nitatively, our results (based on smCE) agree with findings in [GPSW17] (based on binned variants\nofECE), in that temperature scaling appears to be the most effective postprocessing technique.\nsmCEtester. Finally, we implemented our algorithm from Section 3, based on the minimax solver\nfrom [JT23]. We considered two types of augmented Lagrangian relaxations of the hard constraints,\nbased on the sets of constraints Sbase(defined in (20)) and S0(defined in (21)). In each case, we\nscaled up the Lagrangian penalty term by log2(n−1). As described in Section 3.2, when using Sbase,\nthis provably yields a soft-constrained empirical smCEobjective which has the same value as the\nempirical smCElinear program. When using S0, the objective guarantee is no longer provable, but\nwe conduct this experiment because of the sufficiency of S0under hard constraints (see Lemma 7).\nIn Tables 3 and 4, we report the average suboptimality gap (across 5runs) using soft-penalized\nobjectives based on S0andSbaserespectively, varying the sample size and the number of iterations.\nThe dataset is the same synthetic dataset based on Bernoulli samples as described earlier, with\nε⋆= 0.1. We scaled the number of iterations Tas multiples of√n, because this is the ε−1value\nsuggested by our lower bound in Lemma 2. We find that both S0andSbasereliably converge to\nthe empirical smCEcomputed by CVXPY, at the1\nTrate guaranteed by [JT23]. While Sbaseworks\n33T 25+ 1 26+ 1 27+ 1 28+ 1 29+ 1\n4.8√n0.04789 0 .03096 0 .01829 0 .01380 0 .01006\n24√n0.01206 0 .00719 0 .00478 0 .00314 0 .00236\n48√n0.00657 0 .00389 0 .00267 0 .00170 0 .00139\nTable 3: Average suboptimality gap compared to CVXPY in Titerations of [JT23] solver, using\npenalty based on S0constraints, across 5runs. Columns indicate sample size n.\nT 25+ 1 26+ 1 27+ 1 28+ 1 29+ 1\n4.8√n0.04203 0 .02793 0 .01797 0 .01217 0 .00959\n24√n0.00886 0 .00583 0 .00359 0 .00192 0 .00559\n48√n0.00443 0 .00279 0 .00180 0 .00122 0 .00096\nTable 4: Average suboptimality gap compared to CVXPY in Titerations of [JT23] solver, using\npenalty based on Sbaseconstraints, across 5runs. Columns indicate sample size n.\nbetter than S0, there is not a large gap in their performance, suggesting it may be better to use S0\nin practice for efficiency. The constant 48was chosen based on parameters in [JT23]’s algorithm.\nFinally, we note that a limitation of our work is that for these moderate values of n, our preliminary\nunoptimized code is slower than CVXPY’s custom linear program solver for smCE.11We find it\nencouraging that Tables 3, 4 demonstrate high levels of accuracy for small iteration counts ≈100,\nand leave more efficient practical implementations as an important open direction.\n11CVXPY does fail to solve the dCElinear program for moderate n, however, as mentioned earlier.\n34References\n[AAA+16] Dario Amodei, Sundaram Ananthanarayanan, Rishita Anubhai, Jingliang Bai, Eric\nBattenberg, Carl Case, Jared Casper, Bryan Catanzaro, Jingdong Chen, Mike\nChrzanowski, Adam Coates, Greg Diamos, Erich Elsen, Jesse H. Engel, Linxi Fan,\nChristopher Fougner, Awni Y. 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Iyer,†Noah Whelpley,†Juha Tiihonen,‡Paul R. C. Kent,¶Jaron T.\nKrogel,∗,§and Brenda M. Rubenstein∗,†\n†Department of Chemistry, Brown University, Providence, Rhode Island 02912, USA\n‡Department of Physics, Nanoscience Center, University of Jyv¨ askyl¨ a, Jyv¨ askyl¨ a, Finland\n¶Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak\nRidge, Tennessee 37831, USA\n§Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,\nTennessee 37831, USA\nE-mail: krogeljt@ornl.gov; brenda rubenstein@brown.edu\nAbstract\nThe accurate mapping of potential energy surfaces (PESs) is crucial to our un-\nderstanding of the numerous physical and chemical processes mediated by atomic\nrearrangements - such as conformational changes and chemical reactions - and the\nthermodynamic and kinetic feasibility of these processes. Stochastic electronic struc-\nture theories, e.g., Quantum Monte Carlo (QMC) methods, enable highly accurate\ntotal energy calculations which in principle can be used to construct the PES. How-\never, their stochastic nature poses a challenge to the computation and use of forces\nand Hessians, which are typically required in algorithms for minimum-energy pathway\n1arXiv:2402.13189v1  [physics.chem-ph]  20 Feb 2024(MEP) and transition state (TS) identification, such as the nudged-elastic band (NEB)\nalgorithm and its climbing image formulation. Here, we present strategies that utilize\nthe surrogate Hessian line-search method - previously developed for QMC structural\noptimization - to efficiently identify MEP and TS structures without requiring force\ncalculations at the level of the stochastic electronic structure theory. By modifying the\nsurrogate Hessian algorithm to operate in path-orthogonal subspaces and on saddle\npoints, we show that it is possible to identify MEPs and TSs using a force-free QMC\napproach. We demonstrate these strategies via two examples, the inversion of the am-\nmonia (NH 3) molecule and the nucleophilic substitution (S N2) reaction F−+ CH 3F\n→FCH 3+ F−. We validate our results using Density Functional Theory (DFT)-\nand coupled cluster-based NEB calculations. We then introduce a hybrid DFT-QMC\napproach to compute thermodynamic and kinetic quantities - free energy differences,\nrate constants, and equilibrium constants - that incorporates stochastically-optimized\nstructures and their energies, and show that this scheme improves upon DFT accuracy.\nOur methods generalize straightforwardly to other systems and other high-accuracy\ntheories that similarly face challenges computing energy gradients, paving the way for\nhighly accurate PES mapping, transition state determination, and thermodynamic and\nkinetic calculations, at significantly reduced computational expense.\n1 Introduction\nOne of the crowning achievements of modern quantum chemistry is the ability to compute\npotential energy surfaces (PESs) entirely from first principles.1,2With knowledge of these\nsurfaces, researchers can predict the reactivity, dynamics, and kinetics of a wide range of\ntechnologically important and scientifically interesting atomistic systems.3,4For instance,\nthe stable structures of the reactants and products of a catalytic reaction can be determined\nby identifying the minima on its PES using a variety of geometry optimization algorithms.5–7\nReaction mechanisms can, moreover, be predicted by determining the minimum-energy path-\n2way (MEP) that connects the reactant and product states of a reactive system using, for\ninstance, chain-of-states methods8,9or the nudged elastic band (NEB) method10and its vari-\nations.11,12These methods can often be straightforwardly extended - e.g., using the climbing-\nimage nudged elastic band (CI-NEB) method13- to identify the (usually) first-order saddle\npoint of the PES along the MEP, which corresponds to the transition state. Reaction mech-\nanisms predicted in this fashion provide invaluable atomistic information about reaction\npathways, granting researchers key insights into how reactions can be redesigned and poten-\ntially controlled to minimize energy, material, and other resource requirements. Indeed, the\nability to computationally predict reaction mechanisms has become an indispensable tool in\nthe fields of combustion analysis,14catalysis,15,16and chemical reactor design.17\nNonetheless, while virtually all quantum chemical methods can predict the energy of a\nchemical system given the geometries of the species it contains, it can be challenging for some\nmethods to do the inverse, i.e., to very accurately predict equilibrium (reactant/product)\nand transition states, and reaction pathways given the energies of structures sampled in some\nlocal region of the PES. This is because being able to efficiently determine equilibrium struc-\ntures, transition states, and MEPs requires knowledge of energy derivatives, including forces\n(first derivatives) or Hessian matrices (second derivatives). These quantities then guide the\ndynamics of initial structures to the true equilibrium or saddle point structures that are\nconsistent with the given level of ab initio theory. Computing such derivatives in methods\nlike Density Functional Theory (DFT) can be done in a relatively straightforward fashion\nusing the Hellman-Feynman Theorem,18making DFT one of the most popular methods for\ndetermining molecular structures. However, computing these derivatives in more expen-\nsive many-body theories, including Coupled Cluster Theory (particularly with the inclusion\nof single, double, and perturbative triple excitations, CCSD(T)),19,20the Random Phase\nApproximation (RPA),21,22or Quantum Monte Carlo (QMC)23–32methods can be either\napproximate or highly costly. In the case of stochastic many-body theories such as QMC,\nthis also requires formal treatment of the energy derivatives in the presence of inherent sta-\n3tistical errors in the energies.23As a result, instead of determining these geometries or paths\nusing highly accurate many-body methods, many resort to using potentially less accurate\ntheories like DFT to determine structures and then higher-accuracy theories to determine\ntheir corresponding energies.33Although DFT geometries can be sufficiently accurate in the\nmany cases in which electron correlation does not play a critical role in dictating geome-\ntries, computing properties using many-body theories based on single-reference geometries\ncan lead to a fundamental mismatch in theories that can result in inaccurate or inconsis-\ntent predictions. For instance, as the authors have previously shown, lower-accuracy DFT\ngeometries of 2D materials that can be readily stretched/compressed along particular axes\ncan yield misleading predictions of magnetic and other electronic properties when studied\nusing more accurate quantum chemical methods.34,35It would thus be ideal to use the same\nhigh-accuracy method to both predict structures and their energies and other properties.\nRecently, a variety of hybrid approaches for circumventing and reducing the cost of\ndirectly computing energy gradients using relatively expensive high-accuracy methods have\nbeen proposed. Several approaches, particularly machine learning (ML)-based methods for\nmapping PESs and computing QMC forces, have been demonstrated successfully.36–39While\nsuch ML methods open up exciting new avenues for the scalable modeling of large molecules\nand complex chemical reactions,39they are typically highly reliant on the generation of large\ntraining sets. This can limit their practicality for systems for which it is difficult to assemble\nsufficient data, curtailing their generalizability beyond the systems on which they have been\ntrained. Furthermore, the accuracy of ML methods can be significantly improved if they are\ntrained with both energies andforces,37,40which may be challenging to accomplish if forces\nare not readily provided by the methods used to furnish training data.\nAn alternative set of approaches that do not necessitate large training sets or forces are\nso-called surrogate approaches, which use information easier to obtain from less accurate, but\nalso less expensive, theories to guide more accurate, more expensive theories. Some of the\nfirst ideas along these lines were proposed in Ref. 41. More recently, Ref. 42 introduced the\n4surrogate Hessian line-search method, which locates energy minima to within some tunable\nstructural error tolerance via a series of one-dimensional line searches that account for and\nleverage statistical errors in QMC energy calculations. This method was then successfully\napplied to the structural optimization of both molecular42and material systems.34,35\nBuilding upon the success of this approach for identifying equilibrium geometries, in\nthis work, we develop new formalisms and algorithms for generalizing this approach to the\ndetermination of transition state geometries and other structures along the minimum en-\nergy pathways of chemical processes. Key to generalizing these techniques is the realization\nthat reaction pathway identification tasks can be viewed as iterative energy optimization\nsteps. Transition states (saddle points) can be identified by minimizing the energy along\nall directions except one that has negative curvature, along which the energy is maximized\ninstead. MEPs can be identified by minimizing the energy along all directions in the sub-\nspace orthogonal to the tangent to the path at each structure along some guess discretized\npath. We note here that the idea of reformulating MEP identification as a series of energy\noptimization problems instead of force-based dynamical problems has been suggested before,\nnotably in Ref. 43. Additionally, Saccani et al. (Ref. 44) demonstrated the application of\nthe NEB algorithm to QMC-based MEP identification using correlated sampling.26However,\nto the best of our knowledge, ours is the first approach that generalizes MEP identification\nto stochastic electronic structure methods by utilizing Hessian information from lower-level\nsurrogate theories, eliminating the need for computing derivatives at the QMC level. We\ndemonstrate the accuracy and utility of this formalism by determining the transition states\nand minimum energy pathways of two paradigmatic chemical processes: (i) ammonia (NH 3)\ninversion, which involves two structural degrees of freedom, and (ii) and an S N2 nucleophilic\nsubstitution reaction, F−+ CH 3F→FCH 3+ F−,45that involves four degrees of freedom,\nusing Diffusion Quantum Monte Carlo (DMC) informed by DFT Hessian computations. We\ncompare the minimum energy pathways we compute using DMC with those obtained using\nDFT- and Coupled Cluster-based NEB calculations, demonstrating that our surrogate tech-\n5niques can provide at least post-DFT quality pathways at only a small factor of M×(N−1)\ngreater cost than traditional single-point DMC energy calculations, where Mis the num-\nber of points along which a discrete one-dimensional line-search grid is generated and N\nis the number of structural parameters used to parameterize the system. We subsequently\nshow that we can use a combined DFT-QMC approach to more accurately estimate ther-\nmodynamic quantities, such as free energies of reactions, and kinetic quantities, such as rate\nconstants, which can often be more directly compared to experiments. Overall, this work\nprovides clear avenues for determining reaction pathways and transition states with chemi-\ncal (or greater) accuracy at substantially lower computational cost than direct approaches,\nbut without the need for the large training datasets that statistical learning-based methods\nrequire. We believe that these techniques will provide new tools for the study of reactions\nand processes that involve significant amounts of electron correlation and the design of novel\nhomogeneous and heterogeneous catalysts that can accelerate these reactions.\nThis manuscript is organized as follows. In Section 2, we introduce the theoretical foun-\ndations of the surrogate Hessian line-search method and then show how this formalism can\nbe generalized to transition states (Section 2.1.1) and minimum energy pathways (Section\n2.1.2). We also describe how thermodynamic corrections to obtain free energies and equilib-\nrium constants can be performed using the hybrid DFT-QMC approach (Section 2.1.3). We\nthen present our results and benchmarks on our two illustrative chemical processes in Section\n3. Lastly, in Section 4, we conclude with the impacts and potential further generalizations\nof this method.\n62 Methods\n2.1 Theoretical Background\n2.1.1 Surrogate Hessian Line-Search Method for Equilibrium and Transition\nStates\nThe original surrogate Hessian line-search method for identifying equilibrium structures (en-\nergy minima) has previously been introduced in Ref. 42. For brevity, we only summarize\nthe method here while noting the specific modifications necessary to extend the method to\ntransition state identification.\nIn the surrogate Hessian line-search method, a surrogate method (such as DFT) that can\nmore rapidly compute energy gradients is used to calculate Hessians that can then be used\nto determine the directions along which line searches for minima should be performed using\nenergies from a higher-accuracy and costlier theory. This method relies on the assumption\nthat the local curvature of the target, higher-accuracy PES near a critical (0-derivative) point\n- energy minimum or saddle point - is approximated reasonably well by the PES calculated\nusing a surrogate theory. This allows one to compute (a) the optimal directions along which\nto perform the line-search, and (b) the maximum tolerable statistical noise (if a stochastic\nmethod is employed) at each point on the line-search grid in order to be able to determine\nan optimized geometry within some structural tolerance. This information is then used to\nperform a series of one-dimensional line-searches to locate the critical point within some\nconfidence interval.\nWhile the surrogate and higher accuracy approaches that can be employed are highly\ngeneralizable, throughout the rest of this manuscript, we focus on the specific choices of\nDensity Functional Theory (DFT) as our surrogate method and DMC46,47as our higher\naccuracy method. This choice of methods is natural since DFT calculations are often used\nas inputs into DMC calculations. In this context, the key steps of the surrogate Hessian\nline-search method can be summarized as follows:\n71.Initialization and parameterization : To provide an initial seed structure to the\nequilibrium or saddle point line-search, we use an optimized equilibrium or transition\nstate geometry obtained using the surrogate theory (DFT). This may be the result of\na typical structural relaxation or CI-NEB calculation, respectively. In systems where\nsome degrees of freedom are correlated due to structural symmetries, it is often de-\nsirable to define a few structural parameters that capture these symmetries and only\ndescribe the relevant regions of configuration space instead of operating in the full\nspace of 3 NCartesian coordinates (where Nis the number of atoms). We achieve this\nby defining a mapping function FI:R3N→Rd(given dstructural parameters). It is\ndefined to ensure that all structures that share these symmetries map to the same rep-\nresentation in Rd, while its inverse ((FI)−1) maps a structure in Rdto a single structure\ninR3N. When working with solids, it is also reasonable to include the dimensions of\nthe unit cell as additional structural parameters to allow for relaxations of the over-\nall lattice. The structural parameterizations of the NH 3inversion and S N2 reaction\nsystems studied in this manuscript are described in Section 2.2.\n2.Hessian evaluation : Given a reduced structural representation of the initial state, we\nthen compute the Hessian matrix of the PES in the vicinity of that structure using the\nsurrogate theory. Here, since we only define a small number of degrees of freedom, we\nevaluate the Hessian using a finite-difference scheme. For more complex systems, it may\nbe beneficial to evaluate it in generalized coordinates and then map it to the reduced\nspace. Notably, when dealing with equilibrium geometries, we expect all the Hessian\neigenvalues to be positive, while we expect neigenvalues to be negative when dealing\nwith nth-order saddle points. This distinction will become important when defining the\nobjective function of the line-search. Furthermore, the Hessian eigenvectors provide\nnear-optimal directions along which to perform the line-search using the stochastic\ntheory.\n83.Tolerance optimization and line-search : The Hessian eigenvectors now provide us\nwith directions along which we can perform a series of one-dimensional line-search op-\nerations to determine the desired equilibrium or saddle point. This requires evaluating\nenergies using the stochastic theory at a few points along each line-search direction.\nTo determine these points, it is necessary to identify the optimal (minimal) degree of\nstochastic energy sampling required to be able to specify the structural parameters\nwith a certain pre-specified level of precision ( δP). For each line-search direction, this\nis equivalent to evaluating the maximum point-wise energy noise tolerated along each\nline-search direction, σmax, and the corresponding structural window along which the\nenergy grid is sampled, Lmax, both of which are functions of δP. Finally, along each\nline-search direction x, the objective of the line-search for determining the equilibrium\ngeometry can be expressed simply as an energy minimization criterion\nx∗\nmin= arg min E(x). (1)\nThe modification to this objective for locating a saddle point is then relatively straight-\nforward. If a certain line-search direction corresponds to a negative eigenvalue, instead\nof minimizing the energy along that direction, we maximize it. This satisfies the def-\ninition of a saddle point as lying at a local maximum on the PES along some (small\nnumber of) directions while lying at a local minimum along all others. The line-search\nobjective is then modified as\nx∗\nsaddle =\n\narg min E(x) if λx>0\narg max E(x) if λx<0, (2)\nwhere λxis the eigenvalue corresponding to the Hessian eigenvector x.\nWe direct the reader to Ref. 42 for further details regarding the surrogate Hessian line-\nsearch method.\n92.1.2 Path-Orthogonal Subspace Optimization\nWe now describe how the surrogate Hessian line-search method can be adapted to identify\nMEPs. We begin by defining an MEP according to Fukui’s formulation of the intrinsic\nreaction coordinate (see Ref. 48) as a one-dimensional sub-manifold in configuration space\nthat satisfies the following conditions: (1) connects two equilibrium states; (2) passes through\na saddle point (assumed to be first-order for simplicity); and (3) consists exclusively of points\nthat lie in local energy minima in all directions except along the tangent to the path at those\npoints.\nMEPs are typically identified using chain-of-states methods such as NEB (Fig. 1a).10\nThe NEB algorithm begins with the construction of an initial guess pathway, usually a linear\ninterpolation of Mpoints between two equilibrium states (satisfying condition 1). For each\nof these pathway structures (images), tangents are calculated along the path. Then, two\ntypes of forces acting on (each atom of) each image are computed: (1) the “true” force,\ni.e., the negative gradient of the potential energy calculated according to the prescribed\nlevel of theory, and (2) an artificial “spring” force calculated as a function of the excess\ndistance between adjacent images compared to some equilibrium distance, usually set to\nbeD/(M+ 1), where Dis the Euclidean distance between the initial and final states.\n(Note that the result of an NEB calculation ought to be agnostic to the spring constant,\nprovided it is set to a reasonable value, typically ∼0.1 eV/ ˚A2.) These two forces are then\nresolved into components parallel and orthogonal to the path-tangent at each point, after\nwhich the component of the true force along the tangent and the component of the spring\nforce orthogonal to the tangent are set to zero. Finally, the remaining forces are used to\ndynamically propagate the system according to some dynamical rule, such as the BFGS or\nvelocity Verlet algorithms. This process is performed iteratively until a self-consistent path is\nobtained, i.e., the force is converged to within some tolerance (satisfying condition 3). Since\nNEB identifies a discrete path, condition 2 is not immediately satisfied. However, CI-NEB\naddresses this by maximizing the energy of the highest-energy state along the converged\n10NEB path. This is done by setting the path-orthogonal force components to zero and only\nmoving the image along the path.\n(a)\n (b)\nFigure 1: Schematic illustration of the (a) NEB and (b) surrogate Hessian subspace\noptimization algorithms on a contour map of the true PES. Yellow and red contours\nindicate local energy minima and maxima, respectively.\nThe optimization of an NEB path based on force convergence criteria can be understood\nas a constrained geometry optimization of each path image orthogonal to the path. Therefore,\nto identify the MEP given a discrete initial path between two equilibrium structures, we only\nrequire a geometry optimization algorithm capable of operating in the subspace orthogonal\nto the tangent at each point of the path.\nIn the case of the surrogate Hessian line-search algorithm, this translates to calculating\nthe surrogate Hessian matrix in the path-orthogonal subspace at each point along an initial\n11guess pathway and performing the line-search within this subspace. Borrowing a DFT-\nNEB pathway as an initial guess thus permits us to use DMC to obtain a more accurate,\ncorrected MEP. Hereafter, we refer to this as the subspace optimization method. The key\nsteps of subspace optimization for identifying MEPs are the following:\n1.Initialization and parameterization : We begin with a seed MEP that is obtained\nusing an NEB or CI-NEB calculation within the surrogate theory, i.e., DFT in our case.\nSuppose this path is given by {R0≡eA,R1,R2, . . . ,RM,RM+1≡eB}, where eAand\neBare the initial and final equilibria (e.g., the reactants and products or intermediates\nfor a chemical reaction pathway). Each intermediate image between the initial and final\nequilibria is then parameterized using the same scheme FI:R3N→Rdthat is used to\nparameterize the equilibrium and transition states to give {P0, P1, P2, . . . , P M, PM+1},\nwhere Pi≡FI(Ri).\n2.Tangent evaluation : For each intermediate structure along the pathway, we evaluate\nthe tangent to the path at that structure. There are several ways to accomplish this.\nHere, we take the vector difference between the current structure and the adjacent\nstructure that is closer to the saddle point. If the current structure is itself the closest\nstructure to the saddle point, we take the mean of the normalized vector differences\nbetween the current image and each of its two adjacent images. The key difference with\nthe standard implementation is that we compute the vector differences in structural\nparameter space rather than in Cartesian coordinates:\nτi=\n\nPi+1−Pi ifi < i saddle\nPi+1−Pi\n|Pi+1−Pi|+Pi−Pi−1\n|Pi−Pi−1|ifi=isaddle\nPi−Pi−1 ifi > i saddle, (3)\nwhere isaddle is the index of the structure closest to the saddle point.∗\n∗We make the simplifying assumption here that the surrogate CI-NEB MEP (correctly) identifies a path\n123.Subspace vectors and Hessian evaluation : Having computed the normalized path-\ntangents, we then use the Gram-Schmidt process to construct unit vectors that are\northogonal to each tangent. For a given tangent τi, these unit vectors vithen define\nthe path-orthogonal subspace of the overall structural parameter space. At this point,\nwe define secondary mappings FII:Rd→Rd−1that map a point from structural\nparameter space to the path-orthogonal subspace at each intermediate structure, which\nis set to be the origin of the subspace. This mapping, Pi≡FII(Pi|vi), is then used\nas the basis for computing a finite-difference Hessian matrix in the subspace with the\nsurrogate theory. We also construct an inverse mapping (FII)−1:Rd−1→Rdto keep\ntrack of line-search optimized structures in the original space of structural parameters.\n4.Tolerance optimization and line-search : Finally, for each intermediate structure,\nwe optimize the line-search parameters ( Lmax\ni, σmax\ni) in the subspace and perform the\nusual line-search, as described in Section 2.1.1, to locate the local energy minimum\nwithin the subspace. We then map each optimized structure, Popt\ni, back to the original\nspace of structural parameters as (FII)−1(Popt\ni), update each intermediate structure to\nbe the new line-search optimum, and repeat from Step 2 onwards.\nWe summarize the subspace optimization method in Algorithm 1 and show a schematic dia-\ngram in Fig. 1b. We note that, since the tangents are pre-computed for all the intermediate\nstructures in a given iteration of subspace optimization, the aforementioned operations -\ncorresponding to the inner forloop in Algorithm 1 - can be performed on each structure in\nparallel.\nNote that while Algorithm 1 depicts running the outer loop for a predefined number of\niterations ( Niter), this would ideally be replaced with a path convergence-based loop. In\npractice, however, we find that beginning with a CI-NEB initial guess allows us to obtain\nwhose energy increases monotonically until the saddle point and then decreases monotonically. This allows\nus to directly index the structure closest to the saddle point using the surrogate MEP energies. If we don’t\nmake this assumption, the tangent equation would have to be modified slightly to account for the energy of\neach image relative to its neighbors’ energies in any given iteration of the algorithm.\n13Algorithm 1 Surrogate Hessian subspace optimization algorithm\nConsider an N-atom system with two equilibrium states, eAandeB, for which an M-image\nMEP is to be determined with structural accuracy δP={δp1, δp2, . . . , δp M}.\nstart\n{R0=eA,R1,R2, . . . ,RM,RM+1=eB} ←CI-NEB (eA, eB|M)\nP← {FI(R0), FI(R2),. . . ,FI(RM+1)}\n▷FI:R3N→Rdmaps a structure from real space to structural parameter space\nforj←1,2, . . . , N iterdo\nfori←1,2, . . . , M do\nτi←Tangent (Pi|Pi−1, Pi+1)\n▷Computes the path-tangent at image igiven its neighboring images\nvi←GramSchmidt (τi)\n▷Uses the Gram-Schmidt process to identify ( d−1) vectors orthogonal to τi\nPi←FII(Pi|vi)\n▷FII:Rd→Rd−1maps a structure from structural parameter space to a\n(d−1)-dimensional path-orthogonal subspace\n▷Now, we perform the usual surrogate Hessian line-search as described in Ref. 42\nHi←Hessian (Pi) =∇2Esurrogate (Pi)\n(Lmax\ni, σmax\ni)←Optimize (Hi|δP)\nPopt\ni(∆Popt\ni)←LineSearch (Lmax\ni, σmax\ni)\nend for\n{P1, P2, . . . , P M} ← { (FII)−1(Popt\n1), (FII)−1(Popt\n2),. . ., (FII)−1(Popt\nM)}\nend for\nend\n14accurate paths after just a single iteration, due to which we retain the forloop format shown\nin the algorithm.\nThe pathway obtained from this algorithm also presents an alternative approach for\nlocating the saddle point. Instead of computing the surrogate Hessian and initializing the\nline-search at the highest-energy structure of the DFT-NEB pathway as described in Section\n2.1.1, we only compute the Hessian at the highest-energy DFT-NEB structure and initialize\nthe line-search at the highest-energy structure on the DMC subspace optimization pathway .\nThe utility of this approach is that it permits us to initialize the saddle point line-search\nat a point on a more accurate (DMC) pathway while only borrowing the search directions\n(Hessian eigenvectors) from the surrogate theory, thereby providing a more self-consistent\nprotocol for transition state identification. We hereafter refer to this as the self-consistent\nline-search approach for transition states and discuss the application of this approach in\nSection 3.1.\n2.1.3 Hybrid DFT-QMC Approach for Thermodynamics and Kinetics\nHaving obtained equilibrium and transition states, and path images, it is desirable to calcu-\nlate thermodynamic and kinetic quantities using our DMC-optimized geometries and ener-\ngies. In particular, let us suppose we are interested in computing the free energy changes and\nequilibrium and rate constants associated with a molecular transformation along an MEP\nusing data from a 0 K ground state ab initio theory. This can be done in thermochemistry\npackages using molecular statistical mechanical principles by evaluating the translational,\nrotational, and vibrational energies of the structures of interest that compose the enthalpic\nand entropic contributions to the free energy, given the temperature and the pressure. The\nequilibrium constant can then be evaluated as\nKeq= exp\u0012\n−∆G◦\nkBT\u0013\n, (4)\nwhere ∆ G◦is the free energy difference between the final and initial equilibrium states.49\n15Given the transition state, the rate constant for the forward reaction can be obtained\nusing transition state theory as\nk→=kBT\nhexp\u0012\n−∆G⊕\nkBT\u0013\n, (5)\nwhere we use ∆ G⊕to refer to (∆ H⊕−T∆S⊕).49∆S⊕is the entropy change in transforming\nfrom the initial equilibrium state to the transition state, and ∆ H⊕is the activation energy\nof the forward reaction. Keqandk→then automatically determine the backward reaction\nrate constant as k←=k→/Keq.\nPerforming these calculations using DMC, however, is not a straightforward task. This\nis because, in practice, obtaining the vibrational energies requires the evaluation of a finite-\ndifference Hessian matrix in the full R3NCartesian representation of the N-atom system. In\naddition to requiring a number of expensive DMC calculations, the intrinsic statistical nature\nof QMC methods introduces difficulty in stably and reliably diagonalizing the stochastically\nsampled Hessian matrix, which is indeed the original motivation for using the surrogate\nHessian line-search method.\nSurrogate Hessian line-search and subspace optimization are practicable methods due\nto the - in most cases - correct assumption that the local curvature of the surrogate and\nstochastic PESs are similar near the structures of interest to a good approximation. It is\nthen reasonable to suggest that we may rely on DFT to provide only the vibrational energies\nwhile obtaining the non-vibrational contributions to the enthalpy and entropy from DMC.\nIn the ideal gas limit, the contributions to the enthalpy may be decomposed as\nH(T) =Epot+EZPE+ZT\n0CPdT\n=Epot+EZPE+ZT\n0(kB+Ctrans\nV+Crot\nV+Cvib\nV+Celec\nV)dT,(6)\nwhere Epotis the potential energy, EZPEis the zero-point energy, and CPis the heat capacity\nat constant pressure, which can be further broken down into the sum of the translational\n16(Ctrans\nV), rotational ( Crot\nV), vibrational ( Cvib\nV), and electronic ( Celec\nV) heat capacities at constant\nvolume.1We suppose that the zero-point energy and the vibrational component of the heat\ncapacity can be obtained from DFT. Ignoring the electronic heat capacity, it is now clear\nthat the potential energy, Epot, is the only other component of the enthalpy that is sensitive\nto the choice of ab initio method used to optimize the structure and calculate its energy. We\nmay then make an approximation to the enthalpy that leverages the improved accuracy of\nDMC over DFT:\nHhybrid(T) = EDMC\npot|{z}\ncalculated at ⟨R∗\nDMC⟩+EDFT\nZPE+ZT\n0CV,vibdT\n| {z }\ncalculated at R∗\nDFT\n+ZT\n0(kB+Ctrans\nV+Crot\nV)dT,(7)\nwith the statistical error in the DMC potential energy now affecting the enthalpy directly as\nσ(Hhybrid(T)) =σ(EDMC\npot), (8)\nwhich is just the standard error of the DMC energy.\nIt is important to note that EDMC\npotand ( EDFT\nZPE, Cvib\nV|DFT) are calculated using two slightly\ndifferent geometries in Eq. 7. EDMC\npotis taken to be the potential energy of the mean surrogate\nHessian-optimized DMC geometry ( ⟨R∗\nDMC⟩), while the vibrational energies and all ensuing\ncontributions are calculated using the DFT-optimized geometry ( R∗\nDFT). This is crucial for\nthe following reason. Assume we are interested in an equilibrium structure. Then, the local\nfeatures of the PES near this structure will only be reflected in the DFT Hessian matrix if\nthe matrix is calculated around the DFT-equilibrium structure. Since this may be slightly\ndisplaced from the DMC-equilibrium structure, constructing a DFT Hessian matrix centered\nat the latter may capture the curvature of an irrelevant region of the DFT PES. Instead,\nwe combine the curvature (Hessian matrix) near the equilibrium of the surrogate, i.e., DFT,\nPES with the potential energy near the equilibrium of the more accurate, stochastic PES\n17(DMC). The same principle applies to saddle point structures. This strategy is valid because\nthe notion of a critical point within some sufficiently local region of a PES is independent of\nthe exact structural parameters to which it corresponds and the specific level-of-theory that\nwas used to obtain it. This allows us to combine two slightly different structures optimized\nusing different methods because they theoretically refer to the same region of the PES.\nNow, let us turn our attention to the entropy, which can be decomposed as\nS(T, P) =Strans(T, P◦) +Srot(T) +Svib(T) +Selec−kBlnP\nP◦, (9)\nwhere P◦is the pressure at the standard state and Strans,Srot,Svib, and Selecrefer to the\ntranslational, rotational, vibrational, and electronic entropies respectively.1Again, Stransand\nSelecare unaffected by slight differences between structures optimized by different ab initio\nmethods. As earlier, we can rely on DFT to compute Svib. Computing Srotanalytically,\nhowever, is a non-trivial task since it involves the calculation of molecular moments-of-\ninertia - given the uncertainty in the structural parameters of a DMC-optimized molecule,\nit is challenging to predict the propagation of these errors to the moments-of-inertia and\nthe rotational entropy. Not only are the errors in structural parameters correlated, but\nthe propagation of this error also becomes hard to generalize for differently parameterized\nsystems.\nTo overcome this, we employ a Monte Carlo scheme to estimate the rotational entropy\nstochastically. First, we sample a large number of structures using the mean and variance\nof the DMC-optimized structural parameters. After converting these structures back to\ntheir Cartesian representation, we calculate the rotational entropy for each of these sampled\nstructures independently, and finally report the average entropy and variance:\nShybrid(T, P) = Srot(T)|{z}\nMonte Carlo sampling around ⟨R∗\nDMC⟩+ Svib(T)|{z}\ncalculated at R∗\nDFT\n+Strans(T, P◦) +Selec−kBlnP\nP◦,(10)\n18so that\nσ(Shybrid(T)) =sPNMC\ni=1(Si\nrot(T)− ⟨Srot(T)⟩)2\nNMC, (11)\nwhere Si\nrot(T) is the rotational entropy of a structural sample iaround ⟨R∗\nDMC⟩andNMCis\nthe number of Monte Carlo samples.\nFrom here, it is easy to calculate the free energy changes and equilibrium and rate\nconstants from Eqs. 4 and 5.\n2.2 Illustrative Test Cases\nIn this work, we focus on two illustrative chemical processes as test cases: (1) ammonia\n(NH 3) inversion and (2) the nucleophilic substitution (S N2) reaction F−+ CH 3F→FCH 3\n+ F−. Ammonia inversion is a process that occurs ∼30 billion times per second50at room\ntemperature in which the three hydrogen atoms that form a trigonal pyramid around nitrogen\nflip from one side of the N to the other. In the S N2 reaction, one fluorine substitutes for\nanother in CH 3F in a concerted (hence the 2) fashion, resulting in an inversion of the methyl\ngroup to accommodate the bond rearrangement. This and other S N2-type systems have been\nstudied extensively using ab initio methods; see, for example, Ref. 45. We now describe\nexamples of minimal structural parameterizations of the NH 3inversion and S N2 reaction\nsystems that describe the transformations (inversion and bond rearrangement, respectively)\nthat these systems undergo.\nWe define the geometry of the NH 3system using two structural parameters (Fig. 2a): the\nmean N-H bond length in ˚A (p0) and the mean angle in radians ( p1) formed by the N-H\nbonds with the negative x-axis (labeled x′). Without loss of generality, we assume that the\ncentroid of the three H atoms lies on the x-axis in all configurations during the inversion of\nthe molecule. We set the N atom to always coincide with the origin and constrain one of the\nH atoms (labeled H 1) to lie in the x−yplane. We similarly parameterize the S N2 reaction\nsystem (Fig. 2b) using the mean C-H bond length ( p0) and the mean ∡HCx′bond angle\n19(p1). Additionally, we include the two C-F bond lengths, C-F 1(p2) and C-F 2(p3). We set\nthe C atom to coincide with the origin, the H 1atom to lie in the x−yplane, and the C-F 1\nbond to lie along the x′axis. It follows that ∡HCx′=∡HCF 1. Explicit expressions for both\nparameterizations are provided in Section ??of the Supporting Information.\nIn both systems, the initial and final equilibrium geometries are identical and share C3v\nsymmetry. However, grounding the parameterization in a Cartesian reference frame enables\nus to distinguish between the initial and final states in our analysis.\n2.3 Computational Details\nFor the NH 3system, we set error tolerances of 0.01 ˚A for the bond length ( δp0) and 0.05\nrad for the bond angle ( δp1). For the S N2 reaction system, we set the following tolerances:\n(δp0= 0.01˚A, δp1= 0.05 rad , δp2= 0.05˚A, and δp3= 0.05˚A). We choose higher tolerances\nfor the two C-F bond lengths because they vary to a greater extent over the course of the\nreaction, which enables their variation to be captured accurately with lower precision and\nhence lower computational cost.\nWe perform all DFT and Coupled Cluster with Single and Double excitations (CCSD)\ncalculations in PySCF.51,52We use correlation-consistent effective core potentials (ccECPs)\nand basis sets (cc-pVTZ-ccECP).53,54We use the PBE55functional for our DFT calculations\nand use identical DFT settings for computing energies, trial wave functions, and DFT-NEB\npaths.\nFor each QMC calculation, following DFT trial wave function generation, we optimize\none- and two-body Jastrow factors via 8 optimization cycles of 100,000 samples. We then\nfurther optimize the trial wave function with a VMC calculation by taking 1000 samples.\nFinally, we perform a DMC calculation with a timestep of 0.005 a.u. and run it for the\nestimated number of steps required to compute the energies within the stipulated tolerances.\nWe note that translating error tolerances in structural parameters ( δp) into the number of\nDMC steps required to saturate these tolerances involves a series of approximations (see\n20(a)\n(b)\nFigure 2: Structural parameterization of (a) the NH 3inversion system, and (b) the S N2\nreaction system. All bond lengths are measured in ˚A and bond angles in radians.\nRef. 42) and can result in higher or lower structural errors than intended. To mitigate\nthis, we extend the number of DMC steps in some of our calculations by a factor of 16 or\n64 (structure-wise values can be found in Section ??). We perform all QMC calculations\n21using QMCPACK56,57and assemble the DFT-VMC-DMC pipeline using the Nexus workflow\nmanagement system.58\nWe perform all NEB calculations in the Atomic Simulation Environment (ASE)59using\nthe BFGS59algorithm and a force tolerance of 0.01 eV/ ˚A. For the NH 3system, we include\n3 intermediate images between the initial and final equilibrium states, whereas for the S N2\nsystem, we include 5 intermediate images. To locate the saddle point, we perform a climbing\nimage calculation on the structure with the highest energy along the MEP. We also use the\nBFGS algorithm for DFT and CCSD equilibrium geometry optimization. In this case, we\nset the force tolerance to 0.001 eV/ ˚A.\n3 Results and Discussion\n3.1 Equilibrium and Transition State Geometries\nFirst, we report the results of the surrogate Hessian line-search method applied to the opti-\nmization of the equilibrium and transition state geometries of our two test systems. Tables 1\n(NH 3) and 2 (S N2) compare the DFT and CCSD structural relaxations and CI-NEB results\nfor the transition states to those obtained using the surrogate Hessian line-search approach\nusing DMC as described in Section 2.1.1 for the NH 3inversion and S N2 reaction systems.\nAdditionally, to illustrate that our approaches are self-consistent, we include the results for\nsaddle point identification using the self-consistent approach described in Section 2.1.2. For\nthis approach, we compute the line-search directions as the eigenvectors of the surrogate Hes-\nsian at the DFT-NEB saddle point, while the line-search itself is initialized at a high-energy\nimage along the DMC MEP obtained using subspace optimization. Given the structural\nsymmetries of the NH 3and S N2 systems and the odd number of intermediate images we\nhave considered here, the highest-energy image already coincides with the saddle point in\nboth cases. Thus, to make for more challenging test cases, we instead seed the self-consistent\nsaddle point line-search at a point adjacent to the highest-energy image along each path.\n22This corresponds to DMC path image 3 in Fig. 4 for the NH 3case, and DMC path image 2\nin Fig. 6 for the S N2 case.\nTable 1: NH 3structural parameters obtained using: [Row 1] DFT structural relaxation for\nthe equilibrium states and CI-NEB for the transition state; [Row 2] CCSD structural\nrelaxation for the equilibrium states and CI-NEB for the transition state; [Row 3] DMC\nsurrogate Hessian line-search; and [Row 4] DMC self-consistent saddle point line-search\ninitialized at path image 3 in Fig. 4.\nEquilibrium A Transition State Equilibrium B\np0(˚A) p1(rad) p0(˚A) p1(rad) p0(˚A) p1(rad)\nDFT 1.019 1.958 0.998 1.571 1.019 1.184\nCCSD 1.010 1.959 0.990 1.571 1.010 1.183\nDMC (LS) 1.006(1) 1.957(5) 0.991(2) 1.569(14) 1.007(2) 1.202(10)\nDMC (LS-SC) - - 0.992(2) 1.574(34) - -\nFrom Tables 1 and 2, it is clear that the DMC line-search yields geometries that are\nin excellent agreement with CCSD geometries obtained using BFGS structural relaxation,\nwith differences in bond lengths of less than ∼0.02˚A and differences in bond angles of less\nthan∼0.02 rad (or 1.15◦). Additionally, agreement between the direct DMC line-search\n(DMC (LS)) and self-consistent DMC line-search (DMC (LS-SC)), within statistical vari-\nation, indicates the capability of the surrogate Hessian approach to determine transition\nstate structures both in the presence and absence of initial guess transition state geometries\nobtained from surrogate theories such as DFT. We note that, while there are slightly larger\n(∼0.05˚A) differences between DMC and CCSD in the two C-F bond lengths ( p2andp3) in\nthe S N2 example, particularly in the transition state structure, these are a consequence of\nthe very low sensitivity of the potential energy to these structural parameters. Since the sur-\nrogate Hessian line-search method is based on non-linearly combining structural tolerances\ninto an energetic tolerance, locating a critical point to within some energetic tolerance can\nincur - based on how strongly (or weakly) they are coupled - commensurate variability in\nthe structural parameters. This will become clearer when we compare the energy barriers\nalong the MEP obtained by each method. For a more detailed analysis of the coupling of\nenergetic and structural tolerances, we refer the reader to Ref. 42.\n23Table 2: S N2 reaction structural parameters obtained using: [Row 1] DFT structural\nrelaxation for equilibrium states and CI-NEB for the transition state; [Row 2] CCSD\nstructural relaxation for equilibrium states and CI-NEB for the transition state; [Row 3]\nDMC surrogate Hessian line-search; [Row 4 in Transition State block] DMC self-consistent\nsaddle point line-search initialized at path image 2 in Fig. 6.\nEquilibrium A\np0(˚A) p1(rad) p2(˚A) p3(˚A)\nDFT 1.094 1.218 2.575 1.434\nCCSD 1.079 1.243 2.575 1.434\nDMC (LS) 1.079(3) 1.244(16) 2.580(17) 1.412(11)\nTransition State\np0(˚A) p1(rad) p2(˚A) p3(˚A)\nDFT 1.076 1.571 1.863 1.863\nCCSD 1.063 1.571 1.849 1.849\nDMC (LS) 1.066(3) 1.568(14) 1.824(13) 1.816(14)\nDMC (LS-SC) 1.068(6) 1.565(20) 1.851(30) 1.794(28)\nEquilibrium B\np0(˚A) p1(rad) p2(˚A) p3(˚A)\nDFT 1.094 1.923 1.434 2.575\nCCSD 1.079 1.899 1.434 2.575\nDMC (LS) 1.078(2) 1.894(8) 1.412(10) 2.580(14)\n243.2 Minimum-Energy Pathways\nWe now discuss the use of the subspace optimization algorithm to determine the MEPs of\nthe NH 3inversion and the S N2 reaction systems. We begin by defining two equilibrium\nstates, i.e., the DFT-optimized states labeled Equilibrium A and Equilibrium B in Tables\n1 and 2. For example, in the NH 3system, Equilibrium B refers to the molecule shown\nin Fig. 2a with the angle ∡HNx′being acute, while Equilibrium A refers to the inverted\nstructure of this molecule, where the angle ∡HNx′becomes obtuse. In the S N2 case, both\nthe angle ∡HCF 1and the C-F 1and C-F 2bond lengths change in going from Equilibrium A\nto Equilibrium B. For consistency of MEP comparison between the various methods, we use\nthese DFT-optimized configurations as the fixed endpoints for all MEP calculations.\nStarting with a linear interpolation between these two equilibria, we independently per-\nform NEB calculations using DFT and CCSD. Then, we use the DFT-NEB states to compute\nthe subspace surrogate Hessian and its eigenvectors for each intermediate structure along\nthe pathway. To compare a DFT-based surrogate Hessian subspace optimization with the\nDFT-NEB pathway, we displace the DFT-NEB images by a small fixed amount of 0.05 units\nalong each of the subspace eigenvectors. (It is difficult to associate this shift with a partic-\nular unit, such as ˚A or radian, because the subspace is constructed by taking combinations\nof vectors along the individual structural parameters, due to which a certain vector in the\npath-orthogonal subspace has a component along more than one structural parameter with\nno unique unit of measurement.) We then use these displaced states to initialize the DFT-\nbased subspace optimization. We then proceed to identify the DMC MEP by performing\na DMC-based subspace optimization using the DFT-NEB pathway as an initial guess with\nno displacement along the subspace eigenvectors. To illustrate the differences between these\npathways, we plot the NH 3inversion pathway in structural parameter space overlaid on an\nenergy heat map calculated using a fine grid of CCSD calculations in Fig. 3.\nIt is clear from Fig. 3 that the subspace optimization algorithm correctly identifies the\nMEP due to the close agreement between the DFT-NEB and DFT-subspace optimization\n25Figure 3: MEPs for NH 3inversion obtained using subspace optimization and NEB for\ndifferent methods overlaid on a CCSD energy heat map. Energies are measured in eV. The\nlinear interpolation path is used as an initial guess for the DFT-NEB and CCSD-NEB\ncalculations. The displaced DFT guess path (displaced from the fully converged DFT-NEB\npath) is used to seed the DFT subspace optimization calculation. The x- and y-axes\nindicate the NH 3structural parameters, p0andp1, respectively.\npathways. Furthermore, the DMC subspace optimization pathway shows better structural\nagreement with the CCSD-NEB pathway than the DFT-based pathways, indicating the\npotential for more accurate (compared to DFT) and inexpensive (compared to CCSD) MEP\nidentification using surrogate Hessian subspace optimization.\nAt this point, we note that, despite good agreement with NEB, subspace optimization\nand NEB are not strictly equivalent algorithms and therefore cannot always be expected\nto yield fully identical MEPs. The following are a few notable distinctions between typical\nimplementations of NEB and subspace optimization:\n1.Tangent calculation : During an NEB calculation, tangents to the MEP at each inter-\nmediate structure are calculated using vector differences between adjacent structures\nin Cartesian coordinates. Subspace optimization instead computes these tangents in\n26the space of structural parameters. Since the operations that map Cartesian coordi-\nnates to reduced structural parameters may be non-distributive over vector differences\nbetween structures, the tangents - and thereby, the path-orthogonal directions - com-\nputed using NEB and subspace optimization may not be identical. Neither approach\nis necessarily “more correct” since the choice of coordinates is a matter of represen-\ntational convenience. An alternative representation-invariant approach (not addressed\nhere) would be to construct an internal coordinate representation that accounts for\nthe implicit metric change that occurs when computing tangents at points located in\ndifferent regions of the PES.60\n2.Hessian evaluation : In NEB calculations that use force-based dynamical algorithms\nsuch as BFGS to propagate the chain of states toward the MEP, the Hessian matrix\nis evaluated and updated at each step of the dynamics. This results in the possibility\nof multiple Hessian evaluations even within a single NEB iteration. By contrast, the\nsubspace optimization algorithm is currently implemented to only perform a single\nevaluation of the subspace Hessian in a given iteration, which is then used to per-\nform multiple line-search iterations. It is for this reason that providing good starting\nstructures for Hessian evaluation is necessary for this algorithm to operate efficiently.\nWhile multiple Hessian evaluations can be implemented, we do not include this in our\napproach.\n3.Use of springs : Since NEB is based on the resolution of forces along and orthogonal\nto path-tangents, it is reasonable to introduce artificial spring forces to maintain an\neven spacing of images along the path. However, since such resolution of components\nis not possible in a purely energy-based optimization scheme, subspace optimization\ndoes not include spring forces and instead relies on evenly spaced starting images.\n4.Termination/convergence criterion : Implementations of NEB are typically based\non force-based convergence, where the method is set to iterate until the force on each\n27atom falls below some threshold. Subspace optimization, on the other hand, may be\nimplemented using a path-based convergence criterion, where the structural difference\nbetween consecutive paths must fall below a certain threshold for convergence.\nHaving obtained MEPs from each of the methods, we now consider the energy barriers\nalong the MEPs obtained with each method. We plot this for the NH 3system in Fig. 4,\nwith the inclusion of a Gaussian process regression fit to the DMC data points to depict the\nsmooth variation of the mean energy and standard energy error along the MEP. It must be\nnoted that, while we plot each energy barrier with a common x-axis labeled ‘Path image,’\nthese do not correspond to the same reaction coordinate, since the MEP obtained using each\nmethod is slightly different from the rest.\nFigure 4: Energy vs. NH 3inversion path image for different methods. The DFT-NEB and\nDFT subspace optimization points overlap almost completely.\nIt is clear from Fig. 4 that DMC-based subspace optimization is capable of resolving\nenergy barriers to within a few meV and is in very good agreement with the CCSD-NEB\n28energy barrier. We also note that in Fig. 4, the DFT-NEB and DFT subspace optimization\nbarriers are so close that the DFT-NEB data points are not visible.\nWe now turn to the S N2 reaction system, which possesses more degrees of freedom and\nis expected to exhibit greater electron correlation, leading to more significant differences\nbetween the MEPs. Since it is difficult to show the variation of all structural parameters in\nthe same plot, we show the variation of each structural parameter with respect to the path\nimage in Fig. 5. Again, it is evident that subspace optimization accurately captures the\ntrends in the variation of structural parameters across the MEP using both DFT and DMC.\nWe compare the energy barriers obtained by the different methods in Fig. 6. Here, we see\nthat DMC subspace optimization is in excellent agreement with CCSD along the MEP, while\nthe DFT-NEB and DFT subspace optimization differ by less than 2.5 meV, or ∼0.1 mHa,\nfrom each other. Note that the DFT and DMC MEP barrier heights differ by nearly 20 meV\nfrom one another, illustrating that correlation assumes a more significant role in this test\ncase.\nWe note that, due to the distinctions between NEB and subspace optimization mentioned\nabove, the two methods do not yield perfectly identical MEP structures at the DFT level.\nIn our method, the structures along the MEP are determined as a consequence of path-\northogonal energy optimization. Given that several structures can be energetically very\nsimilar, the energy optimization algorithm may rightly identify any one of these structures\nas lying on the MEP, although it may not necessarily coincide with the structure identified\nby a force-minimization algorithm such as NEB. Due to these practically insignificant energy\ndifferences, paths obtained by both methods can be considered to consist of representative\npoints along the MEP despite small structural dissimilarities.\n3.3 Free Energies and Kinetic Parameters\nWe now discuss the evaluation of thermodynamic and kinetic quantities obtained using the\nhybrid DFT-QMC approach described in Section 2.1.3. In particular, we report the free\n29Figure 5: Structural parameters vs. path image of the S N2 reaction system for different\nmethods (see top right panel for legend). The linear interpolation path is used as an initial\nguess for the DFT-NEB and CCSD-NEB calculations. The displaced DFT guess path\n(displaced from the fully converged DFT-NEB path) is used to seed the DFT subspace\noptimization calculation.\nenergy difference between the saddle point and equilibrium structures (∆ G⊕in Eq. 4) and\nbetween the two equilibrium states (∆ G◦in Eq. 5). We also report the associated kinetic\nparameters, k→andKeq. We benchmark these results against DFT and CCSD calculations\nusing the standard approach, in which all energetic and entropic contributions are computed\ndeterministically using the same geometry.1,59\nDuring the Monte Carlo structure sampling to estimate the rotational entropy ( Srot(T)\nin Eq. 10), we sample 2500 structures each for the saddle point and equilibrium states of\nNH 3using the mean and standard error of the structural parameters identified using DMC\nin Table 1. Similarly, in the case of the S N2 reaction, we sample 10,000 structures around\nthe mean and standard error values shown in Table 2.\nThe free energy differences and equilibrium and rate constants are shown in Table 3 for\n30Figure 6: Energy vs. S N2 reaction path image for different methods.\nTable 3: Free energy differences and equilibrium and rate constants for the NH 3inversion\nsystem.\n∆G⊕(eV) k→(×1016s−1)∆G◦(eV) Keq\nDFT -0.199 1.436 0.000 1.000\nHybrid DFT-QMC -0.200(0) 1.483(24) -0.000(1) 1.018(33)\nCCSD -0.222 3.514 0.000 1.000\nthe NH 3inversion system and in Table 4 for the S N2 reaction system. Tables ??and??show\nthe free energy contributions of the various ab initio method-dependent terms in Eqs. 6, 7,\n9, and 10. As can be seen in Table ??, the dominant contribution to ∆ G⊕in the NH 3system\nis the difference in vibrational entropies of the transition state and the equilibrium state,\nwhich DFT underestimates by ∼0.022 eV compared to CCSD. Since the hybrid DFT-QMC\napproach directly borrows the vibrational entropy from the DFT calculation, this explains\nthe high similarity between the DFT and hybrid DFT-QMC values of ∆ G⊕andk→for NH 3\ninversion. In the S N2 system, however, as shown in Table ??, the dominant contribution\n31Table 4: Free energy differences and equilibrium and rate constants for the S N2 reaction\nsystem.\n∆G⊕(eV) k→(×1012s−1)∆G◦(eV) Keq\nDFT 0.004 5.317 0.000 1.000\nHybrid DFT-QMC 0.019(1) 2.918(122) 0.000(0) 0.989(19)\nCCSD 0.020 2.852 0.000 1.000\nnow becomes the difference in potential energies of the transition state and equilibrium state,\nwhich DFT predicts to be ∼0.008 eV, whereas DMC and CCSD predict to be ∼0.022 eV.\nSince DMC computes the potential energies more accurately than DFT, the hybrid DFT-\nQMC approach is able to obtain ∆ G⊕andk→values in much better agreement with the\nCCSD result.\n4 Conclusions\nIn this work, we have demonstrated the extension of the surrogate Hessian line-search method\npreviously employed to determine equilibrium geometries to the identification of transition\nstates and minimum-energy pathways. Modifying the objective function of the line-search to\nmaximize the energy along the negative-eigenvalue directions, as shown in Eq. 2, permits us\nto identify saddle points of arbitrary order. Furthermore, performing the line-search within\nrestricted, path-orthogonal subspaces yields points along the MEP that can be specified to\narbitrary structural accuracy, leveraging the controllable computational cost of DMC and\nother stochastic ab initio theories. Finally, we have shown that it is possible to combine\nstochastic potential energies and Monte Carlo-sampled rotational entropies obtained using\nDMC with vibrational contributions from DFT to compute corrected free energy changes\nand kinetic constants that can improve upon DFT results and, potentially, approach coupled\ncluster accuracy using only energetic information from DMC calculations without the need\nfor gradient evaluation at the DMC level.\nThe surrogate Hessian line-search and subspace optimization methods are both primarily\n32based on energy optimization. Imposing structural tolerances on the geometries that are\nidentified using these methods, therefore, directly influences the precision to which the energy\nis specified. In practice, the structural tolerance must be set large enough so as not to incur\ndisproportionate computational expense while still meeting the level of precision required for\nthe study. If an accurately designed surrogate Hessian-based calculation results in structures\nthat show slight deviations from more accurate methods such as CCSD or CCSD(T), this is\nlikely an indication of the “flatness” of the PES near those structures, which, in turn, reveals\ninteresting information about the physics of the system.\nSubspace optimization, as noted in Section 3.2, features a few methodological departures\nfrom NEB and other chain-of-states methods. However, both classes of methods accomplish\nthe same task and identify slightly different - yet theoretically consistent - points along the\nMEP. Steps can be taken to eliminate the differences between the two approaches, such as\nstandardizing the tangent calculation method, repeated reevaluation of the surrogate Hessian\nat every step taken by the line-search, etc. We leave these modifications for future work.\nHigh-accuracy PES calculations play a significant role in several areas of computational\nchemistry and materials science research. At large system sizes, O(N6−7)- or higher-scaling\ncoupled cluster- and full configuration interaction-based methods remain largely inaccessible.\nStochastic electronic structure theories such as QMC - accelerated using surrogate theories\nsuch as DFT - present opportunities to unravel the physics and chemistry of large, corre-\nlated systems at high accuracy and for potentially lower computational cost, given their\nO(N3−4) scaling. The work presented here also provides a path toward obtaining critical\ninformation about reaction pathways without the explicit use of energy gradients, which can\nbe challenging to obtain in a variety of otherwise promising electronic structure methods.22\nIn the context of DMC, our approach will have particular value for large systems or systems\ncontaining heavy atoms, for which force calculations can be especially cumbersome.\nOverall, our work represents a step further in the expanding suite of capabilities of QMC\nmethods, enabling the accurate description of systems that undergo dynamical transitions\n33along minimum-energy pathways, such as diffusion processes,61chemical reactions,44includ-\ning catalytic reactions,33,62conformational changes, and phase transitions.63\nSupporting Information\nThe Supporting Information document contains the structural parameterization equations\nfor the NH 3and S N2 systems, DMC calculation extension factors, and the breakdown of the\nfree energy contributions. All code associated with this work can be found at\nhttps://github.com/gopal-iyer/surrogate hessian reaction pathway search .\nAcknowledgement\nG.R.I. thanks Shubham Sharma, Benjamin Foulon, and Bjarne Kreitz for useful discus-\nsions. G.R.I. (derivation, implementation, and analysis of the algorithms presented, draft-\ning of the manuscript) was funded by AFOSR Award Number FA9550-19-1-9999 and the\nBrown University Chemistry Department Vince Wernig Fellowship. J.T. (mentorship) was\nself-supported. J.T.K. (concept, mentorship), P.R.C.K. (concept), and B.R. (concept, men-\ntorship, manuscript writing) were supported by the U.S. Department of Energy, Office of\nScience, Basic Energy Sciences, Materials Sciences and Engineering Division, as part of the\nComputational Materials Sciences Program and Center for Predictive Simulation of Func-\ntional Materials. This research was conducted using computational resources and services\nat the Center for Computation and Visualization, Brown University.\nThis manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-\n00OR22725 with the U.S. Department of Energy. The United States Government retains and\nthe publisher, by accepting the article for publication, acknowledges that the United States\nGovernment retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or\nreproduce the published form of this manuscript, or allow others to do so, for United States\nGovernment purposes.\n34References\n(1) Cramer, C. Essentials of Computational Chemistry: Theories and Models ; Wiley, 2013.\n(2) Czak´ o, G.; Szab´ o, I.; Telekes, H. On the Choice of the Ab Initio Level of Theory for\nPotential Energy Surface Developments. The Journal of Physical Chemistry A 2014 ,\n118, 646–654.\n(3) Bowman, J. M.; Czak´ o, G.; Fu, B. 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Goswami1, J. M. Vilchez2, B. Pérez-Díaz2, L. Silva3,4, A. Bressan5, and E. Pérez-Montero2\n1Department of Applied Physics, University of Cádiz, Campus of Puerto Real, 11510 Cádiz, Spain\ne-mail: sabyasachi.goswami@uca.es\n2Instituto de Astrofísica de Andalucía - CSIC, Apdo 3004, E-18080, Granada, Spain\n3INAF-OATs, Via G. B. Tiepolo 11, I-34143 Trieste, Italy\n4IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy\n5SISSA, Via Bonomea 265, I-34136 Trieste, Italy\nFebruary 21, 2024\nABSTRACT\nContext. Recent work presented increasing evidence of high, non-constant S /O abundance ratios observed in star-forming metal-poor\ngalaxies, showing deviations from the constant canonical S /O across a large range of O /H abundance. Similar peculiar high Fe /O\nratios have been also recently detected.\nAims. We investigate whether these high S /O ratios at low metallicities could be explained taking into consideration the process of\nPair Instability Supernovae (PISN) in chemical modelling through which similar behaviour observed for Fe /O ratios was successfully\nreproduced.\nMethods. We use chemical evolution models which take into account the stages of PISN in the yields published by Goswami et al.\n(2022), and adopt a suitable initial mass function (IMF) to characterize this evolutionary stage appropriately.\nResults. The peculiar high values and the behaviour of the observed S /O versus O /H relation can be reproduced when the ejecta of\nvery massive stars that go through the process of PISN are taken into account. Additionally, a bi-modal top-heavy IMF and an initial\nstrong burst of star formation are required to attain the reported high S /O values\nConclusions. We show that the role of very massive stars going through the process of PISN should be taken into account when\nexplaining the chemical enrichment of sulfur and oxygen in metal-poor star-forming regions.\nKey words. galaxies: abundances – galaxies: starburst – galaxies: formation – stars: massive – galaxies: structure\n1. Introduction\nSulfur (S) is produced in massive stars where helium is fused\ninto heavier elements, including S, during the final stages of the\nstar’s life, before the supernova explosion. An oxygen (O) con-\nvective core is formed during the hydrostatic burning of neon,\nwhich leads to the formation of α-elements till S. Sulfur is ad-\nditionally produced during explosive O burning during the su-\npernova (Type II) explosion. Since oxygen is produced through\nthe same processes the study of the S /O ratio can give valuable\nhints on the chemical evolution, star formation history and Ini-\ntial Mass Function (IMF). There is a collection of studies where\nsulfur and oxygen abundances have been derived and studied\nin the context of star-forming galaxies and giant H II regions\n(e.g. Pagel 1978; Vilchez et al. 1988; Garnett et al. 1997; Pérez-\nMontero et al. 2006; López-Sánchez & Esteban 2009; Berg et al.\n2013; Dors et al. 2016; Fernández et al. 2019; Arellano-Córdova\net al. 2020; Dors et al. 2023). However, the debate continues to-\nday regarding the S /O relationship with metallicity as usually\nmeasured in terms of (O /H). Several studies (Garnett et al. 1997;\nIzotov et al. 2006; Guseva et al. 2011; Berg et al. 2020; Rogers\net al. 2021) have shown that the S /O ratio remains constant with\nmetallicity traced by O /H. Since sulfur and oxygen are produced\nthrough the same processes described above, their ratio is ex-\npected to be constant as long as they are produced by similar\nmass ranges of stars. On the contrary, some studies have pointedout that there could be a variation of S /O with decreasing metal-\nlicity (e.g. Vilchez et al. 1988; Dors et al. 2016; Díaz & Zamora\n2022). Recently, Díaz & Zamora (2022) measured sulfur and\noxygen abundances in a large sample of galaxies dominated by\nyoung, massive stars, finding a clear bimodality in the trend of\nthe S /O ratio with metallicity. In their sample, HII regions in\ncompact dwarf galaxies (HII Gal), that are dominated by strong\nstarbursts show S /O ratios mostly below or around solar S /O, and\nan increasing trend of the S /O ratio with increasing metallicity\n(O/H). On the contrary, Giant di ffuse HII regions (DHR), mostly\nin irregular and spiral galaxies, exhibit the opposite trend with\nhigh S /O ratios at low metallicities and a decreasing S /O ratio at\nincreasing O /H. The most peculiar aspect is the extremely high\nS/O ratio observed in DHRs at low metallicities( 12 +log(O /H)\n7.0), for which a theoretical explanation is still lacking.\nSimilarly, high Fe /O ratios at low 12 +log(O /H) values have\nbeen recently reported in extremely metal-poor star-forming\ndwarf galaxies (with 12 +log(O /H)≤7.69; Kojima et al. 2020).\nThe high Fe /O ratios observed in these metal-poor galaxies\nwere explained in the work by Goswami et al. (2022) by\nconsidering in their chemical evolution models very massive\nstars that undergo processes such as PPISN /PISN (Pulsational\nPISN) through their later evolutionary stages. Sulfur is produced\nthrough pair instability supernova which occurs in very massive\nstars, with initial masses greater than around 130 M ⊙. The PISN\nArticle number, page 1 of 8arXiv:2402.13240v1  [astro-ph.GA]  20 Feb 2024A&A proofs: manuscript no. aanda\nstage strongly depends on the mass of the He core at the end of\nthe H burning phase (Heger & Woosley 2002; Takahashi et al.\n2018), which produces large amounts of elements such as Fe,\nS, O, among others. Takahashi et al. (2018) in their work ob-\ntained new yields from PISN nucleosynthesis for rotating and\nnon-rotating zero-metallicity stars. They compared their theoret-\nical models with the abundances of metal-poor stars from the\nSAGA database, and they did not find evidence of PISN metal-\npoor stars in the sample based on [Na /Mg] and [Ca /Mg] abun-\ndances. Since then, there have been multiple observations show-\ning high Fe /O ratios (Kojima et al. 2020), and low [Mg /Fe] ratios\n(Yoshii et al. 2022) both of which have been linked to evidence\nof PISN nucleosynthesis enrichment. Additionally, Xing et al.\n(2023) reported abundances in very metal-poor(VMP) stars with\nextremely low sodium and cobalt abundances, which provide a\nchemical signature of PISN. One of the major di fficulties in get-\nting a direct signature is that PISN explosions do not leave be-\nhind any remnants like CCSN does, and hence the tracers of such\nexplosions can only be indirectly found. Chemical abundances\nof various elements, if available, can provide us with crucial in-\nformation that may allow us to di fferentiate between multiple\npossible nucleosynthesis channels, such as Hypernovae (HNe)\ndepending on which elements are produced. PISN yields have\nbeen used as a possible channel in previous studies to analyse\nthe chemical evolution of di fferent galaxies (Isobe et al. 2022;\nWatanabe et al. 2023). It has been shown that this stage is ex-\npected to be limited up to a threshold metallicity of Z cr≲10−4Z⊙\nor Z cr≲10−2Z⊙depending on e fficient or ine fficient dust cool-\ning, respectively q(Schneider et al. 2006; Salvadori et al. 2008).\nHowever, with current models, the PISN stage is suggested to\nbe important till Z ∼0.5Z⊙(Kozyreva et al. 2014; Langer 2012;\nCosta et al. 2021).\nObservationally, the VLT Tarantula Survey (Schneider et al.\n2006) which studies the 30 Doradus region in the Large Mag-\nellanic Cloud (LMC) inferred very young ages (1-6 Myr) stars\nwith masses of up to 200 M ⊙. Additionally, Crowther et al.\n(2016) estimated star masses that could be as high as 300 M ⊙\nin the same region. Moreover, the IMF derived in specific re-\ngions of NGC 2070 shows an upper mass slope of x=0.65\nand an overall slope of x=0.9. In order to investigate the high\nS/O ratio observed in some metal-poor star-forming galaxies, we\nhave computed new chemical evolution models for these objects,\nincluding the PISN yields. A comparison of these models with\nobservations can thus provide valuable information, firstly, con-\ncerning the nature of the chemical enrichment process at very\nlow metallicity, which is especially interesting for the evolution\nof galaxies in the early universe, and, secondly, on the nature of\nthe IMF and its possible deviations from universality.\nIn summary, sulfur is a very relevant element to be studied\nin the regions hosting very massive star formation, especially\nsignificant for the studies of galaxies in the early universe.\nIts unique nucleosynthesis history and the new data coming\nfrom the JWST for low metallicity galaxies can allow us to\nunderstand the viable role played by very massive stars in the\nchemical evolution of galaxies, and, eventually, help us decipher\nthe properties of the first stars. Thus, the primary goal of this\nwork is to study whether accounting for the PISN stage of\nvery massive stars in chemical evolution modelling could be\na possible channel to replicate the unusually high S /O ratios\nobserved, as shown by the DHR sample of Díaz & Zamora\n(2022) at very low metallicity (12 +log (O /H)⪅7.5). As\npreviously mentioned, PISN is primarily supposed to occur in\nlow-metallicity environments, where high S /O ratios have been\nobserved. Combined with the fact that PISNs produce a largeamount of S, this study of sulphur abundance at low metallicities\ncan o ffer us a unique tracer to reconstruct the early universe’s\nnucleosynthesis history and to comprehend the characteristics\nof its very massive stars.\nThis paper is structured as follows. In Section 2 we describe\nthe chemical evolution model used in this work. In particular, we\ndescribe the PISN yields of oxygen, sulfur and iron. In Section\n3, we present the sample of observational data analysed in this\nwork. In Section 4, we compare the observations to the predic-\ntions of the most suitable chemical evolution models stressing\nthe derived evolutionary constraints. Finally, in Section 5, we\nsummarise our conclusions and outlook.\n2. Methodology\nIn this section, we describe the chemical evolution model we\nhave developed to be used as a tool to understand the high S /O\nratios at low metallicity, as shown by recent studies. We use\nthe code CHE-EVO (Silva et al. 1998) which computes one-zone\nopen chemical evolution models considering the time evolution\nof the gas elemental abundances, and includes the infall of pri-\nmordial gas. This code has been used in several contexts to pro-\nvide the input star formation and metallicity histories to interpret\nthe spectrophotometric evolution of both normal and starburst\ngalaxies (e.g. Vega et al. 2008; Fontanot et al. 2009; Silva et al.\n2011; Lo Faro et al. 2013, 2015; Hunt et al. 2019). The basic\nequation used in this code can be written as follows:\n˙Mg,j=˙MInf\ng,j−˙MSF\ng,j+˙MFB\ng,j (1)\nwhere, for the element j, the first term on the right, ˙MInf\ng,jcor-\nresponds to the infall rate of pristine material. ˙MSF\ng,j, represents\nthe rate of gas consumption by star formation, and ˙MFB\ng,jrefers\nto the rate of gas return to the interstellar medium (ISM) by dy-\ning stars. The latter term also includes the contribution of type\nIa supernovae (SNIa), whose rate is adjusted with the parameter\nASNIawhich corresponds to the fraction of binaries with system\nmasses between 3 M ⊙and 16 M⊙and the right properties to give\nrise to SNIa (Matteucci & Greggio 1986).\nWe used the Schmidt-Kennicutt law (Kennicutt 1998) to model\nthe star formation rate (SFR):\nψ(t)=νMg(t)k(2)\nwhereνis the e fficiency of star formation, Mgis the mass of\nthe gas, and kis the exponent of the star formation law. The\ngas infall law is assumed to be exponential (e.g. Grisoni et al.\n2017, 2018), with an e-folding timescale of τinf. The IMF can be\nwritten as follows:\nϕ(Mi)=dn\ndlog(Mi)∝M−x\ni (3)\n2.1. Stellar Yields\nWe use in this work the stellar yield compilation as in Goswami\net al. (2022). Here we give a brief overview and the reader is\naddressed to the previous work for a more thorough review.\nFor low- and intermediate-mass stars ( Mi<8M⊙), single stars\nand binary systems that give rise to SNe Ia are characterized by\ndifferent yields (Matteucci & Greggio 1986).\nFor single stars with initial masses Mi<6M⊙we adopt the\nyields from calibrated Asymptotic Giant Branch (AGB) models\nArticle number, page 2 of 8S. Goswami et al.: Contribution of Very Massive Stars to the Sulfur Abundance in Star-Forming Galaxies\nby Marigo et al. (2020). For 6 M⊙<Mi<8M⊙we use super-\nAGB yields from Ritter et al. (2018). For SNIa we consider the\nyields provided by Iwamoto et al. (1999). For 8 M⊙≤Mi≤350\nM⊙we adopt the yield compilation by Goswami et al. (2022)\nfor massive stars and very massive objects (VMO).\nA comparison of PISN ejecta with other models in the lit-\nerature has been shown in Figure 1. PISN model predictions\nfrom Goswami et al. (2022) have been shown in green, which\nincludes rotation. These ejecta have been computed by matching\nthe Heger & Woosley (2002) models, which do not include ro-\ntation originally with the corresponding PARSEC (Bressan et al.\n2012; Costa et al. 2021) rotating models. Rotating and non-\nrotating models from Takahashi et al. (2018) are shown in red\nsolid and dotted, respectively. Finally, in magenta, the rotation\nincluded CCSN yields derived by Limongi & Chie ffi(2018) and\nused in Goswami et al. (2022) are shown. This allows us to ex-\nplicitly display the di fference in the production of the relevant\nelements through the two channels, i.e., CCSN and PISN at low\nmetallicities. Looking at the PISN yields in green and red, we\nnotice that the production of sulfur increases as the initial mass\nincreases. As discussed in Sect. 1 the PISN yields strongly de-\npend on the mass of the helium core, nevertheless for the sake\nof simplicity, here we show the yields against the initial masses.\nThis also allows us to compare the PISN yields with the CCSN\nyields simultaneously. The trend of increased production is also\ntrue for iron, where stars with more massive initial masses (as a\nresult of a more massive helium core) produce more iron. How-\never, carbon and oxygen have the opposite trend, as the ejecta\nof both carbon and oxygen starts to decrease as the mass of the\nhelium core increases. The e ffect of rotation is also more clearly\nseen in the case of carbon and oxygen where the production of\nthese elements grows with higher values of rotational velocities.\nFinally, we can see that rotation does not play an important role\nin sulfur production. Comparing the PISN yields to the CCSN\nyields for the same metallicity Zi=0.0001 shown in blue, we\ncan notice that there is relatively minimal sulfur and iron pro-\nduction through the CCSN channel at low metallicities. The pro-\nduction of carbon and oxygen is higher through the CCSN as\ncompared to S and Fe; however, the relative di fference between\nthe CCSN and PISN yields for all four elements is significant in\nthe low metallicities regime.\nTo illustrate this better, in Fig. 2 we show the IMF-averaged\nS/O and Fe /O ratios for the CCSN and PISN yields used in this\nwork. We present the element ratios for two distinct IMFs to\ndemonstrate how, when relying solely on CCSN yields and a\ncanonical IMF, the S /O and Fe /O ratios obtained are not as high\nas they would be if PISN were also taken into consideration. The\ntop-panel of Fig. 2 shows the element ratios averaging over the\nIMF obtained from the CCSN and PISN yields using a top-heavy\nIMF that has an upper mass limit of 300 M⊙and a slope of xUP\n=0.6. Since it has a high upper mass limit and the stepper slope\nincreases the number of such very massive stars.The PISN stage,\ndue to explosive O burning, produces higher amounts of S, Fe,\nthat are taken into account, which causes the S /O and Fe /O ra-\ntios to be high. O production is lower in the high-mass PISN\nregime as compared to low-mass PISN since O is fused to other\nheavier elements. In the bottom panel, the element ratios aver-\naged over a canonical Kroupa IMF with an upper mass limit of\nMUP=100M⊙have been shown. Because of the upper mass cut,\nthe very massive stars going through the PISN stage are not ac-\ncounted for, so the contribution of these ratios comes primarily\nfrom CCSN yields, which is why the two bars match each other.\n0 50 100 150 200 250 300 3500246810Ej[M]\nC12 =0.4 Z=0.0001\n =0.3 Z=0.0001 \nRot Takahashi2018\nNR Takahashi2018\nCCSN Yields \n0 50 100 150 200 250 300 3500.02.55.07.510.012.515.017.5Ej[M]\nS\n0 50 100 150 200 250 300 350\nMi[M]\n01020304050Ej[M]\nO16\n0 50 100 150 200 250 300 350\nMi[M]\n01020304050Ej[M]\nFeFig. 1. The relation between newly produced ejecta ( Ej) in the y-axis of\nthe elements C12, O16, S and Fe versus the initial mass ( Mi) in the x-axis\nfrom Z =0.0001 rotational models of Goswami et al. (2022) in green\n(solid, and dotted lines for 2 rotation rate values), based on Heger &\nWoosley (2002) pure He non-rotating models.CCSN yields are shown in\nblue. In red, yields of rotating (solid) and non-rotating (dotted) models\nby Takahashi et al. (2018) for zero metallicity stars are shown. Since the\nactual S and Fe CCSN yields are small, they are shown multiplied by a\nfactor of 15 for the sake of clarity.\n3. Observational data\nThe abundance data sample used in this work has been taken\nfrom the extensive work by Izotov et al. (2006) and Díaz &\nZamora (2022), with a selection also of H II regions from Yates\net al. (2020). Izotov et al. (2006) examined the chemical compo-\nsition of the emission line galaxies of the Sloan Digital Sky Sur-\nvey (SDSS) Data Release 3. They measured the [SII] λλ6717,31\nand [SIII]λ6312 lines1of sulfur, and used the direct method for\nthe derivation of sulfur and oxygen abundances. The sulfur abun-\ndances they derived in most of these galaxies exhibit a similar\npattern to that obtained for nearby galaxies in the high metal-\nlicity regime; however, at the low metallicity end (12 +log(O /H)\n⪅7.2) the S /O ratios derived exceed the canonical solar value\n(log(S /O)=-1.57; Asplund et al. 2009). Díaz & Zamora (2022)\nin their innovative work derived sulfur abundances for a large\nsample of HII regions in irregular and spiral galaxies includ-\ning the Magellanic Clouds (Giant Di ffuse HII Regions; DHR\nsubsample) and dwarf galaxies with a prominent starburst (HII\nGal subsample), using measurements of [SII] λλ6717,31 and\n[SIII]λλ9069,9532 lines from the literature and re-calculating\nthe abundances following the direct method, based on the deriva-\ntion of the electron temperature. Similarly, abundances derived\nwith the direct method for a set of giant H II regions are also in-\ncluded from Yates et al. (2020) who measured also the [SII] and\n[SIII] lines. Peculiarly high ratios of S /O in the lowest metallic-\nity range were found by Díaz & Zamora (2022) in their DHR\nsample, and confirmed using the Yates et al. (2020) data. Both\n1From S+and S++ions, typically representing common and abundant\nionic species of sulfur in star-forming regions\nArticle number, page 3 of 8A&A proofs: manuscript no. aanda\nS/O Fe/O0.000.020.040.060.080.100.12IMF-Averaged RatioTop-heavy IMF MUP =300 M, xUP=0.6\nCCSN\nCCSN+PISN\nS/O Fe/O0.000.010.020.030.040.050.060.070.08IMF-Averaged RatioKroupa IMF MUP =100 M, xUP=1.5\nCCSN\nCCSN+PISN\nFig. 2. IMF-averaged ratios of S /O and Fe /O shown for two IMFs: a top-\nheavy IMF with a 300 M⊙upper mass limit (PISN stage is accounted\nfor) and a canonical Kroupa IMF with a 100 M⊙limit for the yields sets\nadopted in this work.\nthese samples follow a similar trend of high S /O ratios at low\nO/H and then a decrease in S /O with the increase in metallic-\nity with some dispersion at high O /H. However, in the HII Gal\nsample of Díaz & Zamora (2022) and the majority of the Izo-\ntov et al. (2006) sample the S /O ratios are around the solar S /O\nvalue. In this work, we concentrate on the more anomalous high\nS/O ratios at low metallicities seen in the DHR sample of Díaz &\nZamora (2022), for which there is currently no theoretical expla-\nnation. It is important to emphasize here that invoking possible\ndepletion of oxygen in dust grains ( ≈0.08 dex for this metallic-\nity; Peimbert & Peimbert 2010) could not explain the observed\ntrend, especially at such low abundance.\nWe note that the three abundance studies selected above, irre-\nspective of their specific abundance derivation techniques, re-\nported high S /O values independently, which our models repro-\nduce for a number of star-forming objects at low metallicity. Per-\nforming a study for a larger complete sample with fully homo-\ngeneously derived sulfur abundances is out of the scope of this\nwork.\nIn Figure 3 we show the log(S /O) ratio versus oxygen abun-\ndance, 12 +log(O /H). The red data points correspond to the sam-\nple from Izotov et al. (2006) while the brown and cyan points\nrepresent the abundances derived from Díaz & Zamora (2022)\nfor the DHR and HII Gal samples respectively, and the blue\ntriangles correspond to star-forming regions from Yates et al.\n(2020). As it can be seen in Figure 3, as the O /H abundance\ndecreases, the S /O ratio starts to increase in the DHR and the\nYates et al. (2020) sample shown in the brown and blue points\nrespectively and clearly deviates from the constant solar ratio\nsuggested by various studies mentioned in Sect. 1. On the other\nhand, the data is more dispersed around the solar S /O ratio for\nthe HII Gal and most of the Izotov et al. (2006) sample, which\nare indicated by cyan and red dots, respectively. In the following\nsection, we explore whether chemical evolution models that ac-count for the PISN stage can replicate the high S /O ratios at low\nmetallicity demonstrated by the DHR sample.\n4. Results\nWe have used chemical evolution models to study the evolution-\nary constraints that can be derived from the chemical abundance\nobservations. In this work, we investigate the high values of\nthe S /O ratio and its trend with O /H that have been observed\nat very low metallicity. We have checked if the observed S /O\nratios, which do not follow the expected constant behaviour\nconsidering a similar nucleosynthetic origin for both S and O,\ncould be explained from the point of view of stellar evolution.\nThe peculiarly high S /O ratios seen at low metallicity ( ⪅\n0.1 Z⊙) show a di fferent trend compared to galaxies at high\n12+log(O /H), as presented in Figure 3. To test this behaviour\nwe start by applying a base model as M1 which was previously\nused by Goswami et al. (2021) to explain the data in the high\nmetallicity range following the chemical evolution pattern of the\nMilky Way (MW) disc and other local galaxies. Model M1 has a\ncanonical Kroupa et al. (1993) IMF with an upper mass limit of\n100 M⊙, and includes all the parameters required to reproduce\nthe chemical evolution of the MW (Grisoni et al. 2017, 2018,\n2019, 2020; Spitoni et al. 2021; Goswami et al. 2021). As\nillustrated in Figure 3, model M1, shown in grey dashed line,\ngoes through the HII Gal sample and the majority of the Izotov\nsample with higher O /H, but is unable to achieve the high S /O\nratios shown by the DHR sample at low metallicities. Since\nthe upper mass limit of the IMF in this model is 100 M⊙, and\nas explained in Sect.2.1, till this mass range, CCSN at low\nmetallicities doesn’t produce enough S as shown in Fig.1, to\nincrease the S /O ratios. Most of the points at high O /H resemble\nthe abundances and ratios derived in local group galaxies, and\nour model appears to reproduce this part of the observations.\nWe note that although the HII Gal sample has galaxies with\nvery young ages, this model which doesn’t include a starburst\nphase reaches 12 +log (O /H)∼7.5 at 200 Myr shown by\nthe grey square. This model, if used with an appropriate SFH\n(e.g. increasing the star formation e fficiency) more suitable for\nstarburst-dominated galaxies like in the HII Gal and Izotov et al.\n(2006) samples, can reproduce the cyan and red dots in Fig. 1\nwith the appropriate young ages expected for such galaxies.\nNonetheless, this is not the goal of this work, and in any case,\nthis reference model shows that with a canonical IMF and\nwithout the PISN contribution, the high S /O ratios observed in\nthe DHR sample cannot be obtained.\nA similar problem was recently studied by Goswami et al.\n(2022) where they reproduced peculiarly high Fe /O ratios from\nEMPGs. These EMPGs follow a decreasing trend of Fe /O with\nincreasing metallicity; this trend has also been observed in the\ncase of the DHR subsample’s S /O ratio as shown in Figure 3.\nThese authors explain how the observed high Fe /O ratios and\ntheir decreasing trend with metallicity can be reproduced only\nby using a top-heavy IMF and taking into account the PISN\nstage in the chemical evolution modelling. On the other hand,\nwe comment in Sect.2.1 while comparing the shapes of PISN\nejecta that S and Fe show very similar trends in their ejecta, in\na similar range of masses of stars going through the PISN stage.\nThus, we tested three models M3, M5 and M6 (M5 and M6 each\nwith two di fferent rotational parameters ω=Ω/Ωcrit=0.3 and 0.4)\nused by Goswami et al. (2022) to reproduce the high Fe /O ra-\ntios and investigate if these models can explain the high S /O\nas well. We show in Table 1 the characteristics of each model\nArticle number, page 4 of 8S. Goswami et al.: Contribution of Very Massive Stars to the Sulfur Abundance in Star-Forming Galaxies\n6.0 6.5 7.0 7.5 8.0 8.5 9.0\n12 + log(O/H) 2.5\n2.0\n1.5\n1.0\n0.5\nlog (S/O)\n =0.3 Zcr = 0.01\n =0.4 Zcr = 0.01  \n6.0 6.5 7.0 7.5 8.0 8.5 9.0\n12 + log(O/H) 2.5\n2.0\n1.5\n1.0\n0.5\nlog (S/O)\n =0.3 Zcr = 0.0001\n =0.4 Zcr = 0.0001  \nFig. 3. S/O ratio for two di fferent models with PISN: M5 ( M5ω3 and M5ω4) in the left panel and M6 ( M6ω3 and M6ω4) in the right panel as\na function of the oxygen abundance 12 +log(O /H). The black dotted and solid lines denote the two di fferent values of rotational parameters used.\nThe red points are from Izotov et al. (2006) and the blue triangles are from Yates et al. (2020). The brown and cyan points correspond to the DHR\nand HII Gal subsamples of Díaz & Zamora (2022) respectively. Model M1, in grey dashed line, is a chemical evolution model representative of\nthe MW disc (Goswami et al. 2021) where the grey square corresponds to an age of 200Myr, and the M3 model predictions are shown in yellow\ndashed-dotted line. The six squares on models M5 and M3 represent the model’s age at 5, 15, 30, 60, 100, and 200 Myr. For M6, only ages\ncorresponding to 5 and 15 Myr are shown by the two squares. The values adopted for critical metallicity Zcrand rotational parameter Ωare quoted\nfor the two models. The grey dot-dashed line denotes the solar S /O ratio. See the text and Table 1 for details.\ntested in this work and present them here in brief; however, the\nreader is directed to Goswami et al. (2022) for a detailed de-\nscription of these models. Model M3 uses a top-heavy IMF with\naMUP=300M⊙and a slope of xUP=0.6. This model takes into\naccount the PISN stage due to the high MUPwhich causes the\nS/O ratio to grow due to increased production of S in the low\nmetallicity region as shown by the yellow dashed-dotted line in\nFig. 3. However, this model, although increasing S /O ratio com-\npared to the HII Gal sample and going through the blue sample\nof Yates et al. (2020), it cannot reproduce the highest S /O values\nobserved (around 12 +log(O /H)∼7.1 ; brown dots) which is the\nmain purpose of this work. The age obtained by the model at 12\n+log (O /H)∼7.5 is 30 Myr which is much younger compared\nto M1 at the same metallicity. Although both the HII Gal and\nDHR samples include metal-poor and bursty star-forming re-\ngions, which in principle would be favourable locations for very\nmassive star formation, there could be other reasons for the dif-\nference in S /O ratios operating at the low metallicity end of the\nsample. The exact characteristics associated with the formation\nprocess of these extremely massive stars are still unknown, as are\nthe environmental circumstances that led to their formation. One\npossible reason that may explain the di fference seen between the\nS/O ratios of HII Gal and DHR low-metallicity samples, suppos-\nedly likely for very massive star formation, could be related to\nthe variations in the maximum mass limit of stars that form in\ndifferent galaxies.\nFor instance, a star in the low-mass PISN regime in Figure 1\nwith a mass of 150 M⊙produces more O than S as it doesn’t\nreach high enough temperatures to fuse heavier elements, re-\nsulting in a lower S /O ratio. On the other hand, a star in the\nhigh-mass PISN regime, mass 250 M⊙produces more S than\nit did previously, meaning that the S /O ratio would be higher.\nMoreover, Chru ´sli´nska et al. (2023) has demonstrated that dis-\ntinct trends in the element ratios may be produced associated\nwith specific SFR (sSFR). They discovered a distinct [O /Fe] ra-tio evolution with the sSFR; however, the PISN did not account\nfor this. Such results would point towards a further enhancement\nof this dependency. Additionally, further study of these envi-\nronments could yield significant information about characteris-\ntics other than low-metallicity and bursty star formation regions,\nwhich could be favourable to the creation of VMS. This will al-\nlow for more constraints on the formation of VMS and other nu-\ncleosynthesis channels. Nonetheless, with the current constraints\navailable on the abundance ratios, the ages and star formation\ncharacteristics, the PISN channel could not be ruled out.\nIn models M5 and M6 a bi-modal IMF has been assumed to\nsample the e ffects of VMOs exploding as PISN. To do so, in the\nchemical evolution model, an early phase with a top-heavy IMF,\nsuitable for Pop III-like stars (Wise et al. 2012), has been used\nwhich has the following form:\nϕ(Mi)=dn\ndlog(Mi)∝Mi−1.3×exp− Mchar\nMi!1.6 (4)\nAMcharof 200 M⊙has been adopted for this IMF instead of\nMcharof 40 M⊙used in the original work since the ejecta of S\nand O increase strongly with the initial masses in the high mass\nrange as shown in Sect. 1. We also checked with Mcharof 300 M⊙\nand since the results were similar to Mchar=200 M⊙, we only\nshow here one case. The PISN enrichment phase is limited to\nthe early evolution of the starburst, the duration of which is con-\nstrained by the critical metallicity, Zcr. When this gas metallicity\nis reached, a Kroupa IMF (equat. (3)) with xUP=1.3 and MUP=40\nM⊙is adopted. Since higher mass loss rates cause stars to be un-\nable to reach high enough masses to enter the PISN regime as\nmetallicity grows, an IMF change with a lower MUPis adopted.\nThe critical metallicity is Zcr=10−2for M5 ( M5ω3 and M5ω4)\nandZcr=10−4for M6 ( M6ω3 and M6ω4) depending on whether\ndust cooling is e fficient or not (Schneider et al. 2006; Salvadori\net al. 2008). To model the star formation, a star formation e ffi-\nArticle number, page 5 of 8A&A proofs: manuscript no. aanda\nTable 1. Chemical evolution and IMF parameters of the selected models\nused in this work. These models are shown in Fig. 3.\nModel Chemical evolution parameters IMF\nν kτinf ASNIa Zcr MUP xUP\n(Gyr−1) (Gyr)\nM1 0.8 1.0 6.0 0.04 100 1.5\nM3 0.3 1.0 1.0 0.04 300 0.6\nM5 1.0 1.0 0.1 0.04 0.01 200 bi-modal\nM6 5.0 1.0 0.1 0.04 0.0001 200 bi-modal\nciency ofν=1 Gyr−1has been adopted for M5. In the left panel of\nFigure 3 model M5 is shown in black dotted ( M5ω3) and dashed\nlines ( M5ω4) for two di fferent values of rotation considered for\nthe PISN yields. This model reaches the highest S /O ratios at\naround 15 Myr and to be consistent with the young ages of these\ngalaxies, these models have been shown until 200 Myr of their\nevolution, unlike models M1 and M3 where their entire evolu-\ntion (13 Gyr) is shown.\nThe model can reach the high S /O ratios at lower 12 +(O/H)\nwhere an initial burst is modelled that takes into account the\npresence of VMOs and, once the critical metallicity is reached\nand the e ffect of VMO enrichment ceases and the IMF is\nswitched to Kroupa et al. (1993) like resulting in the decreas-\ning S /O ratio present in the observations. The model starts with\na metal-free gas, Z =10−10and, as the VMOs start to explode,\nthe metallicity rapidly increases and reaches the Zcrvalue in a\nshort period (see Figure 4 of Goswami et al. 2022). During this\nphase, the PISN contributes to increased sulfur production, and\ndue to this, the model can reach high S /O ratios. Once the Zcr\nis reached, the IMF changes to a Kroupa-like IMF where stars\nare only formed till with Mi≤40M⊙. This causes the S /O ratio\nto decrease with increasing metallicity, but the model does not\ngo through the DHR sample points shown in brown at higher\nmetallicities and results in slightly higher ages as expected from\nregions dominated by very massive stars, as shown by Schneider\net al. (2006).\nIn order to explain this decreasing trend of the DHR sample\nas the metallicity increases while being consistent with younger\nages, we use Model M6, which is modelled with two di fferent\nvalues of rotational parameters ( ω=0.3,0.4) and a more e fficient\nstar formation and is shown in the right panel of Figure 3 in the\nblack dotted ( M6ω3) and black solid lines ( M6ω4). This model\ncan not only match the highest S /O ratio shown by the DHR sam-\nple at low O /H but it also shows the decline of the S /O ratio as\nwe go to higher metallicities, although with a steeper slope. To\nreproduce this decline of S /O we adopt this model, where once\nit reaches the Zcrit takes into account the stars that overpro-\nduce oxygen or underproduce sulfur, as shown by the S /O ratio\nof the DHR data in the high metallicity regime. Once the criti-\ncal metallicity is reached, a Kroupa IMF with MUP=150 M⊙is\nused. In this mass range, stars produce large amounts of oxygen\nas compared to sulfur, which causes the S /O ratio to decrease.\nTherefore, we have shown that, by adopting an initial burst of\nVMOs and using a bi-modal IMF both, the high S /O ratios of\nthe DHR subsample and then the decline with increasing metal-\nlicity observed can be explained. Since this model uses a higher\nstar formation e fficiency ofν=5 Gyr−1combined with a top-\nheavy IMF, the ages obtained at the same metallicity compared\nto model M5 are lower. The two black squares correspond to\nages 5 and 15 Myr and this model reaches the highest S /O ratios\nbefore 5 Myr which is consistent with ages derived from regions\nin the Magellanic Clouds hosting very massive stars (Schneider\net al. 2006; Crowther et al. 2016). Hence we demonstrate thatincluding PISN in the models could be a possible scenario to\nexplain these high S /O ratios. Díaz & Zamora (2022) suggested\nin their comprehensive and inspiring work that the decline in\nthe S/O ratio with increasing metallicity might be related to the\nchemical evolution of massive star nucleosynthesis which pro-\nduces more sulfur entwined with variations in the IMF which\nwe have investigated through this model.They also comment that\nsulfur production from SNe Ia is insu fficient to produce the de-\nsired slope for the S /O decline (Weinberg et al. 2019). We note\nthat to model the exact slope of the decline using the two datasets\nshown in the DHR sample of Díaz & Zamora (2022) and Yates\net al. (2020) will require a homogeneous derivation of S /O abun-\ndances between the samples and obtaining more data, especially\nfor the low metallicity regions, which would be the aim of a fu-\nture study. This would allow us to get a constraint on the shape\nof the bi-modal IMF once Zcris reached.\n5. Conclusions and outlook\nIn this paper, we use chemical evolution models to analyze the\nobserved sulfur and oxygen abundances derived in large samples\nof star-forming galaxies which indicate higher S /O ratios at low\nmetallicities, deviating from a constant solar S /O ratio suggested\nby other works. To do so we use chemical evolution models as a\ntool to investigate under which evolutionary scenario these high\nS/O ratios can be explained.\nSimilar peculiar high Fe /O ratios at low metallicities have\nbeen reported for EMPGs (e.g. Kojima et al. 2020; Izotov et al.\n2006). Goswami et al. (2022) reproduced these high Fe /O ra-\ntios taking into account the later evolutionary stages of massive\nstars such as PISN and PPISN in the chemical evolution mod-\nelling. We compare the stellar yields of S and Fe in the VMO\nmass range from the yield set used in their work. In Figure 1 we\nshow that the yields of S and Fe follow a similar trend in this\nmass range where VMOs go through the PPISN /PISN process.\nWe also show that at low metallicities, the sulfur and oxygen pro-\nduction through the CCSN channel is significantly less than the\nPISN channel. Additionally in Fig. 2 we show the IMF-averaged\nratios of S /O and Fe /O obtained for two di fferent IMFs to show\nthe di fferences between only CCSN production and including\nPISN channels. We thus have used the models that reproduce the\nobserved high Fe /O ratio of these galaxies to test if, considering\nthese later evolutionary stages as a possible channel, the high\nS/O ratios shown by the DHR subsample could also be simulta-\nneously explained. Firstly, we show that using model M1 previ-\nously applied to reproduce the MW disc with a canonical IMF,\nthe high S /O ratio at low metallicities cannot be reproduced. This\nmodel does not take into account the PISN stage and has an up-\nper mass limit of MUP=100M⊙does not produce enough sulfur\nthrough the CCSN channel at low metallicities to obtain a high\nS/O ratio. It can, however, explain the S /O ratios shown by the\nHII Gal and Izotov samples derived at higher 12 +(O/H) albeit\nwith a higher age, since in this metallicity range the data appear\nto follow the trend of the local galaxies.\nThereafter, we used models that take into account stars going\nthrough the PISN stage. Model M3 uses a top-heavy IMF with\na higher MUPof 300 M⊙and slope of xUP=0.6, increasing the\nS/O ratios at lower metallicities to go through the Yates et al.\n(2020) sample but not enough to reach the highest S /O values\nobserved in the DHR sample of Díaz & Zamora (2022). Sub-\nsequently, to properly account for the PISN evolutionary stage\nand reach the high S /O ratios, we use a bi-modal IMF where\nthe first part takes into account an IMF devised for Pop III-like\nstars (Wise et al. 2012) with a burst that includes the e ffects of\nArticle number, page 6 of 8S. Goswami et al.: Contribution of Very Massive Stars to the Sulfur Abundance in Star-Forming Galaxies\nVMOs and we limit this early phase using a critical metallic-\nity,Zcrsince VMOs only form in low metallicity environments\naccording to theoretical predictions. Once this Zcris reached, a\ncanonical Kroupa et al. (1993) IMF is adopted. The values of\nZcrused here are Zcr=0.0001 and 0.01 depending on ine fficient\nor efficient dust cooling (Schneider et al. 2006; Salvadori et al.\n2008). In Figure 3, we show that model M5 at low metallicity\ncan reach the high S /O values as indicated in the DHR sample.\nIn this model, the early phase takes into account a strong burst for\nwhich a star formation e fficiency ofν=1 Gyr−1has been used to\nreach a high S /O ratio at young ages. During the PISN stage, sul-\nfur production increases as compared to oxygen in the high-mass\nPISN regime. Hence a high S /O ratio can be obtained. Once the\ncritical metallicity is reached, VMOs stop being formed, leading\nto a reduction in sulfur production, and we can see the decreasing\ntrend of S /O shown by the data being reproduced by the model\nas well, although with a di fferent slope. The e ffect of rotation on\nthe production of sulfur is minimal, as can be seen in Figure 3\nwhere the two models with di fferent rotational ( Ω) values follow\nroughly the same path.\nSince model M5 albeit going through the highest S /O ratios\nat low metallicity, does not go through the DHR sample as the\nmetallicity increases, combined with slightly higher ages as ex-\npected from regions hosting very massive stars, we used model\nM6 which has a higher star formation e fficiency ofν=5 Gyr−1\nto be able to reach the high S /O ratio at a very young age consis-\ntent with previous studies. Model M6 like M5 can reach a high\nS/O ratio at low metallicity because of the increased production\nof sulfur from the PISN stage; then the decreasing trend of the\nS/O starts once the model reaches Zcr. In this case, because of\nthe higher e fficiency in star formation and populating with more\nstars that produce more oxygen than sulfur using an appropri-\nate IMF, for which a second Kroupa et al. (1993) IMF with an\nMUP=150 M⊙andxUP=0.9 has been adopted, the decrease in\nS/O observed is also shown by the model with a stepper slope\nthan observed. In order to accurately replicate the slope, more\nlow metallicity data demonstrating these high S /O ratios would\nneed to be obtained, along with a uniform derivation of the S /O\nratios of the two samples of DHR and Yates et al. (2020). This\nwould be the aim of a subsequent study, and in this instance, we\noffer the theoretical scenario of this decline.\nHence, we show here that by taking into account later evo-\nlutionary stages like the ones of the PPISN /PISN process, and\nadopting an IMF somewhat more suitable for star formation\nevents as those recently studied in the early universe, the ob-\nserved high S /O ratios at low metallicity can be reproduced.\nSuch top-heavy IMF with MUPof 200 M⊙as reported for 30\nDoradus and, in general, in other bursty star-forming regions,\nhas been predicted which favours the formation of such very\nmassive stars (Marks et al. 2012; Je ˇrábková et al. 2018; Zhang\net al. 2018; Crowther et al. 2010; Schneider et al. 2018). Hence\nPISN provide a possible channel to obtain high S /O ratios in\nlow-metallicity star-forming objects.\nHowever, due to the lack of accurate derivations of sulfur\nabundances in the low metallicity range, the IMF slope and the\nupper mass limit are di fficult to constrain robustly. This will also\nallow us to find models that fit these datasets better which is\nthe purpose of a future study, as discussed before. To elucidate\nthe characteristics of this intriguing scenario of chemical\nenrichment from very massive stars, additional observations\nand analysis of other elemental abundance ratios are required.\nMoreover, this would enable us to distinguish between various\npossible nucleosynthesis channels. With more data coming\nfrom JWST to derive chemical abundances, including sulfur, ofyoung star-forming galaxies we could try to constrain the early\nchemical enrichment scenario in a more robust way.\nIn this work, we have shown that with the study of the chemi-\ncal abundance and the S /O ratio of star-forming galaxies, impor-\ntant properties such as the mass limit of the very massive stars\ncan be better theoretically constrained, which could help in e.g.\ncompelling the upper mass limit and tail of the IMF in the early\nuniverse. Similarly, the high Fe /O ratios that were observed in\nthese galaxies at low metallicity have been recently explained by\ntheoretical models that include the PISN stage (Goswami et al.\n2022). However, iron lines are typically faint or undetectable in\nthis class of objects, whereas sulfur lines are frequently seen\nin their spectra even in high redshift galaxies. In addition, sul-\nfur, unlike iron, does not su ffer high depletion in dust grains. To\nthis end, we may encourage JWST observations of IR emission\nlines of sulfur ([SIII], [SIV]) and neon ([NeII], [NeIII]), seen in\nthe spectra of star-forming objects, in order to consistently de-\nrive S /Ne as a possible proxy for S /O. We propose also that the\nderivation of sulfur abundances and S /O ratios of more distant\nstar-forming galaxies (e.g. with JWST given its spectroscopy\nperformance and timing of data gathering), could pave a new,\nunexplored path to understanding the role of very massive stars\nin the early stages of galaxy formation.\nAcknowledgements. S.G. acknowledges financial support from the European\nUnion under the 2014-2020 ERDF Operational Programme and by the De-\npartment of Economic Transformation, Industry, Knowledge, and Universi-\nties of the Regional Government of Andalusia through the FEDER-UCA18-\n107404 grant. S.G. acknowledges funding from Project CRISP PTDC /FIS-\nAST-31546 /2017 funded by FCT. 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McSpadden1,\nDecember 2023v1.3\n1Thomas Jefferson National Accelerator Facility\n2Carnegie Mellon University\nContents\n1 Introduction 2\n2 Project Objectives 3\n2.1 Control of the GlueX Central Drift Chamber . . . . . . . . . . . 4\n2.2 Reduce time commitment required for data monitoring and cali-\nbration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2.3 Reduce delay between data taking and publications . . . . . . . . 4\n2.4 Facilitate automation of other detector systems . . . . . . . . . . 4\n3 Traditional Detector Calibration and Operation 5\n3.1 GlueX Central Drift Chamber . . . . . . . . . . . . . . . . . . . . 5\n4 Development of a CDC control system 7\n4.1 Data analysis & preparation . . . . . . . . . . . . . . . . . . . . . 7\n4.2 Model development . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n4.2.1 Uncertainty Quantification . . . . . . . . . . . . . . . . . 11\n5 Deployments 14\n5.1 Deployment #1: PrimEx 2021 . . . . . . . . . . . . . . . . . . . 15\n5.1.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . 15\n5.1.2 Operational Insights . . . . . . . . . . . . . . . . . . . . . 16\n5.2 Deployment #2: Cosmics Test, March 2022 . . . . . . . . . . . . 16\n5.2.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . 17\n5.2.2 Operational Insights . . . . . . . . . . . . . . . . . . . . . 17\n5.3 Deployment #3: CPP 2022 . . . . . . . . . . . . . . . . . . . . . 18\n∗roark@jlab.org - corresponding author\n†davidl@jlab.org - Principal Investigator\n1arXiv:2402.13261v1  [physics.ins-det]  7 Feb 20245.3.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . 18\n5.3.2 Operational Insights . . . . . . . . . . . . . . . . . . . . . 19\n5.4 Deployment #4: PrimEx 2022 . . . . . . . . . . . . . . . . . . . 19\n5.4.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . 20\n5.4.2 Operational Insights . . . . . . . . . . . . . . . . . . . . . 21\n5.5 Deployment #5: GlueX 2023 . . . . . . . . . . . . . . . . . . . . 21\n5.5.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . 22\n5.5.2 Operational Insights . . . . . . . . . . . . . . . . . . . . . 22\n6 Control System and User Interface 22\n7 Physics results with ML-based calibration constants 26\n7.1 Effect of online control on gain calibration . . . . . . . . . . . . . 26\n7.2 Effect of online control on time-to-distance calibration . . . . . . 26\n8 Other detector Systems 28\n8.1 CLAS12 Drift Chambers and ECAL . . . . . . . . . . . . . . . . 28\n8.2 Forward Drift Chamber . . . . . . . . . . . . . . . . . . . . . . . 28\n8.3 Barrel Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . 29\n8.4 Forward Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . 31\n8.4.1 Forward Calorimeter Gain Calibration Using A Convolu-\ntional Neural Network . . . . . . . . . . . . . . . . . . . . 32\n8.4.2 FCAL Simulated Machine Learning . . . . . . . . . . . . 40\n8.4.3 FCAL Inner Ring Radiation Damage . . . . . . . . . . . . 50\n9 Conclusions and Outlook 51\n10 AIEC Publications & Presentations 54\n10.1 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54\n10.2 Key Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . 54\nAcknowledgements 55\nA Control System Workflows 60\nA.1 Initialize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60\nA.2 AutoOFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61\nA.3 calcHV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62\nA.4 FailSafe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64\nA.5 Lookback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65\nA.6 ProcessEPICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67\nA.7 WatchProcessRecommend . . . . . . . . . . . . . . . . . . . . . . 68\n1 Introduction\nThe Thomas Jefferson National Accelerator Facility (JLab) is home to the Con-\ntinuous Electron Beam Accelerator Facility (CEBAF), four experimental halls,\n2the Superconducting Radio Frequency Institute, and numerous other facilities.\nCEBAF delivers an approximately 12 GeV polarized electron beam to the ex-\nperimental halls, each of which contains unique, complex detector systems. The\nprimary focus of this work involves detector systems that comprise the GlueX\nspectrometer Adhikari et al. [2021] located in Hall-D. Additionally, we include\nexploratory data analysis for detector systems that comprise the CLAS12 spec-\ntrometer Burkert et al. [2020] located in Hall-B.\nTraditional calibrations take place in an offline setting and in most cases,\nafter data has been taken. The timeline to complete calibrations can range\nfrom months to years, which presents a significant delay in publishing. Differ-\nences in calibration procedures and the frequency with which calibrations are\ndone presented a significant roadblock for this project regarding Hall-B, as they\ndo not calibrate every run, a distinction from the Hall-D practice. Since we\nwould ideally train a model to predict calibration constants from previous run\nperiods, the data used for training models for the CLAS12 drift chambers was\nunavailable. In contrast, Hall-D stores calibration constants from every run in\na calibration constants database, which provides a wealth of target values with\nwhich to train a model.\nGiven the novelty of the proposed project, it is important to have a team\nof scientists with diverse expertise. The project team is comprised of members\nfrom the Experimental Physics Software and Computing Infrastructure group\n(EPSCI): Torri Jeske, Thomas Britton, Michael Goodrich, and David Lawrence,\nDiana McSpadden from the newly formed Data Science Department, and Naomi\nJarvis from Carnegie Mellon University. Diana McSpadden (data scientist) and\nTorri Jeske (physicist) were hired for this project in May and April of 2021,\nrespectively. An additional data scientist, Nikhil Kalra, was hired in FY21Q4.\n2 Project Objectives\nThis project was submitted to the Department of Energy in response to the\nannouncement “Data, Artificial Intelligence, and Machine Learning at DOE\nScientific User Facilities”. The main objectives for the project are briefly sum-\nmarized below:\n1. Control the HV of the GlueX CDC and simultaneously generate calibra-\ntion constants autonomously in near real-time. Resulting in:\n(a) Stabilization of the gain to within 5% over a two-week period with\nno measurable degradation of the timing resolution.\n(b) Reduction in time required from experts for data monitoring and\ncalibration.\n(c) Reduction in time before physics analysis can begin\n2. Develop a similar control system for CLAS12 drift chambers\n3The initial objectives regarding the GlueX Central Drift Chamber have been\nmet. As stated previously, a similar control system for the CLAS12 drift cham-\nbers was not developed as the training data is not readily available. For this\nreason, milestones were updated throughout the duration of the project to de-\ntermine other suitable detector systems for ML based calibration and control.\n2.1 Control of the GlueX Central Drift Chamber\nWe proposed to control the high voltage (HV) of the drift chamber anode wires,\ndynamically, using an AI system that incorporates information from various\nsources in the experimental Hall during data taking. The model will predict\na set of calibration values and adjust the HV such that the response of the\ndetector is stable in the presence of varying environmental and experimental\nconditions. Such a system would decrease the time that experts are required to\nspend on calibration. Specifically, the drift chamber gain would be stabilized\nwithin 5% over a two week time period with no measurable degradation of the\ntiming resolution. A successful application with the GlueX CDC would be used\nto develop similar systems for other detectors.\n2.2 Reduce time commitment required for data monitor-\ning and calibration\nAI would reduce the time commitment required from the experts, who would\nthen be free to turn their attention to data analysis and our physics program.\nAn AI would also be able to take over some responsibilities from the shift crew in\ncharge of the detectors and the data acquisition. As the AI would be monitoring\nthe environmental conditions, it would be aware of unusual weather events,\nsuch as the rapidly falling pressure that occurs when a storm front arrives,\nand it would be able to alert the crew of the need to stop a run early or take\nthe appropriate corrective steps on its own and continue running, all without\nrequiring expert intervention.\n2.3 Reduce delay between data taking and publications\nFor this project, the objective is to reduce the number of iterations required\nto calibrate the CDC or reduce the number of runs that must be calibrated.\nStabilizing the detector operation reduces or even eliminates the need to cali-\nbrate the data from the detector. This not only improves the quality of the data\nbut reduces the time spent calibrating the data before the physics analysis can\nbegin. This reduces the time between running the experiment and publishing\nthe results.\n2.4 Facilitate automation of other detector systems\nOther detector systems, namely the Forward Drift Chamber (FDC) and the\nForward Calorimeter (FCAL) in Hall-D and the Electromagnetic Calorimeter\n4Figure 1: Hits per track efficiency as a function of anode voltage. The anode\nvoltage of 2125 V was chosen to give the maximum hit efficiency for small drift\nradii.\n(ECAL) in Hall-B were studied as part of this project.\n3 Traditional Detector Calibration and Opera-\ntion\nThe aim of this section is to describe the traditional calibrations in use for the\noperation of the central and forward drift chambers and the FCAL located in\nHall-D. It is important to understand the existing calibration processes as the\ncalibration constants obtained from them are the target values for the models\nthat are developed.\n3.1 GlueX Central Drift Chamber\nThe GlueX Central Drift Chamber is a cylindrical, straw tube wire chamber\nlocated inside the solenoid of the GlueX detector. It contains a 50:50 mixture\nofAr:CO2gas at atmospheric pressure. The operating anode voltage is held\nat 2125 V. This voltage was determined using a HV scan and calculating the\nhits-per-track efficiency as a function of high voltage. The operating voltage\nwas selected based on the location of the start of the plateau. Fig. 1 shows the\nrelationship between the hits-per-track efficiency and high voltage. The CDC’s\npurpose is to detect and track charged particles with momenta as low as 0.25\nGeV/c; additionally, it provides particle identification via dE/dx.\nThe CDC has two main calibrations performed on a run-by-run basis: the\nchamber gain and drift-time to drift-distance. The chamber gain is sensitive to\nthe atmospheric pressure, gas temperature, and current drawn from the high\nvoltage boards. As the atmospheric pressure varies throughout data taking, this\nfluctuation is accounted for via calibration after data taking. An example of\nthis behavior for the GlueX 2020 run period is shown in Fig. 2.\n5Figure 2: Mean atmospheric pressure (gray) and gain correction factor (blue)\nas a function of run number during the GlueX 2020 Run Period.\nThe drift-time to drift-distance calibration requires many more iterations\nof the calibration software. The six calibration constants correspond to the fit\nparameters of the function:\nd(t) =fδ\u0012d0(t)\nf0P+ 1−P\u0013\n(1)\nwhere\nfδ=a√\nt+bt+ct3, (2)\nwhere\nf0=a1√\nt+b1t+ct3, (3)\nand where\na=a1+a2|δ| (4a)\nb=b1+b2|δ| (4b)\nc=c1+c2|δ| (4c)\nwhere δis the magnitude of the deflection for each straw. P is a piece-wise\nfunction of the form\nP=\n\n0 t > T\nT−t\nTt≤T\nwhere T=250 ns. This value is chosen because drift times less than this are not\nsignificantly affected by the distortion of the electric field due to the straw sag.\nA more detailed description of the calculation regarding the straw deformation\ncan be found in Jarvis et al. [2020].\nThe default calibration values are updated with each iteration of the calibra-\ntions. Typically, time-to-distance calibrations are considered finished when the\ndifference between the fit contours from the current iteration and the previous\n6Figure 3: Resolution of fitted track residuals for 4 iterations as a function of\nrun number.\niteration is minimized. There are about 7-10 iterations of the time-to-distance\ncalibration, representing a substantial amount of computing resources and ex-\npert time. In Fig. 3, a sample of the drift-time to drift distance calibration\nconstants is shown for four iterations. Once the drift time to drift distance\ncalibrations are complete, the chamber gain is calibrated a final time.\nSimilarly to the gain, the time-to-distance fit parameters discussed are corre-\nlated with the gas density. Since the atmospheric pressure and gas temperature\nare measured continuously throughout data taking, it is possible to determine\nstarting values at the beginning (or end) of a data taking session using the\ncorrelations between the fit parameters and gas density. We did not use this\napproach at first, as we did not know how adjusting the HV at the beginning\nof each run would affect the time-to-distance calibration.\nThe chamber gain is calibrated by adjusting the fitted dE/dx peak position\natp= 1.5 GeV/c to match that obtained from simulation. The peak position\nobtained from simulation is 2.02 keV/cm.\n4 Development of a CDC control system\nThis section describes the initial data extraction, feature engineering, and model\ndevelopment for the CDC control system.\n4.1 Data analysis & preparation\nData were extracted from two sources: the Experimental Physics Industrial\nControls System (EPICS) Kozubal et al. [1989], and the Hall-D Calibration\n7Constants Database JeffersonLab. Data specific to the CDC were extracted\nfrom the EPICS archive for use as input variables. A program maintained by\nJefferson Lab called myStats is used to extract data from the EPICS archive.\nThis command line tool outputs the minimum, mean, maximum, and standard\ndeviation values for requested process variables. These process variables corre-\nspond to the input features used for training our model. Initially, the electron\nbeam current, atmospheric pressure, gas temperature measured from four of\nthe five (D1-D5, excluding D2) thermocouples located inside the CDC, current\ndrawn by the CDC’s 72 High Voltage Boards (HVB) in nine rings (A-I rings),\nand additional engineered features were evaluated for use in a model to predict\nthe GCF for an individual run. To correlate the input features with the calibra-\ntion constants, the input features were extracted from EPICS during the time\nperiod in which the experimental data was taken.\nThe CCDB contains the calibration values for all sub-detector systems and\nis the source of our target value, the CDC Gain Correction Factor (GCF), for\neach experimental run. Features from EPICS are readily available throughout\ndata-taking, which is important for online control. Some features, e.g. particle\nmomentum, while correlated with gain, were excluded as their calculation re-\nquires computationally expensive track reconstruction, making them unsuitable\nfor real-time control.\nBalancing the distributions of atmospheric pressure between training and\ntest data sets was motivated by the scarcity of extreme values in atmospheric\npressure. A low pressure threshold of ≤99.27 kPa and a high pressure threshold\nof≥101.57 kPa provided similar pressure distributions and adequate diversity\nof gas temperature and HV board current values for low, medium, and high\npressure data within the pressure-balanced training and test data sets.\n4.2 Model development\nUsing data from experiments performed in 2018 and 2020, a deep feed-forward\nneural network was trained with 122 variables. Linear regression (LR), deep\nfeed-forward artificial neural network (ANN), Random Forest (RF) Breiman\n[2001], XGBoost Chen and Guestrin [2016], and Gaussian process regression\n(GPR) Williams and Rasmussen [1995] algorithms were compared to predict\nGCF using different combinations of input features. Comparisons were con-\nducted using both the 2018 and 2020 data and, alternatively, solely the 2020\ndata set.\nThere is a noticeable offset in the relationships between the input features\nand target values between the 2018 and 2020 run periods. The alcohol solu-\ntion used in the CDC gas system bath was changed from 1-propyl alcohol to\nisopropanol Jarvis et al. [2020] after the 2018 run period. The 2018 data was\nset aside until a procedure was developed to make the GCF values compatible.\nResults from the ML algorithm comparisons, neural architecture search (NAS),\nand hyperparameter search are presented in Table 1. Training and testing with\nonly the 2020 dataset resulted in lower Mean Absolute Percent Error (MAPE)\nand significantly lower max percent error, attributed to the different alcohols\n8used in the CDC during the 2018 and 2020 run periods.\nThe Gaussian Process Regression (GPR) model with 11 features exhibited\nthe lowest MAPE. Notably, two of the input features in this model are sourced\nfrom the Pair Spectrometer (PS). If PS is unused or unavailable, we do not want\nthe CDC control system to be impacted; thus, we deliberately made this choice\nto eliminate this model from further investigation. Additionally, features from\nother sub-detectors were found to be correlated with GCF, but were excluded\nfrom further consideration to ensure the CDC control system is self-contained,\nensuring no data from other detectors is used. This is especially important\nif some sub-detectors are not used in future experiments, due to experimental\ndesign, maintenance, or other other unforeseen reasons.\nShapley Gillies et al. [2007–] values were calculated to measure the average\nmarginal contribution of each feature across possible combinations of features.\nShapley analysis identified three variables to predict GCF. Reassuringly, these\nwere consistent with known drift chamber behavior: the mean atmospheric\npressure, the maximum gas temperature, and an engineered feature: the sum\nof the maximum gas temperatures and mean current drawn by the innermost\nhigh voltage boards on the CDC (a CDC-specific proxy for the rate of charged\nparticles passing through the detector). Again, the statistical calculations are\nfrom the JLab myStats data extraction software. They represent the mean and\nmaximum calculation of EPICS variables for the date and time ranges of the\ndata files used for an experimental run’s traditional calibration. The GPR with\nthree features resulted in the lowest max percent error, and utilized three input\nfeatures:\n1. the mean atmospheric pressure,\n2. the max temperature from the D1 thermocouple,\n3. and an engineered feature of the sum of the max temperature from the\nD1 thermocouple and the max HV board current from the CDC’s A HV\nboard.\nThese three input features agree with the CDC expert’s knowledge of the\noperational behavior of the CDC’s gain in response to experimental and envi-\nronmental conditions. After model comparison and analysis, consultation with\nthe CDC expert, and discussions with Hall-D operations staff regarding imple-\nmentation considerations, three input features were finalized:\n1. MEAN PRESSURE: the mean atmospheric pressure (kPa)\n2. MAX TEMP: maximum gas temperature (K)\n3. MEAN HVBI: mean current drawn by the innermost high voltage (HV)\nboards on the CDC ( µA)\nA pair plot illustrating the final input features from multiple Run Periods is\nshown in Fig. 4.\n9Figure 4: Pair plot displaying the relationship of input features - mean at-\nmospheric pressure, maximum gas temperature, and mean high voltage board\ncurrent - to the target value, GCF, for multiple different run periods\n10Model type Run period(s) # features MAPE Max % err Ratio >1% err\nLR 2018 & 2020 11 1.3% 19.1% 97/164\nLR 2020 11 0.7% 2.0% 30/106\nLR 2018 & 2020 5 2.3% 20.3% 96/164\nLR 2020 5 0.7% 2.6% 26/106\nRF 2018 & 2020 82 1.7% 18.5% 83/164\nANN-7 layers 2018 & 2020 122 1.8% 11.4% 74/164\nANN-4 layers 2018 & 2020 122 1.9% 10.8% 84/164\nANN-3 layers 2018 & 2020 122 1.9% 11.9% 90/164\nXGBoost 2018 & 2020 122 1.6% 10.2% 76/164\nXGBoost 2018 & 2020 82 1.4% 11.8% 68/164\nXGBoost 2018 & 2020 71 1.6% 11.2% 71/164\nXGBoost 2018 & 2020 12 1.8% 11.1% 72/164\nGPR 2018 & 2020 25 1.7% 10.9% 80/164\nGPR 2018 & 2020 14 1.5% 9.7% 66/164\nGPR 2018 & 2020 11 1.5% 10.1% 72/164\nGPR 2020 11 0.5% 4.1% 17/106\nGPR 2018 & 2020 5 1.5% 9.1% 70/164\nGPR 2020 5 0.7% 3.6% 28/106\nGPR 2018 & 2020 3 1.8% 12.0% 74/164\nGPR 2020 3 0.7% 1.9% 25/106\nGPR 2018 & 2020 1 3.4% 18.6% 129/164\nGPR 2020 1 1.1% 4% 49/106\nTable 1: Results from algorithm comparison, neural architecture search, and\nhyperparameter optimization. Mean Absolute Percent Error (MAPE), the max-\nimum percent error, and the ratio of observations with greater than 1% error\nare reported. The features included in the comparisons are presented in Table\n2.\nUsing the predicted GCFs, the HV setting is determined from a second-order\npolynomial fit to the HV as a function of the peak amplitude relative to that\nobtained at the nominal operating voltage. For this, historical data from high\nvoltage scans are used. High voltage scans are taken at the start of each new\nRun Period. Example high voltage scans are shown in Fig. 5.\nBy exploiting the empirically derived correlation of peak height and HV we\nuse the GPR-predicted GCF and an “ideal” GCF to obtain the HV setting that\nstabilizes the CDC gain.\n4.2.1 Uncertainty Quantification\nUncertainty quantification has been identified as a priority research opportu-\nnity by the US Department of Energy ene and is an important step in creating\ntrustworthy AI. Furthermore, the novel use of a data-driven approach to con-\ntrol a particle detector requires the GlueX collaboration and the CDC expert to\ntrust that the system will not interpolate or extrapolate unexpectedly. While a\nphysics-driven algorithm understands the underlying dynamics and even causal-\nity, a data-driven approach only understands training data correlations. At a\nminimum, we require a system that at inference can inform users when current\nconditions are well represented within training data and when the system is\nmerely guessing, i.e., uncertainty. This uncertainty should be distance-aware\nand well-calibrated such that when conditions stray further from training data,\n11Model type # features Description\nLR & GPR 11 Mean and max beam current, Mean and max pair spectrometer rate, Mean and max\natmospheric pressure, Max temperature from D1 thermocouple, Max temperature\nfrom D5 thermocouple, Mean A HV board current, Max A HV board current, (Max\nA HV board current - Max I HV board current)\nLR & GPR 5 Mean beam current, Mean pair spectrometer rate, Mean atmospheric pressure, Max\ntemperature from D1 thermocouple, Max A HV board current, Max A HV board\ncurrent\nRF & XGBoost 82 Features with Pearson correlation coefficient >0.2 with GCF. Features do not in-\nclude variables only available post-track reconstruction but do include beam current,\nengineered features, and features from sub-detectors other than the CDC.\nANN & XGBoost 122 All EPICs-only features. Features do not include variables only available post-track\nreconstruction but do include beam current, engineered features, and features from\nsub-detectors other than the CDC.\nXGBoost 122 All EPICs-only features. Features do not include variables only available post-track\nreconstruction but do include beam current, engineered features, and features from\nsub-detectors other than the CDC.\nXGBoost 71 Features with Pearson correlation coefficient >0.3 with GCF. Features do not in-\nclude variables only available post-track reconstruction but do include beam current,\nengineered features, and features from sub-detectors other than the CDC.\nXGBoost 12 Features with Pearson correlation coefficient >0.4 with GCF. Features do not in-\nclude variables only available post-track reconstruction but do include beam current,\nengineered features, and features from sub-detectors other than the CDC.\nGPR 25 Max, mean, min, and standard deviation of the beam current; max, mean, min, and\nstandard deviation of the pair spectrometer rate; max, mean, min, and standard\ndeviation of the atmospheric pressure; max, mean, min, and standard deviation of\nthe temperature of D1 thermocouple; max, mean, min and standard deviation of the\ntemperature of the D5 thermocouple; max, mean, min, and standard deviation of\nthe current of the CDC A HV board, and the engineered feature that is the difference\nof the max of the current of the A CDC HV board and the max of current of the I\nCDC HV board\nGPR 14 Max and mean beam current; max and mean pair spectrometer rate; max and mean\natmospheric pressure; max and mean temperature from the D1 thermocouple; max\nfrom the D5 thermocouple; engineered feature of the difference between the means\nof the temperatures of the D1 and D5 thermocouples; max and mean current from\nthe A CDC HV board, an engineered feature the difference between the max current\nfrom the A and I CDC HV boards, and an engineered feature the sum of the means\nof the A, B, C, D, E, F, G, H, and I CDC HV boards.\nGPR 3 Mean atmospheric pressure, Max temperature from the D1 thermocouple, engi-\nneered feature of the sum of the max temperature from the D1 thermocouple and\nthe max HV board current from the CDC’s A HV board.\nGPR 1 Mean atmospheric pressure\nTable 2: Descriptions of the input features used for models shown in Table 1\n.\n12Figure 5: Results of high voltage scans for 2018, 2020, and 2021 run periods.\nthe system becomes less certain of predictions.\nThe GPR provides a prediction and the associated uncertainty based on the\nmodel’s training data Williams and Rasmussen [1995]. A well-calibrated GPR\nwill provide high uncertainty estimates in input parameter regions where extrap-\nolation is required Williams and Rasmussen [1995], Bishop and Nasrabadi [2006].\nA control system can act based on uncertainty thresholds. While unlikely due to\noperational constraints of the Thomas Jefferson National Accelerator Facility\naccelerator, if environmental or experimental conditions stray into parameter\nregions resulting in a high or low predicted GCF with high uncertainty, it is\ncritical that we do notadjust the high voltage of the CDC to a correspondingly\nuncertain, or unsafe , value. Moreover, the novel use of a data-driven approach\nto control a particle detector requires physicists to trust the data-driven algo-\nrithm; otherwise, it would not be implemented. A detailed description of the\ncontrol system and the policies established is discussed in Section 6.\nWe investigated the GPR’s predictions and standard deviations, i.e., uncer-\ntainties, for possible environmental and experimental conditions using combi-\nnations of atmospheric pressure between 96.0 kPa and 102.9 kPa, HV board\ncurrent between 0.2 µA and 12.0 µA, and gas temperatures between 297.0 K\nand 300.9 K. The input features were incremented in steps of 0.2. We expected\na correlation of atmospheric pressure to GCF with the correlation varying in-\ndependently with both track rate and gas temperature; this behavior is typical\nof drift chambers Sauli [1977].\nAn example of this investigation is shown in Fig. 6, which illustrates the\nrelationship between atmospheric pressure, gas temperature, and the predicted\nGCF and associated standard deviations while keeping the HVB current con-\nstant (0.4 µA). These plots demonstrate the behavior for in-training (red points)\nvs. out-of-training predictions. The predicted GCF reverts to the mean when\nthe standard deviation is over 6% of the mean GCF. The model standard de-\nviation and illustrations, such as Fig. 6, provide useful visuals of the model\n13behavior. Using the uncertainty information, the detector expert developed\ncontrol protocols for interpolation and extrapolation, such as reverting to the\nnominal high voltage setting if the model is highly uncertain. This allows the\nsystem to collect more training data that could be used for later training to\nexpand the certainty manifold.\nFigure 6: Left: the 2D surface plot of predicted GCF v. atmospheric pressure\nand gas temperature with a high voltage board current of 0.4 µA, including\ntraining set observations (in red dots). Right: the 2D surface plot of the pre-\ndiction standard deviation v. atmospheric pressure and gas temperature with a\ncurrent 0.4 µA, including training set observations (in red dots). Regions of gas\ntemperature v. atmospheric pressure without training data (the surrounding\ngreen areas on the left) correspond to the high uncertainty, yellow areas on the\nright.\n5 Deployments\nA CDC control system was developed using a virtuous cycle of research, model\ndevelopment, model deployment, and evaluation of the in-production model.\nThe virtuous development-delivery cycle is used for the collaboration of research\nand industry in the fields of software and hardware development and data science\nBrodie [2019]. Each step in the cycle informs subsequent phases and leads to\ncontinuous innovation to address the defined problem. As Fig. 7 illustrates,\ndiscoveries in later phases may deliver insights that result in revisiting previous\nphases.\nThis section summarizes the performance of the GP model and the control\npolicies established based on the uncertainty quantification and insights learned\nwhile controlling the GlueX CDC during experiments that took place in Hall-D.\n14Figure 7: The Virtuous Data Science Life Cycle.\n5.1 Deployment #1: PrimEx 2021\nFor the first deployment, the shift crew was provided an executable script before\nthe start of each experimental run. The script read the required input features\nfrom EPICS, ran inference, and a HV setting was recommended, rounded to\nthe nearest 5V, in order to stabilize the gain. For empty target runs, the shift\ncrew was instructed to return the HV to 2125 V. In some instances, these\ninstructions were ignored. Ideally, we would have required the shift crew to\nreturn the CDC to 2125 V prior to running the script to determine the next HV\nsetting. Returning the CDC to 2125 V allows us to obtain the correct reading\nfor the high voltage board current at the traditional operating voltage. The\nshift crew was not asked to do this to save time and to reduce the complexity\nof the instructions.\n5.1.1 Performance\nThe input features used by the GP for inference during the PrimEx 2021 run\nperiod are shown in Fig. 8. The corresponding HV recommendations from the\nscript and the set values are shown in Fig. 9. The predicted gain correction\nfactors for the GP-tuned runs and the typical gain correction factor under similar\nconditions are shown in Fig. 10. The idea behind this figure is to show that\nthe calibration constants predicted from the GP are more stable than those we\nwould have gotten had the traditional calibration been performed.\n15Figure 8: The mean atmospheric pressure (kPa, top row), gas temperature (C,\nmiddle row) and current drawn from the CDC HV boards ( µA, bottom row)\nduring the PrimEx 2021 Run Period.\n5.1.2 Operational Insights\nIt is preferable to not further burden the shift crew with specific operating\ninstructions for specific runs. The shift crews change every 6 hours, and infor-\nmation can be lost from one shift to the next even when clear instructions are\npresent.\n5.2 Deployment #2: Cosmics Test, March 2022\nThe second deployment implemented a fully autonomous control system for a\n2-week collection of cosmic ray data McSpadden et al. [2022]. This made use of\ncosmic rays passing through the detector instead of charged particles emerging\nfrom collisions between the photon beam and target. As such, the HV board\ncurrent given to the model was 9.0 µA in accordance with the statistical mode\nof HV board current in our training data. In order to compare the behavior of\nthe CDC operating at a constant HV to a GPR-tuned HV, software was used to\ndivide the operation of the CDC into two sections: one was held at 2130 V, and\nthe other was updated to the HV setting determined using the GPR prediction\nevery five minutes. Thomas Britton and Torri Jeske monitored data collection\n16Figure 9: The HV recommendations and set values obtained from the script\nduring the PrimEx 2021 Run Period.\nFigure 10: The gain correction factor vs Run Number for runs where the HV\nwas tuned according to the GP versus the gain correction factor obtained from\nthe traditional calibration procedure under similar conditions of atmospheric\npressure.\nfrom home primarily to restart the DAQ if necessary and start new runs. Naomi\nJarvis participated in The Great Bear Chase, a 50k skiathlon at 0 F in the Upper\nPeninsula of Michigan. The control of the CDC was truly autonomous. The\nGP was trained using data from the 2020 and 2021 run periods.\n5.2.1 Performance\nFigure 11 presents the comparison of the two sides of the CDC and the stabi-\nlization of gain for the GPR-controlled side of the CDC during a thunderstorm\nwhich produced significant changes in atmospheric pressure.\n5.2.2 Operational Insights\nThe cosmics test occurred during scheduled downtime, resulting in the temper-\nature of the gas in the CDC straying from the normal range represented in the\ntraining data. We opted to artificially fix the temperature input feature to 299\nor 300K until the Forward Drift Chamber was turned on. This brought the\ntemperature of the gas back into the trained region. The effect of this can be\nseen in Fig. 11 around event number 1.0 ×108, where there is a slight jump in\nthe predicted GCF.\n17Figure 11: GCF for the model-controlled (blue) and constant 2130 HV (orange)\nsections of the CDC for a subset of the cosmic ray experiment corresponding to\nthe largest change in atmospheric pressure. The data was taken over approxi-\nmately four days. The change in atmospheric pressure was due to a large storm\nthat passed through Newport News.\n5.3 Deployment #3: CPP 2022\nDuring the Charged Pion Polarizability (CPP) experiment, the script was run\nautomatically at the start of each run using the DAQ “GO” processes to read\nEPICS and set the HV. Before predicting the GCF, the high voltage board\ncurrent was scaled to what it would have been at the default HV value before\nmaking its prediction. During the CPP run period, a portion of the electronics\nfrom the CDC were removed in order to be used by a different detector system.\nThe target type and position were changed, in addition to running at a much\nlower beam current. Particle identification information from the CDC was not\ncritical for this particular experiment.\n5.3.1 Performance\nThe GCFs for the CPP run period, excluding empty target runs, are shown in\nFig. 12. There are only 9 runs that fall outside of the ±5% metric.\nFigure 12: Gain Correction Factors for the CPP run period. The ideal GCF\nvalue for the run period is indicated by the horizontal dashed line. The green\nbox indicates ±5%. of the ideal GCF value.\n18Figure 13: Atmospheric pressure (kPa), mean gas temperature (K), and mean\nhigh voltage board current ( µA) during the CPP run period.\n5.3.2 Operational Insights\nThis was the first run period where an overall control ON/OFF switch was\nadded to the existing CDC HV GUI. This provided shift takers and detector\nexperts the chance to automatically turn off control of the CDC. The shift crew\nwould have to manually reset the HV back to the nominal HV value of 2125 V.\nAdditionally, the configuration file used by the script was updated to include\nadditional parameters relating to:\n1. how EPICS data is gathered via a lookback windows,\n2. a confidence threshold to determine when to trigger the uncertainty quan-\ntification correction,\n3. and whether to control the CDC using the point of closest confidence.\n5.4 Deployment #4: PrimEx 2022\nThe PrimEx run period took place from August 27, 2022 to December 19, 2022.\nThe input features used for the model during the run are shown in Fig. 14.\n19Figure 14: Mean atmospheric pressure (kPa), maximum gas temperature (K),\nand mean high voltage board current ( µA) during the PrimEx-2022 Run Period.\n5.4.1 Performance\nFigure 15 shows the ratio of the gain correction factor and the ideal gain cor-\nrection factor for the PrimEx-2022 run period. Nearly all of the runs with an\nAI-tuned HV fall within our ±5% threshold.\nFigure 15: Gain correction factor as a function of Run Number. The orange\npoints were runs taken at a constant 2125 V and the blue points correspond to\nruns taken with an AI-tuned HV setting. The dashed line indicates the ideal\nGCF. The green box corresponds to ±5% of the ideal GCF value.\n205.4.2 Operational Insights\nOf note, the mean high voltage board current during this run period is much\nsmaller compared to the initial training set. During this deployment, an auto-off\nswitch was enabled when the target state is not full and ready. This sets the\ndefault operating HV to 2125 V. The shift crew was responsible for changing the\nswitch to on once the target was full again. From September 9th to September\n20th, the AI switch was in the off state to investigate the unexpected model\nbehavior when the atmospheric pressure and recommended HV setting were\nmoving in opposite directions, which was contrary to the expectations of our\ndetector expert. On September 20th, the system was re-initialized and control\nwas resumed. On October 12th, the system crashed in a way that it changed\nthe HV during the empty target runs. On October 14th, the system was unable\nto run inference as the model file was missing. This was rectified and control\nwas resumed. On November 18th, the run coordinator was not aware that the\nsystem would automatically revert to 2125 V for empty target runs and turned\nthe switch to off. Control remained off until Thomas Britton noticed the switch\nwas off during a full target run. As the HV was set to 2125 V, this essentially\nreverted the CDC operation to the traditional method. On December 9th, the\nsystem tried to set the HV on the CDC while one of the HV modules was\nnot communicating. This caused a major alarm, as the setpoint (received by\nEPICS) did not match the setpoint readback, indicating the module did not\nreceive the new setpoint. The instructions for the shift crew were modified\nto provide instructions to rectify the communication errors in the event this\nhappens again.\n5.5 Deployment #5: GlueX 2023\nThe most recent deployment took place during the GlueX 2023 run period which\ntook place from January 12th to March 20th, 2023. The control feature was\nmostly off as a result of complications arising from the atmospheric pressure\nsensor. On February 16th, RoboCDC control was switched on. According to\nEPICS, the AI switch for control was enabled, but the HV remained fixed at\n2125V until run number 120739 (about 6 hours of data taking). This was due to\nthe control script running on a computer without EPICS channel access. This\nwas fixed promptly by the Hall-D slow controls expert on February 16th. On\nFebruary 23rd, the high voltage board current was fixed as it was frequently\nunable to be read from the EPICS archive. On February 24th, during runs\n120822-120830, the control was switched off because it was setting the HV after\nthe run had started.\n215.5.1 Performance\nFigure 16: Gain correction factors for the GlueX 2023 run period. A significant\nnumber of runs used the default 2125 V setting (indicated in orange), whereas\nthe runs where the HV was adjusted are shown in orange. The dashed line\nindicates the ideal GCF. The green box corresponds to ±5% of the ideal GCF\nvalue.\n5.5.2 Operational Insights\nThe GlueX 2023 run period did not utilize roboCDC during the start of the run\nperiod due to a faulty pressure sensor. On March 1st, 2023, for runs 120921-\n120931, the control script failed and the HV remained at 2123 overnight. Fig.\n16 shows the gain correction factors as a function of run number. For runs\n120921-120931, it appears the HV setting was suitable for some, but not all,\nruns during this time. As such, it is fortunate we were at a HV that is close\nto both the default HV setting and what would have been recommended based\non similar environmental conditions. Stable running was achieved during runs\n121122 until run 121214. For some runs, the HV had changed after data taking\nhad started. This is to be avoided, as we do not want the raw data files to\ninclude data taken with multiple high voltage settings in the same raw data\nfile. The runs where the high voltage changed after the start of the run were\nidentified such that the corresponding raw data files could be excluded from the\ncalibrations.\n6 Control System and User Interface\nA modular system, named RoboCDC, was developed to utilize the machine\nlearning model and control the CDC. The complete controls harness is com-\nprised of a control system, which issues requests to change CDC HV settings,\na module which handles the loading and running of the model, and a deci-\nsion engine which gathers input data and performs actions depending on the\nresults of model inferences. The entire system utilizes a MySQL database to\nstore model information as well as the input features, inferences, and control\ndecisions throughout data taking and is configurable by modifying parameters\n22in a single configuration file. A diagram of the system is shown in Fig. 17.\nShift crews can enable/disable the system as deemed necessary by experts by\nusing the standard slow controls GUI Kasemir et al. [2007]. A Grafana dash-\nboard displays visualizations of RoboCDC’s inputs, actions, and performance\nallowing for real-time monitoring of the RoboCDC system by shift crews and\nexperts.\nIn certain situations, such as high beam current tests, the system is disabled\nand begins “trusting the humans”. While in the disabled state, the system will\nnot attempt to change HV settings nor take any controlling actions. The system\nwill, however, continue to take in data and produce predictions. These “non-\ncontrolling” predictions are still recorded in the database for analysis of potential\nmodel behavior. Furthermore, out-of-distribution input parameter sets can be\nidentified in this manner and potentially used to train future models.\nThe configuration file contains many parameters which control RoboCDC’s\nbehavior while running. These variables are able to be edited without impacting\ndata taking. The configuration parameters are as follows:\n•model name - The full path to the saved model which will be used.\n•polltime - The time between each running of the model in minutes.\n•recscale - The number of poll time intervals before the system recom-\nmends a HV change. This number can be less than 0. In the case of\nnegative values of rec scale RoboCDC will run its inference-control loop\nexactly N times where N=−1×recscale with a poll time wait between\neach pass through the inference-control loop. For discrete running, like\nthe typical production configuration of RoboCDC, rec scale was set to -1;\nmaking the system recommend HV settings only once (at run start). In\nthis specific configuration the poll time variable serves little to no purpose.\n•control window - A two element array representing the lower and upper\nbound on the HV voltage recommendation. If RoboCDC recommends set-\nting the HV to a value below the lower bound or above the upper bound it\ninstead recommends setting the HV to the value of the nearest bound. For\nthe CDC the detector expert, Naomi Jarvis, determined [2105.0, 2145.0]\nto be the safe operational envelope.\n•ideal gcf- The ideal gain correction factor. This value is used to calculate\nthe recommended HV setting for the CDC based on the fitted curve in\nFigure 5 and is set during initialization. Alternatively, this value can be\nset manually through this configuration parameter.\n•default hvbi - The high voltage board current that was found when ini-\ntialization was performed. Alternatively, this value can be manually set\nto a negative value. When this value is negative it acts as an override,\ninternally setting the scaled high voltage board current to the fixed value\ngiven by the magnitude of the default hvbi variable. When this value is\n23greater than 0 then the variable is not used and exists as a record of the\nconditions present during initialization only.\n•require human - Either 0 for automatic action or 1 to trigger a protocol\nfor including human-in-the-loop action. Presently, this variable is only\nconnected to a hook in the code and does not impact operation. This was\nleft included in the case where it was deemed necessary to require such\nhuman action. In the course of this work it was unanimously agreed that\nthe involvement of humans in the action/control loop would place undue\nburden on shift crews; The system had enough safe guards in place to\nsafely operate autonomously.\n•control mask - This string dictates what parts of the CDC will actually\nbe under the control of RoboCDC. The variable is passed to the actual\ncontrol script, utilscdc tools.py, alongside the recommended HV and was\nutilized to functionally divide the CDC in half in software for testing.\nDuring normal running it is set to ”ALL”.\n•default temp - The temperature in Kelvin that was found when initial-\nization was performed. Alternatively, this value can be manually set to\na negative value. When this value is negative it acts as an override, in-\nternally setting the temperature used by the model to the magnitude of\nthe default temp variable. When this value is greater than 0 then the\nvariable is not used and exists as a record of the conditions present during\ninitialization only.\n•default pressure - The pressure in kiloPascals that was found when ini-\ntialization was performed. Alternatively, this value can be manually set\nto a negative value. When this value is negative it acts as an override,\ninternally setting the pressure used by the model to the magnitude of the\ndefault pressure variable. When this value is greater than 0 then the vari-\nable is not used and exists as a record of the conditions present during\ninitialization only.\n•baseline V- The nominal HV setting for all HV boards in the CDC.\nThe control system is designed to set, or attempt to set, the CDC to the\nbaseline voltage in the event of any part of the control module throws an\nerror. It is also used in conjunction with ”fail safe” protocols and with\nautoOFF.\n•failsafetimeout - The control system relies on EPICS to obtain the\nmodel’s input parameters. In the event that EPICS fails to return valid in-\nput data for some number of minutes equal to or greater than fail safetimeout\nthe ”FailSafe” protocol is run. This protocol involves setting the CDC\nboards described the control mask to the setting defined by baseline V\nand logging the incident.\n24•confidence threshold - The value of uncertainty in percent of ideal gcf\nthat when crossed above triggers an uncertainty correction when the UQ-\ncorrection variable is enabled.\n•UQcorrection - Either 0 to disable uncertainty correction or 1 to en-\nable Uncertainty correction. This correction involves finding the near-\nest recommended HV, by Euclidean distance, on the surface created by\nthe input parameters which yield uncertainty values equal to the confi-\ndence threshold.\n•autoOFF - This value is either 0 to disable automatically turning off the\ncontrol system or 1 to enable the autoOFF procedure. This procedure\nis run when the system detects the GlueX target is not in the full and\nready state. This procedure was also run when the system desires to set\nthe HV above or below the control window. This was done in efforts to\ngather more training data and extend the “in distribution” set of input\nparameters. When the auto-off procedure is run this action is also logged.\n•lookback dict - Because the model’s input features are engineered (e.g.\nthe mean) the question of look-back time comes into play. This variable\ndefines the differing scales, in seconds, each variable should “look back”.\nWhile able to take in an arbitrary dictionary of look-back times only the\ncurrent is used presently. This was due to the observations of overshoots in\nhigh voltage board currents; correlated with overshoots in beam current,\nespecially after beam trips.\n•max lookback scale - This variable defines how many blocks of lookback\nshould be looked back as a multiple of the length defined in lookback dict.\nFor example, if the value of a variable “I” is 60 in lookback dict and the\nmax lookback scale is 2 then when, say, mean is computed it is computed\nover the last 120 seconds.\nOnce configured appropriately, the control system is initialized when the\nshift crew starts a run via GlueX’s standard data acquisition (DAQ) software.\nAssuming the AI control system is turned on it gathers up the necessary input\nvariables from EPICS and feeds them to the model. The model returns a re-\nsult dictionary, which is appropriately logged in the database, and actions are\ntaken to control the CDC HV in accordance with the model’s output and the\nsystem’s configuration. Once input data is fed into the model a prediction is\nobtained within milliseconds. To obtain the target HV setting a ratio of the\nmodel’s predicted gain correction factor to the ideal gain correction is fed into a\npolynomial fit of HV to the ratio of peak ADC amplitude to the peak amplitude\nat the nominal 2125 Volts (essentially reducing to the ratio of gain correction\nfactors). This polynomial was derived from data taken during HV scans of the\nCDC. The HV of each of the CDC’s 72 boards is adjusted before data-taking\nbegins. The full control workflow of all functions that make up the entirety of\nthe control system can be found in Appendix A.\n25Figure 17: Diagram outlining the main processes for HV control.\n7 Physics results with ML-based calibration con-\nstants\n7.1 Effect of online control on gain calibration\nBy adjusting the HV of the CDC during the run, the gain is stabilized and\nthe correlations between the atmospheric pressure, gas temperature, and HV\nboard current and the calibration values has been removed. This is shown in\nFig. 18. In the top plot of the first column, the atmospheric pressure is no\nlonger correlated with the mean dE/dx position (and thus the gain), except\nfor the runs where the AI system was on but unable to adjust the HV to the\nappropriate value.\n7.2 Effect of online control on time-to-distance calibration\nUp until the GlueX 2023 run period, it was assumed that adjusting the HV of the\ndrift chamber would not affect the time-to-distance calibration in a significant\nway. For the GlueX 2023 run period, we attempt to analyze the effects of\nonline control on the time to distance calibration. In Fig. 19, the width of the\nresiduals obtained before the traditional time to distance calibration are shown\nas a function of run number for runs at 2125 V and those with a voltage set\naccording to ML.\nIn comparison, the residual widths obtained from using starting values from\nthe fits to the gas density are shown in Fig. 20. Utilizing the fits of the calibra-\ntion constants to gas density from the 2020 run period to obtain starting values\nfor the GlueX 2023 run period results in much less variation of the residual\nwidth over the entire run period, regardless of whether the data was taken with\nthe CDC HV set to 2125 V.\nThe time to distance function parameters are correlated with the density of\nthe gas in the CDC. As such, it is possible to obtain starting values for the time\nto distance fit function at the start of each run using the measured atmospheric\npressure and temperature from EPICS. An example of the relationship between\n26Figure 18: Pair plot of the mean dE/dx, atmospheric pressure, gas temperature,\ngas density, and HV setting during the GlueX 2023 run period.\nthe fit parameters and the gas density for the 2020 Run Period is shown in Fig.\n21. Using this, we can generate functions that produce a set of starting values\nfor each of the fit parameters just by measuring the gas density. This could\npotentially save a lot of computing resources as there would be less iterations\nto perform to achieve the desired quality.\nIf the 5% metric on the dE/dx values is obtained from calibration using the\nML-generated calibration values, then the traditional calibration procedure can\nbe replaced. If a future experiment requires more precise calibrations then more\niterations of the software can be used until the designated metric is obtained.\nFigure 19: The residual widths obtained before calibration for runs taken at\n2125 V (red) and a HV setting determined by the ML system (blue).\n27Figure 20: Residual widths obtained using calibration constants obtained from\nthe fits of the calibration constants to the gas density. The runs at 2125 V are\nshown in red, other voltage settings correspond to the blue data points.\n8 Other detector Systems\n8.1 CLAS12 Drift Chambers and ECAL\nThe original proposal highlighted the CLAS12 Drift Chambers (DC) as a natural\napplication for the technology that would first be developed for the GlueX CDC.\nWhen control of the CDC HV was successfully implemented and attention was\nturned towards the CLAS12 DC we found that there was more than an order of\nmagnitude fewer calibration entries for CLAS12 DC than for the GlueX CDC.\nThis was unexpected by the detector experts in both collaborations. Upon\nlooking further, it was seen that this was not unique to just the Drift Chambers,\nbut was a common occurrence across subdetectors (see Fig. 22). This led us to\nconclude that either cultural differences between the collaborations were notable\nwhen it came to determining the frequency of calibration, or the subdetectors\nthemselves were designed differently. Regardless of the reason, the small number\nof entries for CLAS12 made it impractical to train an AI/ML model.\n8.2 Forward Drift Chamber\nThe Forward Drift Chamber (FDC) was not previously calibrated to control for\ndrifts in the chamber gain. ‘There is some initial motivation to calibrate the\nchamber gain in order to improve the PID figure of merit. We did not pursue\ncontrolling the FDC, as we do not have the relevant HV scan data in addition\nto the detector being more complicated in terms of HV settings than the CDC.\nThe first gain calibrations were performed using a similar procedure to that\nused for the CDC. Namely, we obtain the position of the first ionizing peak\natp= 1.5 GeV/c using data from Run Number 11621. In Fig. 23, we show\ndE/dx vs momentum for positive and negative charged tracks followed by the\none-dimensional projections of dE/dx at p= 0.5 GeV/c and p= 1.5 GeV/c.\nThe ideal peak position is obtained from the mean of a Gaussian fit to dE/dx\natp= 1.5 GeV/c. This “ideal” peak position is equal to 1.9335 keV/cm. The\ngain calibration value is then obtained from\n28Figure 21: Pair plot of the time to distance fit parameters and the gas density\nfor the 2020 Run Period.\ngn=mean ideal\nmean actual×gn−1 (5)\nwhere n indicates the iteration number and mean actual corresponds to the mean\nvalue obtained from the fit for the current iteration.\nSimilarly to the CDC gain calibration, the gain calibration values obtained\nfrom this procedure are strongly correlated with the atmospheric pressure.\nThus, it is possible to use a similar system to the CDC for the FDC gain\ncalibrations. We would not initially include a control aspect, but would have\ncalibration constants available at the start of each run.\n8.3 Barrel Calorimeter\nThe Barrel Calorimeter (BCAL) is used to detect photon showers with energies\nof at least 0.05 GeV, within 11-216 degrees in θ, and 0-360 in ϕ. The pedestal\n29Figure 22: Snapshot of the number of entries in calibration DB for CLAS12\nand GlueX. This shows GlueX has roughly 30 times the number of entries as\nCLAS12 across a number of detectors.\nvalues for the BCAL fluctuate with temperature. The readout of the temper-\nature sensors is not fine-grained enough to be useful in identifying correlations\nbetween the pedestal and temperature values. This effort was not pursued fur-\nther as it would have been costly (both in terms of time and equipment) to\ninstall a new temperature sensor with a more fine-grained readout.\n0\n10002000 3000 4000 5000 time since start of run (min)\n400401402403404405406407408409410pedestal mean29.5\n3030.53131.53232.5C)\noCrate temp (BCAL Module=1, Layer=1, Sector=1, End=Up0\n10002000 3000 4000 5000 time since start of run (min)\n400401402403404405406407408409410pedestal mean29.5\n3030.53131.53232.5C)\noCrate temp (BCAL Module=1, Layer=1, Sector=1, End=Up0\n10002000 3000 4000 5000 time since start of run (min)\n400401402403404405406407408409410pedestal mean29.5\n3030.53131.53232.5C)\noCrate temp (BCAL Module=1, Layer=1, Sector=1, End=Up\nFigure 25: BCAL pedestal vs. time for a single channel. The measured pedestal\nvalue(blue) is plotted vs. the left axis and the measured temperature(red) is\nplotted vs. the right axis.\n30Figure 23: Top row: FDC dE/dx for positive (left) and negative (right) tracks vs\nmomentum. Bottom row: The one-dimensional projections of dE/dx at p=0.5\nGeV/c (left) and p=1.5 GeV/c (right).\n8.4 Forward Calorimeter\nHere, we describe work pertaining to Hall-D’s FCAL. First, we discuss a cap-\nstone project conducted with the University of Virginia, which explored a Con-\nvolutional Neural Network (CNN) to predict gain calibration constants for the\nFCAL’s 2800 lead glass detectors. The finding that the gain calibration con-\nstants were only weakly correlated with the PMT measurements of the LED\nmonitoring system’s pulses led to a simulation study. The study found that a\nLong Short-Term Memory neural network is capable of learning from the evolv-\ning time series of a simulated LED pulse and that the gain calibrations can be\npredicted with a mean absolute error of <1%, even with an injection of up to\n7% noise. Finally, we examine the correlation of the change of the gain cali-\nbration constants and luminosity in experimental data to better understand the\n31Figure 24: Top: Gain Correction Factor as a function of Run Number for the\nCPP Run Period. Bottom: Mean atmospheric pressure as a function of run\nnumber.\neffect of radiation damage caused by proximity to the beamline.\n8.4.1 Forward Calorimeter Gain Calibration Using A Convolutional\nNeural Network\nIn partnership with the University of Virginia School of Data Science, a master’s\ncapstone project explored whether the existing LED monitoring system data\ncould be used to train a CNN to produce the gain calibrations for each FCAL\nblock for an experimental run. The self-selected team of four students included\nCullan Bedwell, Abhijeet Chawhan, Julie Crowe, and Diana McSpadden (JLab).\nThe project was conducted over one term; weekly meetings were held with the\nAIEC investigators.\nThe JLab GlueX FCAL is a 2.4 m circular array of 2800 4 cm ×4 cm×45 cm\nlead glass blocks; each block is attached to a photomultiplier tube (PMT) Ad-\nhikari et al. [2021]. Due to variations in the lead glass blocks and the PMTs, their\nresponses to photons are inconsistent. This behavior is expected Leo [2012],\nJones et al. [2006], Adhikari et al. [2021], Hanagaki et al. [2022]. An offline gain\ncalibration is necessary to correct for variations in response to photons.\nTraditionally, a gain calibration constant is applied offline to raw experi-\nmental data to correct for the varying PMT measurements over time and the\nvariations between PMTs. Thus, for each experimental run, a gain calibra-\ntion for each of the FCAL’s 2800 PMTs must be calculated. A reference PMT\nand corresponding reference light source, such as a predictable radioactive light\nsource, are unavailable for the GlueX experiment. When available, reference\nPMTs and radioactive light sources require non-negligible maintenance. Thus,\nthe FCAL calibration procedure must rely on other known quantities or mea-\nsures. For GlueX FCAL calibration, the theoretical π0mass is used.\n32Figure 26: The FCAL, under construction, as viewed looking downstream. Not\npictured are the plexiglass plates, nor the components of the LED monitoring\nsystem.\nThe FCAL gain calibration constant, a unitless scaler, minimizes the differ-\nence in the theoretical π0mass and the measured π0mass summed over run\nevents Jaegl´ e [2009]. To calculate the gain calibrations, an iterative method, the\n“Gain Balancing Procedure” Moriya et al. [2013], has been employed by fitting\nraw experimental FCAL data, applying gain calibrations until the measured\nmass of π0is within a tolerated threshold of the theoretical mass, as shown in\nEq. 6. This method is computationally expensive and time-intensive for scien-\ntists. FCAL experts must also contend with the lack of π0particles in the outer\nblocks of the FCAL due to a lack of π0’s at larger angles from the beam line.\nGCF new=GCF old·\u0012mmeas\nmtheory\u00132\n, (6)\nwhere mmeas is the mass of the neutral pion ( π0) obtained from the experi-\nmental data, and where mtheory = 134 .9786±0.0005 MeV Zyla et al. [2020].\nThe FCAL’s blocks are divided into four quadrants. Plexiglass sheets are\ncoupled to each quadrant. As shown in Fig. 27, ten LED boards are arranged in\ntwo sets of five along the two external edges of each square plexiglass sheet. The\nLED monitoring system utilizes five distinct color-strength flashes to monitor\nthe 2800 PMTs:\n1. the small violet pulse: 390 nm at 12 V,\n2. the large violet pulse: 390 nm at 22 V,\n333. the green pulse: 574 nm at 29 V,\n4. the small blue pulse: 470 nm at 10 V,\n5. and the large blue pulse: 470 nm at 15 V.\nEach LED color and strength pulses for a 10-minute duration. These pulses\nare monitored to ensure that PMT readings are nominal. Additional details\nof FCAL monitoring system, including the verification, spatial variability, and\nmeasures taken to minimize the spatial variability, are available from Anas-\nsontzis et al. [2014]. The system was not designed to be used for calibration;\nhowever, the measurements from the LED monitoring system are stored and\nreadily available for offline or online use, and it is reasonable to investigate if\nthe LED readings could be used for calibration.\nFigure 27: The FCAL monitoring system for one of the quadrants. The four\nLED colors chosen were violet, blue, and green. (Image taken from original\npublication Anassontzis et al. [2014]).\nMeasurements from these pulses include the signal peak fitted with a Gaus-\nsian, the peak’s width, and the statistical mean. A visualization of the ADC\npeak fits for the large blue and green pulses is shown in Fig. 28. Additionally,\ntheχ2values of each of these fits are recorded for each run and each block/PMT.\nIn principle, these LED signals can be correlated to the individual detector gains\nwhen used as inputs to an ML algorithm.\nFCAL blocks are each placed on a grid within a quadrant; additionally, the\nFCAL blocks also have a radial distance from the center beamline and each can\nbe assigned an FCAL “ring number” starting from the center of the FCAL. As\n“ring 1” of the FCAL contains no blocks, “ring 2” is the center-most ring. A\nblock’s ring number is important in two ways related to gain calibration. First,\nit affects the utility of the LED pulses, i.e., the input data. With respect to\nthe monitoring system, the large blue and violet pulses often oversaturate the\n34Figure 28: The peak amplitude recorded during a PrimEx run for the large\nblue, green, and small violet pulses. The x, and y axis are the locations of\nthe FCAL blocks from beamline. Within the boundaries of the FCAL, blocks\nwith zero values were either 1) too near large pulses and were oversaturated, 2)\ntoo distant from small pulses to record, or 3) not responding, possibly due to\ndegrading PMT power supply bases. These plots represent input data for three\nof the five pulse types for a single run.\nblocks in the outer ring of the FCAL. This can be seen in an example of a run’s\nADC channel number for the large blue pulses shown in Fig. 28. Several of the\nouter ring blocks show no value, indicating that the peak amplitude could not\nbe fit due to oversaturation. Conversely, the small blue and violet pulses are\nused to monitor the middle and outer rings, and the pulses often do not reach\nthe inner rings, again leading to no available input data.\nSecond, a block’s radial distance from the beam line leads to changes in the\ngain calibration values throughout a run period. Continued exposure to the\nbeam causes the FCAL’s lead glass to brown, resulting in an increase in the\ngain calibration factor. The correlation between integrated beam current and\nan FCAL block’s gain calibration value holds true only for blocks exposed to\nenough beam current to brown the lead glass. The effect can be seen in Figs.\n29 and 30.\nFigure 29: For the PrimEx runs included the dataset, the average gain for the\nblocks in each FCAL ring is shown sequentially throughout the run period. The\ngain increases throughout the run period for the inner rings and, on average,\nremains stable for the middle and outer rings of the FCAL.\nAnalytically combining the color and strength pulses allows a more complete\npicture of the FCAL, but an ML model needs a mechanism to avoid learning er-\nroneous information from over and undersaturated blocks. Additionally, during\n35(a) First PrimEx run\n (b) Last PrimEx run\nFigure 30: The figures illustrate the average gain calibration factor (unitless)\nfor the FCAL rings for the first PrimEx run, and for the last PrimEx run in the\ncapstone project data set. The average gain calibration factor is 1.0 for the first\nrun, which is treated as a reference run. In the inner rings of the FCAL, the\ngain increases throughout a run period due to the browning of the lead glass\nblocks caused by exposure to the beam.\nthe 2019 PrimEx run period, the PMT’s power supply bases were not frequently\nmonitored, resulting in a progressive degradation in performance. Experts ob-\nserved that the bases failed intermittently, with run failures becoming more\nfrequent as the bases degraded.\nA method was explored to mask blocks that were under or oversaturated or\nhad poorly performing power supplies. To generate the masks for each pulse\ntype, the range of the fitted peak amplitude position was examined from the\ntotal range of peak values. The selected ranges for peak amplitudes for the\npulses were as follows: 100 - 4000 ADC for the large violet and large blue pulses\nand 100 - 2500 ADC for the small violet, small blue, and green pulses. After\nthe fitted peak amplitude cuts, blocks with a negative χ2fit value were masked.\nThe results of the mask are shown in Fig. 31. Notably, 2287 of the 2800 FCAL\nblocks have input from all five types; however, using the mask results in 52\nblocks with no input available for an ML-based gain calibration.\nDue to the spatial nature of the FCAL a CNN approach was explored. 3-D\nconvolutions were implemented. Intuitively, this allows for the mixing of the\ninformation from the colored pulses to predict the gain calibration constants.\nThe input for the CNN contained a matrix of the masked LED data, which was\nrepresented as values per block, per run, and per channel.\nL(N×T) denotes the matrix containing LED pulse data collected from the\n2800 LED blocks, where the rows represent measurements from Nruns, and the\ncolumns are measured for the TLED blocks. The input of a 3-D convolution\nlayer is a 3-D tensor, denoted as X(N, T, c ), where N,T, and crepresent its\nnumbers rows, columns, and channels, respectively. Therefore, we convert the\nmeasurement matrix L(N×T) to a tensor L(N, T, 1) for each channel, so that\nit can be fed to a convolution layer. A convolutional layer consists of several\n36Figure 31: With the mask, 2287 blocks have information from all five pulse\ntypes, 299 blocks from four pulse types, 108 from three pulse types, 38 from two\npulse types, 16 from one pulse type, and 52 blocks with no input data due to\nmask.\nconvolutional kernels. Each kernel is a filter with the size of ( ks, kt, c ), which\nmaps a patch of its input tensor to a scalar, given by\nxout=σ(W(ks, kt, kd )◦X(ks, kt, kd ) +b) (7)\nwhere W(ks, kt, kd ) and bare the parameters of the convolution kernel and\nthe bias term, respectively; σrepresents a nonlinear activation function. A\nconvolutional kernel slides over the run (row) and block (column) dimension of\nthe input tensor with a kddepth and maps to a 1-D output tensor.\nA neural network consisting of 3-D Convolutional layers in conjunction with\n2-D Convolutional layers and finally passing the flattened input through Dense\nlayers was implemented for this experiment. A NAS was conducted to compare\nmodel architecture and hyperparameters. Additionally, the best-performing\nmodel architecture was compared with and without masked input. With only\n375 PrimEx runs available, five-fold cross-fold validation was utilized to evaluate\nthe model performance. Mean squared error (MSE) was used as the training\nloss function. Results are presented in Table 3.\nCalibrations were evaluated based on their utility to calibrate the FCAL’s\nmeasurement of π0mass. By utilizing block masking, we obtained average re-\nsults within 2.37% of the target gain constants, and our best fold’s test set\nwas within 1.86%. However, scientists are interested in whether predicted cal-\nibrations result in physics data worthy of detailed analysis. Thus, the ML\ncalibrations are compared to traditional calibrations using the same criteria.\nMasked blocks were given a predicted gain calibration of 1.0, as is the practice\nof the FCAL expert for blocks that are unable to be calibrated in the traditional\nmethod due to too few detected π0particles.\nFor the best fold’s predictions, we compare the measured π0mass for our\ncalibrations to measurements obtained with traditional calibrations. These com-\nparisons are shown in Fig. 33. The traditional calibrations are more accurate.\n37Table 3: 5-fold performance comparison. Reported metrics are averaged over\nthe FCAL blocks and over a 10% holdout/test set of PrimEx runs.\nDataset fold idx residual MAPE MSE\nmasked 1 0 .025 2 .191 0 .004\nunmasked 1 0 .866 81 .194 25 .103\nmasked 2 0 .025 2 .221 0 .004\nunmasked 2 0 .057 4 .991 0 .015\nmasked 3 0 .031 2 .663 0 .004\nunmasked 3 0 .081 7 .021 0 .033\nmasked 4 0 .033 2 .912 0 .007\nunmasked 4 0 .228 20 .764 0 .747\nmasked 5 0 .021 1 .861 0 .002\nunmasked 5 0 .060 5 .272 0 .016\nmasked avg 0 .027 2 .370 0 .004\nunmasked avg 0 .258 23 .848 5 .183\nFigure 32: Displays the comparison of measured π0mass using the gain\ncalibrations predicted by the best performing CNN (blue) and using stan-\ndard/traditional calibrations (orange). The solid line indicates the theoretical\nπ0mass. The dashed lines indicate the 1% thresholds. The traditional calibra-\ntions are generally more accurate.\nContinued analysis of the LED pulse amplitudes and the traditional gain\ncalibrations makes it clear that other than the radiation-damaged blocks at the\ncenter of the FCAL, there is only weak-to-moderate correlation between the\nmeasured LED pulse amplitudes and an FCAL block’s gain calibration value.\nThis lack of strong correlation is illustrated in Fig. 33 illustrating the Pearson\ncorrelation coefficients for the PrimEx 2019 run period. Moreover, a contempo-\nraneous and independent analysis of correlations between the LED monitoring\nsystem and FCAL gains by a Hall-D scientist did not find a clear relationship.\nThis motivated the following simulation study described in Section 8.4.2.\n38(a) Large violet and small violet pulses\n(b) Large blue, small blue, and green pulses\nFigure 33: The Pearson correlation coefficient (PCC) of the FCAL block gain\ncalibration values and the LED pulse amplitudes. An anti-correlation (values <\n0) is expected; however, the scale of the anti-correlation is smaller than experts\nexpect.\n398.4.2 FCAL Simulated Machine Learning\nThe lack of a strong relationship between LED pulse amplitudes and the gain\ncalibrations constants motivated a simulation study. With simulated data, the\nability to extract the gains from the amplitudes can be verified. This study\ndemonstrates that in the presence of a correlation between the measured LED\npulse amplitude and gain correction factor, a machine-learning approach could\nultimately lead to ML-predicted gain calibration constants. This in turn could\nbe translated into an autonomous control system for the GlueX FCAL and other\ncalorimeter systems.\nFor our simulation, the LED pulse amplitude, A, is modeled by the following\nEq.:\nAb,t=g−1\nb,t·ωb·αt·Rb,t+ϵb,t, (8)\nwhere:\nb is the block index\nt is the time index\nAb,tis the amplitude read for block bat time t\ngb,tis the PMT gain for block bat time t\nωbis the optical coupling of block b\nαtis the LED pulser amplitude at time t\nRb,tis the radiation damage of block bat time t\nϵb,tis a stochastic measurement error for block bat time t\nFor each block, a superposition of two sine functions of arbitrary frequency\nand phase was selected for the simulation of gb,t,\ngb,t= 1 + Cb(sin(ηb,1t+ϕb,1) + sin( ηb,2t+ϕb,2)) (9)\nwhere the block constant Cbscales the sine wave superposition independently\nfor each block and the frequencies( ηb,1, ηb,2) and phases( ϕb,1, ϕb,2) values are\nindependently drawn from a normalized random distribution for each block.\nThe gain correction factor values obtained from Eq. 9 fluctuate around one to\nreflect the experimental data. The optical coupling constant, ωb, is sampled for\neach block and trial from a common normal distribution,\nωb=N(µ, σ2) (10)\nwith µ= 1 and σ= 0.1. For each block and trial, ωwas held constant over t.\nThe amplitude of the LED pulser, αt, is algorithmically modeled as piecewise\nline segments that reverse direction randomly with a probability of 1% and also\nupon reaching the boundaries of a ±5% envelope centered on a value of 1.0.\nRb,tis the radiation damage for each block. For the purposes of this study,\nthe radiation factor, Rb,t, is set equal to one. An additional noise parameter, ϵ,\nis modeled as Zero Mean Gaussian White Noise, and is possibly block dependent\nsuch that when\nϵ←N(0, x) (11)\n40where x is the noise level expressed as a percentage.\nFig. 34 shows an example of αt,gt,andAtthat are used to calculate the\nLED pulse amplitude at each time step without stochastic noise.\nFigure 34: For a single simulated FCAL block, in orange(top), α, the amplitude\nof the LED pulser; in blue(middle), g, the gain correction factor for a single\nblock; and, in green(bottom), A, the ADC readout amplitude for the block.\nThe ADC readout amplitude, A, (example shown at bottom of Fig. 34) is\nthe only input data available for training ML models to predict the PMT gain\ng, in order to be consistent with the approach described in Section 8.4.\nThe first method was to extract the gain calibration values directly from\nEq. 8. The gain calibration values extracted are then given as input features to\na Generalized Linear Model (GLM), Three Layer Perceptron (TLP), Random\nForest (RF), and Long Short Term Memory (LSTM). The approach can be\nsummarized in the steps below:\n•Select some block band use gb,tas the target to model.\n41•For each repeated trial at a selected noise level ( ϵ) use Eq.8 to calculate\nAfor all blocks.\n–Select the two most positively and negatively correlated Ablocks\nwith the target gb, for a total of four predictors, using the training\nportion of the data.\n–Train a selected model type on these Ablocks and report, for each\ntrial, both the % Mean Absolute Error (MAE) and Max Absolute\nError (AE) across the time index using the validation data.\n•Report Mean % MAE and Mean % Max AE over all trials and the corre-\nsponding standard error of the mean values.\nA trial refers to a distinct trained solution obtained and evaluated. All\nmodel results are based on 100 trials for each noise level except for the LSTM\nmodels, which used 25 trials for each noise level, due to differences in the time\nrequired for training. Training for each model used an 80%/20% split for train-\ning/validation without shuffling to preserve the time order of the data. LSTM\nmodels used a history window depth of 50 time unites.\nModel performance for an individual trial was evaluated using the MAE and\nmaximum AE across the time index for the validation portion of each trial. To-\ntal model performance was evaluated using mean MAE and mean Maximum AE\nand associated Standard Errors of the Means over all trial evaluation scores. Ta-\nbles (4) and (5) summarize these results for each noise level and model type. As\nexpected, as the noise level increases, the performance of each model degrades.\nTable 4: Mean % MAE Results from Modeling for PMT Gain (using A).\nModel GLM TLP RF L(50)\nNoise Mean StdErr Mean StdErr Mean StdErr Mean StdErr\n0% 0.66 0.00 0.65 0.00 1.2 0.0027 1.2 0.026\n1% 1.1 0.0025 1.1 0.0032 1.3 0.0036 1.2 0.029\n2% 1.5 0.0054 1.5 0.0068 1.6 0.010 1.2 0.027\n3% 1.8 0.0089 1.8 0.010 1.8 0.014 1.3 0.036\n4% 2.0 0.010 2.0 0.010 2.0 0.014 1.3 0.056\n5% 2.1 0.010 2.1 0.010 2.2 0.011 1.3 0.059\n6% 2.2 0.010 2.3 0.010 2.2 0.010 1.4 0.055\n7% 2.3 0.010 2.3 0.0082 2.3 0.010 1.5 0.063\n8% 2.3 0.010 2.3 0.0094 2.3 0.0079 1.5 0.040\n9% 2.2 0.0069 2.3 0.0067 2.3 0.0067 1.6 0.059\n10% 2.3 0.0061 2.3 0.0056 2.3 0.0059 1.5 0.074\n42Table 5: Mean % Max AE Results from Modeling for PMT Gain (using A).\nModel GLM TLP RF L(50)\nNoise Mean StdErr Mean StdErr Mean StdErr Mean StdErr\n0% 1.3 0.00 1.3 0.00 2.8 0.0094 2.4 0.089\n1% 3.6 0.028 3.8 0.043 3.8 0.028 2.8 0.072\n2% 5.1 0.043 5.1 0.040 5.2 0.048 3.0 0.056\n3% 6.1 0.053 5.7 0.064 5.9 0.050 3.3 0.063\n4% 6.6 0.052 6.1 0.070 6.2 0.040 3.5 0.12\n5% 6.9 0.052 6.3 0.076 6.3 0.036 3.4 0.12\n6% 6.8 0.050 6.2 0.079 6.4 0.038 3.4 0.12\n7% 6.8 0.054 6.1 0.077 6.3 0.034 4.1 0.11\n8% 6.6 0.045 6.0 0.088 6.3 0.033 4.3 0.13\n9% 6.6 0.045 5.9 0.074 6.3 0.034 4.6 0.15\n10% 6.6 0.040 5.9 0.070 6.2 0.030 4.4 0.16\nThe LSTM(50) outperforms all of the other models for nearly all noise levels\nfor both mean and maximum absolute errors. The GLM, TLP and RF models\nhave similar performance across noise levels for both mean and max AE.\nIf we establish a notional acceptance criteria for model performance of 1%\nMean MAE, that is to say average performance over all trials, we see that GLM\nand TLP are acceptable at 0% and marginally at 1% noise (Table 4). If instead\nthe metric is 1% Mean Max AE, or worst-case performance , we see (Table 5)\nthat no models would be acceptable.\nPoor performance of direct modeling efforts on the available block ampli-\ntude data Aas indicated by Tables (4) and (5), suggest pursuing an alternate\napproach using some feature engineering.\nSince mean values across blocks for the block optical coupling constants ω\nand block PMT gains gare very close to unity and likewise since the mean of\nthe overall system noise ϵis close to zero, we expect to benefit from averaging\nover blocks (and in some cases time) in bootstrapping a crude first model into\none that we may be able to improve with machine learning techniques. Thus,\nthe following feature engineering method was implemented as described below.\nInverting Eq. 8 with ωandRconstant for each block and any choice of ϵ,\nwe can write:\nˆαt=1\nBBX\nb=1Ab,t (12)\nWhere ˆ αtis taken as our estimate of the time varying LED pulser amplitude\nαand where the mean is taken across blocks for each discrete time step in a\ntrial. Fig. 35 shows an example of this model with ϵat 10% noise.\n43Figure 35: An Example of Estimated αwith ϵat 10% noise.\nContinuing to bootstrap Eqn. 8, we have for our estimate of mean Abfor\neach block:\nAb=1\nTTX\nt=1Ab,t (13)\nand the mean value of α(for scaling):\nα=1\nTTX\nt=1αt (14)\nwhere these means are calculated for the chosen block over the training\nportion of the data. It is possible to estimate the optical coupling constant ωb\nfor each block as\nˆωb=Ab\nα, (15)\nwhere the mean is taken across the time index.\nFinally, once again inverting Eq. (13) with the previous calculated approxi-\nmations, the estimated gain value for a given block at time t using the averaged\namplitude is written as\nˆgb,t= ˆωb·ˆαt\nAb,t. (16)\nEq. 16 is referred to as an Averaged Amplitude model to extract the gain values.\nFigs 36 - 38 show that the accuracy of the estimated gain ˆ glooks good by eye\nin the absence of noise, but degrades significantly as the noise level increases.\n44Figure 36: An example of estimated PMT gain with ϵ= 0 and\nCorrelation( ˆ gb,t, gb) = 99 .6%\nFigure 37: An example of estimated PMT gain with ϵ=N(0,5%) and\nCorrelation( ˆ gb,t, gb) = 50 .7%\nFigure 38: An example of estimated PMT Gain with ϵ= N(0,10%) and\nCorrelation( ˆ gb,t,gb) = 28.6%\n45Figure 39: An example of estimated PMT Gain versus Actual gwith ϵ=\nN(0,5%).\nFig. 39 shows that at 5% noise level the AA calculation, while a noisy fit\nto the PMT gain (blue line), nonetheless demonstrates no systematic bias in its\nfigure, the centroid (orange line) of which faithfully represents the deterministic\nPMT signal. Other noise levels demonstrate this as well. It is then expected\nthat ML modeling will de-noise this basic result into a final model.\nWe use the left-hand-side of Eq. 16 (the estimated gain for a block at each\ntime index) as the sole feature input to the GLM, TLP, RF, and LSTM models.\nThe GLM, TLP, and RF models had nearly identical performance regardless of\nnoise level, as shown in Fig. 40.\nUsing the same notional acceptance criteria as in the previous method, we see\nthat the AA meets the criteria up to 1% noise, while GLM, and RF models meet\nthe criteria up to 2% noise whereas the LSTM(50) is acceptable up to about\nthe 8% noise level, where LSTM(50) is the LSTM model for history depth of 50\ntime units. Table 6 (Fig. 40) summarizes the mean and standard error of the\n% MAE calculated from the collection of trials as a function of noise level for\neach model.\nFigure 40: Summary of Trials: Mean % MAE for Models versus Noise Level.\n46Table 6: Mean % MAE Results from Modeling for PMT Gain (using ˆ g).\nModel AA GLM TLP RF L(50)\nNoise Mean StdErr Mean StdErr Mean StdErr Mean StdErr Mean StdErr\n0% 0.50 0.00 0.45 0.00 1.0 0.089 0.48 0.00051 0.28 0.0054\n1% 0.83 0.0042 0.69 0.0031 1.3 0.078 0.72 0.0034 0.35 0.010\n2% 1.3 0.0059 1.1 0.0054 1.4 0.056 1.1 0.00047 0.50 0.022\n3% 1.8 0.0092 1.4 0.0056 1.7 0.041 1.4 0.0067 0.64 0.017\n4% 2.3 0.011 1.6 0.0062 1.9 0.031 1.6 0.0062 0.74 0.021\n5% 2.9 0.014 1.8 0.0069 2.0 0.023 1.8 0.0075 0.82 0.023\n6% 3.4 0.017 1.9 0.0061 2.0 0.019 1.9 0.0064 0.91 0.026\n7% 4.0 0.018 2.0 0.0065 2.1 0.014 2.0 0.0065 0.98 0.029\n8% 4.6 0.022 2.0 0.0055 2.1 0.012 2.0 0.0060 1.1 0.036\n9% 5.1 0.024 2.1 0.0053 2.1 0.010 2.1 0.0059 1.2 0.031\n10% 5.7 0.031 2.1 0.0048 2.2 0.0082 2.1 0.0055 1.2 0.048\nIf instead the metric is 1% mean Maximum AE, or worst-case performance ,\nwe see in Table 7 (Fig. 41) that only the LSTM(50) model is acceptable and\nonly for the 0%, 1% noise levels.\nFigure 41: Summary of Trials Mean % Max AE for Models versus Noise Level.\n47Table 7: Mean % Max AE Results from Modeling for PMT Gain (using ˆ g).\nModel AA GLM TLP RF L(50)\nNoise Mean StdErr Mean StdErr Mean StdErr Mean StdErr Mean StdErr\n0% 2.1 0.00 1.6 0.00 2.6 0.16 1.8 0.0041 0.59 0.017\n1% 4.0 0.045 3.6 0.035 4.1 0.069 3.8 0.049 1.1 0.038\n2% 6.3 0.072 5.0 0.050 5.0 0.045 5.2 0.057 1.6 0.051\n3% 8.8 0.090 5.7 0.049 5.5 0.058 6.0 0.054 2.1 0.074\n4% 12 0.12 6.0 0.043 5.6 0.067 6.3 0.049 2.3 0.083\n5% 14 0.15 6.1 0.048 5.7 0.071 6.4 0.047 2.6 0.080\n6% 17 0.20 6.1 0.046 5.6 0.064 6.3 0.045 2.7 0.084\n7% 20 0.20 6.0 0.038 5.5 0.062 6.3 0.046 2.9 0.096\n8% 22 0.22 6.1 0.039 5.6 0.055 6.4 0.051 3.3 0.12\n9% 25 0.25 6.0 0.038 5.5 0.052 6.4 0.051 3.7 0.16\n10% 28 0.33 5.9 0.031 5.5 0.050 6.4 0.049 3.7 0.13\nLift charts for model performance using the engineered inputs shows the\ncomparison to the first method in Fig. 42. For all models, using the feature\nengineered inputs results in improved performance across all noise levels.\nFigure 42: Model performance as a function of percent noise for each model.\nThe circles correspond to using Eqn. 8 directly as inputs whereas the crosses\nutilize the feature engineered inputs .\nThe history depth refers to the amount of past data the model uses to\nmake predictions. To further evaluate the performance of an LSTM model,\nresults were generated for various history depths up to 50. Fig. 43 shows the\nperformance of the LSTM as a function of the history depth size at a 9% noise\nlevel.\nHere at an example 9% noise level, best average performance can be achieved\nat a history window depth of about 25 whereas best worst-case performance\nrequires a history window depth of about 40.\n48Figure 43: The LSTM model performance using ˆ gby history depth at 9% noise\nlevel.\nThus, given a rather tight threshold on MAE of 1%, an LSTM(50) model\nshould be acceptable to predict the gain of individual FCAL blocks given the\npulse amplitudes each block sees over time. It is important to note this work\nignores the effects of radiation damage. Including the radiation damage would\nadd on a spatial dependence and would affect blocks located near the beam line.\nThe simulation work contained within validates a machine learning approach to\nextract the gains assuming there exists a correlation between individual block\ngains and the individual pulse amplitudes detected from LED pulse events. This\ncorrelation is expected but disappointingly has not been observed in the FCAL\nLED pulse data.\n498.4.3 FCAL Inner Ring Radiation Damage\nThe inner blocks of the FCAL suffer radiation damage due to the proximity to\nthe beam line. This effect is seen visually as ”browning” of the lead glass blocks,\nshown in Fig. 44. As the radiation damage is largely localized to blocks located\nin inner rings, we investigate the correlation between the change in the existing\ngain calibration constants from the start of a run period to the luminosity. Hall-\nD does not have real time measurement of the luminosity. As a proxy, we utilize\nthe scalers from the Time-of-Flight (TOF) system in Hall-D.\nFigure 44: Image of the inner blocks during the disassembly of the detector.\nHere the radiation damage is clearly visible as a brown tint on the inner blocks.\nThe normalized average gain by ring as a function of run number is shown in\nFig. 45. The effect is most prominent in ring 1 and gradually declines as the ring\n50Figure 45: Normalized Average Gain by Inner Ring for the PrimEx run period.\nFigure 46: Pearson R correlation coefficients for the integrated TOF scalers and\nintegrated beam current vs the TOF paddle number.\nnumber increases. Ring 2 is a clear outlier, and this behavior is not understood.\nAs a sanity check, we observe the correlations between the integrated beam\ncurrent and the integrated TOF scalers, which should be strongly correlated.\nThis relationship is shown in Fig. 46.\nIn Fig. 47, the correlation coefficients between the normalized average gain\nvalues from the inner FCAL rings and the average scaler values from the TOF\n(averaged over all paddles) is shown. The strongest correlations exist for the\ninner most ring closest to the beam line, as expected. The remaining fluctuations\nin the correlation values for rings 2-10 is unexpected. This investigation should\nbe repeated for other Run Periods for comparison when the calibration method\nutilizing the LEDs is performed.\n9 Conclusions and Outlook\nThe AIEC project resulted in a maintainable, production control system for\nJLab’s Hall-D CDC that autonomously controls the CDC’s HV at the start of\neach run in an uncertainty-aware manner. The system allows for the model to\nbe easily trained with additional data taken from previously unseen environ-\nmental and experimental conditions such that the model increases its certainty\nmanifold. The ML-controlled HV has been shown to stabilize gain without de-\n51Figure 47: Pearson R correlation coefficients for the TOF scalers and FCAL\ngain values. The TOF scalers were averaged by paddle and the FCAL gain\nvalues were averaged by ring. The color of the dots in rings <6 correspond to\nthe same rings shown in Fig. 45.\ngrading timing resolution. Additionally, the project resulted in a calibration\nprocess for the Hall-D FDC detector.\nIt may be obvious, but is important to emphasize that a control system\nrelying on sensors will not function if those sensors are offline, and a contingency\nplan is required. For RoboCDC this occurred in Deployment #5 when the\ncontrol feature was disabled due to the atmospheric pressure sensor being offline.\nThe CDC remained at 2125 V, i.e. uncontrolled, which allowed experiments to\ncontinue as planned, and more training data was collected at the nominal HV.\nSeveral additional detectors in JLab’s Hall-B and Hall-D were investigated\nfor ML-based calibration and control. A barrier to a data-driven control sys-\ntem is the lack of historical calibration data from diverse conditions. While the\noriginal proposal expected the ML techniques applied to the Hall-D CDC to\ntranslate to CLAS12’s Drift Chambers more than an order of magnitude fewer\nhistorical calibration entries made the effort infeasible. Detectors also require\nsensors with measurement frequency and precision required for required cali-\nbration and control tasks. The BCAL detector pedestal is known to fluctuate\nwith crate temperature; however, the only available temperature sensor lacked\nthe precision required.\nA ML method to learn the calibration values for the FCAL’s 2800 blocks\nwas investigated by a UVa capstone project team. There is a lack of correlation\nbetween the FCAL monitoring system’s LED pulses and the gain calibration\nvalues, except for blocks near the center of the FCAL. A concurrent and in-\ndependent analysis agrees with the puzzling lack of correlation. Results from\n52the capstone project led to a simulation study and an investigation of LSTM\nmodels to learn from the time series evolution of a block’s gain.\nInvestigating the varied detectors in Hall-B and Hall-D allowed the project\nstaff to make note of multiple key insights throughout the project. We offer\nthese insights that future projects may benefit.\nKey Insights\n1. The importance of the active participation of the detector expert cannot\nbe overstated. The fields of nuclear physics and data science have their\nown jargon, data visualization preferences, and professional cultures. The\ndetector expert’s inclusion throughout the project planning, data explo-\nration, model development, system implementation, and system evalua-\ntion enabled the detector expert and data scientist to share a common\nproject “language.” The common language facilitated the communica-\ntion of complex requirements, for example, requirements for interpretable\nmodel output. Without the integration of a detector expert within the\nproject team from the inception stage through execution, we expect any\nimplementation of a production ML-based control system to be very chal-\nlenging.\n2. We recommend a “self-contained” control system, with minimal reliance\non additional systems. The system may still experience interruptions in\nsensor availability, as we did during the GlueX 2023 Run Period due to the\nmalfunctioning atmospheric pressure sensor; however, the early decision\nto isolate the CDC control system from other detector subsystems allowed\nus to meet our project objectives without unneeded dependencies.\n3. The inclusion of uncertainty quantification for safe ML-based control in\nautonomous high-risk experimental systems is important, as are the de-\ncision control plans when uncertainty exceeds selected thresholds. Gaus-\nsian process regression is the UQ method used for the CDC control sys-\ntem; additional techniques include ensemble ML methods ZOU [2021], MC\nDropout Gal and Ghahramani [2016], Deep Quantile Regression Koenker\n[2005], and techniques that incorporate GP’s within a deep learning model\nDamianou and Lawrence [2013], Liu et al. [2020], Rasmussen et al. [2006].\n4. The historical record and documentation of traditional calibrations, in-\ncluding a record of the iterations of calibration values, proved useful\nthroughout the CDC process and would have been helpful during the\nevaluation of the FCAL data.\n5. The FCAL simulation exercise yielded several insights:\n(a) First, the importance of feature engineering that takes advantage of\ndomain knowledge and specific knowledge of the underlying dynamics\nof the target signal.\n53(b) Second, the value of recursive deep learning techniques, specifically,\nLSTM modeling Hochreiter and Schmidhuber [1997], particularly as\nthe target signal is considered as a sequence.\n(c) Third, LSTM requires deeper history processing to curtail larger\nworst-case deviations of predicted signals from the target signal than\naverage-case deviations..\nThis body of work represents the first steps towards a “self-driving labora-\ntory”, one in which intelligent machines are able to gather data, perform anal-\nyses, and self-correct. The AIEC program successfully developed and deployed\nan autonomous system for the GlueX CDC, allowing it to control its own HV,\nwhich in turn stabilized the gain correction factor with no significant impact\non time-to-distance calibrations. RoboCDC is now in production, embedded as\npart of a modular control system as part of the standard operating procedure.\nThe work, additionally, took steps beyond the single detector and probed appli-\ncations to other detectors and halls. And even while those steps did not produce\nanother autonomous detector system it did illuminate challenges with further\napplications, which helps inform the field on what needs to be done in the future\nto enable such applications. With the success of this project, another dedicated\nproject, AI for Optimized Polarization (AIOP) is due to begin in 2024. It is the\nhope that this body of work provides both inspiration and technical details to\nothers as we work towards self-driving laboratories and autonomous discoveries.\n10 AIEC Publications & Presentations\n10.1 Publications\nJeske, T., McSpadden, D., Kalra, N., Britton, T., Jarvis, N., & Lawrence, D.\n(2023, February). Using AI to predict calibration constants for the central drift\nchamber in GlueX at Jefferson Lab. In Journal of Physics: Conference Series\n(Vol. 2438, No. 1, p. 012132). IOP Publishing.\nJeske, T., McSpadden, D., Kalra, N., Britton, T., Jarvis, N., & Lawrence,\nD. (2022). AI for Experimental Controls at Jefferson Lab. Journal of Instru-\nmentation , 17(03), C03043.\n10.2 Key Presentations\nAI Driven Experiment Calibration and Control Slides\nBritton, T., Lawrence, D., Jeske, T., McSpadden, D., & Jarvis, N. (2023, May\n8–12). AI Driven Experiment Calibration and Control [Conference presenta-\ntion]. Conference on Computing in High Energy & Nuclear Physics, Norfolk,\nVA, United States. https://indico.jlab.org/event/459/contributions/\n11374/\n54Using Machine Learning to control the GlueX Central Drift Cham-\nberSlides\nJarvis, N., Britton, T., Jeske, T., Lawrence, D., & McSpadden, D. (2023, Jan-\nuary 9–13). Using Machine Learning to control the GlueX Central Drift Cham-\nber [Conference presentation]. Conference on High Energy Physics in LHC Era,\nValpara´ ıso, Chile. https://indico.cern.ch/event/1158681/contributions/\n5192980/\nControl and Calibration of GlueX Central Drift Chamber Using\nGaussian Process Regression Poster Paper\nMcSpadden, D., Jeske, T., Jarvis, N., Britton, T., Lawrence, D., & Kalra,\nN. (2022, December 3). Control and Calibration of GlueX Central Drift Cham-\nber Using Gaussian Process Regression [Poster Presentation]. Machine Learn-\ning and the Physical Sciences workshop at NeurIPS, New Orleans, LA, United\nStates. https://ml4physicalsciences.github.io/2022/\nGaussian process for calibration and control of GlueX Central\nDrift Chamber Slides\nJeske, T., Britton, T., Kalra, N., Jarvis, N., McSpadden, D. &, Lawrence, D.\n(2022, October 23-28). Gaussian process for calibration and control of GlueX\nCentral Drift Chamber [Conference Presentation]. Advanced Computing and\nAnalysis Techniques in Physics Research, Bari, Italy. https://indico.cern.\nch/event/1106990/contributions/4998092/\nAcknowledgements\nThis work is supported by a grant from the U.S. Department of Energy, Office\nof Science, Office of Nuclear Physics under the LAB-20-2261 FOA.\nThe Carnegie Mellon Group is supported by the U.S. Department of Energy, Of-\nfice of Science, Office of Nuclear Physics, DOE Grant No. DE-FG02-87ER40315\nThis research used resources of the Thomas Jefferson National Accelerator Fa-\ncility, which is a DOE Office of Science User Facility supported by the U.S.\nDepartment of Energy, Office of Science, Office of Nuclear Physics under con-\ntract DE-AC05-06OR23177.\n55References\nDepartment of Energy Invests $16 Million in Data-Intensive Scientific Machine Learn-\ning Research and Analysis. Energy.gov .\nEnsemble machine learning paradigms in hydrology: A review. Journal of Hydrology ,\n598:126266, 2021. 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Particle Data Group , 2020:\n083C01, 2020.\n5859A Control System Workflows\nA.1 Initialize\n60A.2 AutoOFF\nAutoOff(def_HV)\ntry to use cdc_tool to set CDC HV to def_HVprint HV\nset errorException\nreturn\n61A.3 calcHV\n6263A.4 FailSafe\nFailSafe(model_name,input_dict,control_mask,baseline_voltage,testing=T rue)\nif testingprint\ntesting\nalertTrue\nset CDC\nHV to\nbaselineFalse\nif testingprint\ntesting\nalertTrue\ninsert action\ninto databaseFalse\nreturn\n64A.5 Lookback\n6566A.6 ProcessEPICS\n67A.7 WatchProcessRecommend\nW atchProcessRecommend(config_file)\nload\nconfiguration\nconfiguration[\"rec_scale\"]<0 return 0 True\nProcessRecommend(config_file)False\nsleep(1.02*60*poll_time)\n68"    },    {        "title": "2402.13313v1.Spin_Triplet_Pairing_in_Heavy_Nuclei_is_Stable_Against_Deformation.pdf",        "content": "Spin-Triplet Pairing in Heavy Nuclei is Stable Against Deformation\nGeorgios Palkanoglou,1, 2Michael Stuck,1and Alexandros Gezerlis1\n1Department of Physics, University of Guelph, Guelph, ON N1G 2W1, Canada\n2TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada\nAny experimental evidence of nucleons paired in spin-triplet states will confirm the existence of\nan exotic phase of nuclear matter. This type of nuclear superfluidity has been hypothesized in heavy\nnuclei, where the antagonizing spin-orbit effects are damped, and there it oftentimes coexists with\ntraditional spin-singlet pairing, leading to the possibility of mixed-spin pairing. Realistic nuclear\ndeformation, not considered in such studies, could make-or-break these proposals, since its effect on\ntriplet-pairing, and the competition (and coexistence) of the two superfluid phases, was expected\nto be crucial. We report on a thorough study on the effect of deformation on triplet-, singlet-,\nand mixed-spin pairing in the relevant region of the nuclear chart. We find that, at low isospin\nasymmetries, spin-triplet pairing is enhanced by deformation, while below the proton-drip line, the\nnovel superfluid phase survives alongside the usual spin-singlet pairing. These results suggest that\nspin-triplet superfluidity exists in realistic nuclei and can be probed in the lab.\nIntroduction Spin-triplet neutron-proton pairs have\nbeen elusive, with the only pairing seen in experimen-\ntally accessible nuclei being of spin-singlet and between\nlike particles [1]. In contrast, our understanding of nu-\nclear forces tells of an increased attraction in the spin-\ntriplet channel of the two-body nuclear interaction, un-\nderlined by the existence of the deuteron. This has gen-\nerated a long-standing question mark on the existence\nof deuteron-like spin-triplet pairs in nuclei. At a mean-\nfield level, the strong spin-orbit field has been identified\nas the suppressor of the spin-triplet superfluidity in nu-\nclei [2, 3], which suggests the possible existence of the\nexotic superfluid phase in heavy nuclei and extended nu-\nclear systems. At large, nuclear superfluidity remains a\nvibrant field probed with many approaches [4–12], with\nties to astrophysics [13, 15–17] and nuclear reactions [18–\n21].\nRobust spin-triplet pairing has been suggested to exist\nclose to the N=Zline of the nuclear chart, A∼130,\nwith a region of an exotic mixed-spin condensate lying in\nthe crossover between spin-triplet and spin-singlet pair-\ning [1, 3, 22]. This region of the nuclear chart is char-\nacterized by a finite deformation [23], so far neglected in\nRefs. [3, 22, 24, 25], which could change this picture dra-\nmatically: with the two pairing phases in general working\nagainst each other, and deformation typically antagoniz-\ning pairing altogether [26], the total effect is non-trivial to\npredict. Furthermore, the existence of spin-triplet pair-\ning in the nuclear ground state relies on the proximity\nof low- jshells to the Fermi surface that happens in the\nA= 130 region; that too will be altered by deformation.\nIn this Letter, we address nuclear deformation, an im-\nportant missing piece which can greatly modify pairing\ncorrelations: we confirm the existence of spin-triplet pairs\nin heavy nuclei, at N=Z, but also within the physical\nregion, where it coexists with spin-singlet pairing, and\ncan be probed in the lab.Context We use a generalization of the Hamiltonian\nused in Refs. [3, 22, 24, 25], which in second quantization\nreads\nH=X\nijϵijc†\nicj+X\ni>j,k>l⟨ij|v|kl⟩c†\nic†\njclck.(1)\nThe one-body part, ϵ, contains the kinetic energy, a\ndeformed Wood-Saxon well, and a deformed spin-orbit\nterm. At the mean-field level the external Wood-Saxon\npotential models the shape of the nucleus’ surface and\nthe spin-orbit term, peaking at the nuclear surface, is\ndefined on the same shape:\nϵ=p2\n2m+VWS(ϖ, z;⃗ α) +C∇[VWS(ϖ, z;⃗ α)]·(σ×p).\n(2)\nThe last term in Eq. (2) is the generalized spin-orbit form\nthat reduces to the familiar (1 /r)dVWS/dr(l·s) in the\nabsence of deformation [28–30]; it’s multiplied by its cou-\npling, C[27]. In lieu of experimental results in the mass\nregion of A∼130, we follow the predictions of Ref. [23]\nwhich prescribe axial deformation for that region with a\nmoderate quadrupole component and small higher-order\nones.\nThe specific shape of the nucleus’ surface is given by a\nvector of deformation parameters, namely, ⃗ α= (ε,⃗β).\nThis contains the elongation of an underlying Cassini\noval, ε, and the additional deformation described on a\nbasis of spherical harmonics with coefficients βλ[31, 32].\nThe vector ⃗ αuniquely defines the nuclear surface,\nϕ(ϖ, z) =h\u0000\n¯z2+ ¯ϖ2\u00012−2εR2\n0\u0000\n¯z2−¯ϖ2\u0001\n+ε2R4\n0i1/4\n−\n−R0\"\n1 +X\nλ>0βλYλ0(x)#\n= 0, (3)\nand the one-body potential this generates,\nVWS(ϖ, z) =V0\u0014\n1 + exp\u00121\naϕ\n|∇ϕ|\u0013\u0015−1\n. (4)arXiv:2402.13313v1  [nucl-th]  20 Feb 20242\n50 55 60 65 70 75\nN505560657075Z\n50 55 60 65 70 75\nNSinglet\nTriplet\nMixed\nDrip-lineNo Deformation\n50 55 60 65 70 75\nN505560657075\n50 55 60 65 70 75\nNβ2=0.1\n50 55 60 65 70 75\nN505560657075\n50 55 60 65 70 75\nNβ2=0.25\n50 55 60 65 70 75\nN505560657075\n50 55 60 65 70 75\nNRealistic  Deformation\n55 60 65 70 75\nN5560657075Z\n55 60 65 70 75\nN∆>0.4 MeV\n∆<0.4 MeV\n55 60 65 70 75\nN5560657075\n55 60 65 70 75\nN55 60 65 70 75\nN5560657075\n55 60 65 70 75\nN55 60 65 70 75\nN5560657075\n55 60 65 70 75\nN\nFIG. 1. Chart of nuclides with Z≤Nfor neutron numbers from 50 to 75. In the top panels, we show the spin character of\nthe nuclear condensate: the green squares, blue circles, and red diamonds, represent nuclei with singlet, mixed-spin, and triplet\npairing, respectively, and the grey squares denote nuclei with Ecorr<0.5 MeV. In the bottom panels, we show strong (black\nsquares) and weak (orange circles) pairing gaps (see Eq. (9)) for the same deformation. The proton-drip line is drawn in black\nin the top panels and in grey in the bottom panels [23].\nThe barred coordinates, ¯ ϖ=f(R, x;ε) and ¯ z=\ng(R, x;ε) are cylindrical coordinates parameterized by\nthe curvilinear coordinates ( R, x) in which Cassini ovals\nspan coordinate surfaces and R0andaare the radius\nand surface diffuseness, respectively, of the undeformed\nnucleus.\nThe two-body part of the Hamiltonian, v, is a contact\npairing interaction,\n⟨ij|v|kl⟩=X\nαvα⟨ij|δ(3)(x−x′)PαPJz=0|kl⟩,(5)\nwhere the projection operator Pαprojects on six spin-\nisospin channels: α= 0,1,2 correspond to the three spin-\nsinglet and isospin-triplet channels, while α= 3,4,5 cor-\nrespond to the three spin-triplet and isospin-singlet chan-\nnels. The projection operator PJz=0takes advantage of\nthe axial symmetry and pairs particles in time reversed\nstates, resulting in axially-symmetric pair-wavefunctions.\nIt conveniently generalizes the common approximation of\npairs with zero orbital angular momentum[1, 3] allowing\nformation of pairs with finite angular momentum which\nare expected to form in ground-states of deformed nu-\nclei. Similar pairing schemes have been used before in\nthe study of spin-singlet and spin-triplet nuclear pair-\ning [33, 34]. The interaction strengths that appear in\nEq. (5) are the same as in earlier works [3, 22, 24, 25].We allow for six pairing channels, the three spin-singlet\nand three spin-triplet ones defined in Eq. (5). Chan-\nnels with the same total spin (or isospin) and different\nprojections are prescribed the same vαand are seen as\nequivalent, that is, we use an interaction with only two\ninteraction strengths.\nHFB treatment and its application The Hartree-Fock-\nBogolyubov (HFB) treatment of the Hamiltonian in\nEq. (1) corresponds to identifying the Bogolyubov trans-\nformation that minimizes the ground-state energy. The\nsystem is then described as a collection of free quasiparti-\ncles, whose vacuum defines the ground-state. In practice,\nthe Bogolyubov transformation is parameterized by the\nmatrices UandV, which mix particle and hole oper-\nators in Fock-space defining the quasi-particles. These\nmatrices also define the normal and anomalous densities,\nρ=V∗VTandκ=V⋆UT, respectively, which are fun-\ndamental to the description, as they in turn define the\nHFB expectation value of the Hamiltonian,\nH00= Tr\u0014\nϵρ−1\n2∆κ∗\u0015\n, (6)\nwhere ∆ ij=1\n2P\nklvijklκkl.\nThe HFB treatment can be implemented very effec-\ntively via the gradient method outlined in Ref. [35], and3\n0 0.05 0.1 0.150123Ecorr [MeV]Singlet\nTriplet\nMixed\n0 0.05 0.1 0.15\nβ200.20.40.60.81y\nTripletSinglet\nMixedy=BHFB\ny=Asinglet0 0.01 0.02 0.030.9911.01\nFIG. 2. The evolution of108\n54Xe’s ground state increasing\nquadrupole deformation. The top panel shows the correlation\nenergy of the nucleus. In the lower panel, Asinglet measures\nthe size of the singlet condensate, relative to the triplet con-\ndensate (see Eq. (7)) and BHFBquantifies the spin-orbit field\n(see Eq.(8)).\nused in a similar fashion as in Refs. [3, 22, 24]: we locate\nthe HFB minimum by performing a gradient descent on\nthe surface of the HFB energy parametrized by the U\nandVmatrices (or κ) in Eq. (6). In this way one can\nstraightforwardly identify solutions of different symme-\ntries by constraining the descent, i.e., introducing the\nappropriate terms with Lagrange multipliers to Eq. (6).\nWe use 8 constraining fields: the neutron and proton\nnumbers and six pairing amplitudes which are defined\nbelow. This allows us to explore different HFB minima,\napart from the global minimum that corresponds to the\nfully-paired ground state.\nThe spin character of an HFB state can be described by\nthe expectation value of the pair operator ¯Pα=PαPJz=0\nthat defines the six pairing channels and appears in\nEq. (5). These six pairing amplitudes can then be used\nas constraining fields to locate states with certain pair-\ning character. Specifically, we use the state where all six\npairing channels are closed, i.e., all pairing amplitudes\nvanish, as a reference state: we measure all energies rel-\native to its unpaired energy, namely, E0, and call the\nresulting quantity correlation energy, Ecorr=E−E0.\nResults We mainly characterize our results by the\nstrength and spin character of their pairing. We quan-\ntify the strength of the pairing correlations via the cor-\nrelation energy. We label the spin character of the con-\ndensate using the expectation values of the pairing fields\nKα=\n¯Pα\u000b\n. We define [22]:\nAsinglet =P\nβ=0,1,2K2\nβP\nαK2α, (7)\nand a nucleus is labeled as spin-singlet if Asinglet >0.8\nand spin-triplet if Asinglet <0.2, otherwise, the nucleus\nis labeled as mixed-spin. Finally, the spin-orbit field isexpected to play a major role in the interplay of the differ-\nent pairing phases, and we quantify it via its expectation\nvalue at the HFB ground state:\nBHFB=⟨HFB|C∇f(ρ, z;⃗ α)·(σ×p)|HFB⟩.(8)\nWe calculate correlation energies and pairing ampli-\ntudes for all nuclei in the interesting region of mass\nA∼130 and we plot the results in the top panels\nof Fig. 1. To visualize how the condensates respond\nto nuclear deformation, we perform these calculations\nfor a sequence of deformation parameters, starting from\nspherical shapes across the region in the leftmost panel,\nwhich reproduces the results of Refs. [3, 22], increasing\nquadrupole deformation for all nuclei progressively in the\ntwo middle panels, and reaching the realistic deformation\nprescribed by M¨ oller et al [23] in the rightmost panel.\nThe bottom panels of Fig. 1 show pairing gaps, i.e., bind-\ning energies of pairs, for the region of interest and for the\nsame deformation parameters. These will be discussed\nlater.\nThe superfluidity’s response to deformation is rich so\nwe start from the N=Zline, where neutrons and pro-\ntons occupy the same orbitals, increasing the odds for\nneutron-proton pairing. First, we investigate quadrupole\ndeformation ( β2), which seems to drive the qualitative\neffects for most nuclei. At very small β2deformation,\nspin-triplet pairing is generally damped, resulting in a\nsmall increase in the correlation energies of singlet nuclei\nand the change of some triplet paired nuclei to mixed-\nspin. This can be seen at A∼125 in Fig. 1 as well\nas the low-deformation end of Fig. 2. As quadrupole\ndeformation approaches the realistic value predicted by\nRef. [23] for the region, spin-triplet completely domi-\nnates the N=Zline inducing a mixed-spin band at\nN−Z= 2 or 3 before the nuclear condensate turns spin-\nsinglet. This dramatic change resembles the results of\nRef. [24], where the spin-orbit interaction was switched-\noff artificially: the enhancement of spin-triplet pairing\ncan be understood as driven by a general suppression of\nthe spin-orbit field at the ground-state. This is clearly\ndemonstrated in Fig. 3 for126\n61Pm where the spin-singlet\ncomponent and the spin-orbit field show similar behav-\nior. The connection can also be seen in Fig. 2 where we\nplot the correlation energy, spin-singlet pairing ampli-\ntude, and spin-orbit field for108\n54Xe, a case of a pure spin-\nsinglet paired ground state at the spherical limit turned\nto spin triplet by deformation. There the transition be-\ntween the two pairing-phases is accompanied by a steady\ndecrease of the spin-orbit field.\nOn the N=Zline, a reduced spin-orbit field corre-\nsponds to a spin-triplet paired ground-state [3, 24]. Mov-\ning along an isobar (i.e., increasing N−Z) the nuclear\ncondensate very quickly changes to spin-singlet. This is\nexpected, as increasing isospin asymmetry translates to\nincreasing mismatch in the proton and neutron Fermi4\n0246810Ecorr [MeV]singlet\nmixed\n00.51yy=Asinglet\ny=BHFB\nTripletMixedSinglet\n0 0.1 0.2 0.3\nβ20 0.03\nβ4-0.03 0\nβ6-20-18-16-14-12-10-8-6\ns.p. levels [MeV]l=4l=2l=0l=5\nFIG. 3. The evolution of126\n61Pm’s ground state from no de-\nformation to the realistic deformation predicted by Ref. [23].\nThe bottom panel shows the evolution of the single-particle\nlevels that lie in the energy window: from bottom to top\nlz= 0 (black), 1 (blue), 2 (green), 3 (red), 4 (orange), and\n5 (yellow). The middle panel shows the singlet pairing am-\nplitude overlaid with the expectation value of the spin-orbit\nfield (at the ground-state), appropriately normalized. The\ntop panel shows the correlation energy of the nucleus along\nwith the shape of the nuclear surface along the evolution.\nsurfaces, making the pairing of a neutron and a proton\nincreasingly unlikely.\nSwitching-on the higher deformation modes, to the val-\nues predicted by Ref. [23], we find that for most nu-\nclei, qualitatively, the picture doesn’t change: spin-triplet\npairing dominates at and around N=Z, mixed-spin nu-\nclei can be found at small isospin asymmetry N−Z, and\nspin-singlet generally takes over for higher isospin asym-\nmetries. Additionally, the cross-over region, between the\ntwo phases has been drastically reduced compared to the\nspherical limit. At a triangular region, close to N=Z,\natA∼130 some triplet- and mixed-spin paired nuclei re-\nemerge, driven by higher-order deformation, with some\nisotopes of Pm ( Z= 61) also being inside the physical\nregion. Specifically, the nuclei125\n61Pm,126\n61Pm, and127\n61Pm\npose an interesting case as they lie within the proton drip-\nline and maintain a sizeable spin-triplet condensate at\nfull deformation. These nuclei also exhibit a comparable\nspin-singlet condensate, co-existing with the spin-triplet,\nmaking them of mixed-spin character.\nIn the survey of the region presented in Fig. 1 we iden-tify some Pm isotopes that lie in the physical region,\ni.e., within the proton drip-line and can support spin-\ntriplet pairing in their ground states. With all three\nnuclei parameterized with similar realistic deformation\nparameters, we plot the evolution of126\n61Pm in Fig. 3\nwhere it is seen that it’s the higher-order deformation\nthat drives this transition. Plotted in the same figure\nis also the evolution of the single-particle levels (bottom\npanel) demonstrating how the overall initial reduction in\nthe correlation energy is driven by the reduction of the\nsingle-particle degeneracy close to the Fermi surface; the\nlatter lies close to the middle of the indicated energy win-\ndow. Finally, ovarlayed on the top panel, are the shapes\nof the nuclear surfaces at various deformations demon-\nstrating a simple connection between correlation energy\n(or pairing) and the geometry of the nucleus.\nThe source of the spin-orbit field’s decrease needs a\ndeeper discussion. We find that the matrix elements of\nthe spin-orbit generally decrease within each jz-shell with\nincreasing quadrupole deformation. Even when some el-\nements increase, the determinant of the average jz-shell\nblock decreases, signaling a decrease of the spin-orbit\nfield’s eigenvalues. This suppression is especially counter-\nintuitive since the spin-orbit field is a surface effect and\ndeformation is moving the nuclear surface away from the\nspherical shape, which has the lowest surface-to-volume\nratio, for a given volume. Indeed, perturbative calcula-\ntions, for very small quadrupole deformation, show that\nan increase in the spin-orbit strength is expected, if the\nnuclear surface is adequately probed by the participating\norbitals. A typical example of such an increase is seen in\nthe low- β2region of Fig. 2. Hence, the existence and/or\nextent of a spin-orbit increase, depends on whether de-\nformation brings higher-j orbitals close to the nucleus’\nFermi surface. This is because high-j wavefunctions lie\ncloser to the nuclear surface and as such can more effi-\nciently probe surface effects. A very pronounced increase\ncan be seen in126\n61Pm, at low β2deformation, where l= 5\nandl= 4 states are brought closer to the Fermi sur-\nface (i.e., the chemical potential). This increase in the\nspin-orbit field’s strength suppresses spin-triplet pairing\nallowing the correlation energy of spin-singlet nuclei to\nincrease. However, the strength of the spin-orbit field\nis then quickly suppressed when increasing deformation\ndrives the l= 4,5 orbitals away from the Fermi sur-\nface. It’s important to clarify that this suppression is\nnotan effect of pairing, as it is seen both in the HFB\nstate and the unpaired state, and it is traced back to the\nreduction of the spin-orbit’s matrix elements, mentioned\nabove. The above suggest a general suppression of the\nspin-orbit mean field by nuclear deformation.\nOutlook This Letter constitutes the first systematic\nstudy of the interplay between spin-singlet pairing, spin-\ntriplet pairing, and deformation in nuclei and it demon-\nstrates that the deformation’s effect on superlfuidity can\nbe understood as ternary. Firstly, it weakens all types of5\npairing correlations by reducing the single-particle den-\nsity of states, as has been shown in the past. Secondly,\nit increases the surface-to-volume ratio of a given nu-\ncleus amplifying the spin-orbit field’s effect, which in turn\ntends to suppress spin-triplet pairing. Finally, by chang-\ning the single-particle level structure close to the Fermi\nsurface, it might induce pairing of particles in low- jor-\nbitals, whose wavefunctions don’t lie very close to the nu-\ncleus’ surface, thus lifting the spin-orbit field’s suppres-\nsion and enhancing spin-triplet pairing. The interplay\nof the three effects, coupled with the well-documented\nantagonism of the two superfluid phases, gives rise to\nan unexpected outcome in the mass region of A∼130:\nrealistic deformation creates a spin-triplet superfluid in\nthe ground states of most nuclei with N=Z, and on\nsome nuclei with small isospin asymmetry where it al-\nmost always coexists with a spin-singlet superfluid phase.\nThe spin triplet phase, be it pure or mixed, was found\nto be moderately strong, yielding correlation energies of\n∼2 MeV for realistically deformed nuclei.\nThe correlation energies are a valuable tool in visual-\nizing pairing correlations since they can be seen as the\nbinding energy that a nucleus gains from the pairing cor-\nrelations. However, they are notexperimentally observ-\nable. Therefore, we also calculated pairing gaps, via the\nfamiliar odd-even staggering, traditionally the smoking\ngun of superfluidity in nuclear systems [1, 37],\n∆(N) =E(N)−1\n2[E(N−1) +E(N+ 1)] ,(9)\nwhere Nis either a proton or a neutron, and an odd\nnumber, while the other nuclear species is kept to a fixed\neven number [38]. We plot the results in the bottom\npanel of Fig. 1. We distinguish between gaps larger and\nsmaller than 0 .4 MeV, while we have excluded results\nwith systematic errors generated by the existence of an\nenergy window (i.e., a hard cut-off) in the single-particle\nspectrum (see Fig.3). That way, the rightmost panel\nof the lower row in Fig. 1 adds nuance to the conven-\ntional wisdom dictating that deformation generally sup-\npresses pairing. That is, deformation of higher multi-\npoles tends to strengthen pairing correlations, compared\nto pure quadrupole deformation, seen both in the pairing\ngaps of Fig. 1 and the correlation energies of Fig. 3. Fi-\nnally, Refs. [3, 22], demonstrated that spin-triplet paired\nground states yielded reduced pairing gaps, but Fig. 1\nshows that deformation partially lifts this suppression.\nThis was accurately predicted in Ref. [1] and an inspec-\ntion of the pairing gaps adds detail: as quadrupole defor-\nmation increases, the drop in the pairing gaps at A∼130\nrises while the rest of the gaps on the N=Zline steadily\ndecrease, resulting in an even distribution of pairing gaps\nalong the N=Zline. A similar effect is seen in the\ncorrelation energies as well, where increasing quadrupole\ndeformation equalizes the relatively large correlation en-\nergies of the non-deformed triplet-paired nuclei, resultingin a less variable distribution of correlation energies at re-\nalistic β2deformation. These effects can be attributed to\nthe fragmentation of the l-shells that is induced by defor-\nmation and demonstrated in the lowest panel of Fig. 3.\nIn general we identify an enhancement of spin-triplet\npairing in the ground-states of realistically deformed nu-\nclei with N∼ZandA∼130. While indirect evidence of\nthis enhancement is available in the literature [26, 39, 40],\nthe complete take-over, or the connection with the spin-\norbit field’s suppression by the scarcely sampled nuclear\nsurface is new here. The latter suppression is encoun-\ntered circumstantially in calculations of hypernuclei [41]\nwhere the spin-orbit strength is routinely quenched to\nachieve the right quadrupole deformation, and in the in-\nvestigations of Ref. [42], where increasing the spin-orbit\ncoupling makes the mean-field ground-state of light nuclei\ntake spherical shape. Similar conclusions were drawn in\nRef. [36] where deformation seems to antagonize the en-\nergy splitting induced by spin-orbit in some heavy nuclei\nin the region of A∼160. Consistently with our argu-\nments, all these nuclei have several low-j orbitals close to\ntheir Fermi surfaces.\nOur results predict the existence of spin-triplet pairs\nin heavy nuclei and invite further accurate yet costly ab\ninitio calculations [12, 43]. Additionally, the existence of\nexotic pairing phases in heavy deformed nuclei may lead\nto other interesting phenomena, e.g., in nuclear dynam-\nics, fission and fusion [19, 20, 44, 45], or nuclear struc-\nture [46–48] and reactions [21]. Finally, our results can\nbe significant for experiments as we have identified some\nnuclei on or below the proton drip-line, which can be\naccessed by experiment most easily, with a sizeable spin-\ntriplet condensate when deformed. These nuclei have a\nmixed-spin pairing character since they exhibit a compa-\nrable co-existing singlet condensate. Various signatures\nof spin-triplet pairing in the nuclear ground state have\nbeen proposed: an enhanced cross-section for nptrans-\nfer [1, 49], the sign of neutron-proton spin-spin corre-\nlations [50] (but also see Ref. [51]), or similarities be-\ntween spectra of odd-odd and even-even nuclei [52]. The\nrest of the spin-triplet condensates lie beyond the proton\ndrip-line where they could be probed in resonance states,\nthough such experiments are more demanding because of\nthe short lifetimes of these states. Hence, these results\nhold the promise of finding this sought-after nuclear pair-\ning in the lab and additionally elucidating the physics of\nmixed-spin paired nuclei.\nWe thank P. Garrett, T. Papenbrock, P. Navratil,\nM. Drissi, L. Jokiniemi, E. Leistenschneider, and A.\nKwiatkowski for useful discussions. This work was sup-\nported by the Natural Sciences and Engineering Research\nCouncil (NSERC) of Canada and the Canada Foundation\nfor Innovation (CFI). 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Let kkkbe an arbitrary field, Abe a standard graded Artinian Goren-\nstein kkk-algebra of embedding dimension four and socle degree three , and π:P→A\nbe a surjective graded homomorphism from a polynomial ring w ith four variables\nover kkkontoA. We give the minimal generators of the kernel of πand the minimal\nhomogeneous resolution of Aby free P-modules. We give formulas for the en-\ntries in the matrices in the resolution in terms of the coeffic ients of the Macaulay\ninverse system for A. We have implemented these formulas in Macaulay2 scripts.\nThe ideal ker πhas either 6, 7, or 9 minimal generators. The number of minima l\ngenerators and the precise form of the minimal resolution ar e determined by the\nrank of a 3 ×3 symmetric matrix of constants that we call SM.\nIf ker πrequires more than six generators, then we prove that ker πis the sum\nof two linked perfect ideals of grade three. If the ker πis six-generated, then we\nprove that Ais a hypersurface section of a codimension three Gorenstein algebra.\nOur approach is based on the structure of Gorenstein-linear resolutions and the\ntheorem that, except for exactly one exception, Ahas the weak Lefschetz property.\nCONTENTS\n1. Introduction. 2\n2. Notation, conventions, and elementary results. 4\n3. The structure of Gorenstein-linear resolutions, genera l case. 7\n4. The structure of Gorenstein-linear resolutions, when d=4 and n=2. 11\n5. A resolution of P/annΦ3by free P-modules. 19\n6. The matrix SM. 23\n6.A. Complexes associated to a symmetric 3 ×3 matrix. 23\n6.B. The complex Cthat is used throughout the paper. 24\n7. The generalized sum of linked perfect ideals. 26\n8. The algebra with no weak Lefschetz element. 28\n9. The case when SM is zero. 30\n10. The case when SM has rank one. 31\n11. Calculations. 34\n12. The seven-generated ideal is a sum of linked perfect idea ls. 37\nDate : February 22, 2024.\n2020 Mathematics Subject Classification. 13C40, 13D02, 13H10, 13E10, 13A02.\nKey words and phrases. Artinian algebra, Gorenstein algebra, Hypersurface secti on. Macaulay\ninverse system, weak Lefschetz property.\nWe made extensive use of Macaulay2 [8]. We are very appreciat ive of this computer algebra\nsystem.\n12 SABINE EL KHOURY AND ANDREW R. KUSTIN\n13. The case when the rank of SM is at least two. 44\n14. Macaulay2 scripts. 54\n14.A. The scripts in SocDeg3.m2. 54\n14.A.1. The script “BasicSocDeg3”. 54\n14.A.2. The script “minres7”. 55\n14.A.3. The script “SumOfLinkedIdeals7”. 56\n14.A.4. The script “SumOfLinkedIdeals9”. 57\n14.A.5. The script “minres6”. 57\n14.B. The scripts. 57\n14.C. We run a few examples. 63\n14.D. A few suitable Φ3’s. 67\n14.D.1. The corresponding Gorenstein ideal has 9 minimal ge nerators. 67\n14.D.2. The corresponding Gorenstein ideal has 7 minimal ge nerators. 67\n14.D.3. The corresponding Gorenstein ideal has 6 minimal ge nerators. 68\nReferences 68\n1. I NTRODUCTION .\nLetkkkbe an arbitrary field, Ube a vector space over kkkof dimension four, U∗be\nHom kkk(U,kkk),Pbe the polynomial ring P=Sym•U, and D•U∗be the corresponding\ndivided power algebra.\nMarques, Veliche, and Weyman [16] describe all Artinian Gor enstein algebras of\nembedding dimension four and socle degree three over an alge braically closed field\nof characteristic zero. Their description uses a classifica tion of homogeneous forms\nof degree three over an algebraically closed field of charact eristic zero. (In charac-\nteristic zero, the Macaulay inverse system of an Artinian Go renstein kkk-algebra of\nsocle degree three may as well be a homogeneous cubic form bec ause D•U∗is\nisomorphic to Sym•U.)\nThe paper [16] expresses the defining ideal of each algebra un der consideration\nas the generalized sum of two linked perfect ideals of grade t hree. There are many\ncases, many ad hoc arguments, and there is no uniform answer. The paper [16]\nincludes an example that indicates that the conclusion migh t fail in characteristic\ntwo.\nThe characteristic two example [16, Ex. 5.6] is very interes ting! After changing\nthe variables, one sees that it is the unique Artinian Gorens teinkkk-algebra of embed-\nding dimension four and socle degree three which does not hav e the weak Lefschetz\nproperty. This algebra has a very pretty resolution and the d efining ideal is indeed\nthe sum of two linked perfect grade three ideals; see Section 8.\nWe proved in [14] that, with one exception, every standard gr aded Artinian\nGorenstein kkk-algebra of embedding dimension four and socle degree three has aARTINIAN GORENSTEIN ALGEBRAS 3\nweak Lefschetz element. The proof in [14] is fairly complica ted: it involves many\ncases and many ad hoc arguments. However, once one has the res ult of [14], then\none is able to prove the results of [16] over an arbitrary field in a uniform manner.\nIn addition to the result of [14], the other main ingredient i n the present paper is\nthe structure of Gorenstein-linear resolutions which is gi ven in [6]. In a sequence of\nthree papers, we gave explicit formulas for the differentia ls in the minimal homo-\ngeneous resolution of P/Iby free R-modules, where P=kkk[x1,..., xd]is a standard-\ngraded polynomial ring over the field kkkandIis a grade dhomogeneous Gorenstein\nideal, provided the resolution is Gorenstein-linear in the sense that the resolution\nhas the form\n0→Fdfd− →Fd−1fd−1− − → Fd−2fd−2− − →...f3− →F2f2− →F1f1− →P\nand all the entries of all of the matrices fi, with 2≤i≤d−1, are linear. The\nformulas are given in terms of the coef���cients of a Macaulay i nverse system for I.\nThe paper [4] proves that the project can be done; [5] carries out the project when\nd=3, and [6] carries out the project for all d.\nIn the present paper, Ais an Artinian Gorenstein kkk-algebra of embedding dimen-\nsion four and socle degree three. Let Φ3∈D3U∗be a Macaulay inverse system for\nA. In other words, Ais isomorphic to P/annΦ3. Fix an element xinUwhich is a\nweak Lefschetz element on A. Then P/annxΦ3has a Gorenstein-linear resolution;\nconsequently, the resolution of P/annxΦ3is given by the formulas of [6]. One\nquickly obtains a resolution of A. Furthermore, there is a symmetric 3 ×3 matrix\nof constants SM which controls the precise form of the minima l resolution of A.\nIf SM has rank zero, then the defining ideal of Ahas nine minimal generators and\nthe minimal resolution of Ais given in Theorem 9.1. If SM has rank one, then the\ndefining ideal of Ahas seven minimal generators. One may change variables to pu t\nSM in the form\n(1.0.1)\nα0 0\n0 0 0\n0 0 0\n,\nfor some nonzero α∈kkk. Once this is done, then the minimal resolution of Ais\ngiven in Theorems 10.3 and 12.1. If SM has rank at least two, th en the defining\nideal of Ahas six minimal generators, Ais a hypersurface section of a codimension\nthree Gorenstein ring, and the minimal resolution of Ais given in Theorems 13.1\nand 13.9.\nOne consequence of the above results is\nCorollary 1.1. If kkk is an arbitrary field, Ais a standard graded Artinian Gorenstein\nkkk-algebra of embedding dimension four and socle degree thre e, and π:P→Ais a\nsurjective graded homomorphism from a polynomial ring with four variables over\nkkk ontoA, then the defining ideal of Ahas6,7, or9minimal generators.4 SABINE EL KHOURY AND ANDREW R. KUSTIN\nCorollary 1.1 is already known in [1] and [16] when kkkhas characteristic zero.\nIndeed, these two papers predict the exact graded Betti numb ers ofA. We obtain\nthe exact same graded Betti numbers in the general case as are obtained in the\ncharacteristic zero case.\n2. N OTATION ,CONVENTIONS ,AND ELEMENTARY RESULTS .\nConvention 2.1. In this paper, kkkis an arbitrary, but fixed, field. Every unadorned\noperation is a functor in the category of kkk-modules. In particular, we write ⊗,∧,/logicalandtexti, Symj,Dj,Li\nj,Ki\nj, to mean ⊗kkk,∧kkk,/logicalandtexti\nkkk, Symkkk\nj,Dkkk\nj,Li\njkkkandKj\nikkk, respectively. We\nwrite U∗to mean Hom kkk(U,kkk),DU∗to mean D(U∗), and x∗(n)to mean(x∗)(n).\n2.2. IfMis a matrix with entries in the ring Pandris a positive integer, then Ir(M)\nis the ideal of Pgenerated by the r×rminors of M.\n2.3. LetIbe a proper ideal in a commutative Noetherian ring R. The grade ofIis\nthe length of a maximal R-regular sequence in I.\n2.4.LetMbe a nonzero finitely generated module over a Noetherian ring R, ann(M)\nbe the annihilator of M, and pdRMbe the projective dimension of M. It is always\ntrue that\ngradeann (M)≤pdRM.\nWhen equality holds, then Mis called a perfect R-module . Let Ibe a proper ideal\nin a commutative Noetherian ring R. The ideal Iis called a perfect ideal ifR/Iis a\nperfect R-module.\nConvention 2.5. For any set of variables {x1,..., xr}and any degree s, we write/parenleftbiggx1,..., xr\ns/parenrightbigg\nfor the set of monomials of degree sin the variables x1,..., xr.\nConventions 2.6. (a) The graded algebra A=/circleplustext\n0≤iAiis astandard graded A0-\nalgebra ifA1is finitely generated as an A0-module and Ais generated as an\nA0-algebra by A1.\n(b) Let kkkbe a field and A=/circleplustext\nAibe a standard graded kkk-algebra. Then Ahas\ntheweak Lefschetz property if there exists a linear form ℓofA1such that the\nkkk-module homomorphism\nµℓ:Ai→Ai+1\nhas maximal rank for each index i, where µℓis multiplication by ℓ. (A homo-\nmorphism ξ:V→Wof finitely generated kkk-modules has maximal rank if rank ξ\nis equal to min {dimV,dimW}.)\n(c) IfA=/circleplustextσ\ni=0Aiis an Artinian standard-graded kkk-algebra, then AisGorenstein\nwith socle degree σifAσis a one dimensional vector space and every nonzero\nideal of Acontains Aσ.ARTINIAN GORENSTEIN ALGEBRAS 5\n2.7. LetA=/circleplustextσ\ni=0Aibe an Artinian standard-graded kkk-algebra. Let Ube the vector\nspaceA1,Pbe the polynomial ring P=Sym•U,Ibe a homogeneous ideal of P\nwithAisomorphic to P/I,U∗be the dual space Hom kkk(U,kkk)ofU, and D•U∗be the\ndivided power kkk-algebra/circleplustext\n0≤iDiU∗, with\nDiU∗=Hom kkk(SymiU,kkk).\nThe rules for a divided power algebra are recorded in [9, Sect ion 7] or [3, Appendix\n2]. (In practice these rules say that ν(n)behaves like νn/(n!)would behave if n!\nwere a unit in kkk, for each element ν∈U∗.)\nThe algebras D•U∗and Sym•Uare modules over one another. If uais in SymaU\nandνb∈DbU∗, then uaνbis the element of\nDb−aU∗=Hom kkk(Symb−aU,kkk)\nwith\nuaνb(ub−a)=νb(uaub−a).\nMacaulay duality guarantees that\nann D•U∗I\nis a cyclic P-submodule of D•U∗generated by an element in DσU∗. (Any generator\nann D•U∗IinDσU∗is called a Macaulay inverse system for A.) Furthermore, if Φσ\nis a Macaulay inverse system for A, then\nI=ann PΦσ.\nAll of the results which are established in this paper use the following data and\nnotation.\nData 2.8. (a) Let kkkbe an arbitrary field, Ube a vector space over kkkof dimension\nfour, U∗be Hom kkk(U,kkk),Pbe the polynomial ring P=Sym•U, and D•U∗be\nthe corresponding divided power algebra. Let Φ3be an element of D3U∗andA\nbe the algebra P/annΦ3. Assume that xinUis a weak Lefschetz element in A\nand ann Φ3does not contain any nonzero linear elements.\n(b) Let p:U→U∗be the homomorphism p(u)=u(xΦ3).\n(c) Fix an element x∗∈U∗with x∗(x)=1. Let U0=ker(x∗:U→kkk). Observe that\nU0is akkk-vector space of dimension d−1 and U=kkkx⊕U0. Let\nU∗\n0={f∈U∗|f(x)=0}.\nObserve that U∗\n0=kkkx∗⊕U∗\n0.\n(d) Define the symmetric bilinear form /an}bracketle{t−,−/an}bracketri}ht: Sym2U0×Sym2U0→kkkby\n/an}bracketle{tu2,0,u′\n2,0/an}bracketri}ht=[u2,0p−1(u′\n2,0Φ3)](Φ3),\nfor elements u2,0andu′\n2,0in Sym2U0.6 SABINE EL KHOURY AND ANDREW R. KUSTIN\n(e) Fix bases ωU0andωU∗\n0for/logicalandtext3U0and/logicalandtext3U∗\n0with\nωU0(ωU∗\n0)=1.\nLet\nωU=x∧ωU0and ωU∗=ωU∗\n0∧x∗\nbe bases for/logicalandtext4Uand/logicalandtext4U∗.\n(f) If a basis y,z,wis chosen for U0, then y∗,z∗,w∗is automatically chosen to be\nthe basis for U∗\n0which is dual to the basis y,z,wforU0. Furthermore, ωU0and\nωU∗\n0are automatically chosen to be\nωU0=y∧z∧wand ωU∗\n0=w∗∧z∗∧y∗.\n(g) Let proj : Sym•U→Sym•U0the projection map induced by U=kkkx1⊕U0.\nRemark. Please notice that the final hypothesis in Data 2.8.(a), that “ann Φ3does\nnot contain any nonzero linear elements” is equivalent to as suming that Ahas em-\nbedding dimension four.\n2.9. LetUbe a finite dimensional vector space over the field kkk. The evaluation map\nDNU∗\n0⊗SymNU0ev− − → kkkis a coordinate-free map; consequently, the dual\nkkkev∗\n−−→ SymNU0⊗DNU∗\n0\nis also a coordinate-free map. Thus,\nev∗(1)=∑\nm∈{m}m⊗m∗\nis an element of SymNU0⊗DNU∗\n0which is completely coordinate-free, where {m}\nis a basis for Uand{m∗}is the corresponding dual basis for U∗.\nObservation 2.10. Adopt Data 2.8. The P-algebra P /annxΦ3has a Gorenstein-\nlinear resolution and the homomorphism p :U→U∗is an isomorphism.\nProof. The fact that Ahas embedding dimension four guarantees that zero is the\nonly linear element in ann Φ. The fact that xis a weak Lefschetz element on\nP/annΦ3guarantees that zero is the only linear element of ann xΦ3. The socle\ndegree of P/xannΦ3is two. Apply [4, Prop. 1.8] to see that P/annxΦ3has a\nGorenstein-linear resolution. /square\n2.11. LetU0be a finite dimensional vector space over the field kkk. The vector spaces\nLa\nbU0andKa\nbU0represent the Schur and Weyl functors for a hook as described , for\nexample, in [4, Data 2.1]. (The translation between our nota tion and the notation of\nBuchsbaum and Eisenbud and the notation of Weyman is contain ed in [4, 2.1 and\n2.5].) In particular,\nLa\nbU0=ker/parenleftig/logicalandtextaU0⊗SymbU0κ− →/logicalandtexta−1U0⊗Symb+1U0/parenrightig\nand\nKa\nbU0=ker/parenleftig/logicalandtextaU0⊗DbU∗\n0η− →/logicalandtexta−1U0⊗Db−1U∗\n0/parenrightig\n,ARTINIAN GORENSTEIN ALGEBRAS 7\nwhere κis a Koszul complex map and ηis an Eagon-Northcott complex map. If\nx1,..., xdandx∗\n1,..., x∗\ndare a pair of dual bases for UandU∗, then\n(2.11.1) κ(θ⊗u)=d\n∑\ni=2x∗\ni(θ)⊗xiuand η(θ⊗ν)=d\n∑\ni=2x∗\ni(θ)⊗xi(ν)\nforθ∈/logicalandtextaU0,u∈SymbU0, and ν∈DbU∗\n0. In light of (2.9), the description of κ\nandηgiven in (2.11.1) is coordinate-free.\n2.12. We often write “after a change of variables” some expression becomes some\nother expression. Indeed, the paper [14] guarantees that (w ith one exception) every\nGorenstein algebra Aunder consideration (as described in Data 2.8) contains a we ak\nLefschetz element. This paper always assumes (except in Sec tion 8) that xis a\nweak Lefschetz element. Often it is necessary to change the v ariables in order to\nmake xbe a weak Lefschetz element. Also, when we find the precise for m of the\nminimal resolution when SM has rank one we assume that one has made a change\nof variables to put “SM” in the form (1.0.1). It is not difficul t to change variables.\nWe show the procedure here.\nSuppose that x,y,z,wandx∗,y∗,z∗,w∗is a pair of dual bases for UandU∗, re-\nspectively and that X,Y,Z,WandX∗,Y∗,Z∗,W∗is another pair of dual bases for U\nandU∗. IfMis an invertible matrix with entries in kkkand\n/bracketleftbigX Y Z W/bracketrightbig\n=/bracketleftbigx y z w/bracketrightbig\nM,\nthen/bracketleftbigX∗Y∗Z∗W∗/bracketrightbig\nMT=/bracketleftbigx∗y∗z∗w∗/bracketrightbig\n.\n3. T HE STRUCTURE OF GORENSTEIN -LINEAR RESOLUTIONS ,GENERAL CASE .\nWe record the main result of [6]. We employ a different point o f view than the\none used in [6]. We work over a field kkkrather than over the ring of integers Z. The\ncoefficients of our Macaulay inverse system Φ2n−2are elements of kkkrather than\nvariables. We deal with the inverse of prather than “ q”, which is essentially the\nclassical adjoint of p. We do not have reason to write det p. In [6], δδδrepresents\nthe determinant of p; in other words, δδδis a polynomial of high degree in the coef-\nficients of Φ2n−2. The results of [6] hold after the coefficients of Φ2n−2have been\nspecialized to elements of kkk, provided the image of δδδis a unit.\nThere is one other difference between the language of [6] and the language we\nuse. In both situations, it is necessary to consider an “inte gral” of Φ2n−2; in other\nwords, we must consider an element Φ2n−1with\n(3.0.1) x(Φ2n−1)=Φ2n−2\nThe plan of [6] was to specify everything and leave no choices . In [6] we chose\nΦ2n−1to have a special property in addition to (3.0.1). We insiste d that when Φ2n−18 SABINE EL KHOURY AND ANDREW R. KUSTIN\nwas written as an element of\nD2n−1U∗\n0⊕D2n−2U∗\n0x∗⊕D2n−3U∗\n0x∗(2)⊕...⊕D0U∗\n0x∗(2n−1),\n(where the decomposition U=kkkx⊕U0had been previously made), then the “con-\nstant term” in D2n−1U∗\n0is zero. (See, for example, [7, Rem. 3.4.(a)] or [6, Data\n2.5.(d)].) Unfortunately, we later realized that this choi ce limited the potential ap-\nplications of [6]. If one uses an arbitrary Φ2n−1rather than the special /tildewideΦ2n−1, with\n“constant term” zero, one gets a different resolution; howe ver, the new resolution is\nisomorphic to the old resolution. When d=3, the argument for this isomorphism\nis given explicitly as [7, Thm. 3.6]. The exact same argument works for all d.\nData 3.1. Letkkkbe a field, Ube a vector space over kkkof dimension d, and n≥2 be\nan integer. Let Φ2n−2be an element of D2n−2U∗. Assume that the homomorphism\np: Symn−1U→Dn−1U∗, given by p(un−1)=un−1Φ2n−2, is an isomorphism.\n(a) Fix elements x1∈Uandx∗\n1∈U∗with x∗\n1(x1)=1.\n(b) Let U0=ker(x∗\n1:U→kkk). Observe that U0is akkk-vector space of dimension\nd−1 and U=kkkx1⊕U0.\n(c) View U∗\n0as the submodule of U∗with U∗\n0(x1)=0.\n(d) Let Φ2n−1be an element of P⊗D2n−1U∗with x1(Φ2n−1)=Φ2n−2inD2n−2U∗.\n(e) Let x2,..., xdbe a basis for U0andx∗\n2,..., x∗\ndbe the corresponding dual basis\nforU∗\n0.\n(f) Let proj : Symn−1U→Symn−1U0the projection map induced by U=kkkx1⊕U0.\nThe following result is the main theorem of [6]. The complex (F,f)is defined\nin [6, Def. 3.1]. The complex (F,f)is shown to be a resolution in [6, Cor. 6.17.g].\nThe universal property of (F,f)is established in [6, Cor. 6.18]. The vector spaces\nKa\nbU0andLa\nbU0are defined in 2.11.\nTheorem 3.2. Adopt Data 3.1. Then the minimal homogeneous resolution of\nP/annΦ2n−2\nby free P-modules is given by\n(F,f): 0→Fdfd− →Fd−1fd−1− − →...f2− →F1f1− →F0,\nas described below.\n(a)Define the free P-module F rby\nFr=\n\nP, if r=0,\nP⊗(Kr−1\nn−1U0⊕Lr−1\nnU0),if1≤r≤d−1,\nP⊗/logicalandtextd−1U0, if r=d.\n(b)The P-module homomorphism f 1:F1→F0=P is defined by\nf1(νn−1,0)=x1p−1(νn−1,0), forνn−1,0∈K0\nn−1U0,\nf1(un,0)=un,0−x1p−1(un,0(Φ2n−1)),for u n,0∈L0\nnU0.ARTINIAN GORENSTEIN ALGEBRAS 9\n(c)For each integer r, with 2≤r≤d−1, the P-module homomorphism\nfr:P⊗Kr−1\nn−1U0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n⊆Fr→Fr−1=\n\nP⊗Kr−2\nn−1U0\n⊕\nP⊗Lr−2\nnU0\nis defined by f r(∑iθr−1,0,i⊗νn−1,0,i)is equal to\n(3.2.1)\n\n−x1⊗∑iη/parenleftbig\nθr−1,0,i⊗[p−1(νn−1,0,i)](Φ2n−1)/parenrightbig\n+∑d\nj=2xj⊗x∗\nj(θr−1,0,i)⊗νn−1,0,i\n−x1⊗∑iκ(θr−1,0,i⊗proj(p−1(νn−1,0,i))),\nforθr−1,0,i∈/logicalandtextr−1U0,νn−1,0,i∈Dn−1U∗\n0, and ∑iθr−1,0,i⊗νn−1,0,i∈Kr−1\nn−1U0.\n(d)For each integer r, with 2≤r≤d−1, the P-module homomorphism\nfr:P⊗Lr−1\nnU0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n⊆Fr→Fr−1=\n\nP⊗Kr−2\nn−1U0\n⊕\nP⊗Lr−2\nnU0\nis defined by f r(∑iθr−1,0,i⊗un,0,i)is equal to\n(3.2.2)\n\nx1⊗∑iη/parenleftbig\nθr−1,0,i⊗[p−1(un,0,i(Φ2n−1))](Φ2n−1)/parenrightbig\n+x1⊗∑iκ/parenleftbig\nθr−1,0,i⊗proj(p−1(un,0,i(Φ2n−1)))/parenrightbig\n+∑d\nj=1∑ixj⊗x∗\nj(θr−1,0,i)⊗un,0,i,\nforθr−1,0,i∈/logicalandtextr−1U0, un,0,i∈SymnU0, and ∑iθr−1,0,i⊗un,0,i∈Lr−1\nnU0.\n(e)The P-module homomorphism f d:Fd→Fd−1is defined by\nfd:P⊗/logicalandtextd−1U0=Fd−→Fd−1=\n\nP⊗Kd−2\nn−1U0\n⊕\nP⊗Ld−2\nnU0,\nwith\nFd(ωd−1,0)=\n\n∑\nm∈(x2,...,xdn)[m−x1p−1(m(Φ2n−1))]⊗η(ωd−1,0⊗m∗)\n− ∑\nm∈(x2,...,xd\nn−1)x1p−1(m∗)⊗κ(ωd−1,0⊗m),\nforωd−1,0∈/logicalandtextd−1U0.\nRemark 3.3. One key feature of the differential given in Theorem 3.2 is th at the\nexpressions\n∑\niη/parenleftbig\nθr−1,0,i⊗[p−1(νn−1,0,i)](Φ2n−1)/parenrightbig\nand\n∑\niη/parenleftbig\nθr−1,0,i⊗[p−1(un,0,i(Φ2n−1))](Φ2n−1)/parenrightbig\nfrom (3.2.1) and (3.2.2) are automatically in Kr−1\nn−1U0, no projection is necessary. Of\ncourse, these expressions are automatically in Kr−1\nn−1U. One could apply a projection10 SABINE EL KHOURY AND ANDREW R. KUSTIN\nmap to look at the component in Kr−1\nn−1U0. However this projection map would be\n“in the way” as one verifies that (F,f)is a complex. It is fortunate that no projection\nmap is needed. To show that\n∑iη/parenleftbig\nθr−1,0,i⊗[p−1(νn−1,0,i)](Φ2n−1)/parenrightbig\nis in Kr−1\nn−1U0it suffices to show that\n(3.3.1) (1⊗x1)/parenleftbig\n∑iη/parenleftbig\nθr−1,0,i⊗[p−1(νn−1,0,i)](Φ2n−1)/parenrightbig/parenrightbig\n=0.\nWe prove (3.3.1) mainly to give the flavor of how the different ial of(F,f)works.\nObserve that the left side of (3.3.1) is equal to\n∑iη/parenleftbig\nθr−1,0,i⊗x1[p−1(νn−1,0,i)](Φ2n−1)/parenrightbig\n=∑iη/parenleftbig\nθr−1,0,i⊗[p−1(νn−1,0,i)](x1Φ2n−1)/parenrightbig\n=∑iη(θr−1,0,i⊗p[p−1(νn−1,0,i)])= ∑iη(θr−1,0,i⊗νn−1,0,i)=0.\nThe first equality holds because the action of 1 ⊗x1on/logicalandtext•U⊗D•U∗commutes\nwith the Eagon-Northcott map η. The second equality holds because D•U∗is a\nmodule over Sym•U. The third equality uses the definition of p, which is given in\nData 3.1. The conclusion follows from the hypothesis that ∑iθr−1,0,i⊗νn−1,0,iis in\nKr−1\nn−1U0.\nRemark 3.4. We also demonstrate that the composition\nP⊗L1\nnU0f2− →F1f1− →P\nfrom the resolution (F,f)of Theorem 3.2 is zero. Our motivation for recording this\nargument here is the same as our motivation for recording Rem ark 3.3. We want to\nconvey some of the tricks involved in studying the resolutio n(F,f).\nTake θ1,0,i∈/logicalandtext1U0andun,0,i∈SymnU0, with ∑iθ1,0,i⊗un,0,i∈L1\nnU0. Observe\nthat(f1◦f2)(∑iθ1,0,i⊗un,0,i)isS1+S2+S3+S4+S5with\nS1=x2\n1·p−1/parenleftig\n∑iθ1,0,i/bracketleftig\n[p−1(un,0,i(Φ2n−1)](Φ2n−1)/bracketrightig/parenrightig\n,\nS2=x1·/bracketleftig\n∑i/parenleftig\nθ1,0,i·proj/parenleftbig\np−1[un,0,i(Φ2n−1)]/parenrightbig/parenrightig/bracketrightig\n,\nS3=−x2\n1·p−1/parenleftig/bracketleftig\n∑i/parenleftig\nθ1,0,i·proj/parenleftbig\np−1[un,0,i(Φ2n−1)]/parenrightbig/parenrightig/bracketrightig\n(Φ2n−1)/parenrightig\n,\nS4=−∑ix1θ1,0,i·p−1[(un,0,i)(Φ2n−1)],and\nS5=∑iθ1,0,i·(un,0,i)=0.\nObserve further that\nS1+S3=x2\n1·p−1/parenleftig/bracketleftig\n∑i/parenleftig\nθ1,0,i·(1−proj)/parenleftbig\np−1[un,0,i(Φ2n−1)]/parenrightbig/parenrightig/bracketrightig\n(Φ2n−1)/parenrightig\nS2+S4=−x1·/bracketleftig\n∑i/parenleftig\nθ1,0,i·(1−proj)/parenleftbig\np−1[un,0,i(Φ2n−1)]/parenrightbig/parenrightig/bracketrightig\n.ARTINIAN GORENSTEIN ALGEBRAS 11\nApply Observation 3.5 to\nun=/bracketleftig\n∑i/parenleftig\nθ1,0,i·(1−proj)/parenleftbig\np−1[un,0,i(Φ2n−1)]/parenrightbig/parenrightig/bracketrightig\nto see that\nS1+S3=x1·/bracketleftig\n∑i/parenleftig\nθ1,0,i·(1−proj)/parenleftbig\np−1[un,0,i(Φ2n−1)]/parenrightbig/parenrightig/bracketrightig\n=−(S2+S4).\nThus, the restriction of f1◦f2toL1\nnU0is identically zero.\nObservation 3.5. Adopt Data 3.1. If u n∈x1Symn−1U, then x 1p−1(un(Φ2n−1))is\nequal to u n.\nProof. Letun=x1un−1. Observe that\nx1p−1(un(Φ2n−1))= x1p−1(x1un−1(Φ2n−1))= x1p−1p(un−1)=x1un−1=un.\n/square\nProposition 7.10 of [6] describes the self-duality of (F,f). We will not describe\nthis self-duality in the situation of Theorem 3.2; however, we exploit the duality in\nSection 4 where we record (F,f)when d=4 and n=2. The duality means that\nwe need only to record f1andf2because f3andf4are obtained from f1andf2by\ntransposing and rearranging.\n4. T HE STRUCTURE OF GORENSTEIN -LINEAR RESOLUTIONS ,WHEN d=4\nAND n=2.\nWe record the resolution (F,f)of Theorem 3.2 when d=4 and n=2. We\ntake advantage of the duality of [6, Prop. 7.10]. We use names for the modules of\nFwhich are less cumbersome than the Schur and Weyl modules La\nbU0andKa\nbU0.\nThe resolution is written using maps in Theorem 4.2 and in ter ms of matrices in\nExample 4.6.\nObservation 4.1. Adopt Data 2.8. Then\nK0\n1U0=U∗\n0, L0\n2U0=Sym2U0, K1\n1U0=ker(U0⊗U∗\n0ev− →kkk),\nL1\n2U0∼=U0⊗U∗\n0\nev∗(1), K2\n1U0∼=D2U∗\n0,and L2\n2U0∼=U0.\nProof. Observe that\nK0\n1U0=ker(/logicalandtext0U0⊗D1U∗\n0→0)=U∗\n0,\nL0\n2U0=ker(/logicalandtext0⊗Sym2U0→0)=Sym2U0,and\nK1\n1U0=ker(/logicalandtext1U0⊗D1U∗\n0ev− →kkk).\nThe Koszul complex\n0→/logicalandtext3U0→/logicalandtext2U0⊗U0→/logicalandtext1U0⊗Sym2U0→/logicalandtext0U0⊗Sym3U0→012 SABINE EL KHOURY AND ANDREW R. KUSTIN\nis split exact; so,\nL1\n2U0=ker(/logicalandtext1U0⊗Sym2U0→/logicalandtext0U0⊗Sym3U0)∼=/logicalandtext2U0⊗U0\nim/logicalandtext3U0.\nIt follows that\n(4.1.1) map1:U0⊗U∗\n0\nev∗(1)→L1\n2U0,\nwhich is induced by\nmap1(u1,0⊗ν1,0)=−κ(ν1,0(ωU0)⊗u1,0),\nforu1,0∈U0andν1,0∈U∗\n0, is an isomorphism. The Eagon-Northcott complex\n0→/logicalandtext3U0⊗D2U∗\n0η− →/logicalandtext2U0⊗D1U∗\n0η− →/logicalandtext1U0⊗D0U∗\n0η− →0\nis also split exact. It follows that\n(4.1.2) map2:D2U∗\n0→K2\n1U0,\nwhich is given by\nmap2(ν2,0)=η(ωU0⊗ν2,0),\nforν2,0∈D2U∗\n0, is an isomorphism. In a similar manner,\n(4.1.3) map3:U0→L2\n2U0,\nwhich is given by\nmap3(u1,0)=κ(ωU0⊗u1,0),\nforu1,0∈U0, is an isomorphism. /square\nWe use the isomorphisms of Observation 4.1 and the duality of [6, Prop. 7.3] to\nrecord the resolution (F,f)of Theorem 3.2 in the following form when d=4 and\nn=2.\nTheorem 4.2. Adopt Data 2.8. The minimal homogeneous resolution of P /annxΦ3\nby free P modules is given by\n(F,f): 0→F4f4− →F3f3− →F2f2− →F1f1− →F0,\nwhere\nF4=P, F3=(P⊗U0)⊕(P⊗D2U∗\n0),\nF2=\n\nP⊗ker(ev :U0⊗U∗\n0→kkk)\n⊕\nP⊗U0⊗U∗\n0\nev∗(1),F1=(P⊗U∗\n0)⊕(P⊗Sym2U0),\nF0=P,\nf4=/bracketleftbiggf∗\n1,1\nf∗\n1,2/bracketrightbigg\n, f3=/bracketleftbigg\nx⊗B∗D∗\nA∗x⊗C∗/bracketrightbigg\n,\nf2=/bracketleftbigg\nA x⊗B\nx⊗C D/bracketrightbigg\n,and f1=/bracketleftbigf1,1f1,2/bracketrightbig\n.ARTINIAN GORENSTEIN ALGEBRAS 13\nIn the following discussion, u i,0is an element of SymiU0andνi,0is an element of\nDiU∗\n0.\n•The mapf1,1:U∗\n0→P sends ν1,0to xp−1(ν1,0).\n•The mapf1,2: Sym2U0→P sends u 2,0to u 2,0−xp−1(u2,0Φ3).\n•The mapA: ker(U0⊗U∗\n0→kkk)→P⊗U∗\n0is induced by the map\nU0⊗U∗\n0→P⊗U∗\n0\nwhich sends\nu1,0⊗ν1,0/mapsto→− x⊗[u1,0p−1(v1,0)](Φ3)+u1,0⊗ν1,0.\n•The mapB:U0⊗U∗\n0\nev∗(1)→U∗\n0sends the class of u 1,0⊗ν1,0to\n/braceleftigg\n−/bracketleftbig\nyp−1/parenleftbig\n(ν1,0(z∧w)u1,0)(Φ3)/parenrightbig/bracketrightbig\n(Φ3)+/bracketleftbig\nzp−1/parenleftbig\n(ν1,0(y∧w)u1,0)(Φ3)/parenrightbig/bracketrightbig\n(Φ3)\n−/bracketleftbig\nwp−1/parenleftbig\n(ν1,0(y∧z)u1,0)(Φ3)/parenrightbig/bracketrightbig\n(Φ3).\n•The mapC: ker(U0⊗U∗\n0→kkk)→Sym2U0is induced by the map\nU0⊗U∗\n0→Sym2U0\nwhich sends u 1,0⊗ν1,0to−u1,0·(proj◦p−1)(ν1,0).\n•The mapD:U0⊗U∗\n0\nev∗(1)→P⊗Sym2U0sends the class of u 1,0⊗ν1,0to\n(4.2.1)\n\n−x⊗w·(proj◦p−1)/parenleftig\n[ν1,0(y∧z)u1,0](Φ3)/parenrightig\n+x⊗z·(proj◦p−1)/parenleftig\n[ν1,0(y∧w)u1,0](Φ3)/parenrightig\n−x⊗y·(proj◦p−1)/parenleftig\n[ν1,0(z∧w)u1,0](Φ3)/parenrightig\n−w⊗ν1,0(y∧z)u1,0+z⊗ν1,0(y∧w)u1,0−y⊗ν1,0(z∧w)u1,0.\nRemarks 4.3. (a) The entries in the matrices for f1,1andf1,2are homogeneous of\ndegree two, for AandDthe entries are homogeneous of degree one, and for B\nandCthe entries are homogeneous of degree zero.\n(b) Theorem 4.2 is implemented in the Macaulay2 script “Basi cSocDeg3”; see Sec-\ntion 14.A.1.\nProof. The maps f1,1,f1,2,A, andCare obviously correct. The map Dis the\ncomposition\nP⊗U0⊗U∗\n0\nev∗(1)map1− −− → P⊗L1\n2U0map4− −− → P⊗Sym2U0,\nwhere map1is given in (4.1.1) and map4is the composition\nP⊗L1\n2U0inclusion− −−−− → F2f2− →F1projection− −−−− → P⊗L0\n2U0=P⊗Sym2U0.\nIfu1,0∈U0andν1,0∈U∗\n0, then map1of the class of u1,0⊗ν1,0inU0⊗U∗\n0\nev∗(1)is equal to\n=−κ(ν1,0(y∧z∧w)⊗u1,0)\n=κ/parenleftbig\n−ν1,0(y)·z∧w⊗u1,0+ν1,0(z)·y∧w⊗u1,0−ν1,0(w)·y∧z⊗u1,0/parenrightbig14 SABINE EL KHOURY AND ANDREW R. KUSTIN\n=\n\n−ν1,0(y)·w⊗zu1,0+ν1,0(z)·w⊗yu1,0\n+ν1,0(y)·z⊗wu1,0−ν1,0(w)·z⊗yu1,0\n−ν1,0(z)·y⊗wu1,0+ν1,0(w)·y⊗zu1,0\n=−w⊗ν1,0(y∧z)u1,0+z⊗ν1,0(y∧w)u1,0−y⊗ν1,0(z∧w)u1,0. (4.3.1)\nHence,Dof the class of u1,0⊗ν1,0is\n=\n\n−x⊗w·(proj◦p−1)/parenleftig\n[ν1,0(y∧z)u1,0](Φ3)/parenrightig\n+x⊗z·(proj◦p−1)/parenleftig\n[ν1,0(y∧w)u1,0](Φ3)/parenrightig\n−x⊗y·(proj◦p−1)/parenleftig\n[ν1,0(z∧w)u1,0](Φ3)/parenrightig\n−w⊗ν1,0(y∧z)u1,0+z⊗ν1,0(y∧w)u1,0−y⊗ν1,0(z∧w)u1,0.\nThe map Bis the composition\nU0⊗U∗\n0\nev∗(1)map1− −− → L1\n2U0map5− −− → U∗\n0,\nwhere\n(4.3.2) map5(∑iu1,0,i⊗u2,0,i)=∑\ni/bracketleftbig\nu1,0,i[p−1(u2,0,i(Φ3))]/bracketrightbig\n(Φ3)\nforu1,0,i∈/logicalandtext1U0andu2,0,i∈Sym2U0with ∑iu1,0,i⊗u2,0,i∈L1\n2U0.\nIfu1,0∈U0andν1,0∈U∗\n0, then apply (4.3.1) to see that Bsends the class of\nu1,0⊗ν1,0inU0⊗U∗\n0\nev∗(1)to\nmap5/braceleftigg\n−ν1,0(y)·w⊗zu1,0+ν1,0(z)·w⊗yu1,0+ν1,0(y)·z⊗wu1,0\n−ν1,0(w)·z⊗yu1,0−ν1,0(z)·y⊗wu1,0+ν1,0(w)·y⊗zu1,0\n=\n\n−ν1,0(y)·/bracketleftbig\nw[p−1(zu1,0)](Φ3)/bracketrightbig\n(Φ3)+ν1,0(z)·/bracketleftbig\nw[p−1(yu1,0)](Φ3)/bracketrightbig\n(Φ3)\n+ν1,0(y)·/bracketleftbig\nz[p−1(wu1,0)](Φ3)/bracketrightbig\n(Φ3)−ν1,0(w)·/bracketleftbig\nz[p−1(yu1,0)](Φ3)/bracketrightbig\n(Φ3)\n−ν1,0(z)·/bracketleftbig\ny[p−1(wu1,0)](Φ3)/bracketrightbig\n(Φ3)+ν1,0(w)·/bracketleftbig\ny[p−1(zu1,0)](Φ3)/bracketrightbig\n(Φ3)\n=\n\n−/bracketleftbig\nyp−1/parenleftbig\n(ν1,0(z∧w)u1,0)(Φ3)/parenrightbig/bracketrightbig\n(Φ3)\n+/bracketleftbig\nzp−1/parenleftbig\n(ν1,0(y∧w)u1,0)(Φ3)/parenrightbig/bracketrightbig\n(Φ3)\n−/bracketleftbig\nwp−1/parenleftbig\n(ν1,0(y∧z)u1,0)(Φ3)/parenrightbig/bracketrightbig\n(Φ3).\nWhen d=4 and n=2, then the self-duality of (F,f)as described in [6, Prop. 7.10]\nis a consequence of the natural perfect pairings\nK0\n1U0⊗L2\n2U0→kkk,\nK1\n1U0⊗L1\n2U0→kkk,and\nK2\n1U0⊗L0\n2U0→kkk.ARTINIAN GORENSTEIN ALGEBRAS 15\nThese perfect pairings are compatible with the isomorphism s of Observation 4.1\nand become the natural perfect pairings\nU∗\n0⊗U0→kkk,\nker(U0⊗U∗\n0→kkk)⊗U0⊗U∗\n0\nev∗(1)→kkk,and\nD2U∗\n0⊗Sym2U0→kkk.\nWhen we record matrices for the resolution (F,f)in Example 4.6, we use the bases\n(4.3.3)ker(U0⊗U∗\n0→kkk)U0⊗U∗\n0\nev∗(1)\ny⊗y∗−w⊗w∗y⊗y∗\ny⊗z∗z⊗y∗\ny⊗w∗w⊗y∗\nz⊗y∗y⊗z∗\nz⊗z∗−w⊗w∗z⊗z∗\nz⊗w∗w⊗z∗\nw⊗y∗y⊗w∗\nw⊗z∗z⊗w∗andF1F3\ny∗y\nz∗z\nw∗w\ny2y∗(2)\nyz y∗z∗\nyw y∗w∗\nz2z∗(2)\nzw z∗w∗\nw2w∗(2).\nEach basis vector is listed next to its dual. The matrix for f1,1is denoted f1,1; for\nf1,2is denoted f1,2; forAis denoted A; forBis denoted B; forCis denoted C; and\nforDis denoted D. The matrices for A∗,B∗,C∗,D∗,f∗\n1,1, and f∗\n1,2automatically\nbecome AT,BT,CT,DT,fT\n1,1, and fT\n1,2, respectively. /square\nRemark 4.4. An alternate description of the map B:U0⊗U∗\n0\nev∗(1)→U∗\n0is available. If\nu′\n1,0,u′′\n1,0andu′′′\n1,0are elements of U0, thenBof the class of u′\n1,0⊗(u′′\n1,0∧u′′′\n1,0)(ωU∗\n0)\ninU0⊗U∗\n0\nev∗(1)is the element of U∗\n0with\n/bracketleftbig\nB/parenleftbig\nu′\n1,0⊗(u′′\n1,0∧u′′′\n1,0)(ωU∗\n0)/parenrightbig/bracketrightbig\n(u1,0)=/an}bracketle{tu1,0u′′\n1,0,u′\n1,0u′′′\n1,0/an}bracketri}ht−/an}bracketle{tu1,0u′′′\n1,0,u′\n1,0u′′\n1,0/an}bracketri}ht,\nfor all u1,0∈U0. The map /an}bracketle{t−,−/an}bracketri}htis defined in Data 2.8.\nProof. We compute\n/bracketleftbig\nB/parenleftbig\nu′\n1,0⊗(u′′\n1,0∧u′′′\n1,0)(ωU∗\n0)/parenrightbig/bracketrightbig\n(u1,0)\n=/bracketleftbig\n(map5◦map1)/parenleftbig\nu′\n1,0⊗(u′′\n1,0∧u′′′\n1,0)(ωU∗\n0)/parenrightbig/bracketrightbig\n(u1,0)\n=−/bracketleftbig\n(map5◦κ)/parenleftbig\n(u′′\n1,0∧u′′′\n1,0)⊗u′\n1,0/parenrightbig/bracketrightbig\n(u1,0) by (4.1.1)\n=−/bracketleftbig\nmap5/parenleftbig\nu′′′\n1,0⊗u′′\n1,0u′\n1,0−u′′\n1,0⊗u′′′\n1,0u′\n1,0/parenrightbig/bracketrightbig\n(u1,0)\n=−/parenleftbig\nu′′′\n1,0[p−1(u′′\n1,0u′\n1,0Φ3)](Φ3)−u′′\n1,0[p−1(u′′′\n1,0u′\n1,0Φ3)](Φ3)/parenrightbig\n(u1,0)by (4.3.2)\n=−u1,0u′′′\n1,0[p−1(u′′\n1,0u′\n1,0Φ3)](Φ3)+u1,0u′′\n1,0[p−1(u′′′\n1,0u′\n1,0Φ3)](Φ3)\n=/an}bracketle{tu1,0u′′\n1,0,u′′′\n1,0u′\n1,0/an}bracketri}ht−/an}bracketle{tu1,0u′′′\n1,0,u′′\n1,0u′\n1,0/an}bracketri}ht.\n/square16 SABINE EL KHOURY AND ANDREW R. KUSTIN\nRemark 4.5. The map B∗:U0→ker(U0⊗U∗\n0→kkk)is given by\nB∗(u′\n1,0)=\n\nw⊗[zp−1(yu′\n1,0)(Φ3)](Φ3)\n−z⊗[wp−1(yu′\n1,0)(Φ3)](Φ3)\n−w⊗[yp−1(zu′\n1,0)(Φ3)](Φ3)\n+y⊗[wp−1(zu′\n1,0)(Φ3)](Φ3)\n+z⊗[yp−1(wu′\n1,0)(Φ3)](Φ3)\n−y⊗[zp−1(wu′\n1,0)(Φ3)](Φ3),\nfor all u′\n1,0inU0.\nProof. A routine calculation shows that the formula given above sat isfies\n[B(u1,0⊗ν1,0)](u′\n1,0)=( ν1,0⊗u1,0)(B∗(u′\n1,0))\nfor all u1,0andu′\n1,0inU0and all ν1,0∈U∗\n0. /square\nExample 4.6. We record the matrices of the resolution (F,f)from Theorem 4.2\nexplicitly. For each matrix, we list the basis of the domain a s the top row and the\nthe target as the left most column. The bases are listed in 4.3 .3. The matrix for X∗\nis the transpose of Xfor each matrix X.\nFor example, the first column of Bshould be read to say:\nB(y⊗y∗)=0y∗+(/an}bracketle{tyw,z2/an}bracketri}ht−/an}bracketle{tyz,zw/an}bracketri}ht)z∗+(/an}bracketle{tyw,zw/an}bracketri}ht−/an}bracketle{tyz,w2/an}bracketri}ht)w∗.ARTINIAN GORENSTEIN ALGEBRAS 17f1,1=y∗z∗w∗\n1 xp−1(y∗)xp−1(z∗)xp−1(w∗),\nf1,2=y2yz yw z2zw w2\n1 y2−xp−1(y2Φ3)yz−xp−1(yzΦ3)yw−xp−1(ywΦ3)z2−xp−1(z2Φ3)zw−xp−1(zwΦ3)w2−xp−1(w2Φ3),\nA=y⊗y∗y⊗z∗y⊗w∗z⊗y∗z⊗z∗z⊗w∗w⊗y∗w⊗z∗\n−w⊗w∗−w⊗w∗\ny∗−[y2p−1(y∗)](Φ3)·x−[y2p−1(z∗)](Φ3)·x−[y2p−1(w∗)](Φ3)·x−[yzp−1(y∗)](Φ3)·x−[yzp−1(z∗)](Φ3)·x−[yzp−1(w∗)](Φ3)·x−[ywp−1(y∗)](Φ3)·x−[ywp−1(z∗)](Φ3)·x\n+[ywp−1(w∗)](Φ3)·x +z +[ywp−1(w∗)](Φ3)·x +w\n+y\nz∗−[yzp−1(y∗)](Φ3)·x−[yzp−1(z∗)](Φ3)·x−[yzp−1(w∗)](Φ3)·x−[z2p−1(y∗)](Φ3)·x−[z2p−1(z∗)](Φ3)·x���[z2p−1(w∗)](Φ3)·x−[zwp−1(y∗)](Φ3)·x−[zwp−1(z∗)](Φ3)·x\n+[wzp−1(w∗)](Φ3)·x+y +[wzp−1(w∗)](Φ3)·x +w\n+z\nw∗−[ywp−1(y∗)](Φ3)·x−[ywp−1(z∗)](Φ3)·x−[ywp−1(w∗)](Φ3)·x−[zwp−1(y∗)](Φ3)·x−[zwp−1(z∗)](Φ3)·x−[zwp−1(w∗)](Φ3)·x−[w2p−1(y∗)](Φ3)·x−[w2p−1(z∗)](Φ3)·x\n+[w2p−1(w∗)](Φ3)·x +y +[w2p−1(w∗)](Φ3)·x+z\n−w −w,\nB=y⊗y∗z⊗y∗w⊗y∗y⊗z∗z⊗z∗w⊗z∗y⊗w∗z⊗w∗\ny∗0 /an}bracketle{tyz,zw/an}bracketri}ht/an}bracketle{tyz,w2/an}bracketri}ht 0 /an}bracketle{tyw,yz/an}bracketri}ht/an}bracketle{tyw,yw/an}bracketri}ht0 /an}bracketle{ty2,z2/an}bracketri}ht\n−/an}bracketle{tyw,z2/an}bracketri}ht−/an}bracketle{tyw,zw/an}bracketri}ht −/an}bracketle{ty2,zw/an}bracketri}ht−/an}bracketle{ty2,w2/an}bracketri}ht −/an}bracketle{tyz,yz/an}bracketri}ht\nz∗/an}bracketle{tyw,z2/an}bracketri}ht 0 /an}bracketle{tz2,w2/an}bracketri}ht/an}bracketle{ty2,zw/an}bracketri}ht 0 /an}bracketle{tzw,yw/an}bracketri}ht/an}bracketle{tyz,yz/an}bracketri}ht 0\n−/an}bracketle{tyz,zw/an}bracketri}ht −/an}bracketle{tzw,zw/an}bracketri}ht−/an}bracketle{tyw,yz/an}bracketri}ht −/an}bracketle{tw2,yz/an}bracketri}ht−/an}bracketle{ty2,z2/an}bracketri}ht\nw∗/an}bracketle{tyw,zw/an}bracketri}ht/an}bracketle{tzw,zw/an}bracketri}ht 0 /an}bracketle{ty2,w2/an}bracketri}ht/an}bracketle{tyz,w2/an}bracketri}ht 0 /an}bracketle{tyz,yw/an}bracketri}ht/an}bracketle{tz2,yw/an}bracketri}ht\n−/an}bracketle{tyz,w2/an}bracketri}ht−/an}bracketle{tz2,w2/an}bracketri}ht −/an}bracketle{tyw,yw/an}bracketri}ht−/an}bracketle{tzw,yw/an}bracketri}ht −/an}bracketle{ty2,wz/an}bracketri}ht−/an}bracketle{tyz,zw/an}bracketri}ht,18 SABINE EL KHOURY AND ANDREW R. KUSTIN\nC=y⊗y∗y⊗z∗y⊗w∗z⊗y∗z⊗z∗z⊗w∗w⊗y∗w⊗z∗\n−w⊗w∗−w⊗w∗\ny2−[y∗p−1(y∗)]−[y∗p−1(z∗)]−[y∗p−1(w∗)] 0 0 0 0 0\nyz−[z∗p−1(y∗)]−[z∗p−1(z∗)]−[z∗p−1(w∗)]−[y∗p−1(y∗)]−[y∗p−1(z∗)]−[y∗p−1(w∗)] 0 0\nyw 0 −[w∗p−1(z∗)]−[w∗p−1(w∗)]0 +[y∗p−1(w∗)] 0 −[y∗p−1(y∗)]−[y∗p−1(z∗)]\nz20 0 0 −[z∗p−1(y∗)]−[z∗p−1(z∗)]−[z∗p−1(w∗)] 0 0\nzw[z∗p−1(w∗)] 0 0 −[w∗p−1(y∗)]0 −[w∗p−1(w∗)]−[z∗p−1(y∗)]−[z∗p−1(z∗)]\nw2+[w∗p−1(w∗)]0 0 0 +[w∗p−1(w∗)]0 −[w∗p−1(y∗)]−[w∗p−1(z∗)],\nand\nD=y⊗y∗z⊗y∗w⊗y∗y⊗z∗z⊗z∗w⊗z∗y⊗w∗z⊗w∗\ny20 0 0 −[y∗p−1(wyΦ3)]·x−[y∗p−1(wzΦ3)]·x−[y∗p−1(w2Φ3)]·x[y∗p−1(yzΦ3)]·x+[y∗p−1(z2Φ3)]·x\n+w −z\nyz[y∗p−1(ywΦ3)]·x[y∗p−1(zwΦ3)]·x[y∗p−1(w2Φ3)]·x−[z∗p−1(ywΦ3)]·x−[z∗p−1(zwΦ3)]·x−[z∗p−1(w2Φ3)]·x−[y∗p−1(y2Φ3)]·x−[y∗p−1(yzΦ3)]·x\n−w +w +[z∗p−1(zyΦ3)]·x+[z∗p−1(z2Φ3)]·x\n+y −z\nyw−[y∗p−1(yzΦ3)]·x−[y∗p−1(z2Φ3)]·x−[y∗p−1(zwΦ3)]·x+[y∗p−1(y2Φ3)]·x[y∗p−1(yzΦ3)]·x[y∗p−1(ywΦ3)]·x+[w∗p−1(yzΦ3)]·x+[w∗p−1(z2Φ3)]·x\n+z −[w∗p−1(wyΦ3)]·x−[w∗p−1(wzΦ3)]·x−[w∗p−1(w2Φ3)]·x\n−y +w\nz2[z∗p−1(ywΦ3)]·x[z∗p−1(zwΦ3)]·x[z∗p−1(w2Φ3)]·x 0 0 0 −[z∗p−1(y2Φ3)]·x−[z∗p−1(yzΦ3)]·x\n−w +y\nzw−[z∗p−1(yzΦ3)]·x−[z∗p−1(z2Φ3)]·x−[z∗p−1(zwΦ3)]·x+[z∗p−1(y2Φ3)]·x+[z∗p−1(yzΦ3)]·x+[z∗p−1(ywΦ3)]·x−[w∗p−1(y2Φ3)]·x−[w∗p−1(yzΦ3)]·x\n+[w∗p−1(ywΦ3)]·x+[w∗p−1(zwΦ3)]·x+[w∗p−1(w2Φ3)]·x −y\n+z −w\nw2−[w∗p−1(yzΦ3)]·x−[w∗p−1(z2Φ3)]·x−[w∗p−1(zwΦ3)]·x+[w∗p−1(y2Φ3)]·x[w∗p−1(yzΦ3)]·x[w∗p−1(ywΦ3)]·x 0 0\n+z −y.ARTINIAN GORENSTEIN ALGEBRAS 19\nObservation 4.7. Let(F,f)be the resolution of Theorem 4.2andP be the quotient\nringP\n(x). Then\nHj/parenleftbig\n(F,f)⊗PP/parenrightbig\n=0,for2≤j.\nIndeed,(F,f)⊗PP is isomorphic to the mapping cone of the zero map\nHomP(E,P)→E,\nwhere E is the Eagon-Northcott resolution of P/I2(M), and M is the matrix\nM=/bracketleftbigg\ny z w 0\n0y z w/bracketrightbigg\n.\nProof. Let ¯ be the functor −⊗ PP. One can read Example 4.6 (or Theorem 4.2) to\nsee that(F,f)⊗PPis the mapping cone of\nHomP(E,P):\n0\n/d15/d150 /d47/d47P(−6)1¯fT\n12/d47/d47\n0\n/d15/d15P(−4)6¯DT/d47/d47\n0\n/d15/d15P(−3)8¯A/d47/d47\n0\n/d15/d15P(−2)3\n0\n/d15/d15\nE 0 /d47/d47P(−4)3¯AT/d47/d47P(−3)8¯D/d47/d47P(−2)6¯f1,2/d47/d47P,\nwhere\n¯AT=\ny0−w\n0y0\n0 0 y\nz0 0\n0z−w\n0 0 z\nw0 0\n0w 0\n,¯D=\n0 0 0 w 0 0 −z0\n−w 0 0 0 w 0 y−z\nz 0 0 −y0 w 0 0\n0−w 0 0 0 0 0 y\n0 z−w0−y0 0 0\n0 0 z 0 0−y0 0\n,\nand\n¯f1,2=/bracketleftbig\ny2yz yw z2zw w2/bracketrightbig\n.\n(Actually, the complex F⊗PPis studied already in [6], where it is called the “skele-\nton” of F.) It is clear that Eis the Eagon-Northcott resolution of the quotient\nP/I2(M)and that HomP(E,P)is the Eagon-Northcott resolution of\nExt3\nP(P/I2(M),P),\nwhich is the canonical module of P/I2(M). It follows that (F,f)⊗PPhas nonzero\nhomology in positions 0 and 1; but otherwise, the homology is zero. /square\n5. A RESOLUTION OF P/annΦ3BY FREE P-MODULES .\nA few slight modifications turn the minimal homogeneous reso lution of Theo-\nrem 4.2 of P/annxΦ3by free P-modules into a resolution of P/annΦ3.\nTheorem 5.1. Adopt Data 2.8. Then the ideal annΦ3is generated by\n{x2p−1(ν1,0)|ν1,0∈U∗\n0}∪{ u2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0}.20 SABINE EL KHOURY AND ANDREW R. KUSTIN\nFurthermore,/parenleftbig\nF,/tildewidef/parenrightbig\n: 0→F4/tildewidef4− →F3/tildewidef3− →F2/tildewidef2− →F1/tildewidef1− →F0\nis a homogeneous resolution of P /annΦ3by free P-modules where\n/tildewidef4=/bracketleftbiggxf∗\n1,1\nf∗\n1,2/bracketrightbigg\n, /tildewidef3=/bracketleftbiggB∗D∗\nA∗x2⊗C∗/bracketrightbigg\n,\n/tildewidef2=/bracketleftbiggA B\nx2⊗C D/bracketrightbigg\n,and /tildewidef1=/bracketleftbigxf1,1f1,2/bracketrightbig\n.\nThe resolution/parenleftbig\nF,/tildewidef/parenrightbig\nis minimal if and only if B =0. Indeed, the minimal number\nof generators of annΦ3is equal to 9−rank B.\nRemark. Theorem 5.1 is implemented in the Macaulay2 script “BasicSo cDeg3”;\nsee Section 14.A.1.\nProof. We first show that\n(5.1.1) /an}bracketle{tx2p−1U∗\n0/an}bracketri}ht+/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0/an}bracketri}ht=annΦ3.\nThe inclusion ⊆is obvious. Indeed,\nxp−1(U∗\n0)⊆annxΦ3=⇒x2p−1(U∗\n0)⊆annΦ3\nand\n[u2,0−xp−1(u2,0Φ3)](Φ3)=u2,0(Φ3)−[p−1(u2,0Φ3)](xΦ3)\n=u2,0(Φ3)−p[p−1(u2,0Φ3)]= 0.\nThe inclusion ⊇requires more work.\nClaim 5.1.2. The elements\nxp−1(x∗),xp−1(y∗),xp−1(z∗),and xp−1(w∗)\nof(P/annΦ3)2are linearly independent. The element x2p−1(x∗)of(P/annΦ3)3is\nnonzero.\nProof of Claim 5.1.2 .Suppose that a,b,c,dare constants with\naxp−1(x∗)+bxp−1(y∗)+cxp−1(z∗)+dxp−1(w∗)=0\ninP/annΦ3. In this case,\nℓ=ap−1(x∗)+bp−1(y∗)+cp−1(z∗)+dp−1(w∗)\nis a linear form in P/annΦ3with xℓ=0. However, xis a weak Lefschetz element\nonP/annΦ3; hence,ℓ=0 inP/annΦ3. The kkk-algebra P/annΦ3has embedding\ndimension four; hence, ℓis zero in P. The elements x∗,y∗,z∗,w∗ofU∗are linearly\nindependent and p−1is an isomorphism. It follows that a,b,c, and dare all zero.\nFor the other assertion, observe that [x2p−1(x∗)](Φ3)=x(x∗)=1, which is not\nequal to 0; hence, x2p−1(x∗)/ne}ationslash=0 in(P/annΦ3)3. This completes the proof of\nClaim 5.1.2.ARTINIAN GORENSTEIN ALGEBRAS 21\nWe finish the proof of (5.1.1) by showing that\n(5.1.3) ann Φ3⊆/parenleftig\n/an}bracketle{tx2p−1U∗\n0/an}bracketri}ht+/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0/an}bracketri}ht/parenrightig\n.\nThe Hilbert series of A=P/annΦ3is(1,4,4,1). Indeed the Hilbert series exhibits\nsymmetry, has dim A1=4 and has a socle of dimension 1 in degree 3. Thus, we see\nfrom Claim 5.1.2, that\n(5.1.4) 1 ,x,y,z,w,xp−1(x∗),xp−1(y∗),xp−1(z∗),xp−1(w∗),x2p−1(x∗)\nis a basis for P/annΦ3. We show (5.1.3) by showing that P0⊕P1⊕P2⊕P3is\ncontained in the vector space spanned by (5.1.4) and\n(5.1.5)/parenleftig\n/an}bracketle{tx2p−1U∗\n0/an}bracketri}ht+/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0/an}bracketri}ht/parenrightig\n.\nWe see that P0andP1are both in the span of (5.1.4). Observe that P2is spanned\nby\n(5.1.6) xp−1(x∗),xp−1(y∗),xp−1(z∗),xp−1(w∗)\nand\n(5.1.7)y2−xp−1(y2Φ3),yz−xp−1(yzΦ3),yw−xp−1(ywΦ3),\nz2−xp−1(z2Φ3),zw−xp−1(zwΦ3),andw2−xp−1(w2Φ3).\nThe first four are a basis for xP1(by Claim 5.1.2) and the last 6 are a basis for\nP2/xP1. The first four are in (5.1.4) and the last six are in (5.1.5).\nUse the basis for P2which is the union (5.1.6) and (5.1.7) to see that P3is spanned\nby\n(a)(x,y,z,w)xp−1(x∗),\n(b)(x,y,z,w)(xp−1(y∗),xp−1(z∗),xp−1(w∗)),\n(c)(x,y,z,w)/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U/an}bracketri}ht.\nObserve that the basis elements of (c) are in (5.1.5) and the b asis elements\nx(xp−1(y∗),xp−1(z∗),xp−1(w∗))\nfrom (b) are in x2p−1U∗\n0; hence these are in (5.1.5). It is clear that x2p−1(x∗)from\n(a) is in 5.1.4. Let ℓbe an element of U0. Consider ℓ·xp−1(x∗)from (a). Observe\nthat\n[ℓp−1(x∗)](xΦ3)=ℓ(x∗)=0;\ntherefore,\nℓp−1(x∗)∈ann(xΦ3)=xp−1U∗\n0+/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0/an}bracketri}ht.\nIt follows that\nxℓp−1(x∗)∈x2p−1U∗\n0+/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0/an}bracketri}ht,\nwhich is (5.1.5).22 SABINE EL KHOURY AND ANDREW R. KUSTIN\nLetℓ1andℓ2be different elements selected from y,z, and w. Consider ℓ1xp−1(ℓ∗\n2)\nfrom (b). Observe that\n[ℓ1p−1(ℓ∗\n2)](xΦ3)=ℓ1(ℓ∗\n2)=0.\nTherefore,\nℓ1p−1(ℓ∗\n2)∈annxΦ3=xp−1U∗\n0+/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0/an}bracketri}ht\nand\nxℓ1p−1(ℓ∗\n2)⊆x2p−1U∗\n0+/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0/an}bracketri}ht,\nwhich is (5.1.5).\nLetℓequal y,z, orw. Consider ℓxp−1ℓ∗from (b). Observe that\n/parenleftig\nℓp−1(ℓ∗)−xp−1(x∗)/parenrightig\n(xΦ3)=ℓ(ℓ∗)−x(x∗)=1−1=0.\nThus,\nℓp−1(ℓ∗)−xp−1(x∗)∈annxΦ3=xp−1U∗\n0+/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0/an}bracketri}ht.\nIn other words, ℓp−1(ℓ∗)is equal to\nxp−1(x∗)+an element of xp−1U∗\n0+/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0/an}bracketri}ht\nandxℓp−1(ℓ∗)is equal to\nx2p−1(x∗)+an element of x2p−1U∗\n0+/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0/an}bracketri}ht.\nThe element x2p−1(x∗)is in (5.1.4) and an element of\nx2p−1U∗\n0+/an}bracketle{tu2,0−xp−1(u2,0Φ3)|u2,0∈Sym2U0/an}bracketri}ht\nis in (5.1.5). This completes the proof of (5.1.1).\nWe prove that the complex (F,/tildewidef)is acyclic by studying the short exact sequence\nof complexes\n0→(F,/tildewidef)γ− →(F,f)→Hom P(E,¯P)[−1]→0.\n(The “[−1]” refers to a shift of homological position.) The complex (F,f)is in-\ntroduced in Theorem 4.2 and the complex Hom P(E,¯P)may be found in Observa-\ntion 4.7. The map γi:(F,/tildewidef)i→(F,f)iis\nγ0=1,γ1=γ2=/bracketleftbigg\nx0\n0 1/bracketrightbigg\n,γ3=/bracketleftbigg\n1 0\n0x/bracketrightbigg\n,γ4=x.\nThe complexes (F,f)and Hom P(E,¯P)are acyclic; thus, (F,/tildewidef)is also acyclic.\nThe resolution (F,/tildewidef)is homogeneous with graded Betti numbers\n0→P(−7)1→P(−4)3\n⊕\nP(−5)6→P(−4)8\n⊕\nP(−3)8→P(−3)3\n⊕\nP(−2)6→P.\nAll of the nonzero maps have positive degree, except possibl y\nB:P(−3)8→P(−3)3and BT:P(−4)3→P(−4)8.ARTINIAN GORENSTEIN ALGEBRAS 23\nIt follows that (F,/tildewidef)is a minimal resolution if and only if Bis zero. Indeed, the\nminimal number of generators of ann Φ3is equal to 9 −rank B. /square\n6. T HE MATRIX SM.\nRetain Data 2.8. In this section, we introduce the 3 ×3 symmetric matrix SM\nwhich controls the rank of the matrix Bfrom Theorems 4.2 and 5.1 and therefore\ncontrols the precise form of the minimal resolution of P/annΦ3. Furthermore, we\nintroduce the complex Cwhich relates SM and B.\n6.A. Complexes associated to a symmetric 3×3matrix.\nContrary to the usual convention 2.8.(a) in this paper, thro ughout Subsection 6.A,\n(−)∗represents Hom R(−,R).\nBackground 6.1. LetRbe a commutative Noetherian ring, U0be a finitely gener-\nated free R-module, and (−)∗represent Hom R(−,R).\n(a) If Γ:U∗\n0→U0is an R-module homomorphism, then Γissymmetric if\n[Γ(w1)](w′\n1)=[ Γ(w′\n1)](w1)\nforw1,w′\n1∈U∗\n0.\n(b) If ωU∗\n0is a fixed basis for/logicalandtextrankU∗\n0U∗\n0andΓ:U∗\n0→U0is an R-module ho-\nmomorphism, then the classical adjoint ofΓis the R-module homomorphism\nΓ∨:U0→U∗\n0with\nΓ∨(u1)=/bracketleftig\n(/logicalandtextrankU∗\n0−1Γ)[u1(ωU∗\n0)]/bracketrightig\n(ωU∗\n0),\nforu1∈U0. (Of course, if Γis invertible, then Γ∨is equal to a unit times Γ−1.)\n(c) If Γ:U∗\n0→U0is a symmetric homomorphism, then the classical adjoint\nΓ∨:U0→U∗\n0\nofΓis also a symmetric homomorphism.\n(d) If β:U0→U∗\n0is a symmetric homomorphism, then there is an element δin\nD2U∗\n0with β(u1)=u1(δ)for all u1∈U0.\nDefinition 6.2. LetRbe a commutative Noetherian ring, U0be a free R-module of\nrank 3,(−)∗represent Hom R(−,R),Γ:U∗\n0→U0be a symmetric homomorphism\nofR-modules, Γ∨:U0→U∗\n0be the classical adjoint of Γ, and δbe the element\nofD2U∗\n0with Γ∨(u1) =u1(δ)foru1∈U0. LetCΓbe the following sequence of\nR-module homomorphisms:\nCΓ: 0→RC3− −− → D2U∗\n0C2− −− →U0⊗RU∗\n0\n(ev∗(1))C1− −− →/logicalandtext2U0→0,\nC3(1)=δ∈D2U∗\n0, for 1∈R,\nC2(ν(2)\n1)=Γ(ν1)⊗ν1, forν1∈U∗\n0, and\nC1(u1⊗ν1)=u1∧Γ(ν1), foru1∈U0andν1∈U∗\n0.24 SABINE EL KHOURY AND ANDREW R. KUSTIN\nLemma 6.3. Adopt the notation of Definition 6.2. Then the following statements\nhold.\n(a)The maps of Cform a complex of free R-modules.\n(b)If I2(Γ)=R, thenCis split exact.\n(c)If I2(Γ)is a proper ideal of R of grade at least 3, thenCand its dual C∗are\nacyclic.\n(d)When the hypotheses of (c)are in effect, then C∗is a resolution of R /I2(Γ)by\nfree R-modules and Cis a resolution of Ext3\nR(R/I2(Γ),R)by free R-modules.\n(e)The ideals I 2(Γ), I3(C1), I5(C2), and I 1(C3)all have the same radical.\nProof. See [11, Thm. 3.1] and [15, 12] for the proof of (a), (b), (c), ( d). We give\nan argument for (e). Let /tildewideRbe the polynomial ring R[{Xi,j|1≤i≤j≤3}],/tildewideΓbe the\ngeneric symmetric homomorphism /tildewideR→/tildewideRwhose matrix is\n\nX1,1X1,2X1,3\nX1,2X2,2X2,3\nX1,3X2,3X3,3\n,\nand/tildewideCbe the complex of Definition 6.2 built using /tildewideCin place of C. The ideal I2(/tildewideΓ)is\nperfect of grade 3 by [15]; hence /tildewideCis exact by (c); thus, I3(/tildewideC1),I5(/tildewideC2), and I1(/tildewideC3)\nall have the same radical by [10, Props. 6.8 and 6.3.(c)]. Let f:/tildewideR→Rbe the R-\nalgebra homomorphism which sends /tildewideΓtoΓ. Observe that f(Ii(/tildewideCj))= Ii(Cj)for all\niandj. The proof is complete because, I1(C3)=I2(Γ). /square\n6.B. The complex Cthat is used throughout the paper.\nWe return to the Data of 2.8. Recall the symmetric bilinear fo rm\n/an}bracketle{t−,−/an}bracketri}ht: Sym2U0×Sym2U0→kkk,\ngiven by\n/an}bracketle{tu2,u′\n2/an}bracketri}ht=[(p−1u2Φ3)u′\n2](Φ3),\nwhich is introduced in Data 2.8.(d).\nDefinition 6.4. Adopt Data 2.8.\n(a) Define\nF:/logicalandtext2U0⊗/logicalandtext2U0→kkk,\nF′:U∗\n0⊗U∗\n0→kkk,and\nΓ:U∗\n0→U0\nby\nF(u1,0∧u′\n1,0⊗u′′\n1,0∧u′′′\n1,0)=/an}bracketle{tu1,0u′′\n1,0,u′\n1,0u′′′\n1,0/an}bracketri}ht−/an}bracketle{tu1,0u′′′\n1,0,u′\n1,0u′′\n1,0/an}bracketri}ht,\nF′(ν1,0⊗ν′\n1,0)=F(ν1,0(ωU0)⊗ν′\n1,0(ωU0)),and\n[Γ(ν1,0)](ν′\n1,0)=F′(ν1,0⊗ν′\n1,0). (6.4.1)ARTINIAN GORENSTEIN ALGEBRAS 25\n(b) The homomorphism Γ∨:U0→U∗\n0is symmetric; so there exists δinD2U∗\n0with\nΓ∨(u1,0)=u1,0δfor all u1,0inU0.\nIn the same manner, if Γis invertible, then there exists a unit λinkkkwith\nΓ−1(u1,0)=u1,0λδ for all u1,0inU0.\n(c) Let SM be the matrix for Γ; use the basis y∗,z∗,w∗forU∗\n0andy,z,wforU:\n(6.4.2) SM =y∗z∗w∗\ny/an}bracketle{tz2,w2/an}bracketri}ht−/an}bracketle{tzw,zw/an}bracketri}ht/an}bracketle{tzw,yw/an}bracketri}ht−/an}bracketle{tzy,w2/an}bracketri}ht/an}bracketle{tyz,zw/an}bracketri}ht−/an}bracketle{tz2,yw/an}bracketri}ht\nz/an}bracketle{tyw,zw/an}bracketri}ht−/an}bracketle{tyz,w2/an}bracketri}ht/an}bracketle{ty2,w2/an}bracketri}ht−/an}bracketle{tyw,yw/an}bracketri}ht/an}bracketle{tyz,yw/an}bracketri}ht−/an}bracketle{ty2,wz/an}bracketri}ht\nw/an}bracketle{tyz,zw/an}bracketri}ht−/an}bracketle{tyw,z2/an}bracketri}ht/an}bracketle{tyw,yz/an}bracketri}ht−/an}bracketle{ty2,zw/an}bracketri}ht/an}bracketle{ty2,z2/an}bracketri}ht−/an}bracketle{tyz,yz/an}bracketri}ht.\n(d) LetCbe the complex CΓ:\n(6.4.3) C: 0→kkkC3− →D2U∗\n0C2− →U0⊗U∗\n0\nev∗(1)C1− →/logicalandtext2U0→0,\nbuilt as described in Definition 6.2 using the data Γof (6.4.1).\nObservation 6.5. Adopt the data of 2.8and the language of Definition 6.4. Let B\nbe the matrix of Theorems 4.2and5.1. Then\n(a)I1(C3)=I2(SM),\n(b)the matrices C1and B have the same rank, and\n(c)if2≤rankSM , then the complex Cof(6.4.3) is split exact.\nProof. (a) Let y,z,wbe a basis for U0. One basis for D2U∗\n0is{m∗|m∈/parenleftbigy,z,w\n2/parenrightbig\n}.\nObserve that\nC3(1)=∑\nm∈(y,z,w\n2)[Γ∨(m)]·m∗.\nIfC3is expressed as a 6 ×1 matrix (see, for example, (13.8.3)), then the entries of\nthis matrix are precisely the collection of two by two minors of the matrix for Γ,\nwhich is SM. We conclude that I1(C3)=I2(SM).\n(b) Let ψ2:/logicalandtext2U0→U∗\n0be the isomorphism with ψ2(θ2)=θ2(ωU∗\n0). We prove that\nthe diagram\n(6.5.1)U0⊗U∗\n0\n(ev∗(1))C1/d47/d47\nB\n/d39/d39❖❖❖❖❖❖❖❖❖❖❖❖❖❖/logicalandtext2U0\n∼= ψ2\n/d15/d15\nU∗\n0\ncommutes. Take u1,0,u′\n1,0,u′′\n1,0,u′′′\n1,0∈U0. We compare the elements\nS1=/bracketleftig\nB/parenleftig\nu′\n1,0⊗(u′′\n1,0∧u′′′\n1,0)(ωU∗\n0)/parenrightig/bracketrightig\n(u1,0)and\nS2=/bracketleftig/parenleftbig\nψ2◦C1/parenrightbig/parenleftig\nu′\n1,0⊗(u′′\n1,0∧u′′′\n1,0)(ωU∗\n0)/parenrightig/bracketrightig\n(u1,0)26 SABINE EL KHOURY AND ANDREW R. KUSTIN\nofkkk. Apply Remark 4.4 to see that\nS1=/an}bracketle{tu1,0u′′\n1,0,u′\n1,0u′′′\n1,0/an}bracketri}ht−/an}bracketle{tu1,0u′′′\n1,0,u′\n1,0u′′\n1,0/an}bracketri}ht.\nOn the other hand,\nS2=/parenleftig\nψ2/parenleftbig\nu′\n1,0∧Γ/bracketleftbig\n(u′′\n1,0∧u′′′\n1,0)(ωU∗\n0)/bracketrightbig/parenrightbig/parenrightig\n(u1,0), by Def. 6.2 ,\n=/bracketleftig/parenleftig\nu′\n1,0∧Γ/bracketleftbig\n(u′′\n1,0∧u′′′\n1,0)(ωU∗\n0)/bracketrightbig/parenrightig\n(ωU∗\n0)/bracketrightig\n(u1,0)\n=/parenleftig\nΓ/bracketleftbig\n(u′′\n1,0∧u′′′\n1,0)(ωU∗\n0)/bracketrightbig/parenrightig/parenleftig\n(u1,0∧u′\n1,0)(ωU∗\n0)/parenrightig\n=F((u′′\n1,0∧u′′′\n1,0)⊗(u1,0∧u′\n1,0)), by Def. 6.4 ,\n=/an}bracketle{tu′′\n1,0u1,0,u′′′\n1,0u′\n1,0/an}bracketri}ht−/an}bracketle{tu′′\n1,0u′\n1,0,u′′′\n1,0u1,0/an}bracketri}ht, by Def. 6.4 .\nObserve that S1=S2.\n(c) Recall that SM is the matrix for Γ. It follows from Lemma 6.3.(b) that Cis split\nexact. /square\nTheorem 6.6. Adopt Data 2.8. The matrix B of Theorems 4.2and5.1has rank 0,\n2, or3. The number of minimal generators of the ideal annΦ3is9,7, or6.\nProof. Every nonzero entry in the matrix Bis plus or minus an entry of the sym-\nmetric matrix (6.4.2). If SM is zero, then B=0. If SM has rank one, then the usual\ntheory of symmetric matrices guarantees that one can choose a pair of dual bases\nforU0andU∗\n0so that SM becomes\n(6.6.1)\nα0 0\n0 0 0\n0 0 0\n,for some unit αinkkk.\nIn this case,\n(6.6.2) B=\n0 0 0 0 0 0 0 0\n0 0 α0 0 0 0 0\n0−α0 0 0 0 0 0\n,\nwhich has rank 2. The matrix SM is a matrix of constants. If 2 ≤rankSM, then\nI2(SM) =P; hence, I1(C3) =PandI3(B) =Pby Lemma 6.3.(e) and Observa-\ntion 6.5.\nWe saw in Theorem 5.1 that the minimum number of generators of annΦ3is\n9−rank B. /square\n7. T HE GENERALIZED SUM OF LINKED PERFECT IDEALS .\nIn this section\n(−)∗is the functor Hom P(−,P).\nObservation 7.1. Let J be a grade g perfect ideal in the Gorenstein ring P. Suppo se\nθ: Extg\nP(P/J,P)→P/J is an injection. Let K be an ideal of P with imθ=K/J. If\ng+1≤grade K, then K is a perfect Gorenstein in P of grade g +1.ARTINIAN GORENSTEIN ALGEBRAS 27\nProof. Let 0→Fg→···→ F0=Pbe a resolution of P/Jby free P-modules, and\n(7.1.1) 0/d47/d47F∗\n0/d47/d47\n/d15/d15.../d47/d47F∗\ng−1/d47/d47\n/d15/d15F∗\ng\n/d15/d15\n0/d47/d47Fg/d47/d47.../d47/d47F1/d47/d47F0\nbe a map of complexes which extends\nF∗\ng/d47/d47Extg\nP(P/J,P)/d47/d47\nθ\n/d15/d150\nF0/d47/d47P/J/d47/d470.\nObserve that the mapping cone of (7.1.1) is a resolution of P/Kof length g+1 and\nthe last Betti number in this resolution is one.\nThe fact that\ngradeann M≤pdM,\nfor all finitely generated P-modules M, ensures that pd P/K=g+1, and that the\nminimal resolution of (P/K)phas length g+1 and last Betti number equal to one,\nfor all prime ideals pofPwhich contain K. Thus Kis a Gorenstein ideal of grade\ng+1. /square\nDefinition 7.2. The ideal Kof Observation 7.1 is called a generalized sum of linked\nideals .\nWe offer two examples of generalized sums of linked ideals. T he first example is\nthe sum of geometrically linked perfect ideals. The second e xample is a hypersur-\nface section of a Gorenstein ideal.\nExample 7.3. Peskine and Szpiro introduced [17, Rem. 1.4] the idea of addi ng\ntwo geometrically linked ideals of grade gto produce a Gorenstein ideal of grade\ng+1. Also, Ulrich’s work [18] along these lines is of significan t interest. Recall\nthat the grade gideals IandJof the Gorenstein (not necessarily local) ring Pare\nlinked , if there exists a regular sequence α=α1,..., αginI∩Jwith(α):I=Jand\n(α):J=I. The ideals IandJaregeometrically linked if they are linked and any of\nthe following three equivalent conditions also occur:\n(1) Ass P/I∩AssP/J=/0,\n(2)I∩J=(α1,..., αg),\n(3)g+1≤grade(I+J).\nWe think of writing a grade 4 Gorenstein ideal as the sum of two grade three Goren-\nstein ideals as a labor-saving device. We are able to describ e an entire length four\nresolution in terms of a length three resolution (and connec ting maps). In particular,\nifIis a perfect grade three ideal geometrically linked to Jand\n(7.3.1) 0 →G3g3− →G2g2− →G1g1− →P→P/I28 SABINE EL KHOURY AND ANDREW R. KUSTIN\nis a resolution of P/I, then\n0→P∗g∗\n1− →G∗\n1g∗\n2− →G∗\n2g∗\n3− →G∗\n3\nis a resolution of\nExt3\nP(P\nI,P)=Hom P(P\nI,P/(α))=α:I\nα=J\nI∩J=I+J\nI\nand the mapping cone of\n0 /d47/d47P∗g∗\n1/d47/d47\nc3\n/d15/d15G∗\n1g∗\n2/d47/d47\nc2\n/d15/d15G∗\n2g∗\n3/d47/d47\nc1\n/d15/d15G∗\n3\nc0\n/d15/d15\n0/d47/d47G3g3/d47/d47G2g2/d47/d47G1g1/d47/d47P\nis a resolution of P/(I+J), where c0:G∗\n3→Pis a map for which\nG∗\n3/d47/d47\nc0\n/d15/d15I+J\nI /d127 /d95\n/d15/d15\nP0/d47/d47P\nI\ncommutes.\nExample 7.4. The same trick works when Iis a grade three Gorenstein ideal and\nais a regular element on P/I. The resolution of P/(I,a)is the mapping cone of\na:G∗→Gwhich covers the injection a:P/I→P/I, where G∼=G∗both resolve\nP/I. Of course, (I,a)is not literally the sum of the linked ideals IandI.\nIn practice the resolution “ G” of (7.3.1) is already well understood. Indeed, for us\nGis the resolution of a grade 3 Gorenstein ideal when ann Φ3has six minimal gen-\nerators; Gis the resolution of a grade three perfect ideal of type 2 with one minimal\nKoszul relation as studied by Anne Brown [2] (see also [13]) w hen ann Φ3has seven\nminimal generators; and Gis a deformation of the Eagon-Northcott resolution of\nthe monomials of degree 2 in three variables when ann Φ3has nine generators.\nThe paper [16] refers to the technique of forming the mapping cone of c:G∗→G,\nwhere H 0(G∗)֒→H0(G), as “doubling”.\n8. T HE ALGEBRA WITH NO WEAK LEFSCHETZ ELEMENT .\nLetkkkbe an arbitrary field and Abe a standard graded Artinian Gorenstein kkk-\nalgebra of embedding dimension four and socle degree three. IfAdoes not have the\nweak Lefschetz property, then it is shown in [14] that kkkhas characteristic two and\nAis isomorphic to\n(8.0.1)kkk[x,y,z,w]\n(xy,xz,xw,y2,z2,w2,x3+yzw).ARTINIAN GORENSTEIN ALGEBRAS 29\nThe Macaulay inverse system of Ais\n(8.0.2) Φ3=x∗(3)+y∗z∗w∗.\nIn this section, we show that the defining ideal of Ais the sum of linked ideals. We\nalso show the algebra of [16, Ex. 5.6] is isomorphic to (8.0.1 ).\nObservation 8.1. Let P be the ring k kk[x,y,z,w], where k kk has characteristic two.\nThe mapping cone of the following map of complexes is the mini mal homogeneous\nresolution of Afrom (8.0.1):\n0 /d47/d47PKT\n1/d47/d47\nLT\n0/d15/d15P5KT\n2/d47/d47\nLT\n1/d15/d15P6KT\n3/d47/d47\nL1/d15/d15P2\nL0/d15/d15\n0/d47/d47P2K3/d47/d47P6K2/d47/d47P5K1/d47/d47P1,\nwhere\nK1=/bracketleftbig\ny2z2xy xz xw/bracketrightbig\n, K2=\nz20x0 0 0\ny2x0 0 0 0\n0 0 y0w z\n0z0w0y\n0 0 0 z y 0\n,\nK3=\nx0\ny20\nz20\n0y\n0z\nyz w\n, L0=/bracketleftbig\nw2x3+yzw/bracketrightbig\n,and\nL1=\n0w20zw 0 0\n0 0 w20yw 0\n0 0 0 x20 0\n0 0 0 0 x20\nw0 0 0 0 x2\n.\nThe proof is obvious.\nExample 8.2. We change the basis to the Macaulay inverse system for [16, Ex . 5.6]\ninto the form of (8.0.2). In this example kkkhas characteristic 2, and the Macaulay\nInverse System for Ais\nΦ3=x∗(3)+x∗y∗z∗+x∗y∗w∗+x∗z∗w∗+y∗z∗w∗∈D3U∗.\nUse the X∗=x∗,Y∗=y∗+w∗,Z∗=z∗+w∗,W∗=w∗forU∗. Notice that\nΦ3=X∗(3)+X∗Y∗Z∗+Y∗Z∗W∗\nDo another change of basis! Replace W∗with/tildewideW∗=X∗+W∗and keep X∗,Y∗, and\nZ∗. The Macaulay inverse system for Ais\nX∗(3)+Y∗Z∗/tildewideW∗.30 SABINE EL KHOURY AND ANDREW R. KUSTIN\n9. T HE CASE WHEN SM IS ZERO .\nTheorem 9.1. Adopt the data of 2.8and the matrix SMof(6.4.2) . IfSMis zero,\nthen annΦ3has nine minimal generators and the minimal resolution of Ais given\nby the complex (F,/tildewidef)from Theorem 5.1. Furthermore, the defining ideal of Ais the\nsum of two linked codimension three perfect ideals.\nRemark. Theorem 9.1 is implemented in the Macaulay2 script “SumOfLi nkedIde-\nals9”; see Section 14.A.4.\nProof. If SM=0, then B=0 and the minimal number of generators of ann Φ3is\n9−rank B=9. It is already shown in the proof of Theorem 5.1 that (F,/tildewidef)is a\nminimal resolution of A.\nWe show that the defining ideal of Ais the sum of two linked codimension three\nperfect ideals. When B=0, then a quick look at Theorem 5.1 shows that the mini-\nmal resolution (F,/tildewidef)ofAis the mapping cone of the following map of complexes:\n0 /d47/d47Pf∗\n1,2/d47/d47\n−xf∗\n1,1\n/d15/d15P⊗D2U∗\n0D∗/d47/d47\nC∗\n/d15/d15P⊗ker(U0⊗U∗\n0)A/d47/d47\n−C\n/d15/d15P⊗U∗\n0\nxf1,1\n/d15/d15\n0 /d47/d47P⊗U0A∗/d47/d47P⊗U0⊗U∗\n0\nev∗(1)D/d47/d47P⊗Sym2U0f1,2/d47/d47P.\nWe complete the proof by observing that\n(9.1.1) 0 /d47/d47P⊗U0A∗/d47/d47P⊗U0⊗U∗\n0\nev∗(1)D/d47/d47P⊗Sym2U0f1,2/d47/d47P\nis the resolution of a perfect codimension three quotient of P. Indeed, the complex\n(9.1.1) is a deformation of the Eagon-Northcott complex Eof Observation 4.7.\nApply Lemma 9.2. /square\nLemma 9.2. Let P=P0[x]be a polynomial ring, F be a complex of finitely gener-\nated free P-modules, and ¯P be the quotient ring P /(x). If\nHi(F⊗P¯P)andHi+1(F⊗P¯P)are both zero,\nthen Hi(F)is zero.\nProof. The long exact sequence of homology that corresponds to the s hort exact\nsequence of complexes\n0→F(−1)x− →F→F⊗P¯P→0\nyields that multiplication by xfrom H i(F)j−1to H i(F)jis an isomorphism for all\nj. (The “(−1)” refers to a shift of internal degree.) However H i(F)j−1is zero for\nallj≪0; consequently, H i(F)=0. /squareARTINIAN GORENSTEIN ALGEBRAS 31\n10. T HE CASE WHEN SM HAS RANK ONE .\nAdopt the data of 2.8 and the matrix SM of (6.4.2). Assume that SM has rank\none. The usual theory of symmetric matrices guarantees that one can choose a pair\nof dual bases for U0andU∗\n0so that SM is given in (6.6.1). Recall the resolution\n(/tildewideF,/tildewidef)ofAwhich is given in Theorem 5.1. The differential /tildewidef2of/tildewideFis the map\nP8\n⊕\nP8/tildewidef2=/bracketleftiggA B\nx2C D/bracketrightigg\n− −−−−−−−− →P3\n⊕\nP6,\nwhere the matrices A,B,C, and Dare given in Example 4.6. When SM is given in\n(6.6.1), then Bis given in (6.6.2).\nThe minimal resolution Ais given in Theorem 10.3. It is shown in Theorem 12.1\nthat the defining ideal of Ais the sum of two linked codimension three perfect ideals.\nDefinition 10.1. Adopt the data of 2.8 and the matrix SM of (6.4.2). Assume that\nSM has rank one. Assume further that SM is given in (6.6.1). Re call the matrices\nA=\na1,1... a1,8\na2,1... a2,8\na3,1... a3,8\n,B=\nb1,1... b1,8\nb2,1... b2,8\nb3,1... b3,8\n,C=\nc1,1... c1,8.........\nc6,1... c6,8\n,and\nD=\nd1,1... d1,8.........\nd6,1... d6,8\n,\nfrom Theorem 4.2 and Example 4.6. The three rows of Aare denoted A1,∗,A2,∗,\nA3,∗. The eight columns of Ddenoted D∗,1,D∗,2,..., D∗,8. The same convention is\nused to denote the rows and columns of all four matrices.\nDefine the invertible matrices θ1andθ2by\nθ1=1 0 0\n0 0 α 03×6\n0−α0\n06×1D∗,2D∗,3I6and θ2=I808×8\nMI8,\nwhere Mis the 8×8 matrix\nM=1\nα\n0−a3,1a2,1 0 0 0 0 0\na3,1 0 a3,3a3,4 a3,5 a3,6 a3,7 a3,8\n−a2,1−a2,20−a2,4−a2,5−a2,6−a2,7−a2,8\n0−a3,4a2,4 0 0 0 0 0\n0−a3,5a2,5 0 0 0 0 0\n0−a3,6a2,6 0 0 0 0 0\n0−a3,7a2,7 0 0 0 0 0\n0−a3,8a2,8 0 0 0 0 0\n.32 SABINE EL KHOURY AND ANDREW R. KUSTIN\nLetJbe the matrix\nJ=/bracketleftbigg\n06×6I6\nI606×6/bracketrightbigg\n.\nDefine the matrices E1,E2,E3,E4by\n•E1is/tildewidef1θ1with columns 2 and 3 deleted;\n•E2isθ−1\n1/tildewidef2θ2with rows 2 and 3 and columns 2 ,3,10,11 deleted;\n•E3=JET\n2; and\n•E4=ET\n1.\nRemark 10.2. In Section 11 we have collected many consequences of the hypo th-\nesis that Bhas the form of (6.6.2). In particular, in (11.1.1) we show th at\na1,3=a2,3=a3,2=a1,2=0 and a3,3=a2,2.\nIt follows that Mis an alternating matrix; row 2 of Mis1\nαA3,∗; and row 3 of Mis\n−1\nαA2,∗. The fact that Mis an alternating matrix ensures that\n(10.2.1) θ2JθT\n2=J.\nTheorem 10.3. Adopt the data of 2.8and the matrix SMof(6.4.2) . Assume that\nSMhas rank one. Assume further that SMis given in (6.6.1) . Let E 1, E2, E3, E4be\nthe matrices of Definition 10.1. Then the minimal homogeneous resolution of Aby\nfree P-modules is given by\nE: 0→PE4− →P7E3− →P12E2− →P7E1− →P.\nRemark. Theorem 10.3 is implemented in the Macaulay2 script “minres 7”; see\nSection 14.A.2.\nProof. Recall the resolution (F,/tildewidef)of Theorem 5.1. Define (F′,f′)to be\n0→Pf′\n4=f′\n1T\n− −−− → P9f′\n3=Jf′\n2T\n−−−− → P16f′\n2− →P9f′\n1− →P,\nwhere\nf′\n2=θ−1\n1/tildewidef2θ2andf′\n1=/tildewidef1θ1.\nUse (10.2.1) in order to observe that\n(F′,f′):\n/d15/d150/d47/d47Pf′\n1T\n/d47/d47P9Jf′\n2T\n/d47/d47\nθ−1\n1T\n/d15/d15P16f′\n2/d47/d47\nθ2/d15/d15P9f′\n1/d47/d47\nθ1/d15/d15P\n(F,/tildewidef): 0/d47/d47P/tildewidef1T\n/d47/d47P9J/tildewidef2T\n/d47/d47P16/tildewidef2/d47/d47P9/tildewidef1/d47/d47P,\nis an isomorphism of complexes.\nClaim 10.3.1. The following statements hold.\n(a)Columns 2and3off′\n1are/bracketleftbig01×2/bracketrightbig\n.ARTINIAN GORENSTEIN ALGEBRAS 33\n(b)Columns 2and3off′\n2are/bracketleftbig09×2/bracketrightbig\n.\n(c)Columns 10and11off′\n2are\n(10.3.2)\n01×2\nI2\n06×2\n.\n(d)Rows 2and3off′\n2are\n/bracketleftbig02×9I202×5/bracketrightbig\n,\nProof of Claim 10.3.1 .(a) Columns 2 and 3 of θ1are equal to columns 2 and 3 of\n/tildewidef2; hence columns 2 and 3 of f′\n1=/tildewidef1θ1are identically zero.\n(b) Column 3 of θ2is equal to1\nαtimes column 2 of /tildewidef3. Column 2 of θ2is equal to\n−1\nαtimes column 3 of /tildewidef3; hence columns 2 and 3 of f′\n2=θ−1\n1/tildewidef2θ2are identically\nzero.\n(c) Observe that\nθ−1\n1=1 0 0\n0 0−1\nα0\n01\nα0\n0−1\nαD∗,31\nαD∗,2I6,(/tildewidef2θ2)∗,10=(/tildewidef2)∗,10=\n0\n0\n−α\nD∗,2\n,and\n(/tildewidef2θ2)∗,11=(/tildewidef2)∗,11=\n0\nα\n0\nD∗,3\n.\nConclude that columns 10 and 11 of f′\n2=θ−1\n1/tildewidef2θ2are given in (10.3.2).\n(d) Rows 2 and 3 of f′\n2=(θ−1\n1/tildewidef2)θ2\n=/bracketleftigg\n−1\nαA3,∗0 1 0 0 1×5\n1\nαA2,∗0 0 1 0 1×5/bracketrightigg/bracketleftbiggI808×8\nMI8/bracketrightbigg\n=/bracketleftigg\n−1\nαA3,∗+M∗,20 1 0 0 1×5\n1\nαA2,∗+M∗,30 0 1 0 1×5/bracketrightigg\n=/bracketleftigg\n−1\nαA3,∗+1\nαA3,∗0 1 0 0 1×5\n1\nαA2,∗−1\nαA∗,20 0 1 0 1×5/bracketrightigg\n=/bracketleftbigg\n01×80 1 0 0 1×5\n01×80 0 1 0 1×5/bracketrightbigg\n,\nas claimed. This completes the proof of Claim10.3.1.\nClaim 10.3.1 establishes that the complex Eis a direct summand of the resolution\nF′and that the complement of EinF′is a split exact complex. It follows that Eis34 SABINE EL KHOURY AND ANDREW R. KUSTIN\na resolution. The resolution Eis homogeneous and has the same Betti numbers as\nthe minimal homogeneous resolution of A; therefore, Eis the minimal resolution of\nA. /square\nRemark 10.4. For future reference, we notice that the graded Betti number s ofE\nare\n0→P(−7)→P(−4)\n⊕\nP(−5)6→P(−3)6\n⊕\nP(−4)6→P(−2)6\n⊕\nP(−3)→P.\n11. C ALCULATIONS .\nWe have already applied (11.1.1) in the proof of Theorem 10.3 ; furthermore, the\ncalculations of this section are used extensively in the pro of of Theorem 12.1.\nLemma 11.1. LetA,B,C, andDbe the maps of Theorem 4.2. Then\nAB∗:P⊗U0→P⊗U∗\n0\nis an alternating map. In particular, if B is given in (6.6.2) , then\nBAT=α\n0 0 0\na1,3 a2,3 a3,3\n−a1,2−a2,2−a3,2\n\nis an alternating matrix. In particular,\n(11.1.1) a1,3=a2,3=a3,2=a1,2=0and a 3,3=a2,2.\nIn other words,\n[y2p−1(w∗)](Φ3)=w∗[p−1(y2Φ3)]= 0, (11.1.2)\n[yzp−1(w∗)](Φ3)=w∗[p−1(yzΦ3)]= 0, (11.1.3)\n[ywp−1(z∗)](Φ3)=z∗[p−1(ywΦ3)]= 0, (11.1.4)\n[y2p−1(z∗)](Φ3)=z∗[p−1(y2Φ3)]= 0, (11.1.5)\n[yzp−1(z∗)](Φ3)=[ ywp−1(w∗)](Φ3),and (11.1.6)\nz∗[p−1(yzΦ3)]= w∗[p−1(ywΦ3)].\nProof. Letu1,0be an element of U0. Use Remark 4.5 and Theorem 4.2 to see that\n−/parenleftbig\n(A◦B∗)(u1,0)/parenrightbig\n(u1,0)\n=A\n\n\n−w⊗[zp−1(yu1,0)(Φ3)](Φ3)+z⊗[wp−1(yu1,0)(Φ3)](Φ3)\n+w⊗[yp−1(zu1,0)(Φ3)](Φ3)−y⊗[wp−1(zu1,0)(Φ3)](Φ3)\n−z⊗[yp−1(wu1,0)(Φ3)](Φ3)+y⊗[zp−1(wu1,0)(Φ3)](Φ3)\n(u1,0)\n=S1+S2,ARTINIAN GORENSTEIN ALGEBRAS 35\nwith\nS1=x⊗\n\n\n/bracketleftig\nwp−1/parenleftig\n[zp−1(yu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n(Φ3)\n−/bracketleftig\nzp−1/parenleftig\n[wp−1(yu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n(Φ3)\n−/bracketleftig\nwp−1/parenleftig\n[yp−1(zu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n(Φ3)\n+/bracketleftig\nyp−1/parenleftig\n[wp−1(zu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n(Φ3)\n+/bracketleftig\nzp−1/parenleftig\n[yp−1(wu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n(Φ3)\n−/bracketleftig\nyp−1/parenleftig\n[zp−1(wu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n(Φ3)\n(u1,0)\n=x⊗\n\n+/bracketleftig/parenleftig\n[zp−1(yu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n[p−1(wu1,0Φ3)]\n−/bracketleftig/parenleftig\n[wp−1(yu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n[p−1(zu1,0(Φ3)]\n−/bracketleftig/parenleftig\n[yp−1(zu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n[p−1(wu1,0Φ3)]\n+/bracketleftig/parenleftig\n[wp−1(zu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n[p−1(yu1,0Φ3)]\n+/bracketleftig/parenleftig\n[yp−1(wu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n[p−1(zu1,0(Φ3)]\n−/bracketleftig/parenleftig\n[zp−1(wu1,0)(Φ3)](Φ3)/parenrightig/bracketrightig\n[p−1(yu1,0Φ3)]\n=x⊗\n\n+/bracketleftig\nz[p−1(wu1,0Φ3)][p−1(yu1,0Φ3)]/bracketrightig\n(Φ3)\n−/bracketleftig\nw[p−1(zu1,0Φ3)][p−1(yu1,0Φ3)]/bracketrightig\n(Φ3)\n−/bracketleftig\ny[p−1(wu1,0Φ3)][p−1(zu1,0Φ3)]/bracketrightig\n(Φ3)\n+/bracketleftig\nw[p−1(yu1,0Φ3)][p−1(zu1,0Φ3)]/bracketrightig\n(Φ3)\n+/bracketleftig\ny[p−1(zu1,0Φ3)][p−1(wu1,0Φ3)]/bracketrightig\n(Φ3)\n−/bracketleftig\nz[p−1(yu1,0Φ3)][p−1(wu1,0Φ3)]/bracketrightig\n(Φ3)\n=0\nand\nS2=\n\n−w·[u1,0zp−1(yu1,0)(Φ3)](Φ3)+z·[u1,0wp−1(yu1,0)(Φ3)](Φ3)\n+w·[u1,0yp−1(zu1,0)(Φ3)](Φ3)−y·[u1,0wp−1(zu1,0)(Φ3)](Φ3)\n−z·[u1,0yp−1(wu1,0)(Φ3)](Φ3)+y·[u1,0zp−1(wu1,0)(Φ3)](Φ3)=0.\n/square\nCorollary 11.2. Let A, B, C, and D be the matrices of Example 4.6. If B is given in\n(6.6.2) , then\n(a)d1,1=d4,4=d4,5=d4,6=d6,7=d1,4+d2,1=d1,7+d3,1=d4,8+d5,5=0,\na1,1+d3,4=a1,4+d1,7=a1,7+d2,1=a2,8+d4,2=a3,8+d4,3=c4,8=0,\n(b)d5,7=d6,4=0,\n(c)d3,7=d6,1=d6,5=0,\n(d)a1,8=d2,4=d4,1=d5,6=0,36 SABINE EL KHOURY AND ANDREW R. KUSTIN\n(e)d4,7=d5,4=0, and\n(f)a1,5=d5,1=d2,7+d3,4=d5,5−d6,6=0.\nProof. One may read (a) directly from Example 4.6; (b) follows from ( 11.1.2); (c)\nfollows from (11.1.3); (d) follows from (11.1.4); (e) follo ws from (11.1.5); and (f)\nfollows from (11.1.6). /square\nLemma 11.3. Let A, B, C, and D be the matrices of Example 4.6. If B is given in\n(6.6.2) , then the following identities all hold in k kk.\n(a)\n0=\n\n[w2p−1(z∗)](Φ3)·[y∗p−1(z2Φ3)]−[zwp−1(z∗)](Φ3)·[y∗p−1(zwΦ3)]\n−α[y∗p−1(z∗)]−[y∗p−1(w2Φ3)]·[w∗p−1(wzΦ3)]\n+[y∗p−1(w2Φ3)]·[y∗p−1(yzΦ3)]−[y∗p−1(zwΦ3)]·[y∗p−1(ywΦ3)]\n+[y∗p−1(zwΦ3)]·[w∗p−1(w2Φ3)],\n(b)\n0=\n\n+αz∗p−1(z∗)−[y∗p−1(ywΦ3)]·[ywp−1(y∗)](Φ3)\n+[y∗p−1(ywΦ3)]·[w2p−1(w∗)](Φ3)−[y∗p−1(w2Φ3)]·[ywp−1(w∗)](Φ3)\n+[z∗p−1(zwΦ3)]·[zwp−1(z∗)](Φ3)−[z∗p−1(zwΦ3)]·[w2p−1(w∗)](Φ3)\n+[z∗p−1(w2Φ3)]·[zwp−1(w∗)](Φ3)+[y∗p−1(y2Φ3)]·[w2p−1(y∗)](Φ3)\n−[z∗p−1(zyΦ3)]·[w2p−1(y∗)](Φ3)+[y∗p−1(yzΦ3)]·[w2p−1(z∗)](Φ3)\n−[z∗p−1(z2Φ3)]·[w2p−1(z∗)](Φ3),\n(c)\n0=\n\n−α[z∗p−1(w∗)]−[y∗p−1(ywΦ3)]·[yzp−1(y∗)](Φ3)\n+[y∗p−1(ywΦ3)]·[wzp−1(w∗)](Φ3)−[z∗p−1(zwΦ3)]·[wzp−1(w∗)](Φ3)\n+[z∗p−1(w2φ3)]·[z2p−1(w∗)](Φ3)+[y∗p−1(y2Φ3)]·[zwp−1(y∗)](Φ3)\n+[y∗p−1(yzΦ3)]·[zwp−1(z∗)](Φ3)−2[z∗p−1(yzΦ3)]·[zwp−1(y∗)](Φ3),\n(d)\n0=\n\n−[x∗p−1(w2Φ3)]+α[z∗p−1(z∗)]+[ w∗p−1(zwΦ3)]·[z∗p−1(w2Φ3)]\n−[y∗p−1(w2Φ3)]·[z∗p−1(yzΦ3)]−[z∗p−1(w2Φ3)]·[z∗p−1(z2Φ3)]\n+[z∗p−1(zwΦ3)]·[z∗p−1(zwΦ3)]−[w∗p−1(w2Φ3)]·[z∗p−1(zwΦ3)],\n(e)\n0=\n\n−α[z∗p−1(w∗)]−[y∗p−1(zwΦ3)][z∗p−1(yzΦ3)]−[x∗p−1(zwΦ3)]\n−[z∗p−1(zwΦ3)]·[z∗p−1(z2Φ3)]+[ z∗p−1(z2Φ3)]·[z∗p−1(zwΦ3)]\n−[w∗p−1(zwΦ3)]·[z∗p−1(zwΦ3)]+[ w∗p−1(z2Φ3)]·[z∗p−1(w2Φ3)],\n(f)\n0=/braceleftigg\n[w∗p−1(wyΦ3)]·[w∗p−1(ywΦ3)]−[y∗p−1(y2Φ3)]·[w∗p−1(ywΦ3)]\n−[x∗p−1(y2Φ3)],\n(g) 0=[y∗p−1(yzΦ3)]·[w∗p−1(ywΦ3)]+[ x∗p−1(yzΦ3)],and\n(h) 0=[w∗p−1(wyΦ3)]·[y∗p−1(ywΦ3)]+[ x∗p−1(ywΦ3)].ARTINIAN GORENSTEIN ALGEBRAS 37\nProof. All of these identities are consequences of the fact that/parenleftbig\nF,/tildewidef/parenrightbig\n, from Theo-\nrem 5.1, is a complex and Bhas the form of (6.6.2). In particular,\n/bracketleftbiggA B\nx2C D/bracketrightbigg/bracketleftbigg\nBTDT\nATx2CT/bracketrightbigg\n=0 and/bracketleftbigx f1,1f1,2/bracketrightbig/bracketleftbiggA B\nx2C D/bracketrightbigg\n=0.\nConsequently,\nx2CBT+DAT=0 and x f1,1B+f1,2D=0.\nThe sum (a) is minus the coefficient of x2in\n(x2CBT+DAT)1,3=0;\n(b) is the coefficient of x2in\n(x2CBT+DAT)2,3=0;\n(c) is the coefficient of x2in\n(x2CBT+DAT)2,2=0;\n(d) is the coefficient of x2zin\n(x f1,1B+f1,2D)3=0;\n(e) is the coefficient of x2zin\n(x f1,1B+f1,2D)2=0;\n(f) is the coefficient of x2win\n(x f1,1B+f1,2D)4=0;\n(g) is the coefficient of x2win\n(x f1,1B+f1,2D)1=0;\nand (h) is the coefficient of x2yin\n(x f1,1B+f1,2D)4=0.\n/square\n12. T HE SEVEN -GENERATED IDEAL IS A SUM OF LINKED PERFECT IDEALS .\nIn this section we prove the following result.\nTheorem 12.1. Adopt the data of 2.8and the matrix SMof(6.4.2) . Assume that\nSMhas rank one. Choose a pair of dual bases for U 0and U∗\n0so that SMis given\nin(6.6.1) . Then the defining ideal of Ais the sum of two linked codimension three\nperfect ideals.\nRemark. Theorem 12.1 is implemented in the Macaulay2 script “SumOfL inke-\ndIdeals7”; see Section 14.A.3.38 SABINE EL KHOURY AND ANDREW R. KUSTIN\nProof. The minimal homogeneous resolution EofAis given in Theorem 10.3. In\nDefinition 12.2 we rearrange the rows and columns of the matri ces of Eand produce\nthe minimal homogeneous resolution E′ofA. In Definition 12.2 we also partition\nE′in the form\nE′\n1=/bracketleftbigL0K1/bracketrightbig\n, E′\n2=/bracketleftbigg\nKT\n3Z\nL1K2/bracketrightbigg\n, (12.1.1)\nE′\n3=/bracketleftbiggZTKT\n2\nK3LT\n1/bracketrightbigg\n,and E′\n4=/bracketleftbiggLT\n0\nKT\n1/bracketrightbigg\n,\nwhere the shape of each component matrix is given by\nmatrix shape\nZ 2×6\nK1 1×5\nK2 5×6\nK3 6×2\nL0 1×2\nL1 5×6.\nIn Lemma 12.4 we prove that the matrix Z(which appears in both E′\n2andE′\n3) is\nidentically zero. It follows immediately that\n(12.1.2) 0/d47/d47P1KT\n1/d47/d47\n−LT\n0/d15/d15P5KT\n2/d47/d47\nLT\n1/d15/d15P6KT\n3/d47/d47\n−L1/d15/d15P2\nL0\n/d15/d15\n0/d47/d47P2K3/d47/d47P6K2/d47/d47P5K1/d47/d47P\nis a map of complexes and E′is isomorphic to the mapping cone of (12.1.2). In\nProposition 12.9 we prove that the bottom row (12.1.2) is a mi nimal homogeneous\nresolution and the image of K1is a perfect ideal of grade three. It follows im-\nmediately that the top row of (12.1.2) is the minimal homogen eous resolution of\nExt3\nP(P/imK1,P). The fact that the mapping cone of (12.1.2) (which is isomorp hic\ntoE′) is acyclic guarantees that the map from the top row of (12.1. 2) to the bottom\nrow of (12.1.2) induces an isomorphism on homology. In parti cular, the induced\ncomplex\n0→Ext3\nP(P/imK1,P)→P/imK1→A→0\nis exact and the defining ideal of Ais the sum of two linked codimension three\nperfect ideals; see Example 7.3. /square\nThe complex E′is obtained by rearranging the rows and columns of the matric es\nof the complex Eof Theorem 10.3.\nDefinition 12.2. Adopt the data of Theorem 12.1. Let Ebe the minimal homo-\ngeneous resolution of Awhich is given Theorem 10.3. We define a new minimalARTINIAN GORENSTEIN ALGEBRAS 39\nhomogeneous resolution\nE′: 0→PE′\n4− →P7E′\n3− →P12E′\n2− →P7E′\n1− →P.\nLetE′\n1be the matrix obtained from E1by arranging the columns in the order\n(12.2.1) 5 ,1,7,6,2,3,4;\nletE′\n2be the matrix obtained from E2by arranging the rows in the order (12.2.1)\nand arranging the columns in the order\n(12.2.2) 12 ,3,4,1,2,5,6,9,10,7,8,11;\nletE′\n3be the matrix obtained from E3by arranging the rows in the order (12.2.2)\nand arranging the columns in the order\n(12.2.3) 7 ,6,2,3,4,5,1;\nand let E′\n4be the matrix obtained from E4by arranging the rows in the order\n(12.2.3). Partition the matrices E′\niin the form (12.1.1).\nThe component matrices are given by\nZ=/bracketleftbigg1\nαa3,8d4,2−1\nαa2,8d4,3+x2c4,8d4,5d4,6d4,1d4,4d4,7\na1,8 0 0 0 0 0/bracketrightbigg\n,\nK1=/bracketleftbig\nw2−xp−1(w2Φ3)zw−xp−1(zwΦ3)y2−xp−1(y2Φ3)yz−xp−1(yzΦ3)yw−xp−1(ywΦ3)/bracketrightbig\nK2=\n1\nαa3,8d6,2−1\nαa2,8d6,3+x2c6,8d6,5d6,6d6,1d6,4d6,7\n1\nαa3,8d5,2−1\nαa2,8d5,3+x2c5,8d5,5d5,6d5,1d5,4d5,7\n1\nαa3,8d1,2−1\nαa2,8d1,3+x2c1,8d1,5d1,6d1,1d1,4d1,7\n1\nαa3,8d2,2−1\nαa2,8d2,3+x2c2,8d2,5d2,6d2,1d2,4d2,7\n1\nαa3,8d3,2−1\nαa2,8d3,3+x2c3,8d3,5d3,6d3,1d3,4d3,7\n,\nK3=\nd4,8 0\n1\nαa3,5d4,2−1\nαa2,5d4,3+x2c4,5a1,5\n1\nαa3,6d4,2−1\nαa2,6d4,3+x2c4,6a1,6\n1\nαa3,1d4,2−1\nαa2,1d4,3+x2c4,1a1,1\n1\nαa3,4d4,2−1\nαa2,4d4,3+x2c4,4a1,4\n1\nαa3,7d4,2−1\nαa2,7d4,3+x2c4,7a1,7\n,\nL0=/bracketleftbig\nz2−xp−1(z2Φ3)x2p−1(y∗)/bracketrightbig\n,and\nL1=\nd6,8h6,5h6,6h6,1h6,4h6,7\nd5,8h5,5h5,6h5,1h5,4h5,7\nd1,8h1,5h1,6h1,1h1,4h1,7\nd2,8h2,5h2,6h2,1h2,4h2,7\nd3,8h3,5h3,6h3,1h3,4h3,7\n,\nwhere\nhi,j=x2ci,j+1\nα/vextendsingle/vextendsingle/vextendsingle/vextendsingledi,2di,3\na2,ja3,j/vextendsingle/vextendsingle/vextendsingle/vextendsingle.40 SABINE EL KHOURY AND ANDREW R. KUSTIN\nDefinition 12.3. Adopt the data of Theorem 12.1. Let\nw1=d6,6=[w∗p−1(ywΦ3)]·x−y\nπ1=d3,4=[y∗p−1(y2Φ3)]·x−[w∗p−1(wyΦ3)]·x−y,\nπ2=d1,7=[y∗p−1(yzΦ3)]·x−z\nπ3=d2,1=[y∗p−1(ywΦ3)]·x−w\nu1,1=d1,6=−[y∗p−1(w2Φ3)]·x\nu1,2=d2,6=−[z∗p−1(w2Φ3)]·x\nu1,3=d3,6=[y∗p−1(ywΦ3)]·x−[w∗p−1(w2Φ3)]·x+w\nu2,1=d1,5=−[y∗p−1(wzΦ3)]·x\nu2,2=d2,5=−[z∗p−1(zwΦ3)]·x+w,and\nu2,3=d3,5=[y∗p−1(yzΦ3)]·x−[w∗p−1(wzΦ3)]·x.\nLetUandΠbe the matrices\nU=/bracketleftbigg\nu1,1u1,2u1,3\nu2,1u2,2u2,3/bracketrightbigg\nand Π=\nπ1\nπ2\nπ3\n.\nFor 1≤k≤3, let ∆kbe(−1)k+1times the determinant of Uwith column kdeleted.\nLemma 12.4. Apply the language of Definition 12.3. The matrices Z, K 2, and K 3\nof Definition 12.2 are equal to Z =0,\nK2=\n1\nαa3,8d6,2−1\nαa2,8d6,3+c6,8x20 w1 0 0 0\n1\nαa3,8d5,2−1\nαa2,8d5,3+c5,8x2w1 0 0 0 0\n1\nαa3,8d1,2−1\nαa2,8d1,3+c1,8x2u2,1u1,1 0−π3π2\n1\nαa3,8d2,2−1\nαa2,8d2,3+c2,8x2u2,2u1,2π3 0−π1\n1\nαa3,8d3,2−1\nαa2,8d3,3+c3,8x2u2,3u1,3−π2π1 0\n,and\nK3=\n−w1 0\n1\nαa3,5d4,2−1\nαa2,5d4,3+c4,5x20\n1\nαa3,6d4,2−1\nαa2,6d4,3+c4,6x20\n1\nαa3,1d4,2−1\nαa2,1d4,3+c4,1x2−π1\n1\nαa3,4d4,2−1\nαa2,4d4,3+c4,4x2−π2\n1\nαa3,7d4,2−1\nαa2,7d4,3+c4,7x2−π3\n.\nProof. Apply Corollary 11.2. /square\nIt follows from Lemma 12.4 that the bottom row of (12.1.2) is a complex, which\nwe denote K:\n(12.4.1) K: 0→P2K3− →P6K2− →P5K1− →P.\nWe show in Proposition 12.9 that Kis a (minimal homogeneous) resolution of a\nperfect module. Indeed we prove that Kis a specialization of the resolution ofARTINIAN GORENSTEIN ALGEBRAS 41\nAnne Brown [2, Prop. 3.6]. The precise formulation of the Bro wn resolution which\nwe use may be found in [13, Prop. 5.8] and also in Proposition 1 2.8. We now put K\ninto the form of the Brown resolution.\nLemma 12.5. Apply the language of Definition 12.3. Let Br1be the matrix\nBr1=/bracketleftbig−(UΠ)1−(UΠ)2w1π1w1π2w1π3/bracketrightbig\n.\nThe matrix K 1of Definition 12.2 satisfies K 1=Br1.\nProof. Use straightforward calculations, including the identiti es (11.1.2)-(11.1.6)\nand the fact that if ℓ∈U, then\nℓ=[x∗ℓ]·x+[y∗ℓ]·z+[z∗ℓ]·y+[w∗ℓ]·w,\nto show that\n(K1)i−(Br1)i=qi·x2,\nfor 1≤i≤5, with qi∈kkkas defined below\nq1=\n\n−[y∗p−1(w2Φ3)]·[y∗p−1(y2Φ3)]+[ y∗p−1(w2Φ3)]·[w∗p−1(wyΦ3)]\n−[z∗p−1(w2Φ3)]·[y∗p−1(yzΦ3)]+[ y∗p−1(ywΦ3)]·[y∗p−1(ywΦ3)]\n−[w∗p−1(w2Φ3)]·[y∗p−1(ywΦ3)]−[x∗p−1(w2Φ3)],\nq2=\n\n−[y∗p−1(wzΦ3)]·[y∗p−1(y2Φ3)]+[ y∗p−1(wzΦ3)]·[w∗p−1(wyΦ3)]\n−[z∗p−1(zwΦ3)]·[y∗p−1(yzΦ3)]+[ y∗p−1(yzΦ3)]·[y∗p−1(ywΦ3)]\n−[w∗p−1(wzΦ3)]·[y∗p−1(ywΦ3)]−[x∗p−1(zwΦ3)],\nq3=/braceleftigg\n−[w∗p−1(ywΦ3)]·[y∗p−1(y2Φ3)]+[ w∗p−1(ywΦ3)]·[w∗p−1(wyΦ3)]\n−[x∗p−1(y2Φ3)],\nq4=−[z∗p−1(yzΦ3)]·[y∗p−1(yzΦ3)]−[x∗p−1(yzΦ3)],and\nq5=−[x∗p−1(ywΦ3)]−[w∗p−1(ywΦ3)]·[y∗p−1(ywΦ3)].\nThe identities of Lemma 11.3: (b) minus (d), (c) minus (e), (f ), (g), and (h) show\nthatq1,q2,q3,q4, and q5are zero, respectively. /square\nProposition 12.6. Apply the language of Definition 12.3. Let θbe the invertible\nmatrix\nθ=\n1 0 0 0 0 0\n1\nα[y∗p−1(w2Φ3)]·x1 0 0 0 0\n−1\nα[y∗p−1(zwΦ3)]·x0 1 0 0 0\n0 0 0 1 0 0\n0 0 0 0 1 0\n0 0 0 0 0 1\n,\nK′be the complex\n0→P2K′\n3− →P6K′\n2− →P5K1− →P,42 SABINE EL KHOURY AND ANDREW R. KUSTIN\nwhere K′\n2=K2θand K′\n3=θ−1K3, and Br3andBr2be the matrices\nBr2=\n1\nα(UΠ)2 0 w1 0 0 0\n−1\nα(UΠ)1w1 0 0 0 0\n0 u2,1u1,10−π3π2\n0 u2,2u1,2π3 0−π1\n0 u2,3u1,3−π2π1 0\nand\nBr3=\n−w1 0\n−1\nα(UΠ)10\n1\nα(UΠ)2 0\n−1\nα∆1−π1\n−1\nα∆2−π2\n−1\nα∆3−π3\n.\nThen K′\n3=Br3and K′\n2=Br2.\nProof. In light of Lemma 12.4, it suffices to study column one of each m atrix. We\ncompute that α(K′\n2)1,1is equal to\n\n\n+[w2p−1(z∗)](Φ3)·[w∗p−1(z2Φ3)]·x2\n−[zwp−1(z∗)](Φ3)·[w∗p−1(zwΦ3)]·x2\n+[w∗p−1(zwΦ3)]·xw+[zwp−1(z∗)](Φ3)·xz\n−zw−α[w∗p−1(z∗)]·x2\n−[w∗p−1(ywΦ3)]·[y∗p−1(zwΦ3)]·x2+[y∗p−1(zwΦ3)]·xy\nandα(Br2)1,1is equal to\n\n\n−[y∗p−1(wzΦ3)]·[y∗p−1(y2Φ3)]·x2+[y∗p−1(wzΦ3)]·[w∗p−1(wyΦ3)]·x2\n−[z∗p−1(zwΦ3)]·[y∗p−1(yzΦ3)]·x2−zw\n+[y∗p−1(yzΦ3)]·[y∗p−1(ywΦ3)]·x2−[w∗p−1(wzΦ3)]·[y∗p−1(ywΦ3)]·x2\n+[y∗p−1(wzΦ3)]·xy+[z∗p−1(zwΦ3)]·xz\n+[w∗p−1(wzΦ3)]·xw.\nThus, α(K′\n2)1,1−α(Br2)1,1=q7·x2for\nq7=\n\n+[w2p−1(z∗)](Φ3)·[w∗p−1(z2Φ3)]−[zwp−1(z∗)](Φ3)·[w∗p−1(zwΦ3)]\n−α[w∗p−1(z∗)]+[ y∗p−1(wzΦ3)]·[y∗p−1(y2Φ3)]\n+[z∗p−1(zwΦ3)]·[y∗p−1(yzΦ3)]−[y∗p−1(yzΦ3)]·[y∗p−1(ywΦ3)]\n+[w∗p−1(wzΦ3)]·[y∗p−1(ywΦ3)]−2[w∗p−1(ywΦ3)]·[y∗p−1(zwΦ3)].\nApply Lemma 11.3.(c) to see that q7=0.\nIn a similar manner, one computes that\nα(K′\n2)2,1−α(Br2)2,1=q8·x2and α(K′\n2)5,1=q9·x2.\nThe constants q8andq9are zero by items (b) and (a), respectively, of Lemma 11.3\nThe entries α(K′\n2)3,1andα(K′\n2)4,1are obviously zero.ARTINIAN GORENSTEIN ALGEBRAS 43\nOne computes that\nα(K′\n3)2,1−α(Br3)2,1=q10·x2,\nα(K′\n3)3,1−α(Br3)3,1=q11·x2,\nα(K′\n3)4,1−α(Br3)4,1=0,\nα(K′\n3)5,1−α(Br3)5,1=q12·x2,and\nα(K′\n3)6,1−α(Br3)6,1=0.\nApply items (b), (c), and (a) of Lemma 11.3 to see that q10=0,q11=0, and q12=0,\nrespectively. /square\nObservation 12.7. The complex K of (12.4.1) is isomorphic to\n0→P2B3− →P6B2− →P5B1− →P,\nfor\nB1=/bracketleftbig(UΠ)1(UΠ)2−w1π1−w1π2−w1π3/bracketrightbig\n.\nB2=\n(UΠ)2 0−w1 0 0 0\n−(UΠ)1w1 0 0 0 0\n0 u2,1−u1,1 0−π3π2\n0 u2,2−u1,2π3 0−π1\n0 u2,3−u1,3−π2π1 0\nand\nB3=\nw1 0\n(UΠ)10\n(UΠ)20\n∆1 π1\n∆2 π2\n∆3 π3\n.\nProof. The complex Kis isomorphic to the complex K′of Proposition 12.6. Fur-\nthermore,\n0/d47/d47P2K′\n3/d47/d47\nβ3/d15/d15P6K′\n2/d47/d47\nβ2/d15/d15P5K1/d47/d47\n1/d15/d15P\n−1\n/d15/d15\n0/d47/d47P2B3/d47/d47P6B2/d47/d47P5B1/d47/d47P,\nwith\nβ3=/bracketleftbigg\n−1\nα0\n0−1/bracketrightbigg\nand β2=\n1\nα0 0 0 0 0\n0 1 0 0 0 0\n0 0−1 0 0 0\n0 0 0 1 0 0\n0 0 0 0 1 0\n0 0 0 0 0 1\n,\nis an isomorphism of complexes. /square44 SABINE EL KHOURY AND ANDREW R. KUSTIN\nProposition 12.8. Let P be a commutative Noetherian ring, and let w 1, z2, ui,jand\nπj, with 1≤i≤2and1≤j≤3, be elements of P. Let U and Πbe the matrices\nU=/bracketleftbigg\nu1,1u1,2u1,3\nu2,1u2,2u2,3/bracketrightbigg\nand Π=\nπ1\nπ2\nπ3\n.\nFor1≤k≤3, let∆kbe(−1)k+1times the determinant of U with column k deleted.\nLetBbe the maps and modules\n0→P2b3− →P6b2− →P5b1− →P,\nwhere\nb3=\nw1 z2\n(UΠ)10\n(UΠ)20\n∆1 π1\n∆2 π2\n∆3 π3\n,\nb2=\n(UΠ)2 0−w1−z2u21−z2u22−z2u23\n−(UΠ)1w1 0 z2u11 z2u12 z2u13\n0 u21−u11 0−π3 π2\n0 u22−u12 π3 0−π1\n0 u23−u13−π2 π1 0\n,and\nb1=/bracketleftbig(UΠ)1(UΠ)2z2∆1−w1π1z2∆2−w1π2z2∆3−w1π3/bracketrightbig\n.\nThenBis a complex and if 3≤grade I1(b1), thenBis a resolution of a perfect\nP-module.\nProof. See [2, Prop. 3.6] or [13, Prop. 5.8]. /square\nProposition 12.9. Adopt the hypotheses of Theorem 12.1. Then the bottom row of\n(12.1.2) is a homogeneous minimal resolution of the perfect module P /imK1.\nProof. Observation 12.7 shows that the complex Kof (12.4.1), which is the bottom\nrow of (12.1.2), is the Anne Brown complex of Proposition 12. 8 with z2=0. In light\nof Proposition 12.8, it suffices to show that 3 ≤gradeim K1. The ideal K1is homo-\ngeneous; so it suffices to prove that 3 ≤grade(imK1)m, wheremis the maximal ho-\nmogeneous ideal of P. The ideal im K1+(x)contains im K1+(xp−1(y∗),xp−1(z∗)),\nwhich is equal to the defining ideal of A. The defining ideal of Ahas grade four;\nhence 4≤grade(imK1+(x))mand 3≤grade(imK1)m. /square\n13. T HE CASE WHEN THE RANK OF SM IS AT LEAST TWO .\nThis section is devoted to the proof of the following Theorem .ARTINIAN GORENSTEIN ALGEBRAS 45\nTheorem 13.1. Adopt the data of 2.8and the matrix SMof(6.4.2) . IfSMhas rank\nat least two, then Ais a hypersurface section in the sense that the defining ideal of\nAis equal to (f)+J for some homogeneous element f of P and some homogeneous\nideal J of P with\n(a)P/J is Gorenstein ring of codimension three, and\n(b)f is regular on P /J.\nRemark. Theorem 13.1 is implemented in the Macaulay2 script “minres 6”; see\nSection 14.A.5.\nThe proof of Theorem 13.1 is given in 13.8. The explicit matri ces in the resolu-\ntion ofAare also given in 13.8. Furthermore, it is shown in 13.8 that t he defining\nideal of Ais the generalized sum of two linked codimension three perfe ct ideals.\nSome of the tools of Section 6 are used in the proof of Theorem 1 3.1. In particu-\nlar, recall the homomorphism\nΓ:U∗\n0→U0\nof Definition 6.4, the complex\n(13.1.1) C: 0→kkkC3− −−− → D2U∗\n0C2− −−− →U0⊗U∗\n0\nev∗(1)C1− −−− →/logicalandtext2U0→0\nfrom (6.4.3), and the element δofD2U∗\n0with\n(13.1.2) Γ∨(u1)=u1(δ),foru1∈U0,\nfrom Definition 6.2. (See (13.8.3) for a concrete version of δ.) Recall, also, from\nObservation 6.5 and Lemma 6.3, that the hypothesis that 2 ≤rankSM guarantees\nthat\n(13.1.3) I2(SM)=I1(C3)=I5(C2)=I3(C1)=I3(B)=kkk.\nObservation 13.2. The hypotheses of Theorem 13.1 guarantee that the minimal\nhomogeneous resolution of Ahas the following graded Betti numbers :\n(13.2.1) 0 →P(−7)→P(−5)6→P(−4)5\n⊕\nP(−3)5→P(−2)6→P.\nFurthermore, the linear syzygies of the defining ideal of Aare given by the restric-\ntion ofDto the kernel of B.\nRemark. The maps f1,2,B, andDfrom Theorem 5.1 all appear in the statement or\nproof of Observation 13.2.\nProof. Recall from (13.1.3) that I3(B)=kkk. It follows that the non-minimal resolu-\ntion/parenleftbig\nF,/tildewidef/parenrightbig\nofA, which is given in Theorem 5.1, has the form\n0→P(−7)→P(−4)3\n⊕\nP(−5)6/bracketleftiggI3∗\n∗ ∗/bracketrightigg\n− −−−− →P(−4)8\n⊕\nP(−3)8/bracketleftigg∗I3\n∗ ∗/bracketrightigg\n− −−−− →P(−3)3\n⊕\nP(−2)6→P.46 SABINE EL KHOURY AND ANDREW R. KUSTIN\nOne may split off a split exact subcomplex to obtain a complex with graded Betti\nnumbers given in (13.2.1).\nThe linear relations on f1,2are induced by\n(13.2.2)U0⊗U∗\n0\nev∗(1)/bracketleftiggB\nD/bracketrightigg\n− −−−−− →P⊗U∗\n0\n⊕\nP⊗Sym2U0.\nThe map Bis a surjection; let σ:P⊗U∗\n0→U0⊗U∗\n0\nev∗(1)be a splitting map in the sense\nthatB◦σis the identity map on P⊗U∗\n0. Once the isomorphism\nimσB|imσ− −− → P⊗U∗\n0\nis split from (13.2.2) one is left with\nkerBD|kerB− −−− → P⊗Sym2U0.\n/square\nObservation 13.3 describes our strategy for proving Theore m 13.1. We look for\na matrix Xand a generating set g1,..., g6for the defining ideal of Asuch that the\nhypotheses of Observation 13.3 are satisfied. If the matrix Xhappens to be an\nalternating matrix, then we are finished.\nObservation 13.3. Adopt the hypotheses of Theorem 13.1. Suppose that g 1,..., g6\nin P is a minimal homogeneous generating set for the defining i deal ofAand X is a\n5×5matrix X of linear forms from P, with linearly independent co lumns, such that\n/bracketleftbigg1g2g3g4g5/bracketrightbig\nX=0.\nLet ggg and d 1be the matrices\nggg=/bracketleftbigg1g2g3g4g5g6/bracketrightbig\nand d 1=/bracketleftbigg1g2g3g4g5/bracketrightbig\n,\nand J be the ideal J =(g1,g2,g3,g4,g5)of P. Then the following statements hold.\n(a)The columns of\n(13.3.1)/bracketleftbiggX−g6I5\n01×5 d1/bracketrightbigg\n,\nwhere I 5is a5×5identity matrix, are a minimal generating set for the module\nof syzygies of g gg.\n(b)The element g 6is regular on P /J.\n(c)The ideal J is a grade three Gorenstein ideal of P.\n(d)If X is an alternating matrix, then the following statements also hold.\n(i)The complex\n0→PdT\n1− →P5X− →P5d1− →P\nis a minimal homogeneous resolution of P /J.ARTINIAN GORENSTEIN ALGEBRAS 47\n(ii)The mapping cone of\n0/d47/d47PdT\n1/d47/d47\ng6\n/d15/d15P5 X/d47/d47\ng6/d15/d15P5d1/d47/d47\ng6/d15/d15P\ng6\n/d15/d15\n0/d47/d47PdT\n1/d47/d47P5 X/d47/d47P5d1/d47/d47P\nis a minimal homogeneous resolution of A.\n(iii) The complex\n0→PgggT\n− →P6g3− →P10g2− →P6ggg− →P\nis a minimal homogeneous resolution of A, whereg2is given in (13.3.1) ,\nand\ng3=/bracketleftbigg\n−g6dT\n1\n−X 0/bracketrightbigg\n.\nProof. (a) Recall that the graded Betti numbers of Aare given in (13.2.1). The\ncolumns of\n(13.3.2)/bracketleftbiggX\n01×5/bracketrightbigg\nare the beginning of a minimal generating set for the linear s yzygies of ggg. It follows\nfrom (13.2.1) that the columns of (13.3.2) generate all of th e linear syzygies of ggg. In\na similar manner, the columns of (13.3.1) are the beginning o f a minimal generating\nset for the module of syzygies of ggg; hence, it follows from (13.2.1) that the columns\nof (13.3.1) are a minimal generating set for the module of all syzygies of ggg.\n(b) The bottom row of (13.3.1) generates the ideal (g1,..., g5):P(g6); hence g6is\nregular on P/J.\n(c) Ifpis any prime ideal of Pwhich contains I1(ggg), then the element g6is regular\non the local ring (P/J)p; hence(P/J)pis Gorenstein if and only if (P/(J,g6))pis\nGorenstein.\n(di) We saw in (a) that\nP5X− →P5d1− →P\nis the beginning of a minimal resolution of P/J. We saw in (c) that Jis a grade\nthree Gorenstein ideal. It follows that the Betti numbers of P/Jare\n0→P1→P5→P5→P.\nLetyyybe a row vector with\n(13.3.3) 0 →P1yyyT\n− →P5X− →P5d1− →P\nexact. The matrix Xis an alternating matrix; hence, XdT\n1=0 and dT\n1=αyyyTfor\nsome α∈P. The quotient P/Jis Gorenstein; so when Hom P(−,P)is applied to\n(13.3.3) one obtains another resolution of P/J. It follows that αis a unit.\nItems (dii) and (diii) now follow immediately. /square48 SABINE EL KHOURY AND ANDREW R. KUSTIN\nThe map “ L” of Definition 13.4 will lead us to the matrix “ X” of Observa-\ntion 13.3.\nDefinition 13.4. Adopt the hypotheses of Lemma 13.1. Define the P-module ho-\nmomorphism\nL:P⊗kkkD2U∗\n0→P⊗kkkSym2U0\nto be the composition\nP⊗kkkD2U∗\n0C2− →U0⊗U∗\n0\nev∗(1)D− →Sym2U0,\nwhereDis given in Theorem 4.2 and C2is given in (13.1.1) and (6.4.3).\nLemma 13.5. Ifν1,0andν′\n1,0are in U∗\n0, then the map Lof Definition 13.4 satisfies\n[L(ν(2)\n1,0)](ν′\n1,0(2))=/braceleftigg\n(ν1,0∧ν′\n1,0)(ωU0)⊗[Γ(ν1,0)](ν′\n1,0)\n−x⊗/bracketleftig\np−1/parenleftbig/bracketleftbig\n(ν1,0∧ν′\n1,0)(ωU0)·Γ(ν1,0)/bracketrightbig\n(Φ3)/parenrightbig/bracketrightig\n(ν′\n1,0).\nProof. We compute\n[L(ν(2)\n1,0)](ν′\n1,0(2))\n=[(D◦C2)(ν(2)\n1,0)](ν′\n1,0(2))\n=[D(Γ(ν1,0)⊗ν1,0)](ν′\n1,0(2))\n=\n\n\n−x⊗w·(proj◦p−1)/parenleftig\n[ν1,0(y∧z)Γ(ν1,0)](Φ3)/parenrightig\n+x⊗z·(proj◦p−1)/parenleftig\n[ν1,0(y∧w)Γ(ν1,0)](Φ3)/parenrightig\n−x⊗y·(proj◦p−1)/parenleftig\n[ν1,0(z∧w)Γ(ν1,0)](Φ3)/parenrightig\n−w⊗ν1,0(y∧z)Γ(ν1,0)\n+z⊗ν1,0(y∧w)Γ(ν1,0)\n−y⊗ν1,0(z∧w)Γ(ν1,0)\n(ν′\n1,0(2)),\nby (4.2.1),\n=\n\n−x⊗w(ν′\n1,0)·(ν′\n1,0)(proj◦p−1)/parenleftig\n[ν1,0(y∧z)Γ(ν1,0)](Φ3)/parenrightig\n+x⊗z(ν′\n1,0)·(ν′\n1,0)(proj◦p−1)/parenleftig\n[ν1,0(y∧w)Γ(ν1,0)](Φ3)/parenrightig\n−x⊗y(ν′\n1,0)·(ν′\n1,0)(proj◦p−1)/parenleftig\n[ν1,0(z∧w)Γ(ν1,0)](Φ3)/parenrightig\n−w⊗(ν′\n1,0∧ν1,0)(y∧z)[Γ(ν1,0)](ν′\n1,0)\n+z⊗(ν′\n1,0∧ν1,0)(y∧w)[Γ(ν1,0)](ν′\n1,0)\n−y⊗(ν′\n1,0∧ν1,0)(z∧w)[Γ(ν1,0)](ν′\n1,0)ARTINIAN GORENSTEIN ALGEBRAS 49\n=\n\n−x⊗w(ν′\n1,0)·p−1/parenleftig\n[ν1,0(y∧z)Γ(ν1,0)](Φ3)/parenrightig\n(ν′\n1,0)\n+x⊗z(ν′\n1,0)·p−1/parenleftig\n[ν1,0(y∧w)Γ(ν1,0)](Φ3)/parenrightig\n(ν′\n1,0)\n−x⊗y(ν′\n1,0)·p−1/parenleftig\n[ν1,0(z∧w)Γ(ν1,0)](Φ3)/parenrightig\n(ν′\n1,0)\n−w⊗(ν′\n1,0∧ν1,0)(y∧z)[Γ(ν1,0)](ν′\n1,0)\n+z⊗(ν′\n1,0∧ν1,0)(y∧w)[Γ(ν1,0)](ν′\n1,0)\n−y⊗(ν′\n1,0∧ν1,0)(z∧w)[Γ(ν1,0)](ν′\n1,0)\n=/braceleftigg\n−x⊗p−1/parenleftig\n[(ν1,0∧ν′\n1,0)(y∧z∧w)Γ(ν1,0)](Φ3)/parenrightig\n(ν′\n1,0)\n−(ν′\n1,0∧ν1,0)(y∧z∧w)⊗[Γ(ν1,0)](ν′\n1,0)\n=/braceleftigg\n−x⊗p−1/parenleftig\n[(ν1,0∧ν′\n1,0)(ωU0)Γ(ν1,0)](Φ3)/parenrightig\n(ν′\n1,0)\n−(ν′\n1,0∧ν1,0)(ωU0)⊗[Γ(ν1,0)](ν′\n1,0)\n=/braceleftigg\n−x⊗/bracketleftig\np−1/parenleftig\n[(ν1,0∧ν′\n1,0)(ωU0)·Γ(ν1,0)](Φ3)/parenrightig/bracketrightig\n(ν′\n1,0)\n+(ν1,0∧ν′\n1,0)(ωU0)⊗[Γ(ν1,0)](ν′\n1,0). /square\nLemma 13.6. Adopt the hypotheses of Theorem 13.1. The following statements\nhold.\n(a)The maps\n0→PC3− −− → P⊗kkkD2U∗\n0L−−→ P⊗kkkSym2U0f1,2−−−→ P→A→0\nform a complex.\n(b)The homomorphism Γ, which is a linear transformation of vector spaces, has\nrank at least two.\n(c)The complex Cof(13.1.1) is split exact.\n(d)The image of Lcontains all of the linear relations on f1,2.\n(e)The element δof D 2U∗\n0of(13.1.2) is nonzero and is in the kernel of L. Ifξis\nany element of Sym2U0with δ(ξ)=1, then the induced map\nP⊗kkkkerξL−−→ P⊗kkkSym2U0\nmaps in a minimal manner onto the linear relations on f1,2.\nProof. Assertions (b) and (c) follow from (13.1.3) and Lemma 6.3.(b ). (Keep in\nmind that SM is the matrix for Γ.) Recall from Observation 13.2 that the linear\nrelations on f1,2are given by the image of\n(13.6.1) D/vextendsingle/vextendsingle/vextendsingle\nker(B).\nApply (6.5.1), followed by (c), to see that\nkerB=kerC1=imC2.50 SABINE EL KHOURY AND ANDREW R. KUSTIN\nCombine the most recent equation with (13.6.1) and Lemma 13. 5 to conclude that\nthe linear relations on f1,2are given by\nim/parenleftbig\nD◦C2/parenrightbig\n=imL.\nAssertion (d) is established. It is clear from the equation L=D◦C2thatL◦C3=0;\nhence (a) is established.\nThe complex Cis split exact; so it is clear that the image of C3is a free kkk-module\nof rank 1. This image is generated by the element δ∈D2U∗\n0of (13.1.2). If ξis any\nelement of Sym2U0with δ(ξ)=1, then\nD2U∗\n0=kerξ⊕kkkδ\nand ker ξis a free kkk-module of rank 5. Combine (d) and the fact that L(δ) =0 in\norder to conclude that the image of\n(13.6.2) P⊗kkkkerξL−−→ P⊗kkkSym2U0\ncontains all of the linear syzygies on f1,2. The map (13.6.2) is minimal because f1,2\nhas a five-dimensional vector space of linear syzygies; see ( 13.2.1). Assertion (e) is\nestablished and the proof is complete. /square\nAs promised in Observation 13.3, the rest of the argument pro ceeds quickly once\nLemma 13.7 is established.\nLemma 13.7. The P-module homomorphism L:P⊗kkkD2U∗\n0→P⊗kkkSym2U0of\nDefinition 13.4 is an alternating map.\nProof. Fix arbitrary elements ν1,0andν′\n1,0inU∗\n0. It suffices to show that\n[L(ν(2)\n1,0)](ν′\n1,0(2))+[L(ν′\n1,0(2))](ν(2)\n1,0)=0.\nApply Lemma 13.5. One quickly sees that it suffices to show tha t\n(13.7.1) 0 =\n\n/bracketleftig\np−1/parenleftbig/bracketleftbig\n(ν1,0∧ν′\n1,0)(ωU0)·Γ(ν1,0)/bracketrightbig\n(Φ3)/parenrightbig/bracketrightig\n(ν′\n1,0)\n+/bracketleftig\np−1/parenleftbig/bracketleftbig\n(ν′\n1,0∧ν1,0)(ωU0)·Γ(ν′\n1,0)/bracketrightbig\n(Φ3)/parenrightbig/bracketrightig\n(ν1,0).\nIt is clear that (13.7.1) holds if ν1,0andν′\n1,0are linearly dependent. So, it suffices\nto establish (13.7.1) when ν1,0andν′\n1,0are linearly independent elements of U∗\n0.\nWe may as well take y,z,wandy∗,z∗,w∗to be a pair of dual bases for U0andU∗\n0\nwith y∧z∧w=ωU0,w∗∧z∗∧y∗=ωU∗\n0,ν1,0=y∗, and ν′\n1,0=z∗. The right side of\n(13.7.1) is now equal to\n/braceleftigg/bracketleftbig\np−1/parenleftbig/bracketleftbig\n(y∗∧z∗)(y∧z∧w)·Γ(y∗)/bracketrightbig\n(Φ3)/parenrightbig/bracketrightbig\n(z∗)\n+/bracketleftbig\np−1/parenleftbig/bracketleftbig\n(z∗∧y∗)(y∧z∧w)·Γ(z∗)/bracketrightbig\n(Φ3)/parenrightbig/bracketrightbig\n(y∗)\n=−/bracketleftbig\np−1/parenleftbig/bracketleftbig\nwΓ(y∗)/bracketrightbig\n(Φ3)/parenrightbig/bracketrightbig\n(z∗)+/bracketleftbig\np−1/parenleftbig/bracketleftbig\nwΓ(z∗)/bracketrightbig\n(Φ3)/parenrightbig/bracketrightbig\n(y∗)\n=−/bracketleftbig/bracketleftbig\nwp−1(z∗)/bracketrightbig\n(Φ3)/bracketrightbig\n[Γ(y∗)]+/bracketleftbig/bracketleftbig\nwp−1(y∗)/bracketrightbig\n(Φ3)/bracketrightbig\n[Γ(z∗)]ARTINIAN GORENSTEIN ALGEBRAS 51\n=\n\n−/bracketleftbig/bracketleftbig\nywp−1(z∗)/bracketrightbig\n(Φ3)/bracketrightbig\n·y∗[Γ(y∗)]+/bracketleftbig/bracketleftbig\nywp−1(y∗)/bracketrightbig\n(Φ3)/bracketrightbig\n·y∗[Γ(z∗)]\n−/bracketleftbig/bracketleftbig\nzwp−1(z∗)/bracketrightbig\n(Φ3)/bracketrightbig\n·z∗[Γ(y∗)]+/bracketleftbig/bracketleftbig\nzwp−1(y∗)/bracketrightbig\n(Φ3)/bracketrightbig\n·z∗[Γ(z∗)]\n−/bracketleftbig/bracketleftbig\nw2p−1(z∗)/bracketrightbig\n(Φ3)/bracketrightbig\n·w∗[Γ(y∗)]+/bracketleftbig/bracketleftbig\nw2p−1(y∗)/bracketrightbig\n(Φ3)/bracketrightbig\n·w∗[Γ(z∗)].\nApply Definition 6.4, especially (6.4.2), to see that\ny∗Γ(y∗)=[ p−1(z2Φ3)](w2Φ3)−[p−1(zwΦ3)](zwΦ3)\ny∗Γ(z∗)=−[p−1(yzΦ3)](w2Φ3)+[p−1(zwΦ3)](ywΦ3)\nz∗Γ(z∗)=[ p−1(y2Φ3)](w2Φ3)−[p−1(ywΦ3)](ywΦ3)\nw∗Γ(z∗)=−[p−1(y2Φ3)](zwΦ3)+[p−1(yzΦ3)](ywΦ3)\nw∗Γ(y∗)=[ p−1(yzΦ3)](zwΦ3)−[p−1(z2Φ3)](ywΦ3).\nIt follows that the right side of (13.7.1) is equal to Y1+Y2+Y3, with\nY1=\n\n[p−1(yw(Φ3))](z∗)·/braceleftigg\n−[p−1(z2(Φ3))](w2(Φ3))\n+[p−1(zw(Φ3))](zw(Φ3))\n+[p−1(yw(Φ3))](y∗)·/braceleftigg\n−[p−1(yz(Φ3))](w2(Φ3))\n+[p−1(zw(Φ3))](yw(Φ3)),\nY2=\n\n[p−1(zw(Φ3))](z∗)·/braceleftigg\n[p−1(yz(Φ3))](w2(Φ3))\n−[p−1(yw(Φ3))](zw(Φ3))\n+[p−1(zw(Φ3))](y∗)·/braceleftigg\n[p−1(y2(Φ3))](w2(Φ3))\n−[p−1(yw(Φ3))](yw(Φ3)),\nand\nY3=\n\n[p−1(w2(Φ3))](z∗)·/braceleftigg\n−[p−1(yz(Φ3))](zw(Φ3))\n+[p−1(z2(Φ3))](yw(Φ3))\n+[p−1(w2(Φ3))](y∗)·/braceleftigg\n−[p−1(y2(Φ3))](zw(Φ3))\n+[p−1(yz(Φ3))](yw(Φ3)).\nWe reconfigure Y3to obtain\nY3=\n\n−[p−1(w2(Φ3))](z∗)·[p−1(yz(Φ3))](zw(Φ3))\n−[p−1(w2(Φ3))](y∗)·[p−1(y2(Φ3))](zw(Φ3))\n+[p−1(w2(Φ3))](z∗)·[p−1(z2(Φ3))](yw(Φ3))\n+[p−1(w2(Φ3))](y∗)·[p−1(yz(Φ3))](yw(Φ3))\n=−[p−1(yu′\n1(Φ3))](zw(Φ3))+[ p−1(u′\n1z(Φ3))](yw(Φ3)),\nforu′\n1=[p−1(w2(Φ3))](z∗)·z+[p−1(w2(Φ3))](y∗)·y. The elements x,y,z,wand\nx∗,y∗,z∗,w∗form a pair of dual bases for UandU∗, respectively; consequently, if\nu1∈U, then\nu1=u1(x∗)·x+u1(y∗)·y+u1(z∗)·z+u1(w∗)·w.52 SABINE EL KHOURY AND ANDREW R. KUSTIN\nIn particular, when u1is taken to be p−1(w2(Φ3))∈U, then\nu′\n1=[p−1(w2(Φ3))]−[p−1(w2(Φ3))](x∗)·x−[p−1(w2(Φ3))](w∗)·w,\nandY3=Y′\n3+Y′′\n3+Y′′′\n3, with\nY′\n3=/braceleftigg\n−[p−1(y[p−1(w2(Φ3))](Φ3))](zw(Φ3))\n+[p−1(z[p−1(w2(Φ3))](Φ3))](yw(Φ3))\n=/braceleftigg\n−/bracketleftbig\ny·[p−1(zw(Φ3))]·[p−1(w2(Φ3))]/bracketrightbig\n(Φ3)\n+/bracketleftbig\nz·[p−1(yw(Φ3))]·[p−1(w2(Φ3))]/bracketrightbig\n(Φ3),\nY′′\n3=/braceleftigg/bracketleftbig\np−1/parenleftbig\ny/parenleftbig/bracketleftbig\n[p−1(w2(Φ3))](x∗)/bracketrightbig\n·x/parenrightbig\n(Φ3)/parenrightbig/bracketrightbig\n(zw(Φ3))\n−/bracketleftbig\np−1/parenleftbig\nz/parenleftbig/bracketleftbig\n[p−1(w2(Φ3))](x∗)/bracketrightbig\n·x/parenrightbig\n(Φ3)/parenrightbig/bracketrightbig\n(yw(Φ3))\n=/bracketleftbig\n[p−1(w2(Φ3))](x∗)/bracketrightbig\n·/parenleftbig\n[p−1(yx(Φ3))](zw(Φ3))−[p−1(zx(Φ3))](yw(Φ3))/parenrightbig\n=/bracketleftbig\n[p−1(w2(Φ3))](x∗)/bracketrightbig\n·/parenleftbig\n(yzw(Φ3))−(zyw(Φ3))/parenrightbig\n=0,\nbecause p−1(u1xΦ3)=p−1(p(u1))= u1foru1∈U, and\nY′′′\n3=/braceleftigg/bracketleftbig\np−1/parenleftbig\ny/parenleftbig/bracketleftbig\n[p−1(w2(Φ3))](w∗)/bracketrightbig\n·w/parenrightbig\n(Φ3)/parenrightbig/bracketrightbig\n(zw(Φ3))\n−/bracketleftbig\np−1/parenleftbig\nz/parenleftbig/bracketleftbig\n[p−1(w2(Φ3))](w∗)/bracketrightbig\n·w/parenrightbig\n(Φ3)/parenrightbig/bracketrightbig\n(yw(Φ3))\n=/bracketleftbig\n[p−1(w2(Φ3))](w∗)/bracketrightbig\n·/braceleftigg\n[p−1(yw(Φ3))](zw(Φ3))\n−[p−1(zw(Φ3))](yw(Φ3))\n=0.\nThus,\nY3=Y′\n3=/braceleftigg\n−/bracketleftbig\ny·[p−1(zw(Φ3))]·[p−1(w2(Φ3))]/bracketrightbig\n(Φ3)\n+/bracketleftbig\nz·[p−1(yw(Φ3))]·[p−1(w2(Φ3))]/bracketrightbig\n(Φ3).\nIn a similar manner\nY1=/braceleftigg\n−/bracketleftbig\nz·p−1[(w2)(Φ3)]·p−1[(yw)(Φ3)]/bracketrightbig\n(Φ3)\n+/bracketleftbig\nw·p−1[(zw)(Φ3)]·p−1[(yw)(Φ3)]/bracketrightbig\n(Φ3)\nY2=/braceleftigg/bracketleftbig\ny·p−1[(w2)(Φ3)]·p−1[(zw)(Φ3)]/bracketrightbig\n(Φ3)\n−/bracketleftbig\nw·p−1[(yw)(Φ3)]·p−1[(zw)(Φ3)]/bracketrightbig\n(Φ3).\nIt is now apparent that Y1+Y2+Y3=0; thus, (13.7.1) is established and the proof\nis complete. /square\n13.8. Proof of Theorem 13.1.Start with the complex\n(13.8.1) P⊗D2U∗\n0L− →P⊗Sym2U0f1,2− − →P→A→0\nof Lemma 13.6. The image of f1,2is equal to the defining ideal of AandLmaps\nonto the linear syzygies of f1,2. Use the basis\n(13.8.2) y∗(2),y∗z∗,y∗w∗,z∗(2),z∗w∗,w∗(2)ARTINIAN GORENSTEIN ALGEBRAS 53\nto identify D2U∗\n0with P6and the basis y2,yz,yw,z2,zw,w2to identify Sym2U0with\nP6. Convert (13.8.1) into the complex\nP(−3)6L− →P(−2)6f1,2− − → P→A→0,\nwhere Lis the matrix of Land f1,2is given in Example 4.6. It is shown in\nLemma 13.7 that Lis an alternating matrix.\nThe nonzero element δ∈D2U∗\n0with δ∈kerLis identified in Lemma 13.6. When\nδis written in terms of the basis (13.8.2), it looks like\n(13.8.3)\nCA(1,1)\nCA(1,2)\nCA(1,3)\nCA(2,2)\nCA(2,3)\nCA(3,3)\n,\nwhere CA is the classical adjoint of SM. Let θbe a 6×6 invertible matrix with\nentries from kkkand last column equal to (13.8.3). Observe that\nP(−3)6θTLθ− −− → P(−2)6f1,2(θT)−1\n− −−−−− → P→A→0\nis a complex which is exact at the positions PandA. Observe also that the alternat-\ning matrix θTLθmaps onto the linear syzygies of f1,2(θT)−1. Write f1,2(θT)−1as/bracketleftbigd1g6/bracketrightbig\n, where d1=/bracketleftbigg1g2g3g4g5/bracketrightbig\n, and write\nθTLθ=/bracketleftbiggX 05×1\n01×5 0/bracketrightbigg\n,\nwhere Xis a 5×5 alternating matrix. None of the linear syzygies of/bracketleftbigd1g6/bracketrightbig\ninvolve g6. The hypotheses of Observation 13.3 are satisfied and the pro of is com-\nplete. Indeed, the minimal homogeneous resolution of Ais the mapping cone of\n(13.8.4)\n0/d47/d47P(−7)dT\n1/d47/d47\ng6\n/d15/d15P(−5)5 X/d47/d47\ng6\n/d15/d15P(−4)5d1/d47/d47\ng6\n/d15/d15P(−2)\ng6\n/d15/d15\n0/d47/d47P(−5)dT\n1/d47/d47P(−3)5 X/d47/d47P(−2)5d1/d47/d47P.\nThe minimal homogeneous resolution of Ais also given by\n0→Pg4− →P6g3− →P10g2− →P6g1− →P.\nwith\ng1=/bracketleftbigd1g6/bracketrightbig\n, g2=/bracketleftbigg\nX−g6\n01×5d1/bracketrightbigg\n,\ng3=/bracketleftbigg\n−g6dT\n1\n−X05×1/bracketrightbigg\n,and g4=/bracketleftbigg\ndT\n1\ng6/bracketrightbigg\n.\n/square54 SABINE EL KHOURY AND ANDREW R. KUSTIN\nTheorem 13.9. Adopt the data of 2.8and the matrix SMof(6.4.2) . If SMhas\nrank at least two, then the defining ideal of Ais the generalized sum of two linked\ncodimension three perfect ideals.\nProof. See (13.8.4) and Example 7.4. /square\n14. M ACAULAY 2SCRIPTS .\nThe package “SocDeg3.m2” contains five Macaulay2 scripts: B asicSocDeg3,\nminres7, SumOfLinkedIdeals7, SumOfLinkedIdeals9, and mi nres6. In this section,\nwe describe each of these scripts, print the package SocDeg3 .m2, run a few ex-\namples, and give a list of Phi’s which are in the “correct form ” in the sense that\nannΦ∩P1={0}, the first listed variable is a weak Lefschetz element on P/an n Phi\nand if rankSM =1, then SM is given by (1.0.1). Recall that SM is given by (1.0. 1)\nif and only if Bis given by (6.6.2).\nRemark 14.1. Recall that if Pis the polynomial ring kkk[x1,..., xn]=Sym•P1, then\nthe graded dual of Pis the Divided Power Algebra D•P∗\n1. The monomials/parenleftbigx1,...,xn\ni/parenrightbig\nform a basis for SymiP1(see Convention 2.5) and the elements\n{x∗(e1)\n1···x∗(en)\nn|e1+···+en=i}\nform a basis for DiP∗\n1. The structure of D•P∗\n1as a Sym•P1- module is given by\n(14.1.1) xf1\n1···xfnn(x∗(e1)\n1···x∗(en)\nn)=/braceleftigg\nx∗(e1−f1)\n1···x∗(en−fn)\nn,iffj≤ejfor all j,\n0, otherwise .\nThe notation that Macaulay2 uses for the element\nx∗(e1)\n1···x∗(en)\nn\nofD•P∗\n1isxe1\n1···xenn. Consequently, in this section, when we write “Let Phi be\nthe element yx2+xzw+xw2+z3+w3ofD3P∗\n1”, we are thinking, “Let Phi be the\nelement y∗x∗(2)+x∗z∗w∗+x∗w∗(2)+z∗(3)+w∗(3)ofD3P∗\n1”.\nThe Macaulay2 command contract( xf1\n1···xfnn,xe1\n1···xenn)produces (14.1.1).\n14.A. The scripts in SocDeg3.m2.\n14.A.1. The script “BasicSocDeg3”. LetPbe a polynomial ring in four variables\nover a field k. For the sake of this introduction P=k[x,y,z,w](although any field\ncan be used and the variables can be given any names). Let Phi b e an element\nofD3P∗\n1, where P1is the subspace of linear forms in P,P∗\n1is Hom k(P1,k), and\nD•P∗\n1is the Divided Power algebra which is the graded dual of the po lynomial ring\nP=Sym•P1The Phi must be chosen so that zero is the only linear element i n\nann(Phi) and the variable x(that is, the first variable in the list of variables) is a\nweak Lefschetz element on P/ann Phi. Run this script by typing:\n(f11,f12,A,B,C,D,SM)=BasicSocDeg3(Phi) .ARTINIAN GORENSTEIN ALGEBRAS 55\nThe output is the seven matrices f1,1,f1,2,A,B,C,D,SM. The first six matrices are\ndescribed in Example 4.6. The last matrix is described in (6. 4.2). According to\nTheorem 4.2, the matrices\n0→P(−6)f4− − →P(−4)3\n⊕\nP(−4)6f3− − →P(−3)8\n⊕\nP(−3)8f2− − →P(−2)3\n⊕\nP(−2)6f1− − → P,\nform a minimal homogeneous resolution of P/ann(xΦ), where\nf4=/bracketleftbiggfT\n1,1\nfT\n1,2/bracketrightbigg\n, f3=/bracketleftbigg\nxBTDT\nATxCT/bracketrightbigg\n,\nf2=/bracketleftbigg\nA xB\nxC D/bracketrightbigg\n,and f1=/bracketleftbigf1,1f1,2/bracketrightbig\n.\nAccording to Theorem 5.1, the matrices\n(14.1.2) 0 →P(−7)F4− − →P(−4)3\n⊕\nP(−5)6F3− − →P(−4)8\n⊕\nP(−3)8F2− − →P(−3)3\n⊕\nP(−2)6F1− − → P,\nform a homogeneous (not necessarily minimal) resolution of P/ann(xΦ), where\nF4=/bracketleftbiggx fT\n1,1\nfT\n1,2/bracketrightbigg\n, F3=/bracketleftbigg\nBTDT\nATx2CT/bracketrightbigg\n,\nF2=/bracketleftbiggA B\nx2C D/bracketrightbigg\n,and F1=/bracketleftbigx f1,1f1,2/bracketrightbig\n.\nThe rank of the 3 ×3 symmetric matrix SM determines the number of minimal\ngenerators in the defining ideal of A. If SM has rank zero, then the defining ideal of\nAhas nine minimal generators and the minimal resolution of Ais given by (14.1.2).\nIf SM has rank one, then the defining ideal of Ahas seven minimal generators. One\nmay change variables to put SM in the form\n\nα0 0\n0 0 0\n0 0 0\n,\nfor some nonzero α∈kkk. Once this is done, then the minimal resolution of Ais\ngiven in Theorem 10.3. This procedure is implemented by the s cript “minres7”. If\nSM has rank at least two, then the defining ideal of Ahas six minimal generators, A\nis a hypersurface section of a codimension three Gorenstein ring, and the minimal\nresolution of Ais given in Theorem 13.1. This procedure is implemented in th e\nscript “minres6”.\n14.A.2. The script “minres7”. LetPbe a polynomial ring in four variables over a\nfield kand Phi be an element of D3P∗\n1. Choose Phi so that zero is the only linear\nelement in ann(Phi) and the first listed variable is a weak Lef schetz element on56 SABINE EL KHOURY AND ANDREW R. KUSTIN\nP/ann Phi. Assume further that the matrix Bof Example 4.6 has rank 2, and has the\nform\n0 0 0 0 0 0 0 0\n0 0 α0 0 0 0 0\n0−α0 0 0 0 0 0\n,\nwhere αis a unit.\nRun this script by typing\n(E1,E2,E3,E4)=minres7(Phi) .\nThe output is the four matrices E1,E2,E3, and E4in the minimal resolution of P/ann\n(Phi), as described in Theorem 10.3.\n14.A.3. The script “SumOfLinkedIdeals7”. LetPbe a polynomial ring in four\nvariables over a field kand Phi be an element of D3P∗\n1. Choose Phi so that zero\nis the only linear element in ann(Phi) and the first listed var iable is a weak Lef-\nschetz element on P/ann Phi. Assume further that the matrix Bfrom Example 4.6\nhas rank 2, and has the form\nB=\n0 0 0 0 0 0 0 0\n0 0 α0 0 0 0 0\n0−α0 0 0 0 0 0\n,\nwhere αis a unit. Run this script by typing\n(B1,B2,B3,L0,L1)=SumOfLinkedIdeals7(Phi) .\nThe output is five matrices B1,B2,B3,L0,L1which sit in the following map of\ncomplexes:\n(14.1.3) 0/d47/d47P(−7)BT\n1/d47/d47\n−LT\n0/d15/d15P(−5)5BT\n2/d47/d47\nLT\n1/d15/d15P(−3)\n⊕\nP(−4)5BT\n3/d47/d47\n−L1\n/d15/d15P(−2)\n⊕\nP(−3)\nL0\n/d15/d15\n0/d47/d47P(−5)\n⊕\nP(−4)B3/d47/d47P(−4)\n⊕\nP(−3)5B2/d47/d47P(−2)5B1/d47/d47P.\nThe bottom row of (14.1.3) is the Brown resolution of P/I1(B1), where I1(B1)is\na perfect grade 3 ideal; see Proposition 12.8. The top row is t he resolution of the\ncanonical module, Ext3\nP(P/I1(B1),P), ofP/I1(B1). The mapping cone of (14.1.3)\nis the minimal resolution of P/annΦ. Thus, the defining ideal of Ais expressed as\nthe sum of linked perfect ideals as described in Theorem 12.1 .ARTINIAN GORENSTEIN ALGEBRAS 57\n14.A.4. The script “SumOfLinkedIdeals9”. LetPbe a polynomial ring in four\nvariables over a field kand Phi be an element of D3P∗\n1. Choose Phi so that zero\nis the only linear element in ann(Phi) and the first listed var iable is a weak Lef-\nschetz element on P/ann Phi. Assume further that the matrix Bfrom Example 4.6\nis identically zero. Run this script by typing\n(K1,K2,K3,L0,L1)=SumOfLinkedIdeals9(Phi) .\nThe output is five matrices K1,K2,K3,L0,L1which sit in the following map of com-\nplexes:\n(14.1.4) 0/d47/d47P(−7)KT\n1/d47/d47\n−LT\n0/d15/d15P(−5)6KT\n2/d47/d47\nLT\n1/d15/d15P(−4)8KT\n3/d47/d47\n−L1/d15/d15P(−3)3\nL0\n/d15/d15\n0/d47/d47P(−4)3K3/d47/d47P(−3)8K2/d47/d47P(−2)6K1/d47/d47P.\nThe bottom row of (14.1.4) is a deformation of the Eagon-Nort hcott resolution of\nP/(y2,yz,yw,z2,zw,w2). The top row is the resolution of the canonical module,\nExt3\nP(P/I1(K1),P), ofP/I1(K1). The mapping cone of (14.1.4) is the minimal res-\nolution of P/annΦ. Thus, the defining ideal of Ais expressed as the sum of linked\nperfect ideals as described in Theorem 9.1.\n14.A.5. The script “minres6”. LetPbe a polynomial ring in four variables over a\nfield kand Phi be an element of D3P∗\n1. Choose Phi so that zero is the only linear\nelement in ann(Phi) and the first listed variable is a weak Lef schetz element on\nP/ann Phi. Assume further that the matrix SM from (6.4.2) is i nvertible. Run this\nscript by typing\n(d1,f,X)=minres6(Phi) .\nThe output is a 1 ×5 matrix d1with entries from P, an element fofP, and a 5×5\nalternating matrix Xwith entries from P. Let Jbe the ideal generated by the entries\nofd1. Then the ideal (J,f)is equal to the ideal ann (Phi) of P; the ring P/Jis a\ncodimension three Gorenstein ring with minimal homogeneou s resolution\n0→P(−5)dT\n1− →P(−3)5X− →P(−2)5d1− →P,\nandfis regular on P/J. Thus P/ann(Phi) is a hypersurface section of a codimension\nthree Gorenstein ring as described in Theorems 13.1 and 13.9 .\n14.B. The scripts.\n--**************************************************\n--**The script BasicSocDeg3 **\n--**Type (f11,f12,A,B,C,D,SM)=BasicSocDeg3(Phi); **\n--**************************************************\nBasicSocDeg3=(Phi) -> (\nRl:=ring(Phi);\nPhil:=Phi;58 SABINE EL KHOURY AND ANDREW R. KUSTIN\nif not isHomogeneous(Phil) then\nerror \"Phi must be homogeneous.\";\nif not (degree(Phil))_0==3 then\nerror \"Phi must have degree 3.\";\nxl:=Rl_0;\nyl:=Rl_1;\nzl:=Rl_2;\nwl:=Rl_3;\nvl:=matrix{{xl,yl,zl,wl}};\nvsql:=matrix{{ylˆ2,yl *zl,yl*wl,zlˆ2,zl *wl,wlˆ2}};\npl:=contract (xl *(transpose vl) *vl,Phil);\nif det pl==0 then\nerror \"One must choose Phi so that the only linear element in a nn(Phi)\nis zero and the variable P_0 is a weak Lefschetz element on P/a nn Phi.\";\npinversel:=inverse (pl);\nblop:=(u)-> (contract(matrix{{u}},matrix{{Phil}})//v l);\nproj:=matrix{{0_Rl,1,0,0},{0,0,1,0},{0,0,0,1}};\nip:=(u,u’)-> (transpose( blop(u)) *pinversel *blop(u’));\n-- If u is an element of Sym_2U, then blop(u) is u(Phi_3), writ ten\n-- as a column vector.\nf11l:=xl *vl*pinversel *matrix{{0,0,0},{1,0,0},{0,1,0},{0,0,1}};\nf12l:=(vsql-xl *vl*pinversel *( matrix{{contract(vsql,Phil)}}//vl));\nc1b212:=\nmatrix{{0},{ip(yl *wl,zlˆ2)-ip(yl *zl,zl*wl)},{ip(yl *wl,zl*wl)-ip(yl *zl,wlˆ2)}};\nc2b212:=\nmatrix{{ip(yl *zl,zl*wl)-ip(yl *wl,zlˆ2)},{0},{ip(zl *wl,zl*wl)-ip(zlˆ2,wlˆ2)}};\nc3b212:=\nmatrix{{ip(yl *zl,wlˆ2)-ip(yl *wl,zl*wl)},{ip(zlˆ2,wlˆ2)-ip(zl *wl,zl*wl)},{0}};\nc4b212:=\nmatrix{{0},{ip(ylˆ2,zl *wl)-ip(yl *wl,yl*zl)},{ip(ylˆ2,wlˆ2)-ip(yl *wl,yl*wl)}};\nc5b212:=\nmatrix{{ip(yl *wl,yl*zl)-ip(ylˆ2,zl *wl)},{0},{ip(zl *yl,wlˆ2)-ip(zl *wl,yl*wl)}};\nc6b212:=\nmatrix{{ip(yl *wl,yl*wl)-ip(ylˆ2,wlˆ2)},{ip(zl *wl,yl*wl)-ip(wlˆ2,yl *zl)},{0}};\nc7b212:=\nmatrix{{0},{ip(yl *zl,yl*zl)-ip(ylˆ2,zlˆ2)},{ip(yl *zl,yl*wl)-ip(ylˆ2,wl *zl)}};\nc8b212:=\nmatrix{{ip(ylˆ2,zlˆ2)-ip(yl *zl,yl*zl)},{0},{ip(zlˆ2,yl *wl)-ip(zl *yl,zl*wl)}};\nb212:=c1b212|c2b212|c3b212|c4b212|c5b212|c6b212|c7b 212|c8b212;\nkb222:=matrix{\n{ 0, 0, 0, wl, 0, 0, -zl, 0 },\n{ -wl, 0, 0, 0, wl, 0, yl, -zl },\n{ zl, 0, 0, -yl, 0, wl, 0, 0 },\n{ 0, -wl, 0, 0, 0, 0, 0, yl },\n{ 0, zl, -wl, 0, -yl, 0, 0, 0 },\n{ 0, 0, zl, 0, 0, -yl , 0, 0 }};ARTINIAN GORENSTEIN ALGEBRAS 59\n-- If l and qu are homogeneous forms in Sym_2U_0 of degree 1 and 2\n-- respectively, then j12(l,qu) is the quadratic form\n-- l times (proj circ pˆ{-1})(qu Phi) in Sym_2U_0. Of course t he\n-- answer is expressed\n-- as a column vector. The entries of the column vector are the\n-- coefficients of yˆ2,yz,yw,zˆ2,zw,wˆ2 in that order.\nprej12:= (l,qu) ->(\nt1:=pinversel *blop(qu);\nreturn(l *matrix{{yl,zl,wl}} *proj*t1));\nj12:= (l,qu) ->( matrix{{prej12(l,qu)}}//vsql);\nb222:=\n(-j12(zl,yl *wl)+j12(wl,yl *zl)|\nj12(wl,zlˆ2)-j12(zl,wl *zl)|\nj12(wl,zl *wl)-j12(zl,wlˆ2)|\n-j12(wl,ylˆ2)+j12(yl,wl *yl)|\n-j12(wl,yl *zl)+j12(yl,wl *zl)|\n-j12(wl,yl *wl)+j12(yl,wlˆ2)|\nj12(zl,ylˆ2)-j12(yl,yl *zl)|\nj12(zl,yl *zl)-j12(yl,zlˆ2));\nkb211:=matrix{{yl, 0, 0, zl, 0, 0, wl, 0},\n{ 0, yl, 0, 0, zl, 0, 0, wl },\n{-wl, 0, yl, 0, -wl, zl, 0, 0}};\nj14:=(mu,nu) ->( above=\nmu*matrix{{yl,zl,wl}} *proj*pinversel *(matrix{{nu}}//vl);\nreturn(matrix{{above}}//vsql)\n);\nb221:=\n(j14(yl,yl)-j14(wl,wl))|\nj14(yl,zl)|\nj14(yl,wl)|\nj14(zl,yl)|\n(j14(zl,zl)-j14(wl,wl))|\nj14(zl,wl)|\nj14(wl,yl)|\nj14(wl,zl);\n-- j14’(mu,nu) is the element proj[ (mu (pˆ{inverse} nu))(P hi)] of U_0\n-- for mu in U and nu in Uˆ *. The answer is written as a column vector.\n-- The input is given in terms of elements of U and Uˆ *.\nj14’:=(mu,nu) ->(\nabove:=contract(mu *vl*pinversel *(matrix{{nu}}//vl),Phil);\nreturn(proj *(matrix{{above}}//vl)));\nb211:=\n(j14’(yl,yl)-j14’(wl,wl))|\nj14’(yl,zl)|\nj14’(yl,wl)|\nj14’(zl,yl)|\n(j14’(zl,zl)-j14’(wl,wl))|\nj14’(zl,wl)|\nj14’(wl,yl)|\nj14’(wl,zl);\nAl:=(-xl *b211+kb211);\nBl:=b212;\nCl:=-b221;\nDl:=(kb222-xl *b222);60 SABINE EL KHOURY AND ANDREW R. KUSTIN\nSMl:= matrix{{ip(zlˆ2,wlˆ2)-ip(zl *wl,zl*wl),\nip(zl*wl,yl*wl)-ip(zl *yl,wlˆ2),\nip(yl*zl,zl*wl)-ip(zlˆ2,yl *wl)},\n{ip(zl*wl,yl*wl)-ip(yl *zl,wlˆ2),\nip(yl*yl,wlˆ2)-ip(yl *wl,yl*wl),\nip(yl*zl,yl*wl)-ip(yl *yl,zl*wl)},\n{ip(yl*zl,zl*wl)-ip(yl *wl,zlˆ2),\nip(yl*wl,yl*zl)-ip(yl *yl,zl*wl),ip(yl *yl,zl*zl)-ip(yl *zl,yl*zl)}};\nreturn(f11l,f12l,Al,Bl,Cl,Dl,SMl)\n);\n--**************************************************\n--**The script minres7 **\n--**Type (E1,E2,E3,E4)=minres7(Phi); **\n--**************************************************\nminres7=(Phi)->(\nRun1:=BasicSocDeg3(Phi);\nf11l1:=(Run1)_0;\nf12l1:=(Run1)_1;\nAl1:=(Run1)_2;\nBl1:=(Run1)_3;\nCl1:=(Run1)_4;\nDl1:=(Run1)_5;\nPl:=ring(Phi);\nxl:=Pl_0;\nalphal:=Bl1_(1,2);\nif alphal==0 then\nerror \"The entry of B in row 2, column 3 must be a unit.\";\nrealB:=matrix {{0,0,0,0,0,0,0,0},{0,0,alphal,0,0,0,0 ,0},\n{0,-alphal,0,0,0,0,0,0}};\nif not (Bl1-realB)==0 then error \"The matrix B has the wrong f orm.\";\nzmat=(r,c)->(map(Plˆr,Plˆc,(i,j)->0));\ntheta1:=(matrix{{1_Pl,0,0},{0,0,alphal},{0,-alphal, 0}}|zmat(3,6))||\n(zmat(6,1)|submatrix(Dl1,,{1,2})|id_(Plˆ6));\nNewM:=(1//alphal) *matrix{\n{0,-Al1_(2,0),Al1_(1,0),0,0,0,0,0},\n{Al1_(2,0),0,Al1_(2,2),Al1_(2,3),Al1_(2,4),Al1_(2,5 ),Al1_(2,6),Al1_(2,7)},\n{-Al1_(1,0),-Al1_(2,2),0,-Al1_(1,3),-Al1_(1,4),-Al1 _(1,5),-Al1_(1,6),-Al1_(1,7)},\n{0,-Al1_(2,3),Al1_(1,3),0,0,0,0,0},\n{0,-Al1_(2,4),Al1_(1,4),0,0,0,0,0},\n{0,-Al1_(2,5),Al1_(1,5),0,0,0,0,0},\n{0,-Al1_(2,6),Al1_(1,6),0,0,0,0,0},\n{0,-Al1_(2,7),Al1_(1,7),0,0,0,0,0}};\nNewtheta2:=(id_(Plˆ8)|zmat(8,8))||(NewM|id_(Plˆ8));\nF1l:=map(Plˆ1, ,(xl *f11l1)|f12l1);\nF2l:= map(source F1l,,(Al1|Bl1)||(xlˆ2 *Cl1|Dl1));\nE1l:=map(Plˆ1,,submatrix’(F1l *theta1,,{1,2}));\nTF2l:=(inverse theta1) *F2l*Newtheta2;\nTE2l:=submatrix’(TF2l,{1,2},{1,2,9,10});\nE2l:=map(source E1l,,TE2l);\nJl:=(zmat(6,6)|id_(Plˆ6))||(id_(Plˆ6)|zmat(6,6));\nE3l:= map(source E2l,,Jl *transpose E2l);\nE4l:=map(source E3l,,transpose E1l);ARTINIAN GORENSTEIN ALGEBRAS 61\nreturn(E1l,E2l,E3l,E4l)\n);\n--*************************************************** *****\n--**The script SumOfLinkedIdeals7 **\n--**Type (B1,B2,B3,L0,L1)=SumOfLinkedIdeals7(Phi); **\n--*************************************************** *****\nSumOfLinkedIdeals7=(Phi)->(\nRun2:=minres7(Phi);\nRun3:=BasicSocDeg3(Phi);\nBl1:=(Run3)_3;\nDl1:=(Run3)_5;\nE1l1:=(Run2)_0;\nE2l1:=(Run2)_1;\nE3l1:=(Run2)_2;\nE4l1:=(Run2)_3;\nE1l’:=submatrix(E1l1,,{4,0,6,5,1,2,3});\nE2l’:=submatrix(E2l1,{4,0,6,5,1,2,3},{11,2,3,0,1,4, 5,8,9,6,7,10});\nE3l’:=submatrix(E3l1,{11,2,3,0,1,4,5,8,9,6,7,10},{6 ,5,1,2,3,4,0});\nE4l’:=submatrix(E4l1,{6,5,1,2,3,4,0},);\nL0l:=submatrix(E1l’,,{0,1});\nK1l:=submatrix(E1l’,,{2..6});\nK2l:=submatrix’(E2l’,{0,1},{0..5});\nL1l:=submatrix(E2l’,{2..6},{0..5});\nK3l:=transpose(submatrix(E2l’,{0,1},{0..5}));\nalphal:=Bl1_(1,2);\nthetal:=matrix{\n{1,0,0,0,0,0},\n{(1//alphal) *Dl1_(1,2),1,0,0,0,0},\n{(-1//alphal) *Dl1_(1,1),0,1,0,0,0},\n{0,0,0,1,0,0},\n{0,0,0,0,1,0},\n{0,0,0,0,0,1}};\nK2l’:=K2l *thetal;\nK3l’:=(inverse thetal) *K3l;\nbeta3:=matrix{{-1//alphal,0},{0,-1}};\nbeta2:=matrix{\n{1//alphal,0,0,0,0,0},\n{0,1,0,0,0,0},\n{0,0,-1,0,0,0},\n{0,0,0,1,0,0},\n{0,0,0,0,1,0},\n{0,0,0,0,0,1}};\nBl1’:=-K1l;\nBl2:=K2l’ *(inverse (beta2));\nBl3:=beta2 *K3l’*( inverse (beta3));\nnewL0l:=L0l *(transpose beta3);\nnewL1l:=-L1l *(transpose(inverse(thetal))) *(transpose(beta2));\nreturn(Bl1’,Bl2,Bl3,newL0l,newL1l)\n);\n--*************************************************** *****\n--**The script SumOfLinkedIdeals9 **\n--**Type (K1,K2,K3,L0,L1)=SumOfLinkedIdeals9(Phi); **\n--*************************************************** *****\nSumOfLinkedIdeals9=(Phi)->(\nRun4:=BasicSocDeg3(Phi);\nf11l1:=(Run4)_0;62 SABINE EL KHOURY AND ANDREW R. KUSTIN\nf12l1:=(Run4)_1;\nAl1:=(Run4)_2;\nBl1:=(Run4)_3;\nCl1:=(Run4)_4;\nDl1:=(Run4)_5;\nPl=ring(Phi);\nxl:=Pl_0;\nif not (Bl1==0) then error \"The matrix B must be zero to use thi s script.\";\nK1l:=f12l1;\nK2l:=Dl1;\nK3l:=transpose Al1;\nL1l:=xlˆ2 *Cl1;\nL0l:=xl*f11l1;\nreturn(K1l,K2l,K3l,L0l,L1l)\n);\n--*************************************************** *****\n--**The script minres6 **\n--**Type (d1,f,X)=minres6(Phi); **\n--*************************************************** *****\nminres6=(Phi)->(\nds’:=(X,r,c)->(return(det submatrix’(X,{r},{c})));\nCA:=(X)->(return (transpose(matrix{{ds’(X,0,0),-ds’( X,0,1),ds’(X,0,2)},\n{-ds’(X,1,0),ds’(X,1,1),-ds’(X,1,2)},\n{ds’(X,2,0),-ds’(X,2,1),ds’(X,2,2)}}))\n);\nRun5:=BasicSocDeg3(Phi);\nf12l1:=(Run5)_1;\nDl1:=(Run5)_5;\nSMl1:=(Run5)_6;\nPl:=ring(Phi);\nif (minors(2,SMl1)==0) then\nerror \"The matrix SM must have rank at least two to use this scr ipt.\";\nSMi:=CA(SMl1);\npredelta:=matrix{{SMi_(0,0)},{SMi_(0,1)},{SMi_(0,2) },{SMi_(1,1)}};\ndelta:=predelta||matrix{{SMi_(1,2)},{SMi_(2,2)}};\nC2:=matrix{\n{SMl1_(0,0),SMl1_(0,1),0,0,-SMl1_(2,1),-SMl1_(2,2)} ,\n{SMl1_(1,0),SMl1_(1,1),SMl1_(1,2),0,0,0},\n{SMl1_(2,0),SMl1_(2,1),SMl1_(2,2),0,0,0},\n{0,SMl1_(0,0),0,SMl1_(0,1),SMl1_(0,2),0},\n{0,SMl1_(1,0),-SMl1_(2,0),SMl1_(1,1),0,-SMl1_(2,2)} ,\n{0,SMl1_(2,0),0,SMl1_(2,1),SMl1_(2,2),0},\n{0,0,SMl1_(0,0),0,SMl1_(0,1),SMl1_(0,2)},\n{0,0,SMl1_(1,0),0,SMl1_(1,1),SMl1_(1,2)}};\nL:=Dl1*C2;\nM:=id_(Plˆ6)|delta;\ni:=#(for i from 0 to 6 when (det submatrix’(M,,{i})==0) list i do i=i+1);\nabove:=submatrix’(id_(Plˆ6),,{i})|delta;\nX:=submatrix’ (transpose(above) *L*above,{5},{5});\nnewf12l:=f12l1 *inverse(transpose above);\nd1:=submatrix’(newf12l,,{5});\nf:=newf12l_(0,5);\nreturn(d1,f,X)\n);ARTINIAN GORENSTEIN ALGEBRAS 63\n14.C. We run a few examples.\nii2 : load \"SocDeg3.m2\"\nii3 : P=QQ[x,y,z,w];\nii4 : Phi=xˆ2 *y+zˆ3+x*z*w+x*wˆ2+wˆ3;\nii5 : (f11,f12,A,B,C,D,SM)=BasicSocDeg3(Phi);\nii6 : -- One can look at the output, for example:\nA\noo6 = | y 0 0 z 0 0 w 0 |\n| 0 y 0 0 x+z -x -x w |\n| -w 0 y -x -w z -x -x |\n3 8\noo6 : Matrix P <--- P\nii7 : -- We form the resolution of P/ann(x Phi):\nf1=map(Pˆ1, ,f11|f12);\n1 9\noo7 : Matrix P <--- P\nii8 : f2= map(source f1,,(A|x *B)||(x*C|D));\n9 16\noo8 : Matrix P <--- P\nii9 : f3top=(x *transpose B)|(transpose D);\n8 9\noo9 : Matrix P <--- P\nii10 : f3bot=(transpose A)|(x *transpose C);\n8 9\noo10 : Matrix P <--- P\nii11 : f3= map(source f2,,(f3top||f3bot));\n16 9\noo11 : Matrix P <--- P\nii12 : f4=map(source f3,,(transpose f11)||(transpose f12 ));\n9 1\noo12 : Matrix P <--- P\nii13 : -- We verify that the f_i form a resolution of P/ann(x(P hi)).\nideal(f1)==ideal(fromDual((contract(x,Phi))))\noo13 = true64 SABINE EL KHOURY AND ANDREW R. KUSTIN\nii14 : (image f2) == (ker f1)\noo14 = true\nii15 : (image f3)==(ker f2)\noo15 = true\nii16 : (image f4)==(ker f3)\noo16 = true\nii17 : -- We form the resolution of P/(ann Phi):\nF1=map(Pˆ1, ,(x *f11)|f12);\n1 9\noo17 : Matrix P <--- P\nii18 : F2= map(source F1,,(A|B)||(xˆ2 *C|D));\n9 16\noo18 : Matrix P <--- P\nii19 : F3top=((transpose B)|(transpose D));\n8 9\noo19 : Matrix P <--- P\nii20 : F3bot=((transpose A)|(xˆ2 *transpose C));\n8 9\noo20 : Matrix P <--- P\nii21 : F3= map(source F2,,(F3top||F3bot) );\n16 9\noo21 : Matrix P <--- P\nii22 : F4=map(source F3,,(transpose (x *f11))||(transpose f12));\n9 1\noo22 : Matrix P <--- P\nii23 : -- We verify that the F_i form a resolution of P/ann(Phi ).\n(ideal F1)==ideal(fromDual(Phi))\noo23 = true\nii24 : (image F2)==(ker F1)\noo24 = true\nii25 : (image F3)==(ker F2)\noo25 = true\nii26 : (image F4)==(ker F3)\noo26 = true\nii27 : SMARTINIAN GORENSTEIN ALGEBRAS 65\noo27 = {-1} | 1 0 0 |\n{-1} | 0 0 0 |\n{-1} | 0 0 0 |\n3 3\noo27 : Matrix P <--- P\nii28 : -- The rank of SM is 1, so ann(Phi) is 7-generated.\n-- The form of SM is correct; so we may apply the script minres7\n-- to obtain the minimal resolution of P/ann(Phi)\n(E1,E2,E3,E4)=minres7(Phi);\nii29 : -- We verify that the E_i form a resolution of P/ann(Phi ).\nideal(E1)==ideal(fromDual(Phi))\noo29 = true\nii30 : (image E2)==(ker E1)\noo30 = true\nii31 : (image E3)==(ker E2)\noo31 = true\nii32 : (image E4)==(ker E3)\noo32 = true\nii33 : -- The Betti numbers of this resolution are:\n(numColumns(E4),numColumns(E3),numColumns(E2),numCo lumns(E1),numRows(E1))\noo33 = (1, 7, 12, 7, 1)\noo33 : Sequence\nii34 : -- These are the expected Betti numbers.\n-- We apply the script SumOfLinkedIdeals7 to write ann Phi as a sum of\n-- linked ideals.\n(B1,B2,B3,newL0,newL1)=SumOfLinkedIdeals7(Phi);\nii35 : -- Notice that the B’s form a Brown resolution of a perfe ct\n-- grade three ideal and the mapping cone of the map of\n-- complexes from Hom_P(B,P) to B, given by\n-- -newL0 transpose, newL1 transpose, -newL1, newL0,\n-- is the minimal homogeneous resolution of P/ann (Phi).\n-- In the next example the ideal ann Phi has nine minimal\n-- generators. The matrices SM and B are both identically\n-- zero. One can read the minimal homogeneous resolution\n-- of P/ann Phi by running\n-- (f11,f12,A,B,C,D,SM)=BasicSocDeg3(Phi). We apply the\n-- script SumOfLinkedIdeals9 in order to write ann(Phi)\n-- as a sum of linked perfect ideals.\nPhi=x*yˆ2+x*zˆ2+xˆ2*w;\nii36 : (K1,K2,K3,L0,L1)=SumOfLinkedIdeals9(Phi);\nii37 : -- In the next example the ideal ann Phi has six minimal g enerators.66 SABINE EL KHOURY AND ANDREW R. KUSTIN\n-- The matrix SM is invertible.\n-- The script minres6 exhibits P/ann Phi as a hypersurface se ction of\n-- a codimension three Gorenstein ring.\nPhi=-3*xˆ3-2*xˆ2*y+2*xˆ2*z+x*zˆ2-xˆ2*w-x*y*w+x*z*w+zˆ2*w;\nii38 : (d1,f,X)=minres6(Phi);\nii39 : ideal(d1):f==ideal(d1)\noo39 = true\nii40 : (ideal(d1)+ideal(f))==(ideal fromDual Phi)\noo40 = true\nii41 : d1=map(Pˆ1,,d1);\n1 5\noo41 : Matrix P <--- P\nii42 : d2=map(source d1,,X);\n5 5\noo42 : Matrix P <--- P\nii43 : d3=map(source d1,,transpose d1);\n5 1\noo43 : Matrix P <--- P\nii44 : image d2==ker d1\noo44 = true\nii45 : image d2==ker d1\noo45 = true\nii46 : (d1,f,X)=minres6(Phi);\nii47 : ideal(d1):f==ideal(d1)\noo47 = true\nii48 : -- Thus f is regular on P/J, where J is generated by the\n-- entries of d1.\n(ideal(d1)+ideal(f))==(ideal fromDual Phi)\noo48 = true\nii49 : -- Thus (J,f) = ann Phi\nd1=map(Pˆ1,,d1);\n1 5\noo49 : Matrix P <--- P\nii50 : d2=map(source d1,,X);\n5 5\noo50 : Matrix P <--- P\nii51 : d3=map(source d1,,transpose d1);ARTINIAN GORENSTEIN ALGEBRAS 67\n5 1\noo51 : Matrix P <--- P\nii52 : image d2==ker d1\noo52 = true\nii53 : image d2==ker d1\noo53 = true\nii54 : -- Thus (d1ˆT, X, d1) are the maps in a resolution of P/J.\n-- In particular, P/J is a codimension three Gorenstein ring .\n14.D. A few suitable Φ3’s.\nLetP=kkk[x,y,z,w]be a polynomial ring in four variables over the field kkk. We list\na few examples of Φ3∈D•P∗\n1with the following properties:\n(a) ann Φ3∩P1=(0),\n(b)xis a weak Lefschetz element on P/annΦ3,\n(c) if SM, from (6.4.2), has rank one, then\nSM=\nα0 0\n0 0 0\n0 0 0\n\nfor some nonzero αinkkk.\nMany of these Φ3were obtained from examples in [16] by changing the variable s in\norder to make (a) – (c) hold. The Φ’s (i), (viii), and (xv) were used in Section 14.C.\nOf course, we had to convert these Φ’s into the notation used by Macaulay2; see\nRemark 14.1.\nThe base field is Qin all of the following examples.\n14.D.1. The corresponding Gorenstein ideal has 9 minimal generator s.\n(i)x∗y∗(2)+x∗z∗(2)+x∗(2)w∗\n(ii) 3 x∗(3)+2x∗(2)y∗+x∗y∗(2)+x∗z∗w∗+y∗z∗w∗+z∗w∗(2)\n(iii)/braceleftigg\n5x∗(3)+2x∗(2)y∗+x∗y∗(2)+x∗(2)z∗+x∗y∗z∗+y∗(2)z∗+x∗(2)w∗\n+x∗w∗(2)+w∗(3)\n(iv)/braceleftigg\n4x∗(3)+x∗(2)y∗+2x∗(2)z∗+x∗y∗z∗+x∗z∗(2)\n+y∗z∗(2)+x∗(2)w∗+x∗y∗w∗+y∗(2)w∗\n(v)x∗(2)y∗+x∗z∗w∗+x∗w∗(2)+w∗(3)\n(vi) 6 x∗(3)+2x∗(2)y∗+x∗y∗(2)+x∗(2)z∗+2x∗(2)w∗+x∗z∗w∗+x∗w∗(2)+z∗w∗(2)\n(vii) x∗(3)+x∗(2)z∗+x∗y∗z∗+y∗(2)z∗+x∗w∗(2)+y∗w∗(2)\n14.D.2. The corresponding Gorenstein ideal has 7 minimal generator s.\n(viii) x∗(2)y∗+z∗(3)+x∗z∗w∗+x∗w∗(2)+w∗(3)\n(ix) 2 x∗(3)+x∗(2)y∗+x∗y∗(2)+y∗(3)+x∗z∗(2)+y∗z∗(2)−x∗w∗(2)−y∗w∗(2)\n(x)x∗(2)y∗+x∗z∗(2)+x∗z∗w∗+z∗w∗(2)68 SABINE EL KHOURY AND ANDREW R. KUSTIN\n(xi)/braceleftigg\nx∗(3)+x∗z∗(2)+y∗z∗(2)−2x∗(2)w∗−2x∗y∗w∗\n−2y∗(2)w∗+3x∗w∗(2)+3y∗w∗(2)−3w∗(3)\n(xii)−x∗(2)y∗−x∗y∗(2)−y∗(3)+x∗z∗w∗+y∗z∗w∗+w∗(3)\n(xiii)/braceleftigg\nx∗(3)+x∗z∗(2)+y∗z∗(2)−x∗(2)w∗−x∗y∗w∗−y∗(2)w∗\n+2x∗w∗(2)+2y∗w∗(2)−3w∗(3)\n(xiv)/braceleftigg\n7x∗(3)+6x∗(2)y∗+6x∗y∗(2)+6y∗(3)+2x∗(2)z∗+2x∗y∗z∗\n+2y∗(2)z∗+2x∗(2)w∗+2x∗y∗w∗+2y∗(2)w∗+x∗z∗w∗+y∗z∗w∗\n14.D.3. The corresponding Gorenstein ideal has 6 minimal generator s.\n(xv)−3x∗(3)−2x∗(2)y∗+2x∗(2)z∗+x∗z∗(2)−x∗(2)w∗−x∗y∗w∗+x∗z∗w∗+z∗(2)w∗\n(xvi)/braceleftigg\n−2x∗(3)−2x∗(2)y∗+2x∗(2)z∗+x∗z∗(2)−x∗(2)w∗−x∗y∗w∗+x∗z∗w∗\n+z∗(2)w∗+x∗w∗(2)+w∗(3)\nREFERENCES\n[1] N. Abdallah and H. Schenck, Free Resolutions And Lefschetz Properties Of Some Artin Gor en-\nstein Rings Of Codimension Four , J. Symbolic Comput. 121(2024), Paper No. 102257, 13 pp.\n[2] A. Brown, A structure theorem for a class of grade three perfect ideals J. Algebra 105(1987),\n308–327.\n[3] D. Eisenbud, Commutative algebra with a view toward algebraic geometry Graduate Texts in\nMathematics, 150. Springer-Verlag, New York, 1995.\n[4] S. El Khoury and A. Kustin, Artinian Gorenstein algebras with linear resolutions , J. Algebra\n420(2014), 402–474.\n[5] S. El Khoury and A. Kustin, The explicit minimal resolution constructed from a Macaula y\ninverse system , J. Algebra 440(2015), 145–186.\n[6] S. El Khoury and A. Kustin, The structure of Gorenstein-linear resolutions of Artinia n alge-\nbras, J. Algebra 453(2016), 492–560.\n[7] S. El Khoury and A. Kustin, Quadratically presented grade three Gorenstein ideals , J. Algebra\n622(2023), 258–290.\n[8] D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geom etry,\nAvailable at http://www2.macaulay2.com\n[9] T. Gulliksen and G. Levin, Homology of Local Rings , Queen’s Papers in Pure and Applied\nMathematics, vol.20, Queen’s University, Kingston, Ont., 1969.\n[10] M. Hochster, Topics in the homological theory of modules over commutativ e rings , CBMS\nReg. Conf. Ser. Math., vol. 24, Amer. Math. Soc., Providence , RI, 1975.\n[11] T. J´ ozefiak, Ideals generated by minors of a symmetric matrix , Comment. Math. Helv. 53\n(1978), 595–607.\n[12] B. Kotzev, Determinantal ideals of linear type of a generic symmetric m atrix , J. Algebra 139\n(1991), 484–504.\n[13] A. Kustin, Perfect modules with Betti numbers (2,6,5,1), J. Algebra 600(2022), 71–124.\n[14] A. Kustin, The weak Lefschetz property for standard graded, Artinian G orenstein algebras of\nembedding dimension four and socle degree three , submitted for publication.\n[15] R. Kutz, Cohen-Macaulay rings and ideal theory in rings of invariant s of algebraic groups ,\nTrans. Amer. Math. Soc. 194(1974), 115–129.\n[16] P. Macias Marques, O. Veliche, and J. Weyman Artinian Gorenstein Algebras Of Embedding\nDimension Four And Socle Degree Three , J. Algebra 638(2024), 788–839.\n[17] C. Peskine and L. Szpiro, Liaison des vari ´et´es alg ´ebriques. I , Invent. Math. 26(1974), 271–\n302.\n[18] B. Ulrich, Sums of linked ideals , Trans. Amer. Math. Soc. 318(1990), 1–42.ARTINIAN GORENSTEIN ALGEBRAS 69\nMATHEMATICS DEPARTMENT , A MERICAN UNIVERSITY OF BEIRUT , RIAD EL SOLH 11-\n0236, B EIRUT , LEBANON\nEmail address :se24aub.edu.lb\nDEPARTMENT OF MATHEMATICS , UNIVERSITY OF SOUTH CAROLINA ,\nCOLUMBIA SC 29208, U.S.A.\nEmail address :kustinmath.sc.edu"    },    {        "title": "2402.13385v1.Regular_Languages_in_the_Sliding_Window_Model.pdf",        "content": "arXiv:2402.13385v1  [cs.FL]  20 Feb 2024REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL\nMOSES GANARDI, DANNY HUCKE, MARKUS LOHREY, KONSTANTINOS MA MOURAS,\nAND TATIANA STARIKOVSKAYA\nAbstract. We study the space complexity of the following problem: For a\nfixed regular language L, test membership of a sliding window of length nto\nLover a given stream of symbols. For deterministic streaming algorithms we\nprove a trichotomy theorem, namely that the (optimal) space complexity is\neither constant, logarithmic or linear, measured in the win dow length n. Addi-\ntionally, we provide natural language-theoretic characte rizations of the space\nclasses. We then extend the results to randomized streaming algorithms and\nwe show that in this setting, the space complexity of any regu lar language is\neither constant, doubly logarithmic, logarithmic or linea r. Finally, we intro-\nduce sliding window testers, which can distinguish whether a window belongs\nto the language Lor has Hamming distance > ǫn toL. We prove that every\nregular language has a deterministic (resp., randomized) s liding window tester\nthat requires only logarithmic (resp., constant) space.\n1.Introduction\n1.1.The sliding window model. Streaming algorithms process a data stream\na1a2a3···of elements aifrom left to right and have at time tonly direct access\nto the current element at. In many streaming applications, elements are outdated\nafter a certain time, i.e., they are no longer relevant. The sliding windo w model is\na simple way to model this. A sliding window algorithm computes for each time\ninstanttavaluethatonlydependsontherelevantpast(theso-calledactive window)\nofa1a2···at. There are several formalizations of the relevant past. One way t o\ndo this is to fix a window size n. Then the active window consists at each time\ninstanttconsists of the last nelements at−n+1at−n+2···at. In the literature this is\nalso called the fixed-size model. Another sliding window model that can be found\nin the literature is the variable-size model, see e.g. [7]. In this model, th e arrival of\nnew elements and the expiration of old elements can happen independ ently, which\nmeans that the window size can vary. This allows to model for instanc e time-based\nwindows, where data items arrive at irregular time instants and the a ctive window\ncontains all data items that arrive in the last nseconds for a fixed n. The special\ncase of the variable-size model, where old symbols do not expire, is th e classical\nstreaming model.\nA general goal in the area of sliding window algorithms is to avoid the ex plicit\nstorage of the active window, which would require Ω( n) space for a window size\nn, and, instead, to work in considerably smaller space, e.g. polylogarit hmic space\nwith respect to the window size n. A detailed introduction into the sliding window\nmodel can be found in [1, Chapter 8].\nThe (fixed-size) sliding window model was introduced in the seminal pa per of\nDatar et al. [26] where the authors considered the basic counting p roblem: Given\na window length nand a stream of bits, maintain a count of the number of 1’s in\nthe window. One can easily observe that an exact solution would requ ire Θ(n) bits.\nIntuitively, the reason is that the algorithm cannot see the bit which is about to\nexpire (the n-th most recent bit) without storing it explicitly; this is in fact the\nmain difficulty in most sliding window algorithms. However, Datar et al. sh ow that\n12 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIKO VSKAYA\nwithO(1\nǫ·log2n) bits one can maintain anapproximatecount up to a multiplicative\nfactor of 1±ǫ. In Section 1.3.6 below we briefly discuss further work on sliding\nwindow algorithms.\nA foundational problem that has been surprisingly neglected so far is the lan-\nguage recognition problem over sliding windows: Given a language L⊆Σ∗and a\nstream of symbols over a finite alphabet Σ, maintain a data structur e which allows\nto query membership of the active window in L.1In other words, we want to devise\na streaming algorithm which, after every input symbol, either accep ts if the current\nactive window belongs to L, or rejects otherwise. This problem finds applications\nin complex event processing, where to goal is to detect patterns in data streams.\nThese patterns are usually described in some languagebased on reg ularexpressions;\nsee e.g. [25, 87] for more details.\nExample 1.Consider the analysis of the price of a stock in order to identify shor t-\nterm upward momentum. The original stream is a time series of stock prices,\nand it is pre-processed in the following way: over a sliding window of 10 s econds\ncompute the linear regression of the prices and discretize the slope into the follow-\ning values: P2(high positive), P1(low positive), Z(zero),N1(low negative), N2\n(high negative). This gives rise to a derived stream of symbols in the a lphabet\nΣ ={P2,P1,Z,N1,N2}, over which we describe the upward trend pattern: (i) no\noccurrence of N2, (ii) at most two occurrences of N1, (iii) and any two occurrences\nofZorN1are separated by at least three positive symbols. The upward tren d\npattern can be described as the conjunction of regular expressio nsei, ¯eiiandeiii\ngiven as follows:\nei= (P2+P1+Z+N1)∗\neii= Σ∗·N1·Σ∗·N1·Σ∗·N1·Σ∗\neiii= (P2+P1)∗·/parenleftbig\n(Z+N1)·(P2+P1)[3,∞)/parenrightbig∗·(ε+Z+N1)·(P2+P1)∗\nWe want to monitorcontinuouslywhether the window ofthe last 5 minu tes matches\nthe upward trend pattern, as this is an indicator to buy the stock a nd ride the\nupward momentum. Our results will show that there is a space efficien t streaming\nalgorithm for this problem, which can be synthesized from the regula r expressions\nei,eii,eiii.\nIn this paper we focus on querying regular languages over sliding wind ows. Un-\nfortunately, there are simple regular languages which require Ω( n) space in the\nsliding window model, i.e. one cannot avoid maintaining the entire window e xplic-\nitly. However, for certain regular languages, such as for the upwa rd trend pattern\nfrom above, we present sublinear space streaming algorithms. Bef ore we explain\nour results in more detail, let us give examples of sliding window algorithm s for\nsimple regular languages.\nExample 2.Let Σ ={a,b}be the alphabet. In the following examples refer to the\nfixed-size sliding window model.\n(i) LetL= Σ∗abe the set of all words ending with a. A streaming algorithm\ncan maintain the most recent symbol of the stream in a single bit, whic h\nis also the most recent symbol of the active window. Hence, the spa ce\ncomplexity of LisO(1) in the sliding window model.\n(ii) LetL= Σ∗aΣ∗be the set of all words containing a. Ifn∈Nis the window\nsize, then a streaming algorithm can maintain the position 1 /lessorequalslanti/lessorequalslantn(from\nright to left) of the most recent a-symbol in the active window or set i=∞\n1For the standard streaming model, where input symbols do not expire, some work on language\nrecognition problems has been done; see Section 1.3.7 below .REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 3\nif the active window contains no a-symbols. Let us assume that the initial\nwindow is bnand initialize i:=∞. On input awe seti:= 1 and on input\nbwe increment iand then set i:=∞ifi > n. The algorithm accepts if\nand only if i/lessorequalslantn. Since the position ican be stored using O(logn) bits,\nwe have shown that Lhas space complexity O(logn) in the sliding window\nmodel.\n(iii) Let L=aΣ∗be the set of all words starting with a. We claim that any\n(deterministic) sliding window algorithm for Land window size n∈N(let\nuscallitPn) muststoreatleast nbits, whichmatchesthe complexityofthe\ntrivial solution where the window is stored explicitly. More precisely, t he\nclaim is thatPnreaches two distinct memory states on any two distinct\nwordsx=a1···an∈Σnandy=b1···bn∈Σn. Suppose that ai/\\e}atio\\slash=\nbi. Then we simulate Pnon the streams xbi−1andybi−1, respectively.\nThe suffixes (or windows) of length nfor the two streams are ai···anbi−1\nandbi···bnbi−1. Since exactly one of the two windows belongs to L, the\nalgorithmPnmust accept exactly one of the streams xbi−1andybi−1. In\nparticular,Pnmust reach two distinct memory states on the words xbi−1\nandybi−1, and thereforePnmust have also reached two distinct memory\nstates on the prefixes xandy, as claimed above. Therefore, Pnmust have\nat least|Σn|= 2nmemory states, which require nbits of memory.\n1.2.Results. Let us now present the main results of this paper. The precise\ndefinitions of all used notions can be found in the main part of the pap er. We\ndenote by FL(n) (resp., VL(n)) the space complexity of an optimal sliding window\nalgorithm for the language Lin the fixed-size (resp., variable-size) sliding window\nmodel. Here, ndenotes the fixed window size for the fixed-size model, whereas\nfor the variable-size model ndenotes that maximal window size among all time\ninstants when reading an input stream.\nOur first result is a trichotomy theorem for the sliding window model, s tating\nthat the deterministic space complexity is always either constant, lo garithmic, or\nlinear. This holds for both the fixed- and the variable-size model. Fur thermore, we\nprovide natural characterizations for the three space classes. For this, we need the\nfollowing language classes:\n•Lenis the class of all regular length languages, i.e., regular languages L⊆\nΣ∗such that for every n/greaterorequalslant0, either Σn⊆Lor Σn∩L=∅.\n•STis the class of all suffix-testable languages, i.e., finite Boolean combina -\ntionsoflanguagesofthe formΣ∗wwherew∈Σ∗(note thatthese languages\nare regular).\n•LIis the class of all regular left ideals, i.e., languages of the form Σ∗Lfor\nL⊆Σ∗regular.\nWe emphasize that the three defined language properties only make sense with\nrespect to an underlying alphabet. If L1,...,Lnare classes of languages over some\nalphabet Σ, then /a\\}⌊ra⌋ketle{tL1,...,Ln/a\\}⌊ra⌋ketri}htdenotes the class of all finite Boolean combinations\nof languages L∈/uniontextn\ni=1Li. We also use the following asymptotic notation in our\nresults: For functions f,g:N→R/greaterorequalslant0,f(n) =¯Ω(g(n)) holds if f(n)/greaterorequalslantc·g(n) for\nsomec >0 and infinitely many n∈N. Furthermore, f(n) =¯Θ(g(n)) holds if\nf(n) =O(g(n)) andf(n) =¯Ω(g(n)). Now we can state our first main result:\nTheorem 3. LetL⊆Σ∗be regular. The space complexity FL(n)is either Θ(1),\n¯Θ(logn), or¯Θ(n). Moreover, we have:\nFL(n) =O(1)⇐⇒L∈/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}ht\nFL(n) =O(logn)⇐⇒L∈/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}ht4 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIKO VSKAYA\nThe space complexity VL(n)is either Θ(1),Θ(logn), orΘ(n). Moreover, we have:\nVL(n) =O(1)⇐⇒L∈{∅,Σ∗}\nVL(n) =O(logn)⇐⇒L∈/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}ht\nTheorem 3 describes which regular patterns can be queried over slid ing windows\nin sublinear space: Regular left ideals over a sliding window express sta tements of\nthe form “recently in the stream some regular event happened”. D ually, comple-\nments of left ideals over a sliding window express statements of the f orm “at all\nrecent times in the stream some regular event happened”.\nMost papers on streaming algorithms make use of randomness. So, it is natu-\nral to consider randomized sliding window algorithms for regular langu ages. Our\nrandomized sliding window algorithms have a two-sided error of 1 /3 (any error\nprobability below 1 /2 would yield the same results). With Fr\nL(n) we denote the\noptimal space complexity of a randomized sliding window algorithm for Lin the\nfixed size model. Our second main result says that the regular langua ges can be\nclassified by four randomized space complexity classes: constant, doubly logarith-\nmic, logarithmic, and linear space. A language Lissuffix-free ifxy∈Landx/\\e}atio\\slash=ε\nimpliesy /∈L. We denote by SFthe class of all regular suffix-free languages.\nTheorem 4. LetL⊆Σ∗be regular. The randomized space complexity Fr\nL(n)of\nLin the fixed-size sliding window model is either Θ(1),¯Θ(loglog n),¯Θ(logn), or\n¯Θ(n). Furthermore:\nFr\nL(n) =O(1)⇐⇒L∈/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}ht\nFr\nL(n) =O(loglogn)⇐⇒L∈/a\\}⌊ra⌋ketle{tST,SF,Len/a\\}⌊ra⌋ketri}ht\nFr\nL(n) =O(logn)⇐⇒L∈/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}ht\nFigure1 comparesthe deterministic and the randomized space comp lexity. With\nRegwe denote the class ofall regularlanguages. We alsoconsider rando mizedalgo-\nrithms with one-sided error and randomized algorithms in the variable -size model.\nIn both settings we obtain the same trichotomy as for deterministic algorithms\n(Theorem 45 and Lemma 48).\nBy Theorems 3 and 4, some (simple) regular languages (e.g. a{a,b}∗) do not\nadmit sublinear space (randomized) algorithms. This gives the motiva tion to seek\nfor alternative approaches in order to achieve efficient algorithms f or all regular\nlanguages. We take our inspiration from the property testing mode l introduced by\nGoldreich et al. [60]. In this model, the task is to decide (with high proba bility)\nwhether the input has a particular property P, or is “far” from any input satisfy-\ningPwhile querying as few symbols of the input as possible. Alon et al. prove that\nevery regular language has a property tester making only O(1) many queries [2].\nThe idea of property testing was also combined with the streaming mo del, yielding\nstreaming property testers, where the objective is not to minimize the number of\nqueries but the required memory [34, 38]. We define sliding window test ers, which,\nusing as little space as possible, must accept if the window (of size n) belongs to\nthe language Land must reject if the window has Hamming distance at least γ(n)\nfrom every word in L. Hereγ(n)/lessorequalslantnis a function that is called the Hamming gap\nof the sliding window testers. We focus on the fixed-size model. Our m ain results\nconcerning sliding window testers are as follows:\nTheorem 5. LetL⊆Σ∗be regular.\n(i) There exists a deterministic sliding window tester for Lwith constant Ham-\nming gap that uses space O(logn).\n(ii) For every ǫ >0there exists a randomized sliding window tester for Lwith\ntwo-sided error and Hamming gap ǫnthat uses spaceO(1/ǫ).REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 5\nReg\n/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}ht\n/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}htReg\n/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}ht\n/a\\}⌊ra⌋ketle{tST,SF,Len/a\\}⌊ra⌋ketri}ht\n/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}htO(n)\nO(logn)\nO(loglogn)\nO(1)space complexity deterministic two-sided error\nFigure 1. The space complexity of regular languages in the fixed-\nsize sliding window model. Reg: regular languages, LI: regular\nleft ideals, ST: suffix testable languages, SF: regular suffix-free\nlanguages, Len: regular length languages. The angle brackets /a\\}⌊ra⌋ketle{t·/a\\}⌊ra⌋ketri}ht\ndenote Boolean closure.\nAdditionally, we study sliding window testers with one-sided error and prove\noptimality of all results by providing matching lower bounds.\n1.3.Related work. This paper builds on four conference papers [42, 43, 44, 46].\nTo keep this paper coherent, we decided to omit some of the results from [42, 43, 44,\n46]. In this section, we briefly discuss these results as well as other related work.\n1.3.1.Uniform setting. In all our results we assume a fixed regular language L.\nThe space complexity is only measured with respect to the window size . It is\na natural question to ask how the space bounds depend on the size of a finite\nautomaton (deterministic or nondeterministic) for L. This question is considered\nin [42]. It is shown that, if Ais a DFA (resp., NFA) with mstates for a language\nL∈/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}ht, thenVL(n) =O(2m·m·logn) (resp., VL(n) =O(4m·logn)).\nFurthermore, for every k/greaterorequalslant1 there exists a language Lk⊆{0,...,k}∗recognized\nby a deterministic automaton with k+ 3 states such that Lk∈ /a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}htand\nFLk(n)/greaterorequalslant(2k−1)·(logn−k).\n1.3.2.Membership in the space classes. In view of Theorem 3 it is natural to ask\nfor the complexity of checking whether a given (non)deterministic fi nite automaton\naccepts a language from /a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}htor/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}ht, respectively. Both problems are\nshown in [42] to be NL-complete for deterministic automata and Pspace-complete\nfor nondeterministic automata. We remark that for the class /a\\}⌊ra⌋ketle{tST,SF,Len/a\\}⌊ra⌋ketri}htfrom\nTheorem 4 the complexities of the corresponding membership proble ms are open.\n1.3.3.Other models of randomness. In Section 1.2 we did not specify the underly-\ning model of randomizedsliding window algorithms(that is used in Theor em4) in a\nprecise way. Let us be a bit more specific: we require that a randomiz ed sliding win-\ndow algorithm for a language Lrunning on an input stream soutputs at every time\ninstantacorrectansweronthequestionwhetherthecurrentwin dowbelongsto Lor\nnot with high probability (say at least 2 /3). This is not the only model of random-\nizedslidingwindowalgorithmsthatcanbefoundintheliterature. Astr ongermodel\nrequires that with high probability the randomized sliding window algorit hm out-\nputs at every time instant a correct answer. So the difference is be tween “∀time in-\nstants: Pr[answer correct] /greaterorequalslant2/3” and “Pr[∀time instants: answer correct] /greaterorequalslant2/3”.6 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIKO VSKAYA\nA randomized sliding window algorithm that fulfills the latter (stronger ) correct-\nness criterion is called strictly correct in [45]. This model is for instanc e implicitly\nused in [13, 26]. In [45] it is shown that every strictly correct rando mized sliding\nwindow algorithm can be derandomized without increasing the space c omplexity.\nThis result is shown in a very general context for arbitraryapprox imation problems.\n1.3.4.Context-free languages. It is natural to ask to which extent our results hold\nfor context-free languages. This question is considered in [48, 41 ]. Let us briefly\ndiscuss the results. In [48] it is shown that if Lis a context-free language with\nFL(n)/lessorequalslantlogn−ω(1) then Lmust be regular and FL(n) = Θ(1). Hence, the\ngap between constant space and logarithmic space for regular lang uages also exists\nfor context-free languages. In contrast, the gap between loga rithmic space and\nlinear space for regular languages does not extend to all context- free languages.\nIn [48], examples of context-free languages LwithFL(n) =¯Θ(n1/c) andVL(n) =\nΘ(n1/c) for every natural number c/greaterorequalslant2 are constructed. These languages are not\ndeterministic context-free, but [48] also contains examples of det erministic one-turn\none-counterlanguages LandL′withFL(n) =¯Θ(log2n) andVL′(n) = Θ(log2n). In\n[41]the spacecomplexityofvisiblypushdownlanguages(alanguagec lassstrictlyin-\nbetween the regularand deterministic context-free languageswit h good closure and\ndecidability properties [5]) is studied. It is shown that for every visibly pushdown\nlanguage the space complexity in the variable-size sliding window model is either\nconstant, logarithmic or linear in the window size. Hence, the space t richotomy\nthat we have seen for regular languages also holds for visibly pushdo wn languages\nin the variable-size model. Whether the visibly pushdown languages als o exhibit\nthe space trichotomy in the fixed-size model is open.\n1.3.5.Update times. In this paper, we only considered the space complexity of\nsliding window algorithms. Another important complexity measure is th e update\ntime of a sliding window algorithm, i.e., the worst case time that is spent p er\nincoming symbol for updating the internal data structures. In [86 ], it is shown that\nfor every regular language Lthere exists a deterministic sliding window algorithm\n(for the fixed-size model) with constant update time. The underlyin g machine\nmodel of the sliding window algorithm is the RAM model, where basic arith metic\noperationsonregistersofbitlength O(logn)(withnthe windowsize)needconstant\ntime. In fact, the algorithm in [86] is formulated in a more general con text for any\nassociative aggregation function. The case of a regular language Lis obtained by\napplying the algorithm from [86] for the syntactic monoid of L. In [47] the result\nof [86] is extended to visibly pushdown languages.\n1.3.6.Further work on sliding windows. We have already mentioned the seminal\nwork of Datar et al. on the sliding window model [26], where the autho rs considered\nthe problem of estimating the number of ones in the sliding window. In t he same\npaper, Datar et al. extend their result for the basic counting prob lem to arbitrary\nfunctions which satisfy certain additivity properties, e.g. Lp-norms for p∈[1,2].\nBraverman and Ostrovsky introduced the smooth histogram fram ework [17], to\ncompute so-called smooth functions over sliding windows, which includ e allLp-\nnorms and frequency moments. Further work on computing aggre gates, statistics\nand frequent elements in the sliding window model can be found in [7, 9, 12, 13,\n16, 27, 35, 54, 55]. The problem of sampling over sliding windows was fir st studied\nin [8] and later improved in [18]. As an alternative to sliding windows, Cohe n\nand Strauss consider the problem of maintaining stream aggregate s where the data\nitems are weighted by a decay function [24].REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 7\n1.3.7.Language recognition in the classical streaming model. Whereas language\nrecognition in the sliding window model has been neglected prior to our work,\nthere exists some work on streaming algorithms for formal languag es in the stan-\ndard setting, where the streaming algorithm reads an input word wand at the end\nhas to decide whether wbelongs to some language. Clearly, for regular languages,\nthis problem can be solved in constant space. Streaming algorithms f or various sub-\nclasses of context-free languages have been studied in [10, 38, 62 , 68, 71]. Related\nto this is the work on querying XML documents in the streaming model [11, 66, 83].\n1.3.8.Streaming pattern matching. Relatedtoourworkistheproblemofstreaming\npattern matching, where the goal is to find all occurrences of a pa ttern (possibly\nwith some bounded number mismatches) in a data stream, see e.g. [6 5, 80, 56, 53,\n19, 20, 21, 22, 23, 57, 59, 58, 78, 84] and search of repetitions in streams [33, 31, 32,\n52, 51, 72, 73].\n1.3.9.Dynamic membership problems for regular languages. A sliding window algo-\nrithm can be viewed as a dynamic data structure that maintains a dyn amic string\nw(the window content) under very restricted update operations. Dynamic mem-\nbership problems for more general updates that allow to change th e symbol at an\narbitrary position have been studied in [6, 39, 40]. As in our work, a t richotomy\nfor the dynamic membership problem of regular languages has been o btained in [6]\n(but the classes appearing the trichotomy in [6] are different from t he classes that\nappear in our work).\n1.4.Outline. The outline of the paper is as follows: In Section 2 we give pre-\nliminary definitions and introduce the fixed-size sliding window model an d the\nvariable-size sliding window model. In Section 3 we study deterministic s liding\nwindow algorithms for regular languages and prove the space tricho tomy and the\ncharacterizations of the space classes (Theorem 3). In Section 4 we turn to ran-\ndomized sliding window algorithms and prove the space tetrachotomy (Theorem 4).\nFinally, in Section 5 we present deterministic and randomized sliding wind ow prop-\nerty testers for regular languages (Theorem 5).\n2.Preliminaries\n2.1.Words and languages. Analphabet Σ is a nonempty finite set of symbols. A\nwordover an alphabet Σ is a finite sequence w=a1a2···anof symbols a1,...,a n∈\nΣ. Thelengthofwis the number|w|=n. Theempty word is denoted by εwhereas\nthe lunate epsilon ǫdenotes small positive numbers. The concatenation of two\nwordsu,vis denoted by u·voruv. The set of all words over Σ is denoted by Σ∗.\nA subset L⊆Σ∗is called a language over Σ.\nLetw=a1···an∈Σ∗be a word. Any word of the form a1···aiis aprefixofw,\nawordoftheform ai···anisasuffixofw, andawordofthe form ai···ajis afactor\nofw. The concatenation of two languages K,LisKL={uv|u∈K, v∈L}. For\na language Lwe define Lninductively by L0={ε}andLn+1=LnLfor alln∈N.\nTheKleene-star of a language Lis the language L∗=/uniontext\nn∈NLn. Furthermore, we\ndefineL/lessorequalslantn=/uniontext\n0/lessorequalslantk/lessorequalslantnLkandL<n=/uniontext\n0/lessorequalslantk<nLk.\n2.2.Automata and regular languages. For good introductions to the theory\nof formal languages and automata we refer to [15, 61, 67].\nThe standard description for regular languages are finite automat a. Let Σ be a\nfinite alphabet. A nondeterministic finite automaton (NFA) is a tuple\nA= (Q,Σ,I,∆,F),8 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIKO VSKAYA\nwhereQis the finite set of states,I⊆Qis the set of initial states, ∆ ⊆Q×Σ×Q\nis the set of transitions , andF⊆Qis the set of final states. A runofAon a word\nw=a1···an∈Σ∗is a finite sequence π=q0a1q1a2q2···qn−1anqn∈Q(ΣQ)∗such\nthat (qi−1,ai,qi)∈∆ for all 1 /lessorequalslanti/lessorequalslantn. We call πsuccessful ifq0∈Iandqn∈F.\nThe language accepted byAis defined as\nL(A) ={w∈Σ∗|there exists a successful run of Aonw}.\nA language L⊆Σ∗isregularif it is accepted by some NFA. The size|A|is defined\nas the number of states.\nA(left-)deterministic finite automaton (DFA) is an NFAA= (Q,Σ,I,∆,F),\nwhereI={q0}has exactly one initial state q0, and for all p∈Qanda∈Σ there\nexists exactly one transition ( p,a,q)∈∆. We view ∆ as a transition function\nδ:Q×Σ→Qand writeAin the formatA= (Q,Σ,q0,δ,F). The transition\nfunction δcan be extended to a right action ·:Q×Σ∗→Qof the free monoid Σ∗\non the state set Qby setting q·ε=qand defining inductively q·ua=δ(q·u,a)\nfor allq∈Q,u∈Σ∗, anda∈Σ. We writeA(w) instead of q0·w. It is known that\nany NFA can be turned into an equivalent DFA by the power set const ruction.\nWe also consider automata with (possibly) infinitely many states as ou r formal\nmodel for streaming algorithms. A deterministic automaton Ahas the same format\nA= (Q,Σ,q0,δ,F) as a DFA but we drop the condition that Qmust be finite. We\nuse the notations from the previous paragraph for general dete rministic automata\nas well.\nIt is well known that for every regular language Lthere exists a unique minimal\nDFAAforL, whose states are the Myhill-Nerode classes of L. This construction\ncan be carried out for every language Land yields a deterministic automaton Afor\nLsuch that for any deterministic automaton BforL,B(x) =B(y) impliesA(x) =\nA(y) for allx,y∈Σ∗[30, Chapter III, Theorem 5.2]. We call this automaton the\nminimal deterministic automaton for L.\n2.3.Streaming algorithms. Astreamis a finite sequence of elements a1···am,\nwhich arrive element by element from left to right. In this paper, the elements are\nalways symbols from a finite alphabet Σ. A streaming algorithm reads t he symbols\nof the input stream from left to right. At time instant tthe algorithm only has\naccess to the symbol atand the internal storage, which is encoded by a bit string.\nThe goal of the streaming algorithm is to compute a function ϕ: Σ∗→Y, where\nΣ is a finite alphabet and Yis a set of output values. For the remainder of this\npaper, we only consider the Boolean case, i.e. Y={0,1}; in other words, ϕis the\ncharacteristic function of a language L. Furthermore, we abstract away from the\nactual computation and only analyze the memory requirement.\nFormally, a deterministic streaming algorithm is the same as a deterministic\nautomatonPand we say thatPis a streaming algorithm for the language L(P).\nThe letterPstands for program. IfP= (M,Σ,m0,δ,F) then the states from Mare\nusually called memory states . We require M/\\e}atio\\slash=∅but allow Mto be infinite. The\nspaceofP(ornumber of bits used by P) is given by s(P) = log|M|∈R/greaterorequalslant0∪{∞}.\nHere and in the rest of the paper, we denote with log the logarithm wit h base two,\ni.e we measure space in bits. If s(P) =∞we will measure the space restricted\nto input streams where some parameter is bounded (namely the wind ow size); see\nSection 2.5.\nWe remark that many streaming algorithms in the literature only prod uce a\nsingle answer after completely reading the entire stream. Also, the length of the\nstream is often known in advance. However, in the sliding window mode l we rather\nassume an input stream of unbounded and unknown length, and nee d to compute\noutput values for everywindow, i.e. at every time instant.REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 9\nIn the following, we introduce the sliding window model in two different v ariants:\nthe fixed-size sliding window model and the variable-size sliding window m odel.\n2.4.Fixed-size sliding window model. We fix an arbitrary padding symbol\n/square∈Σ. Given a stream x=a1a2···am∈Σ∗and awindow length orwindow size\nn∈N, we define lastn(x)∈Σnby\nlastn(x) =/braceleftBigg\nam−n+1am−n+2···am,ifn/lessorequalslantm,\n/squaren−ma1···am, ifn > m,\nwhich is called the window of length n , or theactiveorcurrent window . In other\nwords,lastn(x) is the suffix of length n, padded with /square-symbols on the left. We\nview/squarenas theinitial window ; its choice is completely arbitrary.\nLetL⊆Σ∗be a language. The sliding window problem SWn(L) forLand\nwindow length n∈Nis the language\nSWn(L) ={x∈Σ∗|lastn(x)∈L}.\nNote that for every Land every n,SWn(L) is regular. A sliding window algorithm\n(SW-algorithm) forLandwindowlength n∈Nisastreamingalgorithmfor SWn(L).\nThe function FL:N→R/greaterorequalslant0is defined by\nFL(n) = inf{s(Pn)|Pnis an SW-algorithm for Land window length n}.\nIt is called the space complexity ofLin the fixed-size sliding window model . Note\nthatFL(n)<∞sinceSWn(L) is regular. A subtle point is that the space complex-\nityFL(n) of a language Lin general depends on the underlying alphabet. A simple\nexample is a∗which has complexity FL(n) =O(1) over the singleton alphabet {a}\nwhereas it has complexity FL(n) = Θ(log n) over the alphabet {a,b}(the latter\nfollows from our results).\nWe draw similarities to circuit complexity where a language L⊆{0,1}∗is rec-\nognized by a family of circuits ( Cn)n∈Nin the sense that Cnrecognizes the slice\nL∩{0,1}n. Similarly, the sliding window problem SWn(L) is solely defined by the\nsliceL∩Σn. If we speak of an SW-algorithm for Land omit the window length n,\nthen this parameter is implicitly universally quantified, meaning that th ere exists\na family of streaming algorithms ( Pn)n∈Nsuch that everyPnis an SW-algorithm\nforLand window length n.\nLemma 6. For any language Lwe have FL(n) =O(n).\nProof.AtrivialSW-algorithm PnforLexplicitly storesthe activewindowoflength\nnin a queue so that the algorithm can always test whether the window b elongs to\nL. Formally, the state set of Pnis Σnand it has transitions of the form ( bu,a,ua)\nfora,b∈Σ,u∈Σn−1. Viewed as an edge-labeled graph this automaton is also\nknown under the name de Bruijn graph [28]. Since every word w∈Σncan be\nencoded withO(log|Σ|·n) bits, the algorithm uses O(n) bits. /square\nDepending on the language Lthere are more space efficient solutions. Usually\nsliding window algorithms are devised in the following way:\n(1) Specify some information or property I(w) of the active window wand\nshow that it can be maintained by a streaming algorithm. This means that\ngivenI(bu) anda∈Σ one can compute I(ua).\n(2) Show that one can decide w∈Lfrom the information I(w).\nNotice that the complexity function FL(n) is not necessarily monotonic. For\ninstance, let Lbe the intersection of aΣ∗and the set of words with even length. By\nExample 2(iii), we have FL(2n) = Θ(n) but clearly we have FL(2n+1) =O(1) since\nfor odd window lengths the algorithm can always reject. Therefore , we can only10 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nshowFL(n) =¯Θ(n) (instead of FL(n) = Θ(n) which is false here), where ¯Θ(g(n))\nwas defined in the introduction.\nNote that the fixed-size sliding window model is a nonuniform model: for every\nwindow size we have a separate streaming algorithm and these algorit hms do not\nhave to follow a common pattern. Working with a nonuniform model ma kes lower\nboundsstronger. Incontrast,thevariable-sizeslidingwindowmod elthatwediscuss\nnext is a uniform model in the sense that there is a single streaming alg orithm that\nworks for every window length.\n2.5.Variable-size sliding window model. For an alphabet Σ we define the\nextended alphabet Σ ↓= Σ∪{↓}. In the variable-size model the active window\nwnd(u)∈Σ∗for a stream u∈Σ∗\n↓is defined as follows, where a∈Σ:\nwnd(ε) =ε wnd(u↓) =ε,ifwnd(u) =ε\nwnd(ua) =wnd(u)awnd(u↓) =v,ifwnd(u) =av\nThe symbol↓represents the pop operation . We emphasize that a pop operation\non an empty window leaves the window empty. The variable-size sliding window\nproblemSW(L) of a language L⊆Σ∗is the language\n(1) SW(L) ={u∈Σ∗\n↓|wnd(u)∈L}.\nNote that in general, SW(L) is not a regular language (even if Lis regular). A\nvariable-size sliding window algorithm (variable-size SW -algorithm)PforLis a\nstreaming algorithm for SW(L).\nFor a stream u=a1···am∈Σ∗\n↓letmwl(u) = max{|wnd(a1···ai)||0/lessorequalslanti/lessorequalslantm}\nbe themaximum window length of all prefixes of u. IfP= (M,Σ,m0,δ,F) is a\nstreaming algorithm over Σ ↓we define\nMn={P(w)|w∈Σ∗\n↓,mwl(w) =n}.\nandM/lessorequalslantn=/uniontext\n0/lessorequalslantk/lessorequalslantnMk. Thespace complexity ofPin the variable-size sliding\nwindow model is\nv(P,n) = log|M/lessorequalslantn|∈R/greaterorequalslant0∪{∞}.\nIn other words: that the space complexity of a variable-size SW-alg orithm is\nbounded by f(n) means that the algorithm never has to store more than f(n)\nbits when it processes a stream u∈Σ∗\n↓such that for every prefix of uthe length of\nthe active window never exceeds n.\nNotice that v(P,n) is a monotonic function. To prove upper bounds above log n\nfor the space complexity of Pit suffices to bound log |Mn|as shown in the following.\nLemma 7. Ifs(n)/greaterorequalslantlognis a monotonic function and log|Mn|=O(s(n))then\nv(P,n) =O(s(n)).\nProof.SinceM/lessorequalslantn=M0∪M1∪···∪Mn, we have\nlog|M/lessorequalslantn|= logn/summationdisplay\ni=0|Mi|\n/lessorequalslantlog/parenleftbigg\n(n+1)·max\n0/lessorequalslanti/lessorequalslantn|Mi|/parenrightbigg\n= log(n+1)+ max\n0/lessorequalslanti/lessorequalslantnlog|Mi|\n/lessorequalslantlog(n+1)+ max\n0/lessorequalslanti/lessorequalslantnO(s(i))\n/lessorequalslantlog(n+1)+O(s(n)) =O(s(n)),\nwhich proves the statement. /squareREGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 11\nLemma 8. For every language L⊆Σ∗there exists a space optimal variable-size\nSW-algorithmP, i.e.v(P,n)/lessorequalslantv(Q,n)for every variable-size SW-algorithm Qfor\nLand every n∈N.\nProof.LetPbe the minimal deterministic automaton ASW(L)forSW(L). IfQis\nany deterministic automaton for SW(L) thenQ(x) =Q(y) impliesP(x) =P(y).\nThen, we obtain\nv(P,n) = log|{P(w)|w∈Σ∗\n↓,mwl(w)/lessorequalslantn}|\n/lessorequalslantlog|{Q(w)|w∈Σ∗\n↓,mwl(w)/lessorequalslantn}|=v(Q,n),\nwhich proves the statement. /square\nWe define the space complexity of Lin the variable-sidesliding window model by\nVL(n) =v(P,n) wherePis a space optimal variable-size SW-algorithm for SW(L)\nfrom Lemma 8. It is a monotonic function.\nLemma 9. For any language L⊆Σ∗andn∈Nwe have FL(n)/lessorequalslantVL(n).\nProof.IfPis a space optimal variable-size SW-algorithm for Lthen one obtains\nan SW-algorithm Pnfor window length n∈Nas follows. Let us assume n/greaterorequalslant1\n(forn= 0 we use the trivial SW-algorithm). First one simulates Pon the initial\nwindow/squaren. For every incoming symbol a∈Σ we perform a pop operation ↓in\nP, followed by inserting a. Since the maximum window length is bounded by non\nany stream, the space complexity is bounded by v(P,n) =VL(n). /square\nThe following lemma states that in the variable-size model one must at least\nmaintain the current window size if the language is neither empty nor u niversal.\nThe issue at hand is performing a pop operation on an empty window.\nLemma 10. LetPbe a variable-size SW-algorithm for a language ∅/subsetnoteqlL/subsetnoteqlΣ∗.\nThen,P(x)determines2|wnd(x)|for allx∈Σ∗\n↓and therefore VL(n)/greaterorequalslantlog(n+1).\nProof.Lety∈Σ+be a length-minimal nonempty word such that |{ε,y}∩L|= 1.\nConsider streams x1,x2∈Σ∗\n↓with|wnd(x1)|<|wnd(x2)|=mand assumeP(x1) =\nP(x2). Then, we also have P(x1y↓m) =P(x2y↓m). Butwnd(x2y↓m) =ywhereas\nwnd(x1y↓m) is a proper suffix of y. Now, by the choice of yone these two words\nbelongs to Lwhereas the other does not, which contradicts P(x1y↓m) =P(x2y↓m).\nFor the second statement: if the algorithm reads any stream a1···an∈Σnit\nmust visit n+1pairwise distinct memory states and hence v(P,n)/greaterorequalslantlog(n+1)./square\nAlternative definitions of the variable-size model are conceivable, e .g. one could\nneglect streams where the popping of an empty window occurs, or a ssume that the\nwindow size is always known to the algorithm.\nLemma 11. For any function s(n)andX∈{F,V}, the class{L⊆Σ∗|XL(n) =\nO(s(n))}forms a Boolean algebra.\nProof.LetL⊆Σ∗be a language. Given a SW-algorithm for Lfor some fixed\nwindow size nor in the variable-size model, we can turn it into an algorithm for the\ncomplementΣ∗\\Lbynegatingits output. Clearly, it hasthe samespacecomplexity\nas the original algorithm.\nLetL1,L2⊆Σ∗be two languages. Let P1,P2be SW-algorithms for L1,L2,\nrespectively,eitherforsomefixedwindowsize norinthevariable-sizemodel. Define\nPto be the product automaton of P1andP2which outputs the disjunction of the\noutputs of thePi.\n2In other words, for all x1,x2∈Σ∗\n↓, ifP(x1) =P(x2) then|wnd(x1)|=|wnd(x2)|.12 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nIn the case of a fixed window size n,Phas 2s(P1)·2s(P2)= 2s(P1)+s(P2)many\nstates and hence s(P) =s(P1) +s(P2). This implies FL1∪L2(n)/lessorequalslantFL1(n) +\nFL2(n). For the variable-size model, notice that 2v(P,n)/lessorequalslant2v(P1,n)·2v(P2,n)=\n2v(P1,n)+v(P2,n). Therefore, VL1∪L2(n)/lessorequalslantVL1(n)+VL2(n). /square\n3.Deterministic sliding window algorithms\nIn this section, we will show that the space complexity of every regu lar language\nin both sliding window models is either constant, logarithmic or linear. In Exam-\nple 2 we have already seen prototypical languages with these three space complexi-\nties, namely Σ∗a(constant), Σ∗aΣ∗(logarithmic) and aΣ∗(linear) for Σ ={a,b}.\nIntuitively, for languages of logarithmic space complexity it suffices t o maintain\na constant number of positions in the window. For languages of cons tant space\ncomplexity it suffices to maintain a constant-length suffix of the windo w. Moreover,\nwe describe the languages with logarithmic and constant space comp lexity as finite\nBoolean combinations of simple atomic languages.\n3.1.Right-deterministic finite automata. It turns out that the appropriate\nrepresentation of a regular language for the analysis in the sliding win dow model\nare deterministic finite automata which read the input word, i.e. the w indow, from\nright to left. Such automata are called right-deterministic finite automata (rDFA)\nin this paper. The reason why we use rDFAs instead of DFAs can be ex plained\nintuitively for the variable-size sliding window model as follows. The var iable-size\nmodel contains operations in both “directions”: On the one hand a v ariable-size\nwindow can be extended on the right, and on the other hand the wind ow can be\nshortened to an arbitrary suffix. For regular languages the exten sion to longer\nwindows is “tame” because the Myhill–Nerode right congruences hav e finite index.\nHence, it remains to control the structure of all suffixes with resp ect to the regular\nlanguage, which is best captured by an rDFA for the language.\nFormally, a right-deterministic finite automaton (rDFA) B= (Q,Σ,F,δ,q 0) con-\nsists of a finite state set Q, a finite alphabet Σ, a set of final states F⊆Q, a transi-\ntion function δ: Σ×Q→Q, and an initial state q0∈Q. The transition function δ\nextends to a left action ·: Σ∗×Q→Qbyε·q=qand (aw)·q=δ(a,w·q) for all\na∈Σ,w∈Σ∗,q∈Q. The language accepted by BisL(B) ={w∈Σ∗|w·q0∈F}.\nArunofBon a word w=a1···an∈Σ∗frompntop0is a finite sequence\nπ=p0a1p1a2p2···pn−1anpn∈Q(ΣQ)∗\nsuch that pi−1=ai·pifor all 1 /lessorequalslanti/lessorequalslantn. Quite often we write such a run in the\nfollowing way\nπ:p0a1←−p1a2←−p2···pn−1an←−−pn.\nIf the intermediate states p1,...,p n−1are not important we write this run also as\nπ:p0a1a2···an←−−−−−− pn.\nThe above run πisaccepting ifp0∈F(note that we do not require pn=q0)\nand otherwise rejecting. Itslengthis the length|w|ofw. A run of length zero is\ncalled empty; note that it consists of a single state. A run of length o ne is also\ncalled atransition . Ifπ=p0a1p1···anpnandρ=r0b1r1···bℓrℓare runs such that\npn=r0then their composition πρis defined as πρ=p0a1p1···anr0b1r1···bℓrℓ.\nThis definition allows us to factorize runs in Section 3.3. We call a run πaP-run\nfor a subset P⊆Qif all states occurring in πare contained in P.\nA stateq∈Qisreachable fromp∈Qif there exists a run from ptoq, in which\ncase we write q/√re⌋edesequalBp. We say that qisreachable if it is reachable from the initial\nstateq0. A set of states P⊆Qis reachable if all p∈Pare reachable. The reacha-\nbility relation/√re⌋edesequalBis apreorder onQ, i.e. it is reflexive and transitive. Two statesREGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 13\np q ra\nb\nbaa,b\nFigure 2. An rDFA partitioned into its SCCs.\np,q∈Qarestrongly connected ifp/√re⌋edesequalBq/√re⌋edesequalBp. This yields an equivalence relation\nonQwhose equivalence classes are the strongly connected components (SCCs) of\nB. A subset P⊆Qisstrongly connected if it is contained in a single SCC, i.e. all\np,q∈Pare strongly connected.\n3.2.Space trichotomy. In this section, we state two technical results which di-\nrectly imply Theorem 3. Let B= (Q,Σ,F,δ,q 0) be an rDFA. A set of states P⊆Q\niswell-behaved if for any two P-runsπ1,π2which start in the same state and have\nequal length, either both π1andπ2are accepting or both are rejecting. If every\nreachable SCC in Bis well-behaved then Bis called well-behaved . A state q∈Q\nistransient ifx·q/\\e}atio\\slash=qfor allx∈Σ+. Every transient state in Bforms an SCC\nof size one (a transient SCC ); however, not every SCC of size one is transient. Let\nU(B)⊆Qbe the set of states q∈Qfor which there exists a nontransient state\np∈Qsuch that qis reachable from pandpis reachable from the initial state q0.\nNotice that q∈U(B) if and only if there exist runs of unbounded length from q0\ntoq(hence the symbol Ufor unbounded).\nTheorem 12. LetL⊆Σ∗be regular andBbe any rDFA for L.\n(1) IfBis well-behaved then VL(n) =O(logn)andFL(n) =O(logn).\n(2) IfBis not well-behaved then VL(n) = Ω(n)andFL(n) =¯Ω(n).\n(3) IfU(B)is well-behaved then FL(n) =O(1).\n(4) IfU(B)is not well-behaved then FL(n) =¯Ω(logn).\n(5) IfL∈{∅,Σ∗}thenVL(n) =O(1).\n(6) IfL /∈{∅,Σ∗}thenVL(n) = Ω(log n).\nTheorem 12 implies that FL(n) is either Θ(1), ¯Θ(logn), or¯Θ(n), andVL(n) is\neither Θ(1), Θ(log n), or Θ(n). For the characterizations in Theorem 3 it remains\nto prove:\nTheorem 13. LetL⊆Σ∗be regular.\n(i)FL(n) =O(1)⇐⇒L∈/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}ht.\n(ii)Lis recognized by a well-behaved rDFA ⇐⇒L∈/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}ht.\n3.3.The path summary algorithm. In the following, let B= (Q,Σ,F,δ,q 0)\nbe a right-deterministic finite automaton. We call a run πinternal ifπis aP-\nrun for some SCC P. TheSCC-factorization ofπis the unique factorization π=\nπkτk−1···τ2π2τ1π1, where every πiis an internal (possibly empty) run but cannot\nbe extended to an internal run. The τiare single transitions (i.e., runs from QΣQ)\nconnecting distinct SCCs. Let pk,...,p 1∈Qbe the starting states of the runs\nπk,...,π 1. Then, the path summary ofπis defined as\nps(π) = (|πk|,pk)(|τk−1πk−1|,pk−1)···(|τ2π2|,p2)(|τ1π1|,p1),\nwhich is a sequence of pairs from N×Q. It specifies the first state that is visited\nin an SCC, and the length of the run until reaching the next SCC or th e end of\nthe word, respectively. The leftmost length |πk|can be zero but all other lengths\n|τiπi|= 1+|πi|are strictly positive. We define πw,qto be the unique run of Bon\na wordw∈Σ∗starting from q, and PS B(w) ={ps(πw,q)|q∈Q}.14 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nExample 14.Consider the rDFA Bin Figure 2. For the moment, the final states\nare irrelevant. It consists of two SCCs, namely the blue SCC {p,q}and the red\nSCC{r}. All its runs on the word w=aababbare listed here:\nra← −ra← −qb← −pa← −pb← −qb← −p\nra← −ra← −rb← −ra← −qb← −pb← −q\nra← −ra← −rb← −ra← −rb← −rb← −r\nThen, PS B(w) contains the path summaries (1 ,r)(5,p), (3,r)(3,q) and (6,r).\nThepath summary algorithm for Bis a streaming algorithm overΣ ↓described in\nAlgorithm 1. The data structure at time instant tis denoted by St. The acceptance\ncondition will be defined later.\nLemma 15. Algorithm 1 correctly maintains PSB(w)for the active window w∈\nΣ∗.\nProof.Initially PS B(ε) contains the path summary of every empty run from every\nstate, which is formally {0}×Q.\nAssume St−1= PSB(w) for some window w∈Σ∗and that a∈Σ is the in-\ncoming symbol. The claim is that the algorithm computes St= PSB(wa) from\nSt−1. Suppose that π′is a run inBonwa. It can be factorized as π′=πp1ap0\nwith ps( π)∈St−1. Letπ=πkτk−1πk−1···τ2π2τ1π1be the SCC-factorization\nofπ. Ifp0andp1are strongly connected then the SCC-factorization of π′isπ′=\nπkτk−1πk−1···τ2π2τ1π′\n1whereπ′\n1=π1p1ap0,andotherwise π′=πkτk−1πk−1···τ2π2τ1π1p1ap0.\nIn this way the algorithm computes ps( π′) from ps( π).\nNow, consider the case a=↓. We have w=εif and only if PS B(w) ={0}×\nQ, and in this case the set of path summaries is unchanged, i.e. we set St=\nSt−1. Otherwise assume w=bvfor some b∈Σ. We claim that the algorithm\ncomputes St= PSB(v) fromSt−1. Suppose that π′is a run inBonvwhich ends\nin state p∈Q. Ifq=δ(b,p) inBthen let π=qbpπ′. We have ps( π)∈St−1.\nLetπ=πkτk−1πk−1···τ2π2τ1π1be the SCC-factorization of π. If|πk|/greaterorequalslant1 then\nπ′=π′\nkτk−1πk−1···τ2π2τ1π1is the SCC-factorization of π′whereπk=qbpπ′\nk.\nOtherwise πkis empty and τk−1=qbp. Therefore, π′=πk−1···τ2π2τ1π1is SCC-\nfactorization of π′. In this way the algorithm computes ps( π′) from ps( π)./square\n3.4.Proof of the trichotomy. Using the path summary algorithm we can prove\nTheorem 12(1):\nProposition 16. IfBis well-behaved then the regular language L=L(B)has space\ncomplexity VL(n) =O(|B|2·logn), which isO(logn)for fixedB.\nProof.LetBbe well-behaved. Call a path summary accepting if it is the path\nsummary of some accepting run. The variable-size sliding window algor ithm for\nLis the path summary algorithm for Bwhere the algorithm accepts if the path\nsummary starting in q0is accepting.\nFor the correctness of the algorithm it suffices to show that any ru nπstart-\ning inq0is accepting if and only if ps( π) is accepting. The direction from left to\nright is immediate by definition. For the other direction, consider the path sum-\nmaryps( π) = (ℓk,pk)···(ℓ1,p1) andthe SCC-factorization π=πkτk−1···τ2π2τ1π1.\nSinceps( π) isaccepting, thereisanacceptingrun π′\nkthatstartsin pkandhaslength\nℓk. SinceBis well-behaved, the SCC of pkis well-behaved. Therefore, since π′\nkis\naccepting and|πk|=|π′\nk|=ℓk,πkmust also be accepting and thus πis accepting.\nWe claim that the space complexity of the path summary algorithm is bo unded\nbyO(|B|2·(logn+ log|B|)). Observe that PS B(w) contains|B|path summaries,REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 15\nAlgorithm 1: The path summary algorithm\nInput:sequence of operations a1a2a3···∈Σω\n↓\nS0={0}×Q;\nforeach t/greaterorequalslant1do\nSt=∅;\nifat∈Σthen\nforp0∈Qdo\nletp1=a·p0and (ℓk,pk)···(ℓ1,p1)∈St−1;\nifp0andp1are strongly connected then\nadd (ℓk,pk)···(ℓ2,p2)(ℓ1+1,p0) toSt;\nelse\nadd (ℓk,pk)···(ℓ1,p1)(1,p0) toSt;\nifat=↓then\nifSt−1={0}×Qthen\nSt=St−1;\nelse\nfor(ℓk,pk)···(ℓ1,p1)∈St−1do\nifℓk/greaterorequalslant1then\nadd (ℓk−1,pk)(ℓk−1,pk−1)···(ℓ1,p1) toSt;\nelse\nadd (ℓk−1,pk−1)(ℓk−2,pk−2)···(ℓ1,p1) toSt;\nand a single path summary ps( π) consists of a sequence of at most |B|states and a\nsequence ( ℓk,...,ℓ 1) ofk/lessorequalslant|B|numbers up to|π|. Hence, the path summary ps( π)\ncan be encoded using O(|B|·(log|B|+ log|π|)) bits, which yields the total space\ncomplexityO(|B|2·(logn+log|B|)).\nTo reduce the space complexity to O(|B|2·logn) we need to make a case distinc-\ntion. The algorithm maintains the window size n∈Nand the maximal suffix of the\nwindow of length up to |B|(explicitly) using O(logn+|B|) bits. If n/lessorequalslant|B|then this\ninformationsufficestotestmembershipofthewindowto L. Assoonas nexceeds|B|\nwe initialize PS B(w) and use the path summary algorithm as described above. If\nn >|B|then its space complexity is O(|B|2·(logn+log|B|))⊆O(|B|2·logn)./square\nNext, we will prove a linear lower bound for rDFA which are not well-beh aved.\nLetL⊆Σ∗be a language. We say that Lseparates two words x,y∈Σ∗withx/\\e}atio\\slash=y\nif|{x,y}∩L|= 1. We say that Lseparates two languages K1,K2⊆Σ∗ifK1⊆L\nandK2∩L=∅, orK2⊆LandK1∩L=∅.\nLemma 17. IfBis not well-behaved then there exist words u1,u2,v1,v2,z∈Σ∗\nwhere|ui|=|vi|fori∈{1,2}such that L=L(B)separates u2{u1u2,v1v2}∗zand\nv2{u1u2,v1v2}∗z.\nProof.SinceB= (Q,Σ,F,δ,q 0) is not well-behaved, there are states p∈Q,q∈F,\nr∈Q\\Fand words u=u1u2,v=v1v2,z∈Σ∗such that|u2|=|v2|and\npu1←−qu2←−pz← −q0andpv1←−rv2←−pz← −q0.\nWe can ensure that |u1|=|v1|and hence also|u|=|v|: Ifk=|u|andℓ=|v|we\nreplaceu1byuℓ−1u1andv1byvk−1v1, which preserves all properties above. Then,\nu2{u,v}∗zandv2{u,v}∗zare separated by L. /square\nWe can now show Theorem 12(2):16 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nq0 pq\nrzu2\nu1\nv2v1\nFigure 3. Forbidden pattern for well-behaved rDFAs where\n|u1|=|v1|and|u2|=|v2|.\nProposition 18. IfBis not well-behaved then the language L=L(B)satisfies\nFL(n) =¯Ω(n)andVL(n) = Ω(n).\nProof.Letu1,u2,v1,v2,z∈Σ∗be the words from Lemma 17 and let u=u1u2\nandv=v1v2. Now, consider an SW-algorithm PnforLand window length n=\n|u2|+|u|·(m−1)+|z|for some m/greaterorequalslant1. We prove that Pnhas at least 2mmany\nstates by showing that Pn(x)/\\e}atio\\slash=Pn(y) for any x,y∈{u,v}mwithx/\\e}atio\\slash=y. Notice\nthat|{u,v}m|= 2msinceu/\\e}atio\\slash=vand|u|=|v|.\nRead two distinct words x,y∈{u,v}minto two instances of Pn. Consider the\nright most{u,v}-block where xandydiffer. Without loss of generality assume\nx=x′usandy=y′vsfor some x′,y′,s∈{u,v}∗with|x′|=|y′|. By reading x′z\ninto both instances the window of the x-instance becomes lastn(xx′z) =u2sx′zand\nthe window of the y-instance becomes lastn(yx′z) =v2sx′z. By Lemma 17 the two\nwindows are separated by L, and therefore the algorithm Pnmust accept one of\nthe streams xx′zandyx′z, and reject the other. In conclusion Pn(x)/\\e}atio\\slash=Pn(y) and\nhencePnmust use at least m= Ω(n) bits. This holds for infinitely many n, namely\nallnof the form|u2|+|z|+|u|·(m−1) for some m/greaterorequalslant1.\nThe argument above shows that there exist numbers c,d∈Nsuch that for all\nm/greaterorequalslant1 we have VL(cm+d)/greaterorequalslantFL(cm+d)/greaterorequalslantΩ(m). Ifn/greaterorequalslantdthenm=⌊(n−d)/c⌋=\nΩ(n) satisfies cm+d/lessorequalslantn. Therefore, VL(n)/greaterorequalslantVL(cm+d) = Ω(m) by monotonicity\nofVLand hence VL(n) = Ω(n). /square\nFrom Proposition 16 and Proposition 18 we obtain:\nCorollary 19. LetX∈{F,V}. A regular language L⊆Σ∗satisfies XL(n) =\nO(logn)if and only if Lis recognized by a well-behaved rDFA.\n3.5.Constant space. Next, we study which regular languages have sublogarith-\nmic complexity. Recall that in the variable-size model any such langua ge must\nbe empty or universal because the algorithm must at least maintain t he current\nwindow size by Lemma 10.\nCorollary 20. The empty language L=∅and the universal language L= Σ∗\nsatisfyVL(n) =O(1). All other languages satisfy VL(n) = Ω(log n).\nThis proves points (5) and (6) in Theorem 12. Now, we can turn to th e fixed-size\nmodel and prove the points (3) and (4). Point (3) follows from:\nProposition 21. IfU(B)is well-behaved then L=L(B)has space complexity\nFL(n) =O(|B|), which isO(1)whenBis fixed.\nProof.Letk=|B|. The SW-algorithm PnforSWn(L) maintains lastk(x) for an\ninput stream x∈Σ∗usingO(k) bits. If n/lessorequalslantkthenlastn(x) is a suffix of lastk(x)\nand hencePncan determine whether lastn(x)∈L. Ifn > kthenlastk(x) is aREGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 17\nsuffix oflastn(x), saylastn(x) =slastk(x). We can decide if lastn(x)∈Las follows:\nConsider the run of Bonlastn(x) starting from the initial state:\nrs← −qlastk(x)←−−−−−q0.\nBy the choice of ksome state p∈Qmust occur twice in the run qlastk(x)←−−−−−q0.\nTherefore, pis nontransient and all states in the run rs← −qbelong to U(B). Since\nU(B) is well-behaved, ris final if and only if some run of length |s|=n−kstarting\ninqis accepting. This information can be precomputed for each state qin the\nfixed-size model. /square\nFor the lower bound in Theorem 12(4) we need the following lemma:\nLemma 22. IfU(B)is not well-behaved then there exist words x,y,z∈Σ∗where\n|x|=|y|such that L=L(B)separates xy∗zandy∗z.\nProof.SinceU(B) is not well-behaved, there are U(B)-runsπandρfrom the same\nstarting state q∈U(B) such that|π|=|ρ|and exactly one of the runs πandρis\naccepting. By definition of U(B) the state qis reachable from a nontransient state\npvia some run σsuch that pis reachable from the initial state q0, saypz0←−q0. We\ncan replace πbyπσandρbyρσpreserving the properties of being U(B)-runs and\n|π|=|ρ|. Assume that πandρare runs on words v∈Σ∗andw∈Σ∗, respectively.\nSincepis nontransient, we can construct runs from ptopof unbounded lengths.\nConsider such a run pu← −pof length|u|/greaterorequalslant|v|=|w|. Then,Lseparates vu∗z0and\nwu∗z0. Factorize u=u1u2so that|u2|=|v|=|w|. Notice that all words in u2u∗z0\nreach the same state in Band hence u2u∗z0is either contained in Lor disjoint from\nL. Then,Lseparates either u2u∗z0andvu∗z0, oru2u∗z0andwu∗z0. Hence, Lalso\nseparates ( u2u1)∗u2z0from either vu1(u2u1)∗u2z0or fromwu1(u2u1)∗u2z0. This\nyields the words z=u2z0,y=u2u1andx=vu1orx=wu1with the claimed\nproperties. /square\nProposition 23. IfU(B)is not well-behaved then L=L(B)satisfies FL(n)/greaterorequalslant\nlogn−O(1)for infinitely many n. In particular, FL(n) =¯Ω(logn).\nProof.Letx,y,z∈Σ∗be the words from Lemma 22. Consider an SW-algorithm\nPnforLand window length n=|x|+|y|·m+|z|for some m/greaterorequalslant1. We prove\nthatPnhas at least mmany states by showing that Pn(xyi)/\\e}atio\\slash=Pn(xyj) for any\n1/lessorequalslanti < j/lessorequalslantm. Let 1/lessorequalslanti < j/lessorequalslantm. Then, we have\nlastn(xyiym−iz) =lastn(xymz) =xymz\nand\nlastn(xyjym−iz) =lastn(xym+j−iz) =ym+1z.\nSince exactly one of the words xymzandym+1zbelongs to L, it also holds that\nexactly one of the streams xyiym−izandxyjym−izis accepted byPn. This proves\nthatPnmust reach different memory states on inputs xyiandxyj. In conclusion\nPnmust use log m/greaterorequalslantlogn−O(1) bits, and this holds for infinitely many n./square\n3.6.Characterization of constant space. Next, we prove that a regular lan-\nguageLhas constant space complexity FL(n) if and only if it is a Boolean combi-\nnation of suffix testable languages and regular length languages (Th eorem 13(i)).\nThe language Lis called k-suffix testable if for all x,y∈Σ∗andz∈Σkwe have\nxz∈Lif and only if yz∈L. Equivalently, Lis a Boolean combination of languages\nof the form Σ∗wwherew∈Σ/lessorequalslantk. Clearly, a language is suffix testable if and only if\nit isk-suffix testable for some k∈N. Let us remark that the class of suffix testable\nlanguages corresponds to the variety Dof definite monoids [85]. Clearly, every18 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nfinite language is suffix testable: If kis the maximum length of a word in L⊆Σ∗\nthenLis (k+1)-suffix testable since L=/uniontext\nw∈L{w}and{w}= Σ∗w\\/uniontext\na∈ΣΣ∗aw.\nWe will utilize a distance notion between states in a DFA, which is also stu died\nin [50]. The symmetric difference of two sets AandBisA△B= (A∪B)\\(A∩B).\nWe define the distance d(K,L) of two languages K,L⊆Σ∗by\nd(K,L) =/braceleftBigg\nsupu∈K△L|u|+1,ifK/\\e}atio\\slash=L,\n0, ifK=L.\nNoticethat d(K,L)<∞ifandonlyif K△Lisfinite. ForaDFA A= (Q,Σ,q0,δ,F)\nand a state p∈Q, we defineAp= (Q,Σ,p,δ,F). Moreover, for two states p,q∈Q,\nwe define the distance d(p,q) =d(L(Ap),L(Aq)). If we have two runs pu− →p′and\nqu− →q′wherep′∈F,q′/∈Fand|u|/greaterorequalslant|Q|2then some state pair occurs twice in\nthe runs and we can pump the runs to unbounded lengths. Therefo re,d(p,q)<∞\nimpliesd(p,q)/lessorequalslant|Q|2. In fact d(p,q)<∞impliesd(p,q)/lessorequalslant|Q|by [50, Lemma 1].\nLemma 24. LetL⊆Σ∗be regular andA= (Q,Σ,q0,δ,F)be its minimal DFA.\nWe have:\n(i) for all p,q∈Q,d(p,q)/lessorequalslantkif and only if∀z∈Σk:p·z=q·z,\n(ii)Lisk-suffix testable if and only if d(p,q)/lessorequalslantkfor allp,q∈Q,\n(iii) if there exists k/greaterorequalslant0such that Lisk-suffix testable, then Lis|Q|-suffix\ntestable.\nProof.The proof of (i) is an easy induction. If k= 0, the statement is d(p,q) = 0\niffp=q, which is true because Ais minimal. For the induction step, we have\nd(p,q)/lessorequalslantk+1 iffd(δ(p,a),δ(q,a))/lessorequalslantkfor alla∈Σ iffδ(p,a)·z=δ(q,a)·zfor all\nz∈Σkiffp·z=q·zfor allz∈Σk+1.\nFor (ii), assume that Lisk-suffix testable and consider two states p=A(x) and\nq=A(y). Ifz∈L(Ap)△L(Aq), then|z|< kbecausexz∈Liffyz /∈LandLis\nk-suffix testable.\nNow, assume that d(p,q)/lessorequalslantkfor allp,q∈Qand consider x,y∈Σ∗,z∈Σk.\nSince we have d(A(x),A(y))/lessorequalslantk, (i) impliesA(xz) =A(yz), and in particular\nxz∈Liffyz∈L. Therefore, Lisk-suffix testable.\nPoint (iii) follows from (ii) and from the above cited [50, Lemma 1]. /square\nLemma 25. For any L⊆Σ∗andn/greaterorequalslant0, the language SWn(L)is2FL(n)-suffix\ntestable.\nProof.LetPnbe an SW-algorithm for Land window length nwith space com-\nplexityFL(n). Therefore,Pnhas at most 2FL(n)states. The definition of SWn(L)\ndirectly implies that SWn(L) isn-suffix testable. By Lemma 24(iii) SWn(L) is\n2FL(n)-suffix testable. /square\nNote that Lemma 25 holds for arbitrary languages and not only for r egular\nlanguages.\nProof of Theorem 13 (i).First, let L⊆Σ∗be a regularlanguage with FL(n) =O(1)\nand letk= max n∈N2FL(n). By Lemma 25 the language SWn(L) isk-suffix testable\nfor alln/greaterorequalslant0. We can express Las the Boolean combination\nL= (L∩Σ/lessorequalslantk−1)∪/uniondisplay\nz∈Σk(Lz−1)z= (L∩Σ/lessorequalslantk−1)∪/uniondisplay\nz∈Σk((Lz−1)Σk∩Σ∗z)\nwhere the right quotient Lz−1={x∈Σ∗|xz∈L}is regular [15, Chapter 3, Ex-\nample 5.7]. The set L∩Σ/lessorequalslantk−1is finite and hence suffix testable. It remains to show\nthat each Lz−1forz∈Σkis a length language. Consider two words x,y∈Σ∗ofREGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 19\nthe same length|x|=|y|=n. Since|xz|=|yz|=n+kandSWn+k(L) isk-suffix\ntestable, we have xz∈Liffyz∈L, and hence x∈Lz−1iffy∈Lz−1.\nFor the other direction note that (i) if Lis a length language or a suffix testable\nlanguage then clearly FL(n) =O(1), and (ii){L⊆Σ∗|FL(n) =O(1)}is closed\nunder Boolean operations by Lemma 11. This proves the theorem. /square\n3.7.Characterization of logarithmic space. Recall fromTheorem12that well-\nbehaved rDFAs precisely define those regular languages with logarit hmic space\ncomplexity FL(n) or equivalently VL(n). In the following, we will show that well-\nbehaved rDFAs recognize precisely the finite Boolean combinations o f regular left\nideals and regular length languages (Theorem 13(ii)), which therefo re are precisely\nthe regular languages with logarithmic space complexity. Let us star t with the easy\ndirection:\nProposition 26. Every language L∈/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}htis recognized by a well-behaved\nrDFA.\nProof.LetBbe an rDFA for L. IfLis a length language then for all reachable\nstatesqand all runs π,π′starting from qwith|π|=|π′|we have: πis accepting\nif and only if π′is accepting. If Lis a left ideal, then whenever a final state pis\nreachable, and qis reachable from p, thenqis also final. Hence, for every reachable\nSCCPinBeither all states of Pare final or all states of Pare nonfinal.\nItremainstoshowthatthe classoflanguages L⊆Σ∗recognizedbywell-behaved\nrDFAsisclosedunderBooleanoperations. If Biswell-behavedthenthecomplement\nautomatonBis also well-behaved. Given two well-behaved rDFAs B1,B2, we claim\nthat the product automaton B1×B2recognizing the intersection language is also\nwell-behaved. Suppose that Bi= (Qi,Σ,Fi,δi,q0,i) fori∈{1,2}. The product\nautomaton for the intersection language is defined by\nB1×B2= (Q1×Q2,Σ,F1×F2,δ,(q0,1,q0,2))\nwhereδ(a,(q1,q2)) = (δ1(a,q1),δ2(a,q2)) for all q1∈Q1,q2∈Q2anda∈Σ.\nConsider an SCC SofB1×B2which is reachable from the initial state and let\n(p1,p2),(q1,q2),(r1,r2)∈Ssuch that\n(q1,q2)u← −(p1,p2) and (r1,r2)v← −(p1,p2)\nfor some words u,v∈Σ∗with|u|=|v|. Since for i∈{1,2}we have qiu← −piand\nriv← −pi, and{pi,ri,qi}is contained in an SCC of Bi(which is also reachable from\nthe initial state q0,i), we have\n(q1,q2) is final⇐⇒q1andq2are final\n⇐⇒r1andr2are final\n⇐⇒(r1,r2) is final,\nand thereforeB1×B2is well-behaved. /square\nIt remains to prove that every well-behaved rDFA recognizes a finit e Boolean\ncombination of regular left ideals and regular length languages. With a right-\ndeterministic finite automaton B= (Q,Σ,F,δ,q 0) we associate the directed graph\n(Q,E) with edge set E={(p,a·p)|p∈Q,a∈Σ}. Theperiodg(G) of a directed\ngraphGis the greatest common divisor of all cycle lengths in G. IfGis acyclic we\ndefine the period to be ∞. We will apply the following lemma from Alon et al. [2]\nto the nontransient SCCs of B.\nLemma 27 ([2]).LetG= (V,E)be a strongly connected directed graph with E/\\e}atio\\slash=∅\nand finite period g. Then there exist a partition V=/uniontextg−1\ni=0Viand a constant\nm(G)/lessorequalslant3|V|2with the following properties:20 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\n•For every 0/lessorequalslanti,j/lessorequalslantg−1and for every u∈Vi,v∈Vjthe length of every\ndirected path from utovinGis congruent to j−imodulog.\n•For every 0/lessorequalslanti,j/lessorequalslantg−1, for every u∈Vi,v∈Vjand every integer\nr/greaterorequalslantm(G), ifris congruent to j−imodulog, then there exists a directed\npath from utovinGof length r.\nLemma 28 (Uniform period) .For every regular language there exists an rDFA B\nrecognizing Land a number gsuch that every nontransient SCC CinBhas period\ng(C) =g.\nProof.LetB= (Q,Σ,F,δ,q 0) be any rDFA for L. Letgbe the product of all\nperiodsg(C) over all nontransient SCCs CinB. In the following, we compute in\nthe additive group Zg={0,...,g−1}. We define\nB×Zg= (Q×Zg,Σ,F×Zg,δ′,(q0,0)),\nwhere for all ( p,i)∈Q×Zganda∈Σ we set\nδ′(a,(p,i)) =/braceleftBigg\n(δ(a,p),i+1),ifpandδ(a,p) are strongly connected ,\n(δ(a,p),0),otherwise.\nClearly,L(B×Zg) =L(B). We show that every nontransient SCC of B×Zghas\nperiodg. LetDbe a nontransient SCC of B×Zg. Clearly, every cycle length in D\nis a multiple of g. Take any state ( q,i)∈Dand letCbe the SCC of qinB. Since\nDis nontransient, there exists a cycle in B×Zgcontaining ( q,i), which induces a\ncycle inBcontaining q. This implies that Cis nontransient. Hence, we can apply\nLemma27and obtain acycleoflength k·g(C) inCfor everysufficiently large k∈N\n(k/greaterorequalslantm(C) suffices). Since gis a multiple of g(C),Calso contains a cycle of length\nk·gfor every sufficiently large k. But every such cycle induces a cycle of the same\nlengthk·ginD. Hence, there exists k∈Nsuch that Dcontains cycles of length\nk·gand (k+1)·g. It follows that the period of Ddivides gcd( k·g,(k+1)·g) =g.\nThis proves that the period of Dis exactly g. /square\nProof of Theorem 13 (ii).It remains to show the direction from left to right. Con-\nsider a well-behaved rDFA B= (Q,Σ,F,δ,q 0) for a regular language L⊆Σ∗. We\nprovethat Lis a finite Boolean combination of regularleft ideals and regular length\nlanguages. By Lemma 28 we can ensure that all nontransient SCCs in Bhave the\nsame period g. This new rDFA Bis also well-behaved since in fact any rDFA for\nLmust be well-behaved; this follows from Proposition 18 and Corollary 1 9. Al-\nternatively, one can verify that the transformation from Lemma 2 8 preserves the\nwell-behavedness of B.\nApath description Pis a sequence\n(2) Ck,(qk,ak−1,pk−1),Ck−1,...,(q3,a2,p2),C2,(q2,a1,p1),C1,q1\nwhereCk,...,C 1are pairwise distinct SCCs of B,q1=q0, (qi+1,ai,pi) is a transi-\ntion inBfor all 1 /lessorequalslanti/lessorequalslantk−1,pi,qi∈Cifor all 1 /lessorequalslanti/lessorequalslantk−1, andqk∈Ck.\nA runπinBrespects the path description Pif the SCC-factorization of πis\nπ=πkτk−1···τ2π2τ1π1,πiis aCi-internal run from qitopifor all 1 /lessorequalslanti/lessorequalslantk−1,\nτi=qi+1aipifor all 1 /lessorequalslanti/lessorequalslantk−1, andπkis aCk-internal run starting in qk. Let\nLPbe the set of words w∈Σ∗such that the unique run of Bonwstarting in q0\nrespects the path description P. We can write L=/uniontext\nP(LP∩L) wherePranges\nover all path descriptions. Notice that the number of path descrip tions is finite.\nLet us fix a path description Pas in (2). We provethat LP∩Lis a finite Boolean\ncombination of regularleft ideals and regularlength languages. First , we claim that\nLPis a finite Boolean combination of regular left ideals. Let ∆ = {(a·p,a,p)|p∈\nQ,a∈Σ}be the set of all transition triples and let ∆ P⊆∆ be the set of transitionREGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 21\ntriples contained in any of the SCCs Ck,...,C 1together with the transition triples\n(qi+1,ai,pi) for 1/lessorequalslanti/lessorequalslantk−1. A word w∈Σ∗then belongs to LPif and only if\nwbelongs to the regular left ideal Σ∗LPand the run ofBonwstarting in q0does\nnot use any transitions from ∆ \\∆P. It is easy to construct for every τ∈∆\\∆P\nan rDFADτwhich accepts all words wsuch that the run of Bonwstarting in\nq0uses the transition τ. Clearly, this language is a left ideal. In total we have\nLP= Σ∗LP\\/uniontext\nτ∈∆\\∆PL(Dτ), which proves the claim.\nIfCkis a transient SCC then LP∩Lis either empty or LP, and we are done. For\nthe rest of the proofwe assume that Ckis nontransient. Recall that all nontransient\nSCCs inBhaveperiod g. Furthermore, Ckis well-behavedsince it is reachablefrom\nq0according to the path description P. LetCk=/uniontextg−1\ni=0Vibe the partition from\nLemma 27. We claim that F∩Ckis a union of some of the Vi’s. Towards a\ncontradiction assume that there exist states p,q∈Viwherep∈Fandq /∈F. Let\nr/greaterorequalslantm(C) be any number divisible by g. Then, by Lemma 27 there exist runs\nfromptopand from ptoq, both of length r. This contradicts the fact that Ckis\nwell-behaved.\nLetπ,π′be two runs ofBstarting from q0which respect P. We claim that\n|π|=|π′|(modg)ifandonlyif πandπ′endinthesamepart ViofCk. Considerthe\nSCC-factorizations π=πkτk−1πk−1···τ2π2τ1π1andπ′=π′\nkτk−1π′\nk−1···τ2π′\n2τ1π′\n1.\nFor all 1 /lessorequalslanti/lessorequalslantk−1 the subruns πiandπ′\nistart inqiand end in pi. IfCiis\nnontransient then |πi|=|π′\ni|(modg) by Lemma 27, and otherwise |πi|=|π′\ni|= 0.\nThis implies|τk−1πk−1···τ2π2τ1π1|=|τk−1π′\nk−1···τ2π′\n2τ1π′\n1|(modg). Also by\nLemma 27 we know that |πk|=|π′\nk|(modg) if and only if πandπ′end in the\nsame part Vi. This proves the claim.\nIt follows that we can write LP∩L=LP∩Kwhere\nK={w∈Σ∗|∃r∈R:|w|≡r(modg)}\nfor some R⊆{0,...,g−1}. SinceKis a regular length language, we have proved\nthe claim that Lis a Boolean combination of regular left ideals and regular length\nlanguages. This concludes the proof of Theorem 13(ii). /square\n4.Randomized sliding window algorithms\nMost of the work in the context of streaming uses randomness and /or approx-\nimation to design space- and time-efficient algorithms. For example, t he AMS-\nalgorithm [3] approximates the number of distinct elements in a strea m with high\nprobability inO(logm) space where mis the size of the universe. Furthermore,\nit is proved that any deterministic approximation algorithm and any ra ndomized\nexact algorithm must use Ω( n) space [3]. On the other hand, the exponential his-\ntogram algorithm by Datar et al. [26] for approximating the number o f 1’s in a\nsliding window is a deterministic sliding window approximationalgorithm that uses\nO(1\nǫlog2n) bits. It is proven in [26] that Ω(1\nǫlog2n) bits are necessary even for\nrandomized (Monte Carlo or Las Vegas) sliding window algorithms.\nInthissection, wewillstudyifandhowrandomnesshelpsfortesting membership\nto regularlanguages oversliding windows. The main result of this sect ion is a space\ntetrachotomy in the fixed-size sliding window model, stating that eve ry regular\nlanguage has optimal space complexity Θ(1), ¯Θ(loglog n),¯Θ(logn) or¯Θ(n) if the\nstreaming algorithms are randomized with two-sided error.\n4.1.Randomized streaming algorithms. In the following, we will introduce\nprobabilistic automata [77, 79] as a model of randomized streaming a lgorithms.\nArandomized streaming algorithm or aprobabilistic automaton P= (Q,Σ,ι,ρ,F)\nconsists of a nonempty (possibly infinite) set of states Q, a finite alphabet Σ, an22 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\ninitial state distribution ι:Q→[0,1], a transition probability function ρ:Q×Σ×\nQ→[0,1], and a set of final states F⊆Q, such that\n(i)/summationtext\nq∈Qι(q) = 1,\n(ii)/summationtext\nq∈Qρ(p,a,q) = 1 for all p∈Q,a∈Σ.\nIfιandρmap into{0,1}, thenPcan be viewed as a deterministic automaton.\nFor a random variable Xwith values from Qanda∈Σ we define the random\nvariableX·a(which also takes values from Q) by\nPr[X·a=q] =/summationdisplay\np∈QPr[X=p]·ρ(p,a,q).\nForaword w∈Σ∗wedefinearandomvariable P(w) withvaluesfrom Qinductively\nas follows: the random variable P(ε) is defined such that Pr[ P(ε) =q] =ι(q)\nfor allq∈Q. Moreover,P(wa) =P(w)·afor allw∈Σ∗anda∈Σ. Thus,\nPr[P(w) =q]is the probabilitythat Pis instate qafter reading w. The randomized\nstreaming algorithm Pacceptswwith probability Pr[ P(w)∈F] andrejectswwith\nprobability Pr[P(w)/∈F]. ThespaceofP(ornumber of bits used by P) is given by\ns(P) = log|Q|∈R/greaterorequalslant0∪{∞}. We say thatPis a randomized streaming algorithm\nforL⊆Σ∗with error probability 0 /lessorequalslantλ/lessorequalslant1 if\n•Pr[Pacceptsx]/greaterorequalslant1−λfor allx∈L,\n•Pr[Prejectsx]/greaterorequalslant1−λfor allx /∈L.\nThe error probability λis also called a two-sided error . If we omit λwe choose\nλ= 1/3.\nFor a randomized streaming algorithm P= (Q,Σ,ι,ρ,F) and a number k/greaterorequalslant1\nletP(k)be the randomized streaming algorithm which simulates kinstances ofPin\nparallel with independent random choices and outputs the majority vote. Formally\nthe states ofP(k)are multisets of size koverQ(using multisets instead of ordered\ntuples will yield a better space bound in Section 4.3). Therefore, s(P(k))/lessorequalslantk·s(P).\nLemma 29 (probability amplification) .For all0< λ′< λ <1/2there exists a\nnumberk=O(log/parenleftbig1\nλ′/parenrightbig\n·/parenleftbig1\n2−λ/parenrightbig−2)such that for all randomized streaming algo-\nrithmsPwe have:\n(i) IfPr[Pacceptsx]/greaterorequalslant1−λthenPr[P(k)acceptsx]/greaterorequalslant1−λ′.\n(ii) IfPr[Prejectsx]/greaterorequalslant1−λthenPr[P(k)rejectsx]/greaterorequalslant1−λ′.\nProof.We will choose klater. Let X1,...,X kbe independent Bernoulli random\nvariables with Pr[ Xi= 0] =λand Pr[Xi= 1] = 1−λ. LetX=/summationtextk\ni=1Xiwith\nexpectation µ=k(1−λ). Suppose thatPacceptsxwith probability /greaterorequalslant1−λ, i.e.\nPrejectsxwith probability at most λ. Then,P(k)rejectsxwith probability at\nmost Pr[ X/lessorequalslantk/2]. By the Chernoff bound [74, Theorem 4.5], for any 0 < δ <1 we\nhave\n(3) Pr[ X/lessorequalslant(1−δ)µ]/lessorequalslantexp/parenleftbigg\n−µδ2\n2/parenrightbigg\n.REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 23\nBy choosing δ= 1−1\n2(1−λ)we get (1−δ)µ=k/2. Then, (3) gives the following\nestimate:\nPr[P(k)rejectsx]/lessorequalslantPr[X/lessorequalslantk/2]\n/lessorequalslantexp/parenleftBigg\n−k(1−λ)(1−1\n2(1−λ))2\n2/parenrightBigg\n= exp/parenleftbigg\n−k\n2·(1\n2−λ)2\n1−λ/parenrightbigg\n/lessorequalslantexp/parenleftBigg\n−k\n2·/parenleftbigg1\n2−λ/parenrightbigg2/parenrightBigg\n.\nBy choosing\nk/greaterorequalslant2·ln/parenleftbigg1\nλ′/parenrightbigg\n·/parenleftbigg1\n2−λ/parenrightbigg−2\nwe can bound the probability that P(k)rejectsxbyλ′. Statement (ii) can be shown\nanalogously. /square\n4.2.Space tetrachotomy. Arandomized sliding window algorithm for a language\nLand window size nis a randomized streaming algorithm for SWn(L). Theran-\ndomized space complexity Fr\nL(n) ofLin the fixed-size sliding window model is the\nminimal space complexity s(Pn) of a randomized sliding window algorithm Pnfor\nLand window size n. For this to be well-defined it is important that we require the\nerror probability to be at most 1 /3. Clearly we have Fr\nL(n)/lessorequalslantFL(n). Furthermore,\nwe prove that randomness can reduce the space complexity at mos t exponentially.\nLemma 30. For any language Lwe have FL(n) = 2O(Fr\nL(n)).\nProof.Rabin proved that any probabilistic finite automaton with a so-called is o-\nlatedcut-pointcanbemadedeterministicwithanexponentialsizeinc rease[79]. Let\nP= (Q,Σ,ι,ρ,F) be a probabilistic finite automaton with mstates. Suppose that\nλ∈[0,1] is anisolated cut-point with radius δ >0, i.e.|Pr[Pacceptsx]−λ|/greaterorequalslantδ\nfor allx∈Σ∗. Then,L={x∈Σ∗|Pr[Pacceptsx]/greaterorequalslantλ}is recognized by a DFA\nAwith at most (1+ m/δ)m−1= 2O(mlogm)states [79, Theorem 3].\nNow, letPnbe a minimal probabilistic finite automaton for SWn(L) withm\nstates and error probability /lessorequalslant1/3. SincePnhas 1/2 as an isolated cutpoint with\nradius 1/2−1/3 = 1/6, there exists an equivalent DFA Qnwith|Qn|/lessorequalslant2O(mlogm)\nstates. The statement follows from FL(n)/lessorequalslantlog|Qn|=O(mlogm) =O(2Fr\nL(n)·\nFr\nL(n)), which is bounded by 2O(Fr\nL(n)). /square\nIn this section, we will proveTheorem4, which is a tetrachotomyfor the random-\nized space complexity of regular languages in the fixed-size sliding wind ow model.\nLet us rephrase Theorem 4 and split it into three upper bounds and t hree lower\nbounds.\nTheorem 31. LetL⊆Σ∗be a regular language.\n(1) IfL∈/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}htthenFr\nL(n) =O(1).\n(2) IfL /∈/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}htthenFr\nL(n) =¯Ω(loglog n).\n(3) IfL∈/a\\}⌊ra⌋ketle{tST,SF,Len/a\\}⌊ra⌋ketri}htthenFr\nL(n) =O(loglogn).\n(4) IfL /∈/a\\}⌊ra⌋ketle{tST,SF,Len/a\\}⌊ra⌋ketri}htthenFr\nL(n) =¯Ω(logn).\n(5) IfL∈/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}htthenFr\nL(n) =O(logn).\n(6) IfL /∈/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}htthenFr\nL(n) =¯Ω(n).\nPoints (1) and (5) already hold in the deterministic setting, see Theo rem 13. In\nthe next sections we prove points (2), (3), (4), and (6).24 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nWe first transfer Lemma 11 to the fixed-size model in the randomize d setting:\nLemma 32. For any function s(n), the class{L⊆Σ∗|Fr\nL(n) =O(s(n))}forms\na Boolean algebra.\nProof.Letn∈Nbe a window size. If Pnis a randomized SW-algorithm for Lthen\nPnis a randomized SW-algorithm for Σ∗\\LwherePnsimulatesPnand returns\nthe negated output. Let PnandQnbe randomized SW-algorithms for KandL.\nBy Lemma 29 we can reduce their error probabilities to 1/6 with a cons tant space\nincrease. Then, the algorithm which simulates PnandQnin parallel, and returns\nthe disjunction of the outputs is a randomized SW-algorithm for K∪L. Its error\nprobability is at most 1/3 by the union bound. /square\n4.3.The Bernoulli counter. The crucial algorithmic tool for the proof of The-\norem 31(3) is a simple probabilistic counter. It is inspired by the appro ximate\ncounter by Morris [36, 75], which uses O(loglogn) bits. For our purposes, it suf-\nfices to detect whether the counter has exceeded a certain thre shold, which can be\naccomplished using only O(1) bits.\nFormally, a probabilistic counter is a probabilistic automaton\nZ= (C,{inc},ι,ρ,F)\nover the unary alphabet {inc}. States in Fare called highand states in C\\F\nare called low. We make the restriction that there is a low state c0∈Csuch that\nι(c0) = 1 (and hence ι(c) = 0 for all c∈C\\{c0}); thusZhas a unique initial state\nc0(which must be low) and we write Z= (C,{inc},c0,ρ,F). This restriction is\nnot really important (and can in fact be achieved for every probabilis tic automaton\nby adding a new state), but it will simplify our constructions.\nThe random state of Zreached after kincrements isZ(inck), for which we also\nwriteZ(k); note that it is a random variable. Given numbers 0 /lessorequalslantℓ < hwe say\nthatZis an (h,ℓ)-counter with error probability λ <1\n2if for all k∈Nwe have:\n•Ifk/lessorequalslantℓ, then Pr[Z(k) is high] /lessorequalslantλ.\n•Ifk/greaterorequalslanth, then Pr[Z(k) is low] /lessorequalslantλ.\nIn other words, a probabilistic counter can distinguish with high prob ability values\nbelowℓfrom values above hbut does not give any guarantees for counter values\nstrictly between ℓandh. ABernoulli counter Zpis parameterized by a probability\n0< p <1 and has the state set {0,1}, where 0 is a low state and 1 is a high state.\nInitially the counter is in the state x= 0. On every increment we set x= 1 with\nprobability p; the state remains unchanged with probability 1 −p. We have\nPr[Zp(k) is low] = (1−p)kand Pr[Zp(k) is high] = 1−(1−p)k.\nLet us first show the following claim.\nLemma 33. For allh >0,0< ξ <1and0< ℓ/lessorequalslant(1−ξ)hthere exists 0< p <1\nsuch thatZpis an(h,ℓ)-counter with error probability 1/2−ξ/8.\nProof.We need to choose 0 < p <1 such that\n(i) 1−(1−p)(1−ξ)h/lessorequalslant1/2−ξ/8, or equivalently, 1 /2+ξ/8/lessorequalslant(1−p)(1−ξ)h,\nand\n(ii) (1−p)h/lessorequalslant1/2−ξ/8, or equivalently, (1 −p)(1−ξ)h/lessorequalslant(1/2−ξ/8)1−ξ.\nIt suffices to show\n(4)1\n2+ξ\n8/lessorequalslant/parenleftbigg1\n2−ξ\n8/parenrightbigg1−ξ\n,REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 25\nthen one can pick p= 1−(1/2−ξ/8)1/h∈(0,1). Note that (ii) holds automatically\nfor this value of p. Taking logarithms shows that (4) is equivalent to\nln(4+ξ)−ln8/lessorequalslant(1−ξ)·(ln(4−ξ)−ln8),\nwhich can be rearranged to ln(4 + ξ)/lessorequalslantln(4−ξ) +ξ(ln8−ln(4−ξ)). Since\nln8−ln(4−ξ)/greaterorequalslantln8−ln4 = ln2, it suffices to prove\n(5) ln(4+ ξ)/lessorequalslantln(4−ξ)+ξln2.\nOne can verify 3ln2 ≈2.0794/greaterorequalslant2. We have\n4+ξ/lessorequalslant4+(3ln2−1)ξ\n= 4+(4ln2−1)ξ−ξln2\n/lessorequalslant4+(4ln2−1)ξ−ξ2ln2\n= (4−ξ)(ξln2+1).\nBy taking logarithms and plugging in ln x/lessorequalslantx−1 for all x >0, we obtain\nln(4+ξ)/lessorequalslantln(4−ξ)+ln(ξln2+1) /lessorequalslantln(4−ξ)+ξln2.\nThis proves (5) and hence (4), and thus the lemma. /square\nProposition 34. For allh >0,0< ξ <1,0< ℓ/lessorequalslant(1−ξ)hand0< λ′<1/2\nthere exists an (h,ℓ)-counterZwith error probability λ′which usesO(loglog(1 /λ′)+\nlog(1/ξ))bits.\nProof.Take the ( h,ℓ)-counterZpfrom Lemma 33, whose error probability is λ:=\n1/2−ξ/8. ToZpwe apply Lemma 29, which states that we need to run k=\nO(log(1\nλ′)·1\nξ2) copies to reduce the error probability to λ′. The states ofZ(k)\npare\nmultisets over{0,1}of sizek, which can be encoded with O(logk) =O(loglog1\nλ′+\nlog1\nξ) bits by specifying the number of 1-bits in the multiset. Note that th e unique\ninitial state ofZ(k)\npis the multiset with koccurrences of 0. /square\n4.4.Suffix-free languages. Inthissection,weproveTheorem31(3). ByLemma32\nandsincelanguagesin ST∪Lenhaveconstantspace(deterministic) SW-algorithms\nit suffices to show:\nTheorem 35. IfLis regular and suffix-free then Fr\nL(n) =O(loglogn).\nFix a suffix-free regular language L⊆Σ∗and letB= (Q,Σ,F,δ,q 0) be an rDFA\nforLwhere all states are reachable. Excluding the trivial case L=∅, we assume\nthatBcontains at least one final state. Furthermore, since Lis suffix-free, any run\ninBcontains at most one final state. Therefore, we can assume that Fcontains\nexactly one final state qF, and all outgoing transitions from qFlead to a sink state.\nFor a stream w∈Σ∗define the function ℓw:Q→N∪{∞}by\n(6) ℓw(q) = inf{k∈N|lastk(w)·q=qF},\nwhere we set inf( ∅) =∞(note that{k∈N|lastk(w)·q=qF}is either empty or\na singleton set). Notice that lastn(w)∈Lif and only if ℓw(q0) =n. Also, it holds\nℓw(qF) = 0 for every w∈Σ∗. Adeterministic streaming algorithm can maintain\nthe function ℓwwherew∈Σ∗is the stream prefix read so far: If a symbol a∈Σ is\nread, we can determine\n(7) ℓwa(q) =/braceleftBigg\n0, ifq=qF,\n1+ℓw(a·q),otherwise,\nwhere 1+∞=∞. Since storing ℓw(q) would require storing O(logn) bits, we use\nprobabilistic counters instead.26 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nLetn∈Nbe a window size. The sliding window algorithm PnforLconsists\nof two parts: a constant-space threshold algorithm Tn, which rejects with high\nprobability whenever ℓw(q0)/greaterorequalslant2n, and a modulo-counting algorithm Mn, which\nmaintains ℓwmodulo a random prime number with O(loglogn) bits.\nLemma 36 (Threshold counting) .There exists a randomized streaming algorithm\nTnwithO(1)bits such that for all w∈Σ∗we have:\n•Pr[Tnacceptsw]/greaterorequalslant2/3, ifℓw(q0)/lessorequalslantn, and\n•Pr[Tnrejectsw]/greaterorequalslant2/3, ifℓw(q0)/greaterorequalslant2n.\nProof.By Proposition 34 there is a (2 n,n)-counterZwith error probability 1/3\nwhich usesO(1) space. Let Cbe its state space, c0be its initial state and c∞∈C\nbe an arbitrary high state. The algorithm Tnmaintains for every q∈Qan instance\nZqof the (2n,n)-counterZ. The random variable Zq(w) of the state reached after\nthe input stream w∈Σ∗will satisfy\n(8) Zq(w) =Z(ℓw(q)),\nwhere we setZ(∞) =c∞, which is viewed as the random variable that takes the\nvaluec∞with probability one. We initialize Zqin order to get (8) for w=ε: for\nevery state q∈Q, ifℓε(q)<∞, then we initialize Zqin its initial state c0and then\nexecuteℓε(q) increments, otherwise we set Zqto state c∞(with probability one).\nGiven an input symbol a∈Σ, we compute the new counter values (let us call them\nZ′\nqfor the moment) as follows: First we set Z′\nqFto the initial state c0. This ensures\n(8) forqFsinceℓwa(qF) = 0<∞andZ(0) =c0. For all q∈Q\\{qF}we setZ′\nqto\nZa·q·inc. This ensures again (8):\nZq(wa) =Za·q(w)·inc=Z(1+ℓw(a·q)) =Z(ℓwa(q)).\nThe algorithm accepts if and only if Zq0is low. Correctness follows from the fact\nthatZis a (2n,n)-counter with error probability 1/3:\nPr[Zq0(w) is low] = Pr[Z(ℓw(q0)) is low]/braceleftBigg\n/greaterorequalslant2/3 ifℓw(q0)/lessorequalslantn,\n/lessorequalslant1/3 ifℓw(q0)/greaterorequalslant2n.\nThis proves the lemma. /square\nLemma 37 (Modulo counting) .There exists a randomized streaming algorithm\nMnwithO(loglogn)bits such that for all w∈Σ∗we have:\n•Pr[Mnacceptsw] = 1, ifℓw(q0) =n, and\n•Pr[Mnrejectsw]/greaterorequalslant2/3, ifℓw(q0)<2nandℓw(q0)/\\e}atio\\slash=n.\nProof.Letpibe thei-th prime number. It is known that pi< i·(lni+lnlni) for\ni/greaterorequalslant6 [81, 3.13]. Let kbe the first natural number such that/producttextk\ni=1pi/greaterorequalslantn. Since/producttextk\ni=1pi/greaterorequalslant2k, we have k=O(logn) andp3k=O(logn·loglogn). The algorithm\nMninitially picks a random prime p∈{p1,...,p 3k}, which is stored throughout\nthe run usingO(loglogn) bits. Then, after reading w∈Σ∗,Mnstores for every\nq∈Qabit tellingwhether ℓw(q)<∞and, ifthelatterholds, thevalue ℓw(q) modp\nusing in totalO(|Q|·loglogn) bits. These numbers can be maintained according\nto (7). The algorithm accepts if and only if ℓw(q0)≡nmodp.\nIfℓw(q0) =nthen the algorithm always accepts. Now, assume ℓw(q0)<2nand\nℓw(q0)/\\e}atio\\slash=n. Since−n/lessorequalslantℓw(q0)−n < nand any product of at least k+1 pairwise\ndistinct primes exceeds n, the number ℓw(q0)−n/\\e}atio\\slash= 0 has at most kprime factors.\nTherefore,Mnrejects with probability at least 2 /3. /square\nBy combining the algorithms from Lemma 36 and Lemma 37 we can prove\nTheorem 35. The algorithm Pnis the conjunction of the threshold algorithm Tn\nand the modulo counting algorithm Mn. Recall that lastn(w)∈Lif and onlyREGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 27\nifℓw(q0) =n. Ifℓw(q0) =nthenTnaccepts with probability at least 2 /3 and\nMnaccepts with probability 1; hence Pnaccepts with probability at least 2 /3. If\nℓw(q0)/\\e}atio\\slash=nthenMnorTnrejects with probability at least 2 /3. Hence,Pnrejects\nwith probability at least 2 /3.\n4.5.Lower bounds. In this section, we prove the lower bounds from Theorem 31.\nPoint (2) from Theorem 31 follows easily from the relation FL(n) = 2O(Fr\nL(n))\n(Lemma 30). Since every language L∈Reg\\/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}htsatisfiesFL(n) =¯Ω(logn)\n(Theorem 12 and 13), it also satisfies Fr\nL(n) =¯Ω(loglog n).\nFor(4)and(6) weapplyknownlowerboundsfromcommunicationcom plexityby\nderiving a randomized communication protocol from a randomized SW -algorithm.\nThis is in fact a standard technique for obtaining lower bounds for st reaming algo-\nrithms; see e.g. [82].\nWe present the necessary background from communication comple xity; see [70]\nfor a detailed introduction. We only need the one-way setting where Alice sends a\nsingle message to Bob. Consider a function f:X×Y→{0,1}for some finite sets\nXandY. Arandomized one-way (communication) protocol P= (a,b) consists of\nfunctions a:X×Ra→{0,1}∗andb:{0,1}∗×Y×Rb→{0,1}, whereRaandRb\nare finite sets of random choices of Alice and Bob, respectively. The costofPis\nthe maximum number of bits transmitted by Alice, i.e.\ncost(P) = max\nx∈X,ra∈Ra|a(x,ra)|.\nMoreover, probability distributions are given on RaandRb. Alice computes from\nher input x∈Xand a random choice ra∈Rathe value a(x,ra) and sends it to\nBob. Using this value, his input y∈Yand a random choice rb∈Rbhe outputs\nb(a(x,ra),y,rb). The random choices ra∈Ra,rb∈Rbare chosen independently\nfrom each other. The protocol Pcomputes fif for all ( x,y)∈X×Ywe have\n(9) Pr\nra∈Ra,rb∈Rb[P(x,y)/\\e}atio\\slash=f(x,y)]/lessorequalslant1\n3,\nwhereP(x,y) is the random variable b(a(x,ra),y,rb). Therandomized one-way\ncommunication complexity offis the minimal cost among all one-way randomized\nprotocols that compute f(with an arbitrary number of random bits). The choice\nof the constant 1 /3 in (9) is arbitrary in the sense that changing the constant to\nanyλ <1/2 only changes the communication complexity by a constant (depend ing\nonλ), see [70, p. 30]. We will use established lower bounds on the random ized\none-way communication complexity of some functions.\nTheorem 38 ([69, Theorem 3.7 and 3.8]) .Letn∈N.\n•Theindex function\nIDXn:{0,1}n×{1,...,n}→{0,1}\nwithIDXn(a1···an,i) =aihas randomized one-way communication com-\nplexityΘ(n).\n•Thegreater-than function\nGTn:{1,...,n}×{1,...,n}→{0,1}\nwithGTn(i,j) = 1iffi > jhas randomized one-way communication com-\nplexityΘ(logn).\nThe bounds above also hold for the deterministic one-way communica tion com-\nplexity as witnessed by the trivial deterministic protocols. We also de fine the\nequality function\nEQn:{1,...,n}×{1,...,n}→{0,1}28 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nby EQn(i,j) = 1 if and only if i=j. Its randomized one-way communication com-\nplexity is Θ(loglog n) whereas its deterministic one-way communication complexity\nis Θ(logn) [70].\nWe start with the proof of (6) from Theorem 31, which extends our linear space\nlower bound from the deterministic setting to the randomized settin g.\nProposition 39. IfL∈Reg\\/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}htthenFr\nL(n) =¯Ω(n).\nProof.By Theorem 13(ii) any rDFA for Lis not well-behaved and by Lemma 17\nthere exist words u=u1u2,v=v1v2,z∈Σ∗such that|u1|=|v1|,|u2|=|v2|and\nLseparates u2{u,v}∗zandv2{u,v}∗z. Letη:{0,1}∗→{u,v}∗be the injective\nhomomorphism defined by η(0) =uandη(1) =v.\nNow, consider a randomized SW-algorithm PnforLand window length n=\n|u2|+|u|·m+|z|forsome m/greaterorequalslant1. Wedescribearandomizedone-waycommunication\nprotocol for IDX m.\nLetα=α1···αm∈ {0,1}mbe Alice’s input and i∈ {1,...,m}be Bob’s\ninput. Alice reads η(α) intoPn(using here random choices in order to select the\noutgoing transitions in Pn) and sends the memory state using O(s(Pn)) bits to\nBob. Continuing from the received state, Bob reads uizintoPn. Then, the active\nwindow is\nlastn(η(α)uiz) =sη(αi+1···αm)uiz∈{u2,v2}{u,v}∗z\nwheres=u2ifαi= 0 and s=v2ifαi= 1. Hence, from the output of PnBob can\ndetermine whether αi= 1. The cost of the protocol is bounded by O(s(Pn)) and\nmust be at least Ω( m) = Ω(n) by Theorem 38. We conclude that s(Pn) = Ω(n) for\ninfinitely many nand therefore Fr\nL(n) =¯Ω(n). /square\nNext, we prove point (4) from Theorem 31. For that, we need the f ollowing\nautomaton property. In the following, let B= (Q,Σ,F,δ,q 0) be an rDFA.\nA pair (p,q)∈Q×Qof states is called synchronized if there exist words x,y,z∈\nΣ∗with|x|=|y|=|z|/greaterorequalslant1 such that\nqx← −qy← −pz← −p.\nA pair ( p,q)∈Q×Qis called reachable ifpandqare reachable from q0and\n(p,q) is called F-consistent if either{p,q}∩F=∅or{p,q}⊆F. We remark that\nsynchronized state pairs have no connection to the notion of sync hronizing words.\nLemma 40. A state pair (p,q)is synchronized if and only if pandqare nontran-\nsient and there exists a nonempty run from ptoqwhose length is a multiple of\n|Q|!.\nProof.First assume that ( p,q) is synchronized. Let x,y,z∈Σ+with|x|=|y|=\n|z|=ksuch that qx← −qy← −pz← −p. Then,pandqare nontransient and we have\nqx|Q|!−1y←−−−−− p,\nwherex|Q|!−1yhas length (|Q|!−1)·k+k=|Q|!·k.\nConversely, assume that pandqare nontransient and there exists a nonempty\nrunqy← −pwhose length is divided by |Q|!. Since the states pandqarenontransient,\nthere are words xandzof length at most |Q|withqx← −qandpz← −p. These words\ncan be pumped up to have length |y|. /square\nLetQ=T∪Nbe the partition of the state set into the set Tof transient states\nand the set Nof nontransient states. A function β:N→{0,1}isk-periodic if\nβ(i) =β(i+k) for alli∈N.REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 29\nLemma 41. Assume that every reachable synchronized pair in BisF-consistent.\nThen, for every word v∈Σ∗of length at least |Q|!·(|T|+ 1)there exists a|Q|!-\nperiodic function βv:N→{0,1}such that the following holds: If w∈Σ∗vand\nw·q0∈N, then we have w∈L(B)iffβ(|w|) = 1.\nProof.Letv=ak···a2a1withk/greaterorequalslant|Q|!·(|T|+1), and consider the run\n(10) qkak←−···a2←−q1a1←−q0\nofBonv. Clearly, each transient state can occur at most once in the run. F irst\nnotice that for each 0 /lessorequalslanti/lessorequalslant|Q|!−1 at least one of the states in\nQi={qi+j|Q|!|0/lessorequalslantj/lessorequalslant|T|}\nis nontransient because otherwise the set would contain |T|+ 1 pairwise distinct\ntransient states. Furthermore, we claim that the nontransient s tates in Qiare\neither all final or all nonfinal: Take two nontransient states qi+j1|Q|!andqi+j2|Q|!\nwithj1< j2. Since we have a run of length ( j2−j1)|Q|! fromqi+j1|Q|!toqi+j2|Q|!,\nthese two states form a synchronized pair by Lemma 40, which by as sumption must\nbeF-consistent.\nNow, define βv:N→{0,1}by\nβv(m) =/braceleftBigg\n1,if the states in Qmmod|Q|!∩Nare final,\n0,if the states in Qmmod|Q|!∩Nare nonfinal ,\nwhich is well-defined by the remarks above. Clearly βvis|Q|!-periodic.\nLetw=am···a2a1∈Σ∗vbe a word of length m/greaterorequalslantk. The run ofBonw\nstarting from the initial state prolongs the run in (10):\nqmam←−−···ak+2←−−−qk+1ak+1←−−−qkak←−···a2←−q1a1←−q0\nAssume that qm∈N. As argued above, there is a position 0 /lessorequalslanti < ksuch that\ni≡m(mod|Q|!) andqi∈N. Therefore, there exists a nonempty run from qi\ntoqmwhose length is a multiple of |Q|!. Hence, ( qi,qm) is a synchronized pair by\nLemma 40, which is F-consistent by assumption. Therefore, w∈Liffqm∈Fiff\nqi∈Fiffβv(|w|) = 1. /square\nLemma 42. Assume that every reachable synchronized pair in BisF-consistent.\nThen,L(B)belongs to/a\\}⌊ra⌋ketle{tST,SF,Len/a\\}⌊ra⌋ketri}ht.\nProof.Given a subset P⊆QletL(B,P) :=L(Q,Σ,P,δ,q 0). LetFN=N∩Fand\nFT=T∩F. We disjointly decompose Linto\nL=L(B,FN)∪/uniondisplay\nq∈FTL(B,{q}).\nFirst observe that L(B,{q})∈SFfor allq∈FTbecause a transient state qcan\noccur at most once in a run of B.\nIt remains to show that L(B,FN) belongs to/a\\}⌊ra⌋ketle{tST,SF,Len/a\\}⌊ra⌋ketri}ht. Using the threshold\nk=|Q|!·(|T|+1), we distinguish between words of length at most k−1 and words\nof length at least k, and group the latter set by their suffixes of length k:\nL(B,FN) = (L(B,FN)∩Σ/lessorequalslantk−1)∪/uniondisplay\nv∈Σk(L(B,FN)∩Σ∗v).\nThe first part L(B,FN)∩Σ/lessorequalslantk−1is finite and thus suffix testable. To finish the proof,\nwe will show that L(B,FN)∩Σ∗v∈/a\\}⌊ra⌋ketle{tST,SF,Len/a\\}⌊ra⌋ketri}htfor each v∈Σk. Letv∈Σk\nand letβv:N→{0,1}be the|Q|!-periodic function from Lemma 41. The lemma\nimplies that\nL(B,FN)∩Σ∗v= (Σ∗v∩{w∈Σ∗|β(|w|) = 1})\\L(B,T).30 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nThe language{w∈Σ∗|β(|w|) = 1}is a regular length language, Σ∗vis suffix\ntestable and L(B,T) is a finite union of suffix-free regular languages. /square\nThe following lemma is an immediate consequence of Lemma 42.\nLemma 43. IfL∈Reg\\/a\\}⌊ra⌋ketle{tST,SF,Len/a\\}⌊ra⌋ketri}htthen there exist u,x,y,z∈Σ∗with\n|x|=|y|=|z|/greaterorequalslant1such that Lseparates x∗yz∗uandz∗u.\nNow, we can finally prove point (4) from Theorem 31.\nProposition 44. IfL∈Reg\\/a\\}⌊ra⌋ketle{tST,SF,Len/a\\}⌊ra⌋ketri}htthenFr\nL(n) =¯Ω(logn).\nProof.Considerthewords u,x,y,z∈Σ∗describedinLemma43. Let n=|z|·m+|u|\nfor some m/greaterorequalslant1 and letPnbe a randomized SW-algorithm for L. We describe a\nrandomized one-way protocol for GT m: Let 1/lessorequalslanti/lessorequalslantmbe the input of Alice and\n1/lessorequalslantj/lessorequalslantmbe the input of Bob. Alice starts with reading xmyzm−iintoPn. Then\nshe sends the reached state to Bob using O(s(Pn)) bits. Bob then continues the\nrun ofPnfrom the transmitted state with the word zju. Hence,Pnis simulated\non the word w:=xmyzm−izju=xmyzm−i+ju. We have\nlastn(w) =/braceleftBigg\nxi−1−jyzm−i+ju,ifi > j,\nzmu, ifi/lessorequalslantj.\nBy Lemma 43, lastn(w) belongs to Lin exactly one of the two cases i > jand\ni/lessorequalslantj. Hence, Bob can distinguish these two cases with probability at least 2/3.\nIt follows that the protocol computes GT mand its cost is bounded by s(Pn). By\nTheorem 38 we can conclude that s(Pn) = Ω(log m) = Ω(log n), and therefore\nFr\nL(n) =¯Ω(logn). /square\n4.6.Sliding window algorithms with one-sided error. So far, we have only\nconsidered randomized SW-algorithms with two-sided error (analog ously to the\ncomplexity class BPP). Randomized SW-algorithms with one-sided error (analo-\ngously to the classes RPandcoRP) can be motivated by applications, where\nall “yes”-outputs or all “no”-outputs, respectively, have to be correct. We distin-\nguish between true-biased and false-biased algorithms. A true-biased (randomized)\nstreaming algorithm Pfor a language Lsatisfies the following properties:\n•Ifw∈Lthen Pr[Pacceptsw]/greaterorequalslant2/3.\n•Ifw /∈Lthen Pr[Prejectsw] = 1.\nAfalse-biased (randomized) streaming algorithm Pfor a language Lsatisfies the\nfollowing properties:\n•Ifw∈Lthen Pr[Pacceptsw] = 1.\n•Ifw /∈Lthen Pr[Prejectsw]/greaterorequalslant2/3.\nLetF0\nL(n) (resp.,F1\nL(n)) be the minimal space complexity s(Pn) of any true-biased\n(resp., false-biased) SW-algorithm PnforLand window size n. We have the rela-\ntionsFr\nL(n)/lessorequalslantFi\nL(n)/lessorequalslantFL(n) fori∈{0,1}, andF0\nL(n) =F1\nΣ∗\\L(n).\nWe show that for all regular languages SW-algorithms with one-sided error have\nno advantage over their deterministic counterparts:\nTheorem 45 (One-sided error) .LetLbe regular.\n(i) IfL∈/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}htthenF0\nL(n)andF1\nL(n)areO(1).\n(ii) IfL /∈/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}htthenF0\nL(n)andF1\nL(n)are¯Ω(logn).\n(iii) IfL∈/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}htthenF0\nL(n)andF1\nL(n)areO(logn).\n(iv) IfL /∈/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}htthenF0\nL(n)andF1\nL(n)are¯Ω(n).\nThe upper bounds in (i) and (iii) already hold for deterministic SW-algor ithms\n(Theorem 3). Moreover, the lower bound in (iv) already holds for SW -algorithms\nwith two-sided error (Theorem 31(6)). It remains to prove point ( ii) of the theorem.REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 31\nIn fact we show that any nondeterministic SW-algorithm for a regular language\nL /∈/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}htrequires logarithmic space. A nondeterministic SW-algorithm for a\nlanguage Land window size nis an NFAPnwithL(Pn) =SWn(L), and its space\ncomplexity is s(Pn) = log|Pn|. If we have a true-biased randomized SW-algorithm\nforLwe can turn it into a nondeterministic SW-algorithm by keeping only tho se\ntransitions with nonzero probabilities and making all states qinitial which have\na positive initial probability ι(q)>0. Therefore, it suffices to show the following\nstatement:\nProposition 46. LetL∈Reg\\/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}ht. Then, for infinitely many nevery\nnondeterministic SW-algorithm PnforLhasΩ(√n)many states.\nFor the proof of Proposition 46 we need the following lemma.\nLemma 47. LetL⊆a∗andn∈Nsuch that Lseparates{an}and{ak|k > n}.\nThen, every NFA for Lhas at least√nmany states.\nProof.The easy case is an∈Landak/∈Lfor allk > n. If an NFA for Lhas\nat most nstates then any successful run on anmust have a state repetition. By\npumping one can construct a successful run on akfor some k > n, which is a\ncontradiction.\nNow, assume an/∈Landak∈Lfor allk > n. The proof is essentially the same\nas for [63, Lemma 6], where the statement of the lemma is shown for L=a∗\\{an}.\nLet us give the proof for completeness. It is known that every una ry NFA has\nan equivalent NFA in so-called Chrobak normal form. A unary NFA in Ch robak\nnormal form consists of a simple path (called the initial path in the follo wing)\nstarting in the unique initial state. From the last state of the initial p ath, edges go\nto a collection of disjoint cycles. In [49] it is shown that an m-state unary NFA has\nan equivalent NFA in Chrobak normal form whose initial path consists ofm2−m\nstates. Now, assume that Lis accepted by an NFA with mstates and letAbe\nthe equivalent Chrobak normal form NFA, whose initial path consist s ofm2−m\nstates. If n/greaterorequalslantm2−mthen all states that are reached in Afrom the initial state\nviaanbelong to a cycle and every cycle contains such a state. Since an/∈L, all\nthese states are rejecting. Hence, an+x·d/∈Lfor allx/greaterorequalslant0, where dis the product\nof all cycle lengths. This contradicts the fact that ak∈Lfor allk > n. Hence, we\nmust have n < m2−mand therefore m >√n. /square\nProof of Proposition 46. LetL∈Reg\\/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}ht. By Lemma 22 and the results\nfrom Section 3.6 there are words x,y,z∈Σ∗such that|x|=|y|andLseparates\nxy∗zandy∗z. Note that we must have x/\\e}atio\\slash=y.\nFixm/greaterorequalslant0 and consider the window size n=|x|+m|y|+|z|. LetPn=\n(Q,Σ,I,∆,F) be a nondeterministic SW-algorithm for L, i.e. it is an NFA for\nSWn(L). Notice thatPnseparates{xymz}and{xykz|k > m}. We define an NFA\nAover the unary alphabet {a}as follows:\n•The state set ofAisQ.\n•The set of initial states of Ais{q∈Q|∃p∈I:px− →qinPn}.\n•The set of final states of Ais{p∈Q|∃q∈F:pz− →qinPn}.\n•The set of transitions of Ais{(p,a,q)|py− →qinPn}.\nIt recognizes the language L(A) ={ak|xykz∈SWn(L)}, and therefore L(A)\nseparates{am}and{ak|k > m}. By Lemma 47, Aand thusPnhas at least√m= Ω(√n) many states. /square\nProposition 46 implies F0\nL(n)/greaterorequalslant1/2logn−O(1) on infinitely many nfor all\nL∈Reg\\/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}ht. SinceReg\\/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}htis closed under complement, this32 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nimpliesF1\nL(n) =F0\nΣ∗\\L(n)/greaterorequalslant1/2logn−O(1) on infinitely many nfor allL∈\nReg\\/a\\}⌊ra⌋ketle{tST,Len/a\\}⌊ra⌋ketri}ht.\n4.7.Randomized variable-size model. In this section, we briefly look at ran-\ndomizedalgorithmsinthevariable-sizemodel. Firstwetransferthed efinitionsfrom\nSection 2.5 in a straightforward way. A randomized variable-size sliding window al-\ngorithmPforL⊆Σ∗is a randomized streaming algorithm for SW(L) (defined in\n(1)). Its space complexity isv(P,n) = log|M/lessorequalslantn|∈N∪{∞}whereM/lessorequalslantncontains all\nmemory states in Pwhich are reachable with nonzero probability in Pon inputs\nw∈Σ∗\n↓withmwl(w)/lessorequalslantn. Since the variable-size sliding window model subsumes\nthe fixed-size model, we have Fr\nL(n)/lessorequalslantv(P,n) for every randomized variable-size\nsliding window algorithm PforL.\nAgain we raise the question if randomness can improve the space com plexity\nin the variable-size model. We claim that, in contrast to the fixed-size model,\nrandomness does allow more space efficient algorithms in the variable- size setting.\nClearly, all upper bounds for the deterministic variable-size setting transfer to the\nrandomized variable-size setting, i.e. languages in /a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}hthaveO(logn) space\ncomplexity, and empty and universal languages have O(1) space complexity. For\nevery regular language Lwhich is not contained in /a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}htwe proved a linear\nlower bound on Fr\nL(n) (Proposition 39), which is also a lower bound on the space\ncomplexity of any randomized variable-size sliding window algorithm for L. It\nremains to look at languages ∅/subsetnoteqlL/subsetnoteqlΣ∗, for which we have proved a logarithmic\nlower bound in the deterministic setting (Lemma 10).\nLemma 48. IfPis a randomized variable-size SW-algorithm for a language ∅/subsetnoteql\nL/subsetnoteqlΣ∗thenv(P,n) = Ω(log n).\nProof.Let∅/subsetnoteqlL/subsetnoteqlΣ∗be a language. There is a length-minimal nonempty word\na1···ak∈Σ+such that|{ε,a1···ak}∩L|= 1. By minimality we also have\n|{a1···ak,a2···ak}∩L|= 1. LetPbe a randomized variable-size SW-algorithm\nforL. By Lemma 29 we can assume that the error probability of Pis at most 1/6,\nwhich increases its space complexity v(P,n) by a constant factor.\nFor every n∈Nwe construct a protocol for GT nwith costO(v(P,n)). With\nTheorem 38 this implies that v(P,n) = Ω(log n). Let 1 /lessorequalslanti/lessorequalslantnbe the input of\nAlice and 1 /lessorequalslantj/lessorequalslantnbe the input of Bob. Alice starts two instances of P(using\nindependent random bits) and reads ai\n1into both of them. She sends the memory\nstates to Bob using O(v(P,i))/lessorequalslantO(v(P,n)) bits. Bob then continues from both\nstates, and reads ↓ja2···akinto the first instance and ↓j+1a1a2···akinto the\nsecond instance. Let y1,y2∈{0,1}be the outputs of the two instances of P. With\nhigh probability, namely (1 −1/6)2/greaterorequalslant2/3, both answers are correct, i.e.\ny1= 1⇐⇒wnd(ai\n1↓ja2···ak)∈L\nand\ny2= 1⇐⇒wnd(ai\n1↓j+1a1a2···ak)∈L.\nBob returns true, i.e. he claims i > j, if and only if y1=y2.\nLet us prove the correctness. If i > jthen\nwnd(ai\n1↓ja2···ak) =ai−j\n1a2···ak=wnd(ai\n1↓j+1a1a2···ak)\nand hence Bob returns true with probability at least 2/3. If i/lessorequalslantjthen\nwnd(ai\n1↓ja2···ak) =a2···ak\nand\nwnd(ai\n1↓j+1a1a2···ak) =a1a2···ak.REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 33\nBy assumption, exactly one of the words a1···ak,a2···akbelongs to L, and there-\nfore Bob returns false with probability at least 2/3. /square\nThe lowerbound abovealsoholds forvariable-sizeSW-algorithmswith one-sided\nerror since they are more restricted than algorithms with two-side d error.\n5.Property testing in the sliding window model\nAll settings discussed so far for testing membership of regular lang uages in the\nsliding window model do not allow to use sublinear space for an arbitrar y regular\nlanguage. To be more specific, for any language L∈Reg\\/a\\}⌊ra⌋ketle{tLI,Len/a\\}⌊ra⌋ketri}htit requires\nlinear space to test membership even for randomized sliding window alg orithms\nwith two-sided error. In order to achieve space-efficient sliding wind ow algorithms\nfor all regular languages, we have to allow for a kind of approximative answers. We\nformalize this in the context of the property testing framework. M ore precisely,\nwe introduce in this section sliding window (property) testers , which must accept if\nthe active window belongs to a language Land reject if it is has large Hamming\ndistance from L. We consider deterministic sliding window property testers and\nrandomized sliding window property testers.\nWhile at first sight the only connection between property testers a nd sliding\nwindow property testers is that we must accept the input if it satisfi es a property\nPand reject if it is far from satisfying P, there is, in fact, a deeper link. In\nparticular, the property tester for regular languages due to Alon et al. [2] combined\nwith an optimal sampling algorithm for sliding windows [18] immediately yie lds\nO(logn)-space, two-sided error sliding window property testers with Ham ming gap\nγ(n) =ǫnfor all regular languages. We will improve on this observation.\n5.1.Sliding window testers. TheHamming distance between two words u=\na1···anandv=b1···bnof equal length is the number of positions where uand\nvdiffer, i.e. dist( u,v) =|{i|ai/\\e}atio\\slash=bi}|. If|u|/\\e}atio\\slash=|v|we set dist( u,v) =∞. The\ndistance of a word uto a language Lis defined as\ndist(u,L) = inf{dist(u,v)|v∈L}∈N∪{∞}.\nAdditionally, we define the prefix distance between words u=a1···anandv=\nb1···bnby pdist( u,v) = min{i∈{0,...,n}|ai+1···an=bi+1···bn}. Clearly, we\nhave dist( u,v)/lessorequalslantpdist(u,v). The algorithms presented in this section satisfy the\nstronger property that windows whose prefix distance to the lang uageLis large\nare rejected by the algorithm.\nIn this section, γis always a function γ:N→R/greaterorequalslant0. Adeterministic sliding\nwindow (property) tester with Hamming gap γ(n) for a language L⊆Σ∗and\nwindow size nis a deterministic streaming algorithm Pnover the alphabet Σ with\nthe following properties:\n•Iflastn(w)∈L, thenPnacceptsw.\n•If dist(lastn(w),L)> γ(n), thenPnrejectsw.\nArandomized sliding window tester with Hamming gap γ(n) for a language L⊆Σ∗\nand window size nis a randomized streaming algorithm Pnover the alphabet Σ\nwith the following properties. It has two-sided error if for all w∈Σ∗we have:\n•Iflastn(w)∈L, then Pr[Pnacceptsw]/greaterorequalslant2/3.\n•If dist(lastn(w),L)> γ(n), then Pr[Pnrejectsw]/greaterorequalslant2/3.\nIt istrue-biased if for all w∈Σ∗we have:\n•Iflastn(w)∈L, then Pr[Pnacceptsw]/greaterorequalslant2/3.\n•If dist(lastn(w),L)> γ(n), then Pr[Pnrejectsw] = 1.\nIt isfalse-biased if for all w∈Σ∗we have:34 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\n•Iflastn(w)∈L, then Pr[Pnacceptsw] = 1.\n•If dist(lastn(w),L)> γ(n), then Pr[Pnrejectsw]/greaterorequalslant2/3.\nTrue-biased and false-biased algorithms are algorithms with one-sided error . Again,\nthe success probability 2 /3 is an arbitrary choice in light of Lemma 29. The case\nof Hamming gap γ(n) = 0 corresponds to exact membership testing to Lfrom the\nprevious sections.\nLet us present the main results of this section.\nTheorem 49. For every regular language Lthere exists a deterministic sliding\nwindow tester with constant Hamming gap which uses O(logn)space.\nTheorem 50. For every regular language Land every ǫ >0there exists a ran-\ndomized sliding window tester with two-sided error and Hamm ing gapǫnthat uses\nspaceO(1/ǫ).\nThe log-space bound in Theorem 49 cannot be improved whenever Lis anon-\ntrivialregular language, even for randomized true-biased algorithms. A la nguage\nL⊆Σ∗isγ(n)-trivialif for all n∈NwithL∩Σn/\\e}atio\\slash=∅and allw∈Σnwe have\ndist(w,L)/lessorequalslantγ(n). IfLisO(1)-trivial we say that Listrivial.\nTheorem 51. For every nontrivial regular language Lthere exist ǫ >0and in-\nfinitely many window sizes n∈Nfor which every true-biased (resp., false-biased)\nsliding window tester with Hamming gap ǫnuses space at least logn−O(1)(resp.\nloglogn−O(1)).\nExamples of trivial languages include all suffix testable and all prefix t estable\nlanguages, and also the set of all words over {a,b}which contain an even number of\na’s. Note that Alon et al. [2] call a language Ltrivial if Liso(n)-trivial according\nto our definition, i.e. γ(n)-trivial for some function γ(n) =o(n). In fact, we will\nprove that both definitions coincide for regular languages (Corollar y 68).\nWealsoidentify asmallclassofregularlanguages,whichhavefalse-b iasedsliding\nwindow testers with space complexity O(loglogn):\nTheorem 52. IfLis a finite union of trivial regular languages and suffix-free\nregular languages, then there exists a randomized false-bi ased sliding window tester\nforLwith constant Hamming gap which uses O(loglogn)space.\nWe finally complement Theorem 52 with a matching lower bound:\nTheorem 53. LetLbe a regular language that is not a finite union of regular triv ial\nlanguages and regular suffix-free languages. Then, there exi stǫ >0and infinitely\nmany window sizes n∈Nfor which every false-biased sliding window tester with\nHamming gap ǫnuses at least logn−O(1)space.\n5.2.Deterministic sliding window testers. In this section, we prove Theo-\nrem 49: every regular language has a deterministic sliding window test er with\nconstant Hamming gap which uses O(logn) space. It is based on the path sum-\nmary algorithm from Section 3.3. In the following, we fix a regular langu ageL\nand an rDFAB= (Q,Σ,F,δ,q 0) forL. By Lemma 28 we can assume that every\nnontransient SCC of Bhas the same period g/greaterorequalslant1.\nFor a state q∈Qwe define Acc( q) ={n∈N|∃w∈Σn:w·q∈F}. The\nfollowing lemma is the main tool to prove correctness of our sliding wind ow testers.\nIt states that if a word of length nis accepted from state pandρis any internal run\nfrompof length at most n, then, up to a bounded length prefix, ρcan be extended\nto an accepting run of length n. Formally, a run π t-simulates a runρif one can\nfactorize ρ=ρ1ρ2andπ=π′ρ2where|ρ1|/lessorequalslantt.REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 35\nLemma 54. There exists a number t∈N(which only depends on B) such that\nthe following holds: If ρis an internal run starting from pof length at most n\nandn∈Acc(p), then there exists an accepting run πfrompof length nwhich\nt-simulates ρ.\nBased on this lemma we can prove Theorem 49. Afterwards we prove Lemma 54.\nProof of Theorem 49. Lettbe the constant from Lemma 54. We present a deter-\nministic sliding window tester with constant Hamming gap twhich usesO(logn)\nspace. Let n∈Nbe the window size. By Lemma 15 we can maintain the set of\nall path summaries PS B(w) ={ps(πw,q)|q∈Q}for the active window w∈Σn,\nusingO(logn) bits. In fact, the path summary algorithm works for variable-size\nwindows but we do not need this here.\nIt remains to define a proper acceptance condition. Consider the S CC-factoriza-\ntion ofπw,q0, say\nπw,q0=πmτm−1πm−1···τ1π1\nand its path summary ( ℓm,qm)···(ℓ1,q1). The algorithm accepts if and only if\nthe path summary is accepting, i.e. ℓm=|πm|∈Acc(qm). Ifw∈Lthen clearly\n|πm|∈Acc(qm). On the other hand, if |πm|∈Acc(qm) then the internal run πm\ncan bet-simulated by an accepting run π′\nmof equal length by Lemma 54. The run\nπ′\nmτm−1πm−1···τ1π1is accepting and witnesses that pdist( w,L)/lessorequalslantt. /square\nTo prove Lemma 54 we need to analyze the sets Acc( q) first. For a∈Nand\nX⊆Nwe use the standard notation X+a={a+x|x∈X}. A setX⊆Nis\neventually d-periodic, whered/greaterorequalslant1 is an integer, if there exists a threshold t∈N\nsuch that for all x/greaterorequalslanttwe have x∈Xif and only if x+d∈X. IfXis eventually\nd-periodic for some d/greaterorequalslant1, thenXiseventually periodic .\nLemma 55. For every q∈Qthe setAcc(q)is eventually g-periodic.\nProof.It suffices to show that for all 0 /lessorequalslantr/lessorequalslantg−1 the set Sr={i∈N|r+i·g∈\nAcc(q)}is either finite or co-finite. Consider a remainder 0 /lessorequalslantr/lessorequalslantg−1 where Sr\nis infinite. We need to show that Sris indeed co-finite. Let i∈Srwithi/greaterorequalslant|Q|,\ni.e. there exists an accepting run πfromqof length r+i·g. Sinceπhas length\nat least|Q|, it must traverse a state qin a nontransient SCC C. Choose j0such\nthatj0·g/greaterorequalslantm(C) wherem(C) is the reachability constant from Lemma 27. By\nLemma 27 for all j/greaterorequalslantj0there exists a cycle from qtoqof length j·g. Therefore,\nwe can extend πto a longer accepting run by j·gsymbols for any j/greaterorequalslantj0. This\nproves that x∈Srfor every x/greaterorequalslanti+j0and that Sris co-finite. /square\nTwo sets X,Y⊆Nareequal up to a threshold t∈N, in symbol X=tY, if for\nallx/greaterorequalslantt:x∈Xiffx∈Y. Two sets X,Y⊆Narealmost equal if they are equal\nup to some threshold t∈N.\nLemma 56. A setX⊆Nis eventually d-periodic iff XandX+dare almost\nequal.\nProof.Lett∈Nbe such that for all x/greaterorequalslanttwe have x∈Xif and only if x+d∈X.\nThen,XandX+dare equal up to threshold t+d. Conversely, if X=tX+d,\nthen for all x/greaterorequalslanttwe have x+d∈Xif and only if x+d∈X+d, which is true if\nand only if x∈X. /square\nIf the graph G= (V,E) is strongly connected with E/\\e}atio\\slash=∅and finite period g,\nandV0,...,V g−1satisfy the properties from Lemma 27, then we define the shift\nfromu∈Vitov∈Vjby\n(11) shift( u,v) = (j−i) modg∈{0,...,g−1}.36 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nNotice that shift( u,v) could be defined without referring to the partition/uniontextg−1\ni=0Vi\nsince the length of any path from utovis congruent to shift( u,v) modulo gby\nLemma 27. Also, note that shift( u,v)+shift( v,u)≡0 (modg).\nLemma 57. LetCbe a nontransient SCC in B,p,q∈Cands= shift(p,q). Then,\nAcc(p)andAcc(q)+sare almost equal.\nProof.Letk∈Nsuchthat k·g/greaterorequalslantm(C)wherem(C)istheconstantfromLemma27.\nBy Lemma 27 there exists a run from ptoqof length s+k·g, and a run from q\ntopof length ( k+1)·g−s(the latter number is congruent to shift( q,p) modulo\ng). By prolonging accepting runs we obtain\nAcc(q)+s+k·g⊆Acc(p) and Acc( p)+(k+1)·g−s⊆Acc(q).\nAddings+k·gto both sides of the last inclusion yields\nAcc(p)+(2k+1)·g⊆Acc(q)+s+k·g⊆Acc(p).\nBy Lemma 55 and Lemma 56 the three sets above are almost equal. Als o, Acc(q)+\ns+k·gis almost equal to Acc( q)+sby Lemma 55 and Lemma 56. Since almost\nequality is a transitive relation, this proves the statement. /square\nCorollary 58. There exists a threshold t∈Nsuch that\n(i)Acc(q) =tAcc(q)+gfor allq∈Q, and\n(ii)Acc(p) =tAcc(q)+shift( p,q)for all nontransient SCCs Cand allp,q∈C.\nLet us fix the threshold tfrom Corollary 58 in the following. We can now prove\nLemma 54.\nProof of Lemma 54. Letρbe an internal run starting from pwith|ρ|/lessorequalslantn∈Acc(p).\nWe have to find an accepting run πfrompof length nand factorizations ρ=ρ1ρ2\nandπ=π′ρ2with|ρ1|/lessorequalslantt.\nIf|ρ|/lessorequalslantt, then we canchoosefor πanyacceptingrun from poflength n∈Acc(p).\nOtherwise, if|ρ|> t, then the SCC Ccontaining pis nontransient (since ρis inter-\nnal) and we can factorize ρ=ρ1ρ2so that|ρ1|=twhereρ2leads from pto some\nstateqof the same SCC. Set s:= shift(q,p), which satisfies s+|ρ2|≡0 (modg)\nby the properties in Lemma 27. Since Acc( q) =tAcc(p) +sby Corollary 58(ii),\nn > tandn∈Acc(p), we have n+s∈Acc(q). Finally, since n+s≡n−|ρ2|\n(modg) andn−|ρ2|=n−|ρ|+t/greaterorequalslantt, we know n−|ρ2|∈Acc(q) by Corollary 58(i).\nThis yields an accepting run π′fromqof length n−|ρ2|. Then,ρist-simulated by\nπ=π′ρ2. /square\n5.3.Sliding window testers with two-sided error. Next, we will construct\nfor every regular language a randomized sliding window tester with tw o-sided error\nand Hamming gap ǫnthat usesO(1/ǫ) bits (Theorem 50). We still assume that the\nregularlanguage Lis recognizedby anrDFA B= (Q,Σ,F,δ,q 0) whosenontransient\nSCCs have uniform period g/greaterorequalslant1. Furthermore, we again use the constant tfrom\nCorollary 58.\nFix a parameter 0 < ǫ <1 and a window length n∈N. Ifǫn/4< t+ 1, i.e.\nn <4(t+1)/ǫ, then we use a trivial sliding window tester that stores the window\nexplicitly withO(n) =O(1/ǫ) bits. Now, assume t+ 1/lessorequalslantǫn/4 and define the\nparameters h=n−tandℓ= (1−ǫ)n+t+1, which satisfy\nℓ\nh=(1−ǫ)n+t+1\nn−t/lessorequalslant(1−ǫ)n+ǫ\n4n\nn−ǫ\n4n\n=1−3\n4ǫ\n1−1\n4ǫ=1−1\n4ǫ\n1−1\n4ǫ−1\n2ǫ\n1−1\n4ǫ/lessorequalslant1−1\n2ǫ.(12)REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 37\nqi−1 qi qi+1 q1 qmπi−1 τi−1 πi τi\nciandri(modg)\nFigure 4. A compact summary of a run π.\nLetZbe the (h,ℓ)-counter with error probability 1 /(3|Q|) from Proposition 34,\nwhich usesO(loglog|Q|+ log(1/ǫ)) =O(log(1/ǫ)) space by (12) (as usual, we\nconsider|Q|as a constant). Let Cbe its state set and c0the initial state of Z. The\ncounterZis used to define so-called compact summaries of runs.\nAcompact summary κ= (qm,rm,cm)···(q2,r2,c2)(q1,r1,c1) is a sequence of\ntriples, where each triple ( qi,ri,ci) consists of a state qi∈Q, a remainder 0 /lessorequalslantri/lessorequalslant\ng−1, and a counter state ci∈C. The state c1is always the initial state c0(and\nhence low) and r1= 0. We say that κrepresents a runπif the SCC-factorization\nofπhas the form πmτm−1πm−1···τ1π1, and the following properties hold for all\n1/lessorequalslanti/lessorequalslantm:\n(C1)πistarts in qi;\n(C2)ri=|τi−1πi−1···τ1π1|modg;\n(C3) if|τi−1πi−1···τ1π1|/lessorequalslant(1−ǫ)n+t+1 then ciis a low state;\n(C4) if|τi−1πi−1···τ1π1|/greaterorequalslantn−tthenciis a high state.\nNote that κdoes not restrict πmexcept that the latter must be an internal run\nstarting in qm.\nThe idea of a compact summary is visualized in Figure 4. If m >|Q|then the\nabove compact summary cannot represent a run. Therefore, we can assume that\nm/lessorequalslant|Q|. Foreverytriple( qi,ri,ci), theentries qiandrionlydependontherDFA B,\nand hence can be stored with O(1) bits. Every state ciof the probabilistic counter\nneedsO(log(1/ǫ)) bits. Hence, a compact summary can be stored in O(log(1/ǫ))\nbits. In contrast to the deterministic sliding window tester, we maint ain a set of\ncompact summaries which represent all runs of Bon thecomplete stream read so\nfar (not only on the active window) with high probability.\nProposition 59. For a given input stream w∈Σ∗, we can maintain a set of com-\npact summaries Scontaining for each q∈Qa unique compact summary κw(q)∈S\nstarting in qsuch that for all q∈Qwith probability at least 2/3the runπw,qis\nrepresented by κw(q).\nProof.We show how to maintain for the input w∈Σ∗a set of random compact\nsummaries S={κw(q)|q∈Q}, defined as follows: If πmτm−1πm−1···τ1π1is the\nSCC-factorization of πw,q,πistarts in qi, andℓi=|τi−1πi−1···τ1π1|for 1/lessorequalslanti/lessorequalslantm,\nthen\n(13) κw(q) = (qm,ℓmmodg,Z(ℓm))···(q1,ℓ1modg,Z(ℓ1)).\nNote that the first triple ( q1,ℓ1modg,Z(ℓ1)) is (q,0,c0).\nForw=ε, we initialize S={κε(q)|q∈Q}whereκε(q) = (q,0,c0) forq∈Q.\nIfa∈Σ is the next input symbol of the stream, then Sis updated to the new\nsetS′of compact summaries by iterating over all transitions qa← −pinBand\nprolonging the compact summary starting in qby that transition. To prolong a\ncompact summary κw(q) in the form as in (13) by a transition qa← −pwe proceed\nsimilar to Algorithm 1:\nIfpandq=q1are not in the same SCC then κwa(p) is\n(qm,(ℓm+1) mod g,Z(ℓm+1))···(q1,(ℓ1+1) mod g,Z(ℓ1+1))(p,0,c0),38 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\ni.e. we make a probabilistic increment on each counter state, increme nt all remain-\nders mod g, and append a new triple ( p,0,c0).\nIfpandq=q1belong to the same SCC, then κwa(p) is\n(qm,(ℓm+1) mod g,Z(ℓm+1))···(q2,(ℓ2+1) mod g,Z(ℓ2+1))(p,ℓ1modg,Z(ℓ1)),\ni.e. in all but the last triple we make a probabilistic increment on the coun ter state\nand the remainder mod g, and replace q1byp.\nFinally we claim that for every q∈Q, the compact summary κw(q) from (13)\ncomputed by the algorithm represents πw,qwith probability 2 /3. Properties (C1)\nand (C2) are satisfied by construction. Furthermore, since the le ngth ofκw(q) is\nboundedby|Q|andeachinstanceof Zhaserrorprobability1 /(3|Q|)theprobability\nthat property (C3) or (C4) is violated for some iis at most 1 /3 by the union\nbound. /square\nIt remains to define an acceptance condition on compact summaries . For every\nq∈Qwe define\nAccmod(q) ={ℓmodg|ℓ∈Acc(q) andℓ/greaterorequalslantt}.\nLetκ= (qm,rm,cm)···(q1,r1,c1) beacompactsummary. Since c1isthelowinitial\nstate of the probabilistic counter, there exists a maximal index i∈{1,...,m}such\nthatciis low. We say that κisaccepting ifn−ri(modg)∈Accmod(qi).\nProposition 60. Letw∈Σ∗with|w|/greaterorequalslantnand letκbe a compact summary which\nrepresents πw,q0.\n(i) Iflastn(w)∈L, thenκis accepting.\n(ii) Ifκis accepting, then pdist(lastn(w),L)/lessorequalslantǫn.\nProof.Consider the SCC-factorization of π=πw,q0=πmτm−1···τ1π1and a com-\npact summary κ= (qm,rm,cm)···(q1,r1,c1) representing π. Thus, q1=q0and\nc1=c0. Consider the maximal index 1 /lessorequalslanti/lessorequalslantmwhereciis low, which means\nthat|τi−1πi−1···τ1π1|< n−tby (C4). The run of Bonlastn(w) has the form\nπ′\nkτk−1πk−1···τ1π1for some suffix π′\nkofπkandk/greaterorequalslanti. We have|π′\nkτk−1···πi|=\nn−|τi−1πi−1···τ1π1|> t. By (C2) we know that\nri=|τi−1πi−1···τ1π1|modg=n−|π′\nkτk−1···πi|modg.\nFor (i) assume that lastn(w)∈L. Thus, π′\nkτk−1πk−1···τ1π1is an accepting run\nstarting in q0. By (C1) the run π′\nkτk−1···πistarts in qi. Hence, π′\nkτk−1···πiis an\naccepting run from qiof length at least t. By definition of Acc mod(qi) we have\n|π′\nkτk−1···πi|modg=n−rimodg∈Accmod(qi),\nand therefore κis accepting.\nFor (ii) assume that κis accepting, i.e.\n(n−ri) modg=|π′\nkτk−1···πi|modg∈Accmod(qi).\nRecall that|π′\nkτk−1···πi|> t. By definition of Acc mod(qi) there exists an accepting\nrun from qiwhose length is congruent to |π′\nkτk−1···πi|modgand at least t. By\npoint (i) from Corollary 58 we derive that |π′\nkτk−1···πi|∈Acc(qi). We claim that\n|πiτi−1πi−1···τ1π1|/greaterorequalslant(1−ǫ)n+tby a case distinction. If i=m, then clearly\n|πiτi−1πi−1···τ1π1|=|w|/greaterorequalslantn/greaterorequalslant(1−ǫ)n+t; the latter inequality follows from our\nassumption t+ 1/lessorequalslantǫn/4. Ifi < m, thenci+1is high by maximality of i, which\nimplies|τiπi···τ1π1|>(1−ǫ)n+t+1 by (C3). Since τihas length one, we have\n|πiτi−1πi−1···τ1π1|>(1−ǫ)n+t.\nSince|π′\nkτk−1···πi|∈Acc(qi), we can apply Lemma 54 and obtain an accepting\nrunρof length|π′\nkτk−1···πi|∈Acc(qi) starting in qiwhicht-simulates the internalREGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 39\nrunπi. The prefix distance between ρandπ′\nkτk−1···πi(which we define as the\nprefix distance between the words read along the two runs) is at mo st\n|π′\nkτk−1···πi+1τi|+t=n−|πiτi−1πi−1···τ1π1|+t/lessorequalslantn−(1−ǫ)n=ǫn.\nTherefore, the prefix distance from the accepting run ρτi−1πi−1···τ1π1to the run\nπ′\nkτk−1πk−1···τ1π1is also at most ǫn. This implies pdist( lastn(w),L)/lessorequalslantǫn./square\nWe are now ready to prove Theorem 50.\nProof of Theorem 50. As mentioned at the beginning of this section we use a trivial\nsliding window algorithm whenever ǫn/4< t+1, usingO(1/ǫ) bits. If ǫn/4/greaterorequalslantt+1\nthen we use the algorithm from Proposition 59, which is initialized by rea ding the\ninitial window /squaren. It maintains a compact summary which represents πw,q0with\nprobability 2/3 for the read stream prefix w. The algorithm accepts if that compact\nsummary is accepting. From Proposition 60 (note that ǫn/greaterorequalslantt) we get:\n•Iflastn(w)∈L, then the algorithm accepts with probability at least 2 /3.\n•If pdist(lastn(w),L)> ǫn, then the algorithm rejects with probability at\nleast 2/3.\nThis concludes the proof of Theorem 50. /square\n5.4.Sliding window testers with one-sided error. In the following, weturn to\nsliding window testers with one-sided error and prove Theorem 52, i.e . we present\na false-biased sliding window tester for unions of regular suffix-free languages and\nregular trivial languages with constant Hamming gap using O(loglogn) space. By\nthe following lemma, it suffices to consider the cases when Lis a trivial regular\nlanguage or a suffix-free regular language.\nLemma 61. LetP1andP2be randomized false-biased sliding window testers for\nL1andL2, respectively, for window size nwith Hamming gap γ(n). Then, there\nexists a randomized false-biased sliding window tester for L1∪L2for window size\nnwith Hamming gap γ(n)using spaceO(s(P1)+s(P2)).\nProof.First we reduce the error probability of Pi,i= 1,2, from 1 /3 to 1/9 by run-\nning 2 independent and parallel copies of Piand reject iff one of the copies rejects.\nThen, we run both algorithms in parallel and accept if and only if one of them ac-\ncepts. Ifthewindowbelongsto L1∪L2then eitherP1orP2acceptswithprobability\n1. If the window wsatisfies dist( w,L1∪L2) = min(dist( w,L1),dist(w,L2))> γ(n)\nthen dist( w,Li)> γ(n) for both i∈{1,2}. Hence, both algorithms falsely ac-\ncept with probability at most 1 /9 and the combined algorithm falsely accepts with\nprobability at most 1 /9+1/9/lessorequalslant1/3. /square\nThe case of a trivial regular language is covered by the following resu lt:\nTheorem 62. LetLbe a language and γ(n)be a function. The following state-\nments are equivalent:\n•Lis(γ(n)+c)-trivial for some number c∈N.\n•There is a deterministic sliding window tester with Hamming gapγ(n)+c′\nforLwhich uses constant space for some number c′∈N.\nProof.Assume first that Lis (γ(n) +c)-trivial. Let n∈Nbe a window size. If\nL∩Σn=∅, then the algorithm always rejects, which is obviously correct since any\nactive window of length nhas infinite Hamming distance to L. On the other hand,\nifL∩Σn/\\e}atio\\slash=∅then the Hamming distance between an arbitrary active window\nof length nandLis at most γ(n) +c. Hence, the algorithm that always accepts\nachieves a Hamming gap of γ(n)+c.40 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nWe now show the converse statement. For each window size n∈NletPnbe a\ndeterministic sliding window tester for Lwith Hamming gap γ(n) +c′such that\nthe number of states of Pnis constant. Assume that Pnhas at most sstates for\neveryn. LetN⊆Nbe the set of all nsuch that L∩Σn/\\e}atio\\slash=∅. Note that every\nPnwithn∈Naccepts a nonempty language. The number of deterministic sliding\nwindow testers (DFAs) with at most sstates over the input alphabet Σ is bounded\nby a fixed constant d(up to isomorphism). Hence, at most ddifferent DFAs can\nappear in the list ( Pn)n∈N. We therefore can choose numbers n1< n2<···< ne\nfromNwithe/lessorequalslantdsuch that for every n∈Nthere exists a unique ni/lessorequalslantnwith\nPn=Pni(here and in the following we do not distinguish between isomorphic\nDFAs). Let us choose for every 1 /lessorequalslanti/lessorequalslantesome word ui∈Lof length ni. Now,\ntake any n∈Nand assume that Pn=Pniwhereni/lessorequalslantn. Consider any word\nu∈Σ∗ui. Sincelastni(u) =ui∈L,Pnihas to accept u. Hence,Pnaccepts all\nwords from Σ∗ui. In particular, for every word xof length n−ni,Pnacceptsxui.\nThis implies that dist( xui,L)/lessorequalslantγ(n)+c′for allx∈Σn−ni. Recall that this holds\nfor alln∈Nand that Nis the set of all lengths realized by L. Hence, if we define\nc′′:= max{n1,...,n e}, then every word wof length n∈Nhas Hamming distance\nat most γ(n) +c′+c′′from a word in L. Therefore, Lis (γ(n) +c)-trivial with\nc=c′+c′′. /square\nLet us now turn to the caseofa suffix-free regularlanguage L. We againconsider\nan rDFAB= (Q,Σ,F,δ,q 0) forLwhose nontransient SCCs have uniform period\ng/greaterorequalslant1. Since Lis suffix-free,Bhas the property that no final state can be reached\nfrom a final state by a nonempty run.\nWe adapt the definition of a path description from Section 3.7. In the following,\na path description is a sequence\n(14) P= (qk,ak,pk−1),Ck−1,...,(q2,a2,p1),C1,(q1,a1,p0),C0,q0.\nwhereCk−1,...,C 0is a chain (from right to left) in the SCC-ordering of B,pi,qi∈\nCi,qi+1ai+1←−−−piis a transition in Bfor all 0 /lessorequalslanti/lessorequalslantk−1, andqk∈F. Each\npath description defines a partial rDFABP= (QP,Σ,{qk},δP,q0) by restrictingB\nto the state set QP=/uniontextk−1\ni=0Ci∪{qk}, restricting the transitions of Bto internal\ntransitions from the SCCs Ciand the transitions qi+1ai+1←−−−pi, and declaring qkto\nbe the onlyfinal state. This rDFAis partialsince foreverystate pand everysymbol\na∈Σ there exists at most one transition qa← −pinBP. Since the number of path\ndescriptions Pis finite and L(B) =/uniontext\nPL(BP), we can fix a single path description\nPand provide a sliding window tester for L(BP) (we again use Lemma 61 here).\nFromnowon, wefixapathdescription Pasin(14). TheacceptancesetsAcc P(q)\nare defined with respect to the restricted automaton BP. If allCiare transient,\nthenL(BP) is a singleton and we can use a trivial sliding window tester with space\ncomplexityO(1). Now, assume the contrary and let 0 /lessorequalslante/lessorequalslantk−1 be maximal such\nthatCeis nontransient.\nLemma 63. There exist numbers r0,...,r k−1,s0,...,s e∈Nsuch that the follow-\ning holds:\n(i) For all e+1/lessorequalslanti/lessorequalslantk, the set AccP(qi)is a singleton.\n(ii) Every run from qitoqi+1has length ri(modg).\n(iii) For all 0/lessorequalslanti/lessorequalslante,AccP(qi) =si/summationtextk−1\nj=irj+gN.\nProof.Point (i) follows immediately from the definition of transient SCCs. Let us\nnow show (ii) and (iii). Let 0 /lessorequalslanti/lessorequalslantk−1 and let Nibe the set of lengths of runs\nof the form qi+1ai+1←−−−piw← −qiinBP. IfCiis transient, then Ni={1}. Otherwise,\nby Lemma 27 there exist a number ri∈Nand a cofinite set Di⊆Nsuch thatREGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 41\nNi=ri+gDi. We can summarize both cases by saying that there exist a number\nri∈Nand a set Di⊆Nwhich is eithercofinite or Di={0}such that Ni=ri+gDi.\nThis implies point (ii). Moreover, the acceptance sets in BPsatisfy\nAccP(qi) =k−1/summationdisplay\nj=iNj=k−1/summationdisplay\nj=i(rj+gDj) =k−1/summationdisplay\nj=irj+gk−1/summationdisplay\nj=iDj.\nFor all 0 /lessorequalslanti/lessorequalslantewe get Acc P(qi) =si/summationtextk−1\nj=irj+gNfor some threshold si∈N\n(note that a nonempty sum of cofinite subsets of Nis again cofinite). /square\nLet us fix the numbers riandsifrom Lemma 63. Let pbe a random prime with\nΘ(loglog n) bits. We can obtain Pr[ ℓ≡n(modp)]/lessorequalslant1/3 for all 0 /lessorequalslantℓ < n(see\nalso the proof of Lemma 37). Define the threshold\ns= max{k,k−1/summationdisplay\nj=0rj,s0,...,s e}\nand for a word w∈Σ∗define the function ℓw:Q→N∪{∞}where\nℓw(q) = inf{ℓ∈N|δP(lastℓ(w),q) =qk}\n(we set inf∅=∞). We now define an acceptance condition on ℓw(q). Ifn /∈\nAccP(q0), we always reject. Otherwise, we accept wiffℓw(q0)≡n(modp).\nLemma 64. Letn∈AccP(q0)be a window size with n/greaterorequalslants+|QP|andw∈Σ∗\nwith|w|/greaterorequalslantn. There exists a constant c >0such that:\n(i) iflastn(w)∈L(BP), thenwis accepted with probability 1;\n(ii) ifpdist(lastn(w),L(BP))> c, thenwis rejected with probability at least 2/3.\nProof.Assume first that lastn(w)∈L(BP). SinceL(BP)⊆Lis suffix-free, we have\nℓw(q0) =n(modp) andwis accepted with probability 1.\nConsider now the case when lastn(w)/∈L(BP). By definition, in this case\nℓw(q0)/\\e}atio\\slash=n. In other words, only two cases are possible: either ℓw(q0)< n, or\nℓw(q0)> n. Ifℓw(q0)< n, then by the choice of pwe have ℓw(q0)/\\e}atio\\slash≡n(modp)\nwith probability at least 2 /3. Hence, wis rejected with probability at least 2 /3.\nWe finally consider the case ℓw(q0)> n. We will show that in this case the prefix\ndistance between lastn(w) andL(BP) is bounded by a constant c, which means that\nwe do not have to care about the output of the algorithm. Let πbe the run ofBP\nonlastn(w) starting from the initial state, and let π=πmτm−1πm−1···τ0π0be its\nSCC-factorization. We have |π|=n. Sinceℓw(q0)> n, the run πcan be strictly\nextended to a run to qkand hence we must have m < k. For all 0 /lessorequalslanti/lessorequalslantm, the run\nπiis an internal run in the SCC Cifromqitopi. For all 0 /lessorequalslanti/lessorequalslantm−1 we have\nτi=qi+1ai+1piand|τiπi|≡ri(modg) by point (ii) from Lemma 63. We claim\nthat there exists an index 0 /lessorequalslanti0/lessorequalslantmsuch that the following three properties hold:\n(1)qi0is nontransient,\n(2)|πmτm−1πm−1···τi0πi0|/greaterorequalslants,\n(3)|πmτm−1πm−1···τi0+1πi0+1|/lessorequalslants+m\n(note that|πmτm−1πm−1···τi0+1πi0+1|= 0 is possible).\nIndeed, let 0 /lessorequalslanti/lessorequalslantmbe the smallest integer such that qiis nontransient (recall\nthatn/greaterorequalslant|QP|and hence πmust traverse a nontransient SCC). Then, the run\nτi−1πi−1···τ0π0only passes transient states except for its last state qiand hence\nits length is bounded by |QP|. Therefore, we have\n|πmτm−1πm−1···τiπi|=n−|τi−1πi−1···τ0π0|/greaterorequalslantn−|QP|/greaterorequalslants.42 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nHence, there exists isatisfying properties (1) and (2) (with i0replaced by i). Let\n0/lessorequalslanti0/lessorequalslantmbe the largest integer satisfying properties (1) and (2). We show t hat\nproperty (3) holds for i0.\nIftherun πmτm−1πm−1···τi0+1πi0+1onlypassestransientstates, then itslength\nis bounded by m−i0/lessorequalslants+m, and we are done. Otherwise, let i0+1/lessorequalslantj/lessorequalslantmbe\nthe smallest integer such that qjis nontransient. The run τj−1πj−1···τi0+1πi0+1\nonly passes transient states except for its last state qjand therefore it has length\nj−i0−1. By maximality of i0, we have|πmτm−1πm−1···τjπj|< sand hence\nproperty (3) holds:\n|πmτm−1πm−1···τi0+1πi0+1|=|πm···τjπj|+|τj−1πj−1···τi0+1πi0+1|\n< s+j−i0/lessorequalslants+m.\nLet 0/lessorequalslanti0/lessorequalslantmbe the index satisfying properties (1)-(3). Since qi0is nontransient,\nwe have i0/lessorequalslanteand therefore\n(15) Acc P(qi0) =sk−1/summationdisplay\nj=i0rj+gN\nby Lemma 63(iii) and s/greaterorequalslantsi0. We claim that\n(16) |πmτm−1πm−1···τi0πi0|∈AccP(qi0).\nSincen/greaterorequalslantsandn∈AccP(q0) =s/summationtextk−1\nj=0rj+gNwe have n∈/summationtextk−1\nj=0rj+gN. This\nimplies\n|πmτm−1πm−1···τi0πi0|=n−|τi0−1πi0−1···τ0π0|≡n−i0−1/summationdisplay\nj=0rj≡k−1/summationdisplay\nj=i0rj(modg).\nIn addition, we have |πmτm−1πm−1···τi0πi0|/greaterorequalslantsby property (2). Since s/greaterorequalslant/summationtextk−1\nj=i0rj, we get\n|πmτm−1πm−1···τi0πi0|∈k−1/summationdisplay\nj=i0rj+gN.\nFinally, we obtain (16) from (15).\nByLemma54and(16),thereisanacceptingrun π′fromqi0whicht-simulatesthe\ninternal run πi0and has length|πmτm−1πm−1···τi0πi0|. Here,tis a constant only\ndepending onB. The prefix distance between the runs π=πmτm−1πm−1···τ0π0\nandπ′τi0−1πi0−1···τ0π0is at most tin casei0=m, and at most\n|πmτm−1πm−1···τi0|+t=|πmτm−1πm−1···τi0+1πi0+1|+1+t\n/lessorequalslant1+s+m+t=:c\nin casei0< mdue to property (3): Hence, the prefix distance between lastn(w)\nandL(BP) is bounded by the constant c. /square\nProof of Theorem 52. Letn∈Nbe the window size. From the discussion above, it\nsuffices to give a sliding window tester for a fixed partial automaton BP. Assume\nn/greaterorequalslants+|Q|, otherwise a trivial tester can be used. If n /∈AccP(q0), the tester\nalways rejects. Otherwise, the tester picks a random prime pwith Θ(loglog n) bits\nand maintains ℓw(q) (modp) for all q∈QP, wherewis the stream read so far,\nwhich requiresO(loglogn) bits. When a symbol a∈Σ is read, we can update ℓwa\nusingℓw: Ifq=qk, thenℓwa(q) = 0, otherwise ℓwa(q) = 1+ℓw(δP(a,q)) (mod p)\nwhere 1+∞=∞. The tester accepts if ℓw(q0)≡n(modp). Lemma 64 guarantees\nthat the tester is false-biased. /squareREGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 43\n5.5.Lower bounds for nontrivial regular languages. In this section, weprove\nTheorem 51. For this, we first haveto study trivial regularlanguag esin more detail.\nWe will show a result of independence interest: every o(n)-trivial regular language\nLis already trivial (i.e., O(1)-trivial). Given i,j/greaterorequalslant0 and a word wof length\nat leasti+jwe define cut i,j(w) =ysuch that w=xyz,|x|=iand|z|=j. If\n|w|< i+j, then cut i,j(w) is undefined. For a language Lwe define the cut-language\ncuti,j(L) ={cuti,j(w)|w∈L}.\nLemma 65. IfLis regular, then there are finitely many languages cuti,j(L).\nProof.LetA= (Q,Σ,q0,δ,F) be a DFA for L. Giveni,j/greaterorequalslant0, letIbe the set of\nstates reachable from q0viaisymbols and let F′be the set of states from which\nFcan be reached via jsymbols. Then, the NFA Ai,j= (Q,Σ,I,δ,F′) recognizes\ncuti,j(L). Since there are at most 22|Q|such choices for IandF′, the number of\nlanguages of the form cut i,j(L) must be finite. /square\nLemma 66. Ifcuti,j(L)is a length language for some i,j/greaterorequalslant0, thenLis trivial.\nProof.Assume that cut i,j(L) is a length language. Let n∈Nsuch that L∩Σn/\\e}atio\\slash=∅.\nWe claim that dist( w,L)/lessorequalslanti+jfor allw∈Σn. Ifn < i+jthis is clear. So,\nassume that n/greaterorequalslanti+j. Letw∈Σnandw′∈L∩Σn. Then, cut i,j(w′)∈cuti,j(L)\nand hence also cut i,j(w)∈cuti,j(L). Therefore, there exist x∈Σiandz∈Σjsuch\nthatxcuti,j(w)z∈Lsatisfying dist( w,xcuti,j(w)z)/lessorequalslanti+j. /square\nTherestriction of a language Lto a set of lengths N⊆NisL|N={w∈L|\n|w|∈N}. A language Lexcludes a word was a factor ifwis not a factor of any\nword inL. A simple but important observation is that if Lexcludes was a factor\nandvcontains kdisjoint occurrences of w, then dist( v,L)/greaterorequalslantk: If we change at\nmostk−1 many symbols in v, then the resulting word v′must still contain was\na factor and hence v′/∈L.\nProposition 67. LetLbe regular. If cuti,j(L)is not a length language for all\ni,j/greaterorequalslant0, thenLhas an infinite restriction L|Nto an arithmetic progression N=\nd+eNwhich excludes a factor.\nProof.First notice that cut i,j(L) determines cut i+1,j(L) and cut i,j+1(L): we have\ncuti+1,j(L) = cut 1,0(cuti,j(L)) and similarly for cut i,j+1(L). Since the number\nof cut-languages cut i,j(L) is finite by Lemma 65, there exist numbers i/greaterorequalslant0 and\nd >0 such that cut i,0(L) = cut i+d,0(L). Hence, we have cut i,j(L) = cut i+d,j(L)\nfor allj/greaterorequalslant0. By the same argument, there exist numbers j/greaterorequalslant0 ande >0\nsuch that cut i,j(L) = cut i,j+e(L) = cut i+d,j(L) = cut i+d,j+e(L), which implies\ncuti,j(L) = cut i,j+h(L) = cut i+h,j(L) = cut i+h,j+h(L) for some h >0 (we can take\nh=ed). This implies that cut i,j(L) is closed under removing prefixes and suffixes\nof length h.\nBy assumption cut i,j(L) is not a length language, i.e. there exist words y′∈\ncuti,j(L) andy /∈cuti,j(L) of the same length k. LetN={k+i+j+hn|n∈N}.\nFor anyn∈Nthe restriction L|Ncontains a word of length k+i+j+hnbecause\ny′∈cuti,j(L) = cut i+hn,j(L). This proves that L|Nis infinite.\nLetube an arbitrary word which contains for every remainder 0 /lessorequalslantr/lessorequalslanth−1\nan occurrence of yas a factor starting at a position which is congruent to rmod\nh. We claim that L|Nexcludes aiuajas a factor where ais an arbitrary symbol.\nAssume that there exists a word w∈L|Nwhich contains aiuajas a factor. Then,\ncuti,j(w) contains uas a factor, has length k+hnfor some n/greaterorequalslant0, and belongs\nto cut i,j(L). Therefore, cut i,j(w) also contains hmany occurrences of y, one per\nremainder 0 /lessorequalslantr/lessorequalslanth−1. Consider the occurrence of yin cuti,j(w) which starts at\na position that is divisible by h, i.e. we can factorize cut i,j(w) =xyzsuch that|x|44 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nis a multiple of h. Since|cuti,j(w)|=k+hnand|y|=k, then|z|is also a multiple\nofh. Therefore, y∈cuti+|x|,j+|z|(L) = cut i,j(L), which is a contradiction. /square\nCorollary 68. For every regular language L, the following statements are equiva-\nlent:\n(i)Lis trivial.\n(ii)Liso(n)-trivial.\n(iii)cuti,j(L)is a length language for some i,j/greaterorequalslant0.\nProof.If cuti,j(L) is a length language then Lis trivial by Lemma 66, and thus\nalsoo(n)-trivial. It remains to show the direction (ii) to (iii). Assume that Lis\no(n)-trivial. If (iii) would not hold then some infinite restriction L|NofLexcludes\na factorwby Proposition 67. Hence, if n∈Nis a length with L|N∩Σn/\\e}atio\\slash=∅, then\nany word vof length nwhich contains at least ⌊n/|w|⌋many disjoint occurrences\nofw, has distance dist( v,L)/greaterorequalslant⌊n/|w|⌋toL. Then,Lis noto(n)-trivial, which is\na contradiction. /square\nProof of Theorem 51. We will prove the two lower bounds in Theorem 51 for the\nmore general class of nondeterministic and co-nondeterministic slid ing window\ntesters. A nondeterministic sliding window tester for a language Land window\nsizen∈Nwith Hamming gap γ(n) is a nondeterministic finite automaton Pnsuch\nthat for all input words w∈Σ∗we have:\n•Iflastn(w)∈L, then there is at least one successful run of Pnonw.3\n•If dist(lastn(w),L)> γ(n), then there is no successful run of Pnonw.\nIn contrast,Pnis co-nondeterministic if for all w∈Σ∗we have:\n•Iflastn(w)∈L, then all runs of Pnonwthat start in an initial state are\nsuccessful.\n•If dist(lastn(w),L)> γ(n), then there is a nonsuccessful run of Pnonw\nthat starts in an initial state.\nThe space complexity of Pnis log|Pn|. Clearly, every true-biased (resp., false-\nbiased) sliding window tester is a nondeterministic (resp., co-nondet erministic) one.\nLetLbe a nontrivial regular language. We will prove an log n−O(1) lower\nbound for nondeterministic sliding window testers, and hence also fo r true-biased\nand deterministic sliding window testers. By the power set construc tion one can\ntransformaco-nondeterministicslidingwindowtesterwith mmemorystatesintoan\nequivalent nondeterministic (and in fact, even deterministic) sliding w indow tester\nwith 2mmemory states. Hence, an log n−O(1) lower bound for nondeterministic\nsliding window testers immediately yields an log n(logn−O(1))/greaterorequalslantlognlogn−O(1)\nlowerbound for co-nondeterministic sliding window testers, and hen ce also for false-\nbiased sliding window testers.\nBy Lemma 66, cut i,j(L) is not a length languagefor all i,j/greaterorequalslant0. LetNbe the set\nof lengths from Proposition 67 such that L|Nis infinite and excludes some factor\nwf. Letc=|wf|>0 and choose 0 < ǫ <1/c. SinceNis an arithmetic progression,\nL|Nis regular. Recall that every word vthat contains kdisjoint occurrences of wf\nhas Hamming distance at least kfrom any word in L|N. LetA= (Q,Σ,q0,δ,F)\nbe a DFA for L|N. SinceL(A) is infinite, there must exist words x,y,zsuch that\ny/\\e}atio\\slash=εand forδ(q0,x) =qwe have δ(q,y) =qandδ(q,z)∈F. Letd=|xz|and\ne=|y|>0, which satisfy d+eN⊆N.\nFix a window length n∈Nand consider a nondeterministic sliding window\ntesterPnforLand window size nwith Hamming gap ǫn. Define for k/greaterorequalslant0 the\ninput streams\nuk=wn\nfxykandvk=ukz=wn\nfxykz.\n3Recall from Section 2.2 that a successful run is a run from an i nitial state to a final state.REGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 45\nLetα=cǫ <1. If 0/lessorequalslantk/lessorequalslant⌊(1−α)n−c−d\ne⌋, then the suffix of vkof length ncontains\nat least/floorleftbiggn−d−ek\nc/floorrightbigg\n/greaterorequalslant/floorleftbiggn−d−(1−α)n+c+d\nc/floorrightbigg\n=/floorleftbiggαn+c\nc/floorrightbigg\n=⌊ǫn+1⌋> ǫn\nmany disjoint occurrences of wf. Hence,Pnhas no successful run on an input\nstreamvkwith 0/lessorequalslantk/lessorequalslant⌊(1−α)n−c−d\ne⌋.\nAssume now that the window size nsatisfiesn/greaterorequalslantdandn≡d(mode). Write\nn=d+lefor some l/greaterorequalslant0. We have l=n−d\ne>⌊(1−α)n−c−d\ne⌋. The suffix of\nvl=wn\nfxylzof length nisxylz∈L|N. Therefore, there exists a successful run π\nofPnonvl. Letmbe the number of states of Pn. For 0/lessorequalslanti/lessorequalslantlletpibe the state\non the run πthat is reached after the prefix wn\nfxyiofvl.\nIn order to deduce a contradiction, let us assume that m/lessorequalslant⌊(1−α)n−c−d\ne⌋. Then,\nthere must exist numbers iandjwith 0/lessorequalslanti < j/lessorequalslant⌊(1−α)n−c−d\ne⌋such that\npi=pj=:p. By cutting off cycles at pfrom the run πand repeating this, we\nfinally obtain a run of Pnon an input stream vk=wn\nfxykzwithk/lessorequalslant⌊(1−α)n−c−d\ne⌋.\nThis run is still successful. But this contradicts our previous obser vation thatPn\nhas no successful run on an input stream vkwith 0/lessorequalslantk/lessorequalslant⌊(1−α)n−c−d\ne⌋. We\nconclude thatPnmust have more than ⌊(1−α)n−c−d\ne⌋states. This implies\ns(Pn)/greaterorequalslantlog/parenleftbigg(1−α)n−c−d\ne/parenrightbigg\n/greaterorequalslantlogn−O(1),\nwhich proves the theorem. /square\n5.6.Finite unions of suffix-free and trivial languages. In this section, we\nshow the lowerbound of log n−O(1) from Theorem 53. We start with the following\nobservation.\nLemma 69. Every regular suffix-free language excludes a factor.\nProof.LetB= (Q,Σ,F,δ,q 0) be an rDFA for L. SinceLis suffix-free, we can\nassume that there exists a unique sink state qfail/∈F, i.e.δ(a,qfail) =qfailfor all\na∈Σ, which is reachable from all states. We construct a word wf∈Σ∗such that\nδ(p,wf) =qfailfor allp∈Q. Letp1,...,p mbe an enumeration of all states in\nQ\\{qfail}. We then construct inductively words w0,w1,...,w m∈Σ∗such that for\nall 0/lessorequalslanti/lessorequalslantmand 1/lessorequalslantj/lessorequalslanti:δ(wi,pj) =qfail. We start with w0=ε. Assume\nthatwihas been constructed for some i < m. There is a word xsuch that that\nδ(x,δ(wi,pi+1)) =qfail. We set wi+1=xwi. Then,δ(wi+1,pi+1) =δ(xwi,pi+1) =\nqfailandδ(wi+1,pj) =δ(xwi,pj) =δ(x,qfail) =qfailfor 1/lessorequalslantj/lessorequalslanti. Finally, we\ndefinewf=wm. /square\nLemma 70. Every regular language Lsatisfies one of the following properties:\n•Lis a finite union of regular trivial languages and regular suffi x-free lan-\nguages.\n•Lhas a restriction L|Nwhich excludes some factor and contains y∗zfor\nsomey,z∈Σ∗,|y|>0.\nProof.LetB= (Q,Σ,F,δ,q 0) be an rDFA for L. LetBr= (Q,Σ,Fr,δ,q0) where\nFris the set of nontransient final states and Bq= (Q,Σ,{q},δ,q0) forq∈Q. We\ncan decompose Lasa union of Lr=L(Br) and all languages L(Bq) overall transient\nstatesq∈F. Notice that L(Bq) is suffix-free for all transient q∈Fsince any run to\nqcannot be prolonged to another run to q. IfLris trivial, then Lsatisfies the first\nproperty. If Lris nontrivial, then by Lemma 66 and Proposition 67 there exists an\narithmetic progression N=a+bNsuch that Lr|Nis infinite and excludes some46 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nwordw∈Σ∗as a factor. Let z∈Lr|Nbe any word. Since some nontransient\nfinal state pis reached inBron input z, there exists a word ywhich leads from p\nback top. We can ensure that |y|is a multiple of bby replacing ybyyb. Then,\ny∗z⊆Lr|N⊆L|N. Furthermore, since each language L(Bq) excludes some factor\nwqby Lemma 69, the language L|N⊆Lr|N∪/uniontext\nqL(Bq) excludes any concatenation\nofwand all words wqas a factor. /square\nProof of Theorem 53. LetLbe a regular language that is not a finite union of\nregular trivial languages and regular suffix-free languages. By Lem ma 70,Lhas a\nrestriction L|Nwhich excludes some factor wfand contains y∗zfor some y,z∈Σ∗,\n|y|>0. Letc=|wf|/greaterorequalslant1. We choose 0 < ǫ <1/c. Letd=|z|ande=|y|. Fix\na window length n∈Nand define for k/greaterorequalslant0 the input streams uk=wn\nfykand\nvk=ukz=wn\nfykz.\nWe show the lower bound of log n−O(1) for co-nondeterministic sliding window\ntesters. Consider a co-nondeterministic sliding window tester PnforLand window\nsizenwith Hamming gap ǫn. Letα=cǫ <1 andr=⌊(1−α)n−c−d\ne⌋. If 0/lessorequalslantk/lessorequalslantr,\nthen the suffix of vkof length ncontains at least\n/floorleftbiggn−d−ek\nc/floorrightbigg\n/greaterorequalslant/floorleftbiggn−d−(1−α)n+c+d\nc/floorrightbigg\n=/floorleftbiggαn+c\nc/floorrightbigg\n=⌊ǫn+1⌋> ǫn\nmany disjoint occurrences of wf. Hence,Pnmust reject the input stream vkfor\n0/lessorequalslantk/lessorequalslantr, i.e. there is a run of Pnonvkthat starts in an initial state and ends in\na nonfinal state. Consider such a run πforvr. For 0/lessorequalslanti/lessorequalslantrletpibe the state in\nπthat is reached after the prefix wn\nfyiofvr. Let now mbe the number of states of\nPnand assume m/lessorequalslantr. There must exist numbers iandjwith 0/lessorequalslanti < j/lessorequalslantrsuch\nthatpi=pj=:p. It follows that there is a Pn-run onyj−ithat starts and ends in\nstatep. Using that cycle we can now prolong the run π, i.e. for all t/greaterorequalslant0 there is a\nrun ofPnonvr+(j−i)·t=wn\nfyr+(j−i)·tzthat starts in an initial state and ends in a\nnonfinal state.\nAssume now that the window size satisfies n/greaterorequalslantdandn≡d(mode). Write\nn=d+lefor some l/greaterorequalslant0. Each nwith this property satisfies n∈Nsince the\nwordylzbelongs to L|N. We have l=n−d\ne>⌊(1−α)n−c−d\ne⌋=r. For every k/greaterorequalslantl,\nthe suffix of vk=wn\nfykzof length nisylz∈L. Therefore,Pnacceptsvk, i.e. for\nallk/greaterorequalslantl, every run ofPnonvkthat starts in an initial state has to end in a final\nstate. This contradicts our observation that for all t/greaterorequalslant0 there is a run of Pnon\nvr+(j−i)·tthat goes from an initial state to a nonfinal state. We conclude tha tPn\nhas at least r+1/greaterorequalslant(1−α)n−c−d\nestates. It follows that\ns(Pn)/greaterorequalslantlog/parenleftbigg(1−α)n−c−d\ne/parenrightbigg\n/greaterorequalslantlogn−O(1).\nThis proves the theorem. /square\n5.7.Tradeoff between Hamming gap and space. Comparing Theorem 49 and\nTheorem50leadsto the question whether onecan replacethe Hamm ing gapγ(n) =\nǫnin Theorem 50 by a sublinear function γ(n) while retaining constant space at\nthe same time. We show that this is not the case:\nLemma 71. LetL=a∗⊆{a,b}∗. Every randomized sliding window tester with\ntwo-sided error for Land window size nwith Hamming gap γ(n)< nneeds space\nΩ(logn−logγ(n)).\nProof.We reduce from the greater-than-function GT m, whose randomized one-way\ncommunication complexity is Ω(log m) (Theorem 38). Consider a randomized slid-\ning window tester Pnfora∗and window size nwith Hamming gap γ(n). LetREGULAR LANGUAGES IN THE SLIDING WINDOW MODEL 47\nk:=γ(n)+1 and define m=⌊n/k⌋. Hence, we have n=mk+rfor some r < k.\nWe then obtain a randomized one-way protocol for the greater-t han-function on\nthe interval{1,...,m}: Alice produces from her input i∈{1,...,m}the word\nwi=bkar+(m−i)k. She then runs Pnonwiand sends the memory state to\nBob. Bob continues the run of the randomized sliding window tester, starting\nfrom the transferred memory state, with the input stream ajk. He obtains the\nmemory state reached after the input bkar+(m−i+j)k. Finally, Bob outputs the\nnegated answer given by the randomized sliding window tester. If i/lessorequalslantj, then\nlastn(bkar+(m−i+j)k) =an∈Land Bob rejects with high probability. On the other\nhand, ifi > j, then\n|bkar+(m−i+j)k|=k+r+(m−i+j)k/lessorequalslantk+r+(m−1)k=r+mk=n\nand hence lastn(bkar+(m−i+j)k) contains at least kmanyb’s. Since the Hamming\ndistance between this window and a∗is at least k=γ(n) + 1, Bob accepts with\nhigh probability. Hence, we have a correct protocol for GT m. Therefore, s(Pn) =\nΩ(logm) = Ω(log n−logγ(n)). /square\nFor example, if γ(n)/lessorequalslantncfor some 0 < c <1 then the space complexity in\nLemma 71 is Ω(log n).\n6.Conclusion and future work\nIn this paper we precisely determined the space complexity of regula r languages\nin the sliding window model in the following settings: deterministic, rand omized,\ndeterministic property testing, and randomized property testing . Two important\nrestrictions that made our results possible but that also limit their ap plicability are\nthe following:\n•Our sliding window algorithms only answer boolean queries (does the win -\ndow content belong to a language or not?). In many applications one wants\nto compute a certain non-boolean value, e.g. the number of 1’s in the win-\ndow. This leads to the question whether our automata theoretic fr ame-\nwork for sliding window problems can be extended to non-boolean que ries.\nWeighted automata [29] or cost register automata [4] could be a su itable\nframework for such an endeavor.\n•The incoming data values in our model are from a fixed finite alphabet.\nIn many practical situations the incoming data values are from an infi nite\ndomain (at least on an abstract level) like the natural numbers or re al num-\nbers. Again, the question, whether our automata theoretic appr oach can\nbe extended to such a setting, arises. A popular automata model f or words\nover an infinite alphabet (which are known as data words in this conte xt)\nareregister automata , which are also known as finite memory automata\n[64, 76]. In the context of sliding window streaming, deterministic reg ister\nautomata (DRA for short) [37] might be a good starting point. Bene dikt,\nLey and Puppis [14] proved a Myhill-Nerode-like theorem that charac ter-\nizes the class of data languages recognized by DRA for the case tha t the\nunderlying relational structure Aon the data values is either ( D,=) (where\n= denotes the equality relation) or ( D,<) for a strict linear order <. As\na byproduct of this characterization, they obtain a minimal DRA for any\nDRA-recognizable language. This DRA is minimal in a very strong sense :\nat the same time it has the minimal number of states and the minimal\nnumber of registers among all equivalent DRA. Using these minimal DR A,\none can define space-optimal streaming algorithms for data langua ges anal-\nogously to the case of words over a finite alphabet.48 M. GANARDI, D. HUCKE, M. LOHREY, K. MAMOURAS, AND T. STARIK OVSKAYA\nReferences\n[1] Charu C. Aggarwal, editor. Data Streams - Models and Algorithms , volume 31 of Advances\nin Database Systems . 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ACM, 2014.\nMax Planck Institute for Software Systems (MPI-SWS), Kaise rslautern, Germany,\nganardi@mpi-sws.org\nUniversit ¨at Siegen, Germany, {hucke,lohrey }@eti.uni-siegen.de\nRice University, Houston, USA, mamouras@rice.edu\nDI/ENS, PSL Research University, Paris, France, tat.starikovskaya@gmail.com"    },    {        "title": "2402.13420v1.On_binary_codes_with_distances__d__and__d_2_.pdf",        "content": "arXiv:2402.13420v1  [math.CO]  20 Feb 2024On binary codes with distances dandd+2\nIvan Landjeva,band Konstantin Vorob’evb\naNew Bulgarian University, 21 Montevideo str., 1618 Sofia, Bu lgaria\nbInstitute of Mathematics and Informatics, Bulgarian Acade my of Sciences,\n8 Acad G. Bonchev str., 1113 Sofia, Bulgaria\nAbstract\nWe consider the problem of finding A2(n,{d1,d2}) defined as the maximal size\nof a binary (non-linear) code of length nwith two distances d1andd2. Binary codes\nwith distances dandd+2 of size∼n2\nd\n2(d\n2+1)can be obtained from 2-packings of an\nn-element set by blocks of cardinalityd\n2+1. This value is far from the upper bound\nA2(n,{d1,d2})≤1+/parenleftbign\n2/parenrightbig\nproved recently by Barg et al.\nIn this paper we prove that for every fixed d(deven) there exists an integer\nN(d) such that for every n≥N(d) it holds A2(n,{d,d+2}) =D(n,d\n2+1,2), or, in\nother words, optimal codes are isomorphic to constant weigh t codes. We prove also\nestimates on N(d) ford= 4 and d= 6.\nKeywords: two weight codes, codes with two distances, codes of constant we ight.\n2000 MSC: 05A05, 05A20, 94B25, 94B65.\n1 Preliminaries\nIn this paper we consider binary non-linear codes with two distances . The alphabet\nthroughout the paper will be Ω = {0,1}. A binary code of length nand size M, or an\n(n,M)-code, is a subset Cof Ωn. A code Cis said to be a two-distance code if\n{d(u,v)|u,v∈C,u/\\e}atio\\slash=v}={d1,d2},\nwhere 1≤d1< d2≤nare integers. A two-distance binary code of length n, sizeM,\nand distances d1andd2is referred to as an ( n,M,{d1,d2})-code. If the size of the code\nis not specified we speak of an ( n,{d1,d2})-code. The maximum size Mof a binary code\nof length nwith distances d1andd2is denoted by A2(n,{d1,d2}).\nA natural problem is to determine the exact value of A2(n,{d1,d2}) for given positive\nintegersn,d1andd2. Barg et al. proved in [1] that A2(n,{d1,d2}) is upper bounded by\na quadratic function:\nA2(n,{d1,d2})≤1+/parenleftbiggn\n2/parenrightbigg\n, (1)\nfor alln≥6. The result was proved by embedding a binary code into the sphere Sn−1\nand then using bounds for spherical two-distance sets. In fact, the bound (1) is achieved:\nit is known that A2(n,{2,4}) =/parenleftbign\n2/parenrightbig\n+1 forn≥6. This bound is an improvement on a\nsimilar bound from [6]:\nA2(n,{d1,d2})≤/parenleftbiggn+2\n2/parenrightbigg\n,\nE-mail addresses: i.landjev@nbu.bg (Ivan Landjev), konstantin.vorobev@gm ail.com\n(Konstantin Vorob’ev)proved by a different technique. If we impose some restrictions on d1andd2, it turns out\nthatA2(n,{d1,d2}isboundedbyalinearfunction. So,if d2>2d1,onehas A2(n,{d1,d2})≤\nn+1. Furthermore, for deven and δodd, one has\nA2(n,{d,d+δ})≤/braceleftbigg\nn+1 ifdis even,\nn+2 ifdis odd.\nLet us note that the value of A2(n,{d,d+δ}) is of interest when both dandδare even.\nIn case of δodd the cardinality is bounded by a function which is linear in n, and the\ncase when dis odd, and δ– even is impossible [6].\nI was noticed by Zinoviev [9] (see also [6, 7]) that the characteristic vectors of the\nblocks of a family of k-element subsets with two intersection numbers x < yis a binary\ntwo distance code. In particular, quasisymmetric designs give rise t o two distance codes.\nIf there exists a 2-packing of an n-element set with Mblocks of size k, then there exists\na binary ( n,M,{d,d+2})-code with d= 2k−2 and cardinality M≤n(n−1)\nk(k−1). Equality\nis achieved iff there exists a 2-( n,k,1) design. The maximal size of a 2-( n,k,1) packing\nis known for small values of n(cf. [8]). The maximal size of an ( n,M,{d,d+ 2})-code,\nd-even, is at least equal to the maximal size of a 2-packing by (d\n2+1)-tuples denoted by\nD(n,d\n2+1,2), i.e.\nD(n,d\n2+1,2)≤A2(n,{d,d+2})≤/parenleftbiggn\n2/parenrightbigg\n+1.\nThe goal of this paper is to prove that for large values of nand fixed d, one has\nA2(n,{d,d+ 2}) =D(n,d\n2+ 1,2). The proof relies on a classical result by Erd˝ os and\nHanani [4]. For small values of d(d= 4,6) we give estimates on the smallest nfor which\nequality takes place.\n2 Binary codes with distances dandd+2\nIn this section we consider binary codes with parameters ( n,{d,d+2}) with even d. As\nwe discussed above, for δoddA2(n,{d,d+δ}) is bounded by a linear function, while the\ncasedodd,δeven is impossible (we cannot have a code in which all distances are odd ,\nand do actually appear).\nWe denote by D(n,k,t) the maximal number of blocks in a t-(v,k,1) packing. A\nclassical result by Erd˝ os and Hanani [4] establishes the asymptot ic growth of D(n,k,t)\nasn→∞. We state a special case of their result that will be needed in the seq uel.\nTheorem 1. Letk≥3be a fixed integer. Then\nlim\nn→∞D(n,k,2)k(k−1)\nn(n−1)= 1\nThis result implies immediately the following corollaries.\nCorollary 1. For any even integer d,d≥4,\nlim\nn→∞D(n,d\n2+1,2)(d\n2+1)(d\n2)\nn(n−1)= 1.\n2In view of this statement it is easy to see that the following corollary t akes place.\nCorollary 2. LetCbe an optimal (n,M,{d,d+2})-code. Then for any fixed λ,0< λ <1\nthere exists n0(λ)such that for any n > n0(λ)we have\nM≥λn2\n(d\n2+1)(d\n2).\nProof.ClearlyM≥D(n,d\n2+ 1,2) since the characteristic vectors of the blocks of an\noptimal 2-( n,d\n2+1,2) packing form a ( n,M,{d,d+2}) code. For a fixed constant λselect\n0< ε <1−λ. There exists an integer N1such that for every n > N 1:\n−εn(n−1)\n(d\n2+1)d\n2< D−n(n−1)\n(d\n2+1)d\n2< εn(n−1)\n(d\n2+1)d\n2\nThere exists a positive integer N2such that for n > N 2\n(1−ε−λ)n2−(1−ε)n\n(d\n2+1)d\n2>0.\nThis implies that for all n >max{N1,N2}:\nD(n,d\n2+1,2)>(1−ε)n(n−1)\n(d\n2+1)d\n2>λn2\n(d\n2+1)d\n2.\nFor our main theorem we shall need the following lemma.\nLemma 1. For a fixed even integer d, andnsufficiently large we have that\nD(n+d\n2−1,d\n2+1,2)> D(n,d\n2+1,2)+1.\nProof.LetB={Bi}be an optimal 2-( n,d\n2+ 1,2) packing. Our goal is to show that\nif we addd\n2−1 additional elements, we can construct a new packing with paramet ers\n2-(n+d\n2−1,d\n2+1,1) that contains two more blocks.\nFirst, we shall show that there exists a subset S⊆[n] :={1,...,n}with|S|=d\n2+1\nsuch that any block of BmeetsSin at most two elements. We shall construct such S\nby an iterative procedure. Initially, we set S=∅. We define also a set Lof prohibited\ncoordinate positions. This set Lis also set to be empty, i.e. L=∅. Next, we repeatd\n2\ntimes the following steps:\n1. Select an arbitrary element b∈[n]\\L;\n2.S←S∪{b};\n3. for every element c∈S,c/\\e}atio\\slash=b, check, whether {b,c}is contained in some block\nB∈B.\n4. if this is the case then L←L∪B\n5. otherwise L←L∪{b}.\n3The fact that nis big enough guarantees us that at each step it is possible to select a n\nelementb. Finally, we end up with a set S={a1,a2,a3,...a d\n2+1}such that every block\nfromBcontains at most two elements from S.\nNow we are going to modify Bin such way as to obtain a 2-packing B′of [n+d\n2−1]\nwith the property that no pair of elements from Sis covered by the blocks of B′.\nThis is done in performing at firstd\n2−2 identical steps. For each i= 1,2,...d\n2−2 we\ndo the following:\nFor anyj=i+1,i+2,...,d\n2+1 consider elements ai,aj∈S; if{ai,aj}⊆Bfor some\nblockB∈B, we replace BbyB′:= (B\\{ai})∪{an+i}}.\nThe obtained family of blocks still covers every pair of elements of th e set [n+d\n2−1]\nat most once. Moreover, among the pairs from Sonly{ad\n2−1,ad\n2},{ad\n2−1,ad\n2+1}and\n{ad\n2,ad\n2+1}may still be covered by codewords from C′.\nIf{ad\n2−1,ad\n2}⊆B1for some B1∈BthenB1is replaced by\nB′\n1= (B1\\{ad\n2−1})∪{an+d\n2−1}.\nIf{ad\n2−1,ad\n2+1}⊆B2for some B2∈Bthen we replace it by\nB′\n2= (B2\\{ad\n2+1})∪{an+d\n2−1}.\nFinally, if{ad\n2,ad\n2+1}⊆B3for some B3, then we replace it by\nC′:= (B3\\{ad\n2})∪{an+d\n2−1}.\nThe blocks B1,B2,B3(should there be such blocks) may intersect only in the elements\nad\n2−1,ad\n2,ad\n2+1. The new family of blocks, denoted by B′\n0, is a 2-packing of [ n+d\n2−1]. In\naddition, we have achieved that no pair with elements from Sis covered by a block from\nB′\n0. Hence we can add Sas a block, i.e.B′=B′\n0∪{S}is a 2-(n+d\n2−1,d\n2+1,1) packing.\nSincenisbig enoughwe canstart the whole procedure anew with the only diffe rence is\nthat we must avoid elements that occur together withelements fro m{n+1, n+2, ..., n+\nd\n2−1}in some blocks from B′. So instead of setting L=∅, we set\nL=/uniondisplay\nB′∈B′\nB′∩{n+1,...,n+d\n2−1}/\\e}atio\\slash=∅B′\nIn this way we construct a set S={a′\n1,a′\n2,a′\n3,...a′\nd\n2+1}such that any block from B′\ncontains at most two elements from S. Moreover, blocks that cover some pairs from Sdo\nnot contain elements from {n+1, n+2, ..., n+d\n2−1}. Now we can repeat the argument\ndescribed above to construct a new family B′′that is a 2-packing of [ n+d\n2−1] by blocks\nof sized\n2+1 with|B′′|=|B|+2.\nThis result can be restated in terms of codes.\nLemma 2. Letdbe a fixed integer and let nbe another integer that is big enough. Given\na binary code Cof length n, minimum distance din which all words are of the same\nweightd\n2+1, there exists a binary constant weight code C′′of length n+d\n2−1, the same\nminimum distance, and the same weight of the words for which\n|C′′|=|C|+2.\n4Now we state the main result in this paper.\nTheorem 2. For any even d >2there exists a positive integer N(d)such that for\nn≥N(d)any optimal (n,M,{d,d+2})-codeCis isomorphic to a constant-weight code\nof weightd\n2+1.\nProof.For a fixed even positive integer d, and a fixed constant λ, 0< λ <1, consider\nan optimal code Cwith parameters ( n,M,{d1,d2}) of length n > n 0(λ), where n0(λ) is\nthe constant from Corollary 2. Let us note that both values dandd+2 should occur as\nactual distances between codewords from C; otherwise, Cis equidistant and M≤n+1.\nBy Corollary 2 we have that for n > n0(λ)\nM≥λn2\n(d\n2+1)(d\n2). (2)\nTake two words x0andx1at distance d+2. Without loss of generality, let x0=0and\nwt(x1) =d+2. Since we have two possible distances, all remaining codewords be long to\none of the following sets:\nA1={x∈C:wt(x) =d+2, d(x,x1) =d+2},\nA2={x∈C:wt(x) =d+2, d(x,x1) =d},\nA3={x∈C:wt(x) =d, d(x,x1) =d+2},\nA4={x∈C:wt(x) =d, d(x,x1) =d},\nA5={x0},\nA6={x1}.\nNow let us consider a codeword ysuch that d(x0,y) =d(x1,y) =d\n2+ 1. We define\nthe integers\nsi(y) =|{z∈Ωn:z∈Ai,d(y,z) =d\n2+1}|, i= 1,2,...,6.\nSetB={y∈Ωn:d(x0,y) =d(x1,y) =d\n2+1}. Let us count in two ways the number of\npairs (y,z), where y∈B,z∈C. We have that\n/summationdisplay\ny∈B6/summationdisplay\ni=1si(y) =|A1|+(d\n2+2)|A2|+(d\n2+2)|A3|+\n(d\n2+1)2|A4|+/parenleftbiggd+2\nd\n2+1/parenrightbigg\n|A5|+/parenleftbiggd+2\nd\n2+1/parenrightbigg\n|A6|.\nSinceM=6/summationdisplay\ni=1|Ai|and|B|=/parenleftbiggd+2\nd\n2+1/parenrightbigg\n, there exists a word y0∈Bsuch that\n6/summationdisplay\ni=1si(y0)≥M/parenleftbigd+2\nd\n2+1/parenrightbig\n.\nIn other words, we have that the sphere with center y0and radiusd\n2+1 has at leastM\n(d+2\nd\n2+1)\nwords from C, i.e\n|{x∈C:d(y0,x) =d\n2+1}|≥M/parenleftbigd+2\nd\n2+1/parenrightbig. (3)\n5In the rest of the proof we will work with the code C′=C+y0which is again an\noptimal ( n,M,{d,d+2}) code. Let us define L={x∈C′:wt(x) =d\n2+1}. It is clear\nthat0/\\e}atio\\slash∈C′. By the choice of y0, we have|L|≥M\n(d+2\nd\n2+1), i.e. the size of Lis lowerbounded\nby a quadratic function on n.\nLetz0∈C′be a codeword of maximal possible weight, denoted by w. Clearly,\nw∈{3\n2d+3,3\n2d+1,...,d\n2+3,d\n2+1}.Now we shall consider the different possibilities for\nw.\n1. Letw=3\n2d+3. Evidently, for any x∈Lwe have supp( x)⊂supp(z0). Every pair\nof coordinate positions from supp( z0) is covered at most once by the support of a\ncodeword from L. Hence,\n|L|≤(3\n2d+3)(3\n2d+2)\n(d\n2+1)d\n2.\nConsequently,\n(3\n2d+3)(3\n2d+2)\n(d\n2+1)d\n2≥M/parenleftbigd+2\nd\n2+1/parenrightbig≥λn2\n(d\n2+1)d\n2/parenleftbigd+2\nd\n2+1/parenrightbig, (4)\nwhich does not take place for nbig enough.\n2. Letw=3\n2d+3−2i, 1≤i≤d\n2−1.Clearly, forany x∈L, wehave wt(x∗z0) =d\n2+1−i\norwt(x∗z0) =d\n2+2−i. As before, every pair of coordinate positions from supp( z0)\nis covered at most once by the support of a codeword from L. Consequently,|L|is\nbounded above by the maximal number of codewords of length wof weightsd\n2+1−i\nandd\n2+ 2−iwith the property that any two of them have 0 or 1 common ones.\nTherefore,\n|L|≤(3\n2d+3−2i)(3\n2d+2−2i)\n(d\n2+1−i)(d\n2−i),\nwhence we obtain\n(3\n2d+3−2i)(3\n2d+2−2i)\n(d\n2+1−i)(d\n2−i)≥M/parenleftbigd+2\nd\n2+1/parenrightbig, (5)\nwhich is false for nbig enough.\n3. The case w=d\n2+3. Here we have that for any x∈L, we have wt(x∗z0) = 1 or\nwt(x∗z0) = 2. The number of codewords from Lwith two 1’s in the coordinates\nfrom supp( z0) is bounded above by(d\n2+3)(d\n2+2)\n2due to the fact that two words of\nweightd\n2+ 1 cannot have two common ones. On the other hand, the number o f\nwords from Lthat have exactly one 1 inside supp( z0) isn−(d\n2d+3))(d\n2+3)\nd\n2. This is due\nto the fact that the number of words xwithwt(x∗z0) = 1 where this unit in the\nsame position from supp( z0) is equal to ( n−(d\n2+3))/d\n2Summing up we obtain\n|L|≤(d\n2+3)(d\n2+2)\n2+(n−(d\n2d+3))(d\n2+3)\nd\n2.\nAs a result we get\nM/parenleftbigd+2\nd\n2+1/parenrightbig≤/parenleftBigg\n(d\n2+3)(d\n2+2)\n2+(n−(d\n2+3))(d\n2+3)\nd\n2/parenrightBigg\n, (6)\n6which does not hold for large nsinceMis quadratic in nwhile the function on the\nright is linear in n.\nWe have proved so far that there are no codewords in C′of weight greater thand\n2+1\nfornbig enough. Clearly, there are also no codewords of weight less thand\n2−1, because\nthe minimal distance is d. However, it is possible that C′contains just one vertex of\nweightd\n2−1. Again, if there are at least two of them then the distance betwee n them is\nat mostd−2. Lemma 1 guarantees that in this case we can delete this codeword and after\nappropriate changing of C′construct a code with cardinality M+1 which contradicts to\nthe optimality of C′. Consequently, C′is a constant-weight code. This argument finishes\nthe proof.\nCorollary 3. For fixed even dandnbig enough we have that\nA2(n,{d,d+2}) =D(n,d\n2+1,2).\nFrom this point on N(d) will denote the smallest positive integer with the property\nthat for all n≥N(d) every optimal ( n,M,{d,d+2})-code is a constant weight code.\nA linear code is an almost-constant weight code if the weights of its no n-zero words\nare contained in an interval of length α. It is known that for α= 2 andk≥4 all almost-\nconstant are trivial. They are obtained from several copies of the Simplex code by adding\nor removing of two arbitrary non-zero coordinates (cf. [5]). It is in structive to compare\nthis result with Theorem 2 which says that for large na (non-linear) code with distances\ndandd+2 is a constant weight code.\n3 The cases d= 4andd= 6\nIntheprevioussection, weprovedthatthereexistsaninteger N(d)suchthatfor n≥N(d)\nit holds\nA2(n,{d,d+2}) =D(n,d\n2+1,2).\nHowever the proof relies on an asymptotic result, so that it is difficult to give an explicit\nestimate of N(d). Even if we have one, inequalities (4), (5) and (6) require nto be at\nleast exponential in d. However, for small values of dwe can give reasonable estimates on\nN(d) from the values of D(n,d\n2+1,2). Below we consider the cases d= 4 and d= 6.\nPackingsbytriplesandquadruplesyieldcodeswithparameters( n,{4,6})and(n,{6,8}).\nThe following bounds are well-known (see for instance, [8])):\nD(v,3,2) =\n\nv2−v\n6ifv≡1,3 (mod 6) ,\nv2−2v\n6ifv≡0,2 (mod 6) ,\nv2−2v−2\n6ifv≡4 (mod 6) ,\nv2−v−8\n6ifv≡5 (mod 6) ,\nAlso, for all v /∈{8,9,10,11,17,19}\nD(v,4,2) =\n\nv2−v\n12ifv≡1,4 (mod 12) ,\nv2−3v\n12ifv≡0,3 (mod 12) ,\nv2−2v\n12ifv≡2,8 (mod 12) ,\nv2−2v−3\n12ifv≡5,11 (mod 12) ,\nv2−v−18\n12ifv≡7,10 (mod 12) ,\nv2−3v−6\n12ifv≡6,9 (mod 12) .\n7From these exact values one can easily deduce the following lemma.\nLemma 3. The following inequalities take place\n1. For any integer v≥7we have that D(v+1,3,2)≥D(v,3,2)+2.\n2. For any integer v≥19we have that D(v+3,4,2)≥D(v,4,2)+2.\nThe proof is straightforward.\nThe next theorem exploits the fact that in the cases d= 4 and d= 6 we know the\nexact value of D(n,d\n2+1,2) which enables us to get reasonably small estimates on N(4)\nandN(6).\nTheorem 3. LetN(d)be the smallest integer such that for every n≥N(d)any optimal\n(n,M,{d,d+2})-codeCis (after a corresponding translation) a constant-weight c ode of\nweightd\n2+1. ThenN(4)≤302andN(6) =≤1685.\nProof.The beginning of the proof coincides with the one from Theorem 2. Sin ce we know\nexact values of D(n,d\n2+1,2) ford= 4 and d= 6, instead of inequality (2) we have that\nM≥D(n,d\n2+1,2). (7)\nLet us consider the case d= 4. From inequality (7) follows that M≥n2−2n−2\n6. Therefore,\n(4), (5) and (6) can be rewritten in the following way:n2−2n−2\n6≤240,n2−2n−2\n6≤420 and\nn2−2n−2\n6≤50(n−1) correspondingly. Starting from n= 302 all these inequalities do not\ntake place and an optimal code does not contain codewords of weigh t more than 3. By\nthe fact that Cis optimal and by Lemma 3 it is guaranteed that there are no codewor ds\nof weight 1, too. Therefore, Cis a constant-weight code.\nThe last case is d= 6. Inequality (7) gives us M≥n2−3n−6\n12. Againwe take inequalities\n(4), (5) and (6) (the second one gives us two inequalities for d= 6):n2−3n−6\n12≤770,\nn2−3n−6\n12≤1050,n2−3n−6\n12≤1960 andn2−3n−6\n12≤140n+210. Starting from n= 1685 all\nthese inequalities fail and an optimal code does not contain codewor ds of weight more\nthan 5. Again we use Lemma 3 to show that Cis a constant-weight code.\nIt is expected that this estimates are rather rough and the actua l value of N(d) is\nmuch smaller.\nAcknowledgements\nThe first author was supported by the Bulgarian National Science R esearch Fund under\nContract KP-06-N72/6 - 2023. The research of the second auth or was supported by the\nNSP P. Beron project CP-MACT.\nReferences\n[1] A. Barg, A. Glazyrin, Wei-Jiun Kao, Ching-Yi Lai, Pin-Chieh Tseng, Wei-Hsuan Yu,\nOn the size of maximal codes with 2, 3, and 4 distances, arXiv:2210.07 496, 2022. (to\nappear in IEEE Trans. Inf. Theory)\n[2] P. Boyvalenkov, K. Delchev, D. V. Zinoviev, V. A. Zinoviev On two- weight codes,\nDiscrete Math. , 344(5)(2021), 112318, https://doi.org/10.1016/j.disc.2021.11 2318.\n8[3] P. Boyvalenkov, K. Delchev, D. V. Zinoviev, V. A. Zinoviev On code s\nwith distances dandn,Probl. of Inform. Transmission 58(4)(2022), 62–83,\nhttps://doi.org/10.1134/S0032946022040068.\n[4] P. Erd˝ os, H. Hanani, On a limit theorem in combinatorial analysis, Publ. Math. 10\n(1964), 10–13.\n[5] I. Landjev, A. Rousseva, L.Storme, OnLinear CodesofAlmost Constant Weight and\nthe Related Arcs, Comptes rendus, de l’Academie Bulgare des Science 72(12)(2019),\n1626-1633.\n[6] I. Landjev, A. Rousseva, K. Vorobev, Constructions of bi-\nnary codes with two distances, Discrete Math. 246(2023), 113337,\nhttps://doi.org/10.1016/j.disc.2023.113337.\n[7] N. V. Semakov, V. A. Zinoviev, Constant weight codes and tactic al configurations,\nProbl. Inform. Transmission 5(3)(1969), 29–38.\n[8] D. R. Stinson, Ruizhong Wei, Jianxing yin, Packings, Chapter 40 in; Handbook of\nCombinatorial Designs (eds. Ch. Colbourn, J. Dinitz), Chapman & Ha ll/ CRC, 2007.\n[9] V. A. Zinoviev, The cascade equal-weight codes and maximal pack ings.Probl. of\nControl and Inf. Theory 12(1)(1983), 3–10.\n9"    },    {        "title": "2402.13423v1.Diagonal_poset_Ramsey_numbers.pdf",        "content": "Diagonal poset Ramsey numbers\nMaria Axenovich∗Christian Winter∗†\nFebruary 22, 2024\nAbstract\nA poset ( Q,≤Q) contains an induced copy of a poset ( P,≤P) if there exists an\ninjective mapping ϕ:P→Qsuch that for any two elements X, Y∈P,X≤PY\nif and only if ϕ(X)≤Qϕ(Y). By Qnwe denote the Boolean lattice (2[n],⊆). The\nposet Ramsey number R(P, Q) for posets PandQis the least integer Nfor which any\ncoloring of the elements of QNin blue and red contains either a blue induced copy of\nPor a red induced copy of Q.\nIn this paper, we show that R(Qm, Qn)≤nm−\u0000\n1−o(1)\u0001\nnlogmwhere n≥mand\nmis sufficiently large. This improves the best known upper bound on R(Qn, Qn) from\nn2−n+ 2 to n2−\u0000\n1−o(1)\u0001\nnlogn. Furthermore, we determine R(P, P) where Pis\nann-fork or n-diamond up to an additive constant of 2.\nA poset ( Q,≤Q) contains a weak copy of ( P,≤P) if there is an injection ψ:P→Q\nsuch that ψ(X)≤Qψ(Y) for any X, Y∈Pwith X≤PY. The weak poset Ramsey\nnumber Rw(P, Q) is the smallest Nfor which any blue/red-coloring of QNcontains a\nblue weak copy of Por a red weak copy of Q. We show that Rw(Qn, Qn)≤0.96n2.\n1 Introduction\nAposet is a set Pequipped with a partial order ≤P, i.e., a binary, transitive, reflexive,\nand antisymmetric relation. Usually, we refer to such a poset ( P,≤P) just as Pand call\nthe elements of Pvertices . Let PandQbe two posets. A (strong) embedding ϕ:P→Q\nofPintoQis an injective function where for any two X, Y∈P,X≤PYif and only\nifϕ(X)≤Qϕ(Y). We say that the image of ϕis an (induced) copy ofPinQ. Given a\nnon-empty set Z, we denote by Q(Z) = (2Z,⊆) the Boolean lattice on ground set Zwith\ndimension is|Z|. We use Qnto denote a Boolean lattice with an arbitrary n-element ground\nset. For ℓ∈ {0, . . . ,|Z|}, the ℓ-th layer ofQ(Z) is the set {X∈ Q(Z) :|X|=ℓ}.\nRamsey-type problems in extremal combinatorics are widely studied for various graph\nstructures. Initiated by Neˇ set�� ril and R¨ odl [11], such questions were also considered for\nposets. A blue/red coloring of a poset Qis a mapping c:Q→ {blue,red}. We say that a\nposet is blueif every of its vertices is colored blue, and redif every of its vertices is colored\nred. For an integer kand a poset parameter psuch as size, height, or width, Kierstead\nand Trotter [9] analysed the minimal p(Q) of a poset Qsuch that any blue/red coloring\nc:Q→ {blue,red}contains a monochromatic induced copy of a poset Pwith p(P) =k.\nWhen p(Q) is the dimension of the smallest Boolean lattice containing a copy of Q, this\n∗Karlsruhe Institute of Technology, Karlsruhe, Germany. The research was partially supported by DFG\ngrant FKZ AX 93/2-1.\n†E-mail: christian.winter@kit.edu\n1arXiv:2402.13423v1  [math.CO]  20 Feb 2024problem leads to the poset Ramsey number introduced by Axenovich and Walzer [1]. For\nposets PandQ, let\nR(P, Q) = min {N∈N: every blue/red coloring of QNcontains either\na blue induced copy of Por a red induced copy of Q}.\nThis extremal parameter has been actively studied in recent years with the central question\nbeing the poset Ramsey number R(Qn, Qn) for large n. In this setting, first bounds were\ngiven by Axenovich and Walzer [1] who showed that 2 n≤R(Qn, Qn)≤n2+ 2n. The upper\nbound was improved by Walzer [13] to n2+ 1, and then by Lu and Thompson [10] to the\nbest known value n2−n+ 2. Cox and Stolee [5] improved the lower bound to 2 n+ 1 for\nn≥13, which was extended to all n≥3 by Bohman and Peng [3]. The best known lower\nbound for large nis given by Winter [14], who showed that R(Qn, Qn)≥2.24n.\nNote that any posets PandQare contained as a copy in Qnfor sufficiently large n,\ntherefore R(P, Q) is well-defined. In the off-diagonal setting, Lu and Thompson [10] showed\nthat for n≥m≥4,\nR(Qm, Qn)≤n\u0010\nm−2 +O\u00001\nm\u0001\u0011\n+m+ 3.\nFurther bounds are known if one or two of the parameters mandnare small, e.g. R(Q2, Qn) =\nn+ Θ\u0000n\nlogn\u0001\n, where the upper bound is given by Grosz, Methuku and Tompkins [7] and the\nlower bound is by the authors [2]. Furthermore, R(Q2, Q2) = 4, R(Q2, Q3) = 5, and\nR(Q3, Q3) = 7, see [1],[10], and Falgas-Ravry et al. [6], respectively.\nThe first result of this paper is a strengthened upper bound on the poset Ramsey number\nofQmversus Qnwhen both mandnlarge.\nTheorem 1. Letn, m∈Nwith 225≤m≤n. Then\nR(Qm, Qn)≤n\u0010\nm−\u0000\n1−2√logm\u0001\nlogm\u0011\n.\nMoreover, if n≥mandε∈R,0< ε < 1, such thatn+m\nn·1\n(1−ε) logm+m−ε≤ε, then\nR(Qm, Qn)≤n\u0000\nm−(1−ε)2logm\u0001\n.\nIn this article, log refers to the logarithm with base 2. Theorem 1 is the first result which\nimproves the initial basic upper bound by Axenovich and Walzer [1], see Lemma 6, by a\nsuperlinear additive term. Our result immediately provides an improved upper bound on\nR(Qn, Qn).\nCorollary 2. Forε >0and sufficiently large n∈Ndepending on ε,R(Qn, Qn)≤n2−\n(1−ε)nlogn.\nThe diagonal poset Ramsey number R(P, P) is known exactly for some basic posets P.\nWalzer [13] determined this number for antichains and chains. Chen, Chen, Cheng, Li, and\nLiu [4] found R(P, P) when Pis the poset which consists of two elementwise incomparable\nchains on a given number of vertices, with an added vertex smaller than all other vertices.\nHere, we study the diagonal poset Ramsey number R(P, P) for further posets P. The\nn-fork Vnis the poset consisting of an antichain on nvertices with an added vertex smaller\nthan all other vertices. Similarly, the n-diamond Dnis the poset consisting of an antichain\n2onnvertices and a vertex smaller than all others as well as a vertex larger than all others.\nLetn∈N. We denote by α(n) the minimal dimension Nsuch that QNcontains an\nantichain of size n. We call α(n) the Sperner number ofn. Sperner [12] showed that α(n) is\nthe minimal integer such that\u0000N\n⌊N/2⌋\u0001\n≥n. It is a basic observation that α(n)≤α(2n−1)≤\nα(n) + 2, which we will use repeatedly. Note that α(n) =\u0000\n1 +o(1)\u0001\nlogn, see also an almost\nexact bound on α(n) by Habib, Nourine, Raynaud and Thierry [8].\nTheorem 3. For every n∈N,2α(n)≤R(Dn, Dn)≤α(n) +α(2n−1). In particular,\n2α(n)≤R(Dn, Dn)≤2α(n) + 2 and thus R(Dn, Dn) =\u0000\n2 +o(1)\u0001\nlogn.\nNote that for some values of n,α(2n−1)≤α(n) + 1, in which case the above upper and\nlower bounds on R(Qn, Qn) differ by 1.\nFor the next result, we need two further extremal parameters. Given n, N∈Nwith\nN≥α(n), let β(N, n) and N∗(n) be integers with\nβ(N, n) = min\b\nβ:\u0012N\nβ\u0013\n≥n\t\nand N∗(n) = max\b\nN≥α(n) :N−β(N, n)< α(n)\t\n,\nas illustrated in Figure 3. Note that\u0000N\n⌊α(n)/2⌋\u0001\n≥\u0000α(n)\n⌊α(n)/2⌋\u0001\n≥n >\u0000N\n0\u0001\n, so 1 ≤β(N, n)≤\nα(n)/2. Thus, α(n)≤N∗(n)< β(N∗(n), n) +α(n)≤2α(n), so in particular β(N, n) and\nN∗(n) are well-defined.\nTheorem 4. For every n∈N,N∗(n) + 1≤R(Vn, Vn)≤N∗(n) + 3 .\nLetd=1\n1−cwhere cis the unique real solution of log\u0000\nc−c(1−c)c−1\u0001\n= 1−c, i.e., d≈1.29.\nThen R(Vn, Vn) = (d+o(1)) log( n).\nSimilarly to Theorem 3, by using more careful estimates our proof provides R(Vn, Vn)≤\nN∗(n) + 2 whenever α(2n−1)≤α(n) + 1.\nThe poset Ramsey number R(P, Q) concerns induced copies of posets PandQ. A\nvariation of this Ramsey-type problem deals with so called weak copies . Let PandQbe two\nposets. A weak embedding ψ:P→Qis an injective function such that for any two elements\nX, Y∈P,X≤PYimplies ψ(X)≤Qψ(Y). In this setting, we allow that ψ(X)≤Qψ(Y)\neven if X̸≤PY. We say that the image of ψis aweak copy ofPinQ. Analogously to the\nposet Ramsey number R(P, Q), we define the weak poset Ramsey number for posets Pand\nQas\nRw(P, Q) = min {N∈N: every blue/red coloring of QNcontains either\na blue weak copy of Por a red weak copy of Q}.\nIt is a basic observation that Rw(P, Q)≤R(P, Q) for any posets PandQ. The best\nknown bounds in the diagonal setting Qnversus Qnare a lower bound by Cox and Stolee\n[5] and an upper bound by Lu and Thompson [10],\n2n+ 1≤Rw(Qn, Qn)≤R(Qn, Qn)≤n2−n+ 2.\nGr´ osz, Methuku and Tompkins [7] showed in the off-diagonal setting that Rw(Qm, Qn)≥\nm+n+ 1 for m≥2 and n≥68, and Rw(Qm, Qn)≤n+ 2m−1, where the second bound\nis derived from a result by Cox and Stolee [5].\nOur final result is an improvement of the upper bound on Rw(Qn, Qn).\n3Theorem 5. For sufficiently large n,Rw(Qn, Qn)≤0.96n2.\nIn Section 2, we introduce some notation and definitions, and discuss the Blob Lemmas .\nIn Section 3, we present a proof of Theorem 1. In Section 4, we prove Theorems 3 and 4.\nA proof of Theorem 5 is given in Section 5. We omit floors and ceilings where appropriate.\nWe denote the set {1, . . . , n }of the first nnatural numbers by [ n].\n2 Notation and preliminary results\n2.1 Notation and definitions\nIn this paper we use capital letters to denote subsets of the ground set in a Boolean lattice,\ni.e., the vertices of Boolean lattices. We use bold capital letters for significant ground sets.\nCalligraphic capital letters are used for families of sets and thus for subposets of the Boolean\nlattice. Abstract posets which are not considered as subposets of a specific Boolean lattice\nare denoted by capital letters, for example Qn,DnandVn.\nGiven a set Zwith disjoint subsets S, T⊆Z, we define a blobin a Boolean lattice Q(Z)\nasB(S;T) ={Z⊆Z:S⊆Z⊆S∪T}.We call Tthevariable set of this blob. Note\nthatB(S;T) is a Boolean lattice of dimension |T|. We say that |T|is the dimension of\nthe blob. We remark that every Z∈ B(S;T) has the form S∪TZwhere TZ⊆T. A\nt-truncated blob , denoted B(S;T;t), is the poset {Z∈ B(S;T) :|Z| ≤t}. We also say\nthatB(S;T;t) has dimension |T|. Given a Boolean lattice Q(X) on ground set Xand a\nnon-negative integer t,t≤ |X|, letQ(X)tdenote the t-truncated Boolean lattice , that is the\nsubposet {Z∈ Q(X) :|Z| ≤t}=B(∅;X;t). Given two non-negative integers sandt,\n0≤s≤t≤ |X|, letQ(X)t\nsdenote the ( s, t)-truncated Boolean lattice , that is the subposet\n{Z∈ Q(X) :s≤ |Z| ≤t}. In particular, Q(X)t=Q(X)t\n0.\nLetXandYbe disjoint, non-empty sets. Let ϕ:Q(X)t→ Q (X∪Y) be a (strong)\nembedding. The function ϕisX-good ifϕ(X)∩X=Xfor every X∈ Q(X)t. Note that\nat-truncated blob B(S;T;t) is the image of a T-good embedding of Q(T)t. We say that ϕ\nisredif its image is a red poset, i.e. ϕmaps only to red vertices, and blueif its image is a\nblue poset. If ϕis an embedding of a poset Ainto a Boolean lattice, we use the notation\nϕ(A) for{ϕ(X) :X∈ A} . For a subposet Aof a Boolean lattice, we say that the vol-\nume ofA, denoted Vol( A), is the total number of ground elements in all vertices of A, i.e.,\nVol(A) =|S\nX∈AX|.\n2.2 Blob Lemmas\nIn order to show an upper bound on R(Qm, Qn), we have to find a blue copy of Qmor a\nred copy of Qnin every coloring of a host Boolean lattice. Using our notation, let us briefly\nreiterate the basic approach used for previously known upper bounds, see Lemma 1 in [9],\nBlob Lemma in [1], and Lemma 1 in [10].\nLemma 6 (Blob Lemma) .Letn, m∈NandN=nm+n+m. Any blue/red coloring of\nQ([N])contains a blue copy of Qmor a red copy of Qn.\nProof. Partition [ N] arbitrarily into sets X,Y(0),Y(1), . . . ,Y(n)such that |X|=nand\n|Y(i)|=m,i∈ {0, . . . , n }. We construct a red embedding ϕ:Q(X)→ Q ([N]). For each\nX∈ Q(X), consider the blob BX=B(X∪S|X|−1\ni=0Y(i);Y(|X|)), where we use the convention\n4thatS−1\ni=0Y(i)=∅. If one of the blobs is monochromatic blue, it is a blue copy of Qmas\ndesired.\nSuppose that there is a red vertex ZX∈ BXfor every X. Then the function ϕ:Q(X)→\nQ([N]),ϕ(X) =ZXhas a red image. Observe that ϕ(X)∩X=Xfor every X⊆X. Then\nit is straightforward to verify that ϕis an embedding. Therefore, there is a red copy of\nQn.\nThe general proof idea of our bound on R(Qm, Qn) is to refine Lemma 6 by considering\ntruncated blobs instead of blobs, moreover those chosen based on already embedded layers.\nIn addition, we control the volume of truncated blobs while constructing the embedding.\nFor this we need parts (i),(ii),(iii) of the following variant of Lemma 6. Part (iv) will be\napplied to achieve an upper bound on Rw(Qn, Qn).\nLemma 7 (Truncated Blob Lemma) .Letn, m, t, N ∈N. Let X⊆[N]. Fix a blue/red\ncoloring of the Boolean lattice Q([N]).\n(i) If |X|=n,t≤n, and there is an X-good, red embedding ϕ:Q(X)t→ Q ([N])such\nthatVol(ϕ(Q(X)t)≤N−(n−t)m, then Q([N])contains a blue copy of Qmor a red\ncopy of Qn.\n(ii) If |X|=m,t≤m, and there is an X-good, blue embedding ϕ:Q(X)t→ Q([N])such\nthatVol(ϕ(Q(X)t)≤N−(m−t)n, then Q([N])contains a blue copy of Qmor a red\ncopy of Qn.\n(iii) If |X|=n,t≤n, and there is a set Sdisjoint from Xand a red truncated blob\nB(S;X;t)such that |S∪X| ≤N−(n−t)m, then there is a blue copy of Qmor a red\ncopy of Qn.\n(iv) If 0≤s≤t≤n, and N= (t−s+ 2)n, then Q([N])contains a blue copy of Qnor a\nred copy of Q([n])t\ns.\nProof. (i):We shall extend ϕto a red embedding ϕ′:Q(X)→ Q([N]). As in the proof of\nLemma 6, we select disjoint sets of ground elements and use them to define a blob for each\nnot yet embedded X∈ Q(X). Let U=S\nX∈Q(X)tϕ(X) and note that |U|= Vol( ϕ(Q(X)t)).\nSince ϕisX-good, X⊆U. Thus,\n|[N]\\(U∪X)|=|[N]\\U|=N−Vol(ϕ(Q(X)t))≥(n−t)m.\nFixn−tpairwise disjoint m-element subsets Y(t+1), . . . ,Y(n)of [N]\\U. For every X∈ Q(X)\nwith|X|> t, consider the blob\nBX=B\nX∪(U\\X)∪|X|−1[\ni=t+1Y(i);Y(|X|)\n.\nIfBXis blue, it is a blue copy of Qm, so suppose that there is a red vertex ZX∈ BX. Let\nϕ′:Q(X)→ Q([N]) with\nϕ′(X) =(\nϕ(X) if|X| ≤t,\nZX if|X|> t.\nThe image of ϕ′is red. Now we verify that ϕ′is an embedding. Observe that ϕ′(X)∩X=X\nfor every X⊆X. Let X1, X2⊆X. If ϕ′(X1)⊆ϕ′(X2), then X1=ϕ′(X1)∩X⊆\nϕ′(X2)∩X=X2. Conversely, suppose that X1⊆X2.\n5If|X1| ≤ |X2| ≤t, then ϕ′(X1) =ϕ(X1)⊆ϕ(X2) =ϕ′(X2) because ϕis an embedding.\nIf|X1| ≤t <|X2|, we have ϕ′(X1) =X1∪(ϕ(X1)\\X)⊆X2∪(U\\X)⊆ZX2=ϕ′(X2).\nIft <|X1| ≤ |X2|, then ϕ′(X1)∈ BX1, soϕ′(X1)⊆X1∪(U\\X)∪Y(t+1)∪ ··· ∪ Y(|X1|)⊆\nZX2=ϕ′(X2), as desired.\n(ii): This part is proven analogously to part (i).\n(iii): Observe that B(S;X;t) is the image of an X-good embedding ϕ:Q(X)t→ Q (X),\nϕ(X) =S∪X, with Vol( ϕ(Q(X)t) =|S∪X|.\n(iv): LetX= [n]. Choose disjoint, n-element sets Y(s), . . . ,Y(t)ofY= [N]\\X. For each\nX∈ Q(X) with s≤ |X| ≤t, we define a blob\nBX=B\nX∪|X|−1[\ni=sY(i);Y(|X|)\n,\nwhereS|X|−1\ni=sY(i)=∅for|X|=s. IfBXis blue, it corresponds to a blue copy of Qn.\nSo let ϕ(X) be a red vertex in BX. Then, we see that {ϕ(X) :X∈X}is a red copy of\nQ(X)t\ns.\n3 Upper bound on Ramsey number R(Qm, Qn)\nFirst, we need the following technical proposition.\nProposition 8. Let225≤m≤n. Let ε=1√logm. Thenn+m\nn·1\n(1−ε) logm+m−ε≤ε.\nProof. The bound m≥225implies that ε=1√logm≤1\n5, so in particular 4 ε≤1−ε. Since\nm≤nand log m=ε−2, we obtain that\nn+m\nn·1\n(1−ε) logm≤2n\nn·ε2\n1−ε=ε\n2·4ε\n1−ε≤ε\n2.\nForε≤1\n5, it is straightforward to check that1\nε≥1−logε. Thus, using again that\nlogm=ε−2,\nm−ε= 2−1\nε≤2−1+log ε=ε\n2.\nTherefore,\nn+m\nn·1\n(1−ε) logm+m−ε≤ε\n2+ε\n2=ε.\nNow we show of our main result.\nProof of Theorem 1. Fixnandmsuch that n≥m. Fix an ε∈Rwith 0 < ε < 1 which\nsatifiesn+m\nn·1\n(1−ε) logm+m−ε≤ε. (1)\nLet\nN=n\u0000\nm−(1−ε)2logm\u0001\n.\n6To prove the second statement of the theorem, we shall show that R(Qm, Qn)≤N. The\nfirst statement is a corollary of this: Indeed, if m≥225, then Proposition 8 shows that\ninequality (1) holds for ε=1√logm, thus\nR(Qm, Qn)≤n\u0000\nm−(1−ε)2logm\u0001\n≤n\u0000\nm−(1−2ε) logm\u0001\n=n\u0010\nm−\u0000\n1−2√logm\u0001\nlogm\u0011\n.\nNow we proceed with the proof of the second statement in Theorem 1, i.e., the bound\nR(Qm, Qn)≤N. Let tµ= (1−ε) logmandtη=n\nmtµ. Note that 0 ≤tµ≤nand\n0≤tµ≤m. In this proof we consider tµ-truncated blobs of dimension mandtη-truncated\nblobs of dimension n. It follows that N=n(m−tµ) +εntµ= (n−tη)m+εmtη.\nFix an arbitrary coloring of Q([N]). We shall find a blue copy of Qmor a red copy of\nQnin this coloring. For this, we show that there is a blue embedding ϕofQ([m])tµwhose\nimage has volume at most N−(m−tµ)n, or a red embedding ϕ′ofQ([n])tηwhose image has\nvolume at most N−(n−tη)m. In either case, Lemma 7 gives the desired monochromatic\nBoolean lattice.\nFirst, we suppose that there exist disjoint sets S, T⊆[N] such that |S| ≤εmtη−n\nand|T|=n, for which the truncated blob B(S;T;tη) is monochromatically red. Then\n|S∪T| ≤εmtη=N−(n−tη)m. Thus, part (iii) of Lemma 7 implies the existence of a\nblue copy of Qmor a red copy of Qn, which completes the proof. Thus, from now on we\ncan assume the following:\nProperty (∗):There is a blue vertex in every truncated blob B(S;T;tη) with dimension\n|T|=nand volume |S∪T| ≤εmtη.\nLetXbe an arbitrary subset of [ N] of size m, and let Y= [N]\\X. In the remainder of\nthe proof we construct an X-good, blue embedding ϕ:Q(X)tµ→ Q([N]) such that its im-\nage has a small volume. After stating the complete construction we justify that the defined\nobjects indeed exist.\n3.1 Construction of a blue embedding ϕofQ(X)tµ\nWe shall find a blue ϕ(X) for each X∈ Q(X)tµlayer-by-layer, such that ϕisX-good, i.e.,\nϕ(X)∩X=XforX⊆X. Consider pairwise disjoint subsets Y(0),Y(1), . . . ,Y(tµ)ofY,\nwhere |Y(0)|=nand|Y(i)|= 2i−1tη, fori= 1, . . . , t µ. In our construction, we shall make\nsure that ϕ(X)∩Y⊆Y(0)∪···∪ Y(|X|), which will ensure that the volume of the embedded\nposet Vol( ϕ(Q(X)tµ) is at most |X|+|Y(0)|+···+|Y(tµ)|.\nWe shall use the notation YXto denote ϕ(X)∩Y, i.e., ϕ(X) =X∪YX. To guarantee\nthat the defined function is an embedding, we only need to verify that for any X1⊆X2,\nwe have YX1⊆YX2. Indeed, since ϕ(X)∩X=X, for incomparable X1andX2the images\nϕ(X1) and ϕ(X2) are incomparable.\nLetϕ(∅) be a blue vertex in B(∅;Y(0);tη), that exists by property ( ∗). For embedding\nthei-th layer, we define a blob whose variable set uses a set Y(i)of new ground elements as\nwell as ground elements of Ythat were already used by the embedding of previous layers.\nLeti≥1. Assume that we constructed ϕ(X) for every X⊆X,|X| ≤i−1, such that\n7Q([N])\n[N]\nX[Y(0)[:::[Y(i)\nX[Y(0)[:::[Y(i\u00001)\nX[SX\n\u001e(X)\nX[SX[TX\nX0[YX0=\u001e(X0)\n?\n?[Y?=\u001e(?)\nB(X[SX;TX;t\u0011)Figure 1: The vertex ϕ(X) inB(X∪SX;TX;tη) for i= 3 and |X|= 4\nϕ(X)∩Y⊆Y(0)∪ ··· ∪ Y(i−1).We shall construct ϕ(X) for each X⊆X,|X|=i, so let\nSX=[\nX′⊂Xϕ(X′)∩Y=[\nX′⊂XYX′,\nTX⊆(Y(0)∪ ··· ∪ Y(i))\\SX,|TX|=n.\nLetϕ(X) be a blue vertex in B(X∪SX;TX;tη), that exists by ( ∗). Let YX=ϕ(X)∩Y.\nWe run this procedure for all i≤tµ. For any two X1, X2∈ Q(X)tµwith X1⊆X2,\nit follows from the construction that YX1⊆YX2. Thus indeed, we have an embedding of\nQ(X)tµ.\n3.2 Verification that ϕis well defined and has small image volume\nWe need to make sure that the sets Y(i)andTXexist, i.e., that the sets from which these\nare selected as subsets are large enough. To see that Y(i)exists for i≤tµ, it is sufficient to\nverify that |Y(0)∪ ··· ∪ Y(tµ)| ≤ |Y|. Note that for i≥1,\n|Y(0)∪ ··· ∪ Y(i)|=n+iX\nj=12j−1tη=n+ (2i−1)tη. (2)\nRecalling that tµ= (1−ε) logmandtη=n\nmtµ, we have\n|Y(0)∪ ··· ∪ Y(tµ)|=n+ (2tµ−1)tη\n≤n+m1−εn\nmtµ\n=\u00121\ntµ+m−ε\u0013\nntµ.\n8We picked εsuch thatn+m\nn·1\n(1−ε) logm+m−ε≤ε, which implies1\ntµ+m\nntµ+m−ε≤ε. Thus,\n|Y(0)∪ ··· ∪ Y(tµ)| ≤\u0012\nε−m\nntµ\u0013\nntµ\n=εntµ−m\n=N−n(m−tµ)−m (3)\n≤N−m\n=|Y|. (4)\nIt follows from (4) that Y(i),i≤tµ, exists.\nNext, we shall show that TXexists for every X⊆X,|X|=i, with i∈[tµ]. For that,\nit is sufficient to verify that |(Y(0)∪ ··· ∪ Y(i))\\S\nX′⊂XYX′| ≥n. Observe that in our\nconstruction ϕ(X) is chosen in a tη-truncated blob, in which every vertex is larger than\nSX=S\nX′⊂XYX′. Therefore, |YX\\S\nX′⊂XYX′| ≤tη, i.e., we are introducing at most\ntη“new” elements from Yforϕ(X), compared to the images of proper subsets of X. If\n|X|=i, then Xhas 2i−1 proper subsets X′, and ϕ(X′) uses as most tη“new” elements of\nYcompared to its own subsets, so we have that\n|SX|=\f\f\f\f[\nX′⊂XYX′\f\f\f\f=\f\f\f\f[\nX′⊂X\u0012\nYX′\\[\nX′′⊂X′YX′′\u0013\f\f\f\f≤(2i−1)tη. (5)\nUsing (2) and (5) we have\n\f\f\f\f(Y(0)∪ ··· ∪ Y(i))\\[\nX′⊂XYX′\f\f\f\f≥(n+ (2i−1)tη)−(2i−1)tη=n,\nso we can select a set TXof size nfrom ( Y(0)∪ ··· ∪ Y(i))\\S\nX′⊂XYX′.\n3.3 Completion of the proof\nFinally, we consider the volume of ϕ(Q(X)tµ) and obtain the following bound using (3):\nVol(ϕ(Q(X)tµ))≤ |X|+|Y(0)∪ ··· ∪ Y(tµ)| ≤m+ (N−n(m−tµ)−m) =N−n(m−tµ).\nWe conclude that ϕis a blue X-good embedding of Q(X)t\nµwith Vol( ϕ(Q(X)tµ))≤N−\n(m−tµ)n. Thus, by part (ii) of Lemma 7 with t=tµ, we have a blue copy of Qmor a red\ncopy of Qn.\n4 Diagonal Ramsey numbers for DnandVn\n4.1 Diamonds\nRecall that the Sperner number α(n) is the smallest Nsuch that QNcontains an antichain\nof size n, and Sperner [12] showed that\u0000α(n)\n⌊α(n)/2⌋\u0001\n≥n.\n9Proof of Theorem 3. For the lower bound, color the Boolean lattice Q1:=Q([2α(n)−1])\nsuch that Z∈ Q1is red if |Z|< α(n) and blue if |Z| ≥α(n). Assume that there is a red copy\nDofDnin this coloring with maximal vertex Y. Then |Y|< α(n), soDis contained in the\nsublattice {Z∈ Q1:Z⊆Y}. We know that Dcontains an antichain on nvertices, but by\ndefinition of α(n) there is no antichain on nvertices in {Z∈ Q1:Z⊆Y}, a contradiction.\nSimilarly, we obtain that there is no blue copy of Dn.\nIn order to bound R(Dn, Dn) from above, let N=α(n) +α(2n−1) and consider an\narbitrary coloring of Q2:=Q([N]). Let\nS=n\nZ∈ Q2:|Z|=⌊α(n)/2⌋o\nandT=n\nZ∈ Q2:|Z|=N− ⌊α(n)/2⌋o\n.\nWe distinguish two cases.\nCase 1: At least one of S ∪ {∅}orT ∪ { [N]}is not monochromatic.\nSuppose T ∪{ [N]}is not monochromatic. Let Y∈ Tsuch that Yhas a different color than\nthe vertex [ N], see Figure 2. Let S′={Z∈ S:Z⊆Y}.\nIfS′∪ {∅}is monochromatic, then one of Yor [N] has the same color as S′. Note that\n|S′|=\u0000|Y|\n⌊α(n)/2⌋\u0001\n≥\u0000α(n)\n⌊α(n)/2⌋\u0001\n≥n, where the last inequality follows from Sperner’s theorem.\nThis implies that the vertices S′∪ {∅, Y,[N]}contain a monochromatic copy of Dn.\nIfS′∪ {∅}is not monochromatic, then consider X∈ S′such that Xhas a different\ncolor than the vertex ∅. Note that X⊂Y. The sublattice {Z∈ Q2:X⊆Z⊆Y}has\ndimension |Y| − |X| ≥N−α(n)≥α(2n−1). This implies that there is an antichain Aon\n2n−1 vertices, such that for every Z∈ A,X⊆Z⊆Y. Note that X, Y /∈ A. InAwe\nfindnvertices with the same color, say without loss of generality red. Then these nvertices\ntogether with the red vertex among ∅andXand the red vertex among Yand [ N] form a\nred copy of Dn.\n?\n[N]\nY\nT\nS\nS0\nX\nQ2\nA\nFigure 2: Setting in Case 1 if S′∪ {∅}is not monochromatic\nCase 2: BothS ∪ {∅}andT ∪ { [N]}are monochromatic.\nIfS ∪{∅}andT ∪{ [N]}have the same color, then S ∪{∅,[N]}contains a monochromatic\ncopy of Dnbecause |S|=\u0000N\n⌊α(n)/2⌋\u0001\n≥\u0000α(n)\n⌊α(n)/2⌋\u0001\n≥n. So suppose that S∪{∅}andT ∪{[N]}\n10have distinct colors, say S∪{∅}is red and T ∪{[N]}is blue. Fix the vertex X= [α(n)]∈ Q2.\nIfXhas the same color as S ∪ {∅}, i.e. red, then let S′′={Z∈ S:Z⊆X}. Note that\n|S′′| ≥\u0000|X|\n⌊α(n)/2⌋\u0001\n≥n. Then S′′∪ {∅, X}contains a red copy of Dn. IfXhas the same\ncolor as T ∪ { [N]}, we find a monochromatic copy of Dnby a similar argument.\n4.2 Forks\nLetn, N∈Nsuch that N≥α(n). Recall that\nβ(N, n) = min\b\nβ∈N:\u0012N\nβ\u0013\n≥n\t\nand N∗(n) = max\b\nN≥α(n) :N−β(N, n)< α(n)\t\n,\nas illustrated in Figure 3. Both β(N, n) and N∗(n) are well-defined.\nQ([N\u0003])\n\u0015n\n<n\n<n\nlayer\f\nlayer\f\u00001\nlayerbN\u0003+\f\n2c\n?\n[N\u0003]\nFigure 3: Setting for N∗andβ=β(N∗, n)\nProof of Theorem 4. LetN∗=N∗(n). First, we show the lower bound R(Vn, Vn)> N∗.\nWe construct a coloring of Q1:=Q([N∗]) which contains no monochromatic Vn. Color each\nvertex Zwith|Z|< β(N∗, n) in red and all remaining vertices in blue. Then there is no red\nantichain of size n, so in particular there is no red copy of Vn. Assume towards a contra-\ndiction that there is a blue copy of Vnwith minimal vertex X. Note that |X| ≥β(N∗, n),\nso the sublattice {Z∈ Q1:X⊆Z⊆[N∗]}has dimension at most N∗−β(N∗, n)< α(n),\nthus there is no blue antichain of size n, and in particular no blue copy of Vn, a contradiction.\nFor the upper bound, we define N+to be the smallest integer such that N+−β(N+, n)≥\nα(2n−1). In order to show that R(Vn, Vn)≤N+, we consider an arbitrary coloring of\nQ2:=Q([N+]). We shall find a monochromatic copy of Vn. Without loss of generality, the\nvertex ∅is red. We know that layer β+:=β(N+, n) contains at least nvertices. If layer β+\nis red, we find a red copy of Vn, so suppose that there exists a blue vertex Xwith|X|=β+.\nThen the sublattice Q3:={Z∈ Q2:X⊆Z⊆[N+]}has dimension N+−β+≥α(2n−1).\nThus we find an antichain Aof size 2 n−1 inQ3. Note that X /∈ A. Each vertex in Ais\neither red or blue, so there are nvertices of the same color in this antichain, which together\nwith one of Xor∅form a monochromatic copy of Vnas desired. Therefore, R(Vn, Vn)≤N+.\n11We claim that N+≤N∗+3. To prove this, we first show that β(N∗+1, n)≥β(N∗+3, n).\nRecall that N∗≥α(n), so β(N∗+ 1, n)≤ ⌊α(n)/2⌋ ≤N∗/2. This implies that\n\u0012N∗+ 3\nβ(N∗+ 1, n)\u0013\n≥\u0012N∗+ 1\nβ(N∗+ 1, n)\u0013\n≥n,\nthus indeed β(N∗+ 1, n)≥β(N∗+ 3, n). By definition of N∗(n) we know that ( N∗+ 1)−\nβ(N∗+ 1, n)≥α(n), so\n(N∗+ 3)−β(N∗+ 3, n)≥(N∗+ 1) + 2 −β(N∗+ 1, n)≥α(n) + 2≥α(2n−1).\nRecall that N+is minimal such that N+−β(N+, n)≥α(2n−1), so N+≤N∗+ 3. This\nconcludes the proof of the upper bound. We remark that if α(n) + 1≥α(2n−1), a similar\nargument provides N+≤N∗+ 2.\nIn the Appendix we show that N∗(n) = ( d+o(1)) log n, where d=1\n1−candcis the\nunique real solution of log\u0000\nc−c(1−c)c−1\u0001\n= 1−c, i.e. d≈1.29.\n5 Upper bound on Ramsey number Rw(Qn, Qn)\nProof of Theorem 5. Consider an arbitrary coloring of the Boolean lattice Q([N]) where\nN= 0.96n2. Our goal is to find a monochromatic weak copy of Qn. While an induced copy\nofQnhas a rigid structure, there are many non-isomorphic weak copies of Qn. InQ([N])\nwe shall find a member of a class P(n, s, t ) of special weak copies of Qn.\n5.1 Definition of P(n, s, t )\nIn this section we write P′< P′′for posets P′andP′′if any element of a poset P′is less\nthan any element of a poset P′′. We define the class P(n, s, t ) of posets, see Figure 4 (a),\nsuch that each member of this class is of the form\nP0∪ ··· ∪ Ps−1∪Qt\ns∪P′\nt+1∪ ··· ∪ P′\nn,\nwhere\n•Piis an arbitrary poset with |Pi|=\u0000n\ni\u0001\n,i∈ {0, . . . , s −1},\n•P′\njis an arbitrary poset with |P′\nj|=\u0000n\nn−j\u0001\n=\u0000n\nj\u0001\n,j∈ {t+ 1, . . . , n },\n•Qt\nsis an (induced) copy of an ( s, t)-truncated Qn, i.e., consists of layers s, . . . , t ofQn,\n•P0< P 1<···< Ps−1< Qt\ns< P′\nt+1<···< P′\nn.\nHere, if s= 0 or t=n, we use the convention that P0∪···∪ Ps−1=∅orP′\nt+1∪···∪ P′\nn=\n∅, respectively. Observe that every member of P(n, s, t ) is indeed a weak copy of Qn, where\nlayer iofQncorresponds to Pi, fori∈ {0, . . . , s −1}, layer jofQncorresponds to P′\nj, for\nj∈ {t+ 1, . . . , n }and the remaining layers are contained in the middle part Qt\ns.\n12[N]\n?\nZbqnc\nZ0\nbqnc\n: : :\n: : :\nS1\nS2\nS0\nS0\n0\nS0\n1\nS0\n2\nZ0\nZ1\nZ2\nB\nQ([N])\nZ0\n0\nZ0\n1\nZ0\n2\nP(n; s; t )\nQt\ns\nlayers\nlayert\nP0\nt+1\nP0\nn\nP0\nPs\u00001\n<\n<\n<\n<\n<\n<\n<\nP1\n(a)\n(b)Figure 4: (a) A P(n, s, t ) for s= 4 and t=n−3, (b) Sausage chain in Q([N])\n5.2 Construction of a sausage chain in Q([N])\nLetN= 0.96n2. For our proof we need to define a constant qthat satisfies certain properties.\nFor 0 < p < 1, the binomial entropy function is defined as H(p) =−\u0000\nplogp+(1−p) log(1 −\np)\u0001\n,where log is base 2. Let qbe the real number, 0 < q < 1/2, which minimizes (1 −q) +\n2Rq\n0H(s)ds. It can be verified analytically or numerically that such a qsatisfies H(q) = 1 /2,\ni.e., 0 .11< q < 0.111, andRq\n0H(s)ds≤0.033. In particular, for sufficiently large nwe have\nthat\n(1−q)n2+ 2n2Zq\n0H(s) ds≤0.956n2≤N−εn2, (6)\nfor some constant ε >0.\nNext we define a sausage chain inQ([N]), see Figure 4 (b). Let\nZ0⊂Z1⊂ ··· ⊂ Z⌊qn⌋⊂Z′\n⌊qn⌋⊂Z′\n⌊qn⌋−1⊂ ··· ⊂ Z′\n0\nbe vertices in Q([N]) such that for 0 ≤i≤ ⌊qn⌋,\n|Zi|=iX\nj=0\u0012\u0018\nlog\u0012n\nj\u0013\u0019\n+ 1\u0013\nand |Z′\ni|=N−iX\nj=0\u0012\u0018\nlog\u0012n\nj\u0013\u0019\n+ 1\u0013\n.\nWe will argue later that |Z⌊qn⌋| ≤ |Z′\n⌊qn⌋|, which implies that these vertices are well-defined.\nWe define subposets SiandS′\ni, which we refer to as sausages . LetS0={X∈ Q([N]):X⊂\nZ0}, and\nSi={X∈ Q([N]):Zi−1⊆X⊂Zi}.\nSimilarly, let sausages S′\n0={X∈ Q([N]):Z′\n0⊂X}, and\nS′\ni={X∈ Q([N]):Z′\ni⊂X⊆Z′\ni−1}.\n13Moreover, let\nB={X∈ Q([N]):Z⌊qn⌋⊆X⊆Z′\n⌊qn⌋}.\nThe subposet Bis isomorphic to a Boolean lattice of dimension |Z′\n⌊qn⌋| − |Z⌊qn⌋|. Note that\nS0<S1<···<S⌊qn⌋<B<S′\n⌊qn⌋<···<S′\n0.\nThe subposet S0∪. . .S⌊qn⌋∪ B ∪ S′\n⌊qn⌋∪ ··· ∪ S′\n0ofQ([N]) is referred to as sausage chain .\nNote that the sausage chain is well-defined if all vertices Zi, Z′\ni,i∈ {0, . . . ,⌊qn⌋}, exist.\nNote also that |S0|<···<|S⌊qn⌋|and|S′\n⌊qn⌋|>···>|S0|.\n5.3 Finding a member of P(n, s, t )in the sausage chain\nWe shall find a monochromatic member of P(n, s, t ) for some sandtdepending on the\ncoloring, such that the middle part Qt\nsofP(n, s, t ) is embedded into B, each Piis embedded\ninto its own Sℓ, and each P′\niis embedded in its own S′\nℓ′for some ℓ,ℓ′.\nAssume, without loss of generality, that among all sausages S0, . . . ,S⌊qn⌋,S′\n0, . . . ,S′\n⌊qn⌋,\nmost sausages have majority color red, then at least ⌈qn⌉sausages have this majority color.\nAssume further, that there are ssausages among S0, . . . ,S⌊qn⌋with majority color red, which\nwe denote by Si0, . . . ,Sis−1,i0<···< is−1. Note that possibly s= 0. Since i0≥0, we see\nthati1≥1, and similarly for any j∈ {0, . . . , s −1},ij≥j. For j∈ {0, . . . , s −1}, choose Pj\narbitrarily such that Pj⊆ Sij,|Pj|=\u0000n\nj\u0001\n, and Pjis red. These subposets are well-defined\nsince for any i∈ {0, . . . ,⌊qn⌋},|Si|= 2|Zi|−|Zi−1|−1 = 2⌈log(n\ni)⌉+1−1≥2\u0000n\ni\u0001\n−1,so in\nparticular there are at least\u0000n\nij\u0001\n≥\u0000n\nj\u0001\ndistinct red vertices in Sij.\nSimilarly, we find ⌈qn⌉ −ssausages among S′\n0, . . . ,S′\n⌊qn⌋with majority color red, de-\nnoted by S′\ni′\n0, . . . ,S′\ni′\n⌈qn⌉−s−1,i′\n0<···< i′\n⌈qn⌉−s−1. Here, it is possible that ⌈qn⌉−s= 0. For\nj∈ {0, . . . ,⌈qn⌉ −s−1}, choose P′\nn−jarbitrarily, such that P′\nn−j⊆ S′\ni′\nj,|P′\nn−j|=\u0000n\nn−j\u0001\n,\nandP′\nn−jis red. With a similar argument as before, we see that there are indeed at least\u0000n\nn−j\u0001\ndistinct red vertices in S′\ni′\nj.\nSo far we selected Pjforj∈ {0, . . . , s −1}andP′\njforj∈ {t+ 1, . . . , n }, where t=\nn− ⌈qn⌉+s. It remains to verify that Qt\nsis contained in B. For that we shall show\nthat the dimension of Bis large enough to apply Lemma 7 (iv). Recall that H(p) =\n−\u0000\nplogp+ (1−p) log(1 −p)\u0001\n. Stirling’s formula implies that log\u0000N\npN\u0001\n=\u0000\n1 +o(1)\u0001\nH(p)N\nfor any positive integer Nand 0 < p < 1. Thus\ndim(B) =|Z′\n⌊qn⌋| − |Z⌊qn⌋|\n=N−2⌊qn⌋X\ni=0\u0012\u0018\nlog\u0012n\ni\u0013\u0019\n+ 1\u0013\n≥N−4n−2⌊qn⌋X\ni=1log\u0012n\ni\u0013\n≥N−4n−\u0000\n2 +o(1)\u0001\nn⌊qn⌋X\ni=1H\u0012i\nn\u0013\n. (7)\n14Since His an increasing and continuous function on the interval (0 ,1/2) and is bounded\nby 1, we have\n⌊qn⌋X\ni=1H\u0012i\nn\u0013\n≤Zqn+1\n1H\u0012t\nn\u0013\ndt=Zq+1/n\n1/nH(s)nds≤nZq\n0H(s) ds+ 1,\nThus, using (7) and recalling the bound on Nfrom (6), we have\ndim(B)≥N−4n−\u0000\n2 +o(1)\u0001\u0012\nn2Zq\n0H(s) ds+n\u0013\n≥N−2n2Zq\n0H(s) ds−o(n2)\n≥\u0014\n(1−q)n2+ 2n2Zq\n0H(s) ds+εn2\u0015\n−2n2Zq\n0H(s) ds−o(n2)\n≥(1−q)n2+ 2n= (n−qn+ 2)n.\nIn particular, |Z′\n⌊qn⌋| − |Z⌊qn⌋| ≥0, which implies that ZiandZ′\ni,i∈ {0, . . . ,⌊qn⌋}, are\nwell-defined. Since dim( B)≥(n−qn+ 2)n≥(t−s+ 2)nfort=n− ⌈qn⌉+s, Lemma 7\n(iv) provides that Bcontains either a blue (induced) copy of Qnand we are done, or a red\nQt\ns. If there is a red Qt\nsinB, we conclude that the sausage chain contains a red member of\nP(n, s, t ), and thus a red weak copy of Qn.\n6 Concluding remarks\nIn this paper, we first studied induced poset Ramsey numbers and showed that for n≥m\nand 0 < ε < 1 withn+m\nn·1\n(1−ε) logm+m−ε≤ε,\nR(Qm, Qn)≤n\u0000\nm−(1−ε)2logm\u0001\n.\nWhen applying this result for specific ε, there is a trade-off between the best Ramsey\nbound and the smallest value of mfor which the bound holds. Our main result claims that\nR(Qm, Qn)≤n\u0010\nm−\u0000\n1−2√logm\u0001\nlogm\u0011\nfor 225≤m≤n. In addition, for 1024 ≤m≤n\nor 32≤m≤n\n32, one could obtain the upper bound R(Qm, Qn)≤n\u0000\nm−1\n4logm\u0001\n, using\nε=1\n2.\nTheorem 1 is an improvement of the basic upper bound, Lemma 6, by a superlinear\nadditive term and a step towards the following conjecture raised by Lu and Thompson [10].\nConjecture 9 ([10]).Forn≥mand sufficiently large m,R(Qm, Qn) =o(n2).\nWe propose a stronger conjecture:\nConjecture 10. For any ε >0, there is a large enough m0such that for any two m, n∈N\nwithn≥m≥m0,\nR(Qm, Qn)≤n·mε.\nWe remark that for small m, the authors conjectured the following in [2].\nConjecture 11 ([2]).For fixed m∈Nand large n∈N,R(Qm, Qn) =n+o(n).\n15In the last part of this paper we discussed weak poset Ramsey numbers and improved the\npreviously known upper bound Rw(Qn, Qn)≤R(Qn, Qn)≤n2−o(n2) toRw(Qn, Qn)≤\n0.96n2. It is still open whether or not the weak poset Ramsey number is significantly\nsmaller than the (induced) poset Ramsey number. The second author suggests the following\nconjecture.\nConjecture 12. For any n∈N,R(Qn, Qn)−Θ(n)≤Rw(Qn, Qn)≤R(Qn, Qn).\nAcknowledgments: Research was partially supported by DFG grant FKZ AX 93/2-1.\nReferences\n[1] M. Axenovich, and S. Walzer. Boolean lattices: Ramsey properties and embeddings .\nOrder 34(2), 287–298, 2017.\n[2] M. Axenovich, and C. Winter. Poset Ramsey numbers: large Boolean lattice versus a\nfixed poset . Combinatorics, Probability and Computing, 1–16, 2023.\n[3] T. Bohman, and F. Peng. A Construction for Boolean Cube Ramsey Numbers . Order,\n2022.\n[4] H.-B. Chen, W.-H. Chen, Y.-J. Cheng, W.-T. Li, and C.-A. Liu. Ramsey Properties\nfor V-shaped Posets in the Boolean Lattices . Preprint, available at arXiv:2108.08033v1,\n2021.\n[5] C. Cox, and D. Stolee. Ramsey Numbers for Partially-Ordered Sets . Order 35(3), 557–\n579, 2018.\n[6] V. Falgas-Ravry, K. Markstr¨ om, A. Treglown, and Y. Zhao. Existence thresholds and\nRamsey properties of random posets . Random Structures and Algorithms 57(4), 1097–\n1133, 2020.\n[7] D. Gr´ osz, A. Methuku, and C. Tompkins. Ramsey numbers of Boolean lattices . Bulletin\nof the London Mathematical Society 55(2), 914–932, 2023.\n[8] M. Habib, L. Nourine, O. Raynaud, and E. Thierry. Computational aspects of the 2-\ndimension of partially ordered sets . Theoretical Computer Science 312(2-3), 401–431,\n2004.\n[9] H. Kierstead, and W. Trotter. Ramsey theoretic problem for finite ordered sets . Discrete\nMathematics 63(2-3), 217–223, 1987.\n[10] L. Lu, and C. Thompson. Poset Ramsey Numbers for Boolean Lattices . Order 39(2),\n171–185, 2022.\n[11] J. Neˇ setˇ ril, and V. R¨ odl. Combinatorial partitions of finite posets and lattices - Ramsey\nlattices , Algebra Universalis 19(1), 106–119, 1984.\n[12] E. Sperner. Ein Satz ¨ uber Untermengen einer endlichen Menge , (in German). Mathe-\nmatische Zeitschrift, 27(1), 544–548, 1928.\n[13] S. Walzer. Ramsey Variant of the 2-Dimension of Posets. In: Master Thesis, Karlsruhe\nInstitute of Technology (2015).\n16[14] C. Winter. Erd˝ os-Hajnal problems for posets. Preprint, available at arXiv:2310.02621,\n2023.\n[15] C. Winter. Poset Ramsey number R(P,Qn). I. Complete multipartite posets . Order,\n2023.\n7 Appendix - Calculation for the proof of Theorem 4\nWe shall find dsuch that N∗= (d+o(1)) log n, where\nN∗= max\b\nN≥α(n) :N−β(N, n)< α(n)\t\nand β(N, n) = min\b\nβ∈N:\u0012N\nβ\u0013\n≥n\t\n.\nFor arbitrary N∈Nandqwith 0 < q < 1, Stirling’s formula provides that\nlog\u0012N\nqN\u0013\n=−\u0000\n1 +o(1)\u0001\nN\u0000\nqlogq+ (1−q) log(1 −q)\u0001\n=\u0000\n1 +o(1)\u0001\nH(q)N, (8)\nwhere H(q) =−\u0000\nqlogq+ (1−q) log(1 −q)\u0001\nis the binary entropy function . Let cbe the\nunique solution of 1 −c=H(c), i.e. c≈0.2271. We shall show that N∗=\u00001\n1−c+o(1)\u0001\nlogn.\nLetqsuch that qN∗=β(N∗, n), and let q′such that q′(N∗+ 1) = β(N∗+ 1, n). The\ndefinition of βimplies that\n\u0012N∗\nqN∗−1\u0013\n< n≤\u0012N∗\nqN∗\u0013\nand\u0012N∗+ 1\nq′(N∗+ 1)−1\u0013\n< n≤\u0012N∗+ 1\nq′(N∗+ 1)\u0013\n, (9)\nBy the definition of N∗(n), we know that\n(1−q)N∗< α(n)≤(1−q′)(N∗+ 1). (10)\nIn the following, o(1) always refers to the asymptotic behaviour for large n, so equivalently\nfor large N∗asα(n)≤N∗≤2α(n). Recall that\nα(n) =\u0000\n1 +o(1)\u0001\nlogn. (11)\nWe shall compute q. We label each step of our computation by an inequality from (8)\nto (11) which is being used. For example, ‘(8)=’ means that the equality holds because of (8).\nTo highlight the idea of the upcoming computation, we give a one-line proof, where some\nsteps are not yet justified:\n(1−q)N∗(10)≈\u0000\n1+o(1)\u0001\nα(n)(11)=\u0000\n1+o(1)\u0001\nlogn(9)≈\u0000\n1+o(1)\u0001\nlog\u0012N∗\nqN∗\u0013\n(8)=\u0000\n1+o(1)\u0001\nH(q)N∗,\nwhich would imply q=\u0000\n1 +o(1)\u0001\ncwhere cis the unique solution of 1 −c=H(c). However,\nsome steps in the above computation require significant additional argumentation, which we\ngive in the following.\nObserve that\n(1−q)N∗(10)\n< α(n)(11)=\u0000\n1 +o(1)\u0001\nlogn(9)\n≤\u0000\n1 +o(1)\u0001\nlog\u0012N∗\nqN∗\u0013\n(8)=\u0000\n1 +o(1)\u0001\nH(q)N∗.\n17Thus, 1 −q≤\u0000\n1 +o(1)\u0001\nH(q), which implies that q≤\u0000\n1 +o(1)\u0001\nc. Next we bound q′from\nbelow. We see that\n(1−q′)(N∗+ 1)(10)\n≥α(n)(11)=\u0000\n1 +o(1)\u0001\nlogn(9)\n>\u0000\n1 +o(1)\u0001\nlog\u0012N∗+ 1\nq′(N∗+ 1)−1\u0013\n.\nWe shall show that log\u0000N∗+1\nq′(N∗+1)−1\u0001\n≥\u0000\n1 +o(1)\u0001\nH(q′)(N∗+ 1). For that, we first need a\nrough lower bound on q′.\nWe know from (10) that N∗−qN∗≤α(n)−1. Note that qN∗=β(N∗, n)≤α(n), so\nN∗+ 1≤qN∗+α(n)≤2α(n). Then\n\u0012N∗+ 1\n1\n16(N∗+ 1)\u0013\n≤\u00122α(n)\n1\n8α(n)\u0013\n(8)=\u001222\n(1\n8)1/8(15\n8)15/8\u0013(1+o(1))α(n)\n≤1.6(1+o(1)) log n< n,\nthus q′≥1\n16. This bound implies that\n\u0012N∗+ 1\nq′(N∗+ 1)−1\u0013\n=q′(N∗+ 1)\n(1−q′)(N∗+ 1) + 1\u0012N∗+ 1\nq′(N∗+ 1)\u0013\n≥q′\n2−q′\u0012N∗+ 1\nq′(N∗+ 1)\u0013\n≥1\n31\u0012N∗+ 1\nq′(N∗+ 1)\u0013\n.\nThus,\nlog\u0012N∗+ 1\nq′(N∗+ 1)−1\u0013\n≥ −log(31) + log\u0012N∗+ 1\nq′(N∗+ 1)\u0013\n(8)=\u0000\n1 +o(1)\u0001\nH(q′)(N∗+ 1).\nTherefore, 1 −q′≥\u0000\n1 +o(1)\u0001\nH(q′), which implies that q′≥\u0000\n1 +o(1)\u0001\nc.\nWe obtain that\nα(n)(10)\n≤(1−q′)(N∗+1)≤(1+o(1))(1−c)(N∗+1)≤\u0000\n1+o(1)\u0001\n(1−q)N∗(10)\n≤\u0000\n1+o(1)\u0001\nα(n),\nthus N∗=(1+o(1))\n1−cα(n) =(1+o(1))\n1−clogn,as desired.\n18"    },    {        "title": "2402.13432v1.DrBenchmark__A_Large_Language_Understanding_Evaluation_Benchmark_for_French_Biomedical_Domain.pdf",        "content": "DrBenchmark: A Large Language Understanding Evaluation\nBenchmark for French Biomedical Domain\nYanis Labrak∗1,2, Adrien Bazoge∗3,4, Oumaima El Khettari4\nMickael Rouvier1, Pacôme Constant dit Beaufils5,Natalia Grabar6,Béatrice Daille4\nSolen Quiniou4,Emmanuel Morin4,Pierre-Antoine Gourraud3andRichard Dufour1,4\n1LIA, Avignon Université2Zenidoc\n3Nantes Université, CHU Nantes, Clinique des données, INSERM, CIC 1413, F-44000 Nantes, France\n4Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France\n5Nantes Université, CHU Nantes, Service de Neuroradiologie diagnostique et\ninterventionnelle, CNRS, INSERM, l’institut du thorax, F-44000 Nantes, France\n6UMR 8163 – STL CNRS, Université de Lille\n{first.last}@{univ-avignon.fr, univ-nantes.fr, chu-nantes.fr, univ-lille.fr}\nAbstract\nThe biomedical domain has sparked a significant interest in the field of Natural Language Processing (NLP), which\nhas seen substantial advancements with pre-trained language models (PLMs). However, comparing these models\nhasprovenchallengingduetovariationsinevaluationprotocolsacrossdifferentmodels. Afairsolutionistoaggregate\ndiverse downstream tasks into a benchmark, allowing for the assessment of intrinsic PLMs qualities from various\nperspectives. Althoughstilllimitedtofewlanguages, thisinitiativehasbeenundertakeninthebiomedicalfield, notably\nEnglishandChinese. ThislimitationhamperstheevaluationofthelatestFrenchbiomedicalmodels, astheyareeither\nassessed on a minimal number of tasks with non-standardized protocols or evaluated using general downstream\ntasks. To bridge this research gap and account for the unique sensitivities of French, we present the first-ever\npublicly available French biomedical language understanding benchmark called DrBenchmark. It encompasses\n20 diversified tasks, including named-entity recognition, part-of-speech tagging, question-answering, semantic\ntextual similarity, and classification. We evaluate 8 state-of-the-art pre-trained masked language models (MLMs) on\ngeneral and biomedical-specific data, as well as English specific MLMs to assess their cross-lingual capabilities. Our\nexperimentsrevealthatnosinglemodelexcelsacrossalltasks,whilegeneralistmodelsaresometimesstillcompetitive.\nKeywords: NLP evaluation, Benchmarking, Medical domain, French language, Transformers\n1. Introduction\nFor the past few years, the field of Natural Lan-\nguage Processing (NLP) has witnessed major\nbreakthroughs, particularly in the area of language\nmodeling. Newer approaches such as the self-\nattentionmechanism(Vaswanietal.,2017),Sparse\nTransformer (Child et al., 2019), and Replaced To-\nken Detection (Clark et al., 2020) have emerged.\nThese advancements have enabled the application\nof language models pre-trained on large corpora\nof textual data to a wide range of NLP tasks.\nThe evaluation of proposed models and ap-\nproaches is an essential step in verifying their qual-\nity and performance. In the context of pre-trained\nlanguage models (PLMs), this validation typically\ninvolves assessing their performance on targeted\ndownstream tasks. This task-selection process is\ncrucial, as the performance of models can vary\ndepending on the chosen ones. Consequently, a\nmodelthatperformswellinonecontextmaydeliver\ndisappointing results in another one. To address\nthis issue and validate the models’ generalizabil-\nity, evaluation benchmarks have emerged, typically\nencompassing diverse sets of tasks. Hence, theavailability of evaluation benchmarks plays a vital\nroleindrivingcontinuousprogress,fosteringthede-\nvelopment of community members, and facilitating\nfair comparison between models.\nWhile numerous open benchmarks exist for gen-\neral tasks in NLP across multiple languages, the\nbiomedical field remains an area with relatively few\nproposed benchmarks, mainly for English and Chi-\nnese, facilitatingtheavailabilityofmanybiomedical\nmodels in these two languages. Even if the gap in\nother languages is beginning to narrow with new\nspecialized models, the development of evaluation\nplatforms has been comparatively slower.\nAlthough the French language is generally con-\nsidered as well-endowed, it is notably lacking in\nevaluationresourceswithinthebiomedicalfield. To\naddressthisissue, weintroduceDrBenchmark, the\nfirstcomprehensiveopenbenchmarkfortheFrench\nbiomedical domain, comprising 20 diverse tasks.\nThese tasks encompass part-of-speech (POS) tag-\nging, named-entity recognition (NER), classifica-\ntion, question-answering (QA), and semantic tex-\ntual similarity (STS).\nWe also perform a quantitative study of 8 pre-\ntrained state-of-the-art masked language modelsarXiv:2402.13432v1  [cs.CL]  20 Feb 2024(MLMs) with different configurations (languages\nand domains) on DrBenchmark. Our in-depth anal-\nysis integrates a comparison of these models’ per-\nformance, fine-tuning with limited data and word\ntokenization. Our main contributions are:\n•DrBenchmark, an original evaluation frame-\nwork for French biomedical NLP domain ag-\ngregating a large set of 20 diversified, proven\nand challenging tasks.\n•A quantitative study using our proposed\nbiomedical benchmark on a wide range of\n8 MLMs based on varied architectures, data\nsources and training strategies.\n•The release under CC BY-SA 4.0 license\non HuggingFace1of a new open biomedical\ndatasetwithclinicalcasesmanuallyannotated\ninto the 22 International Classification of Dis-\neases 10th Revision (ICD-10) categories.\n•A modular, reproducible and easily customiz-\nable automated protocol using identical train-\ning and evaluation scripts allowing a simple\nand fair comparison, with, as input, only the\nevaluated language models. DrBenchmark is\nfreely available under MIT license on GitHub,\nHuggingFace and summarized as a leader-\nboard on the website2.\nThe paper is organized as follows: Section 2\nbriefly introduces historical NLP benchmarks, in-\ncluding biomedical ones. Section 3 presents our\nproposed benchmark for French biomedical do-\nmain, with a focus on the downstream tasks. Sec-\ntion4presentstheexperimentalprotocolwhileSec-\ntion 5 details the results and provides an analysis\nof the studied pre-trained models. Section 6 finally\nconcludes the work and opens some perspectives.\n2. Related work\nIn the recent years, several NLP open benchmarks\nhave been created to facilitate direct comparison\nbetween proposed approaches. Among the earli-\nest benchmarks, DecaNLP (McCann et al., 2018)\nand GLUE (Wang et al., 2018), focused on general\nEnglish language understanding tasks rather than\nbeing specific to a particular domain. Thus, GLUE\ngathers nine tasks including text classification (lin-\nguistic acceptability, sentiment analysis, etc.), se-\nmantic analysis (paraphrase verification, sentence\nsimilarity, etc.), QA, coreference detection, and nat-\nural language inference (NLI). Following a similar\nconcept, the French counterpart to GLUE, known\nas FLUE (Le et al., 2020), consists of 7 general-\ndomain tasks in French, covering areas such as\n1https://huggingface.co/DrBenchmark\n2https://github.com/DrBenchmark/DrBenchmarktext classification, paraphrasing, NLI, parsing, and\nword sense disambiguation.\nIn the case of specialized domains, general\nbenchmarks may not adequately evaluate the per-\nformance of in-domain models. Specifically, within\nthe biomedical domain, only few benchmarks have\nbeenproposed,andtheyprimarilyfocusonfewlan-\nguages. For instance, platforms like BLURB (Gu\net al., 2021) and BLUE (Peng et al., 2019) pre-\ndominantly offer benchmarks for English, while\nCBLUE (Zhang et al., 2022a) caters to the Chinese\nlanguage. To provide more specific information,\nBLURB integrates 13 tasks, including NER, infor-\nmation and relation extraction, sentence similarity,\ntext classification, and QA. BLUE encompasses 10\ntasks, such as NER, sentence similarity, relation\nextraction,textclassification,andinference. Onthe\nother hand, CBLUE covers 8 tasks, including NER,\ninformation extraction, text and intent classification,\nsentence similarity, and query relevance.\nTo our knowledge, aside the multilingual bench-\nmark BigBIO (Fries et al., 2022) which includes\nonly 4 corpora for French and is initially intended\nfor generative text completion under zero-shot sce-\nnario,nolargebenchmarkspecializedintheFrench\nbiomedicalfieldexists. Thismakesthecomparison\nofrecentspecializedmodels,suchas(Labraketal.,\n2023; Touchent et al., 2023; Copara et al., 2020;\nBerhe et al., 2023), extremely challenging.\n3. DrBenchmark Overview\nOur proposed benchmark comprises 20 French\nbiomedical language understanding tasks, one of\nwhich is specifically created for this benchmark.\nThe descriptions and statistics of these tasks are\npresented in Table 1. DrBenchmark encompasses\nthe following overall aspects:\n1.A variety of tasks with different require-\nmentsandobjectives: Part-of-Speech(POS)\ntagging,Multi-class,Multi-labelandIntentclas-\nsification, Named-Entity Recognition (NER),\nMultiple-Choice Question-Answering (MCQA),\nand Semantic Textual Similarity (STS).\n2.A diverse range of data origins: Scientific\nliterature, clinical trials, clinical cases, speech\ntranscriptions, and more as described in Ta-\nble 2.\nPleasenotethatwithinDrBenchmark,weinclude\nclassical tasks like NER and POS tagging, as well\nas more specific and challenging tasks like MCQA\nand multi-label classification. In Section 3.1, we\nprovide an overview of the different French down-\nstreamtasks,while,inSection3.2,weofferinsights\ninto the pipeline and its reproducibility.Dataset Task Metric Train Validation Test License\nCAS POS tagging SeqEval F1 2,653 379 758 DUA\nESSAI POS tagging SeqEval F1 5,072 725 1,450 DUA\nQUAERONER - EMEA SeqEval F1 429 389 348 GFDL 1.3\nNER - MEDLINE SeqEval F1 833 832 833 GFDL 1.3\nE3CNER - Clinical SeqEval F1 969 140 293 CC BY-NC\nNER - Temporal SeqEval F1 969 140 293 CC BY-NC\nMorFITT Multi-label Classification Weighted F1 1514 1,022 1,088 CC BY-SA 4.0\nFrenchMedMCQAQuestion-Answering Hamming / EMR 2,171 312 622 Apache 2.0\nMulti-class Classification Weighted F1 2,171 312 622 Apache 2.0\nMantra-GSCNER - EMEA SeqEval F1 70 10 20 CC BY 4.0\nNER - Medline SeqEval F1 70 10 20 CC BY 4.0\nNER - Patents SeqEval F1 35 5 10 CC BY 4.0\nCLISTER Semantic Textual Similarity EDRM / Spearman 499 101 400 DUA\nDEFT-2020Semantic Textual Similarity EDRM / Spearman 498 102 410 DUA\nMulti-class Classification Weighted F1 460 112 530 DUA\nDEFT-2021Multi-label Classification Weighted F1 118 49 108 DUA\nNER SeqEval F1 2,153 793 1,766 DUA\nDiaMed Multi-class Classification Weighted F1 509 76 154 CC BY-SA 4.0\nPxCorpusNER SeqEval F1 1,386 198 397 CC BY 4.0\nMulti-class Classification Weighted F1 1,386 198 397 CC BY 4.0\nTable 1: Descriptions and statistics of the 20 tasks included in DrBenchmark.\nDataset Sources\nCAS Clinical cases\nESSAI Clinical trial protocols\nQUAERO Drug leaflets & Biomedical titles\nE3C Clinical cases\nMorFITT Biomedical abstracts\nFrenchMedMCQA Pharmacy Exam\nMantra-GSC Biomedical abstract / titles, drug labels, & patent\nCLISTER Clinical cases\nDEFT-2020 Clinical cases, encyclopedia & drug labels\nDEFT-2021 Clinical cases\nDiaMed Clinical cases\nPxCorpus Drug prescriptions transcripts\nTable 2: Data sources covered by each datasets.\n3.1. Downstream tasks\nDEFT-2020 (Cardon et al., 2020) contains clini-\ncalcases,encyclopediaanddruglabelsintroduced\nin the 2020 edition of an annual French Text Min-\ning Challenge, called DEFT, and annotated for two\ntasks: (i) textual similarity and (ii) multi-class classi-\nfication. Thefirsttaskaimsatidentifyingthedegree\nof similarity within pairs of sentences, from 0 (the\nless similar) to 5 (the most similar). The second\ntask consists in identifying, for a given sentence,\nthe most similar sentence among three sentences\nprovided.\nDEFT-2021 (Grouin et al., 2021) is a subset of\n275 clinical cases taken from the 2019 edition of\nDEFT. This dataset is manually annotated in two\ntasks: (i)multi-labelclassificationand(ii)NER.The\nmulti-label classificationtask focuseson identifying\nthe patient’s clinical profile based on the diseases,\nsigns, or symptoms mentioned in the clinical cases.\nThe dataset is annotated with 23 axes from Chap-\nter C of the Medical Subject Headings (MeSH).\nThe second task involves fine-grained informationextraction for 13 types of entities (more detail in\nAppendix B.7).\nE3C(Magnini et al., 2020) is a multilingual\ndataset of clinical cases annotated for the NER\ntask. It consists of two types of annotations (more\ndetail in Appendix B.4): (i) clinical entities (e.g.,\npathologies), (ii) temporal information and factual-\nity (e.g., events). While the dataset covers 5 lan-\nguages, only the French portion is retained for the\nbenchmark. Since the dataset does not come with\npre-defined subsets, we performed a 70 / 10 / 20\nrandom split, as described in Table 3.\nSubset Train Validation Test\nClinical 87.38 % of layer 2 12.62 % of layer 2 100 % of layer 1\nTemporal 70 % of layer 1 10 % of layer 1 20 % of layer 1\nTable 3: Description of the sources for E3C.\nThe QUAERO French Medical Corpus (Névéol\net al., 2014), simply referred to as QUAERO in this\npaper, contains annotated entities and concepts\nfor NER tasks. The dataset covers two text genres\n(drug leaflets and biomedical titles), consisting of\na total of 103,056 words sourced from EMEA or\nMEDLINE. 10 entity categories corresponding to\nthe UMLS Semantic Groups (Lindberg et al., 1993)\nwere annotated (more detail in Appendix B.3).. In\ntotal, 26,409 entity annotations were mapped to\n5,797 unique UMLS concepts. Due to the pres-\nence of nested entities in annotations, we simpli-\nfied the evaluation process by retaining only an-\nnotations at the higher granularity level from the\nBigBio(Friesetal.,2022)implementation,following\nthe approach described in Touchent et al. (2023),\nwhich translates into an average loss of 6.06% oftheannotationsonEMEAand8.90%onMEDLINE.\nAdditionally,consideringthatsomedocumentsfrom\nEMEAexceedthemaximuminputsequencelength\nthat most current language models can handle, we\ndecided to split these documents into sentences.\nMorFITT (Labrak et al., 2023) is a multi-label\ndataset annotated with medical specialties. It con-\ntains 3,624 biomedical abstracts from PMC Open\nAccess. It has been annotated across 12 medical\nspecialties (more detail in Appendix B.5), for a total\nof 5,116 annotations.\nFrenchMedMCQA (Labrak et al., 2022) is\na Multiple-Choice Question-Answering (MCQA)\ndataset for biomedical domain. It contains 3,105\nquestions coming from real exams of the French\nmedical specialization diploma in pharmacy, inte-\ngrating single and multiple answers. The first task\nconsists of automatically identifying the set of cor-\nrect answers among the 5 proposed for a given\nquestion. The second task consists of identifying\nthe number of answers (between 1 and 5) suppos-\nedly correct for a given question.\nMantra-GSC (Kors et al., 2015) is a multilingual\ndataset annotated for biomedical NER. From the 5\nlanguages covered, we included only the French\nsubset in this benchmark. The dataset is obtained\nfrom 3 sources which have been partitioned to\nbe evaluated separately by 2 annotation schemes\n(moredetailinAppendixB.6): Medline(11classes),\nand EMEA and Patents (10 classes). The sources\ncover different types of documents (biomedical\nabstracts/titles, drug labels and patents). To en-\nsure evaluation consistency, we randomly split the\ndataset into 3 subsets: 70% for training, 10% for\nvalidation, and 20% for testing.\nCLISTER (Hiebel et al., 2022) is a French clini-\ncal cases Semantic textual similarity (STS) dataset\nof 1,000 sentence pairs manually annotated by\nseveral annotators, who assigned similarity scores\nranging from 0 to 5 to each pair. The scores were\nthen averaged together to obtain a floating-point\nnumber representing the overall similarity. The ob-\njective of this dataset is to develop models that can\nautomatically predict a similarity score that closely\naligns with the reference score based solely on the\ntwo sentences provided.\nCAS(Grabar et al., 2018) comprises 3,790 clini-\ncal cases that have been annotated for POS tag-\nging with 31 classes using automatic annotations\nthrough Tagex3, with an evaluation conducted by\ncomparing the automatic outputs against manual\n3https://allgo.inria.fr/app/tagexannotations. Thisevaluationyielded98%precision.\nSince the dataset does not come with predefined\nsubsets, we made the decision to randomly split it\ninto 3 subsets of 70%, 10% and 20% of the total\ndata for training, validation and test respectively.\nESSAI (Dalloux et al., 2021) contains 7,247 clin-\nical trial protocols annotated in 41 POS tags using\nTreeTagger (Schmid, 1994). As the dataset was\nnot originally divided into 3 subsets, we applied the\nsame procedure as on the CAS corpus.\nPxCorpus (Kocabiyikoglu et al., 2022) is a spo-\nken language understanding dataset in the domain\nof medical drug prescription transcripts. It includes\n4 hours (1,981 recordings) of transcribed and an-\nnotated dialogues focused on drug prescriptions.\nThe recordings were manually transcribed and se-\nmantically annotated. The first task involves classi-\nfying the textual utterances into one of the 4 intent\nclasses (prescribe, replace, negate, none). The\nsecond task is a NER task where each word in a\nsequence is classified into one of 38 classes, such\nas drug, dose, or mode (more detail in Appendix\nB.9).\nDiaMed is an original dataset created specifically\nfor DrBenchmark. It comprises 739 new French\nclinicalcasescollectedfromanopensourcejournal\n(ThePanAfricanMedicalJournal). Thecaseshave\nbeen manually annotated by several annotators,\none of which is a medical expert, into 22 chapters\nof the International Classification of Diseases, 10th\nRevision (ICD-10) (Wor, 2019). These chapters\nprovide a general description of the type of injury\nor disease. To ease the annotation process, only\nlabel at the chapter level were used (more detail in\nAppendix B.8). The inter-annotator agreement be-\ntween the 4 annotators has been computed for two\nannotation sessions (see Table 4), with 15 different\nclinical cases assessed per session.\nSession 1 - 0 to 15 docs Session 2 - 15 to 30 docs\nAnnotator ID κ G κ G\nAnnotator 1 & 2 0.538 0.5660.697 0.705\nAnnotator 1 & 3 0.682 0.7090.697 0.705\nAnnotator 1 & 4 0.397 0.4290.548 0.558\nAnnotator 2 & 3 0.311 0.3571.000 1.000\nAnnotator 2 & 4 0.472 0.4970.672 0.707\nAnnotator 3 & 4 0.311 0.3540.672 0.707\nAverage 0.452 0.4850.714 0.730\nTable 4: Inter-annotator agreement statistics. κis\nreferring to Kappa Cohen and Gto Gwet’s AC1.\n3.2. Reproducibility and usage\nTo facilitate the adoption of DrBenchmark and en-\nsure consistency in implementations, we have de-Model Tokenizer Vocabulary Pretraining Corpus Text Size\nFrench GeneralistCamemBERTa SentencePiece 32K CCNET from-scratch CCNET 4 GB\nCamemBERT SentencePiece 32K OSCAR from-scratch OSCAR 138 GB\nFlauBERT BPE 50K Wiki + Web crawl from-scratch Wiki + Web crawl 71 GB\nFrench BiomedicalDrBERT-FS SentencePiece 32K NACHOS from-scratch NACHOS 7.4 GB\nDrBERT-CP WordPiece 30K PubMed continual pretraining PubMed + NACHOS 21 + 4 GB\nCamemBERT-bio SentencePiece 32K OSCAR continual pretraining OSCAR + biomed-fr 138 + 2.7 GB\nCross-lingual Generalist XLM-RoBERTa WordPiece 30K CC-100 from-scratch CC-100 2.5 TB\nEnglish Biomedical PubMedBERT WordPiece 30K PubMed from-scratch PubMed 21 GB\nTable 5: Summary of the pre-training specifications for the different BERT-based models compared.\nveloped a practical toolkit based on the Hugging-\nFace Datasets library (Lhoest et al., 2021). This\ntoolkit includes data loaders that adhere to nor-\nmalized schemes and predefined data splits. It\nalso provides pre-training and evaluation scripts for\neach of the tasks, utilizing the HuggingFace Trans-\nformers (Wolf et al., 2020) and PyTorch (Paszke\netal.,2019)libraries. Forfurtherguidance,wehave\nintegratedallthetrainingdetails,includinghyperpa-\nrameters, in Appendix A. This information will help\nusers to reproduce and customize the experiments\nconducted with DrBenchmark.\n4. Experimental Protocol\nIn this section, we outline the experimental proto-\ncol used to compare the performance of existing\nlanguage models within DrBenchmark. To guaran-\ntee fair comparison, we focus exclusively on pre-\ntrained masked language models (MLMs) in this\nstudy. These MLMs are based on BERT-like archi-\ntectures (Devlin et al., 2019).\nWe first provide a brief overview in Section 4.1\nof the 8 pre-trained language models that were\nstudied: French generalist models (CamemBERT,\nCamemBERTa and FlauBERT), cross-lingual gen-\neralist model (XLM-RoBERTa), French biomedical\nmodels (DrBERT and CamemBERT-bio), and En-\nglish biomedical model (PubMedBERT). Subse-\nquently, in Section 4.2, we describe the evaluation\nprotocol employed to assess the performance of\nthese models.\n4.1. Pre-trained Masked Language\nModels\nTable 5 summarizes the models and their parame-\nters compared on DrBenchmark.\nCamemBERT (Martinetal.,2020)isaRoBERTa\nbased model for French, pre-trained from-scratch\non the generalist French 138 GB subset of OSCAR\ncorpus (Ortiz Suarez et al., 2019).\nCamemBERTa (Antounetal.,2023)isaDeBER-\nTaV3 (He et al., 2023) based model pre-trainedfrom-scratch on around 30% of the French sub-\nset of CCNETcorpus (Wenzek et al., 2020) used for\nCamemBERT CCNET, that had seen approximately\n133 billion tokens during its pre-training.\nFlauBERT (Le et al., 2020) is a BERT based\nmodel pre-trained from-scratch using a subsam-\nple of 71 GB of the French Common Crawl and\nWikipedia corpora.\nXLM-RoBERTa (Conneauetal.,2020)isacross-\nlingual RoBERTa based model trained on 116 lan-\nguages, including French, by using 2.5 TB of the\nCommonCrawl corpus.\nPubMedBERT (Guetal.,2021)isaBERTbased\nbiomedical-specific model pre-trained from-scratch\non the 3.1 billion words of the PubMedcorpus (21\nGB). This is the only model for English.\nDrBERT-FS and DrBERT-CP (Labrak et al.,\n2023) are French biomedical MLMs built using a\nfrom-scratch pre-training of RoBERTa (DrBERT-\nFS) and continual pre-training of PubMedBERT\n(DrBERT-CP) from the French public biomedical\ncorpus NACHOS(Labraketal.,2023)integrating1.08\nbillionwords(7.4GB)and646millionwords(4GB)\nrespectively.\nCamemBERT-bio (Touchent et al., 2023)\nis a French biomedical language model\nbuilt using a continual pre-training of the\nCamemBERT OSCAR −138GBmodel. It was trained\non the French public corpus biomed-fr (Touchent\net al., 2023) with 413 million words (2.7 GB) and a\nwide range of data collected on the web.\n4.2. Models evaluation\nAll the models are fine-tuned regarding a strict pro-\ntocol using the same hyperparameters for each\ndownstream task. The reported results are ob-\ntained by averaging the scores from four separate\nruns, thus ensuring robustness and reliability. We\nalso report statistical significance computed using\nStudent’s t-test.French Generalist French Biomedical English Biomedical Cross-lingual Generalist\nDataset Task Baseline CamemBERT CamemBERTa FlauBERT DrBERT-FS DrBERT-CP CamemBERT-bio PubMedBERT XLM-RoBERTa\nCAS POS 23.50 95.53** 96.56** 95.22** 96.93 96.46** 95.22** 94.82** 96.91\nESSAI POS 26.31 97.38** 98.08** 97.05* 98.41 98.01** 97.39** 97.42** 98.34\nQUAERONER EMEA 8.37 62.68** 64.86** 74.86 64.11** 67.05 ** 66.59** 53.19** 64.47**\nNER MEDLINE 4.92 55.25** 55.60** 48.98 55.82** 60.10 58.94 53.26** 51.12**\nE3CNER Clinical 4.47 54.70** 55.53 47.61 54.45 56.55 56.96 38.34 52.87**\nNER Temporal 21.74 83.45 83.22 61.64 81.48** 83.43 83.44 80.86** 82.6\nMorFITT Multi-Label CLS 3.24 64.21** 66.28** 70.25 68.70** 70.99 67.53** 68.58** 67.28**\nFrenchMedMCQAMCQA 21.83 /11.5728.53 / 2.25** 29.77 / 2.57** 27.88 / 2.09** 31.07 / 3.22 ** 32.41 / 2.89** 35.3/ 1.45 32.90 / 1.61** 34.74/ 2.09**\nCLS 8.37 66.21 64.44** 61.88 65.38 66.22 65.79 65.41* 64.69*\nMantraGSCNER FR EMEA 0.00 29.14** 40.84** 66.20 66.23 60.88 30.63** 40.14** 52.64*\nNER FR Medline 7.78 23.20** 22.55** 20.69 42.38 35.52 23.66** 27.53* 18.73*\nNER FR Patents 6.20 00.00** 44.16 ** 31.47** 57.34 39.68 00.00** 4.51** 8.58**\nCLISTER STS 0.44 / 0.00 0.55 / 0.33** 0.56 / 0.47** 0.50 / 0.29** 0.62/ 0.57** 0.60 / 0.49* 0.54 / 0.26** 0.70/0.78 0.49 / 0.23**\nDEFT-2020STS 0.49 / 0.00 0.59 / 0.58** 0.59 / 0.43** 0.58 / 0.51** 0.72 / 0.81 * 0.73 /0.86 0.58 / 0.32** 0.78/0.86 0.60 / 0.26**\nCLS 14.00 96.31 97.96 42.37** 82.38 95.71* 94.78* 95.33* 67.66**\nDEFT-2021Multi-Label CLS 24.49 18.04** 18.04** 39.21 34.15** 30.04** 17.82** 25.53** 24.46**\nNER 0.00 62.76** 62.61** 33.51 60.44** 63.43 * 64.36 60.27** 60.32**\nDiaMED CLS 15.36 30.40** 24.05** 34.08** 60.45 54.43** 39.57** 54.96** 26.69**\nPxCorpusNER 10.00 92.89** 95.05** 47.57 95.88 71.38 93.08** 94.66** 95.80\nCLS 84.78 94.41 93.95 93.45* 94.43 94.52 94.49 93.12 93.91\nTable 6: Performance of the studied models over 4 runs. Best model in bold and second is underlined.\nStatistical significance is computed using Student’s t-test: * stands for p < 0.05, ** stands for p < 0.01.\nTo ensure a fair and consistent comparison\namong systems for sequence-to-sequence tasks\nsuch as POS tagging and NER, we chose the Se-\nqEval (Nakayama, 2018) metric in conjunction with\ntheIOB2formatandthetrainingofallthemodelsto\npredictonlythelabelonthefirsttokenofeachword\nas mentioned by Touchent et al. (2023). It provides\na tokenizer-agnostic evaluation and mitigates any\ncorrelation between models’ performances and the\ntokenization process.\nFor STS tasks, the models’ performance was\nassessed using two metrics: (1) the Spearman\ncorrelation, and (2) the mean relative solution dis-\ntance accuracy (EDRM), as defined by the original\nauthors of the DEFT-2020 dataset (Cardon et al.,\n2020).\n5. Experiments and Results\nIn Section 5.1, we compare the results obtained\nby each model within DrBenchmark, which permits\nto position a wide range of state-of-the-art models\nin the biomedical field across various NLP tasks.\nThen, we propose to gain a comprehensive un-\nderstanding of the models’ behavior by examining\nareas such as low-resource fine-tuning scenarios\n(Section 5.2) and the analysis of word tokenization\nof the studied models (Section 5.3).\n5.1.Comparison of models performance\nThe results of the 8 models are reported in Table 6\nand compared to a baseline obtained by consider-\ning the majority class for all predictions. Overall,\nalthough we might anticipate certain models to ex-\ncel in all tasks, we discovered that no single model\noutperforms the rest in all application scenarios.Interestingly, most of the models examined man-\nage to secure the top position in at least one of the\nFrench biomedical downstream tasks studied. The\nonlyexceptionpertainstothecross-lingualgeneral-\nist model ( XLM-RoBERTa ), which manages to reach\nthe second-best position on several tasks.\nDespite this unexpected outcome, we ob-\nserve that French biomedical language models\n(DrBERT-FS ,DrBERT-CP ,CamemBERT-bio ),presumed\nto be the most aligned with the nature of the\ndata of the benchmark, exhibit indeed superior\nperformance across many tasks. More precisely,\nDrBERT-FS achieves the highest performance in 8\ntasks, DrBERT-CP in 5 tasks, and CamemBERT-bio in\n2 tasks. This indicates that domain and language-\nspecialized models achieve the best performance\nin up to 75% of the DrBenchmark downstream\ntasks.\nBiomedical vs. Generalist. The nature of the\ndata appears to have an influence. General-\nist models ( CamemBERT ,CamemBERTa ,FlauBERT and\nXLM-RoBERTa ) are more suitable for tasks that re-\nquire extensive linguistic knowledge but may not\nperform as well as specialized models nor even\nreach their level of performance. We observe that\nall generalist models obtain better performance\nonly on 4 out of the 20 tasks, but still remain com-\npetitive on most tasks. Furthermore, our experi-\nmentswith DrBERT-FS indicatethatbiomedicalmod-\nels may require less pre-training data compared to\ngeneralist ones. However, it is important to note\nthat this observation requires further confirmation.\nInsometasks,biomedicalmodelsthatundergocon-\ntinual pre-training from a generalist model, such as\nCamemBERT-bio , can prove to be the most effective,\nunderscoring the value of pre-training on generalist0.25 0.5 0.75 1.010203040506070\nDrBERT-FS\nCamemBERT a\nFlauBERT\nDrBERT-CP\nPubMedBERT\nCamemBERT\nCamemBERT-BIO\nXLM-RoBERT a(a) MorFITT CLS.\n0.25 0.5 0.75 1.02030405060708090100 (b) DEFT 2020 CLS\n0.25 0.5 0.75 1.0102030405060 (c) E3C NER Clinical.\n0.25 0.5 0.75 1.0010203040 (d) MantraGSC NER\nMedline.\n0.25 0.5 0.75 1.052.555.057.560.062.565.067.570.0\n(e) FrenchMedMCQA\nCLS\n0.25 0.5 0.75 1.0010203040506070(f) MantraGSC NER\nEMEA.\n0.25 0.5 0.75 1.03040506070(g) QUAERO NER\nEMEA.\n0.25 0.5 0.75 1.04045505560657075(h) CLISTER STS.\nFigure 1: Performance with varying training subset sizes (25%, 50%, 75% and 100%). Results are\nreported on the full test set.\ndatasets.\nFrom-scratch vs. Continual Pre-Training.\nDrBERT-CP and CamemBERT-bio , pre-trained from\nPubMedBERT and CamemBERT respectively, demon-\nstrate improved performance compared to their\ninitial models. Notably, DrBERT-CP outperforms\nCamemBERT-bio in 15 out of 20 tasks. These find-\nings suggest that when it comes to continual pre-\ntraining, starting with a specialized model in the\nspecificdomain(here, PubMedBERT )maybeabetter\nchoice than a generalist model (here, CamemBERT ),\neven with different languages. Additionally, we ob-\nserve that DrBERT-FS achieves the highest perfor-\nmance in 8 tasks, suggesting that starting from-\nscratchcan be a competitive strategy compared to\ncontinual pre-training .\nFrench vs. Other language. French models\ngenerally achieve better performance compared\nto English or multilingual ones. When considering\nthe English PubMedBERT model, we observe that its\nperformance in most tasks is comparable to that\nof the French models, with the exception of NER\ntasks where French models demonstrate superior-\nity. Thus, we observe that the language appears to\nbe less prominent when utilized in domain-specific\ntasks, such as those in the biomedical field.\nRoBERTa vs. DeBERTaV3 architectures. De-\nspite being trained on only 30% of the pre-training\ndata used by CamemBERT CCNET,CamemBERTa\nachieves identical or better performances in 68%\nof the tasks (12 out of 20), benefiting from the De-\nBERTaV3 architecture in domain-specific scenar-\nios. However, all the models based on CamemBERTface difficulties in corpora with limited amount of\ndata, such as MantraGSC Patents, where they fail\ntogeneratelabelsotherthan’O’.Ontheotherhand,\nin the same low-resource scenarios, CamemBERTa\nmodels exhibit greater robustness and achieve su-\nperior performance. The architecture on which the\nmodels are based therefore seems to play a role in\nthe performance obtained.\n5.2. Impact of fine-tuning with limited\ndata\nUnliketheprocessoftraininglanguagemodels,the\nfine-tuning approach involves utilizing annotated\ndatatoadaptapre-trainedlanguagemodelforsolv-\ning specific downstream tasks. In the previous sec-\ntion,weobservedthatlanguagemodelspre-trained\non medical data generally achieved better perfor-\nmance on DrBenchmark compared to generalist\nmodels trained on much larger datasets. However,\nwenowquestionthemodels’abilitytobeeffectively\napplied to biomedical tasks when there is limited\nfine-tuningtrainingdataavailable. Forthispurpose,\nwe conducted experiments by varying the amount\nof training data during the fine-tuning process by\nrandomly choosing four percentages of the training\ndata: 25%, 50%, 75% and 100%. To make the\nexperiment as fair as possible, we did four runs for\neach percentage, model and dataset combination.\nThevalidationandtestsetshavenotbeenchanged\nfor the sake of comparison.\nWe observe that on certain datasets, some mod-\nels capture information more quickly than others,\nlike in Figures 1b, 1f and 1a. Unsurprisingly, in\nalmost all scenarios, having the complete training\nset yields better results than having only 25% of\nit. However, we note few exceptions in Figures 1aFrench Generalist French Biomedical English Biomedical Cross-lingual Generalist\nDataset Task CamemBERT CamemBERTa FlauBERT DrBERT-FS DrBERT-CP CamemBERT-bio PubMedBERT XLM-RoBERTa\nCAS POS 1.63 1.64 1.34 1.36 1.81 1.63 1.81 1.8\nESSAI POS 1.55 1.56 1.28 1.29 1.78 1.55 1.78 1.75\nQUAERONER EMEA 1.66 1.67 1.37 1.37 1.73 1.66 1.73 1.77\nNER Medline 2.01 2.01 1.58 1.64 1.97 2.01 1.97 2.18\nE3CNER FR Clinical 1.64 1.65 1.39 1.32 1.80 1.64 1.80 1.78\nNER FR Temporal 1.63 1.63 1.38 1.31 1.80 1.63 1.80 1.76\nMorFITT Multi-Label CLS 1.51 1.51 1.33 1.39 1.91 1.51 1.91 1.73\nFrenchMedMCQAMCQA 1.80 1.80 1.55 1.55 2.03 1.80 2.03 2.00\nCLS 1.80 1.80 1.55 1.55 2.03 1.80 2.03 2.00\nMantraGSCNER FR EMEA 1.50 1.46 1.34 1.37 1.99 1.50 1.99 1.71\nNER FR Medline 2.25 2.25 1.88 2.05 2.47 2.25 2.47 2.49\nNER FR Patents 1.58 1.58 1.41 1.51 2.06 1.58 2.06 1.86\nCLISTER STS 1.76 1.76 1.55 1.55 2.09 1.76 2.09 1.93\nDEFT-2020STS 1.43 1.43 1.31 1.45 1.92 1.43 1.92 1.64\nCLS 1.31 1.32 1.20 1.23 1.75 1.31 1.75 1.51\nDEFT-2021CLS 1.70 1.71 1.48 1.51 2.05 1.70 2.05 1.90\nNER 1.62 1.63 1.35 1.35 1.80 1.62 1.80 1.79\nDiaMED CLS 1.66 1.67 1.45 1.46 1.99 1.66 1.99 1.88\nPxCorpusNER 1.71 1.76 1.63 1.66 2.13 1.71 2.13 1.83\nCLS 1.71 1.76 1.63 1.66 2.13 1.71 2.13 1.83\nAverage 1.67 1.67 1.43 1.47 1.90 1.67 1.90 1.85\nTable7: Averagesub-wordunitsperwordforeachmodelanddataset. Foreachtask, thelowestsub-word\nvalue is shown in bold, and the highest value is underlined. Models are grouped based on their tokenizer\ntype. Cells in green indicate the best model in terms of performance for the task, while cells in red indicate\nthe worst model.\nand 1h with FlauBERT , where we observe the oppo-\nsite trend. For intermediate percentages, 50% and\n75%, we observe a decrease in performance with\ncertain models, such as FlauBERT in Figures 1a\nand 1g, and DrBERT-CP in Figures 1d and 1h. In\nNER tasks (Figures 1a, 1d, 1f and 1g), DrBERT-FS\nachieves the best performance in scenarios with\nvery little data, indicating good model robustness.\n5.3. Analysis of word tokenization\nTokenizers play a crucial role in MLMs by utilizing\nsize-limitedvocabulariestosplittextsintosub-units,\naiming to handle out-of-vocabulary (OOV) words.\nDue to variations in the training data, vocabularies\ndiffer across different models, as illustrated in Fig-\nure 2. As a result, tokenizers segment words in\ndistinct ways, yet remarkably achieve similar per-\nformance levels as previously noted in Table 6.\nSo far, there has been a prevailing notion in the\ncommunity that excessive segmentation of words\nin tokenization leads to a loss of morphological\nform and semantic meaning, introducing noise and\nadversely affecting performance (Church, 2020;\nHofmann et al., 2021; Bostrom and Durrett, 2020).\nHowever, our experiments, as shown in Table 7, re-\nveal that FlauBERT is the model with the least word\nsegmentation (1.43 in average), while DrBERT-CP\ntends to have the highest average segmentation\n(1.90 in average). Surprisingly, when comparing\ntheperformanceofthesetwomodelsonthebench-\nmark tasks, we observe that DrBERT-CP outper-\nforms FlauBERT on 16 out of the 20 tasks, thus\ncontradicting previous conclusions drawn by the\nFigure 2: Vocabularies inter-coverage matrix.\ncommunity. Table 14 in Appendix C provides some\nexamples of the tokenization done by each ana-\nlyzed model, showcasing a list of commonly used\nbiomedical terms. Yet, tokenization, as it is cur-\nrently done in MLMs, seems to play a minor role in\nthe performance of systems.\nTable 8 summarizes the results obtained on av-\nerage by the considered MLMs when aggregat-\ning the tasks into one of the five designated cate-\ngories: POS, NER, MCQA, MCC (Multi-class clas-\nsification), MLC (Multi-label classification), or STS\ntasks. Uponanalyzingtheaverageperformanceby\ntask category, it becomes evident that the leading\nmodel, DrBERT-FS , does not excel in tasks such as\nMLC or STS. For example, the multilingual biomed-\nical model PubMedBERT demonstrates a notable ad-\nvantage, with nearly 18 EDRM points ahead ofCamemBERT-bio in the STS tasks.\nTasks\nModels POS NER MCQA MCC* MLC* STS\nCamemBERT 96.45 51.52 28.53 / 2.25 71.83 41.12 0.57 / 0.45\nCamemBERTa 97.32 58.16 29.77 / 2.57 70.10 42.16 0.57 / 0.45\nFlauBERT 96.13 51.85 27.88 / 2.09 57.94 54.73 0.54 / 0.40\nDrBERT-FS 97.67 64.23 31.07 /3.22 75.66 51.42 0.67/ 0.69\nDrBERT-CP 97.23 59.84 32.41 / 2.89 77.72 50.51 0.66 / 0.67\nCamemBERT-bio 96.30 53.06 35.30/ 1.45 73.65 42.67 0.56 / 0.29\nPubMedBERT 96.12 46.93 32.90 / 1.61 77.20 47.05 0.74/0.82\nXLM-RoBERTa 97.62 54.21 34.74 / 2.09 63.23 45.87 0.54 / 0.24\nTable 8: Average results obtained by the differ-\nent MLMs for each type of task. MLC stands for\nMulti-label classification and MCC for Multi-class\nclassification.\n6. Conclusion\nIn this paper, we introduced DrBenchmark, the first\nlarge language understanding benchmark tailored\nfortheFrenchbiomedicaldomain. Weconducteda\nqualitative evaluation of 8 state-of-the-art masked\nlanguage models (MLMs) on this comprehensive\nbenchmark,encompassing20diversedownstream\ntasks. Our findings illuminate the limitations of\ngeneralist models in tackling complex biomedical\ntasks, emphasizing the importance of employing\ndomain-specific models to achieve peak perfor-\nmance. While the French biomedical models excel\nin most tasks, no single model emerges as univer-\nsally superior. Remarkably, certain out-of-domain\nmodels or models trained in different languages\nexhibit superior performance in specific tasks and\nmaintain competitiveness in others.\nIn conclusion, we have observed that several\nbiomedical tasks in DrBenchmark exhibit relatively\npoor performance, even when utilizing specialized\nbiomedical models. We postulate that the models\nexamined in this study, here state-of-the-art MLMs,\nmay not be the most effective choices for specific\ntasks such as question-answering or multi-label\nclassification. In our future research, we intend\nto shift our focus towards generative approaches,\nsuchasLLaMA(Touvronetal.,2023),OPT(Zhang\net al., 2022b), or GPT-NeoX-20B (Black et al.,\n2022), as well as their instruction-tuned counter-\nparts (Iyer et al., 2023). These alternatives may\noffer more suitable solutions for addressing these\ntypes of tasks.\n7. Acknowledgements\nThis work was performed using HPC resources\nfrom GENCI-IDRIS (Grant 2022-AD011013061R1\nand 2022-AD011013715) and from CCIPL (Centre\nde Calcul Intensif des Pays de la Loire). This work\nwasfinanciallysupportedbyANRMALADES(ANR-23-IAS1-0005), ANR AIBy4 (ANR-20-THIA-0011)\nand Zenidoc.\nEthical considerations\nAll code for DrBenchmark is released under the\nMIT License. The licensing for all datasets re-\nmainsunchangedfromtheoriginalsources,andDr-\nBenchmark has no intention of redistributing these\ndatasets.\nLimitations\nThe quantitative study we conducted on the PLMs\nrequires further in-depth analysis to comprehend\nthe impact of different parameters. Firstly, we in-\nvestigated the influence of tokenizers based on the\naverage number of sub-tokens they produce per\nword. It is important to note that various tokeniza-\ntion algorithms are employed, depending on the\nmodel under examination. Therefore, it is neces-\nsary to specifically assess the impact of these algo-\nrithms on model construction. Additionally, the size\nof the data has not been thoroughly investigated,\nparticularly the significance of the pre-training data\nsize, especially specialized data in the biomedical\nfield. Analyzingtheinfluenceandimportanceofthe\namount of data used would be crucial for gaining\ndeeper insights.\nAlthough the benchmark is easily reproducible\nand customizable, it required a substantial amount\nof computational power to execute all runs. We\nutilized approximately 2,500 hours on V100 GPUs\nfrom the Jean-Zay supercomputer to complete the\nquantitativestudy. AccordingtotheJeanZaysuper-\ncomputer documentation4, the total environmental\ncostisestimatedtobeequivalentto647,500Whor\n36.9 kgCO2eq/kWh, based on the carbon intensity\noftheenergygridmentionedintheBLOOMenviron-\nmentalcoststudyconductedonJeanZay(Luccioni\net al., 2022). While we acknowledge the significant\ncost of our study, we believe it will enable the re-\nsearchcommunitytodirecttheirfuturestudiesmore\nefficiently by providing a comprehensive overview\noftheperformanceandbehaviorofexistingmodels.\nThis will help prevent redundant evaluations of the\nsame models.\n8. 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Proceedings\nof the Seventh Italian Conference on Computa-\ntional Linguistics CLiC-it 2020 .\nAurélie Névéol, Cyril Grouin, Jeremy Leixa, Sophie\nRosset, and Pierre Zweigenbaum. 2014. The\nQUAERO French medical corpus: A ressource\nfor medical entity recognition and normalization.\nInProc of BioTextMining Work , pages 24–30.\nPedro Javier Ortiz Suarez, Benoit Sagot, and Lau-\nrent Romary. 2019. Asynchronous pipelines for\nprocessing huge corpora on medium to low re-\nsource infrastructures. Proceedings of the Work-\nshoponChallengesintheManagementofLarge\nCorpora(CMLC-7)2019.Cardiff,22ndJuly2019,\npages 9 – 16, Mannheim. Leibniz-Institut f\"ur\nDeutsche Sprache.\nGuillaume Wenzek, Marie-Anne Lachaux, Alexis\nConneau, Vishrav Chaudhary, Francisco\nGuzmán, Armand Joulin, and Edouard Grave.\n2020. CCNet: Extracting high quality mono-\nlingual datasets from web crawl data. In\nProceedings of the Twelfth Language Resources\nand Evaluation Conference , pages 4003–\n4012, Marseille, France. European Language\nResources Association.\nAppendices\nA. Hyperparameters\nFor the experiments, we utilize the following hyper-\nparametersthatyieldoptimalperformancefromthe\nmodels. To mitigate overfitting, we locally save the\nbest model based on its validation metric.Hyper-parameter Value\nMax sequence length 512\nEpochs 20\nBatch size 16\nLearning Rate 2e-5\nWeight Decay 0.01\nTable 9: Hyper-parameters for the question-\nanswering experiments.\nHyper-parameter Value\nMax sequence length 512\nEpochs 10 / 25 / 35\nBatch size 16\nLearning Rate 2e-5\nWeight Decay 0.01\nTable 10: Hyper-parameters for the classification\nexperiments. The number of epochs is by default\n10exceptforDEFT-2020(25epochs)andMorFITT\n(35 epochs).\nHyper-parameter Value\nMax sequence length 512\nEpochs 10\nBatch size 16\nLearning Rate 1e-5\nWeight Decay 0.01\nTable 11: Hyper-parameters for the POS tagging\nexperiments.\nHyper-parameter Value\nMax sequence length 512\nEpochs 30\nBatch size 16\nLearning Rate 2e-5\nWeight Decay 0.01\nTable 12: Hyper-parameters for the regression ex-\nperiments.\nHyper-parameter Value\nMax sequence length 512\nEpochs 15\nBatch size 16\nLearning Rate 1e-4\nWeight Decay 0.01\nTable 13: Hyper-parameters for the NER experi-\nments.B. Dataset Classes\nB.1. CAS\nINT,PRO:DEM ,VER:impf ,VER:ppre ,PRP:det,KON,VER:pper ,PRP,PRO:IND ,VER:simp ,VER:con ,\nSENT,VER:futu ,PRO:PER ,VER:infi,ADJ,NAM,NUM,PUN:cit,PRO:REL ,VER:subi ,ABR,NOM,\nVER:pres ,DET:ART ,VER:cond ,VER:subp ,DET:POS ,ADV,SYMandPUN.\nB.2. ESSAI\nINT,PRO:POS ,PRP,SENT,PRO,ABR,VER:pres ,KON,SYM,DET:POS ,VER:,PRO:IND ,NAM,\nADV,PRO:DEM ,NN,PRO:PER ,VER:pper ,VER:ppre ,PUN,VER:simp ,PREF,NUM,VER:futu ,NOM,\nVER:impf ,VER:subp ,VER:infi,DET:ART ,PUN:cit,ADJ,PRP:det,PRO:REL ,VER:cond andVER:subi .\nB.3. QUAERO\nO,GEOG,PHEN,DISO,ANAT,OBJC,PHYS,PROC,DEVI,CHEMandLIVB\nB.4. E3C\nClinical: O, andCLINENTITY\nTemporal: O,EVENT,ACTOR,BODYPART ,TIMEX3 andRML\nB.5. MorFITT\nmicrobiology ,etiology,virology,physiology ,immunology ,parasitology ,genetics,chemistry ,veterinary ,\nsurgery,pharmacology andpsychology\nB.6. MantraGSC\nMedline: ANAT,PROC,CHEM,PHYS,GEOG,DEVI,LIVB,OBJC,DISO,PHENandO.\nEMEA and Patents: ANAT,PROC,CHEM,PHYS,DEVI,LIVB,OBJC,DISO,PHENandO.\nB.7. DEFT-2021\nMulti-label Classification: immunitaire (immunology) ,endocriniennes (endocrinology) ,blessures (in-\njury),chimiques (chemicals) ,etatsosy (signs and symptoms) ,nutritionnelles (nutrition) ,infections (infec-\ntions),virales (virology) ,parasitaires (parasitology) ,tumeur (oncology) ,osteomusculaires (osteomuscular\ndisorders) ,stomatognathique (stomatology) ,digestif (digestive system disorders) ,respiratoire (respiratory\nsystemdisorders) ,ORL(otorhinolaryngologicdiseases) ,nerveux(nervoussystemdisorders) ,oeil(eyedis-\neases),homme (male genital diseases) ,femme (female genital diseases) ,cardiovasculaires (cardiology) ,\nhemopathies (hemic and lymphatic diseases) ,genetique (genertic disorders) andpeau (dermatology) .\nNamed-entity recognition: O,ANATOMY ,DATE,DOSAGE ,DURATION ,MEDICAL EXAM ,FRE-\nQUENCY ,MODE,MOMENT ,PATHOLOGY ,SOSY,SUBSTANCE ,TREATMENT andVALUE\nB.8. DiaMed\n•A00-B99 Certain infectious and parasitic diseases\n•C00-D49 Neoplasms\n•D50-D89 Diseases of the blood and blood-forming organs and certain disorders involving the immune\nmechanism\n•E00-E89 Endocrine, nutritional and metabolic diseases\n•F01-F99 Mental, Behavioral and Neurodevelopmental disorders\n•G00-G99 Diseases of the nervous system•H00-H59 Diseases of the eye and adnexa\n•H60-H95 Diseases of the ear and mastoid process\n•I00-I99 Diseases of the circulatory system\n•J00-J99 Diseases of the respiratory system\n•K00-K95 Diseases of the digestive system\n•L00-L99 Diseases of the skin and subcutaneous tissue\n•M00-M99 Diseases of the musculoskeletal system and connective tissue\n•N00-N99 Diseases of the genitourinary system\n•O00-O9A Pregnancy, childbirth and the puerperium\n•P00-P96 Certain conditions originating in the perinatal period\n•Q00-Q99 Congenital malformations, deformations and chromosomal abnormalities\n•R00-R99 Symptoms, signs and abnormal clinical and laboratory findings, not elsewhere classified\n•S00-T88 Injury, poisoning and certain other consequences of external causes\n•U00-U85 Codes for special purposes\n•V00-Y99 External causes of morbidity\n•Z00-Z99 Factors influencing health status and contact with health services\nB.9. PxCorpus\nIntent classification: MEDICAL PRESCRIPTION ,NEGATE ,NONEandREPLACE\nNamed-entity recognition: O,A,CMA_EVENT ,D_DOS_FORM ,D_DOS_FORM_EXT ,D_DOS_UP ,\nD_DOS_VAL ,DOS_COND ,DOS_UF ,DOS_VAL ,DRUG,DUR_UT ,DUR_VAL ,FASTING ,FREQ_DAYS ,\nFREQ_INT_V1 ,FREQ_INT_V1_UT ,FREQ_INT_V2 ,FREQ_INT_V2_UT ,FREQ_STARTDAY ,FREQ_UT ,\nFREQ_VAL ,INN,MAX_UNIT_UF ,MAX_UNIT_UT ,MAX_UNIT_VAL ,MIN_GAP_UT ,MIN_GAP_VAL ,\nQSP_UT ,QSP_VAL ,RE_UT,RE_VAL ,RHYTHM_HOUR ,RHYTHM_PERDAY ,RHYTHM_REC_UT ,\nRHYTHM_REC_VAL ,RHYTHM_TDTE andROA\nC. Word tokenization\nFrench Generalist French Biomedical English Biomedical Cross-lingual Generalist\nTerm CamemBERTaCamemBERTFlauBERT DrBERT-FSPubMedBERTXLM-RoBERTa\nCamemBERT-bio DrBERT-CP\nasymptomatique a-s-ym-pto-matique a-s-y-mp-to-matique as-ym-ptom-atique ✓ asympt-omat-ique as-y-mp-tomat-ique\nblépharorraphie blé-phar-or-ra-phi-e blé-phar-or-ra-phi-e bl-é-phar-or-raph-ie blé-ph-ar-or-ra-ph-ie ble-pha-ror-ra-phi-e b-lép-har-orra-phi-e\nbradycardie brad-y-cardi-e brad-y-cardi-e bra-dy-car-die ✓ brady-car-di-e bra-dy-card-ie\nbronchographie bronch-ographie bron-ch-ographie bron-cho-graphie bronch-ographie bronch-ograph-ie bron-ch-ographie\nbronchopneumopathie bronch-op-ne-um-opathie bron-cho-p-ne-um-opathie bron-chop-neu-mo-pathie bronchop-neumopathie bronch-op-neum-opath-ie bron-chop-ne-umo-pathi-e\ndysménorrhée dys-mén-or-r-h-ée dys-mén-or-r-h-ée dys-mé-nor-rh-ée dys-m-énorrhée dysm-eno-rr-he-e dys-mén-or-r-hé-e\nglaucome gla-uc-ome gla-uc-ome glau-come ✓ glauc-ome gla-u-come\nIRM ✓ ✓ ✓ ✓ ir-m I-RM\nkystectomie k-yst-ectomie ky-st-ectomie ky-st-ec-tomie kys-tectomie ky-st-ectom-ie ky-st-ecto-mie\nneuroleptique neuro-le-p-tique neuro-le-p-tique neur-ol-ep-tique neur-oleptique neurol-ept-ique neuro-lep-tique\nnicotine ✓ ✓ ✓ ✓ ✓ nico-tine\npoliomyélite poli-om-y-élite poli-om-y-élite poli-omy-élite poli-omyélite poli-omyel-ite poli-om-y-é-lite\nrhinopharyngite rh-ino-phar-y-ng-ite rhin-oph-ary-ng-ite rh-ino-phar-yn-gite rhin-opharyng-ite rhin-oph-aryng-ite r-hin-op-har-y-ng-ite\ntoxicomanie toxico-mani-e toxico-mani-e ✓ ✓ toxic-oman-ie toxic-om-anie\nvasoconstricteur vas-oc-on-strict-eur vas-oc-on-strict-eur vas-o-cons-tri-cteur vasoconstric-teur vasoconstric-te-ur vaso-con-strict-eur\nTable 14: Visual comparison of models’ tokenization on commonly used biomedical terms. A checkmark\nindicates that the word is present as a complete token, while hyphens separate subword units."    },    {        "title": "2402.13451v1.Metrical_results_on_the_Folklore_set.pdf",        "content": "arXiv:2402.13451v1  [math.NT]  21 Feb 2024METRICAL RESULTS ON THE FOLKLORE SET\nMUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WARD\nAbstract. We study the Folklore set of Dirichlet improvable matrices in Rmˆnwhich\nare neither singular nor badly approximable. We show that for “almos t all” positive\ninteger pairs m,n, the Folklore set with respect to the maximum norm is not empty.\nIn most cases, however, we provide significantly stronger results in various aspects. For\nexample, we can often prescribe the exact Dirichlet constant, tak e arbitrary norms,\nestablish Hausdorff dimension bounds, or some combination of the af orementioned gen-\neralisations. In particular, in the mdimensional simultaneous approximation setting we\ncalculate a lower bound of m´2, and in the ndimensional dual approximation setting\nwe calculate a lower bound of n´2. By prescribing the exact Dirichlet constant in cer-\ntain cases we also obtain information on the Dirichlet spectrum in vario us settings. The\nresults of this paper improve on previous results of the middle-name d author (Selecta\nMath. 2023), and of Beresnevich et. al. (Adv. Math. 2023).\nKeywords : Dirichlet spectrum, Folklore set, Uniform approximation, Singular s ets, linear forms, Haus-\ndorff dimension.\nMath Subject Classification 2010: 11J13, 11J82, 11J83, 11K60\n1.Introduction\nIn 1842 Dirichlet proved that for any ξ“ pξ1,...,ξ nq PRnand anyNPNthe system\n|ˆb¨ξ`bn`1|:“ |b1ξ1` ¨ ¨ ¨ `bnξn`bn`1| ăN´n(1)\n}ˆb}:“max\n1ďiďn|bi| ďN\nhas non-zero integer solution b“ pˆb,bn`1q PZn`1. For fixed 0 ăεă1 the set of ε-\nDirichlet improvable numbers , denoted as Dinpεq, are those for which the right-hand side\nof (1) can be “improved” by ε. That is,\nDinpεq:“#\nξPRn:#\n|ˆb¨ξ`bn`1| ăεN´n\n}ˆb} ďNfor allNěN0pε,ξq+\n.\nGenerally ξPRnis said to be Dirichlet improvable , denoted as Din, if there exists some\n0ăεă1 for which ξPDinpεq, andξPRnis said to be Singular, denoted as Singn, if it\nisε-Dirichlet improvable for all 0 ăεă1. Concisely\nDin:“ď\n1ąεą0Dinpεq,andSingn:“č\n1ąεą0Dinpεq.\nNote trivially that for any εą0 we have that SingnĎDinpεq ĎDin.\n12 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WARD\nIn another direction, from Dirichlet’s approximation theorem one ca n deduce that for\nanyξPRn\n(2) |ˆb¨ξ`bn`1| ă }ˆb}´n\nhas infinitely many integer solutions bPZn`1. The set of Badly approximable vectors ,\ndenoted as Badn, consists of the set of ξPRnsuch that the right-hand side of (2) cannot\nbe “improved” by any arbitrarily small constant, otherwise ξis called well approximable.\nThat is,\nBadn:“!\nξPRn: There exists εą0 :|ˆb¨ξ`bn`1| ěε}ˆb}´nfor allbPZn`1zt0u)\n.\nWe have the relations\nBadnĎDinandBadnXSingn“ H.\nThe left was shown by Davenport and Schmidt [13, Theorem 2], the rig ht is obvious. It\nis well known that all of the sets highlighted above are n-dimensional Lebesgue nullsets.\nThis can be proven through a variety of methods, see for example [5 ] for an overview. So\nto give any meaningful notion of size to these sets we consider the H ausdorff dimension\ndimH. See[14]forthedefinition andproperties oftheHausdorff dimensio n. The following\nresults onDinpεq,Din,Singn,Badnhave been proven:\ndimHBadn“n,forn“1 [16] and for ně2 in [28],\ndimHDin“n,[28] with [13, Theorem 2] ,\ndimHSingn“n2\nn`1,forn“2 in [8] and for ně2 in [9].\nThe dimensions of Dinpεqare not yet established, but for ně2 the bounds\nn2\nn`1`εtďdimHDinpεq ďn2\nn`1`Cεn{2\nfor any real number tąnand constant Cą0 were proven in [9, Theorem 1.3].\nThe reader will notice that the dimension results for SingnandDinpεqhold only for\nně2. Forn“1 it was proven by Khintchine [17] that Sing1“Q(furthermore Di1pεq “\nQfor anyεă1\n2, see for example [29, Lemma 4.1] in the case ε“1\n3). By combining a\nresult of Khintchine [17] that reads as Sing1“Qand Theorem 1 of Davenport-Schmidt\n[13], it follows that:\nDi1“QYBad1.\nIn higher dimensions much less was known, until the recent paper of Beresnevich et. al.\n[6], who showed that, for ně2, theFolklore set\nFSn:“DinzpBadnYSingnq\nhas a continuum of points. Beresnevich et. al. mention that this que stion was (for\nně2) first asked in [29] (see the remark immediately following [29, Theore m 4.6]) and\nfurthermore they conjectured that for ně2, dim HFSn“nin [6, Problem 3.1].METRICAL RESULTS ON THE FOLKLORE SET 3\nIn this article we prove a range of results that extend the theorem s given in [6] in two\ndifferent ways:\n‚In Section 2 we consider the Folklore set in the dual approximation se tting and\nadd an additional condition that the point has some specific Dirichlet c onstantc.\nWe prove this set has a positive dimension for all constants cin some small right\nneighbourhood of 0, which allows us to deduce results on the Dirichlet spectrum\nfor linear forms.\n‚In Section 3 we consider the Folklore set in the setting of systems of linear forms\nand, provided certain dimension conditions on the system of linear fo rms are\nsatisfied, we prove non-emptiness (Theorem 3.1) and in a subset of cases we prove\npositive Hausdorff dimension (Theorems 3.2, 3.3, 3.5, 3.6).\nPrior to stating these results we introduce some more notation, fo rmulated in the\ngeneral matrix setting.\n1.1.A refined Folklore set for systems of linear forms. Letn,m PN, let } ¨ }1be\nany norm on Rnand} ¨ }2any norm on Rm. For a given mˆnmatrix Ω PRmˆnwith real\nentries Ω i,jin theith row and jth column, and a parameter tě1 define the piecewise\nconstant approximation function\nψmˆn\nΩp} ¨ }1,} ¨ }2,tq “min\nb“pˆb,˜bqPZn`m:\n1ď}ˆb}1ďt›››Ω¨pb`˜b›››\n2“min\nb“pˆb,b2qPZn`m:\n1ď}ˆb}1ďt›››››nÿ\ni“1Ωibi`˜b›››››\n2,\nwhere Ω idenotes the ith column of the matrix Ω, and\n(3) ˆb“ pb1,...,b nq,˜b“ pbn`1,...,b n`mq,b“ pˆb,˜bq\ninterpreted as column vectors. Note ψmˆn\nΩp}¨}1,}¨}2,tqis precisely how close, with respect\nto} ¨ }2, the matrix Ω can be approximated by the best approximation bPZn`mwith\nheight, with respect to } ¨ }1, bounded by t. See works of Lagarias [20, 21] and the survey\narticles [23, 10] for the history and further details on best approx imations in dimensions\ngreater than 1.\nDefine the Dirichlet constant of a matrix as\nΘmˆnp} ¨ }1,} ¨ }2,Ωq “limsup\ntÑ8tn{mψmˆn\nΩp} ¨ }1,} ¨ }2,tq,\nand further let\nrΘmˆnp} ¨ }1,} ¨ }2,Ωq “liminf\ntÑ8tn{mψmˆn\nΩp} ¨ }1,} ¨ }2,tq.\nThe values Θmˆnp} ¨ }1,} ¨ }2,ΩqandrΘmˆnp} ¨ }1,} ¨ }2,Ωqcan be viewed as how much\nbetter, or worse, the matrix Ω can be approximated relative to the Dirichlet function\nt´n{min the uniform and asymptotic settings, respectively.\nFrom these definitions, by the equivalence of norms and Dirichlet’s Th eorem, we derive\n(4) D“Dmˆnp} ¨ }1,} ¨ }2q:“sup\nΩPRmˆnΘmˆnp} ¨ }1,} ¨ }2,Ωq ă 8.4 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WARD\nIn other words, Dą0 is the smallest constant for which all matrices Ω are Dt´n{m-\nuniformly approximable with respect to the norms } ¨ }1and } ¨ }2. Notice that for } ¨ }1\nand} ¨ }2the standard maximum norms, we have D“1, with the upper estimate coming\nfrom Minkowski’s Theorem for systems of linear forms and this value b eing attained for\nalmost all matrices by well-known metrical results, see [2, 26] for fu rther details on the\nDirichlet spectrum. For 0 ăcăD, define the following sets\nDim,np} ¨ }1,} ¨ }2,cq:“/visualspace\nΩPRmˆn: Θmˆnp} ¨ }1,} ¨ }2,Ωq ďc(\n,\nDim,np} ¨ }1,} ¨ }2q:“/visualspace\nΩPRmˆn: Θmˆnp} ¨ }1,} ¨ }2,Ωq ăD(\n“ď\n0ăcăDDim,np} ¨ }1,} ¨ }2,cq,\nSingm,np} ¨ }1,} ¨ }2q:“/visualspace\nΩPRmˆn: Θmˆnp} ¨ }1,} ¨ }2,Ωq “0(\n“č\n0ăcăDDim,np} ¨ }1,} ¨ }2,cq,\nBadm,np} ¨ }1,} ¨ }2q:“!\nΩPRmˆn:rΘmˆnp} ¨ }1,} ¨ }2,Ωq ą0)\n.\nWhen } ¨ }1,} ¨ }2are the standard maximum norms we will omit them from notation, tha t\nis we will write Dim,np} ¨ }1,} ¨ }2,cq “Dim,npcqand similarly for the other defined sets.\nFor the sets Singm,np} ¨ }1,} ¨ }2qandBadm,np} ¨ }1,} ¨ }2qnote by the equivalence of norms\nand the observation that the sets are essentially independent of c onstants that\nSingm,np} ¨ }1,} ¨ }2q “Singn,m, Bad m,np} ¨ }1,} ¨ }2q “Badm,n.\nWhenn“1 orm“1 we may restrict to } ¨ }1“ | ¨ |and } ¨ }2“ | ¨ |respectively, and for\nease of notation we will write\nDi1,np} ¨ }1,} ¨ }2,cq “Dinp} ¨ }1,cq,\nDim,1p} ¨ }1,} ¨ }2,cq “Di˚\nmp} ¨ }2,cq.\nNote these sets align with our previously defined sets given in the intr oduction.\nAdditional to these sets, for any 0 ăcăD, define\nDIm,np} ¨ }1,} ¨ }2,cq:“/visualspace\nΩPRmˆn: Θmˆnp} ¨ }1,} ¨ }2,Ωq “c(\n.\nThat is, the set of matrices which are Dirichlet improvable by any cons tantc`εbut no\nconstantc´ε, for anyεą0. Note that\n(5) DIm,np} ¨ }1,} ¨ }2,cq ĎDim,np} ¨ }1,} ¨ }2,cq ĎDim,np} ¨ }1,} ¨ }2q.\nAgain, for ease of notation, in the dual setting we will write DI1,np} ¨ }1,} ¨ }2,cq:“\nDInp} ¨ }1,cq, and when } ¨ }1is the max norm this will also be omitted.\nGiven this setup we may define the Folklore set on mˆnmatrices with arbitrary norms\nas\nFSm,np} ¨ }1,} ¨ }2q “Dim,np} ¨ }1,} ¨ }2qz pBadm,nYSingm,nq\nand, for any 0 ăcăD, the smaller set\nFSm,np} ¨ }1,} ¨ }2,cq “DIm,np} ¨ }1,} ¨ }2,cqzBadm,n.\nPrior to stating our metrical results we give the following lemma, which gives a nice\nrelationship between a matrix of FSm,np} ¨ }1,} ¨ }2,cqand its transpose.METRICAL RESULTS ON THE FOLKLORE SET 5\nLemma 1.1. Givenm,n, there exists δ“δm,nso that if ΩPFSm,np} ¨ }1,} ¨ }2,cqfor\nsome0ăcăδ, then its transpose ΩTsatisfiesΩTPFSn,mp} ¨ }2,} ¨ }1q.\nWe will prove the Lemma at the end of Section 10 but should note it is a s imple\nconsequence of German’s Transference principle [15, Theorem 7]. N ote when considering\nthe transpose we lose control over the precise Dirichlet constant .\nItisworthmentioningthatadifferentsetupfordefiningnormedDiric hletimprovability,\nwith respect to a single arbitrary norm on Rm`n, has been recently studied, see for\nexample [4, 18, 19]. It is unclear to us how the two setups are precise ly related, in\nparticular, if our results remain valid in the single norm setting.\n2.The refined Folklore set in the Dual setting\n2.1.Supremum norm. Let us, for now, return to our previous setting of classical dual\napproximation ( m“1,} ¨ }1“max1ďiďn| ¨ |, and } ¨ }2“ | ¨ |). Then we just denote the\nparametric Folklore set above by\nFSnpcq:“DInpcqz pBadnYSingnq, c P p0,1q.\nIt was shown in [25] by the second name author that the set FSnpcqis non-empty for any\ncP p0,1q, refining the aforementioned result from [6].\nWe begin with the following easy-to-state corollary of one of our main results that\ncontribute towards the original conjecture [6, Problem 3.1].\nCorollary 2.1. Letně4. Then\ndimHFSněn´2.\nIndeed, we prove a considerably stronger claim in Theorem 2.2 below, namely that the\nmetrical estimate still holds for the set where we fix any small enoug h positive Dirichlet\nconstant. So we have control over the precise Dirichlet constant .\nTheorem 2.2. Forně4an integer, there exists explicitly computable cnP p0,1qsuch\nthat for any cP r0,cnqwe have\ndimHFSnpcq ěn´2.\nRemark 1. We believe the claim holds for n“3 as well, probably improving on the\nbound that can be obtained from the methods in [26, 25]. However, p roblems occur in\nour proof in this case stemming from auxiliary results from [24], leaving this case open.\nRemark 2. In [26] a lower bound of weaker order p3{8qn`opnqasnÑ 8is shown\nfor according sets with respect to simultaneous approximation. In [25] it is shown that\nFSnpcqfor a linear form as defined above is not empty, and reference to [2 6] is given on\nhow to obtain metrical results in a similar way. However, again these b ounds will turn\nout considerably weaker than in Theorem 2.2 above. The bound of Th eorem 2.2 can in\nfact be further improved a little with some effort, see Remark 14 app earing in Section 5.6 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WARD\nWe may take\ncn“n´3\n8pn´2q3{2¨?π¨Γpn´1\n2q\nΓpnq\nwith Γ the Euler Gamma function. As εÑ0`, we cannot hope for anything better\nthann´1`1\nn`1as Cheung and Chevallier [9] showed this as the asymptotics for the\nHausdorff dimension of the larger set Dinpcq ĚFSnpcq. Hence we consider our bound to\nbe reasonably good. Similar to [26, 25] the expected increase of the Hausdorff dimension\nofFSnpcqas a function of cP r0,1sis not reflected in Theorem 2.2.\nDue to [6, Proposition 1.1 and Lemma 4.9] we have the following easy cor ollary in the\nsimultaneous approximation setting. Note the corollary also follows f rom Lemma 1.1.\nCorollary 2.3. Forně4definingFS˚\nn“Di˚\nnzBad˚\nnwe have\ndimHFS˚\nněn´2.\nMoreover, the Dirichlet spectrum accumulates 0, in particular, it is not finite.\nRemark 3. The accordingly defined sets for simultaneous approximation for th e sameε\nno longer coincide. So in particular we are not able to deduce an equiva lent Theorem for\nany set\nFS˚\nnpcq:“DI˚\nnpcqzBad˚\nn.\n2.2.Arbitrary norms. Our second result extends Theorem 2.2 to arbitrary norms in\nthe dual approximation setting.\nTheorem 2.4. Letně4and } ¨ }be any norm on Rn. Then there is cn“cnp} ¨ }q ą0\nsuch that for any cP r0,cnswe have\ndimHFSnp} ¨ },cq ěn´2.\nIn particular the Dirichlet spectrum\ntΘp} ¨ },ξq:ξPRnu:“ tΘ1ˆnp} ¨ },| ¨ |,ξq:ξPRnu\ncontains some right neighborhood of 0.\nRemark 4. Note that since FSnp} ¨ },cq ĎFSnp} ¨ }qthis immediately gives us a lower\nbound on the Hausdorff dimension of FSnp} ¨ }q.\nRemark 5. Forn“3 and a general norm, even non-emptiness of the set FSnp} ¨ },cq\nfor a given cP p0,Dq, withDas in (4), remains unknown. For certain norms, this can be\nobtained from the method in [25]. Again we expect the Hausdorff dimen sion to rise as a\nfunction of cP r0,Ds, not reflected in the theorem.\nRemark 6. For results on the Dirichlet spectrum with respect to the maximum no rm\nsee [26, Appendix] and more completely [25]. In [25] it is proven that th e spectrum is\nthe unit interval. Theorem 2.4 tells us the weaker statement that th e spectrum contains\nsome small interval starting at the origin, however, our result hold s for any norm. Notice\nTheorem 2.4 is for the dual approximation setting. For results on th e Dirichlet spectrum\nin the simultaneous approximation setting for various norms on R2see the recent article\n[1] for the current state of the art. Briefly, it was shown in [18, Cor ollary 1.2] that theMETRICAL RESULTS ON THE FOLKLORE SET 7\ncorresponding upper bound of D“D2ˆ1p}¨}1,}¨}2q, as defined in (4), is an accumulation\npoint for any norm on R2, and in [1] it is proven that for any arbitrary norm the Dirichlet\nspectrum contains an interval starting at the origin, see [1, Coro llary 1]. Both of these\nresults are for the associated Dirichlet spectrum of simultaneous a pproximation on R2.\nWe should stress that results for specific norms were known prior t o the statements of\n[1, 18], see for example [2, 3, 26].\nRemark 7. By Lemma 1.1 from Theorem 2.4, we may deduce an analogous metrical\nresult on the simultaneous approximation Folklore set for arbitrary norms with mě4,\nthatis, thatsuchsethasHausdorffdimensionatleast m´2. However, weloseinformation\nabout the exact value of the Dirichlet constant, so we prefer not t o state this result here\nas it will be covered by the more general Theorem 3.6 below.\n3.Results for the system of linear forms setting\nWe now come to results in the matrix setting with min tm,n u ą1. We begin with\nthe non-emptiness of the Folklore set provided the matrices satisf y various dimension\nconditions. The following main result will be a consequence of our cons iderably more\ngeneral results below.\nTheorem 3.1. Letm,nbe positive integers satisfying\ntm,n u R tt1,1u,t2,3u,t3,5u,t4,6u,t5,7u,t6,8u,t7,9u,...u.\nThen\nFSm,np} ¨ } 8,} ¨ } 8q ‰ H.\nMoreover, we have that\ndimHFSm,němn´2max tm,n u,\nand we can choose the matrices of full R-rankmintm,n u.\nRemark 8. In the theorem we exclude the cases m“n“1,tm,n u “ t2,3uand all\nm,nwith |m´n| “2 apart from the case tm,n u “ t1,3u. This point of view will be\nconvenient to deduce it from claims below. Note that this means we pr ove non-emptiness\nof the Folklore set for “almost all” pairs pm,n qin a metrical sense. Our result follows\nfrom a combination of Theorems 3.4, 3.6 and the cases m“1,ně2 andmě2,n“1\nsettled in [26, 25].\nRemark 9. Regarding the metrical claim on the dimension appearing in Theorem 3.1 ,\nwe state in advance that it may be conjectured that\ndimHpFSm,np} ¨ }1,} ¨ }2qq “mn,\nand more generally that\ndimHpFSm,np} ¨ }1,} ¨ }2,cqq “mn´op1q\nascÑD´forDfrom (4), for any pair of norms. However, as cÑ0`, we can expect the\nlargest value for the Hausdorff dimension of Dim,np} ¨ }1,} ¨ }2,cq ĚFSm,np} ¨ }1,} ¨ }2,cq\nto bemnp1´1\nm`nq. That is, the Hausdorff dimension of Singm,n, see [11, Theorem 1.1].8 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WARD\nWe again emphasize that we get much more information from the part ial results for-\nmulated below. The refinements will provide us with information on one or more of\nthe following properties: stronger Hausdorff dimension results in se veral cases; arbitrary\nnorms; more information on the Dirichlet spectrum, ideally imposing an exact Dirichlet\nconstant akin to Theorem 2.4; and Q-linear independence of matrix entries together with\nt1u. The separate theorems depend on the dimensions of the matrix, t hat is, onmandn.\nWe split into two cases. Firstly whenm\nnP r1\n2,2s, and secondly whenm\nnP p0,1\n2qYp2,8q.\nThese two cases correspond to when mandnare relatively close (or far) apart, that is,\nnămď2nfor the first case and mě2nfor the second. Within these cases, we again\nsplit into a further two subcases by using Lemma 1.1. For example in th e first case we\nbegin withm\nnP r1,2sand then use Lemma 1.1 to obtain results for whenm\nnP r1\n2,1s.\n3.1.The case m{nP r1{2,2s.We first treat the case m“n. Here we restrict to\nmaximum norms.\nTheorem 3.2. Form“ně2, the setFSn,n“FSn,np} ¨ } 8,} ¨ } 8qsatisfies\ndimHpFSn,nq ě pn´1q2`1.\nMoreover, we can choose the matrices of full R-rankn(i.e. invertible).\nThe construction of the proof gives limited control of the exact Dir ichlet constant here.\nIf it were possible to assume the hypothesis that for every ǫą0 we can find matrices\nBPBadn´1,n´1with Dirichlet constant Θpn´1qˆpn´1qpBq ăǫthen the proof guarantees\nΘnˆnpΩq ăǫfor some Ω PFSn,nas well. In particular then by equivalence of norms,\nwe can choose arbitrary norms in the claim of Theorem 3.2 and get an a ccumulation of\nthe Dirichlet spectrum at 0. The hypothesis seems plausible for ně3 as the according\nhypothesis follows from [2] for certain BPBad2,1. Note however that the hypothesis is\nfalse form“n“2, so thatBPBad1,1, by the result of Khintchine [17] already indicated\nin Section 1.\nNow let us study the case m‰n. Following [26], let us call a norm }¨}onRkexpanding\nif}x} ě }πjpxq}for anyxPRkand every 1 ďjďk, where\nπj:px1,...,x kq Ñ p0,...,0,xj,0,...,0q\nis the projection to the j-th coordinate axis. As remarked in [26] this is a rather mild con-\ndition that in particular includes the maximum norm and all p-norms }x} “ př|xi|pq1{p,\npě1. Let us again call\ntΘmˆnp} ¨ }1,} ¨ }2,Ωq: ΩPRmˆnu Ď r0,Ds\nwithDą0 as in (4), the Dirichlet spectrum in dimensions m,n. We show the following.\nTheorem 3.3. Assumem,nare positive integers with\nně2, m ě4,1ăm\nnď2.\nMoreover assume m´n‰2ifn‰2. Let } ¨ }1be an arbitrary norm on Rnand } ¨ }2\nbe any expanding norm on Rm, unless pm,n q “ p4,2qin which case assume both norms\n} ¨ }1and } ¨ }2to be expanding.METRICAL RESULTS ON THE FOLKLORE SET 9\nThen there exists cm,n“cm,np} ¨ }1,} ¨ }2q ą0so that for any cP r0,cm,ns, we have\nFSm,np} ¨ }1,} ¨ }2,cq ‰ H\nin fact\ndimHpFSm,np} ¨ }1,} ¨ }2,cqq ěm¨ pn´2q `m´2, m ě5\nand\ndimHpFSm,np} ¨ }1,} ¨ }2,cqq ěm¨ pn´2q, m “4.\nIn particular the Dirichlet spectrum contains a right neigh borhood of 0. Moreover, we can\nchoose the matrices of full R-rankn.\nRemark 10. Theconditionof }¨}2beingexpandingcanberelaxedtoaskingthatforsome\nspacespannedbytwocanonicalbasevectors ei,ejtheprojectednormontoitisexpanding.\nFor example, for i“j“1 this means the norm }px1,x2q}1:“ }px1,x2,0,...,0q}2onR2\nis expanding.\nWe can improve the Hausdorff dimension formulas in Theorem 3.3 via app lying [26,\nTheorem 2.3] to estimate from below dim HpFS2,1pcqqforcP r0,1s. This positive value\ncan be added given the method of the proof and using (12). This res embles Remark 2.\nBy equivalence of norms, for arbitrary norms, we have the accord ing result that the\nFolklore set is non-empty and the Dirichlet spectrum accumulates at 0.\nBy Lemma 1.1 we may deduce an analogous result to Theorem 3.3, apar t from notably\nnot fixing exact Dirichlet constant c, in the cases\nně4,1\n2ďm\nnă1.\nTogether with the case m“nfrom Theorem 3.2, we may summarize\nTheorem 3.4. Letm,nbe integers withm\nnP r1\n2,2sandmaxtm,n u ě4, and |m´n| ‰2\nunless if tm,n u “ t2,4u. Then\nFSm,np} ¨ } 8,} ¨ } 8q ‰ H,\nin fact\n(6) dim HpFSm,np} ¨ } 8,} ¨ } 8qq ěmn´2max tm,n u.\nMoreover, we can choose the matrices of full R-rankmintm,n u. If additionally m‰n,\nthen more generally for any norms } ¨ }1onRnand } ¨ }2onRm\nFSm,np} ¨ }1,} ¨ }2q ‰ H\nand the Dirichlet spectrum accumulates at 0, in particular, it is not finite.\nWe remark that (6) can be improved unless tm,n u “ t2,4uwhere the right hand\nside vanishes. In our constructions of all Theorems 3.2, 3.3, 3.4 the real matrices in our\nconstruction have some 0 entries. In particular the entries are no tQ-linearly independent\nas in Theorems 3.5, 3.6 below, which would be desirable.10 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\n3.2.The case m{nR p1{2,2q.Now assume ně2m. In this case, we establish the\nfollowing generalization of Theorem 2.4.\nTheorem 3.5. Letm,nbe positive integers satisfying\n(7)m\nnď1\n2, n ‰m`2.\nLet} ¨ }1be any norm on Rnand } ¨ }2be any norm on Rm. Then there exists cm,n“\ncm,np} ¨ }1,} ¨ }2q ą0such that for any cP r0,cm,ns\ndimHFSm,np} ¨ }1,} ¨ }2,cq ě pn´2qm.\nThe same claims hold when we additionally assume that\n(i) Ωhas fullR-rank (m, so the rows of Ωare linearly independent over R)\n(ii)the set of all its entries Ωi,jtogether with 1is linearly independent over Q.\nIn particular the Dirichlet spectrum contains some right ne ighborhood of 0.\nTheorem 2.4 is the special case m“1 when identifying } ¨ }with } ¨ }1and } ¨ }2“ |.|\nwith the absolute value on R, upon dropping the independence condition of columns. We\nwill use Theorem 2.4 to prove Theorem 3.5.\nThe conditions on m,ncan be stated equivalently as\nně2m, pm,n q R tp1,3q,p2,4qu.\nNote that in the first additional assumption, we impose independenc e overR, notQ. In-\ndependence over Qis equivalent to the matrix giving rise to an infinite (non-terminating)\nsequence of best approximations.\nIn the case pm,n q “ p1,3qwhere the results of Theorem 3.5 are not applicable, we\nstill have that the set Di1,3p} ¨ }1,} ¨ }2,cqzBad1,3ĚFS1,3p} ¨ }1,} ¨ }2,cqis uncountable for\narbitrarily small cą0, since by [25] we have FS1,3p} ¨ } 8,} ¨ } 8,cq ‰ Hfor anycP r0,1s\nand by equivalence of norms. Similar to Section 3.1, using Lemma 1.1 for the reverse\ncasemě2n, we get the following corollary which is new if min tm,n u ��1 even for the\nmaximum norms.\nTheorem 3.6. Letm,nbe positive integers satisfying\n(8)m\nnR`1\n2,2˘\n, pm,n q R tp1,3q,p3,1q,p2,4q,p4,2qu.\nLet} ¨ }1be any norm on Rnand } ¨ }2be any norm on Rm. Then\ndimHFSm,np} ¨ }1,} ¨ }2q ě pmaxtm,n u ´2q ¨mintm,n u “mn´2¨mintm,n u.\nMoreover we may again assume (i), (ii), and the Dirichlet spe ctrum accumulates at 0, in\nparticular it is not finite.\nNote that for the cases tm,n u “ t2,4uortm,n u “ t1,3uexcluded in (8), we have\nresults via Theorem 3.3 and [26, 25].METRICAL RESULTS ON THE FOLKLORE SET 11\nRemark 11. It may be true that Dim,n“Din,mfor every pair m,n. If min tm,n u “1,\none inclusion of the set identity was proved by Davenport and Schmid t [12]. Assuming\nthis property would simplify the proof of Theorem 3.6.\nThe claim is another contribution towards the conjecture formulat ed in [6] indicated\nin the introduction. Notably, we lose control over the precise Dirich let constant when\nm{ně2. In fact then we cannot guarantee for any concrete value in p0,1qthat it belongs\nto the Dirichlet spectrum.\n4.Brief outline of proofs\nIn summary, the claims above are connected in the following way:\nTheorem 2.2Corollary 2.1\nCorollary 2.3\nLemma 1.1\n[25, Theorem 2.2]Theorem 3.2\nTheorem 3.3\nLemma 8.1 Lemma 1.1Theorem 3.4\nTheorem 2.4 Theorem 3.5Lemma 1.1Theorem 3.6Lemma 8.1Theorem 3.1\nTheorem 2.2 is proven in Sections 5 and 6, and Theorem 2.4 is proven in S ection 7. The\ndottedarrow inthe above figurebetween Theorem 2.4 andTheorem 3.5 indicates thatthe\nmethodology of proof is similar, but by no means does Theorem 2.4 imme diately imply\nTheorem 3.5. The complete proof of Theorem 3.5 is given in Section 8. I n Section 9,\nTheorems 3.2 and 3.3 are proven. The arrow between [26, Theorem 2 .2] and Theorem 3.3\nillustrates that [26, Theorem 2.2] is crucial in the proof, but addition al tools including\nespecially Lemma 8.1 (and also Lemma 9.1) are required. Contrary to t he previous\narrows, the final stageof implications linking Theorems 2.2, 3.4, and3 .6 toTheorem 3.1is\nimmediateanddoesnotrequirefurtherdetails. FinallyLemma1.1ispro veninSection10.\nAll proofs of claims from Section 3.2 are based on constructions for the casem“\n1,n“2 of one linear form in two variables. For Theorem 2.2, we can use a con struction12 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nfrom [25], however for the results in arbitrary norms (Theorem 2.4) we follow essentially\na different construction from [27], with some refinements. Then, fo r either result, similar\nas in [27], we use a method of Moshchevitin [24] to extend the claim to lar gern. In\nTheorem 3.5 when mą1, there is some additional variational argument needed to\nextend the m“1,n“2 case first to arbitrary mandn“2 before we use Moshchevitin’s\nargument to be able to arbitrarily increase nas well. As an artifact of the method\nfrom [24], we lose the cases m“1,n“3 andm“2,n“4, on the other hand, ně2m\nis essentially required for the first step m“1,n“2. Note that we gain back the case\nm“2,n“4 via transference from Section 3.1.\nThe proofs of Section 3.1 aresurprisingly easy, given a result from [ 26] onm“2,n“1.\nWe also again employ Moshchevitin’s construction [24] here. In this c ase we lose the case\nmaxtm,n u ă4 due to problems from [26]. The special case m“ně2 of Theorem 3.2\nis not covered by the argument, indeed it does not need either of th ese prerequisite\nresults [24, 26] but uses a different self-contained construction.\nIn either method the cases |m´n| “2 are lost due to a problem rooted in [24].\n5.Proof of Theorem 2.2: Simplifying the problem\nIn short, the proof combines the special two-dimensional case of [25] with an idea of\nMoshchevitin [24]. We first recall the result on linear forms in two varia bles. Throughout\nthis section fix nPNě2and let } ¨ }1“max1ďiďn| ¨ |onRnand } ¨ }2“ | ¨ |onR. For\nζPRnandtě1 for ease of notation write\nψ1ˆn\nζp} ¨ }1,| ¨ |,tq “ψ1ˆn\nζptq.\nTheorem 5.1 ([25]).Letně2andcP r0,1s. Then there exist Q-linearly independent\nwith1real vectors ξ“ pξ1,ξ2q PR2satisfying\n(9) limsup\ntÑ8tnψ1ˆ2\nξptq “c,\nand moreover\n(10) liminf\ntÑ8tnψ1ˆ2\nξptq “0.\nRemark 12. Whenně3, the liminf statement (10) follows from the limsup claim (9),\nfor example by Marnat and Moshchevitin [22]. The linear independence overQofpξ,1q\nis also implied by (9), so it is just stated for completeness.\nObserve that for ną2 this is not the natural order of decay as in Dirichlet’s theorem,\nbut considerably faster, so ξis singular. This rate of decay will be required to increase\nthe dimension to n.\nThe claim is a special case of [25, Theorem 1.1] for n“2 and which treats “exact\nuniform approximation” with respect to more general decay funct ions Φ. As the proof\nof this result was only sketched in [25], we briefly recall the construc tion for convenience\nof the reader, with Φ ptq “ct´n. Define an integer sequence pajqjě1as follows: TakeMETRICAL RESULTS ON THE FOLKLORE SET 13\nthe initial two terms a1,a2large enough with a1|a2. Now iteratively, for kě1 having\nconstructed thefirst 2 ktermsa1,...,a 2k, let thenext twotermsaregivenbytherecursion\na2k`1“aMk\n2k\nand\na2k`2“a2pk`1q“a2k`1¨rc´1an´1\n2k`1sP pan\n2k`1,8q,\nwith integers MkÑ 8that tend to be infinitely fast enough. It turns out that ξ“\npξ1,ξ2q PR2with\nξj“8ÿ\nk“0a´1\n2k`j, j “1,2,\nhas the desired properties of Theorem 5.1. An alternative constru ction forcsmall enough\ncan be derived fromthe method in [27], we will need the latter for our r esults for arbitrary\nnorms.\nNow we want to blow up the vectors in Theorem 5.1 from R2toRnby adding compo-\nnentsξ3,...,ξ n. Let us call the new vector(s) ζPRn. Then the following is an immediate\nobservation.\nProposition 1. Take any ξPR2as in Theorem 5.1. Any ζPRnwith first two coordi-\nnatesξ1“ζ1,ξ2“ζ2, that is any vector ζP tξu ˆRn´2ĎRn, satisfies ζPDinpcqzBadn.\nProof.Let us write ξ˚“ pξ1,ξ2,1qand similarly ζ˚“ pζ1,...,ζ n,1q. Denote ei“\np0,0...,0,1,0,...,0qthei-th canonical base vector of Rn`1.\nOur claim follows from the fact that\n(11) ψ1ˆn\nζptq ďψ1ˆ2\nξptq, t ą0.\nIndeed, any tinduces a best approximation b“ pb1,b2,b3q PZ3with respect to ξthat\nrealizesψξptq. Then the blown-up vector\n˜b:“b1e1`b2e2`b3en`1“ pb1,b2,0,...,0,b3q PZn`1\ninduces the same maximum norm and approximation quality\n}ˆb} “ }ˆ˜b}, |˜b¨ζ˚| “ |b¨ξ˚| “ψ1ˆ2\nξptq.\nso the minimum over all integer points defining ψζptqcannot be larger. Trivially (10) and\n(11) implies\nliminf\ntÑ8tnψ1ˆn\nζptq “0,\nsoζRBadn. Similarly (9) and (11) implies\nlimsup\ntÑ8tnψ1ˆn\nζptq ďc,\nsoζPDinpcq. /square14 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nProposition 1 does not imply that ζRSingn. For example, if ně3 taking ζ“\npξ1,ξ2,ξ1`ξ2,0,...,0q PRnthenζPSingn. However, the following Theorem enables us\nto say that this does not happen for a significantly large subset of Rn´2. In fact, we can\nensure the induced Dirichlet constant is not smaller than the one of ξfor generic vectors\nofRn´2.\nTheorem 5.2. Letně4be an integer. Take any ξPR2as in Theorem 5.1 derived from\nany small enough cP p0,cpnqqas in Theorem 2.2. Then for a set\nY“Yn,ξĎRn´2\nof positive n´2dimensional Lebesgue measure consisting of ζ3,...,ζ n, the vector ζ“\npξ1,ξ2,ζ3,...,ζ nq PRnsatisfies\nΘ1ˆnpζq “limsup\ntÑ8tnψ1ˆn\nζptq “limsup\ntÑ8tnψ1ˆ2\nξptq “c,\nor equivalently\nζPDInpcq.\nAs for Theorem 2.2, we again believe the claim should hold for n“3 as well.\nRemark 13. Probably the pn´2q-dimensional Lebesgue measure of the set Yis full,\nhowever, this is not required for our purposes.\nBy combining Proposition 1 and Theorem 5.2 it is clear that\nζPDInpcqzBadn“FSnpcq.\nLetZcdenote the set of ξPR2satisfying Theorem 5.1. Since λnpYq ą0, for any\ncP p0,cnqwe have that\ndimHpFSnq ědimHpFSnpcqq ědimHpZcˆYq ědimHpZcq `dimHpYq ěn´2.\nThe second to last inequality follows from a standard result on the Ha usdorff dimension\nof Cartesian products\n(12) dim HpAˆBq ědimHpAq `dimHpBq\nforA,Barbitrary sets in Euclidean spaces, see [30]. Hence Theorem 2.2 is pro ven modulo\nTheorem 5.2.\nRemark 14. Notice that in the above calculation, we simply use dim HpZcq ě0. It is\nhere where our lower bound could be improved. For τą2 a parameter, define\nSing2pτq:“\"\nξPR2: limsup\ntÑ8tτψ1ˆ2\nξptq “0*\n.\nIt can be deduced from [11, Theorem 1.4-1.9] by a change of variable s that for any\nτą2`?\n2 we have\ndimHSing2pτq “2\nτ.\nClearlyZcĎSing2pn´εqfor anyεą0, and so\ndimHpZcq ď2\nn´ε.METRICAL RESULTS ON THE FOLKLORE SET 15\nThus forně4ą2`?\n2 the best we could hope for via this method is\ndimHpFSnpcqq ěn´2`2\nn.\nOn the other hand, a positive lower bound for dim HpZcqcan be obtained by a similar\nvariational method as in [27, Theorem 2.1], we do not carry it out.\nTheorem 5.2 remains to be proved, for which we reserve the next se ction.\n6.Proof of Theorem 5.2\nWe have to show the Dirichlet constant of many ζis at leastc, the rest is obvious.\nWe follow ideas from the proof of Moshchevitin [24, Theorem 12] with some twists.\nWriteζ“ pξ1,ξ2,ζ3,...,ζ nqand restrict to pζ3,...,ζ nqin then´2-dimensional unit ball\nBn´2p0,1q ĎRn´2with respect to the Euclidean norm (this will be slightly easier to\nhandle than for the maximum norm). Denote by\nbv“ pb1,v,b2,v,b3,vq PZ3, v ě1,\nthe best approximations with respect to ξ“ pξ1,ξ2qand derive\nMv“max\nj“1,2|bj,v| “ }ˆbv}, v ě1\nand the minimal values\nLv“ |b1,vξ1`b2,vξ2`b3,v|, v ě1.\nClearly pLvqvě1is a strictly decreasing sequence and pMvqvě1is strictly increasing. Fur-\nther, note that we have\nψ1ˆ2\nξptq “Lv, t P rMv,Mv`1q,\nhence\n(13) limsup\ntÑ8tnψ1ˆ2\nξptq “limsup\nvÑ8LvMn\nv`1.\nDenoteei“ p0,0...,0,1,0,...,0qthei-th canonical base vector of Rn`1. To finish the\nproof, it suffices to show the following:\nLemma 6.1. Letně4an integer and ξ“ pξ1,ξ2qas in Theorem 5.1. For a positive\nmeasure set YĎBn´2p0,1q ĎRn´2consisting of γ“ pξ3,...,ξ nq, we have that for\ninfinitely many integers vě1, lettingTv:“Mv`1´1{2, the integer best approximation\nforζ“ pξ,γqatTvis given as\nb1,ve1`b2,ve2`b1,ven`1“ pb1,v,b2,v,0,...,0,b3,vq PZn`1,\nthus the best approximation function at values Tvequals\nψ1ˆn\nζpTvq “ψ1ˆ2\nξpTvq “Lv.16 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nObserve that since Tvis of the same order as Mv`1, forζas in the lemma, (13) and\nthe corresponding property for ζreadily implies\nΘ1ˆnpζq ělimsup\nvÑ8LvTn\nv“limsup\ntÑ8tnψ1ˆ2\nξptq,\nthe reverse inequality is clear by Proposition 1. Hence, assuming the lemma is true, the\nclaim of Theorem 5.2 follows.\nWe finally prove the preceding lemma as in [24].\nProof of Lemma 6.1. We show that for a positive portion of Bn´2p0,1q ĎRn´2of vectors\nγ“ pζ3,...,ζ nq, we have for infinitely many vthat\n(14) minˇˇˇˇˇx1ξ1`x2ξ2`nÿ\ni“3xiζi´yˇˇˇˇˇěLv\nwith minimum taken over all integer points\nz“ px1,...,x n,yq PZn`1zt0u\nwith\n(15) 0 ămax\n1ďjďn|xj| ďTv.\nForvě1, denote by Svthe set of such vectors of Rn´2taken over all integer points xi,y\nobeying (15), i.e.\nSv:“$\n’&\n’%γPBn´2p0,1q: min\npx,yqPZn`1:\n0ămax1ďiďn|xj|ďTvˇˇˇˇˇx1ξ1`x2ξ2`nÿ\ni“3xiζi´yˇˇˇˇˇěLv,\n/.\n/-.\nThen our set is the limsup set of the Sv, or its complement is the set of all vectors that\nlie only in finitely many Sv, that is,\n(16) Bn´2p0,1qzYĎď\niě1č\njěiSc\njĎď\niě1Sc\ni.\nAssume that any complement\nUv:“Sc\nv“$\n’&\n’%γPBn´2p0,1q: min\npx,yqPZn`1:\n0ămax1ďiďn|xj|ďTvˇˇˇˇˇx1ξ1`x2ξ2`nÿ\ni“3xiζi´yˇˇˇˇˇăLv,\n/.\n/-, v ě1,\nhasn´2 dimensional Lebesgue measure less than some δstrictly smaller than the volume\nofBn´2p0,1q. Then this will also be true for Bn´2p0,1qzYby the following easy result.\nProposition 2. Letλbe a measure on a space T. AssumeHiare increasing subsets of\nT, i.e.H1ĎH2Ď ¨ ¨ ¨. Then\nλ˜8ď\nk“1Hk¸\n“lim\nkÑ8λpHkq.\nWe will require the estimate λ`Ť8\nk“1Hk˘\nďlimkÑ8λpHkqonly.METRICAL RESULTS ON THE FOLKLORE SET 17\nProof.WritingGj“HjzHj´1withH0“ H, we see\nλ˜8ď\nk“1Hk¸\n“λ˜8ď\nk“1Gk¸\n“8ÿ\nk“1λpGkq\n“lim\nNÑ8Nÿ\nk“1λpGkq\n“lim\nNÑ8λ˜Nď\nk“1Hk¸\n“lim\nNÑ8λpHNq.\n/square\nIn the sequel denote by λthepn´2q-dimensional Lebesgue measure of a set AĎRn´2.\nAs indicated before, we apply Proposition 2 to\nHi“č\njěiSc\nj“č\njěiUjĎUi, i ě1,\nto see via (16) that\nλpBn´2p0,1qzYq ďlim\nvÑ8λpUvq.\nAssume we have shown that for some uniform ∆ we have\n(17) λpUvq ď∆ăλpBn´2p0,1qq, v ě1.\nThen we conclude that\nλpBn´2p0,1qzYq ď∆ăλpBn´2p0,1qq\nand thus\nλpYq “λpBn´2p0,1qq ´λpBn´2p0,1qzYq ěλpBn´2p0,1qq ´∆ą0,\nas desired.\nTo verify (17), let us estimate λpUvqfor fixedv. For any γ“ pζ3,...,ζ nq PUvit follows\nthat\nmin\npx,yqPZn`1:\n0ămax1ďiďn|xj|ďTvˇˇˇˇˇx1ξ1`x2ξ2`nÿ\ni“3xiζi´yˇˇˇˇˇăLv.\nEquivalently\nx3ζ3` ¨ ¨ ¨ `xnζnPJvpx1,x2,yq\nwhereJvp.qis given as the interval\nJvpx1,x2,yq “ p´y´x1ξ1´x2ξ2´Lv,´y´x1ξ1´x2ξ2`Lvq.18 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nLet\nΩvpx1,...,x n,yq “#\npζ3,...,ζ nq PBn´2p0,1q ĎRn´2:nÿ\ni“3xiζiPJvpx1,x2,yq+\n.\nObserve that Ω vpx1,...,x n,yqis empty if |y| ą |x1ξ1| ` |x2ξ2| `řn\ni“3|xiζi| `Lv. Fix for\nnowεą0. SinceLvÑ0 asvÑ 8and as we may take ξ1,ξ2arbitrarily small positive\nnumbers (the ξfrom Theorem 5.1 are easily seen to be dense in R), say smaller than ε{8,\nand pζ3,...,ζ nq PBn´2p0,1q, Cauchy-Schwarz inequality yields for any εą0 this is true\nif\n|y| ą p?\nn´2`ε{2qTv\nas then\n|y| ą p?\nn´2`ε{2qTv\ně pε{4q ¨Tv`˜nÿ\ni“3|xi|2¸1{2˜nÿ\ni“3|ζi|2¸1{2\n`Lv\ně |ξ1||x1| ` |ξ2||x2| `˜nÿ\ni“3|xi|2¸1{2˜nÿ\ni“3|ζi|2¸1{2\n`Lv\ně |x1ξ1| ` |x2ξ2| `nÿ\ni“1|xiζi| `Lv, v ěv0pεq.\nThe above shows that if we let\n(18) Ω v“ď\nx1,...,xnď\nyΩvpx1,...,x n,yq,\nwith unions are taken over integers pairs x1,x2as in (15) and integers ywith |y| ď\np?n´2`ε{2qTv, then\nUvĎΩv, v ěv0pεq.\nThe distance between the parallel planes defining Ω vpx1,...,x n,yqequals 2 px2\n3` ¨ ¨ ¨ `\nx2\nnq´1{2. Thus each Ω vpx1,...,x n,yqis contained in a cylinder of radius one and height\n2Lv¨ px2\n3` ¨ ¨ ¨ `x2\nnq´1{2. Denoting by λ1thepn´3q-dimensional Lebesgue measure, this\nhas volume\nλpΩvpx1,...,x n,yqq “2λ1pBn´3p0,1qq ¨Lvpx2\n3` ¨ ¨ ¨ `x2\nnq´1{2\nď2Lv¨λ1pBn´3p0,1qq ¨1\nmax3ďiďn|xi|. (19)METRICAL RESULTS ON THE FOLKLORE SET 19\nSince, assuming Tvěε´1, there are at most 2 Tvp?\nn´2`ε{2q `1ďTvp2?\nn´2`2εq\nintegersyin the inner union of (18), in total we get\nλpUvq ďλpΩvq\nďÿ\nx1,x2ÿ\nyÿ\nx3,...,xnλpΩvpx1,...,x n,yqq\nď2p2?\nn´2`2εqLvTvλ1pBn´3p0,1qq ¨ÿ\nx1,x2ÿ\nx3,...,xn1\nmax3ďiďn|xi|\nforvěv0pεq. The outer sum over x1,x2just gives a factor tTvu2ďT2\nvby (15). We split\nthe inner sum into sets where max 3ďiďn|xi| “kP r1,tTvusis constant, which correspond\nto 2pn´2q “2n´4 faces on a hypercube for each kaccording to which variable equals\n˘k, giving in total p2n´4qkn´3integer points for each kto sum 1 {kover. Hence we may\nfurther estimate\nλpUvq ď2p2n´4qp2?\nn´2`2εqλ1pBn´3p0,1qq ¨LvT3\nv¨tTvuÿ\nk“1kn´4\nď2p2n´4qp2?\nn´2`2εqλ1pBn´3p0,1qqˆ1\nn´3`ε2˙\nLvTn\nv, v ěv1pε,ε2q.\nHere the error term ε2ą0 tends to 0 as vÑ 8, and we used n‰3 to avoid a logarithmic\nterm appearing in the last sum. We may assume ε2ăεfor all large v. So ifLvăcT´n\nv\nfor anycP p0,cn´ǫqforǫą0 some small manipulation of εand withcnsatisfying\nλpBn´2p0,1qq ącn4p2n´4q?\nn´2λ1pBn´3p0,1qq\nn´3,\nor equivalently\ncnăn´3\n4p2n´4q?n´2¨λpBn´2p0,1qq\nλ1pBn´3p0,1qq“n´3\n8pn´2q3{2¨?π¨Γpn´1\n2q\nΓpnq,\nindeed (17) holds for some ∆ ăλpBn´2p0,1qq, depending on cn, and allvěv0p∆q “\nv0pcnq. Hence the complement contained in Yhas positive pn´2q-dimensional Lebesgue\nmeasure. Since we can choose εą0 and thusǫą0 arbitrarily small, we may just assume\ncăcnand the proof of Lemma 6.1 is complete. /square\nRemark 15. Improvements of cncanbemadebysharpeningthecontribution2?\nn´2`1\nfor the number of y. Indeed, for most ythe obtained intersections with the unit ball have\na considerably smaller volume than 2 Lv¨ px2\n3` ¨ ¨ ¨ `x2\nnq´1{2, so it seems we can gain some\nfactor by more concise estimates.\n7.Proof of Theorem 2.4\nThe method of proof is similar to that of Theorem 2.2. Namely, the pro of is done in\ntwo steps:20 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nStep I: For small enough c, construct ξ“ pξ1,ξ2qsuch that\nlimsup\ntÑ8tnψ1ˆ2\nξp} ¨ },tq “c.\nStep II: Show that for almost all pξ3,...,ξ nqwith respect to pn´2q-dimensional Lebesgue\nmeasure, the vector ζ“ pξ1,...,ξ nqsatisfies\nΘ1ˆnpζq “limsup\ntÑ8tnψ1ˆn\nζp} ¨ },tq “c\nas well.\n7.1.Proof of Step I. Thisisthemainsubstanceoftheproof. Forthis, weuseadifferent\nconstruction than for the maximum norm where we employed the con struction from [25].\nRather we follow the proof of [27, Theorem 2.1] but with some notable twists.\nWriteξ˚“ pξ,1q PR3anduseasimilarmeaningofstarforotherquantities. Conversely\ndenote by a hat over a vector in R3its restriction to the first 2 coordinates by chopping\noff the last coordinate, i.e.pξ˚“ξ.\nWe construct our real vector. Let năτăµbe parameters to be chosen later related\nby the identity\n(20) τµ´1“nµ ð ñµ“1\nτ´n.\nLetαj,βj,γj,δjbe strictly increasing positive integer sequences and a real number r“\nrp} ¨ }q ą0 all to be fixed later and derive integers\nAj“2αj3γj, B j“5βj7δj,\nso that\n(21) Aj`1“ p1`op1qqrBτ\nj, B j“ p1`op1qqAµ\nj.\nWe first show that such choices are possible. For Aj, we use that tlog2,log3 uis aQ-\nlinearly independent set (easily seen by applying exponential map to a putative vanishing\nlinear form and using unique prime factorization to disprove it) and th e following easy\nProposition 3. Similarly for Bjwe use that tlog5,log7 uisQ-linearly independent.\nProposition 3. Leta,b,cbe real numbers with a,blinearly independent over Q. Then\nordering the set tk1a`k2b`c:kiPNuincreasingly and denoting it by pziqiě1we have\n|zi`1´zi| Ñ0.\nProof.Clearly we may assume c“0 and suppose without loss of generality that băa.\nWe may then equivalently consider\nZ“ tk1`k2b\na:kiPNu\ninstead. Sinceb\nais irrational, Kronecker’s Theorem (see e.g. [7, Section 3.5 Theorem I V])\nshows that the set is dense in the unit interval when considering all pk1,k2q PZˆN. Let\nε“1{Ką0 with a large integer Kand partition r0,1sintoKintervals\nIt“ rpt´1qε,tεs,1ďtďKMETRICAL RESULTS ON THE FOLKLORE SET 21\nof length 1 {K. Then for any Itthere exist pk1\n1ptq,k1\n2ptqq PZˆNso thatk1\n1ptq`k1\n2ptqb\naPIt.\nTakeTε:“maxt|k1\n1ptq|and consider the set\nZTε“/visualspace\npk1ptq `Nq `k2ptqb\na: 1ďtďK, N PNěTε(\nĂZ.\nIt is easy to see that for any MPNěTεthe interval M`Itcontains a point from ZTε.\nThis shows that Zisεdense on some interval r3Tε,8q. Sinceεą0 is arbitrary the claim\nis proved. /square\nWe can now apply the proposition alternatingly to\na“log2, b“log3, c“logr, a “log5, b“log7, c“0\nrespectively to construct iteratively our sequences αj,γjwhich correspond to k1“αj`1´\nαjě0 andk2“γj`1´γjě0, resp.k1“βj`1´βjandk2“δj`1´δj. Thereby, having\nchosen allα.,β.,γ.,δ.up to index j, we construct first pairs αj`1,γj`1and thenβj`1,δj`1,\nand the iterative construction is complete. So having chosen rwe may assume all the\nabove is defined.\nNote that (21) implies\n(22) Aj`1“ p1`op1qqrAτµ\nj, B j`1“ p1`op1qqrµBτµ\nj.\nMoreover, again by (21), upon changing initial terms if necessary, we can assume\n(23) 1 ăA1ăB1ăA2ăB2ă ¨ ¨ ¨.\nFinally let ξ“ pξ1,ξ2qwith\nξ1“ÿ\njě1A´1\nj, ξ 2“ÿ\njě1B´1\nj.\nWe claim that it satisfies the assertions of the theorem upon choosin grappropriately.\nForjě1 put\nσj:“A´1\n1` ¨ ¨ ¨ `A´1\nj, η j:“B´1\n1` ¨ ¨ ¨ `B´1\nj.\nWe clearly have σjÑξ1andηjÑξ2asjÑ 8, and that\nFj“AjσjPN, G j“BjηjPN.\nThensince thesequences pαjq,pβjq,pγjq,pδjqarestrictlyincreasing wehave Fj”1 mod 2 ¨\n3 andGj”1 mod 5 ¨7, which imply the coprimality assertions\n(24) pAj,Fjq “ pFj,6q “1, pBj,Gjq “ pGj,35q “1.\nLet\nvj“ pAj,0,´Fjq,wj“ p0,Bj,´Gjq.\nNote that ˆvj“ pAj,0qandˆwj“ p0,Bjq. Hence their norms } ¨ }satisfy\n(25) }ˆvj} “ }e1}Aj“d1Aj, d 1:“ }e1} “ }p1,0q},\nand\n(26) }ˆwj} “ }e2}Bj“d2Bj(21)“ p1`op1qqd2Aµ\nj, d 2:“ }e2} “ }p0,1q},22 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nThus by (23) clearly\n(27) }ˆvj} ă }ˆwj} ă }ˆvj`1}, j ěj0.\nHerej0depends on d1,d2, thus on the chosen norm. Then by (22) moreover\n|vj¨ξ˚| “ |Aj¨ξ1`0¨ξ2´Fj| “AjpA´1\nj`1`A´1\nj`2` ¨ ¨ ¨ q (28)\n“AjA´1\nj`1p1`op1qq(22)“ p1`op1qqr´1A´pτµ´1q\nj\nand\n|wj¨ξ˚| “ |0¨ξ1`Bj¨ξ2´Gj| “BjpB´1\nj`1`B´1\nj`2` ¨ ¨ ¨ q (29)\n“ p1`op1qqBjB´1\nj`1(22)“ p1`op1qqr´µ¨B´pτµ´1q\nj,\nare small linear forms for jě1. We next show\nLemma 7.1. As soon as ris large enough, all best approximations for ξof large norm\nhave up to sign one of the following forms\nvj,wj,vj`wj,wj´vj,wj`vj`1,vj`1´wj.\nProof.Letbbe any best approximation. Then by (27) there is an index jsuch that either\n}ˆvj} ď }ˆb} ă }ˆwj}or}ˆwj} ď }ˆb} ă }ˆvj`1}. We show that in the first case b“ ˘vjor\n˘pvj˘wjq, and in the latter case b“ ˘wjor˘pwj˘vj`1q. Assume the first case, so\n(30) }ˆvj} ď }ˆb} ă }ˆwj}.\nthe latter works very similarly by symmetry. First, observe that sin cebis the best\napproximation of norm at least }ˆvj}, we know that\n(31) |b¨ξ˚| ď |vj¨ξ˚|.\nWe distinguish two cases.\nCase 1: blies in the two-dimensonal subspace of R3spanned by vj,wj, i.e.bP\nxvj,wjyRXZ3. The special form of Aj,Bjand (24) imply the following crucial result on\ninteger vectors in the two-dimensional lattices xvj,wjyRXZ3.\nProposition 4. Forvj,wjas above, if a linear combination gvj`hwjis an integer\nvector, then in fact gPZandhPZ. In other words, xvj,wjyRXZ3“ xvj,wjyZ.\nProof.Clearly, we must have g,h PQ. If we write pp1{q1qvj` pp2{q2qwjwithpi{qiin\nlowest terms, then it is clear that q1must consist of exclusively non-negative integer\npowers of 2 and 3 and q2of non-negative integer power of 5 and 7 to make the first\ntwo coordinates pp1{q1qAj“ pp1{q1q2αj3γjresp. pp2{q2qBj“ pp2{q2q5βj7δjofgvj`hwj\nintegers. But then by (24) clearly the third coordinate pp1{q1qFj` pp2{q2qGjis not an\ninteger unless q1“q2“1. /square\nWe first show that any integer linear combination b“gvj`hwjrepresenting the best\napproximation as above must have |g| ď1. Assume |g| ě2. It is easily seen that for anyMETRICAL RESULTS ON THE FOLKLORE SET 23\nnorm on R2there is a constant θ“θp} ¨ }qsuch that }w} ďWforw“ pw1,w2q PR2and\nWą0 implies |wi| ďθW,i“1,2. Hence (30) and the special form of vj,wjimply\n|g| ďθ, |h| ďθ\nfor someθindependent of j. On the other hand, since r,µare fixed and AjÑ 8, it is\nclear from (28), (29) and (21) that\n|wj¨ξ˚| “op|vj¨ξ˚|q, j Ñ 8.\nCombination implies\n|b¨ξ˚| “ |pgvj`hwjq ¨ξ˚| ě |g| ¨ |vj¨ξ˚| ´ |h| ¨ |wj¨ξ˚| ě2|vj¨ξ˚| ´ |h| ¨ |wj¨ξ˚|\ně2|vj¨ξ˚| ´θ¨ |wj¨ξ˚| ą |vj¨ξ˚|, j ěj0,\ncontradicting (31). Hence |g| ď1. Now if |h| ě2 then for any |g| ď1 using (27) we get\n}ˆb} “ }gˆvj`hˆwj} ě |h| ¨ }ˆwj} ´ }ˆvj} ě2¨ }ˆwj} ´ }ˆvj} ą }ˆwj}\ncontradicting (30). Thus both g,hare among t´1,0,1u. Hence we are left with the\nvectors of the lemma.\nCase 2:bdoes not lie in the space spanned by vj,wj. For this case, we use an easy\nconsequence of Minkowski’s Second Convex Body Theorem.\nLemma 7.2. There exists a constant cą0such that for any ξPR2and any parameter\nQě1, the system\n|b1| ďQ, |b2| ďQ, |b1ξ1`b2ξ2`b3| ăcQ´2\ndoes not have three linearly independent solutions in integ er vectors b“ pb1,b2,b3q.\nProof.Consider the integer lattice Z3and the box of px1,x2,x3q PR3with coordinates\n|x1| ďQ, |x2| ďQ, |ξ1x1`ξ2x2`x3| ďcQ´2.\nIt has volume 8 c, independent of Qandξ1,ξ2. Hence, by Minkowski’s Second Convex\nBody Theorem, the product of the induced successive minima is !c, hence choosing c\nsmall enough the third successive minimum is smaller than 1. This means there cannot\nbe three linearly independent integer points within the box, which in tu rn is equivalent\nto the claim. /square\nWe first notice that both vjandwjinduce approximations of order greater than two.\nBy (26), (28) and as our choice pτµ´1q{µ“ną2, we get\n(32) |vj¨ξ˚| “ p1`op1qqr´1A´pτµ´1q\nj !r´1B´n\nj!r´1}ˆwj}´năr´1}ˆwj}´2\nand forwjby (29) we have a stronger estimate that also yields\n(33) |wj¨ξ˚| “op1q ¨B´pτµ´1q\nj “op}ˆwj}´nq “op}ˆwj}´2q, j Ñ 8.\nCombined with (27), (30), (31), we have\nmaxt}ˆb},}ˆvj},}ˆwj}u “ }ˆwj},maxt|b¨ξ˚|,|vj¨ξ˚|,|wj¨ξ˚|u !r´1}ˆwj}´2.\nBy the assumptions of Case 2, the three vectors vj,wj,bare linearly independent. So we\nget a contradiction to Lemma 7.2 for rsufficiently large and Q“ }ˆwj}, as soon as }ˆwj}is24 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nsufficiently large. Hence in total Case 2 provides only finitely many bes t approximations,\nof small norm, so Lemma 7.1 is proved. /square\nForstudying ψ1ˆ2\nξ,wemayrestricttointegervectorsoftheformappearinginLemma 7.1.\nNow |wj¨ξ˚|is much larger than |vj`1¨ξ˚|, and |vj¨ξ˚|is much larger than |wj¨ξ˚|\nby (21), (28), (29). Hence by the triangle inequality vj˘wjinduce the same approxi-\nmation quality as vjup to a factor 1 `op1q, and similarly, wj˘vj`1induce the same\napproximation quality up to 1 `op1qaswj, i.e\n(34) |pvj˘wjq ¨ξ˚| “ p1`op1qq ¨ |vj¨ξ˚|,|pvj`1˘wjq ¨ξ˚| “ p1`op1qq ¨ |wj¨ξ˚|.\nMoreover as the norm of ˆwjis much larger than the norm of ˆvjand the norm of ˆvj`1is\nmuch larger than the norm of ˆwj, by (21) and (25), we have by the triangle inequality\nthat\n(35) }ˆvj˘ˆwj} “ p1`op1qq}ˆwj}, }ˆvj`1˘ˆwj} “ p1`op1qq}ˆvj`1}.\nBy (34), (35) and since we have freedom up to a factor 1 `op1qin our claim to be\nproved, we may assume that all best approximations are among vj,wj. Now we choose\nrappropriately. To finish the proof we show that choosing τ,µ,rsuitably we have\n(36) |vj¨ξ˚| “ p1`op1qq ¨c}ˆwj}´n, |wj¨ξ˚| “op1q ¨ }ˆvj`1}´n.\nAssume this is true. Then indeed partitioning the interval r}v1},8qinto consecutive\ndisjoint intervals of the form r}vj},}wj}qand r}wj},}vj`1}qwe see\ntnψ1ˆ2\nξp} ¨ },tq ďcp1`op1qqtÑ 8.\nOn the other hand, since there is no better approximation than the one induced by vj\nfor anytă }wj}, fortj:“ }wj} ´1 we have the reverse inequality\ntn\njψ1ˆ2\nξp} ¨ },tjq ěcp1`op1qqjÑ 8.\nCombining proves the equality\nlimsup\ntÑ8tnψ1ˆ2\nξp} ¨ },tq “c.\nFor the left condition of (36) equivalently by (26), (28) we need\np1`op1qqr´1A´pτµ´1q\nj “c¨d´n\n2A´nµ\nj,\nso we pick\nr“c´1dn\n2Anµ´τµ`1\nj(20)“c´1dn\n2.\nSimilarly, by (25), (29) for the right condition of (36) it suffices to ha ve\nB´pτµ´1q\nj “opd´n\n1A´nτ\nj`1q.\nSinceA´nτ\nj`1—B´nτ2\njby (21) and d1is constant, it suffices to notice that upon (20) we\nmay choose τăµso thatnτ2ăτµ´1. Indeed it suffices to choose τjust slightly larger\nthannso thatµ“1{pτ´nq ąτ2which is equivalent to nτ2ăτµ´1. We leave the\nshort calculation to the reader. We have completed step I. Assumin gcis small enough,\nwe apply Step II below.METRICAL RESULTS ON THE FOLKLORE SET 25\n7.2.Proof of Step II. The proof of step II works analogously to the special case of\nmaximum norm. Our norm } ¨ }onRninduces a projected norm } ¨ }1onR2via\n(37) }px1,x2q}1:“ }px1,x2,0,...,0q}.\nThen for this norm } ¨ }1we apply step I to get ξ“ pξ1,ξ2qwith Dirichlet constant c, i.e.\nlimsup\ntÑ8tnψ1ˆ2\nξp} ¨ }1,tq “c.\nLetbv“ pb1,v,b2,v,b3,vqbe the integer vector sequence constructed in Step I.\nBy construction of } ¨ }1, it is again clear by the same argument as in Proposition 1\nthat any vector ζ“ pξ1,...,ξ nq PRnextending ξhas Dirichlet constant at most cwith\nrespect to } ¨ }, i.e.\nlimsup\ntÑ8tnψ1ˆn\nζp} ¨ },tq ďlimsup\ntÑ8tnψ1ˆ2\nξp} ¨ }1,tq “c.\nIndeed, we can choose the embedded integer vectors ιpbvqvia\nι:py1,y2,y3q Ñ py1,y2,0,...,0,y3q\nfromR3toRn`1, which by (37) induce the same heights and approximation quality\n}zιpbvq} “ }ˆbv}1, |ιpbvq ¨ζ˚| “ |bv¨ξ˚|\nfor anybv, with hat notationaccording to (3) onthe respective spaces. For the non-trivial\nlower estimate ěcwith respect to } ¨ }, define\nLv“ |ιpbvq ¨ζ˚| “ |bv¨ξ˚| “ |b1,vξ1`b2,vξ2`b3,v|, Mv“Mvp} ¨ }q “ } ˆbv}1“ }zιpbvq}\nandTv“Mv´1{2 as in the proof for the maximum norm but where Mv,Tvare now\ndefined with respect to our norm } ¨ }. Then by the equivalence of norms on Rn, for any\ninteger vector of norm }px1,...,x nq} ďTvwe still have\nmax\n1ďiďn|xi| ďCp} ¨ }q ¨Tv\nfor some constant Cdepending on the norm only. We just replace (15) by this modified\ninequality and follow the proof of the special case Lemma 6.1 for the m aximum norm.\nThe twist by the constant Cwill result in some altered value of cndepending on the\nchosen norm for the same conclusion. We omit its explicit calculation in d ependence\nofC.\n8.Proof of Theorem 3.5\n8.1.Proof excluding linear independence claims. We proceed as in §7.1 to con-\nstruct the first two elements of the first line of Ω by ξ“ pξ1,ξ2q “ pΩ1,1,Ω1,2qof our\nmatrix, upon modifying the construction so that (20) is replaced by\nτµ´1“n\nmµ(7)\ně2µ.26 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nDefineψ1ˆ2\nξwith respect to the restriction } ¨ }1\n1of} ¨ }1toR2in the integer vector b, as\nin (37). Then by the same arguments of the proof of Theorem 2.4 we will have\n(38) limsup\ntÑ8tn{mψ1ˆ2\nξp} ¨ }1\n1,tq “c.\nHereby we use our assumption n{mě2 for Case 2, based on Minkowski’s Second Convex\nBody Theorem, to work analogously.\nNow take the matrix VPRmˆ2with allmlines equal to ξ, i.e.\nVj,1“ξ1, V j,2“ξ2,1ďjďm.\nThis matrix gives rise to an approximation function\nψmˆ2\nVp} ¨ }1\n1,} ¨ }2,tq\nwith norms defined as above. First, assume for simplicity that } ¨ }2“ } ¨ } 8is the\nmaximum norm on Rm. We claim that\n(39) ψmˆ2\nVp} ¨ }1\n1,} ¨ }2,tq “ψ1ˆ2\nξp} ¨ }1\n1,tq, t ą0.\nIndeed to see\n(40) ψmˆ2\nVp} ¨ }1\n1,} ¨ }2,tq ěψ1ˆ2\nξp} ¨ }1\n1,tq\nforallt, notethatforarbitrary b“ pb1,...,b m`2q PZm`2andforV˚PRmˆpm`2qobtained\nby glueingVto themˆmidentity matrix to the right, the induced value\n}V˚¨b}2“ }V¨pb`˜b}2“ }V¨ pb1,b2q ` pb3,...,b m`2q}2\nwith vectors interpreted as column vectors, is at least as large as t he modulus of any\nentry, since we are considering } ¨ }2“ } ¨ } 8for now. But for the first entry this equals\n|pb1,b2q ¨ξ`b3| “ |ξ˚¨ pb1,b2,b3q|. Taking the minimum over bas in the definition of\nψmˆ2\nVshows the inequality (40). Conversely, if we extend arbitrary b“ pb1,b2,b3q PZ3\nintoZm`2viaηpbq:“ pb1,b2,b3,...,b 3qwith lastmcoordinates equal to b3, by choice of\nmaximum norm, we have }V˚¨ηpbq}2“ |ξ˚¨ pb1,b2,b3q|, so passing to the minimum over\nbagain we get the reverse inequality\n(41) ψmˆ2\nVp} ¨ }1\n1,} ¨ }2,tq ďψ1ˆ2\nξp} ¨ }1\n1,tq.\nCombining (38), (39) we get\nlimsup\ntÑ8tn{mψmˆ2\nVp} ¨ }1\n1,} ¨ }2,tq “limsup\ntÑ8tn{mψ1ˆ2\nξp} ¨ }1\n1,tq “c.\nTo finish the proof, we show that for a positive measure set of BPRmˆpn´2qinducing\nΩ“ pV,Bqvia putting it to the right of V, we have\nlimsup\ntÑ8tn{mψmˆn\nΩp} ¨ }1,} ¨ }2,tq “limsup\ntÑ8tn{mψmˆ2\nVp} ¨ }1\n1,} ¨ }2,tq.\nAgain the inequality\n(42) limsup\ntÑ8tn{mψmˆn\nΩp} ¨ }1,} ¨ }2,tq ďlimsup\ntÑ8tn{mψmˆ2\nVp} ¨ }1\n1,} ¨ }2,tqMETRICAL RESULTS ON THE FOLKLORE SET 27\nis easy to check for arbitrary } ¨ }2and extensions Ω PRmˆnofVby an arbitrary matrix\nBPRmˆpn´2q. Indeed, for a sufficient estimate\nψmˆn\nΩp} ¨ }1,} ¨ }2,tq ďψmˆ2\nVp} ¨ }1\n1,} ¨ }2,tq, t ą0\nwe embed the integer best approximations of Vby adding 0 coordinates via\nσ:b“ pb1,b2,...,b m`2q Ñ pb1,b2,0,...,0,b3,...,b m`2q PZm`n.\nThese induce the same norm }yσpbq}1“ }pb}1\n1where\nyσpbq “ pb1,b2,...,b nq “ pb1,b2,0,...,0q,pb“ pb1,b2q,\nwith hat again as in (3) on the respective spaces. With Ω˚derived from Ω likewise as for\nV˚, they further induce the same approximation qualities }Ω˚¨σpbq}2“ }V˚¨b}2since\nin fact Ω˚¨σpbq “V˚¨b, proving (42). For the reverse estimate\n(43) limsup\ntÑ8tn{mψmˆn\nΩp} ¨ }1,} ¨ }2,tq ělimsup\ntÑ8tn{mψmˆ2\nVp} ¨ }1\n1,} ¨ }2,tq “c,\nweuseageneralisationofStepIIfrom §7.2formą1asinMoshchevitin[24,Theorem12].\nWe obtain that for small enough cand almost all remaining matrices BPRmˆpn´2qwith\nrespect tompn´2qdimensional Lebesgue measure when added to the right of Vto obtain\nΩ, we have (43). Hereby we require n‰m`2 for similar reasons as ně4 was previously\nneeded in our proof for one linear form, see [24]. Moreover, the ass umptionně2mis\nvital for the volume estimates. See Lemma 8.1 below for a generalisat ion. The proof for\n} ¨ }2the maximum norm is complete.\nNow let the norm } ¨ }2be arbitrary. Again we prove (40), (41), (42), (43). The critical\nestimates where the norm } ¨ }2is relevant for the argument are (40), (41). First notice\nthat by the special form of V, ifbis not of the form b“ pb1,b2,b3,b3,...,b 3q PZm`2\nthen some entry of V˚¨bwill be of modulus at least 1 {2, so }V˚¨b}8ě1{2, hence by\nequivalence of norms, we deduce\n}V˚¨b}2\"1\nwith some absolute constant depending on the norm } ¨ }2only. Hence we only need\nto take into account vectors of the form b“ pb1,b2,b3,b3,...,b 3qwhen studying ψmˆ2\nV.\nBut then the vector V˚¨bPRmlies on the line λ¨ p1,1,...,1q PRmand has norm\n|λ| ¨ }p1,1,...,1q}2proportional to |λ|. Hence we have to minimize |λ|. In§7.1, see in\nparticular Lemma 7.1, we showed that this is achieved when the subve ctor pb1,b2,b3qis\nessentially of the form vj“ pAj,0,´Fjqandwj“ p0,Bj,´GjqforAj,Bj,Fj,Gjdefined\nin the same section (as shown in §7.1, for the other possible vectors of Lemma 7.1 we\nmay only improve the approximation quality by a negligible factor 1 `op1qasjÑ 8).\nThese arguments show that the best integer approximations for Vare essentially of the\nform\n(44) cj:“ pAj,0,´Fj,..., ´Fjq,dj:“ p0,Bj,´Gj,..., ´Gjq28 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nso that for vj,wjPZ3as before, we get error vectors\nV˚¨aj“V¨ˆ\nAj\n0˙\n´Fj¨¨\n˚˚˝1\n1\n...\n1˛\n‹‹‚“vj¨ξ˚¨¨\n˚˚˝1\n1\n...\n1˛\n‹‹‚PRm\nand\nV˚¨bj“V¨ˆ\n0\nBj˙\n´Gj¨¨\n˚˚˝1\n1\n...\n1˛\n‹‹‚“wj¨ξ˚¨¨\n˚˚˝1\n1\n...\n1˛\n‹‹‚PRm.\nThen for Γ : “ }p1,1,...,1q}2they have norms\n}V˚¨cj}2“ |vj¨ξ˚| ¨Γ, }V˚¨dj}2“ |wj¨ξ˚| ¨Γ.\nThus we have to adjust the proof of the special case by the const ant factor Γ in the\ncalculation, so that in order to finalize Step I we now need to pick\nr“c´1Γ´1dn\n2, d 2“ }e2}1\n1“ }p0,1q}1\n1\nto get (41). The reverse inequality (40) holds as well as we noticed t hat the vectors\nin (44) are (essentially) the integer best approximations. We obser ved that (42) holds\nindependent of } ¨ }2.\nFinally, Step II is to again show the reverse inequalities (43) hold. This is done analo-\ngouslytoLemma6.1butinhigherdimension, asinMoshchevitin [24, Theo rem12]. Again\nessentially irrespective of the norm } ¨ }2. We state the according analogue of Lemma 6.1\nthat we will prove analogously but without effort to make constants effective.\nLemma 8.1. Letm,nbe positive integers with ně2andn‰m`2. Let } ¨ }1and } ¨ }2\nbe any norms on RnandRmand} ¨ }1\n1be the restriction of } ¨ }1toR2as in(37). Further\nletVPRmˆ2be any matrix with\n(45) limsup\ntÑ8tn{mψmˆ2\nVp} ¨ }1\n1,} ¨ }2,tq “c\nfor small enough cą0, depending on m,n. LetbvPZm`2be the associated sequence of\nbest approximations, and ˆbv“ pbv,1,bv,2qof norms and approximation qualities\nMv:“ }ˆbv}1\n1, L v:“ }V˚¨bv}2.\nFor a positive measure set YĎBmpn´2qp0,1q ĎRmpn´2qconsisting of matrices BP\nRmˆpn´2q, we have that for infinitely many integers vě1, lettingTv:“Mv`1´1{2, the\ninteger best approximation for Ω“ pV,Bq PRmˆnatTvis given as\nb1,ve1`b2,ve2`b3,ven`1` ¨ ¨ ¨ `bm`2,ven`m“ pb1,v,b2,v,0,...,0,b3,v,...,b m`2,vq PZn`m,\nthus the best approximation function at values Tvequals\nψmˆn\nΩp} ¨ }1,} ¨ }2,Tvq “ψmˆ2\nVp} ¨ }1\n1,} ¨ }2,Tvq “Lv.METRICAL RESULTS ON THE FOLKLORE SET 29\nProof.FixVPRmˆ2. As in the proof of Lemma 6.1 we construct the set Yto be\ncomposed of matrices BPYsuch that the matrices Ω “ pV,Bq PRmˆnsatisfy\nψmˆn\nΩp} ¨ }1,} ¨ }2,Tvq “ψmˆ2\nVp} ¨ }1\n1,} ¨ }2,Tvq “Lv.\nBy the equivalence of norms we have that there exists Cp} ¨ }1q ą0 dependent only on\n} ¨ }1such that\n}ˆx}1ďTv ù ñ }ˆx}8ďCp} ¨ }1qTv.\nThus if again Ω˚“ pV,Bq˚PRmˆpm`nqis obtained by putting the mˆmidentity matrix\nto the right of Ω and we let\nSv:“$\n&\n%BPBmpn´2qp0,1q: min\nxPZn`m\n0ă}ˆx}8ďCp}¨}1qTv}pV,Bq˚¨x}2ěLv,\n.\n-,\nthen\nYĚlimsup\nvÑ8Sv.\nHence if we can show that the set\n(46) λmpn´2q´\nliminf\nvÑ8Sc\nv¯\nďδăλmpn´2q`\nBmpn´2qp0,1q˘\nthen clearly λmpn´2qpYq ą0. Furthermore, by showing that\n(47) λmpn´2qpSc\nvq ďδ v ě1,\nandapplyingProposition2wehave(46),soproving(47)issufficientt overifyλmpn´2qpYq ą\n0.\nLetz“ pz3,...,z nq,y“ pxn`1,...,x n`mqand\nXvpz,x1,x2,yq “$\n&\n%γPBmpn´2qp0,1q:¨\n˝z3γ1,1` ¨ ¨ ¨ `znγ1,n´2......\nz3γm,1` ¨ ¨ ¨ `znγm,n ´2˛\n‚PJvpx1,x2,yq,\n.\n-\nwhere\nJvpx1,x2,yq:“mź\ni“1p´x1ξ1´x2ξ2´xn`i´Cp} ¨ }2qLv,´x1ξ1´x2ξ2´xn`i`Cp} ¨ }2qLvq.\nHere we have again used equivalence of norms, namely that there ex istsCp} ¨ }2q ą0\ndependent only on } ¨ }2so that for any XPRm\nCp} ¨ }2q´1}X}8ď }X}2ďCp} ¨ }2q}X}8.\nConsidering one row of Xvpz,x1,x2,yqat a time (denoted Xpiq\nvpz,x1,x2,yqfor 1 ďiďm)\nwe have the same calculation as in (19), that is\nλn´2`\nXpiq\nvpz,x1,x2,yq˘\nď2Cp} ¨ }2qLvλn´3pBn´3p0,1qq1\n}z}8.30 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nHence\nλmpn´2qpXvpz,x1,x2,yqq ď2mCp} ¨ }2qmLm\nvλn´3pBn´3p0,1qqm1\n}z}m\n8\n“Cpm,n, } ¨ }1,} ¨ }2qLm\nv1\n}z}m8.\nUsing similar calculations as appearing in Lemma 6.1 we have\nλmpn´2qpSc\nvq ďÿ\n0ămaxi“1,2|xi|ďCp}¨}1qTvÿ\n}y}8ďCp}¨}1qTvÿ\n}z}8ďCp}¨}1qTvλmpn´2qpXvpz,x1,x2,yqq\nďCpm,n, } ¨ }1,} ¨ }2qLm\nvTm`2\nvÿ\n}z}8ďCp}¨}1qTv1\n}z}m8\nďC1pm,n, } ¨ }1,} ¨ }2qLm\nvTm`2\nvtCp}¨}1qTvuÿ\nk“1kn´3´m\npn‰m`2q\nďC2pm,n, } ¨ }1,} ¨ }2qLm\nvTm`2\nvT1`n´3´m\nv\nďC2pm,n, } ¨ }1,} ¨ }2qLm\nvTn\nv.\nThus, forany0 ăδăλmpn´2q`\nBmpn´2qp0,1q˘\n,ifLvăcT´n\nmvthenforany cP p0,C3pm,n, }¨\n}1qqwith\nC3pm,n, } ¨ }1,} ¨ }2q ăˆδ\nC2pm,n, } ¨ }1,} ¨ }2q˙1\nm\n,\nwe have that\nλmpn´2qpSc\nvq ăδ\nas required. /square\nThere is nothing essential about Vhaving two columns, it can readily be generalized.\nThe condition n‰m`2 entered in the proof for similar reasons as the requirement ně4\nin Lemma 6.1, see also [24]. All claims of Theorem 3.5 apart from the linear independence\nstatements are proved.\n8.2.Proof of linear independence claims. The first claim (i) on linear independence\nof rows follows fromStep II. It suffices to prove linearly independen t rows canbe arranged\nfor the submatrix Bof Ω “ pV,Bq. Now since n´2ěmby (7), the matrix Bhas not\nless columns than rows, so if Bdoes not have the property it is rank deficient. However,\nthe set of rank deficient matrices has zero Lebesgue measure as a llmˆmminors have\ndeterminant 0, so essentially at least one coordinate is a function of the others, leading to\npositive codimension. On the other hand, we showed that a positive L ebesgue measure\nset of matrices Bgive rise to Ω as in the theorem. Hence removing such rank deficient\nmatrices still leaves us with a set of positive pmˆmq-Lebesgue measure from which we\ncan choose suitable B.\nFor the latter claim (ii) on Qlinearly independent entries, note that in the construction\nof Section 8.1 certain entries of V(thus Ω) are identical, so the condition does not hold.METRICAL RESULTS ON THE FOLKLORE SET 31\nTo fulfill this additional property, let us modifyour construction by varying signs. First\nconstruct the submatrix VPRmˆ2as above by taking for Aj,Bjas before\nΩk,1“Vk,1“8ÿ\nj“1δk,jA´1\nj,Ωk,2“Vk,2“8ÿ\nj“1δ˚\nk,jB´1\nj,p1ďkďmq\nwith sequences δk,jP t´1,1u,δ˚\nk,jP t´1,1uto be chosen later. When δk,j“δ˚\nk,j“1 for\nallk,jwe are in the standard case of the previous sections.\nWhenm“1, all arising matrices/vectors pV1,1,V1,2q “ pξ1,ξ2qshare the properties of\nthe standard case above. For mą1, we get more general error vectors than in (44), now\nof the form\n(48) ˜cj:“V¨ˆ\nAj\n0˙\n´¨\n˚˚˝Fj,1\nFj,2...\nFj,m˛\n‹‹‚,˜dj:“V¨ˆ\n0\nBj˙\n´¨\n˚˚˝Gj,1\nGj,2...\nGj,m˛\n‹‹‚, j ě1,\nwhere\nFj,k“Ajjÿ\nu“1δk,uA´1\nu, G j,k“Bjjÿ\nu“1δ˚\nk,uB´1\nu, p1ďkďmq,\ndepending on the choice of δk,j,δ˚\nk,j. They again satisfy congruence properties\nFj” ˘1 mod 6 and Gj” ˘1 mod 35\nfor the same reasons, sufficient for coprimality. Thus the entries o f the remainders ˜cj,˜dj\nare given as\ncj,k“Aj8ÿ\nu“j`1δk,uA´1\nu, d j,k“Bj8ÿ\nu“j`1δ˚\nk,uB´1\nu, p1ďkďmq.\nHowever, since Aj{Aj`1Ñ0 andBj{Bj`1Ñ0 we again see\ncj,k“δk,j`1¨AjA´1\nj`1p1`op1qq, d j,k“δ˚\nk,j`1¨BjB´1\nj`1p1`op1qq,p1ďkďmq.\nHence, by equivalence of norms, it is clear that the error vectors in (48) all have } ¨ }2\nnorm of order respectively\n(49) }˜cj}2“AjA´1\nj`1Γjp1`op1qq, }˜dj}2“BjB´1\nj`1˜Γjp1`op1qq, j Ñ 8,\nwith Γ j,˜Γjof one of the numbers\nΓj,˜ΓjP t}p˘1,˘1,..., ˘1q}2u, j ě1,\nfor any possible sign choice. The values ˜Γjdo not matter much as the according evalua-\ntions will be of negligible order by the analogue of (36). Now if we let\nΓ :“maxt}p˘1,˘1,..., ˘1q}2u\nwith maximum over all 2msign choices, we can use the argument of Section 8.1. Hereby\nwe may assume that the maximizing sign choice occurs for infinitely man ypδ1,j,...,δ m,jq,\nasotherwiseweredefineΓviathemaximumamongthose mtermsignsequences occurring\ninfinitely often. Moreover, due to (49), the arguments of Section 7.1 applied to each of the32 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nmcoordinatesseparatelystillapplyverysimilarlythatthevectors pAj,0,´Fj,1,..., ´Fj,mq\nandp0,Bj,´Gj,1,..., ´Gj,mqessentially formthe sequence of best approximations for the\ninducedV. We leave the details to the reader. This argument shows that the D irichlet\nconstant ΘmˆnpΩqof each such Ω “Ωpδ,δ˚qequalscas before.\nFinally we show that we can choose δk,j,δ˚\nk,jso that all entries of Ω PRmˆnare linearly\nindependent over Qtogether with t1u. Let us first show this for the submatrix VPRmˆ2.\nLabel the entries Vi,jofVbyv1,...,v 2mwithv2h“Vh,2andv2h´1“Vh,1, 1ďhďm.\nNow the first line of VisQ-linearly independent for any choice of δ1,j,δ˚\n1,jsequences as\notherwise the sequence of best approximations for pV1,1,V1,2qwould terminate, but the\nproof above shows that this is not the case. Hence t1,v1,v2uisQ-linearly independent.\nThen we proceed inductively. Having constructed the first ℓě2 elements v1,...,v ℓ, we\nchoose theδorδ˚sequence for vℓ`1so that we avoid Q-linear dependence of vℓ`1with the\nalready chosen 1 ,v1,...,v ℓ. This is possible since the Q-span of t1,v1,...,v ℓuis count-\nable, but we have uncountably many choices of δorδ˚sequences. These clearly induce\nuncountably many pairwise distinct numbers vℓ`1, the injectivity of the corresponding\nmapsϕ:t´1,1uNÑRresp.ϕ˚:t´1,1uNÑRfollows from the rapid increase of the\nsequencesAjresp.Bj. Repeating this for ℓ“2,3,...,2m´1, finally all entries of Vwill\nbe independent over Qwith t1u, as desired.\nNow Step II of the proof above combined with the following easy argu ment show that\nwe can transition from Vto Ω.\nProposition 5. Letg,hbe positive integers. Assume given real numbers x1,...,x gare\nQ-linearly independent together with 1. Then only for a set of Hausdorff dimension h´1\nof vectors y“ py1,...,y hqinRh, the joint vector z“ px1,...,x g,y1,...,y hqisQ-linearly\ndependent with 1.\nProof.Since thexjare independent, if t1,zuis dependent over Qthen there are integers\nMi,Ni,Nwith\nN`gÿ\ni“1Mixi`hÿ\ni“1Niyi“0\nand not all Niare 0. But this means that ylies in a countable union of affine hyperplanes\nofRh. Hence we see that the set of yinducing dependency over Qindeed has Hausdorff\ndimension at most h´1. /square\nIt suffices to take g“2m,h “ pn´2qmand to choose xithe entries of V(first two\ncolumns of Ω) and yithe remaining entries of Ω (entries of B) in the proposition to\nconclude that a full dimensional set of possible matrices BPRmˆpn´2q–Rmpn´2qremains\n(in fact of positive mpn´2qdimensional Hausdorff measure).\n9.Proofs of Theorems 3.2 and 3.3\nTheorems 3.2 and 3.3 are rather easy. The case m“nis a consequence of the following\nsimple lemma. We consider maximum norms on both RnandRmand simply write\nψmˆn\nΩptqforψmˆn\nΩp} ¨ } 8,} ¨ } 8,tq.METRICAL RESULTS ON THE FOLKLORE SET 33\nLemma 9.1. LetΩPRmˆnbe a block matrix consisting of rectangular blocks Ω1,...,Ωk\nof sizes pmj,njq,1ďjďkalong the diagonal, m“řmj,n“řnj. Then\nψmˆn\nΩptq “min\n1ďjďkψmjˆnj\nΩjptq, t ą0.\nProof.First observe that by the form of Ω, the problem decomposes into kdecoupled\nlinear form approximation problems. Fix tand letb1,...,bkinZmj`njbe the best\napproximation integer vectors inducing ψmjˆnj\nΩjptqfor 1 ďjďk, i.e.b“bjminimizes\n}Ω˚\njb}8“ }Ωjˆb´˜b}8among integer vectors bwith 0 ă }b}8ďt. The inequality\nψmˆn\nΩptq ďmin\n1ďjďkψmjˆnj\nΩjptq\nfollows by taking the integer vector bPZm`nwith entries of bj0at the according places,\nwithj0inducing the minimum of }Ω˚\njbj}8overj, and 0 entries at all other places. By\nchoice of the maximum norm, this will have the same norm\n}Ω˚¨b}8“ }Ω˚\nj0¨bj0}8“ψmj0ˆnj0\nΩj0ptq “min\n1ďjďkψmjˆnj\nΩjptq.\nConversely, since some coordinate of the best approximation vect or for given tis not 0\nand the system is decoupled and we use maximum norm, we must have\nψmˆn\nΩptq ěmin\n1ďjďkψmjˆnj\nΩjptq.\n/square\nThe deduction of the case m“nworks as follows.\nProof of Theorem 3.2. As in Lemma 9.1, take a matrix Ω of the form Ω “diagpΩ1,Ω2q\nwith diagonal blocks Ω 1PBadn´1,n´1and Ω 2“ tηu RBad1,1YQare arbitrary. Keep\nin mind that as n{n“ pn´1q{pn´1q “1{1“1, the Dirichlet exponent (remark: not\nconstant) is 1 for all Ω 1,ηand Ω, for all arguments below.\nBy Lemma 9.1 we have\n(50) ΘnˆnpΩq “limsup\ntÑ8t¨ψnˆn\nΩptq “limsup\ntÑ8t¨mintψpn´1qˆpn´1q\nΩ1ptq,ψ1ˆ1\nηptqu.\nNow since Ω 1PBadn´1,n´1ĎDin´1,n´1we have\nΘpn´1qˆpn´1qpΩ1q “limsup\ntÑ8tn´1\nn´1¨ψpn´1qˆpn´1q\nΩ1ptq “limsup\ntÑ8t¨ψpn´1qˆpn´1q\nΩ1ptq ă1\nso Ω PDin,nas well by (50). Moreover since\nlimsup\ntÑ8tψ1ˆ1\nηptq ě1{2ą0\nfor any irrational ηby Khintchine [17] and\nliminf\ntÑ8tψpn´1qˆpn´1q\nΩ1ptq ą034 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nsince Ω 1PBadn´1,n´1, again by (50) we have ΘnˆnpΩq ą0 and thus Ω RSingn,n. Finally\nagain by Lemma 9.1\n˜ΘnˆnpΩq “liminf\ntÑ8tn{nψΩptq ďliminf\ntÑ8t1{1ψηptq “0\nsinceηRBad1,1, so Ω RBadn,n. Hence Ω PFSn,n.\nIt is well-known that Badn´1,n´1has full Hausdorff dimension pn´1q2and that the\ncomplement of Bad1,1has full 1-dimensional Lebesgue measure, see Section 1. Removing\nthe countable set Qclearly has no effect on the Lebesgue measure. In particular, we\ncan choose regular (full R-rank) matrices Bas the complementary set has strictly lower\nHausdorff dimension, and by the diagonal form and η‰0 the same will be true for any\narising Ω. This argument together with (12) also shows that the dime nsion of the set of\nmatrices Ω constructed above is at least pn´1q2`1. /square\nFor the proof of Theorem 3.2 we combine Lemma 9.1 with the main result from [26]\non simultaneous approximation n“1.\nProof of Theorem 3.3. First note that the assumptions on m,nimply that\n(51)1\n2ďn\nmă1.\nLet us first consider } ¨ }1“ } ¨ } 2“ } ¨ } 8. Then by [26, Theorem 2.2] applied to\nΦptq “ct´n{m(note that conditions (d1)-(d3) of [26] are satisfied by our choice of Φ and\n(51)) we can find ξPR2ˆ1so that\nlimsup\ntÑ8tn{mψ2ˆ1\nξptq “c,liminf\ntÑ8tn{mψ2ˆ1\nξptq “0.\nNow let ζPBadm´2,1arbitrary unless if m“4 thenζ“ξ. Sincem´2ě2 by\nassumption, it is easily seen that in either case\nψ2ˆ1\nξptq ďψpm´2qˆ1\nζ ptq, t ět0.\nHence by Lemma 9.1 the matrix U“diagpξ,ζq PRmˆ2satisfies\n(52) limsup\ntÑ8tn{mψmˆ2\nUptq “c,liminf\ntÑ8tn{mψmˆ2\nUptq “0,\nas well.\nIfn“2 then we must have m“4 due to our assumptionsn\nmě1\n2andmě4, and then\nwe can just take Ω “U. Note also that in this case our claimed bound for Hausdorff\ndimension becomes zero, so there is nothing more to prove.\nIfně3, then in Lemma 8.1 take V“UPRmˆ2. NoticeVsatisfies (45) by (52).\nHence, assuming cą0 is small enough, there exists a set of positive mpn´2q-dimensional\nLebesgue measure YĎRmˆpn´2qof matrices for which any BPY, theRmˆnmatrix\nΩ“ pB,U qsatisfies\nlimsup\ntÑ8tn{mψmˆn\nΩptq “c.METRICAL RESULTS ON THE FOLKLORE SET 35\nFinally, we show Ω is not badly approximable. By (52) there exists a seq uence of best\napproximations such that on the sequence ptjqjě1we have\nlim\njÑ8tn{m\njψmˆ2\nUptjq “0.\nConsidering Ω, take the same sequence of the best approximations and place a zero in all\nentries not related to U(that is starting from the third) to obtain the same statement,\ni.e.\nlim\njÑ8tn{m\njψmˆn\nΩptjq “0.\nCombining these two claims we infer Ω PFSm,npcq. Using (12) the dimension statement\nfollows since we have shown that (when mě5)\ndimFSm,npcq ědimY`dimBadm´2,1“mpn´2q `m´2,\nwhere we used that Badm´1,1has full Hausdorff dimension, and\ndimFSm,npcq ědimY“mpn´2q,\nin the case m“4. The case of maximum norms is done.\nNow let } ¨ }1be arbitrary and still } ¨ }2“ } ¨ } 8. Let } ¨ }1\n1be the projected norm\n}px1,x2q}1\n1:“ }px1,x2,0,...,0q}1.\nLet\nγ“ }e1}1\n1\nthe norm of the first base vector of R2. By [26, Theorem 2.2] pick ξPR2ˆ1such that\n(53) limsup\ntÑ8tn{mψ2ˆ1\nξptq “c1:“cγn{m,liminf\ntÑ8tn{mψ2ˆ1\nξptq “0.\nSuppose we can show that, for any U“diagpξ,ζq PRmˆ2as above, which we can identify\nwith elements in tξu ˆBadm´2,1ĂRm, we have\n(54) ψmˆ2\nUp} ¨ }1\n1,} ¨ }2,tq “ψ2ˆ1\nξp| ¨ |,} ¨ } 8,γ´1tq, t ět0.\nIf this holds then indeed\nlimsup\ntÑ8tn{mψmˆ2\nUp} ¨ }1\n1,} ¨ }2,tq “γ´n{mlimsup\ntÑ8tn{mψ2ˆ1\nξp| ¨ |,} ¨ } 8,γ´1tq “γ´n{mc1“c,\nand\nliminf\ntÑ8tn{mψmˆ2\nUp} ¨ }1\n1,} ¨ }2,tq “γ´n{mliminf\ntÑ8tn{mψ2ˆ1\nξp| ¨ |,} ¨ } 8,γ´1tq “0.\nHence we may apply Lemma 8.1, so for csmall enough we again find some set YĂ\nRmˆpn´2qof positive Lebesgue measure such that for any BPYwe have\nlimsup\ntÑ8tn{mψmˆn\npB,U qp} ¨ }1,} ¨ }2,tq “c,liminf\ntÑ8tn{mψmˆn\npB,U qp} ¨ }1,} ¨ }2,tq “0.\nSopB,U q PFSm,np} ¨ }1,} ¨ }2,cq, and by (12) we have\ndimFSm,np} ¨ }1,} ¨ }2,cq ědimY`dimBadm´2,1“mpn´2q `m´2,\nformě5 and accordingly for m“4.\nThus to complete the proof we need to show (54) holds: For the liftin g from the prime\nnorm to the given norm, it suffices to start with csmall enough to guarantee again via36 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nLemma8.1thatmatrixΩafteradding n´2columnsisDirichletimprovablewithDirichlet\nconstantc.\nFor the inequality\n(55) ψmˆ2\nUp} ¨ }1\n1,} ¨ }2,tq ďψ2ˆ1\nξp| ¨ |,} ¨ } 8,γ´1tq\nobserve that any px,wq PZˆZ2with |x| ďγ´1tgives rise to z“ px,0q “xe1PZ2of\nnorm }z}1\n1ď |x|γďt. Moreover the identity\n(56) }U¨z´ pw,0,...,0q}8“ }ξ¨x´w}8\nholds as\n(57) U¨z´ pw,0,...,0q “ pξ¨x´w,0,...,0q PRm,\nwith the same first two coordinates followed by zeros, where we use d} ¨ }2being the\nmaximum norm. This obviously implies (57).\nFor the reverse inequality\n(58) ψmˆ2\nUp} ¨ }1\n1,} ¨ }2,tq ěψ2ˆ1\nξp| ¨ |,} ¨ } 8,γ´1tq\nlet first pm,n q ‰ p4,2q. Note that if z“ pz1,z2q PZ2with }z}1\n1ďtdoes not lie in the\nspacez2“0, then by equivalence of norms for any yPZm\n(59) }U¨z´y}8\" }z}´1{pm´2q\n8 \" }z}1´1{pm´2q\n1 ět´1{pm´2q\nwhere the most left estimate holds since the system is decoupled and ζPBadm´2,1. By\nour assumptions mě4 and 1 {2ďn{m, in our case pm,n q ‰ p4,2qwe have the strict\ninequality 1 {pm´2q ăn{m. Then, in view of (53), approximation quality (59) is beaten\nbyψ2ˆ1\nξp|¨|,}¨} 8,γ´1tq, so extending accordingly the involved integer best approximation\napproximations to Znby adding 0 entries leads to the same smaller value. So we can\nexclude such vectors. But for vectors zin the linear space z2“0, the integer vector y\nminimizing }U¨z´y}8for given zis easily seen to be of the form y“ pw,0,...,0qwith\nwPZ2, as otherwise some entry of U¨z´yis of modulus at least 1. Then the output\nU¨z´yPRmagain satisfies (56) by (57), settling (58). For m“4,n“2 a very similar\nargument applies using ζ“ξ, however we need } ¨ }1to be expanding to exclude better\napproximations within the space z2“0. However, if the expanding property is assumed,\nthen it is again easy to see that we may exclude this space.\nFinally, let }¨}2beanyexpanding normon Rmandconsider againtheinduced projected\nnorm } ¨ }1\n2onR2by\n}px1,x2q}1\n2:“ }px1,x2,0,...,0q}2.\nIt is easy to see that } ¨ }1\n2is an expanding norm on R2. Thus it follows from [26,\nCorollary2], againapplicableinviewof (51), thatforanysmallenough c, wecansimilarly\narrange some ξPR2ˆ1to satisfy\nlimsup\ntÑ8tn{mψ2ˆ1\nξp| ¨ |,} ¨ }1\n2,tq “c1:“cγn{m,\nand\nliminf\ntÑ8tn{mψ2ˆ1\nξp| ¨ |,} ¨ }1\n2,tq “0.METRICAL RESULTS ON THE FOLKLORE SET 37\nWe can then copy the proof above replacing } ¨ } 8by} ¨ }1\n2whenever considering ψ2ˆ1\nξ.\nAgain by the same argument, it suffices to show the accordingly alter ed variant of (54).\nFor the estimate (55), the only place where we needed } ¨ }2to be the maximum norm\nis to deduce that the projected vectors satisfy (56). However, in view of (57), now the\ndefining property of norm } ¨ }1\n2just means that this equality still holds. The proof of the\nreverse inequality (58) is analogous to the above, only with different implied constants in\n(59). /square\n10.Proof of Lemma 1.1\nFor matrix Ω PRmˆnand real numbers A,B PR`define the convex bodies\nMA,BpΩq:“#\nzPRn`m: max\n1ďjďmˇˇˇˇˇnÿ\ni“1Ωj,izi`zn`jˇˇˇˇˇďA,max\n1ďiďn|zi| ďB+\n,\nxMA,BpΩq:“#\nzPRn`m: max\n1ďiďnˇˇˇˇˇzi´mÿ\nj“1Ωi,jzn`jˇˇˇˇˇďB,max\n1ďjďm|zn`j| ďA+\n.\nThe following theorem proven by German is crucial.\nTheorem 10.1. [15, Theorem 7] LetΩPRmˆnandA,B PR`. If\nMA,BpΩq XZn`mzt0u ‰ H,\nthen\nxMA˚,B˚pΩq XZn`mzt0u ‰ H\nfor\nA˚“CpAnB1´nq1\nn`m´1andB˚“CpA1´mBmq1\nn`m´1,\nwithCą0constant dependent only on nandm.\nEquipped with Theorem 10.1 we can prove Lemma 1.1.\nProof.Since Ω PFSm,np} ¨ }1,} ¨ }2,εq ĎRmˆnthen we have that\nΘmˆnp} ¨ }1,} ¨ }2,Ωq “ε.\nBy the equivalence of norms we have that the same is true up to some constant for the\nsup norm, so\nΘmˆnp} ¨ } 8,} ¨ } 8,Ωq —ε.\nHence, for all sufficiently large tPRand someρą0 there is an integer\nbPMρt´n{m,tpΩq XZn`mzt0u.\nHence, by Theorem 10.1, for all sufficiently large uPRthere exists integer\nbPxMu,δu´m{npΩq XZn`mzt0u,\nforδ“Cm`n\nnρm\nnpn`m´1q. Hence ΩTPDin,mpδq, and by equivalence of norms there exists\nsomeδ˚ą0 such that ΩTPDin,mp}¨}2,}¨}1,δ˚q. IfD˚“Dnˆmp}¨}1,}¨}2qis the Dirichlet\nconstant from (4) for the transpose problem, for small enough δand thusδ˚ăD˚this38 MUMTAZ HUSSAIN, JOHANNES SCHLEISCHITZ, AND BENJAMIN WAR D\nwill induce a Dirichlet improvable matrix ΩT. Conversely, for any 0 ăγăεthere exists\ninfinitely many tPRfor which\nxMt,γt´n{mpΩTq XZn`mzt0u “ H,\notherwise Ω PDim,npγq, which is clearly false since Ω PDIm,npεq. Applying the contra-\npositive of Theorem 10.1 we obtain that there exists infinitely many uPRand constant\nC˚ą0 for which\nMC˚u´n{m,upΩTq XZn`mzt0u “ H.\nHence ΩTRSingn,m. Furthermore, one can similarly show that ΩTRBadn,m, and so\nΩTPDin,mp} ¨ }1,} ¨ }2,δ˚qzpSingn,mYBadn,mq.\nAs previously mentioned Singn,mandBadn,mare independent of norms, and so ΩTP\nFSn,mp} ¨ }2,} ¨ }1qas required. /square\nAcknowledgements. TheresearchofMumtazHussainandBenjaminWardissupported\nby the Australian Research Council Discovery Project (20010099 4). Some of this work\nwas carried out when Johannes Schleischitz visited La Trobe Univers ity supported by an\nInternational Visitor Program of the Sydney Mathematical Resea rch Institute (SMRI)\nand La Trobe University, we are thankful to them for their genero us support.\nReferences\n[1] A. 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Metrical results on the geometry of best appr oximations for a linear form. arXiv\npreprint arXiv:2302.08403 , 2023.\n[28] W. Schmidt. A metrical theorem in diophantine approximation. Canadian Journal of Mathematics ,\n12:619–631.\n[29] F. S¨ uess. Simultaneous diophantine approximation on affine sub spaces and dirichlet improvability.\nPhD Thesis, University of York , 2017.\n[30] C. Tricot, Jr. Two definitions offractional dimension. Math. Proc. Cambridge Philos. Soc. , 91(1):57–\n74, 1982.\nMumtaz Hussain, Department of Mathematical and Physical Sc iences, La Trobe Uni-\nversity, Bendigo 3552, Australia.\nEmail address :m.hussain@latrobe.edu.au\nJohannes Schleischitz, Middle East Technical University, Northern Cyprus Campus,\nKalkanli, G ¨uzelyurt\nEmail address :johannes@metu.edu.tr; jschleischitz@outlook.com\nBenjamin Ward, Department of Mathematical and Physical Sci ences, La Trobe Uni-\nversity, Bendigo 3552, Australia.\nEmail address :Ben.Ward@latrobe.edu.au"    },    {        "title": "2402.13476v1.Exploring_the_dependence_of_the_Hubble_constant_from_the_cluster_lensed_supernova_SN_Refsdal_on_mass_model_assumptions.pdf",        "content": "Exploring the dependence of the Hubble constant from the cluster-lensed supernova\nSN Refsdal on mass model assumptions\nYuting Liu∗1, 2and Masamune Oguri†3, 4, 5\n1Department of Physics, University of Tokyo, Tokyo 113-0033, Japan\n2Department of Astronomy, Beijing Normal University, Beijing 100875, China\n3Center for Frontier Science, Chiba University, Chiba 263-8522, Japan\n4Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan\n5Kavli Institute for the Physics and Mathematics of the Universe,\nUniversity of Tokyo, Kashiwa, Chiba 277-8583, Japan\n(Dated: February 22, 2024)\nThe Hubble constant, H0, which is a crucial parameter in astrophysics and cosmology, is under\nsignificant tension. We explore an independent technique to measure H0based on the time-delay\ncosmography with strong gravitational lensing of a supernova lensed by a galaxy cluster, focusing\non SN Refsdal in MACS J1149.5+2223, the first gravitationally lensed supernova with resolved\nmultiple images. We carefully examine the dependence of constraints on the Hubble constant on\nthe choice of lens mass models, employing 23 lens mass models with different assumptions on dark\nmatter halos and external perturbations. Remarkably, we observe that the dependence on the\nchoice of lens mass models is not significantly large, suggesting the robustness of the constraint on\nthe Hubble constant from SN Refsdal. We combine measurements for the 23 lens mass models to\nobtain H0= 70.0+4 .7\n−4.9km s−1Mpc−1assuming equal weighting. We find that best-fitting Hubble\nconstant values correlate with radial density profiles of the lensing cluster, implying a room for\nimproving the constraint on the Hubble constant with future observations of more multiple images.\nWe also find a clear correlation between best-fitting Hubble constant values and magnification factors\nof supernova multiple images. This correlation highlights the importance of gravitationally lensed\nType Ia supernovae for accurate and robust Hubble constant measurements.\nKeywords: cosmological parameters: individual (Hubble constant) - gravitational lensing: strong - super-\nnovae: individual (SN Refsdal) - galaxies: clusters: individual (MACS J1149.5+2223)\nI. INTRODUCTION\nThe Hubble constant, denoted as H0, quantifies the\ncurrent expansion rate of the Universe and stands as\na key parameter in modern cosmology. The Hubble\nconstant is important, particularly because it is a cor-\nnerstone in determining fundamental properties such as\nthe age, size, and fate of the University [1–5]. Vari-\nous methodologies have been employed to measure H0.\nEarly estimates relied on distance measurements to rela-\ntively nearby galaxies, while contemporary approaches\nincorporate diverse cosmological probes. These ap-\nproaches include Type Ia supernova (SNe Ia), cosmic\nmicrowave background radiation (CMBR), and baryon\nacoustic oscillations (BAO), among others. However, de-\nspite significant observational progress, reconciling dif-\nferent measurements and reducing systematic uncertain-\nties remain ongoing challenges. The quest to precisely\ndetermine H0has led to intriguing controversies [6–11],\nwhich emerge when comparing local H0measurements\nwith those inferred from observations of the early Uni-\nverse. For instance, the distance ladder analysis from\nSH0ES (SNe, H0, for the Equation of State of dark\nenergy) collaboration measures the Hubble constant of\n∗yutingl@mail.bnu.edu.cn\n†masamune.oguri@chiba-u.jpH0= 73 .04±1.04 km s−1Mpc−1in the late Universe\n[12], while CMBR from the Planck satellite releases a\nmuch smaller and tighter measurement with the value of\nH0= 67.4±0.5 km s−1Mpc−1in the early Universe [13].\nThe discrepancies of measured H0values between dif-\nferent measurements are becoming more pronounced.\nGravitational waves (GW) from the coalescence of binary\nsystems are used as a new distance indicator that is in-\ndependent of the local distance ladder, offering a unique\ntechnique to determine H0. For instance, the neutron\nstar merger event GW170817 coupled with its electro-\nmagnetic counterparts AT2017gfo and GRB170817A led\nto the Hubble constant of H0= 66.2+4.4\n−4.2km s−1Mpc−1\n[14]. More recently, Ref. [15] derived a Hubble constant\nvalue of H0= 68.84+15.51\n−7.74km s−1Mpc−1from the first\nthree LIGO/Virgo observing runs and DELVE, specifi-\ncally events GW190924 021846 and GW200202 154313.\nAnalyzing the dispersion measures and redshifts of fast\nradio bursts (FRBs) is another approach to constrain the\nHubble constant [16–18]. Ref. [19] employ such approach\nto derive the value of H0= 73+12\n−8km s−1Mpc−1, us-\ning a sample of 16 Australian Square Kilometre Array\nPathfinder (ASKAP) FRBs detected by the Commensal\nReal-time ASKAP Fast Transients (CRAFT) Survey and\nlocalised to their host galaxies, and 60 unlocalised FRBs\nfrom Parkes and ASKAP. The tension observed among\nvarious measurements raises profound questions regard-\ning the robustness of the standard cosmological model,\nor the potential existence of unknown physics influencingarXiv:2402.13476v1  [astro-ph.CO]  21 Feb 20242\ncosmic expansion. Therefore, exploring alternative pos-\nsibilities is urgent to elucidate the origin of this tension\nand attain a concordant value for the Hubble constant.\nGravitational lensing time delay cosmography, initially\nproposed by Refsdal [20], has evolved into a crucial in-\ndependent method for constraining the Hubble constant,\nthanks to advancements in observations and theoretical\nunderstanding of the dependence on lens models. Tradi-\ntionally, constraints on H0from gravitational lens time\ndelays make use of mainly galaxy-lensed quasar systems\n[21–26]. Recently, Ref. [27] provide a new H0mea-\nsurement ( H0= 73 .3±5.8 km s−1Mpc−1) from seven\ngalaxy-lensed quasar systems using single-aperture kine-\nmatics and maximally flexible models.\nIn addition to galaxy-lensed quasar systems, cluster-\nlensed quasar systems have also been used to con-\nstrain the Hubble constant [28–30]. Even though clus-\nter strong lens mass modeling exhibits relatively large\npositional offsets between observed and model-predicted\nmultiple images, it is still possible to recover mass\nmodels accurately with help of the large number of\nmultiple images [31]. Ref. [32] used the first cluster-\nscale lensed quasar system SDSS J1004+4112 to obtain\nH0= 67.5+14.5\n−8.9km s−1Mpc−1combining sixteen differ-\nent mass models, from which it is concluded that the lens\nmodel dependence is large and needs to be overcome.\nVery recently, the Hubble constant has been derived\nfrom the cluster-lensed supernova system SN Refsdal in\nthe field of MACS J1149+2223 which is the first reported\ncase of strongly lensed supernovae with resolved multiple\nimages [33, 34]. Focusing on this unique and valuable\nsystem, Ref. [35] has carried out Refsdal’s original pro-\nposal to use a multiply imaged supernova to measure H0,\nwith the value of H0= 64+9\n−11km s−1Mpc−1, in which\npreliminary values of time delay measurements from the\nearly data in light curve observations have been used.\nRef. [36] measure H0= 66.6+4.1\n−3.3km s−1Mpc−1based on\naccurate and precise measurements of time delays from\nthe full light curve data. Since their analysis is blinded in\nthe sense that mass models used for the analysis are those\nconstructed before the reappearance of the fifth image of\nSN Refsdal, they do not thoroughly analyze degenera-\ncies between the lens models and the Hubble constant.\nFurthermore, independent measurements of the Hubble\nconstant and the geometry of the Universe from SN Refs-\ndal is presented by Ref. [37], which argues that the mass\nmodel uncertainty is subdominant [38].\nIn this paper, we explore to establish an independent\nmeasurement of H0while meticulously examining the\ninfluence of the assumed lens mass model on H0from\nthis cluster-lensed supernova system SN Refsdal within\nMACS J1149.5+2223. Specifically, we incorporate more\nthan 100 multiple images and the most recent time-delay\nmeasurements between distinct lensed-supernova images\n[39, 40]. Our work analyzes 23 different lens mass models\nwith different complexity to thoroughly explore the de-\npendence of assumptions on mass models on the derived\nH0values. Our approach is complementary to Refs. [36]and [38], and offers a new test of the robustness of the\nHubble constant measurements in previous analysis. Ad-\nditionally, we discuss shapes of critical curves, radial con-\nvergence, tangential shear profiles, and magnifications of\nlensed supernova images, in order to explore how the\nHubble constant measurement can be improved in future\nobservations.\nThis paper is structured as follows. In Sec. II, we pro-\nvide a brief description of the cluster-lensed supernova\nsystem SN Refsdal. Section III introduces the method-\nology employed for mass modeling. Results of our anal-\nysis are presented in Sec. IV. Discussion and conclusions\nare presented in Sec. V and Sec. VI, respectively. In this\npaper, we assume a flat Universe with Ω m= 0.3 and\nΩΛ= 0.7.\nII. OBSERVATIONS OF SN REFSDAL IN MACS\nJ1149.5+2223\nSN Refsdal as the first example of a strongly lensed\nsupernova with resolved multiple images was initially de-\ntected in November 2014 through the Hubble Space Tele-\nscope (HST) observations [33]. This system is composed\nof six multiple images (named S1-S4, SX, and SY in [41])\nof a single supernova at redshift 1.488, where four images\nS1–S4 forming an Einstein-cross configuration around an\nearly-type galaxy were detected by the Grism Lens Am-\nplified Survey from Space (GLASS; GO-13459; PI Treu)\n[42, 43], SX detected in HST imaging on 2015 Decem-\nber 11 (GO-14199; PI Kelly) [34] was the first supernova\nwhose appearance on the sky was predicted in advance\n[41, 44, 45], and a leading image SY appeared in the\nlate 1990s from the lens models predication without ob-\nservation evidence. SN Refsdal is lensed by the cluster\nof galaxies MACS J1149.5+2223 at redshift 0.541 [46].\nMultiple images and their spectroscopic redshifts for this\ncluster have been confirmed and presented in the litera-\nture [47–55].\nIn this study, we utilize the positions of the five su-\npernova images in Ref. [40], which have been refined\nbased on the Hubble Space Telescope (HST) Wide Field\nCamera 3 (WFC3) infrared (IR) detector F125W images\nacquired in 2015 and 2016. Additionally, we incorpo-\nrate positional constraints derived from multiple images\nof seven knots within a lensed face-on spiral galaxy at\nredshift 1.488. For multiple images of other background\ngalaxies, we include data from 28 multiple image sys-\ntems compiled in Ref. [39]. The total number of multiple\nimages in our study is 109, spanning across 36 systems.\nOur analysis includes newly measured time delays be-\ntween lensed supernova images S2–SX and S1, as well\nas recently determined magnification ratios from the full\nlight curve analysis [40]. The magnification ratio S4/S1\nis excluded due to compelling evidence ( >5σ) indicat-\ning that image S4 has undergone chromatic microlensing\n[36].\nTable I provides a summary of observational con-3\nTABLE I: Observational constraints from multiple images\nof SN Refsdal. Source J2000 positions with respect to the\n(RA, DEC)=(177 .3987491, 22 .3985308), the magnification ra-\ntio with respect to the supernova image S1, and the relative\ntime delay are listed in columns 2–5, respectively.\nName ∆ x[′′] ∆ y[′′] ∆ µ ∆t[days]\nS1 1.754 −10.459 ≡1 ≡0\nS2 3.458 −9.908 1 .06±0.25 9 .7±4.4\nS3 4.596 −10.808 0 .91±0.39 7 .9±5.6\nS4 3.168 −12.068 ≡0*19.4±8.0\nSX −4.432 −6.620 0 .30±0.05 376 .0±5.6\n*The magnification ratio S4/S1 is not included because image\nS4 has experienced chromatic microlensing.\nstraints for multiple images of SN Refsdal. While\nRef. [40] offers a detailed depiction of the likelihood\ndistribution for both the time delay and magnification,\nalong with the covariance matrix encompassing these\nmeasurements, in our analysis we adopt a simplified ap-\nproach by assuming symmetric Gaussian distributions for\nboth time delay and magnification measurements with-\nout any covariance between measurements, mainly be-\ncause of the limitation of the software glafic [56, 57]\nthat we use for our analysis. For instance, Ref. [40] report\na relative time delay of 376 .0+5.6\n−5.5days between images SX\nand S1, alongside a magnification ratio SX/S1=0 .30+0.05\n−0.03\n(see Fig.14 and Fig.15 in [40]). We will discuss a possible\nimpact of this simplified assumption in Sec. IV.\nAdditionally, we incorporate information about the po-\nsitions, ellipticities, position angles, and luminosity ratios\n(relative to the brightest cluster galaxy in the MACS\nJ1149.5+2223 field) of 170 cluster galaxy members, as\ndetailed in Ref. [39]. The large number of observational\nconstraints suggests the potential for tightly constraining\nthe Hubble constant through this lens system.\nIII. MASS MODELING\nWe construct lens mass models utilizing the widely em-\nployed software glafic [56, 57], which adopts a paramet-\nric modeling approach.\nA. Dark matter halo\nWe employ the Navarro-Frenk-White (NFW) profile to\ncharacterize the mass distribution of a dark matter halo\n[58]. The projected density profile is defined by six pa-\nrameters, including the mass M, the centroid position x\nandy, the ellipticity e, the position angle θe, and the con-\ncentration parameter c. In our mass model reconstruc-\ntion, we include up to four NFW components to model\nthe complex cluster mass distribution. For NFW compo-\nnents located at the edge or outside the strong lensing re-\ngions, positions are fixed to observed positions of brightcluster member galaxies. In cases where NFW compo-\nnents lack strong observational constraints, the concen-\ntration parameter is fixed to 10. Across all halo compo-\nnents, the halo mass and ellipticity are treated as free\nparameters. We optimize these model parameters, with\nthe ellipticity restricted to be smaller than 0.8 and the\nconcentration parameter restricted to the range between\n1 and 40. We utilize the fast approximation of the lens-\ning calculation of the NFW profile, dubbed as anfw in\nglafic , as proposed in Ref. [57].\nWe also utilize the Pseudo-Jaffe Ellipsoid (PJE,\ndubbed as jaffe inglafic ) profile [59, 60] to charac-\nterize the dark matter halo. In this model, the radial\ndensity profile is defined by seven parameters, including\nthe velocity dispersion σ, the centroid position, the el-\nlipicity, the position angle, the core radius rcore, and the\ntruncation radius rtrun. As in the case of the NFW pro-\nfile, their centroids positions are fixed when located at\nthe edge or outside the strong lensing regions, while al-\nlowing all the other model parameters to be free during\noptimization. The optimization is conducted within the\nspecified range of rtrunfrom 20′′to 500′′andesmaller\nthan 0 .8.\nMoreover, in some cases we simultaneously model the\ndark matter halo components with both the NFW and\nPJE profiles, employing similar construction and opti-\nmization strategies as outlined above.\nB. Member galaxies\nThe member galaxies are modeled by scaled Pseudo-\nJaffe ellipsoids (dubbed as gals inglafic ). This pro-\nfile is parameterized by the scaled velocity dispersion σ∗,\nscaled truncation radius rtrun,∗, and the scaling slope η.\nThroughout our optimization process, all these parame-\nters are allowed to vary, with the range for ηfrom 0 .2 to\n1.5. In addition, we separately model a cluster member\ngalaxy that is responsible for producing the four Einstein\ncross supernova images S1-–S4, using the PJE profile,\nrather than the scaled Pseudo-Jaffe ellipsoids mentioned\nabove. All model parameters for this member galaxy are\ntreated as free parameters during optimization.\nC. Multipole perturbations\nIn order to enhance the flexibility of our modeling, we\nintroduce both an external shear to the lens potential and\nhigher-order perturbations accounting for the potential\nasymmetry in the cluster mass distribution. These per-\nturbations can significantly improve the mass modeling\nprecision. To achieve this, we consider multipole pertur-\nbations ϕ=−(ϵ/m)r2cosm(θ−θϵ−π/2) with orders up\nto 6, i.e., m= 2, 3, 4, 5, and 6 [61–65]. In our optimiza-\ntion process, we optimize the model parameters ϵandθϵ\nwithout imposing any prior ranges. The external shear\ncorresponds to the multipole perturbation with m= 24\n62.5 65.0 67.5 70.0 72.5 75.0 77.5\nH0[km/s/Mpc]M23M22M21M20M19M18M17M16M15M14M13M12M11M10M9M8M7M6M5M4M3M2M1\n60.0 62.5 65.0 67.5 70.0 72.5 75.0 77.5\nH0[km/s/Mpc]9095100105110115120χ2M1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n60.0 62.5 65.0 67.5 70.0 72.5 75.0 77.5\nH0[km/s/Mpc]0.040.050.060.070.080.09σsM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\nFIG. 1: Top: Summary of constraints on the Hubble constant\nacross 23 different lens mass models. Middle: Similar to the\ntop panel, but presenting the correlation between χ2values\nand measurements of the Hubble constant. Bottom: Simi-\nlar to the top panel, but presenting the correlation between\nassumed position errors of supernova images (with small ver-\ntical offsets for clarity) with Hubble constant values.\n[66], dubbed as pert inglafic , while terms with m≥3\nis dubbed as mpole inglafic .D. Model optimization\nIn this paper, we consider 23 different lens mass mod-\nels to fit the cluster-lensed supernova system, aiming to\nexplore the dependence of H0constraints on mass model\nassumptions as well as the dependency between H0and\nother physical properties associated with mass models.\nSpecifically, we consider three variations in the utiliza-\ntion of different models to describe the dark matter halo,\nthe number of dark matter halo components, and differ-\nences in the orders of perturbations. More specifically,\nwe utilize either the NFW profile ( anfw inglafic ) or\nthe PJE profile ( jaffe inglafic ) to characterize the\nmass distribution of dark matter halos. Furthermore, we\nconsider both cases of three dark matter halo compo-\nnents and four components. In terms of perturbations,\nwe consider external shear ( pert inglafic ) and multi-\npole perturbations ( mpole inglafic ) of order up to 6\nat most. We expect that this comprehensive approach\nallows us to thoroughly explore the impact of different\nassumptions on the lens mass model and assess the ro-\nbustness of previous measurements of H0from SN Refs-\ndal [36, 37].\nAll our mass models are summarized in Table II,\nand are labeled from M1 to M23. For instance,\nmodel M1 employs the NFW profile ( anfw) to charac-\nterize three dark matter halo components, utilizes the\nscaled PJE profile ( gals) for most cluster members,\nand includes external shear ( pert). This model is de-\nnoted as M1 ( anfw3 +gals+pert). In contrast, model\nM10 ( anfw4 +gals+pert+mpole(m=3,4,5) ) features\nfour dark matter halo components instead of three dark\nmatter halo components in M1. Additionally, this model\nincorporates not only external shear but also multipole\nperturbations of order up to 5. Furthermore, model\nM21 ( anfw+anfw3 +gals+pert+mpole(m=3) ) signifies\nthe consideration of four dark matter halos with external\nshear and third-pole perturbation. However, this model\nincorporates the NFW profile for one dark matter halo\ncomponent while employing the PJE profile for the oth-\ners. We note that the Oguri-a* model in Ref. [36] corre-\nsponds to M7 in Table II.\nWe follow the standard mass modeling procedure by\nassuming positional errors that are larger than measure-\nment errors. This larger positional error accounts for the\ncomplexities of cluster mass distributions, such as asym-\nmetry and substructures, which may not be fully cap-\ntured in our parametric mass modeling approach. Our\nchoice of positional errors for each mass model is guided\nby the aim to achieve a reduced- χ2value for the best-\nfitting model that is approximately equal to one. Follow-\ning [41] and [34], we assign smaller positional errors for\nthe core and knots of the lensed spiral galaxy, σh, com-\npared to multiple images of other background galaxies,\nσg. Furthermore, we assign even smaller errors for the\nsupernova images, recognizing the importance of accu-\nrately reproducing supernova image positions for precise\npredictions of time delays between multiple images [67].5\n40 50 60 70 80 90 100\nH0[km/s/Mpc]0.000.250.500.751.001.251.501.752.00ProbabilityH0= 70.0+4.7\n−4.9km/s/MpcM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\ncombined\n40 50 60 70 80 90 100\nH0[km/s/Mpc]0.000.020.040.060.08ProbabilityH0= 70.3±4.4km/s/MpcM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\ncombined\nFIG. 2: Top: The probability distribution functions (PDFs)\nof the Hubble constant for all 23 different lens mass models\n(dashed ), along with the PDF after combining all 23 mass\nmodels with equal weighting ( solid). The vertical gray dotted\nlines and shaded region denote the median and 68 .3% confi-\ndence interval for the PDF of the combined result. Bottom:\nSimilar to the top panel, but combining all mass models with\nweighting based on the assumed positional errors of the su-\npernovae images.\nSpecific values of positional errors adopted for the 23 dif-\nferent lens mass models are summarized in Table II.\nIn our mass modeling, we include a total of 109 multi-\nple images originating from 36 systems, resulting in 225\nobservational constraints including magnification ratios\nand time delays for SN Refsdal (see Table I). Throughout\nall calculations and model parameter optimizations, we\nemploy a standard χ2minimization method implemented\nin the glafic software. To enhance computational effi-\nciency, we estimate χ2in the source plane [56].\nIV. CONSTRAINTS ON HUBBLE CONSTANT\nIn the realm of time-delay cosmography, the Hubble\nconstant H0can be directly inferred from time-delay\nmeasurements ∆ tand well-reconstructed lens potential\nϕ. In our analytical framework, we incorporate H0as a\nmodel parameter, varying it in conjunction with all theother model parameters to simultaneously optimize the\nalignment of image positions, magnification ratios, and\ntime delays, as detailed in Table I. The uncertainty as-\nsociated with H0is calculated by systematically varying\nits value around the best-fitting estimate, utilizing a step\nsize of 0 .2 km s−1Mpc−1. Specifically, 68.3% confidence\ninterval is derived from the range where the difference of\nthe best-fitting χ2, denoted as ∆ χ2, is less than 1, under\nthe assumption of Gaussian errors.\nWe present our findings in Table II and Fig. 1. We\nobserve slight differences in the best-fitting Hubble con-\nstant values across different lens mass models. This\nresult is in marked contrast to our previous study in\nRef. [32] focusing on the cluster-lensed quasar system\nSDSS J1004+4112, in which large variations of Hubble\nconstant measurements across different mass models are\nobserved. For instance, when employing the NFW profile\nto model the dark matter halo and incorporating exter-\nnal shear to account for perturbations (M1), we derive\nH0= 73 .5+4.8\n−4.4km s−1Mpc−1. Upon introducing ad-\nditional multipole perturbations, the estimated H0val-\nues become H0= 71 .8+4.4\n−3.4km s−1Mpc−1for M2, and\nH0= 72.6+3.4\n−3.0km s−1Mpc−1for M5. Notably, we ob-\nserve minor differences upon adding more multipole per-\nturbations across all considered mass models, as depicted\nin the top panel of Fig. 1.\nWe find that changing the number of dark matter\nhalo components, particularly from three to four, has a\nmodest impact on the resulting H0value. For instance,\nwithin the lens modeling combination M7, we obtain a\nHubble constant value of H0= 66.9+3.6\n−2.6km s−1Mpc−1.\nThis value is smaller than that derived from the M2\nmodel. Furthermore, changing the mass distribution\nmodeling of the dark matter halo from the NFW pro-\nfile to the PJE profile also influences the results. For\ninstance, H0= 69 .2+2.8\n−2.6km s−1Mpc−1is derived from\nM10, which is smaller than the H0value obtained for\nM20. Additionally, when incorporating four dark mat-\nter halo components with external shear and third-order\nperturbation, but utilizing the NFW profile for two\ndark matter halo components and the PJE profile for\nthe others, we derive a Hubble constant constraint of\nH0= 72.8+3.6\n−3.0km s−1Mpc−1. In general, we do not ob-\nserve significant lens model dependence comparable to\nRef. [32], which examined the cluster-lensed quasar sys-\ntem SDSS J1004+4112. The difference between SN Refs-\ndal and SDSS J1004+4112 presumably originates from\nthe difference in the number of multiple images. In the\ncase of SN Refsdal, the cluster mass distribution is tightly\nconstrained by more than 100 multiple images, leading\nto the robust Hubble constant value against difference\nchoice of mass models.\nThe Hubble constant value of H0 =\n66.9+3.6\n−2.6km s−1Mpc−1for M7 should be com-\npared with the Hubble constant value of H0=\n66.6+4.1\n−3.3km s−1Mpc−1from the combination of the\nOguri-a* model (higher weight) and Grillo-g model\n(lower weight) in Ref. [36]. The very similar best-fitting6\nTABLE II: Summary of assumed positional errors of supernovae images σs, host knots images σh, and galaxy images σg, as\nwell as the Hubble constant with 68.3% confidence interval and reduced- χ2(χ2/dof) values for 23 different lens mass models.\nLabel Model*σs[′′]σh[′′]σg[′′]H0[ km s−1Mpc−1] χ2/dof\nM1 anfw3 +gals+pert 0.09 0.36 0.72 73 .5+4 .8\n−4.4 120.99/108\nM2 anfw3 +gals+pert+mpole (m=3) 0.07 0.28 0.56 71 .8+4 .4\n−3.4 105.63/106)\nM3 anfw3 +gals+pert+mpole (m=3,4) 0.06 0.24 0.48 69 .9+3 .4\n−3.6 105.11/104\nM4 anfw3 +gals+pert+mpole (m=3,4,5) 0.05 0.20 0.40 73 .3+3 .2\n−2.6 116.47/102\nM5 anfw3 +gals+pert+mpole (m=3,4,5,6) 0.05 0.20 0.40 72 .6+3 .4\n−3.0 116.13/100\nM6 anfw4 +gals+pert 0.05 0.20 0.40 68 .5+3 .6\n−3.2 112.07/104\nM7 anfw4 +gals+pert+mpole (m=3) 0.05 0.20 0.40 66 .9+3 .6\n−2.6 101.57/102\nM8 anfw4 +gals+pert+mpole (m=3,4) 0.05 0.20 0.40 66 .7+2 .6\n−3.2 89.66/100\nM9 anfw4 +gals+pert+mpole (m=3,4,5) 0.04 0.16 0.32 70 .0+2 .2\n−3.0 100.30/98\nM10 anfw4 +gals+pert+mpole (m=3,4,5,6) 0.04 0.16 0.32 69 .2+2 .8\n−2.6 100.09/96\nM11 jaffe3 +gals+pert 0.07 0.28 0.56 64 .9+3 .2\n−3.2 107.69/104\nM12 jaffe3 +gals+pert+mpole (m=3) 0.07 0.28 0.56 64 .5+3 .0\n−3.4 104.11/102\nM13 jaffe3 +gals+pert+mpole (m=3,4) 0.07 0.28 0.56 65 .0+3 .2\n−3.6 100.86/100\nM14 jaffe3 +gals+pert+mpole (m=3,4,5) 0.07 0.28 0.56 66 .4+4 .4\n−3.8 99.05/98\nM15 jaffe3 +gals+pert+mpole (m=3,4,5,6) 0.07 0.28 0.56 66 .6+4 .2\n−3.6 99.03/96\nM16 jaffe4 +gals+pert 0.06 0.24 0.48 69 .8+2 .6\n−4.4 98.93/99\nM17 jaffe4 +gals+pert+mpole (m=3) 0.05 0.20 0.40 72 .6+3 .2\n−2.4 114.36/97\nM18 jaffe4 +gals+pert+mpole (m=3,4) 0.05 0.20 0.40 70 .4+2 .8\n−3.6 108.34/95\nM19 jaffe4 +gals+pert+mpole (m=3,4,5) 0.05 0.20 0.40 74 .8+3 .8\n−3.2 89.74/93\nM20 jaffe4 +gals+pert+mpole (m=3,4,5,6) 0.05 0.20 0.40 73 .2+4 .0\n−3.8 88.39/91\nM21 anfw+jaffe3 +gals+pert+mpole (m=3) 0.05 0.20 0.40 72 .5+2 .8\n−2.8 117.08/99\nM22 anfw2 +jaffe2 +gals+pert+mpole (m=3) 0.05 0.20 0.40 72 .8+3 .6\n−3.0 99.43/100\nM23 anfw3 +jaffe1 +gals+pert+mpole (m=3) 0.04 0.16 0.32 70 .0+2 .8\n−2.6 101.64/101\n*For model components, anfw denotes the NFW profile for the halo, jaffe denotes the PJE profile for the halo, gals denotes the\nscaled PJE profile for most member galaxies, pert refers to an external shear, and pert refers to multipole perturbations of order m.\nThe numerical value following anfw orjaffe indicates the number of halos. Please refer to the main text for a detailed explanation.\nFIG. 3: Critical curves for the 23 different lens mass models, computed assuming the source redshift of zs= 6. Left: Critical\ncurves for 12 different lens mass models with best-fitting values of the Hubble constant larger than 70 .0 km s−1Mpc−1.Right:\nCritical curves for 11 different lens mass models with best-fitting values of the Hubble constant smaller than 70 .0 km s−1Mpc−1.\nBlack squares represent lensed supernova images.\nvalues between these result basically support the validity\nof our analysis. We find that the M7 result exhibits a\nslightly tighter constraint, which may partly arise from\nour simplified assumption of Gaussian distributions\nwithout incorporating any covariance between measure-\nments of magnifications and time delays. Another reason\nmay be that the latter constraint contains the contri-bution from the Grillo-g model, albeit with the smaller\nweight. As it will be shown below, the uncertainty of\nour combined constraint on the Hubble constant largely\ncomes from the variation of the best-fitting Hubble\nconstant values among different models, rather than\nstatistical errors for individual mass models, we expect\nthat the impact of our simplified assumption on the final7\n100101102\nr[arcsec]10−210−1100κM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n100101102\nr[arcsec]10−210−1γTM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\nFIG. 4: Radial convergence ( left) and tangential shear ( right) profiles for 23 different lens mass models, computed assuming the\nsource redshift of zs= 1. The Einstein radius for the source redshift of SN Refsdal is displayed by black vertical dashed lines.\n0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02\nκ62.565.067.570.072.575.077.5H0[km/s/Mpc]r=2 arcsecM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82\nκ62.565.067.570.072.575.077.5H0[km/s/Mpc]r=5 arcsecM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n0.46 0.47 0.48 0.49 0.50\nκ62.565.067.570.072.575.077.5H0[km/s/Mpc]r=14 arcsecM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n0.01 0.02 0.03 0.04 0.05 0.06\nκ62.565.067.570.072.575.077.5H0[km/s/Mpc]r=300 arcsecM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\nFIG. 5: The correlation between radial convergence values κat specific radius ( r= 2′′,5′′,14′′,300′′) and Hubble constant\nmeasurements for 23 different lens mass models.\nresult is relatively minor.\nTo integrate findings from various lens mass models, we\nfollow the approach in Ref. [32] to aggregate the posteri-\nors exp( −∆χ2/2) of the Hubble constant from individual\nmass models with equal weighting. The resultant com-\nbined posterior probability distribution function (PDF)\nofH0, alongside PDFs for individual lens mass mod-\nels, is depicted in Fig. 2. The constraint on H0fromthis combined PDF is H0= 70 .0+4.7\n−4.9km s−1Mpc−1\nwhich provides a much tighter constraint compared to\nH0= 67.5+14.5\n−8.9km s−1Mpc−1derived from Ref. [32].\nTo explore the impact of weighting on the result, we\nalso combine all the lens mass models using weight-\ning based on the assumed positional errors of super-\nnovae images σs. Specifically, we adopt the weighting\nvalue of wi= 1/σ2\ns,i, where σs,irepresents the posi-8\n1.00 1.05 1.10 1.15 1.20 1.25\nγT62.565.067.570.072.575.077.5H0[km/s/Mpc]r=2 arcsecM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92\nγT62.565.067.570.072.575.077.5H0[km/s/Mpc]r=5 arcsecM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n0.60 0.61 0.62 0.63 0.64\nγT62.565.067.570.072.575.077.5H0[km/s/Mpc]r=14 arcsecM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10\nγT62.565.067.570.072.575.077.5H0[km/s/Mpc]r=300 arcsecM1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\nFIG. 6: Similar to Fig. 5, but for the tangential shear γT.\ntional errors of individual mass models as listed in Ta-\nble II. With this weighting scheme, we obtain H0=\n70.3±4.4 km s−1Mpc−1(refer to Fig. 2), representing\na slightly tighter, but quite consistent, constraint com-\npared to the equal weighting case.\nOur analysis indicates that there is no significant de-\npendence of the Hubble constant on assumptions on lens\nmass models, and the uncertainty from lens mass model\nassumptions is not unsatisfactory large compared with\ncurrent observational constraints on the Hubble constant\nwith SN Refsdal, which is consistent with Ref. [38].\nV. DISCUSSION\nA. Critical curves\nThe shape and extent of critical curves are highly sen-\nsitive to the mass distribution of the lensing objects.\nTherefore, we begin, by checking critical curves, to exam-\nine potential dependence of Hubble constant values with\ncluster mass distributions. The results are displayed in\nFig. 3. We find that there are some notable differences\nof critical curves, especially in the northern part. These\ndifferences arise largely from the consideration of vary-\ning numbers of dark matter halo components. The largevariation of critical curves in the northern part, how-\never, is seen in both models with large and small values\nof the Hubble constant. Our analysis reveals no signifi-\ncant correlations between critical curve shapes and and\nthe Hubble constant constraints across all examined lens\nmass models.\nB. Radial convergence and tangential shear profiles\nNext, we calculate the azimuthal average of the con-\nvergence at each radius to derive radial convergence and\ntangential shear profiles for all the 23 different lens mass\nmodels. The radial convergence, denoted by κ, is directly\nrelated to the surface mass density of the lensing object\nand provides insights into its overall mass distribution.\nOn the other hand, the tangential shear, denoted by γT,\nmeasures the coherent distortion of background sources\naround the lensing object. Unlike the radial convergence,\nthe tangential shear describes the anisotropic stretching\nof images along the tangential direction with respect to\nthe lens center. The tangential shear is sensitive to the\ngradient of the lens potential and offers valuable informa-\ntion about the internal structure and orientation of the\nlensing mass distribution.\nFig. 4 illustrates the convergence and the tangential9\n10 11 12 13 14 15 16\nµ162.565.067.570.072.575.077.5H0[km/s/Mpc]M1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n10 11 12 13 14 15 16 17\nµ262.565.067.570.072.575.077.5H0[km/s/Mpc]M1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n12 14 16 18 20\nµ362.565.067.570.072.575.077.5H0[km/s/Mpc]M1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n4 5 6 7 8 9\nµ462.565.067.570.072.575.077.5H0[km/s/Mpc]M1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\n3.5 4.0 4.5 5.0\nµX62.565.067.570.072.575.077.5H0[km/s/Mpc]M1\nM2\nM3\nM4\nM5\nM6\nM7\nM8\nM9\nM10\nM11\nM12\nM13\nM14\nM15\nM16\nM17\nM18\nM19\nM20\nM21\nM22\nM23\nFIG. 7: The correlation between magnification factors of five multiple supernova images and Hubble constant measurements\nfor 23 different lens mass models.\nshear profiles for all the 23 different lens mass mod-\nels. While there is a general similarity among the pro-\nfiles across different lens mass models, some differences\nstill persist. Therefore, we concentrate on specific radii\n(r= 2′′,5′′,14′′,300′′) to check the correlation between\nthe Hubble constant and the convergence and the tan-\ngential shear values. We present our results in Fig. 5 and\nFig. 6. Intriguingly, we observe that best-fitting Hubble\nconstant values tend to increase with κandγTin the\ninner regions, while they decrease in the outer regions.\nOur result indicates that best-fitting Hubble constant\nvalues are larger for models with steeper radial densityprofiles. It has been known that time delays sensitively\ndepend on the radial density profile of the lens such\nthat steeper profiles result in larger time delays. (e.g.,\n[68, 69]), and our result is in line with those previous\nfindings. In the case of SN Refsdal, the radial density\nprofile is tightly constrained by many multiple images\nwith different source redshifts, yet such correlation be-\ntween the radial density profile and the Hubble constant\nis seen. This result implies that finding more multiple\nimages in future deep observations can improve the con-\nstraint on the Hubble constant from SN Refsdal.10\nC. Magnification factors\nRef. [70] proposed that the measurement of magnifica-\ntion factors of lensed supernova images, which is available\nwhen the source is Type Ia supernova, breaks the degen-\neracy between the radial density profile and the Hub-\nble constant and significantly improves the constraint on\nthe Hubble constant, because magnification factors are\nsensitive to the radial density profile. It is however not\nimmediately clear whether this method is indeed useful\nfor massive clusters for which the radial profile is already\nconstrained by many multiple images and also their mass\ndistributions are highly complex. Therefore, it is of great\ninterest to explicitly check whether there is a correla-\ntion between best-fitting Hubble constant values and the\nmagnification factors for five multiple images of SN Refs-\ndal, even though SN Refsdal is a core-collapse supernova\nand hence their magnification factors have not been mea-\nsured.\nOur result shown in Fig. 7 indicates that there is a clear\ncorrelation between Hubble constant values and mag-\nnification factors, well demonstrating the usefulness of\nmagnification factor measurements [70], even for massive\nclusters. Our analysis highlights the importance of gravi-\ntationally lensed Type Ia supernova, such as the recently\ndiscovered Supernova H0pe in the galaxy cluster PLCK\nG165.7+67.0 [71], for precise and accurate measurements\nof the Hubble constant.\nVI. CONCLUSION\nIn this paper, we investigate the power of constrain-\ning the Hubble constant through time-delay cosmography\nutilizing cluster-lensed supernova systems. As a specific\nexample, we focus our attention on the strongly lensed\nsupernova system SN Refsdal in MACS J1149.5+2223,\nwhich represents the first gravitationally lensed super-\nnova with resolved multiple images. The background su-\npernova is lensed into six distinct multiple images. Ad-\nditionally, many multiple images of galaxies located be-\nhind the lensing cluster have been identified. In total,\nour analysis encompasses 109 multiple images from 36\nsystems.\nTo explore the dependence of constraints on the Hub-\nble constant on the choice of lens mass models, we uti-\nlize 23 different lens mass models, each making differ-\nent assumptions regarding the profiles of the dark mat-\nter halo, the number of dark matter halo components,and multipole perturbations. We find that the varia-\ntion in the best-fitting values of the Hubble constant\nis relatively small across the 23 lens mass models. By\nsumming the posteriors of the Hubble constant with\nequal weighting, we obtain a combined constraint of\nH0= 70 .0+4.7\n−4.9km s−1Mpc−1. We also try a differ-\nent weighting scheme based on positional errors of su-\npernova images, finding a slightly tighter constraint of\nH0= 70.3±4.4 km s−1Mpc−1. Compared to the con-\nstraint on H0= 64 .8+4.4\n−4.3km s−1Mpc−1derived from\neight cluster lens models in Ref. [36], our results repre-\nsent a comparable precision, although the central value\nis about 1 σhigher in our result.\nOur finding appears to differ from our previous work,\nwhich revealed significant lens mass model dependence in\nthe cluster-lensed quasar system SDSS J1004+4112 and\nyielded a weak constraint on the Hubble constant of H0=\n67.5+14.5\n−8.9km s−1Mpc−1[32]. We argue that the large\nnumber of multiple images for SN Refsdal plays a crucial\nrole in breaking the degeneracy among different lens mass\nmodels and in obtaining the robust measurement of the\nHubble constant.\nIn addition, we examine critical curves, the radial con-\nvergence, the tangential shear, and magnification factors\nof the multiply lensed supernova images, and their cor-\nrelations with the best-fitting Hubble constant values.\nWe observe correlations between the Hubble constant\nand the radial convergence and tangential shear profiles\nas well as magnification factors. 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Freeman,3and Dan Scolnic4\n1School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia\n2Korea Astronomy and Space Science Institute, Yuseong-gu, Daedeok-daero 776, Daejeon 34055, Republic of Korea\n3The Research School of Astronomy and Astrophysics, Australian National University, Stromlo, ACT 2601, Australia\n4Department of Physics, Duke University, Durham, NC 27708, USA\nAuthor for correspondence: Anthony Carr, Email: anthonycarr@kasi.re.kr.\n(Received dd Mmm YYYY; revised dd Mmm YYYY; accepted dd Mmm YYYY; first published online dd Mmm YYYY)\nAbstract\nWe present high-resolution observations of nearby ( z≲0.1) galaxies that have hosted Type Ia supernovae to measure systemic spectroscopic\nredshifts using the Wide Field Spectrograph (WiFeS) instrument on the Australian National University 2.3 m telescope at Siding Spring\nObservatory. While most of the galaxies targeted have previous spectroscopic redshifts, we provide demonstrably more accurate and precise\nredshifts with competitive uncertainties, motivated by potential systematic errors that could bias estimates of the Hubble constant ( H0). The\nWiFeS instrument is remarkably stable; after calibration, the wavelength solution varies by ≲0.5 Å in red and blue with no evidence of a trend\nover the course of several years. By virtue of the 25 ×38 arcsec field of view, we are always able to redshift the galactic core, or the entire galaxy\nin the cases where its angular extent is smaller than the field of view, reducing any errors due to galaxy rotation. We observed 185 southern SN\nIa host galaxies and redshifted each via at least one spatial region of a) the core, and b) the average over the full-field/entire galaxy. Overall, we\nfind stochastic differences between historical redshifts and our measured redshifts on the order of ≲10–3with a mean offset of 4.3 ×10–5, and\nnormalised median absolute deviation of 1.2 ×10–4. We show that a systematic redshift offset at this level is not enough to bias cosmology, as H0\nshifts by +0.1 km s–1Mpc–1when we replace Pantheon+ redshifts with our own, but the occasional large differences are interesting to note.\nKeywords: redshift surveys; observational cosmology\n1. Introduction\nGiven that the discrepancy between the Planck 2018 CMB\nmeasurement of H0, 67.4±0.5 km s–1Mpc–1(Planck Collab-\noration et al., 2020) and most recent SH0ES measurement,\n73.04±1.04 km s–1Mpc–1(Riess et al., 2022) is now at the 5 σ\nlevel, we must ensure we have a comprehensive understanding\nof any possible systematic errors. In the case of local distance\nladder supernova cosmology, these systematics can take many\nforms when measuring Cepheid/SN brightnesses, recession\nvelocities and distances, and may bias our measurements of\ncosmic expansion. An extensive set of systematics was recently\nexplored as part of the recent SH0ES/Pantheon+ collaboration\n(Brout et al., 2022a; Riess et al., 2022, and references therein),\nincluding, but not limited to the geometric–Cepheid distance\ncalibration sample (Yuan et al., 2022); the Cepheid–SN calibra-\ntion sample and Cepheid metallicity dependence (Riess et al.,\n2022); SN photometry calibration (Brout et al., 2022b); SN\ndust and colour (Popovic et al., 2021); SN peculiar velocities\n(Peterson et al., 2022); and SN redshifts (Carr et al., 2022). The\ngeneral conclusion from these analyses is that SN systematics\nare not a solution to the Hubble tension, as each individual\nsystematic can only realistically account for a small fraction of\nthe tension, and in fact, often increase the tension.\nThe most straightforward of the above systematics to test\nare the SN redshifts. A systematic shift to redshift can easily\ninfluence H0in much the same way as a systematic shift to\nmeasured SN magnitudes, as a magnitude shift is degenerate\nwith a shift in H0. A shift in redshift of, e.g. 1 ×10–4, would beequivalent to a magnitude shift of around 9 mmag at z= 0.0233\nand 1.5 mmag at z= 0.15. The effect is smaller at higher\nredshift due to the sub-linear nature of the distance modulus–\nredshift relation. For the same reason, downward shifts in\nredshift have a slightly larger effect on magnitude than the\nsame shift upward. According to Davis et al. (2019), redshift\nerrors on the order of only 5 ×10–4can bias H0by nearly 1\nkm s–1Mpc–1, if the errors are systematic and at low- z(the\nsmaller the redshift, the larger the effect). However, in the case\nof real data, Carr et al. (2022) show that the combined effects\nof existing redshift and peculiar velocity errors amounted to\na negligible shift in H0of –0.12 ±0.20 km s–1Mpc–1. While\nthis systematic has now been thoroughly ruled out as being a\ncomplete solution to the Hubble tension from historical data,\nthere remains a possibility that errors in the measurements of\nthe redshifts are still present.\nCarr et al. (2022) used historical data, so here we measure\nnew redshifts to test whether systematics in the historical data\nare present. Specifically, we target bright, nearby galaxies,\nwhich are the most influential to H0and have the potential to\nbe biased by, e.g. pointing, where a spectroscopic slit or fibre\nmay be placed not on the core, but elsewhere in the galaxy\n(such as the location of the supernova). To combat the effects of\npotential observational bias, we use integral field spectroscopy\nto ensure we can capture the systemic redshift.\nThe paper is set out as follows: in Section 2, we detail\nthe target selection, observation and reduction process for our\nprogram. In Sections 3 and 4 we detail our analysis of thearXiv:2402.13484v1  [astro-ph.CO]  21 Feb 20242 Anthony Carr et al.\ncalibrations we undertook and the performance of the instru-\nment over the course of our program. Finally, we describe our\nredshift results compared to historical data and their impact\non cosmology in Section 5, and then conclude in Section 6.\n2. Observations\nWe take advantage of integral field spectroscopy to measure\nthe spatial variation in redshift across the face of large galax-\nies and gather enough signal-to-noise (S/N) to successfully\nredshift smaller/fainter galaxies. We use the Wide Field Spec-\ntrograph (WiFeS) instrument mounted on the Australian Na-\ntional University 2.3 metre telescope (ANU 2.3m) at Siding\nSpring Observatory (SSO). See Dopita et al. (2007) for the\nfull instrument specifications and Dopita et al. (2010) for the\nmeasured performance.\nWe originally planned to observe using only R= 7000\ngratings, since these offer higher precision redshifts at the cost\nof reduced wavelength coverage compared to the R= 3000\ngratings. The specifications of the grating suite are described\nin Table 1, and the throughput curves measured by Dopita\net al. (2010) are shown in Figure 1. The ANU 2.3m offers\nfull optical spectral coverage at R= 7000 over four individual\ngratings: the ultraviolet and blue gratings are paired with the\nred and infrared gratings respectively. We opted to use only\nthe B7000+I7000 grating pair as the U7000+R7000 did not\noffer useful spectral coverage considering observations would\ntake twice as long and add extra overheads swapping grating\npairs and beamsplitter.\nHowever, after the first two observing runs, we switched\nto observing with full spectral coverage at R= 3000, which\nstill offers excellent precision and better S/N for the same\nexposure time. We made this decision due to the fact that at\nthe low redshifts we were targeting, the I7000 grating was\nsometimes a trade-off between the Calcium triplet and H α\nregion being redshifted out of spectral coverage. We show this\nin Figure 1 with two example spectra at redshifts where the\nI7000 spectral region would be devoid of features. If we have\ngood enough S/N for a late-type galaxy, or the galaxy has a\nparticularly strong Calcium triplet, we would still be able to\nidentify features in I7000 below z≈0.014, and similar for\nearly-type galaxies above z≈0.065. However, for the final\ntwo observation runs, we observed using the B3000+R3000\ngratings since they offer full spectral coverage with a generous\noverlap, and as such have no such restrictions on where typical\noptical galaxy emission and absorption features land.\n2.1 Target Selection\nThe catalogue was created using the Pantheon supernova sam-\nple (Scolnic et al., 2018) since we conducted all of our obser-\nvations before the release of Pantheon+. Galaxies were chosen\nto be easily observable from Siding Spring Observatory (lati-\ntude 149.06◦, longitude –31.27◦), i.e. airmass ≲1.5, and with\nz≲0.1 as these are the redshifts most influential to H0. The\nstrategy was to observe in Nod&Shuffle mode, which results\nin simultaneous science and sky spectra. The target is exposed\non the science CCD pixels, then the telescope is nodded toTable 1. WiFeS grating wavelength ranges and dispersion, as measured from\nour reduced data. Wavelength limits are rounded to the nearest 5 Å. The\nbeamsplitters are named after their dichroic wavelength split in nanometres.\nThe dispersion and upper wavelength limit for each grating varied over time\nby less than 0.02%.\nλmin λmax Dispersion\nGrating Beamsplitter Å Å Å/pixel\nU7000 RT480 3290 4355 0.272\nB7000 RT615 4180 5550 0.347\nR7000 RT480 5290 7025 0.439\nI7000 RT615 6830 9055 0.566\nB3000 RT560 3200 5900 0.775\nR3000 RT560 5300 9565 1.25\nFigure 1. Measured throughput of each grating except for U7000 from Dopita\net al. (2010). The shaded region represents the I7000 wavelength bounds in\npractice (Table 1). The bottom panel shows two examples of spectra that\nmight fail to be redshifted from the I7000 grating. For a redshift of ≲0.014\n(≳0.065 ), none of the H α(Calcium triplet) region is present in I7000 leaving\nonly weak or no features.\nempty sky and the charge already present is ‘shuffled’ across\nthe CCD so that the sky is exposed on a different set of pix-\nels. Sky subtraction is then just the simple case of subtracting\nthe pure 2D sky spectrum from the observed object+sky 2D\nspectrum within the same CCD image during data reduction,\nleaving just the 2D object spectrum. Each galaxy observation\nwas made up of three Nod&Shuffle cycles, with each sky and\nobject exposure being the same length. The sky field was cho-\nsen to be as empty as possible, and close by to reduce the time\nto nod between frames.\nOnce the targets were chosen, we calculated the rough\nexposure time using the WiFeS performance calculator.aThe\naim was a generous S/N of at least 20 in the blue camera after\nthe full 3 ×Nod&Shuffle cycle, which was easily obtained for\nthe brightest galaxies. The red camera naturally gathers more\nsignal for the same observation time, so observation time was\noptimised for the blue camera. Exposure time depends on the\nmoon phase (all observations were done in grey or dark time),\nthe seeing full width at half maximum (FWHM; typically 1.6\nahttps://www.mso.anu.edu.au/rsaa/observing/wifes/perf ormance.shtmlPublications of the Astronomical Society of Australia 3\nFigure 2. DECam g-band image of the galaxy NGC 632, host of SN 1998es. The\nthree estimates of surface brightness were made over the whole WiFeS aper-\nture, within the central pixel of the aperture, and finally, a 5x5 pixel binned\naperture on the outskirts to estimate a lower limit of surface brightness.\narcsec at SSO), as well as the airmass and surface brightness\nof the target. The average total integration time was 485 s\nfor an estimated average g-band surface brightness of 16 mag\narcsec–2. Subexposure times were rounded to the nearest 30\nseconds rather than attempting to save small amounts of time\noptimising to the nearest second.\nSurface brightness was estimated directly from Dark En-\nergy Camera (DECam) images using the US National Sci-\nence Foundation’s NOIRLab Astro Data Lab image cutout\nservice.bMost targets had images; for those that did not, we\nestimated surface brightness by comparison with similar tar-\ngets that did have images. When there were images, we opted\nfor sky-subtracted images in the g-band as the highest prior-\nity, followed by r-band then i-band (with minor corrections\nto account for overestimating the g-band magnitude), and\nstacked images if no sky-subtracted version was available. The\nsurface brightness was estimated from the images using the\nphotutils python packagecwithin different apertures, in-\ncluding the full WiFeS aperture, to estimate a useful average\nfor the whole field of view. See Figure 2 for an example.\nFlux calibration stars were chosen to be CALSPEC starsd\nwhich are the standards used for the Hubble Space Telescope.\nThe only criterion for choosing a CALSPEC flux calibrator\nwas that it was easily observable from SSO. The CALSPEC\nstandard stars were also used to remove Telluric absorption fea-\ntures from the galaxy spectra. We trialled the use of dedicated,\nparticularly smooth-spectrum Telluric standard stars (host,\nmain sequence B stars), but were unable to reliably source\nreference spectra. As such, the Telluric corrections were some-\ntimes lacking, but never resulted in a failure to compute a\nredshift.\nbhttps://datalab.noirlab.edu/sia.php\nchttps://photutils.readthedocs.io/en/stable/\ndhttps://www.stsci.edu/hst/instrumentation/reference-data-f or-calibrati\non-and-tools/astronomical-catalogs/calspecRadial velocity standard stars (stars with well-known ra-\ndial velocities to compare to as another form of instrument\ncalibration) were chosen from Nidever et al. (2002). The stars\nwe used were all chosen to be G- or K-type stars with a prefer-\nence for giants. They were also chosen to primarily be around\nV= 6 mag, similar to the flux calibrators.\n2.2 Reduction\nWe used version 0.7.4 of the default WiFeS reduction pipeline,\nPyWiFeS (Childress et al., 2014), to transform the raw obser-\nvations into calibrated, 3D data cubes. In brief, the reduction\npipeline pre-processes each CCD image (overscan and bias sub-\ntraction, bad pixel repair), then uses spatial calibration frames\nto split the data into the 25 science and 25 sky slitlets. The\ninstrument was designed so that the sky and science slitlets are\ninterleaved on the detectors, and slitlets lie along detector rows.\nOver the entire detector, the slitlets deviate from these rows\nby up to ∼ ±0.5 pixels. This deviation is accounted for by ob-\nserving a uniformly illuminated calibration frame obstructed\nby a straight wire. The shadow of the wire is used to find the\nspatial zero-point of each slitlet along the y-direction (the long\naxis of the aperture) over every CCD column.\nFrom here, the usual steps are performed: finding the wave-\nlength solution, cosmic ray rejection, sky-subtraction, flatfield-\ning, flux calibration and Telluric correction. Finally, the data\nare reformatted to a 3D data cube. We note that while the\nfield of view is 25 ×38 arcsec, in practice we trim the noise-\ndominated outer one or two spaxel rows (depending on the\ngratings and beamsplitter) so we actually use 25 ×35 spaxels\nfor our purposes.\nDue to the excellent stability of the WiFeS instrument, the\nwavelength solution varied only on the sub-pixel level. We\nexpand upon this in Section 3.\n2.3 Spaxel binning and redshifting\nAfter reducing the data, the 3D spectral data cube contains\na wealth of information. For this work, we are mainly in-\nterested in the redshift and its spatial dependence, although\nthere is certainly more that can be done with spatially vary-\ning, medium-resolution ( R= 3000–7000), high S/N data. To\ninvestigate the redshift(s) of the galaxy, we first processed the\ndata further into a format that could be ingested into the Marz\nredshifting tool,ewhich was developed primarily for the use\nof the Australian Dark Energy Survey and Anglo-Australian\nTelescope, also at SSO. This included Voronoi binning the\nspaxels using the vorbin python package (see Cappellari &\nCopin, 2003) to gather signal in the outer regions of the aper-\nture, and to successfully redshift the galaxies that occupied\nvery few spaxels.\nSee Figure 3 for a visualisation of how observations are\nturned into redshifts. It shows the case of a galaxy larger than\nthe aperture, where spaxels further from the bright central\nregion are binned to a common S/N threshold. Each V oronoi\nehttps://samreay.github.io/Marz. The default tool does not include the\ncalcium triplet, so we modified the source code for our own use.4 Anthony Carr et al.\n(a)DECam image of NGC 6928 (r-band).\n(b)Average S/N per spaxel.\n(c)Optimal binning map from vorbin .\n(d)Binned and unbinned S/N from vorbin .\n(e)Redshift map.\nFigure 3. a) Image of NGC 6928 on the sky with the WiFeS field of view in green. SN 2004eo occurred outside the green aperture pictured here, nearly one\narcminute east-north-east of the centre of the galaxy. b)S/N over the WiFeS aperture, as averaged per spaxel over the entire red camera spectra. Evidence of\nthe nearby star can be seen as an increase in S/N around spaxel coordinates (25,23), ∼35 arcsec from spaxel (0,0). c)The Voronoi binning regime that results in\neach bin having roughly 70% the average S/N of the central spaxel. d)The S/N of each unbinned and binned spaxel, with the dashed line showing the S/N target.\ne)Final redshift map, showing rotation along the long axis. The average redshift of all the spaxel bins is zhel= 0.01576 , with standard deviation 2.9 ×10–4.Publications of the Astronomical Society of Australia 5\nbin is then redshifted separately. From this map, the average\nredshift of the whole galaxy, the redshift of the centre and\nthe redshift in the locality of the SN can all be found. The\nparticular example shown does not contain the SN in the\naperture; there is only a moderate probability of the SN being\nwithin the field of view when centred on such large galaxies.\nFor most of the V oronoi binning, we set the S/N target to\n90% of the central pixel S/N by default. This target level was\nsometimes adjusted in the cases where the surface brightness\nprofile was particularly flat (requiring an increase), such as\nMCG-02-02-086, the host of SN 2003ic, or the galaxy as a\nwhole was extremely bright (requiring a decrease to reduce\nthe number of bins from ≫100 to≲100), such as NGC 6928,\nfeatured in Figure 3. In each case, the binning was visually\ninspected to ensure both a decent number of bins and high-\nquality spectra for each.\nIn 24 cases, there was only one bin, necessary for the par-\nticularly small/faint/poor-quality-spectrum galaxies to success-\nfully obtain an accurate redshift. Where possible, we also\nredshift just the central region (estimated from the highest\nS/N spaxels), to compare with the average redshift over all\nV oronoi bins. This often resulted in a strong increase in S/N;\nhowever, for the 24 cases above, a similar or lower S/N spec-trum naturally resulted, since we did not use as many spaxels\nas for the whole galaxy. Of the 185 galaxies we observed, 106\nwere at least roughly as large in the sky as the aperture, and\nnine were small enough to occupy only several spaxels each.\nFor the standard stars, the 3 ×3 spaxel region around the centre\nof the star was used for redshifting.\nThe geocentric-to-heliocentric correction to account for\nour motion around the Sun is automatically handled in Marz\nby including the relevant headers for the telescope location,\nobservation time and object coordinates. The distribution of\ngeocentric corrections between –25 km s–1and +20 km s–1\nonly had a slight positive gradient; however, the mean cor-\nrection was 11.7 km s–1due to a large number of corrections\nfalling between +20 km s–1and +30 km s–1.\n3. Instrument throughput correction\nTransforming the data from photon counts to accurate flux\nas a function of accurate wavelength is a multi-step process.\n‘Dome’ flats using an internal Quartz-Iodine lamp correct\nfor the CCD pixel-to-pixel quantum efficiency. The Quartz-\nIodine lamp is mostly ‘spectrally flat’, in that the wavelength\ndependence can be removed by a moderate order polynomial,\nFigure 4. Illumination corrections for the red and blue camera averaged over each observation run in each operation mode. The R= 3000 andR= 7000\ngrating corrections are basically the same since this is a spatial correction. The corrections are also comparable to those of Childress et al. (2016) indicating the\nstability of the instrument over several years.6 Anthony Carr et al.\nbut does not illuminate the instrument perfectly uniformly.\nTwilight flats using the twilit sky (which are ‘spatially’ flat,\ni.e. uniform illumination, but significant spectral deviation)\nare used to correct for the spatial illumination. Once the pixel-\nto-pixel and large-scale illumination are corrected, one of the\nfinal steps is flux calibration, which corrects for the wavelength\nresponse and instrument+atmosphere throughput.\nChildress et al. (2016) studied the performance of the WiFeS\ninstrument over a three-year period for the ANU WiFeS Su-\nperNovA Programme (A WSNAP), including wavelength so-\nlution, illumination correction, and flux calibration. We have\na similar set of data gathered over a year in two operation\nmodes, five years later, with which to compare. We emulate\ntheir analysis here to observe any trends over nightly to yearly\nscales and detect any need for manual recalibration. We start\nby examining the illumination correction for our different\noperating modes (see Figure 4), and find no significant dif-\nferences (visually) between them and Childress et al. (2016),\nwhich speaks to the stability of the instrument.\n4. Wavelength calibration methods\nIt is well known that temperature fluctuations inside the dome\naffect the wavelength solution of WiFeS spectra, as the gratings\nthemselves thermally expand. We endeavoured to mitigate any\ntemperature effects by taking frequent arc lamp calibration\nframes. The response of the gratings to temperature fluctu-ations may be linear, which can be interpolated over easily,\nbut the temperature fluctuations themselves are not and lag\nbehind the dome internal temperature readings. By making\nfrequent arc lamp observations, temperature variations are ac-\ncounted for since each science observation is calibrated using\nthe nearest arc lamp in time (or if the science observation falls\nbetween two arc lamps, it is calibrated by the average of those\nwavelength solutions). We track the wavelength solution over\nthe course of each night and over the entire observation pro-\ngram, which we detail in this section. In essence, we find that\nthe wavelength solution shows excellent stability and thus our\nredshifts require no spatial or temperature correction.\n4.1 Arc lamp wavelength solutions and temperature depen-\ndence\nSimilar to Childress et al. (2016), we investigated the variation\nof the wavelength solution over the CCD, and over time, and\nwe compare both of these to dome temperature readings to\ncorrelate with the fluctuations.\nFigures 5–7 show our investigation into wavelength solu-\ntion variation as measured from arc lamp observations. Over\nour entire observation program that spanned one year, we\nused two different resolutions each for two runs (approxi-\nmately spanning 6 months each). We find very similar results\nto Childress et al. (2016), in that the average wavelength solu-\ntion generally deviates by ±0.5 Å which for our gratings is\nFigure 5. Difference in the wavelength solution over the CCD for each resolution and camera, compared to the mean for that resolution+camera combination.\nWe show only the middle slitlet, but the others show similar overall trends (see Childress et al., 2016). Each curve is a different arc lamp observation, coloured\nby the temperature recorded at that time. The dashed lines denote the size of a CCD pixel. R= 7000 shows similar levels of variation to R= 3000 (despite the\ndiffering wavelength dispersions), except at the CCD column boundary, where the variation is less extreme because a) throughput is higher at higher resolution\n(especially toward the ends of wavelength coverage where B3000 and R3000 drop to near 0; see Figure 1), and b) the wavelength solution for R= 3000 gratings\nis less stable in the dichroic region at high (low) CCD column numbers for B3000 (R300).Publications of the Astronomical Society of Australia 7\nFigure 6. The same as Figure 5 but now averaged across the CCD column and as a function of observation date. The dependence on temperature is still clear,\nbut now the seasonal dependence is seen by the grouping in different observation runs.\nFigure 7. Average difference in wavelength solution as a function of temperature. The colour map is carried through from Figures 5 and 6, i.e. the x-axis\nhere. Since we observed for a full year, our observations roughly capture the extremes of temperature during a typical year, so it is expected that the average\nwavelength solution will not vary by more than a CCD pixel. According to the linear relations fit to the temperature data, even if the temperature does vary\nmore than expected, it would still be predominantly sub-pixel.8 Anthony Carr et al.\nFigure 8. The redshift error resulting from measuring a spectral feature at λem\nup to±1 Å (λerr) from its true value, due to, e.g. a small residual temperature\ndependence in the wavelength solution. We generally redshift features in\nthe red (H α, Ca triplet), and the average wavelength solution error is less\nthan 0.5 Å over most of the CCD (Figure 5), resulting in a realistic maximum\nredshift error of ∼6×10–5. Sources whose features are primarily at low\nwavelengths could potentially experience larger errors.\nalways less than a CCD pixel (see Table 1), except for a couple\nof B7000 measurements. There are no long-term trends (albeit\nwith only a single year of data) apart from the seasonal (tem-\nperature) difference. Even extreme temperature fluctuations\nare expected to cause an order ∼1 Å fluctuation in wavelength\nsolution.\nTo put into perspective how a 1 Å error would appear while\nredshifting, we show in Figure 8 the severity of measuring\na spectral feature up to ±1 Å off its true value for a source\nz= 0.1,fover the full spectral range of WiFeS. The error is\nmore severe at the blue end, but we generally use features in\nthe red for redshifting. Over the entire CCD, the average\ntemperature-induced shift is less than 0.5 Å which we account\nfor with frequent calibration, so the expected temperature-\ninduced redshift error is much less than the maximum ∼0.5\nÅ or redshift of ∼6×10–5from Figure 8. Below, we show\nthat when we redshift well-known objects/features, we indeed\nsee a smaller average error.\n4.2 Skylines\nSkylines are strong emission lines, mostly in infrared, that come\nfrom our own atmosphere and need to be removed from our\nspectra. However, since they originate on Earth, we can use\nthem as a test of wavelength solution in addition to arc lamps.\nFor every science target, we ‘redshifted’ the sky spectrum\n(which is coincident with but separate from the science spectra\ndue to the Nod&Shuffle mode of operation) in the centre and\ncorners of the aperture to test how the wavelength solution\nvaried across the field of view. For a small subset of the entire\nfWhile we set the source redshift to 0.1, there is no dependency on the\nredshift for this additive offset to the wavelength solution. We could equally\nconsider a stretch to the wavelength solution, and this would act as an additional\nredshift.science sample (one observation from every night), we did\nthe same without any binning, i.e. we tested the wavelength\nsolution of every spaxel since the S/N of the individual skylines\nis always very high. To get the redshift of a sky spectrum,\nwe modified Marz to use a high S/N sky spectrum from the\nEuropean Southern Observatory’s skycorr tool (Noll et al.,\n2014) as a template. Figure 9 shows the average wavelength\nsolution across the aperture, grouped by resolution. The R=\n3000 gratings show more variation, but each only varies by\n< 5×10–5, and the central region is always accurate to within\n��2×10–5.\nFigure 10 shows the results of binning the central and outer\nregions separately for the high and low-resolution configura-\ntions. The R= 7000 gratings show little variability, especially\nin the centre. In every case, the mean offset is < 2 ×10–5,\nwhich corresponds to < 0.1 Å in our observable sky-line spec-\ntral region (much less than any grating’s resolution).\n4.3 Radial velocity standards\nAs a final test of the wavelength solution, we also observed\nradial velocity standard stars. The radial velocities of these\nstars are known precisely (Nidever et al., 2002), so we can\ncompare wavelength solution in a similar way to the known\nzero-redshift wavelength of skylines. Figure 11 shows the red-\nshift difference of each observation of a radial velocity standard\n(some observations are of the same star on different nights),\nsplit by resolution mode. As expected, R= 7000 is much more\nprecise, with an undetectable redshift bias, whereas R= 3000\nhas an offset of 5 ×10–5. Both sets have < 10 observations so it\nis hard to conclude if there is any meaningful correction that\nneeds to be made to the redshifts we obtain for other targets.\nBoth sets are also skewed by a large outlier. When we inves-\ntigated the outlier, we found that the arc lamp frame used to\ncalibrate the spectrum notably influenced the redshift and that\nthe observation ∼0.5 hours after astronomical twilight differed\nin measured redshift by 7 ×10–5from the observation ∼2.5\nhours after twilight. These two effects are seemingly unrelated,\nhowever, so this is an interesting example of how observing\nconditions may have unaccounted effects on redshifts.\nAll the radial velocity standards we chose were spectral\ntype G or K, so we also tested how the stellar template used\nwhile redshifting affected the redshift, regardless of the actual\nspectral type. Thus we redshifted each star with both a G and\na K-star template in Marz. Figure 12 shows the difference in\nredshift offset when each template was used. Note that we now\nmake no distinction between resolution, and the G template\nhistogram is exactly the combined distribution of Figure 11.\nInterestingly, the K-star template resulted in moderately biased\nredshifts, by +7 ×10–5, even if the star was itself K-type.\nNaïvely, the best solution should be to redshift a star with\nthe same spectral type template. However, one would also ex-\npect these two templates, in particular, would agree since the\nabsorption features are similar for K and G. The difference be-\ntween the two templates in Marz is that the G template covers\na broader wavelength range than K and includes the calcium\ntriplet. Assuming the broader wavelength coverage of the G-Publications of the Astronomical Society of Australia 9\nFigure 9. Wavelength solution (effectively redshift) calibration across the aperture per grating, found by means of ‘redshifting’ the z= 0sky spectrum, with no\nspatial binning.\nFigure 10. Wavelength solution/redshift calibration across the aperture with binning shown in the upper right of each panel, from redshifting the z= 0sky\nspectrum. The left panel shows R= 3000 and the right panel shows R= 7000 . These curves have been smoothed over the ∼200 sky-spectrum observations\nfor each individual aperture region. The double-peaked nature of the R= 3000 curve for the outer regions comes from the fact that the upper left corner is\ngenerally blueshifted, and the upper right and lower left redshifted from the centre (see Figure 9). This is also the case for R= 7000 but to a much lesser extent.\nFigure 11. Redshift offset from our observations compared to the published\nradial velocities of radial velocity standards. Again the higher resolution\nperforms better, but both suffer from a large positive outlier.\nFigure 12. Redshift offset of all radial velocity standards regardless of res-\nolution, but redshifted using different stellar templates in Marz. There is a\ndisparity between the average redshift as measured by each template spec-\ntral type.10 Anthony Carr et al.\ntype stellar template is the main reason for the disagreement,\nwe opted to redshift all the radial velocity standards with this\ntemplate. In addition, our redshifts from the G-star template\nshow better agreement with the published radial velocities in\nevery case.\nApart from this moderate disagreement between stellar\ntemplates, none of our investigations into the accuracy of our\nwavelength solution revealed any need to further calibrate our\nredshifts. The stellar template problem is interesting and may\nrequire further investigation, although in the case of galaxies\nit remains important to redshift with the template that best\nmatches. A high S/N galaxy spectrum can be redshifted by an\nearly, intermediate or late-type template, but the redshift may\nshift up to a couple of 10–4depending on which main features\nthe target and template galaxy displays. As such, we always\nredshift galaxies with their matching template in Marz.\nFrom our investigations, we are assured our redshifts are\naccurate to within several 10–5.\n5. Results\n5.1 Redshift performance and comparison\nMost galaxies (161/185) are bright and large enough in the\nsky to obtain at least one redshift at different spatial locations\nwith which we can characterise the average redshift. Others\n(24/185), however, were too small, dim, or otherwise had too\nlittle S/N to obtain multiple redshifts, so instead the entire\ngalaxy was redshifted. The average redshift also reflects the\nsystemic redshift of the galaxy provided that the aperture was\ncentred on the galaxy. Since this is not always the case, we also\nredshift just the core section of the galaxy, where possible, to\ncompare to. This comparison is shown in Figure 13. The mean\noffset is –8.4 ×10–6, and the scatter is of the order of several\n10–5. The agreement on average is as good as we can expect\ngiven our investigations into the accuracy of the wavelength\nsolution, but the scatter is also affected by the S/N, whetherthe galaxy was centred in the aperture, and whether the galaxy\nwas early or late-type. The largest outlier is the host of SN\n2009Y, NGC 5728, which has very strongly double-peaked\nemission lines, even in the central region. In this case, the Ca\ntriplet is much preferred to redshift from but was not always\npresent with high S/N.\nWe present the results of our redshifting in Table A1 and\nFigure 13. We find a mean systematic offset of 4.3 ×10–5with\nnormalised median absolute deviation (NMAD) of 1.2 ×10–4\nbetween our redshifts and Pantheon+, which, as shown by\nCarr et al. (2022) is negligible for SN cosmology. However,\nsurprisingly, there are several redshift discrepancies above the\nlevel of 10–3, and we show the two largest in Figure 14. The\nfeatures in red are shown at the old redshift, while the template\nwe used to redshift is shown at our measured redshift. The\nPantheon+ redshift of the host of SN 2006kf comes from a\nsingle-peaked 21-cm Hydrogen emission feature (Springob\net al., 2005), while SN 2016hpx was redshifted as part of the\nFoundation Supernova Survey from the publicly available SN\nclassification spectrum that showed possible weak host galaxy\nHαemission (Foley et al., 2018; Dimitriadis et al., 2016). Nei-\nther of these historical redshift examples came from optical\nhost galaxy spectra.\nFor SN 2006kf, the redshift originally came from single-\npeaked 21 cm profile according to the NASA/IPAC Extra-\ngalactic Database.gA low S/N double horn distribution could\npotentially be mistaken for a single horn, and therefore could\nbias the redshift determination by the rotation of the galaxy.\nFor SN 2016hpx, of the two publicly available spectra\n(different reductions of the same spectrum) on the Weizmann\nInteractive Supernova Data Repositoryh(WISeREP), one does\nindeed show a faint peak in the wavelength region that would\nbe consistent with a host galaxy around z= 0.033; however,\ngSeen in the comment of the NED record, from Springob et al. (2005).\nhhttps://www.wiserep.org/object/9647\nFigure 13. Left : Self-consistency between the average redshift over all Voronoi spaxel bins in each galaxy and just the core section. Right : Difference in the\nredshift of each galaxy as averaged over all spaxel bins from WiFeS (this work) and Pantheon+, coloured by the standard deviation in redshift over each spaxel\nbin within each galaxy. The mean offset is 4.3 ×10–5, and insensitive to the outliers, which are interesting in themselves. The mean offset is similar in both\npanels, but the scatter in the self-consistency check in the left panel is nearly an order of magnitude smaller than the comparison with Pantheon+ redshifts in\nthe right panel.Publications of the Astronomical Society of Australia 11\nthe other does not. This could be a chance detection of a\nfaint emission line, but it is only a single, weak feature and\nthe spectra are low resolution and quality. The discrepancy\nwith our measurement would be consistent with the original\nredshift being mistaken as it is not a case of galactic rotation\nsince the host, LEDA 762493, is almost face-on and the SN\noccurred only 3′′from the core.\nIntriguingly, the magnitude of the offset with Pantheon+\nis almost exactly that of the average geocentric correction of\n11.7 km s–1(z= 3.9×10–5) we apply to our redshifts. Since\nthe offset is so small, the most likely reason we see it is just due\nto chance. Perhaps it could imply that the historical redshifts\nfrom Pantheon+ did not have a geocentric correction applied,\nbut this is difficult to test, as it requires the exact observation\nlocation and time. In any case, the individual large redshift\ndiscrepancies are potentially more interesting than this small\nsystematic offset.\n5.2 Redshift uncertainty\nSince in most cases we have many spectra for the same galaxy\nand many features in those spectra, we can estimate the varia-\ntion in redshift caused by noise as a measure of precision. We\ncan do this in two ways: the first is to measure the variation\nof wavelength determination of individual features, and the\nsecond is to measure the variation between multiple features.\nBoth can be achieved by redshifting many realisations of each\nspectrum with the flux of each pixel shifted by a Gaussian\nwhose standard deviation is the measured noise of that pixel.\nThe method that utilises multiple features is more appropriate\nfor characterising redshift uncertainty as redshifting from a\nsingle feature is very rarely trustworthy (unless it is particu-\nlarly high S/N and/or has resolved substructure), but here we\nalready have a tight prior on the redshift, and we are mainly\ninterested in its variation rather than its value. Instead of run-\nning each realisation of each spectrum through Marz, we opt\nto fit the features using Gaussians and convert the central wave-\nlength to a redshift. This method is highly scaleable (no user\ninterface and low computation time); however, it may still be\ninteresting to compare the robust correlation method to the\nsimplified Gaussian fitting. Note that we cannot simply use\nthe width of the correlation peak given by Marz to estimate\nredshift uncertainty as it is at least an order of magnitude larger\nthan our actual precision.\nEach spectrum was assigned tags for which features were\npresent with enough S/N to be able to at least somewhat re-\nliably fit Gaussians (both absorption and emission). 500 real-\nisations of each spectrum were generated, and a ±30 Å rest\nwavelength window containing each feature was extracted,\nusing the outer edges to estimate and subtract the continuum.\nA Gaussian was then fit to each feature; the fit was rejected if\nit was more than 10 Å from the known wavelength, if it was\nunreasonably wide or low amplitude, (accounting for the fact\nabsorption lines are generally wider and shallower than emis-\nsion), and if the amplitude was positive (negative) for emission\n(absorption) lines, all of which indicate a failure to capture the\nfeature of interest. A five Å rest wavelength window aroundthe known wavelength was also used to estimate the S/N of\neach feature from the original spectrum.\nThe features we applied this process to were: H α, Hβ\n(both of which can be in both emission and absorption), H γ,\nNa, Mg, the second line of each of O[II], O[III], N[II], S[II],\nH+K absorption, and the Calcium triplet. The NMAD of the\nfitted wavelengths measures how much the redshift can shift\nwithin the bounds of the measured noise, and when plotted\nagainst S/N shows a strong trend of increasing precision with\nincreasing S/N. The mean of all of the fitted wavelengths when\ncompared with the known redshift measures how much the\nredshift can shift according to which features are present or\nmost prominent. This in particular should be a more accurate\nreflection of the total redshift precision. Given the type of\ngalaxy/features and S/N, an estimate of redshift precision can\nbe made. Ideally this would be done on an individual redshift\nbasis, but we save a more thorough treatment for future work.\nFigure 15 shows examples of the methods described above\nand Table 2 shows the numerical results. Figure 15a shows the\n‘intra-line’ variation of the H αemission and Na absorption\nfeatures against their S/N estimates, while Figure 15b shows\nan example of the average Gaussian fit to the 500 realisations of\nthe Calcium triplet of one spaxel bin of NGC 2962 (host of SN\n1995D). The S/N is just an estimate because the five Å window\nused to estimate it is too wide for some emission lines and too\nnarrow for some absorption lines. Occasionally, the noise is\noverestimated and/or the flux is underestimated (e.g. the cases\nof reduction failures), which is why we see S/N < 1 but solid\nfeature detection. Finally, emission lines are naturally higher\nS/N than absorption lines, so 1:1 comparisons between the two\ncan be misleading. In general though, especially at high S/N,\nwhich was the aim of this program, we see excellent precision.\nHowever, some features do not show a trend with S/N (such\nas the hydrogen absorption lines and Calcium H+K), which\nmay be due to their presence at lower S/N in general and/or a\nGaussian fit being less appropriate.\nFigure 15c shows the offset from redshifting a single feature\ncompared to the Marz redshift. These measurements are found\nfrom taking the mean of the mean offsets for each feature and\nspaxel bin compared to their Marz measured redshift. The\nemission lines are generally in much closer agreement with the\nfinal redshift measurement; the reason Mg and Na in particular\nare not in agreement is due to their complex line-profiles.\nSince these lines are (sometimes significantly) asymmetrical\nand deeper in the blue end than the red, the fitted wavelength\nis biased blue. While the Gaussian fit for these lines is biased,\nthe intra-line variation should be robust to the exact location\nof the centre of the Gaussian approximation.\nIn conclusion of this investigation, the high S/N emission\nline galaxy redshifts are precise to better than several 10–5,\nthe high S/N absorption galaxies and low S/N emission line\ngalaxies are precise to better than approximately 1 ×10–4, and\nthe low S/N absorption galaxies are not generally present.12 Anthony Carr et al.\nFigure 14. Comparison of WiFeS redshifts (new) to Pantheon+ redshifts (old) in the cases of the two largest discrepancies. Our redshifts are shown by our\nmeasured spectra (green), obtained at very high S/N from the core section of each galaxy, and by the grey templates, while the relevant feature locations at the\nPantheon+ redshift are shown with red dashed lines.\n(a)Examples of intra-line redshift variation, i.e. the NMAD of the central wave-\nlength of Gaussians fit to 500 Monte-Carlo realisations of each feature of each\nspectrum. The top panel shows each spaxel bin displaying H αemission and the\nbottom shows Na absorption.\n(b)Example of five realisations of the Calcium triplet (central absorption line only)\nin NGC 2962.\n(c)Variation in redshift between the mean of each feature measured individually\nand the corresponding spaxel bin redshift measured in the main analysis.\nFigure 15. a ) shows the variation in fits to the central wavelength of the individual features of H αemission and Na absorption. Each spectrum was realised 500\ntimes, varying the pixel flux by a Gaussian with a width of the measured noise, then Gaussians were fit to those features. Each point in these figures is the\nvariation in those centres, converted to redshift, and each galaxy is a different colour. The mean and median of these are shown in the red dashed and magenta\ndotted lines respectively, and overplotted in black are the binned data. b) shows five of the 500 realisations in solid lines of a particular spaxel bin of the galaxy\nNGC 2962. The red dashed line shows the normalised sum of all 500 fitted Gaussians. The green window is a five Å rest wavelength window about the canonical\nredshift used to estimate the S/N of the feature. c) shows the variation caused by measuring single features when compared to the redshift measured from\nMarz, with emission (green) and absorption (orange) differentiated (H αand Hβoccur in both, sometimes within different regions of the same galaxy). The\ndotted lines are simply to aid in identifying each feature.Publications of the Astronomical Society of Australia 13\nTable 2. Summary of spectral feature fitting precision, converted to redshift.\nRest wavelengths are taken from Marz and converted from vacuum to air,\nexcept for the Calcium triplet which comes from the Vienna Atomic Line\nDatabase.\nFeature Rest wavel. (Å) Mean NMAD Med. NMAD Diff. from zhel\nO[II] 3727.425 4.2×10–44.2×10–49.5×10–6\nH+K 3968.468 6.0×10–45.9×10–4–4.3×10–5\nHγ 4340.469 2.0×10–41.5×10–4–3.9×10–5\nHβ(em.) 4861.325 1.2×10–47.4×10–5–3.2×10–5\nHβ(abs.) 4861.325 5.3×10–45.0×10–49.7×10–5\nO[III] 5006.843 1.5×10–41.1×10–4–1.0×10–5\nMg 5175.3 5.6×10–45.6×10–4–2.8×10–4\nNa 5894.0 3.0×10–42.7×10–4–1.9×10–4\nHα(em.) 6562.80 4.5×10–51.9×10–5–3.4×10–5\nHα(abs.) 6562.80 3.3×10–43.0×10–42.9×10–6\nN[II] 6583.408 7.9×10–55.2×10–5–2.1×10–5\nS[II] 6730.849 1.5×10–41.2×10–4–3.6×10–5\nCa triplet 8542.088 2.0×10–41.8×10–4–4.0×10–5\n5.3 Cosmological Results\nTo quantify the effect of our redshift updates on cosmology,\nwe use the entire SH0ES/Pantheon+ cosmology samplei(pho-\ntometry and Cepheid calibration) and the PIPPINjend-to-end\nSN cosmology analysis pipeline (Hinton & Brout, 2020). This\nmethod allows us to calculate SH0ES/Pantheon+ distance mod-\nuli using the redshifts of this work as well as take advantage of\nthe statistical+systematic covariance matrix Cfor both the orig-\ninal and updated redshift sample. For each of the redshifts we\nremeasure, we transform to the CMB frame then recalculate\npeculiar velocity using pvhubkto correct to the cosmological\nframe ( zHD). The average change in peculiar velocity was\nzero, but individually they varied by up to ±80 km s–1Mpc–1.l\nWith our new set of cosmological redshifts zHD, Pan-\ntheon+ light-curves and SH0ES calibrations, we perform a\nsimultaneous fit for H0andΩmin a flat ΛCDM Universe\n(i.e.ΩΛ= 1 –Ωm,w= –1) by minimising distance modulus\nresiduals defined by\n∆µi=µobs,i–µmodel (zHD,i,H0,Ωm), (1)\nwhere ispans the set of Pantheon+ light-curves. Briefly, PIP-\nPINmakes use of SNANA (Kessler et al., 2009) to take input\nphotometry, redshifts, etc. and calculate µobs,ifrom a modified\nTripp equation of the form\nµobs=m+αx1–βc–M–δbias+δhost, (2)\nwhere light-curve peak magnitude m(equivalent to B-band\nmagnitude), stretch x1and colour care fit with an updated\nSALT2 model (Guy et al., 2010; Brout et al., 2022b), αandβ\nare nuisance parameters, Mis the absolute magnitude of SNe\nihttps://github.com/PantheonPlusSH0ES/DataRelease\njhttps://github.com/dessn/Pippin\nkhttps://github.com/KSaid-1/pvhub\nlThe peculiar velocity field used in pvhub is a discretised grid in redshift-\nspace, and smoothed with a Gaussian kernel of width 4 h–1Mpc. The peculiar\nvelocities only change for the galaxies that are close enough to cell walls that\na small redshift change causes them to fall into a new cell.Ia,δbiasare observational bias corrections estimated from simu-\nlations, and δhostaccounts for the residual correlation between\nSN Ia brightness and host mass. For more details, see Hinton\n& Brout (2020); Brout et al. (2022a). The theoretical distance\nmoduli µmodel are calculated directly from the cosmological\nmodel as\nµmodel (z) = 5 log10(DL(z)) + 25, (3)\nwith luminosity distance DLin Mpc, calculated from\nDL(z) = (1 + z)c\nH0Zz\n0dz′\np\nΩm(1 +z′)3+ 1 –Ωm. (4)\nFor Cepheid calibrated galaxies, µmodel (zi) is replaced with\nµCepheid\ni. With the vector of residuals ∆µ, the best-fit cosmol-\nogy comes from minimising the function\nχ2=∆µTC–1∆µ. (5)\nOf the 185 SN host galaxies we measured, 146 galaxies and\n215 light-curves passed the quality cuts to be used in the fit.\nThis determination of H0is equivalent to fitting the inter-\ncept of the linear distance-redshift relation, as done by SH0ES\n(Riess et al., 2022). The intercept, aB, is constrained by galax-\nies whose motion is dominated by expansion. It is standard\npractice to use the (third-order) cosmographic expansion of\nrecession velocity, which is almost exact in the ‘Hubble Flow’\nredshift range used for fitting H0(0.0233 < zHD< 0.15):\nlog10H0≈log10hcz\n6\u0010\n6 + 3 (1 –q0)z+\u0010\n1 –q0– 3q2\n0+j0\u0011\nz2\u0011i\n+1\n5(–mB+MB+ 25)\n≡aB+ 0.2MB+ 5. (6)\nThe expansion includes the cosmic deceleration parameter\n(q0= –0.55) and jerk ( j0= 1.0), whose values are chosen\nto match the standard ΛCDM model with ( Ωm,ΩΛ,w) =\n(0.3, 0.7, –1). As such, this fitting method has weak cosmo-\nlogical model dependence. Since the dependence is weak,\nthis method can still be used to constrain cosmologies whose\nparameters are somewhat near the input parameters.\nAn input cosmology also enters into our analysis in the\nsimulations used to perform bias corrections, and this depen-\ndence is also weak (Camilleri et al. in prep.). The results of\nour fit are shown in Figure 16; the Hubble diagram of the up-\ndated sample has a best-fit cosmology ( H0,Ωm) = (72.6 ±1.2\nkm s–1Mpc–1, 0.328+0.018\n–0.017), and the weighted average differ-\nence in distance moduli is only –5 ×10–4mag. The error\nbars we show are the statistical and systematic uncertainties\nfrom the covariance matrix, added in quadrature. It should be\nnoted that even when including the 250 km s–1uncertainty\nin peculiar velocities ( ∼8×10–4when converted to redshift),\nredshift uncertainties propagated through to H0are completely\nsubdominant to these distance modulus uncertainties.\nSince we are using magnitudes calibrated with SH0ES, we\nsee a similar central values of H0to SH0ES (73.04 km s–1Mpc–1;\nRiess et al., 2022), but only the shift in H0from our redshift14 Anthony Carr et al.\nFigure 16. Left : full Hubble diagram showing the SNe used to constrain and the residuals of the best-fit flat ΛCDM cosmology (H0,Ωm) = (72.6\nkm s–1Mpc–1, 0.328) .Right : change in individual µ, normalised to the best-fit cosmology of the updated redshift sample. Galaxies with zHD< 0.0233\nare greyed out because they are not used to determine H0. The change from original (green) to updated (orange) in magnitude space is often too small to see.\nThe error bars, shown only for the updated sample, come from the covariance matrix statistical and systematic uncertainty added in quadrature (for display\npurposes only). The weighted average shift in µof–5×10–4mag from the original sample corresponds to a shift in H0of+0.1 km s–1Mpc–1.\nchanges is important here. Thus, the key takeaway from this\nwork is that the shift in H0from the original sample purely\ndue to our redshift updates is only 0.1 km s–1Mpc–1, which\nis negligible compared to the 1.2 km s–1Mpc–1uncertainty\nof each fit. We can also calculate an individual H0directly\nfor each of the host galaxies in our sample from Equation (4)\nwithΩm= 0.3, replacing the left-hand-side with DL(µobs),\nand taking the weighted average before and after the redshift\nupdates gives the same result of 0.1 km s–1Mpc–1. This is not\nunexpected from the magnitude of the changes to redshift,\nand a similar result was found in Carr et al. (2022) although\nopposite in sign. The average redshift correction was an order\nof magnitude larger in Carr et al. (2022) in the same direction,\nso the result of this work is likely just a statistical fluctuation.\nOur result reinforces the general conclusion that while redshift\nerrors have the potential to bias H0, the reality is that any\nrealistic redshift errors are too small to affect H0.\n6. Conclusion\nWe have shown with our new observations that indeed there\nexist errors in the previous measurements of the redshifts of\nnearby SN Ia host galaxies. The differences had a negligible\nsystematic offset, which was reflected in the negligible change\ntoH0.\nWe have shown our redshifts to be accurate to within a\ncouple of 10–5. However, there are several extensions to our\nanalysis that can be made to further investigate the sources of\nthe redshift errors in the interest of mitigating redshift errors\nin the future. Firstly, while we saw good agreement between\naveraging the redshift over all spaxels and just the core sections,\nit would be beneficial to investigate why we still see scatter\nat the level of about 3 ×10–5. In addition, the rest of the cases\nof very large redshift discrepancies (rather than just the two\nlargest) can be examined by comparing with original spectra\nwhere possible. Finally, for many galaxies (about half of our\nsample), we will be able to redshift the region around wherethe SN actually occurred within the galaxy. This would be a\nuseful crosscheck with previous redshifts, as there may be a\ncorrelation between those redshifts and the historical redshifts\nof Pantheon+ in the cases where a slit spectrograph was aligned\nwith the SN but not the core, or a fibre pointing inaccuracy.\nIt is also useful for observing the local SN environment and\nexamining SN Ia brightness and host/environment correla-\ntions, as there is a wealth of information beyond the redshifts\nto explore.\nIn this work, we study the overall accuracy and precision\nof our redshifts, but we have not characterised individual pre-\ncisions for our redshifts, which would be a function of the\nspectrum S/N and number of spaxel bins for a galaxy. Instead\nwe give a general uncertainty class depending on the S/N and\ngalaxy type/prominent features. This can be taken further to\nprovide individual estimates, which would mostly be useful\nto differentiate the lower quality observations, as our high\nS/N data are extremely accurate and precise for an optical\nspectrograph without simultaneous wavelength solution mea-\nsurement (from, e.g., frequency combs). For the purposes of\nmeasuring H0, our redshift data can essentially be taken as\nconstants, at least until the peculiar velocity and distance mod-\nulus uncertainty floor is drastically improved. This has already\nbeen done historically for convenience, but for our sample this\nis a valid assumption; however, it does not necessarily hold for\nother purposes which may be even more sensitive to redshift\nerrors.\nWhile we have confirmed that remeasuring accurate red-\nshifts does not have an effect on H0, we stress it is still important\nto use accurate and precise redshifts for cosmology, especially\nas we gather more and more spectroscopic data. Importantly,\nat least at high-redshift, we will have to start using photomet-\nric redshifts and SN classifications, as spectroscopic follow-up\nbecomes unfeasible from the expected volume of data from\nfuture surveys, such as the Legacy Survey of Space and Time,\nwhich will add new forms of systematic error to investigate.Publications of the Astronomical Society of Australia 15\nData Availability\nThe spectroscopic data obtained as part of this work are avail-\nable from the corresponding author upon reasonable request.\nAcknowledgement\nThe authors thank S. Sweet for useful discussions and advice\non observation optimisation. The authors thank C. Howlett,\nJ. Calcino and K. Said for assistance in carrying out observa-\ntions and M. Craigie for assistance in target selection.\nTMD is the recipient of an Australian Research Council\nAustralian Laureate Fellowship (project number FL180100168)\nfunded by the Australian Government. DS is supported by\nDOE grant DE-SC0010007, DE-SC0021962 and the David\nand Lucile Packard Foundation. DS is supported in part by the\nNational Aeronautics and Space Administration (NASA) un-\nder Contract No. NNG17PX03C issued through the Roman\nScience Investigation Teams Programme.\nThis work is based on data acquired at Siding Spring Obser-\nvatory with the ANU 2.3m Telescope via programs 1200040,\n2200080, 3200100 and 4200059. We acknowledge the tradi-\ntional custodians of the land on which the telescope stands,\nthe Gamilaraay people, and pay our respects to elders past\nand present. This research was also supported by resources\nprovided by the University of Chicago Research Computing\nCenter and used services provided by the Astro Data Lab at the\nUS National Science Foundation’s National Optical-Infrared\nAstronomy Research Laboratory. NOIRLab is operated by\nthe Association of Universities for Research in Astronomy\n(AURA), Inc. under a cooperative agreement with the Na-\ntional Science Foundation. This work has also made use of the\nV ALD database, operated at Uppsala University, the Institute\nof Astronomy RAS in Moscow, and the University of Vienna.\nReferences\nBrout, D., Scolnic, D., Popovic, B., et al. 2022a, ApJ, 938, 110\nBrout, D., Taylor, G., Scolnic, D., et al. 2022b, ApJ, 938, 111\nCappellari, M., & Copin, Y. 2003, MNRAS, 342, 345\nCarr, A., Davis, T. M., Scolnic, D., et al. 2022, PASA, 39, e046\nChildress, M. J., V ogt, F. P. A., Nielsen, J., & Sharp, R. G. 2014, Ap&SS, 349,\n617\nChildress, M. J., Tucker, B. E., Yuan, F., et al. 2016, PASA, 33, e055\nDavis, T. M., Hinton, S. 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M., et al. 2022, ApJ, 934, L7Sako, M., Bassett, B., Becker, A. C., et al. 2018, PASP, 130, 064002\nScolnic, D. M., Jones, D. O., Rest, A., et al. 2018, ApJ, 859, 101\nSpringob, C. M., Haynes, M. P., Giovanelli, R., & Kent, B. R. 2005, ApJS,\n160, 149\nYuan, W., Macri, L. M., Riess, A. G., et al. 2022, ApJ, 940, 6416 Anthony Carr et al.\nAppendix 1. Supplementary Data Tables\nTable A1. Redshift results. For each Type Ia SN we targeted, its Pantheon (source catalogue) ID is listed along with its host galaxy ID if it has one. Also listed\nis the redshift from the main analysis ( zWiFeS\nhel) and the number of spaxels it was measured from ( Nz), along with the redshift from the central region only\n(zWiFeS\nhel,centre). The corresponding redshift from Pantheon+ ( zPantheon+\nhel) is shown along with its difference from zWiFeS\nhel.\nSN Host zWiFeS\nhelNz zWiFeS\nhel,centrezPantheon+\nhelzWiFeS\nhel–zPantheon+\nhel\n1993ae IC 0126 0.019 776 15 0.019 780 0.019 667 1.1 ×10−4\n1994M NGC 4493 0.023 161 23 0.023 160 0.023 083 7.8 ×10−5\n1995ak IC 1844 0.022 815 71 0.022 730 0.022 699 1.2 ×10−4\n1995D NGC 2962 0.006 558 23 0.006 550 0.006 561 −3.2×10−6\n1996Z NGC 2935 0.007 548 72 0.007 540 0.007 565 −1.7×10−5\n1998es NGC 0632 0.010 633 23 0.010 620 0.010 571 6.2 ×10−5\n1999ef UGC 00607 0.038 941 16 0.038 930 0.038 857 8.4 ×10−5\n1999gh NGC 2986 0.007 743 37 0.007 710 0.007 705 3.8 ×10−5\n2001da NGC 7780 0.017 335 15 0.017 330 0.017 381 −4.6×10−5\n2001ic NGC 7503 0.044 123 21 0.044 130 0.044 089 3.4 ×10−5\n2002ck UGC 10030 0.029 827 29 0.029 815 0.029 742 5.8 ×10−5\n2002cr NGC 5468 0.009 452 32 0.009 470 0.009 417 3.5 ×10−5\n2002dj NGC 5018 0.009 375 35 0.009 400 0.009 37 5.1 ×10−6\n2002fk NGC 1309 0.007 185 79 0.007 180 0.007 185 −4.4×10−7\n2002ha NGC 6962 0.014 109 67 0.014 035 0.014 07 6.6 ×10−5\n2003ch UGC 03787 0.028 702 17 0.028 890 0.028 62 8.2 ×10−5\n2003ic MCG -02-02-086 0.055 435 31 0.055 340 0.055 359 7.6 ×10−5\n2003iv UGC 02320 NOTES01 0.034 504 22 0.034 520 0.034 26 2.4 ×10−4\n2004dt NGC 0799 0.019 505 19 0.019 550 0.019 418 8.7 ×10−5\n2004eo NGC 6928 0.015 791 111 0.015 795 0.015 464 3.0 ×10−4\n2004ey UGC 11816 0.015 832 13 0.015 750 0.015 834 −1.7×10−6\n2004gc ARP 327 NED04 0.031 471 16 0.031 490 0.031 92 −4.5×10−4\n2005al NGC 5304 0.012 454 12 0.012 480 0.0124 5.4 ×10−5\n2005am NGC 2811 0.007 095 162 0.007 105 0.007 899 −8.1×10−4\n2005bo NGC 4708 0.013 902 31 0.013 910 0.013 896 5.9 ×10−6\n2005cf MCG -01-39-003 0.006 651 54 0.006 660 0.006 43 2.2 ×10−4\n2005el NGC 1819 0.014 835 64 0.014 840 0.014 83 4.5 ×10−6\n2005eq MCG -01-09-006 0.029 096 25 0.029 130 0.028 952 1.4 ×10−4\n2005ff WISEA J223041.16-004634.2 0.089 810 1 0.089 690 0.089 79 2.0 ×10−5\n2005fn SDSS J204853.04+001129.8 0.095 310 1 0.095 270 0.0951 2.1 ×10−4\n2005hj WISEA J012648.45-011417.0 0.057 517 3 0.057 470 0.057 385 1.3 ×10−4\n2005iq ESO 538- G 013 0.034 126 15 0.034 110 0.034 043 8.3 ×10−5\n2005kc NGC 7311 0.015 125 35 0.015 090 0.015 09 3.5 ×10−5\n2005ki NGC 3332 0.019 584 20 0.019 560 0.019 458 1.3 ×10−4\n2005ktaWISEA J011058.06+001634.1 0.065 360 2 0.065 380 0.065 404 −4.4×10−5\n2005ku WISEA J225942.70-000048.3 0.045 328 22 0.045 440 0.045 248 8.0 ×10−5\n2005lk 2MFGC 16592 0.104 400 1 0.104 220 0.104 161 2.4 ×10−4\n2005lu ESO 545- G 038 0.032 091 22 0.032 080 0.032 189 −9.8×10−5\n2006al WISEA J103928.52+051101.2 0.067 770 1 0.067 760 0.067 802 −3.2×10−5\n2006ax NGC 3663 0.016 722 29 0.016 750 0.016 495 2.3 ×10−4\n2006bh NGC 7329 0.010 894 27 0.010 900 0.010 767 1.3 ×10−4\n2006D MRK 1337 0.008 491 169 0.008 433 0.008 53 −4.6×10−6\n2006ef NGC 0809 0.017 959 14 0.017 940 0.017 812 1.5 ×10−4\n2006ej NGC 0191A 0.020 439 18 0.020 360 0.020 38 5.9 ×10−5\n2006eq 2MASX J21283758+0113490 0.049 472 12 0.049 520 0.049 408 2.6 ×10−5\n2006et NGC 0232 0.022 764 25 0.022 640 0.022 639 1.2 ×10−4\n2006fd 2MASX J20375343+0113100 0.079 965 21 0.080 035 0.079 948 5.3 ×10−5\n2006fy WISEA J232640.11-005025.9 0.082 766 5 0.082 730 0.082 734 3.2 ×10−5\n2006gt WISEA J005618.02-013730.9 0.044 810 4 0.044 710 0.044 799 1.1 ×10−5\n2006hb ESO 552- G 052 0.015 141 43 0.015 140 0.014 957 2.1 ×10−4\n2006hx 2MASX J01135716+0022171 0.045 520 6 0.045 480 0.045 389 1.3 ×10−4\n2006is WISEA J051734.55-234659.7 0.031 380 1 0.031 320 0.0314 −2.0×10−5\n2006kf UGC 02829 0.021 533 25 0.021 540 0.020 037 1.5 ×10−3\n2006lu WISEA J091517.24-253600.6 0.053 303 26 0.053 270 0.0534 −9.7×10−5\n2006oa WISEA J212342.91-005034.7 0.062 510 2 0.062 505 0.062 573 −3.3×10−5\n2006on WISEA J215558.50-010412.9 0.071 820 1 0.071 820 0.071 915 −9.5×10−5\n2006py WISEA J224142.06-000812.7 0.057 855 2 0.057 930 0.057 866 1.0 ×10−4\n2007af NGC 5584 0.005 482 29 0.005 500 0.005 524 −4.2×10−5\n2007al WISEA J095918.72-192823.2 0.012 218 19 0.012 220 0.012 175 4.3 ×10−5\n2007as ESO 018- G 018 0.017 098 43 0.017 140 0.017 572 −4.7×10−4\n2007bd UGC 04455 0.031 035 25 0.031 100 0.030 44 6.0 ×10−4\n2007ca MCG -02-34-061 0.014 104 32 0.014 100 0.014 066 3.8 ×10−5Publications of the Astronomical Society of Australia 17\nRedshift results. (continued)\nSN Host zWiFeS\nhelNz zWiFeS\nhel,centrezPantheon+\nhelzWiFeS\nhel–zPantheon+\nhel\n2007cb ESO 510- G 031 0.036 520 1 0.036 570 0.036 592 −7.2×10−5\n2007cc ESO 578- G 026 0.029 051 14 0.029 040 0.029 125 −7.4×10−5\n2007cq WISEA J221440.71+050442.3 0.025 927 26 0.025 840 0.026 04 −1.1×10−4\n2007fb UGC 12859 0.017 990 23 0.017 920 0.018 026 −3.6×10−5\n2007ht 2MASX J00343398-0112577 0.072 853 3 0.073 040 0.072 753 1.0 ×10−4\n2007jh CGCG 391-014 0.040 891 16 0.040 850 0.040 744 1.5 ×10−4\n2007ks SDSS J204933.00-004543.0 0.096 845 2 0.096 855 0.098 −1.1×10−3\n2007le NGC 7721 0.006 756 49 0.006 750 0.006 721 3.5 ×10−5\n2007nq UGC 00595 0.045 223 32 0.045 150 0.045 21 3.1 ×10−5\n2007om WISEA J235420.72-005501.0 0.105 160 1 0.105 330 0.104 84 3.2 ×10−4\n2007on NGC 1404 0.006 451 76 0.006 460 0.006 248 2.0 ×10−4\n2007pu SDSS J224558.32-003855.9 0.091 350 1 0.091 360 0.0914 −5.0×10−5\n2007ra WISEA J233424.11-005324.7 0.089 158 6 0.089 290 0.089 163 −4.7×10−6\n2007sr NGC 4038 0.005 550 96 0.005 640 0.005 417 1.3 ×10−4\n2007st NGC 0692 0.021 252 31 0.021 250 0.021 181 7.1 ×10−5\n2008051 GALEXASC J151958.89+045417.3 0.037 960 1 0.038 010 0.037 77 1.9 ×10−4\n2008ar IC 3284 0.026 252 14 0.026 230 0.026 173 7.9 ×10−5\n2008bc KK 1524 0.014 828 40 0.014 825 0.015 087 −2.7×10−4\n2008bi NGC 2618 0.013 532 33 0.013 630 0.013 456 7.6 ×10−5\n2008bq ESO 308- G 025 0.034 448 22 0.034 440 0.034 007 4.4 ×10−4\n2008cc ESO 107- G 004 0.010 475 59 0.010 465 0.010 304 2.0 ×10−4\n2008cf LEDA 766647 0.047 290 1 0.047 250 0.046 03 1.3 ×10−3\n2008ff ESO 284- G 032 0.019 165 2 0.019 170 0.019 249 −7.9×10−5\n2008fl NGC 6805 0.020 231 41 0.020 210 0.019 88 3.8 ×10−4\n2008fr LEDA 5069093 0.039 500 1 0.039 480 0.039 5.0 ×10−4\n2008fu ESO 480-IG 021 0.052 121 29 0.052 280 0.052 016 1.0 ×10−4\n2008fw NGC 3261 0.008 522 79 0.008 600 0.008 379 1.4 ×10−4\n2008gg NGC 0539 0.032 091 18 0.032 070 0.032 025 6.6 ×10−5\n2008gl UGC 00881 0.034 226 15 0.034 220 0.0342 2.6 ×10−5\n2008go WISEA J221043.94-204725.9 0.062 215 11 0.062 125 0.062 273 −1.7×10−4\n2008gp MCG +00-09-074 0.033 144 17 0.033 170 0.0335 −3.6×10−4\n2008hj MCG -02-01-014 0.037 609 41 0.037 680 0.037 613 −3.7×10−6\n2008hu ESO 561- G 018 0.049 940 11 0.050 040 0.049 698 2.4 ×10−4\n2008hv NGC 2765 0.012 743 107 0.012 755 0.012 549 1.8 ×10−4\n2008ia ESO 125- G 006 0.022 054 85 0.022 020 0.021 942 1.0 ×10−4\n2008Q NGC 0524 0.008 129 66 0.008 150 0.008 016 1.1 ×10−4\n2008R NGC 1200 0.013 494 42 0.013 490 0.013 296 2.0 ×10−4\n2009aa ESO 570- G 020 0.027 383 25 0.027 380 0.027 052 3.3 ×10−4\n2009ab UGC 02998 0.011 102 25 0.011 120 0.011 178 −7.6×10−5\n2009ad UGC 03236 0.028 356 21 0.028 400 0.0284 −4.4×10−5\n2009ag ESO 492- G 002 0.008 731 67 0.008 740 0.008 686 4.5 ×10−5\n2009al NGC 3388 0.022 063 17 0.022 090 0.022 069 −6.1×10−6\n2009ds NGC 3905 0.019 188 19 0.019 060 0.019 09 9.8 ×10−5\n2009D ESO 549- G 031 0.025 100 42 0.025 250 0.025 097 2.5 ×10−6\n2009ig NGC 1015 0.008 825 33 0.008 820 0.008 77 5.5 ×10−5\n2009kk 2MFGC 03182 0.012 505 17 0.012 620 0.012 859 −3.5×10−4\n2009le ESO 478- G 006 0.017 855 55 0.017 870 0.018 149 −2.9×10−4\n2009Y NGC 5728 0.009 486 87 0.009 635 0.009 743 −1.2×10−4\n2010A UGC 02019 0.020 815 54 0.020 860 0.020 755 6.0 ×10−5\n2010H IC 0494 0.015 257 31 0.015 390 0.015 197 6.0 ×10−5\n2013go . . . 0.073 200 1 0.073 180 0.074 −8.0×10−4\n420100 WISEA J221225.27+005105.3 0.097 830 1 0.097 690 0.097 621 2.1 ×10−4\n530086 2MASX J22112814-0001456 0.051 690 1 0.051 950 0.052 003 −3.1×10−4\nASASSN-15bc LEDA 170061 0.036 928 18 0.036 850 0.036 715 2.1 ×10−4\nASASSN-15hg CGCG 063-098 0.030 056 19 0.029 980 0.029 917 1.4 ×10−4\nASASSN-15il 2MASX J15570808-1240252 0.023 388 19 0.023 350 0.023 316 7.2 ×10−5\nASASSN-15lg CGCG 044-042 0.020 127 32 0.020 180 0.020 151 −2.3×10−5\nASASSN-15nr CGCG 082-031 0.023 168 25 0.023 150 0.023 206 −3.8×10���5\nASASSN-15od MCG -01-07-004 0.017 637 55 0.017 580 0.017 603 3.4 ×10−5\nASASSN-15pr 2MASX J23063962-1234238 0.033 301 17 0.033 300 0.033 093 2.1 ×10−4\nASASSN-15ss MCG -02-16-004 0.035 624 27 0.035 700 0.035 558 6.6 ×10−5\nASASSN-15uw 2MASX J02353437-0603496 0.030 388 12 0.030 320 0.030 811 −4.2×10−4\nASASSN-16aj NGC 1562 0.030 620 22 0.030 630 0.030 745 −1.2×10−4\nASASSN-16ay UGC 03738 0.028 306 24 0.028 290 0.028 343 −3.7×10−5\nASASSN-16bc 2MASX J12052488-2123572 0.032 003 19 0.031 970 0.031 939 6.4 ×10−5\nASASSN-16bq IC 0986 0.024 988 25 0.024 990 0.024 935 5.3 ×10−5\nASASSN-16br 2MASX J15453055-1309057 0.028 661 26 0.028 700 0.028 52 1.4 ×10−418 Anthony Carr et al.\nRedshift results. (continued)\nSN Host zWiFeS\nhelNz zWiFeS\nhel,centrezPantheon+\nhelzWiFeS\nhel–zPantheon+\nhel\nASASSN-16ct SDSS J151354.30+044525.7 0.041 910 1 0.041 910 0.041 91 0.0\nASASSN-16dn GALEXASC J104848.62-201544.1 0.012 920 1 0.012 920 0.012 85 7.0 ×10−5\nASASSN-16dw 2MASX J13300119-2758297 0.034 638 14 0.034 610 0.034 657 −1.9×10−5\nASASSN-16fo 2MASX J13323577-0516218 0.029 234 9 0.029 250 0.0289 3.3 ×10−4\nASASSN-16hz 2MASX J23154564-0120135 0.025 443 17 0.025 410 0.025 308 1.3 ×10−4\nASASSN-16ip ESO 479- G 007 0.017 167 19 0.017 180 0.017 008 1.6 ×10−4\nASASSN-16jf UGCA 430 0.011 440 12 0.011 390 0.011 441 −1.0×10−6\nASASSN-16lg ARK 530 0.021 367 41 0.021 380 0.021 171 2.0 ×10−4\nASASSN-16oz GALEXASC J090013.19-133803.5 0.030 150 1 0.030 110 0.031 −8.5×10−4\nASASSN-17aj MCG -02-30-003 0.021 444 20 0.021 420 0.021 275 1.7 ×10−4\nASASSN-17co UGC 11128 0.018 357 35 0.018 290 0.018 259 9.8 ×10−5\nAT2016aj 2MASX J04422451-2143312 0.067 430 1 0.067 460 0.067 406 2.4 ×10−5\nAT2016htm 2MASX J02320134-2639576 0.043 226 12 0.043 360 0.043 313 −8.7×10−5\nAT2016htn 2MASX J02112819-1630409 0.053 106 17 0.053 110 0.053 117 −1.1×10−5\nAT2017cfc UGC 08783 0.023 840 1 0.023 820 0.024 027 −1.9×10−4\nAT2017lm 2MASX J03013238-1501028 0.030 449 21 0.030 440 0.030 636 −1.9×10−4\nAT2017ns 2MASX J02491020+1436036 0.029 373 11 0.029 370 0.028 766 6.1 ×10−4\nAT2017yk 2MASX J09443215-1218233 0.046 670 14 0.046 760 0.046 439 2.3 ×10−4\nAT2017zd 2MASX J13324217-2148034 0.029 486 14 0.029 650 0.029 47 1.6 ×10−5\nATLAS16dpb CGCG 415-040 0.022 781 22 0.022 790 0.023 083 −3.0×10−4\nATLAS16dqf WISEA J210907.40-180607.8 0.021 049 11 0.021 095 0.021 17 −1.5×10−4\nATLAS17ajn ESO 440- G 001 0.028 725 13 0.028 820 0.028 706 1.9 ×10−5\nATLAS17axb GALEXASC J134322.97-195637.5 0.031 580 1 0.031 630 0.031 652 −7.2×10−5\nGaia16agf . . . 0.025 285 13 0.025 320 0.025 066 2.2 ×10−4\nMASTERJ0134 GALEXASC J013415.00-174836.1 0.044 895 10 0.044 820 0.044 846 4.9 ×10−5\nMASTEROT J08 2MASX J08325728-0351295 0.030 521 18 0.030 570 0.030 584 −6.3×10−5\n080064 GALEXASC J032942.01-275237.5 0.066 290 1 0.066 050 0.066 129 1.6 ×10−4\n100405 . . . 0.103 453 11 0.103 480 0.1034 5.3 ×10−5\nPS15aii CGCG 071-025 0.046 510 23 0.046 500 0.046 549 −3.9×10−5\nPS15bif . . . 0.079 633 3 0.079 673 0.079 37 2.3 ×10−4\nPS15bjg 2MASX J22551005-0024333 0.068 987 8 0.069 020 0.068 888 9.9 ×10−5\nPS15brr GALEXASC J235326.18-153921.5 0.052 845 4 0.052 950 0.051 804 1.0 ×10−3\nPS15bsq MCG -02-60-012 0.034 330 63 0.034 420 0.034 304 −1.7×10−5\nPS15cku 2MASX J01242239+0335168 0.023 446 15 0.023 460 0.023 273 1.7 ×10−4\nPS15cms 2MASX J09583540+0044336 0.064 884 18 0.064 860 0.064 79 9.4 ×10−5\nPS15coh GALEXASC J021558.44+121415.2 0.019 027 4 0.019 040 0.018 837 1.9 ×10−4\nPS15cwx 2MFGC 04279 0.030 310 1 0.030 280 0.030 065 2.4 ×10−4\nPS15cze 2MASX J03472342+0052316 0.039 421 9 0.039 420 0.039 371 5.0 ×10−5\nPS16axi 2MASX J10480747+0010017 0.039 333 15 0.039 440 0.039 299 3.4 ×10−5\nPS16ayd 2MASX J14271887-0140428 0.054 097 7 0.053 910 0.053 997 1.0 ×10−4\nPS16bby 2MASX J14201699-2211186 0.053 550 24 0.053 570 0.053 427 1.2 ×10−4\nPS16bnz UGC 05586 NED02 0.062 893 31 0.062 860 0.0627 1.6 ×10−4\nPS16cqa 2MFGC 12594 0.043 944 30 0.044 060 0.043 857 7.8 ×10−5\nPS16em LCRS B105301.1-030602 0.069 997 8 0.070 100 0.069 815 1.8 ×10−4\nPS16evk 2MASX J22332338-0121266 0.054 530 1 0.054 550 0.054 468 6.2 ×10−5\nPS16fa CGCG 036-091 0.046 043 40 0.046 170 0.046 11 −6.7×10−5\nPS16fbb GALEXASC J000703.01-204149.5 0.052 183 12 0.052 240 0.0525 −3.2×10−4\nPS17akj LCRS B134713.8-024957 0.046 688 9 0.046 770 0.046 808 −1.2×10−4\nPS17bii 2MASX J11253836+0720042 0.073 406 10 0.073 500 0.073 391 1.5 ×10−5\nPSNJ2043531 NGC 6956 0.015 513 43 0.015 540 0.015 497 1.6 ×10−5\n360156 WISEA J100313.52+015343.0 0.045 598 43 0.045 560 0.045 507 9.1 ×10−5\nPTSS-16efw 2MASX J17353788+0848387 0.036 146 19 0.036 230 0.035 573 5.7 ×10−4\nSN2016gmb GALEXASC J003445.02-060936.8 0.058 125 4 0.058 160 0.058 269 −1.4×10−4\nSN2016hpx 2MASXi J0603164-265353 0.031 831 16 0.031 770 0.033 375 −1.5×10−3\nSN2017cjv LCRS B100813.8-033156 0.059 512 4 0.059 580 0.059 528 −1.6×10−5\nSN2017hn UGC 08204 0.023 897 23 0.023 810 0.023 85 4.7 ��10−5\naSN 2005kt was in Pantheon, but not Pantheon+ as the Type Ia classification is not secure (Sako et al., 2018; Carr et al., 2022).\nThus, the reference redshift is actually from the NASA/IPAC Extragalactic Database."    },    {        "title": "2402.13504v1.Field_induced_electric_polarization_and_elastic_softening_caused_by_parity_mixed__d___p__hybridized_states_with_electric_multipoles_in_Ba__2_CuGe__2_O__7_.pdf",        "content": "Field-induced electric polarization and elastic softening caused by parity-mixed d-p\nhybridized states with electric multipoles in Ba 2CuGe 2O7\nR. Kurihara1, Y. Sato2, A. Miyake2, M. Akaki3, K. Mitsumoto4, M. Hagiwara5, H. Kuwahara6, and M. Tokunaga2\n1Department of Physics and Astronomy, Faculty of Science and Technology,\nTokyo University of Science, Noda, Chiba 278-8510, Japan\n2The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan\n3Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan\n4Liberal Arts and Sciences, Toyama Prefectural University, Imizu, Toyama 939-0398, Japan\n5Center for Advanced High Magnetic Field Science, Graduate School of Science,\nOsaka University, Toyonaka, Osaka 560-0043, Japan and\n6Department of Physics, Sophia University, Chiyoda, Tokyo 102-855, Japan\nWe performed high-magnetic-field magnetization, polarization, and ultrasonic measurements in\nBa2CuGe 2O7to investigate field-induced multiferroic properties arising from a cross-correlation\nbetween electric dipoles and electric quadrupoles in addition to cross-correlation between magnetic\ndipoles and electric dipoles. Magnetization Mshows saturation behavior above 20 T for several\nmagnetic field directions, however, electric polarization Pcexhibits an increase, and elastic constants\nshow a softening above 20 T. Based on quantum states with a crystalline electric field for the\nD2dpoint group and d-phybridization between Cu-3 dand O-2 pelectrons, we confirmed that the\nmatrix of an electric dipole Pzwas proportional to that of an electric quadrupole Oxy. Furthermore,\nconsidering the spin-orbit coupling of 3 delectrons and the Zeeman effect, we showed that PzandOxy\nsimultaneously exhibited field-induced responses. These findings indicate that the orbital degrees\nof freedom, in addition to the spin degrees of freedom, contribute to the high-field multiferroicity in\nBa2CuGe 2O7.\nI. INTRODUCTION\nA cross-correlation between magnetic dipoles and elec-\ntric dipoles has been focused on solid-state physics [1].\nBecause this phenomenon describes how magnetic (elec-\ntric) fields generate electric (magnetic) fields in solids,\nscientists and engineers aim for application to some elec-\ntronic devices. Furthermore, a piezoelectric effect due\nto a cross-correlation between electric dipoles and elastic\nquadrupoles, which describes how electric fields (elastic\nstrains) induce elastic strains (electric fields) [2], has also\nbeen applied to a variety of items like motors, injectors,\nspeakers, stages, etc. These phenomena arising from the\ninterplay of different types of multipoles are known as\nmultiferroicity. Understanding the multiferroicity mech-\nanism is an important scientific subject for further appli-\ncations.\nSeveral mechanisms have been proposed to describe\nmultiferroic properties between magnetic dipoles and\nelectric dipoles. One is an exchange striction model de-\nscribed as |Pij| ∝Si·Sj[3–5]. Here, Pijis the polariza-\ntion vector, and Si(Sj) is the spin at site i(j). Another\nis an inverse Dzyaloshinskii-Moriya (DM) model written\nasPij∝eij×(Si×Sj), where eijdenotes the unit vec-\ntor connecting the neighboring site iandj[6]. These two\nmechanisms are based on inter-site interactions. There-\nfore, the magnetic structure plays a key role in such po-\nlarizations. In contrast, a spin-dependent d-phybridiza-\ntion model [7, 8], described as Pl∝(S·rl)2rl, is based on\na cluster consisting of a magnetic ion and non-magnetic\nions. Here, rlis the bonding vector between the transi-\ntion metal and the l-th ligand oxygen ion. This model\nindicates that the rotation of the magnetic moments in\nGeBa\nO\nCu XAYAXBYB\n−κ\n+κ(a) (b)\ndeform\n(c)\nPzEz\n(d)\nOxyxyεxy\nFIG. 1. (a) Top view of the crystal structure of Ba 2CuGe 2O7\nproduced by VESTA [10]. aandbaxes indicate the crystal-\nlographic coordinates. XA(XB) and YA(YB) axes are de-\nfined as the local coordinates at Asite (Bsite).±κindicates\nthe tilting angle of CuO 4tetrahedra about the c axis. (b)\nSchematic view of the CuO 4tetrahedron. Vertical arrows\nindicate the deformation direction of CuO 4. (c) Schematic\nview of the electric dipole Pzand the conjugating external\nfield of the electric field Ez. (d) Schematic view of the elec-\ntric quadrupole Oxyand the conjugate external field of the\ncrystal strain εxy.\nreal space carries a polarization change. As long as mag-\nnetization is increased by magnetic fields, a field-induced\nelectric polarization (FIEP) can be expected in such a\nmodel.\nRecently, the spin-dependent d-phybridization has\nbeen proposed in Ba 2XGe2O7(X= Mn, Co, Cu) to\ndescribe the FIEP of Pcalong the caxis of the crys-\ntallographic orientation accompanied by a magnetizationarXiv:2402.13504v1  [cond-mat.str-el]  21 Feb 20242\nTABLE I. Characters and several basis functions of the point group D2d. Here, we do not consider the time-reversal symmetry.\nlx=−i(y∂z−z∂y),ly=−i(z∂x−x∂z), and lz=−i(x∂y−y∂x) are the angular momentum operators. li(i=x, y, z ) is related\nto the rotation of crystal lattice and electronic systems around the i-axis and the Zeeman effect. Bx,By, and Bzrepresent the\ncomponents of the magnetic fields.\nIrrep E2IC4C22C′\n22σdBasis function: polar axial quadratic producted\nA11 1 1 1 1 z2xyz\nz(lxly+lylx),xlx−yly,yzlx−zxly\nzBxBy,xBx−yBy,xyB xBy,yzBx−zxBy\nlxBx+lyBy\nA21 1 1 −1−1 lz,Bz z\u0000\nx2−y2\u0001\n,xy\u0000\nx2−y2\u0001\nB11−1 1 1 −1 x2−y2\nB21−1 1 −1 1 z xy (lxly+lylx)\nE 2 0 −2 0 0 {x, y} { lx, ly},{Bx, By} {yz, zx}\nprocess [8, 9]. Here, Pcis the sum of the electric dipole\nPzin the solid, and the quantization axis zis set along\nthec-axis. Figure 1(a) shows the crystal structure of\nBa2CuGe 2O7with an ˚ akermanite-type structure belong-\ning to the tetragonal space group P421m. Cu ions re-\nsponsible for magnetic moments are at the center of de-\nformed O 4tetrahedra (see Fig. 1(b)). Thus, a point\ngroup symmetry at Cu-site is D2dwithout the inversion\noperation ( I) [11]. The local coordinates of CuO 4clus-\nters, XiandYi, (i=AandB), are tilted around c-axis\nwith the angle ±κ=±22◦(see Fig. 1(a)). 3 dand 2 por-\nbitals in the CuO 4clusters form valence and conduction\nbands [12]. A magnetic structure below antiferromag-\nnetic (AFM) transition at TN= 3.2 K shows a spiral\nstructure [13] because the DM interaction is active in\nsuch tilting crystals without the inversion operation. In\nmagnetic fields, the magnetic structure changes from an\nincommensurate spiral to a commensurate one [14]. The\nelectric polarization PcforB//[110] exhibits an anomaly\nat such a transition field in addition to the magnetization\ncurve [9].\nOn the other hand, another mechanism describing the\nFIEP has been indicated. In˚ akermanite-type compounds\nSr2CoSi 2O7and Ca 2CoSi 2O7, the FIEP has been ob-\nserved even in spin saturation regions [15, 16]. Since\nthe spin-dependent d-phybridization model is propor-\ntional to ( S·rl)2rl, field-dependent electric polarization\ncannot be expected above the spin saturation fields. Fur-\nthermore, a theoretical study has suggested that such a\nspin-dependent scenario based on the Kramers doublet\nwith spin-1/2 is excluded in terms of the vanishment of\nsingle-site anisotropy [17]. Therefore, we deduce that\nother electronic degrees of freedom also contribute to the\nFIEP.\nOne of the candidate degrees of freedom can be elec-\ntric multipoles. A previous study in Ba 2CuGe 2O7has\nfocused on a cross-correlation between the electric dipole,\nPz∝z, and the spin-nematic operator, SxSy+SySx, that\nis equal to the electric quadrupole, Oxy∝xy(see Figs.\n1(c) and 1(d)) [18]. Because the spatial inversion sym-\nmetry is broken at Cu-sites centered at O 4tetrahedra,\nboth PzandOxybelong to the irreducible representa-tion (irrep) B2of the point group D2d(see Tables I and\nII and Appendix A) [11, 19]. In other words, the matrix\nofPzshould be proportional to the matrix of Oxyfor the\nquantum states under the D2d. This fact indicates that\nthe response of electric quadrupole Oxyinduces the elec-\ntric polarization Pcvia the response of the electric dipole\nPz. Furthermore, due to the coupling between the elec-\ntric quadrupole Oxyand an elastic strain εxy(see Fig.\n1(d)), we can expect the electric polarization mediated\nby the crystal distortion of the CuO 4tetrahedra [20, 21].\nAbove the spin saturation fields, we deduce that the field-\ndependent electric polarization is brought about by the\ncross-correlation described as Pz∝Oxy.\nSince the electric dipole and quadrupole responses are\nexpected, the orbital part of wave functions plays a key\nrole in the FIEP. From the microscopic point of view,\nthe contribution of d-phybridized states consisting of O-\n2pand Cu-3 dorbitals is expected because parity-mixed\nwave functions lead the electric dipole degree of freedom\n[22]. In addition, 3 d-yzandzxorbitals and 2 p-xandy\norbitals in the d-phybridized states also carry the elec-\ntric quadrupole Oxy[23]. Therefore, we focused on the\npolarization and elastic constant measurements in high-\nmagnetic fields to detect the electric dipole and electric\nquadrupole response in Ba 2CuGe 2O7. Furthermore, we\ndiscuss another possible mechanism of the FIEP based\non the d-phybridization and spin-orbit coupling.\nThis paper is organized as follows. In Sect. II, the\nexperimental procedures are described. In Sect. III, we\npresent the experimental results of the high-field magne-\ntization, polarization, and ultrasonic measurements. We\nshow the field-induced elastic softening in addition to the\nfield-dependent electric polarization above the spin sat-\nuration fields. In Sect. IV, we present quantum states\nin high fields and the electric dipole and quadrupole sus-\nceptibilities to describe our experimental results of po-\nlarization and elastic constants. We conclude our results\nin Sect. V.3\nTABLE II. Conjugate fields, electric multipoles, and response functions corresponding to the irreducible representations (irreps)\nof the point group D2dat Cu sites. Ei(i=x,y,z) is an electric field and εΓ(Γ = 3 z2−r2,x2−y2,yz,zx,xy) is a symmetry\nstrain, respectively. The electric dipole Pi(i=x,y,z) and the electric quadrupole OΓ(Γ = 3 z2−r2,x2−y2,yz,zx,xy)\nare written as the electric multipoles. The sign in column Iindicates the spatial inversion property of conjugating fields and\nelectric multipoles (even: +, odd: −). Extra suffixes g (gerade) and u (un-gerade) of the irreps describe the parity of conjugate\nfields, multipoles, and response functions.\nIrrep Conjugating field Electric multipole Response function I\nA1(g) εB=εxx+εyy+εzz OB= 1/r C B= (2C11+ 2C12+ 4C13+C33)/9 +\nεu= (2εzz−εxx−εyy)/√\n3 O3z2−r2=\u0000\n3z2−r2\u0001\n/r2Cu= (C11+C12−4C13+ 2C33)/6 +\nB1(g) εx2−y2= (εxx−εyy)/√\n2 Ox2−y2=\u0000\nx2−y2\u0001\n/(√\n2r2) CT= (C11−C12)/2 +\nB2(g)√\n2εxy Oxy=√\n2xy/r2C66 +\nB2(u) Ez Pz=z/r P c −\nE(g)√\n2εyz Oyz=√\n2yz/r2C44 +√\n2εzx Ozx=√\n2zx/r2C44 +\nE(u) Ex Px=x/r P a −\nEy Py=y/r P b −\nII. EXPERIMENTAL DETAILS\nSingle crystals of Ba 2CuGe 2O7were grown by the\nfloating zone method. The Laue X-ray backscattering\ntechnique was used to cut samples with (100), ( ¯100),\n(001), and (00 ¯1) faces and (110), (1 ¯10), (001), and (00 ¯1)\nfaces. The magnetization Min pulsed magnetic fields\nwas measured by the induction method using coaxial\npickup coils. The FIEP of Pcwas obtained by inte-\ngrating the polarization current as a function of time\n[24]. The ultrasonic pulse-echo method with a numer-\nical vector-type phase detection technique was used for\nthe ultrasonic velocity, v[25, 26]. Piezoelectric trans-\nducers using LiNbO 3plates with a 36◦Y-cut and an\nX-cut (Yamaju Co.) were employed to generate longi-\ntudinal ultrasonic waves with the fundamental frequency\nof approximately f= 30 MHz and the transverse waves\nwith 16 MHz, respectively. Higher-harmonic frequencies\nof 68 MHz and 112 MHz were also employed. The elas-\ntic constant, C=ρv2, was obtained from the ultrasonic\nvelocity, v. Here, the mass density, ρ= 5.07 g/cm3, for\nBa2CuGe 2O7is calculated by the structural parameters\n[13]. The direction of ultrasonic propagation, q, and the\ndirection of polarization, ξ, for the elastic constant, Cij,\nare indicated in figures. gvalues of 2 .04 for the in-plane\nfield direction and 2 .44 for the inter-plane field direction\nbased on the Cu2+ions were estimated by the X-band\nelectron spin resonance (ESR) measurements with the\nfrequency of 9.12 GHz.\nFor high-field measurements up to 60 T, non-\ndestructive pulse magnets with time durations of 25 and\n36 ms installed at The Institute for Solid State Physics,\nThe University of Tokyo were used.\nThe energy scheme and the electric multipole suscep-\ntibility are calculated by the Julia language.III. EXPERIMENTAL RESULTS AND\nDISCUSSIONS\nA. High-field magnetization\n1.4\n1.2\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0M\n (μ\nB\n/Cu)\n60\n50\n40\n30\n20\n10\n0\nB\n (T)\n B\n//[100]\n B\n//[110]\n B\n//[001]\nBa\n2\nCuGe\n2\nO\n7\nB\nc2\nB\nsatB\nsat\nB\n2c\n(a)\nT\n = 1.3 KM\nM\n1.2\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0\n(2/g\n) x M\n (μ\nB\n/Cu)\n75\n60\n45\n30\n15\n0 \n(g\n/2) x B\n  (T)g\nab\n = 2.04\ng\nc\n = 2.44\n0.12\n0.08\n0.04\n0.00M\n (μ\nB\n/Cu)\n3\n2\n1\n0\nB\n (T)B\nc1\n(d) B\n//[001]MdM/dB\n (arb. unit)\n15\n13\n11\n9\n7\n5\nB\n (T)\n(c) B\n//[110]\n up\n downB\nc1B\nc1 MMdM/dB\n (arb. unit)\n30\n26\n22\n18\nB\n (T)B\nc2\nBsatB\nc2\nBsatM M\n(b)\nFIG. 2. (a) Magnetization curve Min Ba 2CuGe 2O7at\n1.3 K for B//[001], B//[100], and B//[110]. The vertical\narrows indicate characteristic fields BM\nc2andBsat. The inset\nin (a) shows the normalized magnetization curves, (2 /g)M,\nas a function of normalized magnetic field, (2 /g)B. (b) First\nderivative of Mwith respect to magnetic field BforB//[001],\nB//[100], and B//[110] in the range of 18 to 30 T. The vertical\narrows indicate transition fields BM\nc2andBsat. The data sets\nare shifted consecutively along the dM/dB axes for clarity. (c)\ndM/dB forB//[110] in the range of 5 to 15 T. The vertical\narrows indicate transition fields BM\nc1for the field up-sweep and\ndown-sweep. The data sets are shifted consecutively along the\ndM/dB axes for clarity. (d) MforB//[001] below 3 T. The\nvertical arrow indicates transition field BM\nc1.\nFigure 2(a) shows the magnetization as a function of\nfields in Ba 2CuGe 2O7at 1.3 K for various field direc-\ntions. We observed several characteristic anomalies and\na magnetization plateau in high fields. The magnetiza-4\ntion curve MforB//[100] and [110] show the increase\nin the field up to BM\nc2= 25.3 T, which is determined by\nthe first derivative of the magnetization with respect to\nthe field, dM/dB , in Fig. 2(b). With further application\nof the fields, Mexhibits a plateau above Bsat= 27 .8\nT up to 56 T. The difference in BM\nc2andBsatbetween\nthe field up-sweep and down-dweep processes was hardly\nnoticeable within our experimental resolution. We also\nobserved the anomaly at BM\nc1= 10.1 T for field up-sweep\nand 7 .1 T for down-sweep in dM/dB (Fig. 2(c)).\nAs discussed later, the anomaly BM\nc1corresponds to\nthe transition field of the incommensurate spiral to the\ncommensurate one. For B//[001], Mshows the increase\nin the field up to BM\nc2= 20 .8 T and a plateau above\nBsat= 21.6 T. The anomaly at BM\nc1= 1.7 T is also ob-\nserved as shown in Fig. 2(d). As shown in the inset of\nFig. 2(a), each magnetization curve is normalized using\nthegvalue of 2 .04 for the in-plane field direction and\n2.44 for inter-plane field direction, which is comparable\nto the previous study [27]. Above Bsat, normalized mag-\nnetizations (2 /g)Mshow almost 1, suggesting that the\npolarized paramagnetic (PPM) state is realized.\nFigures 3 show the magnetization as a function of\nfields in Ba 2CuGe 2O7at several temperatures. We in-\nvestigated the evolution of anomalies with temperature\nchanges. As the temperature is raised, the anomaly at\nBM\nc2shifts to a lower field and the hysteresis loop opens\nfor both field directions of [110] and [001]. The satura-\ntion field Bsatshifts to higher fields with increasing tem-\nperatures for both B//[110] and B//[001]. Here, Bsat\nwas determined from the field down-sweep process be-\ncause of the gradual change in Mfor the up-sweep pro-\ncess. The hysteresis behavior at BM\nc1appears below 4.0\nK for both B//[110] and B//[001]. The difference of BM\nc1\nforB//[001] between the field up-sweep and field down-\nsweep is smaller than that for B//[110]. Above 10 K, the\nloop closes for B//[110]. BM\nc2also shifts to a lower field\nwith increasing temperatures. In addition, the bend in\nMatBM\nc2becomes more gradual, and hysteresis behavior\nbecomes more pronounced. In the magnetization curve\nforB//[110] at 4.2 K, the shape for the field down-sweep\nseems to be similar to the Mat 1.3 K while the shape of\nMfor the field up-sweep can be similar to that at 5 K.\nWe deduce that the hysteresis behavior around Bc2and\nBsatoriginates from temperature changes in the sample\ncaused by magnetocaloric effects under quasi-adiabatic\nconditions [28–30]. The magnetocaloric effect can lead to\na lower sample temperature during the field down-sweep\nprocess compared to the field up-sweep process.\nAs shown above, we observed several anomalies in the\nmagnetization curves. In particular, the observation of\nspin saturation plays a key role in understanding the mul-\ntiferroic mechanism in Ba 2CuGe 2O7. In the following\nSect. III B, we discuss the magnetic phase diagram and\nthe origin of anomalies.\nM\n (μ\nB\n/Cu)\n60\n50\n40\n30\n20\n10\n0\nB\n (T)\n(a)\nB\n//[110]\n1.3 K \n2.0 K\n2.5 K\n3.0 K\n3.2 K\n3.5 K\n3.8 K\n4.0 K\n4.2 K\n5 K\n6 K\n8 K\n10 K\n20 K\n30 K\n50 K\n100 K\n300 K\n0.5B\nc1B\nc2B\nsatM\nM\n60\n50\n40\n30\n20\n10\n0\nB\n (T)\n(b)\nB\n//[001]\n1.3 K\n1.9 K\n2.3 K\n2.7 K\n3.0 K\n3.5 K\n3.9 K\n4.2 K \n4.5 K\n4.7 K\n4.9 K\n0.5\nB\nc1B\nc2\nB\nsat\nBa\n2\nCuGe\n2\nO\n7M\nMFIG. 3. Magnetization curve Min Ba 2CuGe 2O7at several\ntemperatures for (a) B//[110] and (b) B//[001]. The ver-\ntical arrows indicate the characteristic fields BM\nc1,BM\nc2, and\nBsat. Up-arrows (down-arrows) for BM\nc1andBM\nc2indicate the\nanomaly for field up-sweep (down-sweep). The data sets are\nshifted consecutively along the Maxes for clarity.\nB. Phase diagram\nTo compare the characteristic anomalies to those of\nthe previous reports, we summarized several character-\nistic fields of Ba 2CuGe 2O7in the temperature-magnetic\nfield phase diagram in Fig. 4. For B//[110], the phase\nboundary below 10 T seems to be consistent with the pre-\nvious reports (see Fig. 4(a)) [9, 31]. Thus, we conclude\nthat the anomaly at BM\nc1originates from the transition\nof the incommensurate spiral to the commensurate one.\nThe saturation field Bsatis our new observations. Bsat\nseems to be almost independent of temperatures. We\nalso observed BM\nc2obtained from the peak structure of\ndM/dB .BM\nc2decreases with increasing temperatures.\nForB//[001], the low-field phase boundary seems to\nbe consistent with the previous results (see Fig. 4(b))\n[14, 31]. Thus, BM\nc1forB//[001] can correspond to the\nboundary of the incommensurate spiral to the commen-\nsurate one. In contrast, the AF cone state [31] is hardly\nvisible in our high-field measurements. We also observed\nthe anomaly at BM\nc2andBsatforB/[001]. Bsatin-\ncreases with increasing temperature while BM\nc2decreases.\nOur magnetization measurements confirmed the phase\nboundary between the incommensurate spiral to the com-\nmensurate one. We observed the difference in Bc1be-\ntween the field up-sweep and down-sweep processes.\nAs shown in the phase diagram, our results in low fields\nare consistent with those in the previous studies [9, 31].\nIn addition to the previous studies, we demonstrate that\nthe polarized paramagnetic state can be realized above\nBsat.5\nPPM PPM\ncommensurate spiral\nincommensurate spiralincommensurate spiralPM PMcommensurate spiral\nFIG. 4. Temperature-field phase diagram of Ba 2CuGe 2O7\nfor (a) B//[110] and (b) B//[001]. The spin saturation field\nBsatdetermined by the magnetization is shown by the filled\nred triangles. The characteristic field BM\nc1(BP\nc1) determined\nby the magnetization (polarization) measurements is indi-\ncated by filled red (blue) squares for field up-seep and open\nred (blue) squares for down-sweep, respectively. The char-\nacteristic field BM\nc2(BP\nc2) determined by the magnetization\n(polarization) measurements is indicated by filled red (blue)\ncircles for field up-seep and open red (blue) circles for down-\nsweep, respectively. The filled black triangles and filled gray\ntriangles show transition fields in Refs. [31] and [9]. The\nbackground colors are the guide to distinguish each phase.\nC. Field-induced electric polarization\nTo investigate the multiferroicity above the spin sat-\nuration regions, we measured the polarization in high\nfields. Figure 5 shows the magnetic field dependence\nof the electric polarization ∆Pcin Ba 2CuGe 2O7for\nB//[110] at 1.4 K. We also exhibits the first derivative of\n∆Pcwith respect to B,d(∆Pc)/dB, in the inset of Fig.\n5. We observed several anomalies in the magnetization.\n∆Pcandd(∆Pc)/dB exhibit the anomalies at BP\nc1= 9.3\nT for field up-sweep and 7 .2 T for down-sweep. The\nanomaly BP\nc1corresponds to the transition field of the\nincommensurate spiral to the commensurate one. The\nhysteresis behavior around Bc1can be attributed to the\nfield-induced magnetic structure change (see appendix\nB). For further increasing fields, ∆Pcshows the anomaly\natBP\nc2= 25.7 T (26.4 T) for field up-sweep (down-sweep).\nBP\nc2in∆Pcis consistent with BM\nc2in the magnetization\nforB//[110]. Above the saturation field Bsat,∆Pcin-\ncreases in the field up to 56 T.\nTo investigate the temperature dependence of these\nanomalies and the deviation of ∆Pcfrom 1.4 K, we\nalso measured the magnetic-field dependence of ∆Pcin\nBa2CuGe 2O7at several temperatures (see Fig. 6). As\nthe temperature is raised, BP\nc1shifts to lower fields and\nthe hysteresis loop closes. BP\nc2also shifts to a lower field\n50\n40\n30\n20\n10\n0ΔPc\n (μ\nC/m\n2\n)\n60\n50\n40\n30\n20\n10\n0\nB\n (T)\nBa\n2\nCuGe\n2\nO\n7\nB\n//[110]\nT\n = 1.4 K\n \nB\nc2\nB\nc1B\nsatP\nP\nd(ΔP c ) /d B\n (arb. unit)\n30\n25\n20\n15\n10\n5\nB\n (T)\n up-sweep\n down-sweepB\nc1B\nc1\nB\nc2B\nc2PP\nPPFIG. 5. Magnetic field dependence of the electric polar-\nization ∆Pcin Ba 2CuGe 2O7at 1.4 K for B//[110]. The\ndown arrows indicate the characteristic transition fields BP\nc1\nandBP\nc2determined by PcandBsatbyM. The inset shows\nd(∆Pc)/dB.\nwith increasing temperatures. In addition, the hump-\nshaped anomaly around BP\nc2disappears and hysteresis\nbehavior becomes more pronounced. Above 3.8 K, the\nanomaly at BP\nc1is hardly visible in the polarization mea-\nsurements. The hysteresis loop in high fields closes above\n30 K. We added BP\nc1andBP\nc2appeared in Pcto the phase\ndiagram (see Fig. 4). These characteristic fields are con-\nsistent with those in M.\nDespite the spin saturation above Bsat, our polar-\nization measurements revealed that the polarization Pc\nshowed still increase in the fields, indicating another\nmechanism of multiferroicity in Ba 2CuGe 2O7. There-\nfore, we focused on the cross-correlation described as\nPz∝Oxyto understand the quantum states that carry\nthe mechanism of the field-dependent electric polariza-\ntion above Bsat.\nD. Elastic constants in high fields\nTo investigate the cross-correlation between the elec-\ntric dipole and electric quadrupole, we measured elas-\ntic constants in Ba 2CuGe 2O7. Figure 7 shows the\nmagnetic-field dependence of the relative elastic con-\nstants∆Cij/Cijat 1.4 and 4.2 K for B//[110]. In the\nAFM phase at 1.4 K, we observed the elastic softening\nwith increasing fields and characteristic anomalies. At6\nΔPc\n (μ\nC/m\n2\n)\n \n \n60\n50\n40\n30\n20\n10\n0\nB\n (T)\nBa\n2\nCuGe\n2\nO\n7\nB\n//[110]\n1.4 K\n1.5 K\n2.0 K\n2.5 K\n3.0 K\n3.2 K\n3.5 K\n3.8 K\n4.0 K\n4.2 K\n5 K\n6 K\n8 K\n10 K\n4.5 K\n30 K\n100 K\n300 K\n20\nB\nc1B\nc2B\nsatP\nP\nFIG. 6. Magnetic-field dependence of the electric po-\nlarization ∆Pcin Ba 2CuGe 2O7at several temperatures for\nB//[110]. The vertical arrows indicate the characteristic fields\nBP\nc1andBP\nc2determined by PcandBsatbyM. In particu-\nlar, up-arrows (down-arrows) for BP\nc1andBP\nc2indicate the\nanomaly for field up-sweep (down-sweep). The data sets are\nshifted consecutively along the ∆Pcaxes for clarity.\n1.4 K, the longitudinal elastic constants C11in Fig. 7(a)\nandCL[110] = (C11+C12+ 2C66)/2 in Fig. 7(b) with the\nin-plane-type ultrasonic propagating direction show soft-\nening with the increase in the field up to around BM(P)\nc2\naccompanying hysteresis loop. With the further appli-\ncation of the fields, each elastic constant shows a rapid\nincrease with the bending point around Bsat, then elas-\ntic constants exhibit slight softening up to 60 T. In con-\ntrast to the C11andCL[110] , the longitudinal elastic con-\nstant C33with the inter-plane-type ultrasonic propagat-\ning direction in Fig. 7(c) shows hardening with the in-\ncrease in the fields accompanying the anomaly at around\nBsat. The transverse elastic constant C44with inter-\nplane-type ultrasonic propagation in Fig. 7(d) shows\na step-like anomaly around Bsatwhile the field depen-\ndence can be almost field independent. The transverse\nelastic constants CT= (C11−C12)/2 in Fig. 7(e) and\nC66in Fig. 7(f) show softening with the increasig fields\nup to Bsat. With the further application of the fields,\neach elastic constant shows slight softening up to 60 T.\nWe could not observe the elastic anomaly at BM(P)\nc1 in\nall elastic constants within our experimental resolution,\nsuggesting that the contribution of the magnetoelastic\ncoupling around BM(P)\nc1 is small. To study the contribu-\nΔCL[110] / C L[110]\n0.02%\n(b) q\n//ξ\n//[110] ΔCT/C T\n0.02%\n(e) q\n//[110], ξ\n//[110]\n_ΔC44 /C 44\n0.02%\n(d)q\n//[001]\nξ\n//[100] ΔC33 /C 33\n0.02%\n(c)  q\n//ξ\n//[001] ΔC11 /C 11\nBa\n2\nCuGe\n2\nO\n7\n0.02%\n(a) q\n//ξ\n//[100] B\nc2\nBsatM(P)\n 1.4 K (up-sweep)\n 1.4 K (down)\n 4.2 K (up)\n 4.2 K (down)\nB\n//[110]ΔC66 /C 66\n60\n50\n40\n30\n20\n10\n0\nB\n (T)\n0.02%\n(f) q\n//[100], ξ\n//[010]FIG. 7. Magnetic-field dependence of the relative elastic con-\nstants ∆Cij/Cij= [Cij(B)−Cij(B= 0)] /Cij(B= 0) of longi-\ntudinal elastic constants (a) C11, (b) CL[110] , and (c) C33and\ntransverse elastic constants (d) C44, (e)CT= (C11−C12)/2,\nand (f) C66in Ba 2CuGe 2O7at 1.4 and 4 .2 K for B//[110].\nThe ultrasonic propagation direction qand polarization direc-\ntionξare listed. The solid (dotted) lines indicate the elastic\nconstant for the field-upsweep (down-dweep) process. The\ndashed lines in each panel indicate BM(P)\nc2 andBsatdeter-\nmined by the magnetization curve.\ntion of the magnetoelastic coupling to the transition of\nthe incommensurate spiral to the commensurate one, we\nneed high-precision ultrasonic measurements.\nAt 4.2 K, each elastic constant shows a similar field\ndependence to that at 1.4 K except around 27 T, where\nthe anomaly appears. Below 20 T and above 30 T, the\nfield dependence of each elastic constant is almost the\nsame as that at 4.2 K. The anomaly in C11andCL[110]\nchanges from a rapid increase to a gradual slope. C33\ncan be almost temperature independent. The step-like\nanomaly of C44shifts to the lower field side. The anomaly\ninCTandC66changes from the bending-like form to a7\nΔC11 /C 11\n (arb. unit) \nBa\n2\nCuGe\n2\nO\n7\n(a)\n1.4 K B\n//[110] B\nsatΔC\nL[110]\n/C\nL[110]\n (arb. unit)\n(b)\n1.4 KΔPz'\n (arb. unit)\n(c)\n1.4 K−dM /dB\n (arb. unit)\n60\n50\n40\n30\n20\n10\n0\nB\n (T)\n0\n(d)\n1.3 K\nFIG. 8. Magnetic-field dependence of the relative elastic\nconstants (a) C11and (b) CL[110] , (c) the electric polarization\n∆P′\nc, and (d) the first derivative of the magnetization with\nrespect to magnetic fields −dM/dB for the field down-sweep\nprocess in Ba 2CuGe 2O7.dM/dB is calculated after apply-\ning binominal smoothing to M. The horizontal broken lines\nindicating the field-independent factor are shown in panels\n(a) - (c) as guides for the eyes. The horizontal broken line\nin panel (d) represents the zero line of dM/dB indicating\nsaturation magnetization. The vertical broken lines indicate\nBsat. Because the bend in dM/dB becomes gradual due to the\nsmoothing, Bsatmay appear different compared to dM/dB\nbefore smoothing in Fig. 2(b).\ngradual slope.\nWe also observed elastic softening above Bsatindicat-\ning the electric quadrupole response. This fact demon-\nstrates that the quantum states in high fields provide the\nfinite diagonal elements of multipole matrices and the\nexpectation value of multipoles.\nE. Discussions\nWe observed the spin saturation above Bsatand the\ncharacteristic anomaly at BM(P)\nc2 in Ba 2CuGe 2O7for\nB//[110]. Around BsatandBM(P)\nc2 , both the polariza-TABLE III. Longitudinal strains, that of reduction by the\nsymmetry strains, and decomposition for the point group D2d.\nStrain Reduction Decomposition\nεxx εB/3−εu/(2√\n3) +εx2−y2/√\n2 2 A1⊕B1\nεyy εB/3−εu/(2√\n3)−εx2−y2/√\n2 2 A1⊕B1\nεL[110] εB/3−εu/(2√\n3) +√\n2εxy 2A1⊕B2\nεzz εB/3 +εu/√\n3 2 A1\ntion and the elastic constants show step-wise change. We\ndiscuss the origin of this step-wise anomaly in terms of\nthe thermodynamic relation between magnetization and\nelastic constants. Figure 8 shows the magnetic-field de-\npendence of the relative elastic constants, the subtracted\npolarization ∆P′\nc, and the first derivative of the magne-\ntization with respect to magnetic fields, dM/dB . Us-\ning the linear slope f(T) of ∆Pcabove 40 T, we cal-\nculated ∆P′\ncasf(T)−∆Pcto demonstrate the step-\nwise change in the electric polarization. The form of the\nstep-wise anomaly in ∆C11/C11,∆CL[110] /CL[110] , and\n∆Pcseems to be reproduced by −dM/dB , suggesting\nthat the elastic constants and the electric polarization\nare proportional to dM/dB around the spin saturation\nfields. In the low-dimensional spin magnetic compound\nNH4CuCl 3, experimental studies have proposed that the\nelastic anomaly is proportional to dM/dB around the\nmagnetization plateaus [32]. A theoretical study has con-\nfirmed∆C/C∝dM/dB based on the Ehrenfest relation,\nalbeit under special circumstances [33]. Since ∆P′\nccan\nalso be proportional to dM/dB around Bsat, we deduce\nthat this Ehrenfest relation is applicable to polarization.\nTherefore, we conclude that the step-like behavior in the\nelastic constants and the polarization originates from the\nspin saturation of Min Ba 2CuGe 2O7. As shown in Fig.\n7, the little step-like anomaly in C44can also be at-\ntributed to dM/dB .\nIn contrast, our experimental results suggest that the\nalternative mechanism of the FIEP can be realized in\nBa2CuGe 2O7. As shown in Fig. 2, the spin saturation for\nB//[110] above Bsatsuggests that magnetic moments can\nbe aligned parallel to the magnetic field direction. Since\nthe electric dipole Pzis proportional to the square of\nmagnetic moments in the spin-dependent d-phybridiza-\ntion model, the polarization should be independent of\nfields in the spin saturation regions. However, the po-\nlarization for B//[110] increases in the fields above Bsat.\nThis fact indicates that the quantum states describing\nthe polarization still depend on the magnetic fields in the\nsaturation magnetic moment region. In addition, our ul-\ntrasonic measurements revealed the response of electric\nquadrupole in magnetic fields. Therefore, we need to con-\nsider new quantum states and the mechanism of the FIEP\naccompanying the elastic response in magnetic fields.\nTo describe another possible multiferroic mechanism,\nwe focus on active electric multipoles based on the point\ngroup D2d. The irrep of elastic constants and polariza-\ntion provides information on possible quantum states in8\nterms of group theory. As shown in the Table II, the\ntransverse elastic constant C44with the irrep Ereflects\nthe response of the electric quadrupoles OyzandOzxbe-\ncause the ultrasonic waves for C44induces the strains εyz\nandεzx. The monotonic hardening of C44with increasing\nfields indicates that the contribution of the quadrupole-\nstrain coupling between Oyz(Ozx) and εyz(εzx) to elas-\nticity is negligibly small. The elastic softening of CT\nwith the irrep B1originates from the coupling of the\nquadrupole Ox2−y2with the strain εx2−y2. The soften-\ning of C66with the irrep B2is attributed to the Oxyand\nεxy. The elastic constant C11is measured by the longitu-\ndinal ultrasonic waves inducing the strain εxx(εyy). As\nlisted in Table III, since εxx(εyy) includes εx2−y2, the\nsoftening of C11originates from the coupling of Ox2−y2\ntoεx2−y2because little contribution of the coupling be-\ntween OB(O3z2−r2) and εB(ε3z2−r2) with the irrep A1is\nindicated by the hardening of C33. This fact for the inter-\nplane type strain ε3z2−r2is in contrast to magnetostric-\ntion around the spin saturation in Ba 2FeSi 2O7[34]. Also,\nthe elastic softening of CL[110] is attributed to the cou-\npling of Oxytoεxy. In addition to the elastic constants,\nthe increase of polarization Pcindicates the response of\nthe electric dipole Pzwith the irrep B2. Thus, our experi-\nments suggest that the active multipoles in high fields are\nthe electric dipole Pzwith the irrep B2and the electric\nquadrupoles Ox2−y2with B1andOxywith B2.\nSince the above group-theoretical analysis is based on\nthe global coordinates of the crystal lattice, we should\ndiscuss the active multipoles for the local coordinates.\nConsidering the tilting of CuO 4tetrahedra, we deduce\nthat both the response of the electric quadrupole Ox2−y2\nwith the irrep B1and Oxywith B2for the global\ncoordinates originate from the response of an electric\nquadrupole OXYwith B2for the local coordinates. As\na result of this response, the elastic softening of CTwith\nthe irrep B1for the global coordinates is induced by the\nelastic softening of C66with the irrep B2for the local\ncoordinates. The response of the electric dipole Pzfor\nthe global coordinates is the same as that for the local\ncoordinates because the zcomponent is invariant for the\ntilting of CuO 4tetrahedra around the c-axis. Therefore,\nwe conclude that the active multipoles are PzandOXY\nwith the irrep B2of the point group D2d.\nFocusing on the active representation B2obtained by\nthe group-theoretical analysis, we can describe the free\nenergy based on the Landau phenomenological theory for\ntheD2dpoint group. Considering the basis functions\nwith the irrep A1in Table I, the conjugate fields and\nelectric multipoles in Table II, and the product table of\ntheD2dpoint group in Table IV, we can describe the\nfree energy Fcomposed of the minimal elements as a\nfunction of the electric dipole Pz, the electric quadrupole\nOXY, the strain εXY, the electric field Ez, and the in-\nplane magnetic field ( BX, BY,0) for the local coordinateas below:\nF=1\n2αzP2\nz+1\n2αXYO2\nXY+1\n2C′0\n66ε2\nXY\n−gzPzEz−gXYOXYεXY−µZ(lXBX+lYBY)\n−αA1PzOXY−gA1PzBXBY. (1)\nHere, αz,αXY,C′0\n66,gz,gXY,µZ,αA1, and gA1are the\ncoefficients. The terms from the first to the sixth on\nthe right-hand side in Eq. (1) indicate the conventional\nterms incorporated in the free energy. While the angular\nmomentums lxandlyare bases of the irrep E, we include\nthe Zeeman effect for the in-plane magnetic fields as the\nseventh term. Although the eighth and ninth terms are\nconstructed from the products of the basis functions with\ndifferent properties under spatial inversion, each is also\ninvariant for the symmetry operations of D2d. We em-\nphasize that the eighth term can be the characteristic\nterm describing the cross-correlation between the elec-\ntric dipole and electric quadrupole. We can also expect\nfield-induced electric polarization because the contribu-\ntion of the tenth term is allowed under the D2d. Based\non the character table of Table I, we can also include\nother terms, such as the sum of the possible permuta-\ntions of lXOY Z,lYOZX,lXlYPz, and PXBX−PYBY, in\nthe free energy of Eq. (1). However, the contributions of\nOY ZandOZXwith the irrep Ecan be negligible because\nthe elastic hardening of C44indicates no-contribution of\nOyzandOzx. We also deduce that ( lXlY+lYlX)Pz\nis included in the eighth term of αA1PzOXYbecause\nlXlY+lYlXis equivalent to OXYin terms of the Stevens’\noperator [35]. Since PXBX−PYBYis not constructed\nby the electric dipole Pzwith the irrep B2, we do not\ninclude this term in the free energy.\nBased on the free energy of Eq. (1), we can qualita-\ntively describe the cross-correlation between PzandOXY\nand the FIEP and elastic softening. From the equilibrium\nconditions, denoted by ∂F/∂P z= 0 and ∂F/∂O XY= 0,\nthe following relationships are obtained:\nPz=1\nαz(gzEz+αA1OXY+gA1BXBY),(2)\nOXY=1\nαXY(αA1Pz+gXYεXY). (3)\nThese results indicate the cross-correlation between the\nelectric dipole Pzand the electric quadrupole Oxy. In\naddition, we can describe PzandOXYas a function of\nEz,εXY, and ( BX, BY) as\n\u0012\nPz\nOXY\u0013\n=gz\nαzαXY−α2\nA1\u0012\nαXY\nαA1\u0013\nEz\n+gXY\nαzαXY−α2\nA1\u0012\nαA1\nαz\u0013\nεXY\n+gA1\nαzαXY−α2\nA1\u0012\nαXY\nαA1\u0013\nBXBY.(4)\nThis result indicates the FIEP and elastic softening and\npiezoelectric. However, we cannot explain the micro-\nscopic origin of the multipoles and the mechanism of9\nTABLE IV. Product table of the point group D2dcalculated\nby the characters in Table I. The basis functions of the irreps\nA1,A2,B1,B2, and Earez2andxyz,lz,x2−y2,zandxy,\nand{x, y},{yz, zx},{lx, ly}, and{Bx, By}, respectively.\nD2d A1 A2 B1 B2 E\nA1 A1 A2 B1 B2 E\nA2 A2 A1 B2 B1 E\nB1 B1 B2 A1 A2 E\nB2 B2 B1 A2 A1 E\nE E E E E A 1⊕A2⊕B1⊕B2\nthe FIEP and elastic softening. Therefore, we discuss\nthe possible quantum states so as not to contradict our\ngroup-theoretical analysis.\nSince PzandOXYwith the irrep B2can be the ac-\ntive multipoles, the direct product of the irrep of the\nquantum states should contain the decomposition of\nB2(u)(Pz)⊕B2(g)(Oxy). Here, we use the irrep with the\nextra suffix g (gerade) and u (un-gerade) to distinguish\nbetween even and odd parity. As shown in Table IV,\nthe irrep of candidate orbitals constracting the quantum\nstates is E. This result indicates that the quantum states\nare formed by the yzandzxorbitals of Cu-3 delectrons\nandxandyorbitals of O-2 pelectrons. Furthermore,\nbecause the spatial inversion property of these electric\nmultipoles is different from each other as listed in Table\nII, we should describe parity-hybridized wave functions.\nThis fact indicates that the hybridization between yzand\nx,zxandy, and xyandzorbitals can play a key role in\nthe quantum states. The possibility of d-phybridization\nas an origin of such parity-hybridized states is indicated\nby the basis function of xyzwith the irrep A1ofD2d.\nIn addition to the multipole effects, we should con-\nsider the magnetic field effects to describe the quantum\nstates. Since the AFM ordering has been observed in\nBa2CuGe 2O7, the spin degrees of freedom can be neces-\nsary to describe the quantum states. Thus, we consider\nthe spin-orbit coupling. Furthermore, we deduce that\nthe field dependence of the electric multipole response\nis dominated by the Zeeman effect for the orbital part\nof the high-field quantum states because the FIEP and\nelastic softening appear above Bsat. Therefore, we focus\non the angular momentum operators lXandlYfor the\nlocal coordinates with the irrep Eof the D2dpoint group\nfor the in-plane magnetic field direction. To take into ac-\ncount the Zeeman effect, the direct product of the irrep\nof the quantum states should contain the irrep E. This\nfact indicates that the quantum states should contain E\nandB2orbitals (see Table IV). We would like to mention\nthat if the magnetic field is applied along the caxis, lz\nwith the irrep A2contributes to the Zeeman effect. Then,\nwe consider the tilting crystal structure of CuO 4tetra-\nhedra. In our measurements, the magnetic field was ap-\nplied along the [110] direction for the global coordinates.\nThus, based on the tilting angle ±κof CuO 4tetrahedra,\nwe should calculate the high-field quantum states and themagnetic field dependence of the multipole response.\nIn the above discussions, we assumed that the crys-\ntal symmetry and the point group remained conserved\nabove Bsat. Before delving into the analysis of quantum\nstates, it is necessary to evaluate the validity of this as-\nsumption. Below Bsat, we also observed the FIEP and\nelastic softening, indicating that the crystal symmetry\nbreaking characterized by the irrep B2was already in-\nduced by the magnetic field. Due to the crystal symme-\ntry breaking, the symmetry lowering of the point group\nfrom D2dtoC2can be realized above Bsat. Furthermore,\nthe field-independent behavior of the electric polariza-\ntion and the transverse elastic constants are expected\nabove Bsat. However, this contradicts our experimental\nresults. Hence, it can be reasonably assumed that the\ngroup-theoretical analysis based on D2dis approximately\napplicable above Bsat.\nBased on the above discussions, we describe the quan-\ntum states based on the Cu-3 dand O-2 porbitals with\nthed-phybridization, the spin-orbit coupling, and the\nZeeman effect, whose contributions are indicated by our\ngroup-theoretical analysis. We show several theoretical\nstudies of the wave functions in high fields, the electric\nmultipole susceptibility, and group theoretical analysis in\nthe following Sect. IV.\nIV. QUANTUM STATES IN HIGH FIELDS\nAND ELECTRIC MULTIPOLE SUSCEPTIBILITY\nA. Wave functions and field dependence of\neigenenergies\nIn this section, we discuss the origin of the electric\ndipole and the electric quadrupole in Ba 2CuGe 2O7to\ncalculate quantum states based on the crystalline elec-\ntric field (CEF) HCEF, the d-phybridization Hd−p, the\nspin-orbit coupling HSO, and the Zeeman effect HZeeman .\nTaking into account the electric dipole-electric field inter-\naction HL\nDE(electric quadrupole-strain interaction HL\nQS),\nwe calculate the electric dipole (electric quadrupole) sus-\nceptibility for the local coordinate. Furthermore, we dis-\ncuss the possible contribution of inter-site quadrupole in-\nteraction HG\nQQto the high-field quantum states. There-\nfore, total Hamiltonian Htotalis described as below:\nHtotal=H0+Hext+HG\nQQ, (5)\nH0=HCEF+Hd−p+HSO+HZeeman , (6)\nHext=HL\nDE+HL\nQS. (7)\nBased on this Hamiltonian, we demonstrate that the or-\nbital part of the wave functions, in addition to the spins,\ncontributes to the field-induced multiferroicity.\nAt first, we describe the wave functions of Cu-3 delec-\ntrons and the molecular orbitals constructed by the 2 p\nelectrons of O 4tetrahedra. In Ba 2CuGe 2O7, Cu ions are\ncentered at O 4tetrahedron with the point group symme-\ntryD2d. Thus, considering the CEF for Cu-3 delectrons10\ndescribed as [23]\nHCEF=A0\n2\u00123z2−r2\n2r2\u0013\n+A0\n4\u001235z4−30z2r2+ 3r4\n8r4\u0013\n(8)\n+A4\n4 √\n35\n8x4−6x2y2+y4\nr4!\n,\nwe obtain doubly degenerate 3 d-orbitals yzandzxand\nan excited singlet orbital xyas shown in Fig. 9(a). Here,\nA0\n2,A0\n4, and A4\n4denote the CEF parameters. The wave\nfunctions of yz,zx, and xyorbitals and the relationship\nbetween the energy levels and the CEF parameters are\nwritten in Appendix C. Based on the previous studies\nin the ˚ akermanite-type compound [20], we set the en-\nergy levels of xyand degenerate yzandzxorbitals to\n0.1 and −0.7 eV, respectively (see Fig. 9(a)). We de-\nduce that 3 z2−r2andx2−y2orbitals of 3 delectrons do\nnot contribute to the polarization and elasticity because\nthey exist deep in the energy levels [12]. The 2 porbitals\non O 4tetrahedra are also transformed by the symme-\ntry operations of D2d. As shown in Appendix D, a group\ntheoretical analysis provides eight molecular orbitals con-\nsisting of xandyorbitals of O-2 pelectrons. We ignore\nthe molecular orbitals consisting of zorbital of 2 pelec-\ntrons because zorbital with the irrep B2does not carry\nPz,Ox2−y2, and Oxy(see Table IV). Considering the pre-\nvious studies in the ˚ akermanite-type compound [20], we\nset the energy level of degenerate xandyorbitals to\n−4.5 eV. As we will discuss late, our calculations using\nthese values can reproduce the electronic structure near\nthe Fermi level proposed in the previous study [12].\nWe stress that the electric dipole and electric\nquadrupole degrees of freedom are attributed to the 3 d-\nyzandzxand 2 p-xandyorbitals with the irrep E.\nThe matrix elements of the electric dipole and electric\nquadrupoles for the yz,zx, and xyorbitals of 3 delectronsand the xandyorbitals of 2 pelectrons are described as\nOx2−y2=\nyz zx xy x y\n−√\n2\n7r2\nd 0 0 0 0\n0√\n2\n7r2\nd0 0 0\n0 0 0 0 0\n0 0 0√\n2\n5r2\np 0\n0 0 0 0 −√\n2\n5r2\np\n,\n(9)\nOxy=\n0√\n2\n7r2\nd0 0 0√\n2\n7r2\nd0 0 0 0\n0 0 0 0 0\n0 0 0 0√\n2\n5r2\np\n0 0 0√\n2\n5r2\np0\n, (10)\nPz=\n0 0 0 0 p\n0 0 0 p0\n0 0 0 0 0\n0p0 0 0\np0 0 0 0\n. (11)\nHere, we set r2\nd=R∞\n0dr[rf3d(r)]2and r2\np=R∞\n0dr[rf2p(r)]2for the convenience. pinPzof Eq. (11)\nindicates the radial integration of the wave functions for\nthe 3 d-yzandzxand 2 p-xandyorbitals. Using the ma-\ntrices of Eqs. (9) - (11), we can describe the matrices of\nthese electric multipoles for the molecular orbitals with\nthe irrep E(see Eqs. (D2) - (D4) in Appendix D). This\nmatrix analysis is consistent with the group-theoretical\nanalysis (see Table IV).\nThe group-theoretical analysis also indicates the con-\ntributions of the spin-orbit coupling and the Zeemam ef-\nfect for the quantum states in CuO 4clusters. yzandzx\norbitals with the irrep Eandxyorbital with the irrep B2\ncarry the angular momentums lxandlywith the irrep E\nto reduce ( E⊕B2)⊗(E⊕B2) (see Table IV). The matrix\nelements of lxandlyare written as\nlx=\n0 0 0 0 0\n0 0 i0 0\n0−i0 0 0\n0 0 0 0 0\n0 0 0 0 0\n, (12)\nly=\n0 0−i0 0\n0 0 0 0 0\ni0 0 0 0\n0 0 0 0 0\n0 0 0 0 0\n. (13)\nTherefore, both the spin-orbit coupling proportional to\nlxsx+lysyand the Zeeman effect for B//[110] including\nlxandlycause the orbital mixing in quantum states.\nTo obtain the finite expectation value of the electric\ndipole Pz, we consider the parity hybridization between\nthe Cu-3 dand O-2 porbitals. The hybridization between11\ndpH− CEFH\nψSO1±ψSO3±\nψSO2±\nψSO5±ψSO6±\nψSO4±SOH\nE: yz, zxB2: xy\nE: x, yB2: ψ7\nE : ψ3, ψ4\nE : ψ5, ψ6B2: ψ8-0.7 eV0.1 eV\n-4.5 eV-0.0755 eV0.549 eV\n-4.5 eV\n-4.95 eV\n-5.12 eV -5.13 eV-5.12 eV-4.95 eV(a)\n-0.56 eV\n-0.038 eV\n-0.12 eV\nFIG. 9. (a) Energy scheme of the quantum state for the crystalline electric field HCEFwith D2dpoint group symmetry, d-p\nhybridization HCEF+Hd−p, and spin-orbit coupling HCEF+Hd−p+HSO. The red (blue) broken lines indicate the 3 d-orbital\n(2p-orbital) contributions for d-phybridized states. The black broken lines exhibit the contributions of d-phybridized states to\nthe spin-orbit coupled states. The green broken line indicates the origin of the energy (0 eV). Zeeman splitting of the spin-orbit\ncoupled states (b) ψ3±\nSO, (c)ψ2±\nSO, and (d) ψ1±\nSO. The solid (dashed) lines indicate the energy of the quantum state at the A (B)\nsite.\n3dorbitals of Cu ions and the 2 pmolecular orbitals of O 4\nclusters is described by the following Hamiltonian [20]:\nHd−p=X\nl′X\nlX\nσ,σ′tl′,l,σ,σ′\u0010\np†\nl′,σ′dl,σ+ h.c.\u0011\n.(14)\nHere, l=yz, zx, xy andl′=x, yare the indices of the\n3d- and 2 p-electron orbitals, σandσ′=↑,↓are the in-\ndices of spin, tl,l′,σ,σ′is a transfer energy between Cu-3 d\nand O-2 pelectrons [36], dl,σandd†\nl,σare an annihila-\ntion operator and a creation operator of 3 delectrons,\nandpl′,σ′andp†\nl′,σ′are an annihilation operator and a\ncreation operator of 2 pelectrons, respectively. We used\ntx,yz=−0.73 eV, tx,xy=−0.53 eV, and tx,zx=−0.41\neV to estimate the energy scheme (see Appendix D). We\nobtain eleven d-phybridized states and that of eigenen-\nergies to diagonalize HCEF+Hd−pshown in Appendix\nD. The energy scheme and the irrep of hybridized states\nare depicted in Fig. 9(a). The energy levels of ψ3,ψ4,\nandψ7can reproduce the electronic structure around the\nFermi level in the previous study [12]. As shown in Ta-\nble. V, the higher-level hybridized states ψ3andψ4with\nthe irrep Eandψ7with B2are mainly contributed by\n3dorbitals. The difference of the energy levels between\nψ3andψ4states and ψ7state is attributed to the energy\nscheme of yz,zx, and xyorbitals for HCEF. In contrast,TABLE V. Coefficients of ψyz,ψzx,ψE(1+),ψE(1−),ψE(2+),\nψE(2−),ψxy, and ψB2that constitute the wave function ψi\n(i= 3 - 8).\nψyz ψzxψE(1+)ψE(1−)ψE(2+)ψE(2−)ψxy ψB2\nψ3−0.66−0.66 0.31 0 0.17 0 0 0\nψ4−0.66 0.66 0 0.31 0 0.17 0 0\nψ50.25 0.25 0 0.82 0 0.46 0 0\nψ60.25−0.25 0.82 0.46 0 0 0\nψ70 0 0 0 0 0 −0.96 0.29\nψ80 0 0 0 0 0 −0.29−0.96\nthe low-energy states ψ5andψ6with Eandψ8with B2\nare mainly constructed by the 2 porbitals. The other five\nstates consist of the 2 porbitals.\nWe emphasize that the multipoles are active for the\nd-phybridized states. As shown in Eqs. (D26) - (D28) in\nAppendix D, ψ3andψ4states with the irrep Eprovide\nthe multipole matrices with non-zero elements. This re-\nsult is also confirmed in terms of the group theoretical\nanalysis for the ψ3andψ4states that are constructed\nbyyzandzxorbitals with the irrep E(g)andxandy\norbitals with the irrep E(u)because the decomposition\nof the direct product of ( E(g)⊕E(u))⊗(E(g)⊕E(u)) in-\ncludes B1(u)(Pz)⊕B1(g)(Ox2−y2)⊕B2(g)(Oxy). We also12\nTABLE VI. Coefficients of ψ3,↑,ψ4,↑,ψ7,↓,ψ3,↓,ψ4,↓, and ψ7,↑that constitute the wave function ψi±\nSO(i= 1 - 3) at a zero\nmagnetic field.\nψ3,↑ ψ4,↑ ψ7,↓ ψ3,↓ ψ4,↓ ψ7,↑\nψ1+\nSO −0.71 0 .71i 0 0 0 0\nψ1−\nSO 0 0 0 −0.71 −0.71i 0\nψ2+\nSO −0.50 + 0 .50i −0.50−0.50i −0.11 0 0 0\nψ2−\nSO 0 0 0 −0.50−0.50i −0.50 + 0 .50i 0.11\nψ3+\nSO −0.05 + 0 .05i −0.05−0.05i 0.99 0 0 0\nψ3−\nSO 0 0 0 0 .05 + 0 .05i 0.05−0.05i 0.99\nstress that both the matrix PzandOxyare proportional\nto the z-component of Pauli matrix, σz(see Eqs. (D27)\n- (D29)). Therefore, we can conclude that Pzis propor-\ntional to Oxyin a zero field.\nWe also need to reproduce the magnetic-field depen-\ndence of the polarization and elastic constants as exper-\nimentally observed. Thus, we introduce the spin-orbit\ncoupling of Cu-3 delectrons written as\nHSO=λSO(l·s). (15)\nHere, λSOis the coupling constant for 3 delectrons,\nl= (lx, ly, lz) is the vector form of the azimuthal an-\ngular momentums, and s= (sx, sy, sz) is the vector form\nof spin angular momentums. We used λSO=−0.1 eV\nfor calculations [37]. Using the d-phybridized states of\nH=HCEF+Hd−p, the matrix elements of H+HSO\nare written in Appendix E. Diagonalizing H+HSO, we\nobtain the wave functions describing the spin-dependent\nquantum states. The energy scheme of each state is illus-\ntrated in Fig. 9(a). As shown in Tables VI, ψ1+\nSO(ψ1−\nSO)\nconsists of ψ3,↑andψ4,↑(ψ3,↓andψ4,↓).ψ2+\nSO(ψ2−\nSO) is\ndominated by the contribution of ψ3,↑andψ4,↑(ψ3,↓and\nψ4,↓) rather than ψ7,↓(ψ7,↑). In contrast, ψ3+\nSO(ψ3−\nSO) is\nmainly constructed by ψ7,↓(ψ7,↑). We also show that\nthe low-energy states ψ1±\nSOconsist of the 3 d-yzandzx\norbitals and the molecular orbitals with the irrep E(see\nTable X in Appendix E). The ψ2±\nSOstates contain the xy\nand the molecular orbital with the irrep B2in addition to\nthese Eorbitals. In contrast, the high-energy states ψ3±\nSO\nconsist of these B2orbitals orbitals. We deduce that the\nreason why the ψ1+\nSOstates have the lowest energy is at-\ntributed to the energy scheme of yz,zx, and xyorbitals\nforHCEF.\nEach degenerate state formed by HSOmay seem to\ncarry the electric multipole degrees of freedom in terms\nof the contribution of ψ3,↑(3,↓)andψ4,↑(4,↓), however,\nthe diagonal elements are absent because ψ1±\nSO,ψ2±\nSO, and\nψ3±\nSOare each Kramers doublet. From a different per-\nspective, the diagonal elements of the multipole matrices\nbecome zero because the absolute values of the coeffi-\ncients of ψ3,↑(3,↓)andψ4,↑(4,↓)that constitute these three\nstates are equal. On the other hand, as shown in Eqs.\n(E28) - (E30) in Appendix E, ψ3,↑(3,↓)andψ4,↑(4,↓)in\nthese states bring about the off-diagonal elements be-tween these states. This fact indicates that mixing be-\ntween these three states provides multipole contributions\nto polarization and elasticity in fields. Therefore, we fo-\ncus on the Zeeman effect described below:\nHZeeman =−µB(l+ 2s)·B. (16)\nHere, µBis the Bohr magneton and B= (Bx, By, Bz) is\nthe vector form of magnetic fields. We show the matrix of\nH+HSO+HZeeman in Appendix F. Diagonalizing HSO+\nHZeeman for the d-phybridized wave functions, we obtain\nthe field dependence of the eigenenergies for the magnetic\nfields along the crystallographic orientation of [110] as\nshown in Figs. 9(b) - 9(d). The energy levels depend on\nmagnetic fields, but no level crossings or hybridized gaps\nare observed below 60 T, suggesting that such anomalous\nstates cannot be the origin of multipole response in high\nfields.\nTo discuss the origin of the multipole response, for in-\nstance, we show the coefficients of the wave functions at\n50 T for Asite (see Table VII). The proportion of ψ3,↑\n(ψ4,↓) in the lowest energy state ψ1+′\nSO(second-lowest en-\nergy state ψ1−′\nSO) in Fig. 9(d) increases with the applied\nfields. Due to this imbalance of the proportion of ψ3,↑(3,↓)\nandψ4,↑(4,↓), the diagonal elements of the multipole ma-\ntrices become non-zero value (see Table XI in Appendix\nF). In other words, the quantum states in high fields ob-\ntain the multipole degrees of freedom. Such imbalance of\nthe proportion of ψ3,↑(3,↓)andψ4,↑(4,↓)causes the field-\ninduced proportion change of the 3 d-yzandzxorbitals\nand 2 p-xandyorbitals.\nTo confirm the field-induced proportion change, we\ncalculated the field-dependence of the difference in the\ncoefficients of the wave functions, which was denoted\nby∆n(see Fig. 10). In zero magnetic fields, the val-\nues of |∆n|forψyzandψzx,ψE(1+) andψE(1−), and\nψE(2+)andψE(2−)are zero. On the other hand, |∆n|\nfor each wave function show finite values, indicating that\nmagnetic fields induce an anisotropic charge distribution\nbreaking spatial inversion and rotational operations of\ntheD2dpoint group. This anisotropy of charge distribu-\ntion can be the microscopic origin of the electric dipole\nand quadrupoles in high fields.13\nTABLE VII. Coefficients of ψ3,↑,ψ4,↑,ψ7,↓,ψ3,↓,ψ4,↓, and ψ7,↑that constitute the wave function ψi±′\nSO(i= 1 - 3) at 50 T for\nAsite.\nψ3,↑ ψ4,↑ ψ7,↓ ψ3,↓ ψ4,↓ ψ7,↑\nψ1+′\nSO 0.39 + 0 .33i 0.26−0.42i −0.01 −0.31−0.40i −0.34���0.35i −0.01\nψ1−′\nSO 0.42 + 0 .26i 0.33−0.39i 0.01 0 .35−0.34i 0.40−0.31i −0.01\nψ2+′\nSO −0.23 + 0 .44i −0.47−0.16i −0.07 + 0 .03i 0.38 + 0 .32i 0.38−0.33i −0.08\nψ2−′\nSO −0.16 + 0 .47i −0.44−0.23i −0.07 + 0 .03i −0.33−0.38i −0.33 + 0 .38i 0.01\nψ3+′\nSO −0.02 + 0 .05i −0.05−0.02i 0.65−0.26i 0.04 + 0 .04i 0.04−0.04i 0.70\nψ3−′\nSO 0.02−0.05i 0.05 + 0 .02i −0.65 + 0 .26i 0.04 + 0 .04i 0.04−0.04i 0.70\n-0.08\n-0.06\n-0.04\n-0.02\n0.00\n0.02Δn\n60\n50\n40\n30\n20\n10\n0\nB\n (T)\n yz−zx\n (A site)\n yz−zx\n (B site)\n E(1+ ) −E ( 1 − )\n (B site)\n E(2+ ) −E ( 2 − )\n (B site)\n E(2+ ) −E ( 2 − )\n (A site)\n E(1+ ) −E ( 1 − )\n(A site)\nFIG. 10. Magnetic-field dependence of the difference in\nthe coefficients of the wave functions ψyz,↑,ψzx↑,ψE(1+)↑,\nψE(1−)↑,ψE(2+)↑, and ψE(2−)↑that constitute the lowest-\nenergy states ψ1+\nSO. The solid (broken) lines indicate the ∆n\natAsite ( Bsite).\nB. Electric multipole susceptibility without\ninter-site interaction\nHere, we show the field dependence of the multipole\nsusceptibility of Pz,Ox2−y2, and Oxyand the thermal\naverage of Pzwithout inter-site interaction. Our analy-\nsis can explain the field-induced electric polarization and\nelastic softening.\nTo describe multipole properties, we focus on the char-\nacteristic crystal structure of Ba 2CuGe 2O7. We need to\nidentify CuO 4on the corner ( Asite) and the center ( B\nsite) of the crystal shown in Fig. 1(a). Relationship be-\ntween the crystallographic coordinate r= (x, y, z ) and\nthe local coordinate RA= (XA, YA, ZA) for A sites is\ndescribed as\nRA=R(κ)r (17)\nHere, κis a tilting angle of CuO 4tetrahedron at Asites\nandR(κ) is described as\nR(κ) =\ncosκ−sinκ0\nsinκcosκ0\n0 0 1\n. (18)Relation between randRB= (XB, YB, ZB) for B sites\nis also described as\nRB=R(−κ)r (19)\nAs shown in Eqs. (17) and (19), zcomponent is invariant\nfor this tilting structure.\nDue to the relation of Eqs. (17) and (19), the mul-\ntipoles in local coordinates are also described by those\nof the global coordinates. The electric quadrupoles\nOi\nX2−Y2andOi\nXYand electric dipole Pi\nZon the local\nsites i= (A, B) are written by a vector-type coordinate\nof multipoles O(r) =\u0000\nOx2−y2, Oxy, Pz\u0001\nfor the global\ncoordination and O(Ri) =\u0000\nOi\nX2−Y2, Oi\nXY, Pi\nZ\u0001\nfor the\nlocal coordinates as\nO(RA) =R(2κ)O(r), (20)\nO(RB) =R(−2κ)O(r). (21)\nHere, the quadrupoles for each coordinate are defined as\nfollows:\nOx2−y2=1√\n2x2−y2\nr2, (22)\nOxy=√\n2xy\nr2, (23)\nOi\nX2−Y2=1√\n2X2\ni−Y2\ni\nR2\ni, (24)\nOi\nXY=√\n2XiYi\nR2\ni. (25)\nThese quadrupoles for the local coordinates are written\nby the linear combination of Ox2−y2andOxy. Because\nthezcomponent is invariant for the in-plane tilting of\nCuO 4, the electric dipole Pzis also invariant for the ro-\ntational operation R(±2κ). We stress that Ox2−y2with\nthe irrep B1for the global coordinates includes Oi\nXYwith\nB2for the local coordinates.\nTo describe the electric quadrupole and dipole suscep-\ntibility, we also need to consider the quadrupole-strain in-\nteraction and the dipole-electric field interaction for both\nlocal and global coordinates. The rotational transforma-\ntion of Eqs. (17) and (19) provides the strains for local\ncoordinate (see Appendix G). The strains εi\nX2−Y2and14\nεi\nXYand electric field Ezfor the local coordinates are\nwritten by the global coordinate as\nε(RA) =R(2κ)ε(r), (26)\nε(RB) =R(−2κ)ε(r). (27)\nHere, ε(r) =\u0000\nεx2−y2, εxy, Ez\u0001\nand ε(Ri) =\u0000\nεi\nX2−X2, εi\nXY, Ei\nZ\u0001\nare the vector-type coordinates of ex-\nternal fields for the global and local coordinates, respec-\ntively. Thus, each strain for the local coordinates is writ-\nten by the linear combination of εx2−y2andεxy. The\nelectric field Ezis invariant for the rotational operationR(±2κ).\nAs discussed above, we obtained the multipoles and ex-\nternal fields for local coordinates. Thus, the quadrupole-\nstrain interaction for the local coordinates,\nHL\nQS=−X\ni=A,BX\nΓ=X2−Y2,XYgΓOi\nΓεi\nΓ (28)\nalso depends on Ox2−y2,Oxy,εx2−y2, and εxyas written\nin Eqs. (G2) and (G3) (see Appendix G). The partition\nfunction for electric quadrupoles for the site iis described\nas\nZi\u0000\nεi\nX2−Y2, εi\nXY, T, B\u0001\n=X\nΓ=X2−Y2,XYX\nlexp\"\n−Ei\nl\u0000\nεi\nΓ, B\u0001\nkBT#\n(29)\nHere, Ei\nl\u0000\nεi\nΓ, B\u0001\nis the second perturbation energy for the quantum state lin the magnetic fields Batisite of CuO 4\nclusters based on HL\nQSdescribed below:\nEi\nl\u0000\nεi\nΓ, B\u0001\n=Ei\nl\u0000\nεi\nΓ= 0, B\u0001\n−\nl, B|HL\nQS|l, B\u000b\n+X\nl′(̸=l)\nl, B|HL\nQS|l′, B\u000b\nl′, B|HL\nQS|l, B\u000b\nEi\nl\u0000\nεi\nΓ= 0, B\u0001\n−Ei\nl′\u0000\nεi\nΓ= 0, B\u0001 (30)\nHere, Ei\nl\u0000\nεi\nΓ= 0, B\u0001\nis the eigen energy of the non-purturbation Hamiltonian, HCEF+Hd−p+HSO+HZeeman .\nConsidering the free energy of quadrupoles for the local sites,\nF=Flattice +Felectronic (31)\n=X\nΓ1\n2C0\nΓε2\nΓ−X\niNikBTlnZi\u0000\nεi\nX2−Y2, εi\nXY, T, B\u0001\n,\nwhich include the elastic part Flattice for the global strain εΓand electronic part Felectronic , we show that the elastic\nconstants are written as\nC66(T, B) =∂2F\n∂ε2xy(32)\n=C0\n66−N\b\ng2\nXY\u0002\nχA\nXY(T, B) +χB\nXY(T, B)\u0003\ncos22κ+g2\nX2−Y2\u0002\nχA\nX2−Y2(T, B) +χB\nX2−Y2(T, B)\u0003\nsin22κ\n−gXYgX2−Y2\nkBT\u0000\nOA\nXY(T, B)\u000b\nOA\nX2−Y2(T, B)\u000b\n−\nOB\nXY(T, B)\u000b\nOB\nX2−Y2(T, B)\u000b\u0001\nsin 4κ\u001b\n,\nCT(T, B) =∂2F\n∂ε2\nx2−y2(33)\n=C0\nT−N\b\ng2\nXY\u0002\nχA\nXY(T, B) +χB\nXY(T, B)\u0003\nsin22κ+g2\nX2−Y2\u0002\nχA\nX2−Y2(T, B) +χB\nX2−Y2(T, B)\u0003\ncos22κ\n−gXYgX2−Y2\nkBT\u0000\n−\nOA\nXY(T, B)\u000b\nOA\nX2−Y2(T, B)\u000b\n+\nOB\nXY(T, B)\u000b\nOB\nX2−Y2(T, B)\u000b\u0001\nsin 4κ\u001b\n.\nHere, Nis half the number of CuO 4clusters in the unit cell and\nOi\nΓ(T, B)\u000b\ndenotes the thermal average for the\nBoltzmann statistics asP\nl\nl, B|Oi\nΓ|l, B\u000b\nexp\u0002\n−Ei\nl\u0000\nεi\nΓ, B\u0001\n/kBT\u0003\n/Zi\u0000\nεi\nX2−Y2, εi\nXY, T, B\u0001\f\f\fεi\nΓ→0. To calculate C66and\nCT, we used the transformation of the derivative with respect to the strains in Eqs. (G4) and (G5) (see Appendix.\nG). Because\nOA\nXY(T, B)\u000b\nOA\nX2−Y2(T, B)\u000b\n−\nOB\nXY(T, B)\u000b\nOB\nX2−Y2(T, B)\u000b\nin Eqs. (32) and (33) is on the oder of15\n10−6K−1at 60 T, we ignore this term. χi\nΓ(T, B) is the susceptibility of quadrupole OΓatisite described below:\n−g2\nΓχi\nΓ(T, B) =*\n∂2El\u0000\nεi\nX2−Y2, εi\nXY, B\u0001\n∂εi2\nΓ\f\f\fεi\nΓ→0+\n(34)\n−1\nkBT\n\n* \n∂El\u0000\nεi\nX2−Y2, εi\nXY, B\u0001\n∂εi\nΓ\f\f\fεi\nΓ→0!2+\n−*\n∂El\u0000\nεi\nX2−Y2, εi\nXY, B\u0001\n∂εi\nΓ\f\f\fεi\nΓ→0+2\n\n.\nWe also describe the electric dipole susceptibility. The electric dipole-electric field interaction for the local coordi-\nnates is written as\nHL\nDE=−X\ni=A,BgzPi\nzEi\nz. (35)\nThe partition function for electric dipole for the site iis described as\nZi\u0000\nEi\nz, T, B\u0001\n=X\nlexp\"\n−Ei\nl\u0000\nEi\nz, B\u0001\nkBT#\n(36)\nConsidering the second perturbation energy of Eq. (30) for HDEand the free energy for the electric dipole,\nF=−X\niNikBTlnZi\u0000\nEi\nz, T, B\u0001\n, (37)\nwe can describe the thermal average of the electric dipole Pzfor the Boltzmann statistics as below:\n\nPi\nz(T, B)\u000b\n=P\nl\nl, B|Pi\nz|l, B\u000b\nexp\u0002\n−Ei\nl\u0000\nEi\nz, B\u0001\n/kBT\u0003\nZi(Eiz, T, B )\f\fEiz→0. (38)\nHere, Ei\nl\u0000\nEi\nz, B\u0001\nis the second perturbation energy for HL\nDE. The susceptibility of Pzis written as\n−g2\nzχi\nz(T, B) =*\n∂2El\u0000\nEi\nz, B\u0001\n∂Ei2z\f\fEiz→0+\n−1\nkBT\n\n* \n∂El\u0000\nEi\nz, B\u0001\n∂Eiz\f\fEiz→0!2+\n−*\n∂El\u0000\nEi\nz, B\u0001\n∂Eiz\f\fEiz→0+2\n\n.(39)\nOur calculations demonstrate that the multipoles are\nactive for the quantum states in high fields. We show\nthe field dependence of multipole susceptibility for Aand\nBsites at 1.4 K (see Fig. 11(a)). These susceptibili-\nties of the electric dipole Pzand the electric quadrupoles\nOX2−Y2andOXYfor the local coordinates demonstrate\nthe increase in the fields. Furthermore, we show that the\nthermal average of the electric dipole denoted by ⟨Pz⟩for\nAsite (Bsite) at 1.4 K exhibits the increase (decrease) in\nthe fields. The sum of ⟨Pz⟩also shows the increase in the\nfield. These results are qualitatively consistent with the\nfield-induced electric polarization and elastic softening in\nBa2CuGe 2O7.\nUsing these susceptibilities and expectation values, we\nanalyzed our experimental results. Figure 12 shows the\nresults of the analysis. The FIEP above Bsatis described\nbyP0\nc+P1\nc×B+p× ⟨Pz⟩where P0\ncis the constant, P1\nc\nis the coefficient of the first-order term of B, and pis the\nmatrix element of Pzin Eq. (11). We obtaind P0\nc= 2.8\nµC/m2,P1\nc= 0.80µC/m2·T, and p= 4.0×103µC/m2.Furthermore, the transverse elastic constants CTandC66\nabove Bsatare described by Eqs. (32) and (33). We ob-\ntained C0\nT= 3.3664×1010J/m3,C0\n66= 3.3563×1010\nJ/m3,gX2−Y2= 0 K, and gXY= 1.9 K. This result indi-\ncates that the elastic softening of C66is attributed to the\nelectric quadrupole OXYwith the irrep B2gfor the local\ncoordinates. The softening of CToriginates from the re-\nsponse of OXYand the tilting crystal structure of CuO 4\ntetrahedra. Since pandgXYare attributed to the radial\nintegration of the wave functions, estimated values can\nreflect the spatial extent of these wave functions (see Ap-\npendix E.) We stress that this field-induced phenomenon\nis attributed to the proportion change of the 3 d-yzand\nzxorbitals and 2 p-xandyorbitals in the fields.\nWhile our model successfully provides a qualitative\nexplanation of the experimental results, several aspects\ncould be further improved. One is the field dependence\nof⟨Pz⟩. Our experimental and analytical results of FIEP\nshow the contribution of B-linear term in addition to\nour calculated ⟨Pz⟩. For quantitative explanations, it16\n40\n30\n20\n10\n0χ\n (K\n-1\n)\n Pz\n (B site)\n Pz\n (A site)\n Oxy\n (B-site)\n Oxy\n (A site)\n Ox2−y2\n (A site)\n Ox2−y2\n (B site)\n(a)\nT\n = 1.4 K\n2.0\n1.0\n0.0\n-1.0\n10\n3\n x \n〈Pz\n \n〉\n/p\n60\n50\n40\n30\n20\n10\n0\nB\n (T)\n(b)\nA site\nsum\nB site T\n = 1.4 K\nFIG. 11. (a) Magnetic-field dependence of the electric mul-\ntipole susceptibility at AandBsites of CuO 4clusters at 1.4\nK. The solid (broken) lines indicate the susceptibilities at the\nAsite (Bsite). (b) Magnetic-field dependence of the thermal\naverage of the electric dipole Pzfor the Boltzmann statistics\ndivided by pin Eq. (11) at 1.4 K. The dashed (broken) lines\nindicate the thermal average at the Asite (Bsite). The solid\nline shows the sum of the thermal averages for the AandB\nsites.\nmay be necessary to incorporate other interactions such\nas dipole-dipole and spin interactions to calculate the\nquantum state in high fields. Another is the qualita-\ntive explanation of gvalue. The models with the CEF,\nthed-phybridization, and the spin-orbit coupling pro-\nvide a qualitative description of the anisotropy of mag-\nnetization. Based on these wave functions and energy\nschemes, we can estimate an effective gvalue [38, 39] for\nthe [110] ([001]) crystallographic orientation to be 2 .13\n(2). While we can explain the enhancement of the g\nvalue greater than 2 for the in-plane field direction, there\nis room for improvement to reproduce our experimental\nresult of g= 2.44. By adjusting the energy levels of\neach orbital under the CEF as follows: ϵdB2= 0.1 eV,\nϵdE=−0.25 eV, and ϵpE=−2.5 eV, we can reproduce\ng∼2.44. We can also reproduce the gvalue by increas-\ning the magnitude of the d-phybridization. However, the\nenergy levels of the hybridized orbitals calculated from\nthese parameters are narrower compared to the previous\nstudy [12]. Another approach to reproduce g= 2.44 is\nto enhance the strength of the spin-orbit coupling from\nλSO=−0.1 eV to −0.35 eV. Nevertheless, such an en-\nhancement cannot be considered realistic since the mag-\nnitude of the spin-orbit coupling is determined by the\nFIG. 12. Analytical results of the field dependence of (a)\nthe electric polarization ∆Pcand (b) the elastic constant CT\nand (c) C66. The black lines indicate the fit of each physical\nquantity above Bsat.\natomic orbitals. Therefore, we deduce that other inter-\nactions, such as magnetic and anisotropic exchange inter-\nactions, contribute to the observed magnetic anisotropy\nin Ba 2CuGe 2O7. Although a quantitative explanation\nfor the energy levels cannot be achieved, it is possible\nto provide a quantitative explanation for the multipole\ndegrees of freedom originating from the quantum states\nin Fig. 9(a).\nC. Possible contribution of inter-site interaction\nto multipole susceptibility\nIn addition to the quadrupole-strain interaction,\nquadrupole-quadrupole interaction is also needed for the\nquadrupole susceptibility. The quadrupole-quadrupole\ninteraction for the global coordinate is written as\nHG\nQQ=−X\ni∈A,j∈BX\nΓ=x2−y2,xyGij\nΓOi\nΓOj\nΓ. (40)\nHere, Gij\nΓis an interaction coefficient. If we define the\nquadrupole-quadrupole interaction for local coordinates17\nusing the interaction coefficient Gij\nΓ′as\nHL\nQQ=−X\ni∈A,j∈BX\nΓ′=X2−Y2,XYGij\nΓ′Oi\nΓ′Oj\nΓ′, (41)\nHG\nQQis decomposed as below:\nHG\nQQ=HL\nQQ+Hcross. (42)\nHere, Hcross is additional Hamiltonian, which is similar\nto the DM interaction described as\nHcross=−X\ni∈A,j∈BGij\ncross\u0010\nOi\nX2−Y2Oj\nXY−Oi\nXYOj\nX2−Y2\u0011\n=−X\ni∈A,j∈BGij\ncross[O(Ri)×O(Rj)]z. (43)\nThe interaction coefficients Gij\nX2−Y2,Gij\nXY, and Gij\ncross\nare described as\nGij\nX2−Y2=Gij\nx2−y2cos22κ−Gij\nxysin22κ, (44)\nGij\nXY=−Gij\nx2−y2sin22κ+Gij\nxycos22κ, (45)\nGij\ncross=−\u0010\nGij\nx2−y2+Gij\nxy\u0011\nsin 4κ. (46)\nHcrossoriginates from the tilting crystal structure. The\nadditional term of the dipole-dipole interaction for Pz\ndoes not appear because Pzis invariant for the rota-\ntional operation R(κ). Considering the tilting angle κ\nin Ba 2CuGe 2O7, we deduce that sin 4 κinGij\ncross is not\nnegligible. Therefore, if the sum of the interaction coef-\nficients, Gij\nx2−y2+Gij\nxy, has a finite value, we should take\nHcrossinto account for calculating the multipole suscep-\ntibility.\nWe could calculate the quantum states and the\nquadrupole susceptibility with Hcross, however, it is hard\nto obtain analytical results. We desire the theoretical\ncalculations of the elastic constants and the polarization\nwith the inter-site quadrupole-quadrupole interaction in\nEq. (42) and the DM-like quadrupole-quadrupole inter-\naction in Eq. (43). Considering these interactions may\nprovide a more quantitative explanation of the experi-\nmental results. Although we ignore magnetic interac-\ntions for calculations, we believe that our calculations\nsuccessfully demonstrate the importance of the orbital\ncontributions to the field-induced multiferroicity. Taking\ninto account the magnetic interactions, our model can\nalso explain the FIEP and elastic softening below Bsat.\nV. CONCLUSION\nWe investigated magnetization, polarization, and elas-\ntic constants in Ba 2CuGe 2O7under high-magnetic fields\nto elucidate the contribution of electric dipoles and elec-\ntric quadrupoles to the multiferroicity. Above the spin\nsaturation fields, we found that the electric polarizationPcexhibited increasing up to 56 T and the elastic con-\nstants showed softening with the increase in the fields\nup to 60 T. Our theoretical calculation revealed that or-\nbital degrees of freedom of O-2 pand Cu-3 delectrons can\nplay a key role in the multiferroicity between the electric\npolarization and elastic constants in high-magnetic fields.\nThis characteristic phenomenon originates from the crys-\ntalline electric field, d-phybridization between Cu-3 dand\nO-2pelectrons, and the spin-orbit coupling of 3 delec-\ntrons. Cross-correlation between the electric dipole Pz\nand the electric quadrupole Oxysatisfying Pz∝Oxycan\nbe another possible multiferroic mechanism in such quan-\ntum systems.\nACKNOWLEDGMENT\nThe authors thank Yuichi Nemoto and Mitsuhiro\nAkatsu for supplying the LiNbO 3piezoelectric transduc-\ners. The authors also thank Kunihiko Yamauchi and Hi-\nroshi Yaguchi for valuable discussions. This work was\npartly supported by JSPS Grants-in-Aid for early-career\nscientists (KAKENHI JP20K14404, JP22K13999).\nAppendix A: Symmetry operations of the D2dpoint\ngroup for several basis functions\nWe show the representation matrices of the symmetry\noperations of the D2dpoint group for several basis func-\ntions in Table V. Considering the transformation of basis\nfunctions, we can reproduce the character table of Table\nI.\nAppendix B: Field-sweep dependence of the\nanomaly around Bc1\nIn this section, we discuss the origin of the hysteresis\nbehavior around Bc1appearing in the magnetization and\npolarization. Figure 13 shows the magnetic-field depen-\ndence of the electric polarization ∆Pcin Ba 2CuGe 2O7\nat several temperatures measured at several sweep rates\nof magnetic fields. In high-field measurements using the\npulsed magnet, the sweep time of the magnetic field is\nkept constant while varying the maximum value of the\nmagnetic field (see the inset of Fig. 13). As a result, a\nhigher maximum magnetic field corresponds to a faster\nsweep rate of magnetic fields. In these measurements,\nthe hysteresis behavior around BP\nc1seems to be indepen-\ndent of the sweep rate of magnetic fields. If the hystere-\nsis behavior is due to temperature changes induced by\nthe magnetic fields, we can expect that Bc1depends on\nthe time for the temperature to relax. In other words,\nBc1can depend on the sweep rate of the magnetic fields.\nTherefore, we concluded that the origin of this hystere-\nsis behavior is attributed to the field-induced magnetic18\nTABLE VIII. Representation matrix of the symmetry operation of the point group D2dfor the polar bases x,y, and z,\nquadratic bases yz,zx, and xy, the axial bases lx,ly, and lz, and the producted bases. The lines drawn within the matrix are\nused to distinguish between the block matrices of the two- and one-dimensional representations.\nBasis E IC 4 IC−1\n4 C2 Cy\n2 Cx\n2 σy\nd σx\nd \nx\ny\nz\n\n1 0 0\n0 1 0\n0 0 1\n\n010\n1 0 0\n0 0 1\n\n0 1 0\n1 0 0\n0 0 1\n\n1 0 0\n010\n0 0 1\n\n1 0 0\n0 1 0\n0 0 1\n\n1 0 0\n010\n0 0 1\n\n010\n1 0 0\n0 0 1\n\n0 1 0\n1 0 0\n0 0 1\n\n\nyz\nzx\nxy\n\n1 0 0\n0 1 0\n0 0 1\n\n010\n1 0 0\n0 0 1\n\n0 1 0\n1 0 0\n0 0 1\n\n1 0 0\n010\n0 0 1\n\n1 0 0\n0 1 0\n0 0 1\n\n1 0 0\n010\n0 0 1\n\n010\n1 0 0\n0 0 1\n\n0 1 0\n1 0 0\n0 0 1\n\n\nlx\nly\nlz\n\n1 0 0\n0 1 0\n0 0 1\n\n0 1 0\n1 0 0\n0 0 1\n\n010\n1 0 0\n0 0 1\n\n1 0 0\n010\n0 0 1\n\n1 0 0\n0 1 0\n0 0 1\n\n1 0 0\n010\n0 0 1\n\n0 1 0\n1 0 0\n0 0 1\n\n010\n1 0 0\n0 0 1\n\n60\n50\n40\n30\n20\n10\n0ΔPc\n (μ\nC/m\n2\n)\n60\n50\n40\n30\n20\n10\n0\nB\n (T)\nsweep rate of B\nBa\n2\nCuGe\n2\nO\n7\nB\n//[110]\nB\nc1P\n1.5 K\n1.5 K\n1.4 K\n60\n40\n20\n0B\n (T)\n30\n20\n10\n0\ntime (ms)\nFIG. 13. Magnetic-field dependence of the electric po-\nlarization ∆Pcin Ba 2CuGe 2O7at several temperatures for\nB//[110] measured at several sweep rates of magnetic fields.\nThe data sets are shifted consecutively along the ∆Pcaxes\nfor clarity. The inset shows the time dependence of magnetic\nfields.\nstructure change from the incommensurate spiral to the\ncommensurate one.Appendix C: Wave functions of Cu- 3dand O- 2p\nelectrons and CEF parameters\nWe show the wave functions describing Cu-3 delectrons\nand O-2 pelectrons as follows:\nψyz(r) =r\n15\n4πf3d(r)yz\nr2, (C1)\nψzx(r) =r\n15\n4πf3d(r)zx\nr2, (C2)\nψxy(r) =r\n15\n4πf3d(r)xy\nr2, (C3)\nψx(r) =r\n3\n4πf2p(r)x\nr, (C4)\nψy(r) =r\n3\n4πf2p(r)y\nr. (C5)\nHere, f3d(r) and f2d(r) are radial distribution functions\nfor 3dand 2 pelectrons, respectively. Based on the wave\nfunctions of 3 delectrons, we can describe the relationship\nbetween the energy levels ϵE(ϵB2) for the yzandzx(xy)\norbitals and the CEF parameters A0\n2A0\n4andA4\n4in Eq.\n(8) as below:\nϵE=1\n7A0\n2−4\n21A0\n4, (C6)\nϵB2=−2\n7A0\n2+4\n21A0\n4−√\n35\n21A4\n4. (C7)\nAppendix D: Molecular orbitals of CuO 4tetrahedra\nIn this section, we describe the molecular orbitals of\nCuO 4tetrahedra consisting of Cu-3 dand O-2 pelectrons.\nAt first, we describe the molecular orbitals of O 4tetra-\nhedron consisting of O-2 pelectrons. We consider the\nO-2porbitals ( pi\nx, pi\ny, pi\nz) at the O ifori= 1,2,3,4.\nThe symmetry transformations of the point group D2d\nfor these orbitals are summarized in Table IX. The19\nTABLE IX. Transformations of the O-2 porbitals ( pi\nx, pi\ny, pi\nz) at O i(i= 1,2,3,4) by the eight operations R(=E,C2,Cy\n2,Cx\n2,\nIC4,IC−1\n4,σy\nd,σx\nd) of the point group D2dand the characters χ(R).\nR p1\nx p2\nx p3\nx p4\nx p1\ny p2\ny p3\ny p4\ny p1\nz p2\nz p3\nz p4\nz χ(R)\nE p1\nx p2\nx p3\nx p4\nx p1\ny p2\ny p3\ny p4\ny p1\nz p2\nz p3\nz p4\nz 12\nIC4 −p4\ny −p1\ny −p2\ny −p3\ny p4\nx p1\nx p2\nx p3\nx −p4\nz −p1\nz −p2\nz −p3\nz 0\nIC−1\n4 p2\ny p3\ny p4\ny p1\ny −p2\nx −p3\nx −p4\nx −p1\nx −p2\nz −p3\nz −p4\nz −p1\nz 0\nC2 −p3\nx −p4\nx p1\nx −p2\ny −p3\ny −p4\ny −p1\ny −p2\ny p3\nz p4\nz p1\nz p2\nz 0\nCy\n2 −p2\nx −p1\nx −p4\nx −p3\nx p2\ny p1\ny p4\ny p3\ny −p2\nz −p1\nz −p4\nz −p3\nz 0\nCx\n2 p4\nx p3\nx p2\nx p1\nx −p4\ny −p3\ny −p2\ny −p1\ny −p4\nz −p3\nz −p2\nz −p1\nz 0\nσy\nd −p3\ny −p2\ny −p1\ny −p4\ny −p3\nx −p2\nx −p1\nx −p4\nx p3\nz p2\nz p1\nz p4\nz 2\nσx\nd p1\ny p4\ny p3\ny p2\ny p1\nx p4\nx p3\nx p2\nx p1\nz p4\nz p3\nz p2\nz 2\ncharacters χ(R) in Table IX leads the decomposition\n2A1⊕A2⊕B1⊕2B2⊕3E. Considering the projection\noperators for the irrep Γ of D2d, we obtain the twelve\nmolecular orbitals of O 4tetrahedron. Because 2 p-zor-bital does not contribute to the response of the electric\nmultipoles with the irreps B1andB2, we focus on the\neight molecular orbitals consisting of xandyorbitals\ndescribed as follows:\n\nψA1(r1,r2,r3,r4)\nψA2(r1,r2,r3,r4)\nψB1(r1,r2,r3,r4)\nψB2(r1,r2,r3,r4)\nψE(1+)(r1,r2,r3,r4)\nψE(1−)(r1,r2,r3,r4)\nψE(2+)(r1,r2,r3,r4)\nψE(2−)(r1,r2,r3,r4)\n=1\n2√\n2\n1 1 −1 1 −1−1 1 −1\n1−1 1 1 −1 1 −1−1\n1−1−1−1−1 1 1 1\n1 1 1 −1−1−1−1 1\n1 1 1 1 1 1 1 1\n1−1 1 −1 1 −1 1 −1\n1 1 −1−1 1 1 −1−1\n−1 1 1 −1−1 1 1 −1\n\nψx(r1)\nψy(r1)\nψx(r2)\nψy(r2)\nψx(r3)\nψy(r3)\nψx(r4)\nψy(r4)\n(D1)\nWe stress that the molecular orbitals with the irrep Ein Eq. (D1) and 3 d-yzandzxorbitals with the irrep Eprovide\nthe finite value of the matrix elements of the electric multipoles described as follows:\nOx2−y2=\nψyz ψzx ψE(1+) ψE(1−)ψE(2+) ψE(2−)\n−√\n2/7r2\nd 0 0 0 0 0\n0√\n2/7r2\nd 0 0 0 0\n0 0√\n2/5r2\np 0 0 0\n0 0 0 −√\n2/5r2\np 0 0\n0 0 0 0√\n2/5r2\np 0\n0 0 0 0 0 −√\n2/5r2\np\n, (D2)\nOxy=\n0√\n2/7r2\nd 0 0 0 0√\n2/7r2\nd 0 0 0 0 0\n0 0 0√\n2/5r2\np 0 0\n0 0√\n2/5r2\np 0 0 0\n0 0 0 0 0 −√\n2/5r2\np\n0 0 0 0 −√\n2/5r2\np 0\n, (D3)\nPz=\n0 0√\n2p−√\n2p0 0\n0 0√\n2p√\n2p0 0√\n2p√\n2p0 0 0 0\n−√\n2p√\n2p0 0 0 0\n0 0 0 0 0 0\n0 0 0 0 0 0\n. (D4)20\nThed-phybridization between 3 d-yz,zx, and xyorbitals and molecular orbitals in Eq. (D1) leads following matrix,\nHCEF+Hd−p, as\nHCEF+Hd−p\n=\nψyz ψzx ψE(1+) ψE(1−)ψE(2+) ψE(2−) ψxy ψB2 ψA1ψA2ψB1\nϵdE 0√\n2tx,yz√\n2tx,yz√\n2tx,zx√\n2tx,zx 0 0 0 0 0\n0 ϵdE√\n2tx,yz−√\n2tx,yz√\n2tx,zx√\n2tx,zx 0 0 0 0 0√\n2tx,yz√\n2tx,yz ϵpE 0 0 0 0 0 0 0 0√\n2tx,yz−√\n2tx,yz 0 ϵpE 0 0 0 0 0 0 0√\n2tx,zx√\n2tx,zx 0 0 ϵpE 0 0 0 0 0 0√\n2tx,zx−√\n2tx,zx 0 0 0 ϵpE 0 0 0 0 0\n0 0 0 0 0 0 ϵdB2 2√\n2tx,xy 0 0 0\n0 0 0 0 0 0 2√\n2tx,xy ϵpB2 0 0 0\n0 0 0 0 0 0 0 0 ϵpA10 0\n0 0 0 0 0 0 0 0 0 ϵpA20\n0 0 0 0 0 0 0 0 0 0 ϵpB1\n.\n(D5)\nHere, the diagonal elements εdΓandεpΓare the eigenenergies of CEF states, respectively. tx,yz,tx,zx, and tx,xyin\nthe matrix of Eq. (D5) are the transfer energy between 2 porbitals and 3 dorbitals written as [36]\ntx,yz=s2c\u0010√\n3Vpdσ−2Vpdπ\u0011\n, (D6)\ntx,zx=s2c\u0010√\n3Vpdσ−2Vpdπ\u0011\n+cVpdπ, (D7)\ntx,xy=s3\u0010√\n3Vpdσ−2Vpdπ\u0011\n+sVpdπ. (D8)\nHere, s= sin ( κ0+δθ)/√\n2,c= cos ( κ0+δθ), and cos2κ0= 1/3 are the structural parameters describing the\ndistortion of CuO 4tetrahedra. sin( κ0+δθ) = 0 .880 and cos( κ0+δθ) = 0 .474 are estimated by the structural\nparameters in Ba 2CuGe 2O7[13]. We used Vpdσ=−1.5 eV and Vpdπ=−0.45Vpdσ[40, 41] in Eqs. (D6) - (D8) to\nestimate the energy scheme. Diagonalizing the matrix of Eq. (D5), we obtains the eigenenergies as\nHCEF+Hd−p=\nψ1ψ2ψ3ψ4ψ5ψ6ψ7ψ8ψA1ψA2ψB1\nϵpE\nϵpE\nϵ+\nE\nϵ+\nE\nϵ−\nE\nϵ−\nE\nϵ+\nB2\nϵ−\nB2\nϵpA1\nϵpA2\nϵpB1\n. (D9)21\nHere, zeros in the off-diagonal elements are omitted for simplicity. The eigenenergies are described as follows:\nϵ±\nE=ϵdE+ϵpE\n2±δϵE, (D10)\nδϵE=s\u0012∆ϵE\n2\u00132\n+ 4h\n(tx,yz)2+ (tx,zx)2i\n, (D11)\n∆ϵE=ϵpE−ϵdE, (D12)\nϵ±\nB2=ϵdB2+ϵpB2\n2±δϵB2, (D13)\nδϵB2=s\u0012∆ϵB2\n2\u00132\n+ 8 (tx,xy)2, (D14)\n∆ϵB2=ϵpB2−ϵdB2. (D15)\nThus, the energy of six orbitals is shifted due to Hd−p. The eigenfunctions describing CuO 4molecular orbitals and\nthose coefficients are written as\nψ1(r1,r2,r3,r4) =tx,zxq\n(tx,yz)2+ (tx,zx)2ψE(1+)(r1,r2,r3,r4)−tx,yzq\n(tx,yz)2+ (tx,zx)2ψE(2+)(r1,r2,r3,r4),(D16)\nψ2=tx,zxq\n(tx,yz)2+ (tx,zx)2ψE(1−)−tx,yzq\n(tx,yz)2+ (tx,zx)2ψE(2−)(D17)\nψ3=1\n2Cϵ+\nE\u0012∆ϵE\n2−δϵE\u0013\nψyz+1\n2Cϵ+\nE\u0012∆ϵE\n2−δϵE\u0013\nψzx−√\n2tx,yz\nCϵ+\nEψE(1+)−√\n2tx,zx\nCϵ+\nEψE(2+) (D18)\nψ4=1\n2Cϵ+\nE\u0012∆ϵE\n2−δϵE\u0013\nψyz−1\n2Cϵ+\nE\u0012∆ϵE\n2−δϵE\u0013\nψzx−√\n2tx,yz\nCϵ+\nEψE(1−)−√\n2tx,zx\nCϵ+\nEψE(2−)(D19)\nψ5=1\n2Cϵ−\nE\u0012∆ϵE\n2+δϵE\u0013\nψyz+1\n2Cϵ−\nE\u0012∆ϵE\n2+δϵE\u0013\nψzx−√\n2tx,yz\nCϵ−\nEψE(1+)−√\n2tx,zx\nCϵ−\nEψE(2+) (D20)\nψ6=1\n2Cϵ−\nE\u0012∆ϵE\n2+δϵE\u0013\nψyz−1\n2Cϵ−\nE\u0012∆ϵE\n2+δϵE\u0013\nψzx−√\n2tx,yz\nCϵ−\nEψE(1−)−√\n2tx,zx\nCϵ−\nEψE(2−)(D21)\nψ7=1\nCϵB2\u0012∆ϵB2\n2−δϵB2\u0013\nψxy−2√\n2tx,xy\nCϵB2ψB2(D22)\nψ8=2√\n2tx,xy\nCϵB2ψxy+1\nCϵB2\u0012∆ϵB2\n2−δϵB2\u0013\nψB2(D23)\nCϵ±\nE=s\n1\n2\u0012∆ϵE\n2∓δϵE\u00132\n+√\n2 (tx,yz)2+√\n2 (tx,zx)2(D24)\nCϵB2=s\u0012∆ϵB2\n2−δϵB2\u00132\n+ 8 (tx,xy)2.(D25)\nHere, ( r1,r2,r3,r4) in Eqs. (D17) - (D25) is ommited for simplicity.\nConsidering these molecular orbitals with the irrep E, we can describe the matrices of multipoles as follows:\nOx2−y2=C−2\nϵ+\nE\nψ3 ψ4\n0√\n2\n14r2\nd\u0000∆ϵE\n2−δϵE\u00012+2√\n2\n5r2\np\u0000\nt2\nx,yz−t2\nx,zx\u0001\n√\n2\n14r2\nd\u0000∆ϵE\n2−δϵE\u00012+2√\n2\n5r2\np\u0000\nt2\nx,yz−t2\nx,zx\u0001\n0\n,(D26)\nOxy=C−2\nϵ+\nE √\n2\n14r2\nd\u0000∆ϵE\n2−δϵE\u00012+2√\n2\n5r2\np\u0000\nt2\nx,yz+t2\nx,zx\u0001\n0\n0 −√\n2\n14r2\nd\u0000∆ϵE\n2−δϵE\u00012−2√\n2\n5r2\np\u0000\nt2\nx,yz+t2\nx,zx\u0001!\n,(D27)\nPz=C−2\nϵ+\nE\u0012\n−4ptx,yz\u0000∆ϵE\n2−δϵE\u0001\n0\n0 4 ptx,yz\u0000∆ϵE\n2−δϵE\u0001\u0013\n. (D28)22\nUsing the z-component of the Pauli matrix,\nσz=\u0012\n1 0\n0−1\u0013\n, (D29)\nwe obtain the relations between the electric multipoles and the Paul matrix, Oxy∝σzandPz∝σz. Therefore,\nanomalous elasticity as a result of electric quadrupoles in addition to the dipole responses is expected in the CuO 4\nmolecular orbitals. Because CuO 4tetrahedra are compressed along the c-axis, the eigenenergies of the yzandzx\norbitals for HCEFare lower than that of the xyorbital. On the other hand, ψ3andψ4wave functions are mainly\nconstructed by yzandzxorbitals. We deduce that these energy levels for HCEFcan be the reason why the Hilbert\nspace spanned by ψ3andψ4becomes important both in zero and high-magnetic fields.\nAppendix E: Spin-orbit coupling and eigenfunctions\nIn this section, we describe the spin-orbit coupling for the molecular orbitals. First, we write the matrix of spin-orbit\ncoupling for the yz,zx, and xyorbitals as\nHSO=λSO\n2\nψyz,↑ψzx,↑ψxy,↓ψyz,↓ψzx,↓ψxy,↑\n0 i −1 0 0 0\n−i 0 i 0 0 0\n−1−i 0 0 0 0\n0 0 0 0 −i 1\n0 0 0 i 0 i\n0 0 0 1 −i 0\n(E1)\nThus, we can write the matrix of H(=HCEF+Hd−p) +HSOas below:\nH+HSO=\nψ3,↑ ψ4,↑ ψ5,↑ ψ6,↑ ψ7,↓ ψ8,↓\nϵ+\nE −iα+\nE 0 −iβE ei3π\n4γ+ei3π\n4ρ+\niα+\nE ϵ+\nE iβE 0 e−i3π\n4γ+e−i3π\n4ρ+\n0 −iβE ϵ−\nE −iα−\nE ei3π\n4γ−ei3π\n4ρ−\niβE 0 iα−\nE ϵ−\nE e−i3π\n4γ−e−i3π\n4ρ−\ne−i3π\n4γ+ei3π\n4γ+e−i3π\n4γ−ei3π\n4γ−ϵ+\nB20\ne−i3π\n4ρ+ei3π\n4ρ+e−i3π\n4ρ−ei3π\n4ρ−0 ϵ−\nB2\n⊕(C.C.). (E2)\nHere, we set matrix elements for the convenience as below:\nα±\nE=λSO\n4\u0010\nCϵ±\nE\u00112\u0012∆ϵE\n2∓δϵE\u00132\n, (E3)\nβE=λSO\n4Cϵ+\nECϵ−\nE\"\u0012∆ϵE\n2\u00132\n−δϵ2\nE#\n, (E4)\nγ±=λSO\n2√\n2Cϵ±\nECϵB2\u0012∆ϵE\n2∓δϵE\u0013\u0012∆ϵB2\n2−δϵB2\u0013\n, (E5)\nρ±=λSO\nCϵ±\nECϵB2\u0012∆ϵE\n2∓δϵE\u0013\nt1\nx,xy. (E6)\nWe ignore ψ1andψ2because of the absence of 3 dorbitals in these wave functions. We deduce that the matrix\nelements between the high-energy states of ψ3,ψ4, and ψ7and low-energy states of ψ5,ψ6, and ψ8are negligible\nbecause the spin-orbit coupling constant λSO∼0.1 eV is much smaller than the energy gap of ∼6 eV. Thus, we can23\nwrite H+HSOas following simple form:\nH+HSO=\nψ3,↑ ψ4,↑ ψ7,↓ ψ5,↑ ψ6,↑ ψ8,↓\nϵ+\nE −iα+\nE ei3π\n4γ+0 0 0\niα+\nE ϵ+\nE e−i3π\n4γ+0 0 0\ne−i3π\n4γ+ei3π\n4γ+ϵ+\nB20 0 0\n0 0 0 ϵ−\nE −iα−\nE ei3π\n4ρ−\n0 0 0 iα−\nE ϵ−\nE e−i3π\n4ρ−\n0 0 0 e−i3π\n4ρ−ei3π\n4ρ−ϵ−\nB2\n⊕(C.C.). (E7)\nTo diagonalize H+HSO, we obtain the eigenenergies and eigenstates described as follows:\nH+HSO=\nψ1+\nSOψ1−\nSOψ2+\nSOψ2−\nSOψ3+\nSOψ3−\nSOψ4+\nSOψ4−\nSOψ5+\nSOψ5−\nSOψ6+\nSOψ6−\nSO\nϵ1\nSO\nϵ1\nSO\nϵ2\nSO\nϵ2\nSO\nϵ3\nSO\nϵ3\nSO\nϵ4\nSO\nϵ4\nSO\nϵ5\nSO\nϵ5\nSO\nϵ6\nSO\nϵ6\nSO\n(E8)\nψ1+\nSO=1√\n2(−ψ3,↑+iψ4,↑), (E9)\nψ2+\nSO=1\nCϵ2\nSO\"\nγ+(iψ3,↑−ψ4,↑) +ei3π\n4 \nϵ+\nE−ϵ+\nB2+α+\nE\n2−δϵ+\nSO!\nψ7,↓#\n, (E10)\nψ3+\nSO=1\nCϵ3\nSO\"\nγ+(iψ3,↑−ψ4,↑) +ei3π\n4 \nϵ+\nE−ϵ+\nB2+α+\nE\n2+δϵ+\nSO!\nψ7,↓#\n, (E11)\nψ4+\nSO=1√\n2(ψ5,↑−iψ6,↑), (E12)\nψ5+\nSO=1\nCϵ5\nSO\"\nρ−(iψ5,↑−ψ6,↑) +ei3π\n4 \nϵ−\nE−ϵ−\nB2+α−\nE\n2−δϵ−\nSO!\nψ8,↓#\n, (E13)\nψ6+\nSO=1\nCϵ6\nSO\"\nρ−(iψ5,↑−ψ6,↑) +ei3π\n4 \nϵ−\nE−ϵ−\nB2+α−\nE\n2+δϵ−\nSO!\nψ8,↓#\n, (E14)\nψj−\nSO=Tψj+\nSO (E15)24\nϵ1\nSO=ϵ+\nE−α+\nE, (E16)\nϵ2\nSO=ϵ+\nE+ϵ+\nB2+α+\nE\n2+δϵ+\nSO, (E17)\nϵ3\nSO=ϵ+\nE+ϵ+\nB2+α+\nE\n2−δϵ+\nSO, (E18)\nϵ4\nSO=ϵ−\nE−α−\nE, (E19)\nϵ5\nSO=ϵ−\nE+ϵ−\nB2+α−\nE\n2+δϵ−\nSO, (E20)\nϵ6\nSO=ϵ−\nE+ϵ−\nB2+α−\nE\n2−δϵ−\nSO, (E21)\nδϵ+\nSO=vuut \nϵ+\nE−ϵ+\nB2+α+\nE\n2!2\n+ 2 (γ+)2, (E22)\nδϵ−\nSO=vuut \nϵ−\nE−ϵ−\nB2+α−\nE\n2!2\n+ 2 (ρ−)2, (E23)\nCϵ2\nSO=vuut2 (γ+)2+ \nϵ+\nE−ϵ+\nB2+α+\nE\n2−δϵ+\nSO!2\n, (E24)\nCϵ3\nSO=vuut2 (γ+)2+ \nϵ+\nE−ϵ+\nB2+α+\nE\n2+δϵ+\nSO!2\n, (E25)\nCϵ5\nSO=vuut2 (ρ−)2+ \nϵ−\nE−ϵ−\nB2+α−\nE\n2−δϵ−\nSO!2\n, (E26)\nCϵ6\nSO=vuut2 (ρ−)2+ \nϵ−\nE−ϵ−\nB2+α−\nE\n2+δϵ−\nSO!2\n. (E27)\nHere, Tindicates the time reversal operator. We omitted zero elements in this matrix for convenience. These\nwave functions are also written by the yz,zx, and xyorbitals and the molecular orbitals. Taking into account the\nconstruction of the wave functions ψ3,ψ4, and ψ7in Table V, we show the norm of the coefficients of such wave\nfunctions (see Table X).\nTABLE X. Norm of the coefficients of ψyz,↑,ψzx↑,ψE(1+)↑,ψE(1−)↑,ψE(2+)↑,ψE(2−)↑,ψxy↑, and ψB2↑that constitute the\nwave function ψi±\nSO(i= 1 - 3) at 0 T for Asite.\nψyz,↑ ψzx,↑ ψE(1+),↑ ψE(1−),↑ ψE(2+),↑ ψE(2−),↑ ψxy,↑ ψB2,↑\nψ1+\nSO 0.66 0.66 0.22 0.22 0.12 0.12 0 0\nψ1−\nSO 0 0 0 0 0 0 0 0\nψ2+\nSO 0.66 0.66 0.22 0.22 0.12 0.12 0 0\nψ2−\nSO 0 0 0 0 0 0 0.10 0.03\nψ3+\nSO 0.07 0.07 0.02 0.02 0.01 0.01 0 0\nψ3−\nSO 0 0 0 0 0 0 0.95 0.2925\nFor these states, we can calculate the matrix elements of Ox2−y2,Oxy, andPzas below:\nOx2−y2=\nψ1+\nSO ψ1−\nSO ψ2+\nSO ψ2−\nSO ψ3+\nSO ψ3−\nSO\n0 0 −0.15 + 0 .18i 0 −0.02 + 0 .02i 0\n0 0 0 −0.15−0.18i 0 0 .02 + 0 .02i\n−0.15−0.18i 0 0 0 0 0\n0 −0.15−0.18i 0 0 0 0\n−0.02−0.02i 0 0 0 0 0\n0 0 .02−0.02i 0 0 0 0\n,(E28)\nOxy=\n0 0 0 .20 + 0 .16i 0 0 .02 + 0 .02i 0\n0 0 0 0 .20−0.16i 0 −0.02 + 0 .02i\n0.20−0.16i 0 0 0 0 0\n0 0 .20 + 0 .16i 0 0 0 0\n0.02−0.02i 0 0 0 0\n0 −0.02−0.02i 0 0 0 0\n, (E29)\nPz=\n0 0 −0.31−0.25i 0 −0.03−0.03i 0\n0 0 0 −0.31 + 0 .25i 0 0 .03−0.03i\n−0.31 + 0 .25i 0 0 0 0 0\n0 −0.31−0.25i 0 0 0 0\n−0.03 + 0 .03i 0 0 0 0 0\n0 0 .03 + 0 .03i 0 0 0 0\n. (E30)\nTo simplify the calculations, we computed the matrix elements with r2\nd= 1 and r2\np= 1. These real values are considered\nto be incorporated into the experimentally determined pandgXY. These three states carry the off-diagonal elements,\nindicating that the mixing of the wave functions by the Zeeman effect provides the multipole degrees of freedom.\nAppendix F: Quantum states in magnetic fields\nHere, we consider the Zeeman effects for the spin-dependent quantum states. The Zeeman effect HZeeman =\n−µB(l+ 2s)·Bfor the in-plane magnetic field directions B= (B0cosϕ, B 0sinϕ,0) is written in the following form:\nHZeeman =−µBB0(lxcosϕ+lysinϕ)−2µBB0\u0000\ne−iϕs++eiϕs−\u0001\n(F1)\nHere, s+=sx+isyands−=sx−isyare ladder operators. If the strong magnetic field is applied along the [110]\ndirection, we deduce that the point group symmetry lowers from D2dtoD2. In the D2, both electric dipole Pzand\nelectric quadrupole Oxybelong to the irrep B1. Even if the point group symmetry shows indeed lowering, we can\nstill expect the FIEP and elastic softening. For the yz,zx, and xyorbitals of 3 delectrons, the matrix of HZeeman is\nwritten as\nHd\nZeeman =−µBB0\nψyz,↑ ψzx,↑ ψxy,↓ψyz,↓ ψzx,↓ ψxy,↑\n0 0 −isinϕ2e−iϕ0 0\n0 0 icosϕ 0 2 e−iϕ0\nisinϕ−icosϕ 0 0 0 2 e−iϕ\n2eiϕ0 0 0 0 −isinϕ\n0 2 eiϕ0 0 0 icosϕ\n0 0 2 eiϕisinϕ−icosϕ\n. (F2)\nThe matrix of HZeeman for the xandyorbitals of 2 pelectrons is also written as\nHp\nZeeman =−µBB0\nψx,↑ψy,↑ψx,↓ψy,↓\n0 0 2 e−iϕ0\n0 0 0 2 e−iϕ\n2eiϕ0 0 0\n0 2 eiϕ0 0\n. (F3)26\nTABLE XI. Diagonal elements of the electric multipoles Ox2−y2,Oxy, and Pzfor the quantum states ψ1+′\nSO,ψ1−′\nSO,ψ2+′\nSO,ψ2−′\nSO,\nψ3+′\nSO, and ψ3−′\nSOat 50 T for Asite.\nψ1+′\nSO ψ1−′\nSO ψ2+′\nSO ψ2−′\nSO ψ3+′\nSO ψ3−′\nSO\nOx2−y2 −3.20×10−23.17×10−23.32×10−2−3.28×10−2−5.10×10−55.00×10−5\nOxy 9.31×10−3−9.25×10−3−8.12×10−47.45×10−41.05×10−4−0.96×10−4\nPz −1.43×10−21.42×10−21.25×10−3−1.14×10−3−1.61×10−41.47×10−4\nThus, we obtain the matrix of H+HSO+HZeeman as follow:\nH+HSO+HZeeman\n=\nψ3,↑ ψ4,↑ ψ7,↓ ψ3,↓ ψ4,↓ ψ7,↑\nϵ+\nE −iα+\nE ei3π\n4γ+−2µBB0e−iϕ0 iµBB0gϕ−\niα+\nE ϵ+\nE e−i3π\n4γ+0 −2µBB0e−iϕ−iµBB0gϕ+\ne−i3π\n4γ+ei3π\n4γ+ϵ+\nB2−iµBB0gϕ− iµBB0gϕ+−2µBB0e−iϕ\n−2µBB0eiϕ0 iµBB0gϕ− ϵ+\nE iα+\nE −e−i3π\n4γ+\n0 −2µBB0eiϕ−iµBB0gϕ+ −iα+\nE ϵ+\nE −ei3π\n4γ+\n−iµBB0gϕ−iµBB0gϕ+−2µBB0eiϕ−ei3π\n4γ+−e−i3π\n4γ+ϵ+\nB2\n. (F4)\nHere, we set several matrix elements for the convenience as\ng=1\n2Cϵ+\nECϵB2\u0012∆ϵE\n2∓δϵE\u0013\u0012∆ϵB2\n2−δϵB2\u0013\n, (F5)\nϕ±= cos ϕ±sinϕ. (F6)\nH+HSO+HZeeman is hard to diagonalize analytically. So, we obtain the eigenstates and eigenenergies to diagonalize\nthis Hamiltonian numerically by Julia language. The field dependence of eigenenergies for B//[110] is shown in Figs,\n9(b) - 9(d). Based on these calculations, for example, we can describe the diagonal elements of these multipole\nmatrices in a magnetic field of 50 T at Asite (see Table XI).\nAppendix G: Local and global coordinates of strains and quadrupoles\nThe strains for local coordinates are written by that for global coordinates as below:\n\nεA(B)\nXX εA(B)\nXY εA(B)\nXZ\nεA(B)\nY X εA(B)\nY Y εA(B)\nY Z\nεA(B)\nZX εA(B)\nZY εA(B)\nZZ\n\n=\nεxxcos2κ∓εxysin 2κ+εyysin2κ ε xycos 2κ±1\n2(εxx−εyy) sin 2 κ ε zxcosκ∓εyzsinκ\nεxycos 2κ±1\n2(εxx−εyy) sin 2 κ ε xxcos2κ±εxysin 2κ+εyysin2κ±εzxsinκ+εyzcosκ\nεzxcosκ∓εyzsinκ ±εzxsinκ+εyzcosκ ε zz\n (G1)\nHere, we set sign = +( −) for i=A(B). These transformations lead to Eqs. (26) and (27). Because of the\ntransformation for the strains in Eq (G1) and quadrupoles in Eqs (22) - (25), the quadrupole-strain coupling for the\nlocal coordinates in Eq. (28) is described by the global coordinates as below:\nHL\nQS=−gX2−Y2Oi\nX2−Y2εi\nX2−Y2\n=−gX2−Y2\u0000\nOx2−y2εx2−y2cos22κ∓Ox2−y2εxysin 4κ∓Oxyεx−y2sin 4κ+Oxyεxysin22κ\u0001\n, (G2)\nHL\nQS=−gXYOB\nXYεB\nX2−Y2\n=−gX2−Y2\u0000\nOx2−y2εx2−y2sin22κ±Ox2−y2εxysin 4κ±Oxyεx−y2sin 4κ+Oxyεxycos22κ\u0001\n. (G3)27\nIn addition, the derivative with respect to the strains and the electric field for global coordinates are written by those\nfor local coordinates as below:\n\n∂εx2−y2\n∂εxy\n∂Ez\n=R(2κ)\n∂εA\nX2−Y2\n∂εA\nXY\n∂Ez\n, (G4)\n\n∂εx2−y2\n∂εxy\n∂Ez\n=R(−2κ)\n∂εB\nX2−Y2\n∂εB\nXY\n∂Ez\n. (G5)\nThese derivatives lead to the elastic constants of Eqs. (32) and (33) and the quadrupole susceptibility of Eq. (34).\n[1] M. Fiebig, Revival of the magnetoelectric effect, J. Phys.\nD: Appl. Phys. 38R123 (2005).\n[2] W. G. 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Phenom. 92, 97 (1998)."    },    {        "title": "2402.13506v1.Towards_Efficient_Verification_of_Constant_Time_Cryptographic_Implementations.pdf",        "content": "Towards Efficient Verification of Constant-Time\nCryptographic Implementations\nLUWEI CAI, ShanghaiTech University, China\nFU SONG∗†,State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences,\nChina\nTAOLUE CHEN, Birkbeck, University of London, UK\nTiming side-channel attacks exploit secret-dependent execution time to fully or partially recover secrets of\ncryptographic implementations, posing a severe threat to software security. Constant-time programming\ndiscipline is an effective software-based countermeasure against timing side-channel attacks, but developing\nconstant-time implementations turns out to be challenging and error-prone. Current verification approach-\nes/tools suffer from scalability and precision issues when applied to production software in practice. In this\npaper, we put forward practical verification approaches based on a novel synergy of taint analysis and safety\nverification of self-composed programs. Specifically, we first use an IFDS-based lightweight taint analysis to\nprove that a large number of potential (timing) side-channel sources do not actually leak secrets. We then\nresort to a precise taint analysis and a safety verification approach to determine whether the remaining\npotential side-channel sources can actually leak secrets. These include novel constructions of taint-directed\nsemi-cross-product of the original program and its Boolean abstraction, and a taint-directed self-composition\nof the program. Our approach is implemented as a cross-platform and fully automated tool CT-Prover .\nThe experiments confirm its efficiency and effectiveness in verifying real-world benchmarks from modern\ncryptographic and SSL/TLS libraries. In particular, CT-Prover identify new, confirmed vulnerabilities of\nopen-source SSL libraries (e.g., Mbed SSL, BearSSL) and significantly outperforms the state-of-the-art tools.\nCCS Concepts: •Software and its engineering →Formal software verification ;•Theory of computation\n→Program analysis ;•Security and privacy →Formal security models ;Logic and verification .\nAdditional Key Words and Phrases: Timing side-channel, constant-time cryptographic implementation, formal\nverification, taint analysis\nACM Reference Format:\nLuwei Cai, Fu Song, and Taolue Chen. 2024. Towards Efficient Verification of Constant-Time Cryptographic\nImplementations. Proc. ACM Softw. Eng. 1, FSE, Article 46 (July 2024), 24 pages. https://doi.org/10.1145/3643772\n1 INTRODUCTION\nThe security of contemporary software systems and communication heavily depends upon cryp-\ntographic implementations, which are the main focus of the current paper. Timing side-channel\nattacks [ 30,54] can exploit secret-dependent execution time to fully or partially recover secrets\neven remotely, thus posing a severe threat to software security. Over the past few years, numerous\n∗Corresponding author\n†Also with University of Chinese Academy of Sciences, and Nanjing Institute of Software Technology.\nAuthors’ addresses: Luwei Cai, ShanghaiTech University, Shanghai, China, cailw@shanghaitech.edu.cn; Fu Song, songfu@\nios.ac.cn, State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China;\nTaolue Chen, Birkbeck, University of London, London, UK, t.chen@bbk.ac.uk.\nPermission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee\nprovided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and\nthe full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses,\ncontact the owner/author(s).\n©2024 Copyright held by the owner/author(s).\nACM 2994-970X/2024/7-ART46\nhttps://doi.org/10.1145/3643772\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.arXiv:2402.13506v1  [cs.CR]  21 Feb 202446:2 Luwei Cai, Fu Song, and Taolue Chen\ntiming side-channel vulnerabilities have been discovered, allowing adversaries to deduce secrets\nwith very few trials. Notorious examples include the Lucky 13 attack that can remotely recover\nplaintext from the CBC-mode encryption in TLS and DTLS (due to an unbalanced branching\nstatement [ 9]), and Brumley and Boneh’s remote private key recovery attack against the sliding\nwindow exponentiation of the RSA decryption in OpenSSL [ 30,31]. It is vital to implement effective\ncountermeasures to protect cryptographic implementations.\nThere are different means to mitigate the risk of timing side-channels, for instance, via security-\naware system (e.g., [ 56]) and architecture (e.g., [ 39]). A more effective approach is to come up with\nbetter software implementation to eliminate the root cause. Currently, there are two prevailing\ncountermeasures, i.e., constant-time and time-balancing programming disciplines. The former\nrequires that control flow and memory-access patterns of an implementation are independent of\nthe secrets; the latter requires the execution time to be negligibly influenced by secrets which can\nbe considered as a tradeoff between security and enforceability. Both countermeasures have been\nadopted in open-source cryptographic and SSL/TLS libraries such as NaCl [ 24], BearSSL [ 61], Mbed\nTLS [ 72], s2n-tls [ 16], and OpenSSL [ 4]. Writing constant-time or time-balancing implementations\nrequires the use of low-level programming languages or compiler knowledge, and developers\nusually need to deviate from standard programming practices. Furthermore, their correctness\ndepends on global properties across pairs of executions, hence is difficult to reason about. For\ninstance, even though two protections against Lucky 13 were implemented in s2n-tls, the Lucky\nmicroseconds attack, a variant of Lucky 13, can remotely and completely recover plaintext from\nthe CBC-mode cipher suites in s2n-tls, resulting in a complete recovery of HTTP session cookies\nand user credentials such as BasicAuth passwords [8]. Alas, timing side-channel attacks remain a\nlive threat for cryptographic libraries after their discovery over 25 years ago [ 83], and it is essential\nto develop automated reasoning tools for formally verifying these countermeasures.\nIn this paper, we focus on the constant-time programming discipline as it represents a more\nfundamental solution and is more challenging to tackle. Numerous verification approaches have\nbeen proposed (cf. Section 6). A majority of them leverage (lightweight) static analysis, e.g., abstract\ninterpretation, type system, taint analysis and (relational) symbolic execution which are sound\n(and usually efficient) but incomplete (i.e., false positives may occur or depth of explored paths is\nbound). Different from them, the self-composition [ 11] approach reduces constant-time verification\nto safety verification by building a program consisting of two copies of the given program and\nverifying whether the values of the low-security (public) variables in the two copies are identical\nprovided that the public inputs are identical. Self-composition based approaches are sound and\ncomplete in theory, although in practice, the completeness may depend on verification tools to\ncarry out challenging safety property checking on self-composed programs (cf. Section 5.1 for\nconcrete examples).\nSelf-composition based approaches are normally considered as a heavyweight approach which\nis not scalable. The primary reason is that they need to deal with a self-composed program of\nquadratic sizes. As a result, many efforts have been made to ease safety verification of self-composed\nprograms, including self-composition with lockstep execution of loops [ 68] for𝑘-safety properties;\nself-composition with lockstep execution of both loops and branches (called cross-product) [ 11]\nfor constant-time properties; type-directed self-composition [ 71] and lazy self-composition [ 81]\nfor proving information flow properties. The commonality of these approaches is to simplify self-\ncomposed programs, reduce the number of safety checks, and/or keep variables from the two copies\nnear each other.\nDespite these efforts, a noticeable gap exists in verifying constant-time countermeasures in\nsoftware engineering practice. To the best of our knowledge, ct-verif [11] is the only publicly\navailable self-composition based tool that is being actively deployed in the continuous integration\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.Towards Efficient Verification of Constant-Time Cryptographic Implementations 46:3\nof Amazon’s s2n-tls library [ 67], but is significantly less efficient than lightweight static analysis\napproaches (e.g., taint analysis and type system), and often fails to prove constant-time implemen-\ntations (cf. Section 5). In summary, the current status is that the user either chooses an incomplete\napproach but needs to tackle false positives, or chooses a (relatively) complete approach which is\ncostly and may fail on a number of occasions.\nThe main purpose of the current paper is to make the self-composition based approaches scalable\nso they can be used to handle the verification of constant-time cryptographic implementations at\nthe industry level. Our strategy towards both completeness and scalability is to take a stratified\napproach. Technically, starting from potential timing side-channel sources which can be identified\nstraightforwardly (cf. Section 2.2), we devise two different taint analyses and a taint-directed cross-\nproduct, and gradually integrate them to resolve these potential sources, i.e., to determine whether\nthey can actually cause information leakage.\nThe first taint analysis (cf. Section 4.1) leverages the inter-procedural, finite, distributive, subset\nframework (IFDS [ 64]), which is accelerated by propagating the data-flow facts sparsely [ 60]. This\ntaint analysis is flow-, field- and context-sensitive, but path- and index-insensitive. It is often able\nto efficiently prove that a large number of potential (timing) side-channel sources do not actually\nleak secrets, leaving a relatively small number of them unresolved to the next step.\nThe second taint analysis (cf. Section 4.2) uses a novel taint-directed semi-cross-product , which\nreduces the flow-, context-, path-, field- and index-sensitive taint analysis problem to checking\nsafety properties of the cross-product of the given program and its Boolean abstraction. The\nBoolean abstraction is used to track the required information flow from the secrets. This precise\ntaint analysis would be able to further resolve many remaining potential side-channel sources. It is\nworth noting that our cross-product is taint-directed, namely, it is based on the taint information\nfrom the first taint analysis, which greatly reduces the number of safety checks and simplifies the\nproduct program, hence improving the verification efficiency.\nFinally, to resolve the remaining (usually few) potential side-channel sources, we propose a\ntaint-directed cross-product (cf. Section 4.3), which reduces the constant-time security problem to\nthe safety problem of the cross-product of the program where taint information is also used to\nreduce the number of safety checks and simplify the cross-product program.\nWe implement our approaches as a fully automated, cross-platform tool CT-Prover for verifying\n(optimized) LLVM IR implementations. To evaluate CT-Prover , we collect 87 real-world implemen-\ntations from modern cryptographic and SSL/TLS libraries, as well as fixed-point arithmetic libraries.\nThese benchmarks include cryptographic utilities, arithmetic operations, public and private key\ncryptography, and algorithms for encryption, decryption, message authentication code, and dig-\nital signature. The experiment results confirm the effectiveness and efficiency of our approach.\nIn particular, CT-Prover (dis)proves all the (non-)constant-time implementations and find new\nvulnerabilities in open-source libraries Mbed SSL and BearSSL, and is typically significantly faster\nthan state-of-the-art tools.\nIn summary, we make the following main contributions:\n•We provide a novel synergy of taint analysis and self-composition, improving the efficiency\nand scalability of verification of constant-time cryptographic implementations;\n•We develop a fully automated tool to support the verification for production software at the\nindustrial level;\n•We conduct an extensive evaluation on a large set of real-world programs, and identify new,\nconfirmed timing side-channel vulnerabilities of open-source SSL libraries.\nOutline. Section 2 presents the background of the work, including the While language and basics\nof constant-time security. Section 3 gives motivating examples and an overview of our approach.\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.46:4 Luwei Cai, Fu Song, and Taolue Chen\nExpressions: 𝑒::=𝑛|𝑥|𝑒1⊙𝑒2|𝑥[𝑦]\nStatements: 𝑝::=skip|𝑥:=𝑒|𝑥[𝑦]:=𝑧|assert𝑒|assume𝑒|𝑝;𝑝|while𝑥do𝑝od\n|if𝑥then𝑝1else𝑝2fi|𝑥1,···,𝑥𝑚:=𝑓(𝑦1,···,𝑦𝑛)\nProcedures: 𝑓𝑛 ::=def𝑓(𝑥1,···,𝑥𝑛){𝑝;return𝑦1,···,𝑦𝑚;}\nPrograms: 𝑃::=𝑓𝑛+\nFig. 1. The syntax of While .\nSection 4 presents the details of our approaches. Section 5 reports the experiment results. Section 6\ndiscusses the related work. The paper is concluded in Section 7.\n2 PRELIMINARIES\nOur constant-time verification tool works over programs in LLVM intermediate representation (IR).\nThere are three main reasons: (1) LLVM IR is a low-level architecture-independent language so\nverification can be performed on optimized LLVM IR programs, (2) it is more convenient to verify\nand debug LLVM IR programs than binary executables, and (3) the verified compiler CompCert offers\nconstant-time preserving compilation [ 22] and Binsec/Rel [ 36,38] offers bug-finding and bounded\nverification for binary executables. However, for clarity, we use the While language enriched with\narrays, assert/assume statements and procedures, to define the notion of constant-time security\nand formalize our verification approach.\n2.1 The While Language\nSyntax . The syntax of the While language is given in Figure 1. A While program consists of\na sequence of procedures, one of which is the main procedure as the entry of the program. A\nprocedure contains a sequence of formal arguments and a sequence of statements followed by a\nreturn statement. Note that in our While language, the return statement can return a tuple of\nvalues which is required for constructing cross-products. For each procedure 𝑓, we denote by X𝑓\nthe set of variables used by 𝑓, including its formal arguments.\nStatements include skip statements, assert andassume statements, sequential statements,\nif-then -else andwhile -dostatements, assignments, and procedure calls. As in ct-verif [11], we\ninclude the assert andassume statements to simplify the reduction to safety checking. We assume\nthat each statement is annotated by a distinct label ℓ. Expressions include constants, variables,\narithmetic operations and array accesses. We use ⊙to range over binary operators which are\ndeterministic and side-effect free. (Unary operators can be defined similarly and are not presented\nhere.) W.l.o.g., We assume that While programs are given in the single-static assignment (SSA)\nform, array-read 𝑥[𝑦]cannot be used as sub-expressions and all identifies in a program are distinct.\nSemantics . Fix a While programP. LetXbe the set of variables of P,L⊆X∪(X×N)be the set\nof locations comprising scalar variables and pairs (𝑥,𝑖)of array variables 𝑥and indices𝑖, andVbe\nthe set of possible values of variables. A state𝑠:L→Vis a mapping from locations 𝑙to values\n𝑠(𝑙), namely, it maps variables 𝑥(resp. array elements 𝑥[𝑖]) to values𝑠(𝑥)(resp.𝑠(𝑥,𝑖)). An initial\nstate𝑠0is a mapping that only gives the values of the input variables of P, i.e., the parameters of\nthemain procedure. The update of a state 𝑠is written as 𝑠[𝑙↦→𝑛]. We define⊥as a distinguished\nerror state at which the execution of the program is disabled. We denote by 𝑠(𝑒)the value of the\nexpression𝑒in the state𝑠, i.e.,𝑠(𝑛)=𝑛,𝑠(𝑒1⊙𝑒2)=𝑠(𝑒1)⊙𝑠(𝑒2), and𝑠(𝑥[𝑦])=𝑠(𝑥,𝑠(𝑦)).\nAconfiguration 𝑐is a pair⟨𝑠,𝑝⟩consisting of a state 𝑠and a statement 𝑝to be executed. An\ninitial configuration 𝑐0is a pair⟨𝑠0,𝑝⟩such that𝑠0is an initial state and 𝑝is the statement of the\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.Towards Efficient Verification of Constant-Time Cryptographic Implementations 46:5\n𝑠′=𝑠[𝑥↦→𝑠(𝑒)]\n⟨𝑠,𝑥:=𝑒⟩→⟨𝑠′,skip⟩Assign 1𝑠′=𝑠[(𝑥,𝑠(𝑦))↦→𝑠(𝑧)]\n⟨𝑠,𝑥[𝑦]:=𝑧⟩→⟨𝑠′,skip⟩Assign 2\n⟨𝑠,skip ;𝑝⟩→⟨𝑠,𝑝⟩Skip⟨𝑠,𝑝1⟩→⟨𝑠′,𝑝′\n1⟩\n⟨𝑠,𝑝1;𝑝2⟩→⟨𝑠′,𝑝′\n1;𝑝2⟩Seq\n𝑠′=(𝑠(𝑒)=true ?𝑠:⊥)\n⟨𝑠,assert𝑒⟩→⟨𝑠′,skip⟩Assert𝑠(𝑒)=true\n⟨𝑠,assume𝑒⟩→⟨𝑠,skip⟩Assume\n𝑠(𝑥)=false\n⟨𝑠,while𝑥do𝑝od⟩→⟨𝑠,skip⟩While 𝑓𝑠(𝑥)=true\n⟨𝑠,while𝑥do𝑝od⟩→⟨𝑠,𝑝;while𝑥do𝑝od⟩While 𝑡\n𝑠(𝑥)=false\n⟨𝑠,if𝑥then𝑝1else𝑝2fi⟩→⟨𝑠,𝑝2⟩If𝑓𝑠(𝑥)=true\n⟨𝑠,if𝑥then𝑝1else𝑝2fi⟩→⟨𝑠,𝑝1⟩If𝑡\ndef𝑓(𝑥′\n1,···,𝑥′\n𝑛){𝑝;return𝑦1,···,𝑦𝑚;}𝑠in=𝑠[𝑥′\n1↦→𝑠(𝑧1)]···[𝑥′\n𝑛↦→𝑠(𝑧𝑛)]\n⟨𝑠in,𝑝⟩→★⟨𝑠out,skip⟩𝑠ret=𝑠[𝑥1↦→𝑠out(𝑦1)]···𝑠[𝑥𝑚↦→𝑠out(𝑦𝑚)]\n⟨𝑠,𝑥1,···,𝑥𝑚:=𝑓(𝑧1,···,𝑧𝑛)⟩→⟨𝑠in,𝑝⟩and⟨𝑠out,return𝑦1,···,𝑦𝑚;⟩→⟨𝑠ret,skip⟩Call-Ret\nFig. 2. The operational semantics of the While program.\nmain procedure excluding the ending return statement. The semantics of the program Pis defined\nas a transition relation 𝑐→𝑐′between two configurations. We denote by →★the reflexive and\ntransitive closure of the relation →. The transition relation 𝑐→𝑐′is given in Figure 2. For the sake\nof simplicity, array bounds are not checked in our semantics, and we assume that they are checked\nby using assert statements, thus executions are stuck on the error state when indices are out of\nthe range of the array.\nAnexecution𝜌of the program Pis a sequence of configurations 𝑐0𝑐1···𝑐𝑛such that𝑐0is an\ninitial configuration and 𝑐𝑖→𝑐𝑖+1for every 0≤𝑖<𝑛. An execution 𝜌issafeif it does not stick on a\nconfiguration⟨⊥,·⟩with the error state ⊥, and is complete if it ends with a configuration ⟨·,skip⟩.\nThe program Pissafeif all the executions are safe. We remark that in this work, we assume that\nprograms always terminate (i.e., either complete or stick on), because we focus on cryptographic\nimplementations. Termination can be checked by tools, e.g., CPAchecker [25] and T2 [29].\n2.2 Constant-Time Security\nIn practice, execution time variations can create timing side-channel in various forms: (1) unbalanced\nbranching statements may expose the information of the branching condition, (2) non-constant loops\nmay expose the information of the loop condition, (3) memory access patterns (cache hits and misses)\nmay expose the information of the memory address (i.e., indices of arrays in the While programs)\naccessed in load and store instructions, (4) time-variant instructions (e.g., integer divisions) in some\narchitectures may expose the information of operands, and (5) micro-architectural features (e.g.,\nSpectre [ 53] and Meltdown [ 57]) may break conventional constant-time guarantees. In this work, we\nconsider the first three timing side-channels and use the common leakage model [ 11,26,36,83] to\ncharacterize constant-time security. (The last two timing side-channels are not considered because\nthey are architecture-dependent while we consider LLVM IR which is architecture-independent.)\nNote that our methodology is generic and could be adapted to handle the fourth timing side-\nchannel by listing all the timing-sensitive operations [ 11]. Detection, verification and mitigation\ntechniques of the fifth type of timing side-channels have been studied (cf. [ 33] for a survey), most\nof which assume that programs are constant-time without micro-architectural features, and thus\nare orthogonal to our work.\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.46:6 Luwei Cai, Fu Song, and Taolue Chen\nDefinition 2.1 (Constant-time Leakage Model). Given a configuration 𝑐=⟨𝑠,𝑝⟩such that𝑠≠⊥,\nthe observationO(𝑐)is defined as follows.\n(1)if𝑝is a branching statement if𝑥then𝑝1else𝑝2fi, thenO(𝑐)=𝑠(𝑥), namely, the value\nof the branching condition 𝑥is observable to the adversary;\n(2)if𝑝is a loop statement while𝑥do𝑝′od, thenO(𝑐)=𝑠(𝑥), namely, the value of the loop\ncondition𝑥is observable to the adversary;\n(3)if𝑝is an assignment 𝑧:=𝑦[𝑥], thenO(𝑐)=𝑠(𝑥), namely, the value of the index 𝑥in the load\ninstruction is observable to the adversary;\n(4)if𝑝is an assignment 𝑦[𝑥]:=𝑧, thenO(𝑐)=𝑠(𝑥), namely, the value of the index 𝑥in the\nstore instruction is observable to the adversary,\n(5)if𝑝is an sequential statement 𝑝1;𝑝2, thenO(𝑐)=O(⟨𝑠,𝑝1⟩), namely, only the executing\ninstruction𝑝1is considered (Note that 𝑝2will be considered in a subsequent configuration);\n(6) otherwiseO(𝑐)=𝜖, where𝜖denotes an empty observation.\nFor each statement 𝑝with labelℓ(denoted by ℓ:𝑝) as per Definition 2.1(1)-(4) where 𝑥is the\noperand or condition, (ℓ,𝑥)is called a potential (timing) side-channel source . Intuitively, the value\nof𝑥at labelℓis observable to the adversary.\nAn execution 𝜌=𝑐0···𝑐𝑛yields the observation O(𝜌)=O(𝑐0)···O(𝑐𝑛). Two executions 𝜌1and\n𝜌2areindistinguishable (to the adversary with respect to the leakage model O) ifO(𝜌1)=O(𝜌2). It\nis easy to see that two indistinguishable executions 𝜌1and𝜌2must have the same control flow,\ni.e., they execute the same conditional branches and iterations of loops, thus the sequences of\nexecuted statements are the same. This observation is utilized to define cross-product [ 11], i.e.,\nself-composition with lockstep execution of both loops and branches, namely, the copies share the\nsame control flow.\nGiven a program P, we assume that the input variables X𝑖𝑛⊆Xmainare partitioned into public\ninput variables X𝑖𝑛\n𝑙andsecret input variables X𝑖𝑛\nℎ. These sets are to be annotated by users. (For\nthe sake of presentation, an input array variable should be annotated by either public or secret,\nmeaning that all the elements of the array are public or secret.) In our implementation, we provide\nAPI wrappers to precisely annotate elements of input arrays and fields of input structures. The\nadversary knows the implementation details of the program and has access to the values of public\ninput variables at runtime, but does not have any direct access to the values of other variables. The\ngoal of the adversary is to infer the information of secret input variables by analyzing observations\nfrom executions.\nGiven a set of variables 𝑋⊆X, two states 𝑠1and𝑠2are𝑋-equivalent , written as 𝑠1≃𝑋𝑠2, if for\nevery scale variable 𝑥∈𝑋,𝑠1(𝑥)=𝑠2(𝑥)and for every array variable 𝑥∈𝑋and possible index\n𝑖∈N,𝑠1(𝑥,𝑖)=𝑠2(𝑥,𝑖). Two configurations 𝑐1and𝑐2are𝑋-equivalent , written as 𝑐1≃𝑋𝑐2, if\ntheir states are 𝑋-equivalent . For a pair of executions 𝜌=𝑐0𝑐1···𝑐𝑛and𝜌′=𝑐′\n0𝑐′\n1···𝑐′\n𝑛′,𝜌≃𝑋𝜌′\ndenotes that for every 0≤𝑖≤min(𝑛,𝑛′),𝑐𝑖≃𝑋𝑐′\n𝑖.\nDefinition 2.2 (Constant-time Security [ 11]).A safe program Pis(constant-time) secure if for any\npair of complete executions 𝜌=𝑐0𝑐1···𝑐𝑛and𝜌′=𝑐′\n0𝑐′\n1···𝑐′\n𝑛′,\n(𝑐0≃X𝑖𝑛\n𝑙𝑐′\n0)⇒O(𝜌)=O(𝜌′).\nIntuitively, the safe program Pis secure if, for any pair of complete executions that have the\nsame public input values (i.e., the values of public input variables), their observations are the same,\nmeaning that secret inputs are not distinguishable from the observations. Otherwise, there must\nexist a side-channel source (ℓ,𝑥), namely, the values of the variable 𝑥at labelℓdiffer between\ntwo configurations 𝑐𝑖and𝑐′\n𝑖for some𝑖. Thus, a potential timing side-channel source (ℓ,𝑥)does\nnot necessarily leak the secrets (i.e., 𝑥is secret-independent), but leaks the secrets when 𝑥is\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.Towards Efficient Verification of Constant-Time Cryptographic Implementations 46:7\n1uint64_t fixfrac(char* frac) {\n2uint64_t pow10_LUT[20] = {0x1999999999999999, ...,0x0000000000000000};\n3uint64_t pow10_LUT_extra[20] = {0x99, ..., 0x2f};\n4uint64_t result = 0; uint64_t extra = 0;\n5for(int i = 0; i < 20; i++) {\n6 if(frac[i] == '\\0') { break; }\n7 uint8_t digit = (frac[i] - (uint8_t) '0');\n8 result += ((uint64_t) digit) * pow10_LUT[i];\n9 extra += ((uint64_t) digit) * pow10_LUT_extra[i];\n10 }\n11 ... }\nFig. 3. Fragment of the function fixfrac taken from the libfixedtimefixedpoint library.\nsecret-dependent. The security of unsafe programs is undefined, because they are stuck in the error\nstate. (Note that the safety of a program can be verified using standard verification techniques and\ntools, e.g., SMACK [63].)\nStandard library functions malloc ,free ,memcpy andmemset may be used in cryptographic\nimplementations. To handle them, following [ 11], we assume that the address and the length used\nin those functions are observable to the adversary, and the return of free is secret-independent.\n3 MOTIVATION AND OVERVIEW\nIn this section, we present two motivating examples and an overview of our approach.\n3.1 Motivating Examples\nExample 1 . Figure 3 shows a fragment of the function fixfrac , a fixed-point numeric operation\nprovided by the library libfixedtimefixedpoint [ 13]. Given a digit string frac whose length is no\nmore than 20, it computes a 64-bit number which corresponds to\natoi(frac+padding)/1020)∗264\nwherefrac+padding is a digit string obtained from frac by padding some 0’s such that the length\nis20, andatoi coverts a digit string into the corresponding integer. The function fixfrac is\ninvoked by the function fix_pow(𝑥,𝑦)which computes 𝑥𝑦over the fixed-point numbers 𝑥and𝑦.\nThere are five potential side-channel sources in this code snippet, i.e., (6,frac[i]==‘\\0’),(6,i),\n(7,i),(8,i)and(9,i). We can observe that iis secret-independent, thus the last four pairs are not\nside-channel sources. However, it is non-trivial to determine the first potential side-channel source\n(6,frac[i]==‘\\0’). If it is, the information of the secret inputs 𝑥and𝑦can be inferred by the\nadversary via timing side-channels.\nThis example cannot be proved by the self-composition based approach ct-verif [11] unless the\nloop is unrolled or the following loop invariant is added:\n∃𝑖max·0≤𝑖<𝑖max≤20∧frac[𝑖max]==0∧∀𝑗.0≤𝑗≤𝑖max⇒public(frac[𝑗]).\nHowever, loops may be not statically bounded and providing such loop invariants manually is\nnon-trivial, which means that in practice, the self-composition based approach is not fully automatic.\nIn contrast, taint analysis is useful here. Indeed, frac is secret-independent.\nExample 2 . Figure 4 shows a simplified fragment of the function crypto_stream_chacha20_ref\ntaken from the libsodium library, a portable, cross-compilable and installable fork of NaCl with an\nextended API to improve usability. This function implements the ChaCha20 stream cipher [23].\nIn the fragment of crypto_stream_chacha20_ref , variable cpoints to a plaintext to be en-\ncrypted,clen is the length of the plaintext, npoints to an initialization vector, and kpoints to a\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.46:8 Luwei Cai, Fu Song, and Taolue Chen\n1int crypto_stream_chacha20_ref(unsigned char *c, unsigned long long clen,\n2 const unsigned char *n, const unsigned char *k){\n3uint32_t ctx[16];\n4chacha_keysetup(ctx, k); // store k from ctx[0] to ctx[11];\n5chacha_ivsetup(ctx, n, NULL); // store IV and counter into the rest;\n6chacha_encrypt_bytes(ctx, c, c, clen);\n7}\n8static void chacha_encrypt_bytes(uint32_t *x, const u8 *m, u8 *c,\n9 unsigned long long bytes){\n10 j12 = x[12];\n11 if (!j12) {...}\n12}\nFig. 4. Simplified fragment of the function crypto_stream_chacha20_ref taken from the libsodium library.\nClangPre-analysisLightweight taint analysisPointer analysisValue-flow analysisIFDS-based taint analysisPrecise taint analysisTaint-directed semi-cross-product constructionPrecise taint analysisTaint-directed cross-product construction\nLightweight taint analysisPrecisetaint analysisTaint-directed self-composition verification\nAnnotated C program\nResult\nFig. 5. Overview of our approach\nprivate key. The key kis stored at the first 12 positions of the buffer ctx, the initialization vector\nand a counter (initialized as NULL) are stored at the rest 4 positions of the buffer ctx.\nThere is one potential side-channel sources in this simplified fragment, i.e., (11,!j12). It is\nnon-trivial to determine this potential side-channel source, as the value of j12isx[12] while the\nbufferxcontains the secret key k. If it is, the information of the secret key kcan be inferred by the\nadversary via timing side-channels.\nThis example cannot be proved by an index-insensitive taint analysis, because ctxcontains\nboth secret-dependent and secret-independent contents, namely, ctx[0–11] andctx[12–15] .\nAny index-insensitive taint analysis will conservatively taint the whole buffer. In contrast, an\nindex-sensitive taint analysis (e.g., our precise taint analysis) would work in this example.\nThese examples reveal that static analysis (e.g., taint analysis) and self-composition based ap-\nproaches may have complementary strengths even without efficiency considerations. Our method\nprecisely takes advantage of their respective strength for which we provide an overview below.\n3.2 Approach Overview\nAn overview of our approach is shown in Figure 5. In general, for a given annotated C program, the\ntool either outputs proved , suggesting that the program is secure, or outputs (potential) side-channel\nsources and corresponding execution traces for vulnerability localization and repair.\nOur approach works as follows. First, the input program is translated into LLVM intermediate\nrepresentation (IR) with annotations using Clang, the front-end of LLVM. Second, we compute the\ncall graph, interprocedural control-flow graph, points-to information and definition-use chains\nrequired by the subsequent steps via a pre-analysis. Third, a lightweight taint analysis (cf. Section 4.1)\nis performed by leveraging the inter-procedural, finite, distributive, subset (IFDS) framework [ 64].\nOften it is able to determine a large number of potential side-channel sources, leaving few potential\nside-channel sources unresolved. Finally, we resort to a precise taint analysis (cf. Section 4.2) and\na taint-directed self-composition verification (cf. Section 4.3) to resolve the left-over potential\nside-channel sources. The precise taint analysis reduces the flow-, context-, path-, field- and index-\nsensitive taint analysis problem to checking safety properties of a cross-product of the given\nprogram and its Boolean abstraction which tracks the required information flow from the secrets.\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.Towards Efficient Verification of Constant-Time Cryptographic Implementations 46:9\n𝑇′=\u0000𝑦∈𝑇?𝑇∪{𝑥}:𝑇\\{𝑥}\u0001\n𝑥:=𝑦[𝑧]⊢𝑇↩→𝑇′T-Assign 1𝑇′=\u0000var(𝑒)∩𝑇≠∅?𝑇∪{𝑥}:𝑇\\{𝑥}\u0001\n𝑥:=𝑒⊢𝑇↩→𝑇′T-Assign 3\n𝑇′=\u0000𝑧∈𝑇?𝑇∪{𝑥}:𝑇\u0001\n𝑥[𝑦]:=𝑧⊢𝑇↩→𝑇′T-Assign 2𝑝isskip orassert𝑒orassume𝑒\n𝑝⊢𝑇↩→𝑇T-Identity\n𝑝1⊢𝑇↩→𝑇1𝑝2⊢𝑇1↩→𝑇′\n𝑝1;𝑝2⊢𝑇↩→𝑇′T-Seq𝑇′=lfp(𝑝,𝑇)\nwhile𝑥do𝑝od⊢𝑇↩→𝑇′T-While\n𝑝1⊢𝑇↩→𝑇1𝑝2⊢𝑇↩→𝑇2\nif𝑥then𝑝1else𝑝2fi⊢𝑇↩→𝑇1∪𝑇2T-If\ndef𝑓(𝑥′\n1,···,𝑥′\n𝑛){𝑝;return𝑦1,···,𝑦𝑚;}𝑇in=(𝑇\\X𝑔)∪{𝑥′\n𝑖|1≤𝑖≤𝑛∧𝑧𝑖∈𝑇}\ncaller =𝑔 𝑝⊢𝑇in↩→𝑇out𝑇′=(𝑇\\{𝑥1,···,𝑥𝑚}∪𝑇out\\X𝑓∪{𝑥𝑖|1≤𝑖≤𝑚∧𝑦𝑖∈𝑇out}\n𝑥1,···,𝑥𝑚:=𝑓(𝑧1,···,𝑧𝑛)⊢𝑇↩→𝑇′T-Call-Ret\nFig. 6. Taint inference rules for the While language.\nThe taint-directed self-composition consists of two copies of the original program which is a new\nvariant of self-composition. Remarkably, the taint information is utilized in both cross-product\nconstructions to simplify the resulting program and reduce the cost of safety checks. By combining\nlightweight taint analysis and heavyweight safety verification, the overall approach brings the best\nof three worlds: efficiency, soundness and theoretical completeness.\nConsider the motivating examples. The lightweight taint analysis is able to prove that all the\nfive potential side-channel sources in Example 1 actually do not leak secrets, so the next two\nsteps are not needed. In Example 2, the lightweight taint analysis fails to determine the potential\nside-channel source (11,!j12), thus the subsequent analyses have to check (11,!j12), which can\nbe resolved by the precise taint analysis. (The final step is thus not needed.) We remark that the\nprecise taint analysis may fail to prove some constant-time implementations meaning that it is\nsound but incomplete. For instance, when the secret 𝑘is involved in the computation of a potential\ntiming side-channel source 𝑥(e.g.,𝑥=𝑘⊕𝑝⊕𝑘where⊕is Exclusive-OR), and the value of 𝑥is\nindependent upon the secret 𝑘, the precise taint analysis will raise a false positive.\n4 METHODOLOGY\nIn this section, we present the details of the three key components, i.e., lightweight taint analysis,\nprecise taint analysis and taint-directed self-composition.\n4.1 Lightweight Taint Analysis\nIn this subsection, we present a lightweight taint analysis which is designed to be flow-, field- and\ncontext-sensitive, but path-and index-insensitive, for a balance of efficiency and precision. Often it\nis able to prove that a large number of potential side-channel sources do not leak secrets, leaving\nfew potential side-channel sources unresolved.\nTaint source . Fix a safe program P. The taint source is the set X𝑖𝑛\nℎof its secret input variables, each\nof which is a taint fact. We remark that although our implementation supports the element-wise\nannotation of input array variables, this taint analysis is index-insensitive. Thus, if any element of\nan input array variable is annotated by secret, the array variable is regarded as secret, i.e., all the\nelements of the array are tainted.\nTaint inference rule . The taint inference rule is given by the transfer function of the form\n𝑝⊢𝑇↩→𝑇′,\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.46:10 Luwei Cai, Fu Song, and Taolue Chen\nwhere𝑝is a statement, 𝑇and𝑇′are sets of taint facts. The transfer function 𝑝⊢𝑇↩→𝑇′means\nthat the execution of the statement 𝑝with the set of taint facts 𝑇results in the set of taint facts 𝑇′.\nThe taint inference rules of the While language are given in Figure 6, where var(𝑒)denotes the\nset of variables involved in the expression 𝑒, andlfp(𝑝,𝑇)is recursively defined as follows:\nlfp(𝑝,𝑇)=(\n𝑇, if𝑝⊢𝑇↩→𝑇;\n𝑇∪lfp(𝑝,𝑇′),otherwise, where 𝑝⊢𝑇↩→𝑇′.\nThese rules are standard, which, intuitively, propagate taints from the right-hand side variables\nto the left-hand side variable. For example, rule [T- Assign 1] expresses that if the array 𝑦is tainted,\nthen the loaded array element is tainted. Rule [T- Assign 2] expresses that if a tainted value is stored\nin an array, then the array is tainted. (Recall that our taint analysis is index-insensitive.) Rule\n[T-Assign 3] expresses that if any involved variable of an expression 𝑒is tainted, then the result\nof the expression 𝑒is also tainted. Note that if the left-hand side is a scalar variable (i.e., rules\n[T-Assign 1] and [T- Assign 3]), we perform a strong update. Rule [T- While ] computes the least\nfixed point by applying the operator lfpwhich always terminates, because the sequence of sets of\ntaint facts during lfp(𝑝,𝑇)is ascending w.r.t. the order ⊆. Rule [ T-Call-Ret ] is much involved.\nThe set𝑇inof input taint facts at the call-site is obtained by passing actual arguments of the caller\n𝑔to the formal parameters of the callee 𝑓after filtering out the taint facts of the local variables X𝑔\nof the caller 𝑔. The body𝑝of the callee is analyzed using the set 𝑇inof input taint facts, leading to\nthe set𝑇outof output taint facts. The set 𝑇outof output taint facts is merged with the set 𝑇of taint\nfacts at return-site after filtering out the taint facts of the local variables X𝑓of the callee 𝑓and the\nactual return variables are updated accordingly.\nIFDS-based taint analysis . We leverage the inter-procedural, finite, distributive, subset (IFDS)\nframework to implement the lightweight taint analysis. The time complexity and space complexity\nof the vanilla IFDS algorithm are O(|𝐸|·|𝐷|3)andO(|𝐸|·|𝐷|), respectively, where |𝐸|is the number\nof edges in the interprocedural control-flow graph and the size |𝐷|of the domain is the number of\nall possible taint facts, i.e., |X|. The time complexity will increase sharply with |X|in practice [ 55]\nwhich is significant for some cryptographic implementations due to the following reasons.\nOn the one hand, typically, the input of a cryptographic algorithm consists of a key and plaintext;\nthe key is secret and thus tainted, and the key and plaintext are tightly coupled in the computation.\nFor instance, the AES algorithm has multiple modes, with the smallest key size being 128 bits (an\narray with 16 elements) and the smallest encryption process being 10 rounds each of which has\nfour transformations. On the other hand, the SSA form introduces a number of temporary variables.\nConsequently, a large number of taint facts are propagated during the taint analysis. To mitigate\nthis issue, inspired by the sparse data-flow analysis [ 60], we improve the classic IFDS framework\nby directly propagating taint facts of scalar variables via data flow instead of control flow. More\nspecifically, if a scalar variable 𝑥is tainted, this taint fact is directly propagated to the statements\nwhere𝑥is used, using the def-use chains. It avoids the propagation of taint facts of scalar variables\nfor many statements, hence improving efficiency in practice.\nHereafter, for every label ℓof a statement 𝑝, we denote by 𝑇ℓthe set of taint facts at label ℓ(i.e.,\nbefore the execution of the statement 𝑝), obtained by applying the IFDS-based taint analysis. Since\nthe IFDS-based taint analysis is sound, it is straightforward to have that\nLemma 4.1. For any potential side-channel source (ℓ,𝑥)and pair(𝑐0,��′\n0)of initial configurations\nsuch that(𝑐0≃X𝑖𝑛\n𝑙𝑐′\n0), if𝑥is not tainted at the label ℓ, i.e.,𝑥∉𝑇ℓ, then the values of 𝑥are the same at\nthe labelℓin any pair of complete executions 𝜌=𝑐0𝑐1···𝑐𝑛and𝜌′=𝑐′\n0𝑐′\n1···𝑐′\n𝑛′.\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.Towards Efficient Verification of Constant-Time Cryptographic Implementations 46:11\nProof sketch. For a pair of complete executions 𝜌=𝑐0𝑐1···𝑐𝑛and𝜌′=𝑐′\n0𝑐′\n1···𝑐′\n𝑛′with\n𝑐0≃X𝑖𝑛\n𝑙𝑐′\n0, suppose that the values of 𝑥are different at the label ℓ. It suffices to show that 𝑥∈𝑇ℓ,\ni.e.,𝑥is tainted at ℓ.\nNote that the values of public input variables are the same in the initial configurations 𝑐0and𝑐′\n0.\nWe then can deduce that the value of the variable 𝑥at the labelℓdepends upon some secret input\nvariables from X𝑖𝑛\nℎ. By the soundness of the IFDS framework [ 64] and sparse static analysis [ 60],𝑥\nmust be tainted at ℓ. □\nObviously, if for each potential side-channel source (ℓ,𝑥), the variable 𝑥is not tainted at the\nlabelℓ, we can deduce that the safe program is constant-time secure.\n4.2 Precise Taint Analysis via Taint-directed Semi-cross-product\nWhile the lightweight taint analysis is often effective in ruling out a large number of potential\nside-channel sources to be genuinely vulnerable, some can still not be determined due to the\nover-approximation nature of the static analysis, e.g., index-insensitive. In this subsection, we\npropose a precise taint analysis, which reduces the flow-, context-, path-, field- and index-sensitive\ntaint analysis problem to checking safety properties of a novel cross-product, called taint-directed\nsemi-cross-product . We shall first explain the intuition and then present the formal construction.\nIntuition . Given a program P, we construct a semi-cross-product P′of the program Pand its Boolean\nabstraction. Here, ‘semi’ means that one copy in the cross-product P′is a Boolean abstraction of P\ninstead of the original one; ‘cross’ means that P′shares the same control flow of Pand executes\nstatements of two copies in a lockstep manner.\nThe Boolean abstraction has\n•a Boolean variable 𝑏𝑥for each scalar variable 𝑥inPsuch that𝑏𝑥=1iff𝑥is tainted.\n•a Boolean array 𝑏𝑥for each array variable 𝑥inPsuch that𝑏𝑥[𝑖]=1iff𝑥[𝑖]is tainted.\nThe precise taint analysis is to determine whether a scalar variable 𝑥(resp. an array element\n𝑥[𝑖]) is tainted or not. For this purpose, it suffices to check whether there exist inputs to P′such\nthat𝑏𝑥(resp.𝑏𝑥[𝑖]) is1, which is a standard safety verification problem.\nTo reduce the cost of safety verification, we incorporate the results from the lightweight taint\nanalysis into the semi-cross-product P′based on the following observation. The lightweight taint\nanalysis is conservative (i.e., intuitively it may overly taint), consequently, a more precise taint\nanalysis would not taint the variables that have not been tainted by the lightweight taint analysis.\nHence if a variable 𝑥has not been tainted by the lightweight taint analysis, its Boolean abstraction\n𝑏𝑥in the semi-cross-product P′will always be 0. As a result, when verifying P′, it suffices to\nfocus exclusively on the variables that have been tainted by the lightweight taint analysis. For the\nvariables that have not been tainted by the lightweight taint analysis, we can simply assign 0to\nthem, which can further improve the efficiency of safety verification.\nProduct construction . The semi-cross-product P′for a given program Pis constructed by iter-\natively applying the function 𝜋(·)(given in Fig. 7) to each procedure of P. For each procedure,\nformal parameters and return variables are duplicated by the function 𝜋(·)using their Boolean\nabstractions, and the procedure body 𝑝is replaced by 𝜋(𝑝). Moreover, the Boolean abstractions\nof public and secret inputs are respectively initialized by 0and1at the beginning of the main\nprocedure.\nDuring the construction, 𝜉(𝑒)is used to generate the Boolean abstraction of the expression 𝑒\nthat computes the taint value of the result of the expression 𝑒. Specifically, if 𝑒is a constant 𝑛,𝜉(𝑒)\nis the Boolean constant 0; if𝑒a variable𝑥,𝜉(𝑒)is the Boolean abstraction of 𝑥(i.e.,𝑏𝑥); and if𝑒is a\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.46:12 Luwei Cai, Fu Song, and Taolue Chen\n𝜉(𝑛)≜0 𝜉(𝑥)≜𝑏𝑥 𝜉(𝑒1⊙𝑒2)≜𝜉(𝑒1)∨𝜉(𝑒2)\n𝜋(𝑝)≜𝑝if𝑝isskip orassert𝑒orassume𝑒 𝜋 (𝑝1;𝑝2)≜𝜋(𝑝1);𝜋(𝑝2)\n𝜋(ℓ:𝑥:=𝑒)≜(\n𝑥:=𝑒;𝑏𝑥:=0 ifvar(𝑒)∩𝑇ℓ=∅\n𝑥:=𝑒;𝑏𝑥:=𝜉(𝑒)otherwise\n𝜋(ℓ:𝑥:=𝑦[𝑧])≜(\nGuardℓ(𝑧);𝑥:=𝑦[𝑧];𝑏𝑥:=0 if𝑦∉𝑇ℓ\nGuardℓ(𝑧);𝑥:=𝑦[𝑧];𝑏𝑥:=𝑏𝑦[𝑧]otherwise\n𝜋(ℓ:𝑥[𝑦]:=𝑧)≜(\nGuardℓ(𝑦);𝑥[𝑦]:=𝑧;𝑏𝑥[𝑦]:=0 if𝑧∉𝑇ℓ\nGuardℓ(𝑦);𝑥[𝑦]:=𝑧;𝑏𝑥[𝑦]:=𝑏𝑧otherwise\n𝜋(ℓ:while𝑥do𝑝od)≜while𝑥@INV do Guard begin(ℓ)(𝑥);𝜋(𝑝)od;Guardexit(ℓ)(𝑥)\n𝜋(ℓ:if𝑥then𝑝1else𝑝2fi)≜Guardℓ(𝑥);if𝑥then𝜋(𝑝1)else𝜋(𝑝2)fi\n𝜋(𝑥1,···,𝑥𝑚:=𝑓(𝑦1,···,𝑦𝑛))≜𝑥1,𝑏𝑥1,···,𝑥𝑚,𝑏𝑥𝑚:=𝑓(𝑦1,𝑏𝑦1,···,𝑦𝑛,𝑏𝑦𝑛)\n𝜋(def𝑓(𝑥1,···,𝑥𝑛){𝑝;return𝑦1,···,𝑦𝑚;})=def𝑓(𝑥1,𝑏𝑥1···,𝑥𝑛,𝑏𝑥𝑛){𝜋(𝑝);return𝑦1,𝑏𝑦1,···,𝑦𝑚,𝑏𝑦𝑚;}\nGuardℓ(𝑥)≜(𝑥∈𝑇ℓ?assert¬𝑏𝑥:skip)\nFig. 7. The taint-directed semi-cross-product of the While programs, where begin(ℓ)andexit(ℓ)denote\nthe labels of the beginning and exit of the loop body of the loop at the label ℓ.\ncompound expression 𝑒1⊙𝑒2,𝜉(𝑒)is the disjunction 𝜉(𝑒1)∨𝜉(𝑒2)of taint values of sub-expressions,\nnamely, the value of 𝑒is tainted if some variable used in 𝑒is tainted.\nFor each statement 𝑝with labelℓ,𝜋(𝑝)produces a new statement. If 𝑝is askip orassert\norassume statement,𝜋(𝑝)gives𝑝itself, namely, it does not have a Boolean abstraction. If 𝑝is\na sequential statement 𝑝1;𝑝2,𝜋(𝑝)gives the sequential statement 𝜋(𝑝1);𝜋(𝑝2)by recursively\napplying the function 𝜋(·). If𝑝is a standard assignment 𝑥:=𝑒,𝜋(𝑝)is defined to track the\ninformation flow from the operands of the expression 𝑒to the Boolean abstraction 𝑏𝑥of𝑥if some\noperand of𝑒has been tainted by the lightweight taint analysis, otherwise 0.\nIf𝑝is a load statement 𝑥:=𝑦[𝑧],𝜋(𝑝)gives a sequential statement Guardℓ(𝑧);𝑥:=𝑦[𝑧];𝑏𝑥:=0\nif𝑦has not been tainted (i.e., 𝑦∉𝑇ℓ). Otherwise, Guardℓ(𝑧);𝑥:=𝑦[𝑧];𝑏𝑥:=𝑏𝑦[𝑧]is generated\nmeaning that 𝑏𝑥is the same as 𝑏𝑦[𝑧]. The auxiliary function Guardℓ(𝑧)is used to generate an\nassert statement to resolve the taint status of the variable 𝑧if𝑧has been tainted, because (ℓ,𝑧)\nis a potential side-channel source. If 𝑧has not been tainted, the assertion is avoided by replacing\nit with askip statement which can be further removed from the program P′. A store statement\n𝑥[𝑦]:=𝑧is handled similarly, except that 𝑏𝑦=0is checked by inserting an assert statement if 𝑦\nhas been tainted (i.e., 𝑦∈𝑇ℓ) and𝑏𝑥[𝑦]is updated accordingly.\nIf𝑝is awhile -dostatement while𝑥do𝑝od,𝜋(𝑝)inserts an assert statement at the beginning\n(resp. exit) of the loop body to ensure that 𝑏𝑥=0if𝑥has been tainted at the beginning (resp. exit)\nof the loop body. Furthermore, to facilitate safety verification, loop invariants INVare inserted.\nCurrently, we use the following heuristic strategy to generate loop invariants and leave the genera-\ntion of more effective loop invariants as future work. For each variable 𝑦defined in the loop body,\nif it is used to compute the loop condition 𝑥, then the predicate ¬𝑏𝑦is added as a loop invariant.\nHowever, such loop invariants may be invalid in practice. Loop invariants are checked and invalid\nones are removed during safety verification.\nIf𝑝is anif-then -else statement if𝑥then𝑝1else𝑝2fi, similarly,𝜋(𝑝)inserts an assert\nstatement to resolve the taint status of the branching condition 𝑥if it has been tainted.\nFor every potential side-channel source (ℓ,𝑥)such that𝑥∈𝑇ℓ, the program P′must have one or\ntwoassert statements introduced by Guard which checks if 𝑏𝑥=0. By applying a sound safety\nverifier, we can check if such assert statements are valid or not. If they are valid, we can deduce\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.Towards Efficient Verification of Constant-Time Cryptographic Implementations 46:13\nthat(ℓ,𝑥)is not a side-channel source, thus the variable 𝑥can be removed from the set of taint\nfacts𝑇ℓ. We denote by 𝑇′\nℓthe resulting set of taint facts. It is trivial to see that Lemma 4.1 still holds\nwhen𝑇ℓis updated by 𝑇′\nℓfor every potential side-channel source (ℓ,𝑥).\nLemma 4.2. For any potential side-channel source (ℓ,𝑥)and pair(𝑐0,𝑐′\n0)of initial configurations\nsuch that(𝑐0≃X𝑖𝑛\n𝑙𝑐′\n0), if𝑥∉𝑇′\nℓ, then the values of 𝑥are the same at the label ℓin any pair of complete\nexecutions𝜌=𝑐0𝑐1···𝑐𝑛and𝜌′=𝑐′\n0𝑐′\n1···𝑐′\n𝑛′.\nProof sketch. Suppose the values of 𝑥are different at the label ℓin a pair of complete executions\n𝜌=𝑐0𝑐1···𝑐𝑛and𝜌′=𝑐′\n0𝑐′\n1···𝑐′\n𝑛′with𝑐0≃X𝑖𝑛\n𝑙𝑐′\n0. Let𝑏𝑥be the Boolean abstraction of 𝑥. We show\nthat the assert statement assert¬𝑏𝑥for the potential side-channel source (ℓ,𝑥)is not valid.\nAs the values of 𝑥are different at ℓin𝜌and𝜌′, and the values of public input variables are the\nsame in the initial configurations 𝑐0and𝑐′\n0, we can deduce that the value of 𝑥atℓdepends upon\nsome secret input variables from X𝑖𝑛\nℎ. Since the Boolean abstractions of all the secret input variables\nare initialized by 1, the Boolean abstraction of each internal variable is set to 0only if the internal\nvariable is independent of secret input variables, and Boolean abstractions can only be disjunction,\nwe conclude that 𝑏𝑥is1when the semi-cross-product P′takes the inputs from 𝑐0or𝑐′\n0.□\n4.3 Taint-directed Self-composition\nThe precise taint analysis can resolve potential side-channel sources that were left by the lightweight\ntaint analysis, but may still fail on some potential side-channel sources. Indeed, it cannot determine\ngenuine side-channel sources that would leak secrets. Thus, in this subsection, we propose a taint-\ndirected self-composition which improves the original construction [ 11] by incorporating the taint\ninformation. Our construction simplifies the self-composed programs and reduces the number of\nsafety checks. We first describe the intuition and then present the formal construction.\nIntuition . Given a program P, we construct a cross-product ˆPof the program Pcomprising two\ncopies of Pwhich share the same control flow. One copy is the original program, and the other copy\nis referred to as the shadow one which has a shadow variable ˆ𝑥for each variable 𝑥inP. Typically\nwe check whether the variable 𝑥and its shadow ˆ𝑥have the same values when the original and\nshadow counterparts of ˆPare provided with the same public inputs but different secret inputs.\nTo reduce the cost of safety verification, we also incorporate the results from two taint analyses\ninto ˆPbased on the following observation. If a variable 𝑥has not been tainted, 𝑥and its shadow ˆ𝑥\nin the product ˆPmust have the same values when the public inputs of the two copies are the same.\nTherefore, when verifying ˆP, it suffices to focus exclusively on the variables that are still tainted\nafter two taint analyses. Moreover, for those variables that have not been tainted, we can simply\nassign the value of the original variable to the shadow one instead of copying the computation,\nwhich can further improve the efficiency of safety verification.\nProduct construction . The cross-product ˆPfor a given program Pis constructed by iteratively\napplying the function Π(·)(given in Fig. 8) to each procedure of P. Moreover, the shadow counter-\nparts of public inputs are initialized by their original ones at the beginning of the main procedure.\nFor each procedure, Π(·)is defined similar to the function 𝜋(·)in semi-cross-product, where the\nBoolean abstractions are replaced by the shadow counterparts.\nMore specifically, the auxiliary function Ξ(𝑒)used in defining Π(ℓ:𝑥:=𝑒)now generates a\nshadow expression that computes the value of expression 𝑒over shadow variables. Concretely, if 𝑒\nis a constant 𝑛,Ξ(𝑒)is the constant 𝑛; if𝑒is a variable 𝑥,Ξ(𝑒)is the shadow variable ˆ𝑥of𝑥; and if\n𝑒is a compound expression 𝑒1⊙𝑒2,Ξ(𝑒)isΞ(𝑒1)⊙Ξ(𝑒2).\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.46:14 Luwei Cai, Fu Song, and Taolue Chen\nΞ(𝑛)≜𝑛 Ξ(𝑥)≜ˆ𝑥 Ξ(𝑒1⊙𝑒2)≜Ξ(𝑒1)∨Ξ(𝑒2)\nΠ(𝑝)≜𝑝if𝑝isskip orassert𝑒orassume𝑒 Π(𝑝1;𝑝2)≜Π(𝑝1);Π(𝑝2)\nΠ(ℓ:𝑥:=𝑦[𝑧])≜(\nGuard′\nℓ(𝑧);𝑥:=𝑦[𝑧];ˆ𝑥:=𝑥 if𝑦∉𝑇′\nℓ\nGuard′\nℓ(𝑧);𝑥:=𝑦[𝑧];ˆ𝑥:=ˆ𝑦[ˆ𝑧]otherwise\nΠ(ℓ:𝑥[𝑦]:=𝑧)≜(\nGuard′\nℓ(𝑦);𝑥[𝑦]:=𝑧;ˆ𝑥[ˆ𝑦]:=𝑥[𝑦]if𝑧∉𝑇′\nℓ\nGuard′\nℓ(𝑦);𝑥[𝑦]:=𝑧;ˆ𝑥[ˆ𝑦]:=ˆ𝑧 otherwise\nΠ(ℓ:𝑥:=𝑒)≜(\n𝑥:=𝑒;ˆ𝑥:=𝑥 ifvar(𝑒)∩𝑇′\nℓ=∅\n𝑥:=𝑒;ˆ𝑥:=Ξ(𝑒)otherwise\nΠ(ℓ:while𝑥do𝑝od)≜while𝑥@INV′do Guard′\nbegin(ℓ)(𝑥);Π(𝑝)od;Guard′\nexit(ℓ)(𝑥)\nΠ(ℓ:if𝑥then𝑝1else𝑝2fi)≜Guard′\nℓ(𝑥);if𝑥thenΠ(𝑝1)elseΠ(𝑝2)fi\nΠ(𝑥1,···,𝑥𝑚:=𝑓(𝑦1,···,𝑦𝑛))≜𝑥1,ˆ𝑥1,···,𝑥𝑚,ˆ𝑥𝑚:=𝑓(𝑦1,ˆ𝑦1,···,𝑦𝑛,ˆ𝑦𝑛)\nΠ(def𝑓(𝑥1,···,𝑥𝑛){𝑝;return𝑦1,···,𝑦𝑚;})=def𝑓(𝑥1,ˆ𝑥1···,𝑥𝑛,ˆ𝑥𝑛){Π(𝑝);return𝑦1,ˆ𝑦1,···,𝑦𝑚,ˆ𝑦𝑚;}\nGuard′\nℓ(𝑥)≜(𝑥∈𝑇′\nℓ?assert𝑥=ˆ𝑥:skip)\nFig. 8. The taint-directed cross-product of the While programs.\nThe auxiliary function Guard′\nℓ(𝑥)for each left potential side-channel source (ℓ,𝑥)generates an\nassert statement to ensure that the variable 𝑥and its shadow counterpart have the same value\n(i.e.,𝑥=ˆ𝑥) if𝑥has been tainted. It allows us to resolve the taint status of 𝑥.\nThe duplicated Boolean counterpart 𝑏𝑥:=0(resp.𝑏𝑥[𝑦]:=0) is replaced by ˆ𝑥:=𝑥(resp.\nˆ𝑥[ˆ𝑦]:=𝑥[𝑦]) inΠ(ℓ:𝑥:=𝑦[𝑧])(resp. Π(ℓ:𝑥[𝑦]:=𝑧)) if𝑥was untainted, otherwise the\nright-hand side is copied using the corresponding shadow counterparts.\nFor each while𝑥do𝑝od,Π(𝑝)inserts loop invariants INV. Following [ 11], for each variable 𝑦\ndefined in the loop body, if it is used to compute the loop condition 𝑥, then the predicate 𝑦=ˆ𝑦is\nadded as a loop invariant. Similar to the precise taint analysis, such loop invariants are checked\nand invalid ones are removed during safety verification.\nFor every potential side-channel source (ℓ,𝑥)such that𝑥∈𝑇′\nℓ, the program ˆPmust have one or\ntwoassert statements introduced by Guard′which checks if 𝑥=ˆ𝑥. By invoking safety verifiers,\nwe can check if such assert statements are valid or not. If they are valid, we can deduce that (ℓ,𝑥)\nis not a side-channel source, thus the variable 𝑥can be removed from the set of taint facts 𝑇′\nℓ. We\ndenote by ˆ𝑇ℓthe resulting set of taint facts. It is trivial to see that Lemma 4.1 still holds when 𝑇ℓis\nupdated by ˆ𝑇ℓfor every potential side-channel source (ℓ,𝑥).\nLemma 4.3. For any potential side-channel source (ℓ,𝑥)and pairs(𝑐0,𝑐′\n0)of initial configurations\nsuch that(𝑐0≃X𝑖𝑛\n𝑙𝑐′\n0), if𝑥∉ˆ𝑇ℓ, then the values of 𝑥are the same at the label ℓin any pair of complete\nexecutions𝜌=𝑐0𝑐1···𝑐𝑛and𝜌′=𝑐′\n0𝑐′\n1···𝑐′\n𝑛′.\nProof sketch. The correctness of the lemma follows directly from the correctness of the product\nconstruction [11] and Lemmas 4.1—4.2. □\nBy Lemmas 4.1—4.3, we obtain\nTheorem 4.4. Given a safe program P, ifˆ𝑇ℓ=∅for any potential side-channel source (ℓ,𝑥),Pis\nconstant-time secure.\n5 IMPLEMENTATION AND EVALUATION\nWe implement our approach in a fully automated prototype tool, named CT-Prover .CT-Prover\nleverages the static value-flow analysis framework SVF [ 69] for pre-analysis (i.e., computing call-\ngraph, interprocedural-control-flow graph, points-to information and definition-use chains) and\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.Towards Efficient Verification of Constant-Time Cryptographic Implementations 46:15\nPhasar [ 66] for building our lightweight taint analysis. In particular, CT-Prover leverages points-to\ninformation from SVF to cope with dynamic memory accesses and the recursive implementation of\nthe IFDS framework in Phasar is rewritten as a worklist-based loop implementation for efficiency\nconsideration. CT-Prover utilizes the SMACK toolchain [ 63] to translate LLVM IR with taint\ninformation into Boogie IR [ 19] and Bam-Bam-Boogieman [ 58] to construct product programs\non Boogie IR. The Boogie verifier is used as the underlying safety verification engine. Dynamic\nmemory accesses in product programs are handled by SMACK and Boogie.\nResearch questions . We investigate the following research questions:\nRQ1. How effective and efficient is CT-Prover in proving constant-time security?\nRQ2. How efficient is CT-Prover for finding side-channel sources?\nRQ3. What are the respective contributions of the three main steps in CT-Prover ?\nBenchmark . We collect 87 real-world examples which are implementations from widely used\nmodern cryptographic and SSL/TLS libraries (e.g., Tongsuo [ 5], BearSSL [ 61], Mbed TLS [ 72],\nHACL* [ 2], FourQlib [ 59], libsodium [ 3] and OpenSSL [ 4]) and (constant-time) implementations\nfrom fixed-point arithmetic library (e.g., libfixedtimefixedpoint [ 13], curve25519-donna [ 1], and\nMEE-CBC implementations [ 10]). These benchmarks include cryptographic utilities, arithmetic\noperations, public and private key cryptography, and algorithms for encryption, decryption, message\nauthentication code (MAC), and digital signature. Among 87 examples, 58 are explicitly claimed to\nbe constant-time by developers.\nThe statistics of the benchmarks are shown in Table 1, where benchmarks with constant-time\nclaims are marked by†, #Loc shows the number of lines of the analyzed Boogie IR code (similar to\nLLVM bitcode [ 11]), and #Src shows the number of all potential side-channel sources. #Loc ranges\nfrom 28 to 68,502 (667,518 in total) and #Src ranges from 0 to 7,840 (55,060 in total). Interestingly,\nwe found that 7 (out of 87) benchmarks have no potential side-channel source after being translated\ninto Boogie IR, indicating that they avoid the use of branching and load/store-related statements.\nWe include them because they can be used to validate the tool implementation and measure the\nverification efficiency, and also because some of them were verified in ct-verif [11].\nThe experiments were conducted on a machine with Intel Xeon Gold 6342 2.80GHz CPU, 1T\nRAM, and Ubuntu 20.04.1. All the benchmarks were compiled with clang-12 -c -emit-llvm -O0\n-g -Xclang -disable-O0-optnone andopt -mem2reg , the same as ct-verif [11]. In particular,\nO0disables optimizations to avoid compiler-introduced leakage, disable-O0-optnone disables\nthe ‘optnone’ pass which affects ‘-mem2reg’ and mem2reg is used to reduce redundant load/store\ninstructions.\n5.1 Verifying Constant-time Implementations (RQ1, RQ3)\nTo answer RQ1, we verify all 87 programs. CT-Prover returns \" proved \" for 72 benchmarks. (One\nwill see later that the other 15 programs are indeed not constant-time and the constant-time claim\nmade by the developer may be incorrect.) We mainly compare CT-Prover with ct-verif which is\nthe only available tool at the LLVM IR level.\nThe results are reported in Table 2, where verification time is given in seconds and TO denotes\ntime out (1 hour). ct-verif fails to prove 16 constant-time programs: it runs out of time on 14\nprograms without outputting any potential side-channel sources (highlighted in red color) and\noutputs inconclusive verification results on 2 programs within 1 hour (highlighted in blue color).\nThe latter is because of the loop in P7_1 (cf. Example 1) and the type-casting of pointers in P9_4.\nThis indicates that the theoretical completeness of self-composition based approaches may be\ncompromised by the limitation of safety verification. On the other 56 constant-time programs that\ncan be proved by ct-verif andCT-Prover ,CT-Prover is about 50 times faster.\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.46:16 Luwei Cai, Fu Song, and Taolue Chen\nTable 1. Statistics of 87 Benchmarks, where #Loc and#Src show the numbers of lines and potential side-\nchannel sources of the program in Boogie IR, respectively; †indicates that the benchmark is claimed to be\nconstant-time by developers.\nName Lib/Algorithm/Function #Loc #Src Name Lib/Algorithm/Function #Loc #Src\nP1_1 OpenSSL_tls1_cbc_remove_padding 1042 71 P6_1 BearSSL_AES_small_decrypt 1395 310\nP1_2 OpenSSL_ssl3_cbc_digest_record 13089 198 P6_2 BearSSL_AES_small_encrypt 945 54\nP1_3 OpenSSL_ssl3_cbc_remove_padding 351 14 P6_3†BearSSL_ChaCha20_ct_run 2655 316\nP1_4 OpenSSL_ssl3_cbc_copy_mac 452 15 P6_4†BearSSL_GHASH_ctmul32_br_ghash_ctmul 2588 202\np2_1†MAC-then-Encode-then-CBC-Encrypt 22036 396 P6_5†BearSSL_RSA_i15_decrypt 5928 229\nP3_1†Hacl_HMAC_compute_sha2_256 27404 1673 P6_6†BearSSL_EC_p256_m15_api_mul 19535 1895\nP3_2†Hacl_HMAC_compute_blake2b_32 68502 5289 P6_7†BearSSL_EC_p256_m31_api_mul 9560 796\nP3_3†Hacl_HMAC_legacy_compute_sha1 2149 111 P6_8†BearSSL_EC_p256_m64_api_mul 6450 380\nP3_4†Hacl_HMAC_compute_blake2s_32 56465 4309 P6_9†BearSSL_GHASH_ctmul64 1060 10\nP3_5†Hacl_HMAC_compute_sha2_384 34620 2183 P6_10†BearSSL_AES_ct_bitslice_encrypt 1345 60\nP3_6†Hacl_HMAC_compute_sha2_512 34565 2177 P6_11†BearSSL_AES_ct_bitslice_decrypt 1788 72\nP3_7†Hacl_Chacha20_chacha20_decrypt 3303 180 P6_12†BearSSL_AES_ct_key_sched 2161 100\nP3_8†Hacl_Chacha20_chacha20_encrypt 3285 180 P6_13 BearSSL_AES_big_cbc_key_schedule 836 290\nP3_9†Hacl_Curve25519_64_ecdh 1813 70 P6_14 BearSSL_AES_big_cbc_decrypt 1813 565\nP3_10†Hacl_Curve25519_64_scalarmult 1619 67 P6_15 BearSSL_AES_big_cbc_encrypt 1806 565\nP3_11†Hacl_Curve25519_64_secret_to_public 1701 102 P6_16†BearSSL_RSA_i15_pkcs1_sign 6464 264\nP3_12†Hacl_Poly1305_32_poly1305_mac 2240 121 P6_17 BearSSL_DES_table_cbc_decrypt 1514 555\nP3_13†Hacl_Poly1305_128_poly1305_mac 72 0 P6_18 BearSSL_DES_table_cbc_encrypt 1501 555\nP3_14†Hacl_Poly1305_256_poly1305_mac 72 0 P6_19†BearSSL_RSA_i31_pkcs1_sign 5423 235\nP3_15†Hacl_Curve25519_51_ecdh 21158 2313 P6_20 BearSSL_Poly1305_i15_ChaCha20_run 3409 347\nP3_16†Hacl_Curve25519_51_scalarmult 20964 2310 P6_21†BearSSL_Poly1305_ctmul32_ChaCha20_run 4802 425\nP3_17†Hacl_Curve25519_51_secret_to_public 21046 2345 P6_22†BearSSL_EC_p256_m62_api_mul 6617 472\nP4_1 Tongsuo_curve448_ossl_x448 6674 446 P6_23†BearSSL_GHASH_ctmul_br_ghash_ctmul 1947 125\nP4_2 Tongsuo_curve448_derive_pub_key 8994 575 P6_24†BearSSL_AES_ct64_bitslice_encrypt 1348 60\nP4_3 Tongsuo_AES_decrypt 3274 1382 P6_25†BearSSL_AES_ct64_bitslice_decrypt 1791 72\nP4_4 Tongsuo_AES_encrypt 2982 1126 P6_26†BearSSL_AES_ct64_key_sched 2069 119\nP4_5 Tongsuo_constant_time_lookup 339 5 P6_27†BearSSL_RSA_i32_decrypt 4170 179\nP4_6 Tongsuo_curve25519_derive_pub_key 14744 7840 P6_28†BearSSL_Poly1305_ctmul_ChaCha20_run 4318 363\nP4_7 Tongsuo_curve25519_ossl_x25519 3561 163 P6_29†BearSSL_Poly1305_ctmulq_ChaCha20_run 6761 415\nP5_1 Mbed TLS_rsa_decrypt 13797 788 P6_30†BearSSL_DES_ct_cbc_decrypt 1670 49\nP5_2†Mbed TLS_mpi_lt_mpi_ct 411 26 P6_31†BearSSL_DES_ct_cbc_encrypt 1658 49\nP5_3†Mbed TLS_ct_rsaes_pkcs1_v15_unpadding 629 19 P8_1†FourQlib_ECC_double_eccdouble 28 0\nP5_4†Mbed TLS_mpi_safe_cond_assign 729 43 P8_2†FourQlib_ECC_madd_eccmadd 1632 58\nP5_5†Mbed TLS_mpi_core_lt_ct 214 5 P8_3†FourQlib_ECC_norm_eccnorm 2390 79\nP5_6†Mbed TLS_mpi_safe_cond_swap 765 46 P9_1 libsodium_core_salsa208 1126 9\nP5_7†Mbed TLS_ct_memcmp 145 7 P9_2 libsodium_aead_chacha20poly1305_decrypt 9311 397\nP5_8†Mbed TLS_ct_mpi_uint_cond_assign 140 4 P9_3 libsodium_core_salsa20 1094 9\nP5_9†Mbed TLS_ct_memcpy_offset 290 5 P9_4 libsodium_stream_chacha20 6620 259\nP5_10†Mbed TLS_ct_base64_dec_value 309 0 P9_5 libsodium_onetimeauth_poly1305_block 778 25\nP5_11 Mbed TLS_DES_crypt_cbc 1991 546 P9_6 libsodium_hash_sha512_Transformer 29713 2508\nP7_1†libfixedtimefixedpoint_fix_pow_fix_pow 24538 80 P9_7 libsodium_hash_sha256_Transformer 23806 2008\nP7_2†libfixedtimefixedpoint_fix_cmp_fix_cmp 591 0 P10_1 curve25519-donna portable implementation 11737 722\nP7_3†libfixedtimefixedpoint_fix_ln_fix_ln 15200 0 P10_2†curve25519-donna c64 implementation 23936 1599\nP7_4†libfixedtimefixedpoint_fix_eq_fix_eq 153 0\nTo partially answer RQ3, we inspect at which step a program can be proved in the verification\nprocess of CT-Prover . We found that all these programs (except for P1_1 and P9_4) can be proved\nusing the lightweight taint analysis solely. The lightweight taint analysis cannot determine 4 (resp.\n2) potential side-channel sources of P1_1 (resp. P9_4) due to an infeasible branch condition (resp.\nindex-insensitive), which were resolved by the precise taint analysis. While the individual cost of\nthe two taint analyses is lower than ct-verif , the accumulated cost is slightly higher.\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.Towards Efficient Verification of Constant-Time Cryptographic Implementations 46:17\nTable 2. Results of verifying constant-time implementations, where TO denotes time out (1 hour).\nName CT-Prover ct-verif Name CT-Prover ct-verif Name CT-Prover ct-verif Name CT-Prover ct-verif\nP1_1 7.55 7.53 P3_14†0.04 6.93 P6_3†0.10 9.25 P6_30†0.07 8.26\nP1_2 1.12 26.74 P3_15†0.70 TO P6_4†0.20 9.04 P6_31†0.07 8.51\nP1_3 0.04 7.25 P3_16†0.67 TO P6_6†23.24 TO P7_1†0.64 TO\nP1_4 0.05 7.89 P3_17†0.63 TO P6_7†0.45 TO P7_2†0.51 7.42\np2_1†1.44 134.72 P4_1 0.53 TO P6_8†0.21 794.79 P7_3†0.59 21.36\nP3_1†108.80 TO P4_2 0.57 TO P6_9†0.05 7.65 P7_4†0.52 7.37\nP3_2†177.48 TO P4_5 0.04 7.42 P6_10†0.07 7.79 P8_1†0.20 6.87\nP3_3†104.40 TO P4_6 0.35 177.25 P6_11†0.08 8.22 P8_2†0.40 21.69\nP3_4†169.71 TO P4_7 0.34 14.16 P6_12†0.07 8.66 P8_3†0.39 37.73\nP3_5†109.19 TO P5_2†0.08 7.27 P6_20 0.11 12.38 P9_1 0.14 7.86\nP3_6†122.85 TO P5_3†0.08 7.36 P6_21†0.14 14.49 P9_2 25.69 23.02\nP3_7†0.12 30.51 P5_4†1.78 7.49 P6_22†0.24 977.27 P9_3 0.13 7.97\nP3_8†0.13 30.65 P5_5†0.07 7.05 P6_23†0.11 8.40 P9_4 16.05 11.91\nP3_9†0.10 8.94 P5_6†1.86 7.57 P6_24†0.06 7.93 P9_5 0.06 7.25\nP3_10†0.09 8.91 P5_7†0.08 6.91 P6_25†0.08 8.41 P9_6 4.25 44.12\nP3_11†0.09 9.26 P5_8†0.08 6.46 P6_26†0.06 8.94 P9_7 3.34 32.98\nP3_12†0.09 9.28 P5_9†0.07 6.63 P6_28†0.12 12.23 P10_1 0.55 TO\nP3_13†0.04 7.05 P5_10†0.08 7.00 P6_29†0.17 13.86 P10_2†10.89 33.18\nIn summary, CT-Prover can return conclusive results for these benchmarks. Most constant-time\nimplementations can be proved by the lightweight taint analysis solely, whereas the remaining ones\ncan be proved by the precise taint analysis. CT-Prover significantly outperforms the state-of-the-art\ntoolct-verif for real-world constant-time implementations.\n5.2 Verifying Non-constant-time Implementations (RQ2, RQ3)\nTo answer RQ2, we check the remaining 15 cases which are deemed to be vulnerable. The results are\nshown in Table 3, where CT-Prover−refers to CT-Prover without the precise taint analysis (i.e.,\nthe 2nd main step), a:b in Time (s) respectively denote the execution time after the 2nd and 3rd main\nstep (note: the 1st step is very efficient whose time is negligible), TO denotes time out (6 hours),\nx:y:z (resp. x:z) in #Src respectively denote the numbers of side-channel sources reported after the\n1st, 2nd and 3rd (resp. 1st and 3rd) main step, and★indicates that some invalid loop invariants are\nadded by ct-verif and thus may miss side-channel sources. We have manually checked all these final\npotential side-channel sources with their corresponding execution traces produced by CT-Prover ,\nand found that all of them are genuine timing side-channel vulnerabilities (i.e., not false positives).\nIn particular, P6_5, P6_16, P6_19 and P6_27 (from BearSSL) were claimed to be of constant-time.1\nBoth CT-Prover andct-verif ran out of time on 4 programs (P5_1, P6_5, P6_16 and P6_19). We\nnote that on P5_1 our lightweight taint analysis is still able to find 48 potential side-channel sources\nbased on which we manually confirm that P5_1 is not constant-time. CT-Prover finally found\nmore side-channel sources than ct-verif on 9 (out of 15) programs (marked by★in Table 3), because\nct-verif misses side-channel sources when some additional loop invariants cannot be proved, while\nCT-Prover still works after automatically removing such loop invariants. On the other programs,\nCT-Prover andct-verif report the same side-channel sources. In terms of efficiency, CT-Prover is\nslightly slower than ct-verif . It is not surprising as CT-Prover involves three main steps while the\nlast two main steps have to perform safety checking. We remark that ct-verif may miss potential\nside-channel sources since it skips part of the program (e.g., the loop body) when additional loop\ninvariants cannot be proved, which explains its reduced execution time.\n1We have reported all the potential vulnerabilities in Table 3 to the respective developer(s), and received confirmations for\nMbed TLS and BearSSL.\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.46:18 Luwei Cai, Fu Song, and Taolue Chen\nTable 3. Results of verifying non-constant-time implementations, where CT-Prover−refers to CT-Prover\nwithout the precise taint analysis (i.e., the 2nd main step), a:b in Time (s) respectively denote the execution\ntime after the 2nd and 3rd main step (note: the 1st step is very efficient whose time is negligible), TO denotes\ntime out (6 hours), x:y:z (resp. x:z) in #Src respectively denote the numbers of side-channel sources reported\nafter the 1st, 2nd and 3rd (resp. 1st and 3rd) main step, and★indicates that some invalid loop invariants are\nadded by ct-verif and thus may miss side-channel sources.\nNameCT-Prover ct-verif CT-Prover−\nNameCT-Prover ct-verif CT-Prover−\nTime (s) #Src Time (s) #Src Time (s) #Src Time (s) #Src Time (s) #Src Time (s) #Src\nP4_3 24.05:50.84 49:48:48 17.69 32★24.77 49:48 P6_14 13.06:26.57 32:32:32 10.58 0★12.64 32:32\nP4_4 21.03:44.76 49:48:48 16.73 32★21.96 49:48 P6_15 13.06:26.35 32:32:32 10.37 0★12.42 32:32\nP5_1 TO:TO 48:–:– TO – TO 48:– P6_16†2561.81:TO 107:71:53 TO 21★TO107:55\nP5_11 14.10:30.68 26:16:16 11.81 16 15.20 26:16 P6_17 9.75:20.31 11:8:8 9.29 8 9.48 11:8\nP6_1 9.36:19.27 12:1:1 8.46 1 9.13 12:1 P6_18 9.77:20.12 13:8:8 9.26 8 9.63 13:8\nP6_2 8.48:17.24 12:1:1 7.45 1 8.46 12:1 P6_19†3515.77:TO 142:70:39 TO 21★TO142:46\nP6_5†983.86:TO 107:71:50 TO 21★TO107:49 P6_27†131.34:503.23 177:45:45 450.57 22★580.50 177:45\nP6_13 8.85:17.87 6:4:4 8.45 4 8.51 6:4\nTo partially answer RQ3, we analyze the respective number of (potential) side-channel sources\nreported by the three main steps of CT-Prover and two main steps of CT-Prover−(in the form of\nx:y:z and x:z in Table 3). We find that the lightweight taint analysis can determine a large number\nof potential side-channel sources, the precise taint analysis can resolve the remaining few unsolved\nones (i.e., P4_3, P4_4, P5_11, P6_1, P6_2, P6_5, P6_13, P6_16, P6_17, P6_18, P6_19, and P6_27),\nand the left-over ones are often vulnerabilities (i.e., P4_3, P4_4, P5_11, P6_1, P6_2, P6_13, P6_17,\nP6_18 and P6_27). By comparing with CT-Prover−, we can observe that disabling the precise taint\nanalysis in CT-Prover (i.e., the 2nd main step) may both improve (i.e., P6_5) and degrade (i.e.,\nP6_16 and P6_19) the capability of finding side-channel sources while reducing the verification time.\nNevertheless, the precise taint analyses are still useful for finding all the potential side-channel\nsources when the 3rd main step runs out of time (i.e., P6_5, P6_16, and P6_19), and the first two\nmain steps (i.e., the lightweight and precise taint analysis) are often able to find all the side-channel\nsources (i.e., P4_3, P4_4, P5_11, P6_1, P6_2, P6_13, P6_14, P6_15, P6_17, P6_18 and P6_27) with\ncomparable or less execution time than CT-Prover−.\n1uint32_t br_rsa_i15_private(const br_rsa_private_key *sk){\n2const unsigned char *p = sk->p; size_t plen = sk->plen;\n3while (plen > 0 && *p == 0) { p++; plen--;} }\nFig. 9. Simplified fragment of P6_5 taken from the BearSSL library.\nCase study . Fig. 9 shows one side-channel source (4,∗𝑝==0)of P6_5 from BearSSL. Variable p\npoints to a buffer storing a large prime (for RSA) which is secret but the loop condition *p == 0\ndepends upon the content of the buffer. It leaks the number of leading zero of the large prime.\nIn summary, CT-Prover is efficient and effective for finding side-channel sources and significantly\noutperforms ct-verif . Both taint analyses can determine potential side-channel sources, reducing\nthe cost of the subsequent safety verification and manual validation.\nFurther comparison . We also compare CT-Prover with two sound but incomplete tools Ve-\nrasco [ 26] and BINSEC (the latest version of Binsec/Rel [ 38]). Verasco performs taint analysis by\nabstract interpretation with bounded loops, while BINSEC uses relational symbolic execution with\nbounded paths. We use relatively large benchmarks provided by the respective tools with security\nannotations to reduce the engineering efforts of modifying benchmarks, and compile benchmarks\nusing the constant-time preserving compiler [22].\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.Towards Efficient Verification of Constant-Time Cryptographic Implementations 46:19\nWe observe that CT-Prover is significantly more efficient than Verasco on almost all the bench-\nmarks (80.63 seconds vs. 1,127.63 seconds in total). Moreover, Verasco cannot output traces and\nvalues of scalar input variables leading to potential violations, and misses some timing side-channel\nsources on the DES implementation. Compared with BINSEC, CT-Prover is about 2 times faster\nthan BINSEC when BINSEC’s execution time is independent of input size (e.g., curve25519-donna),\nand the speedup becomes more significant when BINSEC’s execution time increases with input size,\ne.g., BINSEC takes 0.36, 10.28 and 102.48 seconds on the libsodium_chacha20_xor implementation\nwhen the input size is 256, 256 ×512 and 256×5120 bits while CT-Prover takes 11.21 seconds without\nlimiting the input size.\n5.3 Threats to Validity\nInternal threats. The major internal threat to our evaluation is the correctness of the imple-\nmentation of CT-Prover which relies on various open-source frameworks and tools. To mitigate\nthis threat, we compared the results of CT-Prover andct-verif , and manually analyzed all the\nside-channel sources discovered by both tools to confirm the correctness of CT-Prover . Moreover,\nwe note that these open-source frameworks and tools have been widely used for years so we would\nhave reasonable confidence in their quality. A more scientific way is to build a verified toolchain,\nwhich however requires more resources and thus is left as interesting future work.\nExternal threats. The major external threat to our evaluation is the benchmarks, as the per-\nformance of CT-Prover may vary with benchmarks. To mitigate this threat, we consider both\nconstant-time and non-constant-time implementations from widely used modern cryptographic\nand SSL/TLS libraries, as well as benchmarks used in prior work, e.g., [ 11,26,38]. Furthermore, the\nbenchmarks are selected to be diverse in terms of both types and sizes. Another external threat is\nthe translation from source code to LLVM IR which may introduce violations of constant-time, as\ndemonstrated by [ 38]. To mitigate this threat, benchmarks are only optimized by -mem2reg which\nreduces redundant address-taken variables in LLVM IR.\n6 RELATED WORK\nTiming side-channels have received considerable attention (cf. [ 48] for a survey). We roughly classify\nthe current approaches into three categories. The first class does not rely on verification; the second\nclass mostly leverages program analysis; the third class is largely based on verification. There\nare verification approaches for other side-channels (e.g., [ 20,41,43–47,62,84]) and time-balance\n(e.g., [14, 15, 28]), which are orthogonal to this work.\nApproaches based on concrete execution. A large number of approaches have been proposed\nfor detecting and/or quantifying timing side-channel leakages in terms of e.g., channel capacity and\nShannon entropy. To this end, for instance, DATA [ 78], ct-fuzz [ 51] and CacheQL [ 83] make use\nof concrete execution. CANAL [ 70], Abacus [ 18], and ENCIDER [ 82] leverage dynamic symbolic\nexecution, symbolic execution on individual concrete execution traces, and concolic execution.\nCaType [ 52] detects timing side-channel vulnerabilities by applying type inference on individual\nconcrete execution traces. In general, this class of approaches are often effective in bug-finding,\nbut cannot prove the absence of timing side-channel leakages.\nApproaches based on program analysis. This class of work is more formal which often is\nable to prove the absence of timing side-channel leakage. CacheAudit [ 40] bounds timing side-\nchannel leakages by over-approximating side-channel observations using abstract interpretation.\nCacheS [ 76] applies abstract interpretation, but its implementation is unsound due to its imprecise\ntreatment of memory. Taint analysis and security type systems have also been applied to detect side-\nchannel leakage by tracking information flow of the secrets [ 21,26,77], varying in accuracy and\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.46:20 Luwei Cai, Fu Song, and Taolue Chen\nefficiency. Moreover, the work [ 17,32,65] bounds the information leakage via symbolic execution\nand model-counting. While these approaches are often sound, they may raise false positives.\nProgram transformations have been proposed to eliminate potential side-channel sources [ 7,35,79].\nPrecise detection approaches can reduce the number of potential side-channel sources but the\nadopted program transformations may bloat the program, making them less efficient to run.\nApproaches based on verification. There are mainly two approaches in this class, i.e., self-\ncomposition [ 12] and relational symbolic execution [ 42]. ct-verif [ 11] uses a variant of self-\ncomposition (i.e., cross-product) to improve the efficiency of safety verification. ct-verif is complete\nassuming the completeness of the underlying safety checker. Binsec/Rel [ 36,38] uses relational\nsymbolic execution enhanced with secret-dependency tracking, untainting and fault-packing to\nimprove the efficiency. Due to the path explosion problem and unsupported dynamic memory\nallocation, it is only complete up to a given depth of paths, and the size of the symbolic input\n(keys, plaintext) has to be fixed. ct-verif targets LLVM IR, a target-independent low-level language,\nwhile Binsec/Rel targets binary executables. (Note that the compilation from LLVM IR to binary\nexecutable may introduce vulnerabilities.)\nOur work generally falls into the third category. Similar to ct-verif, we focus on LLVM IR instead\nof binary executable or source code. Compared with the existing verification approaches, we\nextensively leverage taint analysis to facilitate self-composition, especially for reducing safety\nchecks and simplifying the self-composed program. A similar idea was adopted in Binsec/Rel, but\nin a different way. Our new methodology significantly improves both efficiency and effectiveness.\nConsideration of micro-architecture. The above approaches did not address micro-architectural\nfeatures which have also been studied in literature. For instance, Constantine [ 27] uses dynamic\ntaint analysis; Oo7 [ 75] uses static taint analysis; Binsec/Haunted [ 37] uses relational symbolic\nexecution; Pitchfork [ 34], Spectector [ 49], SpecuSym [ 50] and KleeSpectre [ 74] leverage (dynamic)\nsymbolic execution; Blade [73] uses type system; [80] uses abstract interpretation.\nOur work is orthogonal to these investigations, as they usually tackle the vulnerabilities brought\nby micro-architectural features with the assumption that the program itself is of constant-time.\n7 CONCLUSION\nIn this paper, we have provided practical verification approaches for constant-time implementations\nof cryptographic libraries. Our methods are based on a novel synergy of taint analysis and safety\nverification of self-composed programs. We have implemented a cross-platform and fully automated\ntoolCT-Prover working on LLVM IR. The tool has been extensively evaluated on a large set of real-\nworld benchmarks from modern cryptographic and SSL/TLS and fixed-point arithmetic libraries.\nThe experimental results have confirmed the efficacy of our approaches. In particular, compared\nto the state-of-the-art tool ct-verif ,CT-Prover typically demonstrates 2-3 orders of magnitude of\nimprovement, proving more programs and finding new timing leaks.\nAt the methodology level, we showcase that a combination of lightweight approaches (taint\nanalysis) and heavyweight approaches (self-composition and safety checking) can yield efficiency\nand completeness. In particular, we demonstrate that self-composition based approaches, which are\nnormally considered to be powerful but costly, can be made scalable with the aid of static analysis.\nDATA AVAILABILITY\nTo foster further research, source code and benchmarks are available at [6].\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024.Towards Efficient Verification of Constant-Time Cryptographic Implementations 46:21\nACKNOWLEDGEMENT\nThis work is supported by the Amazon Research Award from Amazon Web Services, National\nNatural Science Foundation of China under grants No. 62072309 and 61872340, CAS Project for\nYoung Scientists in Basic Research (YSBR-040), ISCAS New Cultivation Project (ISCAS-PYFX-\n202201), ISCAS Fundamental Research Project (ISCAS-JCZD-202302), Overseas grants from the State\nKey Laboratory of Novel Software Technology, Nanjing University (KFKT2022A03, KFKT2023A04),\nand Birkbeck BEI School Project (EFFECT).\nREFERENCES\n[1] 2023. curve25519-donna . https://github.com/agl/curve25519-donna.\n[2] 2023. HACL* . https://github.com/hacl-star/hacl-star.\n[3] 2023. libsodium . https://doc.libsodium.org/.\n[4] 2023. OpenSSL . https://www.openssl.org.\n[5] 2023. Tongsuo . https://github.com/Tongsuo-Project/Tongsuo.\n[6] 2024. 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SCInfer: Refinement-Based Verification of Software Counter-\nmeasures Against Side-Channel Attacks. In Proc. of the 30th International Conference on Computer Aided Verification .\n157–177. https://doi.org/10.1007/978-3-319-96142-2_12\nReceived 2023-09-28; accepted 2024-01-23\nProc. ACM Softw. Eng., Vol. 1, No. FSE, Article 46. Publication date: July 2024."    },    {        "title": "2402.13526v1.Pion_nucleus_elastic_scatterings_incorporating_medium_effects_within_the_Eikonal_Glauber_mode.pdf",        "content": "PKNU-NuHaTh-2024\nPion-nucleus elastic scatterings incorporating medium effects\nwithin the Eikonal-Glauber model\nHyeon-dong Han ,1, 2,∗Parada T. P. Hutauruk ,1,†and Seung-il Nam1, 2, 3, ‡\n1Department of Physics, Pukyong National University (PKNU), Busan 48513, Korea\n2Center for Extreme Nuclear Matters (CENuM), Korea University, Seoul 02841, Korea\n3Asia Pacific Center for Theoretical Physics (APCTP), Pohang 37673, Korea\n(Dated: February 22, 2024)\nIn this present investigation, we explore the elastic scattering of pions with nuclei ( π-A), primar-\nily influenced by the ∆(1232) resonance, within the Eikonal-Glauber model. The medium effects\nare incorporated by considering nuclear-density ( ρA) dependent masses of baryons and strong cou-\npling constants. These dependencies are computed and parameterized up to O(ρ2\nA) based on the\nquark-meson coupling (QMC) model. The Wood-Saxon type density profile is utilized for the bound\nnucleons within finite nuclei. The element π+-Nscattering cross section for the Glauber approach is\ndetermined using the conventional effective Lagrangian method. Subsequently, we analyze the total\ncross sections for elastic scattering with4He and12C targets. Our numerical results demonstrate\na favorable agreement with JINR data for the4He target, accurately reproducing the total cross-\nsection. However, when considering the12C target, deviations of approximately ≲10%. We also\nconsider the multiple-scattering effects inside the nucleus approximately, using the single-channel\nmeson-baryon Bethe-Salpeter equation, resulting in the effective width broadening of the ∆ reso-\nnance to reproduce the data better.\nKeywords: Elastic pion-nucleus scattering, Eikonal-Glauber model, effective Lagrangian approach, medium\neffects, Helium and carbon targets, multiple scattering\nI. INTRODUCTION\nThe investigation of low-energy pion-nucleus ( π-A) scattering has been one of the most important subjects within\nthe nuclear physics community since 1950 [1, 2]. This field holds significance as it offers valuable insights into nucleon-\nnucleon ( N-N) interactions and nuclear structure. The π-Ainteractions manifest through elastic scatterings, where\nthe target nucleus remains in its ground state [3–5], inelastic scatterings with the nucleus in excited states upon\npion emission [6], and absorption processes where no pion emerges from the nucleus [7, 8]. Numerous theoretical\nstudies on π-Nandπ-Ascatterings have been conducted using diverse approaches, including the optical model in\ncoordinate space [9, 10], the optical model in momentum space [3], coupled channel theory [11], extended schematic\nmodels [12], cloudy bag models [13], and chiral bag models incorporating quark degrees of freedom [14]. It is noteworthy\nthat theoretical model calculations often account for pion absorption by employing an effective potential, typically\nproportional to the second order of the nuclear density, denoted as O(ρ2\nA).\nThis is akin to the absorption coefficient, signifying that the rate of particle absorption per unit time at a specific\nposition in the nucleus is proportionate to the particle density at that location. This coefficient is a component\nof the Schr¨ odinger equation within the optical potential, manifesting as an imaginary potential with absorptive\ncharacteristics, while the real potential is recognized as dispersive. Alternatively, this absorption coefficient can\nbe derived from the continuity equation, as detailed in Ref. [15]. Nevertheless, contemporary interest in elastic π-\nAscattering has waned due to the intricate nature of nuclear structure. Despite extensive theoretical endeavors,\nnumerous experiments have been conducted, amassing a wealth of data on π-Ascattering cross-sections. However,\nthese experimental findings still bear substantial uncertainties and lack precision owing to limited information about\nnuclear structure. Moreover, the available experimental data pertains only to lower pion energies, approximately\naround 70 MeV. Fortunately, recently, the maximum pion energy has surged to approximately 350 MeV for12C and\n208 MeV for4He, respectively, for instance [16].\nIn this study, we conduct a phenomenological examination of π-Aelastic scattering, emphasizing the dominance\nof the ∆(1232) resonance and incorporating medium effects within the Eikonal-Glauber model (EGM)l, utilizing a\nπ+beam. The medium effects are introduced through nuclear-density ( ρA) dependent masses of baryons and strong\n∗E-mail: monlistor@pukyong.ac.kr\n†E-mail: phutauruk@gmail.com, phutauruk@pknu.ac.kr\n‡E-mail: sinam@pknu.ac.kr, gariwulf@gmail.com\nTypeset by REVT EXarXiv:2402.13526v1  [hep-ph]  21 Feb 20242\ncoupling constants, computed and parameterized based on the quark-meson coupling (QMC) model [17–22] up to\nO(ρ2\nA). QMC computations reveal that effective nucleon masses for A= (4He,12C) gradually increase up to r≃2\nfm and then become saturated. A similar trend is observed for the ∆ mass, albeit with differing magnitudes. For\nthe calculation of π+-Aelastic scattering cross-sections, we utilize the element π+-Nelastic scattering cross-sections\ncomputed via the effective Lagrangian approach at the tree-level Bonn approximation. It is essential to highlight that\nπ+-Nscattering cross-sections are predominantly influenced by the ∆(1232) resonance, and our numerical results\neffectively reproduce existing experimental data, as demonstrated in previous research [23].\nOur findings indicate that the total cross-section for the4He target closely agrees with the data from the Joint\nInstitute for Nuclear Research (JINR) [24]. However, for the12C target, the results tend to overestimate the data\nby less than 10%, particularly when compared to those of the Shanghai Institute of Nuclear Research (SINR) [25],\nattributed to the oversimplification inherent in the current Glauber model for heavier nuclei. To improve the theoretical\nresults, the multiple-scattering effects inside the nucleus are taken into account approximately, using the single-channel\nmeson-baryon Bethe-Salpeter equation. As a result, it is observed that the effective broadening of the decay width of\nthe ∆ resonance occurs, resulting in the data are described better in comparison to that without the multiple-scattering\neffects. We also provide the prediction for the16O target case with the same theoretical framework.\nThis manuscript is structured as follows. Section II provides a brief introduction to the theoretical framework\nfor the π+-Aelastic scatterings. In Section III, we present comprehensive numerical results encompassing various\ntheoretical aspects, such as effective baryon mass, strong couplings, cross sections, and multiple scatterings for nuclei\ntargets mainly, accompanied by detailed discussions. The concluding section is dedicated to summarizing the key\nfindings.\nII. THEORETICAL FRAMEWORK AND NUMERICAL RESULTS\nIn this section, we provide a brief overview of the current theoretical framework governing elastic π+-Ascatterings\nwithin the EGM. We outline the incorporation of medium effects into the model, specifically addressing the deter-\nmination of effective baryon masses and relevant strong couplings. These medium effects are computed using the\nQuark-Meson Coupling (QMC) model up to O(ρ2\nA), with ρArepresenting nuclear density. Additionally, we account\nfor the bound-nucleon density profile within the nuclei using the Wood-Saxon functional form, given in Ref. [26]. His-\ntorically, the Glauber model was initially proposed in Refs. [27, 28], proving highly successful as a phenomenological\nmodel for characterizing nuclear properties, particularly in N-AandA-Ainteractions observed in heavy-ion collision\n(HIC) experiments, as documented in Ref. [29]. This model interprets N-Ascattering in terms of multiple elastic\nN-Nscatterings. Our focus in this study is on the total cross-sections of elastic π-Ascatterings involving different\nnucleus targets, such as4He and12C. To this end, we employ the EGM, an updated version of the Glauber approach\nthat assumes a projectile follows a linear path, known as the Eikonal approximation [30].\nIt is crucial to emphasize that, in our computations employing the current EGM, we rely on the element π-N\nscattering cross-sections as input, where the dominant influence is attributed to the ∆(1232) resonances. Within the\nframework, the expression for the total cross-section is defined as follows:\nσπA(s∗) =Z\nd2bh\n1−exp\u0002\n−˜σπN(E∗\ncm, ρN\nA)TN\nA(b)\u0003i\n, TN\nA(b) =Z∞\n−∞dz ρN\nA\u0010p\n|b|2+z2\u0011\n, (1)\nHere, ˜ σπNrepresents the sum of the π-Nscattering cross-section for N= (n, p) within the nucleus, and TN\nAdenotes\nthe thickness function describing the transverse reaction probability at the two-dimensional impact-parameter vector\nb. Meanwhile, zsignifies the longitudinal spatial position. The combined cross-section, accounting for the numbers\nof neutrons ( n) and protons ( p) within the nuclei, can be expressed as follows:\n˜σπN=\u0012A−Z\nA\u0013\nσπn+\u0012Z\nA\u0013\nσπp. (2)\nIt is worth noting that the ratio between the elementary π-Ncross sections, namely σπpandσπn, is approximately 9:1.\nThis ratio can be understood by their distinct isospin factors in elastic scatterings, as elucidated in prior research [23].\nTo consider the internal density distribution of the nuclei, the Wood-Saxon density profile, presented as a function\nin Eq. (1), is expressed as follows:\nρ(n,p)\nA(r) =\"\nρc\nA\u0002\n1 +c(r/rA)2\u0003\n1 + exp[( r−rA)/d]#\u0014(A−Z, Z)\nA\u0015\n, r=p\n|b|2+z2, (3)\nHere, r,ρc\nA,rA,d, and crepresent the radial distance, center density, average radius, surface thickness, and deformation\nparameter for the target nucleus, respectively. The specific values for these parameters are determined through fitting3\n0 1 2 3 4 5\nr [fm]00.020.040.060.080.10.120.14Density profile per nucleon [fm⁻³]⁴He\n¹²C(a)\n0 1 2 3 4 5\nr [fm]600800100012001400 Effective baryon mass [MeV]N for ⁴He\nΔ for ⁴He\nN for ¹²C\nΔ for ¹²C(b)\nFIG. 1: (a Density profile per nucleon ρN\nA[fm−3] as a function of the radial distance r[fm] for4He (solid) for12C (dashed).\n(b) Effective baryon mass M∗\nB[MeV] for B=N,∆ in the same manner with the panel (a).\nto experimental data, as detailed in Ref. [31, 32], and are compiled for4He,12C, and16O in Table I. The density\nprofile in Eq. (3) satisfies the following normalization condition:\nX\nN=(n,p)Z\ndz d2bρN\nA(p\n|b|2+z2) =A. (4)\nThis implies the conservation of nucleon number density. The density profiles for4He (solid) and12C (dot-dashed)\nare depicted in panel (a) of Fig. 1, showcasing their dependence on the radial distance rfrom the center. Given that\nwe are specifically examining even-even nuclei, it follows that ρn\nA=ρp\nA=ρA.\nAs previously mentioned, in this study, we determine the density-dependent effective masses for baryons, along\nwith the corresponding strong coupling constants, utilizing the QMC model within the mean-field approximation.\nFurther details regarding these calculations can be found in the prior work [23] and the references therein. For\npractical purposes, we parameterize these quantities as functions of ρAup toO(ρ2\nA), akin to the imaginary potential\nforA-Aderived in the Schr¨ odinger equation. Consequently, the parameterization for the effective baryon mass M∗\nB\nis expressed as follows:\nM∗\nB=MB+C1ρA+C2ρ2\nA+O(ρn>2\nA), (5)\nwhere MBstands for the vacuum mass, and the relevant coefficients are given by C1=−1.543 GeV ·fm3andC2=\n2.036 MeV fm3via the QMC calculation.\nThe panel (b) of Fig. 1 illustrates the variation of M∗\nBfor4He and12C with respect to the radial distance r. In\nthe QMC model, the effective nucleon masses M∗\nNexhibit an initial value of approximately 800 MeV at the nuclei’s\ncenter due to the partial restoration of spontaneous chiral symmetry breakdown (SCSB) at finite densities. As one\nmoves radially outward, the medium effects diminish, leading to a convergence toward the nucleon vacuum mass of\napproximately ∼940 MeV. It is noteworthy that the behavior of M∗\n∆for the nuclei mirrors that of the nucleon, given\nthe utilization of the same mass formula in Eq. (5) for both baryons. However, the M∗\nN,∆curves display distinct\ndependencies on rfor different nuclei. This discrepancy arises from their disparate density profiles, as depicted in\npanel (a) of Fig. 1. In addition to examining the influence of the medium effects on masses, we also investigate\nalterations in the strong coupling constants. To address this, we incorporate the Goldberger-Treiman relation (GTR)\nNuclei AZrA(fm) d(fm) ρc\nA(fm−3)c\n4He 421.01 0.327 0.2381 0.445\n12C1262.36 0.522 0.1823 −0.149\n16O1682.608 0.513 0.1701 −0.051\nTABLE I: rA,d,ρc\nA, and cfor4He,12C, and16O from the fitting of experimental data [31, 32].4\n0 0.02 0.04 0.06 0.08 0.1\nρ [fm⁻³]0.511.522.5 Meson-baryon coupling constantπ N N\nπ N Δ(a)\nFIG. 2: (a) Meson-baryon coupling constants fπNN (solid) and fπN∆(dashed) as functions of ρA[fm−3]. (b) ∆-baryon decay\nwidth in Eq. (8), Γ( Ecm, ρA) [MeV] as a function of Ecm[MeV] and ρA[fm−3].\nconcerning the weak axial-vector coupling constant fA, which is expressed as follows:\nfA=fπNNfπ\nMN. (6)\nThe numerical outcomes for M∗\nBas functions of rare presented in panel (b) of Fig. 1 for4He and12C. The effective\nnucleon masses MNcommence at approximately 800 MeV at the nucleus’s center, reflecting the partial restoration of\nthe spontaneous breakdown of chiral symmetry (SBCS) at finite densities within the QMC model. With increasing\nradial distance, medium effects diminish, converging to the nucleon vacuum mass of ∼940 MeV. Notably, M∗\n∆for the\nnuclei exhibits similar behaviors to those of the nucleon, given the employment of the same mass formula in Eq. (5)\nfor both baryons. However, the M∗\nN,∆curves display distinct rdependencies contingent on the nuclei, stemming from\ntheir diverse density profiles as illustrated in panel (a) of Fig. 1.\nf∗\nπNN(ρA) =fπNN−0.633ρA, f∗\nπN∆(ρA) =fπN∆−1.360ρA. (7)\nHere, we adopt the values fπNN = 0.989,fπN∆= 2.127, and fπ= 93.2 MeV, sourced from the Nijmegen potential\nmodel [38] and experimental data [39]. The determination of f∗\nπN∆assumes that the ratio fπNN/fπN∆≈2.15 remains\nconstant within the medium.\nOur findings for the in-medium modifications of the pion-nucleon coupling constants f∗\nπNN, the pion-delta coupling\nconstant f∗\nπN∆, and the pion decay constant with medium correction effects, as functions of ρA, are illustrated in\nFig. 2. In panel (a) of Fig. 2, it is evident that the density dependence of the pion decay constant diminishes with\nincreasing ρA, aligning with results from other theoretical models. Similar trends are observed for f∗\nπNN andf∗\nπN∆,\nboth decreasing as ρAincreases, albeit at a slower rate compared to f∗\nπ. It is important to note that these medium\ncorrections will be utilized as input when calculating the cross-section of pion-nucleus scattering for4He and12C.\nIn our calculations, we also account for the momentum-dependent ∆ decay width Γ, as employed in Ref. [33].\nHowever, in finite nuclei, the decay width should be determined by the nuclear density distribution ρA, taking the\nform:\nΓ(E∗\ncm, ρA) = Γ sp\u0012ρA\nρ0\u0013\n+ Γ0\u0014q(M∗\nN, Mπ, E∗\ncm)\nq(MN, Mπ, M∆)\u00153\u0014M∗\n∆\nE∗cmβ2\n0+q2(M∗\nN, Mπ, M∗\n∆)\nβ2\n0+q2(M∗\nN, Mπ, E∗cm)\u0015\n, (8)\nwhere Γ sp= 80 MeV represents the spreading density dependence of the decay width, determined through fitting to\nthe medium quantity at finite density. Additionally, β0= 200 MeV serves as the cutoff parameter, consistent with\nthe choice made in Ref. [33]. The results for the 3D plot of the decay width for the ∆ baryon in finite nuclei, as a\nfunction of ρAandE∗\ncm, are depicted in Fig. 2. This visualization indicates that the in-medium modifications of the\n∆ baryon decay width differ significantly for4He and12C due to their distinct nuclear charge density distributions of\nρA, as illustrated in Fig. 2. The in-medium modifications of the ∆ baryon decay width increase with rising nuclear\ndensity distribution ρA. Furthermore, it is observed that as Ecmincreases, the in-medium modifications of the ∆\nbaryon resonance decay width Γ∗\n∆also increase, consistent with the findings in the calculation of Ref. [34].5\n1100 1200 1300 1400 1500 1600\nEcm [MeV]020406080100120140160 Total cross section [mb]JINR\n⁴He target with Γ(0,0)\n⁴He target with Γ(Ecm,ρ)(a)\n1100 1200 1300 1400 1500 1600\nEcm [MeV]050100150200250300350 Total cross section [mb]SINR\n¹²C target with Γ(0,0)\n¹²C target with Γ(Ecm,ρ)(b)\nFIG. 3: (a) Total cross-section [mb] for the4He target as a function of the center-of-mass (cm) energy Ecm[MeV]. Here,\nwe consider two cases for the ∆ decay width in Eq. (8) with (dashed) and without (dashed) the ( Ecm, ρA) dependence. The\nexperimental data are taken from JINR. [24]. (b) The same for the12C target. The data are taken from SINR [25].\nNow, we are in a position to present the numerical results for the cross-section of elastic pion-nucleus scattering at\nintermediate energies, where ∆ baryon resonance dominates, for both4He and12C in Fig. 3. For a more in-depth\nexploration of the elastic pion-proton and pion-neutron scatterings, interested readers are referred to Ref. [23]. The\nnumerical results for the elastic cross-sections for the4He target, accounting for medium corrections up to ρ2\nAwith\n˜Γ∗\n∆(Ecm, ρA) as a function of center-of-mass (cm) energy, are presented in the panel (a) of Fig. 3, alongside JINR\nexperimental data[24]. Fig. 3 clearly illustrates the total cross-section for4He, being qualitatively consistent with the\ndata, particularly around the resonance region at Ecm∼1222 MeV. The dashed and solid lines denote the results\nwith and without the energy- and density-dependent decay width, respectively, not showing considerable difference.\nFor the12C, the numerical result, compared with SINR experimental data [25], is shown in the panel (b) of Fig. 3.\nHere, the the numerical result overestimates the data[25] by about 10% percent deviations around the resonance peak,\nin contrast to the result for4He. Again, the modified width in Eq. (8) does not make sizable difference even for the\n12C target. In what follows, we would like to discuss the deviation between the theory and experiment for the12C\ntarget in detail.\nIn principle, the decay width of the resonance can be modified by the multiple scattering (MS) inside the nucleus,\nand this modification can be the cause to explain the deviation observed above. Hence, we would like to estimate the\neffects of the meson-baryon MS inside the nucleus using the single-channel ( π-N) Bethe-Salpeter equation, resulting in\nthe broadening decay width of the ∆ resonance, which dominates the reaction process. The Bethe-Salpeter equation\nfor the scattering can be defined as follows:\n˜MMS\nij≈˜Mij+X\nk˜MikGk˜Mkj+···=˜Mij+X\nk˜MikGk˜MMS\nkj, (9)\nwhere ˜M[mass−1] indicates the reduced amplitude for an element process with the channel indices i, j,···. It relates\nto the invariant amplitude by\n¯ui˜Mijuj=Mij[mass2]. (10)\nHere, Gk=Gk(s=E2\ncm) stands for the meson-baryon propagator in the on-mass-shell approximation for the interme-\ndiate kchannel. In the present work, for simplicity, we only consider the πNelastic channel as mentioned previously,\ni.e., ( i, j) = (πN, πN ). Then, the propagator is given by the dimensional regularization [40] as follows:\nGπN(s) =MN\n8π2h\na+ lnM2\nN\nµ2+s−s′\nπN\n2slnm2\nπ\nM2\nN\n+sπN\n2sln [(sπN+s′\nπN+s) + (sπN−s′\nπN+s)−(sπN+s′\nπN−s)−(sπN−s′\nπN−s)]i\n[mass] ,(11)\nwhere sπNands′\nπNare defined by\nsπN=sπN(s) =p\n[s−(MN−mπ)2][s−(MN+mπ)2], s′\nπN=M2\nN−m2\nπ. (12)6\n1200 1400 1600 1800 2000\n√s [MeV]-30-20-100102030G(s) [MeV]Re[G]\nIm[G](a)\n1100 1200 1300 1400 1500\nEcm [MeV]50100150200250300350 Total cross section [mb]JINR\nSINR\nHe-4\nC-12\nO-16(b)\nFIG. 4: (a) Real (solid) and imaginary (dashed) parts for the propagator GπN(s) in Eq. (11) beyond the π-Nthreshold. (b)\nNumerical results for the elastic cross sections for the4He (solid),12C (dotted), and16O (dashed) targets as functions of Ecm.\nThe data are taken from Refs. [24, 25].\nAs for the π-Nelastic-scattering channel, the renormalization scale µand subtraction parameter aare determined\nby 1.2 GeV and 2, respectively, to reproduce available data for the elementary cross-sections [40, 41]. In the panel\n(a) of Fig. 4, we draw the real (solid) and imaginary (dashed) parts of the π-Npropagator in Eq. (11) beyond its\nthreshold. For instance, the single-channel elastic scattering for the ∆(1232) can be modified by the unitarization of\nthe MS amplitude, due to the multiple scatterings inside the nuclei as follows:\n˜M∆,MS\nπN =˜M∆\nπN\n1−˜M∆\nπNGπN≈ˆM∆\nπN\ns−M2\n∆−iΓ∆M∆ \n1−GπNˆM∆\nπN\ns−M2\n∆−iΓ∆M∆!−1\n, (13)\nwhere, considering the ∆-resonance dominance,\nˆM∆\nπN∝ −if2\nπN∆\nM2π(ki·kf). (14)\nFinally, Eq. (13) turns into\n˜M∆,MS\nπN =ˆM∆\nπN\ns−M2\n∆−iΓ∆M∆+iξGπNf2\nπN∆\nM2π(ki·kf). (15)\nAs understood by Eq. (15), the multiple scatterings change the decay width as well as the pole position. It is easy\nto understand from Eq. (9) that other contributions besides the resonance also get a small but finite effects from the\nmultiple-scattering mechanism. Note that , in Eq. (15), we introduce a free parameter ξ=−15.8×A[MeV] to fit\nthe data. The reasoning for ξ∝Ais that the elementary MS process will be enhanced by the number of the nucleons\ninside the nucleus. In the panel (b) of Fig. 4, we show the numerical results for the elastic cross-sections for the4He\n(solid),12C (dotted), and16O (dashed) targets, being compared with the data. It is observed that the numerical\nresult is considerably improved for the12C, whereas it slightly underestimates the4He data. We show the theoretical\nprediction for the16O target as well. As a consequence, it turns out that the width-broadening due to the MS effect\nis crucial to reproduce the heavier-nucleus data.\nIII. SUMMARY\nIn summary, our investigation focused on the elastic π+-Ascattering process at intermediate energies dominated by\nthe ∆(1232)-resonance, specifically in the I= 3/2 channel, incorporating medium corrections up to ρ2\nAparameterized\nfrom the quark-meson coupling (QMC) model within the Eikonal-Glauber model approach. We proceeded to compute\nthe total cross-sections for4He and12C targets, utilizing elementary π-Ncross-sections as input, which were calculated\nwithin the effective Lagrangian approach at the tree-level Bonn approximation.7\nFor the total cross-section of4He, our results exhibited relative consistency with the JINR data [24], particularly\naround the resonance peak. In contrast, that for12C was found to overestimate the SINR experimental data [25].\nNotably, the resonance peak for12C shifted to a lower cm energy, accompanied by a broadened decay width, compared\nto the corresponding quantities for4He. This observation aligns with findings from other calculations. It was observed\nthat the momentum- and density-dependent decay width does not considerable modifications to the theoretical results.\nFinally, we took into account the multiple-scattering effect inside the nucleus, using the single-channel Bethe-Salpeter\nequation, resulting in the with-broadening of the dominating ∆ resonance. We found that the heavier-nucleus cross-\nsection is improved much by the MS effect, describing the12C data qualitatively well, whereas the light-nucleus data\nfor the4He is reproduced as well with some deviations.\nTo properly assess the results of this study and compare them with existing theoretical calculations, it is imperative\nto acquire new data for the elastic π+-Ascattering reaction process at intermediate energies in future experiments.\nAdditionally, advancing rigorous theoretical approaches is essential for a clearer understanding of the intricate inter-\nactions within both light and heavy nuclei. 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Soc. 45, 1466 (2004)."    },    {        "title": "2402.13538v1.The_Staggered_Mesh_Method__Accurate_Exact_Exchange_towards_the_Thermodynamic_Limit_for_Solids.pdf",        "content": "The Staggered Mesh Method: Accurate Exact\nExchange towards the Thermodynamic Limit for\nSolids\nStephen Jon Quiton,†Hamlin Wu,†Xin Xing,‡Lin Lin,∗,‡and Martin\nHead-Gordon∗,†\n†College of Chemistry, University of California, Berkeley, California 94720, United States\n‡Department of Mathematics, University of California, Berkeley, California 94720, United\nStates\nE-mail: linlin@math.berkeley.edu; mhg@cchem.berkeley.edu\nAbstract\nIn periodic systems, the Hartree-Fock (HF) exchange energy exhibits the slowest\nconvergence of all HF energy components as the system size approaches the thermody-\nnamic limit. We demonstrate that the recently proposed staggered mesh method for\nFock exchange energy [Xing, Li, and Lin, Math. Comp., 2024], which is specifically\ndesigned to sidestep certain singularities in exchange energy evaluation, can expedite\nthe finite-size convergence rate for the exact exchange energy across a range of in-\nsulators and semiconductors when compared to the regular and truncated Coulomb\nmethods. This remains true even for two computationally cheaper versions of this\nnew method, which we call Non-SCF and Split-SCF staggered mesh. Additionally, a\nsequence of numerical tests on simple solids showcases the staggered mesh method’s\nability to improve convergence towards the thermodynamic limit for band gaps, bulk\nmoduli, equilibrium lattice dimensions, energies, and phonon force constants.\n1arXiv:2402.13538v1  [physics.comp-ph]  21 Feb 20241 Introduction\nWavefunction-based methods are becoming increasingly popular for use in electronic struc-\nture calculations of solid-state materials.1–5However, a longstanding, inherent problem with\nthese methods is the finite-size error; specifically, what finite size of the periodic lattice is\nrequired to reach chemical accuracy with respect to the simulation of the infinite lattice,\nthat is, the thermodynamic limit (TDL)? One approach to this problem is to simply in-\ncrease the number of unit cells along the lattice dimensions; however, doing so will increase\nthe number of electrons, atoms, and basis functions, in the unit cell resulting in greater\ncomputational expense. When performing periodic Hartree-Fock (HF) calculations, which is\nused as a reference for wavefunction methods and used in exact exchange for hybrid Density\nFunctional Theory (DFT), many solid-state quantum chemistry codes use an equivalent yet\nmore efficient approach called k-point sampling, where kis a momentum vector in a Bloch\nwavefunction. In this approach, the TDL is defined as the limit when the number of k-points\nsampled over the First Brillouin Zone (FBZ) increases towards infinity.\nThe standard approach in sampling the FBZ is to use the Monkhorst-Pack (MP) mesh,6\nwhich is constructed to evenly sample the FBZ given a number of points to sample along each\ndimension. Using this mesh, the error in the HF exchange energy as one approaches the TDL\ndecreases as O(N−1/3\nk) with an increasing number of k-points Nk. This slow convergence rate\nis primarily due to the excluded divergent term when |q+G|=0in the unscreened Coulomb\nkernel.7To reduce the error due to the Coulomb singularity, a common practice is to include\nthe Madelung constant correction.8,9Interpreting the Madelung constant correction as a\nsingularity subtraction technique allows for a rigorous proof that the finite-size error reduces\ntoO(N−1\nk).7\nTo reduce the number of k-points sampled and thus the computation time, one can\nleverage the symmetry of a unit cell for more efficient sampling. For example, sampling\nallk-points via the MP mesh in a unit cell with high space-group symmetry is redundant;\nonlyk-points in the irreducible Brillouin zone (IBZ) need to be sampled, and therefore the\n2computation time can be reduced by a factor approximately equal to the number of symmetry\noperators in the point group.10–12The use of generalized regular k-point grids,13–15or even\na simple shift of the Γ-centered MP mesh can be used to accomplish this. However, these\nmethods reduce the calculation runtime by a prefactor that is strongly dependent upon\nthe level of symmetry within the unit cell. Modifications to the Coulomb kernel, including\ntruncation16and averaging,17as well as resolution-of-identity schemes18have also been used\nto either reduce the number of k-points sampled increase computational efficiency when\ncomputing exact exchange towards the TDL.\nThe staggered mesh method is a simple alternative approach for accelerating conver-\ngence towards the TDL. Recent studies have demonstrated its effectiveness in improving the\nasymptotic convergence towards TDL for HF Exchange, Second-order Moller-Plesset Per-\nturbation Theory (MP2), and the Random Phase Approximation (RPA).7,19,20Consider the\ntwo-electron integral in reciprocal space; in the integrand of periodic HF exchange is a de-\nnominator that depends on |q|=|kj−ki|, which is the difference between the k-points of\nthe two Bloch orbitals in a pair density. We exploit the fact that if the two sets of k-points\nare staggered by a half k-mesh size, then the term with q=0is no longer included in the\nexpression for the exchange energy. Under the condition of removable discontinuity, the con-\nvergence rate of HF exchange energy to the TDL can be accelerated to O(N−5/3\nk) [Corollary\n6.7, Ref. 7]. This acceleration has been demonstrated using a model potential.7\nIn this work, we compare the performance of the staggered mesh method, along with\ntwo computationally cheaper variants, with the regular method and the Truncated Coulomb\nmethod16,21in reducing finite size errors for computing HF exchange energies, band gaps,\nseveral insulators. We find that all three versions of the staggered mesh method are compet-\nitive in computing HF exchange energies and band gaps, with the original staggered mesh\nmethod showing a distinct advantage in computing phonon force constants, lattice parame-\nters, and bulk moduli. Furthermore, this method of k-point selection is highly appealing in\nthat it requires minimal modification to most solid-state quantum chemistry packages that\n3are capable of band structure calculations.\n2 Theory\nOne way to approach the TDL in solid-state calculations is to use the supercell model, where\none explicitly repeats the unit cell in one or more directions. For this work, however, we use\nk-point sampling of the FBZ in reciprocal space, which is equivalent to the supercell model.\nWe focus primarily on the convergence of the exchange energy EKwith respect to Nk, which\ncontributes most to the finite size error in the HF theory. For notation, AOs are denoted by\nµ, ν, λ, σ , occupied MOs are denoted by i, j, k, l , and virtual MOs are denoted by a, b, c, d .\nWe use the Bloch orbital framework:\nψµk(r) =1√NkX\nR∈Leik·Rϕµ(r−R),\nwhere Ris a vector in the real space Bravais-lattice, L, and ϕµis an atomic orbital (AO).\nConsider a unit cell Ω with volume |Ω|and a corresponding supercell Ωsof volume |Ωs|=\nNk|Ω|. Furthermore, define the pair density as\nρµkµνkν(r) =ϕ∗\nµkµ(r)ϕνkν(r) :=1\n|Ω|X\nG∈L∗ˆρµkµνkν(G)eiG·r,\nwhere ˆ ρµkµνkν(G) is the Fourier transform of the pair density. Then the two-electron integral\nin the AO basis is defined as\n\u0000\nνkνλkλ|σkσµkµ\u0001\n=Z\nΩsdr1Z\nΩsdr2ψ∗\nνkν(r1)ψλkλ(r1) (G)Vcoul(r1,r2)ψ∗\nσkσ(r2)ψµkµ(r2)\n=1\n|ΩS|X′\nG∈L∗ˆρνkνλkλ(G)ˆVcoul(q+G) ˆρσkσµkµ\u0010\nGkν,kσ\nkλ,kµ−G\u0011\n. (1)\nHere, Gkν,kσ\nkλ,kµ:=kλ+kµ−kν−kσ∈L∗by crystal momentum conservation, q=kλ−kνis\nthek-point difference,P′denotes that the term where q+G=0is excluded, and Vcoul(r)\n4andˆVcoul(G) are the Coulomb kernel in real and reciprocal space respectively. We use both\nthe default periodic Coulomb kernel in 3D ˆVcoul(G) =4π\n|q+G|2and the truncated coulomb\nkernel.16,21If we define Ck\nµito be the MO-coefficient matrix,\nψik(r) =X\nµ∈naoCk\nµiψµk(r),\nthen the density matrix at that same k,Pk, is defined as\nPk\nµν=X\ni∈occCk\nµi\u0000\nCk\nνi\u0001∗. (2)\n2.1 Fock exchange energy and its Madelung constant correction\nWith a finite-size mesh K, the Fock exchange energy per unit cell is computed as\nENk\nK=1\nNk\n−1\n2X\ni,jX\nki,kj∈K(iki, jkj|jkj, iki)\n. (3)\nIn the TDL with Kconverging to Ω∗andNk→ ∞ , the energy converges as\nETDL\nK= lim\nNk→∞ENk\nK=Z\nΩ∗dkiZ\nΩ∗dkjX\ni,jFij\nK(kj,ki), (4)\nwhere the integrand is\nFij\nK(kj,ki) =−Nk\n2|Ω∗|2X\ni,j(iki, jkj|jkj, iki).\nNote that each of the ERIs above consists of an N−1\nkfactor and the overall integrand is\nindependent of Nk. From a numerical quadrature perspective, the finite-size energy ENk\nK\napproximates the TDL energy ETDL\nKby discretizing the integral of kiandkjover Ω∗×Ω∗\nby aK × K uniform mesh i.e., a trapezoidal rule. By studying the singularity structure of\n5the integrand Fj,i\nX(kj,ki), Ref. 7 provides a tight estimate of this finite-size error as\n\f\f\fETDL\nK−ENk\nK\f\f\f=O(N−1\n3\nk).\nTo further reduce this error, a common approach is to introduce the Madelung constant\ncorrection8,9,22to the Coulomb kernel involved in the ENk\nKcalculation as\nˆVcoul(G) =\n\n4π\n|G|2 G̸=0\n−|Ω|NkξG=0, (5)\nwhich leads to a Madelung-corrected exchange energy calculation as\nENk,ξ\nK=ENk\nK+noccξ. (6)\nThe Madelung constant ξabove is defined as\nξ=\n|Ω∗|\nNkX\nq∈Kq−Z\nΩ∗dq\n \n1\n(2π)3X′\nG∈L∗4πe−|q+G|2/η\n|q+G|2!\n+X′\nR∈LKqerfc\u0000\nη1/2|R|/2\u0001\n|R|−4π\n|Ω|Nkη,\n(7)\nwhere Kqis a uniform mesh in Ω∗that contains the Γ-point and has the same size as K,\nandLKqis the real space lattice associated with all points q+Gin reciprocal space with\nq∈ K q,G∈L∗. Note that ξis a constant for any arbitrary η > 0. Connecting to the\nequivalent supercell model, LKqis exactly the Bravais lattice of the supercell and q+G\ndefines the associated reciprocal lattice. This method may also be physically interpreted as\nplacing a probe electron at the origin in a uniform charge-compensating background and\ncalculating the potential energy due to interactions with the electron’s periodic images via\nEwald summation.23,24\nThe Madelung constant correction reduces the finite-size error of the exchange energy to\n6O(N−1\nk).7The effectiveness of this correction can be rigorously justified from a numerical\nquadrature perspective. It turns out that qin (7) can be interpreted as the momentum\ntransfer vector ki−kjin the exchange energy calculation (3), which also explains the defi-\nnition of Kqinξ. Plugging the definition of ξinto (6), the two terms in the bracket of (7)\neffectively replace the numerical quadrature of the leading singular term in the integrand\nFij(ki,kj) by its exact integral. As a result, the Madelung constant correction improves the\nintegrand smoothness in the finite-size exchange energy calculation (viewed as a numerical\nquadrature) and thus results in quadrature error reduction. Such a correction is connected\nto a common numerical quadrature technique for singular integrals called the singularity\nsubtraction method.\n2.2 Staggered Mesh Method for Fock exchange\nTo approximate the TDL exchange energy as an integral in (4), a general finite-size calcula-\ntion is to discretize the integrals of kiandkjseparately using two arbitrary uniform meshes\nKiandKj, i.e.,\nENk\nK(Ki,Kj) =|Ω∗|2\nN2\nkX\nki∈Ki,kj∈KjX\ni,jFij\nK(ki,kj)\n=1\nNk\n−1\n2X\nki∈Ki,kj∈KjX\ni,j\u0000\njkjiki|ikijkj\u0001\n. (8)\nIn this setting, the regular exchange energy calculation ENk\nKin (3) requires the two k-meshes\nto be the same, i.e., Ki=Kj=K. The Madelung constant correction in (6) to the regular\nexchange energy calculation can also be generalized accordingly7for the case with two general\nuniform meshes KiandKj.\nThe staggered mesh method for exchange energy calculation is based on the key obser-\nvation that the ERI\u0000\njkjiki|ikijkj\u0001\nwith zero momentum transfer q=ki−kj=0can\nlead to significant finite-size error due to the involved Coulomb singularity at q+G=0,\n7both with and without the Madelung constant correction. Part of this error in a regular\nexchange energy calculation can be traced back to the fact that the same k-mesh is used\nfor both kiandkj. To eliminate this error source, the staggered mesh method applies two\ndifferent meshes KiandKjin (8) so that there is no ERI with zero momentum transfer in\nthe exchange energy calculation. More specifically, we choose Kito be an arbitrary uniform\nmeshKand define KjasK1/2by shifting Kby a half-mesh-size in all directions. See Figure 1\nfor an illustration of such two k-meshes.\nFigure 1: Schematic of the k-point grids used in the staggered mesh method in 2D.\nDue to the two staggered k-meshes ki∈ K andkj∈ K1/2, the minimum images of\nall possible momentum transfer vectors q=ki−kjin Ω∗no longer lie in Kqin (7) that\ncontains the Γ point. For arbitrary K, those momentum transfer vectors instead form another\nuniform mesh K1/2\nqwhich comes from a half-mesh-size shift of Kqin all directions. From this\nobservation and the numerical quadrature explanation about Madelung constant correction\nfor regular exchange energy calculation, the Madelung-like correction for the staggered mesh\n8method can be generalized accordingly as (Remark 5.2 in Ref. 7)\nξ1/2= lim\nη→+∞\n|Ω∗|\nNkX\nq∈K1/2\nq−Z\nΩ∗dq\n1\n(2π)3X\nG∈L∗4πe−|q+G|2/η\n|q+G|2. (9)\nThis generalization consists of the first two terms in the bracket in (6) with Kqreplaced\nbyK1/2\nqwhile dropping the remaining two terms in (6) (these two terms decay to zero with\nη→+∞). Note that the above term in (9) with any η >0 can asymptotically reduce the\nfinite-size error and taking ηto∞is mainly to be consistent with the standard Madelung\nconstant correction. This correction ξ1/2can be numerically computed by increasing η, which\nconverges rapidly.\nFinally, the staggered mesh method with Madelung-like correction for Fock exchange\nenergy is defined as\nENk,ξ1/2\nK\u0000\nK,K1/2\u0001\n=ENk\nK\u0000\nK,K1/2\u0001\n+noccξ1/2. (10)\nThis approach to calculating the exchange energy not only reduces the errors that arise from\nthe Coulomb singularity term in (1) but also further improves the convergence to O(N−5/3\nk)\nunder certain conditions.7Specifically, O(N−5/3\nk) is guaranteed when the unit cell has cubic\nsymmetry. In practice, as will be demonstrated next in the results, to have a cubic unit cell\nis a sufficient but not necessary condition.\n93 Computational Details\n3.1 Implementation of Staggered Mesh for HF Exchange\nIn practical calculations with AOs, the generalized exchange energy in (8) can be formulated\nand computed as\nENk\nK(Ki,Kj) =−1\n2NkX\nki∈Ki,kj∈KjX\nµ,ν,λ,σPkj\nµν\u0000\nνkjλki|σkiµkj\u0001\nPki\nλσ (11)\n=−1\n2NkX\nkj∈KjX\nµ,νPkj\nµνKkj\nνµ, (12)\nwhere the exchange matrix is defined as\nKkj\nνµ=∂EK\n∂Pkj\nµν=−X\nki∈KiX\nσλ\u0000\nνkjλki|σkiµkj\u0001\nPki\nλσ. (13)\nWe introduce the staggered mesh method along with two computationally cheaper ver-\nsions, which essentially all differ in how they compute the densities Pkin equation (11).\n1.(Original) staggered mesh. This delineates the original version of the staggered mesh\nmethod. In this procedure, an SCF calculation is done on the combined set of the\nunshifted and shifted k-points. In other words, this means solving for Pkl, where\nkl∈ K0+1/2=K ∪ K1/2. If the original unshifted k-mesh has Nkk-points, then this\nSCF calculation solves for a total of 2 Nkk-points. Then, we construct Kkjvia (13),\nwhere kj∈ K j=K1/2andki∈ K i=K. Finally, we compute the corrected exchange\nenergy via (12) and (10), As the construction of Kdominates the SCF calculation\ncost with a scaling of O(N2\nk), this approach is expected to cost approximately 4x as\nmuch as a regular HF exchange calculation, making it the most expensive of the three\noptions described here. To keep the subsequent Nkconvergence plots consistent with\nthe other methods, we say that this version of staggered mesh uses Nkpoints if the\nunshifted mesh Kby itself contains Nkpoints.\n102.Split-SCF staggered mesh. In terms of computational expense, an intermediate option\nis to follow the approach above, except we perform two separate SCF calculations,\none on Kand the other on K1/2. This allows us to separately compute PkiµνandPkj\nµν.\nThis is then followed by the exchange energy computation in (12). Therefore, the\ncost is expected to be 2x that of an SCF calculation. Here, there will still be some\ndiscrepancies arising from the fact that the two densities are effectively being optimized\nin different mean-field potentials.\n3.Non-SCF staggered mesh. This is the cheapest of the three options. First, an SCF\ncalculation is done on K, soki,kj∈ K i=K, which gives us Pki. Then, using\nthis density matrix to create an effective potential, we construct Kkjvia (13), where\nki∈ K i=K, but kj∈ K j=K1/2. This exchange matrix is used to build the Fock\nmatrix Fkjwhich is then diagonalized to yield Pkj, that is, the elements of the density\nmatrix on the shifted k-mesh. This is equivalent to performing a band structure\ncalculation on the shifted k-mesh based on the mean-field potential induced by the\ndensities at the unshifted k-mesh. Finally, this new density matrix is used to calculate\nthe staggered mesh exchange energy in (12).\nThe staggered mesh method, along with the Split and Non-SCF versions, are implemented\nin a development version of Q-Chem.25Unless specified otherwise, all geometries were ob-\ntained from the Materials Project,26which were optimized using the PBE functional.27The\nexceptions are the ammonia crystal, which was obtained from the X23 dataset28and the H2\nmolecule, which is two H atoms at the center of a 6-bohr box with an interatomic distance\nof 1.4 bohr. All calculations were performed using the SZV-GTH basis set29except for the\nband gap calculations, which were performed with the uncontracted def2-QZVP-GTH basis\nset.30For computing pair densities in the two-electron integrals, we used the Gaussian and\nPlanewaves (GPW) density fitting scheme, also known as Fast-Fourier-Transform Density\nFitting (FFTDF).29,31\n114 Results\n4.1 Exchange Energies\nA total of twelve crystal systems were tested for the convergence of the exact exchange\nenergy EK. Among these systems, two are molecular (H2and ammonia), and the rest are\nsimple solids made of either one (diamond, silicon, graphite) or two elements (LiH; LiF; SiC;\nhexagonal and cubic BN; AlN; and MgO). Plotting the computed exchange energies versus\nincreasing number of k-points is shown in Fig. 2. For some systems (especially those with\nlarge unit cells and hence a large nAO), certain lines and data points are missing due to\ninsufficient compute capabilities. In particular, for AlN, the Stagger and Stagger Split-SCF\nlines stop at Nk= 43andNk= 53respectively, and for the ammonia crystal, these lines are\nnot available.\nThe trend that stands out in these results is that all staggered mesh methods display accel-\nerated convergence towards the TDL, even outperforming the Truncated Coulomb method.\nThis is a significant result given that the Truncated Coulomb kernel converges exponentially\ntowards the TDL for insulating systems due to the exponential localization of the single-\nparticle density matrix.16,32Amongst the three variants of the staggered mesh method, the\noriginal version performs the best (albeit by a small margin) for most systems as expected,\nwith the notable exceptions of SiC, Si, Cubic BN, and AlN where the Non-SCF and Split-SCF\nversions perform slightly better.\nHowever, the original version of the staggered mesh method has a larger memory foot-\nprint. For large systems such as the ammonia crystal system and AlN, we can only perform\nthe Non-SCF version of the staggered mesh method.\nAsymptotic Convergence In Fig. 3, a power-law curve is fitted to the convergence of\nenergy in diamond and silicon, which both have a rhombohedral primitive cell and exhibit\ncubic symmetry. In addition to a more favorable prefactor than the regular method, these\n12(a) H2\n (b) LiH\n (c) LiF\n(d) Diamond\n (e) SiC\n (f) Si\n(g) Cubic BN\n (h) Hexagonal BN\n (i) Graphite\n13(j) AlN\n (k) MgO\n (l) Ammonia Crystal\nFigure 2: NkConvergence Plots\ncurves demonstrate a convergence rate faster than O(N−1\nk), aligning more closely with the\nO(N−5/3\nk) trend. Conversely, we can expect that the staggered mesh method could become\nless effective (i.e. converge at the same rate as the regular method) in systems where the\nunit cell has lower symmetry.7To highlight this, we compare diamond (Fig. 2d) to graphite\n(Fig. 2i), which both have two carbon atoms in their primitive cells, but graphite has trigonal\nsymmetry instead of cubic. There, we also observe that the staggered mesh method experi-\nences approximately the same degree of oscillation as the regular and Truncated Coulomb\nmethods even when Nk= 63or higher.\nBasis Set Effects We also tested the effect of using the larger GTH-DZVP AO basis set\non the performance of the different versions of the staggered mesh method on LiH, BN,\nsilicon, and diamond. The results, shown in Fig. 4 show similar shapes to their SZV-GTH\ncounterparts, with the three staggered mesh methods giving much more similar curves to\neach other than to the regular method. This points to the use of the Non-SCF version being\nthe most cost-effective variant, while providing similar convergence rates to the original and\nSplit-SCF versions.\n14(a) Diamond\n (b) Si\nFigure 3: Power Law Fits for the Exchange Energy\n(a) LiH. Compare with Fig. 2b\n (b) BN. Compare with Fig. 2g\n15(c) Diamond. Compare with Fig. 2d\n (d) Silicon. Compare with Fig. 2f\nFigure 4: Finite Size Effects of the Exchange Energy at GTH-DZVP\n4.2 Other Physical Observables\nBand Gaps We tested the convergence of band gaps for LiH and diamond towards the\nTDL. While the former has a direct gap at the Xpoint, the latter has an indirect gap. Similar\nto the accelerated convergence demonstrated in the calculation of the exchange energy, the\nstaggered mesh method is also expected to converge faster in computing the band gap.\nThis is because the staggered mesh method also computes orbital energies without running\ninto singular terms containing |q+G|=0, similar to the proof of the O(N−1\nk) (orO(N−1\nk)\nunder the removable discontinuity condition discussed earlier) convergence rates for exchange\nenergy.7We define the exchange contribution to the orbital energy of band jatk-point\nkj∈ K jas\nεK,jkj(Ki) =−1\n2NkX\nki∈KiX\ni\u0000\njkjiki|ikijkj\u0001\n(14)\nwhere Kiis the k-mesh used to converge the SCF calculation. The band gap from the regular\nmethod is defined to be when the k-mesh for sampling the band structure is Kj=Ki. Then,\n16we define the band gap from the staggered mesh method to be when Ki=K1/2\nj, that\nis, the k-mesh used to converge the SCF calculation must be a half shift from the that\nused to sample the band structure. Therefore, to have a direct comparison between the\nregular and staggered mesh methods, the Kjmust be the same for both. To reduce basis\nset incompleteness effects, we used the uncontracted def2-QZVP-GTH basis for band gap\ncalculations.30\nFor LiH, since the direct band gap is at the Xpoint, which corresponds to the point\n[0.5,0.5,0] in the basis of reciprocal lattice vectors, we must set the bands k-mesh Kjas the\nΓ-centered MP mesh shifted whose points are all shifted by [0 .5,0.5,0], and therefore, Ki\nis the Γ-centered MP mesh shifted whose points are all shifted by [0 ,0,0.5]. Following the\ndefinition of the regular and staggered mesh methods above, we demonstrate the relative\nrates of convergence of the band gap calculation in Fig. 5a, with good fits to polynomials of\norder -1 (regular) and -5/3 (staggered) respectively. The staggered mesh method is clearly\nsuperior.\nThe case for the staggered mesh method becomes somewhat weaker for diamond with its\nindirect gap. The indirect gap means it becomes more difficult to sample both the HOMO\nand LUMO in a single MP mesh, which can lead to oscillations as Nkincreases as shown in\nFig. 5b. Here, we use the Γ-centered MP mesh to sample the band structure. Nevertheless,\nabsolute errors are still lower for the staggered mesh method at nearly all Nk.\nLattice Parameters and Bulk Moduli We next demonstrate the effectiveness of the\nstaggered mesh method in computing properties derived from varying the lattice parameter,\na, of the unit cell. In particular, computing the total energy as a function of aaround the\nequilibrium lattice parameter a0and fitting a Birch-Murnaghan equation of state (EOS)33,34\nallows one to estimate a0, the equilibrium total energy E0, the bulk modulus B0, and the\n17(a) LiH\n (b) Diamond\nFigure 5: Band Gaps\nbulk modulus derivative B′\n0= (∂B/∂P )Tas parameters. The EOS is shown below.\nEtotal(a) =E0+9V0B0\n16(\u0014\u0010a0\na\u00112\n−1\u00153\nB′\n0+\u0014\u0010a0\na\u00112\n−1\u00152\u0014\n6−4\u0010a0\na\u00112\u0015)\n(15)\nIn Fig. 6 we show the Birch-Murnaghan curves for LiH and Si using each method with\nNk= 63. The EOS as a function of Nkleads to the data shown in Figs. 7 to 9; the\nconvergence plots of a0,E0, and B0respectively. These plots paint a stronger picture of the\ndifference in accuracy among the staggered mesh methods. In all plots, the original staggered\nmesh method performs significantly better than the Split-SCF and Non-SCF variants. Even\nwhen considering that the original variant requires double the k-points as Non-SCF, a simple\ncomparison of the original staggered mesh at Nk= 43= 64 versus the Split- and Non-SCF\nvariants at Nk= 53= 125 (or the former at Nk= 53vs. the latter at Nk= 63= 216) shows\nthat the original staggered mesh is still more accurate. In fact, in the convergence of B0in\nFig. 7, the Split- and Non-SCF versions appear to do even worse than the regular method\nat low Nk.\nThus, although the Split-SCF and Non-SCF versions of staggered mesh may be more\n18efficient in computing exchange energies, recovering measures of curvature like B0is best done\nthrough the original version of staggered mesh. We expect that the superior performance of\nthe original version is due to (1) the invoking of a single SCF calculation on the combined\nmesh, which means the density matrices used in computing the exchange energy are fully\nself-consistent with each other, and (2) the other components of total energy such as the\ntwo-electron Coulomb matrix Jare also being recovered with the higher density k-mesh.\n(a) LiH\n (b) Si\nFigure 6: Birch-Murnaghan Fits\nPhase Transition Energies The ability to predict the phase transition energy between\ntwo polymorphs of a material also presents a good test for the ability of the staggered\nmesh method to accurately recover measurable properties that are energy differences. For\nthis study, we look at the transition (1) from diamond to rhombohedral graphite35–37and\n(2) from cubic to hexagonal BN.38The diamond-to-graphite transition shows a particularly\nmarked improvement for the original staggered mesh method relative to the regular method.\nThe other two staggered mesh variants, like the regular method, display severe oscillations\ntowards the TDL. The original staggered mesh method attenuates such oscillations, which\njustifies its use of double k-points compared to a normal calculation. For the cubic-to-\nhexagonal BN transition, all three staggered mesh methods perform similarly to each other,\n19(a) LiH\n (b) Si\nFigure 7: Finite Size Effects of Bulk Modulus\n(a) LiH\n (b) Si\nFigure 8: Finite Size Effects of Equilibrium Lattice Parameter\n20(a) LiH\n (b) Si\nFigure 9: Finite Size Effects of Equilibrium Total Energy\nwhile the regular method appears to overshoot the TDL. Thus, just as shown in the con-\nvergence of the Burch-Murnaghan curves, all three versions of the staggered mesh method\noutperform the regular method, with the original version being the most suitable for these\ncalculations.\nPhonon Force Constants We test the staggered mesh method’s ability to recover TDL\npotential energy surfaces (PES) around the equilibrium geometry for LiH and diamond.\nSpecifically, by perturbing the atoms in the direction of a phonon, we can perform a quadratic\nfit of the total energy as a function of mode coordinate. For both LiH and diamond, we picked\nthe highest frequency phonon at the Γ point, which was verified by a Quantum Espresso\ncalculation39and the Phonon Visualizer by Materials Cloud.40For LiH, this corresponds\nto the oscillation of the H atom along the direction of a single lattice vector. We define a\nparametrized mode coordinate λ, so the phonon corresponds to [ λ, 0.5, 0.5] in the basis of\nthe rhombohedral primitive cell lattice vectors for LiH. Similarly, for diamond (which also\nhas a rhombohedral primitive cell), we examine the mode corresponding to the oscillation of\none of the carbon atoms along the direction of a lattice vector and define the parametrized\n21(a) Diamond to Graphite\n (b) Cubic to Hexagonal BN.\nFigure 10: Finite size effects of phase transition energies. For Fig. 10b, the incomplete data\nin the Stagger and Stagger Split-SCF lines is due to insufficient compute capabilities or SCF\nconvergence issues.\ncoordinate of that atom as [ λ, 0.25 0.25]. Subsequently, λ= 0.5 and λ= 0.25 represent the\nequilibrium positions for LiH and diamond respectively.\nFor the 6 ×6×6 case, the PESs and their quadratic fits are shown in Fig. 11, while the\nconvergence of the force constant with Nkfor each method is shown in Fig. 12 with their\ncorresponding modes visualized. The original staggered mesh method shows improvement\nover its two other versions as well as the regular method, especially at low Nk. We note\nthat for diamond specifically in Fig. 12b, the force constant from the regular method may\nappear to be converging to a different TDL than those of the staggered mesh methods. This\nis because the regular method has slightly overshot the TDL value and is still increasing\neven at Nk= 83. When comparing the staggered mesh methods the Split- and Non-SCF\nversions only begin to perform comparably to the original staggered mesh method at high\nNk. Thus, despite its slightly higher cost, the original staggered mesh method shows the\nmost promise in computing properties that are derivatives of the total energy, such as phonon\nforce constants.\n22(a) LiH\n (b) Diamond\nFigure 11: Energy Surfaces Recovered by the Regular and staggered mesh methods\n(a) LiH\n (b) Diamond\nFigure 12: Force Constant Finite Size Effect\n235 Discussion\nThe staggered mesh method presents a promising way to accelerate the convergence of the\nexact exchange energy towards the thermodynamic limit while requiring minimal modifi-\ncation to the SCF and band structure calculation procedures found in many solid-state\nquantum chemistry codes. It therefore has immediate applications in recovering more ac-\ncurate energies obtained from using the increasingly popular hybrid functionals in periodic\ncalculations. We also assessed the effectiveness of two other versions of the staggered mesh\nmethod designed to further reduce its cost: the Non-SCF and Split-SCF variants. We find\nthat using staggered mesh Non-SCF recovers exchange energies and band gaps faster than\nthe regular method as a function of Nk, at the cost of only an extra SCF cycle. In computing\nother properties of interest, including the bulk modulus, equilibrium lattice parameter, and\nequilibrium energy, the original version performs the best.\nBeyond its utility in recovering more accurate energies, the staggered mesh method points\nto a new technique for reducing finite-size errors in periodic calculations by avoiding unnec-\nessary singularities. Although this work discusses only the Hartree-Fock exchange energy,\nthe energy gradients present a natural next step for testing the effectiveness of the staggered\nmesh method for a range of realistic systems. Staggered mesh techniques have also been\ndeveloped for Second Order Moller-Plesset Perturbation theory (MP2),19coupled cluster,\nand random phase approximation (RPA) calculations.20We also aim to extend the staggered\nmesh method to MP3 and its improved variants, in conjunction with acceleration techniques,\nsuch as tensor hypercontraction (THC).41While the present work primarily deals with 3D\nextended systems, studies on 2D (e.g. graphene, hexagonal BN) and 1D extended systems\n(polymers, nanotubes)42–46also present an opportunity for rigorous testing of the staggered\nmesh method in the wavefunction methods mentioned above.\n24Acknowledgement\nThis material is based upon work supported by the National Science Foundation Graduate\nResearch Fellowship Program under Grant No. DGE 2146752 (SJQ) and the U.S. Depart-\nment of Energy, Office of Science, Office of Advanced Scientific Computing Research and Of-\nfice of Basic Energy Sciences, Scientific Discovery through Advanced Computing (SciDAC)\nprogram under Award Number DE-SC0022364 (LL, MHG). 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Chem.\nA2016 ,120, 1051–1064.\n30TOC Graphic\n31"    },    {        "title": "2402.13541v1.Isotope_shift_analysis_with_the__4f__14_6s__2___1S_0__4f__13_5d6s__2__J_2___transition_in_ytterbium.pdf",        "content": "arXiv:2402.13541v1  [physics.atom-ph]  21 Feb 2024Isotope shift analysis with the 4f146s2 1S0−4f135d6s2(J= 2)transition in ytterbium\nAkio Kawasaki,1,∗Takumi Kobayashi,1Akiko Nishiyama,1Takehiko Tanabe,1and Masami Yasuda1\n1National Metrology Institute of Japan (NMIJ), National Ins titute of Advanced Industrial Science and Technology (AIST ),\n1-1-1 Umezono, Tsukuba, Ibaraki 305-8563, Japan\nMeasurements of isotope shifts have recently been attracti ng considrable attentions due to their\npotential in searching for new forces. We report on the isoto pe shifts of the 4 f146s2 1S0−\n4f135d6s2(J= 2) transition at 431 nm in Yb, based on absolute frequency me asurements with\nan accuracy of ∼10 kHz. Utilizing these data, previous reports for other tra nsitions, and theoret-\nical calculation of electronic structure, analyses of isot ope shifts on various aspects are performed,\nincluding determination of the hyperfine constants for173Yb, assessment of nuclear charge radii,\nanalysis on King plots, and a search for new bosons mediating the force between an electron and\na neutron. The analysis motivates further precision measur ements on isotope shifts of the narrow-\nlinewidth transitions in ytterbium, not only for the bosoni c isotopes but also for the fermionic\nisotopes.\nI. INTRODUCTION\nIsotopes, which differs in the number of neutrons Nin\nthe nucleus within the same element, exhibit slightly var-\nied physical properties. Such properties include macro-\nscopic quantities, e.g., phase transition temperatures,\nreaction rates of chemical reactions, and diffusion con-\nstants, as well as nuclear properties specific to nuclei.\nThe nuclear properties manifest themselves particularly\nwell in the difference in the resonant frequency νfor a\ntransition α, which is called an isotope shift. Typically\nranging from 100 MHz to 1 GHz, the isotope shift for an\noptical transition of ∼100 THz is easily observed with\nlaser spectroscopies.\nIsotope shifts are conventionally attributed primarily\nto two sources. One is the mass difference between nuclei\nchanging the reduced mass and correlation of electrons,\nwhich is called the mass shift. The other is the varia-\ntion of the nuclear charge radii distorting the Coulomb\npotential inside the nucleus, referred to as the field shift.\nPrecisely determining the amount of isotope shifts can\npotentially serve as a good probe of nuclear structure.\nIn addition to these two main factors, recent devel-\nopment in analysis of isotope shifts incorporates higher\norder terms as shown in the following equation [1].\nνAA′\nα=Fα/an}bracketle{tr2/an}bracketri}htAA′+KαµAA′+G(4)\nα/an}bracketle{tr4/an}bracketri}htAA′\n+G(2)\nα[(/an}bracketle{tr2/an}bracketri}htA′′)2]AA′+vneDαNAA′(1)\nHere, the superscripted mass number Aspecifies the iso-\ntope, and XAA′≡XA−XA′.αdenotes the transition.\nNote that in this paper transitions are labeled with their\nwavelengths (three-digit number in the unit of nm), and\nall transitions of ytterbium appearing in this paper are\nsummarized in Table I. Coefficients FαandKαcharac-\nterize the field shift and mass shift, respectively. /an}bracketle{trn/an}bracketri}ht\nis thenth moment of the nuclear charge. µA= 1/mA\n∗akio.kawasaki@aist.go.jpis the inverse mass. The third and fourth terms are the\nquadraticfieldshift, where[( /an}bracketle{tr2/an}bracketri}htA′′)2]AA′= (/an}bracketle{tr2/an}bracketri}htAA′′)2−\n(/an}bracketle{tr2/an}bracketri}htA′A′′)2, and the fourth-moment shift, respectively.\nThe fifth term shows the effect of a Yukawa potential\nVne=/planckover2pi1cvneexp(−mφcr//planckover2pi1)/ron the electrons besides\nthe Coulomb potential. vne= (−1)s+1ynye/(4π/planckover2pi1c) char-\nacterizesa hypothetical new force between a neutron and\nan electron, where sis the spin of the boson mediating\nthis force, yn,eare coupling constants to a neutron and\nan electron, and mφis the mass of the boson.\nThe field shift term can be eliminated by a set of iso-\ntope shifts for another transition β. The resulting equa-\ntion is\nνAA′\nβ\nµAA′=Fβ\nFανAA′\nα\nµAA′+Kβ,α+/an}bracketle{tr4/an}bracketri}htAA′\nµAA′G(4)\nβ,α\n+[(/an}bracketle{tr2/an}bracketri}htA′′)2]AA′\nµAA′G(2)\nβ,α+vneNAA′\nµAA′Dβ,α,(2)\nwhereXβ,α=Xβ−XαFβ/FαforX∈ {K,D,G(4),G(4)}.\nThe plot between νAA′\nβ/µAA′andνAA′\nα/µAA′is called\nKing plot and approximately linear [2], as the first two\nterms in the right hand are major. The breakdown of the\nlinearity is proposed to be used for searches for the new\nbosons [3–5], and some advanced analysis techniques are\nalso reported to reduce the effects with large uncertain-\nties [6, 7], along with assessments of which higher order\neffects are significant [8–12]. For heavy atoms, normaliz-\ning both sides with νAA′\nα/µAA′suppresses the influence\nof the uncertainty in mass [7]:\nνAA′\nβ\nνAA′\nα=Fβ\nFα+Kβ,αµAA′\nνAA′\nα+/an}bracketle{tr4/an}bracketri}htAA′\nνAA′\nαG(4)\nβ,α\n+[(/an}bracketle{tr2/an}bracketri}htxA′′)2]AA′\nνAA′\nαG(2)\nβ,α+vneNAA′\nνAA′\nαDβ,α(3)\nAmong various experimental reports on the isotope\nshift measurements aimed at searching for new bosons\n[13–16], ytterbium (Yb) is one of the most extensively\nstudied atomic species for this purpose, as well as nu-\nclear structure, with the nonlinearity of the King plot2\nTABLE I. Transitions in Yb and Yb+. The numbers listed\ninαare the wavelengths of the transitions in the unit of nm.\nThe lower-energy state for all transitions are the groundst ate,\nwhich is 6 s2 1S0for Yb and 6 s2S1/2for Yb+. The 4forbital\nis fully filled with 14 electrons unless otherwise specified. Γ\nis the linewidth of the excited state.\nYb Yb+\nαexcited state Γ αexcited state Γ\n361 5d6s1D2 24 kHz 411 5 d2D5/222 Hz\n399 6s6p1P1 29 MHz 436 5 d2D3/23 Hz\n431 4f135d6s(J= 2) 0.8 mHz 467 4 f136s2 2F7/20.5 nHz\n556 6s6p3P1 184 kHz\n578 6s6p3P0 7 mHz\n[1, 7, 17, 18]. The isotope shifts for the 411, 436, and\n467 nm transitions in Yb+are measured with /greaterorsimilar300 Hz\naccuracy. For Yb, precise isotope shift measurements for\nthe 361 and 578 nm transitions are reported. However,\nthese analyses so far only cover bosonic isotopes, because\nthe nuclear spins in fermionic isotopes potentially add\nextra nonlinearity that can become a background in the\nsearch for new bosons. In fact, except for the 578 nm\ntransition in Yb, no precise isotope shift measurements\nare available for fermionic isotopes; only broad-linewidth\ntransitions utilized for lasercooling have the isotope shift\ndata for the fermionic isotopes. Particularly, regarding\na newly observed narrow-linewidth transition at 431 nm\n[19, 20], the absolute frequency is measured for only a\nsingle fermionic isotope [19], and isotope shifts are mea-\nsured for only bosonic isotopes [20].\nIn this paper, we report absolute frequencies for the\n431 nm transition in170,172,173,174,176Yb. This report\ncompletes the list of isotope shifts of the 431 transition\nfor all stable isotopes at 10 kHz level. Based on these\nnew measurements and some previously reported isotope\nshifts for other transitions, hyperfine structure, nuclear\ncharge radii, and King plot are analyzed. The king plot\nanalysis also leads to a search for new bosons mediating\nadditional force between a neutron and an electron. The\nresults provide some insights on the nuclear structure\nfor the fermionic isotopes for ytterbium, motivating the\nprecise isotope shift measurements for other transitions.\nII. EXPERIMENTAL MEASUREMENTS\nThe experimental setup and sequences are previously\nreported elsewhere [19]. To summarize briefly, ∼105\natomsat ∼30µK are preparedin a magneto-opticaltrap\n(MOT) formed with the 556 nm transition. Atoms are\nthen interrogatedby 431nm probe light at 10mW power\ngenerated by second harmonics generation from the 862\nnm light emitted by a titanium-sapphire laser. The fre-\nquency of the 862 nm light is stabilized to a ultralow-\nexpansion cavity through a frequency comb and a 1064\nnm laser, with all relevant radiofrequency signals refer-\nencedtothe10MHz clocksignalsuppliedfromaphysical3− 2− 1− 0 1 2 3\nFrequency offset from 695 174 917 MHz (MHz)0.40.60.81.01.2Normalized atom number     (a)\n2− 1.5− 1− 0.5− 0 0.5 1 1.5 2\nFrequency offset from 695 175 031 MHz (MHz)0.850.900.951.00 Normalized atom number     (b)\nFIG. 1. Spectra of the depletion of the MOT due to the 431\nnm transition: (a) the F= 7/2 hyperfine state in173Yb with\n0.0866(13) mT bias magnetic field applied along the 431 nm\nprobe beam. (b)174Yb. Black points show the experimental\ndata, and the red curve is the fit of the datapoints. The blue\nvertical line is the average frequency obtained from the fit.\nSee the main text for the form of the fit function.\nrealizationofCoordinated UniversalTime maintainedby\nthe National Metrology Institute of Japan. The 431 nm\nprobe light is retroreflected, so that the atoms are im-\nmune to the Doppler shift.\nThe searchfor the transition is first performed for each\nisotope by chirping the frequency of the 431 nm light\nwhile it is irradiated to the atoms in the MOT. The chirp\nrate varies from 200 kHz/s to 2 MHz/s depending on\nthe expected strength of the transition. Once the initial\nsignal of the transition is observed, further precise mea-\nsurements of the resonant frequency is performed with\nthe frequency of the 431 nm light fixed within a cycle.\nFor theF= 7/2,5/2, and 3/2 hyperfine states in173Yb,\nthe cycle of activating the 431 nm probe light for 3 ms\nwhile the MOTis turned offforashortpreiod is repeated\nover 45 times. For other hyperfine states in173Yb and\nbosonic isotopes, the 431 nm light is continuously irradi-\nated onto atoms for 1 s without turning off the MOT. In\nboth cases, the number of atoms after the irradiation of\nthe 431 nm light normalized by the initial atom number\nis estimated from the amount of fluorescence from the\nMOT. Every time atoms are removed and then reloaded\nto the MOT, the frequency of the 431 nm light is shifted\nto obtain the spectrum of normalized atom number R.\nFig.1(a)and (b) showsample spectraforthe F= 7/2\nstate in173Yb and174Yb, respectively. For the F= 7/2\nstate in173Yb, distinct separation of Zeeman sublevels is\nobserved due to a bias magnetic field of 0.0866(13) mT3\napplied only when the MOT is deactivated, whereas the\nZeeman sublevels in the174Yb spectrum overlap because\nof the inability of applying arbitrary bias magnetic field\nwith the MOT active. The spectra are fitted with the\nfollowing equation.\nR=R0−F/summationdisplay\ni=−FA|i|exp/bracketleftBigg\n−(ν−(ν0+i∆ν))2\n2σ2/bracketrightBigg\n,(4)\nwhereR0is a constant and σis the width for the Gaus-\nsian. The average frequency ν0is regarded as the res-\nonant frequency for the transition, and the amount of\nsplitting ∆ νis induced by the Zeeman shift.\nThe absolute frequency of the transition is calcu-\nlated by compensating systematic shifts from ν0. Ma-\njor sources are the AC Stark shift due to the 431 nm\nprobe beam and the Doppler shift. The former is esti-\nmated for173Yb and the bosonic isotopes separately, due\nto the potential shift in the relative position between the\nMOT and the probe beam. The frequency shift per 1\nmW probe beam is estimated to be 2.40(1.20) kHz and\n0.83(1.41) kHz for173Yb and174Yb as the representative\nof the bosonic isotopes, respectively. The Doppler shift\nis assumed to be zero, because the probe beam is retrore-\nflected. However, the 3.8 kHz offset between the average\nof the recoil-free doublet and the bottom of the Doppler-\nbroadened dip is added in the uncertainty [19]. For the\nbosonic isotopes, the frequency shift of 101.8(7.8) kHz\ndue to the MOT is also taken into account, as estimated\nin Ref. [19].\nObtained absolute frequencies are summarized in Ta-\nbleII. As for168Yb, which cannot be trapped with large\nenough number in the current setup, the absolute fre-\nquency is calculated from that for174Yb and the isotope\nshift reportedinRef. [20]. Differencesofthe absolutefre-\nquencies between bosonic isotopes reported in this work\nagree well within the uncertainty with the isotope shifts\nreported in Ref. [20].\nFor theF= 7/2, 5/2, and 3 /2 states in173Yb, the\nLand´ e’sgfactor is measured by applying bias magnetic\nfield along the direction of the incident probe beam (See\nAppendix for details). The bias magnetic field is cali-\nbrated using the Zeeman splitting of the 556 nm transi-\ntion with a relative uncertainty of 1.5%. The obtained\ngfactors are gF= 0.662(46), 0 .546(24), 0 .196(24) for\nF= 7/2, 5/2, and 3/2, respectively. gJobtained by the\nweighted average of these gFis 1.583(56), which is con-\nsistent with the value in Ref. [19] within an uncertainty,\nbut is off from theoretically expected gJ= 1.5 and the\nvalue in Ref. [20].\nIII. THEORETICAL CALCULATION OF THE\nELECTRONIC STRUCTURE\nThe electronic structure of Yb is numerically cal-\nculated with configuration interaction implemented inTABLE II. Absolute frequencies for the 431 transition in Yb\nisotopes: Regarding171Yb and173Yb, the measured quan-\ntities are the transition frequencies for each Fstate. The\nnumber listed in the line without Frepresents the center of\ngravity of all Fstates.\nAF Absolute frequency [kHz] Reference\n168 695 170 466 295(17) [20]\n170 695 172 220 247(19) this work\n171 695 172 739 827(24)\n3/2 695 171 054 858.1(8.2) [19]\n5/2 695 173 863 140(30) [19]\n172 695 173 850 275(18) this work\n173 695 174 348 875(19)\n1/2 695 175 324 560(47) this work\n3/2 695 175 625 655(17) this work\n5/2 695 175 702 427(13) this work\n7/2 695 174 916 888(12) this work\n9/2 695 172 376 487(17) this work\n174 695 175 030 891(17) this work\n176 695 176 146 678(17) this work\nambit[21]. For Yb, up to n= 19 states for sandpor-\nbitals and n= 14 states for dandforbitals are included\nin the calculation. For Yb+, up ton= 16 states are\nincorporated for all s,p,d, andforbitals. Two electron\nexcitations from states relevant to this analysis and some\nother excited states and one hole excitations from the 4 f\norbital are allowed. For Yb, a hole excitation from the\n5sand 5porbitals are also allowed. Fα,Kα, andDαare\ncalculated by adjusting the initial conditions to observe\nthe resulting energy shifts of each excited state.\nIn the following discussion, the results for Yb are\nmainly used, and the quantities for Yb+are calculatedto\ntest consistency with previous reports. The calculations\nin this work are reasonably consistent with previous re-\nports [1, 6, 7, 22] and experimental values [23], except\nforKαfor Yb+ions and Kβα, as shown in Table VIin\nAppendix.\nIV. DATA ANALYSIS\nA. Hyperfine Structure\nFrom the absolute frequencies listed in Table II, hyper-\nfine constants are calculated. The A constant for171Yb\nis reported in Ref. [19]. For173Yb, hyperfine levels are\ncharacterized by the A, B, and C hyperfine constants\nAh,Bh, andChcorresponding to magnetic dipole, elec-\ntric quadrupole, and magnetic otcupole components (see\nAppendix). The best fits of the absolute frequencies with\nand without the octupole term are summarized in Table\nIII. With the octupole term, the fit quality characterized\nbyχ2/ndf improves. However, the obtained octupole\nmoment has only a significance 1 .62σ, and the shift of\nthe dipole and quadrupole constants induced by the in-\ntroduction of the octupole term is within the standard4\nTABLE III. Hyperfine constants for fermionic Yb isotopes in\nthe unit of kHz: the uncertainties include the inflation by/radicalbig\nχ2/ndf.\nwithout C173\nhwithC173\nh Ref.\n171YbA171\nh1 123 273(13) [19]\n173YbA173\nh-309 446.1(7.0) -309 449.1(5.5) this work\nB173\nh-1 700 631(78) -1 700 652(60)\nC173\nh 7.6(4.7)\nχ2/ndf 12.05 6.711\nTABLE IV. Hyperfine anomalies ∆ 171,173and electric\nquadrupole moments Qfor different states in Yb: references\nprofvide values for the A and B constants.\nState ∆ 171,173/10−3Q (10−31m2) Ref.\n6s6p1P1 13.90(56) 4664.6(2.5) [28]\n4f135d6s2(J= 2) -0.156(55) 404.634(27) this work\n6s6p3P2 -8.96(53) -133.066(56) [29]\n6s6p3P1 -3.856(70) 333.574(35) [30]\n6s7s3S1 -3.50(34) 0.047(233) [29]\n6s5d1D2 -13.0(6.7) 77.0(5.0) [31]\ndeviation. Further improvement in the precision of the\nspectroscopy is desired to determine whether the nucleus\nhas a finite octupole moment. In the following analysis,\ntobeconservative, AhandBhobtainedbythefitwithout\nthe octupole term are used.\nThe ratio of Ahbetween different isotopes equals the\nratioofthe nuclear gfactorsgA. The deviation∆ A,A′de-\nfined by the following equation is referred to as hyperfine\nanomaly [24]:\n∆173,171=A173\nh/A171\nh\ng171/g173−1 =A173\nh/A171\nh\n(µ171n/I171)/(µ173n/I173)−1,\n(5)\nwhereµA\nnandIAare the nuclear magnetic moment and\nnuclear spin, and ( µ171\nn/I171)/(µ173\nn/I173) =−3.6305(1)\n[25–27]. The hyperfine anomaly for the 431 nm and\nother transitions are summarized in Table IV. ∆173,171\nfor the 4 f135d6s2(J= 2) state corresponding to the 431\nnm transition is finite but particularly small compared\nto other states in Yb, as well as anomalies in most alkali\natoms[24]. Also, it is worthnoting that the singlet states\ntend to have large anomaly.\nIn the nonrelativistic limit, AhandBhare written in\nthe following form [24].\nAh=−1\nh16π\n3µ0\n4πµBµN2L(L+1)\nJ(J+1)gA/an}bracketle{tr−3/an}bracketri}htnl(6)\nBh=1\nhe2\n4πǫ02J−1\n2J+2/an}bracketle{tr−3/an}bracketri}htnlQ (7)\nHere,his the Planck constant, µ0andǫ0are the mag-\nnetic and electric constants, LandJare the electronic\norbital and total angular momentum, eis the elementary\ncharge,µBandµNare the Bohr and nulcear magnetons,\nand/an}bracketle{tr−3/an}bracketri}htnlis the average over the wavefunction of theelectronic state ( n,l). The nuclear quadrupole moment\nQcan then be calculated as\nQ=−ǫ0µ0\ne2µBµNgA4L(L+1)\nJ(2J−1)Bh\nAh(8)\nThe obtained Qare summarized in Table IV. The values\ndiffer a lot between different states, and none of them\nare close to the spectroscopic nuclear quadrupole mo-\nmentofQ= 2.80(4)×10−28m2obtainedbyspectroscopy\nof muonic ytterbium [32]. This value is consistent with\nQ∼2.5×10−28m2for174,176Yb obtained from nuclear\nscattering experiments [33]. This suggests that signifi-\ncant portion of the observed quadrupole moment arises\nfromrelativisticeffects, whichissubstantiallylargerthan\nthose observed in alkali atoms [24, 34].\nB. Nuclear charge radius\nWith theoretically calculated FαandKα,/an}bracketle{tr2/an}bracketri}htAA′is\nestimated using Eq. ( 1). The relative uncertainty of\n/an}bracketle{tr2/an}bracketri}htAA′caused by the uncertainty of the spectroscopy is\n∼10���5(see Table VIIin Appendix), and the uncertainty\nfor the obtained /an}bracketle{tr2/an}bracketri}htAA′mainly originates from that of\nthe theoretical calculation of FαandKα, whose amount\nis difficult to estimate. To determine the uncertainty for\n/an}bracketle{tr2/an}bracketri}htAA′,/an}bracketle{tr2/an}bracketri}htAA′\nαis calculated with isotope shift data for\nα= 399 [28], 431, 556 [30], and 578 [18]. The average of\nthem isregardedasthe mostlikely /an}bracketle{tr2/an}bracketri}htAA′, andstandard\ndeviation of them is regarded as the uncertainty.\nObtained /an}bracketle{tr2/an}bracketri}htAA′is plotted in Fig. 2. The values de-\nrived from the 431 nm transition result in this work are\nplotted together with the average values. These values\ntend to be smaller than the numbers in some previous\nreports [32, 35, 36], and closer to those in Ref. [22, 26].\nThe charge radii for Yb isotopes are also measured by\nscattering experiments with electrons [37–39] and pro-\ntons [33]. Ref. [33] explicitly shows /an}bracketle{tr2/an}bracketri}htfor174Yb and\n176Yb estimated by proton scattering. Refs. [37, 38] pro-\nvide experimental values of /an}bracketle{tr2/an}bracketri}htAfor174Yb and176Yb\nfrom electron scattering, as well as values obtained by\ntheoretical fits of their experimental values by DME and\nSKYRME V models. /an}bracketle{tr2/an}bracketri}htAA′obtained with these meth-\nods are 0.08, -0.0532, 0.0426, and 0.0956 fm2, respec-\ntively, all of which supports the relatively small /an}bracketle{tr2/an}bracketri}htAA′\nobtained in this work.\nFrom these data, three point odd-even staggering\n∆/an}bracketle{tr2/an}bracketri}ht(3)is calculated as\n∆/an}bracketle{tr2/an}bracketri}ht(3)=1\n2(/an}bracketle{tr2/an}bracketri}htA+1−2/an}bracketle{tr2/an}bracketri}htA+/an}bracketle{tr2/an}bracketri}htA−1).(9)\nThis is plotted in the inset of Fig. 2. Compared to the\nodd-evenstaggeringobservedin otheratoms[40–43], this\neffect is relatively large for stable isotopes. This could be\nattributed to the fact that both the number of protons\nand neutrons are far from the magic numbers (50 and\n82).5\n168 169 170 171 172 173 174\nA0.5−0.4−0.3−0.2−0.1−)2 (fmAA’〉2r〈Average\n431 nm\n171 172 173\nA0.02−0.01−0.000.010.02)2 (fm(3)〉2r〈 ∆\nFIG. 2. /angbracketleftr2/angbracketrightAA′calculated from isotope shifts with A′=\n176: red circles show the values obtained from the absolute\nfrequencies of the 431 nm transition, and black triangles sh ow\nthe average over the 399, 431, 556, and 578 nm transitions.\nThe inset shows ∆ /angbracketleftr2/angbracketright(3).\n50 55 60 65 70\n)1u1 Hz15 (10578ν/AA’µ10−5−05)4residual (1010× bosonic isotopes\nfermionic isotopes1.25−1.20−578ν/431 ν\n168170170172172174174176\n171174173174\nFIG. 3. Normalized King plot for the 431 nm transition: the\nplot is normalized with the 578 nm transition. Texts near\neach data point show the pair of isotopes that generated the\ndata point. The blue line shows the linear fit of the black\ndata points. See Table VI in Appendix for the fit parameters.\nThe bottom half displays the residual of each data point from\nthe fitted line. The residuals for the bosonic isotopes (blac k\npoints) are magnified by a factor of 10.\nC. King Plot\nThe normalized King plot between the 431 nm transi-\ntion and the 578 nm transition following Eq. 3is shown\nin Fig. 3. In addition to the data points for bosonic\nisotopes, data points for fermionic isotopes are plotted.\nThe representative transition frequency for a fermionic\nisotope is calculated as the center of gravity frequency of\nthe hyperfine levels, and the isotope shift is calculated as\nthe shift from the transition frequency for174Yb, so that10− 5− 0\n)5 (10+ζ5−051015)5 (10ζ467\n361\n411436431\nFIG. 4. Observed nonlinearity in the normalized King plot\ndisplayed on the ( ζ+,ζ−) basis. The blue line is the linear\nfit for the black points with a line on the origin. Best fit is\nζ−=−1.609(14)ζ+, withχ2/ndf = 5.826. The green dashed\nline is the nonlinearity pattern generated by the new bosons .\nTABLE V. Residual δνAfrom the linear fit of the King plot\nfor fermionic isotopes: the residual on the plot is multipli ed\nbyνAA′\n578and displayed in the unit of kHz.\nα δν171(kHz)δν173(kHz)\n399 380(218) 3302(266)\n431 -754(29) -792(25)\n556 1233(79) 455(76)\nthe data points lie between those of bosonic isotopes and\nhave smaller uncertainties. First, to assess the level of\nnonlinearity of the King plot, linear fit for the bosonic\nisotopes is performed by Eq. 3with only the first two\nterms on the right hand side. The plot exhibits slight\nnonlinearitybeyondthestandarddeviationforsomedata\npoints, as characterized by χ2/ndf = 1.264. However, as\ntheprobabilityforthislevelofdeviationfrom χ2/ndf = 1\nis 28.24%, the data points are linear within 95% confi-\ndence level.\nAlthough the amount of the nonlinearity is consis-\ntent with zero, the residual dAfor isotope Acan be\ndecomposed onto the two dimensional space made by\nζ±=d168−d170±(d172−d174), as introduced in Ref. [7].\nThenonlinearityofthe431nmtransitionshowninFig. 4\nis consistent with the one-parameter fit for the 361, 411,\n436 and 467 nm transitions within one standard devia-\ntion. Thisfinding supportstheargumentmadein Ref.[1]\nthatthe majorpartofthe nonlinearitiesin the Kingplots\nfor Yb originates from several nuclear effects. This con-\nsistency observed between the previous results, as well\nas the almost linear behavior on the King plot, ensures\nthat the measurements are not significantly affected by\nany systematic shifts beyond the obtained uncertainties.\ndAforthe fermionicisotopesissignificantlylargerthan\nthe uncertainties on it, as shown in the bottom half of\nFig.3. To investigate the origin of this offset, the nor-\nmalizedKingplotsaremadeforthe399and556nmtran-6\nsitions as well, and δνA=dAνAA′\n578is extracted, as shown\nin Table V. On one hand, it is at most ∼1 MHz and thus\nnot significant in older experiments with large uncertain-\nties[44–46]. Ontheotherhand, with anaccuracyof ∼10\nkHz, this is significant. δν171,δν173, andδν173/δν171are\nplotted against Qin Table IV, ∆171,173/10−3,A171\nh,B173\nh\n,B173\nh/A173\nh, andFα(See Figs. 8,9, and8in Appendix).\nQuantities with largest correlationare B173forδν173and\nFαforδν171, respectively. δν173/δν173does not look to\nhave any good correlationagainst other quantities. How-\never, it is not evident enough to state these quantities are\nrelated,becausethe χ2/ndf forthelinearfitare7.155and\n5.492, respectively, and both fitted lines do not cross the\norigin on the two-dimensional plot. One reason why the\nanalysis is inconclusive is because δν171for the 399 nm\ntransition has a large relative uncertainty. More precise\nspectroscopy, not only limited to the 399, 431, and 556\nnm transitions but also for other narrow-linewidth tran-\nsitions, is desired to figure out the source of the large dA\nfor fermionic isotopes.\nD. Search for new bosons\nTogether with theoretically calculated Dαand the lin-\nearity in the King plot for the 431 nm transition, a con-\nstraintontheexistenceofnewbosonsmediatingtheforce\nbetween an electron and a neutron is set. It should be\nnoted that with the relative uncertainty of ∼1×10−5for\nthe isotope shifts in the 431 nm transition, simple analy-\nsis with a two-dimensional King plot is good enough, as\nother sources of uncertainties such as the ones on µAA′\nis orders of magnitude smaller. To be most conservative,\ntheeffectofthenucleardeformationalongthebluelinein\nFig.4is first subtracted from the data point for the 431\nnm transition, so that all nonlinearity would arise from\nthe new bosons. As shown in Fig. 4, the effect of the new\nboson on the ζ+−ζ−plane has a slope of -0.2585. On\nthis line, −6.685×105< vneD431,578<2.182×106is the\nallowed region for the new bosons with 95% confidence\nlevel, which includes vne= 0. Thus, such bosons are\nexcluded in the region beyond this, as plotted in Fig. 5.\nThe constraint is more than an order of magnitude worse\ncompared to the most stringent constraint reported in\nRef. [1].\nWith an improvement in the accuracy of the isotope\nshift measurement for the 431 nm transition, it is ex-\npected to increase the sensitivity to the region that has\nnever been investigated before. Fig. 5shows the sensi-\ntivity for the improved accuracy by three orders of mag-\nnitude (down to ∼18 Hz), with an assumption that the\nuncertaintyin the isotope shifts for the 431nm transition\nis the primary source of the uncertainty. At this point,\nanalysis with three transitions described in Ref. [6] is\nnecessary.\nch/nyeys+1(1)\n210310410510610\n (eV)φm13−10\n12−10\n11−10\n10−10\n9−10\n8−10\n7−10−\n−\n−\n−\n−\n−\n−13−1012−1011−1010−109−108−107−10\nFIG. 5. Constraint on the existence of new bosons mediat-\ning the force between an electron and a neutron: the black\nregion is excluded by electron g−2 measurements and neu-\ntron scattering experiments [3]. The red region is the most\nstringent exclusion reported so far in Ref. [1], obtained fr om\nthe analysis with the 436, 467, and 578 nm transitions. The\nblue region is the excluded region by the 431 and 578 nm\ntransitions obtained in this work. The blue dashed line is th e\nexpected sensitivity when the accuracy in isotope shifts fo r\nthe 431 nm is improved by a factor of 1000.\nV. CONCLUSION\nThe full list of absolute frequencies for the 431 nm\ntransition in170,172,173,174,176Yb is reported, and various\nanalysesbased on the data are performed. The King plot\nanalysis did not show as much nonlinearity as reported\nfor other transitions. On one hand, this can ensure the\nabsence of any significant systematic shifts in the mea-\nsurement, but on the other hand, further improvement\nin accuracy is desired to investigate the effects induc-\ning the nonlinearity, including the new bosons mediating\nthe force between an electron and a neutron. The hy-\nperfine constants are precisely determined. However, the\nobserved octupole moment is not statistically significant,\nand this also motivates more precise spectroscopy. The\ndifference in root-mean-square nuclear charge radii be-\ntween isotopes is determined by averaging isotope shifts\nof multiple transitions, supporting smaller values com-\npared to some of the previous reports.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI\nJP21K20359, JP22H01161, JP22K04942, JST FOREST\nJPMJFR212S and JST-MIRAI JPMJM118A1. We are\ngrateful to D. Akamatsu, K. Hosaka, H. Inaba, and S.\nOkubo for the development of the frequency comb and\nthe stable laser at 1064 nm. A. K. acknowledges the\npartial support of a William M. and Jane D. Fairbank7\nTABLE VI. Result of the theoretical calculation of the elec-\ntronic structure in ytterbium: the units of να,Fα,Kα,Dα,\nKβ,α, andDβ,αare THz, GHz/fm2, GHz·u, 103THz, GHz ·u,\nand 103THz, respectively. Dαvalues are at mφ= 1 eV. Ref\n[1] values are ones calculated by ambit. Experimental values\nforναare cited from Ref. [23], and the exact value differs by a\nfew GHz according to isotope shifts. Experimental values fo r\nFβ/FαandKβαare obtained by the linear fit of King plots\nfor the bosonic isotopes (see Section IVC).\nthis work Ref. [1] Ref [22] Experiment\nν361 875.19 819.47 829.76\nν399 744.82 751.53\nν411 691.88 707.00 729.48\nν431 810.67 695.17\nν436 649.86 679.86 688.35\nν467 727.06 1051.44 642.12\nν556 524.06 543.18 539.39\nν578 500.53 522.78 522.68 518.29\nF361 -11.988 -13.528\nF399 -7.020\nF411 -15.534 -14.715\nF431 17.771\nF436 -15.770 -14.968\nF467 35.260 36.218\nF556 -9.893 -10.951(21)\nF578 -9.692 -9.719 -10.848(21)\nK361 -926\nK399 412\nK411 -308 -752\nK431 10172\nK436 -158 -661\nK467 10167 12001\nK556 -532 -280(72)\nK578 -527 -288(75)\nD361 5.6683\nD399\nD411 43.37 43.158\nD431 -94.34\nD436 49.74 48.634\nD467 -295.06 -352.38\nD556\nD578 -39.49 -42.855\nF399/F5780.7243 0.4453(12)\nF431/F578-1.8336 -1.62301(8)\nF556/F5781.0207 1.00594(26)\nK399,578 793.7 1332(17)\nK431,578 9205.7 6430.9(1.4)\nK556,578 5.929 33.2(4.0)\nD431,578 -266.7\nPostdoctoral Fellowship of Stanford University for the\nearly stage of this project.\nAppendix A: Summary of the result of theoretical\ncalculation\nThe result of the theoretical calculation described in\nSectionIIIis summarized in Table VI. Also, Ref. [6]\ncalculates K578=−655 GHz ·u.TABLE VII. Summary of /angbracketleftr2/angbracketrightAA′. The second column is the\nvalued obtained from the isotope shift of the 431 nm transi-\ntion. The third column shows the average over the 399, 431,\n556, and 578 transition data. The uncertainties for the sec-\nond column are originated from the spectroscopy only, and\nthe uncertainties in the third column are the standard devia -\ntion over the values derived from four transitions.\nA /angbracketleftr2/angbracketrightAA′(fm2)\n431 nm average\n168 -0.474 791 6(21) -0.442(78)\n170 -0.335 961 7(22) -0.312(54)\n171 -0.286 988 3(25) -0.267(47)\n172 -0.205 025 1(23) -0.189(33)\n173 -0.157 682 6(23) -0.146(25)\n174 -0.100 263 6(22) -0.093(16)\n176 0 0\nAppendix B: Table for /angbracketleftr2/angbracketrightAA′\nValues for /an}bracketle{tr2/an}bracketri}htAA′that were used to generate Fig. 2\nare summarized in Table VII.\nAppendix C: Details of the gfactor determination\nTo determine the gfactors for the F= 7/2 and 5/2\nstates in173Yb, the bias magnetic field B/bardblalong the di-\nrection of the propagation of the 431 nm probe beam is\nvaried to obtain the amount of Zeeman splitting. These\nare shown in Fig. 6. The plot is fitted with the fit func-\ntion ∆ν=gFµB/radicalBig\n(B/bardbl−B0)2+B2\n⊥to accomodate the\ntransverse magnetic field B⊥, wheregF,B0, andB⊥are\nfitting parameters. gFis converted to gJwith the rela-\ntion\ngJ≃2F(F+1)\nF(F+1)−I(I+1)+J(J+1)gF.(C1)\nWith the uncertainty obtained by the fit, the contribu-\ntion of the nuclear spin to gFis negligible.\nAppendix D: Details of the absolute frequency\ndetermination\nSeveral spectra are taken for the absolute frequency\nmeasurement for each isotope and hyperfine state to con-\nfirm the consistency of the data. Weighted average of\nthese data are regarded as the absolute frequencies, as\nshown in Fig. 7. The statistical uncertainty is inflated\nby/radicalbig\nχ2/ndf if/radicalbig\nχ2/ndf>1, and then merged with the\nsystematic uncertainties.8\n0.3 0.4 0.5 0.6 0.7 0.8\n (mT)B0.40.50.60.70.8 (MHz)ν∆ / ndf = 99.33 / 22χ\n 0.006±  = 0.663 Fg\n 0.009±  = 0.064 0B\n 0.002±  = 0.380 B\n(a) The F= 7/2 state0.4 0.5 0.6 0.7 0.8 0.9 1\n (mT)B0.40.50.60.70.8 (MHz)ν∆ / ndf = 0.64 / 22χ\n 0.024±  = 0.546 Fg\n 0.050±  = 0.073 0B\n 0.016±  = 0.414 B\n(b) The F= 5/2 state0.6 0.7 0.8 0.9 1 1.1\n (mT)B0.20.30.4 (MHz)ν∆ / ndf = 3.73 / 32χ\n 0.021±  = 0.196 \nFg\n 0.134±  = 0.262 0B\n 1.110±  = 0.001 B\n(c) The F= 3/2 state\nFIG. 6. The amount of Zeeman splitting ∆ νfor applied bias magnetic field B/bardblfor hyperfine states in173Yb. The uncertainties\non the final values for gFin the main text are inflated by/radicalbig\n��2/ndf. if/radicalbig\nχ2/ndf>1.\n1 2 3 4 5\nData Number15−10−5−051015695 176 146 678 kHz (kHz)Frequency offset from / ndf = 5.005 / 42χ\n 2.451±  = 0.025 0ν\n(a)176Yb1 2 3 4\nData Number5−051015695 175 030 891 kHz (kHz)Frequency offset from / ndf = 0.527 / 32χ\n 3.152±  = 0.184 0ν\n(b)174Yb1 2 3 4 5 6 7 8 9\nData Number10−5−0510695 173 850 275 kHz (kHz)Frequency offset from / ndf = 2.972 / 82χ\n 1.476±  = 0.114 0ν\n(c)172Yb\n1 2 3 4\nData Number20−10−0102030695 172 220 247 kHz (kHz)Frequency offset from / ndf = 2.313 / 32χ\n 8.166±  = 0.312 0ν\n(d)170Yb1 2 3 4\nData Number40−30−20−10−01020695 172 376 487 kHz (kHz)Frequency offset from / ndf = 1.230 / 32χ\n 6.383±  = 0.489 0ν\n(e) theF= 9/2 state in173Yb1 2 3 4 5 6 7 8 9 10 1112\nData Number20−10−010695 174 916 888 kHz (kHz)Frequency offset from / ndf = 13.031 / 112χ\n 1.255±  = 0.010 0ν\n(f) theF= 7/2 state in173Yb\n1 2 3 4 5 6\nData Number10−01020695 175 702 427 kHz (kHz)Frequency offset from / ndf = 19.342 / 52χ\n 2.341±  = 0.245 0ν\n(g) theF= 5/2 state in173Yb1 2 3 4 5 6 7 8 9 10\nData Number20−0204060695 175 625 655 kHz (kHz)Frequency offset from / ndf = 30.274 / 92χ\n 3.647±  = 0.492 0ν\n(h) theF= 3/2 state in173Yb1 2 3 4 5 6 7\nData Number200−100−0100200695 175 324 560 kHz (kHz)Frequency offset from / ndf = 57.195 / 62χ\n 14.250±  = 0.179 0ν\n(i) theF= 1/2 state in173Yb\nFIG. 7. Plots for determining absolute frequencies: ν0in the box shows the weighted average. The uncertainty on ν0is without\nthe inflation by/radicalbig\nχ2/ndf, whereas the green band is with the inflation if/radicalbig\nχ2/ndf>1.9\n0 1\n (MHz)171δ12345)2 m28Q (10\n0 1\n (MHz)171δ5−0510153/10171,173∆\n0 1\n (MHz)171δ024 (GHz)171hA\n0 1\n (MHz)171δ1−0 (GHz)173hB\n0 1\n (MHz)171δ510173h/A173hB\n0 1\n (MHz)171δ10−01020 )2 (GHz/fmαF\nFIG. 8. Plots of the quadrupole moment Qin Table IV, hyperfine anomaly ∆ 171,173/10−3, hyperfine constants A171\nh,B173\nh,\nB173\nh/A173\nh, and field shift Fαagainstδν171: three points correspond to the 399, 431, and 556 nm data. The red line is the\nlinear fit.\n0 2 4\n (MHz)173δ12345)2 m28Q (10\n0 2 4\n (MHz)173δ5−0510153/10171,173∆\n0 2 4\n (MHz)173δ024 (GHz)171hA\n0 2 4\n (MHz)173δ1−0 (GHz)173hB\n0 2 4\n (MHz)173δ510173h/A173hB\n0 2 4\n (MHz)173δ10−01020 )2 (GHz/fmαF\nFIG. 9. Plots of the quadrupole moment Qin Table IV, hyperfine anomaly ∆ 171,173/10−3, hyperfine constants A171\nh,B173\nh,\nB173\nh/A173\nh, and field shift Fαagainstδν171: three points correspond to the 399, 431, and 556 nm data. The red line is the\nlinear fit.10\n0 10 20 30 40\n (MHz)171δ/173δ12345)2 m28Q (10\n0 10 20 30 40\n (MHz)171δ/173δ5−0510153/10171,173∆\n0 10 20 30 40\n (MHz)171δ/173δ024 (GHz)171hA\n0 10 20 30 40\n (MHz)171δ/173δ1−0 (GHz)173hB\n0 10 20 30 40\n (MHz)171δ/173δ510173h/A173hB\n0 10 20 30 40\n (MHz)171δ/173δ10−01020 )2 (GHz/fmαF\nFIG. 10. Plots of the quadrupole moment Qin Table IV, hyperfine anomaly ∆ 171,173/10−3, hyperfine constants A171\nh,B173\nh,\nB173\nh/A173\nh, and field shift Fαagainstδν173/δν171: three points correspond to the 399, 431, and 556 nm data. 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B 11, 2163 (1994) ."    },    {        "title": "2402.13558v1.Mutual_information_scrambling_in_Ising_spin_chain.pdf",        "content": "Mutual information scrambling in Ising spin chain\nSurbhi Khetrapala,b∗and Emil Tore Mærsk Pedersenb,c\naSchool of Physics, University of Hyderabad, Gachibowli 500046 Telangana,\nbVrije Universiteit Brussel, Brussels 1050 Belgium,\ncDepartment of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark\nWe consider a chain of spin-1 /2 particles of a finite length Levolved with the mixed-field Ising\nHamiltonian and impose open boundary condition. We simulate the time evolution of entanglement\nentropy and mutual information following quench from the N´ eel state in this system using tensor\nnetworks. We find that the entanglement entropy for non-integrable systems saturates to a constant\nvalue at late times, however it continues to oscillate for integrable systems. We also find that\nmutual information peaks as a function of distance between intervals decay faster for non-integrable\nsystems compared to integrable systems, in agreement with the conclusion of [1] for XXZ chains.\nWe compare the oscillations in entanglement entropy evolution obtained from simluations in the\nintegrable case with analytic results from quasi-particle picture and find agreement.\nI. INTRODUCTION\nEntanglement has become an increasingly important\nway of characterizing quantum many-body systems. An\nimportant question is how entanglement can be created\nand spreads starting from a system in a non-entangled\nproduct-state. Recent results have shown that the time-\nevolution of entanglement following quench can be an\ninteresting tool to distinguish integrable and chaotic sys-\ntems [1–3]. One way to quantify entanglement is via the\nentanglement entropy and mutual information. The Ising\nmodel is a simple model of a one-dimensional spin-1/2\nchain with only nearest-neighbor interactions, which can\nshow both integrable and chaotic behaviour depending\non values of the free parameters in the Hamiltonian.\nInvestigating the scrambling of quantum information\n(also known as chaos) in many-body quantum systems\n[4, 5], provides valuable insights into the post-quench evo-\nlution of a system. While various methods are available\nto understand the chaotic dynamics of these quantum\nsystems, such as spectral statistics [6] and out-of-time\nordered correlators [7, 8], recently scrambling of mutual\ninformation following quench has also emerged as a di-\nagnostic of chaotic dynamics [1]. In this work we study\nthe scrambling of mutual information following quench\nfor a finite size transverse field Ising spin chain with\nopen boundary condition. The latter boundary condi-\ntion implies that the quasi-particles are reflected from\nthe spin chain boundary and this leads to interesting pat-\nterns such as oscillatory behaviour of the entanglement\nentropy and mutual information at late times in the inte-\ngrable case. We find that mutual information as a func-\ntion of distance between entanglement intervals decays\nfaster in non-integrable systems compared to integrable\nones. These results are in agreement with those obtained\nin [1] for XXZ spin chain.\n∗surbhikhetrapal@uohyd.ac.inA. Setup\nIn this work, we consider a chain of spin-1 /2 particles,\nof length Levolved with the mixed-field Ising Hamilto-\nnian,\nH=LX\ni=1gσ[i]\nx+LX\ni=1hσ[i]\nz+L−1X\ni=1Jσ[i]\nzσ[i+1]\nz, (1)\nwhere σx,σyandσzare the Pauli matrices, Jis the\nstrength of the interaction between neighboring spins, h\nis the strength of the longitudinal magnetic field and gis\nthe strength of the transverse magnetic field. Following\n[2], we consider the following, slightly modified, Hamil-\ntonian\nH=LX\ni=1gσ[i]\nx+L−1X\ni=1hσ[i]\nz+ (h−J)(σ[1]\nz+σ[L]\nz)\n+L−1X\ni=1Jσ[i]\nzσ[i+1]\nz, (2)\nwhere the longitudinal fields at the ends of the chain are\nreduced by Jto keep the ends more similar to rest of the\nchain and avoid edge-effects. Depending on the values\nofhandgthe Ising model can be integrable or non-\nintegrable. With either of the fields off ( h= 0 or g= 0),\nthe model is integrable and analytic results for the entan-\nglement entropy evolution can be obtained. Examples of\nnon-integrable values we use are\n(h, g, J ) = √\n5 + 1\n4,√\n5 + 5\n8,1!\n, (3)\nwhich is used in [2], and\n(h, g, J ) = (0 .5,−1.05,1), (4)\nwhich is highly chaotic [9].\nA common way to bring a quantum system out of equi-\nlibrium is a quantum quench: when a quantum system isarXiv:2402.13558v1  [cond-mat.stat-mech]  21 Feb 20242\nprepared in an eigenstate of a Hamiltonian H0and then\nevolved in time under a different Hamiltonian H1, this\nis called a quantum quench. It is of significant interest\nin quantum information research to study the creation\nand spreading of entanglement following a quench from a\nnon-entangled product state. In this paper we quench the\nsystem from an initial state where every spin is pointing\nin the opposite direction (up or down) from its neighbors,\ncalled the N´ eel state,\n|ΨN⟩=|0⟩ ⊗ |1⟩ ⊗ |0⟩ ⊗ |1⟩ ⊗. . . . (5)\nThe N´ eel state is an eigenstate of the Ising Hamiltonian\nwith the transverse field g= 0 turned off. Hence, evolv-\ning this initial state with the mixed field Ising Hamil-\ntonian with non-zero transverse and longitudinal fields\n(g̸= 0, h̸= 0) is an example of a quench.\nB. Entanglement entropy and mutual information\nFor a pure state |Ψ⟩ ∈HA⊗HBthe density matrix is\nsimply ρ=|Ψ⟩⟨Ψ|. The bipartite von-Neumann entan-\nglement entropy of subsystem Ais defined as,\nS(A) =−Tr (ρAlogρA) (6)\nwhere ρA= Tr B(ρ) is the reduced density matrix of sub-\nsystem A. Performing the trace in (6) in the orthonormal\nbasis of HAwhere ρAis diagonal, with eigenvalues λk,\ngives\nS(A) =−X\nkλklogλk. (7)\nIt can be proven that ρBhas the same eigenvalues as\nρAsoS(A) =S(B). The entanglement entropy S(A) =\nS(B) is a measure of the amount of entanglement be-\ntween subsystems AandB.\nA useful quantity that can be defined in terms of the\nentanglement entropy is the mutual information. The\nmutual information of two subsystems AandBis defined\nas\nIA:B=SA+SB−SA∪B. (8)\nThe mutual information is a measure of the amount of\ninformation subsystems AandBhave in common (due\nto their entanglement).\nC. Quasi-particle picture\nQuasi-particles are excitations in many-body systems,\nthat can propagate through the system behaving like\nparticles. Integrable quantum many-body systems ad-\nmit stable quasi-particles with infinite lifetime, while\nnon-integrable quantum many-body systems only admit\nquasi-particles with finite lifetime. A quantum quench\nFIG. 1: Quasiparticle picture of mutual information\nI(A, B) time evolution\ncan act as a source of quasi-particles. When the ini-\ntial state is a product state, only pairs of quasi-particles\ncreated at the same point are entangled. When these en-\ntangled particles move in opposite directions through the\nsystem they will then spread the entanglement. Consid-\nering a subsystem A, a pair of entangled quasi-particles\nwill then contribute to the entanglement entropy of A\nonly when one of the particles is in Aand the other one\noutside.\nIn case of the mutual information I(A, B), a pair of\nquasi-particles will therefore contribute to the mutual\ninformation when one of the particles is in Aand the\nother in B. Figure 1 shows a space-time diagram of a\nsimple one-dimensional system where entangled pairs of\nquasi-particles (diagonal black world lines) moving in op-\nposite directions, all with the same speed, are produced\nthroughout the system from a quench at time t= 0. The\nthree pairs shown in the figure, all reach a region in time\n(green part of their trajectory) where one of them is in\nAand the other in Band they therefore contribute to\nI(A, B). This will result in a peak in I(A, B) (light-green\nline in the figure, assuming pairs were created uniformly\nthroughout the system) when a maximal amount of pairs\ncontribute. In the more realistic case of a nontrivial dis-\ntribution of quasi-particle species with different veloci-\nties, the mutual information will be a superposition of\npeaks from the different species.\nD. Implementation using Tensor Networks\nTensor network diagrams reviewed in [10–13] are a very\nconvenient and elegant way to compute entanglement in\nquantum spin chains [14–18]. A program was written\nin Python for working with canonical Matrix Product\nStates (MPS), implementing Time Evolving Block Dec-\nimation (TEBD) and computing the entanglement en-\ntropy for different subsystems. A few important consid-\nerations while implementing the algortihms described in\nthese review articles are discussed below:\n1. A maximal bond-dimension D or cutoff value λc=\n10−16was set for Schmidt coefficients λksuch that\nall Schmidt vectors and coefficients with k≥D3\norλk< λc, respectively were discarded efficiently\nwith the NumPy arrays.\n2. Most of the results presented in this work are com-\nputed without limit on the bond dimension D. For\na spin-1\n2chain, where each local site has a local\nstate space of dimension d= 2, with even length\nL, the worst-case bond-dimensions required for an\nexact representation are\nDr=(\n2rwhen r≤L\n2\n2L−rwhen r >L\n2. (9)\nThus the maximum of these bond dimensions is\nD= 2L\n2. For short chains this is not a problem\non a fast processor but for longer chains, such as\nL= 20 or more, computational resources become a\nbottleneck, and efficient truncation is an important\nadvantage of MPS. However, since our program was\nable to simulate these large lengths without trun-\ncation, we continued using exact bond-dimensions\nfor the rest of the results.\n3. Computing the entanglement entropy for a bipar-\ntition of the form ([1 : r],[r+ 1 : N]) is trivial,\nhowever for the more complicated case of com-\nputing the entanglement entropy of two spatially\nseparated intervals SA∪B(reduced density matrix\nρA∪B) with A= [1 : a] and B= [b:N], it is es-\nsential to use an efficient method to contract the\ncentral tensor, which can easily use far too much\ntime or memory. We worked out and encoded the\nmost efficient contraction order.\n4. In order to implement TEBD, it is necessary to\nbring the Ising Hamiltonian into a 2 −site nearest-\nneighbor form. Since the Hamiltonian is time-\nindependent, these h[r:r+1]and their exponentials\nU[r:r+1]=e−i\nℏδth[r:r+1]can be computed from the\nparameters and the time-step δtat the beginning\nof the computation and can then be reused in all\nTEBD steps ( ℏis set to 1). Following this, all time-\nevolution gates are applied in parallel at each even\nor odd step of TEBD. It is crucial to perform the\ntensor contractions efficiently and in the right order\nto minimize memory and time-usage.\nII. SIMULATION RESULTS\nA. Evolution of Entanglement entropy after\nquench from N´ eel state\nWe simulated chains of all even lengths from L= 4\nto 20 with the time step, δt= 0.01 for several values\nofhandgto compare the results. The results for time\nevolution of the entanglement entropy of half-chain S[1 :\nL/2] following quench from N´ eel state and evolution by\nthe Ising Hamiltonian (2) are obtained.1. Integrable case\nAs an example of an integrable case we considered the\ncase with the longitudinal field off and transverse field on\n(h, g, J ) = (0 ,1,1) (for g= 0 the N´ eel state is an eigen-\nstate of the Hamiltonian). The results for all lengths of\nspin-chains L={4,6, . . .20}are shown in Figure 2a. The\ninitial growth is very similar to the non-integrable cases,\nbut afterwards the behaviour differs: the initial growth\nends at a clear peak followed by a rapid (approximately\nlinear) decrease and then rise to a second peak. These\noscillations then continue, becoming more irregular over\ntime. The oscillations remain regular for longer times\nfor longer chains. These oscillations in the evolution of\nentanglement entropy are called revivals and are a sig-\nnature of an integrable system [3] and are not present\nin non-integrable or chaotic systems as we will see next.\nAn analytic understanding of revivals in entanglement\nentropy evolution is provided in section III.\n2. Non-integrable case\nAs examples of non-integrable cases, we consid-\nered the Hamiltonian (2) with parameters ( h, g, J ) =\u0010√\n5+1\n4,√\n5+5\n8,1\u0011\nand ( h, g, J ) = (0 .5,−1.05,1). Fig-\nure 2b shows the results for the former case for spin-\nchain lengths L={4,6, . . .20}. These agree with the\nresults obtained by exact diagonalisation for lengths\nL={4,6,8,10,12}[19]. All lengths start with a sim-\nilar rapid growth of entanglement which saturates near\ntime tL≈L\n2and then continues fluctuating irregularly\naround the saturation value near S(t≫tL)≈L\n4. These\nfluctuations are very pronounced for short lengths and\ndecrease for larger lengths. Similar behaviour is obtained\nfor (h, g, J ) = (0 .5,−1.05,1) however the initial growth\nis slightly faster and saturation values are slightly higher.\nB. Evolution of mutual information after quench\nfrom Neel state\nMutual information scrambling following quench in the\nHeisenberg XXZ chain was studied in [1], comparing the\nbehaviour for integrable versus non-integrable parame-\nters. They look at the mutual information I(A, B) be-\ntween two intervals A= [1 : 2] and B= [L−1 :L] at the\nends of the chain, and consider the behaviour of the mu-\ntual information as a function of the distance d=L−4\nbetween the intervals, arguing that this can be used to\ndistinguish the integrable from non-integrable cases. We\nsimulated the time-evolution of the mutual information\nfor the same intervals A= [1 : 2] and B= [L−1 :L] for\nthe Ising Hamiltonian (2), with both the integrable and\nnon-integrable values of parameters, ( h, g, J ), to see how\ntheir results compare to the Ising chain.4\n(a) Integrable case ( h, g, J ) = (0 ,1,1).\n(b) Non-integrable case ( h, g, J ) = (√\n5+1\n4,√\n5+5\n8,1).\nFIG. 2: Time evolution of the entanglement entropy of\nhalf-chain S([1 :L/2]) in spin chains with lengths L= 4\nto 20.\n1. Integrable case\nFigure 3a shows the results for the time-evolution of\nthe mutual information I([1 : 2] ,[L−1 :L]), for lengths\nL= 6 to 26 in the integrable case ( h, g, J ) = (0 ,1,1).\n(Shorter chains were simulated until time t= 100, but\nlength L= 22 was simulated until time t= 50, L= 24\nuntil t= 40, and L= 26 until t= 10). For all lengths,\nthe mutual information first remains very low, followed\nby a rapid rise to a high peak (the black dots mark the\nmaxima of the first peak in the figures). The time at\nwhich the peak occurs increases approximately linearly\nwith L, while the height of the peak decreases with L.\n(a) Integrable case ( h, g, J ) = (0 ,1,1).\n(b) Non-integrable case ( h, g, J ) =\u0010√\n5+1\n4,√\n5+5\n8,1\u0011\n.\nFIG. 3: Time-evolution of mutual information,\nI([1 : 2] ,[L−1 :L]) in spin chains of lengths L= 6 to\n26. The above plots are in log-log scale for t≤30.\nThe presence of these peaks could be explained by the\nexistence of a main species of quasi-particle with max-\nimal velocity. For longer lengths Lthe particles from\nthe middle of the chain have to travel a longer distance\n≈L\n2−1 before they contribute to the mutual informa-\ntion, which means the peak will occur later in time and,\nif they are scrambled on the way, the height of the peaks\nwill decrease. The first peak is followed by more peaks\nat regular intervals, with smaller peaks scattered in be-\ntween. A striking feature of the main peaks is the regular\nintervals between them: for a fixed L, letting t0(L) be the\ntime of the primary peak, there are always other large\npeaks at times tn(L)≈(2n+ 1)t0(L), n∈N. This tim-\ning pattern is similar to the timings of the peaks in the5\nFIG. 4: Primary peak-value of mutual information\nI([1 : 2] ,[L−1 :L]) in function of distance d=L−4,\nfor different ( h, g, J ). Here, the black dots mark the\nmaxima of the first peak.\nevolution of the entanglement entropy S([1 :L/2]) (Fig-\nure 2a). Another interesting feature, mainly present in\nlonger chains is that, after the big decrease from the pri-\nmary peak to the next main peak, the height of the peaks\ntemporarily rises with time. The regular timings of the\nmain peaks could also be explained if the main species of\nquasi-particle, causing the primary peak, can elastically\nreflect off the edges of the chain and contribute to the\nmutual information again when they reach the opposite\nends of the chain.\n2. Non-integrable case\nFigure 3b shows the results for the time-evolution of\nthe mutual information I([1 : 2] ,[L−1 :L]), for lengths\nL= 6 to 26, with ( h, g, J ) =\u0010√\n5+1\n4,√\n5+5\n8,1\u0011\nand sim-\nilar results were obtained for ( h, g, J ) = (0 .5,−1.05,1).\n(In both cases, shorter chains were simulated until time\nt= 100, but length L= 22 was simulated until time\nt= 50, L= 24 until t= 40, and L= 26 until t= 10).\nIn both the cases, like in the integrable case, the mutual\ninformation first remains very low and then has a peak\n(black dots in the figures). Compared to the integrable\ncase, the time at which the peak occurs is similarly linearinL, but the height of the peak decreases more rapidly\nwith L. Figure 4 shows the heights of these peaks as a\nfunction of the distance d=L−4 between the intervals,\nfor both the non-integrable cases and the integrable case.\nThe presence of these well-defined peaks suggests there\nmight still be a main species of quasi-particle in the non-\nintegrable cases, but the rapid decrease in height (com-\npared to the integrable case) suggests they are scrambled\nsignificantly faster than in the integrable case. After the\nprimary peak there are no other clear regular peaks like\nin the integrable case.\nThe behaviour of the peaks as a function of d(Fig-\nure 4) and the difference between the non-integrable\nand integrable cases agrees well with the results of [1],\nwhere they similarly observed faster decrease in the non-\nintegrable cases, and slower (algebraic ∝d−1/2) decrease\nfor the integrable cases. Therefore they argued that non-\nintegrable systems are better scramblers than integrable\nsystems, and that scrambling is an interesting way of dis-\ntinguishing integrable and chaotic systems. [1] didn’t find\nmany examples of peaks other than the primary ones, this\ncould be due to shorter simulation times than our results.\nIII. ENTANGLEMENT ENTROPY FROM\nQUASI-PARTICLE PICTURE FOR A FINITE\nSYSTEM\nWe briefly review the computation of entanglement en-\ntropy from quasi-particle picture developed in [20]. The\ninitial state |ψ0⟩has very high energy relative to the\nground state of the Hamiltonian, H(λ). Therefore, it\nacts as a source for quasi-particle excitations. These par-\nticles from non-coincident points do not entangle, but\nparticles moving to the left or right from a given point\nare highly entangled. The cross-section to produce pairs\nof particles of momenta ( k′, k′′) isf(k′, k′′). Once these\nquasi-particles separate they move classically. The quasi-\nparticle velocity is v(k) =dϵk\ndk, where ϵkis its dispersion\nrelation. Thus the maximum allowed speed is |v(k)| ≤1.\nA quasi-particle with momentum kproduced at xis at\nx+v(k)tat a given time t. As quasi-particles reach either\nAorAcat time t, they are entangled if the particles\nreaching simultaneously at x′∈Aandx′′∈Acis a\npair emitted from x. Thus the entanglement entropy is\nproportional to the length of the interval in xfor which\nthis can be satisfied,\nSA(t) =Z\nx′∈Adx′Z\nx′′∈Bdx′′Z∞\n−∞dxZ\nf(k′, k′′)dk′dk′′δ(x′−x−v(k′)t)δ(x′′−x−v(k′′)t). (10)\nHere Ais an interval of length l. The total entan-\nglement is twice that between Aand the part of Actothe right of A. Thus restricting to the later implies that\nk′<0, k′′>0. Performing the integral over x, x′andx′′6\ngives, max ( t(v(−k′) +v(k′′)), l). Substituting this, the entanglement entropy becomes,\nSA(t) =Z0\n−∞dk′Z∞\n0dk′′f(k′, k′′) (v(−k′) +v(k′′)) [2t θ(l−(v(−k′) +v(k′′))t) + 2l θ((v(−k′) +v(k′′))t−l)].(11)\nIn [21], it was proposed that for a periodic system, using\nk′′=k, k′=−k, the entanglement entropy is,\nSA(t) = 2 tZ\n2|v|t<ldkv(k)s(k) +lZ\n2|v|t>ldks(k) (12)\nin the limit that t, l→ ∞ with r/tfixed.\nNext, the entanglement entropy from the quasi-particle\npicture is derived for a large system size of length L,\nwhere t, l, L→ ∞ such that t/landl/L(ort/L) are kept\nfixed. The entanglement entropy for periodic boundary\ncondition is [22],\nSl(t) =Z\n{2v(k)t\nL}<l\nLdk\n2πs(k)L\u001a2v(k)t\nL\u001b\n+lZ\nl\nL≤{2v(k)t\nL}<1−l\nLdk\n2πs(k)\n+Z\n1−l\nL≤{2v(k)t\nL}dk\n2πs(k)L\u0012\n1−\u001a2v(k)t\nL\u001b\u0013\n(13)\nwhere {x}means the fractional part of x.\nThe entropy grows linearly up tol\n2v, then exhibits a\nplateau which terminates at t=(L−l)\n2v. The plateau ter-\nminates when the first quasi-particle produced at one\nboundary of the sub-system re-enters it from another\nedge after having turned around the circle. Subsequently,\nmore and more quasi-particles re-enter the system caus-\ning the entropy to decrease till the revival time tR=L\n2v.\nAt the revival time, the dynamics begins again as if the\nsystem were at t= 0. Further computations to obtain\nentanglement entropy by performing the above integrals\nare given in appendix A.\nThe results of simulations presented in section II B 1\nare for open boundary condition. The above computa-\ntion can be modified to give the entanglement entropy in\nthe case of open boundary condition by rescaling both\nL→2Landl→2l. This is because in the case\nof open boundary condition, the quasi-particles reflect\nfrom the boundary of the spin chain and the dynamics\nis the same as if the system size is doubled and peri-\nodic boundary condition are imposed. Thus the revival\ntime in this case is tR=L\nv. Figure 5 shows revivals\nin entanglement entropy following quench for the inte-\ngrable Ising hamiltonian with open boundary condition\nfor (h, g, J ) = (0 ,1,1). The black line depicts the analytic\nresult and the red one depicts the result of simulation.\nFIG. 5: Revivals in entanglement entropy following\nquench for the integrable Ising hamiltonian with\n(h, g, J ) = (0 ,1,1). The black line depicts the analytic\nresult and the red line depicts the result of simulation.\nThe length of the spin chain is L= 20 and the\nentanglement entropy is computed of half-chain of\nlength l= 10 and vm= 2. The difference in the results\nis due to boundary effects.\nThe length of the spin chain is L= 20 and the entan-\nglement entropy is computed of the half-chain of length\nl= 10. The difference in the results is due to boundary\neffects. In the analytic case, the reduction in magnitude\nof the longitudinal field at the boundary points has not\nbeen taken into account. Also, there may be slowing\ndown of quasi-particles due to reflection at the bound-\naries of this open spin chain, this effect is captured in the\nentanglement entropy computed by simulation (as op-\nposed to analytic computation) and hence gives a higher\nvalue at the first peak in entanglement entropy and a\nlonger revival time.\nThese results as well as the ones in [3] (for finite in-\ntegrable spin chains) differ from [20] where the authors\nconsider an infinite integrable system and thus obtain a\nlinear rise followed by saturation of entanglement entropy\nas a function of time. Thus these revivals in evolution of\nentanglement entropy for the integrable spin chains is a\nfinite-size effect.7\nIV. OUTLOOK\nIn this paper, we saw that, mutual information as a\nfunction of distance between the entanglement intervals\ndecays faster in chaotic spin chains compared to inte-\ngrable ones. It is known that the dynamics of an Ising\nmodel at the critical point may be descibed by a certain\nconformal field theory [23, 24]. In a conformal field the-\nory, in order to compute the R´ enyi mutual information\nof two spatially separated intervals, we compute the n-\nth R´ enyi entropies, which are obtained from correlation\nfunctions of twist operators,\nI(n)\nA:B=S(n)\nA+S(n)\nB−S(n)\nA∪B (14)\nwhere\ne(1−n)S(n)\nA=D\nσn\u0010\n−a\n2\u0011\n¯σn\u0010a\n2\u0011E\n(15)\ne(1−n)S(n)\nB=D\nσn\u0010\nd−a\n2\u0011\n¯σn\u0010\nd+a\n2\u0011E\ne(1−n)S(n)\nA∪B=D\nσn\u0010\n−a\n2\u0011\n¯σn\u0010a\n2\u0011\nσn\u0010\nd−a\n2\u0011\n¯σn\u0010\nd+a\n2\u0011E\n.\nHence,\ne(n−1)I(n)\nA:B=\nσn\u0000\n−a\n2\u0001\n¯σn\u0000a\n2\u0001\nσn\u0000\nd−a\n2\u0001\n¯σn\u0000\nd+a\n2\u0001\u000b\n\nσn\u0000\n−a\n2\u0001\n¯σn\u0000a\n2\u0001\u000b\nσn\u0000\nd−a\n2\u0001\n¯σn\u0000\nd+a\n2\u0001\u000b\n(16)\nThus the R´ enyi mutual information of two intervals is\nthe four-point function of twist operators in the orbifold\nCFT [25–27]. This correlator can be written as a sum\nover conformal blocks [25, 28],\nD\nσn\u0010\n−a\n2\u0011\n¯σn\u0010a\n2\u0011\nσn\u0010\nd−a\n2\u0011\n¯σn\u0010\nd+a\n2\u0011E\n=c2\nnx−c\n6(n−1\nn) X\nKαKd2\nkxhKF(hK, hK; 2hK;x)!2\n,\n(17)\nwhere xis the cross-ratio, F(hK, hK; 2hK;x) is the hy-\npergeometric function, summation over Kis over the in-\ndependent quasi-primary operators and other constantsare to be determined from the operator product expan-\nsion of the twist fields.\nWhen using the above formula to compute the four-\npoint function of twist operators, one can only com-\npute sum over a certain finite number of blocks, thus\nthis method provides an approximation to the correla-\ntor. While the correlator is single valued, the sum over\nblocks is multi-valued, this leads to several possible chan-\nnels via which the four twist operators may be contracted\nto obtain an approximation to the above correlator. One\nof these channels will bring the twist operators in out-\nof-time ordering in imaginary time. In this channel we\nexpect the mutual information to be an exponentially\ndecaying function of the distance between entanglement\nintervals, d. In addition, similar to the bound on Lya-\npunov exponent in holographic theories [29], we expect\nthat the rate of exponential decay of mutual information\nas a function of the distance between entanglement inter-\nvals, d, will be bounded by it’s value in Einstein gravity.\nIt will be interesting to find this decay rate and compare\nit with the one obtained for chaotic spin chains in this\npaper.\nACKNOWLEDGMENTS\nThis work began as part of the bachelor project of\nEmil Tore Mærsk Pedersen at Vrije Universiteit Brus-\nsel (VUB). Part of the results in this paper were ob-\ntained during his bachelor project and were presented\nin his (unpublished) bachelor project report [30], 2020.\nWe thank Ben Craps and Charles Rabideau for discus-\nsions during the course of the bachelor project. We also\nthank Diptiman Sen, Nisheeta Desai and Tanay Nag\nfor helpful comments and questions. Research of SK\nduring postdoctoral tenure at VUB was supported by\nFWO-Vlaanderen project G006918N and by Vrije Uni-\nversiteit Brussel through the Strategic Research Program\nHigh-Energy Physics. Currently, SK’s research is sup-\nported by Department of Science and Technology’s IN-\nSPIRE grant DST/INSPIRE/04/2020/001063. ETMP’s\nresearch is currently supported by Novo Nordisk Founda-\ntion (Grant No. NNF20OC0059939 ‘Quantum for Life’).\nAppendix A: Analytic results for entanglement entropy evolution in integrable systems\n1. Entanglement entropy for XXchain\nThe XY chain is described by the Hamiltonian:\nHXY=−LX\nj=1\u00141 +γ\n2Sx\njSx\nj+1+1−γ\n2Sy\njSy\nj+1+hSz\nj\u0015\n(A1)8\nwith periodic boundary conditions, and it can be mapped to the free fermion model given by:\nH=X\nkϵkc†\nkck (A2)\nwhere ϵkrepresents the dispersion relation and is related to the velocity v(k) as follows:\nϵ2\nk= (h−cosk)2+γ2sin2k,\nv(k) =dϵk\ndk. (A3)\nUpon a quench h0→handγ0→γ, the Bogoliubov angle is given by:\ncos ∆ k=hh0−cosk(h+h0) + cos2k+γγ0sin2k\nϵϵ0. (A4)\nUtilizing the Generalized Gibbs Ensemble (GGE), the entanglement entropy can be expressed as,\ns(k) =−1 + cos ∆ k\n2ln\u00121 + cos ∆ k\n2\u0013\n−1−cos ∆ k\n2ln\u00121−cos ∆ k\n2\u0013\n. (A5)\nNow, let’s examine a N´ eel quench in an XXchain. The N´ eel state corresponds to h0=−∞, γ0= 0, while the XX\nchain corresponds to h=γ= 0. Consequently,\nϵ0=h0−cosk, ϵ =−cosk, (A6)\nand the velocity for both dispersion relations is v(k) =vMsink, with vM= 1. The entropy s(k) is constant at\ns= ln 2, for all k.\nFirst, considering the case when t < t R, thenn\n2v(k)t\nLo\n=2v(k)t\nL, so the integral boundaries in (13) become,\nSl(t) =Z\nvMsink<l\n2tdk\n2π2stvMsink+lZ\nl\n2t≤vMsink<L−l\n2tdk\n2πs\n+Z\nL−l\n2t≤vMsink<L\n2tdk\n2πs(L−2tvMsink). (A7)\nThe integration boundaries are illustrated in Figure 6, where k1=(π\n2ift <a\n2vM\nsin−1\u0010\na\n2tvM\u0011\nift≥a\n2vM,k2=\n(π\n2ift <b\n2vM\nsin−1\u0010\nb\n2tvM\u0011\nift≥b\n2vM,k3=π−k2andk4=π−k1.\nUtilizing these boundaries and denoting tR=L\n2vMandtS=l\n2vM, the resulting expression for Sl(t < t R) is,\nSl(t < t R) =\n\n2vMs\nπt if 0≤t < t S\n2vMs\nπt\u0012\n1−q\n1−\u0000tS\nt\u00012\u0013\n+ls\nπ\u0000π\n2−sin−1\u0000tS\nt\u0001\u0001\niftS≤t < t R−tS\n2vMs\nπt\n1−s\n1−\u0012tS\nt\u00132\n−s\n1−\u0012tR−tS\nt\u00132\n\n+ls\nπ\u0012\nsin−1\u0012tR−tS\nt\u0013\n−sin−1\u0012tS\nt\u0013\u0013\n+Ls\nπ\u0012π\n2−sin−1\u0012tR−tS\nt\u0013\u0013iftR−tS≤t < t R.\n(A8)\nWhen tR≤t <2tR, the fractional partn\n2v(k)t\nLo\ntakes on two distinct cases:\n\u001a2v(k)t\nL\u001b\n=(\n2v(k)t\nLifv(k)<L\n2t\n2v(k)t\nL−1 if v(k)≥L\n2t.9\nFIG. 6: Integration boundariesa\n2t≤vMsink <b\n2t\nSplitting the integrals accordingly:\nZ\na\nL≤{2v(k)t\nL}<b\nLf\u0012\u001a2v(k)t\nL\u001b\u0013\ndk\n=Z\n(a\nL≤2v(k)t\nL<b\nL)∧(v(k)<L\n2t)f\u00122v(k)t\nL\u0013\ndk+Z\n(a\nL≤2v(k)t\nL−1<b\nL)∧(v(k)≥L\n2t)f\u00122v(k)t\nL−1\u0013\ndk\n=Z\na\n2t≤vMsink<b\n2tf\u00122v(k)t\nL\u0013\ndk+Z\na+L\n2t≤vMsink<b+L\n2tf\u00122v(k)t\nL−1\u0013\ndk, (A9)\nwhere 0 ≤a≤b≤L. The entanglement entropy is now,\nSl(t∈[tR,2tR]) =2vMs\nπt\n1−s\n1−\u0012tS\nt\u00132\n+ 2s\n1−\u0012tR\nt\u00132\n−s\n1−\u0012tR−tS\nt\u00132\n\n+ls\nπ\u0012\nsin−1\u0012tR−tS\nt\u0013\n−sin−1\u0012tS\nt\u0013\u0013\n+Ls\nπ\u0012\n2 sin−1\u0012tR\nt\u0013\n−sin−1\u0012tR−tS\nt\u0013\u0013\n+\n\n−Ls\nππ\n2iftR≤t < t R+tS\n−2vMs\nπts\n1−\u0012tR+tS\nt\u00132\n−Ls\nπsin−1\u0012tR+tS\nt\u0013\n+ls\nπ\u0012π\n2−sin−1\u0012tR+tS\nt\u0013\u0013 iftR+tS≤t <2tR−tS\n−2vMs\nπt\ns\n1−\u0012tR+tS\nt\u00132\n+s\n1−\u00122tR−tS\nt\u00132\n\n−Ls\nπ\u0012\nsin−1\u0012tR+tS\nt\u0013\n+ 2 sin−1\u00122tR−tS\nt\u0013\n−π\u0013\n+ls\nπ\u0012\nsin−1\u00122tR−tS\nt\u0013\n−sin−1\u0012tR+tS\nt\u0013\u0013if 2tR−tS≤t <2tR.(A10)\nFurther intervals ntR≤t <(n+ 1)tR,n∈N, can be calculated analogously, each time building upon the results of\nthe previous intervals.10\n2. Modifying the above calculation for quench to Ising chain\nThe transverse field Ising model Hamiltonian is,\nHI=−JLX\nj=1\u0000\nσz\njσz\nj+1+gσx\ni\u0001\n. (A11)\nThis is the special case of the XY Hamiltonian HXY=−1\n2PL\nj=1\u00021+γ\n2σx\njσx\nj+1+1−γ\n2σy\njσy\nj+1+hσz\nj\u0003\nwhen γ= 1 and\nh=g, scaled by a factor 2 J, rotated byπ\n2around the y-axis (so σx7→ −σz, σz7→σx). (For J= 1, g= 1, the results\nobtained are the same as the integrable case considered in the main text.) Therefore the dispersion relation (invariant\nunder the rotation) is\nϵk= 2Jp\n1 +g2−2gcosk, (A12)\nand the velocity for this dispersion relation is\nv(k) = 2 Jgsinkp\n1 +g2−2gcosk. (A13)\nWe now consider the N´ eel quench in the Ising model with g= 1. The N´ eel state corresponds to g0= 0, which now\ncorresponds to h0= 0, γ0= 1 in the (rotated) XY chain. So the Bogoliubov angle difference (A4) is cos ∆ k=\f\fsin\u0000k\n2\u0001\f\f\nand therefore,\ns(k) =−1 +\f\fsin\u0000k\n2\u0001\f\f\n2ln \n1 +\f\fsin\u0000k\n2\u0001\f\f\n2!\n−1−\f\fsin\u0000k\n2\u0001\f\f\n2ln \n1−\f\fsin\u0000k\n2\u0001\f\f\n2!\n(A14)\nand, for g= 1,\nv(k) =vMcos\u0012k\n2\u0013\nsgnk , v M= 2J. (A15)\nIn the integrals of (13) the value of k∈[0, π], so we can ignore the absolute values in s(k) and the sgn kinv(k) in\nthe following calculation.\nFor all m∈N,\nmL\n2t≤v <(m+ 1)L\n2t=⇒\u001a2vt\nL\u001b\n=2vt\nL−m, (A16)\nso ifmL\n2t≤v <(m+1)L\n2tthen\n\u0012a\nL≤\u001a2vt\nL\u001b\n<b\nL\u0013\n⇐⇒\u0012a+mL\n2t≤v <b+mL\n2t\u0013\n.\nTherefore,\nZ\na\nL≤{2vt\nL}<b\nLf(k)dk=X\nmZ\na+mL\n2t≤v<b+mL\n2tf(k)dk, (A17)\nwhen 0 ≤a≤b≤L. It is also easily seen that when t≤ntR,n∈N, the only nonzero terms in this sum are those\nwith m < n . So (13) can be written\nSl(t∈[ntR,(n+ 1)tR]) =nX\nm=0(∆Am+ ∆Bm+ ∆Cm)\n∆Am=Z\nmL\n2t≤v<l+mL\n2tdk\n2πs(k)(2tv(k)−mL)\n∆Bm=Z\nl+mL\n2t≤v<L−l+mL\n2tdk\n2πs(k)l\n∆Cm=Z\nL−l+mL\n2t≤v<(m+1)L\n2tdk\n2πs(k)((m+ 1)L−2tv(k)).(A18)11\nFIG. 7: Integration boundariesa+mL\n2t≤vMcos\u0000k\n2\u0001\n<b+mL\n2t\nFor the Ising chain N´ eel quench velocity (A15) these boundariesa+mL\n2t≤v <b+mL\n2tconsist of one interval [ k1, k2],\nas illustrated in Figure 7, where k1= 2 cos−1\u0000b+mL\n2t\u0001\nandk2= 2 cos−1\u0000a+mL\n2t\u0001\n, so\nZ\na+mL\n2t≤vMcos(k\n2)<b+mL\n2tdkf(k) =\n\n0 if t <a+mL\n2vMR2 cos−1\u0010\na+mL\n2tvM\u0011\n0 dkf(k) ifa+mL\n2vM≤t <b+mL\n2vMR2 cos−1\u0010\na+mL\n2tvM\u0011\n2 cos−1\u0010\nb+mL\n2tvM\u0011dkf(k) if t≥b+mL\n2vM.(A19)\nThe integrands in (A18) are all linear combinations of s(k) and s(k)v(k). The antiderivatives Is=R\ndks(k) and\nIvs=R\ndkv(k)s(k) of (A14) and (A15), for the Ising chain N´ eel quench, are\nIs(k) = (2 ln 2 −1)k+ 2i\u0010\nLi2\u0010\n−ie−ik\n2\u0011\n−Li2\u0010\n−ieik\n2\u0011\u0011\n+iπ\n2k+ ln \n1 + sin\u0000k\n2\u0001\n1−sin\u0000k\n2\u0001!\ncos\u0012k\n2\u0013\n,\nIvs(k) =vM \n1−sin\u0000k\n2\u0001\n2!2 \n2 ln \n1−sin\u0000k\n2\u0001\n2!\n−1!\n−vM \n1 + sin\u0000k\n2\u0001\n2!2 \n2 ln \n1 + sin\u0000k\n2\u0001\n2!\n−1!\n,(A20)\nwhere Li 2(z) =−Rz\n0ln(1−t)\ntdtis the dilogarithm (Spence’s function). These antiderivatives were chosen such that\nIs(0) = Ivs(0) = 0. (Despite the complex numbers in the expression, the imaginary part of Is(k) remains 0 for all12\nk∈[0, π]). Denoting Io\ns(x) =Is(2 cos−1(x)) and Io\nvs(x) =Ivs(2 cos−1(x)) the integrals in (A18) become\n∆Am+ ∆Bm+ ∆Cm\n=\n\n0 if t < mt R\nt\nπIo\nvs\u0000mtR\nt\u0001\n−mL\n2πIo\ns\u0000mtR\nt\u0001\nifmtr≤t < mt R+tS\nt\nπ\u0012\nIo\nvs\u0012mtR\nt\u0013\n−Io\nvs\u0012mtR+tS\nt\u0013\u0013\n−mL\n2πIo\ns\u0012mtR\nt\u0013\n+mL+l\n2πIo\ns\u0012mtR+tS\nt\u0013 ifmtR+tS≤t <(m+ 1)tR−tS\nt\nπ\u0012\nIo\nvs\u0012mtR\nt\u0013\n−Io\nvs\u0012mtR+tS\nt\u0013\n−Io\nvs\u0012(m+ 1)tR−tS\nt\u0013\u0013\n+mL+l\n2πIo\ns\u0012mtR+tS\nt\u0013\n+(m+ 1)L−l\n2πIo\ns\u0012(m+ 1)tR−tS\nt\u0013\n−mL\n2πIo\ns\u0012mtR\nt\u0013if (m+ 1)tR−tS≤t <(m+ 1)tR\nt\nπ\u0012\nIo\nvs\u0012mtR\nt\u0013\n−Io\nvs\u0012mtR+tS\nt\u0013\n−Io\nvs\u0012(m+ 1)tR−tS\nt\u0013\u0013\n+mL+l\n2πIo\ns\u0012mtR+tS\nt\u0013\n+(m+ 1)L−l\n2πIo\ns\u0012(m+ 1)tR−tS\nt\u0013\n−(m+ 1)L\n2πIo\ns\u0012(m+ 1)tR\nt\u0013\n−mL\n2πIo\ns\u0012mtR\nt\u0013\n+t\nπIo\nvs\u0012(m+ 1)tR\nt\u0013ift≥(m+ 1)tR.\n(A21)\nInserting this into (A18) then gives the entanglement entropy Sl(t∈[ntR,(n+ 1)tR]) for any n∈N.\n[1] V. 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Pandian3, and \nSandip Dhara1,* \n1Surface and Sensors Studies Division, Indira Gandhi Centre for Atomic Research, A CI of \nHomi Bhabha National Institute, Kalpakkam -603102, India  \n2Surface and Sensors Studies Division, Materials Science Group, Indira Gandhi Centre for \nAtomic Research, Kalpakkam -603102, India  \n3Defects and Damage Studies Division, Materials Science Group, Indira Gandhi Centre for \nAtomic Research, Kalpakkam -603102, India  \n*Corresponding  authors e -mail id: bhuyan.2d@gmail.com, madupu@igcar.gov.in, \ndhara@igcar.gov.in  \nAbstract  \nWe fabricated the 1D nanoscrolled monolayer MoS 2 (1L-MoS 2) with superior characteristics \nfrom 1L -MoS 2 film in a facile route, using a suitable organic solvent with optimum surface \ntension, evaporation rate and dielectric constant, which facilitates the con trolled scroll \nformation.  These nanoscrolls behave as multilayers in morphology and monolayer \nelectronically. The nanoscrolls exhibited a direct optical gap with enhanced \nphotoluminescence quantum yield stemming from the weak interlayer coupling among \nconstituent layers and were corroborated by low -frequency Raman measurements and Kelvin \nprobe force microscopy measurements. Furthermore, enhanced photoluminescence emission \nafter annealing uncovers the thermal stability of nanoscrolls. In addition, conducting  atomic \nforce microscopy results exhibit a significantly higher photocurrent in the nanoscrolled 1L -\nMoS 2 compared to the 1L -MoS 2. We also realized significantly improved field effect \ntransistor device parameters in nanoscrolled 1L -MoS 2 devices. In nanoscro lled devices, we \nreport the highest mobility value of 2400 cm2V-1s-1 reported in any form of 1L -MoS 2. 2 \n Keywords:  Monolayer, Raman, TMDs, mobility, Surface potential, Shear mode, Breathing \nmode  \n1. Introduction  \nUltra -thin monolayer MoS 2 (1L-MoS 2) attracted a great deal of attention among the research \ncommunity because of its interesting electronic and optical properties, with the observation of \ndirect band gap and enhanced photoluminescence (PL) quantum yield (QY). 1 The PL QY of \n1L-MoS 2 can be enhanced by doping,2 surface treatment, 3 and increasing the heat dissipation \narea.4 Recently, it has been reported that PL QY of 1L -MoS 2 can be enhanced with an \nincrease in the flake area owing to the availability of a large heat -dissipating area.4 In \nparticular, large -area 1L -MoS 2 provides one -order higher PL QY than mechanically \nexfoliated 1L -MoS 2 and two -order higher than small -sized as -grown chemical vapour \ndeposition (CVD) grown flakes.4 At the same time,devices fabricated with large channel \ndimensions may not meet the required demand to fabricate miniaturized and compact next -\ngeneration electronic devices. In other words, there is a trade -off between the availability of \nlarge heat -dissipating  areas and the compactness of the devices. Therefore, it is imperative to \neliminate the bottleneck by designing a compact structure that retains the characteristics of \nlarge -area 1L -MoS 2. It is worth mentioning that the optical a bsorption coefficient in th e \nvisible wavelength of the electromagnetic spectrum of 1L -MoS 2 is significantly high \ncompared to that for conventional semiconductors. 4 However, the overall light absorption is \nlimited by the atomic scale thickness of monolayer MoS 2. Thus, the structure containing \nmultiple layers possessing the monolayer behaviour will be necessary for a significant \namount of absorbed light in the visible range.   \nRecently, the one -dimensional (1D) nanostructures designed from 2D monolayers \nhave opened new avenues for tuning their structural, electronic , and optical properties.5,6 3 \n According to the theoretical calculations, the nanoscrolls have a unique topology, which \nmakes them useful in flexible electronics, energy storage, and optical resonator applications.7 \nThe scr olled 1L -MoS 2 gained substantial attention because it exhibited  higher light -matter \ninteractions, which resulted  in higher photo -responsivity than the planar 1L -MoS 2.8 In \nscrolled 1L -MoS 2, the transportation of the carriers occurs in their constituent laye rs, which is \nquite different from their respective multilayers.9 For the first time, amorphous MoS 2 \nnanoscrolls were prepared by the Ar plasma -assisted method.10 The PL properties of \nnanoscrolled MoS 2 vary significantly depending on the fabrication process .8,9,11 -13 Table S1 \nsummarizes the optical properties of reported nanoscrolled MoS 2 samples. So far, the \nnanoscrolls were mostly fabricated by the self -rolling method. The self -rolling was initiated \nby the evaporation of volatile organic solvent  over the m onolayer. Usually, isopropyl alcohol \n(IPA) 12 and acetone11 were used as organic solvents  because of their low surface tension and \nhigh evaporation rates. However, the PL emission studies were absent in nanoscrolls \nfabricated using IPA.8,12 In contrast, the self -rolled nanoscrolls fabricated with the aid of \nacetone showed four times higher PL intensity.11 Unfortunately, PL emission in these \nsamples was quenched after annealing (100 C for 1h). In addition, capillary force -driven \nself-rolls o f vertical van der Waals heterostructures (SnS 2/WSe 2) were also prepared using a \nmixture of ethanol, water, and ammonia.5 However, a reduction in PL emission was observed \nin heterostructured scrolls compared to planar structures, and it was attributed to t heir \nintimate interlayer interactions. Alternatively, the mechanical force -induced scroll fabrication \nhas also been investigated using a mixture of ethanol and pure water.8,9 The strain -induced \nMoS 2 nanoscrolls fabricated using ethanol solution (volume rat io of ethanol: water = 2:1) \nshowed properties of multilayered MoS 2 with reduced PL QY.9,13 Scrolls fabricated by \nwtheater-drag method lack PL measurements.8 In addition, the mechanical force -induced \nrolls are mostly prone to structural deformity due to sta cking disorder and non -uniform 4 \n rolling. So far, PL emission was either not observed or quenched after annealing in earlier \nreported nanoscrolled MoS 2.8 Thus, fabrication of nanoscroll MoS 2 retaining the monolayer \nproperties and maintaining thermal stabilit y also is necessary for device applications.    \nIn this report, we developed and demonstrated the effective and facile route to \nfabricate the thermally stable nanoscrolled 1L -MoS 2. The self -rolled nanoscrolls were \nfabricated by using absolute ethanol. The scrolled 1L -MoS 2 exhibits a high quantum yield of \n2.95% and up to five times more PL intensity than remnant 1L -MoS 2. The enhancement in \nthe PL emission was attributed to negligible interlayer coupling among constituent layers. \nLow-frequency Raman and Kelvi n probe force measurements confirm negligible interlayer \ncoupling. From the PL analysis, it was found that interlayer coupling in nanoscrolls further \nweakened after annealing. The annealed nanoscrolls have shown nearly six times higher PL \nintensity compare d to its counter planar 1L -MoS 2 structures. The nanoscrolled 1L -MoS 2 \nshowed significantly improved electronic and optoelectronic properties compared to their \nplanar counterparts.  \nSelf-rolled nanoscrolled 1L -MoS 2 fabrication  \nThe nanoscrolled 1L -MoS 2 was fabricated from CVD -grown 1L -MoS 2 film. The as -grown \nlarge -area 1L -MoS 2 on SiO 2/Si substrate was taken as the starting material. A little (15 -20 µl) \namount of absolute ethanol (Sigma Aldrich, purity 99.9 %) was dropped on the sample \nsurface containing  1L-MoS 2. The optical microscope captured the step -by-step image of an \nexemplary nanoscrolled 1L -MoS 2 at the formation stage ( Figure  1). In addition, nanoscrolled \n1L-MoS 2 fabrication steps were provided schematically in supplementary ( Figure  S1). Figure \n1a-c illustrates the optical micrographs at each stage of the nanoscroll formation. Figure 1a \ndepicts the ethanol drop on the planar 1L -MoS 2 surface. Subsequently, the size of the ethanol \ndrop shrinks due to evaporation at room temperature (RT) ( Figure  1b). The complete 5 \n evaporation of ethanol generates the rolled structures in a few areas ( Figure  1c). The high \nmagnification image of one of the fabricated nanoscrolls was provided in Figure  1d. \nMoreover, the optical contrast between 1L -MoS 2 and rolled structure  exhibits the successful \nformation of scrolled MoS 2 (Figure  1d). Immediately after the formation of the scroll, sample \nwas placed under an IR lamp for 1 h to remove any ethanol residues from MoS 2 surface. \nInterestingly, scroll formation was only observed i n certain regions of the sample. In \nparticular, scroll formation was initiated at the grain boundaries region of large area MoS 2 \nfilm and edges of triangular flakes. Thus, one can speculate that the scroll formation might \nhave happened because of the penet ration of ethanol between the flake and substrate at the \ngrain boundaries and edge region and subsequent evaporation. Ethanol delaminated the 1L -\nMoS 2 during the evaporation at RT. During the evaporation of ethanol, the heat might be \nextracted from the plan ar 1L -MoS 2 structure, which resulted in reduction of surface energy.9 \nThe decrease in surface energy is compensated by the change of planar MoS 2 structure to its \nscrolled form.  \nIn general, the scroll formation process is governed by solid -liquid interfaci al energy \nand the solvent  surface tension and evaporation rate.5 The interfacial energy is solely \nresponsible for the intercalation of solvent molecules  to delaminate the 1L -MoS 2 from the \nSiO 2 substrate. The surface tension and evaporation rate of the solvent are two important \nparameters as they decide the capillary force (flow direction of the solvent during \nevaporation) at the scroll formation stage.5 In earlier reports, the mixture of ethanol and \nwater, IPA, and acetone solvents were used for the scr oll fabrication. Unfortunately, PL \nemission was either suppressed or reduced after annealing in these scrolls compared to the \nplanar MoS 2.11 Here, we fabricated the scrolls using absolute ethanol because of its optimum \nphysical properties, such as surface tension and evaporation rate. Generally, the mixture of \nethanol and water has a higher surface tension (~72.8 mN m-1 of water -air @ 20 oC) than that 6 \n of pure organic solvents like ethanol (~22.1 mN m-1 of ethanol -air @ 20 oC), IPA (~23 \nmNm-1 of IPA -air @ 2 0 oC) and acetone (~25.2 mN m-1 of acetone -air @ 20 oC). [14] The \nhigh surface tension of the solvent is  prone to bead -up and deteriorates the uniformity of the \nscroll.5 Moreover, ethanol has optimum evaporation rate of 119 mg m-2s-1 and vapour  \npressure ~7.86 kPa@25 oC compared to those of acetone (evaporation rate 504 mg m-2s-1 and \nvapour pressure ~30.8 kPa@25 oC) and IPA (evaporation rate 75.6 mg m-2s-1 and vapour \npressure ~5.69 kPa@25 oC).14 The optimum evaporation rate of ethanol is reason ably \nadvantageous as it allows sufficient time for a uniform scroll formation process.  \nIn addition, it was also reported that the formation of scrolls and their stability is \ngoverned by the thermodynamical parameters. The formation of scrolls is allowed i f the \nenthalpy ( H) decreases upon scrolling. The change in enthalpy, H = U + W, where U \nstands for change in internal energy and W stands for work done due to rolling. Most \nimportantly, the W is dependent on the dielectric constant of the solvents involved in the \nscroll formation. [6] At room temperature, the dielectric constant of ethanol,  IPA and acetone \nare 24.55, 18.3, and 20.7, respectively. The dielectric constant value of ethanol is \ncomparatively higher than IPA and acetone. Thus, the self -rolling process in an ethanol \nmedium is relatively slower than in IPA and acetone.15 Therefore, absolute ethanol is \nadvantageous over other organic solvents for controlled and uniform scroll formation.  \n2. Results and Di scussions  \nThe cross -sectional view of the scrolled 1L -MoS 2 is represented as a schematic in \nFigure  2a. In the present study, we compared the optical properties of the scroll with remnant \n1L-MoS 2 for the fair assessment. The multiple scrolls were formed using a single drop of \nethanol ( Figure  S2). Among them, we carried out a detailed optical analysis on a particular 7 \n scroll ( Figure  2b). FESEM images of other scrolled 1L -MoS 2 structures are shown in  Fig S3.  \nFigure 2c shows the typical FESEM image of the same scroll shown in Figure  2b, and its \nhigh magnification image is provided in Figure  S4a. It reveals that the monolayer is \nuniformly rolled to form the tubular structure. In addition, it is apparen t that the fabricated \nscrolls are clean and wrinkle -free. The length of the given nanoscroll is measured to be ~13.5 \nμm. The lateral size of the scroll across length was in the range of 220 -250 nm. We have \ncarried out an AFM study to estimate the thickness  of the 1L -MoS 2 and the height of the \nnanoscroll. The thickness of 1L -MoS 2 is 0.85 nm ( Figure  S4b,c). The AFM image of the \ntypical nanoscrolled 1L -MoS 2 was provided in Figure  2d. The surface topography of the \nnanoscroll  is uniform. From the height profile, the thickness of the nanoscroll was calculated \nto be 160±10 nm (insets of Figure 2d). We calculated the number of layers present in the \nscroll by assuming it as tubular form , and detailed calculation steps are provided in the \nsupplementary material. In the given  scroll, approximately 28 number of 1L -MoS 2 layers are \nestimated to be present.  \nRaman measurements were carried out to assess the phase and number of layers \npresent in the planar and nanoscrolled MoS 2. In the bac k-scattering configuration, the 1L -\nMoS 2 has two prominent Raman peaks corresponding to E12g and A1g modes.16,17 Raman \nspectra of a typical planar and nanoscrolled MoS 2 are provided in Figure  3a. In planar MoS 2, \nthe E12g and A1g Raman modes are observed wit h ca. 383.9 cm-1 and 403.6 cm-1, respectively. \nAt the same time, E12g and A1g Raman modes are observed at 383.5 cm-1 and 402.6 cm-1, \nrespectively, for nanoscrolled 1L -MoS 2. A small red -shift in E12g mode is attributed to the \ndevelopment of tensile strain due to the curved morphology in the scroll.18 The red -shift of \nA1g (1 cm-1) can be understood in terms of restoring force. It is well known that out -of-plane \nA1g phonon mode frequency is strongly dic tated by the restoring force, whereas in -plane E12g \nmode is dictated by dielectric screening.17,19,20 In the form of a scroll, the bonding between 8 \n flake and SiO 2 happened to be broken, leading to the dis appearance of restoring force \noriginated from the bonding between SiO 2 and MoS 2. Thus, A1g phonon mode red -shifted in \nnanoscrolled MoS 2 as compared to planar MoS 2. Raman analysis was carried out on multiple \nnanoscrolls and the results were summarized in Ta ble S2.  In addition, we compare the \nfrequency difference (∆) between two prominent Raman modes ( E12g and A1g), as it reveals \nthe number of layers present in a MoS 2 flake.16 In planar MoS 2, ∆ is 19.7 cm-1, which \nconfirms that planar flake is indeed monolayer.4,16 Interestingly, the ∆ value in scrolled MoS 2 \nwas 19.1 cm-1. The ∆ value of nanoscrolls was similar to that for monolayer flakes, which \nconfirmed the monolayer characteristics of the scroll.16 As men tioned before, the scroll \ncontains approximately 28 layers. Thus, the ∆ was expected to be of the bulk MoS 2 (>25 cm-\n1).16 It is worth mentioning that scrolls fabricated by many other groups by using different \norganic solvents have shown multilayer characte ristics .8,9,11 -13 In their work, ∆ value was in \nthe range of 21 -24 cm-1, which confirms that the scrolls were indeed multilayer in nature \n(Table S1). The increased ∆ values were attributed to the stacking -induced structural changes \nin scrolled MoS 2.9 As a result, the potential application of the monolayers cannot be fulfilled \nthrough scrolls with interacting constituent layers. In the present study, monolayer \ncharacteristics (  20 cm-1) were observed in nanoscrolled MoS 2. In other words, in contrast \nto the reported works, the fabricated nanoscrolls act as a monolayer even though they contain \nmultiple layers. The monolayer behaviour of nanoscrolled 1L -MoS 2 may be attributed to the \nnon-interacting nature of constituent layers.   \nA 1L -MoS 2, being a direct band gap semiconductor, shows intense PL emission, \nwhich makes it suitable for many optoelectronic applications.1,2 The RT PL spectra of 1L -\nMoS 2 and nanoscrolled 1L -MoS 2 are shown in Figure  3b. The spectra fitted with Gaussian \ncurves were provided in the s upplementary material ( Figure  S5 and analysis in Table S3a). In \n1L-MoS 2, PL spectrum was majorly contributed by A -excitons (1.84 eV) over B -excitons 9 \n (1.96 eV) and trions (1.80 eV) emission. Similarly, PL spectra of nanoscrolled 1L -MoS 2 were \ncontributed by A-exciton (1.83 eV), B -exciton (1.94 eV) and trions (1.79 eV) emission. \nInterestingly, the nanoscrolled 1L -MoS 2 exhibits the direct band gap features, which is a \nprimary signature of 1L -MoS 2. Compared  to 1L -MoS 2, the PL spectra of nanoscrolled 1L -\nMoS 2 were  red-shifted by 10 meV. The observed negligible red -shift may be attributed to the \nreduction in the dielectric screening upon rolling, as it becomes free -standing after being \ndetached from the SiO 2substrate.21 Moreover, it was also reported that the red -shift in exciton \npeak energy was also attributed to the introduction of bending strain.22 Previously, blue -shift \nin exciton peak positions in nanoscrolled MoS 2 were reported, and it was attributed to the \nexistence of solvent  molecules among the interlayer ga ps.11 In the  present work, the \nnegligible shift in exciton peak positions reveals the non -existence of ethanol molecules at \ninterlayer gaps. PL analyses on various spots of a given scroll and other scrolls were provided \nin the supplementary material ( Figur e S6 and analysis in Table S3b).  Furthermore, a low \nemission peak at 1.64 eV was observed in the PL spectrum of scrolled 1L -MoS 2 (Figure  3b). \nThe low energy peak might be attributed to interlayer coupling ( I) peak, and it is usually \nobserved in multilayer  MoS 2 with weak inter -layer coupling and twisting of layers. [23,24] \nHowever, Ipeak in PL spectra of monolayer MoS 2 was also reported earlier under the \napplication of external strain.18,20 Moreover, a small curvature or bending in the monolayer \nMoS 2 film can modulate the band gap by enhancing the interlayer coupling.18 Therefore, the \norigin of Ipeak in scrolled 1L -MoS 2 may be traced from the developed local bending strain. \nCompared to 1L -MoS 2, the mean and maximum enhancement in PL integrated intens ity of \nnanoscrolled 1L -MoS 2is ~ 4 and ~5.2 times, respectively ( Figure  3b and Table S4). The inset \nof Figure  3b shows the PL imaging of scrolled region. The PL imaging was carried out using \nA-exciton intensity. The PL intensity was high over the scroll reg ion compared to remnant \nMoS 2. Here, the overall enhanced PL intensity in nanoscrolls is attributed to the non -10 \n interacting nature of constituent monolayers. The constituent layers of the scrolled MoS 2 are \nquite different from the bulk MoS 2. In general, the bilayer MoS 2 (2L-MoS 2) shows a two -\norder less PL intensity compared to 1L -MoS 2, and the emission intensity further reduces as \nthe number of layers increases.  \nPL quantum yield (QY) was measured using optical absorbance and PL, as described \nin our earlier report.4 Absorbance for the nanoscrolled 1L -MoS 2 and 1L -MoS 2 films were \nmeasured to be 0.62 +0.08 and 0.3 +0.05, respectively, and detailed in the supplementary \ninformation ( Figure  S7). The improved absorbance was attributed to the enhanced light -\nmatter inte raction originating from the existence of multiple layers in the scroll. The PL QY \nof 1L -MoS 2 and nanoscrolled 1L -MoS 2 were calculated to be 1.3% and 2.95%, respectively. \nThe higher QY was attributed to the enhancement of PL emission in nanoscrolled 1L -MoS 2 \ncompared to the planar form.  \nThe interlayer coupling can be analyzed using shear ( S) and layer breathing (LB) \nmodes of MoS 2.25 Usually, shear and breathing modes are not shown in Raman spectra \nbecause of the constraints of the Rayleigh rejection filter.  Low-frequency (LF) Raman \nmeasurements were carried out on fabricated nanoscrolls to investigate the interlayer \ncoupling. The LF Raman spectra of multilayered MoS 2 are dominated by interlayer shear ( S \nor E22g) and layer -breathing (LB or B22g) mode. The S mode represents out -of-phase in -plane \nvibrations of constituent MoS 2 layers, whereas the LB mode represents out -of-phase out -of-\nplane vibrations of the MoS 2 layers.26 Thus, S and LB modes are always absent in 1L -MoS 2 \ndue to the non -existence of an additio nal layer for interactions. Figure  4 shows the typical LF \nRaman spectra of mono -, bi-, and nanoscrolled MoS 2. The bilayer spectrum was provided for \nthe comparison. The S and LB modes were absent in 1L -MoS 2, whereas S (20.6 cm-1) and LB \n(36.5 cm-1) modes we re observed in bilayer (2L) MoS 2. Interestingly, in nanoscrolled 1L -11 \n MoS 2, the existence of insignificantly minute S-mode and the absence of LB mode reveals \nthe existence of weak interlayer coupling.25 Thus, the LF Raman results reinforced the PL \nresults th at interlayer coupling is negligible in nanoscrolled 1L -MoS 2. \nThe surface potential (SP) of a MoS 2 film is quite sensitive to the number of layers \nbecause of the interlayer screening effect.27 We carried out Kelvin probe force microscopy \n(KPFM) measurement s on as -grown MoS 2 flake, which contains the mono -, bi- and \nmultilayer to corroborate our assumption ( Figure  5a). The presence of mono -, bi and \nmultilayer was confirmed by the Raman and PL analysis (not shown in the figure). From the \ncontact potential diff erence map, it is apparent that SP varied depending on the thickness of \nthe flake ( Figure  5b). The SP of the bilayer increased by 100 mV compared to the monolayer \nMoS 2. Thus, SP would change if there is a strongly interacting layer. Subsequently, KPFM \nmeas urements were carried out on nanoscrolled MoS 2 and the remnant MoS 2. Figure 5c \nshows the SP map of the scrolled and monolayer region. It was found that the SP of SiO 2 was \nhigher than 1L -MoS 2 and nanoscrolled MoS 2. The SP values of SiO 2 and 1L -MoS 2 were 520 \nand 280 mV, respectively ( Figure 5d). Interestingly, negligible change in SP was observed for \nthe monolayer and scrolled region except for the edges of scrolls. The unchanged surface \npotential infers a negligible change in electronic properties. I mportantly, the unchanged SP \nvalue among the monolayer and scrolls concludes  the negligible inter -layer coupling among \nthe constituent layers of nanoscrolled 1L -MoS 2.  \nTo investigate the thermal stability of the as -fabricated scroll, we have carried out \nvacuum annealing at 200 °C for 2h. We restricted the annealing temperature to 200 °C to \navoid the defect introduction to the 1L -MoS 2 system.4,28 The PL spectra of the annealed \nsample are shown in Figure  6a. T he integrated PL intensity was increased by ~6.4 times \ncompared to remnant 1L -MoS 2. Moreover, annealed scrolled 1L -MoS 2 shows ~ 5 times 12 \n higher integrated PL intensity compared to as -fabricated scroll ( Figure  3b). The enhancement \nof PL intensity further asc ertains the weakening of interlayer coupling among constituent \nlayers. To date, all the studies have failed to fabricate thermally stable nanoscrolls of 1L -\nMoS 2.8,9,11 -13 Here,  we achieve the formation of thermally stable scrolls because of the \nselection o f absolute ethanol over other solvents.  In this context, as discussed earlier, the \nvacuum annealing of MoS 2 nanoscrolls at 100 °C for 1h with Ar atmosphere was previously \nreported. 11 The PL integrated intensity was quenched ~ 4 times with significant red -shift. In \ncontrast, we found a negligible blue shift of 10 meV in the A -exciton peak position after \nannealing. Thus, our PL results of annealed 1L -MoS 2 nanoscrolls signify that scrolls  are \ndevoid of ethanol residues because of negligible shifts in peak positions and enhanced \nintensity. We also observed that the interlayer peak in the nanoscrolled 1L -MoS 2 disappeared \nafter annealing. It demonstrates further weakening of the interlayer co upling among their \nconstituent layers. The reduction in interlayer coupling after annealing may also contribute to \nenhanced PL intensity.  \n We have carried out Raman analysis on annealed MoS 2 scrolls to further investigate \nthe annealing effect. The Raman pe ak positions of E12g Raman mode are 383.7 and 384.0 cm-\n1 for monolayer and scrolled 1L -MoS 2, respectively. The small blue -shift in E12g mode in \nscrolled 1L -MoS 2 may be because of possible molecular rearrangement in the constituent \nlayers  with the annealing  treatment .16 Similarly, the A1g Raman mode in 1L -MoS 2 and \nscrolled 1L -MoS 2 was found at 402.8, and 404.0 cm-1, respectively ( Figure  6b). After \nannealing, the ∆ value of scrolled 1L -MoS 2 was 20 cm-1 which confirmed the retaining of \nmonolayer nature even after annealing. Thus, the fabricated nanoscrolls were thermally \nstable. The E12g and A1g phonon modes were blue -shifted by 0.5 cm-1 and 1.4 cm-1, \nrespectively, compared to the as -fabricated scroll. T he observed blue -shift was attributed to \nthe relaxation of local bending strain which was developed during the nanoscroll formation 13 \n stage.22 It is also well known that the A1g mode is sensitive to carrier density, and the \nhardening of A1g was reported with  decreased carrier density.17 The blue -shift of A1g mode \nindicates the reduced carrier density after annealing. The increased PL intensity after \nannealing may be attributed to the reduction in carrier density, which decreases the Auger \nrecombination. The effect of annealing on S and LB modes is probed by L F Raman \nmeasurements ( Figure  6c). It was found that S and LB modes were absent in nanoscrolls after \nannealing. Thus, the minor signal of S mode observed in the as -fabricated scroll ( Figure  4) \ndisappeared after annealing. The absence of S and LB modes in an  annealed sample further \nconfirms the negligible effect of annealing on the non-interacting nature of the constituent \nlayers.  \nI-V measurements were carried out using conducting atomic force microscopy (C -\nAFM) under dark and illumination conditions to study  the photo -response of the annealed \nnanoscrolled 1L -MoS 2. The illumination was performed using the feedback laser (wavelength \n~ 650 nm) source. The detailed measurement steps are provided in the experimental section. \nThe inset of Figure  7a represents the s chematic of the C -AFM measurement set -up. I -V \ncharacteristics of the 1L -MoS 2 and nanoscrolled 1L -MoS 2 without illumination are provided \nin Figure  7a. The non -linear behaviour of I -V graphs represents the formation of the Schottky \njunction between the Pt -tip with MoS 2 layers. Pt and Au electrodes form Schottky and Ohmic \njunctions, respectively, with MoS 2 material .29 Further, I -V curves also indicate the p-type \nbehavior of both 1L -MoS 2 and nanoscrolled 1L -MoS 2. The origin of p-type behaviour  \nstemmed from the induced doping by the Pt tip used in C -AFM measurements. In general, the \ndark I -V characteristics are nearly the same for both 1L -MoS 2 and nanoscrolled 1L -MoS 2. \nThe result is obvious because the constituent layers in the nanoscrolls are n ot interacting with \neach other. Under illumination in the forward bias condition, the measured current increases \napproximately from  - 0.5 to - 0.8 nA and  - 0.5 to - 2.24 nA at -4.5 V in 1L -MoS 2 and 14 \n nanoscrolled 1L -MoS 2, respectively. Here, the scrolled 1 L-MoS 2 shows approximately six \ntimes higher photocurrent as compared to the 1L -MoS 2. In addition, under positive bias, the \nmeasured photocurrent in scrolled 1L -MoS 2 is significantly higher as compared to that of 1L -\nMoS 2. The higher net photocurrent in nano scrolled MoS 2 may have originated from its higher \nabsorption cross -section due to the availability of multiple monolayers. Each monolayer \ngenerated the photo -carriers and contributed to enhancement in the overall current. Along \nwith point I -V measurements,  we have also carried out a current map at a constant voltage of \n-3.5 V. Figure 7c and Figure  7d represent the current mapping of the nanoscrolls under dark \nand illumination, respectively. In the dark, the measured current is ~ 100 pA in the region of \n1L-MoS2 (lower right corner) and a relatively higher current over the scroll region (nA). \nThese current values slightly increase under illumination. To sum up, the photocurrent in the \nscrolled region is higher as compared to remnant 1L -MoS 2. The topography of a typical scroll \nis provided in the inset of Figure  7b. \nFET devices were fabricated on planar 1L -MoS 2 and nanoscrolled 1L -MoS 2 to \nevaluate the electrical properties. The detailed fabrication protocols were provided in the \nexperimental section. After fabric ating the multiple FETs, we measured both their output and \ntransfer characteristics. In the output characteristics, the gate voltage (V gs) was varied from 0 \nV to 20 V with an interval of 5 V ( Figure  8a and Figure  8b). In both planar and scrolled 1L -\nMoS 2 based FET devices, I ds were found to raise linearly up to V ds of ~4 V and thereafter \nsaturated. The observed I ds versus V ds results in both the devices are typical behaviour of FET \ndevices. At V gs of 20 V, the saturation drain currents are 3 and 45 µA/µm for  planar and \nnanoscrolled 1L -MoS 2, respectively. The one -order (15 times) higher drain current is \nattributed to the availability of multiple paths for transporting the electrons in constituent \nlayers in scrolled 1L -MoS 2 compared to 1L -MoS 2. For similar devi ce configurations, the \nsaturation drain currents reported in literature are 1.35 and 0.14 µA/µm for planar 1L -15 \n MoS 2.30,31 Moreover, we have also plotted the transfer characteristics of both planar and \nscrolled 1L -MoS 2 to extract the threshold voltage, electron mobility and on/off current ratio \n(Figure  8c and Figure  8d). In addition, we fabricated the multiple devices on planar and \nnanoscrolled 1L -MoS 2 and the transfer characteristics of these devices were provided in \nFigure  S8 and Figure  S9, respectively. Subsequently, the key device parameters are \nsummarized in the Table 1. The threshold voltage (V t) was calculated using Y -function \nmethod.32 The mean V t values are 5.7 and 5.2 V for planar and nanoscrolled 1L -MoS 2, \nrespectively. Earlier it was reported that V t was substantially decreased by increasing the \nnumber of MoS 2 layers.33 However, the minute decrement in V t for nanoscrolled 1L -MoS 2 is \nattributed to weak interlayer coupling that exists among the constituent layers  of the scroll. \nGenerally, the on -state performance of the FET is influenced by the carrier mobility. We \ncalculated the field -effect electron mobility ( ) from peak transconductance using equation \n(Eq. 1) 32 \n                                                           =𝐿\n𝑊.𝐶𝑜𝑥.𝑉𝑑𝑠(∆𝐼ds\n∆𝑉𝑔𝑠)                                               (1)        \nwhere Cox is the gate insulator capacitance per unit area (11.51 × 10-9 Fcm-2 for SiO 2 of 300 \nnm), L and W are the length and width of the channel, respectively. The dimensions ( LW) of \nthe planar and scrolled device are 10 m  10 m and 10 m  0.5 µm, respectively. The \nslope ∆I ds/∆V gs is calculated from the transfer characteristics curve in a linear regime at Vds of \n1 V. The field -effect electron mobility is calculated to be ~ 10.5 and ~2400( +400) cm2V–1s–1 \nfor planar and nanoscrolled 1L -MoS 2, respectively. The on/off current ratios were  106 and \n107 for planar and scrolled 1L -MoS 2, respectively. The above results indicate that \nnanoscrolled 1L -MoS 2 exhibits superior electronic properties compared to planar 1L -MoS 2. \nMoreover, a one -order higher current ratio makes the scrolled 1L -MoS 2 suitable for digital \napplications. The key device parameters of nanoscrolled 1L -MoS 2 FETs were compared with 16 \n the existing literature (Table S5). The two-order  higher on/off current ratio and one-order  \nhigher electron mobility in our nanoscrolled 1L -MoS 2 were observed compared to other \nreported nanoscrolled devices. As a matter of fact, the mobility value is the highest reported \nso far in any form of 1L -MoS 2, including that for top gated planar 1L -MoS 2, showing a \nmobility value of 1090 cm2V–1s–1 (Table S6).34 Thus, compact optoelectronic devices can be \nrealized using nanoscrolled 1L -MoS 2, which retains the monolayer characteristics. Our results \ncan be further extended to other 2D transition metal dichalcogenides monolayer structures to \nfabricate thermally st able scrolls.  \nTable 1: Summary of key device parameters of back -gated FET fabricated on planar and \nnanoscrolled 1L -MoS 2. \nChannel Material  Ion (max)  \n(µA/µm)  Threshold  \nVoltage (V)  Mobility  \n(cm-2V-1s-1) ON/OFF \ncurrent ratio  \n1L-MoS 2 film  3 5.7 ± 0.2  10.5 ± 2.5 106 \nNanoscrolled 1L -\nMoS 2 45 5.2 ± 0.2  2400 ± 400  107 \nAs discussed earlier, countless attempts are made to fabricate the scrolled MoS 2 with \nmultiple layers which retain the monolayer nature.8,9,11-13 Unfortunately, the fabricated scrolls \nbehaved as multilayer MoS 2 where PL emission either disappeared or was suppressed. \nMoreover, the fabricated nanoscr olls have poor thermal stability where PL emission was \nsuppressed after annealing.11 In the present study, nano -scrolls showed enhancement of PL \nintensity and negligible shift compared to the planar 1L -MoS 2. In other words, the result \nsignifies that nano -scrolls behave electronically in a similar manner to monolayers. The \nenhanced PL intensity is attributed to the non -interacting nature of constituent layers in the \nscroll. Interestingly, the monolayer behavior was further improved upon thermal annealing, \nwhich manifested their thermal stability. The properties of nanoscrolls are dependent on the \narrangement of layers inside the scroll at their formation stage. The choice of solvent is most 17 \n important as physical properties such as surface tension, evaporation  rate, and dielectric \nconstant strongly influence the scroll formation. The scrolls fabricated using the solvent with \nhigher surface tension and evaporation rate increase the rolling speed.  As a result, the \nstacking becomes imperfect and closely spaced la yers are expected to form. Therefore, the \ninterlayer interaction increases, and the scroll behaves as a multilayer. However, a solvent \nwith optimum surface tension and evaporation rate can provide sufficient time for the \nformation of a scroll. In addition,  the moderate dielectric constant of the solvent can further \nslowdown the formation of the scroll. In this context, the formation of a scroll using absolute \nethanol is advantageous because of its optimum evaporation rate and moderate dielectric \nconstant, w hich may slowdown the self -rolling process. As a result, one can expect a large \nseparation between the constituent layers. The non -interacting nature of constituent layers \nmight have originated from the large separation. Moreover, the constituent layers al so \nretained the optical properties after a few months of fabrication (not shown here). It suggests \nthat the scrolls have long -term stability in their optical properties. Therefore, our results find \nthat absolute ethanol is a suitable solvent for the fabric ation of nanoscrolls.  \n3. Conclusions   \nWe strategically developed and demonstrated an effective and facile method to \nfabricate the thermally stable nanoscrolled 1L -MoS 2 from CVD -grown 1L -MoS 2 film. A \nsuitable solvent with low surface tension and evaporation rate, ethanol, was used to fabricate \nthe nanoscrolled 1L -MoS 2. The nanoscrolled 1L -MoS 2 retained the monolayer behaviour \neven though multilayer in morphology, which was confirmed by Raman s pectroscopy. The \nnanoscrolled 1L -MoS 2 showed enhanced PL QY because of approximately two times \nincrease in the absorbance and five  times increase in the PL emission, compared to 1L -MoS 2. \nIn addition, low -frequency Raman measurements confirm the negligible interlayer coupling 18 \n among constituent layers in the insignificant presence of S and LB mode. Moreover, the \nKPFM study revealed insignificant variation in SP between  monolayers and scrolled 1L -\nMoS 2, showing similar electronic properties of them. Furthermore, the enhancement of PL \nintensity (approximately six times) after annealing confirmed the thermal stability of \nfabricated nanoscrolls. The PL QY (2.95%) of nanoscroll 1L -MoS 2 was significantly high er \nthan the reported the PL QY of untreated 1L -MoS 2. In addition, C -AFM measurements \ndelineate that the nanoscrolled 1L -MoS 2 shows approximately six times higher photocurrent \nas compared to the 1L -MoS 2. FET device fabricated on nanoscroll 1L -MoS 2 has shown  one \norder higher on/off ratio and two order higher mobility compared to the device fabricated on \nplanar 1L -MoS 2. Therefore, fabricated nanoscrolls can be used for compact and high -\nperformance optoelectronic devices.  \n4. Experimental section  \nThe synthesis o f 1L -MoS 2 flakes was carried out by the atmospheric pressure CVD \ntechnique. The MoS 2 phase was formed by using molybdenum trioxide (MoO 3) and S as \nprecursor materials. The details of the synthesis steps can be found elsewhere.[2] The \ndeposition of 1L -MoS 2 was carried out on SiO 2(300 nm)/Si susbstrate. The growth of planar \nand nanoscrolled 1L -MoS 2 was confirmed by optical microscopy. All morphological \ninformation, including the lateral dimension of the nanoscrolled 1L -MoS 2 was extracted from \nfield emission scanning electron microscopy (FESEM; Supra 55, Zeiss, Germany) images. \nThe thickness of 1L -MoS 2 and the height of the nanoscrolled 1L -MoS 2 were measured by the \natomic force microscopy (AFM) technique (MultiView 4000, Nano nics Imaging Ltd, Israel). \nThe AFM measurements were carried out in intermittent mode. Vibrational and emission \nproperties were studied usingRaman and PL spectroscopy, respectively. Raman and PL \nmeasurements were performed by using a 532 nm laser line with  a power of 0.01 mW. 19 \n Raman scattered light and PL emission light were dispersed through 2400 lines/mm and 1800 \nlines/mm grating, respectively. Subsequently, the dispersed light was collected by using a \ncharged coupled device detector. Low -frequency Raman m easurements were carried out by \nusing a WiTec Alpha 300RA micro -Raman spectrometer. Here, the sample was excited with \na 532 nm laser source, and scattered light was dispersed through 1800 lines/mm grating. A \nBragg grating (RayshieldTM) was employed to cut off the Rayleigh line.  \nCurrent mapping and I -V measurements were carried out using the conducting atomic \nforce microscopy (C -AFM; Model No -NTEGRA II, NT -MDT, Russia). The whole metal Pt \ntip cantilever was used in contact mode. A typical load applied on the  cantilever was ~ 50 nN \nduring electrical measurement. Gold wire with pressure contact served as one electrode, and \nthe AFM tip served as the second electrode. The sample was biased, and the AFM tip was \ngrounded during the measurement. The current mapping was performed under the two -pass \ntechnique.  In first -pass, the current was measured with the illumination of an in -built laser \n(650 nm) which was used for the feedback mechanism. The spill -over light from the feedback \nlaser generated significant photo -carriers, which increased the net current during mapping. In \nthe second pass, the feedback laser was off, and hence, the current mapping was performed in \nthe dark. Similarly, I -V characteristics were also measured with and without the illumination \nof light fr om the feedback laser.  \nPL QY measurements  \nThe PL QY of 1L -MoS 2 and nanoscrolled 1 -MoS 2 was estimated using Rhodamine \n6G (R6G) dye. The R6G shows 95% PL QY under 532 nm excitation. The steps involved in \nthe estimation of PL QY can be found elsewhere.[4] \nNanoscrolled 1L -MoS 2 based FET devices  20 \n Both planar and nanoscrolled 1L -MoS 2-based back -gated FETs were fabricated with \nthe same channel length of 10 m. The FET fabrication protocols can be found elsewhere .[28] \nThe electrodes (Au (200 nm)/Ti (10 0 nm)) were prepared via the lift -off method by direct \nlaser writer (M/s Microtech, Italy, Model No: LW405B) and wet chemical methods. The \nelectrodes were deposited by dual -target DC magnetron sputtering. The FET cha racteristics \nwere measured by source measurement unit (Keysight, Model -B2902A).  \nAcknowledgment  \nWe thank Dr. Suman Paul, attocube  Systems AG, Eglfinger Weg 2, 85540 Haar, for nano \nFTIR measurements. We also thank the Director, MSG , and the Director, IGCAR, for their \nconstant encouragement.  \nConflict of Interest  \nThe authors declare no conflict of interest . \nReferences  \n[1] Mak, K. 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ACS Nano  \n2012 , 6, 10070 . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 23 \n Figures  \nFigure 1  \n \nFigure  1. Step-by-step monitoring the fabrication of nanoscrolled 1L -MoS 2. (a) The surface of 1L -\nMoS 2 under the drop of absolute ethanol.  (b) Shrinkage of ethanol drops during its evaporation \n(boundary marked with red dotted curve and arrow marks indicate the direction of shrinkage). (c) \nMultiple scrolls formed after ethanol evaporation (pointed by arrow  marks). The given scroll selected \nfor further studies was highlighted with a dotted box. Scale bar: 20 μm (d) High magnification image \nof one of the nanoscrolls. Scale bar: 10 μm.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n24 \n Figure 2  \n \nFigure  2. a) Schematic representing a cross -sectional view of 1L -MoS 2 nanoscroll. The optical image \n(b) and its respective FESEM image (c) of the particular scroll. Scale bar: 5 μm. d) The topographic \nimage of the same nanoscroll. Inset shows the height profile acr oss the scroll. The line profile was \ntaken along the green line shown (d).  \n25 \n Figure 3  \n \nFigure  3. Comparative optical characterization of planar and nanoscrolled 1L -MoS 2. (a) Raman and \n(b) PL spectra of planer and scrolled MoS 2. The ∆ value in both planar and nanoscrolled MoS 2 \nconfirms the monolayer nature of MoS 2. Dotted lines are guide to the eye. Insets of (b) show the \nintensity map at a peak energy of A -exciton (1.84 eV). Scale bar (inset figure): 2 μm.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n26 \n Figure 4  \n \nFigure  4. Low-frequency Raman spectra of 1L -, 2L-MoS 2 and nanoscrolled 1L -MoS 2.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n27 \n Figure 5  \n \nFigure  5. Surface potential analysis of 1L -, 2L- and nanoscrolled 1L -MoS 2 using KPFM.  a) KPFM \nimage showing the spatial variation of SP in a flake containing,  mono -, bi- and multi -layered MoS 2. b) \nLine profile across flake. The line profile was taken along the green line (a). c) SP map of \nnanoscrolled 1L -MoS 2 region. d) Line profile of SP provided was taken along the green line in (c).    \n \n \n \n \n \n \n \n \n \n \n28 \n  \nFigure 6  \n \nFigure  6. Raman and PL analysis of vacuum annealed nanoscrolled 1L -MoS 2. Photoluminescence (a), \nRaman (b) and LF Raman (c) spectra of vacuum annealed nanoscrolled 1L -MoS 2. LF Raman \nmeasurements were carried out at three arbitrary positions (P1, P2, and P3)  on nanoscrolled 1L -MoS 2. \n \n \n \n \n \n \n \n \n \n \n29 \n  \n \nFigure 7  \n \nFigure  7. Conductive AFM -based I -V characteristics of 1L -MoS 2 and scrolled 1L -MoS 2 without (a) \nand (b) with spill -over light from feedback laser. Inset of (a) represents the schematic of C -AFM \nmeasurement. T he Inset of (b) corresponds to the topography of the scroll. The C -AFM image of \nscrolled 1L -MoS 2 region under (c) dark and (d) illumination. Scale bar: 1 µm.  \n \n \n \n \n \n \n \n \n30 \n  \n \n \nFigure 8  \n \nFigure  8. Transfer and output characteristics of FET devices fabricated on planar and nanoscrolled \n1L-MoS 2. Output characteristics of (a) planar 1L -MoS 2 FET and (b) nanoscrolled 1L -MoS 2 FET. \nTransfer characteristics of (c) planar 1L -MoS 2 FET and (d) nanoscrolled 1L -MoS 2 FET. The insets of \n(c) and (d) show the FESEM image of typic al FET devices fabricated on planar and scrolled 1L -\nMoS 2, respectively. Scale bar: 5 µm.  \n \n \n \n"    },    {        "title": "2402.13565v1.On_the_application_of_subspace_migration_from_scattering_matrix_with_constant_valued_diagonal_elements_in_microwave_imaging.pdf",        "content": "On the application of subspace migration from scattering\nmatrix with constant-valued diagonal elements in microwave\nimaging\nWon-Kwang Park\nDepartment of Information Security, Cryptology, and Mathematics, Kookmin\nUniversity, Korea.\ne-mail: parkwk@kookmin.ac.kr\nAbstract\nWe consider the application of a subspace migration (SM) algorithm to quickly identify\nsmall objects in microwave imaging. In various problems, it is easy to measure the diagonal\nelements of the scattering matrix if the location of the transmitter and the receiver is the same.\nTo address this issue, several studies have been conducted by setting the diagonal elements\nto zero. In this paper, we generalize the imaging problem by setting diagonal elements of\nthe scattering matrix as a constant with the application of SM. To show the applicability\nof SM and its dependence on the constant, we show that the imaging function of SM can\nbe represented in terms of an infinite series of the Bessel functions of integer order, antenna\nnumber and arrangement, and applied constant. This result enables us to discover some further\nproperties, including the unique determination of objects. We also demonstrated simulation\nresults with synthetic data to support the theoretical result.\n1 Introduction\nThere are considerable and intriguing inverse problems related to retrieving inhomogeneities em-\nbedded in a medium under multistatic measurement circumstances. This is an old but interesting\nproblem due to its diverse applications, which significantly impact human life. For instance, it\ncan be used in biomedical imaging [2, 9], thermal therapy monitoring [20, 22], landmine detection\n[13, 18], damage detection [23, 40], and radar imaging [10, 45]. To this end, several researchers\nhave explored and successfully applied various iterative (quantitative) and non-iterative (qualita-\ntive) reconstruction algorithms to address the different inverse scattering and microwave imaging\nproblems.\nExamples of iterative schemes include the Gauss-Newton method for biomedical imaging [38],\nBorn iterative method for brain stroke detection [21] and compressive sensing imaging [28], Levenverg-\nMarquardt method for reconstructing parameter distribution [17], and level-set method for shape\nreconstruction [16]. For a successful application of iterative-based algorithms, the iteration pro-\ncedure must start with a good initial guess, which is close to the true solution. If not, it will be\nchallenging to obtain good results due to the non-convergence issue, local minimizer problem, and\nrequirement of high computational costs (see [24, 37] for instance).\nAlthough complete information on parameter distribution or unknown targets cannot be re-\ntrieved through the non-iterative scheme, an outline shape of the target can be obtained quickly\nand adopted as a good initial guess. For example, bistatic method for object detection in microwave\n1arXiv:2402.13565v1  [math.NA]  21 Feb 2024imaging [43] and damage detection of civil structure [23], MUltiple SIgnal Classification (MUSIC)\nalgorithm for detecting internal corrosion [5] and object detection in microwave imaging [32], direct\nsampling method for diffusive optical tomography [11] and electrical impedance tomography [12],\ntopological derivative strategy for ultrasonic non-destructive testing [6] and detection of impedance\nobstacles [27], orthogonality sampling method for object detection in microwave imaging [35] and\n3D objects [26], factorization method for breast cancer imaging [15] and electrical impedance to-\nmography [19], and linear sampling method for underwater acoustic imaging [1] and microwave\nimaging [7].\nIn this paper, we explore an application of the SM for the rapid identification of a two-\ndimensional small object with different dielectric permittivity and electric conductivity values\ncompared to the background homogeneous medium. SM is a recently developed and promising\nnon-iterative imaging technique in inverse scattering problems and microwave imaging. It stands\nout for its speed, effectiveness in recognizing target locations and shapes, and robustness against\nrandom noise. As a result, SM has been successfully applied to various problems, including local-\nization of small and extended targets [3, 8, 36], crack imaging [4, 29, 34], and object detection in\nmicrowave imaging [30, 31, 33].\nIn designing an imaging function of SM, it is essential to consider the complete elements of\nthe so-called multistatic response matrix in inverse scattering problems and scattering matrix in\nmicrowave imaging. However, in some microwave imaging applications, collecting the diagonal\nelements of the scattering matrix becomes challenging due to the difficulty in measuring scattering\nparameter data (elements of the scattering matrix) or discerning weak scattered signal from the\nrelatively high antenna reflections when the transmitting and receiving antennas coincide. For a\ndetailed discussion on this topic, refer to [22, 33, 41, 42, 44]. Therefore, applying SM directly\nwithout unknown diagonal elements of the scattering matrix is highly challenging. SM has been\napplied by setting the unknown diagonal elements to zero. Numerical simulations demonstrate that\nsetting the diagonal elements to a nonzero constant yields a poor result. As a result, most studies\non microwave imaging problems have adopted the approach of setting the diagonal elements to\nzero. However, the theoretical rationale behind the effect of converting these elements to a constant\nvalue remains unexplored.\nIn this study, we consider the application of SM for the fast identification of small objects from\na scattering matrix whose elements are scattering parameter data. By converting the unknown\ndiagonal elements of the scattering matrix into a fixed constant, we introduce the imaging function\nof SM and show that it can be represented by an infinite series of the Bessel function of integer\norder, total number of antennas and their arrangement, applied frequency, material properties of\nthe background, and applied constant. Based on the explored structure of the imaging function,\nwe confirm that the imaging performance significantly depends on the applied constant and the\nobject can be identified uniquely if the absolute value of such constant is zero or close to zero. We\npresent simulation results from synthetic data to demonstrate the theoretical result.\nThe remainder of the paper is organized as follows. In Section 2, we briefly present the concept\nof scattering parameter in the presence of small object and introduce the imaging function of SM\nfrom a scattering matrix whose diagonal elements are fixed constant. In Section 3, we construct a\nmathematical theory for the imaging function, discuss various properties of the imaging function,\nincluding ideal and practical conditions for obtaining good results, and explain the unique deter-\nmination of object. Section 4 presents a set of numerical simulation results for single and multiple\nobjects to support the theoretical result. Finally, a short conclusion is presented in Section 5.\n2 Scattering Parameters and Imaging Function of the SM\nThis section briefly introduces the scattering parameters and the imaging function of SM for\nidentifying small object Dwith smooth boundary ∂D. Throughout this paper, we denote Ω\n2as a homogeneous region filled by a matching liquid and Λ nas a dipole antenna located at an\nwith|an|=R,n= 1,2,···, N. We assume that Dand Ω are characterized by their dielectric\npermittivity and electrical conductivity at a given angular frequency ω= 2πf, where fdenotes\nthe ordinary frequency measured in hertz , and time-harmonic dependence e−iωt. Thus, the value\nof the magnetic permeability is set to constant µ(z)≡µb= 4π×10−7H/m for every z∈Ω. The\nvalues of permittivity of Ω and Dare denoted by εbandεa, respectively. Analogously, σbandσa\ndenote the values of the conductivity of Ω and D, respectively. With this, we denote k0andkbas\nthe lossless background and background wavenumbers that satisfy\nk2\n0=ω2εbµband k2\nb=ω2µb\u0010\nεb−iσb\nω\u0011\n,\nrespectively.\nLetS(n, m) be the scattering parameter, which is defined as the following ratio\nS(n, m) =V(n)\nout\nV(m)\nin,\nwhere V(m)\ninandV(n)\noutdenote the input voltages (or incident waves) at Λ mand the output voltages\n(or reflected waves) at Λ n, respectively. We denote Smeas(n, m) as the measurement data obtained\nby subtracting scattering parameters with and without D. Then, based on [20], Smeas(n, m) is\ngiven by\nSmeas(n, m) =−ik2\n0\n4ωµbZ\nD\u0012εa−εb\nεb+iσa−σb\nωεb\u0013\nEinc(am,x)·Etot(x,an)dx, (1)\nwhere Einc(am,x) is the incident field due to the point current density at Λ mthat satisfies\n∇ ×Einc(am,x) =iωµbH(am,x) and ∇ ×H(am,x) = (σb−iωεb)Einc(am,x),\nandEtot(x,an) be the total field owing to the presence of the Dmeasured at Λ nthat satisfies\n∇ ×Etot(x,an) =iωµbH(x,an) and ∇ ×H(x,an) = (σ(x)−iωε(x))Etot(x,an),\nwith transmission conditions at the boundary ∂D. Here, Hdenotes the magnetic field.\nNow, assume that the following relations hold\nωεb≫σband√εadiam( D)<√εbλ,\nwhere diam( D) and λdenote the diameter of Dand wavelength, respectively. Then, based on\n[39], it is possible to apply the first-order Born approximation Etot(x,an)≈Einc(x,an) to (1) and\ncorrespondingly, we can observe that\nSmeas(n, m)≈ −ik2\n0\n4ωµbZ\nD\u0012εa−εb\nεb+iσa−σb\nωεb\u0013\nEinc(am,x)·Einc(x,an)dx. (2)\nLet us emphasize that based on the configuration of the microwave machine, only the z−component\nof the incident and total fields can be handled because the antennas are arranged perpendicular\nto the z−axis, refer to [41, 42]. Then, by letting z−component of incident field Eincas E(z)\nincand\nthe reciprocity property of the Einc, (2) can be written by\nSmeas(n, m)≈ −ik2\n0\n4ωµbZ\nDODE(z)\ninc(am,x)E(z)\ninc(an,x)dx, (3)\n3where\nOD=εa−εb\nεb+iσa−σb\nωεband E(z)\ninc(an,x) =−i\n4H(1)\n0(kb|an−x|). (4)\nHere, H(1)\n0denotes the Hankel function of order zero of the first kind.\nNext, we briefly introduce the imaging function of SM. Note that to introduce the imaging\nfunction, every element Smeas(n, m) of the scattering matrix can be measured, i.e., the following\nscattering matrix must be available\nK=\nSmeas(1,1)Smeas(1,2)··· S meas(1, N−1)Smeas(1, N)\nSmeas(2,1)Smeas(2,2)··· S meas(1, N−1)Smeas(2, N)\n...............\nSmeas(N,1)Smeas(N,2)··· S meas(N, N−1)Smeas(N, N )\n.\nHowever, under a certain simulation configuration, it is impossible to measure Smeas(n, m) when\nn=mbecause when an antenna Λ mis used for signal transmission, the remaining N−1 antennas\nΛn,n= 1,2,···, Nwith n̸=mare used for signal reception. For more discussion on this, refer\nto [41]. Thus, in practice, we can use the following scattering matrix\nK=\nunknown Smeas(1,2)··· S meas(1, N−1)Smeas(1, N)\nSmeas(2,1) unknown ··· S meas(1, N−1)Smeas(2, N)\n...............\nSmeas(N,1)Smeas(N,2)··· S meas(N, N−1) unknown\n.\nand correspondingly, traditional SM cannot be applied directly. Instead of using K, we consider\nthe application of the following scattering matrix. For a constant C∈C, let\nK(C) =\nC Smeas(1,2)··· S meas(1, N−1)Smeas(1, N)\nSmeas(2,1) C ··· S meas(1, N−1)Smeas(2, N)\n...............\nSmeas(N,1)Smeas(N,2)··· S meas(N, N−1) C\n.\nThen, for each search point z∈Ω, based on the (3) and K(C), define a unit test vector\nW(z) =1\n NX\nn=1|E(z)\ninc(an,z)|2!1/2\nE(z)\ninc(a1,z)\nE(z)\ninc(a2,z)\n...\nE(z)\ninc(aN,z)\n(5)\nand by performing SVD of K(C),\nK(C) =NX\nn=1τnUnV∗\nn≈τ1U1V∗\n1,\nthe imaging function can be introduced as\nF(z, C) =|⟨W(z),U1⟩⟨W(z),V1⟩|,where ⟨W(z),U1⟩=W(z)∗U1. (6)\nThen, for some specific selection of C, it is possible to recognize x∈Σ through the map of F(z, C).\n43 Analysis of the Imaging Function\nIn this section, we explore the mathematical structure of the imaging function for a proper selection\nof the constant Cto guarantee a good result. The main result is as follows.\nTheorem 3.1. Letθn=an/R= (cos θn,sinθn),z=|z|(cosφ,sinφ), and z−x=|z−\nx|(cosϕ,sinϕ). If|k(z−an)| ≫0.25forz∈Ω, then F(z, C)can be represented as follows:\nF(z, C) =|Φ1(z) + Φ 2(z) + Φ 3(z, C)|, (7)\nwhere\nΦ1(z) =Nωε be2iRkbOD\n32Rkbτ1πZ\nD\u0012\nJ0(kb|z−x|) +1\nNNX\nn=1X\ns∈Z∗\n0isJs(kb|z−x|)eis(θn−ϕ)\u00132\ndx\nΦ2(z) =−ωεbe2iRkbOD\n32Rkbτ1πZ\nD\u0012\nJ0(2kb|z−x|) +1\nNNX\nn=1X\ns∈Z∗\n0isJs(2kb|z−x|)eis(θn−ϕ)\u0013\ndx\nΦ3(z, C) =C\nτ1\u0012\nJ0(2kb|z|) +1\nNNX\nn=1X\ns∈Z∗\n0isJs(2kb|z|)eisφ\u0013\n.\nHere, Z∗\n0=Z∪ {−∞ ,+∞}\\{ 0}andZdenotes the set of integer number and Jsis the Bessel\nfunction of order sof the first kind.\nProof. Since z∈Ω and |k(z−an)| ≫ 0.25 for all n, the Hankel function can be given as the\nfollowing asymptotic form (see [14, Theorem 2.5] for instance)\nH(1)\n0(kb|z−an|)≈(1 +i)eikb|an|\np\nkbπ|an|e−ikbθn·z.\nBy using this representation, the test vector of (5) and the measurement data (3) can be written\nas\nW(z) =1√\nN\u0012\ne−ikbθ1·z, e−ikbθ2·z, . . . , e−ikbθN·z\u0013T\nand\nSmeas(n, m)≈ −k2\n0e2i|an|kbOD\n32|an|kbωµbπZ\nDe−ikb(θm+θn)·xdx:=AZ\nDe−ikb(θm+θn)·xdx,\nrespectively. Since\nU1V∗\n1=1\nτ1K(C) =1\nτ1\nC Smeas(1,2)··· S meas(1, N−1)Smeas(1, N)\nSmeas(2,1) C ··· S meas(1, N−1)Smeas(2, N)\n...............\nSmeas(N,1)Smeas(N,2)··· S meas(N, N−1) C\n,\n5we can examine that\nW(z)∗U1V∗\n1=1\nτ1√\nN\neikbθ1·z\neikbθ2·z\n...\neikbθN·z\nT\nC Smeas(1,2)··· S meas(1, N−1)Smeas(1, N)\nSmeas(2,1) C ··· S meas(1, N−1)Smeas(2, N)\n...............\nSmeas(N,1)Smeas(N,2)··· S meas(N, N−1) C\n\n=1\nτ1√\nN\nCeikbθ1·z+AZ\nDe−ikbθ1·xX\nn∈N1eikbθn·(z−x)dx\nCeikbθ2·z+AZ\nDe−ikbθ2·xX\nn∈N2eikbθn·(z−x)dx\n...\nCeikbθN·z+AZ\nDe−ikbθN·xX\nn∈NNeikbθN·(z−x)dx\nT\n,\nwhere Nn={1,2,···, N}\\{n}.\nSince the following relation holds uniformly (see [29] for derivation)\nNX\nn=1eikbθn·(z−x)=NX\nn=1eikb|z−x|cos(θn−ϕ)=NX\nn=1\u0012\nJ0(kb|z−x|) +X\ns∈Z∗\n0isJs(kb|z−x|)eis(θn−ϕ)\u0013\n=NJ0(kb|z−x|) +NX\nn=1X\ns∈Z∗\n0isJs(kb|z−x|)eis(θn−ϕ):=J(x,z),\n(8)\nwe can obtain\nW(z)∗U1V∗\n1=1\nτ1√\nN\nCeikbθ1·z+AZ\nDe−ikbθ1·xNX\nn=1eikbθn·(z−x)dx−AZ\nDe−ikbθ1·xeikbθ1·(z−x)dx\nCeikbθ2·z+AZ\nDe−ikbθ2·xNX\nn=1eikbθn·(z−x)dx−AZ\nDe−ikbθ2·xeikbθ2·(z−x)dx\n...\nCeikbθN·z+AZ\nDe−ikbθN·xNX\nn=1eikbθn·(z−x)dx−AZ\nDe−ikbθN·xeikbθN·(z−x)dx\nT\n=1\nτ1√\nN\nCeikbθ1·z+AZ\nDe−ikbθ1·xJ(x,z)dx−AZ\nDe−ikbθ1·xeikbθ1·(z−x)dx\nCeikbθ2·z+AZ\nDe−ikbθ2·xJ(x,z)dx−AZ\nDe−ikbθ2·xeikbθ2·(z−x)dx\n...\nCeikbθN·z+AZ\nDe−ikbθN·xJ(x,z)dx−AZ\nDe−ikbθN·xeikbθN·(z−x)dx\nT\n6and correspondingly,\nW(z)∗U1V∗\n1W(z)\n=1\nτ1N\nCeikbθ1·z+AZ\nDe−ikbθ1·xJ(x,z)dx−AZ\nDe−ikbθ1·xeikbθ1·(z−x)dx\nCeikbθ2·z+AZ\nDe−ikbθ2·xJ(x,z)dx−AZ\nDe−ikbθ2·xeikbθ2·(z−x)dx\n...\nCeikbθN·z+AZ\nDe−ikbθ1·xJ(x,z)dx−AZ\nDe−ikbθN·xeikbθN·(z−x)dx\nT\n\neikbθ1·z\neikbθ2·z\n...\neikbθN·z\n\n=1\nτ1NNX\nn=1\u0012\nCe2ikbθn·z+AZ\nDeikbθn·(z−x)J(x,z)dx−AZ\nDe2ikbθn·(z−x)dx\u0013\n.\nNow, by applying (8) again, we obtain\n⟨W(z),U1⟩⟨W(z),V1⟩=W(z)∗U1V∗\n1W(z)\n=C\nτ1NNX\nn=1e2ikbθn·z+A\nτ1NZ\nDJ(x,z)NX\nn=1eikbθn·(z−x)dx−A\nτ1NZ\nDNX\nn=1e2ikbθN·(z−x)dx\n=A\nτ1NZ\nD\u0012\nNJ0(kb|z−x|) +NX\nn=1X\ns∈Z∗\n0isJs(kb|z−x|)eis(θn−ϕ)\u00132\ndx\n−A\nτ1NZ\nD\u0012\nNJ0(2kb|z−x|) +NX\nn=1X\ns∈Z∗\n0isJs(2kb|z−x|)eis(θn−ϕ)\u0013\ndx\n+C\nτ1N\u0012\nNJ0(2kb|z|) +NX\nn=1X\ns∈Z∗\n0isJs(2kb|z|)eisφ\u0013\n.\nWith this, we obtain the structure (7).\nBased on Theorem 3.1, we can explain some properties of the imaging function and phenomena\nthat can occur in simulation results.\nRemark 3.1 (Detectability of the object) .Since Φ 1(z) and Φ( z) contain the factors J0(kb|z−x|)\nandJ0(2kb|z−x|), respectively, the map of F(z, C) will contain peak of large magnitude when\nz=x∈Dso that it will be possible to recognize the location and shape of D.\nRemark 3.2 (Impact of the constant Cand its best selection) .Since Φ 3(z, C) contains no informa-\ntion of x∈D, this factor contributes to generating several artifacts, and correspondingly, it will\ndisturb the detection of D. Therefore, the application of nonzero constant Cwill make it difficult\nto retrieve objects via the map of F(z, C), so the selection C= 0 will guarantee good imaging\nresults. This is the theoretical reason why the diagonal elements of the scattering matrix are set\nto zero in most studies. In this case, since J0(0) = 1 and F(z, C)≈1 when z∈D,\nF(z,0)≈\f\f\f\fNωε be2iRkbOD\n32Rkbτ1πarea( D)−ωεbe2iRkbOD\n32Rkbτ1πarea( D)\f\f\f\f≈1.\n7Correspondingly, F(z,0) can be represented as\nF(z,0)≈N\n(N−1) area( D)Z\nD\u0012\nJ0(kb|z−x|) +1\nNNX\nn=1X\ns∈Z∗\n0isJs(kb|z−x|)eis(θn−ϕ)\u00132\ndx\n−1\n(N−1) area( D)Z\nD\u0012\nJ0(2kb|z−x|) +1\nNNX\nn=1X\ns∈Z∗\n0isJs(2kb|z−x|)eis(θn−ϕ)\u0013\ndx.\nThis is the same result derived in [31]. Hence, we can say that Theorem 3.1 can be regarded as a\ngeneralized version of the previous study.\nRemark 3.3 (Ideal conditions to obtain good results) .Based on Remarks 3.1 and 3.2, we can obtain\ngood results when Φ 3(z, C) = 0, i.e., C= 0,\n1\nNNX\nn=1X\ns∈Z∗\n0isJs(kb|z−x|)eis(θn−ϕ)= 0,and1\nNNX\nn=1X\ns∈Z∗\n0isJs(2kb|z−x|)eis(θn−ϕ)= 0.(9)\nThe easiest and simplest method is to select N−→+∞; however, this approach is impossible\nconsidering the simulation circumstance. Since the following asymptotic form holds for large z\nJs(z) =r\n2\nπzcos\u0010\nz−sπ\n2−π\n4\u0011\nimplies Js(kb|z−x|)∝s\n2\nkbπ|z−x|,\nit will be possible to make Js(kb|z−x|), Js(2kb|z−x|)−→+∞when|kb| −→ +∞or equivalently\nf−→+∞. However, applying infinite-valued frequency is impossible in practice. Based on these\nobservations, we can confirm that if the total number Nis small or the applied frequency fis low,\nit will not be easy to identify the object.\nRemark 3.4 (Practical condition to obtain good results) .Here, we consider finding a condition\nthat satisfies the following relation: for x̸= 0 and sufficiently large L,\nE(x, L) =NX\nn=1LX\ns=−L,s̸=0isJs(x)eis(θn−ϕ)≈0.\nNote that if N= 16 and all antennas are uniformly distributed on a circle of radius Rsuch as the\nsimulation configuration in Figure 1 of Section 4, then E(x, L)≈0 for small L, refer to [33]. If L\nis sufficiently large, then since (see [25] for instance)\n|JL(x)| ≤b\n3√\nL, b = 0.674885 . . . implies E(x, L)≈0.\nTable 1 presents the values of |E(x, L)|with various xandL. This is the theoretical reason why\nuniformly distributed antenna arrangement was used in most studies.\nBased on Remarks 3.1 and 3.4, we can obtain the following unique identification result.\nCorollary 3.1 (Unique identification of object) .Assume that Nis an even number greater than\n8and all antennas Λn,n= 1,2,···, N, are uniformly distributed on a circle of radius R. Then,\nfor sufficiently high frequency fand sufficiently small constant |C|, the object Dcan be retrieved\nuniquely through the maps of F(z,0)andF(z, C).\n8L= 1 L= 3 L= 5 L= 10 L≥15\nx= 0.11.5266×10−161.3877×10−161.8049×10−162.2053×10−162.2053×10−16\nx= 0.33.3307×10−163.1772×10−164.8908×10−163.1406×10−163.0425×10−16\nx= 0.55.5511×10−174.6230×10−167.4906×10−167.9050×10−166.6733×10−16\nx= 0.73.3307×10−169.8275×10−165.5563×10−169.3646×10−167.0292×10−16\nx= 1.05.5511×10−162.1957×10−164.6286×10−161.5563×10−161.6795×10−16\nTable 1: Values of |E(x, L)|for various xandL.\n4 Simulation Results with Synthetic and Experimental Data\nand Discussions\n4.1 Simulation Results with Synthetic Data\nIn this section, we present simulation results to demonstrate the theoretical result. To perform the\nsimulation, we set the location of dipole antennas for n= 1,2,···, N(= 16),\nan= 0.09 m (cos θn,sinθn), θ n=3π\n2−2π(n−1)\nN.\nThe ROI Ω was selected as the interior of a square region ( −0.08 m,0.08 m)2with material\nproperties ( εb, σb) = (20 ε0,0.2 S/m) at f= 1 GHz. Here, ε0= 8.854×10−12F/m denotes\nthe vacuum permittivity. We selected two balls D1andD2as objects with the same radii\nα= 0.01 m, centers x1= (0.01 m,0.03 m) and x2= (−0.04 m,−0.02 m), and material proper-\nties ( ε1, σ1) = (55 ε0,1.2 S/m) and ( ε2, σ2) = (45 ε0,1.0 S/m). Figure 1 illustrates the simulation\nsettings without and with objects.\nΩ(εb, σb) = (20 ε0,0.2 S/m)\n��D1\n(ε1, σ1) = (55 ε0,1.2 S/m)\nΩD1\n(ε1, σ1) = (55 ε0,1.2 S/m)\nD2\n(ε2, σ2) = (45 ε0,1.0 S/m)\nFigure 1: Illustration of the background (left), single (center) and multiple (right) small objects.\nExample 4.1 (Imaging of single object) .In this example, we consider the imaging of a single object\nD1. Figure 2 shows the distribution of singular values of K(C) with various C. We observe that\nit is very easy to discriminate nonzero singular value τ1when C= 0, 0 .01, 0 .001i, and 0 .01i.\nHowever, if the value of |C|is not close to zero, it is not easy to discriminate nonzero and close to\nzero singular values, e.g., C= 0.1 and C= 0.1i. Fortunately, by regarding differences τn−τn+1,\nn= 1,2,···,15, we selected τ1as the nonzero singular value of K(C) and defined the imaging\nfunction F(z, C) of (6).\n9Figure 3 shows the maps of F(z, C) with various C. Note that if C= 0 or |C|is small\n(|C| ≤ 0.01), the location and shape of D1can be identified accurately through the map of\nF(z, C). Meanwhile, it is not easy to distinguish D1and an artifact with large magnitude at\n(−0.01 m,−0.03 m) if C= 0.1. Although some artifacts are included in the imaging result, it is\npossible to identify D1because the magnitudes of artifacts are small when C= 0.1i. Thus, we can\nconclude that the imaging result of F(z, C) significantly depends on the choice of C. Additionally,\ngood results can be obtained when C= 0 or |C|is sufficiently small ( |C| ≤0.01 in this example)\nfor identifying single object.\n(a)C= 0\n (b)C= 0.01\n (c)C= 0.1\n(d)C= 0.001i\n (e)C= 0.01i\n (f)C= 0.1i\nFigure 2: (Example 4.1) Distribution of the singular values of K(C) atf= 1 GHz with various C.\nExample 4.2 (Imaging of multiple objects) .Now, we use the imaging function F(z, C) to identify\nmultiple small objects D1andD2. In this case, since SVD of K(C) is\nK(C) =NX\nn=1τnUnV∗\nn≈2X\nn=1τnUnV∗\nn,\nthe imaging function of (6) becomes\nF(z, C) =\f\f\f\f\f2X\nn=1⟨W(z),Un⟩⟨W(z),Vn⟩\f\f\f\f\f.\nFigure 4 shows the distribution of singular values of K(C) with various C. Here, it is possible\nto discriminate nonzero singular values τ1andτ2when C= 0 and C= 0.001i. However, if the\nvalue of |C|is not close to zero, it is difficult to discriminate two nonzero singular values. To define\nthe imaging function, we selected τ1as the nonzero singular value of K(C) ifC= 0.01,C= 0.01i,\nand 0 .1i. For C= 0.1, we selected the first six τnas nonzero singular values.\n10(a)C= 0\n (b)C= 0.01\n (c)C= 0.1\n(d)C= 0.001i\n (e)C= 0.01i\n (f)C= 0.1i\nFigure 3: (Example 4.1) Maps of F(z, C) atf= 1 GHz with various C. Black-colored dashed line\ndescribes the ∂D1.\nFigure 5 shows the maps of F(z, C) with various C. Similar to the single object imaging, the\nlocations and shapes of D1andD2can be identified accurately through the map of F(z, C) if\nC= 0, 0001 i, and 0 .01i. It is possible to recognize D1andD2, but several artifacts with small\nmagnitudes are included in the imaging result if C= 0.01iandC= 0.1i. Unfortunately, it is\ndifficult to distinguish D1∪D2and several artifacts with large magnitudes if C= 0.1. Therefore,\nsimilar to the single object imaging, the imaging result of F(z, C) significantly depends on the\nchoice of C. Additionally, good results can be obtained when C= 0 or |C|is sufficiently small for\nidentifying multiple objects.\nExample 4.3 (Imaging of large object) .Although the equation (2) holds for small object, we now\napply the F(z, C) for retrieving a large object. Figures 6 and (7) exhibit the distribution of singular\nvalues of K(C) and corresponding imaging results of K(C) in the presence of a circular object with\nradius α= 0.048 m and material properties ( ε, σ) = (15 ε0,0.5 S/m). Similarly with the previous\nresults in Examples 4.1 and 4.2, it is possible to recognize the outline shape of object if C= 0 and\nC= 0.001i, i.e., if the value of |C|is very small. However, it is impossible to identify the object if\n|C|is not small such that |C| ≥0.01.\n4.2 Simulation Results with Experimental Data\nWe now apply the imaging function of (6) for identifying small objects and demonstrating the effect\nof converted constant C. To this end, the scattering matrix was constructed by using the microwave\nmachine, which was filled by water, manufactured by the research group of the Electronics and\nTelecommunications Research Institute (ETRI), refer to [22]. The ROI Ω was selected as the\n11(a)C= 0\n (b)C= 0.01\n (c)C= 0.1\n(d)C= 0.001i\n (e)C= 0.01i\n (f)C= 0.1i\nFigure 4: (Example 4.2) Distribution of the singular values of K(C) atf= 1 GHz with various C.\ninterior of a circular region centered at the origin with radius 0 .085 m and material properties\n(εb, σb) = (78 ε0,0.2 S/m) at f= 925 MHz. For an illustration of the microwave machine and\nselected objects (cross-section of screw drivers and plastic straw), we refer to Figure 8.\nExample 4.4 (Imaging of small objects with same material properties) .Here, we consider the\nimaging of cross-section of screw drivers Ds,s= 1,2,3. Figure 9 shows the distribution of singular\nvalues of K(C) with various C. Similarly with the synthetic data experiment (Example 4.1), it is\neasy to select first three singular values to define the imaging function when C= 0, 0 .01, 0.001i,\nand 0 .01i. However, it is not easy to select nonzero singular values when C= 0.1 and C= 0.1i.\nFigure 10 shows the maps of F(z, C) with various C. Similar to the results in Example 4.1,\nthe location and outline shape of objects can be identified very accurately through the map of\nF(z, C) but some artifacts are included in the map when |C|= 0.01. Unfortunately, it is very\ndifficult to identify some objects when C= 0.1 and distinguish the objects and artifacts with large\nmagnitude when C= 0.1i. Thus, same as the synthetic data experiments, the imaging result of\nF(z, C) significantly depends on the choice of Cand good results can be obtained when |C|is\nsufficiently small for identifying small objects.\nExample 4.5 (Imaging of small objects with different material properties) .Now, we exhibit a\nresult for identifying cross-section of screw driver and plastic straw. Similar to the previous result\nin Example 4.4, there is no difficult to select nonzero singular value when C= 0, 0 .01, 0.001i, and\n0.01ibut it is still difficult to select when C= 0.1 and C= 0.1i. In this case, we selected τ1as\nthe nonzero singular value of K(C) by regarding differences τn−τn+1,n= 1,2,···,15. It is worth\nto emphasize that the values of permittivity of the plastic straw and screw driver are extremely\nsmall and large, respectively. Thus, the existence of plastic straw does not affect to the Smeas(n, m)\nbecause the value of ODof (4) for screw driver is extremely larger than the one of plastic straw.\n12(a)C= 0\n (b)C= 0.01\n (c)C= 0.1\n(d)C= 0.001i\n (e)C= 0.01i\n (f)C= 0.1i\nFigure 5: (Example 4.2) Maps of F(z, C) atf= 1 GHz with various C. Black-colored dashed lines\ndescribe the ∂D1∪∂D2.\nCorrespondingly, only one singular value of K(C) that is significantly larger than the others has\nbeen appeared.\nFigure 12 shows the maps of F(z, C) with various C. Opposite to the previous results, the\nlocation and outline shape of cross-section of screw driver can be identified very accurately through\nthe map of F(z, C) with |C| ≤0.1. Unfortunately, it is impossible to recognize the existence of the\ncross-section of screw driver for any value C.\nOn the basis of the Theorem 3.1, Remark 3.2, and simulation results with synthetic and exper-\nimental data, we can conclude that applying small value of |C|guarantees good imaging results\nbut the criteria for small value is ambiguous. Hence, converting unknown diagonal data to the\nC= 0 will be the best choice for a proper application of the SM. This is the theoretical reason\nwhy the zero constant was chosen instead of the unknown data in most studies.\n5 Conclusion\nIn this study, we considered the application of SM for the quick identification of small objects by\nconverting the unknown diagonal elements of the scattering matrix into a constant. We explained\nthe effectiveness of this approach by demonstrating that the imaging function can be represented\nusing an infinite series of Bessel functions of integer order, antenna number and arrangement, and\napplied constant. Moreover, we established that small objects can be uniquely retrieved when the\nabsolute value of the applied constant is zero or close to zero.\nIt is important to note that our analysis and application scope in this paper is limited to retriev-\ning small objects in a two-dimensional setting. However, future research focusing on identifying\n13(a)C= 0\n (b)C= 0.01\n (c)C= 0.1\n(d)C= 0.001i\n (e)C= 0.01i\n (f)C= 0.1i\nFigure 6: (Example 4.3) Distribution of the singular values of K(C) atf= 1 GHz with various C.\nlarge objects and extending to three-dimensional microwave imaging would present interesting\nresearch topics.\nReferences\n[1] H. F Alqadah. A compressive multi-frequency linear sampling method for underwater acoustic\nimaging. IEEE Trans. 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Appl. , 128:55–68, 2022.\n17-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.10.20.30.40.50.60.70.80.9(a)C= 0\n-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.10.20.30.40.50.60.70.80.9 (b)C= 0.01\n-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.10.20.30.40.50.60.70.80.9 (c)C= 0.1\n-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.10.20.30.40.50.60.70.80.9\n(d)C= 0.001i\n-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.10.20.30.40.50.60.70.8 (e)C= 0.01i\n-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.050.10.150.20.250.30.350.40.450.50.55 (f)C= 0.1i\nFigure 10: (Example 4.4) Maps of F(z, C) atf= 925 MHz with various C.\n[35] W.-K. Park. A novel study on the orthogonality sampling method in microwave imaging\nwithout background information. Appl. Math. Lett. , 145:Article No. 108766, 2023.\n[36] W.-K. Park. 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Topics Signal Process , 4(1):164–176, 2010.\n19-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.10.20.30.40.50.60.70.80.9(a)C= 0\n-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.10.20.30.40.50.60.70.80.9 (b)C= 0.01\n-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.10.20.30.40.50.60.7 (c)C= 0.1\n-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.10.20.30.40.50.60.70.80.9\n(d)C= 0.001i\n-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.10.20.30.40.50.60.70.8 (e)C= 0.01i\n-0.5 0 0.5\nx-axis (m)-0.8-0.6-0.4-0.200.20.40.60.8y-axis (m)\n0.050.10.150.20.250.30.350.40.450.50.55 (f)C= 0.1i\nFigure 12: (Example 4.5) Maps of F(z, C) atf= 925 MHz with various C.\n20"    },    {        "title": "2402.13592v1.Hypercomplex_structures_arising_from_twistor_spaces.pdf",        "content": "HYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES\nSHUO WANG AND BIN XU\nAbstract. A hyperk¨ ahler manifold is defined as a Riemannian manifold endowed with three\ncovariantly constant complex structures that are quaternionically related. A twistor space\nis characterized as a holomorphic fiber bundle p:Z →CP1possesses properties such as a\nfamily of holomorphic sections whose normal bundle isL2nO(1), a holomorphic section of\nΛ2(NZ)⊗p∗(O(2)) that defines a symplectic form on each fiber, and a compatible real struc-\nture. According to the Hitchin-Karlhede-Lindstr¨ om-Roˇ cek theorem ( Comm. Math. Phys. ,\n108(4):535-589, 1987), there exists a hyperk¨ ahler metric on the parameter space Mfor the\nreal sections of Z. Utilizing the Kodaira-Spencer deformation theory, we facilitate the con-\nstruction of a hypercomplex structure on M, predicated upon more relaxed presuppositions\nconcerning Z. This effort enriches our understanding of the classical theorem by Hitchin-\nKarlhede-Lindstr¨ om-Roˇ cek.\n1.Introduction\nAhypercomplex manifold (M,I,J,K) is a smooth manifold Mequipped with three complex\nstructures I,J, and Kthat satisfy the quaternionic relations:\nI◦J=−J◦I=K.\nAhyperk¨ ahler manifold (M, g, I,J,K) is a hypercomplex manifold ( M,I,J,K) endowed with a\nRiemannian metric gonM, such that ( M, g) becomes a K¨ ahler manifold with respect to the\nthree complex structures I,J, and K, each being covariantly constant under the Levi-Civita\nconnection induced by g.\nLet (M, g, I,J,K) be a hyperk¨ ahler manifold. We define Iζ:=aI+bJ+cKwhere\n(a, b, c ) =\u00121−ζ¯ζ\n1 +ζ¯ζ,ζ+¯ζ\n1 +ζ¯ζ,i(−ζ+¯ζ)\n1 +ζ¯ζ\u0013\n∈S2\nfor each ζ∈C∪ {∞} ∼=CP1. Consequently, the set {Iζ|ζ∈CP1}forms a collection of complex\nstructures on M, and ( M, g, Iζ) constitutes a K¨ ahler manifold for each ζ∈CP1. Let ω1=g(I·,·),\nω2=g(J·,·), and ω3=g(K·,·).\nFrom a hyperk¨ ahler manifold Mof real dimension 4 n, we define the twistor space , denoted as\nZ, as the smooth product manifold Z=M×CP1equipped with an almost complex structure I.\nInT(m,ζ)Z, the almost complex structure Iis given by ( Iζ,I0), where I0represents the operation\nof multiplying by√−1 on the tangent space of CP1. Hitchin, Karlhede, Lindstr¨ om, and Roˇ cek\n[4, Theorem 3.3] have shown that the almost complex structure Iconstitutes a complex structure\nonZ. Moreover, the natural projection p:Z →CP1is a complex fiber bundle over CP1, whose\nfiber over ζ∈CP1is the K¨ ahler manifold ( M, g, Iζ), denoted as Zζ. In simpler terms, for each\nx∈M, there exists a complex projective line CP1\nxwithin Zassociated with the point x∈M,\nB.X. is supported in part by the Project of Stable Support for Youth Team in Basic Research Field, CAS (Grant\nNo. YSBR-001) and NSFC (Grant Nos. 12271495, 11971450 and 12071449).\n†B.X. is the corresponding author.\n1arXiv:2402.13592v1  [math.DG]  21 Feb 20242 SHUO WANG AND BIN XU\nsuch that:\nZ=[\nζ∈CP1(M, g, Iζ) =[\nx∈MCP1\nx.\nFurthermore, Hitchin-Karlhede-Lindstr¨ om-Roˇ cek’s theorem [4, Theorem 3.3] states the following:\n(i) Each CP1\nxhas a normal bundle that is isomorphic toL2nO(1).\n(ii) The holomorphic section ωof Λ2(NZ)∗⊗p∗(O(2)) is defined as follows:\nω=(\n(ω2+iω3) + 2ζ0ω1−ζ2\n0(ω2−iω3) for ζ0∈U0\nζ2\n1(ω2+iω3) + 2ζ1ω1−(ω2−iω3) for ζ1∈U1\nwhere ( U0, ζ0)∪(U1, ζ1)/∼are open charts on CP1as defined in the second paragraph\nof Section 3 and NZrepresents the vertical space of the fiber bundle p. In particular,\nω|Zζis a symplectic form on the fiber ( M, g, I ζ) for each ζ∈CP1.\n(iii) There is an involution mapping τonZinduced by the antipodal map on each CP1\nx.\nConversely, we designate Zas atwistor space if it is a complex manifold of dimension 2 n+ 1\nthat satisfies the following conditions (1), (2), (3), and (4):\n(1)Zis a holomorphic fiber bundle over CP1, denoted by p:Z →CP1.\n(2) There exists a real structure τonZthat is compatible with the antipodal map on CP1.\n(3)Zadmits a family Mof holomorphic sections, where the normal bundle TFsis isomorphic\ntoL2nO(1) for each s∈ M .\n(3′)Zadmits a holomorphic section swhose normal bundle is TFs∼=L2nO(1).\n(4) There exists a holomorphic section ωof Λ2(NZ)∗⊗p∗O(2) such that:\n(a)ωdefines a symplectic form on each fiber of p:Z →CP1.\n(b) For each real section s∈ M ,ω(·, j(τ, s)·) is a negative definite quadratic form on\nH0\u0000\ns(CP1), TFs⊗ O(−1)\u0001∼=C2n.\nwhere j(τ, s) is a quaternionic structure on H0\u0000\ns(CP1), TFs⊗ O(−1)\u0001\ninduced by τ. In [4,\nTheorem 3.3], Hitchin, Karlhede, Lindstr¨ om, and Roˇ cek proposed a method for defining a hy-\nperk¨ ahler manifold Mof real dimension 4 nas the parameter space of real sections of the fiber\nbundle p:Z →CP1.\nTheorem 1.1. [4, Theorem 3.3] LetZbe a complex manifold and σthe antipodal map over\nCP1. Assuming that conditions (1), (2), (3) and (4) are satisfied, and let Mbe the set of\nreal holomorphic sections in M. If M̸=∅, then it constitutes a smooth submanifold of M.\nFurthermore, a hyperk¨ ahler structure exists on M.\nThis establishes a bi-directional correspondence between hyperk¨ ahler manifolds and holomor-\nphic fiber bundles over CP1.\nThe primary aim of this manuscript is to deepen the understanding of the theorem posited by\nHitchin, Karlhede, Lindstr¨ om, and Roˇ cek [4], specifically by incorporating condition (3) into the\ntheoretical framework originally developed by Kodaira [7]. In this study, we explore a complex\nmanifold Wand its compact complex submanifolds V, following the foundational research by\nKodaira [7], which investigates the deformation of complex submanifolds within the overarching\nspace W. These changes are described by paramater space which is a complex manifold, usually\nconsidered simply as B=\b\nt∈Cl| |t|∞< ε\t\n, where εis a tiny positive number. As outlined\nin Definition 2.4, an analytic family of compact complex submanifolds is characterized through\na mapping adhering to four crucial conditions. These conditions ensure that every submanifold\nin the family maintains a connected and compact complex structure with a defined dimension,\nand is closely connected to the parameter space via a complex submanifold of higher dimension.\nThis concept is encapsulated in the natural projection π:V →B, where Vdenotes a complexHYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 3\nsubmanifold of W×B. Here, for each t∈B,Vt:=π−1(t) is identified as a compact complex\nsubmanifold of W, with V=V0marking the initial condition for the deformation. Infinitesimal\ndeformation of VtinWis intuitively understood as a holomorphic section of the normal bundle\nNVtofVt. This concept is rigorously defined by Kodaira [7] through the introduction of the\nKodaira-Spencer map σ:TtB→H0(Vt, NV t). This map associates tangent vectors in the\nparameter space with holomorphic sections of the normal bundle, capturing the deformation’s\nessence. A cornerstone of Kodaira’s findings is the assertion that, provided the cohomology space\nH1(V, NV ) vanishes, an analytic family π:V →Bof compact complex submanifolds VinW\nexists for a sufficiently small parameter space B. This family is distinguished by the Kodaira-\nSpencer map being a linear isomorphism, rendering the deformation maximal in this localized\ncontext. These characteristics lay the groundwork for subsequent analyses, herein referred to\nas the canonical local deformation of V. For a comprehensive discussion and formal definitions,\nrefer to Theorem 2.13 and Definition 2.14.\nBuilding upon Kodaira [7], this work use the concept of regular deformation space , denoted as\ngMX. The notion of a regular deformation space emerges as an intuitively natural framework for\nexploring the deformation processes of compact complex submanifolds. However, the precise ori-\ngins of this concept within the scholarly literature remain unclear to us. This space is envisioned\nas a refined parameter space that facilitates a deeper exploration of the deformation process\nfor a compact complex submanifold Xwithin a complex manifold W. Through this endeavor,\nwe aim to modestly augment Kodaira’s original framework, offering additional insights into the\ncomplexities of deformation. Specifically, let Xbe a compact complex submanifold of W. A\ncompact complex submanifold VofWis termed a deformation ofXif it is possible to transition\nfrom XtoVvia a sequence of local deformations in a finite number of steps. To elaborate, a\ncompact complex submanifold VofWconstitutes a deformation of XinWif there exist ana-\nlytic families πk:Vk→Bkfork= 0, . . . , n −1, characterized by the properties: X=π−1\n0(s0),\nπ−1\nk+1(sk+1) =π−1\nk(tk), and V=π−1\nn−1(tn−1) for certain sk, tk∈Bk. The collection of all such\ndeformations of XinWis represented as MX.\nUnder the assumption that H1(X, NX ) = 0, the regular deformation space ofXinW,\nsymbolized as gMX, is conceptualized as the subset of MXcomprising those deformations Vfor\nwhich H1(V, NV ) = 0. This is formally expressed as:\ngMX=\b\nV∈MX\f\fH1(V, NV ) = 0\t\n.\nFor each deformation VwithinMX, considering the collection of all canonical local deformations\nofVas its open neighborhoods establishes a topology SXon this ensemble. More details are\ndiscussed in Lemma 2.17. We then define the spacegMl\nXas:\ngMl\nX=n\nV∈gMX\f\f\fH0(V, NV ) =lo\n,\nWithin the structured framework of our analysis,gMl\nXis identified as an open subset of the\ntopological space\u0010\ngMX,SX\u0011\n. Furthermore, the construction of gMXis meticulously achieved\nthrough the disjoint union gMX=F\nl∈Z≥0gMl\nX, each component inheriting the topology induced\nfromgMX.\nTheorem 1.2. (Theorem 2.18) The set\u0010gMl\nX,SX\u0011\nis endowed with a complex manifold struc-\nture of complex dimension l, provided thatgMl\nXis non-empty, for every l∈Z≥0.\nBy utilizing condition the condition (1) that p:Z →CP1is a holomorphic fiber bundle\nand suppose sis a holomorphic section of Zgiven by condition (3′), we can regard s(CP1)4 SHUO WANG AND BIN XU\nas a compact complex submanifold in Z. Leveraging the framework we established earlier, it\nis possible to define a complex manifold^M4n\ns(CP1)included in the regular defromation space of\ns(CP1) inZ. Then define M, which serves as an open submanifold^M4n\ns(CP1)as\nM:=(\nV∈^M4n\ns(CP1)|Vis a holomorphic section of Z;TFV∼=2nM\nO(1))\nThe manifold Macts as the parameter space for holomorphic sections of Z. In this case, we\ncan prove that Zsatisfies condition (3), i.e. (1)+(3′)⇒(3). The details of this process are\nfurther elaborated in Proposition 4.6. Furthermore, conditions (1), (2), and (3)′collectively\nfacilitate the identification of a smooth manifold M, endowed with a hypercomplex structure,\nas a smooth submanifold of M. This fact is corroborated by Theorem 4.13. The hypercomplex\nstructure on Mis further elaborated in this article, extending beyond the scope of Hitchin,\nKarlhede, Lindstr¨ om, and Roˇ cek’s original paper. Joyce also mentioned the existence of twistor\nconstruction for hypercomplex manifolds in [5, Subsection 7.5.1]. The details will be discussed\nin Theorem 4.13 as follows:\nTheorem 1.3. (Theorem 4.13) Consider a complex manifold Zequipped with an antipodal map\nσover the projective line CP1. Upon imposing conditions (1), (2), and (3) for Z, letMdenote\nthe collection of real holomorphic sections within M. Should Mbe non-empty, it inherently\nforms a smooth submanifold of M, further endowed with a hypercomplex structure.\nWe conclude the introduction by providing a comprehensive overview of the subsequent four\nsections of this manuscript. Section 2 delves into the deformation theory of compact complex\nsubmanifolds in complex manifolds, extending the foundational work of Kodaira [7]. This section\nalso introduces Lemma 2.18, defining the regular deformation space within the realm of complex\nmanifolds. In Section 3, the focus shifts to examining a real structure onL2nO(1). Proposition\n3.7 introduces a real structure on H0\u0000\nCP1,L2nO(1)\u0001\nand delineates a quaternionic structure on\nH0\u0000\nCP1,CP1×C2n\u0001\n. Section 4 presents the parameter space M, comprising holomorphic sections\nwith the normal bundleL2nO(1). Utilizing theories from Section 2, Mis established as a\ncomplex manifold. This section also demonstrates how the real structure r, introduced in Section\n3, leads to the emergence of a submanifold Mwithin M, and how the quaternionic structure\njbestows a hyperk¨ ahler structure on M. Section 5 sets forth a toy example to illustrate the\nrelationship between hyperk¨ ahler manifolds C2nand holomorphic fiber bundles overL2nO(1)\nonCP1.\n2.Deformation of compact complex submanifolds\nIn this section, we will apply Kodaira’s framework as delineated in [7] to introduce the concept\nof an analytic family of compact submanifolds within a complex manifold W. Consider Xas a\ncompact submanifold of W, meeting specific criteria. We aim to define a regular deformation\nspace of XinW, denoted as gMX, and to establish a complex structure on this space. This\nmethod aligns with the theory of deformation of complex analytic structures, initially developed\nby Kodaira and Spencer in their pioneering work [8].\nKodaira and Spencer, in their seminal article [8], established a foundational framework for\nstudying deformation in complex manifolds. In the following sections, we will revisit and utilize\nseveral key definitions from their work.\nDefinition 2.1. Adifferentiable family of compact complex manifolds is defined as a differ-\nentiable fiber bundle π:V → Nover a differentiable manifold N, where each point of Vhas\na neighborhood U⊂ V satisfying the following conditions: there exists a diffeomorphismHYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 5\nh:U→Cn×π(U) such that, for each point t∈π(U),Vt:=π−1(t) is a compact com-\nplex manifold and the restriction of htoU∩VtsendsU∩Vt,→Cn×{t}holomorphically, where\nCndenotes the space of ncomplex variables\u0000\nz1,···, zn\u0001\nandnis the complex dimension of Vt.\nAcomplex analytic family of compact complex manifolds is defined as a differentiable family of\ncompact complex manifolds π:V → Nwhere π:V → Nis a holomorphic map and both Nand\nVare connected complex manifolds.\nDefinition 2.2. Letπ:V → Nbe a differentiable family of compact complex manifolds. A\ndifferentiable family of πrefers to a differentiable fiber bundle ω:B → V with a structure group\nthat is a complex Lie group G. The fiber bundle ωt:Bt→Vtis holomorphic when ωis restricted\nto each fiber Vt:=π−1(t),t∈NofVandBt:=ω−1(Vt).\nLetπ:V → Nbe a complex analytic family of compact complex manifolds. A complex\nanalytic family of πis defined as a differentiable family Bω→ Vπ→Nofπthat satisfies the\ncondition of Bω→ V being a holomorphic map.\nUtilizing the theoretical framework established in [8], Kodaira and Spencer formulated a piv-\notal theorem, the ’Principle of Upper Semi-continuity.’ This theorem is elaborated and demon-\nstrated in Kodaira’s subsequent paper [6].\nTheorem 2.3. Letπ:V → Nbe a differentiable family of compact complex manifolds as defined\nin 2.1, and suppose the complex vector bundle ω:B → V overVis also a differentiable family\nwith respect to πas defined in 2.2. For each t∈N, letVt:=π−1(t)andBt:=ω−1(Vt). Then\nthere exists a neighborhood Uoft∈Nsuch that, for any t′∈Uandq∈Z,\ndimHq(Vt′, Bt′)≤dimHq(Vt, Bt)\nKodaira’s seminal paper [7] introduces a critical concept in the study of complex manifolds:\nthe analytic family of compact submanifolds. This concept is fundamental to understanding\nvarious aspects of compact complex submanifolds and is defined as follows:\nDefinition 2.4. LetWandNbe complex manifolds with complex dimensions d+randl,\nrespectively. Let j1:W×N→Wandj2:W×N→Nbe the natural projections. An\nanalytic family of compact submanifolds ofWwith dimension dand parameter space Nis a map\nπ:V → Nsatisfying the following conditions:\n(1)Vis a complex submanifold of W×Nwith codimension r.\n(2)πis the restriction of the map j2toV.\n(3) For each t∈N,Vt:=V ∩(W× {t}) is a connected and compact complex submanifold\nofW× {t}with dimension d.\n(4) For each p∈ V, there exists a local chart\np∈\u0010\nUp;w1,···, wr, z1,···, zd, t1,···, tl\u0011\n⊂W×N\nand holomorphic functions f1,···, fronUp, such that\n(i)V ∩Up=n\n(z, w, t )∈Up:f1(z, w, t ) =···=fr(z, w, t ) = 0o\n(ii) rank∂(f1,···, fr)\n∂(w1,···, wr, z1,···, zd)\f\f\f\f\np′=r,∀p′∈Up\nNext, we aim to establish the following proposition, which connects the analytic family of\ncompact submanifolds, defined in Definition 2.4, with the concept of a differentiable family of\ncompact complex manifolds, as detailed in Definition 2.1. This connection allows us to apply the\nPrinciple of Upper Semi-continuity (Theorem 2.3) to our specific scenario.6 SHUO WANG AND BIN XU\nProposition 2.5. Given an analytic family of compact submanifolds π:V → Nof a complex\nmanifold Wwith dimension dand parameter space N, it follows that π:V → Nalso constitutes\na complex analytic family of compact complex manifolds.\nProof. Conditions (1), (2), and (3) imply that π:V → Nis a holomorphic surjective map.\nWithout loss of generality, we can assume that ( z, w, t )(p) = 0 in condition (4). By the Implicit\nFunction Theorem, condition (4- ii) implies that for any fixed t0∈j2(Up) near 0, there exists a\nunique simultaneous equation:\n\n\nw1=φ1(z1,···, zd, t0)\n...\nwr=φr(z1,···, zd, t0)such that\n\nf1(φ1,···, φr, z1,···, zd, t0) = 0\n...\nfr(φ1,···, φr, z1,···, zd, t0) = 0\nfor each ( z1,···, zd) near 0. Shrink Upif necessary, then\u0000\nUp∩Vt0, zr+1,···, zr+d\u0001\nis a\nholomorphic chart on Vt0and\u0000\nUp∩Vt0, zr+1,···, zr+d, t1,···, tl\u0001\nis a holomorphic chart on V.\nThe restriction of πtoUp∩ Vis:\nπ:Up∩ V → j2(Up)\u0010\nzr+1,···, zr+d, t1,···, tl\u0011\n7→\u0010\nt1,···, tl\u0011\nwhich implies that πis a submersion. Furthermore, according to the Ehresmann fibration the-\norem [2], each fiber of πis a compact complex manifold, indicating that π:V → Nforms a\ndifferentiable fiber bundle. □\nIn the context of exploring the complex manifold Wand its compact complex submanifold\nV, it becomes essential to define a specific system of local charts. These charts facilitate the\nexamination of the manifold’s local structure in the vicinity of V. We formalize this concept as\nfollows:\nDefinition 2.6. Define the system of canonical local charts of Win the neighborhood of Vas a\nfinite collection of chartsn\u0000\nWk;w1\nk,···, wr\nk, z1\nk,···, zd\nk\u0001o\nk∈IonW. This collection satisfies the\nfollowing conditions:\n(1)S\nk∈IWkforms an open covering of VinW.\n(2) For each k∈I, the intersection V∩Wkis defined by the conditions\nw1\nk=···=wr\nk= 0.\nThis definition provides a framework for examining the local properties of Waround the\nsubmanifold V.\nDefinition 2.7. The normal bundle NVover a compact complex submanifold VofWis defined\nby the following short exact sequence:\n0→TV→TW|V→NV→0.\nConsidern\u0000\nWk, w1\nk,···, wr\nk, z1\nk,···, zd\nk\u0001o\nk∈Ias the system of canonical local charts in the\nneighborhood of V, as defined in Definition 2.6. Let Vk:=Wk∩Vdenote the intersection of Wk\nandV. The collection\b\nVk\t\nk∈Iforms local charts for V, each with coordinates\u0000\nz1\nk,···, zd\nk\u0001\n. To\nelucidate the structure of the normal bundle NV, we consider the open covering V=S\nk∈IVk\nand define the transformation functions between overlapping charts WiandWkas follows:\n(\nwλ\ni=fλ\nik(wk, zk) for λ= 1,2,···, r\nzµ\ni=gµ\nik(wk, zk) for µ= 1,2,···, dHYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 7\nFigure 1. System of canonical local charts\nThe normal bundle NVis then constructed as NV=S\nk∈I(Vk×Cr)/∼, where ∼denotes the\nequivalence relation defined by the system of transition matrices\b\nBik(zk)\t\n:\nBik(zk) = \n∂wλ\ni\n∂wµ\nk\f\f\f\f\nVi∩Vk!\nλ,µ=1,···,r= \n∂fλ\nik(wk, zk)\n∂wµ\nk\f\f\f\f\nwk=0!\nλ,µ=1,···,r\nHaving established the local structure and normal bundle of a single compact complex sub-\nmanifold VinW, we now extend our discussion to a broader context. We consider an analytic\nfamily of compact submanifolds, represented by the map π:V → N. In this expanded frame-\nwork, we aim to investigate the overall structure of each complex submanifold within this analytic\nfamily, as well as the corresponding structures of their normal bundles.\nLetπ:V → Ndenote an analytic family of compact submanifolds within the complex\nmanifold W. Assume Bl\nε=n\nt∈Cl| ∥t∥∞< εo\nto be a sufficiently small neighborhood around\nthe origin in Cl, which will be considered as an open chart in N. This setting allows us to view\nπ:V →Bl\nεas a parameterized analytic family of compact submanifolds of Wwith dimension d.\nIn this framework, Vtis treated as a submanifold of both Wand the product space W× {t}.\nGiven ε >0, there exists a finite open covering V0:=π−1(0)⊂S\nk∈IUkinW×Bl\nε. Each set\nUkhas local coordinates\u0000\nUk, w1\nk,···, wr\nk, z1\nk,···, zd\nk, t1,···, tl\u0001\nsatisfying:\n(1)Uk=Wk×Bl\nεfor an open set WkinW.\n(2)\u0010\nWk, w1\nk,···, wr\nk, z1\nk,···, zd\nk\u0011\nforms a local chart on W.\n(3) For all t∈Bl\nε,\u0010\nVk,t:=Uk∩Vt, z1\nk,···, zd\nk\u0011\nserves as a local chart on Vt.\nThe intersection of VtandUkis represented by the simultaneous equations:\n\n\nw1\nk=φ1\nk(z1\nk,···, zd\nk, t1,···, tl)\n...\nwr\nk=φr\nk(z1\nk,···, zd\nk, t1,···, tl)\nwhere φλ\nkare holomorphic functions of zkandt. Furthermore, within Uk,V0is defined by\nw1\nk=···=wr\nk= 0.8 SHUO WANG AND BIN XU\nFigure 2. Deformation of compact complex submanifolds\nIn this scenario, the collection of open chartsS\nk∈I(Uk;wk, zk) forms a set of canonical local\ncharts on Wfor the neighborhood of V, as previously defined in Definition 2.7. We introduce\nnew local coordinates on Wkby defining:\n˜wλ\nk,t:=wλ\nk−φλ\nk(zk, t) and ˜ zµ\nk=zµ\nk,\nwhere Vtis represented by ˜ w1\nk,t=···= ˜wr\nk,t= 0 within Uk. Accordingly, the collection of open\nchartsS\nk∈I(Uk; ˜zk,˜wk) then constitutes a set of canonical local charts for Win the neighborhood\nof each fiber Vt, as per Definition 2.7.\nFollowing the same approach as before, the transformation functions between overlapping\nchartsWiandWkare given by:\n(\nwλ\ni=fλ\nik(wk, zk) for λ= 1,2, . . . , r,\nzµ\ni=gµ\nik(wk, zk) for µ= 1,2, . . . , d,\nwhere wk= (w1\nk, . . . , wr\nk) and zk= (z1\nk, . . . , zd\nk). Subsequently, we define the normal bundle NVt\nof each fiber Vtin the analytic family π:V →Bl\nεas follows:\nDefinition 2.8. Define the normal bundle of Vtas a holomorphic vector bundle Btof rank rover\nVt. This vector bundle is characterized by a set of bundle transition functions Bt,ikassociated\nwith the open chart covering Vt=S\nk∈IVk,t. The bundle transition functions Bt,ik(zk) are\nspecified by:\n \n∂˜wλ\ni\n∂˜wµ\nk\f\f\f\f\nVi,t∩Vk,t!\nλ,µ=1,...,r= \n∂fλ\nik(wk, zk)\n∂wµ\nk−∂φλ\ni(zk, t)\n∂zα\ni·∂gα\nik(wk, zk)\n∂wµ\nk\f\f\f\f\nVi,t∩Vk,t!\nλ,µ=1,...,r.\nHaving established the normal bundle Btof each fiber Vtin the family V, we now proceed\nto generalize this concept to the entire family. This allows us to consider the variation of theseHYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 9\nnormal bundles across compact complex submanifold in V, leading to the definition of a holistic\nstructure that encompasses whole V.\nDefinition 2.9. Define the holomorphic function Bik:V ∩Ui∩Uk→GL(r,C) such that\nBik|Vt=Bt,ik. The holomorphic vector bundle BoverVis then defined by the collection of these\nbundle transition functions Bik, corresponding to the open chart covering V=S\nk∈I(Ui∩ V).\nThe projection ω:B → V denotes the complex analytic family of normal bundles on V.\nThe family of normal bundles ω:B → V constitutes a complex analytic family over the\nmapping π:V →Bl\nε, as defined in Definition 2.2. Furthermore, according to Theorem 2.3, for\nanyt∈Bl\nεsufficiently close to 0 and for any integer q≥0, the dimension of the cohomology\ngroup Hq(Vt, Bt) is less than or equal to that of Hq(V0, B0).\nNow that we have described the normal bundles NVtassociated with each fiber Vtof the\nanalytic family π:V →Bl\nε, it is pertinent to delve into the mechanisms of their deformation.\nThis leads us to explore the concept of infinitesimal deformation in the context of complex\nmanifolds, a notion central to understanding the changes in compact complex submanifolds\nunder small perturbations. Kodaira and Spencer’s pioneering work [8] provides a foundation for\nthis exploration. For every fiber Vtofπ:V →Bl\nεand any tangent vector∂\n∂t:=γρ∂\n∂tρattinN,\nletBtdenote the normal bundle of Vt, as defined in Definition 2.8. According to Kodaira [7],\nthe set of holomorphic functions {Ψk,t:Vk,t→Cr}k∈I, defined by\nΨk,t(zk) = \n∂φλ\nk(zk, t)\n∂t\f\f\f\f\nVk,t!\nλ=1,···,r,\nsatisfies the equation Ψ i,t(zi) =Bt,ik·Ψk,t(zk). Thus, {Ψk,t(zk)}k∈Iinduces a holomorphic\nsection of Btover the open cover Vt=S\nk∈IVk,t.\nDefinition 2.10. The holomorphic section of Btinduced by {Ψk,t(zk)}k∈Iis referred to as\ntheinfinitesimal deformation ofVtalong the tangent vector∂\n∂t=γρ∂\n∂tρatt∈N. The linear\nmapping\nσt:TtN→H0(Vt, Bt) given by∂\n∂t7→ {Ψk,t}k∈I\nis known as the Kodaira-Spencer map .\nWe now focus on the specific properties and deformations of a compact complex submanifold\nwithin such a manifold. In the context of a complex manifold Wof dimension r+d, con-\nsider V, a compact complex submanifold of dimension d. Utilizing the canonical local charts\n(Wk, w1\nk,···, wr\nk, z1\nk,···, zd\nk)k∈Inear Vas per Definition 2.6, we denote Vkas the intersection\nWk∩V. Assuming dim CH0(V, NV ) =lwith a basis {β1,···, βl}forH0(V, NV ), each βρin\nVk×Cris represented by\nβρ,k=\u0010\nβ1\nρ,k,···, βr\nρ,k\u0011\nwith βλ\nρ,k(z1\nk,···, zd\nk) as holomorphic functions on Vk. Kodaira’s Theorem [7, Theorem 1] demon-\nstrates the surjectivity of the Kodaira-Spencer map for Vunder specific conditions on NV.\nProposition 2.11. Suppose H1(V, NV ) = 0 . For a sufficiently small positive number ε, there\nexist holomorphic vector functions φk:Vk×Bl\nε→Cr, defined as φk(zk, t) = ( φ1\nk,···, φr\nk),\nsatisfying these transition and boundary conditions:\nTransition Conditions:(\nφλ\ni=fλ\nik(φ1\nk,···, φr\nk, z1\nk,···, zd\nk) λ= 1,···, r,\nzµ\ni=gµ\nik(φ1\nk,···, φr\nk, z1\nk,···, zd\nk) µ= 1,···, d,10 SHUO WANG AND BIN XU\nBoundary Conditions:\n(\nφλ\nk(z1\nk,···, zd\nk,0) = 0 λ= 1,···, r,\n∂φλ\nk\n∂tρ\f\f\f\nt=0=βλ\nρ,k ρ= 1,···, l.\nRemark 2.12.The boundary condition implies that the function system(\u0012∂φλ\nk\n∂tρ\f\f\f\f\nt=0\u0013\nλ=1,···,r:Vk→Cr)\nk∈I\ninduces a holomorphic section of H0(V, NV ), exactly corresponding to βρfor each ρ= 1,···, l.\nLetVtbe defined near Vas\nVt∩Wk=n\nwλ\nk=φλ\nk(zk)|λ= 1,···, ro\ninWk, where φkare the holomorphic functions defined in Definition 2.8. The transition condition\nensures that Vtforms a submanifold of Wfor each t∈Bl\nε. Defining V=F\nt∈BlεVtas a submanifold\nofW×Bl\nε, we obtain an analytic family πV:V →Bl\nε, consisting of compact submanifolds of W.\nThis family has the properties: V0=V,B0=NV, and the Kodaira-Spencer map at t= 0 is an\nisomorphism, thereby constituting the canonical local deformation of V.\nFigure 3. Canonical local deformation\nBuilding upon the foundational concepts we have discussed, Kodaira further extends these\nideas in [7] by presenting a more comprehensive theorem. This theorem not only encompasses\nthe earlier results but also provides deeper insights into the structure of analytic families of\ncompact complex submanifolds.\nTheorem 2.13. IfH1(V, NV ) = 0 anddimCH0(V, NV ) =l, then there exists an analytic family\nπ:V →Bl\nεof compact complex submanifolds of Wsuch that V0=V. This family π:V →Bl\nε\nsatisfies the property that the Kodaira-Spencer map σt:TtBl\nε→H0(V, B t)is an isomorphism for\nallt∈Bl\nε. Additionally, at each point t∈Bl\nε, the family is maximal.\nThe notion of ’maximal’ in this context is further elaborated in the following definition.\nDefinition 2.14. An analytic family π:V → Nof compact submanifolds Vt,t∈N, ofWis\nsaid to be maximal at a point t0ofNif and only if, for any analytic family π′:V′→N′of\ncompact submanifolds V′\ns,s∈N′, ofWsuch that V′\ns0=Vt0for a point s0∈N, there exists a\nneighborhood U′ofs0inN′and a holomorphic map h:U′→Nsending s0intot0such that\nV′\ns=Vh(s)for all s∈U′, where V′\ns=Vtindicates that V′\nsandVtare the same submanifold of\nW.HYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 11\nRemark 2.15.It is crucial to emphasize that the analytic family π:V →Bl\nε, corresponding to\nthe canonical local deformation of Vas defined in Proposition 2.11, fully satisfies the conditions\nstated in Theorem 2.13 when εis sufficiently small.\nLetXbe a compact complex submanifold of Wsuch that H1(X, NX ) = 0. In line with\nthe conditions outlined in Theorem 2.13, we utilize the concept of canonical local deformation as\ndefined in Proposition 2.11. This approach facilitates the exploration of deformations of Xwithin\nthe manifold W, providing a framework to understand how Xcan be varied or transformed while\nremaining within W. To formalize this concept, we introduce the notion of a regular deformation\nspace:\nDefinition 2.16. LetXbe a compact submanifold of Wof dimension d. A compact complex\nsubmanifold VofWis called a deformation ofXinWif there exist analytic families πk:Vk→\nNk,k= 0,···, n−1, with the following properties: X=π−1\n0(s0),π−1\nk+1(sk+1) =π−1\nk(tk), and\nV=π−1\nn−1(tn−1) for some sk, tk∈Nk. The set of all such deformations of XinWis denoted as\nMX.\nAssuming H1(X, NX ) = 0, the regular deformation space ofXinW, denoted as gMX, is\ndefined as the subset of MXconsisting of those deformations Vsatisfying H1(V, NV ) = 0.\ngMX=n\nV∈MX\f\f\fH1(V, NV ) = 0o\n.\nFigure 4. Deformation space\nLemma 2.17. LetXbe a compact complex submanifold of WandH1(X, NX ) = 0 . Then gMX\nhas a C1andT2topology.\nProof. In the regular deformation space gMX, each manifold Vis associated with an analytic\nfamily πV:V →Bl\nε, comprising compact submanifolds of a fixed manifold W. For a suitably\nsmall ε,πVadheres to the criteria:\n(1)V0, the fiber over the origin in Bl\nε, equals V.\n(2) For every t∈Bl\nε, the Kodaira-Spencer map σtrelated to πVis an isomorphism.\n(3)πVis maximal at each point t∈Bl\nε(refer to Definition 2.14).12 SHUO WANG AND BIN XU\nThis family stems from V’s canonical local deformation. Let l:= dim H0(V, NV ) and Vt:=\u0000\nπV\u0001−1(t) represent the fiber for each t∈Bl\nε. With a sufficiently minimized ε, we deduce:\ndimH1(Vt, NV t)≤dimH1(V, NV ) = 0 ,\nin line with the Principle of Upper Semi-continuity (refer to Theorem 2.3). This connotes\ndimH1(Vt, NV t) = 0, confirming that Vtis included in gMX.\nLetUπV\u00001\nn\u0001\n=n\nVt= (πV)−1(t)|t∈Bl\n1\nno\ndenote neighborhoods of VingMX, where nis\nchosen to be sufficiently large. Consider any Y, Z∈gMXsuch that the intersection UπY\u0010\n1\nnY\u0011\n∩\nUπZ\u0010\n1\nnZ\u0011\nis non-empty. For a manifold Vin this intersection, i.e., V∈UπY\u0010\n1\nnY\u0011\n∩UπZ\u0010\n1\nnZ\u0011\n,\nit follows that UπV(ε)⊂UπY\u0010\n1\nnY\u0011\n∩UπZ\u0010\n1\nnZ\u0011\nfor a sufficiently small ε. This inclusion holds\nbecause both πYandπZare maximal at each point in their domains. It is readily observed that\n[\nV∈gMX,n≫1UπV\u00121\nn\u0013\n=gMX.\nTherefore, the collection\b\nUπV\u00001\nn\u0001\t\nn≫1forms a countable basis for the topology at each point\nVingMX, suggesting that gMXis endowed with a C1topology.\nTo establish the T2(Hausdorff) property in the regular deformation space gMX, a more detailed\nexamination of the canonical local deformations is necessary. Consider distinct manifolds Yand\nZingMX, meaning that YandZare separate submanifolds of W. Since both YandZare\ncompact submanifolds, there exist points y∈Y\\Zandz∈Z\\YinW. We select canonical\nlocal charts for Waround YandZas follows:n\u0000\nWY,k;w1\nY,k, . . . , wr\nY,k, z1\nY,k, . . . , zd\nY,k\u0001o\nk∈I,\nn\u0000\nWZ,k;w1\nZ,k, . . . , wr\nZ,k, z1\nZ,k, . . . , zd\nZ,k\u0001o\nk∈I.\nThe corresponding restricted local holomorphic charts on YandZare\nn\u0000\nYk:=WY,k;z1\nY,k, . . . , zd\nY,k\u0001o\nk∈I,n\u0000\nZk:=WZ,k;z1\nZ,k, . . . , zd\nZ,k\u0001o\nk∈I.\nThe canonical local deformations are defined by holomorphic function systems φk:Yk×Bl\nεY→\nCrandψk:Zk×Bl′\nεZ→Cr, such that\nYt:=π−1(t) ={wλ\nY,k=φλ\nk(zY,k, t)}, Z t:=π−1(t) ={wλ\nZ,k=ψλ\nk(zZ,k, t)}\nfor each k∈Iandλ= 1, . . . , r . Here, l= Dim H0(Y, NY ) and l′= Dim H0(Z, NZ ).\nAssuming WY,0andWZ,0are centered at yandz, respectively, we can shrink each WY,kand\nWZ,k, and adjust εYandεZas needed. Without loss of generality, assume\nWY,0∩ [\nk∈IWZ,k!\n=WZ,0∩ [\nk∈IWY,k!\n=∅,\nin accordance with Wbeing a T3topology space. For any t∈Bl\nεYandt′∈Bl′\nεZ, we have\nYt:=\u0000\nπY\u0001−1(t)̸=Zt′:=\u0000\nπZ\u0001−1(t′), demonstrated by\n(\nYt∩WY,0̸=∅, Y t∩WZ,0=∅,\nZt′∩WY,0=∅, Z t′∩WZ,0̸=∅.\nHence, the open neighborhoods UπY(εY) and UπZ(εZ) ofYandZwithin gMXare disjoint. □HYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 13\nThe topology previously defined on the space gMXis hereinafter denoted as SX. We consider\nthe topological space\u0010\ngMX,SX\u0011\nand assume that it satisfies the C2property. Furthermore, we\ndefinegMl\nXas the subset of gMXconsisting of elements Vfor which Dim H0(V, NV ) =l. It is\nnoteworthy thatgMl\nXconstitutes a open subset in the space\u0010\ngMX,SX\u0011\n.\nTheorem 2.18. Given thatgMl\nXis non-empty, the pair (gMl\nX,SX)manifests as a complex\nmanifold with dimension l, for each l∈Z≥0.\nProof. Expanding on the notations introduced in the preceding lemma, consider for each V∈\ngMl\nX, the map\nπV:UπV(ε)→Bl\nε,where Vt=\u0000\nπV\u0001−1(t)7→t∈Bl\nε.\nThis map πVmanifests as a local homeomorphism for sufficiently small ε, justified by the isomor-\nphism of the Kodaira-Spencer map. Consider any two elements Y, Z∈gMl\nX, such that Vbelongs\nto both UπY(εY) and UπZ(εZ). Choose εsmall enough to ensure UπV(ε)⊆UπY(εY)∩UπZ(εZ).\nTo establish the lemma, it suffices to demonstrate that πY◦\u0000\nπZ\u0001−1is holomorphic at the\npoint\u0000\nπZ\u0001−1(V).\nObserve that the family πY:UπY(εY)→Bl\nεYis maximal at the point πY(V). Consequently,\nfor a sufficiently small ε, the family πV:UπV(ε)→Bl\nεis subsumed within πY. Hence, there\nexists a holomorphic function h:Bl\nε→Bl\nεYsatisfying\u0010\u0000\nπV\u0001−1(s)\u0011\n=\u0010\u0000\nπY\u0001−1(h(s))\u0011\n. This\nimplies that πY◦\u0010\u0000\nπV\u0001−1(s)\u0011\n=h(s) defines a biholomorphic map in the vicinity of s= 0. A\nsimilar argument establishes that πZ◦\u0010\u0000\nπV\u0001−1(s)\u0011\nis biholomorphic near s= 0. Therefore,\nπY◦\u0000\nπZ\u0001−1=\u0010\nπY◦\u0010\u0000\nπV\u0001−1\u0011\u0011\n◦\u0010\nπZ◦\u0010\u0000\nπV\u0001−1\u0011\u0011−1\n,\nis holomorphic at the point\u0010\u0000\nπZ\u0001−1(V)\u0011\n. □\n3.real structures onL2nO(1)→CP1\nIn this section, we delve into the Riemann sphere CP1and the vector bundleL2nO(1) over\nCP1. Additionally, we explore CP1, a smooth manifold that is topologically identical to CP1\nbut is equipped with a different system of complex open charts. This distinction is particularly\nsignificant in the context of complex geometry and analysis. In this setting, the antipodal map\nσserves as a biholomorphic map from CP1toCP1. Utilizing σ, we define real structures on the\nvector bundleL2nO(1), corresponding to a quaternionic structure on H0(CP1,(L2nO(1))⊗\nO(−1))∼=C2n. These structures are fundamental for the study of hyperk¨ ahler manifolds, a\ntopic we will explore in Section 4. Our analysis and calculations throughout this section will\nbe grounded in the use of the standard open cover of CP1, ensuring that our approach is both\nsystematic and comprehensive.\nThe Riemann sphere can be depicted as the complex projective line CP1=\b\n[z0:z1]\f\f(z0, z1)∈\nC2\\0\t\n. For analytical convenience, we adopt a standard open covering of CP1, denoted as U0∪U1,\nwhere U0=\b\n[z0:z1]∈CP1\f\fz0̸= 0\t\nandU1=\b\n[z0:z1]∈CP1\f\fz1̸= 0\t\n. The local holomorphic\ncoordinates are defined as ζ0\u0000\n[z0:z1]\u0001\n=z1\nz0onU0andζ1\u0000\n[z0:z1]\u0001\n=z0\nz1onU1. Notably, ζ0∈U0\ncorresponds bijectively to ζ1∈U1if and only if ζ0·ζ1= 1. Furthermore, we define CP1to\npossess the same smooth manifold structure as CP1, with open charts Vi:=Uifori= 0,1. Local\nholomorphic coordinates are introduced as η0\u0000\n[z0:z1]\u0001\n=¯z1\n¯z0onV0, and η1\u0000\n[z0:z1]\u0001\n=¯z0\n¯z1onV1.14 SHUO WANG AND BIN XU\nThe conditions for equivalence between η0∈V0andη1∈V1are met if and only if η0·η1= 1.\nConsequently, these holomorphic charts ( Vi, ηi, i= 0,1) endow CP1with a complex manifold\nstructure.\nIn the framework of smooth manifolds, the identity map, denoted by I, maps the complex\nprojective line CP1to its conjugate CP1. This map is characterized as antiholomorphic and can\nbe articulated in the following manner:\n(\nI0:U0→V0, ζ 07→¯ζ0,\nI1:U1→V1, ζ 17→¯ζ1.\nLikewise, the inverse map I−1:CP1→CP1retains the property of being antiholomorphic.\nIn a broader context, consider ( M,I) as a complex manifold. The conjugate manifold, denoted\nM, is defined to be a smooth manifold that shares the same smooth structure as M, but with the\ncomplex structure altered to −I. This definition implies that the identity map from the complex\nmanifold M= (M,I) to its conjugate M= (M,−I) is an antiholomorphic map.\nIt is well established that there exists a unique complex structure on the 2-sphere S2, implying\nthe existence of a biholomorphic map from CP1toCP1. A notable instance of such a map is\nthe antipodal map on S2, denoted by σ. This map can be delineated by the following system of\nequations:(\nσ0:U0→V1, ζ 07→ −ζ0,\nσ1:U1→V0, ζ 17→ −ζ1.\nDefine ¯ σ:=I−1◦σ◦I−1. It is straightforward to verify that σ◦¯σ= idCP1and ¯σ◦σ= idCP1.\nLetL2nO(1) = ( U0×C2n)∪(U1×C2n)/∼be a holomorphic vector bundle over CP1. The\nlocal coordinates on U0×C2nandU1×C2nare denoted as ( ζ0, x1\n0, . . . , x2n\n0) and ( ζ1, x1\n1, . . . , x2n\n1),\nrespectively. In these coordinates, ( ζ0, x0)∈U0×C2nand ( ζ1, x1)∈U1×C2nare equivalent if\n(\nζ0·ζ1= 1,\nxα\n0=ζ0·xα\n1forα= 1, . . . , 2n.\nSimilarly, we defineL2nO(1) = ( V0×C2n)S(V1×C2n)/∼as a holomorphic vector bundle\noverCP1. It is equipped with local coordinates ( ηi, y1\ni···y2n\ni) on Vi×C2n, i= 0,1. In these\ncoordinates, ( η0, y0)∈U0×C2nand ( η1, y1)∈U1×C2nare equivalent if\n(\nη0·η1= 1,\nyα\n0=η0·yα\n1forα= 1, . . . , 2n.\nWe define an antiholomorphic map I:L2nO(1)→L2nO(1) such that:\nL2nO(1)I−→L2nO(1)\n↓ ⟲ ↓\nCP1 I−→ CP1\nHere, we could express IasI0andI1:\n(\nI0:U0×C2n→V0×C2n,(ζ0, x0)7→(¯ζ0,¯x0)\nI1:U1×C2n→V1×C2n,(ζ1, x1)7→(¯ζ1,¯x1)\nIt is essential to note that the map I:L2nO(1)→L2nO(1) acts as an identity map in the\ncontext of smooth manifolds and possesses antiholomorphic properties. Similarly, the inverse\nmapI−1:L2nO(1)→L2nO(1) also exhibits antiholomorphic characteristics.HYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 15\nDefinition 3.1. A holomorphic mapping f:L2nO(1)→L2nO(1) is considered compatible\nwith σif the following diagram commutes, and the restriction of fto each fiber is a linear\nisomorphism:L2nO(1)f−→L2nO(1)\n↓ ⟲ ↓\nCP1 σ−→ CP1\nWe define ¯f:=I−1◦f◦I−1:L2nO(1)→L2nO(1).\nAfter the definition of a holomorphic mapping compatible with σhas been established, we\nnow turn our attention towards its specific representation. This definition provides us with a\nframework for analyzing such mappings, especially in terms of their linear isomorphism properties\non the fibers. Through this definition, we are equipped to delve deeper into the intrinsic structure\nof the mapping f. In the upcoming lemma, we will specifically explore a form of this mapping,\nshowcasing how it can be represented via a constant matrix A.\nLemma 3.2. Letf:L2nO(1)→L2nO(1)be a holomorphic mapping that is compatible with\nσ. Then fcan be represented as f0andf1defined as follows:(\nf0:U0×C2n→V1×C2n(ζ0, x0)7→(−ζ0,−A·x0)\nf1:U1×C2n→V0×C2n(ζ1, x1)7→(−ζ1, A·x1)\nwhere A∈GL(2n,C)is a constant matrix. Furthermore, if f◦¯f=id, then the matrix Asatisfies\nA·¯A=−I2n.\nProof. Consider a holomorphic map f:L2nO(1)→L2nO(1), which is compatible with σ. In\nthis context, fcan be decomposed into ( f0, f1), where:\n(\nf0:U0×C2n→V1×C2n,(ζ0, x0)7→(−ζ0, A0(ζ0)·x0),\nf1:U1×C2n→V0×C2n,(ζ1, x1)7→(−ζ1, A1(ζ1)·x1).\nHere, A0:U0∼=C→GL(2n,C) and A1:U1∼=C→GL(2n,C) are holomorphic maps. We\nimpose an additional condition that\n(ζ0, x0)∼(ζ1, x1)∈2nM\nO(1)⇐⇒ f(ζ0, x0)∼f(ζ1, x1)∈2nM\nO(1),\nwhich implies that −A1(ζ1) =A0(ζ−1\n1) for any ζ1∈U0∩U1∼=C∗. Consequently, A0:U0∼=C→\nC∗is a holomorphic matrix-valued function with bounded limits at infinity, indicating that A0\nis constant and A1=−A0. Let us denote A1=−A0=A.\nDefine fas the above map. Then, construct a holomorphic map ¯f=I−1◦f◦I−1:L2nO(1)→L2nO(1). ¯fcan be represented by ¯f0and ¯f1as:\n(¯f0:V0×C2n→U1×C,(η0, y0)7→\u0000\n−η0,−¯A·y0\u0001\n,\n¯f1:V1×C2n→U0×C,(η1, y1)7→\u0000\n−η1,¯A·y1\u0001\n.\nThese representations utilize the same trivializations as those defined for the map f. A direct\ncalculation shows that f◦¯f= id if and only if A·¯A=−I2n. □\nConsider a holomorphic map f:L2nO(1)→L2nO(1) that is compatible with σ. Let sbe a\nholomorphic section ofL2nO(1). We aim to examine certain holomorphic sections ofL2nO(1)\nthat are induced by s. Commonly, there are two types of such induced sections: ¯ s, defined as\n¯s:=I◦s◦I−1, and ˜ s, defined as ˜ s:=f◦s◦σ−1. We will analyze these sections in detail as\nfollows.16 SHUO WANG AND BIN XU\nSuppose s:CP1→L2nO(1) is a holomorphic section of the vector bundleL2nO(1). This\nsection can be represented as s0ands1with respect to the trivializationL2nO(1) = ( U0×\nC2n)∪(U1×C2n)/∼, as follows:\n(\ns0:U0→U0×C2n, ζ07→(ζ0, a+b·ζ0),\ns1:U1→U1×C2n, ζ17→(ζ1, a·ζ1+b),\nwhere a, b∈C2n. Considering the trivialization ofL2nO(1) used to represent s, we define a\nC-linear mapping\nφ:C4n→H0 \nCP1,2nM\nO(1)!\n; (a, b)7→s,\nwhich is an isomorphism given that dim CH0(CP1,L2nO(1)) = 4 n. Similarly, a C-linear isomor-\nphism ψ:C4n→H0\u0010\nCP1,L2nO(1)\u0011\ncan be defined. The isomorphisms φandψdepending\non the choice of trivialization.\nLetsbe as defined above and fbe the map given in Lemma 3.2. We define ¯ s:=I◦s◦\nI−1:CP1→L2nO(1) and the section ¯ scan be expressed as ¯ s0and ¯s1as follows:\n(\n¯s0:V0→V0×C2n, η 07→\u0000\nη0,¯a+¯b·η0\u0001\n,\n¯s1:V1→V1×C2n, η 17→\u0000\nη1,¯a·η1+¯b\u0001\n,\nwhere a, b∈C2n. Define ˜ s:=f◦s◦σ−1:CP1→L2nO(1) and the section ˜ scan be expressed\nas ˜s0and ˜s1as follows:\n(\n˜s0:V0→V0×C2n, η 07→(η0, A·(−a·η0+b)),\n˜s1:V1→V1×C2n, η 17→(η1, A·(−a+b·η1)),\nwhere a, b∈C2nandAis the matrix in the lemma. This demonstrates that both ¯ sand ˜sare\nholomorphic sections ofL2nO(1).\nIfs=φ(a, b) represents a holomorphic section ofL2nO(1), then ˜ sand ¯srepresent holo-\nmorphic sections ofL2nO(1) and can be expressed as ψ(A·b,−A·a) and ψ(¯a,¯b), respectively.\nConversely, if s=ψ(a, b) is a holomorphic section ofL2nO(1), then ˜sand¯sare holomorphic\nsections ofL2nO(1), given by ˜s:=¯f◦s◦¯σand¯s:=I−1◦s◦I, which can be expressed as\nφ(¯A·b,−¯A·a) and φ(¯a,¯b), respectively.\nIn summary, we present the following facts:\nLemma 3.3. Letsbe a holomorphic section ofL2nO(1), and let fbe a holomorphic map\ncompatible with σ. Then, finduces a linear automorphism rfonH0\u0010\nCP1,L2nO(1)\u0011\n, defined\nby\ns7→I−1◦f◦s◦σ−1◦I.\nGiven a trivialization ofL2nO(1) = ( U0×C2n)∪(U1×C2n)/∼, corresponding to a standard open\ncover of CP1, we can associate fwith a constant matrix A∈GL(2n,C)as detailed in Lemma\n3.2. Consequently, if s=φ(a, b)with respect to this trivialization, then rf(s) =φ\u0000¯A·¯b,−¯A·¯a\u0001\n.\nRemark 3.4.Letsbe a holomorphic section ofL2nO(1). We define t= ˜s=f◦s◦σ−1as a\nholomorphic section ofL2nO(1). According to the provided definition, it can be observed that\nrf(s) =¯t. Simultaneously, if we define r= ¯s=I◦s◦I−1as a holomorphic section ofL2nO(1),\nthen in this context, rf(s) is equivalent to ˜r.HYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 17\nDefinition 3.5. LetVbe a complex vector space. A conjugate linear map on V, denoted as\nγ:V→V, is defined as follows:\nγ(λ·v+µ·w) =¯λ·γ(v) + ¯µ·γ(w)\nfor all λ, µ∈Candv, w∈V.\n•Areal structure onVis a conjugate linear map r:V→Vthat satisfies r2= id V.\n•Aquaternionic structure onVis a conjugate linear map j:V→Vthat satisfies\nj2=−idV.\nLetVbe a complex vector space with a quaternionic structure denoted by j:V→V. Then,\nVis equipped with a hypercomplex structure characterized by three linear maps: I(v) =√−1·v,\nJ(v) =j(v), and K(v) =√−1·j(v). These maps satisfy the quaternionic relations I2=J2=\nK2=−idVand the compatibility condition K=I◦J=−J◦I. It is important to note that for\nsuch a quaternionic structure to exist, Vmust have an even complex dimension.\nDefinition 3.6. Areal structure onL2nO(1) is a holomorphic map f:L2nO(1)→L2nO(1)\nthat is compatible with σand satisfies f◦¯f= id.\nWe will now establish a correspondence between real structures on the vector bundleL2nO(1)\nand quaternionic structures on\nH0 \nCP1, 2nM\nO(1)!\n⊗ O(−1)!\n=H0\u0010\nCP1,CP1×C2n\u0011\n∼=C2n.\nwhich will play a crucial role in establishing hyperk¨ ahler structures in the following section.\nProposition 3.7.\n\nR:=n\nreal structures onL2nO(1)o\nR:=n\nreal structures on H0\u0010\nCP1,L2nO(1)\u0011o\nJ:=n\nquaternionic structures on H0\u0010\nCP1,\u0010L2nO(1)\u0011\n⊗ O(−1)\u0011o\nThere exists an injection from RtoR. Moreover, there exists a one-to-one correspondence\nbetween RandJ.\nProof. We consider trivializations ofL2nO(1) = ( U0×C2n)S(U1×C2n)/∼andO(−1) =\n(U0×C)S(U1×C)/∼with respect to the standard open cover of CP1.\nIn this trivialization, any holomorphic section ofL2nO(1) can be represented as φ(a, b), and\nany holomorphic section of\u0010L2nO(1)\u0011\n⊗ O(−1) can be represented by the constant x∈C2n.\nSuppose fis a real structure onL2nO(1), defined by f0andf1under the given trivialization.\nWe can find a constant matrix A∈GL(2n,C) such that A¯A=−I2n, and\n(\nf0:U0×C2n→V1×C2n(ζ0, x0)7→(−ζ0,−Ax0),\nf1:U1×C2n→V0×C2n(ζ1, x1)7→(−ζ1, Ax 1).\nThen fcorresponds to the real structure rfonH0\u0010\nCP1,L2nO(1)\u0011\nand the quaternionic struc-\nturejfonH0\u0010\nCP1,\u0010L2nO(1)\u0011\n⊗ O(−1)\u0011\nas follows: For a holomorphic section s=φ(a, b) of\nL2nO(1) and a holomorphic section x∈C2nof the bundle\u0010L2nO(1)\u0011\n⊗ O(−1) with respect\nto the given trivialization, define(\nrf(s) :=f◦s◦σ−1=φ\u0000¯A¯b,−¯A¯a\u0001\n,\njf(x) := ¯A¯x18 SHUO WANG AND BIN XU\nOn the other hand, for any quaternionic structure jonC2n, there exists a matrix A∈GL(2n,C)\nsuch that j(x) is given by ¯A·¯x. We can then construct f0andf1similarly as mentioned above to\nrepresent a real structure fonL2nO(1). The mapping from ftorfis invariant under different\nchoices of trivialization.\nNext, we will demonstrate that the correspondences between fandjfare also unaffected by\nthe choice of trivialization ofL2nO(1) in the following manner:\nIf another trivialization exists forL2nO(1) = ( U0×(C2n)′)S(U1×(C2n)′)/∼with respect\nto the standard open cover of CP1, let ( ζ0, x′\n0) and ( ζ1, x′\n1) be the coordinates on ( U0×(C2n)′)\nand ( U1×(C2n)′). Then there exist matrix-valued holomorphic functions Pi:Ui→GL(r,2n)\nsuch that ( ζi, xi) and ( ζi, x′\ni) represent the same point in CP1if and only if xi=Pi(ζi)·x′\ni,\ni= 0,1. Additionally, ( ζ0, x0)∼(ζ1, x1) if and only if ( ζ0, x′\n0)∼(ζ1, x′\n1), implying that\n(\nζ0·ζ1= 1\nx0= Diag( ζ0,···, ζ0)·x1⇔(\nζ0·ζ1= 1\nx′\n0= Diag( ζ0,···, ζ0)·x′\n1\nThis implies that P0(ζ0)−1·P1(ζ−1\n0) =I2nforζ0∈C∗. Hence, P0(ζ0) is a holomorphic function\nonU0∼=Cwith finite limits at infinity, indicating that P0is a constant function. We denote\nP1=P0:=P.\nUnder this new local trivialization, the real structure falso induces a matrix A′. It can be\nobserved that A′is equal to ¯P·A·P−1. Additionally, in the previous trivialization, x∈Cnis\ntransformed to P·xin the new trivialization. This is further mapped to ¯A′·P·x=P·¯A·¯x.\nNotably, this precisely represents the coordinates of jf(x) in the new trivialization. □\nRemark 3.8.It is worth noting that the definition of jf(s) is not independent of the chosen triv-\nialization of O(−1). Therefore, when discussing jf(s), it is essential to specify the trivialization\nofO(−1) that was used.\nLetfbe a real structure onL2nO(1). We choose a trivialization ( U0×C2n)S(U1×C2n)/∼\nforL2nO(1) and a trivialization for O(−1) based on the standard open cover of CP1. The maps\nrfandjfinduced by fhave the following relationship:\nCorollary 3.9. For any holomorphic section s=φ(a, b)inH0\u0010\nCP1,L2nO(1)\u0011\nunder this\nchosen trivialization, we can express rf(s)asφ(jf(b),−jf(a)). Moreover, we can express fas\nfollows:\n\nf0:U0×C2n→V1×C2n(ζ0, x0)7→\u0010\n−ζ0,−jf(x0)\u0011\n,\nf1:U1×C2n→V0×C2n(ζ1, x1)7→\u0010\n−ζ1,jf(x1)\u0011\n.\n4.Twistor spaces\nIn this section, we examine the sophisticated interconnection between holomorphic fiber bun-\ndles and hyperk¨ ahler manifolds, guided by the framework established by Hitchin, Karlhede,\nLindstr¨ om, and Roˇ cek [4, Section 3.F]. We describe Zas a twistor space, stipulating it as a\ncomplex manifold with dimension 2 n+ 1 that satisfies a set of predefined conditions:\n(1)Zforms a holomorphic fiber bundle over CP1, represented by p:Z →CP1.\n(2) A real structure τexists on Z, compatible with the antipodal map σ.\n(3)Zsupports a family of holomorphic sections M, each with a normal bundle TFs∼=L2nO(1).\n(4) A specific holomorphic section ωof Λ2(NZ)⊗p∗O(2) is present.HYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 19\nIn accordance with conditions (1), we define a complex manifold Mthat acts as the parameter\nspace for these holomorphic sections, each coupled with a normal bundle ofL2nO(1), as elab-\norated in Proposition 4.6. Building upon conditions (1), (2), and (3), the existence of a smooth\nmanifold M, endowed with a hypercomplex structure and integrated as a smooth submanifold\nofM, is established, as validated by Thoerem 4.13. Finally, under the comprehensive conditions\n(1), (2), (3), and (4), Theorem 4.15 presents a methodology for constructing a hyperk¨ ahler man-\nifold Mof real dimension 4 n. This manifold serves as the parameter space for real sections of\nthe fiber bundle p:Z →CP1, elucidating the deep connection between complex geometry and\nhyperk¨ ahler structures.\nIn the fiber bundle Z, the vertical space is denoted as NZand defined as the kernel of\nthe projection map d p:TZ → TCP1. The antipodal map on CP1, denoted by σ, is defined in\nsection 3. Specifically, ¯Zis a complex manifold that is topologically equivalent to Zbut possesses\nanother complex structure denoted as I¯Z:=−IZ. The map I:Z → ¯Zacts as an identity on\nthe smooth manifold on Zand as an antiholomorphic map from the complex manifold Zto¯Z.\nFurthermore, the conjugation from T1,0ZtoT0,1Ztransforms the vertical space NZintoNZ,\nwhich then serves as the vertical space for the projection ¯ p:¯Z →CP1.\nDefinition 4.1. Areal structure onZis a holomorphic map τ:Z → ¯Zsatisfying τ◦¯τ= idZ,\nwhere ¯ τ:=I−1◦τ◦I:¯Z → Z . In this context, ¯ τis an antiholomorphic map from ¯ZtoZ, and\nit aligns with τas a smooth map on Z.\nDefinition 4.2. A real structure τonZiscompatible with σif it satisfies the commutativity of\nthe following diagram:\nZτ−→ ¯Z\n↓ ⟲ ↓\nCP1 σ−→CP1\nLetτdenote a real structure on Z, and consider a holomorphic section s:CP1→ Z . Con-\nsequently, the image s(CP1) forms a complex submanifold within Z. Following this, the normal\nbundle Ns(CP1) ofs(CP1) inZis examined in accordance with Definition 2.7. Specifically, we\nobserve that TZ|s(CP1)equals NZ|s(CP1)⊕Ts(CP1), indicating that Ns(CP1) is equivalent to\nNZ|s(CP1). For brevity, we adopt the notation TFsto denote NZ|s(CP1), which we shall refer to\nas the normal bundle of section s. Ifτrepresents a real structure on Zthat is compatible with\nσ, as indicated by the relation ¯ p◦τ=σ◦p, then it follows that d¯ p◦dτ= dσ◦dp. From this,\nwe can infer that d τ(NZ) =NZ.\nDefinition 4.3. Letτbe a real structure on Z, compatible with σ. A holomorphic section sof\nZis termed a real section if it satisfies the commutativity of the following diagram:\nZτ−→ ¯Z\ns↑ ⟲ ↑¯s\nCP1 σ−→CP1\nwhere ¯ sis identified with sas smooth sections of the smooth fiber bundle Z →CP1and it is\nalso regarded as a holomorphic section of the holomorphic fiber bundle ¯Z →CP1.\nConsider a holomorphic section sofZwith the normal bundle TFs∼=L2nO(1). Define\n˜s:=τ◦s◦σ−1, which is a holomorphic section of ¯Z. In this context, the vector bundle d τ(TFs)\nprecisely aligns with the normal bundle TF˜sover ˜s(CP1) in ¯Z, given that d τ(NZ) equals NZ.\nFurthermore, since τis a biholomorphic map between Zand ¯Z, it follows that TF˜s= dτ(TFs)∼=L2nO(1). Importantly, when sis a real section, this indicates that ˜ scorresponds to both the20 SHUO WANG AND BIN XU\nsmooth section sofZand the holomorphic section ¯ sof¯Z. As a result, d τ(TFs) =TF˜scoincides\nwith TFs.\nLemma 4.4. Letτbe a real structure on Zcompatible with σ, and consider sas a real section\nofZwith the normal bundle TFs∼=L2nO(1). Then, the mapping d τ:TFs→TFsconstitutes\na real structure on TFs, as delineated in Definition 3.6.\nProof. For each point ζ∈CP1, dτmaps a vector in TFsats(ζ) to a vector in TFsatτ(s(ζ)) =\n¯s(σ(ζ)). This demonstrates that d τis compatible with σ. Furthermore, the condition τ◦¯τ= idZ\nimplies that d τ◦dτ= id. □\nBased on this, we can now provide the following definition:\nDefinition 4.5. Given a real structure τonZcompatible with σ, and a real section sofZwith\nthe normal bundle TFs∼=L2nO(1), we define r(τ, s) as the real structure on H0\u0000\ns(CP1), TFs\u0001\n.\nSimilarly, j(τ, s) is defined as the quaternionic structure on H0\u0000\ns(CP1), TFs⊗ O(−1)\u0001\n, both\ninduced by the real structure d τonTFsby proposition 3.7.\nAs highlighted in Remark 3.8, when discussing j(τ, s), it is crucial to specify a fixed trivi-\nalization of O(−1) for each real holomorphic section s. This is accomplished as follows: We\ninitially establish a trivialization of the vector bundle ϖ:O(−1)→CP1. Then, considering\nthe map p:s(CP1)→CP1as a biholomorphism and the pullback bundle p∗(O(−1)) on s(CP1)\nis endowed with a natural trivialization. This trivialization is induced by the original bundle\nϖ:O(−1)→CP1for each real holomorphic section s. In subsequent discussions regarding the\nbundle O(−1) over s(CP1), it is assumed that this particular trivialization is applied.\nWe define sas a holomorphic section of Z, where TFs∼=L2nO(1). The image s(CP1) forms\na compact complex submanifold of Z. Furthermore, we consider the regular deformation space\n^M4n\ns(CP1)within Z, as described in Definition 2.16. It has been established that^M4n\ns(CP1)is a\ncomplex manifold of dimension 4 nin theorem 2.18. We now proceed to define\nM:=(\nV∈^M4n\ns(CP1)|Vis a holomorphic section of Z;TFV∼=2nM\nO(1))\nLemma 4.6. Mconstitutes an open submanifold within^M4n\ns(CP1).\nProof. Letsbelong to M. Consider the standard open cover U0∪U1ofCP1, with each Ui\nisomorphic to Cand possessing a local coordinate ζifori= 0,1. An open cover U0∪U1of\ns(CP1) inZexists, where each Uiis represented as Bi×Ui, withBibeing open subsets of\u0000\nC2n;z1\ni,···, z2n\ni\u0001\nfori= 0,1. Furthermore, Uiserves as local charts on Zwith coordinates\n(z1\ni,···, z2n\ni, ζi). Define zi= (z1\ni,···, z2n\ni). The bundle map pis explicitly represented in these\ncoordinates as: p:Ui→Ui; (zi, ζi)7→ζi, fori= 0,1.\nWithout loss of generality, we assume that s(CP1)∩Ui={zi= 0}for each i= 0,1. The\ntransition functions between U0andU1are given by:(\nζ0·ζ1= 1 on U0∩U1∼=C∗\nzλ\n1=fλ(z0, ζ0) for λ= 1,···,2n\nHere,Videnotes Ui∩s(CP1) for i= 0,1. Considering that H1(s(CP1), TFs) = 0, there exists\nan analytic family π:V →B4n\nεof compact submanifolds which represents canonical local\ndeformations of s(CP1) within Mand is characterized by the following properties:\n(1)V0=s(CP1).\n(2) The Kodaira-Spencer map σtofπis a linear isomorphism for all t∈B4n\nε.HYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 21\n(3)πis maximal at each point t∈B4n\nε.\nMoreover, π−1(B4n\nε) is an open set in^M4n\ns(CP1). When εis sufficiently small, we only need to\nshow π−1(B4n\nε)⊂ M . Vector valued holomorphic functions φi:Vi×B4n\nε→C2n, defined as\nφi(ζi, t1,···, t4n) = (φ1\ni,···, φ2n\ni) for i= 0,1, satisfy the following conditions:\nTransition conditions:(\nζ0·ζ1= 1 on U0∩U1∼=C∗\nφλ\n1=fλ(φ1\n0,···, φ2n\n0, ζ0) for λ= 1,···,2n\nBoundary conditions:(\nφλ\ni(ζi,0) = 0 i= 0,1\n∂φλ\ni\n∂tρ\f\f\f\nt=0=βλ\nρ,iρ= 1,···,4n\nHere, the set {β1,···, β4n}forms a basis of H0(s(CP1), TFs), and\nβρ,i=\u0000\nβ1\nρ,i(ζi),···, β2n\nρ,i(ζi)\u0001\nrepresents the local representation of βρ,iwithin the local trivialization Ui×C2n⊂TFs. For each\nt∈Bε, the complex submanifold Vt:=π−1(t) is locally represented by Vt∩Ui={zλ\ni=φλ\ni(ζi, t)}\ninZ. Define st:CP1→ Z such that st(ζi) ={zλ\ni=φλ\ni(ζi, t), ζi} ∈Uifori= 0,1. Hence,\nVt:=π−1(t) is the image of the section stofZ. Our goal is to demonstrate that sthas a normal\nbundle TFst∼=L2nO(1) for each t∈B4n\nε, provided that εis sufficiently small.\nBy the Grothendieck lemma [3], TFst∼=L2n\nk=1O(nk(t)) as a holomorphic vector bundle. For\nεsufficiently small, we have:\nH0 \nst(CP1),2nM\nk=1O\u0000\nnk(t)\u0001!\n=H0 \ns(CP1),2nM\nk=1O(1)!\n= 4n\nH1 \nst(CP1),2nM\nk=1O\u0000\nnk(t)\u0001!\n≤H1 \ns(CP1),2nM\nk=1O(1)!\n= 0\nwhich implies that nk(t)≥ −1,k= 1,···,2n, and2nP\nk=1nk(t) = 2 n.\nThe system of functions\n\nβρ,i:Ui→C2n, ζi7→ \n∂φλ\ni(ζi, t)\n∂tρ\f\f\f\f\nt=t0!\nλ=1,···,2n\n\n\ni=0,1\ndefines a holomorphic section of TFst0for each ρ= 1,···,4nwhen t0is close to 0, denoted\nasβρ(t0). The set\u0000\nβ1(t0),···, β4n(t0)\u0001\nforms a basis for H0\u0000\nst0(CP1), TFst0\u0001\nthat remains\nholomorphic with respect to t0∈B4n\nε. For any fixed ( γρ)4n\nρ=1̸= 0 inC4n, the section β:=4nP\nρ=1γρβρ\nis a non-trivial holomorphic section of TFs∼=L2n\nk=1O(1) and therefore has at most a simple\nzero. Let β(t0) :=4nP\nρ=1γρβρ(t0) represent a section of TFst0, specifically β(0) = β. Locally on\nUi,β(t0)|Uican be expressed as\u0000\nβ1\ni(ζi, t0), . . . , β2n\ni(ζi, t0)\u0001\nfori= 0,1.\nIf the holomorphic section β(0) never vanishes on s(CP1), then neither does β(t0) fort0in the\nvicinity of 0. If βhas a simple zero, we can assume, without loss of generality, that the simple\nzero point of βisζ0= 0, meaning that βλ\n0(ζ0,0) = 0 for each λ= 1, . . . , 2n. Specifically, we\nassume that ζ0= 0 is a simple zero point of β1\n0(ζ0,0). Let {ζ0∈U0:∥ζ0∥= 1}be a simple loop22 SHUO WANG AND BIN XU\naround ζ0= 0. By Rouch´ e’s Theorem, β1\n0(ζ0, t0) would have exactly one simple zero point in\nthe domain {ζ0∈U0:∥ζ0∥<1}for each t0near 0. This implies that the holomorphic section\nβ0(t0) ofTFst0has at most one zero point in the domain st0\u0000\n{ζ0∈U0:∥ζ0∥<1}\u0001\n⊂st0(U0) for\neach t0near 0. Additionally, since βλ\ni(ζi,0) does not vanish over Ui∩ {ζ0∈U0:∥ζ0∥ ≤1}, then\nβλ\ni(ζi, t0) also has no zero points on Ui∩ {ζ0∈U0:∥ζ0∥ ≤1}fort0near 0. This implies that\nthe holomorphic section β(t0) has at most one simple zero point over st0(CP1).\nBased on the preceding discussion, we can conclude that any non-trivial holomorphic section\nofTFst0∼=L2n\nk=1O(nk(t0)) has at most one simple zero for t0in the vicinity of 0, implying that\nnk(t0)≤1. However, sinceP2n\nk=1nk(t) = 2 n, we have established that when εis sufficiently\nsmall, all nk(t0) for k= 1, . . . , 2nare equal to 1, i.e.,\nTFst∼=TFs∼=2nM\nk=1O(1) for all t∈B4n\nε.\n□\nIn summary, follow the markings above we present the following lemma:\nProposition 4.7. LetZbe a complex manifold, and let σbe the antipodal map over CP1. We\nsuppose the following:\n(1)Zis a holomorphic fiber bundle over CP1, denoted by p:Z →CP1.\n(2)Zadmits a holomorphic section swhose normal bundle is TFs∼=L2nO(1).\nThenZadmits a family of holomorphic sections that are parameterized by a complex manifold\nMof dimension 4n, where the normal bundle TFsis isomorphic toL2nO(1)for each s∈ M .\nFact 1. LetMbe a smooth manifold and let Gbe a finite group acting on Mby diffeomorphisms.\nThen the set of fixed points MG=n\nm∈M:g·m=m,∀g∈Go\nis a smooth submanifold of M.\nMoreover, for any m∈MG⊂M, the tangent vector Xm∈TmMGif and only if dg(Xm) =Xm\nfor any g∈G. (see, e.g. [1, Corollary 2.5, p. 309] .)\nLets∈ M be holomorphic sections of p:Z →CP1such that TFs∼=L2nO(1), and we define\ns:=τ◦s◦σ−1, which is also a holomorphic section of Z. We then obtain a diffeomorphism ι\nonM\nι:M → M s7→s\nwith ι2= idMandsis a real section of Mif and only if ι(s) =s. By the fact we mentioned\nabove, Mencompasses a family of real holomorphic sections, denoted as Mwhich is a smooth\nsubmanifold of Msince Mis the set of fixed points of ι. Suppose s∈MandXis a tangent\nvector in TsM, then X∈TsMif and only if d ι(X) =X.\nLet us recall that r(τ, s) refers to the real structure on H0\u0000\ns(CP1), TFs\u0001\nandj(τ, s) refers to\nthe quaternionic structure on H0\u0000\ns(CP1), TFs⊗ O(−1)\u0001\ninduced by d τ:TFs→TFsaccording\nto Definition 4.5. It is well-established that TsM∼=H0\u0000\ns(CP1), TFs\u0001\n. Consequently, the real\nstructure r(τ, s) on H0\u0000\ns(CP1), TFs\u0001\ndirectly induces the subspace TsMinTsM, as described\nbelow:\nLemma 4.8. For every real holomorphic section sin the manifold M, and for any tangent\nvector X∈TsM=H0(s(CP1), TFs), we obtain d ι(X) =r(τ, s)(X).\nProof. Letsνbe a path in Mforν∈[0,1] such that s0=sandd\ndν\f\f\nν=0sν=X.HYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 23\ndι(X) =d\ndν\f\f\f\f\nν=0ι(sν) = lim\nν→01\nν(τ◦sν◦σ−1−τ◦s◦σ−1)\n=dτ\u0012\nlim\nν→01\nν(sν◦σ−1−s◦σ−1)\u0013\n=dτ◦X◦σ−1\nFurthermore, according to the discussion in proposition 3.7, we have r(τ, s)(X) =dτ◦X◦σ−1=\ndι(X). □\nGiven a trivialization of the tangent bundle of TFswith respect to the standard open cover\nof the base space s(CP1), holomorphic sections of TFscan be represented by the functions\nφs(a, b), where aandbbelong to C2n. Similarly, holomorphic sections of TFs⊗ O(−1) can\nbe represented by the variable x∈C2n. Therefore, according to Corollary 3.9, the real struc-\nturer(τ, s)\u0000\nφs(a, b)\u0001\n=φs\u0010\nj(τ, s)(b),−j(τ, s)(a)\u0011\n. With these considerations, we can state the\nfollowing lemma:\nLemma 4.9. Fix any trivialization of TFswith respect to the standard open cover of s(CP1).\nFor any s∈M,φs(a, b)is a tangent vector in TsMif and only if b=−j(τ, s)(a).\nFor convenience, we will denote j(s, τ) asjif there is no ambiguity and denote TFs⊗ O(n)\nasTFs(n). The above lemma implies that when M̸=∅, it is a smooth submanifold of Mwith\ndimension 4 nbecause the tangent space TsMof any s∈Mis a 2 n-dimensional R-linear subspace\nofTsM. It is worth noting that TsMis not a C-linear space of TsM, and if we want Mto be a\ncomplex manifold, we must first give a complex structure to TsM. Let us do this as follows:\nIfφs(a,−j(a)) is a tangent vector in TsMwhich vanishes at ζ=ζ0, this means that a=\nζ0·j(a). Therefore, we have\nj(a) =j(ζ0·j(a)) =−¯ζ0·a=−∥ζ0∥2j(a)\nwhich implies that j(a) = 0, and thus a= 0 since jis a bijective map. In summary, if we regard\na tangent vector XinTsMas a holomorphic section in TFs, then if Xvanishes at one point on\ns(CP1), it has to vanish everywhere on s(CP1), meaning that X= 0. With this, we can state\nthe following lemma:\nLemma 4.10. For each fiber Zζ:=p−1(ζ)ofp:Z →CP1and any real section s∈M, there\nexists a neighborhood UofsinMsuch that Uis a family of real sections in Zthat intersect\np−1(ζ)at distinct points, provided that Uis sufficiently small.\nFor this lemma, Mcan be locally identified with any one of the fibers in {p−1(ζ)|ζ∈CP1}.\nTherefore, Minherits a complex structure Iζfrom the fiber p−1(ζ) ofZ. Thus, we obtain a family\nof complex structures on Mparameterized by CP1. Furthermore, id M: (M,Iζ)→(M,Iσ(ζ)) is an\nantiholomorphic map. We can also introduce another family of complex structures parameterized\nbyCP1onMas follows:\nLemma 4.11. For every element s∈M, the tangent space TsMconstitutes a real linear subspace\nwithin TsM. Furthermore, there exists a set of R-linear isomorphisms connecting TsMwith\nH0\u0000\ns(CP1), TFs(−1)\u0001\n.\nProof. After fixing a trivialization of TFswith respect to the standard open cover of CP1, we can\ndefine a R-linear map, denoted as Φ α,β:TsM→H0\u0000\ns(CP1), TF(−1)\u0001\n, for constants ( α, β)∈\nC2\\ {0}, as follows:\nΦα,β:φs\u0010\na,−j(τ, s)(a)\u0011\n7→α·a−β·j(τ, s)(a)∈C2n24 SHUO WANG AND BIN XU\nFirstly, the map Φ α,βis an isomorphism. This is demonstrated by the fact that if Φ α,βsends\nφs\u0000\na,−j(τ, s)(a)\u0001\nto zero, then it must be that α·a=β·j(τ, s)(a). Applying j(τ, s) to both\nsides of the equation yields ¯ α·j(τ, s)(a) =−¯β·a, leading to:\n|α|2·a= ¯αβ·j(τ, s)(a) =−|β|2·a\nwhich implies that a= 0, i.e., φs\u0000\na,−j(τ, s)(a)\u0001\nis the zero section. Moreover, both TsMand\nH0\u0000\ns(CP1), TF(−1)\u0001\nhave the same real dimension of 4 n. Secondly, the map Φ α,βis independent\nof the choice of trivialization on TFs. □\nRemark 4.12.Up to now, we have established an equivalence between TsMandH0\u0000\ns(CP1), TF(−1)\u0001\ninduced from the map Φ α,β. Furthermore, Φ α,βendows TsMwith a complex structure derived\nfrom the complex vector space H0\u0000\ns(CP1), TF(−1)\u0001\n. This is a family of complex structures on\nMparameterized by [ α:β]∈CP1.\nLemma 4.10 indicates that TsMinherits a complex structure from each fiber Zζ. We will\nclarify that the complex structures on TsMinherited from Zζcorrespond precisely to those\ncomplex structures on TsMinduced from H0\u0000\ns(CP1), TF(−1)\u0001\nby Φ α,β. To elucidate this fact,\nwe first need to establish a trivialization of TFs.\nConsider the standard open cover U0∪U1ofCP1, where each Uiis isomorphic to Cwith\na local coordinate ζifori= 0,1. There exists a corresponding open cover U0∪U1ofs(CP1)\nwithin Z. EachUican be represented as Bi×Ui, withBibeing a neighborhood of zero in C2n.\nWithout loss of generality, we can assume that s(CP1)∩Uicorresponds to zero in Bi. Since\nTFs∼=L2nO(1), we claim that there exist coordinates zi:= (zλ\ni)λ=1,...,2nonBisuch that\n∂\n∂zλ\n0\f\f\f\f\nz0=0=ζ1·∂\n∂zλ\n1\f\f\f\f\nz1=0for all λ= 1, . . . , 2n.\nWe will demonstrate this fact later. Consequently, we have the following trivialization of TFs:\n\n\nU0×C2n→TFs(ζ0, a1\n0,···, a2n\n0)7→2nP\nλ=1aλ\n0∂\n∂zλ\n0\f\f\f\ns(ζ0)\nU1×C2n→TFs(ζ1, a1\n1,···, a2n\n1)7→2nP\nλ=1aλ\n1∂\n∂zλ\n1\f\f\f\ns(ζ1)\nThis particular trivialization of TFswill be used throughout the remainder of this paper. Con-\nsider a tangent vector Xa\ns:=φs(a,−j(τ, s)) in TsMfor a real section sunder the trivialization\nwe defined above. In detail, we can write Xa\nsas:\n\n\nXa\ns|s(ζ0)=2nP\nλ=1(aλ−ζ0\u0000\nj(τ, s)(a)\u0001λ)∂\n∂zλ\n0\f\f\f\ns(ζ0)\nXa\ns|s(ζ1)=2nP\nλ=1(ζ1aλ−\u0000\nj(τ, s)(a)\u0001λ)∂\n∂zλ\n1\f\f\f\ns(ζ1)(4.1)\nLet us consider the complex structure Iζ0onTsM, which is inherited from Zζ0for some\nζ0∈U0. This complex structure sends Xa\ns|s(ζ0)to√−1·Xa\ns|s(ζ0), and in detail:\n2nX\nλ=1\u0010\naλ−ζ0\u0000\nj(τ, s)(a)\u0001λ\u0011∂\n∂zλ\n0\f\f\f\f\ns(ζ0)7→√\n−1· 2nX\nλ=1\u0010\naλ−ζ0\u0000\nj(τ, s)(a)\u0001λ\u0011∂\n∂zλ\n0\f\f\f\f\ns(ζ0)!\n.\nTherefore, Iζ0maps Xa\nstoXb\ns, where b∈C2nsatisfies the equation:\nb−ζ0·j(τ, s)(b) =√\n−1·\u0000\na−ζ0·j(τ, s)(a)\u0001\n.HYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 25\nIn fact, we can solve this equation and obtain:\nb=√−1(1− |ζ0|2)\n1 +|ζ0|2a−2√−1ζ0\n1 +|ζ0|2j(a).\nOn the other hand, let\nα=1− |ζ0|2\n1 +|ζ0|2andβ=2ζ0\n1 +|ζ0|2.\nthen, the complex structure induced by Φ α,βfrom H0\u0000\ns(CP1), TFs(−1)\u0001\nis just the complex\nstructure inherited from Zζ0. Up to this point, we have successfully obtained a hypercomplex\nstructure on M, as discussed in the following proposition. Notably, Joyce has also explored\nhypercomplex structures in his book [5, Section 7.5].\nTheorem 4.13. Consider a complex manifold Zequipped with an antipodal map σover the\nprojective line CP1. We impose the following conditions for Zto facilitate our study:\n(1)The manifold Zmust be structured as a holomorphic fiber bundle over CP1, formally\nrepresented by the projection p:Z →CP1.\n(2)It is necessary for Zto possess a real structure τ, which aligns with the antipodal map\nσ, ensuring compatibility across the manifold.\n(3)The existence of a holomorphic section sis crucial, where the section’s normal bundle\naligns with TFs∼=L2nO(1), highlighting the manifold’s dimensional complexity.\nGiven these stipulations, let Mdenote the collection of real holomorphic sections within M.\nShould Mbe non-empty, it inherently forms a smooth submanifold of M, further endowed with\na hypercomplex structure.\nProof. Letsbe a real section of Z, and with respect to the trivialization of TFsmentioned\nabove, we can write any tangent vector in TsMin the form φs(a,−j(a)) for some ainC2n. Let\nIα,βbe a family of complex structures on Minduced by Φ α,β:φs\u0000\na,−j(a)\u0001\n7→α·a−β·j(a),\nin more detail, Iα,βsends φs(a,−j(a)) to φs(b,−j(b)) for some binC2nsuch that\nα·b−β·j(b) =√\n−1·\u0000\nα·a−β·j(a)\u0001\nWe can solve this equation for certain pairs ( α, β) and obtain\n(\nI1,0:φs\u0000\na,−j(a)\u0001\n7→φs\u0000√−1·a,−j(√−1·a)\u0001\nI1,i:φs\u0000\na,−j(a)\u0001\n7→φs\u0000\nj(a), a\u0001\nDenote I:=I1,0,J:=I1,i, andK:=I◦J. We note that Iis the complex structure on Minherited\nfrom the fiber Z0andJis the complex structure on Minherited from the fiber Z√−1(−1±√\n2).\nThen, I,J,Kare three complex structures on Mand can check that K=I◦J=−J◦IThen\n(M,I,J,K) is a hypercomplex manifold. □\nTo complete the proof of Theorem 4.13, it is essential to calculate the local charts on Zin the\nvicinity of point s.\nLemma 4.14. Coordinates zi:= (zλ\ni)λ=1,···,2ncan be selected on Bisuch that∂\n∂zλ\n0\f\f\f\nz0=0=\nζ1·∂\n∂zλ\n1\f\f\f\nz1=0holds for every λ= 1,···,2n.\nProof. Select any local coordinates ( z1\ni,···, z2n\ni, ζi) onZwhereUiserves as the local chart\nfori= 0,1. Define zi= (z1\ni,···, z2n\ni) for i= 0,1 and, without loss of generality, assume\ns(CP1)∩Ui={zi= 0}for each i. The transition functions between U0andU1are:26 SHUO WANG AND BIN XU\n(\nζ0·ζ1= 1 on U0∩U1∼=C∗\nzλ\n1=fλ(z0, ζ0) for λ= 1,···,2n\nObserve that fλ(0, ζ0) = 0. Define Vi=Ui∩s(CP1) for i= 0,1. The transition matrix of\nTFsfromV0toV1is:\nF:=\n∂f1\n∂z1\n0···∂f1\n∂z2n\n0.........\n∂f2n\n∂z1\n0···∂f2n\n∂z2n\n0\n\f\f\f\f\f\f\f\f\nz0=0\nSince TFs∼=L2nO(1), there exist matrix-valued holomorphic functions A0:V0→GL(2n,C)\nandA1:V1→GL(2n,C) such that:\nA−1\n1(ζ−1\n0)·F·A0(ζ0) =\nζ−1\n0\n...\nζ−1\n0\n\nLet (wi, ηi) := ( Ai(ζi)·zi, ζi) be new coordinates on Uifori= 0,1. The corresponding transition\nfunctions between U0andU1are:\n(\nη0·η1= 1\nwλ\n1=gλ(w0, η0) :=\u0000\nA−1(η1)\u0001λ\nµ·fµ(A0(η0)·w0, η0)\nThe transition matrix TFsfromV0toV1is:\nG:=\n∂g1\n∂w1\n0···∂g1\n∂w2n\n0.........\n∂g2n\n∂w1\n0···∂g2n\n∂w2n\n0\n\f\f\f\f\f\f\f\f\nw0=0=A−1\n1(ζ−1\n0)·F·A0(ζ0)\nThe refined transition functions between U0andU1are thus:\n(\nη0·η1= 1 where ηi∈V0∩V1\nwλ\n1=η−1\n0·wλ\n0+h(w0, η0)\nHere, hλ(0, η0) =∂hλ\n∂wµ\n0\f\f\f\nw0=0= 0 for each λ, µ= 1,···,2n. Consequently,∂\n∂wλ\n0\f\f\f\nw0=0=η1·\n∂\n∂wλ\n1\f\f\f\nw1=0is valid for every λ= 1,···,2n. □\nFinally, we establish the central theorem of this study, initially put forth by Hitchin, Karlhede,\nLindstr¨ om, and Roˇ cek in their work [4].\nTheorem 4.15. LetZbe a complex manifold and σbe the antipodal map over CP1. Suppose\nthat:\n(1)Zis a holomorphic fiber bundle over CP1, denoted by p:Z →CP1.\n(2)There exists a real structure τonZthat is compatible with σ.\n(3)Zadmits a family Mof holomorphic sections, where the normal bundle TFsis isomor-\nphic toL2nO(1)for each s∈ M .\n(4)There exists a holomorphic section ωofΛ2(NZ)∗⊗p∗O(2)such that:\n(a)ωdefines a symplectic form on each fiber of p:Z →CP1.HYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 27\n(b)For each real section s∈ M ,ω(·, j(τ, s)·)is a negative definite quadratic form on\nH0\u0000\ns(CP1), TFs⊗ O(−1)\u0001∼=C2n.\nLetMbe the set of real holomorphic sections in M. Under these conditions, if M̸=∅, then it\nis a smooth submanifold of M. Additionally, there exists a hyperk¨ ahler structure on M.\nRemark 4.16.(4-b) states that for any x∈H0\u0000\ns(CP1), TFs⊗ O(−1)\u0001\n, we always have ω\u0000\nx, j(τ, s)(x)\u0001\n≤\n0, and ω\u0000\nx, j(τ, s)(x)\u0001\n= 0 if and only if x= 0.\nProof. For any real section s∈ M , the restriction of ωons(CP1) denoted as ω|s(CP1)is a\nholomorphic section of Λ2(TFs)∗⊗ O(2). It can be considered as a 2-form with values in O(2).\nThis indicates that ωis a nondegenerate skew-symmetric 2-form on H0\u0000\ns(CP1), TFs(−1)\u0001\n.\nFor the sake of clarity, our analysis will concentrate exclusively on the complex structure in\nTsMderived from Z0. More precisely, we investigate the complex structure engendered by the\nmapping Φ : TsM→H0\u0000\ns(CP1), TFs(−1)\u0001\n, which is defined such that φs\u0000\na,−j(a)\u0001\n7→a∈C2n.\nIn this framework, the negative definite quadratic form ω, as referenced in condition (4), is\nconsidered as a 2-form on TsM. We denote ω’s restriction on Zζ0for any ζ0∈U0asωζ0.\nAssuming sto be a real section and considering a, b∈C2nas sections of TFs(−1) under the\nspecified trivialization, ω(a, b) is thus rendered a holomorphic function on CP1, implying its\nconstancy. Moreover, ω(a, b)|Zζ0equates to ωZζ0\u0000\na|s(ζ0), b|s(ζ0)\u0001\n. Therefore, it follows that\nω(a, b) =ωZζ0\u0000\na|s(ζ0), b|s(ζ0)\u0001\nfor every ζ0∈U0⊂CP1.\nThe quaternionic structures jonH0(s(CP1), TF(−1)) induce quaternionic structures J:=\nΦ◦j◦Φ−1onTsM. We can represent the quaternionic structure JonTsMas follows: using\nthe trivialization of TFsoutlined in equation 4.1, we have\nJ:TsM→TsM, φ s\u0000\na,−j(a)\u0001\n7→φs\u0000\nj(a), a\u0001\n.\nFor any s∈M, the metric g:TsM×TsM→Ris defined by\u0010\nφs\u0000\na,−j(a)\u0001\n, φs\u0000\nb,−j(b)\u0001\u0011\n7→ −ω\u0000\na, j(b)\u0001\n−ω\u0000\nb, j(a)\u0001\n,\nwith the well-definedness of the Riemannian metric gonMbeing confirmed by condition (4-b).\nLetXa\ns:=φs\u0000\na,−j(a)\u0001\ndenote a tangent vector in TsMand a normal vector field on s(CP1).\nChoose ζ0∈U0⊂CP1and consider a local isomorphism f:U⊂M→U′⊂ Z ζ0:=p−1(ζ0).\nFor every a∈C2nands∈U, utilizing the trivialization of TFsas described in equation 4.1, we\nobtain:\ndf(Xa\ns) =Xa\ns|s(ζ0)=2nX\nλ=1\u0010\na−j(a)ζ0\u0011λ∂\n∂zλ\n0\f\f\f\f\ns(ζ0).\nNext, we define a 2-form ψζ0onUbyψζ0:TsM×TsM→C, where ψζ0(Xa\ns, Xb\ns) := ω\u0000\na−\nj(a)ζ0, b−j(b)ζ0\u0001\n. Here, a−j(a)ζ0andb−j(b)ζ0are viewed as sections of TFs(−1), and\nimportantly, for any ζ0∈U0,\nω\u0010\na−j(a)ζ0, b−j(b)ζ0\u0011\n=ωZζ0 2nX\nλ=1\u0000\na−j(a)ζ0\u0001λ∂\n∂zλ\n0\f\f\f\f\ns(ζ0),2nX\nλ=1\u0000\nb−j(b)ζ0\u0001λ∂\n∂zλ\n0\f\f\f\f\ns(ζ0)!\nThus, ψζ0=f∗\u0000\nω|U′\u0001\nconstitutes a closed form on U, given that ωZζ0is a holomorphic symplectic\nform on Zζ0, in accordance with condition (4-a).\nA complex structure is defined on TsMfor each s∈Mas follows:\n\n\nI:φs\u0000\na,−j(a)\u0001\n7→φs\u0000√−1·a,−j(√−1·a)\u0001\n,\nJ:φs\u0000\na,−j(a)\u0001\n7→φs\u0000\nj(a), a\u0001\n,\nK:φs\u0000\na,−j(a)\u0001\n7→φs\u0000√−1·j(a),−√−1·a\u0001\n.28 SHUO WANG AND BIN XU\nHere, I2=J2=K2=−id and I◦J=K. It suffices to verify that g(I·,I·) =g(J·,J·) =\ng(K·,K·) =g(·,·) and that g(I·,·),g(J·,·),g(K·,·) are closed forms.\nLemma 4.17. g(I·,I·) =g(J·,J·) =g(K·,K·) =g(·,·).\nProof. Given any Xa\ns:=φs\u0000\na,−j(a)\u0001\nandXb\ns:=φs\u0000\nb,−j(b)\u0001\ninTsM, we find that\ng\u0000\nIXa\ns,IXb\ns\u0001\n=g\u0000\nφ(√\n−1·a,√\n−1·j(a)), φ(√\n−1·b,√\n−1·j(b))\u0001\n=−ω\u0000√\n−1·a, j(√\n−1·b)\u0001\n−ω\u0000√\n−1·b, j(√\n−1·a)\u0001\n=ω\u0000\na, j(b)\u0001\n−ω\u0000\nb, j(a)\u0001\n=g(Xa\ns, Xb\ns).\nSimilarly, g(JXa\ns,JXb\ns) =g(Xa\ns, Xb\ns) and g(KXa\ns,KXb\ns) =g(Xa\ns, Xb\ns). □\nLemma 4.18. g(I·,·),g(J·,·), and g(K·,·)are closed 2-forms on M.\nProof. Consider any s0∈Mandζ0∈CP1. Here, ψζ0is a closed 2-form on a neighborhood Uof\ns0inM, defined by\nψζ0(Xa\ns, Xb\ns) :=ω\u0010\na−j(τ, s)(a)ζ0, b−j(τ, s)(b)ζ0\u0011\nfor all s∈U. Therefore,\ng(IXa\ns, Xb\ns) =−ω\u0010√\n−1·a, j(τ, s)(b)\u0011\n−ω\u0010\nb, j(τ, s)(√\n−1·a)\u0011\n=−√\n−1\u0010\nω\u0000\na, j(τ, s)(b)\u0001\n+ω\u0000\nj(τ, s)(a), b\u0001\u0011\n=−√−1\n2\u0000\nψ−1−ψ1\u0001\u0010\nXa\ns, Xb\ns\u0011\n.\nThis shows that g(I·,·) coincides with −√−1\n2(ψ−1−ψ1), a closed 2-form in U. Consequently, as\nclosed forms are defined locally, we conclude that g(I·,·) is a closed 2-form. Similar reasoning\napplies to g(J·,·) and g(K·,·), establishing them as closed 2-forms. □\nWith this, we conclude the proof of the theorem 4.15 : ( M, g, I,J,K) is a hyperk¨ ahler manifold.\n□\n5.the twistor space of C2n\nProposition 5.1. Considering C2nas a hyperk¨ ahler manifold, the twistor space Zconstructed\nfromC2nis biholomorphically equivalent to the vector bundleL2nO(1). Conversely, starting\nfrom the holomorphic vector bundleL2nO(1)equipped with a specific real structure, a defor-\nmation space Mof the zero section 0sin the bundle can be established. Furthermore, Mis\nisomorphic to C4n= (a1, a2, b1, b2), with a1, a2, b1,andb2being elements of Cn. The hyperk¨ ahler\nmanifold M, derived via this approach, can be considered as a smooth submanifold of M, repre-\nsented by M⊂ M as(a1, a2,−i¯a2, i¯a1).\nRemark 5.2.In subsequent analysis, we will focus on the case where n= 1 for simplicity. It is\nimportant to note that scenarios with n >1 can be conceptualized as iterative applications of\nthen= 1 case.\nConsider the complex vector space C2with complex coordinates ( w1, w2). Let M∼=C2be a\nsmooth manifold isomorphic to R4. A natural global coordinate system exists on M, denoted\nby\u0000\nℜw1,ℑw1,ℜw2,ℑw2\u0001\n, and alternatively represented as ( w1, w2). The complexification of\nthe real tangent bundle TM yields a trivial complex bundle TCMof rank four, framed globallyHYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 29\nby\u0010\n∂\n∂w1,∂\n∂w2,∂\n∂¯w1,∂\n∂¯w2\u0011\n. A complex structure on M, defined as a bundle map I:TM→TM\nsatisfying I2=−id, extends to TCMas aC-linear map I:TCM→TCMwith I2=−id.\nEmploying the frame on TCM, we can represent three complex structures I,J,Kas matrices:\nI∼\ni\ni\n−i\n−i\n;J∼\n1\n−1\n1\n−1\n;K∼\ni\n−i\n−i\ni\n.\nNow denote the standard Euclidean metric gonMby:\ng=1\n2\u0010\ndw1⊗d ¯w1+ d ¯w1⊗dw1+ dw2⊗d ¯w2+ d ¯w2⊗dw2\u0011\nWe can verify that ( M, g, I,J,K) is a hyperk¨ ahler manifold.\nLemma 5.3. The twistor space Zis biholomorphically equivalent to the vector bundleL2O(1)\noverCP1.\nRemark 5.4.According to Hitchin-Karlhede-Lindstr¨ om-Roˇ cek’s theorem [4], we can readily es-\ntablish the aforementioned lemma. Indeed, it is feasible to explicitly construct an isomorphism\nϕ:L2O(1)∼− → Z .\nProof. First, we consider the standard open cover of CP1, denoted as U0∪U1, with coordinates\nζ0andζ1. A local trivialization forL2O(1) is provided relative to this cover on CP1. The\nbundle is represented asL2O(1) = ( U0×C2)∪(U1×C2)/∼, with coordinates ( ζ0, z1\n0, z2\n0) and\n(ζ1, z1\n1, z2\n1) on them. The equivalence of ( ζ0, z1\n0, z2\n0)∈U0×C2and ( ζ1, z1\n1, z2\n1)∈U1×C2holds\nifζ0·ζ1= 1, z1\n0=ζ0·z1\n1, and z2\n0=ζ0·z2\n1. For the twistor space Z, constructed from Mand\ndiffeomorphic to M×CP1, an open cover is defined as ( U0×M)∪(U1×M)/∼, with coordinates\n(ζi, w1\ni, w2\ni) onUi×Mfori= 0,1. The points ( ζ0, w1\n0, w2\n0)∈U0×Mand ( ζ1, w1\n1, w2\n1)∈U1×M\nare equivalent in Zifζ0·ζ1= 1,w1\n0=w1\n1, and w2\n0=w2\n1.\nThe complex structure IonZis defined as follows: The restriction of ItoT(·,ζ0)Zis given by\nIζ0⊕I0, where I0represents the standard complex structure on CP1and\nIζ0=1\n1 +ζ0¯ζ0\u0000\n(1−ζ0¯ζ0)I+ (ζ0+¯ζ0)J+\u0000√\n−1(−ζ0+¯ζ0)\u0001\nK\u0001\nIn this context, the complex structure on the fiber Z0over 0 ∈CP1precisely corresponds to I\nas defined above. Furthermore, we can express the complex structure Iζ0onZζ0forζ0∈U0as\na matrix I(ζ0) with respect to the frame\u0000∂\n∂w1,∂\n∂w2,∂\n∂¯w1,∂\n∂¯w2\u0001\n, where I(ζ0) is equal to\n1\n1 +ζ0¯ζ0\n√−1(1−ζ0¯ζ0) 2 ζ0 √−1(1−ζ0¯ζ0) −2ζ0\n2¯ζ0√−1(1−ζ0¯ζ0)\n−2¯ζ0√−1(1−ζ0¯ζ0)\n\nNow, we establish an isomorphism denoted by ϕ:L2O(1)∼− → Z . This isomorphism, rep-\nresented by ϕ0andϕ1with respect to the given charts, is defined as follows: For ϕ0, we map\nU0×C2→U0×Mby\n(ζ0, z1\n0, z2\n0)7→\u0012\nζ0,1\n1 +ζ0¯ζ0\u0000\nz1\n0+√\n−1ζ0¯z2\n0\u0001\n,1\n1 +ζ0¯ζ0\u0000\nz2\n0−√\n−1ζ0¯z1\n0\u0001\u0013\nSimilarly, for ϕ1, we map U1×C2→U1×Mby\n(ζ1, z1\n1, z2\n1)7→\u0012\nζ1,1\n1 +ζ1¯ζ1\u0000¯ζ1z1\n1+√\n−1¯z2\n1\u0001\n,1\n1 +ζ1¯ζ1\u0000¯ζ1z2\n1−√\n−1¯z1\n1\u0001\u001330 SHUO WANG AND BIN XU\nThe final step of our proof is the verification that ϕis a well-defined bundle isomorphism, demon-\nstrating that ϕenables a holomorphic mapping betweenL2O(1) and Z.\nFirst, consider the scenario where ( ζ0, z1\n0, z2\n0) and ( ζ1, z1\n1, z2\n1) represent the same point inL2O(1). In this case, ϕ0(ζ0, z1\n0, z2\n0) and ϕ1(ζ1, z1\n1, z2\n1) correspond to the same point in Z, con-\nfirming that ϕis a smooth map.\nNext, we examine the holomorphic property of ϕ. Given that the complex structure on Zis\ndescribed by Iζ⊕I0, with I0denoting the standard complex structure on CP1, it is necessary to\nverify that the restriction of ϕto each fiber Zζconstitutes a holomorphic map for every ζ∈CP1.\nSpecifically, for ζ0∈U0, the restriction of ϕ0to the fiber {ζ0} ×C2is expressed as\nϕ0\f\f\f\n{ζ0}×C2: (z1\n0, z2\n0)7→1\n1 +ζ0¯ζ0\u0010\nz1\n0+√\n−1ζ0¯z2\n0, z2\n0−√\n−1ζ0¯z1\n0\u0011\nThe Jacobian matrix of ϕ0|ζ0×C2, denoted J(ζ0), is defined as\nJ(ζ0) :=1\n1 +ζ0¯ζ0\n1√−1ζ0\n1 −√−1ζ0\n−√−1¯ζ0 1√−1¯ζ0 1\n\nAfter verifying that det( J) = 1, we conclude that the restriction of ϕ0toζ0×C2is anR-linear\nisomorphism for each ζ0∈U0. Moreover, ϕ0|ζ0×C2is holomorphic since\nI(ζ0)·J(ζ0) =J(ζ0)·\ni\ni\n−i\n−i\n\nThis indicates that Iζ0◦(ϕ0)∗= (ϕ0)∗◦Ifor each ζ0∈U0, meaning the restriction of ϕto each\nfiberZζ0is holomorphic for every ζ0∈U0.\n□\nIn our previous discussion, we explored the twistor spaceL2O(1), derived from the hy-\nperk¨ ahler manifold M∼=C2. We now address the inverse problem of constructing a hyperk¨ ahler\nmanifold from the holomorphic fiber bundleL2O(1). A crucial step in this construction is\nthe introduction of a real structure τonL2O(1). This is achieved by considering a point\np= (x, y) inM, where the section ζ7→(ζ, p) forms a real section in Z. Through the isomor-\nphism ϕ:L2O(1)∼− → Z , this real section corresponds to a holomorphic section sofL2O(1),\nrepresented by s0ands1as:\n\n\ns0:U0→U0×C2, ζ07→\u0010\nζ0, x−√−1ζ0¯y, y+√−1ζ0¯x\u0011\n,\ns1:U1→U1×C2, ζ17→\u0010\nζ1, ζ1x−√−1¯y, ζ1y+√−1¯x\u0011\n.\nThe map τtransforms\n\u0010\nζ, x−√\n−1ζ¯y, y+√\n−1ζ¯x\u0011\n∈U0×C27→\u0010\n−¯ζ,−¯ζx−√\n−1¯y,−¯ζy+√\n−1¯x\u0011\n∈U1×C2,\nthereby establishing a real structure onL2O(1). With the real structure τ, the zero section\n0sbecomes a real section ofL2O(1), as outlined in Definition 4.3. The normal bundle of 0 s\ncoincides withL2O(1), leading to H1(0s,L2O(1)) = 0. Consequently, we can explore the\nregular deformation space]M4n\n0sof 0sin the total space ofL2O(1). In this context, we canHYPERCOMPLEX STRUCTURES ARISING FROM TWISTOR SPACES 31\nconstruct an analytic family of compact submanifolds, as per Definition 2.4. With respect to the\ntrivialization ofL2O(1), any holomorphic section scan be denoted as:\n(\ns0:U0→U0×C2, ζ07→(ζ0, a+bζ0),\ns1:U1→U1×C2, ζ17→(ζ1, aζ1+b),\nand represented by φ(a, b). Furthermore, the real section ofL2O(1), corresponding to the real\nsection ζ7→(ζ, x, y ) ofZdiscussed above, can be expressed as φ(x, y,−√−1¯y,√−1¯x).\nWe aim to construct an analytic family of compact submanifolds ofL2O(1), each of dimen-\nsion one and parameterized by H0\u0010\nCP1,L2O(1)\u0011\n, as described in Definition 2.4. Consider\nV=F\n(a,b)∈C4φ(a, b) as a submanifold ofL2O(1)×H0\u0010\nCP1,L2O(1)\u0011\n, which has a complex\ndimension of five. The natural projection π:V → H0\u0010\nCP1,L2O(1)\u0011\nconstitutes the analytic\nfamily we seek.\nLemma 5.5. Lets=φ(a1, a2, b1, b2)be any holomorphic section ofL2O(1), then the normal\nbundle TFsofs(CP1)is isomorphic toL2O(1).\nProof. Consider the open charts\u0000\nUi:=Ui×C2;ζi, z1\ni, z2\ni\u0001\nonL2O(1). The transition functions\nare given by(\nζ0·ζ1= 1,\nz1\n0=ζ0·z1\n1, z2\n0=ζ0·z2\n1.\nBy the definition of sabove,(\ns∩U0is represented by z1\n0=a1+b1ζ0z2\n0=a2+b2ζ0,\ns∩U1is represented by z1\n1=a1ζ1+b1z2\n1=a2ζ1+b2.\nIntroduce alternative local coordinates ( η0,˜z1\n0,˜z2\n0) onU0by setting η0=ζ0and ˜zα\n0=zα\n0−(aα+\nbαζ0) for α= 1,2. Similarly, on U1, set η1=ζ1and let ˜ zα\n1=zα\n1−(aαζ1+bα), where α= 1,2.\nLet\u0000\nVi=Ui∩s(CP1); ˜z1\ni,˜z2\ni\u0001\nbe open charts on s(CP1). Then, the normal bundle TFsofs(CP1)\nis represented by a transformation matrix from V0toV1, as defined in Definition 2.7,\nB(ζ) =\n∂˜zα\n1\n∂˜zβ\n0\f\f\f\f\f\ns(CP1)\n\nα,β=1,2=\u0012ζ−1\n0\nζ−1\n0\u0013\n.\nwhich implies that TFsis isomorphic toL2O(1). □\nBy Lemma 5.5 above,]M4n\n0sis precisely the parameter space H0\u0010\nCP1,L2O(1)\u0011\n∼=C4, since\nH1(s(CP1), TFs) = 0 for each holomorphic section sofL2O(1), and moreover, we consider\nM:=(\nV∈]M4n\n0s\f\f\fVis a holomorphic section of2M\nO(1) and TFV∼=2M\nO(1))\nto be precisely the entire set π−1\u0010\nH0\u0010\nCP1,L2O(1)\u0011\u0011\n=\b\nφ(a, b)|(a, b)∈C4\t\n. In summary,\nwe have the following equation:\nM=]M4n\n0s=H0 \nCP1,2M\nO(1)!\n∼=C4.\nA holomorphic section φ(a1, a2, b1, b2) is a real section if and only if ( b1, b2) equals ( −√−1¯a2,√−1¯a1).\nBased on this criterion, we can view Mas the complex vector space C4= (a1, a2, b1, b2), and32 SHUO WANG AND BIN XU\nthe hyperk¨ ahler manifold Mcan be considered as a smooth submanifold of M, represented as\nthe set n\n(a1, a2,−√\n−1¯a2,√\n−1¯a1)|(a1, a2)∈C2o\n.\nThis formulation captures the conditions under which a holomorphic section is also a real section,\nthereby characterizing the elements of Mthat correspond to points on the hyperk¨ ahler manifold\nM.\nReferences\n[1] Glen E Bredon. Introduction to compact transformation groups . Academic press, 1972.\n[2] Charles Ehresmann. Les connexions infinit´ esimales dans un espace fibr´ e diff´ erentiable. In Colloque de topologie,\nBruxelles , volume 29, pages 55–75, 1950.\n[3] Alexander Grothendieck. Sur la classification des fibr´ es holomorphes sur la sphere de riemann. American\nJournal of Mathematics , 79(1):121–138, 1957.\n[4] Nigel J Hitchin, Anders Karlhede, Ulf Lindstr¨ om, and Martin Roˇ cek. Hyperk¨ ahler metrics and supersymmetry.\nCommunications in Mathematical Physics , 108(4):535–589, 1987.\n[5] Dominic D Joyce. Compact manifolds with special holonomy . Oxford Mathematical Monographs, 2000.\n[6] K Kodaira. On the variation of almost-complex structure, algebraic geometry and topology. In A Symposium\nin Honor of S. Lefschetz, 1957 , pages 139–150. Princeton University Press, 1957.\n[7] Kunihiko Kodaira. A theorem of completeness of characteristic systems for analytic families of compact sub-\nmanifolds of complex manifolds. Annals of Mathematics , pages 146–162, 1962.\n[8] Kunihiko Kodaira and Donald Clayton Spencer. On deformations of complex analytic structures, i. Annals of\nMathematics , 67(2):328–401, 1958.\nSchool of Mathematical Sciences, University of Science and Technology of China, Hefei 230026\nChina\nEmail address :ws220122@mail.ustc.edu.cn\nCAS Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical\nSciences, University of Science and Technology of China, Hefei 230026 China\nEmail address :bxu@ustc.edu.cn"    },    {        "title": "2402.13594v1.Two_dimensional_Closed_Form_Analytical_Model_of_Laterally_Excited_Film_Bulk_Acoustic_Wave_Resonator_Multiferroic_Antennas.pdf",        "content": "Two-dimensional Closed-Form Analytical Model of Laterally-Excited Film Bulk\nAcoustic Wave Resonator Multiferroic Antennas\nLouis-Charles Ippet-Letembet,1Rui-Fu Xu,1Robin Jeanty,1Zhi\n(Jackie) Yao,2Robert N. Candler,3, 4and Shih-Yuan Chen1, 5,∗\n1Graduate Institute of Communication Engineering,\nNational Taiwan University, Taipei 10617, Taiwan\n2Applied Mathematics and Computational Research Division,\nLawrence Berkeley National Laboratory, Berkeley, California 94720, USA\n3California NanoSystems Institute, Los Angeles, California 90095, USA\n4Department of Electrical and Computer Engineering,\nUniversity of California, Los Angeles, California 90095, USA\n5Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan\n(Dated: February 22, 2024)\nTo overcome the physical limitations of electrically small antennas, strain-mediated magneto-\nelectric antennas have been studied experimentally and theoretically. However, current closed-\nform analytical models include solely one-dimensional approaches. This paper proposes a two-\ndimensional closed-form analytical model of a laterally-excited multiferroic antenna. To model\nthickness-shear vibrations, we propose an extended Mason model to two dimensions. This theory\nsolves unidirectionally-coupled elastodynamics-magnetization dynamics-electrodynamics. Without\nusing the infinitesimal dipole approximation, this model provides a closed-form solution of the ra-\ndiated electromagnetic field with an explicit dependency of the antenna response on the physical\nparameters, enabling their optimization. Furthermore, we compare the efficiency-bandwidth prod-\nuct of different multiferroic antennas with Chu’s limit in the Akhiezer regime, which delimits the\nhighest achievable mechanical quality factor for acoustic resonators at GHz frequencies. We show\nthat phonon-phonon coupling of Akhiezer damping is most likely to limit multiferroic antenna radi-\nation, not Chu’s limit. Moreover, judicious choice of materials can enhance the maximum efficiency\nby more than an order of magnitude.\nI. INTRODUCTION\nThe increasing demand for short-range sensors for\nbiomedical applications, portable wireless devices for\nmonitoring applications, and long-range sensors for ex-\ntremely low-frequency communication systems has led\nthe path for antenna miniaturization. However, tradi-\ntional electrically small resonant antennas suffer from\nphysical limitations such as low efficiency and bandwidth,\nand the platform effect due to the proximity with the\nground plane. These limitations have caused antennas\nto be the constraining factor on many systems over the\npast several decades [1–3].\nIn recent years, mechanical antennas have been devel-\noped based on a mechanically induced radiation mech-\nanism. In this mechanism, static electric charges, elec-\ntric dipoles, or magnetic dipoles are accelerated through\nmechanical motion, potentially alleviating the difficulties\nassociated with traditional electrically small antennas [4–\n6]. Technologies employ rotating permanent magnets [7],\nrotating electrets [8], or ferromagnetic-piezoelectric het-\nerostructures [9–11].\nAmong these new kinds of antennas, the latter, also re-\nferred to as multiferroic antennas, when integrated into\nfilm bulk acoustic wave resonator (FBAR), have shown\n∗shihyuan@ntu.edu.twto be of particular interest for GHz frequencies applica-\ntions [9, 10, 12]. The radiation source is the dynamic\nmagnetization (spin wave) in the ferromagnetic thin film\ngenerated through the piezoelectrically induced magneto-\nelectric effect. Modeling such an antenna implies solving\na system of coupled equations that include elastodynam-\nics, magnetization dynamics, and electrodynamics.\nDomann and Carman first proposed a one-dimensional\nclosed-form model of strain-powered rod antennas, com-\nposed of a single bulk piezoelectric material or a single\nbulk piezomagnetic material [13]. However, radiation-\ninduced from ferromagnetic materials is not in the scope\nof such a model where magnetization dynamics is not\nconsidered. Moreover, the model cannot be applied to\nmultilayer structures and neglects the additional ma-\nterial required for the excitation of ferromagnetic ma-\nterials. Later, an analytical study used the classi-\ncal Mason model to study the radiation by a multi-\nlayer piezoelectric-piezomagnetic structure at Very-Low-\nFrequency [14]. The circuit model is one-dimensional\nand employed for obtaining the impedance characteristic\nof the multiferroic antenna. Nevertheless, magnetization\ndynamics is not included, and the radiation is calculated\nby combining piezomagnetic and piezoelectric constitu-\ntive equations with the dipole approximation. Luong\nand Wang investigated the dynamic magnetoelastic cou-\npling in multiferroic antennas, where for different classes\nof magnetic materials, the impact of the applied stress\norientation was analyzed for optimal coupling [15]. How-arXiv:2402.13594v1  [physics.app-ph]  21 Feb 20242\never, the study neglects elastodynamics, and the magne-\ntization is thus only time-dependent. Furthermore, elec-\ntrodynamics is also not considered and no quantification\nof the radiation is proposed.\nLater, numerical methods have been proposed to\nstudy bulk acoustic wave (BAW) multiferroic anten-\nnas. Yao et al. introduced a three-dimensional uncon-\nditionally stable finite-difference time-domain (FDTD)\nalgorithm for the study of BAW multiferroic anten-\nnas near ferromagnetic resonance (FMR) through the\nsimultaneous solving of the coupled Landau-Lifshitz-\nGilbert/Newton’s/Maxwell’s equations system [16]. Ji\net al. used a one-dimensional FDTD algorithm to in-\nvestigate the impact of magnon-phonon coupling on the\nantenna polarization [17]. However, because the physics\nphenomena considered intervene at very different size\nscales, numerical algorithms require a large computa-\ntional time to solve the fully coupled partial differential\nequations system. A series of Finite-Element-Methods\n(FEM) based studies have later been proposed to ana-\nlyze multiferroic antennas under thickness excitation of\nthe piezoelectric [18–21]. There are methods using the\ncommercially available software COMSOL where mag-\nnetostrictive and piezoelectric effects are analyzed sep-\narately with the magnetostrictive and the piezoelectric\nmodules of the software. The radiated electromagnetic\n(EM) field is then calculated with the dipole approxima-\ntion. Contrary to FDTD algorithms, these FEM-based\nstudies using COMSOL do not solve the system of gov-\nerning equations simultaneously and therefore neglect the\nmutual coupling between the governing equations. In do-\ning so, they do not use the full potential of a 3D solu-\ntion while still keeping its disadvantages, namely the very\nlengthy computational time.\nWe propose here a two-dimensional closed-form an-\nalytical model of multiferroic antennas integrated into\na laterally excited FBAR (XBAR). The XBAR gener-\nates thickness-shear modes, implying mechanical vibra-\ntions in both thickness and lateral directions. A Ma-\nson model extended to two dimensions is proposed to\naccount for the mechanical wave propagation in both di-\nrections. Considering a unidirectional coupling between\nthe governing equations, the knowledge of the mechani-\ncal displacement in the ferromagnetic layer enables us to\nobtain a closed-form analytic solution of the space-time-\ndependent spin wave. The spin wave is plugged into the\nvector potential as a source term and allows us to calcu-\nlate the radiated EM field without infinitesimal dipole ap-\nproximation. The closed-form solutions explicitly spec-\nify the parametric relationships, leading the path for an\noptimization of these parameters. Finally, the efficiency-\nbandwidth product of different multiferroic antennas is\ncompared with Chu’s limit in the Akhiezer regime. The\nAkhiezer regime refers to the phonon-phonon scattering,\nwhich sets for an acoustic resonator, the highest mechan-\nical quality factor achievable in the GHz frequencies. We\nshow that with a proper selection of piezoelectric and fer-\nromagnetic materials, multiferroic antennas operating atthe Akhiezer limit can approach Chu’s limit.\nII. THEORETICAL DESCRIPTION OF\nLATERALLY EXCITED MULTIFERROIC\nANTENNAS\nA. Operating principle of multiferroic antennas\nThe multiferroic antenna consists of a thin-film piezo-\nelectric material deposited on a silicon substrate with a\npair of electrodes placed along the x-axis. On top of\nthe piezoelectric is deposited a thin-film ferromagnetic\nmaterial. The silicon substrate is etched away to create\nan air cavity beneath the piezoelectric, enabling vibra-\ntions in both lateral and thickness directions. As shown\nin Fig. 1(a), the active region studied in this analyti-\ncal model is the free-standing piezoelectric-ferromagnet\nstructure. The active region is delimited by the length\n(L) of the ferromagnet and the thickness of the antenna.\nThe boundary conditions along the thickness and lateral\ndirections are stress-free conditions. For comparison and\nverification, the structure is also analyzed using COM-\nSOL, as shown in Fig. 1(a), where both electrodes and\npiezoelectric material combined with perfectly matched\nlayers (PML) are considered. Aluminum Nitride (AlN),\nis used as piezoelectric material for its high longitudinal\nacoustic velocity. Nickel (Ni) is used as ferromagnetic\nmaterial and Platinum (Pt) for the electrodes. When\na time-varying voltage is applied to the electrodes, the\npiezoelectric is excited and generates acoustic waves in\nthe form of thickness-shear waves. The acoustic waves\npropagate into the ferromagnetic layer, generating time-\nharmonic standing spin waves, the source of EM radia-\ntion.\nThe dynamic response of the multiferroic antenna is\npredicted with a set of equations comprising Maxwell’s\nequations for EM radiation, the Landau-Lifshitz (LL)\nequation for magnetization dynamics, and Newton’s Law\nfor acoustic waves:\n∇ ×E=−∂B\n∂t,∇ ×H=J+∂D\n∂t(1)\n∂m\n∂t=−µ0γ(m×Heff)−αLµ0γm×(m×Heff) (2)\nρ∂2u\n∂t2=∇ ·σ,ε=1\n2\u0010\n(∇u)T+∇u\u0011\n(3)\nIn (1), E,Hare the electric field and the magnetic field,\nD,Bare the electric displacement field and the magnetic\nflux density, and Jis the free current. In (2), mis the\nmagnetization unit vector defined as m=M\nMs, with M\nthe magnetization vector, Msthe saturation magnetiza-\ntion.Heffis the effective magnetic field composed of the\ndifferent physics phenomena that force the magnetization\nvector to rotate. µ0is the vacuum permeability, γis the\ngyromagnetic ratio with a value of −1.76×1011C·kg−1,3\nandαLis a dimensionless damping constant with a value\nof 0.025 [22]. In (3), σ,εare the stress and strain fields.\nuis the displacement vector and ρthe density. The su-\nperscript Trefers to the transpose.\nHooke’s law relates the stress and strain fields through\nthe material elastic stiffness tensor Cand the viscosity\ntensor η. For a non-piezoelectric material:\nσ=C:ε+η∂ε\n∂t(4)\nIn the piezoelectric, Hooke’s law takes the form of the\nso-called piezoelectric constitutive equations:\nσ=C:ε+η∂ε\n∂t−eT·E\nD=e:ε+ϵE(5)\nwhere eis the piezoelectric constant in the stress-charge\nformulation, and ϵis the piezoelectric permittivity.\nThe system of equations is solved in a unidirectional\nway. First, we solve Newton’s law combined with the\nextended two-dimensional Mason model. Only the Vil-\nlari effect (inverse magnetostriction) is considered in this\nwork. Thus, the displacement vectors obtained through\nelastodynamics enable us to find the expression of the\nmagnetoelastic field, the major source of magnetization\nrotations. Subsequently, the LL equation is solved which\ngives the spin wave. Finally, the spin wave is plugged\ninto Maxwell’s equations as a source term to obtain the\nEM far-field.B. Elastic waves\nThe electrodes placed along the x-axis generate parti-\ncle displacement on the xz-plane, namely thickness-shear\nmodes. The dominant stress in the structure is the shear\nstress in the same plane. The shear stress vanishes at the\nlateral boundaries at x= 0 and x=L, which leads to\nthe following general form for the x and z components of\nthe displacement vector in the structure:\nux=∞X\nn=1uxn\u0000\nu+\nxe−jkzz+u−\nxejkzz\u0001\nsin (knx)\nuz=∞X\nn=1uzn\u0000\nu+\nze−jkzz+u−\nzejkzz\u0001\ncos (knx)(6)\nThe propagation constant along the lateral direction x\nis directly fixed in each layer by the boundary conditions\nand its value is kn=(2n−1)π\nLwhere n∈N.uxnand\nuznare the eigenvectors. The piezoelectric and the fer-\nromagnetic materials have different mechanical proper-\nties, rendering the bilayer structure inhomogeneous. As\na consequence, the wavenumber kzalong the thickness\ndirection varies from one layer to another. It will be\ndenoted as kpin the piezoelectric and kfin the ferro-\nmagnetic layer. The two propagation constants along\nthe thickness direction cannot be determined merely by\nthe application of boundary conditions. They are deter-\nmined by the dispersion relations. Inserting uxanduz\nin Newton’s Law and examining individual modes gives\nthe dispersion relations in each layer (See Supplemental\nMaterial [23]).\nω2=\u0010\nCf\n11+Cf\n44\u0011\u0010\nk2\nn+k2\nf\u0011\n2ρ+r\u0010\nCf\n11−Cf\n44\u00112\u0010\nk2\nf−k2n\u00112\n+ 4k2nk2\nf\u0010\nCf\n44+Cf\n12\u00112\n2ρ(7)\nω2=\u0010\nCp\n11+Cp\n44\u0011\nk2\nn+\u0010\nCp\n33+Cp\n44\u0011\nk2\np\n2ρ\n1 +vuuuut1−\u0010\nCp\n11k2n+Cp\n44k2p\u0011\u0010\nCp\n33k2p+Cp\n44k2n\u0011\n−\u0010\nCp\n13+Cp\n44\u00112\nk2nk2p\u0010\nCp\n11+Cp\n44\u0011\nk2n+\u0010\nCp\n33+Cp\n44\u0011\nk2p\n(8)\nIn (7) and (8), ωis the frequency of the AC voltage\napplied at the electrodes. Cf\nij, Cp\nijare the elastic stiff-\nness constants in the ferromagnetic and the piezoelectric\nlayers, respectively, when mechanical damping is consid-\nered:\nCf,p\nij=Cf,p\nij(1 +jηs) (9)\nwhere ηsis the viscosity constant. The dispersion rela-\ntions relate known quantities, the frequency of the AC\napplied voltage ωand the wavenumber knin the lateraldirection, to the unknown wavenumbers along the thick-\nness direction ( kp, kf). Therefore, the latter can be ex-\ntracted from the dispersion relations. However, the two\ndispersion relations taken separately cannot uniquely de-\ntermine the structure’s resonance frequencies. A tool is\nthus needed to obtain the resonance frequencies of such\na multi-layer structure, and this is accomplished by the\nproposed extended Mason model. In the study of BAW\nresonators, conventionally, the Mason model has only\nbeen employed as a 1D approximation for obtaining the\nstructure’s electrical response: the resonance frequencies\nthrough the input impedance or the reflectivity of acous-4\ntic mirrors for instance [24]. The 1D analysis is then\ncombined with COMSOL for the study of mode shapes\n[25]. On the contrary, we show here that extended to\ntwo dimensions and combined with Newton’s law, an ex-\ntended Mason model can give both electrical response\nand mode shapes.\nC. Extended Mason model: electrical response\nWe define an extended Mason model per unit length\nand extend it to two dimensions taking advantage of the\ncoupling in Newton’s law between the x and z compo-\nnents of the displacement vector. Suppose we project\nNewton’s equation onto the z-axis and examine the waves\npropagating along the positive and negative thickness di-\nrections. In that case, one can find the coupling relation\nbetween the magnitudes along x and z. In the piezolayer\nfor instance (See Supplemental Material [23]):\n\u0014\nu+\nz\nu−\nz\u0015\n=jUp\u0014\n−u+\nx\nu−\nx\u0015\n(10)\nwhere Upis the coupling constant defined by:\nUp=kpkn\u0010\nCp\n44+Cp\n13\u0011\n−ρω2+Cp\n33k2p+Cp\n44k2n(11)\nIn other words, the coupling constant Upallows a cir-\ncuit representation of a 2D structure where the wave\npropagation in one direction is considered first and the\nwave propagation in the other direction is directly de-\nduced by (10). Fig. 1(b) is the extended Mason model\nfor the present bilayer resonator under lateral field exci-\ntation. The left part of the circuit with an electrical port\nrepresents the piezolayer, and the right part of the cir-\ncuit is the ferromagnetic layer. The stress-free boundary\nconditions at the top and bottom of the active region are\nrepresented in the circuit by two acoustic shorts. One\nat the bottom of the piezolayer, denoted by the plane\nz= 0, and the other at the top of the ferromagnetic\nlayer denoted by the plane z=z2. The plane z=z1\nis the piezoelectric-ferromagnet interface. In the circuit,\nthe voltages are the modal shear forces denoted as F(z).\nOwing to the orthogonality of the mechanical waves,\nwe modify the definition of the shear force by taking the\nweighted integral along the x-axis of the shear stress. In\nthe piezolayer:\nF(z) =−tpZL\n0sin (knx)σxzdx\n=Zp\u0000\njωu+\nxe−jkpz−jωu−\nxejkpz\u0001\n+2e15V tp\n(2n−1)π\n(12)\nwhere tp=z1is the thickness of the piezolayer, Vthe\ntime-varying voltage applied to the electrodes, and e15\nthe piezoelectric coupling constant for an electric fieldapplied along the x-axis. Defining the modal shear force\nas a weighted integral allows two things. First to quantify\nhow much each mode couples with the applied electric\nfield in the lateral direction, represented by the coefficient\n2\n(2n−1)π. Second to account for the coupling between the\nwaves propagating in the thickness and lateral directions\nin the definition of the mechanical impedance Zp:\nZp=Ltp\n2Cp\n44kp\nω\u0012\n1−knUp\nkp\u0013\n(13)\nIn the circuit, the currents are the magnitude of the\nacoustic velocities along the lateral direction denoted as\nvx(z). Applying Kirchhoff’s laws, we can solve for the\nmagnitudes of the acoustic velocities at the planes z= 0\nandz=z1:\nvx(0) =−jv\u0012\nZftan (kftf) +Zptan\u0012kptp\n2\u0013\u0013\nvx(z1) =jvZptan\u0012kptp\n2\u0013\nv=2e15tp\n(2n−1)πV\nZ2p\u0010\nZftan(kftf)\nZptan(kptp)+ 1\u0011(14)\nwhere tf=z2−z1andZfare the ferromagnetic layer’s\nthickness and mechanical impedance, respectively. (14)\ngives the magnitude of the acoustic velocities of the n-th\npropagating mode at the top and bottom of the piezo-\nlayer with respect to the applied voltage V. The modal\ncurrent is the surface integral of the modal displacement\nfield along the thickness and width of the piezolayer [26]:\nIn=jωZWp\n0Zz1\n0Ddydz (15)\nwhere Wpis the width of the piezolayer. Note that the\nmodal current is a sin ( knx) function. Therefore, we mod-\nify the classic Mason model again by taking the modal\ncurrent’s mean value. It enables to take into account the\ncontribution of the propagation in the lateral direction\nto the modal current’s magnitude:\n˜In=1\nLZL\n0Indx (16)\nThe input admittance Yincan be obtained by express-\ning the total current Ias a function of the applied voltage\nV. The total current Iis defined as the sum of the static\ncurrent, related to the piezoelectric static capacitance,\nand the current flowing in the motional branches. The\nlatter is the sum over all the modes of the mean current\n˜In. The input admittance Yincan now be written as:\nYin=jωC 0 \n1 +∞X\nn=1k2\neff,n!\n(17)\nwhere C0=Wptpϵ\nLis the piezoelectric static capacitance,5\n(a)\n(b)\n(c)\n (d)\nFIG. 1: (a) 2D Schematic representation of the FBAR antenna. (b) Two-dimensional extended Mason model. Com-\nparison of the input admittance and quality factor obtained with COMSOL and the model in (c) and (d).\nandk2\neff,n is the effective electromechanical coupling co-\nefficient of the n-th mode that can be expressed as:\nk2\neff,n =k2\nxK2\nkptp/2tan\u0010\nkptp\n2\u0011\n+Zf\n2Zptan (kftf)\n1 +Zftan(kftf)\nZptan(kptp)(18)\nwhere K2=e2\n15\nϵCp\n44. Note here that k2\nx=8\nπ21\n(2n−1)2cor-\nresponds to the lateral component and follows from the\nBerlincourt’s formula [27]:\nk2\nx=\u0010RL\n0Exsin (knx) dx\u00112\n\u0010RL\n0E2xdx\u0011\u0010RL\n0sin (knx)2dx\u0011\n=8\nπ21\n(2n−1)2(19)\nwhere Exis the electric field induced by the applied volt-\nage along the x-axis. The input admittance is now de-\nfined. The mechanical quality factor Qof the XBAR\nmultiferroic antenna can be expressed as [28]:\nQ=ω\f\f\f\fdS11\ndω\f\f\f\f1\n1− |S11|2(20)where S11can be expressed in terms of the input admit-\ntance as:\nS11=1−YinZs\n1 +YinZs(21)\nwhere Zs= 50 Ω is the system impedance.\nMaterial properties of AlN and Ni layers, along with\nthe Pt electrodes, are tabulated in Table I. The thickness\nof the AlN and Ni layers are 400 and 300 nm respectively.\nThe width of the AlN layer is Wp= 5 mm and the length\nof the active region is L= 400 µm. In COMSOL, the\nelectrodes’ thickness and length are 30 nm and 200 µm\nrespectively. The Anchor losses are modeled in COMSOL\nand this model with a common viscosity constant for the\npiezoelectric and ferromagnetic materials of ηs= 0.001.\nThe dielectric losses are also taken into account with a\nloss tangent of tan δ= 0.01.\nFig. 1(c) shows the frequency responses of the input\nadmittance obtained with this model and with COM-\nSOL simulation. The first two acoustic resonances are\nobserved at 3.49 GHz and 6.63 GHz. Fig. 1(d) compares\nthe quality factor of the XBAR multiferroic antenna ob-\ntained with this model and with COMSOL. At each reso-\nnance, the quality factor reaches a value of approximately6\n1000, corresponding to the inverse of the viscosity con-\nstant.\nD. Extended Mason model: mode shape\nWe show here that the extended two-dimensional Ma-\nson model allows us to describe particle displacement\nin both the thickness and lateral directions. In the\npiezolayer, the relation between the displacement mag-\nnitude ( u+\nx, u−\nx) and the acoustic velocities magnitude\n(vx(0), vx(z1)) is (See Supplemental Material [23]):\njω\u0014\nu+\nx\nu−\nx\u0015\n=1\n2jsin (kptp)\u0014\nejkpz1vx(0)−vx(z1)\n−e−jkpz1vx(0) + vx(z1)\u0015\n(22)\nRecall that (14) defines the acoustic velocities mag-\nnitude and that the coupling condition (10) relates the\nmagnitudes ( u+\nz, u−\nz) in the other direction. Therefore,\nthe particle displacement along the thickness and lateral\ndirections in the piezolayer can be expressed as:\nux=∞X\nn=1juxn\n(2n−1)π(a1sin(kp(z−z1)) +a2sin(kpz))\n×sin(knx)ejωt\nuz=∞X\nn=1uznUp\n(2n−1)π(a1cos(kp(z−z1)) +a2cos(kpz))\n×cos(knx)ejωt\n(23)\nwhere a1, a2are constants defined as:\na1=e15Ex\u0010\nZf\nZptan (kftf) + tan\u0010\nkptp\n2\u0011\u0011\nCp\n44(kp−knUp) sin ( kptp)\u0010\nZftan(kftf)\nZptan(kptp)+ 1\u0011\na2=e15Extan\u0010\nkptp\n2\u0011\nCp\n44(kp−knUp) sin ( kptp)\u0010\nZftan(kftf)\nZptan(kptp)+ 1\u0011(24)\nThe continuity of the stress is automatically consid-\nered in the circuit when Kirchhoff’s laws are applied. As\na consequence, only the continuity of the displacement\nvector at the piezoelectric-ferromagnet interface needs to\nbe verified. Taking into account the stress-free condi-\ntion at the top surface of the ferromagnetic layer leads\nto the following displacement vector components in the\nferromagnetic layer:\nux=∞X\nn=1juxn\n(2n−1)πa3cos (kf(z−z2)) sin ( knx)ejωt\nuz=−∞X\nn=1uznUp\n(2n−1)πa4sin (kf(z−z2)) cos ( knx)ejωt\n(25)where a3, a4are constants defined as:\na3=e15Extan\u0010\nkptp\n2\u0011\nCp\n44(kp−knUp) cos ( kftf)\u0010\nZftan(kftf)\nZptan(kptp)+ 1\u0011\na4=e15Ex\nCp\n44(kp−knUp) sin( kptp)\n×Zf\nZptan(kftf) + tan\u0010\nkptp\n2\u0011\n(1 + cos( kptp))\nsin(kftf)\u0010\nZftan(kftf)\nZptan(kptp)+ 1\u0011(26)\nIn each layer, the shear stress and strain in the xz-plane\nare derived using Hooke’s law:\nεxz=∂ux\n∂z+∂uz\n∂x\nσxz=√\n3Cf,p\n44εxz(27)\nwhere we considered pure shear stress. In the present\nstudy, we considered the active region as a bilayer struc-\nture for simplicity. In the case where a structure with\nmore than two layers is considered, the solution proce-\ndure of the problem is similar. Adding extra layers of\nnon-magnetic buffer materials will shift the resonance fre-\nquencies and similarly to the present case, the continuity\nof the shear force would be automatically considered in\nthe circuit. The continuity condition on the displacement\nvector solely needs to be applied to define in each layer\nthe mode shape.\nFigs. 2(a)-2(d) plot in the active region, at the first\ntwo resonances, the magnitude of the displacement vector\nobtained with this model and with COMSOL. At the first\nresonance or 3.49 GHz, we observe zero displacements\natz≈450 nm. The displacement magnitude starts to\nincrease when leaving the plane z≈450 nm. It indicates\nthat the shear force, proportional to the displacement\ngradient, is higher at this plane. At the second resonance\nor 6.63 GHz, the displacement gradient is higher in the\nmiddle of both layers, indicating that the shear force is\nmaximized in the middle of each layer. This is indeed\nverified by the plots of the stress magnitudes at the first\nand second resonances shown in Figs. 2(e)-2(f). The\nmaximum shear stress in the ferromagnetic layer reaches\na value of 0.35 MPa for the first resonance and 0.42 MPa\nfor the second resonance. As a consequence, the second\nmode will generate higher magnetization rotation in the\nferromagnet through the magnetoelastic field, leading to\nhigher radiation.\nThe knowledge of mechanical displacement enables us\nto define the magnetoelastic field, the principal compo-\nnent of the effective magnetic field that generates the\ntorque that rotates the magnetization. As will be de-\ntailed next, the spin wave is calculated analytically for\nboth time and space dependencies with a unidirectional\ncoupling considered.7\n(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFIG. 2: Displacement magnitude in µmat first resonance with COMSOL in (a) and with this model in (b), and at\nsecond resonance with COMSOL in (c) and with this model in (d). Shear stress magnitude with this model in Pa at\nthe first resonance in (e), and the second resonance in (f).\nE. Standing spin waves\nThe LL equation describes the magnetization preces-\nsion around an effective magnetic field Heff, which can\nbe decomposed into several components:\nHeff=H0+Hexc+Hani+Hme (28)\nwhere H0=H0⃗ xis the bias magnetic field oriented along\nthe x-axis in this study. Hexcthe exchange interaction\nfield, Hanithe magnetocrystalline anisotropy field, and\nHmethe magnetoelastic field.\nFor thin ferromagnetic films, the demagnetizing field\nalong the thickness direction forces the magnetization to\nlie in the xy-plane. In the absence of a strong out-of-plane\nexternal magnetic field that would cancel the demagne-\ntizing effect, the magnetization unit vector mhas a weak\nz component. We will neglect this component and con-\nsider the magnetization unit vector to lie entirely in the\nxy-plane:\nm=mx⃗ x+my⃗ y\n= cos φ⃗ x+ sin φ⃗ y(29)\nwhere φis the angle between mand the x-axis. For a\ncubic crystal, the exchange interaction field, the mag-\nnetocrystalline anisotropy field, and the magnetoelasticfield are written respectively:\nHexc=2Aexc\nµ0Ms\u0000\n∇2mx⃗ x+∇2my⃗ y\u0001\nHani=−2K1\nµ0Ms\u0000\nmx(1−m2\nx)⃗ x+my(1−m2\ny)⃗ y\u0001\nHme=6λ111\nµ0Msσxzmx⃗ z(30)\nAexcis the exchange stiffness constant, K1the cubic\nanisotropy constant and λ111the magnetostriction con-\nstant along the crystallographic direction [111]. The pro-\njection on the x or y-axis of the LL equation gives the\npartial differential equation satisfied by the angle φ:\n∂φ\n∂t=2αLγAexc\nMs∇2φ−αLγK1\n2Mssin 4φ−αLµ0γH0sinφ\n+6γλ111\nMsσxzcosφ (31)\nAcoustically actuated magnetoelectric antennas radi-\nate electromagnetic waves through a perturbation of the\nmagnetization [15]. The magnetization is biased along\nthe x-axis, and the dominant static magnetization is\nalong the x-axis. The magnetization unit vector can\ntherefore be linearized as:\nm=⃗ x+φ⃗ y (32)8\n(a)\n (b)\n(c)\n (d)\nFIG. 3: Magnitude (a),(b) and phase (c),(d) of the time-harmonic standing spin wave φhat the first and second\nresonances.\nThe linearization of the magnetization vector also lin-\nearizes the LL equation:\n∂φ\n∂t=2αLγAexc\nMs∇2φ−αLγ\u00122K1\nMs+µ0H0\u0013\nφ\n+6γλ111\nMsσzx(x, z) (33)\nThe boundary condition verified by φatz=z1and\nz=z2is:\n∇n·m=∂φ\n∂z= 0 (34)\nwhere n=zis the vector normal to the xy-plane. More-over, the ferromagnetic layer thickness is much smaller\nthan the lateral direction tf≪L, so the boundary con-\ndition at the lateral boundaries x= 0 and x=Lcan be\napproximated by a magnetic wall [29]:\nφ= 0 (35)\nThe solution of (33) can be decomposed into a time-\nharmonic term that corresponds to the magnetoelastic\ncontribution and a decay term. The time-harmonic term\nonly plays a role in the radiation and is given by (See\nSupplemental Material [23]):9\nφh=6γλ111\nMs∞X\nn=1σn(\ncos (kftf)−1\nkftfsin (knx)ejωt\njω+αLγ\u0010\n2K1\nMs+µ0H0+2Aexc\nMsk2n\u0011\n+2\nkftf∞X\nk=1(−1)kcos (kftf)−1\n1−\u0010\nkπ\nkftf\u00112sin (knx) cos\u0010\nkπ(z−z2)\ntf\u0011\nejωt\njω+αLγ\u0012\n2K1\nMs+µ0H0+2Aexc\nMs\u0012\nk2n+\u0010\nkπ\ntf\u00112\u0013\u0013) (36)\nwhere σnis the magnitude of the stress in the ferromag-\nnetic layer in (27). The spin wave is complex-valued,\nimplying that both magnitude and phase are used to de-\nscribe it.\nFig. 3 shows the spatial distribution of the magni-\ntude and phase of the time-harmonic standing spin wave\nin the ferromagnetic layer at the first and second me-\nchanical resonances. As expected from (33), the spatial\ndistribution is the same as the shear stress. At the first\nresonance, the magnitude decreases along the thickness\nof the ferromagnetic layer from 2 ×10−3atz= 400 nm\nto 0 at the top of the ferromagnetic layer z= 700 nm.\nAt the second resonance, the magnitude is constant in\nthe middle of the ferromagnet with a normalized value\nof 1.2×10−3. Figs. 3(c)-3(d) show that the spin wave\nat the second resonance has an opposite phase compared\nto the first resonance in the upper part of the ferromag-\nnetic layer. It corresponds to the change of polarity after\neach zero of the displacement vector for a thickness-shear\nmode.\nTABLE I: Material properties in the multiferroic an-\ntenna.\nNi [30]\nMs(A/m) 4 .8×105\nµ0H0(mT) 30\nAexc(J/m) 1 .05×10−11\nλ111 −24×10−6\nK1(J/m3) −4.5×103\n\u0010\nCf\n11, Cf\n12, Cf\n44\u0011\n(GPa) (250 ,160,118)\nρ(kg/m3) 8900\nϵr 1\nαL 0.025\nηs 0.001\nAlN [31]\ne15(C/m2) -0.48\n(Cp\n11, Cp\n13, Cp\n33, Cp\n44) (GPa) (410 ,99,389,125)\nρ(kg/m3) 3300\nϵr 9\ntanδ 0.01\nηs 0.001\nPt [31]\nσ(S/m) 8 .9×106\nρ(kg/m3) 21450\nE (GPa) 168\nν 0.38\nηs 0.001The spin waves being calculated here for both space\nand time dependencies, enables as shown further, to ob-\ntain an analytical expression of the electric vector poten-\ntial in the far-field without the use of the infinitesimal\ndipole approximation. Moreover, since the electric vec-\ntor potential is a volume integral of the time-derivative\nof the magnetization, by examining the stress and spin\nwave spatial distribution, we expect the second mode to\nradiate more than the first mode.\nF. Electromagnetic waves radiation\nRecalling the definition of the magnetic field B=\nµ0(H+M), in a ferromagnetic material, a time varia-\ntion of the magnetization, equivalent to a time variation\nof the magnetic field, generates an electric field through\nthe Maxwell-Faraday equation. The radiation source is:\nµ0∂M\n∂t≈µ0Ms∂φ\n∂t⃗ y (37)\nwhere we used the small angle approximation (32). The\nradiation source is a magnetic current source along y.\nOne obtains the EM field through the computation of\nthe associated electric vector potential in the volume of\nthe ferromagnet. In the far field zone:\nFe=ϵ0µ0Ms\n4πre−jk0r⃗ yZL\n0ZWf\n0Zz2\nz1∂φ\n∂tejk0r′cosψdv′\n(38)\nwhere Wfis the width of the ferromagnetic layer, rthe\ndistance to the observation point, r′spans the ferro-\nmagnet volume and k 0is the free-space wave number.\nψ=⃗ r·⃗r′\nris the angle between the vector that goes from\nthe origin of the coordinate system to the observation\npoint and the vector that goes from the origin to the\nsource point. (See Supplemental Material [23] for the ex-\nplicit expression of the vector potential). Therefore, it is\npossible to obtain a closed-form solution of the radiated\nEM field whose components are expressed by:\nEθ=−jωZ 0Fe·⃗ eϕ\nEϕ=jωZ 0Fe·⃗ eθ(39)\nwhere Z0=p\nµ0/ϵ0is the vacuum impedance and ⃗ eϕ, ⃗ eθ\nthe unit vectors in spherical coordinates. The radiation\npower is given by:\nPrad=Z2π\n0Zπ\n0|Eθ|2+|Eϕ|2\n2Z0r2sinθdθdϕ (40)10\nThe radiation efficiency is calculated by:\nηrad=Prad\nPin×100% (41)\nwhere Pinis the accepted power by the multiferroic an-\ntenna, expressed in terms of the input admittance Yin,\nthe applied voltage V, and the system impedance Zsas\n[16]:\nPin=V2\n2\f\f\f\fY∗\nin−1\nZs\f\f\f\f(42)\nFig. 4(a) shows the radiated EM field of the multifer-\nroic antenna in the xz-plane and yz-plane for the first two\nresonances. The planar dimensions of the multiferroic\nantenna are 400 µm×400µm. The radiated field inten-\nsity at the second resonance is approximately two times\nthat of the first resonance. Fig. 4(b) shows the radi-\nated power and radiation efficiency at the two resonances.\nThe radiated power of the first and second resonances\nare 4 .57×10−14W and 0.16 pW respectively, and the\ncorresponding radiation efficiencies are 5 ×10−10% and\n1.7×10−9%. Indeed, the maximum values of the stress\nin the ferromagnetic layer of 0.35 MPa and 0.42 MPa\nfor the first and second resonances, respectively, already\nindicated a higher radiation from the second resonance.\nHowever, as will be discussed in the subsequent part, the\nthicknesses of the ferromagnetic and piezoelectric layers\nwere not optimized for maximum radiation power.\nIII. OPTIMIZATION AND AKHIEZER LIMIT\nWe show in this section the impact of the viscosity\nconstant and the thicknesses of the piezolayer and fer-\nromagnetic layer, on the performance of the multiferroic\nantenna. Fig. 5(a) plots the radiated power as a function\nof the thickness of the AlN piezolayer for a thickness of\nthe Ni layer of 300 and 500 nm. For a thickness of the\nNi layer of 300 nm, the maximum radiated power is ob-\ntained for a thickness of the AlN piezolayer of 380 nm.\nMoreover, a comparison of the maximum radiated power\nby the 300 and 500 nm cases, reveals that increasing the\nthickness of the Ni layer decreases the value of the maxi-\nmum radiated power from 4 .7×10−14W to 4 .5×10−14W,\nrespectively.\nFig. 5(b) compares the efficiency-bandwidth prod-\nuct of different multiferroic antennas with Chu’s limit.\nTABLE II: Average Gr¨ uneisen’s parameter and f×Qa\nproduct for the different piezoelectric materials.\nγavg[32] f×Qa\nAlN 0.91 2 .5×1013\nLiTaO 3 2.37 2 .9×1013\nLiNbO 3 2.37 3 .086×1013(a)\n(b)\nFIG. 4: (a) Simulated far-field radiation patterns at the\nfirst and second resonances. (b) Radiation power (left\naxis) and radiation efficiency (right axis) at the first and\nsecond resonances.\nTerfenol-D is used as ferromagnetic material for its giant\nmagnetostriction constant λ111= 1.6×10−3and differ-\nent piezoelectric materials including AlN, LiNbO 3and\nLiTaO 3are compared. The viscosity constant consid-\nered is the minimum achievable viscosity constant due to\ninternal frictions, denoted as ηmin. Internal frictions are\nrelated in the GHz regime to the phonon-phonon scatter-\ning, also referred to as Akhiezer damping, which limits\nthe maximum achievable mechanical quality factor Qa\nfor a single crystal-resonator [32–34]:\nQa=ρv4\na\nγ2avgκTω(43)\nwhere ρis the mass density, vathe average Debye acous-\ntic velocity, γavgthe average Gr¨ uneisen’s parameter, κ\nthe thermal conductivity, Tthe ambient temperature and\nωthe frequency of the AC voltage applied to the elec-\ntrodes. For simplicity, we consider the Qaof the bilayer\nstructure to be the Qaof the piezolayer. The minimum11\n(a)\n(b)\nFIG. 5: (a) Radiated power as a function of the thickness of the AlN piezolayer for two thicknesses of the Ni layer. (b)\nComparison with Chu’s limit of the efficiency-bandwidth product of different multiferroic antennas when the Akhiezer\ndamping is considered. k 0a is the product of the free-space wavenumber k 0and the minimum radius a, of a sphere\nenclosing the antenna.\nachievable viscosity constant is calculated taking the in-\nverse of the quality factor ηmin=Q−1\na. To vary the reso-\nnance frequency of the multiferroic antennas, we vary the\nthickness of the piezolayers from 100 to 1000 nm, with a\nfixed thickness of the Terfenol-D layer of 500 nm.\nAt low frequencies, the viscosity constant is low and\nthe bandwidth-efficiency product increases until reach-\ning the maximum value allowed by the structure. Once\nthe maximum value is reached, the bandwidth-efficiency\nproduct decreases along with the increase of the reso-\nnance frequency, and thus along with the increase of the\nviscosity constant.\nThe AlN case has a very low efficiency-bandwidth\nproduct because of its low experimental effective elec-\ntromechanical coupling coefficient k2\nt:\nk2\nt=π\n2fs\nfpcot\u0012π\n2fs\nfp\u0013\n(44)\nwhere fsandfpare the series and parallel resonances.\nLiNbO 3and LiTaO 3with higher k2\ntshow efficiency-\nbandwidth products close to Chu’s limit, with values of\n1.27×10−7and 6 .9×10−8at k 0a = 6 .9×10−3and\nk0a = 5 .38×10−3, respectively. As shown in Fig. 5(b),\nthe choice of LiNbO 3or LiTaO 3enhances the radiation\nefficiency by approximately two orders of magnitudes\ncompared to AlN.\nIV. CONCLUSIONS\nThis paper demonstrates a two-dimensional closed-\nform analytic model of an XBAR multiferroic antenna.\nAn extended Mason model to two dimensions is pro-\nposed to account for the particle displacement along thethickness and the lateral directions. The extended Ma-\nson model enables not only to obtain the electrical re-\nsponse of the structure, as the conventional Mason model\nwould do in one dimension, but also allows to obtain\nthe mechanical response through the mode shape. This\ntheory, developed here based on a bilayer piezoelectric-\nferromagnet heterostructure, is also applicable to struc-\ntures with more than two layers. Indeed, adding ex-\ntra layers of buffer non-magnetic materials only modifies\nthe equivalent impedances seen by the piezoelayer at its\ntop and bottom surfaces, shifting the resonant frequen-\ncies. The displacement vector in the ferromagnetic layer,\nneeded for the computation of the electric vector poten-\ntial, can be obtained similarly as the two-layer case, by\napplication of continuity conditions on the displacement\nvectors.\nMoreover, using a unidirectional coupling between the\ngoverning equations, we found closed-form solutions for\nthe spin wave and radiated EM field without the use\nof infinitesimal dipole approximation. The dependencies\nbetween the antenna response and the parameters that\ncompose it are explicitly specified. As a consequence,\noptimization of these parameters can readily be con-\nducted for the study of radiation enhancement through\nthe definition of the adequate thickness of the piezoelec-\ntric and ferromagnet layers leading to the maximum ra-\ndiated power.\nFinally, the efficiency-bandwidth product of differ-\nent multiferroic antennas is compared with Chu’s limit.\nThe comparison is conducted with the highest achiev-\nable mechanical quality factor in the piezolayer, known\nas the Akhiezer limit. We show that at this funda-\nmental mechanical limit, the efficiency-bandwidth prod-\nucts of LiNbO 3/Terfenol-D and LiTaO 3/Terfenol-D mul-12\ntiferroic antennas approach Chu’s limit. Furthermore,\nthe choice of Terfenol-D ferromagnetic material com-\nbined with LiNbO 3or LiTaO 3piezoelectric materials al-\nlows a radiation efficiency enhancement of approximately\ntwo orders of magnitudes compared to AlN/Terfenol-D.\nThus, the fundamental factor limiting radiation efficiency\nis Chu’s limit. However, when considering the coupled\nphysics nature of multiferroic antennas, phonon-phonon\nscattering in the Akhiezer regime is most likely to limit\nthe antenna radiation for GHz devices, and not Chu’slimit.\nACKNOWLEDGMENTS\nThis work was supported by the National Science and\nTechnology Council, Taiwan, under contracts: NSTC\n112-2918-I-002-011, 112-2218-E-002-031, and 112-2119-\nM-002-015.\n[1] H. A. Wheeler, Fundamental limitations of small anten-\nnas, in Proceedings of the IRE 35, 1479 (1947).\n[2] L. J. Chu, Physical limitation on omni-directional anten-\nnas, J. Appl. Phys. 19, 1163–1175 (1948).\n[3] J. S. 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Cai et al. , Finite element analysis and optimization\nof acoustically-actuated magnetoelectric microantennas,\nIEEE Trans. Antennas. Propag. 71, 4640 (2022).\n[19] X. Guokai, S. Xiao, Y. Li, and B.-Z. Wang, Modeling of\nelectromagnetic radiation-induced from a magnetostric-\ntive/piezoelectric laminated composite, Phys. Lett. A\n385, 126959 (2021).\n[20] N. Li, X. Li, B. Xu, B. Zheng, and P. Zhao, Design\nand optimization of a micron-scale magnetoelectric an-\ntenna based on acoustic excitation, Micromachines 13,\n1584 (2022).\n[21] X. Gan, J. Li, J. Li, H. Dong, Z. Gu, and Y. Gao, Kine-\nmatic model of bulk acoustic wave antenna considering\nmagnetic damping, J. Magn. Magn. Mater. 563, 170039\n(2022).\n[22] S. M. Bhagat and P. Lubitz, Temperature variation of fer-\nromagnetic relaxation in the 3d transition metals, Phys.\nRev. 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Auld, Acoustic Fields and Waves in Solids (Wiley,\nNew York, 1990).\n[36] A. D. Polyanin and V. E. Nazaikinskii, Handbook of Lin-\near Partial Differential Equations for Engineers and Sci-\nentists (2nd ed.) (Chapman and Hall/CRC, New York,\n2016).\n[37] A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear\nPartial Differential Equations (2nd ed.) (Chapman and\nHall/CRC, New York, 2012)."    },    {        "title": "2402.13638v1.Effective_four_dimensional_loop_quantum_black_hole_with_a_cosmological_constant.pdf",        "content": "arXiv:2402.13638v1  [gr-qc]  21 Feb 2024Effective four-dimensional loop quantum black hole with a co smological constant\nJianhui Lin and Xiangdong Zhang∗\nDepartment of Physics, South China University of Technolog y, Guangzhou 510641, China\nIn this paper, we utilize the effective corrections of the ¯µ-scheme in loop quantum black holes to\nobtain a 4-dimensional spherically symmetric metric with a cosmological constant. By imposing the\nareal gauge on the components of Ashtekar variables in the cl assical theory and applying the holon-\nomy corrections, we derive the equations of motion, which ca n be solved to obtain the expression\nfor the effective metric in the Painlevé-Gullstrand coordin ates. A comparison with the Λ = 0 case\nreveals minimal modifications near the outer horizon, while significant differences are observed far\nfrom the outer horizon. Moreover, the physical properties o f these quantum-corrected solutions are\nalso discussed.\nI. INTRODUCTION\nThe prediction of the singularity at the center of black\nholes by classical General Relativity (GR) has led to the\nwidespread belief that classical theory has limitations\nand that quantum gravity effects need to be introduced\nto cure this type of spacetime singularity. Pursuing a\nconsistent quantum gravity theory becames one of the\ngreatest challenges since the 20th century.\nOne approach to investigating how quantum gravity\naffects the spacetime of black holes is to start with a\nspecific quantum gravity theory and determine its model\ncorresponding to spherically symmetric spacetime, and\nthen make physical predictions based on the model. loop\nquantum gravity (LQG) is currently one of the candi-\ndates for a theory of quantum gravity. As a background\nindependent and nonperturbative theory, it has consid-\nerable appeal in this regard (see, e.g., [1–4]). Since the\nlate 1980s, LQG based on Ashtekar variables has seen\nsignificant development up to the present day. This in-\ncludes natural predictions for discrete geometrical spec-\ntrum [5–10] and successful generalizations to metric f(R)\ntheories, higher-dimensional gravity and so on [11–13].\nIt preserves two fundamental principles of general rela-\ntivity: diffeomorphism invariance and background inde-\npendence. The construction of LQG adheres to mathe-\nmatical rigor and physical self-consistency. In situation s\nwhere current experimental conditions cannot be satis-\nfied, it is necessary to theoretically consider whether the\nclassical limit of quantum theory is correct.\nThe application of LQG to cosmology, known as loop\nquantum cosmology (LQC), has been established as an\nappropriate semiclassical state [14], and the expectation\nvalues of the quantum operators in this state match well\nwith their corresponding classical values. It has also led\nto the conclusion that the \"Big Bounce\" replaces the big\nbang, successfully resolving the problem of the cosmo-\nlogical singularity of the big bang [15–17]. LQC is the\nsymmetry-reduced model of LQG [18]. In the classical\nscenario, before quantization, we can utilize the symme-\ntries of spatial homogeneity and isotropy to reduce the\n∗Corresponding author. scxdzhang@scut.edu.cnphase space of gravitational degrees of freedom from infi-\nnite dimensions to finite dimensions. Then, by using the\nmethods and techniques of LQG, it is able to proceed\nwith its quantization. For a detailed review of LQC, see,\ne.g., [19, 20].\nDue to the success of LQC in resolving classical cos-\nmological puzzles, the attempt to apply this technique\nto black holes to address the issue of black hole singu-\nlarity is a very intriguing idea. As the simplest black\nhole solution, the Schwarzschild black hole then serves as\nan ideal arena to implement these ideas. Note that the\ninterior of Schwarzschild black hole is isometric to the\nKantowski-Sachs model [17, 21], thus allowing for the\npotential adaptation of techniques and ideas from LQC.\nTherefore, in the past decade or so, the exploration of\nloop quantum black hole models has been a popular di-\nrection.\nGenerally speaking, according to different quantum\nparameters, these loop quantum black hole models can\nbe classified into three main types: the µ0-scheme [17],\nthe¯µ-scheme [21, 22], and the so-called generalized µ0-\nscheme [23, 24]. Among these, the ¯µ-scheme encounters\nan issue of excessive quantum corrections at the event\nhorizon [26]. In the classical regime, excessive quantum\ncorrections are generally considered unacceptable. How-\never, recently, as suggested by some authors [27], the\nreason of traditional ¯µ-scheme cannot be implemented is\nsimply because we choose the wrong set of coordinates\nwhich becomes null at the horizon. Hence they suggested\nto implement the ¯µ-scheme in terms of another set of co-\nordinates which will not become null at the horizon. The\nuse of everywhere spacelike Painlevé-Gullstrand coordi-\nnates can be employed for the ¯µ-scheme and avoid the\naforementioned problem, leading to an effective frame-\nwork for spherically symmetric vacuum solutions with\nholonomy corrections from loop quantum gravity [27].\nWhile for the generalized µ0-scheme which usually re-\nferred to as the Ashtekar-Olmedo-Singh (AOS) approach\n[23]. In AOS scheme, the quantum regularization param-\neters are set to be Dirac observables, i.e. being constants\nalong each dynamics trajectory but allowed to vary from\none to another. AOS model provides a nice effective de-\nscription for the Kruskal extension of both the exterior\nand interior of the Schwarzschild black holes with large\nmass.2\nUp to now, most of the studies on loop quantum black\nhole models are limited to the Schwarzschild case. Since\nthe quantization scheme is not unique to Schwarzschild\nblack holes, generalizing the quantization scheme to other\ntypes of black holes is crucial to test its universality. One\nof the extensions is to include a cosmological constant\nand to study the quantization of Schwarzschild de-Sitter\n(anti de-Sitter) case. The inclusion of a cosmological con-\nstant in a black hole solution is important both in the\npractical and the theoretical sense. However, general-\nizing the loop quantization scheme to Schwarzschild de-\nSitter(anti de-Sitter) case is not that easy. For the gener-\nalizedµ0-scheme, including a cosmological constant will\nadd a term Λ√qwith√qbeing the determinant of spa-\ntial metric in Hamiltonian constraint and this term does\nnot commute with pbandpc[23]. Hence it is very diffi-\ncult to include a cosmological constant in generalized µ0-\nscheme. Therefore, in this paper, we will follow the line\nof [27] by utilizing ¯µscheme in Painlevé-Gullstrand coor-\ndinates to study the loop quantization of Schwarzschild\nde-Sitter(anti de-Sitter) black holes.\nThis paper will be organized as follows: We will start\nfrom the classical Hamiltonian framework and, in Section\nII, present the specific form of the metric using the com-\nponents of connection and densitized triad as dynamical\nvariables. In Section III, the LQG holonomy corrections\nare employed and obtain corresponding metric solutions.\nAdditionally, a discussion of the physical properties of\nthe solutions is provided in Section IV. Finally, Section\nVconcludes the article with a summary.\nOur conventions are the following: space-time indices\nare denoted by µ,ν,ρ,σ,... ; spatial indices are denoted\nbya;b;c;...; and internal indices are denoted by i,j,k,...\nWe only set c= 1but keep Gand/planckover2pi1explicit.\nII. CLASSICAL THEORY\nThe metric of a 4-dimensional spherically symmetric\nspacetime can be expressed in Painlevé-Gullstrand coor-\ndinates as\nds2=−N2dt2+f2(dx+Nxdt)2+y2dΩ2, (1)\nwhere the lapse N, shift vector Nxandf,yall are the\nfunction of time tand the radial coordinate x, whiledΩ,\ngiven by dΩ2=dθ2+ sin2θdφ2, is the line element on\nthe unit sphere.\nAs the foundation of LQG, connection dynamics is a\ntheory based on the Hamiltonian formulation of GR, de-\nscribed by Ashtekar-Barbero connections and their con-\njugate momentum in terms of triads. In loop quan-\ntum black hole models, we also inherit this feature and\nuse them as basic variables. In this context, the basic\nvariables are consistent with [27], which respectively are\nthe Ashtekar-Barbero connection components given by\nc(x),p(x), and the densitized triad components given byEc(x),Ep(x). The following represent the densitized tri-\nads in terms of metric components [27]\nEx\n1=y2sinθ=Ecsinθ, Eθ\n2=fysinθ=Epsinθ,\nEφ\n3=fy=Ep. (2)\nMoreover, the Ashtekar-Barbero connection are written\nasAi\na= Γi\na+γKi\na, where both the spin-connection Γi\na\nand the extrinsic curvature Ki\naonly have three non-zero\ncomponents [27]\nK1\nx=c\nγ, K2\nθ=p\nγ, K3\nφ=psinθ\nγ\nΓ3\nθ=−∂xEc\n2Ep,Γ1\nφ=−cosθ, Γ2\nφ=∂xEcsinθ\n2Ep,\n(3)\nwhereγis the Barbero-Immirzi parameter. The deter-\nminant of the spatial metric is q=Ec(Epsinθ)2.\nA. Actions and constraints\nThe action of GR plus a cosmological constant reads\nS=1\n16πG/integraldisplay\nd4x√−g(R−2Λ). (4)\nAfter performing canonical analysis, the gravitational ac -\ntion becames\nS=/integraldisplay\ndt/integraldisplay\nΣ/bracketleftBigg˙Ai\naEa\ni\n8πGγ−NHΛ−NaHa/bracketrightBigg\n, (5)\nwhere the dot means the derivative with respect to time t;\nthe Hamiltonian constraint HΛand the diffeomorphism\nconstraintis reads respectively as\nHΛ=−Ea\niEb\nj\n16πGγ2√qǫijk(Fabk−(1+γ2)Ωabk))\n−√qN\n8πGΛ. (6)\nHa=1\n8πGγEb\nkFabk. (7)\nHere the field strength is given by Fabk= 2∂[aAk\nb]+\nǫijkAi\naAj\nband the spatial curvature denotes Ωabk=\n2∂[aΓk\nb]+ǫijkΓi\naΓj\nb. Due to the spherical symmetry, we\ncan integrate over dΩand use the Eq. ( 2), then obtain\nthe following expression\nS=/integraldisplay\ndt/integraldisplay\ndx/bracketleftbigg˙cEc+2˙pEp\n2Gγ−NHΛ−NxHx/bracketrightbigg\n,(8)3\nwhere the Hamiltonian constraint reads\nHΛ=−1\n2Gγ/bracketleftBigg\n2cp√\nEc\nγ+Ep\nγ√\nEc(p2+γ2)−γ(∂xEc)2\n4Ep√\nEc\n−γ√\nEc∂x/parenleftbigg∂xEc\nEp/parenrightbigg\n−γ√\nEcEpΛ/bracketrightbigg\n, (9)\nand the diffeomorphism constraint is\nHx=1\n2Gγ(2Ep∂xp−c∂xEc). (10)\nHere straightforward calculations show only Hxis non-\nzero and Hθ,Hφboth vanish. Moreover, from Eq. ( 8),\nthe symplectic structure of the symmetry-reduced black\nhole models is given by\n{c(x1),Ec(x2)}= 2Gγδ(x1−x2), (11)\n{p(x1),Ep(x2)}=Gγδ(x1−x2), (12)\nwithδbeing the Dirac delta function. We now con-\nsider the smeared constraint function C[N] =/integraltextdxNHΛ\nandD[Nx] =/integraltext\ndxNxHx. Through lengthy but straight-\nforward calculations, the Poission brackets between con-\nstraints algebra read\n{C[N1],C[N2]}=D/bracketleftbiggEc\n(Ep)2(N1∂xN2−N2∂xN1)/bracketrightbigg\n,(13)\n{D[Nx\n1],D[Nx\n2]}=D[(Nx\n2∂xNx\n1−Nx\n1∂xNx\n2)],(14)\n{C[N],D[Nx]}=−C[Nx∂xN]. (15)\nAs compared to the case without the cosmological con-\nstant [27], the symplectic structure and the constraint\nalgebra have not changed. Using the Hamilton evolu-\ntion equations ˙y={y,C[N]+D[Nx]}, the equations of\nmotion for each component with respect to time are de-\ntermined by\n˙Ec=2Np\nγ√\nEc+Nx∂xEc, (16)\n˙Ep=N\nγ√\nEc(cEc+pEp)+∂x(NxEp), (17)\n˙c=N\n2γ√\nEc/bracketleftbiggEp\nEc(p2+γ2)−2cp/bracketrightbigg\n+γ∂x/parenleftbigg∂x(N√\nEc)\nEp/parenrightbigg\n+Nγ\n2√\nEc/bracketleftbigg\n∂x/parenleftbigg∂xEc\nEp/parenrightbigg\n−(∂xEc)2\n4EcEp/bracketrightbigg\n+∂x(Nxc)\n−γ\n2∂x/parenleftbigg\nN∂xEc\nEp√\nEc/parenrightbigg\n+NγEpΛ\n2√\nEc(18)˙p=−N\n2γ√\nEc(p2+γ2)−γ\n2/bracketleftBigg\nN√\nEc/parenleftbigg∂xEc\n2Ep/parenrightbigg2\n−∂x(N√\nEc)∂xEc\n(Ep)2+Nx∂xc/bracketrightbigg\n+Nγ√\nEcΛ\n2(19)\nOnce the lapse and shift vector are chosen, by solving\nthe above equations of motion and the diffeomorphism\nconstraint together with the Hamiltonian constraint, one\ncan obtain the vacuum spherically symmetric metric with\nthe cosmological constant.\nB. Classical solutions\nFollowing the same line that in [27], we also adapt the\nareal gauge for Ecas follows:\nEc=x2. (20)\nOnce the areal gauge is imposed, the diffeomorphism con-\nstraint ( 10) then in turn implies\nc=Ep\nx∂xp. (21)\nThe gauge-fixing condition ζ=Ec−x2is second class\nwith the diffeomorphism constraint Hx, which should be\npreserved by the equations of motion. That is to say,\n˙ζ= 0. Using the condition ˙Ec= 0, we can obtain the\nexpression\nNx=−Np\nγ, (22)\nwhich implies that Nxis no longer a Lagrange multiplier.\nAfter choosing the areal gauge, Nxis determined by the\nlapse function N. Through this process, the action can\nbe simplified and given by:\nSGF=/integraldisplay\ndt/integraldisplay\ndx/bracketleftbigg˙pEp\nGγ−NHΛ/bracketrightbigg\n(23)\nwith\nHΛ=1\n2Gγ/bracketleftbigg3γx\nEp−2γx2\n(Ep)2∂xEp−Ep\nγx∂x[x(p2+γ2)]/bracketrightbigg\n+xEp\n2GΛ. (24)\nSince the Ecis fixed by the areal gauge, the remain-\ning symplectic structure of the symmetry-reduced theory\nreads\n{p(x1),Ep(x2)}=Gγδ(x1−x2), (25)4\nand similarly, the remaining constraint algebra is\n{C[N1],C[N2]}=C/bracketleftbigg\n−1\nγ(N1∂xN2−N2∂xN1)p/bracketrightbigg\n=C[Nx\n1∂xN2−Nx\n2∂xN1], (26)\nwhere the second step utilizes Eq. ( 22), which cor-\nresponds to the Painlevé-Gullstrand coordinates. The\nequations of motion can also be simplified to\n˙Ep=p\nγx(NEp−x∂x(NEp)), (27)\n˙p=γNx\n2(Ep)2+γx2\n(Ep)2∂xN−N\n2xγ∂x/parenleftbig\nxp2+γ2x/parenrightbig\n+γNx\n2Λ, (28)\nwhich can be obtained by substituting areal gauge 20\nand the expression ( 22) into the equations of motion ( 17)\nand (19). Now we set N= 1, and take into account the\nHamiltonian constraint HΛ= 0. Then, ˙Ep= 0and˙p= 0\ncan be solved to obtain\nEp=C1x, (29)\np2=γ2/parenleftbigg1\nC2\n1−1/parenrightbigg\n+γ2x2Λ\n3+C2\nx, (30)\nwhereC1andC2are two constants of integration. Fur-\nthermore, these solutions ( 29) and ( 30) satisfy the Hamil-\ntonian constraint equation HΛ= 0identically. From\nequation ( 2), it is worthy noting that both fandycan\nbe expressed in terms of EcandEpwith\nf2=(Ep)2\nEc, (31)\ny2=Ec. (32)\nUsing the equations ( 22), (31), (32) andN= 1, by in-\nserting the solutions ( 29) and ( 30) into the metric ( 1),\nwe obtain\nds2=−/parenleftbigg\n1−2GM\nx−x2Λ\n3/parenrightbigg\ndt2\n+ 2/radicalbigg\n2GM\nx+x2Λ\n3dxdt+dx2+x2dΩ2,(33)\nwhere we can determine the integration constants C1=\n1andC2= 2GMγ2by compare this solution with\nthe standard Schwarzschild de-Sitter(anti-de Sitter) sol u-\ntions and Mis being the mass of the black holes. More-\nover, we can also confirm that the square root of pis\nnegative. Finally, the solutions to the equations of mo-\ntion are given by\nEp=x, (34)\np=−/radicalbigg\n2GMγ2\nx+γ2x2Λ\n3. (35)III. EFFECTIVE CORRECTION OF LQG\nWe adopt the standard LQC treatment for\nSchwarzschild black holes which requires to intro-\nduce holonomy corrections. The holonomy of an\nSU(2)-connection Ai\nais path-ordering exponential\nintegral along an edge ea\nh(A) =Pexp/integraldisplay\neidtAj\naτjea, (36)\nwithPlabeling path-ordering. Although it is not yet\nfully clear whether the LQG dynamics of a loop quantum\nSchwarzschild black hole can also provide a good approx-\nimation for the full LQG dynamics of the Schwarzschild\nblack hole spacetime, the success of LQC has led people\nto believe that the effective dynamics of the loop quan-\ntum Schwarzschild black hole model can serve as a good\napproximation for the semiclassical quantum dynamics\nof observable objects with physical length scales much\nlarger than the Planck scale. As the basic operators in\nLQG are the holonomies and triads, it is necessary to use\nthe holonomies instead of the Ashtekar-Barbero compo-\nnents in the present paper. In the resulting quantum\neffective Hamiltonian constraints, the holonomy correc-\ntion is simplified by replacing the components of Ashtekar\nconnection pwith [27]\np→x√\n∆sin/parenleftBigg√\n∆\nxp/parenrightBigg\n(37)\nafter the areal gauge is imposed. The effective Hamilto-\nnian can be constructed by replacing all the instances of\npin (24) with ( 37) and the result is\nH(LQG)\nΛ=−1\n2Gγ/bracketleftBigg\nEp\nγx∂x/parenleftBigg\nx3\n∆sin2√\n∆p\nx+γ2x/parenrightBigg\n−3γx\nEp+2γx2\n(Ep)2∂xEp−xγEpΛ/bracketrightbigg\n.(38)\nAnd we can obtain the Poisson brackets between C(LQG)\n{C(LQG)[N1],C(LQG)[N2]}\n=C(LQG)/bracketleftBigg\n−x\nγ√\n∆sin√\n∆p\nxcos√\n∆p\nx/parenleftbig\nN1∂xN2\n−N2∂xN1/parenrightbig/bracketrightbig\n. (39)\nThe equation ( 22) also needs to be changed, and at the\nsame time, in order to give desired constraint algebra\n(39), The equation ( 22) has been modified to read as\nNx=−Nx\nγ√\n∆sin√\n∆p\nxcos√\n∆p\nx. (40)\nFinally just like the classical case, once the effective\nHamiltonian 38is in hand, the modified equations of mo-5\ntion can be obtained\n˙Ep=−x2\n2γ√\n∆∂x/parenleftbiggNEp\nx/parenrightbigg\nsin√\n∆p\nxcos√\n∆p\nx,(41)\n˙p=γNx\n2(Ep)2/parenleftbigg\n1+2x∂xN\nN/parenrightbigg\n−γN\n2x\n−N\n2γ∆x∂x/parenleftbigg\nx3sin2√\n∆p\nx/parenrightbigg\n+xγNΛ\n2.(42)\nWe now turn to solve these equations of motion in the\nPainlevé-Gullstrand coordinates and set N= 1. Tak-\ning into account the Hamiltonian constraint H(LQG)\nΛ, the\ncomplete set of equations is as follows\n˙Ep= 0; ˙p= 0\nN= 1;H(LQG)\nΛ= 0 (43)\nThen by relating ˙Ep= 0andN= 1, Eq. ( 41) implies\nthree possibilities:\n1, ∂x/parenleftbiggEp\nx/parenrightbigg\n= 0, (44)\n2,sin√\n∆p\nx= 0, (45)\n3,cos√\n∆p\nx= 0. (46)\nFor the case 1we assuming the term ∂x/parenleftbigEp\nx/parenrightbig\nin equation\n(41) vanishes, the solutions is\nEp=C3x, (47)\nthen, by using Eq. ( 43), we have\np=x√\n∆arcsin/bracketleftbigg/parenleftbigg1\nC2\n3−1/parenrightbigg\nγ2∆1\nx2+C4\nx3\n+γ2∆Λ\n3/bracketrightbigg1/2\n,(48)\nwhereC3andC4are constants of integration. It is not\nhard to confirm that the solutions ( 47) and ( 48) satisfy\nthe Hamiltonian constraint H(LQG)\nΛ= 0. For case 2and\ncase3, use the same logic of [27] and after careful calcu-\nlations, it is determined that neither of them satisfies the\nHamiltonian constraint. Therefore, the solution has only\none form.\nNext, we need to determine the values of these inte-\ngration constants C3andC4. Noticing equations ( 31),\n(32), (47) andN= 1, the metric ( 1) can now be written\nin the following form\nds2=−/parenleftBig\n1−(C3Nx)2/parenrightBig\ndt2+2C2\n3Nxdxdt\n+C2\n3dx2+x2dΩ2, (49)where we should note that the LQG correction terms are\npresent in Nx. Because we want the target metric to\nrecover the classical case ( 33) when the LQG corrections\nare removed, it is easy to know that C2\n3= 1in this case.\nSubstituting p(48) intoNx(40), we obtain the desired\nresult\nNx=/parenleftbiggx2Λ\n3+C4\nγ2∆x/parenrightbigg1/2/parenleftbigg\n1−γ2∆Λ\n3−C4\nx3/parenrightbigg1/2\n,(50)\ntaking into account that\ncos√\n∆p\nx=−/radicalBigg\n1−sin2√\n∆p\nx\nfor consistency with the classical case. When ∆→0,Nx\nshould revert to the classical case (which can be given by\nusing equations ( 22) and ( 35)). Hence, we can ascertain\nthatC4= 2GMγ2∆. Then, the resulting LQG-corrected\nmetric with the cosmological constant reads\nds2=−F(x)dt2+2Nxdxdt+dx2+x2dΩ2,(51)\nwith\nF(x) = 1−(Nx)2, (52)\nNx=/parenleftbiggx2Λ\n3+2GM\nx/parenrightbigg1/2/parenleftbigg\n1−2GMγ2∆\nx3\n−γ2∆Λ\n3/parenrightbigg1/2\n.(53)\nFrom the above result, it can be easily verified that as\n∆→0, the classical Schwarzschild de-Sitter(anti-de Sit-\nter) solutions are recovered. Another point worth men-\ntioning is that when Λis set to zero, the metric ( 51) can\nbe restored to the case without cosmological constant as\nthat in [27], which further demonstrates the correctness\nof the results of the present paper.\nIV. PHYSICAL PROPERTIES OF THE\nEFFECTIVE METRIC\nA. Geodesic and Effective Range\nAs stated in [25] [27], the metric corrected by LQG is\nonly valid for x > x min. This is because, in the LQC,\nthere is an upper limit to the energy density of any matter\nfield. Therefore, in order to generate a gravitational field\nwith mass M, the corresponding matter field cannot be\ninfinitely small but will have a radius corresponding to\nthe maximum energy density. Now we will search for the\nlower bound xmin.\nInvestigating geodesics provides a direct method for\nunderstanding the structure of spacetime. To simplify6\nthe analysis, we will focus solely on radial geodesics that\nsatisfy the following equation\n−κ=−F(x)/parenleftbiggdt\ndτ/parenrightbigg2\n+2Nx/parenleftbiggdx\ndτ/parenrightbigg/parenleftbiggdt\ndτ/parenrightbigg\n+/parenleftbiggdx\ndτ/parenrightbigg2\n(54)\nwhereτrepresents the geodesic parameter. When κ= 1,\nit corresponds to timelike geodesics, and τrepresents\nproper time. On the other hand, if κ= 0, it refers to\nnull geodesics, and τrepresents a chosen affine param-\neter. It is important to note that in this case of a sta-\ntionary spherically symmetric spacetime, Tµ= (∂/∂t)µ\nis a Killing vector. Setting the tangent vector of the\ngeodesic as ξµ= (∂/∂τ)µ, thenTµξµis constant along\nthe geodesic, which we refer to as the conserved energy\nwritten as\nE=Tµξµ=−F(x)dt\ndτ+Nxdx\ndτ. (55)\nFor timelike geodesics (κ= 1)and considering E= 1, it\ncorresponds to a stationary particle starting its motion\nfrom infinity. The geodesic equation ( 54) can be further\nsimplified as follows\ndx\ndτ=±/radicalbig\n1−F(x), (56)\nwhere it can be observed that xminis atdx/dτ= 0and\nthe solution is\nxmin=/parenleftbigg6GMγ2∆\n3−γ2∆Λ/parenrightbigg1/3\n. (57)\nCompared with the Schwarzschild case [27], it is easy\nto see that the term involving the cosmological constant\ncauses a slight variation on xmin. However, when this\nterm is neglected, we obtain exactly the same value of\nxminas discussed in [27, 28]. Moreover, since the xmin\nshould be greater than zero, then the denominator of Eq.\n(57) implies an upper bound on the cosmological constant\nasΛ<3\nγ2∆.\nBesides, for null geodesic, the situation will be much\nsimpler. Since κ= 0, equation 54can be divided by\ndt/dτ and written as\ndx\ndt=−Nx±1. (58)\nA nontrivial case arises when the outgoing null rays with\ndx/dt=−Nx+ 1have zeros at x >0. However, the\ningoing rays with dx/dt=−Nx−1are always less\nthan zero. One point worth mentioning here is that it\nwas previously stated that Tarepresents a Killing vector\nfield, and its corresponding Killing horizon is given by\nTaTa= 0, which can be simplified as F(x) = 0 . But\nin fact,F(x) = 0 anddx/dt= 0are equivalent to each\nother, since the zeros of dx/dt also represent the Killing\nhorizon. According to [27], M∗= 8γ√\n∆/√\n27Gcorre-\nsponding to the critical mass means that when Λ = 0and, there is only one Killing horizon. More precisely, as\nthe mass is greater than M∗, there exist two Killing hori-\nzons, while if the mass is smaller than M∗, there are no\nKilling horizons. The detailed situation can be visualized\nin the figure 1.\nΛ=-0.1\nΛ=0\nΛ=0.1\n01234567-0.50.00.51.0\nxdx\ndtM=M*\nFIG. 1. For outgoing null geodesics, we have plotted the case\nof the black hole with mass M∗and compared three different\nspacetimes. AdS spacetime is taken to have Λ =−0.1, while\ndS spacetime has Λ = 0.1. It can be observed that the three\nspacetimes exhibit little difference near the Killing horiz on,\nand only significant differences arise far from the horizon. T he\nremaining parameters are taken as {γ= 1,∆ = 0.01,G= 1}.\nFrom Fig 2, in the case of the LQG-corrected AdS black\nhole, the mass exceeds M∗, resulting in the presence of\nboth inner and outer horizons. For LQG-corrected met-\nric, the endpoints of the red curve correspond to the min-\nimum and maximum values of x, wheredx/dt= 1, as can\nbe easily verified by equations ( 52), (56), and ( 58). Con-\nversely, the classical AdS black hole exhibits only one\nhorizon, and dx/dt diverges rapidly as x→0, indicat-\ning the presence of the spacetime singularity. In the case\nof dS black hole, the LQG correction also results in the\nexistence of two Killing horizons, avoiding the singular-\nity issue. Additionally, the cosmological horizon can be\nobserved from the curves.\nB. Curvature scalars and surface gravity\nInvestigating the curvature of this effective metric can\nprovide us with a deeper understanding of the geometric\nstructure of this spacetime. Furthermore, it allows for\na more straightforward comparison to be made with the\nscenario where the cosmological constant is not present.\nWith the corrected metric 51, various curvature scalars\ncan be straightforwardly calculated. The results are sum-\nmarized as follows:\nR= 4Λ−4\n3γ2∆Λ2−24(GMγ)2∆\nx6, (59)7\nΔ=0.01\nΔ=0\n0 1 2 3 4-1.0-0.50.00.51.0\nxdx\ndtM=3M*\nΔ=0.01\nΔ=0\n01234567-1.0-0.50.00.51.0\nxdx\ndtM=3M*\nFIG. 2. For outgoing null geodesics, we have separately plot -\nted the cases of the AdS black holes (first figure) and the dS\nblack holes (second figure) with the mass of 3M∗. In both\nfigures, the red curve represents the LQG-corrected metric,\nwhile the blue curve means the classical metric. For AdS\nblack holes, we consider Λ =−0.1, whereas for dS black\nholes,Λ = 0.1. All the remaining parameters are taken as\n{γ= 1,G= 1}.\nRµνRµν=4\n9Λ2/parenleftbig\n−3+γ2∆Λ/parenrightbig2+1440(GMγ)4∆2\nx12\n+16(GMγ)2/parenleftbig\n−3+γ2/parenrightbig\n∆Λ\nx6,(60)\nRµνρσRµνρσ=8\n27Λ2/parenleftbig\n−3+γ2∆Λ/parenrightbig2\n+320(GM)3γ2/parenleftbig\n−3+2γ2∆Λ/parenrightbig\n∆\nx9\n+32(GMγ)2/parenleftbig\n−3+γ2∆Λ/parenrightbig\n∆Λ\nx6\n+7488(GMγ)4∆2\nx12. (61)\nFor physical reasons, the absolute value of curvature\nscalars should correspond to a maximum when xap-\nproaches xminin vacuum. By substituting xmininto\nvarious curvature scalars, we obtain the following results\nlim\nx→xminR=−6\nγ2∆+8Λ−2γ2∆Λ2, (62)lim\nx→xminRµνRµν\n=2/parenleftbig\n−3+γ2∆Λ/parenrightbig2/parenleftbig\n5−4γ2∆Λ+γ4∆2Λ2/parenrightbig\nγ4∆2,(63)\nlim\nx→xminRµνρσRµνρσ\n=4/parenleftbig\n−3+γ2∆Λ/parenrightbig2/parenleftbig\n10−6γ2∆Λ+γ4∆2Λ2/parenrightbig\nγ4∆2.(64)\nThese are also constants independent of the black hole\nmass. Unsurprisingly, when the terms containing the cos-\nmological constant are discarded, these curvature scalars\nalign with those in [27] [28].\nIn addition, exploring the trapped surface can further\nverify the results of the effective metric we have ob-\ntained. Considering the null geodesic congruence, due\nto the spherical symmetry, on a 2-dimensional sphere S,\nthe radial coordinate xand the time coordinate tare\nconstant. Therefore, there are two independent null tan-\ngent vectors orthogonal to S, referred to as outgoing null\ngeodesics and ingoing null geodesics. These two geodesics\ncorrespond to the outgoing expansion parameter θ+and\ningoing expansion parameter θ−respectively. By defini-\ntion, the trapped surface corresponds to the region where\nbothθ+andθ−are negative. Here, we define the tangent\nvectors of the outgoing null geodesics and the ingoing null\ngeodesics to be the same as [27], denoted by\nlµ=/parenleftbigg∂\n∂t/parenrightbiggµ\n+(1−Nx)/parenleftbigg∂\n∂x/parenrightbiggµ\n, (65)\nand\nkµ=/parenleftbigg∂\n∂t/parenrightbiggµ\n+(−1−Nx)/parenleftbigg∂\n∂x/parenrightbiggµ\n. (66)\nNormalize these two vector fields such that lµkµ=−2\nand the hypersurface metric for Sis\nhµν=gµν+1\n2(lµkν+kµlν). (67)\nBased on the definition of the expansions θ+andθ−, the\nsimple expression can be obtained through straightfor-\nward calculations and the results are\nθ+=hµν∇µlν=2\nx(1−Nx), (68)\nθ−=hµν∇µkν=−2\nx(1+Nx). (69)\nHere,θ−remains negative within the range of x, so con-\nsideringθ+alone is sufficient. The main results are dis-\nplayed in Figure 3and4. Besides, the trapped surface\ntends to increase with the increase of mass. From the\nexpression of the expansion ( 68) and the null geodesic\n(58), it can be observed that they differ only by a factor\nof2/x. Therefore, it is conceivable that the zero points8\nof the null geodesics align with the outgoing expansion\nand the physical interpretation is consistent with the null\ngeodesics.\nΛ=-0.1\nΛ=0\nΛ=0.1\n1234567x\n-0.50.00.51.01.52.0θ+\nM=M*\nFIG. 3. The expansions of the outgoing null geodesics are\ncompared for different cosmological constants. In this case ,\nthe mass is chosen as M∗, indicating that when Λ = 0 , there\nis only one event horizon. More detailed calculations revea l\nwhenΛ =−0.1, there is no event horizon and the curve’s end-\npoint cannot extend indefinitely but reaches a certain point .\nThis also implies the maximum value that xcan take. When\nΛ = 0.1, there are two close inner and outer horizons, and\na cosmological horizon can be clearly observed at the end of\nthe curve. These properties are analogous to those of the out -\ngoing null geodesics. The remaining parameters are taken as\n{γ= 1,∆ = 0.01,G= 1}.\nΛ=-0.1\nΛ=0\nΛ=0.11234567x\n-2-1012θ+\nM=2M*\nFIG. 4. Comparison of the null expansion θ+for the LQG-\ncorrected metric with different values of the cosmological c on-\nstant. The mass is 2M∗, compared to the case with mass M∗,\nthe most notable difference is that regardless of the value of\nthe cosmological constant, the black hole exhibits both inn er\nand outer event horizons. The influence of the cosmological\nconstant is only evident at the endpoint of the curve. The\nremaining parameters are taken as {γ= 1,∆ = 0.01,G= 1}.\nWhen a black hole possesses an outer horizon, it be-\ncomes straightforward to calculate the surface gravity by\nemploying\n��=1\n2∂F(x)\n∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=xout. (70)\nwherexoutrepresents the position of the outer horizon.\nDue to the complexity of the expression of F(x), it isΛ=-0.1\nΛ=0\nΛ=0.1\n510152025300.00.20.40.60.81.0\nMκ\nFIG. 5. By taking the mass of the black hole as the in-\ndependent variable, we compared the surface gravity under\ndifferent values of the cosmological constant. The mass is\nmeasured in units of M∗. The minimum value of the mass\nis set at M∗because when the mass is less than M∗, there\nis no horizon, and any discussion concerning surface gravit y\nbecomes meaningless. The remaining parameters are taken\nas{γ= 1,∆ = 0.01,G= 1}.\nnot practical to calculate the analytical solution of the\nsurface gravity directly. Moreover, if we adopt a first-\norder approximation for ∆, the distinctions between the\nthree types of spacetime will not be apparent and the\nresult of the expression will be the same. Therefore, we\nadopt a numerical approach to solve the problem. The\ndetailed results are shown in Figure 5.\nThe green curve corresponds to the AdS black hole,\nand the red curve represents the dS black hole. At smaller\nmasses, due to the LQG correction, they behave almost\nidentically and both exhibit a peak. However, as the\nmass increases, due to the distinct characteristics of the\nspacetime, the surface gravity of the dS black hole rapidly\ntends to zero and eventually reaches an upper mass limit,\nwhereas the surface gravity of the AdS black hole first\ndecreases and then slowly increases.\nV. CONCLUSIONS AND DISCUSSIONS\nThe main result in this paper is obtained based on\nthe Painlevé-Gullstrand coordinate system and utilizes\nthe components of the Ashtekar-Barbero connection and\ndensity-triad to describe a 4-dimensional spherically sym -\nmetric spacetime with a cosmological constant. We first\nperform calculations in the classical scenario. Start-\ning from the gravitational action, after symmetry reduc-\ntion, we consider the constrained system to consist of\nHamiltonian constraint (scalar constraint) and the diffeo-\nmorphism constraint (vector constraint) due to the con-\nstrained nature of the system. By exploiting the spherical\nsymmetry, further simplification of the constraints can be\nachieved. The equations of motion can then be obtained.\nThen we impose a gauge fixing on the area and se-\nlect a particular Painlevé-Gullstrand coordinate system9\n(N= 1) [27]. As a result, the number of dynamical\nvariables is reduced from four to two, and the equations\nof motion are also reduced to two. By directly solving\nthe equations, the classical Schwarzschild de-Sitter(ant i-\nde Sitter) solutions are obtained as expected.\nWe then modified the variable pof the connection\nby incorporating holonomy corrections and adjusted the\nHamiltonian constraint, symplectic structure, and shift\nvector accordingly. Similarly, by solving the modified\neffective equations of motion, we obtained the LQG-\ncorrected spherically symmetric metric with a cosmolog-\nical constant.\nDue to the LQG correction, the variable xhas a re-\nstricted range of values. For AdS and dS spacetimes,\nwe calculated their common minimum value xminus-\ning timelike geodesics. The dS spacetime exhibits the\nexpected cosmological horizon. The effect of the cos-\nmological constant is only noticeable at distances far\nfrom the event horizon which is consistent with the ex-\npectations. We also investigated the curvature scalars\nR,RµνRµν,RµνρσRµνρσof the LQG-corrected metric,\nwhich are bounded at xmindue to quantum gravity ef-fects and are constants that only depend on Λand∆.\nMoreover, when setting Λ = 0 , all the curvature scalars\ncan be recovered in the form presented in article [27].\nIt is worth mentioning that the computation of xmin\ncan also be directly obtained from F(x). SinceF(x) =\n1−(Nx)2, it can be observed that F(x)≤1, and when\nF(x) = 1, it represents the boundary of applicability for\nF(x). Andxminis one of the solutions to F(x) = 1.\nFurthermore, we plotted the figure of the outgoing ex-\npansion, which shares similar properties with the outgo-\ning null geodesics due to their expressions differing only\nby a factor of 2/x. In addition, we also present the plot\nof surface gravity, it is observed that in the dS spacetime,\nas the black hole mass increases, κrapidly decreases and\napproaches zero, whereas in the AdS spacetime, κcon-\ntinues to slowly increase.\nACKNOWLEDGMENTS\nThis work is supported by National Natural Science\nFoundation of China (NSFC) with Grants No.12275087.\n[1]A. Ashtekar and J. Lewandowski, Background Indepen-\ndent Quantum Gravity: A Status Report, Class. Quan-\ntum Grav. 21, R53 (2004).\n[2]C. Rovelli, Quantum Gravity (Cambridge University\nPress, Cambridge, England, 2004).\n[3]T. 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Vaslin4 \n1 Université  Clermont Auvergne, Clermont Auvergne INP, CNRS, Institut Pascal, \nF-63000 Clermont -Ferrand, France. , e-mail: denis.n -chot@sigma -clermont.fr  \n2 Université Clermont Auvergne, INRAE, UR TSCF, F -63000 Clermont –Ferrand, \nFrance., e -mails: roland.lenain@inrae.fr  ; lama.al -bassit@inrae.fr  \n3 MecaBotiX, At Home / La Cité, 55 avenue  Louis Bréguet, F -31400 Toulouse, \nFrance., e -mail: jc.fauroux@mecabotix.com   \n4 Université Clermont  Auvergne, Clermont Auvergne INP, EMSE, CNRS, LIMOS, \nF-63000 Clermont -Ferrand, France., e -mail: philippe.vaslin@limos.fr  \nAbstract   \nUnmanned ground vehicle obstacle crossing gener ally relies on two strategies:  (i) applying a wheel \ntorque for climbing and (ii) modifying the vehicle shape by using a wheel -leg or wheel -paddle to lift \nthe wheel on top of the obstacle. However, most  of those strategies sacrifice speed in order to have a \nlonger contact duration betw een the wheels and the obstacle . This paper investigates the behaviour of a \n4WD high -speed vehicle while crossing a step  obstacl e using a design of experiment  (DoE) . A 3D \nmultibody vehicle model is equipped with a novel 2 -DoF suspension system , which horizo ntal damp-\ning coefficient is modify  to dampen wheel motion in longitudinal and vertical directions in relation to \nthe chassis , for a given speed and obstacle height . The DoE results allow  to propose a novel high-speed \nobstacle crossing stra tegy based on thr ee metrics : (i) the kinetic energy  variation  of the vehicle, (ii) the \ncontact duration between the wheel  and the obstacle, and (iii)  the pitch rate at the start of the ballistic \nphase.  Experimental  function are proposed to  be able  modify these metric in re al time.  \n \nKey words:  Obstacle crossing, Design of experiment , Fitting function, optimization problem  \n1 Introduction  \nObstacle crossing  of wheeled off -road vehicles  starts from the contact between \nthe wheel and the obstacle . When obstacle slope is sufficien t, the horizontal com-\nponent of the ground reaction force reaches the same order of magnitude as the \nvertical component  [1], which creates dynamic pitch, roll and yaw momentum , re-\nsulting in a high vehicle speed reduction and an increasing risk of vehicle ti pping \nover. Obstacle crossing generally relies on two strategies : (i) to apply sufficient 2 D. N’chot, R . Lenain , J.-C. Fauroux, L. Al  Bassit  and P. Vaslin  \nwheel torque to climb and (ii)  to modify the shape of the vehicle by usin g wheel + \nleg or wheel + blade , for instance [2, 3, 4], with or without passive  or active sus-\npension mechanism , to lift the wheel s on top of the obstacle . These strategies  sac-\nrifice the vehicle speed [2] to maintain contact between the wheel s and the obsta-\ncle. Indeed, the wheel continues to climb over the obstacle when torque and \nfricti on conditio ns are met [3]. In literature, the wheel radius is the maximum \nheight of  an obstacle that can be crossed by a vehicle [4, 5]. Previous work  [1, 6] \nhave  shown that the Susp -4D suspension system  with 2 -DoF: (i) reduces the  de-\nceleration of a vehicle with a ce rtain speed over an obstacle (ii) overcomes obsta-\ncles of greater height than the wheel radius  and (iii) reduce s the risk of tipping \nover. This paper proposes  a study of a step  obstacle crossing at high speed  by a \n4WD vehicle equipped with Susp -4D to discus s novel strategies  based on the con-\ntrol of the horizontal suspensions  to optimize obstacle crossing trough three crite-\nria:  \n- to save kinetic energy of the vehicle;  \n- to have a contact duration long enough to climb the obstacle;  \n- to reduce the pitch rate at the  start of the ballistic phase.  \nSection 2 presents the multi -body model of the vehicle and the  design of experi-\nment . Section s 3, 4 and 5  present  the results of the experiment al design , discuss \nthe control strategy and  finally  conclude with some future work.   \n2 Material and methods  \n2.1 Multi -body model description of the full vehicle  \n \nThe Adams multi -body modelling software was u sed to make a  3D model of a \n4WD  vehicle (Fig. 1). The vehicle consists of a rectangular chassis and four driv-\nen wheels.  The chassis is c onstrained with a plane linkage (P2: Fig. 2(b)) to have a \n2D crossing . Both front wheel are connected to the chassis with Susp -4D, whereas \nboth rear wheels are connected to the chassis with Susp -3D suspensions, which  \nare Susp -4D with locked horizontal susp ensions . The  two first possible motions of \nfront wheels are longitudinal and vertical  with r espect to the chassis, while  rear \nwheels can only move vertically, according to the ki nematics  (Fig.  2(a))  of Susp -\n4D and Susp -3D [6 ], respective ly. The other two d egrees of freedom of the wheel \nare rotation s around Yo and around Zo  (Fig. 1(a)) . The obstacle is a rectangular \nbar of height (hO) fixed to the ground. The vehicle centre of gravity (ZCoG) is at  \n0.13 m  and the wheel radius (wr)  is at 0.0745 m.  \n \n  \n   \n  \n(a)                          (b)            (c)                            (d) \n        Figure 1 3D model of the vehicle (a) before, (b) during and (c) after the obstacle crossing  Word Template for a SYROM’2017  Paper    3 \n \n  \n(a)     (b) \nFigure 2 (a) Kin ematic diagram and (b) linkag e graph of Susp4D. The markers and nomencla-\nture of the solids are given as follows: 2 : chassis  ; 6 : wheel  ; 7 : ground  ; 20 : hub carrier  ; 31 : \nupper shaft; 32 : rear direction shaft; 33 : front  direction shaft; 41, 42, 43, 46, 47, 48, 73, 83 : \ncross -bar; 50 : rudder; 7 1 : vertical body of cylinder ; 72 : vertical rod of cylinder ; 81 : \nlongitudinal body of cylinder ;  82 :  piston  of cylinder; And here the joints : Po7 : ponctual \nlinkage; P70, P80 : prismatic linkage s; R6, R50 :  pivot linkage s; S74, S84 : sph erical  linkage s; \nU41, U42, U43, U46, U47, U48 , U73, U83 : universal joint.  \n    The vehicle tyre is made with an outer rubber envelope filled with elastic  foam  \nand the ground is asphalt . The wheel -obstacle contact  model  is based on the Hertz-\nian contact  theory , which use a spring damper formalism (Fig. 3(a)),  and corre-\nspond s to IMPACT function of Adams : \n   (1) \n \nThe stiffness value (k) of the contact  force  takes into account both the material \nprope rties and the contact geometry,  and ensures the conformity of the simulation \nwith the physical phenomenon [7 ]. The literature propose s several stiffness values \nfor rubber [8, 9] . the chosen value  allows the most realistic simulation with respect \nof the physics. It  is obtaned by  a trial and error approach.  \n \n \n           (a)                                   (b)                           (c) \n       Figure 3 Wheel -ground contact model in Adams (a) impact force (b) exponent selection \ncharts (c) evolution of the dam ping coefficient.  \nThe exponent  (e) of the stiffness force depends on the material. The value  given in \nthe T able 1 corresponds to a good value for rubber . The rec ommended damping \nvalue (cmax) [7 ] in Adams is a round 1% of the stiffness value.  It is chosen  to \nguarantee the penetration depth.  The static and dynamic  coefficients are chosen \namong  existing values in the literature [10 ] and tuned to allow  realistic physic al \nbehaviour in simulation. The stiction values are obtained  using the same method \nas friction ones.  \nTable 1.   Wheel -ground contact parameters used in Adams simulations.  \nStiffness \n(k) \nN/mm  Force \nExponent \n(e)  Damp-\ning \n(cmax)  \nNs/mm   Penetra-\ntion deph \n(d) Static co-\nefficient \n(μs) Dynamic  \ncoefficient \n(μd) Stiction  \nTransition  \nVel.  Friction  \nTransition  \nVel. \n1000  1.1 10  0.01 1 0.95 1500  4000  \n 4 D. N’chot, R . Lenain , J.-C. Fauroux, L. Al  Bassit  and P. Vaslin  \n2.1 Design Of Experiment for obstacle crossing  \nThe obstacle crossing  can be divided into three phases:  (1) Before the crossing : all \nfour whee ls of the vehicle are on the ground and rolling without slipping, the ve-\nhicle is moving toward s the obstacle (Fig. 1(a));  (2) During the crossing : the front \nwheels start from the pre -constrained  position (Cp), the crossing starts at time (t 1) \nwhen the fron t wheels collide with the obstacle  (Fig. 1(b)) , the collision phase \nends when the wheels reach the maximum longitudinal displacement , the crossing \nends at time (t 2) after the rear wheel  has collided with the obstacle or passed over \nthe obstac le without tou ching it (Fig. 1(c )); (3) After the crossing : the vehicle \nreaches the peak of its trajectory at time (t 3) before fall ing back to the ground (Fig. \n1(d)). \n    This paper aims at studying the influence of the influence of the s uspension pa-\nrameters  of the prop osed Susp -4D suspension to optimize obstacle crossing. As \nconsequence, we consider three metrics that qual ifies the crossing:  - the k inetic \nenergy variation ( ΔEc) of its  maximum and minimum value  between (t 1) and (t 2) to \nreduce the speed loss ; - the p itch rate (\n ) at the end of the crossing  (t2) to reduce \nthe pitch rate  and finally ; the w heel-obstacle contact duration  (CDWO ) to climb \nover the obstacle . The contact duration  is the time taken by the wheel to reach the \nmaximum longitudinal shortening .   \n    The parameters  chosen for the design of experiment (Fig. 4) are from three  cat-\negories:  geometric parameter : obstacle  height (hO); kinematic parameter : initial \nvehicle speed (vc) and kinetostatic parameter : front longitudinal damping coeffi-\ncient (cAV). \nWe choose  5 obstacle  height s (hO), 5 vehicle speeds (vc) and finally 5 damping \ncoefficient s of the front longitudinal suspensions (c AV), which corresponds to 243 \ntrials (Table 1) . The torque applied to the wheels (τ) is  monitored  to maintain a \nconstant speed throughout the tests  before crossing  and the stiffness (k) of the sus-\npension is set to 2.26 N/mm . \n \nFigure 4 Synoptic of the  experiment  design . \nTable 1.   Numerical value of the different parameters , with wr = 0.0745 m.  \nParameters         \nhO (m)   25%wr   50%wr  80%wr  90%wr   100%wr  \nvc (m/s)   3  6 9 12  15 \ncAV (Ns/mm)   0.4 0.8 1.6 3.2  6.4 \n3 Results of the experiment  design  \n3.1 Influences of th e damper coefficient when crossing  \nLet us first consider an obstacle crossing of 25% of the wheel radius at a con-\nstant speed of 12 m/s for different damping coefficients (0.4, 1.6 and 6.4) . Three \nsimulations are then run without modifying other parameters.  On each curve, the \nsuccessive points correspond to instants t1, t2 and t3, in chronological order of \ncrossing. During the crossing phase between (t1) and (t2), t he relative veloc ity and Word Template for a SYROM’2017  Paper    5 \nkinetic energy  loss (Fig. 5 (a) and (b) resp. ); the pitch angle and pitch rate at time t 2 \n(Fig. 6 (a) and (b) ); the chassis acceleration  at time t 1 (Fig. 7(a)) decrease, the more \nthe damping coefficient is low. For the maximum torque applied (Fig. 7(b)); the \nvehicle height is at apogee (Fig. 8(a)) and t he horizontal displacement of the wheel \nin relation to the chassis (Fig. 8(b)) increase,  the more the damping coefficient is \nlow.  \n \n(a)      (b) \nFigure 5 Evolution of (a) the chassis s peed vc and  (b) the chassis total kinetic energy (Ec).  \n \n (a) (b) \nFigure 6 Evolution of (a) the  chassis  pitch angle  and (b) the chassis  rate during the cro ssing . \n \n                    (a)                                                (b) \nFigure 7 Evolution of (a) chassis acceleration and (b) torque applied to the wheels during the \ncrossing.  \n 6 D. N’chot, R . Lenain , J.-C. Fauroux, L. Al  Bassit  and P. Vaslin  \n \n(a)                                       (b) \nFigure 8 Evolution of (a) the potential energy of the chassis and (b) the trajec tory of the wheel \non the work space of the mechanism during obstacle crossing . \n \n3.2  Influence  of parameters on each  metrics  when cross-\ning  \nIn this se ction, we study the influence of obstacle height and vehicle speed for \nseveral values of damping coefficient. Several simulations are achieved with the \nparameters values of Table 1, which leads to 243 trials, in order to determine a \nmathematical relationsh ip linking the chosen metrics to parameters. Such relation-\nships are depicted on figures 8(a), (b) and (c) for the three metrics chosen (ΔE c, \n, \nCDWO), respectively , at different obstacle height which correspond to cloud point \nwith different colors (Fig. 8) .  \nThe obstacle height (hO) , the speed (vc), the damping coefficient (cAV), influences \nthe kinetic energy variation (ΔE c) in the same direction. The higher hO is, for a \ngiven vc and cAV; the higher vc is, for a given hO and cAV; or the higher the cAV is, \nfor a given vc and hO, the higher (ΔE c) is (Fig. 8(a)). \n  \n \n                (a)                                                          (b)                                                         (c) \nFigure 9 Fitting function of the responses chosen (a) Kinetic energy variation (b) Pitch rate (c) \nWheel Obstacle Contact Duration. With yellow  dot for Δx_w > 8/3 ΔLmax and ΔL correspond to \nthe mechanism longitudinal displacement . \n    The obstacle height (hO), the speed (vc) influences the pitch rate value (\n) in \nthe same direction. The higher (hO)  is for a given vc and cAV; the higher vc is for a \ngiven hO  and c AV, the higher the pitch rate  is at time t 2 and v ice versa . The damp-\ning coefficient  (cAV) influences (\n) in a parabolic way for a given (vc) and (hO) . \nThe maximum pitch rate value lies in the range of the damping coefficient  be-\ntween 3.2 and 6.4  (Fig. 8(b)). \nThe obstacle height  (hO) influences  the obstacle wheel contact duration (CDWO ) \nin the same direction. The higher (hO)  is for a given vc and cAV, the  higher  \n(CDWO )  is (Fig. 8(c)) . The speed (vc) influences (CDWO) , in the opposite direc-\ntion, at a given (hO) , for (cAV) values higher than 3.2 Ns/mm , on the one hand . In \nthis case, the lower (vc) , the higher (CDWO) is. On the other hand,  for (cAV) lower \nthan 3.2 Ns/mm, in a parabolic way.  In this case, (CDWO) is higher fo r extreme Word Template for a SYROM’2017  Paper    7 \nspeeds (3 and 15 m/s) and  reaches its minimum value for intermedia te speeds. The \ndamping coefficient  (cAV) influences (CDWO) in the opposite direction for a giv-\nen (vc) and (hO),  (CDWO) increases as the (cAV) decreases . We can propose ex-\nperimental  function that link the metrics to (vc) and (c AV) for a given (hO) : \n     (2) \nWith the values of  e00, e10, e01, e20, e11 changing according to the obstacle height.  \n    (3) \nWith the values  p00, p10, p01, p20, p11, p02, p30, p21, p12 changing according to the ob-\nstacle height.  \n         (4) \nWith the values d00, d10, d01, d20, d11, d02, d21, d12, d03 changing according to the ob-\nstacle height.  \n \n4 2D strategies for reducing kinetic energy loss and \nimproving vehicle stability during obstacle crossing  \nThe anticipatory  strategy  of obstacle crossing  is based on the detection of the ob-\nstacle  height to adjust the damping coefficient and torque parameters. Before the \nvehicle is crossing , the damping coefficient is set to its maximum value to limit \nthe load transfer of the chassis during acceleration.  At a maximum speed (vc) of \n15 m/s and minimu m obstacle detection  distance of 1 m.  There is only 0.067 s left \nto apply sufficient torque (τ) and reduce the damping coefficient  (cAV).  \nAt this speed, during the crossing, the vehicle takes off the ground regardless of \nthe applied torque and damping co efficient. In this case, it is necessary to apply an \nadapted torque at t 1 and t 2, for reducing the pitch rate  of the chassis . \nThe novel approach  of high-speed obstacle c rossing  rely on  an optimisation prob-\nlem whose objectives is to minimise (ΔE c) during crossing  while maximising \n(CDWO) in order to apply sufficient torque during  the c rossing  that minimises the \npitch rate at the end of the c rossing  (t2). The problem can be express mathemati-\ncally as follow : \nTo minimise \n , \n  ; to maximise \n  ; \nWith the following contraints  : \n- The damping coefficient must be chosen in such a way as to avoid reach-\ning the stops of the suspensions and to keep the wheel centre horizontal \ndisplacement  in line (Fig. 9).  \n                                                              \n                                                  (4) \n- The torque should be sufficient to climb, but not so strong  as to cause the \ntire to rip on the obstacle:                \n \n                                (5) \nIn addition, there are constraints related to the equipment that will reduce the pa-\nrameters  research area.  The stroke of the cylinders  limit  the horizontal displace-\nment of the wheel centre . The motor  of the vehicle limits the maximum torque . \nThe dimensions of  the cylinder and  characteristics of the valve  limit  in minimum \nand maximum value the damping coefficient.  \n8 D. N’chot, R . Lenain , J.-C. Fauroux, L. Al  Bassit  and P. Vaslin  \n5 Conclusion  \nWe studied in simulation the behaviour of a 3D multi -body  4WD vehicle during a \nstep obstacle  crossing with different obstacle height  (hO) at different speeds  (vc) \nand with different damping coefficients (cAV) of the horizontal suspensions. These  \nthree parameters  with their  five levels chosen for the Design of E xperiment  (DoE)  \nleads to 243 trials . We qualified  the crossing by three metrics:  the kinetic energy \nvariation  (ΔEc), the pitch rate  (\n) at ballistic phase  start t2 and the wheel obstacle \ncontact duration  (CDWO) . \n    In all trials , a lower damping coefficient  result s in less lo ss of speed,  kinetic en-\nergy variation , lower pi tch rate  at t 2, and a long er wheel obstacle  contact duration . \nThis Do E enable to find an empiric relationship between  parameters and metrics, \nwhich  will help, in future work,  to develop a robust control law  of the longitudinal \nsuspensions for better obstac le crossing . \nAcknowledgments  This work has been sponsored by the French government re-\nsearch program \"Investissements d'Avenir\" through the IDEX -ISITE initiative 16 -\nIDEX -0001 (CAP 20 -25) and the IMobS3 Laboratory of Excellence (ANR -10-\nLABX -16-01). \nReferences  \n1. Fauroux, J.C., Bouzgarrou, B.C.: dynamic obstacle -crossing  of a wheeled rover with double  \n    wishbone suspension. In: Field Robotics. pp. 642‑649.  (2011) . Last seen 30/08/22      \nhttp://jc.fauroux.free.fr/PUB/ARTICLES/2011_CLAWAR_Fauroux_Bouzgarrou_DRAFT_Dyn\namic_obstacle -crossing_of_a_wheeled_rover_with_double _wishbone_suspension.pdf  \n2. Lauria , M. Nouveaux concepts de locomotion pour véhicules tout -terrain robotisés . Thesis  \n    EPFL . (2003)  \n3. Abad -Manterola, P., Burdick, J.W., Nesnas, I.A.D., Chinchali,  S., Fuller, C., Xuecheng Zhou:  \n    Axel rover paddle whe el design, efficiency, and sinkage on de formable terrain. In: 2010 IEEE  \n    International Conference on Robotics and Automation. pp. 2821‑2827. (2010)  \n4. Shivesh Kumar: All-terrain mobile robot for extra -terrestrial applications.   \n    Bachelor dissertation , 50 p. (2012 ). Last seen 30/08/22  \n    https://doi.org/10.13140/RG.2.1.2331.1449  \n5. Wada, M.: Studies on 4WD Mobile Robots Climbing Up a Step. In: 2006 IEEE International  \n    Conference on Robotics  and Biomimetics. p. 1529‑1534. (2006)  \n \n6. N’chot, D., Fauroux, J. -C., Al Bassit, L., Lenain, R., Vaslin, P .: Performance Evaluation of an  \n    Innovative Suspension System for Obstacle Crossing. In: Cascalho, J.M., Tokhi, M.O., Silva,  \n    M.F., Mendes, A.,  Goher, K., and  Funk, M. (éd.) Robotics in Natural Settings. pp. 95‑106.  \n    (2023)  \n7. Verheul, C., International, S.: Benelux ADAMS User Meeting.19  p. (2012)  \n8. Gregory, M.J.: Measurement of rubber properties for design. Polymer Testing. Vol 4, pp.  \n    211‑223 (1984).  Last seen 30/08/22  https://doi.org/10.1016/0142 -9418(84)90013 -8 \n9. Yu, W., Zhijun, L., Baolin, L., Mingshuai, G.: 1180. Stiffness characteristic comparison  \n    between metal -rubber and rubber is olator under sonic vibration. Vol 16, Issue 11 (2014)  \n10. J.Y. WONG. Theory of ground vehicles. John Wiley & Sons. 4th Edition. 2008.  \n    https://www.engineersedge.com/ coeffients_of_friction.htm  \n "    },    {        "title": "2402.13679v1.Einstein_s_basement____a_model_for_dark_matter_and_an_expanding_universe_.pdf",        "content": "Einstein’s basement - a model for dark matter and an expanding universe?\nFritz Riehle1and Sebastian Ulbricht1, 2, 3\n1Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100, 38116 Braunschweig, Germany\n2Fundamentale Physik f¨ ur Metrologie FPM, Physikalisch-Technische\nBundesanstalt (PTB), Bundesallee 100, 38116 Braunschweig, Germany\n3Institut f¨ ur Mathematische Physik, Technische Universit¨ at Braunschweig,\nMendelssohnstraße 3, 38106 Braunschweig, Germany\nWe present a model treating the energy-momentum relation of relativistic particles as the upper\nbranch of a generalized energy-momentum relation emerging from a forbidden crossing between the\nconstant energy of a massive particle and the photon line. The lower branch, a regime dubbed as\nEinstein’s basement, gives rise to particles with different kinematics that is analysed in the low-\nvelocity limit. Allowing for gravitational interaction between those particles, and between particles\nof both branches, we discuss whether particles in Einstein’s basement might be suitable dark matter\ncandidates. Moreover, the special dynamics of mixed systems leads to an expansion mechanism for\nregular matter in space. Tests of the presented model by astronomical observations are suggested.\nINTRODUCTION\nThe large scale dynamics of the Universe by regular\nmatter is understood only if we allow for hypothetical\ndark matter and dark energy [1]. Dark matter [2] is a well\nestablished concept, allowing to describe a huge number\nof observations in cosmology [3, 4], besides other explana-\ntions such as “Modified Newtonian Dynamics (MOND)”\n[5, 6]. Whereas regular matter is successfully described\nby the standard model of particles [7], the origin of dark\nmatter is not yet clear. A number of theoretical hy-\npotheses has been offered for the origin of dark matter\nlike primordial black holes [8], weakly interacting mas-\nsive particles (WIMPs) [9], axion-like particles [10], mas-\nsive compact halo objects (MACHOs) [11], dark photons\n[12], erebons [13], Kaluza-Klein particles [14] or others.\nAnother attempt to explain dark matter and dark en-\nergy was based on the assumption of negative masses\n[15]. More than fifty direct or indirect experimental ap-\nproaches for detection of the WIMPs and axions are un-\nder way [16] but up to now none of the proposed candi-\ndate particles have been found to explain dark matter to\nthe extent to which it is present in the universe [17, 18].\nAny attempt to solve the presently unexplained mys-\nteries is stepping back from the known physics in one\npoint or another. A systematic proof of the assump-\ntions of standard model physics therefore is essential for\nconstraining New Physics, and the degree of how strong\nknown principles of physics can be violated. Here, we\nexplore a possible extension where we will relax our re-\nstrictions on what is considered to be a particle.\nWe rely on a more general view of a “particle” as any\nentity with a well defined energy momentum dispersion\nrelation (EMR). Among those are also so-called “quasi\nparticles” or “collective excitations” where a particular\ninteraction dresses the bare particles with modified prop-\nerties. This fruitful concept is used, e.g., in Fermi liquids\n[19], phonons or polaritons [20–22] in solid state physics,\nbut may also apply to the Standard Model where parti-\ncles, normally considered as fundamental, are treated asexcitations of their underlying quantum fields.\nThe paper is organized as follows. First, we find be-\nsides the well known relativistic EMR a second one whose\nenergy is below the former, marking a region that is\ndubbed henceforth as “Einstein’s basement”. Then, as\na logical first step, we apply classical mechanics to the\nnew EMR and derive the corresponding equations of mo-\ntion analogue to the mechanics of special relativity and\nNewtonian physics. We then explore whether such quasi\nparticles might represent candidates for dark matter and\ngive examples of possibly observable phenomena.\nIn special relativity a free particle has the well-known\nEMR\nESR(p) =p\n(mc2)2+p2c2 (1)\nwith mthe rest mass of the particle, cthe speed of light\nin vacuum and p=|p|the modulus of the momentum\nof the particle [23, 24]. This EMR is (the only one) inti-\nmately associated with the geometry of Minkowski space-\ntime governed by Lorentz symmetry. Its properties have\nbeen tested to its best over the last century. This EMR is\nshown in Fig. 1 as the upper two curves for two different\nmasses m1andm2. For low velocities each curve starts\nwith the respective rest mass energy mic2before it turns\ninto a linear behavior for large kinetic energies.\nI. THE MODEL\nOur model starts with the notion that the EMR of spe-\ncial relativity represents the upper branch of an avoided\ncrossing between the EMR of a constant matter bare\nstate Eb1=mc2and the EMR of a photon bare state\nEb2=pc. In a simple form an avoided crossing can\nbe described by a matrix Athat couples these two bare\nstates with the coupling F(p) as\nA=\u0012\npc F (p, m)\nF(p, m)mc2\u0013\n. (2)arXiv:2402.13679v1  [gr-qc]  21 Feb 20242\nFIG. 1. The relativistic energy momentum relations of two\nfree particles (upper two curves) with rest masses m1, m2as\nfunction of the particles’ momenta. They approach the re-\nspective rest energies E=mic2(horizontal lines) for low mo-\nmenta and the energy-momentum relation of a photon (dash-\ndotted line) for high momenta. Furthermore, for each mia\nsecond branch is shown, which, in contrast to the preceding\ncase, asymptotically approaches the photonic line at low mo-\nmenta and the respective mass lines at high momenta. The\ngray area represents what we refer to as Einstein’s basement\nin this paper.\nDiagonalizing this matrix leads to the dressed state E(p)\nwith the characteristic equation\nE(p)2−(pc+mc2)E(p) +pcmc2−F(p, m)2= 0.(3)\nAssuming the first solution to be ESR(p) =p\np2c2+m2c4the second solution is\nEEB(p) = mc2+pc−p\np2c2+m2c4, (4)\nwhere the subscripts SRandEBrefer to special relativity\nand Einstein’s basement, respectively [25].\nThe new EMR of Eq. (4) is displayed also in Fig. 1 for\nthe same two masses m1, m 2. For small momenta, they\napproach the photon line whereas they cling to their rest\nmasses for large momenta.\nWith the coupling F(p), the EMRs ESRandEEBrep-\nresent quasi particles that can be regarded as the exci-\ntations of an (up to now) unknown field. Quasi particles\nare used to describe the physics in a variety of systems in\nsolids, liquids or in quantum field theory. A prominent\nexample describes polaritons in an ionic crystal [20–22],which can be found also in different excitonic materials\n[26]. In the polariton system one observes EMRs that\nstrikingly resemble the two EMRs of Fig. 1 for one par-\nticular mass, where the upper branch EMR SRexactly\nfollows Eq. (1). Observers in this system who are aware\nof the upper branch only, would conclude (like we do\nin special relativity) that spacetime is Minkowskian and\nthat the physical laws have to be the same in each iner-\ntial system. In what follows, we will investigate how this\nchanges when the physics of the lower branch becomes\naccessible [27].\nHaving assumed, that energy is a valid concept in Ein-\nstein’s basement - in the sense of a first integral of motion\n- we will also assume that Hamiltonian mechanics [28]\nwith the Hamiltonian H=H(q,p, t), and the canonical\ncoordinates, position qand momentum p, is suitable to\ndescribe the physics in both branches.\nThe velocity vof a particle is given by the Hamilton\nequation\nv=˙q=∇pH. (5)\nHence, for H=ESRwe obtain\nvSR=pc2\np\n(mc2)2+p2c2ˆv, (6)\nwhere ˆvis the velocity unit vector. In contrast, for the\nlower branch H=EEBwe get\nvEB= \nc−pc2\np\n(mc2)2+p2c2!\nˆv. (7)\nThus, we find that the velocity of a particle behaves\ndifferently in both cases. In special relativity |v|=v\ndepends linearly on the momentum for v≪cand ap-\nproaches the velocity of light for asymptotically high mo-\nmenta. In Einstein’s basement vdecreases from cfor low\nmomenta and goes to zero for p→ ∞ . Both behaviors\nare illustrated in Fig. 2.\nFrom Eq. (6) one derives the well known momentum\nof a relativistic particle\npSR=mvq\n1−\u0000v\nc\u00012ˆv. (8)\nFIG. 2. Velocity of the particle as a function of the momentum\npaccording to Eq. (6) and (7) for ESR(full line) and EEB\n(dashed line), respectively.3\nDoing the same for the lower branch, we obtain\npEB=m(c−v)q\n1−(c−v\nc)2ˆv. (9)\nWe find that both momenta have a similar form, while in\nEq. (9) the velocity vis replaced by c−vin comparison to\nthe relativistic case. Thus, we observe that both branches\nrespect cas an ultimate limit for the motion of particles,\nwhile the roles of v= 0 and v=care interchanged.\nIn the relativistic case, the special relation of the mo-\nmentum (8) and the energy (1), gives rise to the class of\nLorentz transformations, keepingp\nE2\nSR−c2p2\nSR=mc2\ninvariant. However, this does not hold for the lower\nbranch, where no such transformation can be found. In\nparticular,p\nE2\nEB−c2p2\nEBis not Lorentz invariant by\nitself, i.e., has a different value in every Lorentz frame,\ngiving up this principle of relativity in the lower branch\nsolely. We will see that this feature will also translate\ninto the equations of motion, when we investigate the\ndynamics of the newly introduced particles.\nTo study the kinematics of particles in Einstein’s base-\nment, we want to construct the counterpart of the well\nknown relativistic Lagrangian\nLSR=−mc2r\n1−\u0010v\nc\u00112\n(10)\n=−mc2+m\n2v2−. . . .\nfor a free particle, where an expansion for small velocities\nv≪creveals the well-known Newtonian kinetic energy\nterm. Keeping in mind that the relation\npEB=∂LEB\n∂v(11)\nholds, we integrate Eq. (9) to obtain the Lagrangian\nLEB=−mc2+mc2s\n1−\u0012c−v\nc\u00132\n(12)\n=−mc2+m\n2(2c)3/2√v−. . .\nfor a free particle in Einstein’s basement, where the inte-\ngration constant was chosen as −mc2to obtain the La-\ngrangian for a particle at rest. In analogy to the second\nline of Eq. (10), we obtain a kinetic energy term in the\nlow-velocity limit. Remarkably, also in this limit, the\nspeed of light cremains as a parameter in the system, af-\nfecting the motion of such particles. In the next section,\nwe will analyze this expression and discuss the resulting\ndynamics of particles, which qualitatively differs from the\nNewtonian case in many aspects.\nII. NEW DYNAMICS IN EINSTEIN’S\nBASEMENT\nIn the following, we have a closer look at the dynam-\nics in Einstein’s basement by using the free particle La-grangian Eq. (12) in the low-velocity limit and comple-\nment it by an additional potential depending on the par-\nticle’s position in space\n˜L=LEB−V(q). (13)\nThe corresponding Euler-Lagrange equations\nd\ndt∂LEB\n∂v=−∂V(q)\n∂q(14)\nlead to the equations of motion in the form\nma−3m(a·v)\n2v2v=1\n2\u00122v\nc\u00133/2\nF, (15)\nwhere a=¨qis the particle’s acceleration and Fthe\nforce resulting from the potential. We find that the time\nderivative of the momentumd\ndtpEB̸=ma, does not co-\nincide with the acceleration, as it is the case in Newtonian\nphysics. In contrast, mais complemented by an addi-\ntional term, which depends on the particle’s velocity v.\nBy rearranging Eq. (15) in the form\nma=1\n2\u00122v\nc\u00133/2\u0012\nF−3(F·v)\nv2v\u0013\n(16)\nthe equations of motion can be directly compared to New-\nton’s second law. For F= 0 we obtain the result a= 0\nas in the Newtonian case, leading to a uniform motion\nof the particle. In the presence of external forces, how-\never, the dynamics of the particle differs from Newton’s\nsecond law in several aspects. We find that the effect of\nthe external force is drastically reduced for slow particles\ndue to the overall-prefactor, depending on |v|=v≪c.\nHowever, the orientation of vwith respect to Faffects\nthe term in the second brackets. Due to that, the par-\nticle’s response to the force can be positive or negative,\ndepending on the velocity vector. For instance, in the\nspecial case when the force and the velocity are parallel\n(or anti-parallel), we obtain an overall negative sign\nma=−\u00122v\nc\u00133/2\nF, (17)\nsuch that in this case commonly attractive forces, act re-\npulsive on a “basement particle”. We stress, that this\nrepulsive force has its origin in the particle dynamics\nand must not be confused with the assumption of a neg-\native particle mass, as it was assumed in dark energy\nconsiderations by A. Einstein [29], or more recently by\nJ. S. Farnes [15].\nHaving disclosed the unconventional behavior of parti-\ncles in the lower branch of Fig. 1 with respect to external\nforces, it is straightforward to investigate the interaction\nof basement particles with each other and with common\nmatter. In the following, we will consider a purely grav-\nitational interaction between two classical particles, two\nparticles in Einstein’s basement, and between one of each\nkind.4\nA. Dynamics of two classical particles\nIn order to study the gravitational interaction of the\nnewly introduced particles, we first recall the central fea-\ntures of the Keplerian motion of two regular particles of\nmasses m1andm2. The corresponding Lagrangian is\ngiven by\nLSR−SR=m1\n2v2\n1+m2\n2v2\n2+Gm 1m2\n|q1−q2|(18)\nBy utilizing the Euler-Lagrange equations and the defi-\nnition of the relative and center of mass coordinates\nr=q1−q2 (19)\ns=m2\nm1+m2q2+m1\nm1+m2q1, (20)\nthe dynamics of the system is described by\n¨r=−G(m1+m2)\n|r|3r (21)\n¨s= 0. (22)\nTherefore, one finds that the relative motion of the parti-\ncles and their center of mass motion can be described in-\ndependently. The equation (21) leads to the well-known\nKepler orbits while the particle’s center of mass moves\nwith a constant velocity. We will see that neither the\nfirst nor the second condition will hold when particles\nfrom Einstein’s basement are involved.\nB. Dynamics of two slow basement particles\nIn analogy to the latter case we again define the La-\ngrangian\nLEB−EB=m1\n2(2c)3/2√v1+m2\n2(2c)3/2√v2\n+Gm 1m2\n|r|(23)\nof two gravitationally interacting particles using Eq. (12)\nin the low-velocity limit. The equation of relative motion\nfor the two particles reads\n¨r=−√\n2G\u0000\nm1(v2/c)3/2+m2(v1/c)3/2\u0001\n|r|3r\n+3√\n2G\nc3/2|r|3\u0012m2(v1·r)√v1v1+m1(v2·r)√v2v2\u0013\n.(24)\nThis equation contains a number of important and\ncounter-intuitive features. First, it shows that the mo-\ntion of the particles does not only depend on their rel-\native position r, but also on their velocities v1andv2\nwith respect to an absolute space. Second, the first\nterm of Eq. (24) is attractive like in the Newtonian case\nfrom Eq. (21), but with a velocity-dependent mass term\nFIG. 3. Orbit of a basement particle with mass m2and ve-\nlocity v2, orbiting another basement particle of mass m1at\nrest. According to Eq. (24) the moving particle describes a\ncircular orbit in the particular case of v2=R0Ω. Below this\nvalue, for v2< R 0Ω, the second particle comes to a final stop\n(at the red cross) within the circular orbit. For v2> R 0Ω the\nsecond particle increases its distance and eventually escapes\nto infinity.\n√\n2[m1(v1/c)3/2+m2(v2/c)3/2] instead of the total mass\nm1+m2of the system. Thus, for small velocities vi≪c\nthe mass effectively entering the gravitational interaction\nis drastically reduced in comparison to the classical two-\nbody problem. Third, we find the second, also velocity-\ndependent term in Eq. (24) which is absent in the New-\ntonian case. It is always repulsive and can be up to three\ntimes larger than the attractive term. Consequently, the\ninterplay of these two contributions decides whether the\nnet acceleration between the two particles is positive or\nnegative.\nTo study the dynamics of two gravitationally interact-\ning basement particles in more detail we consider the\nthree ultimate cases in Fig. 3. For that purpose we\nassume the first particle to be at rest with respect to ab-\nsolute space, i.e., v1= 0. Further assuming v2·r= 0,\nwe find that there exists a circular orbit of the second\nparticle with |r|=R0= const and a constant frequency\nΩ =2G2m2\n1\nc3R3\n0. The motion of the second particle, regard-\nless of its own mass m2, only depends on the mass m1of\nthe fixed particle and the distance R0. This follows natu-\nrally recalling that the momentum of a basement particle\nwith zero velocity is infinite, cf. Eq. (7) and Fig. 2. The\ncircular orbits with velocity v2=R0Ω are unstable and\nmark the transition between two regimes of distinct be-\nhaviour. If the velocity v2> R 0Ω of the second particle is\nincreased by a very small amount, the particle escapes to\ninfinity, as shown in Fig. 3b. If the velocity v2< R 0Ω of\nthe second particle is reduced, the particle turns into the\nregion of the circular area and ultimately comes to rest\nat a finite distance, as shown in Fig. 3c. Such a stopping\npoint is also encountered when two basement particles5\nof arbitrary mass move towards each other on a straight\nline in absolute space.\nC. Dynamics of a classical and a (slow) basement\nparticle\nHere, we want to discuss the behavior of the relative\nmotion and the non-conservation of the center of mass\nvelocity for the gravitational interaction of classical and\na basement particle. Beginning with the Lagrangian\nLSR−EB=m1\n2v2\n1+m2\n2(2c)3/2√v2+Gm 1m2\n|r|\nand the corresponding Euler-Lagrange equations we find\n¨q1=−Gm 2\n|r|3r (25)\n¨q2=√\n2Gm 1\u0000\nv2\n2r−3(v2·r)v2\u0001\nc3/2|r|3√v2. (26)\nOther than in the Lagrangian, at the level of the equa-\ntions of motion we can now easily perform the limit\nv2/c→0 and find\n¨r=−Gm 2\n|r|3r , ¨q2= 0, (27)\nwhere ¨r=¨q1holds in this case. We find, that the clas-\nsical particle behaves like a test particle attracted by\nthe gravitational potential of a fixed particle of mass\nm2. Within this approximation the basement particle\nis still allowed to move with a constant but small ve-\nlocity v2≪c, such that the total system acts like a\nclassical system with m1≪m2under Galilean transfor-\nmation. Other than that, however, the relation between\nthe masses plays no role if the particle of mass m2is a\nslow basement particle. We see from Fig. 4 that the\nmotion of the particle in Einstein’s basement is virtually\nnot affected by the classical particle whereas the latter\none’s motion encircles the first one. Hence, the center of\nmass velocity of the combined system is not a constant\nof motion.\nIII. ARE PARTICLES IN EINSTEIN’S\nBASEMENT DARK MATTER CANDIDATES?\nWe are now in the position to investigate whether our\nmodel can reproduce the central features of dark matter.\na. The mass of dark matter. In our approach we\ntreat the EMR of regular matter as one of the two\nbranches of a generalized energy-momentum relation that\nemerges from a forbidden crossing of the constant energy\nof a massive particle and the photon line. This leads to\na new zoo of particles, having the same masses as their\nrelativistic counterparts. The emerging new sector couldcomplement the known constituents of the current stan-\ndard model.\nThe existence of dark matter was first deduced by Fritz\nZwicky [30] from observed discrepancies in the dynamics\nof galaxy clusters from the laws of celestial mechanics,\nsuch as the virial theorem. In contrast to the usual ap-\nproach to attribute these observations solely to an in-\nvisible distribution of matter with regular kinematics, in\nour model the emerging particles have a different kine-\nmatics as in the regular relativistic or Newtonian case.\nThis mechanism can be an alternative to the usual dark\nmatter model.\nb. Photons, and why the basement may be dark.\nThere is a striking difference between the regular EMR\nESR(p) and EEB(p) in Einstein’s basement with re-\nspect to the possibility of mass-less particles by setting\nmc2= 0. In the first case we recover ESR(p) =pc,\ni.e., the linear EMR of photons. In contrast, for a mass-\nless particle in Einstein’s basement we find EEB(p) = 0,\nsuch that it never has an energy other than zero. Con-\nsequently, there are no photon-like particles in Einstein’s\nbasement. This, a priori, would not prevent that a par-\nticle in Einstein’s basement could interact with a regular\nphoton. The interaction between a photon and a par-\nticle in Einstein’s basement would require an additional\ncoupling between both branches. Let us look on the prob-\nability of such a hypothetical interaction from a different\nviewpoint. Diabatic electromagnetic transitions are well\nknown to occur in the case of an avoided crossing. Such\ntransitions were modeled long ago by Landau [31], Zener\n[32], St¨ uckelberg [33] and Majorana [34] and have been\nFIG. 4. Simulated trajectories of a regular particle with mass\nm2(black) and a slow basement particle with mass m1(blue),\ninteracting gravitationally. The basement particle is almost\nunaffected by its companion, whereas the regular particle fol-\nlows the former one in a spiraling motion, see (a) and (b).\nThe relative motion of m2with respect to m1, given in (c),\nshows two periodicities: an elliptic orbit comparable to Ke-\npler motion, and an oscillation of the apsidal position.6\ntested experimentally in quantum optics [35]. Assum-\ning that diabatic transitions under the influence of one\nor more of the well known interactions are allowed, they\nwould lead to an increased population in the EEBstate\nwhich has a lower energy than ESR. Consequently, di-\nrect transitions between the two regimes, and, hence, the\ndirect interaction between both states must be extremely\nlow. As a result, we conclude that there is not much room\nfor the interaction of photons with particles in Einstein’s\nbasement.\nc. Low coupling and the predominance of dark mat-\nter over regular matter While we can assume that a\nspontaneous decay of classical particles into the basement\nis highly suppressed at the current age of the universe,\nthe interaction between both branches may have changed\nduring its different phases. However, the basement as the\nenergetically more favorable state can be an explanation\nof the current dominance of dark matter in comparison\nto common matter in the present universe.\nd. Clusters of dark matter. For regular matter, the\nformation of large bodies by accumulation of microscopic\nparticles via gravitational attraction is well understood.\nA possible aggregation of dark matter consisting of par-\nticles from Einstein’s basement would be more intricate\ndue to the repulsive and attractive forces in the game.\nThe velocities of cold dark matter in galaxies and spi-\nral nebula have been determined to be in the range of\n200−550 km/s [36, 37]. As we discussed in Sec. II C,\nregular matter is strongly attracted by such slow base-\nment particles, see Eq. (27), while the potentially repul-\nsive forces between the basement particles are suppressed\nby (v/c)3/2, see Eq. (24). Moreover, given the attractive\nforces between regular matter components, such a mix-\nture could be stable even though the fast constituents of\ndark matter would boil off.\nThe preceding arguments lead us to assume that the\nquasi particles derived from our model of Einstein’s base-\nment can be regarded as possible dark matter candidates.\nHaving discussed the relation of basement particles\nand the observed properties of dark matter, we want\nto emphasize yet another feature of a mixed system of\nbasement particles and regular matter. Under the as-\nsumption of classical gravitational interaction, particles\nin Einstein’s basement - apart from special cases - are re-\npelled by each other, while regular particles follow them\nregardless of their own mass. This mechanism describes\nan expansion of a mixed system in space . Whether this\ncan be a source of the observed accelerated expansion\nof the universe, as it is suggested by the study of type\nIa supernovae candles [38, 39], has to be investigated.\nThe modeling of statistically significant ensembles is in-\ndispensable to answer this question. It is also necessary\nto check if our model can explain other observations al-\nready described reasonably well by the models of ΛCDM\nand MOND at large and small scales, respectively.IV. SUMMARY, OUTLOOK AND\nCONCLUSIONS\nWe have expanded the EMR SRof special relativity by\na second EMR EBin Einstein’s basement derived from a\nforbidden crossing between the constant mass of a parti-\ncle and the photon line. As a result, regular matter and\nmatter originating from Einstein’s basement are consid-\nered as made of quasi particles of a yet unknown field.\nWe studied the kinematics of the newly introduced par-\nticles and found that their velocity vis replaced by c−v\nin comparison to the regular case.\nOur findings might possibly be tested quantitatively\nat different scales, for instance by astrophysical observa-\ntions of stars, nebulae and galaxies and by gravitational\nlensing effects of regular and dark matter components.\nA promising candidate for such an investigation could be\nthe so-called bullet cluster [40]. This cluster comprises of\ntwo components each with a spearhead dark matter con-\ntribution followed by a distribution of hot regular gaseous\nmaterial. In such a geometry our model asks - in con-\ntrast to regular kinematics - for additional acceleration of\nthe two dark matter components away from each other,\nwhich might be observable in successive measurements.\nIt must be checked, whether our model delivers an com-\npetitive explanation for the existing measurement data\nand provides an alternative to the current maps of the\ndark matter distribution in the galaxies [41, 42]. Revis-\niting the huge number of existing data and the new data\nexpected from the EUCLID space mission [43] could give\na hint whether the presented ideas should be pursued\nfurther.\nIf our model can survive the comparison with astro-\nphysical observations there would be many more options\nto explore new physics. In this paper we only considered\nclassical particle dynamics and compared our results to\nthose of classical Newtonian physics. However, with the\nfull Hamiltonian (12) at hand, an extension of our anal-\nysis towards a comparison with special and general rel-\nativity is possible. This is of particular interest, since\nthe concept of relativity does not hold for the sector of\nEinstein’s basement, while it is still preserved for regu-\nlar matter in our model. However, considering interac-\ntions between particles of both sectors, violations of the\nrelativity principle, such as a breaking of local Lorentz\ninvariance, can be expected also for regular matter.\nA field theoretical approach of our model could elu-\ncidate the nature of the field whose excitations are the\nquasi particles, we identified as regular particles and par-\nticles from Einstein’s basement. If we stress the similar-\nities of our model with the description of polaritons in a\ncrystal [20–22], the underlying field would correspond to\nthe crystal lattice in that case. 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Kim,\nRevealing the local cosmic web from galaxies by\ndeep learning, ApJ 913, 76 (2021), also avail-\nable as https://iopscience.iop.org/article/10.3847/1538-\n4357/abf040.\n[43] https://sci.esa.int/web/euclid."    },    {        "title": "2402.13726v1.ExaLogLog__Space_Efficient_and_Practical_Approximate_Distinct_Counting_up_to_the_Exa_Scale.pdf",        "content": "ExaLogLog: Space-Efficient and Practical Approximate\nDistinct Counting up to the Exa-Scale\nOtmar Ertl\nDynatrace Research\nLinz, Austria\notmar.ertl@dynatrace.com\nABSTRACT\nThis work introduces ExaLogLog, a new data structure for approx-\nimate distinct counting, which has the same practical properties\nas the popular HyperLogLog algorithm. It is commutative, idem-\npotent, mergeable, reducible, has a constant-time insert operation,\nand supports distinct counts up to the exa-scale. At the same time,\nas theoretically derived and experimentally verified, it requires 43%\nless space to achieve the same estimation error.\n1 INTRODUCTION\nExact counting of distinct elements in a data set or a data stream is\nknown to take linear space [ 3]. However, the space requirements\ncan be significantly reduced, if approximate results are sufficient.\nHyperLogLog ( HLL) [18] with improved small-range estimation\n[16,21,32,36,42] has become the standard algorithm for approx-\nimate distinct counting. It achieves a relative standard error of\n1.04/√𝑚up to distinct counts in the order of 264≈1.8·1019using\nonly 6𝑚bits [ 21]. Therefore, the query languages of many data\nstores (see e.g. documentation of Timescale, Redis, Oracle Data-\nbase, Snowflake, Microsoft SQL Server, Google BigQuery, Vertica,\nElasticsearch, Aerospike, Amazon Redshift, KeyDB, or DuckDB)\noffer special commands for approximate distinct counting that are\nusually based on HLL. Query optimization [ 20,27], caching [ 40],\ngraph analysis [ 6,31], attack detection [ 8,10], network volume es-\ntimation [ 5], or metagenomics [ 4,7,15,25] are further applications\nof HLL.\nHLL is actually very simple as exemplified in Algorithm 1.\nIt typically consists of a densely packed array of 6-bit registers\n𝑟0,𝑟1,...,𝑟 𝑚−1where the number of registers 𝑚is a power of 2,\n𝑚=2𝑝[21]. The choice of the precision parameter 𝑝allows trading\nspace for better estimation accuracy. Adding an element requires\ncalculating a 64-bit hash value. 𝑝bits are used to choose a register\nfor the update. The number of leading zeros ( NLZ) of the remaining\n64−𝑝bits are interpreted as a geometrically distributed update\nvalue≥1with success probability1\n2, that is used to update the se-\nlected register, if its current value is smaller. Estimating the distinct\ncount from the register values is more challenging, but can also\nAlgorithm 1: Inserts an element with 64-bit hash value ⟨ℎ63ℎ62...ℎ 0⟩2\ninto a HyperLogLog consisting of 𝑚=2𝑝(𝑝≥2) 6-bit registers\n𝑟0,𝑟1,...,𝑟 𝑚−1with initial values 𝑟𝑖=0.\n𝑖←⟨ℎ63ℎ62...ℎ 64−𝑝⟩2 ⊲extract register index\n𝑎←⟨ 0...0\n𝑝ℎ63−𝑝ℎ62−𝑝...ℎ 0⟩2 ⊲mask register index bits\n𝑘←nlz(𝑎)−𝑝+1 ⊲update value 𝑘∈[1,65−𝑝]\n⊲function nlzreturns the number of leading zeros\n𝑟𝑖←max(𝑟𝑖,𝑘) ⊲update registerbe implemented using a few lines of code [ 16,18].HLL owes its\npopularity to the following features, which allow it to be used in\ndistributed systems [17]:\nSpeed: Element insertion is a fast and allocation-free operation\nwith a constant time complexity independent of the sketch size. In\nparticular, given the hash value of the element, the update requires\nonly a few CPU instructions.\nIdempotency: Further insertions of the same element will never\nchange the state. This is actually a natural property every count-\ndistinct algorithm should support to prevent duplicates from chang-\ning the result.\nMergeability: Partial results calculated over subsets can be easily\nmerged to a final result. This is important when data is distributed\nor processed in parallel.\nReproducibility: The result does not depend on the processing\norder, which often cannot be guaranteed in practice anyway. Re-\nproducibility is achieved by a commutative insert operation and a\ncommutative and associative merge operation.\nReducibility: The state can be reduced to a smaller state corre-\nsponding to a smaller precision parameter. The reduced state is\nidentical to that obtained by direct recording with lower precision.\nThis property allows adjusting the precision without affecting the\nmergeability with older records.\nEstimation: A fast and robust estimation algorithm ensures nearly\nunbiased estimates with a relative standard error bounded by a\nconstant over the full range of practical distinct counts.\nSimplicity: The implementation requires only a few lines of code.\nThe entire state can be stored in a single byte array of fixed length\nwhich makes serialization very fast and convenient. Furthermore,\nadd and in-place merge operations do not require any extra memory\nallocations.\nUntil recently HLL was the most space-efficient practical data struc-\nture having all these desired properties. Space-efficiency can be\nmeasured in terms of the memory-variance product ( MVP ) [30],\nwhich is the relative variance of the (unbiased) distinct-count esti-\nmate ˆ𝑛multiplied by the storage size in bits\nMVP :=Var(ˆ𝑛/𝑛)×( storage size in bits), (1)\nwhere𝑛is the true distinct count. If the MVP is asymptotically (for\nsufficiently large 𝑛) a constant specific to the data structure, it can be\nused for comparison as it eliminates the general inverse dependence\nof the relative estimation error on the root of the storage size. Most\nHLL implementations use 6-bit registers [ 21] to support distinct\ncounts beyond the billion range, resulting in a theoretical MVP of\n6.48 [ 30]. A recent theoretical work conjectured a general lower\nbound of 1.98 for the MVP of sketches supporting mergeability and\nreproducibility [29], which shows the potential for improvement.arXiv:2402.13726v1  [cs.DS]  21 Feb 2024Table 1: Comparison of mergeable approximate distinct counting algorithms when estimating the distinct count after in-\nserting 𝒏=106distinct elements. The parameters are chosen to get roughly 1% root-mean-square error ( RMSE ) which\nwas empirically determined from 100k simulation runs. The memory-variance products ( MVP s) are estimated as MVP =\n(average memory/serialization size in bits )×(RMSE)2.\nMVPConstant-time\ninsert operation Algorithm References Source code RMSE memory serialized\nHyperLogLog (HLL, 6-bit registers, 𝑝=13) [1, 21] https://github.com/apache/datasketches-java 1.14% 6.69 6.47 yes\nCompressed probability counting (CPC) ( 𝑝=12) [1, 23] https://github.com/apache/datasketches-java 1.08% 4.93 2.34 no\nHyperLogLog (HLL, 4-bit registers, 𝑝=13) [1] https://github.com/apache/datasketches-java 1.14% 4.72 4.34 no\nUltraLogLog (ULL, ML estimator, 𝑝=12) [17] https://github.com/dynatrace-oss/hash4j 1.18% 4.63 4.60 yes\nHyperLogLogLog ( 𝑝=13) [22] https://github.com/mkarppa/hyperlogloglog 1.15% 4.12 3.84 no\nExaLogLog (ELL, 𝑡=2,𝑑=20,𝑝=10) this work https://github.com/dynatrace-research/exaloglog-paper 1.13% 3.72 3.68 yes\nConjectured lower bound [29] – – 1.98 1.98 not known\n1.1 Related Work\nIn the past, different approaches have been proposed to outperform\nthe space efficiency of HLL. However, many of them sacrificed at\nleast one of the properties listed above [ 9,24,30,34,41]. Therefore,\nin the following, we focus only on algorithms that support at least\nmergeability, idempotency, and reproducibility, which we consider\nto be essential properties for practical applications in distributed\nsystems.\nLossless compression of HLL can significantly reduce the storage\nsize [ 14,23,33]. Since the compressed state prevents random access\nto registers as required for insertion, bulking is needed to realize at\nleast amortized constant update times. The required buffer partially\ncancels out the memory savings. Recent techniques avoid buffer-\ning of insertions. The Apache Data Sketches library [ 1] provides\nan implementation using 4 bits per register to store the most fre-\nquent values relative to a global offset. Out of range values are kept\nseparately in an associative array. Overall, this leads to a smaller\nMVP but also to a more expensive insert operation. Its runtime\nis proportional to the memory size in the worst case, because all\nregisters must be updated whenever the global offset is increased.\nHyperLogLogLog [ 22] takes this strategy to the extreme with 3-bit\nregisters and achieves a space saving of around 40% at the expense\nof an insert operation which, except for very large numbers, has\nbeen reported to be on average more than an order of magnitude\nslower compared to HLL [22].\nCompression can also be applied to other data structures to\nmake them more space-efficient. Probabilistic counting with sto-\nchastic averaging ( PCSA ) [19] is a predecessor of HLL, also known\nas FM-sketch. Although less space-efficient when uncompressed, its\ncompressed state leads to a smaller MVP than that of HLL [23,33].\nThe compressed probability counting ( CPC) sketch as part of the\nApache Data Sketches library [ 1] uses this finding. The serialized\ncompressed representation of the CPC sketch achieves a MVP of\naround 2.31 [ 2] that is already quite close to the conjectured lower\nbound of 1.98 [ 29]. However, this is achieved by a costly consoli-\ndation and compression step, as the in-memory representation is\nmore than twice as large, since an amortized constant runtime of\ninsertions can only be achieved by bulking.\nRecently, data structures have been proposed that are more space-\nefficient than HLL while not giving up constant-time insertions.\nExtendedHyperLogLog ( EHLL ) [26] extends the HLL registers from\n6 to 7 bits to store not only the maximum update value, but alsowhether there was an update with a value smaller by one. This\nadditional information can be used to obtain more accurate esti-\nmates. In particular, the MVP is reduced by 16% to 5.43. Inspired\nby this result, we analyzed, if even more additional bits used to\nstore the information about the occurrence of smaller update values\ncould further improve the MVP [17]. The result, of our theoretical\nanalysis, led to UltraLogLog ( ULL), which uses two extra bits and\nachieves a MVP of 4.63, an improvement of 28% over HLL. Our the-\noretical analysis also showed that further space savings would be\npossible with even more additional bits if, at the same time, success\nprobabilities other than1\n2were used for the geometric distribution\nof the update values. However, as the generation of such update val-\nues is more expensive and therefore less practicable, this approach\nhas not been pursued so far.\nTable 1 summarizes practical state-of-the-art algorithms for ap-\nproximate distinct counting. All MVP values shown were deter-\nmined empirically and may therefore deviate slightly from the\nclaimed theoretical values. They were computed by averaging the\nsquared relative estimation error and also the in-memory/serialized\nsize over 100k simulation cycles. In each cycle, the distinct count\nwas estimated after adding 1M distinct random elements.\nThe object overhead and other auxiliary fields are the reason\nwhy the space required in memory is larger than after serialization.\nThis difference is particularly large for CPC due to the relatively\ncomplex compression applied during serialization. The comparison\nof the serialized MVP s is therefore not entirely fair, as compression\ntechniques could also be applied to the other data structures. For ex-\nample, we have shown that with standard compression techniques,\ntheMVP ofULL can be pushed below 3 [ 17], which would make\nthe gap to CPC much smaller.\n1.2 Summary of Contributions\nWe introduce ExaLogLog ( ELL), which is based on a recently pro-\nposed generalization of earlier data structures such as HLL,EHLL ,\nandPCSA [17]. However, the geometric distribution of the update\nvalues is replaced by a distribution for which it is easier to map\na 64-bit hash value to a corresponding random value. When op-\ntimally configured, ELL achieves a MVP of 3.67 as theoretically\npredicted and experimentally confirmed. Compared to HLL with\n6-bit registers, ELLsupports the same operating range up to the\nexa-scale, but requires 43% less space. In contrast to compressed\nprobability counting ( CPC) [1,23] or HyperLogLogLog [ 22], the\n2insert operation takes constant time independent of the sketch size\nand therefore also independent of the configured precision.\nThe distinct count can be estimated efficiently using the maxi-\nmum likelihood ( ML) method. We show that the corresponding ML\nequation has the same shape as that of HLL andULL and has only\na small number of terms regardless of the chosen precision. The\nMLestimate can therefore be found efficiently with the same solver\nthat we had already developed for HLL [16]. We also considered\nmartingale estimation, which is the standard estimation algorithm,\nif the data is not distributed and merging is not needed [ 11,35].\nAlso in this case, the theory predicts significant reductions of the\nMVP by up to 33% compared to HLL, which was again confirmed\nby our experiments.\nA Java implementation of ELL can be found at https://github.\ncom/dynatrace-research/exaloglog-paper. This repository also con-\ntains all the source code and instructions for reproducing the results\nand figures presented in this work.\n2 DATA STRUCTURE\nRecently, we have introduced and theoretically analyzed a data\nstructure that can be seen as generalization of HLL,EHLL and\nPCSA , which finally led us to ULL, another special case [ 17]. The\ngeneralized data structure consists of 𝑚=2𝑝registers. The update\noperation uses 𝑝bits of a uniformly distributed hash value for\nselecting a register and another geometrically distributed hash\nvalue with probability mass function (PMF)\n𝜌update(𝑘)=(𝑏−1)𝑏−𝑘𝑘≥1,𝑏>1 (2)\nfor updating the selected register. Each register consists of 𝑞+𝑑\nbits. The first 𝑞bits are used to keep track of the maximum update\nvalue𝑢and the remaining 𝑑bits are used to indicate the occurrence\nof update values from {𝑢−𝑑,...,𝑢−1}.\n2.1 Previous Theoretical Results\nWe have recently derived and presented various expressions for\ntheMVP of this generalized data structure [ 17]. If the registers are\ndensely stored in a bit array, the theoretical MVP for an efficient\nunbiased estimator meeting the Cramér-Rao bound is\nMVP≈(𝑞+𝑑)ln𝑏\n𝜁\u0010\n2,1+𝑏−𝑑\n𝑏−1\u0011, (3)\nwhere𝜁is the Hurvitz zeta function as defined in Table 2. For ULL,\nas a special case with 𝑏=2and𝑑=2, we have shown that this\nMVP can also be practically achieved using MLestimation. We have\nalso derived a first-order bias correction [ 13] that can be applied to\nthe ML estimate ˆ𝑛MLaccording to\nˆ𝑛=ˆ𝑛ML\n1+𝑐\n𝑚with𝑐:=(ln𝑏) \n1+2𝑏−𝑑\n𝑏−1!𝜁\u0010\n3,1+𝑏−𝑑\n𝑏−1\u0011\n\u0010\n𝜁\u0010\n2,1+𝑏−𝑑\n𝑏−1\u0011\u00112.(4)Table 2: Notation.\nSymbol Comment\n⌊. . .⌋ floor function, e.g.⌊3.7⌋=3\n⟨. . .⟩2 binary representation, e.g. ⟨110⟩2=6\n| bitwise OR operation, e.g. ⟨1001⟩2|⟨1010⟩2=⟨1011⟩2\n& bitwise AND operation, e.g. ⟨1001⟩2&⟨1010⟩2=⟨1000⟩2\nnlz number of leading zeros if the argument is interpreted as\nunsigned 64-bit value, e.g. nlz(⟨10110⟩2)=59\n𝑛 distinct count\nˆ𝑛 distinct count estimate\n𝑏 base, 𝑏>1, defines distribution of update values, compare (2)\n𝑝 precision parameter\n𝑚 number of registers, 𝑚=2𝑝\n𝑞 number of bits used for storing an update value\n𝑑 number of additional register bits to indicate updates with smaller values\n𝑡 parameter of approximated update value distribution, compare (8)\n𝑟𝑖 value of 𝑖-th register , 0≤𝑖<𝑚,0≤𝑟𝑖≤(65−𝑝−𝑡)2𝑡+𝑑+2𝑑−1\n𝑢𝑖 maximum occurred update value for the 𝑖-th register, 𝑢𝑖=⌊𝑟𝑖/2𝑑⌋,\n0≤𝑢𝑖≤(65−𝑝−𝑡)2𝑡\n𝜌update probability mass function of update values, see (2) and (8)\n𝜌reg probability mass function of register values, see Section 3.1\nL likelihood function, L=L(𝑛|𝑟0. . . 𝑟𝑚−1), see Section 3.2\nΓ gamma function, Γ(𝑥):=∫∞\n0𝑦𝑥−1𝑒−𝑦𝑑𝑦\n𝜁 Hurvitz zeta function, 𝜁(𝑥, 𝑦):=Í∞\n𝑢=0(𝑢+𝑦)−𝑥=1\nΓ(𝑥)∫∞\n0𝑧𝑥−1𝑒−𝑦𝑧\n1−𝑒−𝑧𝑑𝑧\n𝜇 state change probability, see Section 3.3\nIf the state size is measured in terms of the Shannon entropy,\ncorresponding to optimal compression of the state, a different ex-\npression was obtained for the MVP\nMVP≈\u0010\n1+𝑏−𝑑\n𝑏−1\u0011−1\n+∫1\n0𝑧𝑏−𝑑\n𝑏−1(1−𝑧)ln(1−𝑧)\n𝑧ln𝑧𝑑𝑧\n𝜁\u0010\n2,1+𝑏−𝑑\n𝑏−1\u0011\nln 2. (5)\nThis number is also known as Fisher-Shannon ( FISH ) number, for\nwhich a theoretical lower bound of 1.98 was postulated [29].\nIn a non-distributed setting, where merging of sketches is not\nneeded, martingale estimation can be used [ 11,35]. This estimation\nmethod is known to be efficient [ 30] and leads to a smaller MVP\nthan (3). The corresponding asymptotic MVP , if again the registers\nare densely stored in a bit array, is given by\nMVP≈(𝑞+𝑑)ln𝑏\n2 \n1+𝑏−𝑑\n𝑏−1!\n. (6)\nIn contrast, if the registers are optimally compressed, the MVP\nwould be\nMVP≈1+\u0010\n1+𝑏−𝑑\n𝑏−1\u0011∫1\n0𝑧𝑏−𝑑\n𝑏−1(1−𝑧)ln(1−𝑧)\n𝑧ln𝑧𝑑𝑧\n2 ln 2. (7)\nThis expression has a lower bound of 1.63 which corresponds to\nthe theoretical limit [30].\n2.2 Approximated Update Value Distribution\nStandard hash functions usually give a uniformly distributed 64-bit\nhash value. Getting a hash value that is geometrically distributed\naccording to (2)is not straightforward except for 𝑏=2where the\nnumber of leading zeros ( NLZ) of a uniformly distributed hash\nvalue can be taken. For 𝑏≠2a different strategy is needed. One\npossibility is to use the 64-bit hash for seeding a pseudo-random\nnumber generator that is then used to generate values according\nto the desired distribution. However, this would usually involve\nfloating-point operations and slow down the whole insert operation.\n30 5 10 15 202−102−82−62−42−2probabilitygeometric/u1D44F=√\n2\napproximate /u1D461=1\n0 5 10 15 202−72−62−52−42−3probabilitygeometric/u1D44F=4√\n2\napproximate /u1D461=2Figure 1: Comparing the probability mass functions (2)and\n(8)for𝒃=22−𝒕.\nTherefore, we consider other distributions of similar shape that\nallow obtaining the update value directly from the given 64-bit hash\nvalue with only a few simple CPU instructions. We find that the\nrandom values of the distribution with PMF\n𝜌update(𝑘)=1\n2𝑡+1+⌊(𝑘−1)/2𝑡⌋with𝑘≥1,𝑡≥0 (8)\ncan be easily generated by taking 𝑡bits of the hash value and\ndetermining the NLZ of the remaining bits and returning\nupdate value =NLZ×2𝑡+(value of the 𝑡bits)+1. (9)\nIt turns out that this distribution approximates geometric distribu-\ntions (2) with base value 𝑏=22−𝑡quite well as shown in Figure 1.\n2.3 ExaLogLog\nOur new data structure ExaLogLog ( ELL) makes use of this finding.\nIt is basically the generalized data structure as we have recently\nproposed [ 17] and also briefly discussed in Section 2, however,\nusing the approximate distribution (8)instead of the geometric\ndistribution (2)for the update values. Hence, the parameter 𝑏is\nreplaced by 𝑡were the two distributions are similar for 𝑏=22−𝑡.\nFor𝑡=0⇔𝑏=2the two distributions are even identical.\nHoping that the formulas for the MVP presented in Section 2.1\nremain valid when exchanging the update value distribution, we\ncan use them to search for parameters that lead to a good space\nefficiency. As our experimental results will show later, the pre-\ndicted MVP s are indeed very accurate despite the deviation from a\ngeometric distribution.\n2.4 Choice of Parameters\nTo live up its name, ELL should, like HLL with 6-bit registers [21],\nsupport distinct counts up to the exa-scale, which is sufficient for\nany conceivable practical use case. As the maximum supported\ndistinct count is roughly given by 𝑏2𝑞[17],𝑞=6+𝑡must be chosen\nin order to get 𝑏2𝑞=(22−𝑡)26+𝑡=264≈1.8·1019.\nApart from𝑡, that defines the update value distribution (8), there\nare two more parameters we are free to choose. Firstly, the precision\n𝑝which defines the number of registers by 𝑚=2𝑝and allows to\ncontrol the trade-off between space and estimation error. And sec-\nondly,𝑑, the number of additional flags to memorize the occurrence\nof update values that are smaller than the maximum update value.\nTogether with 𝑡, it actually defines the theoretical MVP . Therefore,\n−40%−30%−20%−10%0%\n0 10 20 30 40 50 60\n/u1D4513.54.04.55.05.56.06.5memory-variance product\nHLL\nEHLL\nULL\nELL(1,9)ELL(2,16)ELL(2,20)\nELL(2,24)Figure 2: The memory-variance product ( MVP ) according to\n(3)with 𝒃=22−𝒕and 𝒒=6+𝒕when storing the registers in a\nbit array and using an efficient unbiased estimator. Minima\nare marked by arrows.\n−35%−30%−25%−20%−15%−10%−5%0%\n0 10 20 30 40 50 60\n/u1D4512.62.83.03.23.43.63.84.04.2memory-variance product\nHLL\nEHLL\nULL\nELL(1,9)\nELL(2,16)\nELL(2,20)ELL(2,24)\nFigure 3: The memory-variance product ( MVP ) according to\n(6)with 𝒃=22−𝒕and 𝒒=6+𝒕when storing the registers in\na bit array and using the martingale estimator. Minima are\nmarked by arrows.\nwe will use the notation ELL(𝑡,𝑑) to describe a specific class of ELL\nsketches.\nWe evaluated the MVP s given in Section 2.1 with 𝑞=6+𝑡and\n𝑏=22−𝑡for𝑡∈{0,1,2,3}and𝑑≥0. Figure 2 shows the MVP\naccording to (3)for an unbiased and efficient estimator. Figure 3\nshows the MVP given by (6)when using the martingale estimator\nthat can only be used for non-distributed setups. Figures 4 and 5\nshow the corresponding MVP s when the registers are assumed to\nbe optimally compressed as given by (5) and (7).\nThese figures allow identifying useful configurations. The op-\ntimal setting in Figure 2 is 𝑑=20and𝑡=2, resulting in a total\nregister size of 𝑞+𝑑=6+𝑡+𝑑=28bits and a theoretical MVP of\n3.67which is 43% less than that of HLL with 6-bit registers. Since\ntwo registers can be packed into exactly 7 bytes, register access is\nnot too complicated.\n4−35%−30%−25%−20%−15%−10%−5%0%\n0 10 20 30 40 50 60\n/u1D4512.02.22.42.62.83.0memory-variance product\nHLL\nEHLL\nULL\nELL(1,9)ELL(2,16)\nELL(2,20)\nELL(2,24)Figure 4: The memory-variance product ( MVP ) according to\n(5)with 𝒃=22−𝒕when assuming optimal compression of the\nstate and using an efficient unbiased estimator.\n−18%−15%−12%−10%−8%−5%−2%0%\n0 10 20 30 40 50 60\n/u1D4511.601.651.701.751.801.851.901.952.00memory-variance product\nHLL\nEHLL\nULL\nELL(1,9)ELL(2,16)\nELL(2,20)\nELL(2,24)\nFigure 5: The memory-variance product ( MVP ) according to\n(7)with 𝒃=22−𝒕when assuming optimal compression of the\nstate and using the martingale estimator.\nAn interesting configuration is also 𝑑=24and𝑡=2despite\nthe larger theoretical MVP with a value of 3.78. The registers with\na size of 32 bits allow very fast register access when stored in a\n32-bit integer array. Furthermore, as Figures 4 and 5 indicate, this\nconfiguration is probably more efficient than 𝑑=20or𝑑=16when\nusing compression.\nWhen setting 𝑡=1, the choice 𝑑=9with a MVP of3.90is also\nworth mentioning. Although less space-efficient than the mentioned\nconfigurations with 𝑡=2, this setting also results in byte-aligned\nregisters, as their size is exactly 16 bits. Variants with 𝑡≥3lead to\nlarger MVP s than for𝑑=2. However, as they also lead to large 𝑑\nand therefore to quite large register sizes, we do not consider them\nto be useful in practice.\nIn the case of martingale estimation, which can be used for non-\ndistributed setups, the optimum is achieved for 𝑑=16and𝑡=2(cf.\nFigure 3), which leads to a MVP of2.77, which is 33% below that of\nHLL with 6-bit registers. As the register size is 24 bits and therefore\nfits exactly into 3 bytes, register access is also relatively simple.2.5 Relationship to Other Data Structures\nIn Section 2.3 we already mentioned that ELL with𝑡=0is the\nsame as the generalized data structure with 𝑏=2we have recently\nproposed [ 17]. As this data structure is in turn a generalization\nofHLL with𝑑=0,EHLL with𝑑=1, and ULL with𝑑=2[17],\nthese therefore correspond to ELL(0, 0), ELL(0, 1), and ELL(0, 2),\nrespectively. There is also a direct relationship to PCSA [19] and\nCPC [1,23], as ELL(0, 64) stores exactly the same information,\nalbeit encoded differently. In addition, ELL(𝑡, 0) corresponds to\nHyperMinHash [ 44], whose registers only store the maxima of\nupdate values drawn from a distribution equivalent to (8).\n3 STATISTICAL INFERENCE\nSince 64-bit hash functions are typically used to compute the hash\nvalues of inserted elements, the maximum update value is limited.\nAs𝑝bits are used for register selection and another 𝑡bits are\nconsumed for the update value computation (9), the number of\nleading zeros ( NLZ) must be computed from the remaining 64−𝑝−𝑡\nbits yielding the range [0,64−𝑝−𝑡]. Together with the maximum\nvalue of𝑡bits given by 2𝑡−1, the maximum update value according\nto(9)is given by(64−𝑝−𝑡)2𝑡+(2𝑡−1)+1=(65−𝑝−𝑡)2𝑡.\nFor𝑝≥2the maximum update value always fits into 6+𝑡bits\nas(65−𝑝−𝑡)2𝑡≤63·2𝑡≤26+𝑡−1holds in any case. This was\nalready assumed in Section 2.4 for the calculation of the theoretical\nMVPs.\nAdapting distribution (8)to incorporate the limitation of update\nvalues to the range [1,(65−𝑝−𝑡)2𝑡]gives\n𝜌update(𝑘)=1\n2𝜙(𝑘)with𝑘∈[1,(65−𝑝−𝑡)2𝑡],𝑡≥0, (10)\nwhere we introduced the function\n𝜙(𝑘):=min(𝑡+1+⌊(𝑘−1)/2𝑡⌋,64−𝑝). (11)\nTo simplify the statistical model, we use the common Poisson\napproximation [ 16,18,39] that the number of inserted distinct\nelements is not fixed, but follows a Poisson distribution with mean\n𝑛. As a consequence, since the updates are evenly distributed over\nall registers, the number of updates with value 𝑘per register is\nagain Poisson distributed with mean𝑛\n𝑚𝜌update(𝑘). The probability\nthat a register was updated with value 𝑘at least once, denoted by\nevent𝐴𝑘, is therefore\nPr(𝐴𝑘)=1−𝑒−𝑛\n𝑚𝜌update(𝑘). (12)\nThe probability that ���∈[1,(65−𝑝−𝑡)2𝑡]was the largest update\nvalue, which implies that there were no updates with values greater\nthan𝑢, is given by\nPr©\n«𝐴𝑢∧(65−𝑝−𝑡)2𝑡Û\n𝑘=𝑢+1𝐴𝑘ª®\n¬=Pr(𝐴𝑢)(65−𝑝−𝑡)2𝑡Ö\n𝑘=𝑢+1(1−Pr(𝐴𝑘))\n=\u0010\n1−𝑒−𝑛\n𝑚𝜌update(𝑢)\u0011\nexp©\n«−𝑛\n𝑚(65−𝑝−𝑡)2𝑡∑︁\n𝑘=𝑢+1𝜌update(𝑘)ª®\n¬\n=\u0010\n1−𝑒−𝑛\n𝑚𝜌update(𝑢)\u0011\n𝑒−𝑛\n𝑚𝜔(𝑢), (13)\n5where𝜔is defined as\n𝜔(𝑢):=(65−𝑝−𝑡)2𝑡∑︁\n𝑘=𝑢+1𝜌update(𝑘)=2𝑡(1−𝑡+𝜙(𝑢))−𝑢\n2𝜙(𝑢). (14)\nThis formula can be proven by induction. For 𝑢=(65−𝑝−𝑡)2𝑡\nboth sides are zero. If the identity holds for 𝑢, it can be shown that\nit also holds for 𝑢−1, when using the fact that 𝜙(𝑢)and𝜙(𝑢−1)are\nequal except for the case 𝑢−1≡0(mod 2𝑡)∧𝑢−1≤2𝑡(63−𝑝−𝑡)\nwhich implies 𝜙(𝑢)=𝜙(𝑢−1)+1.\n3.1 Probability Mass Function for Registers\nAs described in Section 2, a register stores the maximum update\nvalue𝑢and also if there were updates in the range [𝑢−𝑑,𝑢−1].\nMultiplying the probability (13)that𝑢was the maximum update\nvalue with Pr(𝐴𝑘)or(1−Pr(𝐴𝑘))(compare (12)) for all𝑘∈[𝑢−\n𝑑,𝑢−1]dependent on whether an update with value 𝑘has occurred\nor not, gives the probability mass function ( PMF ) for a single register\n𝑟:\nCase 𝒓=0:\n𝜌reg(𝑟|𝑛)=𝑒−𝑛\n𝑚𝜔(0)=𝑒−𝑛\n𝑚.\nCase 𝒓=𝒖2𝒅+⟨𝒍1. . . 𝒍 𝒖−1⟩22𝒅+1−𝒖with 1≤𝒖≤𝒅:\n𝜌reg(𝑟|𝑛)=\u0010\n1−𝑒−𝑛\n𝑚𝜌update(𝑢)\u0011\n𝑒−𝑛\n𝑚𝜔(𝑢)·\n·𝑢−1Ö\n𝑗=1\u0010\n𝑒−𝑛\n𝑚𝜌update(𝑢−𝑗)\u00111−𝑙𝑗\u0010\n1−𝑒−𝑛\n𝑚𝜌update(𝑢−𝑗)\u0011𝑙𝑗.\nCase 𝒓=𝒖2𝒅+⟨𝒍1. . . 𝒍 𝒅⟩2with 𝒅+1≤𝒖≤(65−𝒑−𝒕)2𝒕:\n𝜌reg(𝑟|𝑛)=\u0010\n1−𝑒−𝑛\n𝑚𝜌update(𝑢)\u0011\n𝑒−𝑛\n𝑚𝜔(𝑢)·\n·𝑑Ö\n𝑗=1\u0010\n𝑒−𝑛\n𝑚𝜌update(𝑢−𝑗)\u00111−𝑙𝑗\u0010\n1−𝑒−𝑛\n𝑚𝜌update(𝑢−𝑗)\u0011𝑙𝑗.\n3.2 Maximum-Likelihood Estimation\nSince the registers are independent due to the Poisson approxima-\ntion, the log-likelihood function for register states 𝑟0,𝑟1,...,𝑟 𝑚−1\ncan be written as\nlnL=lnL(𝑛|𝑟0,...,𝑟 𝑚−1)=𝑚−1∑︁\n𝑖=0ln𝜌reg(𝑟𝑖|𝑛).\nAccording to (10),𝜌update is always a power of two from the set\n{1\n2𝑡+1,1\n2𝑡+2,...,1\n264−𝑝}. Therefore, the log-likelihood function al-\nways has the shape\nlnL=−𝑛\n𝑚𝛼+64−𝑝∑︁\n𝑢=𝑡+1𝛽𝑢ln(1−𝑒−𝑛\n𝑚2𝑢), (15)\nwhere the sum has at most 64−𝑝−𝑡nonzero terms. The coefficients\n𝛼and𝛽𝑢depend just on the register states 𝑟0,𝑟1,...,𝑟 𝑚−1and can\nbe computed according to Algorithm 2. Since 𝑗is bounded by 64−𝑝,\nall contributions to 𝛼are integer multiples of1\n264−𝑝. Therefore, the\ncoefficients could be calculated with integer arithmetic only, if the\ncontributions to 𝛼are scaled by 264−𝑝.\nFor log-likelihood functions of shape (15), we have already devel-\noped a fast numerical optimizer [ 16] that can be used to calculateAlgorithm 2: Computation of the coefficients of log-likelihood function\n(15) for an ExaLogLog sketch with registers 𝑟0,𝑟1,...,𝑟 𝑚−1.\n𝛼←0\n(𝛽𝑡+1,𝛽𝑡+2,...,𝛽 64−𝑝)←( 0,0,..., 0)\nfor𝑖←0to𝑚−1do ⊲iterate over all 𝑚registers\n𝑢←⌊𝑟𝑖/2𝑑⌋\n𝛼←𝛼+𝜔(𝑢) ⊲see (14) for a definition of function 𝜔\nif𝑢≥1then\n𝑗←𝜙(𝑢) ⊲see (11) for a definition of function 𝜙\n𝛽𝑗←𝛽𝑗+1\nif𝑢≥2then\nfor𝑘←max(1,𝑢−𝑑)to𝑢−1do\n𝑗←𝜙(𝑘)\nif(𝑟𝑖& 2𝑑−𝑢+𝑘)=0then ⊲&denotes bitwise AND\n𝛼←𝛼+1\n2𝑗\nelse\n𝛽𝑗←𝛽𝑗+1\nAlgorithm 3: Incrementally updates the martingale estimate ˆ𝑛martingale\nand the state change probability 𝜇whenever a register is altered from 𝑟\nto𝑟′(𝑟<𝑟′). Initially, ˆ𝑛martingale =0and𝜇=1.\nˆ𝑛martingale←ˆ𝑛martingale+1\n𝜇⊲update estimate\n𝜇←𝜇+(ℎ(𝑟′)−ℎ(𝑟))⊲update state change probability, compare (16)\ntheMLestimate. The powers of two in the exponents allow an\nefficient calculation of all terms occurring in the first derivative\nwithout the need to evaluate transcendental functions. Further-\nmore, we have shown that the corresponding root and thus the ML\nestimate can be found efficiently and robustly after a few iterations\nwhen using the secant method. As an optional last step, the ML\nestimate can be corrected according to (4)with𝑏=22−𝑡to reduce\nthe bias.\n3.3 Martingale Estimation\nIf the data is not distributed and merging of sketches is not needed,\nthe distinct count can be estimated in an online fashion by incre-\nmental updates, which is known as martingale or historic inverse\nprobability ( HIP) estimation [ 11,35] and which even leads to smaller\nestimation errors as already mentioned in Sections 2.1 and 2.4.\nThe martingale estimator keeps track of the state change prob-\nability𝜇. Initially,𝜇=1as the first update will certainly change\nthe state. Whenever a register is changed, the probability of state\nchanges for further elements decreases. The probability, that a new\nunseen element changes the ELL state is given by\n𝜇(𝑟0,...,𝑟 𝑚−1)=𝑚−1∑︁\n𝑖=0ℎ(𝑟𝑖). (16)\nℎ(𝑟𝑖)is the probability that register 𝑟𝑖is changed with the next new\nelement. The function ℎis strictly monotonically decreasing and\ngiven by\nℎ(𝑟)=1\n𝑚©\n«𝜔(𝑢)+𝑢−1∑︁\n𝑘=max(1,𝑢−𝑑)(1−𝑙𝑢−𝑘)·𝜌update(𝑘)ª®\n¬\nwith𝑟=𝑢2𝑑+⟨𝑙1...𝑙𝑑⟩2\n6where𝜔is the function defined in (14).\nThe martingale estimator is incremented with every state change\nby1\n𝜇prior the update as demonstrated by Algorithm 3. 𝜇itself can\nalso be incrementally adjusted, such that the whole update takes\nconstant time. The martingale estimator is unbiased and optimal\n[30], but cannot be used if the data is distributed and mergeability\nis required.\n4 PRACTICAL IMPLEMENTATION\nLike other probabilistic data structures such as HLL,ELLalso relies\non high-quality 64-bit hash values for the elements. Known good\nhash functions are WyHash [ 43], Komihash [ 38], or PolymurHash\n[28]. The update procedure for inserting an element is summarized\nby Algorithm 4. It divides the hash value of an element into three\nparts. The first 64−𝑝−𝑡bits are used to determine the number of\nleading zeros ( NLZ) which is therefore in the range [0,64−𝑝−𝑡].\nThe next𝑝bits address the register and the remaining last 𝑡bits\nare then used in combination with the NLZ of the first part to get\nthe update value according to (9). It is possible to use the bits in a\ndifferent order. For example, the three parts consisting of 64−𝑝−𝑡,\n𝑝, and𝑡bits could be permuted. However, only if the 64−𝑝−𝑡bits\nfor the NLZ and the𝑝bits for the address are adjacent and in this\norder, ELL will be reducible, as discussed below.\n4.1 Mergeability\nIf two ELL data structures are equally configured, hence they have\nequal𝑝,𝑡, and𝑑values, they can be straightforwardly merged\nas shown by Algorithm 5. The result of the merge operation is\nthe same as if the union of all the individual original elements\nthat were previously inserted into one of the two individual data\nstructures had been inserted directly into an empty data structure\nusing Algorithm 4.\nELL sketches are also mergeable, if the parameters are not all\nequal and as long as the sketches share the same 𝑡-parameter.\nIf the sketch parameters are (𝑡,𝑑,𝑝)and(𝑡,𝑑′,𝑝′), respectively,\nthey both can be reduced to an ELL sketch with parameters\n(𝑡,min(𝑑,𝑑′),(𝑝,𝑝′))first. This is useful for migration scenarios, if\nthe precision 𝑝or parameter 𝑑must be changed while mergeability\nwith older records should be preserved.\n4.2 Reducibility\nThe reduction of the 𝑑-parameter is straightforward. Decrementing\nit from𝑑to𝑑′with𝑑≥𝑑′only requires right-shifting all registers\nby𝑑−𝑑′bits. The reduction of the precision from 𝑝to𝑝′is more\ncomplex, as 2𝑝−𝑝′registers need to be combined into one. However,\ndue to the way in which the hash bits are consumed in Algorithm 4,\nthis is also possible in a lossless way, meaning that the result is the\nsame as if all elements would be recorded directly using a sketch\nwith the reduced parameters, as demonstrated by Algorithm 7.\n5 EXPERIMENTAL VERIFICATION\nWe need to verify that the estimation error of ExaLogLog ( ELL)\nreally matches the theoretically predicted error, despite modeling\n(8)by(2)and the number of distinct element insertions by a Poisson\ndistribution. According to (1)we would expect a root-mean-square\nerror ( RMSE ) error of√︁\nMVP/((𝑞+𝑑)𝑚)with𝑞=6+𝑡,𝑏=22−𝑡,Algorithm 4: Inserts an element with 64-bit hash value ⟨ℎ63ℎ62...ℎ 0⟩2\ninto an ExaLogLog with ( 6+𝑡+𝑑)-bit registers 𝑟0,𝑟1,...,𝑟 𝑚−1(𝑚=2𝑝)\nand initial values 𝑟𝑖=0.\n𝑖←⟨ℎ𝑝+𝑡−1ℎ𝑝+𝑡−2...ℎ 𝑡⟩2 ⊲extract register index\n𝑎←⟨ℎ63ℎ62...ℎ 𝑝+𝑡1...1\n𝑝+𝑡⟩2\n𝑘←nlz(𝑎)2𝑡+⟨ℎ𝑡−1ℎ𝑡−2...ℎ 0⟩2+1 ⊲𝑘∈[1,(65−𝑝−𝑡)2𝑡]\n𝑢←⌊𝑟𝑖/2𝑑⌋ ⊲get maximum update value from register\nΔ←𝑘−𝑢\nifΔ>0then\n𝑟𝑖←𝑘·2𝑑+⌊(2𝑑+(𝑟𝑖mod 2𝑑))/2Δ⌋\nelse if Δ<0∧𝑑+Δ≥0then\n𝑟𝑖←𝑟𝑖|2𝑑+Δ⊲|denotes bitwise OR\nAlgorithm 5: Merges two ExaLogLog ( ELL) sketches with registers\n𝑟0,𝑟1,...,𝑟 𝑚−1and𝑟′\n0,𝑟′\n1,...,𝑟′\n𝑚−1, respectively, into an ELLsketch with\nregisters𝑟′′\n0,𝑟′′\n1,...,𝑟′′\n𝑚−1. The input sketches are required to have iden-\ntical parameters 𝑡,𝑑, and𝑝, where𝑚=2𝑝.\nfor𝑖←0to𝑚−1do ⊲iterate over all 𝑚registers\n𝑟′′\n𝑖←MergeRegister( 𝑟𝑖,𝑟𝑖′,𝑑) ⊲see Algorithm 6\nAlgorithm 6: Merges two corresponding registers 𝑟and𝑟′of ExaLogLog\nsketches with identical parameters 𝑡,𝑑, and𝑝.\nfunction MergeRegister( 𝑟,𝑟′,𝑑)\n𝑢←⌊𝑟/2𝑑⌋\n𝑢′←⌊𝑟′/2𝑑⌋\nif𝑢>𝑢′∧𝑢′>0then\nreturn𝑟|⌊(2𝑑+(𝑟′mod 2𝑑))/2𝑢−𝑢′⌋ ⊲|denotes bitwise OR\nelse if𝑢′>𝑢∧𝑢>0then\nreturn𝑟′|⌊(2𝑑+(𝑟mod 2𝑑))/2𝑢′−𝑢⌋\nelse\nreturn𝑟|𝑟′\nAlgorithm 7: Reduces an ExaLogLog ( ELL) sketch with registers\n𝑟0,𝑟1,...,𝑟 𝑚−1and parameters 𝑡,𝑑,𝑝to an ELL sketch with registers\n𝑟′\n0,𝑟′\n1,...,𝑟′\n𝑚′−1and parameters 𝑡,𝑑′,𝑝′, where𝑚=2𝑝,𝑚′=2𝑝′,\n𝑑≥𝑑′≥0, and𝑝≥𝑝′≥0.\n𝑤←(64−𝑡−𝑝)·2𝑡+1\nfor𝑖←0to𝑚′−1do\n𝑟′←0\nfor𝑗←0to2𝑝−𝑝′−1do\n𝑟←⌊𝑟𝑖+𝑗·𝑚′/2𝑑−𝑑′⌋ ⊲right-shift by 𝑑−𝑑′bits\n𝑢←⌊𝑟/2𝑑′⌋\nif𝑢≥𝑤then\n𝑠←((𝑝−𝑝′)−( 64−nlz(𝑗)))· 2𝑡⊲assuming𝑗has 64 bits\nif𝑠>0then\n𝑣←𝑑′+𝑤−𝑢\nif𝑣>0then𝑟←⌊𝑟/2𝑣⌋·2𝑣+⌊(𝑟mod 2𝑣)/2𝑠⌋\n𝑟←𝑟+𝑠·2𝑑′\n𝑟′←MergeRegister( 𝑟,𝑟′,𝑑′) ⊲see Algorithm 6\n𝑟′\n𝑖←𝑟′\nand the MVP either given by (3)or(6), depending on whether the\nML or the martingale estimator is used.\nExtensive empirical tests [ 37] have shown that the output of\nmodern hash functions such as WyHash [ 43], Komihash [ 38], or\nPolymurHash [ 28] can be considered like uniform random values.\n7051015relative error (%)/u1D461=1,/u1D451=9,/u1D45D=4(32 bytes)\n0510/u1D461=2,/u1D451=16,/u1D45D=4(48 bytes)\n0510/u1D461=2,/u1D451=20,/u1D45D=4(56 bytes)\n0510/u1D461=2,/u1D451=24,/u1D45D=4(64 bytes)\n0.02.55.07.5relative error (%)/u1D461=1,/u1D451=9,/u1D45D=6(128 bytes)\n0246/u1D461=2,/u1D451=16,/u1D45D=6(192 bytes)\n0246/u1D461=2,/u1D451=20,/u1D45D=6(224 bytes)\n024/u1D461=2,/u1D451=24,/u1D45D=6(256 bytes)\n024relative error (%)/u1D461=1,/u1D451=9,/u1D45D=8(512 bytes)\n0123/u1D461=2,/u1D451=16,/u1D45D=8(768 bytes)\n0123/u1D461=2,/u1D451=20,/u1D45D=8(896 bytes)\n012/u1D461=2,/u1D451=24,/u1D45D=8(1024 bytes)\n1001031061091012101510181021\ndistinct count012relative error (%)/u1D461=1,/u1D451=9,/u1D45D=10(2048 bytes)\n1001031061091012101510181021\ndistinct count0.00.51.01.5/u1D461=2,/u1D451=16,/u1D45D=10(3072 bytes)\n1001031061091012101510181021\ndistinct count0.00.51.01.5/u1D461=2,/u1D451=20,/u1D45D=10(3584 bytes)\n1001031061091012101510181021\ndistinct count0.00.51.0/u1D461=2,/u1D451=24,/u1D45D=10(4096 bytes)\nML theory ML rmse ML bias martingale theory martingale rmse martingale biasFigure 6: The relative bias and the RMSE for the MLand the martingale estimator for different ELLconfigurations obtained\nfrom 100k simulation runs. The theoretically predicted errors perfectly match the experimental results. Individual insertions\nwere simulated up to a distinct count of 106before switching to the fast simulation strategy.\nOtherwise, field-tested probabilistic data structures like HLL would\nnot work. This fact allows us to simplify the experiments and per-\nform them without real or artificially generated data. Insertion\nof a new element can be simulated by simply generating a 64-bit\nrandom value to be used directly as the hash value of the inserted\nelement in Algorithm 4.\nTo simulate the estimation error for a predefined distinct count\nvalue, the estimate is computed after updating the ELLsketch using\nAlgorithm 4 with a corresponding number of random values and\nfinally compared against the true distinct count. By repeating this\nprocess with many different random sequences, in our experiments\n100k, the bias and the RMSE can be empirically determined.\nAs this approach becomes computationally infeasible for distinct\ncounts beyond 1 million, we switch to a different strategy [ 17].\nAfter the first million of insertions, for which a random value was\ngenerated each time, we just generate the waiting time (the number\nof distinct count increments) until a register is processed with a\ncertain update value the next time. The probability that a register is\nupdated with any possible value 𝑘∈[1,(65−𝑝−𝑡)2𝑡]is given by\n𝜌update(𝑘)/𝑚with𝜌update(𝑘)from (10). Therefore, the number of\ndistinct count increments until a register is updated with a specific\nvalue𝑘the next time is geometrically distributed with correspond-\ning success probability. In this way, we determine the next update\ntime for each register and for each possible update value. Since the\nsame update value can only modify a register once, we do not need\nto consider further updates which might occur with the same valuefor the same register. Knowing these 𝑚×((65−𝑝−𝑡)2𝑡)distinct\ncounts leading to possible state changes in advance, enables us to\nmake large distinct count increments, resulting in a huge speedup.\nThis eventually allowed us to simulate the estimation error for\ndistinct counts up to values of 1021and also to test the presented\nestimators up to the exa-scale.\nFigure 6 shows the empirically determined relative bias and\nRMSE as well as the theoretical RMSE given by√︁\nMVP/((𝑞+𝑑)𝑚)\nfor the MLand the martingale estimator for configurations (𝑡,𝑑)∈\n{(1,9),(2,16),(2,20),(2,24)}and precisions 𝑝∈{4,6,8,10}. For\nintermediate distinct counts, perfect agreement with theory is\nobserved. For small distinct counts the estimation error is even\nmuch smaller. Interestingly, the estimation error also decreases\nslightly at the end of the operating range, which is in the order of\n264≈1.9·1019and thus lies in the exa-scale range. The estimators\nare essentially unbiased. The tiny bias which appears for small 𝑝for\ntheMLestimator can be ignored in practice as it is much smaller\nthan the theoretical RMSE.\n6 FUTURE WORK\nA topic for future research is the compressibility of ELL. According\nto Figures 4 and 5, much lower MVP s could be achieved with optimal\ncompression. For ULL, a special case of ELLwith𝑡=0and𝑑=2\n(compare Section 2.5), a reduction close to the theoretical limit\ncan be achieved with standard compression algorithms [ 17] such\nas Zstandard [ 12]. Unlike ULL, whose register size is exactly one\n8byte, we assume that standard algorithms will work worse in the\ngeneral case. Since the shape of the register distribution is known\n(see Section 3.1), some sort of entropy coding could be a way to\napproach the theoretical limit.\nAs discussed in Section 2.5, HyperMinHash is a special case of\nELL, and PCSA and CPC contain the same information as a ELL(0,\n64) sketch. Therefore, our proposed MLestimation approach, in\nwhich the MLequation is first reduced to the simple form (15)with\na relatively small number of terms, should also work for them. We\nassume that this method could lead to slightly lower estimation er-\nrors than current approaches, as MLestimation is generally known\nto be asymptotically efficient.\n7 CONCLUSION\nWe have introduced a new distinct counting algorithm called Exa-\nLogLog ( ELL), which includes already known algorithms as special\ncases. With the right parameters, the space efficiency can be im-\nproved significantly. 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In the present work, we investigate such thermal systems by using hydro-\nstatic equilibrium thermodynamics. Assuming that the gas between the horizons obeys the Tsallis\nstatistical mechanics, we found that it is possible to obtain the temperature gradient for the classi-\ncal gas. Interestingly, the gas behaves as classical gas near the horizon and behaves like quantum\ngas around flat spacetime with constant temperature. As a result, the multi-horizon black holes in\nhydrostatic equilibrium can be in a stable configuration with the aspect of the q-kinetic theory.\naEmail: maxwelltle@gmail.com\nbEmail: tahpahctar net@hotmail.com\ncEmail: pitbaa@gmail.com\n1arXiv:2402.13742v1  [gr-qc]  21 Feb 2024I. INTRODUCTION\nThe black hole is one of the curious objects predicted by Einstein’s general theory of rela-\ntivity. Classically, the black hole does not allow all particles to escape from its horizon even\nlight. However, by taking the quantum effect into account, the black hole can emit radiation,\ncalled Hawking radiation, like a black body radiation with a certain temperature known as\nHawking temperature [1]. According to the first laws of black hole thermodynamics, the black\nhole entropy called Bekenstein-Hawking entropy is also defined as SBH=A/4 where Ais the\nsurface area at the black hole’s event horizon [1–3]. There have been numerous investigations\nof the thermodynamic properties of black holes (for reviews, see [4–6] and references therein).\nThe significant remark is that the thermodynamic system associated with the black hole can\nbe defined at the horizon. If the black hole has multiple horizons, the Hawking temperatures\nevaluated on distinct horizons are generally different. The multi-horizon black hole is indicated\nas the systems which are not in thermodynamic equilibrium. Consequently, there should exist\na temperature gradient or heat transfer between the aforementioned systems. It is interesting\nto investigate the behaviors of a gas in the curved spacetime between black hole’s horizons.\nIn the statistical mechanic framework, the macroscopic behavior of a thermal gas system is\ndescribed by motions of microscopic constituents based on the mechanical descriptions. The\ncollisions among fine-grained gas particles are simply treated as local interaction and also satisfy\nthe Boltzmann Stosszahlansatz. As a result, the kinetic behaviors of numerous gas particles\ncan be characterized by a one-particle distribution function defined in the phase space. The\nmacroscopic quantities, e.g. the number of particles, energy-momentum tensor, and entropy, are\ndetermined by considering the transfers of suitable fields in the momentum space. According to\nthe transfer equation at equilibrium, or equivalently the balance equation, the Gibbs-Boltzmann\n(GB) entropy flow is proven from the well-known H-theorem that the entropy never decreases.\nOne of the important properties of the GB entropy is additive, S=S(1)+S(2), where S(1)\nandS(2)are the entropy of the subsystems 1 and 2, respectively. The additivity of the entropy\nrefers to ignoring the long-range interaction, such as gravitation. In addition, ignoring the long-\nrange interaction leads to additive internal energy, E=E(1)+E(2), where E(1)andE(2)are,\nrespectively, the internal energy of the subsystem 1 and 2. Consequently, at thermodynamic\nequilibrium, the empirical temperature can be defined by utilizing the principle of the maximum\nentropy, δS= 0 (and the energy is also minimum δE= 0), as 1 /T=∂S(1)/∂E (1)=∂S(2)/∂E (2)\nSuch an empirical temperature is compatible with the zeroth law of thermodynamics, T1=T2,\nleading to the thermal equilibrium.\n2GB statistical mechanics has been applied to many physical phenomena, and it is always\ncompatible with the isothermal system. However, including the long-range interaction, it suffers\nfrom the thermodynamic instability known as Antonov’s instability or gravothermal catastrophe\n[7–11]. This infers that GB statistical mechanics is not suitable to describe the self-gravitating\nsystem due to the long-range interaction. In fact, by taking the long-range interaction into\naccount, the total energy and/or the total entropy lack additivity and extensivity. In order to\ninvestigate the self-gravitating system, statistical mechanics with generalized entropy should\nbe introduced to describe such a system rather than GB statistical mechanics.\nOne of the generalized entropies is Tsallis entropy [12] obeying the pseudo-additive compo-\nsition rule, S=S(1)+S(2)+ (1−q)S(1)S(2)[13, 14], where q >0 is a non-extensive parameter.\nWith the pseudo-additivity, the empirical temperature obtained from the principle of the max-\nimum entropy is not clear how to define. Consequently, the empirical temperature obtained\nfrom Tsallis entropy is not compatible with the zeroth law of thermodynamics [15], leading to\nthe thermodynamic in-equilibrium. As a result, Tsallis statistical mechanics has been widely\ninvestigated in non-isothermal self-gravitating systems. For example, the Tsallis entropy was\napplied to the stellar polytropes of the self-gravitating systems to seek and discuss their stabil-\nity [16–20]. With Tsallis statistical mechanics, the H-theorem, and the Boltzmann transport\nequation were derived [21–24]. The relation between the gradient of the temperature and the\ngradient of the gravitational potential for the self-gravitating systems is obtained, and, using\nthese results, the thermodynamic behaviors of the spatial in-homogeneous temperature are\nanalyzed [21, 25–28]. Besides the investigations as mentioned earlier, statistical mechanics,\nassociated with Tsallis entropy, is also applied to cosmology. In particular, it has been argued\nthat the modified Friedmann equation admits the accelerated expansion without appealing to\nany kind of dark energy models [29–34]. It was pointed out that the non-extensivity in the\ncontext of black hole thermodynamics can infer the generalized uncertainty principle [35, 36].\nRegarding Bekenstein-Hawking entropy as Tsallis one, the thermodynamic properties and the\nstability of the black holes were analyzed [37–46].\nAlthough there are various statistical mechanics approaches dealing with long-range interac-\ntion, the relativistic kinetic theory turns out to be the most efficient and convenient option. In\nfact, it is useful to apply to various contexts ranging from stability theory [47–49], astrophysics\n[50, 51] to cosmology [52, 53]. Moreover, we aim to study gaseous systems in curved spacetime\ncompatible with relativistic systems. Then it is suitable to use the relativistic kinetic theory\nin the present work. There have been firstly developed and generalized the non-relativistic\n3distribution functions for the classical gas and quantum gas to the relativistic ones by J¨ uttner\n[54, 55]. Particularly, the equation of state describing such a case is also derived. The covariant\nkinetic theory of relativistic gases has been constructed by Synge who studied a system of gases\n[56].\nIn curved spacetime, Tolman and Ehrenfest found that relativistic gases can only be in\nequilibrium when the effects of the temperature gradient and the gravitational field cancel out\n[57, 58], and the kinetic theory on curved spacetime was formulated by Tauber and Weinberg\n[59]. The hydrodynamic equations for fluid on curved spacetime were derived from the Boltz-\nmann transport equation by using the Chapman-Enskog method [60], and then applied such\nequations to the gaseous system composed of zero rest mass particles in spherically symmetric\nspacetime [61]. Fourier’s laws for a single gas and Fick’s law for gas mixtures were obtained by\nconsidering the gases in Schwarzschild (Sch) spacetime under the post-Newton approximation\n[62]. For the kinetic theory in the language of modern differential geometry, there are inves-\ntigations such as the geometric formulation on the tangent bundle [63–65] and the cotangent\nbundle [66].\nIn this work, we investigate the possibility of describing the thermal systems for a black hole\nwith multiple horizons. We determine such thermal systems by using the q-kinetic theory in the\npresence of a gravitational field [67, 68]. By assuming that the gas between the horizons obeys\nthe Tsallis statistical mechanics, it is found that there exists the temperature gradient for the\nclassical gas, but it does not for the quantum gas. In fact, the temperature gradient depends\non the gradient of the chemical potential. Due to the similar roles of the chemical potential and\ngravitational potential, we set them to be proportional to each other in order to characterize\nthe behavior of the gases between black hole’s horizons. Interestingly, the gas near the black\nhole horizon behaves as a classical one, while it will be observed the quantum nature of the gas\nfar away from the black hole with the same temperature as that from Hawking radiation. We\nalso found that the similar situation can be obtained for a black hole with a single horizon such\nas the Sch black hole as well as one with an inner horizon such as a charged black hole. As a\nresult, the multi-horizon black holes in hydrostatic equilibrium can be in a stable configuration\nwith the aspect of the q-kinetic theory.\nThis work is organized as follows. In Sec. II, the relativistic Boltzmann transport equation\nbased on GB entropy in the gravitational field is reviewed. The sufficient and necessary condi-\ntions thermodynamic equilibrium of the systems are presented. It is found that GB statistical\nmechanics provides the self-gravitating system being always isothermal. In Sec. III, the Tsallis\n4entropy and the corresponding q-Boltzmann transport equation are introduced. We construct\nthe systems’ q-power law distribution function, and the expressions of physical quantities for\nthe non-extensive parameter, q̸= 1, are derived. We will see from the obtained results that the\ntemperature gradient of the systems does not vanish, specifying a non-isothermal configuration.\nWith those thermodynamic quantities, they have been adopted to construct the thermodynamic\nquantities of the static and spherically symmetric black holes in Sec. IV, In Sec. V, the main\nconclusion and discussion of this work are stated. Throughout this paper, the reduced Planck\nconstant ℏ, Boltzmann constant kB, speed of light c, and Newton’s gravitational constant G\nare set as unity for convenience.\nII. THE RELATIVISTIC BOLTZMANN TRANSPORT EQUATION\nThis section is devoted to recapping the key results of the kinetic theory of the relativistic\ngas in the presence of the gravitational field. Let us start with considering relativistic particles\nof a single non-degenerate gas, i.e., the classical gas. Each particle can be characterized by the\nspacetime coordinates xαand the four-momentum Pα= (P0, Pi). The (curved) spacetime is\ndescribed by metric gαβ. The states of the particle in the phase space ( xα, Pi) is kinetically\ncharacterized by the one-particle distribution function, ρ(xα, Pα) =ρ(xα, Pi). Note that the\ntemporal component of the four-momentum depends on the spatial ones obeying the constraint:\nPαPα=−m2. For the particles with the momentum d3P/P 0passing through a hypersurface\nnαdΣ where nαis the normal vector of the hypersurface, the change in the number of particles\nis defined as d N=ρ(xµ, Pi)dΓ where the invariant volume element of the phase space is given\nby dΓ = PαnαdΣd3P/P 0. The collisions of the gas are assumed to satisfy the Boltzmann\nStosszahlansatz which is composed of three following assumptions. 1) the gas is sufficiently\ndilute so that only the collisions of pairs of particles are considered. 2) the assumption of\nmolecular chaos: pairs of incoming particles (before collision) are not correlated. The pairs of\noutgoing ones (after collision) are also obeyed. 3) the variation of the one-particle distribution\nfunction over time interval is sufficiently slow. Such a time interval is much shorter than the\ntime between two collisions, but much longer than the duration of a collision. As a result, the\nnet rate of change in the number of the particles due to the particles’ local binary collision yields\nthe time evolution of the one-particle distribution function known as the classical Boltzmann\ntransport equation [67, 68]\nPα∂ρ\n∂xα−Γγ\nαβPαPβ∂ρ\n∂Pγ=C[ρ], (1)\n5where Γγ\nαβis the Christoffel symbol. The left-hand side is, in fact, the total derivative of the\ndistribution function with respect to the particles’ proper time. C[ρ] is the collision functional\ngiven by\nC[ρ] =Zd3P∗\nP∗0√−g\u0000\nρ′\n∗ρ′−ρ∗ρ\u0001\nFσdΩ. (2)\nHere, F,σ, dΩ, and gare the invariant flux, the differential cross-section of the collision, the\nelement of the solid angle, and the determinant of the metric tensor, respectively. Due to the\ncurvature of spacetime, the integral is that over the invariant element of the momentum space,\n√−gd3P∗/P∗0. We also have used the abbreviations such that the distribution functions for\nthe incoming particles (before the collision) with the momenta PiandPi\n∗areρ=ρ(xα, Pi)\nandρ∗=ρ(xα, Pi\n∗), respectively. Similarly, the distribution functions ρ′=ρ(xα, P′i) and\nρ′\n∗=ρ(xα, P′i\n∗) stand for those of the outgoing particles (after the collision) with the momenta\nP′iandP′i\n∗, respectively. It is very important to emphasize that there is no contribution due\nto external forces in Eq. (1).\nIn considering the change of an arbitrary scalar function ψ(xα, Pi) over the momentum\nspace, the transfer of the field can be described by [68]\n∇α\u0014Zd3P\nP0√−gPαρ ψ\u0015\n+Zd3P\nP0√−gPαρ\u0012\n−∂αψ+ Γi\nαβPβ∂ψ\n∂Pi\u0013\n=1\n4Zd3P∗\nP∗0d3P\nP0(−g)\u0000\nψ+ψ∗−ψ′−ψ′\n∗\u0001\u0000\nρ′\n∗ρ′−ρ∗ρ\u0001\nFσdΩ. (3)\nThe right-hand side refers to the production of the scalar field due to collisions. It is worthwhile\nto note that the macroscopic description of the gas can be determined by this transfer equation\ncorresponding to the moments of the distribution function. For example, considering the GB\nentropy, which can be expressed in terms of the distribution function ρas follows:\nSGB=−Z\ndΓρlnρ, (4)\nits transfer is simply defined by integrating the integrand in Eq. (4) along the momentum of\nthe gas over the momentum space, i.e.,\nSα=−Zd3P\nP0√−gPαρlnρ. (5)\nThis expression is equivalent to the choice of the arbitrary scalar function given by ψ=−lnρ.\nThe entropy production due to collisions is proved by the well-known Boltzmann’s H-theorem\nto be always non-negative. From the microscopic point of view, the (detailed) balance equation\nfor the entropy four-flow Sαis obtained as ∇αSα≥0 (the details of calculations can be seen\n6in many standard books e.g. [68]). This result remarkably corresponds to the second law of\nthermodynamics.\nAt the equilibrium state, there is no net flux of the number of particles so the collision term\nin the right-hand side of Eq. (1) vanishes. The equilibrium distribution function denoted by\nthat with superscript “(0)”, therefore, satisfies a condition\nρ′(0)\n∗ρ′(0)−ρ(0)\n∗ρ(0)= 0, (6)\nor, equivalently,\nlnρ′(0)\n∗+ lnρ′(0)= lnρ(0)\n∗+ lnρ(0). (7)\nThis is a summational invariant which has been proved that the distribution function must be\ntaken in the form of [68]\nlnρ(0)(xα, Pα)∼A(xα) +Bβ(xα)Pβ, (8)\nwhere A(xα) and Bβ(xα) are arbitrary scalar and vector, respectively. It is noticed that we\nwrite the equilibrium distribution function as ρ(0)(xα, Pα) while keeping in mind that the on-\nshell condition: PαPα=−m2must be held. It has been investigated that ρ(0)is taken in the\nform of the Maxwell-J¨ uttner distribution function [69]\nρ(0)(xα, Pα) =K(xα) exp\u0014µ(xα)−vβPβ\nT(xα)\u0015\n, (9)\nwhere µ(xα),T(xα), and vβare the chemical potential, temperature, and four-velocity of the\ngas particles, respectively. K(xα) is just a normalization factor. The above form of distribution\nfunction satisfied the summational invariant condition for the equilibrium field by replacing the\narbitrary functions in Eq. (8) as A=µ/T andBα=−vα/T.\nBy substituting the distribution function in Eq. (9) into the Boltzmann transport equa-\ntion (1), one straightforwardly obtains\nPαh\n∂αK+K∂α\u0010µ\nT\u0011i\n−1\n2PαPβKh\n∇α\u0010vβ\nT\u0011\n+∇β\u0010vα\nT\u0011i\n= 0. (10)\nSince, on the left-hand side of the above equation, the terms containing different order of the\nmomentum are linearly independent, both the terms in the square bracket must be both zero.\nLet us focus on the quadratic term of the momentum, i.e., ∇α\u0000\nvβ/T\u0001\n+∇β\u0000\nvα/T\u0001\n= 0. It is\nfurther written as\nT\u0000\n∇αvβ+∇βvα\u0001\n=vα∂βT+vβ∂αT. (11)\n7Contracting Eq. (11) with vαvβ, and then employing the property vαvα=−1 and vα∇βvα= 0,\nthe condition on the gradient of the temperature is given by\nvα∂αT= 0. (12)\nFor the rest frame of the particle (i.e., vα= (1,0,0,0)), the above condition leads that tem-\nperature is independent of time. As mentioned in Refs. [25, 26], the non-relativistic gas has no\nspatial gradient in its temperature. In order to keep these facts for any reference frame, we fur-\nther impose that there are no temperature gradients for all components, i.e., ∂αT= 0. Hence,\nin our consideration, the temperature of the classical gas with standard GB statistical mechan-\nics is uniform and constant as time evolves. This means that the equilibrium self-gravitating\nsystem in this scenario is isothermal.\nFurthermore, contracting Eq. (11) with vα, the other thermodynamic relations can be ob-\ntained as\nvα∇αvβ=−1\nT∂βT. (13)\nIt is noticed that the left-hand side refers to the parallel transport of the four-velocity which\nvanishes if the gas particles move along the geodesic in the curved spacetime. Therefore,\nthe temperature gradient on the right-hand side is indeed an external force that governs the\nparticles’ trajectory deviating from the geodesic. This implies that a self-gravitating object\ncan form due to the presence of the temperature gradient which is interpreted as the pres-\nsure counter-balancing the gravitational collapse. According to the aforementioned conclusion,\nthe temperature gradient of the system described by GB statistical mechanics, unfortunately,\nvanishes. The celestial objects are not possible to treat as equilibrium systems.\nIn order to obtain the statistical description for the systems in thermodynamic in-equilibrium,\nstatistical mechanics based on the non-extensive entropy should be taken into account. In the\nnext section, one of the non-extensive entropies known as Tsallis entropy will be introduced.\nThen the Boltzmann transport equation based on such entropy is adopted to derive the ther-\nmodynamic properties of the non-extensive systems.\nIII. THE KINETIC THEORY BASED ON TSALLIS STATISTICAL MECHANICS\nIn this section, the thermodynamic quantities of the N-relativistic particles in the presence of\nthe gravitational fields are investigated by using statistical mechanics based on Tsallis entropy.\nThe Tsallis entropy with one non-extensive parameter, q >0, can be defined in terms of the\n8one-particle distribution function as follows: [12, 13, 70]\nSq=−Z\ndΓρqlnqρ. (14)\nwhere ln qρis the q-logarithmic function of the distribution function. Such a logarithm and its\ninverse (i.e., q-exponential function) of an arbitrary positive function Xcan be, respectively,\nexpressed as\nlnqX=1\n1−q\u0000\nX1−q−1\u0001\n, (15)\nexpqX=\u0002\n1 + (1 −q)X\u00031\n1−q. (16)\nOne of the significant features of the Tsallis entropy is its non-extensivity which is obtained\nfrom the composition identity of the q-logarithmic function\nlnqX+ ln qY= ln q\u0000\nXY\u0001\n−(1−q) lnqXlnqY. (17)\nIt is noticed that the last term on the right-hand side disappears when q= 1. This property\nimplies the pseudo-additive composition rule of the Tsallis entropy: Sq(1+2) =Sq(1)+Sq(2)+\n(1−q)Sq(1)Sq(2), where the subscript (1 + 2) denotes the quantity of the combined system 1\nand 2 while those of subsystems are denoted by (1) and (2). In addition, the Tsallis entropy in\nEq. (14) can recover the GB one by taking the limit of the non-extensive parameter approaching\nunity, i.e., lim\nq→1Sq=SGB. The composition rule also reduces to the additive one in this limit.\nFor the kinetic behavior of the non-extensive systems, such as self-gravitating systems, the\nrelativistic gas can be characterized by q-distribution function, ρq(xα, Pα), in which its time\nevolution obeys the q-Boltzmann transport equation in the presence of the gravitational field\nas [21, 22, 24]\nPα∂ρq\n∂xα−Γα\nβγPβPγ∂ρq\n∂Pα=Cq[ρq]. (18)\nNote that, by comparing to Eq. (1), the non-extensivity does not affect the form of the left-hand\nside of Eq. (18) because the mentioned terms are relevant only to the geometry of spacetime,\nnot to the properties of particles’ collisions. The non-extensivity effect is, thus, incorporated\nthrough the q-generalized collision term:\nCq[ρq] =Zd3P∗\nP∗0√−gRq(ρq, ρ′\nq)FσdΩ, (19)\nwhere Rq(ρq, ρ′\nq) is the difference of correlation functions associated with before- and after-\ncolliding gas particles. As proposed in Ref. [21], the form of Rq(ρq, ρ′\nq) reads\nRq(ρq, ρ′\nq) = expqh\u0000\nρ′\nq∗\u0001q−1lnqρ′\nq∗+\u0000\nρ′\nq\u0001q−1lnqρ′\nqi\n−expqh\u0000\nρq∗\u0001q−1lnqρq∗+\u0000\nρq\u0001q−1lnqρqi\n. (20)\n9It is important to emphasize that the collision term Cq[ρq] with above Rq(ρq, ρ′\nq) is assumed\nto satisfy the q-generalized form of the molecular chaos hypothesis rather than the traditional\nversion. In other words, the recent collision term cannot satisfy Boltzmann’s Stosszahlansatz,\nand the assumption of the collision of the gas described by q-statistical mechanics is then nec-\nessarily modified [21, 23, 71]. At the GB limit, on can check that lim\nq→1Rq(ρq, ρ′\nq) = (ρ′\n∗ρ′−ρ∗ρ)\nor, equivalently, lim\nq→1Cq[ρq] =C[ρ] as expected. Straightforwardly, the transfer of the Tsallis\nentropy is given by\nSα\nq=−Zd3P\nP0√−gPαρqlnqρ. (21)\nIt is found that there exists the inequality condition ∇αSα\nq≥0 which is associated with the\nq-generalization of the H-theorem [21, 23].\nThe equilibrium q-distribution function is defined as a function satisfying the equilibrium\ncondition: Cq[ρq] = 0. From the collision in Eq. (19) together with Eq. (20), the equilibrium\nq-distribution function has to obey the summational invariant:\n\u0000\nρ′(0)\nq∗\u0001q−1lnqρ′(0)\nq∗+\u0000\nρ′(0)\nq\u0001q−1lnqρ′(0)\nq=\u0000\nρ(0)\nq∗\u0001q−1lnqρ(0)\nq∗+\u0000\nρ(0)\nq\u0001q−1lnqρ(0)\nq. (22)\nSimilar to the argument in the previous section, one then obtains\n\u0000\nρ(0)\nq\u0001q−1lnqρ(0)\nq∼Aq(xα) + (Bq)β(xα)Pβ, (23)\nwhere Aqand ( Bq)βare the coefficients associated with the aforementioned summational in-\nvariant. By a few steps of algebraic rearrangement, the equilibrium q-distribution function can\nbe expressed as\nρ(0)\nq(xα, Pα) = Kq(xα)\u0014\n1−(1−q)n\nAq(xα) + (Bq)β(xα)Pβo\u0015−1/(1−q)\n, (24)\nwhere Kq(xα) is the normalization factor. It is found that the above equilibrium distribution\nfunction with q→1 is reduced to the GB one in Eq. (9) when the coefficients are set by\nKq=1=K,Aq=1=Aand\u0000\nBq=1\u0001\nβ=Bβas expected. Plugging this q-distribution function in\ntheq-Boltzmann transport equation (18), the vanishing conditions of both linear and quadratic\nterms in momentum, respectively, yield\n∂αKq\u0002\n1−(1−q)Aq\u0003\n+Kq∂αAq= 0, (25)\n∂αKqh\n−(1−q)\u0000\nBq\u0001\nβi\n+Kq\n2h\n∇α\u0000\nBq\u0001\nβ+∇β\u0000\nBq\u0001\nαi\n= 0. (26)\nIt is possible to solve the above system for a non-trivial condition of Aqand ( Bq\u0001\nβas follows:\n−2(1−q)∂αAq\u0000\nBq\u0001\nβ=\u0002\n1−(1−q)Aq\u0003h\n∇α\u0000\nBq\u0001\nβ+∇β\u0000\nBq\u0001\nαi\n. (27)\n10In the same analysis fashion, the coefficients are interpreted as the standard fields so that\nAq=µ/T qand ( Bq\u0001\nβ=vβ/Tq. First contracting vαvβto Eq. (27) with the aforementioned\nsubstitution and also using the identities: vαvα=−1 and vα∇βvα= 0. As a result, we obtain\nvα∂αTq= (1−q)vα∂αµ. (28)\nObviously, without the non-extensivity (i.e., q= 1), the temperature gradient always vanishes\nas identical to Eq. (12). For the q-generalization, the temperature gradient of the non-extensive\nsystem exists due to the particle transfer characterized by the gradient of the chemical potential.\nConsidering the self-gravitation system, gravity radially attracts all surrounding gas particles\nto form the sphere of the gas. The particles in the system move from the larger radius to\nthe smaller one. This feature is characterized by the gravitational potential Φ( r)∝ −1/r,\nwhere the negative sign refers to the attractive interaction. Therefore, in this sense, the gas\nparticles transfer from a higher gravitational potential to a lower one. In the thermodynamic\nframework, the transfer is indeed characterized by the chemical potential µ. Furthermore,\nthe sign of the chemical potential for a classical system is negative. To illustrate it, let us\nconsider, for example, the system of two distinguishable particles at thermodynamic equilibrium\nwith the internal energy, E= 2ϵ. A number of the microstates for arranging these particles\nin all possible energy levels can be counted as three states so that the possible microstates\nare{(0,2ϵ),(2ϵ,0),(ϵ, ϵ)}. Therefore, the entropy of the system is written as S=S(Ω =\n3). In addition, if we add a particle with zero internal energy into the system, the possible\nmicrostates are given by {(0,0,2ϵ),(0,2ϵ,0),(2ϵ,0,0),(ϵ, ϵ,0),(0, ϵ, ϵ),(ϵ,0, ϵ)}, yielding more\nentropy S′=S′(Ω = 6). At the thermodynamic equilibrium, to keep the entropy constant\nafter adding the particle, the additional particle should have the energy −ϵ, resulting in the\npossible microstates {(ϵ,0,0),(0, ϵ,0),(0,0, ϵ)}. Hence, the total internal energy decreases from\nE= 2ϵtoE′=ϵ. In other words, the change in the internal energy is ∆ E=−ϵ. Using\nthe first law of thermodynamics, the chemical potential of the system can be obtained as\nµ=\u0000\n∂E/∂N\u0001\nS,V=−ϵ. In conclusion, for the classical particles with fixing the entropy and\nthe volume constant, the internal energy of the system must decrease when the number of\nparticles increases. It then indicates that the chemical potential in this scenario is always\nnegative. According to the similar behaviors between the aforementioned potentials, the gas\nsystem in the presence of a gravitational field might be suitably studied (in the next section) by\ninterpreting the chemical potential as the gravitational potential µ(r)∝Φ(r). Consequently,\nfrom Eq. (28), the gradient of the temperature Tqis proportional to that of Φ( r).\nTurning our attention back to another implication of the q-Boltzmann transport equation,\n11let us contract Eq. (27) with vα, and adopt the same setting as previously done. As a result,\none obtains\n−(1−q)µ\nT3\nq∂αTq+2(1−q)\nT2\nq∂αµ\n=1\nT2\nq∂αTq−1\nT2\nq\u0014\n1−(1−q)µ\nTq\u0015\nvαvβ∂βTq+1\nTq\u0014\n1−(1−q)µ\nTq\u0015\nvβ∇βvα.(29)\nEmploying the relation between the gradients of temperature and chemical potential in Eq. (28),\nthe parallel transport of the four-velocity is expressed as\nvβ∇βvα=−\n1 + (1 −q)µ\nTq\n1−(1−q)µ\nTq\n1\nTq∂αTq\n+(1−q)\n2\nTq∂αµ+\u001a\n1−(1−q)µ\nTq\u001b1\nTqvαvβ∂βµ\n1−(1−q)µ\nTq\n. (30)\nIt is complicated and not easy to analyze. In order to capture the physical meaning of this\nequation, let us expand it around q= 1 as\nvβ∇βvα≈ −1\nTq∂αTq+(1−q)\nT2\nq\u0010\n2Tq∂αµ−2µ∂αTq+Tqvαvβ∂βµ\u0011\n. (31)\nThe gas particles described by Tsallis’s statistical mechanics cannot move along the geodesic\nsince the right-hand side does not vanish. This is a key emergent feature when the non-extensive\neffect is taken into account. In addition, the particles are not enforced only by the external\nforce due to the temperature gradient, but there is also that from the non-extensivity. The\ncompact object might be formed undergoing the balance between these forces and gravitational\ncollapse at the thermodynamic equilibrium.\nBefore finishing this section, let us consider the quantum gas in which the gas particles\nare treated as indistinguishable. The q-local equilibrium distribution function in which the\nquantum effect is taken into account is given by [24]\nρ(0)\nq,(quantum)(xα, Pα) =1\n\u0014\n1−(1−q)\u0012µ−vβPβ\nTq,(quantum)\u0013\u00151/(1−q)\n−ξ, (32)\nwhere ξ= 1 for the bosonic gas and ξ=−1 for the fermionic gas. Substituting it into the\nBoltzmann equation (18) and following the same steps of calculation, the condition on the\ntemperature are obtained due to vanishing of the terms with quadratic momentum as follows:\nvα∂αTq,(quantum) = 0. (33)\n12With our assumption, the gradient of the temperature in every component is all zeroes. For the\nstatic and spherically symmetric gas, its temperature is radially constant, i.e., ∂rTq,(quantum) (r) =\n0. Surprisingly, this result is identical to that of the classical gas described by GB statistical\nmechanics. In the recent consideration, the quantum gas cannot be applied to the system of\nnon-uniform temperature such as multi-horizon black holes as will be discussed in the next\nsection.\nIV. THE THERMODYNAMIC PROPERTIES OF THE BLACK HOLES\nA black hole can emit the radiation known as the Hawking radiation. It behaves like the\nblack body and carries thermodynamic states such as temperature and entropy. As a result,\nwe can treat the black hole as a thermal object. Note that the thermodynamic quantities\nof the black hole are evaluated at the event horizons. The black hole with multiple horizons\ncorresponds to multiple thermal objects with generically different temperatures. Therefore, it\nseems like the thermal equilibrium thermodynamics used to describe the thermal behavior of\nthe black hole cannot be fully utilized for such a black hole. In this sense, it is worthwhile to\ntreat the multi-horizon black hole as the thermal object being in hydrostatic equilibrium.\nIn this section, we investigate how two thermal systems evaluated at the black hole’s horizons\nare in hydrostatic equilibrium by assuming that the gas between the horizons obeys the Tsallis\nstatistical mechanics. For concreteness, we consider the Schwarzschild-de Sitter (Sch-dS) black\nhole which is the static and spherical solution of the Einstein field equation with a positive\ncosmological constant, Λ >0. The solution can be written as [72]\nds2=−f(r)dt2+f−1(r)dr2+r2dΩ2,with f(r) = 1−2M\nr−Λ\n3r2, (34)\nwhere Mand Λ are, respectively, the mass parameter and the cosmological constant. Note\nthat we restrict our consideration to the case Λ >0 since it is possible to deal with the black\nhole with two horizons. The horizons of the spherically static black hole can be obtained by\nsetting f(r) = 0. From Eq. (34), it is found that f(r)→ −∞ forr→0+, while f(r)→ −∞\nforr→+∞. The horizon function f(r) is concave, and the maximum point can be obtained\nby solving d f/dr|r=rex= 0. As a result, one obtains the horizon radius for which the horizon\nfunction is maximum as\nrex=\u00123M\nΛ\u00131/3\n. (35)\nSubstituting Eq. (35) into Eq. (34), the extremum value of the horizon function can be obtained\n13as\nf(rex) = 1−\u0000\n9ΛM2\u00011/3. (36)\nBy requiring f(rex)≥0, the condition for having the horizon(s) can be constrained as\n1/51/31/2\n0 2 4 6 8 10 12-0.8-0.6-0.4-0.20.00.20.40.6\nFIG. 1: Behaviors of horizon function f(x) for Sch-dS black hole versus xby fixing α= 1/5\n(red), α= 1/3 (blue) and α= 1/2 (green).\n0< α≡√\nΛM≤1\n3. (37)\nFor convenience, let us define the dimensionless horizon radius as\nx=r\nM. (38)\nBy the definitions in Eqs. (37) and (38), the horizon function in Eq. (34) can be rewritten as\nf(x) = 1−2\nx−α2\n3x2, (39)\nThe horizon function can be illustrated in Fig. 1, from which there exist two horizons, namely,\nthe black hole horizon; rband the cosmic horizon; rc, where 0 < rb≤rc. The horizons can be\nobtained by solving f(xb) = 0 and f(xc) = 0. However, the full expressions of the horizons are\nlengthy and then we do not show them here. However, the approximation for the small value\nofα,|α| ≪1, can be expressed as\nxb,approx =2\n3h\n3 + 4 α2\u0000\n1 + 4 α2\u0001i\n,andxc,approx =−1 +√\n3\u00141\nα−α\n2\u0015\n. (40)\nAs we have discussed, in order to obtain the notion of the hydrostatic equilibrium, we\nsuppose that a system of relativistic particles is in a spherical shell between the black hole\nhorizon and the cosmic horizon. To satisfy the static and spherically symmetric spacetime, the\n14temperature gradient for the non-extensive gas shown in Eq. (28) can be written in terms of\nspherical coordinates as\n∂rTq(r) = (1 −q)∂rµ(r), (41)\nThe difference in the temperature between the black hole horizon and the cosmic horizon can\nbe obtained by integrating Eq. (41) with respect to rfrom rbtorcas follows:\nTq(rc)−Tq(rb) = (1 −q)\u0002\nµ(rc)−µ(rb)\u0003\n. (42)\nIn order to obtain the thermal equilibrium between gases and the thermal system of the black\nhole at the horizon, we suppose that the temperature of gases at each horizon is equal to\nthe thermal system of the black hole. From the above equation, one can see that for q= 1\ncorresponding to Boltzmann gases, the thermal system of the black hole at both horizons will be\nin thermal equilibrium. This situation cannot satisfy the black hole thermodynamics in which\nthe temperature at both horizons is not generically equal. For non-extensive gases, q̸= 1, it is\npossible to obtain the thermal systems which are different temperatures. In this case, we can\nnaturally interpret the gases between the horizon being in the hydrostatic equilibrium or the\nthermodynamic q-equilibrium.\nMoreover, for q̸= 1, one can see that the temperature gradient is related to the chemical\npotential gradient of the particles. Therefore, it is worthwhile to discuss how the chemical\npotential is related to the properties of the black hole. At the extremal case, rb=rc, we see\nfrom Eq. (42) that the difference of the temperature evaluated at each horizon vanishes. In\nthis scenario, the change of the surface gravity also vanishes, i.e., κb=κc. Moreover, the\ntemperature of the black hole evaluated at the horizon is proportional to the surface gravity\nand we also set the temperature of the gas at the horizon to be equal to one of the black holes.\nFurthermore, as we have discussed in Sec. IV, at Newtonian gravity, the chemical potential is\nproportional to the gravitational potential, µ(r)∝Φ(r), in such a way that the tendency of\nparticles to flow from larger radius to the smaller one. Note that the surface gravity can be\ninterpreted as the force per unit mass to hold the particle at the horizon from the observer far\naway [73]\nκbh≡a(rbh) =1\n2\f\f\f\fd\ndrf(rbh)\f\f\f\f. (43)\nAs a result, it is equivalent to the gravitational potential in most situations. From these results,\nwe can interpret the chemical potential to have a similar form of the surface gravity but now\ndoes not need to be evaluated at the horizon. From this argument, we can write the chemical\n15potential in the following form\nµ(r) =−G\f\f\f\fd\ndrf(r)\f\f\f\f, (44)\nwhere Gis the positive proportional constant. Note that the negative sign is adopted from the\nfact that the chemical potential of the classical gas is always negative and the surface gravity\nis always positive. Substituting Eq. (34) into Eq. (44), one obtains\nµ(r) =−G\f\f\f\f2M\nr2−2\n3Λr\f\f\f\f. (45)\nLet us define a dimensionless chemical potential in terms of the dimensionless variable as\nµ(x)≡1√\nΛµ(x) =−2G\n3\f\f\f\fαx−3\nαx2\f\f\f\f. (46)\nThe behavior of the temperature between the black hole horizon and the cosmic horizon can\nbe analyzed by using Eq. (41). Integrating both sides of Eq. (41) with respect to r, it becomes\nTq(r) = (1 −q)µ(r) +C, (47)\nwhere Cis an integrating constant. With GB statistical mechanics, i.e., q→1, a black hole and\na system of particles are at thermal equilibrium with the same temperature. Immediately, at\nthe limit q→1 and at the black hole horizon; r=rb, the integrating constant can be obtained\nas\nC= lim\nq→1Tq(rb) =1\n4π\f\f\f\fd\ndrf(rb)\f\f\f\f. (48)\nSubstituting Eqs. (44) and (48) into Eq. (47), the q-temperature can be obtained as follows:\nTq(r) =λ\f\f\f\fd\ndrf(r)\f\f\f\f+1\n4πd\ndrf(rb), (49)\nwhere λ≡ −(1−q)G. Note that, from Fig. 1, the slope of the horizon function at rbis always\npositive, and hence we have\f\f\f\fd\ndrf(rb)\f\f\f\f=d\ndrf(rb). Substituting the horizon function in Eq. (34)\ninto Eq. (49), the q-temperature and its dimensionless version (using xbfrom Eq. (40)) can be,\nrespectively, expressed as\nTq(r) = λ\f\f\f\f2M\nr2−2\n3Λr\f\f\f\f+1\n4π\u00122M\nr2\nb−2\n3Λrb\u0013\n, (50)\nTq(x)≡1√\nΛT(x) =1\n24α\n81−8α2n\n3 + 4 α2\u0000\n1 + 4 α2\u0001o3\n3πn\n3 + 4 α2\u0000\n1 + 4 α2\u0001o2+ 16λ\f\f\f\fα2x−3\nx2\f\f\f\f\n.(51)\nThe behavior of the temperature evolving between the black hole horizon and the cosmic\nhorizon can be illustrated on the left-hand side in Fig. 2. The temperature changes with x\n160.50.30.1\n0\n4 6 8 10 12 14 160.00.51.01.52.02.53.0\n4 6 8 10 12 14 16-5-4-3-2-1012FIG. 2: The left figure shows the temperature of the system evolving in between xbandxc\nwith the various values of λby fixing α= 1/10. The right figure shows the ratio of the\nchemical potential to the temperature, and the temperature gradient by fixing G= 1,\nα= 1/10, and λ= 1/10.\nbetween the horizons, causing the temperature gradient between the horizons as seen on the\nright-hand side in Fig. 2. Accordingly, the system is generally out of the thermodynamic\nequilibrium and non-isothermal. Interestingly, the temperature evaluated at the black hole\nhorizon with the non-vanishing non-extensive parameter, λ̸= 0, is greater than one with the\nvanishing non-extensive parameter, λ= 0. Between xbandxc, there is a point, xexobtained\nfrom d f/dx|x=xex= 0, at which the temperature evaluated at the black hole horizons with\nλ̸= 0 is equal to one with λ= 0. The mentioned temperature is written as\nTH(b)=81−8α2h\n3 + 4 α2\u0000\n1 + 4 α2\u0001i3\n72παh\n3 + 4 α2\u0000\n1 + 4 α2\u0001i2, (52)\nwhere TH(b)is the Hawking temperature evaluated at the black hole horizon. Although the\ntemperature evaluated at the black hole horizon with λ̸= 0 is generally greater than one with\nλ= 0, at the point xex, these are equal. At the point xex, it may be regarded as a point in\nflat spacetime, in which the temperature of the system is lowest and equal to TH(b)obtained\nfrom the quantum field theory being taken into account for considering around the black hole\nhorizon [1]. According to the mentioned arguments, we can imagine that if there is radiation\nemitted from the black hole to the observer at rest in flat spacetime, xex, the observer can\nmeasure its temperature as TH(b).\nIt is important to note that the behavior of the gases can be considered by using the ratio\nof the chemical potential and temperature, µ/T. The gas will behave as the classical gas where\n17the ratio is much more negative µ/T≪0 and behave as the quantum gas with µ/T→0.\nFrom the right-hand side in Fig. 2, we see that, at the horizons, the ratio is more negative,\ncompared to other points. On the other hand, the ratio approaches zero at the low temperature\nor around flat spacetime, i.e., xex. As a result, around flat spacetime or xex, the description\nof classical gas cannot be fully used since µ/T→0. We need the quantum description to\ncharacterize the behavior of the gas. As we have discussed in Sec. IV, for ideal quantum gases,\nthere is no temperature gradient. In other words, the temperature of the quantum gases is\nalways constant, and the system is at the thermodynamic equilibrium. Note that, there are\ncusps of the temperature profile at xexillustrated from the left-hand side in Fig. 2. Then it\nseems like the temperature is not smooth at this point. However, such a profile at this point is\nnot fully valid since it is the profile of classical gas. In fact, it is to be replaced by the profile\ntemperature of the quantum gas.\nIt is worthwhile to note that, even if we try to explain the thermal objects from the multi-\nhorizon black hole, our notion can be applied to the black hole with one horizon such as the Sch\nblack hole. For example, at the limit xc→ ∞ (orα→0), the Sch-dS case can reduce the Sch\ncase. The point xexalso approaches infinity, meaning that the temperature of the system with\nλ̸= 0 comes closely to the temperature with λ= 0 at the asymptotically flat spacetime, xex.\nTherefore, in this case, the observer stays at rest far away from the black hole and will observe\nthe quantum nature of the gases and the temperature is the same as Hawking’s temperature\nof the black hole.\nFurthermore, we can apply this notion to the black hole with an inner horizon such as a\ncharged black hole. The investigation can describe how the temperature of the gases inside the\nblack hole evolves with r. Note that we realize that it may be not possible to access in formation\nof the matter inside the black hole and the gases must move into the the singularity. However,\nthis investigation will explore how the gases inside the black hole can behave if we require that\nthermal objects evaluated at the horizons must be in hydrostatic equilibrium. Generally, let us\nconsider the Reissner-Nordstr¨ om-de Sitter (RN-dS) black hole where the charge of this black\nhole is denoted by Q. As a result, the horizon function for such a black hole is given by\nf(x) = 1−2\nx−α2\n3x2+β2\nx2, (53)\nwhere the new variables are set as follows x=r/M,α=M√\nΛ, and β=Q/M . The behavior\nof the horizon function is illustrated in Fig. 3. From this figure, we see that, in the range of\nx≥0, there exist three horizons, namely, the inner horizon; xin, the black hole horizon; xband\nthe cosmic horizon; xc, where xin≤xb≤xc.\n180 5 10 15 20-2.0-1.5-1.0-0.50.00.51.0FIG. 3: Behaviors of horizon function f(x) versus xby fixing α= 0.1 and β= 0.6.\nBy using those thermodynamic quantities, the q-temperature and the ratio of the chemical\npotential and temperature evaluating between the horizon from xintoxccan be obtained.\nHowever, their expressions are too lengthy, hence we do not explicitly show them here. However,\nwe confine ourselves to the behavior of the temperature by the plot as illustrated in Fig. 4. The\nleft panel of Fig. 4 shows the temperature profile which changes with xbetween xinandxb,\ndue to the temperature gradient in the black hole’s structure. On the right panel of Fig. 4, the\ntemperature profile behaves like those in the case of a Sch-dS black hole as we have discussed.\nInterestingly, between horizons xinandxb, there exist a cusp about x≈0.4, obtained from\ndf/dx|x=xex= 0, at which the temperature with λ̸= 0 approaches one with λ= 0.\n0.30.20.1\n0\n0.5 1.0 1.502468\n0.30.20.1\n0\n2 4 6 8 10 12 14 160.00.51.01.52.0\nFIG. 4: The left (right) figure shows the temperature of the system evaluating in between the\nhorizons xinandxb(xbandxc) with the various values of λby fixing α= 0.1, and β= 0.6.\nAt this point, the ratio of the chemical potential approaches zero as illustrated in the left panel\nof Fig. 5. Therefore, around this point, the gases become quantum gases and the temperature\ngradient vanishes. This behavior is similar to the one we discussed above. If there exists the\n19flat spacetime (d f/dr= 0), the gases behave as the quantum gases. This is also shown in the\nright panel of the Fig. 5.\n0.5 1.0 1.5-10-8-6-4-202\n2 4 6 8 10 12 14 16-6-5-4-3-2-101\nFIG. 5: The left (right) figure shows the ratio of the chemical potential and the temperature\nevaluating in between the horizons xinandxb(xbandxc) by fixing G= 1,α= 0.1,β= 0.6,\nandλ= 0.1.\nV. CONCLUSION AND DISCUSSION\nBlack holes are known to be so strongly gravitating from which even light cannot escape.\nClassically, they are not treated as thermal objects. However, Hawking demonstrated through\nquantum field theory in a curved space that black holes can emit thermal radiation and act as\nthe black body [1]. This paved the way for numerous studies on black hole thermodynamics\nand allowed physicists to understand such strongly gravitating objects as thermal objects. As\nthe thermal system obtained from the black hole is evaluated at the black hole horizons, the\nblack hole with multiple horizons corresponds to the multiple thermal systems with generically\ndifferent temperatures. Such thermal systems are out of thermal equilibrium and then the\nequilibrium thermodynamics may not be in a position to describe such thermal systems. This\nsuggests that other statistical mechanics beyond Gibbs-Boltzmann might be useful to gain\ninsight into understanding such thermal systems. Moreover, the statistical mechanics based\non Gibbs-Boltzmann entropy cannot be fully used to explain some systems with long-range\ninteraction while such the system will be compatible with the statistical mechanics based on\nthe non-extensive entropy [7–11]. In this work, we used statistical mechanics based on the non-\nextensive Tsallis entropy to describe how the thermal systems of the multiple horizon black\nhole can be in the hydrostatic equilibrium instead of thermodynamic equilibrium.\nBy assuming that there exist relativistic gases between the black hole horizon and the cos-\n20mic horizon, it is possible to investigate the properties of the gases and whether to satisfy the\nhydrostatic equilibrium or not. For GB statistical mechanics, by using the Maxwell-J¨ uttner dis-\ntribution function together with the Boltzmann transport equation, we found that the temper-\nature gradient vanishes for all components, ∂µT= 0. The vanishing of the temporal component\ncorresponds to the fact that the temperature is independent of time in the rest frame of the\nparticles. For the spatial components, it vanishes since in the non-relativistic limit, the spatial\ngradient of the temperature vanishes, ∂iT= 0 [25, 26]. We also found that the self-gravitating\nsystem cannot form since gas particles always move along geodesic without counter-balancing\nterms contributed by the gradient of the temperature. As a result, the gases with GB statistical\nmechanics do not satisfy the presence of two thermal objects obtained from the horizons of the\nSch-dS black hole.\nFor the Tsallis statistical mechanics, we performed in the same step with GB statistical\nmechanics but now use the q-distribution function and q-Boltzmann transport equation instead\nof the Maxwell-J¨ uttner distribution function together with the Boltzmann transport equation\nrespectively. One of the main results is that there exists the gradient of the temperature which\nis proportional to the gradient of chemical potential, ∂αT= (1−q)∂αµ. To satisfy the stationary\nof spacetime, we require that both temperature and chemical potential do not depend on time.\nMoreover, by analyzing the properties of the gravitational potential and the chemical potential,\nwe found that they share the same characteristics. Therefore, the chemical potential is set to\nbe proportional to the gravitational potential to characterize the behavior of the gases between\nthe horizons of the black hole. Furthermore, we also found that the gas particles do not move\nalong geodesics due to the existence of the temperature gradient and non-extensivity, i.e., q̸= 1.\nThis provides us with a possible way to describe two thermal objects of the Sch-dS black hole\nwith the hydrostatic equilibrium. It is worthwhile to note that by using the same step of the\ncalculation, we also found that the quantum relativistic gas does not provide the temperature\ngradient even if the non-extensivity is taken into account.\nDue to the hydrostatic equilibrium of the non-extensive gas, it is worthwhile to apply this\nkind of gas to describe how two thermal objects for the multi-horizon black hole can be in\nequilibrium. In the present paper, we focus on the static and spherically symmetric black\nhole with multiple horizons. For the Sch-dS black hole, there exist two horizons denoted by\nblack hole horizons (smaller one) and cosmic horizon (larger one). According to the spherical\nsymmetry, the temperature is dependent only on radial coordinate r. Then the equation for\nthe temperature gradient can be written as ∂rT= (1−q)∂rµ. In order to obtain the thermal\n21equilibrium between gas and the thermal system of the black hole at the horizon, we suppose\nthat the temperature of gas at each horizon is equal to the thermal system of the black hole.\nThen, the temperature of the gas can be written as T(r) =Tb+λdf(r)\ndrwhere Tbis the Hawking\ntemperature at the black hole horizon and λ=−G(1−q),Gis a proportional constant. Now\nλcharacterizes the non-extensive effect. The temperature profile with various λcan be found\non the left panel of Fig. 2. Moreover, by observing the ratio µ/T on the right panel of Fig. 2,\nwe found that the gas behaves classically near the horizons and acts like quantum gas around\nflat spacetime. If we observe the gas far away from the black hole we will observe the quantum\nnature of the gas and their temperature will be constant and equal to one for the Hawking\nradiation.\nEven though we attempt to describe the thermal properties of the black holes with multiple\nhorizons, this notion can be applied in a black hole with one horizon such as Sch black hole.\nIn this case, we found that the gas near the black hole horizon behaves as a classical one, while\nwe will observe the quantum nature of the gas far away from the black hole with the same\ntemperature as one from Hawking radiation. Moreover, we can apply this notion to a black\nhole with an inner horizon such as a charged black hole. In this case, the gas inside the black\nhole will be classical and there exists a particular radius at which the gas becomes quantum\ngas. 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The algorithm works in the most restrictive low-space setting,\nwhere each machine has nδlocal space for a constant 0< δ < 1.\nThe standard approaches for (∆ + 1)-coloring are ignorant about the graph\ntopology in the following sense: They exploit the property that anypartial coloring\ncan be extended to a feasible (∆+1)-coloring of the whole graph. For most graphs,\nthechromaticnumberismuchsmallerthan ∆+1andwewouldliketofindcolorings\nwith fewer colors. However, as soon as we have fewer than ∆+1colors, completing\npartial colorings might not be possible.\nIn this work, we study the vertex-coloring problem in sparse graphs param-\neterized by their arboricity α, a standard measure for sparsity. We give deter-\nministic algorithms that in constant, or almost constant, time give poly αand\nO(α)-colorings, where αcan be arbitrarily smaller than ∆. A strong and standard\napproach to compute arboricity-dependent colorings is through the Nash-Williams\nforest decomposition, which gives rise to an (acyclic) orientation of the edges such\nthat each node has a small outdegree.\nOur main technical contribution is giving efficient deterministic algorithms to\ncompute these orientations and showing how to leverage them to find colorings\nin low-space AMPC. A key technical challenge is that the color of a node may\ndependon almost all oftheothernodesinthegraphandthesedependenciescannot\nbe stored on a single machine. Nevertheless, our novel and careful exploration\ntechnique yields the orientation, and the arboricity-dependent coloring, with a\nsublinear number of adaptive queries per node.\n1Supported by the Research Council of Finland, Grant 334238\n2Supported by the Austrian Science Fund (FWF), Grant P36280-N\n3Supported in part by Research Council of Finland grant 334238 and Helsinki Institute for Information\nTechnology HIIT.arXiv:2402.13755v1  [cs.DC]  21 Feb 2024Contents\n1 Introduction 1\n1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n2 Sublinear LCA: High Level Challenges and Techniques 5\n2.1 Where should one query? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2.2 Our solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n2.3 Using our LCArecursively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3 Model and Definitons 9\n3.1 The AMPCmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n3.2 Arboricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n3.3 β-Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n3.4 Induced β-partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n3.5 Natural β-partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n4 An LCAfor Partial β-partition 13\n4.1 The Coin Dropping Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n4.2 Game Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14\n4.3 The LCA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16\n5β-partition in AMPC 18\n6 Coloring Graphs in AMPC 20\n6.1 An O(α2+ε)-coloring in O(1/ε)Rounds . . . . . . . . . . . . . . . . . . . . . . . 21\n6.2 An O(α2)-coloring in O(logα)Rounds . . . . . . . . . . . . . . . . . . . . . . . 21\n6.3 A ((2 + ε)α+ 1)-coloring in O(αlogα)rounds . . . . . . . . . . . . . . . . . . . 22\n6.4 Coloring when αis Large . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Introduction\nIn this paper, we study the graph coloring problem in sparse graphs in a modern model of par-\nallel computation, the AMPC-Model [BDE+19]. The Adaptive Massively Parallel Computation\n(AMPC) model of computation captures the challenges in modern platforms of processing mas-\nsive data. We have Pmachines with S=O(nδ)local space each, where 0< δ < 1andnis the\nnumber of nodes in the input graph. In synchronous rounds, the machines can communicate\nthrough a distributed key-value storage. The machines have read access to the storage and, at\nthe end of each round, the machines can write to the storage. A crucial limitation is that the\nlocal space limits the number of reads and writes per round. The AMPCmodel is a practically\nmotivated extension of the standard MPCmodel, where the communication is restricted to\nall-to-all synchronous message-passing [KSV10].\nBefore delving into our results, let us explain the necessary background. The goal of the\nk-vertex-coloring problem is to assign a color out of kcolors to every node of a graph such\nthat no neighboring nodes have the same color. This is a classic and central problem that has\nbeen intensively studied for decades. Vertex-coloring also has close connections to other classic\nproblems such as the maximal independent set and scheduling.\nIn the distributed and parallel setting, the typical goal is to color the graph with ∆ + 1\ncolors, where ∆corresponds to the maximum degree of the input graph. Indeed, any graph can\nbe colored with ∆ + 1colors, and for cliques and odd cycles, this is tight. But many graphs\ncan be colored with significantly fewer than ∆ + 1colors. For example, sparse graphs can have\na high maximum degree but can often be colored with much fewer colors. Sparse graphs often\noccur in practice providing a demand for better coloring algorithms for sparse graphs.\nArboricity. In this work, we study the vertex-coloring problem in graphs parameterized by\narboricity , which is a standard measure of “everywhere” sparsity. The arboricity αcorresponds\nto the minimum number of forests that the edges of a graph can be partitioned into [NW64].\nIt is well-known that any graph can be colored with 2αcolors and in some cases, this is tight.\nAs a straightforward example, a tree clearly has arboricity 1and can be colored with 2colors,\nwhile the maximum degree ∆can be up to n−1. Hence, arboricity captures the number of\ncolors needed to color sparse graphs.\nFinding arboricity-dependent colorings with O(α), poly (α), orexp(α)colors has gained\nattentioninvariousmodelsofcomputationlike LOCAL[BE10b,BE10a,Bar15,FHK16a,MT22,\nMau21], MPC[GS19,FGG23,BCG20], dynamicalgorithms[HNW20,CNR23,CR22,BCPS23],\nand streaming [BCG20]. A common approach is to find an acyclic orientation of the edges such\nthat each node has an outdegree of O(α)or poly α. Starting from sinks, the nodes can then\nbe iteratively colored with outdegree + 1 colors. Typically, the outdegree gives a trade-off\nbetween the runtime and the number of required colors. This approach to coloring highlights a\ncrucial difference between (∆ + 1)-coloring and arboricity-dependent coloring and pinpoints a\nkey challenge. In the former case, any partial solution can be extended to a complete solution\nbut this is not the case for the latter. In this light, the iterative coloring approach can be seen\nas carefully choosing partial solutions that can be extended to a complete solution.\nIn recent work, the problem was studied in the context of dynamic algorithms [CNR23].\nThey showed that givensuch an orientation, there is an efficient Local Computation Algorithm\n(LCA) to find an arboricity-dependent coloring. They also showed how to maintain such an\norientation efficiently. In the LCAmodel, each node can be queried for the ith entry in their\n1adjacency list and their degree [RTVX11]. The goal is to implement an oracle, that returns the\noutput at any node v, say, the color of vin the coloring problem. The crux is that the outputs\n(per node) must yield a consistent and feasible solution over the whole graph.\nIn our setting, the orientation is not given for free and one of our contributions is an LCAto\nfind an acyclic orientation with a small outdegree. As pointed out by the authors in [CNR23],\nthere is a caveat to designing such an LCA. While there is no formal proof, it seems likely\nthe problem has a high volume, i.e., any LCAneeds to do linear (in n) queries in the worst\ncase. To circumvent this obstacle, we design an LCAthat orients almostall of the edges. We\nshow that this LCAis sufficient for designing efficient AMPCalgorithms to compute acyclic low\noutdegree orientations orienting alledges. Lastly, we obtain various AMPCcoloring algorithms\nthat employ different tradeoffs between their runtime and the number of colors used. We next\ndetail these three contributions in order, (1) the LCAfor computing “weak orientations”, (2)\nAMPCalgorithms for computing acyclic low outdegree orientations, and (3) AMPCcoloring\nalgorithms. We emphasize that all of our algorithms are deterministic .\nContribution 1: An LCAfor computing a weak orientation. Aβ-partition of a graph\nis a partition of its vertex set into layers such that every node has at most βneighbors in\nhigher (or the same) layers. The size of the partition is the number of layers. This partition\nnaturally gives rise to the desired orientation required for efficient graph coloring algorithms by\norienting edges from smaller to larger layers while orienting them arbitrarily within the layers.\nOn graphs with arboricity at most α, it is known that such partitions with O(logn)layers exist\nwhen β≥(2+ε)αfor any constant ε >0[BE10b]. Our core building block and also the largest\nchallenge of our approach is the design of an LCAalgorithm that computes a weak version of\nthe desired partition. In Section 2 we underline the technical challenges related to obtaining\nsuch a result. The following simplified version of the main lemma assumes that βandαare\nconstant and that β≥(2 +ε)αholds for any ε >0. We denote by G[S]the subgraph induced\nby node set Sin graph G.\nLemma 1.1 (Simplified version of Lemma 4.7) .For any constant δ >0there is deterministic\nLCAalgorithm that uses at most O(nδ)queries per node on a graph Gwith arboricity αand\nassigns each node a layer from N∪ {∞}such that the following holds:\n▷There exists a subset S⊆Vcontaining at least a 1−1/nO(δ)fraction of vertices such\nthat the layering of Sforms a β-partition of G[S]withO(logβn)layers. ◁\nContribution 2: AMPCAlgorithms for finding acyclic low outdegree orientations.\nOur second contribution is designing low-space AMPCalgorithms to find acyclic orientations in\ngraphs with any arboricity α. Ultimately, our AMPCalgorithm iteratively uses the above LCA\nto find a complete partition and the number of nodes that are not layered (after one application\nof the LCA) dictates the round complexity of the AMPCalgorithm.\nEssentially, Lemma 1.1 can be applied recursively1on the subgraph induced by the nodes\nthatareassignedan ∞layer, yielding O(1/δ) =O(1)recursionlevelsandimplyingthefollowing\ntheorem.\nTheorem 1.2 (β-partitioning) .For any constant ε >0and any β≥(2 +ε)αthere is a deter-\nministic AMPCalgorithm that computes a β-partition of size O(logβ/(2α)n)inO(logβ/(2α)β)\n1This recursion does not work as stated, because we have omitted some details to simplify the exposition.\nWe explain these subtleties in Section 2.3.\n2rounds on any n-node graph with arboricity α. The algorithm requires O(nδ)local space and\nO(n1+δ)total space for any constant δ >0. In particular,\n•when β=O(α), we get size O(logn)inO(logα)rounds and\n•when β=O(α1+ε), we get size O(logαn)inO(1)rounds.\nContribution 3: Coloring Sparse Graphs in AMPC.We design AMPCalgorithms for\ncoloring with different tradeoffs between their runtime and the number of colors used.\nTheorem 1.3 (Coloring Results) .For any constant ε >0on graphs with possibly non-constant\narboricity α, there are deterministic low-space AMPCalgorithms to compute:\n1. An α2+ε-coloring in O(1/ε)rounds.\n2. An O(α2)-coloring in O(logα)rounds.\n3. A ((2 + ε)α+ 1)-coloring in ˜O(α/ε)rounds.2\nAll of our coloring algorithms heavily exploit that we first compute a small outdegree orien-\ntation using Theorem 1.2 with different parameters. While Theorem 1.3 has various tradeoffs\nbetween the runtime and the number of colors, its main strength lies in the following corollary.\nCorollary 1.4. For any constant ε >0there is a constant time AMPCalgorithm to compute\na((2 + ε)α+ 1)-coloring on any graph with bounded arboricity α=O(1).\nThe Quadratic Barrier. One of our easiest results of Theorem 1.3, once we’re given the\nsmall outdegree orientation, is the one coloring with quadratic number of colors. Here, we\nessentially simulate the LOCALmodel algorithm by Linial that, very efficiently, computes a\nβ2-coloring on a graph with an outdegree βand only uses local information. The fact that\nthe algorithm works in graphs with a small outdegree (instead of small maximum degree as\nusually cited) was first observed in [BE10b]. Understanding when one can go below o(α2)colors\nwith low volume and/or round complexity seems fundamentally hard and is an intriguing open\nquestion (see, e.g., [BE13, FHK16b]).\nDeterministic algorithms. We emphasize that, in contrast to most prior algorithms in the\nAMPCmodel and also most efficient LCAalgorithms for large degree graphs, all our algorithms\nare deterministic. We note that deterministic algorithms seem fundamentally more constrained\nthan randomized ones. As an example, in the deterministic setting, even on a tree, it seems\nout of reach to design a sublinear LCAthat finds the nearest leaf of each node. In a closely\nrelated VOLUME model3, it is known that Ω(n)deterministic queries are required to solve\nclosely related problems on trees [RS20]. Furthermore, there is an exponential gap between the\nrandomized and the deterministic complexities.\n2The eO(·)notation hides O(logα)factors.\n3Acrucialdifferencebetween LCAandVOLUME isthatin VOLUME thequeriesmustformasingleconnected\ncomponent.\n3Very High Degrees. A particular challenge in massively parallel computation models is\nto efficiently deal with nodes whose degree is significantly larger than the local space of each\nmachine, especially for deterministic algorithms. As we desire to have efficient algorithms for\nall ranges of parameters, we design an algorithm using orthogonal techniques. We note that the\nfollowingtheoremholdsinthestrictlyweaker[BDE+19]low-spacemassivelyparallelcomputing\nmodel ( MPC).\nTheorem 1.5 (Deterministic Coloring) .For any x > 1, there is a deterministic low-space\nMPCalgorithm to compute an 2x∆-coloring in any n-node graph with maximum degree ∆in\nO(logxn)rounds using O(n+m)polylogntotal space.\nIn the case of α > nδ, we can still compute the low outdegree partitions as claimed in\nTheorem 1.2. Actually, the partition comes with the property that each layer also has a small\nmaximum degree, and hence, we can apply Theorem 1.5 (with x=αO(ε)=nO(1)) to color each\nlayer independently with O(α1+ε)colors in O(logxn) =O(1)time, using a separate color space\nfor each layer. This is one of the ways, how we use Theorem 1.5 in the proof of Theorem 1.3\nin the case of huge arboricity.\n1.1 Related Work\nIn the LCAmodel, there is an algorithm for the classic (∆ + 1)-coloring with poly ∆·logn\nqueries [CFG+19]. Given that the local space is larger than a small polynomial of ∆, we\ncan turn the LCAinto a randomized AMPCalgorithm that runs in constant time. The cur-\nrentdeterministic state-of-the-art originates from the LOCALmodel of distributed message-\npassing. Using the graph-exponentiation technique, we can simulate a known poly log log n\n-round algorithm [CLP18] in O(log log log n)rounds. This can be further derandomized to\nmatch the same runtime deterministically and similar techniques apply to (degree +1)-coloring\nas well [CDP21, CCDPM24]. In AMPC, we can do even better. The aforementioned algorithms\nconsist of two parts: Solving the problem on low-degree instances in O(log log log n)time and\nhigh-degree instances in O(log∗n)time. The former part can easily be simulated in O(1)\nrounds in the low-space AMPCeven though not explicitly stated in the papers. The runtime\nthen reduces to O(log∗n).\nThe coloring problem parameterized by the arboricity αhas enjoyed attention in other re-\nlatedmodelsofdistributedandparallelcomputationaswell. Inthe LOCALmodelofdistributed\nmessage-passing [Lin92], the edges of an input graph correspond to communication links along\nwhich the nodes communicate in synchronous rounds. Barenboim and Elkin gave deterministic\nO(logn)andO(αlogn)round algorithms for computing O(α2)and(2 + ε)α-colorings, re-\nspectively. It is also known that Ω(log n)rounds are required [Lin92] for arboricity-dependent\ncolorings. On the randomized side, it is known how to obtain an O(αlogα)coloring in O(logn)\nrounds and O(α)-coloring in O(logn·min{logα,log log n})rounds [GL17]. We note that there\nare also poly logntime algorithms for finding (1 +ε)α-outdegree orientations, but these orien-\ntations need not to be acyclic [HSV21].\nIn the CONGESTED CLIQUE model, we have nmachines that communicate in an all-to-all\nfashion. Each machine corresponds to a node in the input graph and initially knows its incident\nedges. The CONGESTED CLIQUE model is very close to the non-adaptive MPCmodel with\nlinear memory per machine, i.e., Ω(n)words per machine. There are deterministic algorithms\nforO(α)-colorings in both of the models that run in O(1)rounds [FGG23]. Finally, through the\n4rake-and-compressdecompositionontrees, onecanalsoobtainanorientationwithanoutdegree\nat most 2. In a recent work, it was shown how to obtain this decomposition in O(1)rounds of\nAMPC[HKSS22]. For the special case of forests (α= 1), [GLM+23] give a 3-coloring algorithm\nthat runs in O(log log n)rounds and uses optimal global space. This algorithm is conditionally\noptimal in the MPCmodel.\n1.2 Outline\nIn Section 2, we start with giving a technical overview of our main contributions and the\nchallenges we need to overcome. In Section 3, we define the AMPCmodel and also give several\nkey definitions for structures like partial, induced, and natural β-partitions. In Section 4, we\npresent the LCAfor computing partial β-partition and define a coin dropping game which is\nuseful in describing the query structure of the LCA, proving Lemma 4.7 (formal version of\nLemma 1.1). In Section 5, we show how to leverage the LCAto get a β-partition in the AMPC\nmodel, proving Theorem 1.2. Finally, in Section 6, we show the coloring of sparse graphs,\nproving Theorems 1.3 and 1.5.\n2 Sublinear LCA: High Level Challenges and Techniques\nWe design a deterministic LCAwith a sublinear number of queries to find an acyclic orientation\nwith a small outdegree β= Ω(α)for a very large fraction of the nodes in a graph with arboricity\nα. The informal statement is Lemma 1.1 and the formal one is Lemma 4.7.\nThe ultimate goal for our LCAwould be to compute a so called β-partition , which is a\npartition of its vertex set into layers such that every node has at most βneighbors in higher\n(or the same) layers, with as few layers as possible. This partition naturally gives rise to\nthe desired orientation required for efficient graph coloring algorithms by orienting edges from\nsmaller to larger layers while orienting them arbitrarily within the layers. As discussed in the\nintroduction, it seems unlikely that an LCAwith sublinear query complexity can compute such\na partition. Instead, we design a sublinear LCAthat computes the layers of mostof the nodes\nin the graph, see Figure 1.\n<latexit sha1_base64=\"ShGzq04+UHqe7utsmCLDVesPPW4=\">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KknBDzwVvHisYD+gDWWznbRLN5uwuxFC6I/w4kERr/4eb/4bt20O2vpg4PHeDDPzgkRwbVz321lb39jc2i7tlHf39g8OK0fHbR2nimGLxSJW3YBqFFxiy3AjsJsopFEgsBNM7mZ+5wmV5rF8NFmCfkRHkoecUWOljqAZKuINKlW35s5BVolXkCoUaA4qX/1hzNIIpWGCat3z3MT4OVWGM4HTcj/VmFA2oSPsWSpphNrP5+dOyblVhiSMlS1pyFz9PZHTSOssCmxnRM1YL3sz8T+vl5rwxs+5TFKDki0WhakgJiaz38mQK2RGZJZQpri9lbAxVZQZm1DZhuAtv7xK2vWad1W7fKhXG7dFHCU4hTO4AA+uoQH30IQWMJjAM7zCm5M4L86787FoXXOKmRP4A+fzB8V/jy4=</latexit>layer 1<latexit 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Black\nnodes form the dependency graph D(v)of node v.\n<latexit sha1_base64=\"ShGzq04+UHqe7utsmCLDVesPPW4=\">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KknBDzwVvHisYD+gDWWznbRLN5uwuxFC6I/w4kERr/4eb/4bt20O2vpg4PHeDDPzgkRwbVz321lb39jc2i7tlHf39g8OK0fHbR2nimGLxSJW3YBqFFxiy3AjsJsopFEgsBNM7mZ+5wmV5rF8NFmCfkRHkoecUWOljqAZKuINKlW35s5BVolXkCoUaA4qX/1hzNIIpWGCat3z3MT4OVWGM4HTcj/VmFA2oSPsWSpphNrP5+dOyblVhiSMlS1pyFz9PZHTSOssCmxnRM1YL3sz8T+vl5rwxs+5TFKDki0WhakgJiaz38mQK2RGZJZQpri9lbAxVZQZm1DZhuAtv7xK2vWad1W7fKhXG7dFHCU4hTO4AA+uoQH30IQWMJjAM7zCm5M4L86787FoXXOKmRP4A+fzB8V/jy4=</latexit>layer 1\n<latexit 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The question is easy to answer if we already knew the\nβ-partition ℓthat we will compute. This β-partition implies a subgraph D(v) :=G[D(v)]for\neach node vthat testifies the layer of node vinℓ. In other words, the layer of node vdepends\nonD(v), that’s why we call it the dependency graph ofv. See Figure 2a for an example. Once\nnode vhas learned D(v)with its queries, it can compute its layer. More formally,\nthe dependency graph D(v)of a node vcontains all nodes uthat are reachable from vvia a\npath of nodes with strictly decreasing layers. Again, see Figure 2a. Observe that every node\ninD(v)has at most βneighbors outside of D(v). Of course, the imagined β-partition ℓis\nsomething that we can utilize in the analysis and to build intuition, but we cannot use it to\nguide the nodes to explore the graph, simply because it is not known beforehand.\nObserve that if the dependency graph of a node vis too large, there is no hope for vto\nlearn it in a sublinear number of queries. Hence, our only hope is that at least nodes with small\ndependency graphs ( ≤nδ) manage to learn their respective dependency graphs and compute\ntheir layers. In our analysis, we show that most of the nodes have small dependency graphs.\nEverything boils down to solving an impossible sounding task: guide the graph explo-\nration of a node vsuch that most of what vqueries is within its dependency graph,\nwith no prior information on where to search. Deterministic graph exploration without\nprior knowledge is hard, especially since the graph may look identical in every direction at a\nnode v, but only a small fraction of the directions contain the dependency graph and there is\nno way to identify these directions that we want to explore. Let us briefly explain why the\nclassic search paradigms fail in this setting.\n•Depth-first-search: Recall that node vcan have up to βneighbors outside the dependency\ngraph. If the algorithm happens to initiate the DFS from a neighbor outside of D(v), all\nof the queries might be used on learning something that is not the dependency graph.\n•Breadth-first-search: In order to avoid the pitfalls of DFS, one could employ a BFS with\nthe hope of balancing the exploration of the graph. Indeed, after learning its 1-hop\nsurroundings, at least a 1/βfraction of what is queried is the dependency graph, which\n6sounds promising. Unfortunately, this approach is also flawed. Consider a node vand its\nneighbor uthat is outside v’s dependency graph. The degree of uis unbounded, so if for\nexample deg(u) =nδ, most of the queries in the second level of the BFS will be spent on\nlearning the neighbors of u, which are outside D(v), and hence useless.\nClearly, a more sophisticated and fine-tune exploration procedure is required. Our actual\napproach explores the graph in a \"volume-balanced\" way, instead of \"distance-balanced\" like\nBFS and DFS.\nVolume-basedquerying. Inordertosteerthedirectionofthequeries,considerthefollowing\ncoin dropping procedure. Give nδcoins to node v, who naïvely (this approach will still not\nwork) forwards the coins to all of its neighbors, giving each neighbor an equal share of the\ntotal number of v’s coins. Consider repeating this process recursively for every node holding\ncoins, until coins cannot be further divided, which is in some sense a \"volume-based\" BFS.\nThe forwarding of coins represents querying for nodes, and every node that has received a coin\nhas been queried by vand hence knowntov. Optimally, all nodes of D(v)receive a coin.\nUnfortunately, this approach fails for a large family of counter-examples: skewed dependency\ngraphs (Figure 2b), comprising of a long path where every node has, e.g., nδ/10, neighbors in\nthe lowest layer. In this case, nδcoins will be already spent after only 10 steps of distributing\ncoins, effectively learning only roughly k·nδ/10dependency graph nodes.\n2.2 Our solution\nOur solution uses the same coin dropping procedure, but we devise a more sophisticated for-\nwarding rules that determine how to distribute the coins. These rules utilize the topology of\nthe graph induced by v’s knowledge aka the nodes vhas discovered so far. The forwarding rules\nare also updated frequently during the course of the algorithm to steer the future exploration\nof the graph and making very strong usage of the adaptivity of the queries.\nRecall that every node in D(v)has at most βneighbors outside of D(v). Hence, when coins\nare at a node u∈D(v)anduforwards coins to β+ 1neighbors, we ensure that at least one\nneighbor from D(v)receives coins, or more precisely at least a 1/(β+ 1)-fraction of the coins\ngoes to neighbors in D(v). If we keep recursively applying this approach to all nodes that still\nhave coins available, this rough tells us, that after ksteps of forwarding, at least x/(β+ 1)k\ncoins are at nodes that are contained in the dependency graph. Here xis the number of coins\nthat vstarted with. However, it is not sufficient to choose β+ 1neighbors arbitrarily, since\nit may be that we keep forwarding coins to nodes that we already know, i.e., nodes that have\nseen a coin before. In the end, it may be that all of these coins are just accumulated in a small\npart of D(v)that we already know, or in other words, the process does not lead vto querying\nnew, undiscovered, parts of the dependency graph.\nOur coin forwarding rules. Hence, we need to choose the correct neighbors to whom we\nforward the coins such that enough coins \"flow\" into the part of the dependency graph we\nhave not yet learned, and not too many \"leak out\" outside of the dependency graph or \"get\nstuck\" in a part of the dependency graph that we already know. The rules for choosing the\ncorrect neighbors are chosen adaptively and frequently updated. Node vkeeps simulating an\nestimation of the β-partition (this estimation is called the induced β-partition, formalized in\n7Definition 3.6) for the whole subgraph it has queried so far, which we call Sv(which is only a\nsmall part of the whole graph). In the simulation, every node in Svis assigned a layer by v,\nor if no layer is assigned, we assume it is ∞. During an iteration, nodes forwards coins to the\nβ+1neighbors with the largest simulated layer. If more than β+1neighbors have an ∞layer,\na node can forward the coins to any such β+ 1neighbors. We show that with this forwarding\nrule, enough coins traverse to parts of D(v)that have not received any coins before, and we\nkeep learning more and more nodes of D(v).\nSurprisingly, we cannot halt the coin dropping game at the first point where node vis able\nto compute a layer ̸=∞for itself. It is possible, that after querying most (but not all) of\nits dependency graph, node vcomputes its estimated layer to be, e.g., nδ/10, while its actual\nlayer, if vwould have found the whole dependency graph, would be significantly smaller. This\nis unacceptable, since we ultimately want the number of layers to be logarithmic in n. The\nfinal success of our method depends on the frequency of updating the forwarding rules and the\nduration for which we run the algorithm.\n2.3 Using our LCA recursively\nDue to the sublinear query complexity, our LCAcan only compute the β-partition layers for\na large fraction of nodes, but not all. One would hope to simply recurse on the subgraph\nremaining after removing all nodes with a ̸=∞layer. However, this step turns out to require\nmore care. Taking a closer look at Lemma 1.1, we only get the guarantee that the exists a\nsubset S⊆Vcontaining at least a 1−1/nO(δ)fraction of vertices such that the layering of\nSforms a β-partition of G[S]. The difficulty of recursion stems from the fact that we cannot\nidentify the nodes belonging to this magical set Sand that there may exist nodes outside of set\nSthat have a ̸=∞layer, which are problematic, since including their layering in the partition\nmay result in a globally inconsistent partition. Consider nodes v, u̸∈S; for example, a node\nvmay compute layer 10for itself based on the fact that its neighbor ushould be in layer 9.\nHowever, node u, for some reason, has computed its layer to be 11. If we place nodes vandu\nin their preferred layers, node vmight end up with more that βneighbors in higher layers.\nWe resolve the issue with a slightly stronger LCAthan stated in the informal Lemma 1.1.\nIn addition to outputting a layer ℓu(u), each node u∈Voutputs a proofof its own layer in the\nform of a β-partition ℓuon a small subgraph of G. The proof is basically a set of instructions\nfor layering a part of the graph such that vcan obtain layer ℓu(u).\nHaving access to this proof actually relieves us from needing to know which nodes having\nlayer̸=∞belong to Sand which do not. We can merge all β-partitions ℓuthat are given as\nproofs by nodes u∈Vwith̸=∞via a global minimum function\nℓ(v) = min\nu∈Vℓu(v)\nand get a globally consistent β-partition ℓon the subgraph induced by all nodes with layer\n̸=∞. This is sufficient for using our querying routines for recursion in AMPC.\n83 Model and Definitons\n3.1 The AMPC model\nIn the AMPCmodel, we have an input of size N, which is processed by Pmachines, each with\nspace S. The unit of measure is a word of size O(logn)bits. The total space of all machines is\nT=S·Pand we assume that S= Θ( N1−Ω(1)). In addition, there is collection of distributed\ndata stores (DDS) that we denote by D0, D1, . . .. We assume that all DDS provide a key-value\nstore semantics: they store a collection of key-value pairs, such that given a key, a DDS returns\nthe corresponding value. We require that both key and value consist of a constant number\nof words. The input data is stored in D0and uses a set of keys known to all machines (e.g.\nconsecutive integers). The solution to a problem is stored in the last data store.\nIf there are k >1key-value pairs stored in the DDS with the same key x, the individual\nvalues can be accessed by querying for keys (x,1), . . . , (x, k). Note that the indices from 1to\nkare assigned arbitrarily. A query for a key that does not occur in the DDS, results in an\nempty response. The computation proceeds in rounds. In the ith round, each machine can\nread data from Di−1and write to Di. Within a round, each machine can perform O(S)reads\n(henceforth called queries) and O(S)writes, and carry out arbitrary computation. Each query\nrefers to querying for a single key-value pair from Di−1and each write refers to writing a single\nkey-value pair to Di. The amount of communication that a machine performs per round is\nequal to the total number of queries and writes. The crucial property of the model is that the\nqueries that a machine makes within a round may depend on each other. In particular, say g\nis a function from XtoXand for each x∈X,Di−1stores a key-value pair (x, g(x)). Then, in\nround ia machine can compute gk(y)ifk=O(S).\nThe realism of the AMPCmodel is discussed in more depth in [BDE+19]. The algorithms\npresented in this paper take a graph with nnodes and medges as an input. In this case, the\ninput size is N=n+m. We also assume that S=nδ, where δ∈(0,1)is some constant.\n3.2 Arboricity\nDefinition 3.1 (Arboricity) .The arboricity α(G)of a graph G= (V, E)is defined as\nα(G) = max\nG′⊆G,\n|V(G′)|≥2\u001a\u0018|E(G′)|\n|V(G′)| −1\u0019\u001b\n.\nAn equivalent definition for α(G)is the minimum number of edge-disjoint forests into which\nthe edge set of Gcan be partitioned [NW64]. In this work, we will consider graphs with\narboricity bounded by parameter αthat is α(G)≤αfor some parameter α. To simplify\nnotation we will assume that the arboricity of the graph is α. We now state some simple facts\nabout the arboricity that immediately follow from Definition 3.1 and were also observed in\n[BE10b].\nFact 3.2 ([BE10b]).For any subgraph G′ofG, the arboricity of G′is at most the arboricity\nofG.\nFact 3.3 ([BE10b]).For any ε >0, a graph G= (V, E)with arboricity αhas at least ε|V|/(2+\nε)vertices with degree at most (2 +ε)·α.\n9An equivalent formulation of Fact 3.3 is that, for any ε >0, a graph with arboricity αhas\nless than 2|V|/(2+ε)vertices with degree greater than (2+ε)·α. We generalize this statement\nby parameterizing the degree bound as follows.\nLemma 3.4 (Generalization of Fact 3.3) .For any β≥1, a graph G= (V, E)with arboricity\nαhas less than 2α|V|/βvertices with degree greater than β.\nProof.Suppose for a contradiction that there are at least 2α|V|/βvertices with degree greater\nthan β. It follows that,\n2|E|=X\nv∈Vdeg(v)> β·2α|V|\nβ= 2α· |V|∗\n≥2·|E|\n|V| −1· |V|>2|E|,\nwhich is a contradiction. The inequality ∗follows from Definition 3.1, as |E|/(|V|−1)must be\na lower bound for arboricity.\n3.3 β-Partition\nWe begin with generalizing the graph-theoretic structure known as H-partition , which were\nintroduced in the context of distributed message passing models by Barenboim and Elkin\n[BE10b]. Informally, they define an H-partition as a partition of a graph with arboricity αinto\nO(logn)layers such that every node has at most (2 +ε)·αneighbors in higher or equal layers,\nwhere nis the number of nodes in the graph.\nFor our purposes, we define a β-partition , which is a partition of a graph into layers such\nthat every node has at most βneighbors in higher or equal layers, for any positive value β.\nOur definition also permits an infinity layer: nodes in the infinity layer can have an unbounded\nnumber of neighbors in the infinity layer. If the infinity layer is non-empty, we call the partition\napartial β-partition . Otherwise, we simply call it a β-partition.\nDefinition3.5 ((partial) β-partition) .Consider a graph G= (V, E)and a function λ:V(G)− →\nN∪ {∞}. For i∈N∪ {∞}define Vi={v∈V(G)|λ(v) =i}andV≥i=S\nj≥iVj.\nWe say that λis a β-partition ifdegV≥i(v)≤βfor every node v∈Gwith λ(v)̸=∞\nand|V∞|= 0. If|V∞|>0, we call λapartial β-partition . We refer to Viaslayer iof the\nβ-partition and λ(v)as thelayerof a node v. Thesizeof a (partial) β-partition is the number\nof unique non-infinity layers.\nObserve that if β=non an-node graph, the β-partition would just contain one layer – the\ninput graph. Hence, β-partitions are useful only for certain ranges of β. For example, when\nβ= (2 + ε)αwhere αis the arboricity, we get the well known H-partition by Barenboim and\nElkin [BE10b].\nWe extend the definition of β-partitions further to node-induced subgraphs, where its aprior\nunclear how to deal with neighbors not in the subgraph.\n3.4 Induced β-partition\nBarenboim and Elkin [BE10b] define an algorithm where we iteratively put all nodes with\ndegree at most (2 +ε)αin the top-most layer and delete them from the graph. This algorithm\nhas the property that each node will have at most (2+ε)αneighbors in the same layer or above,\nand therefore, the output of this algorithm is an (2 +ε)α-partition. Using Facts 3.2 and 3.3, it\n10is easy to see that size of the partition produced by this algorithm is at most O(log(2+ε)/2n).\nIn Definition 3.6, we generalize the algorithm of [BE10b] in two ways: first, by making the\ndegree bound βand second, by running it only on a subset Sof the nodes.\nIn the following, we emphasize that all notions refer to the original graph Gif not specified\notherwise, even when we consider an (induced) subgraph of G. In particular, N(u)anddeg(u)\nalways refer to NG(u)anddegG(u), respectively. The following definitions do not assume any\nparticular model of computation.\nDefinition 3.6 (S-induced β-partition) .Consider a graph G= (V, E), a subset S⊆Vand\nanyβ≥1. Consider the partial β-partition σS,β:V(G)− →N∪ {∞}constructed as follows.\n1. Set σS,β(v)← − ∞for every v∈V.\n2. For i= 0, . . . ,|S|:\nSetσS,β(v)← −ifor all nodes v∈Swith at most βnodes u∈N(v)for which σS,β(u) =∞.\nWe refer to σS,β(v)as thelayerof a node v.\nWe often write σorσSinstead of σS,βwhen subset Sand/or parameter βare clear from\ncontext. The following lemma proves many useful properties about function σS,βproduced by\nDefinition 3.6. In particular, part ii.proves that σS,βis indeed a partial β-partition.\nLemma 3.7. Consider a graph G= (V, E), a subset S⊆Vand any β≥1. The partial\nβ-partition σS,βfrom Definition 3.6 satisfies the following properties for a node v∈S.\ni. IfσS,β(v) =∞, then σS,β(w) =∞for at least β+ 1nodes w∈N(v)\nii. If σS,β(v)̸=∞, then σS,β(w)≥σS,β(v)for at most βnodes w∈N(v)\niii. If σS,β(v)̸=∞anddeg(v)≥β+ 1, then σS,β(w)≥σS,β(v)−1for at least β+ 1nodes\nw∈N(v)\nProof.In the following, an iteration refers to one round of Step 2 in Definition 3.6. Let σk\nS(v)\ndenote value σS(v)at the beginning of iteration k. For readability, we write σinstead of σS,β.\ni.Observe that if there is an iteration where no node is assigned a σvalue, none of the\nfollowing iterations assign σvalues. Hence, if value |S|is assigned, σ(v)̸=∞holds for all\nnodes v∈S. Towards a contradiction, consider an iteration i <|S|after which σ(v) =∞\nand there are at most βnodes w∈N(v)with σ(w) =∞. In iteration i+ 1, it must be that\nσ(v)← −i+ 1̸=∞, a contradiction.\nii.Consider σ(v) =σi(v) =i̸=∞. By definition, it holds that σi(w) =∞for at most β\nneighbors w∈N(v). Hence, for said neighbors it will hold that σ(w) =∞orσ(w)∈[i,|S|].\nIn either case, σ(w)≥σ(v).\niii.Consider σ(v) =σi(v) =i̸=∞. Since deg(u)≥β+ 1, it must hold that i >0. Define\nIito be the set of nodes w∈N(v)for which σi(w) =∞. Define Ii−1similarly and observe\nthat Ii⊊Ii−1: since σ(v) =i, it must be that |Ii−1| ≥β+ 1and|Ii| ≤β. Hence, for nodes\nw∈Ii−1\\Iiit holds that σ(w) =i−1. Since σi(w) =∞for nodes w∈Ii, it must be that\nσ(w)≥i. Hence, for |Ik−1| ≥β+ 1nodes w∈N(v)it holds that σ(w)≥i−1 =σ(v)−1.\nThefollowinglemmarevealsthesocallednon-increasingnatureofthe S-induced β-partition.\nVery informally, the intuition for the lemma is that the larger the subgraph used to construct\ntheS-induced β-partition, the smaller the layer assignment for every node in the graph. In\nother words, more information gives a better partition.\n11Lemma 3.8. Consider a graph G= (V, E), subsets S⊆T⊆Vand any β≥1. Then\nσS,β(v)≥σT,β(v)for every node v∈V.\nProof.For readability, we write σSandσTinstead of σS,βandσT,β, respectively. If S=T,\nthen σSequals σTby definition. Hence, consider only S⊂T. The claim holds for all nodes v\nfor which σS(v) =∞. Since σS(v) =∞holds for all nodes v∈V\\Sby definition, we only\nneed to prove the claim for nodes v∈Sfor which σS(v)̸=∞.\nConsider the construction of the induced partitions σSandσTfrom Definition 3.6 side-by-\nside. Towards a contradiction, let ibe the first iteration (side-by-side) after which σS(v)<\nσT(v)holds for a node v∈S. It must be that σS(v) =i̸=∞andσT(v) =∞, which implies\nthatσS(w) =∞for at most βnodes w∈N(v)and that σT(v) =∞for at least β+ 1nodes\nw∈N(v). Hence there must have existed one node xfor which σS(x)< σT(x)after iteration\ni−1, a contradiction.\nThe analysis of our algorithm will crucially rely on the properties of a so called dependency\ngraphof a node, which is defined through a (partial) β-partition.\nDefinition 3.9 (Dependency graph) .Consider a graph G= (V, E), a subset S⊆Vand any\nβ≥1. The dependency graph G[D(σS,β, v)]of a node v∈Vis defined as the output of the\nfollowing recursive procedure.\n•IfσS,β(v) =∞, then D(σS,β, v) =∅\n•IfσS,β(v) = 0, then D(σS,β, v) ={v}\n•otherwise D(σS,β, v) =nS\nw∈N<(v)D(σS,β, w)o\n∪ {v}where N<(v)is the set of nodes\nw∈N(v)such that σS,β(w)< σS,β(v).\nObservation 3.10 (Nested property of dependency graphs) .Consider a graph G= (V, E),\na subset S⊆Vand any β≥1. For all w∈D(σS,β, v)for a node v∈Sit holds that\nD(σS,β, w)⊆D(σS,β, v).\nProof.By Definition 3.9, a node wcan be added to D(σS,β, v)only by adding D(σS,β, w)to\nD(σS,β, v), and D(σS,β, w)contains wby definition.\nLemma 3.11. Consider a graph G= (V, E), a subset S⊆Vand any β≥1. IfσS,β(v)̸=∞\nfor a node v∈S, then |N(v)\\D(σS,β, v)| ≤β.\nProof.We write σinstead of σS,β. By Definition 3.9, set N(v)∩D(σ, v)contains all nodes w∈\nN(v)for which σ(w)< σ(v). Hence, N(v)\\D(σ, v)contains all nodes for which σ(w)≥σ(v).\nBecause σ(v)̸=∞, by Lemma 3.7 it holds that σ(w)≥σ(v)for at most βnodes w∈N(v).\n3.5 Natural β-partition\nWe want to accentuate the case when the whole node set of a graph is used to construct an\ninduced β-partition. For a graph G= (V, E), we call the V-induced β-partition the natural\nβ-partition .\nDefinition 3.12 (Natural β-partition) .Consider a graph G= (V, E), and any β≥1. The\nnatural β-partition ℓβis defined as the V-induced β-partition σV,β.\n12Forβ= (2+ ε)αthe natural β-partition corresponds to the H-partition from [BE10b]. Our\nmotivation behind the natural β-partition definition is that ℓβassigns the lowest possible layer\nto all nodes in the graph among all induced β-partitions. More formally, we have the following\nlemma.\nLemma 3.13. Consider a graph G= (V, E), a subset S⊆Vand any β≥1. It holds that\nσS,β(v)≥ℓβ(v)for every v∈V.\nProof.By Lemma 3.8, it holds that σS,β(v)≥σT,β(v)for any subgraphs S⊆T⊆Vand every\nnode v∈V. When T=V, it holds that σS,β(v)≥σV,β(v) =ℓβ(v).\nLater, in our algorithmic usage of induced β-partitions, we will compute many of these\npartitions in parallel on various induced subgraphs G[S],S⊆Vof the input graph G. The next\nlemma states that the level assignment of a node vinσS,βcorresponds to its layer assignment\nin the natural β-partition if G[S]contains the dependency graph of v.\nLemma 3.14. Consider a graph G= (V, E), a subset S⊆Vand any β≥1. IfD(ℓβ, v)⊆S\nfor a node v∈V, then σS,β(w) =ℓβ(w)for every w∈D(ℓβ, v).\nProof.For readability, we write σSandℓinstead of σS,βandℓβ, respectively.\nLet us first prove the claim for S=D(ℓ, v)by induction on the values of ℓ(v), the base\ncase being ℓ(w) = 0. The base case implies that deg(w)≤β, so by Definition 3.6 it holds that\nσS(w) = 0. Assume that σS(w) =ℓ(w)for all nodes w∈Swith ℓ(w)< i. Consider a node\nu∈Swith layer ℓ(u) =i. Observe that all nodes w∈N(u)with ℓ(w)< ℓ(u) =iare in S\nandσS(w) =ℓ(w)by induction. Since ℓ(u) =i, by Lemma 3.7, there are at most βnodes\nw∈N(u)with ℓ(w)≥ℓ(u), implying that σS(u)will be i.\nWe have established that σD(ℓ,v)(w) =ℓ(w)for every node w∈D(ℓ, v)ifS=D(ℓ, v).\nIfS⊃D(ℓ, v), the previous implies σS(w)≤ℓ(w), because by Lemma 3.8 it holds that\nσS(w)≤σD(ℓ,v)(w). On the other hand, by Lemma 3.13, for any subgraph S⊆Git holds that\nσS(w)≥ℓ(w)for every node w∈S. Hence, if D(ℓ, v)⊆Sit holds that σS(w) =ℓ(w)for every\nnode w∈D(ℓ, v).\n4 An LCAfor Partial β-partition\nIn this section, we present an LCAfor constructing a partial β-partition (Definition 3.5) of the\ninput graph using a coin dropping game. The game is formulated from the perspective of a\nsingle node making local queries to the graph and can be easily implemented as an LCA(see\nLemma 4.7). Later, we will simulate this game for many nodes in parallel in the AMPCmodel\n(see Section 5).\n4.1 The Coin Dropping Game\nThe high-level idea of the coin dropping game for a node vin graph G= (V, E)is as follows.\nDuring the game, we keep track of a certain subset of nodes Sv⊆Vthat initially contains\nonly node v, but grows in every super-iteration. In the beginning of every super-iteration, we\ncompute the Sv-induced β-partition σSv,βfrom Definition 3.6. One can think of σSv,β(v)as\nan approximation of the natural β-partition ℓβ=σV,β(see Definition 3.12). During a super-\niteration, we perform a coin distribution process (consisting of iterations) that uses σSv,βas a\n13guiding mechanism to add certain nodes to Svsuch that σSv,β(v)of the next super-iteration is\na “better” approximation of ℓβ(v).\nThe coin distribution process consist of iterations, in each of which, nodes holding coins\nforward them along the edges of the graph according to certain local forwarding rules . In the\nvery first iteration, only node vis given coins. The whole process can be viewed as a flow of\ncoins that starts at node vand spreads through the graph. The flow stops when it reaches\nnodes outside of Sv, after which all nodes outside of Svthat have received a coin are added to\nSvof the next super-iteration. The local forwarding rules manifest as forwarding sets for every\nnode in Sv, which require computing the Sv-induced β-partition. Informally, the forwarding\nset of a node uis a small subset of u’s neighbors, to whom uwill be forwarding its coins during\nthe super-iteration. More formally, see Definition 4.1. While keeping the size of set Svroughly\nbelow x3for some parameter x, we ensure that eventually σSv,β(v) =ℓβ(v)for all nodes that\nhave small enough dependency graphs, i.e., |D(ℓβ, v)|< x2and a small enough layer in the\nnatural β-partition, i.e., ℓβ(v)≤logβ+1x. We emphasize that the game does not guarantee\nthatD(ℓβ, v)⊆Sv, which complicates the analysis.\nDefinition 4.1 (Forwarding set) .Consider a graph G= (V, E), a subset S⊆Vand any\nβ≥1. Let σS,βbe the S-induced β-partition. The forwarding set F(σS,β, u)for a node u∈S\nis the set of any min{deg(u), β+ 1}nodes w∈N(u)with the highest σS(w)values.\n(x, β, F )-coin dropping game. Consider executing the (x, β, F )-coin dropping game for a\nnode vin a graph G= (V, E). Throughout the game, we keep track of a subset Sv⊆V.\nThe game consists of x2super-iterations, where one super-iteration proceeds as described in\nAlgorithm 1. In the first super-iteration Svis initialized to Sv← − {v}.\nAlgorithm 1 A super-iteration of the (x, β, F )-coin dropping game for a node v∈V.\n1:Compute the Sv-induced β-partition σSv,βand the forwarding set F(σSv,β, u)for all u∈Sv.\n2:Node vis given xcoins.\n3:For|V|iterations: every node u∈Svwith x′≥ |F(σSv,β, u)|coins forwards x′/|F(σSv,β, u)|\ncoins to every node in F(σSv,β, u).\n4:Every node u̸∈Svwith coins is added to Sv.\n4.2 Game Analysis\nLet us analyze the coin dropping game. Recall that Algorithm 1 refers to one super-iteration of\nthe game. Recall that nodes cannot compute the natural β-partition ℓβthemselves. However,\nwe utilize ℓβfor analysis. The following lemma ensures progress, and informally it states that\nif the Sv-induced β-partition assigns a layer to node vwhich does not equal ℓβ(v)(it can be\nanything between ℓβ(v)and∞), a node from the dependency graph D(ℓβ, v)is added to Sv\nwhich was not in Svpreviously.\nLemma 4.2 (Measure of progress) .Consider a graph G= (V, E), any β, x≥1and the\nnatural β-partition ℓβ. Consider the beginning of an arbitrary super-iteration of the (x, β, F )-\ncoin dropping game for a fixed node v∈Vand the corresponding set Sv⊆V. If|D(ℓβ, v)| ≤x2,\n14ℓβ(v)≤logβ+1xandσSv,β(v)> ℓβ(v), a node w∈D(ℓβ, v)\\Svis added to Svat the end of\nthe super-iteration.\nProof.For readability, we write σandℓinstead of σSv,βandℓβ, respectively. Consider a node\nv∈Vas in the lemma statement. Recall that v∈Svby definition. Since σ(v) =∞by\nthe lemma statement, we can assume that deg(v)≥β+ 1(otherwise σ(v) = 0 = σV(v)by\ndefinition). Let P= (w0, w1, . . . , w k)be the longest path path where\nw0=vandwi∈F(σ, w i−1)∩D(ℓ, wi−1)\nBy the guarantees of Lemma 4.3, for every wiit holds that σ(wi)> σ(wi). Observe that\nby Observation 3.10, it holds that P⊆D(ℓ, v). Because Pcannot be any longer, Lemma 4.3\ncannot be applied to wk, so either wk̸∈Svordeg(wk)≤β. There are two possible cases.\n•wk∈Svanddeg(wk)≤β: This implies σ(wk) = 0, contradicting σ(wk)> ℓ(wk), since ℓ\ncannot be negative.\n•wk̸∈Sv: Because wi∈D(ℓ, wi−1), it holds that ℓ(wi−1)> ℓ(wi)by definition for every\ni∈[1, k]. Since ℓ(v)≤logxby the lemma statement, it must hold be k≤logx. In\nAlgorithm 1, v(=w0)forwards x/(β+ 1)coins to w1, who forwards x/(β+ 1)2coins\ntow2, and so on. Since k≤logx, node wkwill receive at least x/(β+ 1)logβ+1x= 1\ncoin(s) and will be added to Svat the end of the super-iteration. Because wk̸∈Sand\nwk∈P⊆D(ℓ, v), it holds that wk∈D(ℓ, v)\\Sv, completing the proof.\nThe following lemma was used in Lemma 4.2. Informally, it ensures that more than half of\ncoins forwarded by a node u∈Svare forwarded to nodes in D(ℓβ, u).\nLemma 4.3. Consider a graph G= (V, E), any β, x≥1and the natural β-partition ℓβ.\nConsider the (x, β, F )-coin dropping game for a fixed node v∈Gand the corresponding set\nSv⊆Vduring any super-iteration. Let u∈Svbe a node with deg(u)≥β+ 1. IfσSv,β(u)>\nℓβ(u), then there exists at least one node w∈F(σSv,β, u)∩D(ℓβ, u)for which σSv,β(w)> ℓβ(w).\nProof.Forreadability, wewrite σandℓinsteadof σSv,βandℓβ, respectively. Consider u∈Svas\nin the lemma statement. Since deg(u)≥β+1, it holds that |F(σ, u)|=β+1by Definition 4.1.\nSince ℓ(u)̸=∞, it holds that |N(u)\\D(ℓ, u)| ≤βby Lemma 3.11. Recall that ℓ(w)< ℓ(u)\nholds for every node w∈D(ℓ, u)\\ {u}by Definition 3.9. Hence, it holds that ℓ(w)<∞for\nevery node w∈D(ℓ, u)\\ {u}.\nLet us consider the case where σ(u) =∞. By Lemma 3.7 there are at least β+ 1nodes\nw′∈N(u)with σ(w′) =∞. Hence, it holds that σ(w) =∞for every node w∈F(σ, u). Since\n|N(u)\\D(ℓ, u)| ≤β,|F(σ, u)|=β+ 1, and F(σ, u)⊆N(u), there must be at least one node\nw∈F(σ, u)∩D(ℓ, u). Because w∈F(σ, u), it holds that σ(w) =∞. Because w∈D(ℓ, u)\\{u},\nit holds that ℓ(w)<∞, proving the claim.\nLet us consider the case where σ(u)̸=∞. By Lemma 3.7, it holds that σ(w)≥σ(u)−1for\nat least β+1nodes w∈N(u). Hence, it holds that σ(w)≥σ(u)−1for every node w∈F(σ, u).\nSince|N(u)\\D(ℓ, u)| ≤β,|F(σ, u)|=β+ 1, and F(σ, u)⊆N(u), there must be at least one\nnode w∈F(σ, u)∩D(ℓ, u). Because w∈F(σ, u), it holds that σ(w)≥σ(u)−1≥ℓ(u).\nBecause w∈D(ℓ, u), it holds that ℓ(u)> ℓ(w). We conclude that σ(w)> ℓ(w).\nThe following two lemmas bring everything together to prove that after x2super-iterations\nof the coin dropping game it holds that σSv,β(v) =ℓβ(v)and|Sv| ≤x3+ 1.\n15Lemma 4.4 (Correctness) .Consider a graph G= (V, E), any β, x≥1and the natural β-\npartition ℓβ. Consider the (x, β, F )-coin dropping game for a fixed node v∈Gand the cor-\nresponding set Sv⊆V. If|D(ℓβ, v)| ≤x2andℓβ(v)≤logβ+1x, then upon termination\nσSv,β(v) =ℓβ(v).\nProof.Recall that the (x, β, F )-coin dropping game comprises of x2super-iterations. Consider\nv∈Vas in the lemma statement. Towards a contradiction, assume that σSv,β(v)̸=ℓβ(v)after\nx2super-iterations. By Lemma 3.13, this implies that σSv,β(v)> ℓβ(v)holds for x2super-\niterations. Observe that Svcan only increase in size during Algorithm 1. Since σSv,β(v)> ℓβ(v)\nin every super-iteration, a node w∈D(ℓβ, v)\\Svis added to Svby Lemma 4.2 in every super-\niteration. Since |D(ℓβ, v)| ≤x2, after x2super-iterations it will hold that D(ℓβ, v)⊆Sv,\nimplying σSv,β(v) =ℓβ(v)by Lemma 3.14, a contradiction.\nRemark4.5 (Nodesmaynotfindtheirdependencygraphs) .In Lemma 4.4 we do not guarantee\nthat eventually D(ℓβ, v)⊆Sv, which would directly yield that σSv,β(v) =ℓβ(v)by Lemma 3.14.\nRather we prove only prove the sufficient statement σSv,β(v) =ℓβ(v)that can also happen even\nwhen D(ℓβ, v)̸⊆Sv.\nLemma4.6 (QueryComplexity) .Consider a graph G= (V, E), any β, x≥1and the (x, β, F )-\ncoin dropping game for a fixed node v∈Vand the corresponding set Sv⊆V. Upon termination,\nit holds that G[Sv]is connected and contains at most x6edges.\nProof.During the game, a node can be added to Svonly if it is adjacent to a node already in\nSv. Hence, G[Sv]is always connected.\nConsider one super-iteration of the (x, β, F )-coin dropping game. Observe that the forward-\ning set of any node u∈Svis of size ≤β+ 1, and only nodes with ≥β+ 1coins forward coins.\nHence, if a node w̸∈Svreceives coins, it receives ≥1coins. Since the total number of usable\ncoins in a super-iteration is x, the number of nodes that can be added to Svin a super-iteration\nis at most x. The claim follows since there are x2super-iterations and G[Sv]is connected.\n4.3 The LCA\nIn this section, we show the LCAfor computing a partial β-partition with a large fraction of\nnodes being assigned a layer ̸=∞. In particular, we prove the following lemma.\nLemma 4.7 (Sublinear LCAfor partial β-partitions) .For any constant ε >0andβ≥(2+ε)α\nthere is a deterministic LCAalgorithm that uses at most x6queries per node on a graph Gwith\narboricity αand assigns each node a layer from N∪ {∞}such that the following holds:\n▷There exists a subset S⊆Vcontaining at least a 1−21−logx/logβ/(2α)(β+1)fraction of\nvertices such that the layering of Sforms a β-partition of G[S]with logβ+1xlayers. ◁\nProof.Consider εandβas in the lemma statement. When questioned about vertex v, the\nLCAperforms the (x, β, F )-coin dropping game and outputs the Sv-induced β-partition layer\nofv. In order to perform the coin dropping game, the LCAneeds to be able to compute the\nSv-induced β-partition (Definition 3.6) locally. This is possible since it only requires knowing\nG[Sv]and the degrees of nodes in Sv. By Lemma 4.6, subgraph G[Sv]contains at most x6\nedges at any point during the game, bounding the query complexity of the LCA.\nBy Lemma 4.4, if |D(ℓβ, v)| ≤x2andℓβ(v)≤logβ+1x, then upon termination the LCA\ncan output ℓβ(v), which is the layer of vin the natural β-partition of graph G. By Lemma 4.9,\nthere are at least |V| ·(1−21−logx/logβ/(2α)(β+1))such nodes, completing the proof.\n16Remark 4.8. Instead of Lemma 4.7, we actually obtain a stronger statement. Our LCA, if\nqueried about a node u, outputs a partial β-partition ℓu:V→N∪ {∞}.\n▷There exists a subset S⊆Vcontaining at least a 1−21−logx/logβ/(2α)(β+1)fraction of\nvertices, and λ:V→N∪ {∞} :v7→minu∈Sℓu(v)is a partial β-partition of Gwith logβ+1x\nlayers ◁\nProof.For a node u∈V, instead of only outputting the Sv-induced β-partition layer ℓu(u)in\nthe proof of Lemma 4.7, the LCAcan simply output the whole partition ℓu. By Lemma 4.10,\nfunction λis a partial β-partition of Gwith the desired properties.\nThe following lemmas are purely combinatorial and do not depend on the coin dropping\ngame nor any model of computation.\nLemma 4.9. Consider a graph G= (V, E)with arboricity αand the natural β-partition ℓβfor\nany constant ε >0andβ≥(2 +ε)α. For any x >1, letX⊆Vsuch that |D(ℓβ, v)| ≤x2and\nℓβ(v)≤logβ+1xforv∈X. It holds that\n|V\\X| ≤ |V| ·21−logx/logβ/(2α)(β+1).\nProof.Divide the nodes of Vinto sets V>andV≤, such that ℓβ(w)>logβ+1xholds for nodes\nw∈V>andℓβ(w)≤logβ+1xholdsfornodes w∈V≤. Furthermore, letusdefine VX\n≤:=V≤∩X.\nObserve that V\\X=V>∪VX\n≤. Let us bound the terms separately.\nVX\n≤: By definition, it holds that |D(ℓβ, w)|> x2for nodes w∈VX\n≤. Towards a contradiction,\nassume that |VX\n≤|>|V|·x−1. Let every node w∈VX\n≤give every node in its dependency\ngraph one coin. Note that all nodes in the dependency graph of ware in V≤. The total\nnumber of coins is hence at least |V| ·x−1·x2=|V| ·x. On the other hand, any node\nuwith ℓβ(u)≤logβ+1xhas at most βneighbors in higher layers, and hence can be\ngiven a coin by at most βlogβ+1x< xnodes in V. Since the total number of coins is\nat least |V≤| ·x, and every node can only be charged at most xcoins, it must be that\n|V|>(|V|·x)/x=|V|,which is a contradiction and hence it holds that |VX\n≤| ≤ |V|·x−1.\nV>: By Lemma 3.4, we get that for each layer ithe number of nodes that will receive a\nlayer larger than ishrinks by a factor of 2α/β. Therefore, the number of nodes with\nℓβ(w)>logβ+1xis\n|V>| ≤ |V| ·(2α/β)logβ+1x=|V| ·(β/(2α))−logβ+1x=|V| ·x−1/logβ/(2α)(β+1).\nHence, it holds that |V\\X| ≤ |V|·x−1+|V|·x−1/logβ/(2α)(β+1)≤ |V|·21−logx/logβ/(2α)(β+1).\nLemma 4.10 (Closure under minimum for partial β-partitions) .Letℓ1, . . . , ℓ kbe partial β-\npartitions of a graph G= (V, E)for any β≥1. The following function λ:V→N∪ {∞}\nassigning a layer to every node v∈Vis also a partial β-partition:\nλ(v) = min\n1≤i≤kℓi(v).\nMoreover, if ℓi(v)̸=∞for any i∈[1, k]then λ(v)̸=∞.\n17Proof.In order to prove that λis a partial β-partition, we need to show that λ(w)≥λ(v)̸=∞\nfor at most βnodes w∈N(v). Let N≥(v)be the set of neighbors w∈N(v)with λ(w)≥λ(v).\nWithout loss of generality, we can assume that λ(v) =ℓ1(v). Let N<\n1(v)be the set of\nneighbors w∈N(v)with ℓ1(w)< ℓ1(v), and let N≥\n1(v)be the set of neighbors w∈N(v)with\nℓ1(w)≥ℓ1(v). Let us count how many nodes from sets N<\n1(v)andN≥\n1(v)could be in N≥(v).\n•Because λ(w) = min {ℓ1(w), . . . , ℓ k(w)}for every node w∈N<\n1(v), it holds that λ(w)≤\nℓ1(w). Hence, no nodes from N<\n1(v)can be in N≥(v).\n•Because ℓ1is a partial H-partition, it holds that |N≥\n1(v)| ≤βso at most βnodes from\nN≥\n1(v)can be in N≥(v).\nWe conclude that |N≥(v)| ≤β, proving that λis a partial β-partition. If ℓi(v)̸=∞for any\ni∈[1, k]then λ(v)̸=∞because function λtakes the minimum and ℓi(v)<∞.\n5β-partition in AMPC\nThis section is dedicated to constructing a β-partition from Definition 3.5 in the AMPCmodel.\nObserve that we mean to construct a β-partition with an empty infinity layer. In particular,\nwe prove Theorem 1.2, which assumes the knowledge of the arboricity. We later remove this\nassumption in Lemma 5.1.\nTheorem 1.2 (β-partitioning) .For any constant ε >0and any β≥(2 +ε)αthere is a deter-\nministic AMPCalgorithm that computes a β-partition of size O(logβ/(2α)n)inO(logβ/(2α)β)\nrounds on any n-node graph with arboricity α. The algorithm requires O(nδ)local space and\nO(n1+δ)total space for any constant δ >0. In particular,\n•when β=O(α), we get size O(logn)inO(logα)rounds and\n•when β=O(α1+ε), we get size O(logαn)inO(1)rounds.\nProof.Consider an graph G= (V, E)with nnodes and arboricity α < n(δ/c)2·log((2+ ε)/2)\n2(2+ε)(this\nrequirement is lifted later in the proof). The algorithm operates on data stores D0, D1, . . .\nsuch that the total number of data stores equals the number of rounds of the algorithm. Let\nus keep track of two types of key-value pairs in every data store Di. The first type being a\nvalue from N∪ {∞}returned for every node v∈V, which represents v’s layer in a (partial)\nβ-partition. The second type is defined using the first type as follows. Let Gi= (Vi, Ei)be\nthe subgraph of G= (V, E), induced by nodes vwith Di(v) =∞, which has arboricity at\nmost αby Fact 3.2. The edges of graph Giare stored in the data store Dias key-value pairs\n(v,degGi(v), j)− →(u,degGi(u)), where j∈[0,degGi(v)−1]and node uis the jth neighbor of\nvinGi, according to some arbitrary ordering. For D0, the values of the first type are set to\n∞, and hence G0=Gis just the input graph.\nLetP=nbe the number of machines4andS= Ω( nδ)be the local space of a machine,\nfor any δ >0. In order to compute Di+1from Di, we have the following protocol. Let Mvbe\nthe machine responsible for node v∈V. Using Di, or more precisely graph Gi, machine Mv\n4[BDE+19] provides the necessary details for employing this number of machines using parallel slackness .\n18performs the LCAof Remark 4.8. The query complexity of the LCAisx6, so when x=nδ/cfor\nsome c >6, the local and total space of our AMPCalgorithm are respected.\nNext, we need to argue that the partial β-partition of Gifrom Remark 4.8 is computable.\nAfter performing the LCAfrom Remark 4.8, every machine Mvhas the Sv-induced β-partition\nσSv,βstored in memory. Machine Mvcan write the key-value pair u− →σSv,β(u)for every\nnode u∈Svfor which σSv,β(u)≤logβ+1xto data store Di+1. The separate set of machines\nthat handles the DDS sorts through all written values and keeps the smallest σ(u)value for\nevery node u∈Vi. Observe that in order to make the (final outputted) β-partition consis-\ntent throughout the algorithm, we need to record σ(u)plus the maximum numbered layer of\nany node in Di, effectively appending the layers. Additionally, using Di, these machines can\ncompute degGi+1(u)for every node u∈Gi+1and port the edges of Gi+1toDi+1.\nLet us analyze the size of |Vi|. By applying Remark 4.8 for every rounds, it holds that\n|Vi| ≤n·2i·(1−logx/logβ/(2α)(β+1))\n= 2i/(logβ/(2α)(β+1))·(logβ/(2α)(β+1)−logx). (1)\nWhen x=nδ/c, it holds that\nlogβ/(2α)(β+ 1)−δ/c·logn <log 2β\nlogβ−log 2α−δ/c·logn\n∗\n≤log(2(2 + ε)α)\nlog((2 + ε)α)−log 2α−δ/c·logn\n= log(2+ε)/2(2(2 + ε)α))−δ/c·logn\n<log(2+ε)/2(α2(2+ε)))−δ/c·logn\n<(δ/c)2·log((2 + ε)/2)·logn\nlog((2 + ε)/2)−δ/c·logn\n=−(δ/c−(δ/c)2)·logn , (2)\nwhere in ∗we use β≥(2 +ε)α. Let d:= (δ/c−(δ/c)2). Observe that since δ/c < 1, it\nholds that d >0. Plugging Equation (2) into Equation (1) gives\n|Vi| ≤2i/(logβ/(2α)(β+1))·(−dlogn).\nHence, after k= logβ/(2α)(β+ 1)/d=O(logβ/(2α)β)rounds, it holds that\n|Vk| ≤n·2−logn= 0.\nBy Remark 4.8, the maximum layer in the partial β-partition computed for every data store\nislogβ+1x. Hence, after O(logβ/(2α)β)rounds, the size of the β-partition is =O(logβ/(2α)n).\nWhen α≥n(δ/c)2·log((2+ ε)/2)\n2(2+ε), we employ a different algorithm altogether. We simulate the H-\npartitionalgorithmof[BE10b]. Foreverydatastore Di, insteadofperformingthe (x, β, F )-coin\ndropping game, every node of degree ≤βsimply places itself in layer i. By Lemma 3.4, for any\nβ≥(2 +ε)α, this algorithm results in a β-partition of size O(logβ/(2α)n)inO(logβ/(2α)n) =\nO(logβ/(2α)β)time.\nTheorem 1.2 assumes the knowledge of the arboricity αof the graph. In Lemma 5.1 we\nremove this assumption.\n19Lemma 5.1. Theorem 1.2 can be applied without the knowledge of the arboricity of the graph.\nProof of Lemma 5.1. Letβbe defined using arboricity α. When αis unknown, we can still\nfind the β-partition with no additional cost in the runtime or memory by employing a guessing\nscheme for α.\nFirst, we perform a series of sequential executions of Theorem 1.2, such that instance i\ncomputes a βi-partition, using the ith guess for arboricity αi= 22i,i∈[0,log log n]. We\nterminate after the first guess αkwhich returns a βk-partition. Due to the double exponential\nguessing and the fact that a correct guess we yield partition, we know that ak< α2.\nNext, we perform a number of parallel executions of Theorem 1.2, such that instance icom-\nputes a βi-partition, where the ith guess for arboricity is αi=√ak·(1 +ε)i,i∈[0,log1+ε√ak]\nfor some ε >0. Observe that for some instance j, it must hold that αj<(1 +ε)α, producing a\nβi-partition of size O(logβi/2αin) =O(logβ/2αn). As our solution, we pick the smallest instance\nthat produces a partition.\nLetusanalyzethetotalruntimewhenemployingtheguessingschemeabove. The sequential\nexecutions take\nkX\ni=0O(logβi/2αiβi)≤∞X\ni=0\u00121\n2i\u0013\nO(logβ/2αα) +O(logβ/2αα2) =O(logβ/2αα) =O(logβ/2αβ)\nrounds in total. The parallel executions take O(logβ/2αα2) =O(logβ/2αβ)rounds. The\nparallel executions require an additional log1+ε√ak=O(logn)factor of space, which can be\nallowed by scaling the constant δ, corresponding to the local space O(nδ).\nIt is worth mentioning that since αjmay differ from αby a (1 + ε)factor, the resulting\nβjmay also differ from βby some ε′>1factor. This can be mended by simply using a\nslightly smaller βvalue to begin with, which is always possible due to the slack in requirement\nβ≥(2 +ϵ)α, ϵ > 0.\n6 Coloring Graphs in AMPC\nIn this section we will prove the three results in Theorem 1.3. We first restate the theorem as\na reference for the reader.\nTheorem 1.3 (Coloring Results) .For any constant ε >0on graphs with possibly non-constant\narboricity α, there are deterministic low-space AMPCalgorithms to compute:\n1. An α2+ε-coloring in O(1/ε)rounds.\n2. An O(α2)-coloring in O(logα)rounds.\n3. A ((2 + ε)α+ 1)-coloring in ˜O(α/ε)rounds.5\nWe will prove the first result in Section 6.1, the second result it Section 6.2, and the third\nresult in Section 6.3. In all three cases we proceed by simulating a LOCALalgorithm in the\nAMPCmodel by showing that in order to simulate xrounds of the LOCALalgorithm, each node\nneedstoknowatmost O(nδ)sizedinformationarounditsneighborhood. Thissimulationcanbe\ndoneinoneroundof AMPCbyassigningeachnodetoadedicatedmachinewhichwilladaptively\n5The eO(·)notation hides O(logα)factors.\n20gather the neighborhood information and locally simulate xrounds of the LOCALalgorithm.\nIn order for the LOCALalgorithm simulation to work, we will assume that α≤nδ/(1+ε)for\nsome constant ε >0. We handle case of α > nδ/(1+ε), separately in Section 6.4.\n6.1 An O(α2+ε)-coloring in O(1/ε)Rounds\nCompute a β-partition using Theorem 1.2 with β=α1+εfor a constant ε >0. This will require\nO(1/ε)AMPCrounds and the β-partition will have O(ε−1logαn)layers.\nTo compute this coloring, we simulate the Arb-Linial coloring algorithm of [BE10b] at each\nnode. The Arb-Linial algorithm takes as input a β-partition and produces an O(β2)-coloring\nwhich using O(log∗n)rounds in the LOCALmodel. Let us orient all edges from lower layer\nto higher layer in the β-partition and edges between nodes in the same layer are oriented\narbitrarily. An important property of the Arb-Linial algorithm is that it is one-sided , that is a\nnode vonly needs to know its out-neighborhood to compute its output.\nIn the algorithm we start with a poly (n)-coloring where each node picks its ID as its color.\nIn one round of algorithm, an m-coloring is reduced to an O(β2logm)-coloring. Therefore,\nafter one round, we have an O(β2logn)-coloring, and this eventually results in an O(β2logβ)-\ncoloring in O(log∗n)rounds. Then in one final round, the O(β2logβ)-coloring is reduced to\nanO(β2)-coloring.\nNote that if αε>logn, we don’t need to simulate all log∗nrounds of the Arb-Linial\nalgorithm. If we just simulate the first round of Arb-Linial, we get an O(β2logn)-coloring.\nSince β=α1+εandlogn < αε,O(β2logn) =O(α2+3ε), which we can say is O(α2+ε), by\nreplacing εbyε/3. We can simulate the first round of Arb-Linial in 1AMPCround by gathering\nthe at most βout-neighbors of each node. We can do this since we assumed α≤nδ/(1+ε), which\nimplies β≤nδ.\nNow we describe how to proceed when αε≤logn. In this case, we simulate all O(log∗n)\nrounds of the Arb-Linial algorithm in 1AMPCround by collecting the βO(log∗n)size out-\nneighborhood at each node v. In order for this to work, we need the entire O(log∗n)-hop\nout-neighborhood to fit into the memory of a single machine, that is, βO(log∗n)≤nδwhich is\ntrue since αε≤logn. Therefore, we get number of colors to be O(β2) =O(α2+2ε), which we\ncan say is O(α2+ε), by replacing εbyε/2.\n6.2 An O(α2)-coloring in O(logα)Rounds\nCompute a β-partition using Theorem 1.2 with β= (2 + ε)αfor a constant ε >0. This will\nrequire O(ε−1logα)AMPCrounds and the β-partition will have O(ε−1logn)layers.\nIfα >2log∗n, we simulate the Arb-Linial algorithm one round at a time. We can simulate\none round of Arb-Linial in 1AMPCround by gathering the at most βout-neighbors of each\nnode. We can do this since we assumed α≤nδ/(1+ε), which implies β≤nδ. The overall\nsimulation will require O(log∗n) =O(logα)AMPCrounds.\nIfα≤2log∗n, we simulate all O(log∗n)rounds of the Arb-Linial algorithm in 1AMPC\nround by collecting the βO(log∗n)size out-neighborhood at each node v. In order for this to\nwork, we need βO(log∗n)≤nδwhich is true since α≤2log∗n. In both cases, we get number of\ncolors to be O(β2) =O(α2).\n216.3 A ((2 + ε)α+ 1)-coloring in O(αlogα)rounds\nCompute a β-partition using Theorem 1.2 with β= (2 + ε)αfor a constant ε >0. This will\nrequire O(ε−1logα)AMPCrounds and the β-partition will have O(ε−1logn)layers.\nWe first construct an initial (improper) coloring of the nodes using (β+ 1)-colors. To do\nthis we use a (∆ + 1)-coloring algorithm by Kuhn and Wattenhofer [KW06] which runs in\nO(∆ log ∆ + log∗n)rounds in the LOCALmodel. This algorithm works in two stages, in the\nfirststagetheycomputean O(∆2)-coloringin O(log∗n)roundsusingLinial’scoloringalgorithm\n[Lin92], and in the second stage, they apply an iterative color reduction technique that reduces\ntheO(∆2)-coloring to a (∆ + 1)-coloring in O(∆ log ∆) rounds. We will use this algorithm to\nfind an initial coloring of the nodes. We first compute a (β+1)-coloring of the subgraph induced\nby nodes in a single layer by simulating the Kuhn-Wattenhofer algorithm independently in each\nlayer in parallel.\nIfα >log∗n, we can simulate the entire algorithm one round at a time in O(αlogα)AMPC\nrounds. Again, we can simulate a single round of the algorithm in 1AMPCround by gathering\nthe at most βout-neighbors of each node. We can do this since we assumed α≤nδ/(1+ε),\nwhich implies β≤nδ.\nIfα≤log∗n, we simulate the two stages separately. First, we compute the O(β2)-coloring\ninO(1)AMPCrounds by having each node collect a O(log∗n)-radius ball around it. In order\nfor the simulation to work, we need that αO(log∗n)≤nδwhich is true for α≤log∗n. Second,\nwe simulate the color reduction stage one round at a time, which requires O(βlogβ)AMPC\nrounds. The initial coloring is not yet a proper coloring of the entire graph, since we can have\nconflicts at edges between nodes in two different layers.\nWe fix these conflicts greedily by simulating the following centralized recoloring process: all\nnodes in the topmost layer fix their initial color to be their final color. Start with the nodes in\nthe second highest layer and go down to the lowest layer, one layer at a time. In each layer i,\nprocess nodes in decreasing order of their initial color and each node picks the highest available\ncolor such that there are no conflicts with neighbors that have finalized their color.\nNote that only nodes in the same or higher layer can finalize their color, and a node vhas\nat most βneighbors in the same or higher layer. So at most βcolors in their palette of size\nβ+ 1are blocked, so there will always be at least one color vcan pick so that there are no\nconflicts with neighbors that have finalized their color.\nIn order to simulate the recoloring process in AMPC, we make the observation that the final\ncolor of a node vonly depends on the final colors of neighbors in the same layer having higher\ninitial color, and the neighbors in higher layers. In other words, if we want to compute the\nfinal color of node v, we first need to compute the final colors of all these neighbors, so that\nwe can find the highest available color for v. We compute the final colors of the neighbors of v\nrecursively, i.e., by computing the final colors of their neighbors in the same layer having higher\ninitial color, and the neighbors in higher layers. Therefore, in order to compute the final color\nof a node v, we will need to compute the final color of all nodes usuch that the each edge in\nthe path from vtoueither goes from one initial color to a higher initial color or goes from one\nlayer to a higher layer.\nSince there are O(ε−1logn)layers and (β+ 1)initial colors, all such nodes uare at most\nO((β/ε) logn)hops away from v. Moreover, all such nodes uare in the same or higher layer\nasv, which means that to compute the final color of vwe need to collect a subgraph of size at\nmost βO((β/ε) logn)=nO((β/ε) logβ)). This is too large to fit in the memory of a single machine,\n22so we process the layers in batches of size (cδ/β) logβn, where cis a constant chosen such that a\nnode in this batch of layers only has to collect a ball of size βO(β·(cδ/β) logβn)=O(nδ), and there\nareO((β/εδ) logβ)batches. Each batch requires 1AMPCround so we use O((β/εδ) logβ) =\nO(αlogα)AMPCrounds for the overall simulation of the recoloring process.\n6.4 Coloring when αis Large\nWe begin with proving Theorem 1.5, which we will use as an important building block. For\nease of reading, we restate the theorem before providing the proof.\nTheorem 1.5 (Deterministic Coloring) .For any x > 1, there is a deterministic low-space\nMPCalgorithm to compute an 2x∆-coloring in any n-node graph with maximum degree ∆in\nO(logxn)rounds using O(n+m)polylogntotal space.\nProof.We will prove this theorem by providing a deterministic, non-component-stable, MPC\nalgorithm that produces the coloring in O(logxn)rounds using O(n+m)polylognglobal\nmemory. Since the AMPCmodel can simulate any MPCalgorithm without any loss in round\ncomplexity, the theorem follows.\nConsider the following randomized trial with pairwise independent random coins: Every\nnode picks a random color from 1, . . . , 2x∆. Let Yebe an indicator random variable for the\nevent that edge eis monochromatic, and let Y=P\neYebe the number of monochromatic\nedges produced by the randomized trial. Since we use pairwise independent random coins, the\nprobability that Ye= 1is1/(2x∆). By the handshaking lemma and linearity of expectation,\nE[Y]≤P\nvP\ne∋vE[Ye] =P\nvdeg(v)/(2x∆)≤ |V|/(2x).\nWe derandomize this process using the method of conditional expectations, which was first\nintroducedinthecontextofthe MPCmodelin[CPS20]. Sinceweusepairwiseindependence, we\ncan reduce the number of random coins required to O(logn)-bits by using a family of pairwise\nindependent hash functions. Then we fix these random bits in batches of size ⌊(δ/3) log2n⌋,\nand we denote the ithbatch by Bi. The invariant we will maintain when fixing the random\nbits in each batch is that E[Y|B1=b1, . . . , B i=bi]≤E[Y].\nWhen fixing batch Bi, let us assume inductively that the invariant is maintained for all\nprevious batches, i.e. we have fixed the random bits such that E[Y|B1=b1, . . . , B i−1=\nbi−1]≤E[Y]. By the law of total expectations, there exists an assignment Bi=bisuch that\nthe invariant is maintained. There are at most nδ/3assignments to Bi, and we will calculate\nthe value of the conditional expectation E[Y|B1=b1, . . . , B i=bi]for each assignment biin\nparallel and pick the minimum.\nLet us see first how to calculate the conditional expectation for a single assignment Bi=bi.\nThe key idea is that we decompose E[Y|B1=b1, . . . , B i=bi]intoP\neE[Ye|B1=b1, . . . , B i=\nbi], so the task is to compute a sum of values where each value can be computed locally by\nthe machine that stores edge e. We create a “broadcast tree” on the set of machines, which\nis a balanced nδ/2-ary tree of depth O(1/δ). Each machine Mlocally computes E[Ye|B1=\nb1, . . . , B i=bi]for all edges estored at Mand adds them up. The leaves in the broadcast tree\nsimply send this sum to their parent. Once a machine in the broadcast tree receives the values\nfrom all its children, it sends the sum of all these values and its locally computed sum to its\nparent. In each round, a machine sends at most one value and receives at most nδ/2values.\nInO(1/δ)rounds, the root of the broadcast tree will calculate the value of the conditional\nexpectation E[Y|B1=b1, . . . , B i=bi].\n23Since there are at most nδ/3assignments to Bi, it is easy to see that the machines can\ncalculate the conditional expectation value for all assignments biin parallel over the same\nbroadcast tree. In each round, a machine sends at most nδ/3values and receives at most n5δ/6\nvalues ( nδ/3values from each of its nδ/2children in the broadcast tree). Therefore the root\nof the tree can pick the assignment biwhich minimizes the conditional expectation, and this\nmaintains the invariant for our assignment to Bi. Since there are O(1/δ)batches, we are able\nto fix all the random bits in O(1/δ2)rounds.\nNow we have a deterministic algorithm that colors nodes with colors 1, . . . , 2x∆, such that\nthereareatmost n/(2x)monochromaticedges. Wefixthecolorofallnodesthatdonothavean\nincident monochromatic edge and all other nodes are uncolored. Let Ube the set of uncolored\nnodes. We have that |U| ≤n/x. If we repeat the same randomized process for the uncolored\nnodes, i.e., each node in Upicks a color in 1, . . . , 2x∆using pairwise independent random bits.\nWe still have that the expected number of monochromatic edges is E[Y]≤ |U|/(2x)≤n/(2x2).\nWe again derandomize this process, and we get a coloring of nodes with colors 1, . . . , 2x∆, such\nthat there are at most n/(2x2)monochromatic edges. The new set of uncolored nodes has size\nat most n/x2. Therefore, if we repeat the process itimes we get a partial coloring of nodes\nwith colors 1, . . . , 2x∆, such that the number of uncolored nodes is at most n/xi. We continue\nuntil no uncolored vertices remain. The number of repetitions is at most O(logxn), and we use\nat most 2x∆colors.\nWe prove the first and second parts of Theorem 1.3 by computing an α1+εcoloring in\nO(1/ε)rounds. Compute a β-partition using Theorem 1.2 with β=α1+εfor a constant ε >0.\nThis will require O(1/ε)AMPCrounds and the β-partition will have O(ε−1logαn) =O(1/ε)\nlayers.\nWe color each layer of the β-partition independently and in parallel with each layer getting\na fresh palette using Theorem 1.5 with x=αε. Note that to run the algorithm, we just need\nto know for each edge what the layer numbers of the two end points are: if both end points\nbelong to layer i, we include this edge in the algorithm for layer i, otherwise we ignore the edge,\nsince it will never be monochromatic. Since the subgraph induced by each layer has maximum\ndegree β, The total number of colors we use in a single layer is O(αεβ). Therefore, the total\nnumber of colors used over all layers is O(ε−1αεβ)≤α1+3ε. We can now adjust εto be ε/3\nto obtain an α1+εcoloring. The coloring using Theorem 1.5 requires O(1/ε)AMPCrounds, so\noverall we require O(1/ε)AMPCrounds.\nWe now prove the third part of Theorem 1.3. First, we compute a β-partition using Theo-\nrem 1.2 with β= (2 + ε)αfor a constant ε >0. This will require O(ε−1logα)AMPCrounds\nand the β-partition will have O(ε−1logn)layers. Then we construct an initial coloring where\nwe color the subgraphs induced by each layer independently and in parallel using Theorem 1.5\nwith with x= 2. Since the max degree of the subgraph induced by nodes in a single layer is\nat most β, this creates a coloring of each layer with colors in {1, . . . , 4β}. Note that to run the\nalgorithm, we just need to know for each edge what the layer numbers of the two end points\nare: if both end points belong to layer i, we include this edge in the algorithm for layer i,\notherwise we ignore the edge, even though it can be monochromatic.\nWe orient all edges from lower layer to higher layer, and for edges with end points in the\nsame layer, we orient them from lower initial color to higher initial color. Since we started with\naβ-partition, each node has at most βoutgoing neighbors. We use this orientation to produce\na proper (β+ 1)-coloring. We iterate through the layers sequentially from the topmost layer\n24to the bottom most layer. Nodes in layer iare processed in decreasing order of their initial\ncolor. Each node vin layer iwith initial color c(in parallel) picks as its final color the smallest\navailable color in {1, . . . , β + 1}. Due to the order in which we process the nodes, only the\noutgoing neighbors of vhave picked their final color before vpicks its final color. Therefore, at\nmost βcolors are blocked for v, so it can always find an available color in {1, . . . , β + 1}. Since\nβcan be much larger than the local space, we need to be a bit careful in finding the smallest\navailable color. In order to do this, we sort in O(1)rounds, the set of outgoing neighbors uof\nnodes vhaving layer iand initial color cwith the key (ID(v),col(u)), where col(u)is the final\ncolor picked by u(constant round deterministic sorting is a well known AMPC/MPCprimitive\n[CDP20, Goo99, GSZ11]). This means that all outgoing neighbors of vare stored in contiguous\nmachinesinincreasingorderoftheirfinalcolor. Fortheverticeswhosesetofoutgoingneighbors\nspans a single machine, the smallest available color can be locally computed by that machine.\nForavertex vwhosesetofoutgoingedgesspanmultiplemachines, wecreateavirtualbroadcast\ntree (which is an nδ/2-ary balanced tree) on the set of machines that store the outgoing edges\nofv, and aggregate the smallest available color at the root. Note that a machine can be part\nof at most two such broadcast trees. It is easy to see that all the aggregations on the virtual\nbroadcast trees can be implemented on a single broadcast tree on the set of all machines. This\nallows us to find in O(1)rounds the smallest available color in {1, . . . , β + 1}for each node\nvin layer iwith initial color c. Therefore, the time required for picking the final colors is\nO(βlogn) =O(αlogα)rounds, which dominates the overall runtime.\n25References\n[Bar15] L. Barenboim. Deterministic ( ∆+ 1)-coloring in sublinear (in ∆) time in static,\ndynamic and faulty networks. In Proc. 34th ACM Symposium on Principles of\nDistributed Computing (PODC) , pages 345–354, 2015.\n[BCG20] Suman K. Bera, Amit Chakrabarti, and Prantar Ghosh. Graph Coloring via\nDegeneracy in Streaming and Other Space-Conscious Models. In Artur Czumaj,\nAnuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on\nAutomata, Languages, and Programming (ICALP 2020) , volume 168 of Leibniz\nInternational Proceedings in Informatics (LIPIcs) , pages 11:1–11:21, Dagstuhl,\nGermany, 2020. 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Tamir, S. Vardi, and N. Xie. Fast Local Computation Algo-\nrithms. In the Proceedings of the Symposium on Innovations in Computer Science\n(ICS), pages 223–238, 2011.\n28"    },    {        "title": "2402.13773v1.Spatial_Domain_Wireless_Jamming_with_Reconfigurable_Intelligent_Surfaces.pdf",        "content": "Spatial-Domain Wireless Jamming with Reconfigurable Intelligent Surfaces\nPhilipp Mackensen∗1, Paul Staat∗2, Stefan Roth1, Aydin Sezgin1, Christof Paar2, and Veelasha Moonsamy1\n1Ruhr University Bochum, Bochum, Germany\n2Max Planck Institute for Security and Privacy, Bochum, Germany\nAbstract\nToday, we rely heavily on the constant availability of wire-\nless communication systems. As a result, wireless jamming\ncontinues to prevail as an imminent threat: Attackers can\ncreate deliberate radio interference to overshadow desired\nsignals, leading to denial of service. Although the broadcast\nnature of radio signal propagation makes such an attack pos-\nsible in the first place, it likewise poses a challenge for the\nattacker, preventing precise targeting of single devices. In\nparticular, the jamming signal will likely not only reach the\nvictim receiver but also other neighboring devices.\nIn this work, we introduce spatial control of wireless jam-\nming signals, granting a new degree of freedom to leverage for\njamming attacks. Our novel strategy employs an environment-\nadaptive reconfigurable intelligent surface ( RIS), exploiting\nmultipath signal propagation to spatially focus jamming sig-\nnals on particular victim devices. We investigate this effect\nthrough extensive experimentation and show that our ap-\nproach can disable the wireless communication of a victim\ndevice while leaving neighboring devices unaffected. In par-\nticular, we demonstrate complete denial-of-service of a Wi-Fi\ndevice while a second device located at a distance as close as\n5 mm remains unaffected, sustaining wireless communication\nat a data rate of 60 Mbit /s. We also show that the attacker can\nchange the attack target on-the-fly, dynamically selecting the\ndevice to be jammed.\n1 Introduction\nWireless communication systems are ubiquitous and seam-\nlessly provide connectivity to smart and interconnected de-\nvices that permanently surround us. Today, we frequently use\ninstant messaging, media streaming, health monitoring, and\nhome automation – all of which build on wireless systems and\ntheir constant availability. Crucially, loss of connectivity im-\npacts the functionality of wireless devices and can thus have\npotentially far-reaching consequences, e.g., when it comes to\n*These authors contributed equally to this work.remote monitoring security, smart grids, transportation, and\nhealthcare systems.\nWireless systems utilize a broadcast medium that is open\nto everyone, inherently exposing a large attack surface. One\nparticularly critical threat is wireless jamming , allowing mali-\ncious parties to rather easily perform denial of service. In the\nclassical jamming attack, the attacker transmits an interfering\nsignal that overshadows another desired signal, thus denying\na victim receiver to correctly decode the desired signal.\nThis basic attack principle has been previously studied by\na large body of research. For example, the attacker can use\nvarious jamming waveforms, such as noise or replayed victim\nsignals [20], and can vary the attack timing, jamming con-\nstantly [56] or only at certain times [45]. However, despite the\nmany existing attack strategies [29, 38, 39, 56], precise target-\ning of single victim devices remains a challenge: While the\nbroadcast nature of radio wave propagation enables wireless\njamming in the first place, it also causes the jamming signal\nto reach devices other than the targeted victim. In particular,\nwhen multiple devices are close to each other, there is no solu-\ntion to direct the jamming to only one device, without affect-\ning other devices in the vicinity. To the best of our knowledge,\ncurrently, the only way to selectively disrupt single devices\nis reactive jamming. Here, the attacker must analyze victim\ntraffic on-the-fly to decide in real-time whether to jam a par-\nticular transmission [3, 40, 45]. For this, however, the attacker\nmust be able to receive every transmission directed to the\nvictim. Furthermore, meaningful protocol-level information\nis required to identify the victim as the intended recipient.\nIn this work, we introduce and evaluate spatial control of\nwireless jamming signals as a new degree of freedom for\nadvanced jamming attacks, aiming to answer the following\nquestion:\nHow to jam a wireless device without affecting neighboring\ndevices at the same time?\nWe tackle this challenge by means of an reconfigurable intelli-\ngent surface ( RIS), enabling sophisticated wireless signal con-\ntrol using low-complexity hardware. In particular, as shownarXiv:2402.13773v1  [cs.CR]  21 Feb 2024by Kaina et al. [24], the RIScan be used to focus wireless\nsignals on certain locations while excluding others. We build\non this mechanism, leveraging the RISto adaptively optimize\nthe attacker’s wireless channel and precisely deliver jamming\nsignals towards a victim, effectively disrupting the target com-\nmunication and leaving other devices untouched. In particular,\nour approach is a true physical-layer targeting mechanism,\nbeing largely independent of protocol-level information.\nUpon publication, we will make the source code available\nas open source. The main contributions of this paper are as\nfollows:\n•We introduce the concept of environment-adaptive wire-\nless jamming to spatially control the strength of the at-\ntacker signal.\n•We present an attack realization based on RISby lever-\naging passive eavesdropping to determine an optimized\nRISconfiguration. Thereby, we are able to focus specific\nvictim devices in the spatial domain while excluding\ndevices that should remain unaffected by the attack.\n•We present a comprehensive experimental evaluation,\ncharacterizing the underlying mechanism to then demon-\nstrate real-world attacks. In particular, we accomplish\nprecise targeting of the jamming effect, disabling victim\nwireless communication while another device at 5 mm\ndistance remains unaffected.\n2 Technical Background\nIn this section, we introduce the necessary background on\nwireless jamming and RISs.\n2.1 Wireless Jamming\nWireless communication’s quality of service, e.g., reliability\nand data throughput, is determined by the quality of received\nsignals. An obvious example for this is low received signal\npower which yields a low signal-to-noise ratio ( SNR ) and\nconsequently, increases the probability of bit errors. A similar\neffect occurs when multiple radios transmit at the same time.\nThen, the receiver observes the superposition of all signals,\ni.e., the desired signal with additional interference, again de-\ngrading performance. A wireless jamming attacker exploits\nthis mechanism by deliberately sending interfering signals to\nremotely increase the error probability and, thus, eventually\nachieve denial of service. Here, a strong jamming signal over-\nshadows legitimate signals, preventing the victim receiver\nfrom correct operation [18, 29, 39].\nThere exist several types of jamming attacks, differing in\nterms of, e.g., the interfering signal type or the mechanism\nto trigger the attack [18]. The signal type can comprise of\nnoise, single or multi-tone carriers, or valid waveforms car-\nrying digital information. While noise jamming reduces the\nSNR , valid signals can enhance attack effectiveness, e.g., by\nmaking the receiver busy [19]. Furthermore, attackers mayconstantly transmit the jamming signal or rely on a trigger,\ne.g., when detecting a certain victim signal. This can be used\nto disable parts of the victim communication, e.g., physical\ncontrol channels [15].\n2.2 Intelligent Reflecting Surfaces\nRecently, the wireless communication research community\nhas investigated RISs, which are specifically engineered sur-\nfaces with reconfigurable radio reflection behavior to realize\nsmart radio environments [28]. In this concept, the propaga-\ntion medium is considered as an additional degree of freedom\nto optimize wireless transmissions apart from the transmitter\nand receiver sides. An RISconsists of many identical sub-\nwavelength unit cell reflector elements, spatially distributed on\na surface. Importantly, the reflection coefficient of each unit\ncell is tunable to manipulate the phase of a reflected signal.\nMany RISimplementations offer 1 bit control, i.e., to select\nfor each unit cell between two reflection coefficients [41].\nThis facilitates low-complexity control, e.g., by directly inter-\nfacing with digital logic signals from a microcontroller. Other\nthan the unit cell elements, no radio frequency ( RF) circuitry\nis required, making RISs rather low-cost devices that can be\nmanufactured as a standard printed circuit board (PCB).\nAs part of the radio propagation environment, the RIS can\nimprove wireless communication, e.g., by enhancing SNR\nat a receiver [5] or spatial diversity [14]. In view of these\ncapabilities, the technology is in discussion to complement\nfuture wireless communication systems such as 6G [11, 23,\n50].\nReflectarray Antenna. In this work, we leverage the RIS\nin a way that resembles an electronically tunable reflectarray\nantenna , where a specifically-engineered planar structure is\nexploited to achieve desired radiation characteristics such as\ndirectivity. This idea came up in 1963 by Berry et al. [6], who\nused a mechanically tunable impedance surface illuminated\nby a feed antenna to adjust the spatial radiation properties of\nthe reflected wave. In the early 2000s, the concept was real-\nized with electronically tunable planar microstrip reflectors\non aPCB [22]. Often, the reflectarray antenna is used to direct\na pencil beam into one particular direction [6, 22]. However,\nbeyond single-beam applications, reflectarray antennas can in-\nvoke arbitrary radiation patterns through so-called beam shap-\ning. For instance, satellites utilize reflectarrays with radiation\npatterns matched to the Earth’s curvature for uniform signal\ncoverage [25]. With the emergence of the RIStechnology,\nsuch reflectarray antennas can now easily be implemented by\nindividuals at low cost.\n3 Our Proposed RIS-based Jamming Attack\nIn this section, we introduce the system and attacker model,\nand outline the general idea behind the adversarial strategy.Further, we describe our experimental setup and present ex-\nperimental results, demonstrating RIS-based jamming.\n3.1 Preliminaries\nAssumptions. In this work, we consider an attacker Eve\nattempting to disrupt the wireless communication of the legit-\nimate parties Alice, Bob, and Carol. We assume that, under\nnormal circumstances, the victim devices actively communi-\ncate, i.e., receiving and transmitting wireless signals. For this,\nthey use time-division duplex ( TDD ) communication, such\nasWi-Fi . We also assume that Eve can eavesdrop wireless\nsignals from a victim party and estimate the signal strength.\nHowever, Eve cannot read encrypted payload information\ncontained in the signals. We further assume that the wireless\nchannels between the attacker and the victim parties are sub-\nject to multipath signal propagation due to typical radio wave\npropagation phenomena, e.g., reflection and scattering [44],\nas found in typical indoor environments.\nTo achieve their goal, we assume that Eve follows a two-\nstep procedure: Firstly, Eve picks a position to launch the\nattack. Here, we do not make restrictions but assume that\nthe aforementioned wireless channel conditions are fulfilled.\nEve can validate this by observing wireless signals from the\nvictim parties and can also adjust their antenna configuration\nto optimize the jamming channel. In this step, Eve remains\nstrictly passive. In the second step, Eve actively transmits a\njamming signal and is capable of adjusting the signal power.\nSignal Model. In the following, we describe the signal\nmodel that applies to the communication between two parties.\nFor simplicity, we use Alice and Bob as an example, but the\nsame model holds for the communication between Alice and\nCarol. We assume that the victim party Bob seeks to receive\na legitimate signal XAfrom Alice. The attacker Eve concur-\nrently transmits a jamming signal JE. Both signals are sent\nvia the respective wireless channels between the parties, i.e.,\nHABfrom Alice to Bob, and HEBfrom Eve to Bob. Thus, Bob\nobserves the superposition of both signals with additional\nwhite Gaussian noise N,i.e.,\nYB=HABXA+HEBJE+N (1)\nFrom this equation, we can define the signal-to-jamming-and-\nnoise ratio (SJNR) as\nSJNR =|HABXA|2\n|HEBJE|2+|N|2(2)\nGenerally, Eve wants to minimize the SJNR and thus strives\nfor maximization of |HEBJE|. Besides increasing JE, Eve can\nalso attempt to increase the channel gain |HEB|. For this, Eve\neavesdrops transmissions XBfrom Bob over the channel from\nBob to Eve, HBE, and receives the signal\nYE=HBEXB (3)Crucially, due to channel reciprocity, HBE=HEBholds and\nEve can passively optimize their reception of Bob’s signals to\nmaximize the effectiveness of a subsequent jamming attack,\nwhich is one of the key mechanisms of our attack described\nin the remainder of this paper.\nReconfigurable intelligent surface\nUnitcellConfigurationcm\nBobEve\ngmdhm\nFigure 1: Illustration of an RISand its components, showing\nits functionality from a theoretical perspective.\nWe assume that Eve employs an RISwhich they can use\nto optimize HEB. An illustration of an RIS and the sub-\ncomponents of HEBis shown in Figure 1. In particular, we\nassume that\nHEB=d+M\n∑\nm=1hmcmgm, (4)\nwhere drepresents the channel components that do not in-\nvolve an RIS. The RISis assumed to consist of Munit cells.\nhmis the channel from the attacker’s feed antenna to the m-th\nunit cell of the RIS,cmis the reflection coefficient of the ac-\ncording unit cell, and gmis the channel from the unit cell to\nBob. Thus, each of the unit cells constitutes an individual sub-\nchannel between the attacker and the victim. In the context\nof this work, we assume that the signal power of the non- RIS\ncomponent dis small compared to the sum of the RISsignal\ncomponents.\nRIS-based Jamming Attack Strategy. With an RIS, the\nattacker can digitally reconfigure their jamming channel HEB\nwith a large number of different configurations. Thus, by pro-\ngramming cm, Eve can quickly alter their radiation behavior\nwithout any mechanical effort, e.g., to direct jamming signals\ntowards or away from certain spatial directions. For this, Eve\nneeds to find appropriate RIS configurations cm.\nIn the context of this work, we concentrate on adaptive\nmethods to optimize cmin situ to match the wireless con-\nditions of a particular environment. For this, Eve utilizes a\ntwo-step attack approach, first eavesdropping signals to ob-\nserve the channel HBEand determine cmbefore switchingto active jamming through the reciprocal channel HEB, as il-\nlustrated in Figure 2. We refer to these steps as preparation\nandjamming , and further demonstrate their application in our\nproposed attack in Section 3.3.\nStep1:\nPreparation\nPassive\nRISoptimizationStep2:\nJamming\nActive\njammingwith\noptimizedRIS\nFigure 2: Two-step framework of the jamming attack using\nthe RIS.\n3.2 Experimental Setup\nExperiment Environment. We conduct our experiments\nin a standard office environment with approximate size,\n9.0×7.5 m. A floor plan is depicted in Figure 3, indicat-\ning the positions of the relevant devices. Furthermore, we\nleverage a commercially-available dual-axis Cartesian robot\nto reposition victim devices in an automated manner with high\nrepeatability at a control resolution of 0.01 mm . The robot\naxes, on which one victim device is mounted, cover an area of\n75×50 cm . The movement area of the robot is also indicated\nin the floor plan.\nYX\nMovement\narea\nAttacker (Eve)\nWi-Fi device (Carol)Wi-Fi device (Bob)\nWi-Fi router (Alice)\nFigure 3: Floor plan of the experiment environment (approx.\n9.0×7.5 m), showing the relevant parties and the robot move-\nment area.\nVictim Devices. To demonstrate and evaluate the feasibility\nof our attack, we utilize low-cost, off-the-shelf Wi-Fi devices.\nFor the access point Alice, we deploy an Asus RT-AC59Uwireless router on channel 112with 40 MHz bandwidth, cor-\nresponding to a channel center frequency of 5560 MHz for\nmost parts of the experiments. Note that some of the experi-\nments require packet rate measurements that depend on the\ncreation of wireless traffic in a controlled manner. Hence,\nwe optionally also employ a TP-Link N750 Wi-Fi router run-\nning OpenWrt with packet injection capabilities for Alice in\nthese parts of the experiments. For the victim devices Bob\nand Carol, we use two Raspberry Pi 4 Model B , connected to\nthe access point created by Alice.\nAttacker Setup. On the attacker side, we use the following\nhardware setup. For eavesdropping in the attack preparation\nstep, the attacker employs a Raspberry Pi 4 Model B with the\nnexmon CSI add-on [17] installed, allowing to obtain the\nreceived signal strength indicator ( RSSI ) value and MAC ad-\ndress for each received Wi-Fi packet. To connect an external\nantenna to the Raspberry Pi, we disconnected the onboard\nPCB antenna and added a coaxial connector.\nAs jamming signals used in the second attack step, we em-\nploy standard IEEE 802.11n/ac Wi-Fi signals, containing 10\nrandomized payload bytes. For convenient signal generation,\nwe use a Signal Hound VSG60 vector signal generator, allow-\ning to transmit signals with a 100 % duty cycle and precisely\ncontrolled power. However, we would like to emphasize that\nthis likewise can be done with ordinary Wi-Fi equipment [45].\nWe use a Mini-Circuits ZX60-V82+ amplifier to obtain a\njamming signal power of up to approx. 20 dBm.\nTo send and receive signals, we employ an inter-\nline PANEL 14 directional antenna which is directed to-\nwards the attacker’s RIS. A photo of the attacker setup is\nshown in Figure 20 in the Appendix. The RIS comprises\nthree combined modules based on the open-source design of\nHeinrichs et al. [21]. The RISmodules are manufactured on\nstandard FR4 PCB and are optimized to operate in the 5 GHz\nWi-Fi frequency range. Each module consists of 256binary-\nphase tunable unit cell elements, yielding a total of 768ele-\nments. The elements can be programmed via USB to select the\nphase of cmto either be 0◦(state ‘ 0’) or 180◦(state ‘ 1’). For\nfurther technical details, we refer to [21]. Finally, to connect\nthe respective devices, we use a Mini-Circuits USB-2SP4T-\n63H RF switch.\n3.3 Attack Implementation\nAs outlined in Section 3.1 and illustrated in Figure 2, the\nattack consists of two steps. First, in the preparation step,\nthe attacker eavesdrops victim signals to optimize the RIS\nconfiguration. Second, in the jamming step, the attacker uses\nthe optimized configuration to transmit a jamming signal\nto the victim device. Finally, we demonstrate this two-step\napproach yields a focused jamming attack effect.3.3.1 Step 1: Preparation\nHere, the attacker passively observes signals from the victim\nparty Bob and, based on this, optimizes the RIS. Recalling our\ndefinition of the SJNR in Equation 2, Eve seeks to maximize\nthe wireless channel HBEto enhance the effect of the wireless\njamming performed in the subsequent attack step. For this,\nEve searches an appropriate RIS configuration cm,i.e.,\nmax\ncm|HBE|. (5)\nAs our RIS is binary-tunable and comprises of 768 el-\nements, we face a vast configuration space of size 2768,\nprohibiting exhaustive search. Therefore, we take advan-\ntage of a greedy RISoptimization algorithm put forward by\nTewes et al. [52]. The algorithm stores a sorted list, ranking\nthe evaluated cost function, e.g., the wireless channel mea-\nsurement results, for a total of BRISconfigurations. Based on\nthis mapping, element-wise empirical probabilities for maxi-\nmizing or minimizing the cost function are calculated. With\neach algorithm step, a new RISconfiguration is generated\nto evaluate the cost function and subsequently, update the\nranking table.\n0 5000 10000 15000\nOptimizer step60\n55\n50\n45\nRSSI (dB)Bob\nFigure 4: RISoptimization progress for maximization of re-\nceived signal power received from a jamming target.\nWe use this procedure to maximize the RSSI value when\nEve eavesdrops the signals stemming from Bob. As the RSSI\nis an estimate for received signal power, it reflects |HBE|.\nSince Eve is passive, the RISoptimization takes place in an\nopportunistic manner, reconfiguring the RISand waiting for a\nsignal from Bob to evaluate the current RISconfiguration. To\nincrease robustness against measurement noise, we average\nover 10consecutive packets received from Bob to evaluate\noneRISconfiguration. We use B=130and run the algorithm\nfor15.000 s teps. The evolution of the RSSI value recorded\nby Eve during the optimization is shown in Figure 4. Here\nwe can see that the received signal power gradually increases\nas the optimization progresses. Initially, the average RSSI\nvalue is at −60 dB , indicated by the dashed line, and even-\ntually reaches −45 dB , amounting to an increase of 15 dB\n(corresponding to a linear factor of 1015/10≈30) in received\nsignal power. This result clearly shows that Eve succeeds topassively maximize the received victim signal power. The\nRISoptimization leads to a coherent superposition of the indi-\nvidual sub-channels in Equation 4. In the spatial domain, this\ncan be viewed as an optimization of the RISradiation pattern\nto match the propagation environment, creating sensitivities\nin angular directions of individual propagation paths between\nBob and Eve.\n3.3.2 Step 2: Jamming\nWith the RIS optimization outlined before, we achieved a\n15 dB signal gain from Bob to Eve. As discussed, due to chan-\nnel reciprocity, this optimization likewise affects the reverse\nsignal direction from Eve to Bob, equally achieving a gain of\n15 dB . Thus, Eve can leverage the optimized RISconfigura-\ntion for the active transmission of a jamming signal, which is\nthe second step of the overall attack effort.\n-20 -10 0 10 20\nJamming power (dBm)02040Received rate (packets / s)15 dBRIS optimized\nRIS random\nFigure 5: Packet receive rates using the optimized RISand\nrandom RIS configurations for several jamming powers.\nTo assess how the RISoptimization aids wireless jamming,\nwe perform the following experiment. Alice sends 50Wi-\nFi packets (each modulated with a modulation and coding\nscheme ( MCS ) value of 7) per second to Bob, who measures\nthe number of successfully received packets per second. At\nthe same time, the attacker Eve sends their jamming signal to\nBob, also consisting of Wi-Fi packets. We plot the received\nrates of Bob versus Eve’s signal power in Figure 5, for both\nrandomized and optimized RISconfigurations. Here, Eve suc-\nceeds to completely disable Bob’s reception of packets from\nAlice. However, as we can see, compared to the randomized\nRISconfigurations, Eve has gained the aforementioned 15 dB\ndue to the RIS optimization, substantially reducing the re-\nquired jamming power to disrupt Bob.\n4 RIS -based Jamming Attacks in the Spatial\nDomain\nIn this section, we introduce spatially selective jamming at-\ntacks which is the main goal of our work. We show that the\nRISprovides minute spatial focusing and exclusion capabili-\nties, which we exploit to demonstrate real-world attacks.-30 0 30\nx (mm)-30\n0\n30y (mm)\n65\n60\n55\n50\nRSSI (dB)Figure 6: Heatmap of RSSI values in a 6 ×6 cm area around\nthe initial victim device position after RIS-based RSSI maxi-\nmization.\n4.1 Targeting a Victim Device\nJamming one particular device without affecting others\naround, i.e., a locally concentrated attack effect, demands\nafocusing mechanism to deliver the jamming signal only\nto the victim device. Fortunately, according to the literature,\ntheRISis capable of creating such an effect. As shown by\nKaina et al. [24], the optimization of a wireless multipath\npropagation channel is valid for one single position and de-\ngrades for other positions. Thus, we expect that the passive\nRISoptimization outlined in Section 3.3 achieves focusing of\nthe victim. To verify this, we utilize our robotic positioning\nsetup. After optimizing the RISto maximize RSSI , we move\nthe victim device Bob in 1 mm steps by 3 cm in each direction\nand measure the RSSI values at each position. A heatmap of\nthe results is depicted in Figure 6, showing the spatial distribu-\ntion of RSSI values. In the center, where the RISoptimization\ntook place, we can clearly see that the RSSI values are great-\nest, while quickly dropping by more than 15 dB when moving\naway from the center. This result confirms that the RIS-based\nRSSI maximization yields a focal point around the victim\ndevice. It is worth noting that this is different from classical\nbeamforming, which rather affects an area and not a particular\nspot.\nTo verify that the focusing effect indeed occurs only for\nthe victim’s position, we additionally record larger RSSI\nheatmaps, shown in Figure 7. Here, we cover the full robot\nmovement area of 75 ×50 cm and indicate the position of\nthe victim device at (x,y) = ( 550,250)mm. Essentially, the\nheatmaps show the relative distribution of signal strength (in\ndB) between the attacker and the victim. First, in Figure 7a,\nwe plot the heatmap resulting from a randomly chosen RIS\nconfiguration (see left side), yielding a random distribution\nofRSSI values. In contrast, Figure 7b shows the heatmap\ncorresponding to the optimized RISconfiguration, where we\nobserve that a maximum only occurs at the victim position. In\nparticular, at positions at least 6 cm away from the focal point,\n0 200 400 600\nx (mm)200\n400y (mm)\n70\n60\n50\nRSSI (dB)(a) Random RIS configuration and corresponding RSSI heatmap.\n0 200 400 600\nx (mm)200\n400y (mm)\n70\n60\n50\nRSSI (dB)\n(b) Optimized RISconfiguration and corresponding RSSI heatmap.\nFigure 7: Spatial distributions of RSSI values for two RIS\nconfigurations. Bob is placed within the black circle.\nas indicated by the black circle, the RSSI values are reduced\nby at least 5 dB and on average by 13 dB . We also obtain\nsimilar results for other victim device positions as depicted in\nthe heatmaps for a total of nine victim device positions shown\nin Figure 21 in the Appendix.\nTheoretical Analysis. Theoretical limits for the focal width\nproduced by the RISare rooted in the correlation of multipath\nchannels at different locations, as shown by Clarke [12]. In\nparticular, in the context of this work, we consider the corre-\nlation of the channel from the attacker to the victim location,\ni.e.,HBE, and a channel towards another location. Their cor-\nrelation can be described in relation to their distance d, using\nthe Bessel function of the first kind, allowing to approximate\nthe half focal width as the first zero point as\nd0=2.4048c\n2πf(6)\nHere, cis the speed of light and fis the signal frequency\n(5560 MHz in our experiments). Thus, d0is approximately\n20.6mm, matching the radius of the bright spot in Figure 6.\nJamming Performance. After having established that the\nattack preparation step yields signal focusing, we now proceed\nto evaluate the jamming step in the spatial domain. For this,\nwe move the victim device in 1 cm steps within the full robot\nmovement area. For each of the 75·50=3750 positions,\nwe repeat the experiment outlined in Section 3.3.2, i.e., Bob\nmeasures the packets received from Alice per second while\nEve increases their jamming power. From this, we plot the\nreceive rate at the focal point and the average receive rate at\nother locations over the jamming signal power in Figure 8.Clearly, the attacker completely disrupts Bob’s reception at\nthe focal point, while the connectivity outside the focal point\nis only marginally affected, reduced to approx. 44packets per\nsecond. This result shows that Eve succeeds to confine the\njamming effect to the victim location.\n4567891011121314\nJamming power (dBm)02550Receive rate (packets / s)\nT arget position\nOther position\nFigure 8: Received packets from Alice per second under jam-\nming when the receiver is positioned at the focal point and at\nother positions.\nThe imperfect selectivity of the jamming, causing other\nlocations than the focal point to be slightly affected, can be\nexplained as follows: The jamming effect is determined by the\nSJNR , recall Equation 2 where the signal powers from Alice\nand Eve contribute to the numerator and denominator, respec-\ntively. Therefore, the spatial distributions of both signals play\na role. First, outside of the focal point, the jamming signal\nis significantly attenuated but not completely cancelled, e.g.,\ndue to channel correlation effects [12], such that devices still\nexperience a low-amplitude jamming signal. Hence, a weak\ndesired signal from Alice at a certain position corresponds to\nan increased susceptibility against the jamming signal. Thus,\nthe communication of devices outside the focal point might\nstill be affected by the jamming when the desired signal power\nfrom Alice is significantly lower than at the focal point.\n4.2 Excluding Neighboring Devices\nIn the previous section, we demonstrated that RIS-based fo-\ncusing enables precisely targeted wireless jamming, confining\nthe attack effect at one spatial location. However, as discussed,\nunder certain conditions, other locations can still be affected\nby the jamming. One attacker-controllable factor for this is\ninsufficient focal selectivity, i.e., reduction of jamming signal\npower outside the focal point. Therefore, we now explore a\nsolution to significantly improve selectivity.\nThe main idea of this solution is as follows: In Section 4.1,\nwe tested a total of 3750 positions within the robot movement\narea to assess the focal selectivity. However, arguably, this\ngoes far beyond typical wireless environments – characterized\nby the presence of devices at single locations. The attacker\ncould take advantage of this, focusing the jamming signal on\nthe victim device while making sure that known other devicesremain unaffected. For this, the attacker can approach the\npreparation step slightly different, simultaneously maximizing\nand minimizing signal power from respective devices.\nTo put this idea to test, we consider the additional party\nCarol, located next to Bob. In the preparation step, Eve now\ngathers RSSI values for transmissions from both parties, al-\nlowing to observe the wireless channels |HBE|and|HCE|from\n(and towards) both devices. Assuming that Eve seeks to jam\nBob while Carol should remain unaffected, we again use the\ngreedy optimization algorithm outlined in Section 3.3, yet\nwith an adjusted optimization goal, i.e.,\nmax\ncm\f\f\f\fHBE\nHCE\f\f\f\f. (7)\nWe plot the optimization progress in Figure 9, where we ob-\nserve an increase of RSSI from Bob by approx. 13 dB . Cru-\ncially, in addition to that, we observe a 20 dB reduction of\nRSSI from Carol. After optimization, in some cases we ob-\nserved that Eve no longer receives signals from Carol any-\nmore, because their signal power falls below Eve’s receiver\nsensitivity. As the smallest RSSI values observable by Eve\nare around −80 dB , we believe that the 33 dB (approx. fac-\ntor2000 ) signal difference in Figure 9 is a minimum value.\nFinally, we found that, despite the additional minimization\nconstraint, a focal point still forms around Bob.\n0 5000 10000 15000\nOptimizer step80\n70\n60\n50\nRSSI (dB)Bob\nCarol\nFigure 9: RISoptimization progress, simultaneous maximiza-\ntion and minimization of the respective received signal powers\nfrom the two devices.\nJamming At Varying Device Distances. The optimization\nresult presented in Figure 9 shows that the RIS can maxi-\nmize the channel towards Bob while minimizing the channel\ntowards Carol. Thus, signals transmitted by Eve will arrive\nwith significant attenuation at Carol, clearly being very useful\nwhen trying to avoid jamming of Carol. However, this result\nwas obtained for one particular placement of Bob and Carol.\nNow, we use our robotic positioning setup to vary their dis-\ntance in the range between 0 mm and600 mm . From this, we\nwere able to verify that the RISoptimization yields satisfac-\ntory results for each device placement. Furthermore, for each0 200 400 600\nDistance (mm)02040Receive rate (packets / s)\nBob\nCarolFigure 10: Packet receive rates of Bob and Carol for various\ndistances using jamming power of 19 dBm.\ndevice distance, we repeat the previously described rate mea-\nsurement experiment. In Figure 10, we report the respective\nreceive rates of Bob and Carol from Alice under jamming.\nHere, we clearly see that the attack affects Bob while Carol\nremains unaffected, regardless of their distance.\nInterestingly, we found that the RSSI optimization even\nworks for very small distances of Bob and Carol, which is\nreflected by the corresponding jamming results in Figure 10.\nWe believe that two effects are at play during the optimiza-\ntion: First, the modified optimization could slightly shift the\nfocal point around the jamming target such that the excluded\ndevice lies in a null of the Bessel function. This would enable\ndistinguishing devices that have a smaller distance than the\naforementioned d0≈20.6mm . Second, while many signal\npropagation components will reach both devices, we believe\nthat some will contribute primarily to just one device and\nthus facilitate device separation. This effect would be caused\nby different victim antenna radiation characteristics w.r.t. the\nattacker and also by environmental factors such as shadowing.\nJamming Close-By Wi-Fi Devices. As we have seen, Eve\ncan configure their RISto yield significant signal power dif-\nferences between Bob and Carol despite them being very\nclose to each other. After having exploited this capability\nwithin our systematic packet reception test, we now put the\nattacker strategy to test in a standard Wi-Fi network. We place\nBob and Carol very close to each other in two geometrical\nconfigurations. In the first (see Figure 11a), we place the de-\nvices directly above each other to minimize their antenna\ndistance while being in the same orientation. In the second\n(see Figure 11c), we place the devices facing each other to\nminimize their antenna distance regardless of the orientation,\nbeing approx. 5 mm . In the attack preparation step, we opti-\nmize the RSSI values from both devices. In particular, we per-\nform three consecutive RISoptimizations to ( i) focus Bob and\nexclude Carol, ( ii) focus Carol and exclude Bob, and ( iii) fo-\ncus Bob and Carol. Then, in the jamming step, we consecu-\ntively configure the RISwith the three optimization results.\nBob\nCarol(a) Parallel aligned antennas.\n0 10 20 30 40\nTime (seconds)0255075Throughput (MBit/s)Jamming\nOff(i) (ii) (iii)Jamming\nOff(i) (ii) (iii)\nBob\nCarol (b) Measured Wi-Fi datarates.\nCarol\nBob\n(c) Facing antennas.\n0 10 20 30 40\nTime (seconds)0255075Throughput (MBit/s)Jamming\nOff(i) (ii) (iii)Jamming\nOff(i) (ii) (iii)\nBob\nCarol (d) Measured Wi-Fi datarates.\nFigure 11: Real-world spatial jamming attack demonstration\nagainst Wi-Fi communication.\nBob and Carol are connected to the Wi-Fi access point and we\nemploy iperf3*to measure the throughput every 0.5 sover a\ntotal duration of 40 swhile Eve performs the jamming attack.\nThe resulting throughput measurements for both device place-\nments are shown as time series in Figure 11b and Figure 11d.\nThe first thing to note is that without jamming, both devices\ninitially have data rates of approx. 60 Mbit /sand86 Mbit /s.\nThen, as Eve starts to jam with the first RISconfiguration, the\ndata rate of Bob drops to zero while Carol remains completely\nunaffected as evident from the unaltered throughput. Eve then\nswitches to the next RISconfiguration, alternating the attack\ntarget. Now, Bob’s throughput is restored to the level with-\nout the attack while Carol’s throughput is zero. Finally, as\nEve switches to the third RISconfiguration, the throughput\nof both Bob and Carol drops to (nearly) zero. Please note\nthat – after activating the jamming – Eve achieves this result\nby merely reconfiguring the RISconfiguration, not changing\nother parameters such as jamming signal type, power, and\nattacker position. Importantly, this experiment highlights the\nattacker’s ability to precisely target victim devices in real-\nworld scenarios despite very close proximity. Finally, it also\ndemonstrates the ad hoc reconfiguration of the RIS, allowing\nto dynamically select the attack target.\n5 Further Evaluation\nIn this section, we present additional results to further char-\nacterize RIS-based spatial jamming. We investigate varying\nattacker positions, scrutinize the RISoptimization, and pro-\nvide comparison with jamming using conventional antennas.\n*https://iperf.fr/5.1 Attacker Position & Channel Conditions\nIn Section 4, we demonstrated spatially controlled jamming\nattacks. In these experiments, Eve was located at the posi-\ntion indicated in Figure 3 which we picked for convenience\nreasons. We would like to stress that other attacker positions\nshowed similar results regarding focusing. As outlined in Sec-\ntion 3, to accomplish focusing, we assume some multipath\nsignal propagation in the channel between Eve and a vic-\ntim party. Here, the intuition of our attack preparation step\nfrom Section 3.3.1 is that with increasing maximization of the\nvictim RSSI , the RISwill collect multipath signals. However,\nif there is a very strong line of sight ( LoS) signal component,\ntheRISoptimization might converge towards a configuration\nthat primarily emphasizes the LoS instead of other propaga-\ntion paths. To investigate this effect, we place the attacker at a\ndistance of 3 mnext to Bob and optimize the RISto maximize\nBob’s RSSI . We then record another RSSI heatmap, shown\ninFigure 12. Here, we see a beam-like pattern instead of a\nfocal point. This can also be seen from far-field radiation pat-\nterns calculated from the resulting RISconfiguration (see Sec-\ntion A.3 in the Appendix), showing that most signal power\nis directed to the angular direction of Bob. Fortunately, the\nobserved effect can easily be detected from looking at the\nresulting RISconfiguration, showing a distinctive concentric\npattern. Finally, we believe that this effect can be mitigated by\nadjusting the optimization formulation, e.g., giving priority\ntoRIS configurations that exhibit more complex radiation\npatterns.\n0 200 400 600\nx (mm)200\n400y (mm)\n50\n40\n30\nRSSI (dB)\nFigure 12: Spatial distributions of RSSI values when the RIS\noptimization converges towards maximization of the LoS\nsignal component.\n5.2 RIS Optimization\nIn the following, we provide further insight on the first step\nof the overall attack effort – finding an adequate RISconfigu-\nration.\nTargeting Multiple Devices In our previous experiments,\nwe considered cases where the attacker optimizes the RSSI\nfrom one (see Section 3.3.1) and two devices (see Section 4.2).\nBeyond that, the RISoptimization can be performed for even\nmore devices. To show this capability, we maximize the RSSIof one device while minimizing the average RSSI from three\nothers. The optimization progress for each device and the re-\nspective starting values (dashed lines) are shown in Figure 13,\nrealizing an RSSI difference of at least 10 dB for each. Please\nnote that the different final RSSI values of the minimization\nare due to their vastly different starting values (depending also\non transmit power) and the optimization formulation based on\ntheir average value. However, apart from the absolute values,\nwe here are rather interested in the signal changes, demon-\nstrating the ability to work with multiple devices and opening\nthe door for more advanced spatial jamming.\n0 5000 10000 15000\nOptimizer step80\n70\n60\n50\nRSSI (dB)Jamming target\nExcluded device 1\nExcluded device 2\nExcluded device 3\nFigure 13: Progress of RIS-based channel optimization with\nfour devices.\nOptimization Algorithm Parameter. Throughout our eval-\nuation in the previous sections we used the greedy optimiza-\ntion algorithm by Tewes et al. [52] with the parameter Bset to\n130(see Section 3.3.1 for further details). To demonstrate the\nimpact of Bon the convergence speed and quality of the re-\nsulting configuration, we repeated the maximization of Bob’s\nRSSI for a range of values and plot the respective optimizer\nprogress for 5.000 s teps in Figure 14. Here, we observe that\nthe convergence speed reduces as Bincreases. At the same\ntime, we see that very quick convergence yields poor results.\nFor more details, we refer to the work of Tewes et al. [52].\nOverall, we note that the optimization parameters can be fine-\ntuned to find a reasonable balance between convergence speed\nand result quality to shorten the attack preparation step. As\nevident from the plot, our initial choice B=130addresses\nthis trade-off already well.\nEffect Of Surface Size. One relevant factor that determines\nthe attacker’s signal gain is the physical size of the RIS[55],\nmotivating the following experiment. We simulate the effect\nof varying RISsizes by setting RISelements to zero, using\nonly a subset of elements around the RIScenter during the\noptimization. In this manner, we increase the size of the sub-\nset of active RISelements from 48to768and for each case\nmaximize the RSSI for Bob and minimize the RSSI for Carol,\nas described in Section 4.2. We plot the difference of Bob’s\nand Carol’s optimized RSSI values over the number of active\nRISelements in Figure 15. Increasing the number of active0 1000 2000 3000 4000 5000\nOptimizer step60\n55\n50\nRSSI (dB)B = 10\nB = 30\nB = 50\nB = 100\nB = 130\nB = 150Figure 14: Variation of optimization parameter Bfor\n5.000 optimization steps.\nelements from 48to200, the RSSI difference quickly rises\nto23 dB . From then on, the difference further increases until\nreaching 32 dB when all 768elements are active. Please note,\nhowever, that the maximum RSSI difference in our setup is\nlimited by path loss in the maximized channel from Bob and\nthe minimum observable signal from Carol due to Eve’s re-\nceiver sensitivity. Overall, the result confirms that a larger\nsurface size enhances the attacker’s signal control capabili-\nties. Importantly, however, it also shows that RISs with fewer\nelements can provide satisfactory performance.\n200 300 600\nNumber of RIS elements102030RSSI difference (dB)\nFigure 15: Signal difference between focused and excluded\ndevice for varying number of RIS elements.\nEffect of Environmental Variation. While we have shown\nthe effect of moving the victim party, another relevant factor\nfor the attack is environmental variation. That is, the attacker\nrelies on the assumption that the result of the attack prepa-\nration step, i.e., the RISoptimization, remains valid for the\nsubsequent active jamming. While this naturally is the case for\nstatic environments, we are interested to assess the robustness\nof the RISoptimized jamming channel against moderate envi-\nronmental variation. As our setup is deployed in an ordinary\nand actively used office environment, our experiments often\nwere subject to environmental variation, e.g., individuals walk-\ning around. Despite that, we did not notice an obvious perfor-\nmance degradation, indicating some degree of robustness. To\nput this claim to test more systematically, we optimize the RISwithout environmental variation to maximize and minimize\nRSSI values of Bob and Carol, respectively (see Figure 11a).\nThen, to assess whether the optimized channel conditions are\naffected, we record the RSSI values of both devices for one\nminute. During this interval, an individual walks through the\nexperiment environment, passing by the victim devices and\ntheRISat a distance of approx 1.5 m. We show the walking\npath and the RSSI values of both devices in Figure 16, indi-\ncating the times at which the individual enters and leaves the\nroom where the victim devices are located and passing by the\nRIS. The first thing to note is that the maximized RSSI values,\ni.e., from Bob, remain completely unaffected. Apart from that,\nthe exclusion of Carol appears to be reduced as the individual\ngets in proximity to the devices, evident by the temporary\nreception of packets from Carol. While a significant RSSI\ndifference of 15 dB remains, the excluded device would tem-\nporarily experience increased jamming signal power which\ncould affect its wireless communication.\n0 20 40 60\nSeconds80\n75\n70\n65\n60\n55\n50\nRSSI (dB)\nEnter roomLeave room\nBefore RISBob\nCarol\nFigure 16: Effect of human motion on attacker’s optimized\nchannel, left: walking path (dashed line), right: RSSI values\nduring walking.\n5.3 Comparison Against Conventional Anten-\nnas\nIn this work, we use the RIS as a means for optimiz-\ning Eve’s jamming channel towards victim devices. How-\never, we acknowledge that Eve could use a conventional\nantenna in the preparation step without relying on the\nRIS. Thus, we repeat some of our experiments using an\nEttus Research VERT2450 omnidirectional antenna and an\nelboxRF TetraAnt 5 19 20 RSLL directional antenna. Here,\ninstead of the automated RIS-based attack preparation, we\nattempt to optimize the victim RSSI manually, i.e., through\nchanging the antenna position and orientation. First of all,\nanalogous to Figure 5, we measure how many packets Bob\nreceives from Alice per second while increasing the jamming\npower. As evident from Figure 17, with both antennas, Eve\nsucceeds to eventually disrupt Bob, however, requiring signif-\nicantly higher jamming power. With the directional and the\nomnidrectional antennas, the required jamming power is 5 dB\nand 18 dB higher than with the optimized RIS.\nApart from the signal gain, we are primarily interested in-20 -10 0 10 20\nJamming power (dBm)02040Received rate (packets / s)RIS optimized\nRIS random\nDirectional\nOmnidirectionalFigure 17: Packet receive rate per second versus increasing\njamming power for RIS and traditional antennas.\nthe spatial separation of jamming effects. Therefore, analo-\ngous to Figure 7, we measure the distribution of victim RSSI\nand plot them as heatmaps for both antennas in Figure 18.\nHere, we see that neither of the antennas achieves a focal\npoint at the victim device location. Instead, the heatmap for\nthe omnidirectional antenna appears rather random while the\ndirectional antenna yields a pattern that is accentuated towards\nthe left side. Due to the lack of any focal selectivity, we expect\nthat spatial jamming is infeasible with these antennas. Thus,\nwe use the directional antenna for jamming while measuring\nBob’s receive rates from Alice at 3750 positions within the\nrobot movement area. As can be seen from Figure 19, unlike\nthe analogous RISresult shown in Figure 8, disrupting the\ncommunication at the target position also strongly affects the\nother positions.\n0 200 400 600\nx (mm)200\n400y (mm)\n70\n60\n50\nRSSI (dB)\n(a) Omnidirectional Antenna\n0 200 400 600\nx (mm)200\n400y (mm)\n70\n60\n50\nRSSI (dB)\n(b) Directional Antenna\nFigure 18: Spatial distributions of RSSI values for two RIS\nconfigurations. Bob is placed within the black circle.\n12131415161718192021\nJamming power (dBm)02550Receive rate (packets / s)\nT arget position\nOther positionFigure 19: Received packets per second over increasing jam-\nming power using the directional antenna.\n6 Discussion\nIn this section, we discuss our experimental setup and the\nresults, limitations of the attack, reason about potential coun-\ntermeasures, and provide directions for future research.\n6.1 Experimental Setup and Results\nVictim Devices. Having demonstrated spatially selective\nwireless jamming, our experiments underscore the attack’s\nreal-world character. In particular, we demonstrated the at-\ntack in an ordinary office environment, where we employed\noff-the-shelf wireless devices on the victim side. Here, we\neavesdropped actual Wi-Fi traffic to optimize the RIS. To\nsystematically evaluate the attack effect, we measured the re-\nceived packet rates with a fixed MCS setting on the transmitter\nside. Apart from that, our end-to-end attack demonstration on\nclose-by Wi-Fi devices (see Figure 11) also covers the case\nof rate-adaptive communication which seeks to balance data\nrate and link robustness.\nAttacker Setup. All components for our attacker implemen-\ntation are either available commercially or open-source, pro-\nmoting reproducibility of the results. Regarding signal recep-\ntion and transmission, the attack can be mounted merely using\ncommodity wireless devices, e.g., with advanced firmware\ncontrol of Wi-Fi chipsets. While we modified the attacker’s\nRaspberry Pi to allow connecting an external antenna, other\ndevices natively offer this ability. While RISs are commer-\ncially marketed [13, 16], there is no broad consumer-level\navailability. However, the RISwe used is based on the well-\ndocumented open-source design by Heinrichs et al. [21] who\nalso provide detailed manufacturing information. In particular,\nwe estimate that RISmodules can be acquired for approxi-\nmately €250 per unit.\nPreparation Time. During RIS optimization, each algo-\nrithm step yields a new RISconfiguration that needs to be\nevaluated by observing at least one packet from the victim. Asshown by our experiments, simple physical-layer quantities\nsuch as RSSI are sufficient for this, making the attack inde-\npendent of upper layer payload data and encryption. However,\nas we rely on opportunistic eavesdropping to gather this data,\nthe victim’s transmission rate is a potential bottleneck for the\nruntime of the RISoptimization. As noted by Zhu et al. [58],\ntypical Wi-Fi devices easily reach transmission rates above\n100 packets per second, which thus would correspond to\n100optimizer steps per second. To tackle low transmission\nrates, the attacker could provoke victim transmissions by send-\ning fake packets [1,2]. In our particular implementation, addi-\ntional delays occur as we average over multiple RSSI values\nand due to reconfiguring the RIS, yielding approx. 13op-\ntimizer steps per second. Thus, the optimization processes\ncomprising of 15.000 s teps takes approx. 20 minutes . How-\never, we would like to emphasize that we did not optimize\nthe implementation for speed. Furthermore, the optimization\nparameters can be adjusted, e.g., reducing the number of steps\nor adjusting B, (see Figure 14) which opens the door for\nsignificantly reduced optimization time, e.g., in the order of\nseconds.\n6.2 Countermeasures\nWe now discuss potential countermeasures, leveraging the\nattacker’s attack preparation step. Please note that the active\njamming attack itself cannot be prevented due to the broadcast\nnature of wireless radio wave propagation. Here, standard\njamming resistant modulation schemes, e.g., spread spectrum\ntechniques, can be used to diminish the attacker’s success.\nMAC Address Randomization. As a raise-the-bar counter-\nmeasure, dynamic randomization of MAC addresses would\nmake it more difficult for the attacker to identify wireless\nsignals coming from the victim party. However, the adversary\ncould utilize physical-layer properties such as channel state\ninformation (CSI) values to distinguish between devices.\nAvoiding Channel Reciprocity. One key mechanism un-\nderlying the attack is channel reciprocity, i.e., the symmetry of\nwireless channels from and towards a victim party, allowing\nthe attacker to passively adapt their jamming channel before\nlaunching the actual attack. Thus, to hamper the attack prepa-\nration, wireless devices may avoid reciprocal channels, e.g.,\nby using sufficiently separated frequencies or antennas for\nreception and transmission.\nRandomizing Signal Power. The attacker’s RISoptimiza-\ntion during attack preparation relies on RSSI values obtained\nfrom eavesdropped wireless signals. Thus, as an ad hoc coun-\ntermeasure, a victim party could randomize their transmit\npower to hamper the RISoptimization. However, this would\nimply a reduced wireless communication quality of service.However, the attacker could also use fine-grained CSI val-\nues for optimization which are mostly independent of the\ntransmission power.\nBeamforming. Transmit beamforming on the victim side\ncan be used to direct radio signals in the direction of the in-\ntended receiver. Ideally, assuming that the attacker is located\nin a different direction, this could prevent an attacker from\neavesdropping. However, under real-world conditions, signals\nalways will emanate in other directions as well, rendering this\napproach infeasible. In fact, this also can be seen from our ex-\nperimental results, where radio signals still reach unintended\nlocations.\n6.3 Limitations\nWith our attack, we have shown that the RISoffers significant\npotential for spatial control and precise targeting of wireless\njamming. However, as certain preconditions must be met for\nthis, the attack also is subject to limitations. First of all, like in\nevery other jamming attack, sufficient jamming signal power\nmust be delivered to the victim. While our scheme offers\nfine-grained spatial jamming control regarding neighboring\nand selected devices, this does not imply that the jamming\neffect completely vanishes at other locations. Given distance-\ndependent path loss, we expect that the jamming signals will\naffect devices in direct vicinity of the attacker. Further, when\nlegitimate signals are weak, i.e., low SNR conditions, even\nlow jamming signal power can lead to a disruption of wire-\nless communication. Therefore, to exploit focal selectivity,\njamming signal power must not be overly high. However, this\nrequirement is mitigated when additionally minimizing the\nsignal power of devices to remain unaffected.\nTo passively optimize the jamming channel towards the\nvictim device, the attacker relies on passive eavesdropping\nand a reciprocal wireless channel. Thus, the attack does not\nwork straightforward with wireless systems that rely on non-\nreciprocal wireless channels, e.g., when transmission and re-\nception employ different signal frequencies or antennas. In-\nstead, the attacker could carefully apply active low-power\njamming, observing slight throughput disruption instead of\nRSSI values as a proxy for the jamming channel quality to\noptimize the RIS accordingly.\nFinally, our attack relies on multipath signal propagation to\nspatially control the attacker’s wireless channel. Our results\nfrom Section 5.1 show that in wireless environments with\nweak multipath effects, i.e., highly directional LoS connec-\ntions, the RISoptimization reinforces the LoS, calling for an\nadjusted RIS optimization to prevent beamforming effects.\n6.4 Future Work\nAlthough our attacker implementation already yields satis-\nfactory results, we believe the hardware setup can be furtherimproved, e.g., realizing the attack with a single-chip wireless\ntransceiver connected to the RIS. Also, it would be interest-\ning to test other RISdesigns, possibly promoting hardware\nminiaturization. It would be interesting to combine spatially\ncontrolled jamming with time-varying jamming techniques\nsuch as reactive jamming to explore the potential of spatio-\ntemporal jamming. We employed a greedy algorithm from\nthe literature to optimize the RIS,i.e., to focus jamming sig-\nnals on one device while excluding others. We believe that\nthis process could be further improved, e.g., by fine-tuning\nthe optimization parameter or studying alternative algorithms,\nincluding machine learning-based approaches. Another way\nto accomplish wireless focusing would be near-field-focused\nantennas [35]. These leverage antenna geometry instead of\nmultipath propagation, yet constraining the focal point to be\nrather close to the antenna. Realizing spatial jamming in this\nmanner provides an interesting opportunity for future work.\n7 Related Work\nWireless jamming is the subject of much prior research. Here,\nwe provide an overview of relevant related work.\nDifferentiation from Previous Work. Previous research\nhas investigated the RISin the context of wireless jamming.\nLyu et al. [31] propose the RISto craft a cancellation sig-\nnal which destructively interferes at the targeted receiver, i.e.,\nreducing received signal power. Staat et al. [48] follow RIS-\nbased fast environmental variation that a victim receiver can-\nnot cope with. Wei et al. [54] describe a similar approach\nand furthermore use the RISto falsify information received\nby a victim party. These three works introduce RIS-based\nmechanisms to disrupt wireless communication passively ,i.e.,\nwithout sending an interfering signal. In contrast, in our work,\nweactively transmit a jamming signal and rather exploit the\nRIS to deliver and precisely target the attack.\nAdversarial RIS Applications. The RIS can be used to\nmanipulate radar readings, as shown by Vennam et al. [42]\nand Chen et al. [9]. Zhu et al. [57] have shown that the RIS\nallows attackers to evade wireless sensing-based physical\nintrusion detection. Other works consider the RISto facilitate\neavesdropping, e.g., Chen et al. [8], Chen and Ghasempour [7],\nand Shaikhanov et al. [46]. Finally, Li et al. [27] have shown\nRIS-based jamming of wireless key generation.\nJamming Attacks An early study on the threat of jamming\nin wireless networks is the work of Xu et al. [56]. They\nstudy several attack strategies, including constant random\nsignal jamming, deceptive jamming based on packets with\nvalid encoding, time-pulsed jamming, and reactive jamming\nto react to certain wireless events. These types are also cov-\nered in various survey and overview works on attacks anddefenses by, e.g., Mpitziopoulos et al. [32], Grover et al. [18],\nPoisel [39], and Lichtman et al. [29]. Some of these works\nrecognize the attacker’s antenna characteristics as a degree of\nfreedom or make distinctions between omnidirectional and\ndirectional antennas. However, a concept like our spatially\nselective jamming is not mentioned. Apart from the general\nthreat, the literature also presents threat analyses for recent\ncellular systems such as 4G [15] and 5G [4], e.g., discussing\nthe impact of disrupting certain control channels. As shown\nby Schulz et al. [45], smartphones can be turned into reac-\ntive jammers. Nashimoto and Nagatsuka [33] demonstrate\njamming of mmWave radar, preventing detection of radar\ntargets. Proano and Lazos [40] describe time-domain selec-\ntive wireless jamming based on real-time packet classification.\nFollowing the same goal, Aras et al. [3] describe a packet clas-\nsification method for LoRaWAN, also allowing time-domain\nseletive jamming.\nA different line of work addresses the detection of jam-\nming attacks [10, 30, 49], more recently also including\nmachine-learning based methods [37, 51]. Other works exam-\nine friendly jamming, directed towards potential adversaries.\nWhile such methods were studied to achieve confidential-\nity,e.g., by Wenboshen et al. [47] and Kim et al. [26], the\nsecurity of such approaches has ben questioned, e.g., by Tip-\npenhauer et al. [53] and Robyns et al. [43]. An interesting\nproposal by Ogen et al. [36] is to exploit jamming attacks as\na covert channel, where the loss of network connectivity can\nbe used to encode information.\n8 Conclusion\nIn this paper, we introduced a novel class of wireless jamming,\nallowing to spatially focus the attack effect. Our approach\nemploys an RISto select which wireless device should be\naffected by the attack and which not. For this, the attacker first\neavesdrops wireless signals from the victim to adapt a RIS. In\na second step, the attacker transmits interfering signals to dis-\nrupt the victim wireless communication. We present analytical\nresults and elaborate the focusing mechanism based on multi-\npath signal propagation. We present extensive experimental\nresults, to demonstrate the effectiveness of the attack under\nreal-world conditions, using commodity Wi-Fi devices and\nan open-source RIS. Our method opens the door for precisely\ntargeted jamming attacks with the ability to differentiate be-\ntween devices that are located only centimeters apart from\neach other. Overall, our work underscores the threat of wire-\nless jamming attacks and recognizes the adversarial potential\nof RISs to enhance the landscape of wireless attacks.\nAcknowledgements\nWe thank Simon Tewes, Markus Heinrichs, and Rainer Kro-\nnberger for providing the RIS prototypes. This work wasfunded by the Deutsche Forschungsgemeinschaft (DFG, Ger-\nman Research Foundation) under Germany’s Excellence Strat-\negy - EXC 2092 CASA - 390781972, RWTÜV-Stiftung\n(project number S0189/10037/2021), Ministry of Economic\nAffairs, Industry, Climate Action and Energy of the State\nof North Rhine-Westphalia, Germany under grant 005-2108-\n0021 (5G-Expo), and Federal Ministry of Education and Re-\nsearch (BMBF) of the Federal Republic of Germany under\nGrant 16KISK037 (6GEM).\nReferences\n[1]Ali Abedi and Omid Abari. WiFi says \"hi!\" back to\nstrangers! In Proceedings of the 19th ACM Workshop\non Hot Topics in Networks , pages 132–138, 2020.\n[2]Ali Abedi and Deepak Vasisht. Non-cooperative Wi-Fi\nlocalization & its privacy implications. In Proceedings\nof the 28th Annual International Conference On Mobile\nComputing And Networking , pages 570–582, 2022.\n[3]Emekcan Aras, Nicolas Small, Gowri Sankar Ramachan-\ndran, Stéphane Delbruel, Wouter Joosen, and Danny\nHughes. 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Internet Society,\n2020.A Appendix\nA.1 Experimental Setup\nHere, in Figure 20, we show a photo of the experimental setup,\ncomprising of the RISthat is illuminated by the attacker’s\nantenna.\nFigure 20: Attacker setup used for experimental evaluation.\nA.2 Heatmaps for Varying Victim Positions\nIn addition to the shown position of (x,y) = ( 550,250)mm\ninFigure 7 from Section 4.1 we performed additional ex-\nperiments at varying positions and measured the correspond-\ning RSSI heatmap as shown in Figure 21.\n0 500250\n500Focal Point: 550, 50\n0 500250\n500Focal Point: 350, 50\n0 500250\n500Focal Point: 150, 50\n0 500250\n500Focal Point: 550, 250\n0 500250\n500Focal Point: 350, 250\n0 500250\n500Focal Point: 150, 250\n0 500250\n500Focal Point: 550, 450\n0 500250\n500Focal Point: 350, 450\n0 500250\n500Focal Point: 150, 450\n70\n60\n50\n70\n60\n50\n70\n60\n50\n70\n60\n50\n70\n60\n50\n70\n60\n50\n70\n60\n50\n70\n60\n50\n70\n60\n50\nFigure 21: RSSI heatmaps measured for different device po-\nsitions during the respective RIS optimization.A.3 Theoretical Radiation Patterns for Focus-\ning and Beamforming\nTheRSSI heatmaps in Figure 7b and Figure 12 show the dif-\nference between RISconfigurations that yield focusing and a\nbeam-like effect. We show the corresponding RISconfigura-\ntions and their theoretical angular radiation patterns, which\ncan be calculated based the spatial distribution of unit-cell\nreflection coefficients and the known feed antenna position.\nFirst of all, in the beamforming case, the RISconfiguration\nexhibits concentric patterns which are in accordance with\ntheory [34]. Further, the calculated radiation pattern shows\nthat signals are mostly directed towards a single direction. In\ncontrast, in the focusing case, the RISconfiguration appears\nmore random and the radiation pattern shows that signals are\ndirected towards many directions.\n(a)RIS configuration and radiation pattern corresponding to the\nheatmap shown in Figure 7b.\n(b)RIS configuration and radiation pattern corresponding to the\nheatmap shown in Figure 12.\nFigure 22: RISconfigurations and calculated radiation pat-\nterns."    },    {        "title": "2402.13786v3.Degree_conditions_for_disjoint_path_covers_in_digraphs.pdf",        "content": "arXiv:2402.13786v3  [math.CO]  28 Feb 2024Degree conditions for disjoint path covers in\ndigraphs\nAnsong Ma1, Yuefang Sun2\n1School of Mathematics and Statistics, Ningbo University,\nZhejiang 315211, China, mas0710@163.com\n2Corresponding author. School of Mathematics and Statistics,\nNingbo University, Zhejiang 315211, China, sunyuefang@nbu.edu.c n\nAbstract\nIn this paper, we study degree conditions for three types of disjo int\ndirected path cover problems: many-to-many k-DDPC, one-to-many\nk-DDPC and one-to-one k-DDPC, which are intimately connected to\nother famous topics in graph theory, such as Hamiltonicity and k-\nlinkage, and have a strong background of applications.\nFirstly, wegettwosharpminimum semi-degreesufficientconditions\nfor the unpaired many-to-many k-DDPC problem and a sharp Ore-\ntype degree condition for the paired many-to-many 2-DDPC proble m.\nSecondly, we obtain a minimum semi-degreesufficient condition for the\none-to-many k-DDPC problem on a digraph with order n, and show\nthat the bound for the minimum semi-degree is sharp when n+kis\neven and is sharp up to an additive constant 1 otherwise. Finally, we\ngive a minimum semi-degree sufficient condition for the one-to-one k-\nDDPC problem on a digraph with order n, and show that the bound\nfor the minimum semi-degree is sharp when n+kis odd and is sharp\nup to an additive constant 1 otherwise.\nKeywords: Minimum semi-degree; Ore-type degree condition;\nDisjoint path covers; Many-to-many k-DDPC; One-to-many k-DDPC;\nOne-to-one k-DDPC\nAMS subject classification (2020) : 05C07, 05C20, 05C38,\n05C70.\n11. Introduction\n1.1 Motivation, terminology and notation\nFor terminology and notation not defined here, we refer to [1] . Note\nthat all digraphs considered in this paper have no parallel a rcs or loops, and\na path always means a directed path. A biorientation of a graph Gis a\ndigraph which is obtained from Gby replacing each edge with two arcs of\nopposite directions. A complete digraph← →Knis a biorientation of a complete\ngraphKn. Acomplete bipartite digraph← →Ka,bis a biorientation of a complete\nbipartite graph Ka,b. We use Ento denote an empty digraph with order n.\nFor a set S, we use|S|to denote the number of elements in S.\nLetD= (V(D),A(D)) be a digraph. An x-ypathinDis a path which\nis fromxtoyfor two vertices x,y∈V(D). For two vertices xandyon\na pathPofDsatisfying xprecedes y, letxPydenote the subpath of P\nfromxtoy. AHamiltonian path is a path containing all vertices of D. A\ndigraphDis said to be Hamiltonian-connected ifDhas anx-yHamiltonian\npath for every choice of distinct vertices x,y∈V(D). Lets∈V(D)\nand letT={t1,t2,...,tk}⊆V(D)\\{s}. LetS={s1,s2,...,sk}and\nT={t1,t2,...,tk}beany two disjoint subsetsof V(D). Adirectedk-linkage\nforSandTinDis a set of kpairwise vertex disjoint paths {P1,P2,...,P k}\nsuch that Piis ansi−tipath for each i∈{1,2,...,k}. An (s,T)-fanis a\ncollection of pairwise internally disjoint paths {P1,P2,...,P k}such that Pi\nis ans−tipath for each i∈{1,2,...,k}.\nLetS={s1,s2,...,sk}andT={t1,t2,...,tk}be any two disjoint\nsubsets of V(D). We call SandTasource set and asink set, respectively.\nA set of kpairwise disjoint paths {P1,P2,...,P k}ofDis amany-to-many\nk-disjoint directed path cover (many-to-many k-DDPCfor short) for Sand\nT, if/uniontextk\ni=1V(Pi) =V(D) and each Piis a path from an element of Sto an\nelement of T. A many-to-many k-DDPC is pairedifPiis ansi−tipath\nfor each i∈{1,2,...,k}, and it is unpaired if, for some permutation σon\n{1,2,...,k},Piis ansi−tσ(i)path for each i∈{1,2,...,k}. Note that\na paired many-to-many k-DDPC can be seen as a special unpaired many-\nto-many k-DDPC. Also, observe that a paired many-to-many k-DDPC for\nSandTis a directed k-linkage for SandTcovering all vertices of D(see\n[1, 4, 5, 6, 7, 8, 12, 15, 20, 22, 25] for more information on the problem of\nk-linkage).\nTherearetwo variantsof many-to-many k-DDPC: one-to-many k-DDPC\nand one-to-one k-DDPC. Let S={s}andT={t1,t2,...,tk}be any two dis-\njoint subsets of V(D). A set of kpaths{P1,P2,...,P k}ofDis aone-\n2to-many k-disjoint directed path cover (one-to-many k-DDPC for short)\nforSandT, if/uniontextk\ni=1V(Pi) =V(D), each Piis ans−tipath and\nV(Pi)∩V(Pj) ={s}fori/\\⌉}atio\\slash=j. Similarly, let S={s}andT={t}. A set of\nkpaths{P1,P2,...,P k}ofDis aone-to-one k-disjoint directed path cover\n(one-to-one k-DDPCfor short) for SandT, if/uniontextk\ni=1V(Pi) =V(D), eachPi\nis ans−tpath and V(Pi)∩V(Pj) ={s,t}fori/\\⌉}atio\\slash=j. By definition, a one-\nto-many k-DDPC for S={s}andT={t1,t2,...,tk}is an (s,T)-fan covering\nall vertices of D, and a one-to-one 1-DDPC for S={s}andT={t}is a\nHamiltonian path from stot.\nWe now introduce the definitions of paired/unpaired many-to -manyk-\ncoverable digraphs and one-to-many/one k-coverable digraphs.\nDefinition 1 (Paired/unpaired many-to-many k-coverable di-\ngraphs). LetDbe a digraph of order n≥2k, wherekis a positive integer.\nIf there is a paired (resp. unpaired) many-to-many k-DDPC in Dfor any\ndisjoint source set S={s1,s2,...,sk}and sink set T={t1,t2,...,tk}, thenD\nis paired (resp. unpaired) many-to-many k-coverable.\nDefinition 2 (One-to-many/one k-coverable digraphs). LetDbe a\ndigraph of order n≥k+1, wherekis a positive integer. If there is a one-\nto-many (resp. one-to-one) k-DDPC in Dfor any disjoint source set S={s}\nand sink set T={t1,t2,...,tk}(resp.T={t}), thenDis one-to-many (resp.\none-to-one) k-coverable.\n1.2 Main results\nRecall that disjoint path cover problems are intimately con nected to\nother famous topics in graph theory, such as Hamiltonicity, k-linkage. In\naddition, they have applications in several areas such as so ftware testing,\ndatabase design and code optimization [16]. Therefore, the se problems have\nattracted much attention from researchers [2, 3, 9, 10, 11, 1 3, 14, 17, 18,\n19, 23, 24, 26]. In this paper, we will focus on the following t hree types of\ndisjoint path cover problems in digraphs: many-to-many k-DDPC problem,\none-to-many k-DDPC problem and one-to-one k-DDPC problem.\n1.2.1 Many-to-many k-DDPC problem\nFor the unpaired many-to-many k-DDPC problem, we will get a sharp\nminimum semi-degree sufficient condition as follows.\n3Theorem 1. LetDbe a digraph of order n≥3k, wherekis a positive inte-\nger. Ifδ0(D)≥⌈(n+k)/2⌉, thenDis unpaired many-to-many k-coverable.\nMoreover, the bound for δ0(D)is sharp.\nIf the condition “ n≥3k” is replaced by “ n= 2k”, then the bound\n“⌈(n+k)/2⌉” forδ0(D) in Theorem 1 can be reduced to “ ⌈(n+k)/2⌉−1”,\nwhich is still sharp. We also prove that, when n= 2k, a complete bipartite\nregular digraph Dwith order nis unpaired many-to-many k-coverable.\nTheorem 2. The following assertions hold:\n(i)LetDbe a digraph of order n= 2k, wherekis a positive integer. If\nδ0(D)≥⌈(n+k)/2⌉−1, thenDis unpaired many-to-many k-coverable.\nMoreover, the bound for δ0(D)is sharp.\n(ii)The digraph← →Km,m(m≥2)is unpaired many-to-many m-coverable,\nbut it is not unpaired many-to-many k-coverable when 1≤k < m.\nFor the paired many-to-many k-DDPC problem, K¨ uhn, Osthus and\nYoung [9] deduced a sharp minimum semi-degree sufficient cond ition.\nTheorem 3. [9] LetDbe a digraph of order n≥Ck9, whereCis a suffi-\nciently large constant and k≥2. Ifδ0(D)≥⌈n/2⌉+k−1, thenDis paired\nmany-to-many k-coverable. Moreover, the bound for δ0(D)is sharp.\nIn this paper, we will obtain a sharp Ore-type degree conditi on for the\npaired many-to-many 2-DDPC problem.\nTheorem 4. LetDbe a digraph of order n. Ifd+\nD(x)+d−\nD(y)≥n+2for\neachxy/\\⌉}atio\\slash∈A(D), thenDis paired many-to-many 2-coverable. Moreover, the\nbound for d+\nD(x)+d−\nD(y)is sharp.\n1.2.2 One-to-many k-DDPC problem\nFor the one-to-many k-DDPC problem, Zhou [26] demonstrated the fol-\nlowing minimum semi-degree condition.\nTheorem 5. [26] Let Dbe a digraph of order n≥Ck9, where Cis a\nsufficiently large constant and k≥2. Ifδ0(D)≥⌈(n+k+1)/2⌉, thenDis\none-to-many k-coverable.\nMa, Sun and Zhang [14] continued to study the one-to-many k-DDPC\nproblem in semicomplete digraphs. In their result, they dec reased the lower\nbounds of the order and the minimum semi-degree condition in this class of\ndigraphs as follows.\n4Theorem 6. [14] LetDbe a semicomplete digraph of order n≥(9k)5, where\nk≥2. Ifδ0(D)≥⌈(n+k−1)/2⌉, thenDis one-to-many k-coverable.\nIn this paper, by Theorem 1, we will make improvements for The orem 5\nby replacing “ n≥Ck9” with “n≥3k” and replacing “ δ0(D)≥⌈(n+k+\n1)/2⌉” with “δ0(D)≥⌈(n+k)/2⌉”, which is sharp when n+kis even and\nis sharp up to an additive constant 1 otherwise.\nTheorem 7. LetDbe a digraph of order n≥3k, wherek≥2. Ifδ0(D)≥\n⌈(n+k)/2⌉, thenDis one-to-many k-coverable. Moreover, the bound for\nδ0(D)is sharp when n+kis even and is sharp up to an additive constant\n1 otherwise.\n1.2.3 One-to-one k-DDPC problem\nFor the one-to-one k-DDPC problem, Cao, Zhang and Zhou [2] proved\nthe following result.\nTheorem 8. [2] LetDbe a digraph of order n≥Ck9, whereCis a suf-\nficiently large constant and k≥2. Ifδ0(D)≥⌈(n+k+ 1)/2⌉, thenDis\none-to-one k-coverable.\nIn this paper, we will improve Theorem 8 by replacing “ n≥Ck9” with\n“n≥k+ 1” and replacing “ δ0(D)≥ ⌈(n+k+ 1)/2⌉” with “ δ0(D)≥\n⌈(n+k−1)/2⌉”, which is sharp when n+kis odd and is sharp up to an\nadditive constant 1 otherwise.\nTheorem 9. LetDbe a digraph of order n≥k+ 1, where k≥2. If\nδ0(D)≥⌈(n+k−1)/2⌉, thenDis one-to-one k-coverable. Moreover, the\nbound for δ0(D)is sharp when n+kis odd and is sharp up to an additive\nconstant 1 otherwise.\n2. Many-to-many k-DDPC problem\nTo prove Theorems 1 and 4, we need the following result.\nTheorem 10. [21] LetDbe a digraph of order n. Ifd+\nD(x)+d−\nD(y)≥n+1\nfor each xy/\\⌉}atio\\slash∈A(D), thenDis Hamiltonian connected.\nIn Theorem 1, we will get a sharp minimum semi-degree sufficien t con-\ndition for the unpaired many-to-many k-DDPC problem. The example for\nthe sharpness of the bound will be given in Proposition 1.\n5Theorem 1. LetDbe a digraph of order n≥3k, wherekis a positive inte-\nger. Ifδ0(D)≥⌈(n+k)/2⌉, thenDis unpaired many-to-many k-coverable.\nMoreover, the bound for δ0(D)is sharp.\nProof.We prove the result by induction on k. The base step of k= 1 holds\nby Theorem 10. Now we assume that k≥2 and prove the inductive step.\nLetS={s1,s2,...,sk}andT={t1,t2,...,tk}be disjoint source set and\nsink set in D, respectively. Let\nS1=S\\{sk},T1=T\\{tk},X=V(D)\\{sk,tk}.\nWe construct a digraph D′of order n′(=n−1) fromD[X] by adding one\nnew vertex rsuch that\nN+\nD′(r) =N+\nD(sk)∩X, N−\nD′(r) =N−\nD(tk)∩X.\nObserve that\nd−\nD′(r)≥d−\nD(tk)−1,d+\nD′(r)≥d+\nD(sk)−1,\nand for every z∈X,\nd−\nD′(z)≥d−\nD(z)−1,d+\nD′(z)≥d+\nD(z)−1.\nHence,\nδ0(D′)≥⌈(n+k)/2⌉−1 =⌈(n+k−2)/2⌉=⌈(n′+k−1)/2⌉.\nBy the induction hypothesis, D′has an unpaired many-to-many ( k−1)-\nDDPC, say P={P1,...,P k−1}, forS1andT1, whereeach Pi(1≤i≤k−1)\nis ansi−tσ(i)path for some permutation σon{1,2,...,k−1}. As\nn′=n−1≥3k−1≥2k+1,\nthere exists a path Pi, sayP1, which contains ras an inner vertex. Let r+\n(resp.r−) be the successor (resp. predecessor) of ronP1. Let\nP∗\n1:=s1P1r−tk,Pk:=skr+P1tσ(1).\nObserve that{P∗\n1,P2,...,P k}is an unpaired many-to-many k-DDPC for S\nandT, that is, Dis unpaired many-to-many k-coverable.\n6Proposition 1. For every k≥1and every n≥3k+ 1, there exists a\ndigraphDonnvertices with minimum semi-degree ⌈(n+k)/2⌉−1which is\nnot unpaired many-to-many k-coverable.\nProof.The proof is divided into the following two cases according t o the\nparity of n+k.\nCase 1. n+kis even.\nLetDbe a digraph of order n≥3ksuch that\nV(D) =A∪B,|A∩B|=k,D[A]∼=D[B]∼=← →K(n+k)/2.\nObserve that\nδ0(D) = (n+k)/2−1 =⌈(n+k)/2⌉−1.\nLet\nS={si: 1≤i≤k}⊆A\\B,T=A∩B={ti: 1≤i≤k}.\nIt can be checked that for any permutation σon{1,2,...,k}, any set of\ndisjoint paths{Pi: 1≤i≤k}does not cover vertices of B\\A, wherePiis\nansi−tσ(i)path. Hence, there is no unpaired many-to-many k-DDPC for\nSandTinD, that is, Dis not unpaired many-to-many k-coverable.\nCase 2. n+kis odd.\nLetD1∼=← →K(n+k−1)/2andD2∼=E(n−k+1)/2. LetDbe a new digraph of\nordern≥3k+1 such that V(D) =V(D1)∪V(D2) and\nA(D) =A(D1)∪A(D2)∪{xy,yx:x∈V(D1), y∈V(D2)}.\nObserve that\nδ0(D) = (n+k−1)/2 =⌈(n+k)/2⌉−1\nand\n|V(D1)|= (n+k−1)/2≥(3k+1+k−1)/2≥2k.\nLetS={s1,s2,...,sk}andT={t1,t2,...,tk}be two disjoint subsets of\nV(D1). It is not hard to check that for any permutation σon{1,2,...,k},\na set of disjoint paths {Pi: 1≤i≤k}covers at most\n(n+k−1)/2−k= (n−k−1)/2(<|V(D2)|)\nvertices of V(D2), where Piis ansi−tσ(i)path. Hence, there is no unpaired\nmany-to-many k-DDPC for SandTinD, that is, Dis not unpaired many-\nto-many k-coverable.\n7If the condition “ n≥3k” is replaced by “ n= 2k”, then the bound\n“⌈(n+k)/2⌉” forδ0(D) in Theorem 1 can be reduced to “ ⌈(n+k)/2⌉−1”,\nwhichisstill sharp. Theexampleforthesharpnessofthebou ndwillbegiven\nin Proposition 2. We also prove that, when n= 2k, a complete bipartite\nregular digraph Dwith order nis unpaired many-to-many k-coverable.\nTheorem 2. The following assertions hold:\n(i)LetDbe a digraph of order n= 2k, wherekis a positive integer. If\nδ0(D)≥⌈(n+k)/2⌉−1, thenDis unpaired many-to-many k-coverable.\nMoreover, the bound for δ0(D)is sharp.\n(ii)The digraph← →Km,m(m≥2)is unpaired many-to-many m-coverable,\nbut it is not unpaired many-to-many k-coverable when 1≤k < m.\nProof. Part (i) . When n= 2k,δ0(D)≥⌈(n+k)/2⌉−1 =⌈3k/2⌉−1.\nThe base case that k= 1 is clear. Now we assume that k≥2. LetS=\n{s1,s2,...,sk}andT={t1,t2,...,tk}be any two disjoint subsets of V(D).\nLetD1=D[S] andD2=D[T]. By deleting A(D1),A(D2) and the arcs\nfromD2toD1, we get a spanning subdigraph, say D3, ofD. Observe that\nD3is a bipartite digraph which only contains the arcs from StoTofD.\nLetGbe the underlying graph of D3.\nWe claim that|NG(Y)|≥|Y|for allY⊆S, whereNG(Y) denotes the\nset of neighbours of vertices of YinG. Supposethat there is a subset Z⊆S\nsuch that|NG(Z)|<|Z|. Note that for any si∈S,\ndG(si) =d+\nT(si)≥⌈3k/2⌉−1−(k−1) =⌈k/2⌉,\nand thus|Z|>|NG(Z)|≥⌈k/2⌉, which means that |Z|≥⌈k/2⌉+ 1. For\nanytj∈T\\NG(Z), we have\ndG(tj) =d−\nS(tj)≤k−|Z|≤k−⌈k/2⌉−1 =⌊k/2⌋−1\nby definitions of ZandNG(Z). Thus, we have\nd−\nD(tj) =d−\nS(tj)+d−\nT(tj)≤⌊k/2⌋−1+k−1≤⌊3k/2⌋−2,\nwhich contradicts with the fact that δ0(D)≥⌈3k/2⌉−1. Thus,|NG(Y)|≥\n|Y|for allY⊆S, this means that Gcontains a perfect matching M, from\nwhich we can obtain a perfect matching M′ofD3, which is also an unpaired\nmany-to-many k-DDPC from StoTinD. Hence, Dis unpaired many-to-\nmanyk-coverable.\n8Part (ii) . LetX,Ybe the bipartition sets of V(← →Km,m). We first consider\nthe case that k=m. LetSandTbe a pair of disjoint source set and sink\nset, respectively, such that |S|=|T|=m. Let\nSX=S∩X, TX=T∩X, SY=S∩Y, TY=T∩Y.\nObserve that|SX|=|TY|and|TX|=|SY|. The subdigraph induced by\nSX∪TYisa complete bipartiteregular digraphwhich hasaperfectm atching\nM1such that each arc e∈M1is fromSXtoTY. Similarly, the subdigraph\ninduced by TX∪SYis a complete bipartite regular digraph which has a\nperfect matching M2such that each arc e∈M2is fromSYtoTX. Clearly,\nM1∪M2constitutes an unpaired many-to-many m-DDPC for SandT.\nHence,Dis unpaired many-to-many m-coverable.\nWe next consider the case that 1 ≤k < m. Suppose that← →Km,mis\nunpaired many-to-many k-coverable, that is, there exists an unpaired many-\nto-many k-DDPC, say{Pi: 1≤i≤k}, for any disjoint source set S=\n{s1,s2,...,sk}and sink set T={t1,t2,...,tk}. We consider the case that\n(T∪{sk})⊆Yand (S\\{sk})⊆X, and will get a contradiction in this\ncase. Without loss of generality, we assume that Pkstarts at sk. By the\nfact that ( T∪{sk})⊆Y, we have|V(Pk)∩X|+1 =|V(Pk)∩Y|. For each\n1≤i≤k−1, we have|V(Pi)∩X|=|V(Pi)∩Y|. By the definition of an\nunpaired many-to-many k-DDPC, we have\n|X|+1 =k/summationdisplay\ni=1|V(Pi)∩X|+1 =k/summationdisplay\ni=1|V(Pi)∩Y|=|Y|,\nwhich contradicts with the fact that |X|=|Y|. Therefore,← →Km,m(m≥2)\nis not unpaired many-to-many k-coverable when 1 ≤k < m.\nProposition 2. For every k≥2and every n= 2k, there exists a digraph\nDonnvertices with minimum semi-degree ⌈(n+k)/2⌉−2which is not\nunpaired many-to-many k-coverable.\nProof.The proof is divided into the following two cases according t o the\nparity of k.\nCase 1. kis odd.\nLetDbe a digraph of order n= 2ksuch that\nV(D) =A∪B,|A∩B|=k−1,D[A]∼=D[B]∼=← →K(n+k−1)/2.\n9Observe that\nδ0(D) =⌈(n+k)/2⌉−2 = (3k−1)/2−1\nand\n|A\\B|=|B\\A|= (3k−1)/2−(k−1) = (k+1)/2.\nLet\nA\\B={s1,s2,...,s(k+1)/2}, B\\A={t1,t2,...,t(k+1)/2}.\nAs now k≥3, we have ( k−1)/2≥1. Choose ( k−1)/2 vertices from\nA∩Band denote them as s(k+3)/2,s(k+5)/2,...,sk. The remaining ( k−1)/2\nvertices in A∩Bare denoted as t(k+3)/2,t(k+5)/2,...,tk. Let\nS= (A\\B)∪{s(k+3)/2,s(k+5)/2,...,sk}\nand\nT= (B\\A)∪{t(k+3)/2,t(k+5)/2,...,tk}.\nSuppose that there is an unpaired many-to-many k-DDPC, say P, for\nSandTinD. Observe that Puses at most ( k−1)/2 vertex-disjoint arcs\nfromA\\Bto{t(k+3)/2,t(k+5)/2,...,tk}and at most ( k−1)/2 vertex-disjoint\narcs from{s(k+3)/2,s(k+5)/2,...,sk}toB\\A, that is, it uses at most k−1\nvertex-disjoint arcs from StoT, a contradiction. Hence, Dis not unpaired\nmany-to-many k-coverable.\nCase 2. kis even.\nLetDbe a digraph of order n= 2ksuch that\nV(D) =A∪B,|A∩B|=k−2,D[A]∼=D[B]∼=← →K(n+k)/2−1.\nObserve that\nδ0(D) =⌈(n+k)/2⌉−2 = (3k)/2−2\nand\n|A\\B|=|B\\A|= (3k)/2−1−(k−2) =k/2+1.\nLet\nA\\B={s1,s2,...,sk/2+1}, B\\A={t1,t2,...,tk/2+1}.\nChoosek/2−1vertices from A∩Banddenote themas sk/2+2,sk/2+3,...,sk,\nthe remaining k/2−1 vertices in A∩Bare denoted as tk/2+2,tk/2+3,...,tk.\nLet\nS= (A\\B)∪{sk/2+2,sk/2+3,...,sk}, T= (B\\A)∪{tk/2+2,tk/2+3,...,tk}.\n10Suppose that there is an unpaired many-to-many k-DDPC, say P, forS\nandTinD. Observe that Puses at most k/2−1 vertex-disjoint arcs from\nA\\Bto{tk/2+2,tk/2+3,...,tk}and at most k/2−1 vertex-disjoint arcs from\n{sk/2+2,sk/2+3,...,sk}toB\\A, that is, it uses at most k−2 vertex-disjoint\narcs from StoT, a contradiction. Hence, Dis not unpaired many-to-many\nk-coverable.\nNow we turn our attention to the paired many-to-many k-DDPC prob-\nlem. In Theorem 4, we will obtain a sharp Ore-type degree cond ition for the\npaired many-to-many 2-DDPC problem. The example for the sha rpness of\nthe bound will be given in Proposition 3.\nTheorem 4. LetDbe a digraph of order n. Ifd+\nD(x)+d−\nD(y)≥n+2for\neachxy/\\⌉}atio\\slash∈A(D), thenDis paired many-to-many 2-coverable. Moreover, the\nbound for d+\nD(x)+d−\nD(y)is sharp.\nProof.LetS={s1,s2}andT={t1,t2}be any disjoint source set and\nsink set of V(D), respectively. Let X=V(D)\\{s2,t1}. We now construct\na digraph D1of order n1(=n−1) by adding a new vertex wsuch that\nN+\nD1(w) =N+\nD(s2)∩X, N−\nD1(w) =N−\nD(t1)∩X.\nFor each xy /∈A(D1), we have\nd+\nD1(x)+d−\nD1(y)≥d+\nD(x)−1+d−\nD(y)−1≥n+2−2 =n=n1+1.\nFor each x∈X\\N−\nD1(w), we have\nd+\nD1(x)+d−\nD1(w)≥d+\nD(x)−1+d−\nD(t1)−1≥n+2−2 =n=n1+1.\nSimilarly, for each y∈X\\N+\nD1(w), we have\nd+\nD1(w)+d−\nD1(y)≥d+\nD(s1)−1+d−\nD(y)−1≥n+2−2 =n=n1+1.\nTherefore, we conclude that\nd+\nD1(x)+d−\nD1(y)≥n1+1 =|V(D1)|+1\nfor each xy/\\⌉}atio\\slash∈A(D1).\nBy Theorem 10, D1is Hamiltonian connected, and so there is a Hamil-\ntonian path, say P′, froms1tot2inD1. Letw+(resp.w−) denote the\nsuccessor (resp. predecessor) of wonP′. Let\nP1:=s1P′w−t1,P2:=s2w+P′t2.\nObserve that{P1,P2}is a paired many-to-many 2-DDPC for SandT, that\nis,Dis paired many-to-many 2-coverable.\n11Proposition 3. There exists a digraph Donn≥9vertices with d+\nD(x) +\nd−\nD(y) =n+ 1for each xy/\\⌉}atio\\slash∈A(D), which is not paired many-to-many\n2-coverable.\nProof.We define a digraph Dof order n≥9 as follows. Let V(D) =\nA∪B∪Ssuch that\nA∩B={z},S={s1,s2,t1,t2},|A|=m≥3,|B|=n−m−3≥3.\nLet the arc set A(D) be defined as follows.\nA(D) ={uv,vu:u,v∈A}∪{uv,vu:u,v∈B}\n∪({uv,vu:u,v∈S}\\{s1t1,s2t2})\n∪{zu,uz:u∈S}\n∪{s1u,us1,t2u,ut2,t1u,us2:u∈A\\{z}}\n∪{t1u,ut1,s2u,us2,t2u,us1:u∈B\\{z}}.\nz\ns1\nt1\ns2\nt2A\\{z} B\\{z}S\nD\nFigure 1. The graph of Proposition 3.\nThe graph is as shown in Figure 1, note that the dashed lines in this\nfigure indicate that s1t1,s2t2/∈A(D[S]). It is not hard to check that for\neachxy /∈A(D), we have d+\nD(x) +d−\nD(y) =n+ 1. Also, observe that\nthere do not exist disjoint s1-t1path and s2-t2path (otherwise, they must\n12containz, a contradiction). Hence, there is no paired many-to-many 2 -\nDDPC from{s1,s2}to{t1,t2}inD, that is, Dis not paired many-to-many\n2-coverable.\n3. One-to-many k-DDPC problem\nIn the following result, we will make improvements for Theor em 5 by\nreplacing “ n≥Ck9” with “n≥3k” and replacing “ δ0(D)≥⌈(n+k+1)/2⌉”\nwith “δ0(D)≥⌈(n+k)/2⌉”, which is sharp when n+kis even and is sharp\nup to an additive constant 1 otherwise, by Proposition 4.\nTheorem 7. LetDbe a digraph of order n≥3k, wherek≥2. Ifδ0(D)≥\n⌈(n+k)/2⌉, thenDis one-to-many k-coverable. Moreover, the bound for\nδ0(D)is sharp when n+kis even and is sharp up to an additive constant\n1 otherwise.\nProof.LetS={s}andT={t1,t2,...,tk}be disjoint source set and sink\nset inD, respectively. As\nδ0(D)≥⌈(n+k)/2⌉≥⌈(3k+k)/2⌉= 2k,\nwe have d+\nV(D)\\T(s)≥k. Let\nS′={s1,s2,...,sk},T={t1,...,tk},\nwheres1=sand{s2,s3,...,sk}⊆N+\nV(D)\\T(s). Sinceδ0(D)≥⌈(n+k)/2⌉\nandn≥3k, by Theorem 1, Dhas an unpaired many-to-many k-DDPC\nforS′andT, say{P1,P2,...,P k}, where Pi(1≤i≤k) is ansi−tσ(i)\npath and σis a permutation on {1,2,...,k}. Recall that ssi∈A(D) for\n2≤i≤k. Therefore,{P1,P∗\n2,...,P∗\nk}is a one-to-many k-DDPC for Sand\nTinD, whereP∗\ni:=ssiPitσ(i)for 2≤i≤k. Hence, Dis one-to-many\nk-coverable.\nProposition 4. For every k≥2and every n≥k+2, there exists a digraph\nDonnvertices with minimum semi-degree ⌈(n+k−1)/2⌉−1which is not\none-to-many k-coverable.\nProof.The proof is divided into the following two cases according t o the\nparity of n+k.\nCase 1. n+kis odd.\n13LetDbe a digraph of order n≥k+2 such that\nV(D) =A∪B,|A∩B|=k−1,D[A]∼=D[B]∼=← →K(n+k−1)/2.\nObserve that\nδ0(D) = (n+k−1)/2−1 =⌈(n+k−1)/2⌉−1.\nLet\nS={s}⊆A\\B,T={ti: 1≤i≤k}⊆B.\nAs|A∩B|=k−1, there is no one-to-many k-DDPC for SandT. Hence,\nDis not one-to-many k-coverable.\nCase 2. n+kis even.\nLetD1∼=← →K(n+k)/2−1andD2∼=E(n−k)/2+1. LetDbe a new digraph of\nordern≥k+2 such that V(D) =V(D1)∪V(D2) and\nA(D) =A(D1)∪A(D2)∪{xy,yx:x∈V(D1), y∈V(D2)}.\nObserve that\nδ0(D) = (n+k)/2−1 =⌈(n+k−1)/2⌉−1.\nLet\nS={s}⊆V(D2),T={t1,t2,...,tk}⊆V(D1).\nIt can be checked that a set of disjoint paths {Pi: 1≤i≤k}covers at most\n((n+k)/2−1)−k+1 = (n−k)/2(<|V(D2)|)\nvertices of V(D2), where Pi(1≤i≤k) is ans−tipath. Hence, Dis not\none-to-many k-coverable.\n4. One-to-one k-DDPC problem\nIn the following result, we will improve Theorem 8 by replaci ng “n≥\nCk9” with “ n≥k+ 1” and replacing “ δ0(D)≥ ⌈(n+k+ 1)/2⌉” with\n“δ0(D)≥⌈(n+k−1)/2⌉”, which is sharp when n+kis odd and is sharp\nup to an additive constant 1 otherwise, by Proposition 5.\nTheorem 9. LetDbe a digraph of order n≥k+ 1, wherek≥2. If\nδ0(D)≥⌈(n+k−1)/2⌉, thenDis one-to-one k-coverable. Moreover, the\nbound for δ0(D)is sharp when n+kis odd and is sharp up to an additive\nconstant 1 otherwise.\n14Proof.We will prove the theorem by induction on k. For the base case\nthatk= 2, we prove the following claim:\nClaim: LetDbe a digraph of order n≥3. Ifδ0(D)≥⌈(n+1)/2⌉, thenD\nis one-to-one 2-coverable.\nProof of the claim: LetS={s}andT={t}such that sandtare\ndistinct vertices of D. By Theorem 10, Dis Hamiltonian connected as\nδ0(D)≥⌈(n+1)/2⌉. Thus, there is a Hamiltonian path Pfromstot. For\neachv∈V(P)\\{s}, we use v−to denote the predecessor of vonP. Let\nX={v−:v∈N+\nD(s)}, Y={u:u∈N−\nD(t)}.\nAst/\\⌉}atio\\slash∈X∪Y,|X∪Y|≤n−1. Hence,\n|X∩Y|=|X|+|Y|−|X∪Y|≥⌈(n+1)/2⌉+⌈(n+1)/2⌉−(n−1)≥2.\nWe choose a vertex w∈X∩Y, clearly sw+,wt∈A(D). Now{P1,P2}is a\none-to-one 2-DDPC for SandT, whereP1=sw+PtandP2=sPwt(see\nFigure 2). Hence, Dis one-to-one 2-coverable.\nsw\nw+t\nFigure 2. The figure of the claim. The s-tpath is a Hamiltonian path. The\nthicker lines express two new s-tpaths:P1=sw+PtandP2=sPwt.\nWe next prove the inductive step. Assume that k≥3. LetS∗=\n{s∗}andT∗={t∗}such that s∗andt∗are distinct vertices of D. As\n|N+\nD(s∗)∪N−\nD(t∗)|≤n,\n|N+\nD(s∗)∩N−\nD(t∗)|=|N+\nD(s∗)|+|N−\nD(t∗)|−|N+\nD(s∗)∪N−\nD(t∗)|\n≥⌈(n+k−1)/2⌉+⌈(n+k−1)/2⌉−n≥2.\nWe choose a vertex h∈N+\nD(s∗)∩N−\nD(t∗). LetD1=D−{h}with order\nn1(=n−1≥(k−1)+1). Observe that\nδ0(D1)≥⌈(n+k−1)/2⌉−1 =⌈(n−1+k−1−1)/2⌉=⌈(n1+(k−1)−1)/2⌉.\n15By the induction hypothesis, D1is one-to-one ( k−1)-coverable. Let\n{P1,P2,...,P k−1}be a one-to-one ( k−1)-DDPC for S∗andT∗inD1.\nRecall that s∗h,ht∗∈A(D). Clearly,{P1,P2,...,P k−1,Pk}is a one-to-one\nk-DDPC for S∗andT∗inD, wherePk:=s∗ht∗. Hence, Dis one-to-one\nk-coverable.\nProposition 5. For every k≥2and every n≥k+1, there exists a digraph\nDonnvertices with minimum semi-degree ⌈(n+k)/2⌉−2which is not\none-to-one k-coverable.\nProof.The proof is divided into the following two cases according t o the\nparity of n+k.\nCase 1. n+kis odd.\nLetDbe a digraph of order n≥k+1 such that\nV(D) =A∪B,|A∩B|=k−1,D[A]∼=D[B]∼=← →K(n+k−1)/2.\nObserve that\nδ0(D) = (n+k−1)/2−1 =⌈(n+k)/2⌉−2.\nLet\nS={s}⊆A\\B, T={t}⊆B\\A.\nAs|A∩B|=k−1, there is no one-to-one k-DDPC for SandT. Hence, D\nis not one-to-one k-coverable.\nCase 2. n+kis even.\nLetDbe a digraph of order n≥k+1 such that\nV(D) =A∪B,|A∩B|=k−2,D[A]∼=D[B]∼=← →K(n+k)/2−1.\nObserve that\nδ0(D) = (n+k)/2−2 =⌈(n+k)/2⌉−2.\nLet\nS={s}⊆A\\B, T={t}⊆B\\A\nAs|A∩B|=k−2, there is no one-to-one k-DDPC for SandT. Hence, D\nis not one-to-one k-coverable.\n16Acknowledgement. We thank Professor Gexin Yu for his helpful sug-\ngestions and constructive comments. This work was supporte d by Na-\ntional Natural Science Foundation of China under Grant No. 1 2371352,\nZhejiang Provincial Natural Science Foundation of China un der Grant No.\nLY23A010011, Yongjiang Talent Introduction Programme of N ingbo under\nGrant No. 2021B-011-G, and Zhejiang Province College Stude nts Science\nandTechnology Innovation Activity PlanandNewSeedlingTa lent Program.\nReferences\n[1] J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorith ms and Ap-\nplications, 2nd Edition, Springer, London, 2009.\n[2] H. Cao, B. Zhang, Z. Zhou, one-to-one disjoint path cover s in digraphs,\nTheoretical Computer Science, 714 (2018) 27-35.\n[3] X. Chen, Unpaired many-to-many vertex-disjoint path co vers of a class\nofbipartitegraphs,InformationProcessingLetters, 110( 2010) 203–205.\n[4] M. Chudnovsky, A. Scott and P.D. Seymour, Disjoint paths in tourna-\nments, Advances in Mathematics, 270 (2015) 582–597.\n[5] M. Chudnovsky, A. Scott and P.D. 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Yu, An improved linear co nnec-\ntivity bound for tournaments to be highly linked, European J ournal of\nCombinatorics, 98 (2021), Article 103390.\n[16] S.-C. Ntafos, S.-L. Hakimi, On path cover problems in di graphs and\napplications to program testing, IEEE Transactions on Soft ware Engi-\nneering, 5(5) (1979) 520-529.\n[17] J. H. Park, Unpaired many-to-many disjoint path covers in restricted\nhypercube-like graphs, Theoretical Computer Science, 617 (2016) 45-\n64.\n[18] J.H. Park, H.S. Lim, Characterization of interval grap hs that are un-\npaired 2-disjoint path coverable, Theoretical Computer Sc ience, 821\n(2020) 71-86.\n[19] J. H. Park, A sufficient condition for the unpaired k-disjoint path cov-\nerability of interval graphs, The Journal of Supercomputin g, 77 (2021)\n6871-6888.\n[20] Y. Sun and A. Yeo, Directed Steiner tree packing and dire cted tree\nconnectivity, Journal of Graph Theory, 102(1) (2023) 86–10 6.\n[21] M. Overbeck-Larisch, Hamiltonian paths in oriented gr aphs, Journal of\nCombinatorial Theory, Series B, 21 (1976) 76-80.\n[22] R. Thomas, P. Wollan, An improved linear edge bound for g raph link-\nages, European Journal of Combinatorics, 26 (2005), 309–32 4\n18[23] F. Wang, W. Zhao, One-to-one disjoint path covers in hyp ercubes with\nfaulty edges, The Journal of Supercomputing, 75(8) (2019) 5 583-5595.\n[24] G. Yu, Covering 2-connected 3-regular graphswith disj oint paths, Jour-\nnal of Graph Theory, 88 (2018), 385-401.\n[25] J. Zhou, Y. Qi and J. Yan, Improved results on linkage pro blems, Dis-\ncrete Mathematics, 346(6) (2023), Article 113351.\n[26] Z. Zhou, Semi-degree conditions for one-to-many disjo int path covers\nin digraph, submitted.\n19"    },    {        "title": "2402.13835v2.Loop_equations_for_generalised_eigenvalue_models.pdf",        "content": "Prepared for submission to JHEP\nLoop equations for generalised eigenvalue models\nEdoardo Vescovi,aKonstantin Zaremboa,1\naNordita, KTH Royal Institute of Technology and Stockholm University,\nHannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden\nE-mail: edoardo.vescovi •su.se,zarembo •nordita.org\nAbstract: We derive the loop equation for the 1-matrix model with generic difference-\ntype measure for eigenvalues and develop a recursive algebraic framework for solving it to\nan arbitrary order in the coupling constant in and beyond the planar approximation. The\nplanar limit is solved exactly for a one-parametric family of models and in the general case\nat strong coupling. The Wilson loop in the N= 2∗super-Yang-Mills theory and the Hoppe\nmodel are used to illustrate our methods.\n1Corresponding author.arXiv:2402.13835v2  [hep-th]  27 Feb 2024Contents\n1 Introduction 2\n2 Loop equation 3\n2.1 Matrix model 3\n2.2 Loop equation 5\n2.3 The ν-model 9\n3 Weak coupling: solution 14\n3.1 Shift of the matrix variables 14\n3.2 Power-series ansatz 16\n3.3 Solving algorithm 17\n4 Weak coupling: applications 20\n4.1 Models with Vandermonde measure 20\n4.2 Models with Gaussian potential 22\n4.3 Circular Wilson loop 23\n4.3.1 N= 4SYM 24\n4.3.2 N= 2∗SYM 27\n4.4 Hoppe model 34\n5 Strong coupling: solution 35\n5.1 Short loops 36\n5.2 Long loops 37\n5.2.1 Shift of the spectral endpoint 42\n6 Strong coupling: Hoppe model 43\n6.1 Short loops 44\n6.2 Long loops 44\n6.3 Comparison to exact solution 45\n7 Conclusion 49\nA Saddle-point equation from loop equation 51\nB Operator identity and orthogonality relations 51\nCU(N)vs.SU(N) 53\nD Derivation of planar equations 54\n– 1 –1 Introduction\nThe loop equations were originally proposed for the Yang-Mills theory [1, 2] with the goal\nto reformulate it as a theory of strings [3]. Their zero-dimensional counterpart [2, 4], in the\nsame vein, gives a non-perturbative definition of quantum gravity in two dimensions [5–7].\nThe starting point in the latter case is a zero-dimensional quantum field theory, a matrix\nmodel. The loop equations then are the Schwinger-Dyson equations for the Wilson loop\nexpectation values that follow from reparametrisation invariance of matrix averages. Their\nperturbative (in 1/N2) solution generates the topological genus expansion of correlation\nfunctions with interpretation in terms of discretised random surfaces [3].\nIn the planar limit of strictly infinite N, matrix models can be solved by a variety\nof techniques such as the saddle-point method [8] or orthogonal polynomials [9, 10]. The\nadded value of the loop equations is an ease in going beyond the planar approximation [11].\nA powerful recursive procedure developed in [12] can be systematically extended to any\ndesired order in 1/N2, and lies at the heart of the topological recursion formulated entirely\nin geometrical terms [13].\nIn this paper we study loop equations for random eigenvalue ensembles of the form\nZ=+∞Z\n−∞NY\ni=1daiY\ni<jµ(ai−aj)e−NP\niV(ai)\n, (1.1)\nwith µ(a)andV(a)subject to the obvious convergence conditions but otherwise arbi-\ntrary. The Hermitian one-matrix model corresponds to the standard Vandermonde measure\nµ(a) =a2. For a one-parameter deformation thereof, µ(a) =a2β, known as the β-ensemble,\nthe topological recursion was formulated in [14, 15] and further studied in [16–19]. Other\nsolvablecasesincludetheHoppemodel µ(a) =a2/(a2+m2)[20,21], andtheChern-Simons\nmatrix model with µ(a) = sinh a[22–24]. Since the model with arbitrary µ(a)andV(a)is\nunlikely to be solvable in general, our goal is more modest – to study the loop equations\nin the regimes where a well-defined approximation scheme can be devised, namely at weak\nand at strong coupling.\nThe eigenvalue integrals of this type arise in many different contexts: quantum D-\nbranes [21], supersymmetric Wilson loops [25, 26], supersymmetric localisation [27], Chern-\nSimons theory [22, 23], knot invariants [28, 29] and many others. To give one example,\nN= 2supersymmetric gauge theories on S4are described by eigenvalue integrals with\nthe Gaussian potential and a fairly complicated measure: for the mass deformation of the\nN= 4super-Yang-Mills known as the N= 2∗theory, the measure is expressed through a\ncombination of the Meijer G-functions [27]. The latter model has been studied by a variety\nof approximate techniques [30–36] but has never been solved in full generality. It will be\nour prime example to illustrate the utility of the loop equations. Localisation integrals\nin superconformal gauge theories [37–44] constitute another set of matrix models where\na complicated measure precludes the use of conventional techniques, and we believe that\nmethods developed in this paper will be useful in this context as well.\nSection 2.1 covers from the basics to the inverse Laplace transforms of n-point resol-\n– 2 –vents, also called n-point Wilson loop operators W(x1, x2, . . . , x n)in this paper. Section 2.2\nderives the (non-linear, integro-differential) loop equation for arbitrary Nand polynomial\npotential. It closes on the single function W(x)in the planar limit and depends also on\nW(x, y)otherwise. In preparation for large- λmodels, section 2.3 converts the equation for a\none-parameter deformation of the Vandermonde measure into an exactly-solvable algebraic\nproblem.\nSection 3.1 operates a change of variables to eliminate the linear term in the potential.\nThis is instrumental in the structure of the perturbative solution in section 3.2 and in the\nalgebraic equations for the series coefficients in section 3.3. We collect and test our results\nforW(x)to high orders against exact formulas in various models in section 4. The key\nobservation that makes contact with physics is the fact that certain (matrix-model) Wilson\nloop operators evaluated at x= 2πcalculate the expectation value of the (gauge-theory)\n1/2-BPS circular Wilson loop in N= 4andN= 2∗SYM theories.\nIn particular section 4.3.2 pays great attention to the latter: an analysis of the perturba-\ntive convergence and a Padé resummation to extrapolate to λ≫1, which build on our\npower-series expansion of the planar Wilson loop to the order λ35. For small mass param-\neter, the ratio test signals a singularity at λ=−π2which also affects other observables\ninN= 2theories. A finite value of the mass slows down the convergence rate and falls\nshort of detecting a mass-dependent convergence radius. The large-mass/flat-space limit\nis incompatible with λ≪1but formally organises the power series in agreement with the\nrenormalisation of N= 2theories.\nSection 5 is about a systematic method to generate the expansion of W(x)in strongly-\ncoupled models with a general deformation of the Vandermonde measure. We distinguish\ntwo regimes: when xis smaller than the characteristic eigenvalue scale in sections 5.1 and\n5.2.1, the leading approximation tracks back to section 2.3, whereas the opposite limit in\nsections 5.2 and 5.2.1 is brought to a form reminiscent of a Wiener-Hopf problem. We\nintersperse remarks for Gaussian models in this part and present a thorough application to\nthe strongly-coupled Hoppe model in section 6.\nSection 7 outlines future advancements and the appendices collate technical proofs. The\nMathematica notebooks attached to the publication automatise the algorithm of section\n3.3 and generate the results of section 4.\n2 Loop equation\n2.1 Matrix model\nThe Wilson loop expectation is an exponential average:\nW(x) =1\nN\ntrexA\u000b\n(2.1)\nin the ensemble defined by the partition function (1.1). The two-point connected correlator\nis defined as\nW(x, y) =\ntrexAtreyA\u000b\n−\ntrexA\u000b\ntreyA\u000b\n. (2.2)\n– 3 –The averages are computed as multiple integrals\n⟨f(a1, . . . , a N)⟩=1\nZ+∞Z\n−∞Y\nidaiY\ni<jµ(ai−aj)f(a1, . . . , a N)e−NP\niV(ai),(2.3)\nover the eigenvalues ai∈R(i, j= 1,2, . . . , N) of the N×Nhermitian matrix A. The\nintegral measure µ(a)is a smooth function with µ(a) =µ(−a)⩾0. The matrix model\n(2.3) encompasses the β-ensemble ( µ(a) =a2β), including the Hermitian matrix model\nwith unitary symmetry U(N)(with Vandermonde measure, β= 1), the real symmetric\nmatrices with orthogonal symmetry SO(N)(β= 1/2) and the real quaternionic matrices\nwith symplectic symmetry Sp(N)(β= 2). We will later impose mild assumptions on\nthe measure to achieve results in different regimes. The probability measure is set by the\npotential V(a), given by a formal sum\nV(a) =8π2\nλ∞X\nn=0Tnan. (2.4)\nWe will need all “times” Tnto generate loop averages by differentiation, upon which we\nassume the potential settles to some fixed polynomial with only finite number of terms\npresent. We factor out the expansion parameter λ, which possibly takes small or large\nvalues, from the “coupling constants” Tn, which are finite and include odd n, thus allowing\nnon-symmetric potentials. When eigenvalues are seen as the coordinates of Nparticles\nalong a line, the i-th particle is subject to the potential V(ai)and each pair repels with\nenergy −logµ(ai−aj). Large values of λspread particles apart and away from the potential\nminima in the equilibrium configuration. For a more detailed account of random matrix\nmodels we refer to reviews [45–47].\nThe loop correlators W(x)andW(x, y)are the first two representatives of the family\nW(x1, . . . , x n) =Nn−2\ntrex1A. . .trexnA\u000b\nc, (2.5)\nwhere the subscript “c” pertains to the connected part [18]. A more commonly used set of\nobservables are correlators of trace resolvents, related to the exponential correlators by the\ninverse Laplace transform. Namely,\nω(x) =1\nN\u001c\ntr1\nx−A\u001d\n. (2.6)\nThe multi-point correlators of resolvents are generating functions of the basic monomial\ncorrelators:\nω(x1, . . . , x n) =Nn−2\u001c\ntr1\nx1−A. . .tr1\nxn−A\u001d\nc(2.7)\n=Nn−2∞X\nk1,...,kn=0x−k1−1\n1 . . . , x−kn−1\nnD\ntrAk1. . .trAknE\nc.\n– 4 –Like the partition function, W(x)andW(x, y)are formal power series in the potential\ncoefficients by the expansion of (2.4) in the vicinity of a minimum. They are generating\nfunctions similar to (2.6) and (2.7):\nW(x) =1\nN∞X\nn=0xn\nn!⟨trAn⟩ (2.8)\nW(x, y) =∞X\nn,m=0xnym\nn!m!(⟨trAntrAm⟩ − ⟨trAn⟩⟨trAm⟩) (2.9)\nand in that they have a 1/N-expansion, called topological expansion [7]. The product\nofntraces enumerates discrete open surfaces with nholes, whose Euler characteristics\ncontrol the power of N, and the variables x1, . . . , x nare fugacities for the lengths of the\nnboundaries [8, 48–50]. The expansion in the couplings is associated with ribbon-graph\ndiagrams [3].\nThe genus expansion begins at N0in our normalisation [14, 51] and odd powers of\nN−1are absent when the measure scales as µ(a)∼Ca2for small eigenvalue separation\n(the constant Cis not important for this conclusion), in accordance with the observation\nthat only the Hermitian model has this property among the β-ensembles. Such measure\ndefines a deformation of the Vandermonde determinant and brings us to coin the name\ngeneralised eigenvalue model for (2.3).\nAlthough the focus of section 2.2 is on an equation for W(x), this has to be fed with\nW(x, y), save for the strict large- Nlimit. The loop-insertion method [52, 53] generates all\nloop correlators by consecutive differentiation:\nW(x1, . . . , x n) =N−2δ\nδV(x1). . .δ\nδV(xn)logZ (2.10)\n=δ\nδV(x1). . .δ\nδV(xn−1)W(xn),\nwhere the loop-insertion operator is defined as1\nδ\nδV(z)=−λ\n8π2∞X\nn=0zn\nn!∂\n∂Tn. (2.11)\nThe simplicity of the recursion comes at the cost of solving the loop equation for an infinite-\ndegree potential (2.4) before choosing a specific polynomial.\n2.2 Loop equation\nWe derive the loop equation (2.26) below: a non-linear integro-differential equation for\nW(x), which depends on an infinite-degree potential (2.4) and a measure function µ. The\n1It is the inverse Laplace transform of the operator −λ\n8π2P∞\nn=0z−n−1∂/∂T n, which similarly acts upon\n(2.6) to generate the tower of multi-point (or multi-loop, in matrix-model terminology) resolvents (2.7).\n– 5 –latter is subject to γ(0) = 2, where we find convenient to define the even function2\nγ(a) =ad\ndalogµ(a). (2.12)\nAs noted below (2.9), this puts a mild constraint on the topological expansion. Far from\nbeing necessary, it is motivated by our interest in supersymmetric theories.\nWe will use also another function\nR(a) =1\n2d\ndalogµ(a) =γ(a)\n2a. (2.13)\nWe assume that these two functions admit regular Fourier representation:\nγ(a) =+∞Z\n−∞dω\n2πe−iωaˆγ(ω), R (a) =+∞Z\n−∞dω\n2πie−iωaˆR(ω). (2.14)\nThe function ˆRis anti-symmetric:\nˆR(−ω) =−ˆR(ω), (2.15)\nand real, with the above definition.\nThe resolvent obeys loop equations, also called Pastur equations in mathematics and\nSchwinger-Dysonequationsinthephysicsliterature. Theloopequationsareself-consistency\nconditionsbasedofftheinvarianceof(2.3)underaninfinitesimallocalchangeofvariablesor\nintegration by parts. We adapt the latter derivation [6, 18] (usually done for the resolvent)\nto the exponential average. The shift of focus draws inspiration from the loop dynamics in\na QCD-like random matrix model [2].\nWe insert an exponential in the form of (2.1) into the average\n1\nZ+∞Z\n−∞Y\nidaiX\np∂\n∂ap\nY\ni<jµ(ai−aj)exape−NP\niV(ai)\n= 0. (2.16)\nThe chain rule delivers three terms. The derivative acting on the potential yields\n−N*X\npV′(ap)exap+\n=−N2V′\u0012d\ndx\u0013\nW(x). (2.17)\n2The condition γ(0) = 2is necessary to have a regular large- Nexpansion with only even powers of 1/N\nappearing. This technical assumption, that immediately excludes all β-ensembles, becomes important once\nwe take into account 1/Ncorrections. All our result at leading planar approximation are independent of it\nand are valid for models with γ(0)̸= 2as well.\n– 6 –We swap the average and the operator\nV′\u0012d\ndx\u0013\n=8π2\nλ∞X\nn=1nTn\u0012d\ndx\u0013n−1\n. (2.18)\nThe derivative acts on µand the exponential as\n*\nxX\npexap+\n+*X\ni̸=pexapµ′(ap−ai)\nµ(ap−ai)+\n. (2.19)\nWe recall (2.1), use γ(a) =γ(−a)to symmetrise in i↔p\nNxW(x) +1\n2*X\ni̸=pexap−exai\nap−aiγ(ap−ai)+\n, (2.20)\ncomplete the sum with the terms with i=p\n\u0012\n1−γ(0)\n2\u0013\nNxW(x) +1\n2*X\ni,pexap−exai\nap−aiγ(ap−ai)+\n, (2.21)\nand notice that the assumption above (2.12) removes the linear dependence on N. The\nnext step is to link up to the functionals (2.2), noting the double sum, and possibly (2.1).\nFor that one has to write the expression as a combination of exai. The way to achieve this\nis not unique3. While the numerator is already in this form, we operate the identity\n1\nap−ai=Zx\n0ds es(ap−ai)(2.22)\non the denominator and trade the measure factor for its Fourier transform\nγ(a) =+∞Z\n−∞dω\n2πe−iωaˆγ(ω), ˆγ(ω) =+∞Z\n−∞da eiωaγ(a). (2.23)\nThis brings (2.21) to the desired form\n1\n2X\ni,p+∞Z\n−∞dω\n2πˆγ(ω)Zx\n0dsD\nesap+(x−s)aieiω(ap−ai)E\n(2.24)\nat the cost of introducing two auxiliary integrations. The comparison with (2.2) yields\n3Any integral with exponential weight, like Fourier and Laplace transforms of a kernel, fits the criteria.\nWe present a simpler version of the loop equation later in this section exploiting this freedom.\n– 7 –explicitly the Wilson loops\n1\n2+∞Z\n−∞dω\n2πˆγ(ω)Zx\n0ds\u0002\nW(s−iω, x−s+iω) +N2W(s−iω)W(x−s+iω)\u0003\n.(2.25)\nThe last step puts together (2.17) and (2.25) and yields the loop equation4\nV′\u0012d\ndx\u0013\nW(x) =1\n2+∞Z\n−∞dω\n2πˆγ(ω)Zx\n0ds\u0014\nW(s−iω)W(x−s+iω) +W(s−iω, x−s+iω)\nN2\u0015\n.\n(2.26)\nThe formula deserves a number of comments.\nFirst, the equation displays a direct dependence on the model (2.3): the parameters\n(number of eigenvalues N, or rank of gauge group U(N), and coupling λ), the integral\nmeasure (via the Fourier transform (2.23) of its logarithmic derivative (2.12)) and potential\n(via the infinitely-many coefficients {Tn}n=1,2,...of (2.18)). The two-point operator W(x, y)\nis linear in the solution through one loop insertion (2.10) and (2.11):\nW(x, y) =δW(y)\nδV(x). (2.27)\nSecond, the equation is to solve for W(x)on the real line. The normalisation of (2.1)\nsets the initial condition W(0) = 1. In section 4 the matrix-model quantity W(2π)maps\nto physical observables (expectation values of gauge-theory Wilson loops), hence one can\nrestrict to the segment x∈[0,2π]. For this to be possible, our derivation prefers (2.22)\namong equivalent representations with domain larger than s∈[0, x]5. The imaginary shifts\n±iωrequire the analytical continuation to the strip Re(x)∈[0,2π]in the complex plane.\nThird, the equation is non-linear (due to the first term in the integrand) and of a\nspecific integro-differential type (thus backed up by little literature). The imaginary shifts\nprevent the interpretation of the s-integral as a convolution, hence precluding a solution in\nLaplace transform. An exception is the subject of section 4.1: the Vandermonde measure\nµ(a) =a2maps to a Dirac delta ˆγ(ω) = 4 πδ(ω), which localises the integrand on vanishing\nshifts.\nFourth, there are two ways to approach the equation perturbatively. At infinite N, the\noperators W(x)andW(x, y)are of order N0and the latter drops out of (2.26). At large\nN, they expand in series of N−2. The suppression factor N−2in (2.26) helps solving order\nby order in an intertwined manner. The knowledge of W(x)up to N−2ncalculates W(x, y)\nup to N−2nby loop insertion, which is necessary in turn to write the equation at the order\nN−2(n+1)and find the next term N−2(n+1)ofW(x).\nThe same argument applies to the expansion in small λ: the left-hand side contains\npowers up to λℓ−1and the right-hand side λℓat the iteration ℓ= 0,1, . . ., hence one\n4This is the first one in a chain of loop equations that intertwines higher-point Wilson loops.\n5It is hard to devise representations with s∈[0,∞]that converge for all orderings ap−ai≶0.\n– 8 –iteratively reads off the order λℓfrom the former. The argument relies on the linearity of\nthe left-hand side to easily adjust the next unknown term.\nThe same reasoning does not go through at large coupling because the right-hand side\nis not linear. Once a truncated solution is plugged into the left-hand side, one would\nhave to disentangle the next order from the integral equation on the right-hand side. The\nobstruction is intrinsic to the loop equation: it is not washed away by large Nnor by a\nchoice of model. Section 5 revisits the equation with a method similar to the Wiener-Hopf\ndecomposition.\nAnother form of the loop equation arises upon Fourier transforming R(a)instead of\nγ(a)and is best formulated in terms of the oscillating loop average:\nW(κ) =1\nN*X\nkeiκak+\n, (2.28)\nrelatedtotheWilsonloopin(2.1)byananalyticcontinuation: W(x) =W(−ix). Repeating\nthe steps following (2.16) with the insertion of eiκapand using the anti-symmetry of ˆR(ω)\nwe find:\ni\n2V′\u0012\n−i∂\n∂κ\u0013\nW(κ) =+∞Z\n−∞dω\n2πˆR(ω)W(κ−ω)W(ω). (2.29)\nThis manifestly real form of the equation (for an even potential) is particularly beneficial\nfor the strong-coupling analysis in section 5. The non-planar corrections, dropped here for\nsimplicity, are easily recovered by replacing WWwith WW +N−2δW/δV.\nThisFourier-spaceformoftheloopequationisslightlyunconventional,ithasnoobvious\nsymmetries, and even for the Vandermonde measure with ˆR(ω) =−πsignωcontains a\nnon-linear integral transform that does not reduce to a simple convolution. In contrast, the\nkernel in (2.26) for the Vandermonde measure becomes the delta-function ˆγ(ω) = 4 πδ(ω)\nand the loop equation assumes the standard convolution form. In the appendix A we\nshow that the alternative form of the loop equation (2.29) is equivalent to the standard\nsaddle-point conditions of the original eigenvalue model.\nBelow we identify a class of models, generalising the usual matrix model with the\nVandermonde measure, for which (2.29) can be solved exactly.\n2.3 The ν-model\nWe start by considering an eigenvalue model with a polynomial potential and the kernel\nˆR(ω) =−πsignω|ω|2ν−2. (2.30)\nThe conventional Vandermonde measure corresponds to ν= 1. Fourier transforming back\nto the a-representation gives a rather baroque-looking measure. Nonetheless the loop equa-\ntion is solvable in this case. This solution will be an integral part of the strong-coupling\nanalysis for generic ˆR(ω). We thus spend some time discussing this type of models which\nwe collectively call the ν-model.\n– 9 –The loop equation with the above kernel can be written as\n−i V′\u0012\n−i∂\n∂κ\u0013\nW(κ) =∞Z\n0dω ω2ν−2W(ω)\u0010\nW(κ−ω)−W(κ+ω)\u0011\n. (2.31)\nThe exact solution of this equation can be found by exploiting the following mathematical\nfact.\nConsider a function f(κ)whose Fourier transform has a compact support:\nf(κ) =1Z\n−1dt eitκˆf(t). (2.32)\nThen the following operator identity holds true for any integer n:\n∞Z\n0dω ω2ν−2Jν+n(ω)\nων+n\u0010\nf(κ−ω)−f(κ+ω)\u0011\n=−iD(ν)\nnf(κ), (2.33)\nwhere D(ν)\nn≡ D(ν)\nn(−i∂/∂κ )is a differential operator of finite degree. For nnegative it\nvanishes identically:\nD(ν)\nn= 0 ( n <0), (2.34)\nand for npositive or zero D(ν)\nnis a differential operator of degree 2n+ 1expressed through\nthe Gegenbauer polynomials:\nD(ν)\nn(a) = (−1)n2ν−n−1Γ(ν−n−1)C(ν−n−1)\n2n+1 (a). (2.35)\nThe proof is surprisingly simple, and goes by direct calculation the details of which are\ngiven in the appendix B.\nWe list here the first few D(ν)\nnpolynomials:\nD(ν)\n0(a) = 2νΓ(ν)a, D(ν)\n1(a) =2νΓ(ν)\n6\u0000\n3a−2νa3\u0001\n,\nD(ν)\n2(a) =2νΓ(ν)\n120\u0002\n15a−20νa3+ 4ν(ν+ 1)a5\u0003\n. (2.36)\nThe general definition and key properties of the Gegenbauer polynomials C(λ)\nkcan be found\nin [54].\nThe integral transform in (2.33) suggests to choose Jν+n(µκ)/(µκ)ν+nas the basic\nbuilding blocks of the Wilson loop and to seek the solution of the loop equation as linear\ncombination of these functions. We thus take an ansatz of the form:\nW(κ) =X\nn⩾0AnJν+n(µκ)\n(µκ)ν+n. (2.37)\n– 10 –The rescaling factor µis necessary for imposing the normalisation condition W(0) = 1.\nPhysically, µcorresponds to the endpoint of the eigenvalue distribution as we detail below.\nThe rationale to take this ansatz is twofold. First, all the functions in the basis have\na Fourier image with a compact support, as follows from the integral representation of the\nBessel function:\nJν+n(κ)\nκν+n=1\n2ν+n√πΓ\u0000\nν+n+1\n2\u00011Z\n−1dt eitκ\u0000\n1−t2\u0001ν+n−1\n2. (2.38)\nThe ansatz therefore satisfies the condition (2.32). At the same time, regarding W(ω)in\n(2.31) as the kernel of an integral operator we can apply the master identity (2.33) valid\nfor all individual Bessel kernels and a function with finite support.\nThe operator identity (2.33) transforms (2.31) into a linear differential equation:\nV′\u0012\n−i∂\n∂κ\u0013\nW(κ) =µ1−2νX\nnAnD(ν)\nn\u0012\n−i\nµ∂\n∂κ\u0013\nW(κ). (2.39)\nBoth sides are differential polynomials acting on one and the same function. It just remains\nto match the coefficients:\nV′(µa) =µ1−2νX\nnAnD(ν)\nn(a). (2.40)\nSolving the loop equation boils down to a simple algebraic problem: given a polynomial\nV′(µa), find its expansion coefficients in the basis (2.35). In each particular case the prob-\nlem is easily solved by hand, just by matching the coefficients one-by-one, but it is also\npossible to derive the general solution by exploiting pseudo-orthogonality of the Gegen-\nbauer polynomials. We first illustrate the matching procedure and then derive the general\nsolution using orthogonality.\nConsider the Gaussian potential:\nV(a) =a2\n2g2. (2.41)\nThe left-hand side of (2.40) is then linear in aand the first polynomial in (2.36) will suffice.\nMatching the coefficient we get:\nA0=µ2ν\n2νΓ(ν)g2, (2.42)\nwhich yields for the Wilson loop:\nW(κ) =1\nΓ(ν)g2\u0010µ\n2κ\u0011ν\nJν(µκ). (2.43)\nThis still contains an unknown parameter µ, and to fix it we need to impose the normali-\nsation condition W(0) = 1.\n– 11 –Taking into account that\nlim\nu→0Jν(u)\nuν=1\n2νΓ(ν+ 1),\nwe get:\u0010µ\n2\u00112ν\n= Γ(ν)Γ(ν+ 1)g2. (2.44)\nThe properly normalised solution thus reads\nW(κ) = Γ( ν+ 1)Jν(µκ)\n(µκ/2)ν. (2.45)\nThe last two equations solve the ν-model with the Gaussian potential.\nWhen ν= 1the relation between µandgis linear: µ= 2g. Upon continuation to\ncomplex κ=−ixwe recover the well-known expression for the Wilson loop in the Gaussian\nmatrix model:\nW(x) =I1(2gx)\ngx. (2.46)\nWe revisit this result in section 4.3.1 from a slightly different perspective.\nReturning to the generic case, an expansion of a given polynomial V′in the basis (2.35)\ncan be derived from pseudo-orthogonality of the Gegenbauer polynomials. The polynomial\nbasis (2.35) does not form an orthonormal set with respect to any measure, for the reasons\nexplained in the appendix B. This can be also verified directly by checking that D(ν)\nndo\nnot satisfy trilinear relations characteristic of orthogonal polynomials, as can be seen by\nexamining the first few cases. The closest counterpart of functional orthogonality is derived\nin the appendix B, and involves a differential operator inside the integral:\n1Z\n−1da(1−a2)ν−n−3\n2D(ν)\nn(a)\u0012\na∂\n∂a+ 2ν−1\u0013\nD(ν)\nm(a) =4πΓ(2ν−1)\n(2n+ 1)!δnm.(2.47)\nApplying the differential operator a∂/∂a +2ν−1to both sides of (2.40) and integrating\nwith the prescribed weight, we get:\nAn=µ2ν−1(2n+ 1)!\n4πΓ(2ν−1)1Z\n−1da(1−a2)ν−n−3\n2D(ν)\nn(a)\u0012\na∂\n∂a+ 2ν−1\u0013\nV′(µa).(2.48)\nTogether with (2.37) this constitutes the full solution of the ν-model with any potential.\nThe real-valued Wilson loop is obtained by analytic continuation to κ=−ixwhich\ntransforms the entries in (2.37) into the modified Bessel functions:\nW(x) =X\nn⩾0AnIν+n(µx)\n(µx)ν+n. (2.49)\nThe width of the eigenvalue distribution µis fixed by imposing the normalisation condition\n– 12 –W(0) = 1:X\nnAn\n2ν+nΓ(ν+n+ 1)= 1. (2.50)\nWe can also find the eigenvalue density directly. The density is formally defined as a\nFourier transform of the Wilson loop:\nW(κ) =µZ\n−µda eiκaρ(a). (2.51)\nUsing the integral representation (2.38) in (2.37) we arrive at\nρ(a) =1√πX\nnAn(µ2−a2)ν+n−1\n2\n(2µ2)ν+nΓ\u0000\nν+n+1\n2\u0001. (2.52)\nWe have checked that for ν= 1this formula, together with the equation for An, boils down\nto the well-known solution of the conventional Vandermonde matrix model:\nρ(a) =−µZ\n−µdb\n2π2V′(b)\nb−as\nµ2−a2\nµ2−b2. (2.53)\nThe eigenvalue density must be positive on the whole interval (−µ, µ)which imposes\ncertain constraints on the coefficients An6. The simplest of those is the positivity of the\ndensity near the endpoints which requires\nA0>0. (2.54)\nIndeed, A0multiplies the smallest power of (µ2−a2)in (2.52). The other terms (those\nwith n >0) are necessarily more suppressed for avery close to µ. IfA0<0the density\nis destined to becomes negative at some a=µ−ϵwith small ϵ. Therefore, the model\nundergoes a transition into a multi-cut phase as soon as A0reaches zero. The multi-cut\nsolutions are not described by our ansatz7. It would be very interesting to explore the\nmulti-cut phases of the ν-model by extending our method to this case, or by some other\ntechnique.\nFor generic V(a)andˆR(ω)the non-linear integro-differential equation (2.29) cannot be\nsolved in a closed form, but the weak and strong coupling expansions can be worked out\nsystematically under very general assumptions. This is what we do in the rest of the paper.\n6The constraints of positivity are very powerful and in conjunction with the loop equations can be used\nto bootstrap matrix models not solvable otherwise [55].\n7We are grateful to D. Bykov for this comment.\n– 13 –3 Weak coupling: solution\n3.1 Shift of the matrix variables\nIn preparation for the perturbative analysis, we eliminate the linear term n= 1in (2.4)\nand change the loop equation accordingly. In quantum models linearities conflict with\nthe interpretation of {ai}i=1,2,...,Nas good perturbative degrees of freedom. The shift\n˜ai=ai−¯aby a constant ¯a, which is a stationary point of the potential V′(¯a) = 0,\ncentres the excitations {˜ai}i=1,2,...,Naround a vacuum. For the model with the Gaussian\npotential (but an arbitrary measure), the centre of mass decouples. The SU(N)andU(N)\nloop averages are then simply related as we show in the appendix C. For arbitrary potential\nthe situation is more complicated but still manageable as we demonstrate below.\nThe ability of operating infinitesimal changes of the potential (2.27), without breaking\nthe stationarity condition at ¯α, is the difficult bit to demonstrate. Let us assume the target\npotential and its deformations to be concave ( V(a)→+∞fora→ ±∞) and to have one\nor many minima ¯a. The new potential is still polynomial\n˜V(˜ai) =V(¯a+ ˜ai) =8π2\nλX\np=0,2,3,...˜Tp˜ap\ni (3.1)\nand parametrised by {¯a,˜T0,˜T2,˜T3, . . .}. The relations\n˜T1=∞X\np=1pTp¯ap−1= 0, ˜Tp′=∞X\np=p′Tp\u0012p\np′\u0013\n¯ap−p′, p′= 0,2,3, . . . (3.2)\ndetermine ¯a= ¯a(T1, T2, . . .), albeit not uniquely, and ˜Tp′=˜Tp′(Tp′, Tp′+1, . . .)respectively.\nThe effect on Wilson loops is\n˜W(x) =e−¯axW(x), ˜W(x, y) =e−¯a(x+y)W(x, y) (3.3)\nby the definitions (2.1) and (2.2). Matrix averages with tilded symbols are averages of the\ntype (2.3) where {˜ai}take the place of {ai}. The operator (2.11) undergoes a major change\nδ\nδ˜V(x)≡e−¯axδ\nδV(x)=λ\n8π2\nx\n2˜T2∂\n∂¯a+X\nm′=0,2,3,... \n(m′+ 1)x\n2˜T2˜Tm′+1−xm′\nm′!!\n∂\n∂˜Tm′\n.\n(3.4)\nThe proof is based on three lemmas. First, the left equation in (3.2) gives\n∂¯a\n∂Tm=−8π2\nλm¯am−1\nV′′(¯a). (3.5)\n– 14 –Second, we repeat on the right equation: use (3.5) with m⩾m′̸= 1\n∂˜Tm′\n∂Tm=\u0012m\nm′\u0013\n¯am−m′−8π2\nλm\nV′′(¯a)∞X\np=m′\u0012p\nm′\u0013\n(p−m′)Tp¯ap−m′+m−2,(3.6)\nuse the inverse of left equation Tp=P∞\np′=p˜Tp′\u0000p′\np\u0001\n(−¯a)p′−p\n∂˜Tm′\n∂Tm=\u0012m\nm′\u0013\n¯am−m′−8π2\nλm¯am−m′−2\nV′′(¯a)∞X\np=m′∞X\np′=p(−)p′−p\u0012p\nm′\u0013\u0012p′\np\u0013\n(p−m′)˜Tp′¯ap′,(3.7)\nswap the sums and resum\n∂˜Tm′\n∂Tm=\u0012m\nm′\u0013\n¯am−m′−8π2\nλm(m′+ 1)¯am−1\nV′′(¯a)˜Tm′+1. (3.8)\nThird, we begin with (2.4) and compare to (3.2)\nV′′(¯a) =8π2\nλ∞X\np=2p(p−1)Tp¯ap−2=16π2\nλ˜T2. (3.9)\nFinally, we collect the lemmas and change variables in (2.11) via the chain rule.\nδ\nδV(x)=−λ\n8π2∞X\nm=0xm\nm!\n∂¯a\n∂Tm∂\n∂¯a+X\nm′=0,2,3,...∂˜Tm′\n∂Tm∂\n∂˜Tm′\n\n=e¯ax\nx\nV′′(¯a)∂\n∂¯a+X\nm′=0,2,3,... \n(m′+ 1)x\nV′′(¯a)˜Tm′+1−λ\n8π2xm′\nm′!!\n∂\n∂˜Tm′\n(3.10)\n=e¯axλ\n8π2\nx\n2˜T2∂\n∂¯a+X\nm′=0,2,3,... \n(m′+ 1)x\n2˜T2˜Tm′+1−xm′\nm′!!\n∂\n∂˜Tm′\n.\nThis ends the proof of (3.4).\nWe are ready to insert (3.1) and (3.3) into (2.26). We pull out e−¯axon both sides\nthanks to the definition (3.4) and the shift property\n˜V′\u0012d\ndx−¯a\u0013\u0010\ne¯ax˜W(x)\u0011\n=e¯ax˜V′\u0012d\ndx\u0013\n˜W(x) (3.11)\nand derive\n˜V′\u0012d\ndx\u0013\n˜W(x) =1\n2+∞Z\n−∞dω\n2πˆγ(ω)Zx\n0ds\"\n˜W(s−iω)˜W(x−s+iω) +˜W(s−iω, x−s+iω)\nN2#\n,\n(3.12)\n– 15 –where the relations (2.18) and (2.27) still hold\n˜V′\u0012d\ndx\u0013\n=8π2\nλ∞X\nn=2n˜Tn\u0012d\ndx\u0013n−1\n, ˜W(x, y) =δ˜W(y)\nδ˜V(x). (3.13)\nThe reader should be aware that this notan independent loop equation. However there\nis more than a formal distinction with (2.26): the averages in (3.12) are written in matrix\nvariables that ensure ˜V′(0)∝˜T1= 0, but artificially complicate the insertion rule (3.4).\n3.2 Power-series ansatz\nThe equation is tractable analytically in the limit of large Nand small λwith the ansatz\n˜W(x) = ˜c000+∞X\nn=0∞X\nℓ=2n2ℓX\nm=1˜cn,ℓ,m\n(16π2)ℓN−2nλℓxm, ˜c000= 1. (3.14)\nThe rationale behind the index ranges becomes clear a posteriori. The power of λputs\nan upper bound on that of xdue to diagrammatical arguments. In (2.3) written in tilded\nvariables, the limit λ→0drives eigenvalues ˜ai→0towards the minimum of the potential.\nOnecanexpandthe“interacting” potential exp(8 π2λ−1NP\niP∞\nn=3˜Tn˜an\ni)andtherestofthe\nintegrand (including a test measure, e.g. µ(a) =a2+a4), then average polynomials of ˜a′\nis\nwith the Gaussian weight exp(8 π2λ−1N˜T2P\ni˜a2\ni).This is doable analytically for each term\nof the series in λ, which is nothing but some Feynman diagrams (integrals) of a quantum\nfieldtheory(matrixmodel)for Nscalarfields ˜aiinzerodimensions. Eachinteractionvertex\n(labeled by n⩾3) brings a power of λ−1and each propagator (after Gaussian integration)\naλ. When one throws xin by choosing f(˜a1, . . . , ˜aN) =P\niexp(x˜ai), the Gaussian integral\nof any monomial is proportional to λℓxmwith m⩽2n. This further restricts to m= 2ℓfor\nGaussian Hermitian models ( ˜Tn⩾3= 0andµ(a) =a2).\nWe also put forward\n˜W(x, y) =∞X\nn=0∞X\nℓ=2n+12ℓ−1X\nm=12ℓ−mX\nm′=1˜dn,ℓ,m,m′\n(16π2)ℓN−2nλℓxmym′. (3.15)\nNecessary conditions for the symmetry x↔yare˜dn,ℓ,m,m′=˜dn,ℓ,m′,mandm+m′⩽2ℓ.\nThe coefficients {˜cn,ℓ,m,˜dn,ℓ,m,m′}in (3.14) and (3.15) constitute the maximal set re-\nquired by arbitrary measure and potential; the simpler these are, the more of the coefficients\nvanish. It is often useful to pad our formulas with extra coefficients though, namely those\nlabeled by indices out of the ranges in (3.14) and (3.15), for example ˜c0,0,1= 0.\nThe coefficients are related via (3.13):\n˜dn,ℓ,1,m′=˜cn,ℓ−1,m′−1\n˜T2+2ℓ−m′+1X\nm′′=2(m′′+ 1)˜Tm′′+1\n˜T2∂˜cn,ℓ−1,m′\n∂˜Tm′′(3.16)\n– 16 –form′= 1, . . . , 2ℓ−1and\n˜dn,ℓ,m,m′=−2\nm!∂˜cn,ℓ−1,m′\n∂˜Tm(3.17)\nform= 2, . . . , 2ℓ−1andm′= 1, . . . , 2ℓ−m. As per our convention, we have ˜d0,1,1,1=˜T−1\n2\nfor example. The validity of the ansatz relies on the stationary point being extremal, i.e.\nV′′(¯a) =˜V′′(0)∝˜T2̸= 0. The apparent asymmetry between (3.16) and (3.17) arises from\nthe partition operated between ˜T1= 0and{˜Tn}n=0,2,3,...and placing xandyas in (3.13).\nThe symmetry ˜dn,ℓ,m,m′=˜dn,ℓ,m′,mis restored on the solution of the loop equation.\nThe coefficients are sensitive to the measure through its Taylor coefficients\nγ(0)=γ(0) = 2 , γ(n)=dnγ\ndxn(0) =+∞Z\n−∞dω\n2π(−iω)nˆγ(ω), n = 2,4,6, . . . , (3.18)\nthe potential through the extremal point ¯aand the Taylor coefficients near this point. The\nindices restrict the dependence:\n∂˜cn,ℓ,m\n∂˜Tm′= 0, m′⩾2ℓ−m+ 3, (3.19)\ntherefore the solution (3.14) and (3.15) up to the order N−2nmaxλℓmaxwith 2nmax⩽ℓmax\ndepends on the finitely-many coefficients {˜T2,˜T3, . . . , ˜T2ℓmax+1}of (3.1). The bound is\ndirectly visible at the level of (3.12) once one plugs (3.1) and (3.14).\n3.3 Solving algorithm\nWe build upon the ansätze (3.14) and (3.15) to turn the loop equation (3.12) into a system\nof algebraic equations for the infinitely-many coefficients {˜cn,ℓ,m}. We take note of (3.16)\nand (3.17), but use the dependent set {˜dn,ℓ,m,m′}too in order not to clutter formulas.\nPlanar limit. The limit of infinite Nwashes away the two-point Wilson loop in (3.12)\nand most coefficients, save for the subset {˜c0,ℓ,m}with ℓ= 1,2, . . .andm= 1,2, . . . , 2ℓ.\nAppendix D details how to spell out (3.14) in the equation. Equating powers of λandx\nleads to two systems of equations:\n2ℓ+1X\nr=2r!˜Trc0,ℓ,r−1= 0 (3.20)\n– 17 –spanned by ℓ= 1,2, . . .and\n1\np!2ℓ−p+1X\nr=2r(p+r−1)!˜Tr˜c0,ℓ,p+r−1 (3.21)\n=X\nq=1−p,3−p,...,p−12ℓ−2X\nm=p−q−1\n22ℓ−2X\nm′=p+q−1\n2ℓ−l\nm′\n2m\n−1X\nℓ′=⌈m\n2⌉X\nd=−p+q−1\n2,−p+q−1\n2+2,...,p+q−1\n2\u0012m\np−q−1\n2\u0013\n×\u0012m′\np+q−1\n4+d\n2\u0013\u0012m′−p+q−1\n4−d\n2\np+q−1\n4−d\n2\u0013(−)m′−p+q−1\n4−d\n2\n3p−q+1\n4−d\n2γ(m+m′−p+1)˜c0,ℓ′,m˜c0,ℓ−ℓ′−1,m′\nbyℓ= 1,2, . . .andp= 1, . . . , 2ℓ−1. The sums with γ(m+m′−p+1)are constrained to have\nm+m′−podd by (3.18). The terms ˜c2inherit the non-linearity of the loop equation.\nThere is a solution strategy that handles only linear equations.\nSuppose the iteration ℓ∗knows the subset {˜c0,ℓ,m}with ℓ= 1,2, . . . , ℓ∗−1andm=\n1,2, . . . , 2ℓ. We recall ˜T2̸= 0below (3.17) and express ˜c0,ℓ∗,1in terms of {˜c0,ℓ∗,m}m⩾2\nthrough (3.20)\n˜c0,ℓ∗,1=−1\n2˜T22ℓ∗+1X\nr=3r!˜Tr˜c0,ℓ∗,r−1. (3.22)\nWe replace this in (3.21) and verify that the equations are linear in {˜c0,ℓ∗,m}m⩾2; in partic-\nular, these coefficients never appear quadratically in the right-hand side, but linearly and\neach multiplied by one found at a previous iteration. Moreover, the equations are inhomo-\ngeneous due to the terms γ(m+m′−p+1), hence the {˜c0,ℓ∗,m}m⩾2turn out to be polynomials\nin the Taylor coefficients of γ(a). Standard techniques can efficiently solve (3.21)8and use\n(3.22) to calculate ˜c0,ℓ∗,1.\nSub-planar corrections. We apply the same logic to the full equation (3.12) to find\n{cn,ℓ,m}and{dn,ℓ,m,m′}. This spells out a (bulky) system of equations, after keeping pow-\ners of Nin appendix D and considering the two-point Wilson loop. We prefer however to\nput together the numerous observations so far and present the algorithmic workflow.\n1. The goal is to find the solution of (3.12) up to order N−2nmaxλℓmaxin the form\n˜W(x) =nmaxX\nn=0ℓmaxX\nℓ=2n˜Wn,ℓ(x)N−2nλℓ, ˜Wn,ℓ(x) =δn,0δℓ,0+2ℓX\nm=1˜cn,ℓ,m\n(16π2)ℓxm(3.23)\n8Existence and uniqueness of the solution fall back on that of the loop equation.\n– 18 –and as a by-product\n˜W(x, y) =nmaxX\nn=0ℓmaxX\nℓ=2n+1˜Wn,ℓ(x, y)N−2nλℓ, ˜Wn,ℓ(x, y) =2ℓ−1X\nm=12ℓ−mX\nm′=1˜dn,ℓ,m,m′\n(16π2)ℓxmym′.\n(3.24)\nWe remind the measure factor obeys γ(0) = 2 and approximate the potential to a\nfinite-degree polynomial\n˜V(x) =8π2\nλ2ℓmax+1X\nn=0˜Tnxn, ˜T1= 0, ˜T2̸= 0, (3.25)\nthanks to the observation below (3.19). The coefficients can take numerical values\nfrom the get-go when the target is the planar limit ( nmax= 0) of (3.23). They\ncannot otherwise, for the sake of the step 2 which extracts (3.24) from (3.23) by\ndifferentiation.\n2. We compute the coefficients in (3.23) and (3.24) iteratively. We loop over n∗=\n0,1, . . . , n maxandℓ∗= 2n∗,2n∗+ 1, . . . , ℓ max9to find the subsets {˜cn∗,ℓ∗,m}mand\n{˜dn∗,ℓ∗,m,m′}m,m′. The iteration starts by collecting terms N−2n∗λℓ∗−1in (3.12)\n˜V′\u0012d\ndx\u0013\n˜Wn∗,ℓ∗(x) =1\n2+∞Z\n−∞dω\n2πˆγ(ω)Zx\n0ds\nn∗X\np=0ℓ∗−1X\nq=0˜Wp,q(s−iω)(3.26)\nטWn∗−p,ℓ∗−q−1(x−s+iω) +1\nN2˜Wn∗−1,ℓ∗−1(s−iω, x−s+iω)\u0015\n.\nThe last term exists at sub-planar level ( n∗⩾1) and involves {˜dn∗−1,ℓ∗−1,m,m′}m,m′\nfound in a previous iteration. The integrand is a polynomial in ω,sandx; each\nmonomial can be integrated via Fourier transform (3.18). The integro-differential\nequation (3.26) turns into a polynomial equation which must hold regardless of the\nvalue of x. The coefficients of x-powers are proportional to one of {˜cn∗,ℓ∗,m}m. There-\nfore equating homonymous powers of xwrites a finite system of linear equations for\n{˜cn∗,ℓ∗,m}m. Once the solution is found, the iteration ends with the computation of\n{˜dn∗,ℓ∗,m,m′}m,m′using (3.16) and (3.17).\n3. At this stage one knows (3.23) and (3.24) for an arbitrary potential (3.25) with ex-\ntremal point in the origin. The analysis of section 3.1 comes in handy when we want\n9The case n∗=ℓ∗= 0is trivial.\n– 19 –to translate it to a generic point ¯a. We translate (3.25) according to (3.1)\nV(a) =˜V(a−¯a) =8π2\nλ2ℓmax+1X\np=0Tpap, (3.27)\nTp=2ℓmax+1X\np′=p˜Tp′\u0012p′\np\u0013\n(−¯a)p′−p, p = 0,1, . . . , 2ℓmax+ 1 (3.28)\nThe solution of (2.26) for this potential is given by (3.3)\nW(x) =e¯ax˜W(x),W(x, y) =e¯a(x+y)˜W(x, y). (3.29)\n4. We assign numeric values to {˜Tn}n=1,...,ℓmaxin order to specialise (3.27) to a target\npotential. The task is already part of step 1 in the special case mentioned therein.\nNotice different potentials correspond to the same solution as long as they share the\nsame shape sufficiently near the extremal point.\n4 Weak coupling: applications\nWe run the algorithm in section 3.3 for a selection of matrix models, perform checks with\nthe literature and derive new results for Wilson loops in SYM theories. We remind that\n(3.23) solves (3.12) with potential ˜V′(0) = 0. Step 3 of the algorithm converts the solution\nto that (3.29) associated to the shifted potential V(a) =˜V(a−¯a). The distinction between\ntilded and untilded variables is immaterial when ¯a= 0and the focus is on the planar limit.\n4.1 Models with Vandermonde measure\nThe Vandermonde measure µ(a) =a2maps to a constant γ(a) = 2. This brings massive\nsimplifications: the Dirac delta ˆγ(ω) = 4 πδ(ω)trivialises one integral in the loop equation\nand all derivatives γ(n)= 2δn,0in (3.18) drop out of step 1. The planar coefficients of (3.23)\nup to λ3are\n˜c0,1,1=−3˜T3\n2˜T2\n2, ˜c0,1,2=1\n2˜T2, ˜c0,2,1=−54˜T3\n3+ 72 ˜T2˜T3˜T4−20˜T2\n2˜T5\n4˜T5\n2,\n˜c0,2,2=18˜T2\n3−8˜T2˜T4\n4˜T4\n2, ˜c0,2,3=−˜T3\n˜T3\n2, ˜c0,2,4=1\n12˜T2\n2,\n˜c0,3,1=−3888 ˜T5\n3+ 9936 ˜T2˜T3\n3˜T4−4320 ˜T2\n2˜T3(˜T2\n4+˜T3˜T5) + 1440 ˜T3\n2(˜T4˜T5+˜T3˜T6)−280˜T4\n2˜T7\n16˜T8\n2,\n˜c0,3,2=3888 ˜T4\n3−6480 ˜T2˜T2\n3˜T4+ 2160 ˜T2\n2˜T3˜T5+ 864 ˜T2\n2˜T2\n4−360˜T3\n2˜T6\n48˜T7\n2, (4.1)\n˜c0,3,3=144˜T3(−˜T2\n3+˜T2˜T4)−30˜T2\n2˜T5\n8˜T6\n2,\n˜c0,3,4=27˜T2\n3−9˜T2˜T4\n12˜T5\n2, ˜c0,3,5=−3˜T3\n16˜T4\n2, ˜c0,3,6=1\n144˜T3\n2\n– 20 –and those at genus one\n˜c1,2,1=−27˜T3\n3+ 48 ˜T2˜T3˜T4−20˜T2\n2˜T5\n8˜T5\n2, ˜c1,2,2=9˜T2\n3−8˜T2˜T4\n8˜T4\n2,\n˜c1,2,3=−˜T3\n4˜T3\n2, ˜c1,2,4=1\n24˜T2\n2, ˜c1,3,1=1\n16˜T8\n2h\n−3402 ˜T5\n3+ 10044 ˜T2˜T3\n3˜T4\n−5220 ˜T2\n2˜T2\n3˜T5+ 432 ˜T2\n2˜T3(−12˜T2\n4+ 5˜T2˜T6) + 80 ˜T3\n2(27˜T4˜T5−7˜T2˜T7)i\n, (4.2)\n˜c1,3,2=3402 ˜T4\n3−7020 ˜T2˜T2\n3˜T4+ 2880 ˜T2\n2˜T3˜T5−720˜T2\n2(−2˜T2\n4+˜T2˜T6)\n48˜T7\n2,\n˜c1,3,3=−126˜T3\n3+ 156 ˜T2˜T3˜T4−40˜T2\n2˜T5\n8˜T6\n2, ˜c1,3,4=177˜T2\n3−60˜T2˜T4\n48˜T5\n2,\n˜c1,3,5=−˜T3\n4˜T4\n2, ˜c1,3,6=1\n72˜T3\n2.\nThe Vandermonde measure is the hallmark of the Hermitian model and plenty of closed-\nformformulaspopulatetheliterature. Thetwo-pointresolventwasconsideredintopological\nexpansion [56] and finite N[57]. We test the solution with the small- λexpansion of (4.3)\nand (4.4) below.\nThe genus-zero and -one resolvents are solvable in closed form [6]. The inverse Laplace\ntransform is worked out in [56], for example: for sixth-order even potentials\nW(x) =2\nµxI1(µx) +3π2\nλ\u0012\n4˜T4+15\n2˜T6µ2\u0013µ3\nxI3(µx) +15π2\n2λ˜T6µ5\nxI5(µx)(4.3)\n+λ\n32π2N2\"\nx2I2(µx)\n3(2˜T2+ 6˜T4µ2+45\n4˜T6µ4)−(4˜T4µ+ 15 ˜T6µ3)xI1(µx)\n(2˜T2+ 6˜T4µ2+45\n4˜T6µ4)2#\n+O(N−4)\nand for general even potentials\nW(x, y) =µxy\n2(x+y)[I0(µx)I1(µy) +I0(µy)I1(µx)] +O(N−2), (4.4)\nwhere Inis the modified Bessel function of the first kind. The coupling dependence enters\nthe branch-cut endpoints [−µ, µ], with µthe solution of the algebraic equation\nRes \nV′\u00001\nz\u0001\nz2p\n1−µ2z2, z= 0!\n=16π2\nλX\np=2,4,...Γ\u0010\np+1\n2\u0011\n√πΓ\u0000p\n2\u0001˜Tpµp= 2. (4.5)\nThe correct solution approximates the Wigner semicircle distribution’s branch point\nµ=1\n2πs\nλ\n˜T2\n1 +∞X\nℓ=1µℓ \nλ\n16π2˜T2\n2!ℓ\n (4.6)\n– 21 –with the corrections\nµ1=−3˜T4, (4.7)\nµ2=1\n2(63˜T2\n4−30˜T2˜T6), (4.8)\nµ3=1\n2(−891˜T3\n4+ 810 ˜T2˜T4˜T6−140˜T2\n2˜T8), (4.9)\nµ4=15\n8(3861 ˜T4\n4−5148 ˜T2˜T2\n4˜T6+ 660 ˜T2\n2˜T2\n6+ 1232 ˜T2\n2˜T4˜T8−168˜T3\n2˜T10).(4.10)\n4.2 Models with Gaussian potential\nThe next-to-simplest models feature the weight\nV(a) =8π2\nλa2(4.11)\nalongside the smooth measure (3.18). We list the non-zero planar coefficients up to λ6\n˜c0,1,2=1\n2, ˜c0,2,2=γ(2)\n4, ˜c0,2,4=1\n12, ˜c0,3,2=6(γ(2))2+ 5γ(4)\n48,\n˜c0,3,4=γ(2)\n12, ˜c0,3,6=1\n144, ˜c0,4,2=18(γ(2))3+ 45γ(2)γ(4)+ 7γ(6)\n288,\n˜c0,4,4=12(γ(2))2+ 7γ(4)\n192, ˜c0,4,6=γ(2)\n96, ˜c0,4,8=1\n2880,\n˜c0,5,2=180(γ(2))4+ 900( γ(2))2γ(4)+ 255( γ(4))2+ 280 γ(2)γ(6)+ 21γ(8)\n5760,\n˜c0,5,4=360(γ(2))3+ 630 γ(2)γ(4)+ 77γ(6)\n8640, ˜c0,5,6=20(γ(2))2+ 9γ(4)\n1920, (4.12)\n˜c0,5,8=γ(2)\n1440, ˜c0,5,10=1\n86400, ˜c0,6,2=1\n172800h\n2700( γ(2))5+ 22500( γ(2))3γ(4)\n+10500( γ(2))2γ(6)+ 4550 γ(4)γ(6)+ 225 γ(2)(85(γ(4))2+ 7γ(8)) + 66 γ(10)i\n,\n˜c0,6,4=450(γ(2))4+ 1575( γ(2))2γ(4)+ 345( γ(4))2+ 385 γ(2)γ(6)+ 24γ(8)\n17280,\n˜c0,6,6=900(γ(2))3+ 1215 γ(2)γ(4)+ 122 γ(6)\n103680, ˜c0,6,8=30(γ(2))2+ 11γ(4)\n34560,\n˜c0,6,10=γ(2)\n34560, ˜c0,6,12=1\n3628800,\ntheN−2-coefficients\n˜c1,2,2=−γ(2)\n4, ˜c1,2,4=1\n24, ˜c1,3,2=−˜c0,3,2, ˜c1,3,4=−γ(2)\n12, ˜c1,3,6=1\n72,\n˜c1,4,2=−˜c0,4,2, ˜c1,4,4=−18(γ(2))2−5γ(4)\n192, ˜c1,4,8=1\n576,\n˜c1,5,2=−36(γ(2))4−180(γ(2))2γ(4)−15(γ(4))2−56γ(2)γ(6)−3γ(8)\n1152,\n– 22 –˜c1,5,4=−126(γ(2))3−135γ(2)γ(4)−7γ(6)\n1728, ˜c1,5,6=−3(γ(2))2+γ(4)\n288, (4.13)\n˜c1,5,8=γ(2)\n576, ˜c1,5,10=1\n8640, ˜c1,6,2=1\n6912h\n−108(γ(2))5−900(γ(2))3γ(4)\n−420(γ(2))2γ(6)+ 70γ(4)γ(6)−45γ(2)(5(γ(4))2+γ(8))i\n,\n˜c1,6,4=−684(γ(2))4−1620( γ(2))2γ(4)+ 15( γ(4))2−224γ(2)γ(6)+ 3γ(8)\n13824,\n˜c1,6,6=−288(γ(2))3−9γ(2)γ(4)+ 35γ(6)\n20736, ˜c1,6,8=12(γ(2))2+ 17γ(4)\n13824,\n˜c1,6,10=7γ(2)\n34560, ˜c1,6,12=1\n207360,\ntheN−4-coefficients\n˜c2,4,4=3(γ(2))2−γ(4)\n96, ˜c2,4,6=−γ(2)\n96, ˜c2,4,8=1\n1920,\n˜c2,5,2=−30(γ(4))2−γ(8)\n960, ˜c2,5,4=90(γ(2))3+ 15γ(2)γ(4)−14γ(6)\n2880,\n˜c2,5,6=−47γ(4)\n5760, ˜c2,5,8=−7γ(2)\n2880, ˜c2,5,10=23\n172800,\n˜c2,6,2=−2250γ(2)(γ(4))2−1050γ(4)γ(6)−75γ(2)γ(8)−11γ(10)\n28800, (4.14)\n˜c2,6,4=540(γ(4))2+ 600( γ(2))2γ(4)−485(γ(4))2−140γ(2)γ(6)−37γ(8)\n23040,\n˜c2,6,6=270(γ(2))3−480γ(2)γ(4)−91γ(6)\n34560, ˜c2,6,8=−210(γ(2))2−77γ(4)\n69120,\n˜c2,6,10=−7γ(2)\n69120, ˜c2,6,12=7\n518400\nand the N−6-coefficients\n˜c3,6,6=45γ(2)(−(γ(2))2+γ(4))−4γ(6)\n17280, ˜c3,6,8=3(γ(2))2−γ(4)\n2304,\n˜c3,6,10=−γ(2)\n7680, ˜c3,6,12=1\n322560. (4.15)\n4.3 Circular Wilson loop\nSupersymmetric localisation [27] calculates the expectation value of the circular Wilson\nloop on S4inN⩾2SYM theories as the matrix-model operator W(2π)in (2.1). Here\none identifies Nandλwith that of the gauge group U(N)and with the ’t Hooft coupling\nrespectively. The potential\nV(a) =8π2\nλa2(4.16)\n– 23 –is generated by the conformal mass of the vector-multiplet scalar in the definition of the\nWilson loop, while the measure comprises the exact one-loop determinants of the field\nfluctuations near the localisation locus.\n4.3.1 N= 4SYM\nThe maximally-symmetric case is captured by the simplest zero-dimensional dynamics with\nµ(a) =a2. (4.17)\nThisfallswithintheapplicabilityrangeofsections4.1and4.2. Hereweset ˜T2= 1,˜Tn⩾3= 0\ndue to (4.16) and γ(n)= 2δn,0due to (4.17). We reproduce the weak-coupling expansion of\n(4.22) and also the N−2-term of (4.23)10below.\nThe planar loop equation is solvable at finite coupling. The Wilson loop appears under\na derivative and a convolution11\nW′(x) =λ\n16π2Zx\n0duW(u)W(x−u), (4.18)\ntherefore it converts to algebraic form by Laplace transform. Since the resolvent (2.6)\ncoincides with ω(s) =Ls(W(x)), (4.18) maps to nothing but the equation for the resolvent\nsω(s)−1 =λ\n16π2ω2(s). (4.19)\nThe correct solution with the decay ω(s)∼1/sat infinity is\nω(s) =8π2\nλ \ns−r\ns2−λ\n4π2!\n(4.20)\nand leads to the famous Wigner semicircle distribution. We transform12\nW(x) =L−1\nx(ω(s)) =4π\nx√\nλI1 \nx√\nλ\n2π!\n(4.21)\nand recover the planar circular Wilson loop [25]\nW(2π) =2√\nλI1(√\nλ). (4.22)\n10An expansion algorithm is presented in [58] and appendix B of [59].\n11W(x)maps to the k-wound circular Wilson loop, with k∈Nanalytically continued to k=x/(2π)∈R.\nUpon a rescaling xandu, the relevant parameter in (4.18) is actually λx2. This matches the gauge-theory\ndependency of the k-wound loop, an example of the coupling-rescaling property of loops on S2[60, 61].\n12The series of (4.20) for s→ ∞matches that of (4.21) for λ→0.\n– 24 –We recall that this gets replaced by [26]13\nW(x) =1\nNeλx2\n32π2NL(1)\nN−1\u0012\n−λx2\n16π2N\u0013\n(4.23)\nat finite N, where L(α)\nnis a generalised Laguerre polynomial.\nIn general, loop equations compute the genus-zero n-point resolvents [12, 62, 63], hence\nthe Wilson loops upon inverse Laplace transforms [56, 64]. High-genus expansion are also\navailable [65] in scaling limits [57]. We reproduce the series of\nW(x, y) =xy√\nλ\n4π(x+y)I0 \nx√\nλ\n2π!\nI1 \ny√\nλ\n2π!\n+xy2λ3/2\n48(2π)3N2I1 \nx√\nλ\n2π!\nI2 \ny√\nλ\n2π!\n+xy2λ2\n192(2 π)4N2X\nk=0,1\"\nyIk \nx√\nλ\n2π!\nIk+2 \ny√\nλ\n2π!\n+x\n2Ik+1 \nx√\nλ\n2π!\nIk+1 \ny√\nλ\n2π!#\n+ (x↔y) +O(N−4). (4.24)\nThe rest of the section is devoted to derive the planar order (4.34) below in a way\nneater, although less general, than [57]. We begin with (2.2)\nW(x, y) =*X\ni,jexaieyaj+\n−N2W(x)W(y). (4.25)\nThe second term is evaluated with (4.23) whereas the first one calls for some ingenuity.\nTaking inspiration from the coincident Wilson loops [26, 57, 59], we represent it as (2.3)\n*X\ni,jexaieyaj+\n=1\nZ+∞Z\n−∞Y\nidaiY\ni<j(ai−aj)2X\ni,jexaieyaje−8π2N\nλP\nia2\ni, (4.26)\nseparate N“diagonal” and N(N−1)“non-diagonal” terms\n1\nZ+∞Z\n−∞Y\nidaiY\ni<j(ai−aj)2h\nNea1(x+y)+N(N−1)ea1x+a2yi\ne−8π2N\nλP\nia2\ni(4.27)\nand rescale ai→λ ai/(8π2N)\nNW(x+y) +N(N−1)\nZ′+∞Z\n−∞Y\nidaiY\ni<j(ai−aj)2eq\nλ\n8π2N(a1x+a2y)e−P\nia2\ni.(4.28)\n13Multiplying this by exp(−λx2\n32π2N2)is the result valid for the SU(N)gauge group.\n– 25 –We apply the method of orthogonal polynomials [26]. The Hermite polynomials\nPn(a) =Hn(a)p\n2nn!√π(4.29)\nare defined by the recurrence relation H0(a) = 1andHn+1(a) = 2 xHn(a)−H′\nn(a). We\nintegrate over the N−2eigenvalues that do not appear explicitly in (4.28)\nNW(x+y) ++∞Z\n−∞dada′N−1X\ni,j=0\u0002\nP2\ni(a)P2\nj(a′)−Pi(a)Pj(a)Pi(a′)Pj(a′)\u0003\neq\nλ\n8π2N(ax+a′y)e−a2−a′2.\n(4.30)\nThe orthogonality relation generalises to the formula in Appendix A of [57]\n+∞Z\n−∞da P i(a)Pj(a)e−\u0012\na−x\n4πq\nλ\n2N\u00132\n=s\nj!\ni!\u0012λx2\n16π2N\u0013i−j\nL(i−j)\nj\u0012\n−λx2\n16π2N\u0013\n.(4.31)\nUsing this and recalling (4.23), (4.30) becomes\neλ(x+y)2\n32π2NL(1)\nN−1\u0012\n−λ(x+y)2\n16π2N\u0013\n+eλ(x2+y2)\n32π2NN−1X\ni=1i−1X\nj=0\u0014\nLi\u0012\n−λx2\n16π2N\u0013\nLj\u0012\n−λy2\n16π2N\u0013\n−j!\ni!\u0012λxy\n16π2N\u0013i−j\nL(i−j)\nj\u0012\n−λx2\n16π2N\u0013\nL(i−j)\nj\u0012\n−λy2\n16π2N\u0013\n+ (x↔y)#\n. (4.32)\nThe large- Nexpansion is hard [65], save for x=y[56] (see section 4.1 of [59])\n*X\ni,jexaiexaj+\n=16π2N2\nλx2\"\nI1 \nx√\nλ\n2π!#2\n+x√\nλ\n4π(4.33)\n×\"\nI0 \nx√\nλ\n2π!\nI1 \nx√\nλ\n2π!\n+1\n6I1 \nx√\nλ\n2π!\nI2 \nx√\nλ\n2π!#\n+O(N−2).\nWhen the dust settles in (4.25), one obtains\nW(x, x) =x√\nλ\n4πI0 \nx√\nλ\n2π!\nI1 \nx√\nλ\n2π!\n+O(N−2). (4.34)\n– 26 –4.3.2 N= 2∗SYM\nThe circular Wilson loop in N= 2∗SYM with hypermultiplet mass mon the sphere S4of\nradius Ris measured by the matrix model W(2π)with14\nµ(a) =a2H2(a)\nH(a−M)H(a+M), H (a) =∞Y\nn=1\u0012\n1 +a2\nn2\u0013n\ne−a2\nn, M =mR.(4.35)\nThe massless/small-radius limit M→0makes the theory flow to the UV theory in section\n4.3.1. The properties of Hare in appendix A of [66].\nSmall-coupling solution. The integral representation15\nγ(a) = 2 + 2 aH′(a)\nH(a)−aH′(a−M)\nH(a−M)−aH′(a+M)\nH(a+M)(4.36)\n= 2 + 4 aZ∞\n0dωsin2(Mω)\nsinh2ωsin(2ωa)\nallows systematic differentiation: we expand\n(sinh ω)−2=∞X\nk=14ke−2kω, (4.37)\nswap x-derivative and ω-integration\n\u0012d\ndx\u0013n\nxsin(2 ωx)\f\f\f\f\nx=0= (−)n\n2+1n(2ω)n−1, (4.38)\nthen integrate and sum over k. The result for n= 2,4, . . .is16\nγ(n)=−2(−)n\n2n!(2ζn−1−ζn−1,1−iM−ζn−1,1+iM−iMζ n,1−iM+iMζ n,1+iM)(4.39)\nin terms of Riemann and Hurwitz zeta functions\nζs=∞X\nk=1k−s, ζ s,a=∞X\nk=0(k+a)−s. (4.40)\nThe result for n= 0complies with the assumption γ(0)= 2.\nA prediction up to N−6andλ6follows from (4.12)-(4.15). We are able to match a\n14Instanton contributions to the measure are dropped because they should become exponentially sup-\npressed at large N. There is also a divergent factor which has no incidence on expectation values and it is\ntherefore removed.\n15The identity holds for real argument and descends from K′′(x)in [66].\n16The case n= 2holds under an operation of limit.\n– 27 –matrix-model prediction [27] and an arduous two-loop calculation in gauge theory [36]\nW(2π)−1\nNeλ\n8NL(1)\nN−1\u0012\n−λ\nN\u0013\n=\u0000\nN2−1\u0001\nλ2\n64π2N2h\nψ(0)(1 +iM) (4.41)\n+ψ(0)(1−iM) +iMψ(1)(1 +iM)−iMψ(1)(1−iM) + 2γi\n+O(λ4),\nwhere ψ(n)(x) =\u0000d\ndx\u0001n+1Γ(x)andγ=−ψ(0)(1)is the Euler’s constant. Further orders\ncan be generated with the Mathematica notebooks.\nLoop equation. We push the solution to very high order thanks to a simpler form of\nthe loop equation in the N= 2∗model\nW′(x) =λ\n16π2Zx\n0ds\u0014\nW(s)W(x−s) +1\nN2W(s, x−s)\u0015\n(4.42)\n−λ\n4π2Z∞\n0dωsin2(Mω)\nsinh2ωIm\u0014\nW(2iω)W(x−2iω) +1\nN2W(2iω, x−2iω)\u0015\n.\nThe proof repeats section 2.2 with the substitution of (2.22) and (2.23) with the single in-\ntegral representation (4.36). The properties W(z∗) = (W(z))∗andW(z∗, w∗) = (W(z, w))∗\nunder complex conjugation pull out the imaginary part Im. The first line is the (finite- N\nversion of the) equation (4.18). In the second line the single integral is due to the mass-\ndependent deformation of the Vandermonde measure (4.36) and offers an edge over the\ngeneral equation (2.26). This could be appreciated in the linear-algebra approach of section\n3.3 or numerically17. Here we explore a shortcut inspired by functional analysis.\nThe equation closes on a single function W(x)in the planar limit. The solution is the\nfixed point\nW(x) =F[W](x),W(0) = 1 (4.43)\nof the functional\nF[g](x) = 1 +λ\n16π2Zx\n0dyZy\n0ds g(s)g(y−s) (4.44)\n−λ\n4π2Zx\n0dyZ∞\n0dωsin2(Mω)\nsinh2ωIm[g(2iω)g(y−2iω)]\nin the space of holomorphic functions on C. The form of the equation restricts the analysis\nto the strip Re(x)∈[0,2π]as noted below (2.26). We construct a sequence of approximants\n{W(n)(x)}n=0,1,...by iteration W(n)=F[W(n−1)]. In perturbation theory we close them on\npolynomials seeded by the normalisation constant W(0)(x) = 1in such a way that the\n17Numerics at strong coupling may benefit from the further step of trading the s-integration with the\nintegral kernel ˆRof (2.29).\n– 28 –fixed-point iteration converges: the approximants are partial sums\nW(n)(x) =nX\nℓ=0Wℓ(x)\u0012λ\n16π2\u0013ℓ\n, (4.45)\ngenerated by\nWℓ(x) =Zx\n0dyZy\n0dsℓ−1X\nℓ′=0Wℓ′(s)Wℓ−ℓ′−1(y−s) (4.46)\n−4Zx\n0dyZ∞\n0dωsin2(Mω)\nsinh2ωℓ−1X\nℓ′=0Im[Wℓ′(2iω)Wℓ−ℓ′−1(y−2iω)].\nEvery approximant is a polynomial of degree ℓinx2. The M-dependence shows up in a\ncomplicated manner through the derivatives (4.39): we perform the ω-integrals as below\n(4.36)\nZ∞\n0dωsin2(Mω)\nsinh2ω(2iω)n=i γ(n+1)\n4(n+ 1), n = 1,3, . . . . (4.47)\nThe imaginary part filters out the cases with even n18.\nSmall- Manalysis. We estimate the radius of convergence of the perturbative solution\nby applying the ratio test to a finite number of its Taylor coefficients in (4.45). We run the\nalgorithm above to collect those with ℓ= 0,1, . . . , 35as exact functions of M.\nAt small Mwe expand19\nWℓ(x) =∞X\nm=0Wℓ,m(x)M2m(4.48)\nand extract the coefficient sets m= 0,1, . . . , 5. The analysis in figure 1 is compatible with\nthe fact that the series at fixed power of M2have a common radius |Rm|, with\n1\nRm=λ\n16π2lim\nℓ→∞Wℓ+1,m(2π)\nWℓ,m(2π)≈ −λ\nπ2, m = 1,2, . . . , 5. (4.49)\nThe ratio means the radius-limiting singularity lies on the negative axis at λ≈ −π2. The\nlack of analytic control on all Taylor coefficients prevents a non-perturbative resummation.\nHowever, the estimated value agrees with the exact critical value λ=−π2that affects the\nfree energy [30, 66], the circular Wilson loop [66] (see (4.51) below) and its phase transi-\ntions in the decompactification limit M→ ∞[32] of N= 2∗SYM. Convergence-limiting\n18A numerical approach to (4.43) and (4.44) could exploit the fast convergence of the exact seed (4.21) at\nzero mass to approximate solutions at small M. Each iteration can analytically continue an approximant\non the real segment x∈[0,2π]to the complex strip Re(x)∈[0,2π]by means of the Cauchy-Riemann\nequations and can perform the ω-integral numerically. The algorithm has the unique advantage to work at\nfinite coupling. However the fixed-point convergence is sensitive to accurate continuations far off the real\naxis.\n19The small-coupling and small-mass limits commute.\n– 29 –0.05 0.10 0.15 0.20 0.25 0.30\n-2.5-2.0-1.5-1.0-0.5 m=0\nm=1\nm=2\nm=3\nm=4\nm=5Figure 1 . Domb-Sykes plot to estimate the radius of convergence of the perturbative solution at\nsmallmass. EachcoloursdenoteasetofTaylorcoefficientsfora M2m-correctiontotheWilsonloop.\nThe points settle on asymptotes (dotted lines) with intercepts r−1\nm=π2/(λRm). The fits of the last\n5 data points give r−1\n2=−0.999,r−1\n2=−1.000,r−1\n3=−0.998,r−1\n4=−0.992andr−1\n5=−0.983.\nThe near-zero value r−1\n0=−0.002detects the infinite radius of the zero-mass solution (4.22). The\ndiscrepancy from 0, due to a finite number of coefficients, may be taken as a rough estimate of the\nerror on the other r−1\nm. Data sets with larger mare bending downwards in the fitting window. This\nfact signals a slower convergence rate and can explain why the estimates of r−1\n4andr−1\n5are slightly\nabove the value −1.\nsingularities are ubiquitous phenomena of perturbative expansions in N= 2theories20,\naffecting the free energy [70], local operators and circular BPS Wilson loops [39, 40, 71–74].\nWe check that the truncated solution at order M2\nM235X\nℓ=0Wℓ,1(2π)\u0012λ\n16π2\u0013ℓ\n(4.50)\nmatches the perturbative expansion of the finite-coupling formula [66]21\nW(2π)−2√\nλI1\u0010√\nλ\u0011\n= 2πM2Z∞\n0dωωJ1\u0010√\nλω\nπ\u0011\n(ω2+π2) sinh2ω(4.51)\n×\"\nπI0(√\nλ)J1 √\nλω\nπ!\n−ωI1(√\nλ)J0 √\nλω\nπ!#\n+O(M4),\n20They are expected for a generic observable in a finite theory [8] including N= 4SYM. Branch-\npoint singularities show up in the single-magnon dispersion relation [67, 68] and the modern quantum\nalgebraic treatment [69] for dimensions of local operators. Convergence properties should be ascribed to\nthe combinatorics of planar graphs and are typically insensitive to the particular observable. The circular\nWilson loop is an exception due to massive diagram cancellations [25], for which the perturbative series has\nan infinite radius of convergence and it is resummed to (4.22). The richer graph content of the observable\ninN= 2∗SYM shifts the radius to a finite value. It may be related to integrability structures yet to be\nfully clarified.\n21Following the ideas in [75], this first term in the Taylor expansion in M2can be related to the integrated\ntwo-point correlator in the N= 4super-Yang-Mills [76, 77].\n– 30 –0.5 1.0 1.5 2.0 2.5 3.0\n-2.×10-132.×10-134.×10-136.×10-13Figure 2 . Analysis of the perturbative solution at finite mass. The Taylor coefficients Wℓ(2π)are\npositive at M= 0, oscillate in a finite region and grow logarithmically to positive infinity. The\nnumber of zeros seems to grow linearly in ℓand their positions to spread over the positive axis.\nNotice two curves are magnified to fit the window.\nwhere InandJnare the Bessel functions of the first kind. To this end we expand (4.51)\nfor small λand integrate\nZ∞\n0dωωn\nsinh2ω=n!\n2n−1ζn n= 2,3, . . . . (4.52)\nThe denominator ω2+π2cancels order by order. The estimated leading singularity agrees\nwith the logarithmic branch point λ=−π2that affects (4.51).\nFinite- Manalysis. The study is hindered by the slow rate of convergence of the\nsolution. Evidence of this trend is appreciated in figure 1. The behaviour of the coefficients,\nsome of them in figure 2, suggests that they are positive for ℓ⩽ℓ∗(M)and alternate in sign\nbeyond a certain order ℓ > ℓ∗(M). The small number of extracted coefficients is insufficient\nto claim that there exists a tail without consecutive sign repetitions22, or to pinpoint a\nradius of convergence by root test applied to such tail. We remind that one expects the\nradius to be a function of Mon general grounds.\nLarge- Manalysis. The Taylor coefficients in (4.45) expand in logMandM−1:\nW0(2π) = 1 ,W1(2π) = 2 π2,\nWℓ(2π) = 2 π2(4 log M)ℓ−1+ℓ−2X\nℓ′=0cℓ′(logM)ℓ′+o(M0) ℓ= 2,3, . . . , (4.53)\nhence the series breaks down by the ratio test. At exponentially-large Mnew singularities\nshould settle in on the positive λ-axis [32]23, although their arise is screened off by the\nnon-commutativity of the limits λ→0andM→ ∞.\n22Aconvergence-limitingsingularityonthenegative λ-axiswouldimplysuchbehaviour. Onlytheinfinite-\nmass limit allows the Wilson loop to undergo phase transitions on the positive axis [32].\n23See figure 2 in [66].\n– 31 –The structure of (4.53) is a corollary of the measure γ(2)= 8 log( eγ+1M) +O(M−1)\nandγ(n)=O(M0)forn̸= 2, and of the solution Wℓ(x), which is made of products of γ(n)\nwith the leading term 2π2(γ(2)/2)ℓ−1. Physically (4.53) connects to the renormalisation-\ngroup properties of N= 2∗SYM [27, 30, 66]. In fact the infinitely-heavy matter can be\nintegrated out, leaving N= 2SYM with one vector multiplet to the leading approximation.\nThe “kinematic” scale Msets the dynamically-generated scale ΛR=Mexp(−4π2/λ+γ+1)\nand the coupling\nλR=λ\n1−βλlog(eγ+1M), (4.54)\nwhere β= (4π2)−1is basically the numerical factor in the beta function of N= 2SYM [27].\nIf one insists on taking λ→0−and later M→ ∞, the Wilson loop admits an expansion\nsimilar to (4.45)\nW(2π) =∞X\nℓ=0Cℓ\u0012λR\n16π2\u0013ℓ\ninN= 2SYM , (4.55)\nwith C0= 1andC1= 2π224, in the renormalised coupling\nλR=∞X\nℓ′=0λℓ′+1\u0000\nβlog(eγ+1M)\u0001ℓ′\n. (4.56)\nReplacing (4.56) into (4.55), the coefficient of (λ/(16π2))ℓreads\nC1\u0000\n16π2βlogM\u0001ℓ−1(4.57)\nand reproduces what observed in (4.53).\nLarge-coupling resummation. Finally we extrapolate the perturbative series be-\nyond the convergence radius by means of Padé resummation [78, 79] in the spirit of\n[39, 71, 72, 74, 80] and compare to independent analytic results at strong coupling. This\nsets an important test of the solution. The Wilson loop was calculated by saddle-point\ntechniques from the matrix model:\nlogW(2π) = 2 πµ+ log\"\n23/2\nπµ3/2\u0012\nˆg(2πi) +1\n25/2π\u0013#\n+o(λ0). (4.58)\nThe leading order is that of the Gaussian model with mass-rescaled coupling [31]\nµ=p\nλ(1 +M2)\n2π(4.59)\n24They match those (4.53) in the massive theory. The first coefficient is the normalisation. The one-loop\nterm is mass independent because in gauge theory it stems solely from the free gluon propagator.\n– 32 –5 10 15\n-4-2246810Figure 3 . Padé approximation of the perturbative solution for M= 0.5. The polynomials (4.61)\nof degrees n= 34andn= 35(blue curves) are indistinguishable in the interval λ≲1and diverge\natλ≈1near the radius of convergence. The blow-up with opposite signs of even- and odd-degree\npolynomials diagnoses a singularity on the negative λ-axis. The Padé approximants (4.62) (orange\ncurves) reconstruct the polynomials up to λ≲1, extend them beyond the perturbative region and\nstabilise around the asymptote (4.58) (green curve). They become unreliable at larger λwhere\nthey asymptote towards horizontal lines (not shown). Round-off errors in the evaluation of the\npolynomials can impair the precision of Padé approximants before reaching the plateaus.\nand the O(1)-correction is the solution of a variant of the Wiener-Hopf problem [33]\nˆg(ω) =i3/2√π\n2ω√\nω+i0+\"\nM2sinh2ω\n2−sin2Mω\n2\nsinh2ω\n2+ sin2Mω\n2+ (M2+ 1)2ωe−iϕω\n2πΓ\u0000M−i\n2πω\u0001\nΓ\u0000\n−M+i\n2πω\u0001\nΓ2\u0000\n−iω\n2π\u0001\n×∞X\nn=1(−)n\nn n!\neiϕn\nM−i\nω−2πn\nM−iΓ\u0010\nM+i\nM−in\u0011\nΓ2\u0010\ni\nM−in\u0011+e−iϕn\nM+i\nω+2πn\nM+iΓ\u0010\nM−i\nM+in\u0011\nΓ2\u0010\n−i\nM+in\u0011\n\n, (4.60)\nϕ= 2Marctan M−log(M2+ 1).\nThe Wilson loop diverges exponentially, hence our analysis should consider the logarithm\nof a truncated solution (4.45)\nlogW(n)(2π) =nX\nℓ=1cℓ\u0012λ\n16π2\u0013ℓ\n+O(λn+1). (4.61)\nForexampleonemeasures c1= 2π2andc2=π2γ(2)−2π4/3withγ(2)in(4.39)25. Thecutoff\nn= 35suffices to deliver good results at moderate values M≲1. When the data is limited\n25The authors of [81] employ a trick to go from the first two terms to large coupling in the case of the\ncusp anomaly of the light-like Wilson cusp. Here it would find the constants C1(M)andC2(M)such that\nC1(M) logW(2)(2π) +C2(M)[logW(2)(2π)]2=λ, regard this as an exactalgebraic equation and invert at\nlarge coupling. Note that such solution would have the same expansion in λ−1/2of the exact formula (4.58).\nHowever the coarse numerical agreement deteriorates when Mnears the zero of C2(M).\n– 33 –by a finite number of coefficients of (4.61), it is useful to define the Padé approximants26\nP[M/K ](λ) =\"35X\nℓ=1cℓ\u0012λ\n16π2\u0013ℓ#\n[M/K ]. (4.62)\nThe square-root scaling and the expansion in fractional powers of (4.58) force a choice\nbetween P[K/K ](λ) =O(λ0)andP[K+1/K](λ) =O(λ). As an example, figure 3 shows that\ndiagonal approximants are an excellent interpolation between the perturbative solution\n(4.61) at λ≲1and the prediction (4.58) in a wide non-perturbative region. In particular,\ntheybetterapproximate(4.58)forincreasingPadédegree Katfixed λ27. Precisenumerical\nresults and the type of the singularity could be investigated by series accelerations and a\nconformal map [72].\n4.4 Hoppe model\nWe also cover the model put forward and solved in the planar limit by [20], with\nµ(a) =a2\na2+ 1. (4.63)\nThe Gaussian case V(a) =a2/(8π2λ)was revisited via a saddle-point integral equation\nfor the planar distribution of the matrix eigenvalues [21]. Our result is (4.12)-(4.15) with\nγ(n)= 2(−)n\n2n!and the values of ˜cn,ℓ,mmultiplied by (8π2)2ℓin order to compensate for\n˜T2= (8π2)−2̸= 1. The non-zero N0-coefficients read up to λ6\n˜c0,1,2= 32π4, ˜c0,2,2=−4096π8, ˜c0,2,4=1024π8\n3, ˜c0,3,2= 1835008 π12,\n˜c0,3,4=−262144 π12\n3, ˜c0,3,6=16384 π12\n9, ˜c0,4,2=−1157627904 π16,\n˜c0,4,4= 46137344 π16, ˜c0,4,6=−2097152 π16\n3, ˜c0,4,8=262144 π16\n45, (4.64)\n˜c0,5,2=−863288426496 π20, ˜c0,5,4=−31675383808 π20, ˜c0,5,6=6308233216 π20\n15,\n˜c0,5,8=−134217728 π20\n45, ˜c0,5,10=8388608 π20\n675, ˜c0,6,2=−712483534798848 π24,\n˜c0,6,4= 24945170055168 π24, ˜c0,6,6=−309237645312 π24, ˜c0,6,8=30064771072 π24\n15,\n˜c0,6,10=−1073741824 π24\n135, ˜c0,6,12=268435456 π24\n14175,\n26Thedegreeof(4.61)putsthebound M+K⩽nonthedegreesofthenumerator Mandthedenominator\nKof the rational approximation.\n27Theyeventuallyfallshortofprovidingasymptoticexpressionsduetothewrongasymptotics, seecaption\nof figure 3.\n– 34 –theN−2-coefficients\n˜c1,2,2= 4096 π8, ˜c1,2,4=512π8\n3, ˜c1,3,2=���1835008 π12, ˜c1,3,4=262144 π12\n3,\n˜c1,3,6=32768 π12\n9, ˜c1,4,2= 1157627904 π16, ˜c1,4,4=−46137344 π16,\n˜c1,4,8=262144 π16\n9, ˜c1,5,2=−695784701952 π20, ˜c1,5,4= 27380416512 π20,(4.65)\n˜c1,5,8=−67108864 π20\n9, ˜c1,5,10=16777216 π20\n135,˜c1,6,2= 241617680203776 π24,\n˜c1,6,4=−12094627905536 π24, ˜c1,6,6=−300647710720 π24\n3,\n˜c1,6,8=15032385536 π24\n3, ˜c1,6,10=−7516192768 π24\n135, ˜c1,6,12=134217728 π24\n405,\ntheN−4-coefficients\n˜c2,4,6=2097152 π16\n3, ˜c2,4,8=131072 π16\n15, ˜c2,5,2=−167503724544 π20,\n˜c2,5,4= 4294967296 π20, ˜c2,5,6=−6308233216 π20\n15, ˜c2,5,8=469762048 π20\n45,\n˜c2,5,10=96468992 π20\n675, ˜c2,6,2= 470865854595072 π24, (4.66)\n˜c2,6,4=−12850542149632 π24, ˜c2,6,6=1228360646656 π24\n3,\n˜c2,6,8=−105226698752 π24\n15, ˜c2,6,10=3758096384 π24\n135, ˜c2,6,12=1879048192 π24\n2025\nand the N−6-coefficients\n˜c3,6,10=536870912 π24\n15, ˜c3,6,12=67108864 π24\n315. (4.67)\n5 Strong coupling: solution\nWe now turn to strong coupling. In what follows we develop a method to systematically\nsolve the loop equation in that regime. This is particularly important for localisation\nmatrix models which often have dual holographic description that becomes simple at strong\ncoupling.\nTo discuss the strong-coupling limit we redefine the potential as\nV(a) =ℏ2ν−2U(ℏa). (5.1)\nIf the potential is quadratic and ν= 1, the new parameter ℏis literally the inverse of the\ncoupling. Indeed, for V(a) =a2/2g2the rescaled potential is U(x) =x2/2, andℏ= 1/g.\nThe anharmonic terms, if present, must follow a particular pattern in the ℏ→0limit.\nIn the general parameterisation (2.4), the individual couplings Tnscale as ℏn, while the\n– 35 –parameter νdefines the overall scaling weight of the potential. We show later that this\nparameter is not arbitrary and is dictated by the behaviour of the integration measure µ(a)\nat large a, in order for the strong-coupling limit to be well-defined.\nThe strong-coupling regime corresponds to ℏ→0. When the potential is quadratic,\nexpanding in ℏis equivalent to expanding in 1/g, exactly opposite to the weak-coupling\nexpansion considered so far. For example, in the N= 2∗SYM theory or any other model\nrelevant for AdS/CFT, V(a) = 8 π2a2/λand thus ℏ= 4π/√\nλ. This coupling coincides with\nthe natural expansion parameter in the dual string theory, making the strong-coupling\nexpansion in the matrix model equivalent to the weak-coupling expansion on the string\nworldsheet.\nWe develop a systematic procedure to generate expansion of the Wilson loop in ℏfor\nthe potential of the form (5.1) and an arbitrary measure. Our motivation comes from\nlocalisation and string theory, but the method is completely general and we expect it to\nhave other applications.\nThe form of the potential (5.1) suggests to rescale the integration variables ai→ai/ℏ.\nThis tacitly assumes that typical ai’s in the eigenvalue integral are very large, of order 1/ℏ.\nThe measure µ(a), it may seem, can be replaced by itslarge-argument asymptotics and then\nexpanded in 1/ato generate the next orders of the expansion. The loop equations, as we\nshall see, do justify this simple reasoning up to a small but important caveat. The measure\ndepends on the eigenvalue difference and the typical distance between nearby eigenvalues\nmaybeoforderoneorsmallerevenifalleigenvaluesarelarge. Thisisespeciallytrueatlarge\nNwhen the eigenvalues form a continuous distribution and can be found arbitrary close to\none another. More care is thus needed, and our goal is to set up a systematic procedure to\ndevelop the strong-coupling expansion that would take this effect into account.\nFor the Wilson loop, the two regimes have to be distinguished at strong coupling:\nshort loops W(x)with x∼ℏand long loops with x∼1. The naive strong-coupling\napproximation holds for short loops, while for long loops (more interesting for applications)\nnearby eigenvalues start to become important and the answer depends on the measure in a\nnon-trivial way. The two regimes match in the overlapping region of validity 1≫x≫ℏ.\n5.1 Short loops\nThe natural variable for short loops is x/ℏ. The integration variables in the loop equation,\nbe it (2.26) or (2.29), need to be rescaled accordingly. The kernels can then be replaced by\ntheir small-argument asymptotics (in the Fourier space) or the large-argument ones in the\noriginal a-variables. To proceed further we need to know how the measure, equivalently the\nkernel in the loop equation, behaves in this limit. We will assume a power-like asymptotics\n(cf. (2.30)), which covers a large class of physically interesting eigenvalue models:\nˆR(ω)ω→0≃ − πβsignω|ω|2ν−2≡ˆR∞(ω). (5.2)\nAll eigenvalue integrals with measure growing not faster than polynomially fall into this\nclass. Indeed, for\nµ(a)a→∞≃C a2β(5.3)\n– 36 –andβ >0,\nˆγ(ω)ω→0≃4πβδ(ω), ˆR(ω)ω→0=−πβsignω, (5.4)\nwhich in our notations corresponds to ν= 1. This behavior is found for example in the\nN= 2∗localisation integral. In the Hoppe model the kernel has a power-like asymptotics\nwith an exponent different from one. We will discuss the Hoppe model in greater detail\nlater.\nDenoting by W∞the leading-order strong-coupling approximation for short loops, we\narrive at the equation\niℏ2ν−1\n2U′\u0012\n−iℏ∂\n∂κ\u0013\nW∞(κ) =+∞Z\n−∞dω\n2πˆR∞(ω)W∞(κ−ω)W∞(ω), (5.5)\nin which all the ℏ-dependence can be absorbed into rescaling κ→ℏκ. Notice that for this\nto happen the exponents in the kernel (5.2) and in the potential (5.1) must agree with one\nanother. Upon the rescaling we get the loop equation for the ν-model from sec. 2.3.\nThe solution (analytically continued to complex κ) is\nW∞(x) =X\nn⩾0AnIν+n\u0000µx\nℏ\u0001\n\u0000µx\nℏ\u0001ν+n. (5.6)\nwith\nAn=µ2ν−1(2n+ 1)!\n4πβΓ(2ν−1)1Z\n−1da(1−a2)ν−n−3\n2D(ν)\nn(a)\u0012\na∂\n∂a+ 2ν−1\u0013\nU′(µa),(5.7)\nwhich differs from (2.48) by an extra factor of βfrom the kernel in (5.4). The parameter\nµis to be fixed by normalisation, which we will discuss later, after taking into account the\nnext correction in ℏ.\n5.2 Long loops\nWe can get an insight into the behaviour of long loops by considering a short loop with\nx≫ℏ, but x≪1. Approximations used in deriving (5.6) then still apply, and we can just\ntake the x→ ∞limit of that expression. All the Bessel functions have the same exponential\nasymptotics, taking into account the power-like prefactor we can see that the term with\nn= 0is the largest one, and we get:\nW∞(x)1≫x≫ℏ≃A0√\n2π\u0012ℏ\nµx\u0013ν+1\n2\neµx\nℏ. (5.8)\n– 37 –The constant A0, given by (5.7), can be simplified taking into account the explicit form of\nthe first Gegenbauer polynomial (2.36) and integrating by parts:\nA0=µ2\n2ν√π βΓ\u0000\nν−1\n2\u0001µZ\n−µda\u0000\nµ2−a2\u0001ν−3\n2U′′(a). (5.9)\nFor yet larger xof order one the short-loop approximation can no longer be trusted,\nbut it is natural to expect that exponential dependence on xwill persist. This suggests an\nansatz:\nW(x) =A0ℏν+1\n2\n√\n2π µν+1\n2eµx\nℏ f(x) (5.10)\nwith some unknown function f(x). Matching to the short-loop solution we find:\nf(x)x→0≃x−ν−1\n2, (5.11)\nwhich will serve as a boundary condition for the equation we are going to derive.\nAccordingly, the oscillating Wilson loop (2.28) behaves as\nW(κ) =A0ℏν+1\n2\n√\n2π µν+1\n2eiµκ\nℏf(κ), (5.12)\nwhere f(κ)and f(x)are related by analytic continuation κ→ −ix. An important remark\nis in order here. For real κthe Wilson loop no longer grows but oscillates and this gives rise\nto subtleties with the analytic continuation. An instructive example is the Bessel function.\nThe modified Bessel function grows exponentially:\nIν(x)x→∞≃1√\n2πxex, (5.13)\nwhile its analytic continuation, the ordinary Bessel function asymptotes to\nJν(κ)κ→∞≃r\n2\nπκcos\u0010\nκ−πν\n2−π\n4\u0011\n, (5.14)\nand contains oscillating exponentials of both signs, so the analytic continuation and the\nlimit of the argument going to infinity do not commute. To suppress the exponential with\nthe wrong sign, and to continue using the single exponent in (5.12), we need to give κa\n(small) negative imaginary part.\nThis example shows that an analytic continuation of f(κ)is only well-defined in the\nlower half-plane of κ, where it coincides with f(x):\nf(−ix) = f(x), Rex >0. (5.15)\nThe function f(κ)can be analytically extended to the real line and to the upper half-plane,\nbut it need not coincide there with the ℏ→0limit of the exact Wilson loop. The latter is\n– 38 –well-defined for any value of the argument, while an analytic continuation of f(κ)into the\nupper half-plane may hit singularities of various kinds.\nOne singularity is actually prescribed by the boundary condition (5.11), according to\nwhich f(κ)has a branch point at zero, shifted slightly into the upper half-plane:\nf(κ)κ→0≃i−ν−1\n2\n(κ−iϵ)ν+1\n2≡f∞(κ). (5.16)\nThe branch of (κ−iϵ)ν+1\n2is defined with a cut along the positive imaginary semi-axis.\nOther singularities may hide further away from the origin.\nKeeping this in mind, we may substitute (5.12) into the loop equation (2.29). As\nemphasisedearlyon[2], anexponentialansatzalwaysgoesthrough. Indeed, theexponential\nfactors re-combine inside the integral and cancel between the two sides so long as the\nexponent is linear in κ. Another simplification occurs because the exponent is large at\nℏ→0, bearing certain similarity to the semi-classical wavefunction in quantum mechanics.\nThe action of the differential operator V′(−i∂/∂κ ) =ℏ2ν−1U′(−iℏ∂/∂κ )on the Wilson\nloop can then be replaced with multiplication by a constant ℏ2ν−1U′(µ), to the leading\norder in ℏ.\nAt a first sight this leads to problems. The power counting does not seem to match.\nThe left-hand side is proportional to ℏ2ν−1×ℏν+1\n2=ℏ3ν−1\n2, while the right-hand side\nscales as ℏν+1\n2×ℏν+1\n2=ℏ2ν+1. Next, the integral over ωdiverges at zero for ν⩽3/2as a\nconsequence of (5.16).\nThe two problems are not unrelated. The integral includes not only ω∼1, but also\nω∼ℏwhere the ansatz (5.16) is not applicable. The short-loop contribution is proportional\ntoℏ2ν−1×ℏν+1\n2=ℏ3ν−1\n2, a factor of ℏ3\n2−νbigger than the contribution of long loops\n(proportional to ℏ2ν+1). To extract the short-loop contribution we can use the solution\n(5.6) inside the integral for the small range of ω∼ℏ, but first it is necessary to isolate this\nleading term in the equation.\nThis can be done by the following trick. In the dangerous region of small ωthe Wilson\nloop is well approximated by W∞and the kernel by ˆR∞(ω). Adding and subtracting those\nsplits the integral into two parts:\niℏ2ν−1\n2U′(µ)W(κ) =+∞Z\n−∞dω\n2πh\u0010\nˆR(ω)−ˆR∞(ω)\u0011\nW(κ−ω)W(ω)\n+ˆR∞(ω)W(κ−ω)\u0010\nW(ω)−W∞(ω)\u0011i\n++∞Z\n−∞dω\n2πW(κ−ω)ˆR∞(ω)W∞(ω).(5.17)\nIn the last term the main contribution comes from ω∼ℏandW(κ−ω)there can be\n– 39 –replaced by W(κ)(forκ∼1). The remainder, one can show, integrates to28\n+∞Z\n−∞dω\n2πˆR∞(ω)W∞(ω) =iℏ2ν−1\n2U′(µ), (5.18)\nand cancels the left-hand side.\nThe integral in the middle of (5.17) converges at ω= 0and thus receives no contribu-\ntion from short loops. The exact Wilson loop can now be replaced by its strong-coupling\nasymptotics (5.12):\n+∞Z\n−∞dω\n2πf(κ−ω)\u0010\nˆR(ω)f(ω)−ˆR∞(ω)f∞(ω)\u0011\n= 0. (5.19)\nThis non-linear integral equation on the scaling function f(κ)describes the strong-coupling\nbehaviour of the Wilson loop for κ∼1.\nOur approach to solving this equation is inspired by the Wiener-Hopf method, in the\nform it was applied to matrix models in [33, 38, 82, 83]. Our results are actually equivalent\nto the Wiener-Hopf solution of N= 2∗super-Yang-Mills [33], given by (4.60), even if our\nderivation is different and also applicable to any generalised eigenvalue model.\nThe key point is the Riemann-Hilbert decomposition of the kernel:\nˆR(ω) =G+(ω)\nG−(ω), (5.20)\nwhere G+(ω)admits an analytic continuation into the upper half-plane and G−(ω)into the\nlower one. This representation exists under very general assumptions and is unique up to\nsimultaneous rescaling of G+andG−by the same constant.\nThe equation (5.19), as it stands, is defined on the real line. The idea is to deform\nthe contour of integration into the complex plane, more precisely into its upper half where\nf(κ−ω)isawelldefinedfunctionof ωforafixedreal κ. Theanalyticformoftheasymptotic\nkernel is\nˆR∞(ω) =−πβ(ω+iϵ)ν−3\n2(ω−iϵ)ν−1\n2, (5.21)\nwhere the two factors have cuts along the imaginary axis in the opposite directions. The\ncomplex function so defined indeed coincides with the real-valued function (5.2) once the\nargument is restricted to the positive half of the real line, while in the complex plane it has\ntwo cuts along the positive and negative imaginary semi-axes. To extend this function to\nnegative semi-axis we need to pass between the two branch points at ω=±iϵ. The phases\nof the two factors rotate in the opposite direction, because their cuts extend in the opposite\nhalf-planes, producing an overall minus sign. This is the origin of the signωfactor in (5.2).\nThe Wiener-Hopf decomposition of the asymptotic kernel, obvious from (5.21), defines\n28This formula can be proven by sending κ→(1−iϵ)∞in the equation (5.5), using (5.8) and replacing\nV′(−iℏ∂/∂κ )withℏ2ν−1U′(µ). It can be also derived by direct integration of the exact solution (5.6).\n– 40 –Figure 4 . Closing the contour of integration in the upper half-plane.\nthe boundary conditions for the exact Wiener-Hopf factors:\nG+(ω)ω→0≃ − πβ(ω+iϵ)ν−3\n2 (5.22)\nG−(ω)ω→0≃(ω−iϵ)1\n2−ν. (5.23)\nCommon normalisation here is a matter of convention. Being made, it fixes the rescaling\nambiguity of the Riemann-Hilbert decomposition.\nWith these data at hand we can proceed with the analytic continuation of (5.19).\nTaking f∞from (5.16) and denoting f/G−=P, we can write the integral equation as\n+∞Z\n−∞dω\n2πf(κ−ω)\"\nG+(ω)P(ω) +πβi−ν−1\n2(ω+iϵ)ν−3\n2\nω−iϵ#\n= 0, (5.24)\nwith Pbeing the new unknown function. Importantly, this function is analytic in the lower\nhalf-plane.\nThe next step is a contour deformation trick. If the contour shown in figure 4 encircles\nnosingularitiesoftheintegrand, theintegralwillautomaticallybezero, andwewillbedone.\nTo achieve this we first notice that the function f(κ−ω)is analytic in the upper half-plane\nofω, and so is G+(ω). There are two possible obstructions to closing the contour. One is\nthe pole at ω=iϵ. Another is possible singularities of P(ω)in the upper half-plane. We are\nin principle agnostic about the latter, however, any singularity of Pthat is not localised at\nω=iεcannot be cancelled by the second term. Barring no accidental cancellations occur,\nsuch singularities better not be there and P(ω)better have a pole at ω=iϵwith no other\nsingularities whatsoever. The problem reduces to matching the residues such that the pole\ncancels in the sum. The integrand in (5.24) is then analytic in the upper half-plane, the\ncontour can be closed and the integral will be zero as requested.\nTaking into account (5.22), the residues will match provided\nP(ω) =i−ν−1\n2\nω−iϵ. (5.25)\n– 41 –We thus find:\nf(κ) =i−ν−1\n2G−(κ)\nκ−iϵ. (5.26)\nThis function is manifestly analytic in the lower half-plane, has the right asymptotics (5.16)\nin virtue of (5.23) and solves (5.19) by the above contour-deformation argument.\nFor the Wilson loop, as defined in (5.10) and (5.12), we get:\nW(x) =i1\n2−νA0ℏν+1\n2\n√\n2π µν+1\n2xeµx\nℏG−(−ix). (5.27)\nThis is our final result. The constant A0is given by (5.9) and G−is determined by the\nRiemann-Hilbert factorisation of the kernel. To close the circuit we need to find µ, which\ncan be done by imposing the normalisation condition W(0) = 1. The asymptotic solution\n(5.27) is unsuitable for this purpose, since it applies to long loops with x≫ℏand cannot\nbe continued down to x= 0. We need to use short loops, but the solution (5.6) is not\nsufficient. Since µenters (5.27) in the ratio µ/ℏit has to be known to O(ℏ). Hence we\nneed to know the short-loop solution to the next order in the strong-coupling expansion.\nFortunately, the first-order corrections are simple and can be easily taken into account.\n5.2.1 Shift of the spectral endpoint\nTo find the short-loop solution (5.6) we replaced the kernel by its asymptotic form (5.2)\nand rescaled ω→ℏω. Since ˆR(ω)is an odd function, see (2.15), at small ω,\nˆR(ω) =ˆR∞(ω)\u0000\n1 +O(ω2)\u0001\n. (5.28)\nThe correction term is of relative order ω2→ℏ2ω2, and can be neglected as long as we are\ninterested in effects linear in ℏ.\nAt the same time, the Riemann-Hilbert factors G±(ω)are neither even nor odd and,\nin general,\nG−(ω)ω→0≃(ω−iϵ)1\n2−ν(1 +iγω+. . .). (5.29)\nThe linear term then affects the solution at O(ℏ).\nThe solution for long loops has to match with (5.8) at small x. Taking into account\nthe first correction, that is, expanding the already known solution (5.27) in x, we find:\nW∞(x)1≫x≫ℏ≃A0√\n2π\u0012ℏ\nµx\u0013ν+1\n2\neµx\nℏ(1 +γx+. . .). (5.30)\nMatching it to the general expression for short loops (5.6) requires to include the n=−1\nterm, absent in the zeroth-order solution, with the coefficient\nA−1=ℏγ\nµA0. (5.31)\n– 42 –The solution for short loops accurate through O(ℏ)is thus\nW∞(x) =X\nn⩾0AnIν+n\u0000µx\nℏ\u0001\n\u0000µx\nℏ\u0001ν+n+ℏγ\nµA0Iν−1\u0000µx\nℏ\u0001\n\u0000µx\nℏ\u0001ν−1. (5.32)\nIt may seem odd that the equation for short loops has not changed at O(ℏ)but the\nsolution does receive an order ℏcorrection. There is no contradiction, however. The reason\nis that the key operatorial identity (2.38) simply gives zero for n <0, according to (2.34).\nAny term with n <0can thus be added without affecting the equation. They cannot arise\nat the zeroth order, as they compromise the boundary conditions at x→ ∞, but at higher\norders they are allowed and do occur by the matching argument above.\nThe normalisation condition W(0) = 1results in\nX\nn⩾0An\n2ν+nΓ(ν+n+ 1)+ℏγA0\n2ν−1µΓ(ν)= 1, (5.33)\nwith an extra correction term compared to (2.50). This equation determines µto the O(ℏ)\naccuracy and thus completes the solution for the Wilson loop at the two first orders in the\nstrong-coupling expansion.\nTo conclude, the solution for the Wilson loop is given by (5.27) where µsi determined\nby (5.33) and the constants Anare taken from (5.7).\nFor the Gaussian model with U(a) =a2/2,\nA0=µ2ν\n2νΓ(ν)β, (5.34)\nall other An= 0and\nµ= 2\u0012\nΓ(ν)Γ(ν+ 1)β\u00131\n2ν\n−ℏγ (5.35)\nto the first order in ℏ. We checked that these equations along with (5.27) reproduce the first\ntwo orders of the strong-coupling expansion in the N= 2∗SYM theory [33]. To further\nillustrate our method we consider the strong-coupling expansion in the Hoppe model.\n6 Strong coupling: Hoppe model\nThe Hoppe model is defined by the Gaussian potential V(a) =a2/2g2and the measure29\nµ(a) =a2\na2+ 1. (6.1)\nThe kernel of the loop equation is determined by the log-derivative of the measure.\nR(a) =1\na−a\na2+ 1, (6.2)\n29A more general measure µ(a) =a2/(a2+m2)does not introduce new parameters, as mcan be scaled\naway by the change of variables a→amwith subsequent redefinition of the coupling.\n– 43 –which gives, upon Fourier transform:\nˆR(ω) =−2π e−|ω|\n2sinhω\n2. (6.3)\nIn contradistinction to (5.4), the kernel vanishes at ω= 0:\nˆR(ω)ω→0≃ − πω≡ˆR∞(ω), (6.4)\nand has a non-trivial index and a slope\nν=3\n2, β = 1. (6.5)\nThe same index controls the scaling of the potential and thus defines the parameter of the\nstrong-coupling expansion. Comparing the potential V(a) =a2/2g2to the general scaling\nform (5.1) we find:\nℏ=g−2/3. (6.6)\nWe thus come to a conclusion, not completely obvious from the model’s definition, that the\nstrong-coupling expansion goes in powers of g−2/3.\n6.1 Short loops\nFor short Wilson loop we can use the solution of the ν-model with the asymptotic kernel,\nwhose explicit form for the Gaussian potential is given by (5.6) with the constants A0and\nµin (5.34), (5.35). Ignoring, for the moment, the O(ℏ)shift of the endpoint, we get:\nµ= (3π)1\n3, A 0= 3rπ\n2, (6.7)\nand\nW∞(x) =r\n3\n2x2I3\n2\u0010\n(3πg2)1\n3x\u0011\n=3\ny3(ycoshy−sinhy)\f\f\f\f\ny=(3πg2)1\n3x.(6.8)\nThis corresponds to the eigenvalue density\nρ(a) =3\n4B3(B2−a2), B = (3πg2)1\n3. (6.9)\ndefined on the interval from −BtoB. The density can be found from (2.52), with A0taken\nfrom (6.7) and µreplaced by Bto account for the rescaling µ→µ/ℏ=µg2/3=Bin (5.6).\nThese results are valid for x∼g−2/3. For longer loops the formulas from section 5.2 should\nbe used instead.\n6.2 Long loops\nThe kernel (6.3) can be readily factorised with the help of an analytic representation of |ω|:\n|ω|=iω\nπ\u0012\nlogω−iϵ\n2π−logω+iϵ\n2π\u0013\n+ω, (6.10)\n– 44 –along with representation of sinhas a product of two gamma-functions. That gives:\nG+(ω) =−π\nΓ\u0000\n1−iω\n2π\u0001e−ω\n4−iω\n2π(logω+iϵ\n2π−1),\nG−(ω) =1\nω−iϵΓ\u0012\n1 +iω\n2π\u0013\neω\n4−iω\n2π(logω−iϵ\n2π−1). (6.11)\nThe solution derived in section 5.2 is expressed in terms of G−, and in addition contains\nanO(ℏ)shift of µ. The latter is determined by a constant that appears in (5.29), the Taylor\nexpansion of G−(ω)at small ω. In the Hoppe model that expansion is contaminated by\nlogarithms:\nG−(ω)ω→0≃1\nω−iϵ\u0012\n1−iω\n2πlogω+. . .\u0013\n. (6.12)\nThis does not have exactly the same form as (5.29), but if we recall that the small- ωexpan-\nsion is matched to the short-loop solution at ω∼ℏ, the log term is essentially equivalent\nto setting\nγ=−1\n2πlogℏ. (6.13)\nThis determines the endpoint shift with the logarithmic accuracy30:\nµ= (3π)1\n3+1\n2πℏlogℏ. (6.14)\nThe next, O(ℏ)term cannot be determined by this simple reasoning.\nTaking this all into account we find from (5.27):\nW(x) =\u00123\n8π2g4x6\u00131\n3\nΓ\u0010\n1 +x\n2π\u0011\neh\n(3πg2)1\n3−1\n6πlog(Cg2x3)i\nx. (6.15)\nThe numerical constant Ccannot be determined within this scheme. Finding it requires a\nmore refined analysis which goes beyond the leading-log approximation.\nIt is no more difficult to repeat the same analysis for an arbitrary even potential, not\nnecessarily Gaussian. We concentrated on the simplest quadratic case because the Gaussian\nmodel is exactly solvable and the results can be directly confronted with the strong-coupling\nasymptotics of the exact solution.\n6.3 Comparison to exact solution\nThe exact solution of the Hoppe model is formulated in terms of a resolvent function G(z)\nwith the following properties31:\n(i) The resolvent is an even function: G(−z) =G(z).\n(ii) It is analytic in the complex plane with two cuts (−B±i/2, B±i/2)(figure 5).\n30The expansion of the full kernel (6.3) also does not match the generic form (5.28), because it starts\nwith a linear term proportional to |ω|and not quadratic in ωas had been assumed in the derivation. This\nentails extra corrections to short loops beyond matching, but those corrections are of order O(ℏ)and in the\nleading-log approximation can be ignored.\n31A derivation can be found in [20, 21]; we just quote the results.\n– 45 –Figure 5 . The resolvent is real on the contour Cand takes values ξiat marked points.\n(iii) The function is real and monotonous along the contour Cshown in the figure.\n(iv) The discontinuity of the resolvent across the cut determines the eigenvalue density:\n2πρ(a) =G\u0012\na+i\n2−iε\u0013\n−G\u0012\na+i\n2+iε\u0013\n; (6.16)\n(v) At infinity the resolvent behaves as\nG(z)z→∞≃z2\n2g2; (6.17)\n(vi) The explicit form of the resolvent can be found by integrating the following differ-\nential equation:\ndz=gdG(G−ξ3)p\n2(G−ξ1)(G−ξ2)(G−ξ4), (6.18)\nwhere ξiare the values of G(z)at the points marked red in figure 5. Monotonicity implies\nthatξ1> ξ2> ξ3> ξ4.\nTheresolventcanexpressedthroughellipticintegralsbutwewillneverneedthisexplicit\nexpression. The parameters of the solution can be also expressed through elliptic integrals,\ntheK≡K(m)andE≡E(m):\nξ1=K[(2−m)K−2E]\n2π2g2,\nξ2=K(K−2E)\n2π2g2,\nξ3=K[(2−m)K−3E]\n2π2g2,\nξ4=K[(1−m)K−2E]\n2π2g2. (6.19)\n– 46 –The elliptic modulus is implicitly determined by the relation:\n6π4g2=K2\u0002\n2(2−m)KE−(1−m)K2−3E2\u0003\n. (6.20)\nWhile the endpoint of the eigenvalue distribution is given by\nπB=KE(φ)−EF(φ), (6.21)\nwhere\nsinφ=K−E\nmK. (6.22)\nA step-by-step derivation of all these results can be found in [20, 21].\nThe strong-coupling limit corresponds to m→1−is when the elliptic Kdevelops a\nlog-divergence. Denoting\nL≡4 log2\n1−m, (6.23)\nwe find:\n6π4g2≃L2(L−3)\n4, (6.24)\nand\nsinφ≃L−2\nL. (6.25)\nLikewise,\nξ1≃L(L−4)\n8π2g2, ξ 2≃L(L−4)\n8π2g2, ξ 3≃L(L−6)\n8π2g2, ξ 4≃ −L\n2π2g2.(6.26)\nCorrections are exponential in L.\nIt then follows from (6.21) that\n2πB≃L−logL−2, (6.27)\nup to corrections of order 1/L. Solving (6.24) for Lwe get, with the same accuracy:\nB= (3πg2)1\n3−1\n6πlog\u0000\n24π4g2e3\u0001\n. (6.28)\nThis agrees with (6.14) in the leading-log approximation, if we recall that B=µ/ℏand\nℏ=g−2/3.\nThe eigenvalue density can be found from (6.16) as the discontinuity across the cut in\nfigure 5. We first notice that ξ3≫ξ4at large L, while ξ2andξ3are approximately equal.\nSince G(z)varies from ξ3toξ2on the lower side of the cut, there it is can be replaced by\na constant\nG\u0012\na+i\n2\u0013\f\f\f\f\nlower side≃ξ2≃L2\n8π2g2≃B2\n2g2, (6.29)\nwhere we have used (6.26) and (6.27).\nOn the upper side, g2Gvaries by a huge amount, from O(L)toO(L2), but both values\n– 47 –Figure 6 . The eigenvalue density is the jump across the branch cut of G(α), equivalently the\ndifference between the two branches for any given α.\nare big and the asymptotic behaviour (6.17) can be used to approximate G(z):\nG\u0012\na+i\n2\u0013\f\f\f\f\nupper side≃a2\n2g2. (6.30)\nSubtracting (6.30) from the (6.29) we get, according to (6.16):\nρ(a)≃B2−a2\n4πg2=3\n4B3(B2−a2). (6.31)\nThis agrees with the solution for short loops (6.9). The Wilson loop itself can be found\nby integrating exawhich makes sense for x∼1/B. For larger xthe exponential is too\nstrong and the integral is dominated by the close vicinity of the endpoint where this crude\napproximation for the density is no longer valid.\nWhile Gitself is very large near the endpoint, O(L2), its variation on the scale O(L)\nbecomes important. It thus becomes necessary to account for the difference between ξ2and\nξ3. At the same time the difference between ξ2andξ1is much smaller and can still be\nneglected. The differential equation (6.18) in this approximation becomes\ndz≃g√2ξ3dGG−ξ3\nG−ξ2, (6.32)\nand can be integrated in elementary functions. Denoting\nB+i\n2−z≡α, (6.33)\nwe find:\nα=g√2ξ3\u0014\n(ξ2−ξ3) logξ2−ξ3\nξ2−G+ξ3−G\u0015\n. (6.34)\nThe integration constant was chosen to ensure G=ξ3atα= 0(figure 5).\nThe function α(G)is displayed in figure 6. It is obvious that its inverse G(α)has a\n– 48 –branch cut starting at α= 0. The density is the discontinuity across the cut and can be\nfound graphically from the definition (6.16), as shown in the plot. The Wilson loop can be\ncomputed as\nW(x) =BZ\n−Bda ρ(a)eax≃eBx∞Z\n0dα ρ(B−α)e−αx=eBx\n2πξ2Z\n−∞dGdα\ndG(G−ξ3)e−αx.\n(6.35)\nUsing (6.32) and (6.34) this can be explicitly written as\nW(x) =eBx\n2πg√2ξ3ξ2Z\n−∞dG(G−ξ3)2\nξ2−Ge−gx√\n2ξ3h\n(ξ2−ξ3) logξ2−ξ3\nξ2−G+ξ3−Gi\n. (6.36)\nIntroducing a “dimensionless” variable\nξ2−G= (ξ2−ξ3)u (6.37)\nand taking into account that\ng√2ξ3(ξ2−ξ3)≃1\n2π, (6.38)\nwe get:\nW(x) =ξ2−ξ3\n4π2e(B+1\n2π)x∞Z\n0du ux\n2π(1−u)2e−ux\n2π=ξ2−ξ3\nx2Γ\u0010\n1 +x\n2π\u0011\nex(B−1\n2πlogx\n2πe).\n(6.39)\nUsing (6.26), (6.24) and (6.28), we finally find:\nW(x) =\u00123\n8π2g4x6\u00131\n3\nΓ\u0010\n1 +x\n2π\u0011\neh\n(3πg2)1\n3−1\n6πlog(3 πg2x3)i\nx, (6.40)\nwhich agrees with the solution of the loop equations for long loops (6.15). Here we have\nkept track of the log normalisation and thus computed the constant Cin (6.15) which was\nleft undetermined in the leading-log approximation.\n7 Conclusion\nThemultipointWilsonloopoperatorsinthispapercangrantaccesstoconnectedcorrelators\nof matrix fields. Repeating common arguments in literature one can show that our loop\nequation too is the top one in a tower of equations that interrelate the n-point and higher-\npoint operators. A special interest arises in N⩾2SYM theories: W(2πk1,2πk2, . . . , 2πkn)\nmeasures the correlator of nsupersymmetric Wilson loops, each wrapping the same circular\npath kitimes.\nOne can venture into finding patterns in perturbative data and engineering closed-\nform solutions. The task would be rewarding given the scarcity of such formulas, but it\n– 49 –may require a dose of ingenuity, judging by the complex nature of the existing results.\nSimplifications occur in particular models or limits.\nWe hint at this prospect in section 4.3.2 when our approximate solution was compared\nto the integral representation of the M2-correction, which is an exact function of λ. The\norder M4, which contains products of two zeta numbers, may descend from the double\nintegral of products of Bessel functions. If such step were repeatable for more orders in M2,\none would be tell the analyticity properties of the mass series and possibly resum it.\nOther interesting questions involve the effect of instantons (to be accounted by an\nN-dependent measure), the large- Nand large- xlimit with x√\nλ∼N(with contributions\nstemming from isolated eigenvalues) for branes in AdS/CFT [57, 84] and the case x1x2<0\nwhich models Wilson loops in complex-conjugate representations.\nSolving the matrix model of circular Wilson loops has been a long-standing dream in\nN= 2theories. The loop equation harbours the potential to move the goalpost further\ninto hardly-accessible regions of the parameter space. One can employ finite-difference\nmethods, as noted in section 4.3.2, and the Monte Carlo method for the matrix-model\naverage [39, 71, 74], which is significantly simpler in matrix models in one dimension.\nLoops in N= 2∗theories feature a rich diagram of quantum phase transitions on the\nline of couplings at infinite mass [32]. The non-commuting limits λ→0−andM→ ∞\nat fixed ΛRare markedly different from the perturbation theory in this paper. One could\nrevisit directly in the loop equation and test the findings against the large- ΛRasymptotics\nin [30].\nThe loop equation is an essential tool to measure observables with a matrix-integral\nrepresentation. Supersymmetric localisation techniques are available for BPS observables\nin any N= 2Lagrangian theories [27]. The prime generalisation is the two circular BPS\nWilson loops in SU(N)×SU(N)quiver theory [37, 44, 82, 83, 85–87]. The model with\nequal couplings is equivalent to the Z2orbifold of N= 4SYM with gauge group SU(2N)\n[88] and the loops have the same expectation value: the planar value coincides with that\ninN= 4SYM and the non-planar expansion was the subject of a recent study [80]. The\nmodel with unequal couplings is the next-to-simplest case where the loop equation could\nclose on and be solvable for all Wilson loops associated to different gauge groups. Likewise\nanother setup is the class of Aq−1circular quiver theories, which are relevant for a number\nof motivations including integrability in N= 2theories. Localisation reduces BPS loops to\nmulti-matrix models for which perturbative algorithms are available [85, 89].\nThe loop equations are applicable to models with fermionic variables, for instance to\nsupereigenvalue models [90–93] or fermionic matrix models [94]. We believe generalisations\nwe discussed here can be extended to this class of models as well.\nAcknowledgments\nWe thank Dmitri Bykov for comments on the ν-model. The work of E. V. was supported by\nthe European Union’s Horizon 2020 research and innovation programme under the Marie\nSklodowska-Curie grant agreement No 895958. The work of K. Z. was supported by VR\ngrant 2021-04578.\n– 50 –A Saddle-point equation from loop equation\nBy the standard argument [8], the saddle-point equations for the eigenvalue integral (2.3)\nare equivalent to an integral equations for the eigenvalue density. The latter can be formally\ndefined as a Fourier transform of the Wilson loop (2.28):\nρ(a) =+∞Z\n−∞dω\n2πe−iωaW(ω). (A.1)\nHere we derive the saddle-point equation for the density from the Fourier-space loop equa-\ntion (2.29).\nThe latter can be written as\ni\n2V′\u0012\n−i∂\n∂κ\u0013\nW(κ) =+∞Z\n−∞dω\n2πˆR(ω)W(ω)e−ω∂\n∂κW(κ), (A.2)\nwhere the exponential operator shifts the argument of W(κ)toκ−ω. Going to the a-\nrepresentation (A.1) and doing the ω-integral with the help of (2.14) we find:\n1\n2V′\u0012\n−i∂\n∂κ\u0013\nW(κ) =Z\nda ρ(a)R\u0012\n−i∂\n∂κ−a\u0013\nW(κ). (A.3)\nDifferential operators on both sides act on the same function W(κ)and should coincide to\ngive the same result. This condition is equivalent to an integral equation for the density:\n1\n2V′(b) =Z\nda ρ(a)R(b−a), (A.4)\nwhere we denoted b≡ −i∂/∂κ. This is the standard saddle-point equation [8] for the\neigenvalue integral (1.1).\nB Operator identity and orthogonality relations\nIn this appendix we derive the operator identity (2.33) valid for functions whose Fourier\ntransform has a compact support, and derive orthogonality relations for the ensuing differ-\nential polynomials.\nSubstituting (2.32) in (2.33) we get:\nD(ν)\nnf(κ) =−2i∞Z\n0dω ω2ν−2Jν+n(ω)\nων+n1Z\n−1dt eitκˆf(t) sintω. (B.1)\n– 51 –Interchanging the order of integration we can evaluate the ωintegral explicitly:\n∞Z\n0dω ω2ν−2Jν+n(ω)\nων+nsintω=2ν−n−1Γ(ν)\nΓ(n+ 1)t2F1\u0012\nν,−n;3\n2;t2\u0013\n. (B.2)\nIt is important to stress that this formula is only valid for |t|<1. For toutside the unit\ninterval the result is way more complicated. The restriction to |t|<1is sufficient for our\npurposes since the tintegration is confined to the (−1,1)interval by definition, because\nwe only consider functions with a Fourier image of finite support. This is not a technical\ncondition, without it the derivation falls apart and the final conclusion does not hold at all.\nThe Gamma-function in the denominator hits the pole at negative integer n. This\nimmediately implies that the whole integral is to zero for n < 0. Ifnis a non-negative\ninteger, the result is not zero but then the Taylor expansion of the hypergeometric function\ntruncates to a finite polynomial [54]:\nt2F1\u0012\nν,−n;3\n2;t2\u0013\n=(−1)nΓ(ν−n−1)Γ(n+ 1)\n2Γ(ν)Cν−n−1\n2n+1(t). (B.3)\nUnder the integral tcan be replaced by −i∂/∂κ, so any polynomial in tcan be replaced\nby a differential operator acting on f(κ). The operator identity (2.33) then immediately\nfollows, with the operator on the right-hand-side given by the differential polynomial (2.35).\nThe polynomials (2.35) do not satisfy any standard orthogonality relations. That per-\nhaps requires some explanation. The Gegenbauer polynomials of course form an orthonor-\nmal set, but that assuming the upper index fixed and the lower one varying. In our case\nboth indices vary with n, so polynomials with different nin fact belong to different series\nwith different orthogonality measures.\nThe closest analogy to classical orthogonality can be derived from the differential equa-\ntion that the Gegenbauer polynomials satisfy32:\n\u0014\n(1−a2)d2\nda2−(2ν−2n−1)ad\nda+ (2ν−1)(2n+ 1)\u0015\nC(ν−n−1)\n2n+1 (a) = 0 .(B.4)\nThe operator in the square brackets is Hermitian on the interval (−1,1)with respect to\nthe measure (1−a2)ν−n−1\n2. We can use this to derive a quadratic integral identity by\nmultiplying both sides of the equation with C(ν−m−1)\n2m+1 (a)(1−a2)ν−n−1\n2and integrating by\nparts:\n1Z\n−1da(1−a2)ν−n−1\n2C(ν−n−1)\n2n+1 (a)\u0014\n(1−a2)d2\nda2−(2ν−2n−1)ad\nda\n+ (2ν−1)(2n+ 1)\u0015\nC(ν−m−1)\n2m+1 (a) = 0 . (B.5)\n32The conventional orthogonality for a fixed upper index is also a consequence of the same differential\nequation.\n– 52 –The Gegenbauer polynomial C(ν−m−1)\n2m+1satisfies (B.4) with nreplaced by m. This can be\nused to simplify the integrand and we finally arrive at an identity\n2(n−m)1Z\n−1da(1−a2)ν−n−1\n2C(ν−n−1)\n2n+1 (a)\u0012\nad\nda+ 2ν−1\u0013\nC(ν−m−1)\n2m+1 (a) = 0 .(B.6)\nWhen m̸=nwe can divide by (n−m)and the integral itself must vanish. If m=n\nwe get do not get any useful relation. The m=nintegral has to be computed by hand,\nfortunately is reduces to table integrals, such that\n1Z\n−1da(1−a2)ν−n−1\n2C(ν−n−1)\n2n+1 (a)\u0012\nad\nda+ 2ν−1\u0013\nC(ν−m−1)\n2m+1 (a)\n=πΓ(2ν−1)\n22ν−2n−4(2n+ 1)! Γ( ν−n−1)2δnm. (B.7)\nTaking into account the normalisation factor in (2.35), we arrive at the orthogonality con-\ndition (2.47).\nCU(N)vs.SU (N)\nTheSU(N)version of the matrix model constrains the centre of mass of the eigenvalues:\nZSU(N)=+∞Z\n−∞NY\ni=1daiδ X\niai!Y\ni<jµ(ai−aj)e−1\n2g2P\nia2\ni, (C.1)\nand the same in the correlation functions. In this appendix we consider the model with the\nGaussian potential but arbitrary measure. We will prove that the U(1)contribution factors\nout despite non-linearity of the measure:\nZU(N)=ZU(1)ZSU(N),WU(N)(x) =WU(1)(x)WSU(N)(x), (C.2)\nwhere\nZU(1)=p\n2πg2,WU(1)(x) =eg2x2\n2. (C.3)\nTo do so we Fourier transform the delta function:\nZSU(N)=Zdω\n2π+∞Z\n−∞NY\ni=1daiY\ni<jµ(ai−aj)e−1\n2g2P\nia2\ni+iωP\niai, (C.4)\nand shift the integration variables:\nai→ai+ig2ω. (C.5)\n– 53 –Thisgeneratesaneffectivepotentialfor ω:V(ω) =g2ω2/2, andhasnoeffectonthemeasure\nsince the shift is common to all the eigenvalues. The integral over ωthus decouples, and\nwe get:\nZSU(N)=1p\n2πg2ZU(N). (C.6)\nThe same manipulations over the Wilson loop give:\nWSU(N)(x) =WU(N)(x)D\neig2xωE\n=WU(N)(x)e−g2x2\n2. (C.7)\nD Derivation of planar equations\nWe derive (3.20) and (3.21). The goal is to insert (3.14) into (3.12) and identify the coeffi-\ncients of λandx, ignoring negative powers of N. We rescale away ˜cn,ℓ,m→(16π2)ℓ˜cn,ℓ,m\nfor simplicity.\nIn the left-hand side\n∞X\nr=2r˜Tr\u0012d\ndx\u0013r−1\n˜W(x) =∞X\nr=2∞X\nℓ=1λℓ2ℓX\np=r−1r p!\n(p−r+ 1)!˜Tr˜c0,ℓ,pxp−r+1(D.1)\nwe rearrange the sums, shift p→p+r−1\n∞X\nℓ=1λℓ∞X\nr=22ℓ−r+1X\np=0r(p+r−1)!\np!˜Tr˜c0,ℓ,p+r−1xp(D.2)\nand pull out the sum over x-powers\n∞X\nℓ=1λℓ2ℓ−1X\np=0xp2ℓ−p+1X\nr=2r(p+r−1)!\np!˜Tr˜cn,ℓ,p+r−1. (D.3)\nIn the right-hand side\nλ\n16π2+∞Z\n−∞dω\n2πˆγ(ω)Zx\n0ds˜W(s−iω)˜W(x−s+iω) (D.4)\nwe apply the binomial theorem on the powers of s−iωandx−s+iωand integrate using\n(3.18)\n1\n16π2∞X\nn=0∞X\nn′=0λn+n′+12nX\nm=0mX\nk1=02n′X\nm′=0m′X\nk2=0m′−k2X\nk3=0(−)m′−k2\u0012m\nk1\u0013\u0012m′\nk2\u0013\u0012m′−k2\nk3\u0013\n×xk1+k2+k3+1\nk1+k3+ 1γ(m+m′−k1−k2−k3)˜c0,n,m˜c0,n′,m′. (D.5)\nA change of indices ℓ=n+n′+ 1brings a homogeneous λ-power. The exponent of xcalls\n– 54 –for more dexterity. We rotate k2andk3into their sum sand difference d\n1\n16π2∞X\nℓ=1λℓℓ−1X\nℓ′=02ℓ′X\nm=02ℓ−2ℓ′−2X\nm′=0mX\nk1=0m′X\ns=0xk1+s+1X\nd=−s,−s+2,...,s(−)m′−s+d\n2\nk1+s−d\n2+ 1\u0012m\nk1\u0013\u0012m′\ns+d\n2\u0013\n×\u0012m′−s+d\n2\ns−d\n2\u0013\nγ(m+m′−k1−s)˜c0,ℓ′,m˜c0,ℓ−ℓ′−1,m′, (D.6)\ncommute the sums over ℓ′andm, m′\n1\n16π2∞X\nℓ=1λℓ2ℓ−2X\nm=02ℓ−2⌈m\n2⌉−2X\nm′=0ℓ−l\nm′\n2m\n−1X\nℓ′=⌈m\n2⌉mX\nk1=0m′X\ns=0xk1+s+1X\nd=−s,−s+2,...,s(−)m′−s+d\n2\nk1+s−d\n2+ 1\u0012m\nk1\u0013\u0012m′\ns+d\n2\u0013\n×\u0012m′−s+d\n2\ns−d\n2\u0013\nγ(m+m′−k1−s)˜c0,ℓ′,m˜c0,ℓ−ℓ′−1,m′, (D.7)\npull out the sums over the indices of x\n1\n16π2∞X\nℓ=1λℓ2ℓ−2X\nk1=02ℓ−2X\ns=0xk1+s+12ℓ−2X\nm=k12ℓ−2X\nm′=sℓ−l\nm′\n2m\n−1X\nℓ′=⌈m\n2⌉X\nd=−s,−s+2,...,s(−)m′−s+d\n2\nk1+s−d\n2+ 1\u0012m\nk1\u0013\u0012m′\ns+d\n2\u0013\n×\u0012m′−s+d\n2\ns−d\n2\u0013\nγ(m+m′−k1−s)˜c0,ℓ′,m˜c0,ℓ−ℓ′−1,m′ (D.8)\nand rotate k1, sintop, qto create xp\n1\n16π2∞X\nℓ=1λℓ2ℓ−1X\np=1xpX\nq=1−p,3−p,...,p−12ℓ−2X\nm=p−q−1\n22ℓ−2X\nm′=p+q−1\n2ℓ−l\nm′\n2m\n−1X\nℓ′=⌈m\n2⌉X\nd=−p+q−1\n2,−p+q−5\n2,...,p+q−1\n2\u0012m\np−q−1\n2\u0013\n×\u0012m′\np+q−1\n4+d\n2\u0013\u0012m′−p+q−1\n4−d\n2\np+q−1\n4−d\n2\u0013(−)m′−p+q−1\n4−d\n2\n3p−q+1\n4−d\n2γ(m+m′−p+1)˜c0,ℓ′,m˜c0,ℓ−ℓ′−1,m′.\n(D.9)\nWe equate the powers of λandxacross (D.3) and (D.9), undo ˜cn,ℓ,m→˜cn,ℓ,m/(16π2)ℓ\nand prove (3.20) and (3.21).\nReferences\n[1] Y. 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It is often used\nwhen traditional integer derivatives models fail to represent cases where the power\nlaw is observed accurately. Fractional calculus must reflect non-local, frequency-\nand history-dependent properties of power-law phenomena. This tool has various\nimportant applications, such as fractional mass conservation, electrochemical analysis,\ngroundwater flow problems, and fractional spatiotemporal diffusion equations. It can\nalso be used in cosmology to explain late-time cosmic acceleration without the need\nfor dark energy. We review some models using fractional differential equations. We\nassume the Einstein-Hilbert action based on a fractional derivative action and add\na scalar field ϕto create a non-minimal interaction theory with the coupling ξRϕ2\nbetween gravity and the scalar field, where ξis the interaction constant. By employing\nvarious mathematical approaches, we can offer precise schemes to find analytical and\nnumerical approximations of the solutions. Moreover, we comprehensively study the\nmodified cosmological equations and analyze the solution space using the theory of\ndynamical systems and asymptotic expansion methods. This enables us to provide a\nqualitative description of cosmologies with a scalar field based on fractional calculus\nformalism.\nKeywords: Fractional calculus; dynamical systems; asymptotic solutions; cosmology.arXiv:2402.13850v1  [gr-qc]  21 Feb 20242\n1. INTRODUCTION\nFractional differential calculus is a mathematical tool that has found applications in\nstudying social and physical behaviours considered \"anomalous\". These phenomena are\noften described empirically using the power law, which is universal. However, some of the\nmathematical models that use integer derivatives, including nonlinear models, need to work\nbetter in many cases where the power law is observed. Alternative tools like fractional\ncalculus must be introduced to accurately reflect power-law phenomena’ non-local, frequency-\nand history-dependent properties.\nFractional calculus has found critical applications in areas such as fractional mass con-\nservation. In such cases, a fractional mass conservation equation is needed to model fluid\nflow when the control volume is not large enough compared to the scale of heterogeneity\nand when the flow within the volume control is not linear. Other areas where fractional\ncalculus is useful include electrochemical analysis and the problem of groundwater flow. In\nthe latter, Darcy’s classical law is generalized by considering the water flow as a function of\na non-integer order derivative of the piezometric height. This generalized law and the law of\nconservation of mass are then used to derive a new equation for groundwater flow.\nOther examples of fractional calculus applications are the fractional advection-dispersion\nequation and fractional spatiotemporal diffusion equations. These models help characterize\nanomalous diffusion processes in complex media. An extension of the fractional derivative is\nthe fractional derivative of variable order. This tool is helpful in structural damping models,\nwhere fractional derivatives are used to model viscoelastic damping in certain materials\nsuch as polymers. In quantum theory, some important examples of fractional calculus\nare the fractional Schrödinger equation and versions such as the variable order fractional\nSchrödinger equation. Recent studies have also demonstrated the usefulness of fractional\ncalculus in cosmology. Modifying the theory using fractional calculus explains late-time\ncosmic acceleration without including dark energy.\n∗Electronic address: kevin.marroquin@alumnos.ucn.cl\n†Electronic address: genly.leon@ucn.cl\n‡Electronic address: alfredo.millano@alumnos.ucn.cl\n§Electronic address: claudio.ramirez@ce.ucn.cl\n¶Electronic address: anpaliat@phys.uoa.gr3\n1.1. Research questions\n•Is it possible to use a combination of local and global variables to qualitatively describe\nscalar field cosmologies based on the fractional calculus formalism?\n•Is it possible to provide precise schemes for finding analytical and numerical approxi-\nmations to solutions by choosing various approaches?\n1.2. Problem Statement\nA formulation of gravity with a scalar field that shows non-minimal or conformal coupling\nto gravity is presented. The new action includes the Ricci scalar R, the scalar field ϕwith\nits potential V(ϕ), the coupling constant ξ, the matter Lagrangian Lm, the scale factor a,\nand the lapse function N. The Gamma function Γ(µ), the fractional parameter µ, and the\nequation of state parameter of matter ware also included. A comprehensive review of the\nstate of the art has been conducted, and it has been confirmed that dynamical systems theory,\nmethods of perturbation theory, and the main tools of fractional calculus can be combined\nto implement qualitative analysis of gravity models with fractional derivatives. That is\nespecially useful in cosmologies with scalar fields and conformal coupling and non-minimal\ncoupling to gravity, thereby generalizing previous results.\n1.3. Goals\n1.3.1. General Objective\n•The general objective of this paper is to analyze new and known results from cosmologies\nwith scalar fields and gravity models with fractional derivatives, especially for a scalar\nfield with conformal coupling and non-minimal coupling to gravity.\n1.3.2. Specific Objectives\n1.Explore the role of dynamical systems theory and the tools of fractional calculus in the\nanalysis of cosmological models.4\n2.Propose cosmologies with scalar fields with conformal and non-minimal coupling to\ngravity in fractional calculus.\n3.Qualitative analysis of models using dynamical systems and asymptotic methods to\nexpand our understanding and allow comprehensive conclusions.\n1.4. Research hypotheses\nAfter conducting a thorough literature review and developing a theoretical framework, a\nresearch hypothesis has been formulated that proposes the possibility of obtaining relevant\ninformation on the flow properties associated with an autonomous system of ordinary\ndifferential equations in the cosmological context through qualitative techniques. The\ndynamical systems theory enables us to provide qualitative descriptions of cosmologies with\na scalar field in fractional calculus. Moreover, it is possible to offer schemes for finding\nanalytical approximations of solutions and exact solutions by selecting various approaches.\n1.5. Methodology\nOur methodology first involves reviewing the literature on fractional calculus methods,\nsolution of fractional differential equations, and their applications. The current literature on\napplying fractional differential calculus to theories of gravity will also be studied, obtaining\nmodifications. The differential equations that describe the dynamics of the different models\nare to be derived and well-posed for mathematical analysis and physical interpretation.\nAt this point, it is expected to select the work tools and become familiar with Caputo’s\nfractional derivative and its application in formulating a fractional cosmological action and\nits respective variational calculation. Secondly, systems of ordinary differential equations will\nbe derived and analyzed from a mathematical point of view. This analysis consists of several\nsteps: finding equilibrium points and invariant sets; defining a linearization matrix for each\nsystem and evaluating it at each critical point to calculate the eigenvalues; classifying the\nstability of the points; generating phase space diagrams and numerical solutions for each\nsystem for appropriate values of the coupling and fractional parameters. It is expected to\nfind each system’s future and past attractors and classify each critical point according to\nits stability and asymptotic behaviour. The dynamical behaviour of the systems will also5\nbe described graphically by plotting orbits that illustrate the qualitative analysis, which\nallows the description of the cosmological evolution. Finally, we generalize previous results\nby considering a conformal and non-minimal coupling to gravity.\n1.6. Scientific Novelty\nWe consider corrections of the Friedmann and Klein-Gordon equations using Caputo’s\nderivative and the formalism of fractional calculus. This approach offers a high level of\napplicability to describe natural phenomena, making it a novel and attractive scenario.\nIn the fractional context, we analyze cosmologies with a scalar field coupled to gravity\nwith conformal and non-minimal coupling. We aim to generalize previous results within\nthe framework of the fractional formulation of gravity and describe the phase space of the\nmodels. We obtain qualitative results and explicit and approximate exact solutions with\nhigh numerical precision from a mathematical point of view.\n1.7. Organization\nThe paper is organized as follows. Section 2 covers the main outcomes of fractional\ncalculus, including different approaches to possible fractional derivatives such as Grünwald-\nLetnikov, Riemann-Liouville, and Caputo derivatives. The emphasis is on defining fractional\nintegration and the fractional derivative. We also discuss fractional differential equations\nand use Wolfram Language 13.3 [21] to solve physical problems formulated.\nIn section 3, we explore fractional cosmologies with non-minimal coupling ( ξ̸= 0), building\nupon previous results with minimal coupling. By studying different dynamical systems,\nincluding the invariant set ΩΛ= 0, we utilized numerical integration methods to analyze\nthe behaviour of solutions to differential equations from the cosmological model. We also\ninvestigated the stability of critical points. Conclusions are given in section 4. In appendix\nA, we define several special functions that appear recurrently in these topics.6\n2. THEORETICAL FRAMEWORK\nFractional calculus is a field of study that deals with the extension of derivatives and\nintegrals to fractional orders. It also focuses on the methods for solving differential equations\nthat involve these fractional derivatives and fractional integrals. It was developed by Newton\nand Leibniz in the 17th century and involves two fundamental operations - differentiation and\nintegration. These fractional operators have memory and are more flexible when describing\nthe dynamic behaviour of phenomena and systems using fractional differential equations. On\nthe other hand, the description with integer differential equations uses local operators and is\nlimited in the differentiation order to a constant. Consequently, the resulting models must\nbe sufficiently accurate in many cases [1].\nAlthough many contributions have been made to this topic over the years, it has not\nbeen applied directly for centuries. However, in recent years, its applications have grown\ntremendously, and it is now used universally as an empirical description of complex social\nand physical phenomena. Research on fractional differentiation spans multiple disciplines\nand has various applications, such as fractional spacetime in quantum mechanics and grav-\nity and fractional quantum field theory [ 2–9]. By utilizing these frameworks, researchers\nhave been able to gain a better understanding of complex systems in classical and quan-\ntum regimes [ 10–20]. This branch of mathematics is gaining more popularity in fluid\ndynamics, control theory, and signal processing. Due to the importance and potential of\nthis topic, support for fractional derivatives and fractional integrals has been added to the\nWolfram Language since the release of Version 13.1 https://blog.wolfram.com/2022/08/\n12/fractional-calculus-in-wolfram-language-13-1/ (revised on February 20th, 2024).\nThis paper will be using Wolfram Language 13.3 [21].\nNiels Henrik Abel is known for contributing to fractional calculus in the early 19th\ncentury. He introduced the integration and differentiation of fractional order and their inverse\nrelationship. Abel also unified the notation for differentiation and integration of arbitrary\nreal order in the \"differintegral\" operation. Abel considered the generalized version of the\ntautochrone problem; formulated as the integral equation\nf(x) =Zx\n0ϕ(s)√x−sds, (1)\nfor the unknown function ϕ(x). After several algebraic manipulations, this integral equation7\ncould be rewritten in the form\nϕ(x) =1\nπ\u0012f(0)√x+Zx\n0f′(τ)√x−τdτ\u0013\n, (2)\nwhich is what we now call Caputo’s fractional derivative of f(x)with respect to x.\nScientists from various areas and backgrounds have been working on fractional calculus\ntheory, approaching it from different perspectives. Some approaches define a fractional\ndifferentiation operation. In contrast, others consider a unified differentiation/integration\noperation, known as the fractional differintegral of order µ, with respect to xand a lower\nlimit a. Differintegrals depend on the function’s value at point a,f(a), using the function’s\n\"history\". In practice, the lower limit is generally taken as 0.\nSummarizing, in fractional calculus, the traditional derivatives and integrals of integer\norder have been generalized to derivatives and integrals of arbitrary order. This concept\nis presented in various sources, including [ 22–33]. The idea is that a derivative operator\nof order n, which is a natural number, has been generalized to a derivative of order µ,\nwhich can be a complex number or even a complex function. The three most popular and\ninfluential definitions widely used in practice are the Grünwald-Letnikov differintegral, the\nRiemann–Liouville fractional derivative and Caputo’s fractional derivative.\n2.1. Grünwald–Letnikov Approach\nThe Grünwald-Letnikov differintegral provides the basic extension of classical deriva-\ntives/integrals and is based on limits:\n0Dµ\nxf(x) = lim\nh→0h−µ∞X\nr=0(−1)r\nµ\nr\nf(x−rh). (3)\nIn practice, this approach is useless since it contains infinite approximations of the function\nat different points.\n2.2. Riemann–Liouville approach\nThe definition of the Riemann–Liouville fractional derivative is:\nRL\n0Dµ\nxf(x) =1\nΓ(n−µ)dn\ndxn\u0012Zx\n0(x−τ)n−µ−1f(τ)dτ\u0013\n, (4)8\nwhere n= max {0,⌊µ⌋}, where ⌊x⌋denotes the floor function, ⌊x⌋= max {m∈Z:m≤x}.\nIt is based on a solid and strict mathematical theory of fractional calculus. This theory is\nwell developed, but the Riemann-Liouville approach has some limitations that make it less\nsuitable for applications in real-world problems.\nFor example, the derivative of a constant is not zero:\nRL\n0Dµ\nxconstant =constant\nΓ(1−µ)x−µ. (5)\nTABLE I: The µ-th fractional derivatives of Riemann-Liouville and n-th ordinary derivatives for\nsome common functions.\nf(x) Dµ\nx(f(x)) f(n)(x)\nex−ex(Q(−µ, x)−1) ex\nx2 2x2−µ\nΓ(3−µ)2x2−n\n(2−n)!\nlog(x)\n\n−(−1)µ(µ−1)!x−µµ∈Z∧µ >0\n−x−µ(ψ(0)(1−µ)−log(x)+γ)\nΓ(1−µ)True\n\n−(−1)n(n−1)!x−nn≥1\nlog(x) True\nsin(x)√π2µ−1x1−µ1˜F2\u0010\n1; 1−µ\n2,3−µ\n2;−x2\n4\u0011\nsin\u0000πn\n2+x\u0001\ntan−1(x)√π2µ−1x1−µ3˜F2\u0010\n1\n2,1,1; 1−µ\n2,3−µ\n2;−x2\u0011√π2n−1x1−n3˜F2\u00001\n2,1,1; 1−n\n2,3−n\n2;−x2\u0001\nJ0(x)√π2µx−µ1˜F2\u0010\n1\n2;1−µ\n2,1−µ\n2;−x2\n4\u0011\n2−nPn\nK1=0(−1)K1J2K1−n(x)\nn\nK1\n\n1F1(a;b;x) Γ( b)x−µ2˜F2(1, a; 1−µ, b;x)\n\nΓ(b)x−n2˜F2(1, a; 1−n, b;x)n≥1\n1F1(a;b;x) True\nTable I shows the µ-th fractional derivatives and n-th ordinary derivatives for some\nstandard functions (see appendix A for definitions of special functions).\n2.2.1. Caputo’s Approach\nCaputo’s definition of fractional derivative is:\nC\n0Dµ\nxf(x) =1\nΓ(n−µ)Zx\n0(x−τ)n−µ−1dn\ndτnf(τ)dτ, n =\n\n⌊µ⌋+ 1µ /∈N\nµ µ ∈N.(6)9\nThere is some similarity between this and the Riemann-Liouville differintegral, and in fact,\nthe Caputo differintegral can be defined via the Riemann-Liouville differintegral by:\nC\n0Dµ\nxf(x) =RL\n0Dµ\nx−⌊µ⌋−1X\nk=0xk−µ\nΓ(k−µ+ 1)dk\ndxkf(0). (7)\nCaputo’s definition of fractional and integral derivatives has many advantages compared\nto those of Riemann-Liouville or Grünwald-Letnikov: first, it takes into account the values of\nthe function and its derivatives at the origin (or, in general, at any lower or upper bound\na), which automatically makes it suitable for solving fractional order initial value problems\nusing Laplace transforms. Furthermore, the Caputo fractional derivative of a constant is 0.\nTherefore, it is more consistent with the classical calculation.\nThe Riemann-Liouville fractional differintintegral has been implemented in the Wolfram\nLanguage, usingtheWolframLanguageVersion13.3[ 21]functionFractionalD .Thisfunction\ncalculates the Riemann-Liouville fractional derivative of order µof the function f(x); the\nlower limit of the integral is considered 0.\nThe Caputo fractional differential integral is CaputoD in Wolfram Language 13.1, which\ngives the Caputo fractional derivative of order µof a function f(x). For negative orders of µ,\nthe output of CaputoD matchesFractionalD .\nTABLE II: Half-order Caputo fractional derivatives of some common mathematical functions.\nf(x) D1\n2x(f(x))\nexexerf(√x)\nx2 8x3\n2\n3√π\nlog(x)log(4)√π√x\nsin(x)2√x1F2\u0010\n1;3\n4,5\n4;−x2\n4\u0011\n√π\ntan−1(x)2√x2F3(1\n2,1,1;3\n4,5\n4;−x2)√π\nJ0(x)1F2\u0010\n1\n2;1\n4,3\n4;−x2\n4\u0011\n−1\n√π√x\n1F1(a;b;x)2F2(1,a;1\n2,b;x)−1√π√x\nTable II presents the half-order Caputo fractional derivatives of some common mathemat-\nical functions.10\n2.2.2. Chain rule for fractional derivatives\nThe Caputo fractional derivative for an arbitrary cis [27]\nC\ncDµ\nxf(t) =1\nΓ(n−µ)Zx\ncdnf(τ)\ndτn(x−τ)n−µ−1dτ, n =\n\n⌊µ⌋+ 1µ /∈N\nµ µ ∈N.(8)\nGiven the square function: f(x) =x2/2, choosing c= 0, and removing the subindex 0,\nand applying the derivative of order 1/2twice, it results: µ= 1/2⇒n:=⌊µ⌋+ 1 = 1,\nCD1/2\nx\u0000\nx2/2\u0001\n=1\nΓ(1\n2)Zx\n0d\ndt\u0014t2\n2\u0015\n(x−t)−1\n2dt=1√π4\n3x3\n2,\nCD1/2\nx\u00121√π4\n3x3\n2\u0013\n=1√π4\n31\nΓ(1\n2)Zx\n0d\ndth\nt3\n2i\n(x−t)−1\n2dt=1√π4\n31√π3πx\n4=x.\nThe first derivative of the square function is obtained through two \"half-order fractional\ndifferentiation\" procedures. One could easily verify that the antiderivative of the square\nfunction can be obtained by two similar half-order integration procedures. So, with this\nexample, we show what fractional calculus is and how it relates to and generalizes the classical\nversion. However, this result holds because of the choice c= 0. In general, the chain rule for\nfractional calculus is:\nDµ\nx\u0002\nDβ\nxf(x)\u0003\n=Dµ+β\nxf(x)−nX\nj=1Dβ−j\nxf(c+)(x−c)−µ−j\nΓ(1−µ−j). (9)\n2.2.3. Fractional derivatives as generalizations of integer order derivatives\nFrom now on, we follow the book [23]. We have the formulas for the n-th derivatives,\ndn\ndxnekx=knekx,dn\ndxnsin (kx) =knsin\u0010\nkx+π\n2n\u0011\n,dn\ndxnxk=k!\n(k−n)!xk−n,(10)\nforn∈Ncan be extended for real and complex µnumbers,\nDµ\nxekx=kµekx, k ≥0, (11)\nDµ\nxsin (kx) =kµsin\u0010\nkx+π\n2µ\u0011\n, k ≥0, (12)\nDµ\nxxk=Γ(k+ 1)\nΓ(k+ 1−µ)xk−µ, x ≥0, k̸=−1,−2,−3, . . . . (13)\nThe definition of the fractional derivative of the exponential function was given by Liouville\nin 1832; those of trigonometric functions were proposed by Fourier in 1822, and those of11\npowers were systematically studied by Riemann in 1847. Still, the first attempts at the latter\nare due to Leibniz and Euler, the latter in 1738. The Riemann-Liouville fractional derivative\nof a constant is given by (13) setting k= 0, we obtain unexpected behaviour\nDµ\nx1 =Dµ\nxx0=1\nΓ(1−µ)x−µ.\nTherefore, as an additional postulate, Caputo in 1967 introduced the condition Dµ\nx1 = 0.\nThese four definitions of fractional derivative (11),(12),(13)and Caputo’s share several\naspects. On the one hand, they satisfy the correspondence principle:\nlim\nµ→nDµ\nxf(x) =dn\ndxnf(x), n = 0,1,2, . . . . (14)\nAnd also the following rules:\nDµ\nxcf(x) =cDµ\nxf(x), Dµ\nx[f(x) +g(x)] =Dµ\nxf(x) +Dµ\nxg(x). (15)\nTherefore, the fractional derivative of an analytic function can be calculated to its series.\nAccording to Liouville:\nf(x) =∞X\nk=0akekx=⇒Dµ\nxf(x) =∞X\nk=0akkµekx. (16)\nAccording to Fourier:\nf(x) =a0+∞X\nk=1aksin (kx) +∞X\nk=1bkcos (kx), (17)\nDµ\nxf(x) =∞X\nk=1akkµsin\u0010\nkx+π\n2µ\u0011\n+∞X\nk=1bkkµcos\u0010\nkx+π\n2µ\u0011\n. (18)\nAccording to Riemann:\nf(x) =xµ−1∞X\nk=0akxkµ, (19)\nDµ\nxf(x) =xµ−1∞X\nk=0ak+1Γ((k+ 2)µ)\nΓ((k+ 1)µ)xkµ. (20)\nAccording to Caputo\nf(x) =∞X\nk=0akxkµ, (21)\nDµ\nxf(x) =∞X\nk=0ak+1Γ(1 + ( k+ 1)µ)\nΓ(1 + kµ)xkµ. (22)12\nCalculating the Caputo fractional derivative of order µto the exponential function, we have\nDµ\nxex=Dµ\nx∞X\nn=0xn\nn!=∞X\nn=1xn−µ\nΓ(1 + n−µ)=x1−µE(1,2−µ, x). (23)\nSo, in contrast to Liouville’s definition (16), Caputo’s definition is not obviously in terms of\nan exponential, but in terms of a generalized Mittag-Leffler function (A14).\n2.2.4. Leibniz’s rule for the fractional derivative of the product\nLet us now analyze whether all the techniques of ordinary differential calculus can be\ntransferred to the field of fractional calculus [ 23]. For example, if we take the ordinary Leibniz\nrule:\nd\ndx(ψχ) =\u0012d\ndxψ\u0013\nχ+ψ\u0012d\ndxχ\u0013\n, (24)\nit could be carelessly inferred that the relation\nDµ\nx(ψχ) = (Dµ\nxψ)χ+ψ(Dµ\nxχ), (25)\nremains in the fractional calculus. However, using Liouville’s definition of a fractional\nderivative, we have\nDµ\nx(ek1xek2x) =Dµ\nxe(k1+k2)x= (k1+k2)µe(k1+k2)x. (26)\nWhile using (25), we obtain\nDµ\nx(ek1xek2x) = (kµ\n1+kµ\n2)(ek1xek2x). (27)\nTherefore, Leibniz’s rule (25) is not valid for fractional derivatives.\nWe start from the rule for successive non-negative integer orders to extend Leibniz’s rule\nto be valid for fractional calculus.\nd\ndx(ψχ) =\u0012d\ndxψ\u0013\nχ+ψ\u0012d\ndxχ\u0013\n,\nd2\ndx2(ψχ) =\u0012d2\ndx2ψ\u0013\nχ+ 2\u0012d\ndxψ\u0013\u0012d\ndxχ\u0013\n+ψ\u0012d2\ndx2χ\u0013\n,\nd3\ndx3(ψχ) =\u0012d3\ndx3ψ\u0013\nχ+ 3\u0012d2\ndx2ψ\u0013\u0012d\ndxχ\u0013\n+ 3\u0012d\ndxψ\u0013\u0012d2\ndx2χ\u0013\n+ψ\u0012d3\ndx3χ\u0013\n,(28)\netcetera. We have the following formula for an arbitrary integer,\ndn\ndxn(ψχ) =nX\nj=0\nn\nj\n\u0012dn−j\ndxn−jψ\u0013\u0012dj\ndxjχ\u0013\n, n ∈N. (29)13\nAccording to (A5), this expression can be extended to an arbitrary µ:\nDµ\nx(ψχ) =∞X\nj=0Γ(µ+ 1)\nΓ(j+ 1)Γ( µ−j+ 1)(Dµ−j\nxψ)(Dj\nxχ), (30)\nSo, this form of Leibniz’s product derivative rule is also valid for fractional derivatives.\n2.3. Fractional integral\nThe Cauchy formula for successive integrations, which can be extended to fractional\nintegrals, is used as a first step. A satisfactory definition of a fractional derivative can be\nobtained after appropriately deriving a fractional integral Iµ. If it is assumed\nDµ\nx=I−µ, (31)\nwhich is equivalent to requiring that a fractional derivative is the inverse of a fractional\nintegral and vice versa; a fractional integral is the inverse of a fractional derivative.\nIfaIis used for the integration operator\naIf(x) =Zx\naf(ξ)dξ, (32)\nIt is observed that the multiple integral is\naInf(x) =Zx\naZxn−1\na···Zx1\naf(x0)dx0···dxn−1, (33)\nwhich, using Cauchy’s repeated integration formula, can be reduced to a single integral:\naInf(x) =1\n(n−1)!Zx\na(x−ξ)n−1f(ξ)dξ. (34)\nThis formula can be easily extended to the fractional case:\naIµ\n+f(x) =1\nΓ(µ)Zx\na(x−ξ)µ−1f(ξ)dξ, x > a, (35)\nbIµ\n−f(x) =1\nΓ(µ)Zb\nx(ξ−x)µ−1f(ξ)dξ, x < b. (36)\nThe constants aandbdetermine the lower and upper limits of the integral, respectively, and\ncan initially be chosen arbitrarily. The two cases are called the left and right, respectively.\nThat can be understood because the left case deals with values ξ < x, values to the left of x.14\nThe right case deals with values ξ > x, values to the right of x. Ifxis a time-type coordinate,\nthen the integral of the left case is causal, and that of the right case is anti-causal.\nThe value of the integral depends on the specific choice of the constants (a, b). That\nwill be the notable difference between the two most widely used definitions of a fractional\nintegral, the so-called Liouville fractional integral and the Riemann fractional integral, which\nare discussed below.\n2.3.1. Liouville fractional integral\nThe Liouville fractional integral is defined by making a=−∞andb= +∞in(35)and\n(36):\nLIµ\n+f(x) =1\nΓ(µ)Zx\n−∞(x−ξ)µ−1f(ξ)dξ, (37)\nLIµ\n−f(x) =1\nΓ(µ)Z+∞\nx(ξ−x)µ−1f(ξ)dξ. (38)\n2.3.2. Riemann Fractional Integral\nThe fractional Riemann integral is defined by making a= 0andb= 0in (35) and (36):\nRIµ\n+f(x) =1\nΓ(µ)Zx\n0(x−ξ)µ−1f(ξ)dξ, (39)\nRIµ\n−f(x) =1\nΓ(µ)Z0\nx(ξ−x)µ−1f(ξ)dξ. (40)\nTo see the difference between Liouville’s definition (37),(38)and Riemann’s definition (39),\n(40) apply both definitions to f(x) =ekx:\nLIµ\n+ekx=k−µekx, x > 0,LIµ\n−ekx= (−k)−µekx, k < 0,\nRIµ\n+ekx=k−µekx\u0012\n1−Γ(µ, kx )\nΓ(µ)\u0013\n, x > 0,RIµ\n−ekx= (−k)−µekx\u0012\n1−Γ(µ, kx )\nΓ(µ)\u0013\n, x < 0,\nwhere Γ(µ, x)is the incomplete Gamma function. Using Liouville’s definition, the function\nekxvanishes for k > 0in the lower limit of the domain of the integral, so there is no\nadditional contribution. On the other hand, when using the Riemann definition, an additional\ncontribution is obtained because, at a= 0, the exponential function does not vanish [23].15\n2.4. Link between integration and fractional differentiation\nUsing the abbreviation\nDµ\nx=Dµ, (41)\nIt is shown that the concepts of fractional integration and fractional differentiation are closely\nrelated. Separating the fractional derivative operator into factors,\nDµ=DmDµ−m=dm\ndxmaIm−µ, m ∈N. (42)\nA fractional derivative can be interpreted as a fractional integral followed by an ordinary\nderivative. Once the fractional integral is specified, the fractional derivative is automatically\ndetermined. We can also reverse the sequence of the operators, resulting in an alternative\ndecomposition:\nDµ=Dµ−mDm=aIm−µdm\ndxm, m ∈N. (43)\nThe decompositions (42)and(43)lead to different results. Thus, we can understand the\nmechanism by which non-locality enters fractional calculus. The standard derivative, of\ncourse, is a local operator, but the fractional integral certainly is not. The fractional\nderivative should be understood as the inverse of fractional integration, a non-local operation.\nConsequently, fractional differentiation and integration produce the same difficulty [23].\nIn the previous section, two different definitions of the fractional integral have been given,\nthe Liouville and the Riemann. For each of these definitions, according to the two different\ndecompositions given above, four different realizations follow for the definition of fractional\nderivative, which are presented below.\n2.4.1. Liouville fractional derivative\nFor the simple case 0< µ < 1, using the Liouville definition (37)and(38), and the\nsequence of operators (42), we have\nLDµ\n+f(x) =d\ndxLI1−µ\n+f(x) =1\nΓ(1−µ)d\ndxZx\n−∞(x−ξ)−µf(ξ)dξ, (44)\nLDµ\n−f(x) =d\ndxLI1−µ\n−f(x) =1\nΓ(1−µ)d\ndxZ+∞\nx(ξ−x)−µf(ξ)dξ. (45)16\n2.4.2. Fractional Riemann derivative\nUsing(39),(40)and the sequence of operators (42), we have the Riemann fractional\nderivative:\nRDµ\n+f(x) =d\ndxRI1−µ\n+f(x) =1\nΓ(1−µ)d\ndxZx\n0(x−ξ)−µf(ξ)dξ, (46)\nRDµ\n−f(x) =d\ndxRI1−µ\n−f(x) =1\nΓ(1−µ)d\ndxZ0\nx(ξ−x)−µf(ξ)dξ. (47)\n2.4.3. Liouville-Caputo fractional derivative\nIn the case that the sequence of operators is inverted according to (43), it is based on the\ndefinition of Liouville fractional integral (37) and (38):\nLCDµ\n+f(x) =LI1−µ\n+d\ndxf(x) =1\nΓ(1−µ)Zx\n−∞(x−ξ)−µd f(ξ)\ndξdξ, (48)\nLCDµ\n−f(x) =LI1−µ\n−d\ndxf(x) =1\nΓ(1−µ)Z+∞\nx(ξ−x)−µd f(ξ)\ndξdξ. (49)\nIt is observed that for a function f(x) =constant, the definition of the Liouville fractional\nderivative (44)leads to a divergent result, while with the Liouville-Caputo definition (48),\nthe result is zero. The conditions for the Liouville fractional derivative to converge are more\nrestrictive than for the Liouville-Caputo counterpart [23].\n2.4.4. Caputo fractional derivative\nThe inverted sequence of operators (43), the definition of the Riemann fractional integral\n(39) and (40), leads to:\nCDµ\n+f(x) =RI1−µ\n+d\ndxf(x) =1\nΓ(1−µ)Zx\n0(x−ξ)−µd f(ξ)\ndξdξ, (50)\nCDµ\n−f(x) =RI1−µ\n−d\ndxf(x) =1\nΓ(1−µ)Z0\nx(ξ−x)−µd f(ξ)\ndξdξ. (51)\nThen, for f(x) =constant, we have that the Riemann fractional derivative isRDµ\n+constant =\nconstant\nΓ(1−µ)x−µ, while when applying the Caputo derivative, we haveCDµ\n+constant = 0. Therefore,\nin this section, we have presented the standard definitions of a fractional derivative, which\nsatisfy the condition\nlim\nµ→1Dµf(x) =d f(x)\ndx. (52)17\nMultiple applications of the first-order derivative can generate higher-order derivatives:\ndn\ndxn=d\ndxd\ndx···d\ndxntimes . (53)\nAfter applying different definitions of the fractional derivative to the same function, it has\nbeen found that they do not give the same result. This can be not comforting, but all these\ndefinitions are equally well-founded. Therefore, physicists can test different definitions of the\nfractional derivative by comparing theoretical results with experimental data. Depending on\nthe specific problem, one of the definitions will show the most significant agreement with the\nexperiment.\n2.5. Fractional Differential Equations\nFractional differential equations (FDE), which are differential equations that involve\nfractional derivatives and generalize ordinary differential equations (ODE), have attracted\nmuch attention. FDEs have been widely used in engineering, physics, chemistry, biology and\nother fields involving relaxation and oscillation models.\nLet us consider, for example, the FDE\nC\n0D21\n10\nty(t) + 10 y(t) = 0 , y(0) = 1 , y′(0) = 0 , y′′(0) = 0 . (54)\nFIG. 1: Solution (55)of the fractional harmonic oscillator of order 2.1given by (54). For µ >2, the\nfractional harmonic oscillator mimics an excited oscillator.\nSolving (54) using the Wolfram Language 13.3 [21] function DSolve we obtain\ny(t) =E\u001221\n10,−10t21/10\u0013\n. (55)18\nThe solution presented in figure 1 is given in terms of the function MittagLefflerE , the\nprimary function for fractional calculus applications. Its role in FDE solutions is similar in\nimportance to the Expfunction for ODE solutions: any FDE with constant coefficients can\nbe solved in terms of Mittag-Leffler functions.\nAn interesting example is the solution of a fractional harmonic oscillator of order 1.9:\nC\n0D19\n10\ntx(t) +x(t) = 0 , x(0) = 1 , x′(0) = 0 . (56)\nSolving(56)using the Wolfram Language 13.3 [ 21] function DSolve we obtain the solution\nfracOsc :x(t) =E\u001219\n10,−t19/10\u0013\n. (57)\nThis type of oscillator behaves similarly to the ordinary damped harmonic oscillator\nx′′(t) +3\n20x′(t) +x(t) = 0 , x(0) = 1 , x′(0) = 0 , (58)\nwith solution\ndampedOsc :x(t) =e−3t/40\u0010\n3√\n1591 sin\u0010√\n1591t\n40\u0011\n+ 1591 cos\u0010√\n1591t\n40\u0011\u0011\n1591.(59)\nThis example highlights how the order of an FDE can act as a control parameter to model\ncomplex systems.\nFigure 2 shows a comparison of the solution of the fractional oscillator (57)with the\ndamping oscillator (59). The error function is drawn in the bottom plot.\nAs a final example of FDE, we consider the fractional friction model\n¨x(t) +1\n4C\n0D1\n2\ntx(t) = 0 , x(0) = x0, x′(0) = v0 (60)\ninvolving the Caputo fractional differintegral of order 1/2. Solving (60)using the Wolfram\nLanguage 13.3 [21] function DSolve we obtain the solution\nx(t) =x0+tv0E\u00123\n2,2,−1\n4t3/2\u0013\n. (61)\nThis solution is given using the functions MittagLefflerE and is represented in figure 3\nby a blue line. The pink line corresponds to the solution\nx(t) =1\n192(t−24)t2+t (62)19\nFIG. 2: Comparison of the fractional oscillator (57)which satisfies the equation of a fractional\nharmonic oscillator of order 1.9(56)with the damping oscillator (59). The error function is drawn\nin the bottom plot. That shows that the error between both solutions goes to zero as tbecomes\nlarge.\nof the classical Newtonian equation with a friction term\n¨x=−1\n4˙x1\n2,˙x >0, (63)\nwhich determines the motion of a point particle influenced by a velocity-dependent force.\nThe thick lines describe the valid motion until the body stops ( v=˙x= 0). The interval is\nextended to values beyond the physical region (thin lines) for both solutions.\nThe fractional equivalent of the Newtonian equation of motion, combined with a classical\nfriction term, allows for a broader range of applications than the classical approach. Damped\noscillations can be accurately described with a suitable set of fractional initial conditions.\nFrom the perspective of fractional calculus, the harmonic oscillator is an example of a\nfractional friction phenomenon. This implies that friction and damping, described using\nseparate approaches in a classical context, can be combined in the fractional approach. This\nfeature is widespread in the fractional approach and can be applied to various phenomena\n[23].20\nFIG. 3: Solution of the FDE (60)for Caputo derivative for µ= 1/2,x0= 0andv0= 1(blue\nline) and comparison with the solution (100)which gives the position of a point particle under the\ninfluence of a velocity-dependent force (pink line).\n2.6. Fractional harmonic oscillator\nThis section shows the behaviour of fractional derivatives with the property.\nlim\nµ→2Dµ=d2\ndx2. (64)\nIn classical mechanics, the equation that describes free oscillations is\n\u0012d2\ndt2+k\nm\u0013\nx(t) = 0 , (65)\nwhose solution is well known:\nx(t) =c1cos (wt) +c2sin (wt), w =p\nk/m. (66)\nThe fractional formulation of the harmonic oscillator is given by\n\u0012d2µ\ndt2µ+k\nm\u0013\nx(t) = 0 . (67)\nthe second-order derivative was directly replaced by one order 2µ, where µis a fractional\nnumber, and the standard harmonic oscillator for µ= 1is recovered.\nThe solutions of this fractional differential equation for different types of fractional\nderivatives are presented below, and the specific differences of the solutions obtained are\nshown.21\n2.6.1. Fourier harmonic oscillator\nWhen the typical ansatzis proposed for this type of equation [23]:\nx(t) =ewt, (68)\nthe solution to the fractional harmonic oscillator reduces to examining the zeros of the\npolynomial\nw2µ+k\nm= 0. (69)\nIn the region 1/2< µ≤3/2, there are exactly two complex conjugate solutions:\nw1,2=\f\f\f\fk\nm\f\f\f\f1\n2µ\u0014\ncos\u0012π\n2µ\u0013\n±isin\u0012π\n2µ\u0013\u0015\n=\f\f\f\fk\nm\f\f\f\f1\n2µ\ne±i(π\n2µ). (70)\nForµ= 1, two pure imaginary solutions are obtained, which correspond to a free and\nundamped oscillation. If µbegins to decrease, an increasing negative real part occurs, which,\nfrom a classical point of view, can be interpreted as increasing damping. For µ >1, we have\nan increasing positive real part corresponding to increasing excitation [23].\nOn the other hand, the ordinary differential equation for a damped harmonic oscillator is\nm¨x(t) =−kx(t)−γ˙x(t), (71)\nwhere mis the mass, kis the spring constant and γis the damping coefficient.\nWith (68), we have a quadratic equation for the frequency w:\nw2=−γ\nmw−k\nm, (72)\nHence, we have two solutions:\nw1,2=−γ\n2m±r\nγ2\n4m2−k\nm. (73)\nComparing with (70), we have to\n−γ\n2m=\f\f\f\fk\nm\f\f\f\f1\n2µ\ncos\u0012π\n2µ\u0013\n, (74)\nand with a Taylor series expansion for cos (π/2µ)around µ= 1, we have\n−γ\n2m≈\f\f\f\fk\nm\f\f\f\f1\n2µ1\n2π(µ−1). (75)22\nNear µ= 1/2, there is no longer any oscillating contribution. The time evolution of this\nsystem is dominated by exponential decay. On the other hand, µ > 1corresponds to a\nnegative classical friction coefficient. Consequently, we have the surprising result that the\nbehaviour of the solutions of the fractional harmonic oscillator under the variation of the\nfractional derivative parameter, µ, can be interpreted from a classical point of view as\ndamping and excitation phenomena, respectively [23].\nThe following initial conditions are introduced:\nx(t= 0) = x0, (76)\nDµx(t)|t=0= ˜v(t= 0) = ˜ v0 (77)\nto compare solutions for different types of fractional derivatives.\nOnly for µ= 1, these initial conditions correspond to the classical initial conditions. For\nall other cases, the physical interpretation of a fractional velocity remains open.\nThe solutions of the fractional harmonic oscillator can be reformulated: A first set with the\nfractional initial conditions x0= 1,˜v0= 0can be considered as a fractional extension of the\nstandard cosine function, while the fractional initial conditions x0= 0,˜v0= 1characterize\nthe fractional extension of the sine function:\nFsin (µ, t) =1\n2i\u0014\ne1\n2(eiπ\n2µ)πt−e1\n2(e−iπ\n2µ)πt\u0015\n, (78)\nFcos (µ, t) =1\n2\u0014\ne1\n2(eiπ\n2µ)πt+e1\n2(e−iπ\n2µ)πt\u0015\n. (79)\nAn appropriately chosen linear combination of these two extended trigonometric functions\nwill again satisfy the classical initial conditions x(t= 0) = x0andv(t= 0) = v0. A graphical\nrepresentation of these solutions (78) and (79) for different µis shown in figure 4.\n2.6.2. Harmonic oscillator according to Riemann\nTo solve the differential equation of the harmonic oscillator, based on the Riemann\nfractional derivative, a series expansion is used according to (19). As\nRDµtnµ=Γ(1 + nµ)\nΓ(1 + ( n−1)µ)t(n−1)µ, (80)23\n0 5 10 15 20-10-505tFSin(μ,t)μ=1.05μ=1.01μ=1.0μ=0.9μ=0.8\n0 5 10 15 20-50510\ntFCos (μ,t)μ=1.05μ=1.01μ=1.0μ=0.9μ=0.8\nFIG. 4: SolutionsFsin (µ, t)andFcos (µ, t)of the fractional harmonic oscillator using the Fourier\nderivative with different values of µ.\nthere are two linearly independent solutions, which are calledRsin (µ, t)andRcos (µ, t), in\nanalogy to trigonometric functions:\nRsin (µ, t) =tµ−1∞X\nn=0(−1)nt(2n+1)µ\nΓ((2n+ 2)µ), (81)\nRcos (µ, t) =tµ−1∞X\nn=0(−1)n t2nµ\nΓ((2n+ 1)µ), (82)\nwith the property\nRDµ\nRsin(µ, wt ) =wµ\nRcos (µ, wt ), (83)\nRDµ\nRcos(µ, wt ) =−wµ\nRsin (µ, wt ). (84)24\nThese functions are related to the Mittag-Leffler function (A14):\nRsin (µ, t) =t2µ−1E(2µ,2µ,−t2µ), (85)\nRcos (µ, t) =tµ−1E(2µ, µ,−t2µ). (86)\nForµ <1, it holds that\nlim\nt→0Rcos (µ, wt ) =∞, (87)\nwhich means that a non-singular solution that satisfies the general initial conditions (76)\ncannot be given. At first glance, this seems to be a serious drawback for practically applying\nthe fractional Riemann derivative. Nevertheless, it must be considered that the solutions\npresented can be useful for problems not determined by the initial conditions but are\nformulated in terms of boundary conditions, as is the case, for example, of the solutions of a\nwave equation [23].\nSolutionsRsin (µ, t)andRcos (µ, t)of the fractional harmonic oscillator using the Riemann\nderivative are presented in figure 5 for different values of µ.\n2.6.3. Harmonic oscillator according to Caputo\nThis solution can be written as a series according to (21). The Caputo derivative for a\npower is given by\nCDµtnµ=\n\nΓ(1+ nµ)\nΓ(1+( n−1)µ)t(n−1)µ, n > 0,\n0, n = 0.(88)\nTherefore, we have two linearly independent solutions, which are called Csin (µ, t)and\nCcos (µ, t), in analogy to trigonometric functions:\nCsin (µ, t) =∞X\nn=0(−1)n t(2n+1)µ\nΓ(1 + (2 n+ 1)µ), (89)\nCcos (µ, t) =∞X\nn=0(−1)n t2nµ\nΓ(1 + 2 nµ). (90)\nThe most important property of this series is:\nCDµ\nCsin (µ, wt ) =wµ\nCcos (µ, wt ), (91)\nCDµ\nCcos (µ, wt ) =−wµ\nCsin (µ, wt ). (92)25\n0 5 10 15 20-3-2-101234\ntRSin(μ,t)μ=1.05μ=1.01μ=1.0μ=0.9μ=0.8\n0 5 10 15 20-4-2024tRCos (μ,t)μ=1.05μ=1.01μ=1.0μ=0.9μ=0.8\nFIG. 5: SolutionsRsin (µ, t)andRcos (µ, t)of the fractional harmonic oscillator using the Riemann\nderivative for different values of µ.\nSolutions Csin (µ, t)andCcos (µ, t)of the fractional harmonic oscillator using the Caputo\nderivative are presented in figure 6 for different values of µ.\nAfter comparing the solutions obtained from different definitions of the fractional derivative\nfor the differential equation of the harmonic oscillator, we notice a remarkable similarity\namong the presented solutions. They all display oscillatory behaviour with exponential decay\nwhen µ <1, which can be interpreted as a classical damping phenomenon. On the other\nhand, for µ >1, the fractional harmonic oscillator acts as an excited oscillator. The damping\nand excitation properties are inherent to the fractional oscillator and do not stem from\nexternal factors. In a study by Stanislavsky [ 34], a multiparticle statistical interpretation\nwas proposed as a physical explanation of the fractional harmonic oscillator.\nMoreover, the frequency of these damped oscillations is quite similar for the various26\n0 5 10 15 20-2024\ntCSin(μ,t)μ=1.05μ=1.01μ=1.0μ=0.9μ=0.8\n0 5 10 15 20-3-2-101234\ntCCos (μ,t)μ=1.05μ=1.01μ=1.0μ=0.9μ=0.8\nFIG. 6: Solutions Csin (µ, t)andCcos (µ, t)of the fractional harmonic oscillator using the Caputo\nderivative for different values of µ.\ndefinitions of fractional derivatives presented, especially for a value of µapproximating 1.\nTherefore, it can be regarded as an independent property of the definition [23].\n2.7. Fractional Friction\nThe evolution of the position x(t)and the velocity v(t) =˙x(t) =d\ndtx(t)of a point particle\nas functions of time caused by an external force Fis determined by Newton’s second law\nF=ma=m¨x(t) =md2\ndt2x(t). (93)\nIn the absence of external forces, the solution continues to integrate twice ¨x= 0:\nx(t) =x0+v0t, (94)27\nwhere the constants of integration are the position x(0) = x0and initial velocity v(0) = v0and\n˙x(t) =v0for all t. That means the point mass maintains its initial velocity forever without\nexternal forces. That is what is known as Newton’s first law. However, daily experience of\nmovement contradicts this result. Sooner or later, without external intervention, all types\nof movement end at rest. Within the framework of Newtonian theory, all these observed\nphenomena are due to friction forces FR.\nAll friction forces point in the opposite direction to the particle’s speed. Empirically, a\npower law function can be proposed:\nFR=−αsgn(v)|v|µ, (95)\nwhere µis an arbitrary real exponent. Some special cases are µ≈0, which is observed for\nstatic and kinetic pressure in solids, while µ= 1corresponds to the Stokes friction of a liquid\nwith high viscosity and µ= 2, which is a general trend towards high speed [23].\nLet us consider the differential equation\nm¨x=−αsgn(v)|v|µ, (96)\nwhere the mass mis measured in kilogram [kg]and the friction coefficient αis measured in\nkg/s2−µ, which describes the dynamics of a point particle under the effect of a friction force\nof the type (95).\nAssuming that the velocity is positive, we have\nm¨x=−α˙xµ,˙x >0. (97)\nImposing the initial conditions\nx(t= 0) = x0, (98)\nv(t= 0) = v0, (99)\nwe obtain the general solution of (97) given by\nx(t) =x0+mv2−µ\n0\nα(2−µ)−mh\n(µ−1)(−v1−µ\n0\n1−µ+α\nmt)i2−µ\n1−µ\nα(2−µ). (100)\nThe equation (100)determines the motion of a point particle under the influence of a\nvelocity-dependent force for an arbitrary µ.28\nThe special cases µ= 0,µ= 1andµ= 2are included as special limits [23]\nlim\nµ→0x(t) =x0+v0t−1\n2α\nmt2, (101)\nlim\nµ→1x(t) =x0+m\nαv0\u0000\n1−e−α\nmt\u0001\n, (102)\nlim\nµ→2x(t) =x0+m\nαln\u0010\n1 +α\nmv0t\u0011\n. (103)\nFinally, the modification of Newton’s equation (97)is studied according to [ 23] where the\nfriction force FR( ˙x) =−α˙x(t)µhas been replaced by the fractional friction force:\nFR=−αDµ\ntx(t), ˙x(t)>0, (104)\nwhere µis the order of the fractional derivative. For µ̸= 1andt >0, we have to\n˙xµ=\u0012d\ndtx(t)\u0013µ\n̸=Dµ\ntx(t), (105)\nso it is expected that the solutions of the fractional equation\nm¨x(t) +αDµ\ntx(t) = 0 (106)\nhave behaviour different from the standard. Note that to compare with the classical damped\nharmonic oscillator given by (71), we have renamed the friction constant α.\nReplacing the ansatz x(t) =ewtinto (106), and applying Liouville’s definition, we have:\nDµ\ntewt=wµewt. (107)\nTherefore, the characteristic polynomial of (106) is\nw2=−α\nmwµ. (108)\nIn addition to the trivial solution, we have the solution\nw= (−1)1\n2−µ\f\f\fα\nm\f\f\f1\n2−µ. (109)\nFor0< µ < 1, and using the abbreviation\nκ=\f\f\fα\nm\f\f\f1\n2−µ, (110)\nThere are two complex conjugate solutions:\nw1,2=κ\u0014\ncos\u0012π\n2−µ\u0013\n±isin\u0012π\n2−µ\u0013\u0015\n. (111)29\nSo the general solution of (106) is\nx(t) =c1+c2ew1t+c3ew2t, 0< µ < 1. (112)\nIt is observed that there are three different constants ci, two of which can be determined using\nthe initial conditions x(t= 0) = x0andv(t= 0) := ˙x(0) = v0. Consequently, the freedom\nto specify an additional reasonable initial condition indicates that the fractional differential\nequation (106) describes a broader range of phenomena than the classical equation [23].\nStarting from (106), the condition can be considered\n¨x(t= 0) = −α\nmvµ\n0. (113)\nTherefore, the set of equations is obtained\n\n1 1 1\n0w1w2\n0w2\n1w2\n2\n\nc1\nc2\nc3\n=\nx0\nv0\n−α\nµvµ\n0\n. (114)\nThen, the solution of the fractional differential equation (106) is\nx(t)≈x0+v0t−1\n2α\nµvµ\n0t2−t3\u0010\n2αcos\u0010\nπ\nµ−2\u0011\nvµ\n0\f\fα\nm\f\f1\n2−µ+mv0\f\fm\nα\f\f2\nµ−2\u0011\n6m+o(t4).(115)\nIf the condition is chosen\n¨x(t= 0) = 0 , (116)\nthe set of equations is obtained\n\n1 1 1\n0w1w2\n0w2\n1w2\n2\n\nc1\nc2\nc3\n=\nx0\nv0\n0\n, (117)\nand in this case, the complete solution is\nx(t) =x0+v0\nκsin\u0010\nπ\n2−µ\u0011\u0014\neκcos\u0010\nπ\n2−µ\u0011\ntsin\u00122π\n2−µ−κsin\u0012π\n2−µ\u0013\nt\u0013\n−sin\u00122π\n2−µ\u0013\u0015\n,(118)\nwhich is valid from t= 0until a final time tfin which the mass comes to rest, which is\ndetermined by the equation\ntf=π\n(2−µ)κsin\u0010\nπ\n2−µ\u0011. (119)30\nLet us compare the classical damped harmonic oscillator given by (71). The two solutions\nof the classical damped harmonic oscillator are:\nw1,2=−γ\n2m±r\nγ2\n4m2−k\nm, (120)\nwhere γis the damping coefficient in (71). By comparing the wiof(111)and(120), the\nrelations are obtained\nk\nm=\u0012α\nµ\u0013 2\n2−µ\n(121)\nand\nγ\nm=−2\u0012α\nµ\u0013 1\n2−µ\ncos\u0012π\n2−µ\u0013\n. (122)\nThese relations describe the behaviour of the solutions of the fractional friction differential\nequation as investigated in [23].\nFIG. 7: Different solutions of the equation of motion (106)with µ= 0.5,x0= 0,v0= 1,m= 1,\nα= 0.25[23].\nIn the figure 7, different solutions of the fractional friction equation (106)and the solution\nof the classical equation (pink) m¨x=−α˙xµgiven by (100)are presented. In red (up to order\n2) and black (up to order 3), the solutions to the fractional friction equation (106)given by\nthe equation (115)are presented. The blue line represents the solution (118)with initial\ncondition ¨x(0) = 0[23].\nIt is observed that the classical solution and the corresponding fractional solution with\nthe initial condition (113)coincide up to the second order in t, and there is a difference only31\nat the end of the movement. The thick lines describe the valid motion until the body stops\n(v=˙x= 0). Furthermore, for both fractional solutions, the plot interval is extended to\nvalues t > t f(thin lines) beyond the physical region.\nWhen the fractional parameter µapproaches zero, the behaviour approaches that of\ndamping. Near µ= 2, fractional friction acts as an acceleration term, and for the ideal case,\nµ= 2, the differential equation for a free particle is recovered.\nUsing fractional calculus, we can expand beyond the classical approach by combining\nthe fractional equivalent of the Newtonian equation of motion with a classical friction term.\nThis allows us to describe a wider range of phenomena, including damped oscillations with\nfractional initial conditions. The harmonic oscillator is an example of a fractional friction\nphenomenon, where friction and damping can be unified.\nIt is important to note that the fractional differential equation represented by (106)\ncombines two different types of classical differential equations. Nonlinear differential equations\ngenerally have complex solutions, whereas a simple second-order differential equation with\nconstant coefficients can be solved using the same fractional equation. This observation\nsuggests that a wide range of complex problems can be solved analytically using fractional\ncalculus with minimal effort. On the other hand, a solution obtained from a classical theory\nmay be more complicated.\nIn this section, we build upon previous research in the field [ 23] and emphasize the\nimportance of fractional calculus in explaining physical phenomena. We then explore the\napplications of this concept in theories of gravity to provide a better understanding of the\ncomplex mathematical structure of the universe. The next section aims to use advanced\nmathematical techniques to create new formulations and tools which will help us better\ncomprehend the cosmos. This is a crucial aspect of our research.\n3. GRAVITY MODELS WITH FRACTIONAL DERIVATIVES\nIn the last section, we discussed the importance of fractional calculus in understanding\nphysical phenomena. Now, we will focus on its applications to theories of gravity for a better\nunderstanding of the universe’s mathematical structure.\nThe Lagrangian density is a mathematical tool used to model the dynamic properties\nof fields [ 35–41]. Fractional Lagrangian densities have gained popularity in addressing32\ncosmological problems. Fractional calculus defines Riemann curvature and the Einstein\ntensor, dependent on the fractional parameter µ. The fractional Einstein tensor equation,\nGαβ(µ) = 8 πGT αβ(µ), where Gαβ(µ)is the fractional Einstein tensor, has been modified to\nstudy various cosmological events. For example, a fractional theory of gravitation has been\ndeveloped for fractional spacetime, leading to new classes of cosmological models. These\nmodels have been studied in great detail [ 14,16,42–76]. In reference [ 60], a joint analysis\nwas performed using cosmic chronometers and type Ia supernovae data. This comparison\nwith observational tests was used to find the best-fit values for the fractional order of the\nderivative. These methods are a robust scheme to investigate the physical behaviour of\ncosmological models [ 60,77–80]. They can be used in new contexts, such as [ 66], where\ndynamical systems were used to analyze a fractional cosmology for different matter contents,\nobtaining a cosmology with late acceleration without including dark energy. References\n[60,81,82] have explored the potential of fractional cosmology to address the H0tension\n[83,84] from the Supernova data and the Planck value for z <1.5and have reported a trend\nofH0that aligns with these values. However, a discrepancy exists between the 1.5< z < 2.5\nrange values, indicating that the tension H0has not been entirely resolved.\nBased on the work presented in [ 60], two research paths were identified for the cosmological\nmodel without a scalar field. The first path involves comparing it with the standard model,\nimposing that the components of the Universe are cold dark matter and radiation. The\nsecond path consists of deducing the equation of state for one of the sources of matter from\ncompatibility conditions [66], which had not been previously analyzed in [60].\n3.1. Cosmological model in the fractional formulation of gravity\nThere are two established methods for incorporating fractional derivatives into cosmology\nunder the classical regime. The simplest method is the modification method, which replaces\nthe cosmological field equations of a given model with corresponding fractional field equations.\nAn example of this can be found in [ 85]. The more fundamental methodology is the first-step\nmodification method. This method involves defining the fractional derivative and establishing\nthe variational principle of fractional action [ 68,86–90] to obtain a modified cosmological33\nmodel. Given the fractional integral action,\nS(τ) =1\nΓ(µ)Zτ\n0L(θ, qi(θ),˙qi(θ),¨qi(θ)) (τ−θ)µ−1dθ, (123)\nwhere Γ(µ)is the Gamma function, Lthe Lagrangian, µthe constant fractional parameter,\nτandθare the physical and intrinsic time respectively.\nVarying the action (123) with respect to qiwe obtain the Euler-Poisson equations [90],\n∂L(θ, qi(θ),˙qi(θ),¨qi(θ))\n∂qi−d\ndθ∂L(θ, qi(θ),˙qi(θ),¨qi(θ))\n∂˙qi+d2\ndθ2∂L(θ, qi(θ),˙qi(θ),¨qi(θ))\n∂¨qi\n=1−µ\nτ−θ\u0012∂L(θ, qi(θ),˙qi(θ),¨qi(θ))\n∂˙qi−2d\ndθ∂L(θ, qi(θ),˙qi(θ),¨qi(θ))\n∂¨qi\u0013\n−(1−µ) (2−µ)\n(τ−θ)2∂L(θ, qi(θ),˙qi(θ),¨qi(θ))\n∂¨qi. (124)\nIn cosmology it is assumed that the flat Friedmann–Lemaître–Robertson–Walker (FLRW)\nmetric\nds2=−N2(t)dt2+a2(t)(dx2+dy2+dz2), (125)\nis the spacetime metric, where a(t)denotes the scale factor and N(t)is the lapse function.\nThis result is supported observationally with data from the Planck’s [ 91] probe. For the\nmetric(125), the Ricci scalar depends on up to second derivatives of aand first derivatives\nofN, which is written as\nR(t) = 6 \n¨a(t)\na(t)N2(t)+˙a2(t)\na2(t)N2(t)−˙a(t)˙N(t)\na(t)N3(t)!\n. (126)\nThus, we can consider the integral action in units where 8πG=c= 1:\nS(τ) =Zτ\n0\"\nR(θ)\n2+˙ϕ2(θ)\n2N2(θ)−V(ϕ(θ)) +ξR(θ)ϕ2(θ) +Lmatter (θ)#\na3(θ)N(θ)dθ,(127)\nwhere R(θ)is the Ricci scalar (126). In cosmology, the Einstein-Hilbert Lagrangian density\nis related to the Ricci scalar. Generically, integration by parts is taken so that a total\nderivative in the action is eliminated, as well as the derivatives ¨a(t)and ˙N(t). However,\nsince we will use a fractional version of the Lagrangian (127), we will not follow the standard\nprocedure and will keep the higher-order derivatives. Using the formulation (123), the\nEuler-Poisson equation (124)is obtained, which involves fractional variational calculus with\nclassical and Caputo derivatives. If Lmatter (θ) =−ρ0a(θ)−3(1+w)forw̸=−1, we recover the34\nusual Lagrangian density of matter of a perfect fluid as in [ 92–94] containing derivatives of\ninteger order in the Lagrangian. To extend the theory given by the effective fractional action\nused in [60], we can assume the Einstein-Hilbert action and add a scalar field ϕto create a\nscalar field theory with coupling ξRϕ2between gravity and the scalar field, with ξbeing the\ncoupling constant. The simplest and most natural case is the minimal coupling where ξ= 0.\nAnother viable option is ξ= 1/6, known as conformal coupling because the action does not\nchange under conformal transformations of the metric. Any value of the coupling parameter\nξ̸= 0is a non-minimal coupling.\nThe resulting action is\nSeff(τ) =1\nΓ(µ)Zτ\n0L\u0010\nθ, N(θ),˙N(θ), a(θ),˙a(θ),¨a(θ), ϕ(θ),˙ϕ(θ)\u0011\n(τ−θ)µ−1dθ, (128)\nwhere Γ(µ)is the Gamma function, and the Lagrangian density is\nL\u0010\nθ, N(θ),˙N(θ), a(θ),˙a(θ),¨a(θ)\u0011\n:=3a(θ)\u0010\nN(θ)\u0000\na(θ)¨a(θ) + ˙a2(θ)\u0001\n−a(θ)˙a(θ)˙N(θ)\u0011\nN2(θ)\n+a3(θ)N(θ) ˙ϕ2(θ)\n2N2(θ)−V(ϕ(θ))!\n−ρ0N(θ)a(θ)−3w(τ−θ)−(µ−1)(w+1)\n−3ξϕ2(θ)a(θ)\u0010\nN(θ)\u0000\na(θ)¨a(θ) + ˙a2(θ)\u0001\n−a(θ)˙a(θ)˙N(θ)\u0011\nN2(θ),\n(129)\nw=p/ρis a constant state equation parameter for matter, ϕis the scalar field, and Vis its\ninteraction potential, depending on the scalar field.\nThe Euler–Poisson equations (124)are derived after varying the action (128)with respect\ntoqi∈ {N, a, ϕ }, obtaining the equations35\n\u0000\n1−ξϕ2(θ)\u0001\"\u0012˙a(θ)\na(θ)\u00132\n+\u0012˙a(θ)\na(θ)\u0013\u00121−µ\nτ−θ\u0013#\n=1\n3ρm0a(θ)−3w−3(τ−θ)−(µ−1)(w+1)+1\n6\u0010\n˙ϕ2(θ) + 2V(ϕ(θ))\u0011\n+ 2ξ˙a(θ)ϕ(θ)˙ϕ(θ)\na(θ),(130)\n\u0000\n1−ξϕ2(θ)\u0001\"\n¨a(θ)\na(θ)+1\n2\u0012˙a(θ)\na(θ)\u00132\n+(1−µ)\n(t−θ)˙a(θ)\na(θ)+(µ−2)(µ−1)\n2(τ−θ)2#\n=−1\n2ρ0wa(θ)−3(1+w)(τ−θ)−(µ−1)(w+1)−1\n4˙ϕ2(θ) +1\n2V(ϕ(θ))\n+ξ\"\n2ϕ(θ)˙a(θ)˙ϕ(θ)\na(θ)+2(1−µ)ϕ(θ)˙ϕ(θ)\nτ−θ+ϕ(θ)¨ϕ(θ) +˙ϕ2(θ)#\n, (131)\n¨ϕ(θ) +3˙a(θ)˙ϕ(θ)\na(θ)+(1−µ)˙ϕ(θ)\nτ−θ+V′(ϕ(θ)) =−6ξϕ(θ)\"\n¨a(θ)\na(θ)+\u0012˙a(θ)\na(θ)\u00132#\n, (132)\nand the conservation of matter equation\n˙ρ(θ) =−3\u0012˙a(θ)\na(θ)+1−µ\n3(τ−θ)\u0013\n(ρ(θ) +p(θ)), (133)\nwhere we have replaced N= 1after the variation of the action (128).\nFor fixed τ, the expressions\nρ(θ) =ρ0a(θ)−3(1+w)(τ−θ)−(µ−1)(w+1), (134)\nand\np(θ) =wρ0a(θ)−3(1+w)(τ−θ)−(µ−1)(w+1), (135)\nwhich define the energy density and isotropic pressure of matter fields.\nTo designate the temporal independent variables, the rule (τ, θ)7→(2t, t)is applied, where\nthe new cosmological time t[69] is used. Hereafter, dots mean derivative with respect to t.\nFurthermore, the parameter Hubble is H≡˙a/a. Therefore, the equations (130)and\n(131), the modified Klein-Gordon equation (132), and the conservation equation (133)can\nbe written as36\n\u0000\n1−ξϕ2(t)\u0001\u0012\n˙H(t) +(1−µ)H(t)\nt+3H2(t)\n2+(µ−2)(µ−1)\n2t2\u0013\n=−1\n2p(t)−1\n4˙ϕ(t)2+1\n2V(ϕ(t)) +ξϕ(t) \n2H(t)˙ϕ(t)−2(µ−1)˙ϕ(t)\nt+¨ϕ(t) +˙ϕ(t)!\n,(136)\n\u0000\n1−ξϕ2(t)\u0001\u0012\nH2(t) +(1−µ)H(t)\nt\u0013\n=1\n3ρ(t) +1\n6˙ϕ2(t) +1\n3V(ϕ(t)) + 2 ξH(t)ϕ(t)˙ϕ(t),(137)\n¨ϕ(t) + 3H(t)˙ϕ(t)−(µ−1)˙ϕ(t)\nt+V′(ϕ(t)) =−ξϕ(t)\u0010\n6˙H(t) + 12 H(t)2\u0011\n, (138)\n˙ρ(t) =−3\u0012\nH(t) +(1−µ)\n3t\u0013\n(p(t) +ρ(t)). (139)\nWe observe that the equation (139)is satisfied for any value of the equation of state of\nmatter parameter. Furthermore, by eliminating the higher order derivatives, we obtain the\nequation (139) and\n˙H=H \n(1−µ)\u0000\nξϕ2−1\u0001\nξ(6ξ−1)tϕ2+t−ξϕ˙ϕ\nξ(6ξ−1)ϕ2+ 1!\n+H2\u0000\n3(1−8ξ)ξϕ2−3\u0001\n2ξ(6ξ−1)ϕ2+ 2−p\n2ξ(6ξ−1)ϕ2+ 2\n+(µ−2)(µ−1)\u0000\nξϕ2−1\u0001\n2t2(ξ(6ξ−1)ϕ2+ 1)+˙ϕ\u0010\n(4ξ−1)t˙ϕ−4(µ−1)ξϕ\u0011\n4 (ξ(6ξ−1)tϕ2+t)\n−ξϕV′(ϕ)\nξ(6ξ−1)ϕ2+ 1+V(ϕ)\n2ξ(6ξ−1)ϕ2(t) + 2, (140)\n¨ϕ=H \n6(µ−1)ξϕ\u0000\nξϕ2−1\u0001\nξ(6ξ−1)tϕ2+t−3\u0000\nξ(4ξ−1)ϕ2+ 1\u0001˙ϕ\nξ(6ξ−1)ϕ2+ 1!\n+3ξH2ϕ\u0000\nξϕ2−1\u0001\nξ(6ξ−1)ϕ2+ 1+3ξpϕ\nξ(6ξ−1)ϕ2+ 1−3(µ−2)(µ−1)ξϕ\u0000\nξϕ2−1\u0001\nt2(ξ(6ξ−1)ϕ2+ 1)\n+˙ϕ\u0010\n2(µ−1) +ξϕ\u0010\n2(µ−1)(12 ξ−1)ϕ+ 3(1−4ξ)t˙ϕ\u0011\u0011\n2 (ξ(6ξ−1)tϕ2+t)\n+\u0000\nξϕ2−1\u0001\nV′(ϕ)\nξ(6ξ−1)ϕ2+ 1−3ξϕV(ϕ)\nξ(6ξ−1)ϕ2+ 1. (141)\nSolving (137) for ρwe obtain\nρ=3H(−tH+µ−1) (ξϕ2−1)\nt−6ξHϕ ˙ϕ−V(ϕ)−1\n2˙ϕ2. (142)\nIn [66] the equations (136)–(139)forξ= 0were studied. Using a procedure similar to\nthat of RG, (137) is required to be conserved over time, i.e.\nd\ndt\u0014\u0000\n1−ξϕ2\u0001\u0012\nH2+(1−µ)H\nt\u0013\n−1\n3ρ−1\n6˙ϕ2−1\n3V(ϕ)−2ξHϕ ˙ϕ\u0015\n= 0. (143)37\nCalculating the corresponding derivatives and substituting (139),(140),(141)and(142),\nwe obtain\n(µ−1)\n12t3(ξ(6ξ−1)ϕ2+ 1)\"\n−6t2H2\u0000\nξϕ2−1\u00012\n+\u0000\nξϕ2−1\u0001\u0010\n−12tH\u0010\n−µ+ξϕ\u0010\n(µ+ 12ξ−3)ϕ+t˙ϕ\u0011\n+ 3\u0011\n−12ξt2ϕV′(ϕ) +t˙ϕ\u0010\n−12(µ−1)ξϕ−t˙ϕ\u0011\u0011\n−2t2p\u0000\nξ(12ξ+ 1)ϕ2−1\u0001\n−12ξt2˙ϕ2\n+ 2t2\u0000\nξ(12ξ+ 1)ϕ2−1\u0001\nV(ϕ) + 6( µ−2)(µ−1)\u0000\nξϕ2−1\u00012#\n= 0. (144)\nThis equation is an identity for µ= 1as expected in standard cosmology. However, for\nµ̸= 1, we acquire the new relation for the fluid pressure\np=−1\n2t2(ξ(12ξ+ 1)ϕ2−1)\"\n6t2H2\u0000\nξϕ2−1\u00012\n−t\u0000\nξϕ2−1\u0001\u0010\n−12H\u0010\n−µ+ξϕ\u0010\n(µ+ 12ξ−3)ϕ+t˙ϕ\u0011\n+ 3\u0011\n−12ξϕ\u0010\n(µ−1)˙ϕ+tV′(ϕ)\u0011\n−t˙ϕ2\u0011\n+ 12ξt2˙ϕ2−2t2\u0000\nξ(12ξ+ 1)ϕ2−1\u0001\nV(ϕ)\n−6(µ−2)(µ−1)\u0000\nξϕ2−1\u00012#\n. (145)\nThen, using a procedure similar to GR, a new Equation (145)is obtained instead of showing\nthat two of three equations are independent. This feature of fractional cosmology leads to\nsome constraints on the matter fields in the Universe that were explored in [ 66]. Therefore,\nthe equation of state is modified\nw(t) =p(t)\nρ(t). (146)\nReplacing the expression for pgiven by (145) into (140), (141) and (139), we obtain\n˙H=\u0000\nξϕ2−1\u0001\u0012(µ−2)(µ−1)\nt2(ξ(12ξ+ 1)ϕ2−1)−2(µ−4)H\nt(ξ(12ξ+ 1)ϕ2−1)\u0013\n−2ξHϕ ˙ϕ\nξ(12ξ+ 1)ϕ2−1+H2\u0000\n3−3ξ(8ξ+ 1)ϕ2\u0001\nξ(12ξ+ 1)ϕ2−1+2ξt˙ϕ2−2ξϕ\u0010\n(µ−1)˙ϕ+tV′(ϕ)\u0011\nt(ξ(12ξ+ 1)ϕ2−1),(147)\n¨ϕ=\u0000\nξϕ2−1\u0001\"\n12(µ−4)ξHϕ\nt(ξ(12ξ+ 1)ϕ2−1)+6ξH2ϕ\nξ(12ξ+ 1)ϕ2−1−t2V′(ϕ) + 6( µ−2)(µ−1)ξϕ\nt2(ξ(12ξ+ 1)ϕ2−1)#\n−3H\u0000\nξ(8ξ+ 1)ϕ2−1\u0001˙ϕ\nξ(12ξ+ 1)ϕ2−1+˙ϕ\u0010\n−µ+ξϕ\u0010\n(µ−1)(24 ξ+ 1)ϕ−12ξt˙ϕ\u0011\n+ 1\u0011\nt(ξ(12ξ+ 1)ϕ2−1), (148)38\nand the auxiliary equation\n˙ρ=\u0000\nξϕ2−1\u0001\"\n3H\nt2(ξ(12ξ+ 1)ϕ2−1)\n×\u0010\n(µ−5)(µ−1) +ξϕ\u0010\n(µ−1)(12( µ−3)ξ−µ+ 5) + 6 t\u0010\n(µ−1)˙ϕ+tV′(ϕ)\u0011\u0011\u0011\n+H2\u0000\n−3µ+ 3ξ(−48µξ+µ+ 120 ξ−13)ϕ2+ 39\u0001\nt(ξ(12ξ+ 1)ϕ2−1)+18H3\u0000\nξ(6ξ+ 1)ϕ2−1\u0001\nξ(12ξ+ 1)ϕ2−1\n−6(µ−1)ξϕ\u0010\n(µ−1)˙ϕ+tV′(ϕ)\u0011\nt2(ξ(12ξ+ 1)ϕ2−1)#\n+\u0000\nξϕ2−1\u00012\u00123(µ−2)(µ−1)2\nt3(ξ(12ξ+ 1)ϕ2−1)−9(µ−2)(µ−1)H\nt2(ξ(12ξ+ 1)ϕ2−1)\u0013\n+3H˙ϕ\u0010\nt\u0000\n6ξ+ξ(6ξ+ 1)ϕ2−1\u0001˙ϕ−4(µ−1)ξϕ\u0000\nξ(6ξ+ 1)ϕ2−1\u0001\u0011\nt(ξ(12ξ+ 1)ϕ2−1)\n+36ξH2ϕ\u0000\nξ(6ξ+ 1)ϕ2−1\u0001˙ϕ\nξ(12ξ+ 1)ϕ2−1−(µ−1)\u0000\n6ξ+ξ(6ξ+ 1)ϕ2−1\u0001˙ϕ2\nt(ξ(12ξ+ 1)ϕ2−1). (149)\nThese equations allow one to deduce an effective equation of state for matter without\nimposing equations of state on each matter field. In the case of a minimal coupling, ξ= 0,\nthis allows the system to be investigated\n˙H=2(4−µ)H\nt−3H2+(µ−2)(µ−1)\nt2, (150)\n¨ϕ=−3H˙ϕ+(µ−1)˙ϕ\nt−V′(ϕ). (151)\nThe first equation is the Riccati ordinary differential equation for µ̸= 1andξ= 0. The\nanalytical solution of (150) is (see an analogous case in [70], equation (36))\nH(t) =1\n3t\u00149−2µ+r\n2−crαr\n0\ncαr\n0+ (H0t)r\u0015\n, (152)\nwhere\nc=−2µ+r−6α0+ 9\n2µ+r+ 6α0−9, r=p\n8µ(2µ−9) + 105 , (153)\nandα0=H0t0, where t0is the value of ttoday. H0is the current value of the Hubble factor,\nandα0is the current age parameter, for which we obtain the best-fit values. For large t, the\nasymptotic scaling factor can be expressed as\na(t)≃t1\n6(+9−2µ+r), (154)39\nThus, the acceleration of the late Universe can be obtained with a scale factor with a power\nlaw and without a cosmological constant. These results are independent of the matter source\nand anisotropy.\n3.2. Minimal Coupling\nIn this section, we discuss the results of [ 81] for minimal coupling ( ξ= 0) and a constant\npotential V(ϕ) = Λ.\nIntroducing the logarithmic independent variable τ=−ln(1 +z) =lna, with τ→ −∞\nwhen z→ ∞,τ→0when z→0andτ→ ∞when z→ −1, and defining the age parameter\nof the Universe as α=tH, we obtain the initial value problem\nα′(τ) = 9−2µ−3α(τ) +(µ−2)(µ−1)\nα(τ), (155)\nt′(τ) =t(τ)/α(τ), (156)\nα(0) = t0H0, t(0) = t0, (157)\nplus the auxiliary equation\nϕ′′(τ) =−(µ−1)ϕ′(τ)(µ−t(τ)ϕ(τ)−2)\nα(τ)2−ϕ′(τ)(−2µ+ 3t(τ)ϕ(τ) + 8)\nα(τ)+ 3ϕ′(τ).(158)\nThe equation (155)gives a one-dimensional dynamical system analyzed in [ 66]. There is an\nasymptotic behavior for large τconsistent with [ 66], in which the attractor solution has an\nasymptotic age parameter limt→∞tH=1\n6(9−2µ+r).\nThe exact solution of this system is\nt(τ) =t0exp\n2\u0012\ntan−1\u0012\n6H0t0+2µ−9√\n8(9−2µ)µ−105\u0013\n−tan−1\u0012\n6α(τ)+2µ−9√\n8(9−2µ)µ−105\u0013\u0013\np\n8(9−2µ)µ−105\n,(159)\nwhere αis obtained implicitly from\n1\n6ln\u0000\n−2µα(τ)−3α(τ)2+ 9α(τ) +µ2−3µ+ 2\u0001\n−(2µ−9) tan−1\u0012\n6α(τ)+2µ−9√\n−16µ2+72µ−105\u0013\n3p\n−16µ2+ 72µ−105\n=1\n6\nln\u0000\n−2H0µt0−3H0t0(H0t0−3) +µ2−3µ+ 2\u0001\n+2(9−2µ) tan−1\u0012\n6H0t0+2µ−9√\n8(9−2µ)µ−105\u0013\np\n8(9−2µ)µ−105−6τ\n,\n(160)40\nwhere the parameter ϵ0has been introduced such that\nϵ0=1\n2lim\nt→∞\u0012t0H0−tH\ntH\u0013\n, α 0=1\n6\u0010\n9−2µ+p\n8µ(2µ−9) + 105\u0011\n(1 + 2 ϵ0).(161)\nϵ0is a measure of the limiting value of the relative error in the age parameter tHwhen\napproximated by t0H0.\nThe (theoretical) Hubble parameter as a function of redshift is tested against cosmological\nobservations. Therefore, for the adjustment, the system (155)and(156)are numerically\nintegrated, which represents a system for the variables (α, t)as a function of τ=−ln (1 + z),\nand for which we consider the initial conditions α(τ= 0)≡α0=t0H0andt(τ= 0)≡t0=\nα0/H0. Then, the parameter Hubble is obtained numerically by Hth(z) =α(z)/t(z).\nFurthermore, for a better comparison, the free parameters of the ΛCDM model are also\nfitted, whose Hubble parameter as a function of redshift is given by [81]\nH(z) =H0q\nΩm,0(1 +z)3+ 1−Ωm,0. (162)\nThe free parameters of the fractional cosmological model are θ={h, µ, ϵ 0}and the free\nparameters of the ΛCDM model are θ={h,Ωm,0}. For the free parameters µ,ϵ0andΩm,0,\nwe consider the following flat priors: µ∈F(1,4),ϵ0∈F(−0.1,0.1)andΩm,0∈F(0,1). It\nis important to mention that due to a degeneracy between H0andM, the SNe Ia data\ncannot constrain the free parameter h(as a reminder, H0= 100km/s\nMpch), contrary to the case\nof OHD and, consequently, in the joint analysis. Therefore, the posterior distribution of h\nfor the SNe Ia data is expected to cover all prior distributions. On the other hand, the prior\nis chosen as ϵ0= 0, which is a measure of the limiting value of the relative error in the age\nparameter tHwhen approximated by t0H0according to the equation (161). For the mean\nvalue ϵ0= 0,α0=1\n6(−2µ+r+ 9)was acquired, which implies c= 0.\nThe analysis of the SNe Ia, OHD data and the joint analysis with SNe Ia + OHD data leads\nrespectively to h= 0.696+0.302\n−0.295,µ= 1.340+2.651\n−0.339andϵ0=\u0000\n1.976+1.709\n−2.067\u0001\n×10−2,h= 0.675+0.041\n−0.021,\nµ= 2.239+1.386\n−1.190andϵ0=\u0000\n0.865+0.793\n−0.773\u0001\n×10−2, and h= 0.84+0.031\n−0.027,µ= 1.840+1.446\n−0.773and\nϵ0=\u0000\n1.213+0.482\n−1.057\u0001\n×10−2, where best-fit values are calculated at 3σCL.\nIn this model, the deceleration parameter is calculated using specific values for α(τ)and\nµ, in closed form as\nq(α(τ)) = 2 +2(µ−4)\nα(τ)−(µ−2)(µ−1)\nα2(τ). (163)41\nBest fit values\nData h Ωm,0 µ ϵ 0×102χ2\nmin\nΛCDMModel\nSNe Ia 0.692+0.209 +0 .296 +0 .307\n−0.120−0.278−0.292 0.299+0.022 +0 .046 +0 .068\n−0.021−0.042−0.059 ··· ··· 1026.9\nOHD 0.706+0.012 +0 .024 +0 .035\n−0.012−0.024−0.036 0.259+0.018 +0 .038 +0 .059\n−0.017−0.033−0.047 ··· ··· 27.5\nSNe Ia+OHD 0.696+0.010 +0 .020 +0 .029\n−0.010−0.020−0.029 0.276+0.014 +0 .030 +0 .043\n−0.014−0.027−0.040 ··· ··· 1056.3\nFractional cosmological model (dust + radiation) [60] (The uncertainties presented correspond to 1σ(68.3%)CL)\nSNe Ia 0.599+0.275\n−0.269 0.160+0.050\n−0.072 2.771+0.161\n−0.214 ··· 54.83\nCC 0.629+0.027\n−0.027 0.399+0.093\n−0.122 2.281+0.492\n−0.433 ··· 16.14\nSNe Ia+CC 0.692+0.019\n−0.018 0.228+0.035\n−0.040 2.839+0.117\n−0.193 ··· 78.69\nFractional cosmological model [81] (The uncertainties presented correspond to 1σ(68.3%),2σ(95.5%), and 3σ(99.7%)CL)\nSNe Ia 0.696+0.215 +0 .293 +0 .302\n−0.204−0.284−0.295 ··· 1.340+0.492 +2 .447 +2 .651\n−0.245−0.328−0.339 1.976+0.599 +1 .133 +1 .709\n−0.905−1.848−2.067 1028.1\nOHD 0.675+0.013 +0 .029 +0 .041\n−0.008−0.015−0.021 ··· 2.239+0.449 +0 .908 +1 .386\n−0.457−0.960−1.190 0.865+0.395 +0 .650 +0 .793\n−0.407−0.657−0.773 29.7\nSNe Ia+OHD 0.684+0.011 +0 .021 +0 .031\n−0.010−0.020−0.027 ··· 1.840+0.343 +1 .030 +1 .446\n−0.298−0.586−0.773 1.213+0.216 +0 .383 +0 .482\n−0.310−0.880−1.057 1061.1\nTABLE III: Best fit values and criteria χ2\nminfor the ΛCDM model with free parameters handΩm,0\nfor the fractional cosmological model (dust + radiation) [ 60] and the fractional cosmological model\nwith free parameters h,µandϵ0[81]. Using the MCMC analysis [ 95], the best-fit values and their\njoint analysis were obtained from the SNe Ia, OHD (or CC) data. The ΛCDM model was used as a\nreference model. (Taken from [82]).\nUsing best-fit estimates from Table III, we show a transition at zt⪆1, with a larger\ntransition redshift than the ΛCDM model. The current deceleration parameter for the\nfractional cosmological model is q0=−0.37+0.08\n−0.11at3σCL.\nWe use the jerk to determine the type of dark energy in the fractional cosmological model.\nIts formula is based on the value of qin(163)using the equation j(τ) =q(τ)(2q(τ)+1)−dq(τ)\ndτ.42\nEntering α(τ), we obtain\nj(α(τ)) =12(µ−4)\nα(τ)+(µ−21)µ+ 50\nα(τ)2−2(µ−3)(µ−2)(µ−1)\nα(τ)3+ 10.(164)\nIfjdeviates from unity, this may suggest a different cosmology with a dynamical equation\nof state during late times.\nThe matter density parameter in the fractional cosmological model has significant uncer-\ntainties due to the absence of the matter equation of state parameter in the Hubble parameter\nused for the reconstruction. The current value of this parameter is Ωm,0= 0.531+0.195\n−0.260at1σ\nCL, which aligns with the asymptotic value Ωm,t→∞= 0.519+0.199\n−0.262determined at the same\nconfidence level through joint analysis.\nThe fractional cosmological model suggests that a larger Ωm,0could explain the smaller\ndeceleration parameter q0and the excess matter at ρfrac= 3(µ−1)t−1Hwith Ωfrac(α(τ)) =\n(µ−1)/α(τ). The current value of Ωfrac,0is0.469, which satisfies Ωm,0+ Ωfrac,0= 1and\npotentially alleviates the Problem of Coincidence. On the other hand, these best-fit values\nlead to an age of the Universe with a value of t0=α0/H0= 25.62+6.89\n−4.46Gyrs [81].\nIn this case, the expressions (142)and(145)are used to calculate ρ(t)andp(t). Substi-\ntuting all expressions in the system (140)–(142)leads to identities. There is an arbitrary\nconstant of integration, and the equations are satisfied identically (no compatibility equations\nare required). Therefore, this is the general solution of the system. This result is generic\nsince it does not require specifying the equation of state parameter of matter. Therefore, the\nequation (152)provides a family of one-parameter solutions that provide a complete solution\nand are independent of matter content.\n3.2.1. Dynamical Systems Analysis\nThis section presents new findings and an alternative approach to the dynamical system\npreviously developed in [ 60,66,81]. Through analyzing dynamical systems, we can identify\nthe model’s asymptotic states and explore the phase space for different fractional derivative\norders and matter models [ 96–101]. This allows us to classify the equilibrium points and\ndetermine the fractional derivative order to produce a power-law accelerated late solution for\nthe scale factor.43\nWith this objective in mind, we introduce the new variables\nx1=t(τ)ϕ′(τ)√\n6α(τ), x 2=t(τ)ϕ(τ), x 3=α(τ), (165)\nthat satisfy\nρ+ Λ = −3x2\n2x3\u0000\n−µξ+ξ+x3\u0000\nξ+x2\n1+ 2√\n6ξx1\u0001\u0001\nt(τ)4+3x3(−µ+x3+ 1)\nt(τ)2,(166)\nand the new time\nf′≡α2d f\ndτ, (167)\nobtaining the dynamical system\nx′\n1=−2µ2x1+µx1(x2+ 4x3+ 6)−x1(3x2x3+x2+ 2(8−3x3)x3+ 4),(168)\nx′\n2=x3\u0010√\n6x1x2\n3+x2\u0011\n, (169)\nx′\n3=x3\u0000\nµ2−3µ−2µx3−3(x3−3)x3+ 2\u0001\n. (170)\nThe equilibrium points of (168), (169) and (170) are:\n1.P1: (x1, x2, x3) := (0 , x2c,0). The eigenvalues are\n{0,(µ−1)(−2µ+x2c+ 4),(µ−2)(µ−1)}. It is normally hyperbolic with a stable 2D\nmanifold for 1< µ < 2, x2c<2µ−4, an unstable 2D manifold for µ <1, x2c<2µ−4,\norµ >2, x2c>2µ−4. It is a saddle for µ <1, x2c>2µ−4, or1< µ < 2, x2c>2µ−4,\norµ >2, x2c<2µ−4.\n2.P2: (x1, x2, x3) :=(x1c,2(µ−2),0). The eigenvalues are {0,0,(−2 +µ)(−1 +µ)}. It\nis non-hyperbolic.\n3.P3: (x1, x2, x3) :=\u0000\n0,0,1\n6(−2µ−r+ 9)\u0001\n, with r=p\n8µ(2µ−9) + 105. The eigen-\nvalues aren\n1\n3(−2µ−r+ 9),1\n6(−2µ−r+ 9),1\n6(−2µ(8µ+r−36) + 9 r−105)o\n.It\nis\n(a) Source for 1< µ < 2.\n(b) Sink for µ <1orµ >2.\n4.P4: (x1, x2, x3) :=\u0010\nµ(−12µ+3r+65)−2(5r+51)\n4√\n6(µ−2)(µ−1)2 ,1\n12(4µ−r−3),1\n6(−2µ−r+ 9)\u0011\n.\nThe eigenvalues aren\n1\n6(−2µ(8µ+r−36) + 9 r−105),\n1\n12\u0010\n−2µ−3√\n2p\n2µ(5µ−27) + (2 µ−9)r+ 93−r+ 9\u0011\n,44\n1\n12\u0010\n−2µ+ 3√\n2p\n2µ(5µ−27) + (2 µ−9)r+ 93−r+ 9\u0011o\n.It is a saddle for all µ̸= 1\nandµ̸= 2.\n5.P5: (x1, x2, x3) :=\u0000\n0,0,1\n6(−2µ+r+ 9)\u0001\n.The eigenvalues aren\n1\n6(−2µ+r+ 9),1\n3(−2µ+r+ 9),1\n6(2µ(−8µ+r+ 36)−3(3r+ 35))o\n. It is a saddle for\nallµ∈R.\n6.P6: (x1, x2, x3) :=\u0010\nµ(−12µ−3r+65)+2(5 r−51)\n4√\n6(µ−2)(µ−1)2 ,1\n12(4µ+r−3),1\n6(−2µ+r+ 9)\u0011\n.\nThe eigenvalues aren\n1\n6(2µ(−8µ+r+ 36)−3(3r+ 35)) ,\n1\n12\u0010\n−2µ−3p\n20µ2−4µ(r+ 27) + 6(3 r+ 31) + r+ 9\u0011\n,\n1\n12\u0010\n−2µ+ 3p\n20µ2−4µ(r+ 27) + 6(3 r+ 31) + r+ 9\u0011o\n.\nIt is a saddle for all µ̸= 1andµ̸= 2.\n3.3. Non-Minimal Coupling\nIn this section, we present the new results of the thesis for the most general case ξ̸= 0\ncorresponding to non-minimal coupling. For this analysis, we will use the age parameter\ngiven by α=tH, and we will use the rules\nd\ndt=Hd\ndτ,d2\ndt2=H2\u0012d2\ndτ2−(1 +q)d\ndτ\u0013\n, q :=−1−˙H\nH2, (171)\nto introduce a new derivative.45\nThus, the system (147)–(148) becomes\nα′(τ) =(µ−2)(µ−1)\u0000\nξϕ2−1\u0001\nα(ξ(12ξ+ 1)ϕ2−1)+α\u0000\n3−3ξ(8ξ+ 1)ϕ2\u0001\nξ(12ξ+ 1)ϕ2−1\n+ξ(−2µ+ 12ξ+ 9)ϕ2+ 2µ−9\nξ(12ξ+ 1)ϕ2−1+2ξt2ϕ2ϕ′2\nα(ξ(12ξ+ 1)ϕ2−1)\n+ϕ′\u0012\n−2(µ−1)ξtϕ2\nα(ξ(12ξ+ 1)ϕ2−1)−2ξtϕ2\nξ(12ξ+ 1)ϕ2−1\u0013\n, (172)\nϕ′′(τ) =12(µ−4)ξϕ\u0000\nξϕ2−1\u0001\nα(ξ(12ξ+ 1)ϕ2−1)−6(µ−2)(µ−1)ξϕ\u0000\nξϕ2−1\u0001\nα2(ξ(12ξ+ 1)ϕ2−1)+6ξϕ\u0000\nξϕ2−1\u0001\nξ(12ξ+ 1)ϕ2−1\n+ϕ′2\n2(µ−1)ξtϕ2\nξ(12ξ+1)ϕ2−1−12ξ2t2ϕ3\nξ(12ξ+1)ϕ2−1\nα2+2ξtϕ2\nα(ξ(12ξ+ 1)ϕ2−1)\n−2ξt2ϕ2ϕ′3\nα2(ξ(12ξ+ 1)ϕ2−1)\n+ϕ′ \n3\u0000\nξ(8ξ+ 1)ϕ2−1\u0001\nξ(12ξ+ 1)ϕ2−1+(µ−1)tϕ(ξ(24ξ+1)ϕ2−1)\nξ(12ξ+1)ϕ2−1−(µ−2)(µ−1)(ξϕ2−1)\nξ(12ξ+1)ϕ2−1\nα2\n+2(µ−4)(ξϕ2−1)\nξ(12ξ+1)ϕ2−1+t(3ϕ−3ξ(8ξ+1)ϕ3)\nξ(12ξ+1)ϕ2−1\nα!\n, (173)\nt′(τ) =t(τ)/α(τ), (174)\nα(0) = t0H0, ϕ(0) = ϕ0, ϕ′(0) = ϕ′\n0, t(0) = t0, (175)\nand for the initial conditions for the scalar field, the speed of the scalar field, we consider a\nmodel with dust-like matter ( p= 0), and we evaluate at τ= 0(today) the expressions (142)\nand (145) to obtain\n−ϕ2\n0ϕ′\n02\n6H2\n0−µ+ξϕ2\n0(−µ+ 2t0ϕ′\n0+ 1)−1\nH0t0−ξϕ2\n0−ΩΛ0−Ωm0+ 1 = 0 , (176)\n−6H2\n0t2\n0ΩΛ0\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001\n+ 6H2\n0t2\n0\u0000\nξϕ2\n0−1\u00012\n+ 12H0t0\u0000\nξϕ2\n0−1\u0001\u0000\nµ\u0000\nξϕ2\n0−1\u0001\n+ξϕ2\n0\u0000\n12ξ+t0ϕ′\n0−3\u0001\n+ 3\u0001\n−6(µ−2)(µ−1)\u0000\nξϕ2\n0−1\u00012+t2\n0ϕ2\n0\u0000\nξ\u0000\nϕ2\n0+ 12\u0001\n−1\u0001\nϕ′\n02\n+ 12( µ−1)ξt0ϕ2\n0\u0000\nξϕ2\n0−1\u0001\nϕ′\n0= 0. (177)\nIn [67] the assumption H=H(ϕ,˙ϕ) =ε˙ϕ/ϕwas explored, corresponding to ϕ′(τ)/ϕ(τ) =ε,\nandα=tε˙ϕ/ϕ.\nIn this paper, we consider a more general case where the new variables are introduced\nx1=t(τ)ϕ′(τ)√\n6α(τ), x 2=t(τ)ϕ(τ), x 3=α(τ), x 4=1\nt(τ)2(ξ(12ξ+ 1)ϕ(τ)2−1),(178)\nthat satisfy\nρ+ Λ = −3x3x4\u0000\n−µ+x2\n2x4\u0000\nx2\n1x3+ 2√\n6ξx1x3−12ξ2(−µ+x3+ 1)\u0001\n+x3+ 1\u0001\n\u0000\nξ(12ξ+ 1)x2\n2x4−1\u00012,(179)46\nand the new time\nf′≡α3d f\ndτ, (180)\nwhich preserves the arrow of time ( α≥0), obtaining the dynamical system\nx′\n1=−24ξx3\n1x2\n2x3\n3x4+ 4√\n6ξx2\n1x2\n2x2\n3x4(µ−3ξx2+x3−1)\n+x1x3h\n−2µ2+ 6µ+ 12ξ2x2\n2x4\u0000\n2µ2−6µ+x2(µ+x3−1)−4µx3−2(x3−8)x3+ 4\u0001\n+µx2−3x2x3−x2+ 6x2\n3+ 4µx3−16x3−4i\n−√\n6ξx2\u0000\n−(µ−2)(µ−1) +x2\n3+ 2(µ−4)x3\u0001\u0000\n12ξ2x2\n2x4−1\u0001\n, (181)\nx′\n2=x2\n3\u0010√\n6x1x2\n3+x2\u0011\n, (182)\nx′\n3=x2\n3h\nµ2−3µ+ 2ξx1x2\n2x3x4\u0010\n6x1x3−√\n6(µ+x3−1)\u0011\n+ 12ξ2x2\n2x4\u0000\n−(µ−2)(µ−1) +x2\n3+ 2(µ−4)x3\u0001\n−3x2\n3−2µx3+ 9x3+ 2i\n, (183)\nx′\n4=−2x2\n3x4\u0010√\n6ξ(12ξ+ 1)x1x2x2\n3x4+ 1\u0011\n. (184)\n1. The set x3= 0is an invariant set of (181), (182), (183) and (184) when\n√\n6(µ−2)(µ−1)ξx2\u0000\n12ξ2x2\n2x4−1\u0001\n= 0. (185)\nThat is, when\n(a)µ∈ {1,2}o\n(b)ξ= 0o\n(c)x2= 0o\n(d)x4x2\n2=1\n12ξ2.\n2. Taking x4= 0, we obtain the dynamical system\nx′\n1=x1x3(6µ+x2(µ−3x3−1) + 2(3 x3−µ)(µ+x3)−16x3−4)\n+√\n6ξx2\u0000\n−(µ−2)(µ−1) +x2\n3+ 2(µ−4)x3\u0001\n, (186)\nx′\n2=x2\n3\u0010√\n6x1x2\n3+x2\u0011\n, (187)\nx′\n3=x2\n3\u0000\nµ2−3µ−2µx3−3(x3−3)x3+ 2\u0001\n, (188)\nThe sytem (186), (187) and (188) supports the following equilibrium points:\n(a)Q1: (x1, x2, x3) := ( x1c,0,0), which is a non-hyperbolic critical point curve.47\n(b)Q2: (x1, x2, x3) :=\u0010\n0,0,1\n6(−r−2µ+ 9)\u0011\n, where r=p\n8µ(2µ−9) + 105 with\neigenvalues {f1(µ,ξ)\n36,f2(µ,ξ)\n108,f3(µ,ξ)\n108}, where fiare complicated expressions that de-\npend on µandξ. Since x3≥0, this point only exists for 1≤µ≤2.In figure 8,\nthe point can show source or saddle behaviour.\n(c)Q3: (x1, x2, x3) :=\u0010\n0,0,1\n6(r−2µ+ 9)\u0011\nwith eigenvalues {g1(µ,ξ)\n36,g2(µ,ξ)\n108,g3(µ,ξ)\n108},\nwhere giare complicated expressions that depend on µandξ. In figure 8, the\npoint is a saddle.\n(d)Q4: (x1, x2, x3) :=\u0010ξ((µ−2)(µ−1)(2µ−5)r+(2−µ)(µ−1)(8µ2−26µ+39))\n4√\n6(µ−2)(µ−1)2\n+3µr−10r−12µ2+65µ−102\n4√\n6(µ−2)(µ−1)2,\n1\n36ξ((−µ(2µ−15)−57)r+µ(8µ2+ 30µ−279) + 549) +1\n36(−3r+ 12µ−9),\n1\n6(−r−2µ+ 9)\u0011\n, with eigenvalues {h1(µ,ξ)\n36,h2(µ,ξ)\n108(µ−2)(µ−1)2,h3(µ,ξ)\n108(µ−2)(µ−1)2},\nwhere, again, hiare complicated expressions that depend on µandξ. Since x3≥0,\nthis point only exists for 1≤µ≤2.In figure 9, the point can show source or\nsaddle behaviour.\n(e)Q5: (x1, x2, x3) :=\u0010ξ(−µ(µ(2µ−11)+19) r+10r+(2−µ)(µ−1)(8µ2−26µ+39))\n4√\n6(µ−2)(µ−1)2\n+−3µr+10r−12µ2+65µ−102\n4√\n6(µ−2)(µ−1)2,\n1\n36ξ((µ(2µ−15) + 57) r+µ(8µ2+ 30µ−279) + 549) +1\n36(3r+ 12µ−9),\n1\n6(r−2µ+ 9)\u0011\n, with eigenvalues {l1(µ,ξ)\n36,l2(µ,ξ)\n108(µ−2)(µ−1)2,l3(µ,ξ)\n108(µ−2)(µ−1)2}, where li\nare complicated expressions that depend on µandξ. In figure 9, the point is a\nsaddle.\n3.3.1. Alternative formulation of the dynamical system.\nWe define the variables\nx=˙ϕ(t)√\n6H(t), y =ϕ(t),ΩΛ=Λ\n3H2(t), (189)\nthat satisfy\n(x+√\n6ξy)2+ξy2(1−6ξ) +(µ−1) (1−ξy2)\nα+ Ω Λ+ Ω m= 1, (190)\nwhere\nΩm=ρm\n3H2, (191)48\nFIG. 8: Real parts of the eigenvalues of Q2(left) and Q3(right). Q2can exhibit source or saddle\nbehaviour, and Q3is a saddle.\nFIG. 9: Real parts of the eigenvalues of Q4(left) and Q5(right). Q4can exhibit either source or\nsaddle behaviour, and Q5is a saddle.\nwhich is the dimensionless energy density of matter, and ΩΛis interpreted as the energy\ndensity of the cosmological constant. The equation of state parameter of matter can be\nexpressed in terms of these variables as49\nFIG. 10: Phase space for system (186),(187)and(188)forξ= 4andµ=1\n2(7±√\n17).The red line\ndenotes the non-hyperbolic set of points Q1.Since x3≥0, only two equilibrium points appear in\nthe right-hand graph.\nFIG. 11: Projection of the dynamics of (186),(187)and(188)in the plane x3= 2forµ=\n1\n2(7±√\n17)andξ= 4.The origin dynamics of the plane (x1, x2) = (0 ,0)(in yellow) shows that\nQ3= (0,0,1\n6(r−2µ+ 9))is an unstable focus. The spiral is generated from the initial conditions\nA= (0.1,0.1),B= (0.1,0.1,4)andC= (1.9,−0.5,1).50\nw=n\nα\u0000\nξ(12ξ+ 1)y2−1\u0001\u0010\n−α+µ+αx2+ξy\u0010\n2√\n6αx+y(α−µ+ 1)\u0011\n+αΩΛ−1\u0011o−1\n×(\nα2h\nx2\u0000\nξ\u0000\ny2+ 12\u0001\n−1\u0001\n+ 2√\n6ξxy\u0000\nξy2−1\u0001\n+ξy2\u0000\n−(12ξ+ 1)Ω Λ+ξy2−2\u0001\n+ Ω Λ+ 1i\n+ 2α\u0000\nξy2−1\u0001h\nµ\u0010\nξy\u0010√\n6x+y\u0011\n−1\u0011\n+ξy\u0010\n−√\n6x+ 12ξy−3y\u0011\n+ 3i\n−(µ−2)(µ−1)\u0000\nξy2−1\u00012)\n. (192)\nIntroducing the new time derivative\nd f\ndτ≡1\nHd f\ndt, (193)\nwe obtain the four-dimensional dynamical system\nα2Gdx\ndτ=−12α2ξx3+ 2√\n6αξx2y(−6αξ+α+µ−1)\n+x\u0000\n3α\u0000\n−µ+ 8(µ−1)ξ2y2+ (µ−3)ξy2+ 3\u0001\n−(µ−2)(µ−1)\u0000\nξy2−1\u0001\u0001\n+√\n6ξy\u0000\nα2+ 2α(µ−4)−(µ−2)(µ−1)\u0001\u0000\nξy2−1\u0001\n, (194)\ndy\ndτ=√\n6x, (195)\nα2GdΩΛ\ndτ= 2Ω Λ(\nα2\u0010\n−12ξx2+ 2√\n6ξxy+ 3ξ(8ξ+ 1)y2−3\u0011\n+ 2α\u0010√\n6(µ−1)ξxy+ (µ−4)\u0000\nξy2−1\u0001\u0011\n−(µ−2)(µ−1)\u0000\nξy2−1\u0001)\n,(196)\nαGdα\ndτ= 3α2+ 2αµ−9α−µ2+ 3µ+ξ\u0010\n12α2x2−2√\n6αxy(α+µ−1)\n+y2\u0000\n−2αµ−3(α−3)α+µ2−3µ+ 2\u0001\u0011\n+ 12(1 −2α)αξ2y2−2, (197)\nwhere\nG:= (ξ(12ξ+ 1)y2−1), (198)\ndefined in phase space\n(x+√\n6ξy)2+ξy2(1−6ξ) +(µ−1) (1−ξy2)\nα+ Ω Λ≤1 (199)\nwhich corresponds to Ωm≥0.\nWe introduce temporal rescaling\nf′:=α2d f\ndτ, (200)51\nFIG. 12: Real parts of the eigenvalues associated with P2. This shows that the point is a sink if\n(µ, ξ)∈Ψ, or a saddle (assuming ξ≥0).\nthat preserves the arrow of time. Sometimes, we use the time variable τ=ln(a/a0)which is\nmore convenient.\nWe obtain the equilibrium points with coordinates (x, y,ΩΛ, α):\n1.The line of equilibrium points P1(yc)with coordinates\u0000\n−√\n6ξyc, yc,0,0\u0001\nparameterized\nbyyc∈R. The equation of state parameter is complex infinity. The eigenvalues\nare\u001a\n0,−2(µ−2)(µ−1)(ξy2\nc−1)\nξ(12ξ+1)y2c−1,−(µ−2)(µ−1)(ξy2\nc−1)\nξ(12ξ+1)y2c−1,(µ−2)(µ−1)(ξy2\nc−1)\nξ(12ξ+1)y2c−1\u001b\n.The curve is a saddle\nbecause it has at least two eigenvalues with different signs.\n2.The equilibrium point P2:\u0000\n0,0,0,1\n6(9−2µ−r)\u0001\n. The equation of state parameter is\nw=r+7\n4−4µ. The eigenvalues are\nn\n1\n3(9−2µ−r),1\n6(−2µ(8µ+r−36) + 9 r−105),1\n36(9(21−4µ)µ−9(µ−3)r−279−s−),\n1\n36(9(21−4µ)µ−9(µ−3)r−279 + s−)o\nwhere r=p\n8µ(2µ−9) + 105 ands−is the\npositive square root of s2\n−= 81( µ(4µ−21) + ( µ−3)r+ 31)2−48ξ(µ(µ(2µ(82µ−933) +\n8355) −17361) + ( µ(5µ(8µ−75) + 1209) −1413) r+ 14481) .It is a saddle for Ψ :=\u001a\nµ≤0.463848 ,0< ξ≤µ(32µ3−8µ2(r+24)+6 µ(5r+83)−51r−567)+18(r+17)\n48(µ−2)(µ−1)(4(µ−4)µ+23)\u001b\n∪\nn\n0.463848 < µ < 1,0< ξ≤µ(2µ(4µ(4µ+r−24)−15r+249)+51 r−567)−18(r−17)\n48(µ−2)(µ−1)(4(µ−4)µ+23)o\n∪\nn\nµ >2,0< ξ≤µ(2µ(4µ(4µ+r−24)−15r+249)+51 r−567)−18(r−17)\n48(µ−2)(µ−1)(4(µ−4)µ+23)o\n.It is a source for\n1\n48\u0012\n−√\n8µ(2µ−9)+105 (µ(8µ2−30µ+51)−18)\n(µ−2)(µ−1)(4(µ−4)µ+23)+38−20µ\n4(µ−4)µ+23+20\nµ−2−7\nµ−1+ 8\u0013\n≤ξ <0.This case52\nFIG. 13: Real parts of the eigenvalues associated with P3. Hence, the point is a saddle.\nis discarded under the assumption ξ≥0. Figure 12 presents the real parts of the\neigenvalues associated with P2. This shows that the point is well for (µ, ξ)∈Ψ, or a\nsaddle (assuming ξ≥0).\n3.The equilibrium point P3:\u0000\n0,0,0,1\n6(9−2µ+r)\u0001\n. The equation of state parameter is\nw=r−7\n4(µ−1). The eigenvalues are\nn\n1\n3(9−2µ+r),1\n6(2µ(−8µ+r+ 36)−3(3r+ 35)) ,1\n36(9(21−4µ)µ+ 9(µ−3)r−279−s+),\n1\n36(9(21−4µ)µ+ 9(µ−3)r−279 + s+)o\nwhere r=p\n8µ(2µ−9) + 105 ands+is the\npositive square root of s2\n+= 81((21 −4µ)µ+ (µ−3)r−31)2+ 48ξ(µ(µ(2(933 −82µ)µ−\n8355) + 17361) + ( µ(5µ(8µ−75) + 1209) −1413) r−14481).The point is always saddle,\nas shown in figure 13.\n4.The equilibrium point P4:\u0010\n0,−1√ξ,0,1\n2\u0011\n. The equation of state parameter is indeter-\nminate. The eigenvalues are\n\b\n−1\n2,1,1\n4\u0000\n2µ−√8µ−19−2\u0001\n,1\n4\u0000\n2µ+√8µ−19−2\u0001\t\n. The eigenvalue 1corre-\nsponds to the coordinate ΩΛ. It is always saddle for ΩΛ̸= 0. In the invariant\nsetΩΛ= 0, they can be attractors if µ < 1. For µ= 1the eigenvalues reduce to\n−1\n2,±i√\n11\n4and as shown in the lower panel of Figure 15 are spiral attractors. The\nanalysis of this invariant set is presented in section 3.4.\n5.The equilibrium point P5:\u0010\n0,1√ξ,0,1\n2\u0011\n. The equation of state parameter is indetermi-53\nnate. The eigenvalues are\n\b\n−1\n2,1,1\n4\u0000\n2µ−√8µ−19−2\u0001\n,1\n4\u0000\n2µ+√8µ−19−2\u0001\t\n. The eigenvalue 1corre-\nsponds to the coordinate ΩΛ. It is always saddle for ΩΛ̸= 0. In the invariant\nsetΩΛ= 0, they can be attractors if µ < 1. For µ= 1the eigenvalues reduce to\n−1\n2,±i√\n11\n4and as shown in the lower panel of Figure 15 are spiral attractors. The\nanalysis of this invariant set is presented in section 3.4.\nThere are additional points that cancel out the numerator and denominator, such as\nthe following points:\n6. The set P6(xc) :\u0010\nxc,−1√ξ,ΩΛ0,0\u0011\n.\n7.The line P7(xc) :\u0010\nxc,−1√ξ,0,0\u0011\n. In both cases, the stability analysis of these point\ncurves will be left for future research since it cannot be implemented with the techniques\ndeveloped in this paper.\n8. The line P8:\u0010\n0,−1√ξ,ΩΛ0,0\u0011\n. The eigenvalues aren\n−2,4,µ−√µ−1−1\nα+o(1),µ+√µ−1−1\nα+o(1)o\nwhen α→0. Therefore, it is a saddle.\n9. The point P9:\u0010\n0,−1√ξ,0,0\u0011\n. The eigenvalues are\nn\n−2,4,µ−√µ−1−1\nα+o(1),µ+√µ−1−1\nα+o(1)o\nwhen α→0. Therefore, it is a sad-\ndle.\n10. The set P10(xc) :\u0010\nxc,1√ξ,ΩΛ0,0\u0011\n.\n11.The line P11(xc) :\u0010\nxc,1√ξ,0,0\u0011\n. In both cases, the stability analysis of these point\ncurves will be left for future research since it cannot be implemented with the techniques\ndeveloped in this paper.\n12. The line P12:\u0010\n0,1√ξ,ΩΛ0,0\u0011\n. The eigenvalues are\nn\n−2,4,µ−√µ−1−1\nα+o(1),µ+√µ−1−1\nα+o(1)o\nwhen α→0. Therefore, it is a sad-\ndle.\n13. The point P13:\u0010\n0,1√ξ,0,0\u0011\n. The eigenvalues are54\nn\n−2,4,µ−√µ−1−1\nα+o(1),µ+√µ−1−1\nα+o(1)o\nwhen α→0. Therefore, it is a sad-\ndle. All have indeterminate parameters of the equation of state of matter.\n14. The line\u0012\n0,−1√\nξ(12ξ+1),ΩΛ0,−µ−p\nµ(2µ−11) + 18 + 4\u0013\n.\n15. The line\u0012\n0,−1√\nξ(12ξ+1),ΩΛ0,−µ+p\nµ(2µ−11) + 18 + 4\u0013\n.\n16. The line\u0012\n0,1√\nξ(12ξ+1),ΩΛ0,−µ−p\nµ(2µ−11) + 18 + 4\u0013\n.\n17.The line\u0012\n0,1√\nξ(12ξ+1),ΩΛ0,−µ+p\nµ(2µ−11) + 18 + 4\u0013\n. Stability can be determined\nnumerically since the Jacobian matrix has infinite entries.\nIn these four cases, the stability analysis of these curves/sets of points will be left for future\nresearch since it cannot be implemented with the techniques developed in this paper. They\nall have infinitely complex parameters of the equation of state of matter.\nTABLE IV: List of initial conditions used in the numerical integration of the system (194)-(197).\nAs shown in figures 15 and 16, the values of ξconsidered are 1/6and0.01while the values chosen\nfor the parameter µare1and1.1.\nSol. x(0) y(0) Ω Λ(0) α(0)\nI 0.011√ξ+0.010.1 0 .01\nII 0.01−1√ξ+0.010.1 0 .01\nThe upper panels of figure 14 show the numerical solution x(τ),y(τ),ΩΛ(τ)andα(τ)of\nthe system (194)-(197), as well as the effective equation of state of matter w(τ)defined in\n(192)forµ <1. Two spiral orbits of the system, starting at the initial conditions labelled I\nandIIin Table IV, asymptotically tend the equilibrium points P4andP5.\nIn figure 15, the top panels show the numerical solution x(τ),y(τ),ΩΛ(τ)andα(τ)of the\nsystem(194)-(197), as well as the effective equation of state of matter w(τ)defined in (192).\nThe values of the parameters ξandµare shown in the figure. In both cases, the equation\nof state w(τ)begins in a phantom regime ( w <−1), crosses the matter-dominated region\n(w= 0) and tends asymptotically to a regime of Sitter ( w=−1). Two spiral orbits of the\nsystem starting at the initial conditions labelled IandIIin Table IV asymptotically tend\nthe equilibrium points P4andP5.55\n0 50 100 150 200250-4-2024\nτξ=1\n6,μ= 0.9 , y (0)=1\nξ+0.012/3-1wxyΩΛα\n0 50 100 150 200250-4-2024\nτξ=1\n6,μ= 0.9 , y (0)=-1\nξ+0.012/3-1wxyΩΛα\nFIG. 14: The upper panels show the numerical solution x(τ),y(τ),ΩΛ(τ)andα(τ)of the system\n(194)-(197), as well as the effective equation of state of matter w(τ)defined in (192)forµ <1.\nTwo spiral orbits of the system, starting at the initial conditions labelled IandIIin Table IV,\nasymptotically tend the equilibrium points P4andP5.\nIn figure 16 the top panels show the numerical solution x(τ),y(τ),ΩΛ(τ)andα(τ)of\nthe system (194)-(197), as well as the effective equation of state w(τ)defined in (192). The\nvalues of the parameters ξandµare shown in the figure. In both cases, the equation of\nstate w(τ)begins in a phantom regime ( w <−1), crosses the matter-dominated region56\n50 100 150 200250-4-2024\nτξ=1\n6,μ= 1, y(0)=1\nξ+0.012/3-1wxyΩΛα\n50 100 150 200250-4-2024\nτξ=1\n6,μ= 1, y(0)=-1\nξ+0.012/3-1wxyΩΛα\nFIG. 15: The upper panels show the numerical solution x(τ),y(τ),ΩΛ(τ)andα(τ)of the system\n(194)-(197), as well as the effective equation of state of matter w(τ)defined in (192)forµ= 1.\nTwo spiral orbits of the system, starting at the initial conditions labelled IandIIin Table IV,\nasymptotically tend the equilibrium points P4andP5.\n(w= 0) and tends asymptotically to a regime of Sitter ( w=−1). The oscillating behaviour\nof the functions x(τ),y(τ),ΩΛ(τ)andα(τ)is still present but exhibits a greater amplitude\ncompared to that shown in figure 15. Two orbits starting at the initial conditions labelled I\nandIIin Table IV tend asymptotically to a limit cycle.57\n50 100 150 200250-4-2024\nτξ=1\n6,μ= 1.1 , y (0)=1\nξ+0.012/3-1wxyΩΛα\n50 100 150 200250-4-2024\nτξ=1\n6,μ= 1.1 , y (0)=-1\nξ+0.012/3-1wxyΩΛα\nFIG. 16: The upper panels show the numerical solution x(τ),y(τ),ΩΛ(τ)andα(τ)of the system\n(194)-(197), as well as the effective state equation w(τ)defined in (192)forµ > 1. Two orbits\nstarting at the initial conditions labelled IandIIin Table IV asymptotically tend to a limit cycle.\nFigure 17 presents the dynamics in the invariant set ΩΛ= 0of the system (194)-(197)\nfor the values of the parameters ξ= 4andµ= 1.1. Two orbits that begin at the initial\nconditions labelled IandIIin Table IV tend to the saddle points P4andP5and then form\nan unstable spiral centred on said points.\nFinally, in Figure 18 phase diagrams are presented for the (194)-(197)system for different58\nFIG. 17: Dynamics in the invariant set ΩΛ= 0of the system (194)-(197)for the values of the\nparameters ξ= 4andµ= 1.1. Two orbits that begin at the initial conditions labelled IandII\nin Table IV tend to the saddle points P4andP5and then form an unstable spiral centred on said\npoints.\nvalues of the parameters ξandµ.\nIn this section, we have numerically examined the invariant set ΩΛ= 0of the(194)-(197)\nsystem. However, to be able to characterize such an invariant set analytically, we will present\na comprehensive analysis of dynamical systems in the section 3.4.\n3.4. Invariant set ΩΛ= 0\nReplacing ΩΛ= 0in (189), and defining the new variables\nx=˙ϕ(t)√\n6H(t), y =ϕ(t), (201)\nwhich satisfies\n(x+√\n6ξy)2+ξy2(1−6ξ) +(µ−1) (1−ξy2)\nα+ Ω m= 1, (202)59\nFIG. 18: Phase diagrams for the system (194)-(197)for different values of the parameters ξandµ.\nwhere\nΩm=ρm\n3H2(203)\nis the dimensionless energy density of matter, we obtain an alternative formulation which\ndescribes the dynamics on the invariant set ΩΛ= 0. The equation of state parameter of\nmatter can be expressed in terms of the new variables as\nw=n\nα\u0000\nξ(12ξ+ 1)y2−1\u0001\u0010\n−α+µ+αx2+ξy\u0010\n2√\n6αx+y(α−µ+ 1)\u0011\n−1\u0011o−1\n×(\nα2h\nx2\u0000\nξ\u0000\ny2+ 12\u0001\n−1\u0001\n+ 2√\n6ξxy\u0000\nξy2−1\u0001\n+ξy2\u0000\nξy2−2\u0001\n+ 1i\n+ 2α\u0000\nξy2−1\u0001h\nµ\u0010\nξy\u0010√\n6x+y\u0011\n−1\u0011\n+ξy\u0010\n−√\n6x+ 12ξy−3y\u0011\n+ 3i\n−(µ−2)(µ−1)\u0000\nξy2−1\u00012)\n. (204)\nIntroducing the derivative (193), we obtain the three-dimensional dynamical system\nα2Gdx\ndτ=−12α2ξx3+ 2√\n6αξx2y(−6αξ+α+µ−1)\n+x\u0000\n3α\u0000\n−µ+ 8(µ−1)ξ2y2+ (µ−3)ξy2+ 3\u0001\n−(µ−2)(µ−1)\u0000\nξy2−1\u0001\u0001\n+√\n6ξy\u0000\nα2+ 2α(µ−4)−(µ−2)(µ−1)\u0001\u0000\nξy2−1\u0001\n, (205)\ndy\ndτ=√\n6x, (206)\nαGdα\ndτ= 3α2+ 2αµ−9α−µ2+ 3µ+ξ\u0010\n12α2x2−2√\n6αxy(α+µ−1)\n+y2\u0000\n−2αµ−3(α−3)α+µ2−3µ+ 2\u0001\u0011\n+ 12(1 −2α)αξ2y2−2,(207)60\nwhere\nG:= (ξ(12ξ+ 1)y2−1), (208)\ndefined on the phase space\n(x+√\n6ξy)2+ξy2(1−6ξ) +(µ−1) (1−ξy2)\nα≤1 (209)\nwhich corresponds to Ωm≥0.\nIntroducing the time-rescaling (200)which preserves the arrow of time; we obtain the\nsystem\nx′=1\nξ(12ξ+ 1)y2−1\"\n−12α2ξx3+ 2√\n6αξx2y(−6αξ+α+µ−1)\n+x\u0000\n3α\u0000\n−µ+ 8(µ−1)ξ2y2+ (µ−3)ξy2+ 3\u0001\n−(µ−2)(µ−1)\u0000\nξy2−1\u0001\u0001\n+√\n6ξy\u0000\nα2+ 2α(µ−4)−(µ−2)(µ−1)\u0001\u0000\nξy2−1\u0001#\n, (210)\ny′=√\n6α2x, (211)\nα′=α\nξ(12ξ+ 1)y2−1\"\n3α2+ 2αµ−9α−µ2+ 3µ+ 12(1 −2α)αξ2y2−2\n+ξ\u0010\n12α2x2−2√\n6αxy(α+µ−1) +y2\u0000\n−2αµ−3(α−3)α+µ2−3µ+ 2\u0001\u0011#\n.(212)\nsometimes is more useful to use the time variable τ= ln( a/a0).\nThe equilibrium points of system (210), (211) y (212) in the coordinates (x, y, α )are:\n1.The equilibrium point curve P1:\u0000\n−√\n6ξyc, yc,0\u0001\nparameterized by yc∈R. The\nequation of state parameter is complex infinity. The eigenvalues are\u001a\n0,−(µ−2)(µ−1)(ξy2−1)\nξ(12ξ+1)y2−1,(µ−2)(µ−1)(ξy2−1)\nξ(12ξ+1)y2−1\u001b\n. The curve is a saddle because it has at least\ntwo eigenvalues with different signs.\n2.The equilibrium point P2:\u0000\n0,0,1\n6(9−2µ−r)\u0001\n. There exists ( α >0) for 1≤µ≤2.\nThe eigenvalues aren\n1\n6(−2µ(8µ+r−36) + 9 r−105),\n1\n72(−72µ2−u−−18rµ+ 378 µ+ 54r−558),\n1\n72(−72µ2+u−−18rµ+ 378 µ+ 54r−558)o\n. Where r=p\n8µ(2µ−9) + 105 and\nu−is the positive square root of\nu2\n−= 324 ( µ(4µ+r−21)−3r+ 31)2\n− 192(µ(3 (403 r−5787) + µ(2µ(82µ+ 20r−933)−375r+ 8355)) −1413r+ 14481) ξ.61\nFIG. 19: Eigenvalues of P2showing that the point is a saddle.\nThe critical point is a saddle in its existence interval, as shown in the figure 19. The\nequation of state parameter is w=−2µ(r+9)+3 r+35\n4µ(2µ−3)−4.\n3. The equilibrium point P3:\u0000\n0,0,1\n6(9−2µ+r)\u0001\n. The eigenvalues aren\n1\n6(2µ(−8µ+r+ 36)−3 (3r+ 35)),\n1\n72(−u+−72µ2+ 18rµ+ 378 µ−54r−558),\n1\n72(u+−72µ2+ 18rµ+ 378 µ−54r−558)o\n, where r=p\n8µ(2µ−9) + 105 andu+is\nthe positive square root of u2\n+= 324\u0000\n4µ2−(r+ 21) µ+ 3r+ 31\u00012\n+ 192 (µ(3 (403 r+ 5787) + µ(2µ(−82µ+ 20r+ 933) −15 (25 r+ 557))) −9 (157 r+ 1609)) ξ.\nP3is a sink for\b\nµ >2,0< ξ≤1\n48\t\n∪\b\nµ= 1,0< ξ≤27\n154\t\n∪\b\nµ= 2,0< ξ≤27\n280\t\n∪\u001a\nµ+1\n4\u0012\n−5−9\n3√\n58+√\n4093+3p\n58 +√\n4093\u0013\n= 0,0< ξ≤ −27(µ−3)\n4(2µ3−3µ2−12µ+90)\u001b\n∪\u001a\n0< µ <1\n4\u0012\n5−9\n3√√\n4093−58+3p√\n4093−58\u0013\n,0< ξ≤1\n48\u001b\n∪\u001a\n1\n4\u0012\n5−9\n3√√\n4093−58+3p√\n4093−58\u0013\n< µ < 1,0< ξ≤1\n48\u001b\n.\nIt is a saddle otherwise.\nThe equation of state parameter is w=2µ(r−9)−3r+35\n4µ(2µ−3)−4.\n4. The equilibrium point P4:\u0010\n0,−1√ξ,1\n2\u0011\n. The eigenvalues aren\n−1\n2,1\n4\u0000\n2µ−√8µ−19−2\u0001\n,1\n4\u0000\n2µ+√8µ−19−2\u0001o\n. The equation of state param-\neter is w=1\n24(ξ3/2+ξ). The point is sink for 0< µ < 1andξ >0, and is saddle for\nµ >1. As shown in the figure 17.62\n5. The equilibrium point P5:\u0010\n0,1√ξ,1\n2\u0011\n. The eigenvalues aren\n−1\n2,1\n4\u0000\n2µ−√8µ−19−2\u0001\n,1\n4\u0000\n2µ+√8µ−19−2\u0001o\n. The equation of state param-\neter is w=1\n24(ξ3/2+ξ). The point is sink for 0< µ < 1andξ >0, and is saddle for\nµ >1. As shown in figure 17.\n6. The equilibrium point P7:\u0010\nxc,−1√ξ,0\u0011\n.\n7. The equilibrium point P9:\u0010\n0,−1√ξ,0\u0011\n. The equation of state parameter is w= 0.\n8. The equilibrium point P11:\u0010\nxc,1√ξ,0\u0011\n.\n9. The equilibrium point P13:\u0010\n0,1√ξ,0\u0011\n. The equation of state parameter is w= 0.\nDue to the complexity of the stability analysis of these four points, it will be left for\nfuture research since it cannot be implemented with the techniques developed in this\npaper.\n10. The equilibrium points\u0012\n0,−1√\nξ(12ξ+1),−µ−p\nµ(2µ−11) + 18 + 4\u0013\n.\n11. The equilibrium points\u0012\n0,−1√\nξ(12ξ+1),−µ+p\nµ(2µ−11) + 18 + 4\u0013\n.\n12. The equilibrium points\u0012\n0,1√\nξ(12ξ+1),−µ−p\nµ(2µ−11) + 18 + 4\u0013\n.\n13. The equilibrium points\u0012\n0,1√\nξ(12ξ+1),−µ+p\nµ(2µ−11) + 18 + 4\u0013\n.\nIn all four cases, the equation of state parameter is complex and infinite. Due to the\ncomplexity of the stability analysis of this set of equilibrium points, numerical solution\nmethods can be used. The stability analysis of these four points is very complex, so it will\nbe left for future research since it cannot be implemented with the techniques developed in\nthis paper.\n3.4.1. Asymptotic expansions for ε:=µ−1→0\nNext, we will study the perturbation problem related to the study of the phase portrait\nof a system of differential equations\n˙x=X(x;ε), x∈R3, ε∼0 (213)63\nin a neighbourhood of the origin, where the unperturbed vector field is X(x; 0)[102–108].\nTo obtain asymptotic expansions for µ→1, the reparametrization ε:=µ−1is introduced\nand the limit ε→0is analyzed using the asymptotic expansion\nρ=ρ0+ερ1+O(ε2), p=p0+εp1+O(ε2), H=H0+εH1+O(ε2), ϕ=ϕ0+εϕ1+O(ε2).\nAfter making the substitution in the field equations and comparing the coefficients with\nequal powers of ε, the following is obtained.\nAtorder 0we have the equations\n˙H0=2tξ˙ϕ02−2tξH 0ϕ0˙ϕ0+ 3H0\u0000\n−ξ((8ξt+t)H0−2)ϕ2\n0+tH0−2\u0001\nt\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001 , (214)\n¨ϕ0=−12tξ2ϕ0˙ϕ02−3tH0\u0000\nξ(8ξ+ 1)ϕ2\n0−1\u0001˙ϕ0+ 6ξH0(tH0−6)ϕ0\u0000\nξϕ2\n0−1\u0001\nt\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001 ,(215)\nand the auxiliary equation\n˙ρ0=3H0\nt\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001\"\nt\u0000\nξ(6ξ+ 1)ϕ2\n0+ 6ξ−1\u0001˙ϕ02+ 12tξH 0ϕ0\u0000\nξ(6ξ+ 1)ϕ2\n0−1\u0001˙ϕ0\n+ 6H0\u0000\nξϕ2\n0−1\u0001\u0000\nξ(12ξ+ (6ξt+t)H0−2)ϕ2\n0−tH0+ 2\u0001#\n, (216)\nand the constraints\nρ0= 3\u0000\n1−ξϕ2\n0\u0001\nH2\n0−6ξϕ0˙ϕ0H0−1\n2˙ϕ02−Λ, (217)\np0=−1\n2t\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001\"\n6tH2\n0\u0000\nξϕ2\n0−1\u00012+ 12H0\u0010\nξϕ0\u0010\n2(6ξ−1)ϕ0+t˙ϕ0\u0011\n+ 2\u0011\u0000\nξϕ2\n0−1\u0001\n+t\u0000\nξϕ2\n0+ 12ξ−1\u0001˙ϕ02+ 2tΛ\u0000\n1−ξ(12ξ+ 1)ϕ2\n0\u0001#\n. (218)\nAtorder 1we have the equations\n˙H1=1\nt2\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u00012\"\n−24ξ2H2\n0ϕ0ϕ1t2+ 4ξ\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001˙ϕ0˙ϕ1t2\n−2H1\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001\u0010\nξϕ0\u0010\nt˙ϕ0−3ϕ0\u0011\n+ 3\u0011\nt\n+ 2H0 \n3tH1\u0000\nξϕ2\n0\u0000\n−ξ(8ξ+ 1)(12 ξ+ 1)ϕ2\n0+ 20ξ+ 2\u0001\n−1\u0001\n+ξn\n−ξ(12ξ+ 1)ϕ4\n0−tξ(12ξ+ 1) ˙ϕ1ϕ3\n0+\u0010\nt(12ξ+ 1)ϕ1˙ϕ0ξ+ 12ξ+ 2\u0011\nϕ2\n0\n+\u0010\n72ξϕ1+t˙ϕ1\u0011\nϕ0+tϕ1˙ϕ0o\n−1!\nt\n+ 2ξ(6ξ+ 1)ϕ2\n0+ξϕ0\u0010\n2t˙ϕ0−ξ(12ξ+ 1)\u0010\nϕ3\n0+ 2t˙ϕ0ϕ2\n0+ 4t2ϕ1˙ϕ02\u0011\u0011\n−1#\n, (219)64\n¨ϕ1=1\nt2\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u00012\"\n6ξ3(12ξ+ 1)(2 t(H0+ (tH0−3)H1) + 1) ϕ5\n0\n+tξ2(12ξ+ 1)\u0010\n6tξϕ1H2\n0−36ξϕ1H0−3t(8ξ+ 1) ˙ϕ1H0+ (24 ξ−3t(8ξ+ 1)H1+ 1) ˙ϕ0\u0011\nϕ4\n0\n+ 12ξ2\u0010\n−2ξ(12ξ+ 1) ˙ϕ0˙ϕ1t2−(6ξ+ 1)(2 t(H0+ (tH0−3)H1) + 1)\u0011\nϕ3\n0\n+ 2tξn\n6tξ(6ξ−1)ϕ1H2\n0+ 3\u0010\n12(1−6ξ)ξϕ1+t(10ξ+ 1) ˙ϕ1\u0011\nH0\n+˙ϕ0\u0010\n6t(12ξ+ 1)ϕ1˙ϕ0ξ2−18ξ+ 3t(10ξ+ 1)H1−1\u0011o\nϕ2\n0\n+ 6ξ\u0010\n2t\u0010\n−3H1+H0\u0010\ntH1−2tξϕ1˙ϕ0+ 1\u0011\n+ 2tξ˙ϕ0˙ϕ1\u0011\n+ 1\u0011\nϕ0\n+t\u0010\n6tξϕ1H2\n0−3\u0010\n12ξϕ1+t˙ϕ1\u0011\nH0+˙ϕ0\u0010\n12tϕ1˙ϕ0ξ2−3tH1+ 1\u0011\u0011#\n, (220)\nthe auxiliary equation\n˙ρ1=1\nt2\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u00012\"\n36t2ξϕ0\u0000\nξ(6ξ+ 1)\u0000\nξ(12ξ+ 1)ϕ2\n0−2\u0001\nϕ2\n0−6ξ+ 1\u0001\nϕ1H3\n0\n+ 3t \n18tH1\u0000\nξϕ2\n0−1\u0001\u0000\nξ(6ξ+ 1)ϕ2\n0−1\u0001\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001\n+ξ(\nξ2(1−36ξ(16ξ+ 1)) ϕ6\n0+ 12ξ2(12ξ+ 1)\u0010\n2(6ξ−1)ϕ1+t(6ξ+ 1) ˙ϕ1\u0011\nϕ5\n0\n+ 3ξ\u0010\n24ξ(8ξ+ 1) + 4 tξ(6ξ+ 1)(12 ξ+ 1)ϕ1˙ϕ0−1\u0011\nϕ4\n0\n−24ξ\u0010\n2(6ξ−1)ϕ1+t(9ξ+ 1) ˙ϕ1\u0011\nϕ3\n0−3\u0010\n8t(3ξ+ 1)ϕ1˙ϕ0ξ+ 12ξ−1\u0011\nϕ2\n0\n+ 12\u0010\n(36ξ−2)ϕ1+t˙ϕ1\u0011\nϕ0+ 12tϕ1˙ϕ0)\n−1!\nH2\n0\n+ 3 \n2\u0000\nξ(6ξ+ 1)ϕ2\n0+ 6ξ−1\u0001\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001˙ϕ0˙ϕ1t2\n+ 24H1\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001\u0010\nξϕ0\u0010\nϕ0\u0000\nξ(6ξ−1)ϕ2\n0−6ξ+ 2\u0001\n+t\u0000\nξ(6ξ+ 1)ϕ2\n0−1\u0001˙ϕ0\u0011\n−1\u0011\nt\n+ξϕ0(\nξ2(12(5−24ξ)ξ+ 7)ϕ5\n0+ 3ξ(8ξ(12ξ−5)−7)ϕ3\n0+ 3(20 ξ+ 7)ϕ0\n−144t2ξ2ϕ1˙ϕ02+ 2t\u0000\nξ\u0000\u0000\nξ−144ξ3\u0001\nϕ2\n0−2\u0001\nϕ2\n0+ 1\u0001˙ϕ0)\n−7!\nH0\n+t(3tH1−1)\u0000\nξ(6ξ+ 1)ϕ2\n0+ 6ξ−1\u0001\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001˙ϕ02#\n, (221)\nand the constraints\nρ1=−6ξϕ0ϕ1H2\n0−3(2tH1−1)\u0000\nξϕ2\n0−1\u0001\nH0\nt−6ξ(H1ϕ0+H0ϕ1)˙ϕ0−\u0010\n6ξH0ϕ0+˙ϕ0\u0011\n˙ϕ1,(222)65\np1=−1\nt2\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u00012\"\n−144t2ϕ0ϕ1˙ϕ02ξ3+ 6t2H2\n0ϕ0\u0000\nξϕ2\n0−1\u0001\u0000\nξ(12ξ+ 1)ϕ2\n0+ 12ξ−1\u0001\nϕ1ξ\n+ 6tϕ0\u0000\nξϕ2\n0−1\u0001\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001˙ϕ0ξ+ 3\u0000\nξϕ2\n0−1\u00012\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001\n+ 6tH1\u0000\nξϕ2\n0−1\u0001\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001\u0010\nξϕ0\u0010\n2(6ξ−1)ϕ0+t˙ϕ0\u0011\n+ 2\u0011\n+t2\u0000\nξϕ2\n0+ 12ξ−1\u0001\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001˙ϕ0˙ϕ1+ 6tH0n\ntH1\u0000\nξ(12ξ+ 1)ϕ2\n0−1\u0001\u0000\nξϕ2\n0−1\u00012\n+ξ \nξ2(12ξ+ 1)ϕ6\n0+ξ2(12ξ+ 1)\u0010\n4(6ξ−1)ϕ1+t˙ϕ1\u0011\nϕ5\n0+ξ\u0010\ntξ(12ξ+ 1)ϕ1˙ϕ0−3(8ξ+ 1)\u0011\nϕ4\n0\n−2ξ\u0010\n4(6ξ−1)ϕ1+t(6ξ+ 1) ˙ϕ1\u0011\nϕ3\n0+\u0010\n2t(6ξ−1)ϕ1˙ϕ0ξ+ 12ξ+ 3\u0011\nϕ2\n0\n+\u0010\n(72ξ−4)ϕ1+t˙ϕ1\u0011\nϕ0+tϕ1˙ϕ0!\n−1o#\n. (223)\nFor simplicity, we will solve the case ξ= 0.\nAtorder 0we have the equations\n˙H0=−3H0(−2 +tH0)\nt, (224)\n˙ρ0=−3H0\u0010\n−6H0(2−tH0)−t˙ϕ2\u0011\nt, (225)\n¨ϕ0=−3H0˙ϕ0. (226)\nand the constraints\nρ0= 3H2\n0−Λ−1\n2˙ϕ02, (227)\np0=6tH2\n0−24H0+ 2Λ t−t˙ϕ02\n2t. (228)\nAtorder 1we have the equations\n˙H1=−1 + 6 tH1+ 2tH0(−1−3tH1)\nt2, (229)\n˙ρ1=1\nt2\u0010\n3tH2\n0(−1−18tH1) +t(−1 + 3 tH1)˙ϕ02+ 3H0\u0010\n−7 + 24 tH1+ 2t2˙ϕ0˙ϕ1\u0011\u0011\n,(230)\n¨ϕ1=(1−3tH1)˙ϕ0−3tH0˙ϕ1\nt. (231)\nand the constraints\nρ1=3H0(2tH1−1)\nt−˙ϕ0˙ϕ1, (232)\np1=−6tH0(−tH1−1) + 12 tH1+t2˙ϕ0˙ϕ1−3\nt2. (233)66\nSolving the resulting systems we obtain\nH0(t) =7t6\n3t7+ 7c1, (234)\nH1(t) =−51t14+ 7c2t7−980c1ln(t)t7+ 49c12\n7t(3t7+ 7c1)2, (235)\nρ0(t) =147t12\n(3t7+ 7c1)2−c32\n2 (3t7+ 7c1)2+c4, (236)\nwhere c1,c2,c3yc4are integration constants.\nϕ0(t) =c5+c3\n77√\n376\n7c16\n7\"\n−2 sin\u0010π\n7\u0011\ntan−1\ncot\u0010π\n7\u0011\n−7q\n3\n7tcsc\u0000π\n7\u0001\n7√c1\n\n+ ln\u0010\n7√\n376\n7t+ 77√c1\u0011\n−cos\u0010π\n7\u0011\nln\u0010\n32\n775\n7t2−27√\n376\n77√c1cos\u0010π\n7\u0011\nt+ 7c12\n7\u0011\n+ ln\u0012\n32\n775\n7t2+ 27√\n376\n77√c1sin\u00123π\n14\u0013\nt+ 7c12\n7\u0013\nsin\u00123π\n14\u0013\n−ln\u0010\n32\n775\n7t2−27√\n376\n77√c1sin\u0010π\n14\u0011\nt+ 7c12\n7\u0011\nsin\u0010π\n14\u0011\n+ 2 tan−1\n7q\n3\n7sec\u00003π\n14\u0001\nt\n7√c1+ tan\u00123π\n14\u0013\ncos\u00123π\n14\u0013\n+ 2 tan−1\n7q\n3\n7tsec\u0000π\n14\u0001\n7√c1−tan\u0010π\n14\u0011\ncos\u0010π\n14\u0011#\n, (237)\nρ1(t) =−1\n7 (3t7+ 7c1)3\"\n−189c7t21+ 3465 t19−1323c1c7t14+ 294(21 c1−c2)t12\n+ 21\u0000\nc3c6−147c12c7\u0001\nt7+ 5145 c12t5+ 14c1c32+c2c32+ 49c1c3c6−2401c13c7\n+ 4\u0000\n10290 c1t12+ 18c32t7+ 7c1c32\u0001\nln(t)#\n, (238)\nwhere c5,c6yc7are integration constants.67\nFinally,\nϕ1(t) =−9c3 3F2\u0012\n8\n7,8\n7,2;15\n7,15\n7;−3t7\n7c1\u0013\nt8\n2744c12+9c3 2F1\u0012\n8\n7,2;15\n7;−3t7\n7c1\u0013\nln(t)t8\n343c12+c2c3t\n343c1\u0000\n3t7+ 7c1\u0001+2c3t\n49\u0000\n3t7+ 7c1\u0001\n−4c3 3F2\u0012\n1\n7,1\n7,2;8\n7,8\n7;−3t7\n7c1\u0013\nt\n49c1+4c3 2F1\u0012\n1\n7,2;8\n7;−3t7\n7c1\u0013\nln(t)t\n49c1+c8+2\u0010\n3\n7\u00116\n7c2c3ln\u0012\n7√\n376\n7t+ 77√c1\u0013\n2401c113\n7\n+4\u0010\n3\n7\u00116\n7c3ln\u0012\n7√\n376\n7t+ 77√c1\u0013\n343c16\n7+c6ln\u0012\n7√\n376\n7t+ 77√c1\u0013\n77√\n376\n7c16\n7−2\u0010\n3\n7\u00116\n7c2c3cos2\u0010\nπ\n14\u0011\nln\u0012\n32\n775\n7t2−27√\n376\n77√c1cos\u0010\nπ\n7\u0011\nt+ 7c12\n7\u0013\n2401c113\n7\n−4\u0010\n3\n7\u00116\n7c3cos2\u0010\nπ\n14\u0011\nln\u0012\n32\n775\n7t2−27√\n376\n77√c1cos\u0010\nπ\n7\u0011\nt+ 7c12\n7\u0013\n343c16\n7−c6cos2\u0010\nπ\n14\u0011\nln\u0012\n32\n775\n7t2−27√\n376\n77√c1cos\u0010\nπ\n7\u0011\nt+ 7c12\n7\u0013\n77√\n376\n7c16\n7\n+3\u0010\n3\n7\u00116\n7c2c3csc\u0010\nπ\n14\u0011\nln\u0012\n32\n775\n7t2+ 27√\n376\n77√c1sin\u0010\n3π\n14\u0011\nt+ 7c12\n7\u0013\nsin2\u0010\nπ\n7\u0011\n4802c113\n7\n+3\u0010\n3\n7\u00116\n7c3csc\u0010\nπ\n14\u0011\nln\u0012\n32\n775\n7t2+ 27√\n376\n77√c1sin\u0010\n3π\n14\u0011\nt+ 7c12\n7\u0013\nsin2\u0010\nπ\n7\u0011\n343c16\n7\n−4\u0010\n3\n7\u00116\n7tan−1 \ncot\u0010\nπ\n7\u0011\n−7q3\n7tcsc\u0010π\n7\u0011\n7√c1!\nc2c3sin\u0010\nπ\n7\u0011\n2401c113\n7−8\u0010\n3\n7\u00116\n7tan−1 \ncot\u0010\nπ\n7\u0011\n−7q3\n7tcsc\u0010π\n7\u0011\n7√c1!\nc3sin\u0010\nπ\n7\u0011\n343c16\n7\n−2 tan−1 \ncot\u0010\nπ\n7\u0011\n−7q3\n7tcsc\u0010π\n7\u0011\n7√c1!\nc6sin\u0010\nπ\n7\u0011\n77√\n376\n7c16\n7−2\u0010\n3\n7\u00116\n7c2c3ln\u0012\n32\n775\n7t2+ 27√\n376\n77√c1sin\u0010\n3π\n14\u0011\nt+ 7c12\n7\u0013\nsin3\u0010\nπ\n14\u0011\n2401c113\n7\n−4\u0010\n3\n7\u00116\n7c3ln\u0012\n32\n775\n7t2+ 27√\n376\n77√c1sin\u0010\n3π\n14\u0011\nt+ 7c12\n7\u0013\nsin3\u0010\nπ\n14\u0011\n343c16\n7−c6ln\u0012\n32\n775\n7t2+ 27√\n376\n77√c1sin\u0010\n3π\n14\u0011\nt+ 7c12\n7\u0013\nsin3\u0010\nπ\n14\u0011\n77√\n376\n7c16\n7\n+2\u0010\n3\n7\u00116\n7c2c3ln\u0012\n32\n775\n7t2−27√\n376\n77√c1cos\u0010\nπ\n7\u0011\nt+ 7c12\n7\u0013\nsin2\u0010\nπ\n14\u0011\n2401c113\n7\n+4\u0010\n3\n7\u00116\n7c3ln\u0012\n32\n775\n7t2−27√\n376\n77√c1cos\u0010\nπ\n7\u0011\nt+ 7c12\n7\u0013\nsin2\u0010\nπ\n14\u0011\n343c16\n7+c6ln\u0012\n32\n775\n7t2−27√\n376\n77√c1cos\u0010\nπ\n7\u0011\nt+ 7c12\n7\u0013\nsin2\u0010\nπ\n14\u0011\n77√\n376\n7c16\n7\n−12\u0010\n3\n7\u00116\n7tan−1 7q3\n7sec\u00103π\n14\u0011\nt\n7√c1+ tan\u0010\n3π\n14\u0011!\nc2c3cos\u0010\nπ\n14\u0011\nsin2\u0010\nπ\n14\u0011\n2401c113\n7−24\u0010\n3\n7\u00116\n7tan−1 7q3\n7sec\u00103π\n14\u0011\nt\n7√c1+ tan\u0010\n3π\n14\u0011!\nc3cos\u0010\nπ\n14\u0011\nsin2\u0010\nπ\n14\u0011\n343c16\n7\n−2\u0010\n3\n7\u00116\n7tan−1 7q3\n7sec\u00103π\n14\u0011\nt\n7√c1+ tan\u0010\n3π\n14\u0011!\nc6cos\u0010\nπ\n14\u0011\nsin2\u0010\nπ\n14\u0011\n7c16\n7−2\u0010\n3\n7\u00116\n7c2c3ln\u0012\n32\n775\n7t2−27√\n376\n77√c1sin\u0010\nπ\n14\u0011\nt+ 7c12\n7\u0013\nsin\u0010\nπ\n14\u0011\n2401c113\n7\n−4\u0010\n3\n7\u00116\n7c3ln\u0012\n32\n775\n7t2−27√\n376\n77√c1sin\u0010\nπ\n14\u0011\nt+ 7c12\n7\u0013\nsin\u0010\nπ\n14\u0011\n343c16\n7−c6ln\u0012\n32\n775\n7t2−27√\n376\n77√c1sin\u0010\nπ\n14\u0011\nt+ 7c12\n7\u0013\nsin\u0010\nπ\n14\u0011\n77√\n376\n7c16\n7\n+\u0010\n3\n7\u00116\n7c6cos2\u0010\nπ\n14\u0011\nln\u0012\n32\n775\n7t2+ 27√\n376\n77√c1sin\u0010\n3π\n14\u0011\nt+ 7c12\n7\u0013\nsin\u0010\nπ\n14\u0011\n7c16\n7\n+4\u0010\n3\n7\u00116\n7tan−1 7q3\n7sec\u00103π\n14\u0011\nt\n7√c1+ tan\u0010\n3π\n14\u0011!\nc2c3cos3\u0010\nπ\n14\u0011\n2401c113\n7+8\u0010\n3\n7\u00116\n7tan−1 7q3\n7sec\u00103π\n14\u0011\nt\n7√c1+ tan\u0010\n3π\n14\u0011!\nc3cos3\u0010\nπ\n14\u0011\n343c16\n7\n+2 tan−1 7q3\n7sec\u00103π\n14\u0011\nt\n7√c1+ tan\u0010\n3π\n14\u0011!\nc6cos3\u0010\nπ\n14\u0011\n77√\n376\n7c16\n7+4\u0010\n3\n7\u00116\n7tan−1 7q3\n7tsec\u0010π\n14\u0011\n7√c1−tan\u0010\nπ\n14\u0011!\nc2c3cos\u0010\nπ\n14\u0011\n2401c113\n7\n+8\u0010\n3\n7\u00116\n7tan−1 7q3\n7tsec\u0010π\n14\u0011\n7√c1−tan\u0010\nπ\n14\u0011!\nc3cos\u0010\nπ\n14\u0011\n343c16\n7+2 tan−1 7q3\n7tsec\u0010π\n14\u0011\n7√c1−tan\u0010\nπ\n14\u0011!\nc6cos\u0010\nπ\n14\u0011\n77√\n376\n7c16\n7, (239)\nwhere c8is an integration constant.\nIn Figure 20, H(t),ϕ(t)andρare presented vs the asymptotic expansions H0(t),ϕ0(t)\nandρ0defined respectively by (234),(237)and(236)for the initial conditions ρm(0.1) =68\n0.5 1.0 1.5 2.0 2.5 3.0024681012H(t)\nH0(t)ϕ(t)\nϕ0(t)\n0.5 1.0 1.5 2.0 2.5 3.00100200300ρ(t)\nρ0(t)\nFIG. 20: Numerical solutions H(t),ϕ(t)andρvs the asymptotic expansions H0(t),ϕ0(t)and\nρ0(t)defined respectively by (234),(237)and(236)(valid for ξ= 0) for the initial conditions\nρm(0.1) = 1 , H(0.1) = 1 ,˙ϕ(0.1) = 1 , ϕ(0.1) = 1and the parameter values c1= 9.571429 ×10−7,\nc3= 7×10−6,c4=−1.5,c5= 0.881801,µ= 1.1, ξ=1\n6, V(ϕ) = Λ = 1 forε= 0.1. The accuracy of\nthe asymptotic approximation is illustrated when ε→0even though the analytical solutions are\ncalculated for ξ= 0.\n1, H(0.1) = 1 ,˙ϕ(0.1) = 1 , ϕ(0.1) = 1(arbitrarily chosen) and the parameter values c1=\n9.571429 ×10−7,c3= 7×10−6,c4=−1.5,c5= 0.881801,µ= 1.1, ξ=1\n6, V(ϕ) = Λ = 1 for\nε= 0.1. The accuracy of the asymptotic approximation when ε→0is illustrated.\n3.4.2. Asymptotic expansions for ξ→0\nWe consider the following expansions\nρ=ρ0+ξρ1+O(ξ2), H=H0+ξH1+O(ξ2), ϕ=ϕ0+ξϕ1+O(ξ2),\nwhere we assume that the non-minimal coupling parameter is a perturbation parameter\n0< ξ≪1.69\nReplacing them in (147), (148) and (149) we have\n−1\nt3h\n−3t3H0˙ϕ02+ 18t3H3\n0+ 3µt2H2\n0−39t2H2\n0−3(µ−1)(4µ−11)tH0\n+ 3(µ−2)(µ−1)2+t3˙ρ0+µt2˙ϕ02−t2˙ϕ02i\n−ξ\nt3h\n54t3H2\n0H1+ 6(µ−13)2H0H1−6t3H0˙ϕ0˙ϕ1+ 18t3H0˙ϕ02\n−18t3H3\n0ϕ2\n0−3t2H2\n0ϕ0\u0010\n(µ−13)ϕ0+ 12t˙ϕ0\u0011\n−6t2H0ϕ0\u0010\n(µ−1)˙ϕ0+ 3tV′(ϕ0)\u0011\n+ 3(µ−1)(4µ−11)tH0ϕ2\n0−3tH1\u0010\n(µ−1)(4µ−11) + t2˙ϕ02\u0011\n+t3˙ρ1+ 2(µ−1)t2˙ϕ0˙ϕ1−6µt2˙ϕ02+ 6µϕ0V′(ϕ0)\n−6t2ϕ0V(ϕ0) + 6t2˙ϕ02−3µ3ϕ2\n0+ 6µ2tϕ0˙ϕ0+ 12µ2ϕ2\n0\n−12µtϕ0˙ϕ0−15µϕ2\n0+ 6tϕ0˙ϕ0+ 6ϕ2\n0i\n= 0, (240)\n1\nt2h\n−t2˙H0−3t2H2\n0−2µtH 0+ 8tH0+ (µ−2)(µ−1)i\n+ξ\nt2h\n−6t2H0H1+ 2t2H0ϕ0˙ϕ0\n−t2˙H1−2(µ−4)tH1+ 2t2ϕ0V′(ϕ0)−2t2˙ϕ02+ 2(µ−1)tϕ0˙ϕ0i\n= 0, (241)\n−1\nt2h\n3t2H0˙ϕ0+t2V′(ϕ0) +t2¨ϕ0−µt˙ϕ0+t˙ϕ0i\n−ξ\nt2h\n3t2H0˙ϕ1+ 6ϕ0((µ−2)(µ−1)−tH0(tH0+ 2µ−8))\n+ 3t2H1˙ϕ0+t2ϕ1V′′(ϕ0) +t2¨ϕ1−µt˙ϕ1+t˙ϕ1i\n= 0. (242)\nAssuming a constant potential V(ϕ) = Λ, the zeroth and first-order equations are:\nZero order, O(ξ0):\n˙ρ0=−(−3tH0+µ−1)\u0010\nt\u0010\nt˙ϕ02−3H0(2tH0+µ−5)\u0011\n+ 3(µ−2)(µ−1)\u0011\nt3,(243)\n˙H0=tH0(−3tH0−2µ+ 8) + ( µ−2)(µ−1)\nt2, (244)\n¨ϕ0=(−3tH0+µ−1)˙ϕ0\nt, (245)\nwith the constraint to order zero\n−(µ−1)H0\nt+H2\n0+1\n6\u0010\n−2(Λ + ρ0)−˙ϕ02\u0011\n= 0. (246)70\nThe last two equations have the general solution\nH0(t) =−2µ−2c1rtr\n0\ntr+c1tr\n0+r+ 9\n6t, (247)\nϕ0(t) =2c2\u0010\nt\nt0\u00111\n2(4µ+r−9)\n2F1 \n1,r+4µ−9\n2r;3r+4µ−9\n2r;−\u0010\nt\nt0\u0011r\nc1!\nc1(4µ+r−9)+ϕc, (248)\nwhere c1,c2andϕcare integration constants. Therefore, the first equation reduces to\n˙ρ0=−\u0010\n2µ+r\u0010\n−1\n2+c1tr\n0\ntr+c1tr\n0\u0011\n−11\n2\u0011\n6t3\n× \n20µ2−75µ−r2(tr−c1tr\n0)2\n(tr+c1tr\n0)2+(µ−3)r(tr−c1tr\n0)\ntr+c1tr\n0+6c22t4µ+r−9t−4µ+r+9\n0\n(tr+c1tr\n0)2+ 90!\n,(249)\nwhich is integrable, giving us the solution\nρ0(t) =r3\n24t2+r2\u0012\n−5µ−8c1trtr\n0\n(tr+c1tr\n0)2+ 14\u0013\n24t2−c22t4µ+r−11t−4µ+r+9\n0\n2 (tr+c1tr\n0)2\n−r\u0010\n16µ2−92µ+8(5µ−12)tr\ntr+c1tr\n0+ 153\u0011\n24t2+5(4µ−11)(µ(4µ−15) + 18)\n24t2−Λ.(250)\nwhere c3=−Λis an integration constant.\nFirst order, O(ξ):\n˙ρ1=1\nt3\"\n3tH1\u0010\nt\u0010\nt˙ϕ02−2H0(9tH0(t) +µ−13)\u0011\n+ (µ−1)(4µ−11)\u0011\n+ (−3tH0+µ−1)\u0010\n−6tϕ0(2tH0+µ−1)˙ϕ0\n+ 3ϕ2\n0((µ−2)(µ−1)−tH0(2tH0+µ−5)) + 2 t2˙ϕ0\u0010\n3˙ϕ0−˙ϕ1\u0011\u0011#\n, (251)\n˙H1=−2\u0010\nH1(3tH0+µ−4)−ϕ0(tH0+µ−1)˙ϕ0+t˙ϕ02\u0011\nt, (252)\n¨ϕ1=−t\u0010\n(3tH0−µ+ 1) ˙ϕ1+ 3tH1˙ϕ0\u0011\n+ 6ϕ0((µ−2)(µ−1)−tH0(tH0+ 2µ−8))\nt2,(253)\nwith the restriction in the first order\nρ1=H1(6tH0−3µ+ 3)−3H0˙ϕ0\u0010\nϕ0(tH0−µ+ 1) + 2 t˙ϕ0\u0011\n−t˙ϕ0˙ϕ1\nt. (254)\nThis expression can be used to define ρ1such that the equation (251)is a compatibility71\ncondition that is satisfied for all t. Hence, we have a reduced system\n¨ϕ1=−6c2H1t1\n2(4µ+r−9)t1\n2(−4µ+r+9)\n0 +˙ϕ1((−4µ+r+ 11) tr−c1(4µ+r−11)tr\n0)\n2t(tr+c1tr\n0)\n+c2t1\n2(4µ+r−9)−2t−2µ−r\n2\n0 2F1\u0010\n1,r+4µ−9\n2r;3r+4µ−9\n2r;−trt−r\n0\nc1\u0011\n3c1(4µ+r−9) (tr+c1tr\n0)2\n× \nt9/2\n0\u0000\n4(69−14µ)µ+r2+ (8µ−30)r−423\u0001\nt2r\n−2c1\u0000\n4µ(14µ−69) + r2+ 423\u0001\ntrtr+9\n2\n0+c12\u0000\n4(69−14µ)µ+r2+ (30−8µ)r−423\u0001\nt2r+9\n2\n0!\n+Λt2\n0\u0012\n56µ2−276µ−4r2t2r\n(tr+c1tr\n0)2−r2+ 8µr+4r(−4µ+r+15)tr\ntr+c1tr\n0−30r+ 423\u0013\n6t2, (255)\n˙H1=H1\u0010\n−1 +r\u0010\n−1 +2c1tr\n0\ntr+c1tr\n0\u0011\u0011\nt\n+2c22t9−4µ\n0t4µ+r−11((4µ+r+ 3)tr−c1(−4µ+r−3)tr\n0)2F1\u0010\n1,r+4µ−9\n2r;3r+4µ−9\n2r;−trt−r\n0\nc1\u0011\n3c1(4µ+r−9) (tr+c1tr\n0)2\n−c2Λt1\n2(4µ+r)−13\n2t1\n2(−4µ+r+13)\n0 ((4µ+r+ 3)tr−c1(−4µ+r−3)tr\n0)\n3 (tr+c1tr\n0)2−2c22t4µ+r−11t−4µ+r+9\n0\n(tr+c1tr\n0)2.\n(256)\nThe second equation is integrable, giving us\nH1(t) =−c22(−4µ+r−3)(4µ+r−9)Γ\u00004µ−9\nr\u0001\nt4µ+r−10t−4µ−r+9\n0 2˜F1\u0012\n1,4µ−9\nr;r+4µ−9\nr;−\u0010\nt\nt0\u0011r\nc1\u0013\n36r\u0010\u0010\nt\nt0\u0011r\n+c1\u0011\n2\n−c22(−4µ+r+ 11)Γ\u0000r+4µ−9\nr\u0001\nt2(2µ+r)−10t−4µ−2r+9\n0 2˜F1\u0012\n1,r+4µ−9\nr;2r+4µ−9\nr;−\u0010\nt\nt0\u0011r\nc1\u0013\n3c1r\u0010\u0010\nt\nt0\u0011r\n+c1\u0011\n2\n+c2t1\n2(4µ+r)−10t−2(2µ+r)\n0\n36\u0010\u0010\nt\nt0\u0011r\n+c1\u0011\n2\"\n2c2t9\n0t1\n2(4µ+r)(12(−4µ+r+ 11) tr+c1(−4µ+r−3)(4µ+r−9)tr\n0)\nc1(4µ+r−9)\n×2F1\n1,r+ 4µ−9\n2r;3r+ 4µ−9\n2r;−\u0010\nt\nt0\u0011r\nc1\n\n−12Λ(−4µ+r+ 11) tr+9\n2t1\n2(4µ+r+13)\n0 + 12c1Λt9/2(4µ+r−11)t1\n2(4µ+3r+13)\n0 +72c2tr+9\n0t1\n2(4µ+r)\n9−4µ#\n+c4\n1 +\u0010\nt\nt0\u0011r\nc1\n−r+1\nr\u0012\n1 +c1\u0012t0\nt\u0013r\u0013\n1\nr−1, (257)72\nwhere c4is a constant of integration. The equation for ϕ1can be solved in quadratures:\nϕ1(t) =Zt\n1e\u0010Rµ3\n1\u0010\nµ−1\nµ1−3H0(µ1)\u0011\ndµ1\u0011\n×\"\nc5+Zµ3\n1e\u0010\n−Rµ2\n1\u0010\nµ−1\nµ1−3H0(µ1)\u0011\ndµ1\u0011\nµ2\n2 \n−3H1(µ2)˙ϕ0(µ2)µ2\n2\n−6((µ−2)(µ−1)−H0(µ2)µ2(2µ+H0(µ2)µ2−8))ϕ0(µ2)!\ndµ2#\ndµ3+c6, (258)\nwhere c5andc6are integration constants and H0(µ1)is calculated using (247),˙ϕ0(µ2)is\ncalculated by taking the derivative with respect to tof(248)andϕ0(µ2)is evaluated using\n(248). Finally, the expression for H1(µ2)is (257).\n4. CONCLUSIONS\nIn this paper, corrections to the Friedman and Klein-Gordon equations based on the\nformalism of fractional calculus using Caputo’s derivative were studied. Thus, we have\npresented fractional calculus as a viable and attractive option for applications to the theory\nof gravity. Based on the knowledge of the different uses that have already been given to\nfractional calculus in other areas of physics and engineering and that are shown in the paper,\na didactic exposition of the main aspects of fractional calculus has been presented, showing\nthe different approaches that exist and definitions of fractional derivative.\nThis paper analyzed the fractional theory in cosmological models with a scalar field with\na coupling constant ξ. We qualitatively analyze the case ξ= 0and mainly the general case\nofξ̸= 0. Using different formulations of dynamical systems, examine their equilibrium\npoints and determine how the fractional term affects their stability—exposing the complex\nbehaviour presented by specific equilibrium points in the systems’ phase space and numerical\nsolutions for different values for ξandµ. We use methods from perturbation theory to\nanalyze the cosmological behaviour obtaining solutions through two asymptotic expansions\nproposed for the system functions around the parameters µ= 1andξ= 0in which the\nmodel has recovered standard cosmology and the theory with minimal coupling to gravity\nwith an excellent approximation to the numerical solutions of the system.\nIn the paper’s introductory section 1, we discussed the challenges of traditional calculations\nin modelling power law phenomena. We then presented the various applications of fractional\ncalculus, highlighting its effectiveness in recent studies in cosmology. This makes it a viable\noption to explore gravity theory further. We also introduced the research questions, posed the73\nfractional formulation of the gravity problem, and outlined the paper’s general and specific\nobjectives. We then described the methodology used to develop the paper. Finally, we\nemphasized the scientific novelty of this research. Specifically, we mentioned that our analysis\nof cosmologies with a scalar field with conformal and non-minimal coupling to gravity in\nthe fractional context seeks to generalize previous results in the framework of the fractional\nformulation of gravity, making it a novel contribution.\nSection 2 summarizes the main results of fractional calculus, mentioning some approaches\nto possible fractional derivatives such as Grünwald–Letnikov, Riemann–Liouville and Caputo.\nWe review the known rules of differentiation in the fractional context under the different\napproaches and show the fractional derivative of known functions. Emphasis is placed on\ntaking integration as an inverse operation of the differentiation to define fractional integration\nby generalizing Cauchy’s formula for iterated integrals, allowing us to define the fractional\nderivative. The Liouville, Riemann, Liouville-Caputo and Caputo derivatives were defined.\nFinally, we discussed what fractional differential equations are, using the tools of Wolfram\nLanguage 13.3 [ 21], and we solved the classical problem of the harmonic oscillator and a\nphysical problem of a point particle that moves under the effect of a friction force in the\nfractional context from different approaches.\nIt is important to emphasize that fractional differential equations, which involve fractional\nderivatives, generalize ordinary differential equations. Fractional differential equations have\nbeen widely used in engineering, physics, chemistry, biology and other fields involving\nrelaxation and oscillation models, and very recently, they have been used in cosmology. For\nthis reason, since the version of Wolfram Language 13.1, two basic operators for fractional\ncalculation were implemented, the functions FractionalD andCaputoD . The algorithms\nof theMittagLefflerE functions have been updated, as they are of crucial importance in\nthe theory of fractional calculus, and the powerful DSolve function was heavily updated\nin version 13.1 to support FDE. This paper solved several fractional differential equations\nusing the DSolve function of Wolfram Language 13.3 [ 21]. The solutions to such equations\nare generally given in terms of the function MittagLefflerE , the primary function for\nfractional calculus applications. Its role in fractional differential equation solutions is similar\nin importance to the Expfunction for ODE solutions: any ODE with constant coefficients\ncan be solved in terms of Mittag-Leffler functions.\nAccording to the analysis carried out in the paper, the behaviour of the fractional harmonic74\noscillator is very similar to the behaviour of the ordinary damped harmonic oscillator. These\nexamples demonstrate that the order of a fractional differential equation can be used as a\ncontrol parameter to model some complicated systems.\nIn section 3, cosmological models were introduced within the fractional formulation of\nthe gravity framework, and previous results were discussed where the coupling is minimal\nξ= 0.The paper’s main result is the application of the tools defined in section 2 to fractional\ncosmologies with non-minimal coupling ξ̸= 0.Different dynamical systems and relevant\ncases were studied, such as the invariant set ΩΛ= 0. Some numerical integration methods\nfrom the specialized Mathematica software were used to study the behaviour of the solutions\nof the systems of differential equations coming from the cosmological model, and the stability\nof the critical points was studied using appropriate tools.\nIn the sections 3.3.1 and 3.4 we found a great technical difficulty when analyzing the\nstability of the curves and critical points P6(xc) :\u0010\nxc,−1√ξ,ΩΛ0,0\u0011\n,P7(xc) :\u0010\nxc,−1√ξ,0,0\u0011\n,\nP8:\u0010\n0,−1√ξ,ΩΛ0,0\u0011\n,P9:\u0010\n0,−1√ξ,0,0\u0011\n,P10(xc) :\u0010\nxc,1√ξ,ΩΛ0,0\u0011\n,P11(xc) :\u0010\nxc,1√ξ,0,0\u0011\n,\nP12:\u0010\n0,1√ξ,ΩΛ0,0\u0011\n,P13:\u0010\n0,1√ξ,0,0\u0011\n. Due to the complexity of the stability analysis of\nthese sets of equilibrium points, numerical solution methods have to be used. However,\npartial stability information can be obtained by computing the eigenvalues using the time\nvariable τ= ln( a/a0)and taking the limit when α→0.\nIn the study of curves\u0012\n0,−1√\nξ(12ξ+1),ΩΛ0,−µ∓p\nµ(2µ−11) + 18 + 4\u0013\nand\n\u0012\n0,1���\nξ(12ξ+1),ΩΛ0,−µ∓p\nµ(2µ−11) + 18 + 4\u0013\nThe technical difficulties lie in the parameter\nof the equation for the state of matter, which is complex and infinite. Furthermore, stability\nhas to be determined numerically since the Jacobian matrix has infinite entries. These\nsituations are due to the non-differentiability of the flow when α→0and G:= (ξ(12ξ+\n1)y2−1)→0.The last case is even more complex since in a neighbourhood of G≡0, the\ndirection of the flow changes, and they enter the context of non-smooth mechanics. In all\nthese cases, the stability analysis of these curves/sets will be left for future research since\nthey cannot be implemented with the techniques developed in this paper.\nNon-smooth mechanics is a modelling approach that does not require the temporal\nevolutions of positions and velocities to be smooth functions. Due to possible impacts, the\nspeeds of the mechanical system may jump at certain moments to comply with kinematic\nrestrictions. Consider, for example, a rigid model of a ball falling to the ground. Just before75\nthe impact between the ball and the ground, the ball has a pre-impact velocity that does not\ndisappear. At the instant of impact, the velocity must jump to a post-impact velocity of\nat least zero, or penetration will occur. Non-smooth mechanical models are often used in\ncontact dynamics.\nA review of research on non-smooth dynamical systems from the point of view of applica-\ntions is provided in reference [ 109]. Two types of lack of smoothness are considered: impact\nand friction. A list of basic models is provided for each type, application areas are specified,\nand research results are reviewed. The literature on non-smooth dynamical systems will be\nreviewed in future work to resolve the difficulties above.\nThe research hypotheses were implemented during this research, and the paper’s general\nand specific objectives were satisfactorily addressed. It was possible to obtain relevant\ninformation on the flow properties associated with several autonomous systems of ordinary\ndifferential equations from the cosmological context through qualitative techniques from the\ndynamical systems theory. The review of the literature on dynamical systems and fractional\ncalculus was exhaustive. The work tools used in the paper were appropriately selected,\nidentifying the use of the Caputo fractional derivative and its application in formulating a\nfractional cosmological action and its respective variational calculation. The main results of\nfractional calculus for the qualitative analysis of cosmological models were presented in a\ndidactic manner.\nThe analysis of cosmologies with a scalar field with conformal and non-minimal coupling to\ngravity in the fractional context is novel, generalizing previous results within the framework\nof the fractional formulation of gravity. From the mathematical point of view, it described\nthe phase space of the models, obtaining qualitative results and explicit and approximated\nexact solutions with high numerical precision. That represents a different and attractive\nscenario due to its high level of applicability to describe natural phenomena.\nAuthor contributions\nConceptualization, K.M., A.D.M. and G.L.; methodology, G.L.; software, K.M., A.D.M.\nand G.L.; validation, K.M., A.D.M. and G.L.; formal analysis, K.M., A.D.M., G.L., C.M.\nand A.P.; investigation, K.M., A.D.M., G.L., C.M. and A.P.; resources, G.L., C.M. and A.P.;\nwriting—original draft preparation, G.L.; writing—review and editing, G.L., C.M. and A.P.;76\nvisualization, K.M., A.D.M. and G.L.; supervision, G.L. and A.M.; project administration,\nG.L. and K.M.; funding acquisition, K.M., A.D.M., G.L., C.M. and A.P. All authors have\nread and agreed to the published version of the manuscript.\nFunding\nK.M. and G.L. were funded by Vicerrectoría de Investigación y Desarrollo Tecnológico\n(VRIDT) at Universidad Católica del Norte (UCN) through Concurso Fondo de Tesis 2023\nUCN, Resolución VRIDT N °063/2023. VRIDT-UCN funded G. L. through Resolución\nVRIDT No. 026/2023 and Resolución VRIDT No. 027/2023. G. L. also acknowledge the\nfinancial support of Proyecto de Investigación Pro Fondecyt Regular 2023, Resolución VRIDT\nNo. 076/2023. A.D.M. was supported by Agencia Nacional de Investigación y Desarrollo -\nANID Subdirección de Capital Humano/Doctorado Nacional/año 2020 folio 21200837, Gastos\noperacionales proyecto de tesis/2022 folio 242220121, and VRIDT-UCN. C. M. was supported\nby ANID Subdirección de Capital Humano/Doctorado Nacional/año 2021- folio 21211604.\nA.P. acknowledges the funding of VRIDT-UCN through Resolución VRIDT No. 096/2022\nand Resolución VRIDT No. 098/2022.\nData availability\nNo new data were created or analyzed in this study. Data sharing is not applicable to\nthis article.\nAcknowledgments\nWe are thankful for the support of Núcleo de Investigación Geometría Diferencial y\nAplicaciones, Resolución VRIDT No. 096/2022. A.P. thanks Dr. Nikolaos Dimakis and the\nUniversidad de La Frontera for the hospitality provided while part of this work was carried\nout. We thank Master’s student Bayron Micolta-Riascos, Dr. Antonella Cid, and Dr. Juan\nMagaña for their helpful comments and suggestions.77\nConflicts of interest\nWe declare no conflict of interest. The funders had no role in the design of the study; in\nthe collection, analyses, or interpretation of data; in the writing of the manuscript; or in the\ndecision to publish the results.\nAppendix A: Special functions\nThis appendix presents several special functions that appear recurrently in these topics.\n1. Gamma Function\nThe Gamma function extends the factorial function; that is, it extends a function of\nintegers to a function of real or complex numbers. Namely, for natural numbers, the factorial\nofnis defined as\nn! = 1×2×3× ··· × n=nY\nj=1j. (A1)\nOn the other hand, the Gamma function can be written as\nΓ(z) =Z∞\n0tz−1e−tdt, (A2)\nwhich works for values of the complex number zwithℜ(z)>0. It can be observed that\nΓ(z+ 1) = zΓ(z), (A3)\nand also\nΓ(n+ 1) = n!. (A4)\nAdditionally, one can write the binomial coefficient in terms of the Gamma function as\n\nz\nv\n=z!\nv!(z−v)!=Γ(z+ 1)\nΓ(v+ 1)Γ( z+ 1−v). (A5)\nThat added to the reflection formula,\nΓ(z)Γ(1−z) =π\nsin (πz), (A6)78\nand to the Incomplete Gamma function,\nΓ(a, z) =Z∞\nzta−1e−tdt, (A7)\nwhere\nΓ(a,0) = Γ( a). (A8)\n2. Mittag-Leffler functions\nFrom the Maclaurin series expansion of the exponential,\nez=∞X\nn=0zn\nn!, (A9)\nWe can replace the factorial with the Gamma function:\nez=∞X\nn=0zn\nΓ(n+ 1). (A10)\nThat can then be extended as follows:\nE(α, z) =∞X\nn=0zn\nΓ(αn+ 1), (A11)\nwhere αis an arbitrary real number greater than zero.\nIn fractional calculus, this function has a similar importance to that of the exponential\nfunction in standard calculus. For some values of αand functions of z, already known\nfunctions can be obtained:\nE(2,−z2) = cos z, E (1/2, z1/2) =ez\u0002\n1 +erf(z1/2)\u0003\n, (A12)\nwhere the Error function, erf (z), is given by\nerf(z) =2√πZz\n0e−t2dt. (A13)\nThe Mittag-Leffler function can also be extended as follows:\nE(α, β, z ) =∞X\nn=0zn\nΓ(αn+β), (A14)\nwhich is known as the generalized Mittag-Leffler function and has several special cases, e.g.\nE(1,2, z) =ez−1\nz, E (2,2, z2) =sinh ( z)\nz. (A15)79\n3. Hypergeometric functions\nThese functions contain many particular cases of special functions and are widely used in\nfractional calculus. From the definition\npFq({ai};{bj};z) =Qq\nj=1Γ(bj)Qp\ni=1Γ(ai)∞X\nn=0Qp\ni=1Γ(ai+n)zn\nQq\nj=1Γ(bj+n)n!, (A16)\nIt follows that the derivative with respect to zis\nd\ndz\u0002\npFq({ai};{bj};z)\u0003\n=Qp\ni=1Γ(ai)Qq\nj=1Γ(bj)\u0002\npFq({ai+ 1},{bj+ 1};z)\u0003\n. (A17)\n[1] West, B.J. (2021) Fractional Calculus and the Future of Science. 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Journal of Applied Mathematics and\nMechanics , 64 (5):765–772."    },    {        "title": "2402.13863v1.How_to_fault_tolerantly_realize_any_quantum_circuit_with_local_operations.pdf",        "content": "How to fault-tolerantly realize\nany quantum circuit with local operations\nShin Ho Choe1,2and Robert König1,2\n1School of Computation, Information and Technology, Technical University of Munich\n2Munich Center for Quantum Science and Technology (MCQST), Munich, Germany\nFebruary 22, 2024\nWe show how to realize a general quantum circuit involving gates between arbitrary\npairs of qubits by means of geometrically local quantum operations and efficient classical\ncomputation. We prove that circuit-level local stochastic noise modeling an imperfect im-\nplementation of our derived schemes is equivalent to local stochastic noise in the original\ncircuit. Our constructions incur a constant-factor increase in the quantum circuit depth and\na polynomial overhead in the number of qubits: To execute an arbitrary quantum circuit on\nnqubits, we give a 3D quantum fault-tolerance architecture involving O(n3/2log3n)qubits,\nand a quasi-2D architecture using O(n2log3n)qubits. Applied to recent fault-tolerance\nconstructions, this gives a fault-tolerance threshold theorem for universal quantum com-\nputations with local operations, a polynomial qubit overhead and a quasi-polylogarithmic\ndepth overhead. More generally, our transformation dispenses with the need for considering\nthe locality of operations when designing schemes for fault-tolerant quantum information\nprocessing.\n1 Introduction\nThroughoutquantumcomplexitytheory,quantumalgorithmsdesignandquantumfault-tolerance,\nthe notion of a quantum operator or operation acting only on a small (i.e., constant) number of\nqubits is ubiquitous. For example, the complexity class QMA [1–3], the natural analog of NP,\nhas been shown to have a natural complete problem, the k-local Hamiltonian problem [2]. Here\nk-local refers to the fact that the considered Hamiltonians involve terms acting on arbitrary\nsubsets of a constant number kof qubits. This notion of locality has occasionally been referred\nto as Kitaev-local to distinguish it from more commonly used geometric locality notions arising,\ne.g., in physics. In quantum compiling, 2-qubit entangling unitaries such as the CNOT gate play\na special role: together with general single-qubit unitaries, such a gate provides computational\nuniversality when it can be applied to arbitrary pairs of qubits. In fault-tolerance theory, special\nattention is paid to low-weight errors as formalized by the notion of code distance, and quan-\ntum low-density parity check (LDPC) codes are of particular interest because their syndrome\nmeasurements involve only a constant number of qubits (see Ref. [4] for a recent review).\nWhileKitaev-localquantumoperationprovideaversatileabstraction, considerationsrelating\nto real-world experiments are typically encumbered by more stringent locality constraints arising\nfrom technological limitations. Corresponding geometric notion of locality typically involve\nan interaction graph formalizing what interactions or operations are available. With an often\nconsiderable amount of effort, such geometric locality notions can sometimes be taken into\naccount successfully: This has led, e.g., to proofs of QMA-completeness of the local Hamiltonian\nproblem on certain lattices of qubits [5]. Similarly, circuit compilation taking into account the\ninteraction graph of an underlying device is an increasingly relevant topic as larger devices are\n1arXiv:2402.13863v1  [quant-ph]  21 Feb 2024becoming available (see e.g., Ref. [6]). In the area of fault-tolerance, the fundamental question\nof whether noisy operations permit universal fault-tolerant quantum computation was addressed\nand answered positively early on, see [7–10]. In a similar vein, the search for and use of codes\nwith geometrically local stabilizers has been major theme for the last few decades.\nWhile the general program of rendering Kitaev-local constructions geometrically local has\nbeen successful in many instances, there are cases where it meets fundamental obstructions.\nFor example, this is captured by trade-off-relations between the number of physical and logical\nqubits, the code distance and the energy barrier (see e.g., [11–13]), or results on the (in)existence\nof protected logical gates in geometrically local stabilizer codes (see e.g., [14–16]). Given these\nlimitations, one may ask to what extent it is reasonable to afford oneself the luxury of thinking\nonly about Kitaev-locality when dealing with the real world.\nOur contribution\nHere we argue that – in a very general sense – it is in fact possible to meaningfully separate quan-\ntum computational questions from geometric locality considerations. Let us consider a general\nadaptive quantum circuit Q. Here adaptivity means that every basic operation of the circuit Q\nsuch as a single- or two-qubit quantum gate can be determined by a (classically) efficiently com-\nputable function from some bits of measurement outcomes associated with previously applied\nmeasurements. (We refer to Section 2.4 for a precise definition.) To this end, we propose a\ntransformation which takes as input (a description of) a general quantum circuit Qinvolving\nnon-local gates, and produces (a description of) a recompiled “localized” quantum circuit Q′\nwith the following properties:\n(i) The circuit Q′is composed of geometrically local gates: Every two-qubit operation acts on\na pair of qubits whose distance is upper bounded by a constant, and all qubits are arranged\non a regular rectangular lattice in 3D. We give two constructions: one using a slab-like\narchitecture of linear dimensions L×L×Θ(L), and one using a quasi-2D architecture of\ndimensions L×L×O(logL), i.e., a thickened square shape.\n(ii) The circuit Q′usesO(nαlog3n)qubits, where α= 3/2orα= 2for the 3D- respectively\nquasi-2D case. Here nis the number of qubits involved in the original circuit Q.\n(iii) The (quantum) circuit depth of Q′is identical to that of Q. Compared to Q, the circuit Q′\nadditionally involves efficient, i.e., polynomial-time classical processing.\n(iv) The circuit Q′realizes the same functionality as Q. In particular, by taking a suitable\nmarginal (i.e., ignoring certain output bits), the output distribution of Q′simulates that\nofQexactly.\nWe note that Properties (i)–(iv) are relatively straightforward to achieve using entanglement\nswapping and suitable routing protocols as discussed below (see Section 2.4). The key new\nproperty we establish for Q′is the following, see below for a formal definition of local stochastic\nnoise and noisy implementations:\nTheorem 1.1. A noisy implementation of Q′with local stochastic noise of strength pis equiv-\nalent to a noisy implementation of Qwith local stochastic noise of strength Cpcfor some con-\nstants C, c > 0. That is, the corresponding output distribution is identical to that obtained by\nrunning Qwith noise (of a related strength).\nTaken together, Properties (i)–(iv) and Theorem 1.1 imply that noise-tolerance consider-\nations of a quantum computational scheme may be studied entirely in the context of general\ncircuits, i.e., quantum circuits involving Kitaev-local (but not necessarily geometrically local)\ngates. This leads to a considerable simplification in settings where the underlying structure is\nsparse but does not necessarily obey operationally meaningful geometric locality constraints.\n2As an application, we consider the problem of approximately sampling from the output\ndistribution of an ideal circuit by means of a circuit made of noisy components. Here we show\nthat this can be achieved by a 3D-local adaptive circuit Qidealwith a polynomial qubit-overhead\nand quasi-polylogarithmic time-overhead. That is, we have the following:\nCorollary 1.2 (3D-local fault-tolerant quantum computation) .There is a threshold p0>0\non error strength such that the following holds for all sufficiently large nand an arbitrary con-\nstant ε∈(0,1). LetQidealbe an adaptive quantum circuit on nqubits of quantum depth T(n) =\nO(poly(n)). There is a circuit Q′with the following properties:\n(i) The circuit Q′usesO(n3/2log3n)qubits and is local when these are arranged on a 3D grid\ngraph.\n(ii) The quantum depth of Q′is of order T(n)·exp\u0000\nO(log2(log(n/ε)))\u0001\n.\n(iii) Any noisy implementation of Q′with local stochastic noise of strength p < p 0produces a\nsample from a distribution whose total variation distance to the output distribution of Qideal\nis upper bounded by ε.\nWe discuss the derivation of this result in Section 4.3. Thanks to our general transformation, it\nis an immediate consequence of the recent work by Yamasaki and Koashi [17]. These authors\ndemonstrate how to achieve time-efficient constant-qubit overhead fault-tolerant quantum com-\nputation by using a concatenation of multiple small-size quantum codes (albeit with non-local\ngates). We note that – alternatively – our transformation can also be applied directly to other\nschemes for fault-tolerant quantum computation such as those relying on constant-rate quan-\ntum LDPC codes [18, 19]. However, this leads to a polynomial quantum circuit depth overhead\nresulting from the sequential application of two-qubit gates in these schemes. It is currently not\nknown whether this depth overhead can be reduced.\nWhat sets our construction apart from earlier work on fault-tolerant quantum computation\nwith local operations [7–10] is its limited quantum circuit-depth overhead: Earlier works rely on\npermuting qubits to achieve locality and consequently incur a polynomial circuit-depth overhead\ncompared to the original ideal (non-local) circuit. In contrast, applying our construction to\na (non-local) fault-tolerant quantum circuit incurs only constant circuit-depth-overhead. This\nresults in the quasi-polylogarithmic quantum circuit scaling when applied to the protocol of [17].\nLet us point out an important difference between our result and the construction of (non-\nlocal) fault-tolerant quantum circuits in [17]. In the latter work, idling (wait) operations in a\nquantum circuit required to wait for the results of classical computation (e.g., for decoding or\ngate teleportation) are explicitly taken into account (i.e., the corresponding notion of circuit\ndepth includes such operations). In contrast, our construction requires additional classical com-\nputations for every timestep of the original quantum circuit. Correspondingly, our notion of\n“quantum depth” only involves the number of timesteps, but not the additional classical (effi-\ncient) computation. In our construction, these computations involve finding a minimal matching\nin each of O(n)slab-like graphs of linear dimension O(n)×O(logn)×O(logn), which is achiev-\nable in polynomial time by Edmond’s algorithm [20]. We leave as a future work to check whether\nthe construction of Ref. [17] is fault-tolerant even when accounting for associated waiting times.\nRelated work\nMost existing constructive work for fault-tolerance with geometrically local operations considers\nthedesignofquantummemories. Thiscanbeseenasaspecialcaseoffault-tolerantcomputation\nwhere the identity operation is approximated using local (noisy) operation.\nGiven an encoded state in a quantum error-correcting code, one can maintain the logical\ninformation for a fixed duration by repeatedly measuring syndromes and applying a suitable\nrecovery map (or “decoding”) involving a classical computation on the syndrome bits. Such\n3protocols only involve local (quantum) processing if a quantum error-correcting code whose\nstabilizers are geometrically local in Euclidean space is used. Let us mention three of the most\nimportant approaches towards storing multiple logical qubits in a quantum memory.\nA first approach is to use multiple quantum error-correcting codes in parallel, with each code\nencoding a fixed number of logical qubits. For example, multiple tiles of 2D surface codes can\nbe used for this purpose [21]. One drawback of using 2D surface codes for this purpose is that\ntheir decoding is not single-shot: In the case where measurements are imperfect, the decoding\nneeds a sequence of (noisy) syndromes obtained from repeated measurements. In three spatial\ndimensions, single-shot decoding is possible with the 3D subsystem toric codes [22].\nA second approach is to use quantum LDPC codes with geometrically local stabilizer gener-\nators. It is known from seminal work by Bravyi and Terhal [11], Bravyi, Poulin and Terhal [12],\nas well as Haah [13] that there are fundamental limitations to quantum codes with geometrically\nlocal stabilizers. These rule out the existence of good (i.e., constant-rate) quantum stabilizer\ncodes with geometrically local generators in low spatial dimensions. In more detail, an [[n, k, d ]]-\ncode with a constant density of qudits in RDand local stabilizer generators necessarily obeys\nthe bounds\nd∈O(nD−1\nD)see Ref. [11] (1)\nkd2\nD−1∈O(n) see Ref. [12]\nk∈O(nD−2\nD)see Ref. [13] . (2)\nIt is natural to try to construct quantum codes coming close to or saturating the trade-off\nbounds (1)–(2). This program has been quite successful recently, resulting in several construc-\ntions derived from quantum LDPC codes: Portnoy [23] has constructed codes with optimal\nscaling of the number of logical qubits and the code distance, up to polylogarithmic factors. Sub-\nsequent work by Lin et al. [24] gave constructions additionally saturating the upper bound of [11]\non the energy barrier. Concurrent work by Baspin and Williamson [25] provides a particularly\nelegant construction of optimal codes in D= 3dimensions: These are [[n,Θ(n1/3),Θ(n2/3)]]-\ncodes on a rectangular lattice of linear extent n1/3. Note, however, that the question of how to\nconstruct and analyze protocols for fault-tolerant computation using such codes remains largely\nopen. In particular, no analytic fault-tolerance threshold results have been established to date.\nA third approach is to go beyond codes with geometrically local stabilizer generators, con-\nsidering either codes with high-weight stabilizers or LDPC codes whose stabilizer generators are\nnot geometrically local when embedded in RD. In either case, this necessitates an analysis of\nthe propagation of errors in local (typically larger-depth) circuits designed to efficiently realize\nnon-local ones. This is the approach pursued in pioneering works on fault-tolerance with local\noperations [7–10]. Following this idea, Pattison et al. [26] introduce so-called hierarchical code\nencoding Θ(n)-logical qubits using Θ(nlog2n)qubits (with O(logn)-weight stabilizer genera-\ntors). The authors showed that measuring all syndromes is possible with a local quantum circuit\nwith depth O(√nlogn)where all qubits are aligned in a bilayer structure. Based on this code\nand its syndrome measurement circuit, they construct a 2D-local quantum memory for nlogical\nqubits using only Θ(nlog2n)qubits.\nOur construction also essentially follows this last approach, but encompasses both the con-\nstruction of local memories as well as fault-tolerant computation. As described above (see The-\norem 1.1), our recompilation procedure renders any quantum circuit Qgeometrically local while\npreserving its quantum depth (up to a constant factor). When applied to a syndrome extraction\nor recovery circuit associated with an LDPC code, this realizes low-weight (but geometrically\nnon-local) measurements in constant depth, and yields a local quantum memory. When applied\nto (non-local) fault-tolerance constructions based on concatenated codes (with high-weight sta-\nbilizers), our construction provides a threshold theorem for fault-tolerant computation with local\noperations, which is depth-efficient (see Corollary 1.2).\n4Converse bounds and optimality\nFor local quantum memories following the third approach, there are known trade-off relations\nbetween the qubit overhead and the circuit depth. Delfosse, Beverland and Tremblay [27] es-\ntablished a trade-off relation between the depth of a syndrome extraction circuit and the qubit\noverhead when using constant-rate quantum LDPC codes whose Tanner graphs are expanders.\nTheir bound applies to geometrically local syndrome extraction circuits that are Clifford circuits\naugmented by Paulis controlled by parities of measurement outcomes. (This allows, in particu-\nlar, for e.g., teleportation within the circuit.) In more detail, consider a constant-rate ( R) LDPC\ncode encoding nlogical qubits into n/Rqubits. Then any such Clifford circuit for syndrome\nextraction, which uses ntot≥n/Rqubits and is geometrically local in RD, must have quantum\ncircuit depth Tobeying\nT= Ω\u0010\nn/(ntot)1−1/D\u0011\nsee [27, Corollary 2]. (3)\nOne of our intermediate constructions saturates this bound when applied to an ideal (non-local)\nsyndrome extraction circuit Qassociated with a constant-rate LDPC codes. (This includes, in\nparticular, expander codes as assumed in the bound (3).) Because of the LDPC property, we\ncan assume that Qis a constant-depth circuit on Θ(n)qubits – here we take into account one\nauxiliary qubit for measuring each constant-weight stabilizer generator. To arrive at the fault-\ntolerant circuit Q′discussed in Theorem 1.1, our construction involves the construction of an\nintermediate circuit Q′′for syndrome extraction with the following properties: The circuit Q′′is\n3D-local, uses ntot=O(n3/2)qubitsandhasdepth T=O(1)(seeLemma2.9below). Withthese\nparameters, the circuit Q′′saturates the bound (3). We note, however, that this circuit Q′′is not\nfault-tolerant. (Our final construction Q′has the fault-tolerance property, and its parameters\nsaturate the bound (3) up to polylogarithmic factors, but the bound does not apply in this case\nbecause our construction uses controls that are not simply parities of measurement results.)\nOur construction focuses on T=O(1), i.e., constant circuit depth. In contrast, the hier-\narchical code [26] constitutes a construction in a different regime. It includes, in particular,\ninstances where\nT= Θ(√n)and ntot= Θ( n)or T= Θ(√nlogn)and ntot= Θ( nlog2n),\nrespectively (depending on the size of the surface codes used in the code concatenation step).\nThe former case saturates the bound (3), but does not provide fault-tolerance. (Similar to\nour construction of Q′, the latter case provide a fault-tolerant construction whose parameters\nsaturate (3). However, the bound does not apply in this case because the code is not an LDPC\ncode as required by the bound.)\nWe emphasize that minimizing syndrome extraction circuit depth is not the only relevant\naspect when considering quantum fault-tolerance: It is equally important to have limited error-\npropagation. Here the bound of Baspin, Fawzi and Shayeghi [28] provides some insight as it\ntakes into account the logical error probability: They establish a trade-off relation between\nqubit overhead and the circuit depth of a recovery map for a local quantum memories built on\ngeneral quantum error-correcting codes. To discuss their bound, let us specialize it to the case\nwhere nlogical qubits are considered and an exponentially small logical failure probability of\nat most exp(−c·n)is assumed. According to their bound, if there is a recovery map achieving\nthis, which is geometrically local in RD, uses ntotqubits and has quantum circuit depth T, then\nT= Ω \nn1+1/D\nntot!\nsee [28, Theorem 28] . (4)\nWe note that our construction provides a local quantum memory as follows: We start with an\nadaptive circuit Qwhich consists in syndrome measurement followed by single-shot decoding\n5(Pauli correction) for a good quantum LDPC code (encoding nqubits). Then the resulting\nfault-tolerant localized circuit Q′is3D-local, achieves exponentially small error probability, and\nhas quantum depth and total number of qubits\nT=O(1)and ntot= Θ( n3/2log3n). (5)\nUnfortunately, the parameters (5) do not saturate the bound (4) with D= 3. The fact that\nthese parameters saturate the syndrome extraction circuit depth bound (3) (up to polyloga-\nrithmic factors) suggests that this may fundamentally be impossible using constructions that\nhave T=O(1): Indeed, should an analog of the bound (3) apply without the restriction to\nparity-controlled Pauli operations, then no LDPC-code based local quantum memory can sat-\nurate both bounds. In other words, our construction may be optimal among constant-depth\nLDPC-code based constructions in terms of qubit overhead.\nLet us now consider the hierarchical code. When the paper [26] was posted to arXiv, the\nsingle-shot decodability of good quantum LDPC codes had not been established. It is, however,\nstraightforward to apply the subsequent result [29] to the hierarchical code construction. This\nresults in a 2D-local quantum memory with exponentially small error probability where the\ncircuit depth and total number of qubits is\nT= Θ(√nlogn)and ntot= Θ( nlog2n). (6)\n(With the original construction discussed in the paper [26], the circuit depth was T(n) =\nΘ(n3/2logn)because it involved d= Θ( n)rounds of (repeated) measurements before decoding\nthe outer LDPC code, following Gottesman’s construction [18]). The parameters (6) saturate\nthe bound (4).\nIn summary, our construction and the hierarchical code target different regimes, for recovery\nmaps as summarized by the following table:\nconstruction qubit overhead quantum depth locality\nhierarchical code [26] polylogarithmic polynomial 2D\nour construction polynomial constant 3D (or quasi- 2D)\nTable 1: Summary on hierarchical codes and quantum memories from our construction.\nOutline\nWe proceed as follows: In Section 2, we discuss parallel routing of qubits on general graphs and\ngive algorithms for grid graphs. We explain how to use these to render a quantum circuit local,\nalbeit without fault-tolerance. In Section 3, we consider the parallel use of a “quantum bus” to\nconcurrently generate several Bell pairs. The main result established there is a parallel repetition\ntheorem that shows robustness against local stochastic noise even in cases where individual\nbuses are affected by correlated errors. In Section 4, we give our circuit transformation which\n“localizes” a quantum circuit while maintaining fault-tolerance properties. The construction\ncombines our efficient parallel routing schemes with our parallel repetition result for quantum\nbuses. Finally, we describe the protocol of 3D-local fault-tolerant quantum computation with\npolynomial qubit-overhead and quasi-polylogarithmic time-overhead.\n2 Parallel routing of qubits in 2D and 3D lattices\nOur construction relies on new schemes for parallel routing of qubits on vertices of a graph.\nWe note that the study of routing on an undirected connected graph G= (V, E)has a long\nhistory in classical computer science. A prominent example is the routing via matching model\n6introduced by Alon et al. [30]: Here labeled pebbles are placed on the vertices initially, and are\nsubsequently moved along edges by means of a sequence of steps consisting of (possibly parallel)\nswaps. The goal is to implement a target permutation of the pebbles using a minimal number\nof steps; the corresponding number is referred to as the routing time (for a given graph and\na permutation). Corresponding routing schemes find direct application to the compilation of\ncircuits which use SWAP-gates to realize long-range gates, see e.g., [6].\n2.1 Parallel routing using entanglement swapping circuits\nOur construction does not use unitary SWAP-gates, but instead relies on entanglement swapping\nrealized by Bell measurements on Bell pairs. In contrast to the unitary case, such measurements\ncan create long-range entanglement (e.g., along a path of Bell pairs of length n) in constant\ntime, up to a Pauli correction that can be computed efficiently (in time O(logn)) by taking\nparities, i.e., classical computation. In particular, this means that with regard to the quantum\ncircuit depth (and assuming efficient classical computation to be free), the usual figures of merit\nsuch as the routing time in the routing via matching problem are not directly relevant in our\nscenario.\nThe following graph-theoretic notions are motivated by our objective: We are interested\nin realizing a set of two-qubit interactions (gates) in each time step. Rather than realizing a\npermutation, thismeansthatweseektoconnectpairsofqubitsaccordingtothepairwisedisjoint\nsupports of a collection of two-qubit operations (e.g., two-qubit gates). The length of any “path\nof travel” is irrelevant (each will be at most of polynomial length) as it does not translate to\nquantum circuit depth. We only need the paths to be pairwise disjoint. This is because we will\nassume that there is a Bell pair on each edge used for entanglement swapping. Most importantly,\nour figure of merit is the number of qubits for which this kind of routing is possible in a given\ngraph. This leads to the following definitions.\nLetG= (V, E)be a connected undirected graph. A subset S=\b\nv1, . . . , v |S|\t\n⊆Vof ver-\ntices is called parallel-routable if its size |S|=: 2kis even and the following holds: For any pair-\ning{(vir, vjr)}k\nr=1of the elements of S, there is a family {πr}k\nr=1ofkpairwise edge-disjoint paths\nsuch that for any r∈ {1, . . . , k }=: [k], the path πrconnects virandvjr. Such a family {πr}k\nr=1\nwill be referred to as a parallel routing scheme associated with the pairing {(vir, vjr)}k\nr=1. The\nparallel routing number k(G) =|S|/2is defined as half the maximal size of a subset S⊆Vof\nvertices which is parallel-routable.\nFor later reference, we make the following obvious observation obtained by restricting a\nrouting scheme:\nLemma 2.1. LetS⊆Vbe a parallel-routable subset of vertices. Let S′⊂Sbe a subset of even\nsize. Then S′is parallel-routable.\nBefore studying these combinatorial notions in detail for grid graphs (see Section 2.2), let\nus state how they give rise to routing protocols of qubits on a general graph G= (V, E). We\nconsider two different routing-tasks: Pairwise entanglement generation as well as qubit pairing.\nLetS={v1, . . . , v 2k} ⊂Vbe a parallel-routable set of vertices. We first discuss pairwise\nentanglement generation. Here we consider a system with 2k+ 2|E|qubits, namely\n(i) A qubit Pjlocated at the vertex vjfor each j∈[2k], and\n(ii) for each edge e={u, v} ∈E, qubits Re\nu,Re\nvlocated at uandv, respectively.\nGiven a pairing {(vir, vjr)}r∈[k]of the vertices S, the goal of pairwise entanglement generation\nis to generate Bell pairs between each pair PirPjrof qubits, for r∈[k]. This is achieved by\nAlgorithm 2, see Fig. 1 for an illustration. The following lemma is an immediate consequence\nof how entanglement swapping works:\n7Figure 1: Three pairs of parallel-routable vertices {(vir, vjr)}3\nr=1and their connecting paths.\nQubits Re\nu,Re\nvassociated with an edge e={u, v}are represented as black dots. (Qubits Pj\nlocated at vj,j∈[n]are not shown.) Bell measurements in Steps 6–9 are represented by\nrectangles.\nLemma 2.2. Given a pairing ({(vir, vjr)})k\nr=1of the vertices S, the quantum algorithm Qentangle\n(see Algorithm 1) creates the state\nkO\nr=1ΦRirRjr. (7)\nIt can be realized by a constant-depth adaptive circuit with local operations on the graph G.\nProof.It is clear (from the properties of the basic entanglement swapping circuit) that Qentangle\ngenerates the state (7). To see that it can be realized by a constant-depth adaptive circuit,\nobserve that Steps 6–9 in the algorithm can be executed in parallel for different r∈[k], because\nthe parallel routing scheme {πr}k\nr=1consists of pairwise edge-disjoint paths. Similarly, Steps 3–4\nand the steps inside iteration 11 can be applied simultaneously for different r∈[k]. Because the\nrelevant parities in Step 9 can be computed by by classical circuit of depth O(logℓ)andℓ≤ |E|\nis polynomial in n, it follows that Qpaircan be realized by a constant-depth quantum circuit\nsupplemented with efficient (logarithmic-time) classical computation.\nIt is straightforward to verify that all two-qubit gates in this circuit are between neighboring\nqubits on the graph G, i.e., the circuit is local.\nNow consider the problem of pairing up qubits: Here we consider a system of qubits that\nhas – in addition to the n+ 2|E|qubits P1, . . . , P nand{Re\nu, Re\nv}e={u,v}∈E,n= 2k“data”\nqubits Q1, . . . , Q 2k, where qubit Qjis located at the vertex vjforj∈[2k]. Given a pair-\ning{(vir, vjr)}r∈[k]of the vertices S, the problem of qubit pairing consists in bringing the\nqubits QirandQjrto the same location, for every r∈[k]. Algorithm 2 achieves this by\ntransferring each qubit QjrtoPirforr∈[k], that is, the qubits Qir,Qjrof the input state\nare located at QirandPirafter application of the algorithm, i.e., at vertex vir. The following\nlemma is an immediate consequence of how the standard teleportation protocol works.\nLemma 2.3. Given a pairing {(vir, vjr)}k\nr=1of the vertices S, the quantum algorithm Qpair(see\nAlgorithm 2) implements a transfer of subsystems that maps each subsystem Qjrto the subsystem\nPirforr∈[k], i.e., a 2k-qubit state Ψon the registers Q1, . . . , Q 2kis mapped according to\nΨQi1Qj1···QikQjk7→ΨQi1Pi1···QikPik. (8)\n8Algorithm 1 Quantum algorithm Qentanglefor pairwise entanglement generation\nInput:A pairing ({(vir, vjr)})k\nr=1of the vertices S={v1, . . . , v 2k}.\nOutput: The Bell state |Φ⟩=1√\n2(|00⟩+|11⟩)on each pair RirRjrof qubits, r∈[k].\n1:Compute a parallel routing scheme {πr}k\nr=1associated with the paring {(vir, vjr)}k\nr=1.\n2:fore={u, v} ∈Edo\n3:Prepare the state |Φ⟩=1√\n2(|00⟩+|11⟩)on the qubits Re\nuRe\nv.\n4:end for\n5:forr∈[k]do ▷Letπr= (vir, w1, . . . , w ℓ, vjr)be the sequence of vertices traversed\nalong πr. For a vertex v∈ {vir, w1, . . . , w ℓ, vjr}, letein(v)andeout(v)denote the in- and\noutgoing edges at vwhen traversing πr.\n6: form∈[ℓ]do\n7: Apply a Bell measurement to the qubits Rein(wm)\nwm Reout(wm)\nwm, getting\noutcomes (am, bm).\n8: end for\n9:Apply XaZbto qubit Rein(vjr)\nvjrwhere a=Lℓ\nm=1amandb=Lℓ\nm=1bm.\n10:end for\n11:forr∈[k]do\n12:Apply the SWAPgate to qubits PirandReout(vir)\nvir.\n13:Apply the SWAPgate to qubits PjrandRein(vjr)\nvjr.\n14:end for\nThe algorithm Qpaircan be realized by an adaptive constant-depth quantum circuit with local\noperations on the graph Gsupplemented by efficient classical computation.\nAlgorithm 2 A quantum algorithm Qpairfor pairing qubits\nInput:A pairing {(vir, vjr)}k\nr=1of the vertices S={v1, . . . , v 2k}.\nA2k-qubit state ΨQ1···Q2kon the qubits Q1, . . . , Q 2k.\nOutput: The permuted state ΨQi1Pi1···QikPikwhere QirPircontains the subsystem QirQjrof the original state Ψforr∈[k].\n1:RunQentanglewith input {(vir, vjr)}k\nr=1to generate kBell pairs, i.e., the\nstateNk\nr=1ΦPirPjr.\n2:forr∈[k]do\n3:Apply a Bell measurement to the qubits QjrandPjrobtaining outcome (ar, br).\n4:Apply XarZbrtoPir.\n5:end for\nProof.After the pairwise entanglement generation step (1), we have k(possibly “long-range”)\nBell states\nkO\nr=1ΦPirPjr. (9)\nSteps 2–5 perform teleportation, transferring system QjrtoPirfor each r∈[k]. It follows\nimmediately that the algorithm Qpairhas the action given by Eq. (8).\nIt remains to argue that the algorithm Qpaircan be realized by a constant-depth adaptive\nquantum circuit which is local on the graph G. It is clear by inspection that all operations\n9involved in the algorithm are local, i.e., they act on one or two qubits that are either located at\nthe same vertex v∈V, or at two vertices u, v∈Vconnected by an edge {u, v} ∈E. It is easy\nto check that Qpaircan be realized by an adaptive constant-depth circuit.\nSlightly abusing notation, we denote by Q−1\npairthe algorithm which, for a given pairing\n{(vir, vjr)}k\nr=1,\nimplementstheinverseofthemap(8)(i.e., maps PirtoQjrforeach r∈[k]). Itisclearthat Q−1\npair\ncan also be realized by an adaptive constant-depth quantum circuit in an analogous fashion by\nproceeding creating long-range entanglement, and then teleporting respectively swapping the\ncorresponding subsystems.\nIn Section 2.4 we explain how to use the routing protocols QpairandQ−1\npairin 3D grid graphs\nin order to realize arbitrary quantum circuits using local operations. We subsequently refine the\nconstruction to incorporate fault-tolerance considerations.\n2.2 Parallel routing on 2D and 3D grid graphs\nIn this section, we study routing schemes in certain graphs: We are interested in architectures\nassociated with (“cuboid”) grid graphs in 2D and 3D. To introduce these, let us first introduce\nthe Cartesian product G×G′of two graphs G= (V, E),G′= (V′, E′)as follows: This is the\ngraph G×G′with vertex set\nV×V′=\b\n(v, v′)|v∈V, v′∈V′\t\n,\nwhere {(u, u′),(v, v′)}is an edge of G×G′if and only if either\n\u0000\nu=vand (u′, v′)∈E′\u0001\nor\u0000\nu′=v′and (u, v)∈E\u0001\n.\nLetPndenote the path on nvertices, that is, the graph with vertex set [n]and edges of the\nform{j, j+ 1}where j∈[n−1]. We call Pn1×Pn2then1×n2grid graph and Pn1×Pn2×Pn3\nthen1×n2×n3grid graph, where n1, n2, n3∈N. These are naturally embedded in R2andR3,\nrespectively, and the distance d(u, v)between two verties u, vin these graphs (i.e., the minimal\nlength of a path connecting uandv) is simply the L1- or “Manhattan”-distance.\nConsider the L×Lgrid graph PL×PL. The following lemma gives a sufficient condition\nfor the existence of a parallel routing scheme associated with a pairing of a subset of vertices of\nthis graph.\nLemma 2.4 (Parallel routing in PL×PL).LetS={vi= (xi, yi)}2k\ni=1be a subset of vertices\nofPL×PLof even size and let {(vir, vjr)}k\nr=1be an associated pairing. For brevity, let us write\n(Xr, Yr) := ( xir, yir) =vir\n(X′\nr, Y′\nr) := ( xjr, yjr) =vjrforr∈[k]\nfor the coordinates of paired vertices. Suppose that\n{Xp, X′\np} ∩ { Xq, X′\nq}=∅\n{Yp, Y′\np} ∩ { Yq, Y′\nq}=∅for all p, q∈[k]withp̸=q . (10)\nThen there exists a parallel routing scheme {πr}k\nr=1associated with the pairing {(vir, vjr)}k\nr=1.\nFurthermore, the length of the path πris equal to d(vir, vjr)≤2Lfor each r∈[k].\nProof.Without loss of generality, we may assume that Xr≤X′\nr. For each r∈[k], define the\npath πrconnecting vir= (Xr, Yr)andvjr= (X′\nr, Y′\nr)by the sequence of traversed vertices as\nπr:=(\n{(Xr, Yr),(Xr+ 1, Yr), . . . , (X′\nr, Yr),(X′\nj, Yr+ 1), . . . , (X′\nr, Y′\nr)}ifYr≤Y′\nr\n{(Xr, Yr),(Xr+ 1, Yr), . . . , (X′\nr, Yr),(X′\nr, Yr−1), . . . , (X′\nr, Y′\nr)}ifYr> Y′\nr.(11)\n10The assumption (10) implies that the paths {πr}k\nr=1are pairwise edge-disjoint. Furthermore,\nthe length of the path πrisd(vir, vjr)for each r∈[k]by construction.\nAn immediate consequence of Lemma 2.4 is the following lower bound on the parallel routing\nnumber of a 2D grid graph.\nCorollary 2.5. LetLbe even. Then k(PL×PL)≥L/2.\nProof.Consider the set S={vi:= (i, i)}L\ni=1of vertices on the “diagonal” of PL×PL. Let\n{(vir, vjr)}k\nr=1be an arbitrary pairing of S. Then ir̸=jrand{ir, jr}∩{is, js}=∅for all r̸=s,\nr, s∈[L]by definition of a pairing. It follows immediately that Condition (10) is satisfied. Since\nthe pairing was arbitrary, this shows that Sis parallel-routable.\nNext, we consider 3D grid graphs. Our main result is the following.\nTheorem 2.6 (Parallel routing in PL×PL×PL).LetLbe even. Consider the subset\n[L]2× {1}={(x, y,1)|x, y∈[L]}\nof vertices of the grid graph PL×PL×P4L. Let{(vir, vjr)}L2/2\nr=1be an arbitrary pairing of the\nvertices belonging to [L]2× {1}. Then there is a parallel routing scheme {πr}L2/2\nr=1associated\nwith{(vir, vjr)}L2/2\ni=1such that each path πrhas length at most 10Lfor each r∈[L2/2].\nIn particular, this immediately implies the following:\nCorollary 2.7. LetLbe even. Then k(PL×PL×P4L)≥L2/2.\nToproveTheorem2.6 weconstructaparallelroutingschemeinagreedyway. More precisely,\nwe use that the graph PL×PL×P4Lcontains 4Lsubgraphs PL×PL× {Z},Z∈[4L]which\nwe refer to as floors. Each of these subgraphs is isomorphic to PL×PL. Our construction\nreduces the problem of finding a parallel routing scheme on PL×PL×P4Lto the 2D case by\napplying Lemma 2.4 to different floors. This is achieved by Algorithm 3, which computes the\ncoordinate Zrof a floor PL×PL×{Zr} ⊂PL×PL×P4Lfor each pair (vir, vjr),r∈[L2/2]. We\nsay that the pair (vir, vjr)is assigned the Zr-th floor by Algorithm 3.\nAlgorithm 3 A greedy algorithm to assign a floor to each pair of vertices in a grid graph\nInput:A pairing ({(vir, vjr)})L2/2\nr=1of the vertices [L]2× {1}.\nWe write vir=: (Xr, Yr,1)andv′\njr=: (X′\nr, Y′\nr,1)forr∈[L2/2].\nOutput: A sequence (Zr)L2/2\nr=1with Zr∈[4L].\n1:FilledRows[1] ← ∅, . . . , FilledRows[4 L]← ∅\n2:FilledCols[1] ← ∅, . . . , FilledCols[4 L]← ∅\n3:forr∈[L2/2]do\n4:\nZr←arg min {Z∈[4L]| {Xr, X′\nr} ∩FilledCols[ Z] =∅and{Yr, Y′\nr} ∩FilledRows[ Z] =∅}.\n5: FilledCols[ Zr]←FilledCols[ Zr]∪ {Xr, X′\nr}\n6: FilledRows[ Zr]←FilledRows[ Zr]∪ {Yr, Y′\nr}\n7:end for\n8:return (Zr)L2/2\nr=1\nThe following lemma shows that the set of pairs assigned to a given floor satisfies the sufficienct\ncondition (10) for the existence of a parallel routing scheme in 2D.\n11Lemma 2.8. For any input pairing {(vir, vjr)}L2/2\nr=1of[L]2× {1}, Algorithm 3 returns a fam-\nily{Zr}L2/2\nr=1⊂[4L]with the following property. Writing vir= (Xr, Yr,1)andvjr= (X′\nr, Yr,1)\nforr∈[L2/2], and defining\nIZ={r∈[L2/2]|Zr=Z}\nas the set of indices of pairs (vir, vjr)assigned to the Z-th floor, we have\n{Xp, X′\np} ∩ { Xq, X′\nq}=∅\n{Yp, Y′\np} ∩ { Yq, Y′\nq}=∅for all p, q∈IZwith p̸=q (12)\nfor every Z∈[4L].\nProof.We first argue that the algorithm is well-defined, i.e., returns a sequence {Zr}L2/2\nr=1⊂[4L]\non any input. To this end, we have to argue that for every r∈[L2/2], the assignment in Step 4\nindeed assigns a value Zr∈[4L]for every r∈[L2/2], i.e., there is some Z∈[4L]such that\n{Xr, X′\nr} ∩FilledCols[ Z]and{Yr, Y′\nr} ∩FilledRows[ Z]are both empty.\nSuppose for sake of contradiction that this is not the case. Let r∈[L2/2]be the minimal\nnumber such that\n\u0000\n{Xr, X′\nr} ∩FilledCols[ Z]̸=∅or{Yr, Y′\nr} ∩FilledRows[ Z]̸=∅\u0001\nfor every Z∈[4L].(13)\nLet us consider the collection of sets {FilledRows[ Z]}Z∈[4L]and{FilledCols[ Z]}Z∈[4L]at the\ncorresponding Step 4 of the algorithm. By definition (see Line 5), FilledCols[ Z]is the set of all\nX-coordinates of pairs (viq, vjq),q≤r−1that have been assigned to the Z-th floor. Similarly\n(see Line 6), FilledRows[ Z]is the set of all Y-coordinates of pairs (viq, vjq),q≤r−1assigned to\ntheZ-th floor. Note that in each iteration, at most one pair is assigned to a hitherto “unused”\nfloor (i.e., one associated with a coordinate Z∈[4L]such that FilledCols[ Z]andFilledRows[ Z]\nare both empty). Since Eq. (13) implies that each of the 4Lfloors is already occupied at the\nr-th iteration, we must have\nr−1≥4L . (14)\nThe definition of the sets {FilledRows[ Z]}Z∈[4L]and{FilledCols[ Z]}Z∈[4L]and (13) also implies\nthat\n\u0000\n{Xr, X′\nr} ∩ {Xq, X′\nq} ̸=∅or{Yr, Y′\nr} ∩ {Yq, Y′\nq} ̸=∅\u0001\nfor each q≤r−1.(15)\nBecause by (15), at least one of the vertices viqorvjqbelongs to the set\nW:=\b\n(X, Y)∈[L]2|X∈ {Xr, X′\nr}orY∈ {Yr, Y′\nr}\t\n,\nthe cardinality of this set must be at least\n|W| ≥4L (16)\nby Eq. (14).\nHowever, since (Xr, Yr)̸= (X′\nr, Y′\nr)∈[L]2, we have from a simple counting argument that\n|W| ≤(\n4L−4ifXr̸=X′\nrandYr̸=Y′\nr\n3L−2ifXr=X′\nrorYr=Y′\nr\nand thus\n|W| ≤4L−4\n12forL≥2. This contradicts Eq. (16), showing that the algorithm returns a sequence {Zr}L2/2\nr=1⊂\n[4L]as claimed.\nTo see that Condition (12) is satisfied for every Z∈[4L]upon completion of the algorithm,\nobserve that a corresponding modified condition is satisfied throughout the algorithm. That is,\nat any point of the algorithm, a new pair ((Xq, Yq),(X′\nq, Y′\nq))only gets assigned to a floor PL×\nPL× {Z}if the latter is either empty, or assigning the pair to the floor does not conflict with\nthe pairs already added to that floor, i.e., does not violate Condition (12). This is ensured by\nthe definition of Zrin Step 4.\nEquipped with Lemma 2.8, we can now give the\nProof of Theorem 2.6. Forr∈[L2/2], we define the path πrconnecting the two vertices\nvir= (Xr, Yr,1)and vjr= (X′\nr, Y′\nr,1)\nas the concatenation of appropriate subpaths πup\nr,πmid\nrandπdown\nr. To construct these paths,\nwe use Algorithm 3 to compute – for each r∈[L2/2]– the coordinate Zr∈[4L]of a floor\nPL×PL× {Zr}of the grid graph PL×PL×P4L. The path πup\nr“vertically” connects virwith\nthe vertex (Xr, Yr, Zr)in the Zr-th floor, and πdown\nrvertically connects the vertex (X′\nr, Y′\nr, Zr)\nin the Zr-th floor with vjr. That is, again defining paths in terms of the vertices traversed, we\nset\nπup\nr := {(Xr, Yr,1),(Xr, Yr,2), . . . , (Xr, Yr, Zr)}\nπdown\nr :={(X′\nr, Y′\nr, Zr),(X′\nr, Y′\nr, Zr−1), . . . , (X′\nr, Y′\nr,1)}.(17)\nWe note that for Zr= 1, these paths are empty, i.e., of length 0, and can be omitted from\nfurther considerations: Here πr=πmid\nr.\nFor each r∈[L2/2], the path πmid\nrconnects the vertices (Xr, Yr, Zr)and(X′\nr, Y′\nr, Zr)within\ntheZr-th floor. To construct these paths, we invoke our construction for the grid graph PL×PL\ngiven in Lemma 2.4. Lemma 2.8 guarantees that the collection of vertices\n\b\n(Xr, Yr, Zr),(X′\nr, Y′\nr, Zr)\t\nr∈IZ\nassociated with pairs assigned to the Zr-th floor satisfies the necessary condition (10). Each\npath πmid\nris thus of the form (11) inside the Zr-th floor, and all path inside a given floor are\npairwise non-intersecting.\nThis constructions satisfies the claimed properties: Denoting by |π|the length of a path π,\nwe have |πup\nr| ≤4Land\f\fπdown\nr\f\f≤4Lby construction, and\f\fπmid\nr\f\f≤2Lby Lemma 2.4 and the\nfact that the diameter of PL×PLis bounded by 2L. It follows that\n|πr|=|πup\nr|+\f\f\fπmid\nr\f\f\f+\f\f\fπdown\nr\f\f\f≤10L\nfor every r∈[L2/2], as claimed.\nFurthermore, the paths {πr}L2/2\nr=1are pairwise edge-disjoint: Consider two such paths πr, πs\nwith r̸=s. Then it is clear from the definition that the “vertical” subpaths πup\nr, πdown\nrand\nπup\ns, πdown\nsare edge-disjoint, as the former are located above virandvjr, respectively, whereas\nthe latter are located above visandvjs, respectively. Finally, the subpaths πmid\nrandπmid\nsare\nedge-disjoint, astheyeitherbelongtodifferentfloors Zr̸=Zs(andarethusclearlyedge-disjoint),\nor they belong to the same floor and are edge-disjoint by Lemma 2.4 as argued above.\n132.3 Definition of a general adaptive quantum circuit\nThe following notion will be useful: By a general adaptive quantum circuit Q(sometimes simply\nreferredtoasa protocol)wemeanonewhichiscomposedsingle-qubitstatepreparations,one-and\ntwo-qubit unitaries, and single-qubit measurements. All these operations can depend or (i.e., be\nclassically controlled by) measurement results obtained in the course of the computation. These\nclassical controls may involve (efficient) classical computation.\nBecause of gate teleportation and associated magic states (see [31]), we may assume without\nloss of generality that Qinvolves only the following quantum operations on a fixed number nof\nqubits:\n(a) preparation of a single qubit in the computational basis state |0⟩.\n(b) preparation of a single qubit in the magic state T|+⟩=1√\n2\u0000\n|0⟩+eiπ/4|1⟩\u0001\n, where T=\n|0⟩⟨0|+eiπ/4|1⟩⟨1|.\n(c) any single- and two-qubit Clifford operation (including the identity, whose inclusion is con-\nvenient for formal purposes)\n(d) measurement of any qubit in the computational basis. This provides a measurement re-\nsultx∈ {0,1}and projects the qubit onto the associated Z-eigenstate, i.e., the computa-\ntional basis state |x⟩.\nAdaptivity refers to the fact that the choice of which of these operations is applied at any\ngiven point may depend on the history of measurement results obtained up to that point. This\ndependence should be given in the form of efficiently computable functions. In our presentation,\nwe leave these functions, the classical computation required to evaluate these functions, as well\nas the corresponding classical registers (keeping a record of the measurement results) implicit. In\nthe context of fault-tolerance, we will further assume that they can be evaluated in an error-free\nmanner.\nWithout loss of generality, we may assume that the algorithm proceeds in an alternating\nfashion. In each time step, a layer of quantum operations from the set (a)–(d) is applied, where\nthe one- respectively two-qubit operations within the layer act on disjoint subsets of qubits\nand can be applied simultaneously. This is followed by a classical computation (based on the\nmeasurement history) to determine the layer of quantum operations applied at the next time\nstep. We refer to the number Tof layers (time steps) needed to implement Qas thequantum\ncircuit depth ofQ, with the understanding that the classical computation involved is efficient\nand can be neglected.\nSuppressing the dependence on classical measurement results, the algorithm Qcan thus be\nwritten as a composition M(T)◦ ··· ◦ M(1)ofn-qubit operations M(t),t∈[T], where each\nof these operations consists in the parallel application of one- and two-qubit operations. For\nsimplicity, let us assume that the number n= 2kof qubits is even, and let us label the qubits\nasQ1···Qn. Possibly combining single-qubit operations into two-qubit operations, we can then\nfind, for each t∈[T], a pairing {(i(t)\nr, j(t)\nr)}k\nr=1of[n](i.e., the set of qubits), and two-qubit\noperations {M(t,r)}k\nr=1such that\nM(t)=kO\nr=1M(t,r)\nQ\ni(t)\nrQ\nj(t)\nr(18)\nis the composition of the operations M(t,r)applied to the two qubits Qi(t)\nrQj(t)\nr, which can be\nexecuted in parallel for each r∈[k].\n142.4 Localizing a general quantum circuit\nIn this section, we show how to render a fully general adaptive quantum circuit local in 3D (re-\nspectively in 2D). That is, let Qbe a general adaptive quantum circuit as defined in Section 2.3.\nCombining the general qubit pairing protocols QpairandQ−1\npairfrom Section 2.1 with the routing\nschemes on grid graphs constructed in Section 2.2, we immediately obtain the following.\nLemma 2.9 (Localizing a general adaptive circuit) .LetQbe a general adaptive quantum circuit\nonn= 2kqubits and of quantum depth Tas described. Then there are adaptive quantum\ncircuits Q′\n2DandQ′\n3Dwith the following properties:\n(i) By taking certain marginals (i.e., tracing out qubits and/or ignoring measurement results),\nthe two circuits exactly simulate Q.\n(ii) Both circuits have quantum depth of order O(T).\n(iii) The circuits Q′\n2DandQ′\n3Dare geometrically local in 2D and 3D, respectively (i.e., only\ninvolve local or nearest-neighbor operations on a corresponding grid graph).\n(iv) The circuits Q′\n2DandQ′\n3Duse a total number of\nntot\n2D=O(n2)and ntot\n3D=O(n3/2) (19)\nqubits, respectively.\nProof.We use the graphs\nG2D=Pn×Pnand G3D=P⌈√n⌉×P⌈√n⌉×P4⌈√n⌉,\nrespectively. For G2Dwe use the parallel-routable subset\nS2D={vi:= (i, i)}n\ni=1,\nsee the proof of Corollary 2.5. For the 3D grid graph G3D, we use an arbitrary subset\nS3D={vi}n\ni=1⊆\u0002\n⌈√n⌉\u00032× {1}\nof size n. Observe that such a set S3Dis parallel-routable by Theorem 2.6 and Lemma 2.1.\nWe construct our circuits Q′\n2DandQ′\n3Dby replacing, in each layer t∈[T]of the original\nadaptive circuit Q, the operation M(t)(see Eq. (18)) by a geometrically local adaptive cir-\ncuitcM(t). To do so, we use – in addition to the “computational” qubits Q1, . . . , Q 2k– a total\nof\nnaux= 2k+ 2|E| (20)\nauxiliary qubits with registers labeled {Pj}j∈[2k]and{Re\nu, Re\nv}e={u,v}∈E. These are arranged as\ndescribed in Section 2.1. We note that the graphs G2DandG3Dhave\n|E2D|= 2n2−2nand |E3D|= 12⌈√n⌉3−9⌈√n⌉2\nedges, respectively, leading (together with (20)) to the claimed total numbers ntot=n+nauxof\nqubits, cf. Eq. (19).\nLet{(i(t)\nr, j(t)\nr)}k\nr=1be the pairing relevant at time step (layer) t∈[T], and let\nQ(t)\npair=Qpair\u0010\n{(i(t)\nr, j(t)\nr)}k\nr=1\u0011\nand (Q(t)\npair)−1=Q−1\npair\u0010\n{(i(t)\nr, j(t)\nr)}k\nr=1\u0011\n(21)\n15be the constant-depth adaptive circuits introduced in Section 2.1. Then the new circuits Q′\n2D\nrespectively Q′\n3Dare obtained by replacing – for each t∈[T]– the operation M(t)in the\ncircuit Qby the composition\ncM(t):= (Q(t)\npair)−1◦ kO\nr=1M(t,r)\nQ\ni(t)\nrP\ni(t)\nr!\n◦ Q(t)\npair.\nIn other words, the qubits paired in layer tare placed next to each other by application of Q(t)\npair.\nThen each operation M(t,r),r∈[k]can be applied locally. Subsequently, the qubits are moved\nback to their original positions by application of (Q(t)\npair)−1. The new circuits then are obtained\nas the composition\nQ′:=cM(T)◦ ··· ◦ cM(1).\nIt is easy to check that this has all the claimed properties. In particular, for each t∈[T], both\ncircuits (21) are adaptive constant-depth circuits, see Lemma 2.3. This immediately implies that\nthe quantum circuit depth of Q′is of order O(T).\n3 A parallel repetition theorem for fault-tolerant quantum buses\nWhile the (ideal) circuits Q′\n2DandQ′\n3Drealize an adaptive circuit Qwith limited overhead in\na local manner (see Theorem 4.5), this is clearly not the case for noisy, i.e., imperfect imple-\nmentations. The most obvious obstacle here concerns the k-fold tensor product of Bell pairs (9)\nassociated with a pairing (This is generated by the algorithm Qentanglewhich is used as a sub-\nroutine): If the implementation (of the subroutines Qpairrespectively Q−1\npair) is subject to noise,\nonly a noisy version of this central resource state can be generated. Aggravating this problem,\nentanglement swapping needs to be performed over extensive (typically polynomial) distances\nin our construction. Even for i.i.d. Pauli noise on each qubit, the resulting states when using\nthe ideal circuit Q′\n2DorQ′\n3Dwill have exponentially small fidelity with a (tensor product of)\nBell state(s) as a function of the distance.\nIn prior work [32], we have introduced a fault-tolerant alternative to entanglement-swapping\nover extensive distances to address this issue. The corresponding protocol establishes long-range\nentanglement between two distant qubits even if all involved operations are noisy. Since this\ncan be combined with teleportation, we refer to the construction as a quantum bus.\nHere we consider a more general class of quantum circuits satisfying a certain condition\nwhich is also satisfied by the quantum bus. We show that any quantum circuit consisting of\nany number of circuits of this class executed in parallel is fault-tolerant. As a result, we have\na characterization of the residual “effective” noise on the generated tensor product of Bell pairs\nwhen several noisy buses are used in parallel.\nIn Section 3.1, we discuss the considered noise model. In Section 3.2, we introduce the\ncentral notion of a noisy implementation of an adaptive circuit. In Section 3.3, we consider a\nclass of fault-tolerant protocol preparing Bell states and check that the quantum bus introduced\nin Ref. [32] belongs to this class. In Section 3.4, we establish a parallel repetition theorem\ncharacterizing effective noise on the output of the parallel use of quantum circuits introduced\nin Section 3.3. In Section 3.5, we show as a corollary of the parallel repetition theorem that\nunder any local stochastic noise of strength below a certain threshold, the parallel repetition\nof the quantum bus generates multiple long-range Bell pairs affected by constant-strength local\nstochastic noise.\n163.1 Local stochastic noise\nThenotionoflocalstochasticnoisewaspioneeredbyGottesman[18]. Itgeneralizestherestricted\nnoise model of i.i.d. Pauli noise, and is physically motivated by the idea that physical sources\nof noise are typically such that errors are exponentially suppressed in their weight. Formally,\nconsider a system of nqubits indexed by [n]. A random variable Ion the power set of [n]is\ncalled a local stochastic set of parameter pif\nPr[F⊆I]≤p|I|for all F⊆[n].\nFor an n-qubit Pauli operator E∈ P n, we write supp( E)⊆[n]for its support, i.e., the set of\nqubits where Eacts non-trivially. For a random variable Eon the n-qubit Pauli group, the\nsupport supp( E)is a random subset of [n]. The random variable Eis called local stochastic\nnoise of strength pif its support supp( E)is a local stochastic set of parameter p, i.e.,\nPr[F⊆supp( E)]≤p|F|for all F⊆[n].\nWe write E∼ N(p)to denote local stochastic noise of strength p.\nWe often consider several errors E1, . . . , E reach acting on nqubits. In this case, we are\ndealing with a joint (not necessarily product) distribution PE1···EronP×r\nn. In this situation, the\nexpression Ej∼ N(pj)is a statement about the marginal distribution of the j-th error Ej.\nLocal stochastic noise is compatible with Clifford circuits as expressed by simple computa-\ntional rules, see e.g., [33, Lemma 11] or [19]. These rules allow to “commute” local stochastic\nerrors forwards or backwards in time through such a circuit. We give quantitative statements\nin Appendix A (see Lemma A.1). There we also discuss consequences e.g., for entanglement\nswapping and teleportation under noise.\n3.2 Noisy implementations of general adaptive circuits\nHere we formally state the notion of noisy implementation of an adaptive quantum circuit.\nRecall from Section 2.3 that a general adaptive quantum circuit Qonnqubits can be written\nas a composition\nM(T)◦ ··· ◦ M(1)\nwhere each n-qubit operation M(t)is composed of the parallel application of one- and two-qubit\noperations. Let {E(t)}T\nt=0be a (T+ 1)-tuple of n-qubit Pauli errors. A noisy implementation\nofQwith errors {E(t)}T\nt=0is then defined as the composition\nUE(T)◦ M(T)◦ ··· ◦ UE(1)◦ M(1)◦ UE(0) (22)\nwhere U(t)\nE(t)(ρ) =E(t)ρ(E(t))†is unitary evolution associated with E(t)for all t∈[T].\nEq. (22) may be interpreted as follows: The error E(0)could be the result of an imperfect\npreparation procedure for preparing the initial state |0n⟩. The error E(t)describes the effect\nof a noisy implementation of M(t)(although some of this effect may also be captured by the\nerror E(t−1)). Recall that Mconsists of one- and two-qubit operations applied simultaneously.\nThe role of the error depends on the operation: When preparing a single qubit state |0⟩orT|+⟩,\nthe corresponding part (factor) of the Pauli error E(t)corresponds to the preparation of a cor-\nrupted version. For single- or two-qubit Cliffords or the identity operation (“wait locations”), the\nerror amounts to an incorrect implementation of the corresponding unitary. Finally, for a com-\nputational basis measurement, the error corresponds to an erroneous bit-flip of the measurement\nresult: Here only Pauli- X-errors are relevant. In summary, errors in a noisy implementation of\nthe circuit Qdescribe fault processes related to all aspects involved in its execution, including\npreparation, unitary evolution, storage and measurements.\n17Anoisy implementation of Qwith local stochastic noise of strength pnow refers to a process\nof the form (22), where E(t)∼ N(p)is local stochastic noise of strength p. In other words, it is\nobtained by randomly drawing an (T+ 1)-tuple {E(t)}T\nt=1ofn-qubit Pauli errors according to\na joint distribution (having constraints on the marginals), inserting these errors between gate\nlayers, i.e., running the corresponding noisy implementation of Qwith errors {E(t)}T\nt=1. As\nmentioned above, the random variables {E(t)}T\nt=1may not be independent.\n3.3 Fault-tolerant preparation of a Bell state\nFor concreteness, let us give the argument for state preparation protocols, which is the case of\ninterest for our purposes. More specifically, let πbe an adaptive quantum circuit on Nqubits\nwhich prepares a target (stabilizer) state Φ∈(C2)⊗ras follows:\n1. First, a constant-depth Clifford circuit Wis applied to\f\f0N\u000b\n. This results in a short-range\nentangled resource state Ψ∈(C2)⊗N.\n2. Subsequently, Pauli measurements (i.e., von Neumann measurements associated with the\neigenbases of different single-qubit Pauli operators) are applied to N−rqubits, resulting\nin a measurement results z∈ {0,1}N−r.\n3. A Pauli correction C(z)∈ Prdepending (deterministically) on the measurement results z\n(i.e., aclassicallycontrolledPauli)isappliedtotheremaining rqubits. Werequirethat C:\n{0,1}N−r→ P rhasthefollowingproperties: First,thereisanefficient(classical)algorithm\ncomputing C(z)from z. Second, if Steps 2 and 3 are applied to the state EΨwhere E∈\nPN, then C(z)is determined by E.\nNow suppose we apply Steps 2 and 3 to the state EΨ. It is straightforward to check that the\noutput of the circuit then is a Pauli-corrupted version Eeff(E)Φof the target state Φ, where the\n“effective error” Eeff(E)depends deterministically on the error E. We are interested in errors E\nthat lead to a non-trivial overall effective error Eeff. To this end, let us define\nFAIL :={E∈ PN|Eeff(E)Φ̸∝Φ}\nas the set of such errors, i.e., errors Efor which neither Eeff(E)nor−Eeff(E)belongs to the\nstabilizer group of Φ. IfEis a stochastic Pauli error, then the probability that the procedure π\ndoes not prepare the desired target state Φis equal to Pr [πfails] = Pr[ E∈FAIL].\nRobustness of πagainst local stochastic noise E∼ N(p)follows from certain combinatorial\nfeatures of the set FAIL. It is useful to formalize this as follows. A circuit πwill be called\n(p0, f)-robust for a threshold error strength p0>0and a function f: [0, p0]→[0,1]if there is a\nfamily {Dm}m∈M⊂2[N]of subsets of [N]such that the following two conditions are satisfied:\n(a) For every E∈FAILthere exists an element m∈ Msuch that Dm⊆supp( E).\n(b) We have\nX\nm∈Mp|Dm|≤f(p)for all p≤p0.\nThe significance of this definition is the following immediate consequence of the union bound.\nLemma 3.1. A(p0, f)-robust circuit πfault-tolerantly prepares the state Φin the presence of\nlocal stochastic noise E∼ N(p)in the sense that\nPr [πfails]≤f(p)for any p≤p0.\n18Proof.Consider local stochastic noise E∼ N(p)with strength p≤p0. Then we have\nPr [E∈FAIL]≤X\nm∈MPr [Dm⊆supp( E)]by the union bound and (a)\n≤X\nm∈Mp|Dm|by the assumption E∼ N(p)\n≤f(p)by (b) .\nInRef.[32], wefollowedtheproofstrategyoutlinedheretoshowthatacertaincircuit πfault-\ntolerantly prepares long-range entanglement, that is, a Bell state between two distant qubits.\nFollowing the seminal work [34], our construction relies on the use of a 3D cluster state Πon\na certain cubic lattice Lcluster⊂R3specified by its linear dimensions d×d×R(here the Bell\nstate is between qubits at opposite ends, i.e., at distance R). For ease of presentation, we will\nuse the grid graph Pd×Pd×PRin the following and identify certain sites of the cluster state\nlattice Lclusterwith vertices of this grid graph. That is, we assume that there is a qubit at each\nvertex of Pd×Pd×PR. Then we can express the result of [32] as follows.\nCorollary 3.2 (Quantum bus architecture) .For any integer R≥2and any ∆∈Nsatisfying\n∆≥8 logR , (23)\nthere is a circuit π(∆, R)with the following properties:\n(i) The circuit π(∆, R)acts on N= ∆2Rqubits arranged on the grid graph P∆×P∆×PR,\nwith one qubit placed on each vertex.\n(ii) In the absence of errors, the circuit π(∆, R)prepares the two-qubit Bell state |Φ⟩=\n1√\n2(|00⟩+|11⟩)on the qubits Q1Q2, where Q1andQ2located at the sites (1,1,1)and\n(1,1, R), respectively.\n(iii) The circuit π(∆, R)follows Steps 1–3 in Section 3.3: It\n(1) first prepares a short-range resource state Ψ∈(C2)⊗Nby applying a Clifford circuit W\nto the product state\f\f0N\u000b\n. The circuit Wis geometrically local on the graph P∆×P∆×\nPRand its depth is upper bounded by 10.\n(2) then performs single-qubit Pauli measurements on all qubits except Q1Q2, obtaining an\noutcome z∈ {0,1}N−2.\n(3) finally applies a two-qubit correction C(z) =Zα(z)Xβ(z)to qubit Q2, where α(z), β(z)∈\n{0,1}are computed by efficient (classical) algorithms from z.\n(iv) The circuit π(∆, R)is(p0, f(p) =p/p0)-robust where p0:= 1/5004.\nCorollary 3.2 is a direct consequence of Ref. [32, Theorem 7.3]. The latter result gives a\nfault-tolerant scheme for preparing a Bell pair on two qubits Q1, Q2at the opposite ends (i.e.,\ndistance R) of a certain lattice Lcluster (d, R)of linear dimensions of the form O(d)×O(d)×R\nspecified by certain pairs of parameters (d, R)(Here dis the distance of an underlying surface\ncode.) The protocol proceeds as described in Section 3.3, that is:\n1. It starts with an N= Θ( d2R)-qubit state\f\f0N\u000b\nwhere each qubit is associated with a\nvertex of Lcluster. Then a depth- 9Clifford circuit W=W(d, R)is applied to prepare a\nresource state Ψ. The state Ψis the cluster state on Lclusterup to local unitaries.\n2. Every qubit except the two qubits Q1andQ2are measured in the computational basis,\nyielding a measurement result z∈ {0,1}N−2.\n193. A Pauli correction Zα(z)Xβ(z)is applied to qubit Q2, where the functions\nα, β:{0,1}N−2→ {0,1}\nare efficiently computable.\nLet us call this protocol πcluster (d, R). The proof of [32, Theorem 2.1] implies the following:\nClaim 3.3. Suppose that R≥3is odd and\n4 logR≤d . (24)\nThen the protocol πcluster (d, R)is(p0, f(p) =p/p0)-robust with p0:= 1/5004.\nProof.The proof of [32, Theorem 2.1] shows that the protocol πcluster (d, R)is(p0, f(p) =p/p0)-\nrobust with p0:=1\n5004andf(p) := 5004 punder the assumption that R≥3is odd, d≥3\nand\nR≤1\nd(10√p0)d−2. (25)\nEq. (25) can be written as\nlogd+ log R≤(d−2) log1\n10√p0=: (d−2)α .\nWith the inequality d/4≤(d−2)α−logd(valid for d≥3) we conclude that Eq. (25) is satisfied\nif\n4 logR≤d ,\nford≥3. Since we assumed R≥3, and thus 4 logR > 4, we can omit the requirement d≥3.\nThis shows that the circuit πcluster (d, R)is(p0, f(p) =p/p0)-robust for any odd R≥3and\nd≥4 logR.\nLet us now argue that Claim 3.3 implies Corollary 3.2.\nProof of Corollary 3.2. Suppose (R,∆)satisfying (23) are given. We separately analyze the\nfollowing cases:\nSuppose R= 2.We give a circuit π(∆, R)which is (1, f(p) = 2 p)-robust, with a unitary W\nof depth 2. The circuit π(∆, R)is non-adaptive, and simply consists of the Clifford\nunitary W=CNOT (H⊗I)applied to the two qubits Q1, Q2. Under local stochastic\nnoise E∼ N (p)applied after the application of W, the resulting state E|Φ⟩is Pauli-\ncorrupted if EQ2ET\nQ1is non-trivial. We can upper bound this by the probability that\neither qubit experiences an error, i.e.,\nPr [πfails]≤Pr [supp(E)∩ {Q1, Q2} ̸=∅]\n≤Pr [Q1∈supp(E)] + Pr [ Q2∈supp(E)]\n≤2p\nwhere we used the union bound and local stochasticity. The claim follows from this.\nSuppose R≥3is odd. We show that there is a protocol π(∆, R)which is (1/5004, f(p) =\n5004p)-robust, with a unitary Wof depth 9. Set d:=\u0004∆+1\n2\u0005\n. Then 2d−1≤∆, and thus\nthe grid graph P2d−1×P2d−1×PRis a subgraph of P∆×P∆×PR. Let π(∆, R)be the\nprotocol obtained by applying πcluster (d, R)to the qubits belonging to this subgraph. To\ncheckthat π(∆, R)is(1/5004, f(p) = 5004 p)-robust, itsufficestocheckthatcondition(24)\nfor Claim 3.3 is satisfied. But this follows immediately from the inequality∆\n2≤ ⌊∆+1\n2⌋\nand assumption (23).\n20Suppose R≥4is even. We give a protocol π(∆, R)which is (1/5004, f(p) = 5004 p)-robust,\nwith a unitary Wof depth 10. Suppose (R,∆)satisfying (23) are given with R≥4\neven. Set R′:=R−1. Then P∆×P∆×PR′is a subgraph of P∆×P∆×PR. To define\nthe circuit π(∆, R)in this case, consider the protocol π(∆, R′)which could be applied to\nthis subgraph (since R′≥3is odd by definition). Let W(∆, R′)be the corresponding\nunitary, and let Q1= (1,1,1)andQ′\n2= (1,1, R′). Applying π(∆, R′)would generate\nentanglement between Q1andQ′\n2, but our objective is to entangle the qubit Q1with the\nqubit Q2= (1,1, R′+ 1) = (1 ,1, R). It is easy to check that this can be achieved by a\nslight modification, leading to the following definition of π(∆, R):\n1. Set W(∆, R) := SWAP Q2Q′\n2W(∆, R′), i.e., succeed the application of the Clifford\nunitary W(∆, R′)with a SWAP-gate between Q2andQ′\n2.\n2. Perform all single-qubit measurements and obtain measurement results z(as in the\nprotocol π(∆, R′)).\n3. Apply the correction C(z) =Zα(z)Xβ(z)to qubit Q2(instead of qubit Q′\n2).\nCondition (24) for R′anddis again satified here because\n4 logR′≤4 logRsince R=R′+ 1\n≤∆\n2by the assumption (23)\n≤\u0016∆ + 1\n2\u0017\n=d .\nBy Claim 3.3, this shows that the protocol π(∆, R)is(1/5004, f(p) = 5004 p)-robust.\n3.4 Fault-tolerant preparation of several Bell states in parallel\nThe result of Lemma 3.1 can be generalized to the case where multiple quantum circuits are run\nin parallel. This is expressed by the following theorem:\nTheorem 3.4 (Parallel repetition) .Forj∈[k], let π(j)be an adaptive quantum circuit on\nN(j)qubits which prepares a target state Φ(j)∈(C2)⊗r(j), and is (p(j)\n0, f(j))-robust. Consider\nthe circuit π=π(1)× ··· × π(k)obtained by running each of these circuits in parallel on N=Pk\nj=1N(j)qubits. Then there are constants C >0,c∈(0,1)such that the following holds. Let\nE∼ N(p)be local stochastic noise on Nqubits with strength p≤min{p(j)\n0}k\nj=1=:p0. Then the\noutput state of πisEeff\u0010Nk\nj=1Φ(j)\u0011\nwhere Eeffis local stochastic noise onPk\nj=1r(j)qubits of\nstrength\nEeff∼ N(f(p))with f(p) =\u0012\nmax\nj∈[k]f(j)(p)\u00131\nr\n, r = max\nj∈[k]r(j). (26)\nIn particular, if there are constants C >0andc∈(0,1)such that\nf(j)(p)≤Cpcfor all p∈[0, p0]andj∈[k], (27)\nthen\nEeff∼ N(C1/rpc/r) (28)\nwhenever p≤p0.\n21Proof.We assume here that we apply the measurement- and correction-parts of each circuit π(j)\nto a state of the form\nE\nkO\nj=1Ψ(j)\nC(j)\n,\nwith local stochastic noise E∼ N(p)acting on all qubits. We can factor Eas\nE=kO\nj=1E(j),\nwherenow E(j)actsonlyonthequbitsassociatedwith C(j), thesetof N(j)qubitsthatcircuit π(j)\nis applied to. Recall from Lemma A.1 (i) that E(j)∼ N(p)is local stochastic noise with the\nsame parameter as E, a fact we will use below.\nLetEeffbe the effective error on thePk\nj=1r(j)“target” qubitsSk\nj=1{Q(j)\n1, . . . , Q(j)\nr(j)}. Let\nL ⊆k[\nj=1{Q(j)\n1, . . . , Q(j)\nr(j)}\nbe a subset of the target qubits. We need to upper bound Pr [L ⊆supp( Eeff)].\nWe can again factor\nEeff=kO\nj=1E(j)\neff,\nwhere each factor E(j)\neffis an r(j)-qubit Pauli operator which depends deterministically on E(j).\nThe event L ⊆ supp( Eeff)means that for at least ℓ:=⌈|L|/(max j∈[k]r(j))⌉of the instances,\nthe desired output is non-trivially corrupted by a Pauli error. Let us assume without loss of\ngenerality (by reindexing if necessary) that this affects the first ℓoutput states Φ(1), . . . , Φ(ℓ).\nIn other words, all the corresponding circuits failed. Thus we obtain the upper bound\nPr [L ⊆supp( Eeff)]≤Prh\nπ(j)fails for every j∈[ℓ]i\n.\nRecall that since π(j)is(p(j)\n0, f(j))-robust for each j∈[k], there is a family {D(j)\nm}m∈M(j)of\nsubsets of supp( E(j))satisfying conditions (a) and (b) in Section 3.3. By condition (a), we have\nthat the event that π(j)fails, which is the event that E(j)∈FAIL(j), implies the existence of\nm(j)∈ M(j)such that D(j)\nm(j)⊆supp( E(j)). Thus we obtain (again using the union bound) that\nPr [L ⊆supp( Eeff)]≤X\nm(1)∈M(1),...,m(ℓ)∈M(ℓ)Prh\nD(j)\nm(j)⊆supp( E(j))for all j∈[ℓ]i\n.(29)\nBut the condition\nD(j)\nm(j)⊆supp( E(j))for all j∈[ℓ]\nis equivalent to\nℓ[\nj=1D(j)\nm(j)⊆ℓ[\nj=1supp( E(j)) = supp( E) (30)\n22since the sets of qubits {D(j)\nm(j)}k\nj=1(respectively {supp( E(j))}k\nj=1) are pairwise disjoint (i.e.,\nbelong to different instances). For the same reason, we have\n\f\f\f\f\f\fℓ[\nj=1D(j)\nm(j)\f\f\f\f\f\f=ℓX\nj=1|D(j)\nm(j)|. (31)\nInserting (30) into (29) gives\nPr [L ⊆supp( Eeff)]≤X\nm(1)∈M(1),...,m(ℓ)∈M(ℓ)Pr\nℓ[\nj=1D(j)\nm(j)⊆supp( E)\n\n≤X\nm(1)∈M(1),...,m(ℓ)∈M(ℓ)pPℓ\nj=1|D(j)\nm(j)|\n=ℓY\nj=1\nX\nm(j)∈M(j)p|D(j)\nm(j)|\n\nwhere we used (31) and the fact that E∼ N(p)is local stochastic. By condition (b), we obtain\nthe upper bound\nPr [L ⊆supp( Eeff)]≤ℓY\nj=1f(j)(p) (32)\nfor all p≤min{p(1)\n0, . . . , p(ℓ)\n0}. Inserting the definition of ℓinto (32), we have\nPr [L ⊆supp( Eeff)]≤\u0012\nmax\nj∈[ℓ]f(j)(p)\u0013l\n|L|\nmaxj′∈[ℓ]r(j′)m\n≤\u0012\nmax\nj∈[ℓ]f(j)(p)\u0013 |L|\nmaxj′∈[ℓ]r(j′)\n.(33)\nThe expression (26) for noise strength of Eefffollows from the upper bound (33) combined with\nthe bounds max j∈[ℓ]f(j)≤max j∈[k]f(j)andmax j′∈[ℓ]r(j′)≤max j′∈[k]r(j′)forℓ≤ktogether\nwith the definition of local stochastic noise. Assuming that (27), the expression (26) for the\nnoise strength immediately follows from Eq. (28).\nWe note that the noise strength of an effective error in Theorem 3.4 can be improved if we\nknow about the stabilizer groups of Φ(j). Recall that the effective error E(j)\neff(E)can be chosen\nup to stabilizers of the j-th target state Φ(j). Let us choose E(j)\neff(E)to be of minimal support,\nand define\n˜r= max\nj∈[k]˜r(j)and ˜r(j)= max\nE∈PN(j)E(j)\neff(E).\nFor example, we have ˜r= 1if all target states Φ(j)are Bell states. Improved noise strength\nofEeffis obtained by replacing rwith ˜rin Eqs. (26) and (28).\n3.5 Generating long-range entangled Bell states using a parallel quantum\nbus\nCombining Corollary 3.2 with Theorem 3.4, we have the following immediate consequence on\nthe fault-tolerance of a parallel quantum bus:\n23Theorem 3.5 (Quantum bus architecture (parallel)) .There is a constant threshold p0>0on\nerror strength and constants C > 0,c∈(0,1)such that the following holds. Let kbe arbitrary,\nand let R1, . . . , R kand∆1, . . . , ∆kbe such that\n∆j≥8 logRjfor all j∈[k].\nConsider the parallel bus π(∆1, R1)× ··· × π(∆k, Rk). Then a noisy implementation of this\nparallel bus with noise strength p≤p0produces kBell states corrupted by local stochastic noise:\nThe output state on qubits {Q1,j, Q2,j}k\nj=1is\nF\nkO\nj=1|Φ⟩Q1,jQ2,j\n\nwhere Fis local stochastic noise on these 2kqubits with F∼ N(Cpc).\n4 Fault-tolerantly localized quantum circuits\nWe saw in Section 2.4 how a general adaptive quantum circuit can be rendered geometrically\nlocal. The new circuit includes auxiliary qubits such that the circuit on the entire system is local\nin a 2D or 3D grid structure. In the new circuit, all “long-range” operations on two qubits with\nlong distance are emulated by consuming short-range Bell pairs aligned on the routing paths.\nSince the length of these paths linearly grows in the linear size of the grid, so does the number\nof short-range Bell pairs for each path. In the case where operations are noisy, this implies that\nthe success probability of this long-range operation decays exponentially with the system size.\nThis problem can be resolved by augmenting the grid structure with quantum buses. We\nwill use the result from Section 3.5 that multiple pairs of noisy Bell states affected by constant-\nstrength noise can be prepared by running multiple noisy quantum buses in parallel. In par-\nticular, such buses can create long-range entanglement, which can be used to (approximately)\nrealize long-range operations.\nIn Section 4.1, we describe the grid graphs we use and explain how to generate pairwise\nentanglement (respectively pair qubits) fault-tolerantly. In Section 4.2, we state our main\nthreshold theorem for fault-tolerantly localizing quantum circuits. As an application, we show\nin Section 4.3 that any quantum computation using polynomial number of qubits and having\npolynomial depth can be fault-tolerantly localized with polynomial qubit overhead and quasi-\npolylogarithmic time overhead.\n4.1 Fault-tolerant pairwise entanglement generation in grid graphs\nIn the following, it will be convenient to use the set ZL={0, . . . , L −1}instead of [L] =\n{1, . . . , L }to label the vertices of PL. The grid graph PL×PL×P4Lthen has vertex set ZL×\nZL×Z4L, and we consider the canonical embedding of this graph into R3. Recall that for L\neven, the subset Z2\nL× {0}can be parallel-routed with paths of length upper bounded by 10L,\nsee Theorem 2.6. One key property of these routing schemes we use below is the following: Each\npath πcan decomposed into a collection (πup, πmid\n1, πmid\n2, πdown)of four straight line segments\nthat are parallel to different coordinate axes, and have the property that adjacent line segments\nare orthogonal. This follows by construction, see Eq. (17) for the definition of πupandπdown,\nand (11) for the definition of the path πmidwhich is the composed of segments πmid\n1andπmid\n2.\nWe will show how to fault-tolerantly route L2qubits in a 3D architecture with additional\nqubits. These additional qubits are placed in such a way that a fault-tolerant quantum bus\nconnecting the endpoints of each path segment can be realized, for any chosen path πoccurring\nin the routing scheme (i.e., for some pairing of the vertices Z2\nL×{0}). Mirroring the requirement\n24Figure 2: Cubes Q(e)associated with edges e. Each cube has mqubits on each side, with\nspacing1\nm. Edges eare highlighted with dashed boxes.\nof edge-disjointness of the paths for a given pairing in a routing scheme, buses associated with\nany edge-disjoint collection of paths (respectively corresponding path segments) do not share\nany qubits, thus enabling parallel operation. Since each path consists of only four segments,\nthe associated four buses can be “connected” using entanglement swapping, thus producing\nentanglement at the endpoints of each path.\nConsider the set\nV(m, R) :=n\u0010x\nm,y\nm,z\nm\u0011\n|x, y∈ {0, . . . , Lm −1}, z∈ {0, . . . , 4Lm−1o\n.\nThese 4(Lm)3points correspond to the vertices of the grid graph PLm×PLm×P4Lm; they\nare the result of using the standard embedding of this graph in R3and subsequently rescal-\ning by a factor1\nm. With a slight abuse of notation, we will refer to this embedded graph\nas1\nm(PLm×PLm×P4m).\nWe note that the vertices of PL×PL×P4Lin the standard embedding are exactly the\nelements of V(m, R)with integer coordinates. Our construction uses three qubits Rred\nℓ,Rgreen\nℓ\nandRblue\nℓat each location ℓ∈V(m, R). We use colors to illustrate these. The main result of\nthis section is the following:\nTheorem 4.1 (Fault-tolerant pairwise entanglement generation in a 3D grid graph) .There\nis a constant threshold error strength p0>0and constants C > 0,c∈(0,1]such that the\nfollowing holds. Let L≥2be an integer, and set m:= 82⌈logL⌉. Consider the embedded graph\n1\nm(PLm×PLm×P4Lm)with three qubits associated with each vertex. Let us write Z2\nL× {0}=\n{v1, . . . , v L2}and let {(vir, vjr)}L2/2\nr=1be an arbitrary pairing of these sites. Then there is an\nadaptive circuit πwith the following properties:\n(i) The circuit πis constant-depth with local gates on1\nm(PLm×PLm×P4Lm).\n(ii) The output of a noisy implementation of πwith noise-strength p≤p0produces L2/2Bell\nstates corrupted by local stochastic noise: The output state on the qubits\n{R(blue)\nvir, R(blue)\nvjr}L2/2\nr=1\nis\nF\nL2/2O\nr=1|Φ⟩R(blue)\nvirR(blue)\nvjr\n\nwhere Fis local stochastic noise on these L2qubits with F∼ N(Cpc).\n25(a) A path πrconnecting two ver-\ntices vir, vjr. The path consists of four\nline segments {π(α)\nr}3\nα=0.\n(b) Four buses {B(α)\nr}generating four Bell\npairs on two qubits located at the endpoints\nof{π(α)\nr}3\nα=1.\nFigure 3: A path πr={π(α)\nr}3\nα=0and four buses {B(α)\nr}3\nα=0for fault-tolerant entanglement\ngeneration.\nProof.Let us associate a cube Q(e)of qubits to each edge eof the graph PL×PL×P4L, see\nFig. 2. We distinguish between three different cases, depending on the orientation of the edge,\ni.e., whether it is parallel to the X-, the Y- or the Z-axis. That is, denoting by Q(x,y,z):=\n[x, x+ 1)×[y, y+ 1)×[z, z+ 1)the “half-open” cube located at (x, y, z ), we set\nQ(e) :=\n\n\b\nRred\nℓ|ℓ∈ Q (x,y,z)∩V(m, R)\t\nife={(x, y, z ),(x+ 1, y, z)}\n\b\nRgreen\nℓ|ℓ∈ Q (x,y,z)∩V(m, R)\t\nife={(x, y, z ),(x, y+ 1, z)}\n\b\nRblue\nℓ|ℓ∈ Q (x,y,z)∩V(m, R)\t\nife={(x, y, z ),(x, y, z + 1)}.\nObserve that for two orthogonal edges e, e′, the two cubes Q(e)andQ(e′)have no intersection\nsince the associated qubits have different colors. For colinear adjacent edges e,e′, the inter-\nsection Q(e)∩ Q(e′)is empty due to half-openness. Therefore, the cubes {Q(e)}eare pairwise\ndisjoint.\nLet now {πr}L2/2\nr=1be a parallel routing scheme associated with a pairing {(vir, vjr)}L2/2\nr=1in\nthe graph PL×PL×P4Las provided by Theorem 2.6; see Fig. 3a for an illustration. By the\ndisjointness of the cubes (and the edge-disjointness of the collection {πr}L2/2\nr=1of paths), any two\ndistinct paths πr̸=πr′in this routing scheme have the property that the collection of qubits in\ncubes associated with their edges are disjoint, i.e.,\n [\ne∈πrQ(e)!\n∩\n[\ne∈πr′Q(e)\n=∅.\nThis implies that we can consider each path πr,r∈[L2/2]separately, and seek to generate a Bell\npair on the qubits Qblue\nvir, Qblue\nvjrlocated at the endpoints using the qubits belonging toS\ne∈πrQ(e).\nInmoredetail, wewillproceedasfollows: Foreach r∈[L2/2], wedecomposethepath πrinto\nfour straight line segments (π(0)\nr, π(1)\nr, π(2)\nr, π(3)\nr)(as discussed above), and use a linear quantum\nbusB(α)\nrfor each α∈ {0,1,2,3}; see Fig. 3b. The bus B(α)\nrcreates a Bell pair on two qubits\n26located at the endpoints ∂π(α)\nr=:{u(α)\nr, w(α)\nr}. Here we denote by u(α)\nrandw(α)\nrthe first and\nlast vertices of π(α)\nr, where the orientation of the path segment π(α)\nris defined by a traversal of\nthe path πrfrom virtovjr. The bus B(α)\nruses the collection\nQ(α)\nr=[\ne∈π(α)\nrQ(e)\nof qubits associated with edges belonging to the line segment π(α)\nr. Recall that all these qubits\nhave the same color: It is the color determined by the coordinate axis the line segment π(α)\nris\nparallel to. We will denote by S(α)\nrthe qubit of this color at the beginning vertex u(α)\nrof the\nline segment, and by T(α)\nrthe qubit of the same color at the ending vertex w(α)\nr.\nNote that the qubits Q(α)\nrare located on the vertices of an embedded version of the grid\ngraph Pm×Pm×PR(α)\nr, where the length R(α)\nr=m·|π(α)\nr|(i.e., number of qubits) is proportional\nto the length |π(α)\nr|of the path segment. (Compared to the standard embedding, this involves\nscale factor1\nmand a rotation such that the graph is parallel to π(α)\nr.) Since this has the form\nof a “standard” (linear) quantum bus on the grid graph P∆×P∆×PR, we can take the proto-\ncolB(α)\nrto be the associated scheme π(∆, R), that is, the protocol discussed in Corollary 3.2.\nBy Corollary 3.2, this protocol generates entanglement at the endpoints of π(α)\nrif\nm≥8 log\u0010\nm· |π(α)\nr|\u0011\n. (34)\nSince|π(α)\nr| ≤ |πr| ≤10Lfor each path segment (see Theorem 2.6), Eq. (34) is satisfied for every\npath segment if\nm≥8 log(10 mL). (35)\nEq. (35) is satisfied because of our definition m:= 82⌈logL⌉andL≥2. Thus each bus B(α)\nris\n(p0, f(p) =p/p0)-robust with p0= 1/5004, and generates a Bell pair at the endpoints of π(α)\nr.\nIn summary, we have a total of 4·L2\n2buses{B(α)\nr}j∈{0,1,2,3},r∈[L2/2]that are each (p0, f(p) =\np/p0)-robust. The parallel repetition Theorem 3.5 thus guarantees that if these protocols are\nrun in parallel the result is a noisy version\nF′\nL2/2O\nr=13O\nα=0|Φ⟩S(α)\nrT(α)\nr\n\nof4(L2/2)-Bell pairs (each spanning a line segment), where F′∼ N(C′pc′)is local stochastic\nfor some constants C′, c′.\nFinally, we apply entanglement-swapping in parallel to each path πr,r∈[L2/2]. That is,\nfor each r∈[L2/2], we perform three Bell measurements on the three qubits pairs (T(0)\nr, S(1)\nr),\n(T(1)\nr, S(2)\nr)and(T(2)\nr, S(3)\nr). This converts the stateN3\nα=0|Φ⟩S(α)\nrT(α)\nrto a Bell state |Φ⟩S(0)\nrT(3)\nr\n(after applying a Pauli correction depending on the measurement result).\nBy definition, the path segments π(0)\nr,π(3)\nrare aligned with the Z-axis, i.e., correspond to\nblue qubits. In particular, (S(0)\nr, T(3)\nr) = (Rblue\nvir, Rblue\nvjr)are the two blue qubits at the endpoints\nofπr, for each r∈[L2/2]. Together with Lemma A.3 (which demonstrates that entanglement\nswapping in parallel is compatible with local stochastic noise), this shows that the final state on\nthese qubits after the entanglement-swapping step is the desired target state\nF\nL2/2O\nr=1|Φ⟩RbluevirRbluevjr\n\n27up to a local stochastic error F∼ N(Cpc)for some constants C, c > 0. This concludes the proof\nof Theorem 4.1.\nBy analogous reasoning, we can give a protocol for fault-tolerant pairwise entanglement\ngeneration in a quasi-2D architecture. Here quasi-2D refers to the fact that the linear extent\nalong one of the coordinate axes is only logarithmic in the linear extent along the other two\naxes.\nTheorem 4.2 (Quasi-2D-local fault-tolerant pairwise entanglement generation) .There is a\nconstant threshold error strength p0>0and constants C > 0,c∈(0,1]such that the following\nholds. Let L≥2be an integer. Set m:= 82⌈logL⌉. Consider the embedded graph1\nm(PLm×\nPLm×Pm)with two qubits associated with each vertex. Let us write {(i, i,0)|i∈ZL}=\n{v1, . . . , v L}and let {(vir, vjr)}L/2\nr=1be an arbitrary pairing of these sites. Then there is an\nadaptive circuit πwith the following properties:\n(i) The circuit πis constant-depth with local gates on1\nm(PLm×PLm×Pm).\n(ii) The output of a noisy implementation of πwith noise-strength p≤p0produces L/2Bell\nstates corrupted by local stochastic noise: The output state on the qubits\n{R(red)\nvir, R(green)\nvjr}L/2\nr=1\nis\nF\nL/2O\nr=1|Φ⟩R(red)\nvirR(green)\nvjr\n\nwhere Fis local stochastic noise on these Lqubits with F∼ N(Cpc).\nProof.The proof is similar to that of Theorem 4.1, so we only provide a sketch. We use the par-\nallel routing {πr}L/2\nr=1of the pairing {(vir, vjr)}L/2\nr=1obtained by Lemma 2.4, instead of that from\nTheorem 2.6. Note that each path πris decomposed into two orthogonal line segments π(0)\nr, π(1)\nr.\nFor fault-tolerance, it suffices to show that the inequality (34) holds. This is the case since\f\f\fπ(α)\nr\f\f\f≤2Lforα∈ {0,1}(see Lemma 2.4).\nAs a consequence of Theorem 4.1, the fault-tolerance of parallel quantum teleportation (see\nLemma A.4), and the fact that local stochastic noise remains local stochastic under a depth- 1\ncircuit, we obtain a fault-tolerant qubit pairing circuit with local operations in 3D. For this, we\nadd two qubits {Qj, Pj}at each site vj∈Z2\nL× {0}in the grid graph1\nmPLm×PLm×P4Lm.\nCorollary 4.3 (3D-local fault-tolerant qubit pairing) .There is a constant threshold error\nstrength p0>0and constants C > 0,c∈(0,1]such that the following holds. Let L≥2\nbe an integer and set m:= 82⌈logL⌉. Consider the embedded graph1\nm(PLm×PLm×P4Lm)\nwith three qubits associated with each vertex and additional qubits {Qj, Pj}at each site vj. Let\nus write Z2\nL× {0}={v1, . . . , v L2}, and let {(vir, vjr)}L2/2\nr=1be an arbitrary pairing of these sites.\nThen there is a constant-depth adaptive circuit QFTpair ,3Dwith the following properties: the\ncircuit QFTpair ,3Dimplements a transfer of subsystems that maps each subsystem Qjrto the\nsubsystem Pirforr∈[L2/2], i.e., an L2-qubit state Ψon the registers Q1, . . . , Q L2is mapped\naccording to\nΨQi1Qj1···QiL2/2QjL2/27→ΨQi1Pi1···QiL2/2PiL2/2.\n28Moreover, for any noisy implementation of the circuit QFTpairwith local stochastic noise of\nstrength p≤p0, the resulting state on the registers Qi1Pi1. . . Q iL2/2PiL2/2is\nFΨQi1Pi1···QiL2/2PiL2/2\nfor local stochastic noise F∼ N(Cpc).\nFollowing the same reasoning, we obtain a fault-tolerant qubit pairing circuit with local\noperations in a quasi-2D architecture as a consequence of Theorem 4.2. Here we add two\nqubits {Qj, Pj}at each site vj∈ {(i, i,0)|i∈ZL}in the grid graph1\nmPLm×PLm×Pm.\nCorollary 4.4 (Quasi-2D-local fault-tolerant qubit pairing) .There is a constant threshold error\nstrength p0>0and constants C > 0,c∈(0,1]such that the following holds. Let L≥2be an\ninteger, and set m:= 82⌈logL⌉. Consider the embedded graph1\nm(PLm×PLm×P4Lm)with two\nqubits associated with each vertex and additional qubits {Qj, Pj}at each site vj. Let us write\n{(i, i,0)|i∈ZL}={v1, . . . , v L}, and let {(vir, vjr)}L/2\nr=1be an arbitrary pairing of these sites.\nThen there is a constant-depth adaptive circuit QFTpair ,quasi2Dwith the following properties: the\ncircuit QFTpair ,quasi2Dimplements a transfer of subsystems that maps each subsystem Qjrto the\nsubsystem Pirforr∈[L/2], i.e., an L-qubit state Ψon the registers Q1, . . . , Q Lis mapped\naccording to\nΨQi1Qj1···QiL/2QjL/27→ΨQi1Pi1···QiL/2PiL/2.\nMoreover, for any noisy implementation of the circuit QFTpair ,quasi2Dwith local stochastic noise\nof strength p≤p0, the resulting state on the registers Qi1Pi1. . . Q iL/2PiL/2is\nFΨQi1Pi1···QiL/2PiL/2\nfor local stochastic noise F∼ N(Cpc).\n4.2 Threshold theorem of quantum circuits localized by a parallel bus\nHaving constructed fault-tolerant qubit pairing protocols, we are now in position to state and\nprove our main result. It states the functionality of a fault-tolerant circuit Q(which uses non-\nlocal gates) can be realized by local circuits.\nTheorem 4.5 (Fault-tolerantlylocalizingageneraladaptivecircuit) .LetQbe a general adaptive\nquantum circuit on n= 2kqubits. Let Tbe the quantum depth of Q. Then there are adaptive\nquantum circuits Q′\nquasi2DandQ′\n3Dwith the following properties:\n(i) By taking certain marginals (i.e., tracing out qubits and/or ignoring measurement results),\nthe two circuits exactly simulate Q.\n(ii) Both circuits have quantum depth of order O(T).\n(iii) The circuits Q′\nquasi2DandQ′\n3Dare geometrically local in quasi-2D and 3D-architectures,\nrespectively (i.e., only involve local or nearest-neighbor operations on a corresponding grid\ngraph).\n(iv) The circuits Q′\nquasi2DandQ′\n3Duse a total number of\nntot\nquasi2D =O(n2log3n)and ntot\n3D=O(n3/2log3n) (36)\nqubits, respectively.\n29(v) There exist a threshold p0>0and some constants C > 0, c > 0such that the following\nholds: Any noisy implementation of the circuits Q′\nquasi -2DandQ′\n3Dunder arbitrary local\nstochastic noise of strength p≤p0are equivalent to a noisy implementation of the circuit Q\nwith local stochastic noise of strength C·pc.\nProof.We use the graphs\nGquasi2D =1\nmPLm×PLm×Pmand G3D=1\nmPLm×PLm×P4Lm\nwith\nL:=(\n⌈n⌉inGquasi2D\n⌈√n⌉inG3D,and m:= 82⌈logL⌉,\nsee Theorem 4.1 respectively Theorem 4.2. For Gquasi2Dwe will use the vertices\nSquasi2D ={vi:= (i−1, i−1,0)}i∈[n]⊂Z2\nL× {0},\nwhereas for the 3D grid graph G3D, we pick an arbitrary subset\nS3D={vi}n\ni=1⊆Z2\nL× {0}\nofZ2\nL× {0}of size |S3D|=n≤L2. (We will attach our “computational” qubits Q1···Qn\nassociated with the circuit Qto these vertices.)\nAssociating two and three auxiliary qubits respectively to each vertex of the graphs Gquasi2D\nandG3D, respectively, and attaching a “computational” qubit Qjand a “register” qubit Pjto\neach vertex vjforj∈[n], we are using a total number of\nntot\nquasi2D = 2n+ 2·L2m3=O(n2log3n)and ntot\n3D= 2n+ 3·4(Lm)3=O(n3/2log3n)\nqubits, respectively. This agrees with Eq. (36).\nWe construct our circuits Q′\nquasi2DandQ′\n3Dby replacing – in each layer t∈[T]of the\noriginal adaptive circuit Q– the operation M(t)(see Eq. (18)) by a geometrically local adaptive\ncircuit cM(t). Let{(i(t)\nr, j(t)\nr)}k\nr=1be the pairing relevant at time step (layer) t∈[T], and let\nQ(t)\nFTpair=QFTpair\u0010\n{(i(t)\nr, j(t)\nr)}k\nr=1\u0011\n,and (Q(t)\nFTpair)−1=Q−1\nFTpair\u0010\n{(i(t)\nr, j(t)\nr)}k\nr=1\u0011\nbetheconstant-depthadaptivecircuitsintroducedinSection4.1, i.e., thecircuit QFTpairiseither\nQFTpair ,quasi2DorQFTpair ,3D. Then the new circuits Q′\nquasi2Drespectively Q′\n3Dare obtained by\nreplacing – for each t∈[T]– the operation M(t)in the circuit Qby the composition\ncM(t):= (Q(t)\nFTpair)−1◦ kO\nr=1M(t,r)\nQ\ni(t)\nrP\ni(t)\nr!\n◦ Q(t)\nFTpair. (37)\nInotherwords, thequbitspairedinlayer tareplacednexttoeachotherbyapplicationof Q(t)\nFTpair.\nThen each (two-qubit) operation M(t,r),r∈[k]can be applied locally. Subsequently, the\nqubits are moved back to their original positions by application of (Q(t)\nFTpair)−1. The new cir-\ncuitsQ′\nquasi2D,Q′\n3Dthen are obtained as the composition\nQ′:=cM(T)◦ ··· ◦ cM(1).\nIt is easy to check that this has all the claimed properties. In particular, for each t∈[T], both\ncircuits (37) are adaptive constant-depth circuits, see Corollaries 4.3 and 4.4. This immediately\nimplies that the quantum circuit depth of Q′is of order O(T).\n304.3 Local quantum fault-tolerance in 3D\nHere we explain how Corollary 1.2 follows from our construction when applied to the fault-\ntolerance scheme of [17]. The latter takes an ideal quantum circuit Qidealand transforms it into\na fault-tolerant (adaptive) circuit QFTwhose output distribution approximates that of the ideal\ncircuit even when it is implemented imperfectly. Paraphrased using our terminology, the main\nresult of Yamasaki and Koashi gives the following.\nTheorem 4.6 (Fault-tolerant quantum computation with non-local gates [17]) .There is a\nthreshold error strength p0>0such that the following holds for all sufficiently large nand an\narbitrary constant ε∈(0,1). LetQidealbe an adaptive quantum circuit using nqubits and having\nquantum depth T(n) =O(poly(n)). 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A , vol. 71, p. 022316, 2 Feb. 2005. doi:10.1103/PhysRevA.\n71.022316 .\n[32] S. H. Choe and R. Koenig, Long-range data transmission in a fault-tolerant quantum bus\narchitecture , Sep. 2022. arXiv: 2209.09774 [quant-ph] .\n[33] S. Bravyi, D. Gosset, R. Koenig, and M. Tomamichel, “Quantum advantage with noisy\nshallow circuits in 3D,” Nature Physics , vol. 16, no. 10, pp. 1040–1045, Oct. 2020, issn:\n1745-2473, 1745-2481. doi:10.1038/s41567-020-0948-z .\n[34] R. Raussendorf, S. Bravyi, and J. Harrington, “Long-range quantum entanglement in noisy\ncluster states,” Phys. Rev. A , vol. 71, p. 062313, 6 Jun. 2005. doi:10.1103/PhysRevA.\n71.062313 .\n33A Local stochastic noise and simple adaptive quantum circuits\nLocal stochastic noise is compatible with Clifford circuits, as expressed by the following Lemma,\n(cf. e.g., [33, Lemma 11] and [19]).\nLemma A.1 (Properties of local stochastic noise) .Consider a system of nqubits. Then the\nfollowing holds.\n(i) Suppose E∼ N(p)andE′is a random Pauli such that supp( E′)⊆supp( E)with probability\n1. Then E′∼ N(p).\n(ii) Suppose E∼ N (p)andF∼ N (q)are random Paulis which may be dependent. Then\nE·F∼ N(q′)where q′= 2 max {√p,√q}.\n(iii) Suppose E∼ N(p)is a random Pauli and let Ube a tensor product of single- and two-qubit\nClifford unitaries. Then UEU†∼ N(q′)where q′=√2p.\n(iv) Let A⊆[n]be a subset of qubits. Suppose E∼ N(p),F∼ N(q)andsupp( E)⊆Aand\nsupp( F)⊆Ac=: [n]\\Awith probability 1. Then E·F∼ N(q′)where q′= max {√p,√q}.\n(v) Let πbe a permutation on [n], and let π(E)be the n-qubit Pauli which acts on the i-th\nqubit as π(i). Then π(E)∼ N(p).\nProof.See [33, Lemma 11] for (i)–(iii). For (iv), observe that for any subset I⊆[n], we have\nPr[I⊆supp( E·F)] = Pr[ I∩A⊆supp( E)andI∩Ac⊆supp( F)]\n≤min{p|I∩A|, q|I∩Ac|}\n≤min{(max{p, q})|I∩A|,(max{p, q})|I∩Ac|}\n≤(max{p, q})|I∩A|+|I∩Ac|\n2\n= (max {√p,√q})|I|,\nwhere we used the definition of local stochastic noise and the fact that min{a, b} ≤√\nabfor any\na, b≥0. For (v), observe that for any subset I={i1, . . . , i k} ⊆[n], we have I⊆supp( π(E))if\nand only if {π−1(i1), . . . , π−1(ik)}=:π−1(I)⊆supp( E). Therefore,\nPr[I⊆supp( π(E))] = Pr\u0002\nπ−1(I)⊆supp( E)\u0003\n≤p|π−1(I)|=p|I|,\nwhere the fact that Eis local stochastic noise is used.\nLemmaA.1allowsonetocommuteerrorsthroughadaptivecircuitsexplicitlyiftheadaptivity\nis expressed by linear functions. We formulate this as follows:\nLemma A.2 (Noisy adaptive quantum circuits) .Letn, n1, n2∈Nwithn=n1+n2and let\nA= (aij), B= (bij)∈Fn1×n2\n2\nbe matrices. Consider a set of qubits A=A1∪ A 2= [n]where A1= [n1]andA2={n1+\n1, . . . , n }. For x∈Fn\n2letX(x) =Qn\nj=1Xxj\njandZ(x) =Qn\nj=1Zxj\njbe products of Pauli- X\nand Pauli- Zoperators with support defined by x. For each x1∈Fn1\n2, let us denote by X(x1)\nandZ(x1)Pauli operators on A1defined analogously.\nLetQbe an adaptive quantum circuit on the qubits Aof the following:\n1. The circuit Qmeasures all qubits belonging to A2in the computational basis, obtaining a\nmeasurement outcome z∈Fn2\n2.\n34z1\n...\nzn2A1\nEZ(Bz) X(Az)\nA2=z1\n...\nzn2A1 Z(Bz) X(Az) F\nA2\nFigure 4: Two different noisy implementations Q1(left) and Q2(right) of an adaptive quantum\ncircuit Qare equivalent if two Paulis E∈ PnandF∈ Pn2satisfies Eq. (39).\n2. Subsequently, an n1-qubit Pauli gate C(z)∈ P n1is applied to the qubits A1, where we\nassume that\nC(z) =X(Az)Z(Bz)for all measurement outcomes z∈Fn2\n2.\nFor any Pauli operator EonA, there exists a Pauli operator F=F(E)onA1determined by E\nsatisfying the following:\n(i) A noisy implementation Q1=Q ◦ U Eof the circuit Qassociated with the error Eacting\nonAbefore the (ideal) circuit Qis applied is equivalent to a noisy implementation Q2=\nUF◦Qof the circuit Qassociated with the error Facting on A2after the (ideal) circuit Q\nis applied (see Fig. 4.)\n(ii) Let w∈Nwith n1·w≤n2, and suppose that all non-zero row vectors of AandBhave\nweight wand all column vectors of AandBhave weight at most 1, i.e.,\nPn2\nj=1aij∈ {0, w},Pn2\nj=1bij∈ {0, w}for all i∈[n1]Pn1\ni=1aij≤ 1,Pn1\ni=1bij≤ 1 for all j∈[n2].\nFor any p∈[0,1], ifE∼ N(p), then F∼ N(p′), where\np′= 4·p1\n4w. (38)\nProof.We construct the Pauli operator Fas follows. Let us write\nE=X(e)Z(f),\nwhere\ne= (e(1), e(2)), f= (f(1), f(2))∈Fn\n2with e(j), f(j)∈Fnj\n2for j∈ {1,2}.\nDefine a Pauli operator F∈ PnonAas\nF:=X(e(1)+Ae(2))Z(f(1)+Be(2)). (39)\n35We show that the Pauli Fsatisfies the property (i). Observe that\n\u0000\nX(Az)Z(Bz)⊗ ⟨z|A2\u0001\nE=\u0010\nFX(A(z+e(2)))Z(B(z+e(2)))\u0011\n⊗D\nz+e(2)\f\f\f\nA2,(40)\nwhere a phase factor from the anti-commutativity of Pauli- Xand Pauli- Zis ignored. Let Qj(ρ)\ndenote the output state of the circuit Qjwith an input state ρonAforj∈ {1,2}. It follows\nfrom Eq. (40) that for any state ρon the system A, we have\nQ1(ρ) =X\nz∈Fm\n2(X(Az)Z(Bz)⊗ ⟨z|)EρE∗((X(Az)Z(Bz))∗⊗ |z⟩)\n=X\nz∈Fm\n2\u0010\nFX(A(z+e(2)))Z(z+e(2))⊗D\nz+e(2)\f\f\f\u0011\nρ(h. c.)\n=Q2(ρ),\nand thus Q1andQ2are equivalent.\nNext, we show that the Pauli Fsatisfies the property (ii). We first claim that\nX(Ae(2))∼ N(4·p1/w)and Z(Be(e))∼ N(4·p1/w). (41)\nHere we only give a proof of the claim (41) for X(Ae(2))since the proof for Z(Be(2))mirrors\nthis. Let I={i1, . . . , i ℓ} ⊆ [n1]be the set of indices such that the i-th row vector of Ais\nnon-zero, and let us denote by j(i)\n1<···< j(i)\nwfor each i∈Ithe indices in [n2]consisting of the\nsupport of the i-th row vector of A, i.e.,\naij(i)\nk= 1for all k∈[w], i∈I ,\nand define\ng(k)=\u0012\ne(2)\nj(i1)\nk, . . . , e(2)\nj(iℓ)\nk\u0013\n∈Fℓ\n2for all k∈[w].\nTheassumptionaboutthecolumnsofthematrix Aimpliesthatforeach k, theindices {j(i)\nk}i∈I⊆\n[n2]are pairwise disjoint. Therefore, there exists some permutation πon[n2]such that g(k)\ncan be obtained by truncating the last n2−ℓbits of the vector π(e(2)) := ( e(2)\nπ(1), . . . , e(2)\nπ(n2)).\nTherefore, by the properties (i) and (v) of local stochastic noise in Lemma A.1, we have that\nX(g(k))∼ N(p)for all k∈[w]. (42)\nBy construction, we have Ae(2)=g(1)+···+g(w), and thus\nX(Ae(2)) =wY\nk=1X(g(k))∼ N\u0010\n4·p2−⌈log2w⌉\u0011\n, (43)\nwhere we applied Lemma A.1 (ii) w−1times together with the fact (42) to compute noise\nstrength of the product of {X(g(k))}w\nk=1in the form of a complete binary tree. The claim (41)\nfollows from Eq. (43) together with the inequality 4·p2−⌈log2w⌉≤4·p1/w.\nFinally, we observe that X(e(1))Z(f(1))∼ N(p)by the property (i) of Lemma A.1. Combin-\ning this with (41), we have\nF=X(Ae(2))Z(Be(2))X(e(1))Z(f(1))∼ N\u0010\n4·p1\n4w\u0011\n,\nwhere Lemma A.1 (ii) is used twice and a phase factor is again ignored.\n36An important special case of Lemma A.2 is the parallel execution of entanglement-swapping\ncircuits.\nLemma A.3 (Entanglement swapping on noisy Bell states) .Consider the following entangle-\nment swapping protocol which takes kℓBell pairs\nΨin=kO\ni=1ℓO\nj=1ΦQ(j)\niQ′\ni(j)\nas an input and generates ℓBell pairs Ψout=Nℓ\nj=1ΦQ(j)\n1Q′\nk(j):\n1. Apply a Bell measurement to the qubits Q′\ni(j), Q(j)\ni+1for each i∈[k−1]andj∈[ℓ], getting\noutcomes (a(j)\ni, b(j)\ni).\n2. Apply Xa(j)Zb(j)to qubit Q′\nk(j)for all j∈[ℓ]where a(j)=Lk−1\ni=1a(j)\niandb(j)=Lk−1\ni=1b(j)\ni.\nLetE∼ N(p)be a2kℓ-qubit Pauli noise which is local stochastic of strength p. If the entangle-\nment swapping protocol takes noisy kℓBell pairs EΨin, then the output state is a noisy ℓ-fold\nBell state FΨoutfor some 2ℓ-qubit local stochastic noise F∼ N(p′), where p′= 4(√2p)1\n4(k−1).\nProof.Note that Step 1 can be decomposed into one layer Uconsisting of two-qubit Clifford\nunitaries followed by single-qubit measurements in computational basis. By commuting Epast\nthe layer U, we see that the protocol with state EΨinis equivalent to the following protocol\nwith the input state (UEU†)UNk\ni=1Nℓ\nj=1ΦQ(j)\niQ′\ni(j):\n1. Apply computational basis measurement to the qubits Q′\ni(j), Q(j)\ni+1for each i∈[k−1]and\nj∈[ℓ], getting outcomes (a(j)\ni, b(j)\ni).\n2. Apply Xa(j)Zb(j)to qubit Q′\nk(j)for all j∈[ℓ]where a(j)=Lk−1\ni=1a(j)\niandb(j)=Lk−1\ni=1b(j)\ni.\nNote that this has the form of an adaptive circuit as discussed in Lemma A.2. By Lemma A.2,\nthe resulting state is a noisy ℓ-fold Bell state affected by a Pauli noise Fwhich is determined\nbyUEU†. By Lemma A.1 (iii), we have\nUEU†∼ N\u0010p\n2p\u0011\n. (44)\nLet us label the qubits by the sets A1andA2defined as\nA1=ℓ[\nj=1{Q(j)\n1, Q′\nk(j)}= [2ℓ]\nA2=k−1[\ni=1ℓ[\nj=1{Q′\ni(j), Q(j)\ni+1}={2ℓ+ 1,2ℓ+ 2, . . . , 2k},\nand construct a string z= (z1, . . . , z (2k−2)ℓ)∈F(2k−2)ℓ\n2by collecting all bits\nk−1[\ni=1ℓ[\nj=1{(a(j)\ni, b(j)\ni)},\nwhere the ordering is consistent with the labeling A2. Then, it is clear by inspection that there\nare matrices A, B∈F2ℓ×(2k−2)ℓ\n2 satisfying\nℓO\nj=1Xa(j)\nQ′\nk(j)Zb(j)\nQ′\nk(j)=X(Az)Z(Bz).\n37Furthermore, all non-zero row vectors of AandBhave weight k−1and all column vectors\nofAandBhave weight at most 1. As a result, we can apply Lemma A.2 (ii). We obtain\np′by inserting√2p, which was obtained by Eq. (44), and k−1respectively into pandwof\nEq. (38).\nAnother consequence of Lemma A.2 is the following statement, which characterizes the effect\nof local stochastic noise on teleportation circuit applied in parallel.\nLemma A.4 (Noisy implementation of the quantum teleportation protocol) .Consider the fol-\nlowing quantum teleportation protocol which takes a 3ℓ-qubit state\nΨ′\nin= Ψ Q1...QℓℓO\nj=1ΦRjR′\nj\nas an input and outputs an ℓ-qubit state Ψ′\nout= Ψ R′\n1...R′\nℓ:\n1. Apply a Bell measurement to the qubits Qj, Rjfor each j∈[ℓ], getting outcomes (aj, bj).\n2. Apply XajZbjto the qubit R′\njfor all j∈[ℓ].\nLetE∼ N (p)be a 3ℓ-qubit Pauli noise which is local stochastic noise of strength p. If the\nquantum teleportation protocol takes a noisy state EΨ′\ninas an input, then the output state is a\nnoisy state FΨ′\noutfor some ℓ-qubit local stochastic noise F∼ N(p′), where p′= 217\n8p1\n8.\nProof.The proof mirrors that of Lemma A.3, so we only give a sketch. We choose√2pand1\nrespectively for the parameters pandwof Eq. (38) to obtain p′.\n38"    },    {        "title": "2402.13879v1.Infinite__continuous__spin_particle_in_constant_curvature_space.pdf",        "content": "arXiv:2402.13879v1  [hep-th]  21 Feb 2024Infinite (continuous) spin particle\nin constant curvature space\nI.L.Buchbinder1,2,3, S.A.Fedoruk1, A.P.Isaev1,5, V.A.Krykhtin2,4\n1Bogoliubov Laboratory of Theoretical Physics, Joint Insti tute for Nuclear Research,\n141980 Dubna, Moscow Region, Russia ,\nbuchbinder@theor.jinr.ru, fedoruk@theor.jinr.ru, isae vap@theor.jinr.ru\n2Department of Theoretical Physics, Tomsk State Pedagogica l University,\n634041, Tomsk, Russia ,\njoseph@tspu.edu.ru\n3Tomsk State University of Control Systems and Radioelectro nics,\nLenin Av. 40, 634050, Tomsk, Russia\n4National Research Tomsk State University,\nLenin Av. 36, 634050, Tomsk, Russia ,\nkrykhtin@tspu.edu.ru\n5Faculty of Physics, Lomonosov Moscow State University, 119 991 Moscow, Russia\nAbstract\nWe present a new particle model that generalizes for curved s pace-time an infinite spin\nparticle in flat space. The model is described by commuting We yl spinor additional\ncoordinates. It is proved that this model is consistent only in an external gravitational\nfield corresponding to constant curvature spaces. A full set of first-class constraints in\nde Sitter and anti-de Sitter spaces is obtained.\nPACS: 11.10.Ef, 11.30.-j, 11.30.Cp, 03.65.Pm, 02.40.Ky\nKeywords: continuous spin particles, gravity, de Sitter an d anti-de Sitter spaces1 Introduction\nModels of relativistic particles in external fields are usef ul tools for studying and clarifying\nvarious properties of more complicated field or string theor ies at classical and quantum levels.\nThere is a huge literature devoted to the formulation of this models and the use of various\naspects of their gauge structure, (super)symmetry, dynami cs, quantization and applications\nfor calculating quantum effects within the world-line forma lism. In this paper, we propose a\nnew model of a 4Drelativistic particle in a gravitational background; ther efore, for review,\nwe note only those works that are related to the description o f (super) particles in various\nexternal (super) gravitational fields (see e.g. [1–6,8–15] and references therein).\nAs is well known from the pioneering papers by Wigner [16] and Bargmann and Wigner\n[17], the irreducible unitary representations of the Poinc ar´ e group are associated with ele-\nmentary particles. Physically non-contradictive irreduc ible representations are divided into\nfinite spin massive and massless, in turn massless ones are di vided into representations with\nfinite helicities and infinite helicity representations, th e latter are called infinite or continuous\nspin representations. Massive and finite helicity represen tations are widely used to describe\nstandard massive and massless elementary particles; as for the infinite spin representations,\ntheir physical meaning is unclear yet. Nevertheless, infini te spin representations attract some\nattention due to the hypothetical possibility of construct ing a consistent massless higher spin\ninteracting theory on such a ground. At present, there is a la rge enough activity devoted to\nvarious aspects of describing infinite spin representation s within field theory (see, e.g., the\nreview [18] and further development in the recent papers [19 –27] and references therein).\nAs pointed out, many features of complicated field theories c an be studied in simplified\nforms within appropriate particle models. The correspondi ng particle models associated\nwith massive or massless helicity irreducible representat ions are well known and broadly\nused for different purposes. As for particle models related t o infinite spin representations,\nsuch models are practically not studied in details. Some asp ects of infinite spin particle\nmodel in 4Dflat space-time were considered in our paper [28]. In this pap er, we generalize\nthe results of the above work to infinite spin particle in curv ed space-time. However, before\ngoing on, we should make some remarks.\nFirst, many years ago, Bargmann and Wigner [17] showed that t he description of an\ninfinite spin particle in flat space must include, in addition to space-time coordinates, some\nauxiliary \"intrinsic\" coordinates, which can be, e.g., a ve ctor or a spinor. It is natural to\nassume that the description in curved space-time must also u se auxiliary coordinates, which\nin this case must somehow be consistent with the geometric st ructure of space-time. For ex-\nample, an auxiliary spinor coordinate should be spinor unde r local Lorentz transformations.\nSecond, the notion of infinite spin relates to irreducible re presentations of the Poincar´ e\ngroup and does not admit direct generalization to arbitrary curved space-time, even to\nnonzero constant curvature space-time. Therefore, when we talk about an infinite spin\nparticle model in curved space-time, we take into account a g eneral covariant model with\nthe same number of degrees of freedom as the corresponding pa rticle model in flat space and\nwith a known flat limit. A number of degrees of freedom in the in finite spin particle model\nin flat space is controlled by a system of first class constrain ts. Therefore, construction of\ninfinite spin particle models basically means finding genera l covariant first class constraints.\nThird, we begin constructing an infinite spin particle model in arbitrary curved space-\ntime, generally speaking with torsion and looking for gener al covariant generalization of flat\n1space constraints that form a closed algebra in terms of Pois son brackets. As we will see,\nsuch consistent requirements are extremely non-trivial an d lead to that space-time must only\nbe a constant curvature space with vanishing torsion.\nThe paper is organized as follows. Section2 is a brief review of our Lagrangian con-\nstruction for a infinite spin particle model in flat space [28] focusing on the structure of\nfirst class constraints. In section3, we discuss a generic sc heme for constructing a Lagrange\nformulation in arbitrary curved space-time with torsion. S ection4 is devoted to deriving\na curved space-time generalization of flat space constraint s and finding conditions yielding\na closed algebra of constraints in curved space. It is proved here that a closed algebra of\nthe constraints is possible only in torsion-free and consta nt curvature space-time. Section5\nsummarizes the results obtained.\n2 Infinite spin particle in flat space\nThe Lagrangian formulation of a free infinite spin particle i n the Minkowski space was\nconstructed in the work [28] within the first-order formalis m using the spinor auxiliary co-\nordinates. As is known (see, e.g., [5]), the Lagrangian for a ny free relativistic particle is\nbasically defined in the first order formalism by a system of co nstraints. In the case under\nconsideration, the Lagrangian is written as follows:\nLflat=pa˙xa+πα˙ξα+¯π˙α˙¯ξ˙α+e0f0+ef+˜e˜f+euu, (2.1)\nwheree0(τ),e(τ),˜e(τ),eu(τ)are the Lagrange multipliers for the Bargmann-Wigner con-\nstraints\nf0∶=papa≈0, (2.2)\nf∶=pa/parenleft.alt1ξσa¯ξ/parenright.alt1−µ≈0, (2.3)\n˜f∶=pa(¯π˜σaπ)−µ≈0, (2.4)\nu∶=N−¯N≈0, (2.5)\nwhere\nN∶=ξαπα,¯N∶=¯π˙α¯ξ˙α. (2.6)\nThe parameter µ∈Rin (2.3), (2.4) is a nonzero constant, xaandpaare the coordinates and\nmomenta of relativistic particles, ξαand¯ξ˙αare the spinor auxiliary coordinates, παand¯π˙α\nare their momenta, and τis an evolution parameter: xa=xa(τ),pa=pa(τ), etc. We use the\nstandard notation for τ-derivatives: ˙xa,˙ξα,˙¯ξ˙α. Constraints (2.3), (2.4) are imposed to get\nan infinite-spin irreducible representation of the Poincar ´ e group. We stress that in the case\nof using spinor auxiliary coordinates, we should introduce the constraint u≈0(2.5) that\ncommutes with all Casimir operators of the Poincar´ e group. This constraint provides an\nequal number of dotted and undotted spinor indices in the dyn amical invariants and hence\nguarantees a possibility to use only vector indices in such q uantities (see details in [28]).\nThe non-vanishing Poisson brackets of the canonical moment apa,πα,¯π˙αand the four-\nvector coordinates xaand auxiliary Weyl spinors ξα,¯ξ˙αare\n{xa,pb}=δa\nb,{ξα,πβ}=δα\nβ,/braceleft.alt1¯ξ˙α,¯π˙β/braceright.alt1=δ˙α\n˙β. (2.7)\n2Therefore, the non-vanishing Poisson brackets of constrai nts (2.2)-(2.5) have the form\n/braceleft.alt1f,˜f/braceright.alt1= −Kf0, (2.8)\nwhere\nK∶=N+¯N (2.9)\nandNand¯Nare given by (2.6). Thus, constraints (2.2)-(2.5) form the c losed Poisson\nbrackets algebra.\nThe main purpose of the paper is to generalize this Lagrangia n approach to the case of\ninfinite spin particle in curved space-time with preservati on of the same number of degrees\nof freedom like in the flat space. The basic requirements for s uch a construction are:\n●General covariant generalization of constraints (2.2)-(2 .5) with a given flat limit.\n●Closure of the algebra of new constraints. One can expect tha t this condition imposes\na restriction on space-time geometry.\nIn the next section, we will consider a realization of the abo ve requirements.\n3 Generalization to curved space-time\nNowxµ(τ)are local coordinates in curved space, where µare vector indices. The two-\ncomponent quantities ξα,¯ξ˙α,πα,¯π˙αare the spinors in target space with the spinor indices\nwhich are raised and lowered by ǫαβand��αβ. It is assumed from the very beginning that\nspace-time geometry is described by a metric and general spi n connection. We also assume\nthat the Lagrangian for a particle in an external gravitatio nal background is constructed ac-\ncording to a general scheme in the form L=p˙x+π˙ξ+¯π˙¯ξ+(terms containing the constraints ).\nThe canonically conjugate coordinate xµ,ξα,¯ξ˙αand momenta pµ,πα,¯π˙αsatisfy the standard\nPoisson brackets (sf. (2.7))\n{xµ,pν}=δµ\nν,{ξα,πβ}=δα\nβ,/braceleft.alt1¯ξ˙α,¯π˙β/braceright.alt1=δ˙α\n˙β, (3.1)\nwhere the momentum pµis the world vector.\nThe Lagrangian Land the constraints, by definition, should be the scalars und er\nthe general coordinate transformations. However, since th e spinors are defined under\nthe local Lorentz transformations in tangent space, their τ-derivatives do not satisfy the\ntrue spinor transformation laws, and it is convenient to use the covariant τ-derivatives:\nDξα=˙ξα+˙xµωµαβξβ,D¯ξ˙α=˙¯ξ˙α+˙xµωµ˙α˙β¯ξ˙β, whereωµαβ=ωµβαandωµ˙α˙β=ωµ˙β˙αare the\nspinor connections. Taking into account these covariant de rivatives, we rewrite identically\nthe kinetic term of the Lagrangian p˙x+π˙ξ+¯π˙¯ξin the formPµ˙xµ+παDξα+¯π˙αDξ˙α, where\nthe quantityPµis defined by the relation\nPµ=pµ+παωµα\nβξβ+¯π˙αωµ˙α˙β¯ξ˙β. (3.2)\nOne can prove that the quantity Pµ(3.2) is the world vector which is consistent with that the\nquantities L,πDξand¯πD¯ξare the world scalars1. Taking into account the above discussion,\n1This statement is not so evident from the very beginning. To s how this, we note that ξ,π,¯ξ,¯πare\ninert under the general coordinate transformations δxµ=ǫµ(x).In this case, δωµαβ=−∂µǫνωναβand\nδωµ˙α˙β=−∂µǫνων˙α˙β. Therefore, under the general coordinate transformations δ(ωµαβ˙xµ)=δ(ωµ˙α˙β˙xµ)= 0.\nHencePµandpµare the world vectors.\n3a natural generalization of the flat space infinite-spin part icle Lagrangian (2.1) is written in\nthe following form:\nL= Pµ˙xµ+πα/parenleft.alt1˙ξα+˙xµωµαβξβ/parenright.alt1+¯π˙α/parenleft.alt2˙¯ξ˙α+˙xµωµ˙α˙βξ˙β/parenright.alt2+λ0F0+λF +˜λ˜F +λuU,(3.3)\nwhere the quantities λ0(τ),λ(τ),˜λ(τ),λu(τ)are the Lagrange multipliers for the constraints\nF0≈0,F≈0,˜F≈0,U≈0. All the constraints must be the scalars under the general co ordinate\ntransformations. We will assume that they are functions of t he variables xµ,Pµ,ξα,¯ξ˙α,πα,\n¯π˙αpresented in the Lagrangian (3.3). First, the constraint U≈0has the same meaning as the\nconstraint u≈0(2.5) in flat space. The main problem is to construct all other constraints.\nFurther, it is convenient to move on to the formulation of geo metrical objects in the local\nLorentz frame. First of all, one rewrites the quantity Pµ(3.2) in the form\nPµ=pµ+1\n2ωµabMab, (3.4)\nwhere\nMab=ξσabπ−¯π˜σab¯ξ. (3.5)\nand\nωµab=(σab)αβωµαβ−(˜σab)˙α˙βωµ˙α˙β(3.6)\nThe Lorentz matrix generators σaband˜σabare defined according to [29].\nUsing the definition of Mab(3.5), the commutation relations (3.1) and the properties\nof the matrices σaband˜σab, one can prove the following relation in terms of the Poisson\nbrackets:\n{Mab,Mcd} =ηacMbd+ηbdMac−ηadMbc−ηbcMad. (3.7)\nAs a result, we obtain the standard commutation relations fo r the Lorentz group generators,\nwhich allows us to identify the Mabwith such generators and therefore to identify the quantity\nωµabwith the Lorentz connection.\nCalculation of the Poisson bracket for the quantities Pµ(3.4) yields\n{Pµ,Pν} = −1\n2RµνabMab, (3.8)\nwhere\nRµνa\nb=∂µωνa\nb−∂νωµa\nb+[ωµ,ων]a\nb (3.9)\nis the curvature tensor in terms of the Lorentz connection. L et us introduce the Lorentz\nvector in tangent space\nPa∶=eµaPµ, (3.10)\nwhereeµa(x)is the inverse matrix to the vierbein matrix eµa(x)2. In this case, the commu-\ntation relations (3.8) yield\n{Pa,Pb}= −1\n2RabcdMcd+2˜TabcPc, (3.11)\nwhere\nRabcd=eµ\naeν\nbRµνcd(3.12)\n2The standard relations are eµaeνa=gµν,eµaeνa=gµν,eµaeνa=δµν.\n4and3\n˜Tabc=−eν\n[a(∂νeµ\nb])eµc(3.13)\nis the connection-free part of the torsion (see (4.5) bellow ).\n4 Construction of the first class constraints at the curved\nbackground\nLet us construct the constraints F0≈0,F≈0,˜F≈0,U≈0which we use in the Lagrangian\n(3.3) and which are a curved space generalization of the flat s pace constraints (2.2)-(2.5).\n4.1 Restrictions on space-time geometry\nTaking into account the form of the constraints (2.2), (2.3) , (2.4), (2.5), we assume that the\nconstraintU≈0should coincide with the flat space constraint u≈0while the other curved\nspace constraints should be constructed from the following quantities:\nℓ0∶=PaPa, ℓ∶=Pa/parenleft.alt1ξσa¯ξ/parenright.alt1,˜ℓ∶=Pa(¯π˜σaπ) (4.1)\nwhich are the natural covariant generalization of the flat sp ace constraints (2.2), (2.3), (2.4).\nThe Poisson brackets of quantities (4.1) are written in the c ompletely covariant form\n/braceleft.alt1ℓ,˜ℓ/braceright.alt1= −Kℓ0−1\n2/parenleft.alt1ξσa¯ξ/parenright.alt1/parenleft.alt1¯π˜σbπ/parenright.alt1RabcdMcd+2/parenleft.alt1ξσa¯ξ/parenright.alt1/parenleft.alt1¯π˜σbπ/parenright.alt1TabcPc,(4.2)\n{ℓ0,ℓ}=/parenleft.alt1ξσa¯ξ/parenright.alt1PbRabcdMcd−4/parenleft.alt1ξσa¯ξ/parenright.alt1PbTabcPc, (4.3)\n/braceleft.alt1ℓ0,˜ℓ/braceright.alt1=(¯π˜σaπ)PbRabcdMcd−4(¯π˜σaπ)PbTabcPc. (4.4)\nHere the quantity Kis given by (2.9) and Tabcis the torsion tensor:\nTµνa=∂[µeν]a+ω[µ,abeν]b, T abµ=eν\naeλ\nbeµ\ncTνλc=−eν\n[a∂νeµ\nb]−eν\n[aωνb]ceµc,(4.5)\nTabc=Tabµeµc, T ab,c=˜Tab,c−ω[a,b]c. (4.6)\nand˜Tab,cis defined by (3.13).\nRelations (4.2), (4.3), (4.4) show that the curved space qua ntities (4.1) can not be treated\nas the first-class constraints since their algebra is not clo sed. We believe that there are\nonly two possible related ways of constructing a closed alge bra: restrictions on space-time\ngeometry and modification of quantities (4.1). As a first step in introducing true curved\nspace constraints, we assume that our curved space-time is t orsionless,\nTµνa=0. (4.7)\nIn this case, the spin connection ωµabis expressed through the vierbein eµain a standard\nway and we arrive at the Riemannian space with the correspond ing curvature tensor Rabcd.\n3Symmetrization and antisymmetrization are defined in the fo rm:A(aBb)=1\n2(AaBb+AbBa),A[aBb]=\n1\n2(AaBb−AbBa), etc.\n5Under assumption (4.7), relations (4.2), (4.3), (4.4) are s implified and take the form\n/braceleft.alt1ℓ,˜ℓ/braceright.alt1= −Kℓ0−1\n2/parenleft.alt1ξσa¯ξ/parenright.alt1/parenleft.alt1¯π˜σbπ/parenright.alt1RabcdMcd, (4.8)\n{ℓ0,ℓ}=/parenleft.alt1ξσa¯ξ/parenright.alt1PbRabcdMcd, (4.9)\n/braceleft.alt1ℓ0,˜ℓ/braceright.alt1=(¯π˜σaπ)PbRabcdMcd, (4.10)\nWe now begin to discuss the possibilities of modifying the fla t space constraints (2.2)-(2.5).\nSince we assumed that the constraint U=u, that is\nU=N−¯N≈0, (4.11)\nwe can use the condition (4.11) in expressions (4.8), (4.9), (4.10). One can prove that the\nfollowing relation takes place:\n/parenleft.alt1ξσ[a¯ξ/parenright.alt1/parenleft.alt1¯π˜σb]π/parenright.alt1/divides.alt1U≈0=−1\n2KMab, (4.12)\nwhere we use the identity σ[a\nα˙ασb]\nβ˙β=−(σabǫ)αβǫ˙α˙β−(ǫ˜σab)˙α˙βǫαβ. Further, we will apply identity\n(4.12) to analyse possible modifications of relations (4.8) , (4.9), (4.10) to determine the true\ncurved space constraints forming a closed algebra.\nDirect calculations show that under identity (4.12), the Po isson bracket (4.8) has the\nform:\n/braceleft.alt1ℓ,˜ℓ/braceright.alt1/divides.alt1N≈¯N=−K/parenleft.alt3ℓ0−1\n4RabcdMabMcd/parenright.alt3. (4.13)\nTherefore, as first approximation in modifying the constrai nt (2.2) is naturally to take instead\nofℓ0≈0the quantity in the right hand side of (4.13):\nℓ0−1\n4RabcdMabMcd≈0. (4.14)\nTo examine whether quantity (4.14) is an appropriate constr ain, let us consider the Poisson\nbrackets of this quantity with ℓand˜ℓ. The result of calculations has the form\n/braceleft.alt3ℓ0−1\n4RabcdMabMcd, ℓ/braceright.alt3=2/parenleft.alt1ξσa¯ξ/parenright.alt1PbRabcdMcd−1\n4/parenleft.alt1ξσh¯ξ/parenright.alt1DhRabcdMabMcd,(4.15)\n/braceleft.alt3˜ℓ0−1\n4RabcdMabMcd,˜ℓ/braceright.alt3=2(¯π˜σaπ)PbRabcdMcd−1\n4/parenleft.alt1¯π˜σhπ/parenright.alt1DhRabcdMabMcd,(4.16)\nwhere\nDhRabcd=∂hRabcd+ωh,agRgbcd+ωh,bgRagcd+ωh,cgRabgd+ωh,dgRabcg(4.17)\nand∂a=eµa∂µ,ωa,bc=eµaωµbc. We see that the above relations do not form a closed algebra\nin the general case. Therefore, we need some restrictions on geometry.\nTo clarify what restrictions we should impose, we turn atten tion that only the quantities\nℓand˜ℓare linear inPa. The right-hand sides of (4.15), (4.16) contain the terms li near inPa\nand the terms that do not contain the Pa. However, the expressions ℓand˜ℓinvolve vector\ncontractionsPa/parenleft.alt1ξσa¯ξ/parenright.alt1andPa(¯π˜σaπ). Therefore, it is required to make such restrictions\n6on the background geometry that would allow the formation of only the terms with the\nabove contractions in the right-hand sides of (4.8), (4.9). Otherwise, it will be necessary\nto introduce new constraints that are linear in Pa, and the resulting theory will not have a\ncorrect flat limit.\nIt is easy to see that the needed terms Pa/parenleft.alt1ξσa¯ξ/parenright.alt1andPa(¯π˜σaπ)will appear on the right-\nhand sides of (4.9), (4.10) in the case when the curvature ten sor (4.17) of the background\ngeometry has the form\nRabcd=κ/parenleft.alt1δc\naδd\nb−δd\naδc\nb/parenright.alt1, (4.18)\nwhereκis a constant. As is well know, relation (4.18) determines th e constant curvature\nspace-time. At zero, positive or negative κwe get the Minkowski, de Sitter and anti de Sitter\nspaces respectively.\nUnder the restriction (4.18), the first terms in the right-ha nd sides of (4.15), (4.16) are\nproportional to/parenleft.alt1ξσa¯ξ/parenright.alt1Paand(¯π˜σaπ)Pa. To see this, we use the identities\n(σa)α˙α(σab)βγ=−(σb)β˙αδγ\nα+1\n2(σb)α˙αδγ\nβ,(σa)α˙α(˜σab)˙β\n˙γ=(σb)α˙γδ˙β\n˙α−1\n2(σb)α˙αδ˙β\n˙γ,\nand obtain\n/parenleft.alt1ξσa¯ξ/parenright.alt1Mab=−1\n2/parenleft.alt1ξσb¯ξ/parenright.alt1K,(¯π˜σaπ)Mab=1\n2(¯π˜σbπ)K. (4.19)\nIt is also well known that for the curvature tensor (4.18) of c onstant curvature spaces the\nfollowing relation takes place:\nDhRabcd=0. (4.20)\nTherefore the second terms in the right-hand sides of (4.15) and (4.16) are nullified.\nNote that due to the equalities\n(σab)αβ(σab)γδ=−2δδ\nαδβ\nγ+δβ\nαδδ\nγ,(˜σab)˙α˙β(˜σab)˙γ˙δ=−2δ˙α\n˙δδ˙γ\n˙β+δ˙α\n˙βδ˙γ\n˙δ,(σab)αβ(˜σab)˙γ˙δ=0,\nfor theσ-matrices, in the case of spaces of constant curvature (4.18 ), the second term in\n(4.14) takes the form:\n−1\n4RabcdMabMcd=1\n2κ/parenleft.alt1N2+¯N2/parenright.alt1. (4.21)\nThus, we see that in the (anti-) de Sitter space (4.18) the qua ntities\nℓ0−1\n4RabcdMabMcd, ℓ, ˜ℓ, (4.22)\nform a closed algebra with respect to the Poisson brackets:\n/braceleft.alt1ℓ,˜ℓ/braceright.alt1= −K/parenleft.alt3ℓ0−1\n4RabcdMabMcd/parenright.alt3, (4.23)\n/braceleft.alt3ℓ0−1\n4RabcdMabMcd, ℓ/braceright.alt3= −2κKℓ, (4.24)\n/braceleft.alt3ℓ0−1\n4RabcdMabMcd,˜ℓ/braceright.alt3=2κK˜ℓ. (4.25)\nHowever, in this closed algebra there are only quantities ℓand˜ℓ. At the same time, if\nwe take into account the form of flat constraints (2.3) and (2. 4), the curved counterparts\nof these constraints should contain terms with constant µ. Therefore, these quantities can\nnot be considered as curved space curved counterparts of flat constraints since they have no\ncorrect flat limit.\n74.2 Final solution to the constraints\nWe move on to the final construction of curved space constrain ts. First of all, a natural step\nis to consider the expressions ℓ−µ≈0,˜ℓ−µ≈0andℓ0−1\n4RabcdMabMcd≈0as constraints.\nHowever, taking into account relations (4.24), (4.25), one sees that the Poisson brackets\namong them do not form a closed algebra. We will show that the q uantities ℓ−µ,˜ℓ−µand\nℓ0−1\n4RabcdMabMcdcan be modified in so as to obtain the true constraints F0≈0,F≈0,\n˜F≈0forming a closed algebra and having a correct flat limit.\nLet us begin with the modification of the quantity ℓ0−1\n4RabcdMabMcd, requiring that the\nmodified quantity commute with ℓ−µand˜ℓ−µ. Using the Poisson brackets\n{N,ℓ}=/braceleft.alt1¯N,ℓ/braceright.alt1=1\n2{K,ℓ}=−ℓ,/braceleft.alt1N,˜ℓ/braceright.alt1=/braceleft.alt1¯N,˜ℓ/braceright.alt1=1\n2/braceleft.alt1K,˜ℓ/braceright.alt1=˜ℓ (4.26)\nand (4.24), (4.25), we see that the quantity, which is equal t o the sum of ℓ0−1\n4RabcdMabMcd\nplus the additional term\n−κ\n2K2or−2κN¯N or−κ(N2+¯N2), (4.27)\nhas the zero Poisson brackets with ℓ−µand˜ℓ−µ. However, if the constraint U=N−¯N≈0\nis valid, the following equalities take place\n−κ\n2K2=−2κN¯N=−κ(N2+¯N2), (4.28)\nwhich allows us to shorten the number of variants in (4.27). W e choose the first option and\ndefine the quantity\nF0,γ∶=ℓ0−1\n4RabcdMabMcd−κ\n2K2+γ≈0 (4.29)\nas the true modified constraint with an arbitrary constant pa rameterγ, which will be fixed\nlater. Let us emphasize once again the obtained important pr operty, that the constraint\n(4.29) has the zero Poisson brackets with ℓ−µand˜ℓ−µ:\n{F0,γ,ℓ−µ}=/braceleft.alt1F0,γ,˜ℓ−µ/braceright.alt1=0. (4.30)\nWe will show that this quantity can be associated with true co nstraintF0at some fixed γ.\nLet us turn to the constraints ℓ−µ≈0,˜ℓ−µ≈0. Relation (4.23) shows that their Poisson\nbracket can not reproduce the quantity F0,γat anyγ. Therefore, if we desire to obtain the\ntrue constraintsFand˜Ffrom the quantities ℓ−µand˜ℓ−µ, latter should be modified in\nsuch a way that their Poisson bracket is equal to a linear comb ination of new constraints,\nunlike the Poisson bracket (4.23).\nThe desired modification is realized by adding the terms cont ainingK2to the constraints\nℓ−µ≈0,˜ℓ−µ≈0. As a result, we arrive at the following functions:\nFβ1∶=ℓ−µ+β1K2≈0, ˜Fβ2∶=˜ℓ−µ+β2K2≈0, (4.31)\nwhereβ1andβ2are some arbitrary constants. In the general case, these con stants are not\nequal to each other, β1≠β2, since the constraints (4.31) are real and are not related to each\nother.\n8Now we require that the quantities F0,γ,Fβ1,Fβ2form a closed algebra. We will show\nthat this requirement completely fixes the constants γ, β1, β2.\nSince{Mab,K}=0, the constraint (4.29) has the zero Poisson brackets with co nstraints\n(4.31):{F0,γ,Fβ1}=/braceleft.alt1F0,γ,˜Fβ2/braceright.alt1=0. The Poisson bracket of quantities (4.31) equals\n/braceleft.alt1Fβ1,˜Fβ2/braceright.alt1=−KF0+4K/parenleft.alt1β2Fβ1+β1˜Fβ2/parenright.alt1−1\n2(κ+16β1β2)K3+(γ+4µ(β1+β2))K.(4.32)\nFor closure of this algebra, the third and fourth terms in the right-hand side must be equal\nto zero. This leads to the following conditions on the consta ntsγ,β1andβ2:\nβ1β2=−1\n16κ, γ=−4µ(β1+β2). (4.33)\nFurther, we consider two cases, κ<0(anti de Sitter space) and κ>0(de Sitter case).\nThe case of anti de Sitter space. In this case, the simplest solution of equations\n(4.33) is\nβ1=β2=β=1\n4/divides.alt0κ/divides.alt01/slash.left2, γ=−2µ/divides.alt0κ/divides.alt01/slash.left2. (4.34)\nDenotingF0=F0,γ=−2µ/divides.alt0κ/divides.alt01/slash.left2,F=Fβ1=1\n4/divides.alt0κ/divides.alt01/slash.left2,˜F=˜Fβ2=1\n4/divides.alt0κ/divides.alt01/slash.left2, we obtain the final constraints\nfor the case of AdS space\nF0= PaPa−1\n4RabcdMabMcd−1\n2κK2−2µ/divides.alt0κ/divides.alt01/slash.left2≈0, (4.35)\nF = P a/parenleft.alt1ξσa¯ξ/parenright.alt1−µ+1\n4/divides.alt0κ/divides.alt01/slash.left2K2≈0, (4.36)\n˜F = P a(¯π˜σaπ)−µ+1\n4/divides.alt0κ/divides.alt01/slash.left2K2≈0, (4.37)\nU =N−¯N≈0. (4.38)\nThe constraints (4.35) and (4.38) have the zero Poisson brac kets with other constraints:\n{F0,F}=/braceleft.alt1F0,˜F/braceright.alt1={U,F}=/braceleft.alt1U,˜F/braceright.alt1={F0,U}=0. (4.39)\nThe only non-zero Poisson bracket of constraints (4.35)-(4 .38) has the form\n/braceleft.alt1F,˜F/braceright.alt1=−KF0+/divides.alt0κ/divides.alt01/slash.left2K/parenleft.alt1F+˜F/parenright.alt1. (4.40)\nWe see that quantities (4.35)-(4.38) form a system of first-c lass constraints, as expected.\nNote, if we take into account (2.9) and (4.21), the constrain t (4.35) is represented in the\nform:\nF0=PaPa−κN¯N−2µ/divides.alt0κ/divides.alt01/slash.left2≈0. (4.41)\nThe case of de Sitter space. In this case, the simplest solution of equations (4.33) is\nβ1=−β2=β=1\n4κ1/slash.left2, γ=0. (4.42)\n9That is, in the case of the background de Sitter space, it is po ssible to select constraints with\nzero constant γ. DenotingF0=F0,γ=0,F=Fβ1=1\n4κ1/slash.left2,˜F=˜Fβ2=−1\n4κ1/slash.left2, we obtain the final\nconstraints for the case of dS space\nF0= PaPa−1\n4RabcdMabMcd−1\n2κK2≈0, (4.43)\nF = P a/parenleft.alt1ξσa¯ξ/parenright.alt1−µ+1\n4κ1/slash.left2K2≈0, (4.44)\n˜F = P a(¯π˜σaπ)−µ−1\n4κ1/slash.left2K2≈0, (4.45)\nU =N−¯N≈0. (4.46)\nThe only non-zero Poisson bracket of constraints (4.43)-(4 .46) is\n/braceleft.alt1F,˜F/braceright.alt1=−KF0−κ1/slash.left2K/parenleft.alt1F−˜F/parenright.alt1. (4.47)\nThe remaining Poisson brackets of the constraints (4.43)-( 4.46) are equal to zero. As in\nthe case of anti-de Sitter space (4.39), constraints (4.43) -(4.46) form a system of first-class\nconstraints.\n5 Summary\nLet us briefly summarize the results. In this paper, we have pr oposed a new classical particle\nmodel in the background gravitational field with Lagrangian (3.3) and constraints (4.35)-\n(4.38) or (4.43)-(4.46) that is a curved space time generali zation of the infinite spin particle\nmodel in flat space [28]. The model under consideration is des cribed by the curved space-\ntime coordinates xµand, like in the corresponding flat-space model, additional Weyl spinor\ncoordinates ξα,¯ξ˙αbelonging to target space of space-time. We work in the first o rder for-\nmalism and the main problem is to construct a system of constr aints forming the closed\nPoisson algebra. We begin with the general enough external g ravitational field with torsion\nand show that the conditions of closure of the constraint alg ebra uniquely fix the background\nto be constant curvature space. The constraints are derived in the explicit form for both\nAdS and for dS spaces. In the flat space limit, we return back to the model proposed in [28].\nThe result is consistent with the modern understanding that the true vacuum for higher spin\nfields is AdS space (see, e.g., the reviews [30–33]).\nAcknowledgements\nI.L.B. and S.A.F. acknowledge the support of the Russian Sci ence Foundation, project No.24-\n12-00116.\n10References\n[1] A.Barducci, R.Casalbuoni, L.Lusanna, Classical spinning particle interactiong with\nexternal gravitational fields , Nucl. Phys. B 124(1977) 521.\n[2] A.P.Balachandran, G.Marmo, B.S.Skagerstam, A.Stern, Spinning Particles in General\nRelativity , Phys. Lett. B 89(1980) 199.\n[3] F.Ravndal, Supersymmetric Direc particle in external fields , Phys. Rev. D 21(1980)\n2823.\n[4] K.Lusanna, B.Milewski, N =2super Yang-Mills and supergravity constraints from\ncoupling to a supersymmetric particl , Nucl. Phys. 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In this paper, we present an alternative solution by employing\nthe quadratic inference approach, a well-established meth od for analyzing correlated data, to estimate the regressio n\ncoefficients. Our approach leverages non-parametrically estima ted basis functions, eliminating the need for choosing\nworking correlation structures. Furthermore, we demonstr ate that our method achieves a parametric√n-convergence\nrate, contingent on an appropriate choice of bandwidth. Thi s convergence is observed when the number of repeated\nmeasurements per trajectory exceeds a certain threshold, s pecifically, when it surpasses na0, with nrepresenting the\nnumber of trajectories. Additionally, we establish the asy mptotic normality of the resulting estimator. The perfor-\nmance of the proposed method is compared with that of existin g methods through extensive simulation studies, where\nour proposed method outperforms. Real data analysis is also conducted to demonstrate the proposed method.\nKeywords: Constant Linear-Effect Models, Functional Principal Component Analysis, Quad ratic Inference,\nSemi-parametric Functional Regression.\n1. Introduction\nLongitudinal data analysis (LDA) involves tracking repeat ed measurements on the same individuals over time,\nallowing us to study changes over time and identify influenci ng factors. Unlike cross-sectional studies, which capture\nonly “between-individual” responses at a single time point , LDA can capture “within-individual” changes through\nrepeated measurements. Longitudinal data is often observe d in clusters, with each cluster representing measurements\nfrom one individual. As data complexity increases with adva ncing technology, functional data analysis (FDA) has\nbecome a vital tool, extending our understanding from finite to infinite dimensions. In LDA, the data are generally\nobserved with noise for measurements at each time-point [6, 14]. Moreover, a few repeated measurements are required\nin LDA, and the data are observed sparsely with noise. On the o ther hand, in FDA, data are densely observed as a\ncontinuous-time stochastic process without or with noise [ 20, 34]. Often, the sampling plan can have an e ffect on the\nperformance of the estimation procedures and inference [12 ]. In some situations, data are typically functions by natur e\nand are observed densely over time. Chiou et al. [4] proposed a class of semi-parametric functional regression models\nto describe the influence of vector-valued covariates on a sa mple of the response curve. When data collection leads to\nexperimental error, smoothing is performed at closely spac ed time-points in order to reduce the e ffect of noise. The\ncurrent developments of functional regression techniques have been rigorously studied in Chen et al. [3], Fan et al.\n[8], Hall et al. [11].\nThe generalized estimating equation (GEE) technique propo sed by Liang and Zeger [21] has been extensively\nused in LDA for estimation of parameters. Although an e fficient technique, the GEE may not be e fficient when the\ncorrelation matrix is not correctly specified. Hence, witho ut requiring the estimation of the correlation parameters,\nthe quadratic inference function (QIF) approach proposed b y Qu et al. [26] is useful for parameter estimation in\nlongitudinal studies [6] and cluster randomized trials [28 ]. By representing the inverse of the working correlation\n∗Corresponding author. Email address: pnyogi1@jhmi.edumatrix in terms of linear combinations of the basis matrices and involving multiple sets of score functions, the QIF\napproach has improved e fficiency over GEE when the working correlation matrix is not sp ecified correctly. Although\nit maintains the same e fficiency as in the situation where the working correlation mat rix is specified correctly, the\nQIF method is not independent of the choice of the working cor relation matrix. A QIF method-based approach to\nvarying-coefficient models for longitudinal data was proposed by Qu and Li [ 25]. The related work of Bai et al. [1] is\nan extension of QIF for the partial linear model. An alternat ive method was presented in Yu et al. [31] where each set\nof score equations was solved separately and their solution s were combined afterward; thereby providing results on\ninference for an optimally weighted estimator and extendin g those insights to the general setting with over-identified\nestimating equations.\nThe fundamental limitations that all the above-mentioned p owerful techniques su ffer from are: (1) all the above\nmethodologies require prior information on the working cor relation structure or choose appropriate basis matrices,\nand (2) the performance of the classical QIF approach is unkn own for dense functional data. Our study is motivated\nby problems from multiple real data applications that invol ve dense functional data when information on the working\ncorrelation structure is lacking. Let us discuss two motiva ting examples that we will use to illustrate the proposed\nmethod in this paper (see Section 5 for details). In the Beijing2017-data example, particulate matter (PM) with a\ndiameter of less than 2.5 micrometers is collected over di fferent time-points in di fferent locations in China. Scientists\nare interested in knowing the linear dependence of the pollu tion factor PM 2.5with other atmospheric chemicals [22].\nFigure 1 (left) in Section 5 pictorially demonstrates the re adings of PM 2.5for the given locations over several hourly\ntime-points; therefore, dense functional data analysis ca n be implemented. In another example from a neuroimaging\nstudyApnea-data , scientists are interested in modeling the change of white m atter structure among voxels in each\nregion of interest (ROI) of the human brain. Xiong et al. [29] investigates white matter structural alterations using\ndiffusion tensor imaging (DTI) in obstructive sleep apnea (OSA) patients. Here, the change of DTI parameters such\nas fractional anisotropy (FA) is investigated by using cons tant linear effect models with the interaction of the count\nof lapses obtained from the Psychomotor Vigilance Task and v oxel locations as predictors. We applied this model to\neach ROI and compared the results obtained from all the ROIs.\nWe propose a data-driven way to select the working covarianc e matrix and express the inverse of the covariance\nfunction in terms of the empirical eigen-functions of the co variance operator. The covariance operator can be estimate d\nas in Hsing and Eubank [16] and other related methods based on functional principal component analysis (FPCA) as\nfound in Dauxois et al. [5], Hall et al. [11], Hall and Hossein i-Nasab [12], Li and Hsing [19], Yao et al. [30], Zhang\nand Li [32]. Note that the estimation of the eigen-functions is nonparametric and it introduces some errors in the\nproposed estimation method. In this article, we answer the f ollowing question: while we estimate the eigen-functions\nnonparametrically from data, is the estimation of coe fficient vectors in a semi-parametric problem√n- consistent\nin dense functional data, and can we achieve asymptotic norm ality? The advantages of our proposed method are\nthe following: First, our method preserves the good propert ies of the QIF method and is easier to implement since\nthe eigen-functions can be estimated using the existing pac kages in statistical software such as R. Second, under\nsome mild conditions, our proposed estimator can obtain the optimal convergence rate and is asymptotically normally\ndistributed with less variance as compared to the classical QIF methods. Third, asymptotic results show the estimation\naccuracy of the coe fficients in semi-parametric functional model, therefore, ma king the influence of the dimension\nreduction step using FPCA redundant. The error in the estima tion of the eigen-functions contributes to the error in\nthe estimation of the parameters. Under some mild bandwidth conditions, the above-mentioned error contribution is\nof the same order of magnitude as an error in parameter estima tion if eigen-functions are known in advance.\nThe rest of the paper is organized as follows. In Section 2, we introduce the basic concept of QIF along with our\nproposed method. The asymptotic results for the proposed es timator are presented in Section 3. In Section 4, we\ndemonstrate the performance for finite samples. We also appl y the proposed method to real data-sets in Section 5. We\nconclude with some remarks in Section 6. All technical proof s are presented in the Appendix of the article.\n2. Functional response model and estimation procedure\n2.1. Basic model\nTo analyze longitudinal data, a straightforward applicati on of a generalized linear model [24] for single response\nvariables is not applicable due to the lack of independence b etween repeated measures. To account for the high\n2correlation in the longitudinal data, some special techniq ues are required. A seminal work by Liang and Zeger [21]\nproposed the use of GLM for the analysis of longitudinal data . The model we consider in this article is commonly\nobserved in spatial modeling, where associations among var iables do not change over the functional domain (see\nZhang and Banerjee [33] and references therein); which is te rmed as constant linear e ffects model. In this paper, the\nvariable time is used as a functional domain variable.\nLety(t) be the response variable at time-point tandx(t) be p-dimensional covariates observed at time t∈ T\nwhere without loss of generality we assume T=[0,1], is the spectrum of the time-points. The stochastic proce ssy(t)\nis square-integrable with marginal mean E{y(t)|x(t)}and finite covariance function; the regression parameter βis\nunknown and is to be e fficiently estimated. Thus, linear models with longitudinal d ata have the following expression.\ny(t)=x(t)Tβ+e(t), (1)\nwhere the stochastic process y(t) is decomposed into two parts: one is the mean function µ(t)=x(t)Tβthat depends on\ntime-varying covariates and vector-valued coe fficient vectorβ, and other is the random error part e(t) whereE{e(t)}=0\nand has finite second-order covariance R(s,t). Let yi(t) be i.i.d. copies of the stochastic process and for each indi vidual,\nthe measurements are taken on midiscrete time-points Ti jforj=1,···,mi;i=1,···,n. Therefore, at time Ti j, we\nobserve a mi×1 response vector yi(Ti j) and corresponding covariates xi(Ti j) for the i-th subject. We assume that\nmi’s are all of the same order as m=nafor some a≥0, thus mi/mis bounded below and above by some constants.\nFunctional data are considered to be sparse depending on the choice of a[12]. Data with bounded mora=0 are\ncalled sparse functional data and if a≥a0where a0is a transition point, are called dense functional data. Mor eover,\nthe regions (0 ,a0) are sometimes referred to as moderately dense. Furthermor e, we denote yi jandxi jasyi(Ti j) and\nxi(Ti j) respectively. ( yi1,···,yimi)Tand (µi1,···,µimi)Taremicomponent vectors, denoted as yiandµirespectively.\nThe derivative ofµ, denoted as ˙µ, is a mi×pmatrix.\nIn the classical problem of GEE, we estimate βby solving the quasi-likelihood equations [21]:\nn/summationdisplay\ni=1˙µT\niV−1\ni(yi−µi)=0. (2)\nWe denote Vi=νA1/2\niRi(ρ)A1/2\niwhere Ri(ρ) is the working correlation matrix, νis an over-dispersion parameter\nandAiis a diagonal matrix where entries are marginal variances Va r(yi1),···, Var( yimi). In this article, we simply\nsetν=1 while the extension to a general νis straightforward. In practice, the prior knowledge of the working\ncorrelation matrix is unknown, and the estimation of the coe fficient is influenced by its choice. Therefore, Qu et al.\n[26] suggested an expansion of the inverse of the working cor relation matrix as R(ρ)−1=/summationtextκ0\nk=1ak(ρ)Mkwhere Mkare\nsome basis matrices. Zhou and Qu [35] modified linear represe ntation by grouping the basis matrices into an identity\nmatrix and some symmetric basis matrices. For example, if th e working correlation matrix is exchangeable /compound\nsymmetric, R(ρ)−1=c1Im+c2Jmwhere Imis the m×midentity matrix and Jmis the m×mmatrix such that 0 is\nin diagonal and 1 is in o ff-diagonal positions. On the other hand, for first-order auto -regressive correlation matrix,\nR(ρ)−1=c1Im+c2J(1)\nm+c3J(2)\nmwhere J(1)\nmis a matrix with 1 on the two main o ff-diagonal positions and 0 otherwise, J(2)\nm\nis a matrix such that 1 is in the corner positions, viz. (1 ,1) and ( m,m) and 0 elsewhere. Here, cks are real constants that\ndepend on the nuisance parameter ρ. Therefore, Equation (2) reduces to the linear combination of the score vectors:\n¯g(β)=n−1n/summationdisplay\ni=1gi(β)=n−1/summationtextn\ni=1˙µT\niA−1/2\niM1A−1/2\ni(yi−µi)\n...\nn−1/summationtextn\ni=1˙µT\niA−1/2\niMκ0A−1/2\ni(yi−µi). (3)\nDue to the higher dimension of g, Qu et al. [26] used the generalized method of moments (GMM) [ 15] for which\nthe method of estimation boils down to minimization of the qu adratic inference function Q(β)=n¯g(β)T/hatwideC(β)−1¯g(β)\nwhere/hatwideC(β)=n−1/summationtextn\ni=1¯gi(β)¯gi(β)Tis the sample covariance matrix of Equation (3). To obtain th e solution ofβ, the\nNewton-Raphson method is used which iteratively updates th e value ofβ.\n32.2. Incorporating eigen-functions in QIF\nNow, due to standard Karhunen-Lo` eve expansions of ei(t)=yi(t)−µi(t) [18, 23]\nei(t)=∞/summationdisplay\nr=1ξirφr(t), (4)\nwith probability 1, where independently distributed rando m variables ξir=<ei,φr>∼(0,λr) for ordered eigen-\nvaluesλrsuch thatλ1≥λ2≥···≥ 0 andφrs are orthonormal eigen-functions such that/integraltext\nφr(t)φl(t)=1(r=l).\nDue to Mercer’s theorem [17], a symmetric continuous non-ne gative definite kernel function Rhas the representation\nR(s,t)=/summationtext∞\nr=1λrφr(s)φr(t) where the sum is absolutely and uniformly continuous. We ex tract the main directions of\nthe variation of the response variables using FPCA. In this s ituation, we take the first κ0terms which provide a good\napproximation of the infinite sum in Equation (4) by consider ing that the majority of the variations in the data are\ncontained in the subspace spanned by few eigen-functions [3 ]. For finite κ0≥1, we therefore consider the rank- κ0\nFPCA model,\nE{y(t)|x(t)}=µ(t)+κ0/summationdisplay\nr=1E{ξr|x(t)}φr(t). (5)\nUnlike existing models in Chiou et al. [4] where the main inte rest of the work is to study how the vector covariates\ninfluence the whole response curve, we study point-wise e ffect of covariates on functional response the influence of\ncovariates vectors indirectly on the functional response t hrough the regression of scores on covariates, viz., E{ξr|x(t)}.\nAn analogue of the truncated empirical version of Equation ( 4) and Mercer’s representation can be provided easily,\nand we discuss the proposed method based on this truncated ve rsion. Moreover, we discuss how to choose κ0in our\nsituation in Section 3 in detail.\nIn this chapter, we propose a data-driven way to compute the b asis matrices to obtain the approximate inverse of\nVas discussed earlier. In this approach, it is enough to find th e eigen-functions to construct a GEE. Let us define\n¯g(β)=n−1/summationtextn\ni=1˙µT\ni/hatwideΦi1(yi−µi)\n...\nn−1/summationtextn\ni=1˙µT\ni/hatwideΦiκ0(yi−µi), (6)\nwhere for k=1,···,κ0, we define/hatwideΦik=/parenleftig\nm−2\ni/hatwideφk(Ti j)/hatwideφk(Ti j′)/parenrightig\nj,j′=1,···,mi. Since the dimension of gin Equation (6)\nis greater than the number of parameters to estimate, instea d of setting gto zero, we minimize quadratic function.\n/hatwideβ=arg minβQ(β) whereQ(β)=n¯g(β)T/hatwideC(β)−1¯g(β) where,/hatwideC(β)=n−1/summationtextn\ni=1gi(β)gi(β)T. For the existence of /hatwideC−1\nwe need the additional restriction: n≥dim(gi)=p×κ0whereκ0is the number of eigen-functions. Under the given\nset-up, by Equation (8) in Qu et al. [26] the estimating equat ion forβwill be ˙Q(β)≈2˙¯g(β)T/hatwideC(β)−1¯g(β). To obtain the\nsolution of the above equation, we use a Newton-like method. In practice, the standard Newton method does not lead\nto a decrease in the objective function, that is, at each step of the iteration, there is no guarantee that Q(βs+1)<Q(βs).\nTherefore, we use the Algorithm 1 to estimate βusing the Quasi-Newton method with halving [10].\n2.3. Estimation of eigen-functions\nEstimation of eigen-functions is an important step in our pr oposed quadratic inference technique. In general,\nFPCA plays an important role as a dimension-reduction techn ique in functional data analysis. Some important the-\nories on FPCA have been developed in recent years. In particu lar, Hall and Hosseini-Nasab [12] proved various\nasymptotic expressions for FPCA for densely observed funct ional data. Later, Hall and Hosseini-Nasab [13] showed\nmore common theoretical arguments, including the e ffect of the gap between eigen-value (a.k.a., spacing) on the\nproperty of eigen-value estimators. In Li and Hsing [19], un iform rates of convergence of the mean and covariance\nfunctions are given, which are equipped for all possible cho ices/scenarios of mis.\nNote that the error process e(t) has mean zero, defined on compact set T=[0,1] satisfying/integraltext\nTE{e2}<∞. The\nfunctional principal components can be constructed via the covariance function R(s,t) (induces the kernel operator\nF) defined as R(s,t)=E{e(s)e(t)}which is assumed to be square-integrable. An empirical anal ogue of the spectral\ndecomposition of Rcan be obtained/hatwideR(s,t)=/summationtext∞\nr=1/hatwideλr/hatwideφr(s)/hatwideφr(t) where the random variables /hatwideλ1≥/hatwideλ2≥···≥ 0 are the\n4Algorithm 1 Estimation ofβusing the Quasi-Newton method with halving.\nSet/hatwideβ0(initial estimates)\nSetǫ0(threshold, a small number)\nSet max.count (maximum number of repetition)\nCalculateQ(/hatwideβ0),˙Q(/hatwideβ0),¨Q(/hatwideβ0) respectively.\nwhile Error>ǫ0do\nCalculate ˙Q(/hatwideβ1) and ¨Q(/hatwideβ1) based on/hatwideβ1\nInitialise r0=1\n/hatwideβ2←/hatwideβ1−r0¨Q(/hatwideβ1)−1˙Q(/hatwideβ1)\nCalculateQ(/hatwideβ1) andQ(/hatwideβ2) based on/hatwideβ1and/hatwideβ2respectively using proposed method\nwhileQ(/hatwideβ2)>Q(/hatwideβ1)do\nr0←r0/2\n/hatwideβ2←/hatwideβ1−r0¨Q(/hatwideβ1)−1˙Q(/hatwideβ1)\nCalculateQ(/hatwideβ1) andQ(/hatwideβ2) based on/hatwideβ1and/hatwideβ2respectively using proposed method\nend while\nCalculate Error=/bardbl/hatwideβ2−/hatwideβ1/bardbl2\n/hatwideβ0←/hatwideβ1\n/hatwideβ1←/hatwideβ2\nend while\nOutputβ\neigen-values of the estimated operator /hatwideFand the corresponding sequence of eigen-functions are /hatwideφ1,/hatwideφ2,···. Further,\nassume that/integraltext\nTφr/hatwideφr≥0 to avoid the issue regarding change of sign [12] for practic al comparison of eigen-functions,\notherwise there is no impact on the convergence rate of eigen -functions and hence the proposed estimators.\nSuppose that Ti jare observational points with a positive density function fT(·). Assume mi≥2 and define\nN=/summationtextn\ni=1Niwhere Ni=mi(mi−1). This approach is based on local linear smoother which is p opular in functional data\nanalysis, including Fan and Gijbels [7], Li and Hsing [19] am ong many others. Let K(·) be a symmetric probability\ndensity function on [ −1,1], which is used as kernel and h>0 be bandwidth, thus re-scaled kernel function is defined\nasKh(·)=h−1K(·). Therefore, for given s,t∈T, choose (/hatwidea0,/hatwideb1,/hatwideb2) be the minimizer of the following equation.\nn−1n/summationdisplay\ni=1N−1\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1\nj1/nequalj2{ei(Ti j1)ei(Ti j2)−a0−b1(Ti j1−s)−b2(Ti j2−t)}2\n×Kh/parenleftig\nTi j1−s/parenrightig\nKh/parenleftig\nTi j2−t/parenrightig\n. (7)\nThus we estimate R(s,t)=E{e(s)e(t)}using the quantity/hatwidea0, viz.,/hatwideR(s,t)=/hatwidea0. The operator/hatwideFis in general positive\nsemi-definite and the estimated eigen-values /hatwideλrare non-negative; indeed, /hatwideRis symmetric.\n3. Asymptotic properties\nIn this section, we study the asymptotic properties of the pr oposed estimator. Let us introduce some notations.\nAssume that mis are all of the same order, viz, m≡m(n)=nafor some a≥0. Define, dn1(h)=h2+hm/m\nanddn2(h)=h4+h3m/m+h2m/m2where m=lim n→∞n−1/summationtextn\ni=1m/miandm=lim n→∞n−1/summationtextn\ni=1(m/mi)2. De-\nnoteδn1(h)=/braceleftig\ndn1(h) log n/(nh2)/bracerightig1/2andδn2(h)=/braceleftig\ndn2(h) log n/(nh4)/bracerightig1/2. Further, νa,b=/integraltext\ntaKb(t)dt. Define,\nW=(φ(t1)T,···,φ(tm)T)Tis a matrix of order m×κ0obtaining after stacking all φks and random components\nξi=(ξi1,···,ξiκ0)T. Further,ξhas mean zero and variance Λwhich is a diagonal matrix with components λ1,···,λκ0.\nThe sign “ /lessorsimilar” indicates that the left-hand side of the inequality is boun ded by the right-hand upto a multiplicative\npositive constant, i.e. for two positive variables f1andf2we define f1/lessorsimilarf2asf1≤C f2where Cis a positive constant\nnot involving n. The following conditions are needed for further discussio n of the asymptotic properties.\n5(C1) Kernel function K(·) is a symmetric density function defined on bounded support [ −1,1].\n(C2) Density function fTofTis bounded above and away from infinity. Also, the density fun ction is bounded below\naway from zero. Moreover, fis differentiable and the derivative is continuous.\n(C3) R(s,t) is twice differentiable, and all second-order partial derivatives are b ounded on [0 ,1]2.\n(C4)E{supt∈[0,1]|e(t)|γ}<∞andE{supt∈[0,1]|xi(t)|2γ}<∞for someγ∈(4,∞).\n(C5) h→0 asn→∞ such that d−1\nn1(logn/n)1−2/γ→0 and d−1\nn2(logn/n)1−4/γ→0 forγ∈(4,∞).\n(C6) Condition for eigen-components.\n(a) for each 1≤k<r<∞max{λk,λr}/|λk−λr|≤C0max{k,r}/|k−r|for non-zero finite generic constant\nC0,.\n(b) For some α>0, with the condition Vrλ−2\nrr1+α→0 asr→∞ where Vr=E{/integraltext\n˙µ(t)φr(t)dt}2.\nThe above two conditions hold if λr=r−τ1Λ(r) and Vr=r−τ2Γ(r) for slowly varying functions ΛandΓ\nwhereτ2>1+2τ1>3 andτ=α+τ1.\n(c)/integraltext\nφ4\nk(t)dtand/integraltext\n˙µ2(t)φ2\nk(t)dtare finite for all k≥1./integraltext\nφ2\nk(t)φ2\nl(t)dtis finite.\n(C7)/hatwideC(β) converges almost surely to an invertible matrix C0=E{g(β0)g(β0)−1}.\n(C8) Conditions for handκ0.\n(a) If a>1/4,κ0=O(n1/(3−τ)) and n−1/4/lessorsimilarh/lessorsimilarn−(a+1)/5\n(b) If a≤1/4,κ0=O(n4(1+a)/5(3−τ)) and h/lessorsimilarn−1/4\nRemark 1. Condition (C1) is commonly used in non-parametric regressi on. The bound condition for the density\nfunction of time-points has the standard Condition (C2) for random design. Similar results can be obtained for a fixed\ndesign where the grid-points are pre-fixed according to the d esign density/integraltextTj\n0f(t)dt=j/mforj=1,···,m, for\nm≥1. Furthermore, it is important to note that this approach do es not involve the requirement to obtain sample path\ndifferentiation when we invoke the estimation of eigenfunction s from Li and Hsing [19]. Therefore the method could\nbe applicable for Brownian motion which has a continuously n on-differentiable sample path. To expand in Taylor\nseries, Condition (C3) is required, and is also common in non -parametric regression. Condition (C4) is required\nfor a uniform bound for certain higher-order expectations t o show uniform convergence. This is a similar condition\nadopted from Li and Hsing [19]. Smoothness conditions in (C5 ) and (C8) is common in kernel smoothing and\nfunctional data to control bias and variance. The first condi tion for tuning the parameters mentioned in (C5) is similar\nto Li and Hsing [19]. The required spacing assumptions for ei gen-values in Conditions (C6)a and (C6)b are similar\nas in Hall and Hosseini-Nasab [13]. Condition (C6)c is a triv ial assumption that frequently arises in functional data\nanalysis literature. In most of the situations, this condit ion automatically holds. Using the weak law of large numbers ,\nCondition (C7) holds for large n. Similar kind of conditions can be invoked such as convexity assumption, i.e.,\nλr−λr+1≤λr−1−λrfor all r≥2. Condition (C8) is determined to control the rate of the num ber of repeated\nmeasurements.\nWe present the following theorem to provide the asymptotic e xpansion and consistency of the proposed estimator for\n/hatwideβ.\nTheorem 1. Letβ0be the true value of β. Under the Conditions (C1)-(C6), for k =1,···,κ0, we have the asymptotic\nmean square error for gk(β0)as\nAMSE{gk(β0)}=O/parenleftig\nn−1+n−1κ3−τ\n0Rn(h)/parenrightig\nalmost surely (8)\nwhere R n(h)=/braceleftig\nh4+(1/n)+(1/nmh)+(1/n2m2h2)+(1/n2m4h4)+(1/n2mh)+(1/n2m3h3)/bracerightig\n. Moreover, under Con-\ndition (C8), AMSE {/hatwidegk(β0)}=O(n−1). Therefore, if in addition, Condition (C7) holds, as n →∞ ,/bardbl/hatwideβ−β0/bardbl=O(n−1/2)\nin probability.\n6The following theorem states the results of asymptotic norm ality.\nTheorem 2. DefineCi=/summationtextκ0\nk1=1/summationtextκ0\nk2=1Φk1XiC−1\nk1,k2XT\niΦk2, where C−1\nk1,k2is a(k1,k2)block of C−1\n0with C0=E/braceleftig\ng(β0)gT\ni(β0)/bracerightig\n.\nAssume that the conditions for Theorem 1 holds. Then√n(/hatwideβ−β0)d− →N(0,Σ)whereΣ=B−1AB−1. where the quanti-\ntiesAandBare, respectively, limits of /hatwideA=n−1/summationtextn\ni=1XT\ni/hatwideCi/hatwideei/hatwideeT\ni/hatwideCiXiand/hatwideB=n−1/summationtextn\ni=1XT\ni/hatwideCiXi, and “d− →”denotes the\nconvergence in distribution.\nRemark 2. Here, the selection of the bandwidth only a ffects the second-order term of the MSE of /hatwideβand has no effect\non the asymptotic result of normality as long as hsatisfies the Conditions (C5) and (C8) along with some restri ctions\nonκ0.\nOutline of the proofs of the proposed theorems is discussed i n the Appendix with additional technical details.\n4. Simulation studies\nWe conduct extensive numerical studies to compare the finite sample performance of our proposed method to that\nof Qu et al. [26] under di fferent correlation structures.\n4.1. Simulation set-up\nConsider the normal response model\nyi(Ti j)=xi(Ti j)Tβ+ei(Ti j). (9)\nForp=2, we set coefficient vectors,β=(β1,β2)Twhereβ1=1 andβ2=0.5. Consider the covariates xik(t)=\nχ(k)\ni1+χ(k)\ni2√\n2 sin (πt)+χ(k)\ni3√\n2 cos (πt)where the coefficientsχ(k)\ni1∼N(0,(2−0.5(k−1))2),χ(k)\ni2∼N(0,(0.85×2−0.5(k−1))2),\nχ(k)\ni3∼N(0,(0.7×2−0.5(k−1))2) andχi js are mutually independent for each trajectories iand each j. In fixed design\nsituations, associated observational times are fixed. Samp le trajectories are observed at m=100 equidistant time-\npoints{t1,···,tm}on [0,1]. Set number of trajectories n∈{100,300,500}. The residual process ei(t) is a smoothed\nfunction with mean zero and unknown covariance function, wh ere each eiis distributed as ei=/summationtext\nk≥1ξiφiandξks are\nindependent normal random variables with mean zero and resp ective variances λk. For numerical computation, we\ntruncate the finite series at k=3 in Karhunen-Lo` eve expansions for Situations (a), (b), an d (c) as described below. In\nSituations (d) and (e), the error process is generated from g iven covariance functions.\n(a) Brownian motion. The covariance function for the Browni an motion is min( s,t),λk=4/π2(2k−1)2and\nφk(t)=√\n2 sin( t/√λk).\n(b) Linear process. Consider the eigen-values be λk=k−2l0andφk(t)=√\n2 cos( kπt). We fix l0∈{1,2,3}.\n(c) Ornstein Uhlenbeck (OU) process. For positive constant sµ0andρ0, we have a stochastic di fferential equation\nfore(t) as∂e(t)=−µ0e(t)∂t+ρ0∂w(t) for the Brownian motion w(t). It can be shown that cov {e(t),e(s)}=\ncexp{−µ0|t−s|}where c=ρ2\n0/2µ0. Here we assume c=1. Thus, by solving the integral equation we have\nφk(t)=Akcos(ωkt)+Bksin(ωkt) andλk=2µ0/(ω2\nk+µ2\n0) whereωis solution of cot( ω)=(ω2−µ2\n0)/2µ0ω. The\nconstants AkandBkare defined as Bk=µ0Ak/ωkwhere Ak=√{2ω2\nk/(2µ0+µ2\n0+ω2\nk)}. Hereµ0is chosen to be\n1 or 3.\n(d) Power exponential. R(s,t)=exp{(−|s−t|/a0)b0}where scale parameter a0=1 and shape parameter b0∈{1,2,5}.\n(e) Rational quadratic. R(s,t)=/braceleftig\n1+(s−t)2/a2\n0/bracerightig−b0where scale parameter a0=1 and shape parameter b0∈\n{1,2,5}.\n74.2. Comparison and evaluation\nFor each of the situations, we perform 500 simulation replic ates. To execute Qu et al. [26]’s approach, we construct\nthe scores using basis matrices as described in Example 1 (ap proximation of the compound symmetric correlation\nstructure, denoted as ldaCS in the tables) and Example 2 (for the first-order autoregress ive correlation structure,\ndenoted as ldaAR in the tables) in their paper. Ordinary least squares estima te (init ) is taken as the initial estimate\nofβfor both ours and Qu et al. [26]’s approach. We indicate fda-k as our proposed method with the number of\nbasis functions k. The estimation procedure in the iterative algorithm conve rges when the square di fference between\nthe estimated values of two consecutive steps is bounded by a small number 10−10or the maximum number of steps\ncrosses 500, whichever happens earlier. To make theoretica l results and numerical examples consistent, we use FPCA\nfunction in R which is available in fdapace packages [9]. Bandwidths are selected using generalized cr oss-validation\nand the Epanechnikov kernel K(x)=0.75(1−x2)+is used for estimation where ( a)+=max( a,0).\nThe means and standard deviations (SD) of the regression coe fficients based on 500 simulations are given as\nsummary measures. We calculate the standard deviation ment ioned in the tables based on 500 estimates from 500\nreplications that can be viewed as the true standard error. M oreover, we also compute absolute bias (defined as\nAB=/summationtext500\nb=1|/hatwideβb−β|/500) and mean square error (defined as MSE =/summationtext500\nb=1(/hatwideβb−β)2/500) to compare the performance\nof estimation, where for b-th replication/hatwideβbbe the estimated value for β, MSEs are reported in the order of 10−2. The\nnumber of selected eigen-functions plays a critical role in the proposed method. We choose κ0based on a scree plot\nwhere the elbow of the graph is found and the components to the left are considered as significant.\n4.3. Simulation results\nSimulation results associated with the Brownian motion are shown in Table 1. In this situation, we observe that\nour approach produces better results in terms of the dispers ion measures. Tables ??,??and??show results for\nlinear processes, our proposed method performs better in si tuations with a working correlation matrix as AR but is\ncomparable for exchangeable structure for l0=1,2,3. Moreover, in our proposed method, as lincreases all dispersion\nmeasures such as MSE decrease. Results based on the OU proces s are documented in Tables ??and??. Our method\noutperforms the existing methods for both situations and as µ0increases, the MSE decreases. For three di fferent\nparameter choices of the power exponential and rational qua dratic covariance structure, the numerical results are\npresented in Tables ??,??,??and??,??,??, respectively. As before, we observe that our proposed meth od is\nfiner than the existing ones in all sub-cases; but interestin gly, when b0increases, the MSE decreases for the power\nexponential, whereas it increases for the rational quadrat ic covariance structure as expected due to the covariance\nstructure. Overall, we observe that for all the above situat ions, as sample size increases, the dispersion measures,\nfor example, SD and MSE decrease. It establishes that as samp le size increases, the parameter estimates get closer\nand closer to the true parameters. In each of the above situat ions, the SDs for the proposed methods decrease as we\nincreaseκ0and stabilize after some value of κ0where the fraction of variance (FVE) is approximately 100%.\n5. Real data analysis\nIn this section, we apply our proposed method to motivating e xamples in two different data-sets.\n5.1. Beijing’s PM 2.5pollution study\nIn the atmosphere, suspended microscopic particles of soli d and liquid matter are commonly known as particulates\nor particulate matter (PM). Such particulates often have a s trong noxious impact toward human health, climate, and\nvisibility. One such common and fine type of atmospheric part icle is PM 2.5with a diameter less than 2 .5 micrometers.\nMany developed and developing cities across the world are ex periencing chronic air pollution, with major pollutants\nbeing PM 2.5; Beijing and a substantial part of China are among such place s. Some studies show that there are many\nnon-ignorable sources of variability in the distribution a nd transmission pattern of PM 2.5, which are confounded with\nsecondary chemical generation. The atmospheric PM 2.5data used in our analysis were collected from the UCI machine\nlearning repository https://archive.ics.uci.edu/ml/datasets/Beijing+Mul ti-Site+Air-Quality+Data\n[22]. The data-set includes daily measurements of PM 2.5and associated covariates at twelve di fferent locations in\nChina, namely Aotizhongxin, Changping, Dingling, Dongsi, Guanyuan, Gucheng, Huairou, Nongzhanguan, Shunyi,\nTiantan, Wanliu, and Wanshouxigong during January 2017. Af ter excluding missing data, there were 608 hourly data\n8Table 1: Performance of the estimation procedure where the residual s are generated from Brownian motion. Mean of the estimated c oefficients,\nstandard deviation, absolute bias, mean square error ( ×100), and FVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 0.9999 0.0373 0.0297 0.1391 0.4995 0.0486 0.0384 0.2354\nldaAR 0.9991 0.0331 0.0265 0.1095 0.5004 0.0445 0.0353 0.1972\nldaCS 0.9987 0.0316 0.0253 0.1000 0.4997 0.0411 0.0322 0.1685\nfda-1 0.9998 0.0564 0.0447 0.3180 0.5006 0.0743 0.0587 0.5516 86. 2672\nfda-2 1.0001 0.0269 0.0213 0.0723 0.4971 0.0362 0.0290 0.1314 96. 3746\nfda-3 0.9998 0.0231 0.0181 0.0532 0.4978 0.0317 0.0251 0.1010 99. 9220\nfda-4 1.0004 0.0052 0.0014 0.0028 0.4994 0.0092 0.0022 0.0085 99. 9979\nfda-5 0.9999 0.0021 0.0008 0.0004 0.4999 0.0051 0.0012 0.0026 100 .0000\nn=300\ninit 1.0002 0.0200 0.0162 0.0401 0.5003 0.0288 0.0226 0.0825\nldaAR 1.0003 0.0184 0.0147 0.0336 0.5000 0.0259 0.0203 0.0670\nldaCS 1.0007 0.0170 0.0134 0.0288 0.4995 0.0242 0.0190 0.0583\nfda-1 1.0002 0.0309 0.0251 0.0955 0.5008 0.0443 0.0350 0.1962 86. 7578\nfda-2 1.0002 0.0142 0.0114 0.0202 0.4998 0.0213 0.0169 0.0451 96. 4747\nfda-3 1.0002 0.0122 0.0098 0.0150 0.4992 0.0179 0.0144 0.0321 99. 9745\nfda-4 1.0002 0.0021 0.0003 0.0004 0.4998 0.0032 0.0005 0.0010 99. 9993\nfda-5 1.0000 0.0002 0.0001 0.0000 0.5000 0.0003 0.0002 0.0000 100 .0000\nn=500\ninit 1.0002 0.0148 0.0117 0.0219 0.5006 0.0223 0.0177 0.0497\nldaAR 1.0005 0.0138 0.0111 0.0189 0.5000 0.0206 0.0162 0.0422\nldaCS 1.0000 0.0128 0.0102 0.0164 0.4992 0.0184 0.0146 0.0340\nfda-1 1.0007 0.0234 0.0185 0.0545 0.5012 0.0348 0.0277 0.1213 86. 7520\nfda-2 0.9996 0.0105 0.0083 0.0110 0.5002 0.0157 0.0126 0.0247 96. 5174\nfda-3 0.9991 0.0091 0.0074 0.0084 0.4999 0.0133 0.0107 0.0177 99. 9851\nfda-4 1.0000 0.0002 0.0001 0.0000 0.5000 0.0003 0.0002 0.0000 99. 9996\nfda-5 1.0000 0.0002 0.0001 0.0000 0.5000 0.0003 0.0002 0.0000 100 .0000\npoints inBeijing2017-data . We assume that the atmospheric measurements are independe nt since they are located\nquite apart. The objective of our analysis is to describe the trend of functional response PM 2.5(as shown in Figure 1\n(left)) and evaluate the e ffect of covariates including the chemical compounds such as s ulfur dioxide (SO 2), nitrogen\ndioxide (NO 2), carbon monoxide (CO) and ozone (O 3) over time. We smooth the covariates and responses to reduce\nvariability and center them. Subsequently, we consider the following model\nYi(t)=β0+SO2(t)β1+NO 2(t)β2+CO(t)β3+O3(t)β4+ei(t). (10)\nWe use Algorithm 1 to estimate the coe fficients of the regression model mentioned above. Through the simulation\nresults, we observe that if the values of κ0increase, the standard deviation of the coe fficients decreases. For small\nFVE such as 50%, the corresponding κ0=1 and the estimation procedure performs poorly; whereas for large FVE\npercentages, the estimation procedure has adequately impr oved in terms of standard error. The estimated values\n90 100 200 300 400 500 6000 200 400 600\ntime−pointsPM 2.5\nFig. 1:Beijing2017-data : (left) Reading of hourly PM 2.5measures for twelve di fferent locations over 608 hourly time-points during January\n2017. (right) Scree plots of a fraction of variance explaine d (FVE).\nforβ0,β1,β2,β3,andβ4produce similar results across di fferent choices of κ0. From the estimated standard error,\nusing scree-plot of standard error, we conclude that the sui table choice of κ0is approximately 10. The estimated\nscaled eigen-values are provided in Figure 1 (right) which c learly shows their decay rate. The estimated coe fficients\nwith standard error are 0.0009 (1.1644), 0.0829 (0.2584), 0 .9503 (0.1586), 0.0196 (0.0037) and 1.1523 (0.1198)\nrespectively.\n5.2. DTI study for sleep apnea patients\nThe diffusion tensor imaging data (DTI) to understand the white matt er structural alteration using di ffusion ten-\nsor imaging (DTI) are used to illustrate the application of t he proposed method and its estimation procedure [29].\nMRI is a powerful technique for investigating the structura l and functional changes in the brain during pathological\nand neuro-psychological processes. Due to the advancement in di��usion tensor imaging (DTI), several studies on\nwhite matter alterations associated with clinical variabl es can be found in recent literature. For our analysis, we use\nApnea-data obtained from one such study on obstructive sleep apnea (OSA ) patients [29]. The data consists of 29\nmale patients between the ages 30-55 years who underwent a st udy for the diagnosis of continuous positive airway\npressure (CPAP) therapy. Among those who have sleep disorde rs other than OSA, night-shift workers, patients with\npsychiatric disorders, hypertension, diabetes, and other neurological disorders were excluded. In this study, the ps y-\nchomotor vigilance task (PVT) was performed in which a light was randomly switched-on on a screen for several\nseconds in a certain interval of time, and subjects were aske d to press a button as soon as they saw the light appear\non screen; such an experiment provides a numerical measure o f sleepiness by counting the number of “lapses” for\neach individual. Psychomotor vigilance task was performed after continuous positive airway pressure treatment which\ncounts the number of lapses in the attention tracking test. I n such a test, subjects were asked to press a button as soon\nas the light appeared on the screen which was turned on random ly for several seconds in a certain interval of time. The\ninteresting measure is sustained attention and provides a n umerical measure of sleepiness by counting the number of\n“lapses” for each individual\nDTI was performed at 3T MRI scanner using a commercial 32-cha nnel head coil, followed by the analysis using\ntract-based spatial statistics to investigate the di fference in fractional anisotropy (FA) and other DTI paramete rs. The\nimage acquisition is as follows. An axial T1-weighted image of the brain (3D-BRA VO) is collected with repetition\ntime (TR)=12ms, echo time (TE) =5,2ms, flip angle=13◦, inversion time=450 ms, matrix=384×256, voxel\n10size=1.2×0.57×0.69mm and scan time =2 min 54 sec. DTI are obtained in the axial plane using a spin-e cho echo\nplanner imaging sequence with TR =4500ms, TE=89.4ms, field of view =20×20cm2, matrix size=160×132,\nslice thickness=3mm, slice spacing =1mm, b-values=0, 1000 s/mm2.\nOur objective is to investigate the structural alteration o f white matter using DTI in patients with OSA over each\nvoxel at various regions of the brain (called ROIs). Thus, ou r response variable is one of the DTI parameters, viz.,\nfractional anisotropy (FA) and we are interested in studyin g the effect of the changes of FA over continuous domains\nsuch as voxels with the interaction of the lapses and the voxe l locations in each ROIs. We consider the following\nmodel for each ROI.\nFAi(s)=β0+β1lapsesi×s+ei(s) (11)\nwhere s∈S, a set of voxels in the considered ROIs. Using the Algorithm 1 , we estimate the coe fficientsβ1andβ2\nas mentioned in the model 11 and the results are presented in T able 2. We find that the coe fficient estimates are close\nenough to their initial estimates and the estimated standar d error is smaller for the coe fficients based on the proposed\nmethod. Here κ0(i.e., the number of eigen-functions) are determined by FVE for simplicity.\n6. Discussion\nIn this article, we propose an estimation procedure for the c onstant linear effects model, which is commonly used\nin statistics [33] especially in spatial modeling. One of th e key factors of this estimation procedure is the fact that it\nis based on the quadratic inference methodology that has ser ved a huge role in the analysis of correlated data since it\nwas discovered by Qu et al. [26]. In contrast with the existin g method, our approach allows the number of repeated\nmeasurements to grow with sample size; therefore, the traje ctories of individuals can be observed on a dense grid of a\ncontinuum domain. Instead of assuming a working correlatio n structure, we propose a data-driven way by estimating\nthe eigen-functions that are obtained by functional princi pal component analysis. Here, we achieve√n−consistency\nof the parametric estimates in the regression model, even th ough the eigen-functions are estimated non-parametricall y.\nAdditionally, our method is easy to implement in a wide range of applications. The applicability of the proposed\nmethod is illustrated by extensive simulation studies. Mor eover, two real-data applications in di fferent scientific\ndomains are provided which confirm the e fficacy of the proposed method.\nAcknowledgement\nWe would like to thank Dr. Xiaohong Joe Zhou of University of I llinois at Chicago for providing the Apnea-data\nused in Section 5.2.\nAppendix A. Some preliminary definitions and concepts of ope rators\nConsider the standard L2[0,1] space that defines the set of square-integrable functions defined on the closed\nset [0,1] that takes values on the real line. The space L2[0,1] is equipped with an inner product and is defined as\n/an}bracketle{tf,g/an}bracketri}ht=/integraltext1\n0f(t)g(t)dtforfandgin that space, and forms a Hilbert space. Moreover, we denote the norm/bardbl·/bardbl 2inL2\nwhich is defined as /bardblf/bardbl2=/braceleftig/integraltext\nf2(u)du/bracerightig1/2. DefineFbe an operator that assigns an element finL2[0,1] to a new\nelementFfinL2[0,1] moreover,Fis linear and bounded. A linear mapping Ff(·)=/integraltext\nR(·,u)f(u)dufor any function\nf∈L2[0,1] and for some integrable function R(·,·) on [0,1]×[0,1]. This function is preferably known as integral\noperator and the bivariate function Ris known as a kernel in statistics and functional analysis li terature. Furthermore,\nunder the assumption that/integraltext/integraltext\nR2(u,v)dudv<∞. It is easy to see that Ff(·) is uniformly continuous and compact for a\nnon-negative definite symmetric kernel R. For someλ, in Fredholm integral equation , Fφ=λφhas non-zero solution\nφthen we call λas eigen-value ofFand the solution of the eigen-equation is called eigen-func tions, altogether, the\npair of eigen-values and eigen-function, viz., ( λ,φ) are called eigen-elements. Due to non-negative definitene ss ofF,\nthe eigen-values are ordered as λ1≥λ2≥···≥ 0 .\nSuppose for self-adjoint compact operator on Hilbert space Hconsider two operators FandG, define perturbation\noperator∆=G−F such thatG=F+∆whereGis an approximation to Fwhere∆amount of error is occurred. Let\nFandGhave kernels FandGrespectively with eigen-elements ( θr,ψr) and (λr,φr). For simplicity, we assume that\nthe eigen-values are distinct. Then the following Lemma pro vides perturbation of the eigen-functions.\n11Lemma 1 (Theorem 5.1.8 in Hsing and Eubank [16]) .Let(λ,φ)be the eigen-components of Fand(θ,ψ)be that of\nGwith multiplicity of all eigen-values are restricted to be 1 . Defineηk=min r/nequalk|λr−λk|. Assume/an}bracketle{tφr,ψr/an}bracketri}ht≥0and\nηk>0. Then\nψk−φk=∞/summationdisplay\nr=1\nr/nequalk(θk−λr)−1Pr∆ψk+Pk(ψk−φk). (A.1)\nThe above equation follows\nψk−φk=∞/summationdisplay\nr=1\nr/nequalk(θk−λr)−1Pr∆ψk+O(/bardbl∆/bardbl2). (A.2)\nRemark 3. Equation (A.2) plays an important role in finding the bound of the proposed estimator introduced in\nSection 2.2. Note that supr≥1|θr−λr|≤/bardbl∆/bardbl≤infr/nequalk|λk−λr|(see Theorem 4.2.8 in Hsing and Eubank [16] for proof).\nThus, it is easy to see, |θr−λr|≤|λk−λr|which implies from Equation (A.1)\nψk−φk=∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1∞/summationdisplay\ns=0{(λk−θr)/(λk−λr)}sPr∆{φk+(ψk−φk)}+Pk(ψk−φk)\n=∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1Pr∆φk+∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1Pr∆(ψk−φk)\n+∞/summationdisplay\nr=1\nr/nequalk∞/summationdisplay\ns=1/braceleftig\n(λk−λs)s/(λk−λr)s+1/bracerightig\nPr∆ψk+Pk(ψk−φk). (A.3)\nMoreover, using Bessel’s inequality, we can bound last thre e terms in the above equation by /bardbl∆/bardbl2.\nAppendix B. Some useful lemmas\nIn this section, we present some useful lemmas. For convenie nce, let us recall the notation. Assume that mis\nare all of the same order, viz, m≡m(n). Define, dn1(h)=h2+hm/manddn2(h)=h4+h3m/m+h2m/m2where\nm=lim n→∞n−1/summationtextn\ni=1m/miandm=lim n→∞n−1/summationtextn\ni=1(m/mi)2. Denoteδn1(h)=/braceleftig\ndn1(h) log n/(nh2)/bracerightig1/2,δn2(h)=\n/braceleftig\ndn2(h) log n/(nh4)/bracerightig1/2andδn(h)=h2+δn1(h)+δ2\nn2(h). Further,νa,b=/integraltext\ntaKb(t)dt. Define, W=(φ(t1)T,···,φ(tm)T)T\nbe matrix of order m×κ0obtained after stacking all φks and random components ξi=(ξi1,···,ξiκ0)T. Further,ξhas\nmean zero and variance Λwhich is a diagonal matrix with components λ1,···,λk0. The sign′/lessorsimilar′indicates that the\nleft-hand side of the inequality is bounded by the right-han d upto a multiplicative positive constant and the constant\ndoes not involves nor function of n.\nLemma 2. Consider Z 1,···,Znbe independent and identically distributed random variabl es with mean zero and\nfinite variance. Suppose that there exists an M such that P (|Zi|≤M)=1for all i=1,···,n. Let T n=n−1/summationtextn\ni=1Zi.\nthen, n−1/summationtextn\ni=1Zi=O((log n/n)1/2almost surely. If√Var(Tn)=O((log n/n)1/2)then T n=O(logn/n)almost surely.\nProof. Bernstein’s inequality states that if Z1,···,Znbe centered independent bounded random variables with prob -\nability 1. Let Tn=n−1/summationtextn\ni=1Zi, then let Var{Tn}=σ2\nn. Then for any positive real number u, we have P(|Tn|≥u)≤\nexp{−nu2/(2σ2\nn+2Mu/3)}where Mis such that P(|Zi|≤M)=1. Moreover, if Tnconverges to its limit in probability\nfast enough, then it converge almost surely in the limit, i.e ., if for any u>0,/summationtext∞\nn=1P(|Tn|≥u)<∞them Tnconverges\nto zero almost surely. Now, choose u=√{4σ2\nnlogn/n}+4Mlogn/3n. Thus,/summationtext∞\nn=1P(|Tn|≥u)</summationtext∞\nn=11/n2which\nis finite. Therefore, Tn=O(u) almost surely. Now let σn≤√{4M2logn/9n}, we have, Tn=O(logn/n) and if\nσn=O(1) then Tn=O((log n/n)1/2) almost surely.\nLemma 3. Suppose T i jare i.i.d. with density f T. Then for fixed i=1,···,n, any k and l≥1, under assumptions (C2)\nand (C6)c, m−1\ni/summationtextmi\nj=1φk(Ti j)φl(Ti j)=1(k=l)+O(/parenleftbiglogmi/mi/parenrightbig1/2)almost surely where 1is the indicator function.\n12Proof. Observe, E/braceleftig\nm−1\ni/summationtextmi\nj=1φk(Ti j)φl(Ti j)/bracerightig\n=/integraltext\nφk(t)φl(t)dt=1(k=l) and,\nVarm−1\nimi/summationdisplay\nj=1φk(Ti j)φl(Ti j)=Em−1\nimi/summationdisplay\nj=1φk(Ti j)φl(Ti j)2\n−1(k=1)\n=m−2\nimi/summationdisplay\nj=1E{φ2\nk(Ti j)φ2\nl(Ti j)}+m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1\nj1/nequalj2E{φk(Ti j1)φk(Ti j2)φl(Ti j1)φl(Ti j2)}\n−1(k=l)\n=m−2\nimi/summationdisplay\nj=1E{φ2\nk(Ti j)φ2\nl(Ti j)}+m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1\nj1/nequalj2E{φk(Ti j1)φl(Ti j1)}E{φk(Ti j2)φl(Ti j2)}\n−1(k=l)\n=m−1\ni/integraltext\nφ4\nk(t)dt+(mi−1)/mi(/integraltext\nφ2\nk(t)dt)2−1 if k=l\nm−1\ni/integraltext\nφ2\nk(t)φ2\nl(t)dt+(mi−1)/mi(/integraltext/integraltext\nφk(t)φl(t)φk(t′)φl(t′)dtdt′) if k/nequall\n=O(1/mi). (B.1)\nTherefore, applying the Lemma 2, the result is immediate.\nLemma 4. Suppose T i jare i.i.d with density f T. Then for fixed i=1,···,n, for any k≥1, under assumptions (C2),\n(C6)c, m−1\ni/summationtextmi\nj=1˙µi(Ti j)φk(Ti j)=/integraltext\n˙µi(t)φk(t)dt+O/parenleftig\n(logmi/mi)1/2/parenrightig\nalmost surely.\nProof. Observe, E/braceleftig\nm−1\ni/summationtextmi\nj=1˙µi(Ti j)φk(Ti j)/bracerightig\n=/integraltext\n˙µi(t)φk(t)dtand,\nVarm−1\nimi/summationdisplay\nj=1˙µi(Ti j)φk(Ti j)≤Em−1\nimi/summationdisplay\nj=1˙µi(Ti j)φk(Ti j)2\n=m−2\nimi/summationdisplay\nj=1E/braceleftig\n˙µ2\ni(Ti j)φk(Ti j)2/bracerightig\n+m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1\nj1/nequalj2E{˙µi(Ti j1)˙µi(Ti j2)φk(Ti j1)φk(Ti j2)}\n=O(1/mi), since/integraldisplay\n˙µ2(t)φ2\nk(t)dt<∞. (B.2)\nTherefore, applying the Lemma 2, the result is immediate.\nLemma 5. Define,Mir=/integraltext\n˙µi(t)φr(t)dt and V r=E{/integraltext\n˙µ(t)φr(t)dt}2for r≥1. Then under Conditions (C6)a and\n(C6)b, for some α>0such that V rλ−2\nrr1+α→0as r→∞ (due to Condition (C6)b,/summationtext∞\nr=1\nr/nequalk(λk−λr)−1n−1/summationtextn\ni=1Mirξik=\nO/parenleftig\n(logn/n)1/2λ1/2\nkk(1−α)/2/parenrightig\nalmost surely.\nProof. It is easy to see that, E/braceleftig/summationtext∞\nr=1\nr/nequalk(λk−λr)−1n−1/summationtextn\ni=1Mirξik/bracerightig\n=0. Using the spacing condition among the eigen-\nvalues in (C6)a, for each 1 ≤k<r<∞and for nonzero finite generic constant C0,\nmax{λk,λr}/|λk−λr|≤C0max{k,r}/|k−r|. (B.3)\nSimilar kind of conditions can be invoked such as the convexi ty assumption, i.e. λr−λr+1≤λr−1−λrfor all r≥2.\n13Thus, using Inequality (B.3), for some α>0, with condition Vrλ−2\nrr1+α→0 asr→∞ , we can write\n∞/summationdisplay\nr=1\nr/nequalkVr(λk−λr)−2/lessorsimilar∞/summationdisplay\nr=1\nr/nequalkVr{max( k,r)/|k−r|max(λk,λr)}2\n=/summationdisplay\nr≤k/2Vrλ−2\nrk2/(k−r)2+/summationdisplay\nr>2kVrλ−2\nkr2/(k−r)2\n+/summationdisplay\nk/2<r<kVrλ−2\nrk2/(k−r)2+/summationdisplay\nk<r<2kVrλ−2\nkr2/(k−r)2\n/lessorsimilar/summationdisplay\nr≤k/2,r>2kVrλ−2\nr+k2/summationdisplay\nk/2<r<2kVrλ−2\nr(k−r)−2\n/lessorsimilar1+k1−α/summationdisplay\nk/2<r<2k(k−r)−2/lessorsimilark1−α. (B.4)\nThis follows the line of proofs in Hall and Hosseini-Nasab [1 3] in different contexts. Thus, using the Inequality (B.4),\nit follows that\nVar/braceleftig∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1n−1n/summationdisplay\ni=1Mirξik/bracerightig\n=n−1λk∞/summationdisplay\nr=1\nr/nequalkVr(λk−λr)−2=O/parenleftig\nn−1λkk(1−α)/parenrightig\n. (B.5)\nTherefore, applying Lemma 2, the proof is immediate.\nLemma 6. ForMir=/integraltext\n˙µ(t)φr(t)dt andηk=min r/nequalk|λk−λr|>0, under Conditions (C6)a and (C6)b,/summationtext∞\nr1/nequal1\nr1/nequalk/summationtextκ0\nr2/nequal1\nr2/nequalk(λk−\nλr1)−1(λk−λr2)−1n−1/summationtextn\ni=1Mir1ξir2=O/parenleftbigg\n(logn/n)1/2κ(3−α)/2\n0λ−1\nκ0/braceleftig/summationtextκ0\nr=1λr/bracerightig1/2/parenrightbigg\nalmost surely.\nProof. It is not difficult to see that,\nE/braceleftig∞/summationdisplay\nr1=1\nr1/nequalkκ0/summationdisplay\nr2=1\nr2/nequalk(λk−λr1)−1(λk−λr2)−1n−1n/summationdisplay\ni=1Mir1ξir2/bracerightig\n=0.\nMoreover, using the spacing condition mentioned in (C6)b, o ne can derive the upper bound of η−1\nkby\nη−1\nk=/braceleftbigg\nmin\nr/nequalk|λk−λr|/bracerightbigg−1\n=max\nr/nequalk|λk−λr|−1\n/lessorsimilarmax\nr/nequalk{max( k,r)/|k−r|max(λk,λr)}≤λ−1\nkk. (B.6)\nDue to the monotonic decreasing property of eigen-values, f or fixed k=1,···,κ0, we have\nκ0/summationdisplay\nr=1\nr/nequalkλr(λk−λr)−2/lessorsimilarη−2\nkκ0/summationdisplay\nr=1λr/lessorsimilarλ−2\nkk2κ0/summationdisplay\nr=1λr/lessorsimilarλ−2\nκ0κ2\n0κ0/summationdisplay\nr=1λr. (B.7)\nTherefore, the following holds under similar conditions to obtain the Inequality (B.4),\nE∞/summationdisplay\nr1=1\nr1/nequalkκ0/summationdisplay\nr2=1\nr2/nequalk(λk−λr1)−2(λk−λr2)−2n−1n/summationdisplay\ni=1Mir1ξir22\n=n−1∞/summationdisplay\nr1=1\nr1/nequalkκ0/summationdisplay\nr1=1\nr1/nequalk(λk−λr1)−2(λk−λr2)−2Vr1λr2\n14/lessorsimilarn−1k1−ακ0/summationdisplay\nr=1\nr/nequalkλr(λk−λr)−2/lessorsimilarn−1κ3−α\n0λ−2\nκ0κ0/summationdisplay\nr=1λr. (B.8)\nTherefore, applying the Lemma 2, the result is immediate.\nRemark 4. Define, dik(Ti j1,Ti j2) :=/hatwideφk(Ti j1)/hatwideφk(Ti j2)−φk(Ti j1)φk(Ti j2). Now replaceG,θandψby/hatwideF,/hatwideλand/hatwideφ\nrespectively since /hatwideFbe the approximation of Fand∆is the corresponding perturbation operator in Equation (A. 1).\nTherefore, Lemma 1 immediately implies the following expan sion, which is the key fact to represent the objective\nfunction in QIF.\n/hatwideφk−φk=∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1/an}bracketle{tφr,∆φk/an}bracketri}htφr+O(/bardbl∆/bardbl2) almost surely , (B.9)\nwhere∆be the integral operator with kernel /hatwideR−R.\nAppendix C. Proof of Theorem 1\nFor the k−th element of gn(β0) (1≤k≤κ0),\ngn,k(β0)=n−1n/summationdisplay\ni=1m−2\ni˙µT\ni/hatwideΦk(yi−µi)=n−1n/summationdisplay\ni=1m−2\ni˙µT\ni/hatwideΦkWξi\n=n−1n/summationdisplay\ni=1m−2\ni˙µT\niΦkWξi+n−1n/summationdisplay\ni=1m−2\ni˙µT\ni(/hatwideΦk−Φk)Wξi:=Jn1\nk+Jn2\nk. (C.1)\nNow, using Lemmas 3 and 4, the first part of the expression of gn,k(β0) becomes\nJn1\nk=n−1n/summationdisplay\ni=1m−2\ni˙µT\niΦkWξi\n=n−1n/summationdisplay\ni=1m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1κ0/summationdisplay\nl=1˙µi(Ti j1)φk(Ti j1)φk(Ti j2)φl(Ti j2)ξil\n/lessorsimilarn−1n/summationdisplay\ni=1/braceleftig\nMik+O((log m/m)1/2)/bracerightig/braceleftig\n1+O((log m/m)1/2)/bracerightig\nξik,\nwhereMik=/integraldisplay\n˙µi(t)φk(t)dt\n=n−1n/summationdisplay\ni=1Mikξik/braceleftig\n1+O/parenleftig\n(logm/m)1/2/parenrightig/bracerightig\n=O/parenleftig\n(logn/n)1/2/braceleftig\n1+(logm/m)1/2/bracerightig/parenrightig\nalmost surely . (C.2)\nOn the other hand, the last part of gn,k(β) can be expressed as\nJn2\nk=n−1n/summationdisplay\ni=1m−2\ni˙µT\ni(/hatwideΦk−Φk)Wξi\n=n−1n/summationdisplay\ni=1m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1κ0/summationdisplay\nl=1˙µi(Ti j1)dik(Ti j1,Ti j2)φl(Ti j2)ξil, (C.3)\nwhere dik(Ti j1,Ti j2) :=/hatwideφk(Ti j1)/hatwideφk(Ti j2)−φk(Ti j1)φk(Ti j2) as defined in Remark 4. Therefore, using the discussion in\nRemark 4, we can obtain the following expression almost sure ly.\ndik(Ti j1,Ti j2) :=/hatwideφk(Ti j1)/hatwideφk(Ti j2)−φk(Ti j1)φk(Ti j2)\n15=φk(Ti j1)+∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1/an}bracketle{tφr,∆φk/an}bracketri}htφr(Ti j1)+O(/bardbl∆/bardbl2)\n×φk(Ti j2)+∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1/an}bracketle{tφr,∆φk/an}bracketri}htφr(Ti j2)+O(/bardbl∆/bardbl2)−φk(Ti j1)φk(Ti j2)\n=∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1/an}bracketle{tφr,∆φk/an}bracketri}ht/braceleftig\nφr(Ti j1)φk(Ti j2)+φk(Ti j1)φr(Ti j2)/bracerightig\n+/summationdisplay\nr1/nequalk/summationdisplay\nr2/nequalk(λk−λr1)−1(λk−λr2)−1/angbracketleftbigφr1,∆φk/angbracketrightbig/angbracketleftbigφr2,∆φk/angbracketrightbigφr1(Ti j1)φr2(Ti j2)\n+O(/bardbl∆/bardbl2)\n:=In1\nik(Ti j1,Ti j2)+In2\nik(Ti j1,Ti j2)+O(/bardbl∆/bardbl2). (C.4)\nThus, almost surely, we can write\nn−1n/summationdisplay\ni=1m−2\ni˙µT\ni(/hatwideΦk−Φk)Wξi\n=n−1n/summationdisplay\ni=1m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1κ0/summationdisplay\nl=1˙µi(Ti j1)dik(Ti j1,Ti j2)φl(Ti j2)ξil\n=n−1n/summationdisplay\ni=1m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1κ0/summationdisplay\nl=1˙µi(Ti j1)dik(Ti j1,Ti j2)φl(Ti j2)ξil\n=n−1n/summationdisplay\ni=1m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1κ0/summationdisplay\nl=1˙µi(Ti j1)/braceleftig\nIn1\nik(Ti j1,Ti j2)+In2\nik(Ti j1,Ti j2)/bracerightig\nφl(Ti j2)ξil+O(/bardbl∆/bardbl2)\n:=Jn2\nk1+Jn2\nk2+O(/bardbl∆/bardbl2). (C.5)\nUnder assumptions (C1)-(C5), by using Theorem 3.3 in Li and H sing [19],/bardbl∆/bardbl2=O(h4+δ2\nn2(h)) almost surely. Now\nobserve that\nJn2\nk1=n−1n/summationdisplay\ni=1m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1κ0/summationdisplay\nl=1∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1˙µi(Ti j1)/braceleftig\nφr(Ti j1)φk(Ti j2)+φk(Ti j1)φr(Ti j2)/bracerightig\n×φl(Ti j2)/an}bracketle{tφr,∆φk/an}bracketri}htξil\n/lessorsimilarn−1n/summationdisplay\ni=1m−1\nimi/summationdisplay\nj1=1∞/summationdisplay\nr=1\nr/nequalkκ0/summationdisplay\nl=1(λk−λr)−1˙µi(Ti j1)φr(Ti j1)/braceleftig\n1(l=k)+O((log m/m)1/2)/bracerightig\n/an}bracketle{tφr,∆φk/an}bracketri}htξil\n+n−1n/summationdisplay\ni=1m−1\nimi/summationdisplay\nj1=1∞/summationdisplay\nr=1\nr/nequalkκ0/summationdisplay\nl=1(λk−λr)−1˙µi(Ti j1)φk(Ti j1)/braceleftig\n1(r=l)+O((log m/m)1/2)/bracerightig\n/an}bracketle{tφr,∆φk/an}bracketri}htξil\n/lessorsimilarn−1n/summationdisplay\ni=1m−1\nimi/summationdisplay\nj1=1∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1˙µi(Ti j1)φr(Ti j1)/an}bracketle{tφr,∆φk/an}bracketri}htξik/braceleftig\n1+O((log m/m)1/2)/bracerightig\n+n−1n/summationdisplay\ni=1m−1\nimi/summationdisplay\nj1=1κ0/summationdisplay\nr=1\nr/nequalk(λk−λr)−1˙µi(Ti j1)φk(Ti j1)/an}bracketle{tφr,∆φk/an}bracketri}htξir/braceleftig\n1+O((log m/m)1/2)/bracerightig\n16:=(Un\nk1+Un\nk2)/braceleftig\n1+O((log m/m)1/2)/bracerightig\nalmost surely . (C.6)\nThen applying the triangle inequality, we have\nUn\nk1=n−1n/summationdisplay\ni=1m−1\nimi/summationdisplay\nj=1∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1˙µi(Ti j)φr(Ti j)/an}bracketle{t∆φk,φr/an}bracketri}htξik\n/lessorsimilar/bardbl∆φk/bardbl∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1n−1n/summationdisplay\ni=1/braceleftig\nMir+O((log m/m)1/2)/bracerightig\nξik\n=/bardbl∆φk/bardbl∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1n−1n/summationdisplay\ni=1Mirξik/braceleftig\n1+O((log m/m)1/2)/bracerightig\n, (C.7)\nwhereMir=/integraltext\n˙µi(t)φr(t)dt. By Lemma 6 of Li and Hsing [19], under conditions (C1)-(C5), for any measurable\nbounded function eon [0,1],/bardbl∆φk/bardbl=O(h2+δn1(h)+δ2\nn2(h))≡O(δn(h)) almost surely, where δn(h)=h2+δn1(h)+\nδ2\nn2(h). Thus, in addition with Inequalities (B.5) in Lemma 5, we ob tainUn\nk1=O/parenleftig\nδn(h)(log n/n)1/2λ1/2\nkk(1−α)/2{1+(logm/m)1/2}/parenrightig\nalmost surely. Next, under the spacing condition mentioned earlier and in assumption (C6)a, using the Inequality\n(B.6), recall η−1\nk/lessorsimilarλ−1\nkk. Thus, observe that\nUn\nk2=n−1n/summationdisplay\ni=1m−1\nimi/summationdisplay\nj=1κ0/summationdisplay\nr=1\nr/nequalk(λk−λr)−1˙µi(Ti j)φk(Tij)/an}bracketle{t∆φk,φr/an}bracketri}htξir\n/lessorsimilar/bardbl∆φk/bardblκ0/summationdisplay\nr=1\nr/nequalk(λk−λr)−1n−1n/summationdisplay\ni=1Mikξir/braceleftig\n1+O((log m/m)1/2)/bracerightig\n/lessorsimilar/bardbl∆φk/bardblη−1\nkκ0/summationdisplay\nr=1\nr/nequalkn−1n/summationdisplay\ni=1Mikξir/braceleftig\n1+O((log m/m)1/2)/bracerightig\n. (C.8)\nUsing condition (C6)b, we also have V1/2\nkη−1\nk/lessorsimilarV1/2\nkλ−1\nkk=O(k(1−α)/2). Finally, combining with the bounds for\nUn\nk1,Un\nk2, we have, almost surely,\nJn2\nk1=O/braceleftig\n(logn/n)1/2δn(h)/parenleftig\nλ1/2\nkk(1−α)/2+η−1\nkV1/2\nkκ0/summationdisplay\nr=1\nr/nequalkλ1/2\nr/parenrightig/parenleftig\n1+(logm/m)1/2/parenrightig/bracerightig\n:=O(ωk1(n,h)), (C.9)\nwhereωk1(n,h)=(logn/n)1/2δn(h)k(1−α)/2/summationtextκ0\nk=1λ1/2\nr/braceleftig\n1+(logm/m)1/2/bracerightig\n.\nIt is easy to see that/summationtextκ0\nr=1λ1/2\nr∼κ−τ1/2+1\n0. Therefore, for τ=α+τ1,ωk1(n,h)∼(logn/n)1/2δn(h)κ(3−τ)/2\n0/braceleftig\n1+(logm/m)1/2/bracerightig\n.\nSimilarly, to the derivation of the bound for Jn2\nk1, we can write\nJn2\nk2=n−1n/summationdisplay\ni=1m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1κ0/summationdisplay\nl=1˙µi(Ti j1)In2\nik(Ti j1,Ti j2)φl(Ti j2)ξil\n=n−1n/summationdisplay\ni=1m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1κ0/summationdisplay\nl=1∞/summationdisplay\nr1=1\nr1/nequalk∞/summationdisplay\nr1=1\nr2/nequalk(λk−λr1)−1(λk−λr2)−1/angbracketleftbigφr1,∆φk/angbracketrightbig/angbracketleftbigφr2,∆φk/angbracketrightbig\n×˙µi(Ti j1)φr1(Ti j1)φr2(Ti j2)φl(Ti j2)ξil\n17/lessorsimilarn−1n/summationdisplay\ni=1mi−1mi/summationdisplay\nj1=1κ0/summationdisplay\nl=1∞/summationdisplay\nr1=1\nr1/nequalk∞/summationdisplay\nr2=1\nr2/nequalk(λk−λr1)−1(λk−λr2)−1/angbracketleftbigφr1,∆φk/angbracketrightbig/angbracketleftbigφr2,∆φk/angbracketrightbig\n×˙µi(Ti j1)φr1(Ti j1)/braceleftig\n1(r2=l)+O((log m/m)1/2)/bracerightig\nξil\n=n−1n/summationdisplay\ni=1mi−1mi/summationdisplay\nj1=1∞/summationdisplay\nr1=1\nr1/nequalkκ0/summationdisplay\nr2=1\nr2/nequalk(λk−λr1)−1(λk−λr2)−1/angbracketleftbigφr1,∆φk/angbracketrightbig/angbracketleftbigφr2,∆φk/angbracketrightbig\n×˙µi(Ti j1)φr1(Ti j1)ξir2/braceleftig\n1+O((log m/m)1/2)/bracerightig\n/lessorsimilar/bardbl∆φk/bardbl2∞/summationdisplay\nr1=1\nr1/nequalkκ0/summationdisplay\nr2=1\nr2/nequalk(λk−λr1)−1(λk−λr2)−1n−1n/summationdisplay\ni=1/braceleftig\nMir1+O((log m/m)1/2)/bracerightig\nξir2\n×/braceleftig\n1+O((log m/m)1/2)/bracerightig\n=/bardbl∆φk/bardbl2∞/summationdisplay\nr1=1\nr1/nequalkκ0/summationdisplay\nr2=1\nr2/nequalk(λk−λr1)−1(λk−λr2)−1×n−1n/summationdisplay\ni=1Mir1ξir2/braceleftig\n1+O((log m/m)1/2)/bracerightig\n. (C.10)\nTherefore, Inequality (C.10) immediately follows using Le mma 6,\nJn2\nk2=O(logn/n)1/2δ2\nn(h)κ(3−α)/2\n0λ−1\nκ0{κ0/summationdisplay\nr=1λr}1/2/braceleftig\n1+(logm/m)1/2/bracerightig\n:=O(ωk2(n,h)), almost surely .\nwhere\nωk2(n,h)=(logn/n)1/2δ2\nn(h)κ(3−α)/2\n0λ−1\nκ0κ0/summationdisplay\nr=1λr/braceleftig\n1+(logm/m)1/2/bracerightig\n.\nWe observe that/summationtextκ0\nr=1λr∼κ−τ1+1\n0under assumption (C6)b. Thus, ωk2(n,h)∼(logn/n)δ2(h)κ(4−τ)/2\n0{1+(logm/m)1/2}.\nSinceωn2=O(ωn1) andδ2\nn(h)=O(ωn1), in summary, for each k=1,···,κ0,\ngk(β)=O/parenleftig\n(logn/n)1/2/braceleftig\n1+(logm/m)1/2/bracerightig\n+ωk1(n,h)/parenrightig\nalmost surely. Since 1 /(nm)=O(1/n),\nAMSE{gk(β0)}=O/parenleftig\nn−1+n−1κ3−τ\n0Rn(h)/parenrightig\n,\nwhere Rn(h)=/braceleftig\nh4+1\nn+1\nnmh+1\nn2m2h2+1\nn2m4h4+1\nn2mh+1\nn2m3h3/bracerightig\n. Combining the above conditions, we find that if\na>1/4,κ0=O(n1/(3−τ)) and n−1/4/lessorsimilarh/lessorsimilarn−(a+1)/5then AMSE{gk(β0)}=O(1/n). On the other hand, if a≤1/4,\nκ0=O(n4(1+a)/5(3−τ)) and h/lessorsimilarn−1/4AMSE{gk(β0)}=O(1/n).\nNote that for a three-dimensional array ( ∂C/∂β 1,···,∂C/∂β p) such that the following is a p×1 vector.\ng(β0)TC−1(β0)˙C(β0)C−1(β0)g(β0) (C.11)\nTherefore,\nn−1˙Q(β0)=2˙g(β0)TC−1(β0)g(β0)−g(β0)TC−1(β0)˙C(β0)C−1(β0)g(β0) (C.12)\nand\nn−1¨Q(β0)=2˙g(β0)TC−1(β0)˙g(β0)+rn1+rn2+rn3+rn4 (C.13)\n18where\nrn1=2¨g(β0)C−1(β0)g(β0)\nrn2=−4˙g(β0)TC−1(β0)˙C(β0)C−1(β0)g(β0)\nrn3=2˙g(β0)C−1(β0)C−1(β0)˙C(β0)C−1(β0)g(β0)\nrn4=−g(β0)TC−1(β0)¨C(β0)C−1(β0)g(β0)(C.14)\nSince g(β0)=OP(n−1/2) and the weight matrix converges almost surely to an inverti ble matrix, g(β0)TC−1(β0)˙C(β0)C−1(β0)g(β0)=\no(n−1) almost surely. Furthermore, rn1=O(n−1/2),rn2=o(n−1/2),rn3=O(n−1/2), and rn4=O(n−1) almost surely.\nCombining these bounds, we have rn=o(1) almost surely. Therefore, /bardbln−1˙Q−2˙g(β0)TC−1(β0)g(β0)/bardbl=oP(n−1) and\n/bardbln−1¨Q−2˙g(β0)TC−1(β0)˙g(β0)/bardbl=oP(1).\nThe following lines are based on common steps in the GEE liter ature that includes Balan et al. [2], McCullagh\nand Nelder [24], Tian et al. [27] among many others. Let βn=β0+δdwhere setδ=n−1/2. We have to show that for\nanyǫ >0 there exists a large constant csuch that\nP{inf\n/bardbld/bardbl=cQ(βn)≥Q(β0)}>1−ǫ (C.15)\nNote that the above statement is always true if ǫ≥1. Thus, we assume that ǫ∈(0,1). Due to Taylor series expansion,\nQ(βn)=Q(β0+δd)=Q(β0)+δdT˙Q(β0)+0.5δdT¨Q(β0)d+/bardbld/bardbl2oP(1) (C.16)\nNow, observe that, using Equation (C.12),\nδdT˙Q(β0)=/bardbld/bardblOP(√nδ)+/bardbld/bardblOP(δ) (C.17)\nand\n0.5δ2dT¨Q(β∗)d=nδ2dT˙g(β0)C−1(β0)˙g(β0)d+nδ2/bardbld/bardbl2oP(1) (C.18)\nTherefore, for given ǫ >0, there exists a large enough csuch that the above equation (C.15) holds. This implies that\nthere exists a ˆβthat satisfies/bardbl/hatwideβ−β0/bardbl=OP(δ). Thus, for large n, with probability 1, Q(β) attains the minimal value at\nˆβand therefore, ˙Q=0.\nAppendix D. Proof of Theorem 2\nRecall,Ci=/summationtextκ0\nk1=1/summationtextκ0\nk2=1Φk1XiC−1\nk1,k2XT\niΦk2, where C−1\nk1,k2is the ( k1,k2) block of C−1\n0. Similarly, we can define\n/hatwideCi=/summationtextκ0\nk1=1/summationtextκ0\nk2=1/hatwideΦk1XiC−1\nk1,k2XT\ni/hatwideΦk2. It is easy to observe that /hatwideCi=Ci+/summationtextκ0\nk1=1/summationtextκ0\nk2=1(/hatwideΦk1−Φk1)XiC−1\nk1,k2XT\ni(/hatwideΦk2−\nΦk2)+2/summationtextκ0\nk1=1/summationtextκ0\nk2=1Φk1XiC−1\nk1,k2XT\ni(/hatwideΦk2−Φk2). Therefore, n−1/summationtextn\ni=1˙µT\ni(/hatwideCi−C)Xi=n−1/summationtextn\ni=1/summationtextκ0\nk1=1/summationtextκ0\nk2=1Pik1C−1\nk1,k2Pik2\nandn−1/summationtextn\ni=1˙µT\ni(/hatwideCi−C)yi=n−1/summationtextn\ni=1/summationtextκ0\nk1=1/summationtextκ0\nk2=1Pik1C−1\nk1,k2Qik2where Pi,k=˙µT\niDikXiandQik=˙µT\niDikyiwith Dikbe\nthe difference matrix with ( j1,j2)-th element is di(Ti j1,Ti j2). Thus, note that, almsot surely we have the following\nrelation,\nPik=m−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1˙µi(Ti j1)di(Ti j1,Ti j2)xi(Ti j2)\n/lessorsimilarm−2\nimi/summationdisplay\nj1=1mi/summationdisplay\nj2=1˙µi(Ti j1)∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1/an}bracketle{tφr,∆φk/an}bracketri}ht/braceleftig\nφr(Ti j1)φk(Ti j2)+φk(Ti j1)φr(Ti j2)/bracerightig\n+O(/bardbl∆/bardbl2)\n/lessorsimilar∞/summationdisplay\nr=1\nr/nequalk(λk−λr)−1/bardbl∆φk/bardbl+O(/bardbl∆/bardbl2)=O(̟),\n19since m−1\nimi/summationdisplay\nj=1˙µ(Ti j)φk(Ti j) and m−1\nimi/summationdisplay\nj=1xi(Ti j)φk(Ti j) are finite\n(D.1)\nwhere̟=/summationtext∞\nr=1\nr/nequalk(λk−λr)−1δn(h)+h2+δ2\nn2(h). A similar result can be obtained for Qik. Combining such results, in\nsummary, we have −2n−1/summationtextn\ni=1XT\ni/hatwideCi(yi−XT\ni/hatwideβ)=−2n−1/summationtextn\ni=1XT\niCi(yi−XT\ni/hatwideβ)+O(̟n). Since, for n→∞ ,Q(β) attains\na minimal value at β=/hatwideβ, we therefore have ˙Q(/hatwideβ)=0. Thus,\n˙Q(/hatwideβ)=−2n−1n/summationdisplay\ni=1XT\ni/hatwideCi(yi−XT\ni/hatwideβ)\n=−2n−1n/summationdisplay\ni=1XT\ni(/hatwideCi−C i)(yi−XT\ni/hatwideβ)−2n−1n/summationdisplay\ni=1XT\niCi(yi−XT\ni/hatwideβ)=0.\n(D.2)\nTherefore, almost surely, we have,\n−2n−1n/summationdisplay\ni=1XT\niCi(yi−XT\ni/hatwideβ)+O(̟n)=0\n−2n−1n/summationdisplay\ni=1XT\niCi(XT\niβ0+ei−XT\ni/hatwideβ)+O(̟n)=0\n√n(/hatwideβ−β0)n−1n/summationdisplay\ni=1XT\niCiXi=n−1n/summationdisplay\ni=1XT\niCiei (D.3)\nNow, using the central limit theorem, we can obtain the follo wing.\nn/summationdisplay\ni=1XT\niCiei/√nd− →N(0,A) (D.4)\nIn addition, by the law of large numbers n−1/summationtextn\ni=1XT\niCiXi→Bin probability. Therefore, using the Slutsky theorem,\nwe complete the proof of Theorem 2.\nReferences\n[1] Y . Bai, Z. Zhu, W. K. 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Qu, Informative estimation and selection of correlation structure for longitudinal data, Journal of th e American Statistical Asso-\nciation 107 (2012) 701–710.\n21Table 2: Estimated values and associated standard errors for the reg ression coefficients are provided upto four decimal places based on the exi sting\nand proposed methods. First line corresponding to each ROI s hows results based on initial estimates and the second line c orresponds to that of\nproposed estimates.\nβ0 β1\nregion # functional points Estimate Std. Error ( ×100) Estimate (×100) Std. Error ( ×100)\nROI.6 659 0.4512 0.1343 -0.0606 0.0130\n0.4512 0.0983 -0.0605 0.0028\nROI.7 1362 0.5048 0.0628 0.0309 0.0061\n0.5050 0.0681 0.0342 0.0007\nROI.8 1370 0.5256 0.0586 -0.0667 0.0057\n0.5271 0.0346 -0.0733 0.0006\nROI.9 690 0.4951 0.0910 0.2904 0.0088\n0.5443 0.0874 0.1660 0.0014\nROI.10 699 0.4951 0.0892 0.3314 0.0086\n0.5262 0.1398 0.4231 0.0014\nROI.11 968 0.4372 0.0979 0.1323 0.0095\n0.4380 0.0637 0.1311 0.0009\nROI.12 968 0.4529 0.0948 0.0965 0.0092\n0.4664 0.0750 0.0504 0.0013\nROI.13 992 0.5448 0.1060 0.3453 0.0103\n0.5449 0.0856 0.3559 0.0011\nROI.14 992 0.5435 0.1068 0.3432 0.0104\n0.5436 0.0754 0.3437 0.0003\nROI.37 1236 0.3695 0.0779 -0.1126 0.0076\n0.3713 0.0669 -0.1175 0.0017\nROI.38 1155 0.3564 0.0819 -0.1356 0.0079\n0.3578 0.0420 -0.1356 0.0009\nROI.39 1124 0.4618 0.0760 0.1972 0.0074\n0.4621 0.0615 0.1996 0.0007\nROI.40 1125 0.4786 0.0658 0.0953 0.0064\n0.4780 0.0369 0.1016 0.0005\nROI.45 380 0.4189 0.1071 0.1647 0.0104\n0.4190 0.0175 0.1648 0.0001\nROI.46 376 0.4074 0.1033 0.1988 0.0100\n0.4074 0.0159 0.1994 0.0002\nROI.47 596 0.4596 0.0932 0.1304 0.0090\n0.4594 0.0191 0.1349 0.0001\nROI.48 600 0.4045 0.0868 0.1100 0.0084\n0.4036 0.0644 0.1067 0.000622arXiv:2402.13907v1  [stat.ME]  21 Feb 2024Supplementary material for “Quadratic inference with dens e functional\nresponses”\nPratim Guha Niyogia,∗, Ping-Shou Zhongb\naDepartment of Biostatistics, Johns Hopkins Bloomberg Scho ol of Public Health, Maryland, USA\nbDepartment of Mathematics, Statistics and Computer Scienc e, University of Illinois at Chicago, Illinois, USA\nAbstract\nIn this Supplementary Paper, we provide the additional resu lts of the simulation studies discussed in Section ??.\n∗Corresponding author. Email address: pnyogi1@jhmi.eduTable S1: Performance of the estimation procedure where the residual s are generated from linear process with l0=1. Mean of the estimated\ncoefficients, standard deviation, absolute bias, mean square err or (×100) and FVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 1.0019 0.0340 0.0267 0.1155 0.5001 0.0456 0.0356 0.2078\nldaAR 1.0007 0.0230 0.0187 0.0528 0.5010 0.0361 0.0285 0.1303\nldaCS 1.0000 0.0006 0.0005 0.0000 0.4999 0.0008 0.0006 0.0001\nfda-1 1.0093 0.1436 0.1150 2.0675 0.5020 0.2010 0.1570 4.0311 73. 0607\nfda-2 1.0036 0.1055 0.0828 1.1123 0.5052 0.1337 0.1070 1.7869 91. 6726\nfda-3 1.0038 0.1024 0.0804 1.0473 0.5056 0.1303 0.1020 1.6969 99. 7657\nfda-4 1.0000 0.0092 0.0021 0.0084 0.5006 0.0096 0.0024 0.0093 99. 9993\nfda-5 1.0000 0.0011 0.0007 0.0001 0.5000 0.0017 0.0010 0.0003 100 .0000\nn=300\ninit 0.9991 0.0181 0.0144 0.0326 0.5006 0.0268 0.0212 0.0715\nldaAR 0.9995 0.0133 0.0105 0.0178 0.5000 0.0212 0.0169 0.0447\nldaCS 1.0000 0.0003 0.0002 0.0000 0.5000 0.0005 0.0004 0.0000\nfda-1 0.9958 0.0767 0.0616 0.5888 0.5037 0.1163 0.0918 1.3523 73. 4907\nfda-2 0.9974 0.0567 0.0458 0.3220 0.5011 0.0804 0.0648 0.6460 91. 7757\nfda-3 0.9974 0.0564 0.0455 0.3182 0.5007 0.0800 0.0643 0.6391 99. 9225\nfda-4 1.0000 0.0005 0.0001 0.0000 0.5000 0.0009 0.0002 0.0001 99. 9998\nfda-5 1.0000 0.0002 0.0001 0.0000 0.5000 0.0003 0.0002 0.0000 100 .0000\nn=500\ninit 0.9999 0.0152 0.0121 0.0230 0.5027 0.0207 0.0163 0.0436\nldaAR 1.0008 0.0100 0.0079 0.0100 0.5019 0.0161 0.0129 0.0263\nldaCS 1.0000 0.0003 0.0002 0.0000 0.5000 0.0004 0.0003 0.0000\nfda-1 0.9990 0.0657 0.0523 0.4303 0.5113 0.0883 0.0698 0.7913 73. 4098\nfda-2 1.0013 0.0468 0.0371 0.2185 0.5078 0.0651 0.0520 0.4292 91. 8501\nfda-3 1.0014 0.0459 0.0364 0.2107 0.5074 0.0650 0.0516 0.4274 99. 9490\nfda-4 1.0000 0.0001 0.0001 0.0000 0.5000 0.0001 0.0001 0.0000 99. 9999\nfda-5 1.0000 0.0001 0.0001 0.0000 0.5000 0.0001 0.0001 0.0000 100 .0000\n2Table S2: Performance of the estimation procedure where the residual s are generated from linear process with l0=2. Mean of the estimated\ncoefficients, standard deviation, absolute bias, mean square err or (×100) and FVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 1.0020 0.0331 0.0261 0.1094 0.4999 0.0451 0.0349 0.2028\nldaAR 1.0002 0.0167 0.0133 0.0278 0.5014 0.0232 0.0183 0.0538\nldaCS 1.0000 0.0003 0.0002 0.0000 0.5000 0.0004 0.0003 0.0000\nfda-1 1.0096 0.1431 0.1144 2.0520 0.5022 0.2003 0.1561 4.0029 92. 5933\nfda-2 1.0014 0.0648 0.0508 0.4188 0.5037 0.0842 0.0667 0.7094 98. 5532\nfda-3 1.0009 0.0535 0.0406 0.2860 0.5018 0.0744 0.0570 0.5524 99. 7251\nfda-4 1.0000 0.0074 0.0017 0.0055 0.4999 0.0103 0.0024 0.0106 99. 9991\nfda-5 1.0000 0.0045 0.0009 0.0021 0.5000 0.0021 0.0009 0.0004 100 .0000\nn=300\ninit 0.9991 0.0175 0.0140 0.0308 0.5006 0.0263 0.0208 0.0689\nldaAR 0.9999 0.0091 0.0071 0.0083 0.4997 0.0127 0.0102 0.0161\nldaCS 1.0000 0.0002 0.0001 0.0000 0.5000 0.0002 0.0002 0.0000\nfda-1 0.9957 0.0767 0.0616 0.5893 0.5038 0.1164 0.0919 1.3541 92. 9525\nfda-2 0.9989 0.0365 0.0295 0.1331 0.4998 0.0511 0.0410 0.2608 98. 7539\nfda-3 0.9992 0.0349 0.0282 0.1218 0.4991 0.0483 0.0385 0.2332 99. 9061\nfda-4 0.9999 0.0017 0.0002 0.0003 0.4999 0.0033 0.0004 0.0011 99. 9997\nfda-5 1.0000 0.0002 0.0001 0.0000 0.5000 0.0002 0.0001 0.0000 100 .0000\nn=500\ninit 0.9998 0.0148 0.0118 0.0219 0.5026 0.0201 0.0158 0.0409\nldaAR 1.0006 0.0073 0.0059 0.0054 0.5008 0.0098 0.0079 0.0097\nldaCS 1.0000 0.0001 0.0001 0.0000 0.5000 0.0002 0.0001 0.0000\nfda-1 0.9990 0.0656 0.0523 0.4295 0.5114 0.0882 0.0698 0.7897 92. 9459\nfda-2 1.0013 0.0296 0.0233 0.0874 0.5039 0.0415 0.0329 0.1735 98. 7957\nfda-3 1.0012 0.0285 0.0224 0.0812 0.5036 0.0407 0.0322 0.1670 99. 9389\nfda-4 1.0000 0.0001 0.0001 0.0000 0.5000 0.0001 0.0001 0.0000 99. 9998\nfda-5 1.0000 0.0001 0.0001 0.0000 0.5000 0.0001 0.0001 0.0000 100 .0000\n3Table S3: Performance of the estimation procedure where the residual s are generated from linear process with l0=3. Mean of the estimated\ncoefficients, standard deviation, absolute bias, mean square err or (×100) and FVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 1.0020 0.0328 0.0260 0.1077 0.4998 0.0451 0.0349 0.2027\nldaAR 1.0000 0.0087 0.0069 0.0076 0.5009 0.0122 0.0096 0.0149\nldaCS 1.0000 0.0001 0.0001 0.0000 0.5000 0.0002 0.0002 0.0000\nfda-1 1.0096 0.1430 0.1143 2.0496 0.5023 0.2001 0.1559 3.9978 97. 9586\nfda-2 1.0000 0.0326 0.0250 0.1058 0.5023 0.0440 0.0344 0.1941 99. 5699\nfda-3 0.9998 0.0144 0.0072 0.0208 0.4986 0.0209 0.0103 0.0436 99. 8941\nfda-4 1.0002 0.0053 0.0013 0.0028 0.5002 0.0067 0.0018 0.0044 99. 9991\nfda-5 1.0003 0.0037 0.0008 0.0014 0.4998 0.0042 0.0011 0.0018 100 .0000\nn=300\ninit 0.9991 0.0174 0.0139 0.0303 0.5006 0.0262 0.0207 0.0684\nldaAR 1.0000 0.0047 0.0037 0.0022 0.4997 0.0066 0.0052 0.0044\nldaCS 1.0000 0.0001 0.0001 0.0000 0.5000 0.0001 0.0001 0.0000\nfda-1 0.9957 0.0767 0.0616 0.5896 0.5038 0.1164 0.0919 1.3543 98. 2262\nfda-2 0.9996 0.0197 0.0158 0.0386 0.4996 0.0274 0.0219 0.0750 99. 7663\nfda-3 1.0002 0.0151 0.0101 0.0227 0.4990 0.0186 0.0125 0.0347 99. 9255\nfda-4 1.0001 0.0022 0.0003 0.0005 0.5001 0.0017 0.0003 0.0003 99. 9997\nfda-5 1.0000 0.0002 0.0001 0.0000 0.5000 0.0002 0.0001 0.0000 100 .0000\nn=500\ninit 0.9998 0.0147 0.0117 0.0216 0.5025 0.0199 0.0157 0.0400\nldaAR 1.0003 0.0038 0.0031 0.0015 0.5003 0.0052 0.0041 0.0027\nldaCS 1.0000 0.0001 0.0001 0.0000 0.5000 0.0001 0.0001 0.0000\nfda-1 0.9990 0.0656 0.0523 0.4293 0.5114 0.0882 0.0698 0.7895 98. 2518\nfda-2 1.0008 0.0159 0.0126 0.0253 0.5015 0.0222 0.0175 0.0495 99. 8021\nfda-3 1.0001 0.0126 0.0091 0.0158 0.5002 0.0179 0.0130 0.0320 99. 9449\nfda-4 1.0000 0.0001 0.0001 0.0000 0.5000 0.0001 0.0001 0.0000 99. 9998\nfda-5 1.0000 0.0001 0.0001 0.0000 0.5000 0.0001 0.0001 0.0000 100 .0000\n4Table S4: Performance of the estimation procedure where the residual s are generated from Ornstein-Uhlenbeck process with µ0=1. Mean of the\nestimated coefficients, standard deviation, absolute bias, mean square err or (×100) and FVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 1.0003 0.0541 0.0434 0.2922 0.4994 0.0711 0.0563 0.5048\nldaAR 1.0001 0.0476 0.0383 0.2261 0.4984 0.0650 0.0513 0.4214\nldaCS 0.9994 0.0398 0.0316 0.1581 0.4978 0.0534 0.0421 0.2849\nfda-1 1.0009 0.0705 0.0563 0.4964 0.5006 0.0947 0.0747 0.8954 79. 4156\nfda-2 1.0001 0.0453 0.0358 0.2044 0.4982 0.0608 0.0482 0.3690 94. 9669\nfda-3 0.9993 0.0386 0.0307 0.1491 0.4978 0.0511 0.0405 0.2613 99. 9949\nfda-4 1.0003 0.0127 0.0081 0.0162 0.4991 0.0242 0.0146 0.0583 99. 9992\nfda-5 0.9997 0.0084 0.0071 0.0071 0.4991 0.0159 0.0124 0.0254 100 .0000\nn=300\ninit 1.0000 0.0288 0.0233 0.0829 0.5003 0.0416 0.0329 0.1728\nldaAR 1.0000 0.0258 0.0206 0.0662 0.4996 0.0368 0.0294 0.1350\nldaCS 1.0002 0.0212 0.0170 0.0449 0.4987 0.0314 0.0254 0.0983\nfda-1 0.9997 0.0388 0.0316 0.1503 0.5014 0.0560 0.0440 0.3130 79. 9233\nfda-2 1.0005 0.0240 0.0193 0.0576 0.4983 0.0352 0.0284 0.1242 95. 0283\nfda-3 0.9999 0.0202 0.0161 0.0409 0.4994 0.0298 0.0240 0.0884 99. 9984\nfda-4 0.9999 0.0072 0.0064 0.0051 0.4997 0.0129 0.0111 0.0166 99. 9998\nfda-5 1.0000 0.0072 0.0064 0.0051 0.4997 0.0128 0.0111 0.0165 100 .0000\nn=500\ninit 1.0000 0.0288 0.0233 0.0829 0.5003 0.0416 0.0329 0.1728\nldaAR 1.0000 0.0258 0.0206 0.0662 0.4996 0.0368 0.0294 0.1350\nldaCS 1.0002 0.0212 0.0170 0.0449 0.4987 0.0314 0.0254 0.0983\nfda-1 0.9997 0.0388 0.0316 0.1503 0.5014 0.0560 0.0440 0.3130 79. 9233\nfda-2 1.0005 0.0240 0.0193 0.0576 0.4983 0.0352 0.0284 0.1242 95. 0283\nfda-3 0.9999 0.0202 0.0161 0.0409 0.4994 0.0298 0.0240 0.0884 99. 9984\nfda-4 0.9999 0.0072 0.0064 0.0051 0.4997 0.0129 0.0111 0.0166 99. 9998\nfda-5 1.0000 0.0072 0.0064 0.0051 0.4997 0.0128 0.0111 0.0165 100 .0000\n5Table S5: Performance of the estimation procedure where the residual s are generated from Ornstein-Uhlenbeck process with µ0=3. Mean of the\nestimated coefficients, standard deviation, absolute bias, mean square err or (×100) and FVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 1.0001 0.0454 0.0363 0.2056 0.4990 0.0591 0.0469 0.3487\nldaAR 0.9996 0.0390 0.0312 0.1521 0.4979 0.0521 0.0414 0.2717\nldaCS 0.9999 0.0429 0.0341 0.1841 0.4981 0.0564 0.0448 0.3176\nfda-1 1.0005 0.0557 0.0445 0.3100 0.5003 0.0743 0.0588 0.5511 59. 1692\nfda-2 1.0005 0.0459 0.0362 0.2107 0.4981 0.0604 0.0480 0.3639 87. 0120\nfda-3 1.0000 0.0436 0.0348 0.1894 0.4975 0.0568 0.0455 0.3221 99. 9908\nfda-4 1.0004 0.0136 0.0058 0.0185 0.4989 0.0237 0.0106 0.0562 99. 9960\nfda-5 1.0001 0.0049 0.0033 0.0024 0.4997 0.0099 0.0061 0.0098 100 .0000\nn=300\ninit 1.0002 0.0239 0.0191 0.0568 0.4998 0.0345 0.0274 0.1190\nldaAR 1.0003 0.0209 0.0167 0.0435 0.4990 0.0303 0.0244 0.0916\nldaCS 1.0003 0.0224 0.0178 0.0500 0.4992 0.0328 0.0265 0.1075\nfda-1 0.9998 0.0305 0.0247 0.0928 0.5011 0.0443 0.0349 0.1957 59. 4418\nfda-2 1.0004 0.0240 0.0192 0.0574 0.4991 0.0347 0.0279 0.1201 86. 9290\nfda-3 1.0003 0.0222 0.0177 0.0492 0.4992 0.0327 0.0264 0.1070 99. 9972\nfda-4 1.0002 0.0043 0.0039 0.0018 0.4998 0.0075 0.0065 0.0057 99. 9987\nfda-5 1.0001 0.0038 0.0035 0.0014 0.4998 0.0068 0.0059 0.0046 100 .0000\nn=500\ninit 1.0003 0.0180 0.0141 0.0323 0.5017 0.0268 0.0213 0.0720\nldaAR 1.0001 0.0155 0.0123 0.0241 0.5016 0.0232 0.0186 0.0541\nldaCS 1.0002 0.0171 0.0134 0.0291 0.5013 0.0251 0.0202 0.0629\nfda-1 1.0005 0.0229 0.0180 0.0523 0.5021 0.0346 0.0276 0.1201 59. 3104\nfda-2 1.0004 0.0177 0.0138 0.0314 0.5022 0.0268 0.0217 0.0724 87. 0183\nfda-3 1.0001 0.0173 0.0136 0.0300 0.5010 0.0249 0.0197 0.0620 99. 9983\nfda-4 1.0000 0.0042 0.0038 0.0017 0.5005 0.0075 0.0065 0.0056 99. 9992\nfda-5 1.0000 0.0038 0.0035 0.0015 0.5004 0.0066 0.0058 0.0044 100 .0000\n6Table S6: Performance of the estimation procedure where the residual s are generated with power exponential covariance function where scale\nparameter a0=1 and shape parameter b0=1. Mean of the estimated coe fficients, standard deviation, absolute bias, mean square err or (×100) and\nFVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 0.9968 0.0525 0.0423 0.2758 0.4961 0.0755 0.0603 0.5705\nldaAR 0.9985 0.0486 0.0387 0.2361 0.4962 0.0702 0.0562 0.4938\nldaCS 0.9978 0.0389 0.0309 0.1514 0.4986 0.0549 0.0438 0.3013\nfda-1 0.9960 0.0708 0.0564 0.5018 0.4951 0.1010 0.0813 1.0195 73. 1399\nfda-2 0.9975 0.0439 0.0347 0.1929 0.4980 0.0629 0.0496 0.3948 87. 5389\nfda-3 0.9975 0.0381 0.0305 0.1453 0.4987 0.0531 0.0425 0.2811 92. 2253\nfda-4 0.9978 0.0392 0.0311 0.1540 0.4977 0.0540 0.0434 0.2914 94. 4211\nfda-5 0.9978 0.0388 0.0302 0.1508 0.4975 0.0527 0.0420 0.2773 95. 6881\nfda-6 0.9979 0.0393 0.0306 0.1546 0.4977 0.0534 0.0424 0.2852 96. 5184\nfda-7 0.9990 0.0396 0.0308 0.1570 0.4986 0.0535 0.0423 0.2857 97. 0882\nfda-8 0.9985 0.0405 0.0317 0.1637 0.4981 0.0540 0.0429 0.2915 97. 5117\nn=300\ninit 0.9975 0.0297 0.0236 0.0885 0.4989 0.0428 0.0352 0.1832\nldaAR 0.9972 0.0281 0.0224 0.0793 0.4992 0.0389 0.0316 0.1509\nldaCS 0.9988 0.0226 0.0180 0.0513 0.4995 0.0300 0.0238 0.0899\nfda-1 0.9967 0.0391 0.0310 0.1539 0.4986 0.0588 0.0482 0.3456 73. 4971\nfda-2 0.9986 0.0254 0.0203 0.0648 0.4990 0.0335 0.0266 0.1122 87. 5190\nfda-3 0.9987 0.0221 0.0173 0.0490 0.4996 0.0293 0.0233 0.0859 92. 1425\nfda-4 0.9986 0.0219 0.0171 0.0479 0.4997 0.0294 0.0233 0.0862 94. 3152\nfda-5 0.9988 0.0213 0.0167 0.0456 0.5000 0.0287 0.0232 0.0823 95. 5616\nfda-6 0.9989 0.0213 0.0166 0.0454 0.5001 0.0289 0.0232 0.0832 96. 3760\nfda-7 0.9988 0.0212 0.0166 0.0449 0.5003 0.0291 0.0234 0.0843 96. 9441\nfda-8 0.9988 0.0212 0.0166 0.0449 0.5001 0.0292 0.0236 0.0852 97. 3641\nn=500\ninit 0.9996 0.0212 0.0169 0.0450 0.4989 0.0316 0.0252 0.0999\nldaAR 0.9993 0.0189 0.0151 0.0356 0.4983 0.0294 0.0237 0.0868\nldaCS 0.9996 0.0170 0.0135 0.0288 0.4998 0.0235 0.0193 0.0553\nfda-1 0.9996 0.0278 0.0222 0.0773 0.4984 0.0422 0.0333 0.1777 73. 6172\nfda-2 0.9992 0.0186 0.0150 0.0347 0.4995 0.0264 0.0216 0.0694 87. 5609\nfda-3 1.0002 0.0164 0.0130 0.0268 0.5001 0.0225 0.0183 0.0505 92. 1497\nfda-4 1.0001 0.0163 0.0129 0.0264 0.4999 0.0226 0.0184 0.0508 94. 3084\nfda-5 1.0000 0.0160 0.0127 0.0255 0.4996 0.0223 0.0182 0.0496 95. 5492\nfda-6 1.0000 0.0161 0.0128 0.0260 0.4995 0.0222 0.0182 0.0493 96. 3576\nfda-7 0.9999 0.0161 0.0128 0.0258 0.4994 0.0219 0.0179 0.0481 96. 9249\nfda-8 0.9999 0.0161 0.0128 0.0258 0.4994 0.0219 0.0177 0.0478 97. 3423\n7Table S7: Performance of the estimation procedure where the residual s are generated with power exponential covariance function where scale\nparameter a0=1 and shape parameter b0=2. Mean of the estimated coe fficients, standard deviation, absolute bias, mean square err or (×100) and\nFVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 0.9967 0.0563 0.0454 0.3170 0.4957 0.0811 0.0649 0.6591\nldaAR 0.9980 0.0506 0.0399 0.2562 0.4970 0.0724 0.0578 0.5234\nldaCS 0.9982 0.0373 0.0294 0.1389 0.4983 0.0531 0.0425 0.2814\nfda-1 0.9957 0.0766 0.0611 0.5872 0.4947 0.1094 0.0881 1.1964 85. 6976\nfda-2 0.9979 0.0440 0.0348 0.1934 0.4975 0.0633 0.0503 0.4010 98. 9964\nfda-3 0.9993 0.0239 0.0187 0.0569 0.4988 0.0343 0.0273 0.1177 99. 9585\nfda-4 0.9984 0.0224 0.0176 0.0505 0.5004 0.0305 0.0242 0.0931 99. 9987\nfda-5 0.9984 0.0224 0.0176 0.0505 0.5004 0.0305 0.0242 0.0931 100 .0000\nn=300\ninit 0.9973 0.0318 0.0253 0.1014 0.4987 0.0461 0.0378 0.2118\nldaAR 0.9972 0.0297 0.0239 0.0888 0.4992 0.0400 0.0323 0.1597\nldaCS 0.9990 0.0216 0.0172 0.0468 0.4995 0.0289 0.0229 0.0835\nfda-1 0.9964 0.0424 0.0336 0.1804 0.4984 0.0637 0.0521 0.4047 86. 0991\nfda-2 0.9987 0.0253 0.0201 0.0642 0.4990 0.0336 0.0266 0.1129 99. 0443\nfda-3 0.9992 0.0146 0.0114 0.0213 0.5003 0.0197 0.0159 0.0388 99. 9593\nfda-4 0.9993 0.0128 0.0100 0.0165 0.5000 0.0176 0.0139 0.0309 99. 9987\nfda-5 0.9993 0.0128 0.0100 0.0165 0.5000 0.0176 0.0139 0.0309 100 .0000\nn=500\ninit 0.9994 0.0226 0.0180 0.0512 0.4987 0.0340 0.0270 0.1153\nldaAR 0.9993 0.0201 0.0161 0.0403 0.4983 0.0309 0.0250 0.0954\nldaCS 0.9993 0.0163 0.0130 0.0266 0.4995 0.0227 0.0186 0.0517\nfda-1 0.9995 0.0301 0.0240 0.0906 0.4983 0.0456 0.0361 0.2081 86. 1915\nfda-2 0.9990 0.0186 0.0150 0.0346 0.4992 0.0265 0.0217 0.0699 99. 0585\nfda-3 1.0004 0.0111 0.0087 0.0122 0.5002 0.0148 0.0121 0.0219 99. 9596\nfda-4 1.0007 0.0100 0.0080 0.0100 0.5007 0.0129 0.0106 0.0168 99. 9987\nfda-5 1.0007 0.0100 0.0080 0.0100 0.5007 0.0129 0.0106 0.0168 100 .0000\n8Table S8: Performance of the estimation procedure where the residual s are generated with power exponential covariance function where scale\nparameter a0=1 and shape parameter b0=5. Mean of the estimated coe fficients, standard deviation, absolute bias, mean square err or (×100) and\nFVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 0.9970 0.0582 0.0468 0.3390 0.4952 0.0841 0.0676 0.7089\nldaAR 0.9977 0.0528 0.0415 0.2789 0.4961 0.0772 0.0621 0.5956\nldaCS 0.9996 0.0274 0.0218 0.0747 0.4979 0.0396 0.0321 0.1572\nfda-1 0.9956 0.0810 0.0646 0.6564 0.4943 0.1157 0.0933 1.3399 92. 2949\nfda-2 0.9995 0.0375 0.0296 0.1400 0.4966 0.0548 0.0441 0.3013 99. 7252\nfda-3 0.9993 0.0283 0.0203 0.0801 0.5004 0.0383 0.0274 0.1462 99. 8787\nfda-4 0.9991 0.0121 0.0053 0.0147 0.4995 0.0165 0.0073 0.0273 99. 9461\nfda-5 1.0001 0.0116 0.0022 0.0134 0.5005 0.0146 0.0030 0.0212 99. 9842\nn=300\ninit 0.9972 0.0328 0.0260 0.1082 0.4986 0.0482 0.0395 0.2317\nldaAR 0.9971 0.0310 0.0251 0.0968 0.4994 0.0426 0.0344 0.1810\nldaCS 0.9995 0.0156 0.0123 0.0244 0.4996 0.0216 0.0171 0.0467\nfda-1 0.9962 0.0449 0.0356 0.2027 0.4982 0.0673 0.0552 0.4524 92. 6741\nfda-2 0.9990 0.0213 0.0168 0.0454 0.4994 0.0291 0.0229 0.0846 99. 8076\nfda-3 0.9995 0.0196 0.0153 0.0384 0.4992 0.0271 0.0213 0.0736 99. 9391\nfda-4 1.0005 0.0075 0.0044 0.0056 0.4994 0.0105 0.0064 0.0111 99. 9716\nfda-5 0.9999 0.0023 0.0006 0.0005 0.5000 0.0037 0.0010 0.0014 99. 9889\nn=500\ninit 0.9994 0.0232 0.0185 0.0539 0.4985 0.0352 0.0280 0.1242\nldaAR 0.9989 0.0209 0.0169 0.0437 0.4981 0.0327 0.0264 0.1073\nldaCS 0.9991 0.0119 0.0095 0.0142 0.4992 0.0168 0.0134 0.0281\nfda-1 0.9995 0.0318 0.0254 0.1011 0.4981 0.0483 0.0382 0.2328 92. 7586\nfda-2 0.9988 0.0158 0.0127 0.0252 0.4988 0.0227 0.0183 0.0517 99. 8272\nfda-3 0.9989 0.0153 0.0123 0.0235 0.4989 0.0218 0.0176 0.0478 99. 9574\nfda-4 0.9997 0.0070 0.0050 0.0049 0.5002 0.0105 0.0072 0.0110 99. 9811\nfda-5 1.0000 0.0024 0.0007 0.0006 0.5001 0.0035 0.0011 0.0012 99. 9923\n9Table S9: Performance of the estimation procedure where the residual s are generated with rational quadratic covariance functio n where scale\nparameter a0=1 and shape parameter b0=1. Mean of the estimated coe fficients, standard deviation, absolute bias, mean square err or (×100) and\nFVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 0.9967 0.0565 0.0455 0.3193 0.4957 0.0814 0.0651 0.6624\nldaAR 0.9982 0.0507 0.0399 0.2567 0.4968 0.0725 0.0580 0.5257\nldaCS 0.9983 0.0349 0.0275 0.1215 0.4986 0.0495 0.0396 0.2444\nfda-1 0.9957 0.0774 0.0617 0.5994 0.4946 0.1105 0.0890 1.2213 87. 2296\nfda-2 0.9980 0.0413 0.0326 0.1706 0.4979 0.0594 0.0471 0.3521 98. 4366\nfda-3 0.9989 0.0265 0.0210 0.0704 0.4988 0.0379 0.0302 0.1434 99. 8285\nfda-4 0.9985 0.0271 0.0212 0.0733 0.4991 0.0372 0.0296 0.1383 99. 9800\nfda-5 0.9987 0.0259 0.0203 0.0671 0.4987 0.0350 0.0276 0.1224 99. 9977\nn=300\ninit 0.9973 0.0319 0.0253 0.1021 0.4987 0.0463 0.0381 0.2144\nldaAR 0.9971 0.0297 0.0239 0.0889 0.4992 0.0404 0.0327 0.1630\nldaCS 0.9991 0.0203 0.0161 0.0412 0.4995 0.0271 0.0215 0.0736\nfda-1 0.9964 0.0429 0.0339 0.1846 0.4984 0.0643 0.0527 0.4132 87. 6273\nfda-2 0.9988 0.0238 0.0190 0.0568 0.4991 0.0317 0.0251 0.1002 98. 4851\nfda-3 0.9991 0.0159 0.0125 0.0255 0.5002 0.0215 0.0173 0.0462 99. 8287\nfda-4 0.9991 0.0157 0.0121 0.0248 0.5001 0.0214 0.0171 0.0457 99. 9800\nfda-5 0.9993 0.0144 0.0113 0.0209 0.5005 0.0202 0.0164 0.0408 99. 9977\nn=500\ninit 0.9995 0.0227 0.0181 0.0514 0.4987 0.0341 0.0271 0.1160\nldaAR 0.9993 0.0200 0.0160 0.0399 0.4982 0.0310 0.0250 0.0962\nldaCS 0.9994 0.0153 0.0122 0.0235 0.4996 0.0213 0.0174 0.0454\nfda-1 0.9995 0.0304 0.0243 0.0924 0.4982 0.0461 0.0365 0.2125 87. 7225\nfda-2 0.9991 0.0176 0.0142 0.0310 0.4993 0.0249 0.0204 0.0620 98. 5022\nfda-3 1.0003 0.0121 0.0096 0.0146 0.5002 0.0163 0.0133 0.0265 99. 8293\nfda-4 1.0005 0.0120 0.0096 0.0145 0.5004 0.0160 0.0131 0.0256 99. 9801\nfda-5 1.0002 0.0114 0.0090 0.0129 0.5000 0.0151 0.0121 0.0227 99. 9977\n10Table S10: Performance of the estimation procedure where the residual s are generated with rational quadratic covariance functio n where scale\nparameter a0=1 and shape parameter b0=2. Mean of the estimated coe fficients, standard deviation, absolute bias, mean square err or (×100) and\nFVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 0.9967 0.0547 0.0441 0.2993 0.4960 0.0788 0.0628 0.6206\nldaAR 0.9983 0.0498 0.0393 0.2473 0.4968 0.0714 0.0571 0.5091\nldaCS 0.9977 0.0425 0.0337 0.1805 0.4981 0.0602 0.0481 0.3618\nfda-1 0.9957 0.0730 0.0583 0.5343 0.4951 0.1042 0.0839 1.0868 78. 3365\nfda-2 0.9972 0.0479 0.0381 0.2297 0.4975 0.0687 0.0544 0.4721 96. 4030\nfda-3 0.9981 0.0368 0.0293 0.1358 0.4982 0.0520 0.0418 0.2699 99. 4758\nfda-4 0.9982 0.0377 0.0299 0.1424 0.4979 0.0528 0.0421 0.2791 99. 9264\nfda-5 0.9986 0.0356 0.0280 0.1266 0.4975 0.0483 0.0384 0.2335 99. 9902\nn=300\ninit 0.9974 0.0309 0.0246 0.0959 0.4988 0.0445 0.0365 0.1977\nldaAR 0.9972 0.0290 0.0233 0.0848 0.4991 0.0393 0.0317 0.1542\nldaCS 0.9987 0.0246 0.0195 0.0606 0.4994 0.0327 0.0259 0.1066\nfda-1 0.9966 0.0403 0.0320 0.1633 0.4986 0.0607 0.0497 0.3682 78. 7132\nfda-2 0.9985 0.0276 0.0220 0.0764 0.4989 0.0364 0.0290 0.1327 96. 4278\nfda-3 0.9985 0.0218 0.0171 0.0477 0.4999 0.0290 0.0232 0.0842 99. 4738\nfda-4 0.9985 0.0218 0.0171 0.0478 0.4998 0.0293 0.0235 0.0858 99. 9259\nfda-5 0.9988 0.0198 0.0157 0.0394 0.5004 0.0271 0.0224 0.0733 99. 9900\nn=500\ninit 0.9995 0.0221 0.0176 0.0490 0.4989 0.0330 0.0263 0.1086\nldaAR 0.9993 0.0197 0.0158 0.0389 0.4983 0.0302 0.0245 0.0910\nldaCS 0.9994 0.0184 0.0146 0.0339 0.4996 0.0257 0.0211 0.0658\nfda-1 0.9996 0.0288 0.0229 0.0826 0.4984 0.0435 0.0344 0.1892 78. 8101\nfda-2 0.9991 0.0201 0.0162 0.0405 0.4993 0.0286 0.0235 0.0819 96. 4486\nfda-3 1.0003 0.0162 0.0128 0.0262 0.5000 0.0222 0.0181 0.0491 99. 4752\nfda-4 1.0003 0.0162 0.0129 0.0263 0.4999 0.0224 0.0182 0.0501 99. 9261\nfda-5 0.9999 0.0151 0.0120 0.0227 0.4994 0.0208 0.0170 0.0434 99. 9900\n11Table S11: Performance of the estimation procedure where the residual s are generated with rational quadratic covariance functio n where scale\nparameter a0=1 and shape parameter b0=5. Mean of the estimated coe fficients, standard deviation, absolute bias, mean square err or (×100) and\nFVE in percentage are summarized.\nMethod β1 β2 FVE %-age\nMean SD AB MSE Mean SD AB MSE\nn=100\ninit 0.9967 0.0504 0.0407 0.2545 0.4965 0.0725 0.0577 0.5251\nldaAR 0.9986 0.0466 0.0369 0.2173 0.4965 0.0671 0.0539 0.4508\nldaCS 0.9970 0.0473 0.0377 0.2238 0.4978 0.0668 0.0531 0.4458\nfda-1 0.9959 0.0648 0.0517 0.4205 0.4960 0.0922 0.0742 0.8505 62. 1253\nfda-2 0.9964 0.0505 0.0405 0.2561 0.4976 0.0721 0.0572 0.5199 89. 2976\nfda-3 0.9970 0.0462 0.0368 0.2136 0.4974 0.0644 0.0513 0.4142 97. 4903\nfda-4 0.9981 0.0464 0.0363 0.2149 0.4957 0.0647 0.0520 0.4201 99. 4795\nfda-5 0.9986 0.0441 0.0345 0.1941 0.4952 0.0621 0.0497 0.3872 99. 9012\nn=300\ninit 0.9977 0.0286 0.0229 0.0820 0.4991 0.0406 0.0332 0.1646\nldaAR 0.9975 0.0271 0.0215 0.0737 0.4991 0.0367 0.0296 0.1346\nldaCS 0.9983 0.0272 0.0216 0.0739 0.4994 0.0362 0.0288 0.1309\nfda-1 0.9972 0.0355 0.0283 0.1264 0.4991 0.0539 0.0441 0.2896 62. 2661\nfda-2 0.9983 0.0289 0.0229 0.0835 0.4990 0.0384 0.0308 0.1473 89. 2161\nfda-3 0.9981 0.0266 0.0209 0.0712 0.4992 0.0355 0.0282 0.1257 97. 4664\nfda-4 0.9981 0.0260 0.0205 0.0677 0.4992 0.0351 0.0278 0.1227 99. 4727\nfda-5 0.9984 0.0244 0.0192 0.0594 0.4996 0.0332 0.0267 0.1097 99. 8990\nn=500\ninit 0.9996 0.0207 0.0165 0.0430 0.4992 0.0304 0.0245 0.0921\nldaAR 0.9993 0.0187 0.0151 0.0351 0.4985 0.0281 0.0229 0.0792\nldaCS 0.9996 0.0201 0.0160 0.0404 0.4997 0.0282 0.0230 0.0792\nfda-1 0.9997 0.0256 0.0204 0.0654 0.4988 0.0385 0.0304 0.1484 62. 3377\nfda-2 0.9992 0.0210 0.0168 0.0440 0.4995 0.0299 0.0244 0.0892 89. 2460\nfda-3 1.0001 0.0193 0.0153 0.0372 0.4999 0.0270 0.0219 0.0728 97. 4696\nfda-4 0.9997 0.0188 0.0150 0.0355 0.4993 0.0269 0.0219 0.0724 99. 4726\nfda-5 0.9994 0.0180 0.0142 0.0322 0.4989 0.0260 0.0211 0.0674 99. 8988\n12"    },    {        "title": "2402.13921v1.Robust_recovery_for_stochastic_block_models__simplified_and_generalized.pdf",        "content": "arXiv:2402.13921v1  [cs.DS]  21 Feb 2024Robust recovery for stochastic block models, simplified and\ngeneralized\nSidhanth Mohanty/asterisk.mathPrasad Raghavendra†David X. Wu‡\nFebruary 22, 2024\nAbstract\nWe study the problem of robust community recovery : efficiently recovering communities in\nsparse stochastic block models in the presence of adversari al corruptions. In the absence of\nadversarial corruptions, there are efficient algorithms whe n the signal-to-noise ratio exceeds\ntheKesten–Stigum (KS) threshold , widely believed to be the computational threshold for this\nproblem. The question we study is: does the computational threshold for robust community reco very\nalso lie at the KS threshold? We answer this question affirmatively, providing an algorith m for\nrobust community recovery for arbitrary stochastic block m odels on any constant number of\ncommunities, generalizing the work of Ding, d’Orsi, Nasser & Steurer [ DdNS22 ] on an efficient\nalgorithm above the KS threshold in the case of 2-community b lock models.\nThere are three main ingredients to our work:\n1. The Bethe Hessian of the graph is defined as /u1D43B/u1D43A(/u1D461)/defines(/u1D437/u1D43A−/u1D43C)/u1D4612−/u1D434/u1D43A/u1D461+/u1D43Cwhere/u1D437/u1D43Ais the\ndiagonal matrix of degrees and /u1D434/u1D43Ais the adjacency matrix. Empirical work suggested that\nthe Bethe Hessian for the stochastic block model has outlier eigenvectors corresponding\nto the communities right above the Kesten-Stigum threshold [KMM+13,SKZ14 ].\nWe formally confirm the existence of outlier eigenvalues for the Bethe Hessian,by explicitly\nconstructing outlier eigenvectors from the community vect ors.\n2. We develop an algorithm for a variant of robust PCA on spars e matrices. Specifically,\nan algorithm to partially recover top eigenspaces from adve rsarially corrupted sparse\nmatrices under mild delocalization constraints.\n3. A rounding algorithm to turn vector assignments of vertic es into a community assignment,\ninspired by the algorithm of Charikar & Wirth [ CW04 ] for 2XOR.\n/asterisk.mathMIT.sidhanth@csail.mit.edu . Much of this work was done while the author was a PhD student a t UC Berkeley.\n†UC Berkeley.raghavendra@berkeley.edu .\n‡UC Berkeley.david_wu@berkeley.edu .Contents\n1 Introduction 1\n1.1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n2 Technical overview 4\n2.1 Outlier eigenvectors for the Bethe Hessian . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2.2 Robust PCA for sparse matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2.3 Rounding to communities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3 Preliminaries 9\n4 Recovery algorithm 10\n4.1 Analysis of algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n5 The number of outlier eigenvalues 13\n5.1 Lower bound on the number of outlier eigenvalues . . . . . . . . . . . . . . . . . . . . 14\n5.2 Outlier eigenspace after degree truncation . . . . . . . . . . . . . . . . . . . . . . . . . 18\n6 Robust recovery of a subspace 20\n6.1 Proof of Item 1: spectrum of cleaned-up matrix . . . . . . . . . . . . . . . . . . . . . . 21\n6.2 Proofs of Items 2 and 3: delocalization and correlation w ith recovered subspace . . . 22\n6.3 Correlation of subspace with communities . . . . . . . . . . . . . . . . . . . . . . . . 24\n7 Rounding algorithm 25\nA On defining weak-recovery 29\nA.1 Interpreting our new notion of weak recovery . . . . . . . . . . . . . . . . . . . . . . . 30\nA.2 Comparing the different notions of weak recovery . . . . . . . . . . . . . . . . . . . . 321 Introduction\nThe stochastic block model (SBM) has provided an enlightenin g lens into understanding a wide\nrange of computational phenomena in Bayesian inference pro blems, such as computational phase\ntransitions & information-computation gaps [ DKMZ11 ,MNS18 ,Mas14 ,HS17 ], spectral methods\nfor sparse matrices [ KMM+13,SKZ14 ,BLM15 ], local message-passing algorithms [ MNS14 ,YP23 ,\nGP23 ], and robustness [ MPW16 ,BMR21 ,DdNS22 ,LM22 ].\nThe SBM is a model of random graphs where the vertices are part itioned into communities ,\ndenoted by /u1D499, and the probability of an edge existing is contingent on the communities that the\ntwo endpoints are part of. The algorithmic task is the community recovery problem: given an input\ngraph/u1D46E, estimate the posterior /u1D499|/u1D46Ewith an efficient algorithm.\nDefinition 1.1 (Informal) .In the stochastic block model , we are given a /u1D458×/u1D458matrix M, a distribution\n/u1D>0Bover[/u1D458], and/u1D451>0, and SBM /u1D45B(M,/u1D>0B,/u1D451)denotes the distribution where an /u1D45B-vertex graph /u1D46Eis\nsampled by:\n1. drawing a color/u1D499(/u1D462) ∼/u1D>0Bfor every/u1D462∈ [/u1D45B],\n2. for each pair of vertices /u1D462,/u1D463, the edge {/u1D462,/u1D463}is chosen with probability M /u1D499(/u1D462),/u1D499(/u1D463)·/u1D451\n/u1D45B.\nIn the community recovery problem, the goal is to give an efficient algorithm that takes /u1D46Eas input and\noutputs a community assignment /hatwide/u1D465approximating /u1D499|/u1D46E(seeDefinition 4.4 for a formal definition).\nComputational thresholds. For a given M and /u1D>0B, increasing /u1D451can only possibly make the problem\neasier. The main question is to understand the computational threshold for community recovery —\ni.e. the minimum value of /u1D451where the problem goes from being intractable to admitting e fficient\nalgorithms.\nThe first predictions for this computational threshold came f rom the cavity method in statistical\nphysics in the work of Decelle, Krzakala, Moore & Zdeborova [ DKMZ11 ]. They posited that the\nlocation of this transition is at the Kesten–Stigum threshold (henceforth KS threshold), a threshold\nfor broadcast processes on trees studied in the works of Kest en & Stigum [ KS66 ,KS67 ]. The\nalgorithmic side of these predictions was confirmed in the cas e of the 2-community block model in\nthe works of Mossel, Neeman & Sly [ MNS18 ] and Massoulié [ Mas14 ], and then for block models\nin increasing levels of generality by Bordenave, Lelarge & Ma ssoulié [ BLM15 ], Abbe & Sandon\n[AS15 ], and Hopkins & Steurer [ HS17 ].\nRobust algorithms. All of these algorithms utilize the knowledge of the distrib ution the input is\nsampled from quite strongly — they are based on Ω(log/u1D45B)-length walk statistics in the stochastic\nblock model. However, the full generative process in infere nce is not always known precisely.\nThus, we would like algorithms that utilize but do not overfit t o the distributional assumptions.\nDemanding that our algorithm be robust , i.e. resilient to adversarial corruptions to the input, is\noften a useful way to design algorithms that are less sensiti ve to distributional assumptions. This\nleads one to wonder: can algorithms that don’t strongly expl oit the prior distribution approach\nthe KS threshold?\nOptimization vs. inference. Earlier approaches to robust recovery in 2-community block models\nwere based on optimization : semidefinite programming relaxations of the minimum bisect ion\nproblem, as in the work of Guedon & Vershynin [ GV16 ]. These approaches have the advantage of\n1being naturally robust, since the algorithms are approxima tely Lipschitz around random inputs,\nbut the minimum bisection relaxation is not known to achieve statistical optimality and only\nsucceeds well above the KS threshold.\nThe following two results point to the suboptimality of opti mization-based strategies. Moitra,\nPerry & Wein [ MPW16 ] considered the monotone adversary in the 2-community setting, where\nthe adversary is allowed to make an unbounded number of edge insertions within communities\nand edge deletions across communities. At an intuitive leve l, this is supposed to only make\nthe problem easier and indeed does so for the minimum bisecti on approach, but to the contrary\n[MPW16 ] proves that the threshold for recovery increases . Dembo, Montanari & Sen [ DMS17 ]\nexactly nailed the size of the minimum bisection in Erdős–Ré nyi graphs, which are complete noise\nand have no signal in the form of a planted bisection — and stri kingly, it is actually smaller than\nthe size of the planted bisection in the detectable regime! T hus, it is conceivable that there are\nbisections completely orthogonal to the planted bisection in a stochastic block model graph that\nnevertheless have the same size.\nThe problem of recovering communities is more related to the task of Bayesian inference , i.e.,\napplying Bayes’ rule and approximating /u1D499|/u1D46E. Optimizing for the minimum bisection is akin to\ncomputing the maximum likelihood estimate , which does not necessarily produce samples represen-\ntative of the posterior distribution of /u1D499|/u1D46E.\nSDPs for inference. The work of Banks, Mohanty & Raghavendra [ BMR21 ] proposed a semidef-\ninite programming-based algorithm for inference tasks tha t incorporates the prior distribution in\nthe formulation, and illustrated that this algorithm can di stinguish between /u1D46Esampled from the\nstochastic block model from an Erdős–Rényi graph of equal av erage degree anywhere above the\nKS threshold while being resilient to Ω(/u1D45B)arbitrary edge insertions and deletions.\nA similar SDP formulation was later studied by Ding, d’Orsi, Nasser & Steurer [ DdNS22 ] in\nthe 2-community setting, and was used to give an algorithm to recover the communities with a\nconstant advantage over random guessing in the presence of Ω(/u1D45B)edge corruptions for all degrees\nabove the KS threshold. They analyze the spectra of matrices a ssociated with random graphs after\ndeleting vertices with large neighborhoods, which introdu ces unfriendly correlations, and causes\ntheir analysis to be highly technical.\nThe main contribution of our work is an algorithm for robust r ecovery, which is amenable to a\nsignificantly simpler analysis. Our algorithm also succeeds at the recovery task for arbitrary block\nmodels with a constant number of communities.\nTheorem 1.2 (Informal statement of main theorem) .Let(M,/u1D>0B,/u1D451)be SBM parameters such that /u1D451is\nabove the KS threshold, and let /u1D46E,/u1D499∼SBM/u1D45B(M,/u1D>0B,/u1D451). There exists /u1D6FF=/u1D6FF(M,/u1D>0B,/u1D451)>0such that the\nfollowing holds. There is a polynomial time algorithm that t akes as input any graph /tildewide/u1D46Ethat can be obtained\nby performing arbitrary /u1D6FF/u1D45Bedge insertions and deletions to /u1D46Eand outputs a coloring /hatwide/u1D499that has “constant\ncorrelation” with /u1D499, with high probability over the randomness of /u1D46Eand/u1D499.\nMany of the ingredients in the above result are of independent interest. First, we exhibit a\nsymmetric matrix closely related to the Bethe Hessian of the graph, such that its bottom eigenspace\nis correlated with the communities. Next, we design an efficie nt algorithm to robustly recover the\nbottom-/u1D45Feigenspace of a sparse matrix in the presence of adversarial corruptions. Finally , we\ndemonstrate a general rounding scheme to obtain community a ssignments from this eigenspace.\n2Remark 1.3 (Robustness against node corruptions) .The node corruption model, introduced by\nLiu & Moitra [ LM22 ], is a harsher generalization of the edge corruption model. In recent work,\nDing, d’Orsi, Hua & Steurer [ DdHS23 ] proved that in the setting of sparse SBM, any algorithm\nthat is robust to edge corruptions can be turned into one robu st to node corruptions in a blackbox\nmanner. Hence, our results apply in this harsher setting too .\n1.1 Related work\nWe refer the reader to the survey of Abbe [ Abb17 ] for a detailed treatment of the rich history and\nliterature on community detection in block models, its stud y in other disciplines, and the many\ninformation-theoretic and computational results in vario us parameter regimes.\nIntroducing an adversary into the picture provides a beacon towards algorithms that utilize\nbut do not overfit to distributional assumptions. Over the years, a variety of adversarial models\nhave been considered, some of which we survey below.\nCorruption models for stochastic block model. Prior to the works of [ BMR21 ,DdNS22 ], Stefan &\nMassoulié [ SM19 ] considered the robust recovery problem, and gave a robust s pectral algorithm\nto recover communities under /u1D442(/u1D45B/u1D>00)adversarial edge corruptions for some small enough /u1D>00 >0.\nLiu & Moitra [ LM22 ] introduced the node corruption model where an adversary gets to perform\narbitrary edge corruptions incident to a constant fraction of corrupted vertices, and gave algorithms\nthat achieved optimal accuracy in the presence of node corru ptions and the monotone adversary\nsufficiently above the KS threshold. Soon after, Ding, d’Orsi, Hua & Steurer [ DdHS23 ] gave\nalgorithms achieving the Kesten–Stigum threshold using al gorithms for the edge corruption model\nin the low-degree setting [ DdNS22 ], and results on the optimization SDP in the high-degree set ting\n[MS16 ] in a blackbox manner.\nSemirandom & smoothed models. Some works have considered algorithm design under harsher\nadversarial models, where an adversarially chosen input un dergoes some random perturbations.\nRemarkably, at this point, the best algorithms for several g raph and hypergraph problems match\nthe performance of our best algorithms for their completely random counterparts. For example,\nat this point, the semirandom planted coloring and clique pr oblems were introduced by Blum &\nSpencer [ BS95 ], and Feige & Kilian [ FK01 ], and a line of work [ CSV17 ,MMT20 ] culminating in\nthe work of Buhai, Kothari & Steurer [ BKS23 ] showed that the size of the planted clique/coloring\nrecoverable in the semirandom setting matches the famed√/u1D45Bin the fully random setting.\nAnother example where algorithms for a semirandom version o f a block model-like problem\nhave been considered is semirandom CSPs with planted soluti ons, where the work of Guruswami,\nHsieh, Kothari & Manohar [ GHKM23 ] gives algorithms matching the guarantees of solving fully\nrandom planted CSPs.\n1.2 Organization\nInSection 2 , we give an overview of our algorithm and proof. In Section 3 , we give some technical\npreliminaries. In Section 4 , we describe our algorithm and show how to analyze it. In Section 5 , we\nprove results about the spectrum of the Bethe Hessian matrix and variants, key to our algorithm.\nInSection 6 , we show how to use a trimming procedure and a spectral algori thm on a Bethe\nHessian variant to recover a subspace which has good correla tion with the true communities in the\n3presence of adversarial corruptions. Finally, in Section 7 , we prove some technical claims relevant\nto analyzing our algorithm.\n2 Technical overview\nAn/u1D45B-vertex graph /u1D46Eis drawn from a stochastic block model and undergoes /u1D6FF/u1D45Badversarial edge\ncorruptions, and then the corrupted graph /tildewide/u1D46Eis given to us as input. For simplicity of discussion, we\nrestrict our attention to assortative symmetric/u1D458-community block models above the KS threshold,\ni.e. the connection probability between two vertices /u1D456and/u1D457only depends on whether they belong\nto the same community or different communities, and the intra -community probability is higher.\nNevertheless, our approach generalizes to any arbitrarily specified/u1D458-community block model\nabove the KS threshold.\nLet us first informally outline the algorithm; see Section 4 for formal details.\n1. First, we preprocess the corrupted graph /tildewide/u1D46Eby truncating high degree vertices, which removes\ncorruptions localized on small sets of vertices in the graph .\n2. We then construct an appropriately defined graph-aware sym metric matrix /u1D440/u1D46E∈R/u1D45B×/u1D45B\nwhose negative eigenvalues contains information about the true communities for the uncor-\nrupted graph. We motivate this construction in Section 2.1 .\n3. We recursively trim the rows and columns of /u1D440/tildewide/u1D46Eto remove small negative eigenvalues in its\nspectrum. Then we use a spectral algorithm to robustly recov er a subspace /u1D448which contains\ninformation about the communities. Both points are describ ed in Section 2.2 .\n4. Finally, we round the subspace /u1D448into a community assignment, using a vertex embedding\nprovided by /u1D448. This is detailed in Section 2.3 .\n2.1 Outlier eigenvectors for the Bethe Hessian\nBordenave, Lelarge & Massoulié [ BLM15 ] analyzed the spectrum of the nonbacktracking matrix and\nrigorously established its connection to community detect ion. The asymmetric nonbacktracking\nmatrix/u1D435/u1D43A∈{0,1}2|/u1D438(/u1D43A)|×2|/u1D438(/u1D43A)|is indexed by directed edges, with\n(/u1D435/u1D43A)(/u1D4621→/u1D4631),(/u1D4622→/u1D4632)/defines1[/u1D4631=/u1D4622]1[/u1D4632≠/u1D4621].\n[BLM15 ] showed that above the KS threshold, the /u1D458outlier eigenvalues for /u1D435/u1D46Ecorrespond to the /u1D458\ncommunity vectors. More precisely, in the case of symmetric /u1D458-community stochastic block models\nabove the KS threshold, [ BLM15 ] proved that for the randomly drawn graph /u1D46E, there is a small\n/u1D>00 >0 for which its nonbacktracking matrix /u1D435/u1D46Ehas exactly /u1D458eigenvalues larger than (1+/u1D>00)√\n/u1D451in\nmagnitude ( Theorem 5.3 ).\nTheBethe Hessian matrix is a symmetric matrix associated with a graph, whose e arly appear-\nances can be traced to the works of Ihara [ Iha66 ] and Bass [ Bas92 ]. The Bethe Hessian of a graph\nwith parameter /u1D461∈Ris defined as\n/u1D43B/u1D43A(/u1D461)/defines(/u1D437/u1D43A−/u1D43C)/u1D4612−/u1D434/u1D43A/u1D461+/u1D43C,\n4where/u1D437/u1D43Aand/u1D434/u1D43Aare the diagonal degree matrix and adjacency matrix of /u1D43A, respectively. For /u1D461in\nthe interval [0,1], it can be interpreted as a regularized version of the standa rd graph Laplacian.\nThe Bethe Hessian for the stochastic block model was conside red in the empirical works [ KMM+13,\nSKZ14 ], where they observed that for some choice of /u1D461, the Bethe Hessian and the nonbacktracking\nmatrix has outlier eigenvectors which can be used for finding c ommunities in block models.\nConcretely, in [ SKZ14 ] they observed that for /u1D46Edrawn from stochastic block models above the KS\nthreshold, there is a choice of /u1D461such that/u1D43B/u1D46E(/u1D461)only has a small number of negative eigenvectors,\nall of which correlate with the hidden community assignment .\nWe confirm this empirical observation in the following propos ition.\nProposition 2.1 (Bethe Hessian spectrum) .Let(M,/u1D>0B,/u1D451)be/u1D458-community SBM parameters such that /u1D451\nis above the KS threshold, and let /u1D46E,/u1D499∼SBM/u1D45B(M,/u1D>0B,/u1D451). Then there exists /u1D>00 >0such that for /u1D461∗=1\n(1+/u1D>00)√\n/u1D451,\nthe Bethe Hessian /u1D43B/u1D46E(/u1D461∗)has at most/u1D458negative eigenvalues and at least /u1D458−1negative eigenvalues.\nConstructing the outlier eigenspace There are two assertions in Proposition 2.1 . To show that\n/u1D43B/u1D46E(/u1D461∗)has at most /u1D458negative eigenvalues, one can relate these negative eigenv alues to the /u1D458outlier\neigenvalues of /u1D435/u1D46Eusing an Ihara–Bass argument and use a continuity argument a s outlined in Fan\nand Montanari [ FM17 , Theorem 5.1].\nThe more interesting direction is to exhibit at least /u1D458−1 negative eigenvalues; we will explicitly\nconstruct a /u1D458−1 dimensional subspace starting with the community vectors to witness the negative\neigenvalues for /u1D43B/u1D43A(/u1D461∗).\nLet1/u1D450denote the indicator vector for the vertices belonging to co mmunity/u1D450and1the all-ones\nvector. We show that every vector in the span of {/u1D434(ℓ)(1/u1D450−1\n/u1D4581)}/u1D450∈[/u1D458]achieves a negative quadratic\nform against /u1D43B/u1D46E(/u1D461∗), where/u1D434(ℓ)is the/u1D45B×/u1D45Bmatrix where the (/u1D456,/u1D457)-th entry encodes the number of\nlength-ℓnonbacktracking walks between /u1D456and/u1D457. This demonstrates a (/u1D458−1)-dimensional subspace\non which the quadratic form is negative. Formally, we show th e following:.\nProposition 2.2. Under the same setting and notations as Proposition 2.1 , forℓ/greaterorequalslant0define\n/u1D440/u1D46E,ℓ/defines/u1D434(ℓ)/u1D43B/u1D46E(/u1D461∗)/u1D434(ℓ).\nForℓ= Θ/parenleftBig\nlog(1//u1D>00)\n/u1D>00/parenrightBig\nand every/u1D450∈ [/u1D458], we have\n/angbracketleftbig\n1/u1D450−1\n/u1D4581,/u1D440/u1D46E,ℓ(1/u1D450−1\n/u1D4581)/angbracketrightbig/lessorequalslant−Ω(/u1D45B).\nHence,/u1D440/u1D46E,ℓhas at most/u1D458negative eigenvalues and at least /u1D458−1negative eigenvalues.\nNonbacktracking powers and related constructions were pre viously studied in [ Mas14 ,MNS18 ],\nbut there they take ℓ= Θ(log/u1D45B), whereas we only consider constant ℓ. Besides simplifying the\nanalysis of the quadratic form, using constant ℓis also critical for tolerating up to Ω(/u1D45B)corruptions.\nAs a consequence of Proposition 2.2 , the negative eigenvectors of /u1D440/u1D46E,ℓare correlated with\nthe centered community indicators {1/u1D450−1\n/u1D4581}/u1D450∈[/u1D458], while the negative eigenvectors of /u1D43B/u1D46E(/u1D461∗)are\ncorrelated with {/u1D434(ℓ)(1/u1D450−1\n/u1D4581)}/u1D450∈[/u1D458]. The upshot is that we can directly use the negative eigenvec tors\nof/u1D440/u1D46E,ℓto recover the true communities in the absence of corruption s.\n5Remark 2.3. Based on the empirical observations in [ KMM+13,SKZ14 ], a natural hope is to\ndirectly use the Bethe Hessian for recovery. However, it tur ns out that the quadratic form of the\ncentered true community indicators/angbracketleftbig\n(1/u1D450−1\n/u1D4581),/u1D43B/u1D46E(/u1D461∗)(1/u1D450−1\n/u1D4581)/angbracketrightbig\nare actually positive close to the\nKS threshold, so the same approach does not establish that the negative eigenvectors of /u1D43B/u1D46E(/u1D461∗)\ncorrelate with the communities.\nWe will now discuss how to recover the outlier eigenspace in t he presence of adversarial\ncorruptions.\n2.2 Robust PCA for sparse matrices\nThe discussion above naturally leads to the following algor ithmic problem of robust recovery:\nGiven as input a corrupted version /tildewide/u1D440of a symmetric matrix /u1D440, can we recover the bottom/top\n/u1D45F-dimensional eigenspace of /u1D440? Since the true communities are constantly correlated with the\noutlier eigenspace of /u1D440=/u1D440/u1D46E,ℓ, recovering the outlier eigenspace of /u1D440from its corrupted version\n/tildewide/u1D440=/tildewide/u1D440/u1D46E,ℓis a major step towards robustly recovering communities.\nThe problem of robustly recovering the top eigenspace, a.k. a. robust PCA has been extensively\nstudied, and algorithms with provable guarantees have been designed (see [ CLMW11 ]). However,\nthe robust PCA problem in our work is distinct from those cons idered in the literature in a couple\nof ways. For us, the uncorrupted matrix /u1D440is sparse and both the magnitude and location of\nthe noisy entries are adversarial. Furthermore, for our pur poses, we need not recover the actual\noutlier eigenspace of /u1D440. Indeed, as we discuss below, it suffices to robustly recover a constant\ndimensional subspace which is constantly correlated with t he true communities.\nWe design an efficient algorithm to robustly recover such a sub space under a natural set of\nsufficient conditions on /u1D440. Before we describe these conditions, let us fix some notation . We will\ncall a vector /u1D465∈R/u1D45Bto be/u1D436-delocalized if no coordinate is large relative to others, i .e.,|/u1D465/u1D456|2/lessorequalslant/u1D436\n/u1D45B/bardbl/u1D465/bardbl2\nfor all/u1D456∈ [/u1D45B]. Delocalization has previously been used in the robust PCA l iterature under the\nname “incoherence” [ CLMW11 ].\nLet/u1D440be a/u1D45B×/u1D45Bmatrix with at most /u1D45Fnegative eigenvalues. In particular, the /u1D45F-dimensional\nnegative eigenspace /u1D449/u1D440of/u1D440is the object of interest. Let /tildewide/u1D440be a corrupted version of /u1D440, differing\nfrom/u1D440in/u1D6FF/u1D45Bcoordinates.\nGiven the corrupted version /tildewide/u1D440, a natural goal would be to recover the /u1D45F-dimensional negative\neigenspace/u1D449/u1D440. It is easy to see that it could be impossible to recover the sp ace/u1D449/u1D440. Instead, we will\nsettle for a relaxed goal, namely, recover a slightly larger dimensional subspace /u1D448that non-trivially\ncorrelates with delocalized vectors in the true eigenspace /u1D449/u1D440. More formally, we will solve the\nfollowing problem.\nProblem 2.4. Given the corrupted matrix /tildewide/u1D440as input, give an efficient algorithm to output a\nsubspace/u1D448with the following properties:\n1.Low dimensional. The dimension of /u1D448is/u1D442(/u1D45F).\n2.Delocalized. The diagonal entries of its projection matrix Π/u1D448are bounded by /u1D442/parenleftbig/u1D45F\n/u1D45B/parenrightbig.\n3.Preserves delocalized part of negative eigenspace. For any/u1D436-delocalized unit vector /u1D466such\nthat/a\\}bracketle{t/u1D466,/u1D440/u1D466/a\\}bracketri}ht<−Ω(1), we have /a\\}bracketle{t/u1D466,Π/u1D448/u1D466/a\\}bracketri}ht/greaterorequalslantΩ(1).\n6In fact, our algorithm will recover a principal submatrix of /tildewide/u1D440whose eigenspace /u1D449for eigen-\nvalues less than −/u1D>02is/u1D442(/u1D45F)-dimensional. Moreover, the eigenspace /u1D449can be processed to another\ndelocalized, /u1D442(/u1D45F)-dimensional subspace /u1D448that satisfies the conditions outlined above.\nAlthough the matrix /u1D440has a constant number of negative eigenvalues, its corrupti on/tildewide/u1D440can\nhave up to Ω(/u1D45B)many. At first glance, it may be unclear how a constant dimensio nal subspace /u1D448\ncan be extracted from /tildewide/u1D440. The crucial observation is that the large negative eigenva lues introduced\nby the corruptions are highly localized. Thus, we will desig n an iterative trimming algorithm that\naims to delete rows and columns to clean up these localized co rruptions. When the algorithm\nterminates, it yields the /u1D442(/u1D45F)-dimensional subspace /u1D449.\nRecovering a principal submatrix. We now describe the trimming algorithm informally and refer\nthe reader to Section 6 for the formal details.\nWe first fix some small parameter /u1D>02 >0 and execute the following procedure, which produces\na series of principal submatrices /tildewide/u1D440(/u1D461)for/u1D461/greaterorequalslant0, starting with /tildewide/u1D440(0)/defines/tildewide/u1D440.\n1. At step/u1D461, if the eigenspace /u1D449of eigenvalues of /tildewide/u1D440(/u1D461)less than −/u1D>02is/u1D442(/u1D45F)-dimensional, we\nterminate the algorithm and output /u1D449.\n2. Otherwise, compute the projection Π(/u1D461)corresponding to the /lessorequalslant−/u1D>02eigenspace of /tildewide/u1D440(/u1D461).\n3. Sample an index /u1D456∈ [/u1D45B]of/tildewide/u1D440(/u1D461)with probability proportional to Π(/u1D461)\n/u1D456,/u1D456.\n4. Zero out row and column /u1D456, and set this new principal submatrix as /tildewide/u1D440(/u1D461+1).\nWe now discuss the intuition behind the procedure and formal ly prove its correctness in\nSection 6.1 . The main idea of step 3 is that one should prefer to delete hig hly localized eigenvectors\nwhich have relatively large negative eigenvalues. This is r easonable because the size of the diagonal\nentries of /tildewide/u1D440(/u1D461)serve as a rough proxy for the level of delocalization.\nAs a concrete illustration of this intuition, suppose that /tildewide/u1D440= Π(0)=−/u1D462/u1D462⊤−/u1D463/u1D463⊤, where/u1D462,/u1D463\nare orthogonal unit vectors. Moreover, suppose /u1D462is/u1D436-delocalized whereas /u1D463=/u1D4521. ThenΠ(0)\n1,1=1\nwhereas |Π(0)\n/u1D456,/u1D456|/lessorequalslant/u1D4362//u1D45Bfor/u1D456>1. Hence, deleting the first row and column of /tildewide/u1D440also deletes the\nlocalized eigenvector /u1D463. In general, whenever one of the eigenvectors of /tildewide/u1D440(/u1D461)is heavily localized\non a subset of coordinates /u1D446, the diagonal entries in Π(/u1D461)\n/u1D446,/u1D446are disproportionately large. This leads\nto a win-win scenario: either we reach the termination condi tion, or we are likely to mitigate the\ntroublesome large localized eigenvectors.\nWe now discuss how we achieve the second and third guarantees inProblem 2.4 .\nTrimming the subspace. The final postprocessing step is simple. Let /u1D449denote the eigenspace\nwith eigenvalues less than −/u1D>02for the matrix /tildewide/u1D440(/u1D447)obtained at end of iterative procedure.\nTo ensure delocalization (condition 2 in Problem 2.4 ), the idea is to take its projector Π/u1D449and\ntrim away the rows and columns with diagonal entry exceeding/u1D>0F\n/u1D45Bfor some large parameter /u1D>0F >0.\nThe desired delocalized subspace /u1D448is obtained by taking the eigenspace of the trimmed Π/u1D449\ncorresponding to the eigenvalues exceeding a threshold tha t is/u1D442(/u1D>02). Since/u1D449is/u1D442(/u1D45F)-dimensional,\nso too is/u1D448.\nThe more delicate part is condition 3 in Problem 2.4 . Namely, we must show that despite\ncorruptions and the repeated trimming steps, /u1D465remains a delocalized witness vector for Π/u1D448, and\nthus has constant correlation with the subspace /u1D448. The key intuition for this is that delocalized\n7witnesses are naturally robust to adversarial corruptions , so long as the adversarial corruptions\nhave bounded row-wise ℓ1norm. In particular, since delocalization is an ℓ∞constraint, Hölder’s\ninequality bounds the difference in value of the quadratic fo rm using/u1D440and/tildewide/u1D440. InSection 6.2 , we\nprove that for sufficiently small constant levels of corrupti on,/u1D465is also a delocalized witness for /tildewide/u1D440\nandΠ/u1D448.\nFinally, we discuss how to round the recovered subspace /u1D448into a community assignment.\n2.3 Rounding to communities\nAt this stage, we are presented with a constant-dimensional subspace/u1D448with the key feature that\nit is correlated with the community assignment vectors {1/u1D450}/u1D450∈[/u1D458]. Our goal is to round /u1D448to a\ncommunity assignment that is “well-correlated” with the gr ound truth. In order to discuss how\nwe achieve this goal, we must make precise what it means to be “ well-correlated” with the ground\ntruth. Notice that a community assignment is just as plausib le as the same assignment with the\nnames of communities permuted, and thus counting the number of correctly labeled vertices is\nnot a meaningful metric.\nA more meaningful metric is the number of pairwise mistakes, i.e. the number of pairs of\nvertices in the same community assigned to different communi ties or in different communities\nassigned to the same community. A convenient way to express t his metric is via the inner product\nof positive semidefinite matrices encoding whether pairs of v ertices belong to the same community\nor not. Given a community assignment /u1D465, we assign it the matrix /u1D44B, defined as\n/u1D44B[/u1D456,/u1D457]=/braceleftBigg\n1 if/u1D465(/u1D456)=/u1D465(/u1D457)\n−1\n/u1D458−1if/u1D465(/u1D456)≠/u1D465(/u1D457).\nFor the ground truth assignment /u1D499and the output of our algorithm /hatwide/u1D465, we measure the correlation\nwith/a\\}bracketle{t/u1D47F,/hatwide/u1D44B/a\\}bracketri}ht. Observe that for any guess /hatwide/u1D44Bthat is oblivious to the input (for example, classifying\nall vertices to the same community, or blindly guessing), th e value of /a\\}bracketle{t/u1D47F,/hatwide/u1D44B/a\\}bracketri}htis concentrated\nbelow˜/u1D442(/u1D45B3/2). On the other hand, if /hatwide/u1D44B=/u1D47F, then this correlation is Ω(/u1D45B2). See Definition 4.4\nandAppendix A for how this notion generalizes to arbitrary block models, a nd subsumes other\nnotions of weak-recovery defined in literature.\nThe projection matrix Π/u1D448satisfies/a\\}bracketle{tΠ/u1D448,/u1D47F/a\\}bracketri}ht/greaterorequalslantΩ(/bardblΠ/u1D448/bardbl/u1D439·/bardbl/u1D47F/bardbl/u1D439)= Ω(/u1D45B). We give a randomized\nrounding strategy according to which E/hatwide/u1D44B/{ollowsequal/u1D450·/u1D45B·Π/u1D448for some constant /u1D450>0. Consequently,\nE/a\\}bracketle{t/u1D47F,/hatwide/u1D44B/a\\}bracketri}ht=/u1D450/u1D45B·/a\\}bracketle{tΠ/u1D448,/u1D47F/a\\}bracketri}ht/greaterorequalslantΩ(/u1D45B2).\nObserve that for any community assignment /u1D465, its matrix representation /u1D44Bis rank-(/u1D458−1),\nwhich lets us write it as /u1D449/u1D449⊤for some/u1D45B×(/u1D458−1)matrix/u1D449. Here, the/u1D456-th row of/u1D449is some vector\n/u1D463/u1D465(/u1D456)that only depends only on the community /u1D465(/u1D456)where vertex /u1D456is assigned.\nOur rounding scheme uses Π/u1D448to produce an embedding of the /u1D45Bvertices as rows of a /u1D45B×(/u1D458−1)\nmatrix/u1D44Awhose rows are in {/u1D4631,...,/u1D463/u1D458}. In the community assignment /hatwide/u1D465outputted by the\nalgorithm, the /u1D456-th vertex is assigned to community /u1D457if the/u1D456-th row of/u1D44Ais equal to/u1D463/u1D457. We then\nshow that E/u1D44A/u1D44A⊤/{ollowsequal/u1D450·/u1D45B·Π/u1D448. Since/hatwide/u1D44B=E/u1D44A/u1D44A⊤, we can conclude E/a\\}bracketle{t/u1D47F,/hatwide/u1D44B/a\\}bracketri}ht/greaterorequalslantΩ(/u1D45B2).\nRounding scheme. Our first step is to obtain an embedding of the /u1D45Bvertices intoR/u1D458−1by choosing\na(/u1D458−1)-dimensional random subspace /u1D448′of/u1D448, then writing its projector as /u1D440′/u1D440′⊤, and choosing\nthe embedding as the rows of /u1D440′:/u1D462′\n1,...,/u1D462′\n/u1D45B. Suppose this embedding has the property that for\n8some/u1D450′>0, the rows of√\n/u1D450′/u1D45B/u1D448′lie inside the convex hull of /u1D4631,...,/u1D463/u1D458, then we can express each /u1D462′\n/u1D456\nas a convex combination/summationtext.1/u1D458\n/u1D457=1/u1D45D(/u1D456)\n/u1D457/u1D463/u1D457and then independently sample /u1D464/u1D456from{/u1D4631,...,/u1D463/u1D458}according\nto the probability distribution (/u1D45D(/u1D456)\n/u1D457)/u1D457∈[/u1D458]. The resulting embedding /u1D44Awould satisfy the property\nthatE/u1D44A/u1D44A⊤/{ollowsequal/u1D450′·/u1D458−1\ndim(/u1D448)·/u1D45B·Π/u1D448, where this inequality holds since the off-diagonal entries are\nequal, and the diagonal of /u1D44A/u1D44A⊤is larger.\nThe reason an appropriate scaling /u1D450′exists follows from the facts that the convex hull of\n/u1D4631,...,/u1D463/u1D458is full-dimensional and contains the origin, which we prove inSection 7 .\n3 Preliminaries\nStochastic block model notation. We write 1to denote the all-ones vector and /u1D452/u1D456to denote the\n/u1D456th standard basis vector, with the dimensions implicit. For a/u1D458-community block model, let\n/u1D>0B∈R/u1D458denote the prior community probabilities, and Π = diag(/u1D>0B), so that/u1D>0B= Π1. The edge\nprobabilities are parameterized by a symmetric matrix M ∈R/u1D458×/u1D458, the block probability matrix. A\ntrue community assignment /u1D499:[/u1D45B] → [/u1D458]is sampled i.i.d. from /u1D>0B. Conditioned on /u1D499, an edge\nbetween/u1D456and/u1D457is sampled with probabilityM/u1D499(/u1D456),/u1D499(/u1D457)/u1D451\n/u1D45B. To ensure that the average degree is /u1D451, we\nstipulate that M /u1D>0B=1.\nWe will also use /u1D47F∈R/u1D45B×/u1D458to denote the one-hot encoding of/u1D499, i.e., the matrix where the /u1D461-th\nrow is equal to /u1D452/u1D499(/u1D461). We will sometimes find it convenient to access the columns of /u1D47F, which are\nthe indicator vectors for the /u1D458different communities; we denote these by 1/u1D450for any community\n/u1D450∈ [/u1D458]. For any/u1D453:[/u1D458] →R, define the lift of /u1D453with respect to the true community assignment by\n/u1D487(/u1D45B)/defines/summationtext.1\n/u1D450∈[/u1D458]/u1D453(/u1D450)·1/u1D450.\nAnother natural matrix that appears throughout the analysi s is the Markov transition matrix\n/u1D447/defines/u1D440Π, which by detailed balance evidently has stationary distri bution/u1D>0B. This is an asymmetric\nmatrix, but since /u1D447defines a time-reversible Markov chain with respect to /u1D>0B,/u1D447is self-adjoint\nwith respect to the inner product /a\\}bracketle{t·,·/a\\}bracketri}ht/u1D>0BinR/u1D458induced by /u1D>0B. Hence/u1D447is diagonalizable with real\neigenvalues and its eigenvalues are 1 =/u1D>061>|/u1D>062|/greaterorequalslant···/greaterorequalslant|/u1D>06/u1D458|, with ties broken by placing positive\neigenvalues before the negative ones. Note that the normali zation condition M /u1D>0B=1translates\ninto/u1D4471=1.\nMatrix notation. We use /√recedesequaland/{ollowsequalto denote inequalities on matrices in the Loewner order.\nFor any/u1D45B×/u1D45Bmatrix/u1D44B, we use Π/lessorequalslant/u1D44E(/u1D44B)andΠ/greaterorequalslant/u1D44E(/u1D44B)to denote the projectors onto the spaces\nspanned by eigenvectors of /u1D44Bwith eigenvalue at most and at least /u1D44Erespectively. We also de-\nfine/u1D44B/lessorequalslant/u1D44E/definesΠ/lessorequalslant/u1D44E(/u1D44B)/u1D44BΠ/lessorequalslant/u1D44E(/u1D44B)and/u1D44B/greaterorequalslant/u1D44E/definesΠ/greaterorequalslant/u1D44E(/u1D44B)/u1D44BΠ/greaterorequalslant/u1D44E(/u1D44B), the corresponding truncations of the\neigendecomposition of /u1D44B.\nFor/u1D446⊆ [/u1D45B], we use/u1D44B/u1D446,/u1D446to denote the matrix obtained by taking /u1D44Band zeroing out all rows\nand columns with indices outside /u1D446.\nNonbacktracking matrix and Bethe Hessian. For a graph /u1D43A, let/u1D435/u1D43Abe its nonbacktracking matrix,\n/u1D434/u1D43Abe its adjacency matrix, /u1D437/u1D43Abe its diagonal matrix of degrees, /u1D434(ℓ)\n/u1D43Abe itsℓ-th nonbacktracking\npower of/u1D434/u1D43A, and/u1D43B/u1D43A(/u1D461)/defines(/u1D437/u1D43A−/u1D43C)/u1D4612−/u1D434/u1D43A/u1D461+/u1D43Cbe its Bethe Hessian matrix. The matrix we use for\nour algorithm is /u1D440/u1D43A,ℓ(/u1D461)/defines/u1D434(ℓ)\n/u1D43A/u1D43B/u1D43A(/u1D461)/u1D434(ℓ)\n/u1D43A. We will drop the /u1D43Afrom the subscript when the graph\n/u1D43Ais clear from context.\nDeterminants. Below, we collect some standard linear algebraic facts that will prove useful.\n9Fact 3.1. Suppose a matrix /u1D44Bhas a kernel of dimension /u1D458, then every (/u1D45B−/u1D457)×(/u1D45B−/u1D457)submatrix of /u1D44Bfor\n/u1D457</u1D458is singular.\nFact 3.2 (Jacobi’s formula) .For any differentiable function /u1D44B:R→R/u1D45B×/u1D45B,\n/u1D451\n/u1D451/u1D462det(/u1D44B(/u1D462))=/u1D45B/summationdisplay.1\n/u1D456=1/u1D45B/summationdisplay.1\n/u1D457=1det/parenleftbig/u1D44B(/u1D462)[/u1D45B]\\{/u1D456},[/u1D45B]\\{/u1D457}/parenrightbig·(−1)/u1D456+/u1D457·/u1D451\n/u1D451/u1D462(/u1D44B(/u1D462))/u1D456,/u1D457.\nLemma 3.3. Let/u1D44B:R→R/u1D45B×/u1D45Band/u1D453:R→Rbe any pair of smooth functions. For any /u1D457/greaterorequalslant0, there exist\nfunctions (/u1D45E/u1D446,/u1D447:R→R)/u1D446,/u1D447⊆[/u1D45B],|/u1D446|=|/u1D447|/greaterorequalslant/u1D45B−/u1D457such that:\n/parenleftbigg/u1D451\n/u1D451/u1D462/parenrightbigg/u1D457\n[det(/u1D44B(/u1D462))·/u1D453(/u1D462)]=/summationdisplay.1\n/u1D446,/u1D447⊆[/u1D45B],|/u1D446|=|/u1D447|/greaterorequalslant/u1D45B−/u1D457det(/u1D44B(/u1D462)/u1D446,/u1D447)/u1D45E/u1D446,/u1D447(/u1D462).\nProof. We prove this by induction. This is clearly true when /u1D457=0, and the induction step is a\nconsequence of Jacobi’s formula. /square\nKesten-Stigum threshold. We say that a stochastic block model is above the Kesten–Stig um (KS)\nthreshold if /u1D>062(/u1D447)2/u1D451>1, where recall that /u1D>062is the second largest eigenvalue in absolute value.\nWe use/u1D45Fto denote the number of eigenvalues of /u1D447equal to/u1D>062(/u1D447).\n4 Recovery algorithm\nLet/u1D46Ebe the graph drawn from SBM /u1D45B(M,/u1D>0B,/u1D451), and let/tildewide/u1D46Edenote the input graph which is /u1D46Ealong\nwith an arbitrary /u1D6FF/u1D45Badversarial edge corruptions. Our algorithm for clusterin g the vertices into\ncommunities proceeds in multiple phases, described formal ly below.\nThe first phase preprocesses the graph by making it bounded deg ree and constructs an ap-\npropriate matrix /u1D440associated to the graph. The second phase cleans up /u1D440and uses a spectral\nalgorithm to robustly recover a subspace containing nontri vial information about the true com-\nmunities. Finally, the third phase rounds the subspace to an actual community assignment.\nAlgorithm 4.1. /tildewide/u1D46Eis given as input, and a community assignment to the vertices is produced as\noutput.\nPhase 1: Deletion of high-degree vertices. For some large constant /u1D435>0 to be specified later,\nwe perform the following truncation step: delete all edges i ncident on vertices with degree larger\nthan/u1D435in/tildewide/u1D46E. This forms a graph /tildewide/u1D46E/u1D435, with corresponding adjacency matrix /u1D434/tildewide/u1D46E/u1D435∈R|/u1D449(/u1D46E)|×|/u1D449(/u1D46E)|. To\navoid confusion, we preserve the vertex set /u1D449(/u1D46E), but it should be understood that the truncated\nvertices do not contribute to the graph.\nFor technical considerations, we also define a (nonstandard) truncated diagonal matrix\n/u1D437/tildewide/u1D46E/u1D435/definesdiag/parenleftbigdeg(/u1D463)1[deg(/u1D463)/lessorequalslant/u1D435]/parenrightbig\n/u1D463∈/u1D449(/u1D46E)(1)\nWith this, we can then define the truncated Bethe Hessian matri x\n/u1D43B/tildewide/u1D46E/u1D435(/u1D461)/defines/u1D43C−/u1D461/u1D434/tildewide/u1D46E/u1D435+/u1D4612(/u1D437/tildewide/u1D46E/u1D435−/u1D43C). (2)\n10Finally, the input matrix to the next phase is\n/u1D440/tildewide/u1D46E/u1D435,ℓ(/u1D461)/defines/u1D434(ℓ)\n/tildewide/u1D46E/u1D435/u1D43B/tildewide/u1D46E/u1D435(/u1D461)/u1D434(ℓ)\n/tildewide/u1D46E/u1D435, (3)\nwhere we also choose the value of /u1D461later.\nRemark 4.2. To reduce any chance of confusion with the notation, we reite rate our conventions\nfor distinguishing between different versions of various ma trices. If a graph is truncated at level\n/u1D435, then we add a subscript /u1D435. We use tilde to denote that we are working with a corrupted gr aph.\nFinally, we use overline to denote that we are working with th e nonstandard version of the Bethe\nHessian after truncation.\nFor example, the matrix /u1D437/u1D46E/u1D435no longer corresponds to the degree matrix of /u1D46E/u1D435, since as stated\nit still counts edges from truncated vertices. This is done t o simplify the analysis of the spectrum\nof/u1D440/u1D46E/u1D435,ℓ(/u1D461)but we do not believe it to be essential.\nPhase 2: Recovering a subspace with planted signal. Define/u1D440/defines/u1D440/tildewide/u1D46E/u1D435,ℓ. We give an iterative\nprocedure to “clean up” /u1D440by deleting a few rows and columns. We then run a spectral algo rithm\non the cleaned up version of /u1D440.\nLet/u1D>02 >0 be a small constant we choose later, and let /u1D43E/defines/u1D4352ℓ+3.\n1. Define/u1D440(0)as/u1D440. Let/u1D461be a counter initialized at 0, and Φ(/u1D44B)as the number of eigenvalues\nof/u1D44Bsmaller than −/u1D>02.\n2. While Φ(/u1D440(/u1D461))>2/u1D43E\n/u1D>02/u1D45F: compute the projection matrix Π(/u1D461)/definesΠ/lessorequalslant−/u1D>02(/u1D440(/u1D461)), choose a random\n/u1D456∈ [/u1D45B]with probabilityΠ(/u1D461)\n/u1D456,/u1D456\nTr(Π(/u1D461)), and define /u1D440(/u1D461+1)as the matrix obtained by zeroing out the\n/u1D456-th row and column of /u1D440(/u1D461). Then increment /u1D461.\nLet/u1D447be the time of termination and /u1D>0F >0 be a large enough constant we choose later. We compute\nΠ(/u1D447), and then compute as the set /u1D446of all indices /u1D456whereΠ(/u1D447)\n/u1D456,/u1D456/lessorequalslant/u1D>0F\n/u1D45B. Define /tildewideΠas/parenleftBig\nΠ(/u1D447)\n/u1D446,/u1D446/parenrightBig\n/greaterorequalslant/u1D>02//u1D43E, and\ncompute its span /u1D448, where we recall that (/u1D44B)/greaterorequalslant/u1D44Edenotes the truncation of the eigendecomposition\nof/u1D44Bfor eigenvalues at least /u1D44E. This subspace /u1D448is passed to the next phase.\nPhase 3: Rounding to a community assignment. Define/u1D45F′as/u1D45F−1 when/u1D>062(/u1D447)>0 and as/u1D45F\nwhen/u1D>062(/u1D447)<0. We first obtain an /u1D45F′-dimensional embedding of the vertices into R/u1D45F′. Compute\na random/u1D45F′-dimensional subspace /u1D448′of/u1D448, and take an orthogonal basis /u1D462′\n1,...,/u1D462′\n/u1D45F′. Place these\nvectors as a column of a matrix /u1D440′inR/u1D45B×/u1D45F′. The rows of /u1D440′gives us the desired embedding.\nOn the other hand, we use the natural embedding of the /u1D458communities into R/u1D45F′induced by the\n/u1D45F′nontrivial right eigenvectors corresponding to the eigenv alue/u1D>062(/u1D447):(/u1D>13/u1D456)1/lessorequalslant/u1D456/lessorequalslant/u1D45F′of/u1D447. In particular,\nletΨ/u1D45F′/defines/bracketleftbig\n/u1D>131···/u1D>13/u1D45F′/bracketrightbig\n∈R/u1D458×/u1D45F′be the matrix of these /u1D45F′nontrivial eigenvectors of /u1D447. Then the\nrow vectors /u1D>191,...,/u1D>19/u1D458∈R/u1D45F′form the desired embedding of communities.\nIn the rounding algorithm, we first find the largest /u1D450such that all the rows of /u1D450·/u1D440′lie in the\nconvex hull of /u1D>191,...,/u1D>19/u1D458. We can find such a value of /u1D450if it exists by solving a linear program, and\nthis/u1D450>0 is guaranteed to exist by Lemma 7.1 . Then, for each /u1D456∈ [/u1D45B]we express each row of /u1D450·/u1D440′\nas a convex combination/summationtext.1/u1D458\n/u1D457=1/u1D464(/u1D457)\n/u1D456/u1D>19/u1D457for nonnegative /u1D464(/u1D457)\n/u1D456such that/summationtext.1/u1D458\n/u1D457=1/u1D464(/u1D457)\n/u1D456=1. Finally, we assign\nvertex/u1D456to community /u1D457with probability /u1D464(/u1D457)\n/u1D456, and output the resulting community assignment /hatwide/u1D499.\n11Remark 4.3. Scaling the rows of /u1D440′so as to lie in the convex hull of {/u1D>19/u1D457}/u1D457∈[/u1D458], is reminiscent of the\nrounding algorithm of Charikar & Wirth [ CW04 ] to find a cut of size1\n2+Ω(/u1D>00\nlog(1//u1D>00))in a graph with\nmaximum cut of size1\n2+/u1D>00: in their algorithm, they scale /u1D45Bscalars to lie in the interval [−1,1].\n4.1 Analysis of algorithm\nOur goal is to prove that the output /hatwide/u1D499of our algorithm is well-correlated with the true community\nassignment /u1D499. We begin by defining a notion of weak recovery for /u1D458-community stochastic block\nmodels.\nDefinition 4.4 (Weak recovery) .LetΨ/defines/bracketleftbig\n/u1D>132···/u1D>13/u1D458/bracketrightbig\n∈R/u1D458×(/u1D458−1)be the matrix of the top- (/u1D458−1)\nnontrivial eigenvectors of the transition matrix /u1D447of a stochastic block model.\nFor/u1D>0C > 0, we say that a (randomized) algorithm for producing commun ity assignments\n/hatwide/u1D47F∈R/u1D45B×/u1D458achieves/u1D>0C-weak recovery if\n/angbracketleftBig\nE/hatwide/u1D47FΨ,/u1D47FΨ/angbracketrightBig\n/greaterorequalslant/u1D>0C/bardblE/hatwide/u1D47FΨ/bardbl/u1D439/bardbl/u1D47FΨ/bardbl/u1D439,\nwhere/u1D435Ψ/defines(/u1D435Ψ)(/u1D435Ψ)⊤for a matrix /u1D435∈R/u1D45B×/u1D458.\nRemark 4.5. Intuitively, this notion is capturing the “advantage” of th e algorithm over random\nguessing, or simply outputting the most likely community. S eeAppendix A for a more detailed\ndiscussion of this notion, how it recovers other previously considered measures of correlation in\nthe case of the symmetric block model, and why it is meaningfu l. In particular, it implies the\nnotion of weak recovery used in [ DdNS22 ].\nOur main guarantee is stated below.\nTheorem 4.6. For any SBM parameters (M,/u1D>0B,/u1D451)above the KS threshold, there is a constant /u1D>0C(M,/u1D>0B,/u1D451)>0\nsuch that the above algorithm takes in the corrupted graph /tildewide/u1D46Eas input and outputs /hatwide/u1D499achieving/u1D>0C(M,/u1D>0B,/u1D451)-\nweak recovery with probability 1−/u1D45C/u1D45B(1)over the randomness of /u1D46E∼SBM/u1D45B(M,/u1D>0B,/u1D451).\nTo prove the above theorem it suffices to analyze/angbracketleftBig\n(E/hatwide/u1D47F)Ψ,/u1D47FΨ/angbracketrightBig\n. To see why, let us first set up\nsome notation. For each vertex /u1D456, we obtain a simplex vector /u1D464/u1D456∈R/u1D458, which we can stack as rows\ninto a weight matrix /u1D44A∈R/u1D45B×/u1D458. We then independently round each vertex so that E/hatwide/u1D47F=/u1D44A.\nTo analyze our rounding scheme, first note that E[/hatwide/u1D47FΨ]is equal to/u1D44AΨoff of the diagonal and\nis larger than /u1D44AΨon the diagonal, and thus E[/hatwide/u1D47FΨ] /{ollowsequal/u1D44AΨ. Since /u1D47FΨis positive semidefinite,/angbracketleftBig\nE[/hatwide/u1D47FΨ],/u1D47FΨ/angbracketrightBig\n/greaterorequalslant/a\\}bracketle{t/u1D44AΨ,/u1D47FΨ/a\\}bracketri}ht. Thus, it suffices to lower bound /a\\}bracketle{t/u1D44AΨ,/u1D47FΨ/a\\}bracketri}ht. By construction, /u1D44AΨis equal\nto/u1D4502·Π/u1D448′, where recall that /u1D448′was a random /u1D45F′-dimensional subspace of /u1D448, the output of Phase\n2 of the algorithm. Thus,\nE\n/u1D448′/u1D44AΨ=/u1D4502·E\n/u1D448′Π/u1D448′=/u1D4502·/u1D45F′\ndim(/u1D448)Π/u1D448.\nInLemma 6.10 , we prove that when (M,/u1D>0B,/u1D451)are above the KS threshold, /a\\}bracketle{tΠ/u1D448,/u1D47FΨ/a\\}bracketri}ht/greaterorequalslantΩ(1) ·\n/bardblΠ/u1D448/bardbl/u1D439·/bardbl/u1D47FΨ/bardbl/u1D439and dim (/u1D448)=/u1D442(1). In Lemma 7.1 , we show that when diag (Π/u1D448)=/u1D442(1//u1D45B),\nwe can take /u1D450= Ω(√/u1D45B); this delocalization condition is guaranteed by phase 2 of t he algorithm.\nCombined with the fact that /bardbl/hatwide/u1D47FΨ/bardbl/u1D439=/u1D442(/u1D45B), it follows that/angbracketleftBig\nE/hatwide/u1D47FΨ,/u1D47FΨ/angbracketrightBig\n/greaterorequalslantΩ(1) ·/bardbl/hatwide/u1D47FΨ/bardbl/u1D439·/bardbl/u1D47FΨ/bardbl/u1D439,\nwhich establishes Theorem 4.6 .\n125 The number of outlier eigenvalues\nIn this section, we pinpoint the number of outlier eigenvalu es of/u1D440/u1D46E,ℓ(/u1D461)/defines/u1D434(ℓ)\n/u1D46E/u1D43B/u1D46E(/u1D461)/u1D434(ℓ)\n/u1D46E. Specifi-\ncally, we will show the following theorem.\nTheorem 5.1. Let(M,/u1D>0B,/u1D451)be a model and let /u1D447be its corresponding Markov transition matrix. Let /u1D45Fbe\nmultiplicity of /u1D>062(/u1D447). There exists constants ℓ,/u1D6FFand/u1D6FC, such that for /u1D461=1+/u1D6FF\n/u1D>062/u1D451, with high probability over\n/u1D46E∼SBM/u1D45B(M,/u1D>0B,/u1D451),/u1D440/u1D46E,ℓ(/u1D461)has at least/u1D45Fnegative eigenvalues of magnitude at most −/u1D6FC, and at most\n/u1D45F+1negative eigenvalues.\nWe first establish the upper bound on the number of outlier eige nvalues, and then the lower\nbound. To establish the theorem, we will harness the Ihara-B ass formula that connects /u1D43B/u1D43A(/u1D461)to\nthe non-backtracking walk matrix, and the work of Bordenave et al. [ BLM15 ] on the spectrum of\nnon-backtracking walk matrix /u1D435in stochastic block models. We begin by recalling these resu lts\nfrom the literature.\nTheorem 5.2 (Ihara–Bass formula) .For any simple, undirected, unweighted graph /u1D43Aon/u1D45Bvertices and\n/u1D45Aedges, we have\ndet(/u1D43C−/u1D461/u1D435/u1D43A)=det(/u1D43B/u1D43A(/u1D461))(1−/u1D4612)/u1D45A−/u1D45B\nFor a matrix /u1D44B, we use/u1D>06/u1D456(/u1D44B)to denote its /u1D456-th largest eigenvalue when sorted by absolute value.\nTheorem 5.3 (Nonbacktracking spectrum of SBM [ BLM15 ]).For/u1D46E∼SBM/u1D45B(M,/u1D>0B,/u1D451):\n/u1D>06/u1D456(/u1D435/u1D46E)=/u1D>06/u1D456(/u1D447)·/u1D451±/u1D45C/u1D45B(1)\nfor all 1/lessorequalslant/u1D456/lessorequalslant/u1D458such that |/u1D>06/u1D456(/u1D447)|>1√\n/u1D451with high probability. All the remaining eigenvalues of /u1D435/u1D46Eare at\nmost√\n/u1D451·(1+/u1D45C/u1D45B(1))in magnitude with high probability.\nThe following is a consequence of the Ihara–Bass formula. Al ong with Theorem 5.3 , this implies\nan upper bound on the number of outlier eigenvalues in Theorem 5.1 .\nLemma 5.4. For any/u1D461∗>0, the number of negative eigenvalues of /u1D43B/u1D43A(/u1D461∗)is at most the number of real\neigenvalues of /u1D435/u1D43Alarger than 1//u1D461∗, and for/u1D461∗<0, the number of negative eigenvalues of /u1D43B/u1D43A(/u1D461∗)is at most\nthe number of real eigenvalues of /u1D435/u1D43Asmaller than 1//u1D461∗.\nProof. We present the proof assuming /u1D461∗>0, since the case when /u1D461∗<0 follows an identical proof.\nFirst, observe that the number of negative eigenvalues of /u1D43B/u1D43A(/u1D461∗)is at most:\n/summationdisplay.1\n0/lessorequalslant/u1D461</u1D461∗:/u1D43B/u1D43A(/u1D461)is singulardim kernel (/u1D43B/u1D43A(/u1D461)),\nsince/u1D43B/u1D43A(0)has no negative eigenvalues, and the eigenvalues are a conti nuous function of /u1D461.\nBy the Ihara–Bass formula, if /u1D43B/u1D43A(/u1D461)is singular, then det (/u1D43C−/u1D461/u1D435/u1D43A)must be 0, and in particular,\n1//u1D461must be an eigenvalue of /u1D435/u1D43A. We now prove that dim kernel (/u1D43B/u1D43A(/u1D461))is at most the multiplicity\nof the eigenvalue 1 //u1D461, from which the desired statement follows.\n13Let/u1D45Fbe the dimension of the kernel of /u1D43B/u1D43A(/u1D461). We will show/parenleftBig\n/u1D451\n/u1D451/u1D462/parenrightBig/u1D457\ndet(/u1D43C−/u1D462/u1D435/u1D43A)/barex/barex/barex\n/u1D462=/u1D461=0 for\n0/lessorequalslant/u1D457/lessorequalslant/u1D45F−1, thereby implying that /u1D461occurs as a root of det (/u1D43C−/u1D462/u1D435/u1D43A), and hence 1 //u1D461as an\neigenvalue, with multiplicity at least /u1D45F. Now, by the Ihara–Bass formula,\n/parenleftbigg/u1D451\n/u1D451/u1D462/parenrightbigg/u1D457\ndet(/u1D43C−/u1D462/u1D435/u1D43A)=/parenleftbigg/u1D451\n/u1D451/u1D462/parenrightbigg/u1D457/bracketleftbig\ndet(/u1D43B(/u1D462))·(1−/u1D4622)/u1D45A−/u1D45B/bracketrightbig\n,\nwhich by Lemma 3.3 is equal to\n/summationdisplay.1\n/u1D446,/u1D447⊆[/u1D45B],|/u1D446|=|/u1D447|/greaterorequalslant/u1D45B−/u1D457det(/u1D43B(/u1D462)/u1D446,/u1D447)·/u1D45E/u1D446,/u1D447(/u1D462)\nfor some collection of scalar functions /u1D45E/u1D446,/u1D447. Since/u1D43B(/u1D461)has a kernel of dimension /u1D45F, byFact 3.1\n/u1D43B(/u1D461)/u1D446,/u1D447is singular for any choice of /u1D446,/u1D447⊆ [/u1D45B]with|/u1D446|=|/u1D447|/greaterorequalslant/u1D45B−/u1D45F+1, and hence the above\nquantity is 0 for all 0 /lessorequalslant/u1D457/lessorequalslant/u1D45F−1, which completes the proof. /square\n5.1 Lower bound on the number of outlier eigenvalues\nIn this section, we lower bound the number of negative eigenv alues of/u1D440/u1D46E,ℓ(/u1D461), where/u1D461>0. The\nsame arguments apply to the case where /u1D461<0, so we suppress the subscript in the dimension /u1D45Fof\nthe outlier eigenspace for ease of notation.\nTo lower bound the number of outliers, we will show that with h igh probability the quadratic\nform of/u1D440/u1D46E,ℓ(/u1D461)is negative on an /u1D45F-dimensional subspace spanned by vectors corresponding to\nthe true community assignment. To do so, we will calculate th e first and second moments of this\nrandom quadratic form, but the bulk of the work comes in calcu lating the first moment. The key\ninsight here is that since these vectors are constructed usi ng the true communities, the expectations\ncan be tractably computed.\nIt is fruitful to give a combinatorial interpretation of the matrix/u1D440/u1D46E,ℓ(/u1D461). As is standard in trace\nmoment calculations, the quadratic form can be decomposed i nto sums over various different\ngraph shapes. Some standard accounting for these shapes the n leads us to the final answer. To\nconcretely articulate what we mean, we first define the notion of apartially labeled shape and its\nassociated matrix.\nDefinition 5.5 (Shape matrix) .Ashape is a graph/u1D43Balong with two distinguished vertices: a left\nvertex/u1D462/u1D43Fand a right vertex/u1D462/u1D445. Given an/u1D45B-vertex graph /u1D43A, its shape matrix of /u1D43A, denoted/u1D440/u1D43B(/u1D43A),\nis an/u1D45B×/u1D45Bmatrix where the (/u1D456,/u1D457)entry is the number of copies of /u1D43Bin/u1D43Asuch that/u1D462/u1D43Fcan be\nidentified with /u1D456and/u1D462/u1D445with/u1D457. Equivalently,\n/u1D440/u1D43B(/u1D43A)/u1D456,/u1D457≔/summationdisplay.1\n/u1D>0F:/u1D449(/u1D43B)→[/u1D45B]\n/u1D>0Finjective\n/u1D>0F(/u1D462/u1D43F)=/u1D456,/u1D>0F(/u1D462/u1D445)=/u1D457/productdisplay.1\n{/u1D44E,/u1D44F}∈/u1D438(/u1D43B)/u1D43A/u1D>0F(/u1D44E),/u1D>0F(/u1D44F)\nRemark 5.6. We only consider shapes /u1D43Bwhose size does not grow with /u1D45B.\nLet us explain at this point how we explicitly construct the /u1D45F-dimensional subspace. The\nidea is to lift the /u1D45Fright eigenvectors /u1D>131,...,/u1D>13/u1D45F∈R/u1D458of/u1D447above the KS threshold using the\nindicator vectors 1/u1D450for/u1D450∈ [/u1D458]. This makes the computation tractable because edges in the S BM are\n14conditionally independent given the true community assign ment. Accordingly, we are interested\nin polynomials of the form:\n1⊤\n/u1D450/u1D440/u1D43B(/u1D43A)1/u1D450′,\nwhere/u1D450and/u1D450′are colors in [/u1D458]and1/u1D450is the indicator vector of vertices with color /u1D450. The above\nquadratic form counts the number of copies of /u1D43Binside/u1D43Awhere the vertex identified with /u1D462/u1D43Fhas\ncolor/u1D450and the vertex identified with /u1D462/u1D445has color/u1D450′.\nWe use the following special case of [ BMR21 , Theorem 6.6].\nLemma 5.7. Suppose/u1D43Bis a connected graph, then for /u1D46E∼SBM/u1D45B(M,/u1D>0B,/u1D451), we have:\n1⊤\n/u1D450/u1D440/u1D43B(/u1D46E)1/u1D450′=/summationdisplay.1\n/u1D>0E:/u1D449(/u1D43B)→[/u1D458]\n/u1D>0E(/u1D462/u1D43F)=/u1D450,/u1D>0E(/u1D462/u1D445)=/u1D450′/productdisplay.1\n/u1D463∈/u1D449(/u1D43B)/u1D>0B/u1D>0E(/u1D463)/productdisplay.1\n/u1D456/u1D457∈/u1D43B/u1D440/u1D>0E(/u1D456)/u1D>0E(/u1D457)·/u1D451|/u1D438(/u1D43B)|·/u1D45B|/u1D438(/u1D43B)|−|/u1D449(/u1D43B)|±/u1D45C(/u1D45B)\nwith probability 1−/u1D45C/u1D45B(1), where the/u1D45C(·)hides factors depending on |/u1D438(/u1D43B)|.\nRemark 5.8. Observe that in the above if /u1D43Bis not a tree, then |1⊤\n/u1D450/u1D440/u1D43B(/u1D46E)1/u1D450′|=/u1D45C(/u1D45B). If/u1D43Bis a tree,\nthen it can be expressed as a path /u1D443=/u1D45D0,...,/u1D45Dℓwhere/u1D45D0=/u1D462/u1D43Fand/u1D45Dℓ=/u1D462/u1D445, along with a subtree\nper vertex on /u1D443. In this case, with high probability:\n1⊤\n/u1D450/u1D440/u1D43B(/u1D46E)1/u1D450′=/angbracketleftbig\n/u1D452/u1D450,/u1D447ℓ/u1D452/u1D450′/angbracketrightbig\n/u1D>0B·/u1D451|/u1D449(/u1D43B)|−1·/u1D45B±/u1D45C(/u1D45B).\nMore generally, we are interested in understanding the quadr atic form of vectors that are\nconstant on each color class with /u1D440/u1D43A(/u1D43B), i.e. lifts of vectors in R/u1D458.\nCorollary 5.9. Let/u1D43Bbe a tree such that /u1D462/u1D43Fand/u1D462/u1D445are distance- ℓapart. For any /u1D453:[/u1D458] →R, define\n/u1D487(/u1D45B)≔/summationtext.1\n/u1D450∈[/u1D458]/u1D453(/u1D450)·1/u1D450. Then with high probability, for all /u1D453,/u1D4<4.alt:[/u1D458] →Rsimultaneously:\n/a\\}bracketle{t/u1D487(/u1D45B),/u1D440/u1D43B(/u1D46E)/u1D488.alt(/u1D45B)/a\\}bracketri}ht=/angbracketleftbig\n/u1D453,/u1D447ℓ/u1D4<4.alt/angbracketrightbig\n/u1D>0B·/u1D451|/u1D449(/u1D43B)|−1·/u1D45B±/u1D45C(/u1D45B).\nThe bulk of the contribution to the value of the quadratic for m comes from tree shapes.\nCorollary 5.10. With high probability over /u1D46E∼SBM/u1D45B(M,/u1D>0B,/u1D451), for any collection of shapes ℋalong with\ncoefficients (/u1D450/u1D43B)/u1D43B∈ℋ, and any functions /u1D453,/u1D4<4.alt:[/u1D458] →R, we have:\n/angbracketleftBigg\n/u1D487(/u1D45B),/summationdisplay.1\n/u1D43B∈ℋ/u1D450/u1D43B/u1D440/u1D43B/u1D488.alt(/u1D45B)/angbracketrightBigg\n=/angbracketleftBigg\n/u1D487(/u1D45B),/summationdisplay.1\n/u1D43B∈ℋ\n/u1D43Btree/u1D450/u1D43B/u1D440/u1D43B/u1D488.alt(/u1D45B)/angbracketrightBigg\n±/u1D45C(/u1D45B).\nTo lighten the notation, for the rest of this section we make t he dependence on the original\ngraph/u1D46Eimplicit. So for our algorithm, we would like to illustrate n egative eigenvalues in the\nspectrum of /u1D440ℓ(/u1D461)=/u1D434(ℓ)/u1D43B(/u1D461)/u1D434(ℓ).\nLet/u1D>131,...,/u1D>13/u1D45Fbe the right eigenvectors of /u1D447corresponding to the largest nontrivial eigenvalue\n/u1D>062(/u1D447)=/radicalBig\n1+/u1D>00\n/u1D451for some/u1D>00 >0. We will show that with high probability for all /u1D463in span{/u1D>131,...,/u1D>13/u1D45F},\nthe quadratic form/angbracketleftbig\n/u1D497(/u1D45B),/u1D440ℓ(/u1D461)/u1D497(/u1D45B)/angbracketrightbig\nis substantially negative, where /u1D461=1+/u1D6FF√\n(1+/u1D>00)/u1D451for some small\nenough/u1D6FFwe choose in posterity.\nLemma 5.11. For allℓsufficiently large and all /u1D463∈span{/u1D>131,...,/u1D>13/u1D45F},/angbracketleftbig\n/u1D497(/u1D45B),/u1D440ℓ(/u1D461)/u1D497(/u1D45B)/angbracketrightbig\n<−/u1D6FC/u1D45B, where\n/u1D6FC >0is an absolute constant. Consequently, /u1D440ℓ(/u1D461)has at least/u1D45Feigenvalues less than −/u1D6FC.\n15Proof. We can write /u1D440ℓ(/u1D461)as a weighted combination of a collection of shape matrices ℋ. The plan\nis to extract out all the tree shapes that occur in ℋalong with their weights and obtain a formula\nfor the quadratic form using Corollary 5.9 .\nWe expand /u1D440ℓ(/u1D461)=/u1D434(ℓ)/u1D434(ℓ)−/u1D461/u1D434(ℓ)/u1D434/u1D434(ℓ)+/u1D4612/u1D434(ℓ)(/u1D437−/u1D43C)/u1D434(ℓ). Each term counts walks following\ncertain simple rules.\nObservation 5.12. We can characterize the walks counted by each term as follows .\n1./u1D434(ℓ)/u1D434(ℓ)[/u1D456,/u1D457]counts walks starting at /u1D456and ending at /u1D457with the phases (i) walk ℓnonback-\ntracking steps, (ii) walk ℓnonbacktracking steps.\n2./u1D434(ℓ)/u1D434/u1D434(ℓ)[/u1D456,/u1D457]counts walks with phases (i) walk ℓnonbacktracking steps, (ii) walk one step,\n(iii) walkℓnonbacktracking steps.\n3./u1D434(ℓ)(/u1D437−/u1D43C)/u1D434(ℓ)[/u1D456,/u1D457]counts walks with phases (i) walk ℓ+1 nonbacktracking steps and then\nbacktrack one step, (ii) walk ℓnonbacktracking steps.\nObservation 5.13. For a walk to give rise to a tree shape, each step must either (i ) visit a vertex not\nvisited in the walk so far, or (ii) use an edge that has already been stepped on.\nUsing this rule, we can deduce all the tree shapes that come fr om each term. The tree shapes\narising in ℋturn out to have quite simple structure. We use /u1D43B/u1D460,/u1D456,/u1D45Fto denote the shape obtained\nby taking a length- /u1D460path where both endpoints are distinguished, and attaching a length-/u1D45Fpath\nat vertex/u1D456.\nAnalysis of /u1D434(ℓ)/u1D434(ℓ).ByObservation 5.12 andObservation 5.13 , any tree shape for this term must\narise as a result of first walking ℓself-avoiding steps, then backtracking /u1D460steps, and then walking\nℓ−/u1D460self-avoiding steps for 0 /lessorequalslant/u1D460/lessorequalslantℓ. The shape this gives rise to is /u1D43B2ℓ−2/u1D460,ℓ−/u1D460,/u1D460. Therefore, by\nCorollary 5.10 , with high probability, for all /u1D463:[/u1D458] →Rwe have:\n/angbracketleftBig\n/u1D497(/u1D45B),/u1D434(ℓ)/u1D434(ℓ)/u1D497(/u1D45B)/angbracketrightBig\n=ℓ/summationdisplay.1\n/u1D460=0/angbracketleftBig\n/u1D497(/u1D45B),/u1D440/u1D43B2ℓ−2/u1D460,ℓ−/u1D460,/u1D460/u1D497(/u1D45B)/angbracketrightBig\n±/u1D45C(/u1D45B)=ℓ/summationdisplay.1\n/u1D460=0/angbracketleftbig\n/u1D463,/u1D4472ℓ−2/u1D460/u1D463/angbracketrightbig\n/u1D>0B·/u1D4512ℓ−/u1D460·/u1D45B±/u1D45C(/u1D45B).\nAnalysis of /u1D434(ℓ)/u1D434/u1D434(ℓ).ByObservation 5.12 andObservation 5.13 , the tree shapes arising here must\nfall into one of the following cases.\n1. Length-ℓself-avoiding walk, then /u1D460backtracking edges, and then a length- (ℓ−/u1D460+1)self-\navoiding walk for 0 /lessorequalslant/u1D460/lessorequalslantℓ. The shapes arising in this case are /u1D43B2ℓ−2/u1D460+1,ℓ−/u1D460,/u1D460.\n2. Length- (ℓ+1)self-avoiding walk, then /u1D460backtracking edges, and then a length- (ℓ−/u1D460)self-\navoiding walk for 1 /lessorequalslant/u1D460/lessorequalslantℓ. The shapes arising in this case are /u1D43B2ℓ−2/u1D460+1,ℓ+1−/u1D460,/u1D460.\n3. Length-ℓself-avoiding walk using edge /u1D452for its final step, then a backtracking step that uses\n/u1D452, and then a backtracking step that uses /u1D452for a third time, followed by a self-avoiding walk\nof length- (ℓ−1). The shape arising in this case is simply a length- (2ℓ−1)path with both\nendpoints distinguished.\n16ByCorollary 5.10 , with high probability, for all /u1D463:[/u1D458] →R:\n/angbracketleftBig\n/u1D497(/u1D45B),/u1D434(ℓ)/u1D434/u1D434(ℓ)/u1D497(/u1D45B)/angbracketrightBig\n=/parenleftBig/angbracketleftbig\n/u1D463,/u1D4472ℓ−1/u1D463/angbracketrightbig\n·/u1D4512ℓ−1−/angbracketleftbig\n/u1D463,/u1D4472ℓ+1/u1D463/angbracketrightbig\n·/u1D4512ℓ+1/parenrightBig\n·/u1D45B+\nℓ/summationdisplay.1\n/u1D460=02/angbracketleftbig\n/u1D463,/u1D4472ℓ−2/u1D460+1/u1D463/angbracketrightbig\n·/u1D4512ℓ−/u1D460+1·/u1D45B±/u1D45C(/u1D45B).\nAnalysis of /u1D434(ℓ)(/u1D437−/u1D43C)/u1D434(ℓ).ByObservation 5.12 andObservation 5.13 , the tree shapes fall into one\nof the cases below.\n1. Length- (ℓ+1)self-avoiding walk, then /u1D460nonbacktracking steps, followed by a length- (ℓ−/u1D460+1)\nself-avoiding walk for 1 /lessorequalslant/u1D460/lessorequalslantℓ+1. The shapes arising in this case are /u1D43B2ℓ−2/u1D460+2,ℓ−/u1D460+1,/u1D460.\n2. Length- (ℓ+1)self-avoiding walk where the last used edge is /u1D452, a backtracking step that uses\n/u1D452, and then a backtracking step that uses /u1D452for a third time, followed by a length- (ℓ−1)\nself-avoiding walk. This shape is simply a length-2 ℓpath with both endpoints distinguished.\nByCorollary 5.10 , with high probability, for all /u1D463:[/u1D458] →R:\n/angbracketleftBig\n/u1D497(/u1D45B),/u1D434(ℓ)(/u1D437−/u1D43C)/u1D434(ℓ)/u1D497(/u1D45B)/angbracketrightBig\n=/angbracketleftbig\n/u1D463,/u1D4472ℓ/u1D463/angbracketrightbig\n·/u1D4512ℓ·/u1D45B+ℓ+1/summationdisplay.1\n/u1D460=1/angbracketleftbig\n/u1D463,/u1D4472ℓ−2/u1D460+2/u1D463/angbracketrightbig\n·/u1D4512ℓ−/u1D460+2·/u1D45B±/u1D45C(/u1D45B).\nEstimating/angbracketleftbig\n/u1D497(/u1D45B),/u1D440ℓ(/u1D461)/u1D497(/u1D45B)/angbracketrightbig\n.By combining the above estimates, the value of/angbracketleftbig\n/u1D497(/u1D45B),/u1D440ℓ(/u1D461)/u1D497(/u1D45B)/angbracketrightbig\nis:\n/u1D45B·/a\\}bracketle{t/u1D463,/u1D45D(/u1D447)/u1D463/a\\}bracketri}ht±/u1D45C(/u1D45B)where/u1D45Dis the univariate polynomial\n/u1D45D(/u1D>06)/defines−/u1D461·/u1D4512ℓ−1/u1D>062ℓ−1·(1−/u1D>062/u1D4512)+/u1D4612/u1D4512ℓ/u1D>062ℓ+ℓ/summationdisplay.1\n/u1D460=0/u1D4512ℓ−/u1D460/u1D>062ℓ−2/u1D460−2/u1D461/u1D4512ℓ−/u1D460+1/u1D>062ℓ−2/u1D460+1+/u1D4612/u1D4512ℓ−/u1D460+1/u1D>062ℓ−2/u1D460.\nTo prove that /a\\}bracketle{t/u1D463,/u1D45D(/u1D447)/u1D463/a\\}bracketri}htis negative for all /u1D463in span{/u1D>131,...,/u1D>13/u1D45F}, it suffices to show it is negative\nfor/u1D463=/u1D>131,...,/u1D>13/u1D45Fsince those are also eigenvectors of /u1D45D(/u1D447). In particular, it suffices to show that\n/u1D45D(/u1D>062)is negative. Towards doing so, we first simplify the expressio n for/u1D45D(/u1D>06).\n/u1D45D(/u1D>06)=(/u1D>06/u1D451)2ℓ·/parenleftbigg\n/u1D4612−/u1D461\n/u1D>06/u1D451·(1−/u1D>062/u1D4512)+/parenleftBig\n1−2/u1D>06/u1D451/u1D461+/u1D451/u1D4612/parenrightBig\n·/parenleftbigg1−(/u1D>062/u1D451)−ℓ−1\n1−(/u1D>062/u1D451)−1/parenrightbigg/parenrightbigg\n.\nWe plug in /u1D>06=/radicalBig\n1+/u1D>00\n/u1D451,ℓ=/u1D43E·/parenleftBig\nlog(1//u1D>00)\n/u1D>00∨1/parenrightBig\nfor some large /u1D43E>0, and/u1D461=1+/u1D6FF√\n(1+/u1D>00)/u1D451for some small\n/u1D6FF >0 we choose in posterity, which yields:\n(/u1D>06/u1D451)2ℓ·/parenleftbigg(1+/u1D6FF)2\n(1+/u1D>00)/u1D451−1+/u1D6FF\n(1+/u1D>00)/u1D451+(1+/u1D6FF)+/parenleftbigg\n1−2(1+/u1D6FF)+(1+/u1D6FF)2\n1+/u1D>00/parenrightbigg\n·1+/u1D>00\n/u1D>00+/u1D45C/u1D43E(1)/parenrightbigg\n=(/u1D>06/u1D451)2ℓ·/parenleftbigg/u1D6FF+/u1D6FF2\n(1+/u1D>00)/u1D451+(1+/u1D6FF)+1+/u1D>00−2(1+/u1D6FF)(1+/u1D>00)+(1+/u1D6FF)2\n/u1D>00+/u1D45C/u1D43E(1)/parenrightbigg\n=(/u1D>06/u1D451)2ℓ·/parenleftbigg\n/u1D6FF+/u1D6FF2\n(1+/u1D>00)/u1D451+(1+/u1D6FF)+/parenleftbigg\n−1−2/u1D6FF+/u1D6FF2\n/u1D>00/parenrightbigg\n+/u1D45C/u1D43E(1)/parenrightbigg\n=(/u1D>06/u1D451)2ℓ·/parenleftbigg\n−/u1D6FF+/u1D6FF\n(1+/u1D>00)/u1D451+/u1D6FF2\n(1+/u1D>00)/u1D451+/u1D6FF2\n/u1D>00+/u1D45C/u1D43E(1)/parenrightbigg\n,\n17where/u1D45C/u1D43E(1)denotes a quantity which vanishes as /u1D43E→ ∞ regardless of the other parameters.\nThe above expression can be made negative by choosing /u1D6FFsmall enough, /u1D43Elarge enough, and\nusing that/u1D451/greaterorequalslant1. Since/u1D>062=/radicalBig\n1+/u1D>00\n/u1D451for some /u1D>00 >0, forℓ=/u1D442/parenleftBig\nlog(1//u1D>00)\n/u1D>00/parenrightBig\nthere exists a choice of /u1D6FF\nsmall enough and an absolute constant /u1D6FC >0 such that /u1D45D(/u1D>062)/lessorequalslant−/u1D6FC. Consequently, for any /u1D463in\nspan{/u1D>131,...,/u1D>13/u1D45F},/a\\}bracketle{t/u1D463,/u1D45D(/u1D447)/u1D463/a\\}bracketri}ht/lessorequalslant−/u1D6FC. /square\n5.2 Outlier eigenspace after degree truncation\nIn this section we show that by picking the truncation thresh old/u1D435to be large enough, the /u1D45F-\ndimensional subspace which witnesses the negative eigenva lues for/u1D440/u1D46E,ℓ(/u1D461)also does the same for\nthe truncated matrix /u1D440/u1D46E/u1D435,ℓ(/u1D461). In this section, we fix /u1D461as specified by Theorem 5.1 and suppress\ndependence on /u1D461, as the same arguments apply regardless of whether we consid er positive or\nnegative/u1D461.\nLemma 5.14 (Truncation doesn’t affect negative witnesses) .Let/u1D46E∼SBM/u1D45B(M,/u1D>0B,/u1D451). For/u1D435>0\nsufficiently large, the following holds true with probability 1−/u1D45C/u1D45B(1). For all unit /u1D465∈span{/u1D>131,...,/u1D>13/u1D45F},\nwe have/angbracketleftBig\n/u1D499(/u1D45B),/u1D440/u1D46E/u1D435,ℓ/u1D499(/u1D45B)/angbracketrightBig\n<−/u1D6FC5.14/u1D45B, where/u1D6FC5.14>0is an absolute constant.\nTo facilitate the discussion, we introduce the following no tion of an affected vertex.\nDefinition 5.15 (Affected vertices) .Let/u1D43Abe a graph. We say that a vertex /u1D463∈/u1D449(/u1D43A)isaffected if its\n2ℓ+1 neighborhood differs in /u1D43Aand/u1D43A/u1D435, otherwise say that /u1D463isunaffected .\nWe also need the following concentration inequality, which follows from [ BLM15 , Lemma 29].\nLemma 5.16. Let/u1D46E∼SBM/u1D45B(M,/u1D>0B,/u1D451). For any vertex /u1D463, let/u∞D4A∃ℓ(/u1D463)denote the distance- ℓneighborhood of\n/u1D463in/u1D46E. Then for any /u1D460/greaterorequalslant0, there exists universal constants /u1D436,/u1D450>0such that\nPr[∃ℓ:|/u∞D4A∃ℓ(/u1D463)|/greaterorequalslant/u1D460/u1D451ℓ]/lessorequalslant/u1D436exp(−/u1D450/u1D460).\nIn other words, the neighborhoods in SBMs look approximately regular. With these ingredients\nassembled, we can now prove Lemma 5.14 .\nProof of Lemma 5.14 .The desired statement follows from Lemma 5.11 once we prove/barex/barex/barex/angbracketleftBig\n/u1D462/u1D462⊤,/u1D440/u1D46E,ℓ−/u1D440/u1D46E/u1D435,ℓ/angbracketrightBig/barex/barex/barex/lessorequalslant\n/u1D45C/u1D435(1), and choose /u1D435as a large enough constant.\nThroughout the proof,we use the following crude estimates: if|/u∞D4A∃2ℓ+1(/u1D463)|/lessorequalslant/u1D43F, then/bardbl/u1D440/u1D46E,ℓ[/u1D463]/bardbl1/lessorequalslant\n2/u1D43F, and/bardblex/bardblex/bardblex/u1D440/u1D46E/u1D435,ℓ[/u1D463]/bardblex/bardblex/bardblex\n1/lessorequalslant/u1D4352ℓ+1.\nBy Hölder’s inequality we have\n/barex/barex/barex/angbracketleftBig\n/u1D465/u1D465⊤,/u1D440/u1D46E,ℓ−/u1D440/u1D46E/u1D435,ℓ/angbracketrightBig/barex/barex/barex/lessorequalslant/bardblex/bardblex/u1D465/u1D465⊤/bardblex/bardblex\nmax/bardblex/bardblex/bardblex/u1D440/u1D46E,ℓ−/u1D440/u1D46E/u1D435,ℓ/bardblex/bardblex/bardblex\n1,\nwhere/bardblex/bardblex/bardblex/u1D440/u1D46E,ℓ−/u1D440/u1D46E/u1D435,ℓ/bardblex/bardblex/bardblex\n1is the entrywise ℓ1norm. Let/u1D463be any vertex in /u1D46E; we first upper bound\nthe expected contribution of the /u1D463th row of/u1D440/u1D46E,ℓ−/u1D440/u1D46E/u1D435,ℓto the norm.\nBy our earlier observation and triangle inequality, we dedu ce\n/bardblex/bardblex/bardblex/u1D440/u1D46E,ℓ−/u1D440/u1D46E/u1D435,ℓ/bardblex/bardblex/bardblex\n1/lessorequalslant2/summationdisplay.1\n/u1D4631[/u1D463is affected ](|/u∞D4A∃2ℓ+1(/u1D463)|+/u1D4352ℓ+1)\n18=2/summationdisplay.1\n/u1D4631[deg(/u1D463)>/u1D435]/summationdisplay.1\n/u1D464∈/u∞D4A∃2ℓ+1(/u1D463)(|/u∞D4A∃2ℓ+1(/u1D464)|+/u1D4352ℓ+1)\n/lessorequalslant2/summationdisplay.1\n/u1D4631[deg(/u1D463)>/u1D435]|/u∞D4A∃4ℓ+2(/u1D463)|(|/u∞D4A∃4ℓ+2(/u1D463)|+/u1D4352ℓ+1),\nsince/u∞D4A∃4ℓ+2(/u1D463) ⊇ /u∞D4A∃ 2ℓ+1(/u1D464). For ease of notation,introduce the quantity /u1D453(/u1D463)/defines|/u∞D4A∃4ℓ+2(/u1D463)|(|/u∞D4A∃4ℓ+2(/u1D463)|+\n/u1D4352ℓ+1). Taking expectations, we see that\nE/bracketleftBig/bardblex/bardblex/bardblex/u1D440/u1D46E,ℓ−/u1D440/u1D46E/u1D435,ℓ/bardblex/bardblex/bardblex\n1/bracketrightBig\n/lessorequalslant2/summationdisplay.1\n/u1D463Pr[deg(/u1D463)>/u1D435]E[/u1D453(/u1D463)|deg(/u1D463)>/u1D435]\nTo proceed, we use the general property that if ℰis an event, /u1D44Bis a random variable with CDF Φ,\nand/u1D453:R→Ris increasing, then E[/u1D453(/u1D44B)|ℰ]/lessorequalslantE[/u1D453(/u1D44B)|/u1D44B/greaterorequalslantΦ−1(Pr[ℰ])] . Takeℰto be the event/braceleftbig\ndeg(/u1D463)>/u1D435/bracerightbig\n, and set/u1D45D/definesPr[deg(/u1D463)>/u1D435]. Then as E[/u1D44B|/u1D44B/greaterorequalslantΦ−1(/u1D45D)]=E[/u1D44B1[/u1D44B/greaterorequalslantΦ−1(/u1D45D)]]\n/u1D45D, we can\nupper bound the conditional expectation by inverting Lemma 5.16 . We see that\nPr[deg(/u1D463)>/u1D435]E[/u1D453(/u1D463)|deg(/u1D463)>/u1D435]/lessorequalslant/uni222B.dsp∞\n/u1D450log(/u1D436//u1D45D)/u1D466/u1D4514ℓ+2(/u1D466/u1D4514ℓ+2+/u1D4352ℓ+1)Pr[|/u∞D4A∃4ℓ+2(/u1D463)|/greaterorequalslant/u1D466/u1D4514ℓ+2]/u1D451/u1D466\n/lessorequalslant/u1D436/uni222B.dsp∞\n/u1D450log(/u1D436//u1D45D)/u1D466/u1D4514ℓ+2(/u1D466/u1D4514ℓ+2+/u1D4352ℓ+1)exp(−/u1D450/u1D466)/u1D451/u1D466\n/lessorequalslant/u1D442(/u1D4358ℓ+4)·/u1D45D(log(1//u1D45D)2+log(1//u1D45D)),\nwhere the last line follows from from an explicit integral co mputation. Note that /u1D45D(log(1//u1D45D)2+\nlog(1//u1D45D))is monotonically increasing for /u1D45D<0.1, so for sufficiently large /u1D435we can simply set /u1D45Dto\nbe the upper bound furnished by Lemma 5.16 . Indeed, since ℓ=/u1D442(1), we conclude that\nPr[deg(/u1D463)>/u1D435]E[/u1D453(/u1D463)|deg(/u1D463)>/u1D435]/lessorequalslant/u1D442(/u1D4358ℓ+6)exp(−/u1D450/u1D435//u1D451)\n=/u1D45C/u1D435(1),\nfor/u1D435= Ω(ℓlogℓ). To summarize, we have just shown that\nE/bracketleftBig/bardblex/bardblex/bardblex/u1D440/u1D46E,ℓ−/u1D440/u1D46E/u1D435,ℓ/bardblex/bardblex/bardblex\n1/bracketrightBig\n=/u1D45C/u1D435(1)/u1D45B\nNext, we bound the variance. Since ℓ=/u1D442(1), an Efron-Stein argument as before shows that\nthe variance of/bardblex/bardblex/bardblex/u1D440/u1D46E,ℓ−/u1D440/u1D46E/u1D435,ℓ/bardblex/bardblex/bardblex\n1is/tildewide/u1D442(/u1D45B). More precisely, consider rerandomizing edge (/u1D456,/u1D457) ∈/u1D46E.\nWith probability 1 −/u1D442(1//u1D45B), the rerandomized edge agrees with the original edge, in whi ch case\ntheℓ1norm doesn’t change. Otherwise, with probability /u1D442(1//u1D45B), we can crudely bound the effect\non theℓ1norm by (log/u1D45B)ℓ. Hence Var/parenleftBig/bardblex/bardblex/bardblex/u1D440/u1D46E,ℓ−/u1D440/u1D46E/u1D435,ℓ/bardblex/bardblex/bardblex\n1/parenrightBig\n/lessorequalslant/tildewide/u1D442(/u1D45B), and the desired concentration\nfollows. /square\nAs a corollary, we obtain the analogous version of Theorem 5.1 for the truncated matrix /u1D440/u1D46E/u1D435,ℓ.\nLemma 5.17 (Spectral gap for truncated matrix) .For sufficiently large /u1D435>0, with high probability over\n/u1D46E∼SBM/u1D45B(M,/u1D>0B,/u1D451), the matrix/u1D440/u1D46E/u1D435,ℓhas at most/u1D45F+1negative eigenvalues, at least /u1D45Fof which are upper\nbounded by −/u1D6FC5.14.\n19Proof. By construction (see Section 4 ),/u1D43Bℓis a principal submatrix of /u1D43Bℓ. This is where we use the\nnon-standard way in which we truncated the degree matrix to g et/u1D43Bℓ. It would not have been true\nif we had naively truncated /u1D437/u1D46E/u1D435.\nSince/u1D43Bℓhas at most /u1D45F+1 negative eigenvalues, by the Cauchy interlacing theorem, its principal\nsubmatrix/u1D43Bℓhas at most /u1D45F+1 negative eigenvalues. The number of negative eigenvalues of\n/u1D440/u1D46E/u1D435,ℓ=/u1D434(ℓ)\n/u1D46E/u1D435/u1D43Bℓ/u1D434(ℓ)\n/u1D46E/u1D435is the same as that of /u1D43Bℓ. Hence/u1D440/u1D46E/u1D435,ℓhas at most /u1D45F+1 negative eigenvalues,\nandLemma 5.14 implies that /u1D440/u1D46E/u1D435,ℓhas at least/u1D45Fnegative eigenvalues. Now using /u1D6FC5.14guaranteed\nby the second part of Lemma 5.14 , we conclude. /square\n6 Robust recovery of a subspace\nIn this section, we consider the following algorithmic prob lem, which we recall from Section 2.2 .\nAssumption 6.1. Let/u1D440be a/u1D45B×/u1D45Bmatrix with at most /u1D45Fnegative eigenvalues. Let /tildewide/u1D440be a corrupted\nversion of/u1D440. Formally, let /tildewide/u1D440=/u1D440+ΔwhereΔis supported on a /u1D6FE/u1D45B×/u1D6FE/u1D45Bsubmatrix, indexed by\n/u1D444⊆ [/u1D45B]. Further, assume that the ℓ1-norm of every row and column of /u1D440andΔare bounded by\n/u1D43E. Observe that this implies that /bardbl/u1D440/bardbl/lessorequalslant/u1D43Eand/bardblΔ/bardbl/lessorequalslant/u1D43E.\nProblem 2.4. Given the corrupted matrix /tildewide/u1D440as input, give an efficient algorithm to output a\nsubspace/u1D448with the following properties:\n1.Low dimensional. The dimension of /u1D448is/u1D442(/u1D45F).\n2.Delocalized. The diagonal entries of its projection matrix Π/u1D448are bounded by /u1D442/parenleftbig/u1D45F\n/u1D45B/parenrightbig.\n3.Preserves delocalized part of negative eigenspace. For any/u1D436-delocalized unit vector /u1D466such\nthat/a\\}bracketle{t/u1D466,/u1D440/u1D466/a\\}bracketri}ht<−Ω(1), we have /a\\}bracketle{t/u1D466,Π/u1D448/u1D466/a\\}bracketri}ht/greaterorequalslantΩ(1).\nWe show that Phase 2 of the algorithm described in Section 4 with/tildewide/u1D440as input indeed solves\nProblem 2.4 , where a more precise quantitative dependence is articulat ed below.\nTheorem 6.2. For every/u1D436,/u1D463,/u1D43E>0, the following holds for all sufficiently small /u1D6FE >0: There is an efficient\nalgorithm that takes in /tildewide/u1D440fromAssumption 6.1 along with/u1D436and/u1D>10as input, and with high probability over\nthe randomness of the algorithm, outputs a subspace /u1D448with the following properties.\n1. The dimension of /u1D448is at most/u1D442/parenleftbig/u1D43E\n/u1D>10·/u1D45F/parenrightbig.\n2. The diagonal entries of the projection matrix Π/u1D448are at most/u1D442/parenleftBig\n/u1D4362/u1D43E3\n/u1D>103·/u1D45F\n/u1D45B/parenrightBig\n.\n3. For any/u1D436-delocalized unit vector /u1D466such that /a\\}bracketle{t/u1D466,/u1D440/u1D466/a\\}bracketri}ht<−/u1D>10, we have /a\\}bracketle{t/u1D466,Π/u1D448/u1D466/a\\}bracketri}ht/greaterorequalslantΩ/parenleftbig/u1D>10\n/u1D43E/parenrightbig.\nTo prove Theorem 6.2 , we analyze Phase 2 of the algorithm in Section 4 . The precise version of\nthe algorithm we analyze is stated below.\nAlgorithm 6.3. The provided input is /tildewide/u1D440according to Assumption 6.1 , and real numbers /u1D>10and/u1D436.\nDefine/u1D>02/defines/u1D>10\n48.\n201. Define /tildewide/u1D440(0)as/tildewide/u1D440. Let/u1D461as a counter initialized at 0, and let Φ(/u1D44B)be the number of eigenvalues\nof/u1D44Bsmaller than −/u1D>02.\n2. While Φ(/tildewide/u1D440(/u1D461))>2/u1D43E\n/u1D>02/u1D45F: compute the projection matrix Π(/u1D461)/definesΠ/lessorequalslant−/u1D>02(/tildewide/u1D440(/u1D461)), choose a random\n/u1D456∈ [/u1D45B]with probabilityΠ(/u1D461)\n/u1D456,/u1D456\nTr(Π(/u1D461)), and define /tildewide/u1D440(/u1D461+1)as the matrix obtained by zeroing out the\n/u1D456-th row and column of /tildewide/u1D440(/u1D461). Then increment /u1D461.\n3. Let/u1D447be the time of termination and let /u1D>0F=2/u1D4362/u1D43E2/u1D45F\n/u1D>022. Compute Π(/u1D447), and define the set of\nindices/u1D446as,\n/u1D446=/braceleftBig\n/u1D456/barex/barex/barexΠ(/u1D447)\n/u1D456,/u1D456/lessorequalslant/u1D>0F\n/u1D45B/bracerightBig\n.\nDefine/tildewideΠas/parenleftBig\nΠ(/u1D447)\n/u1D446,/u1D446/parenrightBig\n/greaterorequalslant/u1D>02//u1D43E, and compute its span /u1D448./one.sup\n4. Output/u1D448.\nWe first bound on the number of eigenvalues of /tildewide/u1D440(/u1D447)smaller than −/u1D>02, thus establishing Item 1\nofTheorem 6.2 .\n6.1 Proof of Item 1 : spectrum of cleaned-up matrix\nLemma 6.4. The matrix /tildewide/u1D440(/u1D447)has the following properties:\n1./tildewide/u1D440(/u1D447)has at most2/u1D43E\n/u1D>02/u1D45Feigenvalues smaller than −/u1D>02.\n2./tildewide/u1D440(/u1D447)is a submatrix /tildewide/u1D440/u1D445,/u1D445where|/u1D445|/greaterorequalslant/parenleftBig\n1−4/u1D43E\n/u1D>02/u1D6FE/parenrightBig\n/u1D45Bwith high probability over the randomness of the\nalgorithm.\nIn service of proving Lemma 6.4 , we prove the following statement about the localization of\noutlier eigenvectors on the indices of the corruptions /u1D444.\nLemma 6.5. Let/u1D445(/u1D461)⊆ [/u1D45B]denote the non-zero rows and columns of /tildewide/u1D440(/u1D461)and let/u1D444(/u1D461)=/u1D444∩/u1D445(/u1D461). Then\nthe following holds:\nTr/parenleftbigg\nΠ/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig\n/u1D444(/u1D461),/u1D444(/u1D461)/parenrightbigg\nTr/parenleftBig\nΠ/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig/parenrightBig/greaterorequalslant/u1D>02\n/u1D43E−/u1D45F\nTr/parenleftBig\nΠ/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig/parenrightBig\nProof. The desired statement follows from the chain of inequalitie s below.\n−/u1D>02·Tr/parenleftBig\nΠ/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig/parenrightBig\n/greaterorequalslantTr/parenleftBig\n/tildewide/u1D440(/u1D461)\n/lessorequalslant−/u1D>02/parenrightBig\n=/angbracketleftBig\n/tildewide/u1D440(/u1D461),Π/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig/angbracketrightBig\n=/angbracketleftBig\n/u1D440/u1D445(/u1D461),/u1D445(/u1D461)+Δ/u1D445(/u1D461),/u1D445(/u1D461),Π/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig/angbracketrightBig\n/one.supRecall that (/u1D44B)/greaterorequalslant/u1D44Edenotes the truncation of the eigendecomposition of /u1D44Bfor eigenvalues at least /u1D44E.\n21/greaterorequalslantTr/parenleftBig/parenleftbig/u1D440/u1D445(/u1D461),/u1D445(/u1D461)/parenrightbig\n/lessorequalslant0/parenrightBig\n−/bardblΔ/bardbl·Tr/parenleftbigg\nΠ/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig\n/u1D444(/u1D461),/u1D444(/u1D461)/parenrightbigg\n/greaterorequalslant−/u1D43E/u1D45F−/u1D43E·Tr/parenleftbigg\nΠ/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig\n/u1D444(/u1D461),/u1D444(/u1D461)/parenrightbigg\n.\nIn the last step, we used the fact that/bardblex/bardblex/u1D440/u1D445(/u1D461),/u1D445(/u1D461)/bardblex/bardblex/lessorequalslant/bardbl/u1D440/bardbl/lessorequalslant/u1D43Eand the number of negative eigenvalues\nof(/u1D440)/u1D445(/u1D461),/u1D445(/u1D461)is at most the number of negative eigenvalues of /u1D440, namely/u1D45F. /square\nWe now prove Lemma 6.4 .\nProof of Lemma 6.4 .By definition of the iterative process, Item 1 is satisfied on termination, and the\nprocess has to terminate since it cannot run for more than /u1D45Bsteps.\nIt remains to prove Item 2 . Towards doing so, first observe that the process terminates i f all\nrows and columns in /u1D444are zeroed out. Indeed, /u1D440itself satisfies the termination condition, and\nif all the rows and columns in /u1D444are zeroed out, then the support is a principal submatrix of /u1D440,\nwhich satisfies the termination condition by Cauchy’s interl acing theorem.\nSuppose the termination condition is not satisfied at iterati on/u1D461, then the/u1D456-th row/column of the\nmatrix is zeroed out where /u1D456is chosen with probabilityΠ/lessorequalslant−/u1D>02(/tildewide/u1D440(/u1D461))/u1D456,/u1D456\nTr/parenleftBig\nΠ/lessorequalslant−/u1D>02(/tildewide/u1D440(/u1D461))/parenrightBig. Consequently, the probability\nthat/u1D456is in/u1D444is equal to:\nTr/parenleftbigg\nΠ/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig\n/u1D444(/u1D461),/u1D444(/u1D461)/parenrightbigg\nTr/parenleftBig\nΠ/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig/parenrightBig,\nwhich by Lemma 6.5 is at least\n/u1D>02\n/u1D43E−/u1D45F\nTr/parenleftBig\nΠ/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig/parenrightBig.\nSince Tr/parenleftBig\nΠ/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440(/u1D461)/parenrightBig/parenrightBig\ncounts the number of eigenvalues smaller than −/u1D>02and the termination\ncondition is not satisfied, the above probability is at least/u1D>02\n2/u1D43E.\nSince the process terminates if all of /u1D444is zeroed out (in which case /u1D445=/u1D444), a standard martingale\nargument shows that the process can last for at most4/u1D43E\n/u1D>02/u1D6FE/u1D45Bsteps with high probability. /square\nWe now proceed to proving Items 2 and3ofTheorem 6.2 .\n6.2 Proofs of Items 2 and3: delocalization and correlation with recovered subspace\nThe proof of Item 2 is fairly straightforward.\nLemma 6.6. All the diagonal entries of Π/u1D448, the projection matrix onto /u1D448, are bounded by2/u1D4362/u1D43E3\n/u1D>023·/u1D45F\n/u1D45B.\nProof. Let/u1D446be the set of indices whose diagonal entries are at most/u1D>0F\n/u1D45B, where/u1D>0F=2/u1D4362/u1D43E2\n/u1D>022·/u1D45F. By\ndefinition, Π/u1D446,/u1D446has entries bounded by/u1D>0F\n/u1D45B, and since Π/u1D446,/u1D446/{ollowsequal(Π/u1D446,/u1D446)/greaterorequalslant/u1D>02\n/u1D43E=/tildewideΠ, the diagonal entries of\n/tildewideΠare at most/u1D>0F\n/u1D45B. All eigenvalues of /tildewideΠare at least/u1D>02\n/u1D43E, and hence/u1D43E\n/u1D>02·/tildewideΠ/{ollowsequalΠ/u1D448, which gives us a bound\nof/u1D43E/u1D>0F\n/u1D>02/u1D45Bon the diagonal entries of Π/u1D448, from which the statement follows. /square\n22Now, we turn our attention to proving Item 3 . We first show that /tildewide/u1D440/u1D445,/u1D445=/tildewide/u1D440(/u1D447)preserves\nnegative correlations with delocalized vectors. Hencefor th, we assume that the high probability\nguarantee from Lemma 6.4 that|/u1D445|/greaterorequalslant/parenleftBig\n1−4/u1D43E\n/u1D>02/u1D6FE/parenrightBig\n/u1D45Bholds.\nLemma 6.7. For unit/u1D436-delocalized vector /u1D466such that /a\\}bracketle{t/u1D466,/u1D440/u1D466/a\\}bracketri}ht<−/u1D>10, we have:/angbracketleftBig\n/u1D466,/tildewide/u1D440/u1D445,/u1D445/u1D466/angbracketrightBig\n<−/u1D>10\n2.\nProof. ByAssumption 6.1 andLemma 6.4 , we can write /tildewide/u1D440/u1D445,/u1D445as/u1D440+/u1D4381+/u1D4382where/u1D4381is supported\non at most/parenleftBig\n4/u1D43E\n/u1D>02+1/parenrightBig\n/u1D6FE/u1D45Brows and/u1D4382is supported on at most/parenleftBig\n4/u1D43E\n/u1D>02+1/parenrightBig\n/u1D6FE/u1D45Bcolumns. Consequently,\nthe entrywise ℓ1norm of/u1D438/defines/u1D4381+/u1D4382is at most10/u1D43E2\n/u1D>02/u1D6FE/u1D45B. The statement follows from the inequality\nbelow for small enough /u1D6FE:\n/a\\}bracketle{t/u1D466,/u1D438/u1D466/a\\}bracketri}ht/lessorequalslant/bardbl/u1D466/u1D466⊤/bardblmax·/bardbl/u1D438/bardbl1/lessorequalslant/u1D442/parenleftbigg10/u1D43E2/u1D4362\n/u1D>02/u1D6FE/parenrightbigg\n</u1D>10\n2./square\nWe have the following immediate corollary of Lemma 6.7 .\nCorollary 6.8. For any/u1D436-delocalized unit vector /u1D466such that /a\\}bracketle{t/u1D466,/u1D440/u1D466/a\\}bracketri}ht<−/u1D>10, we have:\n/angbracketleftBig\n/u1D466,Π/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440/u1D445,/u1D445/parenrightBig\n/u1D466/angbracketrightBig\n/greaterorequalslant/u1D>10/2−/u1D>02\n2/u1D43E−/u1D>02.\nProof. Starting with Lemma 6.7 , we get\n−/u1D463\n2>/angbracketleftBig\n/u1D466,/tildewide/u1D440/u1D445,/u1D445/u1D466/angbracketrightBig\n/greaterorequalslant−/bardblex/bardblex/bardblex/tildewide/u1D440/u1D445,/u1D445/bardblex/bardblex/bardblex·/angbracketleftBig\n/u1D466,Π/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440/u1D445,/u1D445/parenrightBig\n/u1D466/angbracketrightBig\n+(−/u1D>02)·/parenleftBig\n1−/angbracketleftBig\n/u1D466,Π/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440/u1D445,/u1D445/parenrightBig\n/u1D466/angbracketrightBig/parenrightBig\n.\nSince the spectral norm of a matrix is at most its maximum ℓ1norm of a row, /bardbl/tildewide/u1D440/u1D445,/u1D445/bardbl/lessorequalslant2/u1D43E.\nSubstituting in the above inequality and simplifying yield s the desired result. /square\nWe are finally ready to establish Item 3 .\nLemma 6.9. For any unit /u1D436-delocalized vector /u1D466such that /a\\}bracketle{t/u1D466,/u1D440/u1D466/a\\}bracketri}ht<−/u1D>10,\n/a\\}bracketle{t/u1D466,Π/u1D448/u1D466/a\\}bracketri}ht/greaterorequalslantΩ/parenleftBig/u1D>10\n/u1D43E/parenrightBig\n.\nProof. We abbreviate Π/lessorequalslant−/u1D>02/parenleftBig\n/tildewide/u1D440/u1D445,/u1D445/parenrightBig\ntoΠ. Recall/u1D>0F=2/u1D4362/u1D43E2/u1D45F\n/u1D>022from Algorithm 6.3 , and let/u1D446denote\nthe set of indices where the diagonal entries of Πare at most/u1D>0F\n/u1D45B. ByLemma 6.4 , Tr(Π)/lessorequalslant2/u1D43E\n/u1D>02/u1D45F, and\nhence|/u1D446|/lessorequalslant2/u1D43E/u1D45F\n/u1D>02/u1D>0F/u1D45B. By positivity of ΠandCorollary 6.8 ,\n/a\\}bracketle{t/u1D466,Π/u1D446,/u1D446/u1D466/a\\}bracketri}ht+/angbracketleftBig\n/u1D466,Π/u1D446,/u1D446/u1D466/angbracketrightBig\n/greaterorequalslant1\n2/a\\}bracketle{t/u1D466,Π/u1D466/a\\}bracketri}ht/greaterorequalslant1\n2·/u1D>10/2−/u1D>02\n2/u1D43E−/u1D>02.\nThe second term of the LHS is equal to\n/angbracketleftBig\n/u1D466/u1D466⊤\n/u1D446,/u1D446,Π/angbracketrightBig\n/lessorequalslantTr/parenleftBig\n/u1D466/u1D466⊤\n/u1D446,/u1D446/parenrightBig\n·/bardblΠ/bardbl/lessorequalslant/u1D4362|/u1D446|\n/u1D45B/lessorequalslant/u1D>02\n/u1D43E,\n23and consequently,\n/a\\}bracketle{t/u1D466,Π/u1D446,/u1D446/u1D466/a\\}bracketri}ht/greaterorequalslant1\n2·/u1D>10/2−/u1D>02\n2/u1D43E−/u1D>02−/u1D>02\n/u1D43E.\nFinally, since /tildewideΠis obtained by discarding eigenvalues of Π/u1D446,/u1D446of magnitude at most/u1D>02\n/u1D43E:\n/angbracketleftBig\n/u1D466,/tildewideΠ/u1D466/angbracketrightBig\n/greaterorequalslant1\n2·/u1D>10/2−/u1D>02\n2/u1D43E−/u1D>02−2/u1D>02\n/u1D43E/greaterorequalslant/u1D>10/8−3/u1D>02\n/u1D43E/greaterorequalslant/u1D>10\n16/u1D43E.\nThe statement follows since /tildewideΠ/√recedesequalΠ/u1D448. /square\n6.3 Correlation of subspace with communities\nIn the model of adversarial corruptions we are considering, a graph/u1D46Eis sampled from SBM /u1D45B(M,/u1D>0B,/u1D451).\nThe graph then undergoes /u1D6FF/u1D45Badversarial edge insertions and deletions, and the resulti ng graph\n/tildewide/u1D46Eis handed to us.\nIn this section, we prove a guarantee on the subspace /u1D448produced by Phase 2 of the algorithm\ninSection 4 as a special case of Theorem 6.2 .\nLemma 6.10. The subspace /u1D448produced as output at the end of phase 2 of the algorithm describ ed in\nSection 4 has the following guarantees with high probability over the randomness of the stochastic block\nmodel.\n1. The dimension of /u1D448is at most/u1D442/parenleftBig\n/u1D4352ℓ+3\n/u1D6FC5.14/u1D45F/parenrightBig\n.\n2. All the diagonal entries of Π/u1D448are bounded by /u1D442/parenleftbigg\n/u1D4356ℓ+9\n/u1D6FC3\n5.14/u1D>0Bmin·/u1D45F\n/u1D45B/parenrightbigg\n.\n3. For any unit /u1D465inspan{/u1D>131,...,/u1D>13/u1D45F},\n/angbracketleftBig\n/u1D499(/u1D45B),Π/u1D448/u1D499(/u1D45B)/angbracketrightBig\n/greaterorequalslantΩ/parenleftBig/u1D6FC5.14\n/u1D4352ℓ+3·/u1D45B/parenrightBig\n.\nTo prove Lemma 6.10 , we plug in the following into the setup of Theorem 6.2 .\n/u1D440=/u1D440/u1D46E/u1D435,ℓ,Δ/defines/u1D440/tildewide/u1D46E/u1D435,ℓ−/u1D440/u1D46E/u1D435,ℓ,/u1D43E=2/u1D4352ℓ+3,/u1D436=1√/u1D>0Bmin,/u1D6FE=8/u1D4352ℓ+2/u1D6FF, and/u1D>10=/u1D6FC5.14.\nOur proof of Lemma 6.10 is complete once we verify that:\n1.1√/u1D45B/u1D499(/u1D45B)is/u1D436-delocalized and achieves quadratic form less than −/u1D>10with our choice of /u1D440.\n2. The choice of /u1D6FEis indeed a sufficiently small constant depending on /u1D>08,/u1D436, and/u1D43E.\n3./u1D440,Δ, and/u1D43Esatisfy Assumption 6.1 .\nVerifying the assumption on /u1D499(/u1D45B).Note that Lemma 5.14 establishes that the quadratic form of /u1D499(/u1D45B)\nis sufficiently negative, and the delocalization follows fro m the fact that /bardbl/u1D499(/u1D45B)/bardbl∞/lessorequalslant/bardbl/u1D465/bardbl∞/lessorequalslant1√/u1D>0Bmin\nsince/u1D465is a unit vector, along with the fact that /bardbl/u1D499(/u1D45B)/bardblconcentrates around√/u1D45B.\nVerifying the assumption on /u1D6FE.The magnitude of /u1D6FEcan be made sufficiently small by choosing\n/u1D6FFas a sufficient small constant.\nVerifying the assumption on /u1D440,Δ, and/u1D43E.The below lemma proves why our choice of /u1D440,Δand\n/u1D43Esatisfy Assumption 6.1 .\n24Lemma 6.11. The matrix Δ/defines/u1D440/u1D46E/u1D435,ℓ−/u1D440/tildewide/u1D46E/u1D435,ℓis supported on a 8/u1D4352ℓ+2/u1D6FF/u1D45B×8/u1D4352ℓ+2/u1D6FF/u1D45Bprincipal submatrix.\nFurther, each row and column of /u1D440/u1D46E/u1D435,ℓand/u1D440/tildewide/u1D46E/u1D435,ℓhaveℓ1norm bounded by /u1D4352ℓ+3, and consequently each\nrow and column of Δhasℓ1norm bounded by 2/u1D4352ℓ+3.\nProof of Lemma 6.11 .First, observe that /u1D437/u1D46E/u1D435differs from /u1D437/tildewide/u1D46E/u1D435only on rows and columns corre-\nsponding to vertices incident to a corrupted edge, of which t here are at most 2 /u1D6FF/u1D45B.\nNext, we bound the number of vertices whose neighborhoods di ffer between /u1D46E/u1D435and/tildewide/u1D46E/u1D435.\nObserve that if a vertex /u1D463has differing neighborhoods between /u1D46E/u1D435and/tildewide/u1D46E/u1D435, then either /u1D463is\nincident to an edge corruption, or /u1D463has a neighbor in /u1D46E/u1D435∪/tildewide/u1D46E/u1D435that is truncated in one of /tildewide/u1D46E/u1D435or\n/u1D46E/u1D435but not the other.\nThe number of vertices incident to an edge corruption is at mo st 2/u1D6FF/u1D45B. We can bound the number\nof vertices of the second kind by:\n/summationdisplay.1\n/u1D463∈/u1D449(/u1D46E)/summationdisplay.1\n/u1D462∈/u1D441/u1D46E/u1D435∪/tildewide/u1D46E/u1D435(/u1D463)|1[deg/u1D46E(/u1D462)>/u1D435]−1[deg/tildewide/u1D46E(/u1D462)>/u1D435]|\n=/summationdisplay.1\n/u1D462∈/u1D449(/u1D46E)/summationdisplay.1\n/u1D463∈/u1D441/u1D46E/u1D435∪/tildewide/u1D46E/u1D435(/u1D462)|1[deg/u1D46E(/u1D462)>/u1D435]−1[deg/tildewide/u1D46E(/u1D462)>/u1D435]|\n/lessorequalslant/summationdisplay.1\n/u1D462∈/u1D449(/u1D46E)/u1D435·|1[deg/u1D46E(/u1D462)>/u1D435]−1[deg/tildewide/u1D46E(/u1D462)>/u1D435]|\n/lessorequalslant2/u1D6FF/u1D435/u1D45B\nwhere the last inequality is because deg/u1D46E(/u1D462)can differ from deg /tildewide/u1D46E(/u1D462)only when /u1D462is incident\nto an edge corruption. Thus, the number of vertices with diffe ring neighborhoods is at most\n2/u1D6FF(/u1D435+1)/u1D45B/lessorequalslant4/u1D6FF/u1D435/u1D45B.\nFinally,/u1D440/tildewide/u1D46E/u1D435,ℓand/u1D440/u1D46E/u1D435,ℓcan differ only on rows and columns corresponding to vertices that\nare at most 2 ℓ+1 away in either /tildewide/u1D46E/u1D435or/u1D46E/u1D435from a vertex with differing neighborhoods. Since the\ndegrees in both graphs are bounded by /u1D435, there are at most 8 /u1D6FF/u1D4352ℓ+3/u1D45Bsuch vertices.\nTheℓ1norm of any row /u1D463of either/u1D440/tildewide/u1D46E/u1D435,ℓor/u1D440/u1D46E/u1D435,ℓis bounded by the total number of walks\nleaving/u1D463of length at most 2 ℓ+2, which is at most /u1D4352ℓ+3. Thus, the ℓ1norm of any row of Δis at\nmost 2/u1D4352ℓ+3, which completes the proof. /square\n7 Rounding algorithm\nIn this section we fill in the details for Section 4.1 , which analyzed the rounding guarantees. First,\nwe review the rounding procedure described in Section 4 . From phase 2 of the algorithm, we\nobtain a constant-dimensional delocalized subspace /u1D448which has constant correlation with the\ntrue communities. We process /u1D448into a new matrix /u1D440′∈R/u1D45B×/u1D45F′which is directly used in the\nrounding scheme. Consider the matrix Ψ/u1D45F′of the/u1D45F′right eigenvectors of /u1D447corresponding to /u1D>062,\nassumed to be above the KS threshold:\nΨ/u1D45F′/defines/bracketleftbig\n/u1D>131···/u1D>13/u1D45F′/bracketrightbig\n.\nThe rows of Ψ/u1D45F′induce row embeddings {/u1D>19/u1D456}/u1D456∈[/u1D458]intoR/u1D45F′. We then rescale the rows of /u1D440′by a\nuniform factor /u1D450such that we can express each row of /u1D450·/u1D440′as a convex combination of the row\nembeddings {/u1D>19/u1D456}/u1D456∈[/u1D458]. These weights are then used to sample a valid community assi gnment.\n25InLemma 7.1 we prove that the scaling /u1D450used to obtain the rounding weights can be taken to\nbeΩ(√/u1D45B)so long as the subspace /u1D448from phase 2 of the algorithm is delocalized.\nLemma 7.1. Suppose that diag(Π/u1D448)/lessorequalslant/u1D442(1\n/u1D45B). There exists /u1D450= Θ(√/u1D45B)such that the scaled rows of /u1D450·/u1D440′\nlie within the convex hull of the /u1D>19/u1D456’s.\nProof. We first show that we can pick some /u1D450= Ω(1). Since/u1D>131=1is the trivial eigenvector of /u1D447,\nby spectral theory we have that/angbracketleftbig\n1,/u1D>13/u1D457/angbracketrightbig\n/u1D>0B=0 for/u1D457≠1. This implies that the convex combination/summationtext.1/u1D458\n/u1D456=1/u1D>0B(/u1D456)/u1D>19/u1D456=0∈R/u1D45F−1, so the origin is in the convex hull of the row embeddings /u1D>19/u1D456. Note that this\nalone does not suffice to show that we can take /u1D450>0 because the origin might be on the boundary\nof the hull.\nHowever, as we’ll show in Lemma 7.2 below, the origin in fact lies in the interior of the convex\nhull of the /u1D>19/u1D456’s, which permits us to take /u1D450>0. Furthermore, the minimum distance from the\norigin to the boundary is bounded below by a positive constan t (possibly depending on /u1D458), because\n0 lies in the interior and the /u1D>19/u1D456’s are constant-sized objects. So we can take /u1D450= Ω(1). To attain\n/u1D450= Θ(√/u1D45B), first recall that diag (Π/u1D448)/lessorequalslant/u1D442(1\n/u1D45B)by assumption, which implies that diag (Π′\n/u1D448)/lessorequalslant/u1D442(1\n/u1D45B)\nsince/u1D448′⊆/u1D448. By definition, Π/u1D448′=/u1D440′/u1D440′⊤, theℓ2norm of each row of /u1D440′is/u1D442(1√/u1D45B), from which\nthe desired claim holds. /square\nLemma 7.2. Assuming that /u1D>0B >0, we have 0∈int(conv(/u1D>191,...,/u1D>19/u1D458)).\nProof. First, note that since the column eigenvectors /u1D>13/u1D457are orthogonal with respect to /a\\}bracketle{t·,·/a\\}bracketri}ht/u1D>0B, and\n/u1D>0B > 0, the matrix of column eigenvectors has full row rank. On the other hand, we know that\n0∈conv(/u1D>191,...,/u1D>19/u1D458)because/summationtext.1\n/u1D456/u1D>0B(/u1D456)/u1D>19/u1D456=0. Since the /u1D>19/u1D456are linearly independent, they in fact lie\non the boundary of the convex hull and the interior of the conv ex hull is a nonempty open set in\nR/u1D45F. As/u1D>0B >0, we have just shown that 0 lies in the interior of the convex h ull. /square\nAcknowledgments\nWe would like to thank Omar Alrabiah and Kiril Bangachev for di ligent feedback on an earlier\ndraft of this paper.\nReferences\n[Abb17] Emmanuel Abbe. Community detection and stochastic block models: recent develop-\nments. 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In\nAppendix A.1 , we argue that our definition is natural and interpret it in var ious natural settings.\nInAppendix A.2 , we show that our definition implies the other definitions of wea k recovery found\nin the literature.\nFor self-containedness, we reproduce our and previous defini tions of weak recovery below.\nDefinition 4.4 (Weak recovery) .LetΨ/defines/bracketleftbig\n/u1D>132···/u1D>13/u1D458/bracketrightbig\n∈R/u1D458×(/u1D458−1)be the matrix of the top- (/u1D458−1)\nnontrivial eigenvectors of the transition matrix /u1D447of a stochastic block model.\nFor/u1D>0C > 0, we say that a (randomized) algorithm for producing commun ity assignments\n/hatwide/u1D47F∈R/u1D45B×/u1D458achieves/u1D>0C-weak recovery if\n/angbracketleftBig\nE/hatwide/u1D47FΨ,/u1D47FΨ/angbracketrightBig\n/greaterorequalslant/u1D>0C/bardblE/hatwide/u1D47FΨ/bardbl/u1D439/bardbl/u1D47FΨ/bardbl/u1D439,\nwhere/u1D435Ψ/defines(/u1D435Ψ)(/u1D435Ψ)⊤for a matrix /u1D435∈R/u1D45B×/u1D458.\nNext, we recall the notion of weak recovery used in [ DdNS22 ]. Note that their metric is\nspecialized to the symmetric 2-community block model.\nDefinition A.1 (Weak recovery — symmetric 2-community SBM) .We say that an algorithm\nachieves weak recovery for the symmetric 2-community SBM if it outputs a labeling /hatwide/u1D499∈{±1}/u1D45B\nsuch that\nE\n/u1D499,/u1D46E[|/a\\}bracketle{t/u1D499,/hatwide/u1D499/a\\}bracketri}ht|]/greaterorequalslantΩ(/u1D45B).\nMoving beyond the symmetric 2-community setting, we recall a more general notion of weak\nrecovery present in the literature, which applies to arbitr ary/u1D458-community SBMs [ Abb17 ]. To\ndefine it, it is helpful to introduce the shorthand Ω/u1D456={/u1D463∈ [/u1D45B]:/u1D47F/u1D463=/u1D456}for/u1D456∈ [/u1D458].\n29Definition A.2 (Weak recovery — nontrivial partition) .We say that an algorithm achieves weak\nrecovery in the sense of nontrivial partition if it outputs w ith high probability a subset of vertices\n/u1D446⊆/u1D449such that there exists communities /u1D456,/u1D457∈ [/u1D458]with\n|Ω/u1D456∩/u1D446|\n|Ω/u1D456|−|Ω/u1D457∩/u1D446|\n|Ω/u1D457|/greaterorequalslantΩ(1).\nNote that Definition A.2 implies Definition A.1 , by setting /hatwide/u1D499/u1D463=+1 for each/u1D463∈/u1D446and/hatwide/u1D499/u1D463=−1\nfor each/u1D463∈/u1D446.\nAnother reasonable definition would require that the mutual i nformation between /u1D47Fand/hatwide/u1D47Fis\nnontrivially large.\nDefinition A.3 (Weak recovery — mutual information) .We say that an algorithm achieves weak\nrecovery in the sense of mutual information if\n/summationdisplay.1\n/u1D463∈[/u1D45B]I(/u1D47F/u1D463;/hatwide/u1D47F/u1D463)/greaterorequalslantΩ(/u1D45B).\nBy chain rule and monotonicity of entropy,the above guarant ee also implies that I (/u1D47F;/hatwide/u1D47F)/greaterorequalslantΩ(/u1D45B).\nA.1 Interpreting our new notion of weak recovery\nIn this section, we show that if the correlation goes to 1 then we achieve exact recovery on /u1D45F\ncommunities, and if the algorithm uses no information about the graph then we do not achieve\n/u1D>0C-weak recovery for any /u1D>0C >0 as/u1D45B→ ∞ .\nFirst, let us establish some elementary formulas which will prove useful. Since Ψ∈R/u1D458×(/u1D458−1)is\nthe matrix of right eigenvectors of /u1D447with the all-ones vector removed, we have ΨΨ⊤= Π−1−1/u1D4581⊤\n/u1D458.\nWe therefore have\n/a\\}bracketle{t/u1D44AΨ,/u1D47FΨ/a\\}bracketri}ht=Tr/parenleftbigΨ⊤/u1D44A⊤/u1D47FΨΨ⊤/u1D47F⊤/u1D44AΨ/parenrightbig\n=/bardblex/bardblexΨ⊤/u1D44A⊤/u1D47FΨ/bardblex/bardblex2\n/u1D439.\nIt’s not hard to see that conditioned on /u1D47Fand/u1D44A, we have\n(/u1D44A⊤/u1D47F)/u1D456/u1D457=/u1D45BPr/u1D463∼[/u1D45B],/hatwide/u1D47F[/hatwide/u1D47F/u1D463=/u1D456|/u1D47F/u1D463=/u1D457],\nand below we suppress the subscript in the probabilities for sake of conciseness. Notice that the\nabove formula connects our recovery metric to the confusion matrix , a popular metric for multiclass\nclassification in machine learning [ DG06 ]. In particular, the confusion matrix /u1D443∈R/u1D458×/u1D458is defined\nby/u1D443/u1D456/u1D457/definesPr[/hatwide/u1D47F/u1D463=/u1D456|/u1D47F/u1D463=/u1D457], which is exactly the expression on the RHS.\nOn the other hand, we can calculate\n/a\\}bracketle{t/u1D44AΨ,/u1D47FΨ/a\\}bracketri}ht=/angbracketleftbig\n/u1D44A(Π−1−1/u1D4581⊤\n/u1D458)/u1D44A⊤,/u1D47F(Π−1−1/u1D4581⊤\n/u1D458)/u1D47F⊤/angbracketrightbig\n=/angbracketleftbig\n/u1D44AΠ−1/u1D44A⊤,/u1D47FΠ−1/u1D47F⊤/angbracketrightbig\n−/angbracketleftbig\n1/u1D45B1⊤\n/u1D45B,/u1D47FΠ−1/u1D47F⊤+/u1D44AΠ−1/u1D44A⊤/angbracketrightbig\n+/u1D45B2.\nNow using the fact that 1⊤\n/u1D45B/u1D47F=/u1D45B/u1D>0B⊤(1+˜/u1D442(1/√/u1D45B)), the fact that /u1D>0Bis a distribution, and writing\n/hatwide/u1D>0B/defines1\n/u1D45B1⊤\n/u1D45B/u1D44A, we obtain\n/a\\}bracketle{t/u1D44AΨ,/u1D47FΨ/a\\}bracketri}ht=/bardblex/bardblex/bardblexΠ−1/2/u1D44A⊤/u1D47FΠ−1/2/bardblex/bardblex/bardblex2\n/u1D439−/u1D45B2/bardblex/bardblex/bardblexΠ−1/2/hatwide/u1D>0B/bardblex/bardblex/bardblex2\n2+˜/u1D442(/u1D45B3/2)\n30=/u1D45B2/parenlefttpA/parenleftexA\n/parenleftbtA/summationdisplay.1\n/u1D456,/u1D457(/u1D>0B(/u1D456)/u1D>0B(/u1D457))−1Pr[/hatwide/u1D47F/u1D463=/u1D456,/u1D47F/u1D463=/u1D457]2−/bardblex/bardblex/bardblexΠ−1/2/hatwide/u1D>0B/bardblex/bardblex/bardblex2\n2/parenrighttpA/parenrightexA\n/parenrightbtA+˜/u1D442(/u1D45B3/2)\n=/u1D45B2/parenlefttpA/parenleftexA\n/parenleftbtA/summationdisplay.1\n/u1D456,/u1D457/u1D>0B(/u1D456)−1/u1D>0B(/u1D457)Pr[/hatwide/u1D47F/u1D463=/u1D456|/u1D47F/u1D463=/u1D457]2−/summationdisplay.1\n/u1D456/u1D>0B(/u1D456)−1Pr[/hatwide/u1D47F/u1D463=/u1D456]2/parenrighttpA/parenrightexA\n/parenrightbtA+˜/u1D442(/u1D45B3/2) (4)\nwith high probability over /u1D47F.\nExact recovery for /u1D45Fcommunities Suppose there are /u1D45Feigenvalues of /u1D447above the KS threshold.\nLet/u1D47Cbe the subspace spanned by/braceleftBig\n/u1D74D(/u1D45B)\n1,.../u1D74D(/u1D45B)\n/u1D45F/bracerightBig\n. If we achieve perfect correlation with /u1D47C, then\nany orthonormal basis /u1D469of/u1D47Cwith/u1D47C=/u1D469/u1D469⊤exactly recovers at least /u1D45Fof the communities. This\nfollows from the fact that /u1D469has full row rank, so there are /u1D45Frows which are pairwise distinct. From\nthese rows we can exactly recover /u1D45Fcommunities.\nInterpretation of the metric in the symmetric /u1D458-community block model Suppose we are in the\nsymmetric/u1D458-community block model, with parameter /u1D>06, so that/u1D>0B=1\n/u1D4581and/u1D447=/u1D458/u1D>06/u1D43C+(1−/u1D>06)11⊤\nwith/u1D44E≠/u1D44F. The trivial top eigenvector is 1, and for the remaining eigenvectors we can pick any\northonormal basis in 1⊥. ThenΨΨ⊤=/u1D458/u1D43C−11⊤, and from Eq. (4) , when we achieve weak recovery\nwe have\n1\n/u1D45B2/a\\}bracketle{t/u1D44AΨ,/u1D47FΨ/a\\}bracketri}ht=/summationdisplay.1\n/u1D456,/u1D457/parenleftBig\nPr[/hatwide/u1D47F/u1D463=/u1D456|/u1D47F/u1D463=/u1D457]2−Pr[/hatwide/u1D47F/u1D463=/u1D456]2/parenrightBig\n+˜/u1D442(1/√\n/u1D45B)\n/greaterorequalslantΩ(1).\nTherefore, when we succeed at weak recovery in this setting, there exists some /u1D457such that\n/summationdisplay.1\n/u1D456Pr[/hatwide/u1D47F/u1D463=/u1D456|/u1D47F/u1D463=/u1D457]2/greaterorequalslant/summationdisplay.1\n/u1D456Pr[/hatwide/u1D47F/u1D463=/u1D456]2+Ω(1).\nFor example, this correlation is maximized if after permuti ng the columns of /u1D44Awe have/u1D44A=/u1D44B.\nComparing to natural baselines There are two obvious baselines to sanity check our metric fo r:\nlabeling vertices randomly according to the prior /u1D>0B, and deterministically guessing the largest\ncommunity. In fact, we will characterize the correlation fo r any randomized rounding scheme\nwhich is independent of the true assignment. In this case, we have/u1D44A=1/u1D464⊤, where/u1D464is a simplex\nvector which is independent of /u1D47F. Then we have\n/a\\}bracketle{t/u1D44AΨ,/u1D47FΨ/a\\}bracketri}ht=/bardblex/bardblexΨ⊤/u1D4641⊤/u1D47FΨ/bardblex/bardblex2\n/u1D439\n=/u1D45B2/bardblex/bardblexΨ⊤/u1D464/u1D>0B⊤Ψ/bardblex/bardblex2\n/u1D439(1+˜/u1D442(1/√\n/u1D45B))\n=˜/u1D442(/u1D45B3/2),\nwhere the second line holds with high probability over /u1D47F, and the third line holds because we\nremoved 1fromΨ. On the other hand, we know that /bardbl/u1D44AΨ/bardbl/u1D439/bardbl/u1D47FΨ/bardbl/u1D439= Θ(/u1D45B2), so indeed these\nbaselines fail to achieve weak recovery.\n31Indistinguishability and the KS threshold Finally, suppose that two communities are actually\nindistinguishable, in the precise sense that their corresp onding rows in /u1D447are identical. In our\nrounding scheme, notice that the rows for Ψare identical so indeed these two communities are\n“merged” from the perspective of the correlation guarantee . In other words, we could have just\nreduced the weights /u1D44A∈R/u1D45B×/u1D458into weights /u1D44A′∈R/u1D45B×(/u1D458−1)in the obvious way.\nThis is an appealing property because it is a modest step towa rds understanding the more\ngeneral setting where two communities are computationally distinguishable . For example, one could\nconsider a 4-community setup where communities 1 and 2 are co mputationally indistinguishable\n(as well as communities 3 and 4), but the pairs of communities (1,2)and(3,4)are computationally\ndistinguishable from each other.\nA.2 Comparing the different notions of weak recovery\nIn this section, we show that our definition of weak recovery Definition 4.4 implies Definitions A.2\nandA.3.\nLemma A.4 (Weak recovery implies nontrivial partition) .If a recovery algorithm achieves weak recovery\nin the sense of Definition 4.4 , then it also satisfies Definition A.2 .\nProof. If/a\\}bracketle{t/u1D44AΨ,/u1D47FΨ/a\\}bracketri}ht/greaterorequalslantΩ(/u1D45B2), then it follows that\n/summationdisplay.1\n/u1D450,/u1D450′∈[/u1D458]/u1D>0B(/u1D450)−1/u1D>0B(/u1D450′)Pr[/hatwide/u1D47F/u1D463=/u1D450|/u1D47F/u1D463=/u1D450′]2−/summationdisplay.1\n/u1D450/u1D>0B(/u1D450)−1Pr[/hatwide/u1D47F/u1D463=/u1D450]2/greaterorequalslantΩ(1),\nand we conclude that there must exist some /u1D450′such that\n/summationdisplay.1\n/u1D450/u1D>0B(/u1D450)−1(Pr[/hatwide/u1D47F/u1D463=/u1D450|/u1D47F/u1D463=/u1D450′]2−Pr[/hatwide/u1D47F/u1D463=/u1D450]2)/greaterorequalslantΩ(1).\nIt follows that there must exist some /u1D456∈ [/u1D458]for which Pr[/hatwide/u1D47F/u1D463=/u1D456|/u1D47F/u1D463=/u1D450′]/greaterorequalslantPr[/hatwide/u1D47F/u1D463=/u1D456]+Ω(1).\nAfter relabeling the communities for /u1D47F, using any /u1D>0E∈/u1D446/u1D458which swaps /u1D450′and/u1D456, we can assume\nthat in fact\nPr[/hatwide/u1D47F/u1D463=/u1D456|/u1D47F/u1D463=/u1D456]/greaterorequalslantPr[/hatwide/u1D47F/u1D463=/u1D456]+Ω(1) (5)\nMoreover, with /u1D456as defined above, since Pr[/hatwide/u1D47F/u1D463=/u1D456]=/summationtext.1\n/u1D450Pr[/hatwide/u1D47F/u1D463=/u1D456|/u1D47F/u1D463=/u1D450]/u1D>0B(/u1D450), by averaging\nthere must exist some /u1D457≠/u1D456such that\nPr[/hatwide/u1D47F/u1D463=/u1D456|/u1D47F/u1D463=/u1D456]/greaterorequalslantPr[/hatwide/u1D47F/u1D463=/u1D456|/hatwide/u1D47F/u1D463=/u1D457]+Ω(1). (6)\nWith Eq. (6) in hand, it is easy to see how to define the set /u1D446for the partition in Definition A.2 . In\nparticular, set /u1D446=/braceleftbig\n/u1D463:ˆ/u1D44B/u1D463=/u1D456/bracerightbig\n, and pick the same /u1D456and/u1D457as in the above guarantee. /square\nLemma A.5 (Weak correlation implies nontrivial mutual information) .If/a\\}bracketle{t/u1D44AΨ,/u1D47FΨ/a\\}bracketri}ht/greaterorequalslantΩ(/u1D45B2), then/summationtext.1\n/u1D463∈[/u1D45B]I(/u1D47F/u1D463;/hatwide/u1D47F/u1D463)/greaterorequalslantΩ(/u1D45B).\nProof. Notice that Eq. (5) and another averaging argument implies that there exists a s ubset/u1D446of\nΩ(/u1D45B)vertices such that for each /u1D463∈/u1D446,\nPr/hatwide/u1D47F/u1D463,/u1D47F,/u1D46E[/hatwide/u1D47F/u1D463=/u1D456|/u1D47F/u1D463=/u1D456]/greaterorequalslantPr/hatwide/u1D47F/u1D463,/u1D47F,/u1D46E[/hatwide/u1D47F/u1D463=/u1D456]+Ω(1),\n32where we have used the subscript to emphasize that the probab ility no longer samples a random\n/u1D463∼ [/u1D45B]. Now by Pinsker’s inequality it suffices to lower bound /u1D451TV(/u1D45D/u1D47F/u1D463,/hatwide/u1D47F/u1D463,/u1D45D/u1D47F/u1D463/u1D45D/hatwide/u1D47F/u1D463). Since/u1D>0B(/u1D456)/greaterorequalslant\nΩ(1), the above inequality implies that /u1D451TV(/u1D45D/u1D47F/u1D463,/hatwide/u1D47F/u1D463,/u1D45D/u1D47F/u1D463/u1D45D/hatwide/u1D47F/u1D463)/greaterorequalslantΩ(1), so we conclude that\nI(/u1D47F/u1D463;/hatwide/u1D47F/u1D463)/greaterorequalslantΩ(1).\nSumming up over all /u1D463and using the fact that mutual information is nonnegative, w e conclude\nthat I(/u1D47F;/hatwide/u1D47F)/greaterorequalslantΩ(/u1D45B). /square\n33"    },    {        "title": "2402.13923v1.Improved_Lower_Bound_on_the_Number_of_Pseudoline_Arrangements.pdf",        "content": "Improved Lower Bound on the Number of\nPseudoline Arrangements∗\nJustin Dallant /envel⌢pe\nUniversité libre de Bruxelles, Belgium\nAbstract\nWe show that for large enough n, the number of non-isomorphic pseudoline arrangements of order n\nis greater than 2c·n2for some constant c >0.2604, improving the previous best bound of c >0.2083\nby Dumitrescu and Mandal (2020). Arrangements of pseudolines (and in particular arrangements of\nlines) are important objects appearing in many forms in discrete and computational geometry. They\nhave strong ties for example with oriented matroids, sorting networks and point configurations. Let\nBnbe the number of non-isomorphic pseudoline arrangements of order nand let bn:=log2(Bn).\nThe problem of estimating bndates back to Knuth, who conjectured that bn≤0.5n2+o(n2)\nand derived the first bounds n2/6−O(n)≤bn≤0.7924( n2+n). Both the upper and the lower\nbound have been improved a couple of times since. For the upper bound, it was first improved to\nbn<0.6988n2(Felsner, 1997), then bn<0.6571n2by Felsner and Valtr (2011), for large enough n.\nIn the same paper, Felsner and Valtr improved the constant in the lower bound to c >0.1887, which\nwas subsequently improved by Dumitrescu and Mandal to c >0.2083. Our new bound is based on a\nconstruction which starts with one of the constructions of Dumitrescu and Mandal and breaks it into\nconstant sized pieces. We then use software to compute the contribution of each piece to the overall\nnumber of pseudoline arrangements. This method adds a lot of flexibility to the construction and\nthus offers many avenues for future tweaks and improvements which could lead to further tightening\nof the lower bound.\n2012 ACM Subject Classification Theory of computation →Computational geometry; Mathematics\nof computing→Enumeration\nKeywords and phrases pseudoline arrangement, counting, lower bound, construction\nFunding Justin Dallant : Supported by the French Community of Belgium via the funding of a FRIA\ngrant.\n1 Introduction\nA pseudoline arrangement of order nis a collection of nunbounded x-monotone simple\ncurves in the Euclidean plane such that every pair of curves intersects exactly once, and\ncrosses at their intersection. It is said to be simple if no three curves meet at a point. Every\npseudoline arrangement induces a natural labeling on the curves composing it, given by the\norder in which they appear, from bottom to top, to the left of the first intersection (see\nFigure 1). We say two pseudoline arrangements are isomorphic if they can be mapped to\neach other by a homeomorphism of the plane preserving this natural labeling1(or more\nintuitively, they are “combinatorially identical”).\nIsomorphism classes of (simple) pseudoline arrangements (or variations thereof) appear\n∗This manuscript was accepted at SoCG’24 and will be merged with Fernando Cortés Kühnast, Stefan\nFelsner and Manfred Scheucher’s manuscript “An Improved Lower Bound on the Number of Pseudoline\nArrangements” for the proceedings\n1There are notions of pseudoline arrangements and of their isomorphism in the literature which differ\nfrom the one we use here (e.g. working in the projective plane, having explicitly labeled curves or not\nrequiring the mappings to preserve the labelings). In terms of counting the number of non-isomorphic\npseudoline arrangements of order n, these differences only affect lower order factors and have no bearing\non the results of this paper.arXiv:2402.13923v1  [cs.CG]  21 Feb 20242 Improved Lower Bound on the Number of Pseudoline Arrangements\n1234\nFigure 1 A pseudoline arrangement of order 4with its natural labeling.\nin many different forms in discrete geometry and beyond. Among those are configurations of\npoints, wiring diagrams, zonotopal tilings, reorientation classes of oriented matroids of rank\n3 (see e.g. [7] and [6] for more general treatments of the topic).\nn Bn\n1 1\n2 1\n3 2\n4 8\n5 62\n6 908\n7 24 698\n8 1 232 944\n9 112 018 190\n10 18 410 581 880\n11 5 449 192 389 984\n12 2 894 710 651 370 536\n13 2 752 596 959 306 389 652\n14 4 675 651 520 558 571 537 540\n15 14 163 808 995 580 022 218 786 390\n16 76 413 073 725 772 593 230 461 936 736\nTable 1 The known values of Bn. The values for n≤9are from [10, p. 35]. The values for\nn= 10,11,12are from [5], [15] and [13] respectively. The value for n= 16is from [14].\nLetBnbe the number of non-isomorphic simple pseudoline arrangements of order n,\nand let bn:=log2Bn. The values of Bnfor1≤n≤16are known exactly and can\nbe found in Table 1. Here, we are interested in bounding bnas a function of n. In\nhis book [10], Knuth conjectured that bn≤0.5n2+o(n2)and derived the first bounds\nn2/6−O(n)≤bn≤0.7924n2+O(n). The upper bound was derived by showing that the\nnumber of cutpaths in a wiring diagram of order n(which corresponds to the number of\ncombinatorially distinct ways to insert a new curve in a pseudoline arrangement of order\nn) is at most 3n, yielding the bound Bn≤3(n\n2)≤20.7924n2+O(n). The lower bound was\nobtained by a recursive construction, yielding the recurrence Bn≥2n2/8−n/4Bn/2.A more\ngeometric (albeit with smaller constant) recursive construction was later given by Matoušek\nin his book [12]. We review this latter construction below.\nThe upper bound was subsequently improved (for large enough n) tobn<0.6988n2\nby Felsner [5] through a clever encoding of the arrangement, then to bn<0.6571n2by\nFelsner and Valtr [8], through an improved bound on the number of cutpaths. In the sameJ. Dallant 3\n{\nmmm\n{{\nFigure 2 Matoušek’s lower bound construction.\npaper, the lower bound was improved to bn≥0.1887n2by a recursive construction making\nuse of the number of rhombic tilings of a centrally symmetric hexagon. Finally, through\nconstructions which can be seen as generalizations of that of Matoušek, Dumitrescu and\nMandal [3] improved the lower bound to bn≥0.2083n2.\nThe present work builds on one of the constructions of Dumitrescu and Mandal (which\nwe briefly review in the beginning of Section 5), by making the choices throughout less local,\nand with the use of computer software to count the number of choices possible in each “piece”\nof the construction. Our construction yields bn≥0.2604n2, for large enough n.\nMatoušek’s construction.\nMatoušek [12, Sec. 6.2] gave a simple construction yielding a lower bound of 2n2/8on the\nnumber of pseudoline arrangements of order n. One can describe this construction by starting\nwith 2bundles of lines, each consisting of m=n/3horizontal (resp. vertical) lines with unit\ndistance between consecutive lines. This defines a regular grid of m2points. We then insert\na third bundle of mlines with slope−1, which together pass through 3(m2+ 1)/4points of\nthe grid, assuming mis odd (see Figure 2). For each of these, we have two choices leading to\na different arrangement: we can bend the line to pass either above or below the intersection.2\nIt then remains to bound the number of ways in which the lines within each of the bundles\ncan be arranged with respect to each-other. This leads to the recurrence\nB3m≥23m2/4(Bm)3,\nwhich solves to Bn≥2n2/8.\n2 Pseudochord arrangements\nFor our construction, we will work with a generalization of pseudoline arrangements, which\nwe call pseudochord arrangements. Note that similar objects have been considered previously,\n2As described, the obtained arrangement does not consist of x-monotone curves, but this can be remedied\nsimply by rotating the grid slightly. We choose not to do so for consistency with the later constructions\nin the paper.4 Improved Lower Bound on the Number of Pseudoline Arrangements\n0\n1\n23450\n1\n2345\n0\n1\n2345 0\n1\n2\n345(A) (B)\n(C) (D)\nFigure 3 Illustrations (A)and(B)represent two different embeddings of the same pseudochord\narrangement, while (C)is a different arrangement with the same matching. Illustration (D)is an\nembedding of a matching distinct from the three others.\nunder the name of weak pseudoline arrangements [2,4] and partial pseudoline arrangements\n[8,14].\n▶Definition 1. Consider a simple closed curve Cin the plane. Consider also 2kdistinct\npoints on C, labeled 0to2k−1, in counter-clockwise order around C. Consider a perfect\nmatching{a1, b1},{a2, b2}, . . .{ak, bk}on the 2kpoints. For each pair in the matching, add\na simple curve with the corresponding points as endpoints, labeled by these endpoints, with\nthe following restrictions:\nthe curves lie inside the bounded side of C, with the exception of their two endpoints\n(which lie on C);\ntwo curves intersect at most once, and if they do, they cross at the point of intersection.\nWe call this set of curves (including C), together with the 2klabeled points an embedded\npseudochord arrangement of order k. We refer to Cas the bounding curve, to the kother\ncurves as (pseudo-)chords, and to the 2kpoints as endpoints (of the corresponding chords).\nIf no three chords intersect at a common point, the arrangement is said to be simple.\nWe can give a canonical orientation to each chord in an embedded pseudochord arrange-\nment based on the labels of the endpoints: for a chord connecting two endpoints labeled\naandb, with a < b, we orient the chord from atob. This allows us to unambiguously\ndistinguish between the two regions inside the bounding curve Con both sides of the chord,\nwhich we call “above” and “below” the chord.\n▶Definition 2. We call chirotope of a pseudochord arrangement the mapping which to each\ntriplet (c1, c2, c3)of chord labels in a pseudochord arrangement assigns ⊥ifc1andc2do not\nintersect, 1if their intersection is above c3,−1if it is below, and 0if the three chords meet\nat a single point.\nWe say that two embedded pseudochord arrangements are isomorphic (or, more bluntly,\n“the same”) if they have the same chirotope. We call pseudochord arrangement the equivalence\nclass of an embedded pseudochord arrangement under this relationship.\nWe call any embedded pseudochord arrangement in this equivalence class an embedding of\nthe pseudochord arrangement.\nWe further group pseudochord arrangements by the matchings they define.\n▶Definition 3. Given some pseudochord arrangement A, we let MAdenote the matching it\ndefines on its labelled endpoints.J. Dallant 5\nFigure 4 Embedding of a (3,2,4)-matching.\nGiven some perfect matching Mon{0,1, . . .2k−1}, We let arr(M)denote the set of\nsimple3pseudochord arrangements AwithMA=M. We call embedding of Many embedding\nof a pseudochord arrangement in arr(M).\nFigure 3 illustrates these definitions. Note that for two matchings of order kwhich differ\nonly by a relabeling of the endpoints which shifts them cyclically (i.e. a mapping of the labels\nof the form st:a∝⇕⊣√∫⊔≀→a+t[mod 2k]) there is a natural one-to-one correspondence between\ntheir pseudochord arrangements. In particular, when we are only interested in counting the\nnumber of pseudochord arrangements of a matching, we may instead choose any arbitrary\nsuch shift of the matching, or not even specify which of these we consider.\nA family of relevant matchings.\nLet us define and give notations for some matchings which are particularly relevant to the\npresent paper.\n▶Definition 4. Letk1, k2, . . . k rbe strictly positive integers and let kbe their sum. We will\nuse the notation (k1, k2, . . . , k r)-matching to denote the matching corresponding to rgroups\nof chords of size k1, . . . , k rrespectively, such that two chords cross if and only if they belong\nto different groups (see Figure 4 for an illustration). If k1=k2=. . .=kr=k/r, we will\nfurther shorten the notation to (k/r)r-matching.\nIn particular, an embedding of a (1)n-matching has nchords, all pairwise intersecting.\nMoreover, its pseudochord arrangements are in one-to-one correspondence with the pseudoline\narrangements of order n.\nSome basic facts about pseudochord arrangements\nThe following definition and propositions will form the basis of our construction.\n▶Definition 5. Consider an embedding Eof a matching, with bounding curve C. Let C′be\na simple closed curve contained in the union of Cand the bounded side of C. IfCdoes not\nintersect any point where two chords of Ecross, then it naturally defines an embedding of\na matching, up to cyclical shift of the endpoint labels, which we call a subembedding of E.\nWe say that two such subembeddings are independent if the bounded sides of their respective\nbounding curves are disjoint.\n3Throughout the paper, we will only be concerned with counting simple arrangements, and will thus\nomit this qualifier in most places.6 Improved Lower Bound on the Number of Pseudoline Arrangements\n▶Proposition 6. LetEbe an embedding of a matching, and let {E1, E2, . . . , E r}be a set of\npairwise independent subembeddings of E. Then,\n|arr(ME)|≥r/productdisplay\ni=1|arr(MEi)|.\nProof.By perturbing the chords slightly, we can assume without loss of generality that the\ninitial embedded matching Eis simple. For every choice of a simple arrangement in each of\nthe subembeddings, we can choose an embedding with the same bounding curve and same\nchord endpoints as the starting subembedding. Because in every simple arrangement of a\nmatching the same pairs of chords cross, the overall embedding obtained by locally replacing\neach subembedding with the chosen arrangement defines a valid simple arrangement of E.\nMoreover, if two overall arrangements differ in at least one of their subembeddings then they\nare non-isomorphic, as their chirotopes differ for at least one chord triplet. ◀\nThe same recursive procedure as in Matoušek’s recursive construction described above\ngives the following.\n▶Proposition 7. Letr > 0be a positive integer. For any positive integer m, let Dm\nr\ndenote a (m)r-matching. If cis a constant such that log2|arr(Dm\nr)|≥c·m2−O(m), then\nbn≥c\nr(r−1)n2−O(nlogn).\n3 Counting pseudochord arrangements of small matchings\nIn this section we describe a way to compute the number of pseudochord arrangements for\nrelatively small matchings. The general idea follows the straightforward incremental approach\nof adding chords one after the other and generating all possible resulting arrangements.\nThere are however some tricks which can greatly speed up the computation for some specific\nmatchings. We will keep the descriptions somewhat succinct, as this is not the main focus of\nthe paper.\nGeneral approach\nThe general approach we use is straightforward and incremental. Let c1, c2, . . . , c kdenote\nthe chords in a certain order. We incrementally construct embeddings for all pseudochord\narrangements on the first ichords (represented as doubly-connected edge lists, or DCELs)\nfor1≤i < k, by taking all constructed embeddings on the first i−1chords and for each of\nthem inserting chord ciin all combinatorially distinct valid ways. This amounts, for each\nof them, to generating all possible paths in a certain directed acyclic graph (or DAG for\nshort). Once we have all embeddings for the first k−1chords, we count the number of ways\nto insert the chord ckin each of them. This amounts to counting the number of paths in a\ncertain DAG (where previously, we were generating them instead of simply counting them).\nBecause counting the number of paths in a DAG can be done much faster in general than\ngenerating all paths, it makes sense for us to insert the chord with the largest contribution\nto the total number of pseudochord arrangements last. We next discuss how we can push\nthis heuristic a bit further for some types of matchings.\nIndependent chords\nConsider the embedding of a matching represented in Figure 5. A bit of thought should\nreveal that once we have inserted the black chords, the number of combinatorially distinctJ. Dallant 7\nc1c2\nc3\nFigure 5 Example of an embedded matching with independent chords. The blue chord c1is\nindependent from both c2andc3.\nways to insert the blue chord c1(while still representing the same matching) does not depend\non how the red chords c2andc3are inserted (or indeed, on their existence at all), and\nreciprocally the number of ways to insert the red chord c2andc3does not depend on how the\nblue chord c1is inserted. In this sense, we can think of the red chords as being independent\nof the blue chord. We formalize this notion as follows.4\n▶Definition 8. LetMbe an embedded matching. We say that two chords candc′ofMare\nindependent if at least one of the following conditions hold:\nthere exists a chord kofMsuch that candc′lie on different sides of k;\ncandc′do not intersect, and no two other chords kandk′which both intersect candc′\nintersect each other;\nno other chord intersects both candc′.\nOur previous discussion suggests the following algorithm: Given an embedded matching\nM, partition its chords into three sets R,GandB, such that all chords in Rare independent\nfrom all chords in B. Then, for every possible way to insert the chords of G, independently\nand recursively count the number of ways to insert the chords of Rand those of B, and\nmultiply these counts together. The sum over all arrangements of the chords in Ggives the\nfinal count. This procedure, applied to the example of Figure 5, is illustrated in Figure 6.\nWhile the advantage of this method might not be immediately apparent on such a small\nexample, it should be clear that this can offer great performance improvements on larger\nmatchings if there are enough independent chords.\nThe question remains on how to perform the partition. Unfortunately, we do not know how\nto obtain the partition which will most speed up the computation, so in our implementation\nwe have relied purely on heuristics for this. In short, our approach is the following:\nCompute all pairs of independent chords.\nFor each chord ci, estimate its contribution to the total number of line arrangements by\nsampling arrangements of the other chords and counting the number of ways to insert ci\nin these sampled arrangements. This defines a weight wifor the chord.\nGenerate many valid partitions of the chords into sets R, G, B. The weight of a set is the\nproduct of the weights of its chords. Choose the partition which maximizes the minimum\nof the weights of RandB(this captures the intuition that “heavy” chords should be\n4Note that the formal definition we give only constitutes a sufficient condition for chords to be independent\nin the intuitive sense discussed before, but this formal definition is what our algorithm works with.8 Improved Lower Bound on the Number of Pseudoline Arrangements\n(2     ×     3)      +      (1     ×     3)\nFigure 6 Application of the independent chord approach on the example of Figure 5.\nFigure 7 Repeated pattern in the intersection of the three slabs of Matoušek’s construction. The\ndotted line represents the boundary of a basic tile of this pattern.\ninserted last, and that we want to balance the weight as much as possible across the two\nsubproblems).\nRecurse on RandB.\nIn practice, this simple heuristic gave speed-ups of multiple orders of magnitude on some\nof the matchings we considered (compared to not exploiting independent chords at all),\nalthough more work on this could probably offer further improvements.\n4 Warm-up construction\nWe illustrate our method by slightly improving the lower bound obtained by Matoušek’s\nconstruction. Consider the previously mentioned construction (Figure 2) viewed as an\nembedding of an (m)3-matching. We call the region between the two extremal lines of the\nsame slope a slab. We focus on the region in the intersection of all three slabs, which has\narea 3m2/4. Notice that, ignoring the borders, the lines form a repeating pattern inside\nthat region whose basic tile is an axis-aligned unit square, as illustrated in Figure 7. These\nunit squares define a set of identical pairwise independent subembeddings of three pairwiseJ. Dallant 9\nFigure 8 The 20arrangements of the subembedding used in our warm-up improvement of\nMatoušek’s construction.\nintersecting chords. Each of these subembeddings has 2associated pseudochord arrangements.\nBy Proposition 6, this implies that a (m)3-matching has at least 23m2/4−O(m)pseudochord\narrangements. Applying proposition 7, we get a lower bound of 2n2/8−O(nlogn)on the number\nof pseudoline arrangements of order n.\nUp to this point, we have only been describing Matoušek’s construction in different\nterms. Now comes the improvement: instead of partitioning the area in the intersection\nof slabs into unit squares, we partition5it into squares of side-length 2. The number of\nsuch squares inside this area is 3m2/16−O(m)(io roughly 4times fewer than for unit\nsquares, up to border effects). What is the number of pseudochord arrangements for\neach of the associated matchings? If this number was 24= 16, we would recover roughly\nthe previous bound again. But it turns out that this number is 20(see Figure 8 which\nillustrates them all). Thus, by Proposition 6, this implies that a (m)3-matching has at least\n203m2/16−O(m)= 23 log2(20)m2/16−O(m)pseudochord arrangements. Applying proposition 7,\nwe get a lower bound of 2log2(20)n2/32−O(nlogn)on the number of pseudoline arrangements\nof order n, where log2(20)/32>0.135(to be compared with 1/8 = 0 .125for Matoušek’s\noriginal construction). We would get further improvements by considering larger and larger\nsubembeddings (although with this particular construction, we cannot hope to improve the\nconstant past the constant of 0.1887...obtained by Felsner and Valtr [8]).\n5 The main construction\nOur main result is based on the same principle as the warm-up. Instead of starting with\nMatoušek’s construction, we will start with the “rectangular construction with 12 slopes”\nof Dumitrescu and Mandal. We will use many types of constant sized subembeddings, for\nwhich we have explicitly computed the number of pseudochord arrangements (mostly using\n5We use the word “partition” in a loose sense here, as we are ignoring areas near the border of the shape\nto partition.10 Improved Lower Bound on the Number of Pseudoline Arrangements\nFigure 9 Illustration of the extremal lines and slabs of the construction.\nthe method outlined in Section 3).\nRectangular construction with 12 slopes\nWe start by recalling one of the constructions of Dumitrescu and Mandal [3]. The construction\nis based on twelve bundles of parallel lines with respectives slopes 0,∞,±1/3,±1/2,±1,±2,\n±3. Each bundle consists of mequally spaced lines, where mis odd, which are placed such\nthat that the middle line passes through the origin. The two extremal lines of the bundle of\nslope∞are given by the equations x=±(m−1)/2. The other 10extremal lines are given\nbyy=s·x±(m−1)/2fors= 0,±1,±2,±3andy=s·x±s(m−1)/2fors=±1/2,±1/3.\nFrom here on, Dumitrescu and Mandal count the number of points where ilines meet,\nfor3≤i≤12, and argue that for each of these points there are Bipossible choices for the\n“local arrangement” around the intersection point. Multiplying all of these choices together\ngives their bound.\nOur bound\nWe view the previous construction of lines as an embedding of a (12)m-matching (by adding\na bounding curve) and break it into different types of subembeddings, as we did in the\nwarm-up construction.\nWe call the area between the two extremal lines of slope sthe slab of slope s. Figure 9\nillustrates these extremal lines and the corresponding slabs. We group points of the planeJ. Dallant 11\ndepending on which set of slabs they belong to, and we further group them by rotations\naround the origin and reflexions around the lines of slopes 0,∞,−1,1passing through the\norigin (for example, points which belong to exactly the slabs with slopes 1,2and∞would\nbe grouped together with points belonging to exactly the slabs with slopes −1,−2and∞).\nThis grouping gives rise to 19relevant regions, which we will denote by RA, RB, . . . R S.\nFor each of these regions, we will compute a contribution to the overall number of pseudochord\narrangements, by choosing a set of subembeddings. Proposition 6 will then give us a lower\nbound on the number of pseudochord arrangements of a (12)m-matching. To compute this\ncontribution we will need to know the area of each of these 19regions. We refer the reader\nto Dumitrescu and Mandal for how these can be computed.6\nRegion RA\nFigure 10 Extremal lines of the construction, with the region RAhiglighted in red.\nThe region RAis illustrated in red in Figure 10. If we zoom in on some portion of RA,\nwe get a repeating pattern illustrated in Figure 11. On this latter figure a subembedding SA\nis also illustrated by a unit square which is further divided into three subembeddings S1\nA\n(red, top-left), S2\nA(yellow, top-right) and S3\nA(blue, bottom-left).\nThe subembedding S1\nAis an embedding of a (1)12-matching, and thus has exactly\nB12= 2 894 710 651 370 536 pseudochord arrangements. Our computations show that\nsubembedding S2\nAhas1 181 083 068 pseudochord arrangements. The subembedding S3\nAhas\n5 228 739 265 944 pseudochord arrangements. By Proposition 6, the number of pseudochord\narrangements for SAis at least\nnA: = 2 894 710 651 370 536 ·1 181 083 068·5228739265944\n= 17 876 503 929 228 145 018 796 772 391 568 838 912 .\nMoreover, SAcovers an area of 1(recall that there is a unit distance between two consecutive\nhorizontal or vertical lines in the construction), while the region RAhas an area of m2/12.\nThus, SAappears pA:=m2/12−O(m)times (independently) in RA, taking border effects\ninto account.\n6We note here also that the computation of these areas can be fully automated with the use of standard\ncomputer algebra software, thus possibly allowing for computer search of constructions like those of\nDumitrescu and Mandal.12 Improved Lower Bound on the Number of Pseudoline Arrangements\nFigure 11 Illustration of the subdiagram considered in region RA, further partitioned into three\nsubdiagrams.\nRegions RB, and RC\nFigure 12 Region RB, which has area m2/30.\nFigure 13 Region RC, which has area m2/30.\nThe regions RB, and RCare illustrated in red in Figures 12, and 13 respectively. If we\nzoom in on some portion of each of these regions, we get repeating patterns illustrated in\nFigures 14 and 15 respectively (up to symmetries). As previously, subembeddings SBand\nSCare also illustrated by unit squares further divided into two subembeddings S1\nX(red,\ntop-left) and S2\nX(blue, bottom-right) for X∈{B, C}.\nThe subembedding S1\nBis an embedding of a (1)11-matching, and thus has exactly\nB11= 5 449 192 389 984 pseudochord arrangements. The subembedding S1\nCis an embedding\nof a(1)10-matching, and thus has exactly B10= 18410581880 pseudochord arrangements.\nThe subembedding S2\nBandS2\nChave 4 485 362 657 994 086 and6 674 057 692 pseudochord\narrangements respectively. By Proposition 6, the number of pseudochord arrangements forJ. Dallant 13\nFigure 14 Subembeddings considered in RB,\nfurther partitioned into two subembeddings.\nFigure 15 Subembedding considered in RC,\nfurther partitioned into two subembeddings.\nSBandSCare at least\nnB:= 5 449 192 389 984 ·4 485 362 657 994 086 = 24 441 604 062 259 780 293 677 634 624;\nnC:= 18 410 581 880 ·6 674 057 692 = 122 873 285 610 409 820 960 .\nMoreover, both subembeddings cover an area of 1, while the regions RBandRCboth\nhave area m2/30. Thus, SBappears pB:=m2/30−O(m)times (independently) in RB, and\nSCappears pC:=m2/30−O(m)times (independently) in RC.\nRegions RDtoRQ\nRegion n• p•\nRA 17 876 503 929 228 145 018 796 772 391 568 838 912 m2/12−O(m)\nRB 24 441 604 062 259 780 293 677 634 624 m2/30−O(m)\nRC 122 873 285 610 409 820 960 m2/30−O(m)\nRD 145 267 240 140 131 510 094 m2/60−O(m)\nRE 884 854 135 426 438 m2/35−O(m)\nRF 4 354 539 523 065 118 m2/105−O(m)\nRG 134 841 117 561 581 177 808 m2/28−O(m)\nRH 21 027 918 182 m2/35−O(m)\nRI 1 422 375 838 634 144 387 571 m2/84−O(m)\nRJ 36 797 080 857 271 908 723 m2/105−O(m)\nRK 42 961 411 048 824 m2/210−O(m)\nRL 23 454 005 259 745 292 m2/60−O(m)\nRM 15 342 798 480 294 823 m2/60−O(m)\nRN 50 236 135 250 760 2m2/45−O(m)\nRO 50 236 135 250 760 m2/15−O(m)\nRP 104 878 461 268 633 368 974 367 2m2/63−O(m)\nRQ 104 878 461 268 633 368 974 367 m2/315−O(m)\nRR >2349 033m2/5002−O(m)\nRS >2349 033m2/(3·5002)−O(m)\nTable 2The number of arrangements for the chosen subembedding for each region ( n•) and the\nnumber of times said subembedding appears in the corresponding region ( p•).14 Improved Lower Bound on the Number of Pseudoline Arrangements\n(a)RD, with area m2/60.\n (b)RE, with area m2/35.\n (c)RF, with area m2/105.\n(d)RG, with area m2/14.\n (e)RH, with area m2/35.\n (f)RI, with area m2/42.\n(g)RJ, with area m2/35.\n (h)RK, with area 8m2/105.\n (i)RL, with area m2/15.\n(j)RM, with area m2/15.\n (k)RN, with area 4m2/15.\n (l)RO, with area m2/10.\n(m)RP, with area 2m2/3.\n (n)RQ, with area m2/15.\nFigure 16 Regions RDthrough RQ.J. Dallant 15\n(a)Subdiagram used for region RD.\n(b)Subdiagram used for region RE.\n(c)Subdiagram used for region RF.\n (d)Subdiagram used for region RG.\nFigure 17 Subdiagrams used in regions RDthrough RG. The subdiagram used for RGhas area\n2, while the three others have area 1.\nThe regions RDthrough RQare illustrated in Figure 16. Here, we can afford to compute\nthe number of arrangements for a single subembedding each (without further breaking it\ndown into further subembeddings), or even consider larger subembeddings, depending on the\nregion. The chosen subembeddings for regions RDthrough RG, are illustrated in Figure 17\n(the subembeddings chosen for regions RHtoRQcan be found in the appendix). The number\nof simple arrangements for each subembedding, and the number of times each subembedding\nappears in its respective region, are reported in Table 2.\nRegions RRandRS\nUp to area preserving linear transformations, the patterns of lines inside regions RRandRS\nare the same, as illustrated in Figure 20. We will thus consider the pattern illustrated on the\nright of this figure. While for all other regions, we used the approach explained in Section\n3 to count the number of arrangements of a chosen subembedding, the specific pattern in\nthese two regions allows for a different, much more efficient approach.\nWe illustrate it on a small subembedding defined by a square of side length 2(see Figure\n21). We can count the number of arrangements of this subembedding by starting with only\nthe black grid, and counting the number of combinatorially distinct ways to insert three\ncurves starting at s1,s2ands3respectively, and ending at e1,e2e3respectively, such that\nthese three curves do not intersect, and each crosses every black segment at most once. We\ncan in turn translate this into a question about counting the number of ways to have three\npaths in a certain directed acyclic grid graph, starting and ending at prescribed vertices,16 Improved Lower Bound on the Number of Pseudoline Arrangements\nFigure 18 Region RR, which has area m2.\nFigure 19 Region RS, which has area m2/3.\n(a)Pattern inside region RR.\n(b)Pattern inside region RS.\n(c)Pattern obtained from both\nafter vertical shear mappings.\nFigure 20 Illustration of the fact that up to a vertical shear mapping (which preserves areas),\nthe patterns inside regions RRandRScan be viewed as the same regular grid with additional lines\nof slope−1.\nsuch that these paths do not cross (middle illustration in Figure 21). Note that here by “not\ncrossing” we do not mean vertex- or edge-disjoint, but that if a path starts above/right of\nanother, it can at no point go below/left of it. Given three such paths, ordered from top\nright to bottom left, we can shift the second path one unit down and left along the grid and\nthe second path two units down and left along the grid, we get a set of three vertex-disjoint\npaths (right illustration in Figure 21). The reverse is also true: given three vertex-disjoint\npaths on the directed acyclic graph starting at s′\n1, s′\n2, s′\n3and ending at e′\n1, e′\n2, e′\n3, reversing\nthe shifts produces three non-crossing paths starting at s1, s2, s3and ending at e1, e2, e3.\nThus, our question finally becomes counting the number of ways to have three such\nvertex-disjoint paths, whose set of starting vertices is s′\n1, s′\n2, s′\n3and set of ending vertices\nise′\n1, e′\n2, e′\n3(note that the vertex-disjointness ensures that in such a set of paths, the path\nstarting at s′\niwill end at e′\nifor all 1≤i≤3). The Lindström–Gessel–Viennot lemma (or a\nweakened version of it) gives us an efficient method to do so:\n▶Lemma 9 (Lindström [11], Gessel & Viennot [9]) .LetGbe a finite directed acyclic graph.\nConsider starting vertices S={s1, . . . , s k}and destination vertices E={e1, . . . , e k}. For\nany two vertices uandv, letp(u, v)be the number of paths from utov. Assume that for\nany tuple of kvertex-disjoint paths starting in Sand ending in E, the path starting at si\nnecessarily ends at ei, for all 1≤i≤k. Then the number of distinct such tuples is theJ. Dallant 17\ne1\ne2 e3s1 s2\ns3s'1\ns'2\ns'3e'1\ne'2\ne'3s1 s2\ns3 e1\ne2 e3\nFigure 21 Illustration of the correspondence between arrangements of the chord diagram and\ndisjoint paths in the DAG.\ndeterminant of the following matrix:\nM=\np(s1, e1)p(s1, e2). . . p (s1, ek)\np(s2, e1)p(s2, e2). . . p (s2, ek)\n............\np(sk, e1)p(sk, e2). . . p (sk, ek)\n\nIn our specific case, the entries of this matrix are the number of paths going only down\nor right between two specified vertices in a grid, which can be easily expressed as a binomial\ncoefficient. We have\nM=\n/parenleftbig2\n1/parenrightbig /parenleftbig3\n0/parenrightbig\n0/parenleftbig3\n3/parenrightbig /parenleftbig4\n2/parenrightbig /parenleftbig3\n0/parenrightbig\n0/parenleftbig3\n3/parenrightbig /parenleftbig2\n1/parenrightbig\n=\n2 1 0\n1 6 1\n0 1 2\n.\nThe determinant of this matrix is |M|= 20, and we recover the count which was already\nused in the warm-up and illustrated in Figure 8. In general, for a square subembedding of\nsizes×s, the corresponding matrix is M= (mij)1≤i,j≤2s−1, with\nmij=/parenleftbigg2s−|s−i|−|s−j|\n2s−|s−i|−|s−j|+3|i−j|\n2/parenrightbigg\n.\nFor our actual bound, we use a square subembedding of side length s= 500. The number of\nsimple arrangements nR(=nS) of this subembedding is too big to write explicitly here (in\nbase 10, it is 105 070digits long). But it is enough for us to know that log2nR>349 033,\nand that the number of times this subembedding appears independently in regions RRand\nRSis, respectively, pR:=m2/5002−O(m)andpS:=m2/(3·5002)−O(m).\nPutting everything together\nWe can now put everything together by applying Proposition 6. This gives us the following\nlower bound for the number of simple pseudochord arrangements of a (12)m-matching\n(remember that we started with an embedding of such a matching):\n/productdisplay\nX∈{A,B,...S}(nX)pX= 2(/summationtext\nXpXlog2nX).\nComputing this bound yields 2c·m2−O(m)where c >36.65. Thus we have the following:\n▶Theorem 10. For large enough m, the number of simple pseudochord arrangements of a\n(12)m-matching is greater than 2c·m2, for some constant c >34.374.18 Improved Lower Bound on the Number of Pseudoline Arrangements\nBy now applying Proposition 7 we get our main result as a corollary:\n▶Corollary 11. For large enough n, the number of simple line arrangements of order nis\ngreater than 2c·m2, for some constant c >0.2604.\n6 Implementation and computation time\nWe ran all the the code on a 2020 MacBook Pro with an M1 chip. The code for computing\nthe number of pseudochord arrangement for a given matching was written in Rust and can be\nfound on GitHub [1]. It was parallelized to make use of all 8cores available on the machine\n(the algorithm is easy to parallelize, and for large enough chord matchings the speedup is\nclose to linear in the number of cores used). The total CPU time (summed over the 8cores)\nused to compute the number of arrangements for all matchings used in our construction\n(excluding the (1)12,(1)11, and (1)10matchings, for which the numbers of pseudochord\narrangements were previously known) was approximately 85hours.\n7 Discussion\nMany choices in the present construction were made heuristically or due to constraints in\ncomputing power and time. There are thus many avenues for possible improvements of the\nbound we obtain. We list a few here.\nThere is a lot of freedom in the choice of the initial embedding of the matching. Here we\nhave worked with Dumitrescu and Mandal’s “rectangular construction with 12 slopes” but\nit is not unlikely that a different matching or embedding could yield better bounds. The\nbest bound obtained by these authors is based on a different construction (the so-called\n“hexagonal construction with 12 slopes”), but our experiments applying our method on\nthis construction have not yielded better bounds than with the rectangular construction.\nOne can also try more different slopes. In particular, one could try to exploit the recent\nwork of Rote [14], where he counts the number of pseudoline arrangements of order 16\n(although once again, we were unable to do so fruitfully). Note also that the initial\nembedding need not have chords which are straight line segments.\nThe choices of subembeddings we made in each region are largely arbitrary. We have\ntried many different choices, but have no guarantee that the ones we made were good.\nContinuing on the previous point, being able to count the number of pseudochord\narrangements faster or for larger subembeddings should lead to improvements. This could\nbe achieved by better algorithms, better implementations, or simply more computing\npower. In particular, as the algorithm we use is highly parallelizable, running it on a\ncomputing cluster with many cores is an obvious avenue for improvements. One could\nalso try to find and use faster ad-hoc algorithms for different types of matchings.\nReferences\n1Justin Dallant. Counting pseudochord arrangements. https://github.com/justin-dallant/\ncounting_pseudochord_arrangements/ , 2024.\n2Hubert de Fraysseix and Patrice Ossona de Mendez. Stretching of Jordan Arc Contact Systems.\nInGraph Drawing, 11th International Symposium, GD 2003, Perugia, Italy, September 21-\n24, 2003, Revised Papers , volume 2912 of Lecture Notes in Computer Science , pages 71–85.\nSpringer, 2003. doi:10.1007/978-3-540-24595-7\\_7 .\n3Adrian Dumitrescu and Ritankar Mandal. New lower bounds for the number of pseudoline\narrangements. J. Comput. Geom. , 11(1):60–92, 2020. doi:10.20382/JOCG.V11I1A3 .J. Dallant 19\n4David Eppstein, Mereke van Garderen, Bettina Speckmann, and Torsten Ueckerdt. Convex-Arc\nDrawings of Pseudolines. In Proc. 21st Int. Symp. Graph Drawing (GD 2013) , pages 522–523.\nSpringer, 2013.\n5Stefan Felsner. On the Number of Arrangements of Pseudolines. Discret. Comput. Geom. ,\n18(3):257–267, 1997. doi:10.1007/PL00009318 .\n6Stefan Felsner. Geometric Graphs and Arrangements - Some Chapters from Combinatorial\nGeometry . Advanced lectures in mathematics. Vieweg+Teubner, 2004.\n7Stefan Felsner and Jacob E. Goodman. Pseudoline arrangements. In Joseph O’Rourke Csaba\nD. Toth and Jacob E. Goodman, editors, Handbook of Discrete and Computational Geometry,\nThird Edition , chapter 5. Chapman and Hall/CRC, 2017. doi:10.1201/9781315119601 .\n8Stefan Felsner and Pavel Valtr. Coding and Counting Arrangements of Pseudolines. Discret.\nComput. Geom. , 46(3):405–416, 2011. doi:10.1007/S00454-011-9366-4 .\n9Ira Gessel and Gérard Viennot. Binomial determinants, paths, and hook length formulae.\nAdvances in Mathematics , 58(3):300–321, 1985. doi:10.1016/0001-8708(85)90121-5 .\n10Donald E. Knuth. Axioms and Hulls , volume 606 of Lecture Notes in Computer Science .\nSpringer, 1992. doi:10.1007/3-540-55611-7 .\n11Bernt Lindström. On the Vector Representations of Induced Matroids. Bulletin of the London\nMathematical Society , 5(1):85–90, 1973. doi:10.1112/blms/5.1.85 .\n12Jirí Matoušek. Lectures on discrete geometry , volume 212 of Graduate texts in mathematics .\nSpringer, 2002.\n13OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences. http://oeis.org/\nA006245 (accessed November 30, 2023).\n14Günter Rote. Enumeration and counting of pseudoline arrangements. Presentation at the\nConvex and Discrete Geometry Workshop, Erdös Center, Budapest, September 4–8, 2023.\nURL: https://page.mi.fu-berlin.de/rote/Papers/slides/Budapest-2023-Enumeration+\nand+Counting+of+Pseudoline+Arrangements.pdf .\n15Katsuhisa Yamanaka, Shin-Ichi Nakano, Yasuko Matsui, Ryuhei Uehara, and Kento Nakada.\nEfficient enumeration of all ladder lotteries and its application. Theor. Comput. Sci. , 411(16-\n18):1714–1722, 2010. doi:10.1016/J.TCS.2010.01.002 .20 Improved Lower Bound on the Number of Pseudoline Arrangements\nAppendix : Subembeddings chosen for regions RHtoRQ.\nFigure 22 Subembedding chosen for region RH.\nFigure 23 Subembedding chosen for region RI.J. Dallant 21\nFigure 24 Subembedding chosen for region RJ.\nFigure 25 Subembedding chosen for region RK.22 Improved Lower Bound on the Number of Pseudoline Arrangements\nFigure 26 Subembedding chosen for region RL.\nFigure 27 Subembedding chosen for region RM.J. Dallant 23\nFigure 28 Subembedding chosen for regions RNandRO(up to an area preserving linear\ntransformation, the patterns inside these two regions are identical).\nFigure 29 Subembedding chosen for regions RPandRQ(up to an area preserving linear\ntransformation, the patterns inside these two regions are identical)."    },    {        "title": "2402.13944v1.Self_Avoiding_Walks_on_Cayley_Graphs_Through_the_Lens_of_Symbolic_Dynamics.pdf",        "content": "Self-Avoiding Walks on Cayley Graphs Through the Lens of\nSymbolic Dynamics\nNathalie Aubrun, Nicolás Bitar\nAbstract\nWe study dynamical and computational properties of the set of bi-infinite self-avoiding walks on\nCayley graphs, as well as ways to compute, approximate and bound their connective constant. To do\nthis, we introduce the skeleton\n G,Sof a finitely generated group Grelative to a generating set S,\nwhich is a one-dimensional subshift made of configurations on Sthat avoid all words that reduce to\nthe identity. We provide a characterization of groups which have SFT skeletons and sofic skeletons:\nfirst, there exists a finite generating set Ssuch that\n G,Sis a subshift of finite type if and only if Gis\na plain group; second, there exists Ssuch that\n G,Sis sofic if and only if Gis a plain group, ZˆZ{2Z\norD8ˆZ{2Z. We also characterize finitely generated torsion groups as groups whose skeletons are\naperiodic.\nFor connective constants, using graph height functions and bridges, we show that Cayley graphs\nof finitely generated torsion groups do not admit graph height functions, and that for groups that\nadmittransitivegraphheightfunctions, theconnectiveconstantisequaltothegrowthrateofperiodic\npoints of the skeleton. Using a counting argument due to Rosenfeld, we give bounds on the connective\nconstant of infinite free Burnside groups. Finally, we take a brief look at the set of bi-infinite geodesics\nand introduce an analog of the connective constant for the geodesic growth.\nKeywords: Self-avoidingwalk,Cayleygraph,connectiveconstant, subshiftoffinitetype,soficsubshift,\ngraph height function, geodesics, entropy.\n1 Introduction\nIn this article we study bi-infinite self-avoiding walks on Cayley graphs of finitely generated groups from\nthe point of view of symbolic dynamics and group theory. A self-avoiding walk is a path on a graph\nthat visits a vertex at most once, as shown on Figure 1 for the hexagonal grid. They were originally\nintroduced by Flory for the study of long-chain polymers [Flo49]. Although the original setting was\nthe infinite square grid, self-avoiding walks are more generally studied in the context of infinite quasi-\ntransitive graphs, intersecting with areas such as combinatorics, probability and statistical physics. The\nfundamental problem in this area is the study of the asymptotic growth rate of the number of self-avoiding\nwalks of a given length, called the connective constant (see [GL19] for a recent survey). Recently, there\nhas been increasing interest in the study of the set of all self-avoiding walks on edge-labeled graphs from\nthe point of view of formal language theory [LW20; LL23]. We take this study further by focusing on\nboth bi-infinite self-avoiding walks and bi-infinite geodesics on Cayley graphs of finitely generated groups.\nFigure 1: A self-avoiding walk, marked in red, on the hexagonal grid.\n1arXiv:2402.13944v1  [math.CO]  21 Feb 20241. Introduction\nGiven a finitely generated group G, and a symmetric finite generating set S, we introduce the skeleton\nofGwith respect to S, denoted\n G,S, as the set of labels of bi-infinite self-avoiding walks on the Cayley\ngraph ΓpG,Sq. The name skeleton comes from the study the decidability of tiling problems on groups,\nparticularly the infinite snake problem, where this object was originally introduced [AB23]. The skeleton\nG,Sis equivalently defined as the set of bi-infinite words xPSZthat contain no factors representing the\nidentity inG(see Sections 2.3 and 2.4 for the formal definition). This set admits a Z-action through the\nshift operation, making it a symbolic dynamical system, commonly refered to as a subshift. Our present\ngoal is to establish connections between the dynamical properties\n G,S, and geometric and algebraic\nproperties of the underlying group and Cayley graph. The skeleton provides a way to translate results\nfrom groups to shift spaces.\nThe article is divided among three principal lines. First, we look at which groups admit skeletons\nwhicharesubshiftsoffinitetype, soficshiftsoreffectiveshifts, aswellaswhentheactionisminimalityand\nwhat are its periodic points. We completely characterize the first two cases, and provide partial results\nfor the rest. Second, we look at the entropy of the skeleton. Using a result independently established\nby Rosenfeld [Ros22] for subshifts and Grimmet et al. [GHP14] for self-avoiding walks, we show the\ntopological entropy of the skeleton is given by\nhp\nG,Sq“logpµpG,Sqq,\nwhereµpG,Sqis the connective constant of the Cayley graph ΓpG,Sq. This answers one of the questions\nfrom Problem 108 posed by Rufus Bowen in his notebook of problems [Bow17]. With this connection,\nwe give some results on the approximation of connective constants and the existence of graph height\nfunctions. Lastly, we focus our attention on a subset of the skeleton composed of all bi-infinite geodesics,\nwhich we call the geodesic skeleton. We study its dynamic properties and introduce a geodesic analog of\nthe connective constant.\nCharacterizing Classes of Subshifts\nAn important class of subshifts is the class of subshifts of finite type. These subshifts are defined by a\nfinite set of forbidden patterns. We completely classify groups that admit a skeleton that is a subshift of\nfinite type.\nTheorem A. LetGbe a finitely generated group. Then, there exists a finite generating set Ssuch that\nG,Sis an SFT if and only if Gis a plain group.\nPlain groups are a subset of virtually free groups that are defined as free products of a finite number\nof finite groups and a free group (Definition 4.4). Their connective constants were previously studied by\nGilch and Müller [GM17]. We refine their result by showing that, when the skeleton is a subshift of finite\ntype, the connective constant is a non-negative rational power of a Perron number.\nA bigger class of subshifts is the class of sofic shifts. A subshift is said to be sofic if its language of finite\nfactors is regular. For the skeleton, this is equivalent to the language of labels of bi-infinitely extendable\nself-avoiding walks being regular. Lindorfer and Woess showed that the set of labels of finite length self-\navoiding walks on a connected, quasi-transitive, locally finite, deterministically-labeled, infinite graph is\nregular if and only if the graph has only thin ends, all of them of size 1 [LW20]. For Cayley graphs,\nthrough a theorem of Haring-Smith (Theorem 4.6), this can be shown to be exactly the class of plain\ngroups. Nevertheless, when working with bi-infinitely extendable self-avoiding walks, there exist Cayley\ngraphs (in particular the bi-infinite ladder graph) with ends of size 2 and regular language of bi-infinite\nextendable walks, i.e. its skeleton\n G,Sis sofic. We classify all groups that admit such a Cayley graph.\nTheorem B. LetGbe a finitely generated group. There exists Ssuch that\n G,Sis sofic if and only if G\nis a plain group, ZˆZ{2ZorD8ˆZ{2Z.\nThis result should be contrasted with Proposition 5.4 that shows that every group admits a generating\nset such that the skeleton is not sofic. In other words, the property of having a sofic skeleton is dependent\non the Cayley graph. As is the case for subshifts of finite type, when the skeleton is sofic its connective\nconstants is a non-negative rational power of a Perron number.\n21. Introduction\nLastly, fromourstudyofeffectiveskeletons, weprovideanalternativeproofoftheexistenceofeffective\none-dimensional subshift that has no computable configurations (Proposition 4.15).\nConnective Constants and Graph Height Functions\nThe next part of the article is concerned with finding ways to approximate connective constants through\nperiodic configurations and finding lower bounds when they are not available.\nWe show that if a Cayley graph ΓpG,Sqadmits a graph height function (see Definition 6.1), then the\nskeleton contains periodic points. As shown in [AB23], torsion groups can be characterized as groups\nwhose skeletons never contain periodic points, independently of the generating set. This allows us to\nstate the following result.\nTheorem C. The Cayley graphs of infinite torsion finitely generated groups do not admit graph height\nfunctions.\nThis theorem generalizes a result by Grimmet and Li who showed the Grigorchuk group does not ad-\nmit graph height functions, and more generally, Cayley graphs of torsion groups with certain conditions\non the stabilizer of the identity.\nIn contrast, using the General Bridge Theorem by Lindorfer [Lin20], we can extend a result of\nClisby [Cli13] on the approximation of the connective constant through periodic self-avoiding walks for\nthe square lattice, to groups admitting a particular type of graph height function.\nTheorem D. LetGbe a finitely generated group and Sa finite generating set. If ΓpG,Sqadmits a graph\nheight function ph,Hqsuch thatHacts transitively on ΓpG,Sq, then\nµpG,Sq“lim\nnÑ8n?en,\nwhereendenotes the number of periodic points in\n G,Sof periodnPN.\nThe next step is to look at torsion groups to find ways of approximating their connective constants\nwithouttheuseofperiodicpoints. Todothis, weuseacountingmethodpopularizedbyRosenfeld[Ros20;\nRos22], that provides us with a method to find lower bounds on any graph by finding solutions to an\ninequality dependent on the number of simple cycles on the graph (see Proposition 6.8). We use this\nmethod to give a lower bound for the connective constant of infinite free Burnside groups.\nTheorem E. LetBpm,nqbe the free Burnside group for m,ną1. IfBpm,nqis infinite and βą1\nsatisfies\nβ\nβn´1´1`βď2m´1,\nthenµpBpm,nq,Sqěβ. In particular, for ną3andmą1,\nµpBpm,nq,Sqěn´1\nnp2m´1q.\nBi-infinite Geodesics\nA sub-class of self-avoiding walks that are of special interest are geodesics. A geodesic is a shortest path\nbetween two points on the Cayley graph. We define the geodesic skeleton,\ng\nG,Sof a groupGwith respect\nto the generating set Sas the set of all bi-infinite geodesics on the corresponding Cayley graph. We\nobtain partial results on the classification of groups that admit geodesic skeletons that are subshifts of\nfinite type, sofic and effective. In addition, we show that the characterization of torsion groups through\nperiodic points also holds in this case.\nTheorem F. LetGbe a finitely generated group. The following are equivalent,\n•Gis a torsion group,\n•\nG,Sis aperiodic for all (some) generating sets S,\n32. Background and Definitions\n•\ng\nG,Sis aperiodic for all (some) generating sets S.\nFor the entropy of the geodesic skeleton, we introduce an analog of the connective constant. If we take\nΓG,Sthe geodesic growth function of Gwith respect to S, we define the geodesic connective constant of\na Cayley graph as the limit,\nµgpG,Sq“lim\nnÑ8nb\nΓG,Spnq.\nThe geodesic growth of groups has been extensively studied, especially in the case of virtually nilpotent\ngroups [Bri+12; BE22; Bis21; Bod23]. As was the case with the skeleton, the entropy of the geodesic\nskeleton is the logarithm of the geodesic connective constant. We are able to explicitly compute the\ngeodesic connective constants for lattices with known (or well approximated) connective constants such\nas the square grid, ladder graph ( L) and hexagonal grid ( H):\n•µgpZ2q“2,\n•µgpLq“1,\n•µgpHq“?\n2.\nStructure of the Article The paper is organized as follows. Section 2 is devoted to definitions and\nbackground on symbolic dynamics, combinatorial group theory ans self-avoiding walks. Section 3 surveys\ngeneral properties of the skeleton subshift, and shows how its entropy corresponds to the logarithm of the\nconnective constant of the corresponding Cayley graph. In Section 4 we investigate how dynamical and\ncomputationalpropertiesoftheskeletonsubshift–existenceofperiodicconfigurationsin\n G,S, minimality\nof\nG,S, being SFT or effective – relate to properties on the group Gitself. Next, in Section 5 we provide\na characterization of groups that admit sofic skeletons. To do this we also introduce notions from the\nstudy of thin and thick ends of graphs, and automorphisms of graphs. In Section 6 we use the skeleton\nto get new results on entropy and connective constant. We begin by looking at graph height functions\nand bridges, and their relation to periodicity in the skeleton. Then, we use Rosenfeld’s counting method\nto provide lower bounds on the connective constant of infinite free Burnside groups. Finally, Section 7 is\ndevoted to the study of the geodesic skeleton and the geodesic connective constant.\n2 Background and Definitions\nGiven an alphabet A, we denote by Anthe set of words on Aof lengthn,Aďnthe set of words of length\nat mostn, andA˚the set of all finite length words including the empty word ϵ. Furthermore, we denote\nbyA`“A˚ztϵuthe set of non-empty finite words over A. A factorvof a wordwis a contiguous subword\nofw; we denote this by vĎw. For a bi-infinite word xPAZ, giveni,jPZ,xri,jsdenotes the word\nxixi`1... xj,xrj,`8qthe infinite word stating at j, andxp´8,isthe infinite word finishing at i. For a\nwordwPA˚, the expression w8denotes the infinite word obtained by repeating w. We denote the free\ngroup defined by the free generating set of size nbyFn, and FSthe free group generated by S. The\ncommutator of two group elements g,his denoted by rg,hs“ghg´1h´1.\n2.1 Symbolic Dynamics\nGiven a finite alphabet A, we define the full-shift overAas the set of maps AZ“tx:ZÑAu. We\ncall mapsx:ZÑAconfigurations . There is a natural Z-action on the full-shift called the shift,\nσ:AZÑAZ, given byσpxqi“xi`1. The full-shift is also endowed with the prodiscrete topology, making\nit a compact space.\nLetFĎZbe a finite connected subset. A pattern of support Fis an element pPAF. We say a\npatternpPAFappears in a configuration xPAZif there exists kPZsuch thatxk`i“pifor alliPF.\nGiven a set of patterns F, we define the set of configurations where no pattern from Fappears as,\nXF:“txPAZ|@pPF, pdoes not appear in xu.\nAsubshift is a subset of the full-shift XĎAZsuch that there exists a set of patterns Fthat verifies\nX“XF. Subshifts are equivalently defined as closed σ-invariant subsets of the full-shift. We say a\nsubshift XFis\n42.2 Combinatorial Group Theory\n•asubshift of finite type (SFT) if Fis finite,\n•soficifFis a regular language,\n•effective ifFis a decidable language.\nEach class is strictly contained in the next.\nThelanguage of a subshift X, denoted LpXq, is defined as the set of all contiguous patterns that\nappear within some configuration from X. Formally,\nLpXq“twPA˚|DxPX,wĎxu.\nAny subshift can be defined by taking the complement of its language as forbidden patterns, that is,\nX“XLpXqc. Furthermore, a subshift is sofic if and only if its language is regular. Similarly, for a\nshiftX“XFwe define the language of locally admissible patterns ,LlocpXq, as the set of words\nwPA˚which contain no patterns from F. Notice that LpXq ĎLlocpXq, but is often not equal, and\nX“XLlocpXqc.\nAn important measure of the combinatorial properties of the subshift is its complexity ,pX, defined\naspXpnq“|LpXqXAn|, which counts the amount of words of a given length present in the subshift’s\nlanguage. Because the complexity is a submultiplicative function, that is, pXpm`nqďpXpmqpXpnq,\nFekete’s Lemma allows us to define the asymptotic word growth rate as\nα8pXq“lim\nnÑ8na\npXpnq.\nAnalogously, we define the function qpnq “ |LlocpXqXAn|, which is also submultiplicative, and thus\nαpXq“limnÑ8na\nqpnqexists. Rosenfeld showed this quantity is equal to the asymptotic word growth\nrate.\nLemma 2.1 ([Ros22]).ForFĎA`,αpXFq“α8pXFq.\nAn important quantity in symbolic dynamics is the (topological) entropy of the subshift, defined as\nhpXq“lim\nnÑ81\nnlogppXpnqq“ logpα8pXqq.\nFor example, in the case of the full-shift on A,hpAZq “ logp|A|q. Entropy informs many dynamical\nproperties of the subshift, and is invariant under shift-commuting continuous bijections. Furthermore,\nfor two subshifts XandYsuch thatYĎX,hpYq ďhpXq. A classical result by Lind [Lin84] shows\nthat the entropies of SFTs and sofic subshifts are exactly the set of non-negative rational multiple of\nPerron numbers. In contrast, the set of entropies of effective subshifts is the set of right computable real\nnumbers [HS08].\nWe say a configuration xPXisperiodic if there exists kPZzt0usuch thatσkpxq “x. We say\nthe subshift Xisaperiodic if it contains no periodic configurations. If a non-empty subshift is sofic, it\nalways contains periodic configurations.\nWe say a subshift is minimal if it does not contain non-empty proper subshifts. Equivalently, a sub-\nshift is minimal if every orbit under the shift action is dense in the subshift. Finally, a minimal subshift\nwith a periodic configuration is always finite.\nForacomprehensiveintroductiontoone-dimensionalsymbolicdynamicswereferthereaderto[LM21],\nwere proof of our assertions can be found.\n2.2 Combinatorial Group Theory\nLetGbe a finitely generated (f.g.) group and Sa finite generating set. In this article we will only consider\nfinitesymmetric generating sets, that is, generating sets that verify S“S´1, that never contain the\n52.3 The Skeleton, Infinite Domino Snakes and Analogies between Groups and Subshifts\nidentity. Elements in the group are represented as words on the alphabet Sthrough the evaluation\nfunctionwÞÑw. Two words wandvrepresent the same element in Gwhenw“v, and we denote this\nbyw“Gv. We say a word is reduced if it contains no factor of the form ss´1ors´1swithsPS. We\ndenote the identity of a group Gby1G.\nDefinition 2.2. LetGbe a group. We say pS,Rqis apresentation ofG, denotedG“xS|Ry, if the\ngroup is isomorphic to xS|Ry “FS{xxRyy, wherexxRyyis the normal closure of R, i.e. the smallest\nnormal subgroup containing R. We sayGisrecursively presented if there exists a presentation pS,Rq\nsuch thatSis finite and Ris recursively enumerable.\nFor a group Gand a generating set S, we define:\nWPpG,Sq:“twPS˚|w“Gεu.\nDefinition2.3. Thewordproblem ofagroupGwithrespecttoasetofgenerators Saskstodetermine,\ngiven a word wPS˚, ifwPWPpG,Sq.\nWe say a word wPS`isG-reduced ifwcontains no factor in WP pG,Sq. We say a word wPS˚\nis ageodesic if for all words vPS˚such thatw“vwe have|w| ď |v|. For a given group Gand\ngenerating set S, we denote its language of geodesics by GeopG,Sq. The length of an element gPG\nwith respect to Sis defined as}g}S“|w|wherewis any geodesic representing g. This length also defines\naG-invariant metric dS:GˆGÑNgiven bydSpg,hq“}g´1h}S.\nDefinition 2.4. LetGbe a finitely generated group, with generating set S. Thegrowth function\nγG,S:NÑNofGwith respect to Sis defined for a given nPNas the amount of elements of length at\nmostn. In other words, γG,Spnq“|tgPG|}g}Sďnu|.\nWe say an element gPGhastorsion if there exists ně1such thatgn“1G. If there is no such n,\nwe saygistorsion-free . Analogously, we say Gis atorsion group if all of its elements have torsion.\nOtherwise, if the only torsion element is the identity, we say the group is torsion-free .\nThefree product of two groups GandHgiven by presentations xSG|RGyandxSH|RHyrespec-\ntively, is the group given by the presentation,\nG˚H“xSGYSH|RGYRHy.\nFinally, let Pbe a class of groups (for example abelian groups, free groups, etc). We say a group G\nisvirtually P, if there exists a finite index subgroup HďGthat is in P.\n2.3 The Skeleton, Infinite Domino Snakes and Analogies between Groups\nand Subshifts\nLetGbe a finitely generated group with Sa set of generators. We define the skeleton ofGwith respect\ntoSas the subshift,\nG,S“txPSZ|@wĎx, wRWPpG,Squ“XWPpG,Sq.\nIn other words,\n G,Sis the set of all bi-infinite words that do not contain factor that evaluate to\nthe group’s identity. This subshift was originally defined in the context of the infinite snake prob-\nlem[AB23], where its properties where shown to inform the decidability of the problem. In particular,\nif\nG,Sis sofic, then the infinite snake problem for pG,Sqis decidable.\nExample 2.5. TakeZ2with its standard generating set S“ta˘,b˘u. Its skeleton is given by\nZ2,S“txPSZ|@wĎx,|w|a‰0_ |w|b‰0u,\nwhere|w|sis the sum of exponents of the generator s.\n62.4 Cayley Graphs and Self-Avoiding Walks\nExample 2.6. LetD8be the infinite dihedral group. The skeletons of this group can be radically\ndifferent depending on the generating set. For instance, if we take the presentation\nD8“xa,b|a2,b2y,\nthe corresponding skeleton is the finite subshift tpabq8,pbaq8u. On the other hand, if we take the\npresentation,\nD8“xr,s|s2,srsry,\nthe skeleton is infinite: for every nPZit contains a configuration xdefined by xpnq“sandxpkq“r\nfor allk‰n.\nThe skeleton is also present in Rufus Bowen’s notebook of problems [Bow17, Problem 108], where\nhe asks what can be said about\n G,S, what is its entropy and if it is intrisically ergodic. In Section 3.2\nwe tackle the second question. Further still, this subshift is inserted in the larger project of understand-\ning the analogies between multidimensional subshifts and finitely generated groups. Jeandel and Vanier\nobserved [JV19] that the forbidden patterns of a subshift play a similar role to relations of a group.\nA summary of this comparison is shown in Table 1. These comparison has been further strengthened\nGroup Subshift\nGroup with ngenerators Subshift on nsymbols\nWord problem WP pGq co-language LpXqc\nFinitely presented group SFT\nRecursively presented group Effectively closed subshift\nSimple group Minimal subshift\nHis a quotient of G YĎX\nTable 1: A part of the Jeandel-Vanier dictionary between groups and subshifts as introduced in [JV19]\nthrough results such as Higman and Boone-Higman Theorems for subshifts [JV19].\nThe skeleton is an attempt to establish these analogies explicitly, by using the generators as an\nalphabet, and WP pG,Sqas the set of forbidden patterns. We will see that in this case some of these\nanalogies hold and some do not.\n2.4 Cayley Graphs and Self-Avoiding Walks\nLetGbe a finitely generated group along with a finite symmetric generating set S. TheCayley graph\nofGwith respect to S, denoted ΓpG,Sq, is defined by the set of vertices VΓ“Gand the set of labeled\nedgesEΓ“tpg,s,gsq|gPG,sPSuĎGˆSˆG. Each edge e“pg,s,hqPEΓhas an initial vertex\nipeq“g, a terminal vertex tpeq“hand a label λpeq“s. The graph is also endowed with an involution\neÞÑe´1“ ph,s´1,gq PEΓ. If a generator has order 2, that is, if sPSsatisfiess2“1G, we take a\nunique edge between gandgsfor everygPG. Notice that every Cayley graph is |S|-regular, locally\nfinite, transitive and deterministically labeled, that is, for every vertex there is a unique out-going edge\nfor each label S. The group Gacts by translation on ΓpG,Sqby left multiplication, in other words, the\naction ofgPGover a vertex hPVΓis given by g¨h“gh. Through this action, we can identify Gwith\na subgroup of the automorphism group of the Cayley graph.\nWe also consider the undirected Cayley graph ˆΓpG,Sq, where we collapse each edge eande´1to\na single undirected edge between ipeqtotpeq. In other words, ˆΓpG,Sqis the graph with vertex set Gsuch\nthatg,hPGare adjacent if gh´1PS.\nExample 2.7. The hexagonal grid His the Cayley graph of the affine Coxeter group ˜A2given by the\npresentation,\n˜A2“xa,b,c|a2,b2,c2,pabq3,pacq3,pbcq3y.\nThis can be seen in Figure 2.\n72.4 Cayley Graphs and Self-Avoiding Walks\nFigure 2: A Cayley graph of the affine Coxeter group ˜A2. The red edges represent a, blue edges represent\nb, and green edges c.\nExample 2.8. The ladder graph Lis the Cayley graph of ZˆZ{2Zwith the presentation,\nZˆZ{2Z“xt,s|s2,tst´1sy.\nThis can be seen in Figure 3. This is not the only group that admits the ladder graph as a Cayley graph,\nts\nFigure 3: A Cayley graph of the group ZˆZ{2Z. The generator tdefines the horizontal right-pointing\nedges, and the generator sdefines the vertical undirected edges.\nthis is also the case for the groups D8andD8ˆZ{2Z.\nApathonΓpG,Sqis a sequence of edges π“pe1,...,enqsuch that for all iPt1,...,n´1uwe have\nipei`1q “tpeiq. We denote the initial vertex of the path by ipπq “ipe1qand its terminal vertex as\ntpπq“tpenq. The length of the path is given by ℓpπq“n, and its label is λpπq“λpe1q...λpenqPS˚. We\nalso define the sequence of vertices visited by πas the sequence Vpπq“pg0,...,gnqwithgi“ipei`1qfor\nalliPt0,...,n´1uandgn“tpenq. This formalism gives us a one-to-one correspondence between paths\nstartingat 1Gandwordsin S˚. Inparticular, apath πsatisfies ipπq“tpπqifandonlyif λpπqPWPpG,Sq.\nA pathπis aself-avoiding walk (SAW) if it never visits the same vertex twice. We define the\nlanguage of self-avoiding walks over ΓpG,Sqas the set\nLSAWpG,Sq“tλpπq|πis a SAW with ipπq“1Gu.\nRemark that the language remains the same if we change the initial vertex from the identity to any other\ngroup element because the graph is transitive. Furthermore, because Cayley graphs are deterministically\nlabeled, no two SAWs share the same label. A bi-infinite SAW centered at gPGis a sequence of edges\nπ“peiqiPZPEZsuch that ipei`1q“tpeiq, andg“ipe0qsuch thatπnever visits the same vertex twice.\nWe can thus state the following.\nLemma 2.9. LetGbe a group and Sa generating set. Then,\nG,S“tλpπqPSZ|πis a bi-infinite SAW centered at 1Gu,\n“XLSAWpG,Sqc.\nMoreover, Llocp\nG,Sq“LSAWpG,Sq.\n82.5 Connective Constants\nProof.This is a direct consequence of the definitions.\nOnce again, as the graph is transitive, we can change the center for any other element of the group.\nNotice that any finite subwalk of a bi-infinite SAW is a SAW. The converse is not always true, that is,\nthere are SAWs that do not appear in any bi-infinite SAW (see Figure 4).\nFigure 4: On Z2with standard generating set: a finite SAW that does not appear in a bi-infinite SAW.\nRemark 2.10. For the undirected Cayley graph ˆΓpG,Sqwe can define a labelling function λ1from\nthe set of self-avoiding walks over ˆΓpG,SqtoS˚. Take a self-avoiding walk πthat passes through the\nsequence of vertices pg0,g1,...,gnq, its labelλ1pπq “s0... sn´1is such that si“g´1\nigi`1. This way,\nbothLSAWpG,Sqand\nG,Sare equal to the corresponding labels of self-avoiding walks on the undirected\nCayley graph. Because of this, in what follows we do not distinguish between the directed and undirected\nCayley graphs.\n2.5 Connective Constants\nLetcnbe the number of self-avoiding walks of length nin the Cayley graph ΓpG,Sqstarting at the\nidentity. This sequence is submultiplicative, i.e. cn`mďcncmfor allm,nPN, so that by Fekete’s\nLemma, the limit ofn?cnexists:\nµpG,Sq“lim\nnÑ8n?cn“inf\nnPNn?cnPr1,8q.\nThislimitisknownasthe connectiveconstant oftheCayleygraph. Forgeneralquasi-transitivegraphs,\nthis limit was proved to be independent of the starting vertex by Hammersly and Morton [HM54].\nIngeneral, connectiveconstantsarehardtocompute. Nevertheless, theexactvalueofsomeconnective\nconstants is known. For instance, for the hexagonal grid (which as we saw, is a Cayley graph for ˜A2) its\nvalue isa\n2`?\n2[DS12], for the bi-infinite ladder (as in Figure 3) it is the golden mean1\n2p1`?\n5q[AJ90],\nand for some Cayley graphs of free products of finite groups it is the zero of a polynomial [GM17]. On\nthe other hand, giving a closed form for the connective constant of Z2with standard generators is still\nan open problem. The best estimate as of writing is\nµpZ2q«2.63815853032790 p3q,\nobtained by Jacobsen, Scullard, and Guttman [JSG16].\nThere are, however, bounds on the connective constant. We translate the following results from\nGrimmet and Li – which were stated for larger classes of graphs – to our Cayley graph context.\nTheorem 2.11 ([GL14; GL15]) .LetGbe a finitely generated group and Sa generating set.\n•µpG,Sqěa\n|S|´1,\n•Forw‰G1GandNits normal closure in G,µpG{N,SqăµpG,Sq,\n•ForgRSa non-identity element of GandS1“SYtg˘u,µpG,SqăµpG,S1q.\nFor more bounds and details, see [GL19].\n3 General Properties\nLet us begin by establishing properties of the skeleton that are common for all groups and generating\nsets.\n93.1 Bi-infinite SAWs through Group Elements and Computability\n3.1 Bi-infinite SAWs through Group Elements and Computability\nA first observation is that\n G,S“∅for a generating set if and only if Gis a finite group; this is a con-\nsequence of Konig’s Lemma (see [Wat86]). As we only consider infinite finitely generated groups, unless\nexplicitly stated, the skeletons are never empty. Next, if π“peiqiPZis a bi-infinite SAW on the Cayley\ngraph ΓpG,Sq, its inverse π´1“ pe´1\niqiPZis also a bi-infinite SAW. Therefore, for each configuration\nxP\nG,S, its inverse configuration x´1pkq“xpkq´1belongs to\n G,S.\nBy definition, configurations in the skeleton\n G,Savoid words from the word problem WP pG,Sq. But\nis it true that a non trivial group element necessarily appears as a word on SinLp\nG,Sq? For every\nfinitely generated group Gand every group element gPG, one can find a generating set Ssuch that this\nis true (Proposition 3.1).\nProposition3.1. LetGbe a finitely generated group. Then there exists Sa generating set for Gsuch that\nfor every non trivial group element gPG, there exists a word wPS˚such thatw“gandwPLp\nG,Sq.\nProof.A theorem by Seward [Sew14, Theorem 1.8] states that for every finitely generated group G,\nthere exists a finite generating set Ssuch that the Cayley graph ΓpG,Sqhas a regular spanning tree. In\nparticular, this tree has no leaves, thus each path leading to a vertex can be continued (see Figure 5).\nConsider such an Sand one associated regular spanning tree of ΓpG,Sq. Take a non trivial element\ngPGand consider the path connecting the identity 1Gtogin the regular spanning tree. Then this finite\nsimple path can be extended to an infinite simple path inside the spanning tree, leading to an infinite\nsimple path going through 1Gandg. Translating this path into a bi-infinite sequence of elements in S\ngive a configuration of the skeleton\n G,S.\nFigure 5: In blue, an example of regular spanning tree of degree 2– thus a bi-infinite Hamiltonian path\n– forZ2with its standard presentation. In green a path from p0,0qtop2,2qextracted from this regular\nspanning tree. Note that this path is highly non-geodesic.\nThis can be made more precise in the case of one and two ended groups. Cayley graph Γ“ΓpG,Sq\nhaskends ifkis the supremum of the number of infinite connected components of the induced subgraph\nΓrVΓzAsover every finite subset AĎV(see Section 5.1).\nProposition 3.2. LetGbe a finitely generated group with one or two ends, with Sa generating set.\nDefineS1“tgPG|}g}Sď3u. Then, for every non trivial group element gPG, there exists a word\nwPpS1q˚such thatw“gandwPLp\nG,S1q.\nProof.In [Car23, Theorem 1.3] Carrasco-Vargas showed that for one and two ended groups, Sewards’\nTheorem holds for S1, that is, there is a Hamiltonian path on the Cayley graph ΓpG,S1q. This directly\nimplies our statement.\nBy translating and joining Lemmas 4.4 and 3.7 from [Car23], we can state results on the decidability\nof the language of the skeleton for particular sets of generators.\n103.2 Entropy\nProposition 3.3. LetGbe a finitely generated group with one or two ends, with Sa generating set.\nSupposeGhas decidable word problem, and define S1“tgPG|}g}Sď3u. Then, Lp\nG,S1qis computable.\nFor a skeleton, having a computable language means that there is an algorithm that determines if a\nfinite SAW is bi-infinitely extendable. A particular class of subshifts that have computable language are\nsofic subshifts. In Sections 4 and 5 we will explore when skeletons belong to this class.\n3.2 Entropy\nAs seen in Lemma 2.9, the skeleton shift is the set of labels of bi-infinite SAWs over a Cayley graph.\nConsequently, its complexity function counts the number of infinitely bi-extendable SAWs of length n,\nwithα8p\nG,Sqbeing their asymptotic growth rate. Furthermore, the number of locally admissible words\nof lengthnis exactlycn, the number of finite SAWs of length n. Therefore, by Lemma 2.1 we obtain the\nfollowing.\nLemma 3.4. For a finitely generated group Gand a generating set S,hp\nG,Sq“logpµpG,Sqq.\nThis equality can also be deduced from [GHP14], where Grimmet et al. showed that the connective\nconstant is equal to the growth rate of infinitely bi-extendable SAWs.\nLetpG,Sqbe a f.g. group and γG,S:NÑNits growth function. We define its asymptotic growth\nrate as the value,\nHG,S“lim\nnÑ81\nnlogpγG,Spnqq.\nAs the growth function is sub-multiplicative (see [CC10]), HG,Sexists by Fekete’s Lemma.\nRemark 3.5. An alternative way to look at the growth of a group is the strict growth function σG,S,\nwhereσG,Spnqis the number of elements of length exactlyn. As is the case for the growth function, σG,S\nis sub-multiplicative. Its asymptotic growth rate is the same as that of γG,S, namely HG,S. This can be\nseen through their generating functions. Take F,f:CÑCdefined as\nFpzq“ÿ\nnPNγG,Spnqzn, fpzq“ÿ\nnPNσG,Spnqzn.\nBy the Cauchy-Hadamard theorem we know that the asymptotic growth rate of γG,S(resp.σG,S) is\nthe reciprocal of the radius of convergence of F(resp.f). Because strict growth can be expressed as\nσG,Spnq“γG,Spnq´γG,Spn´1q, with the convention that γG,Sp´1q“0, we get that fpxq“p 1´xqFpxq.\nAs the termp1´xqdoes not change the radius of convergence of the series, we have that\nlim\nnÑ8logpσG,Spnqq\nn“HG,S.\nProposition 3.6. LetGbe a f.g. group with Sa finite generating set. Then,\nHG,Sďhp\nG,Sqďlogp|S|´1q.\nProof.Letppnqbe the complexity function for\n G,Sandk“|S|. For the upper bound, notice that the\ntotal number of reduced words of length noverSis exactly the number of elements of length nin free\ngroup Fm, withm“rk\n2s. Therefore,\nppnqďγFmpnq´γFmpn´1q“kpk´1qn´1.\nOn the other hand, every element of length nhas a geodesic representative of length n, which by\ndefinition is G-reduced. In particular, this representative is a SAW of length n. Thus,σG,Spnqďcnand\nHG,S“lim\nnÑ8logpσG,Spnqq\nnďlogpµpG,Sqq.\n114. Dynamic and Computational Aspects\nRemark 3.7. The bounds from Proposition 3.6 are tight in general, as free groups with free generating\nsets satisfy HFm,S“hp\nFm,Sq“logp2m´1q. Nevertheless, by Theorem 2.11 we know that for non-free\ngroupshp\nG,Sqălogp2m´1qfor all generating sets. This same theorem also tells us that for groups\nwith polynomial growth the lower bound is strict, as\n0“HG,Să1\n2logp|S|´1qďhp\nG,Sq.\nAnother straightforward bound we find from algebraic considerations is the following.\nProposition 3.8. TakeGa finitely generated group and Sgenerating set. If ts1,...,snu ĎSis a\nsubset of generators such that there induced semigroup xs1,...,sny`does not contain the identity, then\nhp\nG,Sqělogpnq.\nProof.Ifxs1,...,sny`does not contain the identity, any combination of these generators will give a word\nthat does not contain factors that evaluate to the identity. In other words, the skeleton contains the\nfull-shiftts1,...,snuZ. Consequently, hp\nG,Snqělogpnq.\nExample 3.9. TakeZ2with its standard generating set ta˘,b˘u. Then, the semigroup generated by\naand bdoes not contain the identity. Then, hp\nZ2,ta˘,b˘uqělogp2q. Similarly, if we take the discrete\nHeisenberg group H3with generating set ta˘,b˘,c˘uthrough the presentation,\nH3“xa,b,c|ra,cs,rb,cs,ra,bsc´1y,\nthe semigroup given by the three generators a,band cdoes not contain the identity. Then, by the\nprevious proposition hp\nH3,ta˘,b˘,c˘uqělogp3q.\nRemark 3.10. Given a group G, the entropy of its skeleton can be made arbitrarily large. This can\nbe done be taking bigger and bigger generating sets and using the lower bounda\n|S|´1given by\nTheorem 2.11. This can also be done in torsion-free groups by taking a torsion-free element gPG, a\ngenerating set containing tg,g2,...,gnuand using the previous proposition.\nIn Section 6 we will see methods to approximate entropy and connective constants for different classes\nof groups.\n4 Dynamic and Computational Aspects\nThe goal of this section is to explore the multiple dynamical and computational properties of skeletons,\nand how they interact with the algebraic properties of the underlying group. We look at groups that\nadmits SFT, sofic, effective or minimal skeletons, as well as their periodic points.\nA subshift can be defined by various different sets of forbidden patterns. We saw from its definition\nand Lemma 2.9, that\n G,Sis defined by at least two different sets, namely LSAWpG,Sqand WPpG,Sq.\nWe begin by describing an additional set that will help us better understand the structure of forbidden\npatterns.\nWe begin by looking at the set of patterns that define simple cycles (also called embedded cycles)\nin the Cayley graph. We define the set of labels of simple cycles of a group Gwith respect to a finite\ngenerating set Sas,\nOG,S“twPWPpG,Sq|wdefines a simple cycle in ΓpG,Squ\n“twPWPpG,Sq|@w1Ĺw, w1RWPpG,Squ.\nExample4.1. Consider Z2with its standard presentation xa,b|ra,bsy. Then the word aba´3b´1abab´1\nis in WPpZ2,ta,buqbut not in OZ2,ta,busince there are repeated vertices in the path it represents in\nΓpZ2,ta,buq.\n124.1 SFT Skeletons\nTo extend the self-avoiding perspective, we call elements of OG,Sself-avoiding polygons (SAPs) of\nthe Cayley graph of ΓpG,Sq.\nLemma 4.2. LetF“OG,SYtss´1|sPSu. Then,\n G,S“XF.\nProof.SinceOG,SĎWPpG,Sqandtss´1|sPSuĎWPpG,Sq, we have that FĎWPpG,Sq. So, the\nsubshifts defined by two sets respect the reciprocal inclusion, and we have\n G,SĎXF.\nReciprocally, take some configuration xPXFand assume it contains some pattern wPWPpG,Sq.\nWithout loss of generality we assume that w“s1...snfor somenPN, so thats1...sn“1G. Consider\nthe group elements gidefined bygi“s1...siforiPt1...nuandg0“1G. SincexPXFthe pattern w\ndoes not belong to OG,SYtss´1|sPSu. So necessarily ną1and there are some repetitions among\nthegi’s in addition to g0“gn. Take two indices i,jsuch thatiăjandti,ju‰t 0u,tnu,t0,nu(at least\none of the two indices is neither 0norn) andi,jare minimal. Then the word si...sjdefines a cycle in\nΓpG,Sq, which contradicts our original assumption. Finally, xP\nG,S, which concludes the proof.\nThis alternative set of forbidden patterns for\n G,Swill be particularly helpful in the proof of Theo-\nrem 4.5, where we characterize groups Gwhich admit a generating set Ssuch that\n G,Sis an SFT and\nalso in Section 6.2.\n4.1 SFT Skeletons\nTo find SFTs, we start with a warm-up lemma that contains the central idea used in our classification of\ngroups that admit skeleton SFTs.\nLemma 4.3.\n Zd,Sis not an SFT for dě2and any generating set S.\nProof.LetSbe a generating set for Zdand suppose F1is a finite set of forbidden patterns such that\nZd,S“XF1. Then, as Sgenerated the group, there must exist s,s2PSsuch thatxs1yXxs2y“t 1Zdu,\nandxs1,s2y»Z2. Let us denote N“maxwPF1|w|. Take the SAP defined by the square of length 2N\non the first two generators w“v2N\ni1v2N\ni2v´2N\ni1v´2N\ni2. Notice that no factors of w2of lengthNbelong to\nF1as they are all globally admissible in\n Zd,S. Letx“w8. ClearlyxR\nZd,S, as it contains wwhich\nsatisfiesw“1G. Nevertheless, no factor of xof lengthNis contained in F1. Therefore xPXF1, which\nis a contradiction.\nThe main idea of this lemma is using arbitrarily large cycles that are locally self-avoiding. This way,\nit is not possible to detect that the path eventually crosses itself using a finite window. Which groups\nadmit generating sets that define SFT skeletons then? Let us show that this is the case of a specific class\nof virtually free groups.\nDefinition 4.4. A groupGisplainif there exist finite groups tGiuk\ni“1andmě1such thatGis\nisomorphic to the free product,ˆk˚\ni“1Gi˙\n˚Fm.\nWe say a finite generating set Sfor suchGisstandard if it can be written as the disjoint union\nS“S1Y...YSkYSk`1whereSiis a generating set for GiandSk`1is a free generating set for Fm.\nTheorem 4.5 (Theorem A) .LetGbe a finitely generated group. Then, there exists a finite generating\nsetSsuch that\n G,Sis an SFT if and only if Gis a plain group.\nIn order to prove this theorem we use a characterization of plain groups with respect to their simple\ncycles. The diameter of a simple cycle is the greatest distance between to vertices in the cycle. A vertex\nvinΓpG,Sqis said to be a cut vertex ifΓpG,Sqztvuis disconnected. A graph is said to be 2-connected\nif it contains no cut vertices. A maximal 2-connected subgraph is called a block.\nTheorem 4.6 ([Har83]).LetGbe a group and mPN. Then, the following are equivalent\n•Gadmits a finite generating set Ssuch that all simple cycles in the undirected Cayley graph ΓpG,Sq\nhave diameter at most m,\n•Gadmits a finite generating set Ssuch that all blocks in the undirected Cayley graph ΓpG,Sqhave\ndiamater at most m,\n134.1 SFT Skeletons\n•Gis a plain group.\nProofs of this Theorem can be found in [Har83; EP22].\nProof of Theorem 4.5. LetGbe a plain group decomposed as´˚k\ni“1Gi¯\n˚FmwithS“S1Y...YSk`1a\nstandard generating set. Due to its free product structure, any word wPS˚can be uniquely decomposed\nasw“w1w2....wrwhere,\n•wjPS˚\nlfor somel, for alljPt1, ..., ru;\n•wjandwj`1are words over different alphabets for all jPt1, ...,r´1u.\nIfw“1G, by our decomposition, wj“1Gfor everyj. This means every SAP from Gmust be entirely\ncontained in one of the finite groups Gi, asFmhas no SAPs with its free generating set. Therefore, OG,S\nis finite because the number of SAPs in each finite group is finite. By Lemma 4.2,\n G,Sis an SFT.\nNow, letGbe a finitely generated group with Ssuch that\n G,Sis an SFT, defined by the finite set\nof forbidden patterns F. IfGis not a plain group, by Theorem 4.6, the Cayley graph ΓpG,Sqcontains\narbitrarily large embedded cycles, and therefore arbitrarily large SAPs. Next, we can assume without\nloss of generality that every word in Fhas the same length, say Ně1. IfFĎWPpG,Sq, take a SAP\nWof length greater than N`1. Because SAPs contain no strict factors that belong to WP pG,Sqand\nevery cyclic permutation of the word defining a SAP is itself a SAP, the configuration x“W8does not\ncontain any word from F. Therefore xPXFz\nG,S, which is a contradiction. Suppose there are elements\ninFthat are not in WP pG,Sq. AsFcontains forbidden patterns and XF“\nG,S, for every word wPF\nthere exists NwPNsuch that either for every vPSNwthe wordwvcontains a factor from WP pG,Sq,\nor for every vPSNwthe wordvwcontains a factor from WP pG,Sq. LetM“maxwPFNwand takeW\na SAP of length M`N`2. Once again, because every SAP contains no strict factors that belong to\nWPpG,Sqand every cyclic permutation of the word defining the SAP is itself a SAP, the configuration\nx“W8contains no factors from F. Indeed, if there is a wPFsuch thatwĎWwe can take the cyclic\npermutation of Wsuch thatwis a prefix. Thus, wcan be extended by a word vof lengthM`1such\nthatwvcontains no factor in WP pG,Sq. As a consequence xPXFz\nG,S, which is a contradiction.\nAs plain groups admit SFT skeletons, we have an effective procedure to calculate the connective\nconstant of their Cayley graphs. As mentioned in Section 2.1, entropies of SFTs are non-negative rational\nmultiples of logarithms of Perron numbers (see Theorem 4.4.4 [LM21]). Thus, we can slightly improve\nCorollary 3.4 from [GM17] in the case of (plain) groups.\nCorollary 4.7. LetGbe a plain group with Sa standard set of generators. Then, µpG,Sqis a non-\nnegative rational power of a Perron number.\nLet us sketch how to compute the connective constant using SFTs. Let\n G,Sbe the skeleton of the\nplain group G“p˚k\ni“1Giq˚FmandFbe the finite set of patterns defining it. Recall from Theorem 4.5\nthat this set corresponds to the SAPs on each individual group Gias well as the words ss´1for allsPS.\nLetNbe the length of the biggest word in F. We can extend FtoF1so that all words have length N.\nThe Rauzy graph RNpG,Sqof orderNof\nG,Sis the finite directed graph whose vertices are labeled by\nthe language of size Nof the skeleton LNp\nG,Sq, and edges are labeled with LN`1p\nG,Sq. There is an\nedge labeled by wfromutovifuis the prefix of length Nofwandvis the suffix of length Nofw.\nWe denote the adjacency matrix of the graph RNbyMN, that is, if LNp\nG,Sq“tu1,...,uℓu, the entry\nMNpi,jqrepresents the number of edges in RNfromuitouj. Then the connective constant of ΓpG,Sq\nis the logarithm of the dominant eigenvalue of MN, which exists by Perron-Frobenius’ Theorem.\nExample 4.8. TakeS3the symmetric group on 3 elements with generating set s1“p1 2qands2“p1 3q,\nand the cyclic group Z{3Z“xty. Then, the skeleton of the plain group G“S3˚Z{3Zwith respect to\nS“ts1,s2,t˘1uis defined by the forbidden patterns,\nF“ts2\n1,s2\n2,ps1s2q3,t3,tt´1,t´1tu.\nWe obtain that the connective constant µ“µpG,ts1,s2,t˘1uqis the solution of the polynomial equation\nx7´4x5´8x4´8x3´8x2´8x´4“0obtained from the characteristic polynomial of the matrix\ndescribed above, which is approximately µ«2.8698315.\n144.2 Effective Skeletons\nFigure 6: A portion of the Cayley graph of the plain group S3˚Z{3Z. The two generators for S3are\npictured in red and blue and alternate; the generator for Z{3Zis pictured in green.\nThe skeletons of plain groups with respect to their standard generating sets also have nice dynamical\nproperties. We say a subshift XĎAZisirreducible if for every w1,w2PLpXq, there exists some\nwPLpXqsuch thatw1ww2PLpXq.\nProposition 4.9. LetGbe a plain group with standard generating set S. Then,\n G,Sis irreducible.\nProof.Decompose Gas´˚k\ni“1Gi¯\n˚FmwithS“S1Y...YSk`1a standard generating set. Take\nw1,w2PLp\nG,Sqappearing at position 0 of the configurations xp1q,xp2qP\nG,Srespectively. There is a\nunique decomposition wi“wi\n1wi\n2....wi\nrwhere,\n•wi\njPS˚\nlfor somel, for alljPt1, ..., ru;\n•wi\njandwi\nj`1are words over different alphabets for all jPt1, ...,r´1u.\nIfw1\nr,w2\n2PS˚\ni, take any generator sPSjforj‰iand define x“xp1q\np´8,0sw1sw2xp2q\nr|w2|,`8q. Because\nwe chose a generator that does not belong to Gi, andxp1qandxp2qbelong to the skeleton, xmust also\nbelong to the skeleton. This implies, w1sw2PLp\nG,Sq. If instead w1\nrPS˚\niandw2\n1PS˚\njfori‰j, take\nsPSjands1PSito definey“xp1q\np´8,0sw1ss1w2xp2q\nr|w2|,`8q. As before, ymust belong to\n G,S. This means,\nw1ss1w2PLp\nG,Sq.\nCorollary 4.10. ForGa plain group with standard generating set S, the set of periodic configurations of\nG,Sis dense in\n G,S. In other words, any bi-infinitely extendable SAW on ΓpG,Sqappears in a periodic\nbi-infinite SAW. Furthermore,\nµpG,Sq“lim\nnÑ8n?en,\nwhereendenotes the number of periodic points in\n G,Sof periodnPN.\nThiscorollarystatesageneralpropertyofirreduciblesubshiftsoffinitetype, namely, itssetofperiodic\nconfigurations is dense and its entropy is approximated through its periodic points [LM21]. We obtain\na similar expression for the connective constants of Cayley graphs whose skeletons is not an SFT in\nSection 6.\n4.2 Effective Skeletons\nLet us briefly look at the case of effective skeletons. We know that recursively presented groups have\nrecursively enumerable word problem. WP pG,Sqis thus recursively enumerable for all finite generating\nsets. This enumeration gives us an enumeration of the forbidden patterns of our skeleton.\n154.2 Effective Skeletons\nLemma 4.11 ([AB23]).LetGbe a recursively presented group. Then, for every generating set S, the\nsubshift\n G,Sis an effective subshift. In particular, µpG,Sqis a right computable real number.\nIn order to approach the converse, we give a computational upper bound of the word problem of the\ngroup in terms of the computability of finite SAWs on the Cayley graph.\nLemma 4.12. The word problem for Gwith respect to a generating set Sis decidable given an algorithm\nforLSAWpG,Sqc.\nProof.We describe an procedure to compute the word problem of Ggiven an algorithm that determines\nif a word belongs to L“LSAWpG,Sqc. We begin with an algorithm that computes all words wPSďn\nsuch thatwPWPpG,Sqgivenn. This algorithm, which we call M, is shown in Algorithm 1.\nAlgorithm 1: M\nInput:ně2\nTÐ∅;\nforwPS2do\nifwPLthen\nTÐTYtwu;\nend\nend\nforiPt3,...,nudo\nforwPSido\nifwPLthen\nifwcontains no factors from Tthen\nTÐTYtwu;\nend\nforvPTdo\nDeletevfromwif present, to obtain w1;\nifw1PTthen\nTÐTYtwu;\nend\nend\nend\nend\nend\nreturnT;\nLet us show the output of Monnis WPpG,SqXSďn. LetTibe the setTin the algorithm after the\nfirstiiterations of the forloop, foriPt2,...,nu. We claim Ti“WPpG,SqXSďi. First off, every non-self\navoiding path of length two must represent the identity. Thus, T2“WPpG,SqXS2. Now, suppose we\nhave the equality for Ti. TakewPWPpG,SqXSi`1. This implies wPL, and as seen in Lemma 4.2, it\nmust represent a simple cycle, contain a shorter simple cycle, or a word of the form ss´1. In the first\ncase,wcontains no factors from Tiand is therefore added to Ti`1. In the other two cases, it contains a\nfactor from Tithat after being deleted creates a word that belongs to WP pG,SqXSďi“Ti. Therefore,\nwPTi`1. Conversely, if w1PTi`1zTiwe knowwPL. Ifwwas added to Ti`1because it contains no\nfactors from Ti, it must represent a simple loop and is therefore in WP pG,Sq. On the other hand, if wwas\nadded after deleting a factor from Ti,wis made up of a word representing the identity with a factor rep-\nresentingtheidentityinsertedintoit. Thismeans, wPWPpG,Sqandtherefore Ti`1“WPpG,SqXSďi`1.\nFinally, to determine if a given word belongs to WP pG,Sq, we simply compute Mto the length of\nword and see if it is present.\nAs a consequence, if LSAWpG,Sqis co-recursively enumerable, the word problem of Gmust be in\n∆0\n2on the arithmetical hierarchy. This is the case when\n G˚H,Sis effective for Hany f.g. group, and\nS“SGYSHwithSG,SHgenerating sets for GandHrespectively.\n164.3 Periodic Configurations\nConjecture 4.13. A group is recursively presented if and only if there exists a finite generating set S\nsuch that\n G,Sis effective.\nEven though recursively presented groups define subshifts that are effective, if the structure of the\nunderlying group is computationally complex, the configurations of the skeleton may be uncomputable.\nWe say a confiraguration xiscomputable if there is an algorithm that on input nPZcomputesxn.\nDefinition 4.14. A finitely generated group Gand generating set Sare said to be algorithmically\nfiniteif for every LĎFSrecursively enumerable set, there exist infinitely many pairs of distinct words\nu,vPLsuch thatπpuq “πpvq, whereπ:FSÑGis the canonical projection. We say Gis aDehn\nMonster if it is infinite, recursively presented and algorithmically finite.\nThis class of groups was introduced by Myasnikov and Osin in [MO11], where they showed that\nDehn Monsters exist. Furthermore, they showed that being algorithmically finite does not depend on the\ngenerating set.\nProposition 4.15. LetGbe a Dehn Monster. Then\n G,Sis effective for any finite generating set S, but\nno configuration in\n G,Sis computable.\nProof.As the properties of being infinite, recursively presented and algorithmically finite are independent\nof the generating set, we take any generating set SforG. If there existed a computable configuration\nxP\nG,S, we could recursively enumerate the set of words L“txr0,n´1sPS˚|ně1u. Then for any\nu,vPL,πpuq ‰πpvq. If not, we would arrive at xr0,n´1s“Gxr0,m´1sfor somenąmě1, which\nimpliesxrn,m´1s“Gε. Therefore, any pair of elements in Lmaps to a different element through π, which\ncontradicts the algorithmic finiteness of G.\n4.3 Periodic Configurations\nConfigurations of particular importance in the study of subshifts are periodic configurations. Recall that\na configuration xP\nG,Sis periodic if there exists kPZzt0usuch thatxi`k“xifor alliPZ. Such a\nconfiguration has a rigid structure, if we take w“xr0,k´1sthe configuration is the bi-infinite repetition\nofw, i.e.,x“w8. We will see that the existence of periodic points in the skeleton imposes strong\nrestrictions on the structure of the underlying group.\nIn [AB23] it was shown that for any finitely generated group with a torsion-free element, the skeleton\ncontainsaperiodicpoint. Theperiodicconfigurationwasobtainedbyiteratinganygeodesicofthetorsion-\nfree element with the smallest length in the group. By re-interpreting the proof of [Hal73, Theorem 7]\nwe obtain the following generalization.\nProposition 4.16. LetGbe a finitely generated group. Take a generating set Sand a torsion-free\nelementgPG, andk“argmint}gn}S|ně1u. Then, for any geodesic wPS˚representing gk,\nw8P\nG,Sand is a bi-infinite geodesic.\nProof.Fix a generating set SandgPGtorsion-free. Let kě1be as in the statement of the result,\nand denote h“gk. Take a geodesic wPS˚forhand letπ“pe0,...,en´1qbe the (self-avoiding) walk\nstarting at the identity in the Cayley graph, of label wand|w|“n. Let Πbe the bi-infinite walk made\nby concatenating the paths hm¨π, for allmPZ. Thus,λpΠq“w8. We claim Πis self-avoiding. Suppose\nit is not, and take the smallest mPNsuch thatwmdoes not represent a SAW. Let πidenote the walk\nhi¨πwhere ipπiq“hiandtpπiq“hi`1. Asmis minimal, we know the concatenated walks π0... πm´1\nandπ1...πmare self-avoiding, and therefore the first intersection must occur between π0andπm. Then,\nthere exists v,uĎwprefixes, and fPπ0Xπmsuch thatf“v“hm´1u. Once again, because mis\nminimal,f‰h,hm´1. If we compute the distance,\ndSpf,hm´1fq“dSpu,vqď|w|,\naskwas chosen to minimize }gk}S, the distance between fandhm´1fmust be|w|. As both vertices are\ninπm, this is only possible if f“hmandhm´1f“hm´1. Thushm“1G, which is a contradiction as h\nis torsion-free. Therefore, w8P\nG,S. Finally, as we chose kto minimize the distance to the identity of\npowers ofg,wnmust be a geodesic for all nPN.\n174.4 Minimality\nTheorem 4.17 ([AB23]).LetGbe a finitely generated group. Then, Gis a torsion group if and only if\nG,Sis aperiodic for every (any) generating set.\nProof.SupposeGis a torsion group and let xP\nG,Sbe a periodic configuration that infinitely repeats\nthe wordw. Letg“w. By definition of the skeleton, gn“wn‰1Gfor allnPN. This contradicts the\nfact thatGis a torsion group. Conversely, if Ghas a torsion-element, by Proposition 4.16,\n G,Scontains\na periodic point.\nCorollary 4.18. IfGis a finitely generated torsion group, then for all generating sets\n G,Sis not sofic.\nProof.IfGis a finitely generated torsion group, Theorem 4.17 tells us that none of its skeletons contain\nperiodicconfigurations. Becausenon-emptysoficshiftsalwayscontainperiodicconfigurations, noskeleton\nofGcan be sofic.\n4.4 Minimality\nOur next objective is to find sufficient and necessary properties for the skeleton to be minimal. We begin\nby identifying possible subshifts of\n G,S.\nLemma 4.19. LetGbe a finitely generated group. Then,\n•For a symmetric subset S1ĎSandH“xS1y,\nH,S1is a subshift of\n G,S.\n•ForNIJGa normal subgroup,\n G{N,S1is a subshift of\n G,S, whereS1“SzN.\nProof.The first statement follows from the fact that any configuration from\n H,S1avoids all words from\nWPpG,Sq, asHis a subgroup of G. For the second statement, let xP\nG{N,SandtwiuiĎS˚a set of\ngenerators for N. Then by definition no factor wĎxbelongs to WP pG{N,Sq“xxWPpG,SqYtwiuyy.\nIn particular, it does not belong to WP pG,Sq. Therefore xP\nG,S.\nBecause every non-finite quotient gives us a non-empty subshift of\n G,S, if we want to find a minimal\nskeleton, it is reasonable to look at the class of just infinite groups. A group Gis said to be just infinite\nif it is infinite and every proper quotient is finite.\nProposition 4.20. LetGbe a finitely generated group with a generating set S. If\nG,Sis minimal, then\nGis a just infinite group.\nProof.If\nG,Sis minimal, every subshift of the form\n G{N,Smust be either empty or equal to\n G,S. Let\nNbe a proper normal subgroup, that is, non trivial and not equal to G. By Theorem 2.11, the connective\nconstants satisfy µpG{N,SqăµpG,Sq. Thus, the entropy of\n G{N,Sis strictly less than that of\n G,S, so\nthey cannot be equal. Then\n G{N,S“∅, meaningG{Nis finite. Therefore, Gis just infinite.\nProposition 4.21. LetGbe a finitely generated group with a generating set S. If\nG,Sis minimal, for\nevery symmetric subset S1ĹS, the subgroup xS1yis finite. In particular, torsion-free groups do not admit\nminimal skeletons.\nProof.If\nG,Sis minimal, every subshift of the form\n H,S1, forH“xS1y, must be either empty or equal\nto\nG,S. IfH“G, then by Theorem 2.11, µpG,S1q ăµpG,Sqmeaning\n G,S1is empty, which is a\ncontradiction. Therefore, HňG. Now, take sPSzpS1YHqandxP\nH,S1. Define the configuration\nx1“xp´8,´1ssxr0,`8PSZ. Becausexis inH’s skeleton, we know neither xp´8,´1snorxr0,`8qcon-\ntain subwords from WP pG,Sq. Next, if there exists i,jPNsuch thatxri,´1ssxr0,jsPWPpG,Sq, then\ns“Gpxri,´1sq´1pxr0,jsq´1which implies sPH. This is a contradiction. Therefore, x1P\nG,Sz\nH,S1.\nAs\nG,Sis minimal,\n H,S1“∅and thusHis finite. Finally, if a group is torsion-free, each generator\ngenerates Zwhich is not possible if the skeleton is minimal.\nRemark 4.22. Both conditions are not sufficient to characterize minimal skeletons. Take the group\n˜A2with generating set ta,b,cuas defined in Section 2.4. This group is just infinite [MV24], every\npair of different generators generates a subgroup isomorphic to the finite group S3, and every generator\ngenerates a copy of Z{2Z. Nonetheless, its skeleton is not minimal. Take the periodic configuration\nx“ pabcbq8which belongs to the skeleton. Then, the closure of the orbit of xis finite and contains\nexactly periodic configurations defined by cyclic permutations of abcb. But, the skeleton also contains\nthe periodic configuration y“pbcacq8, which is not one of the cyclic permutations.\n185. Sofic Skeletons\nAs the remark shows, if a minimal skeleton contains periodic configurations, it must be finite. This\nis the case of D8with generating set ta,bu, as seen on Example 2.6, which defines a minimal skeleton.\n5 Sofic Skeletons\nLet us tackle the question of which groups admit skeletons that are sofic. Since SFTs are sofic subshifts,\nfrom Theorem 4.5 we already know that plain groups admit sofic skeletons. But are there groups that\nadmit sofic skeletons which are not SFTs? The first naive strategy would be to ask when the word\nproblem of the group is regular, as this is the set of forbidden patterns used in the definition of the\nskeleton. Unfortunately, Anisimov showed in [An¯ ı71] that WP pG,Sqis regular if and only if Gis a finite\ngroup. We must therefore find other sets of forbidden patterns to study. Lemma 2.9 tells us that we\ncan look at the classes of groups where the language of SAWs is regular. The class of groups with such\nproperty have already been classified.\nTheorem 5.1 ([LW20]).LetGbe a f.g. group with Sa finite generating set. Then, LSAWpG,Sqis\nregular if and only if ΓpG,Sqhas more than one end and all ends are thin of size 1.\nAs Lindorfer and Woess show, if ΓpG,Sqhas only thin ends of size 1 its blocks are finite [LW20,\nLemma 5.3]. Combining this fact with Haring-Smith’s characterization of plain groups (Theorem 4.6),\nwe see that groups where LSAWpG,Sqis regular are exactly plain groups. Nevertheless, when considering\nbi-infinitely extendable SAWs, the situation is different.\nLemma 5.2. The groupG“ZˆZ{2Zgiven by the presentation xs,t|s2,rt,ssyhas a sofic skeleton.\nProof.We will exhibit a regular set of forbidden patterns for\n G,S, withS“ts˘1,t˘1u. Take the set of\nforbidden patterns\nF“tst˘nst¯1|nPNuYtt˘1st¯ns|nPNuYts2,t˘1t¯1u.\nIt is a simple exercise to show that Fis a regular language. Let us show\n G,S“XF. Suppose there is a\nconfiguration xP\nG,SzXF. Becausexis in the skeleton, we know it does not contain factors of the form\ns2ort˘1t¯1. Therefore it must contain a factor of the form st˘nst¯1ort˘1st˘ns. Suppose, xcontains\nthe wordw“stnst´1, for somenPN. There is no way to extend this word to the right, as wscontains\nthe factortst´1swhich evaluates to the identity, extending by t´kwithkěncreates the factor stnst´n\nwhich evaluates to the identity, and extending by t´kswithkďn´1creates the factor tkst´kswhich\nalso evaluates to the identity. This leads to a contradiction. The other cases being analogous, we have\nG,SĎXF.\nNow, suppose there is a configuration xPXFz\nG,S. By Lemma 4.2 and the definition of F,xmust\ncontain a SAP. Nevertheless, all SAPs in Gare cyclic permutations of words of the form stnst´nfor some\nnPN. Thus, each SAP contains a factor from F, leading to a contradiction and proving XFĎ\nG,S.\nThe Cayley graph of ZˆZ{2Zwith respect to the before mentioned generating set is the bi-infinite\nladder, which is a graph with two thin ends of size 2. An analogous proof can be done of the Cayley\ngraph of Z, which is a plain group, with respect to the generating set t˘1,˘2u, which also has thin ends\nof size 2.\nTo characterize groups which admit sofic skeletons we will make use of the fact that the language\nof a sofic subshift is regular. Our main tool in this regard will be the following version of the Pumping\nLemma.\nLemma 5.3 (Pumping Lemma) .LetLbe a regular language. Then, there exists pą0such that every\nwordwPLwith|w|ěpcan be decomposed as w“w1uvwith|u|ą1and|uv|ďp, such that for all\nnPN,w1unvPL.\nThis allows us to show that being sofic is a property of skeletons that depends on the generating set.\nProposition 5.4. Every group Gadmits a generating set Ssuch that\n G,Sis not sofic.\n195.1 Ends and Automorphisms\nts\nFigure 7: The configuration used for the Pumping Lemma (with p“4) depicted in the Cayley graph of\nthe subgroup xs,ty. The blue edges represent sand the red edges t.\nProof.By Corollary 4.18, if Gis a torsion group, no skeleton is sofic. We can therefore suppose Ghas a\ntorsion-free element. Let S1be any generating set for G, andga torsion-free element. We denote s“g2,\nt“g3, and define S“S1Yts,tu. Suppose\n G,Sis sofic. Then, its language Lp\nG,Sqis regular. Take\npą0given by the pumping lemma. The word w“tsp`1t´1s´pis contained in Lp\nG,Sqas it is globally\nadmissible through the configuration s8tsp`1t´1s´pt´1ps´1q8(see Figure 7).\nNow, by the Pumping Lemma we can decompose wasw“w1uvwith|uv|ďp. Thus,u“s´kwith\nkě1. Therefore, the word w1u2v“tsp`1t´1s´pp`kqbelongs to Lp\nG,Sq, which is a contradiction as\ntsp`1t´1s´pp`1q“Gϵ. We conclude that\n G,Sis not sofic.\n5.1 Ends and Automorphisms\nTo go towards a characterization we must make a brief detour through the theory of ends and automor-\nphisms of infinite quasi-transitive graphs. Let us begin by taking a look at the theory of ends of connected\ngraphs as introduced by Halin [Hal64].\nFor a connected graph Γ“pVΓ,EΓq, and a subset of vertices AĎVΓwe denote by ΓzAthe graph\nobtained by removing the vertices from Aand all their incident edges. We define a rayρto be an\ninfinite sequence of distinct vertices π“pv0,v1,...qPVN\nΓsuch that there is an edge between viandvi`1.\nAnalogously, a double ray to be a bi-infinite sequence of distinct vertices π“p...,v´1,v0,v1,...qPVZ\nΓ\nsuch that each successive vertex is connected by an edge. Two rays are said to be equivalent if for any\nfinite setAĎVΓall but finite many of their vertices are contained in the same connected component of\nΓzA. The equivalence classes of this relation are called the endsof the graph. Given an end ωand a\nfinite setAĎVΓ, we define Cpω,Aqto be the connected component of ΓzAwhere all the rays defining\nωeventually end up in.\nAdefining sequence for an end ωis a sequence of finite subsets pAiqiPNsuch that for all iě1,\nAiYCpω,AiqĎCpω,Ai´1q. We say that an end ωisthinif there exist mě1and a defining sequence\npAiqiPNsuch that|Ai|“mfor alliPN. The smallest mverifying this condition is called the sizeof\nω. An end is called thickif its size is infinite. Thomassen and Woess [TW93] showed using Menger’s\nTheorem that an end of size mPNYt8u, seen as an equivalence class of rays, contains a maximum of\nmvertex disjoint rays.\nLet AutpΓqdenote the set of automorphisms of Γ, that is, bijections f:VΓÑVΓthat preserve edge\nadjacency. We say a subgroup GďAutpΓqactsquasi-transitively onΓif the set of orbits of the\nactionGñΓhas finitely many distinct orbits. We say Gacts transitively if there is a unique orbit.\nFreudenthal and Hopf independently showed [Fre44; Hop43] that a quasi-transitive graph has either 0,1,\n2or an infinite amount of ends.\nIn our setting, all Cayley graphs ΓpG,Sqare transitive under the action of the group Gby left\ntranslations. Furthermore, this action preserves the labeling given by the generating set.\nTake Γto be locally finite and connected. We can classify automorphisms of Γinto three classes. An\nautomorphism gPAutpΓqis,\n•ellipticif it fixes a finite subset of VΓ,\n•parabolic if it fixes a unique end, and\n•hyperbolic if it fixes a unique pair of ends.\nHalin showed [Hal73] that for a non-elliptic automorphism gPAutpΓqand vertex vPVthe sequence\npv,g¨v,g2¨v,...quniquely defines and fixes an end which we call the direction ofg, and denote Dpgq.\n205.2 Characterizing Sofic Skeletons\nTheorem 5.5 (Halin, [Hal73] Theorem 9) .Letgbe a non-elliptic automorphism acting on a connected\nlocally finite graph Γ. Then,\n•DpgqandDpg´1qhave the same size m.\n•Dpgq‰Dpg´1qif and only if mă8. In this case gis hyperbolic.\n•There aremdisjoint double rays tπium\ni“1that are invariant by some positive power of g.\n•Ifgis hyperbolic, there exists a set AĎVΓof sizemandkPNsuch thatpgkn¨AqnPNand\npg´kn¨AqnPNare defining sequences for DpgqandDpg´1qrespectively, that intersect each πiin\nexactly one vertex.\nTo precisely understand thin ends, we study the following graphs.\nDefinition 5.6. A connected locally finite graph is called a stripif it is two ended and quasi-transitive.\nWe present general facts about strips that can be found in [LW20] and can be partly deduced from\nTheorem 5.5. For every strip Q, there exits a hyperbolic automorphism gPAutpQqthat fixes both ends\nω`andω´. Both ends have the same size, for instance m, which entails the existence of a finite set A\nof sizemsuch thatpgn¨Aqandpg´n¨Aqare defining sequence for ω`andω´respectively. In addition,\nthere aremdisjoint double rays intersecting every gn¨Aat exactly one vertex. We call such a strip\nag-strip of size m. When working with a g-strip, up to taking a power of g, we can assume that the\nsubgraph induced by Cpω`,AqzCpω`,g¨Aq, which we call Ppω`q, is connected and finite.\nThe following results show that quasi-transitive graphs contain strips, under conditions on their ends\nand automorphisms.\nLemma 5.7 (Lindorfer, Woess, [LW20] Lemma 3.3) .LetΓbe a connected and locally finite graph where\nGďAutpΓqacts quasi-transitively. If Γhas a thin end of size m, then it contains a g-strip of size mfor\nsomegPG.\nLemma 5.8 (Lindorfer, Woess, [LW20] Lemma 3.4) .LetΓbe a connected and locally finite graph where\nGďAutpΓqacts quasi-transitively. If Gcontains a parabolic element, then for every mě1,Γcontains\nag-strip of size at least mfor somegPG.\n5.2 Characterizing Sofic Skeletons\nWe provide the following characterization.\nTheorem 5.9 (Theorem B) .LetGbe a finitely generated group. There exists Ssuch that\n G,Sis sofic\nif and only if Gis a plain group, ZˆZ{2ZorD8ˆZ{2Z.\nThe idea of the proof is as follows. First, we use the same constructions of Lindorfer and Woess [LW20]\nto find ladder-like structures on strips that will allow us to use the Pumping Lemma, and then conclude\nthat all ends of the graph must be thin and of size at least 2. Next, by using similar ideas, we show that\nif the graph has an end of size two and the skeleton is sofic, then the group must be virtually Z. Finally,\nwe characterize virtually Zgroups with sofic skeletons, completing the proof.\nLemma 5.10. LetGbe a finitely group with a generating set S, such that ΓpG,Sqcontains an g-strip\nQfor somegPG. If\nG,Sis sofic, then Qhas size at most 2.\nProof.SupposeQis of size greater or equal than 3. Then,Qcontains three disjoint double rays which\nwe callπ1“pviqiPZ,π2“puiqiPZandπ3“pv1\niqiPZ, that areg-invariant. Recall we took our subgraph\nPpω`qto be connected and finite. Therefore, there is a path p1that connects two of the rays. Suppose\nwithout loss of generality that p1connectsπ1andπ2fromv0tou0with no other vertices from πifor\niPt1,2,3u. Analogously, g¨Ppω`qwill connect π3with another of the ways through a path p2. Up to\nrearranging indices, suppose p2connectsπ2toπ3starting at ukand ending at v1\nk, for somekPNsuch\nthat there are no other vertices from πiforiPt1,2,3u. Because the vertex set of every element of the\nsequencepgn¨Ppω`qqnPNis pairwise disjoint, no walks in tg2n¨p1|nPZuYtg2n¨p2|nPZuintersect.\nThis way, the subgraph induced by the three paths tπiu3\ni“1and allg2n¨p1andg2n¨p2,Q1ĎQis a\nperiodic subdivision of the bi-infinite 3-ladder (see Figure 8).\n215.2 Characterizing Sofic Skeletons\nu0v0\nuk\nv1\nkg¨v0\ng¨u0g¨uk\ng¨v1\nkg3¨v0\ng3¨u0\nFigure 8: The periodic subdivision of the 3-ladder with the configuration xhighlighted in red and blue.\nThe wordλ3λ3\n4λ´1\n3λ´2\n1is marked in blue, whereas the infinite prefix and suffix of xare marked in red.\nNow, let us give names to the labels of the different portions of the subdivision. Denote λ1be the\nlabel fromuktog¨uk,λ2the label from uktog¨u0,λ3the label from u0tov0,λ4the label from v0to\ng¨v0,λ5the label from uktov1\nk, and finally λ6the label from g¨v1\nktov1\nk. Then, for every ně1and\nkănthe configuration\nx“λ8\n1λ2.λ3λn\n4λ´1\n3λ´k\n1λ´1\n2λ5λ8\n6,\nbelongs to the skeleton (See Figure 8). Thus, λ3λn\n4λ´1\n3λ´k\n1PLp\nG,Sqfor everyně1andkăn. Notice\nthat the language L“tλ3λn\n4λ´1\n3λ´k\n1PS˚|k,nPNuis regular. If\n G,Swere sofic, its language Lp\nG,Sq\nwould be regular. Then by the closure properties of regular languages,\nL1“LXLp\nG,Sq“tλ3λn\n4λ´1\n3λ´k\n1PS˚|kănu,\nis regular. By the Pumping Lemma, there exists a pumping length pą0. Takeλ3λp`1\n4λ´1\n3λ´p\n1PL1.\nThis word decomposes as ˜www1such that|ww1|ďp. By the structure of our word, ww1is a suffix of λp\n1.\nNext, ˜ww2w1belongs toL1and therefore has the form\n˜ww2w1“λ3λn\n4λ´1\n3λ´k\n1“λ3λp`1\n4λ´1\n3w1w2w1,\nfor somek,lPNandw1PS˚. Because we are working over a Cayley graph, the labels of different edges\nstarting from u0must be different and thus the first generators for λ4andλ´1\n3are different. Therefore,\nn“p`1. This means, λ´k\n1“w1w2w1. Finally, as λ´k\n1is strictly longer than λ´p\n1,kěp`1. But, this\nwould imply λ3λp`1\n4λ´1\n3λ´k\n1belongs to Lp\nG,Sqand is not self-avoiding, which is a contradiction.\nProposition 5.11. LetGbe a finitely generated group. If there exists Ssuch that\n G,Sis sofic, then G\nhas more than one end, and ΓpG,Sqonly has thin ends of size at most 2.\nProof.LetGbe a finitely generated group with generating set Ssuch that\n G,Sis sofic. By Theorem\n4.17,Gis not a torsion group and therefore contains non-elliptic elements when seen as a subgroup\nof AutpΓpG,Sqq. IfGis one-ended, then ΓpG,Sqhas one end, which is by Lemmas 5.8 and 5.10 is a\ncontradiction. Thus, ΓpG,Sqhas at least one thin end. By Lemma 5.7, every thin end of size m, implies\nthe existence of a strip of size minΓpG,Sq. By Lemma 5.10, these strips – and consequently their\ncorresponding ends – must have size at most 2. Finally, if ΓpG,Sqhad a thick end, from the proof of\nTheorem4.1in[LW20]weknowitcontainsaone-endedsubgraph. Asbefore, thiscontradictsLemmas5.8\nand 5.10.\nThe converse of this proposition is not true: the group F2ˆZ{2Zalong with the generating set\nS“ta˘1,b˘1,su, given by the presentation xa,b,s|s2,ra,ss,rb,ssy, has thin ends of size two, but its\nskeleton is not sofic. Similar to what we did in Proposition 5.4, we can use the Pumping Lemma on the\nwordssan`1sa´n, withnPN, which are in Lp\nG,Sqthrough the configuration b8san`1sa´nb8. The\nnext Lemma captures this idea in the general setting.\nLemma 5.12. LetGbe a finitely generated group. If there exists Ssuch that\n G,Sis sofic and ΓpG,Sq\nhas an end of size 2, then Gis virtually Z.\n225.2 Characterizing Sofic Skeletons\nProof.Suppose ΓpG,Sqhas more than two ends, and take ω`the end of size 2. By Lemma 5.7, there\nexistsgPGandQag-strip of size 2. Then, there exist two g-invariant disjoint double rays π1“pviqiPZ\nandπ2“puiqiPZ. In the induced subgraph Ppω`qwe can find a path plinking, without loss of generality,\nv0andu0with no other vertices from π1andπ2. Furthermore, the walks belonging to tgn¨p|nPZu\ndo not intersect each other. This way, the graph spanned by π1,π2andpis a periodic subdivision of the\ninfinite 2-ladder,Q1ĎQ. Now, take an end ω1‰ω˘andπ3“pv1\niqiPNa ray defining ω1. Asπ3defines\nan end different from ω`there exists a smallest NPNsuch thatv1\niRQ1for alliąN. Because ΓpG,Sq\nis transitive, we can take without loss of generality v1\nNto be equal to some ukwithkPN, placed between\ng¨u0andg2¨u0. This is all represented in Figure 9.\nu0v0\nukg¨v0\ng¨u0uk“v1\nNg3¨v0\ng3¨u0\nπ3\nFigure 9: The periodic subdivision of the 2-ladder with the configuration xhighlighted in red and blue.\nThe wordλ2λ3\n3λ´1\n2λ´2\n1is marked in blue, whereas the infinite prefix and suffix of xare marked in red.\nLet us label the different sections of the bi-infinite ladder. We denote by λ1the label of the path from\nu0tog¨u0,λ2the label from u0tov0,λ3the label from v0tog¨v0,λ4the label from g2¨u0touk, and\nλPSNthe label of the ray pv1\nN`iqiPN. Then, for every nPNandkănthe configuration\nx“λ8\n1.λ2λn\n3λ´1\n2λ´k\n1λ4λPSZ,\nbelongs to the skeleton. Then, λ2λn\n3λ´1\n2λ´k\n1PLp\nG,Sqfor allkăn. Notice that the language given\nbyL“tλ2λn\n3λ´1\n2λ´k\n1|n,kPNuis regular. Therefore, L1“LXLp\nG,Sqis regular as we assume\n G,S\nis sofic. Take pą0the pumping length of L1given by the Pumping Lemma. If we pump the word\nλ2λp`1\n3λ´1\n2λ´p\n1is inL1as we did in the proof of Lemma 5.10, we conclude that there must exist n,kPN\nwithkěnsuch thatλ2λn\n3λ´1\n2λ´k\n1PL1, which is a contradiction as it is not self-avoiding.\nVirtually Zgroups have a very rigid structure. Epstein and Wall [Eps61; Wal67] (see [LG13] for our\ncurrent formulation) showed that a group is virtually Zif and only if it is of one of the following forms:\n1.Z˙ϕF, for some finite group FandϕPAutpFq,\n2.G1˚FG2, forG1,G2andFfinite groups such that rG1:Fs“rG2:Fs“2.\nGroups of the second type, G1˚FG2, can be shown to be isomorphic to D8˙ψFfor some homo-\nmorphismψ:D8ÑAutpFq(see [Gil22, Section 1.3]). Furthermore, every element gPZ˙ϕFcan be\nuniquely expressed as ftnwithfPF,nPZandtthe free generator of Z. Similarly, every element\ngPD8˙ϕFcan be uniquely expressed as frnsbwithfPF,nPZ,bPt0,1u, and rand sgenerators\nforD8“xr,s|s2,rsrsy.\nLemma 5.13. LetG“H˙ϕFbe a group such that Fis a finite group, and His either ZorD8.\nThen, for any generating set Sthe ends of the Cayley graph ΓpG,Sqhave size at least |F|.\nProof.TakeGas in the hypothesis. We will tackle the case when H“ZandH“D8separately.\nCase 1:H“Z:\nLetSbe a generating set for G. Then, there must exist at least one generator that does not belong\ntoF, which we call s. This generator, must have the form s“gtnfor somegPFandnPZ, and is thus\na torsion-free element of the group. For each element fPFwe define the ray πf“pf,fs, ..., fsi, ...q.\nThese rays are all pair-wise disjoint because sis torsion-free. Therefore, the end Dpsqhas size at least |F|.\n236. Approximating Entropy and Connective Constants\nCase 2:H“D8:\nLetSbe a generating set for G. As before, there must exist at least one generator that does not\nbelong toF, which we call s. Ifsis of the form grn, it is a torsion free element, and by the argument for\nthe previous case, Dpsqhas size at least |F|. Suppose then that all elements SzFare of the form grns.\nBecauseSis a generating set, SzFmust contain at least two elements which we will name s“grns\nands1“g1rms. Without loss of generality take nąm. Then,ss1is the torsion-free element g1rn´m\nfor someg1PF. As before, for each fPFdefine the ray πf“ pf,fs,fss1, ..., fpss1qi, ...q. Let us\nprove these rays are disjoint. If fpss1qk“f1pss1qlforf,f1PFandk,lPN, thenpss1qk´lPFhas\ntorsion, which is a contradiction. On the other hand, if fpss1qks“f1pss1qlforf,f1PFandk,lPN, then\nrpn´mqpk`lq`nsPF, which is also a contradiction. Thus, the rays πfare disjoint and therefore Dpss1q\nhas size at least |F|.\nProposition 5.14. LetGbe a virtually Zgroup. Then, there exists Ssuch that\n G,Sis sofic if and only\nifGis either Z,ZˆZ{2Z,D8ˆZ{2ZorD8.\nProof.LetGbe a virtually Zgroup. Then, Gis of the form H˙ϕFforHPtZ,D8uandFa finite\ngroup. Joining Lemma 5.13 and Lemma 5.10, if\n G,Sis sofic for some generating set S,|F| ď2. If\n|F|“1, thenGis either ZorD8. If|F|“2, thenF»Z{2Zandϕis the trivial automorphism. In this\ncaseGis either ZˆZ{2ZorD8ˆZ{2Z.\nConversely, we already know ZandD8admit sofic skeletons as they are plain groups. Similarly, by\nLemma 5.2, ZˆZ{2Zadmits a sofic skeleton. Finally, if we take the presentation for for D8ˆZ{2Zgiven\nbyxa,b,s|a2,b2,s2,psaq2,psbq2ythe corresponding Cayley graph is the bi-infinite ladder, and therefore\nLemma 5.2 can be adapted to show its skeleton is sofic.\nWe now have all the ingredients to characterize groups that admit a sofic skeleton.\nProof of Theorem 5.9. LetGbe a finitely generated group that admits a sofic skeleton through the\ngenerating set S. From Lemma 9, ΓpG,Sqhas only thin ends, all of size at most 2. If all ends are of size\n1,Gis a plain group. Next, if Ghas at least one end of size 2, it is virtually Zby Lemma 5.12. Then,\nby Proposition 5.14 Gis either ZˆZ{2ZorD8ˆZ{2Z. For the other direction, if Gis a plain group by\nTheorem 4.5 it admits a sofic skeleton (as SFTs are sofic). Finally, if Gis either ZˆZ{2ZorD8ˆZ{2Z,\nProposition 5.14 tells us Gadmits a sofic skeleton.\n6 Approximating Entropy and Connective Constants\n6.1 Bridges and Periodic Points\nWe saw in Corollary 4.10 that the connective constant of SFT skeletons of plain groups can be approx-\nimated by their periodic points. This is also the case of irreducible sofic shifts [LM21, Theorem 4.3.6.].\nThe natural question that follows is if this is possible for skeletons that are not sofic. Clisby showed [Cli13]\nthat this is the case for Zdwith standard generating set, which by Theorem 5.9 do not define sofic skele-\ntons. Instead of periodic configurations, Clisby used the term endless SAWs . By using the notion of\na bridge, introduced by Hammersley and Welsh [HW62] and latter expanded upon by Grimmett and\nLi [GL18], we can generalize this result to any Cayley graph admitting a particular kind of graph height\nfunction.\nDefinition 6.1. LetΓbe an infinite, connected, locally finite, quasi-transitive graph. A graph height\nfunctionph,Hqiscomposedofafunction h: ΓÑZandasubgroup HďAutpΓqactingquasi-transitively\nonΓsuch that\n•hp1Gq“0,\n•(H-difference-preserving) for all u,vPΓandgPH\nhpg¨vq´hpg¨uq“hpvq´hpuq,\n•for alluPΓ, there exists v,v1PΓadjacent to usuch thathpvqăhpuqăhpv1q.\n246.1 Bridges and Periodic Points\nAbridgewith respect to the height function ph,Hqis a self-avoiding walk π“pe0, ..., en´1qthat\nverifies\nhpipe0qqăhptpeiqqďhptpen´1qq,\nfor alliPt0,...,n´1u.\nExample 6.2. TakeG“Z2with the standard generating set ta˘,b˘u. If we look at the generators\nasa“p1,0qand b“p0,1q, we define the map hpgq“mforg“pm,nqPZ2. This function defines a\ngraph height function with respect to H“Z2acting by left-translations. Further still, any elementary\namenable group admits a graph height function [GL18].\nLemma 6.3. LetGbe a finitely generated group with generating set S. IfΓpG,Sqadmits a graph height\nfunctionph,Hq, then\nG,Scontains periodic configurations. Moreover, if πis a bridge such that ipπqand\ntpπqlie in the same H-orbit, then λpπq8P\nG,S.\nProof.Letπandπ1be two bridges such that tpπq“ipπ1q. Then, the concatenation of both paths, ππ1,\nis a bridge. Furthermore, for every gPH,g¨πis also a bridge, as hisH-difference-preserving.\nNow, letRbe a finite right transversal for the action of HonΓpG,Sq. Take a bridge πsuch that\nipπq,tpπq PH¨rwithrPR. Ifipπq “h1¨randtpπq “h2¨r, becausehisH-difference-preserving,\nh2h´1\n1¨πis a bridge starting at h2¨r. We can then concatenate πwithh2h´1\n1¨πto create a bridge,\nwhich we denote by π2, whose label is given by λpπq2. This process can be iterated indefinitely to obtain\na bi-infinite SAW whose label is given by λpπq8.\nNext, take a bridge πsuch that ipπqPH¨r1andtpπqPH¨r2, withr1,r2PRdistinct representatives.\nUp to translation by an element from H, we can take any bridge starting at a vertex in H¨r2, sayπ1and\nconcatenate to πto obtain a new bridge ππ1. Such a bridge exists by the definition of a graph height\nfunction as there must exist at least one vertex vnext tor2such thathpr2qăhpvq. Similarly, we can take\nany bridge in the H-orbit of tpπ1q, which we denote π2, and concatenate it –up to translation by H– to\nππ1. Iterating this process, for all nPNwe obtain a bridge ππ1...πn. Because there are a finite number\nofH-orbits, we will have iďjsuch that ipπiq,tpπjqbelong to the same H-orbit. Then, as previously\nstatedπ1“πiπi`1... πjis a bridge that can be iterated to obtain the periodic point λpπ1q8.\nWe saw in Theorem 4.17 that torsion groups have aperiodic skeletons. By the previous lemma, graph\nheight functions imply the existence of periodic points. Combining these two facts we can state the\nfollowing.\nTheorem 6.4 (Theorem C) .The Cayley graphs of infinite torsion f.g. groups do not admit graph height\nfunctions.\nThis generalizes a result from Grimmett and Li who showed that the Grigorchuk group, which is an\ninfinite torsion group, does not admit a graph height function [GL17b]. However, the converse of the\nTheorem does not hold, as they showed the Higman group, which is torsion-free [Hig51], does not admit\ngraph height functions.\nBridges are particularly useful to compute the connective constant of graphs, and have been used to\nobtain exact expressions for the constant (for instance, [DS12]). Let us denote by bn,gthe number of\nbridges of length nstarting at gPG, andbn“mingPGbn,g. As stated in the proof of Lemma 6.3, we\ncan concatenate bridges with corresponding endpoints. Then, bnbmďbn`mand by Fekete’s sub-additive\nLemma, there exists a constant βpΓ,hq, where Γ“ΓpG,Sq, such that\nβpΓ,hq“lim\nnÑ8na\nbn.\nThis process can be done for a larger class of graphs [GL18], and helps us compute connective con-\nstants.\nTheorem 6.5 (General Bridge Theorem [Lin20]) .LetΓbe an infinite, connected, locally finite, quasi-\ntransitive graph. Then, if Γadmits a graph height function ph,Hq,\nµpΓq“maxtβpΓ,hq,βpΓ,´hqu.\n256.2 Lower Bounds with Self-Avoiding Polygons\nUsing this result, we can find conditions under which periodic points approximate the connective\nconstant. In other words, periodic points from\n G,Sapproximate its entropy.\nTheorem 6.6 (Theorem D) .LetGbe a finitely generated group and Sa finite generating set. If ΓpG,Sq\nadmits a graph height function ph,Hqsuch thatHacts transitively on ΓpG,Sq, then\nµpG,Sq“lim\nnÑ8n?en,\nwhereendenotes the number of periodic points in\n G,Sof periodnPN.\nProof.Let us denote by ¯bnthe minimum over all gPGof the number of bridges of length nstarting atg\nfor the graph height function pH,´hq. BecauseHacts transitively on ΓpG,Sq, there is a single H-orbit.\nThus, by Lemma 6.3, every bridge for hand´hcan be iterated to obtain a periodic point. This means,\nmaxtbn,¯bnuďenďcn.\nBy taking the nth root and limit, Theorem 6.5 implies,\nµpG,Sq“maxtβpΓ,hq,βpΓ,´hquď lim\nnÑ8n?enďµpG,Sq.\nExamples of Cayley graphs with a graph height function ph,Hqsuch thatHacts transitively are\ngiven by Cayley graphs that admit stronggraph height function where H“G. Strong graph height\nfunctions are graph functions where we also ask for Hto be a finite index subgroup of G, and to act by\nleft translations [GL17b]. A class of groups that admit such functions are groups with strictly positive\nfirst Betty number [GL17a]. Other sufficient conditions can be found in [GL20].\n6.2 Lower Bounds with Self-Avoiding Polygons\nWhat other methods can we use when graph height functions are not available? We will make use of a\ncounting argument popularized by Rosenfeld [Ros20] to find lower bounds on the connective constant by\nstudying the sets of forbidden patterns defining the skeleton. Rosenfeld found the following criterion for\nsubshifts.\nTheorem6.7 ([Ros22], Corollary12) .LetAbe a finite alphabet and FĎA`a set of connected forbidden\npatterns. If there exists a positive real number βą1such that\n|A|ěβ`ÿ\nně0fnβ1´n,\nthenαpXFqěβ, wherefnis the number of forbidden patterns of length n, that is,fn“|FXAn|.\nTherefore, we can use the different forbidden patterns we have found so far for the skeleton to find\nlower bounds for the connective constant. From Lemma 4.2, we know the set of SAPs along with words\nof the form ss´1define a set of forbidden patterns for the skeleton.\nProposition 6.8. LetpG,Sqbe a finitely generated group. If there exists a positive real number βsuch\nthat\n|S|´1ěβ`ÿ\nně0ρnβ1´n,\nthenµpG,Sqěβ, whereρnthe number of SAPs of length n, that is,ρn“|OG,SXSn|.\nThe proof of the proposition is essentially the same as the one from [Ros22], and the core ideas of the\nproof are also found in the next section.\nThis approach is different from the usual use of self-avoiding polygons to approximate µpG,Sqin the\nliterature. We define the asymptotic growth rate for SAPs through,\nµSAP“lim sup\nnÑ8?ρn.\nIt has been shown that µSAP“µpG,Sqfor Euclidean lattices [Ham61; Kes63], but µSAPăµpG,Sqfor\nmany non-euclidean lattices, including some Cayley graphs of surface groups [Pan19].\n266.3 Free Burnside Groups\n6.3 Free Burnside Groups\nA class of groups where we can use Rosenfeld’s method to obtain lower bounds on the connective constant\nare free Burnside groups. These groups were first introduced in the context of the Burnside Problem,\nwhich asks if there exist infinite finitely generated torsion groups where the order of every element is the\nsame.\nGiven two natural numbers m,ně1, thefree Burnside group Bpm,nq, is the group generated\nbymgenerators S“ ts˘\n1,...,s˘1\nmu, and relations wn“1Bpm,nqfor every word wPS˚. It was shown\nthat these groups are finite for nPt1,2,3,4,6uindependently of m[Bur02; San40; Hal58]. Nevertheless,\nAdian and Novikov showed that Bpm,nqis infinite for odd ně4381and anymě2[NA68]. Since then,\nthis lower bound has been improved multiple times [Iva94; Lys96], and is currently sitting at ně8000\nfor even exponents, and ně557for odd ones [ART23]. It is an open problem to determine the smallest\nnfor which the group is infinite.\nTheorem 6.9. LetBpm,nqbe the free Burnside group for m,ną1. IfBpm,nqis infinite and βą1\nsatisfies\nβ\nβn´1´1`βď2m´1,\nthenµpBpm,nq,Sqěβ.\nProof.LetLkbe the set of SAWs of length kPN. We will prove by induction that |Lk|ěβ|Lk´1|, for\nβą1as in the statement. Notice |L0|“1as it only contains the empty word, and |L1|“|S|“2m. By\nhypothesis, βď2m, and therefore |L1|ěβ|L0|.\nSuppose our statement is true up to some ną0. In particular, for kďn\n|Lk´j|ď|Lk|\nβj.\nNow, because every SAW from Lkcan be extended in |S|´1“2m´1ways, we have that\n|Lk`1|“p2m´1q|Lk|´|B|,\nwhereBis the set of SAWs that when extended generate a path of length k`1that self-intersects. Notice\nthat ifuPB, it can be written in the form u“u1vn. We define the sets Bi“tuPB|u“u1vn,|v|“iu\nto obtain the upper bound |B| ďř\niě1|Bi|. Then, every word in Biis determined by a word from\nLk`1´pn´1qi, namelyu1v. Therefore,\n|Bi|ď|Lk`1´pn´1qi|ď|Lk|\nβpn´1qi´1,\nand consequently,\n|B|ď|Lk|ÿ\niě1β1´pn´1qi\nBecauseβą1, we have a geometric series:\n|B|ď|Lk|β\nβn´1´1.\nFinally, joining all the formulas we obtain:\n|Lk`1|ěˆ\n2m´1´β\nβn´1´1˙\n|Lk|ěβ|Lk|.\nOur induction proven, we can iterate the identity to obtain |Lk|ěβk, and thusµpBpm,nq,Sqěβ.\nCorollary 6.10. IfBpm,nqis infinite and 0ăγă1satisfies\nmě1\n2`1\n2pγp1´γq1\nn´1q´1, (1)\nthenµpBpm,nq,Sqěγp2m´1q. In particular, for ną3andmą1,\nµpBpm,nq,Sqěn´1\nnp2m´1qą?\n2m´1.\n277. Geodesic skeletons\nProof.Let us take 0ăγă1satisfying (1) and denote M“2m´1. By rearranging terms from (1) we\nobtain:\npγ´1qγn´1Mn´1`1ď0.\nThen, multiplying the expression by M, adding˘γ, and rearranging more terms we obtain\nγM`pγ´1qMpγn´1Mn´1´1qď0.\nNext, asγă1satisfies (1), we also know γn´1Mn´1´1ě0. Thus,\nγM\nγn´1Mn´1´1`pγ´1qM“γM`pγ´1qMpγn´1Mn´1´1q\nγn´1Mn´1´1ď0.\nBy Theorem 6.9, µpBpm,nq,Sqěγp2m´1q. Finally,γ“n´1\nnsatisfies (1) and γp2m´1qą?2m´1\nforną3andmą1.\n7 Geodesic skeletons\nA geodesic is always a self-avoiding walk. It is then natural to see what changes when we restrict a\ngroup’s skeleton to bi-infinite geodesics. We define the geodesic skeleton of Gwith respect to Sby,\ng\nG,S“txP\nG,S|@wĎx,w1“Gw:|w|ď|w1|u.\nThis subshift is contained in the skeleton\n G,S, and its locally admissible language is Geo pG,Sq. In\nparticular, it is generated by taking Geo pG,Sqcas the set of forbidden patterns. As was the case with\nthe skeleton,\ng\nG,Sis empty if and only if the group is finite; this is due to Watkins who showed that\nevery transitive infinite graph contains a bi-infinite geodesic [Wat86].\nSFT Geodesics We have a sufficient condition for the geodesic skeleton to be an SFT coming from\na result by Gilman, Hermiller, Holt and Rees [Gil+07] that characterizes virtually free groups. They\nshowed that for a finitely generated group G, there exists a finite generating set Ssuch that Geo pG,Sqis\nk-locally excluding, that is, there exists a set Fof words of length ksuch that a word wPS˚is geodesic\nif no factor of length kbelongs toF, if and only if Gis virtually free. An immediate consequence is the\nfollowing.\nProposition 7.1. LetGbe a virtually free group. Then, there exists Ssuch that\ng\nG,Sis a SFT.\nEffective Geodesics\nLemma 7.2. LetGbe a recursively presented group. Then,\ng\nG,Sis effective for every finite generating\nsetS.\nProof.Wedescribeaco-semi-algorithmforGeo pG,Sq. Byusinganenumerationforthewordproblem, we\ncan test every word w1of length|w1|ă|w|to see if they define the same group element, i.e. w1w´1“G1G.\nIf one suchw1w´1appears in the enumeration, we know wis not geodesic and accept. If wis not geodesic,\nw1w´1will eventually be enumerated, for some w1of shorter length. When wPGeopG,Sqthe algorithm\nnever stops.\nIn other words, the effectiveness of Xg\nG,Sis a consequence of the fact that a recursively enumerable\nword problem implies that the language of geodesics is co-recursively enumerable.\nSofic Geodesics By their definition, we can easily obtain many examples of sofic geodesic skeletons.\nProposition 7.3. LetGbe a finitely generated group and Sa generating set. If GeopG,Sqis regular,\nthen\ng\nG,Sis sofic.\n287.1 Entropy and Connective Constant for Geodesics\nBecause the complement of a regular language is regular, when Geo pG,Sqis regular then\ng\nG,Sis\ndefined by a regular set of forbidden words, and is therefore sofic. We know that Geo pG,Sqis regular for\nall generating sets in abelian groups [NS95] and hyperbolic groups [Eps+92]. Also, there exists at least\none generating set such that Geo pG,Sqis regular for virtually abelian groups [NS95], Coxeter groups\n[How93] and other classes [CM04; HR12; AC16].\nIn order to find a characterization of groups that admit a geodesic skeleton that is sofic, we must look\nat geodesics that are not extendable. These elements are precisely the ones known as dead-ends. An\nelementgPGis adead-end with respect to the generating set Sif there exists ně1such that for all\nwPS˚with|w|ďn,dp1G,gwqďdp1G,gq.\nProposition 7.4. LetGbe a finitely generated group along with a generating set S. Then,\ng\nG,Sis sofic\nand the language of geodesics defining dead-ends is regular if and only if GeopG,Sqis regular.\nProof.Let us denote by Dthe language of geodesics defining dead-ends. The language of prefixes of a\nregular language is regular and the language of inverses of a regular language is regular. Then, if Dis\nregular, the languages pDandpD1denoting the set of prefixes of Dand the set of inverses of prefixes of\nDrespectively, are regular. If Geo pG,Sqis regular, then Dis regular: it suffices to take the finite state\nautomaton for Geo pG,Sqand only keep accepting states with no outgoing transitions. Take the union\nL“Lp\ng\nG,SqYpDYpD1. This language is regular, as the union of regular languages is regular. Let us\nshowL“GeopG,Sq. The first inclusion LĎGeopG,Sqis direct as Lp\ng\nG,SqĎGeopG,Sq, inverses of\ngeodesics are geodesic and prefixes of geodesics are geodesics.\nNext, take wPGeopG,Sq. Ifwis bi-infinitely extendable as a geodesic, then wPLp\ng\nG,SqĎL. Ifw\nis not bi-extendable, it can fail to be extended to the right or to the left. Suppose it is not extendable\nto the right. Then there exists vPS˚such thatwvPGeopG,SqandwvsRGeopG,Sqfor allsPS. This\nmeanswis a prefix of the dead-end wv, and therefore wPpDĎL. Finally, if wis not extendable to the\nleft, there exists vPS˚such thatvwPGeopG,SqandsvwRGeopG,Sqfor allsPS. This is the same\nas,w´1v´1PGeopG,Sqandw´1v´1sRGeopG,Sqfor allsPS. This means wis the inverse of a prefix\nof the dead-end represented by w´1v´1, and thuswPpD1ĎL.\nCorollary 7.5. LetGbe any finitely generated group with generating set S, and Z“ xty. Then,\ng\nG˚Z,SYtt˘1uis sofic if and only if GeopG,Sqis regular.\nProof.The language of dead-ends of G˚Zis empty as any geodesic can be extended by t˘1. Furthermore,\nanygeodesicin G˚Zcanbedecomposedasgeodesicson Gseparatedbyfactorsoftheform t˘n. Therefore,\nGeopG˚Z,SYtt˘1uqis regular if and only if Geo pG,Sqis regular. By Proposition 7.4 this happens if\nand only if\ng\nG˚Z,SYtt˘1uis sofic.\nWe pose the following question for sofic geodesic skeleton.\nQuestion 7.6. Is\ng\nG,Ssofic if and only if Geo pG,Sqis regular?\nPeriodic Geodesics As was the case for the skeleton (Theorem 4.17), the aperiodicity of the geodesic\nskeleton also characterizes torsion groups.\nTheorem 7.7. LetGbe a finitely generated group. Then, Gis a torsion group if and only if\ng\nG,Sis\naperiodic for every (any) generating set S.\nProof.SupposeGcontains a torsion-free element. Then, by Proposition 4.16 for any generating set S,\nthere exists kě1andwPS˚a geodesic for gksuch thatw8P\ng\nG,S, which is a periodic configuration.\nConversely, if there exists a periodic configuration x“w8P\ng\nG,Sfor some generating set S, theng“w\nis a torsion-free element.\n7.1 Entropy and Connective Constant for Geodesics\nThe objective of this section is to define an analog of the connective constant for geodesics. This relies\non finding the asymptotic growth rate of geodesics of a given length. The geodesic growth ofGwith\nrespect toSis the map ΓG,S:NÑNgiven by\nΓG,Spnq“|twPGeopG,Sq||w|ďnu|.\n297.1 Entropy and Connective Constant for Geodesics\nBecause this function is sub-multiplicative we can define the geodesic connective constant of the\nCayley graph ΓpG,Sqas\nµgpG,Sq“lim\nnÑ8nb\nΓG,Spnq.\nAs we saw in Remark 3.5, µgpG,Sqcan be shown to be equal to the growth rate of the number of\ngeodesics of length exactly n. Thus, the geodesic growth is an upper bound on the complexity of\ng\nG,S.\nBecause GeopG,Sqis the set of locally admissible words for the geodesic skeleton, we use Lemma 2.1 to\nobtain an expression for the entropy.\nLemma 7.8. LetGbe a finitely generated group along with a generating set S. Then,\nhp\ng\nG,Sq“logpµgpG,Sqq.\nIn other words, the geodesic connective constant is equal to the connective constant of bi-extendable\ngeodesics.\nSame as with the connective constant, the geodesic version is a non-negative rational power of a\nPerron number when\ng\nG,Sis sofic, and a right-computable number when\ng\nG,Sis effective. It is also\na lower bound of the connective constant, that is, µgpG,Sq ďµpG,Sq. This inequality may be strict:\ngraphs may have geodesic connective constant equal to 1 without being finite. As shown in [Bri+12], the\nvirtually Z2groupH“xa,t|ra,tat´1s,t2y, has geodesic growth of order Opn3qand therefore,\nµgpH,ta,tuq“ 1ă?\n3ďµpH,ta,tuq.\nThis is also the case for lattices with known (or well-approximated) connective constants.\nProposition 7.9. The geodesic connective constants of the square grid, ladder graph and hexagonal grid\nare as follows:\n•µgpZ2q“2,\n•µgpLq“1,\n•µgpHq“?\n2.\nProof. •For the square lattice, we know that ΓZ2,ta,bupnqď2n`3which implies hpXg\nZ2,ta,buq“logp2q,\nasXg\nG,Scontains the full-shift ta,buZ.\n•Recall that the ladder graph Lis the Cayley graph of ZˆZ{2Zwith generating set tt,su, where\ns2“Gεandtis the generator for Z. In this case, the geodesic growth is given by Γpnq“n2`3n\nwhenně2. Thus, the geodesic connective constant is 1.\n•Also recall that the hexagonal grid His the Cayley graph of the Coxeter group ˜A2with generating\nsetta,b,cu(see Example 2.7). From [Ava04] we know that the generating function for the geodesic\ngrowth of ˜A2in this case is given by\nfpzq“2z3`z2`z`1\np1´zqp1´2z2q.\nThus, the geodesic connective constant is given by the reciprocal of the smallest zero of the denom-\ninator, which is?\n2.\nOn the other hand, if we take the infinite dihedral group D8with the generating set S“ta,buas\nseen in Example 2.6, we have that µpD8,Sq“µgpD8,Sq“1.\nQuestion 7.10. Under which conditions µpG,Sq“µgpG,Sq? Under which conditions is the inequality\nstrict?\n30REFERENCES REFERENCES\nAcknowledgments\nWe would like to thank Matthieu Rosenfeld for discussions on the counting method, especially when\napplied ton-power-free words. We would also like to thank Alex Bishop for pointing us towards references\non the geodesic growth of Coxeter groups.\nReferences\n[AJ90] SvenErickAlmandSvanteJanson.“Randomself-avoidingwalksonone-dimensionallattices”.\nIn:Stochastic Models 6.2 (1990), pp. 169–212 (cit. on p. 9).\n[An¯ ı71] A. V. 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In: Journal of Combina-\ntorial Theory, Series B 41.3 (1986), pp. 341–355 (cit. on pp. 10, 28).\nUniversité Paris-Saclay, CNRS, LISN, Gif-sur-Yvette, France\nE-mail address: nathalie.aubrun@lisn.fr\nUniversité Paris-Saclay, CNRS, LISN, Gif-sur-Yvette, France\nE-mail address: nicolas.bitar@lisn.fr\n34"    },    {        "title": "2402.13952v1.Aaronson_Ambainis_Conjecture_Is_True_For_Random_Restrictions.pdf",        "content": "arXiv:2402.13952v1  [cs.CC]  21 Feb 2024AARONSON-AMBAINIS CONJECTURE IS TRUE FOR\nRANDOM RESTRICTIONS\nSREEJATA KISHOR BHATTACHARYA\nAbstract. In an attempt to show that the acceptance probability of a quan-\ntum query algorithm making qqueries can be well-approximatedalmost every-\nwhere by a classical decision tree of depth ≤poly(q), Aaronson and Ambainis\nproposed the following conjecture: let f:{±1}n→[0,1] be a degree dpoly-\nnomial with variance ≥ǫ. Then, there exists a coordinate of fwith influence\n≥poly(ǫ,1/d).\nWe show that for any polynomial f:{±1}n→[0,1] of degree d(d≥2)\nand variance Var[f]≥1/d, ifρdenotes a random restriction with survival\nprobabilitylog(d)\nC1d,\nPr/bracketleftbigg\nfρhas a coordinate with influence ≥Var[f]2\ndC2/bracketrightbigg\n≥Var[f]log(d)\n50C1d\nwhereC1,C2>0 are universal constants. Thus, Aaronson-Ambainis conjec-\nture is true for a non-negligible fraction of random restrictions of t he given\npolynomial assuming its variance is not too low.\n1.Introduction\nOne of the central open problems in the field of quantum query comp lexity is\nfinding if there exists a partial function which is defined on a large fra ction of\nthe Boolean hypercube (say, constant) but whose quantum quer y complexity and\nclassical query complexity are super-polynomially separated. The s eminal result\nof Beals, Burhman et al. [ 4] shows that no such separation is possible when\nthe function is defined on the entire hypercube. On the other hand , functions\nfor which we know such a separation (e.g. - Forrelation [ 1] , Bernstein-Vazirani\n[5]) are defined on an exponentially small fraction of the hypercube. A possible\nexplanationastowhy all known functionsexhibiting largegapsbetwe en quantum\nandclassical querycomplexity haveverysmallsupportsizewouldbe thefollowing\nfolklore conjecture:\nConjecture 1.1. LetQbe a quantum query algorithm with Boolean output\nonnqubits making qqueries. Let P:{±1}n→[0,1] be given by P(x) =\nPr[Qoutputs 1 on x]. For any ǫ >0, there exists a classical query algorithm A\nsuch that E[(A(x)−Q(x))2]≤ǫandAmakes at most poly/parenleftbigg\nq,1\nǫ/parenrightbigg\nqueries.\n12 SREEJATA KISHOR BHATTACHARYA\nItisknownthatif Qmakesatmost qqueries, then Pisgivenbyapolynomialof\ndegree at most 2 q. Although Phas more structure than any arbitrary low degree\nboundedpolynomial, it isfurther conjectured thatsuchstructur e isnotnecessary.\nIn other words, we forget the fact that Parises from a quantum query algorithm\nand instead try to construct a classical query algorithm for anybounded low-\ndegree polynomial. This led to the following conjecture (also folklore) .\nConjecture 1.2. LetP:{±1}n→[0,1] be a degree dpolynomial. For any\nǫ >0, there exists a classical decision tree Tof depth at most poly(d,1/ǫ) such\nthatE[(P(x)−T(x))2]≤ǫ.\nAaronson and Ambainis [ 2] proposed the following query algorithm to estimate\nP: suppose the variance of the function is sufficiently small. Then we te rminate\nthe query algorithm and output the average over the unqueried co ordinates. If\nnot, we query the coordinate with the highest influence and restrict the function\naccording to the response received. We keep doing this until we hav e made too\nmany queries or the variance has become sufficiently low. In order to show that\nthis algorithmgives anaccurate estimate, [ 2] observed that it is sufficient to prove\nthe following conjecture.\nConjecture 1.3. (Aaronson-Ambainis conjecture) Letf:{±1}n→[0,1]\nbe a degree dpolynomial. Then, there exists a coordinate jsuch that Infj[f]≥\npoly(1/d,Var[f])\nAs a side remark, we mention that O’Donnell et al. [ 16] had shown previously\nthat functions which can be approximated by decision trees have a c oordinate\nwith high influence. So conjectures 1.2and1.3are equivalent.\nAaronson-Ambainisconjecturehasreceived significant attention inthepastfew\nyears. A 2006 result of Dinur, Friedgut et al. [ 8] shows that the conjecture is true\nifpoly(d) is replaced by exp( d). In 2012, Montanaro [ 14] proved the conjecture\nin the special case of block-multilinear forms where all coefficients ha ve the same\nmagnitude. In 2016, O’Donnell and Zhao [ 17] showed that it suffices to prove\nthe conjecture for a special class of polynomials known as one-block decoupled\npolynomials . In 2020, Keller and Klein [ 11] claimed to have found a proof for the\nconjecture but their paper had a subtle flaw and turned out to be w rong. More\nrecently, Lovett and Zhang [ 13] initiated a new line of attack using the notions of\nfractional block sensitivity andfractional certificate complexity . In 2022, Bansal,\nSinha, Wolf [ 3] proved that this conjecture is true for completely bounded block\nmultilinear forms - a class of polynomials that captures a special kind of quantum\nquery algorithms.\nIn this work we show that Aaronson-Ambainis conjecture is true fo r a large\nfraction of random restrictions of fassuming Var[f] is not too low. We hope our\nresult gives new insights to the Aaronson-Ambainis conjecture. In particular, this\nopens up a possible line of attack:\n•Assumingasupposedcounterexample f:{±1}n→[0,1],modifyitappro-\npriately (e.g., by composing it with some appropriate gadget or applyin g3\na low noise operator) to get a function ˜f:{±1}˜n→[0,1] such that most\nof its random restrictions remain a counterexample. Combined with o ur\nresult, this will prove Aaronson-Ambainis conjecture. This approa ch is\ndiscussed in a bit more detail in the conclusion.\nOur main result is a new structural restriction about bounded low-d egree poly-\nnomials over the hypercube. While several structural results are known about\nlow-degree booleanfunctions f:{±1}n→{0,1}, such results are rare for low-\ndegreebounded functions f:{±1}n→[0,1]. We show that if f:{±1}n→[0,1]\nhas degree dandρis a random restriction with survival probability O(log(d)/d),\nthen with very high probability fρdepends essentially on ≈poly(d) coordinates,\neven though there are O(nlog(d)/d) alive coordinates on average.\n2.Organization\nWe introduce notations and necessary preliminaries in section 3. We g ive a\nhigh level overview of our proof in section 4. In section 5 we compile so me\nlemmas that will be needed in our main proof. Our main results are prov en\nin section 6. Our main technical tool is Theorem 6.3, which says that most\nrandom restrictions of a bounded low-degree function can be appr oximated by a\nsmall junta. In Theorem 6.4we prove the result mentioned in the abstract (that\nAaronson-Ambainis conjecture is true for a non-negligible fraction of random\nrestrictions).\n3.Notations and preliminaries\nQuery algorithms.\n(1) A classical query algorithm A(or equivalently, a decision tree) for com-\nputing a function f:{±1}n→Rcan access the input x∈{±1}nby\nadaptively issuing queries to its bits. We assume internal computatio ns\nhave no cost. The depth of the query algorithm/decision tree is the maxi-\nmum number of bit queries issued on an input. We say A ǫ-approximates\nfif||f−A||2\n2=Ex∈{±1}n[(f(x)−A(x))2]≤ǫ.\nFor a partial function f:S(⊆{±1}n)→{0,1}, its classical query\ncomplexity D(f) is the smallest dfor which there exists a decision tree T\nof depth dsuch that T(x) =f(x) for allx∈S.\n(2) A quantum query algorithm can access the input x∈{±1}nvia an oracle\nOx. The register has nqubits and some ancilla qubits. The oracle Ox\napplies the following unitary operation on the first nqubits:\nOx|s1s2···sn/a\\}bracketri}ht= (−1)/a\\}bracketle{tx,s/a\\}bracketri}ht|s1s2···sn/a\\}bracketri}ht\nThe quantum query algorithm applies a sequence of unitary operato rs,\nwhere each operator is either Oxor an input-independent unitary opera-\ntorU. Intheend, itmeasuresthefirstqubitandoutputsthemeasurem ent\nresult. The number of queries issued is the number of times Oxis applied.4 SREEJATA KISHOR BHATTACHARYA\nNotice that a quantum query algorithm Qnaturally defines a function\nP:{±1}n→[0,1]:\nP(x) = Pr[Qoutputs 1 on input x]\nIt is well-known that if Qmakesqqueries, then Pis a degree 2 qpolyno-\nmial.\nFor a function f:S(⊆{±1}n)→{0,1}, we define its quantum query\ncomplexity Q(f) to be the smallest qfor which there exists a quantum\nquery algorithm Qmakingqqueries such that for all x∈S,\nPr[Qoutputs 1 on input x]/braceleftBigg\n≥2/3 iff(x) = 1\n≤1/3 iff(x) = 0\nAnalysis of boolean functions. In this section we recall some results from\nanalysis of boolean functions. A good reference is O’ Donnell’s textb ook [15].\n(1) Any function f:{±1}n→Rhas a unique representation as f(x) =/summationdisplay\nS⊆[n]ˆf(S)χS(x)whereχS(x) =/productdisplay\ni∈Sxi. Thecoefficients ˆf(S)aretheFourier\ncoefficients of f. The degree of fis max{|S||ˆf(S)/\\e}atio\\slash= 0}.\n(2) The variance of fis\nVarx∈{±1}n[f(x)] =/summationdisplay\nS/\\e}atio\\slash=φˆf(S)2\n(3) For a cooordinate i, the influence of the i’th coordinate is defined as\nInfi[f] =Ex∈{±1}n/bracketleftBigg/parenleftbiggf(x)−f(x(i))\n2/parenrightbigg2/bracketrightBigg\n=/summationdisplay\ni∈Sˆf(S)2\nThe total influence of fis\nInf[f] =/summationdisplay\ni∈[n]Infi[f] =/summationdisplay\nS|S|ˆf(S)2\nFrom the Fourier expansion it is clear that if deg( f)≤d,Inf[f]≤dVar[f]\n(4) Given two functions f,g{±1}n→R, we sayg ǫ−approximates fif||f−\ng||2\n2=Ex[(f(x)−g(x))2]≤ǫ.\n(5) For a point x∈{±1}nand a subset S⊆[n] and−1≤ρ≤1, we define a\ndistribution Nρ,S(x) on{±1}nas follows:\n•The bits y1,y2,···,ynare independent, and\nPr[yi=xi] =/braceleftBigg\n1 if i/\\e}atio\\slash∈S\n(1+ρ)/2 ifi∈S5\nWhenS= [n], we abbreviate Nρ,S(x) byNρ(x).\n(6) Forf:{±1}n→Rand−1≤ρ≤1 defineTρf:{±1}n→Rby\nTρf(x) =Ez←Nρ(x)[f(z)].\nIt is easy to see that the Fourier expansion of Tρfis given by\nTρf(x) =/summationdisplay\nS⊆[n]ρ|S|ˆf(S)χS(x).\n(7) A function f:{±1}n→Ris ajunta of arity lorl-juntaif there exists\na subset S⊆[n],|S|≤lsuch that fonly depends on the coordinates in\nS. We say fis a (ǫ,l) junta if it can be ǫ-approximated by a l-junta, i.e.,\nthere exists a l-juntagsuch that||f−g||2\n2≤ǫ.\n(8) A restriction ρ= (S,y) off:{±1}n→Ris specified by a subset S⊆[n]\nand an assignment y∈{±1}[n]\\S. Such a restriction naturally induces a\nfunction fρ:{±1}S→R. Sometimes we shall write fyinstead of fρ(note\nthatSis determined by ysincey∈{±1}[n]\\S)\nBy a random restriction with survival probability p, we mean sampling\nρ= (S,y∈{±1}[n]\\S) where each coordinate i∈[n] is included in Swith\nprobability pindependently, and each bit of yis independently set to a\nuniformly random bit.\nRemark 3.1. Throughout the paper, all growing parameters (e.g., the deg reed)\nwill be assumed to be larger than some sufficiently big constan t. This is to make\nthe expressions look neat, as we will be replacing terms like (C1)kpoly(k)by(C2)k\nwhereC2> C1.\n4.Proof Overview\nThe main technical tool in this paper is a structural result for boun ded low-\ndegree functions similar in spirit to Hastad’s switching lemma [ 10]. Letf:\n{±1}n→[0,1] be a polynomial of degree d, and letρdenote a random restriction\nwith survival probabilitylog(d)\nCd. We show that for some constant C,\nPr/bracketleftbig\nfρis a (O(d−C),O(dC)) junta/bracketrightbig\n≥1−1\ndΩ(1).\nOnce this is established, we can prove Aaronson-Ambainis conjectu re for ran-\ndom restrictions as follows: it is easy to see that\nPr/bracketleftbigg\nVar[fρ]≥Var[f]log(d)\n2Cd/bracketrightbigg\n≥Var[f]log(d)\n2Cd.\nThis probability will be significantly more than the failure probability of t he\nswitching lemma/parenleftbigg1\ndΩ(1)/parenrightbigg\n(this is the only place where we need the lower bound\nonVar[f]). So for a≈O(Var[f]log(d)/d) fraction of random restrictions, the\nvariance of fρis highandfρcan be well approximated by a junta with arity6 SREEJATA KISHOR BHATTACHARYA\npoly(d). This means one of the coordinates of the junta must have high infl uence.\nThis concludes the proof.\nNow we give a brief overview of how we prove the switching lemma. The\nstarting point is the work by Dinur, Friedgut et al. [ 8] which states the following:\nTheorem 4.1. For anyf:{±1}n→[0,1], if\n/summationdisplay\n|S|>kˆf(S)2≤exp(−O(k2log(k)/ǫ)),\nthenfis a(ǫ,2O(k)/ǫ2)junta.\nIn other words, if the Fourier tail above a certain level kis bounded, then f\ncan be well approximated by juntas of arity roughly 2O(k).\nWe start with the observation that random restrictions have boun ded Fourier\ntails: if the function has degree dand we make a random restriction with sur-\nvival probabilitylog(d)\nCd, using Chernoff bound we can show that with very high\nprobability the Fourier weight above level log( d) will be low; around the order of\nexp(−Ω(Clog(d))). If we can manage to bring the Fourier weight above log( d)\nsmall enough so that Theorem 4.1applies, then we will get that fρcan be well\napproximated by a poly(d) junta. Unfortunately, if we try this, it turns out that\nwe have to set the survival probability so low that on expectation th e variance\noffρgoes down significantly as well. In other words, while it is true that fρcan\nbe well-approximated by juntas, it is for a trivial reason that its var iance itself\nis very low. (And moreover, this is also true for functions f:{±1}n→Rthat\nare merely L2bounded i.e, E[f2]≤1, so we would not be using any additional\nstructure that arises from the fact that fispointwise bounded.)\nIn order to make this approach work, we need to improve the tail bo und\nexp(−O(k2log(k))/ǫ). The problematic term is the quadratic exp( −O(k2)) in\nthe exponential. If the dominant term were to the order of exp( −O(k)) in-\nstead, the calculations would go through. Can we hope to increase t he tail\nbound to exp(−O(k)) while paying a cost by increasing the junta arity? Un-\nfortunately, again, this is not possible: [ 8] constructs a function which shows\nthat the tail bound is essentially tight upto the log( k) factor - their function has\n||f>k||2\n2≈exp(−Θ(k2)) but approximating it to even 1 /3 accuracy requires read-\ning Ω(n) coordinates. Our key observation is that the function construc ted by [8]\nhas full degree whereas we are working with random restrictions of a low degree\nfunction, so in addition to the fact that Fourier tail of fρabove level log( d) is\nvery small, we also know that fρhas degree d. Can we hope to improve the tail\nbound in Theorem 4.1if we have the additional restriction that the function is\nof degree d? Indeed, this turns out to be true. We prove the following result in\nTheorem 4.2:.\nTheorem 4.2. There exists a constant Csuch that the following is true:7\nIff:{±1}n→[0,1]has degree dand/summationdisplay\n|S|>kˆf(S)2≤ǫ\nCkdC,fis a(ǫ,ǫ−2dCCk)\njunta.\nBelow we briefly discuss how we are able to improve the tail bound unde r the\nadditional degree assumption. Dinur et al [ 8] prove their tail bound by showing\nthe following result (we are omitting the exact quantitative paramet ers here for\nreading convenience).\nTheorem 4.3. Leth:{±1}n→Rbe a degree kfunction with E[h2]≤1(but\nnot necessarily pointwise bounded) which cannot be approxi mated by 2O(k)juntas\nto accuracy µ. Then, for any function g:{±1}n→[0,1],E[(h−g)2]≥ε.\nTo prove Theorem 4.1, [8] applies Theorem 4.3on the truncated function h=\nf≤k=/summationdisplay\n|S|≤kˆf(S)χSand takes gto be the original function f. This then lower\nboundsE[(f−f≤k)2] which is precisely the Fourier tail above weight k. Thus,\nthe distance lower bound εin Theorem 4.3governs the Fourier tail lower bound\nin Theorem 4.1. Since we have the additional information that fis of degree\nd, for our purposes it will suffice to bound the distance from bounded degree\ndfunctions, not necessarily allbounded functions. In Theorem 6.1we prove\na result of the following form (again, we are omitting the exact param eters for\nreading convenience)\nTheorem 4.4. Leth:{±1}n→Rbe a degree kfunction with E[h2]≤1(but not\nnecessarily pointwise bounded) which cannot be approximat ed by2O(k)juntas to\naccuracy µ. Then, for any degree dfunction g:{±1}n→[0,1],E[(h−g)2]≥˜ε.\nThe parameter ˜ εin Theorem 4.4is bigger than the corresponding εparame-\nter in Theorem 4.3because we are only lower bounding the distance of hfrom\nbounded low-degree functions whereas Theorem 4.3lower bounds the distance of\nhfromarbitrary bounded functions. It turns out that the improvement in this\nparameter is sufficiently good for the random restriction approach to go through.\nIn order to prove Theorem 4.4we shall use the main idea of the proof of [ 8]\nalong with a structural restriction for bounded low-degree funct ions discovered\nfirst in [4]. Given any function f:{±1}n→Randx∈{±1}ndefine the block\nsensitivity offatxto be\nbs(f,x) = sup\n/summationdisplay\nj∈[k]/vextendsingle/vextendsinglef(x)−f(x(Bj))/vextendsingle/vextendsingle\n\nwhere the supremum ranges over all partitions ( B1,B2,···,Bk) of the variables\n(x(Bj)denotesxwith the coordinates in Bjflipped). Define the block sensitivity\noffto be bs( f) = supxbs(f,x). We shall use the following fact about bounded\nlow-degree functions:\nTheorem 4.5. [4]Iff:{±1}n→[0,1]has degree d, bs(f)≤6d2.8 SREEJATA KISHOR BHATTACHARYA\nWe give a high-level overview of how we are able to improve upon the bo und\nofεusing the fact that the block sensitivity of a bounded low degree fun ction is\nsmall. Atonepointintheirproof, [ 8]lower boundstheprobabilityofalinearform\nof Rademacher random variables l(x1,x2,···,xt) =a1x1+···+atxtexceeding a\ncertain threshold times its standard deviation, i.e.,\nPr/bracketleftbigg\na1x1+···+atxt≥α/radicalBig\na2\n1+···+a2\nt/bracketrightbigg\n.\nFor each such point xwhere this linear form is high, [ 8] shows that many related\npointsx′must have f(x′)>2. Using this they conclude that fmust deviate\nfrom the interval [0 ,1] too often and therefore cannot be approximated by any\nbounded function.\nWefollowtheproofof[ 8] upuntil thispoint. Insteadofdirectly lower bounding\nthe probability that a1x1+···+atxtis high, we partition the set of variables\n[t] intoLblocksB1,···,BL(Lis an appropriately chosen parameter) such that\neach block gets roughly same total weight: for all j∈[L],\n/summationdisplay\ni∈Bja2\ni≥a2\n1+···+a2\nt\n2L.\nIt will turn out that the ai’s are sufficiently small for such a partition to exist.\nFor each block we lower bound the probability that the linear form res tricted to\nthis block is high:\nPr\n/summationdisplay\nj∈Biajxj≥˜α/radicalBigg/summationdisplay\nj∈Bia2\nj\n.\nNow, onarandomassignment z, thelinearformrestrictedtomanyoftheseblocks\nwill be high. Take such a block Bi:/summationdisplay\nj∈Biajzj≥˜α/radicalBigg/summationdisplay\nj∈Bia2\nj. For each such block\nwe will be able to find a large number of related points zisuch that|f(z)−f(zi)|\nis large. Crucially, these related points will differ from xonly atBi. Thus, we\nwill find many points which differ from zat disjoint sets and whose fdiffer from\nzsignificantly. This will show that fcannot be too close to a bounded low degree\nfunction, because those functions have low block sensitivity.\nOur advantage is that we need to set α′so that we can conclude |f(z)−f(z′)|\nis only somewhat larger than Ω( d2/L) (as opposed to Ω(1) in [ 8]) - by setting L\nlarge enough this allows us to set a much smaller αand get rid of the quadratic\nexponential dependence.\n5.Tools\nIn this section we compile some lemmas that we shall use in our proof.9\nA reverse Markov inequality. We will use the following simple inequality\nthroughout the proof.\nLemma 5.1. LetXbe a random variable such that X≤Mwith probability 1.\nLetE[X] =µ >0. Then,Pr[X≥µ/2]≥µ\n2M\nProof.AssumePr[X≥µ/2]<µ\n2M. Then,\nE[X]≤Pr[X≥µ/2]M+Pr[X≤µ/2]µ\n2< µ,\ncontradiction. /square\nAn anticoncentration inequality for linear forms of Radema cher ran-\ndom variables.\nLemma 5.2. There exists a universal constant Ksuch that the following holds:\nletx1,···,xnbe independentRademacherrandomvariablesand let l(x1,···,xn) =\na1x1+···+anxn. Letσ=/radicalbig\na2\n1+···+a2n. Suppose|ai|≤σ\nKt. Then,\nPr[l(x1,···,xn)≥tσ]≥exp(−Kt2).\nProof.Equation 4.2 in [ 12]. /square\nRandom restrictions have small tail.\nLemma 5.3. Letf:{±1}n→Rhave degree d,C >1be a sufficiently large\nconstant, and let ρ= (S,y∈{±1}[n]\\S)be a random restriction with survival\nprobabilitylog(d)\nCd. Letk= log(d). Then,\nE\n/summationdisplay\n|T|>kˆfy(T)2\n≤exp(−Clog(d)/8)Var[f].\nProof.First suppose ( S,y∈{±1}[n]\\S) is a fixed restriction. Note that for z∈\n{±1}S,\nfy(z) =/summationdisplay\nU⊆[n]ˆf(U)χU(y,z)\nso forT⊆[S],ˆfy(T) =/summationdisplay\nU⊆Sˆf(U∪T)χU(y). By Parseval’s theorem, for a fixed S,\nEy/bracketleftBig\nˆfy(T)2/bracketrightBig\n=/summationdisplay\nU⊆[n]\\Sˆf(U∪T)2.\nTherefore, for a fixed S,\nEy\n/summationdisplay\n|T|>kˆfy(T)2\n=/summationdisplay\nV⊆[n],|V∩S|>kˆf(V)2.10 SREEJATA KISHOR BHATTACHARYA\nRandomizing over Sagain,\nES,y\n/summationdisplay\n|T|>kˆfy(T)2\n=/summationdisplay\nV⊆[n]Pr[|V∩S|> k]ˆf(V)2.\nSincefhas degree d, we only need to worry about the terms where |V|≤d.\nAlso, for|V|≤kthe relevant probability is 0. Since each element is included in\nSwith probabilitylog(d)\nCd, by Chernoff bound, for each Vwith|V|≤d,\nPr[|V∩S|> k]≤exp((C−1)2k/4C)≤exp(−Clog(d)/8).\nThus we get that\nES,y\n/summationdisplay\n|T|>kˆfy(T)2\n≤exp(−Clog(d)/8)/summationdisplay\nT/\\e}atio\\slash=φˆf(T)2= exp(−Clog(d)/8)Var[f].\n/square\nRandom restrictions don’t have low variance.\nLemma 5.4. Letf:{±1}n→Rbe any function and let ρ= (S,y∈{±1}[n]\\S)\nbe a random restriction with survival probability p. Then,E[Var[fρ]]≥pVar[f].\nProof.Fix a restriction ( S,y∈{±1}[n]\\S). For each T⊆S,ˆfy(T) =/summationdisplay\nU⊆[n]\\Sˆf(T∪\nU)χU(y). Thus, by Parseval’s theorem, for a fixed S,\nEy[Var[fy]] =/summationdisplay\nT:T∩S/\\e}atio\\slash=φˆf(T)2.\nRandomizing over Sagain,\nES,y[Var[fy]] =/summationdisplay\nT/\\e}atio\\slash=φPr[T∩S/\\e}atio\\slash=φ]ˆf(T)2\n=/summationdisplay\nT/\\e}atio\\slash=φ/parenleftbig\n1−(1−p)|T|/parenrightbigˆf(T)2\n≥/summationdisplay\nT/\\e}atio\\slash=φpˆf(T)2\n=pVar[f].\n/square\nRandom restrictions with appropriate survival probabilit y put large\nFourier mass on the linear level.\nLemma 5.5. Letf:{±1}n→R,J⊆[n]andkbe such that\n/summationdisplay\n2k≤|T∩Jc|<2k+1ˆf(T)2≥µ.11\nConsider a random restriction ρ= (S,y∈{±1}[n]\\S)where each j∈Jis fixed\nand given an uniformly random assignment, and each i∈Jcis kept alive with\nprobability p= 2−k. Then,\nE/bracketleftBigg/summationdisplay\ni∈Sˆfy({i})2/bracketrightBigg\n≥µ\n20.\nProof.For a fixed ( S,y∈{±1}[n]\\S(note that J∩S=φ) andj∈Swe have\nˆfy({j}) =/summationdisplay\nT⊆[n],T∩S={j}ˆf(T)χT\\{j}(y).\nBy Parseval’s theorem, for a fixed S,\nEy/bracketleftBig\nˆfy({j})2/bracketrightBig\n=/summationdisplay\nT⊆[n],T∩S={j}ˆf(T)2.\nRandomizing over S,\nE/bracketleftBigg/summationdisplay\nj∈Sˆfy({j})2/bracketrightBigg\n=/summationdisplay\nT⊆[n]\n/summationdisplay\nj∈[n]Pr[T∩S={j}]\nˆf(T)2\n≥/summationdisplay\n2k≤|T∩Jc|<2k+1|T|p(1−p)|T|−1ˆf(T)2.\nBy standard inequalities, for n∈[1/p,2/p), np(1−p)n−1≥1/20. It follows that\nEy/bracketleftBig\nˆfy({j})2/bracketrightBig\n≥µ\n20.\n/square\nSome hypercontractive inequalities for low degree functio ns.The proof\nof these lemmas can be found in [ 8].\nLemma 5.6. There exists a universal constant W >0such that the following\nholds:\nLetf:{±1}n→Rbe a degree kfunction. Let E[f2] =σ2. Then,\nE/bracketleftbig\nf(z)21f(z)2≤Wkσ2/bracketrightbig\n≥1\n2E[f(z)2].\nProof.Corollary 2.4 in [ 8]. /square\nLemma 5.7. There exists a universal constant B >0such that the following\nholds:\nLetf:{±1}n→Rbe a degree kfunction. Let ρ∈[−1/2,1/2]be a noise\nparameter, and x0∈{±1}n. Suppose Ez←Nρ(x)[f(z)−f(x0)] =µ≥0. Then,\nPr\nz←Nρ[f(z)−f(x0)≥µ]≥1\nBk.\nProof.Lemma 2.5 in [ 8] /square12 SREEJATA KISHOR BHATTACHARYA\nThe noise lemma. This is the main result of [ 8]. We use a slight variant. First\nwe recall some known results from approximation theory.\nLemma 5.8. For anyk, there exist constants ρ1,ρ2,···,ρk+1∈[−1/2,1/2]with\nthe following property: for any polynomial of degree k,p(x) =a0+a1x+···+akxk,\nthere exists a j∈[k+1]such that p(ρj)≥a1\n2(k+1).\nProof.Page 112 in [ 7]. /square\nNow we state the lemma.\nLemma 5.9. There exists a universal constant B >0such that the following\nholds:\nConsider a degree kpolynomial f:{±1}n→R. LetS⊆[n]andℓ(x) =/summationdisplay\ni∈Sˆf({i})xi. Consider an input x0∈{±1}nsuch that ℓ(x0)≥γ. Sample a\nz←{±1}nby the following procedure:\n(1)Sampleρ←{ρ1,···,ρk+1}uniformly at random.\n(2)Samplez←Nρ,S(x0).\nThen,\nPr/bracketleftbigg\nf(z)−f(x0)≥γ\n2(k+1)/bracketrightbigg\n≥1\n(k+1)Bk.\nRemark 5.1. Observe that zdiffers from xonly in the coordinates of S. This\nwill be crucial later on.\nProof.TakeBto be the same universal constant as in Lemma 5.6. By replacing\nfwith an appropriate restriction if necessary, we can assume S= [n]. Consider\nthe polynomial p(ρ) =Tρf(x0)−f(x0). From the Fourier expansion of noise\noperator, we see that\np(ρ) =/summationdisplay\nS/\\e}atio\\slash=φρ|S|ˆf(S).\nThis is a degree kpolynomial in ρwith linear coefficient l(x0). By Lemma 5.8,\nthere exists a h∈[k+1] such that p(ρh)≥γ/(2k+2). By Lemma 5.6,\nPr\nz←Nρ(x0)/bracketleftbigg\nf(z)−f(x0)≥γ\n2(k+1)/vextendsingle/vextendsingleρ=ρh/bracketrightbigg\n≥1\nBk.\nWe choose ρ=ρhin step (1) with probability 1 /(k+1), so\nPr\nz←Nρ(x0)/bracketleftbigg\nf(z)−f(x0)≥γ\n2(k+1)/bracketrightbigg\n≥1\n(k+1)Bk.\n/square13\nPartitioning a set of numbers in a balanced manner. We need an easy\nlemma about partitioning a set of weights none of which is too large into disjoint\nbuckets where each bucket gets roughly the same total weight. W e will later use\nthis lemma on the set of small linear Fourier coefficients of a function.\nLemma 5.10. Leta1,a2,···,anbe a set of non-negative real numbers and 1≤\nL≤n. Suppose ai≤a1+a2+···+an\n2Lfor all1≤i≤n. Then, there exists a\npartition (B1,B2,···,BL)of[n]such that for all 1≤j≤L,\n/summationdisplay\ni∈Bjai≥a1+···+an\n2L.\nProof.Start with an arbitrary partition ( B1,B2,···,BL). Then, refine it itera-\ntively according to the following algorithm.\nRefinement algorithm:\n(1) Locate a jsuch that the condition is violated for j, i.e.,\n/summationdisplay\ni∈Bjai<a1+···+an\n2L.\nIf no such jexists, terminate.\n(2) Locate a ksuch that\n/summationdisplay\ni∈Bkai≥a1+···+an\nL.\n(3) Take an arbitrary l∈Bksuch that al/\\e}atio\\slash= 0 and place it in Bj;\nBk←Bk\\{l}\nBj←Bj∪{l}\nAn appropriate kalways exists in step (2) by an averaging argument. Since\nal≤a1+···+an\n2L, the size of Bkdoes not go belowa1+···+an\n2Lafter step (3).\nIt is easy to see this procedure must terminate. Formally, notice th at the quantity\n/summationdisplay\nj∈[L]min\na1+···+an\n2L−/summationdisplay\ni∈Bjai,0\n\nreduces by min{ai|ai/\\e}atio\\slash= 0}at each step, so at some point of time it must be 0 at\nwhich point the algorithm terminates and returns a valid partition.\n/square\n6.Main results\n6.1.Improved tail bound for low degree functions. This section is the core\ntechnical part of our work: we show that if we have a function f:{±1}n→R\n(not necessarily bounded) withE[f2]≤1 which cannot be approximated by jun-\ntas, then fcannot be well-approximated by bounded low-degree functions.14 SREEJATA KISHOR BHATTACHARYA\nFor a subset J⊆[n], consider the junta u:{±1}n→Rwhich reads the\ncoordinates of Jand outputs the average over the unqueried coordinates. It\nis easy to see that u(x) =/summationdisplay\nS⊆Jˆf(S)χS(x),so||u−f||2\n2=/summationdisplay\nS/\\e}atio\\slash⊆Jˆf(S)2.Thus,u\napproximates fif and only if/summationdisplay\nS/\\e}atio\\slash⊆Jˆf(S)2is small.\nRemark 6.1. In fact, it is easy to see that there exists a junta udepending only\non coordinates of Jsuch that||f−u||2\n2≤ǫif and only if/summationdisplay\nS/\\e}atio\\slash⊆Jˆf(S)2≤ǫ. This\nimmediately follows from the Fourier expansion of f−u.\nTheorem 6.1. There exists a constant Csuch that the following holds:\nLetf:{±1}n→Rbe a degree kfunction (not necessarily bounded) with E[f2]≤\n1. LetJ={j|Infj[f]≥θ}whereθ=µ2\nCkdC. If/summationdisplay\nS/\\e}atio\\slash⊆Jˆf(S)2≥µ, then for any\ndegreedfunction g:{±1}n→[0,1],E[(f(x)−g(x))2]≥δ=µ\nCkdC.\nRemark 6.2. Notice here that although fis not pointwise bounded, gis.\nProof.LetW,B,K be the universal constants from Lemma 5.6, Lemma 5.9and\nLemma5.2respectively. We take Cto be a constant sufficiently larger than\nB,K,W.\nThere exists a tsuch that/summationdisplay\n2t≤|S∩Jc|<2t+1ˆf(S)2≥µ\nlog(k).\nLetρ= (U,y∈{±1}[n]\\U) be a random restriction where each j∈Jis killed and\ngiven a uniformly random assignment, and survival probability for ea chj/\\e}atio\\slash∈Jis\n2−t. By Lemma 5.5,\nE/bracketleftBigg/summationdisplay\nj∈Uˆfy({j})2/bracketrightBigg\n≥µ\n20log(k).\nFix aUsuch that\nEy∈{±1}[n]\\U/bracketleftBigg/summationdisplay\nj∈Uˆfy({j})2/bracketrightBigg\n≥µ\n20log(k).\nBy Parseval’s theorem we have for all j∈U,\nE/bracketleftBig\nˆfy({j})2/bracketrightBig\n=/summationdisplay\nS∩U={j}ˆf(S)2≤Infj[f].\nFor each y∈{±1}[n]defineSMALL y={j|ˆfy({j})2≤WkInfj[f]}. Observe that\nfor ally,/summationdisplay\nj∈SMALL yˆfy({j})2≤WkInf[f]≤k·Wk≤(2W)k.15\nFor each j∈Uwe have from Lemma 5.7\nE/bracketleftBig\nˆfy({j})21ˆfy({j})2≤WkInfj[j]/bracketrightBig\n≥1\n2E/bracketleftBig\nˆfy({j})2/bracketrightBig\n.\nThus,\nEy∈{±1}[n]\\U\n/summationdisplay\nj∈SMALL yˆfy({j})2\n≥µ\n40log(k),\nso applying Lemma 5.12\nPry∈{±1}[n]\\U\n/summationdisplay\nj∈SMALL yˆfy({j})2≥µ\n80log(k)\n≥µ\n80log(k)(2W)k≥µ\n(3W)k.\nCally∈{±1}[n]\\Ufor which/summationtext\nj∈SMALL yˆfy({j})2≥µ\n40log(k)to begood. Let\nGOOD ={y∈{±1}[n]\\U|yis good}. LetL=/ceilingleftbigg(2B)kd8\nVar[f]/ceilingrightbigg\n. For each good y,\nchoose a partition DIVIDE(y) = (B1,B2,···,BL) ofSMALL ysuch that for all\n1≤i≤L,/summationdisplay\nj∈Biˆfy({j})2≥µ\n80Llog(k).\n(Iftherearemultiplesuchpartitions, chooseanyoneofthemandc allitDIVIDE(y).)\nOurchoiceofparametersensuresthatforall j∈SMALL y,ˆfy({j})2≤µ\n80Llog(k),\nso such a partition exists by Lemma 5.10. Letρ1,ρ2,···,ρk+1be the constants\nfrom Lemma 5.8.\nSuppose, for the sake of contradiction, there exists a degree dpolynomial g:\n{±1}n→[0,1] such that E[(f(x)−g(x))2]≤δ. Throughout the rest of the proof,\nfor a string s1∈{±1}[n]\\Uand a string s2∈{±1}U, the pair ( s1,s2) denotes the\nstrings∈{±1}nwhich agrees with s1on [n]\\Uand with s2onU. Consider the\nfollowing randomized procedure which returns a real number.\nProcedure 1:\n(1) Sample a y∈GOODuniformly at random.\n(2) Sample ρ←{ρ1,ρ2,···,ρk+1}uniformly at random.\n(3) Sample z←{±1}Uuniformly at random.\n(4) LetDIVIDE(y) = (B1,B2,···,BL). Sample ˜ z(i)←NBi,ρ(z) for 1≤i≤L.\n(5) ReturnL/summationdisplay\ni=1|f(y,z)−f(y,˜z(i))|.\nWe estimate the probability that procedure 1 returns a number >15d2in two dif-\nferentways. First, weobtainalowerboundfromourassumptionth at/summationdisplay\nS/\\e}atio\\slash⊆Jˆf(S)2≥\n2See remark 3.1.16 SREEJATA KISHOR BHATTACHARYA\nµ. Then, we obtainanupper boundfrom the assumption that E[(f(x)−g(x))2]≤\nδ. These two bounds will contradict each other - and that will prove t he theorem.\nLower bound: Fix ay∈GOOD. LetDIVIDE(y) = (B1,B2,···,BL).\nLetw=/radicalbiggµ\n80Llog(k). For each i∈[L] we have\n/radicalBigg/summationdisplay\nj∈Biˆfy({j})2≥w.\nChooseαsuch that αw=100d4(2B)k\nL. Our choice of Lensures that α≤1.\nMoreover, our choice for influence threshold θensures that|ˆfy({j})|≤w\nKαfor\nallj∈BiwhereKis the universal constant from the Lemma 5.2.\nTherefore, we can apply Lemma 5.2to obtain that\nPrz∈{±1}Bi/bracketleftBigg/summationdisplay\nj∈Bizjˆfy({j})≥100d4(2B)k\nL/bracketrightBigg\n≥exp(−Kα2)≥1\nK1.\nHereK1= exp(K) is an absolute constant.\nBy Lemma 5.9applied on the restriction fy:{±1}U→[0,1], as we sample\nz←{±1}Uu.a.r,ρ←{ρ1,···,ρk+1}u.a.r, ˜z(i)←Nρ,Bi(z), we have that\nPr/bracketleftbigg\n|f(y,z)−f(y,˜z(i))|≥30d3(2B)k\nL/bracketrightbigg\n≥1\nK1(k+1)Bk≥1\n(2B)k.\nBy linearity of expectation,\nE/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/braceleftbigg\ni∈[L]||f(y,z)−f(y,˜z(i))|≥30d3(2B)k\nL/bracerightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg\n≥L\n(2B)k.\nUsing Lemma 5.1,\nPr/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/braceleftbigg\ni∈[L]||f(y,z)−f(y,˜z(i))|≥30d3(2B)k\nL/bracerightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥L\n2×(2B)k/bracketrightbigg\n≥1\n2L×(2B)k.\nObserve that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/braceleftbigg\ni∈[L]||f(y,z)−f(y,˜z(i))|≥30d3(2B)k\nL/bracerightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥L\n2×(2B)k=⇒/summationdisplay\ni∈[L]|f(y,z)−f(y,˜z(i))|≥15d3.\nWe conclude that for all y∈GOOD, asz,˜z(1),···,˜z(L)are sampled as in Proce-\ndure 1,\nPr\n/summationdisplay\ni∈[L]|f(y,z)−f(y,˜z(i))|≥15d3\n≥1\n2L×(2B)k.\nThus, withprobability atleast1\n2L×(2B)k, procedure1returns anumber greater\nthan 15d3>15d2.17\nUpper bound: SinceE[(f(x)−g(x))2]≤δandPry∈{±1}[n]\\U[yis good]≥µ/(3W)k,\nwe have that\nE[(f(x)−g(x))2|x[n]\\Uis good]≤δ\nµ(3W)k.\nNow consider a uniformly sampled y∈GOOD. Observe that as we sample\nz←{±1}Uu.a.r,ρ←{ρ1,···,ρk+1}u.a.r and ˜ z(i)←Nρ,Bi(z), the marginal\ndistribution of ˜ z(i)is uniform on{±1}U. By Markov’s inequality, we have\nPr/bracketleftbigg\n(f(y,z)−g(y,z))2≥1\nL2/bracketrightbigg\n≤L2δ\nµ(3W)k\nand for all i∈[L],\nPr/bracketleftbigg\n(f(y,˜z(i))−g(y,˜z(i)))2≥1\nL2/bracketrightbigg\n≤L2δ\nµ(3W)k.\nBy union bound, the probability that ( f(y,z)−g(y,z))2≥1\nL2or for some i,\n(f(y,˜z(i))−g(y,˜z(i)))2≥1\nL2is at most ( L+ 1)L2δ\nµ(3W)k≤2L3δ\nµ(3W)k. Our\nchoice of δensures that this quantity is less than <1\n2L×(2B)k. Observe that if\nnone of these bad events holds, since the block sensitivity of gis bounded above\nby 6d2(Theorem 4.5), we have that\n/summationdisplay\ni∈[L]|g(y,z)−g(y,˜z(i))|≤6d2=⇒/summationdisplay\ni∈[L]|f(y,z)−f(y,˜z(i))|≤6d2+1<15d2.\nThus, we conclude\nPr\n/summationdisplay\ni∈[L]|f(y,z)−f(y,˜z(i))|>15d2\n<2L3δ\nµ(3W)k<1\n2L×(2B)k.\nAs promised, we get conflicting lower and upper bounds for the prob ability\nthat procedure 1 returns a number >15d2. This is our desired contradiction. /square\nNow we show that we can improve the tail bound of [ 8] under the additional\nassumption that fhas low degree. This follows straightforwardly from Theorem\n6.1.\nTheorem 6.2. There exists a universal constant C >0such that the following\nis true:\nLetf:{±1}n→[0,1]be a degree dfunction. Let θ=Var[f]2\ndCCkandJ=\n{j|Infj[f]≥θ}. If/summationdisplay\nS/\\e}atio\\slash⊆Jˆf(S)2≥µ,then/summationdisplay\n|S|>kˆf(S)2≥µ\ndCCk.18 SREEJATA KISHOR BHATTACHARYA\nProof.Assume/summationdisplay\n|S|>kˆf(S)2< µ/2 (otherwise we are done). Let ˜Cbe the universal\nconstant from Theorem 6.1.\nThe idea is to apply Theorem 6.1to the truncated function\nf≤k(x) =/summationdisplay\n|S|≤kˆf(S)χS(x).\nNote that while f≤kis not pointwise bounded, it satisfies E[(f≤k)2]≤1 and\nInfj[f≤k]≤Infj[f] for all j(this is clear from the Fourier expressions). Let\nH={j|Infj[f≤k]≥θ}. We have H⊆J, so\n/summationdisplay\nS/\\e}atio\\slash⊆Hˆf≤k(S)2≥/summationdisplay\nS/\\e}atio\\slash⊆Jˆf(S)2−µ\n2≥µ\n2.\nApplyingTheorem 6.1,wegetthatforanyboundeddegree dg:{±1}n→[0,1],\nE[(f(x)−g(x))2]≥µ\n2d˜C˜Ck. Taking gto be our original function f, we get the\ndesired tail lower bound:\nE[(f−f≤k)2]≥µ\n2d˜C˜Ck=⇒/summationdisplay\n|S|>kˆf(S)2>µ\n2d˜C˜Ck.\nTakingCto be a slightly larger constant than ˜C, we get that\n/summationdisplay\n|S|>kˆf(S)2≥µ\ndCCk.\n/square\n6.2.Random restrictions can be approximated by juntas. In this section\nwe use the fact that random restrictions have bounded tails to sho w that they\ncan be approximated by juntas.\nTheorem 6.3. For any constants ˜C1,˜C2>0, there exist constants ˜C3,˜C4,˜C5>0\nsuch that the following holds:\nLetf:{±1}n→[0,1]be a degree dpolynomial and let ρbe a random restric-\ntion with survival probabilitylog(d)\n˜C3d. With probability at least 1−d−˜C2, fρis a\n(d−˜C1Var[f],Var[f]−2d˜C4)junta. Moreover, if Jdenotes the set of coordinates on\nwhich the junta depends, for each j∈Jwe haveInfj[f]≥Var[f]−2d−˜C5.\nProof.We consider a random restriction with survival probabilitylog(d)\n˜C3d.\nBy Lemma 5.3, the expected Fourier tail of fρabove level log( d) is at most\nexp(−˜C3log(d)/8)Var[f] =Var[f]\nd˜C3/8.By Markov’s inequality, with probability at\nleast 1−d˜C3/16, the Fourier tail above log( d) is≤Var[f]\nd˜C3/16. LetCbe the constant19\nfrom Theorem 6.2. Letµ=Var[f]\nd˜C3/16dCClog(d)=Var[f]\nd˜C3/16d2C,θ=µ2\ndCClog(d)=µ2\nd2C\nandJ={j|Infj[fρ]≥θ}. Letu:{±1}n→[0,1] be the junta which reads the\ncoordinates in Jand outputs the average over the coordinates in Jc. Choose\n˜C3large enough so that µ≤d−˜C1Var[f]. Applying Theorem 6.2, we see that u\napproximates fρto accuracy d−˜C1Var[f]. Using the fact that total influence is\nbounded by d, we see that uhas arity≤Var[f]−2dC′˜C3for a universal constant\nC′. Taking ( ˜C4,˜C5) = (C′˜C3,˜C3/32−2C), we are done.\n/square\n6.3.Aaronson-Ambainis conjecture is true for random restricti ons.\nTheorem 6.4. There exist constants C1,C2>0such that the following holds:\nletf:{±1}n→[0,1]be a degree dpolynomial (d≥2)withVar[f]≥1/d. Letρ\ndenote a random restriction with alive probabilitylog(d)\nC1d. Then,\nPr/bracketleftbigg\nfρhas a coordinate with influence ≥Var[f]2\ndC2/bracketrightbigg\n≥Var[f]log(d)\n50C1d.\nProof.LetMbe a large constant. Apply Theorem 6.3with (˜C1,˜C2) = (M,M) to\nget constants ˜C3,˜C4,˜C5. Letρbe a random restriction with survival probability\nlog(d)\n˜C3d. By Lemma 5.4,\nE[Var[fρ]]≥Var[f]log(d)\n˜C3d\nso by Lemma 5.1,\nPr/bracketleftbigg\nVar[fρ]≥Var[f]log(d)\n2˜C3d/bracketrightbigg\n≥Var[f]log(d)\n2˜C3d.\nSinceVar[f]≥1/d,d−M≤Var[f]log(d)\n10˜C3d. By Theorem 6.3and Remark 6.1, with\nprobability at least 1 −d−M, there exists a Jρ⊆[n] such that every coordinate\ninJρhas influence≥Var[fρ]−2d−˜C5and\n/summationdisplay\nS/\\e}atio\\slash⊆Jρˆfρ(S)2≤d−MVar[f].\nSowithprobabilityatleastVar[f]log(d)\n2˜C3d−d−M≥Var[f]log(d)\n4˜C3d, boththeseevents\n(high variance of fρand existence of Jρ) hold and we have that\n/summationdisplay\nS⊆Jρˆfρ(S)2≥Var[fρ]−d−MVar[f]≥Var[f]log(d)\n4˜C3d.\nIn particular, we have that Jρ/\\e}atio\\slash=φ. Since for each j∈Jρwe have Infj[fρ]≥\nVar[fρ]−2d−˜C5, we are done by taking ( C1,C2) = (˜C3,2+˜C5). /square20 SREEJATA KISHOR BHATTACHARYA\n7.Conclusions and further directions\nIn this paper, we showed that if f:{±1}n→{0,1}is a degree dpolynomial,\na large fraction of its random restrictions have an influential coord inate. We\nobserve that this implies one of the results proven in [ 13] about the existence of\nsmall sensitive blocks with a slightly different set of parameters.\nLetf:{±1}n→[0,1]. An input x∈{±1}nis said to be ( r,ǫ) sensitive if there\nexists aysuch that d(x,y)≤rand|f(x)−f(y)|≥ǫ. [13] proves the following:\nTheorem 7.1. Iff:{±1}n→[0,1]has degree d, then at least Ω(Var[f])fraction\nof the inputs are (r,ǫ)sensitive where ǫ=poly(Var[f]/d),r=poly(d,1/ǫ,log(n))\nAnimmediate consequence of ourresult is that atleast Ω( Var[f]/dO(1)) fraction\nof inputs are (1 ,ǫ) sensitive where ǫ= poly(Var[f]/d). Thus, while we lose a bit\nin the fraction of sensitive inputs, we gain by letting our block size be e xactly 1\ninstead of poly( d,1/ǫ,log(n)).\nIt would be interesting to see if we can extend this to the full Aarons on-\nAmbainis conjecture. We describe a potential approach here.\n•Given a degree dpolynomial f:{±1}n→[0,1], we can lift it with a\nBoolean function g:{±1}m→{±1}neach of whose coordinates giis\nunbiased and given by a low degree function. Then, the lifted polynom ial\nf⊙g:{±1}m→[0,1] will be a low degree polynomial. As long as the gi’s\nare pairwise independent, the variance of fwill be preserved as well. Our\nresult shows that a large fraction of random restrictions of f⊙ghave an\ninfluential coordinate. Can we construct g1,g2,···,gnappropriately such\nthat this allows us to conclude fmust have an influential coordinate as\nwell? The gi’s should introduce correlations between the different input\nbits offso that most random restrictions of f⊙gmlook the same in some\nappropriate sense.\nReferences\n[1] Aaronson, S., and Ambainis, A. (2014a). Forrelation: A Problem t hat Optimally Separates\nQuantum from Classical Computing.\n[2] Aaronson, S., and Ambainis, A. (2014b). The Need for Structur e in Quantum Speedups.\n[3] Bansal, N., Sinha, M., and de Wolf, R. (2022). Influence in Complete ly Bounded Block-\nmultilinear Forms and Classical Simulation of Quantum Algorithms.\n[4] Beals, R., Buhrman, H., Cleve, R., Mosca, M., and de Wolf, R. (1998) . Quantum Lower\nBounds by Polynomials.\n[5] Bernstein, E., and Vazirani, U. (1997). Quantum Complexity Theo ry. SIAM Journal on\nComputing, 26(5), 1411–1473. https://doi.org/10.1137/S00975 39796300921\n[6] Bourgain, J. (2002). On the distribution of the Fourier spectru m of Boolean functions.\nIsrael Journal of Mathematics, 131, 269–276. https://doi.org/ 10.1007/BF02785861\n[7] Charafi, A. (1992). Chebyshev polynomials—From approximation theory to algebra\nand number theory: 2nd edition. Theodore J. Rivlin, John Wiley and So ns Limited,\n1990. pp. 249, hardcover. £45.50. Engineering Analysis With Boundary Elements, 9.\nhttps://api.semanticscholar.org/CorpusID:126040229\n[8] Dinur, I., Friedgut, E., Kindler, G., and O’Donnell, R. (2006). On the fourier tails of\nbounded functions over the discrete cube. https://doi.org/10.11 45/1132516.113258021\n[9] Friedgut, E. (1998). Boolean Functions With Low Average Sensit ivity Depend On Few\nCoordinates. Combinatorica, 18, 27–35. https://doi.org/10.1007 /PL00009809\n[10] Hastad, J. (1986). Almost optimal lower bounds for small dept h circuits. Proceed-\nings of the Eighteenth Annual ACM Symposium on Theory of Computin g, 6–20.\nhttps://doi.org/10.1145/12130.12132\n[11] Keller, N., and Klein, O. (2019). Quantum speedups need struct ure.\n[12] Ledoux, Michel, and Talagrand, Michel. (n.d.). Probability in Banac h spaces: isoperimetry\nand processes. http://library.usi.edu/record/1364107\n[13] Lovett, S., and Zhang, J. (2023). Fractional Certificates fo r Bounded Functions. 14th In-\nnovations in Theoretical Computer Science Conference (ITCS 202 3).\n[14] Montanaro, A. (2012). Some applications of hypercontractiv e inequalities\nin quantum information theory. Journal of Mathematical Physics, 53(12).\nhttps://doi.org/10.1063/1.4769269\n[15] O’Donnell, R. (2021). Analysis of Boolean Functions.\n[16] O’Donnell, R., Saks, M., Schramm, O., and Servedio, R. A. (2005). Every decision tree\nhas an influential variable.\n[17] O’Donnell, R., and Zhao, Y. (2015). Polynomial bounds for decou pling, with applications.\nSchool of Technology and Computer Science, Tata Institute o f Fundamental\nResearch, Mumbai\nEmail address :sreejata.bhattacharya@tifr.res.in"    },    {        "title": "2402.13953v2.On_Courant_and_Pleijel_theorems_for_sub_Riemannian_Laplacians.pdf",        "content": "arXiv:2402.13953v2  [math.SP]  28 Mar 2024ON COURANT AND PLEIJEL THEOREMS\nFOR SUB-RIEMANNIAN LAPLACIANS\nRUPERT L. FRANK AND BERNARD HELFFER\nAbstract. We are interested in the number of nodal domains of eigenfunctions\nof sub-Laplacians on sub-Riemannian manifolds. Specifically, we inves tigate the\nvalidity of Pleijel’s theorem, which states that, as soon as the dimens ion is strictly\nlarger than 1, the number of nodal domains of an eigenfunction cor responding to\nthek-th eigenvalue is strictly (and uniformly, in a certain sense) smaller th ankfor\nlargek.\nIn the first part of this paper we reduce this question from the cas e of general\nsub-Riemannian manifolds to that of nilpotent groups.\nIn the second part, we analyze in detail the case where the nilpoten t group is a\nHeisenberg group times a Euclidean space. Along the way we improve k nown bounds\non the optimal constants in the Faber–Krahn and isoperimetric ineq ualities on these\ngroups.\nContents\n1. Introduction 3\n1.1. Short summary 3\n1.2. The H¨ ormander operator 3\n1.3. Courant’s Theorem 4\n1.4. Pleijel’s Theorem 5\n1.5. Organization 8\n1.6. Acknowledgements 8\nPart 1. Courant’s and Pleijel’s bound in the sub-Riemannian case 9\n2. The restriction of a Sobolev function to a nodal domain 9\n2.1. Presentation 9\n2.2. Setting 9\n2.3. The restriction theorem 10\n2.4. Application to Courant’s theorem 12\n3. On nilpotent approximation 13\n4. Main result for sub-Laplacians in the equiregular case 15\n©2024 by the authors. This paper may be reproduced, in its entirety , for non-commercial\npurposes.\nPartial support through US National Science Foundation grant DM S-1954995 (R.L.F.), as well as\nthrough the German Research Foundation through EXC-2111-39 0814868 and TRR 352-Project-ID\n470903074 (R.L.F.) is acknowledged.\n12 RUPERT L. FRANK AND BERNARD HELFFER\n5. Proof of the main sub-Riemannian results 18\n6. Basic examples 23\n6.1. Presentation 23\n6.2. Direct change of variables 24\n6.3. Computation of the canonical privileged coordinates and applica tion 25\n6.4. Conclusion in the setting of the example 26\n6.5. Variations of the example 27\nPart 2. Pleijel’s bound for Hn×Rk29\n7. The Pleijel argument for Hn×Rk29\n8. Computing the constant in the Weyl asymptotics 31\n8.1. The case of Hn 31\n8.2. The Weyl constant on Hn×Rkwithk≥1 34\n9. A first bound on the Faber–Krahn constant 34\n9.1. A bound via the Sobolev constant 34\n9.2. The Pleijel constant γ(Hn) 35\n9.3. A lower bound on the Sobolev constant on Hn×Rkfork≥1 38\n9.4. The Pleijel constant γ(Hn×Rk) fork≥1 40\n10. Continuation by looking at a direct product 43\n10.1. Spectral density 43\n10.2. Sobolev inequality 43\n10.3. Pleijel’s Theorem 45\n11. A second bound on the Faber–Krahn constant 46\n11.1. Preliminary discussion 46\n11.2. Definitions 46\n11.3. Symmetrization 47\n11.4. Pansu’s conjecture and its consequences 50\n11.5. A bound on the isoperimetric constant in Hn 51\n11.6. A bound on the isoperimetric constant in Hn×Rk58\nPart 3. Appendix 62\nAppendix A. Discussion on how to avoid Assumption 1.3 62\nA.1. A first example 62\nA.2. General argument 62\nA.3. Sub-Riemannian geometric construction 63\nA.4. Topological constraints 64\nAppendix B. Weyl law and heat kernel 64\nB.1. Nilpotentization of measure 64\nB.2. Nilpotentization of heat kernel 64\nB.3. Effect of changes of coordinates 65\nB.4. Heat kernel asymptotics and Weyl law 65\nReferences 66PLEIJEL THEOREM FOR SUB-LAPLACIANS 3\n1.Introduction\n1.1.Short summary.\nThe study of nodal domains is a classical topic in spectral geometry . The founding\nwork of R. Courant [21] from 1923 showed that an eigenfunction of the Laplacian cor-\nresponding to the k-eigenvalue hasatmost knodal domains. Inthepast century many\ncontributions have led to a better understanding of nodal domains . An important step\nfor this was a theorem of A. Pleijel [72] from 1956, which shows that asymptotically\nthe number of nodal sets of an eigenfunction becomes significative ly smaller than the\nbound given by Courant when the dimension is strictly larger than 1. T his initially\ninvolved the Dirichlet condition but was more recently extended to ot her boundary\nconditions by I. Polterovich [73] and C. L´ ena [50]; see also [25].\nOn the other hand, it is natural to consider the same question for o ther opera-\ntors and this leads naturally to the consideration of Dirichlet sub-La placians, initially\ncalled H¨ ormander operators [45], which share with the Laplacian the property of hy-\npoellipticity. An important step in their analysis was done by L. P. Roth schild and\nE. M. Stein [78], who proved their maximal hypoellipticity. They procee ded by com-\nparison with operators on nilpotent Lie groups, a technique that is a lso known as\nthe nilpotentization procedure. Using this approximation, G. M´ etiv ier [61] proved a\nbeautiful Weyl formula for the asymptotic behavior of the countin g function of eigen-\nvalues, provided this approximation can be done “uniformly”. Motiva ted by a recent\npaper [26] by S. Eswarathasan and C. Letrouit on Courant’s nodal theorem and many\ndiscussions on related problems with C. Letrouit, our aim in this paper is to try to\nextend Pleijel’s theorem to the sub-Riemannian context.\nOur paper is divided into two, rather independent, parts. In the fir st part, we show\nhow the validity of Pleijel’s theorem in the sub-Riemannian case can be r educed to\nthe specific analysis of sub-Laplacian on nilpotent groups. In the se cond part, we\nanalyze the validity of Pleijel’s theorem in open sets of specific groups related to the\nHeisenberg group. This leads us to the question of finding new explicit and close to\noptimal bounds on the constants for Sobolev inequalities, Faber–K rahn inequalities\nand isoperimetric inequalities.\n1.2.The H¨ ormander operator.\nWe consider in an open set Ω ⊂Rnthe Dirichlet realization of a sub-Laplacian (also\ncalled H¨ ormander’s operator)\n−∆Ω\nX:=p/summationdisplay\nj=1X⋆\njXj,\nwhereX1,...,X pare realC∞vector fields satisfying the so-called H¨ ormander condi-\ntion [45], which reads:\nAssumption 1.1. For somer≥1 theXjand the brackets up to order rgenerate at\neach pointx∈Ω the tangent space TxΩ.4 RUPERT L. FRANK AND BERNARD HELFFER\nNote that the terminology “sub-Riemannian Laplacian” or in short “s ub-Laplacian”\nis posterior to the work of L. H¨ ormander and corresponds to the development of sub-\nRiemannian geometry at the beginning of the nineties; see for examp le [7, 46].\nMore generally, we are given a connected C∞Riemannian manifold Mof dimension\nnwith a given measure µ(with aC∞-density with respect to the Lebesgue measure\nin a local system of coordinates) and a system of p C∞vector fields on Msatisfying\nAssumption 1.1, where p≤n.\nThe sub-Laplacians are known to be hypoelliptic [45] in Ω (meaning that ifω⊂Ω\nis open and uis a distribution in Ω with −∆Xusmooth inω, thenuis smooth in ω).\nMoreover, if the boundary ∂Ω isC∞and non characteristic for X= (X1,···,Xp) at\nthe boundary (i.e. at each point in ∂Ω there exists a vector field Xithat is transverse\nto the boundary), then we have C∞-regularity up to the boundary. We emphasize\nthat we will not need this latter condition for our results; see Sectio n 2.\nThe operator −∆Ω\nXhas compact resolvent provided Ω is bounded and we can con-\nsider for its discrete spectrum all the questions that have been so lved along the years\nfor the Dirichlet realization −∆Ωof the Euclidean Laplacian. These include:\n•Simplicity of the ground state or, more generally, its multiplicity.\n•Local structure of the nodal sets, density of the nodal sets, ...\n•Courant’s theorem: comparison between the minimal labelling kof an eigen-\nvalueλkand the number νkof the nodal domains of the eigenfunction in the\neigenspace corresponding to λk.\n•Pleijel’s theorem.\nWe focus in this paper on the two last items. They will be described mor e deeply in\nthe next subsections.\n1.3.Courant’s Theorem.\nAs is well known, Courant’s theorem states that in the case of the D irichlet Laplacian\nin Ω⋐Rn, an eigenfunction associated with the k-th eigenvalue has at most knodal\ndomains:\nνk≤k.\nIf one looks at the standard proof of Courant’s theorem, this ineq uality mainly ap-\npears as a consequence of a restriction statement (the restrict ion of an eigenfunction\nto its nodal domain is the ground state of the Dirichlet realization of t he Laplacian in\nthis domain), the minimax characterization of the eigenvalues, and t he Unique Con-\ntinuation theorem (UCT). Hence the difficulty is to determine under w hich conditions\nwe can extend these three results to sub-Riemannian Laplacians. C oncerning the re-\nstriction statement, having rather limited information about the no dal sets (i.e. the\nboundary of the nodal domains), we successively adapt to the sub -Riemannian case\na proof proposed in [68], which permits to avoid regularity assumption s on∂Ω. The\nvariational characterization then holds. The UCT was proved by K. Watanabe [81]PLEIJEL THEOREM FOR SUB-LAPLACIANS 5\nin theC∞category in dimension 2, but H. Bahouri [4] gave a discouraging coun ter-\nexample to UCT with two vector fields in R3. Here we are fortunate to know that J.\nM. Bony proved at the end of the sixties [9] that UCT holds when the vector fields\nare analytic1. Hence Courant’s theorem holds in the analytic category [26]. At the\nend of the next section, we will extend statements given in [26] to th e case when the\nboundary is not necessarily non-characteristic.\n1.4.Pleijel’s Theorem.\nIn the same spirit, one can hope for an asymptotic control of νk/kfor largek. In the\ncase of the Dirichlet Laplacian in Ω ⊂Rn, Pleijel’s theorem [72] says that, if n≥2,\nthere exists an Ω-independent constant2γ(n)<1 such that\nlimsup\nk→+∞νk\nk≤γ(n). (1.1)\nLater in the paper we will use the expression ‘Pleijel’s theorem holds’ ( for a given\noperator)tomeantheassertion thatlimsupk→∞νk/k<1uniformly, whenweconsider\nDirichlet realization in open subsets.\nThe proof of Pleijel’s theorem is a nice combination of Weyl’s formula, wh ich gives\nthe asymptotic behavior as λ→+∞of the counting function N(λ,−∆Ω)\nN(λ,−∆Ω)∼ W(Rn)|Ω|λn\n2,\nand Faber–Krahn inequality, which gives the bound\nλ1(−∆Ω)≥ |Ω|−2\nnλ1(−∆B1),\nwhereB1is the ball of unit volume3. One can combine these two ingredients to obtain\ninequality (1.1) with\nγ(n) =W(Rn)−1λ1(−∆B1)−n\n2, (1.2)\nUsing the explicit expressions for the Weyl constant W(Rn) and the lowest Dirichlet\neigenvalueλ1(−∆B1) on the ball, one can establish that\nγ(n)<1 for all n≥2.\nTherefore (1.1) provides an asymptotic improvement over Couran t’s theorem.\n1Note nevertheless that the sub-Laplacians with analytic vector fie lds are not in general hypoelliptic\nanalytic. While the sub-Laplacian on the Heisenberg group Hnis known to be hypoelliptic analytic\n(its fundamental solution is explicitly known and analytic outside the o rigin in the exponential coor-\ndinates; see [27]), the sub-Laplacian on Hn×Rkwithk≥1 is known to be non-hypoelliptic analytic\nas a direct consequence of a result by Baouendi-Goulaouic [5]. We re fer to [62] for a characterization\nof the nilpotent groups of rank 2 whose associated sub-Laplacian is hypoelliptic analytic and to [40]\nfor other counter-examples when the rank of nilpotency is strictly larger than 2. Here we have limited\nour references to the case of sub-Laplacians on nilpotent groups . Starting in the seventies there have\nbeen a lot of contributions on the subject, in particular around an “ evoluting” Tr` eves conjecture.\n2later denoted by γ(Rn)\n3Later on, we will write CFK(Rn) =λ1(−∆B1). For an expression of this constant in terms of Bessel\nfunctions see (11.1) below.6 RUPERT L. FRANK AND BERNARD HELFFER\nIf we want toconsider thegeneralization toour sub-Laplacians, we canbeoptimistic\non the side of Weyl’s formula. Since the pioneering work of G. M´ etivier [61], we are\nrich in results, at least if we add to Assumption 1.1 a certain equiregula rity condition\n(or M´ etivier’s condition), which permits to approximate the Xjat each point xby the\ngenerators of a nilpotent Lie algebra Gx. More precisely, we are given on a connected\nC∞Riemannian manifold M(with or without boundary) of dimension nwith a given\nmeasureµ(with aC∞-density with respect to the Lebesgue measure in a local system\nof coordinates) and a system of p C∞vector fields, where p≤n, satisfying the\nfollowing:\nAssumption 1.2. For eachj≤r, the dimension of the space spanned by the com-\nmutators of length ≤jat each point is constant.\nIn the language of modern sub-Riemannian geometry, this is called an equiregular\ndistribution. The simplest example occurs with r= 2,p= 2 andn= 3 with the\nHeisenberg group Hand more generally in contact geometry (see for example [3]).\nAs above, we consider the operator\n−∆M,µ\nX:=p/summationdisplay\nj=1X⋆\njXj,\nwhereX⋆\njis the formal adjoint obtained by using the L2scalar product with respect\nto the given measure µ. In local coordinates X⋆\nj=−Xj+cjfor a function cj. In the\ncase whenMhas a boundary, we always impose a Dirichlet condition. The operator\n−∆M,µ\nXis hypoelliptic and by Rothschild–Stein [78] we know (through a techniq ue of\nnilpotentization) that it is maximally hypoelliptic, that is, it satisfies\n/ba∇dblXkXℓu/ba∇dbl ≤C(/ba∇dbl∆µ\nXu/ba∇dbl+/ba∇dblu/ba∇dbl),∀k,ℓ,∀u∈C∞\nc(M),\nwhere/ba∇dbl·/ba∇dbldenotes the L2-norm inMwith respect to the measure µ.\nWe denote by Djthe span of all vector fields obtained as brackets of length ≤jof\ntheXk’s. We setnj:= dim(Dj(x)), which by Assumption 1.2 above does not depend\non the point x∈M. A family ( Y1,...,Y n) ofnvector fields is said to be adapted to\nthe flag at x∈Mif for anyj≤r, Span(Y1(x),...,Y nj(x)) =Dj(x). We can then\nintroduce, setting n0:= 0,\nQ:=r/summationdisplay\nj=1j(nj−nj−1). (1.3)\nThis plays the role of an ‘effective dimension’.\nUnder Assumptions 1.1 and 1.2, G. M´ etivier shows (using, in particula r, the tech-\nniques of [78, 77]) that there is a constant c(M,X,µ) such that\nN(λ,−∆M,µ\nX) := #{j:λj(−∆M,µ\nX)≤λ} ∼c(M,X,µ)λQ\n2asλ→ ∞.(1.4)PLEIJEL THEOREM FOR SUB-LAPLACIANS 7\nWe will come back later to the structure of c(M,X,µ) and to its computation in\nparticular cases. Note that in the case r= 2 related results are obtained in [59, 60,\n61, 64], and M´ etivier’s theorem (together with many other results ) has been recently\nrevisited in the light of sub-Riemannian geometry in [18, 19, 20].\nThis concludes our discussion of the first ingredient in Pleijel’s proof, namely an\nanalogue of Weyl’s formula.\nConcerning the second ingredient, namely an analogue of the Faber –Krahn inequal-\nity, our knowledge is rather poor. In the case of the Heisenberg gr oup, one can think of\na result by P. Pansu [70] concerning the isoperimetric inequality. C. L ´ ena’s approach\n[50] for treating the Neumann problem for the Laplacian could be help ful (see [38]) if\nthe set in Ω where the system of the Xjis not elliptic is “small” in some sense, but\nour equiregular assumption excludes this case.\nWe will follow another way by revisiting in the first part the nilpotentiza tion pro-\ncedure. This permits us to deduce Faber–Krahn inequalities for sub -Laplacians from\nFaber–Krahn inequalities for sub-Laplacians on nilpotent groups. T his is done under\nan additional simplifying assumption (see nevertheless the discussio n in Appendix A):\nAssumption 1.3. For eachx∈Mthe tangent vectors Xj(x) (j= 1,...,p) are\nlinearly independent in TxM. In other words, we assume p=n1.\nMore precisely, under the above three assumptions we will prove th at the Faber–\nKrahn inequality holds on subsets of small measure with a constant t hat is arbitrarily\nclose to the infimum of the constants of the Faber–Krahn inequalitie s on the nilpo-\ntent approximations Gx; see Proposition 5.1. This result is in the spirit of a result of\nB´ erard–Meyer [8, Part II, Lemma 16], who have shown that on a Rie mannian man-\nifold manifold the Faber–Krahn inequality holds on subsets of small me asure with a\nconstant that is arbitrarily close to the constant in the Faber–Kra hn inequality on\nRn. We emphasize, however, that in the case of a Riemannian manifold th e model\nspace is the same at each point, namely Rn, while in the sub-Riemannian setting the\napproximating model space Gxmay vary with x. Our techniques are quite different\nfrom those employed in [8] and its generalizations, e.g., in [25].\nCombiningourresultaboutFaber–KrahninequalitieswithM´ etivier’sW eyl-typefor-\nmula, we get asufficient condition forthevalidity ofa Pleijel-type boun d; see Theorem\n4.3, which is the main result of the first part of this paper. The sufficie nt condition in\nthis theorem involves the corresponding Faber–Krahn constants for Dirichlet realiza-\ntions of sub-Laplacians in open set of nilpotent groups.\nThis provides a motivation for the second part of this paper, which is devoted to\nthe validity of a Pleijel-type bound for the nilpotent groups Hn×Rk, whereHnis\nthe Heisenberg group of homogeneous dimension 2 n+ 2 and where k∈N0. While\nwe have not been able to establish a Pleijel-type theorem in the most im portant case8 RUPERT L. FRANK AND BERNARD HELFFER\n(n,k) = (1,0), we have succeeded in proving it if one admits the celebrated conj ecture\nof Pansu concerning the isoperimetric constant on H1. We also have positive results\nunder the assumption that the homogeneous dimension Q= 2n+2+kofHn×Rkis\nsufficiently large.\n1.5.Organization.\nThis paper is divided into two main parts.\nThe first part begins in Section 2 with a somewhat technical result co ncerning the\nrestriction of a Sobolev function to a nodal domain, which plays an imp ortant role in\nthe arguments of Courant and Pleijel and their generalizations. Th e main result of\nthe first part is Theorem 4.3 in Section 4, which gives a sufficient condit ion for the\nvalidity of Pleijel’s theorem via nilpotentization. In the preceding Sect ion 3 we discuss\nthe setting of this theorem and give the proof in the following Section 5. The theorem\nis illustrated by an example in Section 6.\nThe second part of this paper deals with the case of the Heisenberg groupHn\nand, more generally, with Hn×Rkwithk∈N0. The main results of that part are\nsummarized in Section 7; see, in particular, Theorem 7.2 and Proposit ion 7.3. The\nproofs of these results rely on an explicit form of the Weyl asympto tics, treated in\nSection 8, and bounds on the Faber–Krahn constants. For the lat ter, we proceed via\ntwo different techniques that are spread out over Sections 9 (con tinued in Section 10)\nand 11.\nIn two appendices we discuss an assumption appearing in the main res ult of the\nfirst part (Appendix A) and review an approach to Weyl asymptotic s (Appendix B).\n1.6.Acknowledgements.\nThanks to D. Mangoubi for inviting the second author to propose o pen questions in an\nOberwolfach workshop in August 2023 and for previous discussions . Great thanks also\nto C. Letrouit for stimulating discussions about his joint paper with S . Eswarathasan\nand sub-Riemannian geometry when we start to attack these ques tions. Thanks to\nJ. Viola and F. Nicoleau for helping us with the use of Mathematica or Wo lframalpha.\nThanks to V. Colin for his help in topology. We are also grateful to L. H illairet\nand P. Pansu for their help in the understanding of sub-Riemannian r esults and to\nL. Capogna and N. Garofalo for exchanges about the isoperimetric problem on the\nHeisenberg group. Thanks to E. Milman for discussions on Faber–Kr ahn constants.PLEIJEL THEOREM FOR SUB-LAPLACIANS 9\nPart1.Courant’s and Pleijel’s bound in the sub-Riemannian case\n2.The restriction of a Sobolev function to a nodal domain\n2.1.Presentation.\nIn this section we will show that the restriction of a continuous Sobo lev function to\na nodal domain satisfies the boundary values zero in the sense of So bolev functions.\nThis holds under remarkably weak assumptions on the vector fields u sed to define the\nSobolev spaces. The precise formulation can be found in Theorem 2.2 below.\nThat this restriction property requires a proof seems to have bee n overlooked by\nR. Courant in his original proof of the Courant nodal theorem, but this was proved in\nthe case when Mis a compact manifold in [8]. In the case with boundary, the proof\ngoes in the same way when the Dirichlet boundary problem is regular (s ee [2] or more\nrecently [54]). Without this assumption, it was proved in a paper by M¨ uller-Pfeiffer\n[68] that seems to be little known in the spectral theory community. An alternative\nproof, due to D. Bucur, is presented in the book [54]. Here we show t hat the proof\nin [68] can be generalized to the sub-Riemannian setting. Notice that Bucur’s proof\ncould also have been adapted.\n2.2.Setting.\nLetMbe aC1-manifold (without boundary) of dimension nand letµbe aC1nonneg-\native Borel measure on M. (ByC1-measure we mean that in each chart the measure is\nabsolutely continuous with a positive C1-density.) Let X1,...,X pbeC1-vector fields\nonM.\nNotice that in this section our results are established under weaker assumptions\nthan in the rest of this paper. In particular, we emphasize that we d o not make an\nassumption on the span of the vector fields. In fact, even the triv ial casep= 0, where\nthere are no vector fields at all, is formally included in our analysis.\nWe define first-order differential operators X⋆\n1,...,X⋆\npby\n/integraldisplay\nMψXjϕdµ=/integraldisplay\nMϕX⋆\njψdµ for allϕ,ψ∈C1\nc(M).\nWedefineS1(M)tobethesetofallfunctions u∈L2(M)forwhichthereare f1,...fp∈\nL2(M) such that for j= 1,...,pone has\n/integraldisplay\nMuX⋆\njϕdµ=/integraldisplay\nMϕfjdµfor allϕ∈C1\nc(M).\nThefj’s are necessarily unique and we denote them by fj=:Xju, thus extending the\nusual notation in case u∈C1\nc(M). We set\nq[u] :=p/summationdisplay\nj=1/ba∇dblXju/ba∇dbl2\nL2for allu∈S1(M).\nOne easily sees that S1(M) is a vector space that is complete with respect to the\nnorm/radicalbig\nq[u]+/ba∇dblu/ba∇dbl2\nL2. We denote by S1\n0(M) the closure of C1\nc(M) inS1(M). When10 RUPERT L. FRANK AND BERNARD HELFFER\nΩ is an open subset of M, then Ω itself is a manifold and therefore the spaces S1(Ω)\nandS1\n0(Ω) are defined. We also introduce the space S1\nloc(M) of functions such that\nu|Ω∈S1(Ω) for any open Ω ⊂MwithΩ compact.\nWe emphasize that the spaces S1(M) andS1\n0(M) depend on X1,...,X pand, ifM\nis not compact, on µ, even if this is not reflected in the notation. Note that if p= 0,\nthenS1(M) =S1\n0(M) =L2(M). IfM=Rn(or an open set in Rn),p=n, and\nXi=∂xi, then we recover the classical Sobolev spaces.\nWe begin by recording truncation properties, which will play an import ant role in our\narguments.\nLemma 2.1. Ifu,v∈S1\nloc(M), thenmax{u,v},min{u,v} ∈S1\nloc(M)and\nXjmax{u,v}(x) =/braceleftBigg\nXju(x)a.e. in{x∈M:u(x)≥v(x)},\nXjv(x)a.e. in{x∈M:u(x)≤v(x)},\nand\nXjmin{u,v}(x) =/braceleftBigg\nXjv(x)a.e. in{x∈M:u(x)≥v(x)},\nXju(x)a.e. in{x∈M:u(x)≤v(x)}.\nIn particular, if u,v∈S1(M), thenmax{u,v},min{u,v} ∈S1(M).\nProof.This lemma is well known (see, e.g., [34, Lemma 3.5] when the underlying\nmanifold is Rk). For the sake of completeness we give an outline of the main steps o f\nthe proof.\nFirst, one shows that if η∈C1(R) withη′bounded, then u∈S1\nloc(M) implies\nη(u)∈S1\nloc(M) withXjη(u) =η′(u)Xju. (Here one can argue as in [39, Theorem\n1.18].) Next, one applies this result to η(t) =√\nt2+ε2and deduces, after passing to\nthe limitε→0, thatu∈S1\nloc(M) implies |u| ∈S1\nloc(M) with\nXj|u|(x) =/braceleftBigg\nXju(x) a.e. in {x∈M:u(x)≥0},\n−Xju(x) a.e. in {x∈M:u(x)≤0}.\nNote that this shows, in particular, that Xju(x) = 0 a.e. in {x∈M:u(x) = 0}.\n(Here one can argue similarly as in [39, Lemma 1.19].) Since\nmax{u,v}=1\n2(u(x)+v(x)+|u(x)−v(x)|),\nmin{u,v}=1\n2(u(x)+v(x)−|u(x)−v(x)|),\nthis implies the assertion of the lemma. /square\n2.3.The restriction theorem.\nThe following theorem is the main result of this section.\nTheorem 2.2. LetΩ⊂Mbe open and let u∈S1\n0(Ω)∩C(Ω). Ifωis a connected\ncomponent of {x∈Ω :u(x)/\\e}atio\\slash= 0}, thenu|ω∈S1\n0(ω).PLEIJEL THEOREM FOR SUB-LAPLACIANS 11\nThe proof below of this theorem is essentially taken from [68]. It is sim pler than\nBucur’s proof presented in [54] and, in particular, avoids the notion of capacity. Sim-\nilar to that proof, it relies on truncation properties of Sobolev func tions and on the\nfollowing simple lemma.\nLemma 2.3. LetΩ⊂Mbe open and let u∈S1(Ω). Assume that uvanishes outside\nof a compact set of Mand vanishes on ∂Ωin the sense that for any y∈∂Ωand any\nε >0there is a neighborhood UofyinMsuch that |u|< εa.e. inU∩Ω. Then\nu∈S1\n0(Ω).\nProof of Lemma 2.3. This is modeled after [39, Lemma 1.26]; see also [44, Lemma 1].\nWe first note that we may assume that Ω is compact. Otherwise, we consider an open\nset/tildewideΩ with compact closure such that uvanishes almost everywhere in Ω \\/tildewideΩ and apply\nthe result on Ω ∩/tildewideΩ. Further, by considering u+andu−separately (using Lemma 2.1),\nwe may assume that u≥0. Consider uε:= (u−ε)+. We shall show that uε∈S1\n0(Ω).\nSinceuε→uinS1(Ω), this will imply the assertion.\nWe claim that the support of uεis a compact subset of Ω. Indeed, any y∈∂Ω has\nan open neighborhood UyinMwithu<εinUy∩Ω. By compactness of ∂Ω, a finite\nunion of these Uycover∂Ω anduεvanishes almost everywhere in the intersection of\nthis finite union with Ω.\nBy the Meyers–Serrin type result in [34, Theorem 1.13] we see that uεcan be\napproximated in S1(Ω) by functions from C1(Ω)∩S1(Ω). The result in that reference\nis stated for open subsets of Rn, but since it is a local result, the result remains valid\nin our situation by localizing via a partition of unity to coordinate neighb orhoods.\nThe proof in [34] proceeds by convolution with a compactly supporte d function. The\nfact that the support of uεis a compact subset of Ω implies that the approximating\nfunctions belong to C1\nc(Ω). This proves that, indeed, uε∈S1\n0(Ω). /square\nProof of Proposition 2.2. For the sake of concreteness let us assume that u>0 inω,\nthe argument in the opposite case being similar.\nStep 1.Let\nz:=/braceleftBigg\nuinω,\n0 in Ω\\ω.\nWe claim that z∈S1\nloc(Ω).\nLetψ∈C1\nc(Ω). It is easy to see that z∈C(Ω) and consequently also ψz∈C(Ω).\nMoreover,ψz= 0 on (Ω \\ω)∪(Ω\\suppψ). Thusψz|ω=ψu|ωis inS1(ω)∩C(ω)\nand extends continuously to ∂ω, where it vanishes. By Lemma 2.3 this implies that\nψz|ω∈S1\n0(ω). This implies that ψz∈S1\n0(Ω). (Indeed, since ψz|ω∈S1\n0(ω), it can be\napproximated in S1(ω) by functions in C1\nc(ω). Extending these functions by zero to Ω\ngives functions in C1\nc(Ω) and, since ψzvanishes in Ω \\ω, these functions approximate\nψzinS1(Ω), soψz∈S1\n0(Ω).) Since ψ∈C1\nc(Ω) is arbitrary, we deduce z∈S1\nloc(Ω), as\nclaimed.12 RUPERT L. FRANK AND BERNARD HELFFER\nStep 2.Let (ϕj)⊂C1\nc(Ω) such that ϕj→uinS1(Ω) (such functions exist since\nu∈S1\n0(Ω)) and set\nvj:= min{z,(ϕj)+}|ω.\nWe claim that ( vj) is a bounded sequence in S1\n0(ω) and that it converges to u|ωin\nL2(ω).\nTo prove this, we consider wj:= min{z,(ϕj)+}. By Step 1 and truncation prop-\nerties of Sobolev spaces (Lemma 2.1) we deduce that wj∈S1\nloc(Ω). Moreover, these\ntruncation properties also imply that\np/summationdisplay\ni=1/integraldisplay\nω|Xivj|2dµ=p/summationdisplay\ni=1/integraldisplay\nω|Xiwj|2dµ\n=p/summationdisplay\ni=1/integraldisplay\n{u≤ϕj}|Xiu|2dµ+p/summationdisplay\ni=1/integraldisplay\n{u>ϕj>0}|Xiϕj|2dµ\n≤p/summationdisplay\ni=1/integraldisplay\nΩ(|Xiu|2+|Xiϕj|2)dµ.\nSinceu∈S1(Ω)andsince( ϕj)converges in S1(Ω), therightsideisuniformlybounded.\nConcerning the L2-norm, we find/integraldisplay\nω(vj−u)2dµ=/integraldisplay\nω∩{(ϕj)+<u}(u−(ϕj)+)2dµ≤/integraldisplay\nΩ(u−ϕj)2dµ→0.\nWe have shown, in particular, that vj∈S1(ω). It remains to prove that vj∈S1\n0(ω).\nWe note that wjis continuous in Ω. Its restriction vjtoωextends continuously to ∂ω\nand vanishes there. (The vanishing on ∂ω∩Ω comes from the continuity of u. The\nvanishing on ∂ω∩∂Ω comes from the compact support property of ϕj.) Thus, again\nby Lemma 2.3, we have wj|ω∈S1\n0(ω).\nStep 3.We can now finish the proof. Since ( vj) is a bounded sequence in S1\n0(ω),\nafter passing to a subsequence we may assume that it converges w eakly inS1\n0(ω) to\nsomev∈S1\n0(ω). In particular, it converges weakly in L2(ω) tov. Meanwhile, by\nStep 2, it converges strongly in L2(ω) tou|ω, so the uniqueness of the weak limit in\nL2(ω) implies that v=u|ωa.e. In particular, u|ω∈S1\n0(ω), as claimed. /square\n2.4.Application to Courant’s theorem.\nUsing Theorem 2.2 we can recover the results in [26] concerning the D irichlet re-\nalization −∆Ω,µ\nXin an open set Ω of Mbut without having to assume that ∂Ω is\nnoncharacteristic for the system of vector fields Xj(see Assumption 1.5 in [26] or\nAssumption ( HΓ) in [24]). We record this as follows.\nTheorem 2.4. For anyk∈N, any eigenfunction of −∆Ω,µ\nXwith eigenvalue λkhas at\nmostk+mult(λk)−1nodal domains, where mult(λk)denotes the multiplicity of λk.\nIf, moreover, one of the two following assumptions holds\n•n= 2,PLEIJEL THEOREM FOR SUB-LAPLACIANS 13\n•M,µand(X1,...,X p)are real-analytic,\nthen we get an upper bound by k.\nThe first part of the statement relies on a remark of D. Mangoubi t hat permits to\navoid the use of the Unique Continuation Theorem (see [54, Exercise 4.1.15]). For the\nsecond part of the theorem, [26] refers in the first case to [81]. Fo r the second case\ncorresponding to the standard statement of Courant’s nodal Th eorem, the proof is\nbased on a result of J. M. Bony [9].\nLet us further discuss the assumption of being noncharacteristic in [26]. A char-\nacteristic point of ∂Ω relative to D1is a pointxfor which all the elements of D1(x)\nbelong to the tangent space Tx∂Ω. The assumption that ∂Ω is noncharacteristic for\nD1guarantees that eigenfunctions belong to C∞(Ω); see [24], [26] and the book [22]\n(particularly its Chapter 3). This regularity allowed [26] to prove The orem 2.4 (un-\nder their noncharacteristic assumptions). This extra assumption seems to be crucial\nfor the regularity of eigenfunctions at the boundary4. Remarkably, however, it is not\nnecessary for the validity of Theorem 2.2.\nThe condition that ∂Ω is noncharacteristic for D1may appear rather strict. Ex-\namples where this condition holds in the case of the Heisenberg group are given in\n[22, 66]. An example in His given (see [66, Example 3.4] with k= 1) by the domain\n(/radicalbig\nx2+y2−2)2+16t2<1.\nNote that for this example ∂Ω is homeomorphic to T2.\nMeanwhile, by topological considerations the condition that ∂Ω is noncharacteristic\nforD1is never satisfied if ∂Ω is homeomorphic to S2.5\nFinally it was shown in [24] that the measure in the boundary of the cha racteristic\npoints is zero. Other connected results are obtained in the case of the Heisenberg\ngroup by D. Jerison [47].\n3.On nilpotent approximation\nHere we follow the presentation of [77] (based on former assumptio ns and definitions\ngiven by Goodman [35], Folland–Stein [29], Folland [28], M´ etivier [61], Roth schild–\nStein [78]). Since this period in the seventies, a huge literature has de veloped the\nso-called sub-Riemannian geometry (see the appendix in [19] and ref erences therein).\nWe try to combine the two formalisms (see Appendix B).\n4Notice that the question of the analyticity at the boundary (in case the sub-Laplacians are hypoel-\nliptic analytic in Ω) seems open.\n5Many thanks to V. Colin for this remark. If Ω ⊂R3and if we have an equiregular distribution\n(n1=p= 2) and r= 2, then the transversality condition is never satisfied if ∂Ω is homeomorphic to\nS2. The reason is that the non-characteristic condition implies the exis tence of a continuous (with\nrespect to x) unique straight line in Tx∂Ω (this is the intersection of D1(x) withTx∂Ω) and this is\nimpossible when ∂Ω is homeomorphic to the sphere by the Poincar´ e-Hopf theorem [6 3] (the Euler\ncharacteristic is not 0).14 RUPERT L. FRANK AND BERNARD HELFFER\nWe consider the situation presented in the introduction and assume that Assump-\ntions 1.1, 1.2 and 1.3 are satisfied. We consider an adapted flag ( Y1,...,Y n) and we\nimpose in addition that\nYj=Xjforj= 1,···,p (3.1)\n(see the assumption 1.2 above). Given this adapted flag ( Y1,...,Y n) atx∈M, we\ncan define canonical privileged coordinates of the first kind6atxby the mapping θx\ngiven by\nθx(y) :=u= (ui) ify= exp(n/summationdisplay\ni=1uiYi)·x, (3.2)\nwhere exp denotes the exponential map defined in some small neighb orhood ofx. It is\nknown that under our equiregularity assumption and (3.1), it is poss ible, around any\nx∈M, to choose a flag that is adapted at any point in a neighborhood of x. Thus we\nidentify a neighborhood of x∈Mwith a neighborhood of 0 in Rn. It has been shown\nby G. M´ etivier (see below) that everything depends smoothly on x.\nWe now introduce the notion of nilpotentized measure d/hatwideµxatx∈M. We refer to\n[20, Appendix A.5.6] for a definition in the formalism of sub-Riemannian g eometry\nor to our Appendix B and explain here ”by hand” how it can be constru cted for our\nspecific choice of privileged coordinates. On Rnwe have the Lebesgue measure\ndu=/productdisplay\nidui,\nand in these local coordinates the measure dµis in the form\ndµ=a(x,u)du,\nwhere (x,u)/ma√sto→a(x,u) isC∞in both variables xandu. In a small neighborhood of\n0, the nilpotentized measure at xcan be defined by\nd/hatwideµx:=a(x,0)du. (3.3)\nNote that for usmall and locally in xwe have a good control on a(x,u)/a(x,0) and\nits inverse. Then we denote by Yi,x,the image of Yibyθx, which is simply Yiwritten\nin the local canonical coordinates around x. It can be shown that θxis alsoC∞with\nrespect tox.\nOnRn, with coordinates u= (ui), we introduce the family of dilations given by\nδt(ui) = (twiui),\nwherewiisdefined asfollows: thereexists a unique j∈ {1,...,n}such thatnj−1+1≤\ni≤nj, and we set wi=j. We note that the homogeneous dimension Q, defined in\n6Other choices are possible but we only need that some privileged coor dinates exist and will only\nconsider this one, which is actually the one introduced by G. M´ etivier [61] in the proof of his Theo-\nrem 3.1.PLEIJEL THEOREM FOR SUB-LAPLACIANS 15\n(1.3), satisfies\nQ=/summationdisplay\niwi. (3.4)\nVia the family of dilations we have a natural definition of homogeneous functions\nof degreesonRn\\ {0}, a definition of “homogeneous norm” (of degree one) and\ncorresponding notions of vanishing function to order p. A differential operator of the\nformf(u)∂\n∂uiis of order wi−s, providedfis homogeneous of degree s. Whenfis\ndefined in a pointed neighborhood of 0 and can be developed as a sum o f homogeneous\nterms of increasing order, we will say that fis of order ≤sif the term of lowest order\nis homogeneous of degree s.\nG. M´ etivier [61, Theorem 3.1] proves the following theorem (in additio n to the\nregularity of θxalready mentioned above).\nTheorem 3.1. For anyx,Xj,xis of order ≤1. Furthermore,\n•\nXk,x=/hatwideXk,x+Rk,x,\nwhere/hatwideXk,xis homogeneous of order 1andRk,xis of order ≤0.\n•The/hatwideXk,xgenerate a nilpotent Lie algebra Gxof dimension nand rankr.\n•The mapping x/ma√sto→/hatwideXj,xis smooth.\nBy M´ etivier’s theorem (see also [20, Theorem 1] and our Appendix B f or a short\npresentation) we have (as announced in (1.4) in a less precise form) the following Weyl\nformula.\nTheorem 3.2. There exists a continuous, positive function x/ma√sto→κ(x)such that the\ncounting function, of the selfadjoint realization of −∆satisfies, as λ→+∞,\nN(λ,−∆M,µ\nX) := #{j:λj(−∆)≤λ} ∼/parenleftBig/integraldisplay\nMκ(x)dµ(x)/parenrightBig\nλQ\n2.(3.5)\nThis theorem also holds when considering the Dirichlet realization in an o pen set Ω\nofM.\nNote that, when r= 2, there were important contributions on the subject starting\nfrom the end of the seventies [59, 60], [61], [64] and recently develop ed in [18, 19, 20].\nWe refer to the next section and Appendix B for a presentation whic h gives a more\nexplicit way to determine κ(x). In particular, (B.2) gives (3.5) with\nκ(x) = Γ(Q\n2+1)−1/hatwideex(1,0,0), (3.6)\nwhere/hatwideex(t,u,v) is the heat kernel of the nilpotentized sub-Laplacian at x, which will\nbe defined in the following section.\n4.Main result for sub-Laplacians in the equiregular case\nWe work in the setting of the previous section. In particular, we wor k under As-\nsumptions 1.1, 1.2 and 1.3.16 RUPERT L. FRANK AND BERNARD HELFFER\nBythenilpotent approximation, we canassociatewitheach point x∈Manilpotent\ngroupGx(identified with the algebra Gxin theu-coordinates) and a corresponding\nsub-Laplacian\n/hatwide∆x=p/summationdisplay\ni=1/hatwideX2\ni,x\ninU2(Gx) (the elements in the enveloping algebra U(Gx) that are homogeneous of\ndegree 2).\nForanyx∈MwealsohaveaFaber–Krahninequality, thatisthereexistsaconsta nt\nc>0 such that\n/a\\}b∇acketle{t−/hatwide∆xv,v/a\\}b∇acket∇i}htL2(Gx,/hatwideµx)≥c/hatwideµx(Ω)−2\nQ/ba∇dblv/ba∇dbl2\nL2(Gx,/hatwideµx),∀Ω⊂Gxopen,∀v∈C∞\nc(Ω).(4.1)\nWe recall that Qdenotes the homogeneous dimension of Gx; see (1.3). By our as-\nsumption of equiregularity, Qis independent of x∈M.\nBy definition, cFK\nxis the largest constant such that (4.1) holds. We will prove momen-\ntarily the positivity of this constant, even uniformly in x.\nRemark4.1.Having in mind the proof of Pleijel’s theorem, it is important to write the\nabove estimates using the appropriately normalized Lebesgue meas ured/hatwideµx. Note that\nthe Faber–Krahn constant cFK\nxdepends both on Gxand on a specific normalization\nconstant determined by the measure µ; see [20, Appendix A.5.6] and our Appendix B.\nLemma 4.2. We have\ncFK(M,X,µ) := inf\nx∈McFK\nx>0, (4.2)\nProof.The family ˆ∆xcan be indeed analyzed as in Rothschild’s paper [77]. In [77,\nTheorem 4.24] L. Rothschild establishes that given any y∈M, 1<p<Q\n2, there exist\nδ>0 andCy,p>0 such that for any x∈B(y,δ), anyf∈C∞\nc(Rn), we have\n/ba∇dbl(−ˆ∆x)−1f/ba∇dblq≤Cy,p/ba∇dblf/ba∇dblp (4.3)\nwith1\nq=1\np−2\nQ. (This inequality is actually used as an intermediate result toward\nthe proof (Theorem 3.6) that −ˆ∆xhas a (−Q+2)-homogeneous fundamental solution\nkx(u) which depends smoothly on x.)\nAssuming first that Q >2, we can apply (4.3) with p=2Q\nQ+2andq=2Q\nQ−2and\nobtain\n/ba∇dbl(−ˆ∆x)−1\n2f/ba∇dbl2\n2=/integraldisplay\nRdf(−ˆ∆x)−1fd/hatwideµx≤ /ba∇dblf/ba∇dbl2Q\nQ+2/ba∇dbl(−ˆ∆x)−1f/ba∇dbl2Q\nQ−2≤Cy,2Q\nQ+2/ba∇dblf/ba∇dbl2\n2Q\nQ+2\nSince (−ˆ∆x)−1\n2, considered as an operator L2Q\nQ+2→L2and its adjoint L2→L2Q\nQ−2,\ncoincide on L2Q\nQ+2∩L2(as a consequence of the selfadjointness of ( −ˆ∆)−1\n2onL2), we\nobtain\n/ba∇dbl(−ˆ∆x)−1\n2g/ba∇dbl2\n2Q\nQ−2≤Cy,2Q\nQ+2/ba∇dblg/ba∇dbl2\n2.PLEIJEL THEOREM FOR SUB-LAPLACIANS 17\nSubstituting u= (−ˆ∆x)−1\n2gwe obtain the Sobolev inequality\nC−1\ny,2Q\nQ+2/ba∇dblu/ba∇dbl2\n2Q\nQ−2≤ /a\\}b∇acketle{t(−ˆ∆x)u,u/a\\}b∇acket∇i}ht.\nBy H¨ older’s inequality, as in the proof of Proposition 9.1 below, this imp lies that\ncFK\nx≥C−1\ny,2Q\nQ+2. Covering Mby finitely many ball B(y,δ), we obtain the claimed\npositivity when Q>2.\nTo deal with the cases Q= 1,2 we follow an idea of Helffer and Nourrigat already\nused in [77]. We study the operator −ˆ∆x−/summationtexts\ni=1∂2\nionGx×Rswiths∈Nchosen\nsuch thatQ+s>2. Proceeding exactly as before, we obtain the Sobolev inequality\nC−1\ny,2Q\nQ+2/ba∇dblU/ba∇dbl2\n2(Q+s)\nQ+s−2≤ /a\\}b∇acketle{t(−ˆ∆x−s/summationdisplay\ni=1∂2\ni)U,U/a\\}b∇acket∇i}ht.\nfor functions UonGx×Rs. Applying this to a product function U=u⊗ϕwith a\nfixed function ϕ∈C1\nc(Rs), we obtain\ncC−1\ny,2Q\nQ+2/ba∇dblu/ba∇dbl2\n2(Q+s)\nQ+s−2≤ /a\\}b∇acketle{t(−ˆ∆x)u,u/a\\}b∇acket∇i}ht+/ba∇dblu/ba∇dbl2\n2.\nwithc >0 depending only on ϕ. By a simple scaling argument in Gx(see, e.g., [31,\nRemark 2.47]), this inequality can be brought in the form of a Sobolev in terpolation\ninequality where on the right side a geometric mean of /a\\}b∇acketle{t(−ˆ∆x)u,u/a\\}b∇acket∇i}htand/ba∇dblu/ba∇dbl2\n2appears.\nIn this form one can againuse H¨ older’s inequality, similarly in the proof of Proposition\n9.1 below, to obtain a bound on the Faber–Krahn constant and one c oncludes as\nbefore. /square\nAn alternative proof of this lemma could proceed by controlling the ge ometric de-\npendencies of the constants in the Varopoulos’s proof [80] of the S obolev inequality.\nOur main statement concerning nilpotent approximation is the followin g theorem:\nTheorem 4.3. Let−∆M,µ\nX=/summationtext\nℓX⋆\nℓXℓbe an equiregular sub-Riemannian Laplacian\non a closed connected manifold Mwith given measure µ. Then\nlimsup\nk→+∞νk\nk≤/parenleftBig\ncFK(M,X,µ)/parenrightBig−Q\n2·/parenleftBig1\nµ(M)/integraldisplay\nMκ(x)dµ(x)/parenrightBig−1\n,(4.4)\nwhereνkdenotes the maximal number of nodal domains of an eigenfunct ion of−∆M,µ\nX\nassociated with eigenvalue λk.\nCorollary 4.4. If\n/parenleftBig\ncFK(M,X,µ)/parenrightBigQ\n2·/parenleftBig1\nµ(M)/integraldisplay\nMκ(x)dµ(x)/parenrightBig\n>1, (4.5)\nthen Pleijel’s theorem holds.\nRemark 4.5.Theorem 4.3 remains valid when we consider the Dirichlet realization\n−∆Ω,µ\nXof an equiregular sub-Riemannian Laplacian in a relatively compact ope n set\nΩ in a manifold M. It is enough to have Assumptions 1.1, 1.2 and 1.3 satisfied in18 RUPERT L. FRANK AND BERNARD HELFFER\na neighborhood of Ω. We avoid in this case the topological constraints discussed in\nAppendix A, Subsection A.4 .\nExamples are discussed in Section 6.\nIn particular, we can consider an open, relatively compact set Ω in a fi xed stratified\ngroupGwhereXisabasisof G1andµis(intheexponential coordinates)theLebesgue\nmeasure. In this case, the function κis constant. This is further discussed in Part 2.\nRemark 4.6.The weak version of Courant’s theorem (that is, the first part of T heo-\nrem 2.4) yields\nlimsup\nk→+∞νk\nk≤1. (4.6)\nHere we use the fact that Weyl’s formula implies lim k→∞mult(λk)\nk= 0.\nIn view of the previous remark, Theorem 4.3 is only interesting for its application\nin the corollary, that is, when (4.5) holds. In Part 2 we will investigate the validity of\nthis condition in the case of an open subset of Hn×Rk. Other instances where one\nmight be able to prove the validity of (4.5) are, for example, in the for mM3×Tkwith\nklarge enough and with M3a compact 3-dimensional contact manifold.\n5.Proof of the main sub-Riemannian results\nAs in the work of B´ erard and Meyer [8], the idea behind the proof of T heorem 4.3 is\ntoproveaFaber–Krahninequalitywheretheconstantis“almost”t he“good”constant,\nprovided the sets on which the inequality is applied are “small”. For us, the “good”\nconstant is cFK:=cFK(M,µ), defined in (4.2), and “smallness” of sets is understood\nwith respect to their µ-measure. The precise statement is the following:\nProposition 5.1. For anyθ>0there is an η>0such that for any open set D⊂M\nwithµ(D)≤ηand anyv∈C∞\n0(D),\n/a\\}b∇acketle{t−∆M,µ\nXv,v/a\\}b∇acket∇i}ht ≥(1−θ)cFKµ(D)−2\nQ/ba∇dblv/ba∇dbl2. (5.1)\nFor the proof of this proposition we choose a Riemannian structure onMthat is\ncompatible with its smooth structure. (This is always possible, first in local coordi-\nnates and then globally via a partition of unity.) This Riemannian struct ure allows\nus to consider geodesic balls B(x,r) atx∈Mof radiusr>0.\nBy compactness, there is an ε∗>0 such that for any x∈Mthe priviledged\ncoordinates at xare well defined in B(x,ε∗). Moreover, functions with support in\nB(x,ε∗) can be identified via θxwith functions in Gxsupported in a neighborhood of\nthe origin.\nLemma 5.2. There are constants C,ε0>0ands>0such that for any x∈M, any\n0<ε≤ε0and any function v∈C∞\n0(B(x,ε))one has\n/vextendsingle/vextendsingle/vextendsingle/a\\}b∇acketle{t/hatwide∆xv,v/a\\}b∇acket∇i}htL2(Gx0,/hatwideµx)−/a\\}b∇acketle{t∆M,µ\nXv,v/a\\}b∇acket∇i}htL2(M,µ)/vextendsingle/vextendsingle/vextendsingle≤Cεs/a\\}b∇acketle{t−/hatwide∆xv,v/a\\}b∇acket∇i}htL2(Gx,/hatwideµx).PLEIJEL THEOREM FOR SUB-LAPLACIANS 19\nProof of Lemma 5.2. Throughout the proof, we will make use of the uniformity with\nrespecttoxofseveral geometricconstructionsaroundapoint x∈M. Thisisdiscussed\nand proved in [61, 41, 42, 77] and will be used freely in what follows.\nWe give the proof in three steps.\nStep 1: Change of the measure.\nWe denote by ˜∇the sub-Riemannian gradient, so\n−/a\\}b∇acketle{t∆v,v/a\\}b∇acket∇i}htL2(M,µ)=/ba∇dbl˜∇v/ba∇dbl2\nL2(M,µ)=/summationdisplay\nj/ba∇dblXjv/ba∇dbl2\nL2(M,µ).\nDue to the localization of the support of v, we have\n(1−Cε)/ba∇dbl˜∇v/ba∇dbl2\nL2(Gx,/hatwideµx)≤ /ba∇dbl˜∇v/ba∇dbl2\nL2(M,µ)≤(1+Cε)/ba∇dbl˜∇v/ba∇dbl2\nL2(Gx,/hatwideµx).\nStep 2: Comparing Xi,xand/hatwideXi,x.\nDue to the compactness of M, there exists ε0>0 such that, for each x∈Mwe have\na localization function ˜ χxsuch that ˜χx= 1 onB(x,2ε0), supp(˜χx)⊂B(x,4ε0) and\nall the estimates on the derivatives are controlled uniformly. Choos ingε0>0 small\nenough, we can assume that for each x∈Mthe privileged coordinates at xare well\ndefined inB(x,ε0).\nFor technical reasons we also have to introduce another cut-off f unction/hatwideχxof the\nsame type, viz. such that /hatwideχx= 1 onB(x,ε0), supp(/hatwideχx)⊂B(x,2ε0) and with uniform\nbounds on the derivatives.\nWe observe that in the privileged coordinates centered at xwe have7by Lemma 3.1\nthat ˜χx(Xi,x−/hatwideXi,x)˜χxis of degree 0.\nThere is also a notion of type8in [78] (see [77, pp. 654–655]for the adaptationto the\nequiregular situation), which roughly speaking corresponds to the operator of “non\npositive” degree.\nNote that (after localization around 0 in the privileged coordinates) (−/hatwide∆)−1\nxis an\noperator of type 2 and that /hatwideXm,x(−/hatwide∆x)−1is an operator of type 1 [78, Theorem 8].\nThen ˜χx(Xi,x−/hatwideXi,x)˜χxis an operator of type 0. Consequently, for each m= 1,...,p,\nKim,x:=/hatwideχx˜χx(Xi,x−/hatwideXi,x)˜χx(−/hatwide∆x)−1/hatwideXm,x\nis an operator of type 1.\n7See either Helffer–Nourrigat [41, Proposition 5.1], M´ etivier [61], [26, S ubsection 4.2.4], or [77, Section\n1].\n8The authors introduce first a notion of function of type λcorresponding to a function that is\nhomogeneous of degree −Q+λwith respect to the dilation (with an addition condition when λ= 0).\nBy integration on test functions, this defines a distribution of type λ. More generally, a function\nK(x,y) onM×Mis a kernel of type λif for any ℓ≥0, we have\nK(x,y) =s/summationdisplay\ni=1ai(x)k(i)\nx(θx(y))bi(x,y) +Eℓ(x,y),\nwhereai,bi∈C∞\nc(M),k(i)\nxis a kernel of type ≥λwith (x,u)/ma√sto→k(i)\nx(u) smooth away from u= 0,\nandEℓ∈Cℓ(M×M). An operator of type λis a mapping originally defined on C∞\nc(M) whose\ndistribution kernel is of type λ.20 RUPERT L. FRANK AND BERNARD HELFFER\nForv∈C∞\nc(Gx), we have the identity\n/hatwideχx(Xi,x−/hatwideXi,x)˜χxv=−/summationdisplay\nm/hatwideχx˜χx(Xi,x−/hatwideXi,x)˜χx(−/hatwide∆x)−1/hatwideXm,x/hatwideXm,x˜χxv\n=/summationdisplay\nmKim,x/hatwideXm,x˜χxv.\nTo estimate the L2-norm ofKim,x/hatwideXm,x˜χxu, we first compute the Lq-norm with\n1\nq=1\n2−1\nQ. Here we use [78, Theorem 7] and obtain for wwith support in a fixed\ncompact subset in Euclidean space\n/ba∇dblKim,xw/ba∇dblq≤C/ba∇dblw/ba∇dbl2.\nAssume now that v∈C∞\n0(B(x,ε0)) and note that, identifying vwith a function on\nGx,\n/hatwideχx(Xi,x−/hatwideXi,x)˜χxv= (Xi,x−/hatwideXi,x)vand/hatwideXm,x˜χxv=/hatwideXm,xv.\nThus, applying the above inequality to w=/hatwideXm,x˜χxv, which has support in the fixed\ncompact set B(x,ε0), we obtain\n/ba∇dbl(Xi,x−/hatwideXi,x)v/ba∇dblq≤/summationdisplay\nm/ba∇dblKim,x/hatwideXm,x˜χxv/ba∇dblq≤C/summationdisplay\nm/ba∇dbl/hatwideXm,x˜χxv/ba∇dbl2\n≤C√p/parenleftBigg/summationdisplay\nm/ba∇dbl/hatwideXm,xv/ba∇dbl2\n2/parenrightBigg1\n2\n.\nIf, moreover, we have v∈C∞\n0(B(x,ε)) withε≤ε0, then, by H¨ older’s inequality, since\n(Xi,x−/hatwideXi,x)vis supported in B(x,ε),\n/ba∇dbl(Xi,x−/hatwideXi,x)v/ba∇dbl2≤˜Cεn\nQ/ba∇dblv/ba∇dbl2\nfor somen. Here we use the fact that µ(B(x,ε)) is bounded by a constant times εn,\nuniformly in xandε≤ε0.\nStep 3: End of the proof of the lemma.\nWe can now achieve the proof in the following way. Using Step 1, it is eno ugh to use\nthe measure/hatwideµx. We omit this below. Writing\n/ba∇dbl˜∇v/ba∇dbl2+/a\\}b∇acketle{t/hatwide∆xv,v/a\\}b∇acket∇i}ht\n=/summationtext\ni/parenleftBig\n2/a\\}b∇acketle{t(Xi,x−/hatwideXi,x)v,/hatwideXi,xv/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t(Xi,x−/hatwideXi,x)v,(Xi,xj−/hatwideXi,xj)v/a\\}b∇acket∇i}ht/parenrightBig\nand using the inequality from Step 2, we arrive at the inequality in the le mma. /square\nWe are now in position to prove the claimed Faber–Krahn inequality for sets with\nsmall measure.\nProof of Proposition 5.1. For eachε >0, we introduce a family of smooth cut-off\nfunctionsχj:M→Rsuch that\n•/summationtext\njχ2\nj= 1 everywhere,\n•for anyj, there exists xj=xj(ε)∈Mwith supp(χj)⊂B(xj(ε),ε),PLEIJEL THEOREM FOR SUB-LAPLACIANS 21\n•there exists C >0 (independent of ε>0) such that everywhere in M,\n/summationdisplay\nj|∇χj|2≤Cε−2.\nFor anyv∈C∞(M) we have the identity\n−/a\\}b∇acketle{t∆v,v/a\\}b∇acket∇i}htL2(M)=−/summationdisplay\nj/a\\}b∇acketle{tχj∆v,χjv/a\\}b∇acket∇i}htL2(M)\n=/summationdisplay\nj/parenleftbig\n−/a\\}b∇acketle{t[χj,∆]v,χjv/a\\}b∇acket∇i}htL2(M)−/a\\}b∇acketle{t∆(χjv),χjv/a\\}b∇acket∇i}htL2(M)/parenrightbig\n=/summationdisplay\nj/parenleftBig\n−/ba∇dblv˜∇χj/ba∇dbl2\nL2(M)−/a\\}b∇acketle{t∆(χjv),χjv/a\\}b∇acket∇i}htL2(M)/parenrightBig\n. (5.2)\nHere, asinthepreviousproof, ˜∇isnottheEuclidean gradient, butthesubRiemannian\ngradient, and\n/ba∇dblv˜∇χj/ba∇dbl2=/summationdisplay\nℓ/ba∇dblvXℓχj/ba∇dbl2.\nNote first that by our construction of the χjand the compactness of M, there exists\na constantC1such that/summationdisplay\nℓ/ba∇dblvXℓχj/ba∇dbl2≤C1ε−2/ba∇dblv/ba∇dbl2.\nLetη>0 be a parameter that we will be chosen later depending on ε. Assuming that\nv∈C∞\n0(D) withµ(D)≤η, we use the previous bound to get\n/summationdisplay\nℓ/ba∇dblvXℓχj/ba∇dbl2≤C1ε−2(cFK)−1η2\nQcFKµ(D)−2\nQ/ba∇dblv/ba∇dbl2. (5.3)\nWe turn our attention to the last term in (5.2). According to Lemma 5 .2, we have,\nunder the assumption ε≤ε0,/vextendsingle/vextendsingle/vextendsingle/a\\}b∇acketle{t∆χjv,χjv/a\\}b∇acket∇i}htL2(M,µ)−/a\\}b∇acketle{t/hatwide∆xjχjv,χjv/a\\}b∇acket∇i}htL2(Gxj,µxj)/vextendsingle/vextendsingle/vextendsingle\n≤Cεs/a\\}b∇acketle{t−/hatwide∆xj(χjv),χjv/a\\}b∇acket∇i}htL2(Gxj,µxj).\nWe combine this bound with the local Faber–Krahn inequality (4.1),\n−/a\\}b∇acketle{t/hatwide∆xjχjv,χjv/a\\}b∇acket∇i}htL2(Gxj,µxj)≥cFK/hatwideµxj(supp(χjv))−2\nQ/ba∇dblχjv/ba∇dbl2\nL2(Gxj,µxj).(5.4)\nWe also observe that (using θx)\n/hatwideµxj(supp(χjv))≤(1+Cε)µ(supp(χjv))≤(1+Cε)µ(D)\nand\n/ba∇dblχjv/ba∇dbl2\nL2(Gxj,µxj)≥(1−Cε)/ba∇dblχjv/ba∇dbl2\nL2(M).\nThus, we have proved that\n−/a\\}b∇acketle{t∆(χjv),χjv/a\\}b∇acket∇i}htL2(M)≥(1−Cεs)(1+Cε)−2\nQ(1−Cε)cFKµ(D)−2\nQ/ba∇dblχjv/ba∇dbl2\nL2(M).22 RUPERT L. FRANK AND BERNARD HELFFER\nSumming over j, and inserting the resulting bound together with (5.3) into (5.2),\nwe obtain\n−/a\\}b∇acketle{t∆v,v/a\\}b∇acket∇i}htL2(M)≥/parenleftBig\n(1−Cεs)(1+Cε)−2\nQ(1−Cε)−C1ε−2(cFK)−1η2\nQ/parenrightBig\n×cFKµ(D)−2\nQ/ba∇dblv/ba∇dbl2\nL2(M).\nNow given θ>0 we first choose ε>0 such that\n(1−Cεs)(1+Cε)−2\nQ(1−Cε)≤1−θ\n2\nand thenη >0 such that\nC1ε−2(cFK)−1η2\nQ≤θ\n2.\nIn this way we obtain the claimed inequality. /square\nWe are now in position to give the proof of our main result for sub-Lap lacians in\nthe equiregular case.\nProof of Theorem 4.3. Letukbe an eigenfunction associated with λk. Givenθ >0,\nletηbe as in Proposition 5.1 and fix a nodal domain Dkℓofukwithµ(Dkℓ)≤η. We\ndenote byukℓthe restriction of uktoDkℓ, extended by 0 outside Dkℓ. Then,ukℓis\nnot necessarily in the operator domain of −∆, but it is in the form domain, as shown\nin Theorem 2.2. Therefore it can be approximated with respect to th e form-norm by\nC∞\n0(Dkℓ) functions. Therefore we deduce from Proposition 5.1 that\n/a\\}b∇acketle{t−∆ukℓ,ukℓ/a\\}b∇acket∇i}htL2(M)≥(1−θ)cFKµ(Dkℓ)−2\nQ/ba∇dblukℓ/ba∇dbl2\nL2(M).\nHere we are slightly abusing notationby writing /a\\}b∇acketle{t−∆ukℓ,ukℓ/a\\}b∇acket∇i}htL2(M)instead of the more\nprecise/ba∇dbl˜∇ukℓ/ba∇dbl2\nL2(M). Similarly, using the weak formulation of the eigenvalue equation\nand the fact that ukℓ=ukonDkℓwe find\n/a\\}b∇acketle{t−∆ukℓ,ukℓ/a\\}b∇acket∇i}htL2(M)=λk/ba∇dblukℓ/ba∇dbl2\nL2(M).\nCombining the two previous equations and noting that ukℓdoes not vanish identically,\nwe obtain the inequality\nλk≥(1−θ)cFKµ(Dkℓ)−2\nQ. (5.5)\nWe denote by Aηthe family of nodal sets of uksatisfyingµ(Dkℓ)≤η. Raising (5.5)\nto the power Q/2 and summing over ℓwe obtain\nλQ\n2\nkµ(M)≥λQ\n2\nk/summationdisplay\nDkℓ∈Aηµ(Dkℓ)≥(1−θ)Q\n2/parenleftbig\ncFK/parenrightbigQ\n2(#Aη).\nClearly, for the number of nodal sets of ukwithµ-measure exceeding ηwe have\nη#{ℓ:Dkℓ/\\e}atio\\slash∈ Aη} ≤/summationdisplay\nDkℓ/ne}ationslash∈Aηµ(Dkℓ)≤µ(M).PLEIJEL THEOREM FOR SUB-LAPLACIANS 23\nThe two previous relations imply that\nνk\nk≤(1−θ)−Q\n2/parenleftbig\ncFK/parenrightbig−Q\n2λQ\n2\nk\nk+η−1µ(M)1\nk.\nCombining this bound with the Weyl law from Theorem 3.2, we obtain\nlimsup\nk→∞νk\nk≤(1−θ)−Q\n2/parenleftbig\ncFK/parenrightbig−Q\n2/parenleftBig1\nµ(M)/integraldisplay\nMκ(x)dµ(x)/parenrightBig−1\n.\nSinceθ>0 is arbitrary we obtain the bound claimed in Theorem 4.3. /square\n6.Basic examples\nIn this section we give examples of the applicability of Theorem 4.3. Mor e precisely,\nwe will use the version from Remark 4.5, which concerns the result on an open subset\nwith Dirichlet boundary conditions. We will present the same computa tion in various\nforms which, we hope, illustrates the different techniques that can be applied for a\nconcrete operator.\n6.1.Presentation.\nWe denote coordinates on R3by (x,y,z). In an open subset Ω ⊂R3, we consider\nX1=∂\n∂x+K1(x,y)∂\n∂z, X 2=∂\n∂y+K2(x,y)∂\n∂z,\nwith curl/vectorK=∂\n∂xK2−∂\n∂yK1>0. The measure µis simply the Lebesgue measure\ndxdydz.\nOur aimistogive anexplicit criterionfor gettingPleijel’s theoremfor∆ = X2\n1+X2\n2.\nLet (x0,y0,z0)∈Ω. Then in the construction Gx0,y0,z0is the Heisenberg group H\nand the privileged coordinates at ( x0,y0,z0) are given (modulo higher order term if\nwe use the canonical privileged coordinates) in the form\nu1=x−x0, u 2=y−y0, u 3=/hatwideδ(z−z0)+P(x−x0,y−y0),(6.1)\nwherePis a polynomial of order 2 and\n/hatwideδ=1\ncurl/vectorK(x0,y0).\nHence the candidate to be the nilpotentized measure at ( x0,y0,z0) is\ncurl/vectorK(x0,y0)du1du2du3. (6.2)\nIn these coordinates (which are not a priori privileged9but see the detailed discussion\nbelow), we have\n/hatwideX1=∂\n∂u1−1\n2u2∂\n∂u3,/hatwideX2=∂\n∂u2+1\n2u1∂\n∂u3,/hatwideX3=∂\n∂u3.(6.3)\n9We refer to [15, 16] for a complete description of possible privileged c oordinates24 RUPERT L. FRANK AND BERNARD HELFFER\n6.2.Direct change of variables.\nGiven a point ( x0,y0,z0), it is enough to replace K1andK2by their linear approxima-\ntion and it remains to show that this linearization will give us /hatwideX1and/hatwideX2in suitable\ncoordinates.\nModulo a translation, we can assume that ( x0,y0,z0) = (0,0,0)∈Ω.\nIt remains to find the changes of variables transforming\nXlin\n1=∂\n∂x+(K1(0,0))+αx+βy)∂\n∂z, Xlin\n2=∂\n∂y+(K2(0,0)+γx+δy))∂\n∂z,\nNote that\n[Xlin\n1,Xlin\n1] = (γ−β)∂\n∂z\nwith\nγ−β= curlK(x0,y0).\nWe proceed step by step for pedagogical reasons, each change o f variable having the\nform described above.\n•First change to replace K1(0,0) andK2(0,0) by 0.\nFor this, we take a change of variable in the form\n˜x=x,˜y=y,˜z=z−K1(0,0)x−K2(0,0)y,\nand get in the new coordinates\nXlin\n1=∂\n∂˜x+(α˜x+β˜y)∂\n∂˜z, Xlin\n2=∂\n∂˜y+(γ˜x+δ˜y)∂\n∂˜z,\nFrom now on, we forget the tilde.\n•Second change of variables in order to have α= 0 andδ= 0. For this, we take\na change of variable in the form\n˜x=x,˜y=y,˜z=z−1\n2αx2−1\n2δy2,\nand get in the new coordinates\nXlin\n1=∂\n∂˜x+β˜y∂\n∂˜z, Xlin\n2=∂\n∂˜y+γ˜x∂\n∂˜z.\nAgain we forget the tilde.\n•Third change of variables in order to have βnew=−γnew. This time, we we\ntake a change of variable in the form, for some ρ\n˜x=x,˜y=y,˜z=z−ρxy,\nand we impose β−ρ=−(γ−ρ), which leads to ρ= (β+γ)/2. We get in the\nnew coordinates\nXlin\n1=∂\n∂˜x+β−γ\n2˜y∂\n∂˜z, Xlin\n2=∂\n∂˜y−(β−γ)\n2˜x∂\n∂˜z.\nNote that till now we have respected the Lebesgue measure!PLEIJEL THEOREM FOR SUB-LAPLACIANS 25\n•The last step consists in the change of variable\n˜x=x,˜y=y,˜z=/hatwideδz.\nThis time, this changes of variable does not respect the Lebesgue m easure and\ninvolves the curl of /vectorKat the chosen point ( x0,y0). Note that with this change\nof variables (although it is not the canonical privileged coordinate, b ut see\nbelow the second approach) we have obtained the conclusion of Lem ma 3.1\nwhich is what is used in the proof referring to [77].\nMore precisely, if we write ( X1,X2) in the new system of coordinates we get\nX1,(x0,y0,z0)=/hatwideX1+/parenleftbig\nR1,x0,y0/curlK(x0,y0)/parenrightbig∂\n∂u3,\nX2,(x0,y0,z0)=/hatwideX2+/parenleftbig\nR2,x0,y0/curlK(x0,y0)/parenrightbig∂\n∂u3,\nHere, if we write Kj(x,y) =Kj(x0,y0)+∇Kj(x0,y0)·(x−x0,y−y0)+rj,x0,y0(x,y),\nwe have\nRj,x0,y0(u1,u2) :=rj,x0,y0(x0+u1,y0+u2) =O(u2\n1+u2\n2)\n6.3.Computation of the canonical privileged coordinates and ap plication.\nWe start from (3.2) with\nY1:=X1, Y 2:=X2, Y 3:= [X1,X2] = curl/vectorK∂\n∂z.\nThis equation reads with the notation of above and assuming ( x0,y0,z0) = (0,0,0)\n(x,y,z) = exp(/summationdisplay\nuiYi)·(0,0,0). (6.4)\nWe observe that\n/summationdisplay\nuiYi=u1∂\n∂x+u2∂\n∂y+(curl/vectorKu3+K1u1+K2u2)∂\n∂z.\nTo compute the right hand side of (6.4) we have to compute the solut ion att= 1 of\nthe differential system\ndx\ndt=u1\ndy\ndt=u2\ndz\ndt= curl/vectorK(x,y)u3+K1(x,y)u1+K2(x,y)u2.\nwith (x(0),y(0),z(0)) = (0,0,0).\nThis is easy to solve. We have x(t) =tu1,y(t) =tu2and\nz(t) =/integraldisplayt\n0(curl/vectorK(su1,su2)u3+K1(su1,su2)u1+K2(su1,su2)u2)ds.26 RUPERT L. FRANK AND BERNARD HELFFER\nHence we get\n(x,y,z) := exp(/summationdisplay\nuiYi)·(0,0,0)\n=/parenleftbigg\nu1,u2,/integraldisplay1\n0(curl/vectorK(su1,su2)u3+K1(su1,su2)u1+K2(su1,su2)u2)ds/parenrightbigg\n.\nThis is a local diffeomorphism since the differential of the map ( u1,u2,u3)/ma√sto→(x,y,z)\nat (u1,u2,u3) = (0,0,0) is\n\n1 0 0\n0 1 0\nK1(0,0)K2(0,0) curl/vectorK(0,0)\n.\nWe note (for comparison with the first approach) that\n/integraldisplay1\n0(curl/vectorK(su1,su2)u3+K1(su1,su2)u1+K2((su1,su2)u2)ds\n= curl/vectorK(0,0)u3+K1(0,0)u1+K2(0,0)u2\n+1\n2/bracketleftBig∂K1\n∂x(0,0)u2\n1+∂K1\n∂y(0,0)u1u2+∂K2\n∂x(0,0)u1u2+∂K2\n∂y(0,0)u2\n2/bracketrightBig\n+r(u1,u2,u3)\nwithrof degree ≤ −3 for the dilation. Note that if we neglect the remainder r, this\nchange of variable is exactly (6.1).\nFor the inverse we get indeed u1=x,u2=yand\nu3=1\ncurl/vectorK(0,0)/parenleftBig\nz−1\n2/bracketleftbig∂K1\n∂x(0,0)x2+(∂K1\n∂y(0,0)+∂K2\n∂x(0,0))xy+∂K2\n∂y(0,0)y2/bracketrightbig/parenrightBig\n+r(x,y,z).\nThe computation then go in the same way as in the previous approach with this\nchoice of privileged coordinates. Hence we get the same /hatwideXiat (x0,y0,z0), which are\nindependent of ( x0,y0,z0) and given by (6.3). It is clear that (6.2) also holds.\nWe can then apply Lemma 3.1.\n6.4.Conclusion in the setting of the example.\nWe introduce κ(x0,y0,z0), the contribution to the Weyl law (3.5) at ( x0,y0,z0); see\nTheorem 3.2.\nWe use the results of Appendix B. Mis a smooth bounded domain Ω inR3and\nµis the Lebesque measure dxdydz. We have Q= 4. We fix x= (x,y,z)∈Mand\ncompute/hatwideex(1,0,0). We use (6.2) and (6.3). For the heat kernel associated to the\nthree vector fields (6.3) and the measure du1du2du3, we know that /hatwideex(1,0,0) =1\n16:\nthis can be checked with (B.2) and [18, Proposition 3.1]. For the same v ector fieldsPLEIJEL THEOREM FOR SUB-LAPLACIANS 27\nbut a measure given by (6.2) we have, using (B.1),\nκ(x) =1\n2/hatwideex(1,0,0) =1\n32curl/vectorK(x,y).\nWe write this as\nκ(x) =/hatwiderW(H1)\ncurl/vectorK(x,y)\nwith\n/hatwiderW(H1) :=1\n32.\n(see up to renormalization (8.4)).\nWe denote by cFK(H1) the Faber–Krahn constant on the Heisenberg group H1.\nWe now apply Theorem 4.3 or, more precisely, its version for the sub- Laplacian on\nan open set with Dirichlet boundary conditions (Remark 4.5). The con dition (4.5)\nreads\n/parenleftbig/parenleftbig\ncFK(H1)/parenrightbig2/hatwiderW(H1)/parenrightbig\n·inf\nx0,y0curl/vectorK(x0,y0)·/parenleftBig/integraltext\nΩ1\ncurl/vectorK(x,y)dxdydz\n|Ω|/parenrightBig\n>1.\nUnder this condition, Pleijel’s theorem holds.\nThe value of the constant/parenleftbig\ncFK(H1)/parenrightbig2/hatwiderW(H1) will be discussed in the second part\nof this paper, where it is denoted (see (7.4)) by1\nγ(H1). (Note that this constant is\nindependent of the normalization chosen in the definition of the Heise nberg group.\nTherefore the analysis in Part 2 is applicable, even if it uses a different normalization\nthan in the present section.)\nCurrently we have no proof that/parenleftbig\ncFK(H1)/parenrightbig2/hatwiderW(H1)>1, although we will prove\nin Part 2 that this holds provided a well-known conjecture by Pansu c oncerning the\nisoperimetric inequality on the Heisenberg group is true.\nIf it holds that/parenleftbig\ncFK(H1)/parenrightbig2/hatwiderW(H1)>1, then our conclusion is that the same in-\nequality holds in the above example, provided the variation of curl Kis small in a\nquantitative sense.\n6.5.Variations of the example.\nTo have a example leading to a Pleijel’s theorem, we could repeat the ab ove analysis\nwith\nX2\n1+X2\n2+∆w\non Ω⊂R3×Rkwithk≥3. As will be discussed in Part 2, in this case the analogue\n1\nγ(H×Rk)of/parenleftbig\ncFK(H1)/parenrightbig2/hatwiderW(H1) is indeed >1 (see Theorem 7.2 together with (7.4)).\nTherefore Pleijel’s theorem holds in this example, provided the variat ion of curlKis\nsmall in a quantitative sense. We do not carry out the details.\nMore generally, in an open subset of Rn×Rn×R,n∈N, with coordinates ( /vector x,/vector y,z),\n/vector x= (x1,...,x n) and/vector y= (y1,···,yn), we consider the following vector fields for28 RUPERT L. FRANK AND BERNARD HELFFER\nj= 1,...,n,\nX′\nj=∂xj−Kj\n1(xj,yj)∂z,\nX′′\nj=∂yj−Kj\n2(xj,yj)∂z.\nWe assume that curl Kj>0 forj= 1,...,n.\nFor the nilpotentization at ( /vector x,/vector y,z) this leads to the following vector fields on Hn,\nˆX′\nj= curlKj(xj,yj)1/2(∂uj−1\n2vj∂z),\nˆX′′\nj= curlKj(xj,yj)1/2(∂vj+1\n2uj∂z).\ntogether with the Haar measure (/producttextn\nj=1curlKj(xj,yj))−1dudvdz.\nFor the Faber–Krahn part, one has to consider on Hn\nˆ∆/vector x,/vector y=/summationdisplay\njcurlKj(xj,yj)/parenleftBig\n(∂uj−vj\n2∂z)2+(∂vj+uj\n2∂z)2/parenrightBig\n.\nWhen curlKj≡1, it will be clear from our analysis in Part 2 (Theorem 7.2) that\nPleijel’s theorem holds for n≥4 in any open set of Ω. This property remains true if\ncurlKj−1 remains (for j= 1,...,n) small enough in Ω.PLEIJEL THEOREM FOR SUB-LAPLACIANS 29\nPart2.Pleijel’s bound for Hn×Rk\n7.The Pleijel argument for Hn×Rk\nWe work on Hn×Rk, wheren∈N,k∈N0. The case k= 0 corresponds to the\nHeisenberg group Hn. (Everything remains valid for n= 0 as well, that is, for Rk, but\nin this case the results below are well known.) Typically, we will denote c oordinates\ninHnby (x,y,z) withx,y∈Rnandz∈R, and we will denote coordinates in Rk\nbyw. The measure dxdydzdw is the Lebesgue measure on R2n+1+k. For the vector\nfields we will use the following normalization10,\nXj=∂xj+2yj∂z, Y j=∂yj−2xj∂z, W j=∂wj. (7.1a)\nThe sub-Laplacian is\n∆Hn×Rk=n/summationdisplay\nj=1(X2\nj+Y2\nj)+k/summationdisplay\ni=1W2\ni. (7.1b)\nIf Ω⊂Hn×Rkis an open set of finite measure, then the spectrum of the Dirichlet\nrealization of −∆Hn×Rk\nΩis discrete and we can denote its eigenvalues, in nondecreasing\norder and repeated according to multiplicities, by λℓ(Ω),ℓ∈N. We know that eigen-\nfunctions are C∞in Ω ([45]) and therefore the nodal domains are well defined as the\nconnected components of the complement of their zero set in Ω. We denote byνℓ(Ω)\nthe maximal number of nodal domains of eigenfunctions correspon ding to eigenvalue\nλℓ(Ω). We are interested in an upper bound on\nlimsup\nℓ→∞νℓ(Ω)\nℓ\nthat depends only on nandk. The simplest such upper bound is given by one, as\nmentioned in Remark 4.6.\nIn the spirit of Pleijel’s theorem, here we try to improve upon the upp er bound\nby one. Just as Pleijel’s bound, our bound depends on two constant s, namely the\nconstant in the Weyl asymptotics and the Faber–Krahn constant . Let us introduce\nthese constants. The Weyl asymptotics in the case with boundary (which was also\nestablished by G. M´ etivier in [61, Theorem 1.3]) states that, for any open set Ω ⊂\nHn×Rkof finite measure,\nµ−2n+2+k\n2#{ℓ:λℓ(Ω)<µ} → W(Hn×Rk)|Ω|asµ→ ∞.(7.2)\nWe will give a (relatively) explicit expression for the constant W(Hn×Rk) in the next\nsection.\n10In this and the remaining sections it is more convenient to use anothe r normalization than in the\nfirst sections. This simply corresponds to a scaling of the z-variable.30 RUPERT L. FRANK AND BERNARD HELFFER\nThe Faber–Krahn constant CFK(Hn×Rk) is defined to be the largest constant such\nthat for any open Ω ⊂Hn×Rkof finite measure and for any u∈S1\n0(Ω) one has\n/integraldisplay\nΩ(n/summationdisplay\nj=1((Xju)2+(Yju)2)+k/summationdisplay\ni=1(Wiu)2)dxdydzdw\n≥CFK(Hn×Rk)|Ω|−2\n2n+2+k/integraldisplay\nΩu2dxdydzdw. (7.3)\nHereS1\n0(Ω) (see Subsection 2.2) denotes the form domain of the Dirichlet re alization\nof−∆ on Ω or, equivalently, the completion of C1\nc(Ω) with respect to the quadratic\nform\nu/ma√sto→/integraldisplay\nΩ(n/summationdisplay\nj=1((Xju)2+(Yju)2)+k/summationdisplay\ni=1(Wiu)2+u2)dxdydzdw.\nThe defining inequality for the Faber–Krahn constant can also be st ated as\nλ1(Ω)≥CFK(Hn×Rk)|Ω|−2\n2n+2+k\nfor all open Ω ⊂Hn×Rkof finite measure.\nLet us set\nγ(Hn×Rk) :=/parenleftbig\nCFK(Hn×Rk)/parenrightbig−2n+2+k\n2/parenleftbig\nW(Hn×Rk)/parenrightbig−1. (7.4)\nHere is our Pleijel-type bound.\nTheorem 7.1. For any open Ω⊂Hn×Rkof finite measure,\nlimsup\nℓ→∞νℓ(Ω)\nℓ≤γ(Hn×Rk).\nProof of Theorem 7.1. Weconsider aneigenfunction ucorrespondingtotheeigenvalue\nλℓ(Ω). Let (ωα)αbe its nodal domains and let νℓ(u) be their number. (We will see\nshortly that this number is finite.) By Theorem 2.2 with M=Hn×Rk, we know that\nλℓ(Ω) =λ1(ωα) and that u|ωαis the ground state of the Dirichlet realization on ωα.\nThus,\nνℓ(u)\nℓ=λℓ(Ω)2n+2+k\n2\nℓ/summationdisplay\nαλ1(ωα)−2n+2+k\n2≤λℓ(Ω)2n+2+k\n2\nℓ/parenleftbig\nCFK(Hn×Rk)/parenrightbig−2\n2n+2+k/summationdisplay\nα|ωα|\n≤λℓ(Ω)2n+2+k\n2\nℓ/parenleftbig\nCFK(Hn×Rk)/parenrightbig−2\n2n+2+k|Ω|.\nSince this is true for any eigenfunction corresponding to λℓ(Ω), we deduce that\nνℓ(Ω)\nℓ≤λℓ(Ω)2n+2+k\n2\nℓ/parenleftbig\nCFK(Hn×Rk)/parenrightbig−2\n2n+2+k|Ω|.\nTaking the limsup as ℓ→ ∞and with in mind the Weyl asymptotics, we arrive at the\nclaimed bound. /squarePLEIJEL THEOREM FOR SUB-LAPLACIANS 31\nWe recall that by Theorem 2.4 we have limsupℓ→∞νℓ(Ω)\nℓ≤1; see also Remark 4.6.\nIn the remaining sections of this paper we will give sufficient conditions onnandk\nto haveγ(Hn×Rk)<1. We recall that for n= 0 this was shown to be the case for\nk= 2 by Pleijel [72] and for general kby B´ erard and Meyer [8]. Moreover, Helffer and\nPersson Sundqvist [43] showed that, for n= 0, the sequence k/ma√sto→γ(Rk) is decreasing.\nHere we shall prove, among other things, the following.\nTheorem 7.2. Letn≥1andk≥0with(n,k)/\\e}atio\\slash∈ {(1,0),(2,0),(3,0),(1,1)}. Then\nγ(Hn×Rk)<1.\nThe proof of Theorem 7.2 is somewhat long and spread out over seve ral sections.\nHere is a guide. The part concerning k= 0 is proved in Subsection 9.2 and that\nconcerning n≥3 in Subsection 9.4. The part concerning n= 1 andn= 2 is proved\nin Subsection 11.6.\nThere is a well known conjecture, due to Pansu [71], about the shar p isoperimetric\nconstant on the Heisenberg group. It is generally believed to be tru e and supported\nby several partial results. We will discuss this in some detail in Sectio n 11. We shall\nshow that the validity of this conjecture implies Pleijel’s bound.\nProposition 7.3. Letn∈Nand assume that Pansu’s conjecture (11.6)holds. Then\nγ(Hn×Rk)<1for allk∈N0.\nWe will prove this proposition for k= 0 andn= 1,2,3 in Corollary 11.2 and for\nk= 1 andn= 1 in Subsection 11.6. In the remaining cases Theorem 7.2 applies.\n8.Computing the constant in the Weyl asymptotics\n8.1.The case of Hn.\nAs discussed in Theorem 3.2 and (7.2) for the case with boundary, th e Weyl asymp-\ntotics for the Dirichlet realization of the sub-Laplacian in open subse ts Ω ofHnstate\nthat\nµ−2n+2\n2#{ℓ:λℓ(Ω)<µ} → W(Hn)|Ω|asµ→ ∞.\nHansson–Laptev [37] have shown that these asymptotics hold und er the sole assump-\ntion that Ω is an open set of finite measure.\nSince we are interested in a relatively explicit expression of the const antW(Hn) and\nsince Hansson and Laptev use a different normalization from ours, w e repeat part of\ntheir argument. They show that\nN(µ,−∆Hn\nΩ)∼/integraldisplay\nΩ\n/BD(−∆Hn<µ)((x,y,z),(x,y,z))dxdydz asµ→ ∞,(8.1)\nwhere /BD(−∆Hn<µ)((x,y,z),(x,y,z)) is the on-diagonal spectral density of the sub-\nLaplacianonall of Hn. (To bemoreprecise, using coherent states Hansson andLaptev\nshow (8.1) when integrated over µ. Then a Tauberian theorem yields (8.1) as stated.)\nBy translation invariance of the sub-Laplacian on Hn, we know that the diagonal\nof the spectral density /BD(−∆Hn< µ)((x,y,z),(x,y,z)) is independent of the point32 RUPERT L. FRANK AND BERNARD HELFFER\n(x,y,z). Moreover, by dilation covariance, we know that it is proportional toµQ\n2.\nThus, there is a constant W(Hn)>0 such that/BD(−∆Hn<µ)((x,y,z),(x,y,z)) =W(Hn)µQ\n2, (8.2)\nand we obtain the above form of the spectral asymptotics.\nIn the following we are interested in finding an explicit expression for W(Hn). We\nproceed by an explicit diagonalization of the operator −∆Hn. By a Fourier transform\nwith respect to zone arrives at the family of operators\n−n/summationdisplay\nj=1/parenleftbig\n(∂xj+2iyjζ)2+(∂yj−2ixjζ)2/parenrightbig\n,\nwhereζ∈Ris the Fourier variable dual to the variable z.\nNow for each fixed j,\n−/parenleftbig\n(∂xj+2iyjζ)2+(∂yj−2ixjζ)2/parenrightbig\n,\nis a Landau Hamiltonian corresponding to constant magnetic field with intensity 4 |ζ|.\nThesenHamiltonians are independent of each other. The spectrum of each one is\ngiven by 4 |ζ|(2kj+1),kj∈N0={0,1,2,...}.\nThe spectral function of the operator −∆ onHnis then given by/BD(−∆<µ)((x,y,z),(x,y,z)) =/integraldisplay\nRdζ\n2π/parenleftbigg4|ζ|\n2π/parenrightbiggn/summationdisplay\nk∈Nn\n0\n/BD(4|ζ|(2(k1+...+kn)+n)<µ)\n=4n\n(2π)n+1/summationdisplay\nk∈Nn\n02\nn+1/parenleftbiggµ\n4(2(k1+...+kn)+n)/parenrightbiggn+1\n=1\n2(n+1)µn+1\n(2π)n+1/summationdisplay\nk∈Nn\n0/parenleftbigg1\n2(k1+...+kn)+n/parenrightbiggn+1\n=1\n2(n+1)µn+1\n(2π)n+1/summationdisplay\nm∈N0/parenleftbiggm+n−1\nm/parenrightbigg/parenleftbigg1\n2m+n/parenrightbiggn+1\n.\nThus, we have shown that\nW(Hn) =1\n2(n+1)1\n(2π)n+1/summationdisplay\nm∈N0/parenleftbiggm+n−1\nm/parenrightbigg1\n(2m+n)n+1.(8.3)\nNote that\nW(H) =1\n41\n(2π)2/summationdisplay\nm∈N01\n(2m+1)2=1\n41\n(2π)2π2\n8=1\n128(8.4)\nand\nW(H2) =1\n61\n(2π)3/summationdisplay\nm∈N0m+1\n(2m+2)3=1\n61\n(2π)3π2\n48=1\n482π. (8.5)PLEIJEL THEOREM FOR SUB-LAPLACIANS 33\nIt seems like Hansson–Laptev [37] were not aware of a more explicit f orm ofW(Hn)\nforn≥3. We will give explicit formulas below for 3 ≤n≤13 and propose a general\nconjecture.\nLet us set\ncn:=/summationdisplay\nm∈N0/parenleftbiggm+n−1\nm/parenrightbigg1\n(2m+n)n+1. (8.6)\nWe have seen above that\nc1=π2\n8andc2=π2\n48.\nNumerical values for the cn,n= 3,4,5,6, are given in the Hansson–Laptev paper\n[37] and can be completed by using Wolfram Alpha or Mathematica11. This program\nshows first that explicit formulas can be found for cnand then gives numerical values,\nwhich are consequently quite accurate. We get\nc3=π2(12−π2)/768 ≈2.7378·10−2\nc4=π2(15−π2)/17280 ≈2.9303·10−3\nc5=π2(120−100π2+9π4)/368640 ≈2.6027·10−4\nc6=π2(315−105π2+8π4)/29030400 ≈1.9706·10−5\nc7=π2(6720−19600π2+14504π4−1275π6)/2477260800 ≈1.2988·10−6\nc8=π2(1575−1470π2+490π4−36π6)/24385536000 ≈7.5736·10−8\nc9=π2(40320−282240π2+663264π4−439144π6+37975π8)/3329438515200 ≈3.9589·10−9\nc10=π2(3465−6930π2+6006π4−1804π6+128π8)/15450675609600 ≈1.8749·10−10\nContinuing with Mathematica, we get for the quotients:\nc11\nc10=3/parenleftbig1774080 −24393600 π2+129773952 π4−258523760 π6+160227716 π8−13712895 π10/parenrightbig\n10240(3465 −6930π2+6006π4−1804π6+128π8),\nc12\nc11=256/parenleftbig\n2837835 −10405395 π2+18432414 π4−13774761 π6+3835832 π8−265344π10/parenrightbig\n27027(1774080 −24393600 π2+129773952 π4−258523760 π6+160227716 π8−13712895 π10),\nc13\nc12=7/parenleftbig\n2075673600 −49470220800 π2+497175719040 π4−2161554183360 π6+3895229400920 π8−2314322017956 π10+196697984175 π12/parenrightbig\n40960(2837835 −10405395 π2+18432414 π4−13774761 π6+3835832 π8−265344π10).\nThis leads to\nc11≈8.1149·10−12, c12≈3.23369·10−13, c13≈1.1938·10−14.\nAlthough not important for our applications, it is nice to see that this leads to the\nfollowing guess:\nFor anyn, there is a polynomial Pnof degree [n+1\n2] with rational coeffi-\ncients such that\ncn=Pn(π2),\nwhere [x] denotes the largest integer satisfying [ x]≤x.\nThis is related to formulas concerning multi-zeta functions, which ar e recognized by\nMathematica and, as communicated to us by F. Nicoleau, to the Lerc h function [58,\np. 32] and its derivatives.\n11Thanks to J. Viola and F. Nicoleau for their help.34 RUPERT L. FRANK AND BERNARD HELFFER\n8.2.The Weyl constant on Hn×Rkwithk≥1.\nWe show that the constant in the Weyl formula on Hn×Rkcan be expressed in terms\nof that on Hn.\nLemma 8.1. For anyn,k∈N,\nW(Hn×Rk) =W(Hn) (4π)−k\n2Γ(n+2)\nΓ(2n+k+4\n2). (8.7)\nProof.By the same argument as in the previous subsection we know that W(Hn×Rk)\nis given by/BD/parenleftbig\n−∆Hn×Rk<µ/parenrightbig\n(x,y,t,w,x,y,t,w )=W(Hn×Rk)µ2n+2+k\n2.\nSince/BD/parenleftbig\n−∆Hn×Rk<µ/parenrightbig\n(x,y,t,w,x,y,t,w )\n=/integraldisplay\nRdζ\n2π/integraldisplay\nRkdτ\n(2π)k/parenleftbigg4|ζ|\n2π/parenrightbiggn/summationdisplay\nk∈Nn\n0\n/BD(4|ζ|(2(k1+...+kn)+n)+τ2<µ),\nwe find\nW(Hn×Rk) = (2π)−k|Sk−1|/parenleftBig/integraldisplay1\n0(1−ρ2)n+1ρk−1dρ/parenrightBig\nW(Hn)\n= (2π)−k|Sk−1|/parenleftBig\n2−1/integraldisplay1\n0(1−σ)n+1σk−2\n2dσ/parenrightBig\nW(Hn)\n= (2π)−k|Sk−1|2−1Γ(n+2)Γ(k\n2)\nΓ(2n+4+k\n2)W(Hn).\nHere we expressed the beta function integral appearing on the se cond line just above\nin terms of gamma functions. Inserting |Sk−1|= Γ(k\n2)−12πk\n2, we arrive at the claimed\nformula. /square\n9.A first bound on the Faber–Krahn constant\n9.1.A bound via the Sobolev constant.\nWe obtain a bound on the Faber–Krahn constant in terms of the (cr itical) Sobolev\ninequality on Hn×Rk. By definition, CSob(Hn×Rk) is the largest constant such that\nfor allu∈S1\n0(Hn×Rk)\n/integraldisplay\nHn×Rk/parenleftBiggn/summationdisplay\nj=1((Xju)2+(Yju)2)+k/summationdisplay\ni=1(Wiu)2/parenrightBigg\ndxdydzdw\n≥CSob(Hn×Rk)/parenleftbigg/integraldisplay\nHn×Rk|u|2(2n+2+k)\n2n+kdxdydzdw/parenrightbigg2n+k\n2n+2+k\n.\nLemma 9.1. CFK(Hn×Rk)≥CSob(Hn×Rk).PLEIJEL THEOREM FOR SUB-LAPLACIANS 35\nProof.If Ω⊂Hn×Rkis open with finite measure and if u∈S1\n0(Ω), then, by H¨ older,\n/integraldisplay\nΩu2dxdydzdw ≤ |Ω|2\n2n+2+k/parenleftbigg/integraldisplay\nHn×Rk|u|2(2n+2+k)\n2n+kdxdydzdw/parenrightbigg2n+k\n2n+2+k\n.\nBounding the right side by Sobolev, we obtain a Faber–Krahn-type in equality with\nconstantCSob(Hn×Rk). Thisimpliestheclaimedboundfortheoptimalconstants. /square\nAn explicit expression for CSob(Hn) was found by Jerison and Lee [48]; for an\nalternative proof see also [32]. We have\nCSob(Hn) =4πn2\n(22nn!)1\nn+1. (9.1)\n9.2.The Pleijel constant γ(Hn).\nOur goal in this subsection is to prove the part of Theorem 7.2 for k= 0, that is, we\nare going to prove that γ(Hn)<1 forn≥4. To bound γ(Hn), defined in (7.4), we\nuse Lemma 9.1 and the explicit expression for CSob(Hn) from (9.1) to get\nγ(Hn)≤/parenleftbig\nCSob(Hn)/parenrightbig−n−1W(Hn)−1=2n(n+1)!\nn2(n+1)1\ncn=: ˜γn, (9.2)\nwherecnis defined in (8.6). Inspired by [43] we will consider the quotients ˜ γn/˜γn−1.\nIn view of (9.2), the part of Theorem 7.2 for k= 0 is an immediate consequence of\nthe following assertion.\nProposition 9.2. The sequence n/ma√sto→˜γnis decreasing for n≥1. Moreover, ˜γn<1\nforn≥4.\nOur proof of this proposition relies on numerical computations for n≤13. Using\nthe values of cnfrom the previous section, we get\n˜γ1≈3.242\n˜γ2≈1.824\n˜γ3≈1.069\n˜γ4≈6.251·10−1\n˜γ5≈3.628·10−1\n˜γ6≈2.088·10−1\n˜γ7≈1.195·10−1\n˜γ8≈6.808·10−2\n˜γ9≈3.860·10−2\n˜γ10≈2.180·10−2\n˜γ11≈1.227·10−2\n˜γ12≈6.891·10−3\n˜γ13≈3.859·10−336 RUPERT L. FRANK AND BERNARD HELFFER\nIt is also instructive to look at the quotients ˜ γn/˜γn−1, for which we get\n˜γ2/˜γ1≈1.777\n˜γ3/˜γ2≈0.5861\n˜γ4/˜γ3≈0.5848\n˜γ5/˜γ4≈0.5804\n˜γ6/˜γ5≈0.5757\n˜γ7/˜γ6≈0.5721\n˜γ8/˜γ7≈0.5697\n˜γ9/˜γ8≈0.5670\n˜γ10/˜γ9≈0.5648\n˜γ11/˜γ10≈0.5630\n˜γ12/˜γ11≈0.5614\n˜γ13/˜γ12≈0.5601\nThese computations also suggest that the sequence ˜ γn/˜γn−1is decreasing and conver-\ngent, although this remains unproved.\nProof.We are going to show that˜γn\n˜γn−1<1 forn≥13. Since the same holds for n≤12\nby the above numerical computations, we will obtain the claimed mono tonicity. This\nmonotonicity, together with the numerical fact that ˜ γ4<1, implies the corresponding\ninequality for all n≥4.\nTo show that ˜ γn/˜γn−1<1 forn≥13, we note that\n˜γn\n˜γn−1=2(n+1)\nn2(1−1/n)2ncn−1\ncn. (9.3)\nWe write the definition (8.6) of cnin the form\ncn=/summationdisplay\nm∈N0(m+n−1)!\n(n−1)!m!1\n(2m+n)n+1.\nIn view of (9.3) we are mainly interested in a lower bound on the quotien tcn/cn−1.\nObserving that\n(m+n−1)!\n(n−1)!m!1\n(2m+n)n+1=m+n−1\n(n−1)(2m+n)(1−1\n2m+n)n\n×(m+n−2)!\n(n−2)!m!1\n(2m+n−1)n,\nwe get\ncn\ncn−1≥1\nn−1inf\nmm+n−1\n2m+n(1−1\n2m+n)n. (9.4)\nHence we have to analyze inf mθn(m) with\nθn(m) :=m+n−1\n2m+n(1−1\n2m+n)n. (9.5)PLEIJEL THEOREM FOR SUB-LAPLACIANS 37\nWe need to carefully analyze the sequence m/ma√sto→θn(m). Before doing this, let us\nprovide some heuristics coming from limiting regimes. As m= 0, we have θn(0) =\nn−1\nn(1−1\nn)n, which tends to e−1asn→+∞. Asmtends to + ∞, we haveθn(m)→1\n2,\nbut we need uniform lower bounds with respect to mandn.\nAsymptotically as n→+∞, a lower bound for θn(m) is given by considering the\ninfimum of the function (think of the change of variable y=m/n)\n(0,+∞)∋y/ma√sto→1+y\n1+2ye−1/(2y+1).\nAfter the change of variable u=y+1\n2, we have to analyze\n(1\n2,+∞)∋u/ma√sto→1\n2+u\n2ue−1/2u= (1\n2+1\n4u)e−1/2u.\nThis function is increasing and its minimum is at u=1\n2and equals e−1. Hence we\nhavem+n\n2m+ne−n\nn+2m≥e−1. (9.6)\nThis leads to\nlimsup\nn→+∞˜γn\n˜γn−1≤2e−1≈0.735. (9.7)\nThis is closer to the guess. For the lower bound and using an upper bo und for\nlimsupn→+∞θn(m), we get\nliminf\nn→+∞˜γn\n˜γn−1≥4e−2≈0.541.\nThis is quite close to the numerics.\nAfter having discussed these heuristics, we will turn to the proof o f rigorous bounds.\nReturning to θn(m) in (9.5), we write\nθn(m)≥m+n\n2m+n(1−1\n2m+n)n−1\nn.\nUsing\n−log(1−x)≤x+/integraldisplayx\n0t\n1−tdt, x ∈[0,1),\nwe get\nx≤ −log(1−x)≤x+1\n1−xx2\n2, x∈[0,1).\nWithx=1\n2m+n, we obtain\n(1−1\n2m+n)n=enlog(1−1\n2m+n)≥e−n\nn+2me−1\n2(n−1).\nComing back to θn(m) and what we have done for the limsup\nθn(m)≥e−1e−1\n2(n−1)−1\nn.38 RUPERT L. FRANK AND BERNARD HELFFER\nSo we finally get\ncn≥1\nn−1/parenleftBig\ne−1e−1\n2(n−1)−1\nn/parenrightBig\ncn−1.\nComing back to (9.3), we get\n˜γn\n˜γn−1≤2(n+1)(n−1)\nn2(1−1/n)2n/parenleftBig\ne−1e−1\n2(n−1)−1\nn/parenrightBig−1\n≤2e−1(n+1)(n−1)\nn2/parenleftBig\ne−1\n2(n−1)−e\nn/parenrightBig−1\n≤2e−1/parenleftBig\ne−1\n2(n−1)−e\nn/parenrightBig−1\n.\nSo finally, we have shown that\n˜γn\n˜γn−1≤2e−1/parenleftBig\ne−1\n2(n−1)−e\nn/parenrightBig−1\n. (9.8)\nForn= 13, we have\ne−1\n2(n−1)−e\nn≈0.75009, (9.9)\nand, consequently,\n˜γ13\n˜γ12<1.\nLooking atthe bound(9.8) andits monotonicity with respect to n, the bound˜γn\n˜γn−1<1\nholds for anyn≥13. This completes the proof of Proposition 9.2. /square\n9.3.A lower bound on the Sobolev constant on Hn×Rkfork≥1.\nIn this subsection we prove a lower bound on the Sobolev constant CSob(Hn×Rk)\nin terms of the Sobolev constant CSob(Hn) and the constant appearing in a certain\nSobolev interpolation inequality on Rk. Assume 2 ≤q <∞ifk≤2 and 2≤q≤2k\nk−2\nifk>2, and denote by CGN\nq(Rk) the largest possible constant in the inequality, valid\nforu∈H1(Rk),\n/parenleftbigg/integraldisplay\nRk|∇u|2dw/parenrightbiggθ/parenleftbigg/integraldisplay\nRk|u|2dw/parenrightbigg1−θ\n≥CGN\nq(Rk)/parenleftbigg/integraldisplay\nRk|u|qdw/parenrightbigg2\nq\n,(9.10a)\nwhere\nθ=k/parenleftbigg1\n2−1\nq/parenrightbigg\n. (9.10b)\n(The value of θis determined by scaling.) For k= 1 the explicit value of the constant\nCGN\nq(R) is known from a work of Nagy [69]. For k≥2 its explicit value is not known,\nbut we will still be able to derive some results in Subsection 9.4.\nProposition 9.3. For alln,k∈N, settingQ= 2n+2andq=2(Q+k)\nQ+k−2,\nCSob(Hn×Rk)≥CGN\nq(Rk) (CSob(Hn))Q\nQ+kQ+k\nQQ\nQ+kkk\nQ+k.\nThe argument that follows is inspired by the Laptev–Weidl method of lifting in\ndimension [49] and similar to one used in [33].PLEIJEL THEOREM FOR SUB-LAPLACIANS 39\nProof.We begin by applying the inequality\nab≤1\npap+1\np′bp′\nwithp=Q+k\nk, sop′=Q+k\nQ, anda=µ−k\nQ+kgQ+k−2\nQ+k,b=µk\nQ+kg2\nQ+k. We get\ng≤k\nQ+kµ−1gQ+k−2\nk+Q\nQ+kµk\nQg2\nQ.\nWe apply this with\ng=/integraldisplay\nRk|u(ζ,w)|qdwandµ=c/parenleftbigg/integraldisplay\nRk|u(ζ,w)|2dw/parenrightbiggQ\nk\n,\nwhereζ∈Hnis fixed and c>0 is a parameter. We obtain\n/integraldisplay\nRk|u(ζ,w)|qdw≤c−1k\nQ+k/parenleftbig/integraltext\nRk|u(ζ,w)|qdw/parenrightbigQ+k−2\nk\n/parenleftbig/integraltext\nRk|u(ζ,w)|2dw/parenrightbigQ\nk\n+ck\nQQ\nQ+k/parenleftbigg/integraldisplay\nRk|u(ζ,w)|qdw/parenrightbigg2\nQ/integraldisplay\nRk|u(ζ,w)|2dw\n≤c−1k\nQ+k(CGN\nq(Rk))−Q+k\nk/integraldisplay\nRk|∇wu(ζ,w)|2dw\n+ck\nQQ\nQ+k/parenleftbigg/integraldisplay\nRk|u(ζ,w)|qdw/parenrightbigg2\nQ/integraldisplay\nRk|u(ζ,w)|2dw.\nHere we used the Sobolev interpolation inequality on Rk.\nWe integrate this inequality with respect to ζand obtain\n/ba∇dblu/ba∇dblq\nq≤c−1k\nQ+k(CGN\nq(Rk))−Q+k\nk/ba∇dbl∇wu/ba∇dbl2\n2\n+ck\nQQ\nQ+k/integraldisplay\nHn/parenleftbigg/integraldisplay\nRk|u(ζ,w)|qdw/parenrightbigg2\nQ/integraldisplay\nRk|u(ζ,w)|2dwdζ.\nNow by Fubini and H¨ older\n/integraldisplay\nHn/parenleftbigg/integraldisplay\nRk|u(ζ,w)|qdw/parenrightbigg2\nQ/parenleftbigg/integraldisplay\nRk|u(ζ,w)|2dw/parenrightbigg\ndζ\n≤ /ba∇dblu/ba∇dbl2q\nQ\nq/integraldisplay\nRk/parenleftbigg/integraldisplay\nHn|u(ζ,w)|2Q\nQ−2dζ/parenrightbiggQ−2\nQ\ndw.40 RUPERT L. FRANK AND BERNARD HELFFER\nApplying the Sobolev inequality on Hnon the right side and inserting the inequality\nin the above bound, we get\n/ba∇dblu/ba∇dblq\nq≤c−1k\nQ+k(CGN\nq(Rk))−Q+k\nk/ba∇dbl∇wu/ba∇dbl2\n2\n+ck\nQQ\nQ+k(CSob(Hn))−1/ba∇dblu/ba∇dbl2q\nQ\nq/integraldisplay\nRkn/summationdisplay\nj=1/integraldisplay\nHn(|Xju|2+|Yju|2)dζdw.\nWe finally choose csuch that\nc−1k\nQ+k(CGN\nq(Rk))−Q+k\nk=ck\nQQ\nQ+k(CSob(Hn))−1/ba∇dblu/ba∇dbl2q\nQ\nq,\nthat is, we choose\nc=/parenleftbiggk\nQ/parenrightbiggQ\nQ+k\n(CGN\nq(Rk))−Q\nk(CSob(Hn))Q\nQ+k/ba∇dblu/ba∇dbl−2q\nQ+k\nq.\nIn this way we get\n/ba∇dblu/ba∇dblq\nq≤(CGN\nq(Rk))−1(CSob(Hn))−Q\nQ+k/ba∇dblu/ba∇dbl2q\nQ+k\nqQQ\nQ+kkk\nQ+k\nQ+k /parenleftBig\n/ba∇dbl∇wu/ba∇dbl2\n2+/integraltext\nRk/summationtextn\nj=1/integraltext\nHn(|Xju|2+|Yju|2)dζdw/parenrightBig\n.\nThis proves the assertion. /square\nFrom Proposition 9.3 we obtain the following bound on the Pleijel const ant. We\nrecall that ˜γnwas defined in (9.2).\nCorollary 9.4. For alln,k∈N, settingQ= 2n+2andq=2(Q+k)\nQ+k−2,\nγ(Hn×Rk)≤(CGN\nq(Rk))−Q+k\n2QQ\n2kk\n2\n(Q+k)Q+k\n2(4π)k\n2Γ(Q+k+2\n2)\nΓ(Q+2\n2)˜γn.\nProof.According to Lemma 9.1, Proposition 9.3 and (8.7) we have\nγ(Hn×Rk)≤/parenleftbig\nCSob(Hn×Rk)/parenrightbig−Q+k\n2/parenleftbig\nW(Hn×Rk)/parenrightbig−1\n≤(CGN\nq(Rk))−Q+k\n2(CSob(Hn))−Q\n2QQ\n2kk\n2\n(Q+k)Q+k\n2W(Hn)−1(4π)k\n2Γ(Q+k+2\n2)\nΓ(Q+2\n2).\nThis is the claimed inequality. /square\n9.4.The Pleijel constant γ(Hn×Rk)fork≥1.\nOur goal in the present subsection is to prove the following result.\nProposition 9.5. Ifn≥3andk≥1thenγ(Hn×Rk)<1.\nOur proof will rely on the bound on γ(Hn×Rk) from Corollary 9.4. In order to\nexploit this corollary we need a good lower bound on the constant CGN\nq(Rk). As we\nalready mentioned, an explicit expression for this constant is known in dimensionPLEIJEL THEOREM FOR SUB-LAPLACIANS 41\nk= 1 due to the work of Nagy [69]. The idea behind the following result is to reduce\nthe case of Rk=Rk−1×Rto that of Rk−1andR, and to iterate. This argument is\nmost easily expressed in the parametrization\nq=2(γ+k\n2)\nγ+k\n2−1.\nThe theorem will require the assumption γ≥1\n2, which however will not present a\nrestriction in our application. It appears in the proof since the cons tantCGN\n˜q(R) with\n˜q=2(γ+1\n2)\nγ+1\n2−1is only defined for γ≥1\n2.\nLemma 9.6. Letk≥1andγ≥1\n2. Then\nCGN\n2(γ+k\n2)\nγ+k\n2−1(Rk)≥/parenleftBigg/parenleftbiggkπ\n2/parenrightbiggk\n2Γ(γ+1\n2)\nΓ(γ+k+1\n2)(γ+k−1\n2)γ+k−1\n2\n(γ−1\n2)γ−1\n2/parenrightBigg1\nγ+k\n2\nwith the convention (γ−1\n2)γ−1\n2= 1forγ=1\n2.\nAs the following proof will show, the inequality in the lemma is an equality if k= 1.\nProof.Ford∈N, let us introduce the constant\nL(1)\nγ,d=γγ/parenleftbigd\n2/parenrightbigd\n2\n(γ+d\n2)γ+d\n2/parenleftBig\nCGN\n2(γ+d\n2)\nγ+d\n2−1(Rd)/parenrightBig−γ−d\n2.\nThe main step of the proof will be to show that\nL(1)\nγ,k≤k−1/productdisplay\nj=0L(1)\nγ+j\n2,1. (9.11)\nAccepting this for the moment, let us complete the proof. Insertin g Nagy’s explicit\nexpression for the constant CGN\nq(R) [69] (see also [31, Theorem 2.48]), which reads\nCGN\nq(R) =/parenleftbigg(q+2)q+2\n(q−2)q−222(q+2)/parenrightbigg1\n2q/parenleftBigg\n√πΓ(q\nq−2)\nΓ(q\nq−2+1\n2)/parenrightBiggq−2\nq\n, (9.12)\ninto the definition of L(1)\nγ,1, we obtain\nL(1)\nγ,1=1√πΓ(γ+1)\nΓ(γ+1\n2)(γ−1\n2)γ−1\n2\n(γ+1\n2)γ+1\n2;\nsee also [31, Corollary 5.4]. Inserting this into (9.11) and simplifying we o btain the\ninequality in the lemma.\nIt remains to prove (9.11). According to [31, Proposition 5.3], the co nstantL(1)\nγ,d\ncoincides with the so-called one-particle Lieb–Thirring constant. (T his observation is\nessentially due to Lieb and Thirring [56].) We use now the inequality\nL(1)\nγ,k≤L(1)\nγ,1L(1)\nγ+1\n2,k−1(9.13)42 RUPERT L. FRANK AND BERNARD HELFFER\nfrom [33]; see also [31, Proposition 5.14]. This bound can be translated into the\nclaimed lower bound on CGN\nqk(Rk) in terms of the constants CGN\nq1(R) andCGN\nqk−1(Rk−1).\nIt is more convenient, however, to stay in the notation of the Lieb– Thirring constants.\nIterating the above inequality, we arrive at (9.11).\nWe emphasize that (9.13) canbetranslated into a lower bound on CGN\nqk(Rk) in terms\nof the constants CGN\nq1(R) andCGN\nqk−1(Rk−1), whereqd=2(γ+d\n2)\nγ+d\n2−1. This bound could also\nbe proved directly using the method of proof of Proposition 9.3. /square\nIn our application, we need CGN\nq(Rk) withq=2(Q+k)\nQ+k−2whereQ= 2n+ 2. We see\nthat2(γ+k\n2)\nγ+k\n2−1=qifγ=Q\n2. Note also that, since Q≥4, the assumption γ≥1\n2is\nsatisfied. Therefore in our application we have\nCGN\nq(Rk)≥/parenleftBigg/parenleftbiggkπ\n2/parenrightbiggk\n2Γ(Q+1\n2)\nΓ(Q+k+1\n2)(Q+k−1\n2)Q+k−1\n2\n(Q−1\n2)Q−1\n2/parenrightBigg2\nQ+k\n. (9.14)\nWe will deduce our main result from this bound, Corollary 9.4, the know n facts\nabout ˜γnand the monotonicity properties in the following lemma.\nLemma 9.7. For eachQ≥4,\nk/ma√sto→22kΓ(Q+k+1\n2)Γ(Q+k+2\n2)\n(Q+k−1)Q+k−1\n2(Q+k)Q+k\n2\nis decreasing for k≥0.\nProof.Denoting\nαk:= 22kΓ(Q+k+1\n2)Γ(Q+k+2\n2)\n(Q+k−1)Q+k−1\n2(Q+k)Q+k\n2,\nwe have\nαk\nαk−1= 2(Q+k−2)Q+k−2\n2\n(Q+k)Q+k−2\n2,\nwhich we can write as 2(1 −x−1\n0)x0−1withx0=Q+k\n2. We will prove momentarily that\nx/ma√sto→(1−x−1)x−1is decreasing for x≥1. Since it is equal to 1 /2 atx= 2 and since\nx0>2 (provided k−1≥0≥4−Q), we deduce that 2(1 −x−1\n0)x0−1<1, proving that\nαk/αk−1<1.\nIt remains to prove that x/ma√sto→(1−x−1)x−1is decreasing for x≥1 or, what is the\nsame,x/ma√sto→(x−1)ln(1−x−1) is decreasing for x≥1. This follows since the derivative\nof the latter function is ln(1 −x−1)+x−1≤0 forx>1. /square\nWe finally prove the main result of this subsection.\nProof of Proposition 9.5. Inserting the bound (9.14) into Corollary 9.4 we obtain\nγ(Hn×Rk)≤22kΓ(Q+k+1\n2)Γ(Q+k+2\n2)\nΓ(Q+1\n2)Γ(Q+2\n2)(Q−1)Q−1\n2QQ\n2\n(Q+k−1)Q+k−1\n2(Q+k)Q+k\n2˜γn.(9.15)PLEIJEL THEOREM FOR SUB-LAPLACIANS 43\nFirst consider n≥4. According to Lemma 9.7, the right side of (9.15) is decreasing\nwith respect to k. Fork= 0 it is equal to ˜ γn. Since we know that ˜ γn<1 forn≥4,\nwe learn that γ(Hn×Rk)<1 for alln≥4 andk≥0.\nNow letn= 3. Again by Lemma 9.7, the right side of (9.15) is decreasing with\nrespect tok. Fork= 1 it is given by\n4Γ(11\n2)\nΓ(9\n2)77\n2\n99\n2˜γ3= 2/parenleftbigg7\n9/parenrightbigg7\n2\n˜γ3≈8.872·10−1<1.\nHere we used the value ˜ γ3≈1.069 from Subsection 9.2. This proves that γ(H3×Rk)<\n1 for allk≥1. This completes the proof. /square\n10.Continuation by looking at a direct product\nThis section is a short aside from the main topic of this part. We will exp lain that\nmost of the arguments in the previous sections generalize from Hn×Rkto the case of\nG=G1×G2,\nwhereG1andG2are stratified, nilpotent Lie groups. If ∆Gjis the sub-Laplacian on\nGj, then\n∆G= ∆G1⊗I+I⊗∆G2\nis the sub-Laplacian on G.\n10.1.Spectral density.\nIfQ1andQ2denote the homogeneous dimensions of G1andG2, respectively, then\nQ:=Q1+Q2\nis the homogeneous dimension of G. We recall that for G1andG2we have the\nanalogues of formula (8.2)/BD(−∆G1<µ)(ζ1,ζ1) =W(G1)µQ1/2for allζ1∈G1\nand/BD(−∆G2<µ)(ζ2,ζ2) =W(G2)µQ2/2for allζ2∈G2.,\nWe have the same formula for G/BD(−∆G1×G2<µ)(ζ,ζ) =W(G)µQ/2for allζ∈G (10.1)\nand the same argument as in Proposition 8.1 shows that\nW(G) :=W(G1)W(G2)Γ(Q1\n2+1)Γ(Q2\n2+1)\nΓ(Q\n2+1). (10.2)\n10.2.Sobolev inequality.\nWe now discuss the Faber–Krahn part of Pleijel’s proof. In fact, as in the previous\nsection it will be more convenient to work with the Sobolev inequality ra ther than the\nFaber–Krahn inequality. Thus, in this subsection we try to get a goo d lower bound44 RUPERT L. FRANK AND BERNARD HELFFER\non the Sobolev constant CSob(G) in\n/integraldisplay/integraldisplay\nG1×G2|∇Gu|2dζ1dζ2≥CSob(G)/parenleftbigg/integraldisplay/integraldisplay\nG1×G2|u|qdζ1dζ2/parenrightbigg2\nq\n,\nwhere we abbreviate\nq=2(Q1+Q2)\nQ1+Q2−2=2Q\nQ−2.\nFor the validity of the inequality we require Q≥3.\nWe use the Sobolev inequality on G1(assumingQ1≥3),\n/integraldisplay\nG1|∇G1u|2dζ1≥CSob(G1)/parenleftbigg/integraldisplay\nG1|u|2Q1\nQ1−2dζ1/parenrightbiggQ1−2\nQ1,\nas well as the following Sobolev interpolation inequality on G2,\n/parenleftbigg/integraldisplay\nG2|∇G2u|2dζ2/parenrightbiggθ/parenleftbigg/integraldisplay\nG2|u|2dζ2/parenrightbigg1−θ\n≥CGN\nq(G2)/parenleftbigg/integraldisplay\nG2|u|qdζ2/parenrightbigg2\nq\n,\nwhere the parameters qandθare related by\n1\nq=1−θ\n2+θQ2−2\n2Q2.\nWhenQ2≥3, this inequality holds in the range θ∈[0,1], corresponding to q∈\n[2,2Q2\nQ2−2], while ifQ2= 1,2, it holds in the range θ∈[0,Q2\n2), corresponding to q∈\n[2,∞).\nThese inequalities are known to hold; see, for example [28, 80] for t he case of di-\nmensionQ1,Q2≥3. We do not know of an explicit reference for the inequality in case\nQ2= 1,2, but it can easily be deduced from existing results in the literature. One\nway is to proceed as in the proof of Lemma 4.2 by ‘artificially’ adding a fa ctor ofRk\n(with its additive group structure) and applying the result in dimensio n≥3. Another\nway is to apply [6, Corollary 3.6], which reduces the validity of the Sobo lev inequality\nto the validity of another inequality. In the present case we can use [6, Theorem 8.1]\nin view of the validity of the isoperimetric inequality on G2; see, for instance, [36,\nProposition 1.17 and the following remarks].\nWe will apply the Sobolev interpolation inequality with the parameter\nθ12=Q2\nQ,1\nq=1\n2−1\nQ.\nNote that this choice satisfies θ12∈[0,1] ifQ2≥3 andθ12∈[0,Q2\n2) ifQ2= 1,2 (recall\nthat we assume Q1≥3), so the Sobolev interpolation inequality is indeed valid.\nProceeding in the same way as in the proof of Proposition 9.3, we obta in the fol-\nlowing bound for the Sobolev inequality on G:\nCSob(G)≥CGN\nq(G2)/parenleftBig\nCSob(G1)/parenrightBigQ1/QQ\nQQ1/Q\n1QQ2/Q\n2, (10.3)PLEIJEL THEOREM FOR SUB-LAPLACIANS 45\nwhere we recall that qis determined according to1\nq=1\n2−1\nQ.\nWhen both Q1≥3 andQ2≥3, we can apply H¨ older’s inequality to deduce that\nCGN\nq(G2)≥/parenleftBig\nCSob(G1)/parenrightBigQ1/Q\nand obtain the more symmetric bound\nCSob(G)≥/parenleftBig\nCSob(G1)/parenrightBigQ1/Q/parenleftBig\nCSob(G2)/parenrightBigQ2/QQ\nQQ1/Q\n1QQ2/Q\n2.(10.4)\n10.3.Pleijel’s Theorem.\nThe proof of Pleijel’s bound in Theorem 7.1 extends with obvious chang es to the\ncase of general stratified nilpotent groups. Bounding the Faber– Krahn constant that\nappears in this bound by the corresponding Sobolev constant as in L emma 9.1, we\narrive at a Pleijel-type bound with constant\n˜γ(G) :=/parenleftBig\nCSob(G)/parenrightBig−Q\n2W(G)−1.\nIn particular, when both Q1≥3 andQ2≥3, we can insert formula (10.2) and\ninequality (10.4) and arrive at the bound\n˜γ(G)≤˜γ(G1)˜γ(G2)QQ1\n2\n1QQ2\n2\n2\nQQ\n2Γ(Q\n2+1)\nΓ(Q1\n2+1)Γ(Q2\n2+1). (10.5)\nSpecializing to the case G2=Rkwe can get\nTheorem 10.1. LetG1a stratified nilpotent group of homogeneous dimension Q1≥3.\nThen there exists k0(G1)such that for any k≥k0(G1), Pleijel’s Theorem holds for\nthe Dirichlet realization in any domain Ω⊂G1×Rkof the canonical sub-Laplacian\nassociated with the group G1×Rk.\nMore generally, assume that G1a stratified nilpotent group of homogeneous di-\nmensionQ1≥3 and assume that G(k)\n2is a sequence of nilpotent groups such that\nthe homogeneous dimensions Q(k)\n2tend to∞and the corresponding constants ˜ γ(G(k)\n2)\ntend to 0. Then there exists k0such that for k≥k0we have\nγ(G1×G(k)\n2)<1.\nProof.According to Stirling’s approximation, we have\nln/parenleftbig\nx−x+1\n2Γ(x+2\n2)/parenrightbig\n=−x\n2ln(2e)−1\n2lnπ+O(x−1) asx→ ∞.(10.6)\nThis implies that, as k→+∞,\nQQ2/2\n2\nQQ/2Γ(Q/2+1)\nΓ(Q2/2+1)→(1\n2e)Q1/2.\nThe assertion then follows from (10.5) and the assumption ˜ γ(G(k)\n2)→0. /square46 RUPERT L. FRANK AND BERNARD HELFFER\n11.A second bound on the Faber–Krahn constant\n11.1.Preliminary discussion.\nWhile in the previous two sections we have used the Sobolev constant to get bounds\non the Faber–Krahn constant, we will use in this section the isoperim etric constant\nfor this purpose. In those previous two sections we have faced th e difficulty that the\noptimal Sobolev constant on Hn×Rkis not known when k≥1 and our main work\nconsisted in getting bounds on this constant. Similarly, the optimal is operimetric\nconstant on Hn×Rkis not known when k≥0 and we will try to get bounds on it.\nWe emphasize that the isoperimetric constant is even unknown in the case ofHn(that\nis,k= 0), despite a famous conjecture of Pansu [71] and several effo rts to prove it,\nwhich we cite below.\nIt is known that, in a relatively general setting, the validity of an isop erimetric\ninequality implies the validity of a Faber–Krahn inequality; see [12, 6, 8 0]. These\nworks also provide bounds on the Faber–Krahn constant in terms o f the isoperimetric\nconstant, but the numerical values in these bounds have not been the main concern in\nthese works and, as far as we know, they are not sufficient to dedu ce Pleijel’s theorem.\nIncontrast totheseworks, wewillusemethodsthataremorespe cifictotheproblemat\nhand and use the particularities of the Heisenberg group and of Euc lidean space. (For\ninstance, in Lemma 11.4 we use the form of the Green’s function on th e Heisenberg\ngroup, and in Theorem 11.8 we use the isoperimetric inequality on Euclid ean space.)\n11.2.Definitions.\nWe study the isoperimetric inequality in Hn×Rkfrom the point of view of the theory\nof sets of finite perimeter. For an introduction to this theory in the Euclidean setting\nwe recommend [57]. From this theory we recall, for instance, that th e perimeter of a\nmeasurable set E⊂Rkis defined as\nPerRkE:= sup/braceleftbigg/integraldisplay\nEdivϕdw:ϕ∈C1\nc(Rk,Rk),|ϕ| ≤1/bracerightbigg\nand that for bounded sets with C1(or even Lipschitz) boundary this coincides with\nthe standard surface area of the set; see [57, Examples 12.5 and 1 2.6]. The definition\nof sets of finite perimeter goes back to Caccioppoli and their theor y was developed by\nDe Giorgi.\nThere is a natural andwell-known extension ofthis theoryto these tting of stratified\nnilpotent groups, but we limit ourselves here to the case of Hn×Rk. In this case\n(horizontal) perimeter of a measurable set E⊂Hn×Rkis defined to be\nPerHn×Rk(E) :=sup/braceleftBigg/integraldisplay\nE/parenleftBiggn/summationdisplay\nj=1(Xjϕ+Yjϕ)+k/summationdisplay\ni=1Wiϕ/parenrightBigg\ndxdydzdw :\nϕ∈C1\nc(Hn×Rk,R2n+k),|ϕ| ≤1/bracerightbigg\n.PLEIJEL THEOREM FOR SUB-LAPLACIANS 47\nWe denote by I(Hn×Rk) the largest constant such that for every measurable set\nE⊂Hn×Rkof finite measure one has12\nPerHn×Rk(E)≥I(Hn×Rk)|E|2n+1+k\n2n+2+k.\nForbackgroundandfurtherdetailswereferto[34], aswell asthet extbook[11]treating\nthe model case of H1.\n11.3.Symmetrization.\nIn this subsection we show that a bound on the isoperimetric consta nt implies a bound\non the Faber–Krahn constant. Arguments in this spirit are known in the Riemannian\ncontext and appear, for instance in [8, Theorem I.1.5]. As usual, jν,1denotes the first\npositive zero of the Bessel function Jν. For the definition and properties of Bessel\nfunctions, we refer to [1].\nProposition 11.1. Forn≥0andk≥0, we have\nCFK(Hn×Rk)≥I(Hn×Rk)2(2n+2+k)−2j2\n2n+k\n2,1.\nThe proof shows that equality holds in the Faber–Krahn inequality wit h the value\ngiven by the right side, provided that every superlevel set of uattains equality in the\nisoperimetric inequality. This happens in the Euclidean setting where n= 0. In this\ncase, we have\nI(Rk) =kk−1\nk|Sk−1|1\nkandCFK(Rk) =k2k−1\nk|Sk−1|2\nkk−2j2\nk−2\n2,1=ω2\nk\nkj2\nk−2\n2,1,\n(11.1)\nwhereωkis the measure of the unit ball in Rk; see, for instance, [57, Theorem 14.1] or\n[79] for the first equality and [31, Theorem 2.54] or [54, Theorem 5.1.2 ] for the second\none.\nIncidentally, it follows from (11.1) that the inequality in Proposition 11 .1 can be\nstated as\nCFK(Hn×Rk)I(Hn×Rk)−2≥CFK(R2n+2+k)I(R2n+2+k)−2.(11.2)\nProof.We abbreviate D:= 2n+ 2 +k. Our goal is to prove inequality (7.3) for a\ngiven open set Ω ⊂Hn×Rkand a given function u∈C1\nc(Ω). (It will then extend to\nS1\n0(Ω) by density.) By truncation properties of Sobolev functions we m ay assume that\nu≥0. We define u∗to be a nonincreasing function on (0 ,∞) such that\n|{u>τ}|=/integraldisplay\n{u∗(r)>τ}rD−1drfor allτ >0.\n(We could multiply the right side by a constant, for instance |SD−1|, to make it look\nmore like RD, but this does not change the final outcome.) It follows from the lay er\n12More generally, for a nilpotent stratified group Gof homogeneous dimension Qthe isoperimetric\ninequality reads\nPerG(E)≥I(G)|E|Q−1\nQ.48 RUPERT L. FRANK AND BERNARD HELFFER\ncake formula that, for any q>0,/integraldisplay\nHn×Rkuqdxdydzdw =/integraldisplay∞\n0u∗(r)qrD−1dr. (11.3)\nWe are going to show that, for any p≥1,/integraldisplay\nHn×Rk|∇Hnu|pdxdydzdw ≥Ip\nn,kD−pD−1\nD/integraldisplay∞\n0(−u′\n∗(r))prD−1dr, (11.4)\nwhere, for short,\nIn,k:=I(Hn×Rk).\nTo prove this, we note that the coarea formula implies that\n|{u>τ}|=/integraldisplay∞\nτ/parenleftbigg/integraldisplay\n|∇Hn×Rku|−1d|∇Hn×Rk\n/BD{u>t}|/parenrightbigg\ndt. (11.5)\nLet us explain the notation used here. First, we abbreviated\n|∇Hn×Rku|=/radicaltp/radicalvertex/radicalvertex/radicalbtn/summationdisplay\nj=1((Xju)2+(Yju)2)+k/summationdisplay\ni=1(Wiu)2.\nNext, in the proof of the coarea formula one shows that the sets {u > t}have finite\nperimeter for almost every t, and|∇Hn×Rk\n/BD{u>t}|denotes the corresponding perimeter\nmeasure. To recall the definition of the latter, let us note that, by Riesz’s theorem,\nifEis a set of finite perimeter, then there is an R2n+k-valued Radon measure µEon\nHn×Rksuch that\n/integraldisplay\nE/parenleftBiggn/summationdisplay\nj=1(Xjϕ+Yjϕ)+k/summationdisplay\ni=1Wiϕ/parenrightBigg\ndxdydzdw =−/integraldisplay\nEϕ·dµE\nfor allϕ∈C1\nc(Hn×Rk,R2n+k). The perimeter measure |∇Hn×Rk /BDE|:=|µE|is the\ncorresponding total variation measure. Note that, in particular, we have\nPerHn×RkE=/integraldisplay\nd|∇Hn×Rk\n/BDE|.\n(Whenever {u=t}is a sufficiently smooth hypersurface the perimeter measure\n|∇Hn×Rk /BD{u>t}|is concentrated on {u=t}.)\nFor a textbook proof of the coarea formula in the Euclidean case we refer to [57,\nTheorem 13.1]. The corresponding formula in the case of stratified n ilpotent groups\nappears in [34, Theorem 5.2] and [30, Corollary 2.3.5].\nIt follows from (11.5) that\nd\ndτ|{u>τ}|=−/integraldisplay\n|∇Hn×Rku|−1d|∇Hn×Rk /BD{u>τ}|.PLEIJEL THEOREM FOR SUB-LAPLACIANS 49\nBy the assumed isoperimetric inequality and H¨ older’s inequality with1\np+1\np′= 1,\n(in fact, only p= 2 will be relevant), we have\nIn,k|{u>τ}|D−1\nD≤PerHn×Rk{u>τ}=/integraldisplay\nd|∇Hn×Rk\n/BD{u>τ}|\n≤/parenleftbigg/integraldisplay\n|∇Hn×Rku|−1d|∇Hn×Rk /BD{u>τ}|/parenrightbigg1\np′\n×/parenleftbigg/integraldisplay\n|∇Hn×Rku|p−1d|∇Hn×Rk\n/BD{u>τ}|/parenrightbigg1\np\n.\nThe previous two relations combined give\n/integraldisplay\n|∇Hnu|p−1d|∇Hn\n/BD{u>τ}| ≥Ip\nn,k|{u>τ}|pQ−1\nQ/parenleftbigg\n−d\ndτ|{u>τ}|/parenrightbigg−p+1\n.\nBy the coarea formula again, we deduce that/integraldisplay\nHn×Rk|∇Hnu|pdxdydzdw =/integraldisplay∞\n0/integraldisplay\n|∇Hn×Rku|p−1d|∇Hn×Rk /BD{u>τ}|dτ\n≥Ip\nn,k/integraldisplay∞\n0|{u>τ}|pD−1\nD/parenleftbigg\n−d\ndτ|{u>τ}|/parenrightbigg−p+1\ndτ.\nNote that the right side depends on uonly through the function τ/ma√sto→ |{u>τ}|. We\nexpress it through the function u∗.\nFor the sake of simplicity we assume that u∗is strictly monotone. The general case\ncan be treated as well, but the idea becomes clearer in this special ca se. The general\ncan be handled along the lines of [53, Theorem 15.29]. Under the strict monotonicity\nassumption we can then define a function [0 ,/ba∇dblu/ba∇dbl∞]∋τ/ma√sto→Rτ∈(0,∞) by\nu∗(Rτ) =τ.\nIt follows that\n|{u>τ}|=/integraldisplay\n{u∗(r)>τ}rD−1dr=D−1RD\nτ.\nMoreover, one can show that, if uis weakly differentiable, then u∗is absolutely con-\ntinuous (an argument of this type in the Euclidean case can be found in [53, Theo-\nrem 15.23]) and we have\nu′\n∗(Rτ)˙Rτ= 1\nwhere the dot denotes differentiation with respect to τ. It follows that\n/integraldisplay∞\n0|{u>τ}|pQ−1\nQ/parenleftbigg\n−d\ndτ|{u>τ}|/parenrightbigg−p+1\ndτ=D−pD−1\nD/integraldisplay∞\n0RD−1\nτ(−u′\n∗(Rτ))p˙Rτdτ\n=D−pD−1\nD/integraldisplay∞\n0rD−1(−u′\n∗(r))pdr.\nThus, we have completed the proof of (11.4).50 RUPERT L. FRANK AND BERNARD HELFFER\nAtthispointwespecializetothecase p= 2andrecalltheone-dimensionalinequality\n/integraldisplayR\n0U′(r)2rD−1dr≥j2\nD−2\n2,1R−2/integraldisplayR\n0U(r)2rD−1dr,\nvalid for all R >0 and all absolutely continuous functions Uon (0,R) withU(R) =\n0. This follows from the Faber–Krahn inequality in RDwhen restricted to radial\nfunctions; see, e.g., [31, Theorem2.54]. Weapplytheaboveone-dime nsional inequality\ntoU=u∗andR=R0. If we insert\n|Ω|=|{u>0}|=D−1RD\n0,\nwe obtain the claimed inequality. /square\n11.4.Pansu’s conjecture and its consequences.\nThe validity of the isoperimetric inequality in H=H1(that is, the fact that I(H)>0)\nis due to Pansu [70]. In [71] Pansu made a conjecture about the set t hat realizes the\noptimal constant I(H). In the references given below this conjecture is generalized to\nHnwithn≥2 and, after a computation, we find\nI(Hn) =2n\n2n+1(2n+2)2n+1\n2n+2Γ(2n+3\n2)1\n2n+2π2n+1\n2(2n+2)21\nn+1\nΓ(n+1)1\nn+1(Pansu’s conjecture);\n(11.6)\nsee, e.g., [23]. (The extra factor of 21\nn+1compared to their formula comes from the\nfact that they use ∂xj+1\n2yj∂zetc., while we use ∂xj+2yj∂zetc.)\nPansu’s conjecture is generally believed to be true and has been ver ified under\ncertain additional assumptions [75, 23, 76, 74, 65, 67]; see also [52, 51, 13, 14] for a\nsampleofcontributionstothisproblem, aswellasthetextbook[11 ]foranintroduction\nto the isoperimetric problem on the Heisenberg group.\nWe recall that in Subsection 9.2 we have proved Pleijel’s theorem in Hnwhenn≥4.\nNow we will prove it in the remaining dimensions, assuming the validity of P ansu’s\nconjecture.\nCorollary 11.2. Assume that Pansu’s conjecture holds for some n∈ {1,2,3}. Then\nthe Pleijel constant γ(Hn)satisfies for the corresponding value of n\nγ(H)≤0.406114, γ(H2)≤0.155327, γ(H3)≤0.0641172.(11.7)\nProof.We abbreviate Q= 2n+2. The conjectured value for I(Hn) given above leads,\nvia Proposition 11.1, to the bound on the Faber–Krahn constant\nCFK(Hn)≥Q2Q−1\nQ(Q−2)2Γ(Q+1\n2)2\nQπQ−1\nQ42\nQ\n(Q−1)2Γ(Q\n2)4\nQQ−2j2\nQ−2\n2,1\n=(Q−2)2Γ(Q+1\n2)2\nQπQ−1\nQ42\nQ\nQ2\nQ(Q−1)2Γ(Q\n2)4\nQj2\nQ−2\n2,1.PLEIJEL THEOREM FOR SUB-LAPLACIANS 51\nThis leads to the bound on the Pleijel constant\nγ(Hn)≤(CFK(Hn))−Q\n2W(Hn)−1=Q(Q−1)QΓ(Q\n2)2\n(Q−2)QΓ(Q+1\n2)πQ−1\n24j−Q\nQ−2\n2,1W(Hn)−1.\nUsing Subsection 8.1, which gives\nW(H) =1\n128,W(H2) =1\n482π,W(H3) =1\n27·76812−π2\nπ2,\nand the values [1, Table 9.5]\nj1,1∼3.8317, j 2,1∼5.1356, j 3,1∼6.3802,\nwe get (11.7) and the corollary. /square\n11.5.A bound on the isoperimetric constant in Hn.\nIn this subsection we prove a lower bound on I(Hn) which is reasonably good in small\ndimensions n= 1 andn= 2. To state this bound, we need two constants,\nCn:=2n−3nΓ(n\n2)2\nπn+1(11.8)\nand, setting Q= 2n+2,\nC′\nn:=Q1\nQ/parenleftBigg\nπnΓ(2Q−1\n2(Q−1))\nΓ(2Q−1\n2(Q−1)+1\n2)Γ(n\n2+Q\n4(Q−1))\nΓ(n\n2+Q\n4(Q−1)+1\n2)Γ(1+Q\n2(Q−1))\nΓ(n+Q\n2(Q−1))/parenrightBiggQ−1\nQ\n.(11.9)\nForn= 1 we can write the latter constant without Gamma functions, see ( 11.13)\nbelow. In any dimension it can easily be evaluated numerically.\nThese two constants appear in our bound on the isoperimetric cons tant. More\nprecisely, we will prove the following theorem.\nTheorem 11.3. For anyn∈N,\nI(Hn)≥ C−1\nn(C′\nn)−1.\nWe will prove that for any measurable set E⊂Hn/integraldisplay\nEu(ζ)dζ≤ CnC′\nn|E|1\nQ/integraldisplay\nHn|∇Hnu(ζ)|dζ. (11.10)\nHere and below, we denote variables in Hnbyζand we write\n|∇Hnu(ζ)|:=/radicaltp/radicalvertex/radicalvertex/radicalbtn/summationdisplay\nj=1((Xju)(ζ)2+(Yju)(ζ)2).\nIf we prove this inequality for, say, Lipschitz functions uwith compact support, then,\nby a standard approximation argument, it will extend, with the same constant, to\nall functions gof bounded variation satisfying |{|u|> λ}|<∞for allλ >0. (The\nbounded variation condition is understood in the Heisenberg sense, as discussed, for\ninstance, in [11, Section 5.1].) In particular (11.10) is valid for the char acteristic52 RUPERT L. FRANK AND BERNARD HELFFER\nfunction of a set of finite (horizontal) perimeter. Since\n/integraldisplay\nHn|∇Hn\n/BDE(ζ)|dζ= PerHnEand/integraldisplay\nE\n/BDE(ζ)dζ=|E|,\nwe obtain the inequality stated in the theorem. Thus, it remains to pr ove (11.10).\nOur analysis is based on the following well-known representation form ula, where we\nuse the notation\n/ba∇dbl(x,y,z)/ba∇dbl= ((|x|2+|y|2)2+z2)1\n4for (x,y,z)∈Hn.\nLemma 11.4. WithCnfrom(11.8),\nu(ζ) =Cnn/summationdisplay\nj=1/integraldisplay\nHn(Xj/ba∇dbl·/ba∇dbl)(ζ−1η)(Xju)(η)+(Yj/ba∇dbl·/ba∇dbl)(ζ−1η)(Yju)(η)\n/ba∇dblζ−1η/ba∇dblQ−1dη.\nProof.As shown by Folland [27],\nu(ζ) =−˜Cnn/summationdisplay\nj=1/integraldisplay\nHn/ba∇dblζ−1η/ba∇dbl2−Q((X2\nj+Y2\nj)u)(η)dη\nwith\n˜Cn=2n−4Γ(n\n2)2\nπn+1.\nHereζ−1ηdenotes the inverse of ζcomposed with ηin the sense of the Heisenberg\ngroup. The explicit expression for ˜Cncan be found, for instance, in [10, Eq. between\n(1.2) and (1.3)] or [17, Theorem 1.2]. (Meanwhile, there seems to be a computational\nerror in the constant in [11, Theorem 5.15].)\nIntegrating by parts, we find\n−/integraldisplay\nHn/ba∇dblζ−1η/ba∇dbl2−Q((X2\nj+Y2\nj)u)(η)dη\n=/integraldisplay\nHn/parenleftbig\n(Xj/ba∇dblζ−1η/ba∇dbl2−Q)(Xju)(η)+(Yj/ba∇dblζ−1η/ba∇dbl2−Q)(Yju)(η)/parenrightbig\ndη.\nOn the right side, the vector fields XjandYjact with respect to the ηvariable. By\nleft invariance, we find\nXj/ba∇dblζ−1η/ba∇dbl2−Q= (Xj/ba∇dbl·/ba∇dbl2−Q)(ζ−1η) = (2−Q)/ba∇dblζ−1η/ba∇dbl1−Q(Xj/ba∇dbl·/ba∇dbl)(ζ−1η)\nand similarly for Yj. This proves the claimed formula. /square\nWe deduce from Lemma 11.4 that for any measurable set E⊂Hn,\n/integraldisplay\nEu(ζ)dζ=Cn/integraldisplay\nHnn/summationdisplay\nj=1/integraldisplay\nE(Xj/ba∇dbl·/ba∇dbl)(ζ−1η)(Xju)(η)+(Yj/ba∇dbl·/ba∇dbl)(ζ−1η)(Yju)(η)\n/ba∇dblζ−1η/ba∇dblQ−1dζdη.\nHere we used Fubini’s theorem. The arguments given in Lemma 11.5 belo w show that\nthe integrals are absolutely convergent, so the interchange of int egrals is justified.PLEIJEL THEOREM FOR SUB-LAPLACIANS 53\nOur goal is to bound\nn/summationdisplay\nj=1/integraldisplay\nE(Xj/ba∇dbl·/ba∇dbl)(ζ−1η)(Xju)(η)+(Yj/ba∇dbl·/ba∇dbl)(ζ−1η)(Yju)(η)\n/ba∇dblζ−1η/ba∇dblQ−1dζ\nfrom above, for any fixed η.\nAbbreviating aj:= (Xju)(η),bj:= (Yju)(η) andF:=E−1η={ζ−1η:ζ∈E}, we\nneed to boundn/summationdisplay\nj=1/integraldisplay\nFaj(Xj/ba∇dbl·/ba∇dbl)(ξ)+bj(Yj/ba∇dbl·/ba∇dbl)(ξ)\n/ba∇dblξ/ba∇dblQ−1dξ\nfrom above. Note that\n|F|=|E−1η|and|a|2+|b|2=|∇Hng(v)|2.\nLemma 11.5. For anya,b∈Rnand any measurable F⊂Hn,\nn/summationdisplay\nj=1/integraldisplay\nFaj(Xj/ba∇dbl·/ba∇dbl)(ξ)+bj(Yj/ba∇dbl·/ba∇dbl)(ξ)\n/ba∇dblξ/ba∇dblQ−1dξ≤ C′\nn|F|1\nQ(|a|2+|b|2)1\n2\nwithC′\nnfrom(11.9).\nLet us accept this lemma for themoment anduse itto complete thepr oofof (11.10).\nAccording to the above argument, we find, for any η∈Hn,\nn/summationdisplay\nj=1/integraldisplay\nE(Xj/ba∇dbl·/ba∇dbl)(ζ−1η)(Xju)(η)+(Yj/ba∇dbl·/ba∇dbl)(ζ−1η)(Yju)(η)\n/ba∇dblζ−1η/ba∇dblQ−1dζ≤ C′\nn|E|1\nQ|∇Hnu(η)|.\nConsequently, we have shown that/integraldisplay\nEu(ζ)dζ≤ CnC′\nn|E|1\nQ/integraldisplay\nHn|∇Hnu(η)|dη.\nThis is the claimed inequality (11.10). It remains to prove Lemma 11.5.\nProof of Lemma 11.5. Let us abbreviate\nI[F] :=n/summationdisplay\nj=1/integraldisplay\nFaj(Xj/ba∇dbl·/ba∇dbl)(ξ)+bj(Yj/ba∇dbl·/ba∇dbl)(ξ)\n/ba∇dblξ/ba∇dblQ−1dξ.\nWe think of the inequality in the lemma as an optimization problem, where we want\nto maximize I[F] among all sets Fof a given measure. By homogeneity the value of\nthis fixed measure is irrelevant.\nWe know by the bathtub principle (see, for instance, [55, Theorem 1.14]) that there\nis an optimal set F∗for the inequality and that this optimal set is given by\nF∗=/braceleftBig\nξ∈Hn:n/summationdisplay\nj=1aj(Xj/ba∇dbl·/ba∇dbl)(ξ)+bj(Yj/ba∇dbl·/ba∇dbl)(ξ)\n/ba∇dblξ/ba∇dblQ−1>κ/bracerightBig\nfor someκ>0.\nIn fact, as already mentioned, by homogeneity, the value of κis immaterial and we54 RUPERT L. FRANK AND BERNARD HELFFER\nwill set it equal to 1 in what follows. Thus, we have, for any set F⊂Hn,\nI[F]\n|F|1\nQ≤I[F∗]\n|F∗|1\nQ,\nand our task is to compute the right side. More precisely, we want to show that\nI[F∗]\n|F∗|1\nQ=C′\nn(|a|2+|b|2)1\n2\nwithC′\nnfrom (11.9).\nWebeginbybringing I[F]inamoreexplicitform. Wecompute, writing ξ= (x,y,t),\n(Xj/ba∇dbl·/ba∇dbl)(ξ) =/ba∇dblξ/ba∇dbl−3/parenleftbig\n(|x|2+|y|2)xj+yjt/parenrightbig\n,(Yj/ba∇dbl·/ba∇dbl)(ξ) =/ba∇dblξ/ba∇dbl−3/parenleftbig\n(|x|2+|y|2)yj−xjt/parenrightbig\n,\nso\naj(Xj/ba∇dbl·/ba∇dbl)(ξ)+bj(Yj/ba∇dbl·/ba∇dbl)(ξ)\n/ba∇dblξ/ba∇dblQ−1\n=(|x|2+|y|2)(a·x)+(a·y)t+(|x|2+|y|2)(b·y)−(b·x)t\n/ba∇dblξ/ba∇dblQ+2.\nIn particular,\nI[F] =/integraldisplay\nF(|x|2+|y|2)(a·x)+(a·y)t+(|x|2+|y|2)(b·y)−(b·x)t\n/ba∇dblξ/ba∇dblQ+2dξ\nand\nF∗=/braceleftbigg\nξ∈Hn:(|x|2+|y|2)(a·x)+(a·y)t+(|x|2+|y|2)(b·y)−(b·x)t\n/ba∇dblξ/ba∇dblQ+2>1/bracerightbigg\n.\nWe claim that |F∗|andI[F∗] depend on aandbonly through |a|2+|b|2. Indeed,\nfrom the above expression it is obvious that the two quantities do no t change ifaand\nbare simultaneously rotated by the same rotation matrix. This implies t hat the two\nquantities depend on aandbonly through |a|,|b|anda·b. We also see that the\ntwo quantities do not change if we apply a two-dimensional rotation t o (aj,bj). This\nimplies that the two quantities depend on ajandbjonly through a2\nj+b2\nj. This proves\nthe claim.\nAs a consequence, we may set α:=/radicalbig\n|a|2+|b|2and assume that a=αenandb= 0.\nThus, we have\nF∗=/braceleftBig\nξ∈Hn:α(|x|2+|y|2)xn+ynt\n/ba∇dblξ/ba∇dblQ+2>1/bracerightBig\nand we need to compute\n|F∗|andI[F∗] =/integraldisplay\nF∗α(|x|2+|y|2)xn+ynt\n/ba∇dblξ/ba∇dblQ+2dξ.\nTo perform these computations, we introduce coordinates\nxj=r√\nsinθcosϕjωj, y j=r√\nsinθsinϕjωj, t=r2cosθ,PLEIJEL THEOREM FOR SUB-LAPLACIANS 55\nwherer>0,θ∈(0,π),ϕ= (ϕ1,...,ϕ n)∈(−π,π)n=:Tnandω∈Sn−1∩(0,∞)n=:\nΣ. We claim that in this parametrization, the measure is given by\ndxdydt=rQ−1drsinn−1θdθdµ(ω)dϕ, (11.11)\nwheredϕis standard Lebesgue measure on Tnand where dµ(ω) =ω1···ωndωwith\nthe standard surface measure dωonSn−1. More explicitly, if we parametrize ω∈Σ\nby\nω1= sinθn−1sinθn−2···sinθ1,\nω2= sinθn−1sinθn−2···cosθ1,\n...\nωn−1= sinθn−1cosθn−2,\nωn= cosθn−1,\nwith 0<θj<π\n2, then\ndµ(ω) =ω1···ωndω=n−1/productdisplay\nj=1sin2j−1θjcosθjdθj.\nThe latter formula follows immediately from dω=/producttextn−1\nj=1sinj−1θjdθj.\nWe point out that when n= 1, then Σ = {1}andµis trivial in the sense that\nµ({1}) = 1.\nLet us provide the details for formula (11.11). First, for each j, we introduce polar\ncoordinates\nxj=rjcosϕj, y j=rjsinϕj,\nand note that\ndxjdyj=rjdrjdϕj.\nNext, we consider ( r1,...,r n) as an element of Rnand introduce hyperspherical coor-\ndinates in Rn,\nrj=ρωj\nwithρ>0 andω= (ω1,...,ω n)∈Sn−1∩(0,∞)n= Σ. We have\ndr1···drn=ρn−1dρdω,\nso\ndxdy=ρ2n−1dρω1···ωndω=ρ2n−1dρdµ(ω).\nFinally, we set\nρ=r√\nsinθ, t=r2cosθ\nand compute easily\ndρdt=r2(sinθ)−1\n2drdθ.\nInserting this into the above formual for dxdy, we obtain the claimed formula (11.11).56 RUPERT L. FRANK AND BERNARD HELFFER\nIn these coordinates we have\n(|x|2+|y|2)xn+ynt\n/ba∇dblξ/ba∇dblQ+2=√\nsinθ ωnsinθcosϕn+cosθsinϕ\nrQ−1=√\nsinθ ωnsin(θ+ϕn)\nrQ−1.\nIn particular, the constraint α(|x|2+|y|2)xn+ynt\n/bardblξ/bardblQ+2>1 can be written as\nr</parenleftBig\nα√\nsinθωnsin+(θ+ϕn)/parenrightBig1\nQ−1.\nThis allows us to carry out the r-integration and to obtain\n|F∗|=/integraldisplay\nTndϕ/integraldisplay\nΣdµ(ω)/integraldisplayπ\n0dθsinn−1θ/integraldisplay∞\n0drrQ−1/BDF\n=αQ\nQ−11\nQ/integraldisplay\nTndϕ/integraldisplay\nΣdµ(ω)/integraldisplayπ\n0dθsinn−1θ/parenleftBig√\nsinθωnsin+(θ+ϕn)/parenrightBigQ\nQ−1\nand\nI[F∗] =α/integraldisplay\nTndϕ/integraldisplay\nΣdµ(ω)/integraldisplayπ\n0dθsinn−1θ/integraldisplay∞\n0drrQ−1/BDF√\nsinθ��nsin(θ+ϕn)\nrQ−1\n=α/integraldisplay\nTndϕ/integraldisplay\nΣdµ(ω)/integraldisplayπ\n0dθsinn−1θ\n×√\nsinθωnsin(θ+ϕn)/parenleftBig\nα√\nsinθωnsin+(θ+ϕn)/parenrightBig1\nQ−1\n=αQ\nQ−1/integraldisplay\nTndϕ/integraldisplay\nΣdµ(ω)/integraldisplayπ\n0dθsinn−1θ/parenleftBig√\nsinθωnsin+(θ+ϕn)/parenrightBigQ\nQ−1.\nThus,\nI[F∗]\n|F∗|1\nQ=αQ1\nQ/parenleftbigg/integraldisplay\nTndϕ/integraldisplay\nΣdµ(ω)/integraldisplayπ\n0dθsinn−1θ/parenleftBig√\nsinθωnsin+(θ+ϕn)/parenrightBigQ\nQ−1/parenrightbiggQ−1\nQ\n.\nItremainstocomputetheintegralontherightside. Theintegralo ver (ϕ1,...,ϕ n−1)\ncan be computed immediately. Moreover, for fixed θ, we can compute the integral over\nϕnby translation invariance. We obtain\n/integraldisplay\nTndϕ/integraldisplay\nΣdµ(ω)/integraldisplayπ\n0dθsinn−1θ/parenleftBig√\nsinθωnsin+(θ+ϕn)/parenrightBigQ\nQ−1\n= (2π)n−1/integraldisplayπ\n0dϕsinQ\nQ−1ϕ/integraldisplayπ\n0dθsinn−1+Q\n2(Q−1)θ/integraldisplay\nΣdµ(ω)ωQ\nQ−1\nn\nUsing the beta function identity\n2/integraldisplayπ\n2\n0dt(sin2z1−1t)(cos2z2−1t) =B(z1,z2) =Γ(z1)Γ(z2)\nΓ(z1+z2),\nwe compute/integraldisplayπ\n0dϕ(sinQ\nQ−1ϕ) =√πΓ(2Q−1\n2(Q−1))\nΓ(2Q−1\n2(Q−1)+1\n2),PLEIJEL THEOREM FOR SUB-LAPLACIANS 57\n/integraldisplayπ\n0dθ(sinn−1+Q\n2(Q−1)θ) =√πΓ(n\n2+Q\n4(Q−1))\nΓ(n\n2+Q\n4(Q−1)+1\n2)\nand, whenn≥2,\n/integraldisplay\nΣdµ(ω)ωQ\nQ−1\nn=/parenleftBiggn−2/productdisplay\nj=1/integraldisplayπ\n2\n0dθjsin2j−1θjcosθj/parenrightBigg/integraldisplayπ\n2\n0dθn−1sin2n−3θn−1cos1+Q\nQ−1θn−1\n=/parenleftBiggn−2/productdisplay\nj=11\n2Γ(j)\nΓ(j+1)/parenrightBigg\n1\n2Γ(n−1)Γ(1+Q\n2(Q−1))\nΓ(n+Q\n2(Q−1))\n= 2−n+1Γ(1+Q\n2(Q−1))\nΓ(n+Q\n2(Q−1)).\nForn= 1 the same formula remains valid, for in this case the integral is trivia lly equal\nto one. Thus, we have shown that\nI[F∗]\n|F∗|1\nQ=αQ1\nQ/parenleftBigg\nπnΓ(2Q−1\n2(Q−1))\nΓ(2Q−1\n2(Q−1)+1\n2)Γ(n\n2+Q\n4(Q−1))\nΓ(n\n2+Q\n4(Q−1)+1\n2)Γ(1+Q\n2(Q−1))\nΓ(n+Q\n2(Q−1))/parenrightBiggQ−1\nQ\n=αC′\nn.\nThis completes the proof of the lemma. /square\nAt this point the proof of Theorem 11.3 is complete. We now bring the lo wer bound\nin the theorem in a more explicit form in dimensions n= 1 andn= 2.\nCorollary 11.6. Forn= 1,2one has\nI(H)≥8·3−9\n8π1\n4andI(H2)≥π4\n36−1\n6/parenleftbiggΓ(13\n5)\nΓ(11\n10)Γ(9\n5)\nΓ(13\n10)/parenrightbigg5\n6\n.\nLet us discuss the value of the lower bound of I(H). Numerically, one has\n8·3−9\n8π1\n4≈3.09468.\nThis should be compared with Pansu’s conjecture (11.6), which gives the value\n25\n23−3\n4π1\n2≈4.39854. (11.12)\nThus, our value is still quite a bit away from the conjectured sharp v alue. On the\nother hand, it improves over Pansu’s original value\n(8π/3)1\n4≈1.7013.\nSee [70] and also [11, (7.15)]. (The discrepancy to the latter formula by a factor of 41\n4\ncomes from the different normalization of the vector fields used in th at reference.)\nProof.The claimed formula for n= 1 follows from Theorem 11.3, the value C1=\n(4π)−1and\nC′\n1= 2−139\n8π3\n4. (11.13)58 RUPERT L. FRANK AND BERNARD HELFFER\nTo prove the latter formula, we note that, since Q= 4,\nC′\n1= 41\n4/parenleftbigg\nπΓ(7\n6)Γ(5\n6)\nΓ(5\n3)Γ(4\n3)/parenrightbigg3\n4\n.\nBy the reflection formula for the gamma function we deduce\nΓ(7\n6)Γ(5\n6)\nΓ(5\n3)Γ(4\n3)=1\n6Γ(1\n6)Γ(5\n6)\n2\n3Γ(2\n3)1\n3Γ(1\n3)=3\n4Γ(1\n6)Γ(5\n6)\nΓ(1\n3)Γ(2\n3)=3\n4π\nsinπ\n6\nπ\nsinπ\n3=3\n4sinπ\n6\nsinπ\n3=3\n4√\n3\n2\n1\n2=33\n2\n4.\nThis proves the claimed formula (11.13).\nThe formula for n= 2 follows directly from Theorem 11.3, using the definition of\nC2andC′\n2. /square\nRemark11.7.If one uses the lower bounds on I(Hn) from Corollary 11.6 for n= 1,2\nand inserts them into Proposition 11.1, one obtains a lower bound on CFK(Hn) that is\nbetter thanthebound given by Lemma 9.1 andtheJerison–Lee value (9.1). Explicitly,\nweobtainCFK(H)≥8.78829≈2π×1.3987andCFK(H2)≥17.9011≈27/3π×1.13064.\nThis leads to the bounds\nγ(H)≤1.65737 and γ(H2)≤1.26183,\nwhich are both unsatisfactory. We do see, however, that these v alues are better than\nthe bounds from (9.2), which are stated after Proposition 9.2, viz.\nγ(H)≤3.2423 and γ(H2)≤1.8238.\nWhenn= 3, thelower boundfromTheorem11.3, when insertedinto Proposit ion11.1,\ndoes not improve over the bound on CFK(H3) given by Lemma 9.1 and (9.1).\nDespite the negative results in the previous remark, we will see that Theorem 11.3\nwill be useful when dealing with Hn×Rkwithn= 1 andn= 2.\n11.6.A bound on the isoperimetric constant in Hn×Rk.\nWe recall that I(Hn×Rk) denotes the isoperimetric constant on Hn×Rk. Here we\nprove a lower bound on this constant in terms of I(Hn) (and the known isoperimetric\nconstants in Euclidean space).\nTheorem 11.8. For anyn,k∈N,\nI(Hn×Rk)≥I(R2n+2+k)/parenleftbiggI(Hn)\nI(R2n+2)/parenrightbigg2n+2\n2n+2+k\n.\nThe proof of the theorem relies on the following lemma that concerns functionsvof\nbounded variations; see, e.g., [53, Chapters 2 and 7]. By definition, t he distributional\nderivative of vis a measure. By Lebesgue’s decomposition theorem we can write this\nderivative as the sum of a singular measure Dv(s)and a measure that is absolutely\ncontinuous with respect to Lebesgue measure and whose density w e denote by v′.PLEIJEL THEOREM FOR SUB-LAPLACIANS 59\nLemma 11.9. LetN≥2and letv∈LN−2\nN−1(R)be a function of bounded variation.\nThen, for any β >0,\n/integraldisplay\nR/radicalBig\n(βvN−2\nN−1)2+(v′)2dt+|Dv(s)|(R)≥I(RN)/parenleftbiggβ\nI(RN−1)/parenrightbiggN−1\nN/parenleftbigg/integraldisplay\nR|v|dt/parenrightbiggN−1\nN\n.\nIfN= 2, we understand vN−2\nN−1as the characteristic function of {|v| /\\e}atio\\slash= 0}, and we\nunderstand the assumption v∈LN−2\nN−1(R) as requiring that this set has finite measure.\nIf in the first formula in (11.1) we replace |Sk−1|by 2πk\n2Γ(k\n2)−1, then we see that\nI(Rk) can be defined for any (not necessarily integer) real number k >0. With this\ndefinition, Lemma 11.9 remains valid for (not necessarily integer) N≥2. However,\nfor us only the case where Nis an integer case is relevant and we only provide the\nproofin this case. We note that by scaling, if the inequality holds for one valu e of\nβ >0, then it holds for any such value. Thus, we may assume that β=I(RN−1). In\nthis case, the lemma follows from the isoperimetric inequality in RNapplied to the\nset{(x′,xN)∈RN−1×R:ωN−1|x′|N−1< v(xN)}, whereωN−1is the volume of the\nunit ball in RN−1.\nProof of Theorem 11.8. LetE⊂Hn×Rkhave finite measure and finite perimeter and\nset\nv(t) :=|{ζ∈Hn×Rk−1: (ζ,t)∈E}|,\np(t) := Per Hn×Rk−1{ζ∈Hn×Rk−1: (ζ,t)∈E}.\nIt follows from Fubini’s theorem that vis an integrable function with/integraldisplay\nRv(t)dt=|E|.\nMoreover, one can show that vis of bounded variation and that pis integrable and\nthat\nPerHn×RkE≥/integraldisplay\nR/radicalbig\np2+(v′)2dt+|Dv(s)|(R).\n(This requires some work. Similar facts appear in the review paper [7 9] on the isoperi-\nmetric inequality on Rk.) We bound\np(t)≥I(Hn×Rk−1)v(t)2n+k\n2n+k+1for allt∈R\nand apply the lemma with β=I(Hn×Rk−1) andN= 2n+2+kto deduce that\nPerHn×RkE≥I(R2n+2+k)/parenleftbiggI(Hn×Rk−1)\nI(R2n+1+k)/parenrightbigg2n+1+k\n2n+2+k\n|E|2n+1+k\n2n+2+k.\nThus,\nI(Hn×Rk)≥I(R2n+2+k)/parenleftbiggI(Hn×Rk−1)\nI(R2n+1+k)/parenrightbigg2n+1+k\n2n+2+k\n.60 RUPERT L. FRANK AND BERNARD HELFFER\nDroppingnfromthenotationandabbreviating ιk:=/parenleftbig\nI(Hn×Rk)/I(R2n+2+k)/parenrightbig2n+2+k,\nwe can write this bound as ιk≥ιk−1. Thus,ιk≥ι0, which is the assertion of the\ntheorem. /square\nAs an immediate consequence of Theorem 11.8 and Proposition 11.1 (in the form\n(11.2)) we obtain the following lower bound on the Faber–Krahn cons tant.\nCorollary 11.10. For anyn,k∈N,\nCFK(Hn×Rk)≥CFK(R2n+k+2)/parenleftbiggI(Hn)\nI(R2n+2)/parenrightbigg2(2n+2)\n2n+2+k\n. (11.14)\nOne can bring the bound (11.14) into a somewhat more explicit form by using the\nexplicit expressions for CFK(R2n+k+2) andI(R2n+2) from (11.1).\nWith the help of Corollary 11.6 we are ready to prove that γ(Hn×Rk)<1 ifn= 1\nandk≥2 or ifn= 2 andk≥1.\nProof of Theorem 7.2. Cases n= 1andn= 2.\nWe recall the expression for W(Hn×Rk) from (8.7) and write it as\nW(Hn×Rk) =W(R2n+2+k)W(Hn)(4π)n+1Γ(n+2).\nUsing Corollary 11.10 we find that\nγ(Hn×Rk)≤γ(R2n+2+k)/parenleftbiggI(R2n+2)\nI(Hn)/parenrightbigg2n+2\nW(Hn)−1(4π)−n−1Γ(n+2)−1.(11.15)\nNote that the upper bound depends on konly through γ(R2n+2+k). We recall from\n[43, Theorem 5.1] that the sequence d/ma√sto→γ(Rd) is decreasing. Therefore, if we can\nshow that the upper bound is <1 for (n,k) = (1,2) and (n,k) = (2,1), then the\nassertions of the theorem for n= 1 andn= 2 will follow.\nCasen= 1.Recalling from (8.4) that W(H) =1\n128, we get\nγ(H×Rk)≤44/parenleftBigg\nΓ(6+k\n2)\nΓ(3)/parenrightBigg2\n(I(H))−4/parenleftBigg\n1\nj2+k\n2,1/parenrightBigg4+k\n(W(H))−12k\n= 213+kΓ(6+k\n2)2(I(H))−4/parenleftBigg\n1\nj2+k\n2,1/parenrightBigg4+k\n. (11.16)\nSpecializing further to k= 2, this bound becomes\nγ(H×R2)≤21732(I(H))−4/parenleftbigg1\nj2,1/parenrightbigg6\n.\nWe havej2,1≈5.13562 [1, Table 9.5] and, using the lower bound on I(H) from\nCorollary 11.6, we obtain\nγ(H×R2)≤0.701019.\nThis is<1, as desired.PLEIJEL THEOREM FOR SUB-LAPLACIANS 61\nCasen= 2.Recalling from (8.5) that W(H2) =1\n482π, we get\nγ(H2×Rk)≤66/parenleftBigg\nΓ(8+k\n2)\nΓ(4)/parenrightBigg2\n(I(H2))−6/parenleftBigg\n1\nj4+k\n2,1/parenrightBigg6+k\n(W(H2))−12k\n= 212+k36πΓ(8+k\n2)2(I(H2))−6/parenleftBigg\n1\nj4+k\n2,1/parenrightBigg6+k\n.\nSpecializing further to k= 1, this bound becomes\nγ(H2×R)≤25385272π2(I(H2))−6/parenleftBigg\n1\nj5\n2,1/parenrightBigg7\n.\nWe havej5\n2,1≈5.76346 [1, Table 10.6] and, using the lower bound on I(H2) from\nCorollary 11.6, we obtain\nγ(H2×R)≤0.823715.\nThis is<1, as desired. /square\nAs another application of Corollary 11.10 we now show that Pansu’s co njecture\nimplies Pleijel’s theorem.\nProof of Proposition 7.3. InviewofTheorem7.2andCorollary11.2itsuffices toprove\nthat Pansu’s conjecture in Himpliesγ(H×R)<1. Arguing as in the previous proof,\nwe want to show that the right side of (11.16) is <1 whenk= 1 and when I(H) is\nreplaced by the conjectured value on the right side of (11.6), which is 25\n23−3\n4π1\n2; see\n(11.12). Using j3\n2,1≈4.49341 [1, Table 10.6], we obtain\n214Γ(7\n2)2/parenleftBig\n25\n23−3\n4π1\n2/parenrightBig−4\nj−5\n3\n2,1≈2.639·10−1<1,\nas claimed. /square62 RUPERT L. FRANK AND BERNARD HELFFER\nPart3.Appendix\nAppendix A.Discussion on how to avoid Assumption 1.3\nWe recall that in Assumption 1.3 we assumed that, for each x∈M, theXj(x) are\nlinearly independent, In this appendix we further discuss this assum ption and show\nthat it can be locally removed. We discuss in the last paragraph some t opological\nobstructions for doing this globally.\nA.1.A first example.\nTo start with, we just consider an example where\n−∆X=X⋆\n1X1+X⋆\n2X2+X⋆\n3X3,\nassumingr= 2 and, instead of Assumption (2) in Section 3, assuming that at eac h\npointx∈Mthe space generated by X1(x),X2(x),X3(x) is of dimension 2.\nWe explain how to come back to the situation considered in the main tex t. We\nshow that locally, we can consider −∆ in the form\n−∆X=˜X⋆\n1˜X1+˜X⋆\n2˜X2=−∆/tildewideX,\nfor suitable vector fields ˜X1,˜X2inD1, such that ˜X1(x) and˜X2(x) are linearly inde-\npendent.\nTo show this, we proceed as for the proof of a Morse lemma. Let us a ssume that\nX1andX2are linearly independent in some open set UinM. Hence we can write in\nU\nX3(x) =a1(x)X1(x)+a2(x)X2(x).\nThe two desired vector fields are given by\n˜X1=/radicalbig\n1+a2\n1X1+a1a2√\n1+a2\n1X2,\n˜X2=/radicalBig\n1+a2\n1+a2\n2\n1+a2\n1X2.(A.1)\nOur privileged coordinates are then defined by using locally the vecto r fields˜X1and\n˜X2(see (3.2)) and completing by [ ˜X1,˜X2].\nNote that\n[˜X1,˜X2] =/radicalBig\n1+a2\n1+a2\n2[X1,X2]+c1X1+c2X2.\nThen we can take as flag (see Section 3) ˜X1,˜X2,[˜X1,˜X2], so that the group Gxis the\nHeisenberg group.\nA.2.General argument.\nThe general argument is as follows: we consider a sub-Laplacian ∆ = −/summationtextp\ni=1X⋆\niXi\nand the non-negative quadratic form\nq[f] =p/summationdisplay\nj=1/ba∇dblXjf/ba∇dbl2\nL2(M,µ),PLEIJEL THEOREM FOR SUB-LAPLACIANS 63\nand we assume that\nn1<p.\nWe choosen1vector fields, which we can assume to be X1,...,X n1(up to relabeling),\nthat are linearly independent in a neighborhood of some point x0∈M. In the sequel\nxdenotes the variable in this neighborhood. For i=n1+1,...,p, we find coefficients\naijsuch that\nXj=n1/summationdisplay\nℓ=1ajℓ(x)Xℓ.\nExpanding, we obtain for fwith compact support in this neighborhood\nq[f] =/integraldisplayn1/summationdisplay\nℓ,ℓ′=1bℓℓ′(x)Xℓf(x)Xℓ′f(x)dµ(x)\nwith some coefficients bℓℓ′computed in terms of the aij. The matrix\nBx:= (bℓℓ′(x))1≤ℓ,ℓ′≤n1\nis symmetric and definite positive.\nIn this case, we can have a normal form (see the proof of Morse Le mma)\nBx=Tt\nxTx,\nwhereTxis triangular, invertible, depending smoothly on xin the construction. We\nset\n˜Xi(x) :=n1/summationdisplay\nℓ=itiℓ(x)Xℓ(x)\nand get\nq[f] =n1/summationdisplay\ni=1/ba∇dbl˜Xi(x)f(x)/ba∇dbl2\nL2(M,µ).\nThus, locally,\n∆ =−n1/summationdisplay\ni=1˜X⋆\ni˜Xi.\nA.3.Sub-Riemannian geometric construction.\nWe provide13an alternative approach to the local argument of the previous sub section.\nWe refer for example to the book [46]. We introduce the following met ric on the\ndistribution, i.e. for va linear combination of the Xi’s (i= 1,...,p):\ngx(v) := inf\n\n/parenleftBiggp/summationdisplay\ni=1u2\ni/parenrightBigg1/2\n:p/summationdisplay\ni=1uiXi(x) =v\n\n.\n13Discussions with L. Hillairet and C. Letrouit64 RUPERT L. FRANK AND BERNARD HELFFER\nThen we pick a smooth orthonormal frame ( ˜X1(x),...,˜Xn1(x)) for the above metric.\nFinally we check that\np/summationdisplay\ni=1/ba∇dblXif/ba∇dbl2\nL2(M,µ)=n1/summationdisplay\ni=1/ba∇dbl˜Xif/ba∇dbl2\nL2(M,µ). (A.2)\nThis is done as follows: we pick coefficients uijsuch that ˜Xi=/summationtextp\nj=1uijXjwith/summationtextp\nj=1u2\nij= 1. By orthogonality of the /tildewideXi’s, we know that/summationtextp\nj=1uijui′j= 0 fori/\\e}atio\\slash=i′.\nPlugging this into the RHS of (A.2), we obtain its LHS.\nA.4.Topological constraints.\nThe existence for D1ofn1global vector fields giving for each x∈Ma basis ofTxMis\nonlypossible14understrongtopologicalconditionson Mwhichinvolveitsorientability,\nits Euler characteristic, the Euler class of D1and other invariants.\nNote first that the existence of a global non zero vector field implies that the Euler\ncharacteristic of Mis zero by the Poincar´ e-Hopf theorem. When Mhas dimension 3\nand is orientable, this is not an obstruction since this Euler Characte ristic is zero.\nA second obstruction related to the Euler class of the fiber bundle a lso disappears\nwhenM=S3. Ifn1= 2, one can instead of a basis of D1(x) consider the normal to\nthe plane. This gives an application from S3toS2and the associated Hopf invariant\n(which belongs to Z) should be zero, which is not the case in general.\nAppendix B.Weyl law and heat kernel\nThis section is based on notes kindly transmitted to us by C. Letrouit .\nB.1.Nilpotentization of measure.\nWe follow [20, Section A.5.6]. Let µbe a smooth measure on Mand letx∈M. Using\na chartθxof privileged coordinates at xand the associated dilations ( δx\nε), the measure\n/hatwideµxon/hatwiderMx≃Rn(called the nilpotentization of the measure µatx) is given by\n/hatwideµx= lim\nε→0µx\nε,whereµx\nε:=|ε|−Q(δx\nε)∗µ.\nwith convergence in the vague topology.\nB.2.Nilpotentization of heat kernel.\nWe follow [20, Section A.8.3]. The nilpotentized sub-Laplacian at xis defined as an\noperator on functions on /hatwiderMx≃Rnby\n/hatwide∆x=m/summationdisplay\ni=1(/hatwideXi,x)2.\nItisself-adjoint(withusualdomain). Theoperator ∂t−/hatwide∆xishypoellipticandtherefore\nits kernel has a smooth density /hatwideex=e/hatwide∆x,/hatwideµx, defined on (0 ,+∞)×/hatwiderMx×/hatwiderMx. This\n14Discussion with V. Colin.PLEIJEL THEOREM FOR SUB-LAPLACIANS 65\nmeans that for f∈C∞\nc(/hatwiderMx) we have\n(expt/hatwide∆xf)(u) =/integraldisplay\n/hatwiderMxe/hatwide∆x,/hatwideµx(t,u,v)f(v)d/hatwideµx(v)∀u∈/hatwiderMx.\nB.3.Effect of changes of coordinates.\nWe follow [20, second part of Appendix A.8.2]. The above nilpotentizatio ns depend\non the choice of privileged coordinates (we have made a specific choic e in the main\ntext) and on the measure /hatwideµx. But/hatwideex(t,0,0) does not depend on this choice due to [19,\nthird relation in (85)], which says that changing variables both in the o perator and in\nthe measure has no effect on the heat kernel. Indeed, when ϕis a diffeomorphism, we\nhave\neϕ∗∆ϕ∗,ϕ∗µ(t,u,v) =e∆,µ(t,ϕ(u),ϕ(v)).\nThis is written in Mbut also holds in the nilpotentized /hatwiderMx. We will actually only\nuse the particular case considered p. 952, line -8 of Appendix A in [19 ], which reads\nλe/hatwide∆x,λ/hatwideµx=e/hatwide∆x,/hatwideµx,∀λ>0. (B.1)\nB.4.Heat kernel asymptotics and Weyl law.\nWe follow [20, Theorem 4.1 and its proof].\nTheorem B.1. In the equiregular case, for every f∈C∞(M), we have\n/integraldisplay\nMf(x)e∆,µ(t,x,x)dµ(x) =t−Q\n2F(t)\nfor someF∈C∞(R)with\nF(0) =/integraldisplay\nMf(x)/hatwideex(1,0,0)dµ(x).\nIn particular, the eigenvalue counting function satisfies\nN(λ,−∆)∼λ→+∞/integraltext\nM/hatwideex(1,0,0)dµ(x)\nΓ(Q\n2+1)λQ\n2. (B.2)\nThis is due to the fact (see [20, Theorem C.1 and Eq. (96)]) that tQ\n2e∆,µ(t,x,x)\nconverges to/hatwideex(1,0,0) ast→0+, uniformly with respect to x, because we are in the\nequiregular case. The spectral asymptotics (B.2) follows from a Ta uberian theorem.66 RUPERT L. FRANK AND BERNARD HELFFER\nReferences\n[1] M. Abramowitz, I. A. Stegun. 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Hypoellipticit´ e Maximale pour des Op´ erateurs Polynˆ omes de\nChamps de Vecteurs. Progress in Mathematics. Birkh¨ auser. 198 5\n[43] B. Helffer and M. Persson Sundqvist. On nodal domains in Euclidea n balls. Proc. Amer.Math.\nSoc. 144 (11): 4777–4791 (2017).\n[44] J. G. Heywood, E. S. Noussair, and C. A. Swanson. On the zero s of solutions of elliptic inequal-\nities in bounded domains. J. Differential Equations 28(1978), no.3, 345–353.68 RUPERT L. FRANK AND BERNARD HELFFER\n[45] L. H¨ ormander. Hypoelliptic second order differential equation s. Acta Math. 119: 147-171 (1967).\n[46] F. Jean. Control of nonholonomic systems: from sub-Riemann ian geometry to motion planning.\nMonograph. Springer (2014).\n[47] D. Jerison. The Dirichlet problem for the Kohn Laplacian on the He isenberg group. J. of Func-\ntional Analysis 43 (1981), Part I, 97–141, Part II, 224–257.\n[48] D. Jerison and J.M. Lee. Extremal for the Sobolev inequality on t he Heisenberg group and the\nCR Yamabe problem. 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Calgary, C algary, Alta., 1978), 201–\n247, Lecture Notes in Math., 755, Springer, Berlin, 1979.\n[61] G. M´ etivier. Fonction spectrale et valeurs propres d’une class e d’op´ erateurs non elliptiques.\nComm. in PDE 1 (1976), 467–519.\n[62] G. M´ etivier. Hypoellipticit´ e analytique sur des groupes nilpoten ts de rang 2. Duke Math. Journal,\nVol . 47, No 1 (1980), 195–221.\n[63] John W. Milnor. Topology from the differentiable viewpoint. Prince ton University Press, Prince-\nton, NJ, (1997). Revised reprint of the 1965 original.\n[64] A. Mohamed. ´Etude spectrale d’op´ erateurs hypoelliptiques ´ a caract´ eristiqu es multiples I. An-\nnales de l’Institut Fourier, Tome 32 (1982) no. 3, 39–90.\n[65] R. Monti. Heisenberg isoperimetric problem. The axial case. Adv . Calc. Var. 1 (2008), no.1,\n93–121.\n[66] R. Monti and D. Morbidelli. Non-tangentially accessible domains fo r vector fields. Indiana Uni-\nversity Mathematics Journal, Vol. 54, No. 2 (2005).\n[67] R. Monti and M. Rickly. Convex isoperimetric sets in the Heisenbe rg group. Ann. Sc. Norm.\nSuper. Pisa Cl. Sci. (5) 8 (2009), no. 2, 391–415.PLEIJEL THEOREM FOR SUB-LAPLACIANS 69\n[68] E. M¨ uller-Pfeiffer. On the number of nodal domains for eigenfu nctions of elliptic differential\noperators. J. London Math. Soc. (2) 31(1985), no. 1, 91–100.\n[69] B.D.S. Nagy. ¨Uber Integralungleichungen zwischen einer Funktion und ihrer Ableit ung. Acta\nSci. Math. 10, 64–74 (1941).\n[70] P. Pansu. An isoperimetric inequality for the Heisenberg group.C omptes Rendus Acad. Sc. Math.\n295.2 (1982), 127–130.\n[71] P. Pansu. An isoperimetric inequality on the Heisenberg group. C onference on differential geom-\netry on homogeneous spaces (Torino, 1983). Rend. Sem. Mat. Un iv. Politec. Torino 1983, Special\nIssue, 159–174 (1984).\n[72] A. Pleijel. Remarks on Courant’s nodal theorem. Comm. Pure. A ppl. Math., 9: 543–550 (1956).\n[73] I. Polterovich. Pleijel’s nodal domain theorem for free membran es. Proc. Amer. Math. Soc. 137\n(2009), no. 3, 1021–1024.\n[74] M. Ritor´ e. A proof by calibration of an isoperimetric inequality in t he Heisenberg group Hn.\nCalc. Var. Partial Differential Equations 44 (2012), no. 1-2, 47–6 0.\n[75] M. Ritor´ e, C. Rosales. Rotationally invariant hypersurfaces w ith constant mean curvature in the\nHeisenberg group Hn. J. Geom. Anal. 16 (2006), no. 4, 703–720.\n[76] M. Ritor´ e, C. Rosales. Area-stationary surfaces in the Heise nberg group H1. Adv. Math. 219\n(2008), no. 2, 633–671.\n[77] L.P. Rothschild. A criterion for hypoellipticity of operators cons tructed of vector fields. Comm.\nin PDE 4 (6) (1979), 248–315.\n[78] L.P. Rothschild and E.M. Stein. Hypoelliptic differential operators and nilpotent groups. Acta\nMathematica 137, 248–315. (1976)\n[79] G. Talenti. The standard isoperimetric theorem. North-Holland Publishing Co., Amsterdam,\n(1993), 73–123.\n[80] N. Th. Varopoulos. Analysis on nilpotent groups. Journal of Fu nctional Analysis 66, 406–431\n(1986).\n[81] K. Watanabe. Sur l’unicit´ e du prolongement des solutions des ´ e quations elliptiques d´ eg´ en´ er´ ees.\nTohoku Math. Journ. 34 (1982), 239-249.\n(Rupert L. Frank) Mathematisches Institut, Ludwig-Maximilians Universit ¨at M¨unchen,\nTheresienstr. 39, 80333 M ¨unchen, Germany, and Munich Center for Quantum Science\nand Technology, Schellingstr. 4, 80799 M ¨unchen, Germany, and Mathematics 253-37,\nCaltech, Pasadena, CA 91125, USA\nEmail address :r.frank@lmu.de\n(Bernard Helffer) Laboratoire de Math ´ematiques Jean Leray, CNRS, Nantes Univer-\nsit´e, F44000. Nantes, France\nEmail address :Bernard.Helffer@univ-nantes.fr"    },    {        "title": "2402.13998v3.Small_values_and_forbidden_values_for_the_Fourier_antidiagonal_constant_of_a_finite_group.pdf",        "content": "arXiv:2402.13998v3  [math.GR]  31 Mar 2024SMALL VALUES AND FORBIDDEN VALUES FOR THE\nFOURIER ANTI -DIAGONAL CONSTANT OF A FINITE\nGROUP\nYEMON CHOI\n2024-03-23\nAbstract\nFor a finite group G, let AD(G)denote the Fourier norm of the antidiagonal in G×G. It\nwas shown recently in [ Cho23a ] that AD (G)coincides with the amenability constant of the\nFourier algebra of G, and is equal to the normalized sum of the cubes of character d egrees of G.\nMotivated by a gap result for amenability constants from [ Joh94 ], we determine exactly which\nnumbers in the interval [1, 2]arise as values of AD (G).\nAs a by-product, we show that the set of values of AD (G)does not contain all its limit points.\nSome other calculations or bounds for AD (G)are given for familiar classes of finite groups. We\nalso indicate a connection between AD (G)and the commuting probability of G, and use this to\nshow that every finite group Gsatisfying AD (G)<61\n15must be solvable; here the value61\n15is\nbest possible.\nMSC2020: 20C15 (primary); 20D99, 43A30 (secondary)\n1 I NTRODUCTION\n1.1 B ACKGROUND CONTEXT\nGiven a finite group G, the algebra of complex-valued functions on G(equipped with pointwise\nproduct) only depends on the cardinality of G, and does not detect the group structure. However,\nthere is a canonical submultiplicative norm on this algebra , the Fourier norm , such that the result-\ning normed algebra A (G)characterizes the starting group Gup to isomorphism. (More precisely:\ngiven finite groups Gand H, there is an isometric algebra isomorphism between A (G)and A(H)\nif and only if Gand Hare isomorphic groups; this is a a special case of a result of W alter [ Wal72 ].)\nBy identifying a subset of Gwith its indicator function, one can speak of the Fourier nor m of\na subset of G. Calculating Fourier norms of arbitrary subsets is hard (se e [San21 ] for a system-\natic approach), but there is one case where an exact calculat ion is possible and gives interesting\nanswers. Consider the set {(g,g−1):g∈G}. The Fourier norm of this subset of G×G, denoted\nby AD(G)in this paper, is the Fourier anti-diagonal constant mentioned in the title. It was re-\ncently shown by the author [ Cho23a , Theorem 1.4] that we have the following explicit formula\nfor AD(G):\nAD(G) =1\n|G|∑\nφ∈Irr(G)d(φ)3(1.1)\n1where Irr (G)is the set of irreducible complex characters of Gand d(φ)is the degree of φ.\nEquation ( 1.1) implies that AD (G×H) = AD(G)AD(H), and that AD is invariant under\nisoclinism. It can also be shown that AD (H)≤AD(G)whenever H≤G(see Proposition 3.5\nbelow). These hereditary properties suggest that AD (G), viewed as a numerical invariant of G,\ndeserves further study. Furthermore, the sum on the right-h and side of Equation ( 1.1) had already\narisen in earlier work of Johnson [ Joh94 ] on Fourier algebras of compact groups. The results in\n[Joh94 , §4] provide an attractive application of the character the ory of finite groups to obtain new\n(counter-)examples in functional analysis. (For a fuller d iscussion, see [ Cho23a , Section 1].)\nThe following observations, taken from [ Joh94 , Proposition 4.3], are easy consequences of ( 1.1):\n• if Gis abelian, then AD (G) =1;\n• if Gis non-abelian, then AD (G)≥3\n2.\nSince AD (Gn) = AD(G)n, this shows that AD (G)can take arbitrarily large values. However, to\nthe author’s knowledge, nothing further has been done to stu dy the possible values of AD (G)as\nGranges over all non-abelian finite groups. The purpose of the present paper is to make a start\non filling this gap.\n1.2 O UR MAIN NEW RESULTS\nThe following result has probably been noticed independent ly by many readers of Johnson’s pa-\nper, although it is not stated explicitly there. (A proof wil l be given in Section 2for sake of com-\npleteness.)\nPROPOSITION 1.1(implicitly folklore) .Let G be a finite group, and suppose that d (φ)≤2for all\nφ∈Irr(G). Then AD(G)∈ {2−n−1:n∈N}.\nMoreover, every number in {2−n−1:n∈N}is realized as the AD-constant of some (non-\nunique) finite group: this can be seen by considering cyclic g roups and dihedral groups. Our first\nmain result is that these are the only values of AD attained by finite groups in the interval [1, 2].\nTo be precise:\nTHEOREM 1.2(The possible values of AD (G)in[1, 2]).Let G be a finite group, and suppose that\nAD(G)≤2. Then AD(G)∈ {2−n−1:n∈N}.\nCOROLLARY 1.3. The set{AD(G):G a finite group }is not a closed subset of [1,∞).\nTheorem 1.2is an immediate consequence of combining Proposition 1.1with the following\nlower bound for AD (G), which appears to be new.\nPROPOSITION 1.4. Let G be a finite group. If there exists φ∈Irr(G)with d(φ)≥3, then AD(G)≥\n2+|G′|−1.\nThe proof of Proposition 1.4requires some basic character theory, but nothing harder th an\nFrobenius reciprocity. Perhaps surprisingly, while we do n eed character theory for finite groups,\nwe do not rely on any structure theory (we do not even need the S ylow theorems). In contrast,\nour other main result requires the classification of finite si mple groups with characters of small\ndegree.\n2THEOREM 1.5(A threshold ensuring solvability) .Let G be a finite group. If AD(G)<61\n15, then G is\nsolvable.\nA direct calculation shows that AD (A5) =61\n15, and so in this sense Theorem 1.5is sharp. Par-\nticular properties of A5, such as its subgroup structure and its Schur multiplier, pl ay an important\nrole in the proof of Theorem 1.5, since we need to analyze perfect groups that quotient onto A5.\nOne difficulty in proving Theorem 1.5is that AD is not monotone (in either direction) with re-\nspect to taking quotients, and so knowing that a group Hquotients onto A5does not immediately\nimply that AD (H)≥AD(A5). Instead we require a detour through the commuting probability\ncp(G), which has been studied extensively by several authors. Our strategy is inspired by an\nargument of Tong-Viet in [ TV12 ], and indeed the main work needed to prove Theorem 1.5lies in\nestablishing the following stronger version of [ TV12 , Lemma 2.4].\nPROPOSITION 1.6. Let H be a finite non-trivial perfect group satisfying cp(H)>1\n20. Then H ∼=A5or\nH∼=SL(2, 5).\n1.3 O UTLINE OF THIS PAPER\nAfter some preliminary results in Section 2, the proof of Proposition 1.4is given in Section 3.\nSince the paper is intended for a general audience, we have sp elled things out in more detail than\nspecialists in group theory would require.\nIn Section 4.1we calculate the values of AD (G)for some particular families of groups, some\nof which are related to calculations in earlier sections; an d in Section 4.2we present some partial\nresults on the general theme that “small values” of AD (G)imply that Gis close to abelian in some\nsense. Section 5is dedicated to the proof of Proposition 1.6and Theorem 1.5; this is the only part\nof the paper that makes use of the theory of the cp invariant. I n the appendix, we collect some\nproofs of results that are used in the main body of the paper; t hese results are special cases or\nweaker versions of known results, but we take the opportunit y to provide some extra details and\ngive more elementary arguments.\nWe finish this introduction by fixing notation and establishi ng some conventions. Throughout\nthis article, all representations and characters are taken over the complex field. To reduce unnec-\nessary repetition, we adopt the following convention: henc eforth, all groups are assumed to be\nfinite unless explicitly stated otherwise.\nIrr(G)denotes the set of irreducible characters of a group G. We denote the degree of a char-\nacter φbyd(φ); note that this is equal to φ(e). It is convenient to use the notation cd (G)for the\nset{d(φ):φ∈Irr(G)}(note that here we are not counting the multiplicities of the irreducible\ncharacters).\nWe write Irr n(G)for the set of all φ∈Irr(G)that have degree n. If Gis non-abelian, we\ndefine mindeg (G):=min{d≥2 : Irr d(G)is non-empty }. For any G(possibly abelian), we define\nmaxdeg(G):=max{d(ψ):ψ∈Irr(G)}.\nGiven a group G, we equip CGwith the following inner product:\n/a\\}bracketle{tϕ,ψ/a\\}bracketri}htL2(G):=1\n|G|∑\nx∈Gϕ(x)ψ(x).\nWe recall that if ψis a character on G, then it is irreducible if and only if /a\\}bracketle{tψ,ψ/a\\}bracketri}htL2(G)=1.\n3Finally: the derived subgroup of a group G(also known as its commutator subgroup) will be\ndenoted throughout by G′.\n2 S OME EASY LOWER BOUNDS ON AD\nWe start by giving a proof of Proposition 1.1, since it will also serve as a prototype for later argu-\nments. No novelty is claimed.\nProof of Proposition 1.1.Since cd(G)⊆ {1, 2},\nAD(G) =1\n|G|(|Irr1(G)|+8|Irr2(G)|).\nOn the other other hand, basic character theory tells us that\n1=1\n|G|∑\nφ∈Irr(G)d(φ)2=1\n|G|(|Irr1(G)|+4|Irr2(G)|),\nand therefore AD (G)−2=−|Irr1(G)||G|−1.\nSince Irr 1(G)can be identified with the (Pontrjagin) dual of the abelian gr oup G/G′, we have\n|Irr1(G)|=|G:G′|. Hence AD (G) =2−|G′|−1, and since |G′| ∈N, the result follows.\nEXAMPLE 2.1(Dihedral groups) .LetGbe a dihedral group of order 2 k, so that cd G={1, 2}. If\nkis odd then |G′|=k, and if kis even then |G′|=k\n2. By repeating the calculation in the proof of\nProposition 1.1, we see that AD (G) =2−1\nkwhen kis odd and AD (G) =2−2\nkwhen kis even.\nFor general non-abelian G, the proof of Proposition 1.1still suggests a way to proceed. Infor-\nmally: since d(φ)3≥mindeg(G)d(φ)2for all φ∈Irr(G)\\Irr1(G), we can add a correction factor\nto AD(G)to obtain something bounded below by mindeg (G), and the size of the correction fac-\ntor is controlled by the size of |G′|. Making this precise leads us to the following lemma, which\nprovides a convenient tool for dealing with “generic” cases .\nLEMMA 2.2(An all-purpose lower bound) .Let G be non-abelian, and let m ,n∈Nsatisfy mindeg(G)≥\nm and|G′| ≥n. Then\nAD(G)≥1+(m−1)/parenleftbigg\n1−1\nn/parenrightbigg\n. (2.1)\nProof. Since Irr j(G)is empty whenever 2 ≤j≤m−1, we have\nAD(G)−|Irr1(G)|\n|G|=1\n|G|∑\nn≥mn3|Irrn(G)|\nand\n1−|Irr1(G)|\n|G|=1\n|G|∑\nn≥mn2|Irrn(G)|\nHence\nAD(G)−|Irr1(G)|\n|G|≥m/parenleftbigg\n1−|Irr1(G)|\n|G|/parenrightbigg\n.\n4As in the proof of Proposition 1.1,|Irr1(G)|=|G:G′|. Hence|Irr1(G)||G|−1≤n−1, and plugging\nthis into the previous inequality gives\nAD(G)≥m−(m−1)|Irr1(G)|\n|G|≥m−m−1\nn,\ncompleting the proof.\nNote that by Lemma 2.2, if either mindeg (G)≥3 or|G′| ≥3 then AD (G)>3\n2. Therefore, if\nAD(G) =3\n2we must have cd (G) ={1, 2}and|G′|=2; moreover, inspecting the proof of Lemma\n2.2shows that these necessary conditions are also sufficient. W e thus recover Johnson’s lower\nbound ([ Joh94 , Proposition 4.3]) in a slightly more precise form.\nCOROLLARY 2.3(Johnson) .Let G be non-abelian. Then AD(G)≥3\n2, and equality holds if and only if\nthe following two conditions are satisfied: cd(G) ={1, 2}and|G′|=2.\nREMARK 2.4. In [Cho23a ] the present author studied a generalization of AD (G)to the setting of\nvirtually abelian groups, and showed that AD (G) =3\n2if and only if |G:Z(G)|=4. The proof\ngoes via a version of Corollary 2.3, but substantial work is required since Gmay be infinite. It is\ntherefore worth noting that when Gis finite, there is a much simpler proof of this equivalence;\ndetails are given in Appendix A.1.\nIf AD(G)>3\n2, then either mindeg (G)≥3 or|G′| ≥3. The example of S3shows that we can\nhave mindeg (G) = 2 and|G′|=3. In contrast, the next result shows that we can never have\nmindeg(G) =3 and|G′|=2.\nLEMMA 2.5. Let G be a group with |G′|=2. Ifϕ∈Irr(G)and d(ϕ)>1, then d(ϕ)is even.\nLemma 2.5follows from more precise results of Miller, stated in the in troduction of [ Mil38 ].\nHis presentation is rather terse and uses the finiteness of Gin an essential way. We provide a\ndirect proof of Lemma 2.5in Appendix A.2, which also works for (irreducible, finite-dimensional,\nunitary) representations of infinite groups.\nPROPOSITION 2.6. Let G be non-abelian. If 2 /∈cd(G), then AD(G)≥7\n3.\nProof. We split into two cases. If |G′| ≥ 3, then using Lemma 2.2with m=2 and n=3 gives\nAD(G)≥7\n3. If|G′|=2, then 3 / ∈cd(G)by Lemma 2.5, and so mindeg (G)≥4; using Lemma 2.2\nwith m=4 and n=2 gives AD (G)≥5\n2>7\n3.\nIn both cases of the proof, the lower bounds are sharp; see Exa mple 4.1below for details.\n3 T HE PROOF OF PROPOSITION 1.4\nFor a finite set Xand a function f:X→C, we write supp (f)for the support of f, that is, the set\n{x∈X:f(x)/\\e}atio\\slash=0}.\nLEMMA 3.1(The “L-orbit method” for lower bounds) .Letϕ∈Irr(G)and let n =d(ϕ). Let K be\nthe normal subgroup of G generated by supp(ϕ). Then\n|Irrn(G)| ≥ | Irr1(G)||G:K|−1.\n5Proof. To simplify notation, let L=Irr1(G).Lis a group with respect to pointwise product, and\nmultiplication of characters defines a group action L× Irrn(G)→Irrn(G)for each n. TheL-orbit\nofϕis a subset of Irr n(G)and it has size |L||StabL(ϕ)|−1.\nObserve that\nStabL(ϕ) ={γ∈ L:γϕ=ϕ}={γ∈ L:γ(x) =1 for all x∈supp(ϕ)},\nwhich is the set of group homomorphisms G→Cthat factor through G→G/K. Therefore,\nwriting Afor the abelianization of G/K, we have\n|StabL(ϕ)|=|A| ≤ | G/K|=|G:K|,\nand so the L-orbit of ϕhas at least |L||G:K|−1elements. The result now follows.\nCOROLLARY 3.2. Let n∈N. IfIrrn(G)is non-empty, then |Irrn(G)| ≥n−2|Irr1(G)|.\nProof. Pick some ϕ∈Irrn(G), and let Kbe the normal subgroup of Ggenerated by supp (ϕ). Since\nϕis irreducible, /a\\}bracketle{tϕ,ϕ/a\\}bracketri}htL2(G)=1. Therefore, since |ϕ(x)| ��d(ϕ) =nfor all x∈G,\nn2|supp(ϕ)| ≥∑\nx∈G|ϕ(x)|2=|G|.\nHence|G:K| ≤ | G||supp(ϕ)|−1≤n2. Applying Lemma 3.1, the result follows.\nREMARK 3.3. Although the estimates in the proof of Lemma 3.1are potentially wasteful, the\nresulting lower bound in Corollary 3.2is sharp. For if Gis an extraspecial group of order 22k+1, it\nhas exactly 22kdegree-1 characters, and a single irreducible character of degree 2k. On the other\nhand, it will be important later that in certain situations w e can do significantly better (Lemma\n3.10 below).\nPROPOSITION 3.4. If G is non-abelian, then AD(G)≥2+(maxdeg(G)−3)|G′|−1.\nProof. Letd=maxdeg(G). Since|G|=∑d\nn=1n2|Irrn(G)|,\nAD(G)−2=1\n|G|d\n∑\nn=1(n3−2n2)|Irrn(G)|\n≥ −|Irr1(G)|\n|G|+(d3−2d2)|Irrd(G)|\n|G|.\nSince Irr d(G)is non-empty, applying Corollary 3.2gives the desired inequality.\nThe rest of this section deals with cases where cd (G) ={1, 2, 3}. We require a property of\nAD that is not obvious from the definition, but which seems to b e crucial to understanding its\nbehaviour.\nPROPOSITION 3.5(Johnson) .ADis monotone with respect to subgroup inclusion. That is, if H ≤G\nthen AD(H)≤AD(G).\nREMARK 3.6. Proposition 3.5follows from results in [ Joh94 , §4] concerning “amenability con-\nstants” of Fourier algebras, or from the general theory in [ Cho23a , §2]. One can give a direct\nproof, based on considering the induction of characters fro mHtoG, see the author’s MathOver-\nflow question [ Cho23b ] and the comments and answers. It is quite possible that a dir ect proof\nalong these lines was already known to Johnson.\n6PROPOSITION 3.7. Let G be a group such that AD(G)<7\n3. Suppose that cd(G) ={1, 2, 3}; then every\nindex- 2subgroup H ≤G also satisfies cd(H) ={1, 2, 3}.\nThe proof of Proposition 3.7requires some general facts about index-2 subgroups, which we\nstate as a separate lemma for convenience.\nLEMMA 3.8(Character degrees of index-2 subgroups) .Let H≤G be an index- 2subgroup. Then\nmaxdeg(H)≤maxdeg(G)andcd(G)⊆cd(H)∪2 cd(H).\nBoth parts of the lemma are standard results. For completene ss, we quickly sketch their proofs.\nProof. Given ψ∈Irr(H), let ϕ∈Irr(G)be one of the irreducible summands of IndG\nHψ. Then,\nby Frobenius reciprocity, ψis contained in ϕ|H, so d(ψ)≤d(ϕ|H) = d(ϕ)≤maxdeg(G). This\nproves the first claim.\nFor the second claim, let ϕ∈Irr(G). Ifϕ|His irreducible, then d(φ)∈cd(H). If not, then it\nfollows from Clifford theory (or direct arguments using Fro benius reciprocity) that ϕ|Hsplits as\nthe sum of two irreducible characters, say β1and β2, which satisfy ϕ=IndG\nHβ1=IndG\nHβ2. In\nparticular, d(ϕ) =2d(β1)∈2 cd(H).\nProof of Proposition 3.7.By monotonicity of AD (Proposition 3.5), AD(H)≤AD(G)<7\n3. Hence\nby Lemma 3.8, maxdeg (H)≤3 and 3∈cd(H). Since His non-abelian and AD (H)<7\n3, the\ncontrapositive of Proposition 2.6implies that 2 ∈cd(H).\nWe now observe that 2-dimensional irreducible representat ions of Gcan be used to produce\n3-dimensional representations with useful properties.\nLEMMA 3.9. Letπbe a2-dimensional irreducible representation of G, and let π∗denote its contragredient.\n(i)The trivial 1-dimensional representation εoccurs in π⊗π∗with multiplicity 1.\n(ii)Letρbe the summand in π⊗π∗complementary to ε. Suppose that ρis reducible. Then G has an\nindex- 2subgroup.\nThis is surely not a new observation, but since we are unaware of a precise reference, a full\nproof is given below.\nProof. Part (i)follows from Schur’s lemma. (Alternatively, let ψ=Trπ; then the multiplicity of\nthe trivial representation in π⊗π∗can be computed as /a\\}bracketle{tψψ,ε/a\\}bracketri}htL2(G)=|G|−1∑x∈Gψ(x)ψ(x) =1.)\nFor part (ii), it suffices to prove the following claim: ρhas a self-dual 1-dimensional subrep-\nresentation. For if σis such a subrepresentation, then Tr σis a group homomorphism G→ {± 1}\n(which is surjective since σ/\\e}atio\\slash∼ε); hence, the kernel of this homomorphism is an index-2 subgr oup\nofG, as required.\nTo prove the claim: since ρis reducible and 3-dimensional, it has at least one 1-dimens ional\nsubrepresentation, γsay. If γis self-dual, we are done. If not, then since Tr ρis real-valued, γ∗is\nalso a subrepresentation of ρ, and γ/\\e}atio\\slash∼γ∗. Hence ρ∼γ⊕γ∗⊕σ, and σmust be self-dual.\n7IfGhas no index-2 subgroup and 2 ∈cd(G), then by Lemma 3.9, for each ψ∈Irr2(G)the\ncharacter β:=ψψ−εmust be irreducible; and because βis a “small perturbation” of a non-\nnegative character, we can obtain improved lower bounds on |supp(β)|, allowing us to apply\nLemma 3.1more effectively. It turns out that the relevant estimates h ave nothing to do with\ngroup structure, so we present them as a separate lemma.\nLEMMA 3.10. Let X be a finite non-empty set, and let d ≥1. Suppose that f :X→[−1,d]has mean 0\nand variance 1, i.e.\n∑\nx∈Xf(x) =0and∑\nx∈Xf(x)2=|X|.\nThen|supp(f)| ≥d−1|X|.\nProof. Fix some “threshold value” c∈[0,d], to be determined later, and partition supp (f)as\nN∪P∪Rwhere:\n•N:={x∈X:−1≤f(x)<0};\n•P:={x∈X: 0<f(x)≤c};\n•R:={x∈X:c<f≤d}.\nThen, since ∑x∈supp(f)f(x)2=|X|,\n|X|=∑\nx∈Nf(x)2+∑\nx∈Pf(x)2+∑\nx∈Rf(x)2\n≤∑\nx∈N|f(x)|+c∑\nx∈Pf(x)+d∑\nx∈Rf(x).(∗)\nOn the other hand, since ∑x∈supp(f)f(x) =0,\n∑\nx∈Pf(x) =∑\nx∈N|f(x)|−∑\nx∈Rf(x),\nand substituting this into ( ∗) yields\n|X| ≤(c+1)∑\nx∈N|f(x)|+(d−c)∑\nx∈Rf(x)≤(c+1)|N|+d(d−c)|R|\n≤max(c+1,d(d−c))|supp(f)|.\nTaking c=d−1 gives|X| ≤d|supp(f)|as required.\nPROPOSITION 3.11. Suppose that G has no index- 2subgroups but has an irreducible representation of\ndegree 2. Then|Irr3(G)| ≥1\n3|Irr1(G)|.\nProof. Letψ∈Irr2(G)and let β=ψψ−ε. We observe that:\n•βtakes values in [−1, 3], since 0≤ |ψ(x)|2≤4 for all x∈G;\n•/a\\}bracketle{tβ,ε/a\\}bracketri}htL2(G)=0, by Lemma 3.9(i) ;\n•/a\\}bracketle{tβ,β/a\\}bracketri}htL2(G)=1, since βis irreducible by Lemma 3.9(ii) .\nHence, by Lemma 3.10,|supp(β)| ≥1\n3|G|, and applying Lemma 3.1completes the proof.\n8REMARK 3.12. In general, the bound in Proposition 3.11 can not be improved. To see this, take\nG=SL(2, 3). Then cd (G) ={1, 2, 3}and|Irr1(G)|=3=3|Irr3(G)|, while|G:G′|=3 (so that G\ncannot quotient onto the two-element group).\nProof of Proposition 1.4.LetGbe a group with maxdeg (G)≥3. If maxdeg (G)≥4, then AD (G)≥\n2+|G′|−1by Proposition 3.4. So we assume henceforth that maxdeg (G) = 3. Note that this\nimplies|G′| ≥3, by Lemma 2.5. Moreover, if cd (G) ={1, 3}, then by Proposition 2.6, AD(G)≥\n7\n3≥2+|G′|−1.\nIt only remains to deal with the cases where cd (G) ={1, 2, 3}. If AD(G)≥7\n3then we are\ndone, as before. So we may assume that cd (G) ={1, 2, 3}and AD(G)<7\n3. Put H0=Gand\napply the following recursive procedure: if n∈Nand Hn−1has an index-2 subgroup, choose\nHnto be such a subgroup; otherwise, stop. Note that at each stag e, Proposition 3.7ensures that\ncd(Hn) ={1, 2, 3}.\nSince Gis finite this procedure must terminate; let Hbe the last subgroup in this sequence.\nSince cd(H) ={1, 2, 3},\nAD(H) =1\n|H|/parenleftBig\n|Irr1(H)|+8|Irr2(H)|+27|Irr3(H)|/parenrightBig\n, and\n1=1\n|H|/parenleftBig\n|Irr1(H)|+4|Irr2(H)|+9|Irr3(H)|/parenrightBig\n.\nHence AD (H) =2−|H|−1|Irr1(H)|+9|H|−1|Irr3(H)|. On the other hand, since Hhas no index-2\nsubgroups, it satisfies the hypotheses of Proposition 3.11, and so\nAD(H)≥2+2|Irr1(H)|\n|H|=2+2\n|H′|.\nAs AD(G)≥AD(H)(Proposition 3.5) and|G′| ≥ | H′|, we conclude that AD (G)≥2+2|G′|−1,\nwhich completes the proof of Proposition 1.4.\n4 F URTHER EXAMPLES AND IMPLICATIONS OF SMALL VALUES\n4.1 V ALUES OF AD FOR PARTICULAR GROUPS\nWe present three families of groups with rather different pr operties (nilpotent, solvable with triv-\nial centre, and quasi-simple), where one obtains rather sim ple formulas for the AD-constants in\neach family. In each case, the ratio AD (G)maxdeg(G)−1converges to 1 as |G| → ∞.\nEXAMPLE 4.1(Extraspecial p-groups) .Letpbe a prime and let n∈N. IfGis an extraspecial\np-group of order p2n+1, then the degrees of its irreducible characters and their mu ltiplicities are\nwell-documented. Namely, Ghas exactly p2ncharacters of degree 1, and exactly p−1 irreducible\ncharacters of degree pn. Hence\nAD(G) =p2n·13+(p−1)p3n\np2n+1=pn−1(p−1)+1\np. (4.1)\nWe note two particular cases, relevant to Proposition 2.6. Ifp=2 and n=2, then cd (G) ={1, 4}\nand AD(G) =5\n2. Ifp=3 and n=1, then cd (G) ={1, 3}and AD(G) =7\n3.\n9EXAMPLE 4.2(Affine groups of finite fields) .Forqa prime power ≥3, let Fqdenote the finite\nfield with qelements, and consider the natural semidirect product Fq⋊F×\nq(sometimes referred\nto as the affine group or “ ax+bgroup” of Fq). This group has exactly q−1 characters of degree\n1, and a single irreducible character of degree q−1. Hence\nAD(Fq⋊F×\nq) =(q−1)·13+(q−1)3\nq(q−1)=q−2+2\nq. (4.2)\nNote that when q=3 this group is isomorphic to the dihedral group of order 6, an d its AD-\nconstant is5\n3; this matches the calculation in Example 2.1.\nEXAMPLE 4.3(Special linear groups of degree 2) .Letqbe a prime power, and let SL (2,q)denote\nthe special linear group of degree 2 over the finite field with qelements; this has order q3−q.\nThe irreducible character degrees and their multiplicitie s for this group are well-documented; we\nsummarize them below, writing Irr nrather than Irr n(SL(2,q))to ease the notation.\n• If qis even, then: |Irr1|=1;|Irrq−1|=q\n2;|Irrq|=1;|Irrq+1|=q\n2−1.\n• If qis odd, then: |Irr1|=1;|Irr(q−1)/2|=2;|Irr(q+1)/2|=2;|Irrq−1|=q−1\n2;|Irrq|=1;\n|Irrq+1|=q−3\n2.\nBy brute-force calculation, we eventually obtain\nAD(SL(2,q)) =\n\nq3−3\nq2−1=q−1\nq−1+2\nq+1forqeven,\n2q3−q2−9\n2(q2−1)=q−1\n2−2\nq−1+3\nq+1forqodd.(4.3)\nThe next set of examples was suggested to the author by P . Levy .\nEXAMPLE 4.4(Finite subgroups of SO (3)and SU(2)).We ignore the cyclic groups and dihedral\ngroups, and their double covers inside SU (2), since these are covered by previous results. So there\nare only three new examples to consider. In the following lis t, when we refer to the “character de-\ngrees” of a group H, we mean “the degrees of its irreducible characters, listed with multiplicity”.\n(a) The alternating group A4has character degrees 1, 1, 1, 3. Its double cover is the binar y tetra-\nhedral group 2 T∼=SL(2, 3), whose character degrees are 1, 1, 1, 2, 2, 2, 3. Thus\nAD(A4) =30\n12=5\n2and AD(2T) =54\n24=9\n4<AD(A4).\n(b) The symmetric group S4has character degrees 1, 1, 2, 3, 3. Its double cover is the bin ary\noctahedral group 2 O, whose character degrees are 1, 1, 2, 2, 2, 3, 3, 4. Thus\nAD(S4) =64\n24=8\n3and AD(2O) =144\n48=3>AD(S4).\n(c) The alternating group A5has character degrees 1, 3, 3, 4, 5. Its double cover is the bin ary\nicosahedral group 2 I∼=SL(2, 5), whose character degrees are 1, 2, 2, 3, 3, 4, 4, 5, 6. Thus\nAD(A5) =244\n60=61\n15and AD(2I) =540\n120=9\n2>AD(A5).\nREMARK 4.5. It was already known that although AD cannot increase when pa ssing to sub-\ngroups, it can increase when passing to a quotient. For insta nce, in a “note added in proof” in\n[LLW96 ], it is observed that the Schur cover of A6has an AD constant strictly smaller than that of\nthe triple cover of A6. However, Example 4.4(a) shows that they missed a much smaller example.\n104.2 S TRUCTURAL CONSEQUENCES FOR GOF UPPER BOUNDS ON AD\nPROPOSITION 4.6(A cheap lower bound for p-groups) .Let p be a prime. If G is a non-abelian\np-group, then AD(G)≥p−1+1\np. Equality is attained by an extraspecial p-group of order p3.\nProof. Since Gis a p-group, both mindeg (G)and|G′|are powers of p. Therefore both are ≥p,\nsince Gis non-abelian. The rest follows from Lemma 2.2and the calculation in Example 4.1.\nA similar idea can be used to control (sub)groups of odd order whose AD-constant is small.\nThe next result is a slightly stronger version of an observat ion by G. Robinson (personal commu-\nnication).\nLEMMA 4.7. IfAD(G)<7\n3then every odd order subgroup of G is abelian.\nProof. We prove the contrapositive. Suppose Ghas a non-abelian subgroup Hthat has odd order.\nSince d(φ)divides|H|for each φ∈Irr(H), we have mindeg (H)≥3; since|H′|divides|H|, we\nhave|H′| ≥3. Therefore, by monotonicity (Proposition 3.5) and Lemma 2.2,\nAD(G)≥AD(H)≥1+(3−1)2\n3=7\n3,\nas required.\nCOROLLARY 4.8. If G is nilpotent and AD(G)<7\n3, then G is the product of a 2-group and an abelian\ngroup of odd order.\nProof. Ifpis an odd prime, then by Lemma 4.7each p-Sylow subgroup of Gis abelian. But since\nGis finite and nilpotent, it factorizes as the direct product o f its Sylow subgroups.\nREMARK 4.9. We can show by relatively elementary arguments that AD (G)>4 whenever G\nis non-abelian and simple; since AD (A5) =61\n15, this is already quite close to the optimal result.\nAlthough we shall obtain a stronger result in Section 5, we include the proof of the weaker result\nhere as an illustration of our earlier method.\nThe main idea is similar to the proof of Lemma 2.2. Let m=mindeg(G). Since mdivides\n|G|=∑φ∈Irr(G)d(φ)2and since |Irr1(G)|=1,\n∑\nφ∈Irr(G),d(φ)>md(φ)2≡ − 1 mod m.\nHence, there is at least one σ∈Irr(G)with d(σ)≥m+1. Therefore\nAD(G)−m=∑\nφ∈Irr(G)(d(φ)−m)d(φ)2\n|G|\n≥ −m−1\n|G|+(d(σ)−m)d(σ)2\n|G|≥ −m−1\n|G|+(m+1)2\n|G|>0 .\nIfm≥4 this immediately gives AD (G)>4. So it only remains to deal with cases where m=3.\nThe simple finite subgroups of PGL (3,C)were determined by Blichfeldt in [ Bli07 ], and using his\nclassification one can show that the only simple groups with m=3 are A5and PSL(2, 7)(some\nfurther explanation is given in Appendix A.3). We saw in Example 4.4(c) that AD(A5) =61\n15>4,\nand since PSL (2, 7)has character degrees 1, 3, 3, 6, 7, 8 we find that AD (PSL(2, 7)) =563\n84>6.\n115 A SHARP LOWER BOUND ON AD FOR NON -SOLVABLE GROUPS\nOur aim in this section is to prove Theorem 1.5: ifGis non-solvable, then AD (G)≥61\n15. One\ndifficulty if we rely on our existing tools is that although AD behaves well with respect to taking\nsubgroups, it does not behave well with respect to taking quo tients (c.f. Remark 4.5).\nInstead, our proof is inspired by techniques used in [ TV12 ] to prove an analogous “threshold”\nresult for the quantity\nf(G):=1\n|G|∑\nφ∈Irr(G)d(φ).\nThe key in [ TV12 ] is to exploit the inequality f(G)2≤cp(G), where the commuting probability\ncp(G)is equal to |G|−1|Irr(G)|. (Strictly speaking, this is not the original definition of c p, but the\nequivalence with this definition is well known.) The inequal ity relating fwith cp is immediate\nfrom Cauchy–Schwarz; in our setting, we can use H¨ older’s in equality to obtain an analogous\nrelationship between AD and cp, but in the opposite directio n.\nPROPOSITION 5.1. For every G we have 1≤AD(G)2cp(G). Equality is strict if G is non-abelian.\nProof. Applying H¨ older’s inequality with conjugate exponents3\n2and 3 gives\n|G|=∑\nφ∈Irr(G)d(φ)2·1≤\n∑\nφ∈Irr(G)d(φ)3\n2\n3\n∑\nφ���Irr(G)13\n1\n3\n,\nand the inequality is strict unless every φ∈Irr(G)has the same degree, i.e. unless Gis abelian.\nThe result now follows by dividing both sides by |G|and then cubing.\nREMARK 5.2. The invariant cp has been intensively studied, and in partic ular it is shown in\n[GR06 , Theorem 11] that if1\n12>cp(G)>3\n40then Gis solvable. Although this result itself is\nnot strong enough to apply Theorem 1.5, the ideas in its proof can be seen (refracted through the\nprism of [ TV12 ]) in what follows.\nThe key advantage of working with cp, compared with either for AD, is that it behaves well\nwith respect to both taking subgroups and taking quotients. In particular, we make crucial use of\nthe following result.\nLEMMA 5.3(Gallagher, [ Gal70 ]).Suppose that N /triangleleftequalG. Then\nmin(cp(G/N), cp(N))≥cp(G/N)cp(N)≥cp(G).\nIt was observed by Dixon that the largest value of cp on simple non-abelian groups is attained\natA5. We need some information about the second largest value att ained by cp on this class of\ngroups. The following is a slightly stronger version of [ TV12 , Lemma 2.3]\nLEMMA 5.4. There exists1\n28≤δ0≤1\n20with the following property: if S is a finite non-abelian simp le\ngroup and cp(S)>δ0, then S∼=A5(and cp(S) =1\n12).\nBy consulting the classification of finite simple groups (CFS G) and considering the minimal\ndegrees of non-trivial irreducible characters, it can be sh own that one can take δ0=1\n28. (This\nis best possible since cp (PSL(2, 7)) =1\n28.) We can show without resorting to the full CFSG that\n12δ0=1\n20works. Details are given in Appendix A.3: our approach invokes parts of the classification\nof finite subgroups of PGL (3,C)and PGL (4,C), given by Blichfeldt in the 1900s [ Bli05 ,Bli07 ].\nThe following lemma is presumably standard knowledge, but i t seems quicker to give an\nexplanation than to look up a reference.\nLEMMA 5.5. Let H be a perfect group.\n(a)If W is a solvable group and θ:H→W is a homomorphism, then θ(H) ={eW}.\n(b)If X is any set on which H acts, then each H-orbit in X either has size1or size≥5.\nProof. Part (a)follows by induction on the derived series of W. For part (b), observe that an\nH-orbit of size ndefines a homomorphism α:H→Snwhose image acts transitively on the\noriginal orbit. If n≤4, then Snis solvable and so α(H)is trivial by part (a); this is only possible\nifn=1.\nWe now turn to the proof that the only perfect groups with comm uting probability greater\nthan1\n20areA5and SL(2, 5)(Proposition 1.6). Our argument is patterned on the proof of [ TV12 ,\nLemma 2.4], but since we need better bounds than those in Tong -Viet’s paper, we take the oppor-\ntunity to make some simplifications and give a more streamlin ed approach.\nProof of Proposition 1.6.LetSbe the quotient of Hby any maximal proper normal subgroup. Then\nSis simple (by maximality), non-trivial (by properness), an d non-abelian (since His perfect). By\nLemma 5.3cp(S)≥cp(H)>1\n20, so by Lemma 5.4,S∼=A5.\nThus we have a surjective homomorphism H→A5, with kernel Nsay. If Nis trivial, there is\nnothing to prove; so henceforth, we assume |N| ≥2, and aim to prove that H∼=SL(2, 5).\nHis a perfect extension of A5by the group N. Suppose that we can show it is a central\nextension; then by the general theory of central extensions ,Hmust be a quotient of the Schur\ncover of A5, which is isomorphic to SL (2, 5). Since|SL(2, 5)|=2|A5| ≤ | H|, the quotient map\nfrom SL(2, 5)onto Hmust be injective, and we are done.\nTherefore, it suffices to prove that N⊆Z(H). Let kH(N)denote the number of H-conjugacy\nclasses contained in N. As in the proof of [ TV12 , Lemma 2.4], we have the inequality\n12 cp(H)≤kH(N)\n|N|(5.1)\n(We briefly sketch how this works. If Mis any finite group and N/triangleleftequalM, then [ GR06 , Lemma 1(iii)],\nwhich is actually proved in [ KR93 , Remark A2’], tells us that\n|Irr(M)| ≤kM(N)sup\nB≤M/N|Irr(B)|.\nWe then apply this with M=H, noting that |H:N|=60, and appeal to the fact that each\nsubgroup of A5has at most five distinct irreducible characters.)\nSince cp(H)>1\n20, it follows from ( 5.1) that kH(N)>3\n5|N|. On the other hand, kH(N)counts\nthe number of orbits for the conjugation action of HonN. By Lemma 5.5, the size of each non-\nsingleton orbit is at least 5. Therefore, if Fdenotes the set of fixed points of the action,\n|N| ≥5(kH(N)−|F|)+|F|=5kH(N)−4|F|,\n13and combining this with the previous lower bound on kH(N)gives\n|F| ≥1\n4(5kH(N)−|N|)>1\n2|N|.\nNow observe that F=Z(H)∩N. So by the previous inequality |N:Z(H)∩N|<2, which is\nonly possible if Z(H)∩N=N, and this completes the proof.\nFrom this, we can show that on the class of finite perfect group s, the AD-constant is minimized\natA5. In fact, a more precise statement can be made.\nCOROLLARY 5.6. Let H be a non-trivial perfect group which satisfies AD(H)≤2√\n5≅4.47. Then\nH∼=A5andAD(H) =61\n15≅4.07.\nProof. By Proposition 5.1, cp(H)>AD(H)−2≥1\n20. Hence, by Proposition 1.6,His isomorphic to\neither A5or SL(2, 5). But the second possibility is excluded, since we saw in Exam ple4.4(c) that\nAD(SL(2, 5)) =9\n2>2√\n5.\nProof of Theorem 1.5.Since Gis not solvable, its derived series stabilizes at some subgr oup H≤G\nthat is perfect and non-trivial. By monotonicity, AD (G)≥AD(H); and by Corollary 5.6, we have\nAD(H)≥min/parenleftBig\n2√\n5,61\n15/parenrightBig\n=61\n15.\nACKNOWLEDGEMENTS\nThe results presented here are part of a wider programme to un derstand the amenability con-\nstants of Fourier algebras of locally compact groups, some o f which was presented at the 2022\nmeeting of the Canadian Abstract Harmonic Analysis Symposium . I would like to thank the organiz-\ners and participants of that meeting for their interest; par ticular thanks are due to Brian Forrest\nand John Sawatzky.\nCloser to home, thanks are due to Paul Levy for several helpfu l conversations. I also thank Ge-\noff Robinson for his interest in the topic of this paper at an e arlier stage, and for various messages\nof encouragement.\nFinally, I would like to acknowledge the work of the contribu tors and maintainers of the\nGroupPropsWiki , and all those on MathOverflow who have been willing to share ideas and expertise\nwith a Bear of Very Little Brain.\n14A E ASIER PROOFS OF SOME KNOWN RESULTS\nA.1 F INITE GROUPS WITH TWO CHARACTER DEGREES AND DERIVED SUB -\nGROUP OF ORDER 2\nThe groups described in the title are, by Corollary 2.3, those finite non-abelian groups where AD\nattains its minimum value. In this section, we give a quick pr oof that these groups are precisely\nthose in which the centre has index 4.\nLetGbe a finite group with cd (G) ={1, 2}and|G′|=2. We have Irr (G) =Irr1(G)∪Irr2(G);\nletl=|Irr1(G)|and m=|Irr2(G)|. Also, since every conjugacy class injects into G′and|G′|=2,\neach conjugacy class in Ghas size 1 or 2. Let s=|Z(G)|and let nbe the number of conjugacy\nclasses of size 2.\nNote that |G|=∑φ∈Irr(G)d(φ)2=l+4m. Since|G′|=2, we have |G|=2l, and so l=4m. On\nthe other hand, |G|=s+2n, while s+n=l+msince the character table is square. Therefore,\ns+2n=8m,s+n=5m.\nSolving for sand nyields s=2mand n=3m. In particular, we conclude that\n|G:Z(G)|=8m\n2m=4 .\nConversely, suppose that |G:Z(G)|=4. The argument that follows is essentially the same as\nin [Cho23a ], but we include the details for sake of completeness.\nNote that G/Z(G)cannot be cyclic (otherwise by lifting the generator we woul d find that G\nis abelian) and hence it is isomorphic to C2×C2. Pick two generators for G/Z(G)and lift them\ntox,y∈G. Then x2,y2and[x,y]all belong to Z(G). Since G=Z(G)∪xZ(G)∪yZ(G)∪\nxyZ(G), a short case-by-case analysis shows that every commutator inGequals either eor[x,y].\nIn particular |G′|=2.\nMoreover, A=Z(G)∪xZ(G)is an Abelian subgroup of Gwith index 2. Hence, as shown\nin the proof of Lemma 3.8, cd(G)⊆ {1, 2}. Since Gis non-abelian, this inclusion of sets is an\nequality.\nA.2 A SELF -CONTAINED PROOF OF LEMMA 2.5\nWe give a proof of Lemma 2.5, which works even for infinite groups. Thus, for this subsect ion\nonly, we let Gbe a not-necessarily-finite group, and we suppose that |G′|=2.\nLetπbe a finite-dimensional, unitary, irreducible representat ion of Gwith dimension d≥2.\n(When Gis finite, every irreducible representation of Gis automatically finite-dimensional and is\nequivalent to a unitary representation.) Our aim is to show t hatdis even.\nLEMMA A.1. Let G′={e,z}. Then z∈Z(G).\nProof. Ifα∈Aut(G), then α(G′) = G′and α′(e) = e, hence α(z) = z. Now take αto be an\narbitrary inner automorphism of G.)\nLEMMA A.2. Let g∈G. Then g2∈Z(G).\n15Proof. Letg,x∈G. Then gxg−1= [g,x]x. Since[g,x]is central (by Lemma A.1) and[g,x]2=e,\ng2xg−2=g/parenleftbig[g,x]x/parenrightbig\ng−1= [g,x](gxg−1) = [ g,x][g,x]x=x.\nThus g2is central.\nProof that d is even. Recall that every non-trivial commutator in Gis equal to z. Since πis irre-\nducible and d≥2,π(G)is not Abelian, hence π(z)/\\e}atio\\slash=Iπ. By Lemma A.1and Schur’s lemma,\nπ(z)is a scalar multiple of Iπ; since z2=e, it follows that π(z) =−Iπ.\nSince Gis non-abelian, there exist x,y∈Gthat do not commute. Since xyx−1=zy, we have\nπ(x)π(y)π(x)−1=−π(y). On the other hand, by Lemma A.2and Schur’s lemma, π(y2)is a\nscalar multiple of Iπ. Pick λ∈Tsuch that π(y)2=λ2Iπ; then U:=λ−1π(y)is an involution in\nLin(Hπ)and Uis conjugate to −U.\nSince Uis an involution it has exactly deigenvalues counted with multiplicity, and these eigen-\nvalues belong to {−1, 1}; but since Uis conjugate to −U, the eigenvalues −1 and 1 must occur\nwith equal multiplicity, msay. Thus d=2m. (Alternatively, observe that1\n2(Id+U)is an idempo-\ntent which has trace equal tod\n2, which again forces dto be even.)\nA.3 A PROOF OF LEMMA 5.4 WITH δ0=1/20\nWe follow the general strategy seen in the proofs of [ TV12 , Lemma 2.3] and [ GR06 , Theorem 11].\nSuppose Sis simple and non-abelian. Writing mfor mindeg (S), we have\n|S|−1≥ ∑\nφ∈Irr(G),d(φ)>1d(φ)2≥(|Irr(S)|−1)m2=m2cp(S)|S|−m2,\nand rearranging gives m2−1≥(m2cp(S)���1)|S|. If we are given an explicit δ0>0 such that\ncp(S)>δ0, it follows that\n1\n|S|≥m2cp(S)−1\nm2−1>m2δ0−1\nm2−1. (A.1)\nProvided that m2δ0>1, the inequality ( A.1) gives an explicit upper bound on |S|.\nThus, in cases where mis sufficiently large, Sbelongs to some small list of known examples,\nand in each case we can see by inspection that mis actually small (giving a contradiction). Sepa-\nrate ad hoc arguments are then needed to deal with the cases where mis “small”.\nIn [TV12 , Lemma 2,3], this strategy is used with δ0=16\n225, and so the easy part of the argument\nworks for all m≥4; the only remaining cases are those with m=3, and these are covered by the\nfollowing result.\nTHEOREM A.3(Blichfedlt, implicitly) .Let S be a finite simple group with an irreducible representat ion\nof degree 3. Then S∼=A5or S∼=PSL(2, 7).\nInspecting the proof of [ TV12 , Lemma 2.3], the “threshold value” stated there can be impro ved\nfrom16\n225to1\n15, provided that we know the simple groups of order ≤225. However, the methods\nin that paper cannot reach1\n16(since we require m2δ0>1), and for our eventual application to\nProposition 1.6, we require δ0≤1\n20. We therefore need the following additional result.\nTHEOREM A.4(Blichfeldt, implicitly) .Let S be a finite simple group with an irreducible representat ion\nof degree 4. Then S∼=A5.\n16For the reader who wishes to consult the original papers, we p rovide some details of how the\ntheorems stated above are derived from the results stated in [Bli05 ,Bli07 ].\nProofs of Theorem A.3andA.4.Letd∈ {3, 4}, and let Sbe a finite simple group with an irreducible\nrepresentation of degree d. Then the image of Sunder this representation can be identified with\na finite subgroup /tildewideS≤SU(d)that acts irreducibly on Cd. Let S0be the image of /tildewideSin PGL d(C). Of\ncourse, S∼=/tildewideS∼=S0.\nIn the language of [ Bli07 , p. 553] and [ Bli05 , p. 205], S0is primitive: this follows from the fact\nthat simple groups cannot act non-trivially on sets of size ≤4, c.f. Lemma 5.5.\nTHE CASE d=3.The primitive simple finite subgroups of PGL 3(C)are determined up to iso-\nmorphism in [ Bli07 , Section 24] (relying on previous work of Maschke): any such subgroup must\nbe isomorphic to A5, PSL(2, 7)orA6. Moreover, it is observed that in the last case A6cannot be\nlifted from PGL 3(C)up to GL 3(C); thus/tildewideSmust be isomorphic to either A5or PSL(2, 7), and this\ncompletes the proof of Theorem A.3.\nTHE CASE d=4.The primitive simple finite subgroups of PGL 4(C)are determined up to iso-\nmorphism in [ Bli05 , Section III]; the list appears as items 22◦–27◦on pages 225–226 of that paper,\nand consists (in modern notation) of: A5,A6,A7, PSL(2, 7), and PSp (4, 3). Blichfeldt does not state\nfor which of these S0the corresponding “lift” in GL (4,C)is simple, but if we invoke known char-\nacter tables for these groups then we see that the only possib ility for/tildewideSisA5(none of the others\nhave degree-4 irreducible representations), and this comp letes the proof of Theorem A.4.\nProof of Lemma 5.4with δ0=1\n20.LetSbe non-abelian and simple, and let m=mindeg(S). Sup-\npose that cp (S)>1\n20. We start by showing that this forces m≤4. For, if m≥5, taking δ0=1\n20in\n(A.1) gives\n1\n|S|>/parenleftbiggm2\n20−1/parenrightbigg1\nm2−1≥/parenleftbigg25\n20−1/parenrightbigg1\n24=1\n96.\nBut up to isomorphism, the only non-abelian simple group of o rder<96 is A5, which we know\nhasm=3, and this gives a contradiction.\nTherefore m∈ {2, 3, 4}. It is well documented that finite simple groups have no irred ucible\nrepresentations of degree 2 (see e.g. [ JL01 , Corollary 22.13] for an elementary proof), and it follows\nfrom Theorem A.4that m/\\e}atio\\slash=4. The only remaining possibility is that m=3. By Theorem A.3, this\nimplies that S∼=A5orS∼=PSL(2, 7); and since cp (PSL(2, 7)) =1\n28<1\n20, the second case is ruled\nout. We conclude that S∼=A5, as required.\n17REFERENCES\n[Bli05] H. F. B LICHFELDT ,The finite, discontinuous primitive groups of collineation s in four vari-\nables , Math. Ann., 60 (1905), pp. 204–231.\n[Bli07] ,The finite, discontinuous, primitive groups of collineatio ns in three variables , Math.\nAnn., 63 (1907), pp. 552–572.\n[Cho23a] Y. C HOI,An explicit minorant for the amenability constant of the Fou rier algebra , Int. Math.\nRes. Not. IMRN, (2023), pp. 19390–19430.\n[Cho23b] ,Moments of character degrees - is this result new or folklore ?MathOverflow, 2023.\nURL:https://mathoverflow.net/q/437938 (version: 2023-0 1-08).\n[Gal70] P. X. G ALLAGHER ,The number of conjugacy classes in a finite group , Math. Z., 118 (1970),\npp. 175–179.\n[GR06] R. M. G URALNICK AND G. R. R OBINSON ,On the commuting probability in finite groups ,\nJ. Algebra, 300 (2006), pp. 509–528.\n[JL01] G. J AMES AND M. L IEBECK ,Representations and characters of groups , Cambridge Univer-\nsity Press, New York, second ed., 2001.\n[Joh94] B. E. J OHNSON ,Non-amenability of the Fourier algebra of a compact group , J. London Math.\nSoc. (2), 50 (1994), pp. 361–374.\n[KR93] L. G. K OV´ACS AND G. R. R OBINSON ,On the number of conjugacy classes of a finite group ,\nJ. Algebra, 160 (1993), pp. 441–460.\n[LLW96] A. T.-M. L AU, R. J. L OY,AND G. A. W ILLIS ,Amenability of Banach and C∗-algebras on\nlocally compact groups , Studia Math., 119 (1996), pp. 161–178.\n[Mil38] G. A. M ILLER ,Groups whose commutator subgroups are of order two , Amer. J. Math., 60\n(1938), pp. 101–106.\n[San21] T. S ANDERS ,Coset decision trees and the Fourier algebra , J. Anal. Math., 144 (2021),\npp. 227–259.\n[TV12] H. P. T ONG -VIET,On groups with large character degree sums , Arch. Math. (Basel), 99\n(2012), pp. 401–405.\n[Wal72] M. E. W ALTER ,W∗-algebras and nonabelian harmonic analysis , J. Functional Analysis, 11\n(1972), pp. 17–38.\nYEMON CHOI, DEPARTMENT OF MATHEMATICS AND STATISTICS , LANCASTER UNIVERSITY , LANCASTER\nLA1 4YF, U NITED KINGDOM .\ny.choi1@lancaster.ac.uk ,y.choi97@cantab.net\n18"    },    {        "title": "2402.14038v1.Isotropic_and_anisotropic_neutron_star_structure_in_4D_Einstein_Gauss_Bonnet_Gravity.pdf",        "content": "Isotropic and anisotropic neutron star structure in 4D Einstein-Gauss-Bonet Gravity\nGholam Hossein Bordbar∗, Mohammad Mazhari and Ahmad Poostforush\nPhysics Department and Biruni Observatory, Shiraz University, Shiraz 71454, Iran\nWith regards to the coupling constant and the strong magnetic field of neutron stars, we have\nstudied these stars in the 4D Einstein-Gauss-Bonnet (4D EGB) gravity model in order to grasp\na better understanding of these objects. In this paper, we have shown that the neutron stars\nproperties are considerably affected by the coupling constant and magnetic field. We have found\nthat as a consequence of the strong magnetic field and the coupling constant, the maximum mass\nand radius of a neutron star are increasing functions of the coupling constant, while Schwarzschild\nradius, compactness, surface gravitational redshift, and Kretschmann scalar are decreasing functions.\nAdditionally, our study has shown that the physical properties of a magnetized neutron star not\nonly are greatly influenced by the strong magnetic field, but also by the anisotropy. Moreover, we\nhave shown that to obtain the hydrostatic equilibrium configuration of the magnetized material,\nboth the local anisotropy effect and the anisotropy due to the magnetic field should be considered.\nFinally, we have found that in the anisotropic magnetized neutron stars, the maximum mass and\nradius do not always increase with increasing the internal magnetic field.\nI. INTRODUCTION\nA neutron star is an ideal astrophysical laboratory for testing condensed matter physics theories and making\nconnections between nuclear physics, particle physics, and astrophysics [1, 2]. This makes the neutron star as a valuable\nsystem to study its various properties. The purpose of present work is to evaluate the structural characteristics of\nisotropic and anisotropic magnetized neutron stars containing pure neutron matter. Recent observations have revealed\nthat the neutron stars have a mass M≥2.5M⊙[3, 4].\nOwing to the existence of a strong magnetic field, there is anisotropy in the magnetized neutron star pressure;\ntherefore, the pressure is different in the radial ( Pr) direction from the tangential ( Pt) direction. According to\navailable evidence [5–7], it is difficult to obtain the strength of the magnetic field inside the magnetized neutron stars,\nprompting researchers to develop theoretical models that can be used to analyze the influence of strong magnetic\nfields on the physical parameters of magnetized neutron stars. Magnetized neutron stars are classified into two groups\nbased on their surface magnetic field. Pulsars and magnetars have different magnetic fields on their surface. Pulsars\nare found to have surface magnetic fields of approximately 1013Gto 1015G[8], and magnetars are found to have\nsurface magnetic fields of approximately 1016Gto 1017G[9].\nEinstein-Gauss-Bonnet gravity is a natural generalization of general relativity. As the most general torsion-free\ntheory of gravity, this leads to stable second-order equations of motion in higher dimensions. A version of this theory\nwas proposed by Lanczos [10], and later confirmed by David Lovelock [11, 12]. Because this theory shares Einstein’s\nprinciple of gravity, the Einstein-Gauss-Bonnet (EGB) gravity is more suitable than other higher-curvature gravity\ntheories. Einstein’s and GB’s terms are included in Lovelock’s Lagrangian theory. As a result, the famous EGB\ngravity was proposed, which is the most straightforward nontrivial generalization of Einstein’s gravity. It has been\nshown that static wormhole solutions in the EGB theory of gravity with higher dimensions can be found [13, 14].\nA notable feature of EGB gravity in the 4-dimensional case is that the Euler-Gauss-Bonnet term becomes a topo-\nlogical invariant that contributes neither to equations of motion nor gravitational dynamics [15]. A recent paper\nby Glavan and Lin [16] proposes a modified general covariant theory of gravity in 4-spacetime dimensions ( D= 4)\nthat propagates only the massless graviton and bypasses Lovelock’s theorem. The Einstein gravity theory with the\ncosmological constant is the only gravity theory that meets several conditions; metricity, diffeomorphism invariance,\nand second-order equations of motion. Essentially, this theory is formulated in higher dimensions, D > 4, and has\nEinstein-Hilbert terms with cosmological constants, as well as GB coupling rescaled as α/(D−4). 4D theory is defined\nasD→4. As a result, the GB invariance gives rise to the nontrivial contributions in gravitational dynamics, at the\nsame time preserving the graviton degrees of freedom and retaining the freedom from Ostrogradsky instability [15].\nThis new theory is called 4D EGB gravity (see Ref. [16] for more details).\nIn other gravities such as f(R) gravity, it is specifically discussed the mass-radius diagram for static neutron\nstar models obtained by the numerical solution of modified Tolman-Oppenheimer-Volkoff equations [17]. This study\nexplores the impact of different Lagrangians, including quadratic and cubic corrections, on the mass-radius relation.\n∗email address: ghbordbar@shirazu.ac.irarXiv:2402.14038v1  [gr-qc]  21 Feb 20242\nIn addition, it addresses the constraints imposed by the presence of an extra degree of freedom and the stiffness of the\nequation of state, providing valuable insights into the mass-radius relation for neutron stars in the context of f(R)\ngravity. There is also the possibility of discussing strong gravitational fields in local objects in f(R) gravity due to\nits higher-order curvature terms. Neutron stars are frequently studied within this framework [17]. An inflationary\nphase of the universe (the earliest phases of the Universe) can also be unified by f(R) theory [18, 19]. In addition,\nforf(R,T) gravity, the widest parameters and masses are investigated to ensure the stability and stability of static\nwormhole models [20].\nNeutron stars are denser and more compact astronomical bodies than ordinary stars, and are strongly dependent\non their equation of state (i.e., the relationship between the density and pressure). It is unknown what a neutron\nstar comprises, and how it is structured internally. Because of this lack of knowledge, physicists have to resort to\ntheoretical assumptions on different equations of state to determine the characteristics of neutron stars. In view of\nthis, several models have been proposed by other research groups for the equations of state of these objects. Despite\nthis, few of them are stable and realistic for describing the matter inside these compact bodies. The problem of\ncompact stars made of anisotropic material has been studied in Refs. [21, 22, 24, 25].\nOur previous researchs examined neutron star structure properties in the absence and presence of a magnetic field\nusing an isotropic neutron star model. According to the contemporary equation of state, using the equation of state\nbased on the current nuclear potential AV18 [26], we calculated the neutron star structure [27], and a comparison\nwas made between the structure of cold and hot neutron stars with quark cores and those without a quark core\n[28, 29]. In addition, we examined the effects of the cosmological constant Λ and other gravity theories such as (3 +\n1)-dimensional rainbow gravity and spin-2 massive gravitons on the properties of neutron stars [30–33]. Afterwards,\nwe evaluated the properties of cold neutron stars with a quark core [29]. The properties of spin polarized neutron\nmatter in the presence of strong magnetic fields at zero [34–36], and finite temperatures [37] have also been studied.\nIt has been attempted many times to conceive of spherically symmetric fluid spheres with anisotropic matter as\nsuitable internal models for these stars. The Bower and Liang model [38], which considers radial pressure as part of\nthe anisotropic function, strengthened the idea of anisotropic materials. Despite this, Ruderman [39] observed that\nnuclear matter becomes anisotropic at densities significantly greater than 1015g/cm3. Moreover, anisotropic pressure\naffects the mass-radius relation in compact stars [40–42]. A strong magnetic field appears to alter the maximum mass\nof a compact stellar object, but researchers [43–47] remain split on whether it increases or decreases the maximum\nmass, a difficult open question.\nIn the present study, we evaluate the structural properties of an anisotropic neutron star containing pure neutron\nmatter and also an isotropic neutron star that is not magnetized. We use the lowest-order constrained variational\n(LOCV) method to calculate the equation of state of neutron matter for non-magnetized and magnetized neutron\nstars in the absence and presence of a magnetic field. As a final step, we solve the equations in 4D EGB gravity to\ndetermine the maximum mass, radius, and other properties of the system using this equation of state.\nII. EQUATION OF STATE OF ISOTROPIC NON- MAGNETIZED AND ANISOTROPIC\nMAGNETIZED NEUTRON STAR\nThe equation of state is necessary to investigate the properties of isotropic non-magnetic neutron stars and\nanisotropic magnetic neutron stars, allowing us to calculate the structural properties of the star using this equa-\ntion. In this section, we calculate the equation of state for isotropic non-magnetized and anisotropic magnetized\nneutron stars. Here, at first we need to have the total energy of the system. In order to calculate the energy, we use\nthe LOCV method. In the LOCV formalism, we consider the wave function of the interacting system as follows,\nψ=Fϕ. (1)\nHere, ϕis the wave function of Nnon-interacting neutrons and Fis the N-body correlation operator, which is\nconsidered as follows according to the Jastrow approximation,\nF=SY\ni>jf(ij), (2)\nwhere Sandf(ij) are symmetry operator and two-body correlation function, respectively. For calculating the\nHamiltonian expectation value, we use the cluster expansion, and keep only the one-body and two-body terms,\nE([f]) =⟨ψ|H|ψ⟩\nN⟨ψ|ψ⟩=E1+E2. (3)3\nHere, Hdenotes the Hamiltonian of system, and E1is the kinetic energy of Fermi gas which is calculated as follows,\nE1=X\ni=+,−3\n5ℏ2k(i)2\nF\n2mρ(i)\nρ, (4)\nwhere k(i)\nF= (6π2ρ(i))(1\n3)is the Fermi momentum of a neutron with spin projection i. The two-body energy E2is\nalso obtained as follows,\nE2=1\n2NX\nij⟨ij|ν(12)|ij−ji⟩, (5)\nwhere\nν(12) = −ℏ2\n2m[f(12),[▽2\n12, f(12)]] + f(12)V(12)f(12). (6)\nIn the above equation, V(12) is the two-body interaction potential. In the Ref. [26], the detail of calculations for\nnuclear matter equation of state has been given.\nIn the presence of a strong magnetic field, the contribution of magnetic energy for the magnetized neutron matter\nis added to the total energy as follows,\nE(ρ, B) =E1+E2−µnBδ. (7)\nIn the above equation, µn,Bandδare the neutron magnetic momentum, magnetic field, and spin polarization\nparameter, respectively. To obtain the equation of state of the neutron star matter, at each magnetic field, we use\nthe following equation,\nP(ρ, B) =ρ2\u0012∂E(ρ, B)\n∂ρ\u0013\nB. (8)\nIn a strong magnetic field, the pressure of neutron star differs in the radial and tangential directions, indicating that\nthe neutron star undergoes the anisotropy. From the following equation, we can calculate the tangential ( Pt) and\nradial ( Pr) pressure components, respectively [48],\nPt=ρ2\u0012∂E(ρ, B)\n∂ρ\u0013\nB+B2\n8π, (9)\nPr=ρ2\u0012∂E(ρ, B)\n∂ρ\u0013\nB−B2\n8π. (10)\nIn order to examine the effect of magnetic field, we here consider a Gaussian function of the density as follows [43],\nB(ρ) =Bsurf+B0[1−exp(−β(ρ\nρ0)θ)], (11)\nwhere ρis the neutrons density, ρ0is the saturation density, Bsurfis the magnetic field on the star’s surface, which\nis equal to 1015G, and B0is the magnetic field in the dense (central) region of the star. There are two behaviors for\nthe magnetic field based on the remaining parameters βandθ. For fast decay, β= 0.02 and θ= 3.0 are considered,\nandβ= 0.05 and θ= 2.0 are considered for slow decay. In a fast decay (slow decay) of the magnetic field, the field\ndrops rapidly (slowly) from the star center to its surface (see Ref. [45] for more details). Now, using the Gaussian\nmagnetic field, as mentioned previously, we calculate the equation of state of the system.\nIn the absence of a magnetic field ( B= 0), the radial and tangential components of pressure are equal, so there is\nno anisotropy in the neutron star. In addition, the difference in the tangential and radial pressures is not significant\nin the case of weak magnetic fields. Nevertheless, when the magnetic field increases, the anisotropy becomes clear\nunder the pressure of the neutron star. There is a clear indication of this phenomena in Figs. 1 and 2.\nFor different internal magnetic fields B0in two cases of the Gaussian magnetic field decay (fast decay and slow\ndecay), we plot the equations of state in Figs. 1 and 2. It is clear from these graphs that as the internal magnetic\nfield of the neutron star increases, the anisotropy in the magnetized neutron star increases too. As the central density\nincreases, there is a rising difference between the radial and tangential pressures. The pressure in the tangential4\n0.00 .51 .01 .52 .02 .53 .0020406080100120Pressure (1034 dyn/cm2)/s114\n/s32(1015 gr/cm3) B=0\n0.00 .51 .01 .52 .02 .53 .0020406080100120Pressure (1034 dyn/cm2)/s114\n (1015 gr/cm3) SPr for B0=0.5×1018 \nSPt for B0=0.5×1018 \nFPr for B0=0.5×1018 \nFPt for B0=0.5×1018\nFigure 1: The equation of state of non-magnetized neutron star for B= 0 (up panel) and (down panel) the radial pressure and\ntangential pressure vs density of magnetized neutron star for two mode slow-decay (S) and fast-decay (F) with B0= 0.5×1018G.\ncomponent increases at a slower rate in slow decay than in fast decay, and the pressure in the radial component\ndecreases at a slower rate in slow decay than in fast decay for intermediate values of the central density. To use the\nequations of the isotropic state and the anisotropic magnetized state (the radial pressure component of the slow decay\nis considered here), it is necessary to investigate the causality condition. In a physically acceptable model, the speed\nof sound ( ν) should be less than the speed of light ( c) to fulfill the causality conditions for the equation of state of\nneutron stars. Therefore, the condition of causality is given as follows,\n0⩽ν2\nc2=1\nc2(dPr\ndρ)⩽1 (12)\nWe present our results for the speed of sound in terms of the central density in Fig. 3. Accordingly, the magnetized\nneutron star equation of state also fulfills the causality condition for the magnetic fields mentioned above (see Fig.\n3 for details). In accordance with the results presented in Ref. [49], the equation of state of anisotropic magnetized\nneutron stars is relevant for studying their structural properties, such as gravitational mass and corresponding radius.\nAs discussed above, the equations of the state we are considering satisfy the causality conditions, therefore we can\nuse them in the structure calculations.\nIt is well known that to compute the structural properties of a compact object, we must solve the general relativistic5\n0.00 .51 .01 .52 .02 .53 .0020406080100120Pressure (1034 dyn/cm2)/s114\n (1015 gr/cm3) SPr for B0=0.9×1018 \nSPt for B0=0.9×1018 \nFPr for B0=0.9×1018 \nFPt for B0=0.9×1018\n0.00 .51 .01 .52 .02 .53 .0020406080100120140Pressure (1034 dyn/cm2)/s114\n (1015 gr/cm3) SPr for B0=2.1×1018 \nSPt for B0=2.1×1018 \nFPr for B0=2.1×1018 \nFPt for B0=2.1×1018\nFigure 2: The radial pressure and tangential pressure vs density of magnetized neutron star for B= 0.9×1018G(up panel)\nand for two mode slow-decay (S) and fast-decay (F) with B0= 2.1×1018G(down panel).\nhydrostatic equilibrium equations with the desired gravity. In the next section, we discuss the basic formalism and\nmodified hydrostatic equilibrium equations for 4D EGB gravity.\nIII. THE STRUCTURE OF ISOTROPIC NON-MAGNETIZED NEUTRON STAR IN 4D EGB GRAVITY\nIn this section, we compute the structure of an isotropic non-magnetized neutron star in 4D EGB gravity. A\nbrief overview of 4D EGB gravity is given in the following part, followed by a presentation of the modified Tolman-\nOppenheimer-Volkoff (TOV) equations for the stellar structure of isotropic compact stars.\nA. MODIFIED TOV EQUATIONS IN 4D EGB GRAVITY\nTo obtain the field equations for 4D EGB gravity, we consider the D-dimensional action as follows [16],\nS=c4\n16πGZ\nMdDx√−g\u0012\nR+α\nD−4LGB\u0013\n+Smatter . (13)6\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0n2/c2\n (1015 gr/cm3) B=0\n B0=0.5´1018\n B0=0.9´1018\n B0=2.1´1018\nFigure 3: Sound speed vs central density for different magnetic fields.\nAs shown in Eq. (13), we have rescaled the coupling constant α→α/(D−4). Therefore, the above theory is free of\nOstrogradski instability, and a novel 4D EGB gravity can be defined in the limit D→4. In Eq. (13), Ris the Ricci\nscalar and gis the determinant of the metric tensor gµν. The GB coupling constant αhas dimension of [ length ]2,\nandLGBis the GB term defined by\nLGB≡RµνρσRµνρσ−4RµνRµν+R2. (14)\nIn Eq. (13), Smatter represents the distribution of matter lagrangian, which is independent of its derivatives, and\ndepends only on the metric tensor components gµν. The variation of Eq. (13) in terms of the metric tensor gµν, leads\nto the following field equations,\nGµν+α\nD−4Hµν=8πG\nc4Tµν, (15)\nwhere the Einstein tensor Gµνand Lanczos tensor Hµνare defined by\nGµν≡Rµν−1\n2Rgµν, (16)\nHµν≡2\u0012\nRRµν−2RµσRσ\nν−2RµσνρRσρ−RµσρδRσρδ\nν\u0013\n−1\n2gµνLGB. (17)\nThe energy-momentum tensor is given by\nTµν=−2√−gδ(√−gLm)\nδgµν, (18)\nwhere Tµνis the energy-momentum tensor of the matter field, Ris the Ricci scalar, Rµνthe Ricci tensor and Rµσνρ\nbeing the Riemann tensors, respectively. When α= 0, it is easy to see that Eq. (15) reduces to the conventional\nEinstein field equation. The corresponding line element is expressed as follows in the form of a spherically symmetric\n4-dimensional metric describing the interior spacetime of non-rotating neutron stars,\nds2\n4=−e2Φ(r)(cdt)2+e2λ(r)dr2+r2dΩ2\n2, (19)\nwhere the metric functions Φ( r) and λ(r) depend only on the radial coordinate r, and dΩ2\n2represents the metric on\nthe surface of 2-sphere, namely,\ndΩ2\n2=dθ2\n1+sin2θ1dθ2\n2.7\nIn addition, we assume that the matter source is described by an isotropic fluid whose energy-momentum tensor is\ngiven by\nTµν= (ϵ+P)uµuν+Pgµν, (20)\nwhere uµis the 4-velocity of the fluid, ϵ=c2ρis the energy density of standard matter, ρis the mass density, and P\nis the pressure. Here, uµis defined as uµ= (e−Φ(r),0,0,0) with uµuµ=−1.\nAccording to Eq. (19), the effective 4D EGB theory can be obtained in the singular limit D→4 with α→α/(D−4).\nBy carefully performing the limit D→4, we obtain the following dimensionally reduced field equations [22],\n2\nrdλ\ndr=e2λ8πGρ\nc2−(1−e−2λ)\nr2(1−α(1−e−2λ)\nr2)\n(1 + 2 α(1−e−2λ)\nr2), (21)\n2\nrdΦ\ndr=e2λ8πGP\nc4+(1−e−2λ)\nr2(1−α(1−e−2λ)\nr2)\n(1 + 2 α(1−e−2λ)\nr2). (22)\nIn addition, the non-trivial radial component ( ν= 1) of the conservation law of the isotropic energy-momentum tensor\n(continuity equation) obtained from the relation ∇µTµ\nν=∂µTµ\nν+ Γµ\nρµTρ\nν−Γρ\nµνTµ\nρ= 0, is given by the following\nformula,\ndP\ndr=−(ρc2+P)dΦ\ndr. (23)\nTo study the general structure of the solution, we replace the metric function ( e−2λ(r)) by e−2λ(r)= 1−\nr2\n2αG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\n(see Ref. [22], for more details). There are also some authors who convenient adapt\nthe asymptotic function of the metric function e−2λ(r)≈1−2Gm(r)\nc2rfor computing the compact object structure. This\npoint has been fully discussed in Ref. [23]. After some algebra, the generalized TOV equations in 4D EGB gravity\nfor an isotropic neutron star with spherical symmetry in hydrostatic equilibrium are as follows,\ndP\ndr=−r\n2(ρ(r)c2+P(r))\n1−r2\n2αG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\n×8πGP (r)\nc4+1\n2αG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\u0012\n1−1\n2G\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\u0013\n1 +1\nG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013 , (24)\ndm(r)\ndr=−m(r)\nr+c2r2\n2G\u00128πGρ(r)\nc2−1\n2αG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\u0012\n1−1\n2G\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\u0013\n1 +1\nG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\u0013\n+c2r2\n2αG2\u0012r\n1 +8αG2m(r)\nc2r3−1\u0013\n, (25)\nwhere ρ(r) is the energy density, Gis the gravitational constant, αis the the coupling constant of the Gauss-Bonnet\nterm and m(r) is the gravitational mass inside the radius r. Thus, by taking α= 0, we return to the TOV equations\nfor a neutron star in Einstein gravity [50].\nHere, to determine the structure of a neutron star, we use the equation of state for a nonmagnetized isotropic\nneutron star and generalized isotropic TOV equations with boundary conditions ( P(0) = Pcandm(0) = 0). By\nintegrating various coupling constant values from the center of the star r= 0 to its surface r=R, where the pressure\nis zero, we obtain the maximum mass and the corresponding radius of the star. The following section presents our\nnumerical results and discusses the structural properties of neutron stars in 4D EGB gravity.8\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.51.01.52.0\n M[M  ]\nr (1015 gr/cm3) = -3 km2 \n = -2 km2\n = -1 km2\n =0 (GR)\n =1 km2\n =2 km2\n =3 km2\n8 10 12 14 160.00.51.01.52.0\n M[M  ]\nR(km) = -3 km2 \n = -2 km2\n = -1 km2\n =0 (GR)\n =1 km2\n =2 km2\n =3 km2\nFigure 4: M- ρrelation (up panel) and M-R relation (down panel) of isotropic non-magnetized neutron star for different coupling\nconstants ( α) in 4D EGB gravity.\nB. RESULTS AND DISCUSSIONS\nBy solving the stellar structure equations for a range of central density values, we can classify a family of neutron\nstars (NSs) in 4D EGB gravity. Fig. 4 illustrates the results of this study. It is evident that the coupling constant\naffects the maximum mass of isotropic non-magnetized neutron stars. Consequently, as the coupling constant increases,\nso does the gravitational mass and the corresponding radius. For α= 0, the 4D EGB corresponding graphs agree with\nEinstein gravity. In addition, the 4D EGB gravity diagrams satisfy the requirement for neutron star stability, i.e., the\npositiveness of the derivative of the gravitational mass function with respect to the central density. The maximum\nmass and corresponding radius values of neutron stars associated with 4D EGB gravity tend to approach each other\nas the coupling constant increases.\nIn the next step, we examine how the coupling constant affects the maximum mass, radius, and other structural\nproperties of the neutron star. In D= 4, our results are then compared with the structure of a neutron star without\nthe coupling constant. The results of this study are presented in Table I. According to the values in this table, the\nmaximum mass and the corresponding radius increase as the coupling constant increases in the 4D EGB gravity. In\nother words, higher-order curvature terms (here the second order GB term) lead to more massive stars.\nThe next step is to examine how the coupling constant affects other structural properties of neutron stars. In9\nTable I: Properties of isotropic non-magnetized neutron stars for different coupling constants ( α) in 4D EGB gravity.\nα(km2) Mmax(M⊙) R(km) RSch(km) σ(RSch\nR) z K (10−8m−2)\n−3.0 1 .46 8 .18 4 .71 0 .535 0 .575 2 .97\n−2.0 1 .54 8 .38 4 .97 0 .592 0 .567 2 .92\n−1.0 1 .61 8 .56 5 .19 0 .606 0 .594 2 .87\n0.0 1 .68 8 .73 5 .43 0 .621 0 .625 2 .82\n+1.0 1 .75 8 .89 5 .64 0 .635 0 .655 2 .78\n+2.0 1 .81 9 .04 5 .84 0 .646 0 .680 2 .74\n+3.0 1 .86 9 .18 5 .99 0 .653 0 .699 2 .68\nthe selected spacetime Eq. (19), structural properties such as the modified Schwarzschild radius, compactness, and\nsurface redshift are defined as follows,\nRSch=GM\nc2+r\nG2M2\nc4−αG, (26)\nσ=RSch\nR, (27)\nz=1p\ng00(R)−1\n=e−Φ(R)−1\n=1s\n1−R2\n2αG\u0012q\n1 +8αG2M\nc2R3−1\u0013−1. (28)\nThe results of these properties are presented in Table I. According to the results, any increase in the coupling constant\nleads to a greater Schwarzschild radius, compactness, and surface redshift. There is no black hole in the system under\ninvestigation, because the corresponding radius is greater than the Schwarzschild radius, proving that it is definitely\na neutron star.\nThe next property to be examined is the Kretschmann scalar. It is somewhat more complicated to analyze the\nKretschmann scalar in 4D EGB gravity, because it differs from Einstein’s gravity. Initially, we have to decide which of\nthe curvature scalars should be used to measure the neutron star curvature. As we know, the Ricci tensor and Ricci\nscalar components vanish outside the star. In vacuum, the two quantities are zero because of their dependance on\npressure and density. To calculate the space-time curvature, we need to use another quantity. In comparison to the\nRicci scalar and components of the Ricci tensor, the non-zero components of the Riemann tensor are more suitable\nmeasures of space-time curvature. For simplicity, we can evaluate the Kretschmann scalar to measure the vacuum\ncurvature instead of using the Riemann tensor. In Einstein gravity, the surface curvature of a neutron star is given\nby [51],\nK=q\nRµναβRµναβ =4√\n3GM\nc2R3(29)\nIn 4D EGB gravity, we can calculate the Kretschmann scalar as follows:\nK=q\nRµναβRµναβ\n= 2G\u001212m(r)2\nr6c4−4r2m(r)m′(r)\nr6(c4r−2c2Gm(r))\n+r4m′(r)2\nr6(c2r−2Gm(r))2\u00131/2\n. (30)\nThe value of Kis computed at r=R. In Eq. (30), we arrive at the surface curvature relation for Einstein gravity by\nsetting the derivative of mass to zero (equivalent to setting the coupling constant to zero). The results of Kretschmann10\nTable II: Comparison of our results for the mass and radius of isotropic neutron stars with those of observation.\nName M (M⊙) R(km) Our work M (M⊙) R(km)\nPSR J 0740 + 6620 2 .10 12( ±2) For α = 7.7km22.10 9 .76\nPSR J 0348 + 0432 2 .01 13( ±2) For α = 5.9km22.01 9 .55\nPSR J 1614−2230 1 .97 12( ±2) For α = 5km21.97 9 .44\nV ela X-1 1 .80 11( ±2) For α = 2km21.80 9 .04\n4U1608−52 1 .74 9( ±1) For α = 1km21.75 8 .89\nscalar in 4D EGB gravity for different values of coupling constants ( α) confirm that increasing αleads to decreasing\ngravitational strength (See Table I for more details).\nThe observational results of some objects (4 U1608−52 [52], V ela X −1 [53], PSR J 1614−2230 [54], PSR J 0348+\n0432[55] and PSR J 0740 + 6620 [56]) have been compared with the results of our calculations in Table II to consider\ntheir results as constraints in the calculations of the neutron star structure in 4D EGB gravity. The observational\nresults impose some constraints on the values of coupling constant ( α) to match the results of calculations with those\nof the observations. For neutron stars with masses below 1 .68, the coupling constant ( α) in 4D EGB gravity must be\nnegative (See Table I for more details). In order for the neutron star radius to be between ∼(11−13)kmat mass\nof about ∼1.4M⊙, the coupling constant should be in the range 14 ≲α≲43.\nIn the next section, we study the anisotropic magnetized neutron star and calculate its maximum mass and corre-\nsponding radius using generalized anisotropic TOV equations in 4D EGB gravity.\nIV. THE STRUCTURE OF AN ANISOTROPIC MAGNETIZED NEUTRON STAR IN 4D EGB\nGRAVITY\nIn this part of our study, we investigate the structure of anisotropic magnetized neutron stars using a 4D EGB\ngravity model. In the below part, we introduce the momentum energy tensor of anisotropic matter and present\ngeneralized anisotropic TOV equations for 4D EGB gravity.\nA. MODIFIED ANISOTROPIC TOV EQUATIONS IN 4D EGB GRAVITY\nThe matter source is assumed to be anisotropic, whose energy-momentum tensor is given by [24],\nTµν= (ϵ+Pt)uµuν+Ptgµν−∆kµkν, (31)\nwhere Pris the radial pressure, Ptis the tangential pressure, ∆ ≡Pt−Pris the anisotropy factor and kµis a\nunit spacelike 4-vector along the radial coordinate. Here, kµis defined as kµ= (0, e−λ(r),0,0) with kµkµ= +1 and\nkµuµ= 0.\nAccording to metric in Eq. (19) and the anisotropic energy-momentum tensor 31, we have the following anisotropic\nfield equations,\n2\nrdλ\ndr=e2λ8πGρ\nc2−(1−e−2λ)\nr2(1−α(1−e−2λ)\nr2)\n(1 + 2 α(1−e−2λ)\nr2), (32)\n2\nrdΦ\ndr=e2λ8πGP r\nc4+(1−e−2λ)\nr2(1−α(1−e−2λ)\nr2)\n(1 + 2 α(1−e−2λ)\nr2). (33)\nAdditionally, the non-trivial radial component ( ν= 1) of the conservation law of the anisotropic energy-momentum\ntensor (continuity equation) obtained from relation ∇µTµ\nν=∂µTµ\nν+ Γµ\nρµTρ\nν−Γρ\nµνTµ\nρ= 0, can be gotten by the\nfollowing relation,\ndPr\ndr=−(ρc2+Pr)dΦ\ndr+2∆\nr. (34)\nIn the case of anisotropic matter, this is the generalization of Tolman-Oppenheimer-Volkoff (TOV) equations. Eq.\n(34) considers the anisotropic pressure (∆) to describe the distribution of matter and pressure within the stars.11\nThe application of hydrostatic equilibrium equation for anisotropic matter allows researchers to understand how the\nanisotropic nature of the fluid and the presence of a strong magnetic field influence the structural properties of compact\nobjects such as neutron stars.\nIn order to study the anisotropic magnetized neutron star, we need the generalized anisotropic TOV equations.\nHere, we replace e−2λ(r)in the metric function with e−2λ(r)= 1−r2\n2αG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\n(see Ref. [22], for more\ndetails). For 4D EGB gravity after some calculations, the generalized anisotropic TOV equations are obtained as\nfollows,\ndPr\ndr=−r\n2(ρ(r)c2+Pr)\n1−r2\n2αG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\n×8πGP r\nc4+1\n2αG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\u0012\n1−1\n2G\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\u0013\n1 +1\nG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013 +2∆\nr, (35)\ndm(r)\ndr=−m(r)\nr+c2r2\n2G\u00128πGρ(r)\nc2−1\n2αG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\u0012\n1−1\n2G\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\u0013\n1 +1\nG\u0012q\n1 +8αG2m(r)\nc2r3−1\u0013\u0013\n+c2r2\n2αG2\u0012r\n1 +8αG2m(r)\nc2r3−1\u0013\n. (36)\nThe above equations describe the hydrostatic equilibrium in 4D EGB gravity for anisotropic magnetized neutron\nstars. By setting α= 0, these equations are reduced to the ordinary TOV equations for an anisotropic magnetized\nneutron star. In the next section, we introduce a model for the anisotropy (∆) of a magnetized neutron star to solve\nthe anisotropic TOV equations.\nB. Model for anisotropy\nWe need a functional form for anisotropy to close the system of equations so that we can include any effects caused\nby local anisotropy of the fluid and the presence of a strong magnetic field to solve the anisotropic TOV equations\nin 4D EGB gravity. Despite the existence of microscopic theories, there is no explicit and straightforward method to\nexplain the combined effects of anisotropy on fluids and magnetic fields.\nAn anisotropic neutron star can be constructed using the following basic hypotheses; a) At the center of a neutron\nstar, the sum of hydrodynamic and gravitational forces is zero, therefore, to maintain stability through the balance\nof forces, anisotropy must be zero. b) In neutron stars, anisotropy should vary with position and its dependance\non radial pressure should be nonlinear. c) The study conducted in Ref. [21] suggests that the functional form of\nanisotropy includes the effects of local anisotropy of the fluid and strong magnetic fields. Following Bower and Liang\n[38], we consider the anisotropy function (∆) in the following form,\n∆ =βBLG\nc4(c2ρ+Pr)(c2ρ+ 3Pr)e2λr2, (37)\nwhere e−2λ≈1−2Gm\nc2r, and from Eq. (10), we can calculate the radial pressure ( Pr). By setting βBL= 0, we return\nto the isotropic state (see Ref. [38] for more details). In Fig. 5, we plot the anisotropy factor inside the neutron star\nin terms of the distance from the center of the neutron star for a specific value of βBL, and different values of B0in\n4D EGB gravity. In this figure, one can see that the anisotropy function has all the basic assumptions listed above.\nAccording to this figure, as B0increases, the maximum of the anisotropy function occurs at a smaller distance from\nthe center of the star, and its maximum value decreases.12\n0 2 4 6 8 10 12 14 16051015D (1030 dyn/cm2)\nr(km) B0=0.5 ´ 1018 G\n B0=0.9 ´ 1018 G\n B0=2.1 ´1018  G\nFigure 5: Anisotropy functions in 4D EGB gravity vs distance from the neutron star center for different B0.\nTable III: Properties of anisotropic magnetized neutron stars for different anisotropy parameters ( βBL) in 4D EGB gravity, in\nB0= 0.9×1018Gandα= 3km2.\nβBL Mmax(M⊙) R(km) RSch(km) σ(RSch\nR) z K (10−8m−2)\n0(with B = 0) 1 .86 9 .18 5 .99 0 .653 0 .70 2 .68\n+0.05 1 .87 9 .08 6 .03 0 .664 0 .72 2 .79\n+0.10 1 .88 9 .09 6 .06 0 .667 0 .73 2 .80\n+0.15 1 .91 9 .11 6 .16 0 .676 0 .76 2 .82\n+0.20 1 .94 9 .12 6 .26 0 .686 0 .78 2 .86\nC. Effect of anisotropy parameter on magnetized neutron star structure\nFor specific values of the coupling constant ( α= 3km2) and magnetic field ( B0= 0.9×1018G), we examine the\neffects of the anisotropy parameter on the mass-radius and mass-density relations in 4D EGB gravity. The plots in\nFig. 6 present the results of this investigation for different values of the anisotropy parameter βBL. According to\nthe plots in Fig. 6, the anisotropy parameter significantly influences the mass-radius and mass-density curves. As a\nresult, the maximum mass increases with increasing anisotropy.\nAdditionally, we examined how the anisotropy parameter affects the neutron stars maximum mass, radius, and\nother structural properties for βBL≤0.20 (See Table III for more details). The results of our calculations have\nbeen compared to the observational data for objects (PSR 2215+5135 [57], GW190814 [4] in Table IV to consider\nobservational values as constraints in the calculations of the anisotropic neutron star structure in 4D EGB gravity.\nHere we see that in order to have a good agreement between the calculation results with those of observation, the\nobservational results are limited to certain anisotropy parameters ( βBL).\nIn the next section, we concern with the question of whether increasing the anisotropic neutron star internal\nmagnetic field B0always increases its maximum mass and the corresponding radius.\nTable IV: Comparison of our results for the mass of anisotropic neutron stars in 4D EGB with those of observation in B0=\n0.9×1018Gandα= 3km2.\nName M (M⊙) Our work M (M⊙)\nPSR 2215 + 5135 2 .27+0.17\n−0.15 βBL= 0.70 2 .27\nGW190814 2 .50−2.67 βBL= 1.00−1.18 2 .50−2.6713\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.51.01.52.02.53.0\n M[M   ]\nr (1015 gr/cm3) bBL= 0\n bBL=0.5\n bBL=0.7\n bBL=1.0\n bBL=1.18\n8 9 10 11 12 13 14 15 16 17 180.00.51.01.52.02.53.0\n M[M   ]\nR(km) bBL= 0\n bBL=0.5\n bBL=0.7\n bBL=1.0\n bBL=1.18\nFigure 6: Mass-density relation (up panel) and Mass-radius relation (down panel) of magnetized neutron star for different\nvalues of anisotropy parameter, B0= 0.9×1018G, and α= 3km2in 4D EGB gravity.\nD. Effect of internal magnetic field on magnetized neutron star structure\nThis section aims to study the effect of the internal magnetic field B0on the maximum mass and corresponding\nradius of a magnetized neutron star. It can be seen in Fig. 7 that in the presence of the internal magnetic field\nB0of the magnetized neutron star, the maximum mass and the corresponding radius increase by increasing the\nmagnetic field. As shown in Table V, the maximum mass, the corresponding radius and other physical properties of\nthe magnetized neutron star are provided for various values of the internal magnetic field B0in 4D EGB gravity. Here,\nwe examine the behavior of these physical properties. In our study, we found that increasing the internal magnetic\nfieldB0does not always increase the maximum mass and corresponding radius of the magnetized neutron star. These\ntables show that for the magnetic fields B0⩽0.5×1018G, the maximum mass and corresponding radius increase,\nwhile for the internal magnetic fields B0⩾0.5×1018G, the maximum mass and corresponding radius decrease. The\nmodified Schwarzschild radius and compactness have a behavior similar to the maximum mass and corresponding\nradius. The Kretschmann scalar always increases with 4D EGB gravity as the internal magnetic fields B0increase.\nThus, by increasing the internal magnetic fields B0, the strength of gravity increases.14\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.51.01.52.0\n M[M  ]\nr (1015 gr/cm3) B=0\n B0=0.5´1018 G\n8 10 12 14 160.00.51.01.52.0\n M[M  ]\nR(km) B=0\n B0=0.5´1018 G\nFigure 7: Mass-density relation (up panel) and Mass-radius relation (down panel) of magnetized neutron star for B= 0 and\nB0= 0.5×1018G,α= 3km2, and βBL=0.2 in 4D EGB gravity.\nTable V: Properties of anisotropic magnetized neutron star for different magnetic fields in 4D EGB gravity, in α= 3km2and\nβBL= 0.2.\nB0(G) Mmax(M⊙) R(km) RSch(km) σ(RSch\nR) z K (10−8m−2)\nB= 0 1 .86 9 .18 5 .99 0 .653 0 .70 2 .68\n0.5×10181.97 9 .44 6 .35 0 .691 0 .80 2 .83\n0.9×10181.94 9 .12 6 .26 0 .686 0 .78 2 .86\n2.1×10181.76 8 .57 5 .68 0 .662 0 .72 3 .12\nV. CONCLUSIONS\nThis study analyzed and investigated the structure of isotropic non-magnetized and anisotropic magnetized neutron\nstars in 4D EGB gravity frameworks. To accomplish this, we computed the equation of state for a neutron star\ncontaining pure neutron matter in the absence and in the presence of a strong magnetic field. To calculate the\nanisotropic pressure, we used the Gaussian form of the magnetic field, which is dependent on the density of the material.\nOur first step was to investigate the causality of the equation of state of isotropic non-magnetized and anisotropic15\nmagnetized neutron stars. Next, we integrated the generalized TOV equations in 4D EGB gravity to determine the\nmaximum mass of isotropic non-magnetized and anisotropic magnetized neutron stars and their corresponding radii.\nOn the basis of these results, we calculated the Schwarzschild radius, compactness, surface redshift, and Kretschmann\nscalar of the isotropic non-magnetized and anisotropic magnetized neutron stars.\nIn the isotropic case, we found that the mass-radius, mass-energy density relation strongly depend on the coupling\nconstant. Thus, increasing the coupling constant in 4D EGB gravity increases the maximum mass and the corre-\nsponding radius of the neutron star. In fact, the GB term (made from the integration of two orders of Ricci and\nRiemann tensors) creates an additional non-trivial contribution that leads to improved mass-radius and mass-energy\ndensity relations. As the coupling constant increases, the Schwarzschild radius, compactness, and surface redshift also\nincrease. As a final result, increasing the coupling constant decreases the Krischmann scalar.\nIn the anisotropic case, we found that the relation of mass-radius, and mass-energy density, in addition to the\ncoupling constant, depend on the form of the anisotropic function and the value of the internal magnetic field B0.\nAn interesting observed result was that with increasing the internal magnetic field B0, the magnetized neutron star\nmaximum mass and the corresponding radius did not always increase.\nFinally, we observed that both isotropic and anisotropic cases show a good agreement with the observational data,\nwhen physical parameters are properly selected in 4D EGB gravity.\nAcknowledgments\nWe wish to thank Shiraz University Research Council. We also wish to thank S. H. Hendi (Shiraz University) and\nB. Eslam Panah (University of Mazandaran) for their useful comments and discussions during this work.\nData Availability Statement: No Data associated in the manuscript\n[1] A.W. 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O. Box 47415-416, Babolsar, Iran\n4ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy\n5Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,\nChinese Academy of Sciences, Beijing 100049, China.\nWe have investigated the structural properties of strange quark stars (SQSs) in a modified theory\nof gravity known as massive gravity. In order to obtain the equation of state (EOS) of strange\nquark matter, we have employed a modified version of the Nambu-Jona-Lasinio model (MNJL)\nwhich includes a combination of NJL Lagrangian and its Fierz transformation by using weighting\nfactors (1 −α) and α. Additionally, we have also calculated dimensionless tidal deformability (Λ)\nin massive gravity. To constrain the allowed values of the parameters appearing in massive gravity,\nwe have imposed the condition Λ 1.4M⊙≲580. Notably, in the MNJL model, the value of αvaries\nbetween zero and one. As αincreases, the EOS becomes stiffer, and the value of Λ increases\naccordingy. We have demonstrated that by softening the EOS with increasing the bag constant,\none can obtain objects in massive gravity that not only satisfy the constraint Λ 1.4M⊙≲580, but\nthey also fall within the unknown mass gap region (2 .5M⊙−5M⊙). To establish that the obtained\nobjects in this region are not black holes, we have calculated Schwarzschild radius, compactness,\nand Λ MTOV in massive gravity.\nI. INTRODUCTION\nThe gravitational wave (GW) events offer new insights\ninto compact stars. The binary merger GW170817 [1]\nand its electromagnetic counterpart [2] have led to new\nconstraints on the maximum mass of neutron stars (NSs).\nBased on GW170817, the upper bound of MTOV for NSs\nis predicted as ∼(2.3−2.4)M⊙[3, 4]. However, there\nare other observations of pulsars and binary mergers with\nmasses greater than these values such as PSR J0952-0607\n[5], the secondary component of GW190814 [6, 7] and the\nremnants of GW170817 [8] and GW190425 [9] which fall\nwithin the unknown mass gap region (2 .5M⊙≲M≲\n5M⊙). This region was called a mass gap because anal-\nysis of observations of X-ray binaries revealed a small\nnumber of compact objects in this region [10–12]. Re-\ncent observed systems within this interval, show a rela-\ntive gap instead of an absolute gap [13]. There are argu-\nments that the presence or absence of mass gap objects\ncan constrain the properties of the core-collapse super-\nnova engine and the compact objects in the mass gap\nregion can be created by merging lighter compact ob-\njects [14, 15]. There are two crucial reasons why objects\nin the mass gap region are unlikely to be neutron stars.\nThe first reason is that for non-rotating NSs, the equa-\ntion of state (EOS) must be very rigid to get such mas-\nsive objects (the speed of sound is greater than 0 .6 of\n∗J.Sedaghat@shirazu.ac.ir\n†eslampanah@umz.ac.ir\n‡Rahim.Moradi@icranet.org\n§zebarjad@shirazu.ac.ir\n¶ghbordbar@shirazu.ac.irthe speed of light) [16, 17]. Such EOSs contrast dimen-\nsionless tidal deformability (Λ) constraints obtained from\nGW178017, which require softer EOSs. The second rea-\nson is that for a rotating NS, the rotation rate should be\nclose to the keplerian limit [16, 18–20]. Now, the ques-\ntion arises whether these objects are the smallest black\nholes or other forms of compact stars, like hybrid stars\nand self-bounded strange quark stars (SQSs).\nTheoretically, strange quark matter (SQM) is formed\nin two classes of compact stars. i) Hybrid stars with a\nquark core and hadronic shells [21–26] and ii) SQSs com-\nposed of pure quark matter from the core up to the upper\nlayers of the star [27–31]. Ever since Trazawa [32], Wit-\nten [33], and Bodmer [34] proposed SQM as the ground\nstate of QCD, until now when the GWs have found so\nmany binaries, the study of SQM in compact stars has al-\nways been of interest. Many studies on hybrid and quark\nstars have focused on recent discoveries of GWs. Refs.\n[35–37] study hybrid stars with constraints obtained from\nGW190814, GW170817, and PSR J0030+0451 [38, 39].\nIn Ref. [40], the properties of SQM in the bag model and\nnon-newton gravity are investigated by the constraints\nobtained from GW170817 and PSR J0740+6620. In\nRef. [41], the EOS of quark matter is constrained by\nusing the bayesian statistical model and the mass and\nradius measurements of PSR J0030+0451 [42]. Refs.\n[7, 16, 43] investigate whether the secondary component\nof GW190814 could be a SQS. In Refs. [44, 45] the struc-\ntural features of the SQSs have also been investigated\nusing the data obtained from GWs.\nGR is a successful theory of gravity. However, at large\nscales, it cannot explain why our universe is undergoing\nan accelerated cosmic expansion. This is one of the rea-\nsons why GR needs to be modified. Among various mod-arXiv:2402.14040v1  [gr-qc]  21 Feb 20242\nified theories of gravity, massive gravity can describe the\nlate-time acceleration without considering dark energy\n[46–50]. This theory of gravity modifies gravitational ef-\nfects by weakening them at a large scale compared to GR,\nwhich allows the universe to accelerate. However, its pre-\ndictions at small scales seem to be the same as those of\nGR. Massive gravity will result in the graviton having a\nmass of mg, which in the absence of the mass of the gravi-\nton (i.e., mg→0), this theory reduces to GR. In Ref.\n[51], it has been indicated that the mass of a graviton is\nvery small in the usual weak gravity environments but\nbecomes much larger in the strong gravity regimes, such\nas black holes and other compact objects. In addition,\nrecent observations by the advanced LIGO/Virgo collab-\norations have put a tight bound on the graviton mass\n[52, 53]. Also, other theoretical and empirical limits on\nthe graviton’s mass exist [54–57]. In the astrophysics con-\ntext and by considering massive gravity, the properties of\ncompact objects such as relativistic stars [58, 59], neutron\nstars [60, 61], and white dwarfs [62], have been studied.\nIn 2011, de Rham, Gabadadze and Tolley (dRGT) in-\ntroduced a ghost-free theory of massive gravity [63, 64]\nwhich is known dRGT massive gravity. This theory uses\na reference metric to construct massive terms (see Refs.\n[63–65], for more details). These massive terms are in-\nserted in the action to provide massive gravitons. Then,\nVegh extended the dRGT massive gravity theory using\nholographic principles and a singular reference metric in\n2013 [66], which is known as dRGT-like massive gravity.\nRecently, this extended theory of dRGT massive gravity\nattracted many people in various viewpoints of physics.\nIn this paper, we have used a modified version of the\nNambu-Jona-Lasinio (MNJL) model to obtain the struc-\ntural properties of SQS in massive gravity. In GR, the\nsoft EOSs give small masses for quark stars, and the stiff\nEOSs do not satisfy the Λ constraint. We show that in\nmassive gravity, it is possible to have quark stars that\nnot only have masses higher than those in GR, but also\nsatisfy the GW observational constraints well. We have\nused Λ constraint (Λ 1.4M⊙≲580) [67] to restrict the\nmasses of SQSs in massive gravity.\nII. MODIFIED NJL MODEL\nWe have used a modified self-consistent version of the\n(2 + 1) −flavor NJL model in mean field approximation\n[68] to derive the EOS. The modified Lagrangian is a\ncombination of the original NJL Lagrangian and its Fierz\ntransformation [69] by using weighting factors (1 −α) andαas follows [68]\nL= (1−α)LNJL+αLF\n=¯ψ(i∂\u000e\n−m)ψ+ (1−α)G8X\ni=0\u0002\n(¯ψλiψ)2+ (¯ψiγ5λiψ)2\u0003\n−αG\n2h\u0000¯ψγµλC\n0ψ\u00012−\u0000¯ψγµγ5λC\n0ψ\u00012i\n−K\u0000\ndet[¯ψ(1 +γ5)ψ] + det[ ¯ψ(1−γ5)ψ]\u0001\n, (1)\nwhere LNJL is the original Lagrangian in flavor space\nwithout Fierz transformation and LFis the Fierz trans-\nformation of LNJLin color space [68]. ψdenotes a quark\nfield with three flavor (u, d, s) and three color degrees\nof freedom. mis the corresponding mass matrix in fla-\nvor space. In addition, γµs (µ= 0−→ 3) and λis\n(i= 1−→8) are Dirac and Gell-Mann matrices, re-\nspectively. γ5=iγ0γ1γ2γ3andλ0=r\n2\n3I, where Iis\nthe identity matrix. Similar to λ0,λC\n0is equal tor\n2\n3I\nbut in color space [68]. Also, GandKare the four-\nfermion and six-fermion interaction coupling constants,\nrespectively. The parameter αvaries from zero to one.\nBy setting α= 0, the Lagrangian 1 converts to original\nNJL Lagrangian and for α= 1, it converts to Fierz trans-\nformation of LNJL. In Ref. [68], the authors have used\nthe Lagrangian (1) and have shown that this Lagrangian\ngives the EOSs with MTOV≃2M⊙in GR which satisfy\nstructural constraints (mass, radius and Λ) obtained by\nGWs [68]. They have used the first analysis of Λ with\nthe upper bound of Λ 1.4M⊙≤800 [1] and have obtained\na SQS with MTOV≃2.057M⊙and Λ 1.4M⊙= 634. In\nthis paper, we show that by using the Lagrangian (1)\nand massive gravity, SQSs can be obtained with masses\nin gap region (2 .5M⊙≲M≲5M⊙) and simultaneously,\nrespect to the improved analysis of Λ with the upper\nbound of Λ 1.4M⊙≲580 [67].\nIn the following, we use the Lagrangian (1) and obtain\nthe EOS of SQM in mean field approximation. As the\nfirst step, we calculate the quark number densities for\ndifferent choices of α(to know the details of calculations,\nsee the Ref. [68]). Fig. 1 shows quark number densities\nof up, down, and strange flavors as a function of chemical\npotential for different values of α. As one can see from\nthe figure 1, the quark number densities of up and down\nquarks are zero for µ≲200MeV , and the strange quark\nnumber density is zero for µ≲320MeV . This figure\nshows that the quark number densities decrease with in-\ncreasing α. Consequently, such behavior also occurs for\nbaryon number density (for more discussions, see Ref.\n[68]).\nThe following conditions should be imposed to obtain3\nnu,d,α=0.5\nnu,d,α=0.8\nnu,d,α=0.94\nns,α=0.5\nns,α=0.8\nns,α=0.94\n0.0 0.1 0.2 0.3 0.4 0.5 0.60.00.20.40.60.81.01.21.4\nµ(GeV)n(fm-3)\nFIG. 1: Quark number density ( n) versus chemical potential\n(µ). Notably, nu,ddevotes to up and down quark number\ndensities, and nsis related to the strange quark number den-\nsity.\nthe EOS of SQM, which are:\nµs=µd≡µ,&µu=µ−µe,\n2\n3nu−1\n3nd−1\n3ns−ne= 0,\nnB=nu+nd+ns\n3. (2)\nThe first two lines of Eq. (2) come from beta equilib-\nrium and charge neutrality conditions, respectively. In\nthe above conditions, µis (i=u, d, s ) are quark chemical\npotentials with flavor i, and µeis the electron chemical\npotential. nis (i=u, d, s ) are quark number densities\nwith flavor i. In addition, nBis the baryon number den-\nsity. Furthermore, ne=µ3\ne/(3π2) is the electron number\ndensity. Then the pressure and the energy density can\nbe obtained by the following relations\nP=P0+Zµj\n0dµ′\njnj(µ′\nj), (j=u, d, s, e ),\nϵ=−P+µjnj(µj), (3)\nwhere P0is the vacuum pressure, which is taken as the\nvacuum bag constant ( −B). We consider two values for\nBincluding B1/4= 117 MeV andB1/4= 130 MeV . For\nthese values of B, the energy per baryon of three-flavor\nquark matter is lower than that of two-flavor quark mat-\nter for α≤0.94 [68]. This condition is necessary for SQM\nto be the true ground state of the strong interaction [70].\nIn the following, we present our results for the structural\nproperties of SQS in massive gravity for different values\nofBandα. To this end, we first explain the modified\nTOV equation in massive gravity and then derive the\ndifferential equations to calculate the Λ in this theory of\ngravity.III. MODIFIED TOV EQUATION IN MASSIVE\nGRAVITY\nHere, we consider dRGT-like massive gravity, which is\nintroduced in Ref. [66]. Notably, the dRGT-like mas-\nsive gravity is similar to the dRGT massive theory of\ngravity, which is proposed by de Rham-Gabadadze-Tolley\n(dRGT) in 2011 [63]. The dRGT-like massive gravity’s\naction is given by [66]\nI=1\n2κ2Z\nd4x√−g\"\nR+m2\ng4X\niciUi(g, f)#\n+Imatter ,\n(4)\nwhere Randmgare the Ricci scalar and the graviton\nmass, respectively. It is notable that κ2= 8πbecause\nwe consider G=c= 1. Also, fandgare fixed symmet-\nric and metric tensors, respectively. In the above action,\nImatter is related to the action of matter. Moreover, ci’s\nare free parameters of the massive theory, which are ar-\nbitrary constants. Notabley, their values can be deter-\nmined according to theoretical or observational consid-\nerations [71–73]. Ui’s are symmetric polynomials of the\neigenvalues of 4 ×4 matrix Kµ\nν=√gµαfαν(where gµν\nis the dynamical (physical) metric, and fµνis the aux-\niliary reference metric, needed to define the mass term\nfor gravitons) where they can be written in the following\nform\nUi=iX\ny=1(−1)y+1(i−1)!\n(i−y)!Ui−y[Ky], (5)\nwhere Ui−y= 1, when i=y. Also, Kstands for matrix\nsquare root, i.e., Kµ\nν=\u0010√\nK\u0011µ\nλ\u0010√\nK\u0011λ\nυ, and the\nrectangular bracket denotes the trace [ K] =Kµ\nµ.\nWe consider the four-dimensional static spherical sym-\nmetric spacetime\nds2=e2Φ(r)dt2−dr2\ne−2λ(r)−r2hijdxidxj,where i, j= 1,2.\n(6)\nwhere e2Φ(r)ande−2λ(r)are unknown metric functions,\nandhijdxidxj=\u0000\ndθ2+ sin2θdφ2\u0001\n. The equation of mo-\ntion of dRGT-like massive gravity is given in the following\nform\nGν\nµ+m2\ngχυ\nµ= 8πTν\nµ, (7)\nwhere Gν\nµis the Einstein tensor, and Tν\nµdenotes the\nenergy-momentum tensor which comes from the varia-\ntion of Imatter andχµνis the massive term with the\nfollowing explicit form\nχµν=−d−2X\ni=1ci\n2\"\nUigµν+iX\ny=1(−1)y(i!)Ui−y\u0002\nKy\nµν\u0003\n(i−y)!#\n.\nConsidering the quark stars as a perfect fluid, the\nnonzero components of the energy-momentum tensor are4\ngiven by [61]\nT0\n0=ϵ, T1\n1=T2\n2=T3\n3=−P. (8)\nWe follow the reference metric fµνfor four-dimensional\nstatic spherical symmetric spacetime as it is considered\nfor studying neutron stars and white dwarfs in the fol-\nlowing form [61, 62]\nfµν=diag(0,0,−J2,−J2sin2θ), (9)\nin which Jis known as a parameter of the reference met-\nric, which is a positive constant. The explicit functional\nforms of Ui’s are given by [61]\nU1=2J\nr,\nU2=2J2\nr2,\nUi= 0, i > 2. (10)\nUsing the field equation and reference and dynamical\nmetrics, the metric function e−2λ(r), is obtained in the\nfollowing form [61]\ne−2λ(r)= 1−m2\ngJ\u0010c1r\n2+c2J\u0011\n−2M(r)\nr, (11)\nin which M(r) =RR\n04πr2ϵ(r)dr\u0010\nordM(r)\ndr= 4πr2ϵ(r)\u0011\n.\nBy definition C1≡m2\ngc1JandC2≡m2\ngc2J2in Eq. (11),\nthe modified TOV equation in massive gravity is given\nby [61]\ndP(r)\ndr=\u0010\nM(r) + 4πr3P−r2C1\n4\u0011\n(ϵ+P)\nr\u0010\nC1r2\n2+ 2M(r) +r(C2−1)\u0011. (12)\nHere, we obtain Λ in massive gravity. To do this,\nwe need to get a differential equation of metric function\n(H(r)) in massive gravity. In Appendix A, we have shown\nhow we obtained this equation in massive gravity. Also,\nto check the correctness of the calculations, we have in-\nvestigated the limit C1=C2= 0 and shown that in this\nlimit, the differential equation obtained in massive grav-\nity is converted to the differential equation in GR. The\ndifferential equation to get H(r) in massive gravity is\n(A1+A2+A3)H(r) +A4β+dβ\ndr= 0, (13)where β=dH(r)/dr, and Ais are as follows\nA1=6\nr2B1,\nA2=B1\n2r2B2\n1\u001aB2(B2f+B3)\nB1−7B2+ 3B3\n+1\nB4\n(f+ 1)B2\u0000\nB2+C1r\n2\u0001−1+2B1\u0000\nB2+C1r\n2\u0001\nB4\u0010\n2M(r)\nr+ 8πr2ϵ−2\u0011−1\n+16πr2\u0014\nϵ\u0012C1r\n4+C2+ 4πpr2−1\u0013\n+3p\n\u0010\n5C1r\n12+C2+4πpr2\n3−1\u0011−1+M(r) (7p+ 3ϵ)\nr\n\n\n\n,\nA3=4π(p+ϵ)f\nB1,\nA4=2\u00003C1r\n4+ 2πr2(ϵ−p) +C2−1\u0001\n+2M(r)\nr\nrB1, (14)\nwhere B1,B2,B3, and B4are\nB1=C1r\n2+C2−1 +2M(r)\nr,\nB2= 8πpr2−C1r\n2+2M(r)\nr,\nB3=2M(r)\nr−C1r\n2−8πϵr2,\nB4= 1−2M(r)\nr, (15)\nalso, fis defined as dϵ/dP . In addition, Λ is then ob-\ntained by\nΛ =2\n3k2R5, (16)\nwhere k2is dimensionless tidal Love number for l= 2,\nwhich is given as\nk2=16σ5\n5(1−2σ)2h\n1 +σ(y−1)−y\n2i\n×\u001a\n12σ\u0014\n1−y\n2+σ(5y−8)\n2\u0015\n+ 4σ3\u0002\n13−11y+σ(3y−2) + 2 σ2(1 +y)\u0003\n+ 6(1 −2σ)2h\n1 +σ(y−1)−y\n2i\nln (1−2σ)o−1/2\n,\n(17)\nwhere σ=M/R andyis defined as y=Rβ(R)/H(R)−\n4πR3ϵ0/Min which ϵ0is the energy density at the sur-\nface of the quark star [68]. By simultaneously solving\nequations (12) and (13) along with dM/dr = 4πr2ϵ, we5\nobtain the curves of star mass in terms of radius and Λ\nin terms of mass. The results are presented in the next\nsection.\nA. Modified Schwarzschild Radius\nIt is clear that by considering dRGT-like massive grav-\nity, the Schwarzschild radius is modified. Using Eq. (11),\nwe can get the Schwarzschild radius ( RSch) for this the-\nory of gravity. To find the modified Schwarzschild radius,\nwe solve S(r) = 0. So, the modified Schwarzschild radius\nfor dRGT-like massive gravity is given [61]\nRSch=(1−C2)\nC1−q\n(C2−1)2−4C1M\nC1, (18)\nwhere M=M(r=R) is the star’s total mass (or the\nmass of the star on its surface).\nB. Compactness\nThe compactness of a spherical object may be defined\nby the ratio of mass to radius of that object\nσ=M\nR, (19)\nwhich may be indicated as the strength of gravity. In\nthe following, we calculate the thermodynamic proper-\nties of SQM (EOS, speed of sound, and adiabatic in-\ndex) for different values of αandB. We investigate the\nstructural properties of SQS (mass, radius, dimensionless\ntidal deformability, compactness, and Schwarzschild ra-\ndius) for a stiff EOS ( B1/4= 117 MeV ), and a soft EOS\n(B1/4= 130 MeV ) in massive gravity. The value of Λ de-\npends not only on the EOS but also on the gravity used.\nIn GR, the stiff EOSs cause the Λ constraint to be lost\nand the soft EOSs can not cover the objects fall within\nin the mass gap region. We demonstrate that in massive\ngravity, the soft EOSs can not only lead to masses in the\nmass gap region but also satisfy the Λ constraint well.\nIV. EQUATION OF STATES OF SQM FOR\nB1/4= 117 MeV\nIn this section, we present our results for the thermo-\ndynamic properties of SQM in a modified NJL model for\nB1/4= 117 MeV and different values of α. As it was\nmentioned in section II, by choosing this value of bag\nconstant, the parameter αshould be equal or less than\n0.94 to establish the stability condition of SQM. We de-\nrive the EOSs of SQM for different values of αincluding\nα= 0.5,α= 0.8, and α= 0.94. The results are presented\nin Fig. 2. This figure shows that the EOSs become stiffer\nby increasing the αparameter. We will show in sectionIV A that this behavior causes the mass of the star to\nincrease. But it is possible that the structural features of\nthese stars do not satisfy the constraints obtained from\nGW observations. Therefore, at the same time as the\nmass of the star increases, such constraints should also\nbe checked. To check the causality, the speed of sound\nin terms of energy density is plotted in Fig. 3. The plots\nindicate that causality is established well for all αval-\nues. Also, with an increase in the value of α, the speed\nof sound increases, which means that the EOS becomes\nstiffer and subsequently, the mass of the star becomes\ngreater.\nα=0.5\nα=0.8\nα=0.94\n200 400 600 800 10000100200300400\nϵ(MeV /fm3)P(MeV /fm3)\nFIG. 2: EOSs of SQM for B1/4= 117 MeV and different\nvalues of α.\nα=0.5\nα=0.8\nα=0.94\n200 300 400 500 600 700 8000.40.50.60.70.80.91.0\nϵ(MeV /fm3)V\nFIG. 3: The speed of sound ( V) energy density ( ϵ) for different\nvalues of α.\nAs a measure of dynamical stability, the adiabatic in-\ndex\u0000\nΓ =dP\ndϵP+ϵ\nP\u0001\nhas been presented as a function of\nenergy density in Fig. 4. It is known that Γ should be\nhigher than 4 /3 for dynamical stability [61, 74–76, 78].\nAs one can see from this figure, this condition is satisfied\nfor all values of α.\nA. Structural Properties of SQS in Massive\nGravity for B1/4= 117 MeV\nIn this section, we obtain the structural properties of\nSQS, including the M−Rand Λ −Mrelations in dRGT-6\nα=0.5α=0.8α=0.94Γ=4/3200 400 600 800 1000 120051015\nϵ(MeV/fm3)Γ1100 1101 1102 1103 1104 11051.31.41.51.61.71.8\nFIG. 4: Adiabatic index (Γ) versus energy density ( ϵ) for\nB1/4= 117 MeV and different values of α.\nlike massive gravity. We then compare the results with\nthe constraints derived from astronomical observations.\nIn addition, we obtain modified Schwarzschild radius and\ncompactness of quark stars in this theory of gravity.\nPractically, the modified TOV equation (12) is\nsolved simultaneously with the equation, M(r) =RR\n04πr2ϵ(r)dr. To solve these equations, we have to con-\nsider the modified NJL EoS with the obtained pressure\nat the center of the star (i.e. P(r= 0) = Pc, which Pcis\nthe central pressure of the quark star and given by the\nmodified NJL EoS). Notably, the mass of the star at the\ncenter of the star is zero, i.e. M(r= 0) = 0. Then, we\ncontinue to solve the equations up to the star’s surface,\nwhere the pressure is zero ( P(r=R) = 0). The re-\nsult is the mass and corresponding radius of the star (i.e.\nM=M(R)). We do this for different values of central\npressures allowed by the mentioned EOS.\n1.α= 0.5\nNow, we investigate the structural properties of SQS in\nmassive gravity. We first use the obtained EOSs in sec-\ntion IV and the modified TOV equation (12) to derive\nthe mass-radius ( M−R) relation and Λ −Mrelation in\nmassive gravity. As we saw in section IV, we used three\ndifferent values of αincluding 0 .5, 0.8, and 0 .94 for the\nEOSs in the MNJL model. Now, we first examine the\ncase α= 0.5. The equation (12) shows that there are\ntwo parameters, including C1andC2, in massive gravity\n(these parameters are related to the dRGT-like massive\ngravity’s parameters). In the beginning, we set a fixed\nvalue for C1and investigate the structural features of\nSQS for different values of C2. Figures 5 and 6 show the\nM−Rand Λ −Rdiagrams, respectively, for α= 0.5,\nC1= 10−5, and different values of C2. As one can see\nfrom Fig. 5, there are different colored regions in Fig.\n5. The gray and orange areas are the M−Rconstraints\nfrom GW170817. The black boundary corresponds to\nPSR J0740+6620 [79]. The pink area represents the sec-\nondary component of GW190814 [7], and the green area\n8 10 12 14 16 181.01.52.02.53.0\nR(km)M\nM⊙B1\n4=117MeV,α=0.5 &C 1=10-5\nPSR J0740+66202ry component of GW190814\nPSR J2215+5135\nGW170817PSR J095-0607\nPSR J0030+0451PSR J0030+0451C2=-10-3\nC2=-10-2\nC2=-0.1\nC2=-0.14FIG. 5: Mass-Radius diagram for B1/4= 117 MeV ,α=\n0.5, C1= 10−5and different values of C2. The gray and\norange regions are the mass - radius constraints from the\nGW170817 event. The black region shows pulsar J0740+6620,\nThe green and black hatched regions represent pulsars\nJ2215+5135 [77] and PSR J095-0607, respectively. The red\nregion amounts to the secondary component of GW190814.\nThe brown and the purple regions show two different reports\nof the pulsar J0030+0451 [38, 39].\n1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.70500100015002000\nM\nM⊙ΛB1/4=117MeV,α=0.5 &C 1=10-5\nC2=-10-3\nC2=-10-2\nC2=-0.1\nC2=-0.14\nFIG. 6: Dimensionless tidal deformability (Λ) versus mass of\nthe star for B1/4= 117 MeV, α = 0.5, C1= 10−5and different\nvalues of C2. The horizontal black line and the vertical red\nline correspond to Λ = 580 andM\nM⊙= 1.4, respectively.\nshows the remnant mass of GW190425 [9]. As one can see\nfrom this figure, we have not drawn graphs for the values\n|C2|>0.14, because Fig. 6 shows for these values of C2,\nthe Λ condition (Λ 1.4M⊙≲580) is not satisfied. In Fig. 6,\nthe horizontal black line shows the Λ = 580, and the ver-\ntical red line represents M= 1.4M⊙. As one can see from\nthis figure, the constraint Λ 1.4M⊙≲580, is satisfied for\n|C2|≲0.14 which corresponds to the MTOV≲2.13M⊙.\nIndeed, the maximum mass in this case cannot be more\nthan 2 .13M⊙. More details for structural properties of\nSQS for B1/4= 117 MeV ,α= 0.5,C1= 10−5,and dif-\nferent values of C2are presented in Table. I. Previously,\nin the Ref. [68] the value of MTOV forB1/4= 117 MeV\nandα= 0.5 has been calculated in GR. The obvious dif-\nference between their result ( MTOV = 1.70M⊙) with the\nresult obtained in this paper shows the important effect\nof the massive gravity in the mass of the SQS. Table.7\nTABLE I: Structural properties of SQS for B1/4=\n117MeV, α = 0.5,C1= 10−5,and different values of C2.\nC2 −10−3−10−2−0.1−0.14\nR(km) 10.3310.3610.7011.02\nMTOV(M⊙)1.76 1.78 2.02 2.13\nΛ1.4M⊙ 446.76457.64548.92578.14\nσ(10−1) 2.52 2.54 2.79 2.86\nRSch(km)5.20 5.22 5.43 5.53\nI demonstrates that by increasing the absolute value of\nC2, the values of MTOV, Λ1.4M⊙, and the compactness\nof the star are increased too. Additionally, The results\ncalculated for RSchshow that the obtained masses can\nnot be black holes.\nNow, we turn to the next case by choosing B1/4=\n117MeV ,α= 0.5,C2=−0.1 and different values of C1.\nIt is shown in Refs. [61, 76] that changing the C1param-\neter has a negligible effect on the structural properties of\nthe star. Therefore, in this section, we show this feature\nand for other values of αand the bag constant, we only\nchange the value of C2. Figure 7 represents the M−R\nrelation for different values of C1. As this figure shows,\n8 10 12 14 16 181.01.52.02.53.0\nR(km)M\nM⊙B1\n4=117MeV,α=0.5,C 2=-0.1\nPSR J0740+6620PSR J095-0607\nPSR J2215+51352ry component of GW190814\nGW170817\nPSR J0030+0451PSR J0030+0451C1=10-7\nC1=10-6\nC1=10-5\nC1=1.3×10-5\nFIG. 7: Mass-Radius diagram for B1/4= 117 MeV, α = 0.5,\nC2=−0.1, and different values of C1. The gray and orange\nregions are the mass - radius constraints from the GW170817\nevent. The black region shows pulsar J0740+6620, The green\nand black hatched regions represent pulsars J2215+5135 [77]\nand PSR J095-0607, respectively. The red region amounts\nto the secondary component of GW190814. The brown and\nthe purple regions show two different reports of the pulsar\nJ0030+0451 [38, 39].\nchanging the C1parameter has an insignificant effect on\nthe maximum mass of the star. We have not plotted the\nresults for values C1≳1.3×10−5, because as the figure\n8 shows, the condition Λ 1.4M⊙≲580 is not established\nfor these values of C1. The structural properties of SQS\nforB1/4= 117 MeV ,α= 0.5,C2=−0.1, and different\nvalues of C1are in Table. II.\nOn the other hand, our results in Table. II indicate\nthat the obtained objects cannot be black holes because\nthe radii of these compact objects are bigger than the\nSchwarzschild radii. Besides, the strength of gravity in-\ncreases by increasing the value of |C2|.\n1.0 1.2 1.4 1.6 1.8 2.00500100015002000\nM\nM⊙ΛB1/4=117 MeVα=0.5, &C 2=-0.1\nC1=10-7\nC1=10-6\nC1=10-5\nC1=1.3×10-5FIG. 8: Dimensionless tidal deformability (Λ) versus mass of\nthe star for B1/4= 117 MeV, α = 0.5,C2=−0.1, and differ-\nent values of C1. The horizontal black line and the vertical\nred line correspond to Λ = 580 andM\nM⊙= 1.4, respectively.\nTABLE II: Structural properties of SQS for B1/4=\n117MeV, α = 0.5,C2=−0.1, and different values of C1.\nC1 10−710−610−51.3×10−5\nR(km) 10.6110.9210.95 10.99\nMTOV(M⊙)2.01 2.02 2.05 2.06\nΛ1.4M⊙ 473.77480.89548.92 578.25\nσ(10−1) 2.80 2.74 2.77 2.77\nRSch(km)5.41 5.43 5.52 5.54\n2.α= 0.8\nHere, we increase the value of αand investigate its\neffect on the structure properties of the SQS in dRGT-\nlike massive gravity. The figures. 9 and 10 show the\nM−Rand Λ −Mdiagrams for α= 0.8,C1= 10−5, and\ndifferent values of C2, respectively.\n8 10 12 14 16 181.01.52.02.53.0\nR(km)M\nM⊙B1\n4=117MeV, α=0.8 &C 1=10-5\nPSR J0740+66202ry component of GW190814\nPSR J2215+5135PSR J095-0607\nGW170817\nPSR J0030+0451PSR J0030+0451C2=-10-3\nC2=-10-2\nC2=-0.1\nC2=-0.15\nFIG. 9: Mass-Radius diagram for B1/4= 117 MeV, α =\n0.8, C1= 10−5, and different values of C2. The gray and\norange regions are the mass - radius constraints from the\nGW170817 event. The black region shows pulsar J0740+6620,\nThe green and black hatched regions represent pulsars\nJ2215+5135 [77] and PSR J095-0607, respectively. The red\nregion amounts to the secondary component of GW190814.\nThe brown and the purple regions show two different reports\nof the pulsar J0030+0451 [38, 39].\nComparison of figures 9 and 5 shows that the maximum8\n1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.70500100015002000\nM\nM⊙ΛB1/4=117MeV,α=0.8 &C 1=10-5\nC2=-10-3\nC2=-10-2\nC2=-0.1\nC2=-0.15\nFIG. 10: Dimensionless tidal deformability versus mass of the\nstar for B1/4= 117 MeV, α = 0.8, C1= 10−5,and different\nvalues of C2. The horizontal black line and the vertical red\nline correspond to Λ = 580 andM\nM⊙= 1.4, respectively.\nmass of the star increases with the increase of αvalue.\nThis result is obvious because we showed in Fig. 2 that\nthe increase of αmakes the EOS stiffer. However, Fig.\n10 shows that this feature increases Λ and the constraint\nΛ1.4M⊙≲580 is lost for |C2|≳10−2which corresponds\nto the MTOV≲2.33M⊙. Indeed, the maximum mass in\nthis case cannot be more than 2 .33M⊙. The structural\nproperties of SQS for B1/4= 117 MeV ,α= 0.8,C1=\n10−5, and different values of C2are presented in Table.\nIII.\nThere are the same behaviors for the compactness and\nthe modified Schwarzschild radius. They increase by in-\ncreasing the value of |C2|. Also, the radii of these com-\npact objects are more than the modified Schwarzschild\nradii (i.e., R > R Sch).\nTABLE III: Structural properties of SQS for B1/4=\n117MeV, α = 0.8,C1= 10−5, and different values of C2.\nC2 −10−3−10−2−0.1−0.15\nR(km) 10.8010.9211.3311.61\nMTOV(M⊙)1.89 1.92 2.18 2.33\nΛ1.4M⊙ 578.35578.69700.07745.67\nσ(10−1) 2.59 2.60 2.85 2.97\nRSch(km)5.59 5.63 5.87 6.00\n3.α= 0.94\nFor further investigation, we check the effect of massive\ngravity on the structural properties of SQS for α= 0.94.\nWe do not consider values of αgreater than 0 .94. Be-\ncause, as mentioned in section II, the stability condi-\ntion of SQM will be lost for α≳0.94 [68]. BY set-\ntingB1/4= 117 MeV ,α= 0.94,C1= 10−5, and differ-\nent values of C2, we obtain the Fig. 11, and 12 as the\nM−Rand Λ −Mdiagrams, respectively. Here, we also\nsee that with the increase of α, the maximum gravita-\ntional mass increases, but as the EOS becomes stiffer,the Λ constraint is no longer satisfied. Table. IV shows\nthe structural properties of SQS for B1/4= 117 MeV ,\nα= 0.94,C1= 10−5, and different values of C2. Al-\nthough the values obtained for the MTOV forα= 0.94\nare higher than those with α= 0.5 and α= 0.8, Ta-\nble. IV demonstrates that none of the results satisfy the\nconstraint Λ 1.44M⊙≲580.\n8 10 12 14 16 181.01.52.02.53.0\nR(km)M\nM⊙B1\n4=117MeV,α=0.94 &C 1=10-5\nPSR J0740+6620PSR J095-0607\nPSR J2215+51352ry component of GW190814\nGW170817\nPSR J0030+0451PSR J0030+0451C2=-10-3\nC2=-10-2\nC2=-0.1\nC2=-0.15\nFIG. 11: Mass-Radius diagram for B1/4= 117 MeV ,\nα= 0.94,C1= 10−5, and different values of C2. The\ngray and orange regions are the mass - radius constraints\nfrom the GW170817 event. The black region shows pul-\nsar J0740+6620, The green and black hatched regions rep-\nresent pulsars J2215+5135 [77] and PSR J095-0607, respec-\ntively. The red region amounts to the secondary component\nof GW190814. The brown and the purple regions show two\ndifferent reports of the pulsar J0030+0451 [38, 39].\n1.0 1.2 1.4 1.6 1.80500100015002000\nM\nM⊙ΛB1/4=117 MeV, α=0.94 &C 1=10-5\nC2=-10-3\nC2=-10-2\nC2=-0.1\nC2=-0.15\nFIG. 12: Dimensionless tidal deformability versus mass of the\nstar for B1/4= 117 MeV ,α= 0.94,C1= 10−5, and different\nvalues of C2. The horizontal black line and the vertical red\nline correspond to Λ = 580 andM\nM⊙= 1.4, respectively.\nAs an important result of our calculation, we could\nnot describe the mass gap region and observational ob-\njects such as PSRJ0740+6620, the secondary component\nof GW190814, and the remnant mass of GW190425 by\nSQS when we considered the mentioned EOSs of SQM\nwith B1/4= 117 MeV and different values of α. In\nother words, the maximum mass of SQS with the men-\ntioned EOSs in dRGT-like massive gravity is in the range\nMTOV<2.5M⊙.\nTo obtain massive SQSs that satisfy the Λ constraint,9\nTABLE IV: Structural properties of SQS for B1/4=\n117MeV, α = 0.94,C1= 10−5, and different values of C2.\nC2 −10−3−10−2−0.1−0.15\nR(km) 11.4111.4612.0612.23\nMTOV(M⊙)2.11 2.13 2.43 2.60\nΛ1.4M⊙ 728.30733.20747.51887.92\nσ(10−1) 2.74 2.75 2.98 3.09\nRSch(km)6.24 6.30 6.54 6.70\nwe make the EOS soft. Because the value of Λ depends\non both the EOS and the gravity used, simultaneously.\nTherefore, in the next section, we change the value of\ntheBconstant (as we expected) and investigate the re-\nsults for thermodynamic properties of SQM and struc-\ntural properties of SQS in dRGT-like massive gravity.\nV. EQUATION OF STATES OF SQM FOR\nB1/4= 130 MeV\nIn the previous section, we saw that by increasing the\nvalue of α, the EOS becomes stiffer, and subsequently the\nmass of the star increases, but the constraint Λ 1.44M⊙≲\n580 is not satisfied for the SQSs fall within the mass\ngap region and consequently, a small range of the C2\nparameter is reachable. In this section, we increase the\nvalue of the bag constant, B, to make the EOS softer, and\nthe condition of Λ to be satisfied in a larger range of C2\nparameter. Figure 13 shows a comparison between two\nEOSs obtained from two different bag constants ( B1/4=\n117MeV andB1/4= 130 MeV ). From this figure, it can\nbe seen that the EOSs with the same value of αhave\nthe same slope and consequently have the same speed\nof sound. But the diagrams with B1/4= 130 MeV lie\nlower than those with B1/4= 117 MeV . It shows that\nthe EOSs with B1/4= 130 MeV , are softer than those\nwith B1/4= 130 MeV . Although this result is obvious,\nwe will indicate that it has a significant effect on the\nstructural properties of SQS in massive gravity. This\nmakes it possible to obtain quark stars that fall within the\nmass gap region and respect to the constraint Λ 1.4M⊙≲\n580, simultaneously. Now, we set B1/4= 130 MeV and\ninvestigate its effect on the structural properties of SQS\nin massive gravity. In this situation, we consider another\nconstraint too. We assume that the value of Λ for MTOV\nshould not be zero. Because otherwise the obtained mass\nis a black hole [80].\nFigure. 14 represents the adiabatic index for different\nvalues of αandB. As this figure shows, for any choice\nofα, dynamical stability is increased by increasing B.\nA. Structural Properties of SQS in Massive\nGravity for B1/4= 130 MeV\nIn the following, we study the structural properties of\nSQS in the presence of dRGT-like massive gravity for\n200 300 400 500 600 700 800 900050100150200250\nϵ(MeV/fm3)P(MeV/fm3) α=0.5 &B1\n4=117MeV\nα=0.8 &B1\n4=117MeV\nα=0.94 &B1\n4=117MeV\nα=0.5 &B1\n4=130MeV\nα=0.8 &B1\n4=130MeV\nα=0.94 &B1\n4=130MeVFIG. 13: Comparison between EOSs for different values of B.\n200 400 600 800 1000 120051015\nϵ(MeV/fm3)Γ1100 1101 1102 1103 1104 11051.31.41.51.61.71.81.9α=0.5 &B1\n4=117MeV\nα=0.8 &B1\n4=117MeV\nα=0.94 &B1\n4=117MeV\nα=0.5 &B1\n4=130MeV\nα=0.894 &B1\n4=130MeV\nα=0.94 &B1\n4=130MeV\nΓ=4/3\nFIG. 14: Adiabatic index (Γ) for different values αandB.\nB1/4= 130 MeV , and three different values of α= 0.5,\nα= 0.8, and α= 0.94.\n1.α= 0.5\nHere, we derive the structural properties of SQS in\nmassive gravity for α= 0.5 and B1/4= 130 MeV . Simi-\nlar to the previous section, we first set a fixed value for\nC1and investigate M−Rand Λ −Mdiagrams for dif-\nferent values of C2. Figures 15 and 16 show M−R\nand Λ −Mdiagrams, respectively. We can see from\nthe fig. 16 that all the diagrams satisfy the constraint\nΛ1.4M⊙≲580. The results show that by softening the\nEOS, the masses obtained in massive gravity not only\nsatisfy the Λ 1.4M⊙≲580 constraint, but also some of\nthem are located in the mass gap region. As we can see\nfrom these figures, we did not use the values |C2|≳0.84,\nbecause they led to zero value for Λ MTOV. The results\nfor structural properties are given in Table. V.\nAs an important result, our findings indicate that the\nmaximum mass of SQS can be in the range MTOV≲\n4M⊙. Indeed, by using the MNJL model with B1/4=\n130MeV , and α= 0.5, SQSs fall within the mass gap\nregion.\n2.α= 0.8\nAs a further investigation, we change the value of α.\nFigures. 17, and 18 are M−Rand Λ −Mdiagrams\nforB1/4= 130 MeV ,α= 0.8,C1= 10−5and differ-\nent values of C2, respectively. We can see that despite10\nTABLE V: Structural properties of SQS for B1/4= 130 MeV ,α= 0.5,C1= 10−5, and different values of C2. The results\nfalling in mass gap have been separated in different column.\nnone mass gap mass gap\nC2 −10−3−0.1−0.2−0.3−0.4−0.5−0.6−0.7−0.84\nR(km) 9.04 9.46 9.85 10.2410.6611.0611.40 11.7512.34\nMTOV(M⊙)1.56 1.80 2.05 2.31 2.57 2.86 3.15 3.45 3.91\nΛ1.4M⊙ 185.80255.06307.00348.34379.38403.35431.03449.16 480.18\nΛMTOV 59.1126.2012.74 6.20 3.30 1.42 0.64 0.26 0.02\nσ(10−1) 2.55 2.82 3.08 3.34 3.57 3.83 4.09 4.34 4.69\nRSch(km)4.61 4.84 5.06 5.26 5.43 5.64 5.83 6.00 6.29\n8 10 12 14 16 181.01.52.02.53.0\nR(km)M\nM⊙B1\n4=117MeV,α=0.5 &C 1=10-5\nPSR J0740+66202ry component of GW190814\nPSR J2215+5135\nGW170817PSR J095-0607\nPSR J0030+0451PSR J0030+0451C2=-10-3\nC2=-10-2\nC2=-0.1\nC2=-0.14\nFIG. 15: Mass-Radius diagram for B1/4= 130 MeV ,α= 0.5,\nC1= 10−5, and different values of C2. The gray and orange\nregions are the mass - radius constraints from the GW170817\nevent. The black region shows pulsar J0740+6620, The green\nand black hatched regions represent pulsars J2215+5135 [77]\nand PSR J095-0607, respectively. The red region amounts\nto the secondary component of GW190814. The brown and\nthe purple regions show two different reports of the pulsar\nJ0030+0451 [38, 39]. The light blue band denotes the rem-\nnant mass of GW190425.\n1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4050010001500\nM\nM⊙ΛB1/4=130 MeV,α=0.5 &C 1=10-5\nC2=-10-3\nC2=-10-1\nC2=-0.3\nC2=-0.5\nC2=-0.7\nC2=-0.84\nFIG. 16: Dimensionless tidal deformability (Λ) versus mass\nof the star for B1/4= 130 MeV ,α= 0.5,C1= 10−5, and dif-\nferent values of C2. The horizontal black line and the vertical\nred line correspond to Λ = 580 andM\nM⊙= 1.4, respectively.\nthe increase of α, which leads to the stiffer EOS, the\nconstraint Λ 1.4M⊙≲580 is still maintained. Here, the\nobtained masses lie in a larger area of the mass gap re-\ngion compared to the previous case ( B1/4= 130 MeV\nandα= 0.5). Moreover, we checked the Λ of MTOV fordifferent values of C2. We observed that for |C2|≳0.84\nthe values of Λ MTOVbecame zero. The results for struc-\ntural properties are given in Table. VI.\nOn the other hand, our calculation for the compactness\nand the modified Schwarzschild radius show that these\ncompact objects cannot be black holes because their radii\nare bigger than the Schwarzschild radii, i.e., R > R Sch\n(see Table. VI, for more details). Also, the compactness\nincreases by increasing the value of |C2|. In other words,\nthe strength of gravity augments when the value of |C2|\nincreases (see Table. VI, for more details).\n8 10 12 14 16 181.01.52.02.53.03.54.0\nR(km)M\nM⊙B1\n4=130MeV, α=0.8,C 1=10-5\nPSR J0740+66202ry component of GW190814Remnant of GW190425\nPSR J095-0607\nPSR J2215+5135\nGW170817\nPSR J0030+0451PSR J0030+0451C2=-10-3\nC2=-10-1\nC2=-0.3\nC2=-0.5\nC2=-0.7\nC2=-0.82\nFIG. 17: Mass-Radius diagram for B1/4= 130 MeV ,α= 0.8,\nC1= 10−5, and different values of C2. The gray and orange\nregions are the mass - radius constraints from the GW170817\nevent. The black region shows pulsar J0740+6620, The green\nand black hatched regions represent pulsars J2215+5135 [77]\nand PSR J095-0607, respectively. The red region amounts\nto the secondary component of GW190814. The brown and\nthe purple regions show two different reports of the pulsar\nJ0030+0451 [38, 39]. The light blue band denotes the rem-\nnant mass of GW190425.\nIt is notable that by considering the MNJL model with\nB1/4= 130 MeV , and α= 0.8, the maximum mass of\nSQSs can be in the range 1 .68M⊙≲MTOV≲4.2M⊙,\nwhich fall within the mass gap region.\n3.α= 0.94\nAs the final investigation of SQS in dRGT-like mas-\nsive gravity, we consider B1/4= 130 MeV andα= 0.94.\nFigures. 19, and 20 show M−Rand Λ −Mdiagrams11\nTABLE VI: Structural properties of SQS for B1/4= 130 MeV ,α= 0.8,C1= 10−5, and different values of C2. The results\nfalling in mass gap have been separated in different column.\nnone mass gap mass gap\nC2 −10−3−0.1−0.2−0.3−0.4−0.5−0.6−0.7−0.82\nR(km) 9.46 9.94 10.4010.7811.1511.6011.97 12.3612.80\nMTOV(M⊙)1.68 1.94 2.21 2.49 2.79 3.09 3.40 3.73 4.14\nΛ1.4M⊙ 257.98329.76383.76427.33461.80487.14515.31536.06559.21\nΛMTOV 47.0420.43 9.64 2.70 1.95 0.92 0.39 0.14 0.01\nσ(10−1) 2.63 2.89 3.14 3.42 3.70 3.94 4.20 4.47 4.79\nRSch(km)4.97 5.22 5.45 5.67 5.90 6.10 6.29 6.49 6.73\n1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4050010001500\nM\nM⊙ΛB1/4=130MeV, α=0.8,C 1=10-5\nC2=-10-3\nC2=-10-1\nC2=-0.3\nC2=-0.5\nC2=-0.7\nC2=-0.82\nFIG. 18: Dimensionless tidal deformability (Λ) versus mass\nof the star for B1/4= 130 MeV ,α= 0.8,C1= 10−5, and dif-\nferent values of C2. The horizontal black line and the vertical\nred line correspond to Λ = 580 andM\nM⊙= 1.4, respectively.\nforC1= 10−5and different values of C2, respectively.\nFigure 20 shows that for values |C2|≳0.54, the con-\nstraint Λ 1.4M⊙≲580 is no longer satisfied. For this\nreason, compared to the case α= 0.8, a smaller area of\ntheC2parameter is accessible, which results from the\nsharpness of the EOS. The structural properties of SQS\nforB1/4= 130 MeV ,α= 0.94,C1= 10−5and different\nvalues of C2are presented in Table. VII.\nBriefly, the maximum mass of SQSs with B1/4=\n130MeV , and α= 0.94, cannot be more than 3 .45M⊙.\nHowever, the mentioned values of the parameters of our\nsystem can satisfy the mass gap region (see Table. VII,\nfor more details). Besides, there are the same behaviors\nof the compactness and the Schwarzschild radius when\nthe value of |C2|increases. In other words, the strength\nof gravity augments by increasing the value of |C2|.\nAs a final point, by considering the MNJL model with\nB1/4= 130 MeV and different values of α(0.5< α <\n0.94), we indicated that the mass of SQSs could be in\nthe mass gap region. So, it seems reasonable to consider\nSQSs in massive gravity as candidates for the mass gap\nregion.\nVI. CONCLUSION\nWe investigated the structural properties of quark star\nin massive gravity. In this study, we considered three-\n8 10 12 14 16 181.01.52.02.53.03.54.0\nR(km)M\nM⊙B1\n4=130MeV, α=0.94,C 1=10-5\nPSR J0740+6620PSR J095-0607\nPSR J2215+51352ry component of GW190814\nGW170817Remnant of GW190425\nPSR J0030+0451PSR J0030+0451C2=-10-3\nC2=-10-2\nC2=-0.1\nC2=-0.2\nC2=-0.4\nC2=-0.54FIG. 19: Mass-Radius diagram for B1/4= 130 MeV ,\nα= 0.94,C1= 10−5, and different values of C2. The\ngray and orange regions are the mass - radius constraints\nfrom the GW170817 event. The black region shows pul-\nsar J0740+6620, The green and black hatched regions rep-\nresent pulsars J2215+5135 [77] and PSR J095-0607, respec-\ntively. The red region amounts to the secondary component\nof GW190814. The brown and the purple regions show two\ndifferent reports of the pulsar J0030+0451 [38, 39]. The light\nblue band denotes the remnant mass of GW190425.\n1.0 1.5 2.0 2.5 3.00500100015002000\nM\nM⊙ΛB1/4=130MeV,α=0.94,C 1=10-5\nC2=-10-3\nC2=-10-2\nC2=-0.1\nC2=-0.2\nC2=-0.4\nC2=-0.54\nFIG. 20: Dimensionless tidal deformability versus mass of the\nstar for B1/4= 130 MeV ,α= 0.94,C1= 10−5, and different\nvalues of C2. The horizontal black line and the vertical red\nline correspond to Λ = 580 andM\nM⊙= 1.4, respectively.\nflavor quark matter (up, down and strange flavors) in\nMNJL model [68] which is a combination of NJL La-\ngrangian and its Fierz transformation by using weight-\ning factors (1 −α) and α. By imposing charge neutral-\nity and chemical equilibrium, we obtained the EOSs of\nSQM for three different values of α, (α= 0.5,α= 0.8,12\nTABLE VII: Structural properties of SQS for B1/4= 130 MeV ,α= 0.94,C1= 10−5, and different values of C2. The results\nfalling in mass gap region have been separated in a different column.\nnone mass gap mass gap\nC2 −10−3−10−2−0.1−0.2−0.3−0.4−0.54−0.55\nR(km) 9.81 9.85 10.3010.7711.2311.6312.32 12.35\nMTOV(M⊙)1.79 1.81 2.06 2.35 2.65 2.96 3.42 3.45\nΛ1.4M⊙ 326.43333.26398.93452.51499.87532.37578.95580.64\nΛMTOV 32.2532.2615.20 6.51 2.92 1.36 0.35 0.32\nσ(10−1) 2.70 2.72 2.96 3.23 3.49 3.77 4.11 4.13\nRSch(km)5.29 5.30 5.54 5.80 6.03 6.26 6.57 6.59\nandα= 0.94) and two different values of Bag constant\n(B1/4= 117 MeV andB1/4= 130 MeV ). We ensured\nthat the EOSs satisfied causality and dynamical stabil-\nity conditions. We then presented a brief introduction\non the modified TOV equation in massive gravity. We\nexplained the parameters appeared in TOV equation in\nmassive gravity, including C1andC2. We also derived\nthe differential equation for the metric function H(r) in\nmassive gravity to calculate the tidal deformability.\nTo assess the structural properties of SQS, we initially\nconsidered B1/4= 117 MeV in the EOS and then by set-\nting a fixed value for C1(the mass-radius relation is in-\ndependent of C1[61]) and selecting different values of C2,\nwe calculated the structural properties of SQS. The re-\nsults analyzed to ensure compliance with the constraint\nΛ1.4M⊙≲580. We observed that increasing the αpa-\nrameter led to an increase in the maximum gravitational\nmass but also resulted the aforementioned constraint to\nbe satisfied in a smaller range of massive gravity param-\neters. Consequently, the allowed mass range of the star\ndecreased. For the case of B1/4= 117 MeV , we obtained\nthe maximum allowed MTOV approximately 2 .13M⊙.\nSubsequently, we altered, the value of Band consid-\nered B1/4= 130 MeV . This change resulted in a softer\nEOS and a broader range of massive parameters satis-fying the astronomical constraints. We obtained masses\nthat fall within the mass gap region ( M∼(2.5−5)M⊙)\nthat simultaneously satisfied all the observational con-\nstraints. Moreover, we calculated Schwarzschild radius\nto show that compact objects studied in this paper were\nnot black holes.\nIn general relativity, soft EOSs yield small masses for\nquark stars, while stiff EOSs fail to satisfy the the Λ con-\nstraint. 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Enke, and L. Jiarong,, Phys. Rev. D 46,\n3211 (1992).14\nAppendix A\nTo calculate the tidal deformability, we need to obtain\nthe metric function H(r), which satisfies the following\nequation\n\u0012\n−6e2λ\nr2−2(Φ′)2+ 2Φ′′+3\nrλ′\n+7\nrΦ′−2Φ′λ′+f\nr(Φ′+λ′)\u0013\nH\n+\u00122\nr+ Φ′−λ′\u0013\nβ+dβ\ndr= 0, (A1)\nHere β=dH/dr ,fcorresponds to dϵ/dp . Also, Φ′satis-\nfies the following equation\nΦ′=−p′\n(ϵ+p), (A2)\nwhere p′satisfies TOV equation in massive gravity.\np′=(p+ϵ)B2\n2rB1, (A3)\nM′(r) = 4 πr2ϵ, (A4)\nwhere B1=C1r\n2+C2−1 +2M(r)\nr, and B2= 8πpr2−\nC1r\n2+2M(r)\nr.\nBy deriving relations (A2) and (A3) with respect to r\nwe get\nΦ′′=(1 +f)p′2−p′′(ϵ+p)\n(ϵ+p)2, (A5)\nand\np′′=(1 +f)B2p′\n2rB1\n+\u0000\n4πr2(rp′+ϵ) + 8πr2p−C1r\n4−M\nr\u0001\n(p+ϵ)\nr2B1\n−\u0000\nC1r+C2−1 + 8 πr2ϵ\u0001\n(p+ϵ)B2\n2r2B2\n1. (A6)\nIn Eq. (A1), λ′satisfies the following equation\nλ′=s′\n2s, (A7)\nwhere\ns=−1\nB1, (A8)\nand\ns′=C1\n2+2M′(r)\nr−2M(r)\nr2\nB2\n1. (A9)Now, for the convenience of calculations, we write\nequation (A1) as below\n(A1+A2+A3)H+A4β+dβ\ndr= 0, (A10)\nwhere A1,A2,A3, and A4are defined as follows.\nA1≡ −6e2λ\nr2, (A11)\nA2≡ −2 (Φ′)2+ 2Φ′′+3λ′\nr+7Φ′\nr−2λ′Φ′,(A12)\nA3≡(Φ′+λ′)f\nr, (A13)\nand\nA4≡2\nr+ Φ′−λ′. (A14)\nNow using Eqs. (A2) to (A9), we obtain the following\nterms for coefficients A1toA4.\nA1=6\nr2B1, (A15)\nA2=B1\n2r2B2\n1\u001aB2(B2f+B3)\nB1−7B2+ 3B3\n+1\nB4\n(f+ 1)B2\u0000\nB2+C1r\n2\u0001−1+2B1\u0000\nB2+C1r\n2\u0001\nB4\u0010\n2M(r)\nr+ 8πr2ϵ−2\u0011−1\n+ 16πr2\u0014\nϵ\u0012C1r\n4+C2+ 4πpr2−1\u0013\n+3p\n\u0010\n5C1r\n12+C2+4πpr2\n3−1\u0011−1+M(r) (7p+ 3ϵ)\nr\n\n\n\n,\n(A16)\nA3=4π(p+ϵ)f\nB1, (A17)\nA4=2\u00003C1r\n4+ 2πr2(ϵ−p) +C2−1\u0001\n+2M(r)\nr\nrB1.(A18)\nwhere B3=2M(r)\nr−C1r\n2−8πϵr2, and B4= 1−2M(r)\nr.\nTo check the correctness of the calculations, we con-\nsider the case C1=C2= 0 and show that Eq. (A10)\nconverts to the equation obtained in Einstein’s gravity\n20H\nB4 \nπ\u0012\nϵ+9p+ (ϵ+p)f\n10\u0013\n−3\n10r2−\u0000\nB2+C1r\n2\u00012\n20rB4!\n−\u0010\n2M(r)\nr+ 4πr2(ϵ−p)−2\u0011\nH\nr\u0010\n1−2M(r)\nr\u0011′\n+H′′= 0. (A19)15\nBy considering C1=C2= 0, Eqs. (A15) to (A17) are\nconverted to\nA1=−6\nr2B4, (A20)\nA2=4πB4(9p+ 5ϵ)−\u0000\nB2+C1r\n2\u00012\nB2\n4, (A21)\nA3=4π(p+ϵ)f\nB4. (A22)\nIt is clear that the sum of Eqs. (A20) to (A22) gives\nthe coefficient of Hin equation (A19). Furthermore, by\nconsidering C1=C2= 0, Eq. (A17) becomes\nA4=2\u0010\n1 + 2 πr2(p−ϵ)−M(r)\nr\u0011\nrB4, (A23)\nwhich is equivalent to the coefficient of function H′in\nequation (A19).\nAppendix B\nIn the previous sections, we have considered two differ-\nent values for the bag constant, namely B1/4= 117 MeV\nandB1/4= 130 MeV (similar to Ref. [68]). The rea-\nson for choosing these two values of Bis to compare\nthe structural properties of the SQS in massive grav-\nity (which we have done in this paper) with the results\nobtained in Einstein gravity with the same values of B\n(which was done in Ref. [68]). One question that might\narise is the effect of increasing the bag constant on the\nstructural properties of SQS. In this appendix, we in-\ncrease Band check whether the results cover the mass\ngap region. The determination of the range of Bde-\npends on various factors, such as the model used to ob-\ntain the EOS, the number of quark flavors, the mass of\nthe quarks, etc. If we take into account the constraint of\ntidal deformability, determination of the range of Balso\ndepends on the gravity used. This is because to obtain Λ,\nnot only the EOS (in which Bappears) is effective, but\nalso the gravity used is important. For example in Ref.\n[81], by using MIT bag model and considering ms= 0\n(msis the mass of strange quark), Bis constrained in\nthe range 58 .9MeV\nfm3< B < 91.5MeV\nfm3. In [82] by assum-\ningms= 150 MeV , the range of Bis 56−78MeV\nfm3. In\n[83] by studying 20 compact stars, Bis obtained in the\nrange 41 .58−319.31MeV\nfm3. In Ref. [84], using perturba-\ntive QCD and the Λ constraint from GW170817, B1\n4is\nconstrained to a narrow range from 134.1 MeV to 141.4\nMeV (42MeV\nfm3< B < 52MeV\nfm3). In [85] Bis in the range\n100−200MeV4(13MeV\nfm3< B < 208MeV\nfm3). In this paper,\nwe increase the value of Bin the MNJL model for a fixed\nvalue of α(e.g. α= 0.8) and investigate its effect on thestructural properties of SQS in massive gravity. In the\nfollowing, we choose two different values for Binclud-\ningB1\n4= 140 MeV (B= 50MeV\nfm3) and B1\n4= 148 MeV\n(B= 62MeV\nfm3) and check whether the results cover the\nmass gap region. We show that for these values of B,\nthe results fall within the mass gap region. We do not\ntry the values α= 0.8, and B1\n4>148MeV , because\nfor these values, the results do not satisfy the constraint\n70≲Λ1.4M⊙≲580.\n1. The results for α= 0.8and B1\n4= 140 MeV\nNow we increase the value of B1\n4to 140 MeV and show\nthat the results are still in the mass gap region. We first\ninvestigate the stability condition of SQM. Figure. 21\nshows that the energy per baryon of three-flavor quark\nmatter is less than that of two-flavor quark matter for\nB1/4= 140 MeV andα= 0.8 in all baryon number den-\nsities. In the following, we investigate Λ −MandM−R\ndiagrams for B1/4= 140 MeV ,α= 0.8,C1= 10−5,\nand different values of C2parameter. Figure. 22 shows\nthat the condition 70 ≲Λ1.4M⊙≲580 is fulfilled for\nall values of C2. Compared to the Fig. 18 ( α= 0.8\nandB1/4= 130 MeV ), we can see that the values of Λ\ndecrease with the increase of B. This result is obvious,\nbecause as Bincreases, the EOS becomes softer and con-\nsequently Λ decreases.\n2 4 6 8 101050110011501200125013001350\nnB\nn0ϵ\nnB(MeV)2 Flavors\n3 Flavors\nFIG. 21: Energy per baryon versus baryon number density for\nB1/4= 140 MeV andα= 0.8.n0is the nuclear saturation\ndensity which is equal to 0 .16fm−3.\nFigure. 23 represents M−Rdiagram for B1/4=\n140MeV ,α= 0.8,C1= 10−5, and different values of\nC2. We can see that the results still cover the mass gap\nregion. However, compared to the Fig. 17 ( α= 0.8 and\nB1/4= 130 MeV ), the results are obtained in a narrower\nrange of the mass gap. In fact, such a result can be ex-\npected by making the EOS softer.16\n1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4050010001500\nM\nM⊙ΛB1/4=140 MeV,α=0.8 &C 1=10-5\nC2=-10-3\nC2=-10-1\nC2=-0.3\nC2=-0.5\nC2=-0.7\nC2=-0.82\nFIG. 22: Dimensionless tidal deformability (Λ) versus mass\nof the star for B1/4= 140 MeV ,α= 0.8,C1= 10−5, and\ndifferent values of C2. The upper black line, the lower black\nline, and the vertical red line correspond to Λ = 580, Λ = 70,\nandM\nM⊙= 1.4, respectively.\n8 10 12 14 161.01.52.02.53.03.54.0\nR(km)M\nM⊙B1\n4=140MeV, α=0.8,C 1=10-5\nPSR J0740+66202ry component of GW190814Remnant of GW190425\nPSR J2215+5135\nGW170817PSR J095-0607\nGW170817\nPSR J0030+0451PSR J0030+0451C2=-10-3\nC2=-10-1\nC2=-0.3\nC2=-0.5\nC2=-0.7\nC2=-0.82\nFIG. 23: Mass-Radius diagram for B1/4= 140 MeV ,α= 0.8,\nC1= 10−5, and different values of C2. The gray and orange\nregions are the mass - radius constraints from the GW170817\nevent. The black region shows pulsar J0740+6620, The green\nand black hatched regions represent pulsars J2215+5135 [77]\nand PSR J095-0607, respectively. The red region amounts\nto the secondary component of GW190814. The brown and\nthe purple regions show two different reports of the pulsar\nJ0030+0451 [38, 39]. The light blue band denotes the rem-\nnant mass of GW190425.\n2. The results for α= 0.8and B1\n4= 148 MeV\nFor further investigation, we increase the value of B1/4\nto 148 MeV . First of all, we have to check the stability\ncondition of the SQM. Figure. 24 shows that this con-\ndition is well established for all baryon number densi-\nties. In the next step, we examine the Λ behavior. Fig-\nure. 25 represents Λ −Mdiagram for B1/4= 148 MeV ,\nα= 0.8,C1= 10−5, and different values of C2. As we\ncan see from this figure, the results respect to the con-\nstraint 70 ≲Λ1.4M⊙≲580. It is worth noting that for\nα= 0.8 and B1\n4>148MeV the EOS becomes so soft\nthat the Λ 1.4M⊙is below 70 for some values of C2and\nthe constraint 70 ≲Λ1.4M⊙≲580 is lost. Therefore, we\nconsider B1/4= 148 MeV (B= 62MeV\nfm3) as the maximum\n2 4 6 8 101050110011501200125013001350\nnB\nn0ϵ\nnB(MeV)2 Flavors\n3 FlavorsFIG. 24: Energy per baryon versus baryon number density for\nB1/4= 148 MeV andα= 0.8.n0is the nuclear saturation\ndensity which is equal to 0 .16fm−3.\nallowable value of the bag constant in the MNJL model\nforα= 0.8 to study the structural properties of SQM\nin massive gravity. Figure. 26 shows M−Rdiagram\n1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4050010001500\nM\nM⊙ΛB1/4=148 MeV,α=0.8 &C 1=10-5\nC2=-10-3\nC2=-10-1\nC2=-0.3\nC2=-0.5\nC2=-0.7\nC2=-0.82\nFIG. 25: Dimensionless tidal deformability (Λ) versus mass\nof the star for B1/4= 148 MeV ,α= 0.8,C1= 10−5, and\ndifferent values of C2. The upper black line, the lower black\nline, and the vertical red line correspond to Λ = 580, Λ = 70,\nandM\nM⊙= 1.4, respectively.\nforB1/4= 148 MeV ,α= 0.8,C1= 10−5, and different\nvalues of C2. As this figure shows, despite increasing B,\nthe results still fall within the mass gap region. As men-\ntioned above, this value of Bis the maximum possible\nvalue to meet Λ 1.4M⊙constraint in MNJL for α= 0.8 in\nmassive gravity.17\n8 10 12 14 161.01.52.02.53.03.54.0\nR(km)M\nM⊙B1\n4=148MeV, α=0.8,C 1=10-5\nPSR J0740+66202ry component of GW190814Remnant of GW190425\nPSR J2215+5135\nGW170817PSR J095-0607\nGW170817\nPSR J0030+0451PSR J0030+0451C2=-10-3\nC2=-10-1\nC2=-0.3\nC2=-0.5\nC2=-0.7\nC2=-0.82\nFIG. 26: Mass-Radius diagram for B1/4= 148 MeV ,α= 0.8,\nC1= 10−5, and different values of C2. The gray and orange\nregions are the mass - radius constraints from the GW170817\nevent. The black region shows pulsar J0740+6620, The green\nand black hatched regions represent pulsars J2215+5135 [77]\nand PSR J095-0607, respectively. The red region amounts\nto the secondary component of GW190814. The brown and\nthe purple regions show two different reports of the pulsar\nJ0030+0451 [38, 39]. The light blue band denotes the rem-\nnant mass of GW190425."    },    {        "title": "2402.14043v1.Hybrid_Multi_Directional_Quantum_Communication_Protocol.pdf",        "content": "Hybrid Multi-Directional Quantum Communication\nProtocol\nMitali Sisodia1∗, Manoj Kumar Mandal2†, Binayak S. Choudhury2‡\n1Department of Physics, Indian Institute of Technology, Delhi\nNew Delhi 110016, India\n2Department of Mathematics, Indian Institute of Engineering Science and Technology,\nShibpur B. Garden, Howrah - 711103, West Bengal, India\nAbstract\nThe way a new type of state called a hybrid state, which contains more than one degree of\nfreedom, is used in many practical applications of quantum communication tasks with lesser\namount of resources. Similarly, our aim is here to perform multi quantum communication\ntasks in a protocol to approach quantum information in multipurpose and multi-directional.\nWe propose a hybrid multi-directional six-party scheme of implementing quantum teleporta-\ntion and joint remote state preparation under the supervision of a controller via a multi-qubit\nentangled state as a quantum channel with 100% success probability. Moreover, we analyt-\nically derive the average fidelities of this hybrid scheme under the amplitude-damping and\nthe phase-damping noise.\nKeywords: Controlled quantum teleportation, Joint remote state preparation, CNOT, Uni-\ntary operations, Amplitude damping noise, Phase damping noise, Fidelity.\n1 Introduction\nOne of the most surprising consequences in quantum information science is the quantum telepor-\ntation (QT) technique, which has been increasingly investigated as a basic element for various\napplications in quantum technology. The first teleportation scheme was proposed by Bennett\net al. in 1993, in which a two-qubit entangled state (Bell state) has been utilized as a quantum\nresource to teleport a single-qubit unknown quantum state [1]. The essence of teleportation is\nthat the state to be teleported is unknown to the sender. Later, a separate group of protocols\nknown as remote state preparation (RSP) [2] was developed in which the state to be transferred\nis known, joint remote state preparation (JRSP) [3] was advanced in which information of the\nstate to be transferred is divided amongst more than one party. Protocols for the purpose of\nperforming modified and highly efficient quantum communication tasks have been subsequently\ndiscussed in several works, such as bidirectional teleportation [4–6], quantum conferencing [7,8],\ncontrolled quantum teleportation (CQT) [9–11].\n∗e-mail: mitalisisodiyadc@gmail.com\n†e-mail: manojmandaliiest@gmail.com\n‡e-mail : binayak@math.iiests.ac.in\n1arXiv:2402.14043v1  [quant-ph]  21 Feb 2024QT is a one-way quantum communication process in which an unknown quantum state is tele-\nported by the sender to the receiver with the use of an entangled quantum resource. A few years\nlater, variants of QT have been studied which predict that simultaneous two-way communication\nis possible with the use of double QT resource, known as bidirectional quantum teleportation\n(BQT). Further, a three-party QT scheme, i.e., CQT, has been proposed with the use of a three-\nqubit entangled state. In which, a party controls or supervises the roles of different involved\nparties and helps to perform the teleportation task between sender and receiver. So far, a single\nquantum communication task has been performed in a protocol by using a number of entangled\nquantum resources with a single degree of freedom. Later, it was studied that multi-quantum\ncommunication tasks can be performed in a protocol, which is known as hybrid communication\nand the number of degrees of freedom can be used in a quantum state, i.e., hybrid state, to\nincrease the practical applications in quantum information processing [12–15].\nAmong multitasking protocols, hybrid protocols are the multitasking communication processes\nwhere more than one type of quantum protocols are combined to form an integrated process\ndiscussed in [16–21]. For instance, a hybrid protocol in which four-party controlled JRSP and\nCQT has been implemented via a seven-qubit entangled state [16]. Group of Gong et al. re-\nported the bidirectional hybrid protocol [17] and multi-party controlled cyclic hybrid quantum\ncommunication protocol [22] under the effect of different types of noises (bit flip, phase flip,\nbit-phase flip, phase-damping and amplitude-damping noise): the work of Joo et al. developed\nthe hybrid scheme that teleports quantum information from a solid-state qubit to microwave\nphotonic state [21]. In Ref. [22], a hybrid (QT and RSP scheme) double-channel scheme is\nproposed and also checked the effect of noises on the scheme. Recently, a hierarchical controlled\nhybrid quantum communication scheme has been presented for specific IoT (Internet of Things)\napplication scenarios and simulated the results on IBM’s Qiskit Aer quantum computing sim-\nulator [23] and another hybrid scheme which is based on a classical-quantum communications\nprotocols for managing classical blockchains is studied by Liu A. et al. in 2023 [24]. Since\nthen, rapid development in hybrid quantum communication has attracted much attention and\nmotivation from researchers to move forward into it.\nBy motivating and recognizing the several applications and advantages of hybrid communica-\ntion, in the present context, a multitasking hybrid protocol under the supervision of a controller\nis presented where three single-qubit states are transferred through a JRSP-type process while\nthe receivers of the three single-qubit states teleport to the sender one-qubit, two-qubit, and\nthree-qubit states respectively. Also, we assume the distribution of qubits among the parties\nin an open quantum system (interaction with the noisy environment) [25–28], where quantum\nnoise will unavoidably affect the qubits. As a result, the pure state becomes mixed, which has\na significant impact on the scheme’s efficiency (or the integrity of the output state). Keeping\nthis point in mind, in this context, we also study the effect of two Markovian noisy channels\namplitude-damping (AD) and phase-damping (PD) on this hybrid scheme by calculating the\naverage fidelities.\nThe paper is organized as follows: in Sec. 2, a complete hybrid multi-directional CQT pro-\ntocol is discussed. In Sec. 3, the protocol is considered in noisy environments and AD and PD\nnoise is discussed in the subsections. Finally, the work is concluded in the Sec. 4.\n22 Hybrid Multi-directional Controlled Quantum communica-\ntion(HMCQC) Protocol\nIn this HMCQC protocol, we consider six parties: Alice, Bob, Charlie, Davis, Candy, and Simon.\nAlice and Charlie want to create jointly three single-qubit known states in the hands of Bob,\nDavis, and Candy separately, and at the same time, each party Bob, David, and Candy want\nto teleport three different qubit states to Alice under the controller Simon. Suppose, Alice and\nCharlie have three known single-qubit state |ϕ0⟩,|ϕ1⟩, and|ϕ2⟩which are given by\n|ϕ0⟩=x0|0⟩+y0eiθ0|1⟩\n|ϕ1⟩=x1|0⟩+y1eiθ1|1⟩\n|ϕ2⟩=x2|0⟩+y2eiθ2|1⟩.(1)\nwhere xi,yi, and θi(where i= 0,1,2) are real numbers and θj(where j= 0,1,2) is in the range\n[0,2π]. Alice knows the value of xiandyi, while Charlie knows the values of θj. Additionally,\nthe real numbers xiandyisatisfy the normalization condition x2\ni+y2\ni= 1.\nThe state |ϕ0⟩is to be created at Bob’s place, the state |ϕ1⟩is to be created at David’s place\nand the state |ϕ2⟩is to be created at Candy’s place.\nAt the same time, Bob, David, and Candy want to teleport three unknown states |ψ0⟩,|ψ1⟩,\nand|ψ2⟩respectively to Alice. The unknown quantum states are given by\n|ψ0⟩=(a0|0⟩+b0|1⟩)1\n|ψ1⟩=(a1|00⟩+b1|11⟩)23\n|ψ2⟩=(a2|000⟩+b2|111⟩)456.(2)\nwhere aiandbi(where i= 0,1,2) are complex numbers satisfying the condition |ai|2+|bi|2= 1.\nThese coefficients are unknown to all parties.\nAt first, David applies the CNOT gate on his unknown two-qubit state (2 ,3) where qubit\n2 is the controlled qubit and qubit 3 is the target qubit. Candy also applies two CNOT gates.\nThe first CNOT gate is on qubit pair (4 ,5) with qubit 4 as the controlled qubit and qubit 5 as\nthe target qubit. Second, another CNOT gate is applied on qubit pair (5 ,6) with qubit 5 as the\ncontrolled qubit and qubit 6 as the target qubit. Then the state of the qubits given in equations\n(2) is transformed into\n|ψ′\n0⟩=(a0|0⟩+b0|1⟩)1\n|ψ′\n1⟩=(a1|0⟩+b1|1⟩)2⊗ |0⟩3\n|ψ′\n2⟩=(a2|0⟩+b2|1⟩)4⊗ |00⟩56.(3)\nNow Bob, David and Candy will teleport only qubits 1, 2 and 4 to Alice, respectively. To achieve\nthis protocol, all the parties shared a 16-qubit entangled quantum state as a quantum channel,\nwhich is given as\n|τ⟩=1√\n2|W+⟩A0C0B0⊗ |W+⟩A1C1D0⊗ |W+⟩A2C2E0⊗ |Φ00⟩B1A3⊗ |Φ00⟩D1A4⊗ |Φ00⟩E1A5⊗ |0⟩S\n+1√\n2|W−⟩A0C0B0⊗ |W−⟩A1C1D0⊗ |W−⟩A2C2E0⊗ |Φ01⟩B1A3⊗ |Φ01⟩D1A4⊗ |Φ01⟩E1A5⊗ |1⟩S.\n(4)\n3where the qubits A0, A1, A2, A3, A4andA5belong to Alice, qubits C0, C1andC2belong to\nCharlie, B0andB1are in the hands of Bob, qubits D0andD1belong to David and qubits E0\nandE1belong to Candy while qubit Sbelongs to the controller Simon. Quantum circuit for\nthe generation of the entangled state given in Equation (4) is given in Figure 1. Also the states\n|W±⟩=1√\n2(|000⟩ ± |111⟩),|Φ00⟩=1√\n2(|00⟩+|11⟩) and |Φ01⟩=1√\n2(|00⟩ − |11⟩).\nThe initial quantum state of the entire system is given by\n|X⟩= (a0|0⟩+b0|1⟩)1⊗(a1|0⟩+b1|1⟩)2⊗(a2|0⟩+b2|1⟩)4⊗ |τ⟩A0C0B0,···,S, (5)\nThe entire process of the HMCQC protocol is described as follows.\nStep 1: Alice makes single-qubit measurement on her qubits A0, A1andA2with the X-basis\n{|α0\n0⟩,|α0\n1⟩},{|α1\n0⟩,|α1\n1⟩}and{|α2\n0⟩,|α2\n1⟩}respectively, which are given by\n|αj\n0⟩=1√\n2(xj|0⟩+yj|1⟩) (where j = 0,1,2)\n|αj\n1⟩=1√\n2(yj|0⟩ −xj|1⟩),(where j = 0,1,2)(6)\nAfter the measurement, she shares her measurement results {|αj\nl⟩:l= 0,1;j= 0,1,2}with\nCharlie classically. According to the measurement result of Alice, Charlie chooses measurement\nbases for measuring his qubits C0, C1andC2.\nWhen Alice’s measurement results are |αj\n0⟩Aj:j= 0,1,2 then Charlie choose his measurement\nbases {|β0,j\n0⟩,|β0,j\n1⟩}:j= 0,1,2 for measuring qubits Cj:j= 0,1,2 respectively, which are\ngiven as\n|β0,j\n0⟩=1√\n2(|0⟩+e−iθj|1⟩) (where j = 0,1,2)\n|β0,j\n1⟩=1√\n2(|0⟩ −e−iθj|1⟩),(where j = 0,1,2).(7)\nWhen Alice’s measurement outcomes are |αj\n1⟩Aj:j= 0,1,2 then Charlie select his measurement\nbases{|β1,j\n0⟩,|β1,j\n1⟩}:j= 0,1,2 for measuring qubits Cj:j= 0,1,2 respectively, which are given\nas\n|β1,j\n0⟩=1√\n2(e−iθj|0⟩+|1⟩) (where j = 0,1,2)\n|β1,j\n1⟩=1√\n2(−e−iθj|0⟩+|1⟩).(where j = 0,1,2)(8)\nAt the same time, Bob, David and Candy make Bell-basis measurements on the qubit pairs\n(1, B1), (2, D0) and (4 , E0), respectively.\nBell-basis is a two-qubit measurement basis consisting of four linearly independent sets of quan-\ntum states, which are given as\n|Φ00⟩=1√\n2(|00⟩+|11⟩),|Φ01⟩=1√\n2(|00⟩ − |11⟩),\n|Φ10⟩=1√\n2(|01⟩+|10⟩),|Φ11⟩=1√\n2(|01⟩ − |10⟩).(9)\n4Figure 1: Quantum circuit for the generation of the entangled state |τ⟩in equation (4).\n5Step 2: Using all the above basis, we can write the state of the total system as\n|X⟩=1√\n2\"1X\ng0=01√\n2|α0\ng0⟩A0⊗1X\nh0=01√\n2|βg0,0\nh0⟩C0⊗\u0000\n(−1)g0x0|g0⟩+ (−1)h0y0eiθ0|1⊕g0⟩\u0001\nB0#\nO\"1X\ng1=01√\n2|α1\ng1⟩A1⊗1X\nh1=01√\n2|βg1,1\nh1⟩C1⊗\u0000\n(−1)g1x1|g1⟩+ (−1)h1y1eiθ1|1⊕g1⟩\u0001\nD0#\nO\"1X\ng2=01√\n2|α2\ng2⟩A2⊗1X\nh2=01√\n2|βg2,2\nh2⟩C2⊗\u0000\n(−1)g2x2|g2⟩+ (−1)h2y2eiθ2|1⊕g2⟩\u0001\nE0#\nO\"1X\nm0,n0=01\n2|Φm0n0⟩1B1⊗\u0010\na0|m0⟩+ (−1)n0b0|1⊕m0⟩\u0011\nA3#\nO\"1X\nm1,n1=01\n2|Φm1n1⟩2D1⊗\u0010\na1|m1⟩+ (−1)n1b1|1⊕m1⟩\u0011\nA4#\nO\"1X\nm2,n2=01\n2|Φm2n2⟩4E1⊗\u0010\na2|m2⟩+ (−1)n2b2|1⊕m2⟩\u0011\nA5#O\n|0⟩S(10)\n+1√\n2\"1X\ng0=01√\n2|α0\ng0⟩A0⊗1X\nh0=01√\n2|βg0,0\nh0⟩C0⊗\u0000\nx0|g0⟩+ (−1)g0⊕h0⊕1y0eiθ0|1⊕g0⟩\u0001\nB0#\nO\"1X\ng1=01√\n2|α1\ng1⟩A1⊗1X\nh1=01√\n2|βg1,1\nh1⟩C1⊗\u0000\nx1|g1⟩+ (−1)g1⊕h1⊕1y1eiθ1|1⊕g1⟩\u0001\nD0#\nO\"1X\ng2=01√\n2|α2\ng2⟩A2⊗1X\nh2=01√\n2|βg2,2\nh2⟩C2⊗\u0000\nx2|g2⟩+ (−1)g2⊕h2⊕1y2eiθ2|1⊕g2⟩\u0001\nE0#\nO\"1X\nm0,n0=01\n2|Φm0n0⟩1B1⊗\u0010\n(−1)m0a0|m0⟩+ (−1)1⊕m0⊕n0b0|1⊕m0⟩\u0011\nA3#\nO\"1X\nm1,n1=01\n2|Φm1n1⟩2D1⊗\u0010\n(−1)m1a1|m1⟩+ (−1)1⊕m1⊕n1b1|1⊕m1⟩\u0011\nA4#\nO\"1X\nm2,n2=01\n2|Φm2n2⟩4E1⊗\u0010\n(−1)m2a2|m2⟩+ (−1)1⊕m2⊕n2b2|1⊕m2⟩\u0011\nA5#O\n|1⟩S,\nAfter performing the measurements by every party(excluding controller Simon), they shared\ntheir measurement outcomes publicly via the classical channel. Suppose Alice’s measurement\nresults are |α0\ng0⟩A0,|α1\ng1⟩A1and|α2\ng2⟩A2, Charlie’s measurement outcomes are |βg0,0\nh0⟩C0,|βg1,1\nh1⟩C1\nand|βg2,2\nh2⟩C2, Bob’s outcome is |Φm0n0⟩1B1, David’s outcome is |Φm1n1⟩2D1and Candy’s outcome\nis|Φm2n2⟩4E1(here gj, hj, mj, nj∈ {0,1}forj∈ {0,1,2}) then the reduced state of the system\nis given by\n6|R⟩=1√\n2\u0000\n(−1)g0x0|g0⟩+ (−1)h0y0eiθ0|1⊕g0⟩\u0001\nB0⊗\u0000\n(−1)g1x1|g1⟩+ (−1)h1y1eiθ1|1⊕g1⟩\u0001\nD0\n⊗\u0000\n(−1)g2x2|g2⟩+ (−1)h2y2eiθ2|1⊕g2⟩\u0001\nE0⊗\u0000\na0|m0⟩+ (−1)n0b0|1⊕m0⟩\u0001\nA3\n⊗\u0000\na1|m1⟩+ (−1)n1b1|1⊕m1⟩\u0001\nA4⊗\u0000\na2|m2⟩+ (−1)n2b2|1⊕m2⟩\u0001\nA5⊗ |0⟩S\n+1√\n2\u0000\nx0|g0⟩+ (−1)g0⊕h0⊕1y0eiθ0|1⊕g0⟩\u0001\nB0⊗\u0000\nx1|g1⟩+ (−1)g1⊕h1⊕1y1eiθ1|1⊕g1⟩\u0001\nD0\n⊗\u0000\nx2|g2⟩+ (−1)g2⊕h2⊕1y2eiθ2|1⊕g2⟩\u0001\nE0⊗\u0000\n(−1)m0a0|m0⟩+ (−1)1⊕m0⊕n0b0|1⊕m0⟩\u0001\nA3\n⊗\u0000\n(−1)m1a1|m1⟩+ (−1)1⊕m1⊕n1b1|1⊕m1⟩\u0001\nA4⊗\u0000\n(−1)m2a2|m2⟩+ (−1)1⊕m2⊕n2b2|1⊕m2⟩\u0001\nA5⊗ |1⟩S.\n(11)\nStep 3: Controlled Simon examines all the measurement outcomes from all the others involved\nparties. If he satisfies, he measures his qubit Swith the basis {|0⟩,|1⟩}and classically shares\nhis measurement outcome |t⟩S(where t∈ {0,1}) with all the parties.\nFinally, Alice, Bob, David, and Candy apply the unitary operators on their respective qubits to\nget the desired qubits to achieve the task.\nAlice needs to apply the unitary operator on the qubits ( A3, A4, A5) corresponding to the mea-\nsurement results |Φm0n0⟩1B1,|Φm1n1⟩2D1,|Φm2n2⟩4E1and|t⟩Sof Bob, David, Candy and Simon\nrespectively, the unitary operators are given by\nUAlice= (1−t)U(mj,nj)\n0 ⊗U(mj,nj)\n1 ⊗U(mj,nj)\n2 +tV(mj,nj)\n0 ⊗V(mj,nj)\n1 ⊗V(mj,nj)\n2 , (12)\nwhere\nU(mj,nj)\nj =|0⟩⟨mj|+ (−1)nj|1⟩⟨1⊕mj|,\nV(mj,nj)\nj =(−1)mj|0⟩⟨mj|+ (−1)1⊕mj⊕nj|1⟩⟨1⊕mj|.\nNow, Alice introduces three auxiliary qubits A6, A7andA8within the initial state 0 ⟩for each.\nand apply a CNOT gate on each of the qubit pairs ( A4, A6), (A5, A7) and ( A5, A8) where A3\nandA5are the controlled qubit and qubits A6, A7andA8are the targeted qubits. Then the\nstate of the qubits A3, A6, A5, A7andA8transformed into the state given by\n(a0|0⟩+b0|1⟩)A3⊗(a1|00⟩+b1|11⟩)A4A6⊗(a2|000⟩+b2|111⟩)A5A7A8,\nTherefore, we see the three states that Bob, David and Candy want to teleport to Alice are\nsuccessfully teleported into the qubits A3, (A4A6) and ( A5A7A8), respectively.\nBob needs to apply the Unitary operator on his qubit B0to get the desired state corresponding\nto the measurement results of Alice, Charlie and Simon, |α0\ng0⟩A0,|βg0,0\nh0⟩C0and|t⟩Srespectively,\nwhich is\nUBob= (1−t)M(g0,h0)\n0 +tN(g0,h0)\n0 . (13)\nThen the state of the qubit B0becomes ( x0|0⟩+y0eiθ0|1⟩)B0which Alice and Charlie jointly\nwant to prepare in Bob’s place.\n7David needs to apply the Unitary operator on his qubit C0to get the desired state corresponding\nto the measurement results of Alice, Charlie and Simon, |α1\ng1⟩A1,|βg1,1\nh1⟩C1and|t⟩Srespectively,\nwhich is\nUDavid = (1−t)M(g1,h1)\n1 +tN(g1,h1)\n1 . (14)\nThen, the state of the qubit D0evolves ( x1|0⟩+y1eiθ1|1⟩)D0which Alice and Charlie together\ndesire to prepare in David���s place.\nCandy needs to apply the Unitary operator on his qubit C0to get the desired state corresponding\nto the measurement results of Alice, Charlie and Simon, |α2\ng2⟩A2,|βg2,2\nh2⟩C2and|t⟩Srespectively,\nwhich is\nUCandy = (1−t)M(g2,h2)\n2 +tN(g2,h2)\n2 , (15)\nwhere\nM(gj,hj)\nj =(−1)gj|0⟩⟨gj|+ (−1)hj|1⟩⟨1⊕hj|,\nN(gj,hj)\nj =|0⟩⟨gj|+ (−1)1⊕gj⊕hj|1⟩⟨1⊕mj|.\nThen the state of the qubit E0becomes ( x2|0⟩+y2eiθ2|1⟩)E0which Alice and Charlie jointly\ndesire to prepare in Candy’s place.\nTherefore, regardless of the measurement outcomes for all parties, we see that Alice, Bob, David,\nand Candy always reconstructs their intended states. As a result, our protocol successfully fulfills\nits assignment, achieving a 100% success probability or unit fidelity.\n3 Noise Analysis\nIn this section, we would like to analyze the effect of noise on the average fidelity of the proposed\nprotocol over two well-known Markovian channels, i.e., AD and PD. Suppose the controller\n’Simon’ generates the entangled states in his lab, distributes the qubits to respective parties\nthrough a noisy channel and keeps the requisite qubit Swith him. Therefore, the qubits S\nare not affected by the noisy environment, but all the remaining qubits traverse through noisy\nenvironments. Here, we consider the same noise for all the qubits at a time for simplification.\nThe density matrix of the whole initial state |X⟩of the entire system is given as\nρ=|X⟩⟨X|. (16)\nBut specifically, in open quantum system formalism, a quantum state evolves under a noisy\nenvironment and the density matrix( ρ) is transformed into the mixed state, which can be written\nin terms of Kraus operators as follows [25]\nϵ(ρ) =X\njEjρE†\nj, (17)\nwhere Ejis the Kraus operator for the specific noisy channel. Now, all the involved parties\nmake measurements on their respective qubits with corresponding measurement bases described\n8in the main protocol. After the measurement is done, they share the classical information with\none another. According to the classical information, they apply the unitary operation on the\nrespective qubits. Then, the final reduced state of the qubits B0, D0, E0, A3, A4andA5is given\nby\nρout=Tr123A0C0A1C1A2C2B1D1E1S(UΘϵ(ρ)Θ†U†). (18)\nwhere Θ =\u0010N2\ni=0|αi\ngi⟩Ai⟨αi\ngi|\u0011\n⊗\u0010N2\ni=0|βgi,i\nhi⟩Ci⟨βgi,i\nhi|\u0011\n⊗|Φm0n0⟩1B1⟨Φm0n0|⊗|Φm1n1⟩2D1⟨Φm1n1|⊗\n|Φm2n2⟩4E1⟨Φm2n2|⊗|t⟩S⟨t|⊗IB0D0E0A3A4A5andU=UAlice⊗UBob⊗UDavid⊗UCandy⊗I. Here,\nthe partial trace is taken over the qubits 1 ,2,3, A0, C0, A1, C1, A2, C2, B1, D1, E1,andS.\nNow, the fidelity of the process is defined as\nF=⟨Ψin|ρout|Ψin⟩. (19)\nHere the input state is given as |Ψin⟩=N2\ni=0|ϕi⟩N2\ni=0|ψ′\ni⟩.\n3.1 Amplitude-Damping Noisy channel\nAmplitude damping noise involves the energy dissipation in quantum systems and the Kraus\noperators for AD noisy channel are given as [25–28]\nK0=\u00121 0\n0√1−p\u0013\n, K 1=\u00120√p\n0 0\u0013\n. (20)\nwhere p∈[0,1] is the noise intensity or decoherence rate of the amplitude-damping channel.\nSince we consider the effect of noise to be the same for all the qubits, the quantum channel given\nby the pure entangled |τ⟩in (4) is transformed according to linear mapping in (17) into a mixed\nstate which is given by the density matrix\nρAD=(15O\ni=0K0)⊗I|τ⟩⟨τ|(15O\ni=0K0)†⊗I+ (15O\ni=0K1)⊗I|τ⟩⟨τ|(15O\ni=0K1)†⊗I\n=M\u0010\n|W+\n1⟩A0C0B0⊗ |W+\n1⟩A1C1D0⊗ |W+\n1⟩A2C2E0⊗ |Φ′\n00⟩B1A3⊗ |Φ′\n00⟩D1A4⊗ |Φ′\n00⟩E1A5⊗ |0⟩S\n+|W−\n1⟩A0C0B0⊗ |W−\n1⟩A1C1D0⊗ |W−\n1⟩A2C2E0⊗ |Φ′\n01⟩B1A3⊗ |Φ′\n01⟩D1A4⊗ |Φ′\n01⟩E1A5⊗ |1⟩S\u0011\n×\u0010\n⟨W+\n1|A0C0B0⊗ ⟨W+\n1|A1C1D0⊗ ⟨W+\n1|A2C2E0⊗ ⟨Φ′\n00|B1A3⊗ ⟨Φ′\n00|D1A4⊗ ⟨Φ′\n00|E1A5⊗ ⟨0|S\n+⟨W−\n1|A0C0B0⊗ ⟨W−\n1|A1C1D0⊗ ⟨W−\n1|A2C2E0⊗ ⟨Φ′\n01|B1A3⊗ ⟨Φ′\n01|D1A4⊗ ⟨Φ′\n01|E1A5⊗ ⟨1|S\u0011\n+N|000000000000000 ⟩A0C0B0A1C1D0A2C2E0B1A3D1A4E1A5⟨000000000000000 |\n⊗(|0⟩ − |1⟩)S(⟨0| − ⟨1|),\n(21)\n9where\n|W±\n1⟩=1p\n1 + (1 −p)3(|000⟩ ±(p\n1−p)3/2|111⟩),\n|Φ′\n00⟩=1p\n1 + (1 −p)2(|00⟩+ (1−p)|11⟩),\n|Φ′\n01⟩=1p\n1 + (1 −p)2(|00⟩ −(1−p)|11⟩),\nM=(1 + (1 −p)3)3(1 +p2)3\n(1 + (1 −p)3)3(1 +p2)3+p15, N =p15\n(1 + (1 −p)3)3(1 +p2)3+p15.\nAlice, Charlie, Bob, David, Candy and Simon make their respective measurements on the respec-\ntive qubits with the respective measurement bases. Then, they classically share their measure-\nment results with the required parties and lastly, Alice, Bob, David and Candy apply a unitary\noperator on the respective qubits. Suppose Alice’s measurement results are |α0\n0⟩A0,|α1\n0⟩A1and\n|α2\n0⟩A2, Charlie’s measurement outcomes are |β0,0\n0⟩C0,|β0,1\n0⟩C1and|β0,2\n0⟩C2, Bob’s outcome is\n|Φ00⟩1B1, David’s outcome is |Φ00⟩2D1, Candy’s outcome is |Φ00⟩4E1(here gj, hj, mj, nj∈ {0,1}\nforj∈ {0,1,2}) and Simon’s measurement result is |0⟩Sthen after unitary is applied, the final\nreduced state according to (18) is given by\nρAD\nout=M\u0002\n⊗2\ni=0(xi|0⟩+ (1−p)3/2yi|1⟩)⊗(ai|0⟩+ (1−p)bi|1⟩)\u0003\n×\u0002\n⊗2\ni=0(xi⟨0|+ (1−p)3/2yi⟨1|)\n⊗2\ni=0(ai⟨0|+ (1−p)bi⟨1|) +N2O\ni=0(x2\ni|0⟩⟨0|)2O\ni=0(a2\ni|0⟩⟨0|).\n(22)\nAccording to the equation (19), the fidelity is given by\nFAD=M2Y\ni=0(x2\ni+ (1−p)3/2y2\ni)22Y\ni=0(a2\ni+ (1−p)b2\ni)2+N2Y\ni=0(x4\ni)2Y\ni=0(a4\ni), (23)\nhereai= cos(θi\n2),bi= sin(θi\n2)eiξi. Now we calculate average fidelity which is obtained by taking\nthe average over all unknown quantum states, i.e., by computing [27,28].\nFaverage\nAD=M2Y\ni=0(x2\ni+ (1−p)3/2y2\ni)2×2Y\ni=0\u00101\n4πZπ\n0sinθZ2π\n0(cos2θi\n2+ (1−p) sinθi\n2)2dξdθ\u0011\n+N2Y\ni=0(x4\ni)2Y\ni=0\u00101\n4πZπ\n0sinθZ2π\n0cos4θi\n2dξdθ\u0011\n=M2Y\ni=0(x2\ni+ (1−p)3/2y2\ni)2×\u00103−3p+p2\n3\u00113\n+N\n272Y\ni=0(x4\ni).\n(24)\nThe average fidelity depends on the noise intensity parameter ‘ p’ and the known states coefficients\n(xi, yi). When x0=√\n0.3, x1=√\n0.4 and x2=√\n0.5 then the variation in average fidelity\nFaverage\nADwith respect to the noise parameter ‘ p’ is given in figure 2 (a). If we consider that the\nthree known states are the same, i.e., x0=x1=x2, then the average fidelity is given as\nFaverage\nAD(1)=M\n27\u0000\nx2\n0+ (1−p)3/2(1−x2\n0)\u00016\u0000\n3−3p+p2\u00013+N\n27x12\n0. (25)\n10and the variation of Faverage\nAD(1)with respect to the noise parameter ( p) and the coefficient x0is\ngiven in the figure 2 (b).\n(a) (b)\nFigure 2: (Color online) (a) 2D plot shows the variation of Faverage\nADwith respect to the noise\nparameter ‘ p’ when x0=√\n0.3, x1=√\n0.4 and x2=√\n0.5. (b) 3D plot shows the variation of\nFaverage\nAD(1)with respect to the noise parameter pand the coefficient x0.\nIn AD noisy environment, it is evident from figure 2 ( a) and ( b) that the average fidelity is\ndecreasing with respect to an increase in the noise parameter ‘ p’. Also, fidelity is near zero in\nthe interval (0 .6,1).\n3.2 Phase-Damping Noisy channel\nPhase damping is the process of loss of information about the relative phases of a quantum\nstate. In a phase-damping noisy channel, the Kraus operators are given as [25–28]\nK0=\u0012√1−q 0\n0√1−q\u0013\n, K1=\u0012√q0\n0 0\u0013\n, K2=\u00120 0\n0√q\u0013\n, (26)\nwhere q∈[0,1] represents the noise intensity of the phase-damping channel, which lies from 0\nto 1. Since we consider the effect of noise to be the same for all the qubits, the quantum channel\ngiven by the pure entangled |τ⟩in (4) transformed according to linear mapping in (17) into a\n11mixed state which is given by the density matrix\nρPD=2X\nj=0h\n(15Y\ni=0Kj)⊗I|τ⟩⟨τ|(15Y\ni=0Kj)†⊗Ii\n=(1−q)15\nR\u0010\n|W+⟩A0C0B0⊗ |W+⟩A1C1D0⊗ |W+⟩A2C2E0⊗ |Φ00⟩B1A3⊗ |Φ00⟩D1A4⊗ |Φ00⟩E1A5⊗ |0⟩S\n+|W−⟩A0C0B0⊗ |W−⟩A1C1D0⊗ |W−⟩A2C2E0⊗ |Φ01⟩B1A3⊗ |Φ01⟩D1A4⊗ |Φ01⟩E1A5⊗ |1⟩S\u0011\n×\u0010\n⟨W+|A0C0B0⊗ ⟨W+|A1C1D0⊗ ⟨W+|A2C2E0⊗ ⟨Φ00|B1A3⊗ ⟨Φ00|D1A4⊗ ⟨Φ00|E1A5⊗ ⟨0|S\n+⟨W−|A0C0B0⊗ ⟨W−|A1C1D0⊗ ⟨W−|A2C2E0⊗ ⟨Φ01|B1A3⊗ ⟨Φ01|D1A4⊗ ⟨Φ01|E1A5⊗ ⟨1|S\u0011\n+q15\nR|000000000000000 ⟩A0C0B0A1C1D0A2C2E0B1A3D1A4E1A5⟨000000000000000 |\n⊗(|0⟩+|1⟩)S(⟨0|+⟨1|)\n+q15\nR|111111111111111 ⟩A0C0B0A1C1D0A2C2E0B1A3D1A4E1A5⟨111111111111111 |\n⊗(|0⟩ − |1⟩)S(⟨0| − ⟨1|),\n(27)\nwhere R= (1−q)15+ 2q15is the normalization factor.\nAlice, Charlie, Bob, David, Candy, and Simon each conduct measurements on their respective\nqubits using specific measurement bases. Following this, they share their measurement outcomes\nwith the necessary recipients. Subsequently, Alice, Bob, David, and Candy individually apply\na unitary operator to their respective qubits. Let’s denote Alice’s measurement outcomes as\n|α0\n0⟩A0,|α1\n0⟩A1and|α2\n0⟩A2, Charlie’s as |β0,0\n0⟩C0,|β0,1\n0⟩C1and|β0,2\n0⟩C2, Bob’s as |Φ00⟩1B1, David’s\nas|Φ00⟩2D1and Candy’s as |Φ00⟩4E1(here gj, hj, mj, nj∈ {0,1}forj∈ {0,1,2}) and Simon’s\nas|0⟩S. After the application of the unitary operation, the resulting final reduced state can be\ndescribed according to equation (18) as:\nρPD\nout=(1−q)15\nRh\n⊗2\ni=0(xi|0⟩+yi|1⟩)⊗2\ni=0(ai|0⟩+bi|1⟩)ih\n⊗2\ni=0(xi⟨0|+yi⟨1|)⊗2\ni=0(ai⟨0|+bi⟨1|)i\n+q15\nR(2Y\ni−0x2\nia2\ni)|000000 ⟩⟨000000 |+q15\nR(2Y\ni−0y2\nib2\ni)|111111 ⟩⟨111111 |.\n(28)\nAccording to the equation (19), the fidelity is given by\nFPD=(1−q)15\nR+q15\nR2Y\ni=0x4\nia4\ni+q15\nR2Y\ni=0y4\nib4\ni. (29)\nWe consider ai= cos(θi\n2),bi= sin(θi\n2)eiξiwhere θ∈[0, π], ξ∈[0,2π]. Now, the average fidelity\n12over all the unknown states is given by\nFaverage\nPD=(1−q)15\nR+q15\nR2Y\ni=0(x4\ni)2Y\ni=0\u00101\n4πZπ\n0sinθZ2π\n0cos4θi\n2dξdθ\u0011\n+q15\nR2Y\ni=0(y4\ni)2Y\ni=0\u00101\n4πZπ\n0sinθZ2π\n0sin4θi\n2dξdθ\u0011\n=(1−q)15\nR+q15\n27R2Y\ni=0(x4\ni) +q15\n27R2Y\ni=0(y4\ni).(30)\nIf three known states are the same, then the average fidelity is given as\nFaverage\nPD(1)=(1−q)15\nR+q15\n27R(x12\n0+y12\n0). (31)\nFor PD noise, we consider the same known state coefficients as mentioned in the AD noisy\nchannel (see 3.1) and the variation of average fidelity under the PD noisy channel ( Faverage\nPD(1))\nconcerning the noise parameter ‘q’ and the coefficients are given in the figures 3 (a) and (b).\n(a) (b)\nFigure 3: (Color online)(a) Variation of Faveragein the PD noisy channel with respect to the\nnoise parameter ‘q’ when the known state coefficients are x0=√\n0.3, x1=√\n0.4 and x2=√\n0.5.\n(b) 3D plot shows the variation of Faverage\nPD(1)with respect to noise parameter ( q) and the coefficient\nx0.\nIn PD noisy environment, it is evident that the average fidelity approaches unity within the\nrange of p∈[0,0.4] in figures 3 (a) and (b) and diminishes towards zero within the range of\np∈[0.6,1]. Notably, the fidelity experiences a sharp decline within the interval q∈(0.4,0.6).\n4 Discussion and Conclusion\nA hybrid six-party multi-directional CQT protocol is discussed in this paper with the use of\na multi-qubit entangled quantum state. Especially, QT and JRSP quantum communication\nschemes are proposed in a protocol at the same time by using a 16-qubit entangled quantum\nstate under the supervision of a controller ’Simon’. The HMCQC protocol is divided into a few\n13steps to make it easier for new researchers to learn and implement, and it offers many benefits\nover the previous protocol. Several simple operations, including CNOT gate, single-qubit mea-\nsurements, two-qubit measurements, Bell state measurements as well as unitary operations, have\nbeen used to achieve communication tasks successfully. Our work is not only limited to that, but\nwe have also studied the protocol in an open quantum system and analytically studied the effect\nof AD and PD noises on the hybrid communication scheme by calculating average fidelity. We\nbelieve that hybrid multi-directional controlled quantum communication can effectively address\nthe requirements of upcoming quantum communication networks and establish a solid basis for\nmulti-party quantum communication.\nAcknowledgement This work is supported by Indian Institute of Engineering Science and\nTechnology, Shibpur, India. MS thankfully acknowledges the following funding agency: De-\nfence Research and Development Organisation, India (JATC-2 PROJECT # 2(RP03700G)) for\na project grant of Prof. Joyee Ghosh (IIT Delhi, Department of physics) from where MS is\ndrawing the salary while working with Prof. Binayak.\nReferences\n[1] Bennett, C.H., Brassard, G., Cr´ epeau, C., Jozsa, R., Peres, A., Wootters,W.K.: Teleporting\nan unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys.\nRev. Lett. 70, 1895–1899 (1993)\n[2] Bennett, C.H., DiVincenzo, D.P., Shor, P.W., Smolin, J.A., Terhal, B.M., Wootters, W.K.:\nRemote state preparation. Phy. Rev. Lett. 87, 077902 (2001)\n[3] Nguyen, B.A. and Kim, J.: Joint remote state preparation. J PHYS B-AT MOL OPT, 41,\n095501 (2008)\n[4] Yang, Y.Q., Zha, X.W., Yu, Y.: Asymmetric bidirectional controlled teleportation via\nseven-qubit cluster state. Int. J. Theor. 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Process. 16, 1-23\n(2017)\n16"    },    {        "title": "2402.14054v1.Perturbative_reheating_and_thermalization_of_pure_Yang_Mills_plasma.pdf",        "content": "KEK-TH-2602\nTU-1224\nPerturbative reheating and thermalization\nof pure Yang-Mills plasma\nKyohei Mukaida◦,△, Masaki Yamada•,▲\n◦Theory Center, IPNS, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan\n△Graduate University for Advanced Studies (Sokendai),\n1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan\n•FRIS, T ohoku University, Sendai, Miyagi 980-8578, Japan\n▲Department of Physics, T ohoku University, Sendai, Miyagi 980-8578, Japan\nAbstract\nWe investigate the thermalization of high-energy particles injected from the perturbative decay of in-\nflaton during the pre-thermal phase of reheating in detail. In general, thermalization takes a relatively\nlong time in a low-temperature plasma; therefore, the instantaneous thermalization approximation is\nnot justified, even for the reheating of the Standard Model (SM) sector. We consider a pure Yang–\nMills (YM) theory as an approximation of the SM sector or a possible dark sector, considering the\nLandau–Pomeranchuk–Migdal effect, a quantum interference effect in a finite temperature plasma.\nWe perform the first numerical calculation to solve the time evolution of the system, including the\nredshift due to the expansion of the Universe, and show the details of the temperature evolution\nnear the maximum and the behavior of the quasi-attractors at later times. The maximal tempera-\nture𝑇maxand time scale 𝑡maxare determined quantitatively, such as 𝑇max≃0.05×(Γ𝐼𝑀2\nPl/𝑚3\n𝐼)2/5𝑚𝐼\nand𝑡max≃2×103×(Γ𝐼𝑀2\nPl/𝑚3\n𝐼)−3/5𝑚−1\n𝐼in the SM-like system, where 𝑚𝐼andΓ𝐼are the mass and\ndecay rate of inflaton. We also provide a similar formula for pure SU(𝑁)andSO(𝑁)YM theories for\ngeneral values of 𝑁and coupling constant 𝛼, including𝑇max∝𝛼4/5and𝑡max∝𝑁−2𝛼−16/5behaviors\nand their numerical coefficients. The thermalization occurs in a finite time scale, resulting in a lower\nmaximal temperature of the Universe after inflation than that under the instantaneous thermalization\napproximation.arXiv:2402.14054v1  [hep-ph]  21 Feb 2024Contents\n1 Introduction 2\n2 Kinetic equations 3\n2.1 Warmup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n2.2 Qualitative discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2.3 In-medium splitting function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n3 Numerical simulations 8\n3.1 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3.2 Results of the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n3.3 Results of pure YM theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n4 Discussion and conclusions 15\nA Case with other initial conditions 161 Introduction\nThe Big Bang cosmology has been confirmed by the success of the Big Bang nucleosynthesis (BBN)\ntheory [1,2] and the observation of cosmic microwave background (CMB) [3,4]. After the Big Bang, the\ntemperature of the background plasma decreases via the redshift due to the expansion of the Universe. The\nUniverse might experience several phase transitions during cosmological thermal history, including the\nelectroweak and QCD phase transitions. Several models of physics beyond the Standard Model (SM) also\npredict other cosmological phase transitions. In general, phase transitions lead to rich phenomenology\nin cosmology, such as the emission of gravitational waves (GWs) [5–8] and formation of topological\ndefects [9]. The dynamics of topological defects also lead to the emission of GWs [9–21]. These GW\nsignals are outstanding signatures of physics beyond the SM that is not accessible by collider experiments,\nand therefore, understanding the detailed thermal history is important to reveal the physics beyond the\nSM. For this purpose, the maximal temperature of the Universe is an important quantity. If the maximal\ntemperature is higher than the critical temperature of a phase transition, one can expect that the Universe\nexperiences the phase transition as it cools down due to the redshift.\nThe several problems of Big Bang cosmology, related to its initial condition, can be addressed by\ninflation [22–24]. After inflation, the Universe is reheated by the decay of inflaton into radiation. The\ntemperature of radiation reaches its maximum at a specific time and then cools down due to the redshift.\nOne may assume instantaneous reheating , in which the radiation-dominated era is followed by inflation\nwithout an inflaton-dominated era. However, this scenario is considerably simplified in several cases. If\nthe inflaton couples to radiation without any suppression, its decay is considerably fast and might cause\na non-perturbative process called preheating [25,26]. On the contrary, if the inflaton couples to radiation\nfeebly, then the inflaton tends to dominate the Universe for a while and then slowly decays into radiation\nperturbatively. In this paper, we focus on the latter scenario.\nThe slow decay of the inflaton into radiation does not imply that the radiation is thermalized instan-\ntaneously after production. Namely, instantaneous thermalization is not generally justified and the thermal\nhistory during the pre-thermal phase is not trivial. Let us specifically consider a suppressed decay of inflation\ninto gluons, e.g.,via a (Planck-suppressed) higher-dimensional operator. For a typical inflation model, the\ninflaton mass is very large.♮1The decay of such a heavy particle produces very high energy particles, such\nas gluons, in background plasma. The thermalization of high–energy gluons is quite non-trivial because\nthey should lose significant amount of energy to be thermalized and absorbed into thermal plasma [30–33].\nIn fact, our previous studies have shown that the thermalization-time scale is generally significantly longer\nthan the Hubble-time scale in the early stage of reheating, and the maximal temperature of the Universe\nmay be significantly suppressed [34–39]. Detailed thermalization also provides a novel dark matter (DM)\nproduction mechanism in the pre-thermal phase [39–42]. Several previous studies have focused on the DM\nproduction, in which case the particle distribution is reduced to a stationary solution, and the analysis\ncan be significantly simplified.\n♮1Although the decay rate is sufficiently small to justify the perturbative analysis, the temperature of plasma can be greater\nthan the inflaton mass. In this case, the decay rate gets modified by the thermal effects [27–29].\n2The bottleneck process for thermalization is the splitting of a high-energy gluon into low-energy\ndaughter particles. The splitting process experiences an interference between wavepackets, known as\nthe Landau–Pomeranchuk–Migdal (LPM) effect [43–48], and therefore has a strongly suppressed rate of\nΓsplit=𝛼2√︁\n𝑇3/𝑝, where𝑝is the energy of the parent particle, and 𝑇is the (effective) temperature of\nthe background plasma.♮2Comparing this with the Hubble expansion rate, the maximal temperature of\nthe Universe can be estimated as 𝑇max∼𝛼4/5(Γ𝐼𝑀2\nPl/𝑚3\n𝐼)2/5𝑚𝐼, where𝑀Plis the reduced Planck mass,\nΓ𝐼is the inflaton decay rate, and 𝑚𝐼is the inflaton mass [34, 35]. This can be several orders of magni-\ntude smaller than the naive result under the instantaneous thermalization approximation. This qualitative\ndiscussion demonstrates the importance of a detailed investigation of thermalization for inflaton-decay\nproducts to pin down the maximal temperature of the Universe.\nIn this paper, we numerically solve the detailed Boltzmann equation to calculate the time dependence\nof temperature during the pre-thermal phase, considering the LPM effect. To minimize the numerical cost,\na pure Yang–Mills (YM) theory is considered. This is a good approximation for the SM sector and also\nmotivated by a thermalization of the dark sector that may explain DM. We provide a quantitative formula\nfor the maximal temperature and thermalization time scale. This is the first work on the quantitative time\nevolution of thermal plasma during the pre-thermal phase.\nThe organization of this paper is as follows. In Sec. 2, we briefly review the thermalization process of\na high-energy particle under the cosmological expansion. Moreover, a qualitative estimation of thermal\nhistory is explained. In Sec. 3, we explain our numerical method and show our results. The results are\nconsistent with the qualitative discussion, but provide more information, including numerical prefactors.\nWe consider a model that mimics the SM sector and a model of the pure YM dark sector. In the latter\ncase, the gauge group 𝐺is assumed to be SU(𝑁)andSO(𝑁). A formula for maximal temperature is also\nshown in a large 𝑁and small gauge coupling limit. Section 4 is devoted to discussion and conclusions.\n2 Kinetic equations\n2.1 Warmup\nLet us start our discussion by neglecting the thermalization of the produced particles as a warmup. In\nthis case, the relevant Boltzmann equation is given as follows:\n\u0012𝜕\n𝜕𝑡−𝐻𝑝𝜕\n𝜕𝑝\u0013\n𝑓(𝑝,𝑡)=S, (2.1)\nwhere𝑝is the physical momentum, and 𝐻is the Hubble parameter. The distribution function of gluon\nper one degree of freedom is denoted by 𝑓, that is, the total number density of gluon is obtained by\nmultiplying the degrees of freedom. The source term is (approximately) given by a delta function at\n𝑝=𝑝0. Throughout this paper, we consider the case where the primary particles originate from the\ntwo-body decay of a heavy particle, e.g.,inflaton, with number density 𝑛𝐼(𝑡), mass𝑚𝐼, and decay rate Γ𝐼.\n♮2This splitting rate is for the cases with non-Abelian interactions, which we focus on in this paper. The splitting rate for\nAbelian interactions is different; however, a qualitatively similar cosmological thermal history is obtained [39,40].\n3The source term is expressed as follows:\nS=1\n𝜈𝑔dΓ𝐼\nd𝑝2𝜋2\n𝑝2𝑛𝐼(𝑡), (2.2)\ndΓ𝐼\nd𝑝=2Γ𝐼𝛿(𝑝−𝑝0), 𝑝 0=𝑚𝐼/2, (2.3)\nwhere𝜈𝑔is the degrees of freedom of the gluon. If the heavy particle behaves as pressureless matter, we\nhave\n𝑛𝐼(𝑡)=𝑛𝐼(𝑡0)\u0014𝑎(𝑡0)\n𝑎(𝑡)\u00153\n𝑒−Γ𝐼𝑡, (2.4)\nwhere𝑎(𝑡)is the scale factor, and 𝑡0is the reference time.\nEq. (2.1) can be solved, such as\n𝑓(𝑝,𝑡)=𝑓h(𝑝,𝑡)≡𝑓h(𝑝0)\u0014𝑎0𝑝0\n𝑎(𝑡)𝑝\u00153−1/𝑛\n𝑒−Γ𝐼𝑡0h\n𝑎(𝑡)𝑝\n𝑎0𝑝0i1/𝑛\n𝜃(𝑝0−𝑝)𝜃\u0012\n𝑝−𝑎(𝑡inf)\n𝑎(𝑡)𝑝0\u0013\n, (2.5)\n𝑓h(𝑝0)≡6𝜋2𝜈−1\n𝑔\"\n2𝑝0𝑛𝐼(𝑡0)\n3𝐻2\n0𝑀2\nPl# \n𝐻0Γ𝐼𝑀2\nPl\n𝑝4\n0!\n, (2.6)\nwhere𝐻=𝑛/𝑡and𝑎(𝑡)/𝑎0=(𝑡/𝑡0)𝑛. In this paper, we focus on the thermalization in the matter-dominated\nepoch, in which 𝑛=2/3. Hereafter, we neglect the exponential factor in (2.5) because we are primarily\ninterested in the thermalization that occurs in the regime of Γ𝐼<𝐻(𝑡). The momentum 𝑎(𝑡inf)𝑝0/𝑎(𝑡)in\nthe second Heaviside theta function in (2.5) represents the redshifted momentum of gluons generated at\nthe end of inflation 𝑡inf. We take𝑎(𝑡inf)→0for simplicity throughout this paper because its precise value\ndoes not affect our result qualitatively.\n2.2 Qualitative discussion\nWe move on to the discussion on the thermalization of pure YM plasma produced by the decay of a heavy\nparticle, e.g.,inflaton. Before presenting the concrete kinetic equations, we briefly summarize the basic\nassumptions and qualitative discussion of reheating and thermalization. See also Ref. [35]. As emphasized\nin the introduction, we focus on the case in which the decay rate of inflaton, Γ𝐼, is sufficiently small such\nthat the reheating can be well approximated by the perturbative inflaton decay. It is convenient to use the\nfollowing combination for the decay rate Γ𝐼:\nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl\u0012\n=𝜈𝑔𝑝0𝑡0𝑓h(𝑝0)\n32𝜋2\u0013\n, (2.7)\nwhich is assumed to be smaller than unity. This is typically expected at least when the inflaton decay rate\nis smaller than that induced by the dimension-five Planck-suppressed operators, such as 𝜅5𝐼𝐺𝑎\n𝜇𝜈𝐺𝑎𝜇𝜈/𝑀Pl\nand𝜅5𝐼𝐺𝑎\n𝜇𝜈˜𝐺𝑎𝜇𝜈/𝑀Plwith𝜅5≪1, where𝐼represents the inflaton. In this case, the reheating temperature\nis significantly smaller than the inflaton mass. Hence, we generically expect that the temperature of the\n“dilute” plasma, which already exists before the completion of reheating, is smaller than the typical energy\nof gluons just after their production, at least in the later stage of pre-thermal phase.\n4The bottleneck process of thermalization in such cases is given by the splittings of high-energy gluons\ninto the background plasma. In general, a splitter of momentum 𝑝can emit a splittee at a scale of 𝑘 < 𝑝\nonly after𝑡⩾𝑘/𝑘2\n⊥with𝑘⊥being a momentum of 𝑘transverse to the direction of 𝑝because otherwise,\nquantum mechanical interference between the splitter and splittee prevents splittee formation. Suppose\nthere exist soft-thermalized populations of gluons with a temperature 𝑇s, whose condition is specified later.\nThe interactions with the soft thermal plasma induce random diffusions of the transverse momentum,\nwhich is estimated as 𝑘2\n⊥∼ˆ𝑞𝑡∼𝐷𝑠𝛼2𝑇3\ns𝑡.♮3Here, we have included the factor 𝐷sthat depends on the\ndegrees of freedom responsible for the transverse diffusion. One may estimate its dependence as\n𝐷s∼𝐶A∑︁\n𝑖𝜈𝑖\n𝑑𝑖𝑡𝑖, (2.8)\nwhere the summation is taken over species 𝑖contributing to the transverse diffusion, the degrees of freedom\nfor𝑖are denoted by 𝜈𝑖, the dimension and normalization of representation for 𝑖are𝑑𝑖and𝑡𝑖, respectively,\nand the quadratic Casimir for the adjoint representation is 𝐶A. For pure YM plasma, 𝐷s∼𝐶2\nA. Combining\nthese two estimations, we find the formation momentum below which the splittees can be emitted for a\ngiven𝑡\n𝑘 < 𝑘 form(𝑡)≡𝐷s𝛼2𝑇3\ns𝑡2. (2.9)\nEquivalently, the formation time before which the splittees of momentum 𝑘cannot be emitted is given as\nfollows:\n𝑡 >𝑡 form(𝑘)≡√︄\n𝑘\nˆ𝑞∼(𝛼𝑇s)−1√︂\n𝑘\n𝐷s𝑇s. (2.10)\nOnce this condition is met, the splittee of momentum 𝑘can be formed with a probability of 𝛼, which\nleads to the LPM-suppressed splitting rate, ΓLPM(𝑘)∼𝐶A𝛼/𝑡form(𝑘). On the other hand, for 𝑘 > 𝑘 form(𝑡),\nthe interactions with the medium are not sufficient to build up the transverse momentum, whereas a\nlarge-angle emission of 𝜃2\nvac⩾1/(𝑘𝑡)is always allowed quantum-mechanically with a probability of 𝛼(up\nto a logarithmic factor), that is, Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) vacuum shower.\nHence, the splitting rate, where a splitter of momentum 𝑝splits into daughters of 𝑘and(𝑝−𝑘), can be\nobtained as (see also Ref. [49])\nΓsplit(𝑘)∼ \nΓLPM(𝑘)∼𝐶A𝛼2𝑇s√︃\n𝐷s𝑇s\n𝑘for𝑇s< 𝑘 < 𝑘 form(𝑡),\nΓDGLAP(𝑘)∼𝐶A𝛼\n𝑡for𝑘form(𝑡)< 𝑘 <𝑚𝐼\n2,(2.11)\nwhere logarithmic factors are dropped for simplicity.\nThe emitted splittees of momentum 𝑘immediately cascades and participate in the soft thermal plasma\nwithin the Hubble time if ΓLPM(𝑘)>𝐻∼1/𝑡, which reads\n𝑘 < 𝑘 split(𝑡)≡𝐶2\nA𝛼2𝑘form(𝑡)=𝐶2\nA𝐷s𝛼4𝑇3\ns𝑡2. (2.12)\nThe energy conservation implies 𝜌s(𝑡)∼𝑘split(𝑡)Γ𝐼𝑡𝑛𝐼(𝑡)with𝜌sbeing the energy density of the soft\nsector. Assuming the thermalization of the soft sector, i.e.,𝜌s∼𝑔★𝑇4\nswith𝑔★being the effective relativistic\n♮3Here, we neglect the running of 𝛼for simplicity. See the later discussion on how the running modifies the estimation.\n5degrees of freedom in the soft thermal plasma, e.g.,𝑔★=𝜈𝑔=2𝑑Afor pure YM plasma, we obtain\n𝑇s(𝑡)\n𝑚𝐼∼𝑔−1\n★𝐶2\nA𝐷s𝛼4 \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!\n𝑚𝐼𝑡. (2.13)\nNow, the condition for the soft-sector thermalization can be derived. To maintain the thermal distribution,\nthe large-angle scatterings among the soft populations should be much faster than the cosmic expansion\n𝐷s𝛼2𝑇s(𝑡)>𝐻,♮4which leads to\n𝑡 >𝑡 soft≡𝑔1/2\n★𝐶−1\nA𝐷−1\ns𝛼−3 \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!−1/2\n𝑚−1\n𝐼. (2.14)\nThroughout this paper, we restrict ourselves to 𝑡≫𝑡soft.\nWith time, 𝑘splitgrows continuously, and eventually 𝑘split(𝑡)> 𝑚𝐼, or equivalently Γsplit(𝑚𝐼)> 𝐻,\nwhich occurs at\n𝑡 >𝑡 max≡𝑔3/5\n★𝐶−8/5\nA𝐷−4/5\ns𝛼−16/5 \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!−3/5\n𝑚−1\n𝐼. (2.15)\nSubsequently, the gluons generated by the decay of inflaton immediately break up into the soft thermal\nplasma. The temperature of the soft sector is then given by 𝜌s∼𝑚𝐼Γ𝐼𝑡𝑛𝐼, which yields\n𝑇s(𝑡)\n𝑚𝐼∼𝑔−1/4\n★ \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!1/4\n(𝑚𝐼𝑡)−1/4, (2.16)\nfor𝑡max< 𝑡 <Γ−1\n𝐼. The reheating is completed at 𝑡∼Γ−1\n𝐼, and the temperature at that time is often\nreferred to as the reheating temperature ,𝑇R∼𝑔−1/4\n★√𝑀PlΓ𝐼.\nTo sum up, in addition to the hard distribution given in Eq. (2.5), the splittings of high-energy gluons\nyield the cascades towards the soft thermal plasma. Its distribution can be estimated as\n𝜈𝑔𝑓s(𝑘)∼ \n𝜈𝑔𝑇s(𝑡)\n����···𝑘 <𝑇 s(𝑡)\n𝑔★𝐷1/2\ns\n𝐶A\u0010\n𝑡soft\n𝑡\u00112\u0010\n𝑇s(𝑡)\n𝑘\u00117/2\n···𝑇s(𝑡)< 𝑘 < 𝑘 form(𝑡)\n𝐶A𝛼\u0010\n1\n𝑚𝐼𝑡\u0011\u0010\nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl\u0011\u0010\n𝑚𝐼\n𝑘\u00113\n···𝑘form(𝑡)< 𝑘 <𝑚𝐼/2for𝑡soft<𝑡 <𝑡 max, (2.17)\n𝜈𝑔𝑓s(𝑘)∼ \n𝜈𝑔𝑇s(𝑡)\n𝑘···𝑘 <𝑇 s(𝑡)\nΓ𝐼\nΓLPM(𝑚𝐼)𝑛𝐼(𝑡)\n𝑚3\n𝐼\u0010\n𝑚𝐼\n𝑘\u00117/2\n···𝑇s(𝑡)< 𝑘 <𝑚𝐼/2for𝑡max<𝑡 <Γ−1\n𝐼. (2.18)\nNote that the hard population given in Eq. (2.5) vanishes for 𝑡 >𝑡 max. The temperature of the soft sector\nis given by\n𝑇s(𝑡)\n𝑚𝐼∼ \n𝑔−1\n★𝐶2\nA𝐷s𝛼4\u0010\nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl\u0011\n𝑚𝐼𝑡for𝑡soft<𝑡 <𝑡 max,\n𝑔−1/4\n★\u0010\nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl\u00111/4\n(𝑚𝐼𝑡)−1/4for𝑡max<𝑡 <Γ−1\n𝐼,(2.19)\n♮4Here we write the dependence of degrees of freedom for the large-angle scattering rate as 𝐷𝑠for notational simplicity.\nAlthough this is correct at least for pure YM plasma, it is different from 𝐷𝑠for a general case.\n6which is maximized at 𝑡∼𝑡maxas\n𝑇max∼𝑔−2/5\n★𝐶2/5\nA𝐷1/5\ns𝛼4/5 \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!2/5\n𝑚𝐼. (2.20)\nThe rest of this paper is devoted to refining this qualitative understanding through numerical simulations.\n2.3 In-medium splitting function\nThe kinetic equation that describes the reheating and thermalization via the production of high-energy\ngluons after inflation is summarized as follows [39,50]:\n\u0012𝜕\n𝜕𝑡−𝐻𝑝𝜕\n𝜕𝑝\u0013\n𝑓(𝑝,𝑡)=S+C1↔2[𝑓]+C2↔2[𝑓], (2.21)\nwhere the splittings of high-energy gluons are described by C1↔2, and the elastic scatterings are repre-\nsented by C2↔2, which lead to the thermalization of the soft sector. Throughout this paper, we focus on\nthe time scale much longer than the elastic scatterings, i.e.,𝑡≫𝑡soft. Hence, we simply assume that the\nsplittees of momentum 𝑘immediately get thermalized as long as 𝑘IR> 𝑘and impose the thermalization\nby hand without explicitly including C2↔2in the numerical simulations. The results of our numerical\nsimulations should be insensitive to the choice of 𝑘IRas long as𝑇s(𝑡)≲𝑘IR≪𝑘split(𝑡).\nThroughout this paper, we consider the case where the cascades of high-energy gluons are dominated\nby the splittings into low-energy gluons. This restriction is trivially fulfilled for weakly coupled pure YM\ntheories and holds for the high-energy gluons in the SM. The splitting of gluons is encoded in C1↔2as\nC1↔2[𝑓]=(2𝜋)3\n𝑝2𝜈𝑔\u0014\n−∫𝑝\n0d𝑘𝛾𝑔↔𝑔𝑔\u0000𝑝;𝑘,𝑝−𝑘\u0001𝑓(𝑝)+∫∞\n0d𝑘2𝛾𝑔↔𝑔𝑔\u0000𝑝+𝑘;𝑝,𝑘\u0001𝑓(𝑝+𝑘)\u0015\n.(2.22)\nThe splitting function of gluons in the presence of thermal plasma is\n𝛾𝑔↔𝑔𝑔(𝑃;𝑥𝑃,(1−𝑥)𝑃)=1\n2𝑑A𝐶A𝛼\n(2𝜋)4√\n2𝑃(vac)\n𝑔↔𝑔𝑔(𝑥)\n𝑥(1−𝑥)𝜇2\n⊥(𝑃; 1,𝑥,1−𝑥), 𝑃(vac)\n𝑔↔𝑔𝑔(𝑥)≡14+𝑥4+(1−𝑥)4\n𝑥(1−𝑥),\n(2.23)\nwhere the fine structure constant 𝛼is evaluated at a scale 𝑃, the well-known DGLAP splitting function\nfor gluons is denoted as 𝑃(vac)\n𝑔↔𝑔𝑔(𝑥), the degree of freedom is 𝜈𝑔=2𝑑A, the dimension of the adjoint\nrepresentation is 𝑑A, and its quadratic Casimir is 𝐶A; for SU(𝑁)andSO(𝑁)gauge theories,(𝑑A,𝐶A)=\n(𝑁2−1,𝑁)and(𝑁(𝑁−1)/2,2𝑁−4), respectively. The transverse momentum squared at the formation\ntime is denoted by 𝜇2\n⊥, which is developed by interactions with the soft thermal plasma of gluons. It can\nbe computed by solving a self-consistent equation, as done in Refs. [51], whose result at the leading log is\ngiven by\n𝜇4\n⊥(𝑃;𝑥1,𝑥2,𝑥3)=2\n𝜋𝑥1𝑥2𝑥3𝑃𝛼(𝑚D)−𝛼(𝑄⊥)\n−𝑏𝛼/\u000064𝜋3\u0001N𝐶𝐴\n2\u0010\n𝑥2\n1+𝑥2\n2+𝑥2\n3\u0011\n, (2.24)\n7with𝑏𝛼being the one-loop 𝛽-function coefficient, e.g.,𝑏𝛼=−11𝐶A/3in the pure YM theory. The factor\nNis proportional to the number density responsible for the transverse momentum diffusions and is given\nas follows:\nN≡∑︁\n𝑖𝜈𝑖\n𝑑𝑖𝑡𝑖∫d3ℓ\n(2𝜋)3𝑓s(ℓ) (2.25)\n=2𝐶A𝜁(3)\n𝜋2𝑇3\ns for pure YM , (2.26)\nwhere the summation is taken over species 𝑖contributing to the transverse diffusion, the distribution\nfunction of the soft population is denoted as 𝑓s, and the Riemann zeta function at 3is𝜁(3)≃1.20206 .\nNote that the transverse momentum diffusion is dominated by the soft thermal plasma for 𝑡≫𝑡soft. In\nthe second line, we used the fact that the normalization of the adjoint representation fulfills 𝑡A=𝐶A. We\nalso define the following quantities:\n\u0012𝑄⊥\n𝑚𝐷\u00132\n∼\u0012𝑃\n𝑇s\u00131/2\nln1/2\u0012𝑃\n𝑇s\u0013\n, (2.27)\n𝑚2\nD=8𝜋����∑︁\n𝑖𝜈𝑖\n𝑑𝑖𝑡𝑖∫d3ℓ\n(2𝜋)3𝑓s(ℓ)\nℓ(2.28)\n=4𝜋𝛼(𝑇s)𝐶A\n3𝑇2\ns for pure YM . (2.29)\nHere again, the summation is taken over species 𝑖contributing to the transverse diffusion, and we use\nthe fact that the number density is dominated by the soft population 𝑓s(ℓ)for𝑡≫𝑡softand𝑡A=𝐶A. For\npure SU(𝑁)YM theory, 𝑏𝛼=−11𝑁/3,N=2𝑁𝜁(3)𝑇3\ns/𝜋2, and𝑚2\nD=4𝜋𝛼(𝑇s)𝑇2\ns𝑁/3. For pure SO(𝑁)YM\ntheory,𝑏𝛼=−22(𝑁−2)/3,N=4(𝑁−2)𝜁(3)𝑇3\ns/𝜋2, and𝑚2\nD=8𝜋𝛼(𝑇s)𝑇2\ns(𝑁−2)/3. In the case of the\nSM plasma, all the SM particle contributions can be included in the soft sector to the high-energy gluon\ncascades such that 𝑏𝛼=−7,N=15𝜁(3)𝑇3/𝜋2, and𝑚2\n𝐷=8𝜋𝛼𝑇2.\nFinally, we briefly discuss how the equation given in this section is related to the qualitative discussion\nin the previous section. For 𝑘≪𝑝, Eq. (2.24) can be approximated with 𝜇2\n⊥(𝑝;𝑘,𝑝−𝑘)∼√︁\n𝑘ˆ𝑞. It is\nconsistent with the diffused transverse momentum at the formation time given in Eq. (2.10), i.e.,𝑘2\n⊥|𝑡form∼\nˆ𝑞𝑡form∼√︁\n𝑘ˆ𝑞. The splitting function is expressed as 𝛾𝑔↔𝑔𝑔(𝑝;𝑘,𝑝−𝑘)/𝜈𝑔∼𝐶A𝛼√︁\n𝑘ˆ𝑞for𝑘≪𝑝. The\ncorresponding splitting rate is then given as ΓLPM(𝑘)∼𝛾𝑔↔𝑔𝑔(𝑝;𝑘,𝑝−𝑘)/(𝜈𝑔𝑘)∼𝐶A𝛼√︁\nˆ𝑞/𝑘, which is\nconsistent with Eq. (2.11).\n3 Numerical simulations\n3.1 Numerical method\nWe numerically solve Eq. (2.21) without the C2↔2term, under the assumption that the splittees of mo-\nmentum𝑘with𝑘 < 𝑘 IRare thermalized immediately via 2→2elastic scattering process. The comoving\ntemperature of the plasma increases as the energy is injected into the modes 𝑘 < 𝑘 IR. The physical\n8temperature is calculated as follows:\n𝑇s(𝑡)=\u001430\n𝑔★𝜋2\u0000𝜌r, tot(𝑡)−𝜌(𝑡)\u0001\u00151/4\n, (3.1)\nwhere𝑔★is the relativistic degrees of freedom in the thermal plasma. We use 𝑔★=106.75for the SM\nsector and𝑔★=𝜈𝑔for the pure YM theory. The total energy injected into the radiation, 𝜌r, tot(𝑡), is given\nby\n𝜌r, tot(𝑡)=∫𝑡\nd𝑡′2𝑝0Γ𝐼𝑛𝐼(𝑡′)\u0014𝑎(𝑡′)\n𝑎(𝑡)\u00154\n(3.2)\n=6\n5𝑝0Γ𝐼𝑛𝐼(𝑡0)𝑡\u0014𝑎(𝑡0)\n𝑎(𝑡)\u00153\n. (3.3)\nWe define the energy density for non-thermal contributions as\n𝜌(𝑡)=𝜈𝑔∫\n𝑘IRd3𝑝\n(2𝜋)3𝑝𝑓(𝑝,𝑡). (3.4)\nWe take𝑘IR=3𝑇s(𝑡)in our numerical simulations. The results do not change qualitatively when 𝑘IRis\nvaried by a factor of O(1). We rescale the dimensionful parameters and use 𝑇s(𝑡0)≡𝑇0=1in numerical\nsimulations without any loss of generality.\nThe source term Sin Eq. (2.21) can be represented by the initial distribution (2.5). Because the\nnumerical time is limited, our numerical calculation starts from a finite time scale 𝑡0. This implies that\nthe distribution at a small momentum scale is deformed as the second line of Eq. (2.17) ( ∝𝑇−7/2),♮5\nwhereas the hard mode at a high momentum remains in the form of Eq. (2.5) ( ∝𝑇−3/2). To include this\nfact, we adopt the following algorithm in our numerical simulations. We first take an ansatz 𝑓h(𝑝,𝑡0)=\n𝑓h(𝑝0)(𝑝/𝑝0)−3/2for the initial distribution and evolve it by a first time step, 𝑡=𝑡0+𝛿𝑡. Then we determine\n𝑝th,0as a minimal value of momentum satisfying ln[𝑓h(𝑝,𝑡0+𝛿𝑡)/𝑓h(𝑝,𝑡0)]<0.1. Then we perform the\nmain numerical simulations by replacing the initial distribution 𝑓h(𝑝)such that\n𝑓h(𝑝,𝑡0)= \n𝑓h(𝑝th,0)\u0010\n𝑝\n𝑝th,0\u0011−7/2\nfor𝑝 < 𝑝 th,0,\n𝑓h(𝑝0)\u0010\n𝑝\n𝑝0\u0011−3/2\nfor𝑡≥𝑝th,0.(3.5)\nThis initial condition allows the distribution function not to change significantly for the first time step,\nwhich is important to ensure the stability of numerical simulations.\nThe splitting term C1↔2is given by (2.22). We take an IR cutoff for the splitting term as the initial\ntemperature of the system 𝑇0(=1), assuming that temperature does not change by many orders of mag-\nnitude during the simulation time scale. It is convenient to use a comoving momentum ˜𝑝≡𝑎(𝑡)𝑝/𝑎0and\ntemperature ˜𝑇s(𝑡)≡𝑎(𝑡)𝑇s(𝑡)/𝑎0in numerical simulations. We denote\n˜𝑝max(𝑡)≡𝑎(𝑡)\n𝑎(𝑡0)𝑝0, (3.6)\n♮5We neglect the deformation of the distribution of the soft particles for 𝑘 > 𝑘 form(∝𝑘−3) for simplicity.\n9as the comoving momentum injected from the heavy particle decay. To represent the IR cutoff, we dis-\ncretize the comoving momentum such that Δ˜𝑝=𝑇0(=1). We denote 𝑖=(1,2,...,𝑁 grid)and ˜𝑝𝑖=𝑖as\nthe label for the momentum grid, with 𝑁gridbeing its total number. We define 𝑁0such that𝑝0=𝑁0.\nThe time step 𝛿𝑡is taken such that ˜𝑝max(��+𝛿𝑡)−˜𝑝max(𝑡)=1. Namely, a single grid for comoving\nmomentum is added at ˜𝑝max(𝑡+𝛿𝑡)by a single time step. Additionally, we denote 𝑛=(0,1,2,...,𝑁 grid−𝑁0)\nand\n𝑡𝑛=𝑡0\u0012𝑁0+𝑛\n𝑁0\u00133/2\n, (3.7)\nfor the time steps, where we use ˜𝑝max(𝑡𝑛)=(𝑁0+𝑛). For every time step, the splitting function is calculated\nand the distribution function is evolved as\nln𝑓𝑛+1(˜𝑝𝑖)−ln𝑓𝑛(˜𝑝𝑖)=(𝑡𝑛+1−𝑡𝑛)\u0012𝑡0\n𝑡𝑛\u00132/3\u0012˜𝑇𝑠(𝑡𝑛)\n𝑇0\u00133/2(2𝜋)3\n𝜈𝑔˜𝑝2\n𝑖Δ˜𝑝\n×\"\n−𝑖−1∑︁\n𝑘=1𝛾𝑔↔𝑔𝑔\u0000˜𝑝𝑖; ˜𝑝𝑘,˜𝑝𝑖−𝑘\u0001+𝑁0+𝑛−𝑖∑︁\n𝑘=12𝛾𝑔↔𝑔𝑔\u0000˜𝑝𝑖+𝑘; ˜𝑝𝑖,˜𝑝𝑘\u0001𝑓𝑛(˜𝑝𝑖+𝑘)\n𝑓𝑛(˜𝑝𝑖)#\n𝑇→𝑇0,\n(3.8)\nwhere(𝑡0/𝑡𝑛)2/3originates from the redshift. Here, we factorize the temperature dependence for 𝛾𝑔↔𝑔𝑔\nas(𝑇𝑠(𝑡)/𝑇0)3/2×(𝛾𝑔↔𝑔𝑔)𝑇→𝑇0by neglecting its logarithmic dependence.\n3.2 Results of the SM\nTo describe the thermalization in the SM sector, we consider the thermalization of gluon ( i.e.,SU(3)gauge\nfield) into the SM thermal plasma. Namely, we take 𝑔★=106.75,G=SU(3),𝜈𝑔=16,𝑑𝐴=8,𝐶𝐴=3,\n𝑏𝛼=−7,N=15𝜁(3)𝑇3/𝜋2,𝑚2\n𝐷=8𝜋𝛼𝑇2, and𝛼(𝑚𝑍)=0.118with𝑚𝑍being the𝑍-boson mass. We take\n𝑇0=103GeV=1in the numerical simulations, where the dependence on this choice is only because of\nthe renormalization group (RG) running of the gauge coupling constant.\nIn this case, there remains three parameters that should be specified to perform the numerical simu-\nlations:\n𝑓h(𝑝0)(orΓ𝐼), 𝑡 0, 𝑝 0. (3.9)\nHere,𝑓h(𝑝0)should be consistently determined by the first line of (2.19) with 𝑡=𝑡0. This implies that\n𝑓h(𝑝0)∝𝜈−1\n𝑔𝑔★𝛼−4𝑡−2\n0𝑝−3\n0. We are interested in the regime around 𝑡∼𝑡max; thus, we take 𝑡0to be of order\nbut smaller than 𝑡max. Moreover, we need to ensure 𝑡soft≪𝑡0. Specifically, we take\n𝑡0∼𝜖𝑡𝑝1/2\n0\n𝛼2𝑇3/2\n0, (3.10)\nwith √︄\n𝑇0\n𝑝0≪𝜖𝑡≲1, (3.11)\n10��� ��� ��� ���� �������������������Figure 1: Physical temperature as a function of physical time. All curves are rescaled according to (3.15) and (3.16). The\nred and green dotted lines represent the analytic dependence for 𝑇s(𝑡)in the regimes of 𝑡≪𝑡max(𝑇s(𝑡)∝𝑡) and𝑡≫𝑡max\n(𝑇s(𝑡) ∝𝑡−1/4), respectively. The blue solid/dashed curves represent the numerical results for (𝑝0,𝑡0𝑝−1/2\n0, 𝑓h(𝑝0)𝑡2\n0𝑝3\n0)=\n(5×103,2,105),(104,1,105)respectively. The brown solid curve corresponds to (5×103,5,105).\nfrom (2.12), where we use (2.19) to eliminate ˜Γor𝑓h(𝑝0). The lower bound on 𝜖𝑡originates from the\ncondition of 𝑡soft≪𝑡0. In the numerical simulations, we primarily take\n𝑝0=5×103, (3.12)\n𝑡0=2×𝑝1/2\n0, (3.13)\n𝑓h(𝑝0)=105×𝑡−2\n0𝑝−3\n0, (3.14)\nand𝑁grid=2×104. We also perform numerical calculations by changing 𝑡0and𝑝0by a factor of a few to\ndetermine whether the result depends only on physical parameters.\nFigure 1 shows the temperature as a function of time, where the variables are rescaled by\n𝑇𝑠(𝑡)=𝑇res(𝑡) \n𝑓h(𝑝0)\n105𝑡−2\n0𝑝−3\n0!2/5 \n𝑡0\n2𝑝1/2\n0!−2/5\n, (3.15)\n𝑡𝑛=𝑡res \n𝑓h(𝑝0)\n105𝑡−2\n0𝑝−3\n0!−3/5 \n𝑡0\n2𝑝1/2\n0!3/5\u0010𝑝0\n5×103\u00111/2\n. (3.16)\nThese dependencies on the initial parameters are expected from the analytic estimations of Eqs. (2.20),\n(2.15), and Eq. (2.7). The blue solid curve represents the case with the initial condition mentioned above.\nThe brown solid curve represents the case with 𝑝0=5×103,𝑡0=5𝑝1/2\n0,𝑓h(𝑝0)=105𝑡−2\n0𝑝−3\n0, and\n𝑁grid=2×104. The dashed curve represents the case with 𝑝0=104,𝑡0=𝑝1/2\n0,𝑓h(𝑝0)=105𝑡−2\n0𝑝−3\n0, and\n𝑁grid=4×104. All results are in good agreement with each other, except for the regime around 𝑡=𝑡0.\nThe discrepancy results from the fact that the initial distribution (and 𝑓h(𝑝0)) is chosen by hand for soft\nmodes. However, because the energy injection into thermal plasma is dominated by the contribution from\nhard modes, the late-time distribution is not affected, i.e.,implying the basin of attraction. The results for\nother initial conditions are provided in the Appendix.\n11��������������-����-���-���-�������\n��-���-���-�������������Figure 2: Spectra at the end of numerical simulations. (left) The blue/brown solid curves represent the hard/thermal spectra\nrespectively. We also show the initial spectra with the dashed lines in the same color. (right) The red dotted curve represents the\nstationary solution corresponding to Γsplit/𝐻→0given in Eq. (3.21). The color codings for the blue solid/dashed and brown\ncurves are the same as Fig. 1.\nThe red and green dotted lines represent the fitting functions for 𝑇(𝑡):\n𝑇res(𝑡)≃ \n0.006×𝑡resfor𝑡≪𝑡max,\n13×𝑡−1/4\nres for𝑡≫𝑡max.(3.17)\nAll results are in good agreement with the analytic estimations. The asymptotic behavior for 𝑡≫𝑡maxis\nfurther confirmed by simulations with different initial conditions as we show in Fig. 4 in Appendix. From\nFig. 1, the numerical coefficients of the maximal temperature and corresponding time can be determined\nas follows:\n𝑇max≃2.3× \n𝑓h(𝑝0)\n105𝑡−2\n0𝑝−3\n0!2/5 \n𝑡0\n2𝑝1/2\n0!−2/5\n,\n≃0.050× \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!2/5\n𝑚𝐼, (3.18)\n𝑡max≃5.0×102 \n𝑓h(𝑝0)\n105𝑡−2\n0𝑝−3\n0!−3/5 \n𝑡0\n2𝑝1/2\n0!3/5\u0010𝑝0\n5×103\u00111/2\n,\n≃1.6×103 \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!−3/5\n𝑚−1\n𝐼. (3.19)\nThese results confirm the analytic estimations (2.20) and (2.15), and the numerical prefactors are de-\ntermined from our numerical simulations. Note that 𝛼and𝑔★are absorbed into the numerical factors\nbecause we substitute the SM values for them.\nThe blue curves in Fig. 2 show the spectra at the end of numerical simulations. In the left panel, the\ninitial spectrum (3.5) is represented by the blue dashed curve. The solid (dashed) brown curve represents\nthe thermal spectrum at the end (beginning) of numerical simulations. See Eqs. (2.17) and (2.18) for\n12analytic estimations. In the right panel, the distributions are rescaled by ˜Γ(𝑝/𝑝0)−7/2, where\n˜Γ≡4𝜋2Γ𝐼𝑛𝐼(𝑡)\n𝑝5/2\n0𝑇3/2\ns(𝑡)=𝑓h(𝑝0)𝜈𝑔𝐻0𝑝1/2\n0\n˜𝑇3/2\ns(𝑡)𝑡0\n𝑡, (3.20)\nis defined in our previous paper [39]. The three curves correspond to the ones used in Fig. 1 and almost\noverlap with each other. At a late time, Γsplit≫𝐻, in which case the spectrum can be represented by a\nstationary solution of\n𝜈𝑔𝑓(𝑝,𝑡)≃2.6טΓ\u0012𝑝\n𝑝0\u0013−7/2\n, (3.21)\nfor𝑝≪𝑝0. The red dotted curve, which is also overlapped with our numerical results, is the result for\nthe stationary solution of the Boltzmann equation in the limit of Γsplit≫𝐻. All results are consistent with\nthe stationary solution, which justifies 𝑡≫𝑡maxat the end of numerical simulations. The deviation from\nthe dependence of ∝𝑝−7/2at a small𝑝/𝑝0may be due to the IR cutoff used in our numerical simulations.\nFor the stationary solution, the IR cutoff need not to be introduced, as explained in Ref [39,40]; therefore,\nthe stationary solution is almost exactly ∝𝑝−7/2for a small𝑝/𝑝0.\n3.3 Results of pure YM theory\nNow we consider a pure YM theory for G=SU(𝑁)andSO(𝑁)with(𝑁,𝛼)=(3,10−1),(5,10−2),(5,10−3),\n(5,10−4), and(10,10−3)in a dark sector, where the gauge coupling constant is defined at the energy\nscale of𝑚𝑍. We take𝑇0=103GeV (≡1) as an example, though our results depends on its value only\nlogarithmically through the RG running.\nIn the numerical simulations, we take\n𝑝0=𝑝(bm)\n0≡5×103, (3.22)\n𝑡0=𝑡(bm)\n0≡2×𝑝1/2\n0\u0012Γsplit(𝑝0)\nΓsplit,SM(𝑝0)\u0013−1\n, (3.23)\n𝑓h(𝑝0)=𝑓(bm)\nh≡105×𝑡−2\n0𝑝−3\n0\u0012𝜈𝑔\n𝜈SM\u0013−1\u0012𝑔∗\n𝑔∗,SM\u0013 \u0012Γsplit(𝑝0)\nΓsplit,SM(𝑝0)\u0013−2\n, (3.24)\nand𝑁grid=2×104, for(𝑁,𝛼)=(3,0.1), where𝜈SM=16and𝑔∗,SM=106.75. Here, we define\nΓsplit(𝑝0)≡1\n𝜈𝑔𝑝0𝛾𝑔↔𝑔𝑔(𝑝0;𝑝0/2,𝑝0/2). (3.25)\nFor the case with 𝛼∼0.1, including the SM QCD, the difference of the gauge coupling constants at\n𝑝0and at𝑇0due to the RG running is not negligible. For example,\n𝛼(𝑝0)\n𝛼(𝑇0)≃0.5, (3.26)\nfor(𝑁,𝛼)=(3,10−1)with G=SU(𝑁). In this case, 𝛼in𝑡softis relatively larger than that used in 𝑡max.\nThis results in a relatively weaker condition on the initial time, (3.11), by a factor of a few. In contrast,\n13��� ��� ��� ���� �������������������\n��� ��� ��� ���� �������������������Figure 3: Same as Fig. 1 but for the pure SU𝑁(left panel) and SO(𝑁)(right panel) YM theories. The blue curve represents the\nresult of(𝑁,𝛼)=(3,10−1), whereas the brown curves represent the results of (5,10−2),(5,10−3),(5,10−4), and(10,10−3).\nfor a significantly smaller gauge coupling constants, the RG running is negligible. In this case, the lower\nbound on the initial time, given by the first inequality of (3.11), is more severe than the case of the SM.\nTherefore, we instead take 𝑡0=5×𝑡(bm)\n0for(𝑁,𝛼)=(5,10−2),(5,10−3),(5,10−4), and(10,10−3).\nFigure 3 shows the temperature as a function of time in pure YM theories. To show our results, we\nrescale parameters such as\n𝑇(𝑡)=𝑇res(𝑡) \n𝑓h(𝑝0)\n𝑓(bm)\nh!2/5 \n𝑡0\n𝑡(bm)\n0!−2/5\n, (3.27)\n𝑡𝑛=𝑡res \n𝑓h(𝑝0)\n𝑓(bm)\nh!−3/5 \n𝑡0\n𝑡(bm)\n0!3/5 \n𝑝0\n𝑝(bm)\n0!1/2\u0012Γsplit(𝑝0)\nΓsplit,SM(𝑝0)\u0013−1\n. (3.28)\nWe plot the results of SU(𝑁)(left panel) and SO(𝑁)(right panel) gauge theory with (𝑁,𝛼)=(3,10−1)\nas blue solid curve and those with (5,10−2),(5,10−3),(5,10−4), and(10,10−3)as brown solid curves.\nThe brown solid curves overlap and cannot be distinguished from each other. The dotted lines are the\nfitting functions of (3.17). The figure shows that all results are consistent with (3.17) after the rescalings.\nMoreover, the results of (3.18) and (3.19) can be applied to pure YM theories after the correction from\nthe difference of the splitting rate, Γsplit(𝑝0), is included:\n𝑇max≃2.3× \n𝑓h(𝑝0)\n𝑓(bm)\nh!2/5 \n𝑡0\n𝑡(bm)\n0!−2/5\n≃2.2×𝑔−2/5\n∗ \nΓsplit(𝑝0)\n𝑇3/2\ns𝑝−1/2\n0!2/5 \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!2/5\n𝑚𝐼, (3.29)\n𝑡max≃5.0×102× \n𝑓h(𝑝0)\n𝑓(bm)\nh!−3/5 \n𝑡0\n𝑡(bm)\n0!3/5 \n𝑝0\n𝑝(bm)\n0!1/2\u0012Γsplit(𝑝0)\nΓsplit,SM(𝑝0)\u0013−1\n≃3.0×10−3𝑔3/5\n∗ \nΓsplit(𝑝0)\n𝑇3/2\ns𝑝−1/2\n0!−8/5 \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!−3/5\n𝑚−1\n𝐼, (3.30)\n14where we substitute the SM parameters, such as Γsplit(𝑝0)≃5.5×10−4×𝑇3/2\ns𝑝−1/2\n0, and𝑝0=𝑚𝐼/2. This\nis a general result that should be applicable to any pure YM(-like) theories.\nWe also provide useful formulas for 𝑇maxand𝑡maxin large𝑁and small𝛼limits. If𝑁≫1and𝛼≪0.1,\nwe can approximate\nΓsplit(𝑝0)≈(\n(0.01-0.02)×𝑁2𝛼2𝑇3/2𝑝−1/2\n0for G=SU(𝑁),\n(0.03-0.07)×𝑁2𝛼2𝑇3/2𝑝−1/2\n0for G=SO(𝑁),(3.31)\nwhere the prefactors depend on 𝑝0logarithmically. Substituting these into (3.29) and (3.30), we obtain\n𝑇max≈(0.3-0.4)×𝛼4/5 \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!2/5\n𝑚𝐼, (3.32)\n𝑡max≈(2-5)×𝑁−2𝛼−16/5 \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!−3/5\n𝑚−1\n𝐼, (3.33)\nforG=SU(𝑁)and\n𝑇max≈(0.5-0.7)×𝛼4/5 \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!2/5\n𝑚𝐼, (3.34)\n𝑡max≈(0.2-0.8)×𝑁−2𝛼−16/5 \nΓ𝐼\n𝑚3\n𝐼/𝑀2\nPl!−3/5\n𝑚−1\n𝐼, (3.35)\nforG=SO(𝑁), where we use 𝑔∗=𝜈𝑔. These results confirm the analytic estimations of Eqs. (2.15) and\n(2.20), where 𝐷𝑠∼𝐶2\n𝐴∼𝑁2and𝑔★=𝜈𝑔∼𝑁2.\n4 Discussion and conclusions\nWe have obtained a clear dynamical picture of the thermalization of pure YM plasma during perturbative\nreheating after inflation by numerically solving the Boltzmann kinetic equation by appropriately consider-\ning the LPM effect. Our results confirm the results of previous analytic studies based on quasi-equilibrium\nansatz applicable to two regimes of 𝑡soft≪𝑡≪𝑡maxor𝑡max≪𝑡, but also describe the transient epoch\nbetween these regimes, providing a better estimation for the maximal temperature of our Universe. The\nmaximal temperature obtained can be significantly smaller than the estimation based on the instantaneous\nthermalization approximation, highlighting the importance of understanding thermalization. Moreover,\nour results demonstrate the robustness of the quasi-equilibrium solution under the consistent change of\nthe initial conditions. Furthermore, the late-time behavior is stable even when the initial conditions are\napplied far away from the consistent initial conditions.\nThe implications of our results for particle cosmology are broad. In particular, the maximal tem-\nperature of the Universe is the key ingredient for understanding possible cosmological phase transitions.\nRecently, hidden pure YM theories have been gaining attention because they involve the glueballs as a\ncandidate of DM [52–65] and would lead to the first-order phase transitions [66–69], possibly accompa-\nnied by cosmic strings [70–72]. Our results are essential to understanding their implications, such as the\n15prediction of the GW spectrum as well as the condition to obtain the confinement phase transition after\ninflation.\nAs discussed in the main text, we believe our results can be applicable to the SM plasma when the\ninflaton dominantly decays into the SM gluons of SU(3). This expectation is based on our previous\nstudy [39], which showed that the thermalization is dominated by the gluons, although the analysis is\nrestricted to the quasi-equilibrium regime. The complete dynamical analysis of the SM plasma, including\nthe cases in which the inflaton decays into other SM species, is worthwhile to investigate in future studies.\nAcknowledgments\nK. M. was supported by MEXT Leading Initiative for Excellent Young Researchers Grant No. JPMXS0320200430,\nand JSPS KAKENHI Grant No. JP22K14044. M. Y. was supported by MEXT Leading Initiative for Excellent Young\nResearchers, and by JSPS KAKENHI Grant No. JP20H05851 and JP23K13092.\nA Case with other initial conditions\nIn the main part of this paper, we primarily consider the case of thermal plasma generated through the\nthermalization of injected high-energy particles. This requires a consistent initial condition such as (3.14).\nIn this Appendix, we provide results with other initial conditions, particularly for the case with an SM-like\nsystem.\nFigure 4 shows the results for the cases with\n(𝑡0/𝑝1/2\n0, 𝑓h(𝑝0)/(105×𝑡−2\n0𝑝−3\n0))= \n(5,1/2),(10,1/2) ··· (magenta dotted, dashed)\n(2,1),(5,1),(10,1) ··· (blue solid, dotted, dashed)\n(2,2),(5,2),(10,2) ··· (brown solid, dotted, dashed)\n(5,4),(10,4),(20,4) ··· (cyan dotted, dashed, dot-dashed)(A.1)\nwith𝑝0=5×103and𝑁grid=1.5×104. The variables are rescaled according to (3.16).\nNote that the blue dotted curve corresponds to the result of (3.14). The numerical simulations for a\nsmaller𝑓h(𝑝0)/(105×𝑡−2\n0𝑝−3\n0)correspond to the case with a relatively higher initial temperature for the\nambient plasma. This is the case in which ambient plasma is already generated by other mechanisms\nor sources. Once the energy injection from the high-energy particle is sufficiently high, the temperature\nstarts to increase and reaches its maximal value within the time scale of O(𝑡max).\nOn the contrary, the numerical simulations for a larger 𝑓h(𝑝0)/(105×𝑡−2\n0𝑝−3\n0)imply a relatively lower\ninitial temperature for the ambient plasma. This corresponds to cases in which the thermalization via\nthe elastic scatterings in the soft sector or the energy injection into the soft sector are delayed by some\nmechanisms until 𝑡=𝑡0. In these cases, the temperature increases faster to reach the expected behavior\nof (2.19). However, because the simulation time is limited such that 𝑡0and𝑡maxare of the same order\nof magnitude, we cannot confirm that the temperature reaches the attractor of (2.19) before it reaches\n16��� ��� ��� ���� ���� �������������������Figure 4: Same as Fig. 1 but with different initial conditions. See Eq. (A.1) for their color codings.\nthe maximal temperature. Still, all results reach the maximal temperature within the time scale of order\n𝑡max, even if the simulation starts at a later time. All results agree at a later time ( 𝑡≫𝑡max), at which\nhigh-energy particles thermalize within the Hubble-time scale.\nWe expect that the cases with 𝑓h(𝑝0)/(105×𝑡−2\n0𝑝−3\n0)=1(blue curves) are consistent initial conditions\nin which the ambient plasma is generated by the energy injection via the thermalization of high-energy\nparticles. These cases agree with each other except for 𝑡∼𝑡0, even if we change the value of 𝑡0/𝑝1/2\n0. This\nsupports the fact that it consistently starts within the attractor regime of (2.19). This is not the case for\ndifferent values of 𝑓h(𝑝0)/(105×𝑡−2\n0𝑝−3\n0), as shown in the figure. However, we note that the other initial\nconditions may also be interesting in some of the cases mentioned above. Our numerical simulations can\nalso be used to analyze such cases.\nReferences\n[1] D. N. Schramm and M. S. 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B 838\n(2023) 137724, arXiv:2204.13125 [hep-th] .\n21"    },    {        "title": "2402.14082v2.Landau_damping__collisionless_limit__and_stability_threshold_for_the_Vlasov_Poisson_equation_with_nonlinear_Fokker_Planck_collisions.pdf",        "content": "arXiv:2402.14082v2  [math.AP]  26 Feb 2024LANDAU DAMPING, COLLISIONLESS LIMIT, AND STABILITY\nTHRESHOLD FOR THE VLASOV-POISSON EQUATION WITH\nNONLINEAR FOKKER-PLANCK COLLISIONS\nJACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nAbstract. In this paper, we study the Vlasov-Poisson-Fokker-Planck ( VPFP) equation\nwith a small collision frequency 0 < ν≪1, exploring the interplay between the regularity\nand size of perturbations in the context of the asymptotic st ability of the global Maxwellian.\nOurmain result establishes theLandau dampingandenhanced dissipation phenomenaunder\nthe condition that the perturbation of the global Maxwellia n falls within the Gevrey-1\nsclass\nand obtain that the stability threshold for the Gevrey-1\nsclass with s > skcan not be larger\nthanγ=1−3sk\n3−3skforsk∈[0,1\n3]. Moreover, we show that for Gevrey-1\nswiths >3, and for\nt≪ν1\n3, the solution to VPFP converges to the solution to Vlasov-Po isson equation without\ncollision.\nContents\n1. Introduction 1\n2. The working system 11\n3. Proof of the main theorems 15\n4. Density estimates 20\n5. Higher order moment estimates 35\n6. Estimates on the distribution function 40\n7. Zero mode estimates 63\nAppendix A. Elementary inequalities 68\nAppendix B. Properties of S 69\nAppendix C. Properties of the multiplier m(t,∇) 70\nAppendix D. Littlewood-Paley decomposition and paraproducts 73\nReferences 73\n1.Introduction\nIn this paper, we consider the following single-species Vla sov-Poisson-(nonlinear) Fokker-\nPlanck (VPFP) equations with a neutralizing background on Tn\nx×Rn\nv(withTxnormalized\nto length 2π), which models the evolution of the distribution function o f electrons which are\nsubject to the electrostatic force coming from their Coulom b interaction and to a Brownian\nforce which models their collisions:\n(1.1)\n\n∂tF+v·∇xF+E(t,x)·∇vF=ν̺/parenleftbig\nϑ∆vF+∇v·((v−u)F)/parenrightbig\n,\nE(t,x) =−∇x(−∆x)−1(̺−1),\n̺(t,x) =/integraldisplay\nRnF(t,x,v)dv,\n(̺u)(t,x) =/integraldisplay\nRnF(t,x,v)vdv,\n1\n2(̺|u|2)(t,x)+1\n2n(̺ϑ)(t,x) =1\n2/integraldisplay\nRnF(t,x,v)|v|2dv,\nF(t= 0,x,v) =Fin(x,v).\n12 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nThe unknown is the distribution function F(t,x,v), which gives the number density of (say)\nelectrons at location xmoving with velocity v. The electrostatic force is responsible for the\nself-consistent force term E(t,x)· ∇vF. The non-local interaction between E(t,x) and the\ndensity function ̺(t,x) of electrons wouldbethroughCoulomb electrostatic inter actions. The\nBrownian force is modeled by the Fokker-Planck term ν̺/parenleftbig\nϑ∆vF+∇v·((v−u)F)/parenrightbig\nwith the\nsmall coefficient 0 ≤ν≪1 which represents the collision frequency. Here we use the s ame\nnonlinear Fokker-Planck collision operator as that studie d in [1], as opposed to the linear one,\nnamely̺=ϑ= 1, which is also commonly used in studies on weak collisions [28, 30, 48].\nThe nonlinear Fokker-Planck collision operator reflects th at the probability of collisions is\npositively correlated with both the density and temperatur e, which is natural [10, 18, 47].\nTo understand more precisely how the energy in the electric fi eld is converted into heat and\nhow the plasma returns to global thermodynamic equilibrium also motivates us to study the\nnonlinear collision operator. Moreover, mathematically, we introduce new ideas to deal with\nthe nonlinear effect of the collision operator in weak collisi on region, which might be useful\nin the study of other types of collision operators. A direct c alculation gives the following\nconservation laws:\n•Conservation of mass:d\ndt/integraldisplay\nTn̺(t,x)dx= 0.\n•Conservation of momentum:d\ndt/integraldisplay\nTn(̺u)(t,x)dx= 0.\n•Conservation of energy:d\ndtE(t) = 0, where the total energy is defined by\n(1.2) E(t) =1\n2/integraldisplay\nTn/integraldisplay\nRnF(t,x,v)|v|2dvdx+1\n2/ba∇dblE(t)/ba∇dbl2\nL2.\nAssume that the initial condition Finsatisfies\n(1.3)/integraldisplay\nTn×RnFin(x,v)dxdv= (2π)n,/integraldisplay\nTn×RnvFin(x,v)dxdv= 0,E(0) =n\n2(2π)n.\nThe VPFP equation (1.1) admits a steady solution, namely the global Maxwellian,\n(1.4) µ(v) =1\n(2πθ∞)n\n2e−|v|2\n2θ∞,\nwhere the final temperature θ∞is determined by the conservation laws via the energy per\nunit volume:\n(1.5) nθ∞=2\n(2π)nE.\nWe assume θ∞= 1 for the sake of simplicity.\nInthis paper, we study theasymptotic stability of thegloba l Maxwellian µ(v). It is natural\nto introduce the perturbations\nF(t,x,v) =µ(v)+g(t,x,v), ̺(t,x) = 1+ρ(t,x), ϑ(t,x) = 1+θ(t,x), (1.6)\nM1(t,x) =/integraldisplay\nRnvg(t,x,v)dv, M 2(t,x) =/integraldisplay\nRn|v|2g(t,x,v)dv. (1.7)\nThen by using the conservation laws and the initial assumpti ons (1.3), one deduces that\n/integraldisplay\nTnρ(t,x)dx= 0,/integraldisplay\nTnM1(t,x)dx= 0,/integraldisplay\nTnM2(t,x)dx=−/ba∇dblE(t)/ba∇dbl2\nL2. (1.8)\nIn addition, the following relations can bederived from the fourthand fifth equations of (1.1).\n(1+ρ)u=M1, (1.9)\nn(1+ρ)θ=M2−(1+ρ)|u|2−nρ. (1.10)LANDAU DAMPING FOR VPFP 3\nFor notational convenience, define\n(1.11) Mθ= (1+ρ)θ.\nNow we rewrite the system (1.1) in terms of g,ρ,M 1andMθ:\n(1.12a) ∂tg+v·∇xg−νL[g]+E·∇vµ=−E·∇vg+νCµ+νC[g],\nwith initial perturbation g(t= 0,x,v) =gin(x,v), where\nL[g] := ∆vg+∇v·(vg), (1.12b)\nC[g] :=ρ/parenleftbig\n∆vg+∇v·(vg)/parenrightbig\n+Mθ∆vg−M1·∇vg, (1.12c)\nCµ:= (Mθ∆vµ−M1·∇vµ). (1.12d)\nIn many plasma physics settings, the collisions are very wea k, that isν≪1. There\nare many situations when physicists neglect them entirely a nd consider the Vlasov-Poisson\nequation (ν= 0). However, exactly how weak collisions, and collisionle ss effects interact with\nnonlinearity, has been the subject of study and debate in the physics community for nearly\nsix decades [37, 44].\nThe most famous collisonless effect is Landau damping (L.D.) , namely, the rapid de-\ncay of the electric field despite the lack of dissipative mech anisms. Landau damping was\ndiscovered first by Landau in 1946 [29] for the linearized Vla sov-Poisson equations. Landau\ndamping is connected to the mixing in phase-space due to the t ransport operator, and can\nbe considered a variant of velocity averaging. To understan d the nonlinear problem, one\nmust study carefully the important (weakly) nonlinear effect known as the plasma echo ,\nobserved experimentally in [38]. These oscillations are ca used by nonlinear effects exciting\nmodes which are un-mixing in phase space, causing a transien t growth of the electric field,\nwhich can then, in turn, excite further oscillations and cre ate a cascade. The cascade leads to\nderivative loss. The first mathematical justification of the Landau damping for the nonlinear\nVlasov-Poisson equation was given by Mouhot and Villani [43 ], wherenearly analytic regular-\nity for the perturbation is required due to this regularity l oss. A simpler proof was given by\nBedrossian, Masmoudi, and Mouhot [6], where the required re gularity is down to the Gevrey\nthresholds>1\n3, see also [15]. If the perturbation is in lower regularity, w e refer to [34], which\nshows the existence of small BGK waves, and [2] for a mathemat ical study of plasma echoes.\nWe also refer to [7, 19, 20, 25] for the results in the unconfine d setting. As an analogue, the\ninviscid damping (I.D.) of Couette flow for 2D Euler equation was discovered earlier b y\nOrr [45] in 1907. The nonlinear inviscid damping for Couette flow was first proved by [5] for\nthe infinite periodic channel setting, see also [24] for the fi nite channel case. We also refer\nto the recent results [22, 41] for the inviscid damping of sta ble monotone shear flows. If the\nperturbation is less regular, we refer to [11, 33, 49] for asy mptotic instability of shear flows.\nBoth Landau damping and inviscid damping are connected to th e mixing in phase-space due\nto the transport operator, which is common in many other fluid models. One can refer to\n[13, 27, 26, 23, 53, 57, 46, 4, 35, 39, 55, 56] and the reference s therein.\nIn the weak collision case, there is a strong interaction bet ween the transport operator\nand dissipative effect. More precisely, conservative transp ort transfers information to high\nfrequencies where the dissipation is increasingly dominan t, which leads to the enhanced\ndissipation (E.D.) effect, namely the dissipation time scale ν−1\n3is short than the heat\ndissipation time scale ν−1. The enhanced dissipation together with the small enough si ze\nof the perturbation can eliminate the plasma echoes caused b y the nonlinear interaction.\nIn [1], the author studied the interaction between enhanced dissipation and nonlinear effect\nand proved the asymptotic stability of the global Maxwellia n for equation (1.1), when the\nperturbation is in Sobolev spaces with size ν1\n3. We also refer to a recent work [12] where the\nasymptotic stability of the global Maxwellian was proved un der the same assumption on the\ninitial perturbation for the Vlasov-Poisson equation with weak Landau collisions.4 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nIt is natural to ask in the weak collision case, what is the siz e requirement if the pertur-\nbation is smoother? This leads to the stability threshold pr oblem, namely,\nGiven a norm /ba∇dbl·/ba∇dblX, find aγ=γ(X)so that\n/ba∇dblgin/ba∇dblX≤νγ⇒stability, L.D., and E. D. ,\n/ba∇dblgin/ba∇dblX≫νγ⇒instability.(1.13)\nIn the collisionless case ( ν= 0) and in the weak collision case under small perturbation ( γ≥\n1\n3), the key estimate is to study the interaction between one of the linear decay effects with\nplasma echoes. In this paper, we consider the weak collision case, with initial perturbation\nsmoother but larger compared with [1, 12]. We study the plasm a echoes under both Landau\ndamping and enhanced dissipation effect and show a new transie nt growth mechanism that\nis size-regularity dependent.\n1.1.Main result. In this paper, we study the stability threshold problem with initial data\nin Gevrey-1 /sclass fors>skwithsk∈[0,1\n3]. Our first main result states as follows:\nTheorem 1.1. Letsk∈[0,1\n3],λin> c0>0,σ0≥n+ 20, and m≥n+ 10. For any\n1≥s>sk, we define the following weighted Gevrey norm\n/ba∇dblgin/ba∇dbl2\nGs,λin,σ0\nm+2,2ν=/integraldisplay\nTn×Rn/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{t∇x,v/an}b∇acket∇i}htσ0+1eλin/angb∇acketleft∇x,v/angb∇acket∇ights/parenleftBig\n/an}b∇acketle{tv/an}b∇acket∇i}htm+2e2ν|v|2gin(x,v)/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndxdv. (1.14)\nThen the stability threshold γ(Gs,λin,σ\nm+2,2ν) can not be larger than1−3sk\n3−3sk, namely, if\nγ=γ(Gs,λin,σ0\nm+2,2ν)≥1−3sk\n3−3sk, (1.15)\nthen there exists ǫ0>0 such that for any 0 <ǫ<ǫ 0, it holds that\n/ba∇dblgin/ba∇dblGs,λin,σ0\nm+2,2ν≤ǫνγ⇒stability, L.D., and E. D. .\nMore precisely, if /ba∇dblgin/ba∇dblGs,λin,σ0\nm+2,2ν≤ǫνγ, then there exist C >1 andδ1>0, such that for any\nt∈[0,∞), the solution ( g,ρ,M 1,Mθ) to the system (1.12) with initial data gin(x,v) satisfies:\n(1) Enhanced dissipation for non-zero modes\n/ba∇dblP/negationslash=g(t)/ba∇dblL2x,v≤Cǫνγe−δ1ν1/3t. (1.16)\n(2) Dissipation for the zero mode\n/ba∇dblP0g(t)/ba∇dblL2v≤Cǫνγe−νt. (1.17)\n(3) Landau damping\n/ba∇dblρ/ba∇dblL∞+/ba∇dblM1/ba∇dblL∞+/ba∇dblMθ/ba∇dblL∞≤Cǫνγ\n/an}b∇acketle{tt/an}b∇acket∇i}htσ0−10e−δ1ν1/3t. (1.18)\nHereP0g(t,v) =1\n(2π)n/integraltext\nTng(t,x,v)dxandP/negationslash=g(t,x,v) =g(t,x,v)−P0g(t,v), and the con-\nstantsc0,C,δ1,ǫare independent of ν.\nRemark 1.1. We refer to (3.7) for more precise bounds of each quantity.\nRemark 1.2. This kind of ‘stability threshold’ result is entirely analo gous to the line of\nresearch on the two-dimensional Navier-Stokes equations n ear shear flows [32, 40]. Here we\nbriefly introduce our result and compared it with the Navier- Stokes case in the following\nTable 1.LANDAU DAMPING FOR VPFP 5\nTable 1. Stability threshold problem in the fluid and kinetic equatio ns\n2D NS with small viscosity ν\nCouette flowVPFP with small collision frequency ν\nglobal Maxwellian\nSobolev/ba∇dbluin/ba∇dblH3≤ǫν1\n3⇒I.D. and E.D.\nReferences: [42, 52]/ba∇dbl/an}b∇acketle{tv/an}b∇acket∇i}htmgin/ba∇dblHσ≤ǫν1\n3⇒L.D. and E.D.\nReferences: [1, 12]\nGevrey-1\ns/ba∇dbluin/ba∇dblGsf≤ǫνγ⇒I.D. and E.D.\nReferences: [31]/ba∇dbl/an}b∇acketle{tv/an}b∇acket∇i}htme2ν|v|2gin/ba∇dblGs≤ǫνγ⇒L.D. and E.D.\nthis paper\nInviscid\nlimit/ba∇dbluin/ba∇dblG2≤ǫ⇒I.D. and E.D.\nReferences: [8, 31]/ba∇dbl/an}b∇acketle{tv/an}b∇acket∇i}htme2ν|v|2gin/ba∇dblG3−≤ǫ⇒L.D. and E.D.\nthis paper\n•Hereγ(Gsf)≥1−2sf\n3−3sfwithsf∈[0,1\n2]andγ(Gs)≥1−3sk\n3−3skfors > skwithsk∈[0,1\n3].\n•The Gevrey-1\nsnorm is defined by\n/ba∇dblf/ba∇dblGs(Tn×Rn) =\n/summationdisplay\nk∈Zn/integraldisplay\nRne2λ|k,ξ|s|/hatwidef(k,ξ)|2dξ\n1\n2\n,\nwith some λ >0. Here we have used the short hand |k,ξ|=|k|+|ξ|.\n•The constants ǫ >0are independent of ν. The constants σ,mare sufficiently large.\nRemark 1.3. Our estimates also work for the collisionless case ( ν= 0) by simply sending\nνto 0, which gives a slightly better estimate than [6] since th e time growth in the top norm\nof the perturbation of the distribution function in [6] can b e avoided in this paper. This is\nmotivated by [15], see also [3].\nRemark 1.4. It is interesting to discuss the optimality of those indexes in Theorem 1.1.\nTherearefewresultsabouttheoptimality ofthethresholds inbothfluidandkineticequation.\nWe refer to [14, 32] for fluid equations, and refer to [2] for ki netic equations. We also refer\nto section 1.2.4 for further discussions.\nRemark 1.5. By modifying our Fourier multipliers and some estimates, on e may prove that\nthe asymptotic stability holds for initial data which satis fies\n/integraldisplay\nTn×Rn/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{t∇/an}b∇acket∇i}htσ0+1exp/braceleftBig\nλinmin/braceleftbig\n(νγ|∇|)1\nL, ν3γ−1\n2L/bracerightbig/bracerightBig/parenleftBig\n/an}b∇acketle{tv/an}b∇acket∇i}htm+2e2ν|v|2gin(x,v)/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndxdv≤ǫ2ν2γ.\nwith 1≤L<3. See Section 1.2.4 for more discussions.\nRemark 1.6. When we finished this paper, we noticed that an independent wo rk was done\nby Yue Luo [36] for the Vlasov-Poisson with linear Fokker-Pl anck collision operator. After\nreading our abstract1, he improved his result.\nThe second main result is the collisionless limit.\nTheorem 1.2. Lets>1\n3, and letλin,σ0,m,ǫ0,ν0be the same as in Theorem 1.1. Suppose\nthat the initial data ginsatisfies\n/ba∇dblgin/ba∇dblGs,λin,σ0\nm+2,2ν0≤ǫ. (1.19)\nLetg(t) solve (1.12) for any 0 < ν≤ν0and letg0(t) solve (1.12) with ν= 0. There are\nC >1 andδ0>0 independent of ν,tsuch that for any t≤δ0ν1\n3,\n/ba∇dblg(t)−g0(t)/ba∇dblL∞x,v≤Cνt3.\n1The Workshop on the recent progress of kinetic theory and rel ated topics in TSIMF, 15-19th January 2024.6 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n1.2.Main ideas. The nonlinear Fokker-Planck collision operator brings mai nly two effects:\nthe enhanced dissipation from the linear effect and a velocity localization loss in vfrom the\nnonlinear effect. The main ideas in estimating and designing n orms are to take advantage of\nthe enhanced dissipation and overcome the velocity localiz ation loss.\n1.2.1.The time-dependent Gaussian weight. To deal with the collision operator, the classical\nway is to introduce a Gaussian weight, which is commonly used to deal with the Landau\ncollision operator and Boltzmann collision operator. The m ain idea is to use the Gaussian\nweight to absorbthe badterm andrewrite thecollision opera tor as a non-negative self-adjoint\noperator. For the nonlinear Fokker-Planck collision opera tor, this method seems not efficient.\nOnereason is that thekernel of thelinearized Fokker-Planc k collision operator is not thesame\nas the nonlinear Fokker-Planck collision operator, namely ,\nKerl(L+C)/ne}ationslash= Kerl(L).\nIn [1], the author introduced a new coordinate system in freq uency space by observing that\nafter taking theFourier transform, theterm ν∇·(vg) whichcreates thekernel ofthelinearized\nFokker-Planck collision operator becomes a transport term −νξ·∇ξ/hatwidegwhich can be absorbed\nbythenewcoordinate. Suchanideais usefulto control theno nlinear termssuchas νρ∇·(vg),\nwhen the perturbation has enough ν-dependent smallness. In our case, since the perturbation\ncan beν-independent, we cannot apply this method directly.\nOne of the new ideas is to introduce a time-dependent Gaussia n weight in v, namely, we\ndefine\n(1.20) g=e−λ1(t)|v|2gw,\nwith\n(1.21) λ1(t) =ν+ν\n/an}b∇acketle{tt/an}b∇acket∇i}hta0,0<a0≪1.\nWe refer to [21, 51, 54] where a similar time-dependent weigh t is used to study the global\nexistence for the Prandtl system2.\nThe equation for gwcan be written as (see (2.1))\n∂tgw+v·∇xgw−ν∇v·(vgw)\n−ν∆vgw−˙λ1(t)|v|2gw−2νλ1(t)(1−2λ1(t))|v|2gw\n−4νλ1(t)v·∇vgw= other terms .\nBy introducing the time-dependent Gaussian weight, we obta in two new good terms in the\nequation\n−˙λ1(t)|v|2gwand−2νλ1(t)(1−2λ1(t))|v|2gw, (1.22)\nwhich can be used to control low-high interactions in the non linear collision terms such as\nν(ρ+λ1(ρ+Mθ))∇ ·(vgw) andν(2λ1(1−2λ1)ρ+ 4λ2\n1Mθ)|v|2gw, where there is velocity\nlocalization loss. We also note that since e−λ1(t)|v|2is not in the kernel of the linearized\ncollision operator, there are still two bad linear terms\n−ν∇·(vgw) and −4νλ1(t)v·∇vgw.\nFor the first term −ν∇·(vgw), we treat it as a transport term in frequency space and use\nthe idea in [1]. More precisely, after taking the Fourier tra nsform, the problematic term\n−ν∇v·(vgw) becomesνξ·∇ξ/hatwidergw. We introduce the following characteristic curve to absorb\nthe two linear transport terms −k∇ξ/hatwidergw+νξ·∇ξ/hatwidergw:\nd\ndt¯η(t;k,η) =−k+ν¯η(t;k,η),¯η(0;k,η) =η, (1.23)\n2W. Z. would like to thank Professor Zhifei Zhang for mentioni ng those references.LANDAU DAMPING FOR VPFP 7\nwhich gives that\n(1.24) ¯ η(t;k,η) =eνtη−keνt−1\nν.\nThroughout the paper, to simplify the presentation, we intr oduce the notation\ntap=1−e−νt\nν.\nHere ‘ap’ stands for ‘approximate’. It is easy to check that a sν→0+,tap→t. We define\n(/hatwiderfw)k(t,η) = (/hatwidergw)k(t,¯η(t;k,η)),or (/hatwidergw)k(t,ξ) = (/hatwiderfw)k/parenleftbig\nt,e−νtξ+ktap/parenrightbig\n. (1.25)\nWe then get that /hatwiderfwk(t,η) satisfies a better equation (see (2.4))\n∂t(/hatwiderfw)k(t,η)+e−2νt˙λ1(t)(∆η/hatwiderfw)k(t,η)+ν|¯η|2(/hatwiderfw)k(t,η)\n+ν/bracketleftBig\n−(2λ1−4λ2\n1)e−2νt(∆η/hatwiderfw)k(t,η)−4λ1e−νt∇η·/parenleftBig\n¯η/hatwiderfw/parenrightBig\nk(t,η)/bracketrightBig\n= other terms .\nThe second term −4νλ1(t)v·∇vgwcan be controlled by the dissipation term and the new\ngood terms, namely\n−2νλ1(t)(1−2λ1(t))|v|2gwand−ν∆vgw,\nwhenνis small enough.\nRemark 1.7. Since the time-dependent Gaussian weight is also νdependent, we can not use\nit to get aν-uniform integrability in v. Thus we also use the standard polynormial weight,\nsee more discussions in the section 3.\n1.2.2.The time-dependent Fourier multiplier. Now let us consider the main nonlinear in-\nteraction and explain how the enhanced dissipation affects th e plasma echoes through the\nnonlinearity. We have the following formal formula for the d ensity\nˆρ(t,k) =/summationdisplay\nl∈Zn∗/integraldisplayt\n0ˆρ(τ,l)l·k\n|l|2(t−τ)ˆgk−l(τ,kt−lτ)dτ+Easy terms .\nLet us assume that the size of perturbation is ǫνγ. We consider the worst-case scenario\n(taking the 1D example):\n|ˆρ(t,k)|/lessorsimilar/summationdisplay\nl=k±1/integraldisplay\nkt≈lτ|ˆρ(τ,l)|τ\n|l|ǫνγe−δ1ν1\n3τ\n/an}b∇acketle{tkt−lτ/an}b∇acket∇i}ht100dτ+Easy terms\n/lessorsimilarǫt1−3γ\n|k|2/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆρ/parenleftbiggkt\nk+1,k+1/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+Easy terms .(1.26)\nThe density at time τ=kt\nk+1and frequency k+ 1 is strongly amplifying ˆ ρ(t,k). This is a\nplasma echo under the enhanced dissipation effect. This kind o f resonance is the synergism\nof the enhanced dissipation, the transient growth of the pla sma echoes, and the nonlinearity.\nWe can rewrite this plasma echo chain in terms of the distribu tion function g:\n|ˆg(t,k,kt)|/lessorsimilarǫ(|k|t)1−3γ\n|k|3−3γ/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆg/parenleftbiggkt\nk+1,k+1,kt/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+Easy terms .8 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nBy replacing ktbyη, we obtain that the amplification in each step is max/braceleftBig\nǫ|η|1−3γ\n|k|3−3γ,1/bracerightBig\nwhich\ntogether with Stirling’s formula n!∼√\n2πn(n\ne)ngives a total growth\n/productdisplay\nǫ|η|1−3γ\n|k|3−3γ≥1ǫ|η|1−3γ\n|k|3−3γ≈|η|1−3γ\n3−3γ/productdisplay\nk=1ǫ|η|1−3γ\n|k|3−3γ≈ec|η|1−3γ\n3−3γ,\nhere equality holds up to a polynomial correction. We refer t o section 4.1.2 for more precise\nand mathematically rigorous estimates of these plasma echo es that give the relationship\nbetween the regularity index sand the size index γas in Theorem 1.1.\nBased on the above estimate, to capture the growth of different frequencies at different\ntimes, fors>sk=1−3γ\n3−3γ, we introduce the time-dependent Gevrey-1\nstype Fourier multiplier\nAσ0+1,2\n5\nk(t,η) =e1k/negationslash=0δ1ν2\n5teλ(t)/angb∇acketleftk,η/angb∇acket∇ights/an}b∇acketle{tk,η/an}b∇acket∇i}htσ0+1mk(t,η) (1.27)\nwith Sobolev correction /an}b∇acketle{tk,η/an}b∇acket∇i}htσ+1=: (|k|2+|η|2)σ+1\n2. Hereλ(t) is the Gevrey radius, which\nis defined as follows:\n(1.28) λ(t) =λ∞+˜δ\n(1+t)a\nfor positive constants ˜δandachosen sufficiently small, and the multiplier\n(1.29) mk(t,η) =\n\n1,ifk= 0,\nexp/braceleftbig\n−/integraltextt\n0ν1\n3\n1+ν2\n3e2νs|η−ksap|2e2νs|νη−k|2\n/angb∇acketlefteνs(νη−k)/angb∇acket∇ight2ds/bracerightbig\n,ifk/ne}ationslash= 0,\nis themodified ghost-type Fourier multiplier which, togeth er with the collision operator, gives\ntheenhanceddissipation. WerefertoAppendixCformorepro pertiesoftheFouriermultiplier\nm(t,∇). The weight e1k/negationslash=0δ1ν2\n5tgives a suboptimal enhanced dissipation rate. Therefore,\nwe also introduce the time-dependent Gevrey-1\nstype Fourier multiplier with less Sobolev\ncorrection /an}b∇acketle{tk,η/an}b∇acket∇i}htσ1and stronger enhanced dissipation\nAσ1,1\n3\nk(t,η) =e1k/negationslash=0δ1ν1\n3teλ(t)/angb∇acketleftk,η/angb∇acket∇ights/an}b∇acketle{tk,η/an}b∇acket∇i}htσ1mk(t,η), (1.30)\nwith\nσ1+2+n+1\n2≤σ0.\nFor anyσ>0 ande∈ {1\n3,2\n5}, we use the notation\nAσ,ef=Aσ,e(t,∇)f= (Aσ,e\nk(t,ξ)ˆfk(t,ξ))∨\nto simplify the presentation.\nNow let us explain briefly why we only obtain a weaker enhanced dissipation rate for the\nhighestnorm. Weconsiderthelow-high nonlinearinteracti onE·∇vgw, wheretheelectric field\nEhas rapiddecay, but thereis a derivative loss. Thestandard idea is to take advantage of the\ntransportstructureandintroduceacommutator. Theworst- case scenariois( ∇v−t∇x)≈ ∇v.\nThe idea of controlling the derivative loss and paying less s mallness (in terms of ν) is to use\nthe dissipation term and the ‘CK’ term, namely,\nν/ba∇dbl(∇v−t∇x)Aσ,e(vαfw)/ba∇dbl2\nL2and˙λ(t)/ba∇dbl/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2Aσ,e(vαfw)/ba∇dbl2\nL2.\nMore precisely, we have for σ=σ0+1 and e=2\n5, it holds that\n/ba∇dblAσ,e|∇v|1−s\n2(vαfw)/ba∇dblL2≤ /ba∇dblAσ,e|∇v−t∇x|(vαfw)/ba∇dbl2−2s\n2−s\nL2/ba∇dbl/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2Aσ,e(vαfw)/ba∇dbls\n2−s\nL2.LANDAU DAMPING FOR VPFP 9\nThe total loss is hence,\nνγν−1−s\n2−s<νγ+2\n3γ−5≤ν−2\n5,\nthus we need to gain ν2\n5from the commutator, which gives the enhanced dissipation r ate\nfor the highest norm. Note that the commutator e1k/negationslash=0δ1ν2\n5t−1 does not absorb derivatives,\nbut contributes smallness ν2\n5. For the lower norm Aσ1,1\n3, such commutator estimates are not\nnecessary, and thus we can obtain the sharp enhanced dissipa tion ratee−δ1ν1\n3t.\nForσ≥0,δ1<δ<δ0\n16, whereδ0is the constant appearing in (B.1), let us denote\n(1.31) Bσ\nk(t) =eδν1\n3teλ(t)/angb∇acketleftk,ktap/angb∇acket∇ights/an}b∇acketle{tk,ktap/an}b∇acket∇i}htσ.\nThen for the macroscopic quantities ρ,M1,Mθ, the following two Fourier multipliers\n(1.32) Bk(t) =|k|1\n2Bσ0\nk(t),andBσ1\nk(t)\nare involved in the higher and lower norm estimates.\n1.2.3.Three levels of energy functionals. We first recall that there are four main operations:\n(1) Applying the time-dependent Gaussian-type weight.\n(2) Applying the change of coordinate.\n(3) Applying the polynomial weight.\n(4) Applying the time-dependent Fourier multipliers.\nIt is worth pointing out that the order of four operations is c hosen to avoid the bad commu-\ntator [Aσ(t,∇),eλ1|v|2] and derive a good working system since the inverse Gaussian weight\ndoes not commute well with the infinitely many derivatives in A. In [12], the authors study\nthe stability of the Vlasov-Poisson-Landau equation in Sob olev spaces. The order of oper-\nations in their paper is (1) applying the classical Gaussian weight, (2) applying the finite\nnumber of derivatives, (3) applying the polynomial weight a nd the exponential weight eq0|v|ς\n2\nintroduced in [16, 17, 50].\nItisnatural tointroducethefollowing twoenergy function als forthedistributionfunctions:\nEσ0+1,2\n5mandEσ1,1\n3m, where\n/ba∇dblfw(t)/ba∇dbl2\nEσ0+1,2\n5m=\n/summationdisplay\nα∈Nn:|α|≤me−2|α|νt\nK|α|/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5(vαfw(t))/vextenddouble/vextenddouble/vextenddouble2\nL2\n1\n2\n, (1.33)\nis used to control the highest regularity, and\n/ba∇dblfw(t)/ba∇dbl2\nEσ1,1\n3m=\n/summationdisplay\nα∈Nn:|α|≤me−2|α|νt\nK|α|/vextenddouble/vextenddouble/vextenddoubleAσ1,1\n3(vαfw(t))/vextenddouble/vextenddouble/vextenddouble2\nL2\n1\n2\n, (1.34)\nis used to obtain the optimal enhanced dissipation rate. Her e{K|α|}m+2\n|α|=0is an increasing\nsequence withK|α|\nK|α|+1≪1 which is determined in the proof. As mentioned in the previo us\nsection, to control the macroscopic quantity ρ, we use the multiplier Bkwhich is of half\nderivative less than the highest regularity. This loss come s from the low-high interaction in\nˆρ(t,k) =/summationdisplay\n|l|≤1\n4|k|/integraldisplayt/2\n0ˆρ(τ,l)l·k\n|l|2(t−τ)ˆgk−l(τ,kt−lτ)dτ+other terms .\nDuetothisderivativeloss, weneedtocarefully treatthehi gh-low interactions inthenonlinear\ncollision operator, for examples\nνρ∆vgw, νλ1(1−λ1)ρ|v|2gw.10 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nFor these, we work back on the microscopic quantity and use th e identity\nˆρk(t) = ˆgk(t,0) =ˆfk(t,ktap).\nWe refer to Section 6.2.2 for more details. We also face the ve locity localization loss problem\nin the treatment of the high-low interaction in νλ1(1−λ1)ρ|v|2gw. Thus we introduce the\nthird energy functional\n/ba∇dblfw(t)/ba∇dbl2\nEσ1,2\n5\nm+2=\n/summationdisplay\nα∈Nn:|α|≤m+2e−2|α|νt\nK|α|/vextenddouble/vextenddouble/vextenddoubleAσ1,2\n5(vαfw(t))/vextenddouble/vextenddouble/vextenddouble2\nL2\n1\n2\n, (1.35)\nwhich is used to control the velocity localization loss caus ed by the nonlinear collision opera-\ntor. It is worth pointing out that to close the energy estimat e ofEσ1,2\n5\nm+2(t), since it is in lower\nregularity, there is no derivative loss problem. In the ener gy estimates of Eσ1,2\n5\nm+2(t), even the\nhigh-low interaction in νλ1(1−λ1)ρ|v|2gwcan be treated as the low-high interactions where\nwe also take advantage of the two good terms (1.22).\n1.2.4.Further discussion. There is a dual problem to the stability threshold problem (1 .13):\nGiven a real number γ≥0, find a largest function space Xγ,νso that\n/ba∇dblgin/ba∇dblXγ,ν≤νγ⇒stability, L.D., and E. D. ,\n/ba∇dblgin/ba∇dblXγ,ν≫νγ⇒instability.(1.36)\nIn terms of regularity, if the function space is required to b eν-independent, then the function\nspace (1.19) in Theorem 1.1 seems optimal.\nLet us now discuss the case that the function space is ν-dependent. We revisit (1.26) and\nget that\n|ˆρ(t,k)|/lessorsimilar/summationdisplay\nl=k±1/integraldisplay\nkt≈lτ|ˆρ(τ,l)|τ\n|l|ǫνγe−δ1ν1\n3τ\n/an}b∇acketle{tkt−lτ/an}b∇acket∇i}ht100dτ+Easy terms\n/lessorsimilarǫνγte−δ1ν1\n3t\n|k|2/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆρ/parenleftbiggkt\nk+1,k+1/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+Easy terms .(1.37)\nBy a similar argument, we write this plasma echo chain in term s of distribution function g:\n|ˆg(t,k,η)|/lessorsimilarǫνγ|η|e−δ1ν1\n3|η|\n|k|\n|k|3/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆg/parenleftbiggkt\nk+1,k+1,η/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+Easy terms .\nThus we obtain that the amplification in each step is max/braceleftBig\nǫνγ|η|e−δ1ν1\n3|η|\n|k|\n|k|3,1/bracerightBig\n, which gives a\ntotal growth\n/productdisplay\nǫνγ|η|e−δ1ν1\n3|η|/|k|≥|k|3ǫνγ|η|e−δ1ν1\n3|η|\n|k|\n|k|3/lessorsimilareC(νγ|η|)1\n3e−δ1ν1\n3|η|/lessorsimilarexp/braceleftBig\nCmin/braceleftbig\n(νγ|η|)1\n3, ν3γ−1\n6/bracerightbig/bracerightBig\n.\nWe also note that\nmin/braceleftBig\n(νγ/an}b∇acketle{tk,η/an}b∇acket∇i}ht)1\nL, ν3γ−1\n2L/bracerightBig\n/lessorsimilar/an}b∇acketle{tk,η/an}b∇acket∇i}hts\nwiths=3\nL1−3γ\n3−3γ>1−3γ\n3−3γ. Based on the above estimates, we can get Remark 1.5 by modify ing\nour Fourier multipliers and some estimates.\nTo reduce the technical estimates, the ν-independent function space (1.19) is used which\nis of the classical Gevrey type.LANDAU DAMPING FOR VPFP 11\n2.The working system\nIn this section, we derive a good working system as mentioned in section 1.2.3. We first\napply the time-dependent Gaussian weight and introduce the weighted distribution function\ngw(t,x,v) =g(t,x,v)eλ1(t)|v|2. Then we apply the change of coordinate after taking the\nFourier transform to absorb the linear transport part.\n2.1.Reformulation (I): the distribution function. Letgwbe defined in (1.20). Then\nwe infer from (1.12a) that the equation for gwtakes the form of\n∂tgw−˙λ1(t)|v|2gw+v·∇xgw−ν/parenleftBig\nLgw+˜Lλ1\n1[gw]/parenrightBig\n+E·∇vµeλ1(t)|v|2\n=−E·/parenleftBig\n∇vgw+˜Lλ1\n2[gw]/parenrightBig\n+ν(Mθ∆vµ−M1·∇vµ)eλ1(t)|v|2+νρ/parenleftBig\nLgw+˜Lλ1\n1[gw]/parenrightBig\n+νMθ/parenleftBig\n∆vgw+˜Lλ1\n3[gw]/parenrightBig\n−νM1·/parenleftBig\n∇vgw+˜Lλ1\n2[gw]/parenrightBig\n, (2.1)\nwhere\n˜Lλ1\n1[gw] =2nλ1(t)gw−2λ1(t)(1−2λ1(t))|v|2gw−4λ1(t)∇v·(vgw),\n˜Lλ1\n2[gw] =−2λ1(t)vgw,\n˜Lλ1\n3[gw] =2nλ1(t)gw+4(λ1(t))2|v|2gw−4λ1(t)∇v·(vgw).\nTaking Fourier transform on (2.1) yields\n∂t/hatwidergw+˙λ1(t)∆ξ/hatwidergw−k·∇ξ/hatwidergw+νξ·∇ξ/hatwidergw+ν|ξ|2/hatwidergw\n+ν/bracketleftBig\n2nλ1/hatwidergw−(2λ1−4λ2\n1)∆ξ/hatwidergw−4λ1∇ξ·(ξ/hatwidergw)/bracketrightBig\n+ˆEk(t)·F/bracketleftBig\n∇vµeλ1|v|2/bracketrightBig\n(ξ)\n=−F[E·∇vgw]k(t,ξ)+νF/bracketleftBig\n(Mθ∆vµ−M1·∇vµ)eλ1|v|2/bracketrightBig\nk(t,ξ)+νF/bracketleftbig\nC[gw]/bracketrightbig\nk(t,ξ)\n−F/bracketleftBig\nE·˜Lλ1\n2[gw]/bracketrightBig\nk(t,ξ)+νF/bracketleftBig\nρ˜Lλ1\n2[gw]/bracketrightBig\nk(t,ξ)\n+νF/bracketleftBig\nMθ˜Lλ1\n3[gw]/bracketrightBig\nk(t,ξ)−νF/bracketleftBig\nM1˜Lλ1\n2[gw]/bracketrightBig\nk(t,ξ). (2.2)\nRecalling the definition of /hatwiderfwin (1.25), and reorganizing the nonlinearities on the right hand\nside of (2.2) in terms of moments:\nνC[gw]+νρ˜Lλ1\n1[gw]+νMθ˜Lλ1\n3[gw]−νM1˜Lλ1\n2[gw] =M0[gw]+M1[gw]+M2[gw], (2.3)\nwhere\nM0[gw] =ν(ρ+Mθ)∆vgw−νM1·∇vgw+2nνλ1(ρ+Mθ)gw,\nM1[gw] =ν/parenleftbig\nρ−4λ1(ρ+Mθ)/parenrightbig\n∇v·(vgw)+2νλ1M1·vgw,\nM2[gw] =−2νλ1/parenleftbig\n(1−2λ1)ρ−2λ1Mθ/parenrightbig\n|v|2gw,\none deduces that ( /hatwiderfw)k(t,η) solves\n∂t(/hatwiderfw)k(t,η)+e−2νt˙λ1(t)(∆η/hatwiderfw)k(t,η)+ν|¯η(t;k,η)|2(/hatwiderfw)k(t,η)\n+ˆEk(t)·F/bracketleftBig\n∇vµeλ1|v|2/bracketrightBig\n(¯η(t;k,η))+ν/bracketleftBig\n2nλ1(/hatwiderfw)k(t,η)\n−(2λ1−4λ2\n1)e−2νt(∆η/hatwiderfw)k(t,η)−4λ1e−νt∇η·/parenleftBig\n¯η(t;k,η)/hatwiderfw/parenrightBig\nk(t,η)/bracketrightBig\n=−F[E·∇vgw]k(t,¯η(t;k,η))−F/bracketleftBig\nE·˜Lλ1\n2[gw]/bracketrightBig\nk(t,¯η(t;k,η))\n+νF/bracketleftBig\n(Mθ∆vµ−M1·∇vµ)eλ1|v|2/bracketrightBig\nk(t,¯η(t;k,η))\n+F/bracketleftbig\nM0[gw]/bracketrightbig\nk(t,¯η(t;k,η)) +F/bracketleftbig\nM1[gw]/bracketrightbig\nk(t,¯η(t;k,η))+F/bracketleftbig\nM2[gw]/bracketrightbig\nk(t,¯η(t;k,η)), (2.4)12 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nwhere\nF[E·∇vgw]k(t,¯η(t;k,η)) =/summationdisplay\nl∈ZnˆEl(t)·i¯η(t;k,η)(/hatwiderfw)k−l(t,η−ltap),\nF/bracketleftBig\nE·˜Lλ1\n2[gw]/bracketrightBig\nk(t,¯η(t;k,η)) =−2λ1/summationdisplay\nl∈ZnˆEl(t)·ie−νt(∇η/hatwiderfw)k−l(t,η−ltap),\nF/bracketleftbig\nM0[gw]/bracketrightbig\nk(t,¯η(t;k,η)) =−ν/summationdisplay\nl∈Zn/parenleftBig\nˆρl(t)+(/hatwiderMθ)l(t)/parenrightBig\n|¯η(t;k,η)|2(/hatwiderfw)k−l(t,η−ltap)\n−ν/summationdisplay\nl∈Zn(/hatwiderM1)l(t)·i¯η(t;k,η)(/hatwiderfw)k−l(t,η−ltap)\n+2nνλ1/summationdisplay\nl∈Zn/parenleftBig\nˆρl(t)+(/hatwiderMθ)l(t)/parenrightBig\n(/hatwiderfw)k−l(t,η−ltap), (2.5)\nand\nF/bracketleftbig\nM1[gw]/bracketrightbig\nk(t,¯η(t;k,η))\n=ν/summationdisplay\nl∈Zn/parenleftBig\nˆρl(t)−4λ1(ˆρl(t)+(/hatwiderMθ)l(t))/parenrightBig\ne−νt¯η(t;k,η)·(∇η/hatwiderfw)k−l(t,η−ltap)\n+2νλ1/summationdisplay\nl∈Zn(/hatwiderM1)l(t)·ie−νt(∇η/hatwiderfw)k−l(t,η−ltap), (2.6)\nand\nF/bracketleftbig\nM2[gw]/bracketrightbig\nk(t,¯η(t;k,η))\n=2νλ1/summationdisplay\nl∈Zn/parenleftBig\n(1−2λ1)ˆρl(t)−2λ1(/hatwiderMθ)l(t)/parenrightBig\ne−2νt(∆η/hatwiderfw)k−l(t,η−ltap). (2.7)\n2.2.Reformulation (II): the density. Taking Fourier transform in ( x,v) of the equation\n(1.12a), we arrive at\n(2.8)∂tˆgk(t,ξ)−k·∇ξˆgk(t,ξ)+νξ·∇ξˆgk(t,ξ)+ν|ξ|2ˆgk(t,ξ)+ˆEk(t)·iξˆµ(ξ) =Nk(t,ξ),\nwhere\nNk(t,ξ) =−(/hatwiderE·∇vg)k(t,ξ)+ν(/hatwiderCµ)k(t,ξ)+ν(/hatwideCg)k(t,ξ).\nNote that the equations of /hatwidergwand/hatwideghave the same linear transport part. We then use the\nsame change of coordinate and set\n(2.9) ˆfk(t,η) = ˆgk(t,¯η(t;k,η)),\nor equivalently\n(2.10) ˆ gk(t,ξ) =ˆfk/parenleftbig\nt,e−νtξ+ktap/parenrightbig\n.\nThen from (2.8), we find that the equation of ˆfk(t,η) takes the form of\n(2.11) ∂tˆfk(t,η)+ν|¯η(t;k,η)|2ˆfk(t,η)+Lk(t,¯η(t;k,η)) =Nk(t,¯η(t;k,η)),\nwhere\nLk(t,¯η(t;k,η)) =ˆEk(t)·i¯η(t;k,η)ˆµ(¯η(t;k,η)), (2.12)\nNk(t,¯η(t;k,η)) =−/summationdisplay\nl∈Zn∗ˆEl(t)·i¯η(t;k,η)ˆfk−l(t,η−ltap)\n+ν(/hatwiderCµ)k(t,¯η(t;k,η)) +ν(/hatwidestC[g])k(t,k,¯η(t;η)), (2.13)\nwith\n(/hatwiderCµ)k(t,¯η(t;k,η)) =−(/hatwiderMθ)k(t)|¯η(t;k,η)|2ˆµ(¯η(t;k,η))LANDAU DAMPING FOR VPFP 13\n−(/hatwiderM1)k(t)·i¯η(t;k,η)ˆµ(¯η(t;k,η)), (2.14)\nand\n(/hatwidestC[g])k(t,¯η(t;k,η)) =−/summationdisplay\nl∈Zn∗ˆρl(t)|¯η(t;k,η)|2ˆfk−l(t,η−ltap)\n−e−νt/summationdisplay\nl∈Zn∗ˆρl(t)¯η(t;k,η)·(∇ηˆf)k−l(t,η−ltap)\n−/summationdisplay\nl∈Zn(/hatwiderMθ)l(t)|¯η(t;k,η)|2ˆfk−l(t,η−ltap)\n−/summationdisplay\nl∈Zn∗(/hatwiderM1)l(t)·i¯η(t;k,η)ˆfk−l(t,η−ltap). (2.15)\nNow we are in a position to give the macroscopic quantities ρ,M1andM2in terms of f.\nIn fact, from (1.3), (1.6) and (1.7), we have\nˆρk(t) = ˆgk(t,0) =ˆfk(t,ktap), (2.16)\n(/hatwiderM1)k(t) = (/hatwidervg)k(t,0) =ie−νt(∇ηˆf)k(t,ktap), (2.17)\n(/hatwiderM2)k(t) = (/hatwide|v|2g)k(t,0) =−e−2νt(∆ηˆf)k(t,ktap). (2.18)\nLet\n(2.19)Sk(t,τ;η) = exp/parenleftbigg\n−ν/integraldisplayt\nτ|¯η(s;k,η)|2ds/parenrightbigg\n= exp/parenleftBigg\n−ν/integraldisplayt\nτ/vextendsingle/vextendsingle/vextendsingle/vextendsingleeνsη−keνs−1\nν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nds/parenrightBigg\n.\nTakeη=ktapin (2.19), and denote\nSk(t,τ) :=Sk(t,τ;ktap) = exp/parenleftBigg\n−|k|2\nν/bracketleftBigg\nt−τ−2(t−τ)ap+1−e−2ν(t−τ)\n2ν/bracketrightBigg/parenrightBigg\n. (2.20)\nIn particular,\n(2.21) Sk(t) :=Sk(t,0;ktap) = exp/braceleftbigg\n−|k|2\nν/parenleftbigg\nt−2tap+1−e−2νt\n2ν/parenrightbigg/bracerightbigg\n.\nIt is easy to check that Sk(t,τ) =Sk(t−τ). By Duhamel’s principle, the solution ˆfk(t,η) to\n(2.11) can be expressed as\nˆfk(t,η) =Sk(t,0;η)(/hatwiderfin)k(η)−/integraldisplayt\n0Sk(t,τ;η)Lk(τ,¯η(τ;k,η))dτ\n+/integraldisplayt\n0Sk(t,τ;η)Nk(τ,¯η(τ;k,η))dτ. (2.22)\nTakingη=ktapin (2.22), and noting that\n(2.23) ¯ η(τ;k,ktap) =k(t−τ)ap,\nwe get\nˆρk(t) =Sk(t)(/hatwiderfin)k(ktap)−/integraldisplayt\n0Sk(t−τ)ˆEk(τ)·ik(t−τ)apˆµ(k(t−τ)ap)dτ\n+/integraldisplayt\n0Sk(t−τ)Nk(τ,k(t−τ)ap)dτ. (2.24)\nWe postpone the equations for M1andM2to Section 5.14 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nFor convenience, let us denote ∂t\nv=eνt(∇v−tap∇x),whose Fourier symbol is i¯η(t,k,η),\nnamely,\n/hatwide∂tv(k,η) =i¯η(t,k,η) =/hatwide∂tv(k−l,η−ltap), (2.25)\nik(t−τ)ap=ik(t−τ)ap=/hatwider∂τv(k−l,ktap−lτap). (2.26)\n2.3.Relationship between unknowns fandfw.Recalling that g=e−λ1|v|2gw, from\n(2.9) and (1.25), one deduces the relation between ˆfk(t,η) and (/hatwiderfw)k(t,η):\n(2.27) ˆfk(t,η) =/integraldisplay\nRn(/hatwiderfw)k(t,ξ)/parenleftbigg√πeνt\n√λ1/parenrightbiggn\ne−/parenleftbigg\neνt\n2√\nλ1|η−ξ|/parenrightbigg2\ndξ.\nLemma 2.1 (fromfwtof).Let0< s <1,λ >0, andσ≥0. Then for any multi-index\nα∈Nn, there holds\n(2.28) /ba∇dblvαf/ba∇dblGλ,σ;s/lessorsimilar/ba∇dblvαfw/ba∇dblGλ,σ;s,\nand\n(2.29)/vextenddouble/vextenddouble∂t\nv(vαf)/vextenddouble/vextenddouble\nGλ,σ;s/lessorsimilar/vextenddouble/vextenddouble∂t\nv(vαfw)/vextenddouble/vextenddouble\nGλ,σ;s+λ1e−νt/ba∇dblvvαfw/ba∇dblGλ,σ;s.\nProof.From (2.27), we write\n∂α\nηˆfk(t,η) =/integraldisplay\nRn(∂α\nη/hatwiderfw)k(t,ξ)/parenleftbigg√πeνt\n√λ1/parenrightbiggn\ne−/parenleftbigg\neνt\n2√\nλ1|η−ξ|/parenrightbigg2\ndξ.\nTo boundeλ/angb∇acketleftk,η/angb∇acket∇ights/an}b∇acketle{tk,η/an}b∇acket∇i}hts, we use the following partition of unity\n1 =1|η−ξ|≤1\n2|k,ξ|+1|k,ξ|≤2|η−ξ|.\nThen in view of (A.3) and (A.5), we have\neλ/angb∇acketleftk,η/angb∇acket∇ights/an}b∇acketle{tk,η/an}b∇acket∇i}htσ/lessorsimilareλ/angb∇acketleftk,ξ/angb∇acket∇ights/an}b∇acketle{tk,ξ/an}b∇acket∇i}htσecλ/angb∇acketleftη−ξ/angb∇acket∇ights+eλ/angb∇acketleftη−ξ/angb∇acket∇ights/an}b∇acketle{tη−ξ/an}b∇acket∇i}htσecλ/angb∇acketleftk,ξ/angb∇acket∇ights.\nConsequently, noting thateνt√λ1/greaterorsimilar1√ν\neλ/angb∇acketleftk,η/angb∇acket∇ights/an}b∇acketle{tk,η/an}b∇acket∇i}htσ|∂α\nηˆfk(t,η)|\n/lessorsimilar/integraldisplay\nRneλ/angb∇acketleftk,ξ/angb∇acket∇ights/an}b∇acketle{tk,ξ/an}b∇acket∇i}htσ/vextendsingle/vextendsingle/vextendsingle(∂α\nη/hatwiderfw)k(t,ξ)/vextendsingle/vextendsingle/vextendsingleecλ/angb∇acketleftη−ξ/angb∇acket∇ights/parenleftbigg√πeνt\n√λ1/parenrightbiggn\ne−/parenleftbigg\neνt\n2√\nλ1|η−ξ|/parenrightbigg2\ndξ\n+/integraldisplay\nRnecλ/angb∇acketleftk,ξ/angb∇acket∇ights/vextendsingle/vextendsingle/vextendsingle(∂α\nη/hatwiderfw)k(t,ξ)/vextendsingle/vextendsingle/vextendsingleeλ/angb∇acketleftη−ξ/angb∇acket∇ights/an}b∇acketle{tη−ξ/an}b∇acket∇i}htσ/parenleftbigg√πeνt\n√λ1/parenrightbiggn\ne−/parenleftbigg\neνt\n2√\nλ1|η−ξ|/parenrightbigg2\ndξ\n/lessorsimilar/integraldisplay\nRneλ/angb∇acketleftk,ξ/angb∇acket∇ights/an}b∇acketle{tk,ξ/an}b∇acket∇i}htσ/vextendsingle/vextendsingle/vextendsingle(∂α\nη/hatwiderfw)k(t,ξ)/vextendsingle/vextendsingle/vextendsingle/parenleftbigg√πeνt\n√λ1/parenrightbiggn\ne−/parenleftbigg\neνt\n4√\nλ1|η−ξ|/parenrightbigg2\ndξ. (2.30)\nThen(2.28) follows immediately. Toprove (2.29), wedivide ¯η(t;k,η)∂α\nηˆfk(t,η) into two parts:\n¯η(t;k,η)∂α\nηˆfk(t,η) = I+II,\nwhere\nI =/integraldisplay\nRn¯ξ(t;k,ξ)(∂α\nη/hatwiderfw)k(t,ξ)/parenleftbigg√πeνt\n√λ1/parenrightbiggn\ne−/parenleftbigg\neνt\n2√\nλ1|η−ξ|/parenrightbigg2\ndξ, (2.31)\nII =/integraldisplay\nRn(∂α\nη/hatwiderfw)k(t,ξ)eνt(η−ξ)/parenleftbigg√πeνt\n√λ1/parenrightbiggn\ne−/parenleftbigg\neνt\n2√\nλ1|η−ξ|/parenrightbigg2\ndξ\n=2λ1e−νt/integraldisplay\nRn(∂α\nη/hatwiderfw)k(t,ξ)/parenleftbigg√πeνt\n√λ1/parenrightbiggn\n∇ξe−/parenleftbigg\neνt\n2√\nλ1|η−ξ|/parenrightbigg2\ndξLANDAU DAMPING FOR VPFP 15\n=−2λ1e−νt/integraldisplay\nRn(∇η∂α\nη/hatwiderfw)k(t,ξ)/parenleftbigg√πeνt\n√λ1/parenrightbiggn\ne−/parenleftbigg\neνt\n2√\nλ1|η−ξ|/parenrightbigg2\ndξ (2.32)\nThen we bound (2.31) and (2.32) in Gλ,σ;sin the same manner as (2.30), we get (2.29). This\ncompletes the proof of Lemma 2.1. /square\n3.Proof of the main theorems\nInthissection, weproveTheorem1.1andTheorem1.2, starti ngtheprimarystepsaspropo-\nsitions whichareproved inthesubsequentsections. We reca ll themultiplier Aσ,e\nk(t,η) weused\nin this paper and the associated three-level energy functio nals/ba∇dblfw(t)/ba∇dbl\nEσ0+1,2\n5m,/ba∇dblfw(t)/ba∇dbl\nEσ1,2\n5\nm+2,\nand/ba∇dblfw(t)/ba∇dbl\nEσ1,1\n3m, where for σ={σ0+1,σ1}ande={2\n5,1\n3},\nAσ,e\nk(t,η) =e1k/negationslash=0δ1νeteλ(t)/angb∇acketleftk,η/angb∇acket∇ights/an}b∇acketle{tk,η/an}b∇acket∇i}htσmk(t,η),\nand correspondingly\n/ba∇dblf(t)/ba∇dblEσ,e\nm=\n/summationdisplay\nα∈Nn:|α|≤me−2|α|νt\nK|α|/ba∇dblAσ,e(vαf(t))/ba∇dbl2\nL2\n1\n2\n. (3.1)\nBy the definition for the Fourier multiplier Aσ,e(t,∇), when taking the time-derivative of the\nabove functionals, we have the corresponding good terms:\n/ba∇dblfw(t)/ba∇dbl2\nDσ,e\nm=/summationdisplay\nα∈Nn:|α|≤m1\nK|α|/bracketleftbigg\n−e−2|α|νt˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2Aσ,e(vαfw)/vextenddouble/vextenddouble/vextenddouble2\nL2(3.2)\n+e−2|α|νt/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicalbigg\n−∂tm\nmAσ,e(vαfw)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2+νe−2|α|νt/vextenddouble/vextenddouble∂t\nvAσ,e(vαfw)/vextenddouble/vextenddouble2\nL2\n+ν|α|e−2|α|νt/ba∇dblAσ,e(vαfw)/ba∇dbl2\nL2+2nνλ1(t)e−2|α|νt/ba∇dblAσ,e(vαfw)/ba∇dbl2\nL2\n+/parenleftBig\n2νλ1(1−2λ1)−˙λ1(t)/parenrightBig\ne−2(|α|+1)νt/ba∇dblAσ,e(vvαfw)/ba∇dbl2\nL2/bracketrightbigg\n,\nwhere the first two terms arise from the derivative acting on t he time-dependent Fourier\nmultiplier, the remaining terms arise from the good structu res offwequation, facilitated by\nthe introduction of the time-dependent Gaussian weight. Si nce the Fourier multiplier m(t,∇)\nis bounded upper and below, see (C.1), to simplify the presen tation, sometimes we use the\nfollowing notation\nAσ,e\nk(t,η) =e1k/negationslash=0δ1νeteλ(t)/angb∇acketleftk,η/angb∇acket∇ights/an}b∇acketle{tk,η/an}b∇acket∇i}htσ.\nRemark 3.1. We remark that the Fourier multipliers commutes smoothly wi th polynomial\nweights inv. Indeed, we use the following estimates in our proof:\nFor all multi-index α∈Nn,\n(3.3)/vextendsingle/vextendsingle∂α\nηAσ,e\nk(t,η)/vextendsingle/vextendsingle/lessorsimilar1\n/an}b∇acketle{tk,η/an}b∇acket∇i}ht|α|(1−s)Aσ,e\nk(t,η).\nAccordingly, together with Lemma 2.1, we have\ne−mνt/vextenddouble/vextenddouble/vextenddouble(Aσ,eˆf)k(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2\nkHmη/lessorsimilare−mνt/summationdisplay\n|α|≤m/summationdisplay\nβ≤α/parenleftbigg\nα\nβ/parenrightbigg/vextenddouble/vextenddouble/vextenddouble∂α−β\nηAσ,e∂β\nηˆf/vextenddouble/vextenddouble/vextenddouble\nL2\nkL2η\n/lessorsimilarm/summationdisplay\n|α|≤me−|α|νt/vextenddouble/vextenddouble/vextenddoubleAσ,e∂α\nηˆf/vextenddouble/vextenddouble/vextenddouble\nL2η/lessorsimilar/ba∇dblfw(t)/ba∇dblEσ,e\nm, (3.4)16 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nand\nν/summationdisplay\nk∈Zn∗/integraldisplayT∗\n01\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1+/parenleftbigg\ne−mντ/vextenddouble/vextenddoubleF[∂τ\nvAσ,ef/negationslash=]k(τ)/vextenddouble/vextenddouble\nHn−1\n2+\nη/parenrightbigg2\ndτ\n/lessorsimilarν/integraldisplayT∗\n01\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1+/parenleftbigg\ne−mντ/vextenddouble/vextenddouble/vextenddouble¯η(τ;k,η)Aσ,eˆf/negationslash=/vextenddouble/vextenddouble/vextenddouble\nL2\nkHmη/parenrightbigg2\ndτ\n/lessorsimilarν/integraldisplayT∗\n01\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1+/parenleftBig\n/ba∇dblfw(τ)/ba∇dbl2\nEσ,e\nm−1+/ba∇dblfw(τ)/ba∇dbl2\nDσ,e\nm/parenrightBig\ndτ\n/lessorsimilarsup\nt/ba∇dblfw(t)/ba∇dbl2\nEσ,e\nm−1+ν/integraldisplayT∗\n0/ba∇dblfw(t)/ba∇dbl2\nDσ,e\nmdt. (3.5)\n3.1.Bootstrap. A standard argument gives the local wellposedness for the VP FP system\nin Gevrey-1\nsspaces. In this way, we may safely ignore the time interval [0 ,1] by further\nrestricting the size of the initial data.\nThe goal is next to prove by a continuity argument that the thr ee-level energy functionals\n/ba∇dblfw(t)/ba∇dbl2\nEσ0+1,2\n5m,/ba∇dblfw(t)/ba∇dbl2\nEσ1,2\n5\nm+2,/ba∇dblfw(t)/ba∇dbl2\nEσ1,1\n3m(3.6)\n(together with some related quantities) are uniformly boun ded for all time if ǫis sufficiently\nsmall. We define the following controls referred to in the seq uel as the bootstrap hypotheses\nfort≥1:\nsup\nt∈[1,T]/ba∇dblfw(t)/ba∇dbl2\nEσ0+1,2\n5m+/integraldisplayT\n1/ba∇dblfw(t)/ba∇dbl2\nDσ0+1,2\n5mdt≤(4Cfǫνγ)2, (3.7a)\nsup\nt∈[1,T]/ba∇dblfw(t)/ba∇dbl2\nEσ1,1\n3m+/integraldisplayT\n1/ba∇dblfw(t)/ba∇dbl2\nDσ1,1\n3mdt≤(4Cfǫνγ)2, (3.7b)\nsup\nt∈[1,T]/ba∇dblfw(t)/ba∇dbl2\nEσ1,2\n5\nm+2+/integraldisplayT\n1/ba∇dblfw(t)/ba∇dbl2\nDσ1,2\n5\nm+2dt≤(4Cfǫνγ)2, (3.7c)\n/summationdisplay\n|α|≤3e−|α|νt/vextenddouble/vextenddoubleeλ(t)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teνtη/an}b∇acket∇i}htσ0−6(∂α\nηˆf)0(t,η)/vextenddouble/vextenddouble\nL∞η≤4Cfe−νtǫνγ. (3.7d)\nand/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x≤(4C0ǫνγ)2, (3.7e)\n/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBM1/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x≤(4C1ǫνγ)2, (3.7f)\n/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBM2/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x≤(4C2ǫνγ)2, (3.7g)\n/ba∇dblBσ1ρ/ba∇dbl2\nL∞\ntL2x≤(4˜C0ǫνγ)2, (3.7h)\n/ba∇dblBσ1M1/ba∇dbl2\nL∞\ntL2x≤(4˜C1ǫνγ)2, (3.7i)\n/ba∇dblBσ1M2/ba∇dbl2\nL∞\ntL2x≤(4˜C2ǫνγ)2. (3.7j)\nHere C f,Cℓ,˜Cℓ,ℓ= 0,1,2 are positive constants independent of t,νfixed by the proof,\nsatisfying C ℓ<1\n2˜Cℓ.\nProposition 3.2. Letσ0,mbe the same as in Theorem 1.1. Assume that the bootstrap\nhypotheses (3.7)hold on [1,T∗]. Then there exist ǫ0>0andν0>0, such that for any\n0≤ν <ν0and0<ǫ<ǫ 0, the same estimates in (3.7)hold with the occurrences of 4 on the\nright-hand side replaced by 2.LANDAU DAMPING FOR VPFP 17\nTheorem 1.1 follows directly from Proposition 3.2. The rema inder of this paper is devoted\nto the proof of Proposition 3.2. Here we highlight the main id ea of the proof. For the\ndistribution function, we prove the Gr¨ onwall-type inequa lities:\n1\n2d\ndt/ba∇dblfw(t)/ba∇dbl2\nEσ0+1,2\n5m+/ba∇dblfw(t)/ba∇dbl2\nDσ0+1,2\n5m\n≤/parenleftbigg\nc0+C/ba∇dblfw(t)/ba∇dbl\nEσ1,2\n5\nm+2/parenrightbigg\n/ba∇dblfw(t)/ba∇dbl2\nDσ0+1,2\n5m+c0ǫ2ν2γ\n/an}b∇acketle{tt/an}b∇acket∇i}ht2,\n1\n2d\ndt/ba∇dblfw(t)/ba∇dbl2\nEσ1,1\n3m+/ba∇dblfw(t)/ba∇dbl2\nDσ1,1\n3m≤c0/ba∇dblfw(t)/ba∇dbl2\nDσ1,1\n3m+c0ǫ2ν2γ\n/an}b∇acketle{tt/an}b∇acket∇i}ht2,\n1\n2d\ndt/ba∇dblfw(t)/ba∇dbl2\nEσ1,2\n5\nm+2+/ba∇dblfw(t)/ba∇dbl2\nDσ1,2\n5\nm+2≤c0/ba∇dblfw(t)/ba∇dbl2\nDσ1,2\n5\nm+2+c0ǫ2ν2γ\n/an}b∇acketle{tt/an}b∇acket∇i}ht2,\nwherec0>0 is small enough under the bootstrap hypotheses, by choosin gǫ0,ν0,K|α|\nK|α|+1, and\na0small enough. In the energy estimate of /ba∇dblfw(t)/ba∇dbl2\nEσ0+1,2\n5m, we put/parenleftbigg\nc0+C/ba∇dblfw(t)/ba∇dbl\nEσ1,2\n5\nm+2/parenrightbigg\ninstead ofc0to emphasize that a higher moment is required, see (6.66) for more details. We\nrefer to section 6 for the improved estimates of the distribu tion function.\nFor the zero mode of the distribution function, we apply Duha mel’s principle by using\nthe explicit expression of the solution operator S0(t,0,η) corresponding to the Fokker-Planck\noperator.\nFor the density and higher moments, we use the Penrose’s iden tity and take advantage\nof the time-dependent Fourier multiplier. The improved est imates of density and higher\nmoments is in section 4.\n3.2.Proof of Theorem 1.2. The proof of Theorem 1.2 follows a similar energy estimate\nas proving Landau damping for the quasilinear Vlasov-Poiss on equation with a controllable\nforce. Letgδ=g−g0andEδ=E−E0, where (g,E) and (g0,E0) solve (1.12) with ν >0\nandν= 0 respectively. Then ( gδ,Eδ) satisfies\n(3.8)\n\n∂tgδ+v·∇xgδ+Eδ·∇vµ=−Eδ·∇vg0−E0·∇vgδ+ν(L[g]+C[g]+Cµ)\nEδ=−∇x(−∆x)−1ρδ, ρδ=/integraldisplay\nRngδ(t,x,v)dv.\nLet us take the Fourier transform in ( x,v) and define ˆfδ(t,k,η) = ˆgδ(t,k,η−kt). Let\nˆh(t,k,η) = ˆg(t,k,η−kt), then we get that /hatwideρδ(t,k) =ˆfδ(t,k,kt) andˆf(t,k,η) = ˆg(t,k,eνt(η−\nktap)) =ˆh(t,k,eνtη−k(tap−t)). The following lemma follows directly.\nLemma 3.3. Let/hatwiderf0(t,k,η) =/hatwideg0(t,k,η−kt). There isν0such that for any 0<ν <ν 0, and\n0≤t≤ν1\n3, it holds for s>1\n3that\n/ba∇dbleλ∞\n2/angb∇acketleftk,η/angb∇acket∇ights/an}b∇acketle{tk,η/an}b∇acket∇i}htσ0+1(/an}b∇acketle{t∇η/an}b∇acket∇i}htmˆh(t,k,η))/ba∇dbll2\nkL2η≤Cǫ,\n/ba∇dbleλ∞\n2/angb∇acketleftk,η/angb∇acket∇ights/an}b∇acketle{tk,η/an}b∇acket∇i}htσ0+1(/an}b∇acketle{t∇η/an}b∇acket∇i}htm/hatwideg0(t,k,η))/ba∇dbll2\nkL2η≤Cǫ,\nwhich gives that/vextenddouble/vextenddouble/vextenddoubleeλ∞\n2/angb∇acketleftk,η/angb∇acket∇ights/an}b∇acketle{tk,η/an}b∇acket∇i}htσ0+1/bracketleftbig\n/an}b∇acketle{t∇η/an}b∇acket∇i}htm/parenleftbig/hatwidestL[g]+/hatwidestC[g]+/hatwiderCµ/parenrightbig\n(t,k,η−kt)/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\nl2\nkL2η≤Cǫt2,\n/vextenddouble/vextenddouble/vextenddoubleeλ∞\n2/angb∇acketleftk,kt/angb∇acket∇ights/an}b∇acketle{tk,kt/an}b∇acket∇i}htσ0+1/hatwideρ0(t,k)/vextenddouble/vextenddouble/vextenddouble\nl2\nk≤Ce−cts.\nA direct calculation gives that ∂tˆfδ+Lδ\nk(η)+Rδ\nk(η)+Tδ\nk(η)+Fk(η) = 0, where\nLδ\nk(t,η) =i/hatwideEδ(k)(η−kt)ˆµ(η−kt),18 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nRδ\nk(t,η) =i/summationdisplay\nl/hatwideEδ(l)(η−(k−l)t)/hatwiderh0(t,k−l,η),\nTδ\nk(t,η) =i/summationdisplay\nl/hatwiderE0(k−l)(η−lt)/hatwidefδ(t,l,η),\nFk(t,η) =/parenleftbig/hatwidestL[g]+/hatwidestC[g]+/hatwiderCµ/parenrightbig\n(t,k,η−kt).\nWe also get that\n/hatwideρδ(t,k) =−/integraldisplayt\n0/parenleftbig\nLδ\nk(τ,kt)+Rδ\nk(τ,kt)+Tδ\nk(τ,kt)+Fk(τ,kt)/parenrightbig\ndτ.\nWe then define the following energy functionals:\n/ba∇dblfδ/ba∇dblE=/ba∇dbleλ3(t)/angb∇acketleft∇x,v/angb∇acket∇ights/an}b∇acketle{t∇x,v/an}b∇acket∇i}htσ0+1(/an}b∇acketle{tv/an}b∇acket∇i}htmfδ)/ba∇dblL2,\n/ba∇dblfδ/ba∇dblElow=/ba∇dbleλ3(t)/angb∇acketleft∇x,v/angb∇acket∇ights/an}b∇acketle{t∇x,v/an}b∇acket∇i}htσ1(/an}b∇acketle{tv/an}b∇acket∇i}htmfδ)/ba∇dblL2,\nwithλ3(t) =λ∞/3+1\n100(1+t)aλ∞. By following the same argument, we obtain that there is\nǫ1>0 independent of ν,t, such that for any t≤ǫ1ν−1\n3, it holds that\n/ba∇dblfδ/ba∇dblE+/ba∇dblfδ/ba∇dblElow≤Cνt3,\n/an}b∇acketle{tt/an}b∇acket∇i}htb/ba∇dbleλ3(t)/angb∇acketleft∇k,kt/angb∇acket∇ights/an}b∇acketle{tk,kt/an}b∇acket∇i}htσ0|k|1\n2/hatwideρδ/ba∇dblL2\ntl2\nk≤Cνt3.\nWe omit the proof and point out that the estimate of Rδ\nk(t,η) is similar to the treatments\nofNHL\nE;/negationslash=,NHL\nE;0, and NHL\nM, the estimate of Tδ\nk(t,η) is similar to the treatments of NLH\nE;/negationslash=,NLH\nE;0,\nand NLH\nM, and the treatment of the linear part Lδ\nk(t,η) is the same as L.\n3.3.Some estimates on Mθ.The following estimates for Mθderived from the bootstrap\nassumptions (3.7) will be frequently used in the remainder o f this paper.\nProposition 3.4. Letm>n\n2+4. Under the bootstrap hypotheses, it holds that\ne−mνt/vextenddouble/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0Aσ0+1,2\n5\nk(t,ktap)(/hatwiderMθ)k(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk/lessorsimilar/ba∇dblfw\n/negationslash=(t)/ba∇dbl\nEσ0+1,2\n5m, (3.9)\n/ba∇dblBMθ/ba∇dblL2x/lessorsimilar/ba∇dblBM2/ba∇dblL2x+/ba∇dblBM1/ba∇dblL2x+/ba∇dblBρ/ba∇dblL2x, (3.10)\n/ba∇dblBσ1(Mθ)/negationslash=/ba∇dblL2x/lessorsimilar/ba∇dblBσ1(M2)/negationslash=/ba∇dblL2x+/ba∇dblBσ1M1/ba∇dblL2x+/ba∇dblBσ1ρ/ba∇dblL2x, (3.11)\n|(Mθ)0(t)|/lessorsimilarǫ2ν2γe−2δν1\n3t1\n/an}b∇acketle{ttap/an}b∇acket∇i}ht2σ1. (3.12)\nProof.From (1.10), we get\nMθ=1\nnM2−1\nn∞/summationdisplay\nj=0(−1)jρj|M1|2−ρ. (3.13)\nRecalling (2.16)–(2.18), one deduces that\n(/hatwiderMθ)k(t) =−1\nne−2νt∆ηˆfk(t,ktap)−ˆfk(t,ktap)−1\nn∞/summationdisplay\nj=0(−1)jF/bracketleftbig\nρj|M1|2/bracketrightbig\nk(t).\nThen\nAσ0+1,2\n5\nk(t,ktap)(/hatwiderMθ)k(t) =−1\nne−2νt(Aσ0+1,2\n5∆ηˆf)k(t,ktap)−(Aσ0+1,2\n5ˆf)k(t,ktap)\n−1\nn∞/summationdisplay\nj=0(−1)jAσ0+1,2\n5\nk(t,ktap)F/bracketleftbig\nρj|M1|2/bracketrightbig\nk(t). (3.14)LANDAU DAMPING FOR VPFP 19\nNote that\nAσ0+1,2\n5\nk(t,ktap) =eδ1ν2\n5tBσ0+1\nk(t),fork/ne}ationslash= 0,\nwhereBσ0+1\nk(t) is introduced in Lemma A.4 in Section A. Then in view of (A.11 ), and using\n(2.16), (2.17) and (3.4), we find that\n∞/summationdisplay\nj=0/vextenddouble/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5\nk(t,ktap)F/bracketleftbig\nρj|M1|2/bracketrightbig\nk(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk\n/lessorsimilareδ1ν2\n5t/vextenddouble/vextenddouble/vextenddoubleBσ0+1\nk(t)F[M1]k(t)/vextenddouble/vextenddouble/vextenddouble2\nL2\nk\n1+∞/summationdisplay\nj=1/vextenddouble/vextenddouble/vextenddoubleBσ0+1\nk(t)F/bracketleftbig\nρj/bracketrightbig\nk(t)/vextenddouble/vextenddouble/vextenddouble\nL2\nk\n\n/lessorsimilare−δ1ν2\n5t/vextenddouble/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5\nk(t,ktap)F[M1]k(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2\nk\n1+∞/summationdisplay\nj=1Cj−1\n0/vextenddouble/vextenddouble/vextenddoubleBσ0+1\nk(t)ˆρk(t)/vextenddouble/vextenddouble/vextenddoublej\nL2\nk\n\n/lessorsimilare−δ1ν2\n5te−2νt/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0(Aσ0+1,2\n5∇ηˆf)k(t,ktap)/vextenddouble/vextenddouble/vextenddouble2\nL2\nk\n×\n1+∞/summationdisplay\nj=1/parenleftBigg\nC0e−δ1ν2\n5t/vextenddouble/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0(Aσ0+1,2\n5\nkˆf)k(t,ktap)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk/parenrightBiggj\n\n/lessorsimilare−δ1ν2\n5te−2νt/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0(Aσ0+1,2\n5∇ηˆf)k(t,η)/vextenddouble/vextenddouble/vextenddouble2\nL2\nkL∞η\n×\n1+∞/summationdisplay\nj=1/parenleftBigg\nC0e−δ1ν2\n5t/vextenddouble/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0(Aσ0+1,2\n5\nkˆf)k(t,η)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nkL∞η/parenrightBiggj\n\n/lessorsimilar/ba∇dblfw\n/negationslash=(t)/ba∇dbl2\nEσ0+1,2\n5m\n1+∞/summationdisplay\nj=1(C0/ba∇dblfw\n/negationslash=(t)/ba∇dbl\nEσ0+1,2\n5m)j\n/lessorsimilar/ba∇dblfw\n/negationslash=(t)/ba∇dbl2\nEσ0+1,2\n5m, (3.15)\nwhereC0is the constant appearing (A.11). It follows from (3.14) and (3.15) that\ne−mνt/vextenddouble/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0Aσ0+1,2\n5\nk(t,ktap)(/hatwiderMθ)k(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk\n/lessorsimilare−(m+2)νt/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0(Aσ0+1,2\n5∆ηˆf)k(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2\nkHn\n2+\nη\n+e−mνt/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0(Aσ0+1,2\n5ˆf)k(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2\nkHn\n2+\nη+/ba∇dblfw\n/negationslash=(t)/ba∇dbl2\nEσ0+1,2\n5m.\nArguing as (3.4) again, we obtain (3.9) immediately.\nNext, we turn to prove (3.10). One can see from (3.13) that\n(3.16) /ba∇dblBMθ/ba∇dblL2x/lessorsimilar/ba∇dblBM2/ba∇dblL2x+/ba∇dblBρ/ba∇dblL2x+∞/summationdisplay\nj=0/vextenddouble/vextenddoubleBk(t)F/bracketleftbig\nρj|M1|2/bracketrightbig\nk(t)/vextenddouble/vextenddouble\nL2\nk.\nIt suffices to bound the last term on the right-hand side of the a bove inequality. Clearly,\nBk(t)≤eδν1\n3t/an}b∇acketle{tk/an}b∇acket∇i}ht1\n2Bσ0\nk(t) and /an}b∇acketle{tk/an}b∇acket∇i}ht1\n2Bσ0\nk(t)≤Bσ0+1\nk(t).\nThen using (A.12), similar to (3.15), and recalling that ( M1)0= 0, we have\n∞/summationdisplay\nj=0/vextenddouble/vextenddoubleBk(t)F/bracketleftbig\nρj|M1|2/bracketrightbig\nk(t)/vextenddouble/vextenddouble\nL2\nk20 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n/lessorsimilareδν1\n3t/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tk/an}b∇acket∇i}ht1\n2Bσ0\nk(t)(/hatwiderM1)k(t)/vextenddouble/vextenddouble/vextenddouble2\nL2\nk\n1+∞/summationdisplay\nj=1˜Cj−1\n0/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tk/an}b∇acket∇i}ht1\n2Bσ0\nk(t)ˆρk(t)/vextenddouble/vextenddouble/vextenddoublej\nL2\nk\n\n/lessorsimilar/ba∇dblBM1/ba∇dblL2x/parenleftbigg\ne−δ1ν2\n5te−νt/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0(Aσ0+1,2\n5∇ηˆf)k(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2\nkL∞η/parenrightbigg\n(3.17)\n×\n1+∞/summationdisplay\nj=1/parenleftbigg\n˜C0e−δ1ν2\n5t/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0(Aσ0+1,2\n5ˆf)k(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2\nkL∞η/parenrightbiggj\n/lessorsimilar/ba∇dblBM1/ba∇dblL2x.\nSubstituting (3.17) into (3.16) yields (3.10). Similarly, the inequality (3.11) can be obtained\nby using (A.11) instead of (A.12). We omit the details for bre vity.\nTo prove (3.12), from (1.8)–(1.10), we find that\n|(Mθ)0(t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(M2)0(t)−1\nn/parenleftbig\n(1+ρ)|u|2/parenrightbig\n0(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/ba∇dblE(t)/ba∇dbl2\nL2\n(2π)n+1\n(2π)n/integraldisplay\nTn|M1|2−|M1|2ρ\n1+ρdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddoubleˆρk(t)\n|k|/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2\nk+/ba∇dblM1(t)/ba∇dbl2\nL2x/parenleftBigg\n1+/vextenddouble/vextenddouble/vextenddouble/vextenddoubleρ\n1+ρ/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL∞x/parenrightBigg\n/lessorsimilar/ba∇dblρ(t)/ba∇dbl2\nL2+/ba∇dblM1(t)/ba∇dbl2\nL2. (3.18)\nThen (3.12) follows immediately. The proof of Proposition 3 .4 is completed. /square\n4.Density estimates\nIn this section, we improve (3.7e) and (3.7h). Due to the appe arance offin the equation\nofρ, see (2.24), we will use Remark 3.1 frequently to recover the estimates of ffromfw.\nBefore proceeding any further, let us consider the followin g linear Volterra equation:\nˆρ(t,k) =Q(t,k)+/integraldisplayt\n0Kν(t−τ,k)ˆρ(τ,k)dτ, (4.1)\nwithKν(t,k) =−Sk(t)tapˆµ(ktap),t≥0.Note that when νt≤1, ˆµ(ktap) decays faster than\nSk(t) and when νt>1,Sk(t) decays faster than ˆ µ(ktap). Then by (2.21) and (B.2), it is not\ndifficult to verify that there exists a sufficiently small δ2>0 independent of ν, such that\n|Kν(t,k)|/lessorsimilar1\n|k|e−δ2|k|t, (4.2)\n|∂tKν(t,k)|/lessorsimilare−δ2|k|t,|∂ttKν(t,k)|/lessorsimilar|k|e−δ2|k|t. (4.3)\nLet/tildewiderKν(z,k) be the Fourier-Laplace transform of Kν(t,k) with respect to the tvariable:\n/tildewiderKν(z,k) =1\n2π/integraldisplay∞\n0e−ztKν(t,k)dt. (4.4)\nWe have the following estimate of the Fourier-Laplace trans form ofKν(t,k) int, which\nguarantees the Penrose stability condition.\nLemma 4.1. For allν >0sufficiently small, there holds that\ninf\nz∈C:Rez>−cδ2|k|/vextendsingle/vextendsingle/vextendsingle1−/tildewiderKν(z,k)/vextendsingle/vextendsingle/vextendsingle≥κ>0.\nwhereκ>0,c∈(0,1)can be taken independent of ν >0.LANDAU DAMPING FOR VPFP 21\nThe proof of Lemma 4.1 is standard. We highlight the main idea here. On one hand, by\nthe point-wise decay estimate (4.2), we have\n(4.5) |/tildewiderKν(z,k)|/lessorsimilar|k|−2,for allRez >−cδ2|k|.\nHence, we only need to focus on the small wave number case, whi ch is finite. On the other\nhand, by taking the limit ν→0+, we have lim ν→0+Kν(t,k) =−tˆµ(kt) =:K0(t,k), whose\nFourier-Laplace transform satisfies\ninf\nz∈C:Rez>−cδ2|k|/vextendsingle/vextendsingle/vextendsingle1−/tildewiderK0(z,k)/vextendsingle/vextendsingle/vextendsingle≥2κ>0.\nThe lemma then follows immediately from the fact that /tildewiderKν(z,k) is a continuous function in\nν, which is a direct conclusion of the Lebesgue-dominated con vergence theorem. We now\nhave the estimate for the linearized Volterra equation in Ge vrey norms:\nLemma 4.2. For0<δ≪1and allνsufficiently small, we have\n/ba∇dbl/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/ba∇dblL2\nt,x/lessorsimilar/ba∇dbl/an}b∇acketle{tt/an}b∇acket∇i}htbBQ/ba∇dblL2\nt,x.\nProof.We use the idea in [3, 15] to treat the linearized Voltterra eq uation. First, by applying\nintegration by part twice in (4.4), and applying (4.3), toge ther with (4.5), we get that\n|/tildewiderKν(z,k)|/lessorsimilar1\n|k|2+|Imz|2,for allRez >−cδ2|k|. (4.6)\nThanks to Lemma D in [3], we get from Lemma 4.1 and (4.6) that\n/tildewideˆρ(ω,k) =/tildewideQ(ω,k)+H(ω,k)/tildewideQ(ω,k). (4.7)\nwhereH(ω,k) =/tildewiderKν(ω,k)\n1−/tildewiderKν(ω,k). LetR(t,k) be the inverse Fourier-Laplace transform of H(w,k).\nBy using (4.6), we get that |R(t,k)|/lessorsimilar1\n|k|e−δ3|k|tfor someδ3>0 small enough, and then\nˆρ(t,k) =Q(t,k)+/integraldisplayt\n0R(t−τ,k)Q(τ,k)dτ. (4.8)\nApplying the operator /an}b∇acketle{tt/an}b∇acket∇i}htbBto (4.8), and taking νsufficiently small, the exponential decay\nof the kernel R(t,k) enables us to use Young’s inequality to complete the proof o f this lemma\nimmediately. /square\n4.1.L2\nt,xestimate on ρ.TakingQ(t,k) =Sk(t)(/hatwiderfin)k(ktap)+/integraltextt\n0Sk(t−τ)Nk(τ,k(t−τ)ap)dτ,\nin (4.1), by Lemma 4.2, we have\n/ba∇dbl/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/ba∇dbl2\nL2\nt,x/lessorsimilarI+NE+Nµ+Ng, (4.9)\nwhere\nI=/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t)/hatwidergin(k,ktap)/vextendsingle/vextendsingle/vextendsingle2\ndt,\nNE=/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)/integraldisplayt\n0Sk(t−τ)/summationdisplay\nl∈Zn∗ˆρl(τ)l\n|l|2·k(t−τ)apˆfk−l(τ,ktap−lτap)dτ/bracketrightBigg2\ndt,\nNµ=ν2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftbigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)/integraldisplayt\n0Sk(t−τ)(/hatwiderCµ)k(τ,k(t−τ)ap)dτ/bracketrightbigg2\ndt,\nNg=ν2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftbigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)/integraldisplayt\n0Sk(t−τ)(/hatwidestC[g])k(τ,k(t−τ)ap)dτ/bracketrightbigg2\ndt.22 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nRecalling (1.12c),/hatwidestC[g] consists of the following four parts/hatwidestC[g] :=/hatwiderC1g+/hatwiderC2g+/hatwiderC3g+/hatwiderC4g,with\n(/hatwiderC1g)k(τ,k(t−τ)ap) =−/summationdisplay\nl∈Zn∗ˆρl(τ)|k(t−τ)ap|2ˆfk−l(τ,ktap−lτap), (4.10a)\n(/hatwiderC2g)k(τ,k(t−τ)ap) =−/summationdisplay\nl∈Zn∗ˆρl(τ)k(t−τ)ap·e−ντ(∇ηˆf)k−l(τ,ktap−lτap), (4.10b)\n(/hatwiderC3g)k(τ,k(t−τ)ap) =−/summationdisplay\nl∈Zn(/hatwiderMθ)l(τ)|k(t−τ)ap|2ˆfk−l(τ,ktap−lτap), (4.10c)\n(/hatwiderC4g)k(τ,k(t���τ)ap) =−i/summationdisplay\nl∈Zn∗(/hatwiderM1)l(τ)k(t−τ)apˆfk−l(τ,ktap−lτap). (4.10d)\n4.1.1.Iinitial contribution. Let us first treat I, the contribution from the initial data. Note\nthat\n(4.11) t=νt\n1−e−νttap/lessorsimilar/an}b∇acketle{tνt/an}b∇acket∇i}httap.\nCombining this with (B.2), taking the change of variable\n(4.12) t′=|k|tap,\nand then using the Sobolev trace theorem (see Lemma 3.4 in [6] ), thanks to (3.4), we find\nthat\nI/lessorsimilar/summationdisplay\nk∈Zn∗|k|/integraldisplayT∗\n0/an}b∇acketle{tνt/an}b∇acket∇i}ht2be2δν1\n3tS2\nk(t)e2λ(0)/angb∇acketleftk,ktap/angb∇acket∇ights/an}b∇acketle{tk,ktap/an}b∇acket∇i}ht2(σ0+b)|/hatwidergin(k,ktap)|2dt\n/lessorsimilar/summationdisplay\nk∈Nn∗sup\n|ω|=1/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsingleF/bracketleftBig\neλ(0)/angb∇acketleft∇/angb∇acket∇ights/an}b∇acketle{t∇/an}b∇acket∇i}htσ0+bgin(k,ωt′)/bracketrightBig/vextendsingle/vextendsingle/vextendsingle2\ndt′\n/lessorsimilar/vextenddouble/vextenddouble/vextenddoubleF/bracketleftBig\neλ(0)/angb∇acketleft∇/angb∇acket∇ights/an}b∇acketle{t∇/an}b∇acket∇i}htσ0+bgin/bracketrightBig\n(k,η)/vextenddouble/vextenddouble/vextenddouble2\nL2ηHn−1\n2+\nη/lessorsimilar/vextenddouble/vextenddouble/vextenddoubleeλ(0)/angb∇acketleft∇/angb∇acket∇ights/an}b∇acketle{t∇/an}b∇acket∇i}htσ0+b(/an}b∇acketle{tv/an}b∇acket∇i}htmgin)/vextenddouble/vextenddouble/vextenddouble2\nL2/lessorsimilarǫ2ν2γ.\n4.1.2.NEcollisionless contributions. In this section, we treat NE. This is the term that is\ndiscussed in section 1.2.2 and determines the relationship betweensandγ. According to\nwhetherfis at zero frequency or not, we divide NEinto two parts:\nNE=NE;/negationslash=+NE;0.\nTreatment ofNE;/negationslash=.We split NE;/negationslash=into high-low and low-high interactions and write\nNE;/negationslash==NHL\nE;/negationslash=+NLH\nE;/negationslash=,\nwhere the above decomposition is given according to the foll owing partition of unity:\n(4.13) 1 = 1|k−l,ktap−lτap|≤2|l,lτap|+1|l,lτap|<1\n2|k−l,ktap−lτap|.\nTo bound NHL\nE;/negationslash=, on the one hand, in view of (A.5) and (B.2), we findthat for som e 0<c<1,\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)1|k−l,ktap−lτap|≤2|l,lτap|\n/lessorsimilarS3\n4\nk(t−τ)Bl(τ)/an}b∇acketle{tt/an}b∇acket∇i}htbe(λ(t)−λ(τ))/angb∇acketleftk,ktap/angb∇acket∇ightsecλ(τ)/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights/an}b∇acketle{tk−l/an}b∇acket∇i}ht1\n2. (4.14)\nWe also have for k/ne}ationslash=lthat\n|k|(t−τ)ap/lessorsimilareντ/an}b∇acketle{tτap/an}b∇acket∇i}ht|k−l,ktap−lτap|. (4.15)\nTherefore,\nNHL\nE;/negationslash=/lessorsimilar/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg/summationdisplay\nl∈Zn∗,l/negationslash=k/integraldisplayt\n0S3\n4\nk(t−τ)|Bˆρl(τ)|/an}b∇acketle{tt/an}b∇acket∇i}htbLANDAU DAMPING FOR VPFP 23\n×e(λ(t)−λ(τ))/angb∇acketleftk,ktap/angb∇acket∇ights\n|l|ecλ(τ)/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights\n×eντ/an}b∇acketle{tτ/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingleF/bracketleftBig\n/an}b∇acketle{t∇/an}b∇acket∇i}ht3\n2f/bracketrightBig\nk−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle/vextendsingledτ/bracketrightBigg2\ndt. (4.16)\nBy using the Sobolev embedding (3.4), we get that\neντ/an}b∇acketle{tτ/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingleF/bracketleftBig\n/an}b∇acketle{t∇/an}b∇acket∇i}ht3\n2f/bracketrightBig\nk−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1l/negationslash=k\n=/an}b∇acketle{tτ/an}b∇acket∇i}ht1−3γ/bracketleftbigg\neντ/parenleftBig\nν1\n3/an}b∇acketle{tτ/an}b∇acket∇i}ht/parenrightBig3γ\nemντe−δ1ν1\n3τ/bracketrightbigg\ne−λ(τ)/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights\n×ν−γe−mντeδ1ν1\n3τ/vextendsingle/vextendsingle/vextendsingle/vextendsingleF/bracketleftBig\neλ(τ)/angb∇acketleft∇/angb∇acket∇ights/an}b∇acketle{t∇/an}b∇acket∇i}ht3\n2f/bracketrightBig\nk−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1l/negationslash=k\n/lessorsimilare−3\n4δ1ν1\n3τ/an}b∇acketle{tτ/an}b∇acket∇i}ht1−3γe−λ(τ)/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights\n×ν−γsup\nt/parenleftbigg\ne−mνt/vextenddouble/vextenddouble/vextenddoubleAσ1,1\n3f/negationslash=(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2\nkHn\n2+\nη/parenrightbigg\n/lessorsimilare−3\n4δ1ν1\n3τ/an}b∇acketle{tτ/an}b∇acket∇i}ht1−3γe−λ(τ)/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights/parenleftbigg\nν−γsup\nt/vextenddouble/vextenddoublefw\n/negationslash=/vextenddouble/vextenddouble\nEσ1,1\n3m/parenrightbigg\n/lessorsimilarǫe−3\n4δ1ν1\n3τ/an}b∇acketle{tτ/an}b∇acket∇i}ht1−3γe−λ(τ)/angb∇acketleftk���l,ktap−lτap/angb∇acket∇ights. (4.17)\nSubstituting this into (4.16), we find that\nNHL\nE;/negationslash=/lessorsimilarǫ2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg/summationdisplay\nl∈Zn∗/integraldisplayt\n0S3\n4\nk(t−τ)|Bˆρl(τ)|/an}b∇acketle{tt/an}b∇acket∇i}htbe(λ(t)−λ(τ))/angb∇acketleftk,ktap/angb∇acket∇ights\n|l|\n×e−3\n4δ1ν1\n3τ/an}b∇acketle{tτ/an}b∇acket∇i}ht1−3γe−c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ightsdτ/bracketrightBigg2\ndt, (4.18)\nwherec= (1−c)λ∞. To treat the right-hand side of (4.18), we consider two case s.\nCase 1:τ≤t\n2.A direct calculation gives that\n(4.19) λ(τ)−λ(t) =˜δ\n(1+τ)a−˜δ\n(1+t)a/greaterorsimilart−τ\n(1+τ)a(1+t),for allτ <t.\nCombining this with (4.11), (B.2) and the fact t≤2(t−τ) yields, for Γ ≥b+1\ns−a,\nS1\n4\nk(t−τ)/an}b∇acketle{tt/an}b∇acket∇i}htb/an}b∇acketle{tτ/an}b∇acket∇i}ht1−3γe(λ(t)−λ(τ))/angb∇acketleftk,ktap/angb∇acket∇ights\n|l|1k/negationslash=0/lessorsimilarS1\n4\nk(t−τ)/an}b∇acketle{tt/an}b∇acket∇i}htb+1−3γ+aΓ\n/an}b∇acketle{tk,ktap/an}b∇acket∇i}htsΓ1k/negationslash=0\n/lessorsimilar/an}b∇acketle{tν(t−τ)/an}b∇acket∇i}htb+1+aΓS1\n4\nk(t−τ)/an}b∇acketle{ttap/an}b∇acket∇i}htb+1−3γ+aΓ\n/an}b∇acketle{tk,ktap/an}b∇acket∇i}htsΓ1k/negationslash=0/lessorsimilar1. (4.20)\nSubstituting this into (4.18), and using Schur’s test, we ar e led to\nNHL\nE;/negationslash=/lessorsimilarǫ2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg/summationdisplay\nl∈Zn∗/integraldisplayt\n0S1\n2\nk(t−τ)e−1\n2δ1ν1\n3τe−c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights|Bˆρl(τ)|dτ/bracketrightBigg2\ndt\n/lessorsimilarǫ2/ba∇dblBρ/ba∇dbl2\nL2\nt,x\nsup\ntsup\nk/negationslash=0/summationdisplay\nl∈Zn∗/integraldisplayt\n0K1(t,τ,k,l)dτ\n\nsup\nτsup\nl/negationslash=0/summationdisplay\nk∈Zn∗/integraldisplayT∗\nτK1(t,τ,k,l)dt\n, (4.21)24 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nwhereK1(t,τ,k,l) =S1\n2\nk(t−τ)e−1\n2δ1ν1\n3τe−c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights.Taking the change of variable\n(4.22) τ′=|l|τap,\nand using (A.1), we obtain\nsup\nk/negationslash=0/summationdisplay\nl∈Zn∗/integraldisplayt\n0K1(t,τ,k,l)dτ/lessorsimilarsup\nk/summationdisplay\nl∈Zn∗e−csc/angb∇acketleftk−l/angb∇acket∇ights/integraldisplay\nRe−csc/angb∇acketleftktap−l\n|l|τ′/angb∇acket∇ightsdτ′/lessorsimilar1.\nFor the other kernel estimate, the only difference is that we ne ed to use (B.2) to absorb the\nJacobianeνtarising from the change of variable (4.12). Thus, for τ≤t\n2, we arrive at\n(4.23) NHL\nE;/negationslash=/lessorsimilarǫ2/ba∇dblBρ/ba∇dbl2\nL2\nt,x,\nwhich is consistent with Proposition 3.2 provided we set ǫsufficiently small.\nCase 2:t\n2≤τ≤t.The dimensional reducing process in [6, 43] enables us to con sider\nonly the case n= 1. Furthermore, by symmetry, it suffices to investigate the c asek≥1. If\nl<0 or 0≤l<k, we have\n|ktap−lτap| ≥τap.\nThis, together with (4.11), implies that\ne−3\n4δ1ν1\n3τ/an}b∇acketle{tτ/an}b∇acket∇i}ht1−3γe−c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights\n/lessorsimilare−3\n4δ1ν1\n3τ/an}b∇acketle{tντ/an}b∇acket∇i}ht1−3γ/an}b∇acketle{tτap/an}b∇acket∇i}ht1−3γe−1\n2c/angb∇acketleftτap/angb∇acket∇ightse−1\n2c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights\n/lessorsimilare−1\n2δ1ν1\n3τe−1\n2c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights. (4.24)\nThen similar to (4.21), we have\n(4.25) NHL\nE;/negationslash=/lessorsimilarǫ2/ba∇dbl/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/ba∇dbl2\nL2\nt,x.\nFor the remaining case l≥k+1, if|ktap−lτap| ≥tap\n2, then (4.24) still holds. Therefore, it\nsuffices to consider the case\n(4.26) |ktap−lτap| ≤tap\n2withl≥k+1.\nThis in turn shows that\ntap−τap=tap−ktap\nl+ktap\nl−τap=(l−k)tap\nl+ktap−lτap\nl≥tap\n2l.\nWe havetap−τap=e−ντ(t−τ)ap≤e−ντ(t−τ), which together with (4.19) gives\n(4.27) λ(τ)−λ(t)/greaterorsimilare−ν(t−τ)eνttap\n/an}b∇acketle{tτ/an}b∇acket∇i}hta(1+t)|l|/greaterorsimilare−ν(t−τ)\n/an}b∇acketle{tτ/an}b∇acket∇i}hta|l|.\nHence, similar to (4.20), for a fixed large constant Γ 0satisfying\n(4.28) aΓ0+1−3γ=sΓ0,\nthere holds\nS1\n4\nk(t−τ)e−1\n4δ1ν1\n3τ/an}b∇acketle{tτ/an}b∇acket∇i}ht1−3γe(λ(t)−λ(τ))/angb∇acketleftk,ktap/angb∇acket∇ights1k/negationslash=0\n/lessorsimilarΓ0S1\n4\nk(t−τ)e−1\n4δ1ν1\n3τeΓ0ν(t−τ)taΓ0+1−3γ|l|Γ0\n/an}b∇acketle{tk,ktap/an}b∇acket∇i}htsΓ01k/negationslash=0/lessorsimilar|l|Γ0−sΓ0/an}b∇acketle{tk−l/an}b∇acket∇i}htsΓ0. (4.29)\nSubstituting (4.29) into (4.18) and noting that /an}b∇acketle{tt/an}b∇acket∇i}htb≈ /an}b∇acketle{tτ/an}b∇acket∇i}htb, using Schur’s test again, we have\nNHL\nE;/negationslash=/lessorsimilarǫ2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg/summationdisplay\nl∈Zn∗/integraldisplayt\n0/parenleftBig\n/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρl(τ)|/parenrightBig\n|l|Γ0−1−sΓ0S1\n2\nk(t−τ)LANDAU DAMPING FOR VPFP 25\n×e−δ1\n2ν1\n3τe−1\n2c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ightsdτ/bracketrightBigg2\ndt\n/lessorsimilarǫ2/ba∇dbl/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/ba∇dbl2\nL2\nt,x\nsup\ntsup\nk/negationslash=0/summationdisplay\nl/negationslash=0/integraldisplayt\n0K2(t,τ,k,l)dτ\n\n×\nsup\nτsup\nl/negationslash=0/summationdisplay\nk/negationslash=0/integraldisplayT∗\nτK2(t,τ,k,l)dt\n, (4.30)\nwhereK2(t,τ,k,l) =|l|Γ0−1−sΓ0S1\n2\nk(t−τ)e−δ1\n2ν1\n3τe−1\n2c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights.Bythechangeofvariable\n(4.22) and (A.1),\n/summationdisplay\nl/negationslash=0/integraldisplayt\n0K2(t,τ,k,l)dτ/lessorsimilar/summationdisplay\nl/negationslash=0|l|Γ0−2−sΓ0/integraldisplayt\n0/parenleftbigg\ne−δ1\n2ν1\n3τeντ/parenrightbigg\ne−1\n2c/angb∇acketleftk−l,ktap−l\n|l|τ′/angb∇acket∇ightsdτ′\n/lessorsimilar/summationdisplay\nl/negationslash=0e−1\n2csc/angb∇acketleftk−l/angb∇acket∇ights/integraldisplayt\n0e−1\n2csc/angb∇acketleftktap−l\n|l|τ′/angb∇acket∇ightsdτ′/lessorsimilar1,\nprovided\n(4.31) Γ 0−2−sΓ0≤0.\nSimilarly, using the change of variable (4.12), (A.1) and (B .2), we obtain\n/summationdisplay\nk/negationslash=0/integraldisplayt\n0K2(t,τ,k,l)dτ/lessorsimilar/summationdisplay\nk/negationslash=0|k|Γ0−2−sΓ0/integraldisplayt\n0/parenleftbigg\nS1\n2\nk(t−τ)e−δ1\n2ν1\n3τeνt/parenrightbigg\n×/an}b∇acketle{tk−l/an}b∇acket∇i}htΓ0−1−sΓ0e−1\n2c/angb∇acketleftk−l,k\n|k|t′−lτap/angb∇acket∇ightsdt′\n/lessorsimilar/summationdisplay\nk/negationslash=0/an}b∇acketle{tk−l/an}b∇acket∇i}hte−1\n2csc/angb∇acketleftk−l/angb∇acket∇ights/integraldisplayt\n0e−1\n2csc/angb∇acketleftk\n|k|t′−lτap/angb∇acket∇ightsdt′/lessorsimilar1.\nCombining (4.28) with (4.31) yields\n(4.32) s≥1−3γ\n3−3γ+2a\n3−3γ>1−3γ\n3−3γ.\nNow we turn to bound NLH\nE;/negationslash=. Ifτ≤t\n2, similar to (4.20), we have\nS1\n4\nk(t−τ)/an}b∇acketle{tt/an}b∇acket∇i}htbe(λ(t)−λ(τ))/angb∇acketleftk,ktap/angb∇acket∇ights1k/negationslash=0/lessorsimilar1. (4.33)\nIft\n2≤τ≤t,\n(4.34) S1\n4\nk(t−τ)/an}b∇acketle{tt/an}b∇acket∇i}htbe(λ(t)−λ(τ))/angb∇acketleftk,ktap/angb∇acket∇ights1k/negationslash=0/lessorsimilar/an}b∇acketle{tτ/an}b∇acket∇i}htb.\nCombining these two inequalities with (A.3), (B.2) and (4.1 5), we have\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)|k|(t−τ)ap1|l,lτap|<1\n2|k−l,ktap−lτap|1k/negationslash=l\n/lessorsimilar|k|1\n2S1\n2\nk(t−τ)e−1\n2δ1ν2\n5τ/parenleftbigg\n/an}b∇acketle{tτ/an}b∇acket∇i}htb/an}b∇acketle{tτap/an}b∇acket∇i}hteδν1\n3τecλ(τ)/angb∇acketleftl,lτap/angb∇acket∇ights/parenrightbigg\n×Aσ0+1,2\n5\nk−l(τ,ktap−lτap)1k/negationslash=l. (4.35)\nConsequently, using Cauchy-Schwarz yields\nNLH\nE;/negationslash=/lessorsimilar/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗|k|/bracketleftBigg/summationdisplay\nl∈Zn∗,l/negationslash=k/integraldisplayt\n0S1\n2\nk(t−τ)e−1\n4δ1ν2\n5τe−c/angb∇acketleftl,lτap/angb∇acket∇ights/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρl(τ)|26 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n×/vextendsingle/vextendsingle/vextendsinglee−mντ(Aσ0+1,2\n5ˆf)k−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingledτ/bracketrightBigg2\ndt\n/lessorsimilar/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗|k|/summationdisplay\nl∈Zn∗,l/negationslash=k/integraldisplayt\n0e−1\n4δ1ν2\n5τe−c/angb∇acketleftl,lτap/angb∇acket∇ights/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρl(τ)|dτ\n×/summationdisplay\nl∈Zn∗,l/negationslash=k/integraldisplayt\n0Sk(t−τ)e−1\n4δ1ν2\n5τe−c/angb∇acketleftl,lτap/angb∇acket∇ights/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρl(τ)|\n×/vextendsingle/vextendsingle/vextendsinglee−mντ(Aσ0+1,2\n5ˆf)k−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle2\ndτdt. (4.36)\nBy (A.1), one deduces that\n/summationdisplay\nl∈Zn∗/integraldisplayt\n0e−1\n4δ1ν2\n5τe−c/angb∇acketleftl,lτap/angb∇acket∇ights/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρl(τ)|dτ\n/lessorsimilar/summationdisplay\nl∈Zn∗/integraldisplayt\n0e−1\n4δ1ν2\n5τe−csc/angb∇acketleftl/angb∇acket∇ightse−csc/angb∇acketleftlτap/angb∇acket∇ights/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρl(τ)|dτ\n/lessorsimilar\n/summationdisplay\nl∈Zn∗e−2csc/angb∇acketleftl/angb∇acket∇ights/integraldisplayt\n0e−1\n2δ1ν2\n5τe−2csc/angb∇acketleftlτap/angb∇acket∇ightsdτ\n1\n2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble\nL2\nt,x/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble\nL2\nt,x, (4.37)\nwhere we have used the change of variable (4.22). Substituti ng this into (4.36), using Fubini’s\ntheorem, noting that (B.2) implies that Sk(t−τ)/lessorsimilare−ν(t−τ)for sufficiently small ν, and using\nthe change of variable (4.12), the standard Sobolev trace th eorem, see Lemma 3.4 in [6], for\ninstance, and (3.4), we find that\nNLH\nE;/negationslash=/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble\nL2\nt,x/integraldisplayT∗\n0/summationdisplay\nl∈Zn∗e−δ1\n8ν2\n5τe−c/angb∇acketleftl,lτap/angb∇acket∇ights/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρl(τ)|\n×/summationdisplay\nk∈Zn∗/integraldisplayT∗\nτ|k|e−νt/vextendsingle/vextendsingle/vextendsinglee−mντ(Aσ0+1,2\n5ˆf)k−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle2\ndtdτ\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,xsup\n0≤τ≤T∗/summationdisplay\nk∈Znsup\nζ∈Rnsup\n|ω|=1/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsinglee−mντ(Aσ0+1,2\n5ˆf)k/parenleftbig\nτ,ωt′+ζ/parenrightbig/vextendsingle/vextendsingle/vextendsingle2\ndt′\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,xsup\n0≤τ≤T∗/parenleftbigg\ne−mντ/vextenddouble/vextenddouble/vextenddouble(Aσ0+1,2\n5ˆf)k(τ)/vextenddouble/vextenddouble/vextenddouble\nL2\nkHn−1\n2+\nη/parenrightbigg2\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,xsup\n0≤t≤T∗/ba∇dblfw(t)/ba∇dbl2\nEσ0+1,2\n5m. (4.38)\nTreatment of N E;0.We split NE;0into high-low and low-high interactions: NE;0=NHL\nE;0+\nNLH\nE;0,where\nNHL\nE;0=/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg/integraldisplayt\n01|k(tap−τap)|≤2|k,kτap|Sk(t−τ)/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)ˆρk(τ)\n×eντ(tap−τap)ˆf0(τ,k(tap−τap))dτ/bracketrightBigg2\ndt,\nNLH\nE;0=/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg/integraldisplayt\n01|k,kτap|<1\n2|k(tap−τap)|Sk(t−τ)/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)ˆρk(τ)LANDAU DAMPING FOR VPFP 27\n×(t−τ)apˆf0(τ,k(tap−τap))dτ/bracketrightBigg2\ndt.\nForNHL\nE;0, arguing as in (4.33) and (4.34), one deduces that\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)1|k(tap−τap)|≤2|k,kτap|\n/lessorsimilarS3\n4\nk(t−τ)/an}b∇acketle{tt/an}b∇acket∇i}htbBk(τ)e(λ(t)−λ(τ))/angb∇acketleftk,ktap/angb∇acket∇ightsecλ(τ)/angb∇acketleftk(tap−τap)/angb∇acket∇ights\n/lessorsimilar/parenleftBig\n/an}b∇acketle{tτ/an}b∇acket∇i}htbBk(τ)/parenrightBig\necλ(τ)/angb∇acketleftk(tap−τap)/angb∇acket∇ightsS1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2. (4.39)\nIn view of (B.2), we have 1k/negationslash=0S1\n2\nk(t)/lessorsimilare−νt. Then\n(4.40)/integraldisplay∞\n−∞S1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2dτ=/integraldisplay∞\n−∞S1\n2\nk(t)\n/an}b∇acketle{tktap/an}b∇acket∇i}ht2dt/lessorsimilar1.\nThus, by Schur’s test, we obtain that\nNHL\nE;0/lessorsimilar/parenleftbigg\nsup\nτsup\nηeλ(τ)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teντη/an}b∇acket∇i}ht3/vextendsingle/vextendsingle/vextendsingleˆf0(τ,η)/vextendsingle/vextendsingle/vextendsingle/parenrightbigg2\n×/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg/integraldisplayt\n0S1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρk(τ)|dτ/bracketrightBigg2\ndt\n/lessorsimilarsup\nt/vextenddouble/vextenddouble/vextenddoubleeλ(t)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teνtη/an}b∇acket∇i}ht3ˆf0(t,η)/vextenddouble/vextenddouble/vextenddouble2\nL∞ηsup\nk∈Zn∗sup\nt/integraldisplayt\n0S1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2dτ\n×sup\nk∈Zn∗sup\nτ/integraldisplayT∗\nτS1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2dt/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x\n/lessorsimilarǫ2ν2γ/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x. (4.41)\nNext we consider NLH\nE;0. It is worth emphasizing that under the restriction |k,kτap|<\n1\n2|k(tap−τap)|it holds that\n(4.42) eντ<3\n2.\nConsequently, now the analogue of (4.35) reads\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)(t−τ)ap1|k,kτap|<1\n2|k(tap−τap)|\n/lessorsimilar|k|1\n2S1\n2\nk(t−τ)/parenleftbigg\ne−ντ/an}b∇acketle{tτ/an}b∇acket∇i}htbeδν1\n3τecλ(τ)/angb∇acketleftk,kτap/angb∇acket∇ights/parenrightbigg\nAσ0+1,2\n5\n0(τ,k(tap−τap)). (4.43)\nThus, similar to (4.37) and (4.38), we infer from the above tw o inequalities that\nNLH\nE;0/lessorsimilar/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗|k|/bracketleftBigg/integraldisplayt\n0/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρk(τ)|e−c/angb∇acketleftk,kτap/angb∇acket∇ights\n×S1\n2\nk(t−τ)e−mντ(Aσ0+1,2\n5ˆf)0(τ,k(tap−τap))dτ/bracketrightBigg2\ndt\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble\nL2\nt,x/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗|k|/integraldisplayt\n0/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρk(τ)|e−ντe−c/angb∇acketleftk,kτap/angb∇acket∇ights28 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n×Sk(t−τ)/vextendsingle/vextendsingle/vextendsinglee−mντ(Aσ0+1,2\n5ˆf)0(τ,ktap−kτap)/vextendsingle/vextendsingle/vextendsingle2\ndτdt\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,xsup\nτsup\n|ω|=1sup\nζ∈Rn/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsinglee−mντ(Aσ0+1,2\n5ˆf)0/parenleftbig\nτ,ωt′+ζ/parenrightbig/vextendsingle/vextendsingle/vextendsingle2\ndt′\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,xsup\n0≤t≤T∗/ba∇dblfw(t)/ba∇dbl2\nEσ0+1,2\n5m. (4.44)\n4.1.3.Nµcollision contributions. Recalling (2.14) and (2.23), we have\nNµ/lessorsimilarν2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftbigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)/integraldisplayt\n0Sk(t−τ)(/hatwiderMθ)k(τ)(/hatwidest∆vµ)(k(t−τ)ap)dτ/bracketrightbigg2\ndt\n+ν2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftbigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)/integraldisplayt\n0Sk(t−τ)(/hatwiderM1)k(τ)·(/hatwidest∇vµ)(k(t−τ)ap)dτ/bracketrightbigg2\ndt\n=Nµ,1+Nµ,2.\nSince both ∇vµand ∆ vµare smooth and of fast decay, the treatments of Nµ,1andNµ,2\nare the same, here we only bound Nµ,1. To this end, using (4.39), (4.43) and the fact\ntap−τap=e−ντ(t−τ)ap≤(t−τ)ap, we have\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)/lessorsimilar/parenleftBig\n/an}b∇acketle{tτ/an}b∇acket∇i}htbBk(τ)/parenrightBig\neλ(0)/angb∇acketleftk(t−τ)ap/angb∇acket∇ights/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}htσ0+2S1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2.\nThen in view of (4.40), similar to (4.41), one deduces that\nNµ,1/lessorsimilarsup\nη/parenleftBig\neλ(0)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ0+2|(/hatwidest∆vµ)(η)|/parenrightBig2\n×ν2/summationdisplay\nk∈Zn∗/integraldisplayT∗\n0\n/integraldisplayt\n0/an}b∇acketle{tτ/an}b∇acket∇i}htbBk(τ)(/hatwiderMθ)k(τ)S1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2dτ\n2\ndt\n/lessorsimilarν2/summationdisplay\nk∈Zn∗/integraldisplayT∗\n0/integraldisplayt\n0/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tτ/an}b∇acket∇i}htbBk(τ)(/hatwiderMθ)k(τ)/vextendsingle/vextendsingle/vextendsingle2S1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2dτ/integraldisplayt\n0S1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2dτdt\n/lessorsimilarν2/summationdisplay\nk∈Zn∗/integraldisplayT∗\n0/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tτ/an}b∇acket∇i}htbBk(τ)(/hatwiderMθ)k(τ)/vextendsingle/vextendsingle/vextendsingle2/integraldisplayT∗\nτS1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2dtdτ/lessorsimilarν2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBMθ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x.\nThus, we get\n(4.45) Nµ/lessorsimilarν2/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBMθ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x+/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBM1/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x/parenrightbigg\n,\nwhich is consistent with Proposition 3.2 provided νis chosen sufficiently small.\n4.1.4.Ngcollision contributions. Recalling (4.10), it is natural to write\n(4.46) Ng=Ng,1+Ng,2+Ng,3+Ng,4.\nForNg,1, we divide into two cases based on whether fis at zero frequency or not as above:\nNg,1/lessorsimilarNg,1;/negationslash=+Ng,1;0, with\nNg,1;/negationslash==ν2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)/integraldisplayt\n0Sk(t−τ)/summationdisplay\nl∈Zn∗,l/negationslash=kˆρl(τ)|k(t−τ)ap|2\n׈fk−l(τ,ktap−lτap)dτ/bracketrightBigg2\ndt=NHL\ng,1;/negationslash=+NLH\ng,1;/negationslash=,LANDAU DAMPING FOR VPFP 29\nNg,1;0=ν2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)/integraldisplayt\n0Sk(t−τ)ˆρk(τ)|k(t−τ)ap|2\n׈f0/parenleftbig\nτ,e−ντk(t−τ)ap/parenrightbig\ndτ/bracketrightBigg2\ndt=NHL\ng,1;0+NLH\ng,1;0.\nConsider NHL\ng,1;/negationslash=first. It is worth point out that, different from NHL\nE;/negationslash=, now we can use the\ninverse Knudsen number νto absorb the quadratic time growth /an}b∇acketle{tτ/an}b∇acket∇i}ht2with the aid of enhanced\ndissipation, namely, ν2\n3/an}b∇acketle{tτ/an}b∇acket∇i}ht2/lessorsimilare1\n4δ1ν1\n3τ. Thus, by (4.15), similar to (4.14), (4.17) and (4.20),\nwe have\nν/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)|k(t−τ)ap|2/vextendsingle/vextendsingle/vextendsingleˆfk−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle1|k−l,ktap−lτap|≤2|l,lτap|1k/negationslash=l\n/lessorsimilarνe(m+2)ντ/an}b∇acketle{tτ/an}b∇acket∇i}ht2S3\n4\nk(t−τ)/parenleftBig\n/an}b∇acketle{tτ/an}b∇acket∇i}htbBl(τ)/parenrightBig\ne−c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights\n×/vextendsingle/vextendsingle/vextendsingle/vextendsinglee−mντF/bracketleftBig\neλ(τ)/angb∇acketleft∇/angb∇acket∇ights/an}b∇acketle{t∇/an}b∇acket∇i}ht5\n2f/bracketrightBig\nk−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilarν1\n3S1\n2\nk(t−τ)e−1\n2δ1ν1\n3τe−c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights/parenleftBig\n/an}b∇acketle{tτ/an}b∇acket∇i}htbBl(τ)/parenrightBig\n×sup\nτ/parenleftBigg\ne−mντ/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeδ1ν1\n3τF/bracketleftBig\neλ(τ)/angb∇acketleft∇/angb∇acket∇ights/an}b∇acketle{t∇/an}b∇acket∇i}ht5\n2f/negationslash=/bracketrightBig\nk(τ,η)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nkHn\n2+\nη/parenrightBigg\n. (4.47)\nThen by Schur’s test, similar to (4.21), one deduces that\nNHL\ng,1;/negationslash=/lessorsimilarν2\n3+2γǫ2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x\nsup\ntsup\nk/negationslash=0/summationdisplay\nl∈Zn∗/integraldisplayt\n0K1(t,τ,k,l)dτ\n\n×\nsup\nτsup\nl/negationslash=0/summationdisplay\nk∈Zn∗/integraldisplayT∗\nτK1(t,τ,k,l)dt\n/lessorsimilarν2\n3+2γǫ2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x. (4.48)\nNext we turn to NLH\ng,1;/negationslash=. From (2.26) and (4.15), similar to (4.35), for l/ne}ationslash= 0, we find that\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)|k(t−τ)ap|21|l,lτap|<1\n2|k−l,ktap−lτap|1k/negationslash=l\n/lessorsimilar|k|1\n2S1\n2\nk(t−τ)e−1\n2δ1ν2\n5τ/parenleftbigg\n/an}b∇acketle{tτ/an}b∇acket∇i}htb/an}b∇acketle{tτap/an}b∇acket∇i}hteδν1\n3τecλ(τ)/angb∇acketleftl,lτap/angb∇acket∇ights/parenrightbigg\n×(|/hatwider∂τv|Aσ0+1,2\n5)k−l(τ,ktap−lτap)1k/negationslash=l\n/lessorsimilar|k|1\n2S1\n2\nk(t−τ)e−1\n4δ1ν2\n5τe−c/angb∇acketleftl,lτap/angb∇acket∇ights/parenleftbigg\neδν1\n3τeλ(τ)/angb∇acketleftl,lτap/angb∇acket∇ights/an}b∇acketle{tl,lτap/an}b∇acket∇i}htb+3\n2+/parenrightbigg\n×1\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1\n2+e−mντ(|/hatwider∂τv|Aσ0+1,2\n5)k−l(τ,ktap−lτap)1k/negationslash=l. (4.49)\nThen similar (4.36)–(4.38), using the trace theorem and (3. 5), we are led to\nNLH\ng,1;/negationslash=/lessorsimilarν2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗|k|/bracketleftBigg/summationdisplay\nl∈Zn∗/integraldisplayt\n0S1\n2\nk(t−τ)e−1\n4δ1ν2\n5τe−c/angb∇acketleftl,lτap/angb∇acket∇ights\n×eδν1\n3τeλ(τ)/angb∇acketleftl,lτap/angb∇acket∇ights/an}b∇acketle{tl,lτap/an}b∇acket∇i}htb+3\n2+|ˆρl(τ)|\n×1\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1\n2+/vextendsingle/vextendsingle/vextendsingle/vextendsinglee−mντF/bracketleftBig\n∂τ\nvAσ0+1,2\n5f/negationslash=/bracketrightBig\nk−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle/vextendsingledτ/bracketrightBigg2\ndt30 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n/lessorsimilarν2sup\nτsup\nl∈Zn∗/parenleftbigg\neδν1\n3τeλ(τ)/angb∇acketleftl,lτap/angb∇acket∇ights/an}b∇acketle{tl,lτap/an}b∇acket∇i}htb+3\n2+|ˆρl(τ)|/parenrightbigg2\n×/summationdisplay\nl∈Zn∗/integraldisplayT∗\n01\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1+/summationdisplay\nk∈Zn∗|k|/integraldisplayT∗\nτSk(t−τ)e−1\n4δ1ν2\n5τe−c/angb∇acketleftl,lτap/angb∇acket∇ights\n×/vextendsingle/vextendsingle/vextendsingle/vextendsinglee−mντF/bracketleftBig\n∂τ\nvAσ0+1,2\n5f/negationslash=/bracketrightBig\nk−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndtdτ\n/lessorsimilarsup\nt/ba∇dblBσ1ρ(t)/ba∇dbl2\nL2xν2/summationdisplay\nk∈Zn∗/integraldisplayT∗\n01\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1+/parenleftbigg\ne−mντ/vextenddouble/vextenddouble/vextenddoubleF/bracketleftBig\n∂τ\nvAσ0+1,2\n5f/negationslash=/bracketrightBig\nk(τ)/vextenddouble/vextenddouble/vextenddouble\nHn−1\n2+\nη/parenrightbigg2\ndτ\n/lessorsimilarǫ2ν2γ+1/parenleftBigg\nsup\nt/ba∇dblfw(t)/ba∇dbl2\nEσ0+1,2\n5m+ν/integraldisplayT∗\n0/ba∇dblfw(t)/ba∇dbl2\nDσ0+1,2\n5mdt/parenrightBigg\n. (4.50)\nThe treatment of NHL\ng,1;0is similar to (4.41), we just state the result:\n(4.51) NHL\ng,1;0/lessorsimilarν2sup\nt/parenleftbigg/vextenddouble/vextenddouble/vextenddoubleeλ(t)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teνtη/an}b∇acket∇i}ht4ˆf0(t,η)/vextenddouble/vextenddouble/vextenddouble\nL∞η/parenrightbigg2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x.\nAs for the treatment of NLH\ng,1;0, we will use the restriction (4.42) again. In fact, using (4. 42)\nand (B.2), similar to (4.43), one easily deduces that\nν/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)|k(t−τ)ap|21|k,kτap|<1\n2|k(tap−τap)|\n/lessorsimilarν2\n3|k|1\n2S1\n2\nk(t−τ)/parenleftbigg\n/an}b∇acketle{tτ/an}b∇acket∇i}htbeδν1\n3τecλ(τ)/angb∇acketleftk,kτap/angb∇acket∇ights|k|/parenrightbigg\nAσ0+1,2\n5\n0(τ,k(tap−τap)). (4.52)\nThenNLH\ng,1;0can be bounded in the same manner as (4.44):\nNLH\ng,1;0/lessorsimilarν4\n3/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗|k|/bracketleftBigg/integraldisplayt\n0/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρk(τ)|e−c/angb∇acketleftk,kτap/angb∇acket∇ights\n×S1\n2\nk(t−τ)e−mντ(Aσ0+1,2\n5ˆf)0(τ,k(tap−τap))dτ/bracketrightBigg2\ndt\n/lessorsimilarν4\n3/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,xsup\n0≤τ≤T∗/parenleftbigg\ne−mντ/vextenddouble/vextenddouble/vextenddouble(Aσ0+1,2\n5ˆf)0(τ)/vextenddouble/vextenddouble/vextenddouble\nHn−1\n2+\nη/parenrightbigg2\n/lessorsimilarν4\n3/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,xsup\n0≤t≤T∗/ba∇dblfw(t)/ba∇dbl2\nEσ0+1,2\n5m. (4.53)\nNow we turn to Ng,2. Indeed, Ng,2can be bounded by combining the treatments of NE\nandNg,1. More precisely, now the analogue of (4.47) reads\nν/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)|k(t−τ)ap|/vextendsingle/vextendsingle/vextendsinglee−ντ(∇ηˆf)k−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle\n×1|k−l,ktap−lτap|≤2|l,lτap|1k/negationslash=l\n/lessorsimilarν2\n3S1\n2\nk(t−τ)e−1\n2δ1ν1\n3τe−c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights/parenleftBig\n/an}b∇acketle{tτ/an}b∇acket∇i}htbBl(τ)/parenrightBig\n×sup\nτ/parenleftBigg\ne−(m+1)ντ/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeδ1ν1\n3τF/bracketleftBig\neλ(τ)/angb∇acketleft∇/angb∇acket∇ights/an}b∇acketle{t∇/an}b∇acket∇i}ht3\n2vf/negationslash=/bracketrightBig\nk(τ,η)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nkHn\n2+\nη/parenrightBigg\n/lessorsimilarν2\n3S1\n2\nk(t−τ)e−1\n2δ1ν1\n3τe−c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights/parenleftBig\n/an}b∇acketle{tτ/an}b∇acket∇i}htbBl(τ)/parenrightBig\nsup\nt/ba∇dblfw(t)/ba∇dbl\nEσ1,1\n3m\n/lessorsimilarν2\n3S1\n2\nk(t−τ)e−1\n2δ1ν1\n3τe−c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights/parenleftBig\n/an}b∇acketle{tτ/an}b∇acket∇i}htbBl(τ)/parenrightBig\nǫνγ. (4.54)LANDAU DAMPING FOR VPFP 31\nThen similar to (4.48), we have\nNHL\ng,2;/negationslash=/lessorsimilarν4\n3+2γǫ2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x. (4.55)\nForNLH\ng,2;/negationslash=, instead of using the strategy of treating NLH\ng,1;/negationslash=, we just follow the way to treat\nNLH\nE;/negationslash=, see (4.35)–(4.38). Thus,\nNLH\ng,2;/negationslash=/lessorsimilarν2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,xsup\n0≤τ≤T∗/parenleftbigg\ne−(m+1)ντ/vextenddouble/vextenddouble/vextenddouble(Aσ0+1,2\n5∇ηˆf)k(τ)/vextenddouble/vextenddouble/vextenddouble\nL2\nkHn−1\n2+\nη/parenrightbigg2\n/lessorsimilarν2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,xsup\n0≤t≤T∗/ba∇dblfw(t)/ba∇dbl2\nEσ0+1,2\n5m/lessorsimilarǫ2ν2γ+2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x.\nThe rest part Ng,2;0can be treated in the same way as NE;0. In fact, similar to (4.41) and\n(4.44), we have\nNHL\ng,2;0/lessorsimilarν2sup\nt/parenleftBig\ne−νt/ba∇dbleλ(t)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teνtη/an}b∇acket∇i}ht3(∇ηˆf)0(t,η)/ba∇dblL∞η/parenrightBig2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x,\nNLH\ng,2;0/lessorsimilarν2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,xsup\nτsup\n|ω|=1sup\nζ∈Rn/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsinglee−(m+1)ντ(Aσ0+1,2\n5∇ηˆf)0/parenleftbig\nτ,ωt′+ζ/parenrightbig/vextendsingle/vextendsingle/vextendsingle2\ndt′\n/lessorsimilarǫ2ν2γ+2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x.\nNext we consider Ng,3.\nNg,3/lessorsimilar/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)/integraldisplayt\n0Sk(t−τ)ν/summationdisplay\nl∈Zn∗,l/negationslash=k(/hatwiderMθ)l(τ)|k(t−τ)ap|2ˆfk−l(τ,ktap−lτap)dτ/bracketrightBigg2\ndt\n+/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)/integraldisplayt\n0Sk(t−τ)ν(/hatwiderMθ)k(τ)|k(t−τ)ap|2ˆf0/parenleftbig\nτ,e−ντk(t−τ)ap/parenrightbig\ndτ/bracketrightBigg2\ndt\n+/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)/integraldisplayt\n0Sk(t−τ)ν(/hatwiderMθ)0(τ)|k(t−τ)ap|2ˆfk(τ,ktap)dτ/bracketrightBigg2\ndt\n=Ng,3;/negationslash=+Ng,3;0+Ng,3;0/negationslash=.\nClearly,Ng,3;/negationslash=andNg,3;0can be treated in the same way as Ng,1;/negationslash=andNg,1;0, respectively,\nwithρreplaced by ( Mθ)/negationslash=. Here we only estimate Ng,3;0/negationslash=. If fact, now (4.49) reduces to\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)|k(t−τ)ap|2\n/lessorsimilar|k|1\n2S1\n2\nk(t−τ)e−1\n4δ1ν2\n5τ/parenleftbigg\neδν1\n3τ/an}b∇acketle{tτap/an}b∇acket∇i}htb+3\n2+/parenrightbigge−mντ\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1\n2+(|/hatwider∂τv|Aσ0+1,2\n5)k(τ,ktap)1k/negationslash=0. (4.56)\nThen similar to (4.50), using (3.5) and (3.12), together wit h\n/integraldisplayt\n0e−1\n4δ1ν2\n5τ/parenleftbigg\neδν1\n3τ/an}b∇acketle{tτap/an}b∇acket∇i}htb+3\n2+/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(τ)/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\ndτ/lessorsimilarν−2\n5sup\nτ/parenleftbigg\neδν1\n3τ/an}b∇acketle{tτap/an}b∇acket∇i}htb+3\n2+/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(τ)/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n.\nwe have\nNg,3;0/negationslash=/lessorsimilarν2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗|k|/bracketleftBigg/integraldisplayt\n0e−1\n4δ1ν2\n5τ/parenleftbigg\neδν1\n3τ/an}b∇acketle{tτap/an}b∇acket∇i}htb+3\n2+/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(τ)/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n×S1\n2\nk(t−τ)1\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1\n2+e−mντ/vextendsingle/vextendsingle/vextendsingleF/bracketleftBig\n∂τ\nvAσ0+1,2\n5f/negationslash=/bracketrightBig\nk(τ,ktap)/vextendsingle/vextendsingle/vextendsingledτ/bracketrightBigg2\ndt32 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n/lessorsimilarν2/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗|k|/integraldisplayt\n0e−1\n4δ1ν2\n5τ/parenleftbigg\neδν1\n3τ/an}b∇acketle{tτap/an}b∇acket∇i}htb+3\n2+/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(τ)/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\ndτ\n×/integraldisplayt\n0e−1\n4δ1ν2\n5τ/parenleftbigg\neδν1\n3τ/an}b∇acketle{tτap/an}b∇acket∇i}htb+3\n2+/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(τ)/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\nSk(t−τ)\n×1\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1+/vextendsingle/vextendsingle/vextendsinglee−mντF/bracketleftBig\n∂τ\nvAσ0+1,2\n5f/negationslash=/bracketrightBig\nk(τ,ktap)/vextendsingle/vextendsingle/vextendsingle2\ndτdt\n/lessorsimilarν8\n5sup\nτ/parenleftbigg\neδν1\n3τ/an}b∇acketle{tτap/an}b∇acket∇i}htb+3\n2+/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(τ)/vextendsingle/vextendsingle/vextendsingle/parenrightbigg2/summationdisplay\nk∈Zn∗/integraldisplayT∗\n01\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1+\n×/integraldisplayT∗\nτ|k|e−νt/vextendsingle/vextendsingle/vextendsinglee−mντF/bracketleftBig\n∂τ\nvAσ0+1,2\n5f/negationslash=/bracketrightBig\nk(τ,ktap)/vextendsingle/vextendsingle/vextendsingle2\ndtdτ\n/lessorsimilarν3\n5sup\nt/parenleftbigg\neδν1\n3t/an}b∇acketle{ttap/an}b∇acket∇i}htb+3\n2+/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(t)/vextendsingle/vextendsingle/vextendsingle/parenrightbigg2\n×ν/summationdisplay\nk∈Zn∗/integraldisplayT∗\n01\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1+/parenleftbigg\ne−mντ/vextenddouble/vextenddouble/vextenddoubleF/bracketleftBig\n∂τ\nvAσ0+1,2\n5f/negationslash=/bracketrightBig\nk(τ)/vextenddouble/vextenddouble/vextenddouble\nHn−1\n2+\nη/parenrightbigg2\ndτ,\n/lessorsimilarǫ4ν4γ+3\n5/parenleftBigg\nsup\nt/ba∇dblfw(t)/ba∇dbl2\nEσ0+1,2\n5m+ν/integraldisplayT∗\n0/ba∇dblfw(t)/ba∇dbl2\nDσ0+1,2\n5mdt/parenrightBigg\n. (4.57)\nNoting that ( M1)0= 0, the last term Ng,4can be treated in the same way as Ng,2. We\nthus omit the details to avoid repetition.\n4.2.L∞\ntcontrol on ρ.The lower norm estimate on ρis much more straightforward than\nthe higher norm estimate in Section 4.1. From (2.24), we have\n/ba∇dblBσ1ρ(t)/ba∇dbl2\nL2x/lessorsimilarI+L+NE+Nµ+Ng,\nwhere\nI=/summationdisplay\nk∈Zn∗/vextendsingle/vextendsingleBσ1\nk(t)Sk(t)/hatwidergin(k,ktap)/vextendsingle/vextendsingle2,\nL=/summationdisplay\nk∈Zn∗/bracketleftbigg\nBσ1\nk(t)/integraldisplayt\n0Sk(t−τ)ˆρk(τ)(t−τ)apˆµ(k(t−τ)ap)dτ/bracketrightbigg2\n,\nNE=/summationdisplay\nk∈Zn∗/bracketleftBigg\nBσ1\nk(t)/integraldisplayt\n0Sk(t−τ)/summationdisplay\nl∈Zn∗ˆρl(τ)l\n|l|2·k(t−τ)apˆfk−l(τ,ktap−lτap)dτ/bracketrightBigg2\n,\nNµ=ν2/summationdisplay\nk∈Zn∗/bracketleftbigg\nBσ1\nk(t)/integraldisplayt\n0Sk(t−τ)(/hatwiderCµ)k(τ,k(t−τ)ap)dτ/bracketrightbigg2\n,\nNg=ν2/summationdisplay\nk∈Zn∗/bracketleftbigg\nBσ1\nk(t)/integraldisplayt\n0Sk(t−τ)(/hatwidestC[g])k(τ,k(t−τ)ap)dτ/bracketrightbigg2\n.\nClearly, by (B.2) and Sobolev embedding, there holds\nI/lessorsimilar/summationdisplay\nk∈Zn∗/vextendsingle/vextendsingle/vextendsingleeλ(0)/angb∇acketleftk,ktap/angb∇acket∇ights/an}b∇acketle{tk,ktap/an}b∇acket∇i}htσ1/hatwidergin(k,ktap)/vextendsingle/vextendsingle/vextendsingle2\n/lessorsimilar/vextenddouble/vextenddouble/vextenddoubleeλ(0)/angb∇acketleft∇/angb∇acket∇ights/an}b∇acketle{t∇/an}b∇acket∇i}htσ1(/an}b∇acketle{tv/an}b∇acket∇i}htmgin)/vextenddouble/vextenddouble/vextenddouble2\nL2/lessorsimilarǫ2ν2γ.\nTo bound L, using again (B.2), for k/ne}ationslash= 0, we have\nBσ1\nk(t)Sk(t−τ)/lessorsimilareδν1\n3τeλ(τ)/angb∇acketleftk,kτap/angb∇acket∇ights/an}b∇acketle{tk,kτap/an}b∇acket∇i}htσ1ecλ(τ)/angb∇acketleftk(tap−τap)/angb∇acket∇ightsS1\n2\nk(t−τ)LANDAU DAMPING FOR VPFP 33\n+eδν1\n3τecλ(τ)/angb∇acketleftk,kτap/angb∇acket∇ights/an}b∇acketle{tk(tap−τap)/an}b∇acket∇i}htσ1eλ(τ)/angb∇acketleftk(tap−τap)/angb∇acket∇ightsS1\n2\nk(t−τ)\n/lessorsimilarBk(τ)eλ(0)/angb∇acketleftk(t−τ)ap/angb∇acket∇ights/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}htσ1S1\n2\nk(t−τ).\nConsequently, in view of (4.40), one deduces that\nL/lessorsimilar/summationdisplay\nk∈Zn∗\n/integraldisplayt\n0|Bˆρk(τ)|S1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}hteλ(0)/angb∇acketleftk(t−τ)ap/angb∇acket∇ights/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}htσ1+2|ˆµ(k(t−τ)ap)|dτ\n2\n/lessorsimilar/summationdisplay\nk∈Zn∗\n/integraldisplayt\n0|Bˆρk(τ)|S1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}htdτ\n2\nsup\nη/parenleftBig\neλ(0)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ1+2|ˆµ(η)|/parenrightBig2\n/lessorsimilar/summationdisplay\nk∈Zn∗/integraldisplayt\n0|Bˆρk(τ)|2dτsup\nk∈Zn∗/integraldisplayt\n0Sk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2dτ\n/lessorsimilar/ba∇dblBρ(t)/ba∇dbl2\nL2\nt,x/lessorsimilar(C0ǫνγ)2, (4.58)\nwhich is consistent with Proposition 3.2 as long as ˜C0≫C0.\nNext we turn to NE. As in Section 4.1, we split it into two parts based on fis at zero\nfrequency or not. Compared with (4.30) and (4.38), the treat ment of NE;/negationslash=now is much\neasier since the plasma echoes will never happen even when ρis at high frequency. In fact,\nnow one can see from (B.2), (4.13) and (4.15) that\nBσ1\nk(t)Sk(t−τ)|k(t−τ)ap|1k/negationslash=l\n/lessorsimilarBσ1+1\nl(τ)/parenleftbigg\ne−mντeδ1ν2\n5τeλ(τ)/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights/an}b∇acketle{tk−l,ktap−lap/an}b∇acket∇i}htσ1+1/parenrightbigg\n×e(m+1)ντe−δ1ν2\n5τ1k/negationslash=l\n/lessorsimilarBσ1+1\nl(τ)/parenleftbigg\ne−mντAσ0+1,2\n5\nk−l(τ,ktap−lτap)/parenrightbigg\ne−1\n2δ1ν2\n5τ1k/negationslash=l. (4.59)\nMoreover, note that\n/summationdisplay\nl∈Zn∗/integraldisplayt\n0Bσ1+1\nl(τ)|ρl(τ)|dτ/lessorsimilar/summationdisplay\nl∈Zn∗1\n/an}b∇acketle{tl/an}b∇acket∇i}htn\n2+/integraldisplayt\n0/an}b∇acketle{tτ/an}b∇acket∇i}ht1\n2+/an}b∇acketle{tl/an}b∇acket∇i}htn\n2+Bσ1+1\nl(τ)|ρl(τ)|1\n/an}b∇acketle{tτ/an}b∇acket∇i}ht1\n2+dτ\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+Bρ/vextenddouble/vextenddouble/vextenddouble\nL2\nt,x, (4.60)\nprovidedσ1+1+n+1\n2≤σ0. Then it follows from these two inequalities and (3.4) that\nNE;/negationslash=/lessorsimilar/summationdisplay\nk∈Zn∗/bracketleftBigg/integraldisplayt\n0/summationdisplay\nl∈Zn∗,l/negationslash=k|Bσ1+1ˆρl(τ)|e−mντ/vextendsingle/vextendsingle/vextendsingle(Aσ0+1,2\n5ˆf)k−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingledτ/bracketrightBigg2\n/lessorsimilar/summationdisplay\nl∈Zn∗/integraldisplayt\n0|Bσ1+1ˆρl(τ)|dτ/integraldisplayt\n0/summationdisplay\nk∈Zn∗/summationdisplay\nl∈Zn∗,l/negationslash=k|Bσ1+1ˆρl(τ)|\n×/parenleftBig\ne−mντ/vextendsingle/vextendsingle/vextendsingle(Aσ0+1,2\n5ˆf)k−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle/parenrightBig2\ndτ\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+Bρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,xsup\nt/parenleftbigg\ne−mνt/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5ˆf/negationslash=/vextenddouble/vextenddouble/vextenddouble\nL2\nkHn\n2+\nη/parenrightbigg2\n/lessorsimilarǫ2ν2γ/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+Bρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x. (4.61)\nThe rest part NE;0can be bounded in a similar manner to the corresponding term i n high\nnorm estimate. Indeed, compared with (4.39) and (4.43), now it is easier to get\nBσ1\nk(t)Sk(t−τ)|k(t−τ)ap|1|k(tap−τap)|≤2|k,kτap|34 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n/lessorsimilarBσ1\nk(τ)ecλ(τ)/angb∇acketleftk(tap−τap)/angb∇acket∇ights|eντk(tap−τap)|, (4.62)\nand\nBσ1\nk(t)Sk(t−τ)|k(t−τ)ap|1|k,kτap|<1\n2|k(tap−τap)|\n/lessorsimilare−ντe−c/angb∇acketleftk,kτap/angb∇acket∇ightseδν1\n3τeλ(τ)/angb∇acketleftk,kτap/angb∇acket∇ights/parenleftbigg\ne−mνtAσ0+1,2\n5\n0(τ,k(tap−τap))/parenrightbigg\n. (4.63)\nThen by (4.60) and (4.62), one deduces that\nNHL\nE;0/lessorsimilarsup\nτ/vextenddouble/vextenddouble/vextenddoubleeλ(τ)/angb∇acketleftη/angb∇acket∇ights|eντη|ˆf0(τ,η)/vextenddouble/vextenddouble/vextenddouble2\nL∞η/summationdisplay\nk∈Zn∗/bracketleftbigg/integraldisplayt\n0Bσ1\nk(τ)|ˆρk(τ)|dτ/bracketrightbigg2\n/lessorsimilarǫ2ν2γ\n/summationdisplay\nk∈Zn∗/integraldisplayt\n0Bσ1\nk(τ)|ˆρk(τ)|dτ\n2\n/lessorsimilarǫ2ν2γ/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+Bρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x. (4.64)\nFrom (4.37) and (4.63), we are led to\nNLH\nE;0/summationdisplay\nk∈Zn∗/bracketleftBigg/integraldisplayt\n0e−ντe−c/angb∇acketleftk,kτap/angb∇acket∇ights/vextendsingle/vextendsingle/vextendsingle/vextendsingleeδν1\n3τeλ(τ)/angb∇acketleftk,kτap/angb∇acket∇ightsˆρk(τ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n×/vextendsingle/vextendsingle/vextendsingle/vextendsinglee−mνtAσ0+1,2\n5\n0ˆf0(τ,k(tap−τap))/vextendsingle/vextendsingle/vextendsingle/vextendsingledτ/bracketrightBigg2\n/lessorsimilarsup\nt/parenleftBigg\ne−mνt/vextenddouble/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5\n0ˆf0(t,η)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞η/parenrightBigg2/summationdisplay\nk∈Zn∗/bracketleftbigg/integraldisplayt\n0e−ντe−c/angb∇acketleftk,kτap/angb∇acket∇ights|Bˆρk(τ)|dτ/bracketrightbigg2\n/lessorsimilarǫ2ν2γ/ba∇dblBρ/ba∇dbl2\nL2\nt,x. (4.65)\nThe term Nµcan be treated in the same way as L. In fact, similar to (4.58), we have\nNµ/lessorsimilarsup\nη/parenleftBig\neλ(0)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ1+1/parenleftBig\n|(/hatwidest∆vµ)(η)|+|(/hatwidest∇vµ)(η)|/parenrightBig/parenrightBig2\n×ν2/summationdisplay\nk∈Zn∗\n/integraldisplayt\n0Bσ1\nk(τ)(|(/hatwiderMθ)k(τ)|+|(/hatwiderM1)k(τ)|)S1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}htdτ\n2\n/lessorsimilarν2/parenleftBig\n/ba∇dblBMθ/ba∇dbl2\nL2\nt,x+/ba∇dblBM1/ba∇dbl2\nL2\nt,x/parenrightBig\n. (4.66)\nFinally, we turn to Ng, which consists of four parts as before. Here we only treat th e most\ntricky one Ng,3:\nNg,3/lessorsimilarν2/summationdisplay\nk∈Zn∗\nBσ1\nk(t)/integraldisplayt\n0Sk(t−τ)/summationdisplay\nl∈Zn∗,l/negationslash=k(/hatwiderMθ)l(τ)|k(t−τ)ap|2ˆfk−l(τ,ktap−lτap)dτ\n2\n+ν2/summationdisplay\nk∈Zn∗/bracketleftbigg\nBσ1\nk(t)/integraldisplayt\n0Sk(t−τ)(/hatwiderMθ)k(τ)|k(t−τ)ap|2ˆf0(τ,k(tap−τap))dτ/bracketrightbigg2\n+ν2/summationdisplay\nk∈Zn∗/bracketleftbigg\nBσ1\nk(t)/integraldisplayt\n0Sk(t−τ)(/hatwiderMθ)0(τ)|k(t−τ)ap|2ˆfk(τ,ktap)dτ/bracketrightbigg2\n=Ng,3;/negationslash=+Ng,3;0+Ng,3;0/negationslash=.LANDAU DAMPING FOR VPFP 35\nForNg,3;/negationslash=, clearly, the analogue of (4.59) now reads\nBσ1\nk(t)Sk(t−τ)|k(t−τ)ap|21k/negationslash=l/lessorsimilarBσ1+2\nl/parenleftbigg\ne−mντAσ0+1,2\n5\nk−l(τ,ktap−lτap)/parenrightbigg\ne−1\n2δ1ν2\n5τ1k/negationslash=l.\nInaddition, (4.60) still holds with ρreplaced by Mθandσ1+1replaced by σ1+2, respectively,\nas long asσ1+2+n+1\n2≤σ0. Then similar to (4.61), we arrive at\nNg,3;/negationslash=/lessorsimilarǫ2ν2+2γ/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+BMθ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x.\nAs forNg,3;0, we just need to replace |k(t−τ)ap|by|k(t−τ)ap|2in (4.62) and (4.63). Then\nfollowing (4.64) and (4.65), one easily deduces that\nNg,3;0/lessorsimilarν2sup\nτ/vextenddouble/vextenddouble/vextenddoubleeλ(τ)/angb∇acketleftη/angb∇acket∇ights|eντη|2ˆf0(τ,η)/vextenddouble/vextenddouble/vextenddouble2\nL∞η/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+Bρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x\n+ν2sup\nt/parenleftBigg\ne−mνt/vextenddouble/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5\n0ˆf0(t,η)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞η/parenrightBigg2\n/ba∇dblBρ/ba∇dbl2\nL2\nt,x\n/lessorsimilarǫ2ν2+2γ/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+Bρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x. (4.67)\nTo bound Ng,3;0/negationslash=, instead of using (4.56), noting that ( t−τ)ap=eντ(tap−τap)≤eντtap,\nnow we have\nBσ1\nk(t)Sk(t−τ)|k(t−τ)ap|21k/negationslash=0\n/lessorsimilareδν1\n3τe(m+2)ντe−δ1ν2\n5τ/parenleftbigg\ne−mντeδ1ν2\n5τeλ(τ)/angb∇acketleftk,ktap/angb∇acket∇ights/an}b∇acketle{tk,ktap/an}b∇acket∇i}htσ1+2/parenrightbigg\n1k/negationslash=0\n/lessorsimilareδν1\n3τe−1\n2δ1ν2\n5τ/parenleftbigg\ne−mντAσ0+1,2\n5\nk(τ,tkap)/parenrightbigg\n1k/negationslash=0.\nThen by Cauchy-Schwarz inequality and (3.4),\nNg,3;0/negationslash=/lessorsimilarν2/summationdisplay\nk∈Zn∗/bracketleftbigg/integraldisplayt\n0/parenleftbigg\neδν1\n3τ(/hatwiderMθ)0(τ)/parenrightbigg/vextenddouble/vextenddouble/vextenddoublee−mντAσ0+1,2\n5ˆfk(τ,η)/vextenddouble/vextenddouble/vextenddouble\nL∞ηdτ/bracketrightbigg2\n/lessorsimilarν2/integraldisplayt\n0/parenleftbigg\neδν1\n3τ(/hatwiderMθ)0(τ)/parenrightbigg2/summationdisplay\nk∈Zn∗/vextenddouble/vextenddouble/vextenddoublee−mντAσ0+1,2\n5ˆfk(τ,η)/vextenddouble/vextenddouble/vextenddouble2\nL∞ηdτ\n/lessorsimilarǫ2ν2γ+2sup\nt/parenleftbigg\neδν1\n3t/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+(/hatwiderMθ)0(t)/parenrightbigg2\n.\n5.Higher order moment estimates\nIn this section, we improve the estimates (3.7f), (3.7g), (3 .7i) and (3.7j) for M1andM2.\nAs one can see from Section 4.2, the L∞\ntestimates are much more straightforward than the\nL2\ntestimates. We will only sketch the details for the much harde rL2\ntcontrol.\nTo begin with, we give the expressions of ( /hatwiderM1)k(t) and (/hatwiderM2)k(t). As a matter of fact,\nkeeping in mind the relations (2.17) and (2.18), the express ions of (/hatwiderM1)k(t) and (/hatwiderM2)k(t)\ncan be obtained in a similar way to that of ˆ ρk(t), see (2.22)–(2.24). To avoid unnecessary\nrepetition, the expressions of ( /hatwiderM1)k(t) and (/hatwiderM2)k(t) are given below directly, and we refer to\n[1] for more details for the derivations:\n(/hatwiderM1)k(t,η) =ie−νtSk(t)(∇η/hatwidefin)k(ktap)\n−/integraldisplayt\n0Sk(t−τ)ie−νt[∇ηL]k(τ,k(t−τ)ap)dτ36 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n+/integraldisplayt\n0Sk(t−τ)ie−νt[∇ηN]k(τ,k(t−τ)ap)dτ\n−2ν/parenleftbigg/integraldisplayt\n0ie−ν(t−τ)k(t−τ)apdτ/parenrightbigg\nSk(t)(/hatwiderfin)k(ktap)\n+2ν/integraldisplayt\n0/parenleftbigg/integraldisplayt\nτ′ie−ν(t−τ)k(t−τ)apdτ/parenrightbigg\nSk(t−τ′)Lk(τ′,k(t−τ′)ap)dτ′\n−2ν/integraldisplayt\n0/parenleftbigg/integraldisplayt\nτ′ie−ν(t−τ)k(t−τ)apdτ/parenrightbigg\nSk(t−τ′)Nk(τ′,k(t−τ′)ap)dτ′, (5.1)\nand\n(/hatwiderM2)k(t) =−e−2νtSk(t)(∆η/hatwiderfin)k(ktap)+e−2νt/integraldisplayt\n0Sk(t−τ)[∆ηL]k(τ,k(t−τ)ap)dτ\n−e−2νt/integraldisplayt\n0Sk(t−τ)[∆ηN]k(τ,k(t−τ)ap)dτ\n+4νe−νt/parenleftbigg/integraldisplayt\n0e−ν(t−τ)k(t−τ)apdτ/parenrightbigg\n·Sk(t)(∇η/hatwidefin)k(ktap)\n+4νe−νt/integraldisplayt\n0/parenleftbigg/integraldisplayt\nτ′e−ν(t−τ)k(t−τ)apdτ/parenrightbigg\n·Sk(t−τ′)[∇ηL]k(τ′,k(t−τ′)ap)dτ′\n+4νe−νt/integraldisplayt\n0/parenleftbigg/integraldisplayt\nτ′e−ν(t−τ)k(t−τ)apdτ/parenrightbigg\n·Sk(t−τ′)[∇ηN]k(τ′,k(t−τ′)ap)dτ′\n−8ν2/parenleftbigg\ne−2νt/integraldisplayt\n0/integraldisplayτ\n0eντk(t−τ)ap·eντ′k(t−τ′)apdτ′dτ/parenrightbigg\nSk(t)(/hatwiderfin)k(ktap)\n+8ν2/integraldisplayt\n0/parenleftbigg\ne−2νt/integraldisplayt\nτ′′/integraldisplayτ\nτ′′eντk(t−τ)ap·eντ′k(t−τ′)apdτ′dτ/parenrightbigg\n×Sk(t−τ′′)Lk(τ′′,k(t−τ′′)ap)dτ′′\n−8ν2/integraldisplayt\n0/parenleftbigg\ne−2νt/integraldisplayt\nτ′′/integraldisplayτ\nτ′′eντk(t−τ)ap·eντ′k(t−τ′)apdτ′dτ/parenrightbigg\n×Sk(t−τ′′)Nk(τ′′,k(t��τ′′)ap)dτ′′\n+2nν/parenleftbigg/integraldisplayt\n0e−2ν(t−τ)dτ/parenrightbigg\nSk(t)(/hatwiderfin)k(ktap)\n−2nν/integraldisplayt\n0/parenleftbigg/integraldisplayt\nτ′e−2ν(t−τ)dτ/parenrightbigg\nSk(t−τ′)Lk(τ′,k(t−τ′)ap)dτ′\n+2nν/integraldisplayt\n0/parenleftbigg/integraldisplayt\nτ′e−2ν(t−τ)dτ/parenrightbigg\nSk(t−τ′)Nk(τ′,k(t−τ′)ap)dτ′, (5.2)\nwhere we have used the notations\n[∇ηL]k(τ,k(t−τ)ap) =/parenleftbig\n∇η[Lk(τ,¯η(τ;k,η)]/parenrightbig/vextendsingle/vextendsingle\nη=ktap,\n[∇ηN]k(τ,k(t−τ)ap) =/parenleftbig\n∇η[Nk(τ,¯η(τ;k,η)]/parenrightbig/vextendsingle/vextendsingle\nη=ktap,\n[∆ηL]k(τ,k(t−τ)ap) =/parenleftbig\n∆η[Lk(τ,¯η(τ;k,η)]/parenrightbig/vextendsingle/vextendsingle\nη=ktap,\n[∆ηN]k(τ,k(t−τ)ap) =/parenleftbig\n∆η[Nk(τ,¯η(τ;k,η)]/parenrightbig/vextendsingle/vextendsingle\nη=ktap.\nTo improve (3.7f) and (3.7g), now we do not have to use Lemma 4. 2. Instead, we will\napply/an}b∇acketle{tt/an}b∇acket∇i}htbB(t) to (5.1) and (5.2) directly, and then take the L2\nt,xnorms. Note that\n/integraldisplayt\nτ′e−2ν(t−τ)dτ=1\n2ν/parenleftBig\n1−e−2ν(t−τ′)/parenrightBig\n, (5.3)LANDAU DAMPING FOR VPFP 37\n/integraldisplayt\nτ′e−ν(t−τ)k(t−τ)apdτ=k\n2ν2/parenleftBig\n1−e−ν(t−τ′)/parenrightBig2\n, (5.4)\nand\ne−2νt/integraldisplayt\nτ′′/integraldisplayτ\nτ′′eντk(t−τ)ap·eντ′k(t−τ′)apdτ′dτ=|k|2\n8ν4/parenleftBig\n1−e−ν(t−τ′′)/parenrightBig4\n. (5.5)\nCombining (5.4), (5.5) with (B.3) and (B.4), we find that the l ast three term on the right\nhand side of (5.1) and the last six terms on the right hand side of (5.2) can be bounded by\nS1−p\nk(t)/vextendsingle/vextendsingle/vextendsingle(/hatwiderfin)k(ktap)/vextendsingle/vextendsingle/vextendsingle+/integraldisplayt\n0S1−p\nk(t−τ′)/vextendsingle/vextendsingleLk(τ′,k(t−τ′)ap)/vextendsingle/vextendsingledτ′\n+/integraldisplayt\n0S1−p\nk(t−τ′)/vextendsingle/vextendsingleNk(τ′,k(t−τ′)ap)/vextendsingle/vextendsingledτ′, (5.6)\nfor any fixed p∈(0,1). Similarly, in view of (5.4) and (B.3), the first three term s on the right\nhand side of (5.1) and the fourth to sixth terms on the right si de of (5.2) can be bounded by\nS1−p\nk(t)e−νt/vextendsingle/vextendsingle/vextendsingle(∇η/hatwidefin)k(ktap)/vextendsingle/vextendsingle/vextendsingle+/integraldisplayt\n0S1−p\nk(t−τ)e−νt/vextendsingle/vextendsingle[∇ηL]k(τ,k(t−τ)ap)/vextendsingle/vextendsingledτ\n+/integraldisplayt\n0S1−p\nk(t−τ)e−νt/vextendsingle/vextendsingle[∇ηN]k(τ,k(t−τ)ap)/vextendsingle/vextendsingledτ, (5.7)\nfor any fixed p∈(0,1). We would like to remark that, thanks to the property (B.2) possessed\nbySk(·), the index pin (5.6) and (5.7) does not matter and Sk(·)1−pcan serve as Sk(·). In\naddition, we have to deal with the first three terms on the righ t hand side of (5.2), among\nwhich and the six terms in (5.6) and (5.7), the second and thir d terms on the right hand\nside of (5.2) are the most representative and complicated. B efore proceeding any further, let\nus write [∆ ηL]k(τ,k(t−τ)ap) and [∆ ηN]k(τ,k(t−τ)ap) explicitly. From (2.12), one easily\ndeduces that\n[∆ηL]k(τ,k(t−τ)ap) =2e2ντˆρk(τ)k\n|k|2·(∇ηˆµ)(k(t−τ)ap)\n+e2ντˆρk(τ)(t−τ)ap(∆ηˆµ)(k(t−τ)ap). (5.8)\nRecalling the definition of Nk(τ,¯η(τ;k,η)) in (2.13), after lengthy but routine calculations,\nwe obtain\n[∆ηN]k(τ,k(t−τ)ap) = I+II+III , (5.9)\nwhere\nI =−2ieντ/summationdisplay\nl∈Zn∗ˆEl(τ)·(∇ηˆf)k−l(τ,ktap−lτap)\n−i/summationdisplay\nl∈Zn∗ˆEl(τ)·k(t−τ)ap(∆ηˆf)k−l(τ,ktap−lτap) = I1+I2, (5.10)\nII =−2νe2ντ(/hatwiderMθ)k(τ)ˆµ(k(t−τ)ap)\n−4νe2ντ(/hatwiderMθ)k(τ)k(t−τ)ap·(∇ηˆµ)(k(t−τ)ap)\n−νe2ντ(/hatwiderMθ)k(τ)|k(t−τ)ap|2(∆ηˆµ)(k(t−τ)ap)\n−2iνe2ντ(/hatwiderM1)k(τ)·(∇ηˆµ)(k(t−τ)ap)\n−iνe2ντ(/hatwiderM1)k(τ)·k(t−τ)ap(∆ηˆµ)(k(t−τ)ap), (5.11)\nIII =−2νe2ντ/summationdisplay\nl∈Zn∗ˆρl(τ)ˆfk−l(τ,ktap−lτap)38 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n−4νeντ/summationdisplay\nl∈Zn∗ˆρl(τ)k(t−τ)ap·(∇ηˆf)k−l(τ,ktap−lτap)\n−ν/summationdisplay\nl∈Zn∗ˆρl(τ)|k(t−τ)ap|2(∆ηˆf)k−l(τ,ktap−lτap)\n−2ν/summationdisplay\nl∈Zn∗ˆρl(τ)(∆ηˆf)k−l(τ,ktap−lτap)\n−ν/summationdisplay\nl∈Zn∗ˆρl(τ)e−ντk(t−τ)ap·(∇η∆ηˆf)k−l(τ,ktap−lτap)\n−2νe2ντ/summationdisplay\nl∈Zn(/hatwiderMθ)l(τ)ˆfk−l(τ,ktap−lτap)\n−4νeντ/summationdisplay\nl∈Zn(/hatwiderMθ)l(τ)k(t−τ)ap·(∇ηˆf)k−l(τ,ktap−lτap)\n−ν/summationdisplay\nl∈Zn(/hatwiderMθ)l(τ)|k(t−τ)ap|2(∆ηˆf)k−l(τ,ktap−lτap)\n−2νieντ/summationdisplay\nl∈Zn∗(/hatwiderM1)l(τ)·(∇ηˆf)k−l(τ,ktap−lτap)\n−νi/summationdisplay\nl∈Zn∗(/hatwiderM1)l(τ)·k(t−τ)ap(∆ηˆf)k−l(τ,ktap−lτap). (5.12)\nNow we first consider\n(5.13)/summationdisplay\nk∈Zn∗/integraldisplayT∗\n0/bracketleftbigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)e−2νt/integraldisplayt\n0Sk(t−τ)[∆ηL]k(τ,k(t−τ)ap)dτ/bracketrightbigg2\ndt.\nThe treatment of (5.13) is slightly more direct than that of NE;0in Section 4.1. Indeed,\nsimilar to (4.39) and (4.43), it holds that\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)/lessorsimilarS1\n2\nk(t−τ)/an}b∇acketle{tτ/an}b∇acket∇i}htbBk(τ)ecλ(τ)/an}b∇acketle{te−ντk(t−τ)ap/an}b∇acket∇i}hts\n+S1\n2\nk(t−τ)/parenleftbigg\n/an}b∇acketle{tτ/an}b∇acket∇i}htbeδν1\n3τ|k|1\n2ecλ(τ)/angb∇acketleftk,kτap/angb∇acket∇ights/parenrightbigg\n×eλ(τ)/an}b∇acketle{te−ντk(t−τ)ap/an}b∇acket∇i}hts/angbracketleftbig\ne−ντk(t−τ)ap/angbracketrightbigσ0\n/lessorsimilarS1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2/parenleftBig\n/an}b∇acketle{tτ/an}b∇acket∇i}htbBk(τ)/parenrightBig\neλ(0)/angb∇acketleftk(t−τ)ap/angb∇acket∇ights/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}htσ0+2.\nThen similar to (4.41), we arrive at\n/summationdisplay\nk∈Zn∗/integraldisplayT∗\n0/bracketleftbigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)e−2νt/integraldisplayt\n0Sk(t−τ)[∆ηL]k(τ,k(t−τ)ap)dτ/bracketrightbigg2\ndt\n/lessorsimilar/summationdisplay\n|α|≤2/summationdisplay\nk∈Zn∗/integraldisplayT∗\n0/bracketleftBigg/integraldisplayt\n0/an}b∇acketle{tτ/an}b∇acket∇i}htb|Bˆρk(τ)|S1\n2\nk(t−τ)\n/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}ht2eλ(0)/angb∇acketleftk(t−τ)ap/angb∇acket∇ights\n×/an}b∇acketle{tk(t−τ)ap/an}b∇acket∇i}htσ0+3/vextendsingle/vextendsingle(∂α\nηˆµ)(k(t−τ)ap)/vextendsingle/vextendsingledτ/bracketrightBigg2\ndt\n/lessorsimilar\n/summationdisplay\n|α|≤2/vextenddouble/vextenddouble/vextenddoubleeλ(0)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ0+3(∂α\nηˆµ)(η)/vextenddouble/vextenddouble/vextenddouble\nL∞η\n2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x/lessorsimilar(C0ǫνγ)2, (5.14)LANDAU DAMPING FOR VPFP 39\nwhich is consistent with Proposition 3.2 provided we choose C2≫C0.\nNext we turn to investigate\n(5.15)/summationdisplay\nk∈Zn∗/integraldisplayT∗\n0/bracketleftbigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)e−2νt/integraldisplayt\n0Sk(t−τ)[∆ηN]k(τ,k(t−τ)ap)dτ/bracketrightbigg2\ndt.\nOn can see from (5.9)–(5.12) that there are many terms involv ed in (5.15), while these contri-\nbutions can be treated similarly by using the techniques in S ection 4.1. For the collisionless\ncontributions, two terms I 1and I2are involved, see (5.10). Clearly, I 2is the leading term,\nwhich leads to the contribution\n/summationdisplay\nk∈Zn∗/integraldisplayT∗\n0/bracketleftBigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)e−2νt/integraldisplayt\n0Sk(t−τ)/summationdisplay\nl∈Zn∗ˆρl(τ)l\n|l|2·k(t−τ)ap\n×(∆ηˆf)k−l(τ,ktap−lτap)dτ/bracketrightBigg2\ndt.\nThis can be treated exactly in the same as NEin Section 4.1.2 with ˆfk−l(τ,ktap−lτap)\nreplaced by e−2ντ(∆ηˆf)k−l(τ,ktap−lτap). For I 1, let us denote\n˜NE=/summationdisplay\nk∈Zn∗/integraldisplayT∗\n0/bracketleftBigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)e−2νt/integraldisplayt\n0Sk(t−τ)/summationdisplay\nl∈Zn∗ˆρl(τ)\n|l|2l·eντ(∇ηˆf)k−l(τ,ktap−lτap)dτ/bracketrightBigg2\ndt\n=˜NHL\nE;/negationslash=+˜NLH\nE;/negationslash=+˜NE;0.\nDue to the absence of k(t−τ)ap, it is easy to see that now ˜NLH\nE;/negationslash=and˜NE;0can be treated\nin the same way as NLH\nE;/negationslash=andNE;0, respectively, in Section 4.1.2 with ˆfk−l(τ,ktap−lτap)\nreplaced by e−ντ(∇ηˆf)k−l(τ,ktap−lτap). Compared with NHL\nE;/negationslash=in Section 4.1.2, ˜NHL\nE;/negationslash=is\nmuch easier to treat. Indeed, arguing as in Section 4.1.2 bas ed onτ≤t\n2ort\n2≤τ≤t,\nignoring the factor /an}b∇acketle{tτ/an}b∇acket∇i}ht1−3γin (4.20), for k/ne}ationslash=l, we infer from (4.14) that\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)Sk(t−τ)/vextendsingle/vextendsingle/vextendsinglee−ντ(∇ηˆf)k−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle1|k−l,ktap−lτap|≤2|l,lτap|\n/lessorsimilar/an}b∇acketle{tτ/an}b∇acket∇i}htbBl(τ)S1\n2\nk(t−τ)e−1\n2c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights/parenleftBig\neλ(τ)/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights/vextendsingle/vextendsingle/vextendsinglee−ντ(∇ηˆf)k−l(τ,ktap−lτap)/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n/lessorsimilar/an}b∇acketle{tτ/an}b∇acket∇i}htbBl(τ)S1\n2\nk(t−τ)e−1\n2c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ightse−1\n2δ1ν1\n3τsup\nt/vextenddouble/vextenddoublefw\n/negationslash=(t)/vextenddouble/vextenddouble\nEσ1,1\n3m.\nThen similar to (4.21), we immediately have\n˜NHL\nE;/negationslash=/lessorsimilarǫ2ν2γ/integraldisplayT∗\n0/summationdisplay\nk∈Zn∗/bracketleftBigg/summationdisplay\nl∈Zn∗/integraldisplayt\n0S1\n2\nk(t−τ)e−1\n2δ1ν1\n3τe−c/angb∇acketleftk−l,ktap−lτap/angb∇acket∇ights|Bˆρl(τ)|dτ/bracketrightBigg2\ndt\n/lessorsimilarǫ2ν2γ/ba∇dblBρ/ba∇dbl2\nL2\nt,x. (5.16)\nThe collisional contributions stemming from II and III can b e treated by following the way\nof treating the collisions in Section 4.1.2 without additio nal complications. To explain the\nappearance of up to the third derivative on ˆf0(t,η) with respect to ηin (3.7d), here we only\nsketch the estimate of the term that ( ∇η∆ηˆf)0(τ,k(tap−τap)) is involved. To this end, let\nus denote\nIII5;0=−νˆρk(τ)e−ντk(t−τ)ap·(∇η∆ηˆf)0(τ,k(tap−τap)).40 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nSimilar to (4.41) and (4.44), we have\n/summationdisplay\nk∈Zn∗/integraldisplayT∗\n0/bracketleftbigg\n/an}b∇acketle{tt/an}b∇acket∇i}htbBk(t)e−2νt/integraldisplayt\n0Sk(t−τ)III5;0dτ/bracketrightbigg2\ndt\n/lessorsimilarν2/bracketleftBigg\nsup\nt/vextenddouble/vextenddouble/vextenddoublee−3νteλ(t)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teνtη/an}b∇acket∇i}ht3(∇η∆ηˆf)0(t,η)/vextenddouble/vextenddouble/vextenddouble2\nL∞η\n+ sup\n0≤τ≤T∗/parenleftbigg\ne−(m+3)ντ/vextenddouble/vextenddouble/vextenddouble(Aσ0+1,2\n5∇η∆ηˆf)0(τ)/vextenddouble/vextenddouble/vextenddouble\nHn−1\n2+\nη/parenrightbigg2/bracketrightBigg/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x\n/lessorsimilarǫ2ν2γ+2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tt/an}b∇acket∇i}htbBρ/vextenddouble/vextenddouble/vextenddouble2\nL2\nt,x. (5.17)\n6.Estimates on the distribution function\nIn this section, we improve (3.7a)–(3.7c). The following pa rtition of unity\n(6.1) 1 = 1|l,ltap|<1\n2|k−l,η−ltap|+1|k−l,η−ltap|≤2|l,ltap|\nwill be used frequently.\nLemma 6.1. Let(σ,e)∈/braceleftbig\n(σ0+1,2\n5),(σ1,1\n3),(σ1,2\n5)/bracerightbig\n. Then there exists a positive constant\nc∈(0,1)such that\nAσ,e\nk(t,η)1|l,ltap|<1\n2|k−l,η−ltap|/lessorsimilare−(δν1\n3−δ1νe)teδν1\n3tecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ightsAσ,e\nk−l(t,η−ltap), (6.2)\nand\nAσ,e\nk(t,η)1|k−l,η−ltap|≤2|l,ltap|/lessorsimilarecλ(t)/angb∇acketleftk−l,η−ltap/angb∇acket∇ights/parenleftBig\neδ1νeteλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights/an}b∇acketle{tl,ltap/an}b∇acket∇i}htσ/parenrightBig\n. (6.3)\nIn particular, if k=l, and(σ,e) = (σ0+1,2\n5), we have\nAσ0+1,2\n5\nk(t,η)1|η−ktap|≤2|k,ktap|/lessorsimilare−s(δν1\n3−δ1ν2\n5)t/an}b∇acketle{tt/an}b∇acket∇i}htsecλ(t)/angb∇acketleftη−ktap/angb∇acket∇ights|k|s\n2(Bk(t))s\n×/parenleftbigg\neδ1ν2\n5teλ(t)/angb∇acketleftk,ktap/angb∇acket∇ights/an}b∇acketle{tk,ktap/an}b∇acket∇i}htσ0+1/parenrightbigg1−s\n. (6.4)\nProof.Noting that |k,η| ≤ |k−l,η−ltap|+|l,ltap|, using (A.3) and (A.5), we get (6.2) and\n(6.3) respectively. To prove (6.4), using (A.5) again yield s\nAσ0+1,2\n5\nk(t,η)1|η−ktap|≤2|k,ktap|/lessorsimilarecλ(t)/angb∇acketleftη−ktap/angb∇acket∇ights/parenleftbigg\neδ1ν2\n5teλ(t)/angb∇acketleftk,ktap/angb∇acket∇ights/an}b∇acketle{tk,ktap/an}b∇acket∇i}htσ0+1/parenrightbigg1−s\n×/parenleftbigg\ne−(δν1\n3−δ1ν2\n5)teδν1\n3teλ(t)/angb∇acketleftk,ktap/angb∇acket∇ights/an}b∇acketle{tk,ktap/an}b∇acket∇i}htσ0+1/parenrightbiggs\n.\nCombining this with the definition of Bk(t) and the fact 1k/negationslash=0/an}b∇acketle{tk,ktap/an}b∇acket∇i}ht/lessorsimilar|k|/an}b∇acketle{ttap/an}b∇acket∇i}ht/lessorsimilar|k|/an}b∇acketle{tt/an}b∇acket∇i}ht, we\nfind that (6.4) follows immediately. The proof of Lemma 6.1 is completed. /square\nLetA(t,∇)∈/braceleftBig\nAσ0+1,2\n5(t,∇),Aσ1,1\n3(t,∇),Aσ1,2\n5(t,∇)/bracerightBig\n,α∈Nnbe any fixed multi-index.\nFrom (1.27), (1.30) and (2.4), we have the following energy i dentity\n1\n2d\ndt/parenleftBig\ne−2|α|νt/ba∇dblA(vαfw)/ba∇dbl2\nL2/parenrightBig\n−e−2|α|νt˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble2\nL2\n+e−2|α|νt/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicalbigg\n−∂tm\nmA(vαfw)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2+νe−2|α|νt/vextenddouble/vextenddouble∂t\nvA(vαfw)/vextenddouble/vextenddouble2\nL2\n+ν|α|e−2|α|νt/ba∇dblA(vαfw)/ba∇dbl2\nL2+2nνλ1(t)e−2|α|νt/ba∇dblA(vαfw)/ba∇dbl2\nL2LANDAU DAMPING FOR VPFP 41\n+/parenleftBig\n2νλ1(1−2λ1)−˙λ1(t)/parenrightBig\ne−2(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dbl2\nL2\n=δ1νee−2|α|νt/vextenddouble/vextenddoubleA(vαfw\n/negationslash=)/vextenddouble/vextenddouble2\nL2+L+CL+/summationdisplay\n1≤j≤4LEj+NEE+N+CN0+CN1+CN2, (6.5)\nwheree∈ {1\n3,2\n5}is determined by A(t,∇), and\nL=−e−2|α|νtRe/summationdisplay\nk∈Zn/integraldisplay\nηAk(t,η)ˆEk(t)·∂α\nη/parenleftBig\nF/bracketleftBig\n∇vµeλ1|v|2/bracketrightBig\n(¯η(t;k,η))/parenrightBig\nA∂α\nη(/hatwiderfw)k(t,η)dη,\nCL=νe−2|α|νtRe/summationdisplay\nk∈Zn/integraldisplay\nηA∂α\nη/parenleftBig\nF/bracketleftBig\n(Mθ∆vµ−M1·∇vµ)eλ1|v|2/bracketrightBig\nk(t,¯η(t;k,η))/parenrightBig\n×A∂α\nη(/hatwiderfw)k(t,η)dη,\nLE1=2˙λ1(t)e−2(|α|+1)νtRe/summationdisplay\nk∈Zn/integraldisplay\nηA(∇η∂α\nη/hatwiderfw)k(t,η)·∇ηAk(t,η)∂α\nη(/hatwiderfw)k(t,η)dη,\nLE2=−4νλ1(1−2λ1)e−2(|α|+1)νtRe/summationdisplay\nk∈Zn/integraldisplay\nηA(∇η∂α\nη/hatwiderfw)k(t,η)·∇ηAk(t,η)∂α\nη(/hatwiderfw)k(t,η)dη,\nLE3=4νλ1e−νte−2|α|νtRe/summationdisplay\nk∈Zn/integraldisplay\nηA∇η·∂α\nη/parenleftBig\n¯η(t;k,η)/hatwiderfw/parenrightBig\nk(t,η)A∂α\nη(/hatwiderfw)k(t,η)dη,\nLE4=−νe−2|α|νtRe/summationdisplay\nk∈Zn/integraldisplay\nηA[∂α\nη,|¯η|2](/hatwiderfw)k(t,η)A∂α\nη(/hatwiderfw)k(t,η)dη,\nNEE=2λ1e−2|α|νtRe/summationdisplay\nk,l∈Zn/integraldisplay\nηAk(t,η)ˆEl(t)·ie−νt(∇η∂α\nη/hatwiderfw)k−l(t,η−ltap)A∂α\nη(/hatwiderfw)k(t,η)dη,\nN=−e−2|α|νtRe/summationdisplay\nk,l∈Zn/integraldisplay\nηAˆEl(t)·i∂α\nη/bracketleftBig\n¯η(t;k,η)(/hatwiderfw)k−l(t,η−ltap)/bracketrightBig\nA∂α\nη(/hatwiderfw)k(t,η)dη,\nCNℓ=e−2|α|νtRe/summationdisplay\nk∈Zn/integraldisplay\nηA∂α\nη/parenleftbig\nF/bracketleftbig\nMℓ[gw]/bracketrightbig\nk(t,¯η(t;k,η))/parenrightbig\nA∂α\nη(/hatwiderfw)k(t,η)dη, ℓ= 0,1,2.\n6.1.Linear contributions. In this section, we study the linear contributions, namely, we\ntreatL,CL, andLEj, j= 1,2,3,4. Let us consider CLfirst, which is from the linearization of\nthe Fokker-Planck operator. The two terms in CLcan be treated in the same way, it suffices\nto consider\nCLMθ=νe−2|α|νtRe/summationdisplay\nk∈Zn∗/integraldisplay\nηAk(η)(/hatwiderMθ)k(t)∂α\nη/parenleftBig\nF/bracketleftBig\n∆vµeλ1|v|2/bracketrightBig\n(¯η(t;k,η))/parenrightBig\nA∂α\nη(/hatwiderfw)k(η)dη\n+νe−2|α|νtRe/integraldisplay\nηA0(t,η)(/hatwiderMθ)0(t)∂α\nη/parenleftBig\nF/bracketleftBig\n∆vµeλ1|v|2/bracketrightBig\n(eνtη)/parenrightBig\nA∂α\nη(/hatwiderfw)0(t,η)dη\n=CLMθ;/negationslash=+CLMθ;0. (6.6)\nNote that\n∂α\nη/parenleftBig\nF/bracketleftBig\n∆vµeλ1|v|2/bracketrightBig\n(t,¯η(t;k,η))/parenrightBig\n=e|α|νt∂α\nη/parenleftBig\nF/bracketleftBig\n∆vµeλ1|v|2/bracketrightBig/parenrightBig\n(t,¯η(t;k,η)). (6.7)\nThen\nCLMθ;0≤νe−|α|νt/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(t)/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nηeλ(0)/angb∇acketlefteνtη/angb∇acket∇ights/an}b∇acketle{teνtη/an}b∇acket∇i}htσ0+1/vextendsingle/vextendsingle/vextendsingle∂α\nη/parenleftBig\nF/bracketleftBig\n∆vµeλ1|v|2/bracketrightBig/parenrightBig\n(t,eνtη)/vextendsingle/vextendsingle/vextendsingle\n×/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)0(t,η)/vextendsingle/vextendsingle/vextendsingledη42 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n/lessorsimilarνe−n\n2νt/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(t)/vextendsingle/vextendsingle/vextendsingle/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n, (6.8)\nwhere we have used the fact\n(6.9)/parenleftbigg/integraldisplay\nηe2λ(0)/angb∇acketlefteνtη/angb∇acket∇ights/an}b∇acketle{teνtη/an}b∇acket∇i}ht2(σ0+1)/vextendsingle/vextendsingle/vextendsingle∂α\nη/parenleftBig\nF/bracketleftBig\n∆vµeλ1|v|2/bracketrightBig/parenrightBig\n(t,eνtη)/vextendsingle/vextendsingle/vextendsingle2\ndη/parenrightbigg1\n2\n/lessorsimilare−n\n2νt.\nSince/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(t)/vextendsingle/vextendsingle/vextendsingledecays very fast, see (3.12), the estimate (6.8) is consiste nt with Proposition\n3.2 for sufficiently small ν.\nNext turn to CLMθ;/negationslash=. By using the partition of unity\n(6.10) 1 = 1|k,ktap|<1\n2|η−ktap|+1|η−ktap|≤2|k,ktap|,\nwe divide CLMθ;/negationslash=into two parts:\nCLMθ;/negationslash=/lessorsimilarνe−|α|νt/summationdisplay\nk∈Zn∗/integraldisplay\nηAk(t,η)/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)k(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂α\nη/parenleftBig\nF/bracketleftBig\n∆vµeλ1|v|2/bracketrightBig/parenrightBig\n(t,¯η(t;k,η))/vextendsingle/vextendsingle/vextendsingle\n×/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingledη=CLLH\nMθ;/negationslash=+CLHL\nMθ;/negationslash=.\nNow (6.2) reduces to\nAk(t,η)1|k,ktap|<1\n2|η−ktap|/lessorsimilareδ1νetecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ightseλ(t)/angb∇acketleftη−ktap/angb∇acket∇ights/an}b∇acketle{tη−ktap/an}b∇acket∇i}htσ, (6.11)\nforσ∈ {σ0+1,σ1}. It follows this and (6.9) that\nCLLH\nMθ;/negationslash=/lessorsimilarνe−|α|νt/summationdisplay\nk∈Zn∗eδ1νetecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ights/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)k(t)/vextendsingle/vextendsingle/vextendsingle/vextenddouble/vextenddouble/vextenddoubleA∂α\nη(/hatwiderfw)k(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2η\n/lessorsimilarνe−(δ−δ1)ν1\n3t/ba∇dblBMθ/ba∇dblL2x/parenleftBig\ne−|α|νt/vextenddouble/vextenddoubleA(vαfw\n/negationslash=)/vextenddouble/vextenddouble\nL2/parenrightBig\n. (6.12)\nThe estimate of CLHL\nMθ;/negationslash=is tricker when ( σ,e) = (σ0+ 1,2\n5). Indeed, from (6.4), (6.7), and\n(6.9), we use a interpolation technic to obtain\nCLHL\nMθ;/negationslash=/lessorsimilarν/an}b∇acketle{tt/an}b∇acket∇i}htse−s(δ−δ1)ν1\n3te−|α|νt/summationdisplay\nk∈Zn∗/integraldisplay\nη|k|s\n2/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingleAσ0+1,2\n5\nk(t,ktap)(/hatwiderMθ)k(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−s/vextendsingle/vextendsingle/vextendsingleB(/hatwiderMθ)k(t)/vextendsingle/vextendsingle/vextendsingles\n×ecλ(t)/angb∇acketleftη−ktap/angb∇acket∇ights/vextendsingle/vextendsingle/vextendsingle∂α\nη/parenleftBig\nF/bracketleftBig\n∆vµeλ1|v|2/bracketrightBig/parenrightBig\n(t,¯η(t;k,η))/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilarν/an}b∇acketle{tt/an}b∇acket∇i}htse−s(δ−δ1)ν1\n3te−|α|νt/summationdisplay\nk∈Zn∗/vextenddouble/vextenddouble/vextenddouble|k|s\n2A∂α\nη(/hatwiderfw)k(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2η\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingleAσ0+1,2\n5\nk(t,ktap)(/hatwiderMθ)k(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−s/vextendsingle/vextendsingle/vextendsingleB(/hatwiderMθ)k(t)/vextendsingle/vextendsingle/vextendsingles\n/lessorsimilarν/an}b∇acketle{tt/an}b∇acket∇i}hts\n/radicalBig\n−˙λ(t)e−s(δ−δ1)ν1\n3te(1−s)mνt/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n×/ba∇dblBMθ/ba∇dbls\nL2/parenleftBigg\ne−mνt/vextenddouble/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0Aσ0+1,2\n5\nk(t,ktap)(/hatwiderMθ)k(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk/parenrightBigg1−s\n. (6.13)\nNote that\nν/an}b∇acketle{tt/an}b∇acket∇i}hts\n/radicalBig\n−˙λ(t)e−s(δ−δ1)ν1\n3te(1−s)mνt/lessorsimilarν/an}b∇acketle{tt/an}b∇acket∇i}hts+1\n2+e−s\n2(δ−δ1)ν1\n3t/lessorsimilarν2\n3. (6.14)LANDAU DAMPING FOR VPFP 43\nThen it follows from the above two inequalities and (3.9) tha t\n/summationdisplay\n|α|≤m/integraldisplayt\n0CLHL\nMθ;/negationslash=(t′)\nK|α|dt′/lessorsimilarν1\n2+s\n6\n/integraldisplayt\n0/summationdisplay\n|α|≤m′e−2|α|νt′\nK|α|(−˙λ(t′))/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble2\nL2dt′\n1\n2\n×/ba∇dblBρ/ba∇dbls\nL2\nt,x/parenleftbigg\nν1\n6/ba∇dblfw\n/negationslash=(t)/ba∇dbl\nL2\ntEσ0+1,2\n5m/parenrightbigg1−s\n. (6.15)\nBoth (6.12) and (6.15) are consistent with Proposition 3.2 f or sufficiently small ν. When\n(σ,e) = (σ1,1\n3) or (σ1,2\n5),CLHL\nMθ;/negationslash=can be bounded in the same way as CLLH\nMθ;/negationslash=, see (6.12).\nCompared with CL,Lis easier to treat, since1\n|k|hidden in ˆEk(t) will eliminate the extra\nhalf derivation on ρwhenρis at high frequency and σ=σ0+1. Indeed, for σ∈ {σ0+1,σ1}\nandk/ne}ationslash= 0, there holds\n(6.16)1\n|k|/an}b∇acketle{tk,ktap/an}b∇acket∇i}htσ/lessorsimilar|k|1\n2/an}b∇acketle{tk,ktap/an}b∇acket∇i}htσ0/an}b∇acketle{ttap/an}b∇acket∇i}ht.\nThen (6.3) reduces to\n(6.17) Ak(t,η)1|η−ktap|≤2|k,ktap|/lessorsimilarecλ(t)/angb∇acketleftη−ktap/angb∇acket∇ightse−(δ−δ1)ν1\n3t/an}b∇acketle{tt/an}b∇acket∇i}htBk(t).\nCombining this with (6.11), similar to (6.12), we are led to\n(6.18) L/lessorsimilare−(δ−δ1)ν1\n3t1\n/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+/parenleftBig\n/an}b∇acketle{tt/an}b∇acket∇i}ht3\n2+/ba∇dblBρ/ba∇dblL2x/parenrightBig/parenleftBig\ne−|α|νt/vextenddouble/vextenddoubleA(vαfw\n/negationslash=)/vextenddouble/vextenddouble\nL2/parenrightBig\n,\nwhich is consistent with Proposition 3.2 provided we set C f≫C0.\nTo bound LE1andLE2, note that for σ∈ {σ0+1,σ1}, there holds\n∇ηAk(t,η) =/parenleftbigg\nsλ(t)|k,η|∇η|η|\n/an}b∇acketle{tk,η/an}b∇acket∇i}ht2−s+σ|k,η|∇η|η|\n/an}b∇acketle{tk,η/an}b∇acket∇i}ht2+∇ηmk(t,η)\nmk(t,η)/parenrightbigg\nAk(t,η).\nIfk/ne}ationslash= 0, combining this with (C.2) yields\n(6.19) |∇ηAk(t,η)|/lessorsimilar/parenleftbigg1\n/an}b∇acketle{tk,η/an}b∇acket∇i}ht1−s+eνtν1\n3/parenrightbigg\nAk(t,η)/lessorsimilareνtAk(t,η).\nWhereas for k= 0, there holds\n|∇ηA0(t,η)|/lessorsimilar|η|\n/an}b∇acketle{tη/an}b∇acket∇i}ht2−sA0(t,η)/lessorsimilar|η|A0(t,η). (6.20)\nNowwetreat LE1, whichcomes fromthetime-dependentGaussianweight eλ1|v|2. Bychoosing\n|˙λ1(t)|small enough, we gain smallness even for this linear term. Mo re precisely, by the\ndefinition of λ1(t) in (1.21), choosing 0 <a0≪1 but independent of ν, we have\n−˙λ1(t) =a0νt\n/an}b∇acketle{tt/an}b∇acket∇i}hta0+2≪ν.\nIt follows from the above three inequalities that\nLE1/lessorsimilar−˙λ1(t)e−(2|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenleftbig\n/ba∇dblA(vαfw\n/negationslash=)/ba∇dblL2+/vextenddouble/vextenddouble∂t\nvA0(vαfw\n0)/vextenddouble/vextenddouble\nL2/parenrightbig\n≤1\n16(−˙λ1(t))e−2(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dbl2\nL2\n+C(−˙λ1(t))e−2|α|νt/parenleftBig\n/ba∇dblA(vαfw\n/negationslash=)/ba∇dbl2\nL2+/vextenddouble/vextenddouble∂t\nvA0(vαfw\n0)/vextenddouble/vextenddouble2\nL2/parenrightBig\n≤1\n16(−˙λ1(t))e−2(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dbl2\nL2\n+1\n16e−2|α|νt/parenleftBig\nν1\n3/ba∇dblA(vαfw\n/negationslash=)/ba∇dbl2\nL2+ν/vextenddouble/vextenddouble∂t\nvA0(vαfw\n0)/vextenddouble/vextenddouble2\nL2/parenrightBig\n. (6.21)44 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nForLE2, which comes from the commutator [ v,A(t,∇)], the definition of λ1(t) ensures that\nλ1(t)≈ν≪1. Then similar to (6.21), we obtain\nLE2≤1\n16νλ1(1−2λ1)e−2(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dbl2\nL2\n+1\n16e−2|α|νt/parenleftBig\nν1\n3/ba∇dblA(vαfw\n/negationslash=)/ba∇dbl2\nL2+ν/vextenddouble/vextenddouble∂t\nvA0(vαfw\n0)/vextenddouble/vextenddouble2\nL2/parenrightBig\n. (6.22)\nNow we turn to LE3andLE4. Here we gain the smallness from a well-chosen sequence of\nK|α|in the spirit of [1], which appears in the definition of our ene rgy functional.\nIf|α|>1, integrating by parts, LE3can be rewritten as\nLE3=−4νλ1e−νte−2|α|νtRe/summationdisplay\nk∈Zn/integraldisplay\nη¯η(t;k,η)A∂α\nη(/hatwiderfw)k(t,η)·A∇η∂α\nη(/hatwiderfw)k(t,η)dη\n−8νλ1e−νte−2|α|νtRe/summationdisplay\nk∈Zn/integraldisplay\nη¯η(t;k,η)A∂α\nη(/hatwiderfw)k(t,η)·∇ηA∂α\nη(/hatwiderfw)k(t,η)dη\n−4νλ1e−2|α|νt/summationdisplay\n|β|=1/parenleftbigg\nα\nβ/parenrightbigg\nRe/summationdisplay\nk∈Zn/integraldisplay\nηA∂α−β\nη(/hatwiderfw)k(t,η)∂β\nηη·A∇η∂α\nη(/hatwiderfw)k(t,η)dη\n−8νλ1e−2|α|νt/summationdisplay\n|β|=1/parenleftbigg\nα\nβ/parenrightbigg\nRe/summationdisplay\nk∈Zn/integraldisplay\nηA∂α−β\nη(/hatwiderfw)k(t,η)∂β\nηη·∇ηA∂α\nη(/hatwiderfw)k(t,η)dη\n=/summationdisplay\n1≤j≤4LE3;(j).\nIt is easy to see that\nLE3;(1)≤1\n16νλ1(1−2λ1)e−2(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dbl2\nL2\n+1\n16e−2|α|νt/parenleftBig\nν/vextenddouble/vextenddouble∂t\nvA(vαfw)/vextenddouble/vextenddouble2\nL2/parenrightBig\n, (6.23)\nand\nLE3;(3)≤1\n16νλ1(1−2λ1)e−2(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dbl2\nL2\n+CK|α|−1νλ1/summationdisplay\n|β|=1e−2(|α|−1)νt\nK|α|−1/vextenddouble/vextenddouble/vextenddoubleA(vα−βfw)/vextenddouble/vextenddouble/vextenddouble2\nL2. (6.24)\nUsing (6.19) and (6.20), we have\nLE3;(2)≤8νλ1e−2|α|νt/ba∇dbl∂t\nvA(vαf)/ba∇dblL2/parenleftbig\n/ba∇dblA(vαfw\n/negationslash=)/ba∇dblL2+/vextenddouble/vextenddouble∂t\nvA0(vαfw\n0)/vextenddouble/vextenddouble\nL2/parenrightbig\n≤1\n16e−2|α|νt/parenleftBig\nν1\n3/ba∇dblA(vαfw\n/negationslash=)/ba∇dbl2\nL2+ν/vextenddouble/vextenddouble∂t\nvA(vαfw)/vextenddouble/vextenddouble2\nL2/parenrightBig\n, (6.25)\nand\nLE3;(4)/lessorsimilarνλ1e−(2|α|−1)νt/summationdisplay\n|β|=1/ba∇dblA(vα−βfw)/ba∇dblL2/parenleftbig\n/ba∇dblA(vαfw\n/negationslash=)/ba∇dblL2+/vextenddouble/vextenddouble∂t\nvA0(vαfw\n0)/vextenddouble/vextenddouble\nL2/parenrightbig\n≤1\n16e−2|α|νt/parenleftBig\nν1\n3/ba∇dblA(vαfw\n/negationslash=)/ba∇dbl2\nL2+ν/vextenddouble/vextenddouble∂t\nvA(vαfw)/vextenddouble/vextenddouble2\nL2/parenrightBig\n+CK|α|−1νλ1/summationdisplay\n|β|=1e−2(|α|−1)νt\nK|α|−1/vextenddouble/vextenddouble/vextenddoubleA(vα−βfw)/vextenddouble/vextenddouble/vextenddouble2\nL2. (6.26)\nThanks to the fact |α|>|β|= 1, dividing (6.5) by K|α|and summing with respect to α,\nthen the resulting summation arising from the last term on th e right-hand side of (6.24) andLANDAU DAMPING FOR VPFP 45\n(6.26) can be absorbed by\n(6.27) ν/summationdisplay\n1≤|α|≤m′e−2|α|νt\nK|α|/ba∇dblA(vαfw)/ba∇dbl2\nL2,withm′∈ {m,m+2},\nby choosing K|α|≫K|α|−1(even though ignoring the smallness in λ1).\nIf|α|= 0, then LE3;(3)andLE3,(4)will not appear, we just need to estimate LE3by (6.23)\nand (6.25). If |α|= 1, we have\nLE3=LE3;(1)+LE3;(2)+4νλ1e−2|α|νt/ba∇dblA(vαfw)/ba∇dbl2\nL2.\nThe last term above can be absorbed directly by ν|α|e−2|α|νt/ba∇dblA(vαfw)/ba∇dbl2\nL2on the left hand\nside of (6.5).\nForLE4, if|α|>2, we write\nLE4=−2νeνte−2|α|νt/summationdisplay\n|β|=1/parenleftbigg\nα\nβ/parenrightbigg\nRe/summationdisplay\nk∈Zn/integraldisplay\nη¯η(t;k,η)βA∂α−β\nη(/hatwiderfw)k(t,η)\n×A∂α\nη(/hatwiderfw)k(t,η)dη\n−2νe2νte−2|α|νt/summationdisplay\n|β|=2/parenleftbigg\nα\nβ/parenrightbigg\ncβRe/summationdisplay\nk∈Zn/integraldisplay\nηA∂α−β\nη(/hatwiderfw)k(t,η)A∂α\nη(/hatwiderfw)k(t,η)dη\n=LE4;(1)+LE4;(2), (6.28)\nwhere\ncβ=/braceleftBigg\n1,if only one component of βis nonzero,\n0,otherwise.\nThen similar to (6.26), one deduces that\nLE4≤1\n16νe−2|α|νt/parenleftBig/vextenddouble/vextenddouble∂t\nvA(vαfw)/vextenddouble/vextenddouble2\nL2+1|α|>2/ba∇dblA(vαfw)/ba∇dbl2\nL2/parenrightBig\n+CK|α|−1ν/summationdisplay\n|β|=1e−2(|α|−1)νt\nK|α|−1/vextenddouble/vextenddouble/vextenddoubleA(vα−βfw)/vextenddouble/vextenddouble/vextenddouble2\nL2\n+CK|α|−2ν/summationdisplay\n|β|=2e−2(|α|−2)νt\nK|α|−2cβ/vextenddouble/vextenddouble/vextenddoubleA(vα−βfw)/vextenddouble/vextenddouble/vextenddouble2\nL2. (6.29)\nSince|α|>2,|α−β|>0 for|β|= 1,2. Then by choosing K|α|≫K|α|−1≫K|α|−2, the last\ntwo terms on the right hand side of (6.29) can be treated with t he aid of (6.27) analogously\nto (6.26).\nIf|α|= 1, we have\nLE4=−2νeνte−2|α|νtRe/summationdisplay\nk∈Zn/integraldisplay\nη¯ηα(t;k,η)A(/hatwiderfw)k(t,η)A∂α\nη(/hatwiderfw)k(t,η)dη\n≤2ν/vextenddouble/vextenddouble∂t\nvAfw/vextenddouble/vextenddouble\nL2/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n≤1\n16νe−2|α|νt/ba∇dblA(vαfw)/ba∇dbl2\nL2+16K0/parenleftbigg\nν1\nK0/vextenddouble/vextenddouble∂t\nvAfw/vextenddouble/vextenddouble2\nL2/parenrightbigg\n. (6.30)\nThe first term on the right hand side of (6.30) can be absorbed b yν|α|e−2|α|νt/ba∇dblA(vαfw)/ba∇dbl2\nL2\non the left hand side of (6.5). After dividing (6.5) by K|α|and summing with respect to α,\nthe second term on the right hand side of (6.30) can be absorbe d by\n(6.31) ν/summationdisplay\n|α|≤m′e−2|α|νt\nK|α|/vextenddouble/vextenddouble∂t\nvA(vαfw)/vextenddouble/vextenddouble2\nL2,withm′∈ {m,m+2},46 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nas long asK1≫K0.\nIf|α|= 2,LE4;(1)can be treated in the same way as (6.29). Note that LE4;(2)will not\nappear unless α= (j,j),j= 1,2,···n. Integrating by parts, LE4;(2)reduces to\nLE4;(2)=−2νe−2(|α|−1)νtcαRe/summationdisplay\nk∈Zn/integraldisplay\nηA(/hatwiderfw)k(t,η)A∂α\nη(/hatwiderfw)k(t,η)dη\n=2K|α|−1νe−2(|α|−1)νt\nK|α|−1cα/ba∇dblA(vjfw)/ba∇dbl2\nL2\n+4νe−2(|α|−1)νtcαRe/summationdisplay\nk∈Zn/integraldisplay\nη∂η\njAk(t,η)(/hatwiderfw)k(t,η)A∂η\nj(/hatwiderfw)k(t,η)dη\n=LE4;(2,1)+LE4;(2,2), (6.32)\nfor somej∈ {1,2,···,n}. Using (6.19) and (6.20) and the property of mk(t,η) in (C.3), we\nfind that\nLE4;(2,2)/lessorsimilarνe−νtcα/ba∇dblA(vjfw)/ba∇dblL2/parenleftBig/vextenddouble/vextenddoubleAfw\n/negationslash=/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble∂t\nvAfw\n0/vextenddouble/vextenddouble\nL2/parenrightBig\n≤CK1cανe−2νt\nK1/ba∇dblA(vjfw)/ba∇dbl2\nL2+CK0cα1\nK0/parenleftBig\nν/vextenddouble/vextenddoubleAfw\n/negationslash=/vextenddouble/vextenddouble2\nL2+ν/vextenddouble/vextenddouble∂t\nvAfw\n0/vextenddouble/vextenddouble2\nL2/parenrightBig\n≤CK1cανe−2νt\nK1/ba∇dblA(vjfw)/ba∇dbl2\nL2\n+CK0cα1\nK0\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicalbigg\n−∂tm\nmA(fw\n/negationslash=)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2+ν/vextenddouble/vextenddouble∂t\nvA(fw)/vextenddouble/vextenddouble2\nL2\n. (6.33)\nLE4;(2,1)and the first term on the right hand side of (6.33) can be treate d with the aid of\n(6.27). The last term on the right hand side of (6.33) can be tr eated with the aid of the good\nterms (6.31) and\n(6.34)/summationdisplay\n|α|≤m′e−2|α|νt\nK|α|/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicalbigg\n−∂tm\nmA(vαfw)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2withm′∈ {m,m+2}.\n6.2.Nonlinear contributions. In this section, we consider the nonlinear contributions\nwhich contains the collisionless contributions and the col lision contributions. For the colli-\nsionless contributions, we take advantage of the transport structure. For the collision contri-\nbution, the main difficulty is to treat the velocity localizat ion loss.\n6.2.1.Collisionless contributions. To begin with, let us focus on the estimate of NEE. In\nfact, using (6.2), (6.3), (A.8) and (A.9), and (6.16) with kreplaced by l, we have\nNEE=2λ1e−νte−2|α|νt/summationdisplay\nk,l∈Zn/integraldisplay\nη(A∂α\nη/hatwiderfw)k(t,η)Ak(t,η)ˆρl(t)l\n|l|2·(∇η∂α\nη/hatwiderfw)k−l(t,η−ltap)dη\n/lessorsimilarλ1e−νte−2|α|νt/ba∇dblA(vαfw)/ba∇dblL2/ba∇dblA(vvαfw)/ba∇dblL2e−(δ−δ1)ν1\n3t/vextenddouble/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tk/an}b∇acket∇i}htn\n2+eδν1\n3tecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ightsˆρk(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk\n+λ1e−νte−2|α|νt/ba∇dblA(vαfw)/ba∇dblL2e−(δ−δ1)ν1\n3t/an}b∇acketle{ttap/an}b∇acket∇i}ht/ba∇dblBρ/ba∇dblL2x\n×/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tk/an}b∇acket∇i}htn\n2+ecλ(t)/angb∇acketleftk,η/angb∇acket∇ights(∇η∂α\nη/hatwiderfw)k(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2\n≤1\n16νλ1(1−2λ1)e−2(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dbl2\nL2+C/parenleftBig\n/an}b∇acketle{tt/an}b∇acket∇i}ht/ba∇dblBρ/ba∇dblL2x/parenrightBig2/parenleftBig\ne−2|α|νt/ba∇dblA(vαfw)/ba∇dbl2\nL2/parenrightBig\n.LANDAU DAMPING FOR VPFP 47\nNext we turn to the estimates of N, which can be split into two parts:\nN=−e−2|α|νtRe/summationdisplay\nk,l∈Zn/integraldisplay\nηAk(t,η)ˆEl(t)·i/bracketleftBig\n¯η(t;k,η)∂α\nη(/hatwiderfw)k−l(t,η−ltap)/bracketrightBig\n×/parenleftBig\nA∂α\nη(/hatwiderfw)/parenrightBig\nk(t,η)dη\n−ieνte−2|α|νt/summationdisplay\nβ≤α:|β|=1/parenleftbigg\nα\nβ/parenrightbigg\nRe/summationdisplay\nk,l∈Zn/integraldisplay\nηAk(t,η)ˆEβ\nl(t)∂α−β\nη(/hatwiderfw)k−l(t,η−ltap)\n×A∂α\nη(/hatwiderfw)k(t,η)dη=NM+NP.\nWe first treat NM. Two cases will be investigated according to e=2\n5or1\n3. It is worth\npointing out that when ( σ,e) = (σ1,2\n5), we cannot use the Aσ0+1,2\n5weighted high norm to\ncontrol the Aσ1,2\n5weighted low norm, since the latter is assigned more velocit y localizations.\nCase 1:A(t,∇)∈/braceleftBig\nAσ0+1,2\n5(t,∇),Aσ1,2\n5(t,∇)/bracerightBig\n.By using a classical commutator trick and\nfrequency decomposition, we rewrite NMas\nNM=−e−2|α|νtRe/summationdisplay\nk∈Zn/summationdisplay\nl∈Zn∗/integraldisplay\nη(Ak(t,η)−Ak−l(t,η−ltap))\n׈El(t)·/bracketleftBig\ni¯η(t;k,η)∂α\nη(/hatwiderfw)k−l(t,η−ltap)/bracketrightBig\nA∂α\nη(/hatwiderfw)k(t,η)dη=:NLH\nM+NHL\nM,\nwhere the splitting above is given according to the partitio n of unity (6.1).\nTreatment ofNLH\nM. To begin with, it is natural to divide the commutator into th e following\nfour parts:\nAk(t,η)\nAk−l(t,η−ltap)−1\n=eλ(t)(/angb∇acketleftk,η/angb∇acket∇ights−/angb∇acketleftk−l,η−ltap/angb∇acket∇ights)−1\n+eλ(t)(/angb∇acketleftk,η/angb∇acket∇ights−/angb∇acketleftk−l,η−ltap/angb∇acket∇ights)/parenleftbigg/an}b∇acketle{tk,η/an}b∇acket∇i}htσ\n/an}b∇acketle{tk−l,η−ltap/an}b∇acket∇i}htσ−1/parenrightbigg\n+eλ(t)(/angb∇acketleftk,η/angb∇acket∇ights−/angb∇acketleftk−l,η−ltap/angb∇acket∇ights)/an}b∇acketle{tk,η/an}b∇acket∇i}htσ\n/an}b∇acketle{tk−l,η−ltap/an}b∇acket∇i}htσ/parenleftbiggmk(t,η)\nmk−l(t,η−ltap)−1/parenrightbigg\n+eλ(t)(/angb∇acketleftk,η/angb∇acket∇ights−/angb∇acketleftk−l,η−ltap/angb∇acket∇ights)/an}b∇acketle{tk,η/an}b∇acket∇i}htσ\n/an}b∇acketle{tk−l,η−ltap/an}b∇acket∇i}htσmk(t,η)\nmk−l(t,η−ltap)\ne1k/negationslash=0δ1ν2\n5t\ne1k/negationslash=lδ1ν2\n5t−1\n\n=com1+com2+com3+com4,\nforσ∈ {σ0+1,σ1}. Then we denote\nNLH\nM=NLH\nM;1+NLH\nM;2+NLH\nM;3+NLH\nM;4.\nBy virtue of the elementary inequality |ex−1| ≤ |x|e|x|, the mean value theorem, (A.2),\n(A.3), and the fact\n(6.35) |¯η(t;k,η)|=eνt|(η−ltap)−(k−l)tap|/lessorsimilareνt/an}b∇acketle{ttap/an}b∇acket∇i}ht|k−l,η−ltap|,\nwe are led to\n|com1+com2||¯η(t;k,η)|1|l,ltap|<1\n2|k−l,η−ltap|\n/lessorsimilar|l,ltap|ecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ightseνt/an}b∇acketle{ttap/an}b∇acket∇i}ht/an}b∇acketle{tk−l,η−ltap/an}b∇acket∇i}hts\n2/an}b∇acketle{tk,η/an}b∇acket∇i}hts\n2.48 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nCombining this with (4.11), (A.8) and Lemma 2.1 yields for j= 1,2\nNLH\nM;j/lessorsimilare−2|α|νt/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nη/parenleftbigg\neδ1ν1\n3tecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights/an}b∇acketle{ttap/an}b∇acket∇i}ht3+a|ˆρl(t)|/parenrightbigg/parenleftbigg\n/an}b∇acketle{tνt/an}b∇acket∇i}hta+1eνte−δ1ν1\n3t/parenrightbigg\n×1\n/an}b∇acketle{tt/an}b∇acket∇i}hta+1/vextendsingle/vextendsingle/vextendsingle/vextendsingleAF/bracketleftBig\n/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2(vαfw)/bracketrightBig\nk−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleAF/bracketleftBig\n/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2(vαfw)/bracketrightBig\nk(t,η)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilar−e−2|α|νt˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble2\nL2/parenleftBigg\ne−mνt/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeδ1ν1\n3tF/bracketleftBig\neλ(t)/angb∇acketleft∇/angb∇acket∇ightsf/negationslash=/bracketrightBig\n(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nkL∞η/parenrightBigg\n/lessorsimilar−e−2|α|νt˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble2\nL2/ba∇dblfw(t)/ba∇dbl\nEσ1,1\n3m, (6.36)\nwhere we have used the relation ˆ ρl(t) =1l/negationslash=0ˆfl(t,ltap).\nNext we turn to NLH\nM;3. By (A.3) and (6.35), we have\nNLH\nM;3/lessorsimilare−2|α|νt/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights|ˆρl(t)|\n|l|eνt/an}b∇acketle{ttap/an}b∇acket∇i}ht|k−l,η−ltap|\n×/vextendsingle/vextendsingle/vextendsingle/vextendsinglemk(t,η)\nmk−l(t,η−ltap)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingledη\n=/parenleftBig/summationdisplay\nk∈Zn\n∗,l∈Zn\n∗,\nk/negationslash=l+/summationdisplay\nk=0,l∈Zn∗+/summationdisplay\nk∈Zn\n∗,l∈Zn\n∗,\nk=l/parenrightBig/integraldisplay\nη···dη=NLH\nM;3,(1)+NLH\nM;3,(2)+NLH\nM;3,(3). (6.37)\nTo bound NLH\nM;3,(1), we further split it into two parts:\nNLH\nM;3,(1)=e−2|α|νt/summationdisplay\nk∈Zn\n∗,l∈Zn\n∗,\nk/negationslash=l/integraldisplay\nη1|η−ltap|≤2|k−l|/angb∇acketlefttap/angb∇acket∇ight···dη\n+e−2|α|νt/summationdisplay\nk∈Zn\n∗,l∈Zn\n∗,\nk/negationslash=l/integraldisplay\nη1|η−ltap|>2|k−l|/angb∇acketlefttap/angb∇acket∇ight···dη=NLH;x\nM;3,(1)+NLH;v\nM;3,(1).\nIn view of (C.4) and (A.8), similar to (6.36), we find that\nNLH;x\nM;3,(1)/lessorsimilare−2|α|νt/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights|ˆρl(t)|eνt/an}b∇acketle{ttap/an}b∇acket∇i}ht2\n×/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilar1\n/an}b∇acketle{tt/an}b∇acket∇i}ht1+/parenleftBigg\ne−mνt/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeδ1ν1\n3tF/bracketleftBig\neλ(t)/angb∇acketleft∇/angb∇acket∇ightsf/negationslash=/bracketrightBig/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nkL∞η/parenrightBigg/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig2\n/lessorsimilar1\n/an}b∇acketle{tt/an}b∇acket∇i}ht1+/ba∇dblfw(t)/ba∇dbl\nEσ1,1\n3m/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig2\n.\nSimilarly, for NLH;v\nM;3,(1), we use (C.5), (4.11) and (A.8) to get\nNLH;v\nM;3,(1)/lessorsimilare−2|α|νt/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights|ˆρl(t)|\n|l|e2νt/an}b∇acketle{ttap/an}b∇acket∇i}htt/an}b∇acketle{tl,ltap/an}b∇acket∇i}ht\n×/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilar1\n/an}b∇acketle{tt/an}b∇acket∇i}ht1+/ba∇dblfw(t)/ba∇dbl\nEσ1,1\n3m/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig2\n.LANDAU DAMPING FOR VPFP 49\nNext consider NM;3,(2). In fact, from (C.6), we obtain\nNLH\nM;3,(2)/lessorsimilare−2|α|νt/summationdisplay\nl∈Zn∗/integraldisplay\nηecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights|ˆρl(t)|\n|l|eνt/an}b∇acketle{ttap/an}b∇acket∇i}ht(|l|+|η−ltap|)\n×min/braceleftbigg1\n|l|,/an}b∇acketle{tt/an}b∇acket∇i}ht\n|η−ltap|/bracerightbigg/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)0(t,η)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilar1\n/an}b∇acketle{tt/an}b∇acket∇i}ht1+/ba∇dblfw(t)/ba∇dbl\nEσ1,1\n3m/parenleftBig\ne−|α|νt/ba∇dblA(vαfw\n/negationslash=)/ba∇dblL2/parenrightBig/parenleftBig\ne−|α|νt/ba∇dblA(vαfw\n0)/ba∇dblL2/parenrightBig\n.\nSimilarly, we infer from (C.7) that\nNLH\nM;3,(3)/lessorsimilare−2|α|νt/summationdisplay\nk∈Zn∗/integraldisplay\nηecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ights|ˆρk(t)|\n|k|eνt/an}b∇acketle{ttap/an}b∇acket∇i}ht(|η|+|ktap|)\n×min/braceleftbigg1\n|k|,/an}b∇acketle{tt/an}b∇acket∇i}ht\n|η|/bracerightbigg/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)0(t,η−ktap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilar1\n/an}b∇acketle{tt/an}b∇acket∇i}ht1+/ba∇dblfw(t)/ba∇dbl\nEσ1,1\n3m/parenleftBig\ne−|α|νt/ba∇dblA(vαfw\n/negationslash=)/ba∇dblL2/parenrightBig/parenleftBig\ne−|α|νt/ba∇dblA(vαfw\n0)/ba∇dblL2/parenrightBig\n.\nNow we turn to NLH\nM;4, in which case the commutator com4does not gain regularity and\nwe use a interpolation argument. In fact, it is easy to see tha t\n|com4|1|l,ltap|<1\n2/vextendsingle/vextendsingle/vextendsinglek−l,η−l1−eνt\nν/vextendsingle/vextendsingle/vextendsingle/lessorsimilarecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights/parenleftbigg\n1k/negationslash=0,k=lδ1ν2\n5teδ1ν2\n5t+1k=0,k/negationslash=lδ1ν2\n5t/parenrightbigg\n.\nThen\nNLH\nM;4/lessorsimilare−2|α|νt/summationdisplay\nk∈Zn∗/integraldisplay\nηecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ightsν2\n5teδ1ν2\n5t|ˆρk(t)|\n|k|\n×|¯η(t;k,η)|/vextendsingle/vextendsingle/vextendsingleA/hatwider(vαfw)0(t,η−ktap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingledη\n+e−2|α|νt/summationdisplay\nk∈Zn∗/integraldisplay\nηecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ightsν2\n5t|ˆρk(t)|\n|k|\n×|¯η(t;0,η)|/vextendsingle/vextendsingle/vextendsingleA/hatwider(vαfw)−k(t,η−ktap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)0(t,η)/vextendsingle/vextendsingle/vextendsingledη=:NLH\nM;4,(1)+NLH\nM;4,(2).\nIf|η| ≤ |ktap|, then|¯η(t;k,η)|=eνt|η−ktap|/lessorsimilareνt|ktap|.Combining this with (4.11) yields\nNLH\nM;4,(1)/lessorsimilarν2\n5\n/an}b∇acketle{tt/an}b∇acket∇i}hte−2|α|νt/summationdisplay\nk∈Zn∗/integraldisplay\nη/parenleftbigg\neδν1\n3tecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ights|ktap|3|ˆρk(t)|/parenrightbigg\n×/parenleftbigg\n/an}b∇acketle{tνt/an}b∇acket∇i}ht2eνt+δ1ν2\n5t−δν1\n3t/parenrightbigg/vextendsingle/vextendsingle/vextendsingleA/hatwider(vαfw)0(t,η−ktap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilarν2\n5\n/an}b∇acketle{tt/an}b∇acket∇i}ht/ba∇dblBρ(t)/ba∇dblL2x/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig2\n. (6.38)\nIf|η|>|ktap|, then\n|¯η(t;k,η)|/vextendsingle/vextendsingle/vextendsingleA/hatwider(vαfw)0(t,η−ktap)/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilares\n2νt|η|s\n2|¯η(t;k,η)|1−s\n2/vextendsingle/vextendsingle/vextendsingleA/hatwider(vαfw)0(t,η−ktap)/vextendsingle/vextendsingle/vextendsingle\n=es\n2−sνtν−1−s\n2−s/an}b∇acketle{tt/an}b∇acket∇i}hta+1\n2−s|η|s\n2\n/an}b∇acketle{tt/an}b∇acket∇i}hta+1\n2/parenleftBigg\n1\n/an}b∇acketle{tt/an}b∇acket∇i}hta+1\n2/vextendsingle/vextendsingle/vextendsingleF/bracketleftBig\n|∂v|s\n2A(vαfw)/bracketrightBig\n0(t,η−ktap)/vextendsingle/vextendsingle/vextendsingle/parenrightBiggs\n2−s50 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n×/parenleftBig\nν1\n2|¯η(t;0,η−ktap)|/vextendsingle/vextendsingle/vextendsingleA/hatwider(vαfw)0(t,η−ktap)/vextendsingle/vextendsingle/vextendsingle/parenrightBig2−2s\n2−s\n=es\n2−sνtν−1−s\n2−s/an}b∇acketle{tt/an}b∇acket∇i}hta+1\n2−s|η|s\n2\n/an}b∇acketle{tt/an}b∇acket∇i}hta+1\n2(∗), (6.39)\nwhere we have used the fact\n(6.40) ¯ η(t;0,η−ktap) = ¯η(t;k,η).\nThus,\nNLH\nM;4,(1)/lessorsimilarνγ+2\n5−1−s\n2−se−2|α|νt/summationdisplay\nk∈Zn∗/integraldisplay\nην−γ/parenleftbigg\neδ1ν1\n3tecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ights|tap|a+1\n2−s+1|ˆρk(t)|/parenrightbigg\n×/parenleftbigg\n/an}b∇acketle{tνt/an}b∇acket∇i}hta+1\n2−s+1es\n2−sνt+δ1ν2\n5t−δ1ν1\n3t/parenrightbigg\n(∗)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle|η|s\n2\n/an}b∇acketle{tt/an}b∇acket∇i}hta+1\n2A∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilar/parenleftbigg\nν−γ/ba∇dblfw(t)/ba∇dbl\nEσ1,1\n3m/parenrightbigg/parenleftbigg/radicalBig\n−˙λ(t)e−|α|νt/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n×e−|α|νt/bracketleftbigg/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)0/vextenddouble/vextenddouble/vextenddouble\nL2v+ν1\n2/vextenddouble/vextenddouble∂t\nvA(vαfw)0/vextenddouble/vextenddouble\nL2v/bracketrightbigg\n, (6.41)\nwhere we have used the fact that (note that −1−s\n2−sis increasing in s, ands>1−3γ\n3−3γ)\nγ+2\n5−1−s\n2−s>γ+2\n5−1−1−3γ\n3−3γ\n2−1−3γ\n3−3γ=γ+2\n5+2\n3γ−5≥0 for 0 ≤γ≤1\n3.\nThe other term NLH\nM;4,(2)can be treated similarly. In fact, on the one hand, if |η−ktap| ≤\n|ktap|, we have |¯η(t;0,η)|=eνt|η|/lessorsimilareνt|ktap|. ThenNLH\nM;4,(2)can be treated in the same\nmanner as (6.38). On the other hand, if |η−ktap| ≥ |ktap|, then|¯η(t;0,η)|=eνt|η|/lessorsimilar\neνt|η−ktap|. Combining this with the fact ¯ η(t;0,η) = ¯η(t;−k,η−ktap), it is easy to see\nthat (6.39) still holds with ¯ η(t;k,η) andA/hatwider(vαfw)0(t,η−ktap) replaced by ¯ η(t;0,η) and\nA/hatwider(vαfw)−k(t,η−ktap), respectively. Then following (6.41), we can bound NLH\nM;4,(2)immedi-\nately.\nTreatment ofNHL\nM. Different from (6.3) and (6.4), thanks to (A.5), one deduces t hat\nAk(t,η)1|k−l,η−ltap|≤2|l,ltap|/lessorsimilar/parenleftbigg\ne−(δ−δ1)ν1\n3t|l|1\n2/an}b∇acketle{ttap/an}b∇acket∇i}ht/parenrightbigg\nBl(t)ecλ(t)/angb∇acketleftk−l,η−ltap/angb∇acket∇ights. (6.42)\nand\nAk−l(t,η−ltap)1|k−l,η−ltap|≤2|l,ltap|\n/lessorsimilar/parenleftbigg\ne−δν1\n3t|l|1\n2/an}b∇acketle{ttap/an}b∇acket∇i}ht/parenrightbigg/parenleftbigg\neδν1\n3t|l|1\n2/an}b∇acketle{tl,ltap/an}b∇acket∇i}htσ0/parenrightbigg\ne1k/negationslash=lδ1ν2\n5teλ(t)/angb∇acketleftk−l,η−ltap/angb∇acket∇ights.\nIt follows from the above two inequalities, (6.35) and (A.9) that\nNHL\nM/lessorsimilare−(δ−δ1)ν1\n3t+νt/an}b∇acketle{ttap/an}b∇acket∇i}ht2e−2|α|νt/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nη1\n|l|1\n2|Bσ0ˆρl(t)|\n×|k−l,η−ltap|e1k/negationslash=lδ1ν2\n5teλ(t)/angb∇acketleftk−l,η−ltap/angb∇acket∇ights\n×/vextendsingle/vextendsingle/vextendsingle∂α\nη(/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)dη/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilare−2|α|νt/an}b∇acketle{tt/an}b∇acket∇i}ht2/ba∇dblBσ0ρ(t)/ba∇dblL2x/ba∇dblA(vαfw)(t)/ba∇dblL2LANDAU DAMPING FOR VPFP 51\n×/vextenddouble/vextenddouble/vextenddouble/vextenddoubleek/negationslash=0δ1ν2\n5tF/bracketleftBig\n/an}b∇acketle{t∂x/an}b∇acket∇i}htn\n2+|∇|eλ(t)/angb∇acketleft∇/angb∇acket∇ights(vαfw)/bracketrightBig\nk(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk,η\n/lessorsimilar1\n/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+/parenleftBig\n/an}b∇acketle{tt/an}b∇acket∇i}ht5\n2+/ba∇dblBσ0ρ(t)/ba∇dblL2x/parenrightBig/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)(t)/ba∇dblL2/parenrightBig2\n. (6.43)\nCase 2:A(t,∇) =Aσ1,1\n3(t,∇).Now we write NMas\nNM=−e−2|α|νtRe/summationdisplay\nk,l∈Zn/integraldisplay\nηAσ1,1\n3\nk(t,η)ˆρl(t)l\n|l|2·¯η(t;k,η)∂α\nη(/hatwiderfw)k−l(t,η−ltap)\n×/parenleftBig\nAσ1,1\n3∂α\nη(/hatwiderfw)/parenrightBig\nk(t,η)dη.\nThere is no need to use the commutator trick as in Section 6.2, since on the one hand, the\nderivatives landing on fware tolerable. On the other hand, the time weight eδ1ν1\n3twill land\nonρinstead offw. In fact, in view of the partition of unity (6.1), by (A.3), (A .5) and (6.35),\nwe have\nAσ1,1\n3\nk(t,η)|¯η(t;k,η)|\n/lessorsimilare−(δ−δ1)ν1\n3teνt/parenleftbigg\neδν1\n3tecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights/an}b∇acketle{ttap/an}b∇acket∇i}ht/parenrightbigg\nAσ0+1,2\n5\nk−l(t,η−ltap)\n+e−(δ−δ1)ν1\n3teνt/parenleftBig\necλ(t)/angb∇acketleftk−l,η−ltap/angb∇acket∇ights|k−l,η−ltap|/parenrightBig/parenleftbigg\n/an}b∇acketle{ttap/an}b∇acket∇i}hteδν1\n3teλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights/an}b∇acketle{tl,ltap/an}b∇acket∇i}htσ1/parenrightbigg\n.\nConsequently, using (A.8) and (A.9), we arrive at\nNM/lessorsimilar1\n/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+/parenleftBig\n/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2/ba∇dblBρ(t)/ba∇dblL2x/parenrightBig/parenleftBig\ne−|α|νt/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightBig/parenleftBig\ne−|α|νt/vextenddouble/vextenddouble/vextenddoubleAσ1,1\n3(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightBig\n.\nNext we consider NP. Note first that (6.42) holds for all\nA(t,∇)∈/braceleftBig\nAσ0+1,2\n5(t,∇),Aσ1,1\n3(t,∇),Aσ1,2\n5(t,∇)/bracerightBig\n.\nCombining this with (6.2) implies that for l/ne}ationslash= 0, there holds\n1\n|l|Ak(t,η)/lessorsimilare−(δ−δ1)ν1\n3t/an}b∇acketle{tt/an}b∇acket∇i}htBl(t)ecλ(t)/angb∇acketleftk−l,η−ltap/angb∇acket∇ights\n+e−(δ−δ1)ν1\n3teδν1\n3tecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ightsAk−l(t,η−ltap).\nFrom this, (A.8) and (A.9), we infer that\nNP/lessorsimilareνte−2|α|νte−(δ−δ1)ν1\n3t/ba∇dblA(vαfw)/ba∇dblL2/an}b∇acketle{tt/an}b∇acket∇i}ht/ba∇dblBρ(t)/ba∇dblL2x/summationdisplay\nβ���α:|β|=1/vextenddouble/vextenddouble/vextenddoubleA(vα−β)fw/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilar1\n/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2+/parenleftBig\n/an}b∇acketle{tt/an}b∇acket∇i}ht3\n2+/ba∇dblBρ(t)/ba∇dblL2x/parenrightBig/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n×\n/summationdisplay\nβ≤α:|β|=1e−(|α|−1)νt/vextenddouble/vextenddouble/vextenddoubleA(vα−β)fw/vextenddouble/vextenddouble/vextenddouble\nL2\n. (6.44)\nWe would like to remark that, different from the linear contrib utionsLE3andLE4, the last\nfactor on the right hand side of (6.44) can be treated without resorting to (6.27), because we\ncan takeL∞\ntnorm inton it.52 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n6.2.2.Collision contributions (I): A(t,∇) =Aσ0+1,2\n5(t,∇).We first consider CN0. Recalling\n(2.5), it is natural to divide CN0into three parts according to the order of ¯ η(t;k,η):\n(6.45) CN0=CN0;2+CN0;1+CN0;0,\nhere the subscripts ‘2, 1, 0’ after the first subscripts ‘0’ re fers to the order of ¯ η(t;k,η) in (2.5).\nMore precisely, we write CN0;2=CN0;2[ρ]+CN0;2[Mθ], where\nCN0;2[ρ]\n=νe−2|α|νtRe/summationdisplay\nk∈Zn,l∈Zn/integraldisplay\nηA∂α\nη/parenleftBig\n|¯η(t;k,η)|2ˆρl(t)(/hatwiderfw)k−l(t,η−ltap)/parenrightBig\nA∂α\nη(/hatwiderfw)k(t,η)dη\n=−νe−2|α|νtRe/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηA∂α\nη(/hatwiderfw)k(t,η)Ak(t,η)ˆρl(t)\n×|¯η(t;k,η)|2∂α\nη(/hatwiderfw)k−l(t,η−ltap)dη\n−νe−2|α|νteνt/summationdisplay\nβ≤α:|β|=1Re/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηA∂α\nη(/hatwiderfw)k(t,η)Ak(t,η)ˆρl(t)\nׯη(t;k,η)β∂α−β\nη(/hatwiderfw)k−l(t,η−ltap)dη\n−νe−2|α|νte2νt/summationdisplay\nβ≤α:|β|=2Re/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηA∂α\nη(/hatwiderfw)k(t,η)Ak(t,η)ˆρl(t)\n×∂α−β\nη(/hatwiderfw)k−l(t,η−ltap)dη=CN0;2[ρ](1)+CN0;2[ρ](2)+CN0;2[ρ](3),\nandCN0;2[Mθ] is given with ρreplaced by Mθin the definition of CN0;2[ρ] above. Begin with\nCN0;2[ρ](1)and divide into three parts\nCN0;2[ρ](1)=CN0;2[ρ]HL\n(1);0+CN0;2[ρ]HL\n(1);/negationslash=+CN0;2[ρ]LH\n(1), (6.46)\nwhere\nCN0;2[ρ]HL\n(1);0=−νe−2|α|νtRe/summationdisplay\nk∈Zn∗/integraldisplay\nηA∂α\nη(/hatwiderfw)k(t,η)Ak(t,η)ˆρk(t)\n×1|η−ktap|≤2|k,ktap||¯η(t;k,η)|2∂α\nη(/hatwiderfw)0(t,η−ktap)dη,\nCN0;2[ρ]HL\n(1);/negationslash==−νe−2|α|νtRe/summationdisplay\nk∈Zn,l∈Zn\n∗\nk/negationslash=l/integraldisplay\nηA∂α\nη(/hatwiderfw)k(t,η)Ak(t,η)ˆρl(t)\n×1|k−l,η−ltap|≤2|l,ltap||¯η(t;k,η)|2∂α\nη(/hatwiderfw)k−l(t,η−ltap)dη,\nand\nCN0;2[ρ]LH\n(1);/negationslash==−νe−2|α|νtRe/summationdisplay\nk∈Zn,l∈Zn\n∗\nk/negationslash=l/integraldisplay\nηA∂α\nη(/hatwiderfw)k(t,η)Ak(t,η)ˆρl(t)\n×1|l,ltap|<1\n2|k−l,η−ltap||¯η(t;k,η)|2∂α\nη(/hatwiderfw)k−l(t,η−ltap)dη.\nWe first consider CN0;2[ρ]HL\n(1);0. Recalling that ˆ ρk(t) =ˆfk(t,ktap), and noting that a variants\nof (6.14)\nν/an}b∇acketle{tt/an}b∇acket∇i}hts\n/radicalBig\n−˙λ(t)e−s(δν1\n3−δ1ν2\n5)te(1−s)mνte2νt/lessorsimilarν/an}b∇acketle{tt/an}b∇acket∇i}hts+1\n2+e−s\n2(δ−δ1)ν1\n3t/lessorsimilarν2\n3LANDAU DAMPING FOR VPFP 53\nholds, similar to (6.13), we infer from (6.4) that\nCN0;2[ρ]HL\n(1);0/lessorsimilarν/an}b∇acketle{tt/an}b∇acket∇i}htse−s(δ−δ1)ν1\n3te2νte−2|α|νt/summationdisplay\nk∈Zn∗/integraldisplay\nη|k|s\n2/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle\n×/vextendsingle/vextendsingle/vextendsingle(Aσ0+1,2\n5ˆf)k(t,ktap)/vextendsingle/vextendsingle/vextendsingle1−s\n|Bˆρk(t)|s\n×ecλ(t)/angb∇acketleftη−ktap/angb∇acket∇ights|η−ktap|2/vextendsingle/vextendsingle/vextendsingle∂α\nη(/hatwiderfw)0(t,η−ktap)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilarν/an}b∇acketle{tt/an}b∇acket∇i}hts\n/radicalBig\n−˙λ(t)e−s(δ−δ1)ν1\n3te(1−s)mνte2νt/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n×/ba∇dblBρ/ba∇dbls\nL2/parenleftBig\ne−mνt/ba∇dblAσ0+1,2\n5ˆf/negationslash=/ba∇dblL2\nkL∞η/parenrightBig1−s/parenleftBig\ne−|α|νt/vextenddouble/vextenddouble/vextenddoubleecλ(t)/angb∇acketleft∇/angb∇acket∇ights|∇|2(vα(fw)0)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightBig\n/lessorsimilarν2\n3/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n×/ba∇dblBρ/ba∇dbls\nL2/ba∇dblfw\n/negationslash=(t)/ba∇dbl1−s\nEσ0+1,2\n5m/parenleftBig\ne−|α|νt/ba∇dblA(vα(fw)0)/ba∇dblL2/parenrightBig\n. (6.47)\nFurthermore, we can treat (6.47) like (6.15).\nForCN0;2[ρ]HL\n(1);/negationslash=, from (6.3), (6.35), and (A.9), and using the fact ν/an}b∇acketle{ttap/an}b∇acket∇i}ht2/lessorsimilarν1\n3e1\n4δ1ν1\n3t, we\ninfer that\nCN0;2[ρ]HL\n(1);/negationslash=/lessorsimilarνe−2|α|νt/summationdisplay\nk∈Zn,l∈Zn\n∗\nk/negationslash=l/integraldisplay\nη/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(Aσ0+1,2\n5ˆf)l(t,ltap)/vextendsingle/vextendsingle/vextendsingleecλ(t)/angb∇acketleftk−l,η−ltap/angb∇acket∇ights\n×e2νt/an}b∇acketle{ttap/an}b∇acket∇i}ht2|k−l,η−ltap|2/vextendsingle/vextendsingle/vextendsingle∂α\nη(/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilarν1\n3e−1\n2δ1ν1\n3t/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig/parenleftBig\ne−mνt/ba∇dblAσ0+1,2\n5ˆf/negationslash=/ba∇dblL2\nkL∞η/parenrightBig\n×/parenleftbigg\ne−|α|νt/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeδ1ν1\n3tecλ(t)/angb∇acketleft∇/angb∇acket∇ights|∇|2/an}b∇acketle{t∂x/an}b∇acket∇i}htn\n2+(vαfw)/negationslash=/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n/lessorsimilarν1\n3e−1\n2δ1ν1\n3t/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/ba∇dblfw\n/negationslash=(t)/ba∇dbl\nEσ0+1,2\n5m\n×/parenleftBig\ne−|α|νt/vextenddouble/vextenddouble/vextenddoubleAσ1,1\n3(vαfw)/negationslash=/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightBig\n. (6.48)\nAs forCN0;2[ρ]LH\n(1), using (6.2), the fact ¯ η(t;k,η) = ¯η(t;k−l,η−ltap) and (A.9) , we have\nCN0;2[ρ]LH\n(1)/lessorsimilarνe−2|α|νt/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nη/vextendsingle/vextendsingleF[∂t\nvA(vαfw)]k(t,η)/vextendsingle/vextendsingleeδν1\n3tecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights|ˆρl(t)|\n×e−(δ−δ1)ν1\n3t/vextendsingle/vextendsingleF[∂t\nvA(vαfw)]k−l(t,η−ltap)/vextendsingle/vextendsingledη\n/lessorsimilar/parenleftBig\nνe−|α|νt/vextenddouble/vextenddouble∂t\nvA(vαfw)/vextenddouble/vextenddouble\nL2/parenrightBig2\n/ba∇dblBσ1ρ(t)/ba∇dblL2x. (6.49)\nWe would like to emphasize that now we do not treat ρin terms of flike (6.47) because we\ncan not gain extra e−mνtdecay.\nWhen all derivatives are landing on ρ,CN0;2[ρ]HL\n(2)andCN0;2[ρ]HL\n(3)can be treated in the\nsimilar manner as CN0;2[ρ]HL\n(1). In fact, similar to (6.47) and (6.48), we have\nCN0;2[ρ]HL\n(2);0+CN0;2[ρ]HL\n(3);054 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n/lessorsimilarνeνte(1−s)mνt/an}b∇acketle{tt/an}b∇acket∇i}hts\n/radicalBig\n−˙λ(t)e−s(δν1\n3t−δ1ν2\n5)t/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n×/parenleftBigg\ne−mνt/vextenddouble/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5\nk(t,ktap)ˆρk(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk/parenrightBigg1−s\n/ba∇dblBρ(t)/ba∇dbls\nL2x\n×/summationdisplay\nβ≤α:1≤|β|≤2e−(|α|−|β|)νt/vextenddouble/vextenddouble/vextenddoubleecλ(t)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}ht∂α−β\nη(/hatwiderfw)0(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2η\n/lessorsimilarν2\n3e−s\n2(δ−δ1)ν1\n3t/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n×/ba∇dblf/negationslash=(t)/ba∇dbl1−s\nEσ0+1,2\n5m/ba∇dblBρ(t)/ba∇dbls\nL2x/summationdisplay\nβ≤α:1≤|β|≤2e−(|α|−|β|)νt/vextenddouble/vextenddouble/vextenddoubleA(vα−βfw)0/vextenddouble/vextenddouble/vextenddouble\nL2.\nand\nCN0;2[ρ]HL\n(2);/negationslash=+CN0;2[ρ]HL\n(3);/negationslash=\n/lessorsimilarνeνt/an}b∇acketle{tt/an}b∇acket∇i}hte−δ1ν1\n3t/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/ba∇dblAk(t,ktap)ˆρk(t)/ba∇dblL2\nk\n×/summationdisplay\nβ≤α:1≤|β|≤2e−(|α|−|β|)νt/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeδ1ν1\n3tecλ(t)/angb∇acketleftk,η/angb∇acket∇ights/an}b∇acketle{tk,η/an}b∇acket∇i}ht/an}b∇acketle{tk/an}b∇acket∇i}htn\n2+∂α−β\nη(/hatwiderfw)k(t,η)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk,η\n/lessorsimilarν2\n3e−1\n2δ1ν1\n3t/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/ba∇dblfw\n/negationslash=(t)/ba∇dbl\nEσ0+1,2\n5m\n×/summationdisplay\nβ≤α:1≤|β|≤2e−(|α|−|β|)νt/vextenddouble/vextenddouble/vextenddouble/vextenddoubleAσ1,1\n3/parenleftBig\nvα−βfw/parenrightBig\n/negationslash=/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2.\nWhen all derivatives are landing on fw, similar to (6.49), we arrive at\nCN0;2[ρ]LH\n(2)+CN0;2[ρ]LH\n(3)\n/lessorsimilarνe−2|α|νteνt/summationdisplay\nβ≤α:|β|=1/summationdisplay\nk,l∈Zn/integraldisplay\nη|η(t;k,η)|/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\neδ1ν2\n5tecλ(t)/angb∇acketleftl,ltap/an}b∇acket∇i}hts|ˆρl(t)|/parenrightbigg\n×/vextendsingle/vextendsingle/vextendsingleA∂α−β\nη(/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingledη\n+νe−2|α|νte2νt/summationdisplay\nβ≤α:|β|=2/summationdisplay\nk,l∈Zn/integraldisplay\nη/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\neδ1ν2\n5tecλ(t)/angb∇acketleftl,ltap/an}b∇acket∇i}hts|ˆρl(t)|/parenrightbigg\n×/vextendsingle/vextendsingle/vextendsingleA∂α−β\nη(/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilarν1\n2e−(δ−δ1)ν1\n3t/parenleftBig\nν1\n2e−|α|νt/vextenddouble/vextenddouble∂t\nvA(vαfw)/vextenddouble/vextenddouble\nL2/parenrightBig\n/ba∇dblBρ/ba∇dblL2x\n×\n/summationdisplay\nβ≤α:|β|=1e−(|α|−1)νt/vextenddouble/vextenddouble/vextenddoubleA(vα−βfw)/vextenddouble/vextenddouble/vextenddouble\nL2\n\n+νe−(δ−δ1)ν1\n3t/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/ba∇dblBρ/ba∇dblL2x\n×\n/summationdisplay\nβ≤α:|β|=2e−(|α|−2)νt/vextenddouble/vextenddouble/vextenddoubleA(vα−βfw)/vextenddouble/vextenddouble/vextenddouble\nL2\n. (6.50)LANDAU DAMPING FOR VPFP 55\nNext consider CN0;1. Recalling (2.5), we write\nCN0;1=−νe−2|α|νtRe/summationdisplay\nk∈Zn,l∈Zn\n∗/integraldisplay\nηA∂α\nη(/hatwiderfw)k(t,η)Ak(t,η)(/hatwiderM1)l(t)·i¯η(t;k,η)\n×∂α\nη(/hatwiderfw)k−l(t,η−ltap)dη\n−νeνte−2|α|νt/summationdisplay\nβ≤α:|β|=1Re/summationdisplay\nk∈Zn,l∈Zn\n∗/integraldisplay\nηA∂α\nη(/hatwiderfw)k(t,η)Ak(t,η)(/hatwiderM1)β\nl(t)\n×i∂α−β\nη(/hatwiderfw)k−l(t,η−ltap)dη=CN0;1,(1)+CN0;1,(2).\nThen similar to (6.47)–(6.49), we have\nCNHL\n0;1,(1);0+CNHL\n0;1,(2);0\n/lessorsimilarν2\n3e−s\n2(δ−δ1)ν1\n3t/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/ba∇dbl/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/ba∇dblL2/parenrightbigg\n×/parenleftBig\ne−(m+1)νt/ba∇dbl1k/negationslash=0Aσ0+1,2\n5(∇ηˆf)k(t,η)/ba∇dblL2\nkL∞η/parenrightBig1−s\n/ba∇dblBM1(t)/ba∇dbls\nL2x\n×/summationdisplay\nβ≤α:|β|≤1/parenleftBig\ne−(|α|−|β|)νt/ba∇dblA(vα−βfw)/ba∇dblL2/parenrightBig\n/lessorsimilarν2\n3e−s\n2(δ−δ1)ν1\n3t/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/ba∇dbl/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/ba∇dblL2/parenrightbigg\n×/ba∇dblf/negationslash=/ba∇dbl1−s\nEσ0+1,2\n5m/ba∇dblBM1(t)/ba∇dbls\nL2x/summationdisplay\nβ≤α:|β|≤1/parenleftBig\ne−(|α|−|β|)νt/ba∇dblA(vα−βfw)/ba∇dblL2/parenrightBig\n, (6.51)\nCNHL\n0;1,(1);/negationslash=+CNHL\n0;1,(2);/negationslash=/lessorsimilarν3\n5e−1\n2δ1ν2\n5t/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig2\n/ba∇dblf/negationslash=/ba∇dbl\nEσ0+1,2\n5m\n+νe−1\n2δ1ν2\n5t/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/ba∇dblf/negationslash=/ba∇dbl\nEσ0+1,2\n5m\n×/summationdisplay\nβ≤α:|β|=1/parenleftBig\ne−(|α|−1)νt/ba∇dblA(vα−βfw)/ba∇dblL2/parenrightBig\n, (6.52)\nand\nCNLH\n0;1,(1)+CNLH\n0;1,(2)\n/lessorsimilarν1\n2e−(δ−δ1)ν1\n3t/parenleftBig\nν1\n2e−|α|νt/ba∇dbl∂t\nvA(vαfw)/ba∇dblL2/parenrightBig\n/ba∇dblBM1/ba∇dblL2x/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n+νe−(δ−δ1)ν1\n3t/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/ba∇dblBM1/ba∇dblL2x\n/summationdisplay\nβ≤α:|β|=1e−(|α|−1)νt/ba∇dblA(vα−βfw)/ba∇dblL2\n.\nThe treatment for CN0;0is similar and easier, and thus is omitted.\nNow turn to CN1. Recalling (2.6) and (6.5), according to the order of ¯ η(t;k,η), like (6.45),\nwe write CN1=CN1;1+CN1;0. Furthermore, let us denote CN1;1=CN1;1[ρ]−4λ1CN1;1[ρ+\nMθ], where\nCN1;1[ρ] =−νe−νte−2|α|νtRe/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nη/parenleftBig\nA∂α\nη(/hatwiderfw)k(t,η)/parenrightBig\nAk(t,η)ˆρl(t)\n×∂α\nη/parenleftBig\n¯η(t;k,η)·(∇η/hatwiderfw)k−l(t,η−ltap)/parenrightBig\ndη\n=−νe−νte−2|α|νtRe/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nη/parenleftBig\nA∂α\nη(/hatwiderfw)k(t,η)/parenrightBig\nAk(t,η)ˆρl(t)56 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nׯη(t;k,η)·∂α\nη(∇η/hatwiderfw)k−l(t,η−ltap)dη\n−νe−2|α|νt/summationdisplay\nβ≤α,|β|=1/parenleftbigg\nα\nβ/parenrightbigg\nRe/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nη/parenleftBig\nA∂α\nη(/hatwiderfw)k(t,η)/parenrightBig\nAk(t,η)ˆρl(t)\n×∂β\nηη·∇η∂α−β\nη/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=∂αη(/hatwiderfw)k−l(t,η−ltap)dη\n=CN1;1[ρ](1)+CN1;1[ρ](2), (6.53)\nandCN1;1[ρ+Mθ] =CN1;1[ρ] +CN1;1[Mθ] can be given in the same way. Like (6.46),\nCN1;1[ρ](1)can be split as follows:\nCN1;1[ρ](1)=CN1;1[ρ]LH\n(1)+CN1;1[ρ]HL\n(1);0+CN1;1[ρ]HL\n(1);/negationslash=.\nSimilar to (6.47), using (6.4) and (6.35), we are led to\nCN1;1[ρ]HL\n(1);0/lessorsimilarνe−s(δ−δ1)ν1\n3t/an}b∇acketle{tt/an}b∇acket∇i}htse−2|α|νt/summationdisplay\nk∈Zn∗/integraldisplay\nη|k|s\n2/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle\n×/vextendsingle/vextendsingle/vextendsingleAσ0+1,2\n5ˆfk(t,ktap)/vextendsingle/vextendsingle/vextendsingle1−s\n|Bˆρk(t)|sF/bracketleftBig\necλ(t)/angb∇acketleft∇/angb∇acket∇ights|∇|(vvαfw)/bracketrightBig\n0(t,η−ktap)\n/lessorsimilarν/an}b∇acketle{tt/an}b∇acket∇i}hts\n/radicalBig\n−˙λ(t)e−s(δ−δ1)ν1\n3te(1−s)mνteνt/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n×/ba∇dblBρ/ba∇dbls\nL2/parenleftBig\ne−mνt/ba∇dblAσ0+1,2\n5ˆf/negationslash=/ba∇dblL2\nkL∞η/parenrightBig1−s/parenleftBig\ne−(|α|+1)νt/vextenddouble/vextenddouble/vextenddoubleecλ(t)/angb∇acketleft∇/angb∇acket∇ights|∇|(vvα(fw)0)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightBig\n/lessorsimilarν2\n3/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n×/ba∇dblBρ/ba∇dbls\nL2/ba∇dblf/negationslash=/ba∇dbl1−s\nEσ0+1,2\n5m/parenleftBig\ne−(|α|+1)νt/vextenddouble/vextenddouble/vextenddoubleAσ1,2\n5(vvαfw)0/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightBig\n. (6.54)\nMoreover, similar to (6.48), from (6.3), (6.35), and (A.9), we arrive at\nCN1;1[ρ]HL\n(1);/negationslash=/lessorsimilarνe−2|α|νteνt/an}b∇acketle{tt/an}b∇acket∇i}ht/summationdisplay\nk∈Zn,l∈Zn\n∗\nk/negationslash=l/integraldisplay\nη/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleAσ0+1,2\n5ˆfl(t,ltap)/vextendsingle/vextendsingle/vextendsingle\n×F/bracketleftBig\necλ(t)/angb∇acketleft∇/angb∇acket∇ights|∇|(vvαfw)/bracketrightBig\nk−l(t,η−ltap)\n/lessorsimilarνe−δ1ν2\n5teνt/an}b∇acketle{tt/an}b∇acket∇i}hte−2|α|νt/ba∇dblA(vαfw)/ba∇dblL2x,v/ba∇dblAσ0+1,2\n5ˆf/negationslash=/ba∇dblL2\nkL∞η\n×/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeδ1ν2\n5tecλ(t)/angb∇acketleft∇/angb∇acket∇ights/an}b∇acketle{t∂x/an}b∇acket∇i}htn\n2+|∇|(vvαfw\n/negationslash=)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilarν/an}b∇acketle{tt/an}b∇acket∇i}hte(m+2)νte−δ1ν2\n5t/parenleftBig\ne−mνt/ba∇dblAσ0+1,2\n5ˆf/negationslash=/ba∇dblL2\nkL∞η/parenrightBig/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n×/parenleftBig\ne−|α|νt/vextenddouble/vextenddouble/vextenddoubleAσ1,2\n5(vvαfw\n/negationslash=)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightBig\n/lessorsimilarν3\n5e−δ1\n2ν2\n5t/ba∇dblfw\n/negationslash=/ba∇dbl\nEσ0+1,2\n5m/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n×/parenleftBig\ne−(|α|+1)νt/vextenddouble/vextenddouble/vextenddoubleAσ1,2\n5(vvαfw\n/negationslash=)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightBig\n. (6.55)\nAs forCN1;1[ρ]LH\n(1), using (6.2) and (A.8),\nCN1;1[ρ]LH\n(1)/lessorsimilarνe−νte−2|α|νt/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nη/vextendsingle/vextendsingle/vextendsingleˆ∂tvA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\necλ(t)/angb∇acketleftl,ltap/angb∇acket∇ightseδν1\n3t|ˆρl(t)|/parenrightbiggLANDAU DAMPING FOR VPFP 57\n×e−(δ−δ1)ν1\n3t/vextendsingle/vextendsingle/vextendsingleA∂α\nη∇η(/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilar/parenleftBig\nν1\n2e−|α|νt/vextenddouble/vextenddouble∂t\nvA(vαfw)/vextenddouble/vextenddouble\nL2x,v/parenrightBig/parenleftbigg/radicalBig\n−˙λ1(t)e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightbigg\n×/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublee−(δ−δ1)ν1\n3tν1\n2/radicalBig\n−˙λ1(t)/an}b∇acketle{tk/an}b∇acket∇i}htn\n2+ecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ightseδν1\n3tˆρk(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk\n/lessorsimilar/ba∇dblBσ1ρ/ba∇dblL2x/parenleftBig\nν1\n2e−|α|νt/vextenddouble/vextenddouble∂t\nvA(vαfw)/vextenddouble/vextenddouble\nL2/parenrightBig/parenleftbigg/radicalBig\n−˙λ1(t)e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightbigg\n, (6.56)\nwhere we have used\ne−(δ−δ1)ν1\n3tν1\n2/radicalBig\n−˙λ1(t)/lessorsimilar/an}b∇acketle{ttap/an}b∇acket∇i}ht1+a0\n2.\nSimilarly, we can bound CN1;1[ρ](2)as follows\nCN1;1[ρ]HL\n(2);0/lessorsimilarν2\n3/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n×/ba∇dblBρ/ba∇dbls\nL2/ba∇dblfw\n/negationslash=/ba∇dbl1−s\nEσ0+1,2\n5m/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)0/ba∇dblL2/parenrightBig\n, (6.57)\nCN1;1[ρ]HL\n(2);/negationslash=/lessorsimilarνe−δ1\n2ν2\n5t/ba∇dblfw\n/negationslash=/ba∇dbl\nEσ0+1,2\n5m/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig2\n, (6.58)\nand\nCN1;1[ρ]LH\n(2)/lessorsimilar/ba∇dblBρ/ba∇dblL2x/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig2\n. (6.59)\nTo bound CN1;0, we write\nCN1;0=2νλ1e−νte−2|α|νtRe/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηAk(t,η)(/hatwiderM1)l(t)·i(∇η∂α\nη/hatwiderfw)k−l(t,η−ltap)\n×A∂α\nη(/hatwiderfw)k(t,η)dη=CNLH\n1;0+CNHL\n1;0;0+CNHL\n1;0;/negationslash=.\nThen recalling (2.17), similar to (6.54)–(6.56), we have\nCNHL\n1;0;0/lessorsimilarνλ1e−νte−2|α|νt/an}b∇acketle{tt/an}b∇acket∇i}htses(δ1ν2\n5−δν1\n3)t/summationdisplay\nk∈Zn∗/parenleftbigg\nAσ0+1,2\n5\nk(t,ktap)/vextendsingle/vextendsingle/vextendsingle(/hatwiderM1)k(t)/vextendsingle/vextendsingle/vextendsingle/parenrightbigg1−s\n×/vextendsingle/vextendsingle/vextendsingleB(/hatwiderM1)k(t)/vextendsingle/vextendsingle/vextendsingles/integraldisplay\nη/parenleftBig\necλ(t)/angb∇acketleftη−ktap/angb∇acket∇ights/vextendsingle/vextendsingle/vextendsingle(∇η∂α\nη/hatwiderfw)0(t,η−ktap)/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n×/parenleftBig\n|k|s\n2/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle/parenrightBig\ndη\n/lessorsimilarνλ1e−νte−2|α|νt/an}b∇acketle{tt/an}b∇acket∇i}htses(δ1ν2\n5−δν1\n3)t/vextenddouble/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5\nk(t,ktap)(/hatwiderM1)k(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble1−s\nL2\nk/ba∇dblBM1(t)/ba∇dbls\nL2x\n×/vextenddouble/vextenddouble/vextenddoubleAσ1,2\n5(vvαfw\n0)/vextenddouble/vextenddouble/vextenddouble\nL2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilarνλ1/an}b∇acketle{tt/an}b∇acket∇i}hts\n/radicalBig\n−˙λ(t)es(δ1ν2\n5−δν1\n3)te(1−s)mνt/parenleftbigg\ne−(m+1)νt/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5(∇ηˆf)/negationslash=/vextenddouble/vextenddouble/vextenddouble\nL2\nkL∞η/parenrightbigg1−s\n×/ba∇dblBM1(t)/ba∇dbls\nL2x/parenleftBig\ne−(|α|+1)νt/vextenddouble/vextenddouble/vextenddoubleAσ1,2\n5(vvαfw\n0)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightBig\n×/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg58 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n/lessorsimilarν2\n3λ1e−s\n2(δ−δ1)ν1\n3t/vextenddouble/vextenddoublefw\n/negationslash=/vextenddouble/vextenddouble1−s\nEσ0+1,2\n5m/ba∇dblBM1(t)/ba∇dbls\nL2x\n×/parenleftBig\ne−(|α|+1)νt/vextenddouble/vextenddouble/vextenddoubleAσ1,2\n5(vvαfw)0/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightBig/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n, (6.60)\nCNHL\n1;0;/negationslash=/lessorsimilarνλ1e−νte−2|α|νt/summationdisplay\nk∈Zn,l∈Zn\n∗\nl/negationslash=k/integraldisplay\nηeδ1ν2\n5teλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights/an}b∇acketle{tl,ltap/an}b∇acket∇i}htσ+1/vextendsingle/vextendsingle/vextendsingle(/hatwiderM1)l(t)/vextendsingle/vextendsingle/vextendsingle\n×ecλ(t)/angb∇acketleftk−l,η−ltap/angb∇acket∇ights/vextendsingle/vextendsingle/vextendsingle∇η∂α\nη(/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilarνe−δ1ν2\n5t/vextenddouble/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5\nk(t,ktap)(/hatwiderM1)k(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw\n/negationslash=)/ba∇dblL2/parenrightBig\n×/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/lessorsimilarνe−δ1\n2ν2\n5t/parenleftbigg\ne−(m+1)νt/vextenddouble/vextenddouble/vextenddoubleAσ0+1,2\n5(∇ηˆf)k(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2\nkL∞η/parenrightbigg/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw\n/negationslash=)/ba∇dblL2/parenrightBig\n×/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/lessorsimilarνe−δ1\n2ν2\n5t/ba∇dblf/negationslash=/ba∇dbl\nEσ0+1,2\n5m/parenleftbig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw\n/negationslash=)/ba∇dblL2/parenrightbig/parenleftbig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightbig\n, (6.61)\nand\nCNLH\n1;0/lessorsimilarν/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeδ1ν2\n5t/an}b∇acketle{tk/an}b∇acket∇i}htn\n2+ecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ights(/hatwiderM1)k(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk\n×/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightBig/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/lessorsimilarν/ba∇dblBM1/ba∇dblL2x/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightBig/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n. (6.62)\nThe remaining term is CN2. Recalling (2.7), it is natural to write CN2as\nCN2= 2λ1(1−2λ1)CN2[ρ]−4λ2\n1CN2[Mθ] =CN2;ρ+CN2;Mθ,\nwhere\nCN2[ρ] =νe−2(|α|+1)νtRe/summationdisplay\nk∈Zn,l∈Zn/integraldisplay\nηAk(t,η)ˆρl(t)∂α\nη(∆η/hatwiderfw)k−l(t,η−ltap)A∂α\nη(/hatwiderfw)k(t,η)dη,\nandCN2[Mθ] can be given with ρreplaced by Mθas above. CN2;ρandCN2;Mθcan be treated\nin the similar manner. Let us focus on the estimates of CN2;Mθ, which can be split into three\npartsCN2;Mθ=CN2;Mθ;0+CN2;Mθ;/negationslash=/negationslash=+CN2;Mθ;/negationslash=0, where\nCN2;Mθ;0=4νλ2\n1e−2(|α|+1)νt(/hatwiderMθ)0(t)/ba∇dblA(vvαfw)/ba∇dbl2\nL2\n+8νλ2\n1e−2(|α|+1)νt(/hatwiderMθ)0(t)/summationdisplay\nk∈Zn/integraldisplay\nηAk(t,η)(∇η∂α\nη/hatwiderfw)k(t,η)\n·∇ηAk(t,η)∂α\nη(/hatwiderfw)k(t,η)dη=CN2;Mθ;0,(1)+CN2;Mθ;0,(2),\nCN2;Mθ;/negationslash=/negationslash==4νλ2\n1e−2(|α|+1)νt/summationdisplay\nk∈Zn,l∈Zn\n∗\nl/negationslash=k/integraldisplay\nηAk(t,η)(/hatwiderMθ)l(t)(∇η∂α\nη/hatwiderfw)k−l(t,η−ltap)\n·A∇η∂α\nη(/hatwiderfw)k(t,η)dηLANDAU DAMPING FOR VPFP 59\n+8νλ2\n1e−2(|α|+1)νt/summationdisplay\nk∈Zn,l∈Zn\n∗\nl/negationslash=k/integraldisplay\nηAk(t,η)(/hatwiderMθ)l(t)(∇η∂α\nη/hatwiderfw)k−l(t,η−ltap)\n·∇ηAk(t,η)∂α\nη(/hatwiderfw)k(t,η)dη=CN2;Mθ;/negationslash=/negationslash=,(1)+CN2;Mθ;/negationslash=/negationslash=,(2),\nand\nCN2;Mθ;/negationslash=0=−4νλ2\n1e−2(|α|+1)νt/summationdisplay\nk∈Zn∗/integraldisplay\nηAk(t,η)(/hatwiderMθ)k(t)(∆η∂α\nη/hatwiderfw)0(t,η−ktap)\n×A∂α\nη(/hatwiderfw)k(t,η)dη.\nClearly,CN2;Mθ;0,(1)can be absorbed by the left hand side of (6.5). For CN2;Mθ;0,(2), we infer\nfrom (6.19) and (6.20) that\n(6.63) |∇ηAk(t,η)|/lessorsimilareνtAk(t,η).\nThen\nCN2;Mθ;0,(2)/lessorsimilarλ1ν/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(t)/vextendsingle/vextendsingle/vextendsingle/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightBig/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n.\nWe will estimate CN2;Mθ;/negationslash=/negationslash=,(1)by using the partition of unity (6.1). In fact, using (6.2) an d\n(A.8), one deduces that\nCNLH\n2;Mθ;/negationslash=/negationslash=,(1)/lessorsimilarνe−(δ−δ1)ν1\n3t/vextenddouble/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0eδν1\n3t/an}b∇acketle{tk/an}b∇acket∇i}htn\n2+ecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ights(/hatwiderMθ)k(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk\n×/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightBig2\n/lessorsimilarνe−(δ−δ1)ν1\n3t/ba∇dblBσ1(Mθ)/negationslash=(t)/ba∇dblL2x/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightBig2\n. (6.64)\nForCNHL\n2;Mθ;/negationslash=/negationslash=,(1), thanks to (6.3) and (A.9), similar to (6.55), using (3.9), w e have\nCNHL\n2;Mθ;/negationslash=/negationslash=,(1)/lessorsimilarνλ2\n1e−2(|α|+1)νt/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeδ1ν2\n5t/an}b∇acketle{tk/an}b∇acket∇i}htn\n2+ecλ(t)/angb∇acketleftk,η/angb∇acket∇ights(∇η∂α\nη/hatwiderfw\n/negationslash=)k(t,η)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk,η\n×e−δ1ν2\n5t/vextenddouble/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0Aσ0+1,2\n5\nk(t,ktap)(/hatwiderMθ)k(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk/ba∇dblA(vvαfw)/ba∇dblL2\n/lessorsimilarλ1/parenleftBig\nνλ1e−2(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dbl2\nL2/parenrightBig\n/ba∇dblfw\n/negationslash=(t)/ba∇dbl\nEσ0+1,2\n5m. (6.65)\nIn view of (6.63), the other term CN2;Mθ;/negationslash=/negationslash=,(2)can be bounded in a similar manner as\nCN2;Mθ;/negationslash=/negationslash=,(1):\nCN2;Mθ;/negationslash=/negationslash=,(2)/lessorsimilarνλ1e−(δ−δ1)ν1\n3t/ba∇dblBσ1(Mθ)/negationslash=(t)/ba∇dblL2x/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightBig\n×/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n+νλ1/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightBig/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/ba∇dblfw\n/negationslash=(t)/ba∇dbl\nEσ0+1,2\n5m.\nInstead of using the partition of (6.1), we use Bony’s decomp osition (see Section D) to\nboundCN2;Mθ;/negationslash=0. More precisely, we write\nCN2;Mθ;/negationslash=0=−4νλ2\n1e−2(|α|+1)νtRe/summationdisplay\nN≥8/summationdisplay\nk∈Zn∗/integraldisplay\nηAk(t,η)(/hatwiderMθ)k(t)<N\n8(∆η∂α\nη/hatwiderfw)0(t,η−ktap)N\n×A∂α\nη(/hatwiderfw)k(t,η)dη60 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n−4νλ2\n1e−2(|α|+1)νtRe/summationdisplay\nN∈D/summationdisplay\nk∈Zn∗/integraldisplay\nηAk(t,η)(/hatwiderMθ)k(t)N(∆η∂α\nη/hatwiderfw)0(t,η−ktap)<16N\n×A∂α\nη(/hatwiderfw)k(t,η)dη=CNLH\n2;Mθ;/negationslash=0+CNHL\n2;Mθ;/negationslash=0,\nwhere\n(/hatwiderMθ)k(t)N=ϕ/parenleftbiggk,ktap\nN/parenrightbigg\n(/hatwiderMθ)k(t),(/hatwiderMθ)k(t)<N\n8=χ/parenleftbiggk,ktap\nN/8/parenrightbigg\n(/hatwiderMθ)k(t),\n(∆η∂α\nη/hatwiderfw)0(t,η−ktap)N=ϕ/parenleftbiggη−ktap\nN/parenrightbigg\n(∆η∂α\nη/hatwiderfw)0(t,η−ktap),\nand\n(∆η∂α\nη/hatwiderfw)0(t,η−ktap)<16N=χ/parenleftbiggη−ktap\n16N/parenrightbigg\n(∆η∂α\nη/hatwiderfw)0(t,η−ktap).\nFor any fixed N∈D, the restriction on the integrand of CNHL\n2;Mθ;/negationslash=0implies that\nN\n2≤ |k,ktap| ≤3\n2N,|η−ktap| ≤3\n4·16N= 12N≤24|k,ktap|.\nThen by virtue of (A.5), an analogue of (6.4) holds for each N∈D. Consequently, similar\nto (6.47), and using (3.9), we arrive at\nCNHL\n2;Mθ;/negationslash=0/lessorsimilarνλ2\n1e−2νte−2|α|νt/an}b∇acketle{tt/an}b∇acket∇i}htse−s(δν1\n3−δ1ν2\n5)t/summationdisplay\nk∈Zn∗/integraldisplay\nη|k|s\n2/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingleAσ0+1,2\n5\nk(t,ktap)(/hatwiderMθ)k(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−s/vextendsingle/vextendsingle/vextendsingleB(/hatwiderMθ)k(t)/vextendsingle/vextendsingle/vextendsingles\n×ecλ(t)/angb∇acketleftη−ktap/angb∇acket∇ights/vextendsingle/vextendsingle/vextendsingle∆η∂α\nη(/hatwiderfw)0(t,η−ktap)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilarνλ2\n1/an}b∇acketle{tt/an}b∇acket∇i}hts\n/radicalBig\n−˙λ(t)e−s(δν1\n3−δ1ν2\n5)te(1−s)mνt/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n×/parenleftBigg\ne−mνt/vextenddouble/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0Aσ0+1,2\n5\nk(t,ktap)(/hatwiderMθ)k(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk/parenrightBigg1−s\n/ba∇dblBMθ/ba∇dbls\nL2x\n×e−(|α|+2)νt/vextenddouble/vextenddouble/vextenddoubleecλ(t)/angb∇acketleft∇/angb∇acket∇ights(|v|2vαfw)0/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilarν2\n3λ2\n1e−s\n2(δ−δ1)ν1\n3t/parenleftbigg\ne−|α|νt/radicalBig\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{t∇/an}b∇acket∇i}hts\n2A(vαfw)/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightbigg\n×/vextenddouble/vextenddoublefw\n/negationslash=/vextenddouble/vextenddouble1−s\nEσ0+1,2\n5m/ba∇dblBMθ/ba∇dbls\nL2x/parenleftBig\ne−(|α|+2)νt/vextenddouble/vextenddouble/vextenddoubleAσ1,2\n5(|v|2vαfw)0/vextenddouble/vextenddouble/vextenddouble\nL2/parenrightBig\n. (6.66)\nNext, we will use integrating by parts to bound CNLH\n2;Mθ;/negationslash=0:\nCNLH\n2;Mθ;/negationslash=0=4νλ2\n1e−2(|α|+1)νtRe/summationdisplay\nN≥8/summationdisplay\nk∈Zn∗/integraldisplay\nηAk(t,η)(/hatwiderMθ)k(t)<N\n8(∇η∂α\nη/hatwiderfw)0(t,η−ktap)N\n×A∇η∂α\nη(/hatwiderfw)k(t,η)dη\n+8νλ2\n1e−2(|α|+1)νtRe/summationdisplay\nN≥8/summationdisplay\nk∈Zn∗/integraldisplay\nηAk(t,η)(/hatwiderMθ)k(t)<N\n8(∇η∂α\nη/hatwiderfw)0(t,η−ktap)N\n·∇ηAk(t,η)∂α\nη(/hatwiderfw)k(t,η)dη\n+4νλ2\n1e−2(|α|+1)νtRe/summationdisplay\nN≥8/summationdisplay\nk∈Zn∗/integraldisplay\nηAk(t,η)(/hatwiderMθ)k(t)<N\n81\nN(∇ηϕ)/parenleftbiggη−ktap\nN/parenrightbiggLANDAU DAMPING FOR VPFP 61\n·(∇η∂α\nη/hatwiderfw)0(t,η−ktap)A∂α\nη(/hatwiderfw)k(t,η)dη\n=CNLH\n2;Mθ;/negationslash=0,(1)+CNLH\n2;Mθ;/negationslash=0,(2)+CNLH\n2;Mθ;/negationslash=0,(3).\nFor any fixed N∈D, the restriction on the integrand of CNLH\n2;Mθ;/negationslash=0implies that\nN\n2≤ |η−ktap| ≤3N\n2,|k,ktap| ≤3\n4·N\n8≤3\n16|η−ktap|.\nCombining this with (A.3), we find that the analogue of (6.2) w ithl=kholds. Therefore,\nwe have\nCNLH\n2;Mθ;/negationslash=0,(1)/lessorsimilarνe−(δ−δ1)ν1\n3t/ba∇dblBσ1(Mθ)/negationslash=(t)/ba∇dblL2x/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightBig2\n,\nCNLH\n2;Mθ;/negationslash=0,(2)/lessorsimilarνλ1e−(δ−δ1)ν1\n3t/ba∇dblBMθ(t)/ba∇dblL2x/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightBig\n×/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)0/ba∇dblL2/parenrightBig\n,\nand\nCNLH\n2;Mθ;/negationslash=0,(3)/lessorsimilar\n/summationdisplay\nN≥8,N∈D1\nN\nνλ2\n1e−2(|α|+1)νt/vextenddouble/vextenddouble/vextenddouble/vextenddouble1k/negationslash=0eδν1\n3t/an}b∇acketle{tk/an}b∇acket∇i}htn\n2+ecλ(t)/angb∇acketleftk,ktap/angb∇acket∇ights(/hatwiderMθ)k(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\nk\n×/ba∇dblA(vαfw)/ba∇dblL2/ba∇dblA(vvαfw)0/ba∇dblL2\n/lessorsimilarνλ1e−νt/ba∇dblBMθ(t)/ba∇dblL2x/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig/parenleftBig\nλ1e−(|α|+1)νt/ba∇dblA(vvαfw)0/ba∇dblL2/parenrightBig\n. (6.67)\nWe would like to remark that there is a redundant λ1orνin all the estimates for CN2;Mθ.\nThus, the estimates for CN2;Mθ;/negationslash=/negationslash=andCN2;Mθ;/negationslash=0from (6.64) to (6.67) apply to CN2;ρ. We\nomit the details to avoid repetition.\n6.2.3.Collision contributions (II): A(t,∇)∈/braceleftBig\nAσ1,1\n3(t,∇),Aσ1,2\n5(t,∇)/bracerightBig\n.Before proceeding\nany further, it is worth pointing out that in the case under co nsideration, when all the\nderivatives land on fw, i.e.,fwis at high frequency, the collision nonlinearities can be tr eated\nin the same way as in Section 6.2.2. On the other hand, when fwis at low frequency, there is\nno need to use (2.16)–(2.18) to bound the macroscopic quanti tiesρ,M1,Mθin terms of fdue\nto the low regularity. In particular, the key point is to avoi d the loss of velocity localizations\nin (6.54), (6.55), (6.60) and (6.66).\nLet us first estimate (6.54), (6.55), (6.60) and (6.66) in diffe rent way from that in Section\n6.2.2. Indeed, there is no need to distinguish fwis at zero frequency or not when fwis at\nlow frequency. From (6.3), (6.35), and (A.9), one deduces th at\nCN1;1[ρ]HL\n(1)/lessorsimilarνe−(δ−δ1)ν1\n3te−2|α|νt/summationdisplay\nk∈Zn,l∈Zn\n∗/integraldisplay\nη/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle\n×eδν1\n3teλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights/an}b∇acketle{tl,ltap/an}b∇acket∇i}htσ1/an}b∇acketle{ttap/an}b∇acket∇i}ht|ˆρl(t)|\n×e1k/negationslash=lδ1νetF/bracketleftBig\necλ(t)/angb∇acketleft∇/angb∇acket∇ights|∇|(vvαfw)/bracketrightBig\nk−l(t,η−ltap)\n/lessorsimilarνe−(δ−δ1)ν1\n3te−2|α|νt/ba∇dblA(vαfw)/ba∇dblL2/ba∇dblBρ/ba∇dblL2x\n×/vextenddouble/vextenddouble/vextenddoublee1k/negationslash=lδ1νetecλ(t)/angb∇acketleft∇/angb∇acket∇ights/an}b∇acketle{t∂x/an}b∇acket∇i}htn\n2+|∇|(vvαfw)/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilare−1\n2(δ−δ1)ν1\n3t/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/ba∇dblBρ/ba∇dblL2xνe−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2. (6.68)62 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nSimilarly, we have\nCNHL\n1;0/lessorsimilarνλ1e−(δ−δ1)ν1\n3te−νte−2|α|νt/summationdisplay\nk∈Zn,l∈Zn\n∗/integraldisplay\nη/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingle\n×eδν1\n3teλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights/an}b∇acketle{tl,ltap/an}b∇acket∇i}htσ1|(/hatwiderM1)l(t)|\n×e1k/negationslash=lδ1νetF/bracketleftBig\necλ(t)/angb∇acketleft∇/angb∇acket∇ights|∇|(vvαfw)/bracketrightBig\nk−l(t,η−ltap)\n/lessorsimilarνe−(δ−δ1)ν1\n3t/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n/ba∇dblBρ/ba∇dblL2xλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2. (6.69)\nCompared with (6.64) to (6.67), the treatment of CN2;MθwhenMθis at non-zero mode,\ndenoted by CN2;Mθ;/negationslash=below, is more straightforward:\nCN2;Mθ;/negationslash==−4νλ2\n1e−2(|α|+1)νtRe/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηAk(t,η)(/hatwiderMθ)l(t)(∆η∂α\nη/hatwiderfw)k−l(t,η−ltap)\n×A∂α\nη(/hatwiderfw)k(t,η)dη\n=4νλ2\n1e−2(|α|+1)νtRe/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηAk(t,η)(/hatwiderMθ)l(t)(∇η∂α\nη/hatwiderfw)k−l(t,η−ltap)\n·A∇η∂α\nη(/hatwiderfw)k(t,η)dη\n+8νλ2\n1e−2(|α|+1)νtRe/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηAk(t,η)(/hatwiderMθ)l(t)(∇η∂α\nη/hatwiderfw)k−l(t,η−ltap)\n·∇ηAk(t,η)∂α\nη(/hatwiderfw)k(t,η)dη=CN2;Mθ;/negationslash=,(1)+CN2;Mθ;/negationslash=,(2).\nUsing again (6.3) and (A.9), we have\nCNHL\n2;Mθ;/negationslash=,(1)/lessorsimilarνλ2\n1e−2(|α|+1)νt/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηeδ1νeteλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights/an}b∇acketle{tl,ltap/an}b∇acket∇i}htσ1|(/hatwiderMθ)l(t)|\n×ecλ(t)/angb∇acketleftk−l,η−ltap/angb∇acket∇ights/vextendsingle/vextendsingle/vextendsingle(∇η∂α\nη/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∇η∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilarλ1e−(δ−δ1)ν1\n3t/ba∇dblBσ1(Mθ)/negationslash=(t)/ba∇dblL2x/parenleftBig\nνλ1e−2(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dbl2\nL2/parenrightBig\n.\nIn view of (6.2) and (A.8), one deduces that\nCNLH\n2;Mθ;/negationslash=,(1)/lessorsimilarνλ2\n1e−2(|α|+1)νt/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηe−(δ−δ1)ν1\n3teδν1\n3tecλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights|(/hatwiderMθ)l(t)|\n×/vextendsingle/vextendsingle/vextendsingleA(∇η∂α\nη/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∇η∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilarλ1e−(δ−δ1)ν1\n3t/ba∇dblBσ1(Mθ)/negationslash=(t)/ba∇dblL2x/parenleftBig\nνλ1e−2(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dbl2\nL2/parenrightBig\n.\nSimilarly, thanks to (6.63), we obtain\nCNHL\n2;Mθ;/negationslash=,(2)/lessorsimilarνλ2\n1e−(2|α|+1)νt/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nηeδ1νeteλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights/an}b∇acketle{tl,ltap/an}b∇acket∇i}htσ1|(/hatwiderMθ)l(t)|\n×ecλ(t)/angb∇acketleftk−l,η−ltap/angb∇acket∇ights/vextendsingle/vextendsingle/vextendsingle(∇η∂α\nη/hatwiderfw)k−l(t,η−ltap)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA∂α\nη(/hatwiderfw)k(t,η)/vextendsingle/vextendsingle/vextendsingledη\n/lessorsimilarλ1/radicalbig\nνλ1e−(δ−δ1)ν1\n3t/ba∇dblBMθ(t)/ba∇dblL2x/parenleftBig/radicalbig\nνλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightBig\n×/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n, (6.70)LANDAU DAMPING FOR VPFP 63\nand\nCNLH\n2;Mθ;/negationslash=,(2)/lessorsimilarλ1/radicalbig\nνλ1e−(δ−δ1)ν1\n3t/ba∇dblBMθ(t)/ba∇dblL2x/parenleftBig/radicalbig\nνλ1e−(|α|+1)νt/ba∇dblA(vvαfw)/ba∇dblL2/parenrightBig\n×/parenleftBig\ne−|α|νt/ba∇dblA(vαfw)/ba∇dblL2/parenrightBig\n. (6.71)\nThe other ‘HL’ terms that does not involve the loss of velocit y localizations can be treaded\nin a similar manner as the corresponding ‘LH’ terms. To avoid unnecessary duplication, we\nonly show that the representative nonlinear term CN0;2[ρ]HL\n(1)now can be treated analogously\nto (6.49). In fact, using (6.3) and (A.9) yields\nCN0;2[ρ]HL\n(1)/lessorsimilarνe−2|α|νt/summationdisplay\nk∈Zn,l∈Zn∗/integraldisplay\nη/vextendsingle/vextendsingleF[∂t\nvA(vαfw)]k(t,η)/vextendsingle/vextendsingleeδν1\n3teλ(t)/angb∇acketleftl,ltap/angb∇acket∇ights/an}b∇acketle{tl,ltap/an}b∇acket∇i}htσ1|ˆρl(t)|\n×e−(δ−δ1)ν1\n3tecλ(t)/angb∇acketleftk−l,η−ltap/angb∇acket∇ights/vextendsingle/vextendsingleF[∂t\nv(vαfw)]k−l(t,η−ltap)/vextendsingle/vextendsingledη\n/lessorsimilar/parenleftBig\nνe−|α|νt/vextenddouble/vextenddouble∂t\nvA(vαfw)/vextenddouble/vextenddouble\nL2/parenrightBig2\n/ba∇dblBσ1ρ(t)/ba∇dblL2x. (6.72)\n7.Zero mode estimates\nIn this section, we improve (3.7d). To this end, we first give t he equations of ˆf0(t,η),\n(∇ηˆf)0(t,η), (∇2\nηˆf)0(t,η) and (∇3\nηˆf)0(t,η). In fact, direct calculations from (2.11) show that\n(7.1) ∂tˆf0(t,η)+ν|eνtη|2ˆf0(t,η) =N0(t,eνtη),\n∂t(∇ηˆf)0(t,η)+ν|eνtη|2(∇ηˆf)0(t,η) =−2νe2νtηˆf0(t,η)+∇η/parenleftbig\nN0(t,eνtη)/parenrightbig\n, (7.2)\n∂t(∂η\njj′ˆf)0(t,η)+ν|eνtη|2(∂η\njj′ˆf)0(t,η)\n=−2νe2νtηj′(∂η\njˆf)0(t,η)−2νe2νtηj(∂η\nj′ˆf)0(t,η)−2νe2νtδjj′ˆf0(t,η)\n+∂η\njj′/parenleftbig\nN0(t,eνtη)/parenrightbig\n, (7.3)\nand\n∂t(∂η\njj′j′′ˆf)0(t,η)+ν|eνtη|2(∂η\njj′j′′ˆf)0(t,η)\n=−2νe2νtηj′′(∂η\njj′ˆf)0(t,η)−2νe2νtηj′(∂η\nj′′jˆf)0(t,η)−2νe2νtηj(∂η\nj′j′′ˆf)0(t,η)\n−2νδj′j′′e2νt(∂η\njˆf)0(t,η)−2νδj′′je2νt(∂η\nj′ˆf)0(t,η)−2νδjj′e2νt(∂η\nj′′ˆf)0(t,η)\n+∂η\njj′j′′/parenleftbig\nN0(t,eνtη)/parenrightbig\n, (7.4)\nwhere\nN0(t,eνtη) =−/summationdisplay\nk∈Zn∗ˆEk(t)·ieνtηˆf−k(t,η−ktap)+ν(/hatwiderCµ)0(t,eνtη)+ν(/hatwidestC[g])0(t,eνtη). (7.5)\nMore precisely, ( /hatwiderCµ)0(t,eνtη) =−(/hatwiderMθ)0(t)|eνtη|2ˆµ(eνtη) and\n(/hatwidestC[g])0(t,eνtη) =−/summationdisplay\nk∈Zn∗ˆρk(t)|eνtη|2ˆf−k(t,η−ktap)−/summationdisplay\nk∈Zn∗ˆρk(t)η·(∇ηˆf)−k(t,η−ktap)\n−/summationdisplay\nk∈Zn(/hatwiderMθ)k(t)|eνtη|2ˆf−k(t,η−ktap)\n−/summationdisplay\nk∈Zn∗(/hatwiderM1)k(t)·ieνtηˆf−k(t,η−ktap). (7.6)\nWe now introduce the solution operator\n(7.7) S0(t,0;η) =e−1\n2(e2νt−1)|η|2,andS0(t,τ;η) =e−1\n2(e2νt−e2ντ)|η|2.64 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nApplying Duhamel’s principle to (7.1)–(7.4), we get the fol lowing expressions one by one:\nˆf0(t,η) =S0(t,0;η)(/hatwiderfin)0(η)+/integraldisplayt\n0S0(t,τ;η)N0(τ,eντη)dτ, (7.8)\n(∇ηˆf)0(t,η) =S0(t,0;η)(∇η/hatwiderfin)0(η)+/integraldisplayt\n0S0(t,τ;η)∇η(N0(τ,eντη))dτ\n−2ν/integraldisplayt\n0S0(t,τ;η)e2ντηˆf0(τ,η)dτ, (7.9)\n(∂η\njj′ˆf)0(t,η) =S0(t,0;η)(∂η\njj′/hatwiderfin)0(η)+/integraldisplayt\n0S0(t,τ;η)∂η\njj′(N0(τ,eντη))dτ\n−2ν/integraldisplayt\n0S0(t,τ;η)e2ντηj′(∂η\njˆf)0(τ,η)dτ\n−2ν/integraldisplayt\n0S0(t,τ;η)e2ντηj(∂η\nj′ˆf)0(τ,η)dτ\n−2νδjj′/integraldisplayt\n0S0(t,τ;η)e2ντˆf0(τ,η)dτ, (7.10)\nand\n(∂η\njj′j′′ˆf)0(t,η) =S0(t,0;η)(∂η\njj′j′′/hatwiderfin)0(η)+/integraldisplayt\n0S0(t,τ;η)∂η\njj′j′′(N0(τ,eντη))dτ\n−2ν/integraldisplayt\n0S0(t,τ;η)e2ντηj′′(∂η\njj′ˆf)0(τ,η)dτ−2ν/integraldisplayt\n0S0(t,τ;η)e2ντηj′(∂η\nj′′jˆf)0(τ,η)dτ\n−2ν/integraldisplayt\n0S0(t,τ;η)e2ντηj(∂η\nj′j′′ˆf)0(τ,η)dτ−2νδj′j′′/integraldisplayt\n0S0(t,τ;η)e2ντ(∂η\njˆf)0(τ,η)dτ\n−2νδj′′j/integraldisplayt\n0S0(t,τ;η)e2ντ(∂η\nj′ˆf)0(τ,η)dτ−2νδjj′/integraldisplayt\n0S0(t,τ;η)e2ντ(∂η\nj′′ˆf)0(τ,η)dτ. (7.11)\nBy (7.8)–(7.10) and the semigroup property of S0(·,·;η), we have\nν/integraldisplayt\n0S0(t,τ;η)e2ντ/vextendsingle/vextendsingle/vextendsingleˆf0(τ,η)/vextendsingle/vextendsingle/vextendsingledτ\n≤ν/integraldisplayt\n0e2ντdτS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle\n+ν/integraldisplayt\n0/integraldisplayt\nτ′e2ντdτS0(t,τ′;η)/vextendsingle/vextendsingle/vextendsingleN0(τ′,eντ′η)/vextendsingle/vextendsingle/vextendsingledτ′\n/lessorsimilare2νt|η|\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle\n|η|+/integraldisplayt\n0S0(t,τ′;η)/vextendsingle/vextendsingle/vextendsingleN0(τ′,eντ′η)/vextendsingle/vextendsingle/vextendsingle\n|η|dτ′\n, (7.12)\nν/integraldisplayt\n0S0(t,τ;η)e2ντ/vextendsingle/vextendsingle/vextendsingle(∇ηˆf)0(τ,η)/vextendsingle/vextendsingle/vextendsingledτ\n/lessorsimilare2νt/parenleftbigg\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(∇η/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle+/integraldisplayt\n0S0(t,τ′;η)/vextendsingle/vextendsingle/vextendsingle∇η/parenleftBig\nN0(τ′,eντ′η)/parenrightBig/vextendsingle/vextendsingle/vextendsingledτ′/parenrightbigg\n+e2νt|η|/parenleftbigg\nν/integraldisplayt\n0S0(t,τ′;η)e2ντ′/vextendsingle/vextendsingle/vextendsingleˆf0(τ′,η)/vextendsingle/vextendsingle/vextendsingledτ′/parenrightbigg\n, (7.13)\nand\nν/integraldisplayt\n0S0(t,τ;η)e2ντ|η|/vextendsingle/vextendsingle/vextendsingle(∇2\nηˆf)0(τ,η)/vextendsingle/vextendsingle/vextendsingledτLANDAU DAMPING FOR VPFP 65\n/lessorsimilare2νt|η|/parenleftbigg\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(∇2\nη/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle+/integraldisplayt\n0S0(t,τ′;η)/vextendsingle/vextendsingle/vextendsingle∇2\nη/parenleftBig\nN0(τ′,eντ′η)/parenrightBig/vextendsingle/vextendsingle/vextendsingledτ′/parenrightbigg\n+|eνtη|2/parenleftbigg\nν/integraldisplayt\n0S0(t,τ′;η)e2ντ′/vextendsingle/vextendsingle/vextendsingle(∇ηˆf)0(τ′,η)/vextendsingle/vextendsingle/vextendsingledτ′/parenrightbigg\n+e2νt|η|/parenleftbigg\nν/integraldisplayt\n0S0(t,τ′;η)e2ντ′/vextendsingle/vextendsingle/vextendsingleˆf0(τ′,η)/vextendsingle/vextendsingle/vextendsingledτ′/parenrightbigg\n. (7.14)\nIt follows from (7.8)–(7.14) that, for any j∈N, we have\neνt|eνtη|j/vextendsingle/vextendsingle/vextendsingleˆf0(t,η)/vextendsingle/vextendsingle/vextendsingle/lessorsimilar|eνtη|j+1\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(/hatwidefin)0(η)/vextendsingle/vextendsingle/vextendsingle\n|η|+/integraldisplayt\n0S0(t,τ;η)|N0(τ,eντη)|\n|η|dτ\n,\n|eνtη|j/vextendsingle/vextendsingle/vextendsingle(∇ηˆf)0(t,η)/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilar|eνtη|j/parenleftbigg\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(∇η/hatwidefin)0(η)/vextendsingle/vextendsingle/vextendsingle+/integraldisplayt\n0S0(t,τ;η)|∇η(N0(τ,eντη))|dτ/parenrightbigg\n+|eνtη|j+2/parenleftBigg\nS0(t,0;η)|(/hatwiderfin)0(η)|\n|η|+/integraldisplayt\n0S0(t,τ;η)|N0(τ,eντη)|\n|η|dτ/parenrightBigg\n,\ne−νt|eνtη|j/vextendsingle/vextendsingle/vextendsingle(∇2\nηˆf)0(t,η)/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilare−νt|eνtη|j/parenleftbigg\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(∇2\nη/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle+/integraldisplayt\n0S0(t,τ;η)/vextendsingle/vextendsingle∇2\nη(N0(τ,eντη))/vextendsingle/vextendsingledτ/parenrightbigg\n+|eνtη|j+1/parenleftbigg\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(∇η/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle+/integraldisplayt\n0S0(t,τ′;η)/vextendsingle/vextendsingle/vextendsingle∇η/parenleftBig\nN0(τ′,eντ′η)/parenrightBig/vextendsingle/vextendsingle/vextendsingledτ′/parenrightbigg\n+(|eνtη|j+3+|eνtη|j+1)/parenleftBigg\nS0(t,0;η)|(/hatwiderfin)0(η)|\n|η|+/integraldisplayt\n0S0(t,τ;η)|N0(τ,eντη)|\n|η|dτ/parenrightBigg\n,\nand\ne−2νt|eνtη|j/vextendsingle/vextendsingle/vextendsingle∇3\nηˆf0(t,η)/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilare−2νt|eνtη|j/parenleftbigg\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(∇3\nη/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle+/integraldisplayt\n0S0(t,τ;η)/vextendsingle/vextendsingle∇3\nη(N0(τ,eντη))/vextendsingle/vextendsingledτ/parenrightbigg\n+e−νt|eνtη|j+1/parenleftbigg\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(∇2\nη/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle+/integraldisplayt\n0S0(t,τ′;η)/vextendsingle/vextendsingle/vextendsingle∇2\nη/parenleftBig\nN0(τ′,eντ′η)/parenrightBig/vextendsingle/vextendsingle/vextendsingledτ′/parenrightbigg\n+/parenleftbig\n|eνtη|j+2+|eνtη|j/parenrightbig/parenleftBigg\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(∇η/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle\n+/integraldisplayt\n0S0(t,τ′;η)/vextendsingle/vextendsingle/vextendsingle∇η/parenleftBig\nN0(τ′,eντ′η)/parenrightBig/vextendsingle/vextendsingle/vextendsingledτ′/parenrightBigg\n+/parenleftbig\n|eνtη|j+4+|eνtη|j+2/parenrightbig/parenleftBigg\nS0(t,0;η)|(/hatwiderfin)0(η)|\n|η|+/integraldisplayt\n0S0(t,τ;η)|N0(τ,eντη)|\n|η|dτ/parenrightBigg\n.\nCollecting the above four estimates, we arrive at\n/summationdisplay\n|α|≤3e−(|α|−1)νt|eνtη|j/vextendsingle/vextendsingle/vextendsingle(∂α\nηˆf)0(t,η)/vextendsingle/vextendsingle/vextendsingle66 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n/lessorsimilar/summationdisplay\n1≤ℓ≤4|eνtη|j+ℓ\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(/hatwidefin)0(η)/vextendsingle/vextendsingle/vextendsingle\n|η|+/integraldisplayt\n0S0(t,τ;η)|N0(τ,eντη)|\n|η|dτ\n\n+/summationdisplay\n0≤ℓ≤2|eνtη|j+ℓ/parenleftbigg\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(∇η/hatwidefin)0(η)/vextendsingle/vextendsingle/vextendsingle+/integraldisplayt\n0S0(t,τ;η)|∇η(N0(τ,eντη))|dτ/parenrightbigg\n+/summationdisplay\n0≤ℓ≤1e−νt|eνtη|j+ℓ/parenleftBigg\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(∇2\nη/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle\n+/integraldisplayt\n0S0(t,τ;η)/vextendsingle/vextendsingle∇2\nη(N0(τ,eντη))/vextendsingle/vextendsingledτ/parenrightBigg\n+e−2νt|eνtη|j/parenleftbigg\nS0(t,0;η)/vextendsingle/vextendsingle/vextendsingle(∇3\nη/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle+/integraldisplayt\n0S0(t,τ;η)/vextendsingle/vextendsingle∇3\nη(N0(τ,eντη))/vextendsingle/vextendsingledτ/parenrightbigg\n, (7.15)\nfor anyj∈N. By (7.7), for any fixed q∈N, there hold\n|eνtη|qS0(t,0;η) =|eνtη|qS0(t,0;η)(1νt≤1+1νt>1)/lessorsimilar|η|q1νt≤1+e−1\n4|η|21νt>1, (7.16)\n|eνtη|qS0(t,τ;η)/lessorsimilarq/parenleftBig/parenleftbig\n|eνtη|2−|eντη|2/parenrightbigq\n2+|eντη|q/parenrightBig\ne−1\n2(e2νt−e2ντ)|η|2/lessorsimilar/an}b∇acketle{teντη/an}b∇acket∇i}htq. (7.17)\nSubstituting(7.7),(7.16)and(7.17)into(7.15), andtake sumwithrespectto jover{0,1,2,···,σ0−\n6}, we are led to\n/summationdisplay\n|α|≤3e−(|α|−1)νteλ(t)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teνtη/an}b∇acket∇i}htσ0−6/vextendsingle/vextendsingle/vextendsingle(∂α\nηˆf)0(t,η)/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilareλ(0)/angb∇acketleftη/angb∇acket∇ights/summationdisplay\n|α|≤3/an}b∇acketle{tη/an}b∇acket∇i}htσ0−3−|α|/vextendsingle/vextendsingle/vextendsingle(∇|α|\nη/hatwiderfin)0(η)/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle(/hatwidefin)0(η)\n|η|/vextendsingle/vextendsingle/vextendsingle\n+/integraldisplayt\n0eλ(τ)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teντη/an}b∇acket∇i}htσ0−2|N0(τ,eντη)|\n|η|dτ\n+/summationdisplay\n1≤|α|≤3/integraldisplayt\n0eλ(τ)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teντη/an}b∇acket∇i}htσ0−3−|α|/vextendsingle/vextendsingle/vextendsingle∇|α|\nη(N0(τ,eντη))/vextendsingle/vextendsingle/vextendsingledτ. (7.18)\nSince (ˆf)0(t,0) = ˆρ0(t) = 0, one easily deduces that\n/vextendsingle/vextendsingle/vextendsingle(/hatwidefin)0(η)\n|η|/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/vextenddouble/vextenddouble/vextenddouble∇η(/hatwidefin)0(η)/vextenddouble/vextenddouble/vextenddouble\nL∞η/lessorsimilar/ba∇dbl/an}b∇acketle{tv/an}b∇acket∇i}htmfin/ba∇dblL2v.\nThe other initial term can be bounded by/vextenddouble/vextenddoubleeλ(0)/angb∇acketleft∂v/angb∇acket∇ights/an}b∇acketle{t∂v/an}b∇acket∇i}htσ0−3(/an}b∇acketle{tv/an}b∇acket∇i}htmfin)/vextenddouble/vextenddouble\nL2vprovided we choose\nm>n\n2+3.\nOne can see from (7.5) that the two nonlinear terms stemming f rom|N0(τ,eντη)|\n|η|and/vextendsingle/vextendsingle/vextendsingle(∇|α|\nη(N0(τ,eντη))/vextendsingle/vextendsingle/vextendsingleon the right hand side of (7.18) are of the same type. Furtherm ore,\nthere is no need to distinguish the collisionless contribut ions and collision contributions since\nwe are working in a space with low norm. Accordingly, we only s how the details of the\ntreatment of the first nonlinear term. The second nonlinear t erm in (7.18) can be treated\nsimilarly without additional complications. In fact, from (7.5), it is natural to write:\n/integraldisplayt\n0eλ(τ)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teντη/an}b∇acket∇i}htσ0−2|N0(τ,eντη)|\n|η|dτ=IE+Iµ+Ig.LANDAU DAMPING FOR VPFP 67\nNoting that |η| ≤ |kτap|+|η−kτap|, by using the partition of unity 1 = 1|η−kτap|≤2|kτap|+\n1|kτap|≤1\n2|η−kτap|, (3.4) and (4.11), we have\nIE/lessorsimilar/summationdisplay\nk∈Zn∗/integraldisplayt\n0eλ(τ)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ0−2|ˆρk(τ)|e(σ0−1)ντ/vextendsingle/vextendsingle/vextendsingleˆf−k(τ,η−kτap)/vextendsingle/vextendsingle/vextendsingledτ\n/lessorsimilar/summationdisplay\nk∈Zn∗/integraldisplayt\n0eλ(τ)/angb∇acketleftkτap/angb∇acket∇ights/an}b∇acketle{tkτap/an}b∇acket∇i}htσ0−2/an}b∇acketle{tτ/an}b∇acket∇i}ht|ˆρk(τ)|\n×e(σ0−1)ντecλ(τ)/angb∇acketleftη−kτap/angb∇acket∇ights/vextendsingle/vextendsingle/vextendsingleˆf−k(τ,η−kτap)/vextendsingle/vextendsingle/vextendsingle1\n/an}b∇acketle{tτ/an}b∇acket∇i}htdτ\n+/summationdisplay\nk∈Zn∗/integraldisplayt\n0ecλ(τ)/angb∇acketleftkτap/angb∇acket∇ights|ˆρk(τ)|/an}b∇acketle{tτ/an}b∇acket∇i}hte(σ0−1)ντeλ(τ)/angb∇acketleftη−kτap/angb∇acket∇ights\n×/an}b∇acketle{tη−kτap/an}b∇acket∇i}htσ0−2/vextendsingle/vextendsingle/vextendsingleˆf−k(τ,η−kτap)/vextendsingle/vextendsingle/vextendsingle1\n/an}b∇acketle{tτ/an}b∇acket∇i}htdτ\n/lessorsimilarsup\nt/parenleftbigg\ne−mνt/vextenddouble/vextenddouble/vextenddoubleeλ(t)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ0−2ˆfk(t,η)1k/negationslash=0/vextenddouble/vextenddouble/vextenddouble\nL2\nkL∞η/parenrightbigg/integraldisplayt\n0/ba∇dblBˆρk(τ)/ba∇dblL2\nk1\n/an}b∇acketle{tτ/an}b∇acket∇i}htdτ\n/lessorsimilarǫνγ/ba∇dblBρ/ba∇dblL2\nt,x. (7.19)\nBy (3.12), direct calculations give\nIµ/lessorsimilarν/integraldisplayt\n0eλ(τ)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teντη/an}b∇acket∇i}htσ0−2/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(τ)/vextendsingle/vextendsingle/vextendsingleeντ|eντη||ˆµ(eντη)|dτ\n/lessorsimilarsup\nτ/parenleftBig\ne2ντ/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(τ)/vextendsingle/vextendsingle/vextendsingle/parenrightBig/integraldisplayt\n0νe−ντ/parenleftbigg\neλ(0)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{teντη/an}b∇acket∇i}htσ0−1e−|eντη|2\n2/parenrightbigg\ndτ\n/lessorsimilarsup\nt/parenleftBig\ne2νt/vextendsingle/vextendsingle/vextendsingle(/hatwiderMθ)0(t)/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n/lessorsimilarǫ2ν2γ. (7.20)\nForIg, according to (7.6), we write Ig=Ig;(1)+Ig;(2)+Ig;(3)+Ig;(4).\nWe only sketch the treatment of Ig;(2)andIg;(3)since the other two terms can be treated\nsimilarly. Indeed, similar to (7.19) and (7.20), we have\nIg;(2)/lessorsimilarν/summationdisplay\nk∈Zn∗/integraldisplayt\n0eλ(τ)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ0−2e(σ0−1)ντ|ˆρk(t)|/vextendsingle/vextendsingle/vextendsinglee−ντ(∇ηˆf)−k(t,η−kτap)/vextendsingle/vextendsingle/vextendsingledτ\n/lessorsimilarνsup\nt/parenleftbigg\ne−(m+1)νt/vextenddouble/vextenddouble/vextenddoubleeλ(t)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ0−2(∇ηˆf)k(t,η)1k/negationslash=0/vextenddouble/vextenddouble/vextenddouble\nL2\nkL∞η/parenrightbigg\n/ba∇dblBρ/ba∇dblL2\nt,x\n/lessorsimilarǫνγ+1/ba∇dblBρ/ba∇dblL2\nt,x, (7.21)\nand\nIg;(3)/lessorsimilarν/summationdisplay\nk∈Zn∗/integraldisplayt\n0eλ(τ)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ0−1eσ0ντ|(/hatwiderMθ)k(τ)|/vextendsingle/vextendsingle/vextendsingleˆf−k(τ,η−kτap)/vextendsingle/vextendsingle/vextendsingledτ\n+ν/integraldisplayt\n0eλ(τ)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ0−1eσ0ντ|(/hatwiderMθ)0(t)|/vextendsingle/vextendsingle/vextendsingleˆf0(t,η)/vextendsingle/vextendsingle/vextendsingledτ\n/lessorsimilarsup\nt/parenleftbigg\ne−mνt/vextenddouble/vextenddouble/vextenddoubleeλ(t)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ0−1ˆfk(t,η)/vextenddouble/vextenddouble/vextenddouble\nL2\nkL∞η/parenrightbigg\n(7.22)\n×/parenleftbigg\n/ba∇dblBMθ/ba∇dblL2\nt,x+sup\nt/parenleftBig\ne(σ0+1+m)νt|(/hatwiderMθ)0(t)|/parenrightBig/parenrightbigg\n/lessorsimilarǫ2ν2γ.\nWe would like to remark that, for the nonlinear contribution s involving ( ∇|α|\nη(N0(τ,eντη)),\nwhen three derivatives lands on ( ∇ηˆf)−k(t,η−ktap) that comes from the second term on the68 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nright hand side of (7.6), we have to deal with\n˜Ig;(2)=ν/summationdisplay\nk∈Zn∗/integraldisplayt\n0eλ(τ)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ0−6e(σ0−6)ντ|ˆρk(t)||η|/vextendsingle/vextendsingle/vextendsinglee−ντ(∇4\nηˆf)−k(t,η−kτap)/vextendsingle/vextendsingle/vextendsingledτ.\nLike (7.21), using (3.4), we have\n˜Ig;(2)/lessorsimilarνsup\nt/parenleftbigg\ne−(m+4)νt/vextenddouble/vextenddouble/vextenddoubleeλ(t)/angb∇acketleftη/angb∇acket∇ights/an}b∇acketle{tη/an}b∇acket∇i}htσ0−6(∇4\nηˆf)k(t,η)1k/negationslash=0/vextenddouble/vextenddouble/vextenddouble\nL2\nkHn\n2+\nη/parenrightbigg\n/ba∇dblBρ/ba∇dblL2\nt,x\n/lessorsimilarν/ba∇dblfw\n/negationslash=/ba∇dbl\nEσ0+1,2\n5m/ba∇dblBρ/ba∇dblL2\nt,x/lessorsimilarǫνγ+1/ba∇dblBρ/ba∇dblL2\nt,x, (7.23)\nprovided we choose m>n\n2+4.\nAppendix A.Elementary inequalities\nLemma A.1 ([5]).Let0<s<1andx,y≥0.\n(1)We have the triangle inequality in the opposite direction\n(A.1) cs(/an}b∇acketle{tx/an}b∇acket∇i}hts+/an}b∇acketle{ty/an}b∇acket∇i}hts)≤ /an}b∇acketle{tx+y/an}b∇acket∇i}hts,\nfor somecs∈(0,1)depending only on s.\n(2)In general,\n(A.2) |/an}b∇acketle{tx/an}b∇acket∇i}hts−/an}b∇acketle{ty/an}b∇acket∇i}hts|/lessorsimilars|x−y|\n/an}b∇acketle{tx/an}b∇acket∇i}ht1−s+/an}b∇acketle{ty/an}b∇acket∇i}ht1−s.\n(3)Ifx≤y\nKfor someK >1, then we have the improved triangle inequality\n(A.3) /an}b∇acketle{tx+y/an}b∇acket∇i}hts≤ /an}b∇acketle{ty/an}b∇acket∇i}hts+s\n(K−1)1−s/an}b∇acketle{tx/an}b∇acket∇i}hts.\n(4)Forx≥y, we have the following improved triangle inequality\n(A.4) /an}b∇acketle{tx+y/an}b∇acket∇i}hts≤/parenleftbigg/an}b∇acketle{tx/an}b∇acket∇i}ht\n/an}b∇acketle{tx+y/an}b∇acket∇i}ht/parenrightbigg1−s\n(/an}b∇acketle{tx/an}b∇acket∇i}hts+/an}b∇acketle{ty/an}b∇acket∇i}hts).\nCorollary A.2. Let0<s<1and0≤x≤Kyfor someK≥2, then there exist constants\nc∈(0,1)andC >1depending on sandK, such that\n(A.5) /an}b∇acketle{tx+y/an}b∇acket∇i}hts≤C+/an}b∇acketle{ty/an}b∇acket∇i}hts+c/an}b∇acketle{tx/an}b∇acket∇i}hts.\nProof.Ifx≤y\nK, then (A.5) follows from (A.3) immediately. It suffices to con sider the case\ny\nK<x≤Ky. Let us focus on the sub-case y≤x≤Ky. Clearly, now the inequality (A.4)\nholds, and/angb∇acketleftx/angb∇acket∇ight\n/angb∇acketleftx+y/angb∇acket∇ight= 1 whenx=y= 0. Ifx≤1, then\n(A.6) /an}b∇acketle{tx+y/an}b∇acket∇i}hts≤5s\n2.\nIfx≥1, then\n(A.7)/an}b∇acketle{tx/an}b∇acket∇i}ht\n/an}b∇acketle{tx+y/an}b∇acket∇i}ht≤/radicalBigg\n1+x2\n1+x2+y2≤/radicalBigg\n1+x2\n1+(1+1\nK2)x2≤/radicalBigg\n2\n2+1\nK2,\nsince the function φ(z) =1+z\n1+azis strictly decreasing in zfor eacha >1. The treatment of\nrest sub-casey\nK≤x≤yis the same by swapping the positions of xandy. TakingC= 5s\n2\nandc= min/braceleftbigg\ns\n(K−1)1−s,/radicalBig2\n2+1\nK2/bracerightbigg\n, we complete the proof of this corollary. /squareLANDAU DAMPING FOR VPFP 69\nLemma A.3. The following hold for suitable integrable f(k,η),g(k,η)andh(k):\n(A.8)/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nk,l∈Zn/integraldisplay\nRnf(k,η)h(l)g(k−l,η−ltap)dη/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/ba∇dblf/ba∇dblL2\nk,η/ba∇dblg/ba∇dblL2\nk,η/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tk/an}b∇acket∇i}htn\n2+h/vextenddouble/vextenddouble/vextenddouble\nL2\nk,\nand\n(A.9)/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nk,l∈Zn/integraldisplay\nRnf(k,η)h(l)g(k−l,η−ltap)dη/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/ba∇dblf/ba∇dblL2\nk,η/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tk/an}b∇acket∇i}htn\n2+g/vextenddouble/vextenddouble/vextenddouble\nL2\nk,η/ba∇dblh/ba∇dblL2\nk.\nLemma A.4. Letλ>0,s≥0, andσ>n\n2. Denote\n(A.10) Bσ\nk(t) =eλ/angb∇acketleftk,ktap/angb∇acket∇ights/an}b∇acketle{tk,ktap/an}b∇acket∇i}htσ.\nThen there holds\n(A.11)/vextenddouble/vextenddouble/vextenddoubleBσ\nk(t)(/hatwiderφψ)k(t)/vextenddouble/vextenddouble/vextenddouble\nL2\nk/lessorsimilar/vextenddouble/vextenddouble/vextenddoubleBσ\nk(t)ˆφk(t)/vextenddouble/vextenddouble/vextenddouble\nL2\nk/vextenddouble/vextenddouble/vextenddoubleBσ\nk(t)ˆψk(t)/vextenddouble/vextenddouble/vextenddouble\nL2\nk,\nand\n(A.12)/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tk/an}b∇acket∇i}ht1\n2Bσ\nk(t)(/hatwiderφψ)k(t)/vextenddouble/vextenddouble/vextenddouble\nL2\nk/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tk/an}b∇acket∇i}ht1\n2Bσ\nk(t)ˆφk(t)/vextenddouble/vextenddouble/vextenddouble\nL2\nk/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{tk/an}b∇acket∇i}ht1\n2Bσ\nk(t)ˆψk(t)/vextenddouble/vextenddouble/vextenddouble\nL2\nk.\nProof.We use the following partition of unity\n1 =1|k−l,(k−l)tap|≤2|l,ltap|+1|l,ltap|<1\n2|k−l,(k−l)tap|.\nBoth cases can be treated in the same way, so we only consider t he case|k−l,(k−l)tap| ≤\n2|l,ltap|. In fact, thanks to (A.5), we have\n(A.13) Bσ\nk(t)1|k−l,(k−l)tap|≤2|l,ltap|/lessorsimilar1l/negationslash=0Bσ\nl(t)ecλ/angb∇acketleftk−l,(k−l)tap/angb∇acket∇ights,\nfor somec∈(0,1). Accordingly, by using Young’s convolution inequality, we are let to\n/vextenddouble/vextenddouble/vextenddoubleBσ\nk(t)1|k−l,(k−l)tap|≤2|l,ltap|/vextendsingle/vextendsingle/vextendsingle(/hatwiderφψ)k(t)/vextendsingle/vextendsingle/vextendsingle/vextenddouble/vextenddouble/vextenddouble\nL2\nk\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nl∈Zn∗Bσ\nl(t)|ˆφl(t)|ecλ/angb∇acketleftk−l,(k−l)tap/angb∇acket∇ights|ˆψk−l(t)|/vextenddouble/vextenddouble/vextenddouble\nL2\nk\n/lessorsimilar/vextenddouble/vextenddouble/vextenddoubleBσ\nk(t)ˆφk(t)/vextenddouble/vextenddouble/vextenddouble\nL2\nk/vextenddouble/vextenddouble/vextenddoubleecλ/angb∇acketleftk,ktap/angb∇acket∇ights/an}b∇acketle{tk/an}b∇acket∇i}htn\n2+ˆψk(t)/vextenddouble/vextenddouble/vextenddouble\nL2\nk. (A.14)\nThen(A.11)follows immediately. Theinequality (A.12)can beachieved byslightly modifying\nthe above proof. In fact, we have /an}b∇acketle{tk/an}b∇acket∇i}ht1\n2/lessorsimilar/an}b∇acketle{tl/an}b∇acket∇i}ht1\n2/an}b∇acketle{tk−l/an}b∇acket∇i}ht1\n2.Combining this with (A.13), similar\nto (A.14), we get (A.12). This completes the proof of Lemma A. 4. /square\nAppendix B.Properties of S\nLemma B.1 ([1]).For allt≥0and0<ν≤1, there hold\n(1)Sk(t)is strictly decreasing in tand there exists a constant δ0such that\n(B.1) 0 <Sk(t)<exp/braceleftbigg\n−δ0|k|2\nν2min/braceleftbig\n(νt)3,νt/bracerightbig/bracerightbigg\n.\n(2)For allp∈(0,1]and all chosen δ∈(0,pδ0], we have\n(B.2) Sp\nk(t)/lessorsimilarp,δe−δν1\n3|k|2\n3t.\nLemma B.2 ([1]).There holds for any fixed p∈(0,1),\n(B.3) Sp\nk(t−τ)|k|\n2ν2/parenleftBig\n1−e−ν(t−τ)/parenrightBig2\n/lessorsimilarν−2\n3.70 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\nLemma B.3 ([1]).There holds for any fixed p∈(0,1),\n(B.4) Sp\nk(t−τ)|k|2\n8ν4/parenleftBig\n1−e−ν(t−τ)/parenrightBig4\n/lessorsimilarν−4\n3.\nAppendix C.Properties of the multiplier m(t,∇)\nLemma C.1. The following hold for the multiplier m(t,∇):\n(1)There is a universal constant cm>0independent of t,k,ηandν, such that\n(C.1) cm≤mk(t,η)≤1.\n(2)Fork/ne}ationslash= 0, there holds\n(C.2)|∇ηmk(t,η)|\nmk(t,η)/lessorsimilareνtν1\n3.\n(3)Fork/ne}ationslash= 0, there holds\n(C.3) ν1\n3/lessorsimilar−∂tmk(t,η)\nmk(t,η)+ν/vextendsingle/vextendsingle/vextendsingle/vextendsingleeνtη−keνt−1\nν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n.\n(4)Fork/ne}ationslash= 0,l/ne}ationslash= 0andk/ne}ationslash=l, there holds\n(C.4)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/an}b∇acketle{tk−l/an}b∇acket∇i}ht\nmax{|k|,|l|}.\nIn addition, if |ξ| ≥2|l|/an}b∇acketle{ttap/an}b∇acket∇i}ht, we have\n(C.5)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilart/an}b∇acketle{ttap/an}b∇acket∇i}hteνt/an}b∇acketle{tk−l,η−ξ/an}b∇acket∇i}ht\n/an}b∇acketle{tξ/an}b∇acket∇i}ht.\n(5)Fork= 0,l/ne}ationslash= 0, there holds\n(C.6)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarmin/braceleftbigg1\n|l|,/an}b∇acketle{tt/an}b∇acket∇i}ht\n|ξ|/bracerightbigg\n.\n(6)Fork/ne}ationslash= 0,l= 0, there holds\n(C.7)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarmin/braceleftbigg1\n|k|,/an}b∇acketle{tt/an}b∇acket∇i}ht\n|η|/bracerightbigg\n.\nProof.Recalling (1.29), it is easy to check that\n(C.8)∂tmk(t,η)\nmk(t,η)=\n\n0,ifk= 0,\n−ν1\n3\n1+ν2\n3e2νt|η−ktap|2e2νt|νη−k|2\n/angb∇acketlefteνt(νη−k)/angb∇acket∇ight2,ifk/ne}ationslash= 0,andmk(0,η) = 1.\nFrom (1.29) and (C.8), one can get (C.1)–(C.3) directly, see [1] and the associated updated\narXiv version for more details. We only focus on the commutat or estimates (C.4)–(C.7)\nbelow. Before proceeding any further, we claim that for νη/ne}ationslash=k, there holds\n/integraldisplayt\n0ν1\n3\n1+/vextendsingle/vextendsingle/vextendsingleν1\n3eνt′η−ν1\n3keνt′−1\nν/vextendsingle/vextendsingle/vextendsingle2e2νt′|νη−k|2\n/an}b∇acketle{teνt′(νη−k)/an}b∇acket∇i}ht2dt′/lessorsimilar1\n|k|. (C.9)\nIn fact, by using the change of variable τ=/vextendsingle/vextendsingleη−k\nν/vextendsingle/vextendsingleeνt′,dτ=|νη−k|eνt′dt′, we have\n/integraldisplayt\n0ν1\n3\n1+/vextendsingle/vextendsingle/vextendsingleν1\n3eνt′η−ν1\n3keνt′−1\nν/vextendsingle/vextendsingle/vextendsingle2e2νt′|νη−k|2\n/an}b∇acketle{teνt′(νη−k)/an}b∇acket∇i}ht2dt′LANDAU DAMPING FOR VPFP 71\n=/integraldisplay|η−k\nν|eνt\n|η−k\nν|/parenleftBig\n1|ντ|≥|k|\n4+1|ντ|<|k|\n4/parenrightBig\nν1\n3\n1+ν2\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingleη−k\nν\n|η−k\nν|τ+k\nν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2ντ\n/an}b∇acketle{tντ/an}b∇acket∇i}ht2dτ= I1+I2.\nIt is easy to check, if |ντ|>|k|\n4, it holds that\nI1/lessorsimilar1\n|k|/integraldisplay|η−k\nν|eνt\n|η−k\nν|ν1\n3\n1+ν2\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingleη−k\nν\n|η−k\nν|τ+k\nν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2dτ/lessorsimilar1\n|k|.\nFor I2, sinceνη−k/ne}ationslash= 0, we let αk,η=η−k\nν\n|η−k\nν|and further split it into two parts:\nI2/lessorsimilar/integraldisplay|η−k\nν|eνt\n|η−k\nν|1|ντ|<|k|\n4ν1\n3/parenleftBig\n1|αk,η·k|≤|k|\n2+1|αk,η·k|>|k|\n2/parenrightBig\n1+ν2\n3/bracketleftBig/parenleftbig\nτ+k\nν·αk,η/parenrightbig2+|k|2\nν2−/parenleftbigk\nν·αk,η/parenrightbig2/bracketrightBigdτ= I(1)\n2+I(2)\n2.\nIf|αk,η·k| ≤|k|\n2, then|k|2\nν2−/parenleftbigk\nν·αk,η/parenrightbig2≥3\n4|k|2\nν2. Thus,\nI(1)\n2/lessorsimilar/integraldisplay\n|ντ|<|k|\n41\nν1\n3|k|2\nν2dτ/lessorsimilarν2\n3\n|k|.\nIf|αk,η·k|>|k|\n2, using the fact |τ|<|k|\n4ν, we have/vextendsingle/vextendsingleτ+k\nν·αk,η/vextendsingle/vextendsingle≥|k|\n4ν. Thus,\nI(2)\n2/lessorsimilar/integraldisplay|η−k\nν|eνt\n|η−k\nν|1|ντ|<|k|\n4ν1\n31|αk,η·k|>|k|\n2\n1+ν2\n3/parenleftbig\nτ+k\nν·αk,η/parenrightbig2dτ/lessorsimilar/integraldisplay\n|τ+k\nν·αk,η|≥|k|\n4νdν1\n3/parenleftbig\nτ+k\nν·αk,η/parenrightbig\nν2\n3/parenleftbig\nτ+k\nν·αk,η/parenrightbig2/lessorsimilarν2\n3\n|k|.\nSo the claim (C.9) is true.\nCase 1:k/ne}ationslash= 0,l/ne}ationslash= 0andk/ne}ationslash=l.We will consider the following three sub-cases.\nCase 1.1:νη/ne}ationslash=kandνξ/ne}ationslash=l.By virtue of the elementary inequality |ex−1| ≤ |x|e|x|\nand (C.9), we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplayt\n0ν1\n3\n1+/vextendsingle/vextendsingle/vextendsingleν1\n3eνt′η−ν1\n3keνt′−1\nν/vextendsingle/vextendsingle/vextendsingle2e2νt′|νη−k|2\n/an}b∇acketle{teνt′(νη−k)/an}b∇acket∇i}ht2dt′\n+/integraldisplayt\n0ν1\n3\n1+/vextendsingle/vextendsingle/vextendsingleν1\n3eνt′ξ−ν1\n3leνt′−1\nν/vextendsingle/vextendsingle/vextendsingle2e2νt′|νξ−l|2\n/an}b∇acketle{teνt′(νξ−l)/an}b∇acket∇i}ht2dt′\n/lessorsimilar1\n|k|+1\n|l|/lessorsimilar/an}b∇acketle{tk−l/an}b∇acket∇i}ht\nmax{|k|,|l|}\nIn addition, if |ξ| ≥2|l|/an}b∇acketle{ttap/an}b∇acket∇i}ht ≥2|lt′ap|, then\n(C.10)ν1\n3\n/parenleftbig\n1+ν2\n3e2νt′|ξ−lt′ap|2/parenrightbig1\n2/lessorsimilar1\n1+eνt′|ξ|.\nCombining this with the elementary inequality |ex−1| ≤ |x|e|x|, we are led to\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplayt\n0ν1\n3\n1+ν2\n3e2νt′|ξ−lt′ap|21+ν2\n3e2νt′|ξ−lt′ap|2\n1+ν2\n3e2νt′|η−kt′ap|2dt′\n+/integraldisplayt\n0ν1\n3\n1+ν2\n3e2νt′|ξ−lt′ap|2dt′72 JACOB BEDROSSIAN, WEIREN ZHAO, AND RUIZHAO ZI\n/lessorsimilar/integraldisplayt\n0ν1\n3\n1+ν2\n3e2νt′|ξ−lt′ap|2/parenleftBigg\n1+ν2\n3e2νt′/vextendsingle/vextendsingle|ξ−lt′ap|2−|η−kt′ap|2/vextendsingle/vextendsingle\n1+ν2\n3e2νt′|η−kt′ap|2/parenrightBigg\ndt′\n/lessorsimilart\n/an}b∇acketle{tξ/an}b∇acket∇i}ht+ν1\n3/an}b∇acketle{ttap/an}b∇acket∇i}hteνt|k−l,η−ξ|/integraldisplayt\n0ν1\n3\n/parenleftbig\n1+ν2\n3e2νt′|ξ−lt′ap|2/parenrightbig1\n2dt′\n/lessorsimilart/an}b∇acketle{ttap/an}b∇acket∇i}hteνt/an}b∇acketle{tk−l,η−ξ/an}b∇acket∇i}ht\n/an}b∇acketle{tξ/an}b∇acket∇i}ht. (C.11)\nCase 1.2:νη=k.Now we have mk(t,η) = 1. It suffices to consider the case νξ/ne}ationslash=l,\notherwise 1 −mk(t,η)\nml(t,ξ)= 0, there is nothing to prove. Then thanks to (C.9),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−1\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar|1−ml(t,ξ)|\n/lessorsimilar/integraldisplayt\n0ν1\n3\n1+ν2\n3e2νt′|ξ−lt′ap|2e2νt′|νξ−l|2\n/an}b∇acketle{teνt′(νξ−l)/an}b∇acket∇i}ht2dt′/lessorsimilar1\n|l|.\nIf|ξ| ≥2|l|/an}b∇acketle{ttap/an}b∇acket∇i}ht, using (C.10), we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplayt\n0ν1\n3\n1+ν2\n3e2νt′|ξ−lt′ap|2dt′/lessorsimilart\n/an}b∇acketle{tξ/an}b∇acket∇i}ht.\nCase 1.3:νξ=l.Now we have ml(t,ξ) = 1. It suffices to consider the case νη/ne}ationslash=k. In\nview of (C.9), we arrive at\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=|1−mk(t,η)|/lessorsimilar/integraldisplayt\n0ν1\n3\n1+ν2\n3e2νt′|η−kt′ap|2e2νt′|νη−k|2\n/an}b∇acketle{teνt′(νη−k)/an}b∇acket∇i}ht2dt′/lessorsimilar1\n|k|.\nIf|ξ| ≥2|l|/an}b∇acketle{ttap/an}b∇acket∇i}ht, note that when νξ=l, (C.10) reduces to\n(C.12)ν1\n3\n/parenleftBig\n1+ν2\n3e2νt′|ξ−lt′ap|2/parenrightBig1\n2/lessorsimilar1\n1+|l\nν|=1\n1+|ξ|.\nTherefore, similar to (C.11), we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplayt\n0ν1\n3\n1+ν2\n3e2νt′|ξ−lt′ap|21+ν2\n3e2νt′|ξ−lt′ap|2\n1+ν2\n3e2νt′|η−kt′ap|2dt′\n/lessorsimilart/an}b∇acketle{ttap/an}b∇acket∇i}hteνt/an}b∇acketle{tk−l,η−ξ/an}b∇acket∇i}ht\n/an}b∇acketle{tξ/an}b∇acket∇i}ht.\nCase 2:k= 0,l/ne}ationslash= 0.Now it suffices to consider the case νξ/ne}ationslash=l. Using again the\nelementary inequality |ex−1| ≤ |x|e|x|and (C.9), one deduces that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplayt\n0ν1\n3\n1+ν2\n3e2νt′|ξ−lt′ap|2e2νt|νξ−l|2\n/an}b∇acketle{teνt(νξ−l)/an}b∇acket∇i}ht2dt′/lessorsimilar1\n|l|.\nIf|ξ| ≤2|l|/an}b∇acketle{ttap/an}b∇acket∇i}ht, then/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/an}b∇acketle{ttap/an}b∇acket∇i}ht\n|ξ|.\nIf|ξ|>2|l|/an}b∇acketle{ttap/an}b∇acket∇i}ht, using (C.10) again yields\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−mk(t,η)\nml(t,ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplayt\n0ν1\n3\n1+ν2\n3e2νt′|ξ−lt′ap|2dt/lessorsimilart\n/an}b∇acketle{tξ/an}b∇acket∇i}ht.\nThen (C.6) follows from the above three inequalities immedi ately, and (C.7) can be obtained\nin the same way as (C.6). The proof of Lemma C.1 is completed. /squareLANDAU DAMPING FOR VPFP 73\nAppendix D.Littlewood-Paley decomposition and paraproducts\nIn the following, we define the Littlewood-Paley decomposit ion onRn. Letχ∈C∞\n0(Rn)\nbe a radial function such that χ(ξ) = 1 for |ξ| ≤1\n2andχ(ξ) = 0 for |ξ| ≥3\n4and define\nϕ(ξ) =χ/parenleftBig\nξ\n2/parenrightBig\n−χ(ξ), supported in the range/braceleftbig\nξ∈Rn:1\n2≤ |ξ| ≤3\n2/bracerightbig\n. Then we have the\npartition of unity\n(D.1) 1 = χ(ξ)+/summationdisplay\nN∈2NϕN(ξ),\nwhereϕN(ξ) =ϕ/parenleftBig\nξ\nN/parenrightBig\n. Forf∈L2(Rn), let us define\nˆf(ξ)N=ϕN(ξ)ˆf(ξ),ˆf(ξ)1\n2=χ(ξ)ˆf(ξ),\nˆf(ξ)<N=ˆf(ξ)1\n2+/summationdisplay\nK∈2N:K<Mˆf(ξ)K=χ/parenleftbiggξ\nN/parenrightbigg\nˆf(ξ).\nThen the Littlewood-Paley decomposition in the Fourier var iable is defined by\nˆf(ξ) =ˆf(ξ)1\n2+/summationdisplay\nN∈2Nˆf(ξ)N.\nThe Littlewood-Paley decomposition can be given analogous ly when the partition of unity\n(D.1) takes the value of a given function in ξ.\nNow we are in a position to define the paraproduct decompositi on, introduced by Bony\n[9]. Given suitable functions f,g, the product of fgcan be decomposed as follows\nfg=Tfg+Tgf+R(f,g)\n=/summationdisplay\nN≥8f<N\n8gN+/summationdisplay\nN≥8g<N\n8fN+/summationdisplay\nN∈D/summationdisplay\nN\n8≤N′≤8NgN′fN\n=/summationdisplay\nN≥8f<N\n8gN+/summationdisplay\nN∈Dg<16NfN,\nwhere all the sums are understood to run over D=/braceleftbig1\n2,1,2,···,2j,···/bracerightbig\n.\nAcknowledgments JB was supported by the US NSF Award DMS-2108633. 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Zhang ,Longtimewell-posedness ofPrandtl systemwithsmallandanalytic initial\ndata, Journal of Functional Analysis, 270 (2016), pp. 2591– 2615.\n[55]W. Zhao ,Inviscid damping ofmonotone shearflowsfor2Dinhomogeneous Eulerequation with\nnon-constant densityinafinitechannel, arXiv preprint arXiv:2304.09841, (2023).\n[56]W. Zhao and R. Zi ,Asymptotic stability ofcouetteflowinastronguniform magnetic fieldforthe\nEuler-MHD system, arXiv preprint arXiv:2305.04052, (2023).\n[57]C. Zillinger ,Linearinvisciddamping formonotone shearflows, Trans. Amer. Math. Soc., 369 (2017),\npp. 8799–8855.\n(J. Bedrossian) Department of Mathematics, University of California, Los A ngeles, CA 90095,\nUSA\nEmail address :jacob@math.ucla.edu\n(W. Zhao) Department of Mathematics, New York University Abu Dhabi, S aadiyat Island,\nP.O. Box 129188, Abu Dhabi, United Arab Emirates.\nEmail address :zjzjzwr@126.com, wz19@nyu.edu\n(R. Zi)School of Mathematics and Statistics, and Key Laboratory of Nonlinear Analysis &\nApplications (Ministry of Education), Central China Norma l University, Wuhan, 430079, P. R.\nChina.\nEmail address :rzz@ccnu.edu.cn"    },    {        "title": "2402.14089v1.The_origin_of_the_Slow_to_Alfvén_Wave_Cascade_Power_Ratio_and_its_Implications_for_Particle_Heating_in_Accretion_Flows.pdf",        "content": "DRAFT VERSION FEBRUARY 23, 2024\nTypeset using L ATEXtwocolumn style in AASTeX63\nThe origin of the Slow-to-Alfvén Wave Cascade Power Ratio and its Implications for Particle Heating in Accretion Flows\nKAUSHIK SATAPATHY ,1, 2DIMITRIOS PSALTIS ,1AND FERYAL ÖZEL1\n1School of Physics, Georgia Institute of Technology, 837 State St NW, Atlanta, GA 30332, USA\n2Department of Physics, University of Arizona, 1118 E. Fourth Street, Tucson, AZ 85721\nABSTRACT\nThe partition of turbulent heating between ions and electrons in radiatively inefficient accretion flows plays\na crucial role in determining the observational appearance of accreting black holes. Modeling this partition is,\nhowever, a challenging problem because of the large scale separation between the macroscopic scales at which\nenergy is injected by turbulence and the microscopic ones at which it is dissipated into heat. Recent studies of\nparticle heating from collisionless damping of turbulent energy have shown that the partition of energy between\nions and electrons is dictated by the ratio of the energy injected into the slow and Alfvén wave cascades as\nwell as the plasma βparameter. In this paper, we study the mechanism of the injection of turbulent energy into\nslow- and Alfvén- wave cascades in magnetized shear flows. We show that this ratio depends on the particular\n(rϕ) components of the Maxwell and Reynolds stress tensors that cause the transport of angular momentum, the\nshearing rate, and the orientation of the mean magnetic field with respect to the shear. We then use numerical\nmagnetohydrodynamic shearing-box simulations with background conditions relevant to black hole accretion\ndisks to compute the magnitudes of the stress tensors for turbulence driven by the magneto-rotational instability\nand derive the injection power ratio between slow and Alfvén wave cascades. We use these results to formulate\na local subgrid model for the ion-to-electron heating ratio that depends on the macroscopic characteristics of the\naccretion flow.\nKeywords: black-hole, accretion, plasma\n1.INTRODUCTION\nRecent Very Long Baseline Interferometric (VLBI) obser-\nvations by the Event Horizon Telescope of the horizon-scale\nenvironments of the black holes at the centers of the Milky\nWay (Sgr A*) and M87 galaxies have offered a unique op-\nportunity to study plasma astrophysics in low-luminosity ac-\ncretion flows (Event Horizon Telescope Collaboration et al.\n2019; Akiyama et al. 2022). These systems are categorized\nas Radiatively Inefficient Accretion Flows (see Narayan & Yi\n1995) owing to their low luminosities and the characteristics\nof their spectra. They are made up of low density plasmas,\nwith mass accretion rates typically less than 10−3times the\nEddington rate ˙MEdd. The collisional timescales between the\nions and the electrons in these flows are much larger than ac-\ncretion timescale, allowing for the ions and the electrons to\nco-exist in a two-temperature state. In addition, low densities\nalso cause the cooling processes in these systems to be inef-\nficient, resulting in most of the energy carried by the plasma\nto be advected into the black hole.\nDue to the two-temperature nature of the plasma, a first-\nprinciples approach to studying accretion flows involves in-\ndependently evolving the thermodynamics of the ions and\nthe electrons, incorporating physical models for the partition\nof heat between the species (Ressler et al. 2015; S ˛ adowski\net al. 2017). In these accretion flows, heating mechanisms in-\nclude collisionless damping due to wave-particle resonances\n(Quataert 1998; Kawazura et al. 2020), magnetic reconnec-tion (Ball et al. 2018; Rowan et al. 2019), and heating re-\nsulting from instabilities due to velocity-space anisotropies\n(Sharma et al. 2007). These processes are typically domi-\nnant at the length-scales comparable to the gyroradii of the\nions and the electrons, which are much smaller than the scale\nof the system. As a result of the large scale separation, the\nphysics of heating mechanisms studied at the microscopic ki-\nnetic scales need to be incorporated as sub-grid prescriptions\nin global simulations.\nAmong these dissipation processes, magnetic reconnection\nis primarily dominant in localized regions of the flow that\nhave current sheets (Ball et al. 2016) and causes episodic\ndissipation of energy. On the other hand, velocity space\nanisotropies are believed to be driven by mechanisms con-\nnected to the time-evolving large scale magnetic fields. In the\nabsence of current sheets or local mechanisms driving veloc-\nity space anisotropies, the turbulent energy cascades down to\nscales comparable to the ion gyroradius and undergoes col-\nlisionless damping. This is a ubiquitous mechanism of dis-\nsipation, both spatially and temporally and is expected to be\nthe primary driver of electron and ion heating.\nRecent numerical studies show that the partition of heat be-\ntween ions and electrons resulting from collisionless damp-\ning of the turbulent energy (Kawazura et al. 2020) depends\non the plasma β(the thermal to magnetic pressure ratio) and\nthe ratio of the driving power of the compressive slow mag-\nnetosonic wave cascade to that of the Alfvén wave cascade\n(PS/PA). These studies have been performed in the gyroki-arXiv:2402.14089v1  [astro-ph.HE]  21 Feb 20242 S ATAPATHY , PSALTIS , & Ö ZEL\nnetic limit, where the gyromotion of particles around the\nmean magnetic field are averaged out. In such a limit, the\ncompressive slow waves and the Alfvén waves in the turbu-\nlence decouple at the leading order at scales smaller than the\ndriving scales of the turbulence (Schekochihin et al. 2009).\nOwing to this inertial range, the ratio PS/PAdoes not only\nimpact the microphysical partition of heat between ions and\nelectrons but is also evident in the ion and electron tempera-\nture profiles in an accretion flow (Satapathy et al. 2023), un-\nderscoring the importance of studying the nature of the slow\nand Alfvén driving caused by the MRI.\nThe ratio of the driving power of the slow- and the Alfvén-\nwaves, in turn, is determined by the mechanism that drives\nturbulence in the flow. In black hole accretion flows, the\nmagneto-rotational instability (MRI, see Balbus & Hawley\n1991) is believed to drive the turbulence and cause the out-\nward transport of angular momentum. The physics of the\ninjection of energy into the slow and Alfvén wave turbu-\nlent cascades caused by the MRI has not been extensively\nexplored. Only one study, performed by Kawazura et al.\n(2022), addresses the problem numerically using reduced-\nMHD shearing box simulations. These authors studied the\nMRI driven by a near-azimuthal mean magnetic field in a Ke-\nplerian shear flow and showed that the slow-wave cascade is\ndriven approximately at twice the power as the Alfvén-wave\ncascade.\nIn this paper, we study the slow-to-Alfvén wave driving\npower ratio PS/PAand establish its relation to the large-\nscale properties of the turbulence. We show that the ratio\nPS/PAdepends on particular components of the Maxwell and\nReynolds stress tensors that cause the outward transport of\nangular momentum in the disk, the orientation of the mean\nmagnetic field, and the shearing rate. We then employ lo-\ncal shearing box simulations in presence of different mean\nmagnetic field configurations and shearing rates relevant to\naccretion flows to compute the two stress tensors resultant\ndue to the MRI and its parasitic instability in a non-linear\nturbulent state. Subsequently, we infer the slow-to-Alfvén\ndriving power ratio PS/PAfor these configurations. In com-\nbination with hybrid kinetic scale simulations performed by\nKawazura et al. (2020), we formulate a local sub-grid model\nfor the partition of turbulent heat between the ions and the\nelectrons resultant from collisionless damping of slow- and\nAlfvén-waves. We also resolve a confusion in the litera-\nture regarding the compressive nature of slow waves in the\nreduced-MHD limit and establish that the Helmholtz decom-\nposition is not useful in calculating the ratio PS/PA(see Ap-\npendix A).\nIn Section 2, we review the effects of the large scale turbu-\nlent driving on the dissipation scale partition of heat into the\nions and electrons resultant from the collisionless damping\nof the turbulence. In Section 3, we set up equations of the\nshearing box approximation of MHD and derive the energy\nequations for the slow magnetosonic and the Alfvén wave\nfluctuations valid to the first order in the reduced-MHD limit.\nIn Section 4, we discuss the physics of the injection of energy\ninto the slow and the Alfvén wave cascades as caused by the\nFigure 1. A representative figure of the turbulent power spectrum\nin black hole accretion flows (where krepresents the wavenumber\nin units of the gravitational radius Rg). We depict the scales numer-\nically captured by GRMHD, local shearing box, and kinetic sim-\nulations. The large scale dynamics and global angular momentum\ntransport is captured by GRMHD simulations. The properties of tur-\nbulence such as the power spectra and the dynamics of stresses can\nbe studied using local shearing box approaches. Past the injection\nscales, the slow and Alfvén wave cascades of turbulence are en-\nergetically decoupled from each other until reaching kinetic scales\ncompared to the ion Larmor radius. At these dissipation scales, ki-\nnetic studies help in studying the partition of heating between the\nions and the electrons.\nMRI and establish the relation of the injection ratio to the\nMaxwell and Reynolds stress tensors. In Section 5, we com-\npute the value of PS/PAfor a wide range of conditions rel-\nevant to the turbulence driven in black hole accretion flows\nusing a suite of shearing box simulations. We summarize our\nfindings in Section 6.\n2.EFFECTS OF TURBULENT DRIVING ON\nCOLLISIONLESS DAMPING\nThe turbulence in plasmas found in radiatively inefficient\naccretion is believed to be strongly anisotropic in nature,\ncaused by the presence of strong magnetic fields. It is\nuseful to analyze such plasmas in a reduced-MHD limit,\nwhere fluctuation amplitudes, mode anisotropy, and fluc-\ntuation timescales are treated at linear order. In this ap-\nproximation, the slow-wave and Alfvén-wave cascades are\nenergetically decoupled in the inertial range of turbulence\n(Schekochihin et al. 2009). The slow- and the Alfvén-wave\ncascades only undergo mode mixing at length-scales com-\nparable to the ion Larmor radius and undergo collisionless\ndamping, channeling a fraction of the turbulent energy into\nheating the ions. The remaining energy cascades down to\nsmaller length scales and dissipates at the electron Larmor\nradius, heating the electrons. In Figure 1, we represent the\ntypical scales in which the above processes operate.3\nIn a recent paper, Kawazura et al. (2020) studied the depen-\ndence of the heat partition between the ions and the electrons\non the composition of the wave cascades driving the turbu-\nlence. They established that the heating ratio between the\nions and the electrons, Qi/Qe, primarily depends on the ra-\ntio of the driving powers of compressible slow waves and the\nincompressible Alfvén waves PS/PA, and on the plasma β.\nThis dependence can be approximated using the expression\nQi\nQe=PS\nPA+35\n1+\u0000\nβ/15\u0001−1.4. (1)\nIn the limit of plasma β≲1, the value of Qi/Qeis almost en-\ntirely dependent on the composition of the driving-scale tur-\nbulent cascade, with Qi/Qe∼PS/PA. Additionally, in plas-\nmas which are dominantly driven by slow waves, nearly all of\nthe energy carried by the slow waves is channeled into heat-\ning the ions irrespective of the value of plasma β, causing\nQi/Qe∼PS/PA.\nThe second term in equation (1) captures the dependence\nof the heating ratio on the microphysics of collisionless\ndamping of turbulence that occurs at the small lengthscales\ncomparable to the ion gyroradius. On the other hand, the first\nterm involving the ratio of the slow-to-Alfvén driving power,\nPS/PA, is determined by the large scale processes that drive\nturbulence at large scales. Our goal is to explore this ratio in\nthe general setting applicable to RIAFs.\nIn such flows, MRI has been identified as the most likely\nmechanism that drives the turbulence responsible for the\ntransport of angular momentum (Balbus & Hawley 1998).\nThe net outward transport of angular momentum is caused\nby particular off-diagonal components of the Maxwell and\nReynolds stress tensors. The same components of these\nstresses are also responsible for the injection of turbulent en-\nergy into the accretion flow (Hawley et al. 1995). To that\nend, we compute the rates of the injection of turbulent en-\nergy into the individual slow- and Alfvén-wave cascades, and\nshow that we can relate them to the Maxwell and Reynolds\nstress tensors in a local region of the accretion flow.\nIn order to infer properties of the turbulence in accretion\nflows, we will study the turbulence in the shearing-box ap-\nproximation (Goldreich & Lynden-Bell 1965). This approx-\nimation allows us to capture the mechanism of the injection\nof turbulent energy, i.e., the MRI, in a local patch of an accre-\ntion flow. In this paper, we set up a local shearing box with\nmean background magnetic fields ( ⃗B0) and shearing rates ( q)\nrelevant to black hole accretion disks, as we describe in the\nfollowing section.\n3.THE SHEARING BOX ENERGY EQUATIONS IN A\nREDUCED MHD LIMIT\nIn this section, we derive the energy equations for the slow\nmagnetosonic and the Alfvén wave fluctuations in a shearing\nbox MHD system. We first set up the equations of MHD in a\nshearing box configuration. We then expand the equations to\nfirst order in the reduced-MHD approximation. In this limit,\nwe discuss the orientations of the eigenvectors of the normalmodes of MHD and derive the reduced-MHD energy equa-\ntions for the Alfvén- and the slow-wave turbulent cascades.\n3.1. The Shearing Box Approximation in the Reduced MHD\nLimit\nThe equations of MHD in the shearing box approximation\ncan be written as (Goldreich & Lynden-Bell 1965)\n∂ρ\n∂t+(⃗u0+⃗u)·⃗∇ρ+ρ\u0010\n⃗∇·⃗u\u0011\n= 0, (2)\nρ∂⃗u\n∂t+ρh\n(⃗u0+⃗u)·⃗∇i\n(⃗u+⃗u0)=\n−⃗∇\u0012\nP+B2\n2\u0013\n+\u0010\n⃗B·⃗∇\u0011\n⃗B−2ρ⃗Ω×⃗u,(3)\nand\n∂⃗B\n∂t=⃗∇×h\n(⃗u0+⃗u)×⃗Bi\n. (4)\nIn the above equations, ρdenotes the plasma density, Pthe\npressure, ⃗uthe velocity perturbation on the background shear\nflow⃗u0,⃗Bthe magnetic field strength, and ⃗Ωthe angular ve-\nlocity of the system. We set up the equations in a Cartesian\ncomoving coordinate system ( x,y,z) such that the angular ve-\nlocity ⃗Ω≡Ωˆzandˆxrepresents the radial outward orientation\nin a global disk. The background shear flow ⃗u0is given by\n⃗u0=−qΩxˆy, (5)\nwhere q≡d(lnΩ)/d(lnr) is the shearing rate. We assume an\nadiabatic equation of state, P=Kργ, where γis the adiabatic\nindex and Kis a constant.\nIn a magnetized system described by the equations of\nMHD, the linear normal modes consist of the Alfvén, the\nslow magnetosonic, and the fast magnetosonic waves. The\ndirection of the mean magnetic field (denoted by ⃗B0) sets the\norientation of the fluctuations in each of these normal modes.\nThe orientations of the velocity eigenvectors in Fourier space\nfor each of the normal modes can be written as (see Cho &\nLazarian 2003)\nˆξA=−ˆk∥׈k⊥ (6)\nfor the Alfvén wave eigenvector ˆξA,\nˆξS∝(−1+α−√\nD)⃗k∥+(1+α−√\nD)⃗k⊥ (7)\nfor the slow wave eigenvector ˆξS, and\nˆξF∝(−1+α+√\nD)⃗k∥+(1+α+√\nD)⃗k⊥ (8)\nfor the fast wave eigenvector ˆξF. In equations (6)-(8), the\ndirection of the mean magnetic field sets the orientation of\nthe parallel wavevector, with ˆk∥=ˆB0. The perpendicular4 S ATAPATHY , PSALTIS , & Ö ZEL\nk⊥k/bardbl\nB0\nˆξR\nAˆξR\nSˆξAˆξFα→∞\nα→0ˆξS\nα→∞\nα→0\nFigure 2. The velocity eigenvectors for the Alfvén ( ˆξA, green), the\nslow magnetosonic ( ˆξS, blue), and the fast magnetosonic ( ˆξF, red)\nMHD waves. The green dot for the Alfvén eigenvector represents\nan out of plane orientation. The slow and the fast mode eigenvectors\ndepend on the value of α. The α→0 and α→ ∞ limits of these\neigenvectors are indicated by the labels. The reduced-MHD limit\nof the Alfvén eigenvector ( ˆξR\nA) is the same as the MHD eigenvector,\nwhile the same limit of the slow wave eigenvector ( ˆξR\nA) is along the\ndirection of the mean magnetic field (shown in the bottom-right of\nthis figure).\nwavenumber ( ˆk⊥) is defined such that ⃗k=⃗k∥+⃗k⊥. The pa-\nrameter αis defined as the square of the ratio of the sound\nspeed ( cs) to the Alfvén speed ( vA=B0/√4πρ0), i.e.,\nα≡c2\ns\nv2\nA=c2\ns\nB2\n0/4πρ0. (9)\nThe quantity Dis given by D= (1 +α)2−4αcos2θ, where\ncosθ=|⃗k∥|/|⃗k|.\nFigure 2 shows the set of these three orthogonal eigenvec-\ntors in Fourier space. As denoted by the green dot, Alfvén\nwaves are always azimuthally polarized with respect to the\nmean magnetic field (eq. [6]). The slow (shown in blue)\nand fast waves (shown in red) lie on the k∥−k⊥plane with\nthe parameter αdetermining their orientations. In a magnet-\nically dominated plasma ( α→0), the slow waves are po-\nlarized nearly parallel to the mean magnetic field and the\nfast waves are polarized nearly perpendicular to the mean\nmagnetic field. On the other hand, in a pressure-dominated\nplasma where α→ ∞ , the fast waves are polarized along ˆk.\nImportantly, the slow waves in such a limit are nearly incom-\npressible and are polarized in the −ˆθ≡(ˆk⊥׈k∥)׈kdirec-\ntion. They are also referred to as the pseudo-Alfvén waves\n(Maron & Goldreich 2001).The reduced MHD limit of the MHD equations is obtained\nby expanding the equations to first order in mode anisotropy\nk∥/k⊥and in the perturbations of density ( δρ/ρ 0), velocity\n(u/vA), and magnetic fields ( δB/B0) (see Schekochihin et al.\n2009). The frequencies ωof the fluctuations in all the fluid\nquantities are considered to be of the order of the frequen-\ncies of the Alfvén waves ( ω∼k∥vA). As a result, the fast\nwave displacements exhibit timescales much shorter than the\ncharacteristic timescale 1 /ωand, hence, are ordered out.\nWe now examine the polarizations of the Alfvén- and the\nslow-waves in this limit by expanding the eigenvectors ˆξAand\nˆξS, respectively, to the leading order in k∥/k⊥. The polariza-\ntion of the Alfvén modes in the Fourier space are identical to\nthat in a full MHD system, ˆξR\nA=−ˆk∥׈k⊥. The compressible\nslow wave components, on the other hand, undergo displace-\nments parallel to the mean magnetic field ( ˆξR\nS=ˆk∥, see eq. 7\nand also Schekochihin et al. 2009). Henceforth, we denote\nthe slow wave component of the velocity and magnetic fields\nas⃗u∥andδ⃗B∥, respectively, and their Alfvén wave compo-\nnents as ⃗u⊥andδ⃗B⊥. We show the orthonormal reduced\nMHD eigenvectors for the slow and the Alfvén waves in Fig-\nure 2 with the vectors labeled ξR\nSandξR\nA, respectively.1\nExpanding to the first order in the reduced-MHD ordering\ntransforms equations (2)-(4) and the equation of state to (see\nKawazura et al. 2022)\n∂⃗u\n∂t+⃗u⊥·∇⃗u+⃗u0·∇⃗u=−∇⊥\u0012δB2\n2ρ0\u0013\n+1\nρ0\u0010\n⃗B0·∇δ⃗B+δ⃗B⊥·∇δ⃗B\u0011\n+2⃗Ω×⃗u−⃗u·∇(⃗u0)(10)\nand\n\u0012∂\n∂t+⃗u⊥·∇\u0013\u0012\nδ⃗B+1\nαδ⃗B∥\u0013\n+⃗u0·∇\u0012\nδ⃗B+1\nαδ⃗B∥\u0013\n=\nδ⃗B·∇(⃗u0)+\u0010\n⃗B0·∇δ⃗u+δ⃗B⊥·∇δ⃗u\u0011\n.(11)\nIn the above equations, ∇⊥denotes the component of the\ngradient operator ∇perpendicular to the mean magnetic field\nˆB0. Equations (10) and (11) carry the following key physi-\ncal manifestations resulting from the reduced MHD order-\ning: (i)the Goldreich-Sridhar critical balance assumption\nk∥vA∼k⊥u⊥leads the derivatives of the form ⃗x∥· ∇⃗xto\nbecome dynamically unimportant, where ⃗xrepresents veloc-\nity or magnetic field fluctuations; (ii)the fast magnetosonic\nwave fluctuations are ordered at timescales too fast to ex-\ncite, resulting in a perpendicular balance between the pres-\nsure fluctuations and the magnetic tension in the direction\nperpendicular to the magnetic field ( δP=−⃗B0·δ⃗B); and (iii)\nthe induction equation is combined with the continuity equa-\ntion to capture the compressive density fluctuations (which,\n1We note that, as a result of the first order expansion in anisotropy k∥/k⊥,\nthe orientation of the eigenvectors are independent of the quantity α.5\nin turn, is related to the parallel component of the magnetic\nfield via the perpendicular pressure balance).\n3.2. The Energy Equations for Slow- and Alfvén-Waves\nIn order to obtain the energy equation for the Alfvén polar-\nized fluctuations, we take the the dot product of equation (10)\nwithρ0⃗u⊥and of (11) with δ⃗B⊥and add the resulting equa-\ntions. The energy equation of slow wave fluctuations can be\nsimilarly obtained by taking the the dot product of equation\n(10) with ρ0⃗u∥and of (11) with δ⃗B∥and adding the resulting\nequations. We then write the energy equations as\n∂EA\n∂t+∇·(⃗u0EA)+∇∥·n\nρ0⃗vA\u0010\n⃗u⊥·δ⃗B⊥\u0011o\n+∇⊥·\u001a\n⃗u⊥\u0012\nEA+δB2\n2\u0013\n+δ⃗B⊥\u0010\n⃗u⊥·δ⃗B⊥\u0011\u001b\n=RA(12)\nfor the Alfvén waves and\n∂ES\n∂t+∇·(⃗u0ES)+∇∥·n\nρ0⃗vA\u0010\n⃗u∥·δ⃗B∥\u0011o\n+∇⊥·n\n⃗u⊥ES+δ⃗B⊥\u0010\n⃗u∥·δ⃗B∥\u0011o\n=RS\n(13)\nfor the slow waves. In these equations, we denote the total\nAlfvén- and slow-wave energies by EAandES, respectively,\nwhere\nEA=1\n2ρ0u2\n⊥+1\n2δB2\n⊥ (14)\nand\nES=1\n2ρ0u2\n∥+1\n2δB2\n∥\u0012\n1+1\nα\u0013\n. (15)\nThe right hand sides of equations (12) and (13) contain\nthe energy injection rates into the Alfvén waves ( RA) and the\nslow waves ( RS), respectively, and are given by\nRA=−ρ0⃗u⊥·(⃗u⊥·∇⃗u0)+δ⃗B⊥·\u0010\nδ⃗B⊥·∇⃗u0\u0011\n+ρ0⃗u⊥·\u0010\n2⃗Ω×⃗u\u0011\n(16)\nfor the Alfvén waves and\nRS=−ρ0⃗u∥·(⃗u⊥·∇⃗u0)+δ⃗B∥·\u0010\nδ⃗B⊥·∇⃗u0\u0011\n+ρ0⃗u∥·\u0010\n2⃗Ω×⃗u\u0011\n(17)\nfor the slow waves. The first two terms in the energy injec-\ntion rates RAandRSrepresent injection caused by the back-\nground shear and the third term represents the advection of\nenergy caused by the Coriolis force. The left hand side of\nEquations (12) and (13) contains the time derivative of the\nrespective energies and additionally, the divergence of fluxes\ncarried by the shear velocity, the wave fluctuations, and the\nmean magnetic field.\nEquations (16) and (17) give the expressions for the energy\ninjection rates into the slow- and the Alfvén-wave cascades,respectively, correct to the first order in the reduced-MHD\napproximation. Because the slow- and Alfvén-wave cascades\nenergetically decouple at lengthscales smaller than the injec-\ntion scales of the MRI, we can infer the ratio of their relative\ndriving power using numerical shearing-box simulations that\ncapture the physics in the injection scales of the MRI.\nIn the upcoming section, we study the physics of the en-\nergy injection described in equations (16) and (17) and show\nthat energy injection rates in the slow- and Alfvén- wave cas-\ncades can be related to quantities measured at large scales,\nnamely the Maxwell and the Reynolds stress tensors in the\nflow.\n4.PHYSICS OF ENERGY INJECTION INTO SLOW\nAND ALFVÉN WA VES\nThe injection of energy and transport of angular momen-\ntum in a local region in an accretion flow is caused by the\nMaxwell and Reynolds stress tensors in that region. The i j-\nth component of the Maxwell stress tensor Mi jin such a local\ndomain of volume Vin the accretion flow is defined as\nMi j=−Z\nVδBiδBj (18)\nand the i j-th component of the Reynolds stress tensor Ri jin\nthat domain is defined as\nRi j=Z\nVρ0uiuj; (19)\nnote the minus sign in the definition of the Maxwell stress\ntensor. In a local shearing-box configuration, the spatial\nvolume is chosen such that it includes the energy injection\nscales.\nIn a shearing-box system, the background shear velocity\nfield ⃗u0(eq. [5]) is oriented along the azimuthal direction\n(ˆy) and exhibits a gradient along ˆx. Consequently, the x−y\ncomponents of the Maxwell and Reynolds stress tensors are\nresponsible for the net injection of energy and the outward\ntransport of angular momentum in the local shearing domain\n(Hawley et al. 1995; Pessah et al. 2006a). Our aim, however,\nis to compute the rate of energy injection into the individ-\nual slow- and Alfvén-wave cascades, and relate them to the\nMaxwell and Reynolds stress tensors.\nThe total energy injection rates into the Alfvén- and slow-\nwave cascades can be obtained by integrating equations (12)\nand (13), respectively, over the spatial volume of the shear-\ning box. Upon carrying out the integral, the terms involving\nthe divergences vanish as they represent transport of fluxes\nby the background incompressible shear flow and, hence, do\nnot contribute to energy injection. As a result, we obtain the\ninjection powers into the individual slow- and Alfvén-wave\ncascades.\nThe slow wave injection power PSis given by\nPS≡Z\nVdES\ndt=Z\nVqΩh\n⃗u∥·(uxˆy)−δ⃗B∥·(δBxˆy)i\n−Z\nV2Ω⃗u∥·\u0000\nuxˆy−uyˆx\u0001\n,(20)6 S ATAPATHY , PSALTIS , & Ö ZEL\nand the Alfvén wave injection power PAis given by\nPA≡Z\nVdEA\ndt=Z\nVqΩh\n⃗u⊥·(uxˆy)−δ⃗B⊥·(δBxˆy)i\n−Z\nV2Ω⃗u⊥·\u0000\nuxˆy−uyˆx\u0001\n.(21)\nThe background shear flow causes the injection of turbu-\nlent energy into the system, with the injection being captured\nby the terms in the square brackets in equations (20) and (21).\nThe second term in each of equations (20) and (21) describes\nthe effects of the Coriolis force in facilitating advection of\nenergy in the plane perpendicular to the angular velocity of\nthe shearing box (the x−yplane).\nIn order to express PSandPAin terms of components of\nthe Maxwell and Reynolds stress tensors, we now invoke\nthe polarizations of the slow- and Alfvén-wave fluctuations\nin shearing-box configurations relevant to black hole accre-\ntion disks into equations (20) and (21). The orientation of\nthe mean magnetic field determines the polarizations of the\nslow-wave and the Alfvén-wave fluctuations (see discussion\nsurrounding Figure 2).\nIn the disk regions of accretion flows, the radial compo-\nnents of the magnetic field undergo strong shearing in the\nazimuthal direction. As a result, the mean magnetic field in\nthe disk is expected to be primarily along the azimuthal di-\nrection. In addition to large scale azimuthal fields, the tur-\nbulent structure in the disk is expected to develop vertical\ncomponents of the magnetic fields that are determined by the\nlocal dynamics in the flow. The MRI is driven primarily by\nthese weak vertical components of the magnetic field. Mo-\ntivated by these considerations, we setup the mean magnetic\nfield in the shearing box to lie on the y−zplane such that\n⃗B0=By0ˆy+Bz0ˆz.\nUsing expressions for the polarizations of the slow and the\nAlfvén wave fluctuations (equations [7] and [6]) under the\nreduced-MHD approximation, the injection rates PSandPA\ncan be written as\nPS=qΩ\b\u0000\nMxy+Rxy\u0001\ncos2ϕ+(Mxz+Rxz)cosϕsinϕ\t\n−2Ω\u0000\nRxycos2ϕ+Rxzcosϕsinϕ\u0001\n(22)\nand\nPA=qΩ\u0002\u0000\nMxy+Rxy\u0001\nsin2ϕ−(Mxz+Rxz)cosϕsinϕ\u0003\n+2Ω\u0000\nRxycos2ϕ+Rxzcosϕsinϕ\u0001\n,(23)\nrespectively, where the pitch angle ϕis defined such that\ntanϕ≡Bz0/By0≡Γ.\nThese equations lend themselves to a straightforward in-\nterpretation. The azimuthal background shear flow causes\nthe energy injection into the shearing box to be experienced\nby the components of the Alfvén and slow velocities and\nthe magnetic field fluctuations along ˆy. The injection ratesinto these individual wave cascades are proportional to the\nMaxwell and Reynolds stress tensors with factors of sin ϕ\nand cos ϕthat capture the orientation of the wave polariza-\ntions with respect to the shear. At the same time, the ef-\nfect of the Coriolis interactions is apparent on the projec-\ntions of the Alfvén and slow velocity fluctuations on the\nxandzaxes, captured by the Reynolds stress tensors. It\nis noteworthy to observe from equations (23) and (22) that\nthe total energy injection rate into the combined slow and\nAlfvén wave cascade, as expected, depends only on the xy\ncomponents of the Maxwell and Reynolds stress tensors, i.e.\nPS+PA=qΩ(Mxy+Rxy). The x−zcomponents of the stresses\nand the effects of the Coriolis force determine individual in-\njection powers into the slow and Alfvén wave cascade but do\nnot contribute to the total power injected into the system.\nCombining expressions for the injection powers into slow\nand Alfvén wave cascades, we now write the slow-to-Alfvén\ndriving power ratio PS/PAas\nPS\nPA=q\u0002\u0000\nMxy+Rxy\u0001\n−(Mxz+Rxz)Γ2\u0003\n−2\u0000\nRxy+RxzΓ\u0001\nq\u0002\u0000\nMxy+Rxy\u0001\nΓ2−(Mxz+Rxz)Γ\u0003\n+2\u0000\nRxy+RxzΓ\u0001.\n(24)\nEquation (24) represents the ratio PS/PAin a local domain\nof an accretion flow, that captures the lengthscales of en-\nergy injection into the domain. This ratio depends on the\noff-diagonal x−yandx−zcomponents of the Maxwell and\nReynolds stress tensors, the local shearing rate q, and the ori-\nentation of the mean magnetic field in the domain with re-\nspect to the shear (which appears as the factor Γ).\nIn the presence of a predominantly azimuthal mean mag-\nnetic field, the slow wave displacements are nearly along the\ndirection of the shear ( u∥∝ξR\nS∝ˆy) and the Alfvén wave dis-\nplacements lie in the plane perpendicular to the direction of\nthe shear velocity ( u⊥·ˆy= 0). The orientation of the slow\nwave displacements along the shear results in them receiving\nall the energy injection brought into effect by the shear. The\nAlfvén waves do not experience direct injection of energy\nvia the shear velocities but only experience the Coriolis in-\nteraction via the xycomponent of the Reynolds stress tensor.\nHence, in the limit of the magnetic field of the MRI-driven\nsystem to be purely azimuthal, relevant to the midplane re-\ngions of accretion disks, the slow- to Alfvén-wave energy in-\njection ratio depends only on the ratio of the xycomponents\nof the Maxwell and Reynolds stresses and can be written as\nPS\nPA\f\f\f\f⃗B0≈B0ˆy=q\u0012Mxy\nRxy+1\u0013\n−2\n2. (25)\nIn principle, the ratio PS/PAcan be inferred locally in\nglobal simulations of black hole accretion flows using equa-\ntion (24). However, in the interest of examining the depen-\ndence of the PS/PAon the background plasma parameters,\nnamely the plasma βand the shearing rate, we approach the\nproblem by inferring PS/PAusing the Maxwell and Reynolds\nstress tensors using numerical simulations of the shearing-\nbox configuration.7\n5.NUMERICAL COMPUTATION OF\nSLOW-TO-ALFVÉN WA VE INJECTION RATIO\nWe perform a suite of magnetohydrodynamic shearing-\nbox simulations to study the injection of slow and Alfvén\nwaves during the linear and non-linear turbulent stages of the\ngrowth of the MRI. In the previous sections, we derived equa-\ntions for the injection rates into these turbulent cascades that\nare correct to first order in the reduced-MHD approximation.\nWe now infer the value of PS/PAat this order, using numeri-\ncal MHD simulations that resolve the scales of the injection\nof the MRI.\nOur approach is different from the one undertaken by\nKawazura et al. (2022) to compute PS/PA. These authors per-\nformed numerical shearing-box simulations by solving the\ndynamical equations at first order of the reduced-MHD ap-\nproximation. In contrast, we solve the MHD equations with-\nout employing the reduced-MHD approximation, but only in-\nfer the value of PS/PAusing the same approximation.\nBecause Kawazura et al. (2020) solve the reduced-MHD\nequations, the energy cascades of the slow and the Alfvén\nwaves are identically decoupled in their setup past the injec-\ntion scales. In our setup, there can potentially be some cou-\npling between the slow and the Alfvén wave cascades that are\nof higher order in the reduced-MHD approximation. Such\nhigher order terms might also cause non-local exchanges in\nenergy in the Fourier space between the velocity and mag-\nnetic field cascades (Lesur & Longaretti 2011). These effects\non the slow- and Alfvén-wave driving powers in MRI-driven\nMHD turbulence have not been explored and will be the sub-\nject of a later study.\n5.1. Numerical Setup\nWe carry out local MHD simulations of unstratified\nshearing boxes using the Godunov-type finite volume code\nAthena++ (Stone et al. 2020). We set up the box to be\nperiodic in the y(azimuthal) and z(vertical) directions, and\nshear-periodic in the x(radial) direction (see Hawley et al.\n1995, for a discussion on the shear-periodic boundary con-\nditions). The box dimensions are set to a size ( Lx,Ly,Lz)\n= (4L,4L,L), where L=cs/Ω0. The resolutions of the box\nin the x,y,andzdirections are equal, with the number of\ngrid points being ( Nx,Ny,Nz) = (4 N,4N,N). We perform\nour simulations with N= 64. In our simulation set, time-\nintegrations are carried out using the second order van Leer\npredictor-corrector methods and spatial reconstructions are\nperformed using piecewise linear interpolation.\nThe MRI is primarily an instability driven by weak vertical\nmagnetic fields in an accretion disk. The instability amplifies\nvelocity and magnetic field perturbations in the radial direc-\ntion as vertical modes. However, the direction of the mean\nmagnetic field in the accretion disk midplane is expected to\nbe predominantly azimuthal, as a result of the global scale\nshearing caused by the background azimuthal velocity field\non radial magnetic fields. The strong azimuthal fields have\nthe potential to become dynamically important, as they causeTable 1. Background conditions explored in shearing-box simula-\ntions\nParameter Values\nShearing Rate ( q) 0.5, 0.75, 1.0, 1.25, 1.5, 1.75\nVertical β(βz) 800\nAzimuthal β(βy) 2, 5, 10, 20\nEquation of State Isothermal\nthe growth of large-scale azimuthal modes2. These consid-\nerations motivate us to set up the shearing box in our simula-\ntions with a strong magnetic field in the azimuthal direction\ncombined with a weak magnetic field in the vertical direc-\ntion.\nThe midplane regions of accretion disks typically exhibit\nshearing rates equal to the Keplerian value ( q= 1.5). How-\never, in regions away from the midplane, non-Keplerian\nshearing rates are expected. In the interest of allowing our\nstudy to be broadly applicable to global-scale studies of ac-\ncretion flows, we study the local shearing box simulations for\na wide range of shearing rates. We also set the equation of\nstate in the system to be isothermal. In Table 1, we list the\nparameter space of the background conditions of the shearing\nbox explored in this paper.\n5.2. Linear MRI Growth and Turbulence\nThe linear growth of the MRI in the presence of a weak,\nvertical mean magnetic field is well understood both analyt-\nically and numerically (Balbus & Hawley 1991, 1998). In\na numerical simulation, the linear stage of growth is dom-\ninated by the dynamics of the fastest growing mode of the\nMRI. The simulations at this stage take the form of channel\nsolutions where the velocity and the magnetic fluctuations in\nthexyplane undergo exponential growth with a rate equal\nto the growth rate of the fastest growing mode. These chan-\nnel solutions are vertical, axisymmetric modes, and also sat-\nisfy the complete non-linear equations of motion. At later\ntimes of their exponential growth, the structure of the chan-\nnel solutions makes them unstable to Kelvin-Helmholtz-like\nparasitic instabilities, which breaks the solutions into a non-\naxisymmetric turbulent state (Goodman & Xu 1994; Pessah\n2010). In the presence of strong azimuthal magnetic fields,\nlarge wavelength non-axisymmetric modes are also unstable\nto the MRI.\nThe outward transport of angular momentum is brought\ninto effect by the xycomponents of the Maxwell and\nReynolds stress tensors. Pessah et al. (2006b) showed that the\nratio of the volume-integrated Maxwell to Reynolds stress\ntensor at late times of linear MRI growth in the presence of\n2The effects of the curvature of magnetic field lines can be dynamically im-\nportant to the growth of the MRI at strong azimuthal field strengths (Pessah\n& Psaltis 2005). We do not consider these effects in our study.8 S ATAPATHY , PSALTIS , & Ö ZEL\n0.5 1.0 1.5 2.0\nShear (q)100101Mxy/Rxy\nβy= 2\nβy= 5\nβy= 10\nβy= 20\nFigure 3. The ratio of the volume-integrated xycomponent of the\nMaxwell stress tensor to that of the Reynolds stress tensor, plotted\nas a function of shearing rate, for different values of βy, computed\nfrom our simulation set. The triangles indicate the stress ratios at\nthe stage of linear growth of the MRI (channel solutions), while the\nfilled circles represent the same ratio computed during a saturated\nturbulent stage at later times in the simulation ( t= 230 −300Ω−1).\nThe black dotted line represents the analytical calculation (Pes-\nsah et al. 2006b) of the dependence of Mxy/Rxygiven by equation\n(27) on the shearing rate q, for the fastest growing vertical mode\nkM=kmax=√q(4−q)/2. The red dotted line represents the same\nanalytically obtained ratio, but instead assuming that the ratio in the\nnon-linear turbulent stage is captured by an effective wavenumber\nkeff= 0.42kmax.\na weak vertical background magnetic field depends on the\nwavenumber kMof an unstable mode and the shearing rate\n(q), and is given by\nMxy(kM)\nRxy(kM)= 1+2(2−q)\nk2\nM+γ(kM)2. (26)\nIn equation (27), the quantity γ(kM) represents the growth\nrate of the unstable mode with wavenumber kM, and is given\nby\nγ(kM)≡\u001aq\n(2−q)2+k2\nM−\u0002\nk2\nM+(q−2)\u0003\u001b1/2\n. (27)\nAssuming that the growth of the linear MRI is dominated by\nthe fastest growing mode ( kM=kmax=√q(4−q)/2), the ratio\nof the Maxwell to the Reynolds stress tensor in this stage can\nbe approximated as\nMxy\nRxy\f\f\f\f\nlinear MRI≈4−q\nq. (28)\nIn our set of numerical simulations, we observe a period\nof channel-solution-like growth of the linear MRI during the\nearly stages of each simulation. During these periods, we\nnumerically calculate the ratio Mxy/Rxy. In Figure 3, we plotthis ratio as a function of the shearing rate using triangular\ndots. We observe that the ratio is nearly independent of the\nplasma βbut primarily depends on the shearing rate q. Ad-\nditionally, we see that the ratio closely follows the analyti-\ncal result for Mxy/Rxyfor the fastest growing mode given by\nequation (28). These observations establish that the signature\nof the fastest growing vertical mode on the volume integrated\nstress ratio Mxy/Rxyis persistent even in the presence of sig-\nnificant azimuthal background magnetic fields.\nIn order to study the non-linear turbulent stage, we com-\npute the ratio Mxy/Rxyat later times of our simulations ( t=\n230−300Ω−1)3. At this stage, the ratio attains values higher\nthan the corresponding ratio at the the linear stage of growth.\nThis behavior is observed for all values of shearing rate, in-\ndependent of the background plasma β. The numerically\ncomputed ratios are shown as the circular dots in Figure 3,\nwith the overplotted error bars showing the measured stan-\ndard deviation in the ratios. While this behavior is known in\nshearing box systems in the presence of a weak vertical mean\nmagnetic fields (see Figure 3 in Pessah et al. 2006b), the ad-\ndition of strong azimuthal magnetic fields does not seem to\nalter their conclusion.\nWe now aim to provide a simplistic model for the shear-\ndependence of the ratio Mxy/Rxyduring the non-linear tur-\nbulent stage. We posit that at the late times, the dynam-\nics of the turbulence can be captured by an effective MRI-\nunstable mode. We model this effective mode such that its\nwavenumber keff=akmax, where ais a scaling parameter and\nkmax=√q(4−q)/2 is the wavenumber of the fastest growing\nlinear mode. In Figure 3, we plot the theoretical shear depen-\ndence of the ratio Mxy/Rxyfor a wavenumber keffsuch that\na= 0.42, demonstrating that the dependence of Mxy/Rxyon\nthe shearing rate is well captured by this choice.\nWhile the contribution of the azimuthal component of the\nmean magnetic field is subdominant in the dynamics of the\nMRI, it does have important implications on the injection of\nenergy into the slow and the Alfvén wave cascades. First,\nas discussed in §3, the large-scale mean field in the flow\ndetermines the polarizations for the slow- and the Alfvén-\nwave displacements. The relative orientations of the polar-\nizations of modes and the direction of the shear is key to de-\ntermining the energy injection rates into the individual cas-\ncades. Additionally, the persistent large-scale mean field in\nthe disk causes the cascade of turbulence to be anisotropic.\nThe strong anisotropy in the turbulence results in the de-\ncoupling of the wave cascades past the injection scales. In\nthe following subsection, we infer the slow- to Alfvén-wave\ndriving power from the Maxwell and Reynolds stress tensors\ncomputed here.\n5.3. Slow- to Alfvén-Wave Driving Power Ratio\n3In order to compute the Reynolds stress tensor, we use the local fluid den-\nsityρas opposed to the background density ρ0(see eqn. [19]), which is\nconsistent with the order of the calculation.9\n0.5 1.0 1.5 2.0\nShear (q)100101PS/PA\nβy= 2\nβy= 5\nβy= 10\nβy= 20\nFigure 4. The ratio of slow wave injection rate to the Alfvén wave\ninjection rate, PS/PA, plotted as a function of shearing rate, com-\nputed for our simulation set. The triangular dots indicate the in-\nferred ratio at the stage of linear growth of the vertical mode MRI\n(channel solutions), and the circled dots represent the same ratio\ncomputed during a saturated turbulent stage at later times in the\nsimulation ( t= 230 −300Ω−1). The dotted line represents the value\nofPS/PAcalculated analytically for the linear stage using Equation\n(29). The black star indicates the calculation of PS/PAby Kawazura\net al. (2022) who solved the dynamical equations of MHD at the\nfirst order in the reduced-MHD approximation.\nHaving numerically computed the volume integrated\nMaxwell and Reynolds stress tensors in MRI driven turbu-\nlence with predominantly azimuthal mean fields, we now\ntranslate them into the slow-to-Alfvén injection power ratio\nusing equation (25). In Figure 4, we plot the inferred PS/PA\nas a function of shearing rate for all the background magnetic\nfield strengths explored in our set of simulations.\nThe ratio PS/PAassumes a value of near unity during the\nlinear stage of MRI growth, for almost all the our simulations\nspanning various shearing rates and values of plasma β. This\nresult can be analytically explained by combining the theo-\nretical value of Mxy/Rxyfor the fastest growing vertical mode\n(eq. [28]) with equation (25) as\nPS\nPA\f\f\f\f\nlinear MRI ,⃗B0≈B0ˆy=q\u0012Mxy\nRxy\f\f\f\f\nlinear MRI+1\u0013\n−2\n2= 1.(29)\nRemarkably, even though the ratio Mxy/Rxyduring the linear\ngrowth of the MRI carries a strong dependence on the shear-\ning rate q(eq. [28]), the value of PS/PAduring this stage is\nindependent of the shearing rate.\nIn the non-linear stage of the MRI, the ratio PS/PAtakes\nvalues higher than in the linear stage, with a weak depen-\ndence on shear. The values of PS/PArange between ∼2−8\ndepending on the shearing rate. The effects of non-linear in-\nteractions in the turbulence cause the ratio Mxy/Rxyto be sig-\nnificantly higher than the corresponding linear limit at low\n10−1100101102\nplasmaβ10−210−1100Qe/(Qi+Qe)\nq = 0.50q = 0.75q = 1.00q = 1.25q = 1.50q = 1.75RIAF ConditionsFigure 5. The electron heating fraction, Qe/(Qi+Qe), inferred from\neq. (33) as a function of plasma β, plotted for different values of\nthe shearing rate ( q). The electron heating fraction dependence on\nplasma βcorresponding to a Keplerian shearing rate is highlighted\nin blue. The dashed line depicts the electron heating fraction at the\nlimit in which the turbulence is purely driven by Alfvén waves. The\ntypical values of plasma βencountered in GRMHD simulations of\naccretion flows are highlighted in the shaded region.\nvalues of q. Hence, the ratios PS/PAare significantly higher\nat lower values of shearing rates.\nThe background plasma βin the shearing box does not\nhave a strong effect on the ratio PS/PAat a given value of\na shearing rate. This is a result of the fact that the sig-\nnature of MRI-driven turbulence on the ratio of Maxwell\nand Reynolds stress tensors, even in the non-linear stages of\ngrowth, follows the trend determined by the fastest growing\nlinear modes (Figure 3). As discussed in the previous subsec-\ntion, the ratio of the stress tensor depends only on shear and\nis independent of the background magnetic field strength.\nFor shearing rates equal to a Keplerian value, q= 3/2,\nwhich is expected in the midplanes of black-hole accretion\nflows, the value of PS/PAis approximately 2. This computa-\ntion is in agreement with the calculation of the same quantity\nperformed using reduced-MHD simulations by Kawazura\net al. (2022), which we show as a black star in Figure 4. The\nratio computed by Kawazura et al. (2022) is also independent\nof plasma β, in agreement with our findings and physical ex-\nplanation as described in §4.\nWhile our calculation of PS/PAis a first order estimate of\nthe value under the reduced-MHD approximation computed\nfrom MHD simulations, the agreement with Kawazura et al.\n(2022) indicates that the approximation is largely valid in the\nparameter regime that we consider. From our analysis, it is\nevident that irrespective of the background plasma βand the\nshearing rate in an MRI driven turbulent flow, the fraction of\nturbulent energy injected into the slow waves is significantly\nhigher than the energy injected into the Alfvén waves.\n5.4. A Sub-grid Model for the Slow-to-Alfvén Driving\nPower and Ion-to-Electron Heating Ratio10 S ATAPATHY , PSALTIS , & Ö ZEL\nIn the discussion surrounding Figure 3, we suggested that\nthe value of Mxy/Rxyduring the non-linear stage of the tur-\nbulence can be captured by the theoretical estimate of the\nratio for an effective wavenumber keff. We demonstrated that\nan approximate value for keff= 0.42kmax, where kmaxis the\nwavenumber of the fastest growing vertical MRI mode. Us-\ning equation (25), we then write\nPS\nPA\f\f\f\f⃗B0≈B0ˆy=1\n2\u001a\nq\u0014\n2+2(2−q)\nk2\neff+γ(keff)2\u0015\n−2\u001b\n. (30)\nIn equation (30), the value of keffis given by\nkeff= 0.42kmax= 0.42√q(4−q)\n2, (31)\nand the corresponding growth rate γ(keff) for the mode keffis\ngiven by\nγ(keff) =\u0014q\n(2−q)2+k2\neff−\b\nk2\neff+(q−2)\t\u00151/2\n. (32)\nIn the dotted red lines shown in Figure 4, we depict the\nratio PS/PAgiven by equation (30). Clearly, this expression\ncaptures the dependence of the numerically inferred values\nofPS/PAon the shearing rate q. We now combine this with\nmicrophysical studies by Kawazura et al. (2020) for the ion-\nto-electron heating ratio Qi/Qeresultant from collisionless\ndamping of slow- and Alfvén-wave turbulence to write (see\neq. [1]) as\nQi\nQe=1\n2\u001a\nq\u0014\n2+2(2−q)\nk2\neff+γ(keff)2\u0015\n−2\u001b\n+35\n1+(β/15)−1.4.(33)\nThe plasma β-dependent second term captures the effects of\nmode mixing on the heating ratio at microphysical scales\ncomparable to the ion gyroradius. The first term represents\nthe impact of the large scale turbulent driving on the heating\nratio. Notably, this impact of the large-scale driving is inde-\npendent of plasma βbut primarily depends on the shearing\nrateq.\nIn Figure 5, we plot the electron heating fraction,\nQe/(Qi+Qe), as a function of plasma βfor different values\nof the shearing rate q, using equation (33). For comparison,\nwe plot the same fraction for a purely Alfvénically driven\nturbulence. In the shaded region, we depict the values of\nplasma βthat black hole accretion disks typically exhibit in\nGRMHD simulations. Clearly, the shearing rate-dependent\nnature of turbulent driving strongly determines the electron\nheating ratio in the disk regions of accretion flows.\nThe sub-grid model for the ion-to-electron heating ra-\ntio given by equation (33) can be integrated with two-\ntemperature GRMHD simulations in order to model the\nelectron thermodynamics in radiatively inefficient accretion\nflows. It can also be used to infer electron temperatures from\none-temperature GRMHD simulations in a way which is self-\nconsistent with the large-scale turbulent driving. Satapathy\n10−1100101102\nplasmaβ100101102Ti/Teq = 0.50\nq = 0.75\nq = 1.00\nq = 1.25\nq = 1.50\nq = 1.75RIAF ConditionsFigure 6. The temperature ratio, Ti/Te, as a function of plasma β, in\nthe inner region of a radiatively inefficient accretion flow (at ∼10\ngravitational radii), for different values of the shearing rate, calcu-\nlated using the two-temperature, covariant, semi-analytic model of\nSatapathy et al. (2023) and the ratio PS/PAgiven by equation (30).\nFor typical values of the plasma βparameter and for a Keplerian\nshear ( q= 1.5), we find Ti/Te∼5−40, with the ratio increasing to\n∼80 as the shear becomes shallower.\net al. (2023) used a semi-analytic, height- and azimuthally-\naveraged two-temperature model for an accretion flow to\ncalculate the ion-to-electron temperature ratio ( Ti/Te) taking\ninto account the heat partition due to the slow- and Alfven-\nwave turbulent driving as well as compressional heating ef-\nfects (see, eq. [37] in that paper). We use this calculation\ncombined with the shear-dependent model Qi/Qepresented\nabove to estimate the ion-to-electron temperature ratio in the\ninner accretion flow around a black hole and plot it in Fig-\nure 6 as a function of plasma β, for different values of the\nshearing rate. For this plot, we have set the radial distance\nto 10 gravitational radii and the radial profile of the radially\ninfalling velocity to follow a power law with an index -1.5,\nwhich was obtained via calibration with GRMHD simula-\ntions by Özel et al. (2022). For typical values of the plasma\nβparameter and for a Keplerian shear ( q= 1.5), we find\nTi/Te∼5−40, with the ratio increasing to ∼80 as the shear\nbecomes shallower.\nThe primary underlying assumption in the model for\nQi/Qegiven by equation (33) is that the mean magnetic field\nis nearly in the azimuthal direction in the region of interest\nin the accretion flow. This is a reasonable assumption for the\ndisk regions of accretion flows. In certain configurations of\nglobal simulations, however, the disk evolves into a magnet-\nically arrested state (MAD, see Narayan et al. 2012). In such\na state, significant vertical components of magnetic field con-\nfiguration might be observed in the inner regions of the disk.\nAdditionally, the x−zcomponents of stress tensors, which\nare dynamically unimportant in a local shearing box, can po-\ntentially impact the partition of energy into slow and Alfvén\nwave cascades due to the effects of vertical stratification. The11\nimpacts of these effects on driving the slow and Alfvén waves\nneed to be further explored.\n6.DISCUSSION\nIn this paper, we derived an analytical expression for\nthe ratio of the slow-wave to Alfvén-wave driving powers\n(PS/PA) in an MRI-driven turbulence. Working in the linear\norder limit of the reduced-MHD approximation, we estab-\nlished the connection of the ratio PS/PAto the macroscopic\nproperties of the turbulence. We show that the ratio depends\non the components of the volume integrated Maxwell and\nReynolds stress tensors in the turbulence that are responsi-\nble for the outward transport of angular momentum, and on\nthe orientation of the mean magnetic field with respect to the\ndirection of shear. We argue that the underlying physics that\ndetermines this ratio is the impact of shear injection of energy\nby the MRI and the impact of Coriolis force on the projection\nof the slow- and the Alfvén-wave displacements with respect\nto the orientation of the shear.\nUsing numerical shearing box simulations with magnetic\nfield configurations relevant to black hole accretion disks, we\ncompute the ratio PS/PAduring the non-linear saturated tur-\nbulent state of MRI-driven turbulence. During this stage, we\nfind the ratio between the volume-integrated Maxwell and\nReynolds stress tensors to be nearly independent of the back-\nground magnetic field strength. As a result, the ratio of the\ndriving power of the compressive slow-wave cascade to that\nof the Alfvén-wave cascade is nearly independent of plasma\nβ. The only dependence seen in PS/PAis a relatively weak\ndependence on the shearing rate. We combine our inference\nof a shear-dependent model for PS/PAwith hybrid kinetic-\nscale studies of ion and electron heating (Kawazura et al.\n2020) to derive a local sub-grid model for the ion-to-electron\nheating ratio in black hole accretion flows.\nTwo-temperature GRMHD simulations that evolve the\nglobal thermodynamics of the ions and the electrons allow\nfor the direct study of the impact of plasma heating (Ressler\net al. 2015; S ˛ adowski et al. 2017). Recent studies, such asthe ones performed by Chael et al. (2018) and Dexter et al.\n(2020) explore the impact of the electron heating physics on\nthe global thermodynamics in GRMHD simulations and their\nimplications on observational signatures. These studies show\nthat the subgrid models for particle heating have a significant\nimpact on not only the dynamics of the accretion flows but\nalso on their implied images and spectra. However, in all\nof the above studies, subgrid heating of the plasma species\ndue to collisionless damping has been performed assuming\npure Alfvén-wave cascades in the turbulence. Our study of-\nfers a way to accurately model the ion-to-electron heating ra-\ntio considering the injection of energy into slow and Alfvén\nwaves locally in an accretion flow based on components of\nthe Maxwell and Reynolds stress tensors.\nA number of additional dissipation mechanisms can effect\nthe partition of heat between the ions and the electrons. Such\nmechanisms include episodes of magnetic reconnection (Ball\net al. 2018, 2019) and dissipation of the stress resulting from\nanisotropies in velocity space (Sharma et al. 2007; Kunz et al.\n2018). The effects of these mechanisms and their connec-\ntions to global quantities in accretion flows need to be further\nexplored.\nUsing a height- and azimuthally-averaged covariant two-\ntemperature model for an accretion disk, Satapathy et al.\n(2023) showed that the electron temperature in the inner re-\ngions of the disk is determined primarily by the slow waves\nin the turbulence. Incorporating local stress-tensor depen-\ndent models of sub-grid heating in accretion flows will make\nthe small-scale dissipation processes self consistent with the\nlarge scale physics of energy injection.\nACKNOWLEDGMENTS\nThe authors acknowledge support from NASA ATP award\n80NSSC20K0521 and NSF PIRE award OISE-1743747. We\nthank K.G. Klein, M. Golden, M. Avara, T. Trent, and K.\nRoley for useful discussions.\nAPPENDIX\nA.RELATION OF SLOW, FAST, AND ALFVÉN MODES TO THE HELMHOLTZ DECOMPOSITION\nAs discussed in the main text, the partition of turbulent heat between ions and electrons was calculated by Kawazura et al.\n(2020) using hybrid gyrokinetic simulations, in the presence of what these authors called “compressive-” and Alfvén-wave\ncascades. Kawazura et al. (2022) then performed reduced-MHD simulations in a shearing box to compute numerically the\ncompressive-to-Alfvénic driving power ratio and found it to be ≃2. These results have been questioned recently by two parallel\nstudies: Zhdankin (2021) used relativistic particle-in-cell simulations and found only qualitative similarities to the results of\nKawazura et al. (2020); Bacchini et al. (2024) have attempted to compute the compressive-to-Alfvén ratio from kinetic shearing-\nbox simulations of a pair plasma and found it to be significantly smaller than unity, contradicting the results of Kawazura et al.\n(2022).\nIn this Appendix, we show that, even though both sets of studies discuss “compressive” and Alfvénic cascades, they actually\ndecompose the turbulent cascades into different sets of modes. As a result, the only difference between the two sets of studies\nappears to be their definitions of the nature of the “compressive” cascade. Zhdankin (2021) and Bacchini et al. (2024) defined\ncompressive fluctuations in the sense of the Helmholtz decomposition, i.e., following the literal meaning of the word compressive\nand distinguishing modes based on their velocity divergence. Consequently, their compressive-wave cascade consists primarily12 S ATAPATHY , PSALTIS , & Ö ZEL\nk⊥k/bardbl\nB0\nˆk−ˆθ\nˆφ\nSolenoidal Compressivek/bardbl=k⊥\nAlfv´ enFastSlow\nα→∞\nα→0α\n→∞\nα→0\nFigure 7. A representative plot to illustrate the difference between the decomposition of the plasma velocity field into the slow, fast and Alfvén\nmodes, and the Helmholtz decomposition into compressive and solenoidal modes. The slow and fast wave velocity eigenvectors are shown\nin the blue and red arrows respectively. Their orientations are dependent on the parameter α≡c2\ns/v2\nA, where csis the sound speed and vA\nis the Alfvén speed. The Alfvén wave eigenvector is shown in the green circled dot, and is independent of α. The basis for the Helmholtz\ndecomposition is shown for a equal value of k∥/k⊥. The solenoidal components of a vector field resulting from the Helmholtz decomposition\nlie in the ˆθ−ˆϕplane indicated in orange, while the compressive components lie along ˆk, shown in black.\nof fast wave fluctuations, which, as noted by the authors, are expected to result from spiral shock waves in the large-scale system.\nHowever, fast magnetosonic waves are ordered out in the reduced-MHD calculation of Kawazura et al. (2022) so that their label\nof “compressive” cascade consists purely of slow magnetosonic fluctuations. It is this ratio of this slow magnetosonic cascade to\nthe Alfvénic one that determines the partition of heat between ions and electrons.\nIn this paper, we have followed the decomposition of the field into slow, fast, and Alfvén modes derived by Cho & Lazarian\n(2003). We use this to demonstrate the reason why the Helmholtz decomposition is not useful to separate slow and Alfvén modes\nin an MHD system, and to re-iterate that the nature of turbulent driving should be measured by the power in the individual slow,\nfast, and Alfvén wave cascades.\nThe Helmholtz decomposition of a velocity field in Fourier space ⃗˜v(⃗k) can be written as\n⃗˜v(⃗k) =⃗k·⃗˜v(⃗k)\n⃗k·⃗k⃗k+⃗k×{⃗˜v(⃗k)×⃗k}\n⃗k·⃗k. (A1)\nIn the right hand side of equation (A1), the first and the second terms represent the curl-free and the divergence-free components\nof the vector field respectively. If the vector field represents a plasma velocity the curl-free component represents the compressible\npart of the flow, and the divergence-free component represents the incompressible part of the flow.\nIn Figure 7, we use the black arrows to represent the basis of the Helmholtz decomposition. At a given point in the Fourier\nspace defined by the parallel ( k∥) and perpendicular ( k⊥) wavenumbers, the first term in equation (A1) picks out components of\nthe velocity field along ˆk. The second term on the other hand, represents components of the velocity field that lie on the plane\nperpendicular to ˆk, i.e. the ˆθ−ˆϕplane.\nWe now re-examine the eigenvectors of the linear modes of MHD, namely the slow, fast, and Alfvén waves, that we introduced\nin §3. Following the notation of Figure 7, the orientations of the slow and the fast wave eigenvectors are represented by the blue\nand red arrows respectively. These eigenvectors and the basis defined by the Helmholtz decomposition are represented at points\nin the Fourier space such that they have the same k∥/k⊥= 1/3. Depending upon the value of α(equation 9), the slow waves are\noriented between ˆk∥and −ˆθ, and the fast waves are oriented between ˆk⊥andˆk. The Alfvén wave eigenvector orientation, shown\nin the green dot in Figure 7, is independent of αand is always in the ˆϕdirection.\nWe now turn to compare the eigenvectors of the linear MHD modes with the basis defined by the Helmholtz decomposition.\nIt is to be noted that the Helmholtz decomposition only picks out the components of a velocity field along and perpendicular to13\nˆk(shown by the black arrow in Figure 7), and define a solenoidal basis in the plane perpendicular to ˆk(shown in orange). The\nAlfvén waves, being polarized along ˆϕ, however, always lie in the plane perpendicular to ˆk. This is consistent with the physical\npicture of Alfvén waves being incompressible modes.\nOn the other hand, both the slow and fast wave eigenvectors in general have components of velocity along ˆkand in the plane\nperpendicular to ˆk. In the limit of α→ ∞ , the slow waves lie entirely in the plane perpendicular to ˆkand the fast waves are\npolarized along ˆk. These properties of the orientations of the linear MHD modes clarifies that the Helmholtz decomposition in\ngeneral cannot distinguish the Alfvén modes from the components of the slow and the fast waves perpendicular to ˆk. Particularly,\nin anisotropic plasmas where most of the power lies in the region k∥≪k⊥, the fast mode carries most of the compressible power,\nin the ˆkdirection. The slow modes are nearly incompressible, with their polarisation nearly along −ˆθand for this reason, they are\nalso referred to as pseudo-Alfvén modes (Maron & Goldreich 2001).\nIn the reduced-MHD approximation, most of the power is assumed to lie in the k∥≪k⊥region. Additionally, the fast waves are\nordered out, leaving the only normal modes to be the slow and Alfvén waves. Consequently, the Helmholtz decomposition is not\nuseful in separating out these modes and necessitates decomposing the velocity field along the slow and the Alfvén eigenvectors\nto analyze their injection rates. Indeed, because both the linear MRI and its linear order parasitic (Kelvin-Helmholtz) instability\nare fundamentally incompressible in nature, the turbulence is not expected to produce significant power in fully compressive\nmodes (i.e., polarized along ˆk), as found in Bacchini et al. (2024). This, however, is not in contradiction to the result of Kawazura\net al. (2022), i.e., that the slow-to-Alfvénic power in the same turbulence is larger than unity.\nREFERENCES\nAkiyama, K., Alberdi, A., Alef, W., et al. 2022, ApJL, 930, L16\nBacchini, F., Zhdankin, V ., Gorbunov, E. A., et al. 2024, arXiv\ne-prints, arXiv:2401.01399\nBalbus, S. A., & Hawley, J. 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Celso Garcia Cid, Km 380, 86057-970, Londrina, Paran´ a, Brazil\n4Departamento de F´ ısica, Universidad del Valle,\nCiudad Universitaria Mel´ endez, Santiago de Cali 760032, Colombia\n(Dated: February 23, 2024)\nWe introduce a multi-field dark energy model with a non-flat field-space metric, in which one\nfield is dynamical while the others have constant spatial gradients. The model is predictive at the\nbackground level, leading to an early dark energy component at high redshifts and a suppressed\nfraction of late-time anisotropy. Both features have simple expressions in terms of the curvature\nscale of the field-space, and correspond to stable points in the phase space of possible solutions.\nBecause of the coupling between time and space-dependent scalar fields, vector field perturbations\ndevelop tachyonic instabilities at scales below the Hubble radius, thus being potentially observable\nin the number count of galaxies. Overall, the presence of a non-trivial field-space curvature also\nleads to the appearance of instabilities on scalar perturbations, which can impact the matter density\ndistribution at large scales.\nI. INTRODUCTION\nThe existence of an unexplained dark sector continues\nto be one of the main drivers of new ideas in cosmol-\nogy, forcing us to test the limits of the standard Lambda\nCold Dark Matter (ΛCDM) model. Recently this pro-\ngram has gained momentum due to the realization that\nmeasurements of critical cosmological parameters, among\nwhich H0, give different values when done using high\nor low redshift data [1, 2]. Alongside these facts, there\nremain numerous large-angle Cosmic Microwave Back-\nground (CMB) anomalies which also lack an explanation\n— fundamental or not — and that contribute to the sus-\npicion that the ΛCDM model is just a first approximation\nto a more accurate description of the universe.\nAmong the many potential candidates to explain these\nproblems lies the possibility that dark energy results\nfrom new matter degrees of freedom. The most popu-\nlar and paradigmatic example is the quintessence field\nand its many avatars [3]. At first sight, the versatility of\nquintessence models seems to be limited by the imposi-\ntion of translation invariance arising from the Cosmolog-\nical Principle, which forces the fields to be homogeneous.\nHowever, a closer inspection of the Lagrangian from the\nsimplest quintessence model,\nL=−1\n2∂µϕ∂µϕ−V(ϕ),\nreveals that, as far as the kinetic term is concerned, trans-\nlation invariance does not require homogeneity of the\n∗jubeltrana@unal.edu.co\n†aguarnizo50@uan.edu.co\n‡tspereira@uel.br\n§cesar.valenzuela@correounivalle.edu.cofields , but rather the homogeneity of their spatial gradi-\nents. The realization of this possibility has led to many\ninteresting implementations known generically as “solid”\nor “elastic” models, where the dynamics is achieved by\nmeans of space-dependent scalar fields with constant spa-\ntial gradients. To account for a forbidden (symmetry\nbreaking) potential, such models usually re on a non-\ncanonical Lagrangian capable of reproducing an accel-\nerated expansion. This idea has its roots on effective\nfield-theoretic implementations of solid and elastic me-\ndia [4, 5], and has since found fertile ground in cosmo-\nlogical applications, being implemented in inflation [6–8],\ndark matter [9, 10], and dark energy [11, 12].\nIn the standard applications of solids to cosmology, the\ninternal symmetries of the fields, such as SO(3) and shift\ninvariances, are explored when building phenomenologi-\ncally viable cosmological models [7, 13]. In this regard,\none could also envisage the existence of a curvature in\nfield-space, as is usually done in non-linear sigma models\nof quantum field theory. In fact, time-dependent non-\nlinear sigma models have been long explored in the con-\ntext of inflation [14–20] and dark energy [21–25]. Re-\ncently, the authors of Ref. [22] considered the implemen-\ntation of dark energy in one such model, showing that a\nphase of accelerated expansion can be achieved even with\nsteep potentials, provided that trajectories in field-space\nare sufficiently non-geodesic.\nIn this work we ask whether a time-dependent field\ncould coexist with a solid, thus bridging a gap between\nthe solid and non-linear multi-field descriptions. Specif-\nically, we introduce a non-linear dark energy model in\nwhich one field is dynamical, while three other have con-\nstant spatial gradients. The model is predictive at the\nbackground level, and can lead to a percent level frac-\ntion of early dark energy, as well as a suppressed fractionarXiv:2402.14108v1  [astro-ph.CO]  21 Feb 20242\nof late-time anisotropy. We show that the early dark\nenergy component is proportional to the square of the\nfield-space curvature radius, while late-time anisotropies\nhas an upper bound given by the fourth power of this\nradius. Thus, for small and observationally interesting\nradii, the anisotropy is always suppressed compared to\nearly dark energy. When small scale linear perturbations\nare included, we show that the existence of a non-trivial\ncurvature in field space leads to instabilities in transverse\nvector modes, with a characteristic length scale of order\nof the present Hubble radius. Such instabilities will feed\ngravitational vector perturbations, which could, in prin-\nciple, be observed through the abundance of large-scale\nstructures. The same curvature term is responsible for\nthe appearance of gravitational instabilities in scalar per-\nturbations, though the derivation of a dispersion relation\nand the length scales of these instabilities is hard to ob-\ntain analytically. Overall, our findings corroborate the\nclaim that a negative field-space Ricci scalar can lead to\ngravitational instabilities [20].\nThe mathematical formalism appropriate for our de-\nscription, known as harmonic maps, is quickly reviewed\nin Section II. In Section III we discuss the main idea of\nthe curved solid model, and derive its main features at\nthe background level. The dynamics of small scale linear\ninstabilities is analysed in Section IV, after which we con-\nclude and give some perspectives of future developments\nin Section V. Throughout this work we adopt (modified)\nPlanck units in which ℏ=c= 8πG= 1 = M−2\nPland a\nmetric signature ( −,+,+,+).\nII. HARMONIC MAPS\nWe start by giving a concise introduction to the formal-\nism of harmonic maps, which underpins our implementa-\ntion. Further details can be found in [26–28]. Although\nthe mathematics is the same as that used in multi-field\nmodeling (see, e.g., [14, 17, 20]), many physically inter-\nesting insights and possible applications introduced by\nthe formalism seem to be unknown to (most) cosmolo-\ngists. We thus hope that this section can also help draw-\ning attention to these important references. Readers in-\nterested in our implementation can skip to Section III.\nThe formalism of harmonic maps consists in a set of\nscalar fields ϕAevolving over a space-time Mwith metric\ngµν, such that the fields themselves define a Riemannian\nspace N, the field-space , with metric GAB. The dimen-\nsion of Nis arbitrary, although for the applications we\nhave in mind dim N= 4; the field-space indices A, B, . . .\nthus run from 0 to 3.\nBecause the fields live in the space-time, they map\npoints from MtoN, since for any point xµinM\nthere corresponds a point ϕAinNthrough the func-\ntions ϕA(xµ). This identification allows us to pull field-\nspace objects back to the space-time through the ma-\ntrix∂µϕA. This is possible because, under general field\n(ϕA→ψA(ϕC)) and coordinate ( xµ→yµ(xν)) transfor-mations, this matrix transforms linearly:\n∂ψA\n∂yµ=∂ψA\n∂ϕC∂xν\n∂yµ∂ϕC\n∂xν. (1)\nThus, for example, the field-space metric GABand con-\nnection ΓA\nBCcan be pulled back from the field-space to\nthe space-time as\nGµν≡∂µϕA∂νϕBGAB, (2)\nΓA\nµB≡∂µϕCΓA\nCB. (3)\nThis property renders the dynamics of the fields non-\nlinear, and it serves as the central feature from which\nmany of the interesting properties of multi-field cosmo-\nlogical models emerge, as well as many of interesting\nproperties of harmonic maps in general [27].\nGiven this structure, we can take space-time covariant\nderivatives of field-space tensors provided we attach, for\neach field-space index, a corresponding connection term\nΓA\nµB. Thus, for example, the space-time covariant deriva-\ntive of any vector vAinNwill be\n∇µvA=∂µvA+ ΓA\nµBvB, (4)\n=∂µϕB(∂BvA+ ΓA\nBCvC), (5)\n=∂µϕB∇BvA. (6)\nFor a tensor with mixed indices, tABµν..., one proceeds\nas usual, adding the corresponding connection Γµ\nλνfor\neach space-time index, and ΓA\nµBfor each field-space in-\ndex. Note however that ∇µϕAis still given by ∂µϕA, as\nusual. From these definitions and the condition of met-\nric compatibility in field space, ∇AGBC= 0, one readily\nverifies that\n∇µGAB= 0. (7)\nThis is an important property, and ensures that the ma-\nnipulation of field-space indices commutes with space-\ntime derivatives; in particular, it allows us to write\nGAB∇µϕBsimply as ∇µϕA.\nThe fields ϕAare called harmonic when they extremize\nthe action\nS=−1\n2Z\nd4xp\n|g|∇µϕA∇µϕA. (8)\nPhysically, if Mdescribes a rubber and Na body, then S\ngives the elastic energy needed to wrap the rubber around\nthe body, while the fields obey an equation describing the\nconfiguration of zero elastic tension on the rubber [28]:\n∇µ∇µϕA≡1p\n|g|∂µ(p\n|g|∂µϕA) + ΓA\nµB∂µϕB= 0.(9)\nwhere g= det( gµν). This equation makes evident the\nnon-linearity of the dynamics.\nKnown examples of harmonic maps include the Klein-\nGordon equation for a massless scalar field (take\ndimM= 4 and dim N= 1) and the geodesic equation\n(dimM= 1, dim N= 4). A less obvious example is\ngiven by the geodesic deviation equation [27].3\nIII. CURVED SOLID\nThe vanilla harmonic model (8) cannot lead to accel-\nerating cosmologies. For that we also need a potential\nV(ϕA). We thus consider a generalized harmonic model\nwhere the fields are minimally coupled to gravity:\nS=Z\nd4xp\n|g|\u00141\n2R−1\n2∇µϕA∇µϕA−V+Lm\u0015\n,\n(10)\nwithLmbeing the matter Lagrangian. Variation of this\naction then leads to the following non-linear equation\n∇µ∇µϕA−GAB∂BV= 0. (11)\nThe class of models in which we are interested exhibit\nthe most versatility in universes described by homoge-\nneous metrics of the following form:\nds2=−dt2+a2(t)γij(t)dxidxj, (12)\nwith the condition that det( γij) = 1 in order to preserve\ncomoving volumes – see [29] for details. These include\nthe isotropic and spatially flat FLRW model ( ˙ γij= 0),\nas well as the anisotropic Bianchi-I model ( ˙ γij̸= 0). In\nthe latter, a(t) represents the mean scale factor, whereas\nthe anisotropic expansion is characterized by the shear\ntensor\nσij≡1\n2˙γij, (13)\nwhere a dot means d /dtand spatial indices are manip-\nulated with γij. These two metrics can be encapsulated\nby Misner’s parameterization, where one writes (no sum\noveri)\nγij= [e2βi]δij,X\niβi= 0 (14)\nwith the inverse given by γij= [e−2βi]δij. In terms of βi,\nthe shear tensor becomes\nσi\nj= [˙βi]δi\nj. (15)\nThe isotropic case follows naturally by taking βi(t) = 0.\nIn this setup, the energy-momentum tensor of the fields\nacquires its most general form:\nTµν= (ρ+p)uµuν+pgµν+ 2q(µuν)+πµν,\nwhere the heat-flux ( qµ) and anisotropic stress ( πµν) sat-\nisfy the usual constraints: qµuµ=πµ\nµ= 0 = uµπµν.\nFor a comoving observer defined by (12), the non-zero\ncomponents are given by\nρ=1\n2\u0014\n˙ϕA˙ϕA+1\na2∂kϕA∂kϕA\u0015\n+V(ϕA), (16)\np=1\n2\u0014\n˙ϕA˙ϕA−1\n3a2∂kϕA∂kϕA\u0015\n−V(ϕA),(17)\nqi=−˙ϕA∂iϕA, (18)\nπi\nj=1\na2\u0012\n∂iϕA∂jϕA−1\n3∂kϕA∂kϕAδi\nj\u0013\n. (19)Quite generally, expressions (16)-(19) cannot be com-\npatible with the symmetries of (12) for arbitrary fields\nϕA(xµ). In particular, observational data suggests that\nwe live in a non-tilted universe where qi= 0, i.e., where\nfreely-falling observers are comoving with the expand-\ning matter [30]. Thus, we look for field configurations\nin which this condition is satisfied. Given that qicou-\nples time and spatial derivatives, there are three distinct\nclasses of models where it vanishes:\n1. All fields are time-dependent: ϕA=ϕA(t);\n2. All fields are space-dependent: ϕA=ϕA(xi);\n3. A subset of fields are time-dependent, while the\nremaining are space-dependent.\nIncidentally, each one of these configurations automati-\ncally leads to ρ,pandπi\njcompatible with the symme-\ntries of (12). Let us analyse each case individually.\nModels of class 1 consists of the usual multi-field cos-\nmological models that have been used to described ei-\nther inflation and dark energy. In the context of dark\nenergy, it was recently implemented in a two field model\nusing flat [22] and non-flat [25] field-space metrics. Note\nthat in this case not only qibut also πijis zero. Since\nthe latter behaves as a source to the shear, the trace-\nfree part of the Einstein equation [see Eq. (32)] implies\nthat σ2∼a−6. Thus, assuming that the amplitude of\nthe shear is consistent with CMB constraints, typically\n(σ/H)0≲10−10[31, 32], one can safely disregard the\ncontribution of this term to the expansion at late times.\nIn other words, harmonic models with time-dependent\nfields are (virtually) isotropic.\nModels where the fields depend only on spatial coor-\ndinates (class 2) are commonly referred to as “solid” or\n“elastic” models [6, 7, 9]. Fields with an arbitrary spatial\ndependence cannot be generally employed, though, since\nthey violate spatial homogeneity. However, because the\nenergy-momentum tensor of the fields depends only on\ntheir gradients and a potential V, one canbuild cosmo-\nlogically viable models if the fields are given by linear\nfunctions of position and if V= 01. For these mod-\nels, homogeneity also requires that the field-space metric\nGABbe constant. As an example, consider the case of\nthree fields defined as ϕI=xI,I= (1,2,3), and a field-\nspace with flat metric δIJ. In this case ∇I=∂Iand the\nfields ϕIsatisfy Eq. (11) identically. Note however that,\neven though the fields have no dynamics, the matter sec-\ntor of these models is non-trivial, since the stress tensor\nis non-zero\nπi\nj=1\na2\u0012\nγij−1\n3γkkδi\nj\u0013\n. (20)\n1One could also adopt the weaker condition of a constant V, but\nthis is tantamount to a adding a cosmological constant to the\nmodel.4\nMoreover, the dynamics is also non-trivial, since (16) in-\ntroduces an effective curvature term to the expansion\nrate: ρ∼a−2. Thus, for time-dependent γij, the effec-\ntivebackground dynamics is that of a Bianchi-I universe\nwith spatial curvature.\nBefore concluding our discussion on the models of class\n2, it should be noted that space-dependent fields with\nnon-trivial dynamics can be implemented by means of\nan effective field theory approach. In this case, one sub-\nstitutes the kinetic term in (10) with the most general\nfunction of SO(3) invariant operators, which are in turn\nconstructed from products and contractions of the gra-\ndients ∂iϕIwith itself and the metric δIJ. This was ex-\nplored both in the context of inflation [7, 8], dark matter\n[9] and dark energy [11, 12]. At leading order in a deriva-\ntive expansion, the first non-trivial operator is given by\ngij∂iϕI∂jϕIδIJ. In view of the formalism of harmonic\nmaps, this is just the trace of the pushed-forward met-\nricgIJ=gij∂iϕI∂jϕJwith the flat field-space metric\nGIJ=δIJ. Thus, the formalism of harmonic maps pro-\nvides a framework for non-trivial geometric generaliza-\ntions of the standard (flat) solid models.\nClass 3 models represent a novel possibility which, to\nthe best of our knowledge, has not been explored2. Given\nthe preceding discussion, we shall refer to them as curved\nsolid models. As we are going to show, when imple-\nmented as a dark energy component, they predict small\nanisotropies as well as an early dark energy phase. We\nnow give the details of one concrete implementation.\nA. Concrete implementation\nFor concreteness we are going to consider a four field\nmodel in which the first, ϕ0, is dynamical, while the other\nthree, ϕI, are pure spatial gradients. According to our\nprevious discussion, the potential Vcannot depend on\nϕI. We thus define\nϕ0≡φ(t), ϕI≡xI, V ≡V(φ).(21)\nWe shall informally refer to φandϕIas quintessence\nand solid fields, respectively. As for the field-space met-\nricGAB, it can only be fixed by a more fundamental\ntheory. However, similar to the potential V, cosmologi-\ncal applications require that GABbe solely a function of\nφ. Given that we can always choose synchronous coor-\ndinates in which G00= 1, a symmetric and non-trivial\nfield-space metric is that of a flat and “expanding” field-\nspace:\ndℓ2= dφ2+f2(φ)δIJdϕIdϕJ, (22)\n2In the context of flat field-space metrics, time and space-\ndependent fields were considered in [13].where f(φ) is an arbitrary function of φ. The non-zero\nChristoffel symbols for this metric are\nΓ0\nIJ=−ff,φδIJ, ΓI\n0J=f,φ\nfδIJ, (23)\nwhere the “comma φ” notation means d /dφ. Using these\nexpressions and the condition ∂IV= 0, one readily veri-\nfies that the solids ϕIsatisfy Eq. (11). The quintessence\nfield, on the other hand, obeys an effective Klein-Gordon\nequation3,\n¨φ+ 3H˙φ+U,φ= 0, (24)\nwhere\nU(φ, t)≡γ(t)\n2a2f2(φ) +V(φ), (25)\nand\nγ(t)≡tr(γij). (26)\nThis equation also follows from the continuity equation,\n˙ρ+3H(ρ+p) =−σijπij, with the energy density, pressure\nand anisotropic stress given by\nρφ=1\n2˙φ2+γf2\n2a2+V , (27)\npφ=1\n2˙φ2−γf2\n6a2−V , (28)\nπi\nj=f2\na2\u0012\nγij−1\n3γδi\nj\u0013\n, (29)\nrespectively. Because φevolves under the influence of\nthe effective potential U, one is tempted to ask whether a\nphase of accelerated expansion can be achieved by setting\nV= 0 and by letting f(φ) play the role of a slow-roll\npotential. However this is not possible, as one quickly\nfinds by computing the field equation of state:\n1≥pφ/ρφ≥ −1/3, (30)\nwhich, as we see, is not sufficiently negative to accelerate\nthe expansion of the universe.\nThe full background dynamics is given by the following\nequations\n3H2=ρm+ρr+ρφ+σ2\n2, (31)\n(σi\nj)·=−3Hσi\nj+πi\nj, (32)\n¨φ=−3H˙φ−γ\na2ff,φ−V,φ. (33)\nwhere σ2=σijσij. In order to solve this system of equa-\ntions we need to specify the potentials V(φ) and f(φ).\n3Note that H= ˙a/a, where ais the mean scale factor defined\nin (12).5\nFor the purpose of illustrating the features of the model,\nit suffices to consider simple exponentials given by\nV(φ) =V0e−λφ, f(φ) =f0e−µφ, (34)\nwhere λandµmeasure the steepness of the potentials,\nwhile V0andf0measure their amplitudes. It is impor-\ntant to note that while V0is measured in units of [mass]4,\nf0is dimensionless, since it is a component of the field-\nspace metric. However, since the function f(φ) always\ncomes multiplied by combinations of ∂iϕI∂jϕJ, which is\nalso measured in units of [mass]4, the constant f0works\nin practice as a measure of the solid’s energy scale4.\nThus, in what follows, we will interpret f0as an (effec-\ntive) energy scale.\nGiven (34), the field-space metric (22) represents a\nmaximally symmetric manifold with Riemann tensor\ngiven by\nRABCD =R\n4(4−1)(GACGBD−GADGBC).(35)\nwhere Ris the associated Ricci scalar,\nR=−12µ2≡ −12\n(ℓc)2, (36)\nand where, in analogy to the de Sitter metric, we have in-\ntroduced a curvature scale through ℓc=µ−1. Thus, the\nparameter µsets the curvature scale of the field-space. As\nwe will see, the main aspects of the Early Dark Energy\nstage (EDE), including its onset and amplitude, depend\ndirectly on µandf0, and only indirectly on λandV0,\nwhich in turn govern the late-time regime of Dark En-\nergy. Our results are however fairly robust against the\nchoice (34) if compared with other viable cosmological\npotentials. Moreover, the choice we adopt here is more\neasily adapted to a dynamical system analysis, which we\npresent in a separate work [33]. For the numerical analy-\nsis it is also convenient to introduce the usual fractional\ndensities:\nΩα≡ρα\n3H2, α = (m, r, φ ), (37)\nas well as the dark-energy equation of state: wφ=pφ/ρφ.\nNext, we present a concrete numerical implementation\nof this model and discuss some of its main features in a\nsemi-analytical approach.\nB. Early dark energy\nLet us consider first the case of an isotropic FLRW\nuniverse, where βi= 0. Thus\nσi\nj= 0 = πi\nj. (38)\n4To put it differently, we could have defined ϕI=√f0xI, and\nf(φ) =e−µφ, since what matters is the product f(φ)∂iϕI∂jϕJ.\n0:00:20:40:60:81:0\n\n'\n\nm\n\nr\n100101102103104105106\n1 +z0:0\n\u00000:4\n\u00000:8FIG. 1. Top: matter (orange), radiation (green), and dark-\nenergy (blue) fractions as a function of the redshift. The thin\nhorizontal line represents the approximate fraction of dark\nenergy during the EDE phase (Eq. (43)), while the shaded\nregion shows the approximate duration of this regime. Bot-\ntom: dark energy equation of state. For this plot we have\nadopted µ= 2λ= 2, f2\n0/H2\n0= 2V0/H2\n0= 30 (continuous\nlines) and f2\n0/H2\n0= 8V0/H2\n0= 120 (dotted lines).\nSince the matter sector is not coupled to the fields, ρm\nandρrdecays as a−3anda−4, as usual, so that we only\nneed to solve Eqs. (31) and (33). A typical solution is\nshown in Figure 1, where we see an EDE regime start-\ning at z≃1000 and contributing to about 5% of the\ncosmological budget.\nTo understand the qualitative behavior of this figure,\nand in particular how the EDE phase is triggered, we can\nignore the contribution of V(φ) to the dynamics, since its\ncontribution is only relevant at late times ( z≃0). After\nthe radiation era, and assuming EDE is subdominant,\nthe universe is mostly matter dominated. If written in\nterms of the scale factor, Eq. (33) becomes\nd2φ\nda2+5\n2adφ\nda≈γµf2\n0e−2µφ\nH2\n0Ω0ma. (39)\nNote that while γ= 3 in a FLRW universe, the expres-\nsion above, and the ones following from it, hold for ar-\nbitrary γ(t). We thus keep it as γfor latter reference.\nIn what follows we shall consider that µis a non-zero\nand positive constant, although our final results do not\ndepend on its sign.\nThe field φstarts at zero at the top of the potential\nU. During matter domination, while φis slowly rolling\ndown, 2 µφ≈0, and the exponential in the equation\nabove is frozen at 1. The solution satisfying φ(a= 0) = 06\nis given by\nφ=2\n5γµf2\n0\nH2\n0Ω0ma . (40)\nThis approximate solution, valid during matter domina-\ntion, holds until 2 µφ≲1. When 2 µφ≈1 the exponential\nterm, and hence U, decay faster. The field then gains ki-\nnetic energy, and EDE starts. This happens at a redshift\n1 +z∗=a−1\n∗given by\n1 +z∗≈4\n5γµ2f2\n0\nH2\n0Ω0m. (41)\nFor the parameters used in Figure (1) this gives 1 + z∗≈\n103, which is in excellent agreement with the (full) nu-\nmerical solution shown in the same figure. As φcontinues\nto grow, the contribution of the potential f(φ) becomes\nnegligible in comparison to V(φ), and the standard Dark\nEnergy regime starts (this happens at z≃10 in Fig-\nure (1)). The amplitude of Ω φduring this regime, like\nthe onset of EDE, is also controlled by the parameter\nµ. To see how this happens, note from Figure (1) that\nΩφtracks the evolution of Ω mduring matter domina-\ntion. Thus, ˙ φ2∼a−3or, in terms of the scale factor,\ndφ/da=c/a, for some constant c. This constant can\nbe fixed by demanding the continuity of d φ/daata∗.\nFrom the derivative of (40) and Eq. (41) it follows that\nc= 1/(2µ). To proceed, we substitute d φ/da=c/aand\nd2φ/da2=−(1/a)dφ/dainto Eq. (39), which then gives\nγf2\n0e−2µφ≈3\n4H2\n0Ω0\nm\nµ2a. (42)\nIf we now use this result in (27), still neglecting the po-\ntential V, it follows after some algebra that5\nΩφ≈1\n6µ2, (10≲z≲103). (43)\nNote that, in view of (36), Ω φis proportional to the\nsquare of the field-space curvature radius during EDE\nphase: Ω φ∝ℓ2\nc. For µ= 2, this expression gives\nΩφ= 0.042 which is in good agreement with the value\nshown in Figure (1). Of course, one should keep in\nmind that our approximations assumed that dark en-\nergy is subdominant in this regime, which is only true\nforµ≳1. Existing upper bounds on EDE are model-\ndependent, but overall give fEDE≲10% [34] at 95%CL,\nwith fEDE=ρDE/ρtotal. In our case, this translates to\nµ≳1.3, which is consistent with the range of values\nallowed by our solutions.\nIndeed, by studying the whole system via dynamical\nanalysis [33], one can show that EDE regime corresponds\n5We stress that the limit µ= 0 cannot be taken in this expression,\nsince it violates our previous assumption 2 µφ= 1.to a matter dominated point in which the matter density\nscales as\nΩm= 1−1\n6µ2, (44)\nin agreement with the semi-analytical expression derived\nin Eq. (43). This is a saddle point for µ≤ −1/√\n6 and\nλ > 6µorµ≥1/√\n6 and λ < 6µ. The sequence thus\nends with a dark energy attractor satisfying\nΩφ= 1, w φ=λ2\n3−1, (45)\nthat is also in concordance with the numerical results\nshown in Figure (1). This point produces an accelerated\nstage and is stable under the conditions\n−√\n2< λ <√\n2, µ ≶λ2−2\n2λforλ≶0,(46)\nwhich is automatically satisfied within our particular\nchoice µ= 2, λ= 1. The model also presents some inter-\nesting properties as oscillating equation of state for some\nparticular range of parameters. A complete discussion of\nthese properties is presented in [33].\nC. Spatial anisotropies\nAs demonstrated, the gradients of the fields ϕIgen-\nerally lead to the appearance of an anisotropic stress\ntensor, as described in (29). In an expanding universe,\nsmall seeds of anisotropy will be sourced by this ten-\nsor, giving rise to increased anisotropies. These, in turn,\namplify the gradients, which lead to greater anisotropic\nstresses in a self-feeding cycle. This is in fact the con-\ntent of Eq. (32), suggesting that the late-time universe\ncould exhibit significant anisotropies, potentially erasing\nthe EDE phase found before. However, unlike everyday\nsolids, cosmological solids are known to be quite insensi-\ntive to stretches and deformations [8], thus allowing any\ninitial anisotropies to grow in a controlled way. Figure (2)\nshows the evolution of matter, radiation and dark energy\nin a universe with non-zero σi\njandπi\nj. As we can see in\nthis figure, the qualitative background behavior, includ-\ning the estimate (43), is essentially the same as that of\nFigure (1), where σi\nj= 0 = πi\nj. If anything, the pres-\nence of higher initial values of βihave the same effect\nas increasing the field-space metric amplitude f0in the\nisotropic case. This degenerate behavior follows from the\nproduct of γf2appearing in (27).\nPerhaps more surprising is the finding that, for a wide\nrange of initial conditions on βiand˙βi, the fractional con-\ntribution of late-time anisotropy to the cosmic budget,\ngiven by σ2/6H2, can never exceed a limiting and nearly\nconstant value. This feature is shown in Figure (3) for\nseveral values of βiand˙βi. Similarly to Ω φ, this constant\nthreshold is given by a simple function of µ, and can also7\n0:00:20:40:60:81:0\n\f1= 0:0\n\f1= 0:9\n\f1= 1:7\n\f1= 2:5\n100101102103104105106\n1 +z0:0\n\u00000:4\n\u00000:8\nFIG. 2. Top: Background evolution of Ω m(orange), Ω r\n(green) and Ω φ(blue) in a universe containing spatial\nanisotropies. Bottom: the equation of state of dark energy.\nWe have adopted β1(0) = 2 β2(0), and ˙βi(0) = 0 as initial\nconditions. The remaining parameters are the same used in\nthe continuous lines of Figure (1).\n100101102103104105106107\n1 +z10\u00001710\u00001510\u00001310\u00001110\u0000910\u0000710\u00005\n\f1= 0:01\n\f1= 0:1\n\f1= 1:0\n\f1= 2:0\n\f1= 3:0\n\f1= 4:0\nFIG. 3. Background evolution of the anisotropic fraction\nσ2/6H2in a model with a curved solid. For a wide range of\ninitial conditions on βiand ˙βi, the late-time anisotropy has an\nupper bound depending on µ. For this plot we have adopted\nβ1= 2β2,˙β1= 2˙β2= 0 (continuous lines), ˙β1= 2˙β2= 1012\n(dotted lines), and µ= 10. The remaining parameters are the\nsame as in the continuous curves of Figure (1). The horizontal\nthin line represents the upper bound given by the approxima-\ntion (48).be estimated semi-analytically during matter domina-\ntion. For that, and with no loss of generality, we assume\nthat β1≫1 at early times, so that γ= tr( γij)≫γ11.\nDuring matter domination, and recalling (15), equation\n(32) can be approximated as ( i= 1,2)\nd2βi\nda2+5\n2adβi\nda≈ −γf2\n0e−2µφ\n3H2\n0Ω0ma. (47)\nNote the similarity of this expression with (39), which\nshows that the anisotropic scale factors βibehave as ef-\nfective quintessence fields as long as Vis negligible. This\nis possible because the coupling between the field-space\nand spacetime metrics, which leads to a non-zero π1\n1,\nacts as an effective potential for βi. Because of this, ˙β2\ni\ntracks the evolution of Ω m, just like ˙ φ2does. This means\nthat d β/da=c/afor some constant cduring matter\ndomination; see Figure (4). Replacing this relation into\nthe left hand side of (47), and the approximation (42)\non the right, one immediately finds that c=−1/6µ2.\nFinally, using ˙β1=˙β2=−˙β3/2, it follows that\nσ2\n6H2≲1\n36µ4, (10≲z≲103), (48)\nwhere we have used an inequality to account for cases\nwhere βi≈0 (see Figure (3).) Comparing this to (36)\nwe see that σ2/6H2≲ℓ4\nc. In fact, we have found that\nσ2\n6H2≲(Ωφ)2. (49)\nThus, for µ≳1, late-time anisotropies will be subdomi-\nnant in a Bianchi I universe. Of course, these anisotropies\ncan also be subdominant, regardless of µ, if observations\ntell us that βi≈0, in which case we would recover the\nFLRW results of the previous section. The take away\nlesson is that spatial anisotropies, if present , will never\nexceed the dark energy fraction in these models.\nIV. LINEAR PERTURBATIONS\nWe have shown that the curved solid model is pre-\ndictive at the background level, leading to a phase of\nearly dark energy and (possibly) suppressed late-time\nanisotropies. The model is also predictive at the per-\nturbative level, since the introduction of a solid changes\nthe dispersion relation and the sound speed of perturba-\ntions, affecting the gravitational clustering of matter [7].\nA detailed investigation of structure formation is outside\nour present scope, and is left for a separate work. We can\nnonetheless extract the model’s main footprints by con-\nsidering the small scale limit of perturbations, where the\ncosmological effects of the expansion are negligible. In\nthis Minkowskian regime, the field’s perturbations obey\n(see Appendix A)\n∇µ∇µδϕA+RA\nCDB∂µϕC∂µϕBδϕD−(∇AVC)δϕC= 0,\n(50)8\n101102103104105106\n1 +z10\u0000210\u00001100101102103104105\nd\u001e\nda\nd\f\ndaa\u00001\nFIG. 4. Evolution of d ϕ/da(dot dashed), d β/da(dashed),\nand 1 /a(continuous) as a function of redshift, and for many\ninitial values of βi. During matter domination, all curves have\napproximately the same slope.\nwhere δϕArepresents the field’s perturbation and\nRABCD is the field-space Riemann tensor.\nTo simplify, we recycle our background convention and\nwrite δϕ0asδφ. We also introduce a vector notation\nwhere the three components δϕIare written as δ⃗ϕ. Work-\ning in Fourier space and decomposing δ⃗ϕinto its longi-\ntudinal and transverse components,\nδ⃗ϕ=i⃗kδϕL+δ⃗ϕT (⃗k·δ⃗ϕT= 0), (51)\nwe find, after a straightforward computation, the follow-\ning system of equations:\nδ¨φ+\u0002\nk2+M2\neff\u0003\nδφ=−2µf2k2δϕL, (52)\nδ¨ϕL+k2δϕL−2µ˙φδ˙ϕL=−2µδφ , (53)\nδ¨⃗ϕT+k2δ⃗ϕT−2µ˙φδ˙⃗ϕT= 0, (54)\nwhere we have defined M2\neff= 6µ2f2+λ2V.\nThis system presents two new features following from\nthe field-space curvature, and characterized by the pa-\nrameter µ: a coupling between the quintessence and lon-\ngitudinal solid perturbations, δφandδϕL, and an intrin-\nsic damping term in the longitudinal and transverse solid\nperturbations, δϕLandδ⃗ϕT. These features are in effect\ncrucial for the clustering properties of the model’s pertur-\nbations, which can be unveiled by postulating oscillatory\nsolutions and looking for the possible frequencies of os-\ncillation.\nLet us start with the scalar perturbations. Since δφ\nandδϕLare coupled, they share a common harmonic\nsolution eiΩt. The allowed frequencies are given as the\nroots of a fourth order polynomial in Ω,\n[Ω2−(k2+M2\neff)][Ω2−k2+ 2iΩµ˙φ] = 4µ2f2k2,(55)with complex coefficients. Note the presence of cubic and\nlinear terms in Ω, which are generally absent in time-\ndependent multi-field models [17]. In the flat field-space\nlimit ( µ= 0), there are two independent frequencies cor-\nresponding to a light mode, Ω2\n−=k2, and a heavy mode,\nΩ2\n+=k2+M2\neff, both propagating with the speed of\nlight. This is expected because when µ= 0 the model is\nlinear and the perturbations evolve independently. At\nsmall scales, quintessence behave as a massive mode,\nwith the mass being proportional to the second deriva-\ntive of its background potential V(φ). The solid per-\nturbations, however, don’t have a background potential\n(which, we recall, would violate spatial homogeneity) and\nthus evolve as light modes.\nFor arbitrary curvature (i.e., µ̸= 0), the solutions are\ngenerally complex, indicating the presence of instabili-\nties in the scalar sector. Exact solutions to (55) can be\nobtained with algebraic packages, but the resulting ex-\npressions are too long to be useful. Nonetheless, it is\nimportant to note that the imaginary part of the fre-\nquencies result exclusively from the term µ˙φ, and not\nfrom the inhomogeneous term on the right-hand side of\n(55). Indeed, if the former is absent, we find that the\nresulting frequencies,\n1\n2\u0012\n2k2+M2\neff±q\n16µ2f2k2+M4\neff\u0013\n,\nare real since 2 k2+M2\neff≥p\n16µ2f2k2+M4\neff, as one\ncan easily check using the definition of M2\neff. In the pres-\nence of gravity, such small scale instabilities will feed\nscalar perturbations, and can lead to enhanced struc-\ntures at large scales. Moreover, since this is a multif-field\nmodel, one might expect the appearance of isocurvature\nmodes [35], which could be another distinctive signature.\nThe presence of instabilities is clearer in the case of\ntransverse vector perturbations, which decouples from\nthe scalar modes and thus oscillates with its own fre-\nquency. Inserting the ansatz eiωtin (54), the allowed\nfrequencies,\nω±=−iµ˙φ±p\nk2−µ2˙φ2, (56)\nare clearly complex. At distances where gravitational\neffects can be neglected, we expect that k2≫µ2˙φ2. In\nthis limit, ω±simplifies to\nω±≈ −iµ˙φ±k . (57)\nTherefore, small scale transverse perturbations propa-\ngate with the speed of light, and develop instabilities at\ndistances of order ( µ˙φ)−1.\nTo estimate the typical scales involved, and to check\nwhether this description is consistent with our numerical\nsolutions, let us focus on low redshifts at which most\nstructures are found, typically z≲3. Taking µ= 2\nas a reference, our numerical solutions give ˙ φ≃1.26H09\natz= 36, which corresponds to ( µ˙φ)−1≈0.4H−1\n0,\nroughly one half of the present Hubble radius. Adopting\nH−1\n0= 3000 h−1Mpc, this corresponds to a scale deep\nin the linear regime, µ˙φ∼10−3hMpc−1, and justifies a\nposteriori the small scale approximation leading to (57).\nAs is well known, primordial vector modes are not gen-\nerated during inflation at first order in perturbations.\nThis is because, differently from scalar and tensor modes,\nvector modes are not frozen after horizon crossing, but\ndecay away. Moreover, even if they are produced by\nsome mechanism at linear order, they do not act as\nsources of density perturbations, and thus are typically\nneglected in the standard analysis of structure formation.\nHowever, vectors modes can affect the number count of\ngalaxies [36, 37], and their inclusion is also important\nto a proper data analysis of redshift-space distortion ef-\nfects [38]. Here, we have presented a mechanism where\nvector modes generated from sub-horizon perturbations\ncan be amplified at large scales. When gravitational per-\nturbations are included, δ⃗ϕT, will source gravitational\n(transverse) vector modes, which can in principle be de-\ntectable with forthcoming surveys. We leave a thorough\nanalysis of this important question for a future publica-\ntion.\nV. CONCLUSION AND PERSPECTIVES\nWe have presented a non-linear multi-field dark en-\nergy model where the fields evolve under a non-trivial\nand maximally symmetric field-space metric. Differently\nfrom previous models in which the fields are either time\nor space-dependent, we have considered a model in which\none field (quintessence) is time-dependent, while the oth-\ners are given by pure spatial gradients, as in the effective\nfield theory description of solids. The resulting back-\nground model is predictive, and can produce an early\ndark energy (EDE) component, as well as a suppressed\nfraction of late-time anisotropies — both features being\ngiven by simple powers of the field-space curvature ra-\ndius. We have shown that a percent level EDE compo-\nnent, such as the one employed in attempts to solve the\nH0tension, can be easily accomplished in the present\nscenario, and corresponds to a stable point in the space\nof possible solutions.\nAt the perturbative level, we have shown that the cou-\npling between time and space-dependent fields also leads\nto interesting signatures, among which is the appearance\nof complex dispersion relations for the small scale per-\nturbations. In particular, we have shown that small scale\nvector perturbations will develop tachyonic instabilities\nthat can enhance structures at large scales, slightly below\nthe present Hubble radius. Scalar perturbations will also\n6The value of ˙ φvaries little with the choice of µin the range (46)\nsince the field solutions correspond to a stable point.become unstable, although the size of structures affected\ndepend non-trivially on the scale of the perturbation. In-\ncidentally, our findings corroborate earlier claims, made\nin the context of multi-field inflation [20] and dark en-\nergy [22], that a sufficiently negative field-space curvature\ncan lead to tachyonic instabilities, with important obser-\nvational consequences. However, while in the context of\ntime-dependent fields such instabilities arise from the ef-\nfective mass term of fields evolving non-geodesically (in\nfield-space), in the present context they result from nat-\nural couplings between time and space-dependent fields,\nand will always be present if the field-space is curved, re-\ngardless of the field-space trajectories. Overall, our find-\nings represent an interesting observational window to test\nthe curvature of the field-space.\nACKNOWLEDGMENTS\nWe thank Ben Normann and Mikjel Thorsrud for use-\nful insights during the initial stages of this work. JPBA,\nAG and CAVT acknowledge financial support from\nthe Patrimonio Aut´ onomo - Fondo Nacional de Finan-\nciamiento para la Ciencia, la Tecnolog´ ıa y la Innovaci´ on\nFrancisco Jos´ e de Caldas (MINCIENCIAS - COLOM-\nBIA) Grant No. 110685269447 RC-80740-465- 2020,\nprojects 69475, 69553 and 69723. TSP thanks Conselho\nNacional de Desenvolvimento Cient´ ıfico e Tecnol´ ogico\n(CNPq) and Coordena¸ c˜ ao de Aperfei¸ coamento de Pes-\nsoal de N´ ıvel Superior (CAPES) for the financial support\nunder grants 312869/2021-5 and 88881.709790/2022-01.\nJPBA and CAVT thank the Physics Department of Lon-\ndrina State University for the hospitality during the de-\nvelopment of this work.\nAppendix A: Derivation of Eq. (50)\nSuppose that ϕA\nλ(xµ) is a one-parameter family of\nfields on a (flat) manifold M, such that ϕA\n0=ϕAand\n∂ϕA\nλ/∂λ\f\f\nλ=0=δϕA. The equation for the perturbations\nδϕAcan be defined as the first term in a Taylor expan-\nsion of Eq. (9) around λ= 0. The computation can be\nsimplified if we adopt Riemann normal coordinates in N\nsuch that, at λ= 0,GAB=δABand∂λGAB= 0 = ΓA\nBC.\nLet us start with the kinetic term, which gives\n∂\n∂λ∇µ∇µϕA\nλ\f\f\f\f\nλ=0!=∂µ���µδϕA+∂DΓA\nBCδϕD∂µϕB∂µϕC,\n!=∇µ∇µδϕA+∂µΓA\nDCδϕC∂µϕD\n+RA\nCDB∂µϕC∂µϕBδϕD,(A1)\nwhere the exclamation point reminds us that we are\nworking in a specific coordinate system. In the last equal-\nity we have added and subtracted ∂Γ terms needed to\nintroduce the Riemann tensor, and added Γ = 0 terms10\nneeded to rewrite partial derivatives as covariant deriva-\ntives. The potential term can be computed in a similar\nfashion, giving:\n∂\n∂λGAB∂BV\f\f\f\f\nλ=0!=GAB∂C(VB)δϕC,\n!=GAB(∇BVC)δϕC. (A2)Expressions (A1) and (A2) are tensorial, and thus hold in\nany coordinate system. Their difference give (50). Note\nthat, if V= 0, equation (50) is formally equivalent to the\ngeodesic deviation equation (see also [27]).\n[1] E. Abdalla et al. , JHEAp 34, 49 (2022), [2203.06142].\n[2] P. K. 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El-Baklish\nInstitute for Transport Planning and Systems\nDepartment of Civil, Environmental and Geomatic Engineering\nETH Zurich\nZurich, Switzerland\nselbaklish@ethz.ch\nAnastasios Kouvelas\nInstitute for Transport Planning and Systems\nDepartment of Civil, Environmental and Geomatic Engineering\nETH Zurich\nZurich, Switzerland\nkouvelas@ethz.ch\nMichail A. Makridis\nInstitute for Transport Planning and Systems\nDepartment of Civil, Environmental and Geomatic Engineering\nETH Zurich\nZurich, Switzerland\nmmakridis@ethz.ch\nABSTRACT\nAutomated vehicle technologies offer a promising avenue for enhancing traffic efficiency, safety, and\nenergy consumption. Among these, Adaptive Cruise Control (ACC) systems stand out as a prevalent\nform of automation on today’s roads, with their time gap settings holding paramount importance.\nWhile decreasing the average time headway tends to enhance traffic capacity, it simultaneously\nraises concerns regarding safety and string stability. This study introduces a novel variable time gap\nfeedback control policy aimed at striking a balance between maintaining a minimum time gap setting\nunder equilibrium car-following conditions, thereby improving traffic capacity, while ensuring string\nstability to mitigate disturbances away from the equilibrium flow. Leveraging nonlinear H∞control\ntechnique, the strategy employs a variable time gap component as the manipulated control signal,\ncomplemented by a constant time gap component that predominates during car-following equilibrium.\nThe effectiveness of the proposed scheme is evaluated against its constant time-gap counterpart\ncalibrated using field platoon data from the OpenACC dataset. Through numerical and traffic\nsimulations, our findings illustrate that the proposed algorithm effectively dampens perturbations\nwithin vehicle platoons, leading to a more efficient and safer mixed traffic flow.\nKeywords Adaptive Cruise Control (ACC) ·String Stability ·Variable Time Gap ·Nonlinear H∞control\n1 Introduction\nAdaptive Cruise Control (ACC) is a part of the advanced driver assistance systems (ADAS) and is designed specifically\nfor vehicle longitudinal control. In fact, ACC systems are the most widespread Society of Engineers (SAE) level 1\nautomated vehcile (A V) technology in the current markets [ 1]. For instance, 16 out of 20 best-selling cars in the USarXiv:2402.14110v1  [eess.SY]  21 Feb 2024market are equipped with an ACC system [ 2]. More importantly, vehicle automation gives rise to change traffic flow\ndynamics and to alleviate instabilities and congestion.\nAt its center, ACC employs a spacing policy which in turn determines the car-following behavior, platoon stability\nand traffic efficiency [ 3]. A small inter-vehicular spacing results in a higher traffic capacity but undermines safety\nand stability. The notion of stability referred to in this work is string stability. Intuitively, vehicles in a platoon are\nstring stable if small perturbations from an equilibrium flow are dampened as they propagate in the upstream direction.\nSpacing policies in the literature can be categorized into 1) constant spacing (CS), 2) constant time-headway policies\n(CTH), and 3) variable time-headway (VTH) policies [ 4]. String stability was proved unattainable using CS policy\nwithout inter-vehicular communication to provide further information such as leader’s acceleration [ 4,5]. On the other\nhand, CTH and VTH policies can be appropriately designed to be string stable; nevertheless, there exists the conflicting\nobjective of traffic efficiency. Using the OpenACC database for car-following experiments, the authors in [ 6] confirm\nthat a string-stable ACC with large time-headway leads to poor road utilization decreasing capacity. In addition, a large\nheadway threatens drivers’ subjective acceptability due to possibly increased lane changes from adjacent lanes [3].\nThe aim of this work is to develop a VTH policy for ACC which is able to strike a balance between string stability\nand traffic efficiency. Under equilibrium flow, a CTH policy with a minimum time gap is adopted to promote efficient\nroad utilization by ACC vehicles. This relies on the fact that stable driving conditions are most prevalent. This\ncan also be supported by observed trajectory data from the OpenACC database. Figure 1 shows the distributions of\ninstantaneous acceleration and deceleration which are mostly under ±0.5m/s2for all platoons in the AstaZero and\nZalaZone campaigns. However, perturbations/disturbances give rise to sharp acceleration/deceleration; raising safety,\nstability and efficiency concerns. Further investigations by Ciuffo et.al. compared short, medium and large time gap\nsettings for field platoons in the ZalaZone experimental campaign in terms of traffic capacity [ 7]. They showed that\nshort time gap platoons are able to attain the highest flow during stable conditions; while highly deteriorating during\nperturbations to relatively match medium time-gap platoons. Hence, the string instability is detrimental to the potential\nbenefits of ACC systems to traffic flow. Therefore, the CTH policy must to be relaxed to allow for the needed larger\ngaps to attenuate such disturbances. To this end, a feedback H∞-based VTH policy is adopted for guaranteed string\nstability. This scheme provides a compromise between enforcing traffic efficiency during equilibrium car-following\nconditions and attaining string stability during non-equilibrium flow by relaxation to the VTH policy.\n2 Related Work\nThe most widely used ACC model is developed by [ 8] as a linear feedback control algorithm; where the acceleration\nis computed based on deviations away from a target speed (leading vehicle speed in car-following mode) and away\nfrom a target spacing determined by a constant time gap (CTG) policy. This spacing policy assumes that the space\ngap is proportional to the vehicle speed; thus, more appealing to drivers. Furthermore, Gunter et. al. have shown\nthat commercially available ACC systems utilizing CTG policy are string unstable [ 9]. Interestingly, [ 2] presented\nsimulations of string-unstable ACC vehicles platoon with a minimum time gap which enhanced downstream capacity\nrelative to their string-stable counterpart with a maximum time gap. Lin et.al. demonstrated through experiments\nthe need for different time gap setting in different driving scenarios in order to balance safety and driver’s subjective\nacceptance [10].\nOther variable spacing policies assume a nonlinear relationship between the space gap and the vehicle speed; thus,\nimproving stability properties and traffic capacity. The authors in [ 11] proposed a nonlinear spacing policy as a function\nof speed and approaching rate; inspired by Greenshield’s fundamental diagram (FD) and equations. The approaching\nrate is the relative speed of the ego vehicle to the leading vehicle. To enforce such a spacing policy, a sliding mode\ncontroller is implemented where the spacing is considered as the sliding variable. They have reported a 44% decrease\nin travel time compared to a CTG policy in simulations of a pipeline plus on-ramp network. Zhou and Peng have\ndeveloped a quadratic spacing policy as a function of vehicle speed based on their investigation of human drivers’\nspacing policy [ 12]. They have designed the policy parameters to maximize the traffic capacity while imposing linear\nstability constraints. Again, a sliding mode controller is developed on top to implement the proposed spacing policy\nand the control scheme was tested in simulations of a merging scenario. Compared to a modified Greenshields-inspired\nspacing policy suggested by [ 13], the proposed scheme yielded a higher critical density and attenuated upstream\nperturbations in vehicles accelerations. All string stability analyses performed therein were in the strict sense and based\non linear frequency-domain input-output H∞gains. For more details on string stability analysis methods, the reader is\nreferred to [14] and [15].\nVariable time gap (VTG) policies can also be designed to achieve variable spacing. In [ 16], an intelligent driving\nstrategy layer was suggested in order to regulate the ACC parameter settings according to the perceived traffic state.\nThe authors aggregated traffic \"situations\" into five traffic states — free traffic, upstream jam front, congested traffic,\n2Figure 1: Distributions of instantaneous acceleration and deceleration of vehicles per campaign and driving mode. The\nred labels denote the cumulative proportion of acceleration/deceleration that is below/above ±0.5m/s2.\ndownstream jam front, bottleneck sections — according to smoothed measurements of the space-mean speed and\nlocation information of active bottlenecks from the infrastructure. Then, a driving strategy matrix was developed\nto vary the ACC model parameters according to the perceived state. An Intelligent Driver Model (IDM) was used.\nFor instance, the time gap was linearly decreased for the \"downstream jam front\" state. Simulations demonstrate a\nreduction in travel times for small penetration rates of 5%. Another VTG policy was proposed by [ 17]; where the time\ngap was varied linearly with the ratio between the measured space gap and the threshold space gap. The threshold\nspace gap discriminates between cruising and car-following modes. The proposed VTG policy incentives shorter time\ngaps at larger space gaps that indicate higher densities. A two-regime model predictive control (MPC) scheme was\nimplemented on top; where the costs represented safety and cruising and car-following modes efficiency. Spiliopoulou\net. al. proposed adapting the time gap in real-time to the traffic flow state at a motorway section [ 18]. The applied\ntime gap was the minimum of the desired setting of the driver and the suggested one that attempts to avoid congestion\nby increasing the static capacity of the section and, also increase the discharge flow at active bottleneck locations.\nMicroscopic simulation for a motorway stretch with an on-ramp bottleneck show decrease in average vehicle delay and\nfuel consumption. These works showed improved traffic flow, but, without guarantees on string stability.\n3The work by [ 19] adapted the nonlinear spacing policies by [ 11,20] based on macroscopic flow and [ 21] based on\napproaching rate. Hence, the developed ACC time gap strategy combined the vehicle speed, approaching rate, preceding\nvehicle acceleration, free-flow speed and jam density. Simulations using PTV-VISSIM of a single-lane highway with an\non-ramp demonstrated diminished travel time and traffic delay compared to CTG policy and the VTG policy developed\nby [21]. A comparison between CTG and VTG policies for ACC was performed in [ 22]. The VTG policy employed\nvaried with the approaching rate within the time gap bounds. The authors further investigated the linear stability and\nthroughput of the mixed traffic flow. Both investigations and simulations have shown that a VTG policy is more likely\nto stabilize traffic flow and that an increased maximum time gap yields a more stable mixed traffic flow, but with a\nreduced capacity. A macroscopic approach towards designing the VTG policy was introduced in [ 23]. Bekiaris-Liberis\nand Delis assume the time gap of ACC vehicles as a control input to be manipulated to stabilize traffic flow expressed\nby an Aw-Rascle-Zhang (ARZ) mixed traffic model. The feedback VTG control policy was developed for the linearized\nsystem around a uniform, congested equilibrium profile. Numerical simulations reported improved performance in\nterms of fuel consumption, travel time, and comfort compared to the CTG policy; which the authors proved to yield an\nunstable traffic flow.\nIn this work, we employ a VTG policy consisting of both a constant component; which may be a constant desired setting\nby the driver or a minimum time gap for efficient traffic flow, and a manipulated component to be designed in order\nto achieve string stability. This manipulated time gap component is developed as a feedback control policy; similar\nto [23], using nonlinear H∞control in order to achieve L2string stability in the strict sense. A nonlinear analysis is\nnecessary here due to the VTG strategy adopted. Also, formulation through strict string stability facilitates extending\nthe platoon to any number of vehicles; since a platoon can be split to multiple leader-follower subsystems. Another\nimportant contribution presented here is a data-driven string stability analysis of vehicle platoons using empirical\ndata collected from commercially available ACC-operated vehicles and manually driven vehicles. This approach is\ncompletely model-free and, hence, bypasses the need for calibration of car-following models to the experimental\ntrajectory data to overcome any resulting biases.\n3 Preliminaries\nConsider a platoon of N+ 1vehicles. The leader is denoted by the index 0. Each follower vehicle i,∀i= 1, . . . , N, is\ndescribed by the following state-space model.\n˙si(t) =vi−1(t)−vi(t) (1a)\n˙vi(t) =fa,i(si(t), vi(t), vi−1(t)) (1b)\nwhere siandvidenote the space headway and velocity of vehicle irespectively and vi−1denotes the velocity of the\npreceding vehicle. The function fa,i(·)represents the longitudinal dynamics defined by car-following models such as\nthe IDM [24] which may be homogeneous or heterogeneous.\nA signal u(t)∈ L2has, intuitively, bounded energy, i.e.R∞\nt0uT(t)u(t)dt <∞, if it belongs to the Lebesgue space L2\nof square-integrable functions.\nDefinition 1. Letseq,iandveqbe the equilibrium profile of (1),xi(t) = [si(t)−seq,ivi(t)−veq,i]Tbe the deviation\nstates away from the equilibrium profile and δvi−1(t) =vi−1−veq. For any platoon to be strictly input-to-state\nstring-stable, follower vehicle ishould have a local L2-gain equal to or less than γ, γ≤1for any initial state\nxi(0)∈ N if the response xi(t)to any disturbance from the preceding vehicle δvi−1(t)∈ L2[0,∞)satisfies\n∥xi(t)∥2\nL2≤γ2\u0010\nκ(xi(0)) + ∥δvi−1(t)∥2\nL2\u0011\n,∀t≥0, i= 1, . . . , N (2)\nfor some bounded function κ,κ(0) = 0 andN ⊆ X ⊆ R2whereXis the set of admissible states.\nDefinition 2. LetXi(s)and∆Vi−1(s)be the Laplace transform of xi(t)andδvi−1(t). For a linear platoon to be\nstrictly input-to-output string-stable, follower vehicle ishould have an H∞-norm or induced L2-gain equal to or less\nthanγ, γ≤1if the response xi(t)to any disturbance from the preceding vehicle δvi−1(t)∈ L2[0,∞)satisfies\n∥Γi(s)∥H∞= sup\nδvi−1∈L2(0,∞)̸=0∥xi(t)∥2\nL2\n∥δvi−1(t)∥2\nL2≤γ,∀i= 1, . . . , N (3)\nwhereΓi(s)=Xi(s)\n∆Vi−1(s)is the transfer function between follower vehicle iand its predecessor i−1.\nTheL2-gain, or H∞-norm for linear systems, represents the worst-case system gain over all excitations or input\ndisturbances. Thus, if a platoon has an L2-gain of γ≤1, it means that perturbations will dissipate as they propagate\nupstream and the platoon will be string-stable.\n4For a platoon of ACC vehicles [8], the dynamic model can be written as follows.\n˙si(t) =vi−1(t)−vi(t) (4a)\n˙vi(t) =k1,i(si(t)−s0−Li−1−τivi(t)) +k2,i(vi−1(t)−vi(t)) (4b)\nwhere s0is the standstill distance, Li−1is the length of vehicle i−1andk1,i, k2,iandτiare the spacing control\ngain, velocity control gain and the constant time gap for ACC vehicle i. The CTG spacing policy is defined as\nsdes,i=s0+Li−1+τivi(t).\nTo define the error dynamics of a leader-follower subsystem, the equilibrium profile is defined by the constant speed veq\nat which all vehicles are travelling at. The equilibrium space headway is given as seq,i=s0+Li−1+τiveq. Then, the\ndeviation states are expressed as ˜si=si−seq,iand˜v=vi−veqand the deviation dynamics are derived as follows.\n˙˜si=vi−1−vi (5a)\n= (vi−1−veq)−(vi−veq) (5b)\n=−˜vi+δvi−1 (5c)\n˙˜vi=k1,i(si(t)−s0−Li−1−τivi(t)) +k2,i(vi−1(t)−vi(t)) (6a)\n=k1,i(si(t)−s0−Li−1−τi(˜vi+veq)) +k2,i(−˜vi+δvi−1) (6b)\n=k1,i(si(t)−s0−Li−1−τiveq)−k1,iτi˜vi−k2,i˜vi+k2,iδvi−1 (6c)\n=k1,i˜si−k1,iτi˜vi−k2,i˜vi+k2,iδvi−1 (6d)\nwhere δvi−1denotes the deviation of the leader’s velocity away from the equilibrium velocity and represents a\ndisturbance to the considered leader-follower subsystem.\n4 Data-Driven String Stability Analysis of Adaptive Cruise Control\nDue to propriety issues, the true model of ACC controllers is not known. Therefore, researchers proposed many models\nin the literature, most notably the one by [ 8], to analyze ACC controllers and their performance. However, retrieving a\nmeaningful model which calibrates well to trajectory data from commercially available ACC systems is challenging\nrequiring time and expert knowledge. Also, conclusions drawn from calibrated models may be subject to biases acquired\nfrom the optimization and calibration procedure. In this section, a direct data-driven approach for strict string stability\nanalysis with the assumption of linear platoon model is performed using only trajectory data and bypassing the need for\nmodel calibration [25, 26].\n4.1 Data-Driven L2-gain Estimation\nConsider a single-input single-output (SISO) linear time-invariant (LTI) system G(s). In our case, the system input\nis the disturbing leader velocity δvi−1(t)and the system output may be either the deviation of follower velocity\n˜vi(t)or the deviation of space headway ˜si(t). Trajectory data of length NDis assumed to be available. A matrix\nTm(δvi−1)∈R(ND+m−1)×mis formed of mcolumns of the downshifted input signal where m < N D.\nTm(δvi−1) =\nδvi−1(1) 0 . . .\nδvi−1(2) δvi−1(1) . . .\n.........\nδvi−1(ND)......\n0 δvi−1(ND)...\n... 0...\n.........\n(7)\nThen, the estimated auto-correlation matrix is defined as ˆRm(δvi−1) =1\nNDTT\nm(δvi−1)Tm(δvi−1)∈Rm×m. Each\nentry in this matrix represents estimates of the auto-correlation function ˆRuu(T) =1\nNDPND−T\nt=1u(t)u(t+T).\nAssumption 1. A linear time-invariant (LTI) system is persistently excited if the auto-correlation matrix of the input u\ndata trajectory of length NDis full rank, i.e. rank (ˆRm(u)) =1\nNDTT\nm(u)Tm(u) =m.\n5Similar matrices exist for the outputs as well. Provided that Assumption 1 is satisfied, we can define the following\nsemi-definite program to estimate the H∞-norm or L2-gain of the underlying system.\nˆγ= min\nγγ (8a)\ns.t.1\nNDTT\nm(y)Tm(y)−γ21\nNDTT\nm(δvi−1)Tm(δvi−1)≤0,γ≥0 (8b)\nwhere ˆγis the estimated L2-gain and ydenotes the selected system output which can be ˜vi(t)or˜si(t). The estimated\nL2-gain converges to the true value, i.e. ˆγ2→ ∥G(s)∥2\nH∞, as trajectory data length becomes infinite ND→ ∞\nm→ ∞ andm\nND→0[27]. This quantifies the system norm needed to make conclusions about strict string stability for\nlinear platoons, as per Definition 2.\n4.2 Field Data Description\nThe empirical data were taken from two experimental campaigns which took place in AstaZero and ZalaZone proving\ngrounds in Sweden and Hungary respectively. The AstaZero campaign consisted of a fleet of 5 commercially available\nACC-equiped vehicles on the 5.7-km-long Rural Road; while, a fleet of 10 vehicles was deployed on the circular\nDynamic Platform (300-m diameter) in the ZalaZone campaign. Both campaigns were carried out in 2019.\nIn all campaigns, data acquisition was performed using high-accuracy on-board equipment (e.g. U-blox global navigation\nsatellite system (GNSS) receivers) with a sampling frequency at 10 Hz. Maximum and minimum time gap settings\nwere used for the ACC-operated tests in both campaigns. Also, some tests were carried out without ACC operation (i.e.\nmanual driving) in the AstaZero campaign. These data can be freely accessed through the OpenACC database [6].\n4.3 String Stability of Field Platoons\nEach platoon was divided into leader-follower subsystems; where the convex program (8)was solved to estimate the\nL2-gain of the corresponding subsystem and, subsequently, provide numerical thresholds for strict string stability.\nThe speed-to-speed relation was examined; i.e. the output was taken to be the deviation in the follower velocity\ny(t) = ˜vi(t). The equilibrium velocity veqwas taken to be the median velocity of the leading vehicle for 1-minute\nintervals in the duration of the car-following maneuver.\nFigure 2 illustrates the distribution of the estimated L2-gain values of all leader-follower subsystems according to the\nACC settings (’MIN’/’MAX’ for minimum/maximum time gap ACC setting and ’HUMAN’ for manual driving). A\nmaximum time gap ACC setting has consistently lower L2-gains; even for the same make and model of the tested\nvehicle. Audi, Jaguar and Tesla car-makes record the lowest L2-gain values, as observed from Figure 2. In addition,\nmanual driving is shown to be string stable with 87.5% of the L2-gain estimates below the string stability threshold of\nγ= 1. This mainly due to the variable spacing policy adopted by human drivers.\nAs demonstrated in Figure 3, the first vehicles inside the platoon show higher L2-gain estimates which show that\nthey are more susceptible to disturbances in the driving cycle from the platoon leader and, thus, have a crucial role in\ndissipating such disturbances. This case is more notable for the minimum time gap ACC setting.\nThe findings demonstrated in Figures 2 and 3 second those of [ 9] that commercially available ACC vehicles are string\nunstable. These findings are based on the assumption of linearity of the underlying system; hence, possible extensions\nto data-driven nonlinear analysis; specifically for variable spacing policies, are interesting for future research.\n5 Synthesis of H∞-based Variable Time Gap Control Policy\nIn this section, a VTG control policy is designed to guarantee strict string stability during conditions away from the\nequilibrium profile, seq,iandveq, of the ACC model in (4). Consider the following form of the time gap command\nτi=τ⋆\ni+ui(t) (9)\nwhere τ⋆\niandui(t)are the constant and variable time gap components respectively. The deviation dynamics for follower\nvehicle ican be then derived similar to (5) and (6) and are expressed as\n˙˜si(t) = ˜vi(t) +δvi−1(t) (10a)\n˙˜vi(t) =k1,i˜si(t)−(k1,iτ⋆\ni+k2,i)˜vi(t) +k2,iδvi−1(t)−k1,i(veq+ ˜vi(t))ui(t) (10b)\n6Figure 2: Distribution of estimated L2-gains for all leader-follower subsystems per campaign (left) and per vehicle\nmake and model (right). The red dashed line represents the threshold of string stability at γ= 1.\nFigure 3: Distribution of estimated L2-gains for all leader-follower subsystems per vehicle order in platoon for minimum\n(left) and maximum (right) time gap ACC settings. The red dashed line represents the threshold of string stability at\nγ= 1.\nwhere δvi−1(t)is considered as the disturbance from the preceding vehicle to be attenuated by the control input ui(t).\nThe system states are denoted by the vector xi(t) = [˜si(t) ˜vi(t)]T.\n7For strict input-to-state string stability, the VTG control policy for ui(t)needs to be designed to yield an L2-gain of\nγ≤1. Notably,the system is nonlinear due to the last term in (10b) . To this end, a synthesis of nonlinear H∞controller\nensues.\nThe nonlinear system is represented as in the following standard state-space form\n˙x=f(x) +g1(x)w+g2(x)u (11a)\nz=h(x) +K12(x)u (11b)\nwhere the subscript iis dropped for brevity, w=δvi−1denotes the exogenous disturbance, u=uidenotes the control\ninput, and z= [ρs˜siρv˜viρuui]denotes the penalty variables with penalty weights of ρs, ρvandρu. Thus, the\nvector-valued functions are given as follows.\nf(x) =\u0014\n−˜vi\nk1,i˜si−(k1,iτ⋆\ni+k2,i)˜vi\u0015\n(12a)\ng1(x) =\u0014\n1\nk2,i\u0015\n(12b)\ng2(x) =\u0014\n0\n−k1,i(veq+ ˜vi)\u0015\n(12c)\nh(x) = [ρs˜siρv˜vi0]T(12d)\nK12(x) = [0 0 ρu]T(12e)\nhT(x)K12(x) = 0 ,KT\n12(x)K12(x) =R2=ρ2\nu (12f)\nThe problem of designing ufor string stability, as per Definition 1, can be formulated as an infinite-time horizon\nmin-max optimization problem so that the L2-gain from wtozisγ≤1.\nu⋆= arg min\nu∈Umax\nw∈L21\n2Z∞\n0∥z(t)∥2−γ2∥w(t)∥2dtsubject to (11) (13)\nwhere U ⊆Ris the set of admissible control inputs. Here, the objective function balances the minimization of\nthe penalty variables, i.e. performance specifications, against the worst-case exogenous disturbances. Notably, the\npenalty variables z(t)include the deviation of the space headway away from the equilibrium profile seq,iand the\nmanipulated time gap command ui(t). This motivates minimizing the space headway, as an incentive to more efficient\nroad utilization and traffic capacity from a microscopic perspective, and the manipulated time gap command to maintain\nproximal operating conditions to the constant time gap setting τ⋆\ni.\nThis optimization problem is also a differential game; consisting of a two-player zero-sum game. The minimizing\nplayer controls the input u; whereas the maximizing player controls the disturbance w. This game has a saddle-point\nequilibrium solution if it has a value function V:X 7→Rthat is positive definite and satisfies the Hamilton-Jacobi-Isaacs\n(HJI) equation and the optimal feedback policies of both players [28]\nu⋆=−R−1\n2(x)gT\n2(x)Vx (14a)\nw⋆=1\nγ2gT\n1(x)Vx (14b)\nVxf(x) +1\n2hT(x)h(x) +1\n2Vx\u00101\nγ2g1(x)gT\n1(x)−g2(x)R−1\n2gT\n2(x)\u0011\n= 0 (14c)\nwhere Vxis the Jacobian matrix of V(x). A closed form solution for (14c) does not exist, thus, a numerical approxima-\ntion using Taylor’s series is adopted. Then, the HJI equation becomes an algebraic Ricatti equation of the following\n8form to be solved online; since we consider the equilibrium velocity as the subsystem leader velocity veq=vi−1.\nPA+ATP+P \n1\nγ2B1BT\n1−B2R−1\n2BT\n2!\nP+CTC= 0 (15a)\nf(xi) =Axi+ˆf[2+](xi);A=\u0014\n0 ��1\nk1−(k1T⋆\ng+k2)\u0015\n(15b)\ng1(xi) =B1+ ˆg[1+]\n1(xi);B1=\u0014\n1\nk2\u0015\n(15c)\ng2(xi) =B2+ ˆg[1+]\n2(xi);B2=\u0014\n0\n−k1ve\u0015\n(15d)\nh(xi) =Cxi+ˆh[2+](xi);C=\"ρs0\n0ρv\n0 0#\n(15e)\nV(xi) =xT\niPxi+ˆV[3+](xi) (15f)\nThenth-order and higher-order terms of the function (·)(xi)are denoted byˆ(·)[n+](xi)andP⪰0is a symmteric\npositive definite matrix. Note that, in our setting, the only assumptions made are (15d) and(15f) and the rest are exact\nexpressions. On another note, the algebraic Ricatti equation in (15a) is feasible for a prescribed γif the pair (A, B 2)is\nstabilizable and the Hamiltonian matrix\nH=\nA \n1\nγ2B1BT\n1−B2R−1\n2BT\n2!\n−CTC −AT\n\nis dichotomic, i.e. has no eigenvalues on the imaginary axis. This is true for veq>0and for an appropiate choice of\npenalty weights, ρs,ρvandρu. The VTG control strategy can be illustrated in Figure 4.\nFigure 4: Block diagram of the VTG proposed feedback control policy.\nThis VTG policy is robust to velocity disturbances from the leader away from the equilibrium profile; thus it yields a\nstring-stable system in the strict sense with a worst case L2-gain of γ≤1. It also guarantees the internal stability of the\nconsidered vehicle i. Additionally, the constant time gap setting can be kept either at minimum setting for benefits in\ntraffic efficiency or at a desired setting chosen by the driver. Hence, the time gap parameter design is modularized by\nseparating string stability from other considerations.\n6 Numerical Simulations\nIn this section, the performance of the proposed VTG ACC scheme will be investigated against the CTG ACC scheme [ 8]\ncalibrated in field platoon data. The numerical simulations are performed on real driving cycles from the OpenACC\ndataset to demonstrate the efficacy of the proposed VTG ACC scheme versus commercially available ACC systems. An\nimplementation is openly available at https://github.com/shaimaa-elbaklish/vtg-acc .\n96.1 Performance Indices\nIn order to assess the performance of the different ACC algorithms, several indices are introduced to quantify the\nachieved performance. The first performance indicator is used to measure the propagation of perturbations upstream and\nestimate the L2-gain, γ, of the predecessor-follower subsystems at worst-case scenario. It can estimated from simulated\ntrajectories using the approach presented in Section 4. The estimated ˆγcomputed via the semi-definite program in (8)\nrepresents the string stability aspect.\nFurthermore, Time-To-Collision (TTC) is used as a safety surrogate measure and is defined instantaneously as\nTTC i(t) =si(t)\nvi(t)−vi−1(t)(16)\nwhich is valid in car-following situations where the following vehicle is faster than the preceding one, vi(t)> vi−1(t).\nThe last indicator is used to quantify the energy efficiency aspect and is the tractive energy consumption of a vehicle [ 29].\nPi= max\u0010\n0,10−3vi\u0000\nF0+F1vi+F2V2\ni+ 1.03mai+mgsinθ\u0001\u0011\n(17a)\nEi=RT\n0Pidt\n0.036RT\n0vidt(17b)\nTractive energy consumption takes into account only tractive power demand so that it is agnostic to vehicle specifications;\nenabling a fair comparison between different ACC algorithms. The road is assumed horizontal; θ= 0. The road-related\ncoefficients F0= 213 N,F1= 0.0861 Ns/m, F2= 0.0027 Ns2/m2and the vehicle’s mass m= 1500 kg are assumed\nconstant over all vehicles in order to normalize over the different platoons and compare the performance of the utilized\nACC algorithms.\n6.2 Driving Cycles from Field Data\nIn order to assess the performance of the proposed VTG ACC scheme, real driving cycles are used from the AstaZero\ncampaign available in the OpenACC dataset [ 6]. The field platoon data used here is acquired using minimum setting\nfor the ACC systems of all vehicles in the platoon. First, the CTG ACC model by [ 8] is calibrated against each\nleader-follower subsystem’s data trajectories. The calibration process is carried out through the following optimization\nprogram [30]\nβ⋆= arg min\nβγafNRMSE (ai) +γvfNRMSE (vi) +γsfNRMSE (si) (18a)\nsubject to β∈Ωβ (18b)\nβ= [k1,ik2,iτi]T(18c)\nfNRMSE (y) =q\n1\nTPT\nt=0(ysim(t)−yobs(t))2\nq\n1\nTPT\nt=0y2\nobs(t)(18d)\nwhere fNRMSE (y)is the normalized root mean squared error function used as a goodness of fit measure for the signal y\nandΩβis the set of admissible parameters β. The simulated and observed (i.e. from data) trajectories of the signal y\nare denoted by ysim(t)andyobs(t)respectively. The same parameter values for the calibrated CTG ACC model are used\nfor the proposed scheme; i.e. k1,i, k2,iandτ⋆\ni.\nFigure 5 shows an excerpt of the trajectories of the first and last following vehicles in the platoon. The perturbations in\nleader velocity are magnified in the upstream direction. while, the VTG ACC scheme exhibits a robust and resilient\nperformance against these disturbances. Also, the proposed algorithm records significant reduction in space and time\nheadways with respect to the calibrated CTG ACC and the data collected from commercially available ACC systems.\nAlso, it is noteworthy that the proposed scheme adheres to the selected constant time headway policy τ⋆\niduring stable\ncar-following conditions and deviates away from it to attenuate any disturbances as intended.\nThe performance indicators for each following vehicle in the aforementioned experiment are illustrated in Figure 6.\nThe estimated L2-gains for VTG ACC are the lowest compared to the calibrated CTG ACC model and those estimated\nfrom field data; indicating better string stability. However, it should be mentioned that the estimation approach in (8)is\nunder the assumption of LTI systems; whereas, the VTG ACC algorithm is nonlinear. Therefore, the computed gains\nmay be overestimated. The proposed VTG ACC algorithm achieves higher performance in terms of safety and tractive\n10(a) OpenACC Data\n(b) Calibrated CTG ACC\n(c) Proposed VTG ACC\nFigure 5: Simulated trajectories of a platoon of 5 vehicles along a straight road using the calibrated CTG ACC and\nthe proposed VTG ACC. Platoon leader velocity is the black dashed line from AstaZero experimental campaign. All\nvehicles in this platoon are using ACC with minimum time gap settings.\nenergy consumption; where the energy consumption is conformed along the entire platoon and higher TTC values\nare achieved. This difference is quite pronounced for the first follower since it is the immediate following vehicle to\nthe perturbing one (i.e. platoon leader) as the proposed scheme relaxes its time gap policy to attenuate perturbations.\nIn addition, the CTG ACC shows higher energy consumption overall the platoon members due to the disturbances\npropagation along the platoon. Hence, safety and energy efficiency concerns raised due to string instability are relieved\nusing the proposed approach while optimizing road space utilization by adhering to a minimum time gap during stable\ncar-following conditions.\n11Figure 6: Performance indicators for a platoon of 5 vehicles along a straight road using the calibrated CTG ACC and\nthe proposed VTG ACC.\n7 Traffic Simulations\nIn this section, an investigation of the proposed scheme is carried out from a traffic flow perspective to validate our\nfindings and the potential benefits of the proposed algorithm. Simulation of Urban Mobility (SUMO) is used for\nmicroscopic simulations. The proposed algorithm (VTG ACC) will be tested against the CTG ACC [ 8], two string-stable\ncontrollers from the literature [ 12,31], and the IDM models. Again, field platoon data from OpenACC dataset were\nused to calibrate the CTG ACC model for commercially available ACC-equipped vehicle with minimum time gap\nsettings. The same calibrated parameters are used for the VTG ACC as well. Also, manual driving data from AstaZero\ncampaign in the OpenACC dataset is utilized for calibration of IDM to model human driving; which will be termed\n\"IDM Human\" in the subsequent sections.\nThe approach adopted by [ 12] utilizes a quadratic spacing policy inspired from human driving behavior and optimized\nfor string stability and maximum traffic capacity. The desired space headway sdes,i(t)according to the spacing policy\ncan be expressed as\nsdes,i(t) =Li−1+A+Tvi(t) +Gv2\ni(t) (19)\nwhere Ais the standstill distance and T= 0.0019 andG= 0.0448 are coefficients identified by the authors through\noffline constrained optimization. A sliding mode controller is implemented to find the follower acceleration enforcing\nsuch spacing policy. The car-following dynamics in (1) becomes\nfa,i(si(t), vi(t), vi−1(t)) =λ\nT+ 2Gvi(t)(si(t)−sdes,i(t)) +1\nT+ 2Gvi(t)(vi−1(t)−vi(t)) (20)\nwhere λis a controller gain defining the sliding surface dynamics ˙ϵ=−λϵ. The sliding variable ϵis expressed through\ndeviation of space headway away from desired spacing ϵ=si(t)−sdes,i(t). This essentially implements a VTG policy\nand will be referred to as \"VTG SMC\".\nFor the proposed approach, the penalty variables are prescribed as ρs= 0.1,ρv= 2.0,ρu= 1.0andγ= 0.95.\n7.1 Ring Road Scenario\nThe first traffic simulation environment is a ring road of length Lring= 274 m; where N= 10 homogeneous vehicles\nare placed uniformly and initially travelling at an equilibrium flow of veq= 20 m/s. This is done through a calibrated\nminimum time gap of τ⋆= 0.96770 s.\nLring=Ns eq=N(Lveh+s0+τ⋆veq) (21)\nHowever, to establish equilibrium flow at 20m/s for IDM, a length of 348m was needed for the ring road; where the\nequilibrium space headway of IDM vehicles is given as\nsIDM\neq=s0+τIDMveqr\n1−\u0010\nveq\nv0\u0011δ(22)\n12where δis a model parameter.\nThree perturbations are introduced for Vehicle 1in this scenario — deceleration to 15m/s at t= 30 s, then acceleration\nback to 20m/s at t= 340 s and, finally, sudden braking of 2m/s2att= 420 s. The resulting velocity trajectories and\noblique time-space plots are shown in Figure 7. First, the string instability of the CTG ACC model is excited during\nthe second perturbation. This induces shock-waves seen in the oblique trajectories of CTG ACC vehicles in Figure 7;\nwhich eventually cause the vehicles to stop resulting in phantom jams or stop-and-go waves. In contrast, the VTG\nACC and VTG SMC schemes show string stable trajectories and attenuate all disturbances accordingly. Interestingly,\nthe proposed spacing policy VTG ACC is better at achieving a balance in space gaps than the VTG SMC policy,\nas illustrated in Figure 8. The balanced space gaps of the perturbing vehicle \"V1\" and its following vehicle \"V2\"\ndemonstrates a potential of better road space utilization while dissipating disturbances. Regarding human driving, all\nthree perturbations are attenuated; however, at a slower rate than the variable spacing schemes. Generally, the second\nand third perturbations show more aggressive effects.\nFigure 7: Velocity profiles and oblique trajectories of 10 vehicles in a ring road scenario.\nFigure 8: Spacing trajectories of 2 vehicles following the variable spacing policies of VTG SMC and VTG ACC.\nThe ring road simulation scenario is repeated to investigate the performance of the proposed algorithm and the\ncooperative mixed H2/H∞algorithm introduced by [ 31]. Here, we consider homogeneous platoons of ACC-enabled\nvehicles for implementation of the centralized controller by [ 31] for comparison. This test is carried out for a platoon of\nN= 5vehicles in a ring road of length Lring= 137 m as well as the calibrated minimum time gap of τ⋆= 0.96770 s.\nInitial perturbations in the vehicles velocities are introduced as well as a sharp deceleration of −3m/s2lasting 3s for\none random vehicle — ’Vehicle 3’ in Figure 9.\n13As illustrated in Figure 9, both schemes are able to dissipate perturbations as they are string stable by design. However,\nthe centralized ACC algorithm shows lower changes in space and time headway relative to the proposed VTG ACC\nalgorithm. The reason is that the former scheme assumes a centralized cooperative structure; therefore, has global\ninformation about all platoon vehicles. In contrast, our proposed control scheme is completely decentralized via\npredecessor-follower topology and only needs sensor measurements of spacing and lead velocity; thus, it is more\ncomputationally efficient with comparable performance. Interestingly, with the heightened interest in connected vehicles,\nour proposed scheme can be extended for connected vehicles which is left for future research.\n(a) Centralized H2/H∞ACC scheme\n(b) Proposed VTG ACC scheme\nFigure 9: Simulated trajectories of a platoon of 5 vehicles along a ring road.\n7.2 Highway Merging Scenario\nIn this scenario, a single-lane highway of length 1km has an on-ramp merging at position 600m. The maximum flow\nrate of vehicles at the mainline is 2000 veh/h while the on-ramp creates a demand of 100veh/hr. The vehicles’ arrival\non both the highway and the on-ramp are modelled as independent Poisson processes with the corresponding flow\nrate. The vehicles on the mainline have a steady-state speed of 30m/s. The aggressive joining of vehicles from the\non-ramp to the mainline may cause shock-waves downgrading the system performance. The default lane change model\nof SUMO is used in this simulation scenario.\nA mixed traffic flow is created by varying the penetration rate, p, of the ACC-equipped vehicles; i.e. controlled by either\nCTG ACC, VTG ACC or VTG SMC. The calibrated CTG ACC algorithm represents the commercially-available ACC\nsystems and the calibrated IDM algorithm represents manually driven vehicles. Here, the performance of the traffic sys-\ntem is evaluated by average outflow from the network versus penetration rates of p= [0,10%,25%,50%,75%,100%] ;\nwhich is summarized by Figure 10. These are obtained by averaging 10 600 -second runs. The proposed VTG ACC\nscheme for ρu= 0.75yields an improved traffic outflow in comparison to both the CTG ACC and the VTG SMC\nalgorithms. Interestingly, the VTG SMC achieves lower outflow due to larger space gaps required by the adopted\nquadratic spacing policy for speeds higher than around 21m/s; as shown in Figure 11. Thus, the improvement in the\ntraffic outflow exhibited by the proposed algorithm can be attributed to (a) dissipating perturbations created by the\nmerging bottleneck and (b) adhering to a constant minimum time gap policy during stable car-following conditions\nwhich shows better road space utilization and traffic efficiency.\nOn the other hand, for the case of ρu= 1.0, the VTG ACC algorithm is more conservative with respect to string stability\nenforcement and minimizing the manipulated time-gap component ui(t)compared to the case of ρu= 0.75. This can\nbe further observed in Figure 12 where the time-space diagrams are shown for p= 100% and fully manually driven\ntraffic flow. The VTG ACC algorithm with ρu= 1.0shows less perturbations and shock-waves formed compared to\nρu= 0.75. However, the traffic volume yielded after the on-ramp merging bottleneck is higher for ρu= 0.75due to a\n14Figure 10: Average outflow from the highway merge network for different penetration rates of ACC-equipped vehicles.\nFigure 11: Constant time-gap (CTG) and quadratic spacing policies during stable car-following conditions.\nmore relaxed VTG policy. Additionally, the performance indicators for safety and energy consumption are computed\nfor each vehicle in the network and the ratio of improvement with respect to the calibrated CTG ACC averaged over all\nvehicles is computed. As illustrated in Table 1, the maximum improvement in minimum TTC per vehicle is observed\nfor the case of ρu= 1.0as string stability is conservatively enforced; whereas more improvement is seen in tractive\nenergy consumption for ρu= 0.75. This can be attributed to the more relaxed VTG policy which gives rise to a more\nsmooth accommodation of the flow of merging vehicles with relaxed time headways. Hence, an adjustment of the\npenalty variable ρumay result in different behaviors since it allows various degrees of relaxation of the CTG policy and\nstring stability enforcement.\nTable 1: Average ratio of improvement of performance indicators for the proposed VTG ACC scheme with respect to\nthe calibrated CTG ACC scheme.\nPenetration RateVTG ACC, ρu= 0.75 VTG ACC, ρu= 1.0\nminimum TTC Ec minimum TTC Ec\n10% 0.021 0 .025 0.10 0 .014\n25% 0.044 0 .42 0.25 −0.013\n50% 0.057 1 .61 0.42 −0.10\n75% 0.094 5 .33 1.05 −0.0069\n100% 0.26 2 .97 2.59 −0.13\n15(a) Time-Space Diagrams\n(b) Traffic Response\nFigure 12: Time-space diagrams and traffic response in a highway merging network using the calibrated CTG ACC and\nthe proposed VTG ACC.\n8 Conclusion\nIn this article, a variable time gap strategy is designed to guarantee string stability in the strict sense for vehicle platoons.\nThe control architecture is modular; where a constant time gap component τ⋆and a variable time gap component u(t)\nis designed as the control policy for disturbance attenuation. Strict string stability is guaranteed through nonlinear H∞\ncontrol to dissipate perturbations from the leading vehicle to the prescribed penalty variables; space headway and ego\nvehicle’s velocity. The string stability problem is decoupled from equilibrium flow dictated by τ⋆; which may define\ndriver comfort settings or a minimum time gap value for traffic efficiency. The performance of the proposed scheme\nis validated through numerical simulations compared to calibrated models representing commercially available ACC\nsystems and manually driven vehicles; available in the OpenACC dataset, as well as variable spacing policies from\nthe literature. Also, traffic simulations are carried out to demonstrate the efficacy of each model in both a ring road\nscenario and a highway merging scenario. Notably, under a penetration rate of 10%, the proposed VTG ACC policy\nimproved traffic outflow by 1.05% on average compared to the CTG ACC policy (representing commercially available\nACC systems); as well as relieving safety and energy efficiency concerns with an improvement of 2.1%and2.5%\nrespectively.\nAcknowledgments\nThe authors would like to thank Dr. Yifan Zhang for the insightful discussions on car-following models.\nReferences\n[1]SAE, “Society of Automotive Engineers J3016: Taxonomy and definitions for terms related to driving automation\nsystems for on-road motor vehicles,” 2021.\n[2]M. Shang and R. E. Stern, “Impacts of commercially available adaptive cruise control vehicles on highway stability\nand throughput,” Transportation Research Part C: Emerging Technologies , vol. 122, p. 102897, 1 2021.\n[3]C. Wu, Z. Xu, Y . Liu, C. Fu, K. Li, and M. Hu, “Spacing policies for adaptive cruise control: A survey,” IEEE\nAccess , vol. 8, pp. 50149–50162, 2020.\n[4]I. A. 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Green, “Effects of time-gap settings of adaptive cruise control (acc) on driving\nperformance and subjective acceptance in a bus driving simulator,” Safety Science , vol. 47, pp. 620–625, 5 2009.\n[11] J. Wang and R. Rajamani, “Adaptive cruise control system design and its impact on highway traffic flow,”\nProceedings of the American Control Conference , vol. 5, pp. 3690–3695, 2002.\n[12] J. Zhou and H. Peng, “Range policy of adaptive cruise control for improved flow stability and string stability,”\nConference Proceeding - IEEE International Conference on Networking, Sensing and Control , vol. 1, pp. 595–600,\n2004.\n[13] D. Swaroop and K. R. Rajagopal, “Intelligent cruise control systems and traffic flow behavior,” ASME International\nMechanical Engineering Congress and Exposition, Proceedings (IMECE) , vol. 1999-J, pp. 373–380, 1999.\n[14] S. Feng, Y . Zhang, S. E. Li, Z. Cao, H. X. Liu, and L. Li, “String stability for vehicular platoon control: Definitions\nand analysis methods,” Annual Reviews in Control , vol. 47, pp. 81–97, 1 2019.\n[15] M. Montanino and V . Punzo, “On string stability of a mixed and heterogeneous traffic flow: A unifying modelling\nframework,” Transportation Research Part B: Methodological , vol. 144, pp. 133–154, 2 2021.\n[16] A. Kesting, M. Treiber, M. Schönhof, and D. Helbing, “Adaptive cruise control design for active congestion\navoidance,” Transportation Research Part C: Emerging Technologies , vol. 16, pp. 668–683, 2008.\n[17] M. Wang, W. Daamen, S. P. Hoogendoorn, and B. van Arem, “Rolling horizon control framework for driver\nassistance systems. part i: Mathematical formulation and non-cooperative systems,” Transportation Research Part\nC: Emerging Technologies , vol. 40, pp. 271–289, 2014.\n[18] A. Spiliopoulou, D. Manolis, F. Vandorou, and M. Papageorgiou, “Adaptive cruise control operation for improved\nmotorway traffic flow,” Transportation Research Record , vol. 2672, pp. 24–35, 2018.\n[19] J. Chen, Y . Zhou, and H. Liang, “Effects of acc and cacc vehicles on traffic flow based on an improved variable\ntime headway spacing strategy,” IET Intelligent Transport Systems , vol. 13, pp. 1365–1373, 2019.\n[20] J. Wang and R. Rajamani, “Should adaptive cruise-control systems be designed to maintain a constant time gap\nbetween vehicles?,” IEEE Transactions on Vehicular Technology , vol. 53, pp. 1480–1490, 2004.\n[21] D. Yanakiev and L. Kanellakopoulos, “Nonlinear spacing policies for automated heavy-duty vehicles,” IEEE\nTransactions on Vehicular Technology , vol. 47, pp. 1365–1377, 1998.\n[22] J. Dong, J. Wang, L. Chen, Z. Gao, and D. Luo, “Effect of adaptive cruise control on mixed traffic flow: A\ncomparison of constant time gap policy with variable time gap policy,” Journal of Advanced Transportation ,\nvol. 2021, 2021.\n[23] N. Bekiaris-Liberis and A. I. Delis, “Pde-based feedback control of freeway traffic flow via time-gap manipulation\nof acc-equipped vehicles,” IEEE Transactions on Control Systems Technology , vol. 29, pp. 461–469, 2021.\n[24] M. Treiber and A. Kesting, Traffic Flow Dynamics . 2013.\n[25] R. L. Kosut, “Uncertainty model unfalsification: A system identification paradigm compatible with robust control\ndesign,” Proceedings of the IEEE Conference on Decision and Control , pp. 3492–3497, 1995.\n[26] K. V . Heusden, A. Karimi, and D. Bonvin, “Data-driven estimation of the infinity norm of a dynamical system,”\nProceedings of the IEEE Conference on Decision and Control , pp. 4889–4894, 2007.\n17[27] M.-A. Massoumnia and R. L. Kosut, “A family of norms for system identification problems,” IEEE Transactions\non Automatic Control , vol. 39, pp. 1027–1031, 1994.\n[28] J. Huang and C. F. Lin, “Numerical approach to computing nonlinear H∞control laws,” Journal of Guidance,\nControl, and Dynamics , vol. 18, pp. 989–994, 1995.\n[29] T. Apostolakis and K. Ampountolas, “Physics-inspired neural networks for parameter learning of adaptive cruise\ncontrol systems,” 9 2023.\n[30] Y . He, M. Montanino, K. Mattas, V . Punzo, and B. Ciuffo, “Physics-augmented models to simulate commercial\nadaptive cruise control (acc) systems,” Transportation Research Part C: Emerging Technologies , vol. 139,\np. 103692, 2022.\n[31] S. S. Mousavi, S. Bahrami, and A. Kouvelas, “A mixed H2/H∞controller design for a platoon with multiple\nhuman-driven and connected and automated vehicles,” 2023 European Control Conference, ECC 2023 , 2023.\n18"    },    {        "title": "2402.14178v1.Perfect_Tracking_of_Time_Varying_Optimum_by_Extremum_Seeking.pdf",        "content": "Perfect Tracking of Time-Varying Optimum by Extremum Seeking\nCemal Tugrul Yilmaz, Mamadou Diagne and Miroslav Krstic\nAbstract — This paper introduces extremum seeking (ES)\nalgorithms designed to achieve perfect tracking of arbitrary\ntime-varying extremum. In contrast to classical ES approaches\nthat employ constant frequencies and controller gains, our\nalgorithms leverage time-varying parameters, growing either\nasymptotically or exponentially, to achieve desired convergence\nbehaviors. Our stability analysis involves state transformation,\ntime-dilation transformation, and Lie bracket averaging. The\nstate transformation is based on the multiplication of the\ninput state by asymptotic or exponential growth functions. The\ntime transformation enables tracking of the extremum as it\ngradually converges to a constant value when viewed in the\ndilated time domain. Finally, Lie bracket averaging is applied\nto the transformed system, ensuring practical uniform stability\nin the dilated time domain as well as asymptotic or exponential\nstability of the original system in the original time domain.\nWe validate the feasibility of these designs through numerical\nsimulations.\nI. I NTRODUCTION\nExtremum seeking (ES) is a powerful model-free opti-\nmization technique with a rich history of theoretical research\n[3], [12], [14], [20] and practical applications [2], [21],\n[28]. Most schemes in the ES literature assume that the\ncost function remains constant and aim to guide the input\nsignal to ultimately discover an unknown fixed optimum.\nHowever, this assumption is not always valid in practical\nsituations. Real-world processes often face changing external\nconditions, causing the a priori fixed optimal input value\nto shift over time. For instance, renewable energy systems\noperate in fluctuating weather conditions and varying energy\ndemand, necessitating adjustments to controller parameters to\nmaximize power generation [11], [17]. In mineral processing,\nmaintaining optimal air recovery by adjusting aeration rates\nis essential [24]. In CO 2heat pump systems, the efficiency,\nrepresented by the coefficient of performance, varies due to\nenvironmental factors, requiring real-time optimization for\nmaximum efficiency [8].\nSeveral ES designs have emerged for plants exhibiting\nperiodic steady-state outputs with constant optimizers. One\nsuch scheme, detailed in [23], focuses on minimizing the\nsize of the limit cycle by incorporating a block for de-\ntection of the amplitude of the limit cycle and adjusting\na controller parameter to reach a constant optimum value.\nAn ES is designed in [5] to track an optimal orbit of a\nnonlinear dynamical system. This is achieved by exploiting\nthe system’s flatness property. For the plants that exhibit\nperiodic outputs of known periodicity, another ES controller\nC. T. Yilmaz, M. Diagne and M. Krstic are with Department of Mechan-\nical and Aerospace Engineering, University of California, San Diego, La\nJolla, CA, USA. cyilmaz@ucsd.edu, mdiagne@ucsd.edu,\nkrstic@ucsd.eduwith a moving-average filter is presented in [6]. However, the\nperiodic nature of the performance functions in these designs\nposes a limitation on their implementation in a broader range\nof applications, such as those found in [8], [11], [17], [24].\nVarious papers have explored the optimization of arbitrary\ntime-varying cost functions using ES technique. Work on\ntime-varying optimizers is pioneered in [10], where a gen-\neralized ES scheme is developed to track optimizers with\nknown dynamics but unknown coefficients, employing the\ninternal model principle. In [19], a delay-based strategy\nis introduced to extract the gradient signal when dealing\nwith slowly varying optima, which is later extended to\ndynamic systems with input constraints in [25]. The au-\nthors in [22] present an ES approach aimed at achieving\nunknown reference tracking and stabilization for a class of\nunknown nonlinear systems using time-varying nonlinear\nhigh-gain feedback. A model-based adaptive ES algorithm\nis introduced in [16] for a class of unstable nonlinear system\nwith time-varying extremum. The authors in [4] and [7]\nestablish results regarding the local and semi-global practical\nasymptotic stability of the extremum of a dynamic map.\nWhile [7] seeks a constant optimizer by ES to optimize time-\nvarying steady-state plant performance, [4] proves conver-\ngence towards a neighboorhood of a time-varying optimizer\nby extending the Lie bracket approximation method. The\nrobustness of Lie bracket-based ES schemes with respect\nto time-varying parameters is investigated in [13] within the\nframework of input-to-state stability (ISS). Additionally, [18]\nstudies ISS-like properties of fixed-time extremum seeking\nwith a time-varying cost function. In contrast to prior results,\nwhich utilize high-frequency dither signals, [15] develops a\ncooperative ES scheme for tracking moving sources without\nthe need for dither signals. However, it is worth noting that\nall these papers achieve convergence to a small neighborhood\nof the time-varying optimum. To the best of the authors’\nknowledge, the perfect tracking of an unknown time-varying\nextremum remains unaddressed in the ES literature.\nIn this paper, we present a major expansion in the method-\nology of the technique introduced in [26] and [27] for\nachieving unbiased convergence to a constant optimum, to\naddress the problem of tracking time-varying optimum. We\ndevelop two distinct ES algorithms for perfect tracking of\nthe time-varying optimum, achieving both asymptotic and\nexponential convergence. These algorithms are referred to as\nasymptotic ES and exponential ES, respectively. To achieve\nthese results, we depart from the typical ES approach used to\ntrack time-varying extrema, which relies on high-frequency\nsinusoids and constant high gains. Instead, in our ES scheme,\nwe employ time-varying frequencies and controller gainsarXiv:2402.14178v1  [math.OC]  21 Feb 2024that grow monotonically over time. In asymptotic ES, this\ngrowth occurs asymptotically fast, while in exponential ES,\nthe growth is exponential. To analyze the stability of our ap-\nproach, we perform a state transformation using a monoton-\nically increasing function. Following this transformation, we\napply a time-dilation transformation, which converts time-\nvarying frequencies into constants and causes the derivative\nof the optimum to tend to zero in the dilated time domain.\nWe then apply Lie bracket averaging to demonstrate the\npractical uniform stability of the transformed system. This, in\nturn, implies local asymptotic or exponential stability of the\noriginal system in original time domain, depending on the\nchosen algorithm, and guarantees asymptotic or exponential\nconvergence of the output to the extremum, provided that the\ngains are properly selected.\nThis paper is organized as follows: In Section II, we\nprovide essential stability notions and definitions that are\nreferenced throughout the paper. Section III outlines the\nproblem formulation. The designs for asymptotic and expo-\nnential ES are introduced, along with a formal stability anal-\nysis, in Sections IV and V, respectively. Section VI presents\nthe numerical results, and finally, our paper concludes with\nSection VII.\nNotation: Theδ-neighborhood of a set S ⊂Rnis denoted\nbyUS\nδ={x∈Rn: inf z∈S|x−z|< δ}. The Lie bracket\nof two vector fields f, g:Rn×R→Rnwithf(·, t), g(·, t)\nbeing continuously differentiable is defined by [f, g](x, t) :=\n∂g(x,t)\n∂xf(x, t)−∂f(x,t)\n∂xg(x, t). The notation eicorresponds to\ntheith unit vector in Rn.R+denotes the set of non-negative\nreal numbers.\nII. P RELIMINARIES\nConsider a control-affine system\n˙x=f0(x, t) +qX\ni=1fi(x, t)√ωui(ωt), (1)\nwhere x∈ D ⊂ Rn,t∈[t0,∞),t0∈R+,ω > 0,f0:\nD ×R+→Rn,fi:D ×R+→Rn, ui:R+→Rfori=\n1, . . . , q . We compute the Lie bracket system corresponding\nto (1) as follows\n˙¯x=f0(¯x, t) +1\nTqX\ni=1\nj=i+1[fi, fj](¯x, t)ZT\n0Zσ\n0uj(σ)ui(ρ)dρdσ\n(2)\nwhere ¯x(t0) =x(t0),T >0.\nWe recall the following theorem from [3], which charac-\nterizes the notion of practical stability.\nTheorem 1: Consider the system (1) and let the following\nconditions hold:\n•fi∈ C2:D ×R+→Rnfori= 0, . . . , q .\n•The functions |fi(x, t)|,\f\f\f∂fi(x,t)\n∂t\f\f\f,\f\f\f∂fi(x,t)\n∂x\f\f\f,\f\f\f∂2fj(x,t)\n∂t∂x\f\f\f,\f\f\f∂[fj,fk](x,t)\n∂t\f\f\f,\f\f\f∂[fj,fk](x,t)\n∂x\f\f\fare bounded on each com-\npact set x∈ K ⊂ D uniformly in t≥t0, for\ni= 0, . . . , q ,j= 1, . . . , q ,k=j, . . . , q .\n•The functions ujare continuous T-periodic with some\nT >0, andRT\n0uj(σ)dσ= 0 forj= 1, . . . , q .If a compact set S ⊂ D is locally (globally) uniformly\nasymptotically stable for (2), then Sis locally (semi-\nglobally) practically uniformly asymptotically stable for (1).\nTo provide a clear understanding of the concept of prac-\ntical stability, we present the following definition from [3]:\nDefinition 1: A compact set S ⊂Rnis said to be locally\npractically uniformly asymptotically stable for (1) if the\nfollowing three conditions are satisfied:\n•Practical Uniform Stability: For any ϵ >0there exist\nδ, ω0>0such that for all t0∈R+andω > ω 0,\nx(t0)∈US\nδ⇒x(t)∈US\nϵ, t∈[t0,∞). (3)\n•δ-Practical Uniform Attractivity: Let δ > 0. For any\nϵ >0there exist t1≥0andω0>0such that for all\nt0∈R+andω > ω 0,\nx(t0)∈US\nδ⇒x(t)∈US\nϵ, t∈[t0+t1,∞).(4)\n•Practical Uniform Boundedness: For any δ >0there\nexist ϵ >0andω0>0such that for all t0∈R+and\nω > ω 0,\nx(t0)∈US\nδ⇒x(t)∈US\nϵ, t∈[t0,∞). (5)\nFurthermore, if δ-practical uniform attractivity holds for\nevery δ >0, then the compact set Sis said to be semi-\nglobally practically uniformly asymptotically stable for\n(1).\nIII. P ROBLEM STATEMENT\nWe consider the following optimization problem\nmin\nθ∈UJ(θ, ζ(t)), (6)\nwhere θ∈ U ⊂ Rnis the input, ζ(t)∈Rlis an unknown\ntime-varying function, J∈ C2:U ×Rl→Ris an unknown\ncost function. We make the following assumptions:\nAssumption 1: There exists a unique continuously differ-\nentiable function π:Rl→Rnsuch that, the solution of (6)\nis given by\nθ∗(t) =π(ζ(t)) = arg min\nθ(t)∈UJ(θ(t), ζ(t)), (7)\nwhich satisfy the following conditions\nJ(θ∗(t), ζ(t))< J(θ(t), ζ(t)),∀θ(t)̸=θ∗(t),(8)\n∂J(θ∗(t), ζ(t))\n∂θ= 0, (9)\nfort∈[t0,∞), t0≥0.\nAssumption 2: There exist κ1, κ2>0such that\n(θ−θ∗(t))T∂J(θ, ζ(t))\n∂θ≥κ1|θ−θ∗(t)|2, (10)\n\f\f\f\f∂2J(θ, ζ(t))\n∂θ2\f\f\f\f≤κ2I. (11)\nAssumption 3: The time-varying functions ζ(t),˙ζ(t),\nθ∗(t),˙θ∗(t),¨θ∗(t)satisfy the following bound for t∈[t0,∞)\n|ζ(t)|+|˙ζ(t)|+|θ∗(t)|+|˙θ∗(t)|+|¨θ∗(t)| ≤Mθ,(12)J(θ, ζ(t))\n1\ns\ny\nθi\nωiη(t)\nν(t)√αiωicos(·)\nkiφ(t)Fig. 1: Accelerated ES scheme for the ith element θiofθwith\nperfect tracking. The scheme relies on a growing frequency signal\nη(t)as well as increasing gains ν(t)andφ(t). For the correspond-\ning functions, refer to Table I.\nAsymptotic ESν(t) = (1 + β(t−t0))m\nr\nη(t) =t0+ (1 + β(t−t0))r+2\nr(t)−1\nφ(t) = (1 + β(t−t0))2\nr(t)\nExponential ESν(t) =eλp(t−t0)\nη(t) =t0+e2λ(t−t0)−1\nφ(t) =e2λ(t−t0)\nTABLE I: Time-varying functions used in asymptotic and exponen-\ntial ES schemes with any β, r, λ > 0and some m, p > 0.\nwhere Mθis an unknown positive constant. Furthermore,\nthere exists an unknown positive constant MJ>0such that\n|J(θ, ζ(t))|+\f\f\f\f∂J(θ, ζ(t))\n∂ζ\f\f\f\f+\f\f\f\f∂2J(θ, ζ(t))\n∂θ∂ζ\f\f\f\f≤MJ,(13)\nfort∈[t0,∞)and for θ(t)∈UH\nε,ε >0,θ∗(t)∈ H, where\nH ⊂Rnis a compact set.\nAssumption 1 guarantees the existence of a unique mini-\nmum of the function J(θ, ζ(t))atθ(t) =θ∗(t). Assumption\n2 requires the cost function to be strongly convex in θ.\nAssumption 3 ensures the boundedness of the time-varying\nfunctions of ζ∗(t),θ∗(t)and their time derivatives, as well\nas the cost function J(θ, ζ(t))and its partial derivatives.\nWe measure the unknown function J(θ, ζ(t))in real time\nas follows\ny(t) =J(θ(t), ζ(t)), t ∈[t0,∞), (14)\nin which y∈Ris the output.\nOur objective is to develop ES algorithms that utilize\noutput feedback y(t)to achieve perfect tracking of θ∗(t)by\nθboth asymptotically and exponentially, which consequently\nminimizes the value of y(t). This objective is achieved by\nproperly choosing monotonically increasing frequencies and\ncontroller gains. Importantly, the design is performed without\nthe need for prior knowledge of the optimal input θ∗(t)or\nthe function J(θ, ζ(t)). To provide a visual representation,\nthe ES designs to be introduced are depicted schematically in\nFig. 1. The time-varying design parameters ν(t), φ(t), η(t)\nare given in Table I.\nIn practice, to ensure robustness against noise and numer-\nical issues, the gains ν(t)andφ(t), along with the instanta-neous frequency ωidη(t)\ndt, can be constrained to moderately\nlarge values that are sufficient for close tracking of the\nextremum.\nRemark 1: It is worth noting that there exists flexibility in\ndesigning alternative parameters, ν(t), η(t),andφ(t), which\nare capable of achieving the desired convergence results. The\nchoices of the parameters in Table I prioritize conservative\ngrowth in adaptation rate and frequency.\nIV. A SYMPTOTIC ES D ESIGN\nIn this section, we develop an ES referred to as asymptotic\nES, which is designed to achieve perfect asymptotic tracking\nof the time-varying optimum. The key elements of our design\nare the asymptotically growing frequencies and controller\ngains. By carefully selecting a rapid growth rate for the\ntime-varying design parameters, φ(t), η(t),andν(t),we\nguarantee precise tracking of the time-varying optimum.\nThe asymptotic convergence result is stated in the following\ntheorem.\nTheorem 2: Consider the following asymptotic ES design\n˙θ=ξm(t)nX\ni=1√αiωiei\n×cos\u0010\nωi(t0+ξr+2(t)−1) +kiξ2(t)y\u0011\n, (15)\nwith\nξ(t) = (1 + β(t−t0))1\nr, t ∈[t0,∞), (16)\nwhere ωi=ωˆωisuch that ˆωi̸= ˆωj∀i̸=j,t0≥0, and\nβ, r, k i, αi>0,i= 1, . . . , n under Assumptions 1–3. There\nexists ω∗>0such that for all ω > ω∗, the following results\nhold:\n•Ifθ∗is constant, i.e., ˙θ∗(t)≡0,θ(t)asymptotically\nconverges to θ∗for−r\n2≤m≤1andkiαi>2(r+\n2)2β3/(r3κ1),i= 1, . . . , n ,\n•Ifθ∗(t)is time-varying, θ(t)asymptotically converges\ntoθ∗(t)for1\n2< m≤1andkiαi>2(r+ 2)2(2mr−\nr+ 1)β3/(r3κ1),i= 1, . . . , n .\nProof: Let us proceed through the proof step by step.\nStep 1: State transformation. Define the error state\n˜θ(t) =θ(t)−θ∗(t), (17)\nwhich obeys the following dynamics\n˙˜θ=−˙θ∗(t) +ξm(t)nX\ni=1√αiωieicos\u0010\nωi(t0+ξr+2(t)−1)\n+kiξ2(t)J(˜θ+θ∗(t), ζ(t))\u0011\n, (18)\nwith the help of (14). Considering the following transforma-\ntion\n˜θf=ξ(t)˜θ, (19)\nwhich transforms (18) to\n˙˜θf=−ξ(t)˙θ∗(t) +β\nrξr(t)˜θf+ξm+1(t)nX\ni=1√αiωiei×cos\u0010\nω(t0+ξr+2(t)−1) +kiξ2(t)Jf(˜θf, t)\u0011\n,\n(20)\nwhere\nJf(˜θf, t) =J(˜θf/ξ(t) +θ∗(t), ζ(t)), (21)\nwe rewrite (20) by expanding the cosine term as follows\n˙˜θf=−ξ(t)˙θ∗(t) +β\nrξr(t)˜θf\n+ξm+1(t)nX\ni=1√αiωieicos\u0010\nkiξ2(t)Jf(˜θf, t)\u0011\n×cos\u0010\nωi(t0+ξr+2(t)−1)\u0011\n−ξm+1(t)nX\ni=1√αiωieisin\u0010\nkiξ2(t)Jf(˜θf, t)\u0011\n×sin\u0010\nωi(t0+ξr+2(t)−1)\u0011\n. (22)\nStep 2: Time transformation. To carry out our analysis,\nwe introduce the following time dilation and contraction\ntransformations\nτ=t0+ξr+2(t)−1,\n=t0+ (1 + β(t−t0))r+2\nr−1, τ∈[t0,∞),(23)\nt=t0+β−1(τ−t0+ 1)r\nr+2−β−1(24)\nand using the fact that\ndτ\ndt=r+ 2\nrβ(τ−t0+ 1)2\nr+2, (25)\nwe express (22) in the dilated τ-domain as follows\nd˜θf\ndτ=b0(˜θf, τ) +nX\ni=1bc,i(˜θf, τ)√ωicos(ωiτ)\n−nX\ni=1bs,i(˜θf, τ)√ωisin(ωiτ), (26)\nwhere\nb0(˜θf, τ) =−(τ−t0+ 1)1\nr+2dθ∗\nτ(τ)\ndτ\n+1\n(r+ 2)( τ−t0+ 1)˜θf, (27)\nbc,i(˜θf, τ) =r√αi\n(r+ 2)β(τ−t0+ 1)m−1\nr+2ei\n×cos\u0010\nki(τ−t0+ 1)2\nr+2Jf,τ(˜θf, τ)\u0011\n,(28)\nbs,i(˜θf, τ) =r√αi\n(r+ 2)β(τ−t0+ 1)m−1\nr+2ei\n×sin\u0010\nki(τ−t0+ 1)2\nr+2Jf,τ(˜θf, τ)\u0011\n(29)\nwith\nθ∗\nτ(τ) =θ∗(t0+β−1(τ−t0+ 1)r\nr+2−β−1),(30)\nJf,τ(˜θf, τ) =Jf(˜θf, t0+β−1(τ−t0+ 1)r\nr+2−β−1).\n(31)\nNote that after the time transformation from ttoτ, the\nsystem (26) has the form of (1) and consequently, Theorem\n1 is directly applicable.Step 3: Feasibility analysis of (26)for averaging. Before\nperforming Lie bracket averaging, we first show that (26)\nsatisfies the boundedness assumption stated in Theorem 1.\nLetM ⊂ U be a compact set and consider the bound (12)\nin Assumption 3. Then, we get\n(τ−t0+ 1)1\nr+2\f\f\f\fdθ∗\nτ(τ)\ndτ\f\f\f\f≤rMθ\n(r+ 2)β(τ−t0+ 1)1\nr+2,\n(32)\n(τ−t0+ 1)1\nr+2\f\f\f\fd2θ∗\nτ(τ)\ndτ2\f\f\f\f≤r2Mθ\n(r+ 2)2β2(τ−t0+ 1)3\nr+2,\n(33)\nusing (25). From inequalities (32), (33), we obtain the bound-\nedness of |b0(˜θf, τ)|,\f\f∂b0(˜θf,τ)\n∂˜θf\f\f,\f\f∂b0(˜θf,τ)\n∂τ\f\ffor(τ,˜θf)∈\n[t0,∞)× M . In light of the strong convexity property in\nAssumption 2, we obtain the following bound (see. [1, pp.\n461])\n\f\f\f\f∂J(θ, ζ(t))\n∂θ\f\f\f\f≤2κ2|θ−θ∗(t)|=2κ2\nξ(t)|˜θf|. (34)\nAgain, considering Assumptions 2, 3, (34), and noting from\n(19), (23) that\ndθ\nd˜θf=1\nξ(t)=1\n(τ−t0+ 1)1\nr+2, (35)\nwe obtain the following bound\n(τ−t0+ 1)2\nr+2 \f\f\f\f\f∂Jf,τ(˜θf, τ)\n∂˜θf\f\f\f\f\f+\f\f\f\f\f∂2Jf,τ(˜θf, τ)\n∂˜θ2\nf\f\f\f\f\f!\n=ξ(t)\f\f\f\f∂J(θ, ζ(t))\n∂θ\f\f\f\f+\f\f\f\f∂2J(θ, ζ(t))\n∂θ2\f\f\f\f,\n≤2κ2|˜θf|+κ2, (36)\nas well as\nd(τ−t0+ 1)2\nr+2\ndτ \n|Jf,τ(˜θf, τ)|+\f\f\f\f\f∂Jf,τ(˜θf, τ)\n∂˜θf\f\f\f\f\f!\n=2\n(r+ 2)ξr(t)\u0012\n|J(θ, ζ(t))|+1\nξ(t)\f\f\f\f∂J(θ, ζ(t))\n∂θ\f\f\f\f\u0013\n,\n≤2\n(r+ 2)ξr(t)MJ, (37)\nfor(t, τ,˜θf)∈[t0,∞)×[t0,∞)× M . In addition, from\n(11)–(13), (25), (34), (35), the following holds\n(τ−t0+ 1)2\nr+2 \f\f\f\f\f∂Jf,τ(˜θf, τ)\n∂τ\f\f\f\f\f+\f\f\f\f\f∂2Jf,τ(˜θf, τ)\n∂˜θf∂τ\f\f\f\f\f!\n=r\n(r+ 2)β \f\f\f\f∂J(θ, ζ(t))\n∂ζ(t)˙ζ(t)\f\f\f\f+\f\f\f\f\f∂J(θ, ζ(t))\n∂θ\n× \n−˜θf˙ξ(t)\nξ2(t)+˙θ∗(t)!\f\f\f\f\f+1\nξ(t)\f\f\f\f∂2J(θ, ζ(t))\n∂θ∂ζ (t)˙ζ(t)\f\f\f\f\n+1\nξ(t)\f\f\f\f\f∂2J(θ, ζ(t))\n∂θ2 \n−˜θf˙ξ(t)\nξ2(t)+˙θ∗(t)!\f\f\f\f\f!\n,\n≤Md, (38)for some Md>0and for (t, τ,˜θf)∈[t0,∞)×[t0,∞)×M .\nTaking into account the bounds given through (36)–(38)\nand noting that m≤1, we can arrive at the bounded-\nness of |bc,i(˜θf, τ)|,|bs,i(˜θf, τ)|,\f\f∂bc,i(˜θf,τ)\n∂θf\f\f,\f\f∂bs,i(˜θf,τ)\n∂θf\f\f,\n\f\f∂bc,i(˜θf,τ)\n∂τ\f\f,\f\f∂bs,i(˜θf,τ)\n∂τ\f\f,\f\f∂2bc,i(˜θf,τ)\n∂τ∂˜θf\f\fand\f\f∂2bs,i(˜θf,τ)\n∂τ∂˜θf\f\ffor\n(τ,˜θf)∈[t0,∞)× M . Next, the following Lie bracket is\ncomputed\n\u0002\nbc,i(˜θf, τ)bs,i(˜θf, τ)\u0003\n= (τ−t0+ 1)2m\nr+2r2\n(r+ 2)2β2\n×kiαi∂Jf,τ(˜θf, τ)\n∂˜θf,i, (39)\nwhich is bounded due to (36) and m≤1. The bounded-\nness of\f\f∂[bc,i(˜θf,τ),bs,i(˜θf,τ)]\n∂˜θf\f\fand\f\f∂[bc,i(˜θf,τ),bs,i(˜θ,τ)]\n∂τ\f\ffor\n(τ,˜θf)∈[t0,∞)× M is obtained by recalling (36)–(38).\nStep 4: Lie bracket averaging. We derive the Lie bracket\nsystem for (26) as follows\nd¯θf\ndτ=b0(¯θf, τ)−1\n2nX\ni=1\u0002\nbc,i(¯θf, τ)bs,i(¯θf, τ)\u0003\n=−(τ−t0+ 1)1\nr+2dθ∗\nτ(τ)\ndτ+1\n(r+ 2)( τ−t0+ 1)¯θf\n−(τ−t0+ 1)2m\nr+2r2\n(r+ 2)2β2\n×nX\ni=1kiαi\n2ei∂Jf,τ(¯θf, τ)\n∂¯θf,i. (40)\nStep 5: Stability analysis of the Lie bracket system. We\ncan write (40) in t-domain in view of the transformations\n(23), (24) as follows\n˙¯θf=−ξ(t)˙θ∗(t) +β\nrξr(t)¯θf\n−ξ2m+2(t)r2\n(r+ 2)2β2nX\ni=1kiαi\n2ei∂Jf(¯θf, t)\n∂¯θf,i.(41)\nConsider the following Lyapunov function\nV(¯θf) =1\n2|¯θf|2. (42)\nThe computation of the time derivative of (42) along (41)\nleads to the following estimates\n˙V≤ξ(t)|˙θ∗(t)||¯θf|+β\nrξr(t)|¯θf|2−ξ2m+2(t)r2(kα)min\n2(r+ 2)2β2\n×nX\ni=1¯θf,i∂Jf(¯θf, t)\n∂¯θf,i,\n≤ −\u0012r2(kα)minκ1\n2(r+ 2)2β2ξ2m(t)−β\nrξr(t)\u0013\n|¯θf|2\n+ξ(t)|˙θ∗(t)||¯θf|, (43)\nwhere (kα)min= min {kiαi}fori= 1, . . . , n . To obtain\n(43), we have used the following property\n¯θT\nf∂Jf(¯θf, t)\n∂¯θf≥κ1\nξ2(t)|¯θf|2, (44)\nthat is derived using (10) and (19). The stability of (43) is\nexamined based on the following two cases:Case 1: Constant θ∗.In this case, (43) reads\n˙V≤ − 2\u0012r2(kα)minκ1\n2(r+ 2)2β2−β\nr\u0013\nξ2m(t)V, (45)\nand for m≥ −r\n2, the following estimate holds\nV(t)≤e−2\u0012\nr2(kα)minκ1\n2(r+2)2β2−β\nr\u0013Rt\nt0ξ2m(σ)dσV(t0), (46)\nby the comparison lemma [9]. For m=−r\n2, we get ξ2m(t) =\n(1 +β(t−t0))−1, and obtain that\nV(t)≤V(t0)(1 + β(t−t0))−2\nβ\u0012\nr2(kα)minκ1\n2(r+2)2β2−β\nr\u0013\n,(47)\nwhich decays to zero asymptotically, provided that\n(kα)min>2(r+ 2)2β3/(r3κ1). For m >−r\n2, we write\nV(t)≤e−2\u0012\nr2(kα)minκ1\n2(r+2)2β2−β\nr\u0013\n(1+β(t−t0))2m/r+1−1\nβ(2m/r+1)V(t0),(48)\nwhich decays to zero exponentially, provided that (kα)min>\n2(r+ 2)2β3/(r3κ1). This, in turn, implies that the averaged\nsystem (41) is asymptotically stable.\nCase 2: Time-varying θ∗(t).In view of the bound (12), we\ncan rewrite (43) as follows\n˙V≤ −\u0012r2(kα)minκ1\n2(r+ 2)2β2ξ2m(t)−β\nrξr(t)\u0013\n|¯θf|2+Mθξ(t)|¯θf|,\n≤ −\u0012r2(kα)minκ1\n2(r+ 2)2β2−β\nr\u0013\nξ2m(t)|¯θf|2+Mθξ(t)|¯θf|,\n(49)\nform≥ −r\n2. Performing Young’s inequalities for the\nfollowing term\nMθξ(t)|¯θf| ≤cξ\n2ξ(t)|¯θf|2+M2\nθ\n2cξξ(t), (50)\nwhere cξ=r2(kα)minκ1\n2(r+2)2β2−β\nr, we rewrite (49) as\n˙V≤ −\u0012r2(kα)minκ1\n2(r+ 2)2β2−β\nr\u0013\nξ2m(t)V+M2\nθ\n2cξξ(t).(51)\nBy selecting m >1\n2and applying the result from Lemma\n4, we conclude that the averaged system (41) is asymptot-\nically stable given that (kα)min>2(r+ 2)2(2mr−r+\n1)β3/(r3κ1).\nStep 6: Lie bracket averaging theorem. With the asymp-\ntotic stability of the averaged system (41) proved in Step 5,\nwe conclude from Theorem 1 that the origin of the trans-\nformed system (22) is practically uniformly asymptotically\nstable.\nStep 7: Convergence to extremum. Considering the\nresult in Step 6 and recalling from (16), (19) that\nθ(t) =θ∗(t) +1\n(1 +β(t−t0))1\nr˜θf(t), (52)\nwe conclude the asymptotic convergence of θ(t)toθ∗(t).\nThis implies the convergence of the output y(t)to\nJ(θ∗(t), ζ(t))and completes the proof of Theorem 2.V. E XPONENTIAL ES D ESIGN\nIn this section, we take a further step to accelerate the con-\nvergence towards the time-varying optimum. We introduce an\nES, referred to as exponential ES, which relies on gains and\nfrequencies characterized by exponential growth. By care-\nfully selecting the appropriate growth rate for these signals,\nwe achieve perfect exponential tracking of the bounded time-\nvarying optimum. The theorem presented below establishes\nour exponential convergence result.\nTheorem 3: Consider the following exponential ES design\n˙θ=ϕp(t)nX\ni=1√αiωieicos\u0000\nωi(t0+ϕ2(t)−1) +kiϕ2(t)y\u0001\n,\n(53)\nwith\nϕ(t) =eλ(t−t0), t ∈[t0,∞), (54)\nwhere ωi=ωˆωisuch that ˆωi̸= ˆωj∀i̸=j,t0≥0,λ >0,\nunder Assumptions 1–3. There exists ω∗>0such that for\nallω > ω∗, the followings hold:\n•Ifθ∗is constant, i.e., ˙θ∗(t)≡0,θ(t)exponentially\nconverges to θ∗for0≤p≤1andkiαi>4λ2/κ1,\ni= 1, . . . , n ,\n•Ifθ∗(t)is time-varying, θ(t)exponentially converges to\nθ∗(t)for1\n2< p≤1andki>8λ2p/κ1,i= 1, . . . , n .\nProof: Let us proceed through the proof step by step.\nStep 1: State transformation. Taking the derivative of the\nerror state ˜θ(t) =θ(t)−θ∗(t)in view of (53) and recalling\n(14), we get the following error dynamics\n˙˜θ=−˙θ∗(t) +ϕp(t)nX\ni=1√αiωieicos\u0010\nωi(t0+ϕ2(t)−1)\n+kiϕ2(t)J(˜θ+θ∗(t), ζ(t))\u0011\n. (55)\nConsider the following transformation\n˜θf=ϕ(t)˜θ, (56)\nwhich transforms (55) to\n˙˜θf=−ϕ(t)˙θ∗(t) +λ˜θf+ϕp+1(t)nX\ni=1√αiωiei\n×cos\u0010\nω(t0+ϕ2(t)−1) +kiϕ2(t)Jf(˜θf, t)\u0011\n,(57)\nwith\nJf(˜θf, t) =J(˜θf/ϕ(t) +θ∗(t), ζ(t)). (58)\nStep 2: Time transformation. Let us perform the follow-\ning time dilation and contraction transformations\nτe=t0+ϕ2(t)−1,\n=t0+e2λ(t−t0)−1, τ e∈[t0,∞), (59)\nt=t0+1\n2λln(τe−t0+ 1). (60)\nConsidering the following fact\ndτe\ndt= 2λe2λ(t−t0)= 2λ(τe−t0+ 1), (61)we express (57) in the dilated τe-domain as follows\nd˜θf\ndτe=−(τe−t0+ 1)1\n2dθ∗\nτe(τe)\ndτe+1\n2(τe−t0+ 1)˜θf\n+1\n2λ(τe−t0+ 1)p−1\n2nX\ni=1√αiωiei\n×cos\u0010\nωτe+ki(τe−t0+ 1)Jf,τe(˜θf, τe)\u0011\n,(62)\nwithθ∗\nτe(τe) =θ∗(t0+ 1/(2λ) ln(τe−t0+ 1)) and\nJf,τe(˜θf, τe) =Jf\u0012\n˜θf, t0+1\n2λln(τe−t0+ 1)\u0013\n.(63)\nWe rewrite (62) by expanding the cosine term as follows\nd˜θf\ndτe=−(τe−t0+ 1)1\n2dθ∗\nτe(τe)\ndτe+1\n2(τe−t0+ 1)˜θf\n+nX\ni=1√αi\n2λeicos\u0010\nki(τe−t0+ 1)Jf,��e(˜θf, τe)\u0011\n×(τe−t0+ 1)p−1\n2√ωicos(ωiτe)\n−nX\ni=1√αi\n2λeisin\u0010\nki(τe−t0+ 1)Jf,τe(˜θf, τe)\u0011\n×(τe−t0+ 1)p−1\n2√ωisin(ωiτe). (64)\nStep 3: Lie bracket averaging. The feasibility of the\nerror system (64) for Lie bracket averaging can be verified\nanalogously to Step 3 in the proof of Theorem 2. For p≤1,\nwe derive the following average system\nd¯θf\ndτe=−(τe−t0+ 1)1\n2dθ∗\nτe(τe)\ndτe+1\n2(τe−t0+ 1)¯θf\n−1\n4λ2(τe−t0+ 1)pnX\ni=1kiαi\n2ei∂Jf,τe(¯θf, τe)\n∂¯θf,i.\n(65)\nStep 4: Stability analysis. To study the stability of (65),\nwe consider the following Lyapunov function\nV(¯θf) =1\n2|¯θf|2. (66)\nDerivative of (66) with respect to τeusing (10), (56), (65)\nyields\ndV\ndτe≤(τe−t0+ 1)1\n2|¯θf|\f\f\f\fdθ∗\nτe(τe)\ndτe\f\f\f\f+1\n2(τe−t0+ 1)|¯θf|2\n−1\n4λ2(τe−t0+ 1)pnX\ni=1kiαi\n2¯θf,i∂Jf,τe(¯θf, τe)\n∂¯θf,i,\n≤(τe−t0+ 1)1\n2|¯θf|\f\f\f\fdθ∗\nτe(τe)\ndτe\f\f\f\f−\u0012(kα)minκ1\n8λ2\n×(τe−t0+ 1)p−1−1\n2(τe−t0+ 1)\u0013\n|¯θf|2,(67)\nwhere (kα)min= min {kiαi}fori= 1, . . . , n , we use the\nproperty given in (44), except that ξ(t)is replaced by ϕ(t)\nand we recall from (59) that ϕ(t) = ( τe−t0+ 1)1\n2. The\nstability of (67) is examined based on the following two\ncases:Case 1: Constant θ∗.Noting that ˙θ∗(t)≡0, we rewrite (67)\nas\ndV\ndτe≤ − 2\u0012(kα)minκ1\n8λ2−1\n2\u0013\n(τe−t0+ 1)p−1V, (68)\nforp≥0. In the case where p= 0, we compute the solution\nof (68) by comparison principle, as\nV(τe)≤V(t0)(τe−t0+ 1)−2\u0010(kα)minκ1\n8λ2−1\n2\u0011\n, (69)\nwhich asymptotically decays to zero, provided that\n(kα)min>4λ2/κ1. For p >0, we get\nV(τe)≤V(t0)e−2\u0010(kα)minκ1\n8λ2−1\n2\u0011(τe−t0+1)p−1\np, (70)\nwhich decays to zero exponentially, provided that (kα)min>\n4λ2/κ1. This, in turn, implies that the averaged system (65)\nis asymptotically stable.\nCase 2: Time-varying θ∗(t).Recalling the bound (12), we\nrewrite (67) as\ndV\ndτe≤Mθ\n2λ(τe−t0+ 1)1\n2|¯θf| −\u0012(kα)minκ1\n8λ2−1\n2\u0013\n×(τe−t0+ 1)p−1|¯θf|2, (71)\nforp≥0. Performing Young’s inequalities for the following\nterm\nMθ\n2λ(τe−t0+ 1)1\n2|¯θf| ≤1\n(τe−t0+ 1)1\n2\n×\u0012cϕ\n2|¯θf|2+M2\nθ\n8λ2cϕ\u0013\n,(72)\nwhere cϕ=(kα)minκ1\n8λ2−1\n2, we rewrite (71) as\ndV\ndτe≤M2\nθ\n8λ2cϕ(τe−t0+ 1)1\n2−\u0012(kα)minκ1\n8λ2−1\n2\u0013\n×(τe−t0+ 1)p−1V. (73)\nChoosing p >1\n2and applying the result from Lemma 4,\nwe conclude that the averaged system (65) is asymptotically\nstable, provided that (kα)min>8λ2p/κ1.\nStep 5: Lie bracket averaging theorem. Taking into\naccount the asymptotic stability of the averaged system (65)\nproved in Step 4, we can deduce, from Theorem 1, that\nthe origin of the transformed system (64) in τe-domain\n(equivalent to (57) in t-domain) is practically uniformly\nasymptotically stable.\nStep 6: Convergence to extremum. Considering the\nresult in Step 5 and recalling from (54), (56) that\nθ(t) =θ∗(t) +e−λ(t−t0)˜θf(t), (74)\nwe conclude the exponential convergence of θ(t)to\nθ∗(t). This implies the convergence of the output y(t)to\nJ(θ∗(t), ζ(t))and completes the proof of Theorem 3.\n0 5 10 15 20 25 30 35 40\nTime[s]-1.2-1-0.8-0.6\n0 5 10 15 20 25 30 35 40\nTime[s]0.811.21.41.6(a)\n0 5 10 15 20 25 30 35 40\nTime[s]-0.2-0.100.10.20.3\n(b)\nFig. 2: (a)Perfect exponential tracking of the time-varying opti-\nmum θ∗\n1(t), θ∗\n2(t)by the inputs θ1, θ2. (b) Exponential convergence\nof the output y(t)to the time-varying optimum y∗(t).\nVI. N UMERICAL SIMULATION\nIn this section, we conduct a numerical simulation to\nassess the performance of the developed ES algorithms. We\nconsider the following quadratic map\nJ(θ, θ∗(t)) = 0 .2 sin(0 .5t) + (θ1+ 1−0.2 sin(0 .7t))2\n+ (θ2−1−0.3 cos(0 .8t))2, (75)\nwhere θ=\u0002\nθ1θ2\u0003T∈R2is the input, θ∗(t) =\u0002−1 + 0 .2 sin(0 .7t),1 + 0 .3 cos(0 .8t)\u0003Tis the optimum\ninput and y∗(t) = 0 .2 sin(0 .5t)is the optimum output. We\nimplement the exponential ES introduced in (53), employing\nthe following parameters: p= 0.51,α1= 0.015,α2= 0.02,\nω1= 10 ,ω2= 12 ,k1= 10 ,k2= 11 andλ= 0.1. The\ninitial conditions are set to θ1(0) = −0.9,θ2(0) = 0 .9.\nWe present the simulation results in Fig. 2 and 3. In Figure\n2a, we observe that the inputs θ1(t)andθ2(t)converge\ntoward the optimal inputs θ∗\n1(t)andθ∗\n2(t)exponentially at\na rate of λ= 0.1. The exponential growth in frequencies\noccurs at a rate of 2λ= 0.2. Fig. 2b demonstrates the\nexponential convergence of the output y(t)toward the time-0 5 10 15 20 25 30 35 40\nTime[s]010002000300040005000600070008000Instantaneous Frequency (rad/s)Fig. 3: Evolution of the instantaneous frequencies, which corre-\nspond to ω1dϕ2(t)/dtandω2dϕ2(t)/dt, over time.\nvarying optimal y∗(t). In Fig. 3, we depict the instantaneous\nfrequencies of (53), which correspond to ω1dϕ2(t)/dtand\nω2dϕ2(t)/dt, respectively.\nVII. C ONCLUSION\nBy focusing on achieving perfect tracking of arbitrary\ntime-varying extremum, this contribution fills a gap in the\nexisting ES literature. We introduce two unique ES designs\nwith either asymptotic or exponential convergence, achieved\nthrough gains and frequencies characterized by correspond-\ning asymptotic or exponential growth. By carefully tuning\nthe growth rates of these signals, we achieve precise tracking\nwith the desired convergence rate. Our stability analysis re-\nlies on state transformation, time-dilation transformation, and\nLie bracket averaging techniques. We provide a numerical\nsimulation to illustrate the performance of our exponential\nES design in tracking a periodically oscillating extremum.\nAPPENDIX\nA. Additional Lemma\nLemma 4: The system\n˙V=−εaµm1(t)V+εbµm2(t), (76)\nwith\nµ(t) = 1 + β(t−t0), (77)\nfort≥t0, where V ∈R,t0≥0,β > 0,εa> β(m1−\nm2), εb∈R,m1> m 2, and m1≥ −1, is asymptotically\nstable at the origin.\nProof: Consider the following transformation\nVf=µm1−m2(t)V, (78)\nwhich obeys the following dynamics\n˙Vf=β(m1−m2)\nµ(t)Vf−εaξm1(t)Vf+εbξm1(t).(79)\nConsider the following Lyapunov function\nΥ =1\n2V2\nf, (80)whose time derivative yields\n˙Υ =\u0012β(m1−m2)\nµ(t)−εaµm1(t)\u0013\nV2\nf+εbµm1(t)Vf.(81)\nLetεa=β(m1−m2) +εcfor any εc>0. Then, we get\n˙Υ≤ − εcµm1(t)V2\nf+εbµm1(t)Vf, (82)\nform1≥ −1. Performing Young’s inequalities, we rewrite\n(82) as\n˙Υ≤ − εcµm1(t)Υ +ε2\nb\n2εcµm1(t). (83)\nBy comparison principle, we compute from (83) that\nΥ(t)≤e−Rt\nt0εcµm1(σ)dσΥ(t0)\n+ε2\nb\n2εcZt\nt0e−Rt\nςεcµm1(σ)dσµm1(ς)dς,\n≤\n\ne−εc\n(m1+1)β(µm1+1(t)−1)Υ(t0) +ε2\nb\n2ε2c,form1>−1,\nµ−εc/β(t)Υ(t0) +ε2\nb\n2ε2c, form1=−1,\n(84)\nfrom which we deduce the stability of Υand thus, from\n(80), the stability of Vf. In view of this fact, we conclude\nthe asymptotic stability of Vfrom (78).\nREFERENCES\n[1] S. P. Boyd and L. Vandenberghe. Convex optimization . Cambridge\nuniversity press, 2004.\n[2] D. Dochain, M. Perrier, and M. Guay. Extremum seeking control and\nits application to process and reaction systems: A survey. Mathematics\nand Computers in Simulation , 82(3):369–380, 2011.\n[3] H.-B. D ¨urr, M. S. Stankovi ´c, C. Ebenbauer, and K. H. Johansson.\nLie bracket approximation of extremum seeking systems. Automatica ,\n49(6):1538–1552, 2013.\n[4] V . Grushkovskaya, H.-B. D ¨urr, C. Ebenbauer, and A. Zuyev. 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Systems & control letters ,\n56(3):245–252, 2007."    },    {        "title": "2402.14198v1.Tight_Inapproximability_of_Nash_Equilibria_in_Public_Goods_Games.pdf",        "content": "Tight Inapproximability of Nash Equilibria\nin Public Goods Games\nJ´ er´ emi Do Dinh Alexandros Hollender\n´Ecole Polytechnique F´ ed´ erale de Lausanne University of Oxford\njeremi.dodinh@epfl.ch alexandros.hollender@cs.ox.ac.uk\nAbstract\nWe study public goods games, a type of game where every player has to decide whether\nor not to produce a good which is public , i.e., neighboring players can also benefit from it.\nSpecifically, we consider a setting where the good is indivisible and where the neighborhood\nstructure is represented by a directed graph, with the players being the nodes. Papadimitriou\nand Peng (2023) recently showed that in this setting computing mixed Nash equilibria is\nPPAD -hard, and that this remains the case even for ε-well-supported approximate equilibria\nfor some sufficiently small constant ε. In this work, we strengthen this inapproximability\nresult by showing that the problem remains PPAD -hard for any non-trivial approximation\nparameter ε.\nKeywords: public good; game theory; Nash equilibrium; approximation; PPAD\n1 Introduction\nA good is said to be public if everyone can simultaneously benefit from it, and not only the\nperson or organization who expended money (or effort) for its provision. Examples of public\ngoods include clean air and water, information on the internet (e.g., Wikipedia, open-access\nscientific papers), or television broadcasts.\nPublic goods give rise to interesting game theoretic settings due to the free-riding phenomenon:\nan agent can benefit from a good without having to pay for it. Indeed, consider a setting in\nwhich every agent can decide whether to pay for the provision of the public good, but if at least\none agent pays for the good, then everyone has access to it. Clearly, every agent is happier if\nthe public good is available as opposed to unavailable, but ideally they would prefer some other\nagent to pay for its provision.\nIn many applications, the public good does not benefit everyone but rather only the agents\nin the proximity of the good. This is for example the case for clean air or water. A very natural\nway to model this “locality” is by assuming that the agents (or players) are nodes in a graph,\nand the neighborhood structure encodes which agents will benefit from a public good produced\nby a given agent [ BK07 ]. In other words, if a certain agent decides to produce the good, then all\nits neighbors, as well as the agent itself, will benefit from it. Various versions of such public\ngoods games on graphs have been extensively studied [ BK07 ,GGJ+09,DPR11 ,LP13 ,BKD14 ,\nAll15, BK16, YZBV20, GN22, BKS23, KS23, Gil23].\nIn this work, we focus on a particularly simple setting with indivisible public goods. We\nconsider binary action public goods games with the max utility function, which are sometimes\nalso called best-shot games . In such a game, every player ichooses an action ai∈ {0,1}, where\nai= 1 corresponds to producing the public good, and ai= 0 to not producing it. The utility of\n1arXiv:2402.14198v1  [cs.GT]  22 Feb 2024a player is the maximum over the strategies of all the players in its neighborhood (including\nitself) minus the cost (or price) of producing the good. Formally\nui(a1, . . . , a n) = max\nj∈N(i)∪{i}{aj} −p·ai,\nwhere N(i) denotes the neighborhood of player i, and p∈(0,1) is the price of producing the\ngood. Unless stated otherwise, we fix p= 1/2 in the rest of this discussion.\nWhen the graph is undirected, pure Nash equilibria always exist, and it is not hard to see\nthat they exactly correspond to maximal independent sets of the graph [ BK07 ]. In particular, a\npure Nash equilibrium can be computed efficiently.\nIn some settings, interactions are not necessarily symmetric. In that case, it is more suitable\nto model the interactions using a directed graph. A good example is the spread of information\nin social networks, which often have a directed structure. When the graph is directed, a pure\nNash equilibrium is no longer guaranteed to exist [ BK16 ] and Papadimitriou and Peng [ PP23 ]\nhave shown that it is NP-complete to decide whether one exists (and thus also to compute one).\nIn view of the non-existence of pure equilibria, it is natural to consider mixed Nash equilibria,\nwhich are guaranteed to exist even in the directed setting. Unfortunately, it turns out that mixed\nequilibria are also hard to compute, and even to approximate. Papadimitriou and Peng [ PP23 ]\nprove that it is PPAD -hard to compute an ε-well-supported approximate Nash equilibrium, for\nsome sufficiently small (unspecified) constant ε >0. To show this, they introduce a new type\nof game called threshold game and prove that computing an approximate equilibrium in such\na game reduces to the problem of computing an approximate equilibrium of a public goods\ngame. The hardness of computing approximate equilibria for threshold games is then shown\nby reducing from the generalized-circuit problem [ CDT09 ] which is known to be PPAD -hard to\nsolve approximately for some sufficiently small (unspecified) constant ε >0 [Rub18].\nRecent work by Deligkas et al. [ DFHM22 ] has introduced a new tool, called the Pure-\nCircuit problem, for proving PPAD -hardness for explicit large constant approximation factors.\nIn particular, using Pure-Circuit , they were able to show that computing an ε-approximate\nequilibrium of a threshold game is PPAD -hard for any ε <1/6. This is in fact tight, as there\nexists a simple polynomial-time algorithm that computes a 1 /6-approximate equilibrium.\nPlugging this strengthened inapproximability result for threshold games into the reduction\nof Papadimitriou and Peng yields an improved inapproximability result for public goods games.\nNamely, it follows that computing an ε-well-supported approximate Nash equilibrium of a binary\npublic goods game is PPAD -hard for any ε <1/48. In particular, the reduction from threshold\ngames to public goods games loses a factor 8 in the approximation term.\nOur contribution. In this paper, we prove that computing an ε-well-supported approximate\nNash equilibrium of a binary public goods game remains PPAD -hard for any ε <1/2. More\ngenerally, for any price parameter p∈(0,1), we show PPAD -hardness for any ε <min{p,1−p}\n(Theorem 3.1). This result is tight, as a trivial min{p,1−p}-well-supported approximate\nequilibrium can easily be found (see Lemma 2.2).\nTo prove our result, it is necessary to bypass threshold games, since the existing inapprox-\nimability for that problem is already tight. Instead, we provide a direct reduction from the\nPure-Circuit problem to our problem of interest.\n2 Preliminaries\nIn this section, we introduce the notions needed to state and prove our main theorem. Our\nresults are related to two domains of study: algorithmic game theory and the study of PPAD ,\nan important subclass of TFNP . Therefore this section is organized into two subsections. Firstly,\nwe define public goods games, along with the related game theoretic notions and the associated\n2computational problems. Secondly, we define the Pure-Circuit problem, which serves as the\nPPAD -complete problem used to demonstrate the main hardness result.\n2.1 Public Goods Games\nWe now define the important notions concerning public goods games, along with the associated\ngame theoretic notions and the relevant computational problems.\nDefinition 1 (Game) .Agame is defined by a set of players [ n] and a set of actions (pure\nstrategies) Aiavailable to each player i∈[n]. Each player additionally has a utility function\nui:A1× ··· × An→R, by which they obtain a payoff depending on the action choice of all the\nplayers. A pure strategy profile is a tuple a∈A1× ··· × An, representing the choice of actions\nfor each of the players.\nThroughout this work we use the standard game theoretic notation, with:\na−i= (a1, . . . , a i−1, ai+1, . . . , a n),\nand\n(k,a−i) = (a1, . . . , a i−1, k, a i+1, . . . , a n).\nDefinition 2. Anindivisible public goods game in a directed graph is given by a vertex set\nV= [n] corresponding to the players, a set Eof directed edges, as well as a price p∈(0,1).\nEach player has the same action (pure strategy) set A={0,1}, corresponding to the choice of\nwhether or not to produce the good. The payoff of player iwhen playing strategy kis:\nui(k,a−i) =\n\n1−pifk= 1\n1 if k= 0 and ∃j∈N(i) s.t. aj= 1\n0 otherwise,\nwhere the neighborhood for any player i∈Vis defined to be:\nN(i) ={j|(j, i)∈E}.\nDefinition 3 (Mixed strategy) .Amixed strategy is a player’s choice of a distribution over their\nset of actions. In a public goods game, this corresponds to a value si∈[0,1] assigned to each\nplayer representing the probability that player iwill produce the good.\nAstrategy profile is a choice of a mixed strategy (just strategy from hereon) for each of the\nplayers. In a public goods game, it is a tuple s∈[0,1]n= Σ (Σ denoting the set of all possible\nstrategy profiles).\nLemma 2.1. Given a mixed strategy profile s∈[0,1]n, the expected payoff of player iis\n˜ui(s) =si·˜ui(1,s−i) + (1 −si)·˜ui(0,s−i), where for k∈ {0,1}:\n˜ui(k,s−i) =(\n1−p k = 1\n1−Q\nj∈N(i)(1−sj)k= 0.\nProof. We consider the two cases corresponding to the choice of pure strategy kof player i.\nCase: k= 1.In this scenario, player iis unaffected by the choice of strategy of the players in\nN(i), and instead, the player benefits from the good while also incurring the cost of production.\nTherefore, the expected payoff is 1 −p.\n3Case: k= 0.In this scenario, player i’s expected payoff is the probability that at least one of\nthe players in N(i) produces the good. This corresponds to the complement of the probability\nthat none of the players in N(i) produce the good, which isQ\nj∈N(i)(1−sj).\nDefinition 4 (ε-best-response) .Given ε≥0 and a strategy profile for the other players s−i, we\nsay that a strategy s∈[0,1] is an ε-best-response of player itos−i, if it maximizes, up to ε, the\nexpected payoff given that the other players play s−i:\n˜u(s,s−i)≥max\ns′∈[0,1]˜u(s′,s−i)−ε.\nIfε= 0, then we simply use the term best-response . Note that there always exists a pure strategy\nk∈ {0,1}that is a best-response.\nDefinition 5 (Nash equilibria and approximations) .Consider a strategy profile s= (s1, . . . , s n)∈\n[0,1]n. We call saNash equilibrium if no player can improve their expected payoff by choosing\na different strategy. In other words, every player is playing a best-response:\n∀i∈[n],∀s′\ni∈[0,1],˜ui(s)≥˜ui(s′\ni,s−i).\nWe say that sis an ε-Nash equilibrium (ε-NE) if each of the players’ expected payoffs is within\nεof their best-response payoff, i.e., they all play an ε-best-response. That is:\n∀i∈[n],˜ui(s)≥max\nk∈{0,1}˜u(k,s−i)−ε.\nFinally, sis an ε-Well Supported Nash Equilibrium (ε-WSNE) if each player places a positive\nprobability exclusively on pure strategies that are ε-best-responses. That is:\n∀i∈[n],∀k∈supp(si), u i(k,s−i)≥max\nk′∈{0,1}˜u(k′,s−i)−ε,\nwhere supp(si) =n\nj∈ {0,1}: (si)j>0o\n.\nDefinition 6 (Utility difference) .The utility difference for a player iis the difference in the\nexpected payoff for a player between their choice of pure strategies:\n∆ui= ˜ui(1,s−i)−˜ui(0,s−i) =Y\nj∈Ni(1−sj)−p.\nObservation. In an ε-WSNE, ∆ uidictates precisely which strategy player iwould play. In\nparticular:\n•if ∆ui> ε, then strategy 1 is the only pure ε-best-response for player i, and therefore\nsi= 1.\n•if ∆ui<−ε, then strategy 0 is the only pure ε-best-response for player i, and therefore\nsi= 0.\n•if ∆ui∈[−ε, ε], then both strategies 0 and 1 are ε-best-responses for player i, and therefore\nsican be anything in [0 ,1].\nLemma 2.2. Ifε≥min{p,1−p}, the game admits a trivial ε-WSNE.\nProof. To demonstrate this, we consider the two sub-cases of the scenario where ε≥min{p,1−p}\nand show that a trivial equilibrium point exists in these situations. Namely:\n•ifε≥p=min{p,1−p}then it is impossible for ∆ ui<−εand therefore an ε-WSNE is\ntrivially obtained by having all the players produce the good.\n•ifε≥1−p=min{p,1−p}then it is impossible for ∆ ui> εand therefore an ε-WSNE is\ntrivially obtained with none of the players producing the good.\nThus, we obtain a trivial equilibrium in both cases.\n42.2 Pure-Cir cuit\nFor the purposes of the main result presented here, we consider instances of Pure-Circuit\nthat make use of the NOR andPURIFY gates. We now provide the corresponding definition of\nPure-Circuit .\nDefinition 7 (Pure-Circuit [DFHM22 ]).An instance of the Pure-Circuit problem is given\nby a set of nodes V= [n] and a set Gof gates. Each gate g∈Gis of the form g= (T, u, v, w )\nwhere u, v, w ∈Vare distinct nodes and T∈ {NOR,PURIFY }is the type of the gate, with the\nfollowing interpretation.\n•IfT=NOR, then uandvare the inputs of the gate, and wis its output.\n•IfT=PURIFY , then uis the input of the gate, and vandware its outputs.\nWe require that each node is the output of exactly one gate.\nA solution to an instance ( V, G) is an assignment x:V→ {0,1,⊥}that satisfies all the\ngates, i.e., for each gate g= (T, u, v, w )∈Gwe have:\n•ifT=NOR, then xsatisfies\nx[u] =x[v] = 0 = ⇒x[w] = 1\n(x[u] = 1) ∨(x[v] = 1) = ⇒x[w] = 0\n•ifT=PURIFY , then xsatisfies\n{x[v],x[w]} ∩ {0,1} ̸=∅\nx[u]∈ {0,1}=⇒x[v] =x[w] =x[u]\nWe now also state the central theorem from [ DFHM22 ] related to the Pure-Circuit problem,\nand necessary to our main result.\nTheorem 2.3 ([DFHM22 ]).ThePure-Circuit problem with NOR and PURIFY gates is\nPPAD -complete.\n3 Reduction from Pure-Cir cuit\nIn [DFHM22 , Section 4] reductions from the Pure-Circuit problem were demonstrated for\ntheε-generalized-circuit problem and for finding various equilibria points in polymatrix games,\nas well as threshold games. The last reduction is most relevant to this work, firstly because\nthe setting for threshold games is similar to the setting of public goods games (we can think\nof public goods games as a multiplicative version of threshold games), but also because it was\nprecisely threshold games which were used as an intermediate problem to show PPAD -hardness\nof public goods games in [PP23].\nWe proceed to provide the main result, to establish the optimal bound for ε, which constitutes\nthe main contribution of this work.\nTheorem 3.1. For any p∈(0,1)and any constant ε <min{p,1−p}, it is PPAD -hard to find\nanε-WSNE of binary public good games on directed graphs with price p.\nProof. We show that the problem is PPAD -hard for any constant ε <min{p,1−p}through a\nreduction from Pure-Circuit with the use of the NOR andPURIFY gates.\nConstruction. Consider an instance ( V, G) ofPure-Circuit . For each node u∈Vwe\nintroduce a corresponding node u∈V′, where V′is the set of nodes in the public goods graph.\nEach of the gates in the Pure-Circuit instance is implemented by adding corresponding edges\n5u\nvw\nNOR gadget\nu\nc1\nc2 c3 w c4 c5 c6···\nc2i−1 c2i c2l+1 c2l···\n···d v\nPURIFY gadget\nFigure 1: The gadgets used for the public goods instance in Theorem 3.1.\nand additional auxiliary nodes as shown in Figure 1. The NOR gadget is very simple, and it only\nintroduces two edges between the corresponding nodes. The PURIFY gadget is a generalization of\na similar construction appearing in [ DFHM22 ] in the context of threshold games. It introduces\nadditional auxiliary nodes d, c1, . . . , c 2l+1, as well as the edges denoted in the figure. We set\nl=\u0018log(p−ε)\nlog(p+ε)\u0019\n.\nLetsbe an ε-WSNE of the public goods game. For any u∈V′,su∈[0,1] denotes\nthe probability of player/node uproducing the good. We obtain an assignment xto the\nPure-Circuit instance as follows:\n∀u∈V,x[u] =(\nsuifsu∈ {0,1}\n⊥otherwise.\nCorrectness. We will now demonstrate that the assignment xobtained in this manner\nsatisfies the Pure-Circuit gate constraints.\nNOR gates. A (NOR, u, v, w ) gate is simulated by the NOR gadget depicted in Figure 1.\n•Ifsu=sv= 0, then in particular it means that ∆ uw= (1−su)(1−sv)−p= 1−p > ε .\nThis implies that action 1 is the only pure ε-best-response for w, and therefore in an\nε-WSNE we will have that sw= 1.\n•Ifsu= 1 or sv= 1, then we have that ∆ uw= (1−su)(1−sv)−p=−p <−ε. This\nimplies that action 0 is the only pure ε-best-response for w, and therefore in an ε-WSNE\nwe will have that sw= 0.\n6•In any other remaining case, whatever strategy wplays will map back to a value satisfying\ntheNOR gate.\nPURIFY gates. A (PURIFY , u, v, w ) gate is simulated by the PURIFY gadget depicted in\nFigure 1.\n•Ifsu∈ {0,1}, then for all j∈[l] it holds that sc2j=su. This is easy to show by repeatedly\napplying a similar argument to the one used above for the NOR gadget. Similarly, we also\ndeduce that sw=su. Furthermore, it holds that:\n∆ud= (1−su)Y\nj∈[l](1−sc2j)−p= (1−su)l+1−p=(\n1−p > ε ifsu= 0\n−p < −εifsu= 1.\nIt follows from this that sd∈ {0,1}andsd= 1−su, and thus sv=su. Together with\nsw=su, this means that the gate constraint is satisfied for this case.\n•Ifsu/∈ {0,1}, then we show that if sw/∈ {0,1}it must hold that sv= 1. Note that if\nsw/∈ {0,1}, then we must have that ∆ uw∈[−ε, ε]. This in turn implies that sc2l+1∈\n[1−p−ε,1−p+ε]⊂(0,1), since ε <min{p,1−p}. By repeatedly applying this argument\nwe deduce that su, sc2j∈[1−p−ε,1−p+ε], for all j∈[l]. As a result, we have that:\n∆ud= (1−su)Y\nj∈[l](1−sc2j)\n| {z }\n∈[(p−ε)l+1,(p+ε)l+1]−p <(p+ε)l−p≤(p+ε)log(p−ε)\nlog(p+ε)−p=−ε.\nwhere we used the fact that 0 < p−ε≤p+ε <1. It follows from above that 0 is the only\npure ε-best-response for d, and therefore sd= 0, which in turn implies that sv= 1, as\ndesired.\nAs shown above, we have that an ε-WSNE in the game correctly encodes the gate constraints\nimposed by the NOR andPURIFY gates.\n4 Open Problems\nWe conclude with some natural open questions that arise from our work:\n•Is it possible to obtain similar tight inapproximability results for ε-Nash equilibria?\n•The problem of computing an exact Nash equilibrium lies in the class FIXP introduced by\nEtessami and Yannakakis [ EY10 ]. Is it FIXP-complete? Or does the problem always admit\nrational equilibria?\n���Is it possible to prove the hardness result with the use of a gadget whose size does not\ndepend on how close εis to min{p,1−p}? In particular, is it possible to prove hardness\nfor the problem with p= 1/2 and for any ε <1/2 for graphs with bounded degree? Are\npositive results possible?\nAcknowledgements\nWe thank Mika G¨ o¨ os for helpful discussions in the early stages of this project. The second\nauthor was supported by the Swiss State Secretariat for Education, Research and Innovation\n(SERI) under contract number MB22.00026.\n7References\n[All15] Nizar Allouch. On the private provision of public goods on networks. Journal of\nEconomic Theory , 157:527–552, 2015. doi:10.1016/j.jet.2015.01.007 .\n[BK07] Yann Bramoull´ e and Rachel Kranton. Public goods in networks. Journal of Economic\nTheory , 135(1):478–494, 2007. doi:10.1016/j.jet.2006.06.006 .\n[BK16] Yann Bramoull´ e and Rachel Kranton. Games played on networks. In The Oxford\nHandbook of the Economics of Networks , pages 83–112. Oxford University Press,\n2016. doi:10.1093/oxfordhb/9780199948277.013.8 .\n[BKD14] Yann Bramoull´ e, Rachel Kranton, and Martin D’Amours. Strategic interaction and\nnetworks. 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Proceedings of the 34th AAAI\nConference on Artificial Intelligence (AAAI) , pages 2310–2317, 2020. doi:10.1609/\naaai.v34i02.5609 .\n9"    },    {        "title": "2402.14204v1.A_sufficient_condition_for_the_height_function_to_be_constant_in___I_g_times_ρ_mathbb_P__n__.pdf",        "content": "arXiv:2402.14204v1  [math.DG]  22 Feb 2024A SUFFICIENT CONDITION FOR THE HEIGHT FUNCTION TO BE CONSTANT\nINIg×ρPn\nKAIJIAN CAO\n1.abstract\nThis paper makes some modifications to the warped product spa ce. Based on Alias,Impera and Rigoli, a\nwarping function is added to the warped product space. This n ew function affects the Riemannian metric\nof the warped product space. In this new warped product space , we continue to discuss the sufficient\ncondition for calculating the height of the immersed surfac e.\n2.Preparation\n2.1.warped product space Ig×ρPn.\nwhen we talk about Ig×ρPn，amongI⊂Ris an open interval ，Pnis a complete n-dimensional Rie-\nmannian manifold,\nρ:I→R+\ng:Pn→R+\nBoth are positive warping functions.\nThe Riemannian metric for such a warped product space Ig×ρPnis：\n<,>=g2(πP)π⋆\nI(dt2)+ρ2(πI)π⋆\nP(<,>P),\nHereπI:Ig×ρPn→I，πP:Ig×ρPn→Prepresents the natural projection to its two subspaces of\nIg×ρPnrespectively.\n2.2.Some concepts.\nDefinition2.2.1 :Letf: Σn→¯Mn+1be an isometric immersion from Σnto¯Mn+1,and∀p∈Σ,X,Y∈TpΣ\nwe have：\nA:TpΣ→TpΣ\n< AX,Y > =−<¯∇XN,Y >,\nthenAis the shape operator with respect to unit normal field N，¯∇is Riemannian connection of ¯Mn+1.\nLetλ1,...,λ nbe the eigenvalue of A，记e1,...,enare the corresponding orthonormal eigenvectors. The\nsymmetric function is given by the eigenvalues as follows ：\nS0= 1\nSk= Σi1<...ikλi1λi2...λik,1≤k≤n\nSk= 0, k > n.\nDefinition2.2.2 :Under the above assumptions ，k−mean curvature Hkis defined as follows 定义如下：\nHk=Sk\nCkn.\nDate: February 23, 2024.\nKey words and phrases. warped product space; Omori-Yau maximum principle; Newton tensor; 2-mean curvature.\n12 KAIJIAN CAO\nDefinition2.2.3 :Under the above assumptions ，Nweton tensor is defined as follows:\nP0=I,\nPk=SkI−APk−1, k= 1,2,...,n.\nIt’s easy to see that Pk=SkI−Sk−1A+...(−1)kAkΘ and obviously Pkis a symmetric linear map.\nDefiniton2.2.4 :\nLk:C∞(Σ)→C∞(Σ)\nLk=Tr(Pk◦hess).\n2.3.Some properties.\nProposition2.3.1 :∀1≤k≤n, ifH1,H2,...,H n>0, then:\nHk−1Hk+1≤H2\nk.\nProposition2.3.2 :∀1≤k≤n, ifH1,H2,...,H n>0, then:\n(1)H2\n1≥H2;\n(2)H1≥H1\n2\n2≥H1\n3\n3≥ ··· ≥H1\nnn.\nProposition2.3.3 : IfH2>0, thenL1=Tr(P1◦hess) is elliptic operator.\n3.A generation of Omori-Yau maximum principle\n3.1.Omori-Yau Maximum Principle.\nDefinition3.1.1 :Let Σ be n-dimensional Riemannian manifold, if ∀u∈C2(Σ),u⋆= supΣu <+∞,there\nexists{pj} ⊂Σ such that：\n(1)u(pj)> u⋆−1\nj\n(2)||∇u(pj)||<1\nj\n(3)∆u(pj)<1\nj\nthen we call Omori-Yau maximum principle hold for ∆ in Σ.\nEquivalently, we can also describe another equivalent desc ription，if∀u∈C2(Σ),u⋆= infΣu >−∞,\nthere exists {pj} ⊂Σ such that：\n(1)u(pj)< u⋆+1\nj\n(2)||∇u(pj)||<1\nj\n(3)∆u(pj)>−1\nj\nthen Omori-Yau maximum principle hold for ∆ in Σ.\nDefinition3.1.2 :Let Σ ba a n-dimensional Riemannian manifold ，L=Tr(P◦hess) is a semidefinite\nelliptic operator，P:TΣ→TΣ is a semidefinite symmetric operator satisfying supΣTr(P)<+∞. IfA SUFFICIENT CONDITION FOR THE HEIGHT FUNCTION TO BE CONSTAN T INIg×ρPn3\n∀u∈C2(Σ),u⋆= supΣu <+∞,there exists {pj} ⊂Σsuch that：\n(1)u(pj)> u⋆−1\nj\n(2)||∇u(pj)||<1\nj\n(3)Lu(pj)<1\nj\nthen下Omori-Yau maximum principle holds for Lin Σ.\n3.2.A sufficient condition for 下 下 下Omori-Yau maximum principle under L.\nTheorem3.2.1 :Let (Σ,<>) be a Riemannian manifold, L=Tr(P◦hess) is a semidefinite elliptic oper-\nator，P:TΣ→TΣ is a semidefinite symmetric operator satisfying supΣTr(P)<+∞. If there exists\nnon-negative function γ∈C2(Σ) such that:\n(1)γ(p)→+∞, p→ ∞\n(2)∃A >0,s.t.||∇γ|| ≤Aγ1\n2,out side a compact set\n(3)∃B >0,s.t.Lγ≤Bγ1\n2G1\n2(γ1\n2),out side a compact set .\nGis a smooth function in [0 ,+∞) satisfying:\n(i)G(0)>0\n(ii)G′(t)≥0,in [0,+∞)\n(iii)G−1\n2(t)/∈L1[0,+∞)\n(iv)limsup\nt→+∞tG(t1\n2)\nG(t)<+∞.\nthen Omori-Yau maximum principle holds for Lin Σ.\nUsing theorem 3.2.1 ，we want Omori-Yau maximum principle holds in Σ, so naturally wo need to find\nappropriate γandG.\nWe give some notations for convenience, Let(Σ ,<>) be a complete non-compact Riemannian manifold,\no∈Σ is a fixed point, r(p) is the distance function starting from otop,γ(p) =r2(p).\nTheorem3.2.2 :Let (Σ,<>) be a complete non-compact Riemannian manifold, satisying Krad\nΣ≥ −G(r),\nhereKrad\nΣrepresents radial sectional curvature(sectional curvatu re including ∇r),Gis the function which\nsatisfies previous theorems and smooth at 0.\n3.3.Omori-Yau maximum principle in Ig×ρPn.\nHere we consider Pnto be a complete non-compact Riemannian manifold ，o∈Pnis the fixed point ，ˆr\nrepresents distance fuction in Pnstarting from fixed point o.\nFollowing the previous setting for radial sectional curvat ure, we set:\nKrad\nP≥ −G(ˆr),\nhereGis the function which satisfies previous theorems and smooth at 0.\nfis a properly immersed hypersurface, f(Σ)⊂[t1,t2]×Pn.4 KAIJIAN CAO\nTo obtain the Omori-Yau maximum principle under some operat or, we just need to use the previous\ntheorem 3.2.1 to verify that our selected γsatisfies the following three conditions:\n(1)γ(p)→+∞, p→ ∞\n(2)∃A >0,s.t.||∇γ|| ≤Aγ1\n2,outside a compact set\n(3)∃B >0,s.t.Lγ≤Bγ1\n2G1\n2(γ1\n2),outside a compact set .\n其中Gis a smooth function in [0 ,+∞) satisfying:\n(i)G(0)>0\n(ii)G′(t)≥0,in [0,+∞)\n(iii)G−1\n2(t)/∈L1[0,+∞)\n(iv)limsup\nt→+∞tG(t1\n2)\nG(t)<+∞.\n(1) Because fis proper, so f(p)→ ∞(p→ ∞), andf(Σ)⊂[t1,t2]×P, thus ˆr2(πP◦f(p))→+∞(p→ ∞),\nthat isγ(p) = ˜γ(f(p)) = ˆγ(πP◦f(p)) = ˆr2(πP◦f(p))→+∞(p→ ∞).\n(2) We use ˜∇,ˆ∇,∇represent Riemannian connections in Ig×ρPn,P,Σ respectively, γ(p) = ˜γ◦f(p),:\n˜∇˜γ=∇γ+<˜∇˜γ,N > N\nNis unit normal field in immersed hypersurface f.\nAnd ˜γ◦f(p) = ˆγ(πP◦f(p)) has nothing to do with I, so:\n<˜∇˜γ,T >= 0\nhereTis pullback of∂\n∂tfromTItoT(Ig×ρPn).\nFrom the gradient vector field transformation relation:\n<˜∇˜γ,V >=<ˆ∇ˆγ,V > P\nhereVis pullback from TPtoT(Ig×ρPn).\nAnd according to the corresponding relationship of measure ment:\n<˜∇˜γ,V >=ρ2<˜∇˜γ,V > P,\nso˜∇˜γ=1\nρ2ˆ∇ˆγ=2ˆr\nρ2ˆ∇ˆr,thus||ˆ∇ˆr||=ρ||ˆ∇ˆr||P=ρ≥min[t1,t2]ρ >0,that is||∇γ|| ≤ ||˜∇˜γ||=2γ1\n2\nρ≤cγ1\n2.\n(3)\nHessγ(X,X) =Hess˜γ(X,X)+<˜∇˜γ,N >< AX,X >, X ∈TΣ\n˜∇T˜∇˜γ=−H˜∇˜γ+1\nρ4g2<ˆ∇ˆγ,gˆ∇g > T(H=ρ′\nρ)\nHess˜γ(T,T) =g\nρ2<˜∇˜γ,ˆ∇g >=g\nρ4<ˆ∇ˆγ,ˆ∇g > .\nX∈TΣ is decomposed as follows:\nX=X⋆+1\ng2< X,T > T\nhereX=dπP(X).A SUFFICIENT CONDITION FOR THE HEIGHT FUNCTION TO BE CONSTAN T INIg×ρPn5\nthen:\nHess˜γ(X,X) =Hess˜γ(X⋆,X⋆)+2< X,T >\ng2Hess˜γ(X⋆,T)+Hess˜γ(T,T)\ng4< X,T >2\nand:\nHess˜γ(X⋆,T) =−H<∇γ,X > .\nwe use:\n˜∇X⋆˜∇˜γ=1\nρ2ˆ∇X⋆ˆ∇ˆγ−ρ′\nρ3g2<ˆ∇ˆγ,X⋆> T\nand:\nHess˜γ(X⋆,X⋆) =1\nρ2<ˆ∇X⋆ˆ∇ˆγ,X⋆>\n=<ˆ∇X⋆ˆ∇ˆγ,X⋆>P\n=Hessˆγ(X⋆,X⋆).\nSo we simplify above by combining:\nHessγ(X,X) =Hess˜γ(X,X)+<˜∇˜γ,N >< AX,X >\n=Hess˜γ(X⋆,X⋆)+2< X,T >\ng2Hess˜γ(X⋆,T)+Hess˜γ(T,T)\ng4< X,T >2+<,˜∇˜γ,N >< AX,X >\n=Hessˆγ(X⋆,X⋆)−2< X,T >\ng2H<∇,X >+<ˆ∇ˆγ,ˆ∇g >\ng3ρ4< X,T >2+<˜∇˜γ,N >< AX,X > .\nComputing Hessˆγ(X⋆,X⋆) by Hessian comparison theorem:\nHessˆγ(X⋆,X⋆)≤(γG(γ1\n2))1\n2||X||2,\nand|2< X,T >\ng2<∇γ,X >|H ≤2||∇||||X||2\ngHc\ngγ1\n2||X||2.\nAnd we give following restrictions on g:\ng≥1,<,ˆ∇ˆγ,ˆ∇g >= 0\nthenHessγ(X,X)≤c(γG(γ1\n2))1\n2||X||2+<˜∇˜γ,N >< AX,X > .\nThe discussion is divided into the following two operators:\na. When the operator is ∆, suppose supΣ|H|<+∞:\n∆γ=cn(γG(γ1\n2))1\n2+nH <˜∇˜γ,N >,\n|H <˜∇˜γ,N >| ≤supΣ|H|cγ1\n2≤Cγ1\n2≤C(γ1\n2))1\n2γis suffcient large\nSo there must be some compact set, and outside that compact se t there is δγ≤C(γ1\n2))1\n2\nb.When the operator is L, suppose supΣ||A||2<+∞:\n|<˜∇˜γ,N >< AX,X > | ≤ ||˜∇˜γ||||A||||X||2≤C(γ1\n2))1\n2||X||2γsufficient large\nThe following theorem is obtained by summarizing the above d iscussion:\nTheorem3.3.1 :LetPnbe a complete non-compact Riemannian manifold, its radial s ectional curvature\nsatisfiesKrad\nP≥ −G(ˆr)，f: Σ→Ig×ρPis properly immersed hypersurface, and f(Σ)⊂[t1,t2]×Pn,g≥\n1,<ˆ∇ˆγ,ˆ∇g >= 0，if supΣ|H|<+∞, then Omori-Yau maximum principle holds for ∆ in Σ. 。\nTheorem3.3.2 :LetPnbe a complete non-compact Riemannian manifold, its radial s ectional curvature\nsatisfiesKrad\nP≥ −G(ˆr)，f: Σ→Ig×ρPis a properly immersed hypersurface, and f(Σ)⊂[t1,t2]×Pn,g≥6 KAIJIAN CAO\n1,<ˆ∇ˆγ,ˆ∇g >= 0，if supΣ||A||<+∞，then for all semidefinite elliptic operator L=Tr(P◦hess),\nsupΣTr(P)<+∞, Omori-Yau maximum principle holds for Lin Σ.\n4.Conclusion\nProposition4.1.1 :Letf: Σ→Ig×ρPbe an isometric immersion, h=πI◦fis the height function,\nσ(t) =/integraltextt\nt0ρ(u)du,< P k∇h,(ˆ∇1\ng2)T>= 0, then：\nLkh=ρ′\ng2ρ(ckHk−g2< Pk∇h,∇h >)+ckθHk+1\ng2\nLkσ(h) =ρ′\ng2ckHk+ckρθHk+1\ng2\n其中ck= (n−k)Ck\nn= (k+1)Ck+1\nn,θ=< N,T > .\nProof.¯∇πI=T\ng2,then∇h= (¯∇πI)T=1\ng2(T−θN), soTT=T−θN=g2∇h.\nUsing¯∇VT=¯∇TV=1\nρ2g2< V,gˆ∇g > T+(Tlinρ)V, V∈TP.\n¯∇TT=−gˆ∇g\nρ2\nSo\n¯∇XT=¯∇X⋆+1\ng2<X,T>TT\n=−1\nρ2g< X,T > ˆ∇g+1\nρ2g2< X⋆,gˆ∇g > T+(Tlnρ)X⋆\nhereX∈T(Ig×ρP).\nNow we discuss X∈TΣ:\n¯∇X∇h=¯∇X(1\ng2(T−θN))\n=X(1\ng2)(T−θN)+1\ng2¯∇XT−X(θ)N\ng2+θ\ng2AX.\nSo we have:\nhessh(X) =∇X∇h= (¯∇X∇h)T\n=g2X(1\ng2)∇h−1\nρ2g< X,∇h >(ˆ∇g)T+1\nρ2g< X,(ˆ∇g)T>∇h+ρ′\nρg2X⋆+θ\ng2AX.\nSo we can calculate Lkh,Lkσ(h) as follows:\nLkh=Tr(Pk◦hessh) =/summationdisplay\ni< Pk◦hessh(ei),ei>\n=g2< Pk∇h,(ˆ∇1\ng2)T>+θ\ng2Tr(PkA)+ρ′\nρg2Tr(Pk)−ρ′\nρ< Pk∇h,∇h >\n=ρ′\ng2ρ(ckHk−g2< Pk∇h,∇h >)+ckθHk+1\ng2.A SUFFICIENT CONDITION FOR THE HEIGHT FUNCTION TO BE CONSTAN T INIg×ρPn7\nLkσ(h) =Tr(Pk◦hessσ(h))\n=/summationdisplay\ni< ei(ρ)∇h,Pkei>+ρ/summationdisplay\ni<∇ei∇h,Pkei>\n=< Pk∇h,ρ′∇h >+ρLkh\n=ρ′\ng2ckHk+ckρθHk+1\ng2.\n/square\nProposition4.1.2 :Letf: Σ→Ig×ρPbe a hypersurface whose mean curvature is not zero every-\nwhere，f(Σ)⊂[t1,t2]×P, ifH ≥0,<∇h,(ˆ∇1\ng2)T>= 0, and the angle function θdoesn’t change sign ，\nthenafterchoosingasuitabledirection inΣsuchthat H1>0, weknowthat Omori-Yaumaximumprinciple\nholds for ∆ in Σ, then:\n(1) ifθ≤0,thenρ′≥0\n(2) ifθ≥0,thenρ′≤0\nProof.Because Omori-Yau maximum principle holds for ∆ in Σ, We can fi nd a list of points {pj} ⊂Σ\nsatisfying:\nlim\nj→+∞h(pj) =h⋆= inf\nΣh\n||∇h(pj)||2=1\ng4(g2−θ2)<(1\nj)2\n∆h(pj) =ρ′\nρg2(n−g2||∇h(pj)||2)+nθH1\ng2>−1\nj\nthen−nθH1\ng2<1\nj+ρ′\nρg2(n−g2||∇h(pj)||2)\nAnalogously, for h⋆we still have {qj} ⊂Σ satisfying:\nlim\nj→+∞h(qj) =h⋆= sup\nΣh\n||∇h(qj)||2=1\ng4(g2−θ2)<(1\nj)2\n∆h(qj) =ρ′\nρg2(n−g2||∇h(qj)||2)+nθH1\ng2<1\nj\nthen−nθH1\ng2>−1\nj+ρ′\nρg2(n−g2||∇h(qj)||2)\nThere are two conditions to consider:\n(1)θ≤0:\nThen−θ(pj)≥0,H1(pj)>0,using−nθH1\ng2<1\nj+ρ′\nρg2(n−g2||∇h(pj)||2)，letj→ ∞we have ρ′≥0.\n(2)θ≥0:\nThenθ(qj)≥0,H1(qj)>0,using−nθH1\ng2>−1\nj+ρ′\nρg2(n−g2||∇h(qj)||2), letj→ ∞we have ρ′≤0.\n/square\n4.1.A sufficient condition for the height function to be constant in Ig×ρPn.8 KAIJIAN CAO\nDefineξ1= (n−1)ρ′\nρ∆−θL1, soξ1σ(h) =n(n−1)ρ\ng2(ρ′2\nρ2−θ2H2), thenξ1=Tr(ζ1◦hess), hereζ1=\n(n−1)ρ′\nρI−θP1.\nTheorem4.2.1 :Letf: Σ→Ig×ρPbe an compact immersed hypersurface and H2>0, ifH ≥0,θdoesn’t\nchange sign, < P1∇h,(ˆ∇1\ng2)T>=<∇h,()ˆ∇1\ng2T\n>= 0, and for maximum and minimum points pmax,pmin\nofhwe have g(pmax) =g(pmin),θ(pmax) =θmax,θ(pmin) =θmin. Thenhis a constant function.\nProof.(1)θ≤0,ρ′≥0:\nζ1ei= [(n−1)ρ′\nρ−θ(S1−λi)]ei，obviouslythisisasemi-definiteoperator ，||∇h(pmax)||=||∇h(pmin)||=\n0, so:\nθ(pmax) =−g(pmax)\nθ(pmin) =−g(pmin)\nθ(pmax) =θ(pmin).\nAndσis monotonically increasing:\n(σ◦h)⋆= max\nΣσ◦h=σ(h⋆) =σ(h(pmax))\n(σ◦h)⋆= min\nΣσ◦h=σ(h⋆) =σ(h(pmin))\nBecause of the extreme point property:\nHessσ(h)(pmax)≤0\nHessσ(h)(pmin)≥0\nSo:\nξ1σ(h(pmax)) =n(n−1)ρ\ng2(h⋆)(ρ′2\nρ2(h⋆)−θ2(pmax)H2)≤0\nξ1σ(h(pmin)) =n(n−1)ρ\ng2(h⋆)(ρ′2\nρ2(h⋆)−θ2(pmin)H2)≥0\nThusH(h⋆) =H(h⋆)，His a constant function.\nSo we have：\nL1(σ◦h) =n(n−1)ρ\ng2(ρ′\nρH1+θH2)\n≥n(n−1)ρ\ng2(ρ′\nρH1+θminH2)\n=n(n−1)ρ\ng2H1\n2\n2g(pmin)(H1−H1\n2\n2)≥0\nFrom the extreme value principle of elliptic operators, his a constant function.\n(2)θ≥0,ρ′≤0 has the same discussion, we only need to consider −ζ1.\n/square\nTheorem4.2.2 : Letf: Σ→Ig×ρPbe an compact immersed hypersurface and H2>0，Krad\nΣ≥\n−G(r),Krad\nΣrepresents radial sectional curvature, Gis a smooth function that satisfies previous theorems.\nIf supΣ|H1|<+∞,f(Σ)⊂[t1,t2]×P,H′>0,θdoesn’t change sign and attains its maximum and\nminimum, and there exists pmax,pmin∈Σ such that θ(pmax) =θmax,θpmin=θmin,limp→∞g(p) = 1,<\nP1∇h,(ˆ∇1\ng2)T>=<∇h,(ˆ∇1\ng2)T>= 0, then his a constant function.A SUFFICIENT CONDITION FOR THE HEIGHT FUNCTION TO BE CONSTAN T INIg×ρPn9\nProof.(1)θ≤0,ρ′≥0:\nζ1is asemidefinite operator Tr(ζ1) =n(n−1)ρ′\nρ−n(n−1)θH1≤n(n−1)(ρ′\nρ(h⋆)−θminH⋆\n1)<+∞,H⋆\n1=\nsupΣH1<+∞.\nFrom theorem3.2.2, Omori-Yau maximum principle holds for ξ1in Σ.\nSo we can find a list of points {pj} ⊂Σ such that:\nlim\nj→+∞(σ◦h)(pj) = sup\nΣ(σ◦h) =σ(h⋆)<+∞\n||∇(σ◦h)(pj)||=ρ(h(pj))||∇h(pj)||<1\nj\nξ1(σ◦h)(pj)<1\nj\nLetj→+∞we have H(h⋆)≤H2.\nEquivalently from Omori-Yau maximum principle we have a lis t of pioints {qj} ⊂Σ such that:\nlim\nj→+∞(σ◦h)(qj) = inf\nΣ(σ◦h) =σ(h⋆)\n||∇(σ◦h)(qj)||=ρ(h(qj))||∇h(qj)||<1\nj\nξ1(σ◦h)(qj)>−1\nj\nthenH(h⋆)≥H2\nThusHis a constant function, H′>0, sohis a constant function.\n(2)θ≥0,ρ′≤0 is the same as (1) by replacing operator ζ1by−ζ1. /square\nAcknowledgment. I am grateful to Professor Hengyu Zhou(Chongqing Universit y) for many useful dis-\ncussions and suggestions.\nReferences\n[1] Rosenberg H. Hypersurfaces of constant curvature in spa ce forms[J]. Bull. Sci. Math, 1993, 117(2): 211-239.\n[2] Marques Barbosa J L, Colares A G. Stability of hypersurfa ces with constant-mean curvature[J]. Annals of Global Anal ysis\nand Geometry, 1997, 15(3): 277-297.\n[3] Cheng X, Rosenberg H. Embedded positive constant r-mean curvature hypersurfaces in Mm ×R[J]. Anais da Academia\nBrasileira de Ciˆ encias, 2005, 77: 183-199.\n[4] Garc´ ıa-Mart´ ınez S C, Impera D, Rigoli M. A sharp height estimate for compact hypersurfaces with constant k-mean\ncurvature in warped product spaces[J]. Proceedings of the E dinburgh Mathematical Society, 2015, 58(2): 403-419.\n[5] Koutroufiotis D. Elementary geometric applications of a maximum principle for nonlinear elliptic operators[J]. Ar chiv der\nMathematik, 1973, 24: 97-99.\n[6] de Lima E L, de Lima H F. Height estimates and topology at in finity of hypersurfaces immersed in a certain class of\nwarped products[J]. Aequationes mathematicae, 2018, 92: 7 37-761.\n[7] Fontenele F, Silva S L. A tangency principle and applicat ions[J]. Illinois Journal of Mathematics, 2001, 45(1): 213 -228.\n[8] Al´ ıas L, Impera D, Rigoli M. Hypersurfaces of constant h igher order mean curvature in warped products[J]. Transact ions\nof the American Mathematical Society, 2013, 365(2): 591-62 1.\n[9] Brendle S. Constant mean curvature surfaces in warped pr oduct manifolds[J]. Publications math´ ematiques de l’IH ´ES,\n2013, 117(1): 247-269.\n[10] Chen B Y. Differential geometry of warped product manifo lds and submanifolds[M]. 2017.\n1College of Mathematics and Statistics, Chongqing Universi ty, Chongqing, 401331, P. R. China\nEmail address :20175785@cqu.edu.cn"    },    {        "title": "2402.14235v1.Lagrangian_partition_functions_subject_to_a_fixed_spatial_volume_constraint_in_the_Lovelock_theory.pdf",        "content": "Lagrangian partition functions subject to a fixed\nspatial volume constraint in the Lovelock theory\nMengqi LuaRobert B. Manna,b\naDepartment of Physics and Astronomy, University of Waterloo,\nWaterloo, Ontario N2L 3G1, Canada\nbPerimeter Institute for Theoretical Physics,\n31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada\nE-mail: mengqi.lu@uwaterloo.ca\nAbstract: We evaluate the quantum gravity partition function that counts the dimension\nof the Hilbert space of a simply connected spatial region of fixed proper volume in the\ncontext of Lovelock gravity, generalizing the results for Einstein gravity [1]. We find that\nthere exists sphere saddle metrics for a partition function at a fixed spatial volume in\nLovelock theory. Those stationary points take exactly the same forms as in Einstein gravity.\nThe logarithm of Zcorresponding to a zero effective cosmological constant indicates the\nBekenstein-Hawking entropy of the boundary area and the one corresponding to a positive\neffective cosmological constant points to the Wald entropy of the boundary area. We\nalso observe the existence of zeroth order phase transitions between different vacua, a\nphenomenon distinct from Einstein gravity.arXiv:2402.14235v1  [hep-th]  22 Feb 2024Contents\n1 Introduction 1\n2 Towards Lovelock theory 3\n3Imin\nEfor a vanishing cosmological constant 6\n4Imin\nEfor a positive effective cosmological constant 7\n5 Phase transitions between different vacua 8\n6 Conclusions 10\n1 Introduction\nThe Euclidean action approach to the gravitational partition function Zwas originally\nformulated by Gibbons and Hawking [2]. They evaluated the path integral over spacetime\ngeometries by the saddle point approximation and found that the logarithm of Zfor a de\nSitter (dS) spacetime is a quarter of the cosmological horizon area, indicating that (despite\nthe absence of an event horizon) the corresponding thermodynamic potential was the dS\nentropy, analogous to the Bekenstein-Hawking entropy of a black hole [3, 4]. This justified\nthe general holographic nature of gravitational entropy, the concept of which is not limited\nto a black hole horizon.\nHowever interpreting these results involves addressing nuanced inquiries. In the con-\ntext of statistical mechanics, the thermodynamic equilibrium of a system is determined by\nextremizing a thermodynamic potential. Without any constraint, the concept of thermo-\ndynamic equilibrium loses its generality since we can hardly guarantee an unconstrained\nsystem to have extrema in its thermodynamic potentials. This gives rise to the question\nas to what constraint other than fixing the temperature is imposed implicitly in specify-\ning the canonical ensemble for the dS vacuum. Furthermore, the interpretation of the dS\nentropy remains ambiguous [5]. For a Euclidean dS space, the quasi-local energy vanishes\nidentically due to the absence of a boundary. Given the vanishing Hamiltonian, Fischler\n[6] and Banks [7] further argued that the finite de Sitter entropy equates to the logarithm\nof the dimension of the Hilbert space since the partition function reduces to the trace of\nthe identity operator. This assertion has since received several lines of support [8–13].\nFurthermore, the concept of gravitational entropy requires a general description since it\ncan be associated with the area of any boundary separating a region of space [14, 15], and\nnot only to the area of black hole or dS horizons.\n– 1 –Recently, Jacobson and Visser [1] considered some of these issues by adding a spatial\nvolume constraint to the Euclidean path integral. They proposed a partition function with\na Lagrange multiplier λ(τ) of the form\nZ=ZZ\nDλDgexp\u0012\n−IE+Z\ndτλ(τ)ˆC\u0013\n, (1.1)\ninDspacetime dimensions, where the explicit expression ˆCis implemented by\nˆC=Z\ndD−1x√γ−V, (1.2)\nand the action IEis the Euclidean Einstein-Hilbert term plus a bare cosmological constant\nΛ0,\nIE=−1\n16πZ\nMEdτdD−1x√g(R−2Λ0). (1.3)\nThe first term in the exponential in (1.1) is performed on a Euclidean manifold MEof\ndimension Dwith a periodic boundary condition on the imaginary time x0=τ. The\nsecond term imposes the volume constraint on each constant- τslice with induced metric\nγij. The temperature, as the reciprocal of the period βinτ, arises from by eliminating\nthe conical singularity at the horizon. Thus the fixed temperature, or the size of ∂MEin\ntheτdirection, plus the fixed volume define a canonical ensemble. The partition function\n(1.1) should be considered a provisional prescription for manifolds MEwith closed spatial\nsections; otherwise a Hawking-Gibbons-York (HGY) term and counterterms should be\nincluded. Together with the closedness in the τdirection, the desired saddle manifold\nshould have a closed topology similar to the original Euclidean dS space.\nThe partition function (1.1) was shown [1] to admit spherical saddle metrics of the\nform\nds2\nE=h(r)dτ2+1\nf(r)dr2+r2dΩ2\nD−2 (1.4)\nfor both zero and nonzero Λ 0. Here dΩ2\nD−1is the induced metric on a D−2 dimensional\nsphere. Two metric components gττ=handgrr= 1/fare some radial dependent functions\nthat solve the field equations, with hchosen to satisfy the boundary condition\nd√\nh\ndl\f\f\f\f\nl=0= 2πT=2π\nβ(1.5)\nwhere lis the radial proper distance from the Euclidean horizon. When such a stationary\npoint dominates, the partition function can be approximated by the minimum saddle ac-\ntionImin\nE, thus yielding Z≈exp (−Imin\nE) due to the vanishing of ˆCfor any saddles. The\nlogarithm of Zthen turns out to be the Bekenstein-Hawking entropy, i.e., one quarter of\nthe horizon area of the boundary. Hence, the dS partition function can be understood as\na special case of (1.1) where the boundary of Vmatches that of the dS horizon. Further-\nmore, this result not only justifies the interpretation of the dS entropy as the measure of\nthe dimension of Hilbert space, but also generalizes the description of entropy (originally\nassociated with an area) to a volume separated by a boundary.\n– 2 –Motivated by this approach, we investigate here how these conclusions are modified in\nhigher curvature theories of gravity. In the low energy effective action of whatever theory\nprovides the UV completion to General Relativity, such theories are generically expected\nto appear, and are known to modify the entropy of a black hole [16]. It is therefore natural\nto inquire as to how they affect the entropy of a volume of space. For simplicity, we\nconsider Lovelock gravity since the quasi-local energy in Lovelock theory is given by an\nintegral on the system’s boundary [17], affording the same interpretation of dS entropy as\nthe dimension of Hilbert space as for GR [6, 7].\n2 Towards Lovelock theory\nThe only change we make to (1.1) is the replacement of IEby the general Lovelock action\n[18],\nIE=1\n16πX\np=0Z\nMEαp\n(D−2p)!R(p)(2.1)\nwhere the αpare coupling constants. The D-form R(p)is given by the antisymmetrization\nof exterior products of pcurvature 2-forms1Rab, with D−2pvielbeins ea, which reads\nR(p)=ϵa1···aDRa1a2···a2p−1a2p∧ea2p+1···aD, (2.2)\nwhere the total antisymmetric tensor ϵis defined by ϵ0···(D−1)= +1 in an orthonormal\nbasis. For convenience we write\nRa1a2···a2n−1a2n:=Ra1a2∧ ··· ∧ Ra2n−1a2n\nea1a2···an:=ea1∧ ··· ∧ ean.\nThe zeroth and the first order couplings are set to 2Λ 0and−1 respectively for consistency\nwith (1.3), rendering (2.1) a higher curvature modification of GR.\nThe Lovelock action can be easily evaluated for the ansatz (1.4) in the language of\ndifferential forms. In the vielbein frame defined by\ne0=√\nhdt, e1=1√fdr, e2=rdx2, ei=ri−1Y\nξ=2sinxξdxi(i≥3),\nthe curvature 2-forms can be directly obtained by manipulating those vielbeins with Cartan\nformulae for zero-torsion and metric compatible geometries. The non-vanishing and non-\nrepeating components for Rabare then\nR01=−r\nf\nh\u0012r\nf\nhh′\n2\u0013′\ne0∧e1, R0i=−fh′\n2rhe0∧ei, (2.3)\nR1i=−f′\n2re1∧ei, Rij=1−f\nr2ei∧ej\n1Rabis antisymmetric in the two indices.\n– 3 –where′denotes the derivative with respect to r, numbers are used for all coordinate\ndirections and the Latin indices i, jrepresent coordinate indices on the sphere SD−2.\nThe Lovelock density (2.2) of order pcan be rewritten as\nR(p)=X\nσ∈SDsgn(σ)Rσ(0)σ(1)···σ(2p−2)σ(2p−1)∧eσ(2p)···σ(D−1), (2.4)\nwhere the Levi-Civita tensor is replaced by permutations of the cyclic group SDof order D.\nSince the curvature forms Rabare proportional to ea∧eband each index appears exactly\nonce in each permutation, we conclude all indices are distinct in the R···term in (2.4). This\nmeans that there are only five possible pairings of the terms in (2.3), yielding\nR(p)|h,f= (˜RSS)p−2\u0014\nA˜R01˜RSS+B˜R0S˜R1S+C˜R0S˜RSS+D˜R1S˜RSS+E(˜RSS)2\u0015\ne0···D−1,\nfor the on-shell density, where\n˜R01=−r\nf\nh\u0012r\nf\nhh′\n2\u0013′\n,˜R0S=−fh′\n2rh,˜R1S=−f′\n2r,˜RSS=1−f\nr2\nare the four possible coefficients of the curvature forms. The quantities AtoEare\nA= 2\u0000p\n1\u0001\n(D−2)!D=C= (2p)!\u0000D−2\n2p−1\u0001\n(D−2p)!\nB= 8\u0000p\n2\u0001\n(D−2)! E= (2p)!\u0000D−2\n2p\u0001\n(D−2p)!\nand are coefficients counting the duplicates of each building block. For instance, Arepre-\nsents the number of permutations of {0,1}that appear in a single Rab– hence a factor of 2\nshows up for the two different arrangements of {0,1}; the second factor counts all different\nplaces that R01appears; the remaining ( D−2)! term corresponds to all the permutations\nwith 0 ,1 fixed. Other parameters can be obtained in a similar way. Collecting all building\nblocks together, we finally arrive at\nR(p)|h,f=p\n|g|dDx(D−2)!\u00121−f\nr2\u0013p−2\u001a\n−pr\nf\nh\u0012r\nf\nhh′\u0013′\u00121−f\nr2\u0013\n+p(p−1)\u0012ff′h′\nhr2\u0013\n−(D−2p)p\u00121−f\nr3\u0013\u0012fh′\nh+f′\u0013\n+ (D−2p)(D−2p−1)\u00121−f\nr2\u00132\u001b\n.(2.5)\nThe field equations on the GSSS shell (1.4) are obtained through extremizing the\nvolume-fixed action with respect to g00=h(r) and g11=f(r) simultaneously. Given the\ntensor defined as Eab:=δIE[g]/δgab, the relations (2.1), (2.5), together with the volume\nconstraint in (1.1) give rise to\nE00=1p\n|g|δIE[h, f]\nδh= 0,E11=1p\n|g|δIE[h, f]\nδf=1p\n|g|λ\n2f−3\n2rD−2,\n– 4 –which can be equivalently cast into\nE0\n0= 0,E0\n0+E1\n1=λ\n2√\nh.\nThey, respectively, correspond to the explicit expansions\n0 =pmaxX\np=0αp(D−2)!\n2(D−2p−1)![rD−2p−1(1−f)p]′\nrD−2, (2.6)\n8πλ√\nh=pmaxX\np=0αp(D−2)!\n2(D−2p−1)!pr1−2p(1−f)p−1h\u0012f\nh\u0013′\n, (2.7)\nwhich are consistent with the result in [19] but with some sign flips due to the positive\ndefiniteness of the Euclideanized metric (1.4).\nThe metric function fis fully characterized by the first equation, and so is the same\nas the one for unconstrained dS vacua since the volume constraint makes no contribution\nhere. The spatial metric component thus takes the specific form given by\nf(r) = 1−Λr2(2.8)\nwith some positive reduced effective cosmological constant Λ that satisfies the algebraic\nequation\nH(Λ) = 0 , (2.9)\nwhere H(x) is defined as a polynomial of the form\nH(x) :=pmaxX\np=0αp(D−2)!\n(D−2p−1)!xp. (2.10)\nwhere we have taken out a factor of ( D−1) to simplify subsequent expressions.\nThe non-trivial effects of the volume constraint take place in the second equation, which\ncauses hto deviate from its value of unity for unconstrained dS vacua. More precisely, the\nexplicit relation between λand the metric component his given by the direct substitution\nof (2.8) to (2.7) which yields\n16πλ√\nh=h\nr\u0012f\nh\u0013′\nH′(Λ), (2.11)\nwhere H′(x) is the first derivative of Hwith respect to its argument, and λshould be a\npure constant since no factor is τ-dependent in the equation.\nNote that the (2.11) imposes the same structure on has in GR [1], which implies\nthat the solutions for all Lovelock theories (GR included) are qualitatively the same. The\nonly distinction between the various theories is that (2.9) admits multiple solutions that\ndepending on the coupling constants αp. These correspond to various distinct Lovelock\nvacua that are characterized by different effective cosmological constants. These constants\nshould be limited to non-negative values since only those that have closed topologies are\nconsistent with a vanishing Gibbons-Hawking term in the proposed action [1]. The allowed\n– 5 –saddles can be classified into two categories, Λ = 0 and Λ >0, and each contributes to the\npartition function qualitatively differently. In other words, the category of Imin\nEshould be\nclearly specified when approximating the states of density by saddle points. In subsequent\nsections we will explore generic solutions for h(r) and evaluate Zas it is maximized by\neach type.\n3Imin\nEfor a vanishing cosmological constant\nWhenever the saddle of vanishing Λ minimizes the action, the value of Imin\nEcan be obtained\nby substituting the solution of (2.11) into (2.5) under the assumption Λ = 0. In this\nscenario, freduces to 1, and the solution for the ttcomponent of the metric becomes\nh(r) =\u00124πλ\nD−2\u00132\n(r2\nv−r2)2(3.1)\nwhere the factor of D−2 comes from H′(0). The constant of integration rvsatisfying\nh(rv) = 0 locates the horizon at the boundary of the volume ball and is determined by the\nspatial volume constraint\nV= Ω D−2Zrv\n0rD−2dr. (3.2)\nAlong with the metric the range of integration of (2.1) needs to be clarified. The spatial\nrange is limited to the volume Vwhereas the Euclidean time τis restricted to a closed\nloop with period β, whose value is determined from eliminating the conical singularity of\nthe manifold MEatrv. In the vicinity of the horizon the metric is approximately\nds2\nE|l→0≈\u00128πλrv\nD−2\u00132\nl2dτ2+dl2+r2\nvdΩ2\nD−2\nwhere r≈rv−l. It is easy to see that the quantity ϕ:= 8πrvλτ/(D−2) must have a\nperiod of 2 π, or equivalently τhas a period of β= (D−2)/(4λrv), so that the conical\nsingularity is removed. This establishes the relationship between βandλ; in terms of β\nthe metric reads\nds2\nE=(r2\nv−r2)2\n4r2vd\u00122πτ\nβ\u00132\n+dr2+r2dΩ2\nD−2 (3.3)\nand its accompanying Ricci scalar is\nR=4(D−1)\nr2v−r2.\nAn explicit calculation then shows that the action is\nImin\nE=−1\n16πZβ\n0dτZ√gdD−1xR\n=−ΩD−2Z2π\n0dξZrv\n0(D−1)rD−2\n8πrvdr\n=−ΩD−2rD−2\nv\n4=−Av\n4\n– 6 –where\nξ:= 2πτ/β A v:= Ω D−2rD−2\nv.\nWe see that the Ricci scalar is the only contributing term to the Lagrangian density, since\n(2.5) becomes trivial for any p≥2 when f= 1. The cosmological term is necessarily zero\ndue to H(0) = 2Λ 0/(D−1) = 0. The final result clearly indicates that the Bekenstein-\nHawking entropy, which is one quarter of the boundary area Av, is the extremal value of\nthe action in this case for all Lovelock theories.\n4Imin\nEfor a positive effective cosmological constant\nIn the case where the action is minimized by some solution with a positive Λ, the function\nhsatisfying (2.11) takes the form\nh(r) =\u00128πλ\nΛH′\u00132\u0012\n1−√\n1−Λr2\np\n1−Λr2v\u00132\n, (4.1)\nwhere Λ is some positive quantity that solves H= 0. Again, the constant of integrationp\n1−Λr2vimplies the existence of a singularity at the horizon rv. The multiplier λcan\nagain be written in terms of the period βby eliminating the conical singularity, which\nyields\nβ=H′p\n1−Λr2v\n4λrv. (4.2)\nThus the saddle metric for a constrained dS vacuum reads\nds2\nE=1\nΛ\u0014\u0012cosχ−cosχv\nsinχv\u00132\nd\u00122πτ\nβ\u00132\n+dχ2+ sin2χdΩ2\nD−2\u0015\n(4.3)\nwhere the reparametrization sin χ=√\nΛris applied for simplification. If the horizon\nmatches the cosmological one, or equivalently χv=π/2, the metric (4.3) reduces to the\nstatic patch for unconstrained Euclidean dS spacetime, namely\nds2\nEdS=1\nΛ\u0014\ncosχ2dτ2+dχ2+dΩ2\nD−2\u0015\n.\nFor the Lagrangian density (2.5) we obtain\nR(p)= (D−1)!Λp\u0014\nD+2pcosχv\ncosχ−cosχv\u0015√gdτ∧dD−1x, (4.4)\nfor the saddle (4.3). The action then becomes\nImin\nE=Av\n8X\npD(D−2)!\n(D−2p)!αpΛp−1+(D−1)V\n8cosχv\nsinχvΛ−1/2H(Λ), (4.5)\nwhere χvis constrained by\nV= Λ1−D\n2ΩD−2Zχv\n0(sinχ)D−2dχ= Ω D−2Zrv\n0rD−2\n√\n1−Λr2dr, (4.6)\n– 7 –and the horizon area Avis given by\nAv= Λ2−D\n2ΩD−2(sinχv)D−2= Ω D−2rD−2\nv. (4.7)\nNote that the second term in the action vanishes since H(Λ) = 0, and so its value is\nindependent of χv. This means the action is the same as the unconstrained dS action but\nwith Avreplaced by the area of cosmological horizon. The result can be further simplified\nby setting H= 0, for which we rewrite the factor DasD−2p+ 2p,\nImin\nE=Av\n8X\np(D−2)!\n(D−2p−1)!αpΛp−1+Av\n4X\np(D−2)!\n(D−2p)!pαpΛp−1. (4.8)\nFor non-zero Λ, the first term vanishes as it is proportional to H(Λ). The second one is\nthe negative of the Wald entropy for the boundary area Av, consistent with [20].\n5 Phase transitions between different vacua\nAs mentioned above, the approximation for Zonly works for cases where saddle points\nexist. Its validity necessarily confines the system into a certain amount of volume. This\nvalue is bounded by the total spatial volume screened inside the cosmological horizon in\nGR where only one vacuum solution could exist. However, Lovelock gravities extend the\npossibilities for vacuum solutions, and so we expect a discontinuity to appear in Swhen V\nreaches an intermediate value between two cosmological spatial volumes if the saddle point\ndominating for a smaller Vno longer remains a solution due to the volume exceeding its\nmaximum. For a dS vacuum, its free energy reduces to the gravitational entropy due to\nthe lack of quasi-local energy\nF(T, V) =−TlnZ=E−TS=−TS.\nTherefore a discontinuity in Simplies a zeroth-order phase transition at a given tempera-\nture.\nWe take Gauss-Bonnet gravity in D≥5 to illustrate this idea. In this scenario, possible\ndS vaccua are characterized by non-negative solutions of\nH(Λ) =2Λ0\nD−1−(D−2)!\n(D−3)!Λ +α2(D−2)!\n(D−5)!Λ2= 0 (5.1)\nusing (2.9). Consider the parameter space where (5.1) has two non-negative solutions. This\nforces the sum and the product of two solutions to satisfy\nΛ++ Λ−=1\nα2(D−3)(D−4)≥0,Λ+Λ−=2Λ0(D−5)!\nα2(D−1)!≥0, (5.2)\nwhich further reduce to α2>0 and Λ 0≥0. Furthermore, (5.1) simplifies the saddle action\nto be\nIE=Av(V,Λ)\n4\u0014D\nD−4−Υ\u0015\n,\n\nΥ =2(D−2)\n(D−4), if Λ = 0\nΥ =4Λ0\n(D−1)(D−4)Λ,if Λ̸= 0,(5.3)\n– 8 –where Λ is a solution to (5.1), and Avis induced by (4.7) together with the constraint\n(4.6). An immediate conclusion we can draw from (4.6) is that when Vis sufficiently small\nsuch that both saddles exist, a smaller Λ implies a greater rvand thus a larger boundary\narea at a given volume. Note that a smaller Λ vacuum has a larger volume bound; in\norder to construct such a transition, we eliminate cases where the saddle with the smaller\nΛ minimizes the action for any volume smaller than the maximum Vthat corresponds to\nthe other solution. In other words, to have transitions, larger Λ must have smaller action\nwhere this saddle exists.\nIt is easy to see Λ 0= 0 implies one negative and one positive action for Λ = 0 and\nΛ̸= 0, respectively. The solution with zero Λ always dominates the partition function, and\nso yields no interesting transition behaviour. For a positive Λ 0, we divide the discussion\ninto three cases.\n1.D/(D−4)−Υ<0for both vacua In this case, the larger Λ does not dominate\nthe action for any V, since its Avis smaller and the square bracket in (5.3) is less\nnegative. So this case is trivial.\n2.D/(D−4)−Υhas a different sign for each vacuum Here the solution with\nD/(D−4)−Υ>0 has a larger Λ and larger action. Thus this case also produces a\ntrivial effect.\n3.D/(D−4)−Υ>0for both vacua This case likely allows a discontinuity since Avis\nlarger and D/(D−4)−Υ is less positive for the solution with a smaller Λ. However\nwe find that, at least for even dimensions, the smaller Λ solution always dominates\nthe partition function as long as Vis smaller than its cosmological volume. This is\nbecause in order to ensure that (5.3) has the desired properties, we not only need\ntwo degrees of freedom from Λ ±but also that from Λ 0.\nIn summary, a phase transition is generically not allowed in the Gauss Bonnet theory.\nHowever the situation differs in cubic (or higher-curvature) Lovelock gravity, since there\nare more coupling constants but the number of solutions required to make the transition\ntake place is still only two.\nAs an example, we find that zeroth-order transitions can occur in cubic Lovelock\ngravity theory in D≥7. Noting that F/T =−lnZ=I, as shown in figure 1, we can\nsee three different vacua emerge in a particular case of cubic Lovelock gravity where the\nvacuum with a larger Λ has the smaller action. Meanwhile, a larger Λ implies a smaller\ncosmological volume. Thus two zeroth order phase transitions appear in the free energy\ndiagram at a fixed temperature as shown in the figure 1.\nAlthough our examples have been for even spatial dimensions, we expect similar results\nto hold for odd dimensions as well. In this case the integral in volume (4.6) contains a\npart related to an inverse trigonometric function, and solving for rvfor a given Vmust be\ncarried out numerically.\n– 9 –Figure 1 . Zeroth order phase transitions between three vacua of cubic Lovelock gravity with\nΛ0= 0, α2= 1/2, α3= 1/3 in D= 7. Green corresponds to the vacuum with Λ = 1 /2; red\ncorresponds to Λ = 1 /4; blue corresponds to Λ = 0. The figure depicts the free energy as a\nfunction of Vfor the stable phases of the system, showing that at large Vthe Λ = 0 vacuum (blue)\ndominates, at intermediate values of Vthe Λ = 1 /4 vacuum (red) dominates, and at small Vthe\nΛ = 1 /2 vacuum (green) dominates.\n6 Conclusions\nOur results support the interpretation of the dS entropy as the dimension of the Hilbert\nspace of all states constrained in a spatial volume, even when higher-curvature corrections\nto Einstein gravity are taken into account.\nWe have found that such corrections, as described by Lovelock gravity, generally yield\nmetrics of constrained dS vacua that take the same form as that in Einstein gravity [1].\nHowever instead of the unique dS horizon radius given by the bare cosmological constant,\nwe now have multiple vacua characterized by the non-negative zero points of (2.9). As\nlong as the partition function is dominated by one of these saddles, the logarithm of Zis\ngiven by the Wald entropy of the boundary. Once this vacuum no longer minimizes the\nEuclidean action due to volume expansion, a phase transition between vacua will occur.\nAcknowledgements\nThis work was supported in part by the Natural Science and Engineering Research Council\nof Canada. Mengqi Lu was also supported by China Scholarship Council.\nReferences\n[1] T. Jacobson and M.R. Visser, Partition Function for a Volume of Space ,Phys. Rev. Lett.\n130(2023) 221501 [ 2212.10607 ].\n– 10 –[2] G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum\nGravity ,Phys. Rev. D 15(1977) 2752.\n[3] J.D. Bekenstein, Black holes and entropy ,Phys. Rev. D 7(1973) 2333.\n[4] S.W. Hawking, Particle Creation by Black Holes ,Commun. Math. Phys. 43(1975) 199.\n[5] V. Balasubramanian, P. Horava and D. Minic, Deconstructing de Sitter ,JHEP 05(2001) 043\n[hep-th/0103171 ].\n[6] W. 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D 108(2023) L041902 [ 2307.14210 ].\n– 11 –"    },    {        "title": "2402.14243v1.Irregular_Bloch_Zener_oscillations_in_two_dimensional_flat_band_Dirac_materials.pdf",        "content": "Irregular Bloch-Zener oscillations in α-T3lattices\nLi-Li Ye1and Ying-Cheng Lai1, 2,∗\n1School of Electrical, Computer and Energy Engineering,\nArizona State University, Tempe, Arizona 85287, USA\n2Department of Physics, Arizona State University, Tempe, Arizona 85287, USA\n(Dated: February 23, 2024)\nWhen a static electrical field is applied to a two-dimensional (2D) Dirac material, Landau-Zener\ntransition (LZT) and Bloch-Zener oscillations can occur. Employing α-T3lattices as a paradigm\nfor a broad class of 2D Dirac materials, we uncover two phenomena. First, due to the arbitrarily\nsmall energy gaps near a Dirac point that make it more likely for LZTs to occur than in other\nregions of the Brillouin zone, the distribution of differential LZT probability in the momentum\nspace can form a complicated morphological pattern. Second, a change in the LZT morphology as\ninduced by a mutual switching of the two distinct Dirac points can lead to irregular Bloch-Zener\noscillations characterized by a non-smooth behavior in the time evolution of the electrical current\ndensity associated with the oscillation. These phenomena are due to mixed interference of quantum\nstates in multiple bands modulated by the geometric and dynamic phases. We demonstrate that\nthe adiabatic-impulse model describing Landau-Zener-St¨ uckelberg interferometry can be exploited to\ncalculate the phases, due to the equivalence between the α-T3lattice subject to a constant electrical\nfield and strongly periodically driven two- or three-level systems. The degree of irregularity of\nBloch-Zener oscillations can be harnessed by selecting the morphology pattern, which is potentially\nexperimentally realizable.\nI. INTRODUCTION\nThe Landau-Zener transition (LZT) [1, 2] is a fun-\ndamental phenomenon in time-dependent quantum sys-\ntems. The paradigmatic setting for LZT is a two-level\nsystem in which the two energy levels do not cross each\nother and vary adiabatically with time. When the en-\nergy gap between the two levels is sufficiently small, a\nnon-adiabatic transition from one energy level to another\ncan occur - leading to an LZT. The phenomenon of LZT\nis relevant to quantum information science and technol-\nogy, because qubits are essentially two-level systems [3–\n5]. In addition to quantum systems, LZTs can arise in\nother physical situations such as optical lattices [6–8] and\nelectromechanical systems [9, 10]. When a two-level sys-\ntem is periodically driven by an electric field, the tran-\nsition probability will depend on the phase accumulated\nby the two energy bands between subsequent crossings,\nleading to the so-called Landau-Zener-St¨ uckelberg inter-\nferometry [11]. In general, interference among the quan-\ntum states in different energy bands is determined by\ntwo phases: geometric and dynamic, which correspond\nto the adiabatic and Stokes phases, respectively, in the\nadiabatic-impulse model [11, 12] underlying the Landau-\nZener-St¨ uckelberg interferometry, where the sum of the\nadiabatic and Stokes phases gives the St¨ uckelberg phase\n- a concept originated from strongly periodically driven\ntwo-level systems. A generalization from the two-level\nsetting is the three-level LZT model [13] with an addi-\ntional flat band [14, 15].\nIn solid state physics, Bloch oscillations [16, 17] are a\nfundamental phenomenon closely related to LZT, which\n∗Ying-Cheng.Lai@asu.eduoccur when a static electric field is applied to a periodic\nlattice, leading to a linear increase with time in the elec-\ntron momentum and generating a time-dependent quan-\ntum system. The basic periodicity of the momentum\nspace stipulates that the electron must execute oscilla-\ntory motions in the physical space at a frequency de-\ntermined by the lattice constant and the electric field\nstrength (typically in the terahertz regime). In fact, in-\nsofar as the electron moves in a periodic potential, Bloch\noscillations can occur, rendering them a common quan-\ntum phenomenon beyond a solid-state lattice. In the\npast, the oscillations have been observed in diverse sys-\ntems such as semiconductor superlattices [18], photonic\nstructures [19–22] and plasmonic waveguide arrays [23].\nThe phenomenon provides a viable way to convert a di-\nrect current to a high-frequency signal [24, 25].\nBloch oscillations arise from the time evolution of the\nelectron in a single energy band. When there are multi-\nple energy bands, LZTs can occur at the avoided crossing\npoints between the bands. Driven by the static electric\nfield, a quantum state initialized in the lower energy band\nevolves with time. At certain time, the state will reach an\navoided crossing point between distinct energy bands and\npossibly experience an LZT. Thus, in systems with mul-\ntiple energy bands, a combination of LZTs and Bloch os-\ncillations can occur, leading to the so-called Bloch-Zener\noscillations [15, 26–28], which have applications in, e.g.,\nmatter-wave beam splitters and Mach-Zender interferom-\netry [29, 30]. In the past, Bloch-Zener oscillations were\nextensively studied in one-dimensional (1D) gapped peri-\nodic lattices [29–32] and were demonstrated to be sensi-\ntive to the size of the energy gap. In particular, if the gap\nis relatively large, LZTs are inhibited so the Bloch-Zener\noscillations are restricted to within a single energy band.\nFor a smaller gap, interband LZTs can occur, which de-arXiv:2402.14243v1  [cond-mat.mes-hall]  22 Feb 20242\nstroys the periodicity of the Bloch-Zener oscillations [29–\n31, 33, 34]. In a 2D lattice with multiple energy bands,\nin the first Brillouin zone, different subregions can arise\nthat can inhibit or excite LZTs, resulting in irregular\nBloch-Zener oscillations [35, 36]. For example, in pure\ngraphene, when a constant electric field is applied in a\nlattice-rational direction, the oscillation amplitude can\ndecay with time rapidly in a non-adiabatic fashion [37].\nWhen the electric field is in an irrational direction with\nrespect to the lattice structure, Bloch oscillations of com-\nplicated patterns can arise [38].\nIn this paper, we study the effects of a constant and\nuniform electric field on a broad class of 2D Dirac ma-\nterials, the α-T3lattice [39, 40], which has an additional\natom at the center of each unit cell of the honeycomb\ngraphene lattice. The interaction between the central\natom and any of its nearest neighbors is characterized\nby the parameter 0 ≤α≤1 - effectively the strength\nrelative to that between two neighboring atoms at the\nvertices of the graphene cell. For α= 0, the lattice\nreduces to that of graphene with quasiparticles being\npseudospin-1/2 Dirac fermions. As αincreases from\nzero, a flat band through the conic interaction of the\ntwo Dirac cones emerges [39, 40]. The maximal value\nα= 1 gives a pseudospin-1 lattice where, because of the\nextra atom, the low energy excitations need to be de-\nscribed by the pseudospin-1 Dirac-Weyl equation with a\nthree-component spinor [41]. A distinct feature of the\nentire spectrum of α-T3lattices is the existence of two\ndistinct valleys centered about the two non-equivalent\nDirac points of the backbone hexagonal lattice, denoted\nas +Kand−K.\nDepending on the direction and magnitude of the elec-\ntric field, electrons initiated from distinct valleys can ex-\nhibit characteristically different LZTs. As the dynamic\nphases associated with different valleys cancel each other\nexactly, the distinct LZTs are due to the different adia-\nbatic phases of the quantum states in the energy bands\nbetween consecutive crossings. This can be understood\nby considering the reciprocal periodic momentum space\nwith the hexagonal Brillouin zone, as shown in Fig. 1.\nNow apply an electric field in the xdirection. The x\ncomponent of the momentum will then increase linearly\nwith time under the premise of the same energy value. As\na result, the Dirac points + Kand−Kwill shift towards\nthe right. At an original Dirac point (+ Kor−K), the en-\nergy will increase from zero, reach a maximum, and then\ndecrease to zero when the next Dirac point arrives, gener-\nating a time-periodic behavior. Because of the hexagonal\nstructure of the momentum space, the energy variations\nassociated with + Kand−Kare distinct, as indicated\nin Fig. 1(b). The LZT probability depends on the accu-\nmulated phase between subsequent crossings, where the\nadiabatic phase is the integral of the energy variation\nover time. In the specific setting of Figs. 1(a) and 1(b),\nthe integral associated with the Dirac point + Kwill have\na much larger value than that associated with the other\nDirac point −K. As a result, if an electron initiates witha momentum value near + K, the adiabatic phase will be\nnearly constant for a large range of energy gaps deter-\nmined by the momentum deviation from the trajectory\nof Dirac points ±Kin the pydirection. However, if an\nelectron starts with a momentum value near −K, the adi-\nabatic phase will depend sensitively on the energy gap.\nBased on this property, it is possible to generate specific\ndestructive or constructive interference for a large range\nof momentum deviation from the trace of the Dirac point\n+K, whereas there is mixed interference associated with\nall possible phases for electrons with initial momentum\nnear−K.\nOur first finding is the emergence of complicated LZT\nmorphological patterns in the vicinity of distinct Dirac\npoints, which is associated with mixed quantum interfer-\nence among the quantum states in multiple bands. Say\nwe apply an electric field in the xdirection, initialize\nelectrons in the lower Dirac cone (the lower band), and\ncalculate the differential LZT probability, defined as the\ndifference between the probability that an electron is in\nthe upper band and that in the lower band. Different mo-\nmentum values about a Dirac point and the magnitudes\nof the electric field give distinct interference phases. As\na result, in the momentum plane, the differential LZT\nprobability displays different values, giving rise to some\nmorphological pattern that can be complex. During its\ntemporal evolution, the pattern can be maintained for\nsome time but it can change from time to time due to\nthe switching of two two distinct Dirac points ±Kin the\nBrillouin zone as caused by the external electric field and\nthe periodic structure of the momentum space. Specif-\nically, during one period of the Bloch-Zener oscillation,\nthe two valleys go through a complete cycle in the sense\nthat they are switched and then returned to their respec-\ntive original positions.\nThe second finding is that changes in the LZT mor-\nphology can lead to irregularities in Bloch-Zener oscilla-\ntions in α-T3lattice. To explain this, we recall the two\ntypical cases where periodic Bloch oscillations are gener-\nated. One case is a single-band material, such as a normal\nconductor, where the Bloch oscillations are characterized\nby a perfect temporally periodic behavior in the electrical\ncurrent density. Another case is where an LZT causes all\nelectrons initialized in one band to transition completely\nto another band, i.e., the transition probability is one -\nthe so-called ideal LZT. In this case, the resulting Bloch-\nZener oscillations behave as if the electrons were in a\nsingle band. In the three-band α-T3lattice, LZTs are\ntypically not ideal. The mutual switchings of the two\nDirac points ±Kin the Brillouin zone changes the LZT\nmorphology, which can produce an abrupt, nonsmooth\nchange in the current density, thereby leading to ape-\nriodic, irregular Bloch-Zener oscillations. More specifi-\ncally, the coexistence of a variety of LZT possibilities in\nthe momentum space generates complex, mixed quantum\ninterference between the states in the upper, lower, and\nflat bands, disrupting the originally periodic Bloch-Zener\noscillation rhythm before the Dirac point switch. While3\naperiodic Bloch oscillations [29–31, 33, 34] and irregular\nBloch-Zener oscillations [35, 36] have been noted before,\nto our knowledge, the physical mechanisms underlying\nthese irregular behaviors were not clear. Especially, it\nhas not been reported previously that LZTs can form a\ncomplicated morphology in α-T3lattice and a change in\nthe morphology can lead to irregular Bloch-Zener oscil-\nlations.\n(𝑎) 𝒂1 𝒂2\n𝑥𝑦A\nB\nC𝑡𝜖𝛼𝑡𝜖\n(𝑏)\n෨𝑘𝑥෨𝑘𝑦−𝐾\n+𝐾\nFIG. 1. Illustration of the α-T3lattice and its positive en-\nergy band structure. (a) The lattice structure [42, 43] as\ndefined by the three base atoms (A, B and C) in physical\nspace spanned by the two primitive translation vectors aaa1\nandaaa2. The nearest-neighbor hopping energy between A and\nB sites is tϵand that between B and C sites is αtϵ, where\n0≤α≤1 characterizes the coupling strength. (b) Zero-field\nenergy-band structure [42] of the positive dispersion band as a\nfunction of the wave vector kkkfor an arbitrary value of αin the\nhexagonal Brillouin zone. The zero energy points correspond\nto two classes of non-equivalent contact points at the corners\nξKwith the valley index ξ=±1. A reference rectangular re-\ngion (dashed line) for numerical integration is indicated, with\nthe boundaries of the hexagonal Brillouin zone specified by\nthe red solid lines.\nIn Sec. II, we describe the α-T3lattice model and derive\nthe current density associated with the Bloch-Zener os-\ncillations from the adiabatic basis in the hexagonal Bril-\nlouin zone. In Sec. III, we present a general treatment of\nLZTs in the α-T3lattice, display the LZTs morphologyin the long time, and analyze the relationship between\nmorphology and irregular Bloch-Zener oscillations. In\nparticular, in Sec. III A, we linearize the Hamiltonian\nabout the Dirac points to obtain the effective Landau-\nZener Hamiltonian for the two limiting cases: α= 0 and\nα= 1. For 0 < α < 1, we numerically demonstrate\nthe occurrence of LZT. In Sec. III B, we elucidate the\ninterplay between LZT morphological changes and irreg-\nular Bloch-Zener oscillations. In Sec. IV, we focus on the\nLandau-Zener-St¨ uckelberg interferometry in α-T3lattice,\nwhere in Sec. IV A, we establish the equivalence of the α-\nT3lattice to two- or three-level time-dependent systems\nand provide an understanding of the LZT based on the\nSt¨ uckelberg phase. In Sec. IV B, we address the problem\nof harnessing irregular Bloch-Zener oscillations through\nselection of the LZT morphology and discuss the experi-\nmental feasibility of this scheme. A discussion is offered\nin Sec. V.\nII. BASICS OF α-T3LATTICE, LANDAU-ZENER\nTRANSITION, AND BLOCH-ZENER\nOSCILLATIONS\nThe α-T3lattice interpolates between the graphene\nhoneycomb lattice ( α= 0) and the dice lattice ( α= 1)\nwith the parametrization tan φ=α∈[0,1] with the du-\nality [42] α→1/α. The tight-binding Hamiltonian is\ngiven by\nH=\n0 fkkkcosφ 0\nf∗\nkkkcosφ 0 fkkksinφ\n0 f∗\nkkksinφ 0\n, (1)\nwhere\nfkkk=−tϵ\u0010\n1 +e−ikkk·aaa1+e−ikkk·aaa2\u0011\n, (2)\nkkk= (kx, ky), and tϵis the nearest-neighbor hopping\nenergy between A and B sites, as shown in Fig. 1(a).\nThe primitive translation vectors are aaa1=a\u0000√\n3/2,3/2\u0001\nandaaa2=a\u0000\n−√\n3/2,3/2\u0001\nwith abeing the lattice con-\nstant. The corresponding primitive translation vectors\nin the hexagonal Brillouin zone of the reciprocal lattice\narebbb1=\u0000√\n3/3,1/3\u0001\n2π/aandbbb2=\u0000\n−√\n3/3,1/3\u0001\n2π/a.\nThe eigenenergy spectrum of the α-T3lattice is indepen-\ndent of α, which consists of two conic dispersive bands\nϵλ=λ|fkkk|distinguished by the band index λ=±and\na zero energy flat band ϵ0= 0, as shown in Fig. 1(b).\nThe eigenstates of the α-T3lattice in the whole hexago-\nnal Brillouin zone can be obtained through the following\neffective Hamiltonian about the Dirac points:\n|ψ0⟩=\nsinφ eiθkkk\n0\n−cosφ e−iθkkk\n,|ψλ⟩=1√\n2\ncosφ eiθkkk\nλ\nsinφ e−iθkkk\n,\n(3)\nwhere θkkkis the angle of fkkkassociated with the specific\nmomentum.4\nWe describe the Hamiltonian underlying Bloch-Zener\noscillations. Apply a uniform and constant electric field\nto the α-T3lattice in the + xdirection, which is switched\non at t= 0. With the time-dependent vector poten-\ntial [44, 45] AAA(t) = [A(t),0,0], where A(t) =EtΘ (t)/ℏ,\nthe Hamiltonian becomes\nH(t) =\n0 fkkk(t) cosφ 0\nf∗\nkkk(t) cosφ 0 fkkk(t) sinφ\n0 f∗\nkkk(t) sinφ 0\n,(4)\nwhere fkkk(t) is given by\nfkkk(t) =−tϵ \n1 + 2 e−i3\n2ekycos √\n3\n2ekx(t)!!\n,(5)\nwith kx(t)≡kx−eEt/ℏ. For convenience, we define the\ndimensionless quantities\nekx(t)≡kx(t)a, (6)\neky≡kya, (7)\nso that fk(t) has the same dimension as the hopping en-\nergy tϵ.\nIn the presence of the electric field, the eigenenergy\nspectrum of the positive dispersion band in the whole\nhexagonal Brillouin zone is determined by\nϵkkk(t) =|fkkk(t)|\n=tϵp\n1 + 4 cos Xkkk(t) (cos Ykkk+ cos Xkkk(t)) (8)\nwhere Xkkk(t) =ekx(t)√\n3/2 and Ykkk=eky3/2. Because the\nfunction fkkk(t) includes cos( L(t)), where L(t) is a linear\nfunction of time, the Hamiltonian becomes time-periodic.\nConsider the special case of α= 0 (graphene). Expand-\ning the Hamiltonian around the Dirac points yields the\nstandard Landau-Zener Hamiltonian [46]:\neH≈3δeky\n2σx∓3eEet′\n2σz. (9)\nThat is, when the electrons are near the Dirac points,\nLZTs between distinct energy bands can arise.\nThe quantum dynamics are governed by\niℏ∂tψkkk(t) =H(t)ψkkk(t). (10)\nOn the adiabatic basis, the evolution of a quantum state\nis under an infinitesimal electric field [44, 47]:\nU†(t)H(t)U(t) =Szϵkkk(t), (11)\nwhere Szis the z-component of the vector of spin-1 ma-\ntrices. The transformed quantum dynamics are governed\nby\niℏ∂tΦkkk(t) =\u0014\nSzϵkkk(t)−eSxat2\nϵeE\nϵ2\nkkk(t)C0(t)\u0015\nΦkkk(t),(12)through the time-dependent unitary transformation U(t)\ngiven by\n\n1√\n2cosφ eiθkkk(t)sinφ eiθkkk(t) 1√\n2cosφ eiθkkk(t)\n1√\n20 −1√\n2\n1√\n2sinφ e−iθkkk(t)−cosφ e−iθkkk(t) 1√\n2sinφ e−iθkkk(t)\n,\n(13)\nwhere Φ kkk(t) =U†(t)ψkkk(t) and the term incorporating\nC0(t) contributes to the time dependence of U(t) through\n−iℏU†∂tU. Specifically, we have\nC0(t) =√\n3 sinYkkksinXkkk(t) (14)\neSx=Sxsin 2φ−SLcos 2φ, (15)\nwhere\nSx=1√\n2\n0 1 0\n1 0 1\n0 1 0\n, SL≡\n−1\n20−1\n2\n0 1 0\n−1\n20−1\n2\n.(16)\nForα= 1,eSxreduces to Sx, the xcomponent of spin-\n1 matrix and Eq. (12) becomes the quantum evolution\nequation for a dice lattice. This form of Eq. (12) is\nconsistent with that reported in a previous work [45]\nexcept for a periodic factor in C0(t) due to the intrin-\nsic lattice structure. For α= 0,eSxbecome −SLand\nEq. (12) describes the graphene lattice, which is consis-\ntent with a previous work [44] except for the identity\nterm Iexp(−iθ(t)/2)∂texp(iθ(t)/2)/2 due to the differ-\nent form of the basis. Overall, eSxreflects the different\ncoupling strength among the three bands and the peri-\nodic term in C0(t) originates from the intrinsic property\nof the α-T3lattice.\nWe set the initial state as one corresponding fully oc-\ncupied lower band:\nΦkkk(t= 0) = [0 ,0,1]T. (17)\nThe average current density associated with the momen-\ntum⟨Jx⟩kkk(t) in the hexagonal Brillouin zone is given by\n⟨Jx⟩kkk(t)≡Φ†\nkkk(t)Jx, kkk(t) Φkkk(t) (18)\nwhere the current density matrix with the definite mo-\nmentum is\nJx, kkk(t) =−eU†(t)∂kx(t)H(t)U(t) (19)\nand Φ kkk(t) in the adiabatic basis can be written as\nΦkkk(t) =\u0002αkkk(t), γkkk(t), βkkk(t)\u0003T. (20)\nDue to the periodic structure of the energy band (referred\nto as the Bloch band), the average current density will\nexhibit Bloch oscillations.\nThe average current density ⟨Jx⟩kkk(t) can be decom-\nposed into two components, the intraband and interband\ncurrents, respectively [44]:\n⟨Jx⟩kkk(t) =⟨Jx⟩intra\nkkk(t) +⟨Jx⟩inter\nkkk(t), (21)5\nwhich can be written as\n⟨Jx⟩intra\nkkk(t) =J11\nx, kkk(t)\u0010\n|αkkk(t)|2− |βkkk(t)|2\u0011\n, (22)\n⟨Jx⟩inter\nkkk(t) = 2ℜ\u0002\nJ13\nx, kkk(t)α∗\nkkk(t)βkkk(t)\u0003\n+ 2ℜ\u0002\nJ12\nx, kkk(t)α∗\nkkk(t)γkkk(t) +J23\nx, kkk(t)γ∗\nkkk(t)βkkk(t)\u0003\n.(23)\nThe matrix Jx,kkk(t) provides insights into the current den-\nsity⟨Jx⟩kkk(t). In particular, the intraband component\nconsists of both electrons and holes, corresponding to\nJ11\nx, kkk(t)≡J0\nx, kkk(t) cos Θ kkk(t) (24)\nJ33\nx, kkk(t) =−J11\nx, kkk(t), (25)\nrespectively, where the minus sign comes from the op-\nposite sign of the equivalent charge in the electron-hole\npair. The zero group velocity of the flat band results\nin zero intraband contribution. The interband contribu-\ntion arises from the interference between the transition\nfrom the lower to the flat band or the upper band and\nthat from the flat to the upper band, corresponding to\nJ23\nx, kkk(t), J13\nx, kkk(t) and J12\nx, kkk(t), respectively, which are\ngiven by\nJ13\nx, kkk(t)≡iJ0\nx, kkk(t) cos 2 φsin Θ kkk(t),(26)\nJ12\nx, kkk(t) =J23\nx, kkk(t)≡iJ0\nx, kkk(t)/√\n2 sin 2 φsin Θ kkk(t),\n(27)\nwhere\nΘkkk(t)≡θkkk(t) +Ykkk (28)\nandJ0\nx, kkk(t) is the common factor with the dimension of\nthe current density:\nJ0\nx, kkk(t) =−√\n3eatϵsinXkkk(t). (29)\nTo facilitate numerical calculations, we define the dimen-\nsionless quantities:\net=t/t0, (30)\neE=E/E 0, (31)\neϵk(t) =ϵk(t)/ϵ0, (32)\neJx, kkk(t) =Jx, kkk(t)/J0. (33)\nwhere t0≡ℏ/tϵ,E0≡tϵ/(ea),ϵ0=tϵandJ0=eatϵ.\nWe use the fourth-order Runge–Kutta method to calcu-\nlate the adiabatic evolution of the particle. The discrete\nstep sizes in time and momentum are chosen according\nto Eq. (12) with the error tolerance 10−2and normalized\nwavefunction error within 10−4. Figure 1(b) indicates\nthat the ±Kcorrespond to zero energy, which results in\nnumerical divergence of Eq. (12), so we set a minimum\nenergy cut-off to be eϵk≥10−5.\nThe dimensionless current density eJis the result of in-\ntegrating ⟨Jx⟩kkk(t)/J0over the rectangular reference area\nin the hexagonal Brillouin zone, as shown in Fig. 1(b),\nwhich contains three Dirac points and is 3 /2 times larger\nthan the first Brillouin zone. In addition, we divide the\ncurrent density eJby the electric field eEand a constant\n3π2/4, which normalizes the current density in the weak\nfield regime [37] for α= 0.III. MORPHOLOGICAL CHANGES IN LZTS\nAND IRREGULAR BLOCH-ZENER\nOSCILLATIONS\nA. Landau-Zener transition\nTo gain insights, we first consider the special case α= 0\n(graphene), where the Hamiltonian of the α-T3lattice lin-\nearized about the Dirac points ±Kcorresponds to that\nof a standard two-level system. Using the unitary trans-\nformation\nU= exp ( −iπ/4σy) exp (−iπ/4σz), (34)\nwe can write the linearized Hamiltonian as (Appendix C)\nU†eHU≈3δeky\n2σx∓3eEet′\n2σz (35)\nwhere δekyis an infinitesimal deviation from a Dirac point\nandet′= 0 denotes the starting time from the Dirac point.\nRecall the standard Landau-Zener Hamiltonian for a two-\nlevel system [46]:\nHLZ= (g/2)σx+ (st)σz, (36)\nwith the two underlying adiabatic energy levels\neϵ±=±1\n2q\n(2st)2+g2, (37)\nwhere gis the energy gap and sis the slope of the two-\nlevel band about the LZT point. The linearized Hamil-\ntonian for graphene in Eq. (35) can thus be cast in the\nstandard two-level LZT Hamiltonian with the following\nparameter correspondences:\nα= 0, g= 3δeky,ands= 3eE/2. (38)\nFor a finite ∆ ekyinα-T3lattice, the exact energy gap is\ng= 2s\n1−cos2\u00123\n2∆eky\u0013\n. (39)\nThe gap size increases monotonically with the momen-\ntum deviation ∆ ekyfrom a Dirac point. [For α̸= 0 (a\nflat band), the gap between the lower band and the flat\none is half of the gap between the lower and the upper\nbands.] For ∆ eky=π/3, the gap size greaches the maxi-\nmum value of two in the energy unit ϵ0in the hexagonal\nBrillouin zone.\nIn a two-level quantum system, for electrons initial-\nized in the lower band, after the electric field is turned\non, the first LZT to the upper band occurs with the prob-\nability [46]\nPLZ≡ |αkkk|2= exp ( −πr), (40)6\nand the one remaining in the lower band is 1 −PLZ, where\nris the ratio between the energy gap and the slope (in\nthe standard unit ℏ≡1) given by\nr≡(g/2)2/s=3\n2δek2\ny\neE, (41)\nwhich can be treated as a parameter characterizing the\npossible occurrence of LZTs. Note that PLZ≈4% for\nr= 1, so this provides a numerical criterion for deter-\nmining if an LZT can occur: 0 < r≤1.\nWe next consider the opposite extreme case of the α-T3\nlattice: α= 1 (pseudospin-1 lattice). Because of the pres-\nence of the flat band, the Hamiltonian linearized about a\nDirac point can be related to that of the standard LZT\nmodel [13] with three distinct energy levels. In particu-\nlar, employing the unitary transformation\nU= exp\u0012\n−i\nℏπ\n2Sy\u0013\nexp\u0012\n−i\nℏπ\n2Sz\u0013\n, (42)\nwe obtain the pseudospin-1 Hamiltonian as (Appendix C)\nU†eHU≈3δeky\n2Sx∓3eEet′\n2Sz, (43)\nwhere SxandSyare the components of the vector of\nspin-1 matrices. The eigenenergy spectra of the upper\nand lower bands have the same form as Eq. (37), with\nthe addition of the extra flat band in the middle of the\nlower and upper bands. For an electron initialized in the\nlower band, the LZT probabilities for it to transition to\nthe upper band, transition to the flat band and remain\nin the lower band are given by [13]\n|αkkk|2= 1−2p\nPLZ+PLZ, (44)\n|γkkk|2= 2\u0010\n1−p\nPLZ\u0011p\nPLZ, (45)\n|βkkk|2=PLZ, (46)\nrespectively. where PLZis defined in Eq. (40). To appre-\nciate the flat band contribution to the transitions, we set\nPLZ= 1/4 so that |αkkk|2=PLZholds for both two- and\nthree-level systems because 1 −2√PLZ= 0. Figure 2(d)\nshows the numerical result for 0 ≤α≤1, where it can be\nseen that, in this special case, |αkkk|2is independent of the\ncoupling strength between the flat and positive/negative\nbands.\nWe can now analyze the general α-T3lattice as an\ninterpolation between the idealized two-level and three-\nlevel systems, where the coupling between the flat band\nand the other two bands varies in the range 0 < α < 1.\nFor convenience, we again initialize electrons in the lower\nband and examine two transition probabilities: that from\nthe lower to the upper band denoted as |αkkk|2and that\nfrom the lower to the flat band denoted as |γkkk|2. To unveil\nthe effect of increasing the value of αfrom zero, we cal-\nculate the two probabilities for 0 ≤α≤1 and the range\nof momentum deviation ∆ ekyfrom the + KDirac point\n(𝑒)\ntimeenergyFIG. 2. LZT probabilities in the α-T3lattice. For electrons\ninitiated from the lower band, there are two LZT probabilities:\n|αk|2- the transition probability from the lower band to the\nupper band, and |γk|2- the transition probability from the\nlower to the flat bands. (a) Color-coded sum of |αk|2and|γk|2\nin the (∆ eky, α) plane for eE= 0.0336. The initial momenta\nareekx= 3 and eky= ∆ekyabout + K, and the integration time\nstep is det= 0.01. (b) Time evolution of the LZT probabilities\nfor ∆ ekyandαvalues taken from the location of the pentagram\nin (a) with |αkkk|2+|γkkk|2+|βkkk|2= 1. (c) LZT probabilities\nversus ∆ ekyforα= 1, where the arrow indicates the flat-band\ninduced change in the LZT probability. (d) LZT probabilities\nversus αfor ∆ eky= 0.1. (e) LZT in the three-level system.\nIf the initial state is in the lower band, after an LZT about\nthe avoided crossing point, the state is a superposition of the\neigenstates associated with all three bands.\nin the hexagonal Brillouin zone. Figure 2(a) shows the\ncolor-coded sum of the two probabilities in the parame-\nter plane (∆ eky, α), where the range of ∆ ekyto generate a\nhigh LZT probability increases monotonically with αand\nreaches maximum at α= 1, suggesting that the flat band\nenhances LZT. The time evolution of the two probabili-\nties and their sum is shown in Fig. 2(b). Note that, for\nα= 1, the LZT probabilities |αkkk|2and|γkkk|2can be deter-\nmined from Eqs. (44) and (45), respectively. Figure 2(c)\nshows, for α= 1, these two probabilities, together with\ntheir sum, versus ∆ eky, where the horizontal dashed line\nspecifies P= 0.8 and the arrow indicates the enhance-\nment of the LZT by the flat band. As ∆ ekyincreases, the\nvalue of the characteristic parameter rincreases, lead-\ning to a decrease in the LZT probability to the upper\nband. However, even when the LZT probability from the\nlower to the upper bands is effectively zero, there can still\nbe an appreciable transition probability from the lower\nto the flat band. For example, for ∆ eky= 0.2, we have\nr≈1.8. In this case, we have |αkkk|2≈0 but|γkkk|2≈0.25.\nFigure 2(d) shows, for fixed ∆ eky= 0.1, the two probabil-\nities versus α. Note that the probability |γkkk|2increases\nmonotonically with α.7\n(𝑏)෨𝐸Δǁ𝑡=෨𝑘0(𝑎)෨𝐸Δǁ𝑡=0\n(𝑐)\n෨𝐸Δǁ𝑡=2෨𝑘0(𝑑)\n෨𝐸Δǁ𝑡=3෨𝑘0\n෨𝐸෨𝑘𝑥−𝐾\n+𝐾\nFIG. 3. Schematic illustration of Bloch-Zener oscillations. (a)\nTwo nonequivalent Dirac points ±Kin the hexagonal Bril-\nlouin zone at et= 0, corresponding to two distinct valleys.\nDriven by a static electric field in the positive ekxdirection,\nboth Dirac points start to move in the same direction, where\nthe two vertical sides of the gray rectangle denote the periodic\nboundaries in ekx. When the electric field is applied to the α-\nT3lattice, in the laboratory frame we have ekx(t) =ekx−eEet, for\nfixed ekx. In this “static” momentum space, an electron will\nmove toward the left with the momentum ekx(t)∝ − eEet. In\nthe moving frame with ekx(t), the zero energy or Dirac points\nwill move toward the right with the momentum ekx∝eEet, as\nshown in Fig. 3 based on the energy form in Eq. (8). Since\nLZTs occur around the Dirac points, it is convenient to fol-\nlow the movements of the Dirac points. (b-d) The locations\nof±Kafter ∆ et=ek0/eE, 2ek0/eE, and 3 ek0/eE, respectively. For\n∆et= 3ek0/eE, the two Dirac points return to their respective\ninitial starting locations, completing one cycle. The period\nof the Bloch-Zener oscillations is thus etB= 3ek0/eE. During\nthe Bloch period, the Landau-Zener transition occurs twice\nabout ±K, as indicated by the double horizontal color bars.\nB. Morphology and irregular Bloch-Zener\noscillations\nUnder a static electric field, the intraband current den-\nsity will exhibit periodic-like Bloch-Zener oscillations,\nwhere the interband contribution can be neglected in the\nlong time situation [44]. Figure 3 provides a schematic\npicture to explain the origin of the oscillations. Driven\nby a constant electric field in the positive ekxdirection,\nafteret= 0+, the two Dirac points ±Kin the hexagonal\nBrillouin zone start to move in the same direction, where\nthe gray rectangular region denotes a periodic area in\n(𝑎) (𝑏)\n(𝑐)\n෨𝑘𝑥෨𝑘𝑦෨𝑘𝑥෨𝑘𝑦\n෨𝑘𝑥෨𝑘𝑦(𝑑)\n෨𝑘𝑥෨𝑘𝑦\n+𝐊 −𝐊\n෨𝑘𝑥෨𝑘𝑦\n෨𝑘𝑥෨𝑘𝑦(𝑓) (𝑔)(𝑒)FIG. 4. Emergence of distinct LZT morphology and ir-\nregularities in the Bloch-Zener oscillations. (a) Morphol-\nogy of LZT after t= 0+(immediately after the electric\nfield in the ekxdirection is turned on). Shown is the color-\ncoded values of the differential LZT probability, defined as\n∆Pαβ(t)≡ |αkkk(t)|2− |βkkk(t)|2, in the entire Brillouin zone.\n(b-d) Color-coded values of the differential LZT probability\n∆Pαβ(t) at three instants of time after which a morpholog-\nical change in the LZTs occurs: ∆ et=ek0/eE, 2ek0/eEand\n3ek0/eE. (e) Evolution of the intraband current density within\none Bloch period. At the three time instants indicated by\nthe vertical dashed lines and arrows, the curve is nonsmooth,\nwhich correspond to the LZT morphology in (b-d), respec-\ntively, and signify irregularities in the Bloch-Zener oscilla-\ntions. (f,g) Morphology of LZT after 20 periods of Bloch-\nZener oscillations with magnification about + K, -K, respec-\ntively. Simulation parameter values are α= 0, eE= 0.1200,\ndet≈0.01, and dekx≈deky≈0.012.\nthe momentum space, as shown in Fig. 3(a). The edge\nlength of the hexagonal Brillouin zone is ek0= 4π/(3√\n3)\nin units of 1 /a, the inverse of the lattice constant. After\nthe time ∆ et=ek0/eE, the + Kvalley reaches the orig-8\ninal location of the −Kvalley, as shown in Fig. 3(b).\nThe−Kreaches the original location of the + Kafter\nthe time ∆ et= 2ek0/eE, as shown in Fig. 3(c). The oc-\ncurrences of the LZTs associated with the ±Kvalleys\nare indicated by the orange and blue horizontal strips,\nrespectively, where the width of excitation zone depends\non both the gap size and the magnitude of electric field.\nAt the time etB= 3ek0/eE, both Dirac points return to\ntheir original starting locations, completing one cycle of\noscillation during which LZT occurs twice.\nIn the vicinity of a Dirac point (+ Kor−K) in the\nmomentum space, an infinite set of energy gaps exists\nwhere, for different momentum deviations ∆ ekyfrom the\nDirac point, the sizes of the energy gap can be quite\ndistinct from the exact energy gap in Eq. (39). As the\nstatic electric field in the xdirection is turned on, the\nmomentum in the + ekxdirection increases linearly with\ntime, sweeping through all possible values of the xcom-\nponent of the momentum in the Brillouin zone, effectively\neliminating all the original differences in ekxfor different\npoints in the momentum space. However, the various\ndeviations in the ycomponent of the momentum, i.e.,\nthe different ∆ ekyvalues, still matter and in fact per-\nsist because they correspond to different energy gaps.\nAs a result, different values of ∆ ekywill lead to differ-\nent probabilities of LZT, creating a distinct morphology\nwith respect to the LZT probability in the ekydirection\nof the momentum space at any given time. As an exam-\nple, Figs. 4(f) and 4(g) present such a morphology after\n20 periods of Bloch-Zener oscillations with the magnifi-\ncation about the ±KDirac points, respectively, where\nthe color-coded values of the differential LZT probability\n∆Pαβ(t)≡ |αkkk(t)|2− |βkkk(t)|2are shown in the ( ekx,eky)\nplane. An observation is that the LZT morphology, as ex-\nemplified in Figs. 4(f) and 4(g), can undergo changes due\nto the complicated interference pattern between quan-\ntum states in different energy bands in the long time,\ndue to the non-deterministic nature of the LZTs with re-\nspect to the size of the energy gap. Figure 4(a) presents\nsuch a morphology after time t= 0+(immediately after\nthe electric field is turned on) within one period of the\nBloch-Zener oscillation. Three examples are illustrated\nin Fig. 4(b-d), where the color-coded values of ∆ Pαβ(t)\nat three time instants: ∆ et=ek0/eE≈20, 2ek0/eE≈40 and\n3ek0/eE≈60, are displayed. The morphological changes\nin the LZTs are concentrated in the vertical neighbor-\nhoods of the Dirac points, whereas the values of ∆ Pαβ(t)\nin most of the momentum space remain unchanged.\nA remarkable phenomenon is that, when a change in\nthe LZT morphology occurs, some experimentally mea-\nsurable quantities such as the current density can un-\ndergo a sudden change as well. To be concrete, we fo-\ncus on the current density associated with Bloch-Zener\noscillations, which is dominantly determined by the in-\ntraband behaviors [44]. Suppose that, initially, the elec-\ntrons are prepared in the lower band. Due to the change\nin the LZT morphology and the mutual switching be-tween the Dirac points ±K, the dependence of the prob-\nabilities for the electrons to be in the upper band on\nthe momentum will change, and this will lead to a sud-\nden change in the current density that is contributed to\nby all the momenta in the Brillouin zone. Note from\nFigs. 4(b-d) that the changes in the LZT morphology\nare pronounced only near the original Dirac points where\nan LZT is most likely to occur, while there are no such\nchanges for most of the momentum space. Since the cur-\nrent density is the integration over the entire Brillouin\nzone, the resulting change in the current density will be\nquite “subtle” in the sense that it will not be a discon-\ntinuous change in the current density itself but a non-\nsmooth change (or, equivalently, a discontinuous change\nin the time derivative of the current density). Such non-\nsmooth changes have indeed been numerically observed,\nas shown in Fig. 4(a), where the arrows and the verti-\ncal dashed lines indicate the three time instants at which\nsuch a change occurs, corresponding to the distinct LZT\nmorphology in Figs. 4(b-d), respectively. To compare\nirregular and periodic Bloch-Zener oscillations, we also\nstudy the oscillations resulting from near ideal LZTs in\nthe momentum space (Appendix A).\nIV. LANDAU-ZENER-ST ¨UCKELBERG\nINTERFEROMETRY IN α-T3LATTICE\nA. Two- and three-level models\nThe dynamical evolution of the wavefunction in the α-\nT3lattice for α= 0 (1) driven by a static electric field\ncan be described by the Hamiltonian of a strongly period-\nically driven two- or three-level system, as demonstrated\nin Appendix C. According to the adiabatic-impulse the-\nory [11, 12], the quantum evolution can be decomposed\ninto adiabatic evolution and non-adiabatic LZTs, where\nthe former occurs most of the time but the latter occur\non a short time scale. For 0 < α < 1, under the adiabatic\nimpulse approximation, this physical picture still applies.\nSpecifically, for the adiabatic evolution, the eigenenergy\nspectrum is independent of the value of the lattice cou-\npling parameter α, so the adiabatic phase for 0 < α < 1\nis similar to that for α= 1. Insights into the LZTs can be\ngained by numerically calculating the time evolution of\nthe transition probability |αk|2and|γk|2inα-T3lattice,\nas exemplified in Fig. 2.\nThe adiabatic evolution of the wavefunction gener-\nates an adiabatic phase, while the LZTs lead to a non-\nadiabatic phase. To gain insights, consider the double\npassage case in a strongly periodically driven two-level\nsystem, where two successive transitions are required for\na particle initiated from an eigenstate in the lower band\nto reach the upper band with probability one, as exem-\nplified in Fig. 5(a). In this case, after two successive\nLZTs, the transition probability to the upper band is9\n(𝑎)\n(𝑏)\nFIG. 5. Landau-Zener transitions in α-T3lattice for α= 0.\nShown in (a,b) is the time evolution of |αkkk|2, the probability\nof LZT from the lower to the upper band, for four different\ninitial momentum values. The orange traces correspond to\nthe case where the momentum is initiated in the vicinity of\nthe + Kvalley and within one Bloch period, LZT is required\nfor|αkkk|2to reach zero. The purple traces are for the case\nwhere the momentum is initiated in the vicinity of the −K\nvalley. In (a), the initial momentum values for the orange and\npurple traces are eky= 0.07 about + Kand 2 π/3+0.07 around\n−K, respectively. In (b), the corresponding orange and purple\ntraces are the results of setting the initial momentum values\ntoeky= 0.065 (about + K) and 2 π/3 + 0 .065 (around −K),\nrespectively. The transition behaviors displayed can be un-\nderstood in terms of the St¨ uckelberg phase (to be analyzed\nbelow) that tends to be approximately constant when the\ninitial momentum is near the + Kvalley but exhibits large\nvariations when the initial momentum is near the −Kvalley.\nOther parameter values are eE= 0.0317, ekx= 3 and det= 0.01.\ngiven by [11, 48] (Appendix B 1)\nP+= 4PLZ(1−PLZ) sin2(Φst), (47)\nΦst=ζ+φs, (48)\nwhere PLZis the first-time LZT probability given by\nEq. (40) and Φ stis the St¨ uckelberg that consists of two\ncomponents: the adiabatic phase ζbetween two consec-\nutive LZTs and the non-adiabatic phase, i.e., the Stokes\nphase φSat the transition. There are two distinct cases.\nThe first is\nΦst=π/2 +kπ, k ∈Z, (49)\ncorresponding to constructive interference [11] because,\nafter one driving period, the maximum transition prob-\nability to the upper band is P+= 4PLZ(1−PLZ),\nwhich is twice the average transition probability ⟨P+⟩=\n2PLZ(1−PLZ) over one period. The second case is\nΦst=kπ, (50)giving rise to destructive interference [11] as P+= 0 after\none driving period.\nOur equivalence analysis in Appendix C and the treat-\nment of the strongly periodically driven two-level system\nin Appendix B 1 give that, for α= 0 in the α-T3lattice,\nthe adiabatic phase is given by\nζ=Z\nϵkkk(t)dt, (51)\nwhere ϵkkk(t) is defined in Eq. (8) and depends on the elec-\ntric field eEand the momentum deviation ∆ ekyfrom±K.\nThe non-adiabatic phase φsis determined by the lin-\nearized LZT Hamiltonian about the Dirac points ±K:\nφs=π/4 +δ(lnδ−1) + arg Γ(1 −iδ), (52)\nwhere δis determined by the LZT characteristic param-\neter as 2 δ=r, as given by Eq. (41). It can be seen that\nthe St¨ uckelberg phase also depends on the electric field\neEand the momentum deviation ∆ ekyfrom±K.\n(𝑎)\n(𝑏)\nFIG. 6. St¨ uckelberg phase in α-T3lattice for α= 0. (a)\nThe St¨ uckelberg phase versus the momentum deviation from\n+K(orange) and −K(purple) valleys. The range of the\nmomentum deviation is determined by the empirical crite-\nrion: 0 < r≤1, which is ∆ eky∈[0.01,(2eE/3)1/2] for\neE= 0.03 and d∆eky= 0.0001. The integration time step is\ndet= 0.01. (b) The adiabatic phase measured from the Dirac\npoints ±Kas determined by the integral of function f(x) over\nx∈[0,2π/3](−K),x∈[0,4π/3](+K) with ∆ eky= 0.7.\nFigure 6(a) shows the St¨ uckelberg phase for two types\nof initial momentum values for eE= 0.03. For LZTs\nstarting near the Dirac point + K, the St¨ uckelberg phase\nis non-adiabatic and approximately independent of the\nmomentum deviation ∆ eky, while for −K, the phase is\nadiabatic so it depends on the momentum deviation\n[Eq. (52)]. More specifically, for LZTs starting from the10\nDirac points ±K, the adiabatic phase between two suc-\ncessive LZTs is\nζ+K=Z4π\n3\n0f+K(x)dx, (53)\nζ−K=Z2π\n3\n0f−K(x)dx, (54)\nwhere the functions f+K(x) and f−K(x) are given by\nf+K=2√\n3eEs\n1 + 4 a+cos\u00122π\n3−x\u0013\n+ 4 cos2\u00122π\n3−x\u0013\n,\nf−K=2√\n3eEr\n1 + 4 a−cos\u0010π\n3−x\u0011\n+ 4 cos2\u0010π\n3−x\u0011\n(55)\nwith a+≡cos(3∆ eky/2),a−=−a+, ∆ekymeasured from\n±K, and x≡eEet√\n3/2. Figure 6(b) shows the adiabatic\nphase measured from the Dirac points ±K. Since the\nintegration of f+Khas a relatively large value, the adia-\nbatic phase ζ+Kis nearly constant for different momen-\ntum deviation ∆ ekyfor 0 < r≤1, whereas ζ−Kis sen-\nsitive to ∆ eky. The adiabatic phase also depends on the\nelectric field eE. (More elaborate details can be found in\nAppendix D.)\n(𝑎)\n(𝑏)\nFIG. 7. Landau-Zener transitions in α-T3lattice for α= 1.\nBecause of the presence of a flat band, the time evolution\nof two quantities is displayed: (a) |γkkk|2and (b) |αkkk|2, the\ntransition probability from the flat to the upper band and that\nfrom the lower to the upper band, respectively. For the orange\ntraces, the initial momentum is eky= 0.135 around + K. For\nthe purple traces, the initial momentum is eky= 2π/3 + 0.135\naround −K. Other parameters are eE= 0.03,ekx= 3, and\ndet= 0.01.\nForα > 0, a flat band arises in addition to the two\nDirac cone bands. Figure 7 shows the representative timeevolution of the transition probabilities |αk|2and|γk|2\nforα= 1. Exploiting the equivalence of the dice lattice\ndriven by a constant electric field to a strongly period-\nically driven three-level system (Appendix C), we have\nthat, after one Bloch-Zener oscillation period (two suc-\ncessive LZTs), the occupation probability of the upper,\nflat and lower bands are given by (Appendix B 2)\nP+= 16eP2\nLZsin4(ζ/2), (56)\nP0= 2ePLZ((1−4ePLZ)(1−cosζ)2+ sin2ζ),(57)\nP−= (2ePLZcosζ+ (1−2ePLZ))2(58)\nwith the normalization constraint P++P0+P−= 1,\nwhere\nePLZ≡P1/2\nLZ(1−P1/2\nLZ). (59)\nFor the upper band, ζ=π+ 2kπ, k ∈Zcorresponds to\nconstructive interference and ζ= 2kπleads to destruc-\ntive interference [Eq. (56)]. For the flat band, ζ= 2kπ\ngives P0= 0 [Eq. (57)]. For ζ=π+ 2kπ, we have P0̸= 0\nwithePLZ= 1/4. Note that the Stokes phase disappears\nin this case based on the non-adiabatic transition ma-\ntrix [13]. From Eq. (52), for 0 < r= 2δ <1, the Stokes\nphase is a small constant: φs≈0.5. As a result, we have\nΦst≈ζ.\nB. Morphology selection\nSo far, we have numerically observed the complex LZT\nmorphology and found a relationship between morpho-\nlogical changes and irregular Bloch-Zener oscillations. In\naddition, we have analyzed the interference phases in α-\nT3lattice based on the equivalence between the lattice\nsubject to a constant electric field and strongly periodi-\ncally driven two- or three level systems. We have found\nthat the St¨ uckelberg phase (adiabatic phase) of LZTs\nstarting from the Dirac point + Kis nearly independent\nof the momentum deviation ∆ eky, i.e., the energy gap,\nwhile the phase starting from −Kis sensitive to ∆ eky.\nFor Φ st= 0, π, destructive interference among the quan-\ntum states from different bands arises for α= 0. For\nα= 1, only Φ st= 0 corresponds to destructive interfer-\nence. These findings suggest a principle of morphology\nselection for 0 ≤α≤1: setting Φ st= 0 or Φ st=πfor\nLZT starting from the + Kvalley will result in two dis-\ntinct types of LZT morphology, as illustrated in Figs. 8\nand 9, respectively. As analyzed in Appendix A, ideal\nLZTs in the Brillouin zone lead to periodic and regu-\nlar Bloch-Zener oscillations. The destructive interference\nhas a similar effect to that of ideal LZTs. AS a result, the\nresulting morphology pattern can improve the regularity\nof Bloch-Zener oscillations.\nFigure 8 shows, for eE= 0.03 and four different types of\nα-T3lattices, the irregular Bloch-Zener oscillations and\nthe corresponding representative LZT morphology at a11\n𝛼=0.3(𝑎) (𝑏) (𝑐)\n(𝑒) (𝑓) (𝑔)\n෨𝑘𝑥෨𝑘𝑦\n෨𝑘𝑥෨𝑘𝑦\n෨𝑘𝑥෨𝑘𝑦\n෨𝑘𝑥෨𝑘𝑦(𝑑)\n(ℎ)\n𝛼=0.5𝛼=0.7 𝛼=1\nFIG. 8. Mostly regular (or weakly irregular) Bloch-Zener oscillations in general α-T3lattices. (a-d) Oscillations of the intraband\ncurrent density for α= 0.3, 0.5, 0.7, and 1.0, respectively. (e-h) The corresponding LZT morphology revealed by the color-coded\ndifferential LZT probability ∆ Pαβ(t) in the momentum space at a specific time, where the St¨ uckelberg phase Φ st= 0 for LZTs\nstarts from + Kvalley. Simulation parameter values are eE= 0.03,det≈0.001, and dekx≈deky≈0.012.\n𝛼=0(𝑎) (𝑏) (𝑐)\n(𝑑) (𝑒) (𝑓)\n෨𝑘𝑥෨𝑘𝑦\n෨𝑘𝑥෨𝑘𝑦\n෨𝑘𝑥෨𝑘𝑦\n𝛼=0.5 𝛼=1\nFIG. 9. Strongly irregular (“chaotic”) Bloch-Zener oscillations. The static electric field strength is eE= 0.0279. (a) Mostly\nregular oscillations in the intraband current density for α= 0 (graphene). As αincreases from zero, the oscillations become\nstrongly irregular, as shown in (b) and (c) for α= 0.5 and α= 1, respectively. (d-f) The corresponding LZT morphology at\na particular time instant, where the St¨ uckelberg phase Φ st=πfor LZTs starts from the + Kvalley. Integration parameter\nvalues are det≈0.01 and dekx≈deky≈0.012.\ngiven time in the momentum space. It can be seen that\nthe Bloch-Zener oscillations shown in Figs. 8(a-d) are\nsomewhat irregular - they are mostly regular. The de-\nstructive interference pattern of LZTs starting from the\n+Kvalley in the momentum space is shown in Figs. 8(e-\nh) with Φ st= 0. For another type of morphology with\nΦst=π, the Bloch-Zener oscillations are more irregular.\nFor example, for α̸= 0, the oscillations can be strongly\nirregular or “chaotic,” as exemplified in Figs. 9(b) and\n9(c). In this case, for 0 ≤α≤1, the interference pattern\nassociated with the LZTs starting from the + Kvalleychanges from destructive to non-destructive, as shown\nin Figs. 9(d-f). In both cases, the morphology of LZTs\nstarting from the −Kvalley changes little due to the\nsensitivity to the momentum deviation.\nV. DISCUSSION\nBloch or Bloch-Zener oscillations, in addition to be-\ning a fundamental phenomenon in solid state physics,\npractically provides the foundation to convert a direct12\ncurrent into an oscillating current in the terahertz fre-\nquency regime [24]. Research in the past decade or so has\nsuggested that, in 2D multiband materials, Bloch-Zener\noscillations can be vulnerable to Landau-Zener transition\nor tunneling [29, 30, 33–36]. Is this generally true? The\nquestion is important to physics and our present work\nhas provided two-fold answers: yes LZTs can indeed af-\nfect the Bloch oscillations but what is destroyed is not the\noscillations themselves but just the perfect time period-\nicity of the oscillations; no because the irregular oscilla-\ntions can persist even with frequent occurrences of LZTs.\nIn fact, approximately periodic Bloch-Zener oscillations\ncan be maintained if the LZTs are near ideal such that\nthey result in a near-one probability for the electrons to\nswitch into a different band or when near destructive in-\nterference arises between the quantum states in different\nenergy bands. Nonsmooth or irregular behaviors in the\ncurrent density arise when the LZTs are not ideal and\nthe interference is partially destructive.\nOur study encompasses the entire spectrum of a class\nof 2D Dirac materials modeled by α-T3lattices. We have\nfound that the set of points in the 2D momentum space\nnear a Dirac point at which LZTs occur can possess a\ncomplex morphology, and it is the change in the mor-\nphology that results in irregular Bloch-Zener oscillations.\nTheoretically, when driven by a static electric field, an α-\nT3lattice is equivalent the Landau-Zener-St¨ uckelberg in-\nterferometry. Specifically, the α= 0 lattice (graphene) is\neffectively a two-level periodically driven quantum sys-\ntem while the general α-T3lattice for 0 < α≤1 is\nequivalent to a three-level periodically driven system. For\nthe three-level system, we have exploited the concept of\nSt¨ uckelberg phase from the adiabatic impulse theory to\nunderstand the LZTs that occur in the neighborhoods of\nthe Dirac points in distinct valleys.\nIn the α-T3lattice, a non-zero coupling parameter α\ninduces a flat band in between the positive and negative\nenergy bands, effectively resulting a three-level system.\nAfter the first LZT in Fig. 2, the quantum state is a su-\nperposition of the three energy bands and the LZT is\nenhanced by the flat band (compared with the two-level\ncase). For subsequent LZTs (e.g., as shown in Figs. 8 and\n9), the flat band modifies the St¨ uckelberg phase Φ stfor\ndestructive interference in the three-band interferometry\n(compared with the two-band one). More specifically,\nforα= 0, the destructive interference corresponds to\nΦst= 0, πfrom Eq. (47). For α= 1, the destructive\ninterference means Φ st= 0 only and the case Φ st=πis\nexcluded from Eqs. (56-58). For 0 < α < 1, the behav-\nior of destructive interference is obtained numerically, as\nshown in Figs. 8 and 9, which is slightly different from\ntheα= 1 case, especially in Figs. 9(e,f). Taken together,\na nonzero αmodifies the physical picture from two- to\nthree-band quantum interference: it generates a flat band\nin the original two-level system, creates three-band inter-\nferometry, enhances LZT in the momentum space around\nthe Dirac points, and modifies destructive interference.\nTheoretically, the asymmetrical morphology patternaround the ±Kpoints results from the different char-\nacteristics of the St¨ uckelberg phases Φ st. As shown in\nFig. 6(a), Φ stis nearly constant with ∆ ekymeasured from\nthe + KDirac point after two LZTs and Φ stchanges\ngreatly with ∆ ekyfrom−K. Different values of the in-\nterference phase Φ stgive distinct interference patterns,\nsuch as constructive, destructive, and mixed quantum\ninterference (neither constructive nor destructive) in the\nmomentum space, as exemplified in Fig. 4(c).\nFor LZTs with a near unity transition probability,\nwhich can occur for an infinitesimal energy gap, the ±K\npoints make the same contribution to the Bloch oscil-\nlations. However, slightly away from the Dirac points\nwhere the energy gap is no longer infinitesimal, the con-\ntributions differ. The reason why the Bloch oscilla-\ntions around Kare more prominent compared with those\naround the −Kpoint lies in the interference phase Φ st.\nIn particular, the LZTs from the + Kpoint with different\nmomentum deviation ∆ ekyare in phase with each other,\nas shown in Fig. 6(a). However, the LZTs measured from\nthe−Kpoint are out of phase, so the contributions to\nthe Bloch oscillations cancel each other to some degree.\nThe irregular Bloch-Zener oscillations in the average\ncurrent density integrated over the Brillouin zone arise\nfrom the LZTs induced by different energy gaps associ-\nated with the momentum deviation ∆ ekymeasured from\nDirac points. In the α-T3lattice, there is an alternative\nway to induce an energy gap between the valence and\nconduction bands by adding the positive (negative) on-\nsite energy on the A (B) sublattice [49, 50]. Because of\nthe sensitive dependence of the LZT on the size of the\nmomentum-dependent energy gap, irregularities in the\nLZTs are anticipated, so are the irregular Bloch-Zener\noscillations.\nExperimentally, it may be feasible to observe at least\nthe first peak of the Bloch-Zener oscillation in ballistic\ntime [37]. To probe into the momentum-space morphol-\nogy associated with LZTs and to directly observe the\nirregular Bloch-Zener oscillations in a longer time inter-\nval in Dirac material systems remain to be difficult at the\npresent. Alternatively, it may be possible to use quantum\nsimulators [51, 52] by exploring equivalent optical sys-\ntems [20]. In the past, Bloch oscillations have been exper-\nimentally studied in photonic systems [21, 23, 35, 53, 54].\nThe observation of Bloch-Zener oscillations and Landau-\nZener tunneling in photonic graphene [35] is particularly\nrelevant to possible experimental checks of our findings.\nIn this artificial Dirac system, the wave packet of light\nis driven by an index gradient on a non-adiabatic basis\nand the two sublattices are subjected to a potential im-\nbalance. When the momentum deviation is zero from a\nDirac point and the index gradient is applied in a specific\ndirection, perfect (ideal) LZT with transition probability\none can occur due to the zero energy gap. However, when\nthe index gradient is applied in the orthogonal direction,\nthe LZT becomes imperfect (non-ideal) due to the non-\nzero gap. Our work suggests that, if the time evolution\nof the wavefunction in photonic graphene can be approx-13\nimated as constituting two processes: adiabatic evolu-\ntion and non-adiabatic LZT, in principle the St¨ uckelberg\nphase can be calculated to choose the appropriate index\ngradient value to create destructive interference between\nthe quantum states in the upper and lower energy bands.\nIt may thus be possible to generate destructive interfer-\nence to obtain a near-ideal LZT. Our work predicts that,\nin this case, the resulting Bloch-Zener oscillations will\nbecome more periodic. Photonic graphene may be a fea-\nsible experimental testbed for these phenomena.\nACKNOWLEDGMENT\nWe thank Drs. Chen-Di Han and Cheng-Zhen Wang as\nwell as Prof. Liang Huang for valuable discussions. This\nwork was supported by AFOSR under Grant No. FA9550-\n21-1-0186.\nAppendix A: LZT morphology and irregular\nBloch-Zener oscillations\n𝑤=0.5\n෨𝑘𝑥෨𝑘𝑦(𝑐)(𝑎) (𝑏)\n෨𝑘𝑥෨𝑘𝑦\n𝑤=0.05(𝑑)\nFIG. 10. LZTs and Bloch-Zener oscillations. (a,b) For mo-\nmentum integration width w= 0.5 about the ±Ktrace and\ndet≈0.01, the LZT morphology and the resulting Bloch-\nZener oscillation, respectively. (c,d) Similar to (a,b) but for\nw= 0.05 and det≈0.002. The irregularities in the intraband\ncurrent density are indicated by arrows. Other parameter val-\nues are dekx≈deky≈0.012 and eE= 0.1200.\nTo further appreciate the interplay between LZT mor-\nphology and irregular Bloch-Zener oscillations, we con-\nsider two kinds of momentum integration regions about\nthe Dirac points, as shown in Figs. 10(a) and 10(c), re-\nspectively. The resulting time evolution of the current\ndensity is shown in Figs. 10(b) and 10(d), respectively.\nFor the case in Fig. 10(c), the LZTs are near ideal, gener-\nating less irregular Bloch-Zener oscillations in Fig. 10(d).Appendix B: Adiabatic impulse theory\n1. Two-level systems\nThe Hamiltonian under a periodic driving in the non-\nadiabatic basis has the form [11]\nH(t) =−∆\n2σx−ε(t)\n2σz (B1)\nwhere the driving ε(t) =ε0+Asinωtproduces a periodic\ntime evolution of the eigenenergy spectrum:\nϵ±(t) =±1\n2p\n∆2+ε(t)2. (B2)\nIf the bias of the driving is nonzero: ε0̸= 0, one period\nof the evolution of the energy contains two peaks [11]\nthat are in the time intervals [ t1, t2] and [ t2, t1+ 2π/ω],\nrespectively. The quantum dynamical process can be un-\nderstood by using the adiabatic impulse theory [11, 12].\nAccording to this theory, the dynamical process can be\napproximated as the adiabatic evolution from t1tot2\nand from t2tot1+ 2π/ω, which are described by uni-\ntary transformation matrices U1andU2, respectively,\nand non-adiabatic transitions at t2andt1+ 2π/ω that\nare described by the same non-adiabatic transition ma-\ntrixN. After one period, the quantum state under the\nadiabatic basis can be written as [11]\nbbb(t1+ 2π/ω) =NU 2NU 1bbb(t1). (B3)\nWhen the driving signal is linearized about the transition\npoint as ε(t)≈ −vt, the non-adiabatic transition matrix\nin the adiabatic basis is [11]\nN=\u0012√1−PLZe−ieφs−√PLZ √PLZ√1−PLZeieφs\u0013\n, (B4)\nwhere PLZis the first LZT probability of the upper band\ndefined by Eq. (40), eφs=φs−π/2, and φsis the Stokes\nphase. The adiabatic evolution matrix in the adiabatic\nbasis is\nU1=\u0012\ne−iζ10\n0eiζ1\u0013\n, U 2=\u0012\ne−iζ20\n0eiζ2\u0013\n,(B5)\nwhere ζ1andζ2are the adiabatic phases given by\nζ1=Zt2\nt1ϵ+dt,\nζ2=Zt1+2π/ω\nt2ϵ+dt. (B6)\nIn the adiabatic basis, the total transform matrix after\none period is [11]\nNU 2NU 1=\u0012\nα−γ∗\nγ α∗\u0013\n, (B7)14\nwhere the matrix elements are\nγ= 2p\nPLZ(1−PLZ)e−iζ1sin(Φ st), (B8)\nα=e−i(ζ1+eφs)[(1−2PLZ) sin(Φ st) +icos(Φ st)],\nand Φ st=ζ2+φsis the St¨ uckelberg phase. If the initial\nstate is\nbbb(t1) = (0 ,1)T, (B9)\nthe quantum state after one period will be\nbbb(t1+ 2π/ω) = (−γ∗, α∗)T(B10)\nwith the transition probabilities\nP+=|γ|2= 4PLZ(1−PLZ) sin2(Φst), (B11)\nP−=|α|2= 1−4PLZ(1−PLZ) sin2(Φst),(B12)\nwhich satisfy the normalization constraint P++P−= 1.\nFrom Eq. (B11), it can be seen that whether the transi-\ntion is complete depends only on the Stokes phase and the\nadiabatic phase between two consecutive LZTs. Specif-\nically, Φ st=kπ, k ∈Zcorresponds to destructive in-\nterference between the quantum states in the upper and\nlower bands while Φ st=π/2 +kπcorresponds to con-\nstructive interference.\n2. Three-level systems\nA periodically driven three-level system is described by\nH(t) =−∆\n2Sx−ε(t)\n2Sz, (B13)\nwhere the time evolution of the positive and negative\nenergy is given by Eq. (B2) except for the extra flat band\nϵ0= 0. A previous work [13] provided the non-adiabatic\ntransition matrix in the adiabatic basis when the driving\nsignal is linearized about the transition point as ε(t)≈\n−vt:\nN=\nB+ 1A B\nA C A\nB A B + 1\n, (B14)\nwhere all A,BandCare constants:\nA≡ −(2ePLZ)1/2,\nB≡P1/2\nLZ−1,\nC≡1−2P1/2\nLZ, (B15)\nand\nePLZ≡P1/2\nLZ(1−P1/2\nLZ). (B16)The adiabatic evolution matrix of the three-level system\nis\nU1=\ne−iζ10 0\n0 1 0\n0 0 eiζ1\n, U 2=\ne−iζ20 0\n0 1 0\n0 0 eiζ2\n.\n(B17)\nThe total transform matrix M≡NU 2NU 1is\nM=\nM11M12M13\nM∗\n23M22M23\nM∗\n13M∗\n12M∗\n11\n, (B18)\nwhere\nM11=^(B1+ 1)∗^(B2+ 1)∗\n+AeA∗\n1+eB∗\n1eB2,\nM12=A^(B2+ 1)∗\n+AC+AeB2,\nM13=eB1^(B2+ 1)∗\n+AeA1+^(B1+ 1)eB2,\nM22=AeA∗\n2+C2+AeA2,\nM23=eA∗\n2eB1+eA1C+eA2^(B1+ 1), (B19)\nwith\neAi≡Aeiζi,\neBi≡Beiζi,\n^(Bi+ 1)≡(B+ 1)eiζi, (B20)\nfori= 1,2. Suppose the initial quantum state is\nbbb(t1) = (0 ,0,1)T. (B21)\nAfter one period, the quantum state becomes\nbbb(t1+ 2π/ω) = (M13, M 23, M∗\n11)T. (B22)\nThe occupied probabilities of the upper, flat and lower\nbands after one period are given by\nP+= 16eP2\nLZsin4(ζ2/2), (B23)\nP0= 2ePLZ((1−4ePLZ)(1−cosζ2)2+ sin2ζ2),(B24)\nP−= (2ePLZcosζ2+ (1−2ePLZ))2, (B25)\nrespectively, where P+=|M13|2,P0=|M23|2and\nP−=|M11|2with the constraint P++P0+P−= 1.\nAccording to Eq. (B23), for the upper band, we have\nthat ζ2=π+ 2kπ, k ∈Zcorresponds to construc-\ntive interference and ζ2= 2kπto destructive interfer-\nence. For the flat band, ζ2= 2kπcorresponds to de-\nstructive interference, as stipulated by Eq. (B24). For\nζ2=π+ 2kπ, the flat band leads to an LZT with prob-\nability away from zero except for ePLZ= 1/4 The Stokes\nphase defined by Eq. (52) depends only on the parameter\nδ(0< r= 2δ <1). Numerically, we obtain φs≈0.5.\nIn spite of the small phase deviation due to the Stokes\nphase φs, Φst=ζ+φs= 2kπcorresponds approximately\nto destructive interference regardless of the existence of\na flat band. However, Φ st=π+ 2kπdoes not represent\ndestructive interference in the three-level system.15\nAppendix C: Equivalence of α-T3lattice to strongly\nperiodically driven two- or three-level systems\nWhen a static electric field is applied to an α-T3lattice,\nthe time evolution of the wavefunction can be obtained by\nusing the adiabatic impulse theory [11, 12]. For graphene\n(α= 0) and the dice lattice ( α= 1), the tight-binding\nHamiltonians are, respectively,\nH=ℜ[fk(t)]σx− ℑ[fk(t)]σy (C1)\nand\nH=ℜ[fk(t)]Sx− ℑ[fk(t)]Sy, (C2)\nwhere σxand σyare the 2 ×2 Pauli matrices for\npseudospin-1/2 quasiparticles in graphene, SxandSyare\nthe corresponding 3 ×3 matrices from pseudospin-1 quasi-\nparticles in the dice lattice, and\nℜ[fk(t)] =−tϵ\"\n1 + 2 cos\u00123\n2eky\u0013\ncos √\n3\n2ekx(t)!#\n,\nℑ[fk(t)] = 2 tϵsin\u00123\n2eky\u0013\ncos √\n3\n2ekx(t)!\n. (C3)\nThe eigenenergy spectrum of graphene is ϵ±=±|fk|.\nFor the dice lattice, there is a flat band ϵ0= 0. The time\nevolution of nonzero energy band of both graphene and\ndice lattice has the common form for the upper and lower\nbands:\neϵ±(t) =±1\n2q\n∆2\neky+ (ε0,eky+ 4 sin( ωeEet+ϕekx))2,(C4)\nwhere\n∆eky= 2 sin(3 eky/2),\nε0,eky≡2 cos(3eky/2),\nωeE≡√\n3eE/2,\nϕekx=π/2−√\n3ekx/2. (C5)\nEquation (C4) has the same mathematical form as\nthat for a strongly periodically driven two-level system\n[Eq. (B2)].\nA theoretical approach to dealing with a strongly peri-\nodically driven two-level system is the adiabatic and im-\npulse approximation [11, 12], which is valid in the regime\nof strong field\n∆2+A2≫ω2(in units of ℏ= 1). (C6)\nEquations (C4) and (B2) give A= 4 and |ω|=√\n3eE/2<\n0.1, rendering applicable the adiabatic and impulse ap-\nproximation. The idea of the analysis is to decompose\nthe time evolution of system into an adiabatic evolution\nwhen it is far from the points of avoided-crossing and\nnon-adiabatic process in the vicinity of these points.For adiabatic evolution in graphene, the adiabatic\nphase in the wavefunction depends on the integral ζ=R\neϵ+(t)det. The dice lattice has the same energy eϵ+and\nthe flat band corresponds to a zero adiabatic phase ζ= 0.\nThus, for the upper and lower bands in both graphene\nand dice lattice, the adiabatic phase has the same mathe-\nmatical form as that for the periodically driven two- and\nthree-level system, respectively.\nFor the non-adiabatic transition process in graphene\nor dice lattice, the effective Hamiltonian about the Dirac\npoints is the standard or the three-level Landau-Zener\nHamiltonian. Concretely, we can show that the Hamil-\ntonian for graphene and dice lattice, given by Eqs. (C1)\nand (C2), can be written as the standard Landau-Zener\nHamiltonian for two- and three-level systems, respec-\ntively, as\nHLZ= (g/2)σx+ (st)σz, (C7)\nHLZ= (g/2)Sx+ (st)Sz, (C8)\nthrough some unitary transformation. In particular, the\nrequirement is to have σx→σz, σy→ −σxfor graphene\nandSx→Sz, Sy→ −Sxfor dice lattice. These transfor-\nmations can be realized by rotating the original Hamil-\ntonian in two steps since the physical observable does\nnot change after a unitary transformation. First, we ro-\ntate the Hamiltonian Halong the anticlockwise direc-\ntion with π/2 around the axis yto get H→H′, where\nσx→σz, σy→σyfor graphene and Sx→Sz, Sy→Sy\nfor dice lattice. Second, we rotate the Hamiltonian H′\nalong the anticlockwise direction with π/2 around the z-\naxis to obtain H′→H′′, where σz→σz, σy→ − σx\nfor graphene and Sz→Sz, Sy→ −Sxfor dice lattice.\nThe total unitary transformations for graphene and dice\nlattice are\nU= exp ( −iπ/4σy) exp (−iπ/4σz), (C9)\nU= exp\u0012\n−i\nℏπ\n2Sy\u0013\nexp\u0012\n−i\nℏπ\n2Sz\u0013\n, (C10)\nrespectively.\nAs for Hamiltonian expansion about Dirac points,\nfirstly, we expand Hamiltonian in the ekydirection about\nthe Dirac points with\n+K:eky→δeky,\n−K:eky→2π/3 +δeky, (C11)\nleading to\ncos\u00123\n2eky\u0013\n≈ ±1,\nsin\u00123\n2eky\u0013\n≈ ±3\n2δeky. (C12)\nWith the unitary transformation in Eq. (C9) we obtain\nthe approximate Hamiltonian as given by\nU†eHU≈ −ε(t)/2σz−∆ (t)/2σx (C13)16\nwith\nε(t) = 2 \n1±2 cos √\n3\n2ekx(t)!!\n,\n∆ (t) =±6δekycos √\n3\n2ekx(t)!\n. (C14)\nFor the dice lattice, with the total unitary transformation\nin Eq. (C10) we have the approximate Hamiltonian as\nU†eHU≈ −ε(t)/2Sz−∆ (t)/2Sx (C15)\nFrom Eqs. (C13) and (C15), we translate the momentum\nekxto the Dirac points as\n+K:ekx→4π/(3√\n3) +δekx,\n−K:ekx→2π/(3√\n3) +δekx. (C16)\nSetting δekx≡eEet′withet′starting from origin, we obtain,\nabout ±K,\ncos √\n3\n2ekx(t)!\n≈\n\ncos\u0010\n2π\n3−√\n3\n2eEet′\u0011\ncos\u0010\nπ\n3−√\n3\n2eEet′\u0011\n\n\n≈ ∓1\n2+3\n4eEet′. (C17)\nwhere the second-order term δekxδekyhas been neglected.\nFor graphene, the effective Hamiltonian about the Dirac\npoints ±Kis\neH≈3δeky\n2σx∓3eEet′\n2σz, (C18)\nwhich is the standard Landau-Zener Hamiltonian [45].\nFor the dice lattice, the effective Hamiltonian is\neH≈3δeky\n2Sx∓3eEet′\n2Sz, (C19)\nwhich is the Hamiltonian of the three-level Landau-Zener\nmodel [13]. In the vicinity of the Dirac points, the non-\nadiabatic Landau-Zener transition in graphene (dice lat-\ntice) induced by a constant electric field thus shares the\nsame quantum dynamical law as that in the adiabatic\nimpulse theory [11, 12].\nFor the general α-T3forα̸= 0,1, the picture of the\nquantum dynamical evolution as consisting of adiabatic\nevolution and non-adiabatic LZTs is still applicable, be-\ncause the eigenenergy spectrum is independent of the\nlattice coupling parameter α. In fact, adiabatic phase\nwith 0 < α < 1 is the same as that for α= 1. For\nthe non-adiabatic process, the first LZT has been nu-\nmerically calculated, as shown in Fig. 2. Consequently,\nunder the adiabatic impulse approximation, the dynami-\ncal evolution of the α-T3lattice is identical to that of the\nwavefunction of a strongly periodically driven three-level\nsystem.TABLE I. Electric field values at which the St¨ uckelberg phase\nis about πin Fig. 11\netBeE etBeE etBeE eE\nweaker\n100 0.0725 141 0.0514 193 0.0376 0.0072\n121 0.06 157 0.0462 198 0.0366 0.0065\n126 0.0576 162 0.0448 219 0.0332 0.006\n131 0.0554 167 0.0434 229 0.0317 0.0059\n136 0.0533 188 0.0386 260 0.0279 0.0053\nTABLE II. Electric field values at which the St¨ uckelberg phase\nis about zero in Fig. 11\netBeE etBeE eEweaker\n118 0.0614 175 0.0415 0.0069\n144 0.0504 180 0.0403 0.0068\n149 0.0487 185 0.0392 0.0067\n154 0.0471 206 0.0352 0.0062\n211 0.0344 247 0.0294 0.0042\n216 0.0336 257 0.0282 0.0035\n242 0.03 278 0.0261 0.003\nFIG. 11. Near zero or πSt¨ uckelberg phase. Shown is the\naverage St¨ uckelberg phase over the interval of the initial mo-\nmentum deviation ∆ ekyas determined by 0 < r≤1 from the\n+Kvalley for 41 values of the electric field, where the closed\ncircles are for the field values in Tab. I and the closed tri-\nangles correspond to the field values in Tab. II. The integer\nindex Ndenotes these 41 cases, with the electric field ranging\nfrom small to large values for both sets of data points. The\nintegration time step is det= 0.01.\nAppendix D: St¨ uckelberg phase with electric field\nBecause the St¨ uckelberg phase is near constant for\nLZTs starting from the + Kvalley, as exemplified in\nFig. 6(a), we define the average St¨ uckelberg phase over\nthe range of the momentum deviation as determined by\n0< r≤1. Although the St¨ uckelberg or adiabatic phase\nis sensitive to the magnitude of the electric field [Eqs. (53)\nand (55)], we numerically test two sets of electric fields to\nproduce the average St¨ uckelberg phase from the + Kval-\nley around πor zero, as shown in Fig. 11 for the electric17\nfield values listed in Tabs. I and II for α= 0:\n⟨ζ⟩ ≈ ⟨Φst⟩ ≈π+ 2kπor⟨ζ⟩ ≈2kπ. (D1)\nIn Tabs. I and II, the field values are determined based\non the destructive interference pattern of LZTs starting\nfrom the + Kvalley for α= 0 such as the orange traces\nin Figs. 5(a) and 5(b) and are tested in the range of\nmomentum deviation determined by 0 < r≤1. For\nα, two types of behaviors can arise. In the first type,forα > 0, for the electric field values from Tab. I, the\nLZT probability |αk|2can no longer reach zero after two\nsuccessive LZTs, i.e., no destructive interference. In this\ncase, the St¨ uckelberg phase is ⟨Φst⟩ ≈π, as shown by\nthe filled circles in Fig. 6(b). The second type occurs\nfor field values from Tab. II, where |αk|2still displays a\nnear-destructive interference pattern for any α > 0. In\nthis case, the St¨ uckelberg phase is ⟨Φst⟩ ≈0, as shown\nby the filled triangles in Fig. 6(b).\n[1] L. Landau, On the theory of transfer of energy at colli-\nsions II, Phys. Z. Sowjetunion 2, 118 (1932).\n[2] C. Zener, Non-adiabatic crossing of energy levels, Proc.\nR. 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Xu, W. S. Fegadolli, L. Gan, M.-H. Lu, X.-P. Liu,\nZ.-Y. Li, A. Scherer, and Y.-F. Chen, Experimental re-\nalization of Bloch oscillations in a parity-time synthetic\nsilicon photonic lattice, Nat. Commun. 7, 1 (2016)."    },    {        "title": "2402.14271v1.Hyers_Ulam_stability_of_the_first_order_difference_equation_with_average_growth_rate.pdf",        "content": "arXiv:2402.14271v1  [math.DS]  22 Feb 2024Hyers-Ulam stability of the first order\ndifference equation with average growth rate\nYoung Woo Nam\nCollege of Engineering, Konkuk University, 05029 Seoul, Ko rea\nnamyoungwoo@konkuk.ac.kr\nAbstract\nThe first order difference equation induced by the sequence of maps\nonChas Hyers-Ulam stability where the limit of the geometric av erage\nof growth rate is convergent and not equal to one. We show no Hy ers-\nUlam stability where the average growth rate is (pre)period ic even\nthough each periodic growth rate is strictly less than one. E xamples of\ndifference equation generated by time dependent maps which c ontains\ncontracting maps and expanding maps are given.\n1\n1 Introduction and preliminaries\nHyers-Ulam stability has been an approximation theory of fu nctional equa-\ntions, differential equations for decades. See [7], [8]. Hye rs-Ulam stability of\ndifference equation has been studied in some researchers, [1 , 2, 4, 5, 10, 12].\nMany of these topics are concentrated on the difference equat ion generated\nby a single map. For instance, contracting or repelling maps , or linear maps\nwith matrix has been considered. In this paper, we show Hyers -Ulam sta-\nbility of difference equations induced by time dependent map s. Hyers-Ulam\n12010Mathematical Subject Classification : Primary 39A20; Secondary 39A45.\nKey words and phrases : Hyers-Ulam stability; root test; difference equation; ave rage\ngrowth rate.\n1stability is abbreviated as HU stability. Let F:N×C→Cbe a function\nsuch that\n|Fn(u)−Fn(v)| ≤pn|u−v|\nfor every n∈N,u,v∈C. The positive numbers pnis called contracting\nrateofFn. Similarly, if the maps satisfy the inequalities\n|Fn(u)−Fn(v)| ≥pn|u−v|,\npnis called expanding rate . Both contracting and expanding rate are called\ngrowth rate ofFn. Thelimit of the geometric average of growth rate\nof allFnis defined as\nlim\nn→∞/parenleftBiggn/productdisplay\nj=1pj/parenrightBigg1\nn\nwhere all of pnis either contracting or expanding rate of Fn. Before taking\nthe limit, the above product is called (geometric) average growth rate . If\nthe limit exists, then the limit is called exponential growth rate . Note\nthat if the contracting rates of all Fnis bounded above by pless than one,\nthen difference equation has Hyers-Ulam stability.\nLemma 1.1 (Corollary3 in [9]) .LetF:N×C→Cbe a function which\nsatisfies that\n|F(n,u)−F(n,v)| ≤p|u−v|\nfor alln∈N,u,v∈Cand for0< p <1. For a given ε >0, suppose that\nthe complex valued sequence {an}n∈Nsatisfies the inequality\n|an+1−F(n,an)| ≤ε\nfor alln∈N. Then the sequence {bn}n∈Ndefined as bn+1=F(n,bn)satisfies\nthat\n|bn−an| ≤pn−1|b1−a1|+1−pn−1\n1−pε\nforn∈N.\nContracting or expanding maps on higher dimension has simil ar proper-\nties. For various shadowing properties of uniformly contra cting or expanding\ntime varying maps, see [14]. For exponential trichotomy of i nfinite dimen-\nsional maps, see [3]. However, the condition\n|F(n,u)−F(n,v)| ≤p|u−v|\n2for0< p <1in Lemma 1.1 is very sufficient for Hyers-Ulam stability. In\nthis paper, we show Hyers-Ulam stability with exponential g rowth rate in\none dimension. If the limit of the geometric average is expon ential, the dif-\nference equation has Hyers-Ulam stability even though each time dependent\nmap may have various contracting (or expanding) rate. This p aper is a\ncontinuation of [11].\n2 HU stability for exponential growth rate less\nthan one\nLemma 2.1. Letpnbe positive numbers for all n∈N. LetF:N×C→C\nbe a function which satisfies that\n|F(n,u)−F(n,v)| ≤pn|u−v|\nforu,v∈Cand for all n∈N. For a given ε >0, let{an}n∈Nbe a complex\nvalued sequence which satisfies the inequality\n|an+1−F(n,an)| ≤ε\nfor alln∈N. Then the sequence {bn}n∈Ndefined as bn+1=F(n,bn)satisfies\nthat\n|bn−an| ≤/parenleftBiggn−1/productdisplay\nj=1pj/parenrightBigg\n|b1−a1|+/parenleftBiggn−1/summationdisplay\nj=1n−1/productdisplay\ni=j+1pi/parenrightBigg\nε\nforn∈N.\nProof. Suppose for induction that\n|bn−1−an−1| ≤/parenleftBiggn−2/productdisplay\nj=1pj/parenrightBigg\n|b1−a1|+/parenleftBiggn−2/summationdisplay\nj=1n−2/productdisplay\ni=j+1pi/parenrightBigg\nε.\nForn= 1, the inequality trivially holds. The assumption and the abo ve\ninequality imply that\n|bn−an| ≤ |bn−F(n−1,an−1)|+|an−F(n−1,an−1)|\n≤ |F(n−1,bn−1)−F(n−1,an−1)|+|an−F(n−1,an−1)|\n=pn−1|bn−1−an−1|+ε\n3≤pn−1/braceleftBigg/parenleftBiggn−2/productdisplay\nj=1pj/parenrightBigg\n|b1−a1|+/parenleftBiggn−2/summationdisplay\nj=1n−2/productdisplay\ni=j+1pi/parenrightBigg\nε/bracerightBigg\n+ε\n=/parenleftBiggn−1/productdisplay\nj=1pj/parenrightBigg\n|b1−a1|+/parenleftBiggn−2/summationdisplay\nj=1n−1/productdisplay\ni=j+1pi/parenrightBigg\nε+ε\n=/parenleftBiggn−1/productdisplay\nj=1pj/parenrightBigg\n|b1−a1|+/parenleftBiggn−1/summationdisplay\nj=1n−1/productdisplay\ni=j+1pi/parenrightBigg\nε.\nThe proof is complete.\nLemma 2.2 (Lemma 5.4 in [11]) .Let{tn}n∈Nbe the sequence of positive\nreal numbers. Then the limit lim\nn→∞t1\nnn= 1is satisfied if and only if\nlim\nn→∞tnKn\n/summationtextn−1\nj=1tjKj=K−1\nforK >1.\nDefinition 2.3. Let{wn}n∈Nbe a sequence which satisfies the inequality\n|wn+1−gn(wn)| ≤ε\nfor a given ε >0and for all n∈N. The difference equation zn+1=gn(zn)\nis Hyers-Ulam stable if there exists a sequence {zn}n∈Nas follows\nzn+1=gn(zn)and|wn−zn| ≤G(ε)\nfor alln∈NandG(ε)→0asε→0.\nTheorem 2.4. Letpnbe positive numbers for all n∈N. LetF(n,z)be a\nfunction which satisfies that\n|F(n,u)−F(n,v)| ≤pn|u−v|\nforu,v∈Cand for every n∈N. Suppose that\nlim\nn→∞/parenleftBiggn/productdisplay\nj=1pj/parenrightBigg1\nn\n=1\nK<1.\nThen the difference equation bn+1=F(n,bn)has Hyers-Ulam stability.\n4Proof. The assumption is equivalent that\nlim\nn→∞/parenleftBiggn/productdisplay\nj=1p−1\nj/parenrightBigg1\nn\n=K >1.\nThus we may assume that/producttextn\nj=1p−1\nj=tnKnwherelimn→∞t1\nnn= 1.Lemma\n2.2 implies that\nlim\nn→∞/producttextn\nj=1p−1\nj\nn−1/summationtext\nj=1/producttextj\ni=1p−1\ni=K−1>0.\nThis equation is satisfied if and only if\nlim\nn→∞/parenleftBiggn/productdisplay\nj=1pj/parenrightBiggn−1/summationdisplay\nj=1j/productdisplay\ni=1p−1\ni= lim\nn→∞n−1/summationdisplay\nj=1n/productdisplay\ni=j+1pi=1\nK−1>0.\nRecall that/producttextn\ni=j+1piis a positive number for n∈N, that is,/summationtextn\nj=1/producttextn\ni=j+1pi\nis an increasing sequence. Thus we have the inequality as fol lows\nn/summationdisplay\nj=1n/productdisplay\ni=j+1pi=n−1/summationdisplay\nj=1n/productdisplay\ni=j+1pi+1≤1\nK−1+1 =K\nK−1(2.1)\nfor alln∈N. Let{an}n∈Nbe a sequence which satisfies the inequality\n|an+1−F(n,an)| ≤ε (2.2)\nfor alln∈N. Lemma 2.1 and the inequality (2.1) imply that\n|bn−an| ≤/parenleftBiggn−1/productdisplay\nj=1pj/parenrightBigg\n|b1−a1|+/parenleftBiggn−1/summationdisplay\nj=1n−1/productdisplay\ni=j+1pi/parenrightBigg\nε\n≤/parenleftBiggn−1/productdisplay\nj=1pj/parenrightBigg\n|b1−a1|+Kε\nK−1.\nFor each sequence {an}n∈Nwhich satisfies (2.2), define b1is the same as a1.\nHence, the difference equation bn+1=F(n,bn)has Hyers-Ulam stability.\n53 HU stability for exponential growth rate greater\nthan one\nDefinition 3.1. Letqnbe complex numbers for n∈N. LetF:N×C→C\nbe a function. Define qnas follows\nF(n,u)−F(n,v)\nu−v=qn(u,v)\nforu,v∈Cwhereu/ne}ationslash=v. IfF(n,·)is differentiable at u∈C, thenqn(u,u)\nis defined as follows\nqn(u,u) = lim\nv→uF(n,u)−F(n,v)\nu−v\nforu∈Cand forn∈N.\nObserve that qn(u,v) =qn(v,u)for allu,v∈C. IfF(n,·)is differentiable\non the set S⊂C, thenqnis continuous on S×S.\nLemma 3.2. Let{an}n∈Nbe a complex valued sequence which satisfies the\nequation\nan+1−F(n,an) =rn\nfor every n∈N. Then the sequence {bn}n∈Ndefined by the equation bn+1=\nF(n,bn)for every n∈Nsatisfies that\nbn−an=/parenleftBiggn−1/productdisplay\nj=1qj/parenrightBigg\n(b1−a1)−n−1/summationdisplay\nj=1rjn−1/productdisplay\ni=j+1qi\nwhereqj=qj(bj,aj)for each j∈N.\nProof. Forn= 1, the equation trivially holds. Suppose for induction that\nbn−1−an−1=/parenleftBiggn−2/productdisplay\nj=1qj/parenrightBigg\n(b1−a1)−n−2/summationdisplay\nj=1rjn−2/productdisplay\ni=j+1qi.\nThe assumption of the sequence {an}n∈Nand above equation imply that\nbn−an=bn−F(n−1,an−1)−an+F(n−1,an−1)\n6=F(n−1,bn−1)−F(n−1,an−1)−an+F(n−1,an−1)\n=qn−1(bn−1−an−1)−rn−1\n=qn−1/braceleftBigg/parenleftBiggn−2/productdisplay\nj=1qj/parenrightBigg\n(b1−a1)−n−2/summationdisplay\nj=1rjn−2/productdisplay\ni=j+1qi/bracerightBigg\n−rn−1\n=/parenleftBiggn−1/productdisplay\nj=1qj/parenrightBigg\n(b1−a1)−n−2/summationdisplay\nj=1rjn−1/productdisplay\ni=j+1qi−rn−1\n=/parenleftBiggn−1/productdisplay\nj=1qj/parenrightBigg\n(b1−a1)−n−1/summationdisplay\nj=1rjn−1/productdisplay\ni=j+1qi.\nThe proof is complete.\nDenoteFj◦Fj−1◦···◦F1byFjwhereF(j,z)is expressed as Fj(z)forj∈N.\nDefineF0as the identity map, that is, F0= Id.\nLemma 3.3. LetFnbe functions on Cfor every n∈N. For a given ε >0,\nlet{an}n∈Nis a given sequence defined as follows\nan+1−Fn(an) =rn\nwhere|rn| ≤εfor alln∈N. Let{zn}n∈Nis a given sequence defined by\nzn+1=Fn(zn)\nforn∈N. Assume that there exists pn>0which satisfies |Fn(u)−Fn(v)| ≥\npn|u−v|foru,v∈Cwhere the limit as follows\nlim\nn→∞/parenleftBiggn/productdisplay\nj=1pj/parenrightBigg1\nn\n=K >1.\nThen the following sum\n∞/summationdisplay\nj=1j/productdisplay\ni=1|ai−zi|\n|Fi(ai)−Fi(zi)|(3.1)\n7is convergent. The function\nQ(z) =a1+∞/summationdisplay\nj=1rjj/productdisplay\ni=1ai−Fi(z)\nFi(ai)−Fi(z)\nhas the fixed point for all sufficiently small ε >0.\nProof. Firstly, we show the convergence of the series (3.1). The ass umption\nyields the inequality\n|ai−zi|\n|Fi(ai)−Fi(zi)|≤1\npi. (3.2)\nObserve that\nlim\nn→∞/parenleftBiggn/productdisplay\nj=11\npj/parenrightBigg1\nn\n=1\nK<1.\nThus comparison and root tests implies the series (3.1) is co nvergent.\nSecondly, the existence of the fixed point of Q(z)results from the contraction\nmapping theorem. Denoteai−Fi(z)\nFi(ai)−Fi(z)byAitemporarily. Define the norm of\neachAias\n/bardblAi/bardbl= sup\nz/negationslash=ai|ai−Fi(z)|\n|Fi(ai)−Fi(z)|.\nSince the inequality (3.2) is independent of the point zi, the inequality /bardblAi/bardbl ≤\n1\npiholds for every i���N. Thus we obtain that\n|Q(u)−Q(v)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nj=1rjj/productdisplay\ni=1Ai(u)−∞/summationdisplay\nj=1rjj/productdisplay\ni=1Ai(v)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ε∞/summationdisplay\nj=1j/productdisplay\ni=1/bardblAi/bardbl·|u−v|\n≤ε∞/summationdisplay\nj=1j/productdisplay\ni=11\npi|u−v|.\nThe series/summationtext∞\nj=1/producttextj\ni=11\npiis convergent by root test. Then for sufficiently small\nε >0, the function Qis a contracting map. Hence, the contraction mapping\ntheorem implies the existence of the unique fixed point of Q.\n8HU stability depends on the maps Fnand the given ε >0. In many cases\nεmust be small enough positive number. In addition to these co nditions,\nHU stability may depend also on the set which contains the seq uence{an}\nora1. For example, if the sequence of the maps Fn(z) =znkwherenk\nis the positive integers for all k∈Nandsupk{nk}<∞. Assume that\nlimn→∞/producttextn\nk=1nk=K >1. HU stability is expected due to Theorem 3.4.\nHowever, HU stability is satisfied only on the exterior of the disk{z:|z| ≥ρ}\nfor a big enough ρ >0.\nLetSbe a set which contains the sequence {an}n∈Nfor allrjand for a given\nε >0. Assume also that inf\nj∈Ndist(aj,∂S)>0.\nTheorem 3.4. LetFnbe a complex valued function for each n∈N. For a\ngivenε >0, let{an}n∈Nis a given sequence defined as follows\nan+1−Fn(an) =rn\nwhere|rn| ≤εfor alln∈N. Letqn(u,v)be complex numbers for every n∈N\ndefined in Definition 3.1. Suppose that pj≤inf\nz∈S|qj(aj,z)|and the following\nlimit\nlim\nn→∞/parenleftBiggn/productdisplay\nj=1pj/parenrightBigg1\nn\n=K >1\nis satisfied. Then the difference equation bn+1=F(n,bn)has Hyers-Ulam\nstability.\nProof. Lemma 3.2 implies that\nbn−an=/parenleftBiggn−1/productdisplay\nj=1qj/parenrightBigg\n(b1−a1)−n−1/summationdisplay\nj=1rjn−1/productdisplay\ni=j+1qi\nwhereqn=qn(bn,an)forn∈N. Then Lemma 3.3 implies that b1can be\nchosen as follows\nb1=a1+∞/summationdisplay\nj=1rjj/productdisplay\ni=11\nqi\nwhereqi=qi(bi,ai). The difference bn−anis as follows\nbn−an=/parenleftBiggn−1/productdisplay\nj=1qj/parenrightBigg∞/summationdisplay\nj=1rjj/productdisplay\ni=11\nqi−n−1/summationdisplay\nj=1rjn−1/productdisplay\ni=j+1qi\n9=/parenleftBiggn−1/productdisplay\nj=1qj/parenrightBigg/parenleftBigg∞/summationdisplay\nj=1rjj/productdisplay\ni=11\nqi−n−1/summationdisplay\nj=1rjj/productdisplay\ni=11\nqi/parenrightBigg\n=/parenleftBiggn−1/productdisplay\nj=1qj/parenrightBigg/parenleftBigg∞/summationdisplay\nj=nrjj/productdisplay\ni=11\nqi/parenrightBigg\n=∞/summationdisplay\nj=nrjj/productdisplay\ni=n1\nqi.\nThen the absolute value |bn−an|is bounded as follows\n|bn−an| ≤ε∞/summationdisplay\nj=n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglej/productdisplay\ni=n1\nqi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ε∞/summationdisplay\nj=n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglej/productdisplay\ni=n1\npi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=ε∞/summationdisplay\nj=n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen−1/productdisplay\nj=1pjj/productdisplay\ni=11\npi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=ε/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen−1/productdisplay\nj=1pj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigg∞/summationdisplay\nj=n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglej/productdisplay\ni=11\npi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightBigg\n(3.3)\n=ε/parenleftbig\ntn−1Kn−1/parenrightbig∞/summationdisplay\nj=n1\ntjKj. (3.4)\nLetLis a positive number in (1,K), that is, K > L > 1. Denote tn/parenleftbigL\nK/parenrightbign\nbysnforn∈N. Since the inequality1\nKj−n+1<1\nLj−n+1is satisfied for all\nj≥n−1, the following inequality holds\n∞/summationdisplay\nj=ntn−1Kn−1\ntjKj<∞/summationdisplay\nj=ntn−1Ln−1\ntjLj\n=tn−1/parenleftbiggL\nK/parenrightbiggn−1\nKn−1∞/summationdisplay\nj=n1\ntj/parenleftbigL\nK/parenrightbigjKj\n=sn−1Kn−1∞/summationdisplay\nj=n1\nsjKj.\n10Observe that limn→∞sn= 0andlimn→∞(sn)1\nn=L\nK<1. In order to estimate\nε(tn−1Kn−1)/summationtext∞\nj=n1\ntjKj, we define C1function on the set of positive real\nnumbers, say g, which satisfies the following properties.\n1.g(x)>0for allx >0,\n2.g(j) =1\nsjfor every j∈N,\n3.limn→∞g(x) =∞, and\n4.limx→∞{g(x)}1\nx=K\nL>1.\nThe forth property is equivalent to the limit limx→∞ln(g(x))\nx= ln/parenleftbigK\nL/parenrightbig\nor\nlim\nx→∞g′(x)\ng(x)= lnK−lnL >0 (3.5)\nby L’Hospital’s rule. Consider the function g(x)K−xand calculate its deriva-\ntive/parenleftbig\ng(x)K−x/parenrightbig′=/bracketleftbiggg′(x)\ng(x)−lnK/bracketrightbigg\ng(x)K−x.\nThe equation (3.5) implies that there exists big enough x0such that the\ninequality|g′(x)|\ng(x)−lnK <0for allx > x 0. Thusgis decreasing on the\ninterval{x|x > x0}. Moreover, there exists a positive integer nas follows\n|g′(x)|\ng(x)≤1\n2lnK\nfor allx≥n−1≥x0with a suitable choice of L >√\nK. Then we may use\nthe comparison between infinite series and definite integral as follows\n∞/summationdisplay\nj=n1\nsjKj≤/integraldisplay∞\nn−1g(x)K−xdx. (3.6)\nThe definite integral on [n−1,∞)is estimated as follows\n/integraldisplay∞\nn−1g(x)K−xdx=/integraldisplay∞\nn−1g(x)e−xlnKdx\n=/bracketleftbigg\n−1\nlnKg(x)e−xlnK/bracketrightbigg∞\nn−1+1\nlnK/integraldisplay∞\nn−1g′(x)e−xlnKdx\n11≤1\nlnKg(n−1)e−(n−1)lnK+1\n2/integraldisplay∞\nn−1g(x)K−xdx. (3.7)\nThus an upper bound of the integral (3.7) is as follows\n/integraldisplay∞\nn−1g(x)K−xdx≤2\nlnK1\nsn−1K−(n−1).\nThe estimations (3.6) are applied to the difference |bn−an|in the equation\n(3.4). Then we have that\n|bn−an| ≤ε/parenleftbig\ntn−1Kn−1/parenrightbig∞/summationdisplay\nj=n1\ntjKj\n< ε/parenleftbig\nsn−1Kn−1/parenrightbig∞/summationdisplay\nj=n1\nsjKj\n≤ε/parenleftbig\nsn−1Kn−1/parenrightbig/integraldisplay∞\nn−1g(x)K−xdx\n=ε/parenleftbig\nsn−1Kn−1/parenrightbig2\nlnK1\nsn−1K−(n−1)\n=2ε\nlnK,\nthat is, we obtain an upper bound of difference between bnandanas follows\n|bn−an| ≤2ε\nlnK(3.8)\nfor all sufficiently large n∈N. An upper bound of |bN−aN|is estimated for\nN= 1,2,...,n−1by the equation (3.3) and the estimation (3.8) as follows\n|bN−aN| ≤ε∞/summationdisplay\nj=Nj/productdisplay\ni=N|q−1\ni| ≤ε∞/summationdisplay\nj=Nj/productdisplay\ni=N|p−1\ni|\n=ε/parenleftBiggn/summationdisplay\nj=Nj/productdisplay\ni=N|p−1\ni|+∞/summationdisplay\nj=n+1j/productdisplay\ni=N|p−1\ni|/parenrightBigg\n=ε/bracketleftBiggn/summationdisplay\nj=Nj/productdisplay\ni=N|p−1\ni|+/parenleftBiggn/productdisplay\ni=N|p−1\ni|/parenrightBigg∞/summationdisplay\nj=n+1j/productdisplay\ni=n+1|p−1\ni|/bracketrightBigg\n12≤ε/bracketleftBiggn/summationdisplay\nj=Nj/productdisplay\ni=N|p−1\ni|+/parenleftBiggn/productdisplay\ni=N|p−1\ni|/parenrightBigg\n2\nlnK/bracketrightBigg\nforN= 1,2,...,n−1. Hence, the difference equation bn+1=F(n,bn)has\nHyers-Ulam stability.\nRemark 3.5. The unpper bound of |bn−an|in Theorem 3.4 would be better\nwith a particular assumption of tn. For example, if we assume the bounded-\nness oftn, that is, 0< m= infn∈N{tn} ≤supn∈N{tn}=M <∞, then an\nupper bound of |bn−an|is as follows\n|bn−an| ≤ε/parenleftbig\ntn−1Kn−1/parenrightbig∞/summationdisplay\nj=n1\ntjKj\n≤εMKn−1∞/summationdisplay\nj=n1\nmKj\n=Mε\nmKn−11\nKn·1\n1−1\nK\n=Mε\nm(K−1),\nwhich is a smaller upper bound than that of (3.8)for large K >1.\n4 No HU stability for periodic average growth\nrate\nThe sequence {an}ispreperiodic where there is a positive integer psuch that\nal+m=alfor every l≥N. IfN= 1, then the sequence is called the periodic\nsequence. The number mis said to be the period of sequence.\nLemma 4.1. Letpnbe positive numbers for every n∈N. LetF(n,z)be a\nfunction which satisfies that\n|F(n,u)−F(n,v)| ≤pn|u−v|\nforu,v∈Cand for every n∈N. Suppose that the values/parenleftBig/producttextn\nj=1pj/parenrightBig1\nn\nare preperiodic and the maximum of the values is less than one . Then the\ndifference equation bn+1=F(n,bn)has no Hyers-Ulam stability.\n13Proof. Let1\nK1,1\nK2,...,1\nKmbe all possible values of/parenleftBig/producttextn\nj=1pj/parenrightBig1\nnfor every n >\nN. Letm≥2be the period of the sequence/braceleftBig\n1\nKn/bracerightBig\nn≥0. Denote nbykm+l\nwherel= 1,2,...,m for some k∈N. Letk1be the least positive number\nwhich satisfies that k1m+l > N . ThusKl=Kkm+lfor every k= 0,1,2,....\nThen we may assume that\nkm+l/productdisplay\nj=N+1pj=1\n(Kl)km+l−N\nfor every non negative integer ksatisfying km+l > N . We claim that\n/summationtextn\nj=1/producttextn\ni=j+1piis unbounded for n∈N. It suffice to consider/parenleftBig/producttextn\nj=1pj/parenrightBig1\nnis\nperiodic, that is, N= 0, because the first finite terms of the series does not af-\nfect convergence or divergence of the given series. Denote max/braceleftBig\n1\nK1,1\nK2,...,1\nKm/bracerightBig\nby1\nKpwhich is less than one. Thus there exists1\nKqless than1\nKpwherep/ne}ationslash=q.\nThe sum,/summationtextn\nj=1/producttextn\ni=j+1pi, saysn, wheren=km+pfork∈Nis estimated\nby the rearrangement of each terms as follows\nkm+p/summationdisplay\nj=1km+p/productdisplay\ni=j+1pi=km+p/summationdisplay\nj=1(Kj)j\n(Kp)km+p=1\n(Kp)km+pkm+p/summationdisplay\nj=1(Kj)j\n=1\n(Kp)km+p/bracketleftBiggk/summationdisplay\nj=0(K1)jm+1+k/summationdisplay\nj=0(K2)jm+2+···+k/summationdisplay\nj=0(Kp)jm+p\n+k−1/summationdisplay\nj=0(Kp+1)jm+p+1+···+k−1/summationdisplay\nj=0(Km)jm+m/bracketrightBigg\n≥1\n(Kp)km+pk−1/summationdisplay\nj=0(Kq)jm+q\n=1\n(Kp)km+p·(Kq)q((Kq)km−1)\n(Kq)m−1\n>1\n(Kp)km+p·(Kq)q(Kq)km\n(Kq)m\n=/parenleftbiggKq\nKp/parenrightbiggkm(Kq)q\n(Kp)p+m. (4.1)\n14The fact that Kq> Kpimplies the above inequality (4.1) is unbounded for\nk∈N. Since a subsequence {skm+p}k∈Nof{sn}n∈Nis unbounded, so is the\nsequence {sn}n∈N. Let{an}n∈Nbe a sequence which satisfies the inequality\nan+1−F(n,an) =rn\nfor every n∈Nwhere|rn| ≤εfor alln∈N. Let{bn}n∈Nbe the sequence\ndefined by the equation bn+1=F(n,bn)for every n∈N. Lemma 3.2 implies\nthat\nbn−an=/parenleftBiggn−1/productdisplay\nj=1qj/parenrightBigg\n(b1−a1)−n−1/summationdisplay\nj=1rjn−1/productdisplay\ni=j+1qi\nwhereqj=qj(bj,aj)for each j∈N. We may choose rn=εfor alln∈N.\nThen\n|bn−an|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen−1/productdisplay\nj=1qj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle|b1−a1| ≥/parenleftBiggn−1/summationdisplay\nj=1n−1/productdisplay\ni=j+1pi/parenrightBigg\nε. (4.2)\nObserve that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen−1/productdisplay\nj=1qj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen−1/productdisplay\nj=1pj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftbigg1\nKp/parenrightbiggn−1\n<1,\nwhich is bounded for all n∈N. Thus for any choice of b1, the second term\nof the left side of equation (4.2) is bounded. However, the eq uation (4.1)\nimplies that/vextendsingle/vextendsingle/vextendsingle/summationtextn−1\nj=1/producttextn−1\ni=j+1pi/vextendsingle/vextendsingle/vextendsingleis unbounded for n∈N. Then|bn−an|is also\nunbounded for n∈N. Hence, the difference equation bn+1=F(n,bn)has no\nHyers-Ulam stability.\nTheorem 4.2. Letpnbe positive numbers for all n∈N. LetF(n,z)be a\nfunction which satisfies that\n|F(n,u)−F(n,v)| ≤pn|u−v|\nforu,v∈Cand for every n∈N. Denote nbykm+lwherekis a non-\nnegative number, m≥2is some positive integer and l= 1,2,...,m . Suppose\nthat the values/parenleftBig\nCn/producttextn\nj=1pj/parenrightBig1\nnforn∈Nis preperiodic with period mwhere\npositive constants Ckm+l=Clis for all non-negative integer k. Suppose also\nthat the maximum of preperiodic values is less than one. Then the difference\nequation bn+1=F(n,bn)has no Hyers-Ulam stability.\n15Proof. The proof is similar to that of Lemma 4.1. We use the same notio ns\nfor the proof. Thus we obtain that\nClkm+l/productdisplay\nj=N+1pj=1\n(Kl)km+l−N(4.3)\nfor every k= 0,1,2,.... Let the numbers1\nK1,1\nK2,...,1\nKmbe all possible\npositive values of (4.3) and denote the maximum of these numb ers by1\nKp.\nSincem≥2, there exists Kq> Kpfor some q/ne}ationslash=p. We claim that/summationtextkm+p\nj=N+1/producttextkm+p\ni=j+1piis unbounded for k. We may assume that the preperi-\nodic values is periodic, that is, N= 0because the first finite terms of the\npartial sums does not affect the convergence or divergence of the given series.\nThe similar calculation of (4.1) in Lemma 4.1 yields the foll owing estimations\nkm+p/summationdisplay\nj=1km+p/productdisplay\ni=j+1pi=1\nCpkm+p/summationdisplay\nj=1(Kj)j\n(Kp)km+p=1\nCp1\n(Kp)km+pkm+p/summationdisplay\nj=1(Kj)j\n≥1\nCp(Kp)km+pk−1/summationdisplay\nj=0(Kq)jm+q\n>1\nCp(Kp)km+p·(Kq)q(Kq)km\n(Kq)m\n=/parenleftbiggKq\nKp/parenrightbiggkm(Kq)q\nCp(Kp)p+m, (4.4)\nwhich is unbounded for k∈N. For any given ε >0, let{an}n∈Nbe a sequence\nwhich satisfies the inequality\nan+1−F(n,an) =ε\nfor alln∈N. Lemma 3.2 and triangular inequality imply that\n|bn−an|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen−1/productdisplay\nj=1qj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle|b1−a1| ≥/parenleftBiggn−1/summationdisplay\nj=1n−1/productdisplay\ni=j+1pi/parenrightBigg\nε (4.5)\nwhereqj=qj(aj,bj)for each j∈N. The right side of the equation (4.5) is\nunbounded for n∈Nby the estimation (4.4). Hence, the difference equation\nbn+1=F(n,bn)has no Hyers-Ulam stability.\n165 Applications\nExample 5.1. Letf(n,z)be the functions as follows\nf(n,z) =/braceleftBigg\n2z, n= 1,3,5,...\n1\n3z, n= 2,4,6,....\nThus|f(n,u)−f(n,v)| ≤pn|u−v|where\npn=/braceleftBigg\n2, n= 1,3,5,...\n1\n3, n= 2,4,6,....\nThus we obtain that lim\nn→∞/parenleftBiggn/productdisplay\nj=1pj/parenrightBigg1\nn\n=/radicalbigg\n2\n3because\nlim\nk→∞/parenleftBigg2k/productdisplay\nj=1pj/parenrightBigg1\n2k\n= lim\nk→∞/parenleftBiggk/productdisplay\nj=12\n3/parenrightBigg1\n2k\n=/radicalbigg\n2\n3\nforn= 2k, k= 1,2,3,...and moreover\nlim\nk→∞/parenleftBigg2k+1/productdisplay\nj=1pj/parenrightBigg1\n2k+1\n= lim\nk→∞/parenleftBigg\n2k/productdisplay\nj=12\n3/parenrightBigg1\n2k+1\n=/radicalbigg\n2\n3\nforn= 2k+ 1, k= 0,1,2,.... Theorem 2.4 implies that the difference\nequation bn+1=f(n,bn)has Hyers-Ulam stability. Let us choose that b1=a1.\nThen we obtain that\n|bn−an| ≤/radicalBig\n3\n2ε\n/radicalBig\n3\n2−1=√\n3ε√\n3−√\n2=���\n3/parenleftBig√\n3+√\n2/parenrightBig\nε=/parenleftBig\n3+√\n6/parenrightBig\nε <6ε\nfor alln∈N. Since|bn−an|<6εfor alln∈N, the difference equation\nbn+1=f(n,bn)has Hyers-Ulam stability.\n17Example 5.2. Letg(n,z)be the functions as follows\ng(n,z) =/braceleftBigg\n3nz, n= 1,3,5,...\n1\n2nz, n= 2,4,6,....\nThus|g(n,u)−g(n,v)| ≤pn|u−v|where\npn=/braceleftBigg\n3n, n= 1,3,5,...\n1\n2n, n= 2,4,6,....\nThus we obtain that\nn/productdisplay\nj=1pj=\n\n3k\n2k·1·3·5···(2k−1)\n2·4·6···(2k), n = 2k, k = 1,2,3,...\n3k+1\n2k·1·3·5···(2k−1)(2k+1)\n2·4·6···(2k), n= 2k+1, k= 1,2,3,....\nIn order to calculate the limit of/producttextn\nj=1pj, we use the following inequalities\n1√\n4k+1≤1·3·5···(2k−1)\n2·4·6···(2k)≤1√\n3k+1(5.1)\n3√\n2√\n2k+1≤3(2k+1)√\n4k+1≤31·3·5···(2k−1)\n2·4·6···(2k)(2k+1)≤3(2k+1)√\n3k+1≤3√\n2k+1\nfor every k∈N. Observe that for any rational expression Q(k), the limit\nlimk→∞/parenleftBig/radicalbig\nQ(k)/parenrightBig1\nk= 1is satisfied. Since the squeezing of the limits as\nfollows\nlim\nk→∞/parenleftBigg\n1√\n4k+1/parenleftbigg3\n2/parenrightbiggk/parenrightBigg1\n2k\n≤lim\nk→∞/parenleftBigg2k/productdisplay\nj=1pj/parenrightBigg1\n2k\n≤lim\nk→∞/parenleftBigg\n1√\n3k+1/parenleftbigg3\n2/parenrightbiggk/parenrightBigg1\n2k\nlim\nk→∞/parenleftBigg\n3√\n2√\n2k+1/parenleftbigg3\n2/parenrightbiggk/parenrightBigg1\n2k+1\n≤lim\nk→∞/parenleftBigg2k+1/productdisplay\nj=1pj/parenrightBigg1\n2k+1\n≤lim\nk→∞/parenleftBigg\n3√\n2k+1/parenleftbigg3\n2/parenrightbiggk/parenrightBigg1\n2k+1\nwe obtain the limit\nlim\nn→∞/parenleftBiggn/productdisplay\nj=1pj/parenrightBigg1\nn\n=/radicalbigg\n3\n2.\n18Then Theorem 3.4 implies that the difference equation bn+1=g(n,bn)has\nHyers-Ulam stability.\nExample 5.3. Leth(n,z)be the functions as follows\nh(n,z) =\n\n2nz, n= 1,3,5,...\n1\n2n+3z, n= 2,4,6,....\nThus|h(n,u)−h(n,v)| ≤pn|u−v|where\npn=\n\n2n, n= 1,3,5,...\n1\n2n+3, n= 2,4,6,....\nThus we obtain the product of pnas follows by straightforward calculation\nn/productdisplay\nj=1pj=\n\n1\n2n−2n= 1,3,5,...\n1\n22n, n= 2,4,6,....\nThen we have that\n/parenleftBigg\n42k−1/productdisplay\nj=1pj/parenrightBigg1\n2k−1\n=1\n2and/parenleftBigg2k/productdisplay\nj=1pj/parenrightBigg1\n2k\n=1\n4\nfor allk∈N. The maximum of the set/braceleftbigg/parenleftBig\n4/producttext2k−1\nj=1pj/parenrightBig1\n2k−1,/parenleftBig/producttext2k\nj=1pj/parenrightBig1\n2k/bracerightbigg\nis1\n2, which is less than one. Hence, Theorem 4.2 implies that the d ifference\nequation bn+1=h(n,bn)has no Hyers-Ulam stability.\nExample 5.4. Letf(n,x)be the function from RtoRdefined as follows\nf(n,x) = 3x+1\nnsin/parenleftBigx\nn/parenrightBig\nforn∈N. Thusqnoff(n,x)defined in Definition 3.1 is as follows\nqn(u,v) = 3+1\nn2sin/parenleftbigu\nn/parenrightbig\n−sin/parenleftbigv\nn/parenrightbig\nu\nn−v\nn\n19forn∈N. Thus Mean Value Theorem implies that\nqn(u,v)≥3−1\nn2\nfor every n∈N. Denote 3−1\nn2bypn. Since the product of numbers\n1\n3n/producttextn\nj=1/parenleftbig\n3−1\nn2/parenrightbig\nis convergent as n→ ∞, we obtain that\nlim\nn→∞/parenleftBiggn/productdisplay\nj=1pj/parenrightBigg1\nn\n= 3>1.\nHence, Theorem 3.4 implies that the difference equation bn+1=f(n,bn)has\nHyers-Ulam stability.\nNote and discussion\nEquation (5.1) in Example 5.2 can be shown by induction. The r atio of\nproduct of even and odd integers can be expressed as the ratio of two gamma\nfunctions. For various and refined evaluation, see [13].\nIf the complex analytic function Fsatisfies|F(u)−F(v)| ≤C|u−v|for\nsomeC >0and for all u,v∈C, thenFis a polynomial of degree one by\nLiouville’s theorem. Thus in most cases we require a domain Swhich is not\nC, which is (forward) invariant under all given functions, th at is,Fn(S)⊂S\nfor alln∈N. For example, let fandgbe polynomials of which degree is\ngreater than or equal to two. Assume that the filled Julia set o ffcontains\nthat ofg. Thus we may choose the Fatou component of ∞as an invariant\nset under fandg, sayS, rather than exterior of the disk with sufficiently\nbig radius. If we choose Fnas the arbitrary composition of functions fand\ngfor every n∈N, then the inequality |Fn(u)−Fn(v)| ≥pn|u−v|for all\nn∈Nandinf{pn}>1. The positive number εmay depend on the distance\nbetween a1and∂S. For the dynamics of polynomial semigroup, see [15].\nHU stability of difference equation induced one dimensional map may be\ngeneralized on higher dimension. For instance, the matrix d ifference equation\nis defined as follows\n/bracketleftbiggzn+1\nwn+1/bracketrightbigg\n=/bracketleftbiggpn0\n0qn/bracketrightbigg/bracketleftbiggzn\nwn/bracketrightbigg\n20wherepnandqnare coefficients of maps defined in Example 5.1 and Exam-\nple 5.2. Thus the above matrix difference equation has HU stab ility because\ndifference equations zn+1=fn(zn)andzn+1=gn(zn)have HU stability.\nHowever, decomposed equations are not induced by either con tracting or ex-\npanding maps.\nReferences\n[1] D. Anderson, M. Onitsuka, Best constant for Hyers-Ulam s tability of\nsecond order h-difference equations with constant coefficien ts, Results\nin Mathematics (2019)\n[2] D. Anderson, M. Onitsuka, J.M.Rassias, Best constant fo r Ulam stability\nof first order h-difference equations with periodic coefficien t, Journal of\nMathematical Analysis and Applications 491 (2020),no.2,1 24363, 15pp\n[3] L. Backes, D. Dragičević, Shadowing for infinite dimensi onal dynamics\nand exponential trichotomies, Proceedings of the Royal Soc iety of Ed-\ninburg Section A:Mathematics,(2021) Volume 151, Issue 3, p p. 863-884,\nDOI:https://doi.org/10.1017/prm.2020.42\n[4] A.-R. Baias, F.Blaga and D.Popa, On the best Ulam constan t of a first\norder linear difference equation in Banach spaces, Acta Math ematica\nHungaria 163(2021), 563-575.\n[5] J. Brzdęk, K. Cieplinski, Z. Lesniak, On Ulam’s type stab ility of the\nlinear equation and related issues, Discrete Dynamics in Na ture and\nSociety (2014) Article ID 536791.\n[6] C. Buşe, D. O’Regan, O. Saierli, Hyers-Ulam Stability fo r Linear Differ-\nences with Time Dependent and Periodic Coefficients:The Case When\nthe Monodromy Matrix Has Simple Eigenvalues, Symmetry (201 9),\n11(3), 339, https://doi.org/10.3390/sym11030339\n[7] D. H. Hyers, On the stability of the linear functional equ ation, Proceed-\ning of National Academy Sciences of the United States of Amer ica 27\n(1941) 222–224.\n21[8] J. Huang, S.-M. Jung, Y. Li, On Hyers-Ulam stability of no nlinear dif-\nferential equations, Bulletin of Korean Mathematical Soci ety 52 (2015),\nNo.2, 685-697, http://dx.doi.org/10.4134/BKMS.2015.52 .2.685\n[9] S.-M. Jung, Y. W. Nam, On the Hyers-Ulam stability of the\nfirst order difference equation, Journal of Function Spaces ( 2016)\nhttp://dx.doi.org/10.1155/2016/6078298.\n[10] Y.W. Nam, Hyers-Ulam stability of loxodromic Möbius di fference equa-\ntion, Applied Mathematics and Computation, 356(1) (2019), 119-136,\nhttps://doi.org/10.1016/j.amc.2019.03.033\n[11] Y.W. Nam, Hyers-Ulam stability of the first order differ-\nence equation generated by linear maps, Preprint 2022,\nhttp://arxiv.org/abs/2101.02364v6\n[12] D. Popa, Hyers-Ulam stability of the linear recurrence with constant\ncoefficients, Advances in Difference Equations (2005) 101–10 7.\n[13] F. Qi, Q.M. Luo, Bounds for the ratio of two gamma functio ns:\nfrom Wendel’s asymptotic relation to Elezović-Giordano-P ečarić’s\ntheorem, Journal of Inequalities and Applications 2013, 20 13:542,\nhttp://www.journalofinequalitiesandapplications.com/ content/2013/1/542\n[14] J. N. Sarkooh, Various shadowing properties for time va rying\nmaps, Bull. Korean Math. Soc. 59(2022), No 2, pp. 481-506,\nhttps://doi.org/10.4134/BKMS.b210359\n[15] R. Stankewitz, H. Sumi, Dynamical properties and struc ture of Julia\nsets of postcritically bounded polynomial semigroups, Tra nsactions of\nthe american mathematical society, Volume 363, Number 10, O ctober\n2011, Pages 5293-5319\n22"    },    {        "title": "2402.14331v1.Convergence_of_cones_of_metric_measure_spaces_and_its_application_to_Cauchy_distribution.pdf",        "content": "arXiv:2402.14331v1  [math.MG]  22 Feb 2024CONVERGENCE OF CONES OF METRIC MEASURE SPACES\nAND ITS APPLICATION TO CAUCHY DISTRIBUTION\nSYOTA ESAKI, DAISUKE KAZUKAWA, AND AYATO MITSUISHI\nAbstract. We prove that the sequence of cones of metric measure spaces co nverges if the\nsequence of base spaces converges in Gromov’s box, concentrat ion, and weak topologies.\nAs an application, we show that the generalized Cauchy distribution w ith suitable scaling\nconverges to a half line in the concentration topology as the dimensio n diverges to infinity.\nThis is a new example distinguished from previously known examples suc h as Gaussian\ndistributions and typical closed Riemannian manifolds with constant R icci curvature.\n1.Introduction\nTheconcentrationofmeasurephenomenonhasbeenattractinga ttentioninrecentyears. It\nwas first discovered by P. L´ evy for the unit spheres, and then wa s put forward by V. Milman\nin the 1970’s. [ 10,11,14] are listed for reference. Roughly speaking, the concentration o f\nmeasure phenomenon is explained as “any 1-Lipschitz function is almo st a constant”, which\nis oftenobserved for high-dimensional spaces. For example, onth en-dimensional unit sphere\nSninRn+1, we have\n(1.1) σn({θ∈Sn| |f(θ)−mf| ≥ε})≤2e−n−1\n2ε2\nfor any 1-Lipschitz function fwith median mfand anyε>0, whereσnis the uniform prob-\nability measure on Sn. This phenomenon strongly reflects the geometry of high dimension s.\nTaking a 1-Lipschitz function fonSnas the distance from a Borel subset Awithσn(A)≥1\n2,\n(1.1) implies\nσn(Uε(A))≥1−2e−n−1\n2ε2,\nwhereUε(A) is the open ε-neighborhood of A. More generally, this phenomenon is observed\nfor closed Riemannian manifolds whose Ricci curvature diverges to in finity (see [ 6]).\nFor the study of the concentration of measure phenomenon, Gro mov provided a revolu-\ntionaryconcept in[ 5, Chapter 3.1\n2+]. Heintroduced a distance function, calledthe observable\ndistance, on the set Xof isomorphism classes of metric measure spaces. The induced topo l-\nogy is called the concentration topology . The classical concentration of measure phenomenon\ncorresponds to convergence to a one-point metric measure spac e with respect to the con-\ncentration topology. Such a sequence of metric measure spaces is called aL´ evy family . For\nexample, the sequence {Sn}∞\nn=1of unit spheres is a L´ evy family. Thus the concentration\ntopology gives a characteristic notion of convergence of metric me asure spaces capturing the\nhigh-dimensional aspects. It is also interesting as a generalization o f the celebrated mea-\nsured Gromov-Hausdorff convergence. Indeed, any measured G romov-Hausdorff convergent\nsequence concentrates (i.e., converges with respect to the conc entration topology).\nFurthermore, Gromov also introduced a natural compactification , denoted by Π, of X\nwith respect to the concentration topology at the same time, which is one of powerful tools\nto study the concentration topology. The topology of this compac tification is called the\nDate: February 22, 2024.\n2020Mathematics Subject Classification. 53C23, 60B12, 53C21, 60A10.\nKey words and phrases. metric measure space, concentration of measure phenomenon, c oncentration\ntopology, cone, Cauchy distribution.\n12 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\nweak topology and each element of Π is called a pyramid. The space Π contains many\ninfinite-dimensional objects, for example, the (virtual) infinite-dim ensional Gaussian space\nand infinite-dimensional cube. Pyramids themselves are also worth s tudying.\nIn recent years, the study of the concentration and weak topolo gies is rapidly growing.\nEspecially, the discovery of new concentrating sequences is signific ant as the generalization\nof the concentration of measure phenomenon. Shioya [ 20,21] proved that the n-dimensional\nsphereSn(√n) inRn+1with radius√nconverges to the infinite-dimensional Gaussian space\nin the compactification Π. In addition, the limit spaces of such as proj ective spaces, ellip-\nsoids, and Stiefel manifolds whose dimension diverges to infinity have been determined under\nappropriate scales (see [ 8,9,22]).\nIn this paper, we study the convergence of cones of metric measu re spaces and pyramids\nandapply theresult to find new examples ofconcentrating sequenc es. Westart by describing\nour setting and main results.\nDefinition 1.1 (κ-Cone).Theκ-coneKκ(X) of a metric space ( X,dX) forκ∈Ris the\nquotient space\nKκ(X) :=Iκ×X/∂Iκ×X\nwith metric dKκ(X), whereIκis the interval given by\nIκ:=/braceleftBigg\n[0,π/√κ] ifκ>0,\n[0,∞) ifκ≤0,\nand∂Iκis the set of end points of Iκ, that is,∂Iκ={0,π/√κ}ifκ >0 and∂Iκ={0}if\nκ≤0. Letq:Iκ×X→Kκ(X) be the quotient map. The metric dKκ(X)is given by\n(1.2) cκ(dKκ(X)(rx,r′x′)) =cκ(r)cκ(r′)+κsκ(r)sκ(r′)cos(min {dX(x,x′),π})\nforrx=q(r,x),r′x′=q(r′,x′)∈Kκ(X), where sκis the unique solution to\ns′′+κs= 0, s(0) = 0, s′(0) = 1,\nandcκis defined by cκ:=s′\nκ. Note that sκ,cκare explicitly expressed as\nsκ(r) :=\n\n1√κsin(√κr) ifκ>0,\nr ifκ= 0,\n1√−κsinh(√−κr) ifκ<0,cκ(r) :=\n\ncos(√κr) ifκ>0,\n1 if κ= 0,\ncosh(√−κr) ifκ<0.\nIn the case of κ= 0, we interpret ( 1.2) as the limit formula\ndK0(X)(rx,r′x′)2=r2+(r′)2−2rr′cos(min{dX(x,x′),π})\nasκ→0. Moreover, forametricmeasurespace( X,dX,mX)andaBorelprobabilitymeasure\nµonIκ, we define a metric measure space Kκ,µ(X) by the metric space ( Kκ(X),dKκ(X))\nequipped with the push-forward measure q#(µ⊗mX) and call it the κ-cone ofXwith\nrespect toµ.\nFurthermore, foranypyramid P ∈Π, wecandefinethe κ-coneKκ,µ(P)∈ΠofPnaturally.\nSee Definition 3.4for the precise definition of Kκ,µ(P).\nThe following theorem is the main result in this paper.\nTheorem 1.2. Letκ∈R. Assume that a sequence {Pn}∞\nn=1of pyramids converges weakly\nto a pyramid Pand that a sequence {µn}∞\nn=1of Borel probability measures on Iκconverges\nweakly to a Borel probability measure µ. Then the sequence {Kκ,µn(Pn)}∞\nn=1ofκ-cones\nconverges weakly to Kκ,µ(P).\nIn particular, we obtain the following as a corollary of Theorem 1.2directly.CONVERGENCE OF CONES OF METRIC MEASURE SPACES 3\nTheorem 1.3. Letκ∈Rand let{Xn}∞\nn=1be a L´ evy family. Assume that a sequence\n{µn}∞\nn=1of Borel probability measures on Iκconverges weakly to a Borel probability measure\nµ. Then the sequence {Kκ,µn(Xn)}∞\nn=1ofκ-cones concentrates to the interval (Iκ,|·|,µ).\nNote that the interval ( Iκ,|·|,µ) is isomorphic to the κ-cone of a one-point metric measure\nspace with respect to µ. In Section 4, we also give another proof to Theorem 1.3in a very\nsimple way that is limited to this case. In this proof, we prove that “an y 1-Lipschitz function\nonKκ,µn(Xn) is almost a radial function if nis large”. This is exactly a philosophy based\non the concentration of measure phenomenon.\nAs an application of our results, we discover a very natural new exa mple of concentrating\nsequences. This isthegeneralized Cauchy distribution withone-par ameter forconcavity. For\na givenβ >0, we define a Borel probability measure νn\nβon the Euclidean space ( Rn,/ba∇dbl·/ba∇dbl)\nby\n(1.3) νn\nβ(dx) :=Γ(n+β\n2)\nπn\n2Γ(β\n2)dx\n(1+/ba∇dblx/ba∇dbl2)n+β\n2\nand call it the n-dimensional generalized Cauchy distribution (ormeasure)with exponent\nβ. This name is quoted from [ 2] and this is also called the multivariate t-distribution in\nsome contexts. From now on, we call them the Cauchy distributions simply unless otherwise\nstated. We denote by Cn\nβthen-dimensional Euclidean space equipped with the Cauchy\ndistribution νn\nβand callCn\nβthen-dimensional Cauchy space with exponent β.\nThe Cauchy distribution with β= 1 is well-known along with the Gaussian distribution.\nThen-dimensional Cauchy distribution νn\n1is the law of random vector ( X1/X0,...,Xn/X0),\nwhere random variables Xi,i= 0,1,...,n, are independent and identically distributed by\nthe standard Gaussian distribution. The distribution νn\n1is one of the stable distributions,\nas is the Gaussian distribution. Moreover, it is also used in statistics a nd physics, so is of\nparticular important in recent years.\nWe obtain the following result. Here, we denote by tXthet-scaled space ( X,tdX,mX) of\na metric measure space X= (X,dX,mX) fort>0.\nTheorem 1.4. The sequence {1√nCn\nβ}∞\nn=1concentrates to the half line ([0,∞),|·|)equipped\nwith the Borel probability measure\n(1.4) νβ(dt) :=21−β\n2\nΓ(β\n2)1\ntβ+1e−1\n2t2dt.\nSurprisingly, the sequence {1√nCn\nβ}∞\nn=1concentrates to a one-dimensional space, neither to\naone-pointnortoaninfinite-dimensional space. BehindTheorem 1.4, thereisthephilosophy\nthat “any (1 /√n)-Lipschitz function on Cn\nβis almost a radial function if nis large”. The\nlimit measure νβis also of interest. If β= 1, the limit measure ν1is called the absolutely\nreciprocal normal distribution, that is, ν1is the law of the random variable 1 /|X|provided\nthat the law of Xis the standard one-dimensional Gaussian distribution.\nTheorem 1.4can be stated more precisely as follows.\nTheorem 1.5. Let{rn}∞\nn=1be a sequence of positive real numbers. Then we have the fol-\nlowing(1) – (3).\n(1)Ifrn√n→0, then the sequence {rnCn\nβ}∞\nn=1box-converges to a one-point metric mea-\nsure space.\n(2)Ifrn√n→λ∈(0,∞), then the sequence {rnCn\nβ}∞\nn=1concentrates to the half line\n([0,∞),|·|)equipped with the Borel probability measure\nνβ,λ(dt) :=21−β\n2\nΓ(β\n2)λβ\ntβ+1e−λ2\n2t2dt.4 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\n(3)Ifrn√n→ ∞, then the sequence {rnCn\nβ}∞\nn=1infinitely dissipates.\nThis theorem means that the n-dimensional Cauchy space has the phase transition prop-\nerty with the critical scale order 1 /√n. See [18] for the definition of the phase transition\nproperty. The box-convergence induced by the box distance fun ction onXis a slight modifi-\ncationofthemeasuredGromov-Hausdorffconvergence. Thebox distancewasalsointroduced\nby Gromov in [ 5]. Any measured Gromov-Hausdorff convergent sequence box-co nverges and\nany box-convergent sequence concentrates. The infinite dissipa tion is the opposite notion of\nthe L´ evy family. For details on this notion, see Subsection 2.5or refer to [ 5,21].\nAs a corollary, we obtain a statement similar to the normal law ` a la L´ e vy.\nCorollary 1.6. Letfn:Rn→R,n= 1,2,..., be(1/√n)-Lipschitz functions. Assume that,\nfor a subsequence {fni}of{fn}, the push-forward measure (fni)#νni\nβconverges weakly to a\nBorel probability measure µonR. Then, there exists a 1-Lipschitz function α: [0,∞)→R\nsuch that\nα#νβ=µ.\nThe usual normal law requires that αis a monotone function. Unfortunately, our result\ndoes not lead to the monotonicity of α. This law is deeply related to the isoperimetric\nproblem (see [ 4]).\nMoreover, we obtain the asymptotic behavior of the observable dia meter, which is one\nof the most fundamental invariants for the concentration of met ric measure spaces. The\ndefinition of the observable diameter is in Definition 2.15.\nCorollary 1.7. We have\nlim\nn→∞ObsDiam(1√nCn\nβ;−κ) = ObsDiam(([0 ,∞),|·|,νβ);−κ)\nfor anyκ>0.\nOn Theorem 1.4, it is also significant to know whether the sequence {1√nCn\nβ}∞\nn=1converges\nwith respect to a topology stronger than the concentration topo logy. We also obtain the\nfollowing theorem.\nTheorem 1.8. Thesequence {1√nCn\nβ}∞\nn=1has nobox-convergent (henceno measuredGromov-\nHausdorff convergent )subsequence.\nAs the final part of the introduction, we discuss the relation betwe en Theorem 1.4and\nthe weighted Ricci curvature. For the precise definition, see Sect ion7. We see that the\n(−β)-weighted Ricci curvature Ric 1√nCn\nβ,−βof the scaled Cauchy space1√nCn\nβis nonnegative,\nor equivalently,1√nCn\nβis a CD(0,−β) space. The study of convergence of CD( K,N) spaces\nforN <0 is performed by Magnabosco-Rigoni-Sosa [ 12] and Oshima [ 17]. Theorem 1.4\nprovides a natural example of the main result in [ 17]. Moreover, we observe the asymptotic\nbehavior of weighted Ricci curvature as n→ ∞. At any point x∈Rn\\{o}, we have\nRic1√nCn\nβ,−β(x\n/ba∇dblx/ba∇dbl,x\n/ba∇dblx/ba∇dbl) =n(n+β)\n(1+n/ba∇dblx/ba∇dbl2)2→1\n/ba∇dblx/ba∇dbl4\nand, for any w∈Rn\\{o}withw⊥x,\nRic1√nCn\nβ,−β(w\n/ba∇dblw/ba∇dbl,w\n/ba∇dblw/ba∇dbl) =n(n+β)\n1+n/ba∇dblx/ba∇dbl2→ ∞\nasn→ ∞. As mentioned at the beginning, the divergence of the Ricci curvat ure to infinity\nimplies the L´ evy family (see [ 17, Corollary 4.3] for N <0). Therefore Theorem 1.4canCONVERGENCE OF CONES OF METRIC MEASURE SPACES 5\nbe interpreted as that the ( n−1)-dimensional spherical directions at each point of1√nCn\nβ\nconcentrates to a single point leaving only the 1-dimensional radial d irection.\nThe organization of this paper is as follows. After the preliminaries se ction, in Section\n3, we prove Theorem 1.2. The proof there is close to the discussion in [ 7,8]. Moreover,\nin Section 4, we give another proof to Theorem 1.3, which restricts Theorem 1.2to L´ evy\nfamilies. In the proof of Theorem 1.3, we exactly prove that “any 1-Lipschitz function on a\ncone is almost a radial function if nis large” in ( 4.3) based on the concentration of measure\nphenomenon. From Section 5to8, we mainly consider the Cauchy distribution. At first,\nin Section 5, we describe fundamental properties of Cauchy distribution and p rove Theorem\n1.4applying Theorem 1.3. The refinement from Theorem 1.4to Theorem 1.5is a simple\nargument based on previous works. In Section 6, we prove Theorem 1.8. The key is the\ncontinuity of the partial diameter with respect to the box distance . After the proof of our\nresults, in Section 7, we calculate several geometric and analytic quantities for the Cau chy\nspace and the limit space. Finally, in Section 8, we remark a relation to the ergodic theory.\nWe give some examples of non-ergodic systems generated by the Ca uchy distribution.\nAcknowledgement. The authors would like to thank Professors Shouhei Honda, Atsus hi\nKatsuda, Kazuhiro Kuwae, Hiroki Nakajima, Takashi Shioya, and A suka Takatsu for their\nvaluable comments and encouragement. S.E. is supported in part by JSPS KAKENHI Grant\nNumbers JP22H04942 and JP23K03158. D.K. is supported in part by JSPS KAKENHI\nGrantNumbersJP22K20338. A.M.issupportedinpartbyJSPSKAKE NHIGrantNumbers\nJP20K03598 and JP21H00977.\n2.Preliminaries\nIn this section, we describe the definitions and some properties of m etric measure space,\nthe box distance, and the observable distance. We use most of the se notions along [ 21]. As\nfor more details, we refer to [ 21] and [5, Chapter 31\n2+].\n2.1.Metric measure spaces. Let (X,dX) be a complete separable metric space and mXa\nBorel probability measure on X. We call the triple ( X,dX,mX) ametric measure space , or\nanmm-space for short. We sometimes say that Xis an mm-space, in which case the metric\nand the measure of Xare respectively indicated by dXandmX.\nDefinition 2.1 (mm-Isomorphism) .Two mm-spaces XandYaresaidtobe mm-isomorphic\nto each other if there exists an isometry f: suppmX→suppmYsuch thatf#mX=mY,\nwheref#mXis the push-forward measure of mXbyf. Such an isometry fis called an\nmm-isomorphism . Denote by Xthe set of mm-isomorphism classes of mm-spaces.\nNote that an mm-space Xis mm-isomorphic to (supp mX,dX,mX). We assume that an\nmm-spaceXsatisfies\nX= suppmX\nunless otherwise stated.\nDefinition 2.2 (Lipschitz order) .LetXandYbe two mm-spaces. We say that X(Lips-\nchitz)dominatesYand writeY≺Xif there exists a 1-Lipschitz map f:X→Ysatisfying\nf#mX=mY. We call the relation ≺onXtheLipschitz order .\nThe Lipschitz order ≺is a partial order relation on X.6 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\n2.2.Box distance and observable distance. For a subset Aof a metric space ( X,dX)\nand for a real number r>0, we set\nUr(A) :={x∈X|dX(x,A)<r},\nwheredX(x,A) := infa∈AdX(x,a).\nDefinition 2.3 (Prokhorov distance) .TheProkhorov distance dP(µ,ν) between two Borel\nprobability measures µandνon a metric space Xis defined to be the infimum of ε >0\nsatisfying\nµ(Uε(A))≥ν(A)−ε\nfor any Borel subset A⊂X.\nThe Prokhorov metric dPis a metrization of the weak convergence of Borel probability\nmeasures on Xprovided that Xis a separable metric space. Note that if a map f:X→Y\nbetween two metric spaces XandYis 1-Lipschitz, then we have\n(2.1) dP(f#µ,f#ν)≤dP(µ,ν)\nfor any two Borel probability measures µandνonX.\nDefinition 2.4 (Ky Fan metric) .Let (X,µ) be a measure space and ( Y,dY) a metric space.\nFor twoµ-measurable maps f,g:X→Y, we define dµ\nKF(f,g) to be the infimum of ε≥0\nsatisfying\nµ({x∈X|dY(f(x),g(x))>ε})≤ε.\nThe function dµ\nKFis a metric on the set of µ-measurable maps from XtoYby identifying\ntwo maps if they are equal to each other µ-almost everywhere. We call dµ\nKFtheKy Fan\nmetric.\nLemma 2.5 ([21,Lemma1.26]) .LetXbe a topologicalspacewith a Borel probabilitymeasure\nµandYa metric space. For any two Borel measurable maps f,g:X→Y, we have\ndP(f#µ,g#µ)≤dµ\nKF(f,g).\nDefinition 2.6 (Parameter) .LetXbe an mm-space. A map ϕ: [0,1]→Xis called a\nparameter ofXifϕis a Borel measurable map such that\nϕ#L1=mX,\nwhereL1is the one-dimensional Lebesgue measure on [0 ,1].\nNote that any mm-space has a parameter (see [ 21, Lemma 4.2]).\nDefinition 2.7 (Box distance) .We define the box distance /square(X,Y) between two mm-spaces\nXandYto be the infimum of ε≥0 satisfying that there exist parameters ϕ: [0,1]→X,\nψ: [0,1]→Y, and a Borel subset J⊂[0,1] withL1(J)≥1−εsuch that\n|dX(ϕ(s),ϕ(t))−dY(ψ(s),ψ(t))| ≤ε\nfor anys,t∈J.\nTheorem 2.8 ([21, Theorem 4.10]) .The box distance function /squareis a complete separable\nmetric on X.\nLemma 2.9 ([21, Proposition 4.12]) .LetXbe a complete separable metric space. For any\ntwo Borel probability measures µandνonX, we have\n/square((X,µ),(X,ν))≤2dP(µ,ν).\nThefollowingnotiongivesoneoftheconditionsthatareequivalent to thebox-convergence.CONVERGENCE OF CONES OF METRIC MEASURE SPACES 7\nDefinition 2.10 (ε-mm-Isomorphism) .LetXandYbe two mm-spaces and f:X→Ya\nBorel measurable map. We say that fis anε-mm-isomorphism for a real number ε≥0 if\nthere exists a Borel subset /tildewideX⊂Xsuch that\n(1)mX(/tildewideX)≥1−ε,\n(2)|dX(x,x′)−dY(f(x),f(x′))| ≤εfor anyx,x′∈/tildewideX,\n(3)dP(f#mX,mY)≤ε.\nWe call/tildewideXanon-exceptional domain off.\nLemma 2.11 ([21, Lemma 4.22]) .LetXandYbe two mm-spaces.\n(1)If there exists an ε-mm-isomorphism f:X→Y, then/square(X,Y)≤3ε.\n(2)If/square(X,Y)<ε, then there exists a 3ε-mm-isomorphism f:X→Y.\nGiven a metric space X, we denote by Lip1(X) the set of 1-Lipschitz functions on X.\nMoreover, for a map p:S→Xfrom a setS, we set\np∗Lip1(X) :={f◦p|f∈ Lip1(X)}.\nNote that, for any parameter ϕ: [0,1]→Xof an mm-space X, the setϕ∗Lip1(X) consists\nof Borel measurable functions on [0 ,1].\nDefinition 2.12 (Observable distance) .We define the observable distance dconc(X,Y) be-\ntween two mm-spaces XandYby\ndconc(X,Y) := inf\nϕ,ψdH(ϕ∗Lip1(X),ψ∗Lip1(Y)),\nwhereϕ: [0,1]→Xandψ: [0,1]→Yrun over all parameters of XandYrespectively, and\ndHis the Hausdorff distance with respect to the metric dL1\nKF. We say that a sequence {Xn}∞\nn=1\nof mm-spaces concentrates to an mm-space XifXndconc-converges to Xasn→ ∞.\nTheorem 2.13 ([21, Theorem 5.13, Proposition5.5]) .The observable distance function dconc\nis a metric on X. Moreover, we have\ndconc(X,Y)≤/square(X,Y)\nfor any two mm-spaces XandY.\nLemma 2.14 ([21, Lemma 5.25]) .Letp:X→Ybe a Borel measurable map. Then we have\ndconc(X,Y)≤2dP(p#mX,mY)+dH(Lip1(X),p∗Lip1(Y)),\nwheredHis the Hausdorff distance with respect to the metric dmX\nKF.\n2.3.L´ evy family. The observable diameter is one of the most fundamental invariants of\nan mm-space.\nDefinition 2.15 (Partial and observable diameter) .LetXbe an mm-space. For a real\nnumberα≤1, we define the partial diameter diam(X;α) ofXto be the infimum of diam A,\nwhereA⊂Xruns over all Borel subsets with mX(A)≥αand diamAis the diameter of A.\nFor a real number κ>0, we define the observable diameter ofXto be\n(2.2) ObsDiam( X;−κ) := sup\nf∈Lip1(X)diam((R,|·|,f#mX);1−κ).\nThe observable diameter isan invariant under mm-isomorphism. Note that diam(X;1−κ)\nand ObsDiam( X;−κ) are nonincreasing in κ>0.\nDefinition 2.16 (L´ evy family) .A sequence {Xn}∞\nn=1of mm-spaces is called a L´ evy family\nif\nlim\nn→∞ObsDiam(Xn;−κ) = 0\nfor anyκ>0.8 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\nWe denote by ∗the one-point mm-space equipped with the trivial distance and the D irac\nmeasure.\nProposition 2.17 ([21, Corollary 5.8]) .A sequence {Xn}∞\nn=1of mm-spaces is a L´ evy family\nif and only if it concentrates to ∗asn→ ∞.\nDefinition 2.18 (Concentration function) .LetXbe an mm-space. We define the concen-\ntration function αXofXto be\nαX(r) := sup\nA(1−mX(Ur(A)))\nforr>0, whereA⊂Xruns over all Borel subsets with mX(A)≥1/2.\nProposition 2.19 ([10, Proposition 1.12], [ 21, Remark 2.28]) .The following (1)and(2)\nhold.\n(1) ObsDiam( X;−κ)≤2inf{r>0|αX(r)≤κ/2}for anyκ>0.\n(2)αX(r)≤sup{κ>0|ObsDiam(X;−κ)≥r}for anyr>0.\nIn particular, a sequence {Xn}∞\nn=1of mm-spaces is a L´ evy family if and only if\nlim\nn→∞αXn(r) = 0\nfor anyr>0.\nExample 2.20 ([10, Theorem 2.3]) .LetSn−1be the (n−1)-dimensional unit sphere in Rn.\nAssume that Sn−1is equipped with the Riemannian distance or the restriction of Euclidea n\ndistance and the uniform probability measure σn−1. Then we have\n(2.3) αSn−1(r)≤e−n−2\n2r2\nfor anyr>0.\nDefinition 2.21 (Median and L´ evy mean) .LetXbe a measure space with probability\nmeasureµandf:X→Ra measurable function. A real number m∈Ris called a median\noffif it satisfies\nµ({x∈X|f(x)≥m})≥1\n2andµ({x∈X|f(x)≤m})≥1\n2.\nIt is easy to see that the set of medians of fis a nonempty bounded closed interval. The\nL´ evy mean lm(f;µ) offwith respect to µis defined to be\nlm(f;µ) :=m+m\n2,\nwheremis the minimum of medians of f, andmthe maximum of medians of f.\nProposition 2.22 ([21, Section 2.3]) .A sequence {Xn}∞\nn=1of mm-spaces is a L´ evy family\nif and only if for any fn∈ Lip1(Xn),\nlim\nn→∞dmXn\nKF(fn,lm(fn;mXn)) = 0.\nProposition 2.23 ([10, Proposition 1.3]) .LetXbe an mm-space and let fbe aL-Lipschitz\nfunction on X. Then we have\n(2.4) mX({x∈X| |f(x)−lm(f;mX)| ≥ε})≤2αX(ε\nL)\nfor anyε>0.CONVERGENCE OF CONES OF METRIC MEASURE SPACES 9\n2.4.Pyramid.\nDefinition 2.24 (Pyramid) .A nonempty box-closed subset P ⊂ Xis called a pyramidif it\nsatisfies the following (1) and (2).\n(1) IfX∈ Pand ifY≺X, thenY∈ P.\n(2) For any Y,Y′∈ P, there exists X∈ Psuch thatY,Y′≺X.\nWe denote the set of pyramids by Π.\nFor an mm-space X, we define\nPX:={Y∈ X |Y≺X},\nwhich is a pyramid. We call PXthepyramid associated with X.\nWeobserve that Y≺Xif andonlyif PY⊂ PX. We seethat {∗}andXaretheminimum\nand maximum pyramids with respect to the inclusion respectively, tha t is,{∗} ⊂ P ⊂ X for\nany pyramid P.\nProposition 2.25 ([21, Lemma 6.10]) .LetDbe a subset of Xsatisfying the condition (2)\nof Definition 2.24. Then/uniondisplay\nX∈DPX/square\nis a pyramid.\nWe define the weak convergence of pyramids as follows. This is also kn own as the Kura-\ntowski convergence as subsets of ( X,/square).\nDefinition 2.26 (Weak convergence) .We say that a sequence {Pn}∞\nn=1of pyramids con-\nverges weakly to a pyramid Pasn→ ∞if the following (1) and (2) are both satisfied.\n(1) For any mm-space X∈ P, we have\nlim\nn→∞/square(X,Pn) = 0.\n(2) For any mm-space X∈ X \\P, we have\nliminf\nn→∞/square(X,Pn)>0.\nTheorem 2.27 ([5,21]).There exists a metric, denoted by ρ, onΠsuch that the following\n(1) – (4) hold.\n(1)ρis compatible with weak convergence.\n(2)The mapι:X ∋X/ma√sto→ PX∈Πis a1-Lipschitz topological embedding map with\nrespect todconcandρ.\n(3) Πisρ-compact.\n(4)ι(X)isρ-dense in Π.\nIn particular, (Π ,ρ) is a compactification of ( X,dconc). We often identify XwithPX.\n2.5.Dissipation. Dissipation is the opposite notion to concentration. We omit to state the\ndefinition of the infinite dissipation (see [ 21, Definition 8.1] for the definition). Instead, we\nstate the following proposition. Let {Xn}∞\nn=1be a sequence of mm-spaces.\nProposition 2.28 (see [21, Proposition 8.5(2)]) .The sequence {Xn}∞\nn=1infinitely dissipates\nif and only if {PXn}∞\nn=1converges weakly to Xasn→ ∞.\nAn easy discussion using [ 18, Lemma 6.6] leads to the following.\nProposition 2.29. The sequence {Xn}∞\nn=1infinite dissipates if and only if\nlim\nn→∞ObsDiam(Xn;−κ) =∞\nfor anyκ∈(0,1).10 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\n3.Convergence of cones\nWe first recall some properties of the cone metric. For κ∈R, we set\nIκ:=/braceleftBigg\n[0,π/√κ] ifκ>0,\n[0,∞) ifκ≤0.\nFor a metric space X, we denote\nrx:=q(r,x)∈Kκ(X),0 :=q(0,x)∈Kκ(X)\nforr∈Iκandx∈X, whereq:Iκ×X→Kκ(X) is the quotient map. Moreover, we recall\nsκ(r) :=\n\n1√κsin(√κr) ifκ>0,\nr ifκ= 0,\n1√−κsinh(√−κr) ifκ<0,cκ(r) :=\n\ncos(√κr) ifκ>0,\n1 if κ= 0,\ncosh(√−κr) ifκ<0,\nand put\n˜sκ(r) :=smin{κ,0}(r)\nforr∈Iκ. By the definition of the cone metric ( 1.2) and the geometry of surface with\nconstant curvature, we obtain the following two lemmas immediately.\nLemma 3.1. LetXbe a metric space and let κ∈R. Then we have\n(1)dKκ(X)(rx,r′x) =|r−r′|for anyr,r′∈Iκandx∈X,\n(2)dKκ(X)(rx,rx′)≤˜sκ(r)min{dX(x,x′),π}for anyr∈Iκandx,x′∈X.\nIn particular, we have\n(3.1) dKκ(X)(rx,r′x′)≤ |r−r′|+min{˜sκ(r),˜sκ(r′)}min{dX(x,x′),π}\nfor anyrx,r′x′∈Kκ(X).\nLemma 3.2. LetXandYbe two metric spaces and let κ∈R,ε>0, andR>0. For any\nx,x′∈Xandy,y′∈Ywith\n|dX(x,x′)−dY(y,y′)|<ε\nand for any r,r′∈Iκ∩[0,R], we have\n|dKκ(X)(rx,r′x′)−dKκ(Y)(ry,r′y′)|<˜sκ(R)ε.\nAs the first step to prove Theorem 1.2, we prove the following.\nTheorem 3.3. Letκ∈R. Assume that a sequence {Xn}∞\nn=1of mm-spaces box-converges to\nan mm-space Xand that a sequence {µn}∞\nn=1of Borel probability measures on Iκconverges\nweakly to a Borel probability measure µ. Then{Kκ,µn(Xn)}∞\nn=1box-converges to Kκ,µ(X).\nBoth the assumption and conclusion of this theorem are stronger t han those of Theorem\n1.2.\nProof of Theorem 3.3.Take any small number ε>0. Since {µn}∞\nn=1converges weakly to µ,\nthere exists R>0 such that\nsup\nn∈Nµn((R,∞))<εandµ((R,∞))<ε.\nNote thatR=π/√κcan be assumed if κ >0. Since {Xn}∞\nn=1box-converges to X, there\nexists anεn-mm-isomorphism fn:Xn→Xfor eachnwithεn→0 by Lemma 2.11. We\ndefine a map gn:Kκ,µn(Xn)→Kκ,µ(X) by\ngn(rx) :=rfn(x), rx∈Kκ,µn(Xn).\nThen we prove that the map gnis a 3ε-mm-isomorphism from Kκ,µn(Xn) toKκ,µ(X) for any\nsufficiently large n, which implies that /square(Kκ,µn(Xn),Kκ,µ(X))<9εby Lemma 2.11again.CONVERGENCE OF CONES OF METRIC MEASURE SPACES 11\nLet/tildewideXnbe a non-exceptional domain of fnand put\n/tildewideKn:=q([0,R]×/tildewideXn)⊂Kκ(Xn) and /tildewideK:=q([0,R]×X)⊂Kκ(X).\nThen we have\nmKκ,µn(Xn)(/tildewideKn) =µn⊗mXn([0,R]×/tildewideXn)>(1−ε)(1−εn)>1−(ε+εn).\nWe consider the metric\nd((r,x),(r′,x′)) :=|r−r′|+˜sκ(R)min{dX(x,x′),π}\non the product space Iκ×X. By [7, Lemma 4.5], we have\nµn⊗(fn)#mXn(A)≤µ⊗mX(Uε(A))+ε\nfor any Borel subset AonIκ×Xif\n2dP(µn,µ)+2˜sκ(R)dP((fn)#mXn,mX)<ε.\nMoreover we have\nUε(q−1(B))⊂q−1(Uε(B))\nfor anyB⊂/tildewideK. Indeed, for any ( r,x)∈Uε(q−1(B)), there exists ( r′,x′)∈q−1(B) such that\nd((r,x),(r′,x′)) =|r−r′|+˜sκ(R)min{dX(x,x′),π}<ε.\nBy Lemma 3.1, we havedKκ(X)(rx,r′x′)<ε. Therefore, for any Borel subset B⊂Kκ,µ(X),\nwe have\n(gn)#mKκ,µn(Xn)(B)≤(gn)#mKκ,µn(Xn)(B∩/tildewideK)+ε+εn\n=q#(µn⊗(fn)#mXn)(B∩/tildewideK)+ε+εn\n≤(µ⊗mX)(Uε(q−1(B∩/tildewideK)))+2ε+εn\n≤(µ⊗mX)(q−1(Uε(B∩/tildewideK)))+2ε+εn\n≤mKκ,µ(X)(Uε(B∩/tildewideK))+3ε,\nwhich implies that dP((gn)#mKκ,µn(Xn),mKκ,µ(X))≤3εfor any sufficiently large n.\nFor anyrx,r′x′∈/tildewideKn, since\n|dX(fn(x),fn(x′))−dXn(x,x′)| ≤εn,\nLemma3.2implies that\n|dKκ(X)(rfn(x),r′fn(x′))−dKκ(Xn)(rx,r′x′)| ≤˜sκ(R)εn<3ε\nfor any sufficiently large n. Therefore gnis a 3ε-mm-isomorphism for any sufficiently large\nnand then {Kκ,µn(Xn)}∞\nn=1box-converges to Kκ,µ(X). /square\nWe now define the κ-cone for a pyramid and prove Theorem 1.2.\nDefinition 3.4 (κ-Cone for a pyramid) .We define the κ-coneKκ,µ(P) of a pyramid Pby\nKκ,µ(P) :=/uniondisplay\nX∈PPKκ,µ(X)/square\n.\nTheκ-coneKκ,µ(P) is a pyramid by Proposition 2.25since the operation X/ma√sto→Kκ,µ(X)\npreserves the Lipschitz order.\nIn order to prove Theorem 1.2, we prove the following (1) and (2) under the assumptions.\n(1) For any Y∈Kκ,µ(P), we have lim n→∞/square(Y,Kκ,µn(Pn)) = 0.\n(2) If an mm-space Ysatisfies liminf n→∞/square(Y,Kκ,µn(Pn)) = 0, then Y∈Kκ,µ(P).\nThe proof of (1) of Theorem 1.2is an easy work using Theorem 3.3.12 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\nProof of (1) of Theorem 1.2.Take anyY∈Kκ,µ(P) and any ε >0. Then there exist\nX∈ PandY′≺Kκ,µ(X) with/square(Y,Y′)<ε. SincePnconverges weakly to P, there exists\na sequence {Xn}∞\nn=1withXn∈ Pnbox-converging to X. By Theorem 3.3, the sequence\n{Kκ,µn(Xn)}∞\nn=1box-converges to Kκ,µ(X). By [21, Lemma 6.10(1)], there exists a sequence\n{Yn}∞\nn=1withYn≺Kκ,µn(Xn)∈Kκ,µn(Pn) box-converging to Y′. Therefore we obtain\nlimsup\nn→∞/square(Y,Kκ,µn(Pn))<lim\nn→∞/square(Y′,Yn)+ε=ε.\nAsε→0, we finish the proof. /square\nThe proof of (2) of Theorem 1.2is harder than (1). We mimic the proof of [ 8, Theorem\n1.2]. We prepare a term and a lemma.\nDefinition 3.5 (1-Lipschitz up to an additive error) .LetXbe an mm-space and Ybe a\nmetric space. A map f:X→Yis said to be 1- Lipschitz up to (an additive error )ε≥0 if\nthere exists a Borel subset /tildewideX⊂Xsuch that\n(1)mX(/tildewideX)≥1−ε,\n(2)dY(f(x),f(x′))≤dX(x,x′)+εfor anyx,x′∈/tildewideX.\nWe call/tildewideXanon-exceptional domain off.\nLemma 3.6 ([8, Corollary 4.7]) .LetPbe a pyramid and Yan mm-space. Assume that, for\nanyε>0, there exist an mm-space Xε∈ Pand a Borel measurable map fε:Xε→Ysuch\nthatfεis1-Lipschitz up to εanddP((fε)#mXε,mY)≤εholds. Then Y∈ P.\nMoreover, we define a norm /ba∇dbl·/ba∇dbl∞onRnby\n/ba∇dblx/ba∇dbl∞:= max\ni|xi|, x= (x1,...,xn)∈Rn.\nProof of (2) of Theorem 1.2.We first prove the case κ= 0. We denote Kµ=K0,µin this\nproof. Choosing a subsequence, we can assume that\nlim\nn→∞/square(Y,Kµn(Pn)) = 0.\nThen there exist Xn∈ PnandYn≺Kµn(Xn) for every nsuch that {Yn}∞\nn=1box-converges\ntoY. SinceYn≺Kµn(Xn), there exists a 1-Lipschitz map fn:Kµn(Xn)→Ynwith\nfn#mKµn(Xn)=mYn. Since{Yn}∞\nn=1box-converges to Y, there exists an εn-mm-isomorphism\ngn:Yn→Ywithεn→0. Let/tildewideYnbe a non-exceptional domain of gn. Here, ifµ({0})>0,\nthen we can assume that fn(0)∈/tildewideYnfor any sufficiently large n. Indeed, in this case, we see\nthatfn(0)∈Uε(/tildewideYn) for any fixed ε >0 and any sufficiently large n. By [21, Lemma 3.5],\nthere exists a Borel measurable map πn:Yn→/tildewideYnsuch thatπn|/tildewideYn= id/tildewideYnand\ndYn(πn(y),y)≤dYn(y,/tildewideYn)+εn\nfor anyy∈Yn. Then we see that the map gn◦πnis a (4εn+ 2ε)-mm-isomorphism with\nnon-exceptional domain Uε(/tildewideYn). Thus we can replace gnwithgn◦πn.\nTake any 0 <ε<1. We find finitely many open subsets B1,...,BNinYsuch that\nmax\njdiamBj<ε,min\njmY(Bj)>0,N/summationdisplay\nj=1mY(Bj)>1−ε,\nandδ′:= min\nj/ne}ationslash=j′dY(Bj,Bj′)>0CONVERGENCE OF CONES OF METRIC MEASURE SPACES 13\n(cf. [19, Lemma 42]). In the case of µ({0})>0, we assume that gn◦fn(0)∈/unionsqtextN\nj=1Bj. Let\nB0:=Y\\N/unionsqdisplay\nj=1Bj\nand take a point yj∈Bjfor eachj= 0,1,...,N. IfB0is empty, we ignore the case j= 0.\nWe define a finite mm-space ˙Yas\n˙Y:= ({y0,y1,...,yN},dY,N/summationdisplay\nj=0µY(Bj)δyj).\nNote that the inclusion map ι:˙Y→Yis anε-mm-isomorphism.\nSince{µn}∞\nn=1converges weakly to µ, there exists R>1 such that µ({R}) = 0 and\nµ((R,∞))≤sup\nn∈Nµn((R,∞))<ε\nN.\nTake any sufficiently small number η >0 as 2η<min{δ′,ε}. We find finitely many disjoint\nintervalsI1,...,IM⊂[0,R] such that\nmax\nidiamIi<η,min\niµ(Ii)>0,M/summationdisplay\ni=1µ(Ii) =µ([0,R]),and max\niµ(∂Ii) = 0.\nHere, max iµ(∂Ii) = 0 follows from the fact that the set of atomic points of µis countable.\nNote thatµn(Ii)→µ(Ii) asn→ ∞sinceµ(∂Ii) = 0. Moreover, we assume that 0 /\\e}atio\\slash∈/unionsqtextM\ni=1Ii\nifµ({0}) = 0. Ifµ({0})>0, then there exist unique numbers i∈ {1,...,M}andj∈\n{1,...,N}with 0∈Iiandfn(0)∈g−1\nn(Bj)∩/tildewideYn. We denote them by i0andj0.\nWe define subsets An\nijofXnfori= 1,...,Mandj= 1,...,Nby\nAn\nij:=/braceleftBig\nx∈Xn/vextendsingle/vextendsingle/vextendsinglethere exists s∈Iisuch thatsx∈f−1\nn(g−1\nn(Bj)∩/tildewideYn)/bracerightBig\n.\nNote thatAn\ni0j0=Xnifµ({0})>0 andnis large. We define a 1-Lipschitz map Φ n:Xn→\n(RMN,/ba∇dbl·/ba∇dbl∞) by\nΦn(x) := (min {dXn(x,An\nij),π})i=1,...,M,j=1,...,N.\nThen, since supp(Φ n)#mXnis in the compact set/braceleftbig\nz∈RMN/vextendsingle/vextendsingle/ba∇dblz/ba∇dbl∞≤π/bracerightbig\n, the sequence\n{(Φn)#mXn}∞\nn=1is tight. Thus there exists a subsequence of {(Φn)#mXn}∞\nn=1converging\nweakly to a Borel probability measure νon/braceleftbig\nz∈RMN/vextendsingle/vextendsingle/ba∇dblz/ba∇dbl∞≤π/bracerightbig\n. Defining an mm-space\nXby\nX:= (suppν,/ba∇dbl·/ba∇dbl∞,ν),\nwe haveX∈ Psince (supp(Φ n)#mXn,/ba∇dbl · /ba∇dbl∞,(Φn)#mXn)∈ Pnand{Pn}∞\nn=1converges\nweakly to P.\nNow our goal is to find a Borel measurable map Ψ: Kµ(X)→˙Ysuch that Ψ is 1-\nLipschitz up to 3 εanddP(Ψ#mKµ(X),m˙Y)≤2ε. If we prove this, then we see that the map\nι◦Ψ:Kµ(X)→Yis 1-Lipschitz up to 4 εand satisfies dP((ι◦Ψ)#mKµ(X),mY)≤3ε, which\nleads toY∈Kµ(P) by Lemma 3.6.\nFor anyi∈ {1,...,M}, we define a map proji:RMN→RNby\nproji((zij)i=1,...,M,j=1,...,N) := (zi1,...,ziN),(zij)i=1,...,M,j=1,...,N∈RMN\nand putνi:= (proji)#νand Φn,i:= proji◦Φn. Note that (Φ n,i)#mXnconverges weakly to\nνiasn→ ∞. Moreover, we define a disjoint family {Zj}N\nj=1of subsets of RNas\nZj:=/braceleftbigg\n(z1,...,zN)/vextendsingle/vextendsingle/vextendsingle/vextendsinglezj= 0 and min\nj′/ne}ationslash=jzj′≥δ/bracerightbigg14 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\nfor a fixed positive real number δ>0 with\nδ <min{δ′\n3R,π}.\nClaim 3.7. For everyi= 1,...,Mand everyj= 1,...,N, we have\nΦn,i(An\nij)⊂Zj\nifnis sufficiently large.\nProof.Take anyx∈An\nijand let (z1,...,zN) = Φn,i(x). It is obvious that\nzj= min{dXn(x,An\nij),π}= 0.\nWe prove that zj′= min{dXn(x,An\nij′),π} ≥δifj′/\\e}atio\\slash=j. Take any x′∈An\nij′. There exist\nr,r′∈Iisuch that\nrx∈f−1\nn(g−1\nn(Bj)∩/tildewideYn) andr′x′∈f−1\nn(g−1\nn(Bj′)∩/tildewideYn).\nThen, by Lemma 3.1, we have\nδ′≤dY(Bj,Bj′)≤dY(gn◦fn(rx),gn◦fn(r′x′))\n≤dK0(Xn)(rx,r′x′)+εn≤ |r−r′|+RdXn(x,x′)+εn\n≤η+RdXn(x,x′)+εn,\nwhich implies that dXn(x,x′)≥δifnis sufficiently large. Thus we obtain zj′≥δ. The proof\nis completed. /square\nBy this claim and the portmanteau theorem of weak convergence of measures, we have\nνi(Zj)≥limsup\nn→∞(Φn,i)#mXn(Zj)≥limsup\nn→∞mXn(An\nij)\nfor anyj= 1,...,N. Now we define a Borel measurable map Ψ: Kµ(X)→˙Yby\nΨ(rx) :=/braceleftBigg\nyjifr∈Iiand proji(x)∈Zj,\ny0otherwise\nforrx∈Kµ(X). Here, ifB0=∅, then we replace y0with an arbitrary point of {y1,...,yN}.\nNote that if µ({0})>0, then we have νi0(Zj0) = 1 and then\nX= suppν⊂proj−1\ni0(Zj0),\nwhich implies that Ψ(0) = yj0. Thus Ψ is well-defined on Kµ(X). From now on, we denote\ndKn:=dK0(Xn),dK:=dK0(X),mKn:=mKµn(Xn), andmK:=mKµ(X)in this proof for\nsimplicity.\nWe first prove that dP(Ψ#mK,m˙Y)≤2ε. For anyj= 1,...,N, we have\nm˙Y({yj}) =mY(Bj)≤liminf\nn→∞(gn)#mYn(Bj)\n≤liminf\nn→∞(mYn(g−1\nn(Bj)∩/tildewideYn)+εn)≤limsup\nn→∞mKn(f−1\nn(g−1\nn(Bj)∩/tildewideYn))\n= limsup\nn→∞µn⊗mXn(q−1(f−1\nn(g−1\nn(Bj)∩/tildewideYn)))\n≤limsup\nn→∞M/summationdisplay\ni=1µn⊗mXn(q−1(f−1\nn(g−1\nn(Bj)∩/tildewideYn))∩(Ii×Xn))+µn((R,∞))\n≤limsup\nn→∞M/summationdisplay\ni=1µn(Ii)mXn(An\nij)+ε\nN≤M/summationdisplay\ni=1µ(Ii)νi(Zj)+ε\nN.CONVERGENCE OF CONES OF METRIC MEASURE SPACES 15\nOn the other hand, we have\nΨ#mK({yj}) =mK(M/unionsqdisplay\ni=1q(Ii×proj−1\ni(Zj)))\n≥µ⊗ν(M/unionsqdisplay\ni=1Ii×proj−1\ni(Zj)) =M/summationdisplay\ni=1µ(Ii)νi(Zj).\nThese imply that\nm˙Y({yj})≤Ψ#mK({yj})+ε\nN\nfor anyj= 1,...,N. Takingm˙Y({y0})<εinto account, we obtain\ndP(Ψ#mK,m˙Y)<ε+N·ε\nN= 2ε.\nWe next prove that Ψ is 1-Lipschitz up to 3 ε. Letting\n/tildewideK:=q(M/unionsqdisplay\ni=1(Ii×proj−1\ni(suppνi∩N/unionsqdisplay\nj=1Zj))),\nwe have\nmK(/tildewideK)≥M/summationdisplay\ni=1N/summationdisplay\nj=1µ(Ii)νi(Zj)≥N/summationdisplay\nj=1(m˙Y({yj})−ε\nN)≥1−2ε.\nTake anysx,s′x′∈/tildewideKand fix them. There exist i,i′∈ {1,...,M}andj,j′∈ {1,...,N}\nsuch that\ns∈Ii,proji(x)∈Zj,ands′∈Ii′,proji′(x′)∈Zj′.\nThen, since (Φ n)#mXnconverges weakly to ν, there exist xn∈An\nijandx′\nn∈An\ni′j′for eachn\nsuch that\n/ba∇dblΦn(xn)−x/ba∇dbl∞,/ba∇dblΦn(x′\nn)−x′/ba∇dbl∞→0 asn→ ∞.\nWe prove\n(3.2) dY(yj,yj′)≤/radicalBig\ns2+(s′)2−2ss′cos(min{dXn(xn,An\ni′j′),π})+εn+3ε.\nIndeed, taking any ˜ x′∈An\ni′j′, there exist t∈Iiandt′∈Ii′such that\ntxn∈f−1\nn(g−1\nn(Bj)∩/tildewideYn) andt′˜x′∈f−1\nn(g−1\nn(Bj′)∩/tildewideYn).\nThus we have\ndY(yj,yj′)≤dY(Bj,Bj′)+2ε≤dY(gn◦fn(txn),gn◦fn(t′˜x′))+2ε\n≤dKn(txn,t′˜x′)+εn+2ε≤dKn(sxn,s′˜x′)+|s−t|+|s′−t′|+εn+2ε\n<dKn(sxn,s′˜x′)+2η+εn+2ε<dKn(sxn,s′˜x′)+εn+3ε.\nTherefore we obtain ( 3.2). Moreover, since\nmin{dXn(xn,An\ni′j′),π}= min{dXn(xn,An\ni′j′),π}−min{dXn(x′\nn,An\ni′j′),π}\n≤ /ba∇dblΦn(xn)−Φn(x′\nn)/ba∇dbl∞,\nwe have\ndY(yj,yj′)≤lim\nn→∞/radicalbig\ns2+(s′)2−2ss′cos(/ba∇dblΦn(xn)−Φn(x′n)/ba∇dbl∞)+εn+3ε\n=/radicalbig\ns2+(s′)2−2ss′cos(/ba∇dblx−x′/ba∇dbl∞)+3ε\n=dK(sx,s′x′)+3ε.\nTherefore Ψ is 1-Lipschitz up to 3 ε.16 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\nWe apply Lemma 3.6to the map ι◦Ψ and then obtain Y∈Kµ(P). This completes the\nproof of the case κ= 0.\nIn the case of κ<0, taking\nδ <min{δ′\n3sκ(R),π},\nwe obtain Claim 3.7by Lemma 3.1and\ndY(yj,yj′)≤c−1\nκ(cκ(s)cκ(s′)+κsκ(s)sκ(s′)cos(min {dXn(xn,An\ni′j′),π}))+εn+3ε\ninsteadof ( 3.2). ThuswecanprovethatΨis1-Lipschitzupto3 εanddP(Ψ#mKκ,µ(X),m˙Y)≤\n2ε. In the case of κ >0, we have to pay attention to the handling of the two points\n0,π/√κ∈Kκ(X). Eventually, it can be proved similarly by Lemma 3.1. We now finish the\nproof of Theorem 1.2. /square\n4.Another simple proof for L ´evy families\nIn this section, we give another simple proof to Theorem 1.3.\nProposition 4.1. Letfbe a1-Lipschitz function on the κ-cone of a metric space X. For\nanyr∈Iκ, we define a function fr:X→Rby\n(4.1) fr(x) :=f(rx), x∈X.\nThenfris˜sκ(r)-Lipschitz with respect to min{dX,π}.\nProof.This proposition follows from Lemma 3.1directly. /square\nProposition 4.2 (cf. [21, Proof of Proposition 7.32]) .Letfbe a1-Lipschitz function on the\nκ-cone of an mm-space X. We define a function ¯f:Iκ→Rby\n(4.2) ¯f(r) := lm(fr;mX), r∈Iκ,\nwherefris defined as (4.1)forr∈Iκ. Then the function ¯fis1-Lipschitz on Iκ.\nProof.We denote by m(r) andm(r) the minimum and maximum of medians of frwith\nrespect tomX, respectively. Note that\n¯f(r) =m(r)+m(r)\n2.\nThen, for any r,r′∈Iκ, since\n|fr(x)−fr′(x)| ≤dKκ(X)(rx,r′x) =|r−r′|,\nwe have\nmX({x∈X|fr′(x)≤m(r)+|r−r′|})≥mX({x∈X|fr(x)≤m(r)})≥1\n2,\nwhich together with the minimality of m(r′) implies\nm(r′)≤m(r)+|r−r′|.\nExchanging randr′, we obtain\n|m(r)−m(r′)| ≤ |r−r′|.\nThusmis 1-Lipschitz on Iκ. Similarly, mis also 1-Lipschitz on Iκ, and then ¯fis 1-Lipschitz\nonIκ. The proof is completed. /squareCONVERGENCE OF CONES OF METRIC MEASURE SPACES 17\nProof of Theorem 1.3.For anyf∈ Lip1(Kκ(Xn)), the functions fr:Xn→R,r∈Iκ, and\n¯f:Iκ→Rare defined as ( 4.1) and (4.2), respectively. Propositions 4.1and4.2imply that\nfris˜sκ(r)-Lipschitz with respect to min {dXn,π}and¯fis 1-Lipschitz on Iκ. Moreover we\nhave\n|fr(x)−¯f(r)| ≤diamfr(Xn)≤π˜sκ(r)\nfor anyr∈Iandx∈Xn.\nIn order to prove Theorem 1.3, by Lemma 2.14and the assumption, it is sufficient to prove\nthat, for any ε>0, there exists a number N∈Nsuch that any f∈ Lip1(Kκ(Xn)) satisfies\n(4.3) mKκ,µn(Xn)(/braceleftbig\nrx∈Kκ(Xn)/vextendsingle/vextendsingle|f(rx)−¯f(r)| ≥ε/bracerightbig\n)≤ε\nifn≥N. This together with the 1-Lipschitz continuity of pn(rx) :=ronKκ(Xn) implies\ndH(Lip1(Kκ(Xn)),(pn)∗Lip1(Iκ))≤ε,\nwhere the Hausdorff distance is defined with respect to dmKκ,µn(Xn)\nKF .\nSince{µn}∞\nn=1converges weakly, there exists R>0 such that\nsup\nn∈Nµn((R,∞))<ε.\nWe assume that R=π/√κifκ>0. Thus we have\nmKκ,µn(Xn)(/braceleftbig\nrx∈Kκ(Xn)/vextendsingle/vextendsingle|f(rx)−¯f(r)| ≥ε/bracerightbig\n)\n≤mKκ,µn(Xn)(/braceleftbig\nrx∈Kκ(Xn)/vextendsingle/vextendsingle|f(rx)−¯f(r)| ≥εandr≤R/bracerightbig\n)+ε.\nHere, by the Markov inequality, we calculate\nmKκ,µn(Xn)(/braceleftbig\nrx∈Kκ(Xn)/vextendsingle/vextendsingle|f(rx)−¯f(r)| ≥εandr≤R/bracerightbig\n)\n=mKκ,µn(Xn)(/braceleftbig\nrx∈Kκ(Xn)/vextendsingle/vextendsingle1{r≤R}(rx)|f(rx)−¯f(r)| ≥ε/bracerightbig\n)\n≤1\nε/integraldisplay\nq([0,R]×Xn)|f(rx)−¯f(r)|dmKκ,µn(Xn)(rx)\n=1\nε/integraldisplayR\n0/parenleftbigg/integraldisplay\nXn/vextendsingle/vextendsinglefr(x)−¯f(r)/vextendsingle/vextendsingledmXn(x)/parenrightbigg\ndµn(r).\nBy Proposition 2.23, we have\nmXn/parenleftbig/braceleftbig\nx∈Xn/vextendsingle/vextendsingle|fr(x)−¯f(r)| ≥ε2/bracerightbig/parenrightbig\n≤2αXn(ε2\n˜sκ(r)).\nThus we estimate\n/integraldisplay\nXn/vextendsingle/vextendsinglefr−¯f(r)/vextendsingle/vextendsingledmXn\n=/integraldisplay\n|fr−¯f(r)|≤ε2/vextendsingle/vextendsinglefr−¯f(r)/vextendsingle/vextendsingledmXn+/integraldisplay\n|fr−¯f(r)|>ε2/vextendsingle/vextendsinglefr−¯f(r)/vextendsingle/vextendsingledmXn\n≤ε2+π˜sκ(r)mXn({|fr−¯f(r)|>ε2})≤ε2+2π˜sκ(r)αXn(ε2\n˜sκ(r)).18 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\nThese imply that\nmKκ,µn(Xn)(/braceleftbig\nrx∈Kκ(Xn)/vextendsingle/vextendsingle|f(rx)−¯f(r)| ≥ε/bracerightbig\n)\n≤1\nε/integraldisplayR\n0/parenleftbigg\nε2+2π˜sκ(r)αXn(ε2\n˜sκ(r))/parenrightbigg\ndµn(r)+ε\n≤2ε+2π\nε/integraldisplayR\n0αXn(ε2\n˜sκ(r))˜sκ(r)dµn(r)\n≤2ε+2π˜sκ(R)\nεαXn(ε2\n˜sκ(R)),\nwhere the last inequality follows from the fact that r/ma√sto→αXn(ε2/r) is non-decreasing. Since\n{Xn}∞\nn=1isaL´ evyfamily, theconcentrationfunction αXnconverges pointwise to0as n→ ∞.\nThus, we obtain\nlimsup\nn→∞mKκ,µn(Xn)(/braceleftbig\nrx∈Kκ(Xn)/vextendsingle/vextendsingle|f(rx)−¯f(r)| ≥ε/bracerightbig\n)≤2ε.\nTherefore we obtain ( 4.3). We finish the proof of Theorem 1.3. /square\nRemark4.3.By [10, Proposition 1.8], ( 4.3) also holds if ¯fis replaced with\n¯f(r) :=/integraldisplay\nXnf(rx)dmXn(x).\nWe also obtain the following corollary about the Hopf action. Let F=R,C, orH, where\nHis the algebra of quaternions, and let d:= dim RF. The multiplicative group\nUF:={t∈F| /ba∇dblt/ba∇dbl= 1}\nacts on the unit sphere Sdn−1inFnby\nSdn−1×UF∋((x1,...,xn),t)/ma√sto→(tx1,...,txn)∈Sdn−1,\nwhere (x1,...,xn)∈Sdn−1⊂Fnwithxi∈F. This action extends to the cone Kκ(Sdn−1)\nas\nKκ(Sdn−1)×UF∋(rx,t)/ma√sto→r(tx)∈Kκ(Sdn−1)\nand it is isometric and preserves the measure mKκ,µ(Sdn−1)for anyµ, that is, the mm-action.\nThus we have the quotient mm-space\nKκ,µ(Sdn−1)/UF\nbut, after all, this coincides with the cone of the quotient of sphere s\nKκ,µ(Sdn−1/UF).\nCorollary 4.4. LetF=R,C, orHand letd:= dim RFandκ∈R. Assume that a sequence\n{µn}∞\nn=1of Borel probability measures on Iκconverges weakly to a Borel probability measure\nµ. Then the sequence {Kκ,µn(Sdn−1)/UF}∞\nn=1concentrates to the interval (Iκ,|·|,µ).\nProof.This directly follows from Theorem 1.3since\nKκ,µn(Sdn−1)/UF=Kκ,µn(Sdn−1/UF)\nand{Sdn−1/UF}∞\nn=1is a L´ evy family. /square\nIn this type of concentration, the limit space is independent of the H opf action.\nRemark 4.5.In [15], the concentration with group action, called the equivariant concen-\ntration, has been studied. The equivariant concentration implies the conce ntration of the\nquotient spaces. Actually, we can prove that the pair ( Kκ,µn(Sdn−1),UF) equivariant con-\ncentrates to (( Iκ,|·|,µ),{idIκ}) from our proof of Theorem 1.3.CONVERGENCE OF CONES OF METRIC MEASURE SPACES 19\n5.Cauchy distribution\nIn this section, we deal with the Cauchy distribution and prove Theo rems1.4and1.5by\napplying Theorem 1.3. We recall the Cauchy measure νn\nβonRnas in (1.3);\nνn\nβ(dx) :=Γ(n+β\n2)\nπn\n2Γ(β\n2)dx\n(1+/ba∇dblx/ba∇dbl2)n+β\n2,\nand the Borel probability measure νβon [0,∞) as in (1.4);\nνβ(dt) :=21−β\n2\nΓ(β\n2)1\ntβ+1e−1\n2t2dt.\nWe denote Cn\nβ:= (Rn,/ba∇dbl·/ba∇dbl,νn\nβ) and call it the n-dimensional Cauchy space.\nProposition 5.1. Letπn\nk:Rn→Rk,n>k, be the natural projection given by\nπn\nk(x1,...,xn) = (x1,...,xk).\nThen we have\n(5.1) ( πn\nk)#νn\nβ=νk\nβ.\nIn particular, we have\nC1\nβ≺C2\nβ≺ ··· ≺Cn\nβ≺ ···.\nProof.It is sufficient to prove the case of k=n−1. Take any Borel measurable function f\nonRn−1. Changing of variable\nxn= (1+x2\n1+···+x2\nn−1)1\n2y,\nwe have\n/integraldisplay\nRn−1fd(πn\nn−1)#νn\nβ=Γ(n+β\n2)\nπn\n2Γ(β\n2)/integraldisplay\nRnf(x1,...,xn−1)\n(1+x2\n1+···+x2n)n+β\n2dx1···dxn\n=Γ(n+β\n2)\nπn\n2Γ(β\n2)/integraldisplay∞\n−∞dy\n(1+y2)n+β\n2/integraldisplay\nRn−1f(x1,...,xn−1)dx1···dxn−1\n(1+x2\n1+···+x2\nn−1)n−1+β\n2\n=Γ(n+β\n2)\nπn\n2Γ(β\n2)√πΓ(n−1+β\n2)\nΓ(n+β\n2)/integraldisplay\nRn−1f(x1,...,xn−1)dx1···dxn−1\n(1+x2\n1+···+x2\nn−1)n−1+β\n2\n=/integraldisplay\nRn−1fdνn−1\nβ.\nTherefore we obtain ( πn\nn−1)#νn\nβ=νn−1\nβand thenCn−1\nβ≺Cn\nβ. /square\nForβ >0, the Borel probability measure ˜ νn\nβonRnis defined by\n(5.2) ˜ νn\nβ(dx) :=nn\n2Γ(n+β\n2)\nπn\n2Γ(β\n2)dx\n(1+n/ba∇dblx/ba∇dbl2)n+β\n2.\nProposition 5.2. Let/tildewideCn\nβ:= (Rn,/ba∇dbl·/ba∇dbl,˜νn\nβ). Then1√nCn\nβis mm-isomorphic to /tildewideCn\nβ.\nProof.We define a map Ln:Rn→Rnby\nLn(x) =x√n, x∈Rn.\nThe mapLnis an isometry from1√nCn\nβto/tildewideCn\nβand (Ln)#νn\nβ= ˜νn\nβholds. Therefore1√nCn\nβand\n/tildewideCn\nβare mm-isomorphic to each other. /square20 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\nThe following is a key lemma to apply Theorem 1.3to the Cauchy distribution.\nLemma 5.3. Letpn:Rn→[0,∞)be the function defined by\npn(x) :=/ba∇dblx/ba∇dbl, x∈Rn.\nThen(pn)#˜νn\nβconverges weakly to the Borel probability measure νβasn→ ∞.\nProof.Take any bounded (not necessarily continuous) function fon [0,∞). By the polar\ntransformation, we have\n(5.3)/integraldisplay∞\n0f(t)d(pn)#˜νn\nβ(t) =/integraldisplay∞\n02Γ(n+β\n2)\nΓ(n\n2)Γ(β\n2)f(t)\nt(1+nt2)β\n2/parenleftbiggnt2\n1+nt2/parenrightbiggn\n2\ndt.\nBy Stirling’s formula, we have\nlim\nn→∞Γ/parenleftbign+β\n2/parenrightbig\nΓ/parenleftbign\n2/parenrightbig1\nt(1+nt2)β\n2/parenleftbiggnt2\n1+nt2/parenrightbiggn\n2\n=1\n2β\n2tβ+1e−1\n2t2\nfor anyt>0. Moreover, since\nsup\nn∈NΓ(n+β\n2)\nΓ(n\n2)nβ\n2<∞and 0<nβ\n2\nt(1+nt2)β\n2/parenleftbiggnt2\n1+nt2/parenrightbiggn\n2\n<1\ntβ+1\nfor anyt>0, there exists a constant C >0 depending only on fsuch that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2Γ(n+β\n2)\nΓ(n\n2)Γ(β\n2)f(t)\nt(1+nt2)β\n2/parenleftbiggnt2\n1+nt2/parenrightbiggn\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<Cmin{1,1\ntβ+1}.\nTherefore, taking the limit of ( 5.3) by the dominated convergence theorem, we obtain this\nlemma. /square\nProof of Theorem 1.4.LetSn−1be the unit sphere in Rncentered at origin. Assume that\nSn−1isequipped withtheRiemanniandistanceandtheuniformprobabilityme asureσn−1. It\niswell-known thattheEuclideanspace ( Rn,/ba∇dbl·/ba∇dbl)isisometric totheEuclidean cone K0(Sn−1).\nMoreover, by the polar transformation, we have\n˜νn\nβ=q#/parenleftbig\n(pn)#˜νn\nβ⊗σn−1/parenrightbig\n.\nThus we obtain\n/tildewideCn\nβ=K0,(pn)#˜νn\nβ(Sn−1).\nTherefore, by Theorem 1.3together with Lemma 5.3and (2.3), the sequence {/tildewideCn\nβ}∞\nn=1con-\ncentrates to the half line ([0 ,∞),|·|,νβ). This completes the proof. /square\nRemark5.4.In the case of Rn=K0(Sn−1), Theorem 1.3is also expressed as follows. Let ˆ µn\nbe a radially symmetric Borel probability measure on Rnand letpn(x) :=/ba∇dblx/ba∇dblforx∈Rn.\nAssume that the measure µn:= (pn)#ˆµnconverges weakly to a Borel probability measure µ\non [0,∞). Then the sequence {(Rn,/ba∇dbl·/ba∇dbl,ˆµn)}∞\nn=1concentrates to the half line ([0 ,∞),|·|,µ).\nCorollary 4.4leads to the concentration of the following concrete example.\nCorollary 5.5. LetF=R,C, orHand letd:= dim RF. Then the quotient space\n1√nCdn\nβ/UF\nconcentrates to the half line ([0,∞),|·|,νβ)asn→ ∞.\nWe next prove Corollaries 1.6and1.7.CONVERGENCE OF CONES OF METRIC MEASURE SPACES 21\nProof of Corollaries 1.6.This followsfromTheorem 1.4and[21, Proposition6.2(1)]directly.\n/square\nProof of Corollary 1.7.In this proof, we denote\ndiam(νβ;1−κ) := diam(([0 ,∞),|·|,νβ);1−κ)\nfor simplicity. By the definition of the observable diameter, we see th at\nObsDiam(([0 ,∞),|·|,νβ);−κ) = diam(νβ;1−κ).\nTheorem 1.4and the limit formula of the observable diameter in [ 18, Theorem 1.1] imply\ndiam(νβ;1−κ)≤liminf\nn→∞ObsDiam(1√nCn\nβ;−κ)\n≤limsup\nn→∞ObsDiam(1√nCn\nβ;−κ)≤lim\nε→0+diam(νβ;1−(κ−ε))\nfor anyκ>0. We prove\n(5.4) lim\nε→0+diam(νβ;1−(κ−ε)) = diam(νβ;1−κ)\nfor anyκ>0. Fixκ>0 and take any δ >0 with diam( νβ;1−κ)<δ. There exists a Borel\nsubsetAof [0,∞) such that\nνβ(A)≥1−κand diam A<δ.\nLetη:=δ−diamA>0. Then, since\nνβ(Uη\n3(A))>νβ(A)≥1−κ,\nwe have\nlim\nε→0+diam(νβ;1−(κ−ε))≤diamUη\n3(A)≤diamA+2\n3η<δ,\nwhich implies ( 5.4). We obtain this corollary. /square\nWe now prove Theorem 1.5.\nProof of Theorem 1.5.Let{rn}∞\nn=1be a sequence of positive real numbers.\n(1) Assume that rn√n→0. Take any ε>0. Lemma 5.3implies\nliminf\nn→∞νn\nβ(Uεr−1\nn(o)) = liminf\nn→∞(pn)#˜νn\nβ([0,ε\nrn√n))≥νβ([0,∞)) = 1,\nwhereois the origin of Rn. Thus, for sufficiently large n, we have\nνn\nβ(Uεr−1\nn(o))≥1−ε,\nwhich implies that dP(νn\nβ,δo)≤ε, wheredPis defined with respect to rn/ba∇dbl·/ba∇dblonRn.\nTherefore, by Proposition 2.9, we obtain\nlimsup\nn→∞/square(rnCn\nβ,∗)≤2ε.\nAsε→0, the sequence {rnCn\nβ}∞\nn=1box-converges to the one-point mm-space ∗.\n(2) Assumethat rn√n→λ∈(0,∞). ByTheorem 1.4and[7, Theorem1.4],thesequence\n{rnCn\nβ}∞\nn=1concentrates to the half line ([0 ,∞),λ|·|,νβ) scaled with λ. Let\nνβ,λ(dt) :=21−β\n2\nΓ(β\n2)λβ\ntβ+1e−λ2\n2t2dt.\nThen ([0,∞),λ|·|,νβ) and ([0,∞),|·|,νβ,λ) are mm-isomorphic to each other. There-\nfore we obtain the conclusion.22 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\n(3) Assume that rn√n→ ∞. By Corollary 1.7, we have\nliminf\nn→∞ObsDiam(rnCn\nβ;−κ) = liminf\nn→∞(rn√n)ObsDiam(1√nCn\nβ;−κ) =∞.\nTherefore {rnCn\nβ}∞\nn=1infinitely dissipates by Proposition 2.29.\nTherefore we obtain Theorem 1.5. /square\nRemark5.6.In[9], then-dimensionalGaussianspaceΓn\n{(n−1)2}=n−1Γn\n{12}withvariance1 /n2\nconcentrates to ∗asn→ ∞but it does not box-converge, that is, there exist scaling orders\nof non-box-convergent L´ evy family. On the other hand, for the Cauchy space, {rnCn\nβ}∞\nn=1is\na L´ evy family if and only if it box-converges.\n6.Proof of Theorem 1.8\nIn this section, we prove Theorem 1.8. We need the following two lemmas.\nLemma 6.1 ([21, Lemma 5.43]) .If a sequence {Xn}∞\nn=1of mm-spaces box-converges to an\nmm-spaceX, then we have\ndiam(X;s)≤liminf\nn→∞diam(Xn;s)≤limsup\nn→∞diam(Xn;s)≤lim\nδ→0+diam(X;s+δ)\nfor anys>0.\nLemma 6.2 ([9, Lemma 4.5]) .Let{Xn}∞\nn=1be a box-convergent sequence of mm-spaces\nand{Yn}∞\nn=1a sequence of mm-spaces with Yn≺Xn. Then, {Yn}∞\nn=1has a box-convergent\nsubsequence.\nProof of Theorem 1.8.LetR>0 be the real number with\nνβ([0,R]) =1\n3.\nWe define maps ϕn:Rn→Rnandϕ: [0,∞)→[0,∞) by\nϕn(x) :=/braceleftBigg\nxif/ba∇dblx/ba∇dbl ≤R,\nR\n/bardblx/bardblxif/ba∇dblx/ba∇dbl ≥R,andϕ(r) :=/braceleftBigg\nrifr≤R,\nRifr≥R.\nThenϕnandϕare 1-Lipschitz and /ba∇dblϕn(x)/ba∇dbl=ϕ(/ba∇dblx/ba∇dbl) for anyx∈Rn.\nClaim 6.3. The sequence {(Bn\nR(o),/ba∇dbl·/ba∇dbl,(ϕn)#˜νn\nβ)}∞\nn=1concentrates to ([0,R],|·|,ϕ#νβ)as\nn→ ∞, whereBn\nR(o)is the closed ball in Rncentered at the origin owith radius R.\nProof.Letpn(x) :=/ba∇dblx/ba∇dblforx∈Rn. Then we have\ndP((pn)#(ϕn)#˜νn\nβ,ϕ#νβ) =dP(ϕ#(pn)#˜νn\nβ,ϕ#νβ)≤dP((pn)#˜νn\nβ,νβ)\nsinceϕis 1-Lipschitz and ( 2.1). Thus, by Theorem 1.3and Lemma 5.3, the sequence of\nK0,(pn)#(ϕn)#˜νn\nβ(Sn−1) = (Bn\nR(o),/ba∇dbl·/ba∇dbl,(ϕn)#˜νn\nβ)\nconcentrates to ([0 ,R],|·|,ϕ#νβ) asn→ ∞. /square\nNow we suppose that {1√nCn\nβ}∞\nn=1has a box-convergent subsequence. Then, by Lemma\n6.2, up to subsequence, {(Bn\nR(o),/ba∇dbl · /ba∇dbl,(ϕn)#˜νn\nβ)}∞\nn=1box-converges to ([0 ,R],| · |,ϕ#νβ) as\nn→ ∞. Sinceϕ#νβ({R}) =2\n3, we have\ndiam(ϕ#νβ;2\n3) = diam {R}= 0.\nMoreover, if a Borel subset A⊂Bn\nR(o) satisfies (ϕn)#˜νn\nβ(A)≥1\n2, then\nA∩{x∈Rn| /ba∇dblx/ba∇dbl=R} /\\e}atio\\slash=∅CONVERGENCE OF CONES OF METRIC MEASURE SPACES 23\nsince (ϕn)#˜νn\nβ({/ba∇dblx/ba∇dbl< R}) = ˜νn\nβ({/ba∇dblx/ba∇dbl< R})→1\n3asn→ ∞. In addition, if diam A < R,\nthentheset Aisinahalfspaceof Rn, which implies( ϕn)#˜νn\nβ(A)<1\n2. Thisisacontradiction.\nThus we have\nliminf\nn→∞diam((ϕn)#˜νn\nβ;1\n2)≥R>0.\nOn the other hand, Lemma 6.1implies\nlimsup\nn→∞diam((ϕn)#˜νn\nβ;1\n2)≤lim\ns→0+diam(ϕ#νβ;1\n2+s)≤diam(ϕ#νβ;2\n3) = 0.\nThus{1√nCn\nβ}∞\nn=1must have no box-convergent subsequence. The proof is complet ed./square\n7.Several quantities for Cauchy space\nIn this section, we compute several geometric and analytic invarian ts for the Cauchy space\nand the limit space. We have developed the general theory of invaria nts for mm-spaces in\nour previous paper [ 3].\nDefinition 7.1 (N-weighted Ricci curvature, cf. [ 13,16]).For ann-dimensional weighted\nRiemannian manifold ( M,g,µ) withµ=ρvolgfor some positive smooth function ρ, and for\na real number N∈(−∞,∞)\\{n}, theN-weighted Ricci curvature is given by\nRicµ,N= Ricρ,N:= Ricg+Hess(−logρ)−∇logρ⊗∇logρ\nN−n.\nProposition 7.2. We have\n(7.1) Ric ˜νn\nβ,−β=n(n+β)\n(1+n/ba∇dblx/ba∇dbl2)2((1+n/ba∇dblx/ba∇dbl2)Id−nxxt)\nat any point x∈Rn, whereIdis the identity matrix and xxt= (xixj)ij, under the natural\nidentification of the Ricci tensor with a matrix, and\n(7.2) Ric νβ,−β=1\nt4+1\n(β+1)t6\nat any point t∈[0,∞).\nProof.Let\nρn(x) :=nn\n2Γ(n+β\n2)\nπn\n2Γ(β\n2)1\n(1+n/ba∇dblx/ba∇dbl2)n+β\n2.\nWe calculate\n∇(−logρn) =n+β\n22nx\n1+n/ba∇dblx/ba∇dbl2=n(n+β)\n1+n/ba∇dblx/ba∇dbl2x,\nand then\nHess(−logρn) =n(n+β)\n(1+n/ba∇dblx/ba∇dbl2)2((1+n/ba∇dblx/ba∇dbl2)Id−2nxxt),\n∇(logρn)⊗∇(logρn)\n−β−n=−n(n+β)\n(1+n/ba∇dblx/ba∇dbl2)2nxxt.\nTherefore we obtain ( 7.1). We next prove ( 7.2). Let\nρ(t) :=21−β\n2\nΓ(β\n2)1\ntβ+1e−1\n2t2.\nWe calculate\n∇(−logρ) =−1\nt3+β+1\nt,24 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\nand then\nHess(−logρ) =3\nt4−β+1\nt2,\n∇(logρ)⊗∇(logρ)\n−β−1=−1\nβ+1/parenleftbigg1\nt6−2(β+1)\nt4+(β+1)2\nt2/parenrightbigg\n.\nTherefore we obtain ( 7.2). The proof is completed. /square\nDefinition 7.3 (Variance) .LetXbe an mm-space. The ( observable )varianceV(X) ofX\nis defined by\nV(X) := sup\nf∈Lip1(X)1\n2/integraldisplay\nX/integraldisplay\nX|f(x)−f(y)|2dm⊗2\nX(x,y) (≤ ∞).\nNote thatV:X →[0,∞] is 2-homogeneous, lower semicontinuous with respect to the box\nand concentration topologies, and monotone with respect to Lipsc hitz order (see [ 3, Section\n4.1]).\nProposition 7.4. Ifβ >2, then we have\n(7.3) V(1√nCn\nβ)≤1\nβ−2\nfor anyn. Moreover, we have\n(7.4) liminf\nn→∞V(1√nCn\nβ)≥V(([0,∞),|·|,νβ)) =1\nβ−2−Γ(β−1\n2)2\n2Γ(β\n2)2>0.\nProof.Since\n/integraldisplay\nRn/integraldisplay\nRn/a\\}b∇acketle{tx,y/a\\}b∇acket∇i}htd(νn\nβ)⊗2(x,y) =n/integraldisplay\nR/integraldisplay\nRxyd(ν1\nβ)⊗2(x,y) =n/parenleftbigg/integraldisplay\nRxdν1(x)/parenrightbigg2\n= 0,\nwe have\nV(1√nCn\nβ)≤1\n2/integraldisplay\nRn/integraldisplay\nRn1\nn/ba∇dblx−y/ba∇dbl2d(νn\nβ)⊗2(x,y) =1\nn/integraldisplay\nRn/ba∇dblx/ba∇dbl2dνn\nβ(x) =1\nβ−2.\nIndeed, we verify the last equality as follows.\n1\nn/integraldisplay\nRn/ba∇dblx/ba∇dbl2dνn\nβ(x) =Γ(n+β\n2)\nnπn\n2Γ(β\n2)vol(Sn−1)/integraldisplay∞\n0rn+1\n(1+r2)n+β\n2dr\n=Γ(n+β\n2)\nπn\n2Γ(β\n2)2πn\n2\nΓ(n\n2)Γ(n\n2+1)Γ(β\n2−1)\n2Γ(n+β\n2)=1\nβ−2.\nMoreover, since the variance Vis lower semicontinuous with respect to the concentration\ntopology, Theorem 1.4implies\nliminf\nn→∞V(1√nCn\nβ)≥V(([0,∞),|·|,νβ)).\nand we have\nV(([0,∞),|·|,νβ)) =/integraldisplay∞\n0t2dνβ(t)−/parenleftbigg/integraldisplay∞\n0tdνβ(t)/parenrightbigg2\n=1\nβ−2−Γ(β−1\n2)2\n2Γ(β\n2)2.\nThus we obtain ( 7.4). The proof is completed. /squareCONVERGENCE OF CONES OF METRIC MEASURE SPACES 25\nDefinition 7.5 (Poincar´ e constant) .An mm-space Xsatisfies the (2 ,2)-Poincar´ e inequality\nfor constant C >0 provided that for any bounded Lipschitz function fonX,\n1\n2/integraldisplay\nX/integraldisplay\nX|f(x)−f(y)|2dm⊗2\nX(x,y)≤C2/integraldisplay\nXlipa(f)2dmX,\nwhere lipa(f) is theasymptotic Lipschitz constant defined by\nlipa(f)(x) := lim\nr→0sup\ny/ne}ationslash=y′∈Ur(x)|f(y)−f(y′)|\ndX(y,y′).\nThe infimum of such a constant Cis denoted by C2,2(X) and is called the (2 ,2)-Poincar´ e\nconstant ofX.\nNote thatC2,2:X →[0,∞] is 1-homogeneous, lower semicontinuous with respect to the\nbox and concentration topologies, and monotone with respect to t he Lipschitz order (see\n[3, Section 4.2]).\nLemma 7.6. Cn\nβand([0,∞),|·|,νβ)do not satisfy the (2,2)-Poincar´ e inequality, that is,\nC2,2(Cn\nβ) =C2,2(([0,∞),|·|,νβ)) =∞.\nProof.By Muckenhoupt’s criterion [ 1, Theorem 4.5.1] and Proposition 5.1, it is sufficient to\nprove\nsup\nx>0ν1\nβ([x,∞))/integraldisplayx\n0π(1+t2)dt=∞, (7.5)\nsup\nx>mνβ([x,∞))Γ(β\n2)\n21−β\n2/integraldisplayx\nmtβ+1e1\n2t2dt=∞, (7.6)\nwheremis a median of νβ.\nWe first prove ( 7.5). For anyx>1, we have\nν1\nβ([x,∞))/integraldisplayx\n0π(1+t2)dt≥/integraldisplay∞\nxdt\n2t2/integraldisplayx\n0(1+t2)dt=1\n2+1\n6x2,\nwhich implies ( 7.5). We next prove ( 7.6). We estimate\nνβ([x,∞))≥21−β\n2\nΓ(β\n2)e−1\n2x2/integraldisplay∞\nxdt\ntβ+1=21−β\n2\nΓ(β\n2)e−1\n2x21\nβxβ\nand\nΓ(β\n2)\n21−β\n2/integraldisplayx\n1tβ+1e1\n2t2dt≥Γ(β\n2)\n21−β\n2/integraldisplayx\n1tβ+1dt=Γ(β\n2)\n21−β\n21\nβ+2(xβ+2−1).\nThus we have\nνβ([x,∞))Γ(β\n2)\n21−β\n2/integraldisplayx\n1tβ+1e1\n2t2dt≥1\nβ(β+2)e−1\n2x2/parenleftbigg\nx2−1\nxβ/parenrightbigg\n→ ∞asx→ ∞.\nOn the other hand, since νβis a Borel probability measure and mis finite, we have\nlim\nx→∞νβ([x,∞))Γ(β\n2)\n21−β\n2/integraldisplaym\n1tβ+1e1\n2t2dt= 0.\nThese together imply that\nlim\nx→∞νβ([x,∞))Γ(β\n2)\n21−β\n2/integraldisplayx\nmtβ+1e1\n2t2dt=∞,\nwhich implies ( 7.6). The proof is completed. /square26 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\n8.Relation to the ergodic theory\nIn this section, we construct an infinite sequence of random variab les that does not satisfy\nthe law of large number while are identically distributed and have a finite moment, using the\nCauchy distribution. We start with recalling the Birkhoff ergodic theo rem in this section.\nTheorem 8.1 ([23, Theorem 1.14]) .Let(X,Σ,µ)be a probability measure space and let\nT:X→Xbe a measure-preserving map, i.e., T#µ=µ. For anyf∈L1(X,µ), the time\naverage\nˆf(x) := lim\nn→∞1\nnn−1/summationdisplay\ni=0f(Tix)\nexists for almost all x∈X, whereTi=T◦Ti−1andT0= idX. Moreover, ˆfis given by the\nconditional expectation\nE[f| I]\nalmost everywhere on X, whereI:={A∈Σ|T−1(A) =A}called the invariant σ-field of\nT.\nDefinition 8.2 (Ergodicity) .Let (X,Σ,µ) be a probability measure space and let T:X→\nXbe a measure-preserving map. The map Tis said to be ergodicprovided that for any\nA∈Σ withµ(T−1(A)\\A) =µ(A\\T−1(A)) = 0,µ(A) is either 0 or 1.\nTheorem 8.3 ([23, Remark just below Theorem 1.14]) .Under Theorem 8.1, ifTis er-\ngodic, then for any f∈L1(X,µ), the time average ˆfcoincides with the mean E[f]almost\neverywhere on X.\nExample 8.4. Letµbe a Borel probability measure on R. We consider the measure space\n(RN,B(RN),µ⊗∞). The map Θ is defined by the shift\nΘ(x1,x2,...) = (x2,x3,...),(x1,x2,...)∈RN.\nThe shift Θ is ergodic with respect to µ⊗∞. Assume that\nE[|x|] =/integraldisplay\nR|x|dµ(x)<∞.\nLettingf(x1,x2,...) =x1, we have\nlim\nn→∞1\nnn/summationdisplay\ni=1xi=/integraldisplay\nRxdµ(x)\nfor almost all ( x1,x2,...)∈RNby Theorem 8.3. This is exactly the strong law of large\nnumbers. All coordinates xiofRNare independently and identically distributed by µwith\nfinite moment.\nAs a corollary of Lemma 5.3and Proposition 7.4, we obtain the following theorem.\nTheorem 8.5. Letν∞\nβ,β >0, be the Borel probability measure on RNcharacterized by\n(πn)#ν∞\nβ=νn\nβ,\nwhereπn:RN→Rn,n∈N, are natural projections. If β >2, then the shift\nΘ(x1,x2,...) = (x2,x3,...)\nis not ergodic with respect to ν∞\nβ.CONVERGENCE OF CONES OF METRIC MEASURE SPACES 27\nProof.Suppose that the shift Θ is ergodic with respect to ν∞\nβforβ >2. We define a function\nf:RN→Rby\nf(x1,x2,...) =x2\n1.\nNote thatfis integrable with respect to ν∞\nβby Proposition 7.4. Then Theorem 8.3implies\nthat\nlim\nn→∞1\nnn/summationdisplay\ni=1x2\ni=V(C1\nβ) =1\nβ−2\nfor almost all ( x1,x2,...)∈RNwith respect to ν∞\nβ. This means that\n(pn)#˜νn\nβ→δ1√β−2weakly\nasn→ ∞, which contradicts Lemma 5.3. The proof is completed. /square\nRemark 8.6.The existence and uniqueness of ν∞\nβfollows from the Kolmogorov extension\ntheorem and ( 5.1). All coordinates xiofRNareidentically distributed by ν1\nβand the measure\nν1\nβhas a finite variance if β >2. On the other hand, since ν∞\nβis not a product measure,\ncoordinates are not independently distributed to each other.\nReferences\n[1] D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of Markov diffusion operators , Grundlehren\nder mathematischen Wissenschaften [Fundamental Principles of Ma thematical Sciences], vol. 348,\nSpringer, Cham, 2014.\n[2] S. G. Bobkov and M. Ledoux, Weighted Poincar´ e-type inequalities for Cauchy and other convex mea-\nsures, Ann. Probab. 37(2009), no. 2, 403–427.\n[3] S. Esaki, D. Kazukawa, and A. Mitsuishi, Invariants for Gromov’s pyramids and their applications .\npreprint (2023), arXiv:2308.01620 , to appear in Adv. Math.\n[4] M. Gromov, Isoperimetry of waists and concentration of maps , Geom. Funct. Anal. 13(2003), no. 1,\n178–215.\n[5] M. Gromov, Metric structures for Riemannian and non-Riemannian space s, Reprint of the 2001 English\nedition, Modern Birkh¨ auser Classics, Birkh¨ auser Boston, Inc., B oston, MA, 2007.\n[6] M. Gromov and V. D. Milman, A topological application of the isoperimetric inequality , Amer. J. Math.\n105(1983), no. 4, 843–854.\n[7] D. Kazukawa, Concentration of product spaces , Anal. Geom. Metr. Spaces 9(2021), no. 1, 186–218.\n[8] ,Convergence of metric transformed spaces , Israel J. Math. 252(2022), no. 1, 243–290.\n[9] D. Kazukawa and T. Shioya, High-dimensional ellipsoids converge to Gaussian spaces . preprint (2020),\narXiv: 2003.05105, to appear in J. Math. Soc. Japan.\n[10] M. Ledoux, The concentration of measure phenomenon , Mathematical Surveysand Monographs,vol.89,\nAmerican Mathematical Society, Providence, RI, 2001.\n[11] P. L´ evy, Probl` emes concrets d’analyse fonctionnelle. Avec un comp l´ ement sur les fonctionnelles analy-\ntiques par F. Pellegrino , Gauthier-Villars, Paris, 1951 (French). 2d ed.\n[12] M. Magnabosco, C. Rigoni, and G. Sosa, Convergence of metric measure spaces satisfying the CD\ncondition for negative values of the dimension parameter ,NonlinearAnal. 237(2023),PaperNo.113366,\n48.\n[13] E. Milman, Beyond traditional curvature-dimension I: new model space s for isoperimetric and concen-\ntration inequalities in negative dimension , Trans. Amer. Math. Soc. 369(2017), no. 5, 3605–3637.\n[14] V. D. Milman, The heritage of P. L´ evy in geometrical functional analysis , Ast´ erisque 157-158 (1988),\n273–301. Colloque Paul L´ evy sur les Processus Stochastiques (P alaiseau, 1987).\n[15] H. Nakajima and T. Shioya, Convergence of group actions in metric measure geometry . preprint (2021),\narXiv:2104.00187 , to appear in Commun. Anal. Geom.\n[16] S.-i. Ohta, ( K,N)-convexity and the curvature-dimension condition for nega tiveN, J. Geom. Anal. 26\n(2016), no. 3, 2067–2096.\n[17] S.Oshima, Stability of curvature-dimension condition for negative d imensions under concentration topol-\nogy, J. Geom. Anal. 33(2023), no. 12, Paper No. 377, 37.\n[18] R. Ozawa and T. Shioya, Limit formulas for metric measure invariants and phase tran sition property ,\nMath. Z. 280(2015), no. 3-4, 759–782.28 S. ESAKI, D. KAZUKAWA, AND A. MITSUISHI\n[19] R. Ozawa and T. Yokota, Stability of RCD condition under concentration topology , Calc. Var. Partial\nDifferential Equations 58(2019), no. 4, Paper No. 151, 30.\n[20] T.Shioya, Metric measure limits of spheres and complex projective spa ces,Measuretheoryinnon-smooth\nspaces, Partial Differ. Equ. Meas. Theory, De Gruyter Open, War saw, 2017.\n[21] ,Metric measure geometry , IRMA Lectures in Mathematics and Theoretical Physics, vol. 25,\nEMS Publishing House, Z¨ urich, 2016. Gromov’s theory of converge nce and concentration of metrics and\nmeasures.\n[22] T. Shioya and A. Takatsu, High-dimensional metric-measure limit of Stiefel and flag m anifolds, Math.\nZ.290(2018), no. 3-4, 873–907.\n[23] P. Walters, An introduction to ergodic theory , Graduate Texts in Mathematics, vol. 79, Springer-Verlag,\nNew York-Berlin, 1982.\nDepartment of Applied Mathematics, Fukuoka University, Fu kuoka 814-0180, JAPAN\nEmail address :sesaki@fukuoka-u.ac.jp\nFaculty of Mathematics, Kyushu University, Fukuoka 819-03 95, JAPAN\nEmail address :kazukawa@math.kyushu-u.ac.jp\nDepartment of Applied Mathematics, Fukuoka University, Fu kuoka 814-0180, JAPAN\nEmail address :mitsuishi@fukuoka-u.ac.jp"    },    {        "title": "2402.14346v1.Dependable_Distributed_Training_of_Compressed_Machine_Learning_Models.pdf",        "content": "Dependable Distributed Training\nof Compressed Machine Learning Models\nFrancesco Malandrino∗†, Giuseppe Di Giacomo‡, Marco Levorato§, Carla Fabiana Chiasserini‡∗†\n∗CNR-IEIIT, Italy –†CNIT, Italy –‡Politecnico di Torino, Italy –§UC Irvine, USA\nAbstract —The existing work on the distributed training of\nmachine learning (ML) models has consistently overlooked the\ndistribution of the achieved learning quality, focusing instead\non its average value. This leads to a poor dependability of the\nresulting ML models, whose performance may be much worse\nthan expected. We fill this gap by proposing DepL, a framework\nfordependable learning orchestration , able to make high-quality,\nefficient decisions on (i) the data to leverage for learning, (ii)\nthe models to use and when to switch among them, and (iii)\nthe clusters of nodes, and the resources thereof, to exploit. For\nconcreteness, we consider as possible available models a full\nDNN and its compressed versions. Unlike previous studies, DepL\nguarantees that a target learning quality is reached with a target\nprobability , while keeping the training cost at a minimum. We\nprove that DepL has constant competitive ratio and polynomial\ncomplexity, and show that it outperforms the state-of-the-art by\nover 27% and closely matches the optimum.\nI. I NTRODUCTION\nMachine learning (ML) models, and deep neural networks\n(DNNs) in particular, are becoming more capable, but also\nharder and more costly to train. This issue has been tackled\nthrough two complementary approaches: distributed training\nand model compression. The former exploits the resources\n(e.g., computational) and data available at different nodes,\nthus better distributing the training burden. The latter includes\na wealth of different techniques (e.g., model pruning and\nknowledge distillation) allowing for the transfer of knowledge\nfrom a model to a (usually, simpler) one.\nDistributed training and model compression have been\nvariously combined together in the literature, allowing the\nused resources to adapt to the chosen model [1], selecting\nthe best model given the available resources [2], [3], and even\nperforming mutual adaptation between the two [4]. However,\nan aspect that has been overlooked so far, which we aim to\ninvestigate in this work, is the dependability of DNN training.\nIndeed, ML is increasingly used for mission-critical (and\neven safety-related) applications, hence traditional approaches\nfocusing on the expected learning quality may be inadequate\nto present needs.\nIn this work, we aim at providing the right network support\nto make DNN training dependable , i.e., to guarantee that a\ntarget learning quality (e.g., loss or accuracy) is reached by\na target time and with a target probability – at the minimum\ncost. To this end, we make the key observation that learning\ndependability is affected by two main factors: (i) the used\nmodels (i.e., full or compressed versions), and (ii) the quality\noftraining (including the data generated by data sources as\nwell as the nodes and the resources thereof that are used).\nFig. 1. DepL enables the optimization of dependable, distributed training of\nneural networks over heterogeneous nodes, datasets, and model architectures.\nThe optimal configuration is designed to control the uncertainty in the loss\nprogression along training epochs.\nOur work is the first to account for all these contributions in a\nholistic manner, and to leverage them all to make high-quality,\nefficient training decisions. Importantly, although we focus\non model compression as the source of alternative models,\nthe problem we pose and the solution we envision extend to\narbitrary models.\nMore specifically, we consider a distributed learning sce-\nnario where nodes in different parts of the network (e.g.,\nfar edge, edge, and cloud) cooperatively train a DNN model\naided by a learning orchestrator. In this setting, we develop\nand evaluate a novel DNN training scheme called DepL (for\ndependable learning), allowing the learning orchestrator to\nmake joint decisions about: (a) the data sources (hence, the\ndatasets) to exploit for learning; (b) the models to train for the\nML task at hand, and when to switch from one to another; (c)\nthe node clusters to use at different stages of training. Unlike\nprevious work, DepL accounts for the mutual interaction\nbetween such decisions, and the effect that each of them has on\nthe expected learning quality as well as its distribution . Thanks\nto this ability, DepL can meet the challenges of present-\nand next-generation ML applications, including safety-critical\nones. Also, DepL makes near-optimal decisions, minimizing\nthe cost incurred by the learning process, while keeping a re-\nmarkably low (polynomial) computational complexity. Finally,\nDepL can be combined with emerging approaches based upon\nconformal prediction sets [5], [6], seeking to add reliability a\nposteriori , i.e., starting from an already-trained model. Indeed,\nmodels trained in a dependable manner through DepL also\nhave smaller conformal sets.\nThe main contributions of our work are thus as follows:\n• We propose a comprehensive system model, capturing all\nrelevant aspects of distributed ML training, including learningarXiv:2402.14346v1  [cs.LG]  22 Feb 20240 10 20 30 40 50 60\nepoch0.51.01.52.02.53.0lossF1=0.5\nF2=0.75expected\n1-99 pctFig. 2. Example of evolution of the test loss during training of the AlexNet\nCNN model pruned with 0.5 and 0.75 factor. Lines represent the expected\nloss, while shaded areas are delimited by the loss 1st and 99th percentiles.\nquality distribution;\n• In light of the problem complexity, we propose a solution\nstrategy called DepL, allowing to achieve the target learning\nquality distribution at the minimum cost;\n• We analyze DepL’s complexity and competitive ratio,\nproving the former to be polynomial and the latter to be both\nlow and constant;\n• We show that DepL closely matches the optimum and\nconsistently outperforms the state of the art.\nThe rest of the paper is organized as follows. Sec. II further\nmotivates our approach, considering a real-world example and\nshowing why solely focusing on the expected learning quality\nmay be suboptimal. Sec. III formally introduces our system\nmodel and problem formulation. The DepL solution is then\npresented in Sec. IV and analyzed in Sec. V. Finally, after\nevaluating DepL’s performance in Sec. VI and discussing\nrelated work in Sec. VII, we conclude the paper in Sec. VIII.\nII. T HEIMPORTANCE OF DEPENDABLE TRAINING\nWe now illustrate the reasons why it is important to take a\ndependable approach to ML model training. As an example,\nassume that the popular AlexNet CNN model [7] for image\nclassification has to be trained at the network edge, using\nthe CIFAR-10 dataset [8]. To make it more amenable to the\nlimited computational capability of edge servers, the model\ncan be compressed by pruning it with a given factor. Clearly,\nthe larger the pruning factor, the smaller and the less complex\nthe model becomes, thus leading not only to a less demanding\nmodel for inference but also to a lower cost during training.\nOn the other hand, a more compressed version of the model\nexhibits higher sensitivity to a drift in the distribution of\nthe input data and may provide less accurate output during\ninference due to the reduced number of model weights [9],\n[10]. An aspect that however has been widely overlooked so far\nis how the dependability of training, i.e., the confidence level\nin the accuracy achieved through training, may be severely\nimpacted by the version of the model and the dataset that\nare used. Importantly, these, on their turn, depend upon the\ncomputational capability of the nodes executing the training\nas well as the data sources used for the training.An example of experimental evidence of the above effect\nis given by the results shown in Fig. 2, obtained using two\ndifferent versions of the AlexNet model, one with pruning\nfactor F1=0.5and the other with pruning factor F2=0.75,\nand adopting the gradient descent optimizer with learning rate\nand momentum, respectively, equal to 10−3and0.9, and batch\nsize equal to 64. As a proxy metric for the achieved accuracy,\nthe figure presents the evolution of the test loss through the\ntraining epochs. Three important facts can be noted:\n1) The trajectories of the test loss evolution over the epochs\nare not deterministic but stochastic in nature, as high-\nlighted by the shaded areas;\n2) Their behavior visibly depends on the model that is used;\n3) Additionally, the average and the empirical distribution\nof the loss values may exhibit a different trend, since,\ne.g., as the average gets smaller, the variance may instead\nincrease (as in the case of F2, blue curve).\nThus, upon establishing learning strategies for a given ML\ntask, it is critical to account for all relevant aspects, including\nthe dependability of the trained models, besides the training\ncomputational cost or the robustness of the compressed model.\nTo properly tackle training dependability, in the following we\ndevelop a system model accounting for the stochastic behavior\nof the test loss evolution, and envision a solution strategy that\nselects the best learning strategy to achieve a target accuracy\nwith a target probability .\nIII. S YSTEM MODEL AND PROBLEM FORMULATION\nSystem model. We tackle a distributed learning scenario\nwith a set of network nodes, including cloud, edge, and far-\nedge nodes, that can contribute to a learning task, and a set of\ndata sources that can transfer data to them. All such nodes are\nassisted by a learning orchestrator located at an edge server,\nwhich, given an ML task at hand, selects (i) the data to be\nleveraged for the model training, (ii) the ML model to be\nused, as well as (iii) the set of nodes, hereinafter referred\nto as learning nodes, that should contribute to the training\nof such a model. To encompass a wide range of distributed\nlearning paradigms, including federated learning (FL) and\nsequential learning, we consider that the orchestrator identifies\nclusters of learning nodes, with each cluster performing the\nvanilla FL [11], and, possibly, contributing to the training\nprocess for a given number of epochs before handing the\nmodel over to the next cluster. Additionally, each cluster may\nuse a different variation of the ML model (i.e., a full or a\ncompressed version). The scenario thus includes the following\nmain elements:\n• Clusters of learning nodes, n∈N, that, for simplicity and\nwithout loss of generality, are assumed to include homoge-\nneous nodes with equal communication and computational\ncapability. We thus denote with r(n)the computational ca-\npability of the generic node in cluster nand with b(n)the\nbandwidth available on the link between the FL coordinator\nwithin cluster nand each learning node. Each unit of comput-\ning resource has a cost,κ(n), which depends on the equipment\navailable at the cluster nodes;• Datasets, d∈D, each generated by a data source, that can\nbe conveyed to node clusters. Let β(d, n)be the total network\ncost to transfer dataset dfrom the source to the nodes1of\ncluster nbefore the cluster starts training the model;\n• ML models suitable for the task at hand, in their full\nor compressed versions. They are represented as elements\nof the models set, m∈M. Each model is associated with a\nsizes(m)and a complexity factor. It follows that, at each\nepoch, the communication latency within cluster ndue to\nthe learning nodes sending their updated model mto their\nFL coordinator iss(m)\nb(n). Also, given the model complexity,\nletc(k, m, d )indicate the quantity of computational resources\nneeded by a learning node to train an epoch kof model m\nusing data from dataset d.\nGiven a model m∈M and being cluster n∈N the one cur-\nrently training mwith dataset d⊆D, we define as lrun(k, d, m )\nthe global change in the loss achieved by training model m\nover dfor the k-th epoch. Notice that training aims at\nminimizing the loss, hence, lrunwill be negative if the training\ndoes progress. Instead, switching from model m(k−1)to\nmodel m(k)changes the loss by lsw(k, d, m (k−1), m(k)),\nwithkbeing the current epoch and d⊆ D the dataset being\nused. Notice that lswis typically positive, as model switching\noften results – in the short term – in an increase of the\nloss. Importantly, all these values are input information to our\nproblem; as discussed in Sec. VI-A, estimating them is an\northogonal problem to our own.\nDecisions and problem formulation. The orchestrator has\nto make the following decisions:\n• Whether cluster n∈N contributes to the learn-\ning at epoch kor not, expressed through binary vari-\nabley(k, n)∈{0,1};\n• Whether cluster n∈N uses dataset d∈D for learning,\nexpressed through binary variable z(d, n)∈{0,1};\n• The amount of resources allocated for the model training\nby a node of a selected cluster, expressed through the real\nvariable x(k, n)≤y(k, n)r(n);\n• The total number Kof epochs to run;\n• For each epoch, the model version m(k)∈M to use.\nGiven the above decisions and neglecting the one-time data\ntransfer from sources to clusters, the duration of epoch k\nis given by the computation and communication time of the\nselected cluster, i.e.,\nT(k) =c(k, d, m (k))\nx(k, n)+s(m(k))\nb(n). (1)\nConcerning the test loss ℓ(k), it can be computed recursively\nby accounting for the two l-contributions, i.e.,\nℓ(k)=ℓ(k−1)+lrun(k, d, m (k))+lsw(k, d, m (k−1), m(k)).\nwhere ℓ(0)is the initial loss value. Note how the last term\nin the above equation reduces to zero if no model switching\noccurs from epoch k−1to epoch k.\n1Again, for simplicity and without loss of generality, we consider that,\ngiven d, each cluster node receives an equal portion of the dataset.The objective of the orchestrator is to minimize the total\ncost, subject to the fact that the learning quality target is\nachieved within the target time:\nmin\nx,y,z,K,mKP\nk=1P\nn∈N\u0012P\nd∈Dβ(d, n)z(d, n) +κ(n)x(k, n)\u0013\n(2)\ns.t. ℓω(K)≤ℓmax\nω (3)PK\nk=1T(k)≤Tmax(4)\nx(k, n)≤y(k, n)r(n)∀k, n (5)\nz(d, n)≤y(k, n)∀d, n, k. (6)\nIt is important to highlight that in (3) ω∈[0,1]indicates a spe-\ncific quantile of the loss, as opposed to a loss expected value.\nBy tweaking ω, we are therefore able to enforce different levels\nof risk, depending upon the scenario and application at hand.\nAlso, notice how (6) implies that inactive nodes neither receive\nnor use datasets.\nAs proven in Property 1 later, directly optimizing the\nproblem above is prohibitively complex, as the problem is NP-\nhard. Accordingly, to efficiently and effectively solve large-\nscale problem instances, we propose a heuristic solution named\nDependable Learning (DepL) , as described below.\nIV. T HEDEPL S OLUTION\nThe DepL solution effectively tackles two critical issues of\nthe system under study. First, the need for dependable training,\nseeking to keep a target quantile ωof the loss distribution\nbelow a threshold value ℓmax\nωthat is deemed to be acceptable\nfor the application at hand. Second, the complexity of the\ndecisions to make, which, along with the sheer amount of\nexisting options to explore, makes finding the optimal solution\nimpractical for realistic-sized problem instances. To address\nthe first issue, DepL makes three main decisions:\n1) selecting the datasets to use for learning;\n2) choosing the model to use at each epoch, switching\nmodels as needed;\n3) identifying the cluster of learning nodes to call for at\neach epoch, and the resources they should commit.\nThen, to tackle the second issue, DepL efficiently makes\norchestration decisions by decoupling them, while iterating\nas needed so as to account for their mutual influence.\nTo this end, it is critical to establish the order in which\ndecisions are made. DepL properly accounts for the decisions’\nmutual influence by making decisions that strongly influence\neach other sequentially. We observe that the mutual influence\nis strongest between (i) dataset and model selection, since\ndifferent models may call for different data, and (ii) model\nselection and resource allocation, as insufficient resources may\ncripple the model performance. Additionally, DepL seeks to\nmake first decisions with the deepest impact on the overall\ntraining performance. To do so, we remark that:\n• Dataset selection is related to the model selection and\naffects the resources needed at the nodes, as more data\ninvariably means more processing at the nodes;\n• Model selection is less critical, yet very relevant as it\nimpacts both the learning progress (hence, how many epochschoose datasets\nfrom choose models\nfrom choose clusters\nfrom , resources\ninfeasibleinfeasibleFig. 3. The main steps of the DepL solution strategy.\nwe need) and the required computational resources (hence,\nhow long each epoch will take);\n• The impact of the resources and nodes to use is limited\nto the epoch duration and cost.\nBased on the above, DepL takes the approach depicted in\nFig. 3, making dataset selection decisions in the outermost\nloop, then model selection, and finally resource allocation.\nImportantly, a further reason to make decisions in the order\nhighlighted in Fig. 3 has to do with how difficult it is to\nestimate their consequences. Indeed, choosing the data sources\nto use for learning requires sophisticate statistics [12] and\nfeedback loops [13]. The impact of model selection on the\noverall learning is somehow better understood, though with\nsignificant uncertainty [4], and finding the right resources\nto run a given model can be mapped to many well-studied\nproblems [14], [15]. By making harder-to-estimate decisions\nless often, we can reduce the impact of estimation errors on\nthe overall solution quality.\nFurthermore, in view of the above observations, we make\neach decision by considering the least restrictive option for\nthe subsequent ones. As an example, while choosing the\ndata, we assume that the most complex (i.e., full) version of\nthe model is chosen in the subsequent step, and then that\nthe cluster with the largest amount of available resources\nis used and they are fully allocated to the model training.\nIndeed, since adding more data is the most consequence-\nfraught decision to make, we want to avoid doing that, unless\nit is utterly necessary; similarly, a simpler model is used only\nif the available computational and networking resources are\ninsufficient to make a more complex model work.\nA. Dataset selection\nAdding new datasets to improve learning quality may have\nundesirable consequences: (i) adding a new dataset implies\nthat more training should be performed (hence, more resources\nare needed at the nodes using the dataset); (ii) more datasets\ntypically imply more network resources to consume for their\ncollection and, hence, a larger network. Accordingly, we begin\nwith no selected datasets and add them as needed. Importantly,\nat this stage, we take the learning time as proxy of the overall\ntraining cost, as we do not directly tackle resource allocation.\nLet us denote the set of datasets selected for learning with\nbD={d∈D:∃n∈N:z(d, n)=1}. Let also bK(bD)denote the\nnumber of epochs needed to reach convergence if the datasets\ninbDare used, and by bT(bD)the duration of each epoch. Notice\nthat considering at this stage a fixed resource allocation implies\nthat the epoch duration depends only on the selected datasets.\nWe remark that adding a new dataset to bDhas two con-\ntrasting effects on the training epochs. On the one hand, morelearning is performed at each epoch – hence, fewer epochs\nare needed to reach the target learning quality. On the other\nhand, more data is collected and processed at each epoch –\nhence, each epoch lasts longer. Our key observation is that\nthese two effects do not have the same evolution: (i) Adding\nmore data decreases bK(bD)) according to a logarithm law [16],\n[17]; (ii) Processing and time to transmit model updates, bT(bD)\ngrow instead linearly with the quantity of data. It follows that,\nintuitively, if adding a new dataset does not help (specifically,\nit does not make convergence faster), adding another one will\nalso not help. More formally, bK(bD)bT(bD)issubmodular .\nRecalling that here we take the learning time as proxy of the\ncost, and that the only decision we have to make is selecting\nthe datasets, we write the problem to solve as:\nmin\nbD⊆DbK(bD)bT(bD), (7)\nwhich is an unbounded submodular minimization prob-\nlem [18]. Such problems can be solved to optimality in\npolynomial time via several different algorithms. Examples\nrange from the traditional ellipsoid method [18] (based on the\nnotion of Lov ´asz extension [18], linking submodular combi-\nnatorial problems to convex continuous ones) to more modern\napproaches [19] that achieve strongly polynomial runtime, only\ndepending upon the problem size.\nIn summary, regardless of the concrete algorithm used to\nsolve (7), it is possible to choose the datasets in such a way\nthat the learning time (hence, cost) is minimized. This means\nthat DepL solves to optimality the first step of the solution\nstrategy, although not original problem in (2), which is NP-\nhard (as per Property 1).\nB. Model selection\nThe high-level goal is now to select a model in Mto use at\neach epoch. To cope with the complexity and the combinatorial\nnature of the decision to make, we leverage an expanded graph\napproach, predicated upon:\n1) Creating a graph whose vertices represent the possible\nstates of the system;\n2) Adding edges therein representing the possible decisions\nand their effects;\n3) Seeking the path (hence, a sequence of decisions) that\nconnects the current state with a feasible one and has the\nlowest weight (i.e., cost).\nAs mentioned above, at this stage the datasets to use are given,\nand we assume that the most capable node cluster, and all the\nresources therein, can be used.\nTo build the expanded graph, let ηbe an integer parameter\ndetermining the accuracy with which we discretize learning\ntime and test loss. Each vertex of the graph is then associated\nwith a 4-uple (m(k), k, T (k), ℓω(k)), where:\n•m(k)is a model in M;\n•kis an epoch in 1, . . . , K ;\n•T(k)=i\nηTmax, with i=1, . . . , η , is the elapsed learning\ntime in the state represented by the vertex;•ℓω(k)=j\nηℓ(0), with j=1, . . . , η andℓ(0)being the initial\nloss value, reflects, in a similar way, the ω-quantile of the\ncurrent loss.\nThe total number of vertices is then |M|Kη2. Such vertices\nrepresent a subset of all possible states of the system (e.g.,\nthere is no vertex with T(k) =0.5\nηTmax), while their indices\nkeep track of the corresponding elapsed training time. This\nallows the size of the graph to be manageable, at the expense\nof precision.\nConcerning edges, we add an edge from\nvertex (m(k), k, T (k), ℓω(k))to vertex (m(k+ 1) , k+\n1, T(k+ 1), ℓω(k+ 1)) if (i) it is possible to switch\nfrom m(k)tom(k+ 1), (ii) it takes at most T(k+ 1)−T(k),\nand (iii) the loss changes by at most ℓω(k+ 1)−ℓω(k). The\nweight of such edge represents the cost associated with the\nswitch. Edges are also created between vertices corresponding\nto the same model, i.e., indicating no model switch.\nFinally, we add a virtual vertex Ωand a zero-weight edge\nfrom any vertex representing a feasible solution (i.e., with\nan elapsed time lower than Tmaxand a loss quantile lower\nthan ℓmax\nω) toΩ. Since vertices connected to Ωcorrespond\nto feasible states, and edges in the graph are a conservative\nrepresentation of the outcome of model-switching decisions,\nany path connecting the current state to Ωcorresponds to a\nfeasible set of decisions. The lowest-weight among such paths\nrepresents then the lowest-cost, i.e., optimal , decisions we can\nmake, given the graph representation. Such decisions may not\nbe optimal solutions to the original problem, due to the fact\nthat not all states are represented by vertices in the graph.\nHowever, by increasing η, we can reduce this issue and get\narbitrarily close to the optimum, at the cost if increasing the\nsize of the graph and the solution complexity.\nThere is an additional problem to deal with, stemming from\nthe fact that expanded-graph approaches work with additive\nconstraints [20], [21], i.e., constraints where the effect of a\nsequence of decisions (in our case, model selection at different\nepochs) can be expressed as the sum of the effects of individual\ndecisions (in our case, the single lruncomponents). Such a\nproperty is required in order to ensure that the features of a\npath on the expanded graph (i.e., the associated learning time\nor loss quantile) can be expressed as the sum of features of\nindividual edges therein. The property holds for the elapsed\ntime, and it would be true for the expected loss, but not for\nthe loss quantile. Indeed, we know the effects of model selec-\ntion decisions as distributions, e.g., through their probability\ndensity function (pdf); however, the pdf of the sum of two\nrandom variables is not the sum of the individual pdfs, but\ntheir convolution.\nWe face this hurdle by adopting the following approach to\ndeal with the evolution of the value of loss-quantile, based on\nfunction transforms:\n1) We move to the transformed domain, replacing pdfs with\ntheir Laplace transforms;\n2) Convolutions between pdfs are thus replaced by products\nof transformed pdfs;3) We further compute the logarithm of the transformed\npdfs, so that products can be computed as sums.\nIt is important to highlight that the aforementioned approach\nworks in both cases (i) when pdfs of the test loss are known\nexactly (in which case the transforms and logarithms can be\ncomputed symbolically), and (ii) when they are not (in which\ncases all operations are performed numerically). In the latter\ncase, the fast Fourier transform (FFT) may be used in lieu of\nthe Laplace transform, owing to the vast availability of fast,\nhigh-quality implementations.\nFinally, if no model can be selected that meets the target\nlearning time and quality, DepL goes back to the previous\nstage: the current dataset choice is blacklisted and the datasets\nselection is repeated, still considering the most complex\nmodel. Optimized model selection will then be attempted again\nunder the new data selection.\nC. Node and resource allocation\nOnce the datasets and models are chosen, the only remain-\ning decisions to make concern: (i) the clusters to use for\nlearning, and (ii) which resources each cluster has to commit.\nImportantly, neither decision affects how many epochs are\nneeded for convergence, nor the distribution of the resulting\nloss value; on the other hand, clusters and resource selection\nhave a very strong influence on the learning time (i.e., the\nduration of each epoch) and the resulting cost.\nWe make the key observation that the resulting problem\n– including its decisions and their effects – can be mapped,\none-to-one, to the classic problem of VNF placement, also\nreferred to as service embedding. Specifically: (i) clusters\nare mapped to servers, which can run VNFs; (ii) models\nare mapped to the VNF themselves; (iii) model complexity\nis mapped to VNF requirements. As a consequence, we can\nuse any VNF placement strategy to solve this stage of our\nown problem. VNF placement is a very well studied problem,\nwith several efficient and effective schemes existing for it. In\nparticular, we take [22] as a reference, and obtain the same\nO(1 +ε)competitive ratio and O(1/ε)complexity.\nAgain, in the case where no cluster or resource allocation\ncan be found that can make the model be trained within the\ntarget time, DepL goes back to the previous decision stage,\ni.e., model switching, again blacklisting the model choices that\nhave proven insufficient and considering that any cluster and\nresource can be used. The overall process depicted in Fig. 3\nmay therefore be iterated till a feasible solution is found, or\nthe existing options have been exhausted.\nV. P ROBLEM AND SOLUTION ANALYSIS\nIn the following, we formally prove several important prop-\nerties about the problem we solve and our DepL solution. Our\nresults can be summarized as follows:\n•The problem we solve is NP-hard (Property 1), hence, a\nheuristic solution is required;\n•The DepL solution has low, namely, polynomial compu-\ntational complexity (Property 2);•The solutions yielded by DepL are provably very close\nto the optimum (Property 3).\nBeginning from the problem, we prove its NP-hardness via\na reduction from the knapsack problem.\nProperty 1. The problem of optimizing (2) subject to con-\nstraints (3)–(6) is NP-hard.\nProof: We prove NP-hardness by reduction from the\nknapsack problem, which is known [23] to be NP-hard. To\ndo so, we provide a polynomial-time procedure transforming\nany instance of the knapsack problem into an instance of\nour problem. Recall that the input to the knapsack problem\nconsists of a set I={i}of items, each with weight wiand\nvalue vi, along with a maximum weight wmax. We have to\ndecide whether to take each item iwith the goal of maximizing\nthe total value, subject to the constraint that the total weight\ndoes not exceed wmax. Given the above, we can create an\ninstance of our problem where: (1) items correspond to models\ninM; (2) the learning improvement yielded by each model\nis deterministic and equal to the value of the corresponding\nitem; (3) the requirements of each model are equal to the cost\nof the corresponding item; (4) there is only one dataset and\nonly one cluster, with sufficient capabilities to run any model;\n(5) all data transfer costs β(d, n)are set to zero. It can be\nseen by inspection that reduction is polynomial in complexity\n– indeed, constant, as it has no loops. As we have successfully\nreduced an instance of an NP-hard problem to an instance of\nour own, we can conclude that our problem is NP-hard.\nIt is also worth remarking that the instance of our problem\ngenerated by the reduction is a very simple one; this suggests\nthat our problem is significantly more complex than the knap-\nsack problem in practice. It also implies that general-purpose\nheuristic approaches for the knapsack problem (or any other\nknown problem) will work poorly with ours, rather, domain-\nand problem-specific strategies like DepL are warranted.\nConcerning DepL, we can prove that it has a remarkably\nlow, namely, quadratic worst-case computational complexity:\nProperty 2. The DepL procedure has worst-case polynomial\ncomplexity of O\u0010\n|D|2+|M|2K21\nη4+1\nε\u0011\n.\nProof: The total complexity is obtained by considering\nthe three steps depicted in Fig. 3 and described in Sec. IV.\nFor dataset selection (Sec. IV-A), we leverage the results\nin [19], solving unbounded submodular minimization prob-\nlems to the optimum with a computational complexity that is\nquadratic in the size of the set over which to optimize, i.e., D.\nFor model switching (Sec. IV-B), we have to compute the\nminimum-cost path over the expanded graph. The maximum\nnumber Vof vertices in the expanded graph is given by\nall combinations of model, epoch, time, and loss, hence,\nV≤ |M| KTmax\nηℓmax\nη. Recalling that minimum-cost path\nalgorithms, e.g., Dijkstra’s, have a complexity of O(V2), it\nfollows that the complexity of the model selection procedure\nisO\u0010\n|M|2K21\nη4\u0011\n. Finally, for cluster selection (Sec. IV-C),\nwe rely on [22], which has a complexity of O\u00001\nε\u0001\n, where εisa configurable optimality parameter. Combining all contribu-\ntions, we obtain O\u0010\n|D|2+|M|2K21\nη4+1\nε\u0011\n, which proves\nthe thesis.\nThe last aspect we look at is the effectiveness of the\ndecisions made by DepL. We can prove that the competitive\nratio (i.e., the ratio of DepL’s solution cost to the optimal one)\nis constant (i.e., it does not depend upon the problem size) and\nremarkably small.\nProperty 3. The DepL solution has a constant competitive\nratio, namely,1\nη(1 +ε).\nProof: Similarly to the proof of Property 2, we need to\nconsider the competitive ratio of each of the steps described in\nSec. IV, and then multiply them. Dataset selection (Sec. IV-A)\nis optimal, hence, the corresponding competitive ratio is 1. For\nmodel selection, instead, the only source of sub-optimality is\nthe parameter η; specifically, as per [21, Theorem 4.3], the\nresulting competitive ratio is1\nη. Finally, the approach in [22],\nwhich we consider as a reference for node selection, yields a\ncompetitive ratio of 1 +ε.\nA constant competitive ratio like the one proven in Prop-\nerty 3 implies that the quality of the solutions found by DepL\nis not influenced by the size of the problem instance being\nsolved; intuitively, DepL is robust to large problem sizes.\nVI. N UMERICAL RESULTS\nIn this section, we first (Sec. VI-A) demonstrate how\nthelrunandlswparameters can be estimated through real-world\nexperiments. Then (Sec. VI-B), we leverage the resulting data\nto compare the performance of DepL, the optimum, and the\nstate-of-the-art.\nA. Estimating the loss evolution\nAs observed in Sec. II, the evolution of the test loss over\nthe training epochs, k, cannot be predicted with certainty,\nrather, a probabilistic representation thereof should be used.\nPrevious work [24] has shown that the loss evolution can\nbe well approximated by an inverse-square-root law, i.e., it\nis proportional to1√\nk. We generalize such an expression as:\nℓrun(m, k) =Am√k+Bm·x, (8)\nwhere x∼ Xmis a sample drawn from distribution Xm, which\nis specific to the considered model and data. The parameters\nin (8) can be estimated as follows: (i) AmandBmare\nobtained through traditional parameter fitting techniques, e.g.,\nleast-squares, under the assumption that x=1; (ii) given Am\nandBm,Xmis obtained through distribution fitting.\nFig. 4(left) shows the empirical pdf of xfor an example test\nloss curve, along with the fitted pdf Xm, in the case where two\nclusters of nodes sequentially train the AlexNet CNN using the\nCIFAR-10 dataset. The two clusters are located, respectively,\nin the network edge and the far-edge, and perform FedAvg.\nThe order in which the clusters contribute to the training goes\nfrom the highest (edge) to the lowest (far-edge) computing\ncapability, as ultimately the model needs to be personalized0.9 1.0 1.1 1.2 1.3\nx051015202530pdfEmpirical\nFitted: Inv-Gamma\nMean: 1.0\nStd: 0.015\np-value: 0.823\n0 0.2 0.4 0.6 0.8 1\np-value0100200300400Number of occurencesUnsatisfying fit\nSatisfying fit\n−0.2 0.0 0.2 0.4 0.6 0.8\nlsw02468101214pdfEmpirical\nFitted: Student's t\nMean: 0.053\nStd: 0.033\np-value: 0.9372Fig. 4. Example distributions of Xas defined in (8) along with the Inverse-Gamma fit (left); distribution of the p-values highlighting fitting quality (center);\nexample distribution of lswas defined in (2) along with the Student’s t fit (right).\n6 8 10 12 14\nGuaranteed (99th pct.) loss0100200300400500600Incurred cost [units]\noptimal\nDepL\nbest-exp\n6 8 10 12 14\nGuaranteed (99th pct.) loss0.860.880.900.920.940.960.981.00Norm. (to the optimum) expected loss\nDepL\nbest-exp\n6 8 10 12 14\nGuaranteed (99th pct.) loss0100200300400500600Incurred cost [units]\noptimal\nbest-exp\nDepL, η=100\nDepL, η=20\nDepL, η=10\nFig. 5. AlexNet: performance of DepL and alternative benchmarks: cost (left); normalized expected loss (center); effect of η(right).\n6 8 10 12 14\nGuaranteed (99th pct.) loss050100150200250300350Incurred cost [units]\noptimal\nDepL\nbest-exp\n6 8 10 12 14\nGuaranteed (99th pct.) loss0.750.800.850.900.951.00Norm. (to the optimum) expected loss\nDepL\nbest-exp\n6 8 10 12 14\nGuaranteed (99th pct.) loss050100150200250300350Incurred cost [units]\noptimal\nbest-exp\nDepL, η=100\nDepL, η=20\nDepL, η=10\nFig. 6. MobileNet: performance of DepL and alternative benchmarks. Cost (left); normalized expected loss (center); effect of η(right).\nwith the data owned by IoT or user devices (which have\nindeed lower compute capabilities). At each training stage,\nthe model is further pruned, so as to make it amenable\nfor training at the lower-end nodes. Let K1(K2) be the\nnumber of epochs performed at the first (second) stage, using\na model pruned with factor F1(F2); for Fig. 4(left) we set\nK1=17,K2=60,F1=0.5, and F2=0.75. The empirical pdf,\nand, more specifically, the standard deviation, highlight that,\ngiven a combination of learning parameters, the prediction\nof the next loss value exhibits significant uncertainty, thus\nfurther confirming the need for a probabilistic representation\nof the loss trajectory. Importantly, one can notice that the\nInverse-Gamma distribution provides an excellent goodness\nof fit. The latter is measured by computing the p-value\nusing the Kolmogorov-Smirnov (KS) test, with p-values higher\nthan 0.05corresponding to a good fit2. The histogram in\n2Ap-value higher than 0.05indicates that our null hypothesis, i.e., data are\ndistributed according to an Inverse-Gamma distribution, cannot be rejected.Fig. 4(center) shows the p-values for the distributions obtained\nusing all possible combinations of the number of epochs\nK1∈[1,20]andK2∈{1,5,15,25,40,60}, and of the pruning\nfactors F1, F2∈{0.25,0.36,0.5,0.75}: in almost all cases they\nare larger than the 0.05threshold, again indicating the good-\nness of fit of the Inverse-Gamma distribution on the empirical\ndistributions.\nModel switching, e.g., compression, has a different effect:\nnamely, a one-time increase in the loss (hence, a reduction in\naccuracy). We modeled this effect by lsw, i.e., the difference\nbetween the test loss values after and before the model\nswitching. Clearly, each combination of learning parameters\nyields a different distribution. Fig. 4(right) shows an example\noflswdistribution, representing the empirical and the fitted\npdfs for F1=0.36,F2=0.5. The empirical pdf, in Fig. 4(right),\nshows that the support of lswdistribution is composed of\npositive values, indicating an increase of the test loss after\nmodel switching. Again, such an increase cannot be predicted6 8 10 12 14\nGuaranteed (99th pct.) loss051015202530354045Epochs performedfast-unsteady\nintermediate\nslow-steady\n6 8 10 12 14\nGuaranteed (99th pct.) loss051015202530354045Epochs performedfast-unsteady\nintermediate\nslow-steady\n6 8 10 12 14\nGuaranteed (99th pct.) loss051015202530354045Epochs performedfast-unsteady\nintermediate\nslow-steadyFig. 7. Epochs spent with each model by the optimal solution (left), DepL (center), and best-exp (right).\nwith certainty; hence, we need a probabilistic representation\noflsw. In this case, the KS test reveals that satisfying p-values\ncan be obtained using the Student’s t-distribution.\nBelow, we use the above experimental estimation of the loss\nevolution to assess and benchmark the performance of DepL.\nB. Performance evaluation\nSince, as discussed in Sec. IV-A, the dataset selection\nis provably optimal, our performance evaluation focuses on\nmodel and node selection, i.e., steps 2 and 3 in Fig. 3. To this\nend, we benchmark our DepL solution described in Sec. IV-B\nagainst two alternatives:\n•Optimal decisions, obtained by brute force (labelled as\noptimal in plots);\n•A state-of-the-art solution, aiming to optimize the ex-\npected loss, inspired to [4] (labelled as best-exp in plots).\nThe best-exp scheme makes decisions of the same nature as\nour DepL; however, it accounts for the expected loss instead\nof any quantile thereof. Thus, comparing against best-exp is\na valuable way to gauge the impact of accounting for the\ndistribution of the loss besides its expected value. Finally,\nunless otherwise specified, for DepL we set η=200 .\nFor all strategies, we consider as items of M:\n•either three versions of the AlexNet DNN pruned, respec-\ntively, with factors F1=0.5,F2=0.75, and F3=0.9,\n•or three versions of the MobileNet network, pruned\naccording to the same factors.\nTab. I summarizes the features of the DNNs we use. Specifi-\ncally, as the name assigned to the models suggests, the “fast-\nunsteady” model provides a fast learning on average, but it\nis liable to occasionally yield poor test loss performance;\n“slow-steady”, on the other hand, provides slower but more\nreliable convergence, while “intermediate” yields balanced\nperformance.\nWe consider a network infrastructure including two types\nof server (hence, clusters nodes in N): A-class nodes, with\ncapabilities set to r(n)=100 and processing cost κ(n)=1, and\nB-class nodes, with r(n)=50 and processing cost κ(n)=1.2.\nFor simplicity, we assume that clusters contain one node\neach, and set β(n, d) = 0 for all nodes and sources. For all\nstrategies, node assignment and resource allocation decisions\nare made following the methodology in [22].TABLE I\nALEXNET AND MOBILE NET VERSIONS USED FOR PERFORMANCE\nEVALUATION :EXPECTED AND VARIANCE OF LOSS IMPROVEMENT PER\nEPOCH ,AND RESOURCE DEMAND\nModel Exp.\nloss\nim-\nprov.Var.\nof\nloss\nimp.Norm.\nneeded\nre-\nsourcesExp.\nloss\nim-\nprovVar.\nof\nloss\nimp.Norm.\nneeded\nre-\nsources\nAlexNet MobileNet\nFast-unsteady 1.02 0.031 2.6 1.02 0.029 1.8\nIntermediate 1.00 0.010 1.4 0.95 0.012 1.0\nSlow-steady 0.98 0.007 1 0.92 0.006 0.6\nThe first aspect in which we are interested is the relative\nperformance of the strategies we compare, i.e., how good\nthey are at minimizing the objective (2). The results for the\nAlexNet model are summarized in Fig. 5(left), showing the\ncost incurred for different target values of the loss 99th per-\ncentile. As one might expect, tighter loss requirements require\nmore training, hence, a higher cost. Most interestingly, DepL\noutperforms best-exp and virtually matches the optimum. The\ndifference is especially large when the requirements are less\nstrict, hence, there is more margin for learning strategies that\noptimize cost over sheer performance.\nFig. 5(center), depicting the loss achieved by DepL and\nbest-exp, normalized to the optimum, sheds additional light\non this effect. All strategies meet the learning quantile con-\nstraint (3) with an equality sign, i.e., they provide exactly the\nrequired value of the loss 99th percentile. On the other hand,\nFig. 5(center) shows that both best-exp and, to a much smaller\nextent, DepL yield a lower (i.e., better) expected loss than\nthe optimum. Clearly, this is a waste of resources: since our\nconstraint is on the 99th percentile of the loss, improving its\nexpected value brings no benefit.\nIn Fig. 5(right), we assess the effect of decreasing the\nparameter η, which we use when building the expanded graph\nin Sec. IV-B, on the performance of DepL. We observe how\nthere is indeed a significant margin to pursue different trade-\noffs between DepL’s computational efficiency and the quality\nof the decisions it yields, as indicated by the analysis in Sec. V.\nThe same behavior can be observed from Fig. 6, presenting\nthe behavior when MobileNet versions are used. We can\nobserve that, compared to Fig. 5, costs tend to be lower, a\ntestament to the better efficiency of MobileNet compared to\nAlexNet. Furthermore, costs decrease more steeply as the loss6 8 10 12 14\nGuaranteed (99th pct.) loss0100200300400500600Incurred cost [units]fast-unsteady\nintermediate\nslow-steady\n6 8 10 12 14\nGuaranteed (99th pct.) loss0100200300400500600Incurred cost [units]fast-unsteady\nintermediate\nslow-steady\n6 8 10 12 14\nGuaranteed (99th pct.) loss0100200300400500600Incurred cost [units]fast-unsteady\nintermediate\nslow-steadyFig. 8. AlexNet: cost incurred with each model by the optimal solution (left), DepL (center), and best-exp (right).\n0 10 20 30 40 50\nEpoch246810121416Lossoptimal\nDepL\nbest-expexpected\n1-99 pct.\n0 5 10 15 20 25 30\nEpoch789101112131415Lossoptimal\nDepL\nbest-expexpected\n1-99 pct.\n1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00\nEpoch12.2512.5012.7513.0013.2513.5013.7514.0014.25Lossoptimal\nDepL\nbest-expexpected\n1-99 pct.\nFig. 9. AlexNet: evolution of the expected and guaranteed loss when the guaranteed loss is 5(left), 9(center), and 13(right).\ntarget increases, showing that MobileNet struggles more for\nmore complex learning tasks. Besides these differences, it is\nimportant to remark how – exactly like in Fig. 5 – DepL\nsignificantly outperforms best-exp, and closely matches the\noptimum.\nMoving back to AlexNet, in Fig. 7 we take a closer look at\nthe decisions made by each strategy, specifically, at the number\nof epochs for which each of the models in Tab. I is used for\ntraining. DepL (center plot) makes very similar choices to the\noptimum (left plot), consistently with Fig. 5; in both cases,\nall models are used as required by the loss quantile target.\nInterestingly, when the target is loose, both strategies can\nonly use the “intermediate” model, and avoid expensive model\nswitches. On the contrary, as the target gets tighter, switches\nbecome necessary and all models are used. The right plot,\ndepicting the decisions made by best-exp, shows a completely\ndifferent picture. As best-exp optimizes the expected loss, it\nutilizes the “fast-unsteady” model as much as possible, which\nresults in a higher number of epochs to run, hence, the higher\ncosts presented in Fig. 5(left).\nFig. 8 shows the total cost of each strategy, broken down\naccording to the model they use. As in Fig. 5(left), best-exp\nrequires the most computational resources, hence, incurs the\nhighest cost; also, in accordance with Tab. I, much of such cost\nis due to the “fast-unsteady” model. This confirms how solely\nfocusing on the expected loss means performing more epochs\n(Fig. 7) and spending more in each of them (Fig. 8), while\nobtaining a needlessly low expected loss (Fig. 5(center)).\nFinally, Fig. 9 shows the evolution of the loss across\ndifferent epochs, for different strategies and values of the\ntarget learning quality; peaks correspond to model switches.\nIn all plots, lines represent the expected loss, while the shadedarea is delimited by the loss 1st and 99th percentiles. We\ncan immediately observe that the lines corresponding to DepL\n(and the optimum) are shorter, i.e., fewer epochs are necessary\nto reach the target learning quality. Further, it is evident\n(especially in the middle plot) how best-exp is associated with\nwider shaded areas, i.e., a larger distance between the loss\n1st and 99th percentiles. This is consistent with Fig. 7 and\nFig. 8: from those figures, we can observe that best-exp tends\nto use more the “fast-unsteady” model, which is associated\nwith a larger variance (as per Tab. I). Furthermore, Fig. 9\nconfirms how DepL virtually always matches the optimum,\nmaking model switching decisions that account for both the\nexpected value and the distribution of the loss.\nVII. R ELATED WORK\nDepL is related to three main research areas: resource-\naware distributed DNN training, DNN model compression,\nand robust DNN training. In the former case, the goal is to\nlocate and exploit the best resources for a given training task.\nThe main decision is often choosing the nodes to involve in\ntraining, based on their speed [1], [2], the quantity [1], [25]\nand quality [3] of their data, the speed and reliability of their\nnetwork [2], [26], as well as trust [27]. In all cases, the model\nto train is assumed to be given and fixed. Recent works [28]\ntarget more complex scenarios, where layers of the DNN can\nbe duplicated at different learning nodes if needed. The studies\nin [4], [29] also seek to adapt the learning task to the available\nresources, choosing from different existing models.\nModel compression is a set of techniques to transfer the\nknowledge from a model to a different (usually simpler)\none. Prominent among such techniques is model pruning,\nwhich consists in removing some weights from a DNN model,assuming that very few of the parameters of a model actually\nimpact its performance. Choosing the right parameters to\nprune is a challenging problem itself; many solutions adopt\nan iterative approach [30], by observing the evolution of\nthe parameter values. Other compression techniques include\nknowledge distillation [31], where a “student” model is trained\nto mimic the decisions of a “teacher” model. Later studies\nhave tackled how distillation can be combined with generative\nmodels [32] and domain adaptation [31].\nRobustness in distributed DNN training is an emerging\ntopic. Most works set in a FL scenario and focus on node\nselection. Such a robustness objective is balanced against fair-\nness consideration in [33], aiming at distributing the training\nload as evenly as possible. Robustness considerations are also\ncombined with communication efficiency in [34] and data\nquality. Importantly, all the aforementioned works deal with\nnode selection; the reliability of the DNN model and the\ntraining process are never accounted for.\nAn alternative approach to adding reliability to DNN results\nis the so-called conformal prediction [5], [6]. Conformal\napproaches create a set of possible results of a given DNN\n(e.g., predicted class) such that the correct result (e.g., the\ntrue class) lies in the set with a target probability; intuitively,\nbetter models will have smaller conformal sets. Conformal\nprediction techniques are applied after training, hence, they\ncan be seamlessly combined with DepL, and indeed benefit\nfrom the high-quality training it yields.\nVIII. C ONCLUSIONS\nWe presented, analyzed and evaluated DepL, a strategy for\nthe dependable training of distributed ML models. Unlike pre-\nvious work, DepL can guarantee that the target learning quality\n(e.g., accuracy) is reached with a target probability . DepL\nachieves this goal by making joint, high-quality decisions\nabout the data, nodes, and models to use, including the time\nat which model switching (through, e.g., compression) shall\nbe performed. We have formally proved that DepL achieves\nnear-optimal decisions with a constant competitive ratio and\nlow complexity. We have further shown that DepL closely\nmatches the optimum and significantly outperforms the state-\nof-the-art.\nACKNOWLEDGMENT\nThis work was supported by the SNS-JU-2022 project\nADROIT6G under the European Union’s Horizon Europe\nresearch and innovation programme under Grand Agreement\nNo. 101095363.\nREFERENCES\n[1] A. M. Abdelmoniem, A. N. Sahu, M. Canini, and S. A.\nFahmy, “Resource-Efficient Federated Learning,” arXiv preprint\narXiv:2111.01108 , 2021.\n[2] S. Wang, T. Tuor, T. Salonidis, K. K. Leung, C. Makaya, T. He, and\nK. Chan, “Adaptive federated learning in resource constrained edge\ncomputing systems,” IEEE JSAC , 2019.\n[3] H. Wu and P. Wang, “Fast-convergent federated learning with adaptive\nweighting,” IEEE TCCN , 2021.[4] F. Malandrino, G. Di Giacomo, A. Karamzade, M. Levorato, and\nC. 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We consider the damped nonlinear Klein-Gordon equation wit h a\ndelta potential\n∂2\ntu−∂2\nxu+2α∂tu+u−γδ0u−|u|p−1u= 0,(t,x)∈R×R,\nwherep >2,α >0, γ <2, andδ0=δ0(x) denotes the Dirac delta with the\nmass at the origin. When γ= 0, Cˆ ote, Martel and Yuan [7] proved that any\nglobal solution either converges to 0 or to the sum of K≥1 decoupled solitary\nwaves which have alternative signs. In this paper, we first pr ove that any\nglobal solution either converges to 0 or to the sum of K≥1 decoupled solitary\nwaves. Next we construct a single solitary wave solution tha t moves away from\nthe origin when γ <0 and construct an even 2-solitary wave solution when\nγ≤ −2. Last we give single solitary wave solutions and even 2-sol itary wave\nsolutions an upper bound for the distance between the origin and the solitary\nwave.\nContents\n1. Introduction 2\n1.1. Setting of the problem 2\n1.2. Main results 3\n1.3. Previous results 5\n2. Local well-posedness and uniform bound 6\n2.1. Linear evolution in the energy space 6\n2.2. Proof of Local well-posedness of the Cauchy problem 8\n2.3. Uniform bound on global solutions 9\n3. Soliton resolution 11\n4. Global dynamics below the ground state energy 15\n4.1. Minimizing problems and variational structure 15\n4.2. Classification of global dynamics 17\n4.3. Proof of Theorem 1.2 18\n5. Dynamics close to solitary waves 20\n5.1. Basic properties of the ground state 20\n5.2. Modulation of the center 21\n5.3. Estimates of the center 23\n5.4. Energy estimates 26\n5.5. Estimate of the instability term 30\n5.6. Proof of Theorem 1.3 32\nDate: February 27, 2024.\nKey words and phrases. Nonlinear Klein-Gordon equation, Solitons, Soliton resol ution, Dirac\ndelta potential.\n12 K. ISHIZUKA\nAcknowledgement 33\nReferences 33\n1.Introduction\n1.1.Setting of the problem. We considerthe followingdamped nonlinearKlein-\nGordon equation with a delta potential\n/braceleftBigg\n∂2\ntu−∂2\nxu+2α∂tu+u−γδ0u−f(u) = 0,(t,x)∈R×R,\nu(0,x) =u0(x), x ∈R,(DNKG)\nwheref(u) =|u|p−1u,p >2,α >0, γ <2, andδ0=δ0(x) denotes the Dirac delta\nwith the mass at the origin. We define the operator Aas\nA=/parenleftbigg0 1\n∂2\nx−1−2α/parenrightbigg\n,\nwith domain\nD={(u,v)∈H2,∗×H1:u′(0+)−u′(0−) =−γu(0)} ⊂ H,(1.1)\nwhere\nH2,∗=H1(R)∩H2(R\\{0}),\nH=H1(R)×L2(R).\nAgenerates a strongly continuous semigroup on H, which allows us to establish the\nlocal well-posedness of (DNKG). We will show it in Section 2. Moreover defining\nthe energy of a solution /vector u= (u,∂tu) by\nEγ(/vector u) =1\n2/parenleftbig\n/bardblu/bardbl2\nH1+/bardbl∂tu/bardbl2\nL2−γ|u(0)|2/parenrightbig\n−1\np+1/bardblu/bardblp+1\nLp+1,\nit holds\nEγ(/vector u(t2))−Eγ(/vector u(t1)) =−2α/integraldisplayt2\nt1/bardbl∂tu(t)/bardbl2\nL2dt. (1.2)\nFurthermore the equation (DNKG) enjoys several invariances:\nu(t,x)/mapsto→ −u(t,x),\nu(t,x)/mapsto→u(t,−x).(1.3)\nWhenγ= 0, it is well-known that up to sign and translation, the only stationar y\nsolution of (DNKG) is ( Q,0), where Qis\nQ(x) =/parenleftBigg\np+1\n2cosh2/parenleftbigp−1\n2x/parenrightbig/parenrightBigg1\np−1\n,\nwhich solves the equation\nQ′′−Q+Qp= 0 on R\n(see [32]). From the expression of Q, we obtain that as |x| → ∞,\n/vextendsingle/vextendsingle/vextendsingleQ(x)−cQe−|x|/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−2|x|,/vextendsingle/vextendsingle/vextendsingle/vextendsingleQ′(x)+x\n|x|cQe−|x|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−2|x|, (1.4)3\nwherecQ= (2p+2)1\np−1. When γ= 0, it is proved in [7] that any global solution\neither converges to 0 or to the sum of K≥1 decoupled solitary waves which have\nalternative signs, namely for any global solution /vector u∈C([0,∞),H) of (DNKG) there\nexistK≥0,σ=±1 and functions zk: [0,∞)→Rfor allk= 1,···,Ksuch that\nlim\nt→∞/parenleftBigg\n/bardblu(t)−σK/summationdisplay\nk=1(−1)kQ(·−zk(t))/bardblH1+/bardbl∂tu(t)/bardblL2/parenrightBigg\n= 0,\nlim\nt→∞(zk+1(t)−zk(t)) =∞for k= 1,2,···,K−1.(1.5)\nRemark 1.1. IfK= 0 then the terms/summationtextK\nk=1are 0.\nOn the other hand, when γ/\\e}atio\\slash= 0, there may exist a solution of (DNKG) whose\nasymptotic behavior is different from the γ= 0 case thanks to the potential-soliton\nrepulsion(or attraction). In fact, when γsatisfies|γ|<2 andγ/\\e}atio\\slash= 0, the positive\nstationary solution of (DNKG) is ( Qγ,0), where Qγis\nQγ(x) =/parenleftBigg\np+1\n2cosh2/parenleftbigp−1\n2|x|+tanh−1(γ\n2)/parenrightbig/parenrightBigg1\np−1\n,\nwhich is the solution of\nQ′′\nγ−Qγ+γδ0Qγ+Qp\nγ= 0.\nIts asymptotic behavior is different from (1.5).\n1.2.Main results. We are interested in the long-time asymptotic behavior of\nglobal solutions of (DNKG). When γ/\\e}atio\\slash= 0, it is difficult to describe precisely the\nasymptotic behavior as t→ ∞of all global solutions of (DNKG) as [7] because\nthe potential-soliton repulsion (or attraction) is delicate. However by using the\nconcentration-compactness arguments we obtain the soliton res olution as the fol-\nlowing:\nTheorem 1.1. For any global solution /vector uof(DNKG) , there exist K≥0, signs\nσ= 0,±1,σk=±1for anyk∈ {1,2,···,K}, and functions zk: [0,∞)→Rfor\nanyk∈ {1,2,···,K}such that\nlim\nt→∞/parenleftBigg\n/bardblu(t)−σQγ−K/summationdisplay\nk=1σkQ(·−zk(t))/bardblH1+/bardbl∂tu(t)/bardblL2/parenrightBigg\n= 0,\nlim\nt→∞(zk+1(t)−zk(t)) =∞fork= 1,···,K−1,\nlim\nt→∞|zk(t)|=∞fork= 1,···,K.(1.6)\nRemark 1.2. Whenσ= 0 and K= 0,/vector uconverges exponentially to 0 in Has\nt→ ∞. We prove this in Section 4.2.\nRemark 1.3. Ifγ≤ −2, we may assume σ= 0 because Qγdoes not exist.\nTheorem 1.1 implies that (1.6) exhausts all solutions in the energy spa ce. On\nthe other hand, it is difficult to determine signs ( σk)1≤k≤Kof (1.6). For example,\nwe do not know whether there exists a global solution /vector uof (DNKG) satisfying (1.6)\nwithK= 3 and ( σ,σ1,σ2,σ3) = (0,1,1,−1). This is due to the potential-soliton\nrepulsion (attraction). When γ= 0, we only consider the interaction between\nsolitary waves and (1.5) exhausts all solutions. On the other hand, whenγ/\\e}atio\\slash= 0 we4 K. ISHIZUKA\nneed to reveal whether the interaction between solitary waves is s tronger than the\npotential-soliton repulsion (attraction). However, the potential- soliton repulsion\n(attraction) is delicate and it prevents from determining signs of so litary waves.\nRemark 1.4. By [7] and (1.1), for any K∈Nandσ=±1 there exist functions\nzk: [0,∞)→Rfork= 1,2,···,Kand an odd global solution /vector u∈C([0,∞),H) of\n(DNKG) such that\nlim\nt→∞/parenleftBigg\n/bardblu(t)−σK/summationdisplay\nk=1{Q(·−zk(t))−Q(·+zk(t))}/bardblH1+/bardbl∂tu(t)/bardblL2/parenrightBigg\n= 0,\nlim\nt→∞zk(t) =∞for k= 1,2,···,K,\nlim\nt→∞(zk+1(t)−zk(t)) =∞for k= 1,2,···,K−1.\nAs above, it is difficult to determine signs σand (σk)1≤k≤Kof (1.6). On the\nother hand, when (1.6) with σ= 0,K= 1 and γ <0 or (1.6) with σ= 0,K= 2\nandγ≤ −2, thanks to the structure of the ground state energy we can co nstruct a\nglobal solution whose asymptotic behavior is different from (1.5).\nTheorem 1.2. (1)Letγ <0. Forσ=±1, there exists a global solution\n/vector u∈C([0,∞);H)of(DNKG) such that\nlim\nt→∞(/bardblu(t)−σQ(·−z(t))/bardblH1+/bardbl∂tu(t)/bardblL2) = 0,\nlim\nt→∞|z(t)|=∞,(1.7)\nfor some z: [0,∞)→R.\n(2)Letγ≤ −2. Forσ=±1, there exists an even global solution /vector u∈\nC([0,∞);H)of(DNKG) such that\nlim\nt→∞(/bardblu(t)−σ(Q(·−z(t))+Q(·+z(t)))/bardblH1+/bardbl∂tu(t)/bardblL2) = 0,\nlim\nt→∞z(t) =∞,(1.8)\nfor some z: [0,∞)→R.\nRemark 1.5. Whenγ= 0, a solution /vector uof (DNKG) satisfying (1.5) with K= 1\nconverges to the stationary solution ±Q(·−y) for some y∈Rast→ ∞. However,\nwhenγ <0, a solution /vector uof (DNKG) satisfying (1.7) behaves asymptotically as\nt→ ∞so that±Qmoves far away from the origin.\nTo establish our next result, we define a solution as (1.7) or (1.8).\nDefinition 1.1. (1) A solution /vector uof (DNKG) is called a single solitary wave\nif there exist a sign σ=±1 and a function z: [0,∞)→Rsuch that (1.7)\nis satisfied.\n(2) A solution /vector uof (DNKG) is called an even 2-solitary wave if/vector uis even and\nthere exist a sign σ=±1 and a function z: [0,∞)→Rsuch that (1.8) is\nsatisfied.\nTheasymptoticbehaviorofasinglesolitarywaveappearsas Qmovingawayfrom\nthe origin. Furthermore, an even 2-solitary wave asymptotes to t he superposition\nof two solitary waves of the same sign, which implies the potential-solit on repulsion\nis stronger than the soliton-soliton attraction. Our main result is th at if a solution\n/vector uof (DNKG) is a single solitary wave or an even 2-solitary wave, the dist ance\nbetween the center of the solitary wave and the origin has an upper bound.5\nTheorem 1.3. (1)Letγ <0. If a global solution /vector uis a single solitary wave,\nthen we have\n|z(t)|−1\n2logt/lessorsimilar1.\n(2)Letγ≤ −2. If a global solution /vector uis an even 2-solitary wave, then we have\nz(t)−1\n2logt/lessorsimilar1.\nGathering Theorem 1.2 and Theorem 1.3, if γ <0 there exists a solution of\n(DNKG)satisfying (1.7)bythepotential-solitonrepulsion, butthe p otential-soliton\nrepulsion is not strong so that the distance between origin and the c enter of the\nsolitary wave is roughly less than1\n2logt. Furthermore, if γ≤ −2, there exists a\nsoliton of (DNKG) satisfying (1.8) by the potential-soliton repulsion, which means\nthat the potential-soliton repulsion is stronger than the soliton-so liton attraction.\n1.3.Previous results. There have been intensive study on the global behavior of\ngeneral (large) solutions for the nonlinear dispersive equations, w here the guiding\nprinciple is soliton resolution conjecture , which claims that generic global solutions\nare asymptotic to the superposition of solitary waves for large time . In particular,\nthe conjecture has been studied for the energy-critical wave eq uation. We refer to\n[3, 11, 13, 14, 28], where the conjecture has been proved for all r adial solutions in\nspace dimension N= 3,5,7,···orN= 4,6. Similar results have been obtained in\nthe case with a potential [30, 35], with a damping term [20], as well as fo r the wave\nmaps [12, 26], all under some rotational symmetry.\nOn the other hand, the conjecture is widely open for the nonlinear K lein-Gordon\nequation, namely the α= 0 and γ= 0 case. Nakanishi and Schlag [39] have proved\nthe conjecture as long as the energy of the solution is not much larg er than the\nground state energy in space dimension N= 3.\nHowever, for the damped nonlinear Klein-Gordon equation, namely t heγ= 0\ncase, the energy decay (1.2) makes the analysis simpler than the un damped case.\nHistoricallyKeller[31]constructedstableandunstablemanifoldsinan eighborhood\nof any stationary solution of (DNKG), and Feireisl [15] proved the conjecture along\na time sequence. Burq, Raugel and Schlag [2] proved the conjectu re in the full\nlimitt→ ∞for all radial solutions. More recently, Cˆ ote, Martel and Yuan [7 ]\nproved the conjecture without size restriction and radial assump tion. To the best\nof our knowledge, this is the first complete proofof the soliton reso lution conjecture\nfor a non-integrable equation without any size or symmetry restric tion and with\nmoving solitons. Other results for the damped nonlinear Klein-Gordo n equation\nare in [8, 10, 23]. Their arguments are useful for (DNKG).\nIn addition, there are many results about multi-solitary waves for t he nonlinear\ndispersive equations. For the nonlinear Klein-Gordon equation, Cˆ o te and Mun˜ oz\n[6] have constructed multi-solitary waves. Note that multi-solitary waves move at\na constant speed. More recently, Aryan [1] has constructed 2-s olitary waves whose\nspeed converges to 0 as t→ ∞. Strictly in [1] the distance between each solitary\nwave is 2(1 + O(1))logtast→ ∞. As [1], there are several results about multi-\nsolitary waves with logarithmic distance; see e.g.[18, 36, 37, 38]. We note that for\nthe damped nonlinear Klein-Gordon equation, multi-solitary waves of (DNKG) are\nimpossible to move at a constant speed by the damping α. See also [24, 25, 26, 27,\n28, 29] for works about two-bubble solutions.6 K. ISHIZUKA\nThere are few results about (DNKG) with α= 0 because of the singularity of δ0.\nCsobo, Genoud, Ohta and Royer [9] have proved the stability of st anding waves for\n(DNKG). On the other hand, there are many results for the nonline ar Schr¨ odinger\nequation with a delta potential; see e.g.[16, 17, 18, 19, 21]. Notably, we have\nseveral technical tools in [16]. We also see the potential-soliton rep ulsion in [18].\n2.Local well-posedness and uniform bound\nInthissectionwediscussthelocalwell-posednessoftheCauchypr oblem(DNKG).\nFirst we show that the operator Agenerates a C0-semigroup on H. Then (DNKG)\ncan be rewritten as\n\nd\ndtU(t)−AU(t) =F(U(t)),\nU(0) =U0,(2.1)\nwhereU= (u,v)∈ Hand\nF(U) =/parenleftbigg\n0\nf(u)/parenrightbigg\n.\nThus we only need to show the local well-posedness of (2.1), and this can be proven\nby the standard arguments (see [4]).\n2.1.Linear evolution in the energy space. In this subsection we show that\nthe operator Agenerates a strongly continuous group on H. SinceDdefined by\n(1.1) is dense in H, we can consider AonH(See [9]). Next, for µ≥0 we introduce\nonHthe quadratic form defined by\n/bardbl(u,v)/bardblH,µ,γ=/bardblu′/bardbl2\nL2+µ2/bardblu/bardbl2\nL2−γ|u(0)|2+/bardblv/bardbl2\nL2.\nWe denote by /a\\}bracketle{t·,·/a\\}bracketri}htH,µ,γthe correspondingbilinear form. Then by direct calculation\nforU= (u,v)∈ D,\n/a\\}bracketle{tAU,U/a\\}bracketri}htH,1,γ=−2α/bardblv/bardbl2\nL2≤0.\nIn addition the following lemma holds.\nLemma 2.1. Forγ <2there exists Cγ>1such that for all u∈H1, we have\nC−1\nγ/bardblu/bardbl2\nH1≤ /bardblu/bardbl2\nH1−γ|u(0)|2≤Cγ/bardblu/bardbl2\nH1. (2.2)\nProof.See [9, Lemma 2.3]. /square\nProposition 2.2. The operator Agenerates a C0-semigroup on H.\nProof.Let 0< µ <1 be close to 1. For U= (u,v)∈ D, we have\n/a\\}bracketle{tAU,U/a\\}bracketri}htH,µ,γ= (µ2−1)/a\\}bracketle{tu,v/a\\}bracketri}htL2−2α/bardblv/bardbl2\nL2,\nand\n|/a\\}bracketle{tAU,U/a\\}bracketri}htH,µ,γ|/lessorsimilar/bardblu/bardbl2\nL2+/bardblv/bardbl2\nL2.\nMoreover by Lemma 2.1,\n/bardblU/bardbl2\nH,µ,γ/greaterorsimilar/bardblu/bardbl2\nL2+/bardblv/bardbl2\nL2.\nHence, fixing β >0 large enough, we have\n/a\\}bracketle{t(±A−β)U,U/a\\}bracketri}htH,µ,γ≤0.7\nTherefore the operators ±A−βare dissipative. In particular, for λ > βwe have\n/bardbl(±A−λ)U/bardbl2\nH,µ,γ≥ /bardbl(±A−β)U/bardbl2\nH,µ,γ+(λ−β)2/bardblU/bardbl2\nH,µ,γ,(2.3)\nso that±A−λare injective with close range. Next we show that A−λis surjective.\nLetF= (f,g)∈ H. ForU= (u,v)∈ D, we have\n(A−λ)U=F⇔/braceleftBigg\nv=λu+f,\nu′′−(λ2+2αλ+1)u=g+(λ+2α)f.\nThen by [9, Lemma 2.5], the operator A −λis surjective. Hence, A −λhas a\nbounded inverse and by (2.3),\n/bardbl(A−λ)−1/bardblL(H)≤1\nλ−β.\nThereforebytheHille-YoshidaTheorem,thisprovesthat Ageneratesa C0-semigroup\nonH. Now the same holds with Areplaced by −A, and the proof is complete. /square\nFurthermore we estimate the strongly continuous group ( etA)t≥0.\nProposition 2.3. There exist C >0andκ >0such that, for all t≥0\n/bardbletA/bardblL(H)≤Ce−κt,/bardbletA/bardblL(L2×H−1)≤Ce−κt. (2.4)\nProof.Consider the linear equation/braceleftBigg\n∂2\ntu−∂2\nxu+2α∂tu+u−γδ0u= 0,(t,x)∈R×R,\nu(0,x) =u0(x), x ∈R,\nand define Eas\nE(t) =1\n2/parenleftbig\n/bardbl∂tu(t)/bardbl2\nL2+/bardblu(t)/bardbl2\nH1−γ|u(t,0)|2/parenrightbig\n+ǫ/parenleftbigg/integraldisplay\nu(t)∂tu(t)+α/bardblu(t)/bardbl2\nL2/parenrightbigg\n,\nwhereǫ >0 small enough. Then by direct calculation, Esatisfies\nE′(t) =−2α/bardbl∂tu(t)/bardbl2\nL2+ǫ/parenleftbig\n/bardbl∂tu(t)/bardbl2\nL2−/bardblu(t)/bardbl2\nH1+γ|u(t,0)|2/parenrightbig\n=−ǫ\n2/parenleftbig\n/bardbl∂tu(t)/bardbl2\nL2+/bardblu(t)/bardbl2\nH1−γ|u(t,0)|2/parenrightbig\n−ǫ\n2/parenleftbig\n/bardblu(t)/bardbl2\nH1−γ|u(t,0)|2/parenrightbig\n−(2α−5\n4ǫ)/bardbl∂tu(t)/bardbl2\nL2.\nSinceǫis small enough, we have\nǫ2/parenleftbigg/integraldisplay\nu(t)∂tu(t)+α/bardblu(t)/bardbl2\nL2/parenrightbigg\n≤(α+1\n2)ǫ2/bardblu(t)/bardbl2\nH1+ǫ2\n2/bardbl∂tu(t)/bardbl2\nL2\n≤(α+1\n2)C−1\nγǫ2/parenleftbig\n/bardblu(t)/bardbl2\nH1−γ|u(t,0)|2/parenrightbig\n+ǫ2\n2/bardbl∂tu(t)/bardbl2\nL2\n≤ǫ\n2/parenleftbig\n/bardblu(t)/bardbl2\nH1−γ|u(t,0)|2/parenrightbig\n+(2α−5\n4ǫ)/bardbl∂tu(t)/bardbl2\nL2.\nTherefore we have\nE′(t)<−ǫE(t),\nwhich implies\nE(t)< e−ǫtE(0).8 K. ISHIZUKA\nFurthermore, we define γ′=γ\n1−ǫ. Thenγ < γ′<2 holds because ǫis small enough.\nTherefore we have\nE(t)≥1\n2(1−ǫ)/parenleftbigg\n/bardblu(t)/bardbl2\nH1−γ\n1−ǫ|u(t,0)|2/parenrightbigg\n+(1\n2−ǫ\n2)/bardbl∂tu(t)/bardbl2\nL2\n≥1\n2(1−ǫ)C−1\nγ′/bardblu(t)/bardbl2\nH1+(1\n2−ǫ\n2)/bardbl∂tu(t)/bardbl2\nL2.\nSo we get E(t)∼ /bardbl/vector u(t)/bardbl2\nH, and\n/bardbl/vector u(t)/bardbl2\nH/lessorsimilare−ǫt/bardbl/vector u(0)/bardbl2\nH.\nThis means that /bardbletA/bardblL(H)/lessorsimilare−κtfor some κ >0. Now the same holds with H\nreplaced by L2×H−1, and the proof is complete.\n/square\n2.2.Proof of Local well-posedness of the Cauchy problem. We now turn\nto the equation (2.1). We show that the equation (2.1) is locally well-po sed inH\nby using the standard arguments as [4].\nProposition 2.4. The following properties hold.\n(1)For any U0∈ H, the equation (2.1)has a unique maximal solution U∈\nC([0,T(U0)),H), with initial data U0.\n(2)IfT(U0)<∞, thenlimt→T(U0)−0/bardblU(t)/bardblH=∞.\n(3)T:H →(0,∞]is lower semicontinuous: given the initial conditions\nU0,U0,n∈ Hsuch that U0,nconverges to U0inH, we have that\nT(U0)≤liminf\nn→∞T(U0,n).\nMoreover if U0,n→U0and ifT < T(U0), thenUn→UinC([0,T],H),\nwhereUnandUare the solutions of (2.1)corresponding to the initial data\nU0,nandU0.\nWe prove Proposition 2.4 in the same argument as [9]. To show the same as [9],\nwe only need to see the nonlinearity Fand the continuous semigroup ( etA)t≥0.\nProof.To apply the arguments as [9, Section 2], we estimate the nonlinearit yF\nand the continuous semigroup ( etA)t≥0.\nFirst, we estimate F. we note that\n|f(u)−f(v)|/lessorsimilar(1+|u|p−1+|v|p−1)|u−v|, (2.5)\nand so for any R >0, there is a constant L(R) such that, for any U,V∈ Hwith\n/bardblU/bardblH< Rand/bardblV/bardblH< R, we have\n/bardblF(U)−F(V)/bardblH≤L(R)/bardblU−V/bardblH. (2.6)\nSecond we estimate ( etA)t≥0. This has been estimated by (2.4). Thus by (2.4)\nand (2.6) we complete the proof to apply [9, Section 2].\n/square9\n2.3.Uniform bound on global solutions. By subsection 2.2, we get the local\nwell-posedness of (DNKG). In this subsection, we get a uniform bou nd for global\nsolutions of (DNKG). Essentially, this proof is the same as [7, Theore m 2.2]. To\nprove the following theorem, we only need to note the effect of a delt a potential.\nTheorem 2.5. There exists a function F: [0,∞)→[0,∞)such that F(0) = 0\nand for any global solution /vector uof(DNKG) , we have\nsup\nt∈[0,∞)/bardbl/vector u(t)/bardblH≤ F(/bardbl/vector u(0)/bardblH).\nProof.Let/vector ube a global solution of (DNKG). We define E,M,W : [0,∞)→Ras\nE(t) =Eγ(/vector u(t)),\nM(t) =1\n2/bardblu(t)/bardbl2\nL2+α/integraldisplayt\n0/bardblu(s)/bardbl2\nL2ds,\nW(t) =1\n2/bardbl/vector u(t)/bardbl2\nH−γ\n2|u(t,0)|2.\nBy (2.4) we note that for all ( u,v)∈ H\n/bardblu/bardbl2\nH1+/bardblv/bardbl2\nL2∼ /bardblu/bardbl2\nH1+/bardblv/bardbl2\nL2−γ|u(0)|2. (2.7)\nThen by direct computations, we have\nM′(t) =/integraldisplay\nu(t)∂tu(t)dx+α/bardblu(t)/bardbl2\nL2 (2.8)\n=/integraldisplay\nu(t)∂tu(t)dx+2α/integraldisplayt\n0/integraldisplay\nu(s)∂tu(s)dxds+α/bardblu(0)/bardbl2\nL2, (2.9)\nM′′(t) =/bardbl∂tu(t)/bardbl2\nL2−/parenleftbig\n/bardblu(t)/bardbl2\nH1−γ|u(t,0)|2/parenrightbig\n+/bardblu(t)/bardblp+1\nLp+1 (2.10)\n=p+3\n2/bardbl∂tu(t)/bardbl2\nL2+p−1\n2/parenleftbig\n/bardblu(t)/bardbl2\nH1−γ|u(t,0)|2/parenrightbig\n−(p+1)E(t),(2.11)\nW′(t) =−2α/bardbl∂tu(t)/bardbl2\nL2+/integraldisplay\nf(u(t))∂tu(t)dx. (2.12)\nFurthermore by (2.7), (2.8), and the Cauchy-Schwarzinequality, there exists c1>0\nsuch that for all t≥0\n|M′(t)| ≤c1W(t). (2.13)\nMoreover by (1.2) and (2.11), there exists c2>0 such that for all t≥0\nM′′(t)≥c2W(t)−(p+1)E(0). (2.14)\nIn addition by (2.9) and the Cauchy-Schwarz inequality,\n|M′(t)| ≤ /bardblu(t)/bardblL2/bardbl∂tu(t)/bardblL2\n+2α/parenleftbigg/integraldisplayt\n0/bardblu(s)/bardbl2\nL2ds/parenrightbigg1\n2/parenleftbigg/integraldisplayt\n0/bardbl∂tu(s)/bardbl2\nL2ds/parenrightbigg1\n2\n+α/bardblu(0)/bardbl2\nL2.10 K. ISHIZUKA\nLetǫ >0 to be chosen later, we estimate\n|M′|2≤(1+ǫ)/bracketleftBigg\n/bardblu/bardblL2/bardbl∂tu/bardblL2+2α/parenleftbigg/integraldisplayt\n0/bardblu(s)/bardbl2\nL2ds/parenrightbigg1\n2/parenleftbigg/integraldisplayt\n0/bardbl∂tu(s)/bardbl2\nL2ds/parenrightbigg1\n2/bracketrightBigg2\n+/parenleftbigg\n1+1\nǫ/parenrightbigg\nα2/bardblu(0)/bardbl4\nL2,\nbecause for x >0,y >0,ǫ >0 the inequality x+y≤/radicalBig\n(a+ǫ)x2+(1+1\nǫ)y2holds.\nBy using the Cauchy-Schwarz inequality again, we have\n|M′(t)|2≤(1+ǫ)M(t)/bracketleftbigg\n2/bardbl∂tu(t)/bardbl2\nL2+4α/integraldisplayt\n0/bardbl∂tu(s)/bardbl2\nL2ds/bracketrightbigg\n+/parenleftbigg\n1+1\nǫ/parenrightbigg\nα2/bardblu(0)/bardbl4\nL2.(2.15)\nFurthermore by (1.2) and (2.11),\nM′′(t) = 2/bardbl∂tu(t)/bardbl2\nL2+(p−1)W(t)\n+2α(p+1)/integraldisplayt\n0/bardbl∂tu(s)/bardbl2\nL2ds−(p+1)E(0).\nNow we fix ǫsuch that\n0< ǫ </parenleftbiggp+7\n8/parenrightbigg1\n3\n−1. (2.16)\nThenM′′satisfies\nM′′(t)≥(1+ǫ)3/bracketleftbigg\n2/bardbl∂tu(t)/bardbl2\nL2+4α/integraldisplayt\n0/bardbl∂tu(s)/bardbl2\nL2ds/bracketrightbigg\n+p−1\n2W(t)−(p+1)E(0).(2.17)\nWe assume lim t→∞M′(t) =∞. Then by (2.15) and (2.17), for all tlarge enough,\n(1+ǫ)|M′(t)|2< M′′(t)M(t). (2.18)\nThis inequality (2.18) impliesd2\ndt2[M−ǫ(t)]<0, and there exists t1>0 such that\nd\ndt[M−ǫ(t1)]<0 and for all t≥t1,\n0≤M−ǫ(t)≤M−ǫ(t1)+(t−t1)d\ndt/bracketleftbig\nM−ǫ(t1)/bracketrightbig\n, (2.19)\nwhich is a contradiction. Therefore we obtain\nliminf\nt→∞M′(t)<∞. (2.20)\nNext we prove that there exists C3>0 such that\nsup\nt∈[0,∞)|M′(t)|< C3(|E(0)|+|M′(0)|). (2.21)\nBy (2.13) and (2.14), we obtain\nM′′(t)≥c4|M′(t)|−(p+1)E(0), (2.22)11\nwherec4=c2\nc1. Thus the following two inequalities hold :\nd\ndt[c4M′(t)−(p+1)E(0)]≤c4[c4M′(t)−(p+1)E(0)],\nd\ndt[−c4M′(t)+(p+1)E(0)]≤ −c4[−c4M′(t)+(p+1)E(0)].\nBy (2.20), we obtain (2.21). In detail, see [7, Theorem 2.2].\nLast, we prove the global bound\nsup\nt∈[0,∞)W(t)<∞. (2.23)\nBy integrating (2.14) on ( t,t+τ), where τ∈(0,1], there exists C5>0 such that\nfor allt≥0 and 0< τ≤1\n/integraldisplayt+τ\ntW(s)ds < C 5(|E(0)|+|M′(0)|). (2.24)\nWe note that C5is independent of τ. By (2.12), there exists C6>0 such that for\nallt≥0\nW′(t)≤ −2α/bardbl∂tu(t)/bardbl2\nL2+/integraldisplay\n|u(t)|p|∂tu(t)|dx≤C6/bardblu(t)/bardbl2p\nL2p.\nForτ∈(0,1) andt≥τ, integrating on ( t−τ,t), we have\nW(t)≤W(t−τ)+C6/parenleftbigg/integraldisplayt\nt−τ/bardblu(s)/bardbl2p\nL2pds/parenrightbigg\n.\nUsing (2.7), (2.24), and the Sobolev inequality, there exists C7>0 such that for\nall 0< τ≤1 andt≥1\nW(t)≤W(t−τ)+C7/parenleftbigg/integraldisplayt\nt−τW(s)ds/parenrightbiggp\n. (2.25)\nIf 0≤t≤1, by observing (2.25) with t=τ, we have\nW(t)≤W(0)+C7/parenleftbigg/integraldisplayt\n0W(s)ds/parenrightbiggp\n≤W(0)+C7{C5(|E(0)|+|M′(0)|)}p.(2.26)\nIft≥1, by integrating (2.25) on τ∈(0,1), we have\nW(t)≤/integraldisplay1\n0W(t−τ)dτ+C7{C5(|E(0)|+|M′(0)|)}p\n≤C5(|E(0)|+|M′(0)|)+C7{C5(|E(0)|+|M′(0)|)}p.(2.27)\nThus we obtain (2.23). In particular, by (2.26) and (2.27) Whas an upper bound\nwhich depends only on /bardbl/vector u(0)/bardblH, and by (2.7) we complete the proof.\n/square\n3.Soliton resolution\nIn this section, we show Theorem 1.1. Before proving Theorem 1.1, w e introduce\nthe concentration-compactness argument.12 K. ISHIZUKA\nProposition 3.1. Let(un)n⊂H1be a bounded sequence satisfying\nlim\nn→∞/bardbl−∂2\nxun+un−γδ0(x)un−|un|p−1un/bardblH−1= 0. (3.1)\nThen there exist a subsequence (vn)n⊂(un)n,K≥0, signsσ= 0,±1,σk=±1\nfor anyk∈ {1,2,···,K}, and a sequence (ξk,n)k∈{1,2,···,K}⊂RKsuch that\nlim\nn→∞/bardblvn−σQγ−K/summationdisplay\nk=1σkQ(·−ξk,n)/bardblH1= 0,\nlim\nn→∞(ξk+1,n−ξk,n) =∞,fork= 1,···,K−1,\nlim\nn→∞|ξk,n|=∞fork= 1,···,K.\nConcentration-compactness arguments are well-known. In fact , whenγ= 0,\nProposition 3.1 follows from [33, Appendix A] and [34, Theorem III.4]. T o prove\nProposition 3.1, we use the same arguments as them. We only need to observe the\neffect of a potential.\nWe denote /a\\}bracketle{t·,·/a\\}bracketri}httheL2scalar product.\nProof.Since (un)nis bounded in H1, there exist (˜ un)n⊂(un)n, (ξn)n⊂R, ˜u∈\nH1(R),c≥0 andξ∈ {−∞,0,∞}such that\nlim\nn→∞/bardbl˜un/bardblp+1\nLp+1=c,\n˜un(·−ξn)⇀˜uinH1(R),\nlim\nn→∞ξn=ξ.\nIfc= 0, ˜unconverges to 0 as n→ ∞. We assume c >0.\nFirst we consider ξ= 0. Then ˜ usatisfies\n−∂2\nx˜u+ ˜u−γδ0˜u−|˜u|p−1˜u= 0. (3.2)\nTherefore if |γ| ≥2 there does not exist a solution of (3.2) and if |γ|<2 ˜usatisfies\n˜u=σQγ, whereσ=±1. We define wn= ˜un−σQγ. Then we have as n→ ∞\nwn⇀0 inH1(R).\nFurthermore by the Brezis–Lieb lemma, we have\nlim\nn→∞/bardblwn/bardblp+1\nLp+1=c−/bardblQγ/bardblp+1\nLp+1.\nFor every ϕ∈H1(R), we have\n/a\\}bracketle{t−∂2\nxwn+wn−γδ0wn−f(wn),ϕ/a\\}bracketri}ht=/a\\}bracketle{t−∂2\nx˜un+ ˜un−γδ0˜un−f(˜un),ϕ/a\\}bracketri}ht\n−/a\\}bracketle{tf(wn)−f(˜un)+σf(Qγ),ϕ/a\\}bracketri}ht.\nSincef(wn)−f(˜un)+σf(Qγ) satisfies\n|f(wn)−f(˜un)+σf(Qγ)|/lessorsimilare−|x|(|wn|p−1+|wn|),\nwe have lim n→∞/a\\}bracketle{t−∂2\nxwn+wn−γδ0wn−f(wn),ϕ/a\\}bracketri}ht= 0. Therefore we obtain\nlim\nn→∞/bardbl−∂2\nxwn+wn−γδ0wn−f(wn)/bardblH−1= 0.\nSecond we consider ξ=±∞. Then ˜usatisfies\n−∂2\nx˜u+ ˜u−f(˜u) = 0.13\nTherefore we may assume that ˜ u=σQ, whereσ=±1. Then we define Qn=\nσQ(·+ξn) andw′\nn= ˜un−Qn. Then by the Brezis–Lieb lemma, we have\nlim\nn→∞/bardblw′\nn/bardblp+1\nLp+1=c−/bardblQ/bardblp+1\nLp+1.\nNow we fix ϕ∈H1(R). Then we have\n/a\\}bracketle{t−∂2\nxw′\nn+w′\nn−γδ0w′\nn−f(w′\nn),ϕ/a\\}bracketri}ht=/a\\}bracketle{t−∂2\nx˜un+ ˜un−γδ0˜un−f(˜un),ϕ/a\\}bracketri}ht\n+/a\\}bracketle{tγδ0Qn,ϕ/a\\}bracketri}ht\n−/a\\}bracketle{tf(w′\nn)−f(˜un)+f(Qn),ϕ/a\\}bracketri}ht.\nBy direct computation, we have\n|f(w′\nn)−f(˜un)+f(Qn)|/lessorsimilar|Qn||˜un|p−1+|Qn|p−1|˜un|.\nTherefore for any R >0, we obtain\n|/a\\}bracketle{tf(w′\nn)−f(˜un)+f(Qn),ϕ/a\\}bracketri}ht|/lessorsimilar/bardblϕ/bardblL2(|x|>R)+e−|xn|eR/bardblϕ/bardblH1,\nwhich implies lim n→∞/a\\}bracketle{t−∂2\nxw′\nn+w′\nn−γδ0w′\nn−f(w′\nn),ϕ/a\\}bracketri}ht= 0. Thus we obtain\nlim\nn→∞/bardbl−∂2\nxw′\nn+w′\nn−γδ0w′\nn−f(w′\nn)/bardblH−1= 0.\nGathering these arguments, there exist Q ∈ {±Qγ,±Q}and (ξn)n⊂Rsuch that\nlim\nn→∞/bardbl˜wn/bardblp+1\nLp+1≤lim\nn→∞/bardbl˜un/bardblp+1\nLp+1−min(/bardblQ/bardblp+1\nLp+1,/bardblQγ/bardblp+1\nLp+1),\nlim\nn→∞/bardbl−∂2\nx˜wn+ ˜wn−γδ0˜wn−f(˜wn)/bardblH−1= 0,\nwhere ˜wn= ˜un−Q(·+ξn). We note that if |γ| ≥2, min(/bardblQ/bardblp+1\nLp+1,/bardblQγ/bardblp+1\nLp+1) =\n/bardblQ/bardblp+1\nLp+1. Therefore by repeatedly applying this argument, we complete the proof.\n/square\nNow we prove Theorem 1.1. First we define\nE0,R=R,\nEK,R={z∈RK:|zk|> Rfor 1≤k≤Kandzk+1−zk> Rfor 1≤k≤K−1},\nwhereK≥1 andR >0. Furthermore we define\ndK,R,ς,σ(u) = inf\nz∈EK,R/bardblu−ςQγ−K/summationdisplay\nk=1σkQ(·−zk)/bardblH1,\nNK,R,ς,σ,ǫ={u∈H1(R) :dK,R,ς,σ(u)< ǫ},\nwhere\nς∈ {0,±1}, σ= (σk)1≤k≤K∈ {±1}K, K≥0, R >0, ǫ >0.\nWe note that when R >0 is large enough and ǫ >0 is small enough, the family\nof sets (NK,R,ς,σ,ǫ) is disjoint.\nProof of Theorem 1.1. Let/vector ube a global solution of (DNKG). First, we prove\nlim\nt→∞/bardbl∂tu(t)/bardblL2= 0, (3.3)\nlim\nt→∞/bardbl−∂2\nxu(t)+u(t)−γδ0u(t)−|u(t)|p−1u(t)/bardblH−1= 0.(3.4)14 K. ISHIZUKA\nLet\n/vector v(t) =/parenleftbigg\nv(t)\n∂tv(t)/parenrightbigg\n=/parenleftbigg\n∂tu(t)\n∂2\ntu(t)/parenrightbigg\n.\nBy Proposition 2.3, we obtain\n/vector v(t) =etA/vector v(0)+/integraldisplayt\n0e(t−s)A/parenleftbigg\n0\n|u(s)|p−1∂tu(s)/parenrightbigg\nds, (3.5)\nand for all t≥0,\n/bardbletA/vector v(0)/bardblL2×H−1/lessorsimilare−κt.\nBy (1.2), v=∂tu∈L2([0,∞)×R), and by Theorem 2.5, u∈L∞([0,∞),H1).\nTherefore we have\n/bardbl/integraldisplayt\n2\n0e(t−s)A/vector w(s)ds/bardblL2×H−1/lessorsimilar/integraldisplayt\n2\n0e−κ(t−s)/bardblu/bardblp−1\nL∞([0,∞),H1)/bardblv(s)/bardblL2ds\n/lessorsimilare−κt\n2/bardblu/bardblp−1\nL∞([0,∞),H1)/bardblv/bardblL2([0,∞)×R),\n/bardbl/integraldisplayt\nt\n2e(t−s)A/vector w(s)ds/bardblL2×H−1/lessorsimilar/integraldisplayt\nt\n2e−κ(t−s)/bardblu/bardblp−1\nL∞([0,∞),H1)/bardblv(s)/bardblL2ds\n/lessorsimilar/bardblu/bardblp−1\nL∞([0,∞),H1)/bardblv/bardblL2((t\n2,∞)×R),\nwhere\n/vector w(t) =/parenleftbigg\n0\n|u(t)|p−1∂tu(t)/parenrightbigg\n.\nThus we obtain (3.3) and (3.4).\nThen, (u(n))n⊂H1(R) is bounded and satisfies (3.1). Therefore by Proposition\n3.1, there exist ˜K≥0, (˜tn)n, ˜ς∈ {0,±1}, ˜σ= (˜σk)1≤k≤˜K∈ {±1}˜Ksuch that for\nanyR >0,\nlim\nn→∞d˜K,R,˜ς,˜σ(u(˜tn)) = 0.\nNext we prove that for any sequence tn→ ∞,usatisfies\nlim\nn→∞d˜K,R,˜ς,˜σ(u(tn)) = 0. (3.6)\nWe assume that there exist a sequence tn→ ∞andR >0 such that\nlimsup\nn→∞d˜K,R,˜ς,˜σ(u(tn))>0.\nThen there exist a subsequence ( t′\nn)n⊂(tn)nand sufficiently small ǫ >0 such that\nfor alln∈N\nu(t′\nn)/∈ N˜K,R,˜ς,˜σ,ǫ. (3.7)\nSince (u(t′\nn))nis bounded and satisfies (3.1), by Proposition 3.1 there exist a sub-\nsequence ( t′′\nn)n⊂(t′\nn)n,ˆK,ˆς, and ˆσsuch that\nlim\nn→∞dˆK,R,ˆς,ˆσ(u(t′′\nn)) = 0.15\nWe note that by (3.7), ( ˆK,ˆς,ˆσ)/\\e}atio\\slash= (˜K,˜ς,˜σ) holds. Therefore when n,m∈Nsatisfy\nu(tn)∈ N˜K,R,˜ς,˜σ,ǫandu(t′\nm)∈ NˆK,R,ˆς,ˆσ,ǫ, there exists s∈Rbetween tnandt′\nm\nsuch that\nu(s)/∈/uniondisplay\nK≥0,ς∈{0,±1},σ∈{±1}KNK,R,ς,σ,ǫ.\nBy this argument, there exists a sequence sn→ ∞such that for all n∈N,\nu(sn)/∈/uniondisplay\nK≥0,ς∈{0,±1},σ∈{±1}KNK,R,ς,σ,ǫ. (3.8)\nThen, (u(sn))nis bounded and u(sn) satisfies (3.1). Therefore by Proposition 3.1,\nthere exist a subsequence ( s′\nn)n⊂(sn)n,˙K,˙ς,˙σsuch that\nlim\nn→∞d˙K,R,˙ς,˙σ(u(s′\nn)) = 0,\nwhich contradicts (3.8). Therefore for any tn→ ∞,usatisfies (3.6), and therefore\nwe have\nlim\nt→∞d˜K,R,˜ς,˜σ(u(t)) = 0.\nWe prove Theorem 1.1. We define z(t) = (zk(t))∈R˜Kas\n/bardblu(t)−˜ςQγ−˜K/summationdisplay\nk=1˜σkQ(·−zk(t))/bardblH1= inf\nz∈R˜K/bardblu(t)−˜ςQγ−˜K/summationdisplay\nk=1˜σkQ(·−zk)/bardblH1.\nWe notethat forany R >0, the minimum is attainedfor z(t) satisfying z(t)∈ER,˜K\nwhentis large enough. Therefore we obtain (1.6) with K=˜K, σ= ˜ς, andσk= ˜σk\nfor 1≤k≤˜K, and we complete the proof.\n/square\n4.Global dynamics below the ground state energy\n4.1.Minimizing problems and variational structure. First we collect prop-\nerties of the ground state energy. First we define Kγ,Jγ:H1(R)→Ras\nKγ(ϕ) =/bardblϕ/bardbl2\nH1−γ|ϕ(0)|2−/bardblϕ/bardblp+1\nLp+1,\nJγ(ϕ) =1\n2/parenleftbig\n/bardblϕ/bardbl2\nH1−γ|ϕ(0)|2/parenrightbig\n−1\np+1/bardblϕ/bardblp+1\nLp+1,\nand we consider the following minimizing problems.\nnγ= inf{Jγ(ϕ) :ϕ∈H1(R)\\{0},Kγ(ϕ) = 0},\nrγ= inf{Jγ(ϕ) :ϕ∈H1\nrad(R)\\{0},Kγ(ϕ) = 0},\nwhereH1\nrad(R) :={ϕ∈H1(R) :ϕ(x) =ϕ(−x)}. In [16], we get the following\nstatements about nγandrγ.\nProposition 4.1. The following properties hold.\n(1)if0≤γ <2,Qγis a minimizer of nγandrγ.\n(2)if−2< γ <0,Qγis a minimizer of rγ. Furthermore nγ=J0(Q)has no\nminimizer.\n(3)ifγ≤ −2,nγ=J0(Q)has no minimizer and rγ= 2J0(Q)has no mini-\nmizer.16 K. ISHIZUKA\nFurthermore we get a uniform bound on Kγ.\nProposition 4.2. (1)For any ϕ∈H1(R)withJγ(ϕ)< nγ, we have\nKγ(ϕ)/greaterorsimilarmin{nγ−Jγ(ϕ),/bardblϕ/bardbl2\nH1}or Kγ(ϕ)/lessorsimilar−(nγ−Jγ(ϕ)).(4.1)\n(2)For any ϕ∈H1\nrad(R)withJγ(ϕ)< rγ, we have\nKγ(ϕ)/greaterorsimilarmin{rγ−Jγ(ϕ),/bardblϕ/bardbl2\nH1}or Kγ(ϕ)/lessorsimilar−(rγ−Jγ(ϕ)).(4.2)\nThis uniform bound is useful to classify the global behavior of solutio ns whose\nenergy is less than ground state energy for the nonlinear dispersiv e equations. To\nprove Proposition 4.2, we refer to [21, Proposition 2.18] and [22, Lem ma 2.12].\nProof.First we prove (1). We may assume ϕ/\\e}atio\\slash= 0. We define s(λ) =Jγ(eλϕ). Then\nwe have\ns(λ) =e2λ\n2/parenleftbig\n/bardblϕ/bardbl2\nH1−γ|ϕ(0)|2/parenrightbig\n−e(p+1)λ\np+1/bardblϕ/bardblp+1\nLp+1, (4.3)\ns′(λ) =e2λ/parenleftbig\n/bardblϕ/bardbl2\nH1−γ|ϕ(0)|2/parenrightbig\n−e(p+1)λ/bardblϕ/bardblp+1\nLp+1, (4.4)\ns′′(λ) = 2e2λ/parenleftbig\n/bardblϕ/bardbl2\nH1−γ|ϕ(0)|2/parenrightbig\n−(p+1)e(p+1)λ/bardblϕ/bardblp+1\nLp+1.(4.5)\nBy (4.4) and (4.5), we obtain s′(0) =Kγ(ϕ) and\ns′′(λ)≤2s′(λ). (4.6)\nFirst we consider Kγ(ϕ)<0. Then we have s′(0)<0 ands′(λ)>0 for sufficiently\nsmallλ <0. Therefore there exists λ0<0 such that s′(λ0) = 0. By integrating\n(4.6) on [ λ0,0], we have\ns′(0)−s′(λ0)≤2(s(0)−s(λ0)). (4.7)\nBy (4.7) and s′(λ0) = 0, we obtain\nKγ(ϕ) =s′(0)≤ −2(nγ−Jγ(ϕ)).\nSecond we consider Kγ(ϕ)≥0. Since ϕ/\\e}atio\\slash= 0, we consider Kγ(ϕ)>0. If\nKγ(ϕ)≥1\n3(p−1)/bardblϕ/bardblp+1\nLp+1,\nwe have\n(1+1\n3(p−1))Kγ(ϕ)≥1\n3(p−1)/parenleftbig\n/bardblϕ/bardbl2\nH1−γ|ϕ(0)|2/parenrightbig\n.\nBy (2.2), we obtain Kγ(ϕ)/greaterorsimilar/bardblϕ/bardbl2\nH1. If\nKγ(ϕ)<1\n3(p−1)/bardblϕ/bardblp+1\nLp+1,\nthere exists λ1>0 such that\n0< s′(λ)<1\n3(p−1)e(p+1)λ/bardblϕ/bardblp+1\nLp+1 (4.8)\nfor 0< λ < λ 1. Then by direct calculation, we have\ns′′(λ) = 2s′(λ)−(p−1)e(p+1)λ/bardblϕ/bardblp+1\nLp+1\n<{2−3(p−1)2}s′(λ)\n<−s′(λ)17\nfor 0< λ < λ 1. In particular, as long as (4.8) holds, s′decreases. Furthermore\nlimλ→∞s′(λ) =−∞holds and we may assume s′(λ1) = 0. Then by integrating\ns′′<−s′on [0,λ1], we have\ns′(λ1)−s′(0)<−(s(λ1)−s(0)).\nThus we obtain\nKγ(ϕ)> nγ−Jγ(ϕ).\nGathering these arguments, we obtain (4.1).\nWe note that we only use the variational structure to obtain (4.1). Therefore we\nobtain (4.2) by the same argument.\n/square\n4.2.Classification of global dynamics. Inthis subsection, weclassifythe global\nbehavior of solutions whose energy is less than the ground state en ergy. To classify\nthe global behavior of solutions, we use the argument as [40].\nProposition 4.3. Let/vector ube a solution of (DNKG) .\n(1)If/vector usatisfies Eγ(/vector u(0))< nγandKγ(u(0))≥0, then/vector uis global and con-\nverges exponentially to 0inHast→ ∞.\n(2)If/vector usatisfies Eγ(/vector u(0))< nγandKγ(u(0))<0, then/vector ublows up in finite\ntime.\n(3)If/vector uis even and satisfies Eγ(/vector u(0))< rγandKγ(u(0))≥0, then/vector uis global\nand converges exponentially to 0inHast→ ∞.\n(4)If/vector uis even and satisfies Eγ(/vector u(0))< rγandKγ(u(0))<0, then/vector ublows up\nin finite time.\nBefore we prove Proposition 4.3, we define P:H →Ras\nP(ϕ1,ϕ2) =/integraldisplay\nϕ1ϕ2+α/bardblϕ1/bardbl2\nL2.\nProof.First we prove (1). Since Jγ(u(0))< nγ, for allt≥0Kγ(u(t))≥0 holds.\nThen we have for t≥0\nEγ(/vector u) =1\np+1Kγ(u)+1\n2/bardbl∂tu/bardbl2\nL2+p−1\n2(p+1)/parenleftbig\n/bardblu/bardbl2\nH1−γ|u(0)|2/parenrightbig\n≥1\n2/bardbl∂tu/bardbl2\nL2+p−1\n2(p+1)/parenleftbig\n/bardblu/bardbl2\nH1−γ|u(0)|2/parenrightbig\n,\nand therefore we obtain\nEγ(/vector u(t))∼ /bardbl/vector u(t)/bardbl2\nH (4.9)\nfor allt≥0. By direct calculation and (4.9) and P(/vector u)/lessorsimilar/bardbl/vector u/bardblH, there exists C >0\nsuch that\n∂t[P(/vector u)+CEγ(/vector u)] = (1−2αC)/bardbl∂tu/bardbl2\nL2−Kγ(u)≤ −1\nC[P(/vector u)+CEγ(/vector u)],\n/bardbl/vector u(t)/bardblH∼ P(/vector u(t))+CEγ(/vector u(t)).\nThus we obtain\n/bardbl/vector u(t)/bardblH/lessorsimilare−t\nC/bardbl/vector u(0)/bardblH,\nand so the desired estimate.18 K. ISHIZUKA\nNext we prove (2). We assume that /vector uis global. By Proposition 4.2 and (1.2),\nthere exists c >0 such that\n∂tP(/vector u) =/bardbl∂tu/bardbl2\nL2−Kγ(u)\n≥c(nγ−Jγ(u))\n≥c(nγ−Eγ(/vector u))\n≥c(nγ−Eγ(/vector u(0))).\nBythe aboveestimate, lim t→∞P(/vector u(t)) =∞holds and thereforelim t→∞/bardbl/vector u(t)/bardblH=\n∞holds, which contradicts Theorem 2.5. Thus we have proved (2).\nLast, we prove(3) and (4). Let /vector ube an even. Then (4.2) holds, and we can prove\nthem by the same arguments as (1) and (2). Therefore we complet e the proof.\n/square\nRemark 4.1. We assume that a solution /vector uof (DNKG) satisfies lim t→∞/bardbl/vector u(t)/bardblH=\n0. Then /vector usatisfies /bardbl/vector u(t)/bardblH≪1 for some t∈R, and therefore /vector u(t) satisfies\nEγ(/vector u(t))< nγandKγ(u(t))≥0. Therefore By (1) of Proposition 4.3, /vector uconverges\nexponentially to 0 as t→ ∞. By this argument, When a global solution /vector usatisfies\n(1.6) with σ= 0 and K= 0,/vector uconverges exponentially to 0 as t→ ∞.\nWe define Hradas\nHrad={(ϕ1,ϕ2)∈ H:ϕ1(x) =ϕ1(−x), ϕ2(x) =ϕ2(−x)}.\nFurthermore we define B,Brad,V,Vradby\nB={/vector ϕ∈ H:/vector ublows up in finite time },\nBrad={/vector ϕ∈ Hrad:/vector ublows up in finite time },\nV={/vector ϕ∈ H:/vector uis global and lim\nt→∞/bardbl/vector u(t)/bardblH= 0},\nVrad={/vector ϕ∈ Hrad:/vector uis global and lim\nt→∞/bardbl/vector u(t)/bardblH= 0},\nwhere/vector uis the solution of (DNKG) with initial data /vector ϕ. Then by Proposition 4.3,\nVandVradare open. On the other hand, by Proposition 2.4 and Theorem 2.5, we\nobtain that the set of global solutions of (DNKG) is closed. Therefo reBandBrad\nare open.\n4.3.Proof of Theorem 1.2. In this subsection, we prove Theorem 1.2. Let z >0\nbe large enough. We define\nQς=Q(·−z)+ςQ(·+z),\nQλ,ς=eλQς,\n/vectorQλ,ς= (Qλ,ς,0),\nAς={/vectorQλ,ς:−1≤λ≤1},\nwhereς= 0,1 and−1≤λ≤1. Furthermore we define\nxς(λ) =e2λ\n2/bardblQς/bardbl2\nH1−e(p+1)λ\np+1/bardblQς/bardblp+1\nLp+1−γe2λ\n2|Qς(0)|2.\nThen by direct calculation, we have\nx′\nς(λ) =e2λ/bardblQς/bardbl2\nH1−e(p+1)λ/bardblQς/bardblp+1\nLp+1−γe2λ|Qς(0)|2,\nx′′\nς(λ) = 2e2λ/bardblQς/bardbl2\nH1−(p+1)e(p+1)λ/bardblQς/bardblp+1\nLp+1−2γe2λ|Qς(0)|2.19\nNow we start to prove Theorem 1.2.\nProof of Theorem 1.2. First we prove(1) of Theorem 1.2. We consider ς= 0. Since\nK0(Q) = 0 holds, we have\nx′\n0(0) =−γ|Q(z)|2,\nx′′\n0(0) =−(p−1)/bardblQ/bardblp+1\nLp+1−2γ|Q(z)|2.\nWe recall that zis large enough. Thus there exists λ′<0 close to 0 such that\nx′\n0(λ′) = 0. Moreover there exists C >0 and 0< δ <1 such that for −δ≤λ≤δ\n|x′\n0(λ)−x′\n0(λ′)| ≥C|λ−λ′|.\nWe note that δis independent of zifzis large enough. Then for −δ≤λ < λ′,\nx′\n0(λ)>0 holds and for λ′< λ≤δ,x′\n0(λ)<0 holds. Therefore for −δ≤λ≤δ, we\nhave\nx0(λ′)−x0(λ) =/integraldisplayλ′\nλx′\n0(s)−x′\n0(λ′)ds\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayλ\nλ′|x′\n0(s)−x′\n0(λ′)|ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≥C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayλ\nλ′|s−λ′|ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=C\n2|λ−λ′|2.\nThus we have\nx′\n0(−δ)>0, x′\n0(δ)<0,\nx0(−δ)< nγ, x0(δ)< nγ,\nbecause λ′is close to 0. Therefore by Proposition 4.3, /vectorQ−δ,0∈ Vand/vectorQδ,0∈ B.\nSinceAis connected and BandVare open, there exists −1≤λ⋆≤1 such that\n/vectorQλ⋆,0/∈ Band/vectorQλ⋆,0/∈ V. Moreover we have\nEγ(/vectorQλ⋆,0)< Eγ(/vectorQλ′,0). (4.10)\nWe notethat Eγ(/vectorQλ′,0)is closeto nγ. Thus byTheorem1.1and (4.10), the solution\n/vector u⋆,0of (DNKG) with initial data /vectorQλ⋆,0satisfies\nlim\nt→∞(/bardblu⋆,0(t)−σ′Q(·−z(t))/bardblH1+/bardbl∂tu⋆,0(t)/bardblL2) = 0,\nwhereσ′=±1 and lim t→∞|z(t)|=∞. This completes the proof of (1) of Theorem\n1.2, since (1.3) holds.\nNext we prove (2) of Theorem 1.2. We consider ς= 1. Then A1⊂ Hradholds.\nFurthermore by Proposition 4.1 and direct computation, we have\n|x1(0)−rγ|/lessorsimilare−z,\n|x′\n1(0)|/lessorsimilare−z,\n/vextendsingle/vextendsingle/vextendsinglex′′\n1(0)+2(p−1)/bardblQ/bardblp+1\nLp+1/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−z.\nThus by the same argument as (1) of Theorem 1.2, we complete the p roof of (2) of\nTheorem 1.2.20 K. ISHIZUKA\n/square\n5.Dynamics close to solitary waves\n5.1.Basic properties of the ground state. In this subsection, we collect some\nbasic properties. First, the linearized operator LaroundQwrites\nL=−∂2\nx+1−pQp−1. (5.1)\nWe define ν,φas\nν=/radicalbigg\n(p−1)(p+3)\n4,\nφ(x) =/parenleftBigg\n1\ncosh(p−1\n2x)/parenrightBiggp+1\np−1\n.\nWe note that\nφ=/parenleftbigg2\np+1/parenrightbiggp+1\n2(p−1)\nQp+1\n2\nholds. We introduce the following properties. In detail, see [5].\nLemma 5.1. (1)Lhas a unique negative eigenvalue −ν2and its eigenspace\n(L+ν2)−1({0}) = span {φ}, and its kernel is span{Q′}.\n(2)There exists c >0such that, for all ϕ∈H1(R),\n/a\\}bracketle{tLϕ,ϕ/a\\}bracketri}ht ≥c/bardblϕ/bardbl2\nH1−1\nc/parenleftbig\n/a\\}bracketle{tϕ,φ/a\\}bracketri}ht2+/a\\}bracketle{tϕ,Q′/a\\}bracketri}ht2/parenrightbig\n.\nNow we consider the linearized operator around ( Q(· −z),0), where z≫1.\nWhen we ignore the potential-soliton repulsion, the linearized evolutio n around\n(Q(·−z),0) is given by\nd\ndtv=/parenleftbigg0 1\n−(−∂2\nx+1−pQp−1(·−z))−2α/parenrightbigg\nv. (5.2)\nWe define ν±andY±as\nν±=−α±/radicalbig\nα2+ν2,\nY±=/parenleftbigg1\nν±/parenrightbigg\nφ(·−z).\nSinceν+>0, the solution v=eν+tY+of (5.2) exhibits the exponential instability\nof (Q(· −z),0). In particular, we see that the presence of the damping α >0 in\nthe equation does not remove the exponential instability.\nSecond We recall properties of soliton interactions. In detail, see [8 ]. Before\nwe establish the lemma about soliton interactions, we define some not ations. We\nassume that z1,z2∈Rsatisfy|z1−z2| ≫1 and we define\nQ1(x) =Q(x−z1), Q2(x) =Q(x−z2).\nLemma 5.2. The following properties hold for |z1−z2| ≫1.\n(1)The following equation holds:\n/integraldisplay\nQpe−x= 2cQ. (5.3)21\n(2)For any 0< m′< m,\n0</integraldisplay\nQm\n1Qm\n2/lessorsimilare−m′|z1−z2|, (5.4)\n0</integraldisplay\nQm\n1Qm′\n2/lessorsimilare−m′|z1−z2|. (5.5)\n(3)For any m >1,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nQ1Qm\n2−cme−|z1−z2|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−3\n2|z1−z2|,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n(∂xQ1)Qm\n2−z1−z2\n|z1−z2|cme−|z1−z2|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−3\n2|z1−z2|,\nwherecm=cQ/integraltext\ne−xQm.\n(4)The following estimate holds:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/a\\}bracketle{tf(Q1+Q2)−f(Q1)−f(Q2),∂xQ1/a\\}bracketri}ht−z1−z2\n|z1−z2|·2c2\nQe−|z1−z2|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−θ|z1−z2|,\nwhereθ= min(p−1,3\n2).\nProof.First we prove (1). Let R >0. By integration by parts, we obtain that as\nR→ ∞,\n/integraldisplayR\n−RQpe−x=/integraldisplayR\n−R(−Q′′+Q)e−x\n=cQ+/integraldisplayR\n−R(−Q′+Q)e−x+O(e−2R)\n= 2cQ+O(e−2R).\nThereforewe obtain (5.3). (2), (3) and (4) areprovedby [7, Lemm a 3.2], [8, Lemma\n2.1] and (5.3). /square\n5.2.Modulation of the center. Now we assume that a solution /vector uof (DNKG)\nis a single solitary wave or an even 2-solitary wave. We have some free dom for the\nchoice of centers z. So we choose the center of the solitary wave.\nLemma 5.3. (1)Letσ=±1. There exists δ >0small enough such that for\nanyϕ= (ϕ1,ϕ2)∈ Handz∈Rsatisfying\n|z|>1\nδ,/bardblϕ1−σQ(·−z)/bardblH1+/bardblϕ2/bardblL2≤δ,\nthere exists a unique ˜z∈Rsatisfying |z−˜z| ≤δand\n/integraldisplay\n{ϕ2+2α(ϕ1−σQ(·−˜z))}Q′(·−˜z) = 0.\nMoreover. ˜zisC1with respect to (ϕ,z).\n(2)Letσ=±1. There exists δ >0small enough such that for any ϕ=\n(ϕ1,ϕ2)∈ Hradandz∈Rsatisfying\nz >1\nδ,/bardblϕ1−σ(Q(·−z)+Q(·+z))/bardblH1+/bardblϕ2/bardblL2≤δ,22 K. ISHIZUKA\nthere exists a unique ˜z∈Rsatisfying |z−˜z| ≤δand\n/integraldisplay\n{ϕ2+2α(ϕ1−σ(Q(·−˜z)+Q(·+ ˜z)))}Q′(·−˜z) = 0.\nMoreover. ˜zisC1with respect to (ϕ,z).\nProof.We note that if ϕis an even function, for any z >0,/integraldisplay\nϕQ′(·−z) =−/integraldisplay\nϕQ′(·+z)\nholds. Then by [23, Lemma 4.1] we complete the proof.\n/square\nBy Lemma 5.3, if a global solution /vector uis a single solitary wave, there exist σ=±1\nandz(t) such that\nlim\nt→∞(/bardblu(t)−σQ(·−z(t))/bardblH1+/bardbl∂tu(t)/bardblL2) = 0,\nlim\nt→∞|z(t)|=∞,\n/integraldisplay\n{∂tu(t)+2α(u(t)−σQ(·−z(t)))}Q′(·−z(t)) = 0,\nand if a global solution /vector uis an even 2-solitary wave, there exist σ=±1 andz(t)\nsuch that\n/vector uis an even solution ,\nlim\nt→∞(/bardblu(t)−σ(Q(·−z(t))+Q(·+z(t)))/bardblH1+/bardbl∂tu(t)/bardblL2) = 0,\nlim\nt→∞z(t) =∞,\n/integraldisplay\n{∂tu(t)+2α(u(t)−σ(Q(·−z(t))+Q(·+z(t))))}Q′(·−z(t)) = 0.\nFurthermore by (1.3), we only need the following proposition to prov e Theorem\n1.3.\nProposition 5.4. (1)Letγ <0. If a global solution /vector usatisfies\n\n\nlim\nt→∞(/bardblu(t)−Q(·−z(t))/bardblH1+/bardbl∂tu(t)/bardblL2) = 0,\nlim\nt→∞z(t) =∞,\n/bardblu(0)−Q(·−z(0))/bardblH1+/bardbl∂tu(0)/bardblL2≪1, z(0)≫1,\n/integraldisplay\n{∂tu(t)+2α(u(t)−Q(·−z(t)))}Q′(·−z(t)) = 0,(5.6)\nthenz(t)−1\n2logt/lessorsimilar1.\n(2)Letγ≤ −2. If a global solution /vector usatisfies\n\n\n/vector uis an even solution ,\nlim\nt→∞(/bardblu(t)−Q(·−z(t))−Q(·+z(t))/bardblH1+/bardbl∂tu(t)/bardblL2) = 0,\nlim\nt→∞z(t) =∞,\n/bardblu(0)−Q(·−z(0))−Q(·+z(0))/bardblH1+/bardbl∂tu(0)/bardblL2≪1, z(0)≫1,\n/integraldisplay\n{∂tu(t)+2α(u(t)−Q(·−z(t))−Q(·+z(t)))}Q′(·−z(t)) = 0,(5.7)23\nthenz(t)−1\n2logt/lessorsimilar1.\n5.3.Estimates of the center. We assume that a solution of (DNKG) satisfies\n\n\nlim\nt→∞(/bardblu(t)−Q(·−z(t))−σQ(·+z(t))/bardblH1+/bardbl∂tu(t)/bardblL2) = 0,\nlim\nt→∞z(t) =∞,\n/bardblu(0)−Q(·−z(0))−σQ(·+z(0))/bardblH1+/bardbl∂tu(0)/bardblL2≪1, z(0)≫1,\n/integraldisplay\n{∂tu(t)+2α(u(t)−Q(·−z(t))−σQ(·+z(t)))}Q′(·−z(t)) = 0,(5.8)\nwhereσ= 0,1. We define Q±,/vectorQ±, andφ±as\nQ±(x) =Q(x∓z(t)),/vectorQ±=/parenleftbigg\nQ±\n0/parenrightbigg\n, φ±(x) =φ(x∓z(t)).\nFurthermore we define /vector ǫ= (ǫ,η) as\n/vector ǫ=/parenleftbiggǫ\nη/parenrightbigg\n=/vector u−/vectorQ+−σ/vectorQ−.\nThen (DNKG) can be rewritten as\n\n\n∂tǫ=η+z′(∂xQ+−σ∂xQ−),\n∂tη=−/braceleftBig\n−∂2\nxǫ+ǫ−(pQp−1\n++σpQp−1\n−)ǫ/bracerightBig\n−2αη\n+γδ0u+(f(u)−f(Q+)−f(σQ−)−f′(Q+)ǫ−f′(σQ−)ǫ).(5.9)\nIn addition we define LtandRas\nLt=−∂2\nx+1−pQp−1\n+−σpQp−1\n−,\nR=f(u)−f(Q+)−f(σQ−)−f′(Q+)ǫ−f′(σQ−)ǫ.\nThen we rewrite (5.8) and (5.9) as\n\n\nlim\nt→∞/bardbl/vector ǫ(t)/bardblH= 0,lim\nt→∞z(t) =∞,/bardbl/vector ǫ(0)/bardblH≪1, z(0)≫1,\n/integraldisplay\n(η+2αǫ)∂xQ+= 0,(5.10)\nand\nd\ndt/parenleftbigg\nǫ\nη/parenrightbigg\n=/parenleftbigg\n0 1\n−Lt−2α/parenrightbigg/parenleftbigg\nǫ\nη/parenrightbigg\n+/parenleftbigg\nz′(∂xQ+−σ∂xQ−)\nγδ0u+R(t)/parenrightbigg\n.(5.11)\nFurthermore we define R1as\nR=pσQp−1\n+Q−+R1. (5.12)\nFirst we give some estimates about RandR1.24 K. ISHIZUKA\nLemma 5.5. Letθ= min(2 p−2,3). Then the following estimates hold:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR1∂xQ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−θz+/bardbl/vector ǫ/bardbl2\nH, (5.13)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR1φ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−θz+/bardbl/vector ǫ/bardbl2\nH, (5.14)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR∂xQ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−2z+/bardbl/vector ǫ/bardblH, (5.15)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRφ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−2z+/bardbl/vector ǫ/bardbl2\nH, (5.16)\n/bardblR/bardblL2/lessorsimilare−2z+/bardbl/vector ǫ/bardbl2\nH. (5.17)\nProof.FirstR1satisfies\nR1=f(u)−f(Q+)−f(σQ−)−f′(Q+)ǫ−f′(σQ−)ǫ−σf′(Q+)Q−\n=f(Q++σQ−+ǫ)−f(Q++σQ−)−f′(Q++σQ−)ǫ\n+(f′(Q++σQ−)−f′(Q+))ǫ\n+f(Q++σQ−)−f(Q+)−f(σQ−)−f′(Q+)(σQ−)\n−f′(σQ−)ǫ.\nBy direct computation we have\n|f(Q++σQ−+ǫ)−f(Q++σQ−)−f′(Q++σQ−)ǫ|/lessorsimilar|ǫ|2+|ǫ|p,\n|(f′(Q++σQ−)−f′(Q+))ǫ|/lessorsimilar/parenleftbig\n|Q+|p−2|Q−|+|Q−|p−1/parenrightbig\n|ǫ|,\n|f(Q++σQ−)−f(Q+)−f(σQ−)−f′(Q+)(σQ−)|/lessorsimilar|Q+|p−2|Q−|2+|Q+||Q−|p−1,\n|f′(σQ−)ǫ|/lessorsimilar|Q−|p−1|ǫ|.\nTherefore by Lemma 5.2 and e−2z/bardbl/vector ǫ/bardblH/lessorsimilare−3z+/bardbl/vector ǫ/bardbl3\nH, we obtain (5.13) and (5.14).\nNext we estimate (5.17). Rsatisfies\nR=f(Q++σQ−+ǫ)−f(Q++σQ−)−f′(Q++σQ−)ǫ\n+(f′(Q++σQ−)−f′(Q+)−f′(Q−))ǫ\n+f(Q++σQ−)−f(Q+)−f(σQ−)−f′(Q+)(σQ−).\nFurthermore, we have\n|(f′(Q++σQ−)−f′(Q+)−f′(Q−))ǫ|/lessorsimilare−z|ǫ|,\nand (5.17) holds. Last, we prove (5.15) and (5.16). By Lemma 5.2, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nQp−1\n+Q−∂xQ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−2z,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nQp−1\n+Q−φ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−2z.\nThus we obtain (5.15) and (5.16).\n/square\nNext we estimate the distance between the origin and the center of the solitary\nwave.25\nProposition 5.6. Let/vector ube a global solution of (DNKG) satisfying (5.10)for some\nz(t)andσ= 0,1. Thenz′satisfies\n2α/bardblQ′/bardbl2\nL2z′=−γcQe−zu(t,0)−2σc2\nQe−2z−R2+O(e−θz+/bardbl/vector ǫ/bardbl3\nH)\n={−γ(1+σ)−2σ}c2\nQe−2z−γcQe−zǫ(t,0)−R2+O(e−θz+/bardbl/vector ǫ/bardbl3\nH),\n(5.18)\nwhereθ= min(2 p−2,3)and\nR2=/integraldisplay\nR1∂xQ+.\nProof.Differentiating the second equation of (5.10), we obtain\n0 =/integraldisplay\n(∂tη+2α∂tǫ)∂xQ+−z′/integraldisplay\n(η+2αǫ)∂2\nxQ+\n=z′/parenleftbigg\n2α/bardblQ′/bardbl2\nL2−2ασ/integraldisplay\n∂xQ−∂xQ+−/integraldisplay\n(η+2αǫ)∂2\nxQ+/parenrightbigg\n+γ/integraldisplay\nδ0(u·∂xQ+)+σp/integraldisplay\nQp−1\n−∂xQ+ǫ+/integraldisplay\nR∂xQ+.(5.19)\nBy Lemma 5.2 and (5.12), we have\n/integraldisplay\nR∂xQ+=σ/integraldisplayd\ndx(Qp\n+)Q−+R2\n= 2σc2\nQe−2z+R2.(5.20)\nFurthermore by (1.4), (5.4), (5.19) and (5.20), we have\n0 =z′/parenleftBig\n2α/bardblQ′/bardbl2\nL2+O(e−3\n2z+/bardbl/vector ǫ/bardblH)/parenrightBig\n+γcQe−zu(t,0)+2σc2\nQe−2z+R2+O(/bardbl/vector ǫ/bardbl3\nH+e−3z).\nTherefore we obtain (5.18).\n/square\nRemark 5.1. By (5.13), R2satisfies\n|R2|/lessorsimilar/bardbl/vector ǫ/bardbl2\nH+e−θz, (5.21)\nwhereθ= min(2 p−2,3).\nBy Proposition 5.6 and (5.21), zsatisfies\n|z′|/lessorsimilare−2z+/bardbl/vector ǫ/bardbl2\nH. (5.22)\nRemark 5.2. ToproveTheorem1.3, weonlyneed (5.22). However,byProposition\n5.6 and (5.21) we conjecture that if σ= 1,γ <−1 orσ= 0,γ <0,zsatisfies\nz∼1\n2logt.\nNext we define some notations to extract eigenmodes with real eige nvalues\naround the solitary wave. We define a+,a−,a0as\na±(t) =/integraldisplay\n(η−ν∓ǫ)φ+,\na0(t) =/integraldisplay\nη∂xQ+.26 K. ISHIZUKA\nProposition 5.7. Let/vector ube a solution of (DNKG) satisfying (5.10)for some z(t)\nandσ= 0,1. Thena±anda0satisfy\n/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndta±−ν±a±/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−2z+/bardbl/vector ǫ/bardbl2\nH, (5.23)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndta0+2αa0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−2z+/bardbl/vector ǫ/bardbl2\nH. (5.24)\nProof.As the proof of Proposition 5.6, we compute a±anda0. First, we have\nd\ndta±=/integraldisplay\n(∂tη−ν∓∂tǫ)φ+−z′/integraldisplay\n(η−ν∓ǫ)∂xφ+. (5.25)\nWe note that\n∂tη+ν∓∂tǫ=−(−∂2\nxǫ+ǫ−pQp−1\n+ǫ)−(2α+ν∓)η\n+σpQp−1\n−ǫ+γδ0u+R(t)−ν∓z′(∂xQ′\n+−σ���xQ−).(5.26)\nBy (5.16), (5.22), (5.25), and (5.26), we obtain\nd\ndta±=/integraldisplay/braceleftBig\n−(−∂2\nxǫ+ǫ−pQp−1\n+ǫ)−(2α+ν∓)η/bracerightBig\nφ++O(/bardbl/vector ǫ/bardbl2\nH+e−2z),\n/integraldisplay/braceleftBig\n−(−∂2\nxǫ+ǫ−pQp−1\n+ǫ)−(2α+ν∓)η/bracerightBig\nφ+\n=/integraldisplay\nǫ·/parenleftBig\n−∂2\nxφ++φ+−pQp−1\n+φ+/parenrightBig\n−(2α+ν∓)φ+\n=/integraldisplay\n(ν2ǫ+ν±η)φ+\n=ν±a±.\nTherefore we obtain (5.23). Second we have\nd\ndta0=−2α/integraldisplay\nη∂xQ++/integraldisplay\n(pσQp−1\n−ǫ+γδ0u+R)∂xQ+.\nBy the same argument as the proof of Proposition 5.6, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n(pσQp−1\n−ǫ+γδ0u+R)∂xQ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−2z+/bardbl/vector ǫ/bardbl2\nH.\nThus we obtain (5.24).\n/square\nBy Proposition 5.7 and |a0|+|a+|+|a−|/lessorsimilar/bardbl/vector ǫ/bardblH, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt(a±)2−2ν±(a±)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar|a±|(e−2z+/bardbl/vector ǫ/bardbl2\nH), (5.27)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt(a0)2+4α(a0)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−3z+/bardbl/vector ǫ/bardbl3\nH. (5.28)\n5.4.Energy estimates. Forµ >0 small enough we denote ρ= 2α−µ. We define\nEas\nE(t) =1\n2/integraldisplay\n{(∂xǫ)2+(1−ρµ)ǫ2+(η+µǫ)2−(pQp−1\n++pσQp−1\n−)ǫ2}−γ\n2|u(t,0)|2.\n(5.29)27\nLemma 5.8. Let/vector ube a solution of (DNKG) satisfying (5.10)for some z(t)and\nσ= 0,1. Then for t >0,\nE′(t) =−2µE(t)−(ρ−µ)/bardblη+µǫ/bardbl2\nL2−µγcQ(1+σ)e−zu(t,0)+O(e−3z+/bardbl/vector ǫ/bardbl3\nH).\n(5.30)\nProof.By formal calculation, we have\nE′(t) =/integraldisplay\n∂xǫ∂x∂tǫ+(1−ρµ)ǫ∂tǫ+(η+µǫ)(∂tη+µ∂tǫ)−(pQp−1\n++pσQp−1\n−)ǫ∂tǫ\n+p−1\n2z′/braceleftbigg/integraldisplay\n(Qp−2\n+∂xQ+−σQp−2\n−∂xQ−)ǫ2/bracerightbigg\n−γu(t,0)∂tu(t,0).\nFurthermore by (5.9) and integration by parts, we have\n/integraldisplay\n∂xǫ∂x∂tǫ+(1−ρµ)ǫ∂tǫ+(η+µǫ)(∂tη+µ∂tǫ)−(pQp−1\n++pσQp−1\n−)ǫ∂tǫ\n=/integraldisplay\n(−∂2\nxǫ+(1−ρµ)ǫ+µ(η+µǫ)−(pQp−1\n++pσQp−1\n−)ǫ)∂tǫ+(η+µǫ)∂tη,\n/integraldisplay\n(−∂2\nxǫ+(1−ρµ)ǫ+µ(η+µǫ)−(pQp−1\n++pσQp−1\n−)ǫ)∂tǫ\n=/integraldisplay\n(Ltǫ−µ(ρ−µ)ǫ+µη)∂tǫ,\n/integraldisplay\n(η+µǫ)∂tη\n=/integraldisplay\n(η+µǫ)(−Ltǫ−2αη+γδ0u+R(t)).\nBy (5.17) and (5.22), we have\n/integraldisplay\n(Ltǫ−µ(ρ−µ)ǫ+µη)∂tǫ=/integraldisplay\n(Ltǫ−µ(ρ−µ)ǫ+µη)η+O(e−3z+/bardbl/vector ǫ/bardbl3\nH),\n/integraldisplay\n(η+µǫ)(γδ0u+R(t)) =γ∂tu(t,0)u(t,0)+µγǫ(t,0)u(t,0)+O(e−3z+/bardbl/vector ǫ/bardbl3\nH).\nGathering these estimates we obtain\nE′(t) =/integraldisplay\n[{(Ltǫ−µ(ρ−µ)ǫ+µη)η}+(η+µǫ)(−Ltǫ−2αη)]\n+µγǫ(t,0)u(t,0)+O(e−3z+/bardbl/vector ǫ/bardbl3\nH)\n=−2µE(t)−(ρ−µ)/bardblη+µǫ/bardbl2\nL2+µγ(ǫ(t,0)−u(t,0))u(t,0)+O(e−3z+/bardbl/vector ǫ/bardbl3\nH)\n=−2µE(t)−(ρ−µ)/bardblη+µǫ/bardbl2\nL2−µγcQ(1+σ)e−zu(t,0)+O(e−3z+/bardbl/vector ǫ/bardbl3\nH),\nand we complete the proof. /square\nFurthermore by (5.18), we obtain\nE′=−2µE −µγc2\nQ(1+σ)2e−2z−µγcQ(1+σ)e−zǫ(t,0)\n−(ρ−µ)/bardblη+µǫ/bardbl2\nL2+O(e−3z+/bardbl/vector ǫ/bardbl3\nH)\n=−2µE(t)+C◦z′+C•e−2z−µ(1+σ)R2\n−(ρ−µ)/bardblη+µǫ/bardbl2\nL2+O(e−3z+/bardbl/vector ǫ/bardbl3\nH),(5.31)28 K. ISHIZUKA\nwhere\nC◦= 2(1+σ)α/bardblQ′/bardbl2\nL2µ, C•= 2σ(1+σ)c2\nQµ.\nWe define\nb(t) =/parenleftbigg/integraldisplay\nǫ∂xQ−/parenrightbigg2\n+/parenleftbigg/integraldisplay\nǫφ−/parenrightbigg2\n.\nLemma 5.9. Let/vector ube a solution of (DNKG) satisfying (5.10)for some z(t)and\nσ= 0,1. Then there exists C >0large enough such that for all t≥0,\nC/braceleftbig\na+(t)2+a−(t)2+a0(t)2+σb(t)/bracerightbig\n+E(t)+γ\n2|u(t,0)|2≥1\nC/bardbl/vector ǫ(t)/bardbl2\nH.(5.32)\nFurthermore if γ <0, there exists ˜C >0large enough such that for all t≥0\nE+C/parenleftbig\na2\n++a2\n−+a2\n0+σb/parenrightbig\n+e−2z\n≤˜C/braceleftbig\n(Eγ(/vector u)−(1+σ)J0(Q))+(a2\n++a2\n−+a2\n0+σb2)+e−2z/bracerightbig\n.(5.33)\nRemark 5.3. Cand˜Care independent of µ >0.\nIn detail, see [8, Lemma 2.6].\nProof.First we note that\n/a\\}bracketle{tǫ,φ+/a\\}bracketri}ht=a+−a−\nν+−ν−,/a\\}bracketle{tǫ,∂xQ+/a\\}bracketri}ht=−a0\n2α\nhold. Therefore we have\n/a\\}bracketle{tǫ,φ+/a\\}bracketri}ht2+/a\\}bracketle{tǫ,∂xQ+/a\\}bracketri}ht2/lessorsimilara2\n++a2\n−+a2\n0. (5.34)\nWhenσ= 0, by Lemma 5.1 and µ >0 small enough, there exists C >0 large\nenough such that (5.32) is satisfied.\nWeconsider σ= 1. Let χbeasmoothfunctionsatisfyingthefollowingproperties\nχ= 1 on [0 ,1], χ= 0 on [2 ,∞), χ���≤0 onR.\nForλ=|z|\n2≫1, let\nχ+(x) =χ/parenleftbigg|x−z|\nλ/parenrightbigg\n, χ−(x) = (1−χ2\n+(x))1\n2.\nWe define ǫ±=ǫ·χ±. Then we have\nǫ2=ǫ2\n++ǫ2\n−. (5.35)\nTherefore we have\n/bardblǫ+/bardbl2\nL2+/bardblǫ−/bardbl2\nL2=/bardblǫ/bardbl2\nL2, (5.36)\n/bardbl∂xǫ/bardbl2\nL2=/bardbl∂xǫ+/bardbl2\nL2+/bardbl∂xǫ−/bardbl2\nL2+O/parenleftbigg/bardblǫ/bardbl2\nH1\n|z|/parenrightbigg\n, (5.37)\n|/a\\}bracketle{tǫ−ǫ−,∂xQ−/a\\}bracketri}ht|+|/a\\}bracketle{tǫ−ǫ−,φ−/a\\}bracketri}ht|/lessorsimilare−z/bardblǫ/bardblL2, (5.38)\n|/a\\}bracketle{tǫ−ǫ+,∂xQ+/a\\}bracketri}ht|+|/a\\}bracketle{tǫ−ǫ+,φ+/a\\}bracketri}ht|/lessorsimilare−z\n2/bardblǫ/bardblL2. (5.39)\nBy Lemma 5.1, we have\n/a\\}bracketle{tLtǫ+,ǫ+/a\\}bracketri}ht+/a\\}bracketle{tpQp−1\n−,ǫ2\n+/a\\}bracketri}ht ≥c/bardblǫ+/bardbl2\nH1−1\nc/parenleftbig\n/a\\}bracketle{tǫ+,φ+/a\\}bracketri}ht2+/a\\}bracketle{tǫ+,∂xQ+/a\\}bracketri}ht2/parenrightbig\n,(5.40)\n/a\\}bracketle{tLtǫ−,ǫ−/a\\}bracketri}ht+/a\\}bracketle{tpQp−1\n+,ǫ2\n−/a\\}bracketri}ht ≥c/bardblǫ−/bardbl2\nH1−1\nc/parenleftbig\n/a\\}bracketle{tǫ−,φ−/a\\}bracketri}ht2+/a\\}bracketle{tǫ−,∂xQ−/a\\}bracketri}ht2/parenrightbig\n.(5.41)29\nFurthermore\n|/a\\}bracketle{tpQp−1\n−,ǫ2\n+/a\\}bracketri}ht|+|/a\\}bracketle{tpQp−1\n+,ǫ2\n−/a\\}bracketri}ht|/lessorsimilare−z\n2/bardblǫ/bardbl2\nL2 (5.42)\nholds. We note that by (5.35) we have\n/a\\}bracketle{tLtǫ,ǫ/a\\}bracketri}ht=/a\\}bracketle{tLtǫ+,ǫ+/a\\}bracketri}ht+/a\\}bracketle{tLtǫ−,ǫ−/a\\}bracketri}ht+O/parenleftbigg/bardblǫ/bardbl2\nH1\n|z|/parenrightbigg\n. (5.43)\nGathering (5.36)-(5.43), we obtain\n/a\\}bracketle{tLtǫ,ǫ/a\\}bracketri}ht ≥c\n2/bardblǫ/bardbl2\nH1−1\nc/summationdisplay\n∗=±/parenleftbig\n/a\\}bracketle{tǫ,∂xQ∗/a\\}bracketri}ht2+/a\\}bracketle{tǫ,φ∗/a\\}bracketri}ht2/parenrightbig\n.\nFurthermore by (5.34) and µ >0 small enough, there exists C >0 large enough\nsuch that (5.32) is satisfied.\nNext we estimate (5.33). We denote E=J0(Q). By direct computation, we\nhave\nEγ(/vector u) = (1+ σ)E−σ/integraldisplay\nQp\n+Q−+1\n2/a\\}bracketle{tLtǫ,ǫ/a\\}bracketri}ht+1\n2/bardblη/bardbl2\nL2−γ\n2|u(t,0)|2\n−/integraldisplay\n{f(Q++σQ−)−f(Q+)−f(σQ−)}ǫ\n−1\n2/integraldisplay\n{f′(Q++σQ−)−f′(Q+)−f′(σQ−)}ǫ2\n−1\np+1/integraldisplay/braceleftBig\n(Q++σQ−)p+1−Qp+1\n+−σQp+1\n−/bracerightBig\n+O(/bardbl/vector ǫ/bardbl3\nH+e−2z).\nWe note that\n/vextendsingle/vextendsingle{f′(Q++σQ−)−f′(Q+)−f′(σQ−)}ǫ2/vextendsingle/vextendsingle/lessorsimilar/parenleftBig\n|Qp−2\n+Q−|+|Q+Qp−2\n−|/parenrightBig\nǫ2\nholds. Therefore we obtain\nEγ(/vector u) = (1+ σ)E+1\n2/a\\}bracketle{tLtǫ,ǫ/a\\}bracketri}ht+1\n2/bardblη/bardbl2\nL2−γ\n2|u(t,0)|2+O(/bardbl/vector ǫ/bardblτ\nH+e−2z),(5.44)\nwhereτ=4\n2−min(2(p−2),2\n3)>2. By (5.29), we have\nEγ(/vector u)−(1+σ)E=E+µ\n2/braceleftbig\n(ρ−µ)/bardblǫ/bardbl2\nL2−2/a\\}bracketle{tǫ,η/a\\}bracketri}ht/bracerightbig\n+O(/bardbl/vector ǫ/bardblτ\nH+e−2z).\nWe recall that µ >0 is small enough and that γis negative. By (5.32), we have\nEγ(/vector u)−(1+σ)E≥1\n2E −C\n2/parenleftbig\na2\n++a2\n−+a2\n0+σb2/parenrightbig\n−C′e−2z(5.45)\nfor some C′>0. By (5.45), we obtain (5.33). /square\nNow we assume that /vector usatisfies (5.6) or (5.7). Then /vector usatisfies (5.10) for some\nz(t) andσ= 0,1. Furthermore, when /vector usatisfies (5.7), we have\n/a\\}bracketle{tǫ,∂xQ−/a\\}bracketri}ht=−/a\\}bracketle{tǫ,∂xQ+/a\\}bracketri}ht,/a\\}bracketle{tǫ,φ+/a\\}bracketri}ht=/a\\}bracketle{tǫ,φ−/a\\}bracketri}ht,\nbecauseǫis even. Thus we get the following result.\nCorollary 5.10. We assume that /vector usatisfies (5.6)or(5.7). Then there exists\nC >0such that\nC/braceleftbig\na+(t)2+a−(t)2+a0(t)2/bracerightbig\n+E(t)+γ\n2|u(t,0)|2/greaterorsimilar/bardbl/vector ǫ/bardbl2\nH.(5.46)30 K. ISHIZUKA\nFurthermore if γ <0, there exists ˜C >0large enough such that\nE+C/parenleftbig\na2\n++a2\n−+a2\n0/parenrightbig\n+e−2z\n≤˜C/braceleftbig\n(Eγ(/vector u)−2J0(Q))+(a2\n++a2\n−+a2\n0)+e−2z/bracerightbig\n.\nLetLbe large enough. We define Gas\nG(t) =E(t)+L/parenleftbig\na2\n−(t)+a2\n0(t)/parenrightbig\n. (5.47)\nBy Corollary 5.10, if γ <0 and/vector usatisfies (5.6) or (5.7), we have\ncG/bardbl/vector ǫ/bardbl2\nH≤ G+La2\n+≤CG/braceleftbig\n(Eγ(/vector u)−(1+σ)J0(Q))+a2\n++a2\n−+a2\n0+e−2z/bracerightbig\n,\n(5.48)\nfor some constants cG>0 andCG>0. In addition Gsatisfies\n|G|/lessorsimilar/bardbl/vector ǫ/bardbl2\nH+e−2z. (5.49)\nLemma 5.11. Letγ <0. We assume that /vector usatisfies (5.6)or(5.7). Then there\nexistL1>0andδ >0such that\nG′≤ −δG+L1a2\n++1\nδ/parenleftbig\ne−2z+/bardbl/vector ǫ/bardbl3\nH/parenrightbig\n. (5.50)\nProof.By (5.27), (5.28), (5.31) and (5.47), we have\nG′=−2µE −µγcQ(1+σ)e−zǫ(t,0)−(ρ−µ)/bardblη+µǫ/bardbl2\nL2\n+2Lν−a2\n−−4αLa2\n0+O(e−2z+/bardbl/vector ǫ/bardbl3\nH).\nNow we choose ˆδ >0 small so that\n2(µ−ˆδ)E(t)+2µL/parenleftbig\na2\n−(t)+a2\n+(t)+a2\n0(t)/parenrightbig\n+γ\n2|u(t,0)|2≥ˆδ|ǫ(t,0)|2,\nν−+µ <−ˆδ,−2α+µ <−ˆδ.\nWe note that the above estimate holds by (5.46). Then there exists C >0 such\nthat\nG′≤ −2µE+ˆδ|ǫ(t,0)|2+2Lν−a2\n−−4αLa2\n0+C/parenleftbig\ne−2z+/bardbl/vector ǫ/bardbl3\nH/parenrightbig\n≤ −2ˆδG+2µLa2\n++C/parenleftbig\ne−2z+/bardbl/vector ǫ/bardbl3\nH/parenrightbig\n.\nTherefore by choosing L1= 2µLandδ= min(2 ˆδ,1\nC), we obtain (5.50).\n/square\n5.5.Estimate of the instability term. In this subsection, we estimate the in-\nstability term a+.\nProposition 5.12. Letγ <0. We assume that /vector usatisfies (5.6)or(5.7). Then\nfor anyM >0, there exists TM>0such that for all t≥TM\n|a+(t)| ≤1\nM/parenleftbig\nG+e−2z/parenrightbig1\n2. (5.51)\nProof.We fixM >0. We may assume that there exist 0 < δM≪1 andTM>0\nsuch that for t≥TM\ne−z(t)+|G(t)|+|a+(t)|< δM. (5.52)\nIn addition we assume that there exists T1> TMsatisfying\n1\nM/parenleftBig\nG(T1)+e−2z(T1)/parenrightBig1\n2<|a+(T1)|.31\nWe introduce the following bootstrap estimate\n1\nM1/parenleftbig\nG+e−2z/parenrightbig1\n2≤ |a+| ≤δM, (5.53)\nwhereM1satisfies\n(M2\n1−CG)ν+−2CG−1>0, M1> M.\nWe define T2≥T1as\nT2= sup{t∈[T1,∞) such that holds (5.53) on [ T1,t]}.\nSinceδMis sufficiently small, a+satisfies for t≥TM\nd\ndt|a+|2≥ν+|a+|2. (5.54)\nFurthermore by (5.22), (5.27), and (5.28) there exists C1>0 such that\nd\ndta2\n0≤C1/parenleftbig\n/bardbl/vector ǫ/bardbl3\nH+e−3z/parenrightbig\n,\nd\ndta2\n−≤C1/parenleftbig\n/bardbl/vector ǫ/bardbl3\nH+e−3z/parenrightbig\n,\n|z′| ≤C1/parenleftbig\n/bardbl/vector ǫ/bardbl2\nH+e−2z/parenrightbig\n.\nSinceδMis small enough, we have\nd\ndt/braceleftbig\n(a0)2+(a2\n−)/bracerightbig\n≤a2\n+, (5.55)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndte−2z/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤a2\n+. (5.56)\nWe define\nI=M2\n1a2\n+−CG/parenleftbig\nE(/vector u)−2J0(Q)+a2\n++a2\n−+a2\n0+e−2z/parenrightbig\n−e−2z.\nThen by (1.2), (5.55), and (5.56) we have\nI′≥/braceleftbig\n(M2\n1−CG)ν+−2CG−1/bracerightbig\na2\n+>0.\nTherefore as long as (5.53) holds, we have\ne−2z+CG/parenleftbig\nE(/vector u)−2J0(Q)+a2\n++a2\n−+a2\n0+e−2z/parenrightbig\n< M2\n1a2\n+.\nThus by (5.48), we obtain\nG+e−2z< M2\n1a2\n+. (5.57)\nThis means that as long as (5.53) holds, we have (5.57). Therefore f orT1≤t≤T2,\n(5.57) holds and a2\n+increases exponentially. Thus a+(T3) =δMholds for some\nT3>0, which contradicts (5.52). Therefore for all t≥TM,\n|a+| ≤1\nM/parenleftbig\nG+e−2z/parenrightbig1\n2.\n/square\nWe note that since /bardbl/vector ǫ(0)/bardblHis small enough we can replace TMwith 0 in Propo-\nsition 5.12.32 K. ISHIZUKA\n5.6.Proof of Theorem 1.3. In this subsection we prove Theorem 1.3. First we\norganize some estimates of G. Gathering (5.48)-(5.51), there exist C >0 andδ >0\nsatisfying\nG′≤ −δG+Ce−2z, (5.58)\nG+e−2z∼ /bardbl/vector ǫ/bardbl2\nH+e−2z. (5.59)\nTherefore by (5.22), there exists ˆC >0 such that\nz′≤ˆC/parenleftbig\nG+e−2z/parenrightbig\n. (5.60)\nBefore we prove Theorem 1.3, we prove Proposition 5.4.\nProof of Proposition 5.4. We define w(t) =e2z(t). Then by (5.60) we have\nw′≤2ˆCGw+2ˆC. (5.61)\nSince lim t→∞w(t) =∞and lim t→∞G(t) = 0 hold, there exist ˆδ >0 small enough\nandT >0 such that for t≥T,\nw′≤ˆδw.\nTherefore we have for t≥s≥T,\nw(t)≤eˆδ(t−s)w(s). (5.62)\nBy (5.58) and (5.62), we have for t > T\nG(t)≤e−δ(t−T)+C/integraldisplayt\nTeδ(s−t)·1\nw(s)ds (5.63)\n≤e−δ(t−T)+C\nw(t)/integraldisplayt\nTe(δ−ˆδ)(s−t)ds (5.64)\n≤e−δ(t−T)+C\nδ−ˆδ·1\nw(t). (5.65)\nGathering these arguments, we have\nw′/lessorsimilar1. (5.66)\nIntegrating (5.66) on [0 ,t], there exists C⋆>0 such that for all t >0,\nw(t)≤C⋆(t+1),\nwhich implies\nz(t)−1\n2logt/lessorsimilar1. (5.67)\n/square\nProof of Theorem 1.3. By Proposition 5.4 and (1.3), we complete the proof.\n/square\nRemark 5.4. By (5.65), G/lessorsimilare−2zholds. Furthermore by (5.23), (5.24), and\n(5.59), there exists C′>0 such that/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndta−−ν−a−/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C′e−2z, (5.68)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndta0+2αa0/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C′e−2z. (5.69)33\nBy (5.68), we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftBig\ne−ν−ta−/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C′e−ν−t−2z. (5.70)\nIntegrating (5.70) on ( T,t), we obtain\n/vextendsingle/vextendsingle/vextendsinglee−ν−ta−(t)−e−ν−Ta−(T)/vextendsingle/vextendsingle/vextendsingle≤C′/integraldisplayt\nTe−ν−s−2z(s)ds\n=C′e−2z(t)/integraldisplayt\nTe−ν−s·e−ˆδ(s−t)ds\n≤C′\n−ν−−ˆδe−ν−t−2z(t),\nsince (5.62) holds. Hence we have |a−|/lessorsimilare−2z. By the same argument, we obtain\n|a0|/lessorsimilare−2z. Furthermore by |a−|+|a0|/lessorsimilare−2z, (5.46), (5.51), and (5.59), we have\nE+γ\n2|u(t,0)|2/greaterorsimilar/bardbl/vector ǫ/bardbl2\nH.\nNow we estimate E. When |u(t,0)|2≪e−2z,/bardbl/vector ǫ/bardbl2\nH/greaterorsimilare−2zholds and therefore\nE/greaterorsimilare−2zholds. On the other hand, when |u(t,0)|2/greaterorsimilare−2zholds,Esatisfies\nE/greaterorsimilar/bardbl/vector ǫ/bardbl2\nH+|u(t,0)|2/greaterorsimilare−2zsinceγ <0. Thus we obtain\nE ∼e−2z.\nIf we get the sharp estimate of u(t,0), we may improve the estimate of /bardbl/vector ǫ/bardblH.\nHowever it is difficult to get a sharp estimate of u(t,0) since a delta potential is\ndelicate.\nAcknowledgement\nThe author wishes to thank Kenji Nakanishi for many helpful comm ents, discus-\nsions and encouragementon the present paper. The author also w ould like to thank\nNobu Kishimoto, Masaya Maeda and Takahisa Inui for their helpful d iscussions.\nReferences\n[1] S. 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Math. 22(1975), 272-303.\n(K.Ishizuka) Research Institute for Mathematical Sciences, Kyoto Univer sity, Kyoto\n606-8502, JAPAN\nEmail address :ishizuka@kurims.kyoto-u.ac.jp"    },    {        "title": "2402.14442v1.X_ray_observations_of_Isolated_Neutron_Stars.pdf",        "content": "X-ray observations of Isolated Neutron Stars\nMichela Rigoselli𝑎,∗\n𝑎INAF, IASF Milano,\nVia Alfonso Corti 12, I-20133 Milano, Italy\nE-mail: michela.rigoselli@inaf.it\nPulsarsarerapidlyspinningneutronstars,thatradiateattheexpenseoftheirstrongmagneticfield\nandtheirhighsurfacetemperature. Fivedecadesofmulti-wavelengthobservationsshowedalarge\nvariety of physical parameters, such as the spin period, the magnetic field and the age, and of\nobservational properties, especially in the radio and X-ray band. Isolated neutron stars have been\nclassified according to the presence of thermal or non-thermal emission, and whether they show\na constant flux, rapid flares and bursts or long-standing outbursts. One of the current challenges\nin the study of such objects is to explain these different manifestations in the context of a unified\nevolutionary picture. On the other hand, recent findings show that the classes of isolated neutron\nstars are more connected than previously thought, and that non only magnetars hold a complex\nmagnetic field topology in the crust and above the surface.\nMultifrequency Behaviour of High Energy Cosmic Sources - XIV\n12–17 June 2023\nPalermo, Italy\n∗Speaker\n©Copyright owned by the author(s) under the terms of the Creative Commons\nAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/arXiv:2402.14442v1  [astro-ph.HE]  22 Feb 2024X-ray observations of INS Michela Rigoselli\n1. The Isolated Neutron Stars zoology\nNeutron stars have been detected for the first time as radio pulsating sources in 1967 [1], and\nin the subsequent fifty years more than 3300 have been registered 1[2]. They have mainly been\ndiscovered thanks to the detection of their pulsed non-thermal emission, at wavelengths spanning\nfrom radio to 𝛾-rays, and they can be isolated stars or members of a binary system.\nThe energy that sustains pulsar emission is supplied by their fast rotation (Section 1.1), via\nthe braking operated by their intense magnetic field, that is assumed to be dipolar. Under these\nvery simple assumptions, three characteristic quantities can be inferred from the period 𝑃and its\nderivative¤𝑃: the so-called spin-down luminosity\n¤𝐸rot=4𝜋2𝐼NS¤𝑃𝑃−3, (1)\nwhere𝐼NS≈1045g cm2is the moment of inertia, the dipolar magnetic field on the surface\n𝐵dip≈3.2×1019(𝑃¤��)1/2G, (2)\nand the characteristic age\n𝜏𝑐=𝑃\n2¤𝑃, (3)\nthat takes this form if the current period is way larger than the initial period 𝑃0.\nWith the advent of X-ray satellites and all-sky surveys, several thermally emitting isolated\nneutronstars(INSs)havebeendiscoveredatthecenterofsupernovaremnants(SNRs)(Section1.3),\nor as serendipitous sources with a very soft X-ray stable spectrum (Section 1.4), or as extremely\nflaring sources (Section 1.2).\n𝑃and¤𝑃play a fundamental role in characterizing the pulsar properties, and the neutron star\npopulation is usually represented in the 𝑃−¤𝑃diagram (shown in Figure 1), as the ordinary stars\nare represented in the Hertzsprung-Russell diagram; the different classes of INSs are placed on\ndifferent zones of the diagram.\nInthefollowing,IwillreviewthemainX-raypropertiesofINSsandthelinksthatareemerging\namong the different classes. On this topic, see also [3–6].\n1.1 Rotation-powered pulsars\nThe rotation-powered pulsars (RPPs) are the bulk of INSs and are detected because of their\npulsed emission. Nowadays more than 3000 RPPs are known, and they have been detected from\nthe radio band to the very-high energy 𝛾-rays (see e.g. the recent result on Vela pulsating emission\nat 20 TeV [8]). They also show optical/UV/X-ray thermal emission from the cooling surface and\nfrom hot spots heated by returning currents [9, and references therein].\nRPPs fill the central region of the 𝑃−¤𝑃diagram (Figure 1, black dots); a newborn pulsar\nappearsinthetop-leftcornerand,accordingtothemagnetic-dipolebrakingmodel,itevolvesalong\nthe𝐵dipconstantlines(dashed),crossingthe 𝜏𝑐constantlines(dotted). Thebottom-rightcornerof\ntheplotisemptybecausewhenpulsarscrosstheso-called“deathline”[7],theyaretoooldandtoo\nslow to maintain the required potential difference for pair production in the vacuum gap [10, 11].\n1https://www.atnf.csiro.au/research/pulsar/psrcat/.\n2X-ray observations of INS Michela Rigoselli\nFigure 1:𝑃−¤𝑃diagram of pulsars. The bulk of the pulsar population is made up of the rotation-powered\npulsars(blackdots),sub-dividedintohigh-Bpulsars(greendiamonds)andmillisecondpulsars(orangedots).\nA significant fraction of the pulsars ( 10%) are in a binary system (grey circles). The other classes of INSs\nplottedherethemagnetars(bluedots),theXDINSs(redsquares)andtheCCOs(lightbluetriangles). Lines\nofequalcharacteristicage(dotted, 104−1010yr)andequaldipolemagneticfield(dashed, 108−1014G)are\nindicated. The radio pulsar death line 𝐵/𝑃2=1.7×1011G s−2[7] is also shown. The data are taken from\nthe ATNF Pulsar Catalogue [2].\nActually, there is a population of RPPs that have 𝜏𝑐≳1Gyr that are still detected: they are\nknown as millisecond pulsars (MSPs) because of their short and stable period ( 𝑃≲16ms [12],\n¤𝑃≲10−18ss−1,seeFigure1,orangedots). Theyareoldpulsarsthathavebeenspunupthroughthe\naccretionofmatterfromthebinarycompanionatsometimeinthepast,aspartofabinarysystem.\nThe majority of them still have a companion star and they emit powerful non-thermal emission in\nthe radio and X-ray band, plus thermal X-rays from heated polar caps (see e.g. [13] for a review).\nAmong the young RPPs, there is a sub-sample of objects having magnetic fields larger than\n𝐵QED=𝑚2\n𝑒𝑐3\n𝑒ℏ≈4.4×1013G (4)\n(see Figure 1, green diamonds). They are usually referred to as high-B pulsars (see the review\n[14]), and they do not constitute a separated class of INSs, but they act as ordinary pulsars most of\nthe time, and suddenly show bursting phenomena as the magnetars (see Section 1.2) or share the\nspectral properties of the XDINS (see Section 2.1).\n3X-ray observations of INS Michela Rigoselli\n1.2 Magnetars\nMagnetars 2areINSsthathavebeendiscoveredthankstotheirrepeatedburstingactivity 3[16]\nandpulsatingemission( 𝑃∼2−12s,¤𝑃∼10−9−10−12ss−1)intheX-rayband. Itwassoonclear\nthat they could not be powered by rotation, as their X-ray luminosity exceeds ¤𝐸rotby some orders\nof magnitude, but by their very strong magnetic field [17–19] ( 𝐵dip∼1014−1015G, see Figure 1,\nbluedots). Recentreviewsonmagnetarsare[20,21],seealsoAndreiBeloborodov’scontribution.\nThe persistent X-ray emission of magnetars is characterized by soft thermal spectra ( 𝑘𝑇∼\n0.2−0.6keV) plus a power-law component in the hard X-ray band with photon index Γ =1−4,\ncommonly interpreted as resonant Compton scattering of thermal photons by charged particles\nflowing along the closed field lines of the twisted magnetosphere [22]. The continuum is often\nabsorbedbyabsorptionfeatures[23,24,e.g.] that,ifinterpretedasprotoncyclotronfeaturesprovide\na magnetic fields\n𝐵cyc,p=𝐸abs𝑚𝑝𝑐\nℏ𝑒(1+𝑧)−1≈𝐸abs\n1 keV1.3×1014G, (5)\nwhere𝑧≈0.2is the gravitational redshift.\nRecently,polarizationpatternsfromthethreebrightestmagnetarsinquiescenceweredetected\nbetween 2−8keV by the IXPEsatellite [25–27], see also Roberto Taverna’s contribution. In the\nradio band only few magnetars are detected, and they are characterized by large variability both\nin flux and pulse profile shape on timescale of days, by a very flat spectrum ( 𝛼 >−0.5) and high\npolarization [28, 29].\nRecent discoveries challenged the long standing belief that magnetars must posses supercrit-\nical magnetic fields (Eq. 4): there are radio pulsars that sometimes show bursting activity (PSR\nJ1846−0258[30],PSRJ1119 −6127[31])andthereareneutronstarsactinglikemagnetarsbutwith\nadipolarmagneticfieldlowerthan 4×1013G(SGRJ0418+5729[32],SwiftJ1822 −1606[33]and\n3XMMJ1852+0033[34]). Thistopicwasconsideredinmoredetailsin[35]. Detailedcalculations\nshowthatalocalmagneticfield >1014Gshouldbenecessarytobreakthecrust,inordertoproduce\nbursts and outbursts [36, 37]; these values can be provided by the internal toroidal components\nand higher-order multipoles [e.g. 38]. For this reason, objects with similar dipolar magnetic field\nstrength as inferred from their timing parameters can display very different behaviors.\n1.3 Central Compact Object\nThecentralcompactobjects(CCOs)formaclassofINSsthatarefoundinthecenterofyoung\n(0.3−7kyr) SNRs, that emit only in the X-rays and have no counterparts at any wavelength [39].\nA review on these objects can be found in [40].\nAt present, the class counts a dozen objects, three of which are pulsators [41–43]. The CCOs\naredetectedinthesoftX-rayrange,andtheirspectraareexclusivelythermal,withhightemperatures\n(0.2−0.5keV) and very small emitting radii (ranging from 0.1 to a few km). Two sources show\nabsorption features at 0.7−0.8keV [44, 45], that have been interpreted as electron cyclotron lines\ndue to a magnetic field of the order of\n𝐵cyc,e=𝐸abs𝑚𝑒𝑐\nℏ𝑒(1+𝑧)−1≈𝐸abs\n0.1 keV7.2×109G. (6)\n2A catalog of magnetars can be found at http://www.physics.mcgill.ca/ ∼pulsar/magnetar/main.html, see also [15].\n3See also http://magnetars.ice.csic.es/#/outbursts for a complete catalogue of outbursting magnetars.\n4X-ray observations of INS Michela Rigoselli\nThethreepulsatingCCOs(seeFigure1,lightbluetriangles)haveperiodsof 0.1−0.4sandspin\nderivatives of about 10−17s s−1[46, 47], from which weak dipole magnetic fields ( 𝐵dip∼1010G)\nandhighcharacteristicages( 𝜏𝑐∼108yr)arederived. ThisisatvariancewiththeSNRassociations,\nand the reason could be that the approximation of Eq. 3 is no longer valid because these sources\nhave𝑃≈𝑃0.\nAlso the picture of CCOs as weakly magnetized INSs has issues: the high pulsed fraction\n(up to 64% [46]) and the steep surface temperature distribution cannot be easily explained without\ninvoking magnetic fields of 1014−1015G [48] (see also Section 4.3). To explain this tension, the\n“buried B scenario” was invoked: the actual strong field is screened by the matter which fallback\nonto the star after the supernova explosion [49, and references therein]. One remarkable CCO\nin SNR RCW 103 shows a 6.7-hr X-ray periodicity of yet unknown origin as well as distinctly\nmagnetar-like behavior [50, 51]. Such long spin periods in young non-accreting objects can be\nexplained in a model where a strong magnetic field of the star interacts with a fallback disc (see\nSection 3.2).\n1.4 X-ray-dim Isolated Neutron Stars\nAmong INSs, the so-called X-ray-dim isolated neutron stars (XDINSs) represent a peculiar\nclass of nearby sources, characterized by their thermal emission in the X-ray band, with faint\noptical/UV counterparts [52] and no confirmed detection of radio pulsations [53]. Reviews on\nXDINSs can be found, e.g., in [54, 55].\nThey are seven INSs discovered in the nineties by the ROSATsatellite and soon gained the\nnickname of “Magnificent Seven”. The XDINSs have spin periods in the range 𝑃∼3−17s and\nperiod derivatives of a few 10−14ss−1(see Figure 1, red squares), which result in characteristic\nages of𝜏𝑐∼1−4Myr and magnetic fields of the order of a few 1013G.\nTheir very soft ( 𝑘𝑇≲0.1keV) X-ray spectra are well reproduced by a simple blackbody with\nlittleinterstellarabsorption,withtheadditionalpresenceofbroadabsorptionlinesat 0.2−0.4keV\nin most sources [56–59], and narrow, phase-variable ones in few cases [60, 61]. If these lines are\ninterpreted as proton cyclotron features or atomic transitions [62, e.g.], and if the variable X-ray\nradiationismodeledtakingintoaccountanisotropicthermalconductivity[63],themagneticfields\nestimatedareofthesameorderasthosederivedfromthespin-downrateassumingmagneticdipole\nbraking.\n1.5 The magneto-thermal unification\nThevarietyoftheobservationalappearanceofyoungINSsmustfindanexplanation,giventhat\ntheycannotbedifferentclasses: thesumoftheirbirthratesexceedstherateofGalacticcore-collapse\nsupernovae, which is about 1/60yr−1[62, 64].\nThe idea is to find a combination of initial distributions and evolutionary laws that allows\nto unite all known types of sources in one framework; this must also include transitions between\ndifferent types of activity and appearance of hybrid behavior. Nowadays it is believed that such\nscenariomustincludemagneticfieldevolution,namelyitsdecayandapossiblere-emergenceona\ntime scale∼104−105yr, and a significant contribution of non-dipolar fields.\n5X-ray observations of INS Michela Rigoselli\nIn a few words, the magnetic field evolves in the solid crust decaying due to the Ohmic\ndissipation and the Hall drift [65–67]. The timescales of these two processes are:\n𝜏Ohm=4𝜋𝜎𝐿2\n𝑐2≈4.4×106\u0010𝜎\n1024s−1\u0011\u0012𝐿\n1 km\u00132\nyr (7)\n𝜏Hall=4𝜋𝑛𝑒𝑒𝐿2\n𝑐𝐵≈6.4×104\u0010𝑛\n1034cm−3\u0011\u0012𝐵\n1013G\u0013−1\u0012𝐿\n1 km\u00132\nyr (8)\nwhere𝜎is conductivity and 𝐿is the characteristic length of magnetic field variations.\nOn the other hand, the surface temperature lowers because of the neutron star cooling. Given\nthat these two quantities are related to each other and evolve together [68], they have to be studied\nin a comprehensive auto-consistent model, called “magneto-thermal evolution” [69, and reference\ntherein]: temperature affects crustal electrical resistivity, which in turn affects magnetic field\nevolution,whilethedecayofthefieldcanproduceheatthatthenaffectsthetemperatureevolution.\nThereisquitegeneralagreementthatthemagneticfieldofrelativelyyoungneutronstarsdecays\nontimescalesof 104−106years,dependingontheconductivity,thicknessofthecrust,andstrength\nand structure of the initial field [70]. On the other hand, population synthesis studies [71] suggest\nthatoldpulsarsshownosignificantmagneticfielddecayovertheirlifetime(timescaleslargerthan\n109−1010years [72]). This issue remains somewhat open, but can be tentatively solved by the\nassumptionthatthemagneticfieldismaintainedbytwocurrentsystems: longlivingcurrentsinthe\nsuperconducting core support the large scale dipolar field and are responsible for the spin down of\nold pulsars, while currents in the crust support the short living part of the field [73].\nRecent X-ray observations provided new evidences of link between the classes of INSs, and\nchallenge the current theoretical models.\n2. Links between XDINS and RPP\nThelinkbetweenXDINSandRPPhasbeenreinforcedwhenaradiohigh-Bpulsarwasobserved\nsharingtimingandspectralpropertieswiththeXDINSs(Section2.1),andwhenweaknon-thermal\nemission was detected in the cumulative X-ray spectra of two XDINSs (Section 2.2).\n2.1 The case of PSR J0726 −2612\nPSR J0726−2612 is slowly rotating ( 𝑃=3.4s), highly magnetized ( 𝐵dip=3×1013G)\nradio pulsar: its timing parameters are in the range of those of the XDINSs, but it does show radio\npulsations. ThesimilaritywiththeXDINSswasreinforcedbyX-rayobservationswiththe Chandra\nsatellite[74],thatrevealedasoftthermalspectrumandpulsationswithasinusoidal,double-peaked\nprofile with a pulsed fraction of 30%.\nRigoselli et al. 2019 [75] analyzed deep XMM-Newton data of PSR J0726−2612 and reported\nthat the soft X-ray spectrum can be well fit by two blackbodies (temperatures 𝑘𝑇1=74+6\n−11eV and\n𝑘𝑇2=140+40\n−20eV, radii𝑅1=10.4+10.8\n−2.8𝑑kpckm and𝑅2=0.5+0.9\n−0.3𝑑kpckm) plus a Gaussian line in\nabsorption with the line placed at 𝐸=0.39+0.02\n−0.03keV and with a broadening of 𝜎=0.08+0.03\n−0.02keV.\nThe inferred magnetic field is 𝐵≃5×1013G (see Eq. 5) is in good agreement with the dipole\nmagnetic field.\n6X-ray observations of INS Michela Rigoselli\nAssuming a distance of 1 kpc, the luminosity of PSR J0726 −2612 is𝐿=(4.0+4.4\n−1.0)×1032\nergs−1. This is greater than its spin-down luminosity, as for the XDINSs, but is in reasonable\nagreement with the expected thermal luminosity of a ∼200kyr-old pulsar.\nRigoselli et al. 2019 [75] also tried to model the double-peaked pulse profile, but no simple\nmodel based on blackbody emission could reproduce the observed pulse profile. Whatever the\nmechanism responsible for the surface emission, the presence of a strong magnetic field results in\nsome degree of anisotropy in the emitted radiation. In the case of a magnetized atmosphere, more\ncomplicated energy-dependent beaming patterns are produced. The best match with the data was\nobtainedassumingemissionfromtwoantipodalhotspotswithaneffectivetemperatureof0.5MK,\nand(Ω−𝜇)≈(Ω−LOS),whereΩand𝜇aretherotationandthemagneticaxes,respectively,and\nthe LOS is the direction of the line of sight.\nWhilemostoftheXDINSshavesingle-peakedpulseprofiles,twoofthem(RXJ0720.4 −3125\n[76]andRXJ1308.6 +2127[77])showdouble-peakedprofilessimilartoPSRJ0726 −2612,although\nwith smaller pulsed fractions ( 18%and11%, respectively). The inferred geometries for these two\nobjects are(Ω−𝜇)≈90◦and(Ω−LOS)≈45◦; with the usual assumption that the radio beam\ncoincideswith,oriscloseto,themagneticdipoleaxis,suchalargeimpactparametercannaturally\naccount for the fact that their radio emission is not visible from the Earth. Contrary to the two\nXDINSs, PSR J0726 −2612 should lie inside this region and this might explain why it is detected\nin the radio band, while the two XDINSs are not (see Figure 2, left panel).\nFigure 2: Left panel: Visibility of a radio beam with aperture of ∼8◦as a function of the angles between\nΩ−𝜇,andΩ−LOS.TheestimatedpositionsforPSRJ0726 −2612,RXJ0720.4−3125andRXJ1308.6 +2127\nare shown (see Section 2.1). Adapted from [75]. Right panel: Total X-ray spectrum of RX J1856.5 −3754\nobtained with XMM-Newton data (see Section 2.2). The best-fitting model with two blackbody (blue and\nred) and a power-law (green) components is superimposed in the top panel. The two bottom panels show\nthe residuals for the two-blackbody plus power law fit (2BB+PL) itself and for the two blackbody (2BB) fit,\nwhichyieldunsatisfactoryresidualsat 𝐸 >1keV.Onthecontrary,theresidualsbelow0.6keVareatalevel\n<2%, well below the instrument effective area uncertainties. Adapted from [78].\n7X-ray observations of INS Michela Rigoselli\n2.2 The case of RX J1856.5 −3754\nThesoftX-raypulsarRXJ1856.5 −3754isthebrightest( 𝐹𝑋≃1.5×10−11ergs−1cm−2[79])\nand closest ( 𝑑=123+11\n−15pc, [80]) member of the XDINS class. Its pulsation, at 𝑃=7.05s [81]\nand¤𝑃=3×10−14ss−1[82], was detected despite a very small pulsed fraction PF≃1.2%. The\ntiming parameters yield ¤𝐸rot=3.4×1030ergs−1,𝜏𝑐=3.7×106yr and𝐵dip=1.5×1013G. Its\nX-ray spectrum resembles a pure blackbody emission with temperature 𝑘𝑇≃60eV, even though\ntheemissionintheopticalband( 𝑉∼25.7[83])liesabovetheextrapolationofasingle-temperature\nX-ray blackbody at low energies.\nThe brightness, simple spectrum, and steadiness of its emission make RX J1856.5 −3754 an\nidealtargetforthecalibrationof XMM-Newton ,thatobserveditabouteverysixmonthssince2002.\nUsing all the 2002–2022 data from the XMM-Newton EPIC-pn camera, De Grandis et al. 2022\n[78] obtained a 0.3−7.5keV spectrum having 1.43 Ms of net exposure time, that showed a hard\nexcess with respect to the pure blackbody emission [see also 84, 85]. They report a best fit made\nup of the sum of two blackbodies (temperatures 𝑘𝑇1=61.9±0.1eV and𝑘𝑇2=138±13eV, radii\n𝑅1=4.92+0.04\n−0.06kmand𝑅2=31+8\n−16m)andapowerlaw(photonindex Γ=1.4+0.5\n−0.4,fluxinthe 2−8\nkeV band of(2.5+0.7\n−0.6)×10−15ergs−1cm−2(see Figure 2, right panel).\nTheluminosityofthenon-thermalcomponentcorrespondsto 10−3timesthespin-downpower\n¤𝐸rot. This value is consistent with what is observed in rotation powered X-ray pulsars with higher\n¤𝐸rot[e.g.86],sothatamagnetosphericoriginforthiscomponentappearsasthemostnaturaloption.\nHintsforpulsationsabove2keVwerereported,andthiswouldreinforcethemagnetosphericorigin\nand the link to the RPPs.\nDeGrandisetal.2022[78]analyzedalso20yearsof XMM-Newton dataofRXJ0420.0−5022,\ntheXDINShavingthesecondhighest ¤𝐸rotoftheclass. Theyfoundahardexcessthatcanbefiteither\nwith a second blackbody of a power law, or their sum. Also in this case, the putative non-thermal\ncomponent would have an efficiency 𝐿PL/¤𝐸rot∼10−3. Can we expect that the other five XDINSs\nhave a non-thermal component too? If we assume a similar efficiency and we consider the ¤𝐸rotof\ntheremainingstars,theexpectedX-rayfluxshouldbe 𝐹PL≲10−16ergs−1cm−2. Thislevelofflux\nis beyond the sensitivity of the current facilities, but maybe Athenacould achieve it 4.\n3. New (?) flavors of INSs\nThankstoX-ray(Section3.1)andradio(Section3.2)currentfacilities,newflavorsofINShave\nbeendiscovered. Theeffortisnowtomakethemfitintotheclassicalmagneto-thermalevolutionary\nmodel.\n3.1 Thermally emitting INSs\nCalvera (1RXS J141256.0 +792204) is an enigmatic X-ray pulsar with properties that do not\nfit easily with those of the known classes of INSs. It was discovered in the ROSATall sky survey\nas a soft X-ray source with high X-to-optical flux ratio, qualifying it as an INS candidate [87]. Its\nspectral properties (thermal X-ray only, absence of radio or 𝛾-ray counterpart) resemble those of\nXDINSs (Section 1.4) and CCOs (Section 1.3), and for these reasons it was initially considered as\n4https://www.cosmos.esa.int/web/athena/study-documents.\n8X-ray observations of INS Michela Rigoselli\na possible new member of the “Magnificent Seven” class and nicknamed “Calvera”. However, it\nwas later discovered that Calvera has a spin period of 59 ms [88] and is spinning down at a rate\n¤𝑃=3.2×10−15ss−1[89]. Thesetimingparametersgiveacharacteristicage 𝜏𝑐=2.9×105yrand\na dipole magnetic field 𝐵dip=4.4×1011G, similarly to middle-aged RPPs.\nA magnetized hydrogen atmosphere model, covering the entire star surface and having an\nanisotropictemperaturemap,providesagooddescriptionofthephase-resolvedspectraandenergy-\ndependentpulsedfraction[90]. Theinferreddistance 𝑑≃3.3kpc,coupledwithaGalacticlatitude\n𝑏=+37◦, provide an unusually high height above the Galactic disk ( 𝑧≃3kpc). This supports\nthe idea that Calvera was born in the Galactic halo, most likely from the explosion of a run-away\nmassive star or, possibly, in a more unusual event involving a halo star, such as, e.g., the accretion\ninduced collapse of a white dwarf.\nZaneetal.2011[88]reportedthepresenceofdiffuseX-rayemissionabout13′westofCalvera,\nwithspectralpropertiesconsistentwithaSNR;recently,radio[91]and 𝛾-ray[92,93]counterparts\nof this remnant were discovered. These findings might imply that Calvera is younger than inferred\nfrom its timing parameters. The refined measurement of the proper motion (coming soon) could\nconfirm or discard the association between Calvera and this supernova remnant.\nThe storage of several hundreds of ks of XMM-Newton andChandra pointings, and the in-\ncreased sensitivity of the all sky survey in the soft X-ray band provided by eROSITA allowed\nus to discover new INS candidates even in the absence of pulsations. This could be caused by\nan unfavorable orientation of the rotation and magnetic axes, or by the intrinsic lack, or faint-\nness, of non-thermal magnetospheric emission. The most promising INS candidates are 2XMM\nJ104608.7−594306 [94, 95], 4XMM J022141.5 −735632 [96, 97], eRASSU J065715.3 +260428\nand eRASSU J131716.9 −402647 [98], but with the data release of the first eROSITA catalog we\nexpect a significant enlargement of the sample [99, 100].\n3.2 Long Period Pulsars\nFinally, there is a new emergent class of pulsars characterized by a long period: PSR\nJ2251−3711[𝑃=12.1s,101],PSRJ1903+0433[𝑃=14.0s,102],PSRJ0250+5854[𝑃=23.5s,\n103],andPSRJ0901 −4046[𝑃=75.9s,104]. TheyarelocatednearmagnetarsandXDINSsonthe\n𝑃−¤𝑃diagram,thereforetheyarecharacterizedbyhigh 𝐵dip∼1013−1014Gandlong𝜏𝑐∼107yr,\nbuttheyareradiosourceswithoutanyX-raycounterpart[105,106]. Anevenmoreexoticobjectis\nwithoutanydoubtsGLEAM-XJ162759.5 −523504.3(hereafterGLEAM-X),thatshowedpulsating\nradio emission with a period of ∼18minutes [107]. The period derivative is not measured, yet,\npreventingrobustdeterminationofthesourcenature,buttheupperlimitof ¤𝑃<2×10−9ss−1sets\nan incredibly strong 𝐵dip<3×1016. Differently from the others long period pulsars observed so\nfar, GLEAM-X is a transient source with a very variable flux, showing periods of “radio outburst”\nlasting a few months, a 90% linear polarization, and a very spiky and variable pulse profile. The\nproperties of GLEAM-X resemble in some sense those of the 6.7-hr pulsating source at the center\nof SNR RCW 103, see Section 1.3.\nThisnew(?) classofradiopulsarchallengesourcomprehensionofradioemissionmechanism,\nas they lie well below the “death line” obtained in the standard framework of inner vacuum-gap\n9X-ray observations of INS Michela Rigoselli\ncurvature radiation 5, see Figure 1, and they have a radio luminosity ≳¤𝐸rot, indicating that the\nemission is not generated purely by spin down but that an additional source of energy (magnetic?)\nis needed. The upper limits on the X-ray luminosity confirm that they are rather old neutron stars\nand do not discard the similarities with the magnetar family [108].\nThelocationofthesepulsarsonthe 𝑃−¤𝑃diagramishardtoreconcileintheclassicalmagneto-\nthermal evolution, because such strong magnetic fields would decay in timescales smaller than the\ninferred𝜏𝑐(for fields∼1015G,𝜏Hallcan be as small as ∼100yr, see Eq. 8). Possible solutions\nare either real ages of these objects are much smaller than their characteristic ages due to large\ninitial spin periods, or some mechanism prevents the crustal magnetic field from decaying in time\n[see109,andrefrencestherein]. Associationswithwhitedwarfsinsteadofneutronstarshavebeen\nconsidered and discarded [108], but not because of the spin periods: white dwarfs with 𝑃≲70s\nhave been found both in binary systems [110, 111] and as isolated sources [112].\n4. Multipolar magnetic field\nIn this section I will report the growing evidences that the magnetic field has complicated\nfield lines (multipolar and toroidal components) in the crust and above the surface also in ordinary\nRPPs: they concern the polar cap size (Section 4.1), the presence of absorption features in the soft\nX-rayband(Section4.2),andthethermalsurfacemapthatcanbeinferredfromtimingandspectral\nanalysis (Section 4.3).\n4.1 Polar cap size\nWhentheneutronstarhasmorethanonemillionyears,thermalX-raysareproducedfromhot\nspots, which are heated by backward-accelerated magnetospheric particles. The relation between\nnon-thermal and thermal emission mechanisms is of particular interest for the study of particle\nacceleration in the pulsar magnetosphere [113–116].\nIn the classical dipolar scenario, the hot spots are identical and antipodal, are situated in\ncorrespondence of the magnetic poles and have a size 𝑅dip=√︁\n2𝜋𝑅3∗/𝑃𝑐≈145𝑃−1/2\n1 sm. Several\nXMM-Newton andChandra observations have been devoted to measure the characteristics of the\nhot spots of the brightest old pulsars. Currently, there are only eight pulsars older than 1 Myr with\nahotspotemissiondetectedwithenoughconfidence,whileinthemajorityofcasesthepresenceof\na thermal component is still under debate [117–119]. The emitting radii, inferred with blackbody\nfits, are a few meters only, and not all the pulse profiles are symmetric.\nIt was proposed [120, 121] that such a small emitting areas can be explained if we consider\nthat, close to the star surface, the magnetic field is stronger than the dipolar field and described by\nmultipolar components:\n𝐵PC=𝐵dip\u0012𝑅dip\n𝑅PC\u00132\n≈102𝐵dip (9)\nand the conservation of the magnetic flux through the polar cap area explain why we observe\n𝑅PC∼10−40m. The asymmetry of the pulse profile also denotes that the geometry of the\n5Although this tension can be reconciled if a space-charge-limited flow model with a multipolar magnetic field\nconfiguration is considered [104].\n10X-ray observations of INS Michela Rigoselli\nmagneticfieldclosetothesurfacemaybeoff-centred,andthiswouldimplytwocapsnon-identical,\nnon-antipodal, and out of phase with respect to the radio main peak. Recent results obtained with\nNICER[122–125] also report evidences for non-dipolar magnetic fields, but they refer to the older\nMSPs that have 𝐵dip∼109G.\nThesediscrepanciescanalsobeaddressedusingmorerealisticthermalmodels,accountingfor\nthe effects of magnetic field on either a condensed surface or an atmosphere. It is well known that\natmospheric models are harder than blackbodies, thus yielding best-fit temperatures a factor about\ntwo lower and, as a consequence, larger emitting areas. For three pulsars, the latter are consistent\nwith the dipole polar cap (PSR B0950 +08 [126], PSR B0823 +26 [127], PSR B0943 +10 [128]).\nMoreover,differentlyfromtheisotropicblackbodyemission,themagneticfieldimprintsabeaming\nontheemergentradiation,evenifthesurfacetemperatureisuniform,andtheresultingpulseprofiles\narenaturallynon-symmetrical[75]. However,thesemodelsdependonmanymoreparametersthan\nasimpleblackbody,suchasthechemicalcomposition,thestarmassandradius,themagneticfield,\nthe system geometry, that in most cases are unknown and are difficult to constrain in these rather\nfaint old pulsars.\n4.2 Absorption lines\nUsually, cyclotron features are detected in the (thermal) X-ray spectra of particular classes\nof INSs (Sections 1.2, 1.3, 1.4), that have the appropriate magnetic field to produce electron or\nprotonabsorptionlines,respectively. Onthecontrary,fortheordinarypulsars(with 𝐵dip∼1012G)\nneither of this conditions is satisfied at the star surface and hence no analogous cyclotron features\nare expected in soft X-rays (Eqq. 5 and 6).\nNevertheless,inthelastfewyearssomeordinaryRPPs(PSRJ1740 +1000[129],PSRB1133 +16\n[117],andPSRB0656 +14[119])showedevidenceforthepresenceoffeaturesintheirX-rayspectra\nat about∼0.5keV. The case of PSR J1740 +1000 is extremely intriguing because, after a glitch\noccurredin2012,theX-rayspectrumhaschangedandtheabsorptionlinesseemtohavedisappeared\n[130].\nIftheelectronsareresponsibleforsuchfeatures,theymustbelocatedhighinthemagnetosphere,\natseveralstellarradiiabovethestellarsurfacewherethedipolefieldisweaker( 𝐵∝(𝑟/𝑅)−3). Thus,\ntheX-rayphotonscanbeproducedonthesurfaceorinthemagnetosphere. Alternatively,ifthelines\nareattributedtoprotonsclosetothestarsurface,thenamagneticfield 𝐵=4.5×1013G≈10×𝐵dip\nisneeded. Thisisnotimplausibleifweconsiderthat,onthesurface,thedipoleapproximationmay\nbenolongervalid,butasourceofsoftX-rayphotonsonthesurface,withanextensionappropriate\nfor this context (i.e. 𝑅PC=𝑅dip/√\n10≈40m, see Eq. 9), is needed.\n4.3 Thermal emission\nThe phase-resolved spectral analysis of the X-ray thermal component can help to interpret\nthe temperature map of the neutron star surface, that is directly influenced by the magnetic field\nstructure. The bulk of thermal energy is stored in the star interior, that is essentially isothermal,\nand arises from the envelope, that is only 100 m thick and has a density up to 1010g cm−3. In\nthe presence of a strong magnetic field, the electron transport in directions perpendicular to the\nfieldisheavilysuppressed,whileitisenhancedparalleltothefield,resultinginananisotropicheat\ntransport and a non-uniform temperature distribution on the surface [154, e.g.].\n11X-ray observations of INS Michela Rigoselli\nFigure3: Pulsedfractionvsage(leftpanel)andratioofthetwoemittingradiivsratioofthetwotemperatures\nderivedfrom2BBfitsofthermalspectra(rightpanel). TheblackdotsrepresentRPPs(#1PSRJ0108 −1431\n[131],#2PSRB0823+26[127],#3PSRB0943+10[128],#4PSRB0950+08[126],#5PSRB1929+10[132],\n#6 PSR J0007+7303 [133], #7 PSR J0357+3205 [134], #8 PSR J0538+2817 [135], #9 PSR J0633+0632\n[136],#10PSRJ0633+1746/Geminga[137],#11PSRB0656+14[119],#12PSRB0833 −45/Vela[138],#13\nPSR B1055−52 [130], #14 PSR J1357 −6429 [139], #15 1RXS J141256.0 +792204/Calvera [90], #16 PSR\nB1706−44[140],#17PSRJ1740+1000[130],#18PSRJ1741 −2054[141],#19PSRB1822 −09[142],#20\nPSR J1836+5925 [143], #21 PSR J1957+5033 [144], #22 PSR J2021+4026 [64]), the green dots represent\nhigh-B pulsars (#23 PSR J0726 −2612 [128], #24 PSR J1119 −6127 [145], #25 PSR J1819 −1458 [146]),\nthe red dots represent XDINSs (#26 RX J0420.0 −5022, #27 RX J0720.4 −3125, #28 RX J0806.4 −4123,\n#29 RX J1308.6+2127, #30 RX J1605.3+3249, #31 RX J1856.5 −3754 and #32 RX J2143.0+0654, [147]),\nthe light blue dots represent CCOs (#33 RX J0822.0 −4300 [45], #34 1E 1207.4 −5209 [44], #35 CXOU\nJ185238.6+004020 [148], #36 1WGA J1713.4 −3949 [149]; #37 XMMU J172054.5 −372652 [150]; #38\nXMMU J173203.3 −344518 [151], #39 CXOU J232327.9+584842, [152]). Both panels show the overall\nexpected regions for an isotropic emission and a dipolar magnetic field [153] (in magenta) and the hot spots\nfrom returning currents (in orange).\nTherearetwoquantitiesthataredirectlycorrelatedtothetemperaturemap: thepulsedfraction\nofthethermalcomponent,andtheratiobetweenthemultiplecomponentsthatareneededtofitthe\nthermalspectrum. BothofthesequantitiesarerepresentedinFigure3forallthethermallyemitting\nRPPs (black), high-B pulsars (green), XDINSs (red) and CCOs (light blue) known.\nThe left panel of Figure 3 shows the pulsed fraction as a function of the INS age. There is\nno correlation between the PF and the age, nor between the INS class. While for the older ( >106\nyr) RPPs the high pulsed fraction can be explained with the small size of the emitting area and a\nfavorable orientation of Ω,𝜇and the LOS (orange area), there are at least four objects younger\nthan 104yr (PSR J2021+4026 [64], PSR J1119 −6127 [145], PSR J1357 +6429 [139] and PSR\nJ1852+0040 [148]) for which we expect that the thermal emission comes from a large fraction\nof the cooling surface. In case of isotropic emission, several authors [75, e.g.] showed that the\nmaximumPFexpectedshouldbe ≲0.3(magentaarea),eveniftheemittingareaissmallerthanthe\nentiresurface[155]andevenifthetemperaturedistributionisinfluencedbyadipolarmagneticfield\n[153]. We need a non-isotropic emission mechanism, as in the case of magnetized atmosphere,\nor a highly inhomogeneous temperature distribution, that can be produced only with multipolar\n12X-ray observations of INS Michela Rigoselli\ncomponent and/or toroidal component [156–158].\nTherightpanelofFigure3showsinsteadalltheINSswhosethermalspectrumcanbefittedwith,\natleast,twoblackbodies,thatmimicthetwoextremesofamorecomplextemperaturedistribution.\nTherefore in this plot all the old RPPs are excluded because the surface has cooled down and it\nis not longer visible in the X-rays. The figure shows the ratios of emitting radii ( 𝑅hot/𝑅cold) and\ntemperatures ( 𝑇hot/𝑇cold), values that are model and distance independent.\nIt is very striking to observe that, for all these INSs except three (Geminga [137], RX\nJ0420.0−5022andRXJ1856.5 −3754[Section2.2]),theratio 𝑅hot/𝑅coldisintherange 0.03−0.3.\nThusitisdifficulttointerpretthehotterandcoldercomponentsintermsofemissionfromhotspots\nand from the whole surface ( 𝑅≲200m and𝑅∼12−16km, respectively, see orange area). We\nalso note that all different INS classes are placed in the same area of this space parameter: this\nmeansthat,despitethedifferences,theyhaveasimilarthermalmap. Thisisindeedquitesurprising\nsincethesurfacetemperatureisexpectedtoevolveintimeandtobesensitivetotheinitialmagnetic\nfieldconfigurationinthestarcrust(see[159]forareview). Yakovlev2021[160]andRigosellietal.\n2022 [130] showed that the simplest case of a blackbody emission with a temperature distribution\nresulting from a dipolar magnetic field [153] gives values of the 𝑅hot/𝑅coldand𝑇hot/𝑇coldratios\nclearly inconsistent with the observed ones (magenta area).\n5. Final remarks\nRecentfindingsshowthattheXDINSclassisprobablymorevariegatethanpreviouslythought:\ntwo of the softer objects show non-thermal emission, that could have a magnetospheric origin as\nfor the ordinary pulsars, while the absence of radio emission that was the main difference between\nXDINSs and RPPs could be explained by orientation effects.\nTheanalysisoftimingandspectralX-raydataofdifferentclassesofINSssupportthemounting\nevidence that ordinary RPPs may have complicated thermal maps and magnetic fields capable to\ninduce cyclotron absorption in the keV energy range, as already seen for XDINSs and magnetars\nin quiescence. 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N., Nature Astronomy 5(2021) 145.\n17"    },    {        "title": "2402.14444v1.Laser_patterning_of_magnonic_structure_via_local_crystallization_of_Yittrium_Iron_Garnet.pdf",        "content": "  \n1 \n Title : Laser patterning of magnonic structure via local crystallization  of Yittrium Iron \nGarne t \n \nA. Del Giacco, F. Maspero,  V. Levati,  M. Vitali, E. Albisetti, D. Petti, L. Brambilla, V. \nPolewczyk, G.Vinai, G. Panaccione, R. Silvani, M. Madami, S. Tacchi,  R. Dreyer, S. R. Lake, \nG. Woltersdorf, G . Schmidt, Riccardo Bertacco*  \n \nAndrea Del Giacco, Federico Maspero, Valerio Levati, Matteo Vitali, Edoardo Albisetti, \nDaniela Petti, Riccardo Bertacco * \nDipartimento di Fisica, Politecnico di Milano, Via G. Colombo 81 , 20133 Milano (Italia)  \n* E-mail: riccardo.bertacco@polimi.it  \n \nLuigi Brambilla  \nDipartimento di Chimica, Materiali e Ingegneria Chimica Giulio Nat ta, Politecnico di Milano, \nP.za L. da Vinci 32, 20133 Milano (Italia)  \n \nVincent Polewczyk, Giovanni Vinai, Giancarlo Panaccione  \nIstituto Officina dei Materiali del CNR (CNR -IOM), SS. 14, km 163,5, 34149 – Trieste, Italy  \n \nRaffaele Silvani, Marco Madami  \nDipartimento di Fisica e Geologia, Università di Perugia, Via A. Pascoli , 06123 Perugia, \nItaly  \nSilvia Tacchi  \nIstituto Officina dei Materiali del CNR (CNR -IOM), Unità di Perugia, Via A. Pascoli , 06123  \nPerugia, Italy  \n \nRouven Dreyer, Stephanie R. Lake, Georg Woltersdorf, Georg Schmidt  \nMartin Luther University Halle -Wittenberg, Institute of Physics, Von -Danckelmann -Platz 3, \n06120 Halle (Saale), Germany  \nGeorg Schmidt also:  \nMartin Luther University Halle -Wittenberg, Interdisziplinäres Zentrum für \nMaterialwissenschaften, Heinrich -Damerow Straße 4, 06120 Halle (Saale), Germany  \n \n   \n2 \n This is the pre -peer reviewed version of the following  article:  \n\"Patterning  magnonic  structures via laser induced crystallization of  Yittrium Iron  \nGarnet\", by A. Del Giacco et al.  https://doi.org/10.1002/adfm.202401129 , \nwhich has been accepted for publication in Advanced Functional Materials. This article may \nbe used for non -commercial purposes in accordance with Wiley Terms and Conditions \nfor Use of Self -Archived Versions.  \n \n \nKeywords: Yittrium Iron Garnet, magnonics, laser annealing  \n \nAbstract  \n \nThe fabrication and integration of high -quality structures of Yttrium Iron Garnet (YIG) is \ncritical  for magnonics. Films with excellent properties are obtained  only on single crystal \nGadolinium Gallium Garnet (GGG) substrates  using high -temperature processes . The \nsubsequent realization of magnonic  structures  via lithography and etching is not straightforward \nas it require s a tight control of the edge roughness , to avoid magnon scattering , and \nplanarization in case of multilayer devices . \nIn this work we describe a different approach based on  local laser annealing of amorphous YIG \nfilms , avoiding the need for subjecting the entire sample to high thermal budgets and for \nphysical etching . Starting from amorphous and paramagnetic YIG films grown by pulsed laser \ndeposition at room temperature  on GGG , a 405 nm laser is used for patterning arbitrary shaped \nferrimagnetic structures  by local  crystallization.  In thick films (160 nm) the laser induced \nsurface corrugation prevents the propagation of sp in-wave modes  in patterned conduits . For \nthinner films (80 nm) coherent propagation is observed in 1.2 m wide conduits displaying  an \nattenuation length  of 5 m which is  compatible with a damping coefficient of about 5· 10-3. \nPossible routes to achieve damping coefficients compatible with state -of-the art epitaxial YIG \nfilms  are discussed .   \n \n1. Introduction  \nMagnonics  is nowadays considered a promising technology for implementing wave -computing \nstrategies and high -frequency analog signal processing. [1] The much shorter wavelength of \nspin-waves with respect to free -space propagation of electromagnetic waves paves the way to \nminiaturization of devices working in the GHz and THz regime .[2][3][4][5][6] Non-linear \nphenomena can be easily exploited to modulate spin -wave transmission and implement logic, \ncomputing or signal processing functionalities. [7][8][9][10][11] Furthermore, spin currents \nassociated to coherent magnon propagation can be used to efficiently drive domain -wall motion,   \n3 \n thus opening the way to the local storage of the information carried by spin waves. [12] \nNevertheless some bottlenecks still hamper the development of a mature magnonic  technology \nwith industrial applications. High insertion losses, the need for bulky electromagnets to bias \ndevices and the difficulties related to the fabrication of devices with long propagation length \nfor SWs ar e the main obstacles to be overcome . \nThe low damping coefficient ( ) of Yttrium Iron Garnet (YIG) is the main reason for the wide \nusage of this materials in magnonic structures. Record values of 3·10-5 and 6.5·10-5 have been \nreported for bulk single crystals and thin films grown on Gadolinium Gallium Garnet  (GGG) \nsubstrates. [13][14] Epitaxial films grown by liquid phase epitaxy on GGG, can reach values of \n on the order of 1· 10-4 . [15][16] Similar values are found in films deposited by Pulsed Laser \nDeposition on GGG, displaying  as low as 2·10-4.[17][18][19] Record values of  =5.2·10-5 \nhave been recently reported for 75 nm YIG films grown  by sputtering  on GGG  followed by \npost-annealing up to 900°C. [20] \nNoteworthy, these numbers can be achieved only on GGG(111) single crystal substrates, due \nto the extremely small lattice mismatch which promotes pseudomorphic growth.  So far , \nattempts to integrate high quality YIG films on silicon using optimized templates failed in \nreproducing the quality of films grown on GGG single crystals.  In the best case ,  on the order \nof 2·10-3  are reported  for YIG films on Si/SiO x.[21] Furthermore, processes for the synthesis \nof epitaxial YIG films  require high temperatures  ( > 600°C) , either during the growth or post \nannealing, which are not compatible with the thermal budget acceptable for integration on \ncomplementary metal oxides semiconductor ( CMOS ) platforms.  \nAnother critical aspect to be considered  is the fabrication of YIG micro or nanostructures for \nspin wave propagation. Conventional top -down processes based on lithography and etching \nmust be carefully optimized to avoid YIG degradation and the introduction of extrinsic magnon \nscattering by edge roughness. [22] In addition, for relatively thick structures, planarization \nprocesses are needed to avoid the formation of steps in the metallic antennas at the edges of the \nconduits or in case of devices requiring additional overlayers . Some solutions to overcome these \ndifficulties have been proposed. Nanochannels for SW propagation have been demonstrated in \nYIG via dipolar coupling to ferromagnetic metal nanostripes. [23] Recently, a method for the \nfabrication of free -standing YIG structures has been reported , based on a post -growth annealing \ninducing the crystallization of three -dimensional amorphous structures defined by room -\ntemperature PLD and lift off using electron beam lithography with only one or two contact \npoints with the GGG substrate .[24] Low values of damping coefficients (2 ·10-4) have been \nreported, essentially due to the expulsion of dislocation at the bended portions of the suspended   \n4 \n structures in proximity to the anchor points on the GGG substrate. The crystallized suspended \nstructures can be released from the GGG substrate and transferred on another substrate such as \na silicon wafer, in view of integration with CMOS electronics. [25] \n \nHere  we propose a different method for patterning magnonic structures with arbitrary shape \nembedded in a  continuous film of amorphous YIG, by local  laser-induced  crystallization. Under \nthe right conditions of irradiation, the area exposed to the laser beam undergoes a phase \ntransition from the amorphous (paramagnetic) to the crystalline (ferr imagnetic) phas e which \nallow s for spin wave propagation. Arbitrary shaped ferrimagnetic magnonic structures can be \npatterned by scanning the laser beam according to a p re-defined layout, without need of \nphysical etching of the paramagnetic YIG out of the patterned area and avoiding the exposure \nof the entire sample (wafer) to a high thermal budget  not compatible with  integration.  Our \nmethod is complementary to other s starting from epitaxial YIG and producing a local  change \nof the magnetic properties, e.g. via ion -irradiation, so as to introduce a sort of grey -scale \nmagnonics. [26] However, we stress here that in our case we don’t need any physical fabrication \nstep of pre -patterning as the local crystallization intrinsically defines the geometry of the \nmagnonic structure  embedded in a non -magnetic amorphous film .  \nThe paper is organized as follows.  We first describe the  laser writing  technique by pointing out \nthe critical parameters influencing the performances  of patterned structures. Then we report on \nthe optimization of the writing parameters with reference to two YIG  thickness es (160 and 80 \nnm), showing that only for 80 nm the laser annealing gives rise to corrugation -free patterned \nareas with  uniform magnetic properties  leading to a well -defined FMR precession  mode . \nFinally,  we demonstrate  the suitability of the proposed technique for patterning magnonic \nstructures by investigating spin -wave propagation in rectangular conduits . For conduits \npatterned in 80 nm thick YIG films we found an attenuation length on the order of 5 microns, \ncompatible with a damping coefficient of about 5·10-3. Possible routes to overcome critical  \nissues currently limiting the propagation length are discussed . \n  \n2. Experimental results  \n \n2.1 Critical  parameters  for laser patterning  \nLaser patterning is performed using the Nanofrazor Explore apparatus by Heidelberg, featuring \na 405 nm laser , with minimum spot size 2·w0 = 1.2 m (where w0 is the  waist  radius) , which is \nshone  onto a substrate mounted on an interferometric stage. A sketch of the process is shown    \n5 \n in Figure 1. The intensity  (I) at the sample surface can be tuned by varying both the laser power \n(P) and focus distance  (z), corresponding to the sample height with respect to that ensuring the \noptimal focus (minimum spot size  w0). The pattern is created using a raster scanning mode, \nwhose essential parameters are: (i) pixel size dx,y (minimum step of the grid used for the image \ndiscretization), (ii) pixel time p (exposure time for each point in the grid). In this work dx=dy \nso that, from now on, we will just use a unique value for the pixel size: d p=dx=dy. The laser is \noperated in the continuous mode, so that for a given intensity  the dose (D) and time-dependent \nintensity  (I(t))  for each pixel size in a line of the  pattern can be easily evaluated (see Methods  \nfor details ) as a function of the writing parameters . \nOf course, the local maximum temperature and time profile strongly depend on the optical and \nthermal properties of film and substrate. In the present case, at 405 nm wavelength the photon \nenergy  (3.06 eV) is higher than the gap of YIG and smaller than that of GGG ( 2.8 eV and 5.6 \neV, respectively ) so that efficient energy absorption takes place only in the volume of the YIG \nfilm. [27][28][29] Since  the thickness  of films under consideration  is smaller ( 80-160 nm) than \nthe photon penetration depth  in YIG at 405 nm (about 0.2 m) crystallization is expected to \nstart from the GGG substrate, acting as a template for the achievement of single phase structures \nwith (111) orientation.  Noteworthy , the maximum temperature reached in the film during the \nlocal heating cycle also depends on the film thickness. For film thickness es lower than the \nabsorption length  and transparent substrates, as in our case,  the maximum temperature increases \nwith thickness as more heat is absorbed in the film while dissipation channels are unchanged , \napart from the lateral heat flow which is not so relevant in  this thickness regime . [30]  \nFinally notice that the shape of the patterned area also influences the laser induced \nmodifications in the film.  The ultimate dose in each pixel depends on the superposition of the \nspots along a single writing line but also on the superpositions of adjacent lines spaced by dy. \nFor the same laser power pixel s of a simple one -dimensional line undergo  a single heating cycle, \nwhile pixels of a true two -dimensional patterned area undergo  multiple annealing due to the \nsuperposition of adjacent writing lines. With the same laser writing conditions,  we thus expect \na much stronger impact in circles, squares  or wide conduits t han in narrow  conduits . This must \nbe taken into account when comparing data taken with different experimental techniques on \npatterned areas adapted to the specific experiment.  \n \n2.2. Optimization of patterning conditions  \nA careful investigation of the impact of the different parameters on the capability of inducing a \nlocal crystallization has been carried out by creating 2D arrays of circular and square dots with   \n6 \n diameter  and side of 20 m, respectively, spanning different ranges of writing parameters along \nrows and columns.  Their properties have been investigated by optical microscopy, microRaman , \nelectro n back scatter diffraction (EBSD),  and AFM. In the following we present this analysis \nfor two representative thickness, 160 nm and 80 nm, corresponding to two distinct regimes: (i) \na high-thickness regime where the laser irradiation induces a peculiar  corrugation with \nassociated spatial modulation of the magnetic properties, (ii) a low-thickness regime where said \ncorrugation disappears and the patterned areas  display more uniform  magnetic properties .  \n \n2.2.1 . 160 nm thick films  \nIn Figure 1b, we show a typical 2D array created during the optimization of the writing \nparameters , where we scanned the laser power  in the 25 -55 mW range on the horizontal axis \nand dp, between 10 and 90 nm, on the vertical axis . The other parameters were  p=70s, z= 0.  \nIn these conditions the line-dose (average intensity) var ies from 0.23 to 0.51 mJ/ m2 (13.8 to \n30.3 mW/ m2) when going from left to right in the first row of Figure 1b and from 0.23 to \n0.026 mJ/ m2 (13.8 to 13.6 mW/ m2) when going from top to bottom in the first column, while \nthe average duration t of the heating cycle  (constant in each row) goes from 17 to 3.4 ms when \nmoving from the first to the last row. We report this case as it turned out that the power and \npixel size are the more effective parameters inducing changes in the local crystal properties . \nThe change in color in the optical microscopy images under white light illumination, from grey \nto pink and finally  light purple , is associated to amorphous -crystalline  phase transition,  as \nconfirmed by microRaman  (see below) . Interestingly enough, in the array of Fig. 1b we notice \nthat the induced crystallization is not always uniform in the dot. For some combinations of \npower and pixel size (e.g. 45mW - 40 nm) we observe that only the upper part of the dots is \ncrystallized , corresponding to the light -purples regions with a tip pointing to  the bottom . This \nis more evident in Fig. 1c, where we report some patterns obtained for P= 47.5 mW, dp=30 nm, \np=70 s, representing a sort of “threshold condition”  for crystallization . The phase transition \nstarts at the apex of the tip  which corresponds to the initial nucleation center for crystallization \n(randomly placed in the different dots and probably corresponding to defects increasing local \nabsorption) and then propagates towards the upper part, as the raster scan of the laser proceeds \nfrom left to right and from the bottom to the top. Above th is threshold , however, the \ncrystallization is more uniform in the dot , as indicated by the quite uniform light-purple  color \nobserved in optical microscopy on the right upper part of the array of Figure 2a.  \nIn Figure 1b we observe that local crystallization is favored at high laser power and small pixel \nsize, as these conditions  lead to a higher average laser intensity (𝐼𝑎𝑣=𝐷/∆𝑡) during the heating   \n7 \n cycle (see Methods ). As the crystallization  is driven by thermally activated  nucleation and \ngrowth of crystal grains, the process is mainly determined by the temperature profile during \nlaser heating. The evaluation of the actual temperature evolution in our YIG films requires  a \ncareful modelling starting from accurate estimates of the thermal and optical parameters of \namorphous  or crystalline YIG and GGG,  [31] which is beyond the scope of this work. Here we \njust point out that the maximum temperature  is more directly related to  the average intensity  \nthan to the dose . For this reason, I av can be used as an empirical parameter describing the \nefficiency of laser induced crystallization  for a fixed shape of the patterned area , in agreement \nwith the trend of Figure 1b . This is confirmed also by the fact that  we did not find a sizable \ndependence of crystallization on the pixel time p, as Iav does not depends on p even though  \nthe dose  depends linearly  on it (See Methods ). \n \nFigure 1 d shows Micro -Raman spectra from dots written in the same conditions of those of the \nfirst row of Fig. 1b. As the laser power increases above 40 mW, i.e. in conditions leading to a \nclear transition towards a light -purple color in the optical images, a YIG T 2g feature  (C) at about \n187.5 cm-1 appears , which is a fingerprint of crystalline YIG. [32] The prominent T2g peak s \n(A,B) at 168.5 and 178.5  cm-1 are instead arising  from the GGG substrate . The intensity of the \nYIG T 2g peak  (C), normalized to that of the adjacent GGG T 2g peak  (B), is shown in the inset \nof Figure 1d. These intensities have been  calculated by subtracting from  each spectrum a \nreference from an amorphous YIG film  and fitting the curve with three peaks to obtain the \ncorresponding amplitudes.  We observe a  relative increase of the weight of crystalline YIG with \nrespect to the signal from GGG up to  60 mW . For higher laser power the average intensity is \nso high that crystallization is accompanied by other phenomena, leading to  evident \nmorphological transformation (see Figure 2)  and chemical modifications (Figure 3) .  \nAn intriguing phenomenon observed in 160 nm thick films  of YIG is the creation  of a peculiar \ncorrugation in areas patterned above threshold, i.e. in conditions leading to crystallization. A \nregular ripple appears with parallel valleys always perpendicular to the writing directions, as \nreported in Figure 1e,f,g, where the AFM topography of square areas patterned with a writing \ndirection at 45, 0 and 90 degrees with respect to the [1-10] direction of GGG is shown . This \ncorrugation appears only upon crystallization, as demonstrated in Figure 1e, where we can \ndistinguish the full square area first patterned in the raster -scan mode with P=45 mW ( dp=30 \nnm, p=70 s, z=0 ) and then with a second partial raster scan starting from the top -left corner. \nFor this threshold value of writing power, below which we do not observe crystallization and \nabove which crystallization becomes deterministic, t he regular ripple appears only in the region   \n8 \n patterned twice , where the second localized annealing is capable to induce  full crystallization .  \nThe effect of the first irradiation is still visible in the AFM topography  as a reduction of the \nfilms thickness by 5 nm due to the increase of YIG density upon  laser annealing , but we have \nno evidence for crystallization as no magnetic signal was detected in this area (see below) . Even \nthough the front of the fully crystallized area is not linear , most probably due to the non -uniform \ndistribution of nucleation centers, the ripple is quite regular, with a peak-to-peak distance  of \nabout 1 micron and a n amplitude of 10-20 nm. The same kind of corrugation  is observed a lso \nin figures 1f,g where the writing direction is differently oriented with respect to the GGG crystal \naxes and we used a power level well above threshold ( P=55 m W), while the other parameters  \nare unchanged. Noteworthy, magnetic force microscopy (MFM) (see figure 1, panels h,i,l) \nindicates that a magnetic contrast is found only in corrugated areas, thus confirming  that the \nripple is associated to crystallization, as amorphous YIG is paramagnetic. Because the MFM is \nperformed in the lift mode at 200 nm heig ht, we can rule out a strong influence of morphology \non the magnetic signal, which must instead be ascribed to the stray field from magnetic domains \ncorrelated with the corrugation . This is fully consistent with the magnetic dynamics observed \nin 160 nm thick films, showing a non -uniform behavior  which can be associated with the \nsuperposition  of magnetic excitations originating from the spatial corrugation  (see section \n2.3.1 ). The appearance of this ripple in laser irradiated films has been already reported for other \nmaterials  and is usually  ascribed to the formation of capillary waves due to continuous melting \nand crystallization of the film during irradiation. [33] The observed corrugation is similar to  \nLaser Induced Periodic Surface Structures which are mainly investigated using fs -lasers but can \nbe also produced with continuous lasers leading to quasi -periodic ripples with low spatial \nfrequency on the order of the laser wavelength. [34][35]  \nA more detailed analysis of the film transformation induced by laser irradiation at increasing \npower  is reported in Figure 2 . In the top row  we present optical an d AFM images showing the \nmorphology of the patterned areas  on YIG films with 160 nm thickness , in the 40 -110 mW \npower range. Below 50 mW the main effect is a laser cleaning of the sample surface which \npresented some spurious particles . At 47.5 mW  (Fig. 2a) we are at threshold so that only a \nsecond laser scan can induce the formation of a clear ripp le, as reported in Figure 1 e,h. From \n50 to 60 mW (Fig. 2b) we see the appearance of a long-range  corrugation decorated with some \ncrystallites of about 200 nm lateral size and 10 nm height, whose areal density increases with \npower. At 70 mW (Fig. 2 c) we observe a fully developed ripple with perfect coalescence of \ncrystallites and the appearance of elongated protrusions of about 10 -15 nm height perpendicular \nto the laser writing directions, mainly parallel to each other but with some bifurcations ,   \n9 \n separating regions with very low roughness (< 1 nm rms) . These are probably the result of the \nformation of capillary waves during the solidification of molten material upon irradiation .[33] \nFrom 80 to 100 mW (Fig. 2d and 2e corresponding to 80 and 95 mW) we see the appearance \nof another regime, with holes having increasing depth (50 -100 nm) which tend to align  in rows . \nAt 110 mW instead  (Fig. 2f) , we see some elongated bumps with typical height of 200 nm,  \nalways presenting some preferential orientat ion. X-ray absorp tion spectroscopy (XAS) on \nirradiated areas of the 160 nm thick film (see Figure 3) reveals that also at 100 mW, the Fe L 2,3 \nedge is clearly visible, thus signaling  the presence of YIG. However, the line -shape of the L 2,3 \nfeatures reported in Figure 3 clearly shows an evolution of the oxidation state of Fe  with \nannealing. At 40 mW and 60 mW the XAS spectrum is identical to that of the as -deposited \namorphous film, displaying a ratio between the A and B peaks of the L 3 edge which is on the \norder of 0.5, in nice agreement with the expected value of 0.46 for nominal YIG. [37]. At 80 \nmW this ratio increases while at 100 mW it becomes  larger  than one, th us indicat ing the \npresence  of Fe2+, as confirmed by the appearance of the pre -edge feature of L 2 (indicated with \nan arrow in figure 3) in the spectra corresponding to 80 mW and 100 mW. [38] This can be \nattributed to oxygen vacancies formation at the high temperatures reached in thick YIG films \nirradiated with 80 -110 mW, where we observe a sort of de -wetting of the GGG substrate and \nYIG accumulation on GGG terraces, associated to a major chemi cal change of the film which \nprobably involves a surface tension modification.  \nTo summarize this part, writing conditions preserving  the YIG stoichiometry  and maximiz ing \nthe crystallinity for 160 nm thick YIG correspond to a pixel size of about 30 nm and a laser \npower of 60-70 mW, but in this condition the film develops a ripple which results in magnetic \ninhomogeneity.  \n \n2.2.2. 80 nm thick YIG  \nMoving to smaller thickness, namely 80 nm, the formation of the characteristic ripple discussed \nabove is strongly suppressed. The AFM and optical images of amorphous YIG films 80 nm \nthick , irradiated  in the same conditions as for  160 nm thick films , are presented in the bottom \nrow of Figure 2, for a direct comparison. It is immediately cl ear from the optical microscope \nimages that in this case the irradiated areas with power up to 130 mW never display the same \nblack contrast as the patterns written with 110  mW in 160 nm thick YIG . All transition s are \nsomehow shifted to higher power s. The AFM images shown in panels 2g,h,i, corresponding to \n60, 80 and 100 mW of lase r power, just show some crystallites with 5 -10 nm height on a flat \nsurfac e and  some spurious particles already present on the  surface of the film before patterning .   \n10 \n At 110mW we see the appearance of some holes with depth of about 80 nm, compatible with \nthe YIG thickness, which tend to align to the GGG terraces as it happens in the case of thicker \nfilms (see panels 2d,e). At 130 mW the roughness  definitely increases and we see some YIG \nbumps with 200 nm height similar to those found for 160 nm thick YIG and reported in panel \n2f. \nThe observed shift at higher laser power of the  phase transitions  is not surprising as in this \nthickness range, much smaller than the absorption length, the absorbed power is essentially \nproportional to the film thickness while thermal dissipation channels are unchanged, so that a \nlower temperature is expected for the same laser power  in 80 nm thick YIG .[30] On the other \nhand, the formation of the ripple is not simply delayed but mainly suppressed, due to the \nmodification of the peculiar laser absorption, thermodynamic  and hydrodynamic conditions \nleading to this phenomenon in thin films. [33][35]   \nAlso for 80 nm thickness we checked by  microRaman  that above 70 mW of writing power, the \ncharacteristic peak of crystalline YIG at about 187.5  cm-1 appears , even though the reduced \nthickness makes the signa l less intense (data not shown). Overall, the laser induced \ncrystallization of 80 nm thick YIG requires a higher laser intensity with respect to thicker films, \nbut the topography of patterns  is much more uniform. Furthermore, EBSD (see Figure 5 below) \nshows that a uniform crystallization can be achieved within a patterned area with 20 m \ndiameter , thus indicating that thin films are good candidates for an effective excitation and \npropagation of magnetic perturbations.  \n \n2.3 Magnetization dynamics in patterned dots  \nTo assess the magnetization dynamics in patterned dots on 160 and 80 nm thick YIG films we \ndesigned a frequency -resolved Magneto Optical Kerr Effect  (Super-Nyquist sampling MOKE  \nor SNS-MOKE ) experiment with cw RF-excitation provided by a coplanar wave guide \n(CPW). [39] Different groups of circular dots with 20 µm diameter have been patterned with \ndifferent writing conditions in the gap region of the CPW (see figure 4a). A RF-current is fed \nin the CPW to excite magnetization dynamics within the dots via an out -of-plane RF-field in \nthe GHz range and SNS -MOKE is used to sample the  amplitude and phase information of the  \ndynamics with its diffraction limited spot size of 300 nm.  \n \n2.3.1 Magneti zation  dynamics of structures patterned in 160 nm thick YIG   \nIn Figures 4d,e,f,g,h,i the  locally obtained  Kerr signals proportional to the real and imaginary \npart of the dynamic magnetic susceptibility  are presented  versus the in -plane bias field . In the   \n11 \n measurements the magnetic field is aligned  perpendicular to the signal line of the CPW  while \na fixed RF frequency of 2.001 GHz  is applied . We investigated  patterned  dots which correspond \nto different power  levels  (50, 60, 70, 80, 90, 100 mW) in an amorphous film with 160 nm \nthickness as a function of the magnetic bias field applied in plane . The other parameters used \nfor patterning are p=70 s , d p=30 nm, z=0.   Below 60 mW no magnetic signal  has been \ndetected , in fair agreement with Raman spectra showing the appearance of a sizable T2g YIG \npeak only at 60 mW (see Figure 1). Starting from 60 mW of laser power some  coherent  \nmagnetic signal s appear, with increasing intensity up to 80 mW, while above 80 mW the MOKE \nsigna l vanishes again. This is consistent with the XAS analysis presented above, showing a \nsubstantial modification of the film chemistry starting to develop above 80 mW, with the \nappearance of a prominent Fe2+ signal. However, also from 60 to 80 mW  laser power, the line \nshape  of the dynamic MOKE signal is quite complex and does not display a unique resonance, \nas could be expected for a homogeneous  dot excited close to its ferromagne tic resonance \nconditions  (about 22 mT for an excitation frequency of 2 GHz ). In fact,  the spatially resolved \nphase images  reconstructed from the real and imaginary part data  in Figures 4b,c from a dot \npatterned  with 80 mW, taken at 2 GHz  with 23 and 27 mT, show a quite inhomogeneous  phase  \ndistribution which is not compatible with a uniform FMR mode.  In both cases the phase of the \nsignal  varies in the range of 2𝜋 across the patterned dot.  Noteworthy some regions with uniform \nphase aligned vertically, along the same direction of the ripple  described above as the writing \ndirection here is horizontal  (see the left part of Figure 4c ). Clearly the FMR un iform mode is \nsuppressed by  such morphological and magnetic non-uniformity.  \nThis picture is confirmed by micro -focused Brillouin Light Scattering (micro -BLS) \ninvestigation of  SWs  propagation  in magnonic conduits patterned in 160 nm thick films by laser \nirradiation with typical parameters  leading to sizable ripple, as discussed below . \n \n2.3.2 Magneti zation  dynamics of structures patterned in 80  nm thick YIG  \nIn a second set of experiments, we repeated the measurements shown in the previous section \nfor a 80 nm thick YIG layer. The real and imaginary parts of the magnetic susceptibility \nmeasured by SNS-MOKE  are shown in Figures 5a -f, as a function of in -plane field for different \nlaser power levels between 55 and 80 mW . As in the last set of measurements, a n RF-frequency \nof 2 GHz has been applied to the CPW. The other writing parameters were p=70 s , d p=30 \nnm, z=0.  A tiny signal with the shape of a single resonance appears at 60 mW, while at 65 mW \nthe clear signature of FMR resonance is visible. Here, the real (dispersive) and imaginary \n(dissipative) parts of the MOKE signal, proportional to the magnetic susceptibility,  show the   \n12 \n 90 degrees phase shift  expected for FMR . As the writing power becomes larger, the signal starts \nto deviate from the ideal FMR line shape , while at 80 mW it appears as the superposition of \nmultiple modes.  This has a counterpart in the spatial ly resolved phase images  presented in the \ninset s of panels 5c and 5f. While in the case of 65 mW the phase of the MOKE signal is \nconsistent with the FMR uniform mode, for the dot patterned at 80 mW we clearly see that the \nphase is highly non -uniform , possibly arising fro m the superposition of higher order modes \ncoupled to some sample inhomogeneity.  The analysis of the FMR curves can be used to estimate  \nindividual linewidths as well as  the saturation magnetization and the  Gilbert  damping \ncoefficient  in a frequency dependent manner . Figure 5g reports the FMR frequency vs the in -\nplane applied field for the dot patterned with 65 mW, together with a Kittel  fit; by assuming  \n= 28 GHz/T we find that 0Ms is equal to 0.183±0.001 T, in nice  agreement with nominal \nvalues for YIG. F rom the linear fit of the corresponding linewidth of the resonance vs applied \nfield shown in Figure 5h we can extract the damping  = (5.8±0.4)·10-3. As we compare this \npatterned structure with the one irradiated with 75 mW, we obtain just a slight ly larger 𝛼 = \n(6.3±0.4 )·10-3, as depicted in Fig. 5i. However, by investigating  the zero -frequency linewidth \n𝜇0Δ𝐻0 for these two irradiation powers we see a marked  increase , from 34.5±30.7 µT to \n174.2±41.2 µT as we enhance the laser power from 65 mW to 75 mW. These values are \nobtained in the low -frequency regime up to 3.2 GHz since for higher frequencies the \ninterpretation of distinguishable Lorentzian peaks becomes challenging due to the onset of \ndifferent spin wave modes superimposing the quasi -uniform mode. T he enhanced \ninhomogeneous line broadening for larger laser intensities is in agreement with the observation  \nof the inhomogenous phase distribution in the dot patterned with 75 mW (see Figure 3f) arising \nfrom laser induced roughness or even tiny damages in the YIG layer.  \nInterestingly enough, the dot patterned  with 65 m W and displaying an ideal FMR lineshape \nappear s essentially black in the EBSD image of Figure 5j, thus indicating  the absence of a well -\ndefined crystal orientation. The fact that it displays a magnetic resonance, however, rules out \nthe possibility that it is just amorphous, as we never measured  a SNS-MOKE signal on the \nunpatterned film , displaying a paramagnetic behavior.  We conclude that for 65 mW the film is \npolycrystalline , with small  magnetic crystal grains randomly oriented. Also  at 75 mW, when \nthe FMR resonance is well visible, the EBSD image ( Figure  5k) shows a polycrystalline \ncomposition, with a few  small  areas having  well-defined  orientation. A uniform (111) \norientation of YIG, compatible with an epitaxial film on  the GGG(111) substrate, is found only \nin the dot patterned with 80 mW, which displays a superpositio n of modes  in MOKE analysis. \nHowever, i n this  case the presence of different  spin wave  modes  superimposing the uniform   \n13 \n response  prevents the  determination of a Gilbert damping parameter 𝛼 due to indistinguishable  \nresonance lines  in the spectrum.  \n \n2.4 Spin wave propagation in patterned magnetic conduits  \nTo assess  the potential of our technique for fabricating magnonic structures  we investigated the \npropagation of SWs excited by a RF -antenna in conduits patterned by local laser annealing . For \n160 nm thick YIG films the patterning creates a characteristic ripple  which strongly affects  SW \npropagation  (data not shown) . SWs display a sort of channeling in regions aligned to the ripple, \nwith a longer propagation distance when the valleys of the ripple are aligned to the wave vector \nof the excited spin waves. However, in the best case the propagation length is on the order of 1 \nmicron, so that the ripple appears as a severe obstacle to the p ropagation of SWs. These findings \nsuggests that the corrugation is associated to a magnetic inhomogeneity in agreement with SNS -\nMOKE measurements of section 2.3.1.  \nBetter results are obtained on 80 nm thick YIG films , as in this case we can achieve full \ncrystallization without sizable corrugation . To avoid  the formation of the characteristic ripple \nperpendicular to the writing direction seen in thicker films, we first patterned el ongated conduits \nwith 1.2 m width  and 100 m length  in the 80 nm thick YIG film , using a raster -scan mode \nwith longitudinal writing lines .  \nDifferent patterning conditions have been investigated, by varying the focus and the laser power, \nwhile keeping d p=30 nm and t p=70 s fixed . By changing the focus around the optimal \ncondition (z = 0) by ± 2 m we did not find a sizable change apart from what is expected from \nthe increase of the beam spot size, leading to a decreased intensity and thus to the need to \nincrease the power to retrieve similar results with respect to the optimal focusing conditions.  \nThe impact of the laser power is summarized in Figure 6. Panel 6a shows  the EBSD map taken \non a group  of four conduits written with decreasing power, from 100 to 8 5 mW. The central \nporti on of the conduit displays a single crystal structure with (111) orientation, corresponding \nto epitaxial YIG on GGG(111), whose width decreases with the laser power until it disappears \nat 85 mW.  Noteworthy, there is a second effect of laser annealing which is visible also at 85 \nmW. This is evident in Figure 6b, where the EBSD signal (color scale) is super imposed to  the \nsecondary electron microscopy ( SEM ) morphological signal  (grey scale) . A rectangular conduit , \ncontaining the crystallized area and whose wid th scales with the laser power,  clearly emerges \nfrom the surroundings, most probably reflecting an amorphous -polycrystalline phase  transition. \nDue to the gaussian profile of the laser beam, only in the central part of the conduit the laser \nintensity overcomes the threshold for full crystallization while on the sides there are two stripes   \n14 \n where the intensity is still enough to promote the creation of polycrystalline grains with random \norientation. Of course, the width of these regions scales with the laser power , as the same \nthreshold is achieved for larger distances from the center of the conduit  at higher power . This \npicture is corroborated by the fact that the total width of the conduit seen in SEM images is \nlarger than the beam spot -size, thus indicating that the tails of the laser beam play a role.  \nIn order to investigate SW propagation m icro-stripline  antennas were fabricated on both sides \nof the conduits, with groups of 4 conduits crossed by the same stripline connecting the signal \nand ground of the RF circuit in the ground -signal -ground ( GSG ) configuration. SW dynamics \nhave been studied by means of micro -BLS measurements applying an external magnetic field  \nof μ0Hext= 100 mT parallel  to the antenna and to the short axis of the conduits , corresponding  \nto the Damon Eschbach (DE) geometry.  \nFirst micro-BLS characterization was performed measuring the SW intensity as a function of \nexcitation frequency  in the range between 3 and 12 GHz . A micro-BLS signal is not detected \nfor the conduits written using a power from 40 mW to  70 mW. Interestingly enough, for \nconduits written with a laser power in the 75 -85 mW  range , where a crystalline phase with well -\ndefined orientation is not observed in the EBSD , a sizable  BLS signal  is found . This confirms \nthat also the polycrystalline phase is magnetic and can  sustain SW propagation , as already \ndemonstrated by SNS-MOKE measurements . On the other hand, we find that the lowest  power \nfor measuring  a BLS signal in these conduits (75 mW) is larger than that needed (65 mW) to \nobserve a sizable  signal in SNS-MOKE experiments on YIG films with the same thickness \n(Figure 5). This can be explained by the different shape of the patterned area: the total local  \ndose in a dot with 20 𝜇𝑚 diameter is larger than that in a conduit with 1.2 𝜇𝑚 nominal width, \ncorresponding to the laser spot size , due to the multiple superposition of writing lines in the \nraster -scan mode . We thus expect that the threshold condition for crystallization can be \nachieved at lower laser power in a large dot with respect to a narrow  conduit.  \nIn BLS spectra we  observe a single peak whose frequency  is about 4.3±0.3 GHz  for writing \npower in the 75-105 mW range and decreases to about 3.6 GHz for a power of 110 and 120 \nmW (Fig.6 h ) As it can be seen in Fig. 6f, where the micro -BLS intensity measured at fixed \nfrequency (4.22 GHz) along the transverse x-axis with a step size of 250 nm for the 100 mW \nsample  is reported , this peak correspond s to the fundamental mode of the conduits .  \nTwo dimensional maps of the BLS intensity of the fundamental mode acquired at fixed \nfrequency as a function of the distance from the antenna  over an area of about 3×1 2 μm2 with \n250 nm step size for the conduits written at 100, 95 and 90 mW are plotted in Figures 6c,d,e, \nrespectively. As it can be seen  a micro -BLS signal is  detectable up to a distance of  about 12   \n15 \n m.  The decay length for the SW intensity, has been estimated from the fit of the micro -BLS \nintensity profile taken along the y direction at the center of the mode, as shown in figure 6g for \nthe conduit patterned with 100 mW, using  the equation  𝐼(𝑦)=𝐼1exp (−2𝑦\n𝜆𝐷)+𝐼0, where  𝑦 is \nthe position along the waveguide, 𝐼1 the initial intensity , and I0 the offset intensity .   \nAs visible  in Fig. 6 h the large st values of the decay length (approx imately 5 m) are observed \nfor the conduits  patterned with a laser power between 90 and 105 mW,  while the decay length \nslightly decreases for lower writing power, consistent  with the decreasing width of the single \ncrystal phase in the central portion of the conduit s observed in EBSD measurements . Much \nsmaller values of  the decay length (down to 1.3 m) are found  for the conduits patterned with  \n110 and 120 mW. This, together with the decrease of the frequency of the fundamental mode, \nsuggests a major degradation of the magnetic properties at high laser power which can be \nascribed both to the chemical modifications observed by XAS and to an increased density of \ndefects and edge c orrugation due to grain boundaries. As a matter of fact, EBSD on conduits \npatterned at 120 mW (data not shown) indicates a larger width of the inner crystal region \nassociated to some interruptions of the conduits, probably arising from local re -amorphization \nupon over -heating, similarly to what happens in phase -change materials.  Notice that the \nchemical modification seen by XAS in Figure 3 are already evident at 100 mW, where we \nmeasured  the maximum attenuation length . This apparent inconsistency results fr om the fact \nthat the laser induced modification does not only depend on the laser power, but also on the \nshape of the patterned area. F or XAS experiments we patterned square pads with 50 m side, \nso that the effective heating for the same nominal power was larger than in case of 1.2 m wide \nconduits due to the superposition of multiple adjacent lines in the raster scan mode used for \nwriting.  \n \n3. Discussion   \nUpon optimization of the writing conditions, good crystalline quality and uniform magnetic \nproperties have been achieved only in thin YIG films (80 nm), as thicker films (160 nm) tend \nto develop a morphological corrugation which is associated also to a mag netic non -\nhomogeneity. Well-defined FMR lines were  found by SNS-MOKE in dots patterned in 80 nm \nthick YIG films, from which a damping parameter  = (5.8±0.4)·10-3 has been estimated , with \na low zero -frequency linewidth of (34.5±30)  µT, and a saturation mag netization \n0Ms=(0.183±0.001)T corresponding to Ms = (146±0.7)  kA/m . Spin waves  propagation in the   \n16 \n DE configuration has been observed by micro -BLS in 1.2 m wide conduits  patterned on 80 \nnm thick YIG,  with an attenuation length of about  5 microns in optimized conditions.  \nTo correlate these experimental findings, we simulated the SWs spectrum for a YIG conduit \nhaving a thickness of 80 nm and a width of 1.5 m (larger than the nominal width of 1.2 m \nbut corresponding to the width of the crystallized region reported in Fig. 5b for the conduit \nwritten with 100 mW) using nominal values for the saturation magnetization  and exchange \nstiffness:  Ms =146 kA/m (in agreement with SNS -MOKE) and A=0.42 10-6 erg/cm . The \nsimulated frequency of the uniform mode turns out to be 4.4 GHz, in nice agreement with the \nexperimental values  (4.3±0.3 GHz for writing power in the 75 -105 mW range) . This indicates  \nthat YIG structures patterned with optimized conditions  are characterized by a saturation \nmagnetization and exchange constant within the typical range for high quality Y IG films . \nThe damping parameter within patterned structures can be also extracted from micro -BLS data \nand compared to that estimated by SNS -MOKE . First,  we calculated the SWs group velocity \n𝑣𝑔 at k=0 from the simulated dispersion curve of the fundamental SW mode in the YIG \nwaveguide, obtaining a value of 683 m/s. Using the relation 𝜆𝐷=𝑣𝑔\n𝛼𝜔  and the experimental \nvalues for the decay length and frequency measured for the sample irradiated at 100mW, we \nobtain  = 5.610-3 in good agreement with that found by SNS -MOKE  (5.8±0.3 10-3). This \nvalue is above the best values reported for epitaxial films grown on GGG by liquid -phase \nepitaxy and PLD [13-17] or free -standing structure obtained by re -crystallization [22], both in \nthe low 10-4. However , this work represents just a first proof -of-concept indicating the \nfeasibility of local patterning of YI G by laser induced local crystallization , while there is still \nplenty of room for further optimization.   \nHere we have shown that, b y properly tuning the critical writing parameters suitable conditions \nfor inducing the crystallization without altering the stoichiometry can be found, thus inducing \nthe transition towards a polycrystalline  or single -crystal phase , which can sustain magnetic \nexcitations. Noteworthy, t hese conditions strongly depend on the Y IG thicknes s. In thick films  \nthe crystallization is associated to the development of a ripple  which is detrimental to the \ncoherent propagation of spin waves  but could be engineered to tune the YIG anisotropy and \nexploit spin -wave channeling . In thinner films shapes with flat topography and uniform \nmagnetic properties sustaining coherent propagation of spin -waves  can be patterned.  Next steps \nrequire a careful investigation of the quality of patterned crystalline structures , which can be \nobtained in YIG films even t hinner than 80 nm and in writing condition avoiding sizable \nsuperposition of adjacent writing lines in the currently used raster -scan mode for writing   \n17 \n arbitrary shapes. A fine analysis of the edge roughness of patterned areas is another key \nparameter to avoid extrinsic scattering of spin waves during propagation in conduits. Finally, \ndifferent combination of doping and substrates should be carefully investigated to tune the \nmicromagnetic properties of patterned structures.  \n \n4. Conclusion  \nTo summarize , in this paper we investigated the patterning of magnonic structures  via laser \ninduced local crystallization of amorphous YIG films on GGG(111). For thick films (160 nm) \nthe writing conditions leading to the formation of a crystalline and ferrimagnetic phase without \nsizable modification of the stoichiometry  also give rise to a surface ripple  with valley and ridges \naligned perpendicularly to the writing direction. This in turns causes a magnetic non -\nhomogeneity which is detrimental for uniform magnetic excitations. At reduced  thickness th is \ncorrugation is suppressed . On 80 n m thick films we  patterned single crystal and poly crystalline \ndots suitable for FMR measurements , from which a damping coefficient  = (5.8±0.4)·10-3 was \nestimated. Magnonic conduits have been patterned on 80 nm films and investigated by micro -\nBLS, showing an attenuation length on the order of 5 m for the fundamental DE mode , which \nis compatible with the damping parameter estimated from FMR. Beyond these promising \nresults, next steps involve a careful optimization of the laser -induced crystallization process in \nfilms with controlled thickness below 100  nm, pushing the damping parameter towards the \nvalues of epitaxial films on GG G in this thickness range .   \n \n5. Methods   \nLaser patterning with Nanofrazor Explore:  For local crystallization we exploited the laser -\nwriting option featured by the Nanofrazor Explore tool provided by Heidelberg.  The laser is \noperated in the continuous mode, so that for a given intensity the dose (D)  for each point of the \ngrid in the scan of a single line along the x direction can be calculated as follows, assuming a \ngaussian beam profile : 𝐷=∑2𝑃\n𝜋𝑤(𝑧)2+𝑁\n−𝑁 𝑒𝑥𝑝 (−2𝑑𝑥2 𝑛2\n𝑤(𝑧)2)𝜏𝑝. Here t he summation over n considers \nthat each point sees an intensity which first increases and then decreases during the motion of \nthe laser beam. N can be safely chosen in such a way that Ndx= 2w(z) , while  w(z) is the radius \nof the laser beam where the intensity is reduced by a factor 1/e2 with respect to the intensity at \nthe center (I 0).  \nThis dose is delivered in a time t on the order of 2N p, so that the average intensity is \nIav=D/(2N p) while the actual power versus time displays a gaussian profile replicating the   \n18 \n spatial profile in the time domain: 𝐼(𝑡)=𝐼(𝑛𝜏𝑝)=2𝑃\n𝜋𝑤(𝑧)2 𝑒𝑥𝑝 (−2(𝑑𝑥𝜏𝑝⁄)2 𝑡2\n𝑤(𝑧)2 ). As we are \ndealing with a laser annealing process, this means that each point undergoes a heating cycle of \nduration Δ𝑡=4𝑤(𝑧)𝜏𝑝𝑑𝑥⁄. Notice that the average intensity increases when decreasing d x, as \nin the ratio D/ t the exponential dependence of the dose on −2(𝑑𝑥𝑤(𝑧) ⁄ )2 dominates over the \nincrease of t which is just proportional to 𝑤(𝑧)𝑑𝑥⁄.These estimates are  valid for a single line, \nwhile in the patterning of an arbitrary shape by raster scan we must consider also the dose \narising from the overlap of the laser spot of the neighbor lines spaced by d y, which in this work \nis choses equal to d x. This can happen after a considerable time from the heating cycle due the \nwriting of the first line, so that for a two-dimensional  area each point undergoes many \nsubsequent thermal cycles . The microstructure of the crystallized phase  thus depends also on  \nthe shape as discussed in the main text . \nMicro -RAMAN : The phase transition induced on  amorphous YIG films by laser annealing was \nconfirmed with micro -Raman analysis using a Horiba Jobin Yvon LabRam HR800 Raman \nspectrometer, coupled to an Olympus BX41 microscope. The spectra were recorded in back -\nscattering geometry, using a 50X objective an d a Laser Quantum diode -pumped solid -state laser \nTorus emitting at 532nm. We set the power of the laser line at 30 mW which allowed us to \nobtain reliable data without inducing changes or photodegradation of the samples durin g the \nmeasures. Each spectrum has been obtained as an average of three acquisitions lasting 180 \nseconds each in the range 150 -650 cm-1. \nX-ray Absorption : XAS measurements at Fe L2,3 edges were taken a t the NFFA APE -HE \nbeamline ( Elettra  synchrotron, Italy ) [40] at room temperature  in total electron yield (TEY) \nmode  in linear horizontal polarization , with the sample surface at 45° with respect to the \nincident photon beam . Two orthogonal mechanical slits in front of the endstation were used to \nminimize the spot dimensions on the patterned samples . \nSNS-MOKE : In order to  map the dynamic properties of the laser written structures within the \nYIG layer we employ a special type of time -resolved magneto -optical Kerr microscopy (TR -\nMOKE). In this approach with continu ous wave RF-excitation we sample the GHz dynamics \nwith ultrashort laser pulses (with a wavelength of 515 nm) at a repetition rate f rep of 80 MHz \nlocally. The diffraction limited spot size is on the order of 300 nm. In this type of experiment , \nwe u se undersampling effects to generate alias frequencies at |f rf – n*f rep| whereas the lowest \norder frequency component is the difference of the rf -frequency with the nearest comb line n \nof the ultrashort laser pulse train. This frequency within the lowest order Nyquist zone is \naccessible by a balanced photodetector with MHz ba ndwidth and can be analyzed by means of   \n19 \n Lock -In detection. The direct demodulation at alias frequencies corresponds to an intrinsic \nfrequency modulation and thus no further modulation technique is required. The in -phase and \nquadrature signals which are demodulated simultaneously at the alias fre quency represent the \nreal and imaginary part of the dynamic susceptibility and yield amplitude and phase information. \nSince we overcome the Nyquist criterion we refer to this technique as Super -Nyquist -Sampling \nMOKE (SNS -MOKE) .[39]  \nBrillouin Light Scattering : Micro -BLS measurements are performed by focusing a single -mode \nsolid -state laser (with a wavelength of 532 nm ) at normal incidence onto the sample using an \nobjective with numerical aperture of 0.75, giving a spatial resolution of about 300 nm. A (3+3) -\npass tandem Fabry -Perot interferometer is used to analyze the inelastically scattered light. A \nnanopositioning stage allows us to position the sample with a precision down to 10 nm on all \nthree axes. A micro -stripline antenna, havi ng a width of 3 m, is used to excite SWs with a \nmicrowave source generator.  A dc/ac electrical probe station ranging from dc up to 20 GHz is \nused for spin -wave pumping. The microwave power is set at +16 dBm on the rf generator output. \nMicro -BLS measurements were carried out in Damon -Eshbach (DE) geometry applying a \nmagnetic field parallel to the antenna.  \n \nAcknowledgements  \nA. Del Giacco, F . Maspero, R . Bertacco and S. Tacchi  acknowledge funds from the EU project \nMandMEMS, grant  101070536. E. Albisetti acknowledges  funding from the European Union's \nHorizon 2020 research and innovation programme under grant agreement number 948225 \n(project B3YOND)  and from the FARE programme of the Italian Ministry for University and \nResearch (MUR) under grant agreement R20FC3PX8R (project NAMASTE) . E. Albisetti and \nS. Tacchi acknowledge  funding  from the European Union - Next Generation EU - “PNRR - \nM4C2, investimento 1.1 - “Fondo PRIN 2022”  - TEEPHANY - ThreEE -dimensional \nProcessing tecHnique of mAgNetic crYstals for magnonics and nanomagnetism  ID \n2022P4485M CUP D53D23001400001”. D. Petti acknowledges funding from the European \nUnion - Next Generation EU - “PNRR - M4C2, investimento 1.1 - “Fondo PRIN 2022” - PATH \n- Patterning of Antiferromagnets for THz operation id 2022ZRLA8F – CUP D53D23002490006”  \nand from Fondazione Cariplo and Fondazione CDP, grant n° 2022 -1882.  M. Madami, R. \nSilvani and S.Tacchi acknowledge  financial support from the Italian n ational project \nTEEPHANY -(PRIN2022P4485M), and NextGenerationEU National Innovation Ecosystem \ngrant ECS00000041−VITALITY (C UP B43C22000470005 and CUP   J97G22000170005 ),   \n20 \n under the Italian Ministry of University and Research (MUR).  S. Lake and G. 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Panaccione et al. , ‘Advanced photoelectric effect experiment beamline at Elettra: A \nsurface science laboratory coupled with Synchrotron Radiation’, Review of Scientific \nInstruments , vol. 80, no. 4, p. 043105, Apr. 2009, doi: 10.1063/1.3119364.  \n \n \n    \n24 \n  \n \n \n \n \n \n \nFigure 1:   a) Sketch of the laser patterning process; b) Optical microscopy of a 2D matrix of \n20 microns circular dots patterned with   different conditions on 160 nm thick amorphous YIG; \nc) Zoom on dots patterned at threshold conditions for crystallization. d) mic roRaman spectra \non areas patterned with different laser power; e -g ) AFM images and h -l MFM images  showing \nthe peculiar ripple   perpendicular to the writing direction.  \n \n  \n25 Figure 2:  Optical (top) and AFM (bottom) images from square patterns with 20 microns side, \nwritten on 160 nm (top row ) and 80 nm thick (bottom row) amorphous YIG,  as a function of \nthe laser power (P) used for writing. The writing direction is horizontal for all patterned areas. \n  \n26 \n  \n \n \n \nFigure 3:  XAS spectra from areas patterned on 160 nm thick amorphous YIG at different laser \npower . Vertical dashed lines correspond to the photon energies of A and B peaks at L 3 edge. \nThe vertical arrow indicates the shoulder of the L 2 edge which is associated to Fe2+. \n  \n704 712 720 728Fe-L2Fe-L3\nBXAS signal (arb.u)\nPhoton energy (eV) 100mW\n 80mW\n 60mW\n 40mW\n 0mW\nA  \n27 \n  \n \n \nFigure 4:   TRMOKE analysis on circular dots patterned on 160 nm thick YIG. a) Optical image \nof the coplanar waveguide (Au/Ti – yellow) fabricated on top of YIG (brown); at the bottom a \nzoom in the region between signal and ground where series of 5 circular dots wit h 20 m \ndiameter (outlined by red circles) have been patterned for each laser power. The bias field was \napplied along the horizontal x direction. b -c) two -dimensional map of the phase of MOKE \nsignal recorded at 2 GHz on a dot patterned with 80 mW, for 23 a nd 27 mT of applied field, \nrespectively. d -i) real and imaginary part of the MOKE signal as a function of the applied \nmagnetic field for a fixed excitation frequency of 2 GHz as a function of the writing power  \nreported on the top -right of each panel .  \n \n  \n  \n28 \n  \n \n \n \n \n \n \n \nFigure 5:  TRMOKE data on circular dots patterned at different laser power on 80 nm thick \nYIG, taken from experiments carried out using the same layout of figure 4a.  a -f) real and \nimaginary part of the TRMOKE signal as a function of the applied magnetic field for a  fixed \nfrequency of 2 GHz in the excitation waveguide; in the inset of panels c and f representative \nmaps of the instantaneous phase recorded in the dots are reported. g) resonance frequency vs. \napplied field of for the dot patterned with 65 mW and showing  a clear FMR line -shape; h) plot \nof the FMR linewidth vs. frequency for the dots patterned with 65 mW; the damping parameter \n() has been estimated from the slope of the linear fit (red line). i) same analysis as in panel h \nfor a dot written with 75 mW (panel e). j -k-l) EBS images taken from dots written at 65, 75 and \n80 mW.    \nj) \n k) \nl) \n65 mW  \n 75 mW  \n 80 mW  \n10 m \n 10 m \n 10 m \n80 mW  \n001 \n111 \n101   \n29 \n  \n \n \n \n \n \nFigure 6 : BLS analysis of magnonic conduits with nominal width of 1.2 m patterned on 80 \nnm thick YIG.  a) EBSD image from conduits patterned with a laser power between 85 and 100 \nmW, the blue color indicates a crystalline zone with (111) orientation; b) EBSD image (red \ncolor for (111) orientation), superposed to a conventional SEM image (grey scale)  where white \nsegments are used to highlight the borders of the conduits containing the crystallized areas; c-\nd-e) two -dimensional maps of the BLS intensity on co nduits patterned with 100, 95 and 90 \nmW; on the y axis the distance from the antenna ( corresponding to the red line  at the top) along \nthe propagation direction (SW wave -vector k) is reported, while x gives the transverse \ncoordinate; f) transverse section of the map in panel c at 6 m distance from the antenna; g) \nexponential fit of the BLS intensity decay reported in panel c; h) attenuation length (black dots) \nand frequency of the propagating mode (red dots) vs. the laser power used for patterning.  \n \n"    },    {        "title": "2402.14479v1.Existence_and_upper_semicontinuity_of_pullback_attractors_for_Kirchhoff_wave_equations_in_time_dependent_spaces.pdf",        "content": "arXiv:2402.14479v1  [math.AP]  22 Feb 2024Existence and upper semicontinuity of pullback attractors for Kirchhoff wave\nequations in time-dependent spaces\nBin Yang1Yuming Qin2,∗Alain Miranville3Ke Wang2\n1College of Information Science and Technology, Donghua Uni versity, Shanghai, 201620, P. R. China.\n2Department of Mathematics, Institute for Nonlinear Scienc e, Donghua University, Shanghai, 201620,\nP. R. China.\n3Laboratoire de Math´ ematiques et Applications, Universit ´ e de Poitiers, UMR CNRS 7348-SP2MI,\nBoulevard Marie et Pierre Curie-T´ el´ eport 2, F-86962, Cha sseneuil Futuroscope Cedex, France.\nAbstract\nIn this paper, we shall investigate the existence and upper semicon tinuity of pullback attractors\nfor non-autonomous Kirchhoff wave equations with a strong dampin g in the time-dependent space\nXt. After deriving the existence and uniqueness of solutions by the Fa edo-Galerkin approximation\nmethod, we establish the existence of pullback attractors. Later on, we prove the upper semi-\ncontinuity of pullback attractors between the Kirchhoff-type wav e equations with δ≥0 and the\nconventional wave equations with δ= 0 by a series of complex energy estimates.\nKeywords: Kirchhoff wave equations; strong damping; pullback attract ors; upper semiconti-\nnuity; the time-dependent space.\n2020 MSC: 35B40, 35B41, 35L05.\n1 Introduction\nIn this paper, we will consider the following non-autonomou s Kirchhoff wave equations\n\n\nε(t)∂ttu−(1+δ/ba∇dbl∇u/ba∇dbl2)∆u−∆∂tu+λu=g(u)+h(x,t)inΩ×(τ+∞),\nu(x,t) = 0 on ∂Ω×(τ+∞),\nu(x,τ) =u0\nτ, ∂tu(x,τ) =u1\nτ, x ∈Ω,(1.1)\n*Corresponding author: yuming@dhu.edu.cn\nE-mails: binyangdhu@163.com, Alain.Miranville@math.un iv-poitiers.fr, kwang@dhu.edu.cn\n1where Ω ⊂Rn(n≥3) is a bounded domain with smooth boundary ∂Ω,ε(t) is a time-dependent\nfunction, −∆utis a strong damping, g(u) is a nonlinear function, h(x,t) is an external force and\nδ,λ≥0 are constants.\nSuppose the time-dependent function ε(t)∈C1(R) is decreasing and satisfies\nlim\nt→+∞ε(t) =αǫ≥1, (1.2)\nand there exists a constant L≥αǫsuch that\nsup\nt∈R(|ε(t)|+|ε′(t)|)≤L. (1.3)\nMoreover, assume g∈C1(R;R) withg(0) = 0 and satisfies\n/vextendsingle/vextendsingleg′(u)/vextendsingle/vextendsingle≤C(1+|u|p), (1.4)\ng′(u)≤k, (1.5)\nlimsup\n|u|→∞ug(u)−γG(u)\nu2≤0 (1.6)\nand\nlimsup\n|u|→∞G(u)\nu2≤0, (1.7)\nwherep=4\nn−2,C,k,γ > 0 are constants and G(u) =/integraltextu\n0g(s)ds.\nBesides, let h(x,t),∂th(x,t)∈L2\nloc/parenleftbig\nR;L2(Ω)/parenrightbig\nand there exists a parameter σ>0 such that\n/integraldisplayt\n−∞eσs/ba∇dblh(x,s)/ba∇dbl2ds<+∞,∀t∈R. (1.8)\nOur phase space is the time-dependent space Xt=H1\n0(Ω)×L2(Ω), which is equipped with the\nfollowing norm\n/ba∇dbl(u0,u1)/ba∇dblXt=/ba∇dbl∇u0/ba∇dbl2+ε(t)/ba∇dblu1/ba∇dbl2. (1.9)\nAdditionally, the norm of Xtcan be equivalently written as\n/ba∇dbl(u,∂tu)/ba∇dbl2=/ba∇dbl∇u/ba∇dbl2+ε(t)/ba∇dbl∂tu/ba∇dbl2. (1.10)\nIn 1883, Kirchhoff [ 6] first proposed problem utt−/parenleftbig\n1+ǫ/ba∇dbl∇u/ba∇dbl2/parenrightbig\n∆u+f(u) =g(x) for anyǫ >\n0. Subsequently, scholars discovered that the Kirchhoff-typ e wave equations play a crucial role in\nseismic imaging, oil and gas exploration, and other fields. T herefore, many mathematicians have\ndevoted themselves to studying attractors for such equatio ns (see [3,7,10,12]). It is noted that the\n2Kirchhoff term is non-local, which brings some difficulties to derive the well-posedness of solutions.\nIn addition, if a Kirchhoff wave equation is non-autonomous a nd contains a damping, the difficulties\nwill be greatly increased. So far, few results have been obta ined about the attractors for the non-\nautonomous Kirchhoff wave equations with a damping.\nFan and Zhou [ 5] proved the existence of compact kernel sections for the pro cess generated by\nproblemutt−α∆ut−/parenleftbig\nβ+γ/parenleftbig/integraltext\nΩ|∇u|2dx/parenrightbigρ/parenrightbig\n∆u+h(ut)+f(u,t) =g(x,t) inH1\n0(Ω)×L2(Ω), where\nα,β >0,ρ >−1 andγ≥0 are constants. Moreover, [ 5] derived a precise estimate of upper bound\nof Hausdorff dimension of kernel sections, which decreases a s the strong damping grows for the large\nstrongdamping, particularly intheautonomouscase. ThenW angandZhong[ 16]consideredtheupper\nsemicontinuity of pullback attractors for problem utt−∆ut−/parenleftbig\n1+ǫ1/ba∇dbl∇u/ba∇dbl2/parenrightbig\n∆u+f(u) =g(x,t) with\nǫ1>0 inH1\n0(Ω)×L2(Ω). Furthermore, Yang and Li [ 17] concerned with the existence and stability\nof pullback exponential attractors for problem utt−M/parenleftbig\n/ba∇dbl∇u/ba∇dbl2/parenrightbig\n∆u+ (−∆)α1ut+f(u) =g(x,t) in\nH1+α1(Ω)×Hα1(Ω) forα1∈(1/2,1). Later on, Li and Yang [ 8] proved the robustness of pullback\nattractors and pullback exponential attractors for proble mutt−/parenleftbig\n1+ǫ2/ba∇dbl∇u/ba∇dbl2/parenrightbig\n∆u−σ/parenleftbig\n/ba∇dbl∇u/ba∇dbl2/parenrightbig\n∆ut+\nf(u) =g(x,t) in/parenleftbig\nH1\n0(Ω)∩Lp+1(Ω)/parenrightbig\n×L2(Ω), where ǫ2∈[0,1] is an extensibility parameter.\nYang and Li [ 18] obtained the process generated by problem utt−M/parenleftbig\n/ba∇dbl∇u/ba∇dbl2/parenrightbig\n∆u+(−∆)α2ut+\nf(u) =g(x,t) has pullback attractors for α2∈(1/2,1) and the family of pullback attractors is upper\nsemicontinuous in H1\n0(Ω)∩Lp+1(Ω)×L2(Ω) when the nonlinearity f(u) is of supercritical growth\np: 1≤p < pα2≡N+4α2\n(N−4α2)+withN≥3. In addition, Ma, Wang and Xie [ 11] verified the existence\nof pullback attractors for problem utt−∆ut−φ/parenleftbig\n/ba∇dbl∇u/ba∇dbl2/parenrightbig\n∆u+f(u) =h(x,t) inH1\n0(Ω)×L2(Ω).\nFurthermore, Li, Yang and Feng [ 9] established the existence and continuity of uniform attra ctors for\nproblemutt−/parenleftbig\n1+ǫ3/ba∇dbl∇u/ba∇dbl2/parenrightbig\n∆u−∆ut+f(u) =g(x,t) inH1\n0(Ω)∩Lp+1(Ω)×L2(Ω) whenf(u) satisfies\noptimal supercritical growth condition p:N+2\nN−2=p∗<p<p∗∗=N+4\n(N−4)+withN≥3 andǫ3∈[0,1].\nMoreover, we will state the main difficulties and innovations of this paper.\n(i) It is well-known that Kirchhoff wave equations have a wide range of applications in actual life.\nHowever, since each non-autonomous Kirchhoff wave equation contains a non-local Kirchhoff term and\na time-dependent external force function, which bring lots of computational difficulties to investigate\nthe long-time behavior of solutions. Therefore, there are o nly a dozen papers about the existence and\nupper semicontinuity of pullback attractors for Kirchhoff w ave equations. Meanwhile, no paper has\nstudied pullback attractors of Kirchhoff wave equations in t ime-dependent space Xt=H1\n0(Ω)×L2(Ω),\n3thus our paper is a new attempt.\n(ii) In addition, the strong damping −∆∂tumakes the application of problem ( 1.1) more com-\nprehensive. But −∆∂tudirectly makes it necessary to overcome more difficulties in c alculations and\nestimates. We deal with these difficulties by undertaking a la rge number of meticulous energy esti-\nmates and by adjusting the range of parameters in test functi ons.\n(iii) Whenδ= 0, problem ( 1.1) is an ordinary wave equation, in which case we obtain that th ere\nexists a pullback attractor A0. Moreover, when δ≥0, problem ( 1.1) is a Kirchhoff wave equation\nwith a strong damping, in this case we derive that there exist s a pullback attractor Aδ. We verify the\nupper semicontinuity of A0andAδin Section 4, which is helpful for learning the relationship between\nKirchhoff wave equations and ordinary wave equations.\nThis paper is organized as follows. In Section 2, we recall so me definitions and lemmas about\npullback attractors. Furthermore, by a series of elaborate energy estimates, in Section 3 we derive\nthe existence of pullback attractors for the process genera ted by problem ( 1.1) in the time-dependent\nspaceXt. In Section 4, we establish the upper semicontinuity of pull back attractors between the\nnon-autonomous Kirchhoff wave equations ( 1.1) withδ≥0 and the conventional wave equations with\nδ= 0 inXt.\n2 Preliminaries\nIn this section, we will recall some necessary definitions an d lemmas about pullback attractors and\nupper semicontinuity of pullback attractors in detail.\nSupposeXis a Banach space or a closed subset of a Banach space, which is endowed with the\nnorm/ba∇dbl·/ba∇dblX.\nDefinition 2.1. ([4,14])A process or a two-parameter semigroup on Xis the family {U(t,τ)}t≥τof\nmappingU(t,τ) :X→Xsatisfying that U(τ,τ) =Idis the identity operator and U(t,s)U(s,τ) =\nU(t,τ)for anyt≥s≥τ.\nDefinition 2.2. ([4,14])The Hausdorff semi-distance of two nonempty sets A1,A2⊂Xis defined by\ndistX(A1,A2) = sup\nx1∈A1inf\nx2∈A2/ba∇dblx1−x2/ba∇dblX.\nDefinition 2.3. ([1,13,19])A family D={D(t)}t∈Ris pullback absorbing for the process U(·,·)in\n4X, if for any t∈Rand any bounded subset D⊂X, there exists a constant T=T(t,D)>0such that\nU(t,t−τ)D⊂D(t)for anyτ≥T.\nDefinition 2.4. ([1,13,19])A processU(·,·)is pullback D-asymptotically compact in X, if for any\nt∈R, any sequence τi→ −∞ andxi∈D(t−τi), the sequence {U(t,t−τi)xi}i∈Nis relatively\ncompact in X.\nDefinition 2.5. ([1,13,19])A family of compact sets A={A(t)}t∈Ris called a pullback attractor\nfor the process U(·,·), if it fulfills the following properties\n(i)Ais invariant, i.e., U(t,τ)A(τ) =A(t)for anyτ≤t;\n(ii)Ais pullback attracting, i.e.,\nlim\nτ→−∞distX(U(t,t−τ)D,A(t)) = 0\nfor any bounded D⊂X.\nLemma 2.6. ([2])Suppose the family D={D(t)}t∈Ris pullback absorbing and the process U(·,·)is\npullback D-asymptotically compact in X, if the family A={A(t)}t∈Rsatisfies\nA(t) = Λ(D,t) =/intersectiondisplay\ns≥0/uniondisplay\nτ≥sU(t,t−τ)D(t−τ)X\n,∀t∈R, (2.1)\nthenAis a pullback attractor for U(·,·)inX. In addition, if for any t∈R, there exists a T=T(t)>0\nsuch that\nU(t,t−τ)D(t−τ)⊂D(t),∀τ >T, (2.2)\nthen\nlim\nτ→−∞distX(U(t,t−τ)D(t−τ),A(t)) = 0. (2.3)\nDefinition 2.7. ([1,13,19])Suppose the set D⊂Xis bounded, then Ψ(·,·) :D×Dis called a\ncontractive function, if for any sequence {xi}i∈N⊂D, there exists a subsequence {xik}k∈N⊂ {xi}i∈N\nsuch that\nlim\nk→+∞lim\nl→+∞Ψ(xik,xil) = 0.\nLemma 2.8. ([15])Suppose the family D={D(t)}t∈R⊂Xand satisfies (2.2), and for any constant\nζ >0and anyt∈R, there exists a time τ=τ(t,ζ,D)>0and a contractive function Ψt−τ(·,·) :\nD(t−τ)×D(t−τ)depending on t−τsuch that\n/ba∇dblU(t,t−τ)x−U(t,t−τ)y/ba∇dblX≤ζ+Ψt−τ(x,y),∀x,y∈D(t−τ),\n5thenU(·,·)is pullback D-asymptotically compact in X.\nBelow we will introducethe correlation lemma, which is used to prove the process U(·,·) is pullback\nD-asymptotically compact in X, as well as the lemmas to derive the upper semicontinuity of p ullback\nattractors, which have been systematically proved in [ 16].\nLetXaandXbbe two Banach spaces, which are endowed with the norms /ba∇dbl · /ba∇dblXaand/ba∇dbl · /ba∇dblXb,\nrespectively. Then assume X=Xa×Xbis equipped with the norm /ba∇dbl·/ba∇dblX. In addition, suppose the\nprocessU(·,·) defined in Xadmits that U(·,·) : (u0;u1)→(Ua(·,·)u0;Ub(·,·)u1), whereUa(·,·) and\nUb(·,·) are continuous and defined in XaandXb, respectively.\nLemma 2.9. ([16])Suppose the process Ua(·,·) =Ua1(·,·)+Ua2(·,·)and the family D={D(t)}t∈R⊂\nX, if for any t∈Rthe following properties hold\n(i) there exists a function φ:R×R→R+satisfying\nlim\nτ→−∞φ(t,τ) = 0\nand /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/uniondisplay\n(u0;u1)∈D(t−τ)Ua1(t,t−τ)u0/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nXa≤φ(t,τ)\nfor anyτ >0;\n(ii)/uniontext\n(u0;u1)∈D(t−τ)Ua2(t,t−τ)u0is relatively compact in Xafor anyτ >0;\n(iii)/uniontext\n(u0;u1)∈D(t−τ)Ub(t,t−τ)u1is relatively compact in Xbfor anyτ >0;\nthenU(·,·)is pullback D-asymptotically compact in X.\nLemma 2.10. ([16])Assume the process Uδ(·,·)has a family of pullback absorbing sets Dδ={Dδ(t)}t∈R\nsatisfying (2.2)for anyδ∈I,Aδ={Aδ(t)}t∈Rsatisfies Lemma 2.6, and the following properties hold\n(i) there exists a δ0∈Isatisfying that\n/uniondisplay\nδ∈IDδ(t)X\n⊂Dδ0(t); (2.4)\n(ii) for any sequences δi∈Iwithδi→δ0∈I,ti>αwithti→t0,xi∈Xwithxi→x0andτ >0, it\nfollows\nlim\ni→+∞/ba∇dblUδi(ti,α−τ)xi−Uδ0(t0,α−τ)x0/ba∇dblX= 0; (2.5)\n6(iii) for any sequence δi∈I, any bounded {ti}i∈N⊂R,τi>0withτi→ −∞andxi∈D(α−τi),\n{Uδi(ti,α−τi)xi}i∈Nis relatively compact in X; (2.6)\nthen\nlim\nδ→δ0sup\nt∈[α,β]distX(Aδ(t),Aδ0(t)) = 0 (2.7)\nfor any[α,β]⊂R.\nThe following lemma is used to prove Lemma 2.10(iii).\nLemma 2.11. ([16])Assume [α,β]×I⊂R×Rand there exists a constant W=W(t,T)>0such\nthat the pullback absorbing family Dδ={Dδ(t)}t∈Rsatisfying\nUδ(t,t−τ)Dδ(t−τ)⊂Dδ(t)\nfor anyt∈Randτ >W. Additionally, let Uδ(·,·) = (Ua(·,·);Ub(·,·))andUa(·,·) =Ua1(·,·)+Ua2(·,·),\nif the following properties hold\n(i) there exists a function Φ[α,β](·) :R→R+satisfying\nlim\nτ→−∞Φ[α,β](τ) = 0\nand /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/uniondisplay\nt∈[α,β]/uniondisplay\nδ∈I/uniondisplay\n(u0;u1)∈Dδ(α−τ)Ua1(t,α−τ)u0/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nXa≤Φ[α,β](τ);\n(ii)/uniontext\nt∈[α,β]/uniontext\nδ∈I/uniontext\n(u0;u1)∈Dδ(α−τ)Ua2(t,α−τ)u0is relatively compact in Xafor anyτ >0;\n(iii)/uniontext\nt∈[α,β]/uniontext\nδ∈I/uniontext\n(u0;u1)∈Dδ(α−τ)Ub(t,α−τ)u1is relatively compact in Xbfor anyτ >0;\nthen Lemma 2.10(iii) holds.\n3 Existence of pullback attractors Aδ={Aδ(t)}t∈R\nIn this section, we first derive the existence and uniqueness of weak solutions for problem ( 1.1)\nin the time-dependent space Xt. Then through a series of elaborate and complex calculation s and\nestimates, we establish the existence of the pullback attra ctorsAδ={Aδ(t)}t∈Rfor the process Uδ(·,·)\ngenerated by problem ( 1.1) inXt.\nUsing the Faedo-Galerkin approximation method, the follow ing theorem can be easily verified.\n7Theorem 3.1. Under the assumptions of ε(t),g(u),h(x,t),δandλin Section 1, let/parenleftbig\nu0\nτ,u1\nτ/parenrightbig\nbe\ngiven, then for any δ≥0and anyT > τ, there exists a unique weak solution uto problem (1.1)\nsatisfying\n(u,∂tu)∈C([τ,T];Xt)\nin the time-dependent space Xt, which continuously depends on its initial data in Xt.\nFor any weak solution uto problem ( 1.1), letϕt= (u(t),∂tu(t)), then we can define the continuous\nprocessUδ(t,τ) =Xτ→Xtgenerated by problem ( 1.1) as follows\nUδ(t,τ)ϕτ= (u(t),∂tu(t))\nfor anyt≥τ,δ≥0 andϕτ∈Xτ.\nLemma 3.2. Under the assumptions of ε(t),g(u),h(x,t),δandλin Section 1, then it follows that\n/ba∇dbl∇u(t)/ba∇dbl2+ε(t)/ba∇dbl∂tu(t)/ba∇dbl2≤Ce−σ1(t−τ)/parenleftBig/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble2+ε(τ)/vextenddouble/vextenddoubleu1\nτ/vextenddouble/vextenddouble2+/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble2n\nn−2+δ/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble4/parenrightBig\n+Ce−σ1t/integraldisplayt\n−∞eσ1s/ba∇dblh(x,s)/ba∇dbl2ds+C(3.1)\nfor anyδ≥0andτ≤t∈R, whereC/ne}ationslash=C(t,τ,δ)is a positive constant.\nProof.Taking inner product between ∂tu+ρuand (1.1)1inL2(Ω), we obtain\n1\n2d\ndt/parenleftbigg\nε(t)/ba∇dbl∂tu+ρu/ba∇dbl2−ρ2ε(t)/ba∇dblu/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2+δ\n2/ba∇dbl∇u/ba∇dbl4+ρ/ba∇dbl∇u/ba∇dbl2+λ/ba∇dblu/ba∇dbl2−2(G(u),1)/parenrightbigg\n−1\n2ε′(t)/ba∇dbl∂tu/ba∇dbl2−ρε′(t)(∂tu,u)−ρε(t)/ba∇dbl∂tu/ba∇dbl2+ρ/ba∇dbl∇u/ba∇dbl2+δρ/ba∇dbl∇u/ba∇dbl4\n+/ba∇dbl∇∂tu/ba∇dbl2+λρ/ba∇dblu/ba∇dbl2−ρ(g(u),u) = (h(x,t),∂tu+ρu), (3.2)\nwhereρ>0 is a constant.\nThen from ( 3.2), we arrive at\nd\ndtE(t)−ε′(t)/ba∇dbl∂tu/ba∇dbl2−2ρε′(t)(∂tu,u)−2ρε(t)/ba∇dbl∂tu/ba∇dbl2+2ρ/ba∇dbl∇u/ba∇dbl2\n+2δρ/ba∇dbl∇u/ba∇dbl4+2/ba∇dbl∇∂tu/ba∇dbl2+2λρ/ba∇dblu/ba∇dbl2−2ρ(g(u),u) +4ρ2ε(t)(∂tu,u)\n= 2/parenleftbig\nh(x,t),∂tu+ρu)+4ρ2ε(t)(∂tu,u), (3.3)\nwhere\nE(t) =ε(t)/ba∇dbl∂tu+ρu/ba∇dbl2−ρ2ε(t)/ba∇dblu/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2+δ\n2/ba∇dbl∇u/ba∇dbl4+ρ/ba∇dbl∇u/ba∇dbl2\n+λ/ba∇dblu/ba∇dbl2−2(G(u),1) +2c0(3.4)\n8andc0>0 is a constant.\nThanksto( 1.2), (1.3), thePoincar´ e inequality andnoticing that ε(t) is decreasing, thenif ρsatisfies\n√\n2λ≤ρ≤min/braceleftbigg\nλ1\n4L,√\n(λ1+4λ)L\n2L/bracerightbigg\n, it follows that\n−ε′(t)/ba∇dbl∂tu/ba∇dbl2−2ρε′(t)(∂tu,u)−2ρε(t)/ba∇dbl∂tu/ba∇dbl2+ρ\n2/ba∇dbl∇u/ba∇dbl2+/ba∇dbl∇∂tu/ba∇dbl2\n+2λρ/ba∇dblu/ba∇dbl2+4ρ2ε(t)(∂tu,u)\n≥2ρε(t)/ba∇dbl∂tu+ρu/ba∇dbl2. (3.5)\nFurthermore, by the Young inequality, we conclude\n(h(x,t),∂tu+ρu)≤1\n2ρ/ba∇dblh(x,t)/ba∇dbl2+ρ\n2/ba∇dbl∂tu+ρu/ba∇dbl2(3.6)\nand\n2ρ2(∂tu,u)≤4ρ2\nλ1/ba∇dbl∂tu/ba∇dbl2+ρ2λ1\n4/ba∇dblu/ba∇dbl2. (3.7)\nInserting ( 3.5)−(3.7) into (3.3), we deduce\nd\ndtE(t)+1\n2/ba∇dbl∇∂tu/ba∇dbl2+ρ\n2/ba∇dbl∇u/ba∇dbl2+2δρ/ba∇dbl∇u/ba∇dbl4−2ρ(g(u),u) +ρ(2ε(t)−ρ)/ba∇dbl∂tu+ρu/ba∇dbl2\n+1\n2/ba∇dbl∇∂tu/ba∇dbl2+ρ/ba∇dbl∇u/ba∇dbl2−8ρ2ε(t)\nλ1/ba∇dbl∂tu/ba∇dbl2−ρ2λ1ε(t)\n2/ba∇dblu/ba∇dbl2\n≤1\nρ/ba∇dblh(x,t)/ba∇dbl2, (3.8)\nwhereχ>0 is a constant.\nSuppose\nI(t) =ρ\n2/ba∇dbl∇u/ba∇dbl2+2δρ/ba∇dbl∇u/ba∇dbl4−2ρ(g(u),u) +ρ(2ε(t)−ρ)/ba∇dbl∂tu+ρu/ba∇dbl2−χE(t) (3.9)\nand\nK(t) =1\n2/ba∇dbl∇∂tu/ba∇dbl2+ρ/ba∇dbl∇u/ba∇dbl2−8ρ2ε(t)\nλ1/ba∇dbl∂tu/ba∇dbl2−ρ2λ1ε(t)\n2/ba∇dblu/ba∇dbl2, (3.10)\nthen we derive\nd\ndtE(t)+1\n2/ba∇dbl∇∂tu/ba∇dbl2+I(t)+K(t)+χE(t)≤1\nρ/ba∇dblh(x,t)/ba∇dbl2. (3.11)\nUsing (3.10) and the Poincar´ e inequality, then if ρfurther satisfies ρ≤min/braceleftBig\n2\nL,λ1√\nL\n4L/bracerightBig\n, we obtain\nK(t)≥0. (3.12)\nThanks to ( 1.6) and (1.7), we conclude that there exist constants c1,c2,c3,c4>0 such that\n−(g(u),u)≥ −γ(G(u),1)−c1/ba∇dblu/ba∇dbl2−c2 (3.13)\n9and\n−(G(u),1)≥ −c3/ba∇dblu/ba∇dbl2−c4. (3.14)\nMoreover, if ρ,L,c3satisfyλ1+ρλ1−ρ2L−2c3+λ1≥0 and letc0< c4, then by the Poincar´ e\ninequality, we deduce\n−ρ2ε(t)/ba∇dblu/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2+ρ/ba∇dbl∇u/ba∇dbl2+λ/ba∇dblu/ba∇dbl2−2(G(u),1) +2c0≥0, (3.15)\nwhich leads to\nE(t)≥0. (3.16)\nInserting ( 3.4) into (3.9), we obtain\nI(t) =/parenleftBigρ\n2−χ−χρ/parenrightBig\n/ba∇dbl∇u/ba∇dbl2+/parenleftbigg\n2δρ−χδ\n2/parenrightbigg\n/ba∇dbl∇u/ba∇dbl4+(ρ(2ε(t)−ρ)−χε(t))/ba∇dbl∂tu+ρu/ba∇dbl2\n+/parenleftbig\nχρ2ε(t)−χλ)/ba∇dblu/ba∇dbl2−2ρ(g(u),u/parenrightbig\n+2χ(G(u),1)−2χc0.(3.17)\nThus, from ( 3.13), (3.14) and (3.17), ifχfurther fulfills χ<ργ, we arrive at\n−2ρ(g(u),u) +2χ(G(u),1)\n≥ −(2ργc3−2χc3+2ρc1)/ba∇dblu/ba∇dbl2−(2ργ−2χ)c4−2ρc2.(3.18)\nInserting ( 3.18) into (3.17), we conclude\nI(t) =/parenleftBigρ\n2−χ−χρ/parenrightBig\n/ba∇dbl∇u/ba∇dbl2+/parenleftbigg\n2δρ−χδ\n2/parenrightbigg\n/ba∇dbl∇u/ba∇dbl4+/parenleftbig\n2ρε(t)−ρ2−χε(t)/parenrightbig\n/ba∇dbl∂tu+ρu/ba∇dbl2\n+/parenleftbig\nχρ2ε(t)−χλ−2ργc3+2χc3−2ρc1/parenrightbig\n/ba∇dblu/ba∇dbl2−(2ργ−2χ)c4−2ρc2−2χc0.(3.19)\nFrom (3.19), we derive that if the following inequalities hold\n\n\nρ\n2−χ−χρ≥0,\n2δρ−χδ\n2≥0,\n2ρε(t)−ρ2−χε(t)≥0,\nχρ2ε(t)−χλ−2ργc3+2χc3−2ρc1≥0,\n−(2ργ−2χ)c4−2ρc2−2χc0≥0,(3.20)\nthen there exists a constant c5>0 such that\nI(t)≥ −c5. (3.21)\nNoticingc0<c4and by (1.2), (1.3) and (3.20), then (3.21) holds when c3≤λ\n2,√\n(λ−2c3)L\nL≤ρ≤2\nand\nmax/braceleftbigg2ργc3+2ρc1\nρ2−λ+2c3,2ργc4+2ρc2\n2c4−2c0/bracerightbigg\n<χ<min/braceleftbiggρ\n2(1+ρ),4ρ,ργ,2ρ−ρ2/bracerightbigg\n.(3.22)\n10Therefore, from ( 3.11), (3.12), (3.16) and (3.21), we obtain\nd\ndtE(t)≤ −χE(t)+1\nρ/ba∇dblh(x,t)/ba∇dbl2+c5. (3.23)\nThen taking 0 <σ1<χand by (3.23), we conclude\nd\ndt/parenleftbig\neσ1tE(t)/parenrightbig\n≤ −(χ−σ1)eσ1tE(t)+eσ1t\nρ/ba∇dblh(x,t)/ba∇dbl2+eσ1tc5. (3.24)\nIntegrating ( 3.24) fromτtot, we derive\nE(t)+(χ−σ1)e−σ1t/integraldisplayt\nτeσ1sE(s)ds≤e−σ1(t−τ)E(τ)+1\nρeσ1t/integraldisplayt\nτeσ1s/ba∇dblh(x,s)/ba∇dbl2ds+c5\nσ1.(3.25)\nCarrying out some calculations similar to ( 3.7), we deduce\n2ρε(t)|(∂tu,u)| ≤4ρ\nλ1ε(t)/ba∇dbl∂tu/ba∇dbl2+ρλ1\n4ε(t)/ba∇dblu/ba∇dbl2. (3.26)\nNoticing that ρ≤λ1\n4and by (1.2), (1.3) and the Poincar´ e inequality, we obtain if L≥64\nλ1,ρfurther\nsatisfiesρ≥λ1L+/radicalBig\n(λ2\n1L−64λ1)L\n8L, then there exists a constant\nc6≥max/braceleftbiggλ1\nλ1−4ρ,4λ1+ρ2L−ρLλ1\n4λ1/bracerightbigg\nsuch that\n/ba∇dbl∇u/ba∇dbl2+ε(t)/ba∇dbl∂tu/ba∇dbl2≤c6/parenleftBig\n/ba∇dbl∇u/ba∇dbl2+ε(t)/ba∇dbl∂tu+ρu/ba∇dbl2/parenrightBig\n. (3.27)\nInserting ( 3.14) into (3.4), we arrive at\nE(t)≥ε(t)/ba∇dbl∂tu+ρu/ba∇dbl2−ρ2ε(t)/ba∇dblu/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2+δ\n2/ba∇dbl∇u/ba∇dbl4+ρ/ba∇dbl∇u/ba∇dbl2+λ/ba∇dblu/ba∇dbl2−2c3/ba∇dblu/ba∇dbl2.(3.28)\nBesides, inserting ( 3.27) into (3.28), we conclude if c3≤λ−ρ2L\n2andρfurther satisfies ρ≤√\nλL\nL,\nthen\nc−1\n6/parenleftbig\n/ba∇dbl∇u/ba∇dbl2+ε(t)/ba∇dbl∂tu/ba∇dbl2/parenrightbig\n≤E(t). (3.29)\nSimilarly, by ( 3.26), we obtain\nε(t)/ba∇dbl∂tu+ρu/ba∇dbl2−ρ2ε(t)/ba∇dblu/ba∇dbl2≤/parenleftbigg\n1+4ρ\nλ1/parenrightbigg\nε(t)/ba∇dbl∂tu/ba∇dbl2+ρλ1\n4ε(t)/ba∇dblu/ba∇dbl2. (3.30)\nMoreover, from ( 1.4) andG(u) =/integraltextu\n0g(s)ds, we deduce that there exist constants c8>c7>0 such\nthat\n−2(G(u),1)≤2c7/parenleftbigg/integraldisplayu\n0|u|n+2\nn−2ds,1/parenrightbigg\n+2c7(u,1)\n≤c8/ba∇dblu/ba∇dbl4\nn−2.(3.31)\n11Then by ( 3.4), (3.30) and (3.31), we derive there exist constants\nc9≥max/braceleftbigg\n1+4ρ\nλ1,ρL\n4+λ\nλ1+1+ρ,1\n2,c8/bracerightbigg\n>0\nandc10>c0such that\nE(t)≤c9/parenleftBig\n/ba∇dbl∇u/ba∇dbl2+ε(t)/ba∇dbl∂tu/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2n\nn−2+δ/ba∇dbl∇u/ba∇dbl4/parenrightBig\n+2c10. (3.32)\nHence, combining ( 3.29) with (3.32), we obtain\nc−1\n6/parenleftbig\n/ba∇dbl∇u/ba∇dbl2+ε(t)/ba∇dbl∂tu/ba∇dbl2)≤E(t)≤c9/parenleftBig\n/ba∇dbl∇u/ba∇dbl2+ε(t)/ba∇dbl∂tu/ba∇dbl2+/ba∇dbl∇u/ba∇dbl2n\nn−2+δ/ba∇dbl∇u/ba∇dbl4/parenrightBig\n+2c10.(3.33)\nFrom (3.16), (3.25) and noticing 0 <σ1<χ, we deduce\nc6E(t)≤c6e−σ1(t−τ)E(τ)+c6\nρeσ1t/integraldisplayt\nτeσ1s/ba∇dblh(x,s)/ba∇dbl2ds+c5c6\nσ1. (3.34)\nUsing (1.2), (1.3) and (3.33), we derive\nE(τ)≤c9/parenleftBig/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble2+ε(τ)/vextenddouble/vextenddoubleu1\nτ/vextenddouble/vextenddouble2+/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble2n\nn−2+δ/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble4/parenrightBig\n+2c10. (3.35)\nThen inserting ( 3.33) and (3.35) into (3.34), we conclude\n/ba∇dbl∇u/ba∇dbl2+ε(t)/ba∇dbl∂tu/ba∇dbl2≤c6c9e−σ1(t−τ)/parenleftBig/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble2+ε(τ)/vextenddouble/vextenddoubleu1\nτ/vextenddouble/vextenddouble2+/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble2n\nn−2+δ/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble4/parenrightBig\n+c6\nρeσ1t/integraldisplayt\nτeσ1s/ba∇dblh(x,s)/ba∇dbl2ds+2c6c10e−σ1(t−τ)+c5c6\nσ1.(3.36)\nLetc11=c6c9,c12=c6\nρandc13= 2c6c10+c5c6\nσ1, then we obtain\n/ba∇dbl∇u/ba∇dbl2+ε(t)/ba∇dbl∂tu/ba∇dbl2��c11e−σ1(t−τ)/parenleftBig/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble2+ε(τ)/vextenddouble/vextenddoubleu1\nτ/vextenddouble/vextenddouble2+/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble2n\nn−2+δ/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble4/parenrightBig\n+c12e−σ1t/integraldisplayt\nτeσ1s/ba∇dblh(x,s)/ba∇dbl2ds+c13.(3.37)\nConsequently, ( 3.1) holds directly. /square\nLemma 3.3. Under the assumptions of ε(t),g(u),h(x,t),δandλin Section 1, then for any δ≥0,\nthe process {Uδ(t,τ)}t≥τgenerated by problem (1.1)has a pullback absorbing family Dδ={Dδ(t)}t∈R\nthat satisfies (2.2).\nProof.Let\nB(t) =/parenleftbigg\nc14e−σ1t/integraldisplayt\n−∞eσ1s/ba∇dblh(x,s)/ba∇dbl2ds+c14/parenrightbigg1\n2\n, (3.38)\nwherec14≥max{2c12,2c13}andσ1>0 is the same as in Lemma 3.2.\n12In addition, suppose Dδ={Dδ(t)}t∈Rsatisfies\nDδ(t) =/braceleftBig/parenleftbig\nu0;u1/parenrightbig\n∈Xt/vextendsingle/vextendsingle/vextenddouble/vextenddouble/parenleftbig\nu0;u1/parenrightbig/vextenddouble/vextenddouble\nXt≤B(t)/bracerightBig\n. (3.39)\nAssume/parenleftBig\nu0\n˜t−τ,u1\n˜t−τ/parenrightBig\n∈Dδ/parenleftbig˜t−τ/parenrightbig\nfor any˜t∈R, then by ( 3.37)−(3.39), we deduce there exists a\nconstantσ2satisfyingσ1<σ2<χsuch that\n/vextenddouble/vextenddouble/vextenddoubleUδ/parenleftbig˜t,˜t−τ/parenrightbig/parenleftBig\nu0\n˜t−τ,u1\n˜t−τ/parenrightBig/vextenddouble/vextenddouble/vextenddouble2\nXt=/ba∇dbl∇u(˜t)/ba∇dbl2+ε(˜t)/ba∇dbl∂tu(˜t)/ba∇dbl2\n≤c11e−σ2τ/parenleftBig\n1+/parenleftbig\nB(˜t−τ)/parenrightbig2n\nn−2+δ/parenleftbig\nB/parenleftbig˜t−τ/parenrightbig/parenrightbig4/parenrightBig\n+c12e−σ1˜t/integraldisplay˜t\n˜t−τe(σ2−σ1)seσ1s/ba∇dblh(x,s)/ba∇dbl2ds+c13\n≤c14e−σ2τ/parenleftBig/parenleftbig\nB(˜t−τ)/parenrightbig2n\nn−2+δ/parenleftbig\nB/parenleftbig˜t−τ/parenrightbig/parenrightbig4/parenrightBig\n+c15e−σ1˜t/integraldisplay˜t\n−∞eσ1s/ba∇dblh(x,s)/ba∇dbl2ds+c13, (3.40)\nwherec14≥2c11andc15≥2c12are constants.\nHence, it follows from ( 3.40) that for any ˜t∈Rthere exists a T=T(˜t)>0 such that\nUδ(˜t,˜t−τ)Dδ(˜t−τ)⊂Dδ(˜t) (3.41)\nfor anyτ >Tand anyδ≥0. /square\nLemma 3.4. Under the assumptions of ε(t),g(u),h(x,t),δandλin Section 1, assume/vextenddouble/vextenddouble∇u0\nτ/vextenddouble/vextenddouble2+\nε(τ)/ba∇dblu1\nτ/ba∇dbl2≤/tildewideC, then the following inequality hold\nε(t)/ba∇dbl∂ttu(t)/ba∇dbl2\n−1+/ba∇dbl∇∂tu(t)/ba∇dbl2+/integraldisplayt+1\nt/ba∇dbl∂ttu(s)/ba∇dbl2ds\n≤Z/parenleftbigg1\n(t−τ)2/integraldisplayT\nτ(s−τ)2/ba∇dbl∂th(x,s)/ba∇dbl2ds+/integraldisplayT\nτ/ba∇dbl∂th(x,s)/ba∇dbl2ds/parenrightbigg\n,(3.42)\nwheret∈(τ,T)andZ(·) =Z(L,λ,ρ,/tildewideC)>0.\nProof.Integrating ( 3.23) fromτtot, we conclude\nmax\nt∈[τ,T]/ba∇dbl∇u(t)/ba∇dbl2+/integraldisplayT\nτ/ba∇dbl∇∂tu(t)/ba∇dbl2dt≤c16, (3.43)\nwherec16>0 is a constant.\nTaking the derivative of ( 1.1)1with respect to tand assuming w=∂tu, we obtain\nε′(t)∂tw+ε(t)∂ttw−2δ(∇u,∇w)∆u−/parenleftbig\n1+δ/ba∇dbl∇u/ba∇dbl2/parenrightbig\n∆w−∆∂tw+λw\n=g′(u)w+∂th(x,t).(3.44)\nChoosing ( −∆)−1∂tw+ρwas a test function of ( 3.44), we derive\n1\n2d\ndt/parenleftbigg\nε(t)/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2\n+2ρε(t)(∂tw,w)+/ba∇dblw/ba∇dbl2+ρ/ba∇dbl∇w/ba∇dbl2+λ/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2w/vextenddouble/vextenddouble/vextenddouble2/parenrightbigg\n13+1\n2ε′(t)/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2\n−ρε(t)/ba∇dbl∂tw/ba∇dbl2+2δ(∇u,∇w)/parenleftbig\n−∆u,(−∆)−1∂tw/parenrightbig\n+λρ/ba∇dblw/ba∇dbl2\n+2δρ(∇u,∇w)2+δ/ba∇dbl∇u/ba∇dbl2/parenleftbig\n−∆w,(−∆)−1∂tw/parenrightbig\n+ρ/parenleftbig\n1+δ/ba∇dbl∇u/ba∇dbl2/parenrightbig\n/ba∇dbl∇w/ba∇dbl2+/ba∇dbl∂tw/ba∇dbl2\n=/parenleftbig\ng′(u)w,(−∆)−1∂tw/parenrightbig\n+ρ/parenleftbig\ng′(u)w,w/parenrightbig\n+/parenleftbig\n∂th(x,t),(−∆)−1∂tw+ρw/parenrightbig\n. (3.45)\nThanks to the Cauchy and Young inequalities, we deduce\n2δ(∇u,∇w)/parenleftbig\n−∆u,(−∆)−1∂tw/parenrightbig\n=δ/ba∇dbl∇u/ba∇dbl2/parenleftBig\n2(∇w,(−∆)−1\n2∂tw)/parenrightBig\n≤δ/ba∇dbl∇u/ba∇dbl2/parenleftbigg\n/ba∇dbl∇w/ba∇dbl2+/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2/parenrightbigg\n(3.46)\nand\nδ/ba∇dbl∇u/ba∇dbl2/parenleftbig\n−∆w,(−∆)−1∂tw/parenrightbig\n=δ/ba∇dbl∇u/ba∇dbl2/parenleftBig\n∇w,(−∆)−1\n2∂tw/parenrightBig\n≤δ\n2/ba∇dbl∇u/ba∇dbl2/parenleftbigg\n/ba∇dbl∇w/ba∇dbl2+/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2/parenrightbigg\n. (3.47)\nBesides, from ( 1.4) and the Young inequality, we conclude there exists a consta ntc17>0 such\nthat\n/vextendsingle/vextendsingle/parenleftbig\ng′(u)w,(−∆)−1∂tw/parenrightbig/vextendsingle/vextendsingle≤c17/parenleftBig\n1+/ba∇dbl∇u/ba∇dbl4\nn−2/parenrightBig/parenleftbigg\n/ba∇dbl∇w/ba∇dbl2+/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2/parenrightbigg\n.(3.48)\nFurthermore, it follows from ( 1.5) that\n−ρ(g′(u)w,w)≤ρk/ba∇dblw/ba∇dbl2. (3.49)\nThanks to the Young inequality, we derive\n/parenleftbig\n∂th(x,t),(−∆)−1∂tw+ρw/parenrightbig\n≤1\n2/ba∇dbl∂th(x,t)/ba∇dbl2+/vextenddouble/vextenddouble(−∆)−1∂tw/vextenddouble/vextenddouble2+ρ2/ba∇dblw/ba∇dbl2. (3.50)\nLet\nL(t) =ε(t)/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2\n+2ρε(t)(∂tw,w)+/ba∇dblw/ba∇dbl2+ρ/ba∇dbl∇w/ba∇dbl2+λ/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2w/vextenddouble/vextenddouble/vextenddouble2\n,(3.51)\nthen from Lemmas 3.2and3.3, we deduce there exist constants 0 < c18≤min{1,ρ}andc19≥\nmax{1,ρ}depending on L,λandρsuch that\nc18/parenleftbigg\nε(t)/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2\n+/ba∇dbl∇w/ba∇dbl2/parenrightbigg\n≤L(t)≤c19/parenleftbigg\nε(t)/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2\n+/ba∇dbl∇w/ba∇dbl2/parenrightbigg\n.(3.52)\n14Inserting ( 3.51) into (3.45), then by ( 1.2), (1.3) and (3.46)−(3.50), we arrive at\nd\ndtL(t)+2/ba∇dbl∂tw/ba∇dbl2≤(L+2)/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2\n+2ρL/ba∇dbl∂tw/ba∇dbl2\n+2c17/parenleftBig\n1+/ba∇dbl∇u/ba∇dbl4\nn−2/parenrightBig/parenleftbigg\n/ba∇dbl∇w/ba∇dbl2+/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2/parenrightbigg\n+/ba∇dbl∂th(x,t)/ba∇dbl2. (3.53)\nMoreover, from Lemma 3.2and noticing n≥3, we obtain that there exists a constant c20>0 such\nthat\nd\ndtL(t)+/ba∇dbl∂tw/ba∇dbl2≤(L+2)/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2\n+2ρL/ba∇dbl∂tw/ba∇dbl2\n+c20/parenleftbigg\n/ba∇dbl∇w/ba∇dbl2+/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2/parenrightbigg\n+/ba∇dbl∂th(x,t)/ba∇dbl2.(3.54)\nHence, there exists a constant c21>max{L+2+c20,2ρL}depending on /tildewideC,L,ρandλsuch that\nd\ndtL(t)+/ba∇dbl∂tw/ba∇dbl2≤c21L(t)+/ba∇dbl∂th(x,t)/ba∇dbl2. (3.55)\nNoticingw=∂tuand inserting ( 3.51) into (3.55), we obtain\nd\ndtL(t)≤c21L(t)+/ba∇dbl∂th(x,t)/ba∇dbl2. (3.56)\nBy the Gronwall inequality, we deduce that there exist const antsc22> c23>0 depending on\n/tildewideC,L,ρandλsuch that\nL(t)≤c22L(τ)+c23/integraldisplayT\nτ/ba∇dbl∂th(x,s)/ba∇dbl2ds. (3.57)\nTherefore, there exists a constant c24≥max{c22,c23}depending on /tildewideC,L,ρandλsuch that\nε(t)/ba∇dbl∂ttu(t)/ba∇dbl2\n−1+/ba∇dbl∇∂tu(t)/ba∇dbl2+/integraldisplayt+1\nt/ba∇dbl∂ttu(s)/ba∇dbl2ds\n≤c24/parenleftbigg\nL(τ)+c23/integraldisplayT\nτ/ba∇dbl∂th(x,s)/ba∇dbl2ds/parenrightbigg\n.(3.58)\nThen combining ( 3.51) with (3.58), it follows that there exists a constant c25>0 such that\nL(t)≤c25/parenleftBig\nε(t)/ba∇dbl∂ttu(t)/ba∇dbl2\n−1+/ba∇dbl∇∂tu(t)/ba∇dbl2/parenrightBig\n. (3.59)\nThus, thanks to ( 3.58) with (3.59), we conclude there exists a function Z(·) =Z(L,λ,ρ,/tildewideC)>\nc24c25>0 such that for any t∈[τ,T−1],\nε(t)/ba∇dbl∂ttu(t)/ba∇dbl2\n−1+/ba∇dbl∇∂tu(t)/ba∇dbl2+/integraldisplayt+1\nt/ba∇dbl∂ttu(s)/ba∇dbl2ds\n≤Z/parenleftbigg\nε(τ)/ba∇dbl∂ttu(τ)/ba∇dbl2\n−1+/ba∇dbl∇∂tu(τ)/ba∇dbl2+/integraldisplayT\nτ/ba∇dbl∂th(x,s)/ba∇dbl2ds/parenrightbigg\n.(3.60)\n15From (3.52) and (3.56), we derive\nd\ndt/parenleftbig\n(t−τ)2L(t)/parenrightbig\n+(t−τ)2/ba∇dbl∂tw/ba∇dbl2≤2(t−τ)c19/parenleftbigg\nε(t)/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2\n+/ba∇dbl∇w/ba∇dbl2/parenrightbigg\n+c21(t−τ)2L(t)+(t−τ)2/ba∇dbl∂th(x,t)/ba∇dbl2.(3.61)\nBy (3.51) and the embedding theorem, we obtain that there exist const antsc26,c27>0 such that\n2(t−τ)c19/parenleftbigg\nε(t)/vextenddouble/vextenddouble/vextenddouble(−∆)−1\n2∂tw/vextenddouble/vextenddouble/vextenddouble2\n+/ba∇dbl∇w/ba∇dbl2/parenrightbigg\n≤c26/parenleftbig\n1+(t−τ)2/ba∇dbl∇∂tu/ba∇dbl2L(t)/parenrightbig\n+1\n2(t−τ)2/ba∇dbl∂tw/ba∇dbl2+c27.(3.62)\nThen inserting ( 3.62) into (3.61), it follows that there exist constants c28,c29>0 such that\nd\ndt/parenleftbig\n(t−τ)2L(t)/parenrightbig\n≤c28(t−τ)2L(t)+c29(t−τ)2/ba∇dbl∂th(x,t)/ba∇dbl2. (3.63)\nUsing the Gronwall inequality, we deduce that there exists a constantC >0 such that\n(t−τ)2L(t)≤c29/integraldisplayt\nτe/integraltextt\nτc28dr(s−τ)2/ba∇dbl∂th(x,s)/ba∇dbl2ds\n≤c29C/integraldisplayT\nτ(s−τ)2/ba∇dbl∂th(x,s)/ba∇dbl2ds.(3.64)\nHence, it follows that\nL(t)≤c29C\n(t−τ)2/integraldisplayT\nτ(s−τ)2/ba∇dbl∂th(x,s)/ba∇dbl2ds. (3.65)\nInserting ( 3.65) into (3.60), then (3.42) follows directly. /square\nNext, we establish in Lemma 3.5the existence of a pullback absorbing family in Xtfor problem\n(1.1), which is helpful to derive the existence of pullback attra ctors.\nLemma 3.5. Under the assumptions of Lemma 3.4, if there exists a τδ>0such that\nUδ(t,t−τ)Dδ(t−τ)⊂Dδ(t) (3.66)\nand\nUδ(t−1,t−τ)Dδ(t−τ)⊂Dδ(t−1) (3.67)\nfor anyτ≥τδandt∈R, and assume /tildewideDδ={/tildewideDδ(t)}t∈Rsatisfies\n/tildewideDδ(t) =/uniondisplay\nτ≥τδUδ(t,t−τ)Dδ(t−τ)Xt\n, (3.68)\nthen the following properties hold\n16(i)/tildewideDδ={/tildewideDδ(t)}t∈Ris a pullback absorbing family in Xtand satisfies\nUδ(t,t−τ)/tildewideDδ(t−τ)⊂/tildewideDδ(t),∀τ≥T; (3.69)\n(ii)/tildewideDδ(t)is bounded in H1\n0(Ω)×H1\n0(Ω).\nProof.SupposeB1⊂Xtis bounded and choose a time τ1>0 large enough, then noticing\nDδ={Dδ(t)}t∈Ris a pullback absorbing family in Xt, we derive\nUδ(t−τδ−1,t−τ)B1⊆Dδ(t−τδ−1),∀τ≥τ1. (3.70)\nBy (3.68) and (3.70), then we conclude for any τ≥τδ,\nUδ(t,t−τ)B1=Uδ(t,t−τδ−1)Uδ(t−τδ−1,t−τ)B1\n⊆Uδ(t,t−τδ−1)Dδ(t−τδ−1)\n⊆/tildewideDδ(t). (3.71)\nSimilarly, we deduce\n/tildewideDδ(t)⊂Dδ(t),∀t∈R. (3.72)\nTherefore, it follows that\nUδ(t,t−τ)/tildewideDδ(t−τ)⊆Uδ(t,t−τ)Dδ(t−τ)⊆/tildewideDδ(t),∀τ≥τδ. (3.73)\nThen from ( 3.66)−(3.68), we obtain\n/tildewideDδ(t)⊆Uδ(t,t−τ)Dδ(t−τ)\n=Uδ(t,t−1)Uδ(t−1,t−τ)Dδ(t−τ)\n⊆Uδ(t,t−1)Dδ(t−1) (3.74)\nfor anyτ≥τδandt∈R.\nConsequently, by ( 3.43), it follows that /tildewideDδ(t) satisfies Lemma 3.5(ii). /square\nBefore proving the process Uδ(·,·) generated by problem ( 1.1) is pullback Dδ-asymptotically com-\npact in the time-dependent space Xt, we verify the following lemma in advance.\nLemma 3.6. Under the assumptions of Lemma 3.5, suppose /tildewideDδ={/tildewideDδ(t)}t∈Rsatisfies (3.68)and the\nprocessUδ(·,·)is pullback /tildewideDδ-asymptotically compact in the time-dependent space Xt, thenUδ(·,·)is\nalso pullback Dδ-asymptotically compact in Xt. In addition, it follows that\nΛ(/tildewideDδ,t) = Λ(Dδ,t). (3.75)\n17Proof.Leti∈Nandassumethesequences ri→+∞,τi→+∞areincreasingand xi∈Dδ(t−τi).\nThen from ( 3.68), we conclude\n/tildewideDδ(t−ri) =/uniondisplay\nτ≥τδ,t−riUδ(t−ri,t−τ)Dδ(t−τ)Xt\n. (3.76)\nBesides, let j∈N, and select a sequence τij∈ {τi}i∈Nsatisfiesτij≥τδ,t−rj,τij≥ττij−1and\nτij→+∞for anyrj∈ {ri}i∈N.\nSupposeaj=Uδ/parenleftbig\nt−rj,t−τij/parenrightbig\nxij, then byxi∈Dδ(t−τi) and (3.68), we derive\naj∈Uδ/parenleftbig\nt−rj,t−τij/parenrightbig\nDδ/parenleftbig\nt−τij/parenrightbig\n⊆/tildewideDδ(t−rj) (3.77)\nand\nUδ/parenleftbig\nt,t−τij/parenrightbig\nxij=Uδ(t,t−rj)Uδ/parenleftbig\nt−rj,t−τij/parenrightbig\nxij\n=Uδ(t,t−rj)aj.(3.78)\nFrom (3.77) and (3.78), we obtain\nUδ/parenleftbig\nt,t−τij/parenrightbig\nxij=Uδ(t,t−rj)aj⊆Uδ(t,t−rj)/tildewideDδ(t−rj). (3.79)\nNoticing the process Uδ(·,·) is pullback /tildewideDδ-asymptotically compact in the time-dependent space\nXt, it follows that/braceleftbig\nUδ/parenleftbig\nt,t−τij/parenrightbig\nxij/bracerightbig\nis precompact in Xt.\nTherefore, by Definition 2.4, we conclude that Uδ(·,·) is pullback Dδ-asymptotically compact in\nXt.\nFurthermore, thanks to ( 3.68), we deduce\n/tildewideDδ(t)⊂Dδ(t),∀t∈R, (3.80)\nthen from Lemma 2.6, we arrive at\nΛ(/tildewideDδ,t)⊂Λ(Dδ,t),∀t∈R. (3.81)\nSupposea∈Λ(Dδ,t), then there exist sequences τi→+∞andxi∈Dδ(t−τi) such that\nUδ(t,t−τi)xi∈Uδ(t,t−τi)Dδ(t−τi)→aasi→+∞. (3.82)\nSimilarly, we can choose sequences rj→+∞andτij∈ {τi}i∈Nsuch that\naj=Uδ/parenleftbig\nt−rj,t−τij/parenrightbig\nxij⊂/tildewideDδ(t−rj). (3.83)\n18Then by ( 3.82), we derive\nUδ(t,t−rj)aj=Uδ(t,t−rj)Uδ/parenleftbig\nt−rj,t−τij/parenrightbig\nxij\n=Uδ/parenleftbig\nt,t−τij/parenrightbig\nxij→a. (3.84)\nFurthermore, from ( 3.76) and (3.77), we obtain\nUδ(t,t−rj)aj∈Uδ(t,t−rj)/tildewideDδ(t−rj)⊆Λ(/tildewideDδ,t). (3.85)\nHence, thanks to ( 3.83) and (3.84), we conclude\nUδ(t,t−rj)aj→a∈Λ(/tildewideDδ,t). (3.86)\nConsequently, ( 3.75) holds immediately. /square\nTheorem 3.7. Under the assumptions of Lemma 3.6, then for any δ≥0, there exists a pullback\nattractor /tildewideAδ={/tildewideA(t)}t∈Rfor the process Uδ(·,·)generated by problem (1.1)in the time-dependent\nspaceXt, which satisfies\n/tildewideAδ(t) = Λ(/tildewideDδ,t),∀t∈R. (3.87)\nProof.From Lemmas 2.6and3.3, we only need to prove the process Uδ(·,·) is pullback Dδ-\nasymptotically compact in Xt.\nLetδ∈[0,1], then by Lemma 3.3and (3.43), we deduce\nmax\nt∈[t0−τ,t0]/ba∇dbl∇u(t)/ba∇dbl2+/integraldisplayt\nt0−τ/ba∇dbl∇∂tu(t)/ba∇dbl2dt≤c16 (3.88)\nfor any/parenleftbig\nu0\nt0−τ,u1\nt0−τ/parenrightbig\n∈/tildewideDδ(t0−τ), where the constant c16is the same as in ( 3.43).\nDecompose the solutions to problem ( 1.1) as\nUδ(t,t0−τ)/parenleftbig\nu0\nt0−τ;u1\nt0−τ/parenrightbig\n=/parenleftbig\nUa,δ(t,t0−τ)u0\nt0−τ;Ub,δ(t,t0−τ)u1\nt0−τ/parenrightbig\n= (u(t);∂tu(t)), (3.89)\nwhere\nUa,δ(t,t0−τ)u0\nt0−τ=Ua1,δ(t,t0−τ)u0\nt0−τ+Ua2,δ(t,t0−τ)u0\nt0−τ (3.90)\nwith\nUa1,δ(t1,t0−τ)u0\nt0−τ=u1(t) (3.91)\n19and\nUa2,δ(t,t0−τ)u0\nt0−τ=u2(t). (3.92)\nThenUa1,δ(t,t0−τ)u0\nt0−τandUa2,δ(t,t0−τ)u0\nt0−τsatisfy the following equations\n\n\n−/parenleftbig\n1+δ/ba∇dbl∇u/ba∇dbl2/parenrightbig\n∆u1−∆∂tu1+λu1=ϕ(u1+u2)−ϕ(u2) in Ω ×(τ,+∞),\nu1(x,t) = 0 on ∂Ω×(τ,+∞),\nu1(x,t−τ) =u0\nt−τ, x ∈Ω,(3.93)\nand \n\n−/parenleftbig\n1+δ/ba∇dbl∇u/ba∇dbl2/parenrightbig\n∆u2−∆∂tu2+λu2=ϕ(u2)+ψin Ω×(τ,+∞),\nu2(x,t) = 0 on ∂Ω×(τ,+∞),\nu2(x,t−τ) = 0, x ∈Ω,(3.94)\nrespectively, where ϕ(s) =g(s)−ksandϕ′(s)≤0 for anys∈R,ψ=−ε(t)∂ttu+ku+h(x,t), andk\nis the same as in ( 1.5).\nChoosingu1as a test function of ( 3.93)1, we derive\nd\ndt/ba∇dbl∇u1/ba∇dbl2+2/ba∇dbl∇u1/ba∇dbl2+2δ/ba∇dbl∇u/ba∇dbl2/ba∇dbl∇u1/vextenddouble/vextenddouble2+2λ/vextenddouble/vextenddoubleu1/ba∇dbl2≤0. (3.95)\nThen by the Gronwall inequality, ( 3.91) and (3.93)3, we obtain\n/ba∇dbl∇u1(t)/ba∇dbl2=/vextenddouble/vextenddouble∇Ua1,δ(t,t0−τ)u0\nt0−τ/vextenddouble/vextenddouble2≤e−2(t−t0+τ)/vextenddouble/vextenddouble∇u0\nt0−τ/vextenddouble/vextenddouble2. (3.96)\nIn addition, choosing −∆u2as a test function of ( 3.94)1, we conclude\n2/ba∇dbl∆u2/ba∇dbl2+2δ/ba∇dbl∇u/ba∇dbl2/ba∇dbl∆u2/ba∇dbl2+d\ndt/ba∇dbl∆u2/ba∇dbl2+2λ/ba∇dbl∇u2/ba∇dbl2\n= 2(ϕ(u2),−∆u2)+2(−ε(t)∂ttu+ku+h(x,t),−∆u2).(3.97)\nThanks to ( 1.4),ϕ(s) =g(s)−ksand the Young inequality, we arrive at\n2|(ϕ(u2),−∆u2)| ≤4/ba∇dblg(u2)/ba∇dbl2+1\n2/ba∇dbl∆u2/ba∇dbl2+4k2/ba∇dblu2/ba∇dbl2\n≤C/parenleftBig\n1+/ba∇dblu2/ba∇dbl2n+4\nn−2/parenrightBig\n+1\n2/ba∇dbl∆u2/ba∇dbl2+4k2/ba∇dblu2/ba∇dbl2.(3.98)\nSimilarly, by ( 1.2), (1.3) and the Young inequality, we deduce\n2(−ε(t)∂ttu+ku+h(x,t),−∆u2)\n≤2L2/ba∇dbl∂ttu/ba∇dbl2+3\n2/ba∇dbl∆u2/ba∇dbl2+2k2/ba∇dblu/ba∇dbl2+2/ba∇dblh(x,t)/ba∇dbl2.(3.99)\n20Inserting ( 3.98), (3.99) into (3.97), we obtain\n2δ/ba∇dbl∇u/ba∇dbl2/ba∇dbl∆u2/ba∇dbl2+d\ndt/ba∇dbl∆u2/ba∇dbl2+2λ/ba∇dbl∇u2/ba∇dbl2\n≤C/parenleftBig\n1+/ba∇dblu2/ba∇dbl2n+4\nn−2/parenrightBig\n+4k2/ba∇dblu2/ba∇dbl2+2L2/ba∇dbl∂ttu/ba∇dbl2+2k2/ba∇dblu/ba∇dbl2+2/ba∇dblh(x,t)/ba∇dbl2.(3.100)\nUsing Lemma 3.3and noticing u=u1+u2, we derive there exist constants c30,c31,c32>0\ndepending on δandλsuch that\nd\ndt/ba∇dbl∆u2/ba∇dbl2+c30/ba∇dbl∆u2/ba∇dbl2≤c31+c32/ba∇dblh(x,t)/ba∇dbl2. (3.101)\nLetc33= max{c31,c32}, then by the Gronwall inequality, we obtain\n/ba∇dbl∆u2/ba∇dbl2≤e−/integraltextt\nt0−τc30dsu2(t0−τ)+c33/integraldisplayt\nt0−τe/integraltexts\ntc30dy/ba∇dblh(x,s)/ba∇dbl2ds\n=c33/integraldisplayt\nt0−τe−c30(t−s)/ba∇dblh(x,s)/ba∇dbl2ds,∀t∈[t0−τ,t0].(3.102)\nThanksto( 3.60)and(3.102),wededucethereexistsaconstant W1>0dependingon c30,c31,c32,c33,t0,τ\nand/ba∇dbl/tildewideDδ(t0−τ)/ba∇dbl2\nXtsuch that\n/ba∇dbl∆u2/ba∇dbl=/vextenddouble/vextenddouble∆Ua2,δ(t,t0−τ)u0\nt0−τ/vextenddouble/vextenddouble2≤W1<+∞ (3.103)\nfor anyt∈[t0−τ,t0] and/parenleftbig\nu0\nt0−τ;u1\nt0−τ/parenrightbig\n∈/tildewideDδ(t0−τ).\nFrom (3.42) andUb,δ(t,t0−τ)u1\nt0−τ) =∂tu(t), it follows that there exists a constant W2>0\ndepending on τ,TandZ(·) such that\n/vextenddouble/vextenddouble∇Ub,δ(t,t0−τ)u1\nt0−τ/vextenddouble/vextenddouble2≤W2<+∞ (3.104)\nfor anyτ >0 and any/parenleftbig\nu0\nt0−τ;u1\nt0−τ/parenrightbig\n∈/tildewideDδ(t0−τ).\nHence, from Lemma 2.9, we deduce the process Uδ(·,·) generated by problem ( 1.1) is pullback\nDδ-asymptotically compact in Xt.\nThen by Lemma 2.6, we derive there exists a pullback attractor /tildewideAδ={/tildewideA(t)}t∈RinXt, which\nsatisfies ( 3.87). /square\nRemark 3.8. From Lemmas 3.3and3.5and Theorem 3.7, then it follows that the process Uδ(·,·)\ngenerated by problem (1.1)is pullback Dδ-asymptotically compact in Xtand/tildewideAδ(t) =Aδ(t) := Λ(Dδ,t)\nfor anyt∈Rand anyδ≥0.\n214 Upper semicontinuity of pullback attractors\nIn this section, by using Lemmas 2.10and2.11, we shall investigate the relationship between Aδ(t)\nandA0(t), which are pullback attractors of processes generated by p roblem (1.1) forδ>0 andδ= 0\nin the time-dependent space Xt, respectively.\nThe following theorem is the main result of this section.\nTheorem 4.1. Under the assumptions of Theorem 3.7, then the pullback attractor Aδ={Aδ(t)}t∈R\ngiven by Remark 3.8satisfies\nlim\nδ→0+sup\nt∈[α,β]distXt(Aδ(t),A0(t)) = 0 (4.1)\nfor any[α,β]⊂Randδ≥0.\nIn order to prove Theorem 4.1, it is necessary to establish the following three lemmas, th en from\nLemmas 2.10and2.11, we can easily derive Theorem 4.1.\nLemma 4.2. Under the assumptions of Theorem 4.1, assume the family /tildewideDδ={/tildewideDδ(t)}t∈Rsatisfies\n(3.68), then there exists a δ∈[0,1]such that\n/uniondisplay\nδ∈[0,1]Dδ(t)Xt\n⊂D0(t) (4.2)\nfor anyt∈R.\nProof.By Lemmas 3.3and3.5, we conclude there exists a δ∈[0,1] such that ( 4.2) holds. /square\nLemma 4.3. Under the assumptions of Theorem 4.1, assumeτ >0,ǫi→0+,xi→x0in the\ntime-dependent space Xtandti>αwithti→t0for anyi∈N, then\nUǫi(ti,α−τ)xi→U0(t0,α−τ)x0 (4.3)\nin the time-dependent space Xt.\nProof.Supposeui(t) =Uδi(t,α−τ)xiis a weak solution to problem ( 1.1) with respect to δ=δi\nand initial value/parenleftbig\nu0\nα−τ;u1\nα−τ/parenrightbig\n= (u0\nα−τ,i;u1\nα−τ,i) =xi.\nSimilarly, let u0=U0(t,α−τ)x0be a weak solution to problem ( 1.1) with respect to δ= 0 and\ninitial value/parenleftbig\nu0\nα−τ;u1\nα−τ/parenrightbig\n=/parenleftbig\nu0\nα−τ,0;u1\nα−τ,0/parenrightbig\n=x0.\n22Assumeti∈(α,T) and\n/vextenddouble/vextenddouble∇u0\nα−τ,i/vextenddouble/vextenddouble2+ε(α−τ)/vextenddouble/vextenddoubleu1\nα−τ,i/vextenddouble/vextenddouble2≤V1,∀i∈N, (4.4)\nwhereV1>0 is a constant.\nFrom (3.88), we deduce there exists a constant V2=V2(α−τ,T,V1)>0 such that\n/ba∇dbl∇ui(t)/ba∇dbl2+/integraldisplayT\nα−τ/ba∇dbl∇∂tui(s)/ba∇dbl2ds≤V2,∀t∈[α−τ,T], i∈N. (4.5)\nLetz=ua−ubandδ=δa−δb, thenzsatisfies\nε(t)∂ttz−∆z−δa/ba∇dbl∇ua/ba∇dbl2∆ua+δb/ba∇dbl∇ub/ba∇dbl2∆ub−∆∂tz+λz\n=g(ua)−g(ub).(4.6)\nThen inserting\n−1\n2/parenleftBig\nδa/ba∇dbl∇ua/ba∇dbl2+δb/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n∆z−1\n2δa/parenleftBig\n/ba∇dbl∇ua/ba∇dbl2−/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n∆(ua+ub)\n−1\n2(δa−δb)/ba∇dbl∇ub/ba∇dbl2∆(ua+ub)\n=−δa/ba∇dbl∇ua/ba∇dbl2∆ua+δb/ba∇dbl∇ub/ba∇dbl2∆ub (4.7)\ninto (4.6), we obtain\nε(t)∂ttz−∆z−1\n2/parenleftBig\nδa/ba∇dbl∇ua/ba∇dbl2+δb/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n∆z−1\n2δa/parenleftBig\n/ba∇dbl∇ua/ba∇dbl2−/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n∆(ua+ub)\n−1\n2(δa−δb)/ba∇dbl∇ub/ba∇dbl2∆(ua+ub)−∆∂tz+λz\n=g(ua)−g(ub). (4.8)\nAdditionally, the initial data of zis\n/parenleftbig\nz0\nα−τ;z1\nα−τ/parenrightbig\n=/parenleftbig\nu0\nα−τ,a−u0\nα−τ,b;u′\nα−τ,a−u′\nα−τ,b/parenrightbig\n. (4.9)\nChoosing∂tz+ξzas a test function to ( 4.8), we conclude\n1\n2d\ndt/parenleftbig\nε(t)/ba∇dbl∂tz/ba∇dbl2+2ξε(t)(∂tz,z)+/ba∇dbl∇z/ba∇dbl2+ξ/ba∇dbl∇z/ba∇dbl2+λ/ba∇dblz/ba∇dbl2/parenrightbig\n−1\n2ε′(t)/ba∇dbl∂tz/ba∇dbl2\n−ξε′(t)(∂tz,z)−ξε(t)/ba∇dbl∂tz/ba∇dbl2+ξ/ba∇dbl∇z/ba∇dbl2+ξ\n2/parenleftBig\nδa/ba∇dbl∇ua/ba∇dbl2+δb/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n/ba∇dbl∇z/ba∇dbl2\n+ξδa\n2/parenleftBig\n/ba∇dbl∇ua/ba∇dbl2−/ba∇dbl∇ub/ba∇dbl2/parenrightBig2\n+1\n2/parenleftBig\nδa/ba∇dbl∇ua/ba∇dbl2+δb/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n(−∆z,∂tz)\n+δa\n2/parenleftBig\n/ba∇dbl∇ua/ba∇dbl2−/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n(−∆(ua+ub),∂tz)+/ba∇dbl∇∂tz/ba∇dbl2+λξ/ba∇dblz/ba∇dbl2\n23+δa−δb\n2/ba∇dbl∇ub/ba∇dbl2(−∆(ua+ub),∂tz)+ξ(δa−δb)\n2/ba∇dbl∇ub/ba∇dbl2/parenleftBig\n/ba∇dbl∇ua/ba∇dbl2−/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n= (g(ua)−g(ub),∂tz+ξz), (4.10)\nwhereξ>0 is a constant.\nFurthermore, let\nE0(t) =−1\n2ε′(t)/ba∇dbl∂tz/ba∇dbl2−ξε′(t)(∂tz,z)−ξε(t)/ba∇dbl∂tz/ba∇dbl2+ξ/ba∇dbl∇z/ba∇dbl2\n+ξ\n2/parenleftBig\nδa/ba∇dbl∇ua/ba∇dbl2+δb/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n/ba∇dbl∇z/ba∇dbl2+/ba∇dbl∇∂tz/ba∇dbl2+λξ/ba∇dblz/ba∇dbl2\n+ξδa\n2/parenleftBig\n/ba∇dbl∇ua/ba∇dbl2−/ba∇dbl∇ub/ba∇dbl2/parenrightBig2\n, (4.11)\nE1(t) =1\n2/parenleftBig\nδa/ba∇dbl∇ua/ba∇dbl2+δb/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n(−∆z,∂tz)\n+δa\n2/parenleftBig\n/ba∇dbl∇ua/ba∇dbl2−/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n(−∆(ua+ub),∂tz),(4.12)\nE2(t) =δa−δb\n2/ba∇dbl∇ub/ba∇dbl2(−∆(ua+ub),∂tz)\n+ξ(δa−δb)\n2/ba∇dbl∇ub/ba∇dbl2/parenleftBig\n/ba∇dbl∇ua/ba∇dbl2−/ba∇dbl∇ub/ba∇dbl2/parenrightBig (4.13)\nand\nE3(t) = (g(ua)−g(ub),∂tz+ξz). (4.14)\nThen inserting ( 4.11)−(4.14) into (4.10), we derive\n1\n2d\ndt/parenleftBig\nε(t)/ba∇dbl∂tz/ba∇dbl2+2ξε(t)(∂tz,z)+/ba∇dbl∇z/ba∇dbl2+ξ/ba∇dbl∇z/ba∇dbl2+λ/ba∇dblz/ba∇dbl2/parenrightBig\n+E0(t)+E1(t)+E2(t) =E3(t). (4.15)\nBy (4.5), the Cauchy and Young inequalities, we deduce\n|E1(t)| ≤1\n2/parenleftBig\nδa/ba∇dbl∇ua/ba∇dbl2+δb/ba∇dbl∇ub/ba∇dbl2/parenrightBig/parenleftbigg1\n2/ba∇dbl∇z/ba∇dbl2+1\n2/ba∇dbl∇∂tz/ba∇dbl2/parenrightbigg\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingleδa\n2/parenleftBig\n/ba∇dbl∇ua/ba∇dbl2−/ba∇dbl∇ub/ba∇dbl2/parenrightBig\n[(∇ua,∇∂tz)+(∇ub,∇∂tz)]/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤1\n4(δa+δb)V2\n2+1\n4(δa+δb)V2/ba∇dbl∇∂tz/ba∇dbl2\n+δa\n2/vextendsingle/vextendsingle/vextendsingle/ba∇dbl∇ua/ba∇dbl2[(∇ua,∇∂tz)+(∇ub,∇∂tz)]/vextendsingle/vextendsingle/vextendsingle\n≤3\n4δaV2\n2+1\n4δbV2\n2+3\n4δaV2/ba∇dbl∇∂tz/ba∇dbl2+1\n4δbV2/ba∇dbl∇∂tz/ba∇dbl2\n≤2(δa+δb)V2\n2+2(δa+δb)V2/ba∇dbl∇∂tz/ba∇dbl2(4.16)\n24and\n|E2(t)| ≤|δa−δb|\n32ξ/ba∇dbl∇ub/ba∇dbl2/ba∇dbl∇ua/ba∇dbl2+|δa−δb|\n4/ba∇dbl∇ub/ba∇dbl2/ba∇dbl∇∂tz/ba∇dbl2\n+ξ|δa−δb|\n2/ba∇dbl∇ub/ba∇dbl2/ba∇dbl∇ua/ba∇dbl2\n≤ |δa−δb|(1+δ)V2\n2+|δa−δb|V2/ba∇dbl∇∂tz/ba∇dbl2, (4.17)\nwheret∈[α−τ,T] anda,b∈N.\nUsing (1.4), (1.5), the Young inequality and the embedding H1\n0(Ω)֒→L2n\nn−2(Ω), we derive there\nexists a constant C=C(k,ξ,V2)>0 such that\nE3(t)≤C/parenleftBig\n/ba∇dbl∇z/ba∇dbl2+/ba∇dbl∂tz/ba∇dbl2/parenrightBig\n+1\n2/ba∇dbl∇∂tz/ba∇dbl2. (4.18)\nLet\n/tildewideE(t) =ε(t)/ba∇dbl∂tz/ba∇dbl2+2ξε(t)(∂tz,z)+/ba∇dbl∇z/ba∇dbl2+ξ/ba∇dbl∇z/ba∇dbl2+λ/ba∇dblz/ba∇dbl2, (4.19)\nthen from ( 4.15)−(4.19), we obtain there exists a constant /tildewideC=/tildewideC(δ,k,L,V 2)>0 such that\nd\ndt/tildewideE(t)≤/tildewideC/tildewideE(t)+/tildewideC(δa+δb)/parenleftbig\n/ba∇dbl∇∂tz(t)/ba∇dbl2+1/parenrightbig\n. (4.20)\nThen by the Gronwall inequality, we arrive at\n/tildewideE(t)≤e/tildewideC(T−α+τ)/tildewideE(α−τ)+/tildewideC(δa+δb)/integraldisplayT\nα−τe/tildewideC(T−s)/parenleftbig\n/ba∇dbl∇∂tz(s)/ba∇dbl2+1/parenrightbig\nds. (4.21)\nThanks to ( 4.5) and (4.19), then there exists a constant C >0 such that\n/tildewideC(δa+δb)/integraldisplayT\nα−τe/tildewideC(T−s)/parenleftbig\n/ba∇dbl∇∂tz(s)/ba∇dbl2+1/parenrightbig\nds≤C. (4.22)\nMoreover, noticing ui(t) =Uδi(t,α−τ)xiandz=ua−ub, we derive\nua(t) =Uδa(t,α−τ)xa (4.23)\nand\nub(t) =Uδb(t,α−τ)xb. (4.24)\nFrom (4.19), (4.21), (4.23) and (4.24), we deduce\n/tildewideE(t) =/ba∇dblUδa(t,α−τ)xa−Uδb(t,α−τ)xb/ba∇dbl2\nXt+ξ(/ba∇dbl∇z(t)/ba∇dbl2+2ε(t)(∂tz(t),z(t)))\n+λ/ba∇dblz(t)/ba∇dbl2.(4.25)\n25From (4.21) and (4.25), we obtain\n/ba∇dblUδa(t,α−τ)xa−Uδb(t,α−τ)xb/ba∇dbl2\nXt+ξ(/ba∇dbl∇z(t)/ba∇dbl2+2ε(t)(∂tz(t),z(t)) +λ/ba∇dblz(t)/ba∇dbl2\n≤e/tildewideC(T−α+τ)/tildewideE(α−τ)+/tildewideC(δa+δb)/integraldisplayT\nα−τe/tildewideC(T−s)/parenleftbig\n/ba∇dbl∇∂tz(s)/ba∇dbl2+1/parenrightbig\nds.(4.26)\nSuppose /tildewideC1≥max{2ξLV2,/tildewideC}, then by ( 4.5) and (4.26), we conclude\n/ba∇dblUδa(t,α−τ)xa−Uδb(t,α−τ)xb/ba∇dbl2\nXt\n≤e/tildewideC1(T−α+τ)/bracketleftBig\nε(α−τ)/vextenddouble/vextenddoubleu1\nα−τ,a−u1\nα−τ,b/vextenddouble/vextenddouble2\n+2ξε(α−τ)/parenleftbig\nu1\nα−τ,a−u1\nα−τ,b,u0\nα−τ,a−u0\nα−τ,b/parenrightbig\n+(1+ξ)/vextenddouble/vextenddouble∇/parenleftbig\nu0\nα−τ,a−u0\nα−τ,b/parenrightbig/vextenddouble/vextenddouble2+λ/vextenddouble/vextenddoubleu0\nα−τ,a−u0\nα−τ,b/vextenddouble/vextenddouble2/bracketrightBig\n+/tildewideC1(δa+δb)/integraldisplayT\nα−τe/tildewideC1(T−s)/parenleftbig\n/ba∇dbl∇∂tz(s)/ba∇dbl2+1/parenrightbig\nds. (4.27)\nUsing (4.22) and (4.27), we derive there exists a constant V3=V3(/tildewideC1,V2,ξ,λ)>0 such that\n/ba∇dblUδa(t,α−τ)xa−Uδb(t,α−τ)xb/ba∇dbl2\nXt\n≤V3/parenleftBig/vextenddouble/vextenddouble∇/parenleftbig\nu0\nα−τ,a−u0\nα−τ,b/parenrightbig/vextenddouble/vextenddouble2+/vextenddouble/vextenddoubleu1\nα−τ,a−u1\nα−τ,b/vextenddouble/vextenddouble2+δa+δb/parenrightBig (4.28)\nfor anyt∈[α−τ,T] anda,b∈N.\nFurthermore, it follows from ( 4.28) that there exists a Nδ∈Nsuch that\n/ba∇dblUδa(t,α−τ)xa−Uδb(t,α−τ)xb/ba∇dbl2\nXt≤η, (4.29)\nwhereη>0 is a constant, t∈[α,T] anda,b∈N.\nTherefore, we obtain {Uδi(t,α−τ)xi}i∈Nis a Cauchy sequence in Xtfor anyt∈[α,T].\nWithout loss of generality, let/braceleftbig\nUδi(˜t,α−τ)xi/bracerightbig\ni∈Nbe a convergent sequence in Xt, that is, for\nsome ˜y∈Xt,\nUδi(˜t,α−τ)xi→˜yasi→+∞. (4.30)\nThen it follows that\n/vextenddouble/vextenddoubleUδi(ti,α−τ)xi−Uδi(˜t,α−τ)xi/vextenddouble/vextenddouble\nXt\n=/ba∇dblUδi(ti,α−τ)xi���Uδi(˜t,α−τ)xi−Uδb(ti,α−τ)xb+Uδb(ti,α−τ)xb\n−Uδb(˜t,α−τ)xb+Uδb(˜t,α−τ)xb/ba∇dblXt\n≤ /ba∇dblUδi(ti,α−τ)xi−Uδb(ti,α−τ)xb/ba∇dblXt+/vextenddouble/vextenddoubleUδb(ti,α−τ)xb−Uδb(˜t,α−τ)xb/vextenddouble/vextenddouble\nXt\n26+/vextenddouble/vextenddoubleUδb(˜t,α−τ)xb−Uδi(˜t,α−τ)xi/vextenddouble/vextenddouble\nXt. (4.31)\nNext, we will take turns to analyze each item in ( 4.31).\nNoticingti→t0asi→+∞, then we can find suitable ˜b,n1∈Nsuch that\n/vextenddouble/vextenddoubleUδi(ti,α−τ)xi−Uδi(˜t,α−τ)xi/vextenddouble/vextenddouble\nXt\n≤/vextenddouble/vextenddouble/vextenddoubleUδi(ti,α−τ)xi−Uδ˜b(ti,α−τ)x˜b/vextenddouble/vextenddouble/vextenddouble\nXt+/ba∇dblU˜δ˜b(ti,α−τ)x˜b−Uδi(˜t,α−τ)xi/ba∇dblXt\n≤/vextenddouble/vextenddouble/vextenddoubleUδi(ti,α−τ)xi−Uδ˜b(ti,α−τ)x˜b/vextenddouble/vextenddouble/vextenddouble\nXt+/vextenddouble/vextenddouble/vextenddoubleUδ˜b(˜t,α−τ)x˜b−Uδi(˜t,α−τ)xi/vextenddouble/vextenddouble/vextenddouble\nXt\n≤η\n2. (4.32)\nBy Theorem 3.1andui(t) =Uδi(t,α−τ)xi, we deduce Uδ/tildewidek(·,α−τ)∈C([α−τ,T];Xt), which\ncombines with ti→˜tasi→+∞to yield that there exists a n2≥n1such that\n/vextenddouble/vextenddouble/vextenddoubleUδ˜b(ti,α−τ)x˜b−Uδ˜b(˜t,α−τ)x˜b/vextenddouble/vextenddouble/vextenddouble\nXt≤η\n2(4.33)\nwhenn≥n2.\nFrom (4.31)−(4.33), we obtain there exists a nη∈Nlarge enough such that\n/vextenddouble/vextenddoubleUδi(ti,α−τ)xi−Uδi(˜t,α−τ)xi/vextenddouble/vextenddouble\nXt≤η,∀n≥nη. (4.34)\nThus by ( 4.30) and (4.34), we arrive at\nUδi(ti,α−τ)xi→˜yasi→+∞ (4.35)\ninXt.\nIn addition, if let the constant V2in (4.5) be large enough, then ( 4.34) also holds for u0.\nSupposev=ui−u0, thenvsatisfies\nε(t)∂ttv−∆v−δi/ba∇dbl∇ui/ba∇dbl2∆ui−∆∂tv+λv=g(ui)−g(u0) (4.36)\nwith initial data\n/parenleftbig\nv0\nα−τ;v1\nα−τ/parenrightbig\n=/parenleftbig\nu0\nα−τ,i−u0\nα−τ,0;u1\nα−τ,i−u1\nα−τ,0/parenrightbig\n. (4.37)\nFrom Lemma 3.2, we obtain\n/vextenddouble/vextenddouble∇/parenleftbig\nUδi(˜t,α−τ)xi−U0(˜t,α−τ)x0/parenrightbig/vextenddouble/vextenddouble2+ε(˜t)/vextenddouble/vextenddouble∂t/parenleftbig\nUδi(˜t,α−τ)xi−U0(˜t,α−τ)x0/parenrightbig/vextenddouble/vextenddouble2→0.(4.38)\n27Then by ( 4.30) and (4.35), we conclude\nUδi(ti,α−τ)xi→U0(t0,α−τ)x0 (4.39)\ninXt. /square\nInspired by Lemmas 2.10and2.11, if we can verify the three properties (i) −(iii) in Lemma 2.11,\nthen the following lemma holds immediately. To this end, we w ill perform some calculations similar\nto those in the proof of Theorem 3.7.\nLemma 4.4. Under the assumptions of Theorem 4.1, then for any δi∈[0,1],ti∈[α,β],τi→+∞\nandxi∈Dδi(ti−τi)withi∈N, the sequence {Uδi(ti,ti−τi)xi}i∈Nis relatively compact in Xt.\nProof.Thanks to ( 3.96), we deduce\n/vextenddouble/vextenddouble∇va1,δ(t,t0−τ)u0\nt0−τ/vextenddouble/vextenddouble2≤e−2τ/vextenddouble/vextenddouble∇u0\nt0−τ/vextenddouble/vextenddouble2, (4.40)\nwhich leads to\nsup\nδ∈[0,1]/vextenddouble/vextenddouble∇Ua1,δ(t,α−τ)u0\nα−τ/vextenddouble/vextenddouble2≤e−2τ/vextenddouble/vextenddouble∇u0\nα−τ/vextenddouble/vextenddouble2. (4.41)\nFrom (3.103), we obtain there exists a constant /tildewiderW1>0 such that\nsup\nδ∈[0,1]/vextenddouble/vextenddouble∆Ua2,δ(t,α−τ)u0\nα−τ/vextenddouble/vextenddouble2≤/tildewiderW1. (4.42)\nFurthermore, by ( 3.104), we deduce there exists a constant /tildewiderW2>0 such that\nsup\nδ[0,1]/vextenddouble/vextenddouble∇Ub,δ(t,α−τ)u1\nα−τ/vextenddouble/vextenddouble2≤/tildewiderW2. (4.43)\nThe above inequalities ( 4.40)−(4.42) hold for any t∈[α,β],τ >0 and/parenleftbig\nu0\nα−τ;u1\nα−τ/parenrightbig\n∈Dδ(t−τ),\n/tildewiderW1and/tildewiderW2depend onα−τand/ba∇dblDδ(α−τ)/ba∇dblXt.\nAs a result, by Lemmas 2.10and2.11, then it follows that {Uδi(ti,ti−τi)xi}i∈Nis relatively\ncompact in Xt. /square\nAcknowledgment\nWe would like to express our sincere gratitude to the editors and anonymous reviewers for their\ninvaluable feedback and insightful comments on the initial draft of this paper. Their thorough review\nand constructive suggestions have played a crucial role in i mproving the quality of this manuscript.\nWe greatly appreciate their dedication and support through out the review process.\n28Funding\nYang was supported by the China Scholarship Council with num ber 202206630048. Qin was sup-\nported by the National Natural Science Foundation of China w ith contract numbers 12171082 and\n12226403, thefundamentalresearchfundsforthecentral un iversitieswithcontract numbers2232022 G-\n13, 2232023 G-13 and a grant from science and technology commission of Sha nghai municipality. Wang\nwas supported by Science and Technology Commission of Shang hai Municipality with contract num-\nbers 23ZR1402100.\nConflict of interest statement\nThe authors have no conflict of interest.\n29References\n[1] A. V. Babin and M. I. Vishik, Attractors of evolutionary e quations, North-Holland, Amsterdam,\n1992.\n[2] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attra ctors for asymptotically compact non-\nCautonomous dynamical systems, Nonlinear Anal., 64(3)(20 06), 484-498.\n[3] I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J.\nDiff. Equ., 252(1)(2012), 1229-1262.\n[4] M. Conti, V. Pata and R. 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Jorge Silva, Attractors and their proper ties for a class of Kirchhoff models\nwith integro-differential damping, Appl. Analy., 101(9)(20 20), 3284-3307.\n[11] H. Ma, J. Wang and J. Xie, Pullback attractors for nonaut onomous degenerate Kirchhoff equa-\ntions with strong damping, Adv. Math. Phy., (2021), 7575078 , DOI: 10.1155/2021/7575078.\n[12] H. Ma and C. Zhong, Attractors for the Kirchhoff equation s with strong nonlinear damping,\nAppl. Math. Lett., 7(1)(2017), 127-133.\n[13] X. Peng, Y. Shang and X. Zheng, Pullback attractors of no nautonomous nonclassical diffu-\nsion equations with nonlocal diffusion, Zeitschrift f¨ ur Ang ewandte Mathematik und Physik,\n69(4)(2018), 1-14.\n[14] J. C. Robinson, Infinite-dimensional dynamical system s, Cambridge University Press, England,\n2011.\n[15] Y. Wang, Pullback attractors for nonautonomous wave eq uations with critical exponent, Non-\nlinear Anal., TMA, 68(2)(2008), 365-376.\n[16] Y. Wang and C. Zhong, Upper semicontinuity of pullback a ttractors for nonautonomous Kirch-\nhoff wave models, Disc. Conti. Dyn. Sys., 33(7)(2013), 3189- 3209.\n[17] Z. Yang and Y. Li, Criteria on the existence and stabilit y of pullback exponential attractors\n30and their application to non-autonomous kirchhoff wave mode ls, Disc. Contin. Dynam. Syst.,\n38(5)(2018), 2629-2653.\n[18] Z. Yang and Y. Li, Upper semicontinuity of pullback attr actors for non-autonomous Kirchhoff\nwave equations, Disc. Contin. Dynam. Syst., 24(9)(2019), 4 899-4912.\n[19] K. Zhu and C. Sun, Pullback attractors for nonclassical diffusion equations with delays, J. Math.\nPhy., 56(9)(2015), 1-20.\n31"    },    {        "title": "2402.14512v1.Running_Coupling_and_Beta_Functions_for_HQCD_with_Heavy_and_Light_Quarks.pdf",        "content": "Running Coupling and Beta-Functions for\nHQCD with Heavy and Light Quarks\nIrina Ya. Aref’evaa, Ali Hajiloua, Pavel Slepovaand Marina Usovaa\naSteklov Mathematical Institute, Russian Academy of Sciences,\nGubkina str. 8, 119991, Moscow, Russia\nE-mail: arefeva@mi-ras.ru ,hajilou@mi-ras.ru ,slepov@mi-ras.ru ,\nusovamk@mi-ras.ru\nAbstract: We consider running coupling constant and beta-function in holographic\nmodels supported by Einstein-dilaton-Maxwell action for heavy and light quarks. We\nobtain a significant dependence of the running coupling constant αandβ-function on\nchemical potential and temperature. At first-order phase transitions, the functions α\nandβ-functions undergo jumps depending on temperature and chemical potential. For\nheavy quarks these jumps depend more sharply on thermodynamic parameters. The\nmagnitude of the jumps increases with decreasing energy scales for both light and heavy\nquarks.\nKeywords: AdS/QCD, holography, beta-function, renormalization group flow, heavy\nquark, light quarkarXiv:2402.14512v1  [hep-th]  22 Feb 2024Contents\n1 Introduction 2\n2 Preliminary 6\n2.1 Holographic model for light quarks 6\n2.1.1 Solution and background 6\n2.1.2 Phase structure for light quarks model 8\n2.2 Holographic model for heavy quarks 14\n2.2.1 Solutions and background 14\n2.2.2 Phase structure for heavy quarks model 15\n3 Running coupling constant 19\n3.1 Running coupling constant for light quarks model 19\n3.2 Running coupling constant for heavy quarks model 25\n4 Beta-Function 32\n4.1 Beta-Function for light quarks model 32\n4.1.1 β-function as a function of thermodynamic parameters at fixed\nenergy scale 32\n4.1.2 βfunction as function of αat different thermodynamic parameters 37\n4.2 Beta-Function for heavy quarks model 39\n4.2.1 β-function as a function of thermodynamic parameters at fixed\nenergy scale 39\n4.2.2 β-function as a function of αat different thermodynamic param-\neters 41\n5 RG flow 45\n5.1 RG flow equation for zero temperature and zero chemical potential 45\n5.1.1 RG flow for light quarks model 46\n5.1.2 RG flow for heavy quarks model 47\n5.2 RG flow equations for non-zero temperature and non-zero chemical po-\ntential 48\n5.2.1 RG flow for light quarks model 49\n5.2.2 RG flow for heavy quarks model 52\n6 Conclusion and Discussion 54\n– i –7 Acknowledgements 56\nA EOM at zero temperature 57\nA.1 Preliminary about the reconstruction method 57\nA.2 Reconstruction of the potential 59\nA.2.1 Potential of light quarks model 59\nA.2.2 Potential of heavy quarks model 61\nA.3 The blackening function g(z) and the dilaton potential V(φ) 63\nA.3.1 Light quarks model 63\nA.3.2 Heavy quarks model 64\nA.4 Reconstruction of the gauge kinetic function f0(φ) 65\nA.4.1 Light quarks model 65\nA.4.2 Heavy quarks model 65\nB Approximation coefficients of potential and gauge kinetic function 66\nC Beta-function β(α)-2D plot 70\nC.1 Beta-function β(α) for light quarks model 70\nC.2 Beta-function β(α) for heavy quark model 73\n– 1 –1 Introduction\nThe renormalization group (RG) is the approach that refers to changing the physical\nsystem as we considered different scales of energy [1–3]. In fact, RG approach provides\nsystematic picture to describe the physical system with many physical parameters.\nThe dependence of coupling constants of physical system on the energy scale can be\ndescribed by βfunction of the theory [2, 4–6]. The RG flows are the variation of fields\nversus energy scale of the theory. The RG is widely used in various area of the modern\ntheoretical physics [7, 8].\nHolographic duality describes a correspondence between a class of strongly cou-\npled field theories and weakly-coupled gravitational theories [9–11]. Then, one can\ninvestigate the strongly coupled regime of gauge theories via holography. In particu-\nlar, holographic RG flow sets up an explanation of RG flow in terms of gravitational\nmodel coupled to dilaton field [12–24]. The RG flow in this approach has geometrical\ndescription and is dual with gravitational solution with specific asymptotic character-\nistics in such a way that the holographic coordinate corresponds with the energy scale\nof gauge theory. Therefore, the Hamiltonian evolution in the holographic direction is\ncorrespondence with the evolution of the physical system under changing the RG scale.\nThe studies of holographic RG flow was extensively performed in [19–24] in context\nof the QCD applications. The RG flows for the physical systems including chemical\npotential and asymmetry properties of quark–gluon plasma (QGP) are investigated in\n[25–27]. Special interest present the exact holographic RG flows, that exist for higher-\ndimensional cases [28] as well as for low ones [29–31].\nQCD phase transitions diagram in (temperature, chemical potential)-plane is a\nchallenging and very important task in high energy physics. This is not only an issue\nof fundamental interest but has also important implications on the description of the\nearly evolution of the Universe [32] and on the study of the interior of compact stellar\nobjects [33, 34]. One of the goal of the experiments on LHC, RHIC, NICA and FAIR\nis to study the phase structure of the strong-interaction matter under the extreme\nconditions. Standard theoretical methods for performing QCD calculations, such as\nperturbation theory no longer works for the strong coupling regime. The lattice theory\nhas a sign problem for non-zero chemical potential calculations, and at its current level\ncannot provide the suitable information. Therefore to describe physics of the strongly\ncoupled QGP produced in heavy ion collisions (HIC) at RHIC, LHC and NICA (see\nfor example [35]), and future experiments, we need a non-perturbative approach [10, 11].\n– 2 –A B\nFigure 1 . Phase structure for light (A) and heavy quarks model (B). Hadronic, QGP and\nquarkyonic phases are denoted by brown, blue and green regions, respectively. The magenta\nstar shows the critical end point (CEP).\nIt is expected that QCD phase diagram structure essentially depends on the quark\nmasses, in particular from lattice calculations [36–39], and some effective phenomeno-\nlogical approach, see for example [35, 40, 41], it is expected that the QCD phase dia-\ngrams have the form presented in Fig. 1. Here regions of the hadronic phase are filled\nby brown color, the regions where quark- gluon phase is realized is filled by blue color.\nThe intermediate regions are filled by the green color. The hardon phase is bounded\nfrom above mostly by the first order phase transition for light quark model (light green\nline in Fig. 1A) and by both the first order phase transition line (light green line in Fig.\n1B) and the line dividing confinement and deconfinement phases (blue line in Fig. 1)\nfor heavy quarks. From these plots we see essential difference for light and heavy quarks.\nThe holographic QCD models for heavy and light quarks constructed in [25, 42–46]\nreproduce these phase diagram features at small chemical potential and predict new\nphenomena for finite chemical potential, in particular, the locations of the critical end\npoints. The goal of this paper to study behaviour of running coupling αandβ-functions\nat different phases: quark confinement phases and QGP phase as well as near critical\nlines.\nThe scattering amplitudes of particles formed during collisions of heavy ions and\nbelonging to the sector of strong interactions significantly depend on the running QCD\n– 3 –A B\nFigure 2 . Coupling constant α(z;µ, T) for light and heavy quarks at energy scales z= 1.5\n(A) and z= 0.5 (B), respectively; [ µ] = [T] = [z]−1= GeV.\ncoupling constants. Typical behavior of the running coupling αat a certain fixed en-\nergy scale, different temperatures and chemical potentials is presented in Fig. 2 for\nlight (A) and heavy quarks (B), respectively. Here, in accordance with the color code\nadopted in Fig. 1, the region of parameters ( µ, T), where the hadronic phase is re-\nalized, is shown in brown, and the region where the quark-gluon phase is realized,\nshown in blue. Intermediate phases are indicated in green. We see that the running\ncoupling α(z) for light and heavy quarks has jumps on first-order phase transitions.\nWith increasing temperature and chemical potential, the coupling constants for QGP\nphases increase faster than for hadronic phases. This behaviour of running coupling is\nin agreement with β-function behaviour presented in Fig. 26 and Fig. 33, see Sect.4.1\nand Sect.4.2. Here we also see jumps located exactly on the first order phase transition.\nThis paper is organized as follows:\nIn Sect. 2 we present 5-dim holographic models for heavy and light quarks and\ndescribe thermodynamic properties of these models.\nIn Sect. 3 we describe running coupling constant for light and heavy quarks models.\nIn Sect. 4 we describe beta-function for light and heavy quarks models.\nIn Sect. 5 we describe RG flows constant for light and heavy quarks models.\nIn Sect. 6 we review our main results.\nThis work in complemented with:\n– 4 –Appendix A where we describe reconstruction method and solve EOMs,\nAppendix B where we present approximation coefficients of potential and gauge\nkinetic function,\nAppendix C with 2D plots of beta-functions on coupling constant.\n– 5 –2 Preliminary\n2.1 Holographic model for light quarks\n2.1.1 Solution and background\nWe consider Einstein-Maxwell-scalar (EMS) system with the action [42, 43]\nS=1\n16πG5Z\nd5x√−g\u0014\nR−f0(φ)\n4F2−1\n2∂µφ∂µφ− V(φ)\u0015\n, (2.1)\nwhere G5is the 5-dimensional Newtonian constant, Fµνis the electromagnetic tensor\nof the gauge Maxwell field Aµ,Fµν=∂µAν−∂νAµ,φis the scalar (dilaton) field, f0(φ)\nis the gauge kinetic function associated to the Maxwell field, V(φ) is the potential of\nthe scalar field φ.\nCorresponding to (2.1) equations of motion in general take the following form:\n∇2φ=∂V\n∂φ+F2\n4∂f0\n∂φ,∇µ[f0(φ)Fµν] = 0 , (2.2)\nRµν−1\n2gµνR=f0(φ)\n2\u0012\nFµρFρ\nν−1\n4gµνF2\u0013\n+1\n2\u0014\n∂µφ∂νφ−1\n2gµν(∂φ)2−gµνV(φ)\u0015\n.\n(2.3)\nTo solve this system of equations, we consider the ansatz for the metric, scalar field\nand Maxwell field as [42, 43]\nds2=B2(z)\u0014\n−g(z)dt2+d⃗ x2+dz2\ng(z)\u0015\n, (2.4)\nB(z) =eA(z)\nz, φ =φ(z), A µ=\u0010\nAt(z),⃗0,0\u0011\n, (2.5)\nwhere ⃗ x= (x1, x2, x3) and A(z) is a scale (or warped) factor. Then equations of motion\nare [25, 42–44, 46–50]\nφ′′+\u0012g′\ng+ 3A′−3\nz\u0013\nφ′+\u0012z2e−2AA′\ntf0,φ\n2g−e2AVφ\nz2g\u0013\n= 0, (2.6)\nA′′\nt+\u0012f′\n0\nf0+A′−1\nz\u0013\nA′\nt= 0, (2.7)\nA′′−A′2+2\nzA′+φ′2\n6= 0, (2.8)\ng′′+\u0012\n3A′−3\nz\u0013\ng′−e−2Az2f0A′2\nt= 0, (2.9)\nA′′+ 3A′2+\u00123g′\n2g−6\nz\u0013\nA′−1\nz\u00123g′\n2g−4\nz\u0013\n+g′′\n6g+e2AV\n3z2g= 0. (2.10)\n– 6 –here all functions depend on the omitted holographic coordinate zandV(z) =V(φ(z)),\nVφ(z) =Vφ(φ(z)) and f0(z) = f0(φ(z)).\nTherefore in isotropic case we have five nontrivial equations: three the Einstein\nequations, one equation for the scalar field φ, one equation for the component of vector\npotential At. Eq. (2.6) is the consequences of eq. (2.7) - (2.10) due to the Bianchi\nidentity. Also we have six nonzero functions: two nonzero diagonal components of the\nmetric gMN, one scalar field φ, one component of vector potential At, the potential of\nscalar field Vand one coupling function for the Maxwell tensor f0(φ).Therefore we\nshould fix two functions f0(z) and b(z) to solve the system of EOM. In this case we\nobtain four equations (2.7) -(2.10) for four unknown functions ( φ,At,g(z),V(z)).\nTo solve the equations of motion (2.6)-(2.10) the usual boundary conditions (B.C)\nare used\nAt(0) = µ, A t(zh) = 0 , (2.11)\ng(0) = 1 , g (zh) = 0 . (2.12)\nFor dilaton field we use\nφ(zh) = 0 . (2.13)\nNote, that if we change B.C. for φthe Regge spectrum does not change. Indeed, ob-\ntaining linear Regge trajectory one solves Schrodinger equation, i.e (eq. 10 in [51]),\nthat has no dependence on dilaton and just depends on its first and second derivatives.\nAt zero temperature and zero chemical potential, i.e. T=µ= 0, the vector meson\nspectrum should satisfy the linear Regge trajectories [51],[42, 43]. Assuming that the\nMaxwell field Aµalso provides linear Regge trajectories, it is necessary to choose a\ngauge kinetic function in the form [42]\nf0(z) =e−c z2−A(z), (2.14)\nwhere the scale factor A(z) for the light quarks has the form\nA(z) =−alog(bz2+ 1) , (2.15)\nanda,bandcare parameters that can be fitted with experimental data as a= 4.046,\nb= 0.01613 GeV2andc= 0.227 GeV2[42].\n– 7 –Under boundary conditions (2.11), (2.12), analytical solutions of the system of\nequations of motion (2.6)-(2.10) are\nφ′(z) =vuut−6 \nA′′−A′2+2\nzA′!\n, (2.16)\nAt(z) =µecz2−ecz2\nh\n1−ecz2\nh, (2.17)\ng(z) = 1−1Rzh\n0y3e−3Ady\"Zz\n0y3e−3Ady−2cµ2\n(1−ecz2\nh)2\f\f\f\f\fRzh\n0y3e−3AdyRzh\n0y3e−3Aecy2dyRz\nzhy3e−3AdyRz\nzhy3e−3Aecy2dy\f\f\f\f\f#\n,\n(2.18)\nV(z) =−3z2ge−2A\"\nA′′+ 3A′2+ \n3g′\n2g−6\nz!\nA′−1\nz \n3g′\n2g−4\nz!\n+g′′\n6g#\n. (2.19)\n2.1.2 Phase structure for light quarks model\nFor metric (2.4) temperature can be written as:\nT=|g′|\n4π\f\f\f\nz=zh. (2.20)\nFor metric (2.4) and the warp factor (2.5) entropy becomes\ns=b3/2(zh)\n4z3\nh. (2.21)\nThe entropy monotonically decreases with horizon growth.\nThe BH-BH (black hole - black hole, or Hawking-Page-like) phase transition caused\nby non-monotonicity of the temperature function produces a jump of density ρ, that is\na coefficient in Atexpansion:\nAt=µ−ρz2+···=µ−cµz2\n1−ecz2\nh+. . . , ρ =−cµ\n1−ecz2\nh. (2.22)\nTo get BH-BH phase transition transition line we need to calculate free energy as\na function of temperature:\nF=Zzh2\nzhs T′dz. (2.23)\nWhile T≥0, i.e for small chemical potentials, we integrate to zh2=∞. When\nsecond horizon where T= 0 appears, one should integrate to it’s value, i.e. to zh2=\n– 8 –4.609. These conditions determine the end-point of the phase diagram, i.e. maximum\npermissible chemical potential µmax.\nIn Fig. 3 the phase diagram of the light quarks model describes two different types\nof phase transitions. It is very important to note the green line that corresponds to the\n1-st order phase transition is obtained using thermodynamics of the theory, i.e. free\nenergy calculations and the blue line corresponds to the confinement/deconfinement\nphase transition is obtained via Wilson loop calculations [42]. In fact, the phase diagram\nshows three different phases, i.e. hadronic, quarkyonic and QGP phases are indicated\nby brown squares, green disks and blue triangles, respectively. All these points are\nmentioned in an attached table in Fig. 4. We have concentrated on three different fixed\ntemperatures, i.e. T= 0.08,0.11 and 0 .2 and different values of chemical potential\nµ= 0.30,0.31,0.32,0.36,0.394,0.43,0.557 and 2 to investigate the running coupling\nconstant and beta-function in different phases of the theory.\n⋆ ⋆\n● ●● ●\n■ ■■ ■ ▲ ▲\n▲ ▲▲ ▲ ▲ ▲\n⋆ ⋆\n0.0 0.5 1.0 1.5 2.0 2.5μ(GeV)0.000.050.100.150.20T(GeV)\nFigure 3 . Phase diagram of the light quarks model indicating the points at T= 0.08,0.11\nand 0 .2. Confinement/deconfinement is depicted by blue line and 1-st order phase transition is\ndepicted by green line (including magenta stars). The hadronic, quarkionic and QGP phases\nare represented by brown squares, green disks and blue triangles, respectively.\nIn Fig. 4, the dots corresponded to different phases of light quarks phase diagram\nare depicted in ( zh, T)-plane. We see that in this figure there are a fewer brown squares,\nonly two of them related to different sizes. This is due to the fact that for very small\nvalues of chemical potential the transition temperature occurs at values higher than\nT= 0.08,0.11 and 0 .2. Using this plot, we can obtain the proper zhvalues associated\nto definite µandTin such a way that we can obtain suitable B.C. for dilaton field as\nφ\f\f\f\nz=zh(T,µ)= 0 for light quarks. Then, we can calculate the running coupling constant\nand beta-function for light quarks in the next sections.\nThe phase transition for small chemical potential in details is presented in Fig. 5.\nThere is no phase transition for µ <0.048 for light quarks model.\n– 9 –⋆ ⋆\n⋆ ⋆● ●● ●\n■ ■■ ■ ▲ ▲\n▲ ▲▲ ▲▲ ▲μ=0μ=0.3μcr=0.31μ=0.32μ=0.36μcr=0.394μ=0.43μ=0.557μ=2T=0.08T=0.11\n0 5 10 15 20 25zh,1/GeV0.000.050.100.150.20T,GeV\nFigure 4 . Temperature T(zh) for light quarks model with the same points mentioned in Fig.\n3. The coordinates of the points are presented in the attached table.\n4 5 6 7 8zh,1/GeV 0.15740.15760.15780.15800.15820.15840.15860.15880.1590T,GeV\nμ=0.0\nT=0.1588\nμ=0.01\nT=0.1587\nμ=0.02\nT=0.1586\nμ=0.03\nT=0.1584\nμ=0.04\nT=0.15813μ=0.045\nT=0.15794\nμ=0.048\nT=0.15782\nμ=0.05\nT=0.157725\nμ=0.055\nT=0.15748\nFigure 5 . Detailed zoom of Fig. 4 for light quarks model. The plot shows that there is no\n1-st order phase transition for small chemical potential µ.\nFig. 4 shows that in isotropic case for µ= 0 temperature is a monotonically\ndecreasing function of horizon. Increasing chemical potential one local minimum for\ntemperature as a function of zhappears. As we will see below, this is directly related to\nthe Hawking-page-like phase transition. Indeed, in the isotropic case first-order phase\ntransition for light quarks shouldn’t exist near zero chemical potential and we should\nsee a crossover, Fig. 5. The larger chemical potential is the lesser temperature value\nat this local minimum becomes. For µ≈0.557 local minimum temperature Tmin= 0\nand second horizon appears.\n– 10 –In Fig. 6 Hawking-Page-like phase transition for isotropic case is depicted. In\nisotropic case BH-BH phase transition starts from a critical point µc= 0.04779, Tc=\n0.1578.\nFor the Hawking-Page-like phase transition the free energy should be a multi-valued\nfunction of temperature. Graphically it is displayed as a “swallow-tail”. The point\nwhere the free energy curve intersects itself determines the Hawking-Page-like phase\ntransition temperature. The larger µbecomes the more pronounced the “swallow-tail”\nis (Fig. 6).\nμ=0\nμ=0.04\nμ=0.04779\nμ=0.050.1575 0.1580 0.1585 0.1590T\n-1.×10-8-8.×10-9-6.×10-9-4.×10-9-2.×10-9F\nμ=0.05\nμ=0.06\nμ=0.07\nμ=0.08\nμ=0.09\n0.155 0.157 0.159T\n-2.×10-92.×10-94.×10-96.×10-98.×10-91.×10-8F\nA B\nFigure 6 . Free energy as a function of temperature F(T) in isotropic case of the light\nquarks model for different µbefore phase transition (A) and after phase transition (B).\nMagenta points denote the Hawking-Page-like phase transition temperature.\nTo describe in detail the procedure that is used to construct the 2D ( µ, zh)-plane\nphase transition diagram, Fig. 7, for light quarks model1we consider the following\nsteps:\n•We extract coordinates ( µ, T) of the points corresponding to the 1-st order and\nconfinement/deconfinement phase transition lines from the original phase dia-\ngram, i.e. Fig. 3.\n•For each chosen point ( µ, T) on the phase transition lines, we produce Tas a\nfunction of zh, i.e. T(zh) plot, for fixed but different µand a line of corresponding\nconstant T.\n1This procedure to obtain 2D ( µ, zh)-plane is also the same as for heavy quarks model.\n– 11 –•For each point ( µ, T) of original phase diagram Fig. 3, we extract suitable coor-\ndinate zhcorresponding to the intersection point between constant Tand graph\nofT(zh) at fixed µin this way:\n–for the confinement/deconfinement transition - We consider the first inter-\nsection at the stable branch of plot T(zh) associated with the large black\nhole. Corresponding result in Fig. 7 are blue lines.\n–for the 1-st order phase transition - We consider the first intersection at the\nstable branch of plot T(zh) associated with the large black hole and the last\nintersection at the unstable branch of plot T(zh) associated with the small\nblack hole that produce the light magenta line and dark magenta line in Fig.\n7, respectively.\n–forT= 0 (the 2nd horizon) - We consider different fixed µand extract\ncorresponding coordinates zhwhere the plots T(zh) cross T= 0 line. Cor-\nresponding result in Fig. 7 is the red line.\n•Finally, we use the coordinates of the resulted points ( µ, zh) to find approxima-\ntions of functions zh(µ) for confinement/deconfinement transition line, 1-st order\nphase transition line and the 2nd horizon to plot Fig. 7. Also, we can produce\n3D plot ( µ, zh, T) as in Fig. 8.\nThe phase structure of light quarks model in 2D ( µ, zh)-plane in Fig. 7 shows\ndifferent domains of phases, i.e. QGP, quarkyonic and hadronic correspond to blue,\ngreen and brown regions, respectively. The solid blue lines in Fig. 7 correspond to the\nconfinement/deconfinement phase transition line in Fig. 3 obtained via Wilson loop\ncalculations [42]. The solid magenta lines in Fig. 7 correspond to the 1-st order phase\ntransition line in Fig. 3 obtained via free energy calculations [42] and the solid red line\ncorresponds to the second horizon where T= 0. All these lines have been obtained\nvia approximation procedure. We see that for µ <0.048 there is crossover region and\natT= 0.154 the phase transition (without any jump) occurs between hadronic and\nQGP phases. But for 0 .0101 < µ < 0.560 when we change the temperature there is\na 1-st order phase transition between hadronic and quarkyonic phases with a jump\nthat is clear in Fig. 7. Also, for µ > 0.101 by changing the temperature there is the\nconfinement/deconfinement transition between quarkyonic and QGP phases without\nany jump. Black solid lines show the temperature indicated in yellow squares and the\nred square corresponds to the intersection of the confinement/deconfinement transition\nline and 1-st order phase transition which is denoted by the blue stars. The magenta\nstar indicates the end of the 1-st order phase transition.\n– 12 –Figure 7 . 2D plot in ( µ, zh)-plane for light quarks model with different phases, i.e. QGP,\nquarkyonic and hadronic corresponding to blue, green and brown regions, respectively. Solid\nblack lines show the temperature indicated in yellow squares and the intersection of the\nconfinement/deconfinement and 1-st order phase transition lines is denoted by the blue stars.\nThe magenta star indicates CEP; [ µ] = [zh]−1= GeV.\nTo have a better intuition the 3D phase structure ( µ, zh, T) of light quarks in our\nmodel is depicted in Fig. 8.\n– 13 –Figure 8 . 3D plot ( µ, zh, T) for light quarks model explaining different phase transitions\nmentioned in 2D plot in Fig. 7; [ µ] = [T] = [zh]−1= GeV.\n2.2 Holographic model for heavy quarks\n2.2.1 Solutions and background\nThe holographic models describing heavy quarks is described by the same action (2.1)\n[43, 54], but the warp factor is given by the scalar factor A(z) different from (2.15),\nA(z) =−s\n3z2−p z4, (2.24)\ns and pare parameters that can be fitted with the experimental data as s = 1 .16GeV2\nandp= 0.273 GeV4. To respect the linear Regge trajectories the gauge kinetic function\nfor heavy quarks model is chosen in the form [43]\nf0(z) =e−sz2−A(z), (2.25)\ncompare with (2.14). As in the case of light quarks, the analytic solution for heavy\nquarks model is obtained by solving the system of equations of motion (2.6)-(2.10) with\nthe boundary conditions (2.11) and (2.12). Therefore, analytical solutions for heavy\nquarks model are functionally the same, i.e. are given by equations (2.16)-(2.19), only\nthe scale factor A(z) is different as well as constant cis replaced by the constant s.\n– 14 –2.2.2 Phase structure for heavy quarks model\nThe phase diagram of the heavy quarks model describes two different types of phase\ntransitions Fig. 9. It is very important to note the green line that corresponds to the\n1-st order phase transition is obtained using thermodynamics of the theory, i.e. free\nenergy calculations and the blue line corresponds to the confinement/deconfinement\nphase transition is obtained via Wilson loop calculations [43, 54]. As a matter of fact,\nthe phase diagram shows three different phases, i.e. hadronic, quarkyonic and QGP\nphases are indicated by brown squares, green disk and blue triangles, respectively. In\nthe phase diagram of the heavy quarks Fig. 9, we concentrated on three different fixed\ntemperatures, i.e. T= 0.2,0.532 and 0 .574 and different values of chemical potential\nµ= 0.10,0.35,0.443,0.55,0.64,0.70,0.76 and 1 .5 to investigate the running coupling\nconstant and beta-function in different phases of the theory. The points on the 1-\nst order phase transition line are indicated by magenta stars. All these points are\nmentioned in a table in Fig. 10C.\n⋆ ⋆⋆ ⋆■ ■■ ■● ●▲ ▲▲ ▲\n■ ■ ▲ ▲\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4μ(GeV)0.20.40.6T(GeV)\nFigure 9 . Phase diagram of the heavy quarks model indicating the points at T= 0.2,0.532\nand 0 .574. Confinement/deconfinement is depicted by blue line and 1-st order phase transition\nis depicted by green line (including magenta stars). The hadronic, quarkyonic and QGP\nphases are represented by brown squares, green disk and blue triangles, respectively.\nThe behavior of the temperature Tas a function of zhfor heavy quark model for\ndifferent chemical potentials µincluding different points correspond to different phases\nis shown in Fig. 10A and its zoom at 0 .5≤zh≤1.3 and 0 .52≤T≤0.595 in Fig.\n10B. For µ≥0.76 there is no phase transition. Using this plot we can obtain the zh\nvalues associated to definite µandTin such a way that we can obtain suitable B.C.\nfor dilaton field as φ\f\f\f\nz=zh(T,µ)= 0 for heavy quarks. Then we can calculate the running\ncoupling constant and beta-function for heavy quarks in the next sections.\nThe phase structure of heavy quarks model as 2D ( µ, zh)-plane is plotted in Fig. 11\nwith different domains of phases, i.e. hadronic, quarkyonic and QGP in correspondence\n– 15 –⋆ ⋆⋆ ⋆\n■ ■■ ■ ● ●\n▲ ▲▲ ▲\n■ ■ ▲ ▲μ=0.1μ=0.35μcr=0.443μ=0.55μ=0.64μcr=0.7μ=0.76μ=1.4T=0.2T=0.532T=0.574\n0.0 0.5 1.0 1.5zh0.00.20.40.60.81.0T\n⋆ ⋆⋆ ⋆\n■● ●\n▲ ▲▲ ▲\n0.6 0.8 1.0 1.2zh0.520.530.540.550.560.570.580.59TA B\nC\nFigure 10 . Temperature T(zh) for heavy quarks model with different µ(A) and a zoom\nof panel (A) in a region (B). The coordinates of the points are presented in the table (C);\n[µ] = [T] = [z]−1= GeV.\nwith brown, green and blue regions, respectively. The solid thick blue lines in Fig. 11\ncorrespond to the confinement/deconfinement phase transition in Fig. 9 obtained via\nWilson loop calculations [43, 54]. The solid thick magenta lines in Fig. 11 correspond to\nthe 1-st order phase transition in Fig. 9 obtained via free energy calculations [43, 54] and\nthe solid thick red line corresponds to the second horizon where T= 0. All these lines\nhave been obtained via approximation procedure. For 0 < µ < 0.694 when we change\nthe temperature in Fig. 11 there is a jump between hadronic and quarkyonic phases that\n– 16 –Figure 11 . 2D plot ( µ, zh)-plane for heavy quarks model with different phases, i.e. hadronic,\nquarkyonic and QGP are in correspondence with brown, green and blue regions, respectively.\nDifferent temperatures are shown in yellow squares with associated lines. The intersection of\nthe confinement/deconfinement transition line and 1-st order phase transition is denoted by\nthe blue stars. The magenta star indicates CEP; [ µ] = [zh]−1= GeV.\nrepresents the 1-st order phase transition, while there is the confinement/deconfinement\nphase transition between quarkyonic and QGP phases without any jump. Also, for\n0.694< µ < 1.181 there is a confinement/deconfinement phase transition between\nhadronic and QGP without any jump. The temperature is shown by black solid lines\nindicated in yellow squares. The intersection of the confinement-deconfinement and 1-\nst order phase transition lines is denoted by the blue stars. The magenta star indicates\nCEP that is the end of the 1-st order phase transition. To have a better intuition the\n3D phase structure ( µ, zh, T) of light quarks model is depicted in Fig. 12.\n– 17 –Figure 12 . 3D plot ( µ, zh, T) for heavy quarks model explaining different phase transitions\nmentioned in 2D plot in Fig. 11; [ µ] = [T] = [zh]−1= GeV.\n– 18 –3 Running coupling constant\nThe running coupling constant is defined as [19, 20, 53]\nα(z) =eφ(z), (3.1)\nwhere zis the energy scale of the theory and φis the dilaton field that solve the\nsystem of equation (2.6)-(2.10). To obtain the running coupling for light and heavy\nquarks model we have to fix the boundary conditions. We take the following boundary\ncondition for the dilaton field\nφ\f\f\f\nz=zh(T,µ)= 0, (3.2)\nwhere zh(T, µ) is the value of the horizon for the stable black hole corresponding to\ngiven chemical potential µand temperature Tthat can be obtained via T(zh) plot\ndescribed in subsection 2.1.2.\n3.1 Running coupling constant for light quarks model\nThe dilaton field φ(z) is depicted in Fig. 13 for T= 0.08,0.11 and 0 .2 with different\nvalues of the chemical potential corresponding to the points of different regions of light\nquarks phase diagram in Fig 3.\nCoupling constant α(z;µ, T) for light quarks model is depicted in Fig. 14 for two\nscales of energy z= 1, GeV−1(A,B) and z= 1.5, GeV−1(C,D). Hadronic, QGP and\nquarkyonic phases are denoted by brown, blue and green, respectively. At both scales,\nμ=0.3, T=0.11\nμcr=0.31, T=0.11\nμ=0.32, T=0.11\nμ=0.557, T=0.11\nμ=2, T=0.11\nμ=0.36, T=0.08\nμcr=0.394, T=0.08\nμ=0.43, T=0.08\nμ=0.557, T=0.08\nμ=2, T=0.085 10 15 20 25z\n-30-25-20-15-10-5φ\nμ=0.32, T=0.2\nμ=2, T=0.20.5 1.0 1.5 2.0z\n-3.0-2.5-2.0-1.5-1.0-0.50.0φ\nA B\nFigure 13 . The dilaton field φ(z) for light quarks model at T= 0.08,0.11 (A) and T= 0.2\n(B) for different µand different B.C; [ z]−1= GeV.\n– 19 –coupling constant in hadronic phase is less than in QGP and quarkyonic phases. For\nvery small values of µ, i.e. crossover region αincreases continuously, while for µ >0.048\nthere is an increment with a jump that shows 1-st order phase transition.\nA B\nC D\nFigure 14 . Coupling constant α(z;µ, T) for light quarks at two energy scales z= 1 (A,B)\nandz= 1.5 (C,D). Hadronic, QGP and quarkyonic phases are denoted by brown, blue and\ngreen, respectively. Panel (B) is a zoom of panel (A) and panel (D) is a zoom of panel (C);\n[µ] = [T] = [z]−1= GeV.\nComparison of coupling constant α(z) for light quarks model in two energy scales\nz= 1 GeV−1(bright images) and z= 1.5 GeV−1(dark images) are shown in Fig.\n15. Hadronic, QGP and quarkyonic phases are denoted by brown, blue and green,\nrespectively. At fixed µandT(see panels (A), (B) and (D)) the value of αdecreases\n– 20 –by increasing energy of system (decreasing z) and it is compatible with the running\ncoupling of QCD [3]. The jump of αbetween hadronic and quarkyonic phases is larger\nfor lower energy (larger z), see panel (C).\nA B\nC D\nFigure 15 . Comparison of coupling constant α(z;µ, T) for light quarks in two energy scales,\nz= 1 (lighter surfaces) and z= 1.5 (darker surfaces). Hadronic, QGP and quarkyonic phases\nare denoted by brown, blue and green, respectively; [ µ] = [T] = [z]−1= GeV.\nCoupling constant α(z;µ, T) of light quarks at fixed T= 0.08 (A), T= 0.11 (B)\nandT= 0.2 (C) for different scales of energy z= 0.4 (thin lines) and z= 1 (thick\nlines) are depicted in Fig. 16. Hadronic, QGP and quarkyonic phases are denoted by\nbrown, blue and green lines, respectively. For different fixed Tthe coupling increases\nby increasing µwhen our physical system enjoys phase transition from hadronic to\nquarkyonic and then from quarkyonic to QGP in (A-B). The important result is that\n– 21 –there is a jump in panels (A) and (B) when 1-st order phase transition occurs between\nhadronic and quarkyonic phases anticipated from 3d plots in Fig. 14 and Fig. 19.\nFrom (A-B) it is clear that for higher values of energy scale (smaller z) the jump is\nbigger in comparison to the lower values of energy scale. For panel (C) there is no\nphase transition because system is in the QGP phase for all values of µ.\n0.38 0.42μ0.0.01α\n0.5 1.0 1.5 2.0μ 0.00.20.40.60.81.0αT=0.08\n0.26 0.3 0.34μ00.01α\n0.0 0.5 1.0 1.5 2.0μ 0.00.20.40.60.81.0αT=0.11\nA B\nz=1z=0.4\n0.5 1.0 1.5 2.0μ0.20.40.60.81.01.2αT=0.2\nC\nFigure 16 . Coupling constant α(z;µ, T) of light quarks at fixed T= 0.08 (A), T= 0.11 (B)\nandT= 0.2 (C) for different scales of energy z= 0.4 (thin lines) and z= 1 (thick lines) and\ndifferent µ. Hadronic, QGP and quarkyonic phases are denoted by brown, blue and green\nlines, respectively. [ µ] = [T] = [z]−1= GeV.\nTo confirm the results in Fig. 16, the 3D plots for coupling constant α(z;µ, T) for\nlight quarks in Fig. 17 are depicted at fixed temperature T=0.08 (A,B), T=0.11 (C,D),\nT=0.2 (E). Fix Tis defined by intersection of the plot in Fig. 14 with the planes shown\nby the red color and for all we set z= 1. Hadronic, QGP and quarkyonic phases are\ndenoted by brown, blue and green lines, respectively.\nCoupling constant α(z;µ, T) and log α(z;µ, T) for light quarks at fixed µ= 0.024\n(A-B), µ= 0.3 (C-D), µ= 0.8 (E-F) at different energy scales z= 1 (thin lines)\n– 22 –A B\nC D E\nFigure 17 . 3D representation of Fig. 16, i.e. coupling constant α(z;µ, T) for light quarks\nat fixed temperatures T=0.08 (A,B), T=0.11 (C,D), T=0.2 (E) are defined by intersection of\nthe plot in Fig. 14 with the planes with fixed temperature shown by the red color. Hadronic,\nQGP and quarkyonic phases are denoted by brown, blue and green lines, respectively. We\nset energy scale z= 1; [ µ] = [T] = [z]−1= GeV.\nandz= 0.3 (thick lines) and different Tare shown in Fig. 18. Hadronic, QGP and\nquarkyonic phases are denoted by brown, blue and green lines, respectively. Magenta\narrows in (D) at µ= 0.3 show the jumps at the 1-st order phase transition between\nhadronic and quarkyonic phases, while there is a phase transition without any jump\nbetween hadronic and QGP µ= 0.024 (A,B), and between quarkyonic and QGP µ= 0.8\n(E,F). For different fixed µthe coupling increases when we increase the temperature\nand physical system experiences different phase transitions from hadronic to quarkyonic\nand then from quarkyonic to QGP.\n– 23 –0.10 0.15 0.20 0.25 0.30 0.35T0.20.40.60.81.0α\nμ=0.024\n0.10 0.15 0.20 0.25 0.30 0.35T-30-25-20-15-10-50Logα\nμ=0.024A B\n0.05 0.10 0.15 0.20 0.25 0.30 0.35T0.20.40.60.81.0α\nμ=0.3\n0.05 0.10 0.15 0.20 0.25T\n-25-20-15-10-50Logα\nμ=0.3\nC D\n0.05 0.10 0.15 0.20 0.25 0.30T0.20.40.60.81.0α\nμ=0.8\n0.05 0.10 0.15 0.20 0.25 0.30T\n-4-3-2-1Logα\nμ=0.8\nE F\nFigure 18 . Coupling constant α(z;µ, T) and log α(z;µ, T) for light quarks at fixed µat\ndifferent energy scales z= 1 (thin lines) and z= 0.3 (thick lines). Hadronic, QGP and\nquarkyonic phases are denoted by brown, blue and green lines, respectively. Magenta arrows\nin D show the jumps at the 1-st order phase transition. [ µ] = [T] = [z]−1= GeV.\nCoupling constant α(z;µ, T) for light quarks model at two fixed values of µ= 0.3 in\nFig. 19A and µ= 0.8 in Fig. 19B are plotted. Hadronic, QGP and quarkyonic phases\n– 24 –are denoted by brown, blue and green lines, respectively. For all values of changing in\nenergy scale zand fixed µ= 0.3αchanges continuously between quarkyonic and QGP,\ni.e. phase transition occurs with no jump, while there is a jump between hadronic and\nquarkyonic phases that is 1-st order phase transition. Therefore, coupling constant is\nsensible to the 1-st order phase transition. At fixed energy scale zby increasing the\ntemperature, αincreases from hadronic to quarkyonic and then from quarkyonic to\nQGP phase. At fixed µ= 0.8 there is a continuous phase transition between just two\nphases, i.e. QGP and quarkyonic.\nIt is very important to note that to obtain the physical results for coupling constant\nαfor light quarks model as a function of different parameters of the theory in Fig. 14 -\n19, we must respect the physical domains of different phases that has already obtained\nin Fig. 7.\nA B\nFigure 19 . Coupling constant α(z;µ, T) for light quarks at µ= 0.3 (A) and µ= 0.8\n(B) Hadronic, QGP and quarkyonic phases are denoted by brown, blue and green lines,\nrespectively; [ µ] = [T] = [z]−1= GeV.\n3.2 Running coupling constant for heavy quarks model\nThe dilaton field φ(z) is depicted in Fig. 20 for T= 0.2,0.532 and 0 .574 with different\nvalues of the chemical potential corresponding to the points of different regions of heavy\nquarks phase diagram in section 2.2.2.\nCoupling constant α(z;µ, T) for heavy quarks model at two energy scales z= 0.5\nGeV−1(dark images) (A), z= 0.3 GeV−1(bright images) (B) and their comparison\n(C,D) is presented in Fig. 21. Hadronic, QGP and quarkyonic phases are denoted\nby brown, blue and green, respectively. For 0 < µ < 0.7 there is transition between\nhadronic and quarkyonic phases as a jump that shows 1-st order phase transition.\n– 25 –0.5 1.0 1.5z\n-15-10-5φ\nFigure 20 . The dilaton field φ(z) for heavy quarks model at T= 0.2,0.532 and 0 .574 for\ndifferent µand different B.C; [ z]−1= GeV.\nµ= 0.7 denotes the intersection of confinement/deconfinement and 1-st order phase\ntransition lines. For 0 .7< µ < 1.18 the transition occurs continuously between hadronic\nand QGP phases. At fixed µandT(panels (C) and (D)) the value of αincreases by\ndecreasing the energy of system (increasing z) the same as light quarks model and it\nis compatible with the running coupling of QCD [3]. The jump of αbetween hadronic\nand quarkyonic phases is larger for lower energy (larger z), see panel (C).\nThe dependence of coupling constant α(z;µ, T) on the µfor heavy quarks at dif-\nferent energy scales z= 0.3 (thin line) and z= 0.5 (thick line) and different fixed\ntemperatures T= 0.574 (A), T= 0.532 (B), T= 0.2 (C) is plotted in Fig. 22.\nHadronic, QGP and quarkyonic phases are denoted by brown, blue and green, respec-\ntively. In (A) the jump is obtained between hadronic and quarkyonic phases and in\n(B) there is a jump between hadronic and QGP phases.\n3D representation of Fig. 22, i.e. coupling constant α(z;µ, T) for heavy quarks\nat fixed temperature T= 0.574 (A), T= 0.532(B), T= 0.2 (C) is depicted in Fig.\n23. The temperature is defined by intersection of the plot in Fig. 14 with the planes\nwith fixed temperature shown by the light pink color. Hadronic, QGP and quarkyonic\nphases are denoted by brown, blue and green lines, respectively. We set energy scale\nasz= 0.5. The jumps are shown for (A) between hadronic and quarkyonic for (B)\nbetween hadronic and QGP phases.\nCoupling constant α(z;µ, T) and log α(z;µ, T) for heavy quarks at fixed µ= 0.3\nand 0 .8 at different energy scales z= 0.3 (thin lines) and z= 0.2 (thick lines) is\ndepicted in Fig. 24. Hadronic, QGP and quarkyonic phases are denoted by brown,\nblue and green lines, respectively. Magenta arrows in (B) with µ= 0.3 show the jumps\n– 26 –A B\nC D\nFigure 21 . Coupling constant α(z;µ, T) for heavy quarks at two energy scales z= 0.5 (dark\nimages) (A) and z= 0.3 (bright images) (B). Hadronic, QGP and quarkyonic phases are\ndenoted by brown, blue and green, respectively. Comparison of coupling constant for two\nscales of energy (C,D); [ µ] = [T] = [z]−1= GeV.\nat the 1-st order phase transition between hadronic and quarkyonic phases while at\nµ= 0.8 (C,D) there is a phase transition between hadronic and QGP phases without\nany jump. The coupling increases when the temperature increases and the physical\nsystem undergoes a transition from hadronic to quarkyonic and from quarkyonic to\nQGP phases.\nCoupling constant α(z;µ, T) for heavy quarks model at µ= 0.3 in Fig. 25A and\nµ= 0.8 in Fig. 25B is presented. Hadronic, QGP and quarkyonic phases are denoted\nby brown, blue and green lines, respectively. At fixed µ= 0.3 (in fact 0 < µ < 0.7)α\n– 27 –0.4 0.5μ0.10.20.3α\n0.2 0.4 0.6 0.8 1.0 1.2 1.4μ0.20.40.60.8αT=0.574\n0.65 0.75μ0.050.15α\n0.2 0.4 0.6 0.8 1.0 1.2 1.4μ0.20.40.60.8αT=0.532A B\n0.2 0.4 0.6 0.8 1.0 1.2 1.4μ0.010.020.030.04αT=0.2\nC\nFigure 22 . Coupling constant α(z;µ, T) for heavy quarks for different fixed temperatures\nT= 0.574 (A), T= 0.532 (B), T= 0.2 (C) at different energy scales, z= 0.3 (thin lines) and\nz= 0.5 (thick lines). Hadronic, QGP and quarkyonic phases are denoted by brown, blue and\ngreen, respectively; [ µ] = [T] = [z]−1= GeV.\nchanges continuously between quarkyonic and QGP , i.e. phase transition occurs with\nno jump, while there is a jump between hadronic and quarkyonic phases that is 1-st\norder phase transition for any range of the energy scale z. It shows that the coupling\nconstant is sensible to the 1-st order phase transition. For each phases in Fig. 25A\nat fixed T, decreasing the energy (increasing z) coupling constant increases and vise\nversa that is compatible with running coupling of QCD [3]. At fixed µ= 0.8 there is a\ncontinuous phase transition between just two phases, i.e. QGP and hadronic phases.\nIt is important to note that to obtain the physical results for heavy quarks model for\ncoupling constant α(z;µ, T) as a function of different parameters of the theory in Fig.\n21 - 25, we considered the physical domains of different phases heavy quarks that has\nbeen already obtained in Fig. 11. The very important result is that coupling constant\nas a physical quantity feels the 1-st order phase transition line and undergoes a jump at\n– 28 –A B C\nFigure 23 . 3D representation of Fig. 22, i.e. coupling constant α(z;µ, T) for heavy quarks at\nfixed temperature T= 0.574 (A), T= 0.532 (B), T= 0.2 (C) is defined by intersection of the\nplot in Fig. 14 with the planes with fixed temperature shown by the light pink color. Hadronic,\nQGP and quarkyonic phases are denoted by brown, blue and green lines, respectively. We\nset the energy scale z= 0.5; [µ] = [T] = [z]−1= GeV.\nthis line and this jump for heavy quarks are more sharply depends on thermodynamic\nparameters.\n– 29 –0.2 0.4 0.6 0.8 1.0T0.20.40.60.81.0α\nμ=0.3\n0.2 0.4 0.6 0.8 1.0 1.2T\n-10-8-6-4-20Logα\nμ=0.3A B\n0.2 0.4 0.6 0.8 1.0T0.20.40.60.81.0α\nμ=0.8\n0.2 0.4 0.6 0.8 1.0T\n-8-6-4-2Logα\nμ=0.8\nC D\nFigure 24 . Coupling constant α(z;µ, T) and log α(z;µ, T) for heavy quarks at fixed µat\ndifferent energy scales z= 0.3 (thin lines) and z= 0.2 (thick lines). Hadronic, QGP and\nquarkyonic phases are denoted by brown, blue and green lines, respectively. Magenta arrows\nin B show the jumps at the 1-st order phase transition. [ µ] = [T] = [z]−1= GeV.\n– 30 –A B\nFigure 25 . Coupling constant α(z;µ, T) for heavy quarks at µ= 0.3 (A) and µ= 0.8\n(B). Hadronic, QGP and quarkyonic phases are denoted by brown, blue and green lines,\nrespectively; [ µ] = [T] = [z]−1= GeV.\n– 31 –4 Beta-Function\nBeta-function, β(α), encodes the dependence of coupling constant αon the energy scale\nof the physical system. Holographic β-function is given by [12, 19, 52]\nβ(α) = 3 α X , (4.1)\nwhere α(z) is defined in (3.1) and Xis a new dynamical variable defined as [26–28]\nX(z) =˙φB\n3˙B, (4.2)\nwhere B(z) is defined in (2.5) and φfor light quarks model is obtained in (A.25).\n4.1 Beta-Function for light quarks model\n4.1.1 β-function as a function of thermodynamic parameters at fixed en-\nergy scale\nThe main result of the light quarks is presented as the dependence of β-function on the\ntemperature and the chemical potential at fixed coupling constant α= 1 in Fig. 26.\nThe choice of fixed coupling constant α= 1 can be obtained considering energy scale at\nthe horizon i.e. z=zh. Clearly, we see the jumps in the values of the β-function at the\n1-st order phase transition between hadronic and quarkyonic phases. Therefore, this\nresult shows a very important point that β-function as a physical quantity can feel the\n1-st order phase transition. Fig. 26 shows that for very small values of µin crossover\nregion there is continuous phase transition between hadronic and QGP phases and also\na continuous phase transition between quarkyonic and QGP.\nThe dependence of the light quarks β-function on µfor fixed T= 0.08,0.11 and 0 .2\nat fixed coupling constant α= 1 is plotted in Fig. 27. Hadronic, QGP and quarkyonic\nphases are denoted by brown, blue and green colors, respectively. In T= 0.08 (A) and\nT= 0.11 (B) there is a jump between hadronic and quarkyonic phases and physical\nsystem experiences the 1-st order phase transition at µ= 0.4 and µ= 0.32, respectively.\nIn addition, in (A,B) β-function is almost fixed until the jump and then the β-function\nincreases monotonically and phase transition occurs between quarkyonic and QGP\nphases without any jump. In T= 0.2 (C) just QGP phase dominates and there is no\nphase transition as well as increasing µincreases the β-function monotonically.\nIn Fig. 28 the 3D representation of Fig. 27, i.e. β(z;µ, T) for light quarks at fixed\ntemperatures T= 0.08 (A), T= 0.11(B), T= 0.2 (C) is plotted. The constant T\nare defined by intersection of the plot in Fig. 26 with the planes at fixed temperature\nshown by the light orange color. Hadronic, QGP and quarkyonic phases are denoted\nby brown, blue and green lines, respectively. We fixed coupling constant α= 1.\n– 32 –Figure 26 . Beta-function β(z;µ, T) for light quarks at fixed coupling constant α= 1.\nHadronic, QGP and quarkyonic phases are denoted by brown, blue and green colors, respec-\ntively; [ µ] = [T] = [z]−1= GeV.\n0.5 1.0 1.5 2.0μ\n-2.5-2.0-1.5β\nT=0.08\n0.5 1.0 1.5 2.0μ\n-2.5-2.0-1.5β\nT=0.11\n0.5 1.0 1.5 2.0μ\n-2.2-2.0-1.8-1.6-1.4-1.2-1.0β\nT=0.2\nA B C\nFigure 27 . The dependence of the light quarks β-function on µfor fixed T= 0.08 (A), 0 .11\n(B) and 0 .2 (C) at fixed coupling constant α= 1. Hadronic, QGP and quarkyonic phases are\ndenoted by brown, blue and green colors, respectively; [ µ] = [T] = [z]−1= GeV.\nThe temperature dependence of β-function, β(z;µ, T) and log β(z;µ, T) for light\nquarks model at fixed µ= 0.024 (A,B), µ= 0.3 (C,D) µ= 0.8 (E,F) and different\nenergy scales z= 1 (thin lines) and z= 0.3 (thick lines) is depicted in Fig. 29.\nHadronic, QGP and quarkyonic phases are denoted by brown, blue and green lines,\n– 33 –A B C\nFigure 28 . 3D representation of Fig. 27, i.e. β(z;µ, T) for light quarks at fixed temperatures\nT= 0.08 (A), T= 0.11(B), T= 0.2 (C) that are defined by intersection of the plot in Fig.\n26 with the planes with fixed temperature shown by the light orange color. Hadronic, QGP\nand quarkyonic phases are denoted by brown, blue and green lines, respectively. We fixed\ncoupling constant α= 1; [ µ] = [T] = [z]−1= GeV.\nrespectively. Magenta arrows in D show that β-function feels the jumps at the 1-st\norder phase transition between hadronic and quarkyonic phases. In µ= 0.024 (A) and\nµ= 0.8 (E) there is a phase transition without any jump between hadronic to QGP\nand between quarkyonic to QGP, respectively. β-function decreases with increasing T\nfor fixed µ. In addition, for larger energy scale (smaller z) in the system, the β-function\ndecreases more sharply.\n– 34 –0.10 0.15 0.20 0.25 0.30 0.35T\n-1.4-1.2-1.0-0.8-0.6-0.4-0.2β\nμ=0.024\n0.10 0.15 0.20 0.25 0.30 0.35T-30-25-20-15-10-50Log|β|\nμ=0.024A B\n0.10 0.15 0.20 0.25 0.30T\n-1.4-1.2-1.0-0.8-0.6-0.4-0.2β\nμ=0.3\n0.05 0.10 0.15 0.20 0.25 0.30T\n-30-25-20-15-10-50Log|β|\nμ=0.3\nC D\n0.05 0.10 0.15 0.20 0.25 0.30T\n-1.4-1.2-1.0-0.8-0.6-0.4-0.2β\nμ=0.8\n0.05 0.10 0.15 0.20 0.25 0.30T\n-3.0-2.5-2.0-1.5-1.0-0.50.5Log|β|\nμ=0.8\nE F\nFigure 29 . Beta-function β(z;µ, T) and log β(z;µ, T) for light quarks at fixed µat different\nenergy scales z= 1 (thin lines) and z= 0.3 (thick lines). Hadronic, QGP and quarkyonic\nphases are denoted by brown, blue and green lines, respectively. Magenta arrows in D show\nthe jumps at the 1-st order phase transition. [ µ] = [T] = [z]−1= GeV.\nIn Fig. 30 the 3D representation for the dependence of the light quarks β-function\nonTare obtained by intersections of 3D plot presented in Fig. 26 with gray planes\n– 35 –at fixed µ= 0.4 (A) and µ= 2 (B). Hadronic, QGP and quarkyonic phases are\ndenoted by brown, blue and green colors, respectively. In (A) the jump occurs at\nTjump = 0.078, µ= 0.4. We fixed coupling constant α= 1. The β-function at a\nfixed µ= 0.4 first remains almost constant in the hadronic phase until a jump at\nTjump= 0.078, then increases with increasing temperature T. In (B) there is a continues\nphase transition between quarkyonic and QGP phases by increasing the temperature\nandβ-function increases as well.\nA B\nFigure 30 . 3D representation for the dependence of the light quarks β-function on Tare\nobtained by intersections of 3D plot presented in Fig. 26 with gray planes at fixed µ= 0.4 (A)\nandµ= 2 (B). Hadronic, QGP and quarkyonic phases are denoted by brown, blue and green\ncolors, respectively. In (A) the jump occurs at Tjump = 0.078, µ= 0.4. We fixed coupling\nconstant α= 1; [ µ] = [T] = [z]−1= GeV.\n– 36 –4.1.2 βfunction as function of αat different thermodynamic parameters\nTo see these jumps in the values of the β-function during the 1-st order phase transition\non different scales parameterized by the holographic coordinate z, we represent the\nbeta-function β(α;µ, T) for light quarks at fixed µ= 0.02 (A), its zoom (B) and\nµ= 0.3 (C), its zoom (D) in Fig. 31. Hadronic, QGP and quarkyonic phases are\ndenoted by brown, blue and green colors, respectively. Red solid lines denote the fixed\ntemperatures. In Fig. 31 A and B, β(α;µ, T) for light quark model is plotted at\nµ= 0.02 in the crossover region where the 1-st order phase transition does not occur,\nwhile confinement/deconfinement phase transition occurs with out any jump between\nhadronic and QGP phases. In Fig. 31. C and D, β(α;µ, T) is plotted at µ= 0.3. For\nthis value of the chemical potential the 1-st order phase transition occurs at T= 0.1145.\nBeta-function β(α;µ, T) for light quarks at fixed µ= 0.3 (A) and its zoom at the\nfirst order phase transition temperature T= 0.113 (B) is plotted in Fig. 32. Hadronic,\nQGP and quarkyonic phases are denoted by brown, blue and green colors, respectively.\nThe plane with T= 0.113 is shown by light grey. The jump that discussed in Fig. 31\nand the phase transition can be seen in detail in Fig. 32. Both figures show the linear\ndependence of β-function on αat different Tfor hadronic and QGP phases. In Fig.\n31-Fig. 32, we see that the dependence of β-function on αis mainly linear with a slope\ndepending on the parameters Tandµ. To make this relationship more clear, β-function\nas a function of αfor different values of Tandµon separate graphs is plotted, see Fig.\n58 - Fig. 60 in Appendix C.\nIt is needed to emphasize that to obtain the physical results for β-function for light\nquarks model as a function of different parameters of the theory in Fig. 26 - 32, we\nmust respect the physical domains of different phases for light quarks that has already\nbeen obtained in Fig. 7.\n– 37 –T=0.1594 GeV\nT=0.160 GeV\nβ=β(α), T=0.160 GeV\nβ=β(α), T=0.1594 GeVA B\nT=0.145 GeV\nT=0.155 GeV\nβ=β(α), T=0.145 GeV\nβ=β(α), T=0.155 GeV\nC D\nFigure 31 . Beta-function β(α;µ, T) for light quarks at fixed µ= 0.02 (A) with its zoom\n(B) and µ= 0.3 (C) with its zoom (D). Hadronic, QGP and quarkyonic phases are denoted\nby brown, blue and green colors, respectively. Red solid lines denote the fixed temperatures;\n[µ] = [T] = GeV.\n– 38 –T=0.113 GeV\nA B\nFigure 32 . Beta-function β(α;µ, T) for light quarks at fixed µ= 0.3 (A) and its zoom at\nthe first order phase transition temperature T= 0.113 (B). Hadronic, QGP and quarkyonic\nphases are denoted by brown, blue and green colors, respectively. The plane with T= 0.113\nis shown by light grey; [ µ] = [T] = GeV.\n4.2 Beta-Function for heavy quarks model\n4.2.1 β-function as a function of thermodynamic parameters at fixed en-\nergy scale\nThe main result of the heavy quarks model is presented as the dependence of β-function\non the temperature and the chemical potential at fixed coupling constant α= 1 in Fig.\n33. The choice of fixed coupling constant α= 1 can be obtained considering energy\nscale at the horizon i.e. z=zh. Clearly, we see the jumps in the values of the β-function\nat the 1-st order phase transition between hadronic and quarkyonic phases. Therefore,\nthis result shows a very important point that β-function as a physical quantity can feel\nthe 1-st order phase transition.\nThe dependence of the heavy quarks β-function on µfor fixed T= 0.2 (A), T=\n0.532 (B) and T= 0.574 (C) at fixed coupling constant α= 1 is plotted in Fig.\n34. Hadronic, QGP and quarkyonic phases are denoted by brown, blue and green\ncolors, respectively. In T= 0.2 (A) there is a phase transition between hadronic and\nQGP phases without any jump. In addition, increasing µdecreases the β-function\nmonotonically. In T= 0.532 (B) there is a jump between hadronic and QGP phases\nand in T= 0.574 (C) there is a jump between hadronic and quarkyonic phases. In\n(B,C) the physical system experiences the 1-st order phase transition at µ= 0.7 and\nµ= 0.45, respectively. In addition, in (B,C) the β-function decreases by increasing µ\n– 39 –Figure 33 . Beta-function β(z;µ, T) for heavy quarks at fixed coupling constant α= 1.\nHadronic, QGP and quarkyonic phases are denoted by brown, blue and green colors, respec-\ntively; [ µ] = [T] = [z]−1= GeV.\n0.2 0.4 0.6 0.8 1.0 1.2 1.4μ\n-3.0-2.9-2.8-2.7-2.6-2.5-2.4-2.3-2.2β\nT=0.2\n0.2 0.4 0.6 0.8 1.0 1.2 1.4μ\n-3.0-2.8-2.6-2.4-2.2-2.0β\nT=0.532\n0.2 0.4 0.6 0.8 1.0 1.2 1.4μ\n-3.0-2.8-2.6-2.4-2.2-2.0β\nT=0.574\nA B C\nFigure 34 . The dependence of the heavy quarks β-function on µfor fixed T= 0.2 (A),\nT= 0.532 (B) and T= 0.574 (C) at fixed coupling constant α= 1. Hadronic, QGP and\nquarkyonic phases are denoted by brown, blue and green colors, respectively; [ µ] = [ T] =\n[z]−1= GeV.\nuntil the jump and then increases. In (A) the β-function decreases by increasing µ.\nIn Fig. 35 the 3D representation of Fig. 34, i.e. β(z;µ, T) for heavy quarks at\nfixed temperatures T= 0.2 (A), T= 0.532 (B), T= 0.574 (C) that are defined by\nintersection of the plot in Fig. 33 with the planes with fixed temperature shown by the\nlight orange color. Hadronic, QGP and quarkyonic phases are denoted by brown, blue\nand green lines, respectively. We fixed coupling constant α= 1.\nThe temperature dependence of β-function, β(z;µ, T) and log β(z;µ, T) for heavy\nquarks model at fixed µ= 0.3 (A,B), µ= 1.4 (C,D) at different energy scales z= 0.3\n– 40 –A B C\nFigure 35 . 3D representation of Fig. 34, i.e. β(z;µ, T) for heavy quarks at fixed tempera-\ntures T= 0.2 (A), T= 0.532 (B), T= 0.574 (C) that are defined by intersection of the plot\nin Fig. 33 with the planes with fixed temperature shown by the light orange color. Hadronic,\nQGP and quarkyonic phases are denoted by brown, blue and green lines, respectively. We\nfixed coupling constant α= 1; [ µ] = [T] = [z]−1= GeV.\n(thin lines) and z= 0.2 (thick lines) is depicted in Fig. 36. Hadronic, QGP and\nquarkyonic phases are denoted by brown, blue and green lines, respectively. Magenta\narrows in (B) show that the jumps at the 1-st order phase transition between hadronic\nand quarkyonic phases. In µ= 1.4 (C) there is no phase transition because the QGP\nphase is dominated. β-function is negative and decreases with increasing Tat fixed µ.\nIn addition, for smaller energy scale (larger z) in the system, the β-function decreases\nmore sharply.\nIn Fig. 37 3D representation for the dependence of the heavy quarks β-function\nonTare obtained by intersections of 3D plot presented in Fig. 33 with gray planes at\nfixed µ= 0.4 (A) and µ= 1.4 (B). We fixed coupling constant α= 1. Hadronic, QGP\nand quarkyonic phases are denoted by brown, blue and green colors, respectively. In\n(A) the jump occurs at Tjump= 0.58, µ= 0.4 between hadronic and quarkyonic phases\nthat shows the 1-st order phase transition. In (A) before jumping the β-function almost\nis constant and after jump increases with increasing the temperature. In (B) the QGP\nphase is dominated with no phase transition and β-function has a local minimum.\n4.2.2 β-function as a function of αat different thermodynamic parameters\nTo investigate the jumps in the values of the β-function during the 1-st order phase\ntransition on different scales parameterized by the holographic coordinate z, we rep-\nresent the beta-function β(α;µ, T) for heavy quarks in Fig. 38. Hadronic, QGP and\nquarkyonic phases are denoted by brown, blue and green colors, respectively. Red solid\nlines denote the fixed temperatures. In Fig. 38 A there is confinement/deconfinement\nphase transition occurs without any jump between quarkyonic and QGP phases and\n– 41 –0.2 0.4 0.6 0.8 1.0T\n-1.2-1.0-0.8-0.6-0.4-0.2β\nμ=0.3\n0.2 0.4 0.6 0.8 1.0 1.2T\n-10-8-6-4-20Log|β|\nμ=0.3A B\n0.2 0.4 0.6 0.8 1.0T\n-1.2-1.0-0.8-0.6-0.4-0.2β\nμ=1.4\n0.2 0.4 0.6 0.8 1.0T\n-4-3-2-1Log|β|\nμ=1.4\nC D\nFigure 36 . Beta-function β(z;µ, T) and log β(z;µ, T) for heavy quarks at fixed µat different\nenergy scales z= 0.3 (thin lines) and z= 0.2 (thick lines). Hadronic, QGP and quarkyonic\nphases are denoted by brown, blue and green lines, respectively. Magenta arrows in D show\nthe jumps at the 1-st order phase transition. [ µ] = [T] = [z]−1= GeV.\nthere is a jump between hadronic and quarkyonic phases as a 1-st order phase transi-\ntion. To see the jump in detail Fig. 38B is depicted and the light grey surface shows\ntheTjump= 0.585 where the 1-st order phase transition occurs.\nIn Fig. 38, the dependence of β-function on αis mostly linear with a slope de-\npending on the parameters Tandµ. To have more clarification, we plotted β-function\nas a function of αfor different values of Tandµon separate graphs, see Fig. 61 and\nFig. 62 in Appendix C.\nIt is very important to note that our results show at the 1-st order phase transition\nβ-function experiences the jumps depending on the parameters of theory, i.e Tand\nµ. In fact, the β-function can feel the the 1-st order phase transition. In particular,\n– 42 –A B\nFigure 37 . 3D representation for the dependence of the heavy quarks β-function on Tare\nobtained by intersections of 3D plot presented in Fig. 33 with gray planes at fixed µ= 0.4\n(A) and µ= 1.4 (B). Hadronic, QGP and quarkyonic phases are denoted by brown, blue\nand green colors, respectively. In (A) the jump occurs at Tjump = 0.58, µ= 0.4. We fixed\ncoupling constant α= 1; [ µ] = [T] = [z]−1= GeV.\nto obtain the physical results for β-function for heavy quarks model as a function of\ndifferent parameters of the theory in Fig. 33 - 38, we must respect the physical domains\nof different phases for heavy quarks that has already been obtained in Fig. 11.\n– 43 –T=0.585 GeVA B\nFigure 38 . Beta-function β(α;µ, T) for heavy quarks at fixed µ= 0.3 (A) and its zoom at\nthe first order phase transition temperature T= 0.585 (B). The plane with T= 0.585, is\nshown by light grey. Hadronic, QGP and quarkyonic phases are denoted by brown, blue and\ngreen colors, respectively; [ µ] = [T] = GeV.\n– 44 –5 RG flow\n5.1 RG flow equation for zero temperature and zero chemical potential\nLet us consider zero temperature and zero chemical potential case. One defines a new\ndynamical variable [19, 20, 26–28]\nX(z) =˙φB\n3˙B, (5.1)\nand consider Xsuch that\nX(z) =X(φ(z)). (5.2)\nThe function X(φ) satisfies the the following equation\ndX\ndφ=−4\n3\u0012\n1−3\n8X2\u0013\u0012\n1 +1\nX∂φV(φ)\nV(φ)\u0013\n. (5.3)\nthat following from EOM (2.6)-(2.10). One calls this equation the RG flow equation\n[19, 20], since Xis related with β-function defined as\nβ= 3α X, (5.4)\nwhere Xis given by (5.1) and αis the running coupling that is given by equation (3.1).\n-5 5z\n-0.50.5X\nφ0= -10φ0=0\n-20 -15 -10 -5 5 10φ\n-0.50.5\nA B\nFigure 39 . The behavior of X=X(z) for light quarks (A) and X=X(φ) with different\nB.C. φ0=−10 (lime line) and φ0= 0 (green line) for light quarks (B). The dotdashed line\nrepresents an unphysical region z <0.\n– 45 –-10 -8 -6 -4 -2 0-1.5-1.0-0.50.0\nφ\nFigure 40 . RG flows for light quark in isotropic background with T= 0 and µ= 0 corre-\nsponding to V(φ) with B.C. φ0=−10. The green dashed lines represent X=−p\n8/3 fixed\n(attractor) line. The green solid lines depict the function X(φ) presented in Fig. 39B.\n5.1.1 RG flow for light quarks model\nThe plots of X(z) given by equation (5.1) and X=X(φ) are presented in Fig. 39A\nand Fig. 39B, respectively. Fig. 39B is recovered from equations (5.1) and (A.25).\nIn Fig. 40 RG flows for light quark in isotropic background with T= 0 and µ= 0\ncorresponding to V(φ) with different B.C. φ0=−10 (A) and φ0= 0 (B) are presented.\nThe blue and red lines depict the function Xpresented in\nIn Fig. 40 RG flows for the case T= 0, µ= 0 corresponding to V(φ) with different\nboundary conditions: φ0=−10 (A) and φ0= 0 (B) are shown. The blue and green\nlines depict the function Xpresented in Fig. 39B and recovered from (5.1) and (A.25).\nParts of these lines with φ > φ 0act as a repulsor and parts with φ < φ 0as an attractor\n(this is actually the non-physical part). We see that in Fig. 39A X(z)→0 for z→0.\nIn Fig. 39B X(φ)→0 for φ→φ0. This happends because (5.1) X(z)∼˙φB/ ˙B,\n˙φ→const andB/˙B→0 for z→0.\nIn Fig. 40 we see the comparison between our solution and phase portrait of\n(5.8). Also X(φ)→0 for φ→φ0. Here X(φ)→const forφ→0. In Fig. 40 the\napproximation for the potential is checked only for −23.5≤φ≤3.5 see Fig. 49. So\nwe can trust the stream plot only for |φ+ 10| ≤13.\n– 46 –5.1.2 RG flow for heavy quarks model\nThe plots of X(z) given by equation (5.1) and X=X(φ) for heavy quarks model are\npresented in Fig. 41A and Fig. 41B, respectively. RG flows for heavy quarks at T= 0\nandµ= 0 is depicted in Fig. 42. In this plot the approximation for the potential is\nvalid only for −23.5≤φ≤3.5 for φ0=−10, see Fig. 51.\n-5 5z\n-1.0-0.50.51.0X\nφ0= -10\nφ0=0\n-20 -15 -10 -5 510φ\n-1.0-0.50.51.0\nA B\nFigure 41 . The behavior of X=X(z) for heavy quarks (A) and X=X(φ) with different\nB.C. φ0=−10 (khaki line) and φ0= 0 (olive line) for heavy quarks (B). The dotdashed line\nrepresents an unphysical region z <0.\n– 47 –5.2 RG flow equations for non-zero temperature and non-zero chemical\npotential\nIntroducing new variables\nX=˙φ\n3B\n˙B, (5.5)\nY=1\n4˙g\ngB\n˙B, (5.6)\nH=˙At\nB2(5.7)\none obtains the following RG equations [25, 27]\ndX\ndφ=−4\n3\u0012\n1−3\n8X2+Y\u0013\u0012\n1 +1\nX2∂φV(φ)− H2∂φf0\n2V(φ) +H2f0\u0013\n, (5.8)\ndY\ndφ=−4Y\n3X\u0012\n1−3\n8X2+Y\u0013\u0012\n1 +3\n2YH2f0\n2V(φ) +H2f0\u0013\n, (5.9)\ndH\ndφ=−\u00121\nX+∂φf0\nf0\u0013\nH, (5.10)\nX(φ),Y(φ) and H=H(φ) are related with X,YandHasX=X(φ(z)),Y=Y(φ(z))\nandH=H(φ(z)).\n-10 -8 -6 -4 -2 0-1.5-1.0-0.50.0\nφ\nFigure 42 . RG flows for heavy quarks in isotropic background with T= 0 and µ= 0\ncorresponding to V(φ) with B.C. φ0=−10. The green dashed lines represent X=−p\n8/3\nfixed (attractor) line. The olive line depict the function Xpresented in Fig. 41B .\n– 48 –The RG flow equations (5.8), (5.9) and (5.10) are be obtained from the definition\n(5.5), which actually gives X=X(z), and the given set φ(z, φ 0), which is determined\nfrom the equation (A.8) up to boundary condition specified by the constant φ0. We\ncan also find solutions to the system of first order differential equations (5.8)-(5.10).\nSolutions to the latter system may contain more solutions compared to the solutions\ndetermined by the reconstruction method, compare with [28].\n5.2.1 RG flow for light quarks model\na) RG flow for our solution\nWe can find exact expressions for our solutions to the system of first order differ-\nential equations (5.5)-(5.7). We subsitute expessions from (2.16)-(2.18) to (5.5)-(5.7).\nTherefore we can present for fixed parametes of model (fixed chemical potential and\ntemperature) solution for light and heavy quarks models in the 3D plots in ( X,Y,H)\ncoordinates.\nIn Fig. 43 The 3D plot of ( X,Y,H) defined by equations (5.5)-(5.7) for the light\nquarks model. The green lines with decreasing thickness represent the case with zh= 1\nandµ= 1.5,1.25,1,0.75,0.5,0. The darker orange lines represent the case with zh= 2\nand the same set of µ. We see that for small values of zbrown and green lines almost\ncoincides. This happens because X(φ)→0 for φ→φ0as in previous case for zero\ntemperature and chemical potential. This happens because (5.1) X(z)∼˙φB/ ˙B, ˙φ→\nconst andB/˙B→0 for z→0. Also YandH → 0 for z→0.\nFigure 43 . The 3D plot of ( X,Y,H) for the light quarks model. The green lines with\ndecreasing thickness represent the case with zh= 1 and µ= 1.5,1.25,1,0.75,0.5,0.25,0; the\ndarker orange lines represent the case zh= 2 with the same set of µ.\n– 49 –In Fig. 44 the 3D plot ( X,Y,H) for light quarks model is presented. The green and\norange lines with decreasing thickness represent the cases with µ= 1.5,1.25,1,0.75,0,5,0\nand different horizons zh= 1 and zh= 2. The plot legends are the same as in Fig. 43.\nFigure 44 . The 3D plot ( X,Y,H) for light quarks model. The green and darker orange lines\nwith decreasing thickness represent the cases µ= 1.5,1.25,1,0.75,0.5,0.25,0 with different\nhorizons zh= 1 and zh= 2,respectively. The plot legends are the same as in Fig. 43.\n– 50 –b) RG flow for other solutions\nA B\nFigure 45 . The 3D plots of ( X,Y, φ) for light quarks model representing dependence of X\nandYon the dilaton field φwith different boundary conditions. These boundary conditions\nφ|z=zh= 0, correspond to different values of zh= 0.6,2.5,4 and 6 are shown by magenta,\ncyan, blue and orange, respectively (A) and combination of our solutions in left panel with\n3D RG flows of ( X,Y, φ) (B).\nIn Fig. 45A there are 3D RG plots with trajectories for fixed zhin (X,Y, φ) coordi-\nnates for light quarks model. These trajectories are obtained for solutions constructed\nfrom potential reconstruction method. The values of zh= 0.6,2.5,4 and 6 correspond\nto magenta, cyan, blue and orange curves, respectively.\nIn Fig. 45B we see other solutions of RG equations (5.8),(5.9) for zero chemical\npotential (i.e. H= 0) for the same dilaton potential V(φ). Note that solutions obtained\nvia potential reconstruction method are unstable for light quarks model. For fixed\ndilaton potential V(φ) we solve (5.8) and (5.9) and obtain different solutions that\ncorresponds to different factors b(z),φ(z) and V(z) but the same V(φ). When we\nuse reconstruction potential method we have fixed b(z),f0(z) and fixed boundary\nconditions for scalar field φand blackening function g, and obtain only one solution of\nEOM. Different solutions correspond to different boundary conditions of scalar field φ.\nTherefore this means that the obtained solution is sensitive to change of b(z).\n– 51 –5.2.2 RG flow for heavy quarks model\na) RG flow for our solution\nThe similar plots we can present for heavy quarks model. In Fig. 46 the 3D plot of\n(X,Y,H) defined by equations (5.5)-(5.7) for the heavy quarks model. The olive lines\nwith decreasing thickness represent the case with zh= 1 and µ= 1.5,1.25,1,0.75,0.5,0.\nThe chocolate lines represent the case with zh= 2 and the same set of µ. We can see\nthe similar behaviour for asymptotic as for the light quarks case. Also X(φ)→0 for\nφ→φ0.\nFigure 46 . The 3D plot of ( X,Y,H) for the heavy quarks model. The olive lines with\ndecreasing thickness represent the case with zh= 1 and µ= 1.5,1.25,1,0.75,0.5,0; the\nchocolate lines represent the case with zh= 2 and the same set of µ.\n– 52 –b) RG and other solutions\nA B\nFigure 47 . The 3D plots of ( X,Y, φ) for heavy quarks model representing dependence of X\nandYon the dilaton field φwith different boundary conditions. These boundary conditions\nφ|z=zh= 0, correspond to different values of zh= 0.45,1.0,1.4 and 1 .65 are shown by magenta,\ncyan, blue and orange, respectively (A) and combination of our solutions in left panel with\n3D RG flows of ( X,Y, φ) (B).\nIn Fig. 47A the 3D plots in ( X,Y, φ) coordinates for heavy quarks model are\npresented for different boundary conditions. These boundary conditions φ|z=zh= 0,\ncorrespond to different values of zh= 0.45,1.0,1.4 and 1 .65 that are shown by magenta,\ncyan, blue and orange lines, respectively (A) and combination of our solutions in left\npanel with 3D RG flows in ( X,Y, φ) coordinates (B).\nIn Fig. 47B we see other solutions of RG equations for the same dilaton potential\nV(φ). The solutions obtained via potential reconstruction method are unstable for\nheavy quarks also as for light quarks case. For fixed dilaton potential V(φ) we solve\n(5.8) and (5.9) and obtain different solutions that corresponds to different factors b(z),\nφ(z) and V(z) but the same V(φ). When we use reconstruction potential method we\nhave fixed b(z),f0(z) and fixed boundary conditions for scalar field φand blackening\nfunction g, and obtain only one solution of EOM. Therefore this means that the ob-\ntained solution is sensitive to change of b(z).\n– 53 –6 Conclusion and Discussion\nIn the paper we have considered the running coupling constant and β-function in\nholographic models supported by Einstein-dilaton-Maxwell action for heavy and light\nquarks. We obtain a significant dependence of the running coupling constant αand\nβ-function on mass of quarks, chemical potential and temperature. At the first-order\nphase transitions, the functions αandβfunctions undergo jumps depending on tem-\nperature and chemical potential. For heavy quarks these jumps depend more sharply\non thermodynamic parameters. The magnitude of the jumps increases with decreasing\nenergy scales for both light and heavy quarks.\n•The dependence of running coupling αfor light quarks on thermodynamic pa-\nrameters µandTis presented in Fig. 14. We see that\n–in the region of the hadronic phase, the running coupling increases very\nslowly with increasing chemical potential and temperature.\n–in the region of the QGP phase, the running coupling increases significantly\nfaster than in the hadronic phase with increasing chemical potential and\ntemperature.\n–the slope of increment in coupling constant is larger for lower energy scales\n–on the 1-st order transition line, the running coupling has a jump. the\nmagnitude of the jump is zero at the CEP and increases with increasing µ\nnear the CEP to a value that remains almost the same along the entire line\nof the 1-st order phase transition.\n–the magnitude of the jump increases with decreasing energy scale.\n•The dependence of running coupling αfor heavy quarks on thermodynamic pa-\nrameters µandTis presented in Fig. 21. We see that\n–in the region of the hadronic phase, the running coupling increases very\nslowly with increasing chemical potential and almost does not change with\nincreasing temperature.\n–in the region of the QGP phase, the running coupling increases rather fast\nthan the temperature increases and not so fast with increasing chemical\npotential.\n–the slope of increment in coupling constant is larger for lower energy scales\n–on the 1-st order transition line, the running coupling has a jump. The\nmagnitude of the jump is greatest at µ= 0 and decreases along the line of\nthe 1-st order transition to zero in the CEP.\n– 54 ––the magnitude of the jump increases with decreasing energy scale.\n•The dependence of the β-function for light quarks at a coupling constant equal\nto 1 on the thermodynamic parameters µandTis shown in Fig. 26. We see that\n–in all regions the β-function is negative.\n–in the region of the hadronic phase, the β-function is practically independent\nof the chemical potential and log |β|increases with increasing temperature.\n–in the region of the QGP phase, the β-function increases with increasing\nchemical potential and decreases with increasing temperature.\n–on the 1-st order transition line, the β-function has a jump; the magnitude\nof the jump is zero at the CEP and becomes negative values along the 1-st\norder phase transition line and takes the maximum absolute value at T=0.\n–the jump of log |β|decreases with increasing energy scale.\n•The dependence of the β-function for heavy quarks at a coupling constant equal\nto 1 on the thermodynamic parameters µandTis shown in Fig. 33. We see that\n–in all regions the β-function is negative.\n–in the region of the hadronic phase, the β-function is practically independent\nof the chemical potential and log |β|increases with increasing temperature.\n–in the region of the QGP phase, the βfunction increases with increasing\nchemical potential and decreases with increasing temperature.\n–on the 1-st order transition line, the β-function has a jump; the magnitude\nof the jump is zero at the CEP and becomes negative values along the 1-st\norder phase transition line and takes the maximum absolute value at T=0.\n–the jump of log |β|decreases with increasing energy scale.\n•We also studied the RG flow near our solution. Our solution is unstable, that\nis inevitably associated with a negative potential, that in turn is associated with\nthe existence of the hardonic phase.\n•The RG flow for light and heavy quarks are approximately the same although the\nassociated warp factor is completely different.\n•Choosing different boundary conditions to obtain the RG flows for heavy and\nlight quarks, just exert a shift in the solution without changing significantly.\n– 55 –It is important to note that to obtain the RG flows for heavy and light quarks\nwe utilized some approximations for dilaton potential and gauge kinetic function. In\nfact, the solution of dilaton field is as a function of z, i.e. V(z) and gauge kinetic\nfunction f0(z). Therefore, we have no direct formula for V(φ) and f0(φ). Then using\napproximations functions, dilaton field and gauge kinetic function as a function of φ\ncan be obtained.\nWe would like to emphasize that to respect the null energy condition (NEC) we\nshould consider the blackening function and the dilaton potential for light and heavy\nquarks in holographic model for different values of µandzhin such a way that zhto be\nless than values of second horizon zh2. At the second horizon zh2temperature equals\nto zero. Therefore blackening function g(z) should be non-negative at the interval\n0≤z≤zh.\nOther physical quantities also have jumps at the 1-st order phase transition in\nisotropic and anisotropic models. Namely, entanglement entropy [62, 63], electric con-\nductivity and direct photons emission rate [50, 64] and energy loss [65, 66] have these\njumps at the 1-st order phase transition.\nIt is known how the primary (spatial) anisotropy affect the QCD phase transition\ntemperature [25, 59]. Also that there is another type of anisotropy due to a magnetic\nfield and its effect on the QCD phase diagram too, see [49, 50, 56–58, 60, 61]. The\ncorresponding anisotropy also changes the running coupling αandβ-function. We will\nstudy these changes in a separate paper.\n7 Acknowledgements\nThe work of I.A., P.S. and M.U. is supported by RNF grand 20-12–00200. The work of\nA.H. was performed at the Steklov International Mathematical Center and supported\nby the Ministry of Science and Higher Education of the Russian Federation (Agreement\nNo.075-15-2022-265).\n– 56 –A EOM at zero temperature\nLet us consider zero temperature case. At zero temperature the blackening function\ng(z) = 1. Then we obtain the following simplifications for the metric (2.4)\nds2=e2A(z)\nz2\u0002\n−dt2+d⃗ x2+dz2\u0003\n(A.1)\nand for the EOMs (2.6)-(2.10)\nφ′′+\u0012\n3A′−3\nz\u0013\nφ′+\u0012z2e−2AA′\ntf0,φ\n2−e2AVφ\nz2\u0013\n= 0, (A.2)\nA′′\nt+\u0012f′\n0\nf0+A′−1\nz\u0013\nA′\nt= 0, (A.3)\nA′′−A′2+2\nzA′+φ′2\n6= 0, (A.4)\ne−2Az2f0A′2\nt= 0, (A.5)\nA′′+ 3A′2−6\nzA′+4\nz2+e2AV\n3z2= 0. (A.6)\nWe see that equation (A.5) requires A′\nt= 0 and we left with\nφ′′+\u0012\n3A′−3\nz\u0013\nφ′−e2AVφ\nz2= 0, (A.7)\nA′′−A′2+2\nzA′+φ′2\n6= 0, (A.8)\nA′′+ 3A′2−6\nzA′+4\nz2+e2AV\n3z2= 0. (A.9)\nA.1 Preliminary about the reconstruction method\nLet us consider the following action\nS=1\n16πG5Z\nd5x√−g\u0014\nR−1\n2∂µφ∂µφ− V(φ)\u0015\n. (A.10)\n– 57 –The Einstein equations that follow from the action (A.10) for the ansatz for metric\n(A.1) and scalar field φ=φ(z) have the form\nφ′′+ (3A′−3\nz)φ′−e2AVφ\nz2= 0, (A.11)\nA′′−A′2+2\nzA′+φ′2\n6= 0, (A.12)\nA′′+ 3A′2−6\nzA′+4\nz2+e2AV\n3z2= 0. (A.13)\nFrom equation (A.8) we get the expression for dilaton field\nφ′(z) =vuut−6 \nA′′−A′2+2\nzA′!\n, (A.14)\nand from equation (A.13) we get the expression for potential depending on zcoordinate\nV(z) =−3z2e−2A\"\nA′′+ 3A′2+ \n−6\nz!\nA′−1\nz \n−4\nz!#\n. (A.15)\nFor B.C. φ(z0) = 0 we define\nφ(z, z0) =Zz\nz0vuut−6 \nA′′−A′2+2\nzA′!\ndz (A.16)\nSo, our solution depends on z0. We can denote our solution as φz0(z). We define\nV(φ, z 0) =V(Z(φ, z 0)) (A.17)\nwhere V(z) is defined from eq.(A.15) and Zis the inverse to the function φz0=φz0(z)\nZ:φ→z, such that φz0(Z(φz0)) =φz0 (A.18)\nWe can draw the form of V=V(φz0) using the parametric plot.\nIntroducing new variable\nX=˙φ\n3B\n˙B, (A.19)\none obtains the following RG equation [19, 20]\ndX\ndφ=−4\n3\u0012\n1−3\n8X2\u0013\u0012\n1 +1\nX∂φV(φ)\nV(φ)\u0013\n(A.20)\n– 58 –Now we solve equation (A.20) with boundary condition\nB.C. :X\f\f\f\nφ=φ1=X1 (A.21)\nWe take X=X(z) and solution φz0(z) with fixed z0and draw\nX=X(φ) (A.22)\nusing parametric plot. Now we take some value of φ=φ1and in the parametric plot\nfindX1\nX1=X(φ1) (A.23)\nWe can solve the 1-st order diff. equation (A.20) with B.C.(A.23). We denote solution\nX=X(φ,X1, φ1). Then we can find X=X(z) as\nX(z) =X(φz0(z),X1, φ1) (A.24)\nthe result should be independent of z0and coincide with (A.19).\nA.2 Reconstruction of the potential\nA.2.1 Potential of light quarks model\nSubstituting the scale factor (2.15) into (2.16) and (2.19), we obtain the solution for\ndilaton field as\nφ(z, φ 0) =φ0+ 2√\n3a \n√\n2a+ 1 arcsinh r\nb(2a+ 1)\n3z!\n(A.25)\n−p\n2(a−1) arctanh s\n2b(a−1)\n(2a+ 1)bz2+ 3z! !\nand the potential of dilaton\nV(z) =−6\u0000\nbz2+ 1\u00012a−2\u0002\nbz2\u0000\n(a(6a+ 7) + 2) bz2+ 5a+ 4\u0001\n+ 2\u0003\n. (A.26)\nThe functions φ(z) and V(z) given by (A.25) and (A.26) are presented in Fig. 48.A\nand Fig. 48.B, respectively. The function V(φ) that we recover from given functions\nV(z), eq.(A.26), and φ(z) depending on φ0, eq.(A.25), is presented in Fig. 49. We see\nthatV(φ) depends on φ0.\n– 59 –φ0= -10φ0=0-1.0 -0.5 0.5 1.0z\n-12-10-8-6-4-22φ\n-1.0 -0.5 0.5 1.0z\n-18-17-16-15-14-13-12VA B\nFigure 48 . The scalar field for the light quarks as a function of zfor different B.C. φ0=−10\n(light green) and φ0= 0 (dark green) (A) The potential Vas a function of zfor light quarks\n(B).\nφ0=-10φ0=0-15 -10 -5 5φ\n-140-120-100-80-60-40-20\nFigure 49 . The light quarks potential Vas a function of the scalar field φwith B.C.\nφ0=−10 (solid green line) and φ0= 0 (solid green line). The green dashed lines correspond\nto the approximation functions (A.27) and (A.28), respectively.\nWhen the boundary condition for the scalar field is selected as φ0=−10 at z= 0,\nwe can approximate the light quarks potential function Fig. 49 by\nVapprox, −10(φ) =18X\ni=0ciφi, (A.27)\nand by fitting the points of the graph on the domain −23.5≤φ≤3.5, we obtain the\nfollowing coefficients that is given in the Table 1 in Appendix B.\nWith boundary condition φ0= 0 at z= 0, the approximation for the light quarks\n– 60 –potential takes the form\nVapprox, 0(φ) =18X\nj=0bjφj, (A.28)\non the domain −13.5≤φ≤13.5 with fitting coefficients that is given in the Table 1\nin Appendix B.\nA.2.2 Potential of heavy quarks model\nThe solution for dilaton field by inserting the scale factor (2.24) into the expression\n(2.16) as\nφ(z, φ 0) =√\n6Zz\n0dξs\u0012\n−4p ξ3−2 sξ\n3\u00132\n+ 2\u0012\n4p ξ2+2 s\n3\u0013\n+ 12 p ξ2+2 s\n3\n+φ0 (A.29)\nand the potential of dilaton\nV(z) =−2e1\n3(6p z4+2 sz2) \n6 + 5 s z2+ 2(9 p+ s2)z4+ 24 psz6+ 72 p2z8!\n.(A.30)\nThe potential of heavy quarks as a function of the scalar field is presented in Fig.\n51. Considering the boundary condition for the scalar field as φ0=−10 at z= 0, one\ncan approximate the heavy quarks potential function Fig. 51 (khaki dashed line) by\nVapprox, −10(φ) =18X\ni=0ciφi, (A.31)\nφ0= -10φ0=0-1.0 -0.5 0.5 1.0z\n-15-10-55φ\n-1.0 -0.5 0.5 1.0z\n-18-17-16-15-14-13-12V\nA B\nFigure 50 . The scalar field φfor heavy quarks as a function of zfor different boundary\nconditions φ0=−10 (khaki line) and φ0= 0 (olive line) (A) and the potential Vas a\nfunction of zfor heavy quarks (B).\n– 61 –and by fitting the points of the graph on the domain −23.5≤φ≤3.5, we obtain the\nfollowing coefficients that is given in the Table 2 in Appendix B. Also, for choosing b.c\nasφ0= 0 at z= 0, one can approximate the heavy quarks potential function Fig. 51\n(olive dashed line) by\nVapprox, 0(φ) =18X\ni=0biφi, (A.32)\nand by fitting the points of the graph on the domain −13.5≤φ≤13.5, we obtain\nthe following coefficients that is given in the Table 2 in Appendix B. It is important to\nnote that our valid domain in these approximations for zis−8.5≤z≤8.5.\nφ0=-10φ0=0-15 -10 -5 5φ\n-140-120-100-80-60-40-20\nFigure 51 . The heavy quarks potential Vas a function of the scalar field φwith B.C.\nφ0=−10 (solid khaki line) and φ0= 0 (solid olive line). The khaki and olive dashed lines\ncorrespond to the approximation functions (A.31) and (A.32), respectively.\n– 62 –A.3 The blackening function g(z)and the dilaton potential V(φ)\nA.3.1 Light quarks model\nThe blackening function and the dilaton potential depicted for the light quarks model\ncorrespondingly for different values of µandzhin Fig. 52 and Fig. 53, respectively.\nTo respect the null energy condition (NEC) we should consider holographic model for\nzhless than values second horizon zh2. At the second horizon zh2temperature equals\nto zero. Therefore blackening function g(z) should be non-negative at the interval\n0≤z≤zh.\nμ=0,zh=1\nμ=1,zh=1\nμ=1.5,zh=1\n0.0 0.2 0.4 0.6 0.8 1.0z 0.00.20.40.60.81.0g\nμ=0,zh=2\nμ=1,zh=2\nμ=1.5,zh=2\n0.0 0.5 1.0 1.5 2.0z 0.00.20.40.60.81.0g\nA B\nFigure 52 . The blackening function for the light quarks model for different values of µand\nzh.\nμ=0,z h=1\nμ=1,z h=1\nμ=1.5,z h=1\n0.0 0.5 1.0 1.5φ -17-16-15-14-13-12\nμ=0,z h=2\nμ=1,z h=2\nμ=1.5,z h=2\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5φ-35-30-25-20-15\nA B\nFigure 53 . The dilaton potential for the light quarks model for different values of µandzh.\n– 63 –A.3.2 Heavy quarks model\nThe blackening function and the dilaton potential depicted for the heavy quarks model\ncorrespondingly for different values of µandzhin Fig. 54 and Fig. 55, respectively.\nμ=0,z h=1\nμ=1,z h=1\nμ=1.5,z h=1\n0.0 0.2 0.4 0.6 0.8 1.0z 0.00.20.40.60.81.0g\nμ=0,z h=2\nμ=1,z h=2\nμ=1.5,z h=2\n0.0 0.5 1.0 1.5 2.0z 0.00.20.40.60.81.0g\nA B\nFigure 54 . The blackening function for the heavy quarks model for different values of µand\nzh.\nμ=0,z h=1\nμ=1,z h=1\nμ=1.5,z h=11 2 3 4 5 6φ\n-300-250-200-150-100-500\nμ=0,z h=2\nμ=1,z h=2\nμ=1.5,z h=25 10 15 20 25φ\n-8×108-6×108-4×108-2×1080\nA B\nFigure 55 . The dilaton potential for the heavy quarks model for different values of µand\nzh.\n– 64 –A.4 Reconstruction of the gauge kinetic function f0(φ)\nA.4.1 Light quarks model\nThe gauge kinetic function for light quarks model takes the form [42]\nf0(z) =e−cz2+alog(bz2+1), (A.33)\nwhere the parameters a,bandcare introduced in (2.14) and (2.15). Using the explicit\nform of the solution (A.25) in Fig. 56 we plot f0(φ). This function can be approximated\nas\nf0,approx (φ) =1\nm3√\n2πexp\"\n−1\n2\u0012φ−φ0\nm1\u00132#\n+m2, (A.34)\nwhere the coefficients are the following: m1= 2.70895, m2= 8.26922 ∗10−6and\nm3= 0.39931.\n0\n0,approx\n-25 -20 -15 -10 -5 0 5φ0.20.40.60.81.00\nFigure 56 . The gauge kinetic function f0for light quarks as a function of φproduced for\nB.C. φ0=−10. The dashed line corresponds with the approximation function (A.34).\nA.4.2 Heavy quarks model\nThe gauge kinetic function for heavy quarks takes the form [43, 54]\nf0(z) =e−2\n3sz2+p z4, (A.35)\nwhere sandpare the same parameters are introduced in (2.24). We utilized the form\nof the solution (A.29) to plot f0(φ) in Fig. 57. When the boundary condition for the\nscalar field is selected as φ0=−10 at z= 0, this function can be approximated by\nf0,approx (φ) =30X\ni=0wiφi, (A.36)\n– 65 –and by fitting the points of the graph on the domain −25≤φ≤6, we obtain the\nfollowing coefficients that is represented in Table 3 Appendix B.\n0\n0,approx\n-25 -20 -15 -10 -5 0 5φ0.20.40.60.81.00\nFigure 57 . The gauge kinetic function f0for heavy quarks as a function of φproduced for\nB.C. φ0=−10. The dashed line corresponds with the approximation function (A.36).\nB Approximation coefficients of potential and gauge kinetic\nfunction\nThe coefficients of approximation of light and heavy quarks for potential function with\ndifferent b.c, i.e. φ0=−10 and φ0= 0 are given in the Table 1 and 2, respectively.\nThe coefficients of approximation of heavy quarks for gauge kinetic function with\nboundary condition φ0=−10 is given in the Table 3.\n– 66 –φ0=−10 φ0= 0\nc0=−8333.49 b0=−11.996\nc1=−6769.87 b1=−2.86057 ∗10−12\nc2=−2734.34 b2=−1.50457\nc3=−731.531 b3= 2.544∗10−14\nc4=−145.923 b4=−0.082475\nc5=−23.1662 b5= 4.19564 ∗10−15\nc6=−3.05115 b6=−0.0020461\nc7=−0.343354 b7=−1.87468 ∗10−16\nc8=−0.0337693 b8=−0.0000216737\nc9=−0.00294892 b9= 3.45919 ∗10−18\nc10=−0.000229452 b10=−2.21701 ∗10−7\nc11=−0.0000156801 b11=−3.4441∗10−20\nc12=−9.12457 ∗10−7b12=−2.48019 ∗10−10\nc13=−4.34389 ∗10−8b13= 1.91673 ∗10−22\nc14=−1.61884 ∗10−9b14=−7.31265 ∗10−12\nc15=−4.4908∗10−11b15=−5.53005 ∗10−25\nc16=−8.662∗10−13b16= 1.20875 ∗10−14\nc17=−1.03328 ∗10−14b17= 6.33555 ∗10−28\nc18=−5.74044 ∗10−17b18=−5.74045 ∗10−17\nTable 1 . The coefficients of the approximation for light quarks potential function with two\nboundary condition, i.e. φ0=−10 and φ0= 0.\n– 67 –φ0=−10 φ0= 0\nc0=−22808 .2 b0=−11.9744\nc1=−21257 .3 b1=−2.20242 ∗10−12\nc2=−9781.21 b2=−1.52795\nc3=−2964.73 b3=−4.47153 ∗10−13\nc4=−666.91 b4=−0.0811821\nc5=−118.904 b5= 4.56467 ∗10−14\nc6=−17.5253 b6=−0.00460869\nc7=−2.20197 b7=−1.74355 ∗10−15\nc8=−0.241625 b8=−0.0000276665\nc9=−0.0235074 b9= 3.37277 ∗10−17\nc10=−0.00202414 b10=−1.26397 ∗10−6\nc11=−0.00015091 b11=−3.58649 ∗10−19\nc12=−9.40185 ∗10−6b12= 4.41746 ∗10−9\nc13=−4.70369 ∗10−7b13= 2.13034 ∗10−21\nc14=−1.81389 ∗10−8b14=−7.61899 ∗10−11\nc15=−5.14567 ∗10−10b15=−6.64785 ∗10−24\nc16=−1.00593 ∗10−11b16= 2.05593 ∗10−10\nc17=−1.20764 ∗10−13b17= 8.49863 ∗10−27\nc18=−6.7091∗10−16b18=−6.70907 ∗10−16\nTable 2 . The coefficients of the approximation for heavy quarks potential function with two\nboundary condition, i.e. φ0=−10 and φ0= 0.\n– 68 –φ0=−10\nw0= 0.649979 w16= 1.31403 ∗10−12\nw1= 0.0505165 w17= 6.87886 ∗10−14\nw2= 0.0106962 w18= 7.47132 ∗10−16\nw3=−0.00133661 w19=−6.25931 ∗10−17\nw4=−0.00089918 w20=−8.64631 ∗10−19\nw5= 0.000254561 w21= 1.02579 ∗10−19\nw6= 0.000146991 w22= 1.07317 ∗10−21\nw7=−0.0000111608 w23=−1.69210 ∗10−22\nw8=−0.0000108994 w24=−1.05002 ∗10−24\nw9=−4.23684 ∗10−7w25= 2.89729 ∗10−25\nw10= 3.80199 ∗10−7w26= 2.86221 ∗10−27\nw11= 4.82338 ∗10−8w27=−4.96868 ∗10−28\nw12=−4.28331 ∗10−9w28=−2.10423 ∗10−29\nw13=−1.28932 ∗10−9w29=−3.46181 ∗10−31\nw14=−6.18659 ∗10−11w30=−2.15509 ∗10−33\nw15= 8.43119 ∗10−12\nTable 3 . The coefficients of the approximation of gauge kinetic function for heavy quarks\nwith boundary condition φ0=−10.\n– 69 –C Beta-function β(α)-2D plot\nC.1 Beta-function β(α)for light quarks model\nBeta-function β=β(α) for light quarks at T= 0.08 with different µcorresponding to\npoints in Fig. 3 and Fig. 4 is plotted in Fig. 58. The points are indicated by blue\ntriangle ( µ= 2, zh= 0.9637), two green dots µ= 0.43, zh= 2.816 and µ= 0.557, zh=\n3.652, magenta star µ= 0.394, zh= 4.415 and brown square µ= 0.36, zh= 25.8. From\nFig. 58, we see that the blue line showing β(α) for the QGP phase deviates from the\nlinear relationship for small α. Also the line order (top to bottom) of the thick green\nand brown line changes as we move from small αto larger one.\n0.96 0.98 1.α -2.45-2.32-2.2β\n0.36 0.38 0.4α -0.92-0.89-0.86βμ=0.36, T=0.08μ=0.394, T=0.08μ=0.43, T=0.08μ=0.557, T=0.08μ=2, T=0.080.2 0.4 0.6 0.8 1.0α\n-4-3-2-10β\nFigure 58 . Beta-function β=β(α) for light quarks at T= 0.08 with different µcorrespond-\ning to points in Fig. 3 and Fig. 4 indicated by blue triangle ( µ= 2, zh= 0.9637), two green\ndotsµ= 0.43, zh= 2.816 and µ= 0.557, zh= 3.652, magenta star µ= 0.394, zh= 4.415 and\nbrown square µ= 0.36, zh= 25.8; [µ] = [T] = [z]−1= GeV.\nBeta-function β=β(α) for light quarks at T= 0.11 with different µcorresponding\nto points in Fig. 3 and Fig. 4 is plotted in Fig. 59. The points are indicated by blue\ntriangle ( µ= 2, zh= 1.0025), two green dots µ= 0.32, zh= 4.12 and µ= 0.557, zh=\n2.5, magenta star µ= 0.31, zh= 4.64 and brown square µ= 0.3, zh= 17.205. Also, for\nT= 0.11 there is a linear relationship for βas a function of α.\n– 70 –μ=0.3, T=0.11\nμ=0.31, T=0.11\nμ=0.32, T=0.11μ=0.557, T=0.11\nμ=2, T=0.11\n0.36 0.39 0.42α -0.92-0.87-0.83β\n0.96 0.98 1.α -2.45-2.32-2.2β0.2 0.4 0.6 0.8 1.0α\n-4-3-2-10βFigure 59 . Beta-function β=β(α) for light quarks at T= 0.11 with different µcorrespond-\ning to points in Fig. 3 and Fig. 4 indicated by blue triangle ( µ= 2, zh= 1.0025), two green\ndots µ= 0.32, zh= 4.12 and µ= 0.557, zh= 2.5, magenta star µ= 0.31, zh= 4.64 and\nbrown square µ= 0.3, zh= 17.205; [ µ] = [T] = [z]−1= GeV.\nBeta-function β=β(α) for light quarks at different fixed T= 0.2,T= 0.11,\nT= 0.08 with different µcorresponding to the points in Fig. 3 and Fig. 4 is depicted\nin Fig. 60. In all cases the β-function decreases with increasing α.\n– 71 –0.96 0.98 1.α -2.45-2.32-2.2β\n0.9766 0.9769α -2.395-2.39βμ=0.36, T=0.08μ=0.394, T=0.08μ=0.43, T=0.08μ=0.557, T=0.08μ=2, T=0.08μ=0.3, T=0.11μ=0.31, T=0.11μ=0.32, T=0.11μ=0.557, T=0.11μ=2, T=0.11μ=0.32, T=0.2μ=2, T=0.20.2 0.4 0.6 0.8 1.0α\n-4-3-2-10βFigure 60 . Beta-function β=β(α) for light quarks at different fixed T= 0.2,T= 0.11 and\nT= 0.08 with different µcorresponding to the points in Fig. 3 and Fig. 4; [ µ] = [T] = [z]−1=\nGeV.\n– 72 –μ=0.35,T 1=0.574\nμcr=0.443,T 1=0.574\nμ=0.55,T 1=0.5740.2 0.4 0.6 0.8 1.0α\n-3.0-2.5-2.0-1.5-1.0-0.50.0β\nμ=0.64,T 2=0.532\nμcr=0.7,T 2=0.532\nμ=0.76,T 2=0.5320.2 0.4 0.6 0.8 1.0α\n-3.5-3.0-2.5-2.0-1.5-1.0-0.50.0βA B\nμ=1.5,T 1=0.574\nμ=3,T 1=0.574\nμ=1.5,T 2=0.532\nμ=3,T 2=0.5320.2 0.4 0.6 0.8 1.0α\n-2.5-2.0-1.5-1.0-0.50.0β\nμ=0.35,T 1=0.574\nμcr=0.443,T 1=0.574\nμ=0.55,T 1=0.574\nμ=0.64,T 2=0.532\nμcr=0.7,T 2=0.532\nμ=0.76,T 2=0.532\n0.35 0.40 0.45 0.50 0.55α-2.4-2.2-2.0-1.8-1.6-1.4-1.2-1.0β0.2 0.4 0.6 0.8 1.0α\n-3.0-2.5-2.0-1.5-1.0-0.50.0β\nC D\nFigure 61 . Beta-function β=β(α) for heavy quarks model at fixed T= 0.574, small µ(A)\nT= 0.532 (B), T= 0.574, large µ(C) and combination of (A,B) in (D) corresponding to the\npoints in Fig. 9 and Fig. 10; [ µ] = [T] = [z]−1= GeV.\nC.2 Beta-function β(α)for heavy quark model\nBeta-function β=β(α) for heavy quarks model at fixed T= 0.574, small µ(A)\nT= 0.532 (B), T= 0.574, large µ(C) and combination of (A,B) in (D) is plotted in\nFig. 61. All the lines are corresponding to the points in Fig. 9 and Fig. 10. Similar to\nlight quark model, beta-function has the linear dependence on α.\nIn Fig. 62 the beta-function β=β(α) for heavy quarks model at fixed T= 0.2 (A)\nµ= 0.1 (B), µ= 1.5 (C) is depicted. All lines are corresponding to the points in Fig.\n9 and Fig. 10. The linear dependence of βonαis also seen and increasing αincreases\ntheβ-function.\n– 73 –μ=1.5, T=0.2μ=0.64, T=0.2μ=0.1, T=0.2\n0.38 0.4 0.42α\n-1.2-1.1-1β0.2 0.4 0.6 0.8 1.0α\n-4-3-2-10β\n0.4 0.401α\n-1.079-1.077-1.075βμ=0.1, T=0.574μ=0.1, T=0.532μ=0.1, T=0.20.2 0.4 0.6 0.8 1.0α\n-4-3-2-10βA B\nμ=1.5, T=0.574\nμ=1.5, T=0.532\nμ=1.5, T=0.20.2 0.4 0.6 0.8 1.0α\n-4-3-2-10β\nC\nFigure 62 . Beta-function β=β(α) for heavy quarks model at fixed T= 0.2 (A) µ= 0.1\n(B),µ= 1.5 (C) corresponding to the points in Fig. 9 and Fig. 10; [ µ] = [T] = [z]−1= GeV.\n– 74 –References\n[1] N.N. Bogoliubov and D.V. Shirkov, The Theory of Quantized Fields. GTTL, Moscow\n(1957); English trans.: New York, NY: Interscience (1959).\n[2] K. G. Wilson and J. B. Kogut, “The Renormalization group and the epsilon\nexpansion,” Phys. Rept. 12, 75-199 (1974).\n[3] Schwartz M. D. Quantum field theory and the standard model. – Cambridge university\npress, 2014.\n[4] M. Gell-Mann, M. and F. E. 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Slepov, “Energy Loss in Holographic Anisotropic\nModel for Heavy Quarks in External Magnetic Field”, [arXiv:2012.05758 [hep-th]]\n– 79 –"    },    {        "title": "2402.14513v1.Chern_Simons_gravity_and_PT_Symmetry.pdf",        "content": "KCL-PH-TH/2024- 06\nChern-Simons gravity and PTSymmetry\nN E Mavromatosa,band Sarben Sarkarb\naPhysics Division, School of Applied Mathematical and Physical Sciences,\nNational Technical University of Athens, Zografou Campus, Athens 157 80, Greece and\nbTheoretical Particle Physics and Cosmology Group,\nDepartment of Physics, King’s College London, London, WC2R 2LS, UK\nThis paper considers a string inspired effective axion anomalously coupled to Abelian gauge fields\nand gravity. By considering the renormalisation group flows in the flat space limit it is observed\nthat a Hermitian parity symmetric phase of the theory can flow into a non-Hermitian parity-time\nsymmetric phase. This behavior has implications for Chern-Simons gravity. The repulsive nature\nof gravity, usually attributed to the existence of a positive cosmological constant, is reinterpreted\nat large scales as a flow from a Hermitian (attractive) gravitational theory, to a PT-symmetric\n(repulsive) gravity in the infrared. The discussion here is presented in the context of a Chern-Simons\ngravitational theory but it may be valid more generally in gravity with torsion. The validity of such\na scenario in realistic theories might alleviate the need for de Sitter phases in the current epoch of\nthe cosmological evolution, thus avoiding their associated conceptual and technical complications.\nI. INTRODUCTION AND MOTIVATION\nNonHermitian PT-symmetric field theories [1–16] are field theory extensions of PT-symmetric quantum mechan-\nics [17–23], which has a plethora of applications, both, experimental [24, 25] and theoretical. Pis a linear operator\n(such as parity) and Tis an anti-linear operator (such as time-reversal). A quantum mechanical system with unbro-\nkenPTsymmetry has a completely real spectrum which leads to unitary dynamics [17, 18, 22] . The antilinearity,\nrather than Hermiticity, that characterises PT-symmetric theories, provides a novel guiding principle for quantum\ntheory [26, 27]. PT- symmetric field theories are effective field theories, which may describe aspects of Beyond-the-\nStandard Model physics (BSM), including massive neutrinos [13, 28–45].\nOur aim in this work is to pursue a potential gravitational aspect of PT-symmetric quantum physics related to\nthe observed accelerated expansion of the universe [46, 47]. In particular, we shall conjecture that a PT-symmetric\nphase of gravity, in which the gravitational interaction appears as repulsive, can arise within Chern-Simons (CS)\ngravity [48, 49] inspired from string theory [50–54]. This phase arises due to a specific singular behaviour under\nrenormalization-group (RG) flows of specific couplings in the model, related to the interactions of axionic degrees of\nfreedom, that are known [54, 55] to characterise string theory spectra [50–53], with CS terms; such terms arise in the\ntheory as a result of the Green-Schwarz mechanism [56] for the cancellation of gauge and gravitational anomalies in\nthe extra-dimensional spaces of compactified strings. We shall consider string effective theories in which gravitational\nand gauge CS anomalous terms survive in the (3+1)-dimensional spacetime after sting compactification. The flat\nspacetime limit of such constructions leads to an axion electrodynamics model, once the gauge sector CS terms are\nrestricted to Abelian U(1) (electromagnetic) gauge groups. We shall reinterpret existing works in the literature on\nnon-perturbative renormalisation group (RG) studies of such axion-electrodynamics models [57], in terms of potential\nphase transitions of the Hermitian system to a nonHermitian but PT-symmetric one. The transition occurs at the\nsingularities of the RG beta functions in the infrared (IR) region, which such systems exhibit.\nOn coupling the system to dynamical gravity, we conjecture that the PT-symmetric nonHermitian IR phases may\ncorrespond to repulsive gravitational interactions at large (cosmological) scales. Should such a feature be valid in\nrealistic cosmologies, such phases might describe the observed accelerated expansion of the universe at late eras,\nthus avoiding the technical and conceptual issues associated with de Sitter vacua, especially from the point of view\nof string theory. Positive cosmological-constant (de Sitter) gravitational backgrounds appear in conflict with both\nperturbative [58, 59] and non-perturbative string theory frameworks [60–62]. The presence of (observer-dependent)\nde Sitter horizons, prevents the existence of a well-defined scattering S-matrix, due to the lack of asymptotic “in” and\n“out” states. The failure of the decoupling of infinite towers of massive string states at specific points of the moduli\nspaces of string theory in de Sitter backgrounds, leads to the incompatibility of such backgrounds with a consistent\nembedding in a quantum gravity setting. Thus the latter models appear to belong to the so-called “swampland\nregions” of quantum gravity theories. Such frameworks allow therefore only quintessesnce models for dark energy as\nthe cause of the currently observed accelerating expansion of the universe.\nMotivated by these issues, we embark on a speculative journey in which PT-symmetric gravitational theories may\noffer alternative explanations of the current-era cosmological data. Our investigation is prompted by the intriguing\nbehaviour noticed in a recent study of a field theory developed for gravitational axion phenomenology and dynamical\nmass generation [40, 41, 63–65]. We noticed a RG flow [64] from Hermitian values of the coupling to those of aarXiv:2402.14513v1  [hep-th]  22 Feb 20242\nnon-Hermitian but PT-symmetric version of the field theory in a one-loop analysis, which might be interpreted as\na dynamical appearance of PTsymmetry, as in the nonHermitian Nambu-Jona-Lasinio model, for instance, where\nthe nonHermiticity is claimed to appear spontaneously [15]. We further examined the robustness of these findings by\nworking with beta functions with non-zero ϵ(where spacetime dimension D= 4−ϵ) and by working to three loops\nin the Yukawa coupling and two loops in the quartic coupling‘[66, 67]. The idea that the effects of renormalisation\ncan lead from a Hermitian theory to a PTsymmetric theory was demonstrated in a different context in the Lee\nmodel [68, 69], a solvable toy model. The concepts of PTsymmetry are most developed in the context of quantum\nmechanics however [17]. PTsymmetry is realised not just in the form of the Hamiltonian but implies also a different\nphase of the system [1, 70, 71] with an energy spectrum which cannot be obtained continuously from a Hermitian\ntheory by analytic continuation in the couplings of the theory [72–74]. Although the phase is well understood for the\nquantum mechanical anharmonic oscillator (with negative coupling) it can only be seen approximately in quantum\nfield theoretic systems through, for example, singularites in renormalisation group beta functions (calculated in some\napproximation).\nThe possibility of classical repulsive PT-symmetric gravity was noted in [75] at a speculative level but with no\nconcrete proposal. Clearly gravity is attractive at everyday scales but at very large scales it is not. One way of\nunderstanding scales is through the renormalisation group in quantum theory. In this article we consider a gravi-\ntating system that exhibit singularities in their phase diagrams for large length scales. Such systems involve axion\n(pseudoscalar) degrees of freedom coupled to topological Pontryagin anomalous densities of both gravitational and\ngauge nature. For simplicity we consider the case of Abelian gauge sectors and the absence of gravitational anomalous\nterms. For this case, which coincides with axion electrodynamics, we show that the singularities in the phase diagram\nof the quantum theory lead to inconsistencies that can be remedied (by appropriate phase transitions) on replacing\nthe Hermitian theory by a nonHermitian but PT-symmetric version.\nFirst let us consider RG flows in the simpler context of chiral Yukawa interactions of axion fields ϕ(x) coupled to\nfermion fields ψ(x) with coupling constant g[40, 64, 65],\nLYukawa =ig ϕψ γ5ψ. (1)\nThe one-loop RG βfunction of ginDspacetime dimensions as ϵ→0+leads to1\nd\ndt(g2) =5\n8π2(g2)2−ϵg2, (2)\nwhere d/dt≡µ d/dµ , and µis a transmutation mass scale. The solution to (2) with ϵt≪0 is\ng2≈ −1\nC+5\n8π2t, (3)\nwhere Cis an integration constant. For Hermitian couplings [where gin (1) is real] at t= 0 and C <0, we see from\n(3) that g2increases with increasing µuntil g2hits a pole at finite t=tp= 8π2|c|/5>0 [76]. This is similar to the\nLandau pole in Abelian U(1) gauge theories and may be a sign of new physics. In terms of an infrared (IR) cutoff\nm,t= log( µ/m). Here, t= 0 corresponds to an IR (large-distance) regime and t→ ∞ (µ≫m) corresponds to an\nultraviolet (UV) (short-distance) fixed point.\nPerturbation theory breaks down at the pole, and one must decide how to go around the pole. Formally, for values of\nt > t pthe theory becomes non-Hermitian because g2<0. However, this theory is PT-symmetric andasymptotically\nfree[77] because g2→0−ast→ ∞ . The βfunction of the Hermitian theory ends up at a pole (starting from\nsome IR regime), while the βfunction of the PT-symmetric non-Hermitian theory starts at that pole, and then\nexhibits asymptotic freedom in the UV. This indicates that one could indeed go around such RG β-function poles by\n“analytically continuing” the theory to the asymptotically free non-Hermitian PT-symmetric version, which has the\nsame observables.2\nA conjectural prescription based on analysis of scalar field theories in D= 1 is given in Ref. [78] that allows the\nanalytic continuation of a Hermitian (but ill-defined) self-interacting scalar theory in Dspacetime dimensions to a\nwell-defined non-Hermitian PT-symmetric theory. The conjecture in higher D(which is not proven) states that if\nZ(λ) is the partition function of the path integral of the Hermitian self-interacting scalar theory with coupling λ, then\nthe corresponding partition function ZPT(g) of the PT-symmetric theory with self-interaction coupling g, is given by\nlnZPT(g) = Re\u0002\nlnZ(λ=−g+i0+)\u0003\n. (4)\n1An analysis with respect to the coupling grather than g2was made in Refs. [64, 65] but in this paper we restrict our attention to RG\nflows of g2. In [67] the analysis is extended to higher loops where the renormalization group flows show the same qualitiative features.\n2Since the PTtheory has in general a different dynamical behavior, and a different phase space than the initial theory, the word “analytic\ncontinuation” here should be interpreted as meaning the substitution of the pertinent coupling gby its purely imaginary image, ig,\ng∈R. This is to be understood throughout the current article.3\nThis conjecture suggests a mechanism for avoiding the aforementioned poles in the RG βfunctions by analytically\ncontinuing the theory to a PT-symmetric theory with analytically continued couplings.\nRecently, the conjecture was applied in [79] to treat the UV pole that appears in the RG βfunction of the O(N)\nscalar models at large Nwith Hermitian self-interactions of the form (in Euclidean formalism)\nLint=λ\nN(⃗ϕT·⃗ϕ)2, N≫1, λ∈R, (5)\nwhere ⃗ϕdenotes the N-component scalar field and Tindicates matrix transposition [76]. The avoidance of the pole\nin the conjecture of [78] is accomplished under the condition that the physical observables of the conjectured PT-\nsymmetric theory in the IR and the original Hermitian theory in the UV remain the same and the poles in the β\nfunction do not affect the physical observables.3\nThe Hermitian large- Ntheory is known to be characterized by a positive RG βfunction and a Landau pole\nin the UV [81], as happens for the renormalised coupling (3) of the Yukawa interaction (1) in (3+1)-dimensional\nmodels [64, 65, 67]. The PT-symmetric theory is obtained by replacing λ→ −gin the Euclidean formalism (taking\ncare of the Riemann sheet structure in complex gshown in detail for the quantum anharmonic oscillator [82]),\nwhere gdenotes the PT-theory coupling; following the conjecture (4) of Ref. [78], one analytically continues λ→\n−g+i0+in the corresponding expressions of the initially Hermitian theory, such as thermodynamic quantities at\nfinite temperature [79]. The nonperturbative (large N) renormalized coupling gR(µ) ofPTsymmetry (in dimensional\nregularization in D= 4−ϵdimensions, ϵ→0+), exhibits a negative βfunction [79],\nβ=d\ndlnµgR(µ) =−8π2\nln2(µ2\nΛ2c)=−g2\nR(µ)\n2π2<0, (6)\nwhere µis a transmutation mass scale. The scale Λ cdenotes the scale at which the running coupling gR(µ) develops\na pole as µincreases from µ= Λctowards the UV:\ngR(µ) =4π2\nln(µ2\nΛ2c). (7)\nNotably, the expressions (6) and (7) are exact in the large- Nlimit. As in the Yukawa case (3), the renormalised\ncoupling of PTsymmetry becomes small in the UV; that is, it exhibits asymptotic freedom .\nIn Ref. [83] the authors apply the conjecture of [78] to a (4 −ϵ)-dimensional theory, ϵ→0+, with fermionic\nself-interactions (the large- NGross-Neveu model of a four-fermion interaction among N-component fermions):\nLint=\u0010α\nN\u00111\nD−1\u0010\nψfψf\u0011D\nD−1, (8)\nwhere ψT\ns= (ψ2, ψ2, . . . , ψ N), represents the N-component fermion fields and αis the coupling. The Hermitian\nversions of such theories are asymptotically free in the UV and exhibit IR poles in their RG βfunctions (a situation\nopposite to the Yukawa coupling of [64, 65, 67] examined above. The avoidance of the β-function pole is similar).\nThe “analytic continuation” (or rather switch ) in Ref. [78] to the counterpart PT-symmetric nonHermitian theories\nin the IR leads to interesting phase transitions in a finite-temperature analysis. Indeed, the renormalized coupling of\nthe Hermitian theory α(µ) has a negative β-function\nβα=d αR(µ)\ndlnµ=−α2\nR(µ)\n2π2, (9)\nwhich is opposite in sign from the corresponding β-function of the large- Nscalar theory [79]. Its solution leads to\nthe existence of a pole at some cutoff scale Λ cin the IR, and asymptotic freedom in the UV ( µ/Λc→0+):\nαR(µ) = 4 π2/ln(µ2/Λ2\nc). (10)\nOn the other hand, for large Nthe nonperturbatively renormalized coupling of the PTtheory, αPT\nR(µ), obtained\nby the analytic continuation procedure of Ref. [78], has a positive βfunction, which decays into the IR regime,\nβPT\nα=d\ndlnµαPT\nR(µ) =αPT\nR(µ)2\n2π2⇒ αPT\nR(µ) =−4π2\nln(µ2\nΛ2c), µ < Λc. (11)\n3There are unresolved subtleties in defining a PT-symmetric path integral for N-component fields. Moreover, the conjecture of [78] has\nbeen scrutinised in [80], where, within a WKB approximation it was argued that the conjecture is valid under certain conditions.4\nIt is hoped that such simple models may lead to a nonperturbative understanding of the rich phase structure such\nsystems may posses.4\nIn this article we consider gravitating systems that exhibit singularities in their phase diagrams for large length\nscales. These systems involve axion degrees of freedom coupled to topological Pontryagin anomalous densities of\nboth gravitational and gauge nature. These couplings are related. Such self-gravitating axion systems have a natural\ninterpetation as low-energy systems of string-inspired gravity, where the axion plays the rˆ ole of the dual of the field\nstrength of the Kalb-Ramond spin-one field of the massless gravitational multiplet of the string. We can simplify to\nthe case of Abelian gauge sectors but no gravitational anomalous terms. This simplification allows us to make use of\nresults from axion electrodynamics. We demonstrate that there are singularities in the phase diagram of the theory,\nwhich lead to inconsistencies that can be remedied (by appropriate phase transitions) involving a replacement of the\nHermitian theory by a nonHermitian but PT-symmetric version. When embedded in curved space time, such models\nlead to gravitational models owing to negative values of the Newton constant [84–86].\nAxions may also be viewed as totally antisymmetric torsion in the geometry of the string-inspired effective theory\n[54, 87], at least up to quartic order of target-spacetimes derivatives in a Regge slope α′expansion [50–53]. Such a\nlink could also lead to the conjecture that the aforementioned behavior may characterize more general gravitational\ntheories with torsion [88–92], such as Einstein-Cartan theory [93] and not just standard Einstein gravity.\nThis paper is organized as follows: In section II, we briefly review some results from the asymptotic safety ap-\nproach to quantum gravity, as regards UV versus IR properties of the running of the gravitational constant with\nthe renormalization group (RG) scale; this running will be correlated with the running of the coupling in our axion\nelectrodynamics. In Sec. III we give details of the superstring-inspired gravitational model and explain the origin of\nthe axion and its anomalous gauge and gravitational couplings in this context. In Sec. IV, based on a reduction of the\ngauge structure of the string-inspired model to Abelian electromagnetic interactions only, we discuss the nonpertur-\nbative renormalization of the coupling in (a flat-spacetime) axion electrodynamics, the prototypical Hermitian model.\nThe RG flow shows a singular infrared behaviour which lies near the nontrivial infrared fixed point of the model.\nAlthough this singularity is much more complicated than a pole (noted in earlier examples), we propose to bypass such\na singular flow behavior by replacing the theory by its non-Hermitian PT-symmetric counterpart, which has a trivial\nIR fixed point but a nontrivial UV fixed point. In Sec. V, we connect this flat spacetime model to our (string-inspired)\ngravitational theory (in a dynamically curved spacetime) and provide arguments on how the analytically extended\nPTversion of the model in the infrared results in a phase transition to a repulsive- gravity phase at large scales.\nWe also discuss potential subtleties concerning the validity of this conjecture in the quantum gravity sector, which\nwill itself lead to a RG running of the gravitational coupling (Newton’s “constant”). Our conjecture rests on the\npresence of the (singularly) running axion coupling; the gravitational coupling changes sign at the infrared and leads\nto a repulsive-gravity phase at large scales. Our conclusions and outlook are given in Sec. VI.\nII. ASYMPTOTIC SAFETY AND QUANTIZATION OF GRAVITY\nThe quantization of Einstein gravity as an ordinary field theory leads to the well-known asymptotic safety of\nEinstein (3+1)-dimensional gravity [94], and the existence of a UV fixed point for the running Newton constant\n[84, 95]. For our purposes we follow the treatment in [84], and references therein (in particular [85, 86]), which is\ncloser in spirit to the work here. The approach is based on a quantization of gravity on a Euclidean lattice, using a\ndiscrete higher-derivative action whose naive classical continuum limit corresponds to the action\nS=Z\nd4x√−gh\n−1\n2κ2R+b RµνρσRµνρσ+1\n2(a−4b)CµνρσCµνρσ+ Λi\n, (12)\nwhere aandbare dimensionless constants, Cµνρσ is the Weyl tensor, and Λ is a cosmological constant term. The\nhigher-derivative terms in the action help to avoid unboundedness (from below) of the Einstein-Hilbert action on the\nlattice [96, 97]. For string-inspired quantum theories of gravity, such higher-derivative additions are natural.\nAccording to that approach to gravity quantization, the RG running of Netwon’s constant G = 8 π κ2in the vicinity\nof the UV fixed point is described by\nG(k2) = 8 π κ2= G ch\n1 +a0\u0010m2\nIR\nk2\u00111/2ν\n+O\u0010\u0010m2\nIR\nk2\u00111/ν\u0011i\n, (13)\n4The thermodynamical analysis of Ref. [83] reveals also an interesting first-order phase transition at the pole, separating stable, meta-\nstable, and unstable phases.5\nwhere k2is a running mass scale, and m2\nIRis an IR mass scale, which can be taken to be of order of today’s Hubble\nparameter H0, if we are to explain the current-epoch acceleration of the universe. The detailed computations of [85, 86]\nhave shown that a0>0 and the critical exponent ν= 1/3. The running Newton’s coupling flows to G c, asymptotically\nfor large k≫mIR(short distances); G ccan be identified with the physical value of Newton’s “constant”, measured\nat laboratory scales, i.e.(in units ( ℏc= 1):√8πGc=p\n8πGphys∼M−1\nPl, with the reduced Planck mass scale\nMPl= 2.4×1018GeV.\nThe UV behavior (13) is model dependent, but the approach of G to an UV fixed-point value G cask2/m2\nIR→ ∞ [94]\nseems to be model independent. On the other hand, the expression (13) is ill-behaved in the IR, k2/m2\nIR→0+, where\nit develops a polethat needs to be regulated . Such a regularisation is highly model dependent. In [84], the following\nregularization has been suggested, by means of which the pole 1 /k2is replaced simply by 1 /(k2+m2\nIR). Under such\na scheme one may write for the IR behavior of the running Netwon’s coupling in the models of [84–86]\nG(k2)≃Gch\n1 +a0\u0010m2\nIR\nk2+m2\nIR\u00111\n2ν+. . .i\n. (14)\nIf this assumption is correct, this would imply the following IR (large (cosmological) distance) behavior of the\nrunning Netwon’s coupling in (3+1)-dimensions:\nlim\nk2/m2\nIR→0+G(k2) = G ∞h\n1−\u0010a0\n2ν(1 +a0)+. . .\u0011k2\nm2\nIR+O(k4\nm4\nIR)i\n, (15)\nwhere G ∞= G c(1 +a0+. . .), independent of the IR cutoff mIR.\nIn the above formulas the UV cutoff Λ UV, which can be identified with the (reduced) Planck scale MPl, appears\ninside the expression for G c= Λ−2\nUVeGc, with eGca dimensionless coupling. In this scheme for regulating the IR pole\nin the large-distance behavior of the renormalized Newton’s coupling G, the latter is always positive , thus gravity\nappears as attractive during the entire RG flow from UV to IR. In such a formalism, the addition of a positive\ncosmological constant Λ >0 can be interpreted as leading to an effectively repulsive gravity that leads to the current-\nera acceleration of the universe [46]. However, in such a de Sitter Universe, the asymptotic Newtonian limit cannot be\nrigorously defined, and hence such an interpretation is not strictly valid [98] unless the de Sitter phase is metastable.\nBypassing this IR pole will be of concern to us below, where we argue that such a task can also be achieved by\npassing to a version of gravity to its non-Hermitian but PT-symmetric version that we shall define properly within the\ncontext of CS gravity [48, 49] inspired from strings [50–53]. The nonHermitian theory will be argued to characterise\na repulsive phase of the anomalous CS gravity. This model requires a detailed discussion not commonly found in\ntreatments of ordinary Einstein gravity theory, as we see in the next section.\nIII. STRING-INSPIRED MODELS OF CHERN-SIMONS GRAVITY\nIn superstring theory [50–53] after compactification to four space-time dimensions, the bosonic ground state of the\nclosed-string sector consists of massless fields in the so-called gravitational multiplet , which contains a spin-0 (scalar)\ndilaton Φ( x), a spin-2 traceless symmetric tensor field, gµν(x), which is uniquely identified as the (3+1)-dimensional\ngraviton, and a spin-1 antisymmetric tensor gauge field Bµν(x) =−Bνµ(x), known as the Kalb-Ramond (KR) field.\nIn what follows, for brevity and concreteness, we set the four-dimensional dilaton field to a constant, Φ( x) = Φ 0. This\nfixes the string coupling\ngs= exp(Φ) = exp(Φ 0). (16)\nThere are always consistent solutions of the four-dimensional string theory with such a configuration, and this suffices\nfor our purposes here.\nThere is a U(1) gauge symmetry of the closed-string (3+1)-dimensional target-space-time effective-field-theory\naction, associated with the KR B-field transformations\nBµν(x)→Bµν(x) +∂[µθν](x), µ, ν = 0, . . .3, θ µ(x)∈R, (17)\nwhere Greek indices from now on denote space-time indices, taking on the values 0 , . . .3, and the symbol [ . . .] denotes\nantisymmetrization of the respective indices. The U(1) gauge symmetry of the closed-string sector implies that the\ncorresponding effective action will be expressed only in terms of the field strength of the B-field:\nHµνρ(x) =∂[µBνρ](x). (18)6\nThis is subject to the following Bianchi identity\nH[νρσ;µ]=∂[µHνρσ]= 0. (19)\nFrom now on, we omit the space-time-coordinate arguments of the fields for brevity. The semicolon denotes covariant\nderivative with respect to the standard Christoffel connection Γα\nµν= Γα\nνµof the metric gµν.\nWe next make the important remark that in superstring theory anomaly cancellation requirements imply a modifi-\ncation of the KR field strength (18) by appropriate gauge (Yang-Mills (Y)) and Lorentz (L) (gravitational) CS terms\n(Green-Schwarz mechanism) [51]\nH=dB+α′\n8κ\u0010\nΩ3L−Ω3Y\u0011\n,\nΩ3L=ωa\nc∧dωc\na+2\n3ωa\nc∧ωc\nd∧ωd\na,Ω3Y=A∧dA+A∧A∧A, (20)\nwhere α′=M−2\ns, with Msthe string mass scale, which is in general different from the four-dimensional Planck mass\nscale MP= 1.22×1019GeV≡√\n8π κ−1. We used standard differential form notation for brevity [99]. Above, A\nis the Yang-Mills gauge field one-form, and ωa\nbthe spin-connection one-form (the Latin indices a, b, c, d are (3+1)-\ndimensional tangent-space ( i.e. Lorentz-group-SO(1,3)) indices, referring to the Minkowski manifold which is tangent\nto the space-time manifold at a coordinate point x). The addition of the CS terms to (20) leads to a modification of\nthe Bianchi identity (19), which can now be written as [51]\nεµνρσH[νρσ;µ]=εµ\nabcHabc\n;µ=α′\n32κ√−g\u0010\nRµνρσeRµνρσ−FµνeFµν\u0011\n, (21)\nwhere the right side denotes the mixed anomaly, due to chiral fermions in the theory circulating in the anomalous\nloop [100, 101], gdenotes the determinant of the metric tensor, F=dA+A∧Ais the two-form corresponding to the\nYang-Mills field strength (we use form notation for brevity here), Rµνρσ is the Riemann space-time curvature tensor5\nand\nεµνρσ =√−g ϵµνρσ, εµνρσ=sgn(g)√−gϵµνρσ, (22)\nwith ϵ0123= +1, etc., are the gravitationally covariant Levi-Civita tensor densities, totally antisymmetric in their\nindices. The tilde symbol above the curvature and gauge-field tensors denotes the corresponding dual, defined as\neRµνρσ =1\n2εµνλπRλπ\nρσ,eFµν=1\n2εµνρσFρσ. (23)\nThe nonzero quantity on the right hand side of (21) is the “mixed (gauge and gravitational) quantum anomaly” [100,\n101], which is known to be a total divergence of a topological current Kµ(containing the CS forms in (20)). In general,\nthe parameters α′andκ2are independent in generic string models [52, 53], especially in view of the possibility of\nlarge-extra-dimension compactifications,\nImplementing the Bianchi identity (21) in a path integral of the low-energy theory, we obtain an effective target-\nspace gravitational action for the ground state of the bosonic sector of superstring theory, of interest to us here. To\nlowest order in the string Regge slope α′, the (3+1)-dimensional effective action of the closed-string bosonic sector is\nthen given by [50–54, 102]:\nSeff(I)\nB=Z\nd4x√−gh\n−1\n2κ2R+1\n2∂µb ∂µb+r\n2\n3α′\n96κb(x)\u0010\nRµνρσeRµνρσ−FµνeFµν\u0011\n+. . .i\n, (24)\nwhere the dots . . .denote gauge, as well as higher derivative, terms appearing in the string effective action, that\nwe ignore for our discussion here. In this construction, b(x) is the KR axion, which appears initially as a standard\npseudoscalar Lagrange multiplier field , and eventually becomes dynamical, after path-integration of the Hµνρfield [54],\nwith a canonically-normalised kinetic term with the correct sign [102] relative to the space-time curvature (Eisntein-\nHilbert) terms in (24).\n5Our conventions and definitions used throughout this work are: signature of metric (+ ,−,−,−), Riemann Curvature tensor Rλ\nµνσ=\n∂νΓλ\nµσ+ Γρ\nµσΓλ\nρν−(ν↔σ), Ricci tensor Rµν=Rλ\nµλν, and Ricci scalar R=Rµνgµν.7\nAt this point we should remark that, in (3+1) spacetime dimensions, the KR axion field is the dual of the KR field\nstrength Hµνρ:\n∂σb∝εσµνρHµνρ. (25)\nGiven that in string theory, up and including O(α′) in the effective target-space action, Hµνρplays the rˆ ole of\nspacetime torsion, one may observe that the considerations in our article regarding the model (24), apply also to\nEinstein-Cartan theories [93], in which the axionic degrees of freedom are dual (in the above sense (25)) to the totally\nantisymmetric part of the torsion [88–90, 92]. For completion at this stage we also note that, because of (25), the\nEuclidean version of the string effective action (to O(α′)) [50–53]:\nSB=−Z\nd4x√−g\u00101\n2κ2R+1\n6HλµνHλµν+. . .\u0011\n, (26)\nthere are ambiguities when we analytically continue (26), with (25), back to Minkowski-signature spacetimes. The\nlatter are associated with the order we implement (25) in the path integral [102]. Indeed, if we firstuse the property\nof the Levi-Civita tensor density in four space-time dimensions with Euclidean metric,\nε(E)\nµνρλεµνρσ (E)= +6 δσ\nλ, (27)\nwhere δσ\nλdenotes the Kronecker delta, and then analytically continue the action (26) to Minkowskiian-signature space\ntime, we obtain the real (Hermitian) effective gravitational action (24). However, if we first analytically continues\n(26) to a Minkowski-signature space time , and then use the Minkowski-signature version of (27), that is the relation:\nεµνρλεµνρσ=−6δσ\nλ, (28)\nthen, we obtain an effective action in which the kinetic terms of the bfield have the wrong sign relative to the space-\ntime-curvature terms in the effective action, and thus the KR axion would behave like a ghost (phantom) field. As\ndiscussed in [41], one can map such an action into a nonHermitian PTversion of attractive CS gravity [48, 49]. In\nour article we stay within the normal axion field case. Our PTnonHermitian gravity theory, that represents the\nphase transition of the Hermitian system at the infrared-region singularities of the RG flow, corresponds to a repulsive\ngravitational force at large sditances.\nThe KR axion is the so-called string-model independent axion [103], associated with the gravitational string mul-\ntiplet as explained above, which is to be distinguished from other stringy axions arising from compactification. We\nalso observe that, in view of the mixed anomaly appearing in (21), the KR axion field couples to the gravitational\nand gauge fields. This latter interaction is PandTviolating, and hence in view of the overall CPT invariance of the\nrelativistic, local and unitary (quantum) field theory (24), also CPviolating ( Cdenotes the Dirac charge-conjugation\noperator). This is the standard approach to deriving the effective theories of string gravitational axions and their\nmixing with the axions coming from compactifications [54, 103], in which the axions are pseudoscalar fields. As is\nwell known, this approach leads to a rich phenomenology [104–110].\nAn important property of the non-Abelian axion-gauge-Pontryagin-index interactions in (24) is periodicity of the\neffective actions under shifts of the axion field b(x). Indeed, in view of the Pontryagin-topological-class interpretation\nof the gauge CS anomalous action term (in the language of forms for brevity) [99]:\nχ=1\n16π2Z\nd4xTr(F∧F)∈Z, (29)\nwhere Fthe non-Abelian gauge field strength two form, we observe that the Hermitian action term involving the CS\ncoupling of the b-field:\nS∋1\n16π2fbZ\nd4x b(x) Tr(F∧F) (30)\nexhibits a periodicity in the bfield, b(x)→b(x) + 2π fb, where fbdefines the so-called axion coupling, which in the\ncase of (24) is given by\nf−1\nb=π2\n2r\n1\n6α′\nκ. (31)\nThis implies the breaking of the shift-symmetry b(x)→b(x) + constant (due to the total derivative nature of the\nanomalous Pointryagin terms in (24), by instanton effects in the non-Abelian gauge theories for which the index χ̸= 08\n(29), and leads to the well-known nonperturbative axion potentials V(b) = Λ4\ninst\u0002\n1−cos(b/fb)\u0003\n, and axion masses of\norder mb∼Λ2\ninst/fb, where Λ instis the scale of the non-Abelian gauge group instanton effects.\nFor our purposes we reduce the gauge sector to only Abelian ( U(1) electrrodynamics) terms. In what follows we\nshall ignore for the moment the gravitational dynamics placing the rest of the terms in (24) in flat target spacetimes.\nThe so-obtained part of the action (24) then leads to an axion electrodynamics model discussed in Sec. IV, which is\nquantised using path-integral and RG analysis nonperturbatively for the axion coupling [57]. This reveals a crucial\nsingular behavior (in infrared regions of the energy scale) in the phase space, which we bypass by means of replacing\n(near an appropriate infrared fixed point) the Hermitian model by a suitable non-Hermitian but PT-symmetric model,\nobtained via the replacement of the axion coupling fb(31) by a purely imaginary one:\nfb→ ifb. (32)\nThe reader should notice that the resulting nonHermitian field theory no longer exhibits a periodicity in the axion\nfield b(x). Axion masses in such a case may be generated dynamically, e.g. through their Yukawa interactions\nwith appropriate chiral fermions [40, 41, 63–65], including heavy right-handed neutrinos [111], that may appear in\nextensions of the Standard Model. The replacement (32) signifies a new phase. We then use such flat-spacetime\narguments as support for our conjecture on the emergence of a repulsive phase of gravity at large scales (infrared),\nonce we embed the model in a dynamical curved spacetime.\nIV. A REINTERPRETATION OF THE NONPERTURBATIVE RENORMALIZATION GROUP\nANALYSIS OF AXION ELECTRODYNAMICS\nSome of the analysis and conjectures in this paper are based on the nonperturbative renormalisation group (RG)\nflow, given in [57], but the interpetation in terms of PTphases given below has not been discussed or implied in that\nwork. We commence our discussion by first reviewing a nonperturbative RG analysis [57] of the Euclidean Lagrangian\nLErelated to (30) in the massless ( m→0) limit:\nLE=1\n4FµνFµν+1\n2∂µa∂µa+1\n2m2a2+1\n4iga Fµν˜Fµν, (33)\nwith gauge fixing ∇.A= 0 and A 0= 0. The mass dimension of gis−1 and so the theory is not perturbatively\nrenormalisable.6\nThis theory, however, can be treated using the functional renormalisation group (FRG) and the Wetterich equation\n[112, 113].\n∂tΓk=1\n2Tr[∂tRk(Γ(2)\nk+Rk)−1] (34)\nwhere∂\n∂t≡kd\ndk, with ka running RG scale. Γ kis the effective average action. Γ k→Λ=SΛis the bare action and\nΛ is the UV cut-off. Γ k→0= Γ is the full quantum effective action. The Wetterich equation is an integro-differential\nnonlinear equation and cannot be solved exactly. Progress in obtaining solutions requires an ansatz for Γ k. A standard\nansatz restricts Γ kto\nΓk=Z\nd4x[ZF\n4(Fµν(x))2+Za\n2(∂µa)2+m2\nk\n2a(x)2+igk\n4a(x)Fµν(x)˜Fµν(x)] +ZFZ\nd4x LGF (35)\nwhere LGFdenotes a gauge fixing piece to the Lagrangian, whose details can be found in [57]. The beta functions\ncalculated using this scheme are not universal but are scheme dependent [57]. This is a feature of nonrenormalisability\nin the perturbative sense.\nIt is shown in [57] that from the above ansatz\n∂\n∂tm2\nk=∂\n∂tgk= 0. (36)\nThe physical observables are expressed in terms of dimensionful quantities:\nm2\nR=m2\nk\nZa,g2\nR=g2\nk\nZ2\nFZa(37)\n6At this stage the reader should recall, for future use, that in the context of our string theory (24), the axion field a(x) corresponds to\nthe KR massless axion b(x), and the coupling gis given by g=q\n2\n3α′\n24κ=1\n4π2fb, where fpis defined in (31).9\nwhere\n∂tZa=1\nΩ\u0014\n∂2\n∂q2Z\nd4pδ2\nδa(p)δa(−q)∂tΓk\u0015\na,A,q→0, (38)\n∂tZF=1\nΩ\u0014\n4\n3∂2\n∂q2Z\nd4p nκnλδ2\nδAκ(p)δAλ(−q)∂tΓk\u0015\na,A,q→0(39)\nThis is a standard procedure in applying the functional renormalisation group. Associated dimensionless quantities\nare\nm2≡m2\nR\nk2, g2≡g2\nRk2. (40)\nWe define the field anomalous dimensions as\nγa=−∂tlogZ a, γ F=−∂tlogZ F. (41)\nThis leads to the following nonperturbative beta functions, which determine the running of the coupling constants:\n∂tg2= (2 + 2 γF+γa)g2, (42)\n∂tm2= (γa−2)m2. (43)\nwhere\nγa=g2\n6 (4π)2\u0010\n2−γF\n4\u0011\n, (44)\nγF=g2\n6(4π)2\u0012(2−γa\n4)\n(1 +m2)2+(2−γF\n4)\n1 +m2\u0013\n(45)\nThese equations, derived in [57], are the starting point of our analysis. They have not been carefully analysed for\nthe full range of mandg. However the analysis of various limiting cases is still relevant for us in conjunction with\nrelations of the form given in (14); these will be discussed in the next subsection.\nA. Behavior of renormalisation group flows in limiting cases\nForm≫1 and g≪1 (weak coupling),\n∂tg2= \n2 +g2\n3 (4π)2!\ng2(46)\nwhich has a solution g2\nR(k) =g2\nR(Λ)\n1+1\n6(4π)2(Λ2−k2)g2\nR(Λ). Also m2\nR(k) =m2\nR(Λ)\n1+1\n6(4π)2(Λ2−k2)g2\nR(Λ).\nLet us consider the massless theory, which is appropriate for the flat spacetime and Abelian gauge sector limit\nof our string-inspired grtavitational axion theory (24), discussed in the previous section. In this case, from (42) we\ndeduce that m= 0 is an infrared fixed point. For m= 0 and g≪1\n∂tg2= \n2 +5g2\n3 (4π)2!\ng2. (47)\nThis leads to\ng2\nR(k) =g2\nR(Λ)\n1 +5\n6(4π)2(Λ2−k2)g2\nR(Λ), (48)\na result similar to that found for the case m≫1, examined above. Both these results do not indicate any interesting\nnonanalyticity in the flow to the infrared. We move away from these limits to see whether there is any nonanalytic\nstructure.10\nFor arbitrary values of gwe are able to solve (44) for γaandγF(as discussed in detail in [57], where we refer the\ninterested reader for further study)\n∂tg2=βg2= 2g213g4−384π2g2\u0000\n21 + 17 m2+ 4m4\u0001\n−147456 π4\u0000\n1 +m2\u00012\ng4−384π2g2(1 +m2)−147456 π4(1 +m2)2(49)\n∂tm2=βm2= 6m2−g4+ 128 π2g2\u0000\n3 + 7 m2+ 4m4\u0001\n−49152 π4\u0000\n1 +m2\u00012\n−g4+ 384 π2g2(1 +m2) + 147456 π4(1 +m2)2(50)\nThis form of the beta functions is manifestly nonperturbative. However our application of the solution of these\nequations to Chern-Simons gravity and an emergence of PTsymmetry is new. For m= 0 the expressions simplify.\nForm= 0 we are able to solve these equations which is crucial for our analysis.\nB. Massless case: infrared singularities in the non-perturbative renormalisation group flows and a\nPTsymmetric phase\nFor our purposes, we consider below the massless case, m= 0, but arbitrary g, studied in [57], which corresponds\nto the flat spacetime and Abelian gauge limit of the string effective action (24). The corresponding beta function\nreads:7\n∂tg2= 2g213g4−8064 π2g2−147456 π4\ng4−384π2g2−147456 π4(51)\nThe beta function is plotted in Fig. 1 as a function of g2. The values of gfor which the denominator vanishes (thus\nleading to a singular behavior of the beta function) have the property that they are either real or purely imaginary\nand come in conjugate pairs:\nSingularity of βgoccurs for gsing: (±48.3728i,±78.2689) . (52)\nThis is significant and leads to a new PTsymmetric interpretation which crucial for our analysis. Hence, presumably,\nthe beta function of a nonHermitan theory would also show such singular behavior, as we shall show later. Of interest\nin what follows will be the singularity of the Hermitian theory, occurring at ( cf.(52)):\ng2\nsing= (gsing R (k)k)2\f\f\f\nk=ksing= 6126 .0207 , (53)\nwhich corresponds to a RG momentum scale k=ksing>0.\nNext we look at the fixed points of the beta function of the Hermitian theory, i.e.the values of the coupling at\nwhich the beta function vanishes. Notably, (51) has a trivial IR attractive fixed point at zero coupling g=gIR\n⋆= 0, as\nwell as two real and two purely imaginary nontrivial fixed points , but no fixed points with both real and imaginary\nparts. Explicitly the nontrivial fixed points denoted by g⋆are:\ng⋆= (±13.2389 i,±79.3174) . (54)\nFor the Hermitian theory therefore, the nontrivial IR stable fixed point occurs at\ng⋆2= 6291 .2499 . (55)\nThis is an IR fixed point as seen from the form of the beta function in the neighbourhood of this fixed point (passing\nfrom negative to positive, as g2increases). Thus, the Hermitian theory has an IR attractive fixed point at zero coupling\ng=gIR\n⋆= 0, followed (as g2increases) by a singular behavior, which occurs at a RG momentum scale k=ksing>0,\ncorresponding to the value (53) of g.\nWe now examine the scale k=ksingcorresponding to the singularity (53) of the RG βfunction near the non-trivial\nIR fixed point (see fig. 1). We estimate the value of ksingby solving the RG equation (51) to obtain g=g(t).\n7This equation can be solved using initial conditions determined, e.g. from astrophysical data for axion physics [114], if we accept the\nrˆ ole of the KR axion as a standard axion. This might not be the case within the most general framework of string theory, so other\nconstraints might be applicable. Nonetheless, as we shall see below, for our purposes, such constants are not relevant, as we are dealing\nwith ratios of couplings between appropriate points in the RG flow.11\n00βgg2\nFIG. 1. The beta function βg≡∂tg2of the Hermitian theory, given in Eq. (51), exhibiting the proximity of a singular behavior\nat a finite running coupling (53), as well as a fixed point behavior ( βg(g)\f\f\ng=g⋆= 0), characterised by the existence of a trivial\nfixed point at zero coupling, and a non-trivial one at strong coupling, g2\n⋆≫1. A similar figure appears in [57].\nOn integrating (51) from the nontrivial IR fixed point (55), corresponding to a RG scale k0≡mIR, to a point\ncorresponding to an RG scale k, we obtain8\n|g(k)|(2281 + 13 g2(k))−0.46|27.6−4.4·10−3g2(k)|0.0013=aet−t0=ak\nk0. (56)\nwhere ( cf.(55)) a≡ |g⋆|(2281 + 13 g⋆2)−0.46|27.6−4.4·10−3g⋆2|0.0013≃0.4, and thus, the RG running scale ksing,\ncorresponding to the singularity (53), is given by:\nksing= 1.01k0. (57)\nThe solution in (56) is just the principal branch of a function g(k) with an infinite Riemann sheet structure and so is\nextremely complicated. The singularity structure and nonanalyticity indicate that the undelying approximations are\ninadequate, and so the field theory may exist in a different phase (as occurs in an anharmonic oscillator in quantum\nmechanics where a much more complete analysis is possible). From (56) we deduce at k=ksingimmediately a very\nsimilar relation for gsing. This result is consistent since in this range of g2the beta function is negative; hence in the\nRG flow from the region of the singularity (53) to the non-trivial IR fixed point (55), an increasing running g2(k)\ncorresponds to a decreasing RG running scale k. Formally, we can take k0→0, indicating that the singularity of the\nHermitian theory lies very near the non trivial IR fixed point. Physically k0could be taken to be the energy scale\ncorresponding to the inverse size of the Universe, for our purposes.\nAsg2> g2\nsingthe beta function turns to (infinite) negative values until the (nearby lying) non-trivial IR fixed point\ng⋆2> g2\nsingis reached, at which the beta function vanishes. After this second IR fixed point the beta function is\npositive, increasing monotonically (in an unbounded way) with increasing g. As g2→ ∞ the beta function exhibits\nanother singular (infinite) behavior. In the region of very large positive g2, the beta function equation (51) can be\napproximated by\n∂tg2≃26g2, g2≫g⋆2= 6291 .2499 . (58)\nIn this region of g→ ∞ , the above equation can be easily integrated to give analytically the scaling [57]:\ng2(k∞)\ng2(k)∼e26(t∞−t)=\u0010k∞\nk\u001126\n, g2(k)≫1, k < k ∞, (59)\n8In our analysis we denotep\ng2(k) by|g(k)|.12\nwhere k∞∼et∞is the RG scale at which g(k→k∞)→ ∞ , which implies that the coupling diverges highly non-\nlinearly with the RG scale (large anomalous dumension). From (59), then, it immediately follows that this singular\nbehavior for infinite goccurs at ultraviolet RG scales k∞→ ∞ .\nWe next remark that, if we adopt the point of view of effective field theory, that no RG scale can exceed the UV\ncutoff Λ, then in Eq. (59) we must identify k∞with Λ. In our string-inspired model, we may identify Λ = Ms, where\nMsthe string scale, which is assumed not renormalised, i.eit does not run with the scale k. In the pertinent part of\nthe diagram of fig. 1, then, the RG scale kflows from k0∼0 toMs, in agreement with the positive signature of the\ncorresponding beta function.\nThe plot in Fig. 1 is problematic below the nontrivial IR fixed point, due to the singular behavior of the beta\nfunction which lies close to the IR fixed point. We would like to connect the interacting theory to the free theory at\ng= 0 smoothly via an appropriate RG flow from an UV to IR fixed point. This appears not to be possible if we stay\nexclusively within the Hermitian theory, as a result of the beta-function singularity at (53).\nIn the following we treat the singular behavior of the Hermitian beta function at k=ksingas a phase transition\nin the system by considering the corresponding non-Hermitian theory obtained via g→ig.9Notably, this is notan\nanalytic continuation. This is due to the different RG behavior of this theory. We therefore replace the part of the\ndiagram in Fig. 1 to the left of and including the singularity until the trivial IR fixed point by the corresponding flow\ndiagram pertaining to the non-Hermitian PT symmetric theory.\nThe non-Hermitian theory can be studied explicitly by writing g=i˜g. In this case, we obtain [57]\n∂t˜g2=β˜g= 2˜g213˜g4+ 8064 π2˜g2−147456 π4\n˜g4+ 384 π2˜g2−147456 π4. (60)\nThe beta function exhibits a singularity at g2\nsing NH = 2339 .93 and nontrivial fixed points at ˜ g∗\nNH=±13.2456 ⇒˜g∗2\nNH=\n175.4459. We give plots of β˜gas a function of ˜ g2(and ˜ g) in Fig. 2. The nontrivial fixed point, which is UV in nature\ndue to the form of the beta function (turning from positive to negative, as g2increases, see fig. 2), is asymptotically\nfree for this Abelian case. Thus, the RG scale at the nontrivial fixed point can be identified with the UV cutoff\nk⋆= Λ. The trivial fixed point is an infrared (IR) attractor.\nAs in the Hermitian case, by integrating the beta function from the UV fixed point (corresponding to k=k⋆\nNH= Λ)\nto the scale kNH sing , corresponding to the singularity of the beta function ( cf.fig. 2), we obtain:\n|˜gsing NH ||2281−13 ˜g2\nsing NH |−0.46(27.6 + 4 .4·10−3˜g2\nsing NH )0.0013=aNHet−t⋆=aNHkNH sing /k⋆\nNH, (61)\n(where aNH≡ |˜g⋆\nNH||2281−13 ˜g⋆2\nNH|−0.46(27.6 + 4 .4·10−3˜g⋆2\nNH)0.0013) which implies that the singularity of the beta\nfunction of the nonHermitian theory occurs at:\nkNH sing ≃0.34Λ (62)\nThis is a consistent result. The beta function is negative for g2> g⋆\nNH2, which implies that g2should increase for\ndecreasing k, in agreement with the above result. But the singularity here occurs in the UV regime.\nThe singularity of the beta function of the Hermitian theory, which corresponds to a RG scale ksing, (57), is assumed\nin our approach to correspond to a phase transition of the system, which for k≲ksingis described by the nonHermitian\ntheory. The reader should bear in mind that the scale ksing≪kNHsing = 0.34Λ = 0 .34k⋆\nNH. Since the beta function\nof the nonHermitian theory is positive, this implies that in the region of the RG scales k≤ksing,gNH(k)< gNH sing ,\nand thus the corresponding portion of the nonHermitian theory connects smoothly (via a RG flow) to the trivial fixed\npoint g= 0 with g̸= 0.\nV. INTERPRETATION IN TERMS OF A REPULSIVE GRAVITY PHASE\nWe now make inferences for the scale dependence of Newton’s constant from the above results. Our findings, using\naxion electrodynamics, are embedded into the discussion of string-inspired gravity with effective action given by (24).\nThe analogue of the dimensionless running (renormalised) axion coupling (defined in (33)) is given by:\ng=gRk=1\n24r\n2\n3\u0010MPl,R(k)\nM2\ns,R(k)\u0011\nk , (63)\n9We note for completion that, the non-perturbative RG study of this nonHermitian axion electrodynamics model has been discussed in\nthe Appendix of [57], without, however, making any attempt to give that theory any physical singificance, or the connection with the\nPTphase of gravity we are conjecturing in the present article.13\n00βgg2~\n~\n00βgg2~~\n~\n~00\nFIG. 2. Nonperturbative RG beta function for non-Hermitian case as a function of ˜ g2, given in Eq. (60) is qualitatively different\nfrom the Hermitian beta function. The area included in the red rectangle includes a nontrivial fixed point (apart from the\ntrivial one at zero coupling), which is indicated explicitly in the middle panel. The lower panel is added for completeness, and\nshows the behavior of the beta function as a function of ˜ g >0, in the neighborhood of its fixed points. The middle figure is\nsimilar to that in [57].\nwhere the index Rdenotes a renormalised quantity, and kis a “running” (RG) momentum scale. Our assumption\nis that when we embed this theory into a quantum string-inspired gravity version, the effective low-energy theory\nwill also be characterized by a renormalized combination of scales\u0010\nMPl(k)\nMs(k)\u0011\nR, running with the scale k. Since the\nstring mass scale Msis associated with the UV momentum cutoff of the effective low-energy theory, it is reasonable\nto assume that Msis not renormalized, i.e.it does not run with the RG scale k. The effective four-dimensional\nMPlis renormalized and runs since it is defined as the effective gravitational scale of the (3+1)-dimensional effective\ntheory.10Hence the square of the dimensionless running (renormalised ( R))) axion coupling (63) is g2=1\n8641\neGNk2\nM2s\nwhere the dimensionless quantity eGN≡GNM2\ns, with G Nthe (3+1)-dimensional gravitational (Netwon’s) coupling,\n10In conventional string theory MPlis proportional to the string-compactification volume and runs with k.14\nIn the case of the effective string-inspired low energy theory, we define t≡log\u0010\nk\nMs\u0011\nwhich implies that the\nultraviolet (UV) region is described by the limit k→Ms,i.e. tUV→0. By contrast, the infrared (IR) region\ncorresponds to k→0 ( or k→k0=mIR≪Ms, with mIRan infrared mass cutoff).11The RG running of eGN(k)\nwith k, or equivalently with t, is obtained by differentiating both sides of (63) with respect to t. This leads to:\nd\ndtlog\u0010\ng2eGN\u0011\n= 2 ⇒ g2eGN=Dexp\u0010\n2t\u0011\n, (64)\nwhere Dis a positive integration constant to be fixed by the boundary conditions of the RG flow. Considering the\nequation in the UV regime, t→0, we have D=g2(t= 0)eGN(t= 0). For finite k=Ms<∞, the value of\ng(k=Ms)≈g(t= 0) is large but finite, while the dimensionless eGN(t= 0) cannot be directly constrained from the\nrunning of g, which scales with kas (k/M s)26as follows from the non-perturbative renormalisation flow of g2(t). We\ncan therefore assume it also finite.\nThus, from (64), we arrive that at the non-trivial infrared fixed point of the RG flow, in which lim t→−∞ g(t) =\ng⋆<∞, finite, we obtain for the Newton’s gravitational parameter in the IR region of this flow:\nlim\nt→−∞eGN(t) = lim\nt→−∞g2(t= 0)\ng⋆2eGN(t= 0)e2t= 0,\nindicating that the gravitational constant at the singularity goes to zero. The finite value eGN(t= 0) is identified with\nthe Newton’s constant in the UV regime (short distances). In (24) the axions couple to both gravitons and gauge\nfields; the presence of gravitons is expected to modify the axion self energy, and through this their nonperturbative\nrunning. However, the topological (metric independent) nature of the Pontryagin density term of (24), b FeF, is\nexpected not to alter the singularity structure of the axion coupling, thus leading to the aforementioned properties of\nthe running gravitational coupling, in particular its vanishing in the infrared.\nTransitioning to the non-Hermitian CS theory, is achieved by replacing gwith ig, which, in the context of our string\ninspired theory (33), (63), is equivalent to the replacement of the gravitational constant in the effective action (24)\nby a purely imaginary one,\nκ→i˜κ,˜κ∈R. (65)\nUnder this very restrictive complexification of the gravitational coupling, the effective action becomes a PT-\nsymmetric nonHermitian action,12describing repulsive (anti)gravity dynamics:13\nSeff anti\nB =Z\nd4x√−gh1\n2eκ2R+1\n2∂µb ∂µb−ir\n2\n3α′\n96eκb(x)\u0010\nRµνρσeRµνρσ−FµνeFµν\u0011\n+. . .i\n, (66)\nIn the nonHermitian theory, the running of the gravitational coupling with talso follows from (63), and we have the\nfollowing relation connecting the values of the couplings at the trivial IR fixed point with their values in the UV one\nof the pertinent RG flow:\nlim\nt→−∞g2(t)eGN(t→ −∞ ) = lim\nt→−∞g2(t= 0)\ng⋆2eGN(t= 0)e2t= 0. (67)\nBut at the trivial IR fixed point lim t→−∞ g2(t)→0 (see fig. 1), while at the non trivial UV one g2(t= 0) = g⋆2<∞,\na fixed finite value. With eGN(t= 0) fininte, as discussed above, this leaves eGN(t→ −∞ ) also undetermined, and we\nassign it a finite value, which is that of a Newton’s gravitational constant at large scales.\nThis scenario may lead to a reinterpretation of the dark-energy sector of the universe observed at large distances.\nThe issue is to observe the possibility that unbroken PTsymmetry is a low energy symmetry of the real world. In\nthis sense repulsive gravity would correspond to a low-energy (large distance) phase of the universe, which would be\ndescribed effectively by the nonHermitian PT-symmetric gravitational effective action (66) of an underlying string\ntheory. In summary, upon assuming the phase transition at ksingto a phase whose dynamics is described by the\n11When we discussed nonperturbative renormalisation of the axion electrodynamics field theory (33), we gave an alternative definition of\nthe running RG parameter t, in terms of an IR scale k0:t= log( k/k0), with k0=mIR, which eventually can be taken to zero, unless\nthere are IR divergencies. In such a case the UV region corresponds to large t≫1 (formally t→ ∞ ), whilst the IR regime is described\nbyt→0. The connection with our case is to subtract from tthe (formally infinite) constant log( k0/Ms).\n12Our approach does not consider issues related to the conjecture that there are no global symmetries in quantum gravity [115].\n13As already mentioned, the presence of imaginary CS terms implies the loss of periodicity of the effective axion potential, in contrast to\nthe Hermitian case ( cf.(29), (30)). In the nonHermitian case, one may have though dynamical generation of the axion mass, as in the\nscenarios exploited in [40, 41, 63–65, 111]. We hope to come back to such scenarios in a future publication.15\nnonHermitian theory for k < k sing, one can connect smoothly a zero gravitational constant G( ksing) = 0 to a negative ,\nnon-zero and finite Netwon’s constant G( mIR)<0. This behaviour that characterises our string-inspired Chern-\nSimons gravity model is different from the standard RG analysis in asymptotic safety model ( cf.Eqs. (13), (15)). In\nour analysis, loop corrections in the gravity sector have not been considered, and so we can only provide arguments\non this behavior for gravity in this CS model, but not a proof.\nThe complete answer on the running of Newton’s gravitational parameter in our PT-symmetric-nonHermitian-\ngravity approach can only be determined when the full quantum dynamics of the gravity sector in the model is\ntaken into account. Nonetheless, we make the important remark that the topological nature of the gauge Pontryagin\ndensity (but not of the Hirzebruch term in (24), whose variation with respect to the metric field yields the Cotton\ntensor [48, 49]) is encouraging for the maintenance of the crucial singularity structure of the axion coupling in such\ncases. Further analysis therefore in the gravitational sector is necessary to fully understand this behavior but this is\nbeyond the scope of the current article.\nVI. CONCLUSIONS AND OUTLOOK\nThis article presents a novel perspective on repulsive gravity at large cosmological distances, potentially explaining\nthe observed accelerated expansion of the universe. It contrasts the conventional interpretation of this accelerated\nexpansion in terms of a positive cosmological constant or dark energy, with the proposal of a PT-symmetric phase of\ngravity. The gravitational system, which leads to this conjecture, is a string-inspired Chern-Simons gravity, involving\nthe coupling of axions to anomalous Chern-Simons gauge and gravitational terms. Thus the conjecture may be specific\nto such string-inspired systems.\nOur conjecture is based on having a a running Newton’s coupling (i.e. the inverse of the square of the reduced\nPlanck mass), which in the ultraviolet regime exhibits an asymptotic safety behavior, (13), with the running coupling\ntending to a nontrivial UV fixed-point value as the scale k2/m2\nIR→ ∞ . In the IR, however, the running leads to the\nHermitian system’s beta function developing a singularity at a running RG scale ksing∼1.01mIR→0. There is a\nrelation (63) between the axion couplings in the string effective action (24) and the coupling in field theory for massless\naxion electrodynamics, which is the limit of the string-effective gravitational theory for flat-spacetime and Abelian-\ngauge-group. We refer to this connection as an embedding. The IR singularity corresponds to a phase transition in\naxion electrodynamics. The Hermitian version of the embedded model exhibits an RG flow ( cf.fig. 1) characterised\nby a singularity of the beta function (53) in the IR region of the running RG scale (57). We propose that such a\nregion corresponds to a phase transition of the system, such that the theory that describes the singularity, as well as\nthe region k < k sing, is the nonHermitian model obtained by g→i˜g, ˜g∈R. This theory exhibits a smooth behavior\nin the RG running from ksingof the Hermitian model to the trivial IR fixed point at ˜ g= 0 ( cf.fig. 2). Bypassing\nthe singular behavior is a highly model-dependent issue. In most of the approaches so far, the attractive nature of\ngravity is maintained when the IR behavior of gravity is regulated (15). In this work we have given a prescription\nfor avoiding the IR singular behavior of the Newton’s coupling, as the running RG scale k2/m2\nIR→0+, within the\ncontext of the string-inspired gravity model with axions. The presence of the anomalous topological couplings are\ncrucial for inducing the repulsive PTphase. In our case, the embedded nonHermitian axion electrodynamics theory\nleads in the IR region of the RG running, to a repulsive phase of gravity upon using (65), in which the axion coupling\ngis related to the gravitational scales of string theory (63). This gives a prescription for PT-symmetric version of\nrepulsive gravity at large (cosmological) scales. For a more complete justification of the conjecture of the repulsive\nphase of gravity, we need a proper (non-perturbative) quantisation of the effective gravitational theory obtained from\nstrings. This is very challenging.\nFinally, we stress that ordinary quantized Einstein gravity does not appear to exhibit repulsive behavior. We\nonly see this effect in anomalous string-inspired gravity with gravitational axions (24) ([50, 51]). Such features do\ncharacterize strings, but one can alsopostulate local gravity theories that share them, most notably modified gravity\ntheories with torsion [88–93], known to be associated with propagating axionic degrees of freedom. Apart from the\npossible connection with repulsive gravity, our scenario of the emergence of PTsymmetry is also connected to a\nchange in the generalised symmetries [116] of axionic electrodynamics. This deserves closer attention per se which we\nhope to consider in a future publication.16\nACKNOWLEDGEMENTS\nWe thank Carl M. Bender for discussions on PT-symmetric quantum mechanics. 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Rept.\n1051 , 1 (2024), arXiv:2307.07547 [hep-th]."    },    {        "title": "2402.14543v1.Low_frequency_Resonances_in_Grid_Forming_Converters__Causes_and_Damping_Control.pdf",        "content": " \n \n \nLow\n-\nfrequency Resonance\ns\n \nin\n \nGrid\n-\nForming \nConverter\ns\n: Causes and \nDamping \nControl\n \n \nF\nangzhou Zhao\n,\n \nMember, \nIEEE\n, \nTianhua Zhu\n,\n \nMember, IEEE\n, \nZejie Li, \nStudent m\nember, IEEE\n,\n \n \nand \nXiongfei Wang, \nFell\now\n, IEEE\n \n \nAbstract\n—\nGrid\n-\nforming voltage\n-\nsource converter (GFM\n-\nVSC) \nmay\n \nexperience\n \nlow\n-\nfrequency resonances,\n \nsuch as\n \nsynchronous \nresonance (SR) and sub\n-\nsynchronous resonance (SSR)\n,\n \nin\n \nthe \noutput \npower\n. \nThis paper\n \noffers\n \na comprehen\nsive study on the \nroot \ncause\ns\n \nof \nlow\n-\nfrequency \nresonances \nwith GFM\n-\nVSC \nsystems \nand\n \nthe \ndamping control\n \nmethod\ns\n.\n \nThe typical GFM control structures \nare introduced first\n, along\n \nwith a mapping between the \nresonances\n \nand control loops. Then, the causes of SR and SSR\n \nare \ndiscuss\ned\n,\n \nhighlighting\n \nthe \nimpacts\n \nof control\n \ninterac\ntions on the resonances.\n \nFurther\n, \nthe \nrecent \nadvancements in \ns\ntabiliz\ning control \nmethod\ns\n \nfor SR and SSR\n \nare \ncritically reviewed with \nexperimental \ntests of \na GFM\n-\nVSC \nunder \ndifferent\n \ngrid conditions.\n \nIndex \nTerms\n—\nControl interactions, g\nrid\n-\nforming control\n, sub\n-\nsynchronous resonance\n, synchronous resonance\n, stabilization\n.\n \nI.\n \nI\nNTRODUCTION\n \nHE control stability\n \nof \nconverter\n-\nbased resources \nhas \nbecome critical to a reliable operation of power systems, \ndue to the \nwidespread use of converters in renewable \npower generation, flexible dc or ac power transmission systems, \nand energy\n-\nefficient power loads \n[1]\n, \n[2]\n. \nThe stability issues \nhave \nstimulated \na \nsignificant\n \ndemand\n \nfor \nthe \ngrid\n-\nforming \n(GFM) \ncapabilities \nof voltage\n-\nsource converters (VSC\ns\n)\n \n[3]\n.\n \nGFM\n-\nVSCs\n \ntypically employs\n \na cascaded control structure \nwith distinct \ntimescale\ns \nand \ncontrol \nfunction\nalities\n \n[4]\n.\n \nThe \no\nuter \ncontrol \nloops, \nmanag\ning \nthe\n \nsynchronization\n, power\n \nand \nvoltage regulation, \nmaintain a \nnearly constant \ninternal voltage \nphasor in the sub\n-\ntransient time frame\n \n[5]\n. This is \nan\n \nessential\n \nspecification \nfor GFM\n-\nVSCs\n \nto\n \ndeliver instantaneous voltage \nand frequency support\n \n[6]\n. \nIn contrast, the i\nnner control loops,\n \nsuch as\n \nvector voltage control (VVC) and vector current control \n(VCC), \nactively shape the output impedance of GFM\n-\nVSC, to \nmitigate \nvoltage harmonics\n \n[7]\n,\n \nlimit fault current\n \n[8]\n, \nand\n \nto\n \ndamp\nen\n \nthe \n \nresonance \n[9]\n.\n \nWhile\n \nsuch cascaded control \nhas \nbeen \nwidely used\n,\n \nlow\n-\nfrequency \nresonance\ns\n \nare increasingly \nreported\n \non\n \nGFM\n-\nVSC\ns under\n \ndifferent \ngrid\n \nconditions\n \n[10]\n. \n \nRecent\n \nstudies indicate that \nGFM\n-\nVSC\ns\n \nexhibit\n \ntwo \ntypical\n \nlow\n-\nfrequency resonances\n \nin the output power, i.e.,\n \nthe\n \ns\nyn\n-\nchro\nnous \nr\neso\nnance (SR)\n \n[11]\n, \n[12]\n \nand \nthe \ns\nub\n-\nsynchronous \nr\nesonance (SSR)\n \n[13]\n, \n[14]\n.\n \nThe frequency of SR in power is \nslightly lower than the fundamental frequency \n \n[15]\n, \noften\n \napproxi\nmated to \n \nfor\n \nsimplicity. In contras\nt\n, the SSR is \ntypically found \nin the \nfrequency \nrange of \n0 ~ 0.5\n \n[16]\n.\n \n \nThe causes of \nsuch low\n-\nfrequency \nresonances are \ncomplex \nand multifaceted.\n \nFor SR, it originates from the plant model of \npower\n-\nbased synchro\nnization control (PSC) in the outer loop \n[17]\n. \nNevertheless\n, \nit is \nalso found \nthat \nthe inner\n \ncontrol\n \nloops \ncan \ninteract with\n \nthe\n \nPSC\n, introducing \ndistinct impacts on SR \n–\n \nthe inner current \nloop\n \ndampens the SR\n,\n \nwhile\n \nthe inner voltage \nloop can \nexacerbate\n \nthe\n \nSR\n \n[16]\n.\n \nFurther, \nthe cause\ns\n \nof SSR \nin \nGFM\n-\nVSC\n \nsystems\n \ndiffer from\n \nthat of\n \nthe \ntorsional interaction\ns\n \nin\n \nsynchronous generator\ns\n \n(SG\ns\n) \n[18]\n. \nThe studies in \n[13]\n, \n[19]\n \nreveal that \nthe \ninteraction between the PSC and\n \ninner voltage \ncontrol \ncan\n \nc\naus\ne\n \nthe \nSSR.\n \nThe virtual synchronous generator \n(VSG) control in the outer loops can also manifest the SSR \n[20]\n.\n \nDespite the different causes\n \nof SSR\n,\n \nthe studies in \n[21]\n, \n[22]\n \nindicate\n \nthat\n \nGFM\n-\nVSC\ns\n \nare generally \nprone to \nlow\n-\nfrequency \nresonances \nwhen connected to stiff grids\n \nor \nconnected in \nparallel with other GFM sources,\n \nsince\n \nparalleled operation of \nstiff \nvoltage sources \ntend to be unstable\n.\n \nH\nowever,\n \nw\nith the \nadvancement \nof GF\nM control\n, \nthe stability robustness of GFM\n-\nVSCs against varying grid strength \n–\n \nboth weak and stiff\n \ngrids \n–\n \ncan be enhanced with \nsta\nbiliz\ning control\n \napproache\ns\n.\n \nIn general, t\nhe principles of stabiliz\ning control approaches \ncan be \nclassified\n \ninto\n \nseveral \ncategories\n: \n1) \nthe \nfeedforward \ncontrol\n \nfor\n \nimprov\ned\n \ndynami\ncs \nof PSC \n[23]\n; 2\n)\n \nthe\n \nimpedance \nshaping\n \ncontrol, either\n \nproviding \na \nvirtual \nresistance to \nsuppress\n \nSR \n[24]\n, \nor\n \nintroducing a\n \nvirtual\n \nimpedance between the VSC \nand \nthe\n \nstiff grid \n[25]\n;\n \n3\n) \nthe \ndynamics decoupling\n \ncontrol to\n \nmitigat\ne\n \nthe \ninteractions\n \nof different control\n \nloops \n[26]\n, \n[27]\n;\n \n4\n) \nunification\n \nof grid\n-\nfollowing (GFL) \nand GFM\n, \nhybridiz\ning\n \nthe \ncontrol loops to take th\ne best\n \nof \nboth \nGFM and GFL\n \ncontrol\n \n[28]\n, \n[29]\n, \n[30]\n,\n \n[31]\n,\n \nsuch as\n \nthe robust\n \nstability\n \nof GFM \nunder\n \nweak grid\n \nconditions\n \n[32]\n \nand \nthe \nrobust\n \nstability\n \nof GFL \nunder\n \nstiff grid\n \nconditions\n \n[33]\n.\n \n \nT\nhis paper \nprovides\n \nan in\n-\ndepth exploration on the causes of \nlow\n-\nfrequency \nresonance issues\n \nin GFM\n-\nVSCs\n, and\n \na particular \nattention \nis given to the control\n-\nloop interactions based on two \ngeneral GFM control structure\ns\n.\n \nA mapping between specific \ncontrol loops and resonance phenomena is \nestablished. \nMore \nimportantly\n, a\n \nbenchmarking study on the \nactive \nstabilization \nmethods \nof the two general GFM control structures \nis \npresented\n \nwith experimental \ntest \nresults\n, focusing on the low\n-\nfrequency \nresonances issues\n. \nThe study provides insights into how these \ncontrols mitigate resonances. \nFurther, \ncomparisons of \nmethods \nin respect to their \ndamping effects on SR and SSR under\n \nthe\n \ndifferent grid conditions \nare p\nrovided, \nfollowed by the\n \ndetailed \ndesign considerations\n, which has \nyet\n \nbeen holistically \naddressed \nin the recent literature \n[4]\n, \n[10]\n, \n[21]\n.\n \nThe remainder of this paper is\n \nstructured \nas follows. \nSection \nII introduces two GFM control structures\n, i.e.,\n \nopen\n-\nloop \nVVC \nand closed\n-\nloop VVC, \nalong with \nthe low\n-\nfrequency resonance \nphenomena\n. In Section III, the causes of SR and SSR are \npresented. \nSection IV \npresent\ns \nthe \ndamping\n \nmethods \nwith\n \nopen\n-\nloop VVC, while Section V \ndiscusse\ns \nthe\n \nmethods with closed\n-\nloop VVC.\n \nFinally, Section VI concludes this paper.\n \nT \n \n \nII.\n \nG\nENERA\nL \nC\nONTROL \nS\nTRUCTURES AND \nL\nOW\n-\nF\nREQUENCY \nR\nESONANCE \nP\nHENOMENA\n \nA.\n \nGeneral Control Structures of GFM\n-\nVSC\n \nFig. \n1\n \nshows a single\n-\nline diagram of \na\n \nthree\n-\nphase\n \nVSC\n \nwith \ntwo \ngeneral GFM control structures\n. They \ncompris\ne\n \nouter\n \nand\n \ninner\n \ncontrol \nloops\n.\n \nPOC denotes the point of connection, and \nthe grid is \ndenot\ned by an infinite bus with \na series\n \nimpedance. \nThe outer \ncontrol \nloops\n \nserve three purposes, i.e.,\n \nthe \ngrid \nsynchroni\nzation\n,\n \nthe \nvoltage\n \ncontrol, \nand \nthe \nactive power\n \nregulation\n \nof GFM\n-\nVSC\n. \nT\nhe GFM\n-\nVSC commonly employs\n \nthe \nPSC\n \nto\n \nform\n \nits own\n \nphase\n \nrather than follow\ning\n \nthe \nvector \nof \nPOC\n \nvoltage\n.\n \nDifferent \napproache\ns can be used \nwith the \nPSC\n, \ne.g., \nthe \ndroop\n \ncontrol \n[34]\n, \n[35]\n, \nthe \nvirtual synchronous \nma\n-\nchine\n \n(VSM)\n \n[36]\n, \n[37]\n \ncontrol\n, \nthe \ndc\n-\nlink \nvoltage \ncontrol\n \n[38]\n, \n[39]\n, \netc.\n \nThe \nvoltage regulation \naims to \nform\n \nthe\n \na\nlternating\n \nvoltage magnitude\n \nof \nGFM\n-\nVSC\n. It is often realized by reactive \npower control (RPC)\n \nan\nd\n/or\n \nalternat\ning voltage control (AVC)\n \n[15]\n, \n[40]\n, \n[4]\n \ndepend\ning on \nspecific \n \napplication\ns\n.\n \nThe outer \nloops \nmake GFM\n-\nVSC\n \nbehave \nas a\n \nvoltage source\n,\n \nbut only \nwithin \nthe\nir\n \ncontrol \nbandwidth\ns\n.\n \nIn contrast, \nthe \ninner loops \nare \ncommonly designed to be faster,\n \nin order to\n \nlimit\n \nthe\n \nfault\n \ncurrent\n \nand\n \nto actively shape the output impedance of GFM\n-\nVSC\n. \nT\nhere\n \nare\n \nmany\n \nchoice\ns \nfor\n \ninner loops\n,\n \nand different \ndesig\nns\n \ncan result in\n \ndistinct \nstability\n \nbehavior\ns\n.\n \nIn respect to \nthe configuration of inner loops,\n \nt\nhe \nGFM control structures are \nfurther \nc\nlassifi\ned into two \ngroups\n,\n \nas \nshown in \nFig. \n1\n(a) and (b)\n, \nwhich are the open\n-\nloop and closed\n-\nloop vector voltage control \n(VVC)\n.\n \n1)\n \nOpen\n-\nloop \nVVC\n \nFig. \n1\n(a) shows the \nstructure \nof \nopen\n-\nloop VVC\n \n–\n \nt\nhere is no \nVVC to follow the \nreference of voltage vector generated by the \nouter loops. \nThe inner\n \ncurrent\n \nloops are \noften \nused to\n \nlimit the \nfault current\n \nthrough\n \nan \nembedded\n \ncurrent control\n \n(CC) \n[15]\n, or\n \nan \nadaptive virtual impedance\n \n(VI)\n \n[41]\n.\n \nHowever\n, \nduring\n \nthe\n \nnormal operation, these loops remain \ninactive and \ntransparent\n, \nand hence, they \ndo not affect\n \nthe\n \nsmall\n-\nsignal stability\n \nof GFM\n-\nVSC\n.\n \nThey are \nshown by\n \nthe block with dashed lines in \nFig. \n1\n(a).\n \nExamples of\n \nGFM controls using open\n-\nloop VVC \nstructure \ncan be found\n \nin \n[15]\n, \n[24]\n, \n[42]\n, \n[43]\n.\n \nThis \nopen\n-\nloop VVC\n \nis easy to implement, and i\nt \nfeatures \nminimal coupling between the inner and outer loops.\n \nHence, \ni\nt \nis \noften preferred\n \nto \nmeet\n \nthe \nGFM \nfunctional \nspecifications \n[6]\n. \nHowever, \nsince \nthere is no closed\n-\nloop\n \nVVC, th\nis structure \nmay \nfail\n \nto effectively \nreject disturbances\n.\n \n \n2)\n \nClosed\n-\nloop VVC\n \nFig. \n1\n(b) shows the structure of closed\n-\nloop VVC\n.\n \nThe VVC \ntracks the reference \nwith a closed\n-\nloop feedback control\n, and \nis \ntypically cascaded with \nthe \nvector current control (\nVCC\n)\n \n[34]\n. \nFig. \n2\n \nshows the block diagram of \nthe \nclassic VVC and VCC \n[44]\n, \n[34]\n, \n[45]\n \nusing the \ncomplex\n-\nvector \nrepresentation \n[46]\n. \nA\n \nfast\n \nVVC\n \ncan\n \nhel\np\n \nto \nreject disturbances, e.g., \nmitigat\ning\n \nvoltage harmonics \nacross\n \na broader frequency range\n \n[7]\n, \n[4]\n.\n \nT\nhe \ninner \nVCC \ncan effectively prevent VSC from\n \novercurrent \nunder\n \noverloading conditions\n, where the saturation (SAT) block \nbetween the VVC and VCC is \nutilized\n \n[8]\n.\n \nNevertheless, during\n \nnormal operation\ns\n, the SAT has no impact on \nthe \nsmall\n-\nsignal \nstability, thus it is \nshown\n \nin the block with dashed lines in \nFig. \n1\n(b).\n \nExamples of\n \nGFM controls \nwith\n \nthe\n \nclosed\n-\nloop VVC \nstructure \ncan be found in\n \n[47]\n,\n \n[34]\n, \n[7]\n, \n[35]\n, \n[42]\n, \n[37]\n, \n[9]\n, \n[22]\n.\n \nNote that the \nVVC and VCC \naffect the output\n \nimpedance \nof \nthe VSC \n[48]\n, \n[45]\n, and\n \nintroduc\ne\n \nmore\n \ndynamic\n \ncoupling\ns with\n \nthe outer loop\n \ncontrol\n. \nSuch couplings\n \ncan\n \nbe detrimental to \nthe \nsystem \nstability\n, as will be\n \ndiscussed\n \nin the following\n.\n \n(\na)                                                                                                                    (b)\n \nFig. \n1\n. \nSingle\n-\nline diagram of a \ngrid\n-\nconnected \nthree\n-\nphase \nVSC\n \nwith two general GFM control structures\n.\n \n(a) Open\n-\nloop vector voltage control \n(VVC), including power synchronization control (PSC), reactive power control (RPC), alternating voltage control (AVC), refere\nnce generator \n(Ref gen), CC (embedded current control) or adaptive virtual impedance (VI). (b\n) Closed\n-\nloop VVC, including PSC, RPC, AVC, Ref gen, VVC, \nsaturation block (SAT), and vector current control (VCC).\n \n \n \n \nB.\n \nLow\n-\nFrequency Resonance Phenomena \n \nFig. \n3\n \nshows the instability risks of GFM\n-\nVSC with mapping \nbetween th\ne d\nynamics and control loops.\n \nThe power \n(\nP\n,\n \nQ\n) and \ncurrent\n \n(or voltage)\n \ndy\nnamics \nare depicted separately since their \nresonance frequencies are different\n. This is due to \nthe fact \nthat \npower is the \nproduct of\n \nvoltage and current.\n \nThe low\n-\nfrequency \nresonance (\n, \nwhere \n \nis the fundamental frequency\n) can \nbe observed\n \nin output power, current and voltage. Nevertheless, \nthe high\n-\nfrequency resonance (HFR) mainly presents in \nthe \ncurrent and voltage, but not in \nthe \noutput power, due to the low\n-\npass filters (LPFs) in power measurement and the slow response \nof outer loops. T\nhis paper focuses on the low\n-\nfrequency reso\n-\nnance\ns\n, \nhence\n \nthe HFR is not discussed in the rest of this \nwork\n. \nFor more details about HFR, please refer to \n[49]\n, \n[9]\n, \n[50]\n.\n \nA \nrecognized\n \nlow\n-\nfrequency resonance \nissue of GFM\n \ncontrol \nis SR\n \n[15]\n, \n[11]\n,\n \n[27]\n,\n \n[12]\n,\n \n[51]\n,\n \n[16]\n, \n[52]\n, where\n \nthe resonance \nfrequency \nof \noutput \npower \nor voltage magnitude of GFM\n-\nVSC \nclosely aligns with\n \nthe fundamental fre\nquency\n \n(in fact slightly \nbelow\n \n \n[15]\n, \n[16]\n)\n.\n \nSR \narise\ns from \nthe \nGFM control\n \nplant\n \nof \nthe \nouter loops\n, such as the plant of PSC\n \n[15]\n.\n \nIn addition, \nthe \ninteractions\n \nbetween the outer and inner \ncontrol \nloops can also \naffect\n \nSR\n, since the inner loop also affect the equivalent plant \nof the outer loops\n \n[16]\n.\n \n \nIt \nis worth noting that the \nSR \nin the output \npower\n \nof GFM\n-\nVSC leads to \na\nn\n \nalmost 0 Hz resonance\n \nat the output current, \ndue to \nthe \nfrequency coupling\n \neffect\n \n[53]\n, \n[52]\n.\n \nFig. \n4\n \nshows a typical experimental waveform of S\nR in \nthe \nstep response\n \nof active power\n.\n \nParameters of the experimental \nsetup are listed in Appendix. \nThe GFM converter uses open\n-\nloop VVC\n \nand PSC \n[24]\n,\n \nwhile\n \nthe active\n \nresistance\n \nis \ndisabled\n.\n \n \nFollowing the power reference step change, the active power \nshows an\n \nevident\n \noscillation at approximately 50 Hz\n, and\n \nthe \ncurrent \n(\nphase \na\n) show\ns an oscillation \nclose to \n0 Hz\n \n(DC\n-\noffset)\n.\n \nA\nnother\n \nacknowledged\n \nproblem\n \nof GFM \ncontrol \nis\n \nthe\n \nSSR\n \n[34]\n,\n \n[13]\n, \n[26]\n, \n[22]\n,\n \n[37]\n, \n[39]\n,\n \n[14]\n,\n \n[19]\n, \n[16]\n, \n[20]\n,\n \n[42]\n,\n \n[54]\n, \nwhere\nin\n \nthe resonance fre\nquency \nin the output \npower\n \nof GFM\n-\nVSC\n \nis \nbelow\n \nthe \nfundamental fre\nquency.\n \nThe\n \nSSR\n \nmay \nresult \nfrom\n \nthe \nPSC\n \nwith \nin\nsufficient\n \nphase margin\n \n[37]\n, \nundesired \ncontrol\n-\nloop interactions\n \n[26]\n,\n \nor \nthe\n \ninter\nconnect\nion\n \nof \nGFM\n-\nVSC to\n \nseries\n-\ncompensated \ntransmission \nlines \n[14]\n, \netc\n.\n \nFurther, \nt\nhe SSR is \nalso\n \nobserved\n \nwhen GFM\n-\nVSC operates\n \nunder stiff\n-\ngrid conditions \n[13]\n, \n[26]\n, \n[22]\n, \n[20]\n.\n \nD\nue to the frequency coupling effect, \nwhen the frequency of \nSSR in power is \n< \n, the \noutput \ncurrent\n \nof VSC\n \nexhibits \ntwo resona\nting components\n, i.e., \n \nand \n \n[49]\n.\n \nHence, both SSR and near\n-\nsynchronous resonance (NSR\n, \noscillation \nfrequency higher than \n) \n \ncan be \nobserved\n \nin\n \nthe \noutput\n \ncurrent\n, as indicated in \nFig. \n3\n \n(red dots).\n \nFig. \n5\n \nshows a\nn\n \nexperimental waveform of S\nS\nR \nin the\n \nstep \nresponse of active power. \nThe \nVSC u\nses closed\n-\nloop VVC and \nPSC \n[34]\n \nand connects to a\n \nstiff\n \ngrid, with a short circuit ratio \n(SCR) \nof\n \n10.\n \nThe \noscillation frequency of power is \nnear\n \n10 Hz, \nwh\nereas\n \nthe current\n \nshows frequencies of\n \n40 Hz and 60 Hz.\n \nWhile most studies on power dynamics \nfocus on frequencies \nbelow \nand near\n \nthe fundamental frequency\n, t\nhere is \nno\n \nreport on \nNSR in power\n \nyet\n. One possible reason is that the PSC\n \nbandwidth\n \nis \nnormally \nkept \nlow\n,\n \nas required by grid \ncode\ns\n \n[55]\n.\n \nFig. \n3\n. \nM\napping between the dynamics and \ncontrol loops\n.\n \n \nFig. \n4\n. \nExample experimental waveforms of SR.\n \n \nFig. \n5\n. Example experimental waveforms of SSR.\n \n \nFig. \n2\n. Block diagram of classic VVC and VCC using complex\n-\nvector \nrepresentation (POC voltage vector \n, output current vector\n, etc.).\n \n \n \n \nI\nI\nI.\n \nC\nAUSES OF\n \nL\nOW\n-\nF\nREQUENCY\n \nR\nESONANCES\n \nTa\nble I\n \nsummarizes \nthe\n \ngrid \nconditions\n \nand\n \ncauses \nof\n \nSR and \nSSR\n.\n \nDetails are given as follows.\n \nA.\n \nCause\ns\n \nof SR\n \n1)\n \nOpen\n-\nloop VVC\n \nIn the open\n-\nloop VVC in Fig. 1(a), \nthe SR \nis caused by\n \nthe \nplant of\n \nthe \nGFM outer \ncontrol \nloops.\n \n \nFig. \n6\n \nshows\n \nthe small\n-\nsignal model\n \nof \nthe \nouter loops\n.\n \nThe plant matrix \n \ncan be \nexpressed by \n[11]\n \n \n \n(\n1\n)\n \nw\nhere \n \ndenotes the \nfour \nnumerator\n \nelements\n \nof \nthe transfer \nfunction matrix\n, and they share the same denominator \n. \nClearly, \n \nhas two conjugate poles \n[15]\n \n \n \n(\n2\n)\n \nw\nhere \nR\n \nand \nL\n \ndenote the resistance and inductance between \nthe VSC and grid\n.\n \nT\nhe natural frequency\n \nof the conjugate poles, \ni.e., the imaginary part in \n(\n2\n)\n, \nis exactly the fun\ndamental \nangular \nfrequency\n \n.\n \nT\nhe \nparasitic \nresistance \nin the \nsystem\n, including\n \nthe\n \nequivalent series resistance (ESR) of filter,\n \nthe parasitic resistance of power \ncable\ns, transmission \nline\ns, \nor \nof the\n \ntransformer, \ncollectively\n \ncontribute to \n \nin \n(\n2\n)\n, which \nplays a\n \ncritical \nrole \nin suppressing the \nSR\n.\n \nWhen the system is \nmainly \ninductive, e.g., transmission system\n \nwith a high \n \nratio\n \n[18]\n, the damping ratio of \n \ntend to be\n \ninsufficient, and \nit may induce a high risk of SR \n[16]\n.\n \nIn contrast,\n \nfor \nthe \nlow\n-\nvoltage \ndistribution networks\n, \ne.g.,\n \nmicrogrids with\n \na \nlow \n \nratio \n[56]\n, \nthe SR \ncan be\n \nnaturally dampened.\n \nS\nince \nSR\n \nin the plant\n \nis \nclose to\n \n, \na simple way to avoid \nSR\n \nis \nto introduce an\n \nLPF\n \nwith \na \nlow cut\n-\noff frequency, \ne.g.,\n \n5\n \nHz\n \n[34]\n, i\nnto the outer\n-\nloop controller\n.\n \nThe VSM \ncontrol \nwith the\n \ninertia\n \nemulation \nalso help\ns to mitigate SR,\n \nas it is equiv\nalent to \nthe \ndroop control \nwith an\n \nLP\nF\n \n[36]\n, \n[57]\n.\n \nIn this case, the open\n-\nloop VVC does not have the SR issue.\n \n2)\n \nClosed\n-\nloop VVC\n \nA\ns for \nthe \nclosed\n-\nloop VVC in \nFig. \n1\n(b), \nit is proved in\n \n[16]\n, \n[48]\n, \n[45]\n \nthat \nthe inner VCC\n \nacts as\n \na\n \nvirtual \nresistance, which \nprovide\ns additional damping to \nthe \nSR.\n \nHence, there is\n \na low\n \nrisk of SR when \nthe \nVCC \nis\n \nadopted\n.\n \nIn contrast, \nthe VVC \nhas \na \nnegative damping effect\n \nto SR, which \ncan be a contributing \nfactor to SR if VCC is not used\n \n[16]\n. \n \nB.\n \nCauses of SSR\n \n1)\n \nOpen\n-\nloop VVC\n \nAs\n \ndiscussed earlier, using VSM control or equivalent LPFs \nin PSC \ncan\n \nmitigate SR\n \nin \nthe \nopen\n-\nloop VVC\n. However, it also \nintroduces a risk \nto\n \nSSR. \nTaking the \nsmall\n-\nsignal model of PSC\n \nas an example, \nwhen the cut\n-\noff frequency \nof LPF is \nfar below \nthe fundamental frequency\n, \nthe plant from \n \nto \n \nin \n(\n1\n)\n \ncan\n \nbe\n \napproximated to a con\nstant \nwhen \nthe power angle is small \n[35]\n \n \n \n(\n3\n)\n \nw\nhere \n \nand \n \nare grid and converter voltage magnitudes, and \n \ni\ns the \nsystem \nreactance\n.\n \nConsidering a droop controller with \ngain \n \nand \na \nLPF \nwith cut\n-\noff frequency \n \nin\n \nPSC \n[34]\n, then \nthe closed\n-\nloop transfer function of PSC is given by\n \n \n \n(\n4\n)\n \n \nis a second order system and the damping ratio \n \nof \nthe conjugate poles can be \nderived as\n \n[57]\n \n \n \n(\n5\n)\n \nI\nt is clear that increasing the droop gain or decreasing the cut\n-\noff frequency of LPF \nhas negative effects on\n \nthe damping ratio\n,\n \ndegrading\n \nthe \nstability margin \n[37]\n, \n[57]\n.\n \nT\nhe reac\ntance \n \nalso \naffects the \ndamping ratio\n. Under the stiff grid conditions\n,\n \nwhere \n \ntends to be\n \nsmall, the system \nshows a\nn increased\n \nrisk of SSR\n \n[20]\n, \n[58]\n.\n \nFig. \n7\n \ns\nhows\n \nthe\n \nexperimental waveform of SSR \ncaused by \nthe \nLPF and \ndroop \ncontroller \nin PSC\n \nunder stiff grid conditions\n \n(\n=0.05\n, \n=\n31.4 rad/s\n,\n \nand\n \nSCR=\n2\n0\n)\n. The resonance \nfrequency in power is around \n9\n \nHz\n.\n \n2)\n \nCl\nosed\n-\nloop\n \nVVC\n \nT\nhe inner loops are \nused to regulate fast dynamics of voltage \nand current and shape the impedance\n \n[45]\n, \n[25]\n.\n \nTypically, the \ninner loops cover the bandwidth of \nthe \nouter loops, \ntherefore\n \ninfluencing\n \nthe\n \ndynamics of\n \nouter loops\n \nas well\n.\n \nThis \ncontrol\n-\nloop interaction\n \nintroduce\ns\n \nnew oscillation mode\ns\n,\n \npotentially\n \ninduc\ning\n \nSSR\n \n[13]\n, \n[26]\n, \n[22]\n,\n \n[19]\n, \n[16]\n.\n \nFig. \n6\n. Example of small\n-\nsignal model of outer loops.\n \n \nFig. \n7\n. Example experimental waveforms of SSR \ncaused by the \nLPF \nand \ndroop controller in PSC under stiff grid conditions\n.\n \n \n \n \nOne recognized cause o\nf SSR is the \nVVC in \nFig. \n1\n(b)\n \nwhen \nthe converter is connected to a\n \nstiff grid\n \n[16]\n, \n[13]\n, \n[22]\n, \n[19]\n, \n[42]\n, \n[54]\n.\n \nThe \ncommonly \nused I controllers in \nVVC \n[34]\n \n \naffect\n \nthe \noutput impedance of \nthe\n \nGFM\n-\nVSC\n \nin the form of \na \nvirtual \nresist\nor\n \nin parallel with \na \nvirtual induct\nor\n \n[48]\n. This \nvirtual \nimpedance \ncan \naffect the plant of outer loops in \n(\n1\n)\n. \nIt \nhas been\n \ndemonstrat\ned in \n[16]\n, \n[22]\n, \n[31]\n \nthat the \nI\n \ncontrollers in VVC \nbring a \nnew \npair of conjugate poles to the closed\n-\nloop PSC \nsystem\n, \nwhich results in the\n \nSSR \nas the \ngrid strength\n \nincreases.\n \nBased on Nyquist criterion, t\nhe studie\ns in \n[13]\n, \n[19]\n \nshowed that \nthe stability margin decreases \nwhen the grid is st\ni\nff\n,\n \nand\n \nthe SSR \nis \nclose\nly \nrelated to\n \nthe VVC.\n \nIn \n[54]\n,\n \nthe \nimpacts of inner loops \nare evaluated by the damping torque\n. The \nfinding\ns \nindicate\n \nthat \nthe \nVVC exerts negative damping torque\n, which\n \ninduce\ns\n \nlow\n-\nfrequency \nresonance\ns\n \nunder stiff grid conditions.\n \nFig. \n5\n \nshows \nthe \nwaveforms of SSR discussed above.\n \nAs for the VCC, it is \nfound in \n[16]\n \nthat the P controllers of V\nC\nC also pose negative \ndamping to SSR.\n \nI\nn s\nummary\n,\n \nthe \ninner loops \nhave an \nobvious \nimpact on\n \nthe \nlow\n-\nfrequency dynamics of outer loops\n,\n \nwhich may\n \ncause SSR. \nA \nnoteworthy observation\n \nis that \nthe \nmajority of \nSSR\n \nissues\n \nin\n \nthe\n \nliteratures \n(e.g., \n[16]\n, \n[13]\n, \n[26]\n, \n[22]\n, \n[19]\n, \n[42]\n, \n[54]\n)\n \noccur\n \nunder stiff grid conditions\n. However, it \nis important\n \nto\n \nclarif\ny\n \nthat \nthe SSR is not a universal \nissue\n \nof GFM\n-\nVSCs\n \nwhen \nconnected to stiff grids.\n \nT\nhe open\n-\nloop VVC\n \nwithout inertia\n \ndoes not have the SSR issue a\ncross varying\n \nSCR\n \nconditions.\n \nIn \nthe following \nstudies\n \nof \nadvanced \nstabilization \nmethod\ns, more \nexample\ns \nof closed\n-\nloop VVC \nwill be explored\n \nwhich\n \npresent\n \nrobust\n \nstability\n \nregardless\n \nof grid strength.\n \nI\nV\n.\n \nA\nCTIVE \nD\nAMPING \nC\nONTROLS\n \nFOR \nGFM\n-\nVSC\n \nWITH \nO\nPEN\n-\nL\nOOP \nVVC\n \nTa\nble II\n \nsummarizes the active \ndamping controls\n \nfor GFM\n-\nVSC\n \nemploying open\n-\nloop VVC, which is shown in \nFig. \n1\n(a).\n \nA.\n \nOuter loop c\nontroller tuning and \ndesign\n \nAs discussed earlier, the GFM control with open\n-\nloop VVC \nhas \nthe risk of SR. To mitigate this risk, a simple approach is to \nslow down the \nPSC \nloop\n, ensuring that\n \nits control\n \nbandwidth \nremains well below the fundamental frequency. This can be \nachieved by reducing the \ngain of the d\nroop \ncontroller\n \nor \nadopting different controllers in the PSC loop.\n \n1)\n \nLimit\ning droop gain\n \nIt has been proved in \n[16]\n \nthat the droop gain \n \nin PSC \nshow\ns a\n \nnegative \ndamping\n \neff\nect\n \nto the \nclosed\n-\nloop \npoles\n \nthat\n \nare related to SR\n,\n \ni.e.,\n \n \n \n(\n6\n)\n \nw\nhere \n \nand \n \ndenote the steady\n-\nstate points of magnitude of \ncontrol voltage \n \nand \n-\naxis voltage of POC voltage in \nFig. \n1\n, \nand \n \nis a scaling factor. For peak\n-\nvalue scaling, \n=1.5, and for \np.u. normalization of the quantities, \n=1 \n[24]\n.\n \n(\n6\n)\n \nindicates that \nwhen the droop gain is sufficiently low, it ensures that the poles \nhave a \nnegative\n \nreal part\n.\n \nThe maximum p.u. value of \n \nis \napproximately twice the resistance value in the system to ensure \nthe system stability \n[16]\n, \n[59]\n. Hence\n, when the GFM\n-\nVSC\n \nis \nconnected to a\n \ntransmission\n \npower system with a high \n \nratio \n[18]\n, \nthe selection range for \n \nis significantly restricted. \nThe parameter should be chosen based on the worst\n-\ncase \nscenario, namely the grid with the lowest grid resistance \n–\n \noften \na stiff grid.\n \nI\nt should be noted that the bandwidth of PSC varies with the \nchange of grid impedance. An increase in grid impedance leads \nto a decrease in the bandwidth of PSC \n[24]\n. Therefore, when a \nlow droop gain is selected under the stiff grid with a low grid \nimpedance, it may \nresult in an excessively slow response in \nactive power control\n \nin the weak grid with a high impedance\n. \nWhile this may be suitable for GFM\n-\nVSCs in the \nstatic \nsynchronous compensators (STATCOM\ns\n), it may fall short of \nmeeting the requirements for \nphotovoltaics\n \ninverters and wind \nturbines,\n \nwhere\n \na rapid response to \nthe \npower reference \ndispatch \nmay be demanded\n.\n \n2)\n \nUse of LPF\n \nAn alternative \napproach\n \nto decreas\ning\n \nthe bandwidth of outer \nloops, without compromising the droop gain, \nis\n \nusing\n \nan \nLPF in \nseries with the controller, or equivalently, the \nVSM\n \ncontrol \n \n[36]\n, \n[57]\n.\n \nChoosing a low cut\n-\noff frequency for the LPF, \ne.g.,\n \n5 Hz, or opting for a high inertia constant in the VSM, can \neffectively mitigat\ne\n \nSR in the plant\n \n[12]\n.\n \nH\nowever, as \nintroduced in Section III\n-\nB, the \nuse of LPF may\n,\n \nin turn\n,\n \ncause SSR \nunder stiff grid conditions\n \n[37]\n, \n[20]\n, \n[58]\n.\n \nTABLE\n \nI\n \nGrid conditions and causes of SR and SSR of the two general GFM control structures\n \nControl\n \nResonance\n \nGrid c\nondition\n \nCause\n \nOpen\n-\nloop \nVVC\n \nSR\n \n(\n)\n \n–\n \nConnection to a grid with a h\nigh \n \nratio\n \n–\n \nUnder\n-\ndamped oscillation mode in the plant of outer \nloops\n \nwhen \nLPF \nor virtual inertia \nin PSC \nis not used\n \nSSR\n \n(\n)\n \n–\n \nConnection to a s\ntrong grid (low \ngrid impedance)\n \n–\n \nPotential i\nnsufficient stability margin \nwhen \nLPF or \nvirtual inertia in \nPSC \nis used\n \nClosed\n-\nloop \nVVC\n \nSR\n \n(\n)\n \n–\n \nConnection to a grid with a h\nigh \n \nratio\n \n(when VCC is not used)\n \n–\n \nI\nf VCC is not used, SR is caused by u\nnder\n-\ndamped \noscillation mode in the plant of outer loops\n \n–\n \nIf VCC is used, SR can be mitigated\n \nSSR\n \n(\n)\n \n–\n \nConnection to a s\ntrong grid (low \ngrid impedance)\n \n–\n \nControl interactions between the outer loops and \ninner loops (I \ncontrollers of VVC)\n \n  \n \n \nThis can be easily seen from the damping ratio\n \nin \n(\n5\n)\n \n[57]\n, \nand \nfrom the SSR phenomenon in the step response of active power \nin \nFig. \n7\n. In this case, the grid is strong (SCR=20), the cut\n-\noff \nfrequency of LPF is set at 5 Hz, and the oscillation frequency \nof active power is around 9 Hz. Further, if the cut\n-\noff frequency \nof LPF is higher (e.g., close t\no \n),\n \nthe damping to SR will be \ndecreased\n. Cons\nequently, this poses a trade\n-\noff in the parameter \nselection between the damping of SR and SSR when the grid \nimpedance varies across a wide range.\n \nT\no further mitigate the SSR caused by the LPF, the studies \nin \n[60]\n, \n[59]\n, \n[61]\n \nintroduce a lead\n-\nlag compensator into the \nPSC. The method enhances the system stability and preserves \nthe droop and inertial characteristics. It can balance the \ndamping for both SR and SSR \n[59]\n, and offer the required \ninertial power as per the grid code requirement \n[55]\n, \n[43]\n. Yet,  \nthe design of lead\n-\nlag compensator coefficients is dependent on \nthe grid impedance\n \n[60]\n.\n \nConsequently, it should be formulated \nbased on\n \nthe worst\n-\ncase scenarios, including the grid conditions \nwith the lowest resistance for\n \nthe\n \nSR, and with the lowest \nreactance\n \nfor\n \nthe\n \nSSR\n, as shown in \n(\n5\n)\n.\n \nB.\n \nImpedance shaping\n \nIn addition to adjusting outer loop controllers, an alternative \napproach to mitigat\ning\n \nSR \nis \nto \nemulate\n \na \nvirtual resistance\n \n(VR)\n \na\nt\n \nthe fundamental frequency\n \n[15]\n, \n[12]\n, \n[24]\n. The principle is \nto \nshape\n \nthe impedance \nand the plant\n, \nthereby\n \nenhancing the \ndamping ratio \nin \n(\n2\n)\n. \nFig. \n8\n \nshows the control diagram of VR in \n-\nframe and the PSC, while other outer loops are not plotted \nfor simplicity. The VR controller includes a proportional gain \n \nand a high\n-\npass filter (HPF) \n[26]\n. The gain \n \nis used to \nactively dampen the SR, while the HPF prevents the VR from \naffecting the steady\n-\nstate point (0 Hz in \n-\nframe) and from a \nstatic coupling between active and reactive power \n[62]\n. The cut\n-\noff frequency of the HPF can be selected between 5 ~ 10 Hz. If \nVR is implemented in stationary\n-\nframe, a notch filter at the \nfundamental frequency is required to replace the HPF. \nFig. \n9\n \nshows a comparison of the plant from \n \nto \n \nin \n(\n1\n)\n \nwithout \nand with the VR. It is clear that the SR is well mitigated.\n \nT\nhe studies in \n[24]\n \npresent \na\nn analytical parameter design of \nVR. The findings indicate that selecting the droop gain of PSC \n \nas follows\n \n \n \n(\n7\n)\n \nensures robust stability (gain margin higher than 2) under \ndifferent SCR and operating conditions, where \n \nis the same as \nin \n(\n6\n)\n, and \n \nis the magnitude of POC voltage. However, the \ninfluence of HPF is not considered in the analysis. Eq. \n(\n7\n)\n \nlinks \nthe VR to the droop gain. It is recommended in [29] that \nselecting \n \n= \n0.2 p.u.\n \ncan dampen the SR, which results in \nTABLE\n \nII\n \nSummary of active \ndamping control\ns of GFM\n-\nVSC\n \nbased on open\n-\nloop VVC\n \nStabilization\n \nReference\n \nSR and SSR issues\n \nDesign consideration\n \nO\nuter\n-\nloop controller \ntuning and design\n \nLimit\ning droop gain \n[16]\n, \n[59]\n, etc.\n \n–\n \nSuf\nficient damping to SR\n \n–\n \nN\no SSR issue\n \n–\n \nTrade\n-\noff between low droop gain for SR \nmitigation and required response time to active \npower reference\n \n–\n \nI\nnteractions with voltage/reactive power control\n \nUse of LPF\n/VSM \n[36]\n, \n[57]\n, \nor \nlead\n-\nlag \n[60]\n, \n[59]\n, \n[61]\n, etc.\n \n–\n \nMedium\n \ndamping to SR\n \n–\n \nPotentially cause SSR under \nstiff grid conditions\n \n–\n \nTrade\n-\noff in parameter selection between \ndamping to SR and SSR (SSR only occurs in \nstiff grids)\n \n–\n \nI\nnteractions with voltage/reactive power control\n \nI\nmpedance shaping\n \nV\nirtual resistance (VR) \n[15]\n, \n[12]\n, \n[24]\n, etc.\n \n–\n \nSuf\nficient damping to SR\n \n–\n \nPotentially cause SSR under \nstiff grid conditions\n \n–\n \nTrade\n-\noff in droop gain and VR selection\n \nbetween damping to SR and SSR (SSR only \noccurs in stiff grids)\n \n–\n \nI\nnteractions with voltage/reactive power control\n \nP\nower reference \nfeedforward (PRF) \ncontrol\n \n[23]\n \n–\n \nSuf\nficient damping to SR\n \n–\n \nMedium damping to\n \nSSR \nunder stiff grid conditions\n \n–\n \nI\nnteractions with voltage/reactive power \ncontrol\n \nPower decoupling \ncontrol (PDC)\n \n[26]\n,\n \n[27]\n, \n[51]\n, etc.\n \n–\n \nSuf\nficient damping to SR\n \n–\n \nSuf\nficient damping to S\nS\nR\n \n–\n \nDependency of decoupling controllers on grid \ninductance and resistance\n \n \n \nFig. \n8\n. Block diagram of VR in \n-\nframe and PSC\n \n[15]\n, \n[24]\n.\n \n \nFig. \n9\n. Bode plot (magnitude) of plant \n \nwithout VR (black line) \nand with VR (red line).\n \n \n \n \n=0.2 p.u. when voltage \n≈\n1 p.u. This design is beneficial for \nthe stability, yet it potentially leads to an overly rapid power \ncontrol (\n \nis large), which may not be suitable for all GFM\n-\nVSC applications. The latest grid code requirement \n[55]\n \nand \nfunctional specifications \n[6]\n \nsuggest to employ a slow PSC so \nthat the GFM converter provides near\n-\ninstantaneous responses \nto grid disturbances. This often leads to a droop gain \n \nless \nthan 0.05 p.u. under stiff grid conditions \n[43]\n.\n \nH\nowever, \nas pointed out \nby\n \n[26]\n,\n \nwhen a typical\nly\n \nlow value \nof \n \nis used \n(\ne.g.,\n \n0.03 p.u.\n)\n \nand \n \n= \n0.2 p.u\n. \n[24]\n, the system\n \nmay present SSR when connected to stiff grids. \nThis new oscil\n-\nlation mode is \ncause\nd by the HPF in VR\n, which can \nresult in\n \nSSR with a frequency lower than 10 Hz \n[26]\n.\n \nIn the plant of \nouter loops \n(\n1\n)\n, when VR is considered, the denominator \n \nis approximated to\n \n \n(\n8\n)\n \nw\nhere \n \nis the HPF, and \n \nis the cut\n-\noff frequency.\n \nBy \nsubstituting \n \ninto \n,\n \nit \nis\n \nseen\n \nthat there \nexists\n \nanother \npair of conjugate poles related to \nthe \nSSR\n \n[26]\n,\n \nin addition to \nthe SR mode in \n(\n2\n)\n.\n \nApart from\n \nthe VR, \nthe \nreactive power and \nvoltage control also \naffect the power\n \ndynamics, which are not \nconsidered in \n[24]\n.\n \nFig. \n10\n \ncompares the \nstep responses of \nactive power \nin \nstiff grid\n \nconnections\n \n(SCR=2\n0\n) \nw\nith \n=0.03 \np.u. and \n=\n=0.2 p.u. in \n(\n7\n)\n, re\nspectively. \nBesides PSC and \nVR (\n \n= \n0.2 p.u\n.), t\nhe AVC and RPC in \n[26]\n \nare \nalso \nincluded. \nFig. \n10\n \nindicates that when \nthe \ndroop gain and VR are not \nmatched\n \n(\n)\n,\n \nserious\n \nSSR may appear when connected to \na stiff grid. \nThis\n \nissue\n \ncan be \nmitigate\nd by \nproperly \nreduc\ning the \ncut\n-\noff frequency of \nthe \nHPF or decreasing \n.\n \nC.\n \nPower reference feedforward control\n \nB\nased on the PSC and VR, the research in \n[23]\n \nintroduces a \npower reference feedforward (PRF) control to further improve \nthe dynamic performance of active power step response. \nFig. \n11\n \nshows the control diagram with PRF loop. The \n-\naxis current \nstill forms the VR \n[15]\n, whereas the \n-\naxis current is governed \nby a closed\n-\nloop control with a P gain \n \nto track the ref\nerence \n. The rationale behind this design is  \n \nin \nsteady state when the \n-\naxis voltage \n \nis negligible, and thus, \nthe reference of \n-\naxis current can be directly calculated based \non \n. The PRF does not affect the closed\n-\nloop poles of PSC.\n \nInstead, it shap\nes\n \nthe\n \nzeros that approximately cancel out the \npoles\n \n[23]\n. As a result, the negative effect of the poles is \nsuppressed, leading to enhanced dynamic performance from \n \nto \n, such as reduced overshoot and shorter settling time.\n \nI\nt is worth noting that the effect of pole\n-\nzero cancellation is \nmore obvious when the control parameters are selected from \n(\n7\n)\n \nin \n[24]\n. Hence, choosing a low droop gain \n \nmay degrade the \nperformance. In addition, the interaction between RPC/AVC \nand PSC can also contribute to SSR, which is not considered in \n[23]\n. \nFig. \n12\n \nshows the step responses of GFM control with PRF. \nAll parameters are the same as \nFig. \n10\n. While a slight SSR still \npresents when GFM\n-\nVSC is connected to a stiff grid, the PRF \nhelps to mitigate it compared to \nFig. \n10\n, \nirrespective \nof\n \nwhether \nthe droop gain\n \n(\n)\n \nand VR\n \n(\n)\n \nare matched or not.\n \nD.\n \nPower decoupling control\n \nTh\ne plant of outer loops in \n(\n1\n)\n \nis a MIMO system. Therefore, \nthe control interactions between the active and reactive power, \ncoming from the non\n-\ndiagonal entries, can also affect the power \ndynamics. Such effect\ns\n \nare\n \nnot considered in the aforemen\n-\ntioned controls.\n \nBased on the frequency range of interest, \nthe \npower coupling \ncan be categorized into two types: 1) static coupling \n[56]\n, \n[62]\n, \n[63]\n \nand 2) dynamic coupling\n \n[26]\n, \n[64]\n, \n[27]\n, \n[51]\n. \nThe static \ncoupling is derived from the quasi\n-\nstatic model\n \nof the plant \n[63]\n,\n \nwhere it characterizes the \nstatic\n \ngain of the transfer functions\n. It \nprimarily focuses on a narrow frequency range around 0 Hz, \ntherefore cannot address the SR issue. The dynamic coupling \ninvolves the full\n-\norder transfer functions of non\n-\ndiagonal \nentries in the plant \n[26]\n. It offers a more accurate description of \nresonance behaviors, which can better guide the control design \nfor stability enhancements.\n \n \n(\na)\n \n \n(\nb)\n \nFig. \n10\n. Comparison of responses\n \nof GFM control with PSC and VR \nwhen SCR=20. (a) low droop gain \n=0.03 p.u. (b) high droop gain \n=\n=0.2 p.u. following the design \n(\n7\n)\n \nin \n[24]\n.\n \n \nFig. \n11\n. Block diagram of PRF, VR and PSC\n \n[23]\n.\n \n \n \n \nFig. \n13\n \nshows a general control diagram of power decoupling \ncontrol (PDC), including the small\n-\nsignal plant. The controllers \n \nand \n \nare designed to cancel out the effect of the \ncouplings in the plant. Thus, the basic design rules are given by\n \n \n \n(\n9\n)\n \nT\nhen the small\n-\nsignal model is equivalent to \nFig. \n14\n, where the \ncross\n-\ncoupling loops are eliminated. \nT\nhe\n \nstudies in\n \n[26]\n,\n \n[27]\n, \n[51]\n \npresent the \ndetailed\n \ntransfer functions of \nthe \ndecoupling \ncontrollers. The method can effectively improve the dynamic \nperformances. Furthermore, the diagonal entries of the plant are \nshaped by the PDC as follows:\n \n \n \n(\n10\n)\n \nI\nt is found in \n[27]\n \nthat the equivalent plants in \n(\n10\n)\n \ncan be \napproximated to constants, indicating that the SR mode is also \ncancelled out in the diagonal entries. Thus, the stability margin \nis significantly enhanced, and a fast power control is achieved. \nYet, the main issue of \n[27]\n, \n[51]\n \nlies in the strong dependency \nof the decou\npling controllers\n \non the grid resistance \n \nand the \ngrid inductance \n. Hence, the a\nccurate estimation of \n \nand \n \nis cr\nitical\n \nfor\n \nthe\n \nPDC, \nwhich \npos\nes\n \na challenge, particularly \nunder\n \nstiff grid\n \nconditions,\n \nwhere \n \nand \n \nvalues \nare\n \nsmall\n. \nThis issue can challenge the effectiveness of power decoupling \ncontrol in practical applications.\n \nT\no reduce the dependency of PDC on the grid impedance, the \nwork in \n[26]\n \nactively includes both the filter inductance \n \nand \nthe VR \n \n[24]\n \ninto the decoupling controllers. Thus, when the \ngrid impedance is small, \n \nand \n \n(known variables) can \ndominate the plant of PDC, tolerating estimation error of grid \nimpedance. This method effectively enhances the accuracy and \nperformance of power decoupling when connected to stiff grids. \nThe work in \n[26]\n \nis capable of integrating \n \ninto PDC due to \nits adoption of the open\n-\nloop VVC, whereas the works in \n[27]\n, \n[51]\n \nadopt closed\n-\nloop VVC in \nFig. \n1\n(b)\n. The closed\n-\nloop VVC \nshapes a zero impedance at fundamental frequency at the POC, \nwhile the open\n-\nloop VVC can leverage the filter impedance, \nwhich provides additional damping to the system \n[42]\n. The \napproach demonstrates robust stability \nwith no occurrence of \nSR or SSR in stiff\n-\ngrid connections\n \n[26]\n, albeit it still needs a \nrough estimation of grid inductance when the grid impedance \nvaries across a wide range.\n \nV\n.\n \nA\nCTIVE \nD\nAMPING\n \nC\nONTROL\nS FOR \nGFM\n-\nVSC\n \nWITH \nC\nLOSED\n-\nL\nOOP \nVVC\n \nTa\nble II\nI\n \nsummarizes the active stabilization approaches of \nGFM controls based on the closed\n-\nloop VVC in \nFig. \n1\n(b).\n \nA.\n \nInner loop controller design\n \nFig. \n2\n \nshows the block diagram of the classic inner control \nloops in \n-\nframe \n[34]\n, \n[35]\n. Compared to the open\n-\nloop VVC, \nthe inner loops of closed\n-\nloop VVC shape the output impedance \nof VSC \n[48]\n. \nIn this \ncase\n, \nthe\n \nouter\n-\nloop controller tuning \nmethod\ns \ndesigned for the\n \nopen\n-\nloop VVC (Section V\n-\nA) may \nbe unnecessary. \nFor \nexample\n, \nreduc\ning the \ndroop gain\n \nof PSC\n \nto dampen the\n \nSR may not be \nneed\ned, given\n \nthe fact\n \nthat the\n \n \nFig. \n13\n. Block diagram of general PDC, PSC and RPC with small\n-\nsignal model of plant \n[26]\n, \n[27]\n,\n \n[51]\n.\n \n \nFig. \n14\n. Equivalent small\n-\nsignal model with PDC.\n \n \n(\na)\n \n \n(\nb)\n \nFig. \n12\n. Comparison of responses\n \nof GFM control with PRF, VR and \nPSC when SCR=20. (a) low droop gain \n=0.03 p.u. (b) high droop \ngain \n=\n=0.2 p.u. following the design \n(\n7\n)\n \nin \n[24]\n.\n \n \n \n \nVCC can also \ncontribute \nto the \ndamping\n \n[45]\n. Therefore, i\nt is \nimportant\n \nto \nconsider\n \nthe inner loops \nwhen\n \nanalyzing the \nlow\n-\nfrequency \ndynamics of outer loops\n. They may have both \npositive and negative impacts on different oscillation modes \nand ignor\ning them can \ncause\n \ninaccu\nracies\n \nstability predictions\n.\n \n1)\n \nLow\ner\n \nSR risk \nwith\n \ninner control loops\n \nIt\n \nis shown in \n[48]\n, \n[45]\n, \n[16]\n, \n[52]\n \nthat the P\n-\n \ncontroller gain \nof VCC acts as resistance, effectively dampen\ning the SR, \naccording to Eq. \n(\n2\n)\n. This is similar to the VR \n[15]\n, \n[24]\n \nused in \nthe open\n-\nloop VVC. In contrast, the P\n-\ncontroller gain of VVC \nbehaves as negative resistance, which deteriorates the damping \nto SR \n[16]\n.\n \nN\note that the VCC bandwidth is typically in the range from \nhundreds of Hz to kHz. This often results in a high P\n-\ncontroller \ngain of VCC, e.g., near 1 p.u. or even higher \n[65]\n. This gain is \nsufficiently high to dampen the SR, comparing with the \nrecommended VR value of 0.2 p.u. in \n[24]\n. Consequently, the \nuse of VCC significantly lowers the risk of SR.\n \n2)\n \nHigher SSR risk with inner control loops\n \nI\nt is reported in \n[54]\n,\n \n[13]\n, \n[16]\n, \n[19]\n, \n[45]\n \nthat the VVC can \nmanifest the SSR when the GFM\n-\nVSC is connected to stiff grid. \nThe SSR mode is introduced by the I controller of the VVC and \nit is closely associated with the value of I gain. The damping \neffects of different controllers on the SSR are sum\nmarized as \nfollows:\n \na)\n \nThe P\n-\ncontroller gain of VVC helps to dampen SSR\n \n[54]\n. \n \nb)\n \nT\nhe I\n-\ncontroller gain of VVC degrades damping to SSR\n \n[16]\n.\n \nc)\n \nThe P\n-\ncontroller gain of VCC degrades damping to SSR\n \n[45]\n.\n \nd)\n \nT\nhe I\n-\ncontroller gain of VCC is typically low and, hence, \nnot considered in the analysis \n[65]\n, \n[28]\n.\n \nThe P\n-\ncontroller of VVC plays a critical role in dampening \nthe SSR, and thus increasing the P gain of VVC is beneficial for \nmitigating the SSR, leading to a fast VVC. \nFig. \n15\n \nshows the \nwaveforms after the P\n-\ncontroller gain of VVC decreases when \nconnected to a stiff grid (SCR=20), where the system becomes \nunstable with the SSR in the power waveform.\n \nI\nn addition, it is interesting to find that the P\n-\ncontroller gain \nof VVC exhibits opposite damping effects on SR and SSR. If \nVCC is not used (no damping to SR), a low P\n-\ncontroller gain of \nVVC is needed, giving a slow VVC. If VCC is used (a lower \nrisk of SR)\n, a high P\n-\ncontroller gain of VVC is suggested for \nthe SSR mitigation. It is worth noting that in such dual\n-\nloop \nvector control structure, the bandwidth of the \nVVC \nis designed \nbelow that\n \nof the VCC\n \n[28]\n.\n \nB.\n \nImpedance shaping\n \nAs discussed earlier, the inner control loops in \nFig. \n2\n \nmay \nencounter the SSR issue when the GFM\n-\nVSC is connected to a \nstiff grid \nwith \na \nlow grid impedance\n. Therefore, a direct solution \nis to add a virtual impedance (VI) \n[25]\n, \n[60]\n, \n[66]\n, \n[67]\n \nbetween \nTABLE\n \nIII\n \nSummary of active damping controls \nof GFM\n-\nVSC\n \nbased on \nclosed\n-\nloop VVC\n \nStabilization\n \nReference\n \nSR and SSR \nissues\n \nDesign consideration\n \nInner\n-\nloop controller \ntuning and design\n \nR\neducing I\n-\ncontroller \ngain or increasing P\n-\ncontroller gain of VVC \n[16]\n, \n[54]\n, \netc.\n \n–\n \nSuf\nficient damping to SR\n \n–\n \nPotentially cause SSR under \nstiff grid conditions\n \n–\n \nTrade\n-\noff in parameter selection of VVC and \nVCC between damping to SR and SSR (SSR \nonly occurs in stiff grids)\n \nI\nmpedance shaping\n \nVirtual impedance (VI) \ncontrol \n[66]\n, \n[67]\n, \netc.\n \n–\n \nSuf\nficient damping to SR\n \n–\n \nSufficient damping to SSR\n \n–\n \nAVC with I or PI controller is needed to avoid an \nobvious static error of POC voltage magnitude\n \n–\n \nLPF is needed for VI to avoid amplifying high\n-\nfrequency noises\n \nV\nirtual admittance (VA) \ncontrol \n[25]\n, \n[60]\n, \netc.\n \n–\n \nSuf\nficient damping to SR\n \n–\n \nSufficient damping to SSR\n \n–\n \nAVC with I or PI controller is needed to avoid an \nobvious static error of POC voltage magnitude\n \nUnified GFM and \nGFL controls\n \nHybrid synchronization \ncontrol and hybrid inner \nloops \n[28]\n, \n[31]\n, \n[74]\n \n–\n \nSuf\nficient damping to SR\n \n–\n \nSufficient damping to SSR\n \n–\n \n-\nFF loop uses the same \n-\nframe angle as PSC\n \n–\n \nWhen current limiter is activated during faults, \nthe hybrid inner loops may become ineffective\n \nActive susceptance \n[30]\n, \n[73]\n \n–\n \nSuf\nficient damping to SR\n \n–\n \nSufficient damping to SSR\n \n–\n \nWhen current limiter is activated during faults, \nthe hybrid inner loops may become ineffective\n \nGFM\n-\nVCC\n \n[29]\n \n–\n \nSuf\nficient damping to SR\n \n–\n \nSufficient\n \ndamping to SSR\n \n–\n \nWhen current limiter is activated during faults, \nthe hybrid inner loops may become ineffective\n \n \n \nFig. \n15\n. Example experimental waveforms of instability (SSR) \ncaused \nby \nreducing P\n-\ncontroller gain of VVC under stiff grid conditions.\n \n \n \n \nthe converter and the grid. There are typically two ways to \nsynthesize the VI \n[68]\n:\n \n1)\n \nCurrent\n-\nfeedback\n \nVI\n \ncontrol\n \nFig. \n16\n \nshows the conceptual diagram of VI control based on \nfeeding back the output current of GFM\n-\nVSC \n[66]\n, \n[67]\n, which \nis represented by complex\n-\nvectors. The resistance (\n) and \ninductance (\n) are emulated between the converter and the \ngrid within the bandwidth of VVC and VCC \n[68]\n. This method \nequivalently softens the interconnection of VSC with the grid, \nconsequently mitigating the SSR.\n \nN\note that the derivative controller \n \nin \nFig. \n16\n \nis rarely \nused in practice, as it amplifies the high\n-\nfrequency noise. An \nLPF is often used in series with \n \n[67]\n. Since the SSR \nfrequency is below \n, the cut\n-\noff frequency of LPF for the VI \nemulation in \n-\nframe can be set in the frequency range \n~\n.  \nIn the cases where the SSR frequency approaches 0 Hz, the \nmagnitude of \n \nis much smaller than that of the coupling \nterm\n. Therefore, \n \nmay\n \nbe omitted for simplicity\n \n[69]\n.\n \n2)\n \nVoltage\n-\nfeedback \nv\nirtual admittance (VA) control\n \nFig. \n17\n \nshows the block diagram of VA control based on \nfeeding back the ac output voltage of GFM\n-\nVSC \n[25]\n, \n[60]\n, and \nthe complex vector representation is used. The VA is realized \nby calculating the current reference as\n \n \n \n(\n11\n)\n \nw\nhere \n \nis the voltage reference vector, which emulates the \nelectromotive\n-\nforce (EMF) vector \n[60]\n, \n[70]\n. Hence, within the \nbandwidth of VCC, \n, the virtual resistance \n \nand virtual \ninductance \n \nare effectively emulated between the virtual \nEMF point (\n) and the POC\n \n(\n)\n. Thereby, the physical \nmeaning of VI in \nFig. \n16\n \nand VA in \nFig. \n17\n \nis different. The \nformer adds \n \nand \n \nbetween the\n \nPOC and the grid, while \nthe latter emulates \n \nand \n \nbetween the VSC and the POC. \nDespite this difference, both methods increase the coupling \nimpedance between the VSC and the grid, which can mitigate \nthe SSR under stiff grid conditions.\n \nThe VA control does not involve any derivative controller for \nemulating virtual inductance \n[68]\n, and it only modifies the PI \ncontroller of VVC as an LPF, preserving the cascaded vector \nvoltage and current control structure. Therefore, the VA control \nis easy to implement and it does not alter the voltage source \ncharacteristic of GFM control.\n \nFig. \n18\n \nshows the step response of GFM control with VA (\n=0.1 p.u. and \n=0.3 p.u. \n[60]\n) when SCR=20. Compared to \nthe conventional VVC shown in \nFig. \n5\n, the SSR is suppressed.\n \nThe values of \n \nand \n \nshould be deliberately selected. \n \nis typically lower than \n \nto maintain a high \n \nratio \n[52]\n.  \nWhen a high\n \nvalue of \n \nis \nused\n, it may result in a\n \nconsiderable \nsteady\n-\nstate POC voltage error, particularly \nunder\n \nweak\n-\ngrid \ncon\nditions\n. This is because 1) in the VI control, the actual \nvoltage reference of VVC is no longer 1 p.u. 2) in the VA \ncontrol, the VVC uses an LPF, \nintroducing steady\n-\nstate error in \ntracking the POC \nvoltage\n \nreference. \n \nFig. \n19\n \ncompares the step response (\n=0.1 p.u. and \n=0.3 \np.u. \n[60]\n) when SCR=1.5. \nAs the converter injects active power \n(0.5 p.u.), the \nPOC \nvoltage magnitude decreases to below 0.9 \np.u., clearly \nrevoking\n \nthe grid code\n \nrequirement \n[71]\n. In the \nworst case, the emulated impedance can even cause the loss of \nsteady\n-\nstate points under the ultra\n-\nweak grid condition. To \nsolve this issue, an additional ac\n-\nbus voltage magnitude control \n \n(\na)\n \n \n(\nb)\n \nFig. \n16\n. Inner loops with VI control \n[66]\n, \n[67]\n. (a) \nC\nonceptual block \ndiagram. (b) \nE\nquivalent circuit diagram of VI.\n \n \n(\na)\n \n \n(\nb)\n \nFig. \n17\n. Inner loops with VA control \n[25]\n, \n[60]\n. (a) \nC\nonceptual block \ndiagram. (b) \nE\nquivalent circuit diagram of VA.\n \n \nFig. \n18\n. Response of GFM control with VA when SCR=20.\n \n \nFig. \n19\n. Response of GFM control with VA when SCR=1.5.\n \n \n \n \nwith a PI or a P controller is needed, which adjusts the vector \nvoltage reference for the VA, maintaining the POC voltage \nclose to 1 p.u. across a wide range of grid conditions \n[52]\n.\n \nC.\n \nUnified GFM \nand GFL \ncontrols\n \nIt is well known that the traditional GFL control has a high \nstability robustness when VSC is connected to stiff grids, yet it \ntends to be unstable with weak grid interconnections \n[22]\n, \n[72]\n. \nIn contrast, the GFM control with the closed\n-\nloop VVC shows \nthe opposite stability robustness \n–\n \nit furnishes the VSC a higher \nstability robustness under weak grid conditions than operating \nin stiff grids \n[16]\n, \n[13]\n. T\nhis duality serves as \nthe \nmotivation to \nunify\n \nGFM and GFL controls, \ntaking\n \nthe \nbest of both \nto achieve \nhigh \nstability robust\nness\n \nagainst variation of grid strength \n[73]\n. \nFollowing the control principles, the unifi\ned\n \nof GFM and GFL \ncontrols \nare listed as follows\n:\n \n1)\n \nHybrid synchronization and inner loops\n \n[28]\n \nT\nhe general idea is to hybridize the traditional GFM and GFL \ncontrol loops. \nFig. \n20\n \nillustrates the block diagram of the hybrid \nscheme, which can be divided into two parts \n–\n \nhybrid synchro\n-\nnization control (HSC) \n[28]\n, \n[74]\n \nand hybrid inner loops \n[28]\n, \n[31]\n.\n \nT\nhe HSC \nadds a \n-\naxis voltage feedforward (\n-\nFF) loop to \nPSC. Note that \n-\nFF loop uses the same \n-\nframe angle as PSC. \nIt is distinct from the control structure of paralleled PLL and \nPSC \n[75]\n, \n[76]\n, where separate angles are used.\n \nIt is shown in \n[74]\n \nthat the \n-\nFF loop \noffers positive damping to SR under \nstiff \ngrid conditions. H\nowever, it can \nalso \ninduce a negative \ndamping effect on SR \nwhen connected to\n \nultra\n-\nweak grids\n \n[74]\n.\n \nThis behavior is similar to that of PLL. \nA\n \nmoderate \nP gain of \n-\nFF loop\n \nis suggested \nto ensure a robust design.\n \nT\nhe hybrid inner control loops are \ncritical to stability. The \nVCC remains unchanged, as both GFM control in \nFig. \n2\n \nand \nGFL control \n[65]\n \nuse the identical VCC. The damping effect of \nVCC on SR \n[16]\n \npersists\n. The VVC is modified, and the hybrid \ninner loops can be divided into two parts:\n \na) VVC of GFM control in \nFig. \n2\n \n[34]\n, \n[35]\n, which is a \nsymmetric structure \n[31]\n, including\n \nloops from \n \nto \n \n(\n-\nto\n-\n) \nand from \n \nto \n \n(\n-\nto\n-\n).\n \nThe two loops use PI controllers.\n \nb) \nA\nctive \npower control loop of GFL \n[35]\n, which \ncalculate\ns \nthe \n-\naxis current reference \nby dividing \n \nby voltage \n. \nIt\n \nis \nsimilar to \nPRF\n \n[23]\n \nin Section IV\n-\nC. Another loop is the AVC \nof GFL \n[65]\n, \n[21]\n, from \n \nto \n \n(\n-\nto\n-\n). It is\n \noften used in \nGFL\n-\nSTATCOMs\n \n[77]\n.\n \nThis \n-\nto\n-\n \ncoupling loop \nuses an I \ncontroller\n.\n \nFrom the standpoint of GFM, the \n-\nto\n-\n \nloop \nfrom\n \nGFL \nplays a\nn\n \nimportant\n \nrole in effectively dampening the SSR mode \nwhen \nc\nonnected \nto\n \nstiff grids \n[31]\n. Conversely, when viewed \nfrom the perspective of GFL, the \n-\nto\n-\n \nloop originating from \nGFM \neffectively\n \nenhances the\n \nstability \nin\n \nweak grid conditions \n[78]\n. Consequently, these \nhybrid\n \nloops \ncan\n \nsupport each other\n \nto\n \nachieve\n \nrobust stability \nacross a wide range of SCRs\n.\n \nFig. \n21\n \nshows the step response of hybrid VVC and HSC in \n[28]\n \nwhen \nSCR=1.5 and SCR=20. \nThe dynamic performances of power \nunder both ultra\n-\nweak and stiff grid conditions are satisfactory.\n \nN\note that the\n \n-\nto\n-\n \ncoupling loop shares the same integrator \nwith the \n-\nto\n-\n \nloop\n \nas shown \nin \n[28]\n.\n \nThis configuration avoid\ns\n \nthe saturation issue compared to using two separate integrators.\n \nThe method does not add virtual impedance to the converter, in \ncontrast to \nFig. \n16\n \nor \nFig. \n17\n, therefore preserving the same \nsteady\n-\nstate points and the controllability of \nPOC\n \nvoltage.\n \n2)\n \nA\nctive susceptance\n \n(AS)\n \n[30]\n, \n[73]\n \nBased on the\n \nhybrid \ninner loops in\n \n[28]\n, the studies in \n[30]\n \nfurther uses a\n \nloop from \n \nto \n \n(\n-\nto\n-\n), acting as susceptance, \nas shown in \nFig. \n22\n. \nThe AS builds a connection between \n \nand \n, which enables the potential of using PLL to mimic the \n \n(a)\n \n \n(\nb)\n \nFig. \n20\n. Block diagram of hybrid synchronization control (HSC) and \nhybrid inner control loops \n[28]\n. (a) HSC. (b) Hybrid inner loops.\n \n \n(\na)\n \n \n(\nb)\n \nFig. \n21\n. Response of GFM control with hybrid VVC and HSC when \nSCR=20 and SCR=1.5. (a) SCR=20. (b) SCR=1.5.\n \n \n \n \nswing\n-\nequation \n[30]\n. \nThe studies in \n[73]\n \nsimplifie\nd\n \nthe hybrid \ncontrol structure, and\n, more importantly,\n \nfound \nth\nat the AS loop \ncan \nprovide\n \ndamping to reject power disturbances.\n \nThe control in \n[73]\n \nis shown in \nFig. \n23\n. \nIt preserves the \noriginal PSC \n(no \nPLL\n)\n, and the hybrid \ninner loops\n \nuse\ns 1) \nPRF\n \n(\n \nto\n \n), 2) \n-\nto\n-\n \nloop from GFL, and 3) \nAS\n, a loop from \n \nto\n \n \n(\n-\nto\n-\n). \nConsidering an active power disturbance \n, \nt\nhe\nn the\n \ndisturbance in \n-\naxis current (\n) enters the grid, \ncausing a \n-\naxis\n \nvoltage disturbance\n \n \nacross the \ngrid impedance \n.\n \nWhen using the AS\n \nloop\n, the\n \n-\naxis current \nreference \ncan directly respond to \n, as follows\n \n[73]\n \n \n \n(\n12\n)\n \nw\nhere \n \nis the P gain of AS loop. This results in a negative \nchange of\n \n-\naxis current\n \nreference to resist \n, which helps to \ndampen power disturbance \n. Th\nis\n \ndamping effect to SSR is \nmore obvious under stiff grid conditions \n[73]\n.\n \nFig. \n24\n \nshows the \nstep response of active power when SCR=20. The dynamic \nresponse is \nas\n \ngood as \nthe universal controller \n[28]\n \nin \nFig. \n21\n.\n \n3)\n \nGFM\n-\nVCC\n \n[29]\n \nFig. \n25\n \nshows the basic control concept.\n \nThe method intends \nto shape the converter \nas\n \na virtual synchronous condenser (SC) \nin parallel with a virtual current source\n \n(CS)\n \n[29]\n. The virtual \nSC provides GFM \ncharacteristics\n \nincluding voltage \nstiffness \nand frequency \nresponse\n, while the virtual \nCS\n \nrealizes fast power \nsetpoints tracking\n,\n \nacting as a GFL source\n.\n \nFig. \n26\n \nshows the \ndetailed control block diagram. \nThe inner loops\n \nhybridize\n \nthe \nVA\n \ncontrol in \nFig. \n17\n \nand \nPRF\n \nloop (\n \nto \n)\n. \nThe \nVA\n \naims \nto mimic the stator impedance of S\nC\n, and \nthe swing equation is \nemulated by PLL with specially designed parameters\n \n[29]\n. The \nPRF\n \noperate\ns as a virtual \nCS\n \nparallel to the \nPOC\n, \ninject\ning\n \nactive power\n \ngiven by the setpoints. \n \nUnder weak grid conditions, the virtual SC maintains \nthe\n \nvoltage for PLL\n. This is helpful to mitigate the SSR issue of the\n \nv\nirtual CS\n \n(GFL)\n \nfor stable\n \npower \ninjection\n. While \nunder\n \nstiff \ngrid con\nditions\n,\n \nthe VA control \nprovide\ns necessary coupling \nimpedance\n \n(stator impedance of virtual SC)\n \nbetween \nthe \nEMF \npoint and POC, \nwhich can also \nmitigat\ne\n \nthe\n \nSSR \nissue \nof GFM.\n \nThe method demonstrates desired dynamic performances in \nboth islanded operation and grid\n-\nconnected operation over an \nextensive range of SCRs. The responses are similar to \nFig. \n21\n \nand \nFig. \n24\n; therefore, they are not repeated here.\n \nVII.\n \nC\nONCLUSION\n \nThis paper \nhas given a comprehensive review of the low\n-\nfre\n-\nquency issues in GFM\n-\nVSC.\n \nThe root causes of SR and SSR\n, \nincluding the origins of oscillation mode and impacts of control \ninteractions,\n \nhave been\n \ndiscussed in detail. \nIt has been pointed \nout that the interactions between outer and inner loops \ncan\n \ncause \n \nFig. \n22\n. Block diagram of GFM control based on PLL and AS\n \n[30]\n.\n \n \nFig. \n23\n. Block diagram of GFM control with AS\n \n[73]\n.\n \n \nFig. \n24\n. \nResponse of GFM control with AS\n \n[73]\n \nwhen SCR=20.\n \n \n \nFig. \n25\n. Basic control concept of GFM\n-\nVCC \n[29]\n \n–\n \nparalleled virtual \nsynchronous condenser (SC) and virtual current source (CS).\n \n \nFig. \n26\n. Block control diagram of GFM\n-\nVCC \n[29]\n.\n \n \n \n \nSSR under \nspecific\n \nconditions. \nNotably\n, systematic \nstudie\ns of \nthe advanced \ndamping\n \ncontrol \nmethods have been \ngiven\n \nbased \non two types of \ngeneral \nGFM control structures\n. Comparisons, \nadvantages, and \ndesign consideration\ns have been \nprovided\n.\n \nT\nhis \npaper highlight\ns\n \nthat \nemploying appropriate \nactive\n \nstabilization \nmethod\ns can effectively \ndampen\n \nthe low\n-\nfrequency \nresonance\ns\n \nof GFM\n-\nVSC \nunder high SCR\n \nconditions\n.\n \nTh\nis\n \nensures a robust \nand stable\n \nconnection\n \n(in response to small\n-\nsignal disturbances)\n \nwith the grid, ranging from ultra\n-\nweak to stiff\n \ngrids\n.\n \nVIII.\n \nA\nPPENDIX\n \nT\nhe parameters of \nthe experimental setup used in this study\n \nare listed in Table I\nV.\n \nTABLE I\nV\n \nParameters of \ne\nxperimental \ns\netup\n \nS\nymbol\n \nD\nescription\n \nV\nalue\n \n \nRated voltage \n(\nL\n-\nL, RMS)\n \n190.5 \nV (1 p.u.)\n \n \nR\nated capacity\n \n3\n \nk\nW (1 p.u.)\n \n \nNominal frequency\n \n314 rad/s\n \n(50 Hz)\n \n \nB\nase impedance\n \n1\n2.1 \nΩ\n \n \nFilter\n \ninduct\nance\n \n3\n \nmH (\n0\n.\n078\n \np.u.)\n \n \nF\nilter capacitance\n \n1\n0 μF (0.038 p.u.)\n \n \nD\nC\n-\nlink voltage\n \n6\n00 V (1 p.u.)\n \n \nS\nwitching frequency\n \n1\n0 kHz\n \nR\nEFERENCES\n \n[1]\n \n“Grid\n-\nForming Technology in Energy Systems Integration.” \nReston, VA: Energy Systems Integration Group, Mar. 2022. \n[Online]. 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We also show how to construct,\nfor any k, an O(klogD)-purely additive spanner of Pwith 2O(D)nSteiner vertices and\n2O(D)n·λk(n) edges, where λk(n) is the kth-row inverse Ackermann function. Finally, we\nshow how to construct an approximate Voronoi diagram for Pof size 2O(D)n. It allows us\nto answer approximate near-neighbor queries in 2O(D)+O(logn) time, with additive error\nO(logD). These constructions can be done in 2O(D)nlogntime.\n1 Introduction\nThe Poincar´ e halfplane H2is perhaps the most common model of hyperbolic spaces, together\nwith the Poincar´ e disk which it is isometric to. The points of H2are the points ( x, z),z >0\nof the upper halfplane of R2. The shortest paths in H2, called geodesics, are arcs of circles\northogonal to the x-axis (Figure 1a) and the arc length is given by integrating the relation\nds2= (dx2+dz2)/z2. More generally, the Poincar´ e halfspace is a D-dimensional model of\nhyperbolic space, that consists of points ( x, z) where x∈RD−1andzis a positive real number.\nInHD, the expression of ds2is the same, but xis now a point in RD−1, and geodesics are arcs\nof circles orthogonal to the hyperplane z= 0.\nHyperbolic spaces behave very differently from Euclidean spaces in some respects. For\ninstance, in fixed dimension, the volume of a hyperbolic ball grows exponentially with its radius,\nwhile the radius of a Euclidean ball grows polynomially. A triangle, formed by connecting three\npoints by the geodesic between each pair of these points, is thin in the sense that from any\npoint on an edge, there is a point on another edge at distance bounded by a constant.\n∗This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea\ngovernment (MSIT) (No. 2022R1F1A107586911).\n†Corresponding author\n1arXiv:2402.14604v2  [cs.CG]  1 Apr 2024z\nx\nO\n1\n1\nq\np\nz\nx\nO\n1\n1\nq\np\n(b)\n(c)\nz\nx\nO\nq\np\n(a)Figure 1: Three models of hyperbolic spaces. (a) The Poincar´ e halfplane H2. (b) The first\ndiscrete model ( B2, d1), with a shortest path of length d1(p, q) = 5. (c) The second discrete\nmodel ( B2, d2), with a shortest path of length d2(p, q) = 6.\nAs a consequence, hyperbolic spaces are sometimes more suitable than Euclidean spaces to\nrepresent some types of data. It has been shown, for instance, that there are better embeddings\nof the internet graph into hyperbolic spaces, compared with its embeddings into Euclidean\nspaces [12]. There has also been recent interest in hyperbolic spaces in the context of artificial\nneural networks [5].\nIn this paper, we present embeddings of finite subsets of HDinto graph metrics with a\nlinear number of edges, and additive distortion O(logD). As an application, we present an\napproximate near-neighbor data structure with O(logD) additive distortion. These two results\nhave no multiplicative distortion.\n1.1 Our results.\nGiven two metric spaces ( M, d) and ( M′, d′), we say that a mapping φ:M→M′is an embedding\nwith additive distortion ∆ if for any two points p, q∈M, we have\nd(p, q)−∆⩽d′(φ(p), φ(q))⩽d(p, q) + ∆ .\nOur first result (Theorem 14) is an embedding of HDwith additive distortion O(logD) into\na discrete metric space ( BD, d1) that is obtained from a binary tiling of HD. In two dimensions,\nthis binary tiling uses boxes whose sizes increase exponentially with the z-coordinate. (See\nFigure 1b.). A point is placed at the center of each box, and the distance between two points p\nandqis the minimum number of box boundaries that are crossed when going from ptoq. (See\nSection 2 for a detailed description.)\nGiven a subset Pofnpoints of a metric space ( M, d), apurely-additive spanner ofPwith\ndistortion ∆ and n′Steiner points is a graph G(V, E) where S⊂M,|S|=n′, the points in S\nare called Steiner points, V=P∪S, the length of any edge pqisd(p, q), and the shortest path\ndistance dG(·) in this graph satisfies\nd(p, q)⩽dG(p, q)⩽d(p, q) + ∆ for all p, q∈P.\nWe show, for any subset Pofnpoints of ( BD, d1), how to construct in 2O(D)nlogntime a\npurely-additive spanner of Pwith 2O(D)nedges and Steiner vertices, and distortion 2. (Theo-\nrem 16.)\n2Based on the two results above, we first obtain an embedding of any subset Pofnpoints of\nHDinto a graph metric with 2O(D)nvertices and edges, and additive distortion O(logD). We\nalso obtain an O(klogD) purely additive spanner of Pwith 2O(D)nvertices and 2O(D)n·λk(n)\nSteiner vertices and edges, where λk(n) is the kth-row inverse Ackermann function.\nGiven a subset Pofnpoints of a metric space ( M, d), and a query point q∈P, anearest\nneighbor ofqis a point p∈Psuch that d(p, q) is minimum. An approximate nearest neighbor\n(ANN) with additive distortion ∆ is a point p′∈Psuch that d(p′, q)⩽d(p, q) + ∆.\nWe give data structures for answering ANN queries in ( BD, d1) and in HDwith query time\n2O(D)+O(logn) and construction time 2O(D)nlogn. For ( BD, d1), the additive distortion is\n2, and for HD, it is O(logD). (See Theorem 21 and Corollary 22.) These data structures are\nin fact Approximate Voronoi Diagrams (AVD): They give a partition of the space into 2O(D)n\nregions, each region being associated with 2O(D)representative points, and such that for any\nquery point in one of these regions, one of the representative points is an approximate near\nneighbor.\n1.2 Comparison with previous work.\nSpanners have been studied in the more general context of an arbitrary weighted graph met-\nric [1]. For instance, one can find a t-multiplicative spanner of total weight (1 + 1 /t) times the\nweight of a minimum spanning tree.\nFor Euclidean metrics, it was shown that there are spanners with a linear number of edges,\nand multiplicative distortion arbitrarily close to 1 [2]. One difference with this paper is that we\nconsider an additive distortion. In the worst case, one cannot hope to find a non-trivial additive\nspanner for the Euclidean metric, because the additive error for a given graph can be made\narbitrarily larger by scaling up the input point set.\nGromov-hyperbolicity is a notion of hyperbolicity that applies to any metric space ( M, d),\nincluding discrete metric spaces. These spaces have the property that for any 4 points p, q, r, s ∈\nM, the largest two sums of distances among d(p, q) +d(r, s),d(p, r) +d(q, s),d(p, s) +d(q, r)\ndiffer by at most a constant δ. As shown by Gromov [6], any metric space of npoints with\nhyperbolicity δcan be embedded into a tree-metric with additive distortion O(δlogn). The\nPoincar´ e half-space has hyperbolicity O(1), so this result applies to a more general type of\nhyperbolic spaces than our embedding of HDinto a graph metric with O(n) edges. On the\nother hand, we obtain an additive distortion O(1) in constant dimension D=O(1), while\nGromov’s construction gives O(logn) in our case. Chepoi et al. [4] give additive O(δlogn)\nspanners with O(δn) edges for unit graph metrics (i.e. metrics for graphs where each edge\nweight is equal to 1) that are δ-hyperbolic. Compared with our result, it allows arbitrary\nhyperbolicity, but it is restricted to unit graphs, and the distortion is logarithmic. Lee and\nKrautghamer [10] present multiplicative spanners with O(n) edges for locally doubling, geodesic\nGromov-hyperbolic spaces.\nThere has also been some work on problems other than spanners and embeddings in hy-\nperbolic spaces. Lee and Krautghamer [10] presented an ANN data structure for geodesic and\nlocally doubling Gromov-hyperbolic spaces, with O(1) additive error, O(n2) space usage and\nO(log2n) query time. Kisfaludi-Bak et al. gave an algorithm for the TSP problem in H2[8].\nMore recently, Kisfaludi-Bak and van Wordragen [9] gave an ANN data structure and Steiner\nspanners for HDwith multiplicative error (1 + ε). Their approach is based on the same binary\ntiling that we use, but they derive from it a non-trivial type of quadtree that is taylored for\nproviding multiplicative guarantees in hyperbolic spaces. The lower bound that they present\non spanners without Steiner points imply that in order to an achieve additive error as we do,\nSteiner points are also required.\n31.3 Our approach.\nWe use an approximation of the Poincar´ e halfspace ( HD, dH) by a discrete metric space ( BD, d1)\nwhere the points are centers of hypercubes (called cells) whose sizes grow exponentially with y.\n(See Figure 1b.) The distance between two points p, qis the minimum number of cells that are\ncrossed when going from ptoq. This discrete model was mentioned, for instance, by Cannon\net al. [3]. We present these models of hyperbolic spaces in Section 2.\nWe introduce a different distance function d2overBD: we go upwards from pandquntil\nwe reach adjacent squares, and then we connect the subpaths using a single horizontal edge.\n(See Figure 1c.). This model is not a metric as it does not satisfy the triangle inequality. In\nSection 3, we show that d1andd2differ by at most 2. Then in Section 4, we show that over\nBD, the metric dHdiffers from d1andd2by an additive error O(logD).\nThese discrete models lend themselves well to the use of quadtrees, and the efficient con-\nstructions of our embeddings, spanners and ANN-data structure, as well as the bound on their\nsize, are based on compressed quadtrees. In Section 5, we present the compressed quadtree data\nstructure that we use.\nThis structure allows us, in Section 6, to show that the overlay of the shortest paths according\ntod2has linear complexity, and hence gives a 2-additive spanner of linear size with respect to\nd1. (See Figure 7.) In Section 7, we use the spanner we constructed in the discrete model\nto obtain an embedding with O(logD) additive distortion into the metric of a graph that has\n2O(D)nvertices and edges. Then using a transitive-closure spanner, we turn it into a spanner\nwith respect to dH. This spanner is embedded in HD, and has 2O(D)nedges and Steiner points.\nOur data structure for approximate near neighbor (ANN) searching is presented in Section 8.\nThis data structure is an approximate Voronoi diagram for the discrete model BD, that is\nbased on compressed quadtrees. This construction is done in two steps. We first compute the\nsubdivision induced by the (minimal) quadtree that records the input point set P. Then we\nrefine this subdivision by adding additional quadtree boxes where it is needed.\n2 Models of hyperbolic spaces\nFigure 1 shows the three models of hyperbolic spaces that we consider in this paper, when\nD= 2. The Poincar´ e halfplane ( H2, dH) is shown in Figure 1a. It is a model of 2-dimensional\nhyperbolic space with constant negative curvature −1. The halfplane H2consists of the points\np= (x(p), z(p)) where z(p)>0. The geodesics in ( H2, dH) are arcs of semi-circles that are\northogonal to the x-axis. (See Figure 1a.)\nMore generally, when D⩾2, the Poincar´ e halfspace HDconsists of the points p= (x(p), z(p))\nwhere x∈RD−1andzis a positive real number. The distance dH(p, q) between two points p\nandqis the length of the geodesic from ptoq, where the arc-length is given by the relation\nds2= (dx2+dz2)/z2. This distance is given by the expression\ndH(p, q) = 2 arsinh \n1\n2∥pq∥p\nz(p)z(q)!\nwhere ∥ · ∥ is the Euclidean norm. In the special case where x(p) =x(q) and z(q)> z(p), it\nis simply dH(p, q) = ln( z(q)/z(p)). The geodesics in ( HD, dH) are arcs of semi-circles that are\northogonal to the hyperplane z= 0.\nThe first discrete model ( BD, d1) is defined using a binary tiling of HDwith the hypercubes,\ncalled cells,\nC= [k12i,(k1+ 1)2i]×[k22i,(k2+ 1)2i]× ··· × [kD−12i,(kD−1+ 1)2i]×[2i,2i+1]\n4p\nU(p)\nq1\nq2\nq3\nq4\npFigure 2: Possible moves in B3. (left) Vertical moves U(p) and D(p)∈ {q1, q2, q3, q4}. (right)\nThe 8 Horizontal moves H(p), seen from above.\nwhere k1, k2, . . . , k D−1, i∈Z. (See Figure 1b for an example when D= 2.) The level of this\ncellCis the integer lev( C) =i, and its width is 2i.\nThe parent of a cell Cat level iis the cell at level i+ 1 whose bottom facet contains the\nupper facet of C. The children ofCare the cells whose parent is C, hence Chas 2D−1children.\nAn ancestor of Cis either the parent of C, or a parent of an ancestor of C. A descendant of Cis\neither a child of C, or a descendant of a child of C. The horizontal neighbors ofCare the cells\nat level idifferent from Cthat intersect Calong its boundary, hence Chas 3D−1−1 horizontal\nneighbors.\nWe denote by b(C) the center of the cell C, so the z-coordinate of b(C) isz(b(C)) = 3·2i−1when\nlev(C) =i. Then BDis the set of the points b(C) for all the cells C. The level lev( b(C)) of the point\nb(C) is the level of C. The parent, children, ancestors, descendants and horizontal neighbors of\nCare the point b(C′) where C′is a parent, child ancestors, descendants and horizontal neighbors\nofC, respectively.\nFrom a point p∈BD, we allow the following types of moves:\n•Anupward move to the parent of p.\n•Adownward move to a child of p.\n•Ahorizontal move to a horizontal neighbor of p. (Hence, when moving horizontally, we\nallow to move along diagonals.)\n(See Figure 2.)\nThen for any p, q∈BD, we define d1(p, q) to be the minimum number of moves that are\nneeded to reach qfrom p, and we define d2(p, q) to be the length of the shortest path from pto\nqthat has at most one horizontal move. Figure 1(b) and (c) show examples of shortest paths in\nthese models. A path from ptoqconsisting of d1(p, q) such moves is called a d1-path . Similarly,\nthe path from ptoqconsisting of d2(p, q) moves, at most one of which being horizontal, is called\nad2-path . For any p, q, there is only one d2-path from ptoq: Ifpis neither a descendant nor\n5p\nq\nr\n(a)\np\nq\n(b)Figure 3: (a) A counterexample that shows that d2does not obey the triangle inequality: we\nhave d2(p, q) = 1, d2(q, r) = 1 and d2(p, r) = 3 (b) An example where d2(p, q) = 5 = 2+ d1(p, q).\nan ancestor of q, then this path bends at the lowest ancestors of pandqthat are horizontal\nneighbors.\nThe space ( BD, d2) is not a metric space, but a semi-metric space, as it does not satisfy the\ntriangle inequality. (See counterexample in Figure 3a).\n3 Shortest paths in the discrete model\nIn this section, we study the structure, and give bounds on the length of d1-paths and d2-paths.\nLemma 1. Letp, q∈BD, and suppose that there is a d1-path from ptoqcontaining at least\none upward move. Then there is a d1-path from ptoqwhose first move is upward.\nProof. Letρ= (p0, p1, . . . , p k) be a path of length k=d1(p0, pk) from p=p0toq=pk,\ncontaining at least one upward move. If the first move of ρis upward, then we are done.\nOtherwise, Let ( pi, pi+1) be the first upward move, hence ( pi−1, pi) is horizontal or downward.\nIf (pi−1, pi) is downward, then we have pi−1=pi+1, so the path obtained from ρby deleting pi\nandpi+1is a path from ptoqof length d1(p, q)−2, a contradiction.\nNow suppose that ( pi−1, pi) is horizontal. As pi−1andpiare horizontal neighbors, their\nparents either are equal, or are horizontal neighbors. So the parent of pi−1is either pi+1, or is\na horizontal neighbor of pi+1. The parent of pi−1cannot be pi+1, because if it were the case,\nwe would obtain a path from ptoqof length d1(p, q)−1 by deleting pifrom ρ. So the parent\nofpi−1is a horizontal neighbor p′\niofpi+1. Hence, we obtain a path ρ′from ptoqof length\nd1(p, q) by replacing piwith p′\ni.\nThe first upward move of ρ′is now pi−1p′\ni, hence it has been moved one position to the left.\nBy repeating this process i−1 times, we find a d1-path from ptoqthat starts with an upward\nmove.\nLemma 2. For any p, q∈BD, there is a d1-path from ptoqthat consists of m1upward moves,\nfollowed by m2horizontal moves, and finally m3downward moves, where m1, m2, m3⩾0.\nProof. By applying Lemma 1 repeatedly, we obtain a d1-path from ptoqconsisting of a sequence\nαof upward moves, followed by a sequence βof horizontal or downward moves. Let γbe the\npath obtained by following βbackward, hence γconsists of upward or horizontal moves. We\napply Lemma 1 to γ, until we obtain a path γconsisting of upward moves followed by horizontal\nmoves. Then the path obtained by following α, and then following γbackwards, has the desired\nproperty.\n6Letp, q∈BDbe two points at the same level i. Their horizontal distance λ(p, q) is the\nlength of a shortest path from ptoqconsisting of horizontal moves only. It is given by the\nexpression λ(p, q) =∥x(p)x(q)∥∞/2iwhere ∥ · ∥∞is the L∞norm.\nLemma 3. Ifpandqare two points of BDandlev(p) = lev( q), then 2λ(p′, q′)−1⩽λ(p, q)⩽\n2λ(p′, q′) + 1 , where p′andq′are the parents of pandq, respectively\nProof. Let ∆ i={−2i−1,0,2i−1}D−1, then we have {x(p′)−x(p), x(q′)−x(q)} ⊂∆i. By the\ntriangle inequality, it implies that\n∥x(p′)x(q′)∥∞−2i⩽∥x(p)x(q)∥∞⩽∥x(p′)x(q′)∥∞+ 2i.\nThe result follows by dividing these inequalities by 2i.\nWe now obtain a recurrence relation for d1(p, q) when pandqare at the same level.\nLemma 4. Ifpandqare two points of BDsuch that lev(p) = lev( q), and if λ=λ(p, q)is their\nhorizontal distance, then\nd1(p, q) =(\nλ ifλ⩽4,and\n2 +d1(p′, q′)ifλ⩾5.\nwhere p′andq′are the respective parents of pandq.\nProof. By Lemma 2, there is a shortest path from ptoqthat either consists of horizontal moves\nonly (type 1), or that goes through p′andq′(type 2).\nAny path of type 2 has length at least 3. So if λ⩽3, there is a shortest path of type 1, and\nthus d1(p, q) =λ.\nIfλ= 4, then by Lemma 3, we have λ(p′, q′)⩾2, so any path of type 2 has length more\nthan 3. So there is a shortest path of type 1, and thus d1(p, q) =λ.\nNow suppose that λ⩾5. Then by Lemma 3, we have λ(p′, q′)⩽(1 + λ)/2⩽λ−2, and\nthus there is a shortest path which is of type 2. It follows that d1(p, q) = 2 + d1(p′, q′).\nLemma 5. Ifpandqare two points of BDandlev(p) = lev( q), then d2(p, q)⩽d1(p, q) + 2 .\nProof. Letp′andq′be the parents of pandq, respectively. Let p′′andq′′be the parents of\np′andq′, respectively. Let λ=λ(p, q),λ′=λ(p′, q′) and λ′′=λ(p′′, q′′).We make a proof by\ninduction on λ.\nWe first handle the basis cases.\n•Ifλ= 0, then p=qand thus d2(p, q) =d1(p, q) = 0.\n•Ifλ= 1, then pandqare horizontal neighbors, and thus d2(p, q) =d1(p, q) = 1.\n•Ifλ= 2, then λ′= 1 by Lemma 3. It follows that d2(p, q) = 3 and d1(p, q) = 2.\n•Ifλ= 3, then 1 ⩽λ′⩽2, and λ′′⩽1 by Lemma 3. It follows that d1(p, q) = 3 and\nd2(p, q)⩽5.\n•Ifλ= 4, then λ′= 2 and λ′′= 1. It follows that d1(p, q) = 4 and d2(p, q) = 5.\nNow suppose that λ⩾5. It follows from Lemma 3 that 3 ⩽λ′⩽(1 +λ)/2< λ. So by the\ninduction hypothesis, we have d2(p′, q′)⩽d1(p′, q′)+2. As λ⩾2, we have d2(p, q) =d2(p′, q′)+2,\nand thus d2(p, q)⩽d1(p′, q′) + 4. By Lemma 4, we also have d1(p, q) = 2 + d1(p′, q′). It follows\nthatd2(p, q)⩽d1(p, q) + 2.\n7We can now prove the main result of this section.\nTheorem 6. For any two points p, q∈B2, we have d1(p, q)⩽d2(p, q)⩽d1(p, q) + 2.\nProof. We have d1(p, q)⩽d2(p, q) because d1(p, q) is the length of a shortest path from ptoq\ninBD, and d2(p, q) is the length of some path from ptoq. We now prove the other side of the\ninequality.\nIfpandqare at the same level, then the result is given by Lemma 5. So we may assume that\npandqare not at the same level. Without loss of generality, we assume that the lev( p)⩽lev(q).\nLetrbe the ancestor of pthat is at the same level as q, and let i= lev( r)−lev(p).\nBy definition, we have d2(p, q) =i+d2(r, q). By Lemma 2, there is a shortest path from pto\nqgoing through r, and hence d1(p, q) =i+d1(r, q). By Lemma 5, we have d2(r, q)⩽d1(r, q)+2.\nIt follows that d2(p, q)⩽d1(p, q) + 2.\nThe bound in Theorem 6 is tight: In Figure 3b, we have d2(p, q) =d1(p, q) + 2 = 5. Finally,\nwe give a property of d2-paths that will be needed later.\nLetp, q∈BDbe such that pis neither a parent nor a descendant of q, and p̸=q. Then the\nd2-path ρfrom ptoqcontains exactly one horizontal move, from a point ¯ pto ¯q. We call the\nedge ¯ p¯qthebridge ofρ. For instance, in Figure 3b, the bridge is the top edge of the blue path.\nThelevel of the bridge is lev(¯ p) = lev(¯ q), which we denote lev( p, q).\nWhen p=q, orpis a descendant of q, we let lev( p, q) = lev( q, p) = lev( q). The lemma\nbelow allows us to approximate the level of a bridge.\nLemma 7. Letpandqbe two distinct points in BDsuch that pis neither a parent nor a\ndescendant of q. Let ℓ=⌊log2∥x(p)x(q)∥∞⌋. Then lev(p, q)−1⩽ℓ⩽lev(p, q).\nProof. Without loss of generality, we assume that lev( p)⩽lev(q). Let i= lev( p, q), and let ¯ p¯q\nbe the bridge of the d2-path from ptoq. AsC(¯p) and C(¯q) are horizontal neighbors at level\ni, and x(p) and x(q) lie in the vertical projection of these cells onto the hyperplane z= 0, we\nhave∥x(p)x(q)∥∞<2i+1, and thus ℓ⩽i.\nSuppose that lev( q) =i, and hence q= ¯q. As x(p) is not in the projection of C(q) onto the\nhyperplane z= 0, we have ∥x(p)x(q)∥∞⩾2i−1. It follows that ℓ⩾i−1.\nThe remaining case is when lev( p)⩽lev(q)⩽i−1. Let p′be the first point after the\nbridge vertex ¯ pon the d2-path from ¯ ptop, hence p′is a child of ¯ p. We define q′in the same\nway. The points x(p) and x(q) are in the vertical projection of the cells C(p′) and C(q′) onto\nthe hyperplane z= 0. As p′andq′are at level i−1 and are not neighbors, it follows that\n∥x(p)x(q)∥∞⩾2i−1, and thus ℓ⩾i−1.\n4 Embedding HDinto the discrete models\nIn this section, we give an embedding of the Poincar´ e half-space HDinto the discrete models\n(BD, d1) and ( BD, d2) with additive distortion O(logD). Our embedding maps any point p∈HD\nto the center b(C) of the cell CofHDthat contains p. If several cells contain p, we take b(p) to\nbe the center of a cell at the maximum level.\nWe first observe that for all x⩾1,\narsinh( x) = ln( x+p\nx2+ 1)⩽ln(x+√\n2x2)⩽ln(x) + ln(1 +√\n2)<ln(x) + 1. (1)\nWe can then bound the hyperbolic distance between pandb(p).\nLemma 8. For any p∈HD, we have dH(p, b(p))<lnD.\n8p\nq\np′\nq′\nz\nh\np\nq=q′=h\np′\nz\nz\nβ\nα\np\nq\np′\nO\n(a)\n(b)\n(c)\nR\nc\nαFigure 4: Lemma 9\nProof. Leti= lev( p). We have ∥pb(p)∥⩽2i−1√\nD,z(p)⩾2iandz(b(p))⩾2i. Therefore\ndH(p, b(p)) = 2 arsinh \n1\n2∥pb(p)∥p\nz(p)z(b(p))!\n⩽2 arsinh \n1\n22i−1√\nD\n2i!\n= 2 arsinh √\nD\n4!\nso by Inequality (1),\ndH(p, b(p))⩽2 ln(√\nD/4) + 2 = ln( D) + 2−4 ln 2 <lnD.\nLetp, q∈HDand let hbe the highest point on the geodesic from ptoq. Let p′andq′be\nthe points vertically above pandq, respectively, that are on the horizontal line through h. (See\nFigure 4a and 4b.)\nLemma 9. For any p, q∈HD, we have 0⩽dH(p, q)−dH(p, p′)−dH(q, q′)<2.\nProof. We first consider the special case where the geodesic from ptoqis an arc of a circle\nof radius Rcentered at the origin O, and the geodesic from ptoqdoes not cross the z-axis.\nLetαandβbe the angles that OpandOqmake with the z-axis, respectively. Without loss of\ngenerality, we assume that α > β . (See Figure 4c.)\nBy the triangle inequality, we have\ndH(p, q)−dH(p, p′)⩽dH(p′, q) = 2 arsinh \n1\n2s\nR2(sinα−sinβ)2\nR2cos2β!\n⩽2 arsinh \n1\n2s\n(1−sinβ)2\ncos2β!\n⩽2 arsinh\n1\n2s\n1−sin2β\ncos2β\n\n= 2 arsinh(1 /2)<1.\nThe inequality above applies whenever qis the highest point on the geodesic arc pq, as in\nFigure 4b. When the geodesic arc pqhas a local maximum hin its interior (See Figure 4a), we\napply the inequality above twice through the point h, which yields the result. (See Figure 4a.)\n9Still using the same notation as in Figure 4, we have the following two observations.\nLemma 10. Letˆp=b(p′)andˆq=b(q′). Then d1(ˆp,ˆq)⩽5.\nProof. LetRbe the radius of the circle containing the geodesic arc pq. Suppose that p′q′is\ntangent to this geodesic arc at h. (See Figure 4a.) Then z(h) =R, and the cells intersected by\np′q′are at a level isuch that 2i⩽R <2i+1. So we have i= lev(ˆ p) = lev(ˆ q) =⌊log2R⌋. Since\n∥p′q′∥⩽2R, we have ∥x(ˆp)x(ˆq)∥∞⩽2R+ 2i, and since the width of the cells at level iis at\nleast R/2, the horizontal distance λ(ˆp,ˆq) is at most 5. It follows that d1(ˆp,ˆq)⩽5.\nNow suppose that h=qand the center of the circle containing the arc pqis at the origin\nO. (See Figure 4c.) Let αandβbe the angles that OpandOq, respectively, make with the\nz-axis. Then the Euclidean length of p′q′=p′qisR(sinα−sinβ) and the cells intersected by\np′q′have width at least Rcos(β)/2. Since\nR(sinα−sinβ)\nR(cosβ)/2⩽21−sinβ\ncosβ⩽21−sin2β\ncos2β= 2,\nby the same argument as above, d1(ˆp,ˆq)⩽3.\nThe lemma below gives us a lower bound on the level of the bridge ¯ p¯qwhen p′q′is tangent\nto the geodesic arc pq, as in Figure 4a.\nLemma 11. Suppose that pandqare two distinct points in BDsuch that pis neither a parent\nnor a descendant of q, and his between pandqalong the geodesic arc pq. Then we have\nlev(p, q)⩾log2(z(h))−log2(D)/2−1.\nProof. The geodesic semi-circle through pandqhas radius z(h) and center c= (x(h),0). Let\nαbe the angle between the segment cpand the segment ch.\nIfα⩽π/3, then z(p)⩾z(h)/2. Since lev( p, q)⩾lev(p) and z(p) = (3 /2)2lev(p), it follows\nthat\nlev(p, q)⩾lev(p) = log2(z(p))−log2(3) + 1 ⩾log2(z(h))−log2(3).\nIfα⩾π/3, then ∥x(p)x(q)∥⩾√\n3z(h)/2, so\n∥x(p)x(q)∥∞⩾√\n3z(h)/(2√\nD),\nand thus by Lemma 7,\nlev(p, q)⩾log2(√\n3z(h)/(2√\nD))⩾log2(z(h))−log2(D)/2−1.\nWe now prove that dH,d1andd2differ by at most O(logD) over BD. We begin with two\nspecial cases, and then we handle the general case.\nLemma 12. For any p, q∈BDsuch that qis an ancestor of p, we have\nln(2)·d1(p, q)⩽dH(p, q)⩽ln(2)·d1(p, q) + ln( D) + 2 + ln 4 .\nProof. Leti= lev( p) and j= lev( q). Let r= (x(q), z(p)). Then we have\ndH(p, q)⩾dH(r, q) = ln( z(q)/z(p)) = ln(2j−i) = (j−i) ln 2.\nand thus\ndH(p, q)⩾(j−i) ln 2. (2)\n10p\nq\np′\nq′\nˆp\nˆq\np′′\nq′′\nhFigure 5: Proof of Lemma 13\nWe now prove an upper bound on dH(p, q). Observe that ∥pq∥⩽2j+1√\nD.\ndH(p, q) = 2 arsinh \n1\n2∥pq∥p\nz(p)z(q)!\n⩽2 arsinh \n2j+1√\nD\n3·2(i+j)/2!\n⩽2 arsinh\u0010\n2(j−i+2)/2√\nD\u0011\nThen it follows from Inequality 1 that\ndH(p, q)⩽2 ln\u0010\n2(j−i+2)/2√\nD\u0011\n+ 2\n= ln(2j−i+2) + ln( D) + 2 = ( j−i+ 2) ln(2) + ln( D) + 2\nand thus dH(p, q)⩽(j−i) ln(2) + ln( D) + 2 + ln 4.\nTogether with (2), it shows that\n(j−i) ln 2⩽dH(p, q)⩽(j−i) ln(2) + ln( D) + 2 + ln 4 .\nThe result follows from d1(p, q) =j−i.\nLemma 13. For any p, q∈BD, we have\n−7 ln 2 < dH(p, q)−ln(2)·d1(p, q)⩽ln(D) + 2 + 6 ln 2 .\nProof. Without loss of generality, we assume that lev( p)⩽lev(q). The case where qis an\nancestor of pis handled by Lemma 12, so we assume that qis not an ancestor of p. Let p′, q′be\ndefined as above. (See Figure 5.) Let ˆ p=b(p′), ˆq=b(q′),p′′= (x(p),ˆz(p)) and q′′= (x(q),ˆz(q)).\nLeti= lev( p) and j= lev(ˆ p) = lev(ˆ q).\nBy Lemma 10, we have d1(ˆp,ˆq)⩽5, and thus\nd1(p, q)⩽d1(p,ˆp) +d1(q,ˆq) + 5. (3)\nSuppose that q= ˆq, and thus d1(q,ˆq) = 0. Then the d1-path from ptoqgoes through ˆ p,\nand we have d1(p, q)⩾d1(p,ˆp), therefore d1(p, q)⩾d1(p,ˆp) +d1(q,ˆq).\nNow suppose that q̸= ˆq. It implies that p̸= ˆp, and that his between pandqalong the\ngeodesic arc pq. So by Lemma 11, we have lev( p, q)⩾log2z(h)−1\n2log2D−1. As z(h)⩾2j, it\nimplies that lev( p, q)⩾j−1−1\n2log2D. Hence the d2-path pqfollows the path pˆpuntil ˆ p, or\n11a descendant of ˆ pat most 1 +1\n2log2Dlevels below ˆ p. So the portion of the d2-path pqbefore\nthe bridge has length at least d1(p,ˆp)−1\n2log2D−1. Similarly, the portion after the bridge has\nlength at least d1(q,ˆq)−1\n2log2D−1. Therefore, d2(p, q)⩾d1(p,ˆp) +d1(q,ˆq)−2−log2D. By\nTheorem 6, it implies that d1(p, q)⩾d1(p,ˆp) +d1(q,ˆq)−4−log2D.\nTogether with Equation (3), it implies that, whether q= ˆqor not,\n−5⩽d1(p,ˆp) +d1(q,ˆq)−d1(p, q)⩽4 + log2D.\nWe also have d1(p,ˆp) =j−iand thus\ndH(p, p′′) = ln( z(p′′)/z(p)) = ln\u0000\u0000\n3·2j−1\u0001\n/\u0000\n3·2i−1\u0001\u0001\n= (j−i) ln 2 = ln(2) ·d1(p,ˆp).\nSimilarly, we have dH(q, q′′) = ln(2) ·d1(q,ˆq), and thus\n−5 ln 2⩽dH(p, p′′) +dH(q, q′′)−ln(2)·d1(p, q)\n⩽ln(2)·log2(D) + 4 ln 2 = ln( D) + 4 ln 2 .\nWe also have dH(p′, p′′) =|ln(z(p′)/z(p′′)|⩽ln 2 and, similarly, dH(q′, q′′)⩽ln 2. It follows\nthat\n−7 ln 2⩽dH(p, p′) +dH(q, q′)−ln(2)·d1(p, q)⩽ln(D) + 6 ln 2 .\nThe result follows from Lemma 9.\nIt follows from Theorem 6, Lemma 8 and Lemma 13 that our embedding b:HD→BDgives\nO(logD) additive distortion with respect to d1andd2.\nTheorem 14. For any p, q∈HD, we have |dH(p, q)−ln(2)·d1(b(p), b(q))|⩽3 ln(D) +O(1)\nand|dH(p, q)−ln(2)·d2(b(p), b(q))|⩽3 ln(D) +O(1).\n5 Compressed quadtrees\nIn this section, we present compressed quadtrees , which will be needed for our graph metric\nembeddings and for approximate near neighbor searching. Compressed quadtrees are presented\nin the book by Har-Peled [7]. Our quadtrees will record points in BD, which correspond to\nquadtree boxes in RD−1. (See the discussion below.) Hence, our quadtrees record a set of\nquadtree boxes instead of recording a set of points in RD−1. This does not affect the time and\nspace bounds [7, Lemma 2.11].\nLetCbe a cell of our binary tiling of HD. The vertical projection CxofCis the set of the\nprojections x(p) of all the points p∈ C. Thus, Cxis a box in RD−1, and more precisely, Cxis of\nthe form\n[k12−i,(k1+ 1)2−i]× ··· × [kD−12−i,(kD−1+ 1)2−i] where k1, . . . , k D−1∈Z.\nA box Cxof this form is called a quadtree box . Thus, each point b∈BDcorresponds to exactly\none quadtree box Cx(b)⊂RD−1, and the corresponding cell C(b) of our binary tiling of HD.\nIt allows us extend the notions of children, parents, ancestors, descendants and levels to\nquadtree boxes. So for any two cells CandC′of our binary tiling, we say that Cxis a child\n(resp. parent, ancestor, descendant) of C′\nxifCis a child (resp. parent, ancestor, descendant) of\nC′. The level lev( Cx) of the quadtree cell Cxis lev( C).\nLetQbe a set of npoints in BD, such that x(Q)⊂(0,1)D−1. A compressed quadtree\nTstoring Qis a tree such that each node νofTrecords one cell C(ν) of our binary tiling,\n12ν\nν1\nν2\nCx(ν)\nCx(ν1)\nCx(ν2)\n(a)\nν\nν1\nCx(ν)\n(b)\nCx(ν1)\nrg(ν)Figure 6: Nodes of a compressed quadtree. (a) An ordinary node ν, with two children cells ν1\nandν2. (b) A compressed node ν, with its child ν1. The region rg( ν) is shaded in red.\nand, equivalently, the quadtree box Cx(ν) and the center point b(ν) =b(C)∈BD. This tree is\nconstructed as follows. A cell Cthat contains no point of Q, and such that no descendant of C\ncontains a point of Q, is said to be empty . A cell C(ν) where νis a node of Tis aleaf cell if\nit contains at most one point of P. IfC(ν) is a leaf cell, then νis a leaf node of T. The root\nrecords the cell [0 ,1]D−1×[1,2]. The children of a node νthat is not a leaf are constructed as\nfollows:\n•If two or more children cells of C(ν) are non-empty, then we create a node νicorresponding\nto each such child cell, and make νia child of ν. In this case, νis an ordinary node . (See\nFigure 6a.)\n•If only one child of C(ν) is non-empty, let ν1be the descendant of νsuch that Cx(ν1)∩Q=\nCx(ν)∩Qand the level of ν1is minimum. Then ν1is the only child of ν, and νis a\ncompressed node . The region associated with νis rg( ν) =Cx(ν)−Cx(ν1). (See Figure 6b.)\nThe quadtree boxes stored at the leaf nodes and the regions of the compressed nodes form\na partition of [0 ,1]D−1. This compressed quadtree ThasO(n) nodes, it can be constructed\ninO(Dnlogn) time, and the node corresponding to the leaf cell or compressed node region\ncontaining a query point q∈RD−1can be found in time O(Dlogn) time [7, Chapter 2].\nIn addition, we can answer cell queries inO(Dlogn) time: Given a query quadtree box Cx,\nwe can find the largest box C−\nxstored in T′such that C−\nx⊂ Cx, and we can find the smallest\nboxC+\nxstored in T′that contains Cx.\n6 Spanner in the discrete model\nIn this section, we give a 2-additive spanner Sin the first discrete model ( BD, d1), for a set\nP⊂BDofnpoints. Our approach is the following: We overlay the d2-paths of all pairs of\npoints in P, and add a new vertex, called a Steiner vertex, at each point where such a path\nbends—in other words, we add a Steiner vertex at each endpoint of each bridge, if this endpoint\nis not in P. (See Figure 7.) We also add a Steiner vertex whenever two such d2-paths merge.\nThe length of an edge is the number of cell boundaries that it crosses.\nThe resulting graph Scontains, by construction, a path of length d2(p, q) between any two\npoints p, q∈P. By Theorem 6, it follows that:\nProposition 15. For any p, q∈P, there is a path from ptoqin the graph Sof length at least\nd1(p, q)and at most d1(p, q) + 2 .\n13p1\np2\np3\nv1\nv3\nv4\nv2\nv5\nv6\np5\ne1\ne2\ne3\np6\np4Figure 7: Our construction of a 2-additive spanner Sfor the points p1, . . . , p 6with respect\nto the metric d1. The Steiner vertices are v1, . . . , v 6. The vertices v1andv4are the bending\npoints of the d2-path from p1top2, hence v1v4is the bridge between p1andp2. The vertex\nv3is the bending point of the d2-path from p2top3. The distance from p3top6within Sis\ndS(p3, p6) = 6, as the edge v6p6has length 2. On the other hand, we have d1(p3, p5) = 4. The\nvertex v6is the Steiner vertex at which the d2-paths from p5andp6top3merge.\nUsing compressed quadtrees (Section 5), we now give a bound on the size of S, as well its\nconstruction time. Let Pbe our input set of points in HD, and let x(P) ={x(p) :p∈P}be its\nvertical projection. Without loss of generality, we assume that x(P)⊂[0,1]D−1. If it were not\nthe case, we could apply to Pa scaling transformation centered at Ofollowed by a horizontal\ntranslation so that the vertical projection of Pis in [0 ,1]D−1. The hyperbolic distance dH(·) is\ninvariant under these transformations. We construct the compressed quadtree Tthat records\nthe points in P.\nLetνbe a node in T, and let rbe the point in BDthat corresponds to C(ν), soCx(r) =Cx(ν).\nGiven a horizontal neighbor r′ofr, we can check whether rr′is a bridge as follows. First check\nusing cell queries in Twhether the cells C(r) and C(r′) are non-empty. If so, check whether,\namong the children of C(r) and C(r′), there is a pair of non-empty cells that are not horizontal\nneighbors. For a given horizontal neighbor r′orr, we can check it using 2O(D)cell queries, and\nsince there are 2O(D)horizontal neighbors r′, and we can do it using 2O(D)queries in total. We\nperform this operation for each node νofT. As there are O(n) nodes in T, and cell queries\ncan be answered in O(Dlogn) time, this process generates 2O(D)nbridges in 2O(D)nlogntime.\nWe still need to find bridges that do not correspond to any node ν∈ T, that is, bridges rr′\nsuch that neither C(r) norC(r′) is recorded in T. As the quadtree boxes stored at leaf cells and\nthe regions associated with the compressed nodes form a partition of [0 ,1]D−1, such a bridge\nrr′must satisfy x(r)∈rg(ν) and x(r′)∈rg(ν′) for some compressed nodes ν, ν′. Then rr′must\nconnect the points of BDcorresponding to the cell of the children of νandν′. Given νandν′,\nwe can determine the level of this bridge, if it exists, in O(1) time, using Lemma 7. We check\nthe existence of such a bridge for all pairs of compressed nodes ν, ν′whose cells are adjacent.\nThere are 2O(D)nsuch pairs to check, because any cell Cx(ν) is adjacent to 2O(D)cells of size\nat least the size of Cx(ν). So again, this process generates 2O(D)nbridges.\nThe construction above also gives us the vertical edges descending from every Steiner point.\nIt follows that:\nTheorem 16. Given a set P⊂BDofnpoints, we can compute in time 2O(D)nlogna2-\n14additive spanner SofPthat has 2O(D)nSteiner vertices and edges. More precisely, Sis a\nweighted graph embedded in BD, any of its edge uvhas length d1(u, v), and for any p, q∈P,\nthere is a path in Sfrom ptoqthat has length d2(p, q), where d1(p, q)⩽d2(p, q)⩽d1(p, q) + 2.\n7 Embedding into a graph metric and spanner for HD\nWe can embed any set Pofnpoints in HDinto a graph metric as follows. First we map each\np∈Pto the point b(p)∈BD. Then we construct the spanner Sfor these points with respect to\nd1as in Theorem 16. We multiply all the edge lengths of Sby ln(2), thus obtaining a graph S′\nsuch that the length dS′(p, q) of a shortest path in S′between any two points p, q∈BDsatisfies\nln(2)·d1(p, q)⩽dS′(p, q)⩽ln(2)·d2(p, q).\nBy Theorem 14, this path length approximates dH(p, q) within an additive error O(logD).\nIn summary:\nCorollary 17. Given a set P⊂HDofnpoints, we can compute in time 2O(D)nlogna positively\nweighted graph G(V, E)that has 2O(D)nvertices and edges, and a mapping b:P→V, such\nthat|dH(pi, pj)−dG(b(pi), b(pj))|=O(logD)for any pi, pj∈P.\nThe graph from Corollary 17 is not a spanner in the sense that the length of an edge\nb(p)b(q),p, q∈HDis ln(2) ·d1(b(p), b(q)) instead of dH(p, q). If we set the edge weights to be\ndH(b(p), b(q)) for each edge b(p)b(q), then we introduce a constant additive error for each edge\nofS, and since a shortest path in Smay consist of Ω( n) edges, we will no longer have O(logD)\nadditive error.\nIn order to obtain an additive spanner for P⊂HDthat is embedded in HD, we will add more\nedges, which will act as shortcuts. So let Tbe the forest obtained by removing the horizontal\nedges from S, and let kbe a fixed integer. We orient all the edges of Tupwards. We construct a\nk-transitive closure spanner TkofS[11, 13]. This graph TkhasO(n·λk(n)) edges, where λk(n)\nis the k-th row of the inverse Ackermann function. The vertices of Tkare the points b(p),p∈P\nand the edges of Tkare a superset of the edges of T, also oriented upwards. The key property\nofTkis that, if there is a path from vertex ptoqinTk, then there exists such a path from\nptoqof length at most k. The k-transitive closure spanner can be computed in O(n·λk(n))\ntime [13, Algorithm L].\nOur Spanner Skis obtained from Tkby adding all the horizontal edges of S, and by adding\neach point p∈Pas a vertex, together with the edge pb(p). Each edge pqofSkis assigned the\nweight dH(p, q). In particular, an edge pb(p) has weight O(logD) by Lemma 8.\nLetp, q∈P. We now prove that there is a path of length dH(p, q) +O(klogD) inSk. By\nconstruction, there is a path in Sfrom b(p) tob(q) of length d2(b(p), b(q)) that consists of a\nvertical path ρup from b(p) to a vertex ¯b(p), then a horizontal edge from ¯b(p) to a vertex ¯b(q),\nfollowed by a vertical path ρ′down from ¯b(q) tob(q).\nInTk, the path ρcan be replaced with a path ρkof at most kedges. The length of each\nedge vwinρkisdH(v, w), which is ln(2) ·d2(v, w) +O(logD) by Theorem 14. So the length\nofρk(i.e. the sum of the weights of its edges) is ln(2) ·d2(b(p),¯b(p)) +O(klogD). Similarly,\nthere is a path ρ′\nkfrom ¯b(q) tob(q) inTkwith length ln(2) ·d2(¯b(q), b(q)) +O(klogD). It\nfollows that the length of the path in Sconsisting of pb(p) followed by ρk,¯b(p)¯b(q),ρ′\nkand\nqb(q) has length ln(2) ·d2(b(p), b(q)) +O(klogD). By Lemma 8 and Theorem 14, this length is\ndH(p, q) +O(klogD).\nTheorem 18. Letkbe an integer, and let Pa set of npoints in HD. We can construct in\n2O(D)nlogntime an O(klogD)purely additive spanner of Pwith 2O(D)nSteiner vertices and\n2O(D)n·λk(n)edges.\n158 Approximate Voronoi diagram (AVD)\nIn this section, we first show how to answer exact near-neighbor queries in ( BD, d2), and then\nwe briefly explain how this result yields ANN data structures for ( BD, d1) andHDwith additive\nerror 2 and O(logD), respectively.\nSo let P={p1, . . . , p n}be a set of npoints in BD. For a query point q∈BD, we want to\nfind a point n2(q)∈Psuch that d2(q, n2(q)) is minimum. As there can be several closest point\ntoq, we break the ties by taking n2(q) to be the point pisuch that d2(pi, q) = min jd2(pj, q)\nandiis minimum.\nWithout loss of generality, we assume that x(P)⊂[1/4,1/2]D−1. Ifx(q)/∈[0,1]D−1, then\nn2(q) is the highest point in P, which the data structure described below can return in constant\ntime.\nThis data structure records an approximate Voronoi diagram (AVD), which is a partition of\nBDin 2O(D)nregions V1, . . . , V msuch that each region Viis associated with a set Ri⊂Pof at\nmost 2O(D)representative points . For any point q∈Vi, we have n2(q)∈Ri.\nIn the definition above, the AVD allows us to find an exact nearest neighbor with respect to\nthe second discrete model ( BD, d2). In the first discrete model ( BD, d1) and in HD, it will allow\nus to find an approximate near-neighbor, with additive error 2 and O(logD), respectively.\nAs in Section 6, we first compute the compressed quadtree Tthat records our input point\nsetP. This quadtree ThasO(n) nodes, it can be computed in time O(Dnlogn) time, and it\nallows us to perform cell queries in O(Dlogn) time.\nRefinement step Suppose that νis an ordinary node of T. Let b(ν) be the corresponding\npoint of BD, then b(ν) is the only point of BDcontained in C(ν). So we associate νwith a\nVoronoi region consisting only of b(ν), and with a single representative n2(b(ν)).\nOn the other hand, if νis a leaf node of T, then the descendants of C(ν) do not correspond\nto nodes of T. So we would like νto be associated with a Voronoi region Vithat covers b(ν)\nand all its descendants. The issue here is that we may need Ω( n) representatives for Vi, if for\ninstance, Cx(ν) is adjacent to Ω( n) smaller, disjoint cells of T.\nIn order to solve this problem, we will break Cx(ν) into smaller cells, by inserting more\nboxes into our quadtree. We call this step of the construction the refinement step . For each\nnode νofTsuch that C(ν) is not empty, we insert into our compressed quadtree all the boxes\nthat are horizontal neighbors of C(ν), thus obtaining a quadtree T′that is associated with a\nfiner subdivision of RD−1. (See Figure 8.) In particular, for a compressed node ν, we insert\nthe neighbors of the outer cell and the inner cell, that is, the horizontal neighbors of C(ν) and\nC(ν1), where ν1is the child of ν.\nWe say that two quadtree boxes in RD−1areadjacent if they intersect along their boundaries,\nbut not in their interior. The two lemmas below will help us bound the number of representative\npoints for each cell of our AVD.\nLemma 19. Letν′be a leaf cell of T′. Then the number of compressed nodes νofTsuch that\nCx(ν)is adjacent to Cx(ν′)is2O(D).\nProof. The leaf cells and the regions of the compressed nodes of Tform a subdivision of RD−1.\nThe leaf cells and the regions of the compressed nodes of T′form a refinement of this subdivision.\nSo there are at most 2O(D)such cells Cx(ν) adjacent to Cx(ν′) of size larger than or equal to the\nsize of Cx(ν′). So in the following, we consider cells Cx(ν) whose size is smaller than the size of\nCx(ν′).\nFor the sake of contradiction, suppose that there are at least 3D−1such nodes νsuch that\nthe size of Cx(ν) is smaller than that of Cx(ν′). Then two of them, Cx(ν1) and Cx(ν2), must\n16Figure 8: (Left) The subdivision corresponding to the initial quadtree T. (Right) The subdivi-\nsion after refinement, corresponding to the quadtree T′.\nCx(ν′)\nC′′\nx\nCx(ν1)\nCx(ν2)\nFigure 9: Proof of Lemma 19. The node ν2is a compressed node.\n17be contained in a quadtree box C′′\nxthat has the same size as Cx(ν) and is adjacent to it. (See\nFigure 9.) This is impossible, as it would have caused Cx(ν′) to be partitioned at the refinement\nstep.\nLemma 20. Letν′be a compressed node of T′. Then the number of compressed nodes νofT′\nsuch that Cx(ν)is adjacent to rg(ν′)is2O(D).\nProof. The same proof as for Lemma 19 shows that there are 2O(D)such cells that are adjacent\nto rg( ν′) along its outer boundary. A similar argument again shows that there are 2O(D)such\ncells adjacent along the inner boundary.\nThe approximate Voronoi diagram Our AVD consists of one region V(ν′) for each node\nν′ofT′:\n•Ifν′is an ordinary node, then the Voronoi region V(ν′) isC(ν′).\n•Ifν′is a leaf node, then the Voronoi region V(ν′) is the union of C(ν′) and all its descen-\ndants. In other words, V(ν′) is the set of all the points that are in or below C(ν′).\n•Ifν′is a compressed node, let ν′\n1be its child. Then V(ν′) is the set of all the points that\nare on or below C(ν′), and are not on or below C(ν′\n1).\nPreprocessing We first compute, for every node ν′ofT′such that C(ν′) is not empty, the\npoint h(ν′) inPwith highest z-coordinate that is recorded in the subtree rooted at ν′. (Again,\nwe break ties by taking the point piwith smallest index iamong these highest points.) We can\ncompute all these points in linear time by traversing the tree T′from bottom to top.\nNext, we compute the d2-nearest neighbor n2(ν′) =n2(b(ν′)) for all the nodes ν′ofT′\nas follows. We compute these points by traversing the tree from top to bottom, so when we\ncompute n2(ν′), we already know n2(ν′′) where ν′′is the parent of ν′. Let ρbe the d2-shortest\npath from b(ν′) ton2(ν′). The 4 possible cases are listed below. We compute the nearest point\nfor each case, and record the nearest one as n2(ν′).\n1. If ρgoes through b(ν′′), then n2(ν′) =n2(ν′′), and we are done as we already know n2(ν′′).\n2. If ρis a downward path, or if b(ν)∈P, then n2(ν) =h(ν).\n3. If ρstarts with a horizontal move from b(ν′), then n2(ν′) is the highest point that is on\nor below a cell C′that is a horizontal neighbor of C(ν′). We can find this highest point in\nO(Dlogn) time by performing a cell query in T, and since there are 2O(D)such cells C′,\nwe find n2(ν) in 2O(D)logntime.\n4. Otherwise, ρfirst goes upward from b(ν′), then follows a bridge r′r, and then goes down-\nward to n2(ν′). As ρdoes not go through b(ν′′), it follows that ν′′is a compressed node.\nThenC(r) is not stored in T′, because if it were the case, we would have created a node\nforC(r′) inT′, and ν′′could not be the parent of ν′. Soris inV(ν) for some compressed\nnode νofT, and we have n2(b(ν)) = h(ν). As Cx(ν′) and Cx(ν) must be adjacent, by\nLemma 20, there are 2O(D)candidates for ν, so we can find it in 2O(D)logntime.\n18Finding the representative points We now explain how we choose the representative points\nfor each region V(ν′). For each node ν′ofT′, we pick n2(ν′) as a representative point. If ν′is\nan ordinary cell, then we do not add any other representative point.\nIfν′is a leaf node or a compressed node, then for each compressed node νofTsuch that\nCx(ν) is adjacent to Cx(ν′) or rg( ν′), respectively, we add h(ν) as a representative point. By\nLemma 19 and Lemma 20, there are 2O(D)such points.\nWe now prove that for each point qin a Voronoi cell V(ν′), one of these representative points\nisn2(q). So suppose that n2(q)̸=n2(ν′), and thus ν′is not an ordinary node. Let ρbe the\nd2-path from qton2(q). As ν′is either a leaf node or a compressed node, there is no point in\nPbelow q, soρcannot be a downward path. As n2(q)̸=n2(ν′), it cannot be an upward path\neither. So it must have a bridge r′r, where ris on or above n2(q). IfC(r) is recorded in a node\nν(r) ofT, then at the refinement stage, we must have inserted into T′the node ν(r′) such that\nC(ν(r′)) =C(r′), and then n2(q) =n2(ν′), a contradiction. Therefore, C(r) is not recorded in\nT. Hence, ris in rg( ν) for some compressed node νofT, such that Cx(ν) is adjacent to Cx(ν′).\nIt follows that n2(q) =h(ν), which is one of our representative points.\nResult AsT′records 2O(D)nnodes, it can be computed in 2O(D)nlogntime, and it can be\nqueries in O(D+ log n) time. Since our data structure returns an exact nearest neighbor with\nrespect to d2, by Theorem 6, it returns an approximate near neighbor with respect to d1, with\nadditive error 2. So we obtained the following result:\nTheorem 21. LetPbe a subset of BDof size n. Then we can compute in 2O(D)nlogntime an\nAVD of Pwith2O(D)nregions, and 2O(D)representative points per region. Using this diagram,\ngiven a query point q∈HD, a point n2(q)∈Psuch that d2(q, n2(q))is minimum can be returned\nin2O(D)+O(logn)time. This point satisfies d1(q, n2(q))⩽d1(q, n1(q)) + 2 , where n1(q)is a\npoint in Psuch that d1(q, n1(q))is minimum.\nThe discrete AVD presented above can also be turned into an AVD for a set of points Pin\nHD, by constructing the d2-AVD for the points {b(p) :p∈P}, and replacing each point pin a\nVoronoi region by the whole box C(p), and thus we obtain a partition of [0 ,1]D−1×(0,2]⊂HD.\nThen for a query point q∈HD, we return a point in Psuch that b(p) =n2(b(q)). Theorem 14\nimplies that this point is an O(logD)-additive approximate near neighbor:\nCorollary 22. LetPbe a subset of HDof size n. Then we can compute in 2O(D)nlogn\ntime an AVD of Pwith 2O(D)nregions, and 2O(D)representative points per region. Using this\ndiagram, for any query point q, we can return in 2O(D)+O(logn)time a point r∈Psuch that\ndH(q, r)⩽dH(q, nH(q)) +O(logD), where nH(q)is a the point in Psuch that dH(q, nH(q))is\nminimum.\nReferences\n[1] Ingo Alth˝ ofer, Gautam Das, David P. Dobkin, Deborah Joseph, and Jos´ e Soares. On\nsparse spanners of weighted graphs. Discret. Comput. Geom. , 9:81–100, 1993. doi:10.\n1007/BF02189308 .\n[2] Sunil Arya, Gautam Das, David Mount, Jeffrey Salowe, and Michiel Smid. Euclidean\nspanners: short, thin, and lanky. In Proc. 27th ACM Symposium on Theory of Computing ,\npages 489–498, 1995. doi:10.1145/225058.225191 .\n[3] James Cannon, William Floyd, Richard Kenyon, and Walter Parry. Hyperbolic geometry.\nIn Silvio Levy, editor, Flavors of Geometry , volume 31, pages 167–196. MSRI Publications,\n1997.\n19[4] Victor Chepoi, Feodor F. Dragan, Bertrand Estellon, Michel Habib, Yann Vax` es, and Yang\nXiang. Additive spanners and distance and routing labeling schemes for hyperbolic graphs.\nAlgorithmica , 62(3-4):713–732, 2012. doi:10.1007/s00453-010-9478-x .\n[5] Octavian-Eugen Ganea, Gary B´ ecigneul, and Thomas Hofmann. Hyperbolic neural net-\nworks. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicol` o\nCesa-Bianchi, and Roman Garnett, editors, Advances in Neural Information Processing\nSystems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS\n2018, pages 5350–5360, 2018.\n[6] M. Gromov. Hyperbolic Groups , chapter 6, pages 75–263. Springer New York, 1987.\n[7] Sariel Har-peled. Geometric Approximation Algorithms . American Mathematical Society,\n2011.\n[8] S´ andor Kisfaludi-Bak. A Quasi-Polynomial Algorithm for Well-Spaced Hyperbolic TSP.\nInProc. 36th International Symposium on Computational Geometry (SoCG 2020) , pages\n55:1–55:15, 2020.\n[9] S´ andor Kisfaludi-Bak and Geert van Wordragen. A quadtree for hyperbolic space, 2023.\narXiv:2305.01356 .\n[10] Robert Krauthgamer and James R. Lee. Algorithms on negatively curved spaces. In 2006\n47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06) , pages\n119–132, 2006. doi:10.1109/FOCS.2006.9 .\n[11] Sofya Raskhodnikova. Transitive-closure spanners: A survey. In Oded Goldreich, editor,\nProperty Testing - Current Research and Surveys , volume 6390 of Lecture Notes in Com-\nputer Science , pages 167–196. Springer, 2010. doi:10.1007/978-3-642-16367-8\\_10 .\n[12] Y. Shavitt and T. Tankel. On the curvature of the internet and its usage for overlay\nconstruction and distance estimation. In IEEE INFOCOM 2004 , volume 1, page 384,\n2004. doi:10.1109/INFCOM.2004.1354510 .\n[13] Mikkel Thorup. Parallel shortcutting of rooted trees. J. Algorithms , 23(1):139–159, 1997.\ndoi:10.1006/jagm.1996.0829 .\n20"    },    {        "title": "2402.14607v1.Real_Time_Seedless_Post_Processing_for_Quantum_Random_Number_Generators.pdf",        "content": "Real-Time Seedless Post-Processing for Quantum Random Number Generators\nQian Li1and Hongyi Zhou2,∗\n1Shenzhen lnternational Center For Industrial And Applied Mathematics,\nShenzhen Research Institute of Big Data, Shenzhen, China.\n2State Key Lab of Processors, Institute of Computing Technology,\nChinese Academy of Sciences, 100190, Beijing, China.\nQuantum-proof randomness extraction is essential for handling quantum side information possessed by a\nquantum adversary, which is widely applied in various quantum cryptography tasks. In this study, we introduce\na real-time two-source quantum randomness extractor against quantum side information. Our extractor is\ntailored for forward block sources, a novel category of min-entropy sources introduced in this work. These\nsources retain the flexibility to accommodate a broad range of quantum random number generators. Our\nonline algorithms demonstrate the extraction of a constant fraction of min-entropy from two infinitely long\nindependent forward block sources. Moreover, our extractor is inherently block-wise parallelizable, presenting\na practical and efficient solution for the timely extraction of high-quality randomness. Applying our extractors\nto the raw data of one of the most commonly used quantum random number generators, we achieve a simulated\nextraction speed as high as 64 Gbps.\nKeywords: Quantum random number generators; quantum adversary; online two-source extractors; constant\nextraction rate; infinitely long sources.\nI. INTRODUCTION\nRandomness extraction is a fundamental concept in information theory and cryptography [ 1], playing a key role\nin various applications, from secure communication to cryptographic key generation. Traditionally, randomness\nextractors have been designed to efficiently distill uniform randomness from weak-randomness sources against\nclassical side information. However, in the era of quantum technology, the threat posed by quantum adversaries has\nattracted lots of attention. A quantum adversary has a quantum memory that shares quantum correlations with the\nrandomness source. The information can be extracted by performing various measurements on the quantum memory,\nwhich is called quantum side information. An extractor dealing with classical side information may fail in the case of\nquantum side information [ 2]. Therefore, the need for quantum-proof randomness extractors that are able to provide\ninformation-theoretic security in the presence of a quantum adversary [3] has emerged as a critical concern.\nIt has been shown that no deterministic extractor can extract even one bit of randomness from one general\nweak-randomness source with large min entropy, even not from a Santha-Vazirani source [ 4] that is more structural.\nTherefore, two main approaches to extracting randomness from weak-randomness sources have been explored: seeded\nextractors and two-source extractors. Seeded extractors rely on a relatively short sequence of random numbers called\nrandom seed, which may not be uniform [ 5] but should be truly random. During the extraction procedure, the seed\nmust be periodically updated to maintain randomness and security, even for strong extractors where the seed can be\nreused [ 6]. Managing and updating the seed securely can be a logistical challenge, especially in long-term deployments\nor scenarios where the system may be physically tampered with. On the other hand, two-source extractors leverage\ntwo independent weak-randomness sources to extract high-quality random bits, which avoids the issues above.\nSubstantial researches have been dedicated to two-source extractors. The first explicit construction of a two-source\nextractor appeared in [ 7], where the authors proposed that randomness can be extracted from two independent sources\nwith an entropy rate of at least 1/2based on the Lindsey’s lemma. The following researches mainly aim at relaxing\nthe entropy rate requirement [ 8–13]. In 2016, Chattopadhyay and Zuckerman [ 13] provided an explicit two-source\nextractor that works for two independent sources of 𝑛bits with an entropy rate of log𝐶𝑛/𝑛for a large enough\nconstant𝐶, which is nearly optimal. The two-source extractor against quantum adversaries was first considered in\n[14], which shows the two-source extractor proposed in [ 15] is still secure if the quantum side information about the\ntwo sources is in the product form. This result was further improved in [ 16,17] for more general quantum adversaries.\n∗zhouhongyi@ict.ac.cnarXiv:2402.14607v1  [quant-ph]  29 Jan 2024The existing two-source extractors against quantum adversary either output just one bit or suffer from limitations\non the real-time applicability, i.e., the extraction procedure cannot begin until all raw data are generated. The demand\nfor such real-time extractors arises from the fact that the generation rate of random bits is constrained by the\npost-processing speed [ 18,19], especially for trusted-device quantum random number generators [ 20], which can\nhinder the timely extraction of high-quality randomness for applications like quantum key distribution. In response to\nthis challenge, there is a pressing demand for the development of online algorithms capable of extracting randomness\nin real time. Besides, even worse, we are not aware of any existing quantum-proof two-source extractor that can\nextract a constant fraction of min-entropy from the two independent sources.\nIn this research, we propose two real-time two-source quantum-proof randomness extractors. In contrast with\nconventional quantum-proof randomness extractors characterizing the input raw data as min-entropy sources, we find\nthat a large class of quantum random number generators, including one of the most commonly used trusted-device\nquantum random number generators [ 21], can be characterized by the so-called forward block sources [22]. This\ninsight allows us to develop more efficient online extraction algorithms. Both of our extractors process input raw\ndata on-the-fly, partitioning it into blocks sequentially based on the time order of arrival: they employ the inner\nproduct function in specific finite fields on each block. In our first extractor, all blocks share a uniform length that\nis logarithmic in the overall input raw data length. This property makes the first extractor hardware-friendly and\nachieve a high extraction speed. While our second extractor utilizes incremental block lengths, rendering it capable\nof handling infinitely long input raw data. Both extractors can extract a constant fraction of min-entropy from the\nraw data. Additionally, the independence of applying the inner product function to each block facilitates natural\nparallel implementation of our extractors.\nII. PRELIMINARIES\nThroughout the paper, we use capital and lowercase letters (e.g., 𝑋and𝑥) to represent random variables and their\nassignments, respectively. We use 𝑈𝑚to represent the perfectly uniform random variable on 𝑚-bit strings and 𝜌𝑈𝑚to\nrepresent the 𝑚-dimensional maximally mixed state. The terms “quantum adversary” and “quantum side information”\nwill be used interchangeably.\nDefinition II.1 (Conditional min-entropy) .Let𝑌be a classical random variable that takes value 𝑦with probability 𝑃𝑦\nandEbe a quantum system. The state of the composite system can be written as 𝜌𝑌E=Í\n𝑦𝑃𝑦|𝑦⟩⟨𝑦|⊗𝜌𝑦\nE, where{|𝑦⟩}𝑦\nforms an orthonormal basis. The conditional min-entropy of 𝑌givenEis𝐻min(𝑌|E)𝜌𝑌E=−log2𝑝guess(𝑌|E), where\n𝑝guess(𝑌|E)is the maximum average probability of guessing 𝑌given the quantum system E. That is,\n𝑝guess(𝑌|E)=max\n{𝐸𝑦\nE}𝑦\"∑︁\n𝑦𝑃𝑦Tr\u0010\n𝐸𝑦\nE𝜌𝑦\nE\u0011#\n, (1)\nwhere the maximization is taken over all positive operator-valued measures (POVMs) {𝐸𝑦\nE}𝑦onE. When𝜌𝑌Eis clear\nfrom the context, we will denote the conditional min-entropy as 𝐻min(𝑌|E)for brevity.\nIn this paper, we call the raw data generated by a QRNG a random source. A general random source is the\nmin-entropy source, where the conditional min-entropy is lower bounded. We consider how to extract randomness\nfrom two sequences of raw data generated by two separated QRNGs. Here, we assume that the two separated QRNGs\nare in product, then the quantum side information is also in the product form. This scenario corresponds to two\nindependent quantum adversaries, each tampering with one of the sources, and then jointly trying to guess the\nextractor’s output. Formally, we have the following definition.\nDefinition II.2 (Product quantum side information [ 14]).Let𝑋and𝑌be two independent sources and Ebe the\nquantum side information. We say that the quantum side information is product if the joint state 𝜌𝑋𝑌E=𝜌𝑋E1⊗𝜌𝑌E2.\nNote that then 𝐻min(𝑋|E)=𝐻min(𝑋|E1)and𝐻min(𝑌|E)=𝐻min(𝑌|E2). .\nDefinition II.3 (Min-entropy quantum-proof two-source extractor) .A function Ext :{0,1}𝑡×{0,1}𝑡→{0,1}𝑚\nis a(𝑡,𝑘,𝜖)min-entropy quantum-proof two-source extractor, if for every sources (𝑋,𝑌)and product quantum side\ninformationEwhere𝐻min(𝑋|E)≥𝑘and𝐻min(𝑌|E)≥𝑘, we have\n1\n2∥𝜌Ext(𝑋,𝑌)E−𝜌𝑈𝑚⊗𝜌E∥≤𝜖, (2)\n2where∥·∥ denotes the trace norm defined by ∥𝐴∥=Tr√\n𝐴†𝐴.\nIII. MAIN RESULT\nWe use𝑋=𝑋1𝑋2···𝑋𝑁∈({0,1}𝑏)𝑁and𝑌=𝑌1𝑌2···𝑌𝑁∈({0,1}𝑏)𝑁to denote the raw data generated by two\nQRNGs, both consisting of 𝑁samples each of 𝑏bits. For a set 𝐼⊂N+, we write𝑋𝐼for the restriction of 𝑋to the\nsamples determined by 𝐼. For example, if 𝐼={1,3,7}, then𝑋𝐼=𝑋1𝑋3𝑋7. We useEto denote the quantum system\npossessed by the quantum adversary.\nA. Model the raw data as forward block sources\nIn contrast with conventional quantum-proof randomness extractors characterizing the input raw data as min-\nentropy sources, we model the input raw data as forward block sources , defined as follows.\nDefinition III.1 (Forward block source) .A string of random variables 𝑋=𝑋1···𝑋𝑁∈({0,1}𝑏)𝑁is a(𝑏,𝑁,𝛿)-forward\nblock source given a quantum system Eif for every 1≤𝑘≤𝑖≤𝑁and every𝑥1,𝑥2,···,𝑥𝑘−1,\n𝐻min(𝑋𝑘,𝑋𝑘+1,···,𝑋𝑖|𝑋1=𝑥1,···,𝑋𝑘−1=𝑥𝑘−1,E)≥(𝑖−𝑘+1)𝛿·𝑏. (3)\nIntuitively, Eq. (3)implies each sample 𝑋𝑖brings at least 𝛿𝑏new randomness. The notion of forward block sources is\nmore special than the min-entropy sources, so it is possible to design extractors with better performance. Fortunately,\nthe notion of forward block sources is still quite general so that it can characterize the input raw data generated\nby a large class of QRNGs. When the independent and identically distributed (i.i.d.) assumption holds, the raw\ndata straightforwardly satisfies the forward block source. For example, the trusted-devices QRNGs based on single\nphoton detection [ 23–25], vacuum fluctuations [ 26–28], and photon arrival time [ 29–31] can generate independent\nraw data satisfying the forward block source. Besides, the raw data of the (semi-)device-independent QRNG under\nthe independent and identically distributed (i.i.d.) assumption also satisfies the forward block source if non-zero\nmin-entropy lower bound is calculated in test rounds. Moreover, for QRNGs with correlated raw data, one can\nconstruct appropriate physical models to check whether Eq. (3)is satisfied. For example, the ones based on homodyne\ndetection [ 21] with finite bandwidth can both be shown to be a forward block source [ 22]. Finally, we remark that\nthere does not exist any nontrivial deterministic extractor for a single block source.\nB. Gadget: a construction of min-entropy quantum-proof two-source extractor\nIn this subsection, we provide a construction of min-entropy quantum-proof two-source extractor (see Definition\nIII.2), which will be used as a gadget. Let F2𝑞denote the finite field on 2𝑞elements. We identify F2𝑞with{0,1}𝑞and\nF𝑛\n2𝑞with({0,1}𝑞)𝑛where the addition operation “+𝑞”in the field F2𝑞is the bit-wise parity operation in {0,1}𝑞.\nDefinition III.2 (Inner-product extractor) .The(𝑛,𝑞)inner-product extractor Ext𝑛,𝑞\n𝐼𝑃:({0,1}𝑞)𝑛×({0,1}𝑞)𝑛→{0,1}𝑞\nis defined as Ext𝑛,𝑞\n𝐼𝑃(𝑥,𝑦)=Í𝑛\n𝑖=1𝑥𝑖𝑦𝑖. Here,𝑥=𝑥1···𝑥𝑛∈({0,1}𝑞)𝑛and𝑦=𝑦1···𝑦𝑛∈({0,1}𝑞)𝑛, and the addition\noperations and multiplication operations are in the field F2𝑞.\nTheorem III.3. Ext𝑛,𝑞\n𝐼𝑃is a(𝑞𝑛,𝛿𝑞𝑛,√\n3·2−1\n4−(𝛿\n4−1\n8)𝑞𝑛+2𝑞)min-entropy quantum-proof two-source extractor.\nTheorem III.3 immediately follows from the following two lemmas, namely Lemma III.4 and Lemma III.5.\nLemma III.4 (Classical-Quantum XOR-Lemma [ 14]).Let𝑍be a classical random variable of 𝑞bits, andEbe a quantum\nsystem. Then\n∥𝜌𝑍E−𝜌𝑈𝑞⊗𝜌E∥≤2𝑞∑︁\n𝑆∈{0,1}𝑞\\{0𝑞}∥𝜌(𝑆·𝑍)◦E−𝜌𝑈1⊗𝜌E∥.\nHere𝑆·𝑍:=Í𝑛\n𝑖=1𝑆𝑖·𝑍𝑖(mod 2) is the inner product over F2.\n3The following lemma will also be used, whose proof can be found in Appendix A.\nLemma III.5. Let𝑋,𝑌 be random variables on ({0,1}𝑞)𝑛andE=E1⊗E2be the product quantum side information\nwhere𝐻min(𝑋|E1)≥𝛿𝑞𝑛and𝐻min(𝑌|E2)≥𝛿𝑞𝑛. Let𝑍=Ext𝑛,𝑞\n𝐼𝑃(𝑋,𝑌). Then for any 𝑆∈{0,1}𝑞\\{0𝑞}, we have\n1\n2∥𝜌(𝑆·𝑍)◦E−𝜌𝑈1⊗𝜌E∥≤√\n3·2−1\n4−(𝛿\n4−1\n8)𝑞𝑛.\nC. Online Seedless Extractors for forward block sources\nIn this subsection, we design two online seedless quantum-proof extractors that can both extract a constant fraction\nof the min-entropy from forward block sources. The two extractors both proceed in the following fashion: partition\nthe input raw data 𝑋,𝑌 into blocks, and apply the inner-product extractor Ext𝐼𝑃to each block separately. Thus the\ntwo extractors can be paralleled naturally. The two extractors are described in Algorithms 1 and 2 respectively. Given\nthe input raw data 𝑋,𝑌 each of length 𝑁, the first extractor, named Exteq\nbs, evenly partitions the input raw data into\nblocks each of size 𝑂(log𝑁), and the second extractor, named Extneq\nbs, requires incremental block lengths. Compared\nto the first extractor, the second one is less hardware-friendly and sacrifices the extraction speed in general. On the\nother hand, it enjoys the property that it can deal with infinite raw data, without the need to determine the raw data\nlength prior to extraction.\nAlgorithm 1: Exteq\nbs\n1Input : Two independent (𝑏,𝑁,𝛿)-forward block sources 𝑋=𝑋1···𝑋𝑁and𝑌=𝑌1···𝑌𝑁. And 0<𝜖<1;\n2Let𝑛:=\u000624\n2𝛿−1\u0007\n,𝑞:=𝑏l\nlog2\u0010\n𝑁\n𝜖𝑛\u0011\n/𝑏m\n, and𝑖:=1;\n3forℓ=1to𝑁𝑏/𝑞𝑛do\n4 Let𝐼ℓ:=[𝑖,𝑖−1+𝑞𝑛/𝑏];\n5 Compute𝑍(ℓ):=Ext𝑛,𝑞\n𝐼𝑃(𝑋𝐼ℓ,𝑌𝐼ℓ);\n6 Let𝑖:=𝑖+𝑛;\n7 Output𝑍(ℓ).\nAlgorithm 2: Extneq\nbs\n1Input : Two(𝑏,∞,𝛿)-forward block sources 𝑋=𝑋1···𝑋∞and𝑌=𝑌1···𝑌∞independent conditioned on quantum\nadversaryE;\n2Parameter: 𝑞1,Δ∈N+;\n3Let𝑖:=1and𝑛:=\u000624\n2𝛿−1\u0007\n;\n4forℓ=1to∞do\n5 Let𝐼ℓ:=[𝑖,𝑖−1+𝑞ℓ𝑛/𝑏];\n6 Compute𝑍(ℓ):=Ext𝑛,𝑞ℓ\n𝐼𝑃(𝑋𝐼ℓ,𝑌𝐼ℓ);\n7 Let𝑖:=𝑖+𝑞ℓ𝑛/𝑏and𝑞ℓ+1:=𝑞ℓ+Δ·𝑏;\n8 Output𝑍(ℓ).\nNote that if we impose Δ=0and let𝑞1=𝑏\u0006\nlog2\u0000𝑁\n𝜖𝑛\u0001/𝑏\u0007\n, then Extneq\nbsbecomes Exteq\nbs. We first analyze the second\nextractor Extneq\nbs.\nTheorem III.6. The extractor Extneq\nbssatisfies the following properties. For any 𝑘∈N,\n1\n2∥𝜌𝑍(1)◦𝑍(2)◦···◦𝑍(𝑘)E−𝜌𝑈𝑚𝑘⊗𝜌E∥≤√\n3·𝑘∑︁\nℓ=12−1\n4−(𝛿\n4−1\n8)𝑞ℓ𝑛+2𝑞ℓ≤√\n3·𝑘∑︁\nℓ=12−1\n4−𝑞ℓ<√\n3·2−1\n4−𝑞1\n1−2−Δ𝑏, (4)\nwhere𝑚���=Í𝑘\nℓ=1𝑞ℓ=𝑘𝑞1+(𝑘−1)𝑘Δ𝑏\n2.\n4FIG. 1: Illustration of Extneq\nbs.\nProof. Fix𝑘∈N. For convenience of presentation, for any 0≤ℓ≤𝑘, we use𝑍>ℓto represent 𝑍(ℓ+1)◦𝑍(ℓ+2)◦···◦𝑍(𝑘),\n𝑋≤ℓto represent 𝑋𝐼1𝑋𝐼2···𝑋𝐼ℓ, and𝑌≤ℓto represent 𝑌𝐼1𝑌𝐼2···𝑌𝐼ℓ. Besides, let ˜𝑞ℓ:=𝑞ℓ+1+𝑞ℓ+2+···+𝑞𝑘.\nIn fact, we will prove that for any 0≤ℓ≤𝑘, it has\n1\n2\r\r𝜌𝑍>ℓ𝑋≤ℓ𝑌≤ℓE−𝜌𝑈˜𝑞ℓ⊗𝜌𝑋≤ℓ𝑌≤ℓE\r\r≤√\n3·𝑘∑︁\n𝑗=ℓ+12−1\n4−(𝛿\n4−1\n8)𝑞𝑗𝑛+2𝑞𝑗, (5)\nwhich implies the theorem immediately by letting ℓ=0.\nThe proof is by an induction on ℓ. The base case when ℓ=𝑘is trivial. The induction proceeds as follows. Assume\nEq. (5) is true. Then due to the contractivity of trace-preserving quantum operations, we have\n1\n2\r\r𝜌𝑍>ℓExt𝐼𝑃(𝑋𝐼ℓ,𝑌𝐼ℓ)𝑋≤ℓ−1𝑌≤ℓ−1E−𝜌𝑈˜𝑞ℓ⊗𝜌Ext𝐼𝑃(𝑋𝐼ℓ,𝑌𝐼ℓ)𝑋≤ℓ−1𝑌≤ℓ−1E\r\r≤√\n3·𝑘∑︁\n𝑗=ℓ+12−1\n4−(𝛿\n4−1\n8)𝑞𝑗𝑛+2𝑞𝑗. (6)\nOn the other hand, by Definition III.1, for any assignment 𝑥≤ℓ−1of𝑋≤ℓ−1and𝑦≤ℓ−1of𝑌≤ℓ−1, we have\n𝐻min(𝑋𝐼ℓ|𝑋≤ℓ−1=𝑥≤ℓ−1,E)≥𝛿𝑞ℓ𝑛,and𝐻min(𝑌𝐼ℓ|𝑌≤ℓ−1=𝑦≤ℓ−1,E)≥𝛿𝑞ℓ𝑛. (7)\nThen, recalling that Ext𝐼𝑃is a min-entropy quantum-proof two-source extractor (Theorem III.3), it follows that\n1\n2\r\r\r𝜌Ext𝐼𝑃(𝑋𝐼ℓ,𝑌𝐼ℓ)𝑋≤ℓ−1𝑌≤ℓ−1E−𝜌𝑈𝑞ℓ⊗𝜌𝑋≤ℓ−1𝑌≤ℓ−1E\r\r\r≤√\n3·2−1\n4−(𝛿\n4−1\n8)𝑞ℓ𝑛+2𝑞ℓ.\nThus,\n1\n2\r\r𝜌𝑈˜𝑞ℓ⊗𝜌Ext𝐼𝑃(𝑋𝐼ℓ,𝑌𝐼ℓ)𝑋≤ℓ−1𝑌≤ℓ−1E−𝜌𝑈˜𝑞ℓ⊗𝜌𝑈𝑞ℓ⊗𝜌𝑋≤ℓ−1𝑌≤ℓ−1E\r\r≤√\n3·2−1\n4−(𝛿\n4−1\n8)𝑞ℓ𝑛+2𝑞ℓ. (8)\nFinally, combining inequalities (6) and (8) and applying the triangle inequality, we conclude that\n1\n2\r\r𝜌𝑍>ℓExt𝐼𝑃(𝑋𝐼ℓ,𝑌𝐼ℓ)𝑋≤ℓ−1𝑌≤ℓ−1E−𝜌𝑈˜𝑞ℓ⊗𝜌𝑈𝑞ℓ⊗𝜌𝑋≤ℓ−1𝑌≤ℓ−1E\r\r≤√\n3·𝑘∑︁\n𝑗=ℓ2−1\n4−(𝛿\n4−1\n8)𝑞𝑗𝑛+2𝑞𝑗(9)\nThat is,\n1\n2\r\r𝜌𝑍>ℓ−1𝑋≤ℓ−1𝑌≤ℓ−1E−𝜌𝑈˜𝑞ℓ−1⊗𝜌𝑋≤ℓ−1𝑌≤ℓ−1E\r\r≤√\n3·𝑘∑︁\n𝑗=ℓ2−1\n4−(𝛿\n4−1\n8)𝑞𝑗𝑛+2𝑞𝑗,\nSo Eq. (5) also holds for ℓ−1. The proof is finished. □\n5As can be seen from the proof, to extract infinitely long randomness, the parameter ΔofExtneq\nbsmust be strictly\npositive, since the upper bound√\n3·Í𝑘\nℓ=12−1\n4−(𝛿\n4−1\n8)𝑞ℓ𝑛+2𝑞ℓon the error converges if and only if Δ>0. Moreover,\nthe infinitely long output string 𝑍(1)◦𝑍(2)◦··· can be arbitrarily close to the uniform distribution by choosing a\nsufficiently large constant 𝑞1. In addition, Extneq\nbsextracts𝑞ℓ\n2𝛿𝑞ℓ𝑛≈2𝛿−1\n48𝛿fraction of the min-entropy of the raw data\n𝑋𝐼ℓ,𝑌𝐼ℓ. Finally, we remark there is a tradeoff between the error and the computational complexity by setting Δ. AsΔ\nincreases, the error becomes smaller, but the size of block increases, which leads to heavy computations.\nWe intuitively explain Theorem III.6, i.e., how our algorithms work for the forward block sources. When the input\nraw data𝑋,𝑌 are forward block sources, the block 𝑋𝐼ℓand𝑌𝐼ℓare independent and each has large min-entropy\nconditioned on any assignment (𝑥<ℓ,𝑦<ℓ)of previous blocks (𝑋<ℓ,𝑌<ℓ), which implies that the output 𝑍(ℓ)of this\nblock is (approximately) uniform conditioned on any assignment (𝑥<ℓ,𝑦<ℓ). Note that𝑧<ℓcan be fully determined by\n(𝑥<ℓ,𝑦<ℓ). So𝑍(ℓ)is (approximately) uniform conditioned on any assignment 𝑧<ℓof𝑍<ℓ. By induction on ℓ, we can\nconclude that the whole output 𝑍(1)◦···◦𝑍(𝑘)◦··· is (approximately) uniform.\nAs a corollary of Theorem III.6, we have the following result for Exteq\nbs.\nTheorem III.7. The output𝑍(1)◦···◦𝑍(𝑁𝑏/𝑞𝑛)ofExteq\nbssatisfies that\n1\n2∥𝜌𝑍(1)◦𝑍(2)◦···◦𝑍(𝑁𝑏/𝑞𝑛)◦E−𝜌𝑈𝑚⊗𝜌E∥≤𝜖, (10)\nwhere𝑚:=𝑁𝑏\n𝑛.\nIV. SIMULATIONS OF THE REAL-TIME RANDOMNESS GENERATION RATE\nIn this section, we provide an analysis of the computational efficiency of our extractors. For simplicity of discussion,\nwe focus on the first extractor Exteq\nbs.\nA. Asymtoptic analysis.\nSuppose we apply Exteq\nbson two independent (𝑏,𝑁,𝛿)-forward block sources. The extractor will set 𝑛=𝑂(1)\nand𝑞=𝑂(log𝑁), and then compute the inner product function Ext𝑛,𝑞\n𝐼𝑃for𝑁𝑏/𝑞𝑛=𝑂(𝑁𝑏/log𝑁)times. The\ncomputation of Ext𝑛,𝑞\n𝐼𝑃involves(𝑛−1)additions ‘+𝑞’ and𝑛multiplications ‘·𝑞’ over the finite field F2𝑞. The addition\n‘+𝑞’ is the bit-wise parity in {0,1}𝑞and thus can be implemented by 𝑞parity gates. The multiplication ‘ ·𝑞’ can be\nimplemented by a 𝑂(𝑞log𝑞log log𝑞)-size Boolean circuit (see the last inequality in Section 2 and the third paragraph\non Page 3 in [ 32]). So Ext𝑛,𝑞\n𝐼𝑃can be computed by a 𝑂(𝑛𝑞log𝑞log log𝑞)-size Boolean circuit. Therefore, it requires\n𝑂(𝑁𝑏/log𝑁)·𝑂(𝑛𝑞log𝑞log log𝑞)=𝑂(𝑁𝑏log log𝑁log log log𝑁)logic operations in total to execute Exteq\nbs.\nWe compare the computational complexity of Exteq\nbswith the seeded extractor proposed in [ 22], which evenly\npartitions the input raw data into blocks each of size 𝑂(log𝑁)and applies the Toeplitz-hashing extractor to each\nblock separately with the same seed of 𝑂(log𝑁)bits. Note that it needs 𝑂(log2𝑁)logic operations to multiply a\n𝑂(log𝑁)×𝑂(log𝑁)dimensional Toeplitz matrix with a 𝑂(log𝑁)dimensional vector. Thus, it requires 𝑂(𝑁𝑏/log𝑁)·\n𝑂(log2𝑁)=𝑂(𝑁𝑏log𝑁)logic operations in total to execute the seeded extractor. So Exteq\nbsis asymptotically faster\nthan the seeded extractor.\nB. Simulations.\nIn the following, we make a simulation estimating the extraction speed of the first extractor Exteq\nbsimplemented\nin the Xilinx Kintex-7 XC7K480T Field Programmable Gate Array (FPGA), a common application in industry. We\nuse the raw data from Ref. [ 21] as the random source, which is a forward block source with parameters 𝑏=16and\n𝛿=10.74/16≈0.67. We consider a raw data length of 𝑁=251bits and a total security parameter 𝜖=2−30, which\nmeans the final output data is 2−30-close to a uniform distribution. Correspondingly, the algorithmic parameters 𝑛,𝑞\nofExteq\nbsare specified to 𝑛=⌈24/(2𝛿−1)⌉=71and𝑞:=𝑏⌈log2(𝑁/𝜖𝑛)/𝑏⌉=80. By Theorem III.7, Exteq\nbsuses no\nseed and will output about 0.45PB random bits.\n6Precisely, Exteq\nbssimply divides the input raw data 𝑋,𝑌 into blocks each containing 71×80bits, and then apply\nExt71,80\n𝐼𝑃on each block(𝑋𝐼ℓ,𝑌𝐼ℓ). The inner product function Ext71,80\n𝐼𝑃involves 71−1=70addition operations and 71\nmultiplication operations in the field F280. The addition operation in F280is the bit-wise parity operation in {0,1}80,\nand so can be computed in 80‘⊕’ operations. Each multiplication operation in F280can be computed by at most 4885\n‘⊕’ or ‘∧’ operations [ 32]. Thus, each Ext71,80\n𝐼𝑃can be implemented by at most 80×(71−1)+4885×71=352435\n‘⊕’ or ‘∧’ operations.\nThe parameters of the Xilinx Kintex-7 XC7K480T FPGA are as follows. The clock rate is set to be 200MHz; the\nnumber of Look-Up-Tables (LUTs) is 3×105; each LUT can perform 5basic logical operations (e.g., ‘ ⊕’ or ‘∧’)\nsimultaneously. To make full use of the FPGA, we can perform the matrix multiplications of ⌊3×105×5/352435⌋=4\nblocks in parallel. Therefore, the extraction speed of Exteq\nbsis200×106×4×80=64Gbps, which is significantly\nimproved by one order of magnitude compared to the state-of-the-art result with a speed of 18.8 Gbps [ 19]. 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Wu, H.-Q. Ma, and G.-J. Zhai, Journal of the Optical Society of America A 30, 124 (2013).\n[31] Y.-Q. Nie, H.-F. Zhang, Z. Zhang, J. Wang, X. Ma, J. Zhang, and J.-W. Pan, Applied Physics Letters 104, 051110 (2014).\n[32] J. Rif `a and J. Borrell, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 9th International Symposium,\nAAECC-9, New Orleans, LA, USA, October 7-11, 1991, Proceedings , Lecture Notes in Computer Science, Vol. 539, edited by H. F.\nMattson, T. Mora, and T. R. N. Rao (Springer, 1991) pp. 352–359.\nAppendix A: Proof of Lemma III.5\nIn this section, we prove Lemma III.5. Let 𝑋,𝑌 be random variables on ({0,1}𝑞)𝑛andE=E1⊗E2be the\nproduct quantum side information where 𝐻min(𝑋|E1)≥𝛿𝑞𝑛and𝐻min(𝑌|E2)≥𝛿𝑞𝑛. Let𝑍=Ext𝑛,𝑞\n𝐼𝑃(𝑋,𝑌), and fix\n𝑆∈{0,1}𝑞\\{0𝑞}arbitrarily. We will use “ ×” to denote the multiplication operation in the field F2𝑞.\nNote that for any 𝑆∈{0,1}𝑞, there exists a 𝑎∈F2𝑞such that𝑆·𝑍corresponds exactly to the first bit of 𝑎×𝑍\nviewed as a vector in {0,1}𝑞. Moreover this correspondence between 𝑆and𝑎is a bijection in the sense that different\n𝑆∈{0,1}𝑞correspond to different 𝑎∈F2���. The special case when 𝑆=0𝑞corresponds to the case 𝑎=0. Thus, for any\n𝑆∈{0,1}𝑞\\{0𝑞}, there exists some non-zero 𝑎∈F2𝑞such that𝑆·𝑍=[𝑎×𝑍]1. Here,[𝑎×𝑍]1is used to denote the\nfirst bit of𝑎×𝑍viewed as a vector in {0,1}𝑞.\nFirst, we show that the function [𝑎×Ext𝑛,𝑞\n𝐼𝑃(·,·)]1is a hadamard function.\nDefinition A.1 (hadamard function) .A function𝑓:{0,1}𝑡×{0,1}𝑡→{0,1}is called a hadamard function if for any\ntwo distinct 𝑥,𝑥′∈{0,1}𝑡, it hasÍ\n𝑦∈{0,1}𝑡(−1)𝑓(𝑥,𝑦)+𝑓(𝑥′,𝑦)=0.\nProposition A.2. For any non-zero 𝑎∈F2𝑞,[𝑎×Ext𝑛,𝑞\n𝐼𝑃(·,·)]1is a hadamard function.\nProof. For simplicity of notations, we let 𝑓𝑎(𝑥,𝑦):=[𝑎×Ext𝑛,𝑞\n𝐼𝑃(·,·)]1. Given any two distinct 𝑥=𝑥1···𝑥𝑛,𝑥′=\n𝑥′\n1···,𝑥′\n𝑛∈({0,1}𝑞)𝑛, w.l.o.g. we assume 𝑥1≠𝑥′\n1. Recalling that the addition operation “+𝑞”in the field F2𝑞\ncorresponds to the bitwise-parity operation in {0,1}𝑞, we have\n∑︁\n𝑦∈({0,1}𝑞)𝑛(−1)𝑓𝑎(𝑥,𝑦)+𝑓𝑎(𝑥′,𝑦)=∑︁\n𝑦∈({0,1}𝑞)𝑛(−1)𝑓𝑎(𝑥,𝑦)−𝑓𝑎(𝑥′,𝑦)=∑︁\n𝑦∈({0,1}𝑞)𝑛(−1)𝑓𝑎(𝑥−𝑥′,𝑦)\n=∑︁\n𝑦∈({0,1}𝑞)𝑛(−1)[𝑎×Ext𝑛,𝑞\n𝐼𝑃(𝑥−𝑥′,𝑦)]1=∑︁\n𝑦∈({0,1}𝑞)𝑛(−1)[𝑎×(𝑥1−𝑥′\n1)×𝑦1]1+[Í𝑛\n𝑖=2𝑎×(𝑥𝑖−𝑥′\n𝑖)×𝑦𝑖]1\n=©\n«∑︁\n𝑦1∈{0,1}𝑞(−1)[𝑎×(𝑥1−𝑥′\n1)×𝑦1]1ª®\n¬·©\n«∑︁\n𝑦2,···,𝑦𝑛∈{0,1}𝑞(−1)[Í𝑛\n𝑖=2𝑎×(𝑥𝑖−𝑥′\n𝑖)×𝑦𝑖]1ª®\n¬.\n8Note that𝑎×(𝑥1−𝑥′\n1)≠0, so\n\f\f\b\n𝑦1∈{0,1}𝑞|\u0002\n𝑎×(𝑥1−𝑥′\n1)×𝑦1\u0003\n1=1\t\f\f=\f\f\b\n𝑦1∈{0,1}𝑞|\u0002\n𝑎×(𝑥1−𝑥′\n1)×𝑦1\u0003\n1=0\t\f\f=2𝑞−1.\nThus we get the conclusion immediately by Definition A.1. □\nDefinition A.3 (One-bit-output two-source extractor) .A function𝑓:{0,1}𝑡×{0,1}𝑡→{0,1}is a(𝑡,𝑘,𝜖)one-bit-\noutput two-source extractor, if for every pair of independent sources (𝑋,𝑌)with min-entropy≥𝑘each, we have\n(1−𝜖)/2≤𝑃[𝑓(𝑋,𝑌)=1]≤(1+𝜖)/2.\nLemma A.4 ([7]).Let𝑓:{0,1}𝑡×{0,1}𝑡→{0,1}be a hadamard function. Then 𝑓is a(𝑡,𝑘,21−2𝑘−𝑡\n2)one-bit-output\ntwo-source extractor for any 𝑘≤𝑡.\nIn other words, let 𝑋,𝑌 be independent random variables on {0,1}𝑡with min-entropy≥𝑘each. Then(1−𝜖)/2≤\n𝑃[𝑓(𝑋,𝑌)=1]≤(1+𝜖)/2where𝜖=21−2𝑘−𝑡\n2.\nSo by the Lemma A.4, we know that the function [𝑎×Ext𝑛,𝑞\n𝐼𝑃(·,·)]1is a(𝑞𝑛,𝑘, 21−2𝑘−𝑞𝑛\n2)one-bit-output two-source\nextractor for any 𝑘≤𝑞𝑛.\nFinally, Lemma III.5 follows from the Lemma A.5 immediately by letting 𝑘=\u0010\n1\n4+𝛿\n2\u0011\n𝑞𝑛+1\n2and𝜖=21\n2−(𝛿\n2−1\n4)𝑞𝑛.\nLemma A.5 (Corollary 27 of [ 14] for𝑚=1).Let𝑓be a(𝑡,𝑘,𝜖)one-bit-output two-source extractor. Then it is a\n(𝑡,𝑘+log𝜖−1,√︁\n3𝜖/2)-min-entropy quantum-proof two-source extractor. That is, for any random variables 𝑋,𝑌 on\n{0,1}𝑡and any product quantum side information Ewhere𝐻min(𝑋|E)≥𝑘+log𝜖−1and𝐻min(𝑌|E)≥𝑘+log𝜖−1,\nwe have1\n2∥𝜌𝑓(𝑋,𝑌)E−𝜌𝑈1⊗𝜌E∥≤√︁\n3𝜖/2.\n9"    },    {        "title": "2402.14636v1.Production_of_singly_charm_pentaquark_from_meson_baryon_interaction_in_B_decay.pdf",        "content": "arXiv:2402.14636v1  [hep-ph]  22 Feb 2024Production of singly charm pentaquark from meson-baryon in teraction in B\ndecay\nNa Li,∗Ye Xing,†and Ke-Fan Jiang‡\nSchool of Materials Science and Physics, China University o f Mining and Technology, Xuzhou 221000, China\nIn the paper, we discuss the possible interpretation of the JP= 1/2−singly charm pen-\ntaquark as hadronic molecules. With the effective Lagrangian metho d, we further analyze\nthe production properties of singly charm pentaquark from decay s ofBmeson, including\nstrong coupling constants and production branching ratios of the charm pentaquark. Our\nnumerical results show that the branching ratio of production of p entaquark from B meson\ncan reach to order of 10−6.\nI. INTRODUCTION\nRecently, there is resurgent interest in terms of pentaquar k, triggered by the progress in\nthe experiment. Since 2015, LHCb collaboration has claimed the discoveries of different pen-\ntaquark candidates, hidden charm pentaquark Pc(4380),Pc(4312),Pc(4440),Pc(4457),Pcs(4459)\nandPc(4337) [1–4]. The observations stimulate extensive theore tical studies of pentaquark struc-\ntures using different scenarios and frameworks [5–19]. In par ticular, the composite molecular-state\ninterpretation is quite consistent with the LHCb observati on. In Ref [20–24], the authors found\nPc(4440),Pc(4457) and Pc(4380) can be well interpreted as ¯D(∗)Σ(∗)\ncmolecules, In Ref [25–29],\nPc(4459) and Pc(4337) are believed to be ¯D∗Ξcand Ξc¯Dmoleculars respectively. Nevertheless,\nthere are still many other possible proposals to understand the origin of these states, such as com-\npact pentaquark states [30–32] and kinematical effects [15, 3 3, 34]. Upon the assumptions, the\nproperties of the pentaquark, such as the various interacti ons, are investigated with the applica-\ntion of different approaches including the effective approach [ 27, 35, 36], potential models [37–39]\nand QCD sum rules [10, 13, 17, 40]. Though there are still many controversial arguments about\nproperties of the hidden charm pentaquark. Moreover, the po ssible unobserved pentaquark states\npossessing single charm quark are also quested for with the e xpectation to be observed in future\nexperiments.\nInspired by the observation of Pc(4337) in the analysis of Bs→J/Ψp¯p, the present work will\nfocus on the production of pentaquark via B meson decay. Depe nding on the mass spectrum of\nsinglycharmpentaquarkbasedontriquark-diquarkmodel[4 1], effective Lagrangianmethod[35,36]\nand Quark Delocalization Color Screening model [42], we fou nd the masses are around thresholds\n∗Electronic address: TS22180005A31@cumt.edu.cn\n†Electronic address: Corresponding author. xingye˙guang@ cumt.edu.cn\n‡Electronic address: TS23180004A31@cumt.edu.cn2\nof pseudoscalar meson baryon, Ξ D,NDsand ΣD. In the approaches of QCD sum rules [40] and\nlight-meson exchange model [43], they are near thresholds o f vector meson baryon, Ξ D∗,ND∗\nsand\nΣD∗. So it is naturally to follow the assumption that the pentaqu ark is a meson-baryon molecule\nstate. In order to fully investigate the production of singl y charm pentaquark from Bmeson\ndecay, we will use the effective Lagrangian approach to descri be the interaction of hadrons, and\nsimultaneously discussing both molecule scenarios, inclu ding pseudoscalar meson baryon molecular\nand vector meson baryon molecular in the present considerat ion.\nThe building blocks and related evaluation strategy of effect ive Lagrangian approach to the\nhadronic molecular, can be sketched out as follows: First, a gauge invariant phenomenological La-\ngrangian is established to describes the interaction of the molecular with its constituents. In which,\nthe molecular and constituents are described by standard fie ld operators. By the compositeness\ncondition [44], an effective way that representing the probab ility to find the hadronic molecular\nas a bare state, the coupling strengths of the hadronic molec ular to its constituents can be deter-\nmined. Using interaction Lagrangian in hadronic level one c an further construct transition matrix\nelements for hadronic processes involving the hadronic mol ecular. Accordingly, the remaining step\nof the evaluation of matrix elements will completely reveal the hadronic transition process. We\nexpect our research to provide assistance for the study of si ngly charm pentaquark in experiments.\nThe paper is organized as follows. In Sec. II, we discuss the p ossible meson-baryon molecular of\npentaquark: pseudoscalar meson baryon BPand vector meson baryon BV. The coupling constant\nof pentaquark and its constituent is given in the following. Sec. III contains the details of effective\nLagrangian approach to calculate the production of singly c harmed pentaquark via B meson decay.\nWe present numerical analysis for branching ratios of the pr oduction processes in Sec. IV. Last\nsection gives a short conclusion.\nII. SINGLY CHARM PENTAQUARK\nThe singly charm pentaquark can be decomposed by SU(3) flavor symmetry to form ground\n15-plet labeled as Pc, in which the states with constituents c¯udss,c¯sudd,c¯udds,c¯duss,c¯suudare\nexclusively selected as the possible stable pentaquark can didates [41]. We arrange Pcto be S-wave\nstates always with spin-parity JP=1\n2−. In previous papers [35, 36, 40–43], the masses of Pcbased\non different assumptions of their internal structures were st udied. It is found that the JP=1\n2−\nstates with constituents c¯nssn,c¯snnnandc¯nsnn(n=u,d) in the triquark-diquark model [41],\neffective Lagrangian method [35] and Quark Delocalization Co lor Screening Model [42] are close\nto the pseudoscalar meson baryon thresholds Ξ D,NDsand ΣDrespectively, indicating existence\nof a S-wave loosely molecular with meson-baryon BPcomponents. However, under the QCD sum\nrules [40] and light-meson exchange [43], the masses of c¯snnnandc¯nsnnwithJP=1\n2−are in\nproximity to the vector meson baryon thresholds Ξ D∗,ND∗\nsand ΣD∗, suggesting the possibility\nofPcforming a meson-baryon BVmolecular. So we consider both possible channels relevant t o the3\npentaquark in the work, which explicitly expressed as pseud oscalar meson baryon BPand vector\nmeson baryon BV.\nIn general, the mass of the meson-baryon type singly charm pe ntaquark can be expressed in\nterms of the constituent meson mass mM, baryon mass mBand the binding energy ǫas,\nmPc=mM+mB−ǫ. (1)\nNote that ǫis a variable quantity in our calculation, here we vary it fro m 5 to 50 MeV. Once\nthe mass of the composite pentaquark state is fixed, the coupl ings of pentaquark to meson and\nbaryon can beextracted dependingon thecompositeness cond ition. Inthe following, wediscussthe\nconstituent of singly charm pentaquark, respectively rega rding as hadronic meson-baryon molecule\nBPof 1/2+baryon and pseudoscalar meson interaction or meson-baryon BVmolecular coupled\nby the interaction 1 /2+baryon and vector meson.\nA. Meson-baryon S1: Pcas hadronic molecule BP\nThe framework is based on an effective interaction Lagrangian describing the couplings of the\npentaquark to their constituents, the nonlocal strong Lagr angian involving JP= 1/2−pentaquark\nstate(Pc), a 1/2+baryon(B) and pseudoscalar meson( P) is expressed as follows[24]:\nLPcBP(x) =gPcBPPc(x)/integraldisplay\ndyΦ(y2)B†(x+y\n2)P(x−y\n2). (2)\nThe correlation function Φ( y2) characterize the finite size of the pentaquark formed as BPbound\nstates. A basic requirement for the choice of an explicit for m of the correlation function is that its\nFourier transform vanishes sufficiently fast in the ultravio let region of Euclidean space to render\nthe Feynman diagrams ultraviolet finite. We adopt the Gaussi an form [45, 46],\nΦ(y2) =/integraldisplayd4p\n(2π)4e−ipy/tildewideΦ(−p2) with /tildewideΦ(p2).= exp(−p2/Λ2\nPc), (3)\nwhere the usual Gaussian regulator cutoff is taken as Λ Pc= 1.0 GeV [27]. The coupling constant\ngPcBPcan be easily extracted from the Weinberg compositeness con dition [44], which implies the\nrenormalization constant ZPcof the hadron wave function is set equal to zero [47, 48]:\nZPc= 1−g2\nPcBPdΣPc(M2\nPc)\ndp //vextendsingle/vextendsingle/vextendsingle/vextendsingle\np /=mPc= 0. (4)\nThemassoperatorofpentaquarkΣ Pc(M2\nPc)canbedescribedbytheself-energydiagramofFig.1(a),\nwhich deduced as,\nΣPc(M2\nPc) =/integraldisplayd4k\n(2π)4/tildewideΦ2(−k2)(k /+p /\n2+mB)\n((k+p\n2)2−m2\nB)((k−p\n2)2−m2\nP)2, (5)4\nFollowing the renormalization constant in Eq.(4), the coup ling constant can be expressed in the\nform,\n1\ng2\nPcPB=1\n16π21/integraldisplay\n0dα∞/integraldisplay\n0dββ\n(1+β)2/tildewideΦ2(∆)\n×/parenleftBig1+2β(1−α)\n2(β+1)+βp /(β+1)−β2p /(2α−1)2\nΛ2\n1(β+1)(p /+2βp /(1−α)\n2(β+1)+mB)/parenrightBig\n,(6)\nwhere\n∆ =β(m2\nP−αm2\nP+m2\nB−p2\n4+βp2\n4+4β−4αβp2\n4+4β+4α2βp2\n4+4β). (7)\nFurther determination obtain the numerical result of the co uplings, gathered into Tab.I.\nB. Meson-baryon S2: Pcas hadronic molecule BV\nTheeffectiveinteraction Lagrangianforthecouplingsof1 /2−pentaquark( Pc), a1/2+baryon(B)\nand vector meson( V) is given as [27]:\nLPcBV(x) =gPcBVPc(x)γ5(gµν−pµpν\nm2\nPc)γν/integraldisplay\ndyΦ(y2)B†(x+y\n2)V(x−y\n2). (8)\nTheself-energy diagram of Pcastheboundstate BVisillustrated inFig.1(b). Directly thecoupling\nconstant can be obtained as:\n1\ng2\nPcBV=d\ndp //integraldisplayd4k\ni(2π)4Φ2(−k2)γν(k /+p /\n2−mB)γρ\n((k+p\n2)2−m2\nB)((k−p\n2)2−m2\nV)\n(gµν−pµpν\nm2\nPc)(gσρ−pσpρ\nm2\nPc)(−gµσ+(k−p\n2)µ(k−p\n2)σ\nm2\nV) (9)\nwherek+p\n2isthemomentumofbaryon Bandk−p\n2isthemomentumofvector meson Vrespectively.\nIII. THE PRODUCTION FROM B MESON DECAY\nUsing the effective Lagrangian method, we could further study the production properties of\npentaquarkfromBmesondecays. Thenaivetwobodyproductio nB→Pc¯Bcanbedividedintotwo\nsubprocesses under the phenomenological approach, by inse rting possible hadron complete states,\nexpressed as a weak transition matrices /angbracketleftλ|Heff|B/angbracketrightand a strong coupling matrices /angbracketleftPcB|Hλ|λ/angbracketright.\n/angbracketleftPcB|Heff|B/angbracketright=/summationdisplay\nλ/angbracketleftPcB|Hλ|λ/angbracketright/angbracketleftλ|Heff|B/angbracketright, (10)\nwhereBcan be charm anti-baryon ¯ c¯q¯qor light anti-baryon ¯ q¯q¯q. The possible intermediate hadron\nstates(λ) of charm-light mesons and charm-charm mesons ( λ=D(∗)M, D(∗)D) have the dominant5\nB\nPPc Pc\n(a)B\nVPc Pc\n(b)\nFIG. 1: The self-energy diagrams of singly charm pentaquark( Pc), respectively regarded as pseudoscalar\nmeson and baryon molecular BP(a) or vector meson baryon molecular BV(b).\ncontribution, in light of the leading factorizable process of weak decays of B meson shown in Fig.3.\nHλis the possible Lagrangian of strong interaction in hadroni c level, which wound be discussed in\nthe following. Under the framework of SU(3) light quark symm etry [49–51], we could select several\ngolden production channels, corresponding to quark transi tionsb→c¯cd/sandb→c¯ud/s[41],\nB0\ns→P++\nc¯suudΣ−−\n¯c,B0→P−\nc¯udssΞ+,B0\ns→P+\nc¯suddp, B−→P−\nc¯uddsΛ0,B0→P+\nc¯dussΣ−.(11)\nThe processes can be described by chain B→M1M2B− →PcBon account of effective Lagrangian\nmethod, pictured as triangle diagrams in Fig. 2. Here, two in termediate mesons M1M2can be\ncharm-light mesons D(∗)Mor charm-charm mesons D(∗)D, they exchange by baryon( B) to produce\nthe expected pentaquark.\nA. Effective Lagrangian and Form Factors\nTo describe the chain process B→M1M2, the effective Hamiltonian of weak decays of B meson\nis ready\nHeff=GF√\n2VCKM(C1(µ)O1(µ)+C2(µ)O2(µ))+h.c. (12)\nHereGFis the Fermi constant, C1,2(µ) andO1,2(µ) are the tree level Wilson coefficients and\nfermion operators of transition b→c¯ud/sorb→c¯cd/s, for instance, O1= (¯cαbβ)V−A(¯dβuα)V−A,\nO2= (¯cαbα)V−A(¯dβuβ)V−Afor the transition b→c¯ud. The weak transition matrices /angbracketleftλ|Heff|B/angbracketright\ncan be further expressed as\nM(B→D(∗)M) =GF√\n2VCKMa1/angbracketleftD(∗)|(¯cb)V−A|B/angbracketright/angbracketleftM|(¯qq)V−A|0/angbracketright, (13)\nwhere ¯qq=¯du, ¯suor ¯sc.a1appears as the combinations of effective Wilson coefficient a1=\nC1+C2/NcwithNcthe number of colors.\nThe parameterized forms of B→DandB→D∗are [52–54]\n/angbracketleftD(k2)|(¯cb)V−A|B(p)/angbracketright=F1(k2\n1)(pµ+kµ\n2−m2\nB−m2\nD\nk2\n1kµ\n1)+F2(k2\n1)(m2\nB−m2\nD\nk2\n1kµ\n1),6\nΞ,Σ,N(k3)\n(a)D(k2)B(p)K/π(k1)¯B(p1)\nPc(p2)Ξ,Σ,N\n(b)DsBD/π¯B\nPc\nΞ,Σ,N\n(c)DBK∗/ρ¯B\nPcΞ,Σ,N\n(d)DsBD∗/ρ¯B\nPc\nΞ,Σ,N\n(f)D∗\nsBD/π¯B\nPc\nΞ,Σ,N\n(h)D∗\nsBD∗/ρ¯B\nPcΞ,Σ,N\n(e)D∗BK/π¯B\nPc\nΞ,Σ,N\n(g)D∗BK∗/ρ¯B\nPc\nFIG. 2: The triangle diagrams of singly charm pentaquark produced by two body decays of B meson. In\n(a-d), pentaquark is assumed to be pseudoscalar meson baryon BPmolecular, while in (e-h) it exists as\nvector meson baryon BVmolecular. The exchanged particles can be light baryons N, Ξ and Σ.\n/angbracketleftD∗(k2)|(¯cb)V−A|B(p)/angbracketright=2iA0(k2\n1)\nmB+mD∗εµνρσǫ∗\nνk2ρpσ−2mD∗A1(k2\n1)ε∗(k2)·k1\nk2\n1kµ\n1\n−(mB+mD∗)A2(k2\n1)(ε∗µ(k2)−ε∗(k2)·k1\nk2\n1kµ\n1)\n+A3(k2\n1)ε∗(k2)·k1\nmB+mD∗(pµ+kµ\n2−m2\nB−m2\nD∗\nk2\n1kµ\n1). (14)7\nb c\n¯q ¯qd/s¯u/¯c\nFIG. 3: The dominant contribution of weak decay of B meson coming f rom the factorizable W-emission\ndiagram.\nWherepandk2represent momentum of BandD(∗)mesons. ε∗denotes the polarization vector of\nvector meson D∗, andk1=p−k2is transition momentum.\nThe matrix elements involving pseudoscalar or vector meson s and the vacuum exhibit the fol-\nlowing definitions [55],\n/angbracketleftV(k1,ε∗\nµ)|(¯qq)V−A|0/angbracketright=mVfVε∗\nµ,/angbracketleftP(k1)|(¯qq)V−A|0/angbracketright=ifPk1µ. (15)\nWherein fPandfVare decay constants of pseudoscalar Pand vector Vmesons respectively.\nThe description about chain M1M2→Pc¯Bcan be accomplished by the effective Lagrangian\nincluding two baryons and a meson, given as [38]:\nLBBP=gBBP\nmP/angbracketleftBγ5γµ∂µPB/angbracketright, (16)\nLBBV=gBBV/angbracketleftBγµVµB/angbracketright+fBBV\n2mV/angbracketleftBσµν∂µVνB/angbracketright. (17)\nThe coupling constants between meson and two baryons can be d etermined by the generic SU(4)\nsymmetry, we deduce them into Tab.II.\nB. Decay width\nArmed with the effective Lagrangian and form factor, the produ ction process of pentaquark is\ndirect. We show amplitudes of process B0\ns→P+\nc¯sudd¯pcorresponding with Fig.2(b,d,f,h), given as\nMb=ifπ/integraldisplayd4k3\n(2π)4gNNπgPcNDs¯uPc(p2)(/ k3+mN)γ5k /1νp(p1)\n(k2\n1−m2\nπ) (k2\n2−m2\nDs) (k2\n3−m2\nN)F2(k2\n3)\n/parenleftBig\n(2p2+p1−k3)·(p1+k3)F1(k2\n1)−(m2\nB−m2\nDs)F1(k2\n1)+(m2\nB−m2\nDs)F2(k2\n1)/parenrightBig\n,(18)8\nMd=fρmρ/integraldisplayd4k3\n(2π)4gNNρgPcNDs¯uPc(p2)(/ k3+mN)(γν−fNNρσξνkξ\n1/4mρ)νp(p1)\n−(k2\n1−m2\nρ) (k2\n2−m2\nDs) (k2\n3−m2\nN)\n(−gνµ+kν\n1kµ\n1\nm2ρ)F2(k2\n3)/parenleftBig\n(p+k2)µF1(k2\n1)/parenrightBig\n, (19)\nMf=ifπ/integraldisplayd4k3\n(2π)4gNNπgPcND∗\ns¯uPc(p2)γβ(/ k3−mN)k /1ν¯p(p1)(gαβ−p2αp2β\nm2\nPc)(−gαµ+kµ\n2kα\n2\nm2\nD∗s)\n(k2\n1−m2π)(k2\n2−m2\nD∗s)(k2\n3−m2\nN)\n/parenleftBig\n−2mD∗sA1(k2\n1)k1µ+A3(k2\n1)k1µ\nmB+mD∗s(p·k1+k2·k1−m2\nB+m2\nD∗s)/parenrightBig\nF2(k2\n3),(20)\nMh=fρmρ/integraldisplayd4k3\n2π4gNNρgPcND∗\ns¯uPc(p2)γ5γβ(/ k3+mN)(γδ−fNNρσηδkη\n1/4mρ)νp(p1)\n−(k2\n3−m2\nN)(k2\n2−m2\nD∗s)(k2\n1−m2ρ)\n/parenleftBig2iA0(k2\n1)\nmB+mD∗sεµρσνkρ\n2pσ−A2(k2\n1)gνµ(mB+mD∗s)+A3(k2\n1)k1ν\nmB+mD∗s(p+k2)µ/parenrightBig\nF2(k2\n3)(gαβ−p2αp2β\nm2\nPc)(−gαν+kα\n2kν\n2\nmD∗s)(−gµδ+kµ\n1kδ\n1\nm2ρ). (21)\nWheremNandk3represent the mass and momentum of exchanged baryon N. The momentum\ndefinitions can be found in Fig.2. To describe the structure a nd off-shell effect, a monopole form\nfactor is introduced for the exchanged baryon B(Ξ,Σ, N) [56, 57],\nF(k2\n3) =m2\nE−Λ2\nk2\n3−Λ2.\nmEandk3represent the mass and momentum of exchanged baryons B. We adopt the cutoff value\nΛ =mE+α[51, 56–60].\nThe two-body decay width of BmesonB→Pc¯Bcan be formulated as\nΓ(B→Pc+B) =|Pf|\n8πm2\nB|M|2,\nhere,Mrepresents amplitude of singly charm pentaquark produced f romBmeson, and Pfis the\nmagnitude of the three-momentum of final state, known as |Pf|=1\n2mB/radicalBig\nλ(m2\nB,m2\nPc,m2\nB), with\nλ(a,b,c) =a2+b2+c2−2ab−2bc−2ac.\nIV. NUMERICAL ANALYSIS\nIn the numerical analysis, the form factors F1(k2),F2(k2) andAi(k2)(i=1,2,3) of transition\nB→D(D∗) can be parameterized using the Bourrely-Caprini-Lellouc h (BCL) method [53, 54]\nF1(k2) =1\n1−k2/m2\nBc(a0z0+a1z1+a2z2),\nF2(k2) =1\n1−k2/m2\nB∗c/parenleftbig\na0z0+a1(z1−z3\n3)+a2(z2+2z3\n3)/parenrightbig\n,\nAi(k2) = (/radicalbig\n(mB+mD∗)2−k2+/radicalBig\n(mB+mD∗)2−m2\npole\n/radicalbig\n(mB+mD∗)2−k2−/radicalBig\n(mB+mD∗)2−m2\npole)(a0+a1z1+a2z2\n1+a3z3\n1),(22)9\nTABLE I: The fitted expansion parameters aiof form factors for transitions B→DandB→D∗.\na0a1a2a3mpole(GeV)\nB→DF1(0) 0.666−0.260−0.106 0.00 −\nF2(0) 0.666−3.236−0.075 0.00 −\nB→D∗A0(0) 0.100−0.180−0.006 0.00 6.335\nA1(0) 0.105−0.430−0.100−0.03 6.275\nA2(0) 0.055−0.010−0.030 0.06 6.745\nA3(0) 0.059−0.110−0.250−0.05 6.745\nTABLEII:Thecouplingconstantsofsinglycharmpentaquarkandt hemolecularmeson-baryoncomponents,\nobtained by compositeness condition. The couplings between meson and two baryons, derived from SU(4)\nflavor symmetry, where the generic coupling parameters are take n asgBBP= 0.989,gBBV= 3.25 and\nfBBV= 6.1 [38].\ngPcΞDgPcNDsgPcΣDgPcΞD∗gPcND∗\nsgPcΣD∗\n2.280 2.209 2.282 0.337 0.293 0.328\ngΞΞπgΣΛπgΣcNDgΞΣKgNNπgΞΞcDs\n3.78 9.97 5.34 19.09 13.5 13.41\ngNNρgΞΞcD∗sgΞΣK∗gΞΞρgΛΣρgΣcND∗\n3.25 1.88 4.60 3.25 3.75 4.60\nfNNρfΞΞcD∗sfΞΣK∗fΞΞρfΛΣρfΣcND∗\n6.1 3.52 8.63 6.1 7.04 8.63\nwith the expressions of z- andz1-coefficient\nz=√\n(mB+mD)2−k2−√\n(mB+mD)2√\n(mB+mD)2−k2+√\n(mB+mD)2, z1=√\n(mB+mD∗)2−k2−√\n(mB+mD∗)2−(mBs−mD∗s)2√\n(mB+mD∗)2−k2+√\n(mB+mD∗)2−(mBs−mD∗s)2.(23)\nWherempoledenotes the physical pole masse of Bcmeson. The fitted parameters a0,1,2,3andmpole\nare collected into Tab. I. Additionally, we need decay const ants of mesons [38],\nfπ= 0.130GeV, fK= 0.156GeV, fD= 0.212GeV, fDs= 0.249GeV,\nfK∗= 0.210GeV, fD∗= 0.220GeV, fρ= 0.216GeV, fDs= 0.230GeV,\nas well as CKM matrix elements and effective Wilson coefficient [ 61, 62],\n|Vcd|= 0.221,|Vud|= 0.974,|Vus|= 0.2243,|Vcd|= 0.221,|Vud|= 0.974,\nGF= 1.166×10−5, a1= 1.07, τB= 1.638×10−12s.\nWe adopt cutoff parameter αwith a central value 250 MeV to determine the branching ratio s of\ncharm pentaquark Pcproduced from Bmeson, respectively at the binding energy ǫ= 5,20 and 50\nMeV. The branching ratios are achieved at the order of 10−7∼10−6for the meson-baryon scenario\nBPmolecular, and 10−10∼10−9for the meson-baryon BVmolecular, the results are shown in10\nTABLE III: The production processes of pentaquark( B→PcB) in B meson decays can be discussed\nseparately in two scenarios, S1: pentaquark as the hadron molecu le ofBP, S2: pentaquark as the hadron\nmolecule of BV. The branching ratios are calculated at a central value Λ = mE+ 0.25 GeV, within the\nvalue of binding energy ǫ= 5, 20, 50 MeV. The exchanged particles (EP) can be Ξ ,N,Σ. For definiteness,\nwe show the dominant intermediate hadron states in the last column.\nB→PcBBranching ratio( ×10−7)EPDominant hadronsǫ= 5MeV ǫ= 20MeV ǫ= 50MeV\nB0→P−\nc¯udssΞ+6.84 15.49 28.51 Ξ ρD\nB0\ns→P+\nc¯suddp18.88 36.48 57.92 N ρDs\nS1B−→P−\nc¯uddsΛ06.80 17.16 34.52 Σ ρD\nB0→P+\nc¯dussΣ−22.20 27.27 29.06 Ξ KD\nB0\ns→P++\nc¯suudΣ−−\n¯c6.17 14.43 23.74 N DDs\nB0→P−\nc¯udssΞ+0.014 0.045 0.094 Ξ ρD∗\nB0\ns→P+\nc¯suddp0.009 0.045 0.100 N ρD∗\ns\nS2B−→P−\nc¯uddsΛ00.020 0.018 0.056 Σ ρD∗\nB0→P+\nc¯dussΣ−0.002 0.012 0.025 ΞKD,K∗D\nTab. III. For completeness, we draw branching ratios changi ng with cutoff parameter αin range\nof 100−300 MeV in Fig. 4 and Fig. 5.\nAmongtheprocesses withattention, thebranchingratiound erpentaquarkas hadronicmolecule\nBP(S1) holds the higher possibility than the meson-baryon BVmolecular (S2). It is noteworthy\nthat the process B0\ns→P++\nc¯suddΣ−−\n¯cis unallowed under the latter molecular case owing to the effec t\nof phase space. The results of branching ratios obviously sh ow that, the production channels\nB0\ns→P+\nc¯suddpandB0→P+\nc¯dussΣ−possess considerable order of branching ratio in two molecu lar\nscenarios, for instance, in BPmolecular at the binding energy ǫ=20MeV,\nBr(B0\ns→P+\nc¯suddp) = 3.648×10−6,br(B0\ns→ρDsN− →P+\nc¯suddp) = 2.59×10−6,\nBr(B0→P+\nc¯dussΣ−) = 2.727×10−6,br(B0→KDΞ− →P+\nc¯dussΣ−) = 2.60×10−6.(24)\nIn particularly, the dominant contributions among the two p ossible sub processes come from B0\ns→\nρDsN− →P+\nc¯suddpas Fig. 2.(d) and B0→KDΞ− →P+\nc¯dussΣ−as Fig. 2.(a).\nV. CONCLUSIONS\nIn this work, we force on the properties and productions of si ngly charm pentaquark. Under the\nassumption of pentaquark molecular composition, it is mean ingful to investigate the productions\nof singly charm pentaquark by decays of B meson B→PcB. According to the effective Lagrangian\nmethod, the selected suitable processes can then be studied with the benefit of hadronic triangle\ndiagrams. We deducebranchingratios of the productionseve ntually, what deserves our attention is\nthe productions of pentaquark P+\nc¯suddandP+\nc¯duss. Our calculations suggest that branching ratios of11\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■\n◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼●B0→Pcu\n_dss-\u0002+\n■Bs0→Pcs\n_udd+p\n◆B-→Pcu\n_dds-\u00030\n▲B0→Pcd\n_\nuss+Σ-\n▼Bs0→Pcs\n_uud++Σc--\n0.000.050.100.150.200.250.300510152025\nαBranchingratio ×10-7\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■\n◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼●B0→Pcu\n_dss-\u0002+\n■Bs0→Pcs\n_udd+p\n◆B-→Pcu\n_dds-\u00030\n▲B0→Pcd\n_\nuss+Σ-\n▼Bs0→Pcs\n_uud++Σc--\n0.000.050.100.150.200.250.30010203040\nαBranchingratio ×10-7\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■\n◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼●B0→Pcu\n_dss-\u0002+\n■Bs0→Pcs\n_udd+p\n◆B-→Pcu\n_dds-\u00030\n▲B0→Pcd\n_\nuss+Σ-\n▼Bs0→Pcs\n_uud++Σc--\n0.000.050.100.150.200.250.30010203040506070\nαBranchingratio ×10-7\nFIG. 4: In the configuration of meson-baryon BPmolecular(S1), the branching ratios of B→PcBvary with\nthe cutoff parameter α. The graphs correspond with binding energy ǫ=5 MeV(left), 20 MeV(middle) and\n50 MeV(right).\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲●B0→Pcu_dss-\u0000+\n■Bs0→Pcs_udd+p\n◆B-→Pcu_dds-Λ0\n▲B0→Pcd_\nuss+Σ-\n0.000.050.100.150.200.250.300.00.51.01.52.0\nαBranchingratio ×10-9\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲●B0→Pcu_dss-Ξ+\n■Bs0→Pcs_udd+p\n◆B-→Pcu_dds-Λ0\n▲B0→Pcd_\nuss+Σ-\n0.000.050.100.150.200.250.300123456\nαBranchingratio ×10-9\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲●B0→Pcu_dss-Ξ+\n■Bs0→Pcs_udd+p\n◆B-→Pcu_dds-Λ0\n▲B0→Pcd_\nuss+Σ-\n0.000.050.100.150.200.250.30051015\nαBranchingratio ×10-9\nFIG. 5: In the configuration of meson-baryon BVmolecular(S2), the branching ratios of B→PcBvary\nwith the cutoff parameter α. The graphs correspond with binding energy ǫ=5 MeV(left), 20 MeV(middle)\nand 50 MeV(right).\nthetwo Cabibboallowed processes can reach to 10−6and 10−8, whichneeds to befurtherconfirmed\nby experiments.\nAcknowledgments\nY.X. acknowledges the support by National Natural Science F oundation of China under Grant\nNo. 12005294, and the Fundamental Funds for Key disciplines in Physics with No.2022WLXK05.\nAppendix A: Decay amplitudes\nThe amplitude of B0→P−\nc¯udssΞ+is\nMa=ifπ/integraldisplayd4k3\n(2π)4gΞΞπgPcΞD¯uPc(p2)(/ k3+mΞ)γ5k /1νΞ(p1)\n(k2\n1−m2\nπ) (k2\n2−m2\nD) (k2\n3−m2\nΞ)12\nF2(k2\n3)/parenleftBig\n(2p2+p1−k3)·(p1+k3)−m2\nB+m2\nD)F1(k2\n1)+(m2\nB−m2\nD)F2(k2\n1)/parenrightBig\n,(A1)\nMc=fρmρ/integraldisplayd4k3\n(2π)4gΞΞρgPcΞD¯uPc(p2)(/ k3+mΞ)(γν−fΞΞρσξνkξ\n1/4mρ)νΞ(p1)\n−(k2\n1−m2\nρ) (k2\n2−m2\nD) (k2\n3−m2\nΞ)\n(−gνµ+kν\n1kµ\n1\nm2ρ)F2(k2\n3)/parenleftBig\n(p+k2)µF1(k2\n1)/parenrightBig\n, (A2)\nMe=ifπ/integraldisplayd4k3\n(2π)4gΞΞπgPcΞD∗¯uPc(p2)γβ(/ k3−mΞ)k /1νΞ(p1)(gαβ−p2αp2β\nm2\nPc)(−gαµ+kµ\n2kα\n2\nm2\nD∗)\n(k2\n1−m2π)(k2\n2−m2\nD∗)(k2\n3−m2\nΞ)\n/parenleftBig\n−2mD∗A1(k2\n1)k1µ+A3(k2\n1)k1µ\nmB+mD∗(p·k1+k2·k1−m2\nB+m2\nD∗)/parenrightBig\nF2(k2\n3),(A3)\nMg=fρmρ/integraldisplayd4k3\n2π4gΞΞρgPcΞD∗¯uPc(p2)γ5γβ(/ k3+mΞ)(γδ−fΞΞρσηδkη\n1/4mρ)νΞ(p1)\n−(k2\n3−m2\nΞ)(k2\n2−m2\nD∗)(k2\n1−m2ρ)\n/parenleftBig2iA0(k2\n1)\nmB+mD∗εµρσνkρ\n2pσ−A2(k2\n1)gνµ(mB+mD∗)+A3(k2\n1)k1ν\nmB+mD∗(p+k2)µ/parenrightBig\nF2(k2\n3)(gαβ−p2αp2β\nm2\nPc)(−gαν+kα\n2kν\n2\nmD∗)(−gµδ+kµ\n1kδ\n1\nm2ρ). (A4)\nThe amplitude of B−→P−\nc¯uddsΛ0\nMa=ifπ/integraldisplayd4k3\n(2π)4gΛΣπgPcΣD¯uPc(p2)(/ k3+mΣ)γ5k /1νΛ(p1)\n(k2\n1−m2\nπ) (k2\n2−m2\nD) (k2\n3−m2\nΣ)\nF2(k2\n1)/parenleftBig\n((2p2+p1−k3)·(p1+k3)−m2\nB+m2\nD)F1(k2\n1)+(m2\nB−m2\nD)F2(k2\n1)/parenrightBig\n,(A5)\nMc=fρmρ/integraldisplayd4k3\n(2π)4gΛΣρgPcΣD¯uPc(p2)(/ k3+mΣ)(γν−fΛΣρσξνkξ\n1/4mρ)νΛ(p1)\n−(k2\n1−m2\nρ) (k2\n2−m2\nD) (k2\n3−m2\nΣ)\n(−gνµ+kν\n1kµ\n1\nm2ρ)F2(k2\n3)/parenleftBig\n(p+k2)µF1(k2\n1)/parenrightBig\n, (A6)\nMe=ifπ/integraldisplayd4k3\n(2π)4gΛΣπgPcΣD∗¯uPc(p2)γβ(/ k3−mΣ)k /1νΛ(p1)(gαβ−p2αp2β\nm2\nPc)(−gαµ+kµ\n2kα\n2\nm2\nD∗)\n(k2\n1−m2π)(k2\n2−m2\nD∗)(k2\n3−m2\nΣ)\n/parenleftBig\n−2mD∗A1(k2\n1)k1µ+A3(k2\n1)k1µ\nmB+mD∗(p·k1+k2·k1−m2\nB+m2\nD∗)/parenrightBig\nF2(k2\n3),(A7)\nMg=fρmρ/integraldisplayd4k3\n2π4gΛΣρgPcΣD∗¯uPc(p2)γ5γβ(/ k3+mΣ)(γδ−fΛΣρσηδkη\n1/4mρ)νΛ(p1)\n−(k2\n3−m2\nΣ)(k2\n2−m2\nD∗)(k2\n1−m2ρ)\n/parenleftBig2iA0(k2\n1)\nmB+mD∗εµσρνkρ\n2pσ−A2(k2\n1)gνµ(mB+mD∗)+A3(k2\n1)k1ν\nmB+mD∗(p+k2)µ/parenrightBig\nF2(k2\n3)(gαβ−p2αp2β\nm2\nPc)(−gαν+kα\n2kν\n2\nmD∗)(−gµδ+kµ\n1kδ\n1\nm2ρ). (A8)\nThe amplitude of B0→P+\nc¯dussΣ−\nMa=ifπ/integraldisplayd4k3\n(2π)4gΞΣKgPcΞD¯uPc(p2)(/ k3+mΞ)γ5k /1νΣ(p1)\n(k2\n1−m2\nK) (k2\n2−m2\nD) (k2\n3−m2\nΞ)\nF2(k2\n1)/parenleftBig\n((2p2+p1−k3)·(p1+k3)−m2\nB+m2\nD)F1(k2\n1)+(m2\nB−m2\nD)F2(k2\n1)/parenrightBig\n,(A9)13\nMc=fρmρ/integraldisplayd4k3\n(2π)4gΞΣK∗gPcΞD¯uPc(p2)(/ k3+mΞ)(γν−fΞΣK∗σξνkξ\n1/4mρ)νΣ(p1)\n−(k2\n1−m2\nK∗) (k2\n2−m2\nD) (k2\n3−m2\nΞ)\n(−gνµ+kν\n1kµ\n1\nm2\nK∗)F2(k2\n3)/parenleftBig\n(p+k2)µF1(k2\n1)/parenrightBig\n, (A10)\nMe=ifπ/integraldisplayd4k3\n(2π)4gΞΣKgPcΞD∗¯uPc(p2)γβ(/ k3−mΞ)k /1νΣ(p1)(gαβ−p2αp2β\nm2\nPc)(−gαµ+kµ\n2kα\n2\nm2\nD∗)\n(k2\n1−m2\nK)(k2\n2−m2\nD∗)(k2\n3−m2\nΞ)\n/parenleftBig\n−2mD∗A1(k2\n1)k1µ+A3(k2\n1)k1µ\nmB+mD∗(p·k1+k2·k1−m2\nB+m2\nD∗)/parenrightBig\nF2(k2\n3),(A11)\nMg=fρmρ/integraldisplayd4k3\n2π4gΞΣK∗gPcΞD∗¯uPc(p2)γ5γβ(/ k3+mΞ)(γδ−fΞΣK∗σηδkη\n1/4mρ)νΣ(p1)\n−(k2\n3−m2\nΞ)(k2\n2−m2\nD∗)(k2\n1−m2\nK∗)\n/parenleftBig2iA0(k2\n1)\nmB+mD∗εµρσνkρ\n2pσ−A2(k2\n1)gνµ(mB+mD∗)+A3(k2\n1)k1ν\nmB+mD∗(p+k2)µ/parenrightBig\nF2(k2\n3)(gαβ−p2αp2β\nm2\nPc)(−gαν+kα\n2kν\n2\nmD∗)(−gµδ+kµ\n1kδ\n1\nm2\nK∗). (A12)\nThe amplitude of B0\ns→P++\nc¯suudΣ−−\n¯c\nMb= (ifDs)/integraldisplayd4k3\n(2π)4gΣcNDgPcNDs¯uPc(p2)(/ k3+mN)γ5k /1νΣ¯c(p1)\n(k2\n1−m2\nDs) (k2\n2−m2\nD) (k2\n3−m2\nN)\nF2(k2\n3)(2p2+p1−k3)·(p1+k3)−m2\nB+m2\nD)F1(k2\n1)+(m2\nB−m2\nD)F2(k2\n1)/parenrightBig\n, (A13)\nMd= (ifDs)/integraldisplayd4k3\n(2π)4gΣcND∗gPcNDs¯uPc(p2)(/ k3+mN)(γν−fΣcND∗σmνkm\n1/4mD∗)νΣ¯c(p1)(−gµν+kµ\n2kν\n2\nm2\nD∗)\n(k2\n1−m2\nDs) (k2\n2−m2\nD∗) (k2\n3−m2\nN)\nF2(k2\n3)/parenleftBig\n−2mD∗A1(k2\n1)k1µ+A2(k2\n1)k1µ\nmB+mD∗(p·k1+k2·k1−m2\nB+m2\nD∗)/parenrightBig\n. (A14)\nWheremΞ,mΣ,mNandk3represent the mass and momentum of exchanged baryon(Ξ,Σ, N)\nrespectively.\n[1] R. Aaij et al.[LHCb], Phys. Rev. Lett. 115(2015), 072001 doi:10.1103/PhysRevLett.115.072001\n[arXiv:1507.03414 [hep-ex]].\n[2] R. Aaij et al.[LHCb], Phys. Rev. Lett. 122(2019) no.22, 222001 doi:10.1103/PhysRevLett.122.222001\n[arXiv:1904.03947 [hep-ex]].\n[3] R. Aaij et al.[LHCb], Sci. 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To elucidate this defect-\ncontrolled process, we have developed a Neural Network Potential (NNP) trained on first-principle\ncalculations of Na-feldspar (Albite) and its charged defects. This new force field reproduces various\nexperimentally known properties of feldspar, including its lattice parameters, elastic constants as\nwell as heat capacity and DFT-calculated defect formation energies. A new type of dumbbell\ninterstitial defect is found to be most favorable and its free energy of formation at finite temperature\nis calculated using thermodynamic integration. The necessity of including electrostatic corrections\nbefore training an NNP is demonstrated by predicting more consistent defect formation energies.\nI. INTRODUCTION\nFeldspar is the most abundant mineral in the Earth’s\ncrust and an important ingredient for ceramics [1]. It\nforms a solid-solution between a sodium (NaAlSi 3O8-\nalbite), calcium (CaAl 2Si2O8- anorthite) and a potas-\nsium (KAlSi 3O8- K-feldspar) end-member component.\nThe most prominent feldspar groups are represented by\nthe plagioclase (NaAlSi 3O8- CaAl 2Si2O8) and the alkali\nfeldspar (NaAlSi 3O8- KAlSi 3O8) solid-solution series.\nAt high temperatures typical for magmatic and metamor-\nphic environments, both solid-solution series show com-\nplete miscibility. Towards lower temperatures miscibil-\nity gaps open, and feldspars of intermediate composition\ntend to exsolve, typically producing lamellar intergrowth\nof feldspars with different compositions. The exsolution\nmicrostructures are of particular interest for reconstruct-\ning the thermal history of magmatic and metamorphic\nrocks [2, 3]. In addition, elastic strain associated with\nexsolution microstrutures may lead to pseudocleavage,\nwhich substantially enhances the ice nucleation activ-\nity of exsolved feldspar making the corresponding aerosol\nparticles important players in the glaciation of clouds [4].\nExsolution of an initially homogeneous feldspar implies\nsegregation of the different cations on the alkali and al-\nkali earth sub-lattices, which occurs by intracrystalline\ndiffusion. Due to the long timescales over which diffusive\nphase transformations in feldspar proceed, the underly-\ning diffusion mechanisms and their possible anisotropy\nstill hold unresolved questions, such as the recently dis-\ncussed systematic differences between experimentally de-\ntermined Na-K interdiffusion coefficients and those the-\noretically predicted from measured tracer diffusion coef-\nficients [5–10].\nOn the computational side, the diffusion of alkali de-\nfects in feldspar has been studied with kinetic Monte\nCarlo simulations [11–13] using empirical rate constants.\nInteratomic potentials capturing all constituent elementsof the material and its defects are needed for studying\nthe microscopic mechanism of diffusion without referring\nto laboratory experiments. It has been demonstrated\nthat classical force-fields are capable of reproducing the\noverall properties of various kinds of silicates and alumi-\nnosilicates [14], and they have been applied for study-\ning diffusion barriers of sodium vacancies in Na- and K-\nfeldspar [15]. These classical force fields, however, fall\nbehind density functional theory (DFT) methods, which\nhave been demonstrated to predict various properties of\nfeldspar accurately [16]. Yet, DFT and other ab initio\nmethods are computationally considerably more expen-\nsive than classical force fields, drastically limiting the\naccessible system sizes and simulation times. Recently,\nmachine learned force fields (MLFF) have been shown\nto offer accuracy comparable to first-principle electronic-\nstructure calculations at a cost comparable to that of\nclassical force-field calculations [17, 18]. Such MLFFs\nhave already been used to study a vast array of systems\nranging from organic molecules and inorganic crystals to\nsurfaces, aqueous systems and biomolecules [19–21]. In\nparticular, MLFFs have been used for studying diffusion\nin lead and cadmium telluride [22] and an MLFF for\nstudying ice nucleation on K-feldspar has been developed\nrecently [23].\nIn this paper, we develop a Behler-Parrinello type neu-\nral network potential (NNP) for sodium feldspar, paying\nparticular attention to the accurate representation of de-\nfects. Trained on a reference data set obtained with DFT\nusing the PBE functional [24], our NNP accurately repro-\nduces forces and energies and it can be used to carry out\nextensive molecular dynamics simulations. We demon-\nstrate its applicability by computing the heat capacity\nas well the elastic constants and the geometry of the\nunit cell of Na-feldspar over a wide range of tempera-\ntures, finding good agreement with experimental data.\nIn addition, we investigate the structure and diffusion of\ncharged interstitial and vacancy defects, taking electro-\nstatic finite size corrections into account. Such FrenkelarXiv:2402.14640v1  [cond-mat.mtrl-sci]  22 Feb 20242\npairs, consisting of Na+interstitials and Na+vacancies,\nare expected to be the majority carrier of alkali defects,\nsince Schottky defects are energetically improbable due\nto the high Si-O bonding energy [25]. The computational\nefficiency of the NNP allowed us to determine the defect\nformation free energy, and from it the defect concentra-\ntions for temperatures from 0 K up to 1400 K using ther-\nmodynamic integration. Furthermore, our simulations\nrevealed a new interstitial configuration, in which two\nNa+cations share an alkali lattice site. This dumbbell-\nlike defect is thermodynamically more favorable than the\nNa+\n(001\n2)interstitial that was considered previously [26].\nInterestingly, the energetically favourable orientation of\nthe dumbbell changes discontinuously at a temperature\nof about 752 K. The current version of our NNP can be\nused to model sodium feldspar, but it can be readily ex-\ntended to potassium feldspar and the solid solution of\nalkali feldspar by retraining it with an extended traning\nset.\nThe remainder of the paper is organized as follows.\nIn Sec. II we describe the computational methods used\nin our work, including the preparation of the reference\ndata and the training of the NNP. Results are presented\nand discussed in Sec. III and conclusions are provided in\nSec. IV.\nII. METHODS\nA. Ab-Initio Calculations\nAll our DFT calculations were performed using the\nVienna Ab-initio Simulation Package (VASP) [27–29]\nwith the projector augmented wave method [30] utilizing\nthe Perdew-Burke-Ernzerhof (PBE) exchange-correlation\nfunctional [24]. The conventional cell containing four for-\nmula units of NaAlSi 3O8, initially constructed using ex-\nperimental data of Armbruster and colleagues [31] with\nAl always kept at the T 1O-sites, was sampled using a\n6×4×6 Γ-centered k-point mesh. The same k-point den-\nsity in the Brillouin zone was kept across all calculations\nand the energy cutoff was set to 560 eV. The conver-\ngence criteria were 10−5eV for the self-consistent cycle\nand 10−3eV˚A−1for ionic relaxations. When varying the\nnumber of electrons, a neutralizing jellium background\nwas included.\nB. Defect Formation Energies and their Correction\nTo determine the equilibrium concentrations of point\ndefects in Na-feldspar it is necessary to calculate de-\nfect formation energies. Since vacancies and interstitials\ncarry a charge, their energy in a periodicically replicated\nsystem suffers from strong finite size effects that require\nproper corrections. For charged defects in a periodic su-percell the defect formation energy is given by [32, 33]\nEform= (E[Defect] + Ecorr[Defect]) −E[Bulk]\n−/summationdisplay\niniµi+q(ϵVBM+ ∆EFL).(1)\nHere, E[Defect] is the total energy obtained from a su-\npercell calculation containing the defect and E[Bulk] is\nthe total energy of the pristine supercell. The term\nEcorr[Defect] is a finite-size correction explained in more\ndetailed below. The integer niis the number of atoms of\ntype iadded to ( ni>0) or removed from ( ni<0) the\nsystem and µiis the corresponding chemical potential in\nthe reservoir with which the atoms are exchanged. The\nlast term in the above equation represents the energy re-\nquired to introduce or remove the electronic charge q,\nwhich is q=−1 for Na+vacancies and q= +1 for Na+\ninterstitials. Here, ϵVBM is the electron energy at the\nValence Band Maximum (VBM) and ∆ EFLthe Fermi\nlevel. As is common for 0 K DFT calculations, we iden-\ntify ∆ EFL= 0.\nAs mentioned above, when calculating the formation\nenergy of charged defects in the supercell approach one\ninvariably introduces interactions between the defect, its\nperiodic images and the neutralizing background charge,\nnecessitating the correction Ecorr. The most important\ncontributions to this correction is the spurious Coulomb\ninteraction, which remains significant for any practical\nsupercell size. Several methods exist to subtract this in-\nteraction from the defective supercells. Apart from the\nextrapolative scheme, in which the defect formation en-\nergy is determined for a sequence of supercells of increas-\ning size whose limiting value can be identified with the\ndilute limit, there have been major developments in for-\nmally removing the electrostatic interactions by correc-\ntions [34]. Here, we use the correction of Kumagai and\nOba [34]. To compute this correction, we first determined\nthe dielectric tensor of albite using DFT perturbation\nroutines in VASP [35], see Tab. I. Then, we calculated\nthe correction from pristine and relaxed defective super-\ncells of the various systems (listed in Tab IV) using the\nSpinney package [36]. The potentials involved in the cor-\nrection are provided in the Supplementary Information.\nC. Architecture of the NNP-Committee\nIn this work, the potential energy surface is represented\nby artificial neural networks as proposed by Behler and\nParrinello in 2007 [17]. Local atomic environments are\nexpressed by radial and angular atom-centered symmetry\nfunctions with a cutoff of 6 ˚A. The atomic feed forward\nneural networks consist of an input layer containing 256\nnodes in the case of aluminum, 272 for silicon, 273 for\nsodium and 276 for oxygen. All four networks further\nconsist of two hidden layers with 25 nodes each and one\nsingle output node. The weights and biases were initial-\nized randomly and optimized using a parallel Kalman-\nFilter [38] to minimize the root mean squared deviation3\nTABLE I. Ionic ( ϵ0) and electronic ( ϵ∞) contributions to\nthe static dielectric tensor ϵij(0) of Na-feldspar computed in\nGGA. In order to compare with experimental data, we esti-\nmated the dielectric constant of a polycrystalline sample by\nϵpoly= (λ1+λ2+λ3)/3, where λ1, λ2, λ3are the eigenvalues\nof the dielectric tensor. The result is an ϵpolyequal to 6.18\nversus values of 7 and 8 from experiments of Olhoeft [37] and\nJones et al. [15] respectively.\nϵ∞ϵ0Σ\nϵxx2.404 2 .602 5 .006\nϵyy2.441 4 .664 7 .105\nϵzz2.415 4 .019 6 .434\nϵyz−0.005−1.576−1.581\nϵzx−0.007−0.273−0.280\nϵxy0.001 0 .110 0 .111\nbetween predicted energies and forces and the reference\nvalues. The n2p2 package [39] was used to train the po-\ntentials and to interface them with the molecular dynam-\nics simulation package LAMMPS [40].\nWe used several independent NNPs, trained with dif-\nferent initial weights, in a committee machine, i.e.pre-\ndictions for a certain configuration Aare calculated as\nan average over Nindependent predictions yi:\ny(A) =1\nN/summationdisplay\niyi(A). (2)\nFor all our calculations we used the committee method\nimplemented in n2p2 by K´ yvala et al. [41] with a com-\nmittee size of N= 4. Committee machines are known to\nincrease the precision of machine-learning models, reduce\noverfitting and they have been used to improve predic-\ntions of various physical properties [42, 43]. Furthermore,\nthe variance σ(A) of the individual predictions can serve\nas a measure for the uncertainty of the averaged predic-\ntiony(A) [44–46]. As explained below, we use this uncer-\ntainty in our active learning strategy for the generation\nof the reference data.\nD. Dataset Generation\n1. Active Learning\nTo generate the data set needed to train and test our\nNNP, we used a two-stage strategy. In the first stage,\nwe carried out an on-the-fly machine learning simula-\ntion using VASP [47, 48]. In this approach, an ab initio\nmolecular dynamics simulation is started and a Gaussian\nprocess regression (GPR) model is continuously trained\non the energies and forces computed from DFT. As the\nsimulation proceeds and the accuracy of the GPR model\nimproves, the expensive electronic structure calculations\nare successively replaced by the computationally inex-\npensive GPR model. DFT calculations are only carriedout if the accuracy of the GPR model, as determined from\nBayesian error estimation, falls below a given threshold.\nSince with time the relevant part of configuration space\ngets explored by the simulation, fewer and fewer DFT\ncalculations become necessary, speeding up the simula-\ntion considerably.\nWith this on-the-fly approach, a relaxed 2 ×1×2 super-\ncell was driven from 0 to 1300 K in 20 000 steps at atmo-\nspheric pressure in the NPT ensemble. In these and all\nsubsequent on-the-fly simulations a timestep of 1 .5 fs was\nused. Whenever an electronic structure calculation was\ncarried out, the respective configuration together with\nits energy and forces was added to the reference data set.\nThe simulation was continued at 1300 K until there were\nno new uncertain configurations added to the reference\nset for a span of at least 20 000 steps. The same proce-\ndure was repeated for a relaxed 2 ×1×2 supercell in which\na sodium atom was introduced and an electron removed\nto create an Na+\niinterstitial defect. This system took the\nlongest to reach acceptable predictive accuracy at 1300 K\nwith 100 000 steps necessary. Finally, the procedure was\ncarried out for a relaxed supercell in which a sodium was\nremoved and an additional electron introduced to create\na V–\nNavacancy. Throughout all of these simulations a\ntotal of 220 000 timesteps were carried out, long enough\nsuch that several intracrystalline hopping events of the\ndefects were already observed therein. Of the total num-\nber of timesteps, 1679 were calculated using forces deter-\nmined ab initio .\nWhile MD simulations can be carried out with the\nGPR model directly, it is known that NNPs are superior\nto GPR models in terms of speed. In the second stage of\nour strategy, we have therefore used energies and forces\ncomputed ab initio for the 1679 configurations collected\non-the-fly to train an NNP. More specifically, a commit-\ntee of four neural-network potentials (their architecture\nis given in Sec. II C) was trained with n2p2. The com-\nmittee was then used to run MD simulations with 2 ×2×2\nsupercells in a pristine system and with interstitial or va-\ncancy defects in the NPT ensemble up to 1400 K with\na timestep of 1 fs. Every 5th timestep the force uncer-\ntainty for each atom was determined. Since the force\nuncertainty is a local quantity in contrast to the uncer-\ntainty of the total energy, we expect it to be a more suit-\nable measure of uncertainty. Configurations where any\natom-wise force uncertainty was above the threshold of 5\ntimes the RMSE (analogous to Ref. [49]) were collected.\nConfigurations that were at least 100 fs apart were recal-\nculated with VASP and put into the reference data set.\nWith this augmented reference data set a new commit-\ntee was trained and the process was repeated. After four\nsuch iterations the committee produced a low rate of un-\ncertain configurations and the procedure was stopped. A\ntotal of 670 additional DFT calculations were necessary\nin this procedure. An increase in the required training\nset size between the initial GPR model of VASP and the\nfinal NNP had to be expected as GPR models are known\nto require less data than NNPs, see e.g. Ref. [50]. In to-4\ntal, the training set contained 2351 configurations with\nenergies and forces.\nTo test the prediction of the NNP, a test set was cre-\nated by sampling random structures throughout all sim-\nulations. In addition, with the final committee, MD sim-\nulations exceeding 3 ns of the 2 ×2×2 systems were per-\nformed and 400 structures each of the pristine system and\nthe system containing an Na+\nias well as the system with\na V–\nNawere calculated with first principles and added to\nthe test set. In total, the test set contained 1302 con-\nfigurations with energies and forces. Let us denote the\ndataset obtained so far by\nA1={Ai,(Ei,fi)},\nwhere Aiare the configurations, and Eiandfiare the\ncorresponding potential energies and forces, respectively,\nobtained in the supercell-calculations.\n2. Including Corrections in the Dataset\nThe dataset A1does not contain the finite-size cor-\nrections stipulated in Equation (1) to calculate defect\nformation energies. Since machine-learning force-fields\nallow us to go to large system sizes at which the spuri-\nous long-range Coloumb interaction between a charged\ndefect and its periodic images may actually become neg-\nligible, it was not clear if electrostatic finite-size effects\nshould be included or corrected for in the reference data.\nTo ensure a proper force-field that treats these charged\ndefects consistently, we created a second labeled dataset:\nA2={Ai,(Ei+Ecorr(Ai),fi)},\nwhere Ecorr(Ai) is the electrostatic finite size correction\nthat depends on the configuration Ai. That is, corre-\nsponding to the system size and defect type, we add\nthe correction before training to incorporate it into the\nneural-network potential. Note that the Kumagai-Oba\ncorrection is only defined for relaxed systems. We there-\nfore identified Ecorr(Ai) for any finite temperature con-\nfiguration Aiby the correction of the relaxed system of\nthe same supercell-size and defect type.\nE. Heat Capacity, Elastic Constants and Phonons\nTo assess the performance of the NNP compared\nto its DFT-reference as well as to experimental data\nwe calculated the elastic constants and heat capacity.\nTo calculate the elastic constants at the DFT-level we\nused the stress-strain relationship [51] implemented in\nVASP. To calculate the elastic constants with NNPs\nwe used the stress-strain relationship as implemented in\nLAMMPS/examples/ELASTIC. The isochoric heat ca-\npacity CVwas determined through the phonon density\nof states obtained using the phonopy [52] codes for DFTand the NNPs respectively. To compare with experi-\nments the isochoric heat capacity CVwas transformed\nto the isobaric heat capacity CPthrough the relation\nCP=CV+TV α2K. Values for the expansion coefficient\nαand the bulk modulus K were taken from literature\n[53, 54]. A finite temperature anharmonic renormaliza-\ntion of the phonon band structure was performed using\nnormal-mode decomposition [55] as implemented in dy-\nnaphopy [56].\nF. Thermodynamic Integration and Parallel\nTempering\nOne of the goals of this work is to compute the equilib-\nrium concentrations c(T) of interstitial and vacancy de-\nfects as a function of temperature. We accomplish this by\ndetermining the free energy of formation of Frenkel pairs\nGFP(T), from which their concentration follows [57],\nc(T) = exp/parenleftbigg−GFP(T)\n2kBT/parenrightbigg\n. (3)\nThe formation free energy GFP(T) is calculated via ther-\nmodynamic integration following Cheng and Ceriotti [58]\nusing i-PI [59] and LAMMPS [40]. To achieve sufficient\nsampling also at low temperatures, we employ a paral-\nlel replica scheme [60] coupling isothermal-isobaric MD\nsimulations at different temperatures, from 100K up to\nthe melting point of 1400 K [61–63]. Note that this ap-\nproach takes into account full anharmonic effects, which\ncan become relevant at the higher temperatures [64, 65].\nDetails about this free energy calculation are included in\nthe Supporting Information.\nIII. RESULTS\nA. Performance of the Neural Network Potentials\nThe accuracy of the neural network potential com-\nmittees trained without charge corrections (NNPw/o)\nand with charge corrections (NNPw/corr) was first as-\nsessed by comparing the predicted forces and total en-\nergies with the reference data for the test set. The test\nset of NNPw/corr was appropriately shifted just like its\ntraining set A2.\nAs can be seen in Fig. 1 and Tab. II, the NNP pre-\ndictions are close to the reference data with an accuracy\nthat is on par with similarly constructed neural-network\npotentials [22] or other types of machine-learning force\nfields [48]. In particular, the errors for NNPw/o and\nNNPw/corr are essentially the same.\nB. Heat Capacity, Elastic Constants and Unit Cell\nIn Fig. 2 (a) the elastic constants obtained from DFT\nas well as with the NNP with (NNPw/corr) and without5\nFIG. 1. a) Deviation of the predicted energy from the\nreference energy vs. reference energy for NNPw/o and\nNNPw/corr, b) Predicted forces vs. reference forces for\nNNPw/o and NNPw/corr.\nTABLE II. Root mean squared errors for the test set ob-\ntained from the committee trained without charge correction\n(NNPw/o) and and with charge corrections (NNPw/corr).\nRMSE NNPw/o NNPw/corr\n∆E/bracketleftbigmeV\natom/bracketrightbig\n0.40 0.39\n∆F/bracketleftbigeV\n˚A/bracketrightbig\n0.065 0.065\n(NNPw/o) corrections are shown. The figure also in-\ncludes results of DFT calculations of Kaercher et al. [16]\nand experimental results of Brown et al. [53] and Rhy-\nzova [66]. Note that in the calculations of Kaercher et\nal. [16] the volume of the system was fixed at the exper-\nimental density during relaxation, making their results\ncorrespond to a high pressure. The results for NNPw/o\nand NNPw/corr almost fall on top of each other and both\nreproduce the DFT generated constants well overall.\nIn Fig. 2 (b) the isobaric heat capacity obtained from\nDFT, NNPw/o, NNPw/corr and the experiments of\nBenisek et al. [67] and Salje et al. [68] are shown. The\nheat capacities we obtained with three methods trace\nSalje’s data, which are marginally higher than Benisek’s\nexperiments.\nIn Tab. III we list the unit cell parameters obtained\nfrom DFT, NNPw/o, and NNPw/corr as well as the ex-\nperimental results of Brown et al. [53]. As is common\nwith GGA-DFT, the lattice parameters are slightly over-\nestimated, and this is reproduced in both NNPs. In Fig. 3\nFIG. 2. a) Elastic constants obtained from the NNP, DFT\nand experiments. Here we use the orientation convention\nZ∥c,Y∥a×c,X∥Y×Z),b) Isobaric heat capacity CPvs.\ntemperature.\nwe plotted the change in unit cell dimensions as a func-\ntion of temperature for the NNPw/corr. As can be in-\nferred from the figure, the lattice parameters increase\nanisotropically with temperature. The change of the box\nangles is in the correct direction as feldspars become more\nmonoclinic as temperature increases [69].\nTABLE III. Unit cell parameters of Na-feldspar for the ex-\nperiment of Brown and coworkers [53] and fully relaxed unit\ncells of GGA and the two NNPs.\na [˚A] b [ ˚A] c [ ˚A]α[deg] β[deg] γ[deg]\nGGA 8.260 12.933 7.250 94.210 116.489 87.555\nNNPw/corr 8.254 12.929 7.250 94.171 116.474 87.556\nNNPw/o 8.248 12.935 7.250 94.165 116.471 87.573\nBrown et al. 8.137 12.786 7.158 94.253 116.605 87.756\nC. Defect Formation Energies at 0K\nIn the feldspar structure illustrated in Fig. 4 a) an\ninterstitial atom can occupy only a restricted number\nof positions. We started by placing a sodium cation at\nthe (0 ,0,1\n2) position, as proposed by Petrovic [26], and\nrelaxed the system. This interstitial is refereed to as\nNa+\n(001\n2)and is shown in Fig. 4 b). In this configuration,\nthe inserted cation lies on the same (010) plane as its6\nFIG. 3. Percent changes of the edge lengths a, b,andc(lefty-\naxis) of the unit cell as a function of temperature with respect\nto the unit cell at 0K for NNPw/corr. Also shown are the\nangles α, βandγof the unit cell (right y-axis).\nneighboring sodium ions, to which it has a large separa-\ntion. The two nearest sodium neighbors are 4 .17˚A away\nand the two next-nearest are at a distance of 4 .34˚A,\nsuch that the defect is at an electrostatically favorable\nposition as argued by Petrovic. While this defect is sta-\nble at low temperatures, on heating the system as de-\ncribed in Sec. II D 1 we observed that at a temperature\nof roughly 600 K the defect transitions to the dumbbell-\ntype defect Na+\nDBin which two Na+share a single alkali\nlattice site, as shown in Fig. 4 c). Such a dumbbell con-\nfiguration is rare for ionic crystals due to electrostatic\nrepulsion, but the large cavities of the feldspar frame-\nwork seem to allow for this extra ion. The two Na+ions\nlie outside the (010) plane spanned by the other sodiums.\nThey are only 2 .67˚A apart but have a comparable sep-\naration to their next-nearest neighbors as conventionally\noccupied Na-sites. We also initialized a supercell with a\nsodium interstitial between two neighboring Na-sites in-\nside the folded eight-membered ring as was suggested by\nBehrens and coworkers [25]. However, this configuration\nwas unstable and reconfigured to the dumbbell configu-\nration during relaxation. The final defect we discuss is\nthe sodium vacancy V–\nNapictured in Fig. 4 d).\nThe conventional unit cell of Na-feldspar contains 4\nformula units in a C 1 space group. A single formula unit\nsupports only half of one (0 ,0,1\n2)-site whereas the dumb-\nbell and vacancy can appear once for every sodium site.\nThe conventional unit cell therefore supports 2 (0 ,0,1\n2)-\nsites, 4 vacancy-sites and 4 dumbbell-sites, each of them\nequivalent up to symmetry, respectively. We convinced\nourselves of this explicitly for the dumbbell configuration\nand noticed that its axis remains the same on all possible\nlocations. This implies that the two Na+are equidistant\nto the centre of inversion, which is situated in the middle\nof the eight-membered ring of tetrahedrons.\nIn Tab. IV we show the formation energies calculated\nfor the defects in individual supercells as well as in sys-tems containing both interstitial and vacancy defects\ntogether. The defect energies, calculated according to\nEqu. (1), were obtained for GGA, GGA including elec-\ntrostatic corrections, the NNP committee trained on a\nreference without corrections (NNPw/o) as well as the\nNNP committee trained on a reference including correc-\ntions (NNPw/corr). The KO finite size charge correction\nfor the charged defects is relevant in the 2 ×1×2 and only\nmarginally less so in the 2 ×2×2 system. The corrected for-\nmation energy does however seem to be converged rather\nwell already in the 2 ×2×2 system, decreasing only by\n0.03 eV compared to the smaller system for both V–\nNaand\nNa+\n(001\n2).\nTo calculate the defect energy according to Equ. (1)\nwe use the chemical potential of sodium bcc computed\natT= 0 and P= 0 with GGA. The electron energy at\nthe valence band maximum, ϵVBM, of the perfect feldspar\ncrystal was also determined with GGA under the same\nconditions. Note that in Ref. [70], GGA formation ener-\ngies of sodium cation vacancies were computed without\nseparating elemental and electronic contributions to the\nformation energy. Instead, the authors used the energy\nof a charged sodium ion in vacuum as reference, resulting\nin a defect formation energy that is 9 eV higher than the\ncorresponding energy shown in Tab. IV.\nThe necessity of including NNPw/o and the\nNNPw/corr in this study becomes apparent when looking\nat the defect formation energies shown in the rightmost\ncolumns of Tab IV. Generally, NNPw/o and NNPw/corr\nseem to reproduce the uncorrected and corrected DFT\nreference, respectively. The precision at which this is ac-\ncomplished, however, favors NNPw/corr. In particular,\nfor the formation energy of the Frenkel pair, shown in the\nlast row of Tab. IV, NNPw/corr predicts a value quite\nFIG. 4. Unit cell of albite (a) and three possible defects (b-\nd). White tetrahedrons are [SiO 4]–4, black ones are [AlO 4]–5\nand yellow spheres are Na+. The defects are shown relative to\nthe central eight-membered ring of tetrahedrons. The three\ndefects, shown in side- and top-view, are: (b) the Na+\n(001\n2)\ndefect, (c) the dumbbell Na+\nDBand (d) the vacancy V–\nNa. The\nconfigurations shown above were relaxed using NNPw/corr.\nNote that the unit cell used here is shifted by (1\n21\n21\n2) with\nrespect to the origin of the unit cell usually used for feldspar.7\nclose to the corrected reference, while NNPw/o does nei-\nther predict the uncorrected energy nor the corrected one\nwith acceptable precision. Note that this formation en-\nergy of the Frenkel pair is consistent with the upper limit\nof∼2.4 eV of the enthalpic contribution to the forma-\ntion free energy, which was deduced by El Maanaoui and\ncoworkers [10] using ionic conductivity measurements in\nK-rich feldspar.\nTABLE IV. Defect formation energies at 0K for various de-\nfects after full ionic and volumetric relaxation. The defect\nenergies were calculated according to Equ. (1) using the in-\nternal energy per atom of sodium bcc, µNa=−1.31 eV, and a\nvalence band maximum of ϵVBM = 1.18 eV, both determined\nin GGA. The Fermi level was set at the VBM, ∆ EFL= 0.\nThe defect energy in the last row is that of the lowest energy\nFrenkel pair in the 2 ×2×2 system resulting from summation\nof the energies in rows 1 and 3.\nFormation Energies Eform[eV]\nGGA NNP\nDefect Type System qw/o w/corr w/o w/corr\nV–\nNa 2×2×2−15.11 5 .29 4.91 5 .20\nV–\nNa 2×1×2−15.03 5 .32 4.91 5 .21\nNa+\nDB 2×2×2 +1−3.34−3.16−3.38−3.13\nNa+\n(001\n2) 2×2×2 +1−3.00−2.82−3.10−2.83\nNa+\n(001\n2) 2×1×2 +1−3.06−2.79−3.08−2.81\nV–\nNa+Na+\nDB 2×2×2 +0 1.87 - 1.49 2 .03\nV–\nNa+Na+\n(001\n2) 2×2×2 +0 2.27 - 1.79 2 .35\nLowest energy Frenkel pair +0 1.77 2 .13 1.53 2 .07\nTABLE V. Defect Formation Energies calculated with GGA\nwithout altering the number of electrons in the supercells.\nDefect Type System qEform[eV]\nV0\nNa 2×2×2 +0 5.15\nNa0\nDB 2×2×2 +0 1.53\nNa0\n(001\n2) 2×2×2 +0 1.91\nNa0\n(001\n2) 2×1×2 +0 1.92\nLowest energy Frenkel pair +0 6.68\nThe need for corrections is most visible in the defect\nformation energy of the Frenkel pairs in a single supercell,/parenleftbig\nV–\nNa+Na+\nDB/parenrightbig\nand/parenleftbig\nV–\nNa+Na+\n(001\n2)/parenrightbig\n. As the Frenkel defect\npair has no total charge, no charge correction is needed in\nthis case. However, due to the finite size of the cell there\nremains an attraction between the interstitial defect and\nthe vacancy, which decreases the formation energy com-\npared to the sum of formation energies of V–\nNaand Na+\nDB\nor Na+\n(001\n2). In both cases, the NNPw/corr formation en-\nergy is closer to the reference than the NNPw/o result.\nSince no electrostatic correction needs to be applied in\ncalculating the single Frenkel pair, and no configuration\nof a Frenkel pair in a single system was included in thetraining data, the good agreement of the NNPw/corr cor-\nroborates the validity of the electrostatic correction and\nindicates that it is a necessary ingredient in the construc-\ntion of the training set.\nTo ensure that we correctly identified the lowest en-\nergy charge state of the Frenkel pair defect, we also\ncalculated the defect formation energies without alter-\ning the numbers of electrons and listed these energies in\nTab. V. The last row shows the sum of uncharged V0\nNa\nand Na0\nDB, which corresponds to a Frenkel pair without\ncharge transfer between vacancy and interstitial. The for-\nmation energy of this Frenkel pair is much higher than its\ncounterpart with charge transfer shown in the last row\nof Tab. IV. Since the latter is also similar in value to the\ndirect Frenkel pair V–\nNa+Na+\nDBin the single supercell, we\nconclude that it is the appropriate lowest energy charge\nstate.\nThe energies shown in Tabs. V and IV hold informa-\ntion on the transitions between charge states. The differ-\nence in formation energy of the interstitial in its neutral\nand charged state is 4 .69 eV in the case of the dumbbell\nand 4 .73 eV in the case of Na+\n(001\n2). An optical transi-\ntion/parenleftbig\nNa0\nDB+h Na+\nDB+ℏω/parenrightbig\nwith hbeing a hole in\nthe valence band would yield a photon with an energy\nof 4.65 eV, where slight dissimilarities in the local lattice\nrelaxations of Na0\nDBand Na+\nDBare taken into account.\nPossibly, this transition is related to a prominent peak in\na recent XEOL spectrum of Na-rich feldspar at 4.5 to 4.3\neV (for 300 to 100 K) that has not been categorized yet\n[71]. Moreover, Garcia and coworkers [72] have detected\na peak at 4 .3 eV and speculated that it originates from\nNa+at interfaces. Finally, a peak is also visible at 4 .5 eV\nin the phosphorescence spectrum of Na-feldspar [73]. An\nargument against a connection between Na0\nDB Na+\nDB\nand those peaks is that the latter study also reports a\nsmall peak at 4 .4 eV in the phosphorescence spectra of 2\nout of 4 specimens of K-rich feldspar. All these systems,\nhowever, contain also some amount of Na (as do all nat-\nural occurring Alkali feldspars) and we suspect that in\nAlkali feldspars of any composition, the dumbbell defect\nshould be the most favorable interstitial defect as both\nK-K dumbbells and Na-K dumbbells would be sterically\nunfavorable compared to Na+\nDB.\nD. Finite Temperature Behaviour\n1. Free Energy of Defect Formation\nUsing thermodynamic integration starting from a har-\nmonic reference model we calculated the free energies at\nfinite temperatures and atmospheric pressure for a pris-\ntine system, a system containing V–\nNaand a system con-\ntaining Na+\nDB. Details of the thermodynamic integration\nprocedure, combined with parallel tempering, and the re-\nsulting free energies are provided in the Supplementary\nInformation. Based on these individual free energies we\ndetermined the free energy of formation of the Frenkel8\npair as a function of temperature\nGFP=GV–\nNa+GNa+\nDB−2Gpristine ,\nwhich is plotted in Fig. 5. Interestingly, the free en-\nergy decreases monotonically with increasing tempera-\nture, but goes through an inflection point at around\n650 K. Below this temperature the free energy of for-\nmation decreases in a convex fashion and above it it de-\ncreases concavely.\nWe note that at temperatures around and above the\ninflection point we observe transitions of the interstitial\nin its Na+\nDBconfiguration to the Na+\n(001\n2)defect. This de-\nfect state occurs very infrequently and when it does it is\nshort-lived, quickly transitioning back to Na+\nDB. Hence,\nthe inflection point in the defect free energy might be\nrelated to additional configuration space volume that be-\ncomes accessible at higher temperatures.\n2. Discontinuity in the Dumbbell Orientation and a\nDynamical Instability\nFurther insights into the nature of the dumbbell defect\ncan be obtained by analyzing the normal mode frequen-\ncies of the system with and without defect. In Fig. 6 we\nshow the lowest normal mode frequency at the Γ-point of\nsystems containing a dumbbell and of the pristine crystal\nas a function of temperature. We calculated the normal\nmode frequencies using the harmonic approximation or\nnormal-mode decomposition, as described in Sec. II E,\nfor the box dimensions corresponding to the respective\ntemperatures (cf. Fig. 3). The box dimensions as a func-\ntion of temperature where determined from parallel tem-\npering simulations. Close to the instability temperature\ndiscussed below additional box sizes were obtained by in-\nterpolation and at higher temperature by extrapolation.\nFIG. 5. Formation free energy GFPwith error bars of the\nFrenkel pair as a function o temperature T. The estimation of\nthe error bars is described in the Supplementary Information.\nAlso shown in the resulting defect concentration cobtained by\napplying Eq. (3). Defect concentrations are given in defects\nper formula unit (d.p.f.u).In the system with the dumbbell, at a temperature\nof about 752 K a frequency vanishes and reappears at\na different value. Such a discontinuity implies that a\nminimum in the potential energy landscape disappears,\nafter which the system relaxes to a different minimum.\nSince we do not observe the same discontinuity in the\npristine system, we conclude that this effect is related to\na mechanical instability of the dumbbell defect.\nA closer analysis reveals that the instability at 752 K\nis related to the orientation of the axis of the dumbbell.\nFigure 7 (upper right) shows this axis at the potential en-\nergy minimum for all box dimensions in a stereographic\nprojection. At 0 K it has an almost 45 deg angle to the\n[010]-direction (cf. Fig. 4 c). With increasing tempera-\nture this angle slightly decreases and at 752 K it jumps\nby about 22 .5 deg. Upon further temperature increase\nthe angle decreases to almost 0 deg at 1400 K.\nThe lower parts of Fig. 7 show the potential energy\nlandscape as a function of the axis orientation for differ-\nent temperatures. To determine these energy landscapes,\nwe held the axis of the two Na+in the dumbbell fixed\nat random values and relaxed all other degrees of free-\ndom (including the distance of the dumbbell atoms to\neach other). Only samples with a separation of the two\nNa+smaller than 3 .8˚A are counted as dumbbell defects.\nConfigurations with a larger separation correspond to a\nNa+\nDBat a different alkali site or even an Na+\n(001\n2)defect.\nAcross all box dimensions the dumbbell axis shows\nhigh anisotropy and energetically favors orientations be-\ntween [010] and [00 1]. Orientations between [010] and\n[100] are energetically much more costly. As the box\ndimensions increase with temperature, the well between\n[010] and [00 1] flattens and at 752 K a new minimum\nemerges. As only one alkali site is considered, the po-\nFIG. 6. The lowest normal mode frequency at the Γ-\npoint. Here anharmonic renormalization refers to normal-\nmode-decomposition [56]. The first minimum at 751 .91 K\nis related to a discontinuity in the orientational axis of the\ndumbbell. The soft mode at 1325 .60 K disappears when in-\ncluding anharmonic effects.9\ntential energy landscape would have to be appropriately\ninverted for an Na+\nDBat the alternative alkali site.\nThe necessity for using thermodynamic integration to\nincorporate anharmonic effects in the free energies of\nFig. 5 is evidenced by the appearance of a soft mode\nat about 1325 K in the harmonic approximation in both\npristine and interstitial systems, as shown in Fig 6. Be-\nlow 1300 K the crystal is dynamically stable, but between\n1300-1400 K an optical mode almost vanishes and an\nacoustic band exhibits a dynamical instability. An ex-\ncerpt of the bandstructure at 1325 .60 K is shown in Fig. 8.\nCuriously, the crystal regains its harmonic stability if we\nFIG. 7. The orientation of the dumbbell axis is shown using\na stereographic projection, visualized in the upper left. To\nobtain this projection, the axis at Pwhich lives on the upper\nhemisphere is mapped into the point P′of the image plane.\nThe [010] direction is kept at 0 rad latitude and the [100] at\n0 deg longitude. In the upper right panel, the axis orientation\ncorresponding to the energy minimum at all temperatures of\nFig. 6 is plotted and the discontinuity in latitude occurring at\n751.91 K is evident. The lower part shows the potential energy\nlandscapes sampled for different box dimensions. The axis\norientation corresponding to the energy minimum is indicated\nby a star.\nFIG. 8. Phonon bandstructure in the harmonic approxima-\ntion (blue) and with anharmonicity included using normal-\nmode decomposition (red) at 1325 .60 K. Only a step of the\nband path suggested in [74] is shown. The complete band\nstructure is included in the Supporting Information.\nextrapolate the box dimensions beyond 1400 K. The dy-\nnamical instability in the bandstructure disappears com-\npletely after renormalizing the bands by incorporating\nfinite temperature effects using normal-mode decompo-\nsition. This shows that the dynamical instability is a\nharmonic artifact and that anharmonic effects are essen-\ntial in modelling Na-feldspar at these high temperatures.\nIV. CONCLUSIONS\nIn this work we presented an NNP, purpose-built for\nNa-feldspar (Albite) and its defects, and demonstrated\nits applicability to simulate this system by predicting\nrelevant physical properties. In constructing the poten-\ntial we made use of on-the-fly machine learning initially\nand then expanded the dataset using an NNP commit-\ntee machine. This way to proceed required less human\nintervention than typically needed for NNPs and should\nbe widely applicable.\nWe have shown that electrostatic corrections need to\nbe included into the training dataset for the NNP. This\nis particularly important, when defect formation energies\nare of interest. Including corrections after training pro-\nduces a potential that cannot transfer to systems that are\nmade up of multiple defects that neutralize each other,\nsuch as Frenkel pairs in Na-feldspar.\nBased on our NNP, we have found a new kind of inter-\nstitial defect in the dumbbell configuration Na+\nDB, where\ntwo Na+cations share a single site on the alkali sub-\nlattice. This defect is the energetically most favorable,\nbut the Na+\n(001\n2)interstitial predicted previously also ap-\npears at elevated temperatures. These different defect\nconfigurations as well as the instability in the axis orien-\ntation of the dumbbell defect might affect the free energy\nof formation of the Frenkel pair, which we found to go10\nthrough an inflection at elevated temperatures.\nA promising future application of the NNP develped\nin our work will be the direct determination of diffusion\ncoefficients of defects using MD. Combined with the free\nenergy of defect formation, it will be possible to com-\npare these with experimentally determined tracer diffu-\nsion coefficients and elucidate the underlying diffusion\nmechanism. A second topic of interest is the extension\nof this NNP potential to include also potassium feldspar\nto study mixed systems of both sodium and potassium\nfeldspar, if possible including disorder in the sites oc-\ncupied by silicon and aluminum. These points will be\naddressed in future research.V. ACKNOWLEDGMENTS\nWe would like to thank Ferenc Karsai and the VASP\nteam for supporting us in using the on-the-fly method-\nodology in its then development stage. We further ac-\nknowledge financial support from the Austrian Science\nFund (FWF) through Grant No. I 4404 and through the\nSFB TACO, Grant No. F-81. The computational results\npresented were achieved using the Vienna Scientific Clus-\nter (VSC).\nVI. DATA AVAILABILITY\nThe training and test data, the NNPw/corr,\na template for molecular dynamics as well as\nLAMMPS-DATA configurations of the pristine, Na+\n(001\n2),\nNa+\nDB, V–\nNa, V–\nNa+Na+\nDBand V–\nNa+Na+\n(001\n2)systems\nare openly available through the Zenodo repository\n(https://doi.org/10.5281/zenodo.10630971).\n[1] M. Dondi, Feldspathic fluxes for ceramics: Sources, pro-\nduction trends and technological value, Resources, Con-\nservation and Recycling 133, 191 (2018).\n[2] I. Parsons, J. D. Gerald, and M. R. Lee, Review. 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Rojo, Ultraviolet-blue ionic luminescence of alkali\nfeldspars from bulk and interfaces, Physics and Chem-\nistry of Minerals 26, 658 (1999).\n[73] M. R. Baril and D. J. Huntley, Infrared stimulated lu-\nminescence and phosphorescence spectra of irradiated\nfeldspars, Journal of Physics: Condensed Matter 15,\n8029 (2003).\n[74] Y. Hinuma, G. Pizzi, Y. Kumagai, F. Oba, and I. Tanaka,\nBand structure diagram paths based on crystallography,\nComputational Materials Science 128, 140 (2017).1\nSupplemental Materials: Structure and thermodynamics of defects in Na-feldspar\nfrom a neural network potential\nS-I. ELECTROSTATIC CORRECTION OF KUMAGAI AND OBA\nThe potentials involved in the KO-correction, defined in [1], are plotted using the Spinney package [2] in Fig. 1\nfor V–\nNaand Na+\n(001\n2)in the 2×2×2system. In both cases the difference between the defect-induced potential Vq/bto\nthe point-charge potential VPC,qreaches a constant value inside the sampling region. Both charges seem to possess a\nsimilar degree of localization as their respective defect-induced potentials mirror each other.\nFIG. 1. Values of the different potentials involved to calculate the correction as functions of their distance to the defect. Colored\nin blue is the region used in sampling.\nS-II. THERMODYNAMIC INTEGRATION\nThermodynamic integration was performed using the approach described by Cheng and Ceriotti [3], except that we\nhave performed it at atmospheric pressure and have coupled the simulations at different temperatures with parallel\ntempering. Without parallel tempering, the free energy of defect formation at low temperatures did not converge\nproperly. We provide a short account of the process below, which we repeated for a pristine system measuring 2×2×3\nunit cells, the same with a vacancy V–\nNaand once more with the dumbbell interstitial Na+\nDB. The real/anharmonic\npotential of interest is given by NNPw/corr.\n1.Ionic and box degrees of freedom were relaxed at 0 K . Relaxing the triclinic box required some care.\nWe used a conjugated gradient descent until the energy varied less than 10−10eV and continued with a steepest\ngradient descent until the energy varied less than 10−12eV. The minimization was done with LAMMPS at\natmospheric pressure, resulting in free energies G(P= 1.013 25 bar , T= 0 K).\n2.NPT simulations were performed between 100 K to 1400 K at atmospheric pressure with parallel\ntempering . The replicas were equilibrated at the chosen temperatures in Tab. I with parallel tempering in the\nNPT ensemble [4] for 1 ns. This corresponded to about twice the time the simulation that started at 841 .80 K\ninitially, took to reach the lowest temperature through exchanges. The run continued for at least 4 ns more in\nwhich produced energies and boxes were saved every 10 fs. The acceptance rate was kept approximately equal\nfrom 100 K to 841 .80 K by distributing these target temperatures exponentially. From 900 K to 1400 K they were\ndistributed linearly instead, since simulations at these temperatures can be expected to be well-equilibrated by\nthemselves already and to allow for a more dense distribution at the high temperatures which are of increased\ninterest to study diffusion. Swaps were attempted every 0 .1 ps, the timestep was 1 fs and the center of mass was\nconstrained in all replicas. The anharmonic enthalpy, defined in [3], was determined and plotted in Fig. 2.\n3.Equilibrium configuration and phonon modes at 100 K The ionic degrees of freedom of a system with the\naveraged box dimension hfor the replica at 100 K were relaxed, and then the Hessian matrix at this minimumarXiv:2402.14640v1  [cond-mat.mtrl-sci]  22 Feb 20242\nwas determined with i-PI [5]. The vibrational modes of the Hessian were then used to describe a reference\nharmonic crystal with a known free energy to allow determining the free energy of the anharmonic potential\nNNPw/corr by thermodynamic integration from the reference.\n4.Integration from the reference harmonic to the real anharmonic potential in NVT. We switched\nfrom the reference harmonic hamiltonian Hhto the anharmonic NNPw/corr Haat 100 K by increasing λfrom\n0 to 1 in the effective Hamiltonian:\nH(λ) = (1 −λ)Hh+λHa.\nThe integration was done in steps of ∆ λ= 0.1 and the results are illustrated in Fig. 3.\n5.Putting everything together. We used the annotated script provided in the SI of [3] to obtain the Helmholtz\nfree energy at 100 K. To transform it to the Gibbs free energy at the same temperature we had to extend the\nprocedure in the script to triclinic systems. The Gibbs free energy is obtained from:\nG(P, T) =A(h, T) +Tdet(h) +kBTlnρ(h|P, T),\nwhere ρ(h|P, T) is the probability to observe the cell parameters hin the ensemble of PandT. To determine this\nprobability from the results of recorded box sizes we first determined the deviations of recorded box dimension\nat the NPT replica at 100 K to their average hfor each of the 6 degrees of freedom, resulting in different\nrealizations of\n∆h=\n∆XX 0 0\n∆XY ∆Y Y 0\n∆XZ ∆Y Z ∆ZZ\n.\nThe probability was then determined by calculating the Frobenius norm ||∆h||=/radicalig\nΣij∆h2\nijfor each realization\nand approximating ρas:\nρ(h|P, T)≈N/summationdisplay\ni1x<r(∥∆hi∥)\nN·V6 ball(r),\nin which 1x<r(x) is the identity for values xsmaller than rand zero otherwise. The expression is a histogram\nin which we count deviations whose Frobenius norm is smaller than ras realizations of h, normalized by all\npossible realizations. This is identical to the procedure taken in [3] for the orthogonal box, except with the\naddition of the three additional box degrees of freedom and the appropriate V6 ball instead of V3 ball. The radius\nof the ball rin which we count the deviation as a realization was varied until a local maximum was found.\nUsing the 2-norm instead of the Frobenius produced an almost identical result. The difference of the term\nkBTlnρ(h|P, T) between pristine, V–\nNaand Na+\nDBwas less than 2 meV in any case.\nWe could then carry on with the process [3] by including a correction of the constrained center of mass, integrating\nthe free energies over temperature resulting in Fig. 4 and finally generating the free energies of formation for\nthe Frenkel pair given in Tab. I with the relation:\nGfrenkel pair =Gvacancy +Ginterstitial −2Gpristine ,\nwhich does not contain temperature dependent chemical potentials, which would be needed to calculate the\nfree energy of formation for the individual defects separately.\nS-III. NORMAL MODE DECOMPOSITION\nAnharmonic features of the phonon bandstructure for the pristine system were determined using the method of\nnormal-mode decomposition [6] implemented in dynaphopy [7]. Supercells of size 2×2×2were first equilibrated in\nthe NVT-ensemble for 250 ps using a Langevin-thermostat. Then they were equilibrated in the NVT-ensemble for\nanother 250 ps using a Nose-Hoover thermostat. After which a trajectory was produced for 3 ns in the NVT-ensemble\nusing the same Nose-Hoover thermostat. A timestep of 1 fs was used and configurations containing the velocities of3\nFIG. 2. Anharmonic contribution to the enthalpy.\nFIG. 3. Average difference between potential energies of harmonic and anharmonic hamiltonians in the ensemble of the effective\nHamiltonian determined by λ.\nall particles were saved each timestep.\nFigure 5 shows the complete anharmonic bandstructure together with the harmonic approximation at the box di-\nmensions showing a minimum in an optical mode. The imaginary frequencies in the harmonic approximation do not\ndisappear for additional Q-points and the plotted harmonic bandstructure results from a 6 ×6×6 supercell. The fact\nthat the soft mode disappears below 1300 K and if we extrapolate above 1400 K in the harmonic approximation is\nevidenced in the bandstructures of these respective temperatures in Fig. 6 and Fig. 7. The imaginary frequencies have\nalmost disappeared in both cases.4\nFIG. 4. Gibbs free energies.\nTABLE I. Gibbs free energies of formation of the frenkel pair at atmospheric pressure.\nT[K]GFP[eV] Uncertainty [eV] T[K]GFP[eV] Uncertainty [eV]\n105.614 2.0015 0.0007 370.990 1.9432 0.0117\n111.544 1.9995 0.001 391.819 1.9407 0.0126\n117.807 1.9973 0.0013 413.817 1.9389 0.0136\n124.421 1.9952 0.0016 437.051 1.937 0.0146\n131.407 1.9931 0.0019 461.589 1.9344 0.0157\n138.784 1.9909 0.0022 487.505 1.932 0.0169\n146.576 1.9883 0.0025 514.875 1.9296 0.0182\n154.806 1.9856 0.0028 543.783 1.9271 0.0195\n163.497 1.9831 0.0031 574.313 1.9246 0.021\n172.677 1.9806 0.0035 606.558 1.922 0.0225\n182.372 1.9781 0.0039 640.613 1.9195 0.0242\n192.611 1.9756 0.0043 676.580 1.9175 0.0259\n203.425 1.9731 0.0047 714.566 1.9142 0.0278\n214.846 1.9707 0.0051 754.685 1.9094 0.0298\n226.909 1.9681 0.0056 800 1.9053 0.032\n239.648 1.9655 0.0061 841.807 1.9018 0.0342\n253.103 1.9629 0.0067 900 1.8947 0.037\n267.314 1.9603 0.0073 1000 1.8807 0.0413\n282.322 1.9576 0.0079 1100 1.8636 0.0457\n298.173 1.9549 0.0086 1200 1.844 0.05\n314.914 1.9522 0.0092 1300 1.8228 0.0544\n332.594 1.9492 0.01 1400 1.7975 0.05885\nFIG. 5. The phonon bandstructure in the harmonic approximation (blue) and using normal-mode-decomposition (red) at the\ninterpolated box dimensions corresponding to 1325 .60 K. The path is the one suggested in [8] for the reciprocal “reduced” cell\nhaving all-acute interaxial angles.6\nFIG. 6. 1300 K.\nFIG. 7. 1400 K.7\n[1] Y. Kumagai and F. Oba, Electrostatics-based finite-size corrections for first-principles point defect calculations, Physical\nReview B 89, 195205 (2014).\n[2] M. Arrigoni and G. K. Madsen, Spinney: Post-processing of first-principles calculations of point defects in semiconductors\nwith python, Computer Physics Communications 264, 107946 (2021).\n[3] B. Cheng and M. Ceriotti, Computing the absolute gibbs free energy in atomistic simulations: Applications to defects in\nsolids, Physical Review B 97, 054102 (2018).\n[4] T. Okabe, M. Kawata, Y. Okamoto, and M. Mikami, Replica-exchange monte carlo method for the isobaric–isothermal\nensemble, Chemical Physics Letters 335, 435 (2001).\n[5] V. Kapil, M. Rossi, O. Marsalek, R. Petraglia, Y. Litman, T. Spura, B. Cheng, A. Cuzzocrea, R. H. Meißner, D. M. Wilkins,\nB. A. Helfrecht, P. Juda, S. P. Bienvenue, W. Fang, J. Kessler, I. Poltavsky, S. Vandenbrande, J. Wieme, C. Corminboeuf,\nT. D. K¨ uhne, D. E. Manolopoulos, T. E. Markland, J. O. Richardson, A. Tkatchenko, G. A. Tribello, V. V. Speybroeck,\nand M. Ceriotti, i-pi 2.0: A universal force engine for advanced molecular simulations, Computer Physics Communications\n236, 10.1016/j.cpc.2018.09.020 (2019).\n[6] T. Sun, D.-B. Zhang, and R. M. Wentzcovitch, Dynamic stabilization of cubic CaSiO 3perovskite at high temperatures and\npressures from ab initio molecular dynamics, Physical Review B , 094109.\n[7] A. Carreras, A. Togo, and I. Tanaka, Dynaphopy: A code for extracting phonon quasiparticles from molecular dynamics\nsimulations, Computer Physics Communications 221, 221 (2017).\n[8] Y. Hinuma, G. Pizzi, Y. Kumagai, F. Oba, and I. Tanaka, Band structure diagram paths based on crystallography, Com-\nputational Materials Science 128, 140 (2017)."    },    {        "title": "2402.14643v1.The_Spatial_Whitham_Equation.pdf",        "content": "The Spatial Whitham Equation\nJohn D. Carter∗1, Diane Henderson2, and Panayotis Panayotaros3\n1Mathematics Department, Seattle University, 901 12th Ave, Seattle, WA 98122, USA\n2Department of Mathematics, Penn State University, 218 McAllister Building, University\nPark, PA 16802\n3Instituto de Investigaciones en Matem´ aticas Aplicadas y en Sistemas, Universidad\nNacional Aut´ onoma de M´ exico, Apdo. Postal 20-716, 01000 M´ exico D.F., M´ exico\nFebruary 23, 2024\nAbstract\nThe Whitham equation is a nonlocal, nonlinear partial differential equation that models the tem-\nporal evolution of spatial profiles of surface displacement of water waves. However, many laboratory\nand field measurements record time series at fixed spatial locations. In order to directly model data of\nthis type, it is desirable to have equations that model the spatial evolution of time series. The spatial\nWhitham equation, proposed as the spatial generalization of the Whitham equation, fills this need. In\nthis paper, we study this equation and apply it to water-wave experiments on shallow and deep water.\nWe compute periodic traveling-wave solutions to the spatial Whitham equation and examine their\nproperties, including their stability. Results for small-amplitude solutions align with known results for\nthe Whitham equation. This suggests that the systems are consistent in the weakly nonlinear regime.\nAt larger amplitudes, there are some discrepancies. Notably, the spatial Whitham equation does not\nappear to admit cusped solutions of maximal wave height. In the second part, we compare predictions\nfrom the temporal and spatial Korteweg-deVries and Whitham equations with measurements from lab-\noratory experiments. We show that the spatial Whitham equation accurately models measurements\nof tsunami-like waves of depression and solitary waves on shallow water. Its predictions also compare\nfavorably with experimental measurements of waves of depression and elevation on deep water. Ac-\ncuracy is increased by adding a phenomenological damping term. Finally, we show that neither the\nspatial nor the temporal Whitham equation accurately models the evolution of wave packets on deep\nwater.\n1 Introduction\nWe investigate the spatial Whitham (sWhitham) equation, proposed by Trillo et al. (2016), for gravity\nwaves propagating on the surface of water. This equation was motivated by the success of the temporal\nKorteweg-de Vries (tKdV) and temporal Whitham (tWhitham) equations in modeling the evolution of\nunidirectional waves in shallow water. The tWhitham equation uses a more accurate linear dispersive\nterm than the tKdV equation and therefore may be useful for depths ranging from shallow to deep.\nThe tWhitham equation can also approximate some singularity formation effects, see Whitham (1967)\nand Ehrnstr¨ om & Wahl´ en (2019). Comparisons with experimental data suggest the tWhitham equation\nimproves on the tKdV equation in the small-amplitude long-wave regime, see Trillo et al. (2016) and\nCarter (2018).\nA limitation of temporal models is that they describe how a given surface displacement profile evolves\nin time. However, many field and laboratory experiments provide temporal profiles of the surface dis-\nplacement at fixed spatial locations and ask how these temporal profiles evolve in space. This problem\n∗Corresponding author: carterj1@seattleu.edu\n1arXiv:2402.14643v1  [physics.flu-dyn]  22 Feb 2024arises in many wave phenomena, e.g. in optics, and there is a long use of “spatial evolution” equations\nthat describe how a signal, e.g. an optical image, changes along some direction in space. Despite their mo-\ntivation, spatial equations have not been used systematically for nonlinear dispersive water wave models.\nHerein, we examine a recently proposed model, the sWhitham equation of Trillo et al. (2016), and show\nevidence that this equation can improve on predictions of the tWhitham, tKdV, and sKdV equations. We\nconsider this equation mathematically, numerically, and experimentally and compare its properties and\npredictions with those of the tKdV, tWhitham, and sKdV equations. Our results yield new information\non the properties of the sWhitham equation and its strengths and limitations as a model of unidirectional\nwater waves.\nThe motion of a two-dimensional, irrotational, incompressible, inviscid fluid on a horizontal, imperme-\nable bed can be modeled by\nϕxx+ϕzz= 0,for−h0< z < η, (1a)\nϕt+1\n2|∇ϕ|2+gη= 0,atz=η, (1b)\nηt+ηxϕx=ϕz,atz=η, (1c)\nϕz= 0,atz=−h0, (1d)\nwhere ϕ(x, z, t ) represents the velocity potential of the fluid, η(x, t) represents the surface displacement of\nthe fluid, grepresents the acceleration due to gravity, h0represents the uniform depth of the fluid at rest,\nandx,z,trepresent the horizontal, vertical, and temporal coordinates, respectively. See Johnson (2001)\nfor details of this system. Since this system is a nonlinear free-boundary problem, approximate models\nare often used. Linear theory gives the dispersion relation\nω2=gκtanh( κh0), (2)\nwhere κandωare the wavenumber and angular frequency of the linear wave, respectively.\nThe dimensional temporal Korteweg & de Vries (1895) (tKdV) equation,\nηt+p\ngh0ηx+1\n6h2p\ngh0ηxxx+3\n2h0p\ngh0ηηx= 0, (3)\nis a small-amplitude, long-wave approximation of (1). The tKdV equation has been well studied math-\nematically (e.g. Miles (1981); Ablowitz & Segur (1981); Lannes (2013)) and experimentally (e.g. Russell\n(1844); Zabusky & Galvin (1971); Hammack (1973); Hammack & Segur (1974)). It has dispersion relation\nωKdV=p\ngh0\u0010\nκ−1\n6h2\n0κ3\u0011\n. (4)\nEquation (4) is a unidirectional approximation of (2) that is valid in the long-wave (i.e. κh0→0) limit.\nIn order to obtain a weakly nonlinear approximation to (1) that is valid for a wider range of κh0values,\nWhitham (1967, 1974) proposed the following dimensional equation\nηt+rg\nh0K(κ)∗η+3\n2h0p\ngh0ηηx= 0, (5)\nas a model for the evolution of small-amplitude waves on shallow water. We refer to (5) as the dimensional\ntemporal Whitham (tWhitham) equation. The convolution term is defined by\nK(κ)∗η=˜F−1\u0010\nK(κ)˜F(η)\u0011\n, (6)\nwhere K(κ) is the nondimensional Fourier multiplier given by\nK(κ) =isgn(κ)p\nκh0tanh( κh0), (7)\n2and ˜Fand ˜F−1represent the Fourier and inverse Fourier transforms in x, respectively. We write the\ntWhitham equation and Kin forms different than those used by Whitham in order to simplify the work\nbelow. The linear dispersion relation for the tWhitham equation is\nωW= sgn( κ)p\ngκtanh( κh0). (8)\nEquation (8) is one root of (2), which means that the tWhitham equation exactly models the wave speed\nof unidirectional linear waves for any κh0.\nGiven an initial spatial profile of the surface displacement (i.e. given ηfor all values of xatt= 0),\nequations (3) and (5) provide predictions for the spatial profiles of ηfort >0. Because of this requirement\nfor initial spatial data, these equations are known as evolution-in-time or “temporal” equations. Many\nlaboratory and field water wave measurements are made by recording time series at fixed spatial locations\n(i.e. using stationary gauges or buoys to collect time series). Evolution-in-space or “spatial” equations\nare required to directly compare with measurements of this type. In these situations, time series at the\nfirst measurement location are used as initial conditions in spatial equations that are solved to obtain\npredictions for the time series at the downstream measurement locations. Spatial equations are commonly\nused in nonlinear optics, e.g. in the the study of pulse propagation in optical fibers, see for example\nManakov (1974) and Agrawal (2019).\nTwo common methods used for deriving approximate spatial equations from equation (1) include: (i)\ninterchanging the roles of xandtin the derivation of the temporal equations or (ii) starting with an\napproximate temporal equation and then applying the change of variables ηt∼ −√gh0ηxfor waves in\nshallow water. Using either (i) or (ii) gives the dimensional spatial KdV (sKdV) equation\nηx+1√gh0ηt−√gh0\n6g2ηttt−3\n2h0√gh0ηηt= 0. (9)\nTrillo et al. (2016) proposed the following dimensional spatial Whitham (sWhitham) equation\nηx+s\nh0\ngK−1(ω)∗η−3\n2h0√gh0ηηt= 0, (10)\nwhere the convolution term is defined by\nK−1(ω)∗η=F−1\u0010\nK−1(ω)F(η)\u0011\n, (11)\nusingFandF−1to represent the Fourier and inverse Fourier transforms in t, respectively, and K−1(ω) as\nthe inverse of K(κ). Although a closed-form expression for K−1(ω) is not known, the inverse is guaranteed\nto exist since K(κ) is one-to-one and onto for κ∈R. In general, if the dispersion is monotonic and onto,\nthen the linear initial-value problem can be written in spatial form. In other words, one can recover the\nspatial initial condition from time series at x= 0 and vice versa. Note that if weak surface tension were\nincluded, then K(κ) would not be one-to-one and a unique inverse would not exist.\nIn the following, we consider both the mathematical properties of the sWhitham equation and its\napplications. Section 2 includes a summary of the mathematical properties including its conserved quan-\ntities, traveling-wave solutions, and their stability. Section 3 contains comparisons between measurements\nfrom laboratory experiments and predictions obtained from the tKdV, sKdV, tWhitham, and sWhitham\nequations, and their dissipative generalizations. Section 4 contains a summary of the results.\n2 Properties of the spatial equations\nIn order to gain an understanding of the sWhitham equation, we present its properties along with the\nproperties of the sKdV equation for comparative purposes. We define nondimensional variables u,χ, and\nτby\nu=η\nh0, χ =x\nh0, τ =rg\nh0t. (12)\n3The corresponding nondimensional sKdV equation is\nuχ+uτ−1\n6uτττ−3\n2uuτ= 0, (13)\nand the nondimensional sWhitham equation is\nuχ+K−1(w)∗u−3\n2uuτ= 0, (14)\nwhere w=p\nh0/g ω is the nondimensional circular frequency and Kis the nondimensional Fourier\nmultiplier defined by\nK(k) =isgn(k)p\nktanh( k), (15)\nwhere k=h0κis the nondimensional wavenumber.\n2.1 Conserved quantities\nThe sKdV equation is known to have an infinite number of conservation laws, see for example Ablowitz\n& Segur (1981). The first three are conservation of mass, momentum, and the Hamiltonian,\n˜Q1=Z∞\n−∞u dτ, (16a)\n˜Q2=Z∞\n−∞u2dτ, (16b)\n˜Q3=1\n2Z∞\n−∞\u0012\nu2+1\n6u2\nτ−1\n2u3\u0013\ndτ, (16c)\nrespectively. The sWhitham equation conserves mass, ˜Q1, momentum, ˜Q2, and its Hamiltonian\n˜Q4=1\n2Z∞\n−∞\u0012\nu\u0012K−1(w)\ni w\u0013\n∗u−1\n2u3\u0013\ndτ. (17)\nWe emphasize that the quantities ˜Q1through ˜Q4are constant in χ, the nondimensional spatial vari-\nable. The periodic generalizations of these conserved quantities, ˜Q1,˜Q2,˜Q3,˜Q4, for the tKdV, sKdV,\ntWhitham, and sWhitham equations are included in Appendix A.\n2.2 Traveling-wave solutions\nThe sKdV equation admits a two-parameter family of traveling-wave solutions given by\nu(χ, τ) =U0+U2cn2\u00122E1(m)\nT(τ−γχ), m\u0013\n, (18)\nwhere\nU0=−16\n3T2E1(m)\u0000\nE2(m) + (m−1)E1(m)\u0001\n, (19a)\nU2=16m\n3T2\u0000\nE1(m)\u00012, (19b)\nγ= 1 +8\n3T2(m−2)\u0000\nE1(m)\u00012+8\nT2E1(m)E2(m). (19c)\nHere Tis the (temporal) period of the solution, U2is the wave height of the solution, m∈[0,1) is\nknown as the elliptic parameter of the Jacobi elliptic function cn( ·, m), and E1(m) and E2(m) are the\ncomplete elliptic integrals of the first and second kinds, respectively. See Byrd & Friedman (1971) for\n4Figure 1: Plots of four 2 π-periodic, zero-mean, traveling-wave solutions to (a) the sKdV equation and (b)\nthe sWhitham equation. The γvalues and wave heights, H, are included in the legends.\ndetails of elliptic functions. The integration constant, U0, was chosen so that the solutions have zero mean\n(i.e.Q1= 0) because such solutions are the most physically relevant. This two-parameter family (the\nfree parameters are Tandm) comprises all zero-mean, traveling-wave solutions to the sKdV equation. A\nprofile of the form u(τ−γχ) of the spatial equation is interpreted as the u(−γ(x−γ−1t)) traveling-wave\nprofile of the temporal evolution. This means that the nondimensional real parameter γcorresponds to\nthe inverse of the nondimensional wave speed.\nFigure 1(a) contains plots of four 2 π-periodic solutions of the sKdV equation. The wave heights, H,\nandγvalues for these solutions are included in the legend. The sKdV equation does not admit a solution\nwith maximal height, nor does it admit a solution with minimal γvalue. As m→1,Hincreases without\nbound, and γdecreases without bound. Once the height of the solution becomes large enough, the γvalue\nbecomes negative. For solutions with period T= 2π, this sign transition occurs for solutions with height\nH≈4.28. Since γis the inverse of wave speed, γgoing through zero corresponds to the wave speed going\nto infinity. This nonphysical result may be due to the fact that the sWhitham equation is a generalization\nof the sKdV equation, which is a model of small-amplitude waves and these waves are well outside the\nsmall-amplitude regime.\nEhrnstr¨ om & Kalisch (2009) proved that the tWhitham equation admits small-amplitude, periodic\ntraveling-wave solutions and computed some of these solutions. Ehrnstr¨ om & Wahl´ en (2019) proved\nWhitham’s conjecture that the tWhitham equation admits a traveling-wave solution with maximal wave\nheight and that this solution is cusped. Carter (2023) numerically examined the properties of solutions to\nthe tWhitham equation close to the maximal height.\nWe consider periodic traveling-wave solutions of the sWhitham equation of the form\nu(χ, τ) =f(τ−γχ) =f(ξ), (20)\nwhere γis a real constant and fis a smooth, real-valued function of ξ=τ−γχwith nondimensional\ntemporal period T. Substituting (20) into (14) and integrating with respect to ξonce gives\n5−γf+\u0012K−1(w)\ni w\u0013\n∗f−3\n4f2=B, (21)\nwhere Bis the constant of integration. This equation is invariant under the transformation\nf→f+ν, γ →γ−3\n2ν, B →B+ν(γ−3\n4ν), (22)\nwhere νis any real constant. Therefore, without loss of generality, we only consider traveling-wave solutions\nof the sWhitham equation that have zero mean.\nEquation (21) can be solved approximately by assuming fhas a Fourier expansion of the form\nf(ξ) =NX\nj=−Nˆf(j) exp\u00122πijξ\nT\u0013\n, (23)\nwhere Nis a large positive integer and the ˆfare complex constants. Since fhas zero mean, ˆf(0) = 0.\nFor simplicity, we assume that the solutions are real and even. Thus, ˆf(−j) = ˆf(j) for j= 1, . . . , N .\nSubstituting (23) into (21) gives\n−γˆf(j) +TK−1(2πw\nT)\n2πi wˆf(j)−3\n4NX\nl=−N+jˆf(j−l)ˆf(l) = 0 , forj= 1,2, . . . , N . (24)\nWe solved this system of nonlinear algebraic equations for the ˆf’s using Newton’s method, see Ehrnstr¨ om\n& Kalisch (2009) and Carter et al. (2022) for the details in closely related problems.\nFigure 1(b) includes plots of four 2 π-periodic, zero-mean, traveling-wave solutions to the sWhitham\nequation. As the value of γdecreases, the height of the solution increases. We do not see evidence of a\nwave of maximum height that is analogous to the cusped wave seen in the tWhitham equation. For the\nsWhitham equation, the γvalue for 2 π-periodic solutions becomes negative when H≈3.70.\nFigure 2 includes plots of four traveling-wave solutions to the sWhitham equation with period T= 10π.\nJust as in the T= 2πcase, there does not appear to be a solution with maximal height, nor a solution\nwith minimal γvalue. However, once the height becomes large enough, the solutions no longer increase\nmonotonically on ξ∈(−5π,0). This is demonstrated in the inset plot in Figure 2. This nonmonotonic\nbehavior is not exhibited by solutions to the sKdV equation. For clarity, we define height by H=\nmax( u)−min(u) regardless if the solution is monotone on ξ∈(−T\n2,0) or not.\n2.3 Stability of traveling-wave solutions\nBenjamin & Feir (1967) showed that small-amplitude periodic traveling-wave solutions to (1) are stable\nwith respect to the modulational instability (long-wavelength perturbations) when2π\nL<1.363 and are\nunstable with respect to the modulational instability when2π\nL>1.363, where Lis the spatial wavelength.\nBottman & Deconinck (2009) and Deconinck & Nivala (2010) proved that alltraveling-wave solutions of\nthe tKdV equation are stable regardless of their height or (spatial) period. Due to the similarities between\nthe tKdV and sKdV equations, these results also establish that all traveling-wave solutions of sKdV are\nstable regardless of their height or period. Hur & Johnson (2015) proved that traveling-wave solutions\nto the tWhitham equation with sufficiently small amplitude are stable with respect to the modulational\ninstability if 2 π/L < 1.146 and are unstable with respect to the modulational instability otherwise. The\nmore general work of Binswanger et al. (2021) also shows that the small-amplitude modulational instability\ncutoff occurs at 2 π/L= 1.146 in the tWhitham equation. Sanford et al. (2014) numerically corroborated\nthese results, numerically showed that all periodic traveling-wave solutions of the tWhitham equation with\nlarge enough amplitude are unstable to perturbations of all wavelengths regardless of their wavelengths,\nand showed that the instability growth rate increases monotonically with the solution height.\nApplying formula (23) of Binswanger et al. (2021) to the sWhitham equation establishes that all\nsmall-amplitude periodic traveling-wave solutions are stable with respect to the modulational instability\n6Figure 2: Plots of four traveling-wave solutions to the sWhitham equation with period T= 10π. The inset\nplot shows that the solutions are not monotonic on ξ∈(−5π,0) when the solution height is sufficiently\nlarge.\nregardless of their period. This result is qualitatively similar to the tKdV and sKdV results, but is\nqualitatively different than the tWhitham result.\nIn order to numerically study the stability of traveling-wave solutions to the sWhitham equation, we\nemploy the Fourier-Floquet-Hill method of Deconinck & Kutz (2006). First, enter a coordinate frame\nmoving with the solution via the change of variables\nξ=τ−γχ. (25)\nThis converts (14) to\nuχ−γuξ+K−1(w)∗u−3\n2uuξ= 0, (26)\nand converts traveling-wave solutions of (14) into stationary (i.e. ∂χu= 0) solutions of (26). Next, consider\nperturbed solutions of the form\nupert(ξ, χ) =u(ξ) +ϵv(ξ, χ) +O(ϵ2), (27)\nwhere uis a periodic traveling-wave solution of the sWhitham equation, ϵis a small real constant, and ϵv\nis the leading-order part of the perturbation. Substituting (27) into (26) and linearizing gives\nvχ−γvξ+K−1(w)∗v−3\n2u′v−3\n2uvξ= 0, (28)\nwhere “prime” means derivative with respect to ξ. Without loss of generality assume\nv(ξ, χ) =V(ξ)eλχ+c.c., (29)\nwhere Vis a complex-valued function, λis a complex constant whose real part corresponds to the growth\nrate of the instability, and c.c.stands for complex conjugate. Substituting (29) into (28) and rearranging\ngives\n7γV′−K−1(w)∗V+3\n2u′V+3\n2uV′=λV. (30)\nAll bounded solutions to this equation (i.e. solutions with anyperiod) have the form V(ξ) =eiµξW(ξ),\nwhere Wis aT-periodic function and µ∈[−π\nT,π\nT] is a constant known as the Floquet parameter, see\nDeconinck & Kutz (2006). This gives\nV(ξ) = eiµξ∞X\nj=−∞ˆW(j)e2πijξ/T, (31)\nwhere the ˆW(j) are complex numbers. In our numerical computations, we use the truncation\nV(ξ) = eiµξNX\nj=−NˆW(j)e2πijξ/T, (32)\nwhere Nis a large positive integer. If there exists a bounded solution to (30) with λthat has a positive\nreal part, then the perturbation grows exponentially in χand the corresponding solution to the sWhitham\nequation is said to be unstable. If all solutions of (30) have λvalues that are purely imaginary, then the\ncorresponding solution is said to be spectrally stable.\nTraveling-wave solutions to the sWhitham equation with period T= 2π, height H < 0.842, and\nγ > 1.14 are spectrally stable. As the wave height surpasses H= 0.842 (and γfalls below 1 .14), the\nsolutions become unstable. The spectra corresponding to solutions with heights just above the critical\nvalue are oval-like shapes centered at the origin. The oval-like shapes correspond to perturbations with\nµvalues near ±0.5. (A perturbation with µ= 0.5 has a period that is twice that of the unperturbed\nsolution.) As the height increases further, more µvalues lead to instability and the ovals transition\ninto figure infinities centered at the origin. All nonzero µvalues lead to instability when the complete\nfigure infinity is formed. For solutions with period 2 π, the complete figure infinity is first observed near\nH= 0.845. These sWhitham stability results are qualitatively different than the tWhitham results where\nthe first unstable solutions have spectra with figure eights centered at the origin. Such spectra are created\nby perturbations with µvalues near zero and are representative of the modulational instability.\nFigure 3(a) includes plots of the stability spectra for the sWhitham solutions shown in Figure 1(b).\nFigure 3(b) includes plots of max( ℜ(λ)), i.e. the maximal instability growth rate, versus µfor the same\nsolutions. The solution with smallest height (colored blue in the plots) is spectrally stable since it has\na purely imaginary spectrum. The fact that this solution is stable is consistent with the Binswanger\net al. (2021) asymptotic result that all traveling-wave solutions of the sWhitham equation with sufficiently\nsmall amplitudes are stable with respect to the modulational instability. Additionally, this solution does\nnot exhibit the “bubble” instabilities seen in the Euler equations, see Deconinck & Oliveras (2011), or in\nbidirectional generalizations of the tWhitham equation, see Deconinck & Trichtchenko (2015). The other\nthree solutions plotted in Figure 1 are unstable.\nThe solution with second smallest height (colored orange) is unstable. Its stability spectrum is an oval-\nlike shape centered at the origin that is transitioning into a figure infinity, see Figure 3(a). This sWhitham\nsolution is only unstable with respect to perturbations with approximately |µ| ∈[0.1,0.5]. Since µnear\nzero does not lead to instability, this solution is stable with respect to the modulational instability. The\nsolution is most unstable with respect to perturbations with µ= 0.5. This means that perturbations\nwith period T= 4π, i.e. twice the period of the underlying solution, grow fastest. These results are\nqualitatively different than those obtained for moderate-amplitude 2 π-periodic traveling-wave solutions of\nthe tWhitham equation, see Sanford et al. (2014).\nThe solution with the third smallest height (colored green in the plots) is unstable with respect to\nperturbations with any nonzero value of µ. The stability spectrum is a figure infinity centered at the\norigin. The instability growth rates for this solution are larger than those for the solution with the second\nsmallest height. It is most unstable with respect to the perturbation with µ= 0.5, a perturbation with\nT= 4π. Finally, the solution with the largest height (colored magenta) is also unstable with respect to\nperturbations with any nonzero µand its stability spectrum is a figure infinity centered at the origin.\n8Figure 3: Plots of the stability results for the sWhitham solutions with T= 2πshown in Figure 1(b). The\ncolored curves in this figure correspond to the solutions of the same colors in Figure 1(b). Plot (a) shows\nthe stability spectra and (b) shows the maximum growth rate versus the Floquet parameter.\nHowever, the growth rates of the instabilities of this solution are smaller than the growth rates of the\nsolution with third smallest wave height, see Figure 3(b).\nAs the wave height continues to increase, there are alternating bands of stability and instability. Figure\n4 shows regions of ( T, H)-space for which periodic traveling-wave solutions to the sWhitham equation are\nunstable with respect to µ= 0.5 perturbations (black dots). For this plot, we only examined µ= 0.5\nperturbations because for all ( T, H) pairs we examined, the µ= 0.5 perturbation had the largest instability\ngrowth rate. There does not appear to be a simple relationship between ( T, H) and stability. For example,\nall four of the 10 π-periodic solutions shown in Figure 2 are stable, but not all 10 π-periodic solutions are\nstable. The approximate values of ( T, H) where γturns negative are shown by the red curve. This curve\ndoes not appear to have a simple relationship with stability. Figure 4 shows that there are bands of\ninstability and stability. This is qualitatively different than what happens in the tWhitham case where all\nlarge-amplitude solutions are unstable and the growth rate increases monotonically with wave height. It\nis unintuitive that some large-amplitude solutions are stable. This unintuitive result may be attributed to\nthe fact that the sWhitham equation is a model for small-amplitude waves and these solutions are outside\nof that range of validity.\nFinally, we did not find any solutions to the sWhitham equation that are unstable with respect to\nthe superharmonic instability, a perturbation that has the same period as the solution (i.e. µ= 0).\nAdditionally, neither the momentum nor the Hamiltonian oscillate for the sWhitham equation as solution\nheight increases. These results are qualitatively different than the tWhitham case where traveling-wave\nsolutions with large-enough amplitude are unstable with respect to the superharmonic instability, see\nCarter et al. (2023).\n3 Comparisons with experiments\nThe tKdV equation has been shown to compare favorably with experiments, see for example Russell\n(1844); Zabusky & Galvin (1971); Hammack (1973); Hammack & Segur (1974). Trillo et al. (2016) showed\nthat both the sKdV and sWhitham equations accurately model the evolution of experimental waves of\n9Figure 4: Plot of the regions of ( T, H)-space for which periodic traveling-wave solutions of the sWhitham\nequation are unstable (black dots) with respect to the µ= 0.5 instability. Solutions corresponding to\n(T, H) in the white bands are stable with respect to µ= 0.5 perturbations. The green stars represent the\napproximate location of the most unstable solutions for that period. The red curve shows the approximate\nvalues of ( T, H) where γbecomes negative.\ndepression on shallow water. Carter (2018) showed that the tWhitham equation more accurately predicts\nthe evolution of experimental waves of depression than do the tKdV and Serre equations.\nIn this section, we present comparisons between predictions obtained from numerical simulations of the\nmodel equations and measurements from four different series of laboratory experiments including waves\nof depression on shallow water (Sections 3.2 and 3.3), solitons on shallow water (Section 3.4), waves of\ndepression and elevation on deep water (Section 3.5), and wave packets on deep water (Section 3.6).\n3.1 Numerical methods\nIn order to make comparisons with measurements from laboratory experiments, the model equations need\nto be solved numerically. The tKdV and tWhitham equations are solved numerically using fourth-order\noperator splitting in time (see Yoshida (1990)) and periodic boundary conditions in space. The nonlinear\nparts of the equations are solved using fourth-order Runge-Kutta in time and a Fourier pseudospectral\nevaluation of the nonlinearity on a uniform grid (pointwise multiplication in space and spectral computation\nof the derivative). The linear part of the equations is solved exactly in Fourier space. We use the fast\nFourier transform (FFT) to move between the spatial and spectral variables. The length of the numerical\ntank must be large enough that waves do not leave one end of the domain, wrap around, and impact waves\non the other end of the domain. Additionally, the spatial gridpoints must be selected to line up with the\nexperimental gauge locations. As a check on the results, the numerical preservation of the conserved\nquantities of the equations, see Appendix A, was monitored.\nThe sKdV and sWhitham equations are solved using the same methods except that space and time\nare interchanged. The inverse K−1(w) is evaluated numerically using Newton’s method. As a check on\nthe results, the preservation of the conserved quantities, see Appendix A, was monitored.\n103.2 Waves of depression on shallow water\nHammack & Segur (1974) conducted a series of water-wave experiments in a long, narrow tank with\nh0= 10 cm, and a wavemaker at one end. The wavemaker was a rectangular, vertically-moving piston\nlocated on the bottom of the tank next to a rigid wall at one end of the tank. The piston spanned the\nwidth of the tank and had a length of 61 cm, so the initial wavelength was 122 cm. This experiment is in\nthe shallow-water regime because the ratio r=h0\nL= 0.082≪1 and tanh(2 πr) = 0 .47. The experiments\nwere initialized by rapidly moving the piston downward a prescribed amount. Time series were collected\nby wave gauges located 61 + 500 jcm for j= 0, . . . , 4 from the upstream end of the tank. This means that\nthe first gauge was located at the downstream edge of the wavemaker. The tank was long enough that\nwaves reflecting from the downstream end of the tank did not impact the time series.\nWe compare predictions from the tKdV, sKdV, tWhitham, and sWhitham equations. The time series\ncollected by the first gauge were used as initial conditions for the simulations of the spatial equations. The\ninitial conditions for the temporal equations require knowledge of the surface displacement for all values\nofxat the initial time, but that information was not recorded in these experiments. To approximate it,\nwe used the following function as initial condition for the temporal equations\nη(x,0) =\n\n0 −7869≤x <−183,\n−1\n2A0+1\n2A0sn(0.0925434 x,0.99992) −183≤x≤61,\n0 61 < x≤7747,(33)\nwhere A0is the amplitude of the piston motion in centimeters, xis measured in centimeters, and sn( ·, m)\nis a Jacobi elliptic function with elliptic modulus m, see Byrd & Friedman (1971). This function represents\na trough of 122 cm centered at x=−61 cm.\nFigure 5 contains plots comparing the experimental time series with the predictions obtained from the\ntKdV and sKdV equations for the experiment with A0= 1.5 cm. Both equations do a reasonable job\npredicting the experimental measurements. However, the sKdV equation more accurately predicts the\nphase speed. (This is especially visible at the downstream gauges.) It is important to note that part of the\nerror in the tKdV prediction is due to the fact that the initial surface is estimated. This is a shortcoming\nof the temporal equations. Both models overpredict the amplitudes at the downstream gauges and this\noverprediction increases as the waves travel down the tank. This overprediction is due to the fact that\nboth tKdV and sKdV are conservative models and the experiment contains dissipation. See Section 3.3\nfor a discussion of the role dissipation plays.\nFigure 6 contains plots comparing the experimental time series with the predictions obtained from the\ntWhitham and sWhitham equations for the same experiment. The differences between the tWhitham and\nsWhitham predictions are smaller than the differences between the tKdV and sKdV predictions. Note\nthat part of the error in the tWhitham prediction is due to the fact that the initial surface displacement\nis estimated. Again, this highlights a shortcoming of the temporal equations. Both the tWhitham and\nsWhitham equations overpredict the amplitudes of the waves at the downstream gauges due to their\nconservative nature; see Section 3.3.\nThe predictions from all four models (tKdV, sKdV, tWhitham, and sWhitham) for the Hammack\n& Segur (1974) experiment with A0= 0.5 cm (plots omitted for conciseness) are more accurate than\nthose in the A0= 1.5 cm case. However, the results were qualitatively the same: the spatial equations\nprovide more accurate predictions than do the temporal equations, the Whitham equations provide more\naccurate predictions than do the KdV equations, and all four equations overpredicted the wave amplitudes.\nFinally, we note that linear theory is not sufficient to model the time series from either the A0= 0.5 cm\norA0= 1.5 cm experiments. Predictions obtained from linear theory (plots omitted for conciseness) are\nsignificantly worse than any of the models examined herein.\n3.3 Dissipative models of waves of depression on shallow water\nFigures 5 and 6 show that the tKdV, sKdV, tWhitham, and sWhitham equations overpredict the ampli-\ntudes of the waves measured in the experiments. This overprediction is due to the fact that the equations\nare conservative, while the experiments contain dissipation. Figure 7 shows that the dimensional quantity\n11Figure 5: The experimental time series (blue curve) from the Hammack & Segur (1974) experiment using\nA0= 1.5 cm with the predictions obtained from the tKdV (orange curve) and sKdV (red curve) equations.\nThe plots are ordered from top to bottom by increasing distance from the wavemaker.\nFigure 6: The experimental times series (blue curve) from the Hammack & Segur (1974) experiment\nusing A0= 1.5 cm with the predictions obtained from the tWhitham (light green curve) and sWhitham\n(dark green curve) equations. The plots are ordered from top to bottom by increasing distance from the\nwavemaker.\n12M(x) =1\nτZτ\n0η(x, t)2dt, (34)\nwhere τis the dimensional length of the time series, decays nearly exponentially as the waves travel down\nthe tank for both the A0= 0.5 cm and A0= 1.5 cm experiments. In order to address this, we consider\nthe dimensional dissipative sKdV equation\nηx+1√gh0ηt−√gh0\n6g2ηttt−3\n2h0√gh0ηηt+δη= 0, (35)\nand propose the following ad-hoc, dimensional dissipative generalization of the sWhitham equation\nηx+s\nh0\ngK−1(ω)∗η−3\n2h0√gh0ηηt+δη= 0. (36)\nHere δis a nonnegative constant representing the sum total of all forms of dissipation in the experiment.\nWe refer to equations (35) and (36) as the dissipative sKdV (dsKdV) and the dissipative sWhitham\n(dsWhitham) equations, respectively. These equations predict that Mwill decay exponentially as x\nincreases (i.e. as the waves travel down the tank). The single free parameter, δ, is determined by best\nfitting the measured exponential decay of M. The δvalues for the A0= 0.5 cm and A0= 1.5 cm\nexperiments are 7 .14∗10−5cm−1and 2 .07∗10−4cm−1, respectively. This form of dissipation assumes\nthat waves of all periods decay with the same rate. It is not a wavenumber dependent form of dissipation.\nFigure 8 shows comparisons between the experimental time series and predictions from the dsKdV\nand dsWhitham equations. The dsKdV equation does a reasonable job, but incorrectly models the phase\nspeeds. The dsWhitham equation does an excellent job modeling the A0= 1.5 cm experimental time\nseries. The predictions obtained from the dsWhitham equation are much better than those obtained from\nthe conservative models. The results for the A0= 0.5 cm experiment (not shown) are similarly excellent.\nFigure 7: Plots of Mversus xfor the Hammack & Segur (1974) experiments with (a) A0= 0.5 cm and (b)\nA0= 1.5 cm. The blue dots represent the experimental measurements, the magenta curves represent the\nbest-fit exponentials of the experimental data, and the black circles represent the dsWhitham predictions.\n3.4 Solitons on shallow water\nWe conducted experiments on solitons in a 1,524 cm long (reflected waves did not impact the evolving\nsoliton), 25.4 cm wide wave channel in the W.G. Pritchard Fluid Mechanics Laboratory. The tank is\ndescribed in detail in Vasan et al. (2017). Briefly, the channel was cleaned with alcohol and filled with\nwater to a desired depth. A wind was blown over the water surface along the length of the channel, creating\na surface current that carried surface contaminants to the other end, where they were vacuumed with a wet-\nvac. The depth was then measured to be 5.30 cm. We generated solitons using a horizontal displacement\nof a piston: a vertical plate that spanned the width and height of the channel. The piston was programmed\n13Figure 8: The experimental time series (blue curve) from the Hammack & Segur (1974) experiment using\nA0= 1.5 cm and the predictions obtained from the dsKdV (gray curve) and dsWhitham (black curve)\nequations.\nusing the approach of Goring & Raichlen (1980), to take into account the real-time displacement of the\nplate. This approach and the details for our wavemaker are spelled out in Hammack et al. (2004). Herein,\nthe wavemaker produced a repeatable KdV soliton with the desired height of 2.00 cm. Hammack et al.\n(2004) also generated a KdV soliton with that height in water of depth 5.00 cm. They measured the spatial\nwavelength of the soliton to be about 80 cm. We did not measure a spatial wavelength, but the ratio in the\npresent experiments is nearby this value. Therefore, the ratio of depth to wavelength, r=h0/λ≈0.066,\nmeaning these experiments are in the shallow-water regime. A capacitance-type wave gauge was used to\nmeasure the surface displacement. For a fixed soliton amplitude, we conducted seven experiments with\nthe wave gauge moved to the different x-locations, x=50, 150, 250, 350, 450, 550, 650 cm away from the\npiston in its rest position. In these seven experiments, the start time of the paddle was not synced to\nthe start time of data collection, so that the time series had to be shifted by hand to match comparisons;\nhence, we did not test accuracy of wave speed.\nFigure 9 shows a comparison of the experimental time series and the predictions obtained from the\ndsWhitham equation for all seven gauges. The predictions lineup very closely with the experimental\nmeasurements. Figure 10 shows comparisons of the experimental time series and the predictions from\nthe sKdV, dsKdV, sWhitham, and dsWhitham equations for the last gauge. Although these experiments\ninvolve waves of elevation instead of waves of depression, the results are similar to those presented in\nthe previous two subsections. The sWhitham predictions are more accurate than the sKdV predictions.\nThe dsWhitham equation more accurately models the experimental time series than did the sWhitham\nequation. Other plots (omitted for conciseness) show that the dsWhitham equation is more accurate than\nthe tKdV and sKdV equations. Additionally, the spatial equations more accurately predict the wave\nevolution than do the temporal equations.\n3.5 Waves of depression and elevation on deep water\nThe tWhitham and sWhitham equations were proposed as models for waves on shallow water. However,\nthey accurately reproduce the phase speeds of all unidirectional linear waves, regardless of the (uniform)\ndepth of water. Because of this, it is reasonable to ask whether these equations accurately predict the\n14Figure 9: The experimental time series (blue curve) from the experiment described in Section 3.4 along\nwith the prediction obtained from the dsWhitham equation (black curve). The plots are ordered by\nincreasing distance from the wavemaker.\nFigure 10: The experimental time series (blue curve) from the final gauge of the experiment described in\nSection 3.4 along with the predictions obtained from (a) the sKdV (red curve) and sWhitham (dark green\ncurve) equations and (b) the dsKdV (gray curve) and dsWhitham (black curve) equations.\n15evolution of waves on deep water.\nIn this section, we address this question by comparing predictions from the tWhitham and sWhitham\nequations with time series from our experiments of waves on deep water that are similar to those discussed\nabove in Sections 3.2 and 3.3. In particular, we created an initial, localized, positive surface displacement\nand an initial, localized negative surface displacement. These surface displacements were created in the\nsame tank used for the soliton experiments (see Section 3.4) but with a horizontally aligned piston that\nhad a vertical displacement. The piston for the experiments in this section was a horizontal plate that\nwas 25.4 cm long in the x-direction, 1 cm tall in the vertical direction, and spanned the width of the tank.\nThe initial wave had a length of λ= 50.8 cm, and the depth was approximately h0= 20 cm. The resulting\nratio of fluid depth to wavelength was r=h0/λ= 0.393, with tanh(2 πh0/λ) = 0 .986, so that the waves\nwere effectively in the deep-water regime.\nThe plate was impulsively lowered or raised 0.75 cm. To obtain the motion, the operator moved a\nTwo-Servo Joystick, which sent a signal to a Servo Travel Tuner, in which we had programmed the desired\nplate displacement, and then to a Hi-Tec Linear Servo, which provided the plate motion. (All parts were\nfrom www.servocity.com). There was a vertical barrier behind the plate so that the resulting waves were\nforced to travel in the x >0 direction.\nFor the positive initial displacement, the tank was filled, and the surface was cleaned as described above\nto a depth of 20.18 cm. The plate was leveled parallel to the quiescent water surface and submerged 0.25 cm.\nIt was dropped 0.75 cm impulsively, creating a localized positive surface displacement. Four in-situ,\ncapacitance wave gauges, located at x= 4.0, 225.5, 461.4, 679.0 cm from the edge of the plate, provided\ntime series. Figure 11 shows the results. The data from the first gauge are used as the initial conditions for\nthe spatial equations, so the model output and measurements agree exactly at that location. At the second\ngauge site, all three models predict well the oscillations that develop near the initial displacement and the\nwave packet that forms further downstream. Dissipation becomes important in the surface displacement\nevolution by the third wave gauge. The conservative predictions agree qualitatively with the data, while the\ndsWhitham agrees both qualitatively and quantitatively, matching quite well even the radiation. By the\nfourth gauge, the agreement between the conservative predictions and the data worsens. The sWhitham\nprediction more accurately models the phase velocities than does the sKdV equation. The dsWhitham\nequation continues to produce a good quantitative comparison with the data.\nFor the negative initial displacement, the tank was filled, and the surface was cleaned as described\nabove to a depth of 20.22 cm. The plate was leveled parallel to the quiescent water surface and submerged\n0.75 cm. It was lifted 0.75 cm impulsively, creating a localized negative surface displacement. The\nfour in-situ, capacitance wave gauges, located at x= 4.0, 225.6, 462.5, 679.0 cm from the edge of the\nplate, provided time series. Figure 12 shows the results. The data from the first gauge is used as the\ninitial condition, so the model output and measurements agree. Similar to the negative displacement\nexperiments, all three models predict reasonably well the measured surface displacement at the second\ngauge site. Results from the sKdV and sWhitham equations are in qualitative agreement with data at the\nthird and fourth gauge sites, while the dsWhitham equation predicts quite well the measured time series\nat the third and fourth gauge sites.\nThe sKdV predictions are less accurate than the sWhitham predictions due to phase velocity issues\nComparisons with sKdV and sdKdV are similar, although they do not align with the experimental data\nas well.\n3.6 wave packets on deep water\nIn order to further test the range of validity of the sWhitham and dsWhitham equations, we compare their\npredictions with experimental measurements of wave packets on deep water. Segur et al. (2005) showed that\nthe dissipative NLS (dNLS) equation provided much more accurate predictions, both quantitatively and\nqualitatively, than did the NLS equation for the experiments discussed in this section. They also showed\nthat although plane-wave solutions to the NLS equation are unstable with respect to the modulational\ninstability (also known as the sideband instability), uniform-amplitude solutions to the dNLS equation are\nstable with respect to the modulational instability. Finally, they showed that although the dNLS equation\nis stable with respect to the modulational instability, the sidebands may grow a limited amount.\n16Figure 11: Comparisons of predictions from the (a) sKdV equation, (b) sWhitham equation, and (c)\ndsWhitham equation when the initial surface displacement is positive. The blue curves correspond to the\nmeasured surface displacement, the red, dark green, and black curves correspond to the sKdV, sWhitham,\ndsWhitham predictions, respectively. The plots are ordered from top to bottom by increasing distance\nfrom the plate/wavemaker.\nFigure 12: Comparisons of predictions from the (a) the sKdV equation, (b) sWhitham equation, and (c)\ndsWhitham equation when the initial surface displacement is negative. The blue curves correspond to the\nmeasured surface displacement, the red, dark green, and black curves correspond to the sKdV, sWhitham,\ndsWhitham predictions, respectively. The plots are ordered from top to bottom by increasing distance\nfrom the plate/wavemaker.\n17Figure 13: Plots comparing the experimental measurements (blue dots) from the deep-water experiment\ndescribed in Section 6.2 of Segur et al. (2005) with the predictions from the sWhitham equation (dark\ngreen dots). The top plot is the carrier wave and the other six plots are the six most dominant sidebands.\nIn the Segur et al. (2005) experiments, the wave tank was 1,311 cm long, 25.4 cm wide, had glass\nsidewalls and bottom, and had a constant water depth of h0= 20 cm. A plunger-type wavemaker that\nspanned the width of the tank, had an exponential cross-section, and oscillated vertically was located\nat one end of the tank. The wavemaker created slowly modulated wavetrains with waves of wavelength\nλ≈15 cm. Since the ratio r=h0/λ≈1.33 and tanh(2 πh0/λ)≈1.00, this experiment is in the deep-water\nregime.\nFigure 13 shows comparisons between the time series recorded in an experiment from Section 6.2\nof Segur et al. (2005) and the predictions from the sWhitham equation. Due to the complexity of the\ntime series, we compare the magnitudes of the dominant Fourier coefficients instead of the time series\nthemselves. The carrier wave frequency was f0= 3.33 Hz. The perturbation frequency was fp= 0.17 Hz.\nSo the sideband frequencies were f±n=f0±nfp, forn= 1, 2, 3. The subplots show the evolution of the\ncarrier wave (top plot) and the six most dominant sidebands. The plots show that the sWhitham equation\ndoes not accurately predict the evolution of these Fourier coefficients. This discrepancy appears to be\nrelated to the modulational instability. The experimental sidebands grow in magnitude as the waves travel\ndown the tank (the experimental parameters were chosen so that the sidebands would grow), while the\nsWhitham equation predicts no such growth. This is consistent with the fact that the sWhitham equation\nis not unstable with respect to the modulational instability. An accurate model of this experimental data\nmust allow for the sidebands to grow.\nFigure 14 shows comparisons between the same experimental data and the predictions from the\ndsWhitham equation. These plots show that the dsWhitham equation does not accurately model the\nexperimental data either. Though, note that the dsWhitham equation accurately models the evolution of\nthe carrier wave. Similar results are obtained when the other three deep-water experiments from Carter\net al. (2018) are examined. Therefore, it does not appear that the sWhitham or the dsWhitham equation\ncan accurately model the evolution of wave packets on deep water.\n18Figure 14: Plots comparing the experimental measurements (blue dots) from the deep-water experiment\ndescribed in Section 6.2 of Segur et al. (2005) with the predictions from the dsWhitham equation (black\ndots). The top plot is the carrier wave and the other six plots are the six most dominant sidebands.\n4 Summary\nWe examined both the mathematical properties and applications of the sWhitham equation. The sWhitham\nequation can be used to model how time series of surface displacement evolve as waves travel in space.\nOur analysis shows that the sWhitham equation admits periodic traveling-wave solutions, but does not\nappear to admit a periodic traveling-wave solution with maximal height. Small- and moderate-amplitude\ntraveling-wave solutions to the sWhitham equation are stable with respect to the modulational instability.\nSome larger-amplitude traveling-wave solutions are stable while others are unstable depending on wave\nperiod and wave height. There does not appear to be a simple relation between wave period and height\nthat determines stability or instability.\nOur comparisons with experiments show that for waves of depression and solitons on shallow water, the\nsWhitham equation provides more accurate predictions for experimental time series than do the tWhitham,\ntKdV, and sKdV equations. Part of the reason that the spatial predictions are more accurate than the\ntemporal predictions is that the initial conditions need to be approximated in the temporal case. Although\nthe sWhitham equation was proposed as a model for shallow-water waves, we show that the sWhitham\nequation accurately models the evolution of initial waves of depression and elevation on deep water. The\npredictions obtained from the sWhitham equation are improved by adding dissipation. Finally, neither\nthe sWhitham nor sdWhitham equation accurately models the evolution of wave packets on deep water.\n5 Acknowledgements\nJDC and PP thank the Direcci´ on General de Asuntos del Personal Acad´ emico (DGAPA) at the Universidad\nNacional Aut´ onoma de M´ exico (UNAM) for a Programa de Estancias de Investigaci´ on (PREI) grant that\nsupported JDC’s sabbatical stay at UNAM. PP acknowledges partial support by PAPIIT IG100522. We\nthank Robert Geist for constructing the laboratory apparati, including the mechanical components, the\nmotion control, and the computer interfaces, for the shallow-water soliton and all of the deep-water\nexperiments. We thank Vishal Vasan for writing the wavemaker program for experiments on the shallow-\n19water soliton and the deep-water wave-packets. We thank Robert Crable for building and maintaining the\ncapacitance wave gauges. We thank Knox Hammack for digitizing the experimental data from Hammack\n& Segur (1974). We thank Eleanor Byrnes for helpful discussions.\nA Conserved quantities\nThe conserved quantities for the nondimensional temporal KdV equation are\nQ1=ZL\n0u dχ, (37a)\nQ2=ZL\n0u2dχ, (37b)\nQ3=1\n2ZL\n0\u0012\nu2+1\n6u2\nχ−1\n2u3\u0013\ndχ, (37c)\nwhere Lis the χ-period of the solution. The conserved quantities for the nondimensional temporal\nWhitham equation are Q1,Q2, and\nQ4=1\n2ZL\n0\u0012\nu\u0012K(k)\ni k\u0013\n∗u−1\n2u3\u0013\ndχ. (38)\nThese quantities are conserved in τ.\nThe conserved quantities for the nondimensional spatial KdV equation are\nQ5=ZT\n0u dτ, (39a)\nQ6=ZT\n0u2dτ, (39b)\nQ7=1\n2ZT\n0\u0012\nu2+1\n6u2\nτ−1\n2u3\u0013\ndτ, (39c)\nwhere Tis the τ-period of the solution. The conserved quantities for the nondimensional spatial Whitham\nequation are Q5,Q6, and\nQ8=1\n2ZT\n0\u0012\nu\u0012K−1(w)\ni w\u0013\n∗u−1\n2u3\u0013\ndτ. (40)\nThese quantities are conserved in χ.\nReferences\nAblowitz, M.J. & Segur, H. 1981 Solitons and the Inverse Scattering Transform . SIAM, Philadelphia.\nAgrawal, G. 2019 Nonlinear Optics , Sixth edn. London: Academic Press.\nBenjamin, T.B. & Feir, J.E. 1967 The disintegration of wave trains on deep water: Part I. Theory.\nJournal of Fluid Mechanics 27, 417–430.\nBinswanger, A.L., Hoefer, M.A., Ilan, B. & Sprenger, P. 2021 Whitham modulation theory for\ngeneralized Whitham equations and a general criterion for modulational instability. 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Physics Letters A 378, 2100–2107.\nSegur, H., Henderson, D., Carter, J.D., Hammack, J., Li, C., Pheiff, D. & Socha, K. 2005\nStabilizing the Benjamin-Feir instability. Journal of Fluid Mechanics 539, 229–271.\nTrillo, S., Klein, M., Clauss, G.F. & Onorato, M. 2016 Observation of dispersive shock waves\ndeveloping from initial depressions in shallow water. Physica D 333, 276–284.\nVasan, V.V., Oliveras, K., Henderson, D. & Deconinck, B. 2017 A method to recover water-wave\nprofiles from pressure measurements. Wave Motion 75, 25–35.\nWhitham, G.B. 1967 Variational methods and applications to water waves. Proceedings of the Royal\nSociety of London, A 299, 6–25.\nWhitham, G.B 1974 Linear and Nonlinear Waves . John Wiley & Sons, Inc., New York.\nYoshida, H. 1990 Construction of higher order symplectic integrators. Physics Letters A 150, 262–268.\nZabusky, N.J. & Galvin, C.J. 1971 Shallow-water waves, the Korteweg-de Vries equation and solitons.\nJournal of Fluid Mechanics 47, 811–824.\n22"    },    {        "title": "2402.14675v1.Constant__Q__curvature_metrics_on_a_product_Riemannian_manifold.pdf",        "content": "arXiv:2402.14675v1  [math.DG]  22 Feb 2024CONSTANT Q-CURV ATURE METRICS ON A PRODUCT RIEMANNIAN MANIFOLD\nSALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\nABSTRACT . Let(M,g)be an analytic Riemannian manifold of dimension n≥5. In this paper, we\nconsider the so-called constant Q-curvature equation\nε4∆2\ngu−ε2b∆gu+au=up, inM, u > 0, u∈H2\ng(M)\nwherea,bare positive constants such that b2−4a >0,pis a sub-critical exponent 1< p <2#−1 =n+4\nn−4,\n∆g= div∇denotes the Laplace-Beltrami operator and ∆2\ng:= ∆g(∆g)is the bilaplacian operator on M.\nWe show that, if ε >0is small enough, then positive solutions to the above consta ntQ-curvature\nequation are generated by a maximum or minimum point of the fu nctionτg, given by\nτg(ξ) :=n/summationdisplay\ni,j=1∂2gii\nξ\n∂z2\nj(0),\nwheregij\nξdenotes the components of the inverse of the metric gin geodesic normal coordinates. This result\nshows that the geometry of Mplays a crucial role in finding solutions to the equation abov e and provides a\nmetric of constant Q-curvature on a product manifold of the form (M×X,g+ε2h)where(M,g)is flat\nand closed, and (X,h)anym-dimensional Einstein Riemannian manifold, m≥3.\nCONTENTS\n1. Introduction 1\n1.1. Motivation 1\n1.2. Setting of the problem and main result 3\n2. Background 6\n3. Finite dimensional reduction 8\n4. Asymptotic Expansion 15\n5. The reduced problem 18\n6. Appendix 21\nReferences 23\n1. I NTRODUCTION\n1.1. Motivation. Given a Riemannian manifold (M,g)of dimension n≥5, theQ-curvature of the\nmetric is defined as\nQg=−1\n2(n−1)∆gScalg−2\n(n−2)2/ba∇dblRicg/ba∇dbl2+n3−4n2+16n−16\n8(n−1)2(n−2)2Scal2\ng,\nwhere∆gu= divg(∇gu)is the Laplace operator on (M,g)and/ba∇dblA/ba∇dbl2= trg(AAt).\nSimilarly to the Yamabe problem, a key question is whether a c onformal metric to gcan be found with\nconstantQ-curvature. This particular problem involves solving a fou rth-order elliptic partial differential\nequation, where the solution is the conformal factor. To defi ne the equation, the Paneitz operator Pg\n2010 Mathematics Subject Classification : 53C21, 58J05, 35J60, 35B33.\nKey words :Q-curvature, spaces of constant curvature, bilaplacian ope rator.\nS. Alarc´ on is partially supported by Fondecyt Project 1211 766. S. Masn´ u is partially supported by ANID through Beca\nde Doctorado Nacional 21241432. P. Montero is partially sup ported by Fondecyt Projects 1231214 and 1240101. C. Rey is\npartially supported by Fondecyt Project 3200422.\n12 SALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\nis introduced. This operator is a fourth-order operator tha t is defined in terms of the Q-curvature and\npossesses a conformal invariance similar to that of the conf ormal Laplacian in the Yamabe problem.\nWhen considering a local g-orthonormal frame (ei)n\ni=1, the Paneitz operator can be expressed as\nPgψ= ∆2\ngψ+4\nn−2divg(Ricg(∇ψ,ei)ei)−n2−4n+8\n2(n−1)(n−2)divg(Scalg∇ψ)+n−4\n2Qgψ.\nFix a metric g0inMand denote by [g0]its conformal class. Writing conformal metrics g∈[g0]as\ng =u4\nn−4g0, withu:M→R>0, the Paneitz operator satisfies the property of conformal in variance, ie,\nfor anyψ:M→R,\nPgψ=u−n+4\nn−4Pg0(uψ).\nThus,Qg=2\nn−4Pg(1) =2\nn−4u−n+4\nn−4Pg0(u), and we have\nPg0u=n−4\n2Qgun+4\nn−4.\nTherefore, the constant Q-curvature equation for the metric g =u4\nn−4g0reads\nPg0u=λun+4\nn−4, λ∈R. (1)\nWe also call this equation the Paneitz-Branson Equation.\nThe constant Q-curvature problem is also well defined when Mis3or4-dimensional, but it has a\ndifferent expression. Many authors studied the four-dimen sional equation, for example, A. Chang and P.\nC. Yang in [ CY95 ], S. Brendle in [ Bre03 ], Z. Djadli and A. Malchiodi in [ DM08 ], J. Li, Y . Li, and P. Liu\nin [LLL12 ], among others. In dimension 3some existence theorems were proved in [ HY16a ].\nWe are interested in studying equation ( 1) on manifolds of dimension bigger than 5. This problem\nhas been widely considered in the literature. For example, J . Qing and D. Raske established a positive\nsolution of the Paneitz-Branson equation in [ QR06 ] on a locally conformally flat manifold with positive\nscalar curvature and a Poincar´ e exponent smaller thanN−4\n2. In [ GM15 ], M. J. Gursky and A. Malchiodi\ndemonstrated the existence of a positive solution provided thatQgis non-negative and positive at some\npoint inM. See also [ DHL00 ,DM08 ,ER02 ,GHL16 ,Rob03 ,HY16b ,HY15 ] for further existence\nresults. In the case of the round sphere, C. S. Lin [ Lin98 ] used the moving-plane approach to classify\nconformal metrics with constant Q-curvature and found out an explicit multi-dimensional fam ily of\nconformal metrics with constant Q-curvature.\nIn addition, E. Hebey and F. Robert in [ HR04 ] proved that the Paneitz-Branson equation is compact\nwhen the Paneitz operator is of strong positive type. There a re other significant results concerning the\ncompactness of the space of solutions, such as [ Li19 ,LX19 ,Mal06 ]. On the other side, J. Wei and C.\nZhao in [ WZ13 ] built examples of non-compactness of the space of positive solutions in high dimensions.\nThere is also an important result about the uniqueness of met rics of constant Q-curvature in its con-\nformal class. It is evident that when gis Einstein, it has a constant Q-curvature. J. V´ etois showed in\n[V´ et23 ] that ifgis Einstein and it is not isometric to the constant curvature metric on the sphere, then it\nis the only metric with that property, up to multiplication b y a constant. This result is a counterpart to\nthe classical theorem of M. Obata [ Oba72 ] in the Yamabe problem.\nThere are instances where we get multiple conformal metrics with constant Q-curvature, such as\nin Riemannian products (see [ APR23 ]). This result also arises in the context of the Yamabe probl em\n(see [ Pet19 ]). Moreover, in [ RR21 ] the authors built multi-peaks solutions to the Yamabe equa tion in a\nRiemannian product of two manifolds, with the conformal fac tor concentrated around an isolated local\nminimum or maximum point of the scalar curvature.\nIn this paper, we built a solution to ( 1) on a Riemannian product that concentrates on a certain poin t of\nthe first factor. Concretely, we prove that if (M,g)and(X,h)are closed Riemannian manifolds, where\n(M,g)is analytic and flat, and (X,h)is an Einstein manifold with positive scalar curvature, the n, for\nsufficiently small ε >0, the N-dimensional Riemannian product (M×X,g+ε2h)has a conformal3\nmetric of the form uN\nN−4(g+ε2h), whereu:M→Rconcentrates at an isolated minimum or maximum\npoint of a suitable function τg∈C∞(M)(see Theorem 1.2).\n1.2. Setting of the problem and main result. Let(M,g)be any closed analytic n-dimensional Rie-\nmannian manifold which is flat (ie, Ricg= 0 = Scal g), and(X,h)be an m-dimensional Riemannian\nEinstein manifold, with Rich= Λ0>0andScalh=mΛ0.We will be interested in positive solutions\nof the constant Q-curvature equation ( 1) for the product manifold (M×X,g+ε2h), which is\nPg+ε2hu=λεun+m+4\nn+m−4, λε∈R. (2)\nNote thatn+m+4\nn+m−4<n+4\nn−4. So, when studying solutions which depend only on one of the v ariables, the\nequation is sub-critical. Therefore, assuming that u:M→R, we obtain the following expression for\nthe constant Q-curvature equation (for details, see [ APR23 ]).\nε4∆2\ngu−ε2b∆gu+au=up, inM, (3)\nwhere1<p<2♯−1 =n+4\nn−4, and the constant coefficients are\na=an,m=Λ2\n0m(N−4)\n2(N−2)2/bracketleftbigg\n−2+(N3−4N2+16N−16)\n8(N−1)2m/bracketrightbigg\n(4)\nb=bn,m=N2−4N+8\n4(N−1)(N−2)mΛ0, (5)\nwhereN=m+n.\nRemark 1.1.Recall from [ APR23 , Lemmas 2.2 and 2.3] that the constants in ( 2), (4) and ( 5) satisfy the\nnext conditions.\n•Ifm= 2, thenλε<0forN= 5,6,7,8andλε>0forN≥9. Ifm≥3, thenλε>0.\n•aN,m>0ifm≥3orm= 2andN≥9.\n•Ifm≥3, orm= 2andN≥9, thenbn,m>2√an,m.\nFrom now on, we will assume that the dimension of the base mani fold(M,g)isn≥5and the dimension\nof the fiber (X,h)ism≥3.\nThe compactness of the embedding H2(M)֒→Lp+1(M)ensures that\ninf\nu∈C∞(M)\\{0}/integraltext\nMuPg+ε2h(u)dvg\n/parenleftbig/integraltext\nM|u|p+1dvg/parenrightbig2\np+1\nis achieved, and so problem ( 3) always has a solution for any p∈(1,2♯−1). Moreover, in [ APR23 ], the\nauthors showed that there are at least cat(M)+1 non-trivial positive solutions to equation ( 3) provided\nεis small enough. Here cat(M)denotes the Lusternik–Schnirelmann category of M, which means that\nthe amount of solutions to ( 3) depends on the topological properties of the manifold M. In this paper,\nwe show that, if ε >0is small enough, then a positive solution to the problem ( 3) is generated by an\nisolated minimum or maximum points of the function τg, which is strongly related with the geometry of\nthe manifold.\nJust as the topology of Mgives us a lower bound for the number of solutions to the equat ion, we will\nshow that its geometry allows us to find solutions that are con centrated at a geometrically important point\nof the manifold. Furthermore, we believe that it is possible to show that, given a C1-stable critical point\nofτg, there is a solution that is concentrated around that point, as M. Micheletti and A. Pistoia proved in\n[MP09 ] for a Yamabe-type equation.\nIn this paper, we prove a concentration result for Eq. ( 3) withε >0small enough. In consequence,\nwe will obtain for each ε >0small enough, a metric in (M×X)conformal to g+εhwith constant\nQ-curvature for any Einstein Riemannian manifold (X,h)of positive scalar curvature. To achieve this,\nwe applied a well-known technique, known as the Lyapunov-Sc hmidt reduction procedure, which was\nintroduced in [ BC88 ,FW86 ,LN98 ] and has been used in many articles. Here we follow the approa ch4 SALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\nemployed by [ MP09 ,DP11 ,RR21 ], among others. We now briefly describe this method and state the\nresults we have obtained.\nLetH2\nεbe the Hilbert space H2\ng(M)equipped with the norm\n/ba∇dblu/ba∇dbl2\nε:=1\nεn/parenleftBig\nε4/integraldisplay\nM∆2\ngudVg+bε2/integraldisplay\nM|∇u|2dVg+/integraldisplay\nMau2dVg/parenrightBig\n. (6)\nWe also denote by Lq\ng,εthe Banach space Lq\ng(M)furnished with the norm\n|u|q,ε:=/parenleftBig1\nεn/integraldisplay\nM|u|qdVg/parenrightBig1/q\n. (7)\nWe will split the space H2\nεinto two subspaces, one finite-dimensional and its orthogon al complement.\nFirst, we introduce the embedding ιε:H2\nε֒→Lp+1\ng,ε, in order to rewrite the problem ( 3) asu=ι∗\nε(up\n+)\n(details in Section 2). Then, we introduce the following equation in the Euclidea n space, which is the\nlimit equation for our problem.\n∆2u−b∆u+au=upinRn, (8)\nwitha,b >0, and for 2< p+ 1 = 2♯\nN<2♯, where2♯\nN=2N\nN−4. Solutions to Eq. ( 8) have been\nextensively studied in [ BCdSN18 ], where they are found as critical points of the energy funct ionalE:\nH2(Rn)→R, given by\nE(u) =1\n2/integraldisplay\nRn/parenleftbig\n(∆u)2+b|∇u|2+au2/parenrightbig\ndz−1\np+1/integraldisplay\nRn|u|p+1dz.\nThey prove that if b≥2√a, then the infimum\nµ= inf\nu∈H2(Rn)\\{0}E(u)\nis achieved by a positive function that is radially symmetri c around a certain point and strictly radially\ndecreasing. We denote this minimizer by Uand note that it naturally satisfies ( 8).\nAssumption 1. We assume that Uis nondegenerate in H2(M), which means that for any solution vof\nthe linearized equation\n∆2v−b∆v+av=pUp−1vinRn,\nthere existsξ∈Rnsuch thatv(x) =ξ·∇U(x). In other words, vis a linear combination of the functions\nΨi=∂U\n∂zi,withi= 1,...,n.\nIn [BCdSN18 , Theorem 1.3], the authors prove the above property for the g round state solutions to\nthe equation\nγ∆2u−b∆u+au=|u|p−1uinRn,\nunder the hypothesis that 0<p−1<4/Nandγ >0is small enough. They conjecture that the property\nholds probably without the smallness assumption on γ, which is our case.\nMoreover, they show that the function Uand its derivatives are exponentially decaying at infinity,\nnamely, there is a positive constant Csuch that\nU(x)<C√\nb2−4ae/parenleftBigg√\nb−√\nb2−4a√\n2−δ/parenrightBigg\n|x|\n(9)\nfor anyδ >0.The same kind of argument also implies that each derivative o fUhas an exponential\ndecay. Note that if we write Uε(x) =U(x\nε), thenUεis a solution of\nε4∆2u−bε2∆u+au=upinRn.\nOur goal is to find an approximation for the solution to equati on (3) by transforming the function U,\nwhich is defined in Rn, into a function in Mthat concentrates on a specific point.5\nLetrbe the injectivity radius of M, and letχrbe a positive smooth cutoff function such that\nχr(z) =/braceleftBigg\n1 ifz∈B(0,r/2)\n0 ifz∈Rn\\B(0,r),\nwith bounded derivatives up to fourth order, i.e., there are positive constants c1,c2,c3,c4such that\n|∂iχr(z)| ≤c1,|∂2\nijχr(z)| ≤c2,|∂3\nijkχr(z)| ≤c3,and|∂4\nijkℓχr(z)| ≤c4. (10)\nFor any point ξ∈Mand for any positive real number λ, we define the approximate solution as the\nfunctionWε,ξonMgiven by\nWε,ξ(x) :=/braceleftbiggUε(exp−1\nξ(x))χr(exp−1\nξ(x)) ifx∈Bg(ξ,r),\n0 otherwise,(11)\nwhereexpξ:TξM→Mis the geometrical exponential map, Bg(ξ,r)denotes the ball in Mcentered\natξwith radiusrwith respect to the distance induced by the metric g, andvε(x) =v(x\nε)for allv∈\nH2(Rn). Fori= 1,...,n we denote Ψi\nε(z) := Ψi(ε−1z),and define on Mthe functions\nZi\nε,ξ(x) :=/braceleftbiggΨi\nε(exp−1\nξ(x))χr(exp−1\nξ(x)) ifx∈Bg(ξ,r),\n0 otherwise.(12)\nWe will look for a solution u∈H2\ng(M)to (3), or equivalently to\nu−ι∗\nε(up\n+) = 0\nof the formu=Wε,ξ+φ,with/ba∇dblφ/ba∇dblε→0.So we are looking for a point ξ∈Mand a function φsuch\nthat\nWε,ξ+φ−ι∗\nε(f(Wε,ξ+φ)) = 0. (13)\nTo this end, for each ε>0andξ∈M, we consider the spaces\nKε,ξ:=span{Zi\nε,ξ:i= 1,...,n}\nK⊥\nε,ξ:={φ∈H2\nε:/angbracketleftbig\nφ,Zi\nε,ξ/angbracketrightbig\nε= 0,for alli= 1,...,n},\nand the associated orthogonal projections Πε,ξ:H2\nε→Kε,ξandΠ⊥\nε,ξ:H2\nε→K⊥\nε,ξ. Then, equation ( 13)\nis equivalent to the system\nΠ⊥\nε,ξ{Wε,ξ+φ−ι∗\nε(f(Wε,ξ+φ)}= 0, (14a)\nΠε,ξ{Wε,ξ+φ−ι∗\nε(f(Wε,ξ+φ)}= 0. (14b)\nThe goal is to demonstrate, using the Inverse Function Theor em, that given ε >0small enough, and\nξ∈M, there exists a unique small φ=φε,ξ∈K⊥\nε,ξthat fulfils ( 14a). This is the essence of Proposition\n3.4. Additionally, we need to find a point ξ∈Mthat solves the finite-dimensional problem ( 14b). This\nis the main idea behind Proposition 5.2.\nOur main result reads as follows.\nTheorem 1.2. Let(M,g)be a closed (compact boundaryless) analytic Riemannian man ifold, andξ0∈\nMbe an isolated minimum or maximum point for the function τg:M→R, which is given by\nτg(ξ) :=n/summationdisplay\ni,j=1∂2gii\nξ\n∂z2\nj(0), (15)\nwheregij\nξdenotes the components of the inverse of the metric gin geodesic normal coordinates. Then,\nthere existsε0>0such that for any ε∈(0,ε0)problem (3)has a solution uεwhich concentrates at ξ0\nasεgoes to zero.\nThis Theorem allows us to find metrics on a product manifold wi th constantQ-curvature. So, as a\nCorollary we have6 SALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\nCorollary 1.3. Let(M,g)be ann−dimensional analytic Riemannian manifold, that is flat and c losed,\nand(X,h)anm−dimensional Einstein Riemannian manifold, with n≥5andm≥3. Ifξ0∈M\nis an isolated minimum or maximum point for the function τg, then there exists ε0>0such that for\nanyε∈(0,ε0)there is a conformal metric to g+ε2hin the product manifold M×Xwith constant\nQ-curvature.\nThe structure of the paper is as follows. In Section 2, we provide the necessary notation and prelim-\ninary results. In Section 3, we establish bounds for the involved operators and we show h ow to apply\nthe Implicit Function Theorem to solve equation ( 14a). In Section 4, we examine the energy functional\nassociated with ( 3) and illustrate its expansion around the approximate solut ion. In Section 5, we focus\non solving the finite-dimensional problem ( 14b) and we prove the Theorem 1.2. Finally, in Section ( 6)\nwe compute some auxiliary estimations.\n2. B ACKGROUND\nIn this section, we establish some notation and define the fun ctional framework needed to study our\nproblem.\nLetH2\nεbe the Hilbert space H2\ng(M)equipped with the inner product\n/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}htε:=1\nεn/parenleftBig\nε4/integraldisplay\nM∆gu∆gvdVg+bε2/integraldisplay\nM∇u∇vdVg+/integraldisplay\nMauvdV g/parenrightBig\n,\nwhich induces the norm defined in ( 6). Similarly, on Rnwe define the inner product for u,v∈H2(Rn)\nas\n/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}ht:=/integraldisplay\nRn∆u∆vdz+b/integraldisplay\nRn∇u∇vdz+a/integraldisplay\nRnuvdz,\nwhich induces the norm /ba∇dblu/ba∇dbl:=/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}ht.We also denote by Lq\ng,εthe Banach space Lq\ng(M)furnished with\nthe norm ( 7). As usual, we identify the dual space (Lq\ng,ε)′withLq′\ng,εwith the pairing\n/a\\}b∇acketle{tϕ,ψ/a\\}b∇acket∇i}htq,q′=1\nεn/integraldisplay\nMϕψ,\nforϕ∈Lq\ng,ε,ψ∈Lq′\ng,ε. The Sobolev embedding theorem (see e.g. [ Heb96 ]) asserts that H2\ng(M)is\ncontinuously embedded in Lq\ng(M)for1< q≤2#, and this embedding is compact when q <2#. It\nfollows that there exists a constant cindependent of εsuch that\n|u|q,ε≤c/ba∇dblu/ba∇dblεfor anyu∈H2\nε. (16)\nIn consequence, the embedding ιε:H2\nε֒→Lp+1\ng,ε, and the adjoint operator ι∗\nε:Lp+1\npg,ε→H2\nε, satisfy\nu=ι∗\nε(v)⇔ /a\\}b∇acketle{tι∗\nε(v),ϕ/a\\}b∇acket∇i}htε=/a\\}b∇acketle{tv,ι(ϕ)/a\\}b∇acket∇i}htp+1,p+1\np=1\nεn/integraldisplay\nMvϕ, for allϕ∈H2\nε,\nwhich means\nu=ι∗\nε(v)⇔ε4∆2\ngu−ε2b∆gu+au=v(weakly) on M. (17)\nLetf(u) :=up\n+.Now, the problem ( 3) is equivalent to\nu=ι∗\nε(f(u)), u∈H2\nε. (18)\nMoreover, from ( 17) and ( 16), we have the inequality\n/ba∇dblι∗\nε(v)/ba∇dblε≤c|v|p+1\np,εfor anyv∈Lp+1\npg,ε and somec∈Rindependent of ε. (19)\nLet us recall an important property of the exponential map expξ:TξM→M. There exists an\ninjectivity radius r>0such that for any ξ∈M, the restriction\nexpξ|B(0,r):B(0,r)→Bg(ξ,r)7\nis a diffeomorphism. Here, B(0,r)denotes the ball in Rncentered at 0with radius r, andBg(ξ,r)\ndenotes the ball in Mcentered atξwith radiusrwith respect to the distance\ndg(x,y) = exp−1\nx(y).\nThis fact ensure us that the functions Wε,ξandZi\nε,ξ,i= 1,...,n , given by ( 11) and ( 12), are well\ndefined. Moreover, it allows us to consider the pullback of th e metric inM.\nLetgξdenote the Riemannian metric read in B(0,r)through the normal coordinates defined by the\nexponential map expξatξ. We denote |gξ(z)|:= det(gξ(z)ij)and/parenleftBig\ngij\nξ(z)/parenrightBig\nis the inverse matrix of\ngξ(z). In particular, it holds\ngij\nξ(0) =δij and∂gij\nξ\n∂zk(0) = 0 for anyi,j,k,\nwhereδijis the Kronecker symbol. Let ξ0∈Mbe fixed, and ξ(z)∈Bg(ξ0,r). Recall the expression of\nthe scalar curvature Sg(ξ0)ofgatξ0, given by\nSg(ξ0) =Sg(ξ(0)) =N/summationdisplay\ni,j=1∂2gii\nξ\n∂z2\nj(0)−N/summationdisplay\ni,j=1∂2gij\nξ\n∂zi∂zj(0).\nAssumming Mto be flat, then we have Sg(ξ) = 0 for allξ∈M, and\nτg(ξ) :=n/summationdisplay\ni,j=1∂2gii\nξ\n∂z2\nj(0) =n/summationdisplay\ni,j=1∂2gij\nξ\n∂zi∂zj(0). (20)\nNow we will discuss the asymptotical behaviour of the approx imate solution.\nRemark 2.1.Since the function Uis radial it follows that if i/\\e}atio\\slash=jthen/angbracketleftBig\nΨi\nε,Ψj\nε/angbracketrightBig\nε= 0. Then, it is easy\nto see that for any ξ∈Mit holds\nlim\nε→0/angbracketleftBig\nZi\nε,ξ, Zj\nε,ξ/angbracketrightBig\nε=δij/integraldisplay\nRn/parenleftbig\n|∇Ψi|2+|Ψi|2/parenrightbig\ndz.\nLet us callC=/integraldisplay\nRn(|∇Ψi|2+(Ψi)2)dz. Fory∈B(0,r)⊂Rnwe setξ(y) = expξ(y)∈Bg(ξ,r).\nThen, around ξit holds\nlim\nε→0ε/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂Wε,ξ(y)\n∂yk/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nε=C,lim\nε→0ε/angbracketleftbigg\nZi\nε,ξ(y),∂Wε,ξ(y)\n∂yk/angbracketrightbigg\nε=δikC, (21)\nand\nlim\nε→0ε/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂Zi\nε,ξ(y)\n∂yk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nε=/integraldisplay\nRn/parenleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇∂Ψi\n∂zk/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/parenleftbigg∂Ψi\n∂zk/parenrightbigg2/parenrightBigg\ndz. (22)\nTo set the asymptotic expansion of the exact solution in Sect ion4, we give the well-known expansion\nof the functions gijin normal coordinates, which is proved, for instance, in [ Gra73 ].\nLemma 2.2. Let(M,g)be an analytic Riemannian manifold of dimension n. In a normal coordinates\nneighborhood of ξ0∈M, the Taylor’s series of garoundξ0is given by\n/radicalBig\n|gξ(εz)|= 1−ε2\n4n/summationdisplay\nl,r,k=1∂2gll\nξ\n∂zr∂zk(0)zrzk+O(ε3|z|3),\ngij\nξ(εz) =δij+ε2\n2n/summationdisplay\nr,k=1∂2gij\nξ\n∂zr∂zk(0)zrzk+O(ε3|z|3).8 SALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\n3. F INITE DIMENSIONAL REDUCTION\nIn this section, we use a fixed point theorem and the implicit f unction theorem to solve equation ( 14a).\nLetLε,ξ,Nε,ξ:K⊥\nε,ξ→K⊥\nε,ξbe the following operators\nLε,ξ(φ) := Π⊥\nε,ξ/braceleftbig\nφ−ι∗\nε/bracketleftbig\nf′(Wε,ξ)φ/bracketrightbig/bracerightbig\nNε,ξ(φ) := Π⊥\nε,ξ/braceleftbig\nι∗\nε/bracketleftbig\nf(Wε,ξ+φ)−f(Wε,ξ)−f′(Wε,ξ)φ/bracketrightbig/bracerightbig\n.\nAndRε,ξ∈K⊥\nε,ξgiven by\nRε,ξ:= Π⊥\nε,ξ{ι∗\nε[f(Wε,ξ)]−Wε,ξ}, (23)\nso the equation ( 14a) can be rewritten as\nLε,ξ(φ) =Nε,ξ(φ)+Rε,ξ.\nWe will start this section by proving the invertibility of Lε,ξ.\nProposition 3.1. There exists ε0>0andc>0such that for any ξ∈Mand for anyε∈(0,ε0)\n/ba∇dblLε,ξ(φ)/ba∇dblε≥c/ba∇dblφ/ba∇dblε,for anyφ∈K⊥\nε,ξ.\nProof. We will proceed by contradiction. Since Mis compact, we may assume that there are sequences\n{εj}j∈N, withεj→0and{ξj}j∈N, withξj→ξfor someξ∈M, with{φj} ⊂K⊥\nεj,ξj, such that\nLεj,ξj(φj) =ψj, with/ba∇dblφj/ba∇dblεj= 1and/ba∇dblψj/ba∇dblεj→0.\nLetζj:= Πεj,ξj{φj−i∗\nεj[f′(Wεj,ξj)φj]}. Hence,\nφj−i∗\nεj[f′(Wεj,ξj)φj] =ψj+ζj. (24)\nThat is, for each j,ψj∈K⊥\nεj,ξjandζj∈Kεj,ξj. Now, letuj:=φj−(ψj+ζj).\nWe will prove the following contradictory consequences of t he existence of such series\n1\nεn\nj/integraldisplay\nMf′(Wεj,ξj)u2\njdVg→1, (25)\nand\n1\nεn\nj/integraldisplay\nMf′(Wεj,ξj)u2\njdVg→0, (26)\nand then, such sequences {ξj},{φj},{εj}cannot exist. We divide the proof in five steps.\nStep 1. We start by proving\n/ba∇dblζj/ba∇dblεj→0asj→ ∞. (27)\nSinceζj∈Kεj,ξj, letζj:=n/summationdisplay\nk=1ak\njZk\nεj,ξj. For anyl∈ {1,2,...,n}we multiply ψj+ζj(eq. ( 24)) by\nZl\nεj,ξj, and we find\nn/summationdisplay\nk=1ak\nj/angbracketleftBig\nZk\nεj,ξj, Zl\nεj,ξj/angbracketrightBig\nεj=−/angbracketleftBig\nι∗\nεj[f′(Wεj,ξj)φj], Zl\nεj,ξj/angbracketrightBig\nεj. (28)\nOn the other hand, we have that\nn/summationdisplay\nk=1ak\nj/angbracketleftBig\nZk\nεj,ξj, Zl\nεj,ξj/angbracketrightBig\nεj=Cal\nj+o(1).9\nIndeed\n/angbracketleftBig\nZk\nεj,ξj, Zl\nεj,ξj/angbracketrightBig\nεj=1\nεn\nj/parenleftBig\nε4\nj/integraldisplay\nM∆gZk\nεj,ξj∆gZl\nεj,ξjdVg\n+bε2\nj/integraldisplay\nM∇Zk\nεj,ξj∇Zl\nεj,ξjdVg+a/integraldisplay\nMZk\nεj,ξjZl\nεj,��jdVg/parenrightBig\n=I1+I2+I3\nwhere, because of the Taylor expansion of gin normal coordinates given in ( 2.2), we have\nI1=ε4−n\nj/integraldisplay\nB(0,r)\n1/radicalBig\n|gξj(z)|n/summationdisplay\nα,β=1∂α/parenleftbigg\ngαβ\nξj(z)/radicalBig\n|gξj(z)|∂β/parenleftbigg\nΨk/parenleftbiggz\nεj/parenrightbigg\nχr/parenleftbiggz\nεj/parenrightbigg/parenrightbigg/parenrightbigg\n\n×\n1/radicalBig\n|gξj(z)|n/summationdisplay\nα,β=1∂α/parenleftbigg\ngαβ\nξj(z)/radicalBig\n|gξj(z)|∂β/parenleftbigg\nΨl/parenleftbiggz\nεj/parenrightbigg\nχr/parenleftbiggz\nεj/parenrightbigg/parenrightbigg/parenrightbigg\n/radicalBig\n|gξj(z)|dz\n=/integraldisplay\nRn∆Ψk∆Ψldz+o(1).\nAnalogously, we have\nI2=b/integraldisplay\nRn∇Ψk∇Ψldz+o(1) andI3=a/integraldisplay\nRnΨkΨldz+o(1).\nSumming up these three terms, we obtain\nI1+I2+I3=/integraldisplay\nRn∆Ψk∆Ψl+b∇Ψk∇Ψl+aΨkΨldz+o(1) =/angbracketleftBig\nΨk,Ψl/angbracketrightBig\n+o(1) =δk,l+o(1).\nbecause of the radial symmetry of the solution U.\nOn the other hand, combining this and ( 28), we get\nCal\nj+o(1) =1\nεn\nj/integraldisplay\nM[ε4\nj∆gZl\nεj,ξj∆gφj+bε2\nj∇Zl\nεj,ξj∇φj+aZl\nεj,ξjφj−f′(Wεj,ξj)φjZl\nεj,ξj]dVg.(29)\nLet\n˜φj(z) =/braceleftbiggφj(expξj(εjz))χr(εjz) ifz∈B(0, r/εj),\n0 otherwise .\nThen we have that for some constant ˜c,/ba∇dbl˜φj/ba∇dblH1(Rn)≤˜c/ba∇dbl˜φj/ba∇dblεj≤˜c. Therefore, we can assume that ˜φj\nconverges weakly to some ˜φinH2(Rn)and strongly in Lq\nloc(Rn)for anyq∈[2,2♯). Then, by eqs. ( 29),\nwe have the following\nCal\nj+o(1) =1\nεn\nj/integraldisplay\nM[ε4\nj∆gZl\nεj,ξj∆gφj+bε2\nj∇Zl\nεj,ξj∇φj+aZl\nεj,ξjφj−f′(Wεj,ξj)φjZl\nεj,ξj]dVg.\n=/integraldisplay\nRn(∆Ψl∆˜φ+b∇Ψl∇˜φ+aΨl˜φ−f′(U)Ψl˜φ)dz+o(1) =o(1). (30)\nFrom(30), we get that al\nj→0asj→ ∞ , and then (27)follows.\nStep 2. Now we are ready to prove ( 25). Recall that uj=φj−(ψj+ζj).\nSince/ba∇dblφj/ba∇dblεj= 1,/ba∇dblψj/ba∇dblεj→0and/ba∇dblζj/ba∇dblεj→0byStep 1 , then\n/ba∇dbluj/ba∇dblεj→1. (31)\nMoreover, by ( 24) we know that uj=ι∗\nεj[f′(Wεj,ξj)φj], and hence, by ( 17), it satisfies weakly\nε4\nj∆2\nguj−bε2\nj∆guj+auj=f′(Wεj,ξ)uj+f′(Wεj,ξ)(ψj+ζj)inM. (32)10 SALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\nMultiplying ( 32) byuj, and integrating over M, we get\n/ba∇dbluj/ba∇dbl2\nεj=1\nεn\nj/integraldisplay\nMf′(Wεj,ξ)u2\njdVg+1\nεn\nj/integraldisplay\nMf′(Wεj,ξ)(ψj+ζj)ujdVg. (33)\nBy H¨ older’s inequality and eq. ( 16), it follows that\n|1\nεn\nj/integraldisplay\nMf′(Wεj,ξj)(ψj+ζj)ujdVg| (34)\n≤/parenleftBigg\n1\nεn\nj/integraldisplay\nM(f′(Wεj,ξj)uj)2dVg/parenrightBigg1\n2/parenleftBigg\n1\nεn\nj/integraldisplay\nM(ψj+ζj)2dVg/parenrightBigg1\n2\n(35)\n≤c/ba∇dbluj/ba∇dblεj/ba∇dblψj+ζj/ba∇dblεj=o(1). (36)\nFinally, since /ba∇dblψj/ba∇dblεj→0,/ba∇dblζj/ba∇dblεj→0,and/ba∇dbluj/ba∇dblεj→1asj→ ∞ , then we get eq. ( 25).\nStep 3. In this Step, for each j, we define the auxiliar functions\n˜uj=/braceleftBigg\nuj/parenleftBig\nexpξj(εjz)/parenrightBig\nχr/parenleftBig\nexpξj(εjz)/parenrightBig\nz∈B(0, r/εj),\n0 otherwise ,\nWe claim that ˜uj→˜uweakly inH2(Rn)and strongly in Lp+1(Rn), where˜uis a weak solution to the\nequation\n∆2˜u−b∆˜u+a˜u=f′(U)˜u inRn. (37)\nFirst note that each ujis compactly supported in Bg(ξj, r)and in consequence, ˜ujis well defined.\nMoreover, it satisfies\n/ba∇dbl˜uj/ba∇dbl2\nH2(Rn)≤c/ba∇dbluj/ba∇dbl2\nεj≤c.\nHence, up to a subsequence, ˜uj→˜uweakly inH2(Rn)and strongly in Lq\nloc(Rn)for anyq∈[1,2♯), for\nsome˜u∈H2(Rn). For anyϕ∈C∞\n0(Rn), we define\nϕj(x) :=ϕ/parenleftBiggexp−1\nξj(x)\nεj/parenrightBigg\nχr/parenleftBig\nexp−1\nξj(x)/parenrightBig\n, x∈Bg(ξj,εjR)⊂M.\nLet us takeRsuch that suppϕ⊂B(0,R), andjlarge enough such that Bg(ξj,εjR)⊂Bg(ξj,r/2).\nMultiplying (32)byϕjand integrating over M, we have\n1\nεn\nj/integraldisplay\nM/parenleftbig\nε4\nj∆guj∆gϕj+bε2\nj∇guj∇gϕj+aujϕj/parenrightbig\ndVg\n=1\nεn\nj/integraldisplay\nMf′(Wεj,ξj)ujϕjdVg+1\nεn\nj/integraldisplay\nMf′(Wεj,ξj)(ψj+ζj)ϕjdVg. (38)\nBy settingx= expξj(εjz), we can integrate over B(0,R)⊂Rnin the following way\n1\nεn\nj/integraldisplay\nMf′(Wεj,ξj)ujϕjdVg=/integraldisplay\nB(0,R)f′(U(z)χr(εjz))˜ujϕ|gξj(εjz)|1/2dz\nand\n1\nεn\nj/integraldisplay\nMf′(Wεj,ξj)(ψj+ζj)ϕjdVg=/integraldisplay\nB(0,R)f′(U(z)χr(εjz))(˜ψj+˜ζj)ϕ|gξj(εjz)|1/2dz,\nwhere˜ψj(z) :=ψj(expξj(εjz))and˜ζj(z) :=ζj(expξj(εjz))forz∈B(0,R/εj).\nOnce again, H¨ older’s inequality comes to the rescue and we g et\n/integraldisplay\nB(0,R)˜ujϕ|gξj(εjz)|1/2dz≤ε2\nj/parenleftBigg/integraldisplay\nB(0,R)˜u2\njdz/parenrightBigg1/2/parenleftBigg/integraldisplay\nB(0,R)ϕ2|gξj(εjz)|dz/parenrightBigg1/2\n≤cε2\nj/ba∇dbl˜uj/ba∇dblH2(Rn)=o(εj).11\nwithcan upper bound for/integraltext\nB(0,R)ϕ2|gξj(εjz)|dz. Recall also that ujis bounded independently of jin\nH2(Rn). Hence, taking the limit in ( 38) we have\nlim\nj→∞1\nεn\nj/integraldisplay\nM/parenleftbig\nε4\nj∆guj∆gϕj+bε2\nj∇guj∇gϕj+aujϕj/parenrightbig\ndVg=/integraldisplay\nRnf′(U(z))˜uϕdz, (39)\nsince˜ψj,˜ζj→0weakly inH2(Rn)and strongly in Lp+1(Rn). Eq. ( 39) proves the claim.\nStep 4. We now claim that for any k∈ {1,2,...n},˜usatisfies also\n/integraldisplay\nRn/parenleftBig\n∆Ψk∆˜u+∇Ψk∇˜u+Ψk˜u/parenrightBig\ndz= 0. (40)\nSinceφj,ψj∈K⊥\nεj,ξj, from eq. ( 27) we have\n|/angbracketleftBig\nZk\nεj,ξj, uj/angbracketrightBig\nεj|=|/angbracketleftBig\nZk\nεj,ξj,φj−ψj−ζj/angbracketrightBig\nεj|=|/angbracketleftBig\nZk\nεj,ξj, ζj/angbracketrightBig\nεj| ≤ /ba∇dblZk\nεj,ξj/ba∇dblεj/ba∇dblζj/ba∇dblεj=o(1).(41)\nOn the other hand, we have\n/angbracketleftBig\nZk\nεj,ξj, uj/angbracketrightBig\nεj=1\nεn\nj/integraldisplay\nM/parenleftBig\nε4\nj∆gZk\nεj,ξj∆guj+bε2\nj∇Zk\nεj,ξj∇guj+aZk\nεj,ξjuj/parenrightBig\ndVg\n≤c/integraldisplay\nRn/parenleftBig\n∆Ψk∆˜u+∇Ψk∇˜u+Ψk˜u/parenrightBig\ndz+o(1). (42)\nFrom(41)and(42)we prove the claim of eq. ( 40). Therefore, (37)and(40)imply that ˜u= 0, because\nwe assumeUto be nondegenerate.\nStep 5. In this final Step, we deduce eq. ( 26). Since˜u= 0byStep 4 , we know that\n1\nεn\nj/integraldisplay\nMf′(Wεj,ξj)u2\njdVg=/integraldisplay\nB(0, εjr)f′(U(z)χr(εjz))˜u2\nj(z)|gξj(εjz)|1/2dz=o(1).\nThis last equation proves ( 26), which contradicts ( 25) and concludes the proof. /square\nNext, we study the term estimate of Rε,ξ= Π⊥\nε,ξ{i∗\nε[f(Wε,ξ)]−Wε,ξ}.\nRemark 3.2.Letv(z) :=u(expξ(z)),z∈B(0,r),u∈H2\ng(M). From [ MP09 , Remark 3.2] we know\nthat\n∆gξv=−∆v+Aij∂2\nijv+Bk∂kv\nwhere\nAij(z) :=−/bracketleftbig\ngij(z)−δij(z)/bracketrightbig\n,andBk(z) :=gij(z)Γk\nij(z).\nHere∆is the Euclidean Laplacian and we are using the Einstein summ ation convention. Then, we have\n∆2\ngξv= ∆gξ[−∆v+Ast∂2\nstv+Bh∂hv/bracketrightBig\n=−∆/bracketleftBig\n−∆v+Ast∂2\nstv+Bh∂hv/bracketrightBig\n+Aij∂2\nij/bracketleftBig\n−∆v+Ast∂2\nstv+Bh∂hv/bracketrightBig\n+Bk∂k/bracketleftBig\n−∆v+Ast∂2\nstv+Bh∂hv/bracketrightBig\n.\nWe also need to recall the formula for the bi-laplacian of a pr oduct function, which is\n∆2(Uεχr) = ∆2(Uε)χr+2∆(Uε)∆χr+4∂3\nijjUε∂iχr+4∂2\nijUε∂2\nijχr+4Uε∂3\nijjχr+Uε∆2(χr).\nNow, we are ready to prove the following result.\nLemma 3.3. There existε0>0andc>0such that for any ε∈(0,ε0), it holds\n/ba∇dblRε,ξ/ba∇dblε≤cε2.12 SALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\nProof. Asι∗\nε:Lp+1\npg,ε→H2\nεis a surjective map, we define the function Vε,ξonMbyWε,ξ=ι∗\nε(Vε,ξ)\nand consider ˜Vε,ξ(z) :=Vε,ξ/parenleftbig\nexpξ(z)/parenrightbig\nonB(0,r)⊆Rn. In other words, it satisfies\n˜Vε,ξ(z) =ε4∆2\ngWε,ξ/parenleftbig\nexpξ(z)/parenrightbig\n−ε2b∆gWε,ξ/parenleftbig\nexpξ(z)/parenrightbig\n+aWε,ξ/parenleftbig\nexpξ(z)/parenrightbig\n.\nDenote also ˜Wε,ξ(z) =Wε,ξ(expξz) =Uε(z)χr(z), wherevε(z) :=v(ε\nz)(and in consequence\n∂iUε(z) =1\nε∂iU(z\nε)). By Remark 3.2, we can compute\nε2∆gξ˜Wε,ξ(z) =ε2∆(Uεχr)(z)−ε2Aij∂2\nij(Uεχr)(z)+ε2Bk∂k(Uεχr)(z)\n= ∆U/parenleftBigz\nε/parenrightBig\nχr(z)+2ε2∂iUε(z)∂iχr(z)+ε2Uε(z)∆χr(z)\n+ε2Bk∂k(Uεχr)(z)−ε2Aij∂2\nij(Uεχr)(z).\nIn the same spirit, we compute\nε4∆2\ngξ˜Wε,ξ(z) =ε4∆2(Uεχr)(z)−ε4∆/parenleftbig\nAst∂2\nst(Uεχr)/parenrightbig\n(z)−ε4∆/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n(z)\n−ε4Aij∂2\nij(∆(Uεχr))(z)+ε4Aij∂2\nij/parenleftbig\nAst∂2\nst(Uεχr))/parenrightbig\n+ε4Aij∂2\nij/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n−ε4Bk∂k(∆(Uεχr))+ε4Bk∂k/parenleftbig\nAst∂2\nst(Uεχr)/parenrightbig\n+ε4Bk∂k/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n= ∆2U/parenleftBigz\nε/parenrightBig\nχr(z)+2ε4∆Uε(z)∆χr(z)+4ε4∂3\nijjUε(z)∂iχr(z)\n+4ε4∂2\nijUε∂2\nijχr(z)+4ε4Uε(z)∂3\nijjχr(z)+ε4Uε(z)∆2χr(z)\n−ε4∆/parenleftbig\nAst∂2\nst(Uεχr)/parenrightbig\n(z)−ε4∆/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n(z)\n−ε4Aij∂2\nij(∆(Uεχr))(z)+ε4Aij∂2\nij/parenleftbig\nAst∂2\nst(Uεχr))/parenrightbig\n+ε4Aij∂2\nij/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n−ε4Bk∂k(∆(Uεχr))+ε4Bk∂k/parenleftbig\nAst∂2\nst(Uεχr)/parenrightbig\n+ε4Bk∂k/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n.\nTaking into account the fact that the function Usatisfies the limit equation and the previous computations,\nit follows that\n˜Vε,ξ(z) =Up\nε(z)χr(z)+Rε(Uε,χr),\nwhere\nRε(Uε,χr) = 2ε4∆Uε(z)∆χr(z)+4ε4∂3\nijjUε(z)∂iχr(z)\n+4ε4∂2\nijUε(z)∂2\nijχr(z)+4ε4∂iUε(z)∂3\nijjχr(z)+ε4Uε(z)∆2χr(z)\n−ε4∆/parenleftbig\nAst∂2\nst(Uεχr)/parenrightbig\n(z)−ε4∆/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n(z)\n−ε4Aij∂2\nij(∆(Uεχr))(z)+ε4Aij∂2\nij/parenleftbig\nAst∂2\nst(Uεχr))/parenrightbig\n(z)+ε4Aij∂2\nij/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n(z)\n−ε4Bk∂k(∆(Uεχr))(z)+ε4Bk∂k/parenleftbig\nAst∂2\nst(Uεχr)/parenrightbig\n(z)+ε4Bk∂k/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n(z)\n−2bε2∂iUε(z)∂iχr(z)−bε2Uε(z)∆χr(z)−bε2Bk∂k(Uεχr)(z)+bε2Aij∂2\nij(Uεχr)(z).\nOn the other hand, by ( 23) and ( 19), we get there exists a positive constant Csuch that for ε>0small\nand for any point ξ∈M, there holds,\n/ba∇dblRε,ξ/ba∇dblε=/ba∇dblι∗\nε(f(Wε,ξ)−Vε,ξ)−Wε,ξ+ι∗\nε(Vε,ξ)/ba∇dblε≤C|f(Wε,ξ)−Vε,ξ|p+1\np,ε. (43)13\nWe need to point out that for some positive constant cwe have/integraldisplay\nBg(ξ,r)|f(Wε,ξ)−Vε,ξ|p+1\npdVg≤c/integraldisplay\nB(0,r)|f(˜Wε,ξ)−˜Vε,ξ|p+1\npdz (44)\n≤c/integraldisplay\nB(0,r)|Up\nε(z)(χp\nr(z)−χr(z))|p+1\npdz\n+c/integraldisplay\nB(0,r)|Rε(Uε,χr)|p+1\npdz.\nWe are led to estimate each term on the right-hand side in ( 44). However, by arguing as in [ MP09 ], it\nis easy to see that/integraldisplay\nB(0,r)|Rε(Uε,χr)|p+1\npdz=o(εn+2p+1\np). (45)\nWe write some of the estimates in the Section 6. Hence, from ( 43), (44) and ( 45), the following estimates\nhold\n/ba∇dblRε,ξ/ba∇dblε≤C\nεn/integraldisplay\nB(0,r)|Up\nε(z)(χp\nr(z)−χr(z))|p+1\npdz+o(ε2p+1\np).\nUsing again the exponential decay of Uand the definition of χr, we have/integraldisplay\nB(0,r)|Up\nε(z)(χp\nr(z)−χr(z))|p+1\npdz≤/integraldisplay\nB(0,r)\\B(0,r/2)Up\nε(z)dz≤o/parenleftBig\nεn+2(p+1)/p/parenrightBig\n.\nSince1<p<p♯thenε2p+1\np<ε2and we get the estimation claimed. /square\nNext, we prove the main result of this section.\nProposition 3.4. There exists ε0>0andc >0such that for any ξ∈Mand for allε∈(0,ε0)there\nexists a unique φε,ξ=φ(ε,ξ)∈K⊥\nε,ξwhich solves (14a). Moreover,\n/ba∇dblφε,ξ/ba∇dblε<cε2,\nandξ/ma√sto→φε,ξis aC1map.\nProof. We argue exactly as in [ MP09 , Proposition 3.5]. For the reader’s convenience, we briefly recall\nthe main steps. We solve ( 14a) by a fixed point argument. Recall that eq. ( 14a) is equivalent to\nLε,ξ(φ) =Nε,ξ(φ)+Rε,ξ. (46)\nWe define the operator Tε,ξ:K⊥\nε,ξ→K⊥\nε,ξby\nTε,ξ(φ) =L−1\nε,ξ(Nε,ξ(φ)+Rε,ξ).\nBy Proposition 3.1,Tε,ξis well defined and it holds\n/ba∇dblTε,ξ(φ)/ba∇dblε≤c/parenleftbig\n/ba∇dblNε,ξ(φ)/ba∇dblε+/ba∇dblRε,ξ/ba∇dblε/parenrightbig\nfor some constant c >0. Moreover, by the linearity of L−1\nε,ξ, the properties of ι∗, and the Mean Value\nTheorem we get\n/ba∇dblTε,ξ(φ1)−Tε,ξ(φ2)/ba∇dblε≤c/parenleftbig\n/ba∇dblNε,ξ(φ1)−Nε,ξ(φ2)/ba∇dblε/parenrightbig\n=cΠ⊥\nε,ξ/braceleftbig\nι∗\nε/bracketleftbig\nf(Wε,ξ+φ1)−f(Wε,ξ+φ1)−f′(Wε,ξ)φ1+f′(Wε,ξ)φ2/bracketrightbig/bracerightbig\n≤c/vextendsingle/vextendsinglef(Wε,ξ+φ1)−f(Wε,ξ+φ2)−f′(Wε,ξ)φ1+f′(Wε,ξ)φ2/vextendsingle/vextendsinglep+1\np,ε\n≤c/vextendsingle/vextendsinglef′(Wε,ξ+φ2+t(φ1−φ2))−f′(Wε,ξ)/vextendsingle/vextendsinglep+1\np,ε/ba∇dblφ1−φ2/ba∇dblp+1\np,ε\n≤c/vextendsingle/vextendsinglef′(Wε,ξ+φ2+t(φ1−φ2))−f′(Wε,ξ)/vextendsingle/vextendsinglep+1\np,ε/ba∇dblφ1−φ2/ba∇dblε.\nHere,cdenotes any positive constant.14 SALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\n|f(Wε,ξ+φ1)−f(Wε,ξ+φ2)−f′(Wε,ξ)(φ1−φ2)|p+1\np,ε\n≤C|(f′(Wε,ξ+φ2+τ(φ1−φ2))−f′(Wε,ξ))(φ1−φ2)|p+1\np,ε\n≤C|f′(Wε,ξ+φ2+τ(φ1−φ2))−f′(Wε,ξ)|p+1\np−1,ε|φ1−φ2|p+1\np,ε.\nRecall (from [ Li98 , Lemma 2.2]) that for all a>0, b∈R, we have\n||a+b|q−aq| ≤/braceleftbigg\nC(q)min/braceleftbig\n|b|q, aq−1|b|/bracerightbig\nifq∈(0,1)\nC(q)/parenleftbig\naq−1|b|+|b|q/parenrightbig\nifq≥1,\nwhich implies\n|f′(Wε,ξ+φ)−f′(Wε,ξ)| ≤/braceleftBiggC(q)|φ|p−1ifp∈(1,2)\nC(q)/parenleftBig\nWp−2\nε,ξ|φ|+|φ|p−2/parenrightBig\nifp≥2.(47)\nMoreover, it is easy to check that\n||φ|p−1|p+1\np≤c|φ|p−1\np,ε,\nand, ifp≥2\n|Wp−2\nε,ξφ2|p+1\np,ε|φ|2\np+1,ε≤C/ba∇dblφ/ba∇dbl2\nε.\nFrom these facts and from ( 47), it follows that\n|f′(Wε,ξ+φ2+τ(φ1−φ2))−f′(Wε,ξ)|p+1\np−1,ε≤C/ba∇dblφ1−φ2/ba∇dblε.\nMoreover, with the same estimates, we get\n/ba∇dblNε,ξ(φ)/ba∇dblε≤c/parenleftbig\n/ba∇dblφ/ba∇dbl2\nε+/ba∇dblφ/ba∇dblp−1\nε/parenrightbig\n.\nSo, we deduce\n/ba∇dblTε,ξ(φ1)−Tε,ξ(φ2)/ba∇dblε≤ /ba∇dblNε,ξ(φ1)−Nε,ξ(φ2)/ba∇dblε≤c/ba∇dblφ1−φ2/ba∇dblε, (48)\nforc∈(0,1), provided /ba∇dblφ1/ba∇dblεand/ba∇dblφ2/ba∇dblεare small enough. This fact, combined with Lemma 3.3, gives\nus\n/ba∇dblTε,ξ(φ)/ba∇dblε≤c/parenleftbig\n/ba∇dblNε,ξ(φ)/ba∇dblε+/ba∇dblRε,ξ/ba∇dblε/parenrightbig\n≤c/parenleftbig\n/ba��dblφ/ba∇dbl2\nε+/ba∇dblφ/ba∇dblp−1\nε+ε2/parenrightbig\n.\nSo, there exists c∈(0,1)such thatTε,ξmaps a ball of center 0 and radius cε2inK⊥\nε,ξinto itself and\nit is a contraction. So, there exists a fixed point φε,ξwith the norm /ba∇dblφε,ξ/ba∇dblε≤ε2.\nFinally, to prove that the map ξ→φε,ξis in fact aC1map, we use the Implicit Function Theorem\napplied to the function\nG:M×H2\nε−→R,(ξ,u)/ma√sto−→G(ξ,u) :=Tε,ξ(u)−u.\nAs mentioned previously, equation ( 48) ensures the existence of a φε,ξsuch thatG(ξ,φε,ξ) = 0 .\nMoreover, when restricted to a sufficiently small ball, Tε,ξ(φ)is differentiable and has a differentiable\ninverseLε,ξ(φ), by ( 46). Consequently, according to the Implicit Function Theore m, the mapping ξ→\nφε,ξisC1.\nWe have that G(ξ,φε,ξ) = 0 and that the operator∂G\n∂u(ξ,φε,ξ) :H2\nε→H2\nεis invertible. This\nconcludes the proof. /square15\n4. A SYMPTOTIC EXPANSION\nIn this section, we will prove some important properties reg arding the functional Jε:H2\nε→R,\ndefined by\nJε(u) :=1\nεn/integraldisplay\nM/parenleftbiggε4\n2|∆gu(x)|2+ε2\n2b|∇u(x)|2+a\n2|u(x)|2−1\n(p+1)(u+(x))p+1/parenrightbigg\ndVg(x),\nwhereu+(x) = max{u(x),0}. It is well known that any critical point of the functional Jεcorresponds\nto a solution of the problem ( 18).\nProposition 4.1. Forξ∈M, andε>0small, the following expansion holds.\nJε(Wε,ξ) =α+βε2τg(ξ) +o(ε2),\nwhere\nα:=1\n2/integraldisplay\nRn/parenleftbig\n|∆U(z)|2+|∇U(z)|2+U2(z)/parenrightbig\ndz−1\np+1/integraldisplay\nRnUp+1(z)dz, (49)\nβ:=/integraldisplay\nRn/parenleftbiggU′(|z|)\n|z|−U′′(|z|)/parenrightbigg2\nz2\n1z2\n2dz+b\n2/integraldisplay\nRnU′(|z|)2z2\n1z2\n2dz, (50)\nandτgas in (15).\nProof. We will start by estimating ε4−n/integraltext\nM|∆gWε,ξ(x)|2dVg(x).Straightforward calculations show\nε4−n/integraldisplay\nM|∆gWε,ξ(x)|2dVg(x)\n=ε4−n/integraldisplay\nB(0,r)1/radicalbig\n|gξ(z)|\nn/summationdisplay\ni,j=1∂i/parenleftbigg\ngij\nξ(z)/radicalBig\n|gξ(z)|/parenleftbigg1\nε∂jU/parenleftBigz\nε/parenrightBig\nχr(z)+U/parenleftBigz\nε/parenrightBig\n∂jχr(z)/parenrightbigg/parenrightbigg\n2\ndz\n=ε4−n/integraldisplay\nB(0,r)\n1/radicalbig\n|gξ(z)|n/summationdisplay\ni,j=1∂i/parenleftbigg\ngij\nξ(z)/radicalBig\n|gξ(z)|/parenrightbigg/parenleftbigg1\nε∂jU/parenleftBigz\nε/parenrightBig\nχr(z)+U/parenleftBigz\nε/parenrightBig\n∂jχr(z)/parenrightbigg\n+1/radicalbig\n|gξ(z)|n/summationdisplay\ni,j=1gij\nξ(z)/radicalBig\n|gξ(z)|∂i/parenleftbigg1\nε∂jU/parenleftBigz\nε/parenrightBig\nχr(z)+U/parenleftBigz\nε/parenrightBig\n∂jχr(z)/parenrightbigg\n2/radicalBig\n|gξ(z)|dz\n=ε4−n/integraldisplay\nB(0,r)\n1/radicalbig\n|gξ(z)|n/summationdisplay\ni,j=1∂i/parenleftbigg\ngij\nξ(z)/radicalBig\n|gξ(z)|/parenrightbigg/parenleftbigg1\nε∂jU/parenleftBigz\nε/parenrightBig\nχr(z)+U/parenleftBigz\nε/parenrightBig\n∂jχr(z)/parenrightbigg\n+n/summationdisplay\ni,j=1gij\nξ(z)/parenleftbigg1\nε2∂2\nijU/parenleftBigz\nε/parenrightBig\nχr(z)+2\nε∂jU/parenleftBigz\nε/parenrightBig\n∂iχr(z)+U/parenleftBigz\nε/parenrightBig\n∂2\nijχr(z)/parenrightbigg\n2/radicalBig\n|gξ(z)|dz\n=ε4−n/integraldisplay\nB(0,r)\n1/radicalbig\n|gξ(z)|n/summationdisplay\ni,j=1∂i/parenleftbigg\ngij\nξ(z)/radicalBig\n|gξ(z)|/parenrightbigg/parenleftbigg1\nε∂jU/parenleftBigz\nε/parenrightBig\nχr(z)/parenrightbigg\n+n/summationdisplay\ni,j=1gij\nξ(z)/parenleftbigg1\nε2∂2\nijU/parenleftBigz\nε/parenrightBig\nχr(z)/parenrightbigg\n2/radicalBig\n|gξ(z)|dz+o(ε2),16 SALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\nwhere the error term arises from disregarding the derivativ es ofχr. By implementing a variable trans-\nformation, we obtain\nε4−n/integraldisplay\nM|∆gWε,ξ(x)|2dVg(x)\n=ε4−n/integraldisplay\nB(0,r)\nn/summationdisplay\ni,j=11/radicalbig\n|gξ(z)|∂i/parenleftbigg\ngij\nξ(z)/radicalBig\n|gξ(z)|/parenrightbigg/parenleftbigg1\nε∂jU/parenleftBigz\nε/parenrightBig\nχr(z)/parenrightbigg\n2/radicalBig\n|gξ(z)|dz\n+2ε4−n/integraldisplay\nB(0,r)\nn/summationdisplay\ni,j=11/radicalbig\n|gξ(z)|∂i/parenleftbigg\ngij\nξ(z)/radicalBig\n|gξ(z)|/parenrightbigg/parenleftbigg1\nε∂jU/parenleftBigz\nε/parenrightBig\nχr(z)/parenrightbigg\n×\n\nn/summationdisplay\ni,j=11\nε2gij\nξ(z)∂2\nijU/parenleftBigz\nε/parenrightBig\nχr(z)\n/radicalBig\n|gξ(z)|dz\n+ε4−n/integraldisplay\nB(0,r)\nn/summationdisplay\ni,j=11\nε2gij\nξ(z)∂2\nijU/parenleftBigz\nε/parenrightBig\nχr(z)\n2/radicalBig\n|gξ(z)|dz+o(ε2)\n=/integraldisplay\nB(0,r/ε)\nn/summationdisplay\ni,j=1gij\nξ(εz)∂2\nijU(z)χr(εz)\n2/radicalBig\n|gξ(εz)|dz+o(ε2).\nHere, the final term is the sole term with an order greater than ε2. Consequently, from the Taylor\nexpansions in Lemma 2.2we obtain\nε4−n/integraldisplay\nM|∆gWε,ξ(x)|2dVg(x)\n=/integraldisplay\nB(0,r/ε)\n∆U(z)+ε2\n2n/summationdisplay\ni,j,r,k=1∂2gij\nξ\n∂zr∂zk(0)zrzk∂2\nijU(z)\n2\nχ2\nr(εz)\n×\n1−ε2\n4n/summationdisplay\nl,r,k=1∂2gll\nξ\n∂zr∂zk(0)zrzk\ndz+o(ε2)\n=/integraldisplay\nRn(∆U(z))2dz+ε2/integraldisplay\nRn∆U(z)\nn/summationdisplay\ni,j,r,k=1∂2gij\nξ\n∂zr∂zk(0)zrzk∂2\nijU(z)\n−1\n4n/summationdisplay\nl,r,k=1∂2gll\nξ\n∂zr∂zk(0)zrzk∆U(z)\ndz+o(ε2).\nTo deal with the remaining terms, we proceed as in [ MP09 , Lemma 5.3], and we have\nε2−nb\n2/integraldisplay\nM|∇Wε,ξ|2dVg=b\n2/integraldisplay\nRn|∇U|2dz+ε2b\n4n/summationdisplay\ni,j,h,k=1∂2gij\nξ\n∂zh∂zk(0)/integraldisplay\nRn/parenleftbiggU′(|z|)\n|z|/parenrightbigg2\nzizjzhzkdz\n−ε2b\n8n/summationdisplay\ni,h=1∂2gii\nξ\n∂z2\nh(0)/integraldisplay\nRn|∇U|2z2\nhdz+o/parenleftbig\nε2/parenrightbig\n, and\n1\np+1/integraldisplay\nMWp+1\nε,ξdVg=1\np+1/integraldisplay\nRnUp+1dz−ε2\n4(p+1)n/summationdisplay\ni,j=1∂2gii\nξ\n∂z2\nj(0)/integraldisplay\nRnUp+1z2\njdz+o/parenleftbig\nε2/parenrightbig\n.17\nFinally, we get\nJε(Wε,ξ)−/integraldisplay\nRn/parenleftbigg1\n2|∆U(z)|2+b\n2|∇U|2+a\n2U2−1\np+1Up+1/parenrightbigg\ndz\n=ε2\n1\n2n/summationdisplay\ni,j,h,k=1∂2gij\nξ\n∂zh∂zk(0)/integraldisplay\nRn∂2\nijU(z)∆U(z)zhzkdz−1\n4/summationdisplay\nl,h,k∂2gll\nξ\n∂zh∂zk(0)/integraldisplay\nRn(∆U(z))2zhzkdz\n+b\n4n/summationdisplay\ni,j,h,k=1∂2gij\nξ\n∂zh∂zk(0)/integraldisplay\nRn/parenleftbiggU′(|z|)\n|z|/parenrightbigg2\nzizjzhzkdz\n−n/summationdisplay\nl,j=1∂2gll\nξ\n∂z2\nh(0)/integraldisplay\nRn/parenleftbiggb\n8|∇U|2−1\n4(p+1)Up+1−a\n8U2/parenrightbigg\nz2\nhdz\n+o(ε2).\n(51)\nAsUis a radial function, we have\n/summationdisplay\nl,h,k/parenleftBigg\n∂2gll\nξ\n∂zh∂zk(0)/integraldisplay\nRn(∆U(z))2zhzkdz/parenrightBigg\n=\n/summationdisplay\ni,k∂2gii\nξ\n∂z2\nk(0)\n/integraldisplay\nRn(∆U(z))2z2\n1dz.\nOn the other side, for each k= 1,...,n , there holds\n/integraldisplay\nRn/parenleftbiggb\n8|∇U|2−1\n4(p+1)Up+1−a\n8U2/parenrightbigg\nz2\nhdz=/integraldisplay\nRn/parenleftbiggb\n8|∇U|2−1\n4(p+1)Up+1−a\n8U2/parenrightbigg\nz2\n1dz.\nFrom there, it follows that the second and last terms in the r. h.s. of ( 51) cancel each other out. To\ncompute the remaining terms in the equation ( 51) we use the auxiliary Lemmas 4.2and4.3and we get\nJε(Wε,ξ)−/integraldisplay\nRn/parenleftbigg1\n2|∆U(z)|2+b\n2|∇U|2+a\n2U2−1\np+1Up+1/parenrightbigg\ndz=βε2τg(ξ) +o(ε2)\nand the result follows. /square\nIn the following, we will present two auxiliary Lemmas that w e have used in the previous proof. We\nfollow the ideas of [ AM99 ].\nLemma 4.2. There holds\nn/summationdisplay\ni,j,h,k=1/parenleftBigg\n∂2gij\nξ\n∂zh∂zk(0)/integraldisplay\nRn∂2\nijU(z)∆U(z)zhzkdz/parenrightBigg\n=\n/summationdisplay\ni,k∂2gii\nξ(0)\n∂z2\nk\n/integraldisplay\nRn/parenleftbigg\n−U′(|z|)\n|z|3+U′′(|z|)\n|z|2/parenrightbigg\n∆U(z)z2\n1z2\n2dz\nProof. Note that, as Uis a radial function, it satisfies\n/integraldisplay\nRn∂2\nijU(z)∆U(z)zhzkdz=/integraldisplay\nRn/parenleftbigg\n−U′(|z|)\n|z|3+U′′(|z|)\n|z|2/parenrightbigg\n∆U(z)zizjzhzkdz.\nLet us denote I(p(z)) =/integraldisplay\nRn/parenleftbigg\n−U′(|z|)\n|z|3+U′′(|z|)\n|z|2/parenrightbigg\n∆U(z)p(z)dz.18 SALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\nLet us turn our attention to the term I(zizjzkzl): it is different from zero only when i=jandl=k,\nor wheni=kandl=j, or wheni=landj=k. Hence, there holds\n/summationdisplay\ni,j,k,l∂2gij\nξ(0)\n∂zk∂zlI(zizjzkzl)\n=/summationdisplay\ni=j,k=l∂2gij\nξ(0)\n∂zk∂zlI(zizjzkzl)+/summationdisplay\ni=k,j=l∂2gij\nξ(0)\n∂zk∂zlI(zizjzkzl)\n+/summationdisplay\ni=l,j=k∂2gij\nξ(0)\n∂zk∂zlI(zizjzkzl)−2/summationdisplay\ni=j=k=l∂2gij\nξ(0)\n∂zk∂zlI(zizjzkzl)\n=/summationdisplay\ni,k∂2gii\nξ(0)\n∂z2\nkI/parenleftbig\nz2\niz2\nk/parenrightbig\n+2/summationdisplay\ni,j∂2gij\nξ(0)\n∂zi∂zjI/parenleftbig\nz2\niz2\nj/parenrightbig\n−2/summationdisplay\ni∂2gii\nξ(0)\n∂z2\niI/parenleftbig\nz4\ni/parenrightbig\n.\nSinceI/parenleftbig\nz4\nk/parenrightbig\n= 3I/parenleftbig\nz2\nkz2\nl/parenrightbig\n,k/\\e}atio\\slash=l, then\n/summationdisplay\ni,k∂2gii\nξ(0)\n∂z2\nkI/parenleftbig\nz2\niz2\nk/parenrightbig\n=/summationdisplay\ni,k∂2gii\nξ(0)\n∂z2\nkI/parenleftbig\nz2\n1z2\n2/parenrightbig\n+2\n3/summationdisplay\ni∂2gii\nξ(0)\n∂z2\niI/parenleftbig\nz4\ni/parenrightbig\nand\n/summationdisplay\ni,j∂2gij\nξ(0)\n∂zi∂zjI/parenleftbig\nz2\niz2\nj/parenrightbig\n=/summationdisplay\ni,j∂2gij\nξ(0)\n∂zi∂zjI/parenleftbig\nz2\n1z2\n2/parenrightbig\n+2\n3/summationdisplay\ni∂2gii\nξ(0)\n∂z2\niI/parenleftbig\nz4\ni/parenrightbig\n.\nThe last two equalities and equation ( 20) imply\n/summationdisplay\ni,j,k,l∂2gij\nξ(0)\n∂zk∂zlI(zizjzkzl) =/summationdisplay\ni,j∂2gii\nξ(0)\n∂z2\njI/parenleftbig\nz2\n1z2\n2/parenrightbig\n+2/summationdisplay\ni,j∂2gij\nξ(0)\n∂zi∂zjI/parenleftbig\nz2\n1z2\n2/parenrightbig\n=τg(ξ)I/parenleftbig\nz2\n1z2\n2/parenrightbig\n.\nSo we get the desired identity. /square\nUsing the same techniques, we get the following result.\nLemma 4.3. There holds\nn/summationdisplay\ni,j,h,k=1/parenleftBigg\n∂2gij\nξ\n∂zh∂zk(0)/integraldisplay\nRn/parenleftbiggU′(|z|)\n|z|/parenrightbigg2\nzizjzhzkdz/parenrightBigg\n=\nn/summationdisplay\ni,j=1∂2gii\nξ\n∂z2\nk(0)\n/integraldisplay\nRn/parenleftbiggU′(|z|)\n|z|/parenrightbigg2\nz2\n1z2\n2dz.\nProof. DenoteI(p(z)) =/integraldisplay\nRn/parenleftbiggU′(|z|)\n|z|/parenrightbigg2\np(z)dz,and follow the same steps in the proof of the previ-\nous lemma. /square\n5. T HE REDUCED PROBLEM\nIn this section, we study the problem ( 14b). Forξ∈Mwe consider the unique φε,ξ∈K⊥\nε,ξgiven by\nProposition 3.4that solves problem ( 14a) and define the function Jε:M→R, by\nJε(ξ) =Jε(Wε,ξ+φε,ξ).\nLemma 5.1. It holds\nJε(ξ) =Jε(Wε,ξ+φε,ξ) =Jε(Wε,ξ)+o(ε2)\nC0-uniformly in compact sets of M.19\nProof. Sinceφε,ξ∈K⊥\nε,ξand satisfies ( 14a), we have\n0 =/a\\}b∇acketle{tφε,ξ,Wε,ξ+φε,ξ−ι∗\nε(f(Wε,ξ+φε,ξ))/a\\}b∇acket∇i}htε\n=/ba∇dblφε,ξ/ba∇dbl2\nε+/a\\}b∇acketle{tφε,ξ,Wε,ξ/a\\}b∇acket∇i}htε−1\nεn/integraldisplay\nMf(Wε,ξ+φε,ξ)φε,ξdVg(x)\nTherefore, if we set F(u) =1\np+1(u+)p+1, then we get\nJε(Wε,ξ+φε,ξ)−Jε(Wε,ξ) =1\n2/ba∇dblφε,ξ/ba∇dbl2\nε+/a\\}b∇acketle{tφε,ξ,Wε,ξ/a\\}b∇acket∇i}htε−1\nεn/integraldisplay\nMf(Wε,ξ)φε,ξdVg(x)\n−1\nεn/integraldisplay\nM[F(Wε,ξ+φε,ξ)−F(Wε,ξ)−f(Wε,ξ)φε,ξ]dVg(x)\n=−1\n2/ba∇dblφε,ξ/ba∇dbl2\nε+1\nεn/integraldisplay\nM[f(Wε,ξ+φε,ξ)−f(Wε,ξ)]φε,ξdVg(x)\n−1\nεn/integraldisplay\nM[F(Wε,ξ+φε,ξ)−F(Wε,ξ)−f(Wε,ξ)φε,ξ]dVg(x).\nBy the Mean Value Theorem we know that for some t1,t2∈[0,1]it holds\n1\nεn/integraldisplay\nM[f(Wε,ξ+φε,ξ)−f(Wε,ξ)]φε,ξdVg(x) =1\nεn/integraldisplay\nMf′(Wε,ξ+t1φε,ξ)φ2\nε,ξdVg(x),\nand\n1\nεn/integraldisplay\nM[F(Wε,ξ+φε,ξ)−F(Wε,ξ)−f(Wε,ξ)φε,ξ]dVg(x) =1\n2εn/integraldisplay\nMf′(Wε,ξ+t2φε,ξ)φ2\nε,ξdVg(x).\nMoreover we have for any t∈[0,1]\n1\nεn/integraldisplay\nM|f′(Wε,ξ+tφε,ξ)|φ2\nε,ξdVg(x)≤c1\nεn/integraldisplay\nMWp−1\nε,ξφ2\nε,ξdVg(x)+c1\nεn/integraldisplay\nMφp+1\nε,ξdVg(x)\n≤c1\nεn/integraldisplay\nMφ2\nε,ξdVg(x)+c1\nεn/integraldisplay\nMφp+1\nε,ξdVg(x)≤C(/ba∇dblφε,ξ/ba∇dbl2\nε+/ba∇dblφε,ξ/ba∇dblp+1\nε) =o(ε2).\nIn the last inequality we use ( 16) and by Proposition 3.4we get/ba∇dblφε,ξ/ba∇dblε=o(ε2). /square\nIt is well known that any critical point of Jεis solution to problem ( 3). Now, we prove the following:\nProposition 5.2. For anyξ∈Mwe have\nJε(ξ) =α+βε2τg(ξ)+o(ε2), (52)\nC0-uniformly with respect to ξin compact sets of Masε→0, withαandβgiven in (49)and(50), and\nthe function τgis defined in (15). Moreover, if ξεis a critical point of Jε, then the function Wε,ξε+φε,ξε\nis a solution to the problem (14b).\nProof. Proposition 4.1and Lemma 5.1prove ( 52). We are left to prove that if ξεis a critical point\nofJε, then the function Wε,ξε+φε,ξεis a solution to problem ( 3). Fory∈B(0,r)⊂Rnwe let\nξ(y) = expξε(y)∈Bg(ξ,r)⊂M. We remark that ξ(0) =ξε. Sinceξεis a critical point of Jε,\n∂\n∂yiJε(ξ(y))/vextendsingle/vextendsingle/vextendsingle\ny=0= 0,fori= 1,...,n. (53)\nNote that Equation ( 3) is equivalent to ∇Jε(u) = 0 , where∇Jε:H2\nε→H2\nε. Moreover, we can write\n∇Jε/parenleftbig\nWε,ξ(y)+φε,ξ(y)/parenrightbig\n= Π⊥\nε,y∇Jε/parenleftbig\nWε,ξ(y)+φε,ξ(y)/parenrightbig\n+Πε,y∇Jε/parenleftbig\nWε,ξ(y)+φε,ξ(y)/parenrightbig\n,\nwhere the first term on the right is 0by the construction of φε,ξ(y). Then, we write the second term as\nΠε,y∇Jε/parenleftbig\nWε,ξ(y)+φε,ξ(y)/parenrightbig\n=/summationdisplay\nkCk\nεZk\nε,y,20 SALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\nfor some functions Ck\nε:B(0,r)→R. We have to prove that for each k= 1,...,n andε >0small,\nCk\nε(0) = 0 . Indeed, ifiis fixed, then we have\n0 =∂\n∂yiJε(ξ(y))/vextendsingle/vextendsingle/vextendsingle\ny=0=/angbracketleftbigg\n∇Jε(Wε,ξε+φε,ξε),∂\n∂yi/vextendsingle/vextendsingle/vextendsingle\ny=0/parenleftbig\nWε,ξ(y)+φε,ξ(y)/parenrightbig/angbracketrightbigg\nε\n=/angbracketleftBigg/summationdisplay\nkCk\nε(0)Zk\nε,ξε,∂\n∂yi/vextendsingle/vextendsingle/vextendsingle\ny=0/parenleftbig\nWε,ξ(y)+φε,ξ(y)/parenrightbig/angbracketrightBigg\nε. (54)\nSinceφε,ξ(y)∈K⊥\nε,ξ(y), for anyk, we have that/angbracketleftbig\nZk\nε,y,φε,ξ(y)/angbracketrightbig\nε= 0. Then\n/angbracketleftbigg\nZk\nε,ξε,∂\n∂yi/vextendsingle/vextendsingle/vextendsingle\ny=0φε,ξ(y)/angbracketrightbigg\nε+/angbracketleftbigg∂\n∂yi/vextendsingle/vextendsingle/vextendsingle\ny=0Zk\nε,y,φε,ξε/angbracketrightbigg\nε= 0.\nIt follows that\nliminf\nε→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg\nZk\nε,ξε,∂\n∂yi/vextendsingle/vextendsingle/vextendsingle\ny=0φε,ξ(y)/angbracketrightbigg\nε/vextendsingle/vextendsingle/vextendsingle/vextendsingle= liminf\nε→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/angbracketleftbigg∂\n∂yi/vextendsingle/vextendsingle/vextendsingle\ny=0Zk\nε,ξ(y), φε,ξε/angbracketrightbigg\nε/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤liminf\nε→0/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂\n∂yi/vextendsingle/vextendsingle/vextendsingle\ny=0Zk\nε,ξ(y)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nε/ba∇dblφε,ξε/ba∇dblε= 0,\nwhere the last equality follows from Proposition 3.4and ( 22). On the other side, we have from ( 21) that\nlim\nε→0ε/angbracketleftBigg/summationdisplay\nkCk\nε(0)Zk\nε,ξε,∂\n∂yi/vextendsingle/vextendsingle/vextendsingle\ny=0Wε,ξ(y)/angbracketrightBigg\nε=Ci\nε(0)C.\nAnd then it follows from ( 54) thatCi\nε(0) = 0 for alli= 1,...,n . /square\nNow, we are ready to prove our main result.\nProof of Theorem 1.2.Let us assume that ξ0is an isolated maximum for the function τg. Then, there is\nρ>0such thatξ0is the only maximum point of τginBg(ξ0,ρ). AsJε∈C0(M), we know that there\nis aξε∈Bg(ξ0,ρ)which satisfies that\nJ(ξε)≥J(ξ),for allξ∈Bg(ξ0,ρ). (55)\nBy Proposition 5.2we have\nJε(ξε) =α+βε2τg(ξε)+o(ε2). (56)\nNow, consider an auxiliary point, given by ξ∗\nε= expξ0(√εz), for somez∈Rnwith|z|= 1. Then\nξ∗\nε∈Bg(ξ0,ρ)forεsmall enough as dg(ξ0,ξ∗\nε)2≤ε. Using the Taylor expansion of τgaroundξ0, we\nhave\nτg(ξ∗\nε) =τg(ξ0)+o(1)\nand it follows from there and Proposition 5.2that\nJε(ξ∗\nε) =α+βε2τg(ξ0)+o(ε2). (57)\nReading at the equations ( 55), (56) and ( 57) we deduce\nβ(τg(ξε)−τg(ξ0))≥o(1), (58)\nand asξ0is the only maximum point of τginBg(ξ0,ρ)andβ >0, then we have by ( 56) and ( 58) that\nlim\nε→0τg(ξε)−τg(ξ0) = 0 andlim\nε→0ξε−ξ0= 0.\nMoreover, by Proposition 5.2, the function uε=Wε,ξε+φε,ξεis a solution to problem ( 3) and by\nProposition 3.4, we have\n/ba∇dbluε−Wε,ξε/ba∇dblε=/ba∇dblφε,ξε/ba∇dblε=o(ε). (59)\nSo, we proved that the solution uεto Equation ( 3) is concentrated around ξ0asε→0. /square21\n6. A PPENDIX\nIn this Section, we compute the estimations omitted in ( 45).\nLemma 6.1. It holds/integraldisplay\nB(0,r)|Rε(Uε,χr)|p+1\npdz=o/parenleftBig\nεn+2p+1\np/parenrightBig\n.\nProof. Recall that in the following expression, we are using the Ein stein notation.\nRε(Uε,χr) = 2ε4∆Uε(z)∆χr(z)+4ε4∂3\nijjUε(z)∂iχr(z)\n+4ε4∂2\nijUε(z)∂2\nijχr(z)+4ε4∂iUε(z)∂3\nijjχr(z)+ε4Uε(z)∆2χr(z)\n−ε4∆/parenleftbig\nAst∂2\nst(Uεχr)/parenrightbig\n(z)−ε4∆/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n(z)\n−ε4Aij∂2\nij(∆(Uεχr))(z)+ε4Aij∂2\nij/parenleftbig\nAst∂2\nst(Uεχr))/parenrightbig\n(z)+ε4Aij∂2\nij/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n(z)\n−ε4Bk∂k(∆(Uεχr))(z)+ε4Bk∂k/parenleftbig\nAst∂2\nst(Uεχr)/parenrightbig\n(z)+ε4Bk∂k/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n(z)\n−2bε2∂iUε(z)∂iχr(z)−bε2Uε(z)∆χr(z)−bε2Bk∂k(Uεχr)(z)+bε2Aij∂2\nij(Uεχr)(z).\nConsidering the decay of Uand the boundedness of χr, as given in ( 9) and ( 10), we get that\n2ε4p+1\np/integraldisplay\nB(0,r)|∆Uε|p+1\np(z)|∆χr(z)|p+1\npdz≤cε4p+1\np/integraldisplay\nB(0,r)\\B(0,r/2)|∆Uε|p+1\np(z)dz=o/parenleftBig\nεn+4p+1\np/parenrightBig\n4ε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle∂3\nijjUε(z)∂iχr(z)/vextendsingle/vextendsinglep+1\npdz≤cε4p+1\np/integraldisplay\nB(0,r)\\B(0,r/2)/vextendsingle/vextendsingle∂3\nijjUε/vextendsingle/vextendsinglep+1\np(z)dz=o/parenleftBig\nεn+4p+1\np/parenrightBig\n4ε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle∂2\nijUε(z)∂2\nijχr(z)/vextendsingle/vextendsinglep+1\npdz≤cε4p+1\np/integraldisplay\nB(0,r)\\B(0,r/2)/vextendsingle/vextendsingle∂2\nijUε/vextendsingle/vextendsinglep+1\np(z)dz=o/parenleftBig\nεn+4p+1\np/parenrightBig\n4ε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle∂iUε(z)∂3\nijjχr(z)/vextendsingle/vextendsinglep+1\npdz≤cε4p+1\np/integraldisplay\nB(0,r)\\B(0,r/2)|∂iUε|p+1\np(z)dz=o/parenleftBig\nεn+4p+1\np/parenrightBig\nε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingleUε(z)∆2χr(z)/vextendsingle/vextendsinglep+1\npdz≤cε4p+1\np/integraldisplay\nB(0,r)\\B(0,r/2)|Uε|p+1\np(z)dz=o/parenleftBig\nεn+4p+1\np/parenrightBig\n2bε2p+1\np/integraldisplay\nB(0,r)|∇Uε(z)·∇χr(z)|p+1\npdz≤cε2p+1\np/integraldisplay\nB(0,r)\\B(0,r/2)|∇Uε|p+1\np(z)dz=o/parenleftBig\nεn+2p+1\np/parenrightBig\nand\nbε2p+1\np/integraldisplay\nB(0,r)|Uε(z)∆χr(z)|p+1\npdz≤cε2p+1\np/integraldisplay\nB(0,r)\\B(0,r/2)|Uε|p+1\np(z)dz=o/parenleftBig\nεn+2p+1\np/parenrightBig\n.\nBy the standard properties of the exponential map establish ed in Lemma 2.2, we know that there exists a\npositive constant Csuch that for any point z���B(0,z)and any indices i,j,andk,it holds\n|Aij(εz)|=/vextendsingle/vextendsinglegij(εz)−δij(εz)/vextendsingle/vextendsingle≤C|εz|2,and|Bk(εz)|=|gij(εz)Γk\nij(εz)| ≤C|εz|.22 SALOM ´ON ALARC ´ON, SIM ´ON MASN ´U, PEDRO MONTERO, AND CAROLINA REY\nThen we have\nε2p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingleAij(z)∂2\nij(Uεχr)(z)/vextendsingle/vextendsinglep+1\npdz\n=ε2p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle/vextendsingle/parenleftBig\ngij\nξ(z)−δij/parenrightBig/parenleftbig\n∂2\nijUεχr+2∂iUε∂jχr+Uε∂2\nijχr/parenrightbig\n(z)/vextendsingle/vextendsingle/vextendsinglep+1\npdz\n≤cεn/integraldisplay\nB(0,r/ε)/vextendsingle/vextendsingle/vextendsingle/parenleftBig\ngij\nξ(εz)−δij/parenrightBig\n∂2\nijU(z)/vextendsingle/vextendsingle/vextendsinglep+1\npdz\n+cε4p+1\np/integraldisplay\nB(0,r)\\B(0,r/2)/vextendsingle/vextendsinglez2Uε(z)/vextendsingle/vextendsinglep+1\npdz+cε4p+1\np/integraldisplay\nB(0,r)\\B(0,r/2)/vextendsingle/vextendsinglez2(∂iUε)(z)/vextendsingle/vextendsinglep+1\npdz\n=O/parenleftBig\nεn+2p+1\np/parenrightBig\n.\nSimilarly, since Mis an analytic manifold, we obtain bounds for the derivative s ofAijfrom the\nestimates given in Lemma 2.2as well. More precisely, we have that\nε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingleAij∂2\nij(∆(Uεχr))(z)/vextendsingle/vextendsinglep+1\npdz\n=cεn/integraldisplay\nB(0,r/ε)/vextendsingle/vextendsingleAij∂2\nij∆U(z)/vextendsingle/vextendsinglep+1\npdz+o/parenleftBig\nεn+4p+1\np/parenrightBig\n≤O/parenleftBig\nεn+2p+1\np/parenrightBig\n.\nMoreover\nε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle∆/parenleftbig\nAst∂2\nst(Uεχr)/parenrightbig\n(z)/vextendsingle/vextendsinglep+1\npdz\n≤cεn/integraldisplay\nB(0,r/ε)/vextendsingle/vextendsingleAst∆∂2\nstU(z)/vextendsingle/vextendsinglep+1\npdz+ε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle∆/parenleftbig\nAst/parenrightbig\n∂2\nst(Uεχr)(z)/vextendsingle/vextendsinglep+1\npdz\n+2ε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle∇/parenleftbig\nAst/parenrightbig\n∇∂2\nst(Uεχr)(z)/vextendsingle/vextendsinglep+1\npdz=O/parenleftBig\nεn+2p+1\np/parenrightBig\nAnalogously, as the derivatives of Bhare bounded, we have\nε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle/vextendsingle∆/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n(z)/vextendsingle/vextendsingle/vextendsinglep+1\npdz=\nεn+p+1\np/integraldisplay\nB(0,r/ε)/vextendsingle/vextendsingle/vextendsingleBh(εz)∂h∆U(z)/vextendsingle/vextendsingle/vextendsinglep+1\npdz+ε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle/vextendsingle∆/parenleftBig\nBh(z)/parenrightBig\n∂h(Uεχr)(z)/vextendsingle/vextendsingle/vextendsinglep+1\npdz+\nε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle/vextendsingle∇/parenleftBig\nBh(z)/parenrightBig\n∇∂h(Uεχr)(z)/vextendsingle/vextendsingle/vextendsinglep+1\npdz=O/parenleftBig\nεn+2p+1\np/parenrightBig\nand\nε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle/vextendsingleAij∂2\nij/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n(z)/vextendsingle/vextendsingle/vextendsinglep+1\npdz=O/parenleftBig\nεn+4p+1\np/parenrightBig\nand\nε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle/vextendsingleBk∂k(∆(Uεχr))(z)/vextendsingle/vextendsingle/vextendsinglep+1\npdz=O/parenleftBig\nεn+2p+1\np/parenrightBig\nand\nε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle/vextendsingleBk∂k/parenleftbig\nAst∂2\nst(Uεχr)/parenrightbig\n(z)/vextendsingle/vextendsingle/vextendsinglep+1\npdz=O/parenleftBig\nεn+4p+1\np/parenrightBig\nand\nε4p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle/vextendsingleBk∂k/parenleftBig\nBh∂h(Uεχr)/parenrightBig\n(z)/vextendsingle/vextendsingle/vextendsinglep+1\npdz=O/parenleftBig\nεn+4p+1\np/parenrightBig23\nand\nbε2p+1\np/integraldisplay\nB(0,r)/vextendsingle/vextendsingle/vextendsingleBk∂k(Uεχr)(z)/vextendsingle/vextendsingle/vextendsinglep+1\npdz=O/parenleftBig\nεn+2p+1\np/parenrightBig\n.\nThe result follows. /square\nREFERENCES\n[AM99] Antonio Ambrosetti and Andrea Malchiodi. 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Partial Differential Equations , 46(1-2):123–164, 2013.\nDEPARTAMENTO DE MATEM ´ATICAS , UNIVERSIDAD T´ECNICA FEDERICO SANTA MAR´IA, VALPARA ´ISO, CHILE\nEmail address :salomon.alarcon@usm.cl\nDEPARTAMENTO DE MATEM ´ATICAS , UNIVERSIDAD T´ECNICA FEDERICO SANTA MAR´IA, VALPARA ´ISO, CHILE\nEmail address :simon.masnu@sansano.usm.cl\nDEPARTAMENTO DE MATEM ´ATICAS , UNIVERSIDAD T´ECNICA FEDERICO SANTA MAR´IA, VALPARA ´ISO, CHILE\nEmail address :pedro.montero@usm.cl\nDEPARTAMENTO DE MATEM ´ATICAS , UNIVERSIDAD T´ECNICA FEDERICO SANTA MAR´IA, VALPARA ´ISO, CHILE\nEmail address :carolina.reyr@usm.cl"    },    {        "title": "2402.14684v1.Adaptive_time_series_forecasting_with_markovian_variance_switching.pdf",        "content": "ADAPTIVE TIMESERIES FORECASTING WITH MARKOVIAN\nVARIANCE SWITCHING\nBaptiste Abélès\nUniversidad Pompeu Fabra\nbaptistabeles@gmail.comJoseph de Vilmarest\nViking Conseil\njoseph.de-vilmarest@vikingconseil.fr\nOlivier Wintemberger\nSorbonne Université\nolivier.wintenberger@sorbonne-universite.fr\nABSTRACT\nAdaptive time series forecasting is essential for prediction under regime changes. Several classical\nmethods assume linear Gaussian state space model (LGSSM) with variances constant in time.\nHowever, there are many real-world processes that cannot be captured by such models. We consider a\nstate-space model with Markov switching variances. Such dynamical systems are usually intractable\nbecause of their computational complexity increasing exponentially with time; Variational Bayes\n(VB) techniques have been applied to this problem. In this paper, we propose a new way of estimating\nvariances based on online learning theory; we adapt expert aggregation methods to learn the variances\nover time. We apply the proposed method to synthetic data and to the problem of electricity load\nforecasting. We show that this method is robust to misspecification and outperforms traditional expert\naggregation.\nKeywords Time Series ·Online Learning ·Kalman Filter ·Markov Chains\n1 Introduction\nTime series forecasting is a fundamental issue. A few examples are weather forecasts for farmers, sales and inventory\nforecasting in retail, price and cost forecasting for industrials. In this paper, a special motivation is electricity load\nforecasting, a crucial task for grid operators as the production must balance the consumption in real time [4, 15].\nState-space models have been widely employed to model the temporal behavior of data. In particular, linear Gaussian\nstate-space models (LGSSMs) yield tractable posterior distributions obtained by the Kalman filter (KF) [19]. Many\nsituations are filled with non linearity both in dynamics and observation equation. A vast literature has been devoted\nto deal with non-linear dynamical systems, e.g. the extended Kalman filter (EKF) [10] which essentially consists to\nlinearize the model using Taylor first-order expansion, and the unscentend Kalman filter (UKF) [30] which is radically\ndifferent and consists to apply the unscented transformation to both prediction and filtering step.\nAlthough the inference in LGSSMs is known [19], the Kalman filter crucially depends on the variances of the\nobservation and state noises. The choice of these variances may be seen as the parametrization of a second-order\ngradient descent algorithm, once we remark the equivalence of the Kalman recursions and gradient updates [8, 26]. The\nmost widely-employed setting of these variances is time-invariant [6,10,11], and the natural choice of these variances is\nmaximum-likelihood. However, constant variances mean a smooth evolution of the environment, while in some cases\nthere are changes of regime. To tackle that problem, various methods have been introduced, often under the paradigm\nadaptive Kalman filter [16, 17, 22].\nAlternatively the switching Kalman filter (SKF) [24] considers several linear dynamic models and try to combine them\nwith a switching mechanism (such as a Markov chain), selecting over time one among several regimes. This latter\napproach is of a particular interest in many real-world problems that cannot be approximated with a single regime, andarXiv:2402.14684v1  [stat.ML]  22 Feb 2024where multiple behaviors are needed instead. For instance, time series forecasting during a period covering stable as\nwell as unstable moments motivates switches between different regimes. Our application to electricity load forecasting\nis one of these, as we include very unstable periods such as lockdowns due to the coronavirus in 2020, during which it\nwas best to switch to a more reactive method [25].\nThe complexity of these hybrid dynamical systems, i.e. systems that combine discrete and continuous valued latent\nvariables, increases exponentially with time. State-of-the art approaches propose to learn a switching linear dynamical\nsystem using Variational Bayesian (VB) techniques [2]. However, VB approaches have some limits under model\nmisspecification [31].\nIn this paper, we propose to consider the SKF as an aggregation of experts, where each expert corresponds to a different\nregime. We apply online learning techniques to track the best expert as in [5, 14]. After introducing various methods in\nSection 2, we define our algorithm in Section 3, and we provide a regret bound in expectation. Then, in Section 4, we\nconduct various experiments on synthetic data, as well as a real data set, forecasting the French national electricity load.\nWe study well-specified and misspecified experiments and we show that this new framework is robust to misspecification\nand outperforms Kalman aggregation as in [1].\n2 Background\nNotations Given probability measures pandqwith a common dominating measure µ, we denote the Kullback-\nLeibler divergence as D(p||q) :=R\nplog(p/q)dµ. For any positive integer twe denote [t] :={1, ..., t}andFt:=\nσ(x1, y1, ...,xt, yt,xt+1)is the sigma algebra generated by the pair (xi, yi)t\ni=1andxt+1. For some µ, σ∈R, the\nnotation N(.|µ, σ)refers to as the normal distribution with mean µand variance σ2(we keep the same notation if\nµis a vector and Σis the covariance matrix associated). We denote ∆(α, β, γ )as the diagonal matrix in R3×3with\ncoefficients α, β, γ .\nLetyt∈Rbe the variable of interest, xt∈Rda vector of explainable variables and θt∈Rdthe vector of latent\nvariables at time t. The classical LGSSM writes :\nyt=θT\ntxt+ϵt,where ϵt∼ N(0, σ2\nt), (1)\nθt+1=θt+νt,where νt∼ N(0, Qt), (2)\nOur goal is to estimate jointly the state and variances knowing the past that we denote ˆθt|t−1=E[θt|Ft−1]and\nPt|t−1=E[(θt−ˆθt|t−1)(θt−ˆθt|t−1)T|Ft−1]. The well-known KF gives the exact updates if σ2\ntandQtare known :\nTheorem 1. (Kalman Filter)\nPt|t=Pt|t−1−Pt|t−1xtxT\ntPt|t−1\nxT\ntPt|t−1xt+σ2\nt,\nˆθt|t=ˆθt|t−1−Pt|t\nσ2\nt\u0010\nxt(ˆθT\nt|t−1xt−yt)\u0011\n,\nPt+1|t=Pt|t+Qt,\nˆθt+1|t=ˆθt|t.\nUsual choice in the literature is to assume that the variances are time-invariant [6, 10] and apply the Expectation-\nMaximisation (EM) algorithm. A second paradigm which has frequently been called Adaptive Kalman Filter (AKF) [22]\nconsists in estimating variances over time and applying the Kalman recursive updates using such adaptive variances.\nA more recent approach consists in finding the complete posterior of both state and variances [9]. The authors assume\nthatQt, σ2\ntare random variables and try to estimate the joint distribution of the state and the variances in a dynamical\nway. They assume a model on the dynamics of the variances; then, starting from a prior on (θ1, σ2\n1, Q(1)), they propose\nto recursively estimate the posterior distribution of (θt, σ2\nt, Qt). However there is no natural parametric class of\ndistributions on the joint distribution such that the posterior remains in the class considered. To handle this issue, they\napply the Variational Bayesian (VB) approach [27]; it consists in approximating the posterior with the best factorized\nform in the sense of the minimal Kullback-Leibler divergence. We believe that the limits of such methods lie on too\nmuch degree of freedom and we aim to constrain the model.\n2To reduce the degree of freedom of the problem, one can assume the existence of a hidden Markov chain (Zt)∈[K]\nparameterizing the dynamics of the classical LGSSM. At each time t,σ2\nt=σ2\nZt, Qt=QZt. This model writes :\nyt=θT\ntxt+ϵt,where ϵt∼ N(0, σ2\nZt), (3)\nθt+1=θt+νt,where νt∼ N(0, QZt), (4)\nP(Zt+1=k) =KX\ni=1MikP(Zt=i), (5)\nwhere Mik=P(Zt+1=k|Zt=i)fori, k∈[K]is an entry of the transition matrix M∈RK×K.\nHowever, the complexity of the posterior distribution of such dynamical systems grows exponentially in time, making it\nintractable. Indeed the posterior writes :\np(θ1:t, Z1:t|Ft)∝π(Z1)N(θ1|ˆθ1, P1)tY\ns=1N(ys|θT\nsxs, σ2\nZs)\nt−1Y\ns=1N(θs+1|θs, QZs)MZs+1,Zs,\nwhere π(Z1)denotes the prior law on Z1. Several approaches, referred to as Generalized Pseudo Bayesian algorithm\n[24], approximate this exponential mixture of Gaussians with a smaller mixture of Gaussians. Recent works use VB\ntechniques to approximate the posterior [2]. The main idea is to look for a tractable approximation of the posterior. If\nwe denote pFtthe posterior we look for (q∗\nZ, q∗\nθ)∈ Dtsuch that :\n(q∗\nZ, q∗\nθ)∈arg min\n(qZ,qθ)∈DtD(qZ×qθ||pFt),\nwhereDtis the parametric class considered. We provide the derivation and results of this method that performs poorly\nin our experiments in supplementary materials. An important limitation of this method is the lack of robustness to\nmisspecification [31].\nTo overcome this problem we propose a new approach to adaptive variance estimation relying on expert aggregation.\nAggregation of experts is a well-established framework in machine learning [5, 29]. In particular, it yields excellent\nresults in the context of electricity load forecasting [1]. Our work follows this classical setting, we consider a learner, K\nexperts and a loss function ℓ:E×Y → RwhereEis the space of experts predictions and Yis the space of observation.\nAt each time t, each expert k∈[K]predicts ek,t∈E. The learner picks a distribution ptover the set of experts’\npredictions, predicts et=PK\nk=1pk,tek,t∈Eand observes his loss ℓt:=ℓ(et, yt). The learner has also access to the\nloss of each expert at time t, that we denote ℓk,t:=ℓ(ek,t, yt). The goal of the learner is to minimize his cumulative\nloss on sequence of length T. A first way to measure his performance is to compare his cumulative loss with other\nexpert during the sequence.\nDefinition 1. We define the regret with respect to the expert kfor a sequence of length Tas :\nRT(k) =TX\nt=1ℓt−ℓk,t\n2.1 Exponential-Weight-Average (EWA) algorithm for Kalman Filter\nTo better understand our framework, we present the famous EWA to show how a learner can optimize his strategy to\ncontrol his regret.\nWe make a small remark here that is we will essentially focus on the variance of the state Equation 4. Indeed Kalman\nFilter gives exactly the same update for the pair (σ2, Q)and(1, Q∗)withQ∗=Q\nσ2(refer to [3]).\nLet’s consider the notations of the state space model and assume each expert is a KF derived with Q∗\nt=Q∗(k)which\nstarting from some ˆθ0|0yields the prediction ˆθ(k)\nt|t−1of the state at time t, fort∈[T]. In this case, let’s identify the\nspace of experts as the observation space , i.eE=Yand each expert predicts ek,t= ˆyk,twhere ˆyk,t=ˆθ(k)T\nt|t−1xt. We\nconsider the quadratic loss ℓk,t= (yt−ˆyk,t)2.\n3Algorithm 1 EWA Algorithm\nSetp1=K−11K\nfort∈[T]do\nselectpt, forecast ˆyt=PK\nk=1pt(k)ˆyk,t\nderive loss ℓtandℓk,tfork∈[K]\nfork∈[K],update: pt+1(k) =pt(k)e−ηℓk,t\nPK\nj=1pt(j)e−ηℓj,t\nend for\nA first important result is given by the following proposition.\nProposition 1. Assume that the loss function ℓis convex and takes value in [0,1]:\nR∗\nT≤log(K)\nη+ηT\n8.\nRefer to [5], Theorem 2.2 for a proof.\nAlgorithm 1 belongs to the so-called class of Static Expert algorithm which tend to converge to the best expert. However\nin our case since we assume switches; we look for aggregation algorithms able to switch from one expert to the other.\n2.2 Fixed-Share (FS) algorithm\nFor this reason we should consider a more refined measure of performance.\nDefinition 2. LetkT= (k1, ..., k T)be a sequence of experts we can define the regret with respect to this sequence as:\nRT(kT) =TX\nt=1ℓt−ℓkt,t\nand\nR∗\nT= sup\nkTRT(kT)\nA first and naïve approach would be to consider each sequence as a new expert and try to recover the best expert on\nthe set of every possible sequence of expert using EWA algorithm starting from a prior distribution p1(kT)over this\nset. However in practice we will face a problem of intractability since we need to store and update O(KT)weights to\ncompare to RT(kT). The problem of controlling the regret with respect to sequences of experts, known as tracking\nthe best expert, was introduced by Herbster and Warmuth [14], who proposed the simple FS algorithm with good\nregret guarantees. A key fact highlighted in [28] (section 3.3) is that FS, can be interpreted as the EWA algorithm on\nsequences of experts under a markov prior.\nBelow we present a generalization of FS named as Markov-Hedge (MH) algorithm which starting from markov chain\npriorp1(k1, ..., k T) =π(k1)Mk2|k1...M kT|kT−1collapses to efficient algorithm with bounded regret [23]. For t∈[T],\nMkt|kt−1correspond to the entry of the transition matrix M∈RK×Kof the markov chain.\n4Algorithm 2 Markov-Hedge Algorithm\nRequire: M= (Mkt|kt−1),p1=K−11K\nfort∈[T]do\nselectpt, forecast ˆyt=PK\nk=1pk,tˆyk,t\nderive loss ℓtandℓk,tfork∈[K]\nfork∈[K],update: pt+1(k) =pt(k)e−ηℓk,t\nPK\nj=1pt(j)e−ηℓj,t\nupdate: pt+1=Mpt+1i.efork∈[K]:\npt+1(k) =KX\nj=1Mk|jpt+1(j)\nend for\nWe have the following proposition :\nProposition 2. Algorithm Markov-Hedge, with initial distribution p1(k1, ..., k T) =π(k1)Mk2|k1...M kT|kT−1and\ntransition matrix M∈RK×K, guarantees the following regret bound. For T≥1and any sequence kT= (k1, ..., k T)\n:\nRT(kT)≤1\nηlog1\nπ(k1)+1\nηTX\nt=2log1\nMkt|kt−1+ηT\n8.\nRefer to [5] Theorem 5.1 for a proof\n3 Methodology\nFollowing the remark made in Subsection 2.1 we will first consider σ2\nt= 1fort∈[T].\n3.1 Well-specified setting\nAs we saw in Section 2 what is usually done [1] is to perform an aggregation over the space of observations. This\nmeans that we consider our Kexperts as KKalman Filters derived with constant variances Qt=Q(k), σ2\nt= 1. The\nexpert kpredicts ˆyk,tfor all t∈[T]and updates his weights using Algorithm 2. The main contribution of this paper is\nto apply the aggregation framework over the space of variances, as displayed by diagrams in Figure 1. The algorithm\nthat we design is a Kalman filter that competes the best covariance matrix at each time tusing expert aggregation on\nQ={Q(1), ..., Q(K)}. We consider the space of experts E=Qso that at time t, each expert kcomputes P(k)\nt|t−1,ˆθ(k)\nt|t−1\nusing equations of Theorem 1 and predicts:\nˆyk,t=ˆθ(k)T\nt|t−1xt.\nThe learner observes the loss ℓk,t= (yt−ˆyk,t)2of each expert and updates his weights as in Algorithm 2. Then he\nselects Q(k)with probability pt(k)to compute Pt+1|t,θt+1|tgiven his previous predictions Pt|t−1,θt|t−1and predicts:\nˆyt+1=ˆθT\nt+1|txt+1.\nThis yields the randomized version of our algorithm, Kalman Filter Markov-Hedge (KFMH), detailed in Algorithm 3.\nThe main parameters of this algorithm are ηandM. To reduce the complexity of the problem we will assume in our\nexperience that Monly depends of a parameter αsuch that :\nMα=\n1−αα\nK−1. . .α\nK−1\nα\nK−1.........\n.........α\nK−1α\nK−1. . .α\nK−11−α\n\nWe can then calibrate ηandαby grid-search based on the best cumulative loss.\n5Figure 1: Aggregation of Kalman (left) versus Kalman with aggregation of variances (right).\nAlgorithm 3 Randomized KFMH\nRequire: p1=K−11K,M= (Mi,j)K\ni,j=1\nfort∈[T]do\nReveal ytandxt−1\nfork∈[K]do\ncompute P(k)\nt|t−1,ˆθ(k)\nt|t−1, and ˆyk,t=ˆθ(k)T\nt|t−1xt\nderive loss ℓk,t\nend for\nupdate: pt+1(k) =pt(k)e−ηℓk,t\nPK\nj=1pt(j)e−ηℓj,t\nupdate: pt+1=Mpt+1\nSelect Qt=Q(k)with proba pt+1(k)\nCompute Pt+1|t,ˆθt+1|t\nForecast ˆyt+1=ˆθT\nt+1|txt+1\nend for\nProposition 3. Algorithm KFMH randomized, with transition matrix Mαand fined tuning for ηandα, satisfies:\nE[R∗\nT]≤ O(p\nSTlogT), T ≥1,\nwhere Sis the number of regime switching of the SKF competitor.\nProof. Without loss of generality we consider that the random selection at step tfollowing the distribution ptis\nindependent of Ftbut included in Ft+1. The tower property yields the identity\nEhTX\nt=1ℓti\n=EhTX\nt=1E[ℓt| Ft]i\n=EhTX\nt=1KX\nk=1pt(k)ℓt,ki\n.\nRewriting the regret bound of Proposition 2 for the linearized loss and for the specific transition matrix Mα, we obtain\nE[RT]≤S+ 1\nηlogK\nα+T−S\nηlog\u00101\n1−α\u0011\n+ηT\n8.\n6The desired result follows after an optimisation of the regret bound in αandη.\n3.2 Misspecified setting\nIn well-specified cases, the SKF competitor is optimal and has small cumulative quadratic losses, see e.g. [1]. However,\nin real-world applications one needs to be more robust to specification. We introduce a new setting where the prediction\nis provided by a Kalman filter acting on a state equation driven by the covariance matrix\nˆQt=KX\nk=1pt(k)Q(k).\nThe interpretation is to transfer the confidence on each experts computed by ptdirectly on the state Equation 4.\nThe gain is to design an estimation of the state variance and a forecaster that is a KF and not an aggregation. One\nalso has to design each expert kin a different way than pure independent KF. Indeed, while in Algorithm 3 each\nexpert was computing his prediction independently from the learner; now the expert starts from a prior knowledge\nPt−2|t−2,ˆθt−1|t−2of learner’s previous predictions and derives new KF updates as follows:\nP(k)\nt−1|t−2=Pt−2|t−2+Q(k),\nP(k)\nt−1|t−1=P(k)\nt−1|t−2−P(k)\nt−1|t−2xt−1xT\nt−1P(k)\nt−1|t−2\nxT\nt−1P(k)\nt−1|t−2xt−1+ 1,\nˆθ(k)\nt−1|t−1=ˆθt−1|t−2−P(k)\nt−1|t−1xt(ˆθT\nt−1|t−2xt−yt),\nˆθ(k)\nt|t−1=ˆθ(k)\nt−1|t−1,\nto predict:\nˆyk,t=ˆθT(k)\nt|t−1xt.\nThen the learner observes the loss ℓk,tof each expert, updates the weights and computes ˆθt+1|t, Pt+1|tapplying KF\nupdates of Theorem 1 with ˆQt+1.\n3.3 Choice of loss and optimization over σ\nIn Section 2 we considered the square loss. However, standard methods optimizing time-invariant variances in state-\nspace models optimize the likelihood as the natural objective function [6,10]. This leads us to consider the log-likelihood\nas loss function in our algorithm. Using the chain rule, the log-likelihood writes:\nlogp(y1:T,x1:T|Q1:T−1, σ2\n1:T)\n=TX\nt=1logp(yt|xt, y1:t−1,x1:t−1, Q1:(t−1), σ2\n1:t) +c\n=TX\nt=1ℓt(ˆθt|t−1, Pt|t−1, σ2\nt, yt) +c .\nwhere cis a constant term and the loss function writes:\nℓt(ˆθt|t−1, Pt|t−1, σ2\nt, yt)\n= logN(yt|ˆθT\nt|t−1xt, σ2\nt+xT\ntPt|t−1xt).\nThe dependence of this loss to Qt−1is hidden in the estimated state distribution represented by ˆθt|t−1, Pt|t−1. The\npeculiarity of the log-likelihood is that it strongly depends of σ2\nt. This means that the assumption σ2\nt= 1 is too\nrestrictive. In our experiments we observed that it did not perform well to apply the expert framework to both variances.\nWe believe that it’s due to the fact that Qt−1andσ2\ntare strongly related. Furthermore, estimating observation noise\nvariance is an easier task compared to the estimation of the covariance matrix of a latent process. Therefore we directly\napply a gradient descent algorithm, namely ADAM [20]. Equation (∗)in Algorithm 4 and function f(.)refers to this\n7ADAM gradient descent which involves 4 hyper-parameters (namely α1, β1, β2, ϵ)and the gradient of the loss over σ\nwhich writes :\n∂ℓt\n∂σ=−σ(y−ˆθT\nt|t−1xt)2\nσ2+xT\ntPt|t−1xt+σ\nσ2+xT\ntPt|t−1xt.\nNote that by decoupling the estimation of both variances, we don’t need a priori on the possible values of σ2\ntand their\nrelation to the possible values of Qt.\n3.4 Sliding Window\nWe propose another improvement of Algorithm 4 motivated by [17]. The main intuition behind this is to better\ndiscriminate between the different experts. We note that Pt−1|t−1depends on Q1, Q2, . . . Q t−2. Our initial algorithm\nproposes that expert kdepends on Q1, Q2, . . . Q t−3, Q(k), that is, we replace only Qt−2. Instead, we propose to replace\nthe last τcovariance matrices by Q(k)for each expert k. This replacement of a window τis also motivated by the fact\nwe assume α≪1in our experiments (typically α= 0.01), that is, the covariance matrix is not switching too frequently\nfrom one mode to another. Considering this sliding window helps capturing the dynamics by discriminating better\nbetween the experts.\nFormally, instead of starting from ˆθt−2|t−2, Pt−2|t−2, we start from ˆθt−τ−1|t−τ−1, Pt−τ−1|t−τ−1. For each expert k,\nwe derive Kalman updates using Qs=Q(k)fors≥t−τ−1, yielding ˆθ(k)\nt|t−1, P(k)\nt|t−1. Consequently, this sliding\nwindow is integrated in Algorithm 4 as the introduction of a new loss function ℓ(τ)\nk,t=N(yt|ˆθ(k)T\nt|t−1xt, σ2\nt+xT\ntP(k)\nt|t−1xt)\nwhere the dependency in τrefers to the way we create ˆθ(k)\nt|t−1, P(k)\nt|t−1.\nAlgorithm 4 KFMH\nRequire: p1=K−11K,M= (Mi,j)K\ni,j=1, σ0\nfort∈[T]do\nReveal ytandxt−1.\nfork∈[K]do\ncompute P(k)\nt|t−1,ˆθ(k)\nt|t−1, and ˆyk,t=ˆθ(k)T\nt|t−1xt\nderive loss ℓ(τ)\nk,t\nend for\nfork∈[K]update: pt+1(k) =pt(k)e−ηℓk,t\nPK\nj=1pt(j)e−ηℓj,t\nupdate: pt+1=Mpt+1\nSelect Qt=PK\nk=1pt(k)Q(k)\nCompute Pt+1|t,ˆθt+1|t\nForecast ˆyt+1=ˆθT\nt+1|txt+1\nObserve loss ℓt(ˆθt|t−1, Pt|t−1, σ2\nt, yt)\nUpdate σt=σt−1+f\u0010∂ℓt(ˆθt|t−1,Pt|t−1,yt)\n∂σ\u0011\n(*)\nend for\n4 Experiments\nWe compare the different methods introduced on various data sets. We begin with synthetic data; first we generate the\ndata following the state-space model introduced in Section 2, we call that well-specified and it is the most favourable\nsetting; then, we generate data not following the state-space assumption anymore, we call that experiment misspecified\nand it is an intermediate step before the application to real data. We conclude by a real data set motivating our setting:\nelectricity load forecasting. We show that our method outperforms the standard Kalman aggregation as long as the\nexperts are well-known even in the misspecified context.\n4.1 Experiments on Synthetic Data\nIn the following we consider K= 2.\n84.1.1 Well-Specified (WS) Data with σ2\nt=σ2\nData Generation\nWe consider the model given by Equations (3),(4) and (5) and generate data as follow :\nP(Zt+1=Zt) = 1−α,\nθt+1−θt∼ N(0R3, QZt),\nyt−θT\ntxt∼ N(0,1),\nwithQZt∈ {Q(1), Q(2)}where:\nQ(1)= ∆(10−4,10−1,10−2),\nQ(2)= ∆(10−1,10−4,10−2).\nWe distinguish three ways of generating xt: i.i.d Gaussian design: xt∼ N(0Rd,1Rd),i.i.d uniform design: xt∼\nU(0Rd,1Rd)and non-i.i.d design: x1is generated as before. Then for j∈[ [1,3] ]andt≥2, we consider at,j=\nxt−1,j+ϵt,jwhere ϵt,j∼ N(0R3,10−3)and we use xt,j=at,jif0≤at,j≤1, otherwise xt,j=⌈at,j⌉ −at,j. We\ngenerate sequence of length T= 1000 with a transition parameters α= 10−2.\nAlgorithm Parameterization\nFor both Algorithm 2 and 4 we set η= 1, and take latence τ= 5for KFMH and initial σ0= 0.8. For ADAM optimizer\nf(.)we take the step-size α1= 0.01and the others standard hyper-parameters as in [20] (β1= 0.9, β2= 0.999, ϵ=\n10−8).\n4.1.2 Misspecified (MS) Data\nIn this subsection we forget the Markov representation and consider that we play against a sequence of expert which is\ngenerated by some specific function .\nData generation : Sinusoidal\nAs discussed before, we assume here that σ2\ntandQtare independent. We generate Qtandσ2\ntin the following way :\nQt= cos2\u00123πt\nT\u0013\nQ(1)+ sin2\u00123πt\nT\u0013\nQ(2), (6)\nσt=σ(1)cos2\u00125πt\nT\u0013\n+σ(2)sin2\u00125πt\nT\u0013\n, (7)\nwhere Q(1), Q(2)are as in Subsection 4.1.1 and σ(1)= 1, σ(2)= 2.\n4.2 Real Data\nLastly we present our results on Real Data. We apply our methodology to the case study of [25]. We forecast the\nhalf-hourly French national electricity consumption published by Réseau de Transport d’Électricité1, as well as\nmeteorological data from MétéoFrance2. We consider the adaptation of the following generalized additive model, built\nindependently for the 48 half-hours of the day:\nyt=f(Toyt) +7X\ni=1αi1DayType t=i (8)\n+f(TempS 95t) +f(TempS 99t) +f(t, Temp t)\n+s(CPWind t) +f(t) +βLag 1t+γLag 7t+εt\nwhere at each time t,ytis the consumption, DayType tis the day of the week between 1and7,Temp tis the\ntemperature, TempS 95tandTempS 99tare exponential smoothings of the temperature of exponential parameters\n0.95and0.99respectively, CPWind tis the cooling power of wind (c.f. [21], Section 2.2), Lag1tandLag7tare the\ndelayed consumption (one day ago and seven days ago, respectively).\n1https://opendata.reseaux-energies.fr/\n2https://donneespubliques.meteofrance.fr/\n9We follow the framework of [25]: we consider the adaptation of a linear combination of the frozen effects. Formally,\nEquation (8)yields a GAM of 9 effects, and we define xt∈R9the vector composed of these linear and non-linear\neffects. Our state-space paradigm is integrated naturally as\nyt=θT\ntxt+ϵt, ϵ t∼ N(0, σ2\nt)\nθt+1=θt+νt, ν t∼ N(0, Qt)\nOur data ranges from January 2012 to December 2022. It has been extensively studied that the coronavirus crisis has\nimplied a brutal drop in the electricity consumption in many countries, and France in particular [12, 18]. State-space\nmodels have provided a very efficient framework to tackle the poor performances of traditional forecasting methods\nduring this period [7, 25]. We propose to apply a switching strategy, in order to win in stable as well as unstable periods.\nIntuitively, defining a slow covariance matrix Q(slow )and a fastcovariance matrix Q(fast ), a switching Kalman-based\nalgorithm is able to use Q(slow )inslow mode and Q(fast )infastmode, yielding improvements in both worlds.\nTo define Q(slow )andQ(fast ), we rely on an optimization of constant variances introduced in [25], Section 2.A.2.\nHowever, we optimize the variances on different time periods. Optimizing on pre-covid data (2012-2019) yields the\nslow mode, and optimizing on covid times (the year 2020) yields the fastmode. We apply the same optimization method\nto define σ2(slow )andσ2(fast )associated. We compute our test performances on the period ranging from January 2021\nto December 2022.\n4.3 Results\nFigure 2: Evolution of the weights ptfor WS (left) and MS Data (right).\nThe results of our experiments on synthetic data are gathered in table 1 (see Supplementary materials). We generate\nniter= 1000 sequences of length Tand compare our algorithm KFMH to differents methods using the Mean Square\nError (MSE). First we consider the KF-adaptive which is the optimal Kalman filter knowing Qt, σ2\nt; this is the oracle or\nbest possible forecast. Then we observe the performance of experts 1and2which are Kalman filters with constant\ncovariance matrix Qt=Q(1)orQt=Q(2); we denote them by KF- Q(1)and KF- Q(2). We consider another Kalman\nfilter with constant covariance matrix, KF- Q(mean ), which applies Qt=Q(1)+Q(2)\n2. This is a better containder for\nour algorithm since in both WS and MS scenarios we generate data such that we approximately spend the same time\nin both regimes. Finally we present the results of KF- aggthe method that executes Algorithm 2. Except for our\nalgorithm, every method has access to the true σ2\nt. We see that our algorithm outperforms the basic aggregation over\nthe space of observation in every case. We observe that in the MS setting, KF- Q(mean )with knowledge of σ2\ntbeats our\nalgorithm when generating i.i.d uniform and non-i.i.d xt, although we are very close. Overall we observe that Q(mean )\nyields good performances on synthetic data; this was to expect because the covariance matrix of the latent process\nisQ(1)half of the time and Q(2)the other half. The results on real data are presented in table 2 (see Supplementary\nmaterials). For our evaluation, we consider the first ten instants of the day over the 2021−2022 period. Since we did\nnot generate the data, KF-adaptive exists no longer and we only compare KFMH to the Kalman filters of constant\nvariances (KF- Q(slow ), KF-Q(fast, KF-Q(mean )), as well as the standard expert aggregation. This time we don’t have\naccess to the knowledge of σ2\ntso each constant expert assumes σ2\nt=σ2where σ2is respectively σ2(slow ),σ2(fast )\n10Figure 3: Evolution of σtstarting from 2 different initial points for WS (left) and MS Data (right).\nandσ2(mean )=σ2(slow )+σ2(fast )\n2. We find that we always outperform every constant expert, and beat the standard\naggregation in average.\nFigure ??shows the evolution of the weights ptobtained by Algorithm 4 (in blue) and Algorithm 2 (in green) in\nwell-specified as well as misspecified settings. We also display the evolution of σtestimated with Algorithm 4 starting\nfrom two different initial points; respectively σ0= 0.5(blue) and σ0= 2(green) with respect to the true σt(black)\nboth in WS case ( σ2\nt= 1) and MS context where σtis generated with Equation 7. The figures suggest the consistency\nof the method and its robustness towards misspecification.\nBelow we provide the results of the experiments described in Section 4. We see that our algorithm is always better than\na standard aggregation when performing on synthetic datasets, and always very close to the optimal.\nTable 1: Comparison of MSE for each method on synthetic datas in both WS and MS case.\nMethod KF-adpative KF- Q(1)KF-Q(2)KF-Q(mean )KF-agg KFMH\ni.i.d gauss WS 1.802 5 .091 5 .523 1 .990 2 .519 1.916\ni.i.d unif WS 1.796 4 .021 4 .393 1 .931 2 .301 1.891\nnon-i.i.d WS 1.251 1 .471 1 .415 1 .267 1 .301 1.266\ni.i.d gauss MS 1.908 5 .192 5 .526 1 .988 2 .799 1.986\ni.i.d unif MS 1.370 2 .367 2 .343 1.393 1.653 1 .402\nnon-i.i.d WS 2.758 3 .040 2 .964 2.768 2.823 2 .777\nTable 2: Comparison of RMSE over 2021-2022 period for each method on real data for 10 instants of the day.\nMethod KF-Q(slow )KF-Q(fast )KF-Q(mean )KF-agg KFMH\nInstant 1 1005.572 805 .620 796 .485 848 .549 788.284\nInstant 2 888.951 808 .081 797 .431 790 .475 786.788\nInstant 3 896.954 838 .322 827 .544 810.124 812.846\nInstant 4 820.090 876 .641 827 .933 812.816 813.703\nInstant 5 842.068 848 .340 844 .974 849 .140 838.871\nInstant 6 842.665 846 .738 845 .461 842 .816 840.530\nInstant 7 854.612 908 .600 858 .957 895 .452 849.253\nInstant 8 853.396 926 .782 860 .292 843.292 847.000\nInstant 9 863.809 935 .049 871 .084 855.107 859.937\nInstant 10 958.609 922 .263 912 .158 937 .550 901.185\n115 Conclusion\nIn this paper, we have studied LGSSMs under unknown variances. We relied on online learning to estimate the variances.\nWe considered the paradigm of SKF and used aggregation of experts to infer the variance of the process noise; to\napproximate the observation noise variance we applied an online gradient descent algorithm. Our experiments show\ngood performances not only in the favourable well-specified setting but also under misspecification on both synthetic\nand real data, illustrating the robustness of our method.\nThis paper opens different leads for future research. First, we specified on KF algorithms but we could apply our\nframework to other forecasting algorithms, involving the expert aggregation confidence weights at the right level, not\nnecessarily on the predictions. Second, the choice of the diversity among the experts is crucial in aggregation [13], and\nit is challenging to design different KF algorithms in an optimal way.\nReferences\n[1]Eric Adjakossa, Yannig Goude, and Olivier Wintenberger. 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In Proceedings\nof the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No.\n00EX373) , pages 153–158. Ieee, 2000.\n[31] Yixin Wang and David Blei. Variational bayes under model misspecification. Advances in Neural Information\nProcessing Systems , 32, 2019.\n13A VB Derivation\nIn this section we provide the details of the derivation of the VB approach, mentioned in Section 2, that we found\nineffective.\nAs mentioned VB approach looks for the pair (q∗\nθ, q∗\nZ) = (q∗\nθ(θ1:t), q∗\nZ(Z1:t))that is the closest to the complete ( i.e\ntaken over the whole sequence) joint posterior pFt=p(θ1:t, Z1:t|Ft):\n(q∗\nθ, q∗\nZ)∈argmin(qθ,qZ)∈DKL(qθ×qZ||pFt)\nOne can show that :\nlnq∗\nθ(θ1:t) =EqZ(Z1:t)[lnpFt] +c1 (E-θstep) (9)\nlnq∗\nZ(Z1:t) =Eqθ(θ1:t)[lnpFt] +c2 (E-Zstep) (10)\nThe aim is then to iteratively approximate the marginal q∗\nθandq∗\nZ, assuming the other is known and fixed. We provide\nfurther the closed form of what we call respectively E- θstep and E- Zstep.\nA.1 E- θstep\nProposition 4. (E-θstep)\nFor some t >1, for any qZ(Z1:t)if we define for s∈[ [1, t] ]:\nσ−1\ns=vuutKX\nk=1qZ(Zs=k)1\nσ2\nk\nQ−1\ns=KX\nk=1qZ(Zs=k)Q−1\nk\nthen the distribution q∗\nθwhich minimizes KL(qθ×qZ||pFt)forqZfixed writes:\nlnq∗\nθ(θ1:t) =tX\ns=1 \nlnN(ys|θT\nsxs,σ2\ns) + lnN(θs|θs−1,Qs−1)!\n+c (11)\nwhere we denote θ0=ˆθ1andQ0=P1. .\nProof. Fors∈[1 :t]we have:\nEqZ\u0002\nlnN(ys|θT\nsxs, σ2\nZs)\u0003\n=−1\n2X\n1≤k1,...,k t≤KqZ(Z1:t=k1, . . . , k t) \nθT\nsxsxT\nsθs\nσ2\nks−2ysθT\nsxs\nσ2\nks!\n+c\n=−1\n2KX\nks=1qZ(Zs=ks) \nθT\nsxsxT\nsθs\nσ2\nks−2ysθT\nsxs\nσ2\nks!\n+c\nEqZ[lnN(θs|θs−1, QZs)] =−1\n2KX\nks=1qZ(Zs−1=ks−1)\u0010\nθT\nsQ−1\nks−1θs−2θT\nsQ−1\nks−1θs−1+θT\ns−1Q−1\nks−1θs−1\u0011\n+c\nThus by denoting ¯σ−1\ns=qPK\nk=1qZ(Zs=k)1\nσ2\nkand¯Q−1\ns=PK\nk=1qZ(Zs=k)Q−1\nk, we therefore have:\nlnq∗\nθ(θ1:t) =tX\ns=1lnN(ys|θT\nsxs,¯σ2\ns) + lnN(θs|θs−1,¯Qs−1) +c\n14This last proposition shows that we are exactly in the Kalman setting where the covariance matrix at each time step is\nthe inverse of the average of the inverse covariance matrix over the law of Z. Indeed if we define the latent space model\nas :\nState : θt+1=θt+νt,νt∼ N(0,Qt) (12)\nSpace : yt=θT\ntxt+ϵt, ϵ t∼ N(0,σt) (13)\nThe log-likelihoof of the model writes as in (11).\nProposition 5. For some t >1ands∈[ [1 :t] ], if we denote the qF\nθ(θs)the filtering approximation of p(θs|x1:s, y1:s)\nthen we have:\nqF\nθ(θs) =N(θs|ˆθs|s, Ps|s) (14)\nwith the following updates :\nPs|s−1=Ps−1|s−1+Qs−1\nˆθs|s−1=ˆθs−1|s−1\nPs|s=Ps|s−1−Ps|s−1xsxT\nsPs|s−1\nxTsPs|s−1xs+σ2\ns\nˆθs|s=ˆθs|s−1−Ps|s\nσ2\ns\u0010\nxs(ˆθT\ns|s−1xs−ys)\u0011\nThe proof is immediate considering Theorem 1 and the previous remark.\nA.2 E- Zstep\nProposition 6. (E-Zstep)\nFor some t >1, for any qθ(θ1:t), the distribution q∗\nZwhich minimizes KL(qθ×qZ||pFt)forqθfixed writes:\nq∗\nZ(Z1:t)∝π(Z1)N(µ1|ˆθ1, P1)N(y1|µT\n1x1, σ2\nZ1) exp\u0012\n−1\n2Tr\u0012\u0014\nP−1\n1+x1xT\n1\nσ2\nZ1\u0015\nV1\u0013\u0013\ntY\ns=2\"\nN(ys|µT\nsxs, σ2\nZs)N(µs|µs−1, QZs−1) exp\u0012\n−1\n2Trh\nQ−1\nZs−1(Vs+Vs−1−2Ws−1)i\u0013\nMZs|Zs−1#\nwhere, for s∈[ [1, t] ]:µs=Eqθ(θ1:t)[θs], Vs=Eqθ(θ1:t)[θsθT\ns]−µsµT\nsandWs=Eqθ(θ1:t)[θsθT\ns+1]−µsµT\ns+1\nwhich can be rewritten as:\nqZ(Z1:t) =ρ1(Z1)tY\ns=2ρs(Zs)γZs|Zs−1 (15)\nwhere for s >1:\nρ1(Z1)∝π(Z1)N(µ1|ˆθ1, P1)N(y1|µT\n1x1, σ2\nZ1) exp\u0012\n−1\n2Tr\u0012\u0014\nP−1\n1+x1xT\n1\nσ2\nZ1\u0015\nV1\u0013\u0013\nρs(Zs)∝ N(ys|µT\nsxs, σ2\nZs) exp\u0012\n−1\n2Tr\u0012xsxT\ns\nσ2\nZsVs\u0013\u0013\nγZs|Zs−1∝ N(µs|µs−1, QZs−1) exp\u0012\n−1\n2Trh\nQ−1\nZs−1(Vs+Vs−1−2Ws−1)i\u0013\nτZs|Zs−1\nThis last expression (15) is equivalent to a Hidden Markov Model (HMM) . We define the filtration Hs=\nσ(x1:s, y1:s, µ1:s, V1:s, W 1:s−1)andp′such that qZ(Z1:t) =p���(Z1:t|Ht). Therefore, by computing the standard\nforward-backward algorithm for HMMs, one can easily obtain the forward qF\nZ(Zs) =p′(Zs|Hs)≈p(Zs|Fs)and back-\nward qB\nZ(Zs) =p′(Zs|Hs+1:t)≈p(Zs|xs+1:t, ys+1:t)and obtain the posterior approximation qZ(Zs)≈p(Zs|Ft)\n:\nqZ(Zs)∝qF\nZ(Zs)qB\nZ(Zs) (16)\nProposition 7. The forward-backward recursions write for some t >1ands∈[ [1, t] ]:\nqF\nZ(Zs)∝ρs(Zs)⊙(qF\nZ(Zs−1)γZs|Zs−1) (17)\nqB\nZ(Zs)∝γZs+1|Zs\u0000\nqB\nZ(Zs+1)⊙ρs+1(Zs+1)\u0001\n(18)\n15Proof. Starting from the joint distribution we have :\nαs(Zs) =p(Zs, x1:s, y1:s, µ1:s, V1:s, W 1:s−1)\n=KX\nk=1p(Zs, Zs−1=k, y 1:s, x1:s, µ1:s, V1:s, W 1:s−1) (marginalizing over Zs−1)\n=KX\nk=1p(ys|Zs, Zs−1=k, y 1:s−1, x1:s, µ1:s, V1:s, W 1:s−1)p(Zs, Zs−1=k, y 1:s−1, x1:s, µ1:s, V1:s, W 1:s−1)\n=KX\nk=1p(ys|Zs, Zs−1=k, xs, µs, Vs, Ws−1)p(Zs|Zs−1=k)\np(xs|Zs−1=k, x 1:s−1, y1:s−1, µ1:s−1, V1:s−1, W 1:s−2)p(Zs−1=k, x 1:s−1, y1:s−1, µ1:s−1, V1:s−1, W 1:s−2)\n∝KX\nk=1ρs(Zs)γZs|Zs−1=kαs−1(Zs−1=k)\n∝ρs(Zs)⊙\u0000\nαs−1(Zs−1)γZs|Zs−1\u0001\n=⇒qF\nZ(Zs) =p(Zs|x1:s, y1:s)\n∝ρs(Zs)⊙(qF\nZ(Zs−1)γZs|Zs−1)\nqB\nZ(Zs) =p(ys+1:t, xs+1:t|Zs)\n=KX\nk=1p(ys+1:t, xs+1:t, Zs+1=k|Zs) (marginalizing over Zs+1)\n=KX\nk=1p(ys+1|xs+1:t, ys+2:t, Zs+1=k, Zs)p(xs+1:t, ys+2:t, Zs+1=k, Zs)\n=KX\nk=1p(ys+1|xs+1, Zs+1=k, Zs)p(Zs+1=k|Zs, xs+1:t, ys+2:t)p(xs+1:t, ys+2:t, Zs+1=k)\n∝KX\nk=1ρs+1(Zs+1=k)qB\nZ(Zs+1=k)γZs+1=k|Zs (àp(xs+1|.)près)\n∝γZs+1|Zs(qB\nZ(Zs+1)⊙ρs+1(Zs+1))\nA.3 VB Algorithm\nFinally the VB algorithm can writes as follow. The parameter nitercorresponds to the number of iteration we need to\nconverge and usually is of order 5.\nAlgorithm 5 VB\nRequire: q(0)\nZ,(µ(0)\ns)t\ns=1,(V(0)\ns)t\ns=1,(W(0)\ns)t−1\ns=1(xs)t\ns=1,(ys)t\ns=1\nfork∈[ [1, niter] ]do:\nFors∈[ [1, t] ], compute ¯σs,¯Qs−1.\nFors∈[ [1, t] ], derive qF\nθ(θs)with Proposition 5. ▷E-θstep\nFors∈[ [1, t] ]compute µs, Vs, Ws.\nFors∈[ [1, t] ], derive qZ(θs)∝qF\nZ(Zs)qB\nZ(Zs)with Proposition 6. ▷E-Zstep\nend for\n16"    },    {        "title": "2402.14712v1.Gilbert_Varshamov_Bound_for_Codes_in__L_1__Metric_using_Multivariate_Analytic_Combinatorics.pdf",        "content": "1\nGilbert–Varshamov Bound for Codes in L1Metric\nusing Multivariate Analytic Combinatorics\nKeshav Goyal∗, Duc Tu Dao∗, Mladen Kova ˇcevi´c†, and Han Mao Kiah∗\n∗School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore\n†Faculty of Technical Sciences, University of Novi Sad, Serbia\nEmails: {keshav002, daoductu001, hmkiah }@ntu.edu.sg, kmladen@uns.ac.rs\nAbstract\nAnalytic combinatorics in several variables refers to a suite of tools that provide sharp asymptotic estimates\nfor certain combinatorial quantities. In this paper, we apply these tools to determine the Gilbert–Varshamov lower\nbound on the rate of optimal codes in L1metric. Several different code spaces are analyzed, including the simplex\nand the hypercube in Zn, all of which are inspired by concrete data storage and transmission models such as the\nsticky insertion channel, the permutation channel, the adjacent transposition (bit-shift) channel, the multilevel flash\nmemory channel, etc.\nI. I NTRODUCTION\nCodes in L1(or Manhattan)1metric arise as appropriate constructs for error correction in a surprisingly diverse\nset of applications, in models that at first glance do not have much in common. We list below only a few examples\nthat served as motivation for the present work.\n•The sticky insertion (or repetition, or duplication) channel, as well as the 0-insertion channel that it is equivalent\nto, were introduced as models for communication in the presence of certain types of synchronization errors.\nCodes for these channels [20], [2], [29], [23] can equivalently be described in the space [13], [28]\n∆+\nn,r=(\n(u1, u2, . . . , u r)∈Zr:ui⩾1,rX\ni=1ui=n)\n(1)\nunder L1metric, meaning that every code correcting up to tsticky insertions can be obtained from codes\nin the simplex ∆+\nn,rhaving minimum L1distance >2t. In this description, the parameter ncorresponds to\nthe length of the input sequences and the parameter rto the number of runs of identical symbols in those\nsequences. A very similar characterization holds also in more general channels with uniform tandem duplication\nerrors/mutations that are relevant for in vivo DNA-based data storage systems [9], [14], [18].\n•The permutation channel, which randomly reorders the transmitted symbols, has been studied as a model for\nnetworks that do not guarantee in-order delivery of packets, as well as for unordered data storage systems, in\nparticular those based on DNA. Multiset codes [15], [13] that are appropriate for some channels of this kind\ncan equivalently be described in the space\n∆n,r=(\n(u1, u2, . . . , u r)∈Zr:ui⩾0,rX\ni=1ui=n)\n(2)\nThe paper was presented in part at the 2023 IEEE International Symposium on Information Theory (ISIT) [6].\nMK was supported by the Secretariat for Higher Education and Scientific Research of the Autonomous Province of V ojvodina (project\nnumber 142-451-2686/2021).\n1Recall that the L1distance between two vectors u= (u1, . . . , u n),v= (v1, . . . , v n)∈Rnis defined as D(u,v) =Pn\ni=1|ui−vi|.arXiv:2402.14712v1  [cs.IT]  22 Feb 20242\nunder L1metric, meaning that a multiset code correcting (e.g.) up to tsymbol deletions is equivalent to a\ncode in ∆n,rhaving minimum L1distance >2t. Here the parameter nrepresents the number of transmitted\nsymbols, i.e., code length, and the parameter rthe size of the alphabet the symbols take values from.\n•The binary channel in which input sequences may be impaired by adjacent transpositions (or bit-shifts) was\nanalyzed quite extensively as a model of some magnetic recording devices [27]. Codes for such a channel\n[31], [22], [17], [11] can equivalently be described in the space [12]\n∇n,r=\b\n(u1, u2, . . . , u r)∈Zr: 1⩽u1< u 2<···< ur⩽n\t\n(3)\nunder L1metric, meaning that every code correcting up to tadjacent transpositions can be obtained from codes\nin∇n,rhaving minimum L1distance >2t. In this description, the parameter ncorresponds to the length of\nthe input sequences and the parameter rto the Hamming weight of those sequences.\n•Various types of channels in which the symbols are represented by different voltage/charge levels are of interest\nin practice, e.g., in digital communication systems employing Pulse Amplitude Modulation, in multilevel flash\nmemories, etc. The set of all possible inputs in such channels can be represented as\nZn\nq=\b\n(u1, u2, . . . , u n)∈Zn: 0⩽ui⩽q−1\t\n, (4)\nwhere the parameter ncorresponds to the code length and the parameter qto the size of the alphabet, i.e.,\nthe number of different voltage levels. It is easy to see that a code correcting a total voltage change of up to\ntis equivalently described as a code in Zn\nqhaving a minimum L1distance >2t. See for example [1] for an\napplication of the ternary case Zn\n3.\nIn this paper, we study the highest attainable asymptotic rates of codes in the spaces (1)–(4) having minimum\nL1distance d=δn, for any fixed δandn→ ∞ . In particular, our main contributions include lower bounds on\nthese rates, which, as mentioned in the above examples, can be directly translated into lower bounds on the rates\nof optimal codes correcting aδ\n2-fraction of sticky insertions, transpositions, voltage jumps/drops, etc. Almost all\nprior works that discussed bounds on codes for relevant channel models focused on the regime d=const, n→ ∞ .\nTo the best of our knowledge, only the work [11] studied the problem under the assumption d=δn. As we shall\ndemonstrate in Section IV-C, our bound significantly improves upon the bound from [11].\nThe bounds we derive are versions of the well-known Gilbert–Varshamov (GV) bound [4], [30] or, more precisely,\nof the generalization thereof obtained by Gu and Fuja [8], which states that the maximum cardinality of a code of\nminimum distance dis lower-bounded by the ratio of the size of the input space and the average volume of a ball\nof radius d−1in that space. We shall also follow the approach suggested by Marcus and Roth [24] for further\nimproving this bound. In order to compute the average ball volume, we employ the tools of multivariate analytic\ncombinatorics (see [25] for an introductory text and [26] for a survey of combinatorial applications). We remark\nthat the use of generating functions in determining the GV bound, and in coding theory more generally, is not new.\nFor example, in one of the pioneering papers, Kolesnik and Krachkovsky [10] employed generating functions to\ncompute the GV bound for runlength-limited codes. However, studies of the multivariate case are very recent [12],\n[16], [19]. The present paper continues this line of work and we hope it will contribute to inspiring further research\non the applications of multivariate analytic combinatorics in coding theory.3\nNotation Remark Formula\nΣ Alphabet\nS Constrained (Ambient) Space\nSn S∩Σn\nd Metric defined on S\nd Distance d=⌊δn⌋\nA(Sn, d) Maximum code size\nα(S, δ) Highest attainable rate lim supn→∞logA(Sn,⌊δn⌋)\nn\nCap( S) Capacity of S lim supn→∞log|Sn|\nn\nT(Sn, d−1) Total ball {(u,v)∈Sn×Sn:v∈V(u, d−1)}\neT(S, δ) lim supn→∞log|T(Sn,⌊δn⌋)|\nn\nRGV(S, δ) Standard GV bound 2Cap( S)−eT(S, δ)\nS(τ) A subset of the Constrained Space\nRMR(S, δ) GV-MR bound max τ∈I\u0002\n2Cap( S(τ))−eT(S(τ), δ)\u0003\nu,v,x,y,z,w,k Vectors\nzkMonomial in z,k∈Zℓ\n⩾0Qℓ\ni=1zki\ni\nF(z), G(z), H(z) Functions of the vector z\nz∗Root of a function\nD(u,v) L1distance between u,v∈ZnPn\ni=1|ui−vi|\n∆n,r Standard simplex {(u1, u2, . . . , u r)∈Zr:ui⩾0,Pr\ni=0ui=n}\n∆+\nn,r Positive simplex {(u1, u2, . . . , u r)∈Zr:ui⩾1,Pr\ni=0ui=n}\n∇n,r Inverted simplex {(u1, u2, . . . , u r)∈Zr: 1⩽u1< u 2. . . < u r⩽n}\nZn\nq Hypercube {(u1, u2, . . . , u n)∈Zn: 0⩽ui⩽q−1}\nH(p) Binary entropy function −plogp−(1−p) log(1 −p)\nTABLE I\nThe paper is organized as follows. In Section II we recall the statement of the GV bound and its improvements\nfor general code spaces. In Section III we give an overview of the results from multivariate analytic combinatorics,\nand state several refinements thereof, that will be needed in the derivations that follow. Our main results, GV bounds\nfor codes in the L1metric, are given in Section IV for the spaces (1)–(3) and in Section V for the space (4). For\neasier reference, the notation used throughout the paper is summarized in Table I.4\nII. P RELIMINARIES\nLetΣbe an alphabet, Σnthe set of all words of length noverΣ, and Σ∗=S∞\nn=1Σnthe set of all finite-length\nwords over Σ. Let S⊆Σ∗be a constrained space and Sn=S∩Σn. Let d:S×S→Z⩾0∪ {∞} be a metric\ndefined on S. A subset C⊆Snsuch that d(c1,c2)⩾dfor all distinct c1,c2∈Cis called an (Sn, d)-code. The\nmaximum cardinality of a code having a given minimum distance, denoted\nA(Sn, d)≜max\b\n|C|:C⊆Sn,d(c1,c2)⩾dfor all c1,c2∈C,c1̸=c2\t\n, (5)\nis the quantity of central importance in coding theory. In particular, one is interested in determining the highest\nattainable asymptotic rate,\nα(S, δ) = lim sup\nn→∞logA(Sn,⌊δn⌋)\nn, (6)\nfor any fixed δ. An exact characterization of this rate remains elusive in all nontrivial models. We next describe\nthe best known general lower bound on α(S, δ), which is the main object of study in this paper.\nThe Gilbert–Varshamov Bound\nForu∈Sn, denote by V(u, r) ={v∈Sn:d(u,v)⩽r}the ball of radius rcentered at u. If|V(u, r)|is\nconstant over all u∈Sn, the GV bound states that A(Sn, d)⩾|Sn|/|V(u, d−1)|. In non-uniform spaces, however,\nin which |V(u, r)|depends on u, the bound needs to be adapted. Kolesnik and Krachkovsky [10] showed that the\nGV lower bound can be generalized to |S|/4V(d−1), where V(d−1) =1\n|Sn|P\nu∈Sn|V(u, d−1)|is the average\nball volume. This was further improved by Gu and Fuja [8] to |Sn|/V(d−1). For convenience, we consider the\ncollection of word pairs\nT(Sn, d−1)≜\b\n(u,v)∈Sn×Sn:v∈V(u, d−1)\t\n. (7)\nHence, |T(Sn, r)|=|Sn|·V(r)represents the “total ball size”, and the above-mentioned result of Gu and Fuja can\nbe restated as\nA(Sn, d)⩾|Sn|2\n|T(Sn, d−1)|. (8)\nIn terms of asymptotic rates when n→ ∞ , the bound (8) asserts that there exists a family of (Sn,⌊δn⌋)-codes\nsuch that their rates approach\nRGV(S, δ) = 2Cap( S)−eT(S, δ) (9)\nwhere\nCap( S)≜lim sup\nn→∞log|Sn|\nn(10)\nand\neT(S, δ)≜lim sup\nn→∞log|T(Sn,⌊δn⌋)|\nn. (11)\nNote that Cap( S) =eT(S,0). Therefore, in order to determine RGV(S, δ), we need to compute eT(S, δ).\nLater, Marcus and Roth [24] improved the GV bound (9) by considering certain subsets of the constrained space\nS, which are denoted S(τ)for some parameter τin a bounded interval I; we shall refer to this bound as the GV-MR\nbound. Let Sn(τ)be the set of all words of length ninS(τ)and define Cap( S(τ)) = lim supn→∞1\nnlog|Sn(τ)|.\nSimilar to before, define also eT(S(τ), δ) = lim supn→∞1\nnlog|T(Sn(τ),⌊δn⌋)|. Since Sn(τ)is a subset of Sn,\nit follows from the usual GV argument that there exists a family of (Sn,⌊δn⌋)-codes whose rates approach5\n2Cap( S(τ))−eT(S(τ), δ)for every τ. Therefore, we have the following lower bound on achievable asymptotic\ncode rates:\nRMR(S, δ) = max\nτ∈I\u0002\n2Cap( S(τ))−eT(S(τ), δ)\u0003\n. (12)\nA key remark from [24] is that both Cap( S(τ))andeT(S(τ), δ)can be obtained via different optimization problems.\nWe refer the reader to [5] for a discussion of efficient numerical procedures for solving the optimization problems\nappearing in the evaluation of the GV and GV-MR bounds.\nIII. A NALYTIC COMBINATORICS IN SEVERAL VARIABLES (ACSV)\nIn many cases of interest, generating functions provide a concise description of the combinatorial quantity\n|T(Sn,⌊δn⌋)|that is needed for determining the GV bound. As most of these generating functions involve several\nvariables, we will need to borrow tools from multivariate analytic combinatorics to provide the required asymptotic\nestimates.\nLet the number of variables be ℓand let zdenote the ℓ-tuple (z1, . . . , z ℓ). For k∈Zℓ\n⩾0, let zkdenote the\nmonomialQℓ\ni=1zki\ni. Consider a multivariate array of non-negative integers {ak}k∈Zℓ\n⩾0with the generating function\nF(z) =P\nkakzk. The following theorem is crucial for this paper.\nTheorem 1 ([26, Theorem 1.3]) .LetF(z) =P\nkakzk=G(z)\nH(z)where GandHare both analytic, H(0)̸= 0,\nandak>0. For each k= (k1, k2, . . . k ℓ)>0there is a unique solution z∗= (z∗\n1, z∗\n2, . . . , z∗\nℓ)>0satisfying the\nequations\nH(z) = 0\nkℓzj∂H(z)\n∂zj=kjzℓ∂H(z)\n∂zℓfor1⩽j⩽ℓ−1.(13)\nFurthermore, if G(z∗)̸= 0,\nak∼(2π)−(ℓ−1)/2H−1/2 G(z∗)\n−zℓ∂H(z)\n∂zℓ|z=z∗k−(ℓ−1)/2\nℓ(z∗)−k, (14)\nwhere His the determinant of the Hessian of the function parametrizing the hypersurface {H= 0}in logarithmic\ncoordinates.\nFor a detailed calculation of the Hessian matrix, we refer the reader to [25, Lemma 5]. More general asymptotic\nresults are available in [25, Theorems 5.1–5.4]. In this paper, we are interested in the case when all coordinates of\nkgrow linearly with n, i.e., ki=nrifor fixed ri,1⩽i⩽ℓ. In this case all terms in (14) tend to constants except\nk−(ℓ−1)/2\nℓ(z∗)−k, and the asymptotic behavior of the sequence akcan be simplified to:\nak=anr1,nr2,...,nr ℓ= Θ \nn(ℓ−1)/2ℓY\ni=1(z∗\ni)−nri!\n(15)\nlim\nn→∞loganr1,nr2,...,nr ℓ\nn=−ℓX\ni=1rilogz∗\ni (16)\nTo illustrate the theorem, we modify an example from [26].\nExample 1 (Binomial coefficients [26, Section 4.1]) .Consider the bivariate ( ℓ= 2) sequence an,k=\u0000n\nk\u0001\n. The\nfollowing recursion holds an,k=an−1,k−1+an−1,kfor all (n, k)̸= (0,0), from which the generating function6\nF(z) =1\n1−z1−z1z2can be derived, namely:\nF(z) =X\nn,k⩾0an,kzn\n1zk\n2\n= 1 +X\n(n,k)̸=(0,0)(an−1,k−1+an−1,k)zn\n1zk\n2\n= 1 +X\n(n,k)̸=(0,0)an−1,k−1zn\n1zk\n2+X\n(n,k)̸=(0,0)an−1,kzn\n1zk\n2\n= 1 + ( z1z2+z1)F(z).\nHence, G(z) = 1 andH(z) = 1−z1−z1z2. Further, we can explicitly solve the system of equations (13), which\nin this example has the form:\n1−z1−z1z2= 0,\nkz1∂H\n∂z1=nz2∂H\n∂z1.\nFrom the first equation, we have z1+z1z2= 1. Since z1∂H\n∂z1=−z1−z1z2=−1andz2∂H\n∂z1=−z1z2, we have\nk=nz1z2. Hence, the solution is z∗=\u0000\n1−k\nn,k/n\n1−k/n\u0001\n. When we fix k=rn, we obtain from (16) the well-known\nfact that\nlim\nn→∞logan,rn\nn=−log(1−r)−rlogr\n1−r=−(1−r) log(1 −r)−rlogr=H(r).\nFurther, we note that it is possible to reduce the number of variables ℓinvolved in Theorem 1 using a symmetry\nargument. We say two dummy variables z1andz2are symmetric if k1=k2and interchanging z1andz2will have\nno effect on the generating function F(z). Sometimes the symmetric variables can be identified before even finding\nthe generating function just by observing certain symmetry properties of the quantity of interest. Specifically, for\nthe computation of total ball size, it is possible to reduce the number of variables almost by a factor of 1/2, as\nwill be demonstrated in the next section.\nProposition 2. Consider a multivariate polynomial H(z)such that in each monomial zmwe have that m1=m2.\nConsider k∈Zℓ\n>0withk1=k2. Ifz∗is a solution to (13), then we have z∗\n1=z∗\n2. We define another multivariate\npolynomial ¯Hthat involves ℓ−1variables. Specifically, we set ¯H(¯z) =¯H( ¯z1,¯z3, . . . , ¯zℓ)≜H( ¯z1,¯z1,¯z3, . . . , ¯zℓ).\nIfz∗is a solution to (13), then ¯z∗≜(z∗\n1, z∗\n3, . . . , z∗\nℓ)is a solution to the following set of equations\n¯H(¯z) = 0\nkℓ¯z1∂¯H(¯z)\n∂¯z1= 2k1¯zℓ∂¯H(¯z)\n∂¯zℓ\nkℓ¯zj∂¯H(¯z)\n∂¯zj=kj¯zℓ∂¯H(¯z)\n∂¯zℓfor3⩽j⩽ℓ−1.\nProof. Since H(z)is a multivariate polynomial, we have H(z) =P\nmcmQℓ\ni=1zmi\ni. We differentiate H(z)with\nrespect to z1andz2and multiply by k2z1andk1z2, respectively. Further using m1=m2andk1=k2, we have\nk2z1∂H(z)\n∂z1=k2X\nmm1cmℓY\ni=1zmi\ni=k1X\nmm2cmℓY\ni=1zmi\ni=k1z2∂H(z)\n∂z2.\nHence, z∗\n1=z∗\n2. Furthermore, since we set ¯H(¯z) =¯H( ¯z1,¯z3, . . . , ¯zℓ)≜H( ¯z1,¯z1,¯z3, . . . , ¯zℓ), we have that ¯H(¯z) =P\nmcm¯z12m1Qℓ\ni=3¯zimi. Note that, since z∗is a solution to (13), we have H(z∗) = 0 or equivalently ¯H(¯z∗) = 0 .7\nNow, we differentiate ¯H(¯z)with respect to ¯z1and¯zℓand multiply by kℓ¯z1andk1¯zℓrespectively:\nkℓ¯z1∂¯H(¯z)\n∂¯z1=kℓX\nm2m1cm¯z12m1ℓY\ni=3¯zimi\n= 2kℓX\nmm1cm¯z12m1ℓY\ni=3¯zimi\n= 2kℓ¯z1∂H( ¯z1,¯z1,¯z3, . . . , ¯zℓ)\n∂¯z1\n= 2k1¯zℓ∂¯H(¯z)\n∂¯zℓ.\nRecall that from (13) we have\nkℓ¯z1∂H( ¯z1,¯z1,¯z3, . . . , ¯zℓ)\n∂¯z1=k1¯zℓ∂H( ¯z1,¯z1,¯z3, . . . , ¯zℓ)\n∂¯zℓ\nand hence\nkℓ¯z1∂¯H(¯z)\n∂¯z1= 2k1¯zℓ∂¯H(¯z)\n∂¯zℓ.\nThe rest of the proof follows directly from (13). ■\nOnce we determine the unique solution ¯z∗= (z∗, z∗, z∗\n3, z∗\n4, . . . , z∗\nℓ)from Proposition 2, we have that\n−r1logz∗\n1−r2logz∗\n2=−2rlogz∗\n1. (17)\nand therefore, when k1=k2orr1=r2, (16) becomes\nlim\nn→∞loganr1,nr1,nr3,...,nr ℓ\nn=−ℓX\ni=1rilogz∗\ni=−2rlogz∗\n1−ℓX\ni=3rilogz∗\ni. (18)\nNotice that the unique solution z∗in (13) is completely represented by k= (k1, . . . , k ℓ) =n(r1, . . . , r ℓ) =nr.\nHence, the right hand side of (16) is a function that depends on r. In what follows, we fix r1, . . . , r ℓ−1and optimize\nthe expression with respect to rℓ. Let Φ(z∗, rℓ)be the corresponding function. Let z∗(rℓ) = ( z∗\ni(rℓ))i∈[ℓ]be the\ncorresponding unique solution in (13) while rℓvaries. The next theorem characterizes the ℓ-th component of z∗\nwhen Φ(z∗, rℓ)is maximized.\nTheorem 3. SetΦ(z∗, rℓ)≜−Pℓ\ni=1rilogz∗\ni(rℓ). Then∂\n∂rℓΦ(z∗, rℓ) =−log(z∗\nℓ). Therefore, the quantity Φ(z∗, rℓ)\nis maximized if and only if z∗\nℓ(rℓ) = 1 .\nProof. As we vary rℓ, since riare assumed to be fixed, we have from Theorem 1 that the function H(z∗(rℓ))is\nidentically zero. This implies that its partial derivative with respect to rℓis also zero. Then applying Chain Rule\nand (13), we have that\n∂H(z∗(rℓ))\n∂rℓ=ℓX\ni=1∂H(z∗(rℓ))\n∂z∗\ni∂z∗\ni\n∂rℓ=ℓX\ni=1riz∗\nℓ\nrℓz∗\ni∂H(z∗(rℓ))\n∂z∗\nℓ∂z∗\ni\n∂rℓ=z∗\nℓ\nrℓ∂H(z∗(rℓ))\n∂z∗\nℓℓX\ni=1ri\nz∗\ni∂z∗\ni\n∂rℓ= 0.\nNow, since zℓ∗>0and (13) implies that the partial derivative with respect to z∗\nℓis nonzero, we have that\nℓX\ni=1ri\nz∗\ni∂z∗\ni\n∂rℓ= 0.8\nBy taking the partial derivative of Φ(z∗, rℓ)with respect to rℓ, we have\n∂Φ(z∗, rℓ)\n∂rℓ=−ℓX\ni=1ri\nz∗\ni∂(z∗\ni)\n∂rℓ−log(z∗\nℓ) =−log(z∗\nℓ).\n■\nExample 2 (Example 1 continued) .Recall that an,k=\u0000n\nk\u0001\n. Suppose we fix r1= 1and set k=r2nwithr2varying.\nThen the quantity Φ(r2) =H(r2). Theorem 3 states that Φ(r2)is maximized when the solution z∗\n2=r2/(1−r2) = 1 ,\ni.e., when r2= 1/2, as expected.\nIn the previous section we mentioned that the computation of the GV-MR bound [24] involves different opti-\nmization problems. Nevertheless, when we have the explicit generating functions for the constrained space and\ncorresponding ball sizes, we are able to characterize certain component for the solution in (13).\nCorollary 4. Consider a constrained system Sand its subsets parameterized by τ. Let Sn(τ)andT(Sn(τ), δn)be\ndefined as before, and suppose that the generating functions\nFS(z) =X\nk⩾0|Sn(τ)|zk, F T(y) =X\nj⩾0|T(Sn(τ), δn)|yj,\nhave ℓandmvariables, respectively. Further, suppose that kℓ=jm=τnand that the other components ki\n(i∈[ℓ−1]) and ji(i∈[m−1]) are all independent of τ. Then when the optimization program (12) is maximized,\nwe have that either (z∗\nℓ)2=y∗\nmorτis at the end-points of I.\nProof. Recall from (12) that RMR(δ) = max τ∈I[2Cap( S(τ))−eT(S(τ), δ)]. Furthermore, Cap( S(τ)) = Φ( z∗, τ)\nandeT(S(τ), δ) = Φ( y∗, τ), as defined in Theorem 3. It follows from Theorem 3 that\n∂\n∂τΦ(z∗, τ) =−log(z∗\nℓ),∂\n∂τΦ(y∗, τ) =−log(y∗\nm).\nHence, to maximize the right-hand side of (12), we need 2∂\n∂τΦ(z∗, τ)−∂\n∂τΦ(y∗, τ) = 0 or, equivalently, (z∗\nℓ)2=\ny∗\nm. ■\nIV. GV B OUNDS FOR SIMPLEX SPACES\nA. The Standard Simplex\nIn this subsection, we consider the code space given by the simplex of dimension rand weight n, namely\n∆n,r≜(\nu= (u1, u2, . . . , u r)∈Zr:ui⩾0,rX\ni=0ui=n)\n. (19)\n∆n,rcan be understood as the set of all multisets of cardinality nover an alphabet of size r. As noted in the\nIntroduction, the motivation for studying codes in this space comes from certain types of permutation channels and\nunordered (e.g., DNA-based) data storage channels.\nWe shall consider the asymptotic regime where n→ ∞ andr=ρn, for an arbitrary constant ρ⩾0. Denote the\nfamily of simplices satisfying this relation by ∆ρ, i.e.,\n∆ρ=∞[\nn=1∆n,⌊ρn⌋. (20)\nThe following statement is well-known and follows directly from the fact that |∆n,r|=\u0000n+r−1\nr−1\u0001\n.9\nProposition 5. For fixed ρ⩾0,\nCap(∆ ρ)≜lim\nn→∞log|∆n,⌊ρn⌋|\nn= (1 + ρ)H\u0010ρ\n1 +ρ\u0011\n.\n1) Evaluation of the GV Bound: We denote the L1-ball with center u∈∆n,rand radius sbyV(u, s)≜{v∈\n∆n,r:D(u,v)⩽s}. To determine the total ball size, we first consider the number of pairs (u,v)withL1-distance\ns, denoted by\nN(n1, n2, r, s) =|{(u,v)∈∆n1,r×∆n2,r:D(u,v) =s}|. (21)\nThis quantity can be recursively characterized as follows.\nLemma 6.\nN(n1, n2, r, s) =X\ni⩾0N(n1−i, n2−i, r−1, s)\n+X\ni⩾0X\nj⩾1N(n1−i, n2−i−j, r−1, s−j)\n+X\ni⩾0X\nj⩾1N(n1−i−j, n2−i, r−1, s−j).\nProof. Letu= (u1, . . . , u r)∈∆n1,randv= (v1, . . . , v r)∈∆n2,r. Consider truncating the last coordinate ur\nandvr. Ifur=vr=i, we get the first sum where the distance between the truncated vectors remains the same.\nOtherwise, suppose ur=iandvr=i+j, for i⩾0andj⩾1. In this case, the truncated vectors belong to\n∆n1−i,r−1and∆n2−i−j,r−1, respectively, and the distance between them is s− |ur−vr|=s−j, which gives the\nsecond term. The last term is obtained similarly when ur> vr. ■\nWe are ready to derive the generating function F(x1, x2, y, z) =P\nn1,n2,r,s⩾0N(n1, n2, r, s)xn1\n1xn2\n2yrzs. In order\nto apply Theorem 1, we present F(x1, x2, y, z)as a ratio of two multivariate polynomials.\nLemma 7. F(x1, x2, y, z) =G(x1,x2,y,z)\nH(x1,x2,y,z), where G(x1, x2, y, z)is some multivariate polynomial and\nH(x1, x2, y, z) = (1 −x1)(1−x2)\u0002\n(1−x1x2)(1−x1z)(1−x2z)−y(1−x1x2z2)\u0003\n.\nProof. Applying Lemma 6, we obtain\nF(x1, x2, y, z) =X\nn1,n2,r,s⩾0N(n1, n2, r, s)xn1\n1xn2\n2yrzs\n=X\nn1,n2,r,s⩾0X\ni⩾0N(n1−i, n2−i, r−1, s)xn1\n1xn2\n2yrzs\n+X\nn1,n2,r,s⩾0X\ni⩾0X\nj⩾1N(n1−i, n2−i−j, r−1, s−j)xn1\n1xn2\n2yrzs\n+X\nn1,n2,r,s⩾0X\ni⩾0X\nj⩾1N(n1−i−j, n2−i, r−1, s−j)xn1\n1xn2\n2yrzs\n= 1 +X\ni⩾1xi\n1+X\ni⩾1xi\n2+F(x1, x2, y, z)\u0012\nyX\ni⩾0(x1x2)i\u0013\u0012\n1 +X\nj⩾1(x1z)j+X\nj⩾1(x2z)j\u0013\n.10\nHence,\nF(x1, x2, y, z) =1 +P\ni⩾1xi\n1+P\ni⩾1xi\n2\n1−\u0000\nyP\ni⩾0(x1x2)i\u0001\u0000\n1 +P\nj⩾1(x1z)j+P\nj⩾1(x2z)j\u0001\n=(1 +P\ni⩾1xi\n1+P\ni⩾1xi\n2)(1−x1x2)(1−x1z)(1−x2z)\n(1−x1x2)(1−x1z)(1−x2z)−y(1−x1x2z2).\n=(1−x1x2)2(1−x1z)(1−x2z)\n(1−x1)(1−x2)[(1−x1x2)(1−x1z)(1−x2z)−y(1−x1x2z2)].\n■\nAs we are interested in the pairs residing in the simplex with the same weight nand dimension ρn, and having\ndistance δn, the quantity of interest is N(n, n, ρn, δn ). Since n1=n2=n, we can reduce the number of variables\nin the generating function using Proposition 2 and set x1=x2=x. More specifically, we define the generating\nfunction\nF(x, x, y, z ) =X\nn,r,s⩾0N(n, n, r, s )x2nyrzs=(1 +x)2(1−xz)\n(1−x2)(1−xz)−y(1 +xz). (22)\nBy Theorem 1 and Proposition 2, with r=ρn, s =δn, we need to solve the following system of equations:\nH= 0 andxHx\n2=yHy\nρ=zHz\nδ,where H= (1−x2)(1−xz)−y(1 +xz). (23)\nHere Hxdenotes the partial derivative∂H\n∂x, and the function H(x, y, z )is written as Hfor simplicity.\nLemma 8. For fixed ρ, the solution of (23) with respect to δis\nx∗(δ) =r\n1−2ρ\n2 + 2 ρ−δ,\nz∗(δ) =p\nρ2+δ2−ρ\nx∗δ,\ny∗(δ) = 2p\nρ2+δ2−δ\n2−δ+ 2ρ.\nProof. Appendix A. ■\nApplying (16), we have that\nlim\nn→∞logN(n, n, ρn, δn )\nn=−2 logx∗(δ)−ρlogy∗(δ)−δlogz∗(δ). (24)\nThe total ball size is\n|T(∆n,ρn, δn)|=δnX\ns=0N(n, n, ρn, s )\nand hence\neT(∆ρ, δ)≜lim\nn→∞log|T(∆n,ρn, δn)|\nn\n= max\n0⩽δ1⩽δ−2 logx∗(δ1)−ρlogy∗(δ1)−δ1logz∗(δ1)\n=\n\n−2 logx∗(δ)−ρlogy∗(δ)−δlogz∗(δ), δ⩽δmax,\n2(1 + ρ)H(ρ\n1+ρ), δ ⩾δmax.\nHere, δmax= 2(1 + ρ)/(2 +ρ). Details of the derivations for δmaxare also given in Appendix A.11\nIn conclusion, we have the following explicit formulas for the asymptotic total ball size and the GV lower bound.\nProposition 9. For fixed ρ⩾0, setδmax= 2(1 + ρ)/(2 +ρ). When 0⩽δ⩽δmax, we have\neT(∆ρ, δ) =−ρ+δlog(δ)−ρlog\u0000p\nρ2+δ2−δ\u0001\n−δlog\u0000p\nρ2+δ2−ρ\u0001\n+ (1 + ρ−δ/2) log(2 + 2 ρ−δ)−(1−δ/2) log(2 −δ).\nOtherwise, when δ⩾δmax, we have eT(∆ρ, δ) = 2(1 + ρ)H(ρ\n1+ρ).\nTheorem 10. For fixed ρ, δ⩾0, we have α(∆ρ, δ)⩾RGV(∆ρ, δ), where\nRGV(∆ρ, δ)≜2(1 + ρ)H\u0010ρ\n1 +ρ\u0011\n−eT(∆ρ, δ).\n2) Evaluation of Marcus and Roth’s Improvement of the GV Bound: In this subsection, we further improve the\nbound obtained above by introducing an additional parameter that constrains the code space, then deriving the GV\nbound for the resulting space, and finally optimizing this bound over all values of the introduced parameter. This\napproach was first suggested in [24].\nThe additional parameter we introduce, denoted p, is the number of zeros in each vector. More precisely, we set\n∆n,r(p) =(\nu= (u1, u2, . . . , u r)∈Zr:ui⩾0,rX\ni=0ui=n,|{i:ui= 0}|=p)\n. (25)\nWe allow pto grow linearly with nand set p=τn. Let ∆ρ(τ)denote the family of constrained spaces satisfying\nthis relation, i.e., ∆ρ(τ) =S\nn∆n,⌊ρn⌋(⌊τn⌋).\nProposition 11. For fixed ρ⩾0,0⩽τ⩽ρ, we have\nCap(∆ ρ(τ))≜lim\nn→∞log|∆n,⌊ρn⌋(⌊τn⌋)|\nn=ρH(τ/ρ) +H(ρ−τ).\nProof. Follows directly from |∆n,r(p)|=\u0000r\np\u0001\u0000n−1\nr−p−1\u0001\n. ■\nWe perform the same analysis for total ball size to obtain the bound. We consider balls centered at u∈∆n,r(p)\nand having radius s, that is V(u, s, p) ={v∈∆n,r(p) :D(u,v)⩽s}.\nTo estimate the ball size, we consider the number of pairs (u,v)with L1-distance exactly s, denoted by\nN(n1, n2, r, s, p 1, p2) =|{(u,v)∈∆n1,r(p1)×∆n2,r(p2) :D(u,v) =s,|{i:ui= 0}|=p1,|{i:vi= 0}|=p2}|.\nThe following lemma gives a recursive expression for this quantity.\nLemma 12.\nN(n1, n2, r, s, p 1, p2) =N(n1, n2, r−1, s, p 1−1, p2−1)\n+X\nj⩾1N(n1, n2−j, r−1, s−j, p1−1, p2)\n+X\nj⩾1N(n1−j, n2, r−1, s−j, p1, p2−1)\n+X\ni⩾1N(n1−i, n2−i, r−1, s, p 1, p2)\n+ 2X\ni⩾1X\nj⩾1N(n1−i, n2−i−j, r−1, s−j, p1, p2)\n+ 2X\ni⩾1X\nj⩾1N(n1−i−j, n2−i, r−1, s−j, p1, p2).12\nProof. Letu= (u1, . . . , u r)∈∆n1,r(p1)andv= (v1, . . . , v r)∈∆n2,r(p2). Truncate the last coordinate urand\nvr, and consider the following possibilities: (1) ur=vr= 0, (2)ur= 0,vr=j, (3)ur=j,vr= 0, (4)ur=i,\nvr=i, (5)ur=i,vr=i+j, (6)ur=i+j,vr=i. ■\nAs before, the quantity of interest is N(n, n, ρn, δn, τn, τn ). Therefore, we define the generating function\nF(x, y, z, w ) =P\nn,r,s,p ⩾0N(n, n, r, s, p, p )x2nyrzsw2pwith four instead of six variables (see Proposition 2).\nWe next show that Fis rational and that Theorem 1 can be applied to it.\nLemma 13. F(x, y, z, w ) =G(x,y,z,w )\nH(x,y,z,w ), where G(x, y, z, w )is some multivariate polynomial and\nH(x, y, z, w ) = (1 −x2)(1−xz)−y(w2(1−xz)(1−x2) + 2wxz(1−x2) +x2(1 +xz)).\nProof.\nF(x, y, z, w ) =X\nn,r,s,p ⩾0N(n, n, r, s, p, p )x2nyrzsw2p\n=X\nn,r,s,p ⩾0N(n, n, r −1, s, p−1, p−1)x2nyrzsw2p\n+X\nn,r,s,p ⩾0X\nj⩾1N(n, n−j, r−1, s−j, p−1, p)x2nyrzsw2p\n+X\nn,r,s,p ⩾0X\nj⩾1N(n−j, n, r−1, s−j, p, p−1)x2nyrzsw2p\n+X\nn,r,s,p ⩾0X\ni⩾1N(n−i, n−i, r−1, s, p, p )x2nyrzsw2p\n+X\nn,r,s,p ⩾0X\ni⩾1X\nj⩾1N(n−i, n−i−j, r−1, s−j, p, p )x2nyrzsw2p\n+X\nn,r,s,p ⩾0X\ni⩾1X\nj⩾1N(n−i−j, n−i, r−1, s−j, p, p )x2nyrzsw2p\n= 1 + 2X\ni⩾1xi+F(x, y, z, w )y \nw2+ 2wX\nj⩾1(xz)j+X\ni⩾1(x2i) + 2X\ni⩾1X\nj⩾1(x2i)(xz)j!\n.\nHence,\nF(x, y, z, w ) =1 + 2P\ni⩾1xi\n1−y\u0000\nw2+ 2wP\nj⩾1(xz)j+P\ni⩾1(x2i) + 2P\ni⩾1P\nj⩾1(x2i)(xz)j\u0001\n=(1−xz)(1 + x)2\n(1−x2)(1−xz)−y\u0000\nw2(1−xz)(1−x2) + 2wxz(1−x2) +x2(1 +xz)\u0001.\n■\nFrom Theorem 1 and Proposition 2, we need to solve the following system of equations:\nH= 0 andxHx\n2=yHy\nρ=zHz\nδ=wHw\n2τ, (26)\nwhere the function H(x, y, z, w )is written as Hfor simplicity.\nLetλ1=xzandλ2=x2.13\nLemma 14. When ρis fixed, the solution of (26) with respect to δis:\nx∗(δ) =p\nλ∗\n2,\nz∗(δ) =λ∗\n1\nx∗,\nw∗(δ) =λ∗\n2(δ(1−(λ∗\n1)2)−2λ∗\n1(ρ−τ))\nλ∗\n1(1−λ∗\n2)(2(ρ−τ)−δ(1−λ∗\n1)),\ny∗(δ) =(1−λ∗\n1)(1−λ∗\n2)\n(w∗)2(1−λ∗\n1)(1−λ∗\n2) + 2w∗λ∗\n1(1−λ∗\n2) +λ∗\n2(1 +λ∗\n1)\nwhere λ∗\n1is the root of the equation\nδ2(1 +ρ)λ5\n1+ (−δ2(1 +τ) + 4δ(ρ−τ)(1 + ρ))λ4\n1+ (−δ2(ρ+ 2) + 2 δ(ρ−τ)(ρ−τ−2) + 4( ρ−τ)2(1 +ρ))λ3\n1\n+ (−δ2(ρ−2τ−2)−2δ(ρ−τ)(ρ+τ+ 2)−4(ρ−τ)2(1−ρ))λ2\n1+ (δ2−4δ(ρ−τ)(ρ−τ−1))λ1+δ2(ρ−τ−1) = 0 .\nandλ∗\n2is given by\nλ∗\n2(δ) = 1−(1 +λ∗\n1)2(ρ−τ)−δ(1−λ∗\n1)\n(2−δ).\nProof. Appendix B. ■\nApplying (16), we have that\nlim\nn→∞logN(n, n, ρn, δn, τn, τn )\nn=−2 logx∗(δ)−ρlogy∗(δ)−δlogz∗(δ)−2τlogw∗(δ). (27)\nThe total ball size is |T(∆n,ρn(τn), δn)|=Pδn\ns=0N(n, n, ρn, s, τn, τn ). Hence, we have\nProposition 15. For fixed ρ⩾0,0⩽τ⩽ρ, and 0⩽δ⩽2,\neT(∆ρ(τ), δ) = max\n0⩽δ1⩽δ−2 logx∗(δ1)−ρlogy∗(δ1)−δ1logz∗(δ1)−2τlogw∗(δ1)\n=−2 logx∗(δ)−ρlogy∗(δ)−δlogz∗(δ)−2τlogw∗(δ).\nNote that setting w∗= 1, or equivalently τ=τinitial =ρ2\n1+ρ, to evaluate the expression 2Cap(∆ ρ(τ))−\neT(∆ρ(τ), δ), we obtain a similar expression as for the GV bound. But to obtain a better lower bound than the GV\nbound, we need to evaluate RMR(∆ρ, δ) = max τ∈I\u0002\n2Cap(∆ ρ(τ))−eT(∆ρ(τ), δ)\u0003\n. For this purpose, we will be\nconsidering δas a parametric function in terms of λ1and we iterate over λ1to evaluate the GV-MR bound in the\nfollowing proposition.\nTheorem 16. For fixed ρ⩾0,0⩽λ∗\n1⩽1, we have α(∆ρ, δ(λ∗\n1))⩾RMR(∆ρ, δ(λ∗\n1)), where\nRMR(∆ρ, δ(λ∗\n1))≜2Cap(∆ ρ(τopt))−eT(∆ρ(τopt), δ(λ∗\n1))\nand\nτopt=ρ2\n1 +ρ−λ∗\n1,\nδ(λ∗\n1) =2λ∗\n1ρ\n1−(λ∗\n1)2+λ∗\n1ρ.\nProof. Appendix B. ■\n3) Numerical Results: In this subsection, we plot the GV and GV-MR lower bounds from Theorems 10 and 16.\nFor comparison purposes, we first derive a sphere-packing bound in Proposition 17.14\nSphere-Packing Bound . For any u∈∆n,r, consider the set u+ ∆ b,r={u+w:w∈∆b,r} ⊆∆n+b,r. It is easy\nto see that, if C⊆∆n,ris a code of minimum distance d, the sets u+ ∆ b,randv+ ∆ b,r, with b=⌊d−1\n2⌋, have\nto be disjoint for any two distinct codewords u,v∈C. The cardinality of the largest such code can therefore be\nupper-bounded by\nA(∆n,r, d)⩽|∆n+b,r|\n|∆b,r|⩽\u0000n+b+r−1\nr−1\u0001\n\u0000b+r−1\nr−1\u0001,\nwhere b=⌊d−1\n2⌋. The following claim is a direct consequence.\nProposition 17. For fixed ρ⩾0and0⩽δ⩽2, we have that α(∆ρ, δ)⩽RSP(∆ρ, δ), where\nRSP(∆ρ, δ)≜(1 +δ/2 +ρ)H\u0010ρ\n1 +δ/2 +ρ\u0011\n−(δ/2 +ρ)H\u0010ρ\nδ/2 +ρ\u0011\n.\nLower Bound from Binary Constant Weight Codes . In deriving the GV-MR bound, note that we are optimizing\nover the parameter τwhere τnis the number of zeroes in the constrained codewords. In the special case where\nτ=ρ−1, we observe that the space ∆n,r(p)is equivalent to constant weight binary codes of length rand weight\nn. So, for this instance, the bound corresponds to the GV lower bound for constant weight binary codes of length\nρnand weight n(see for example [7]). For completeness, we derive the bound here.\nProposition 18. For fixed ρ⩾1andτ=ρ−1, setδmax= 2ρ/(ρ−1). When 0⩽δ⩽δmax, we have\neT(∆ρ(τ), δ) =ρH\u00101\nρ\u0011\n+H\u0010δ\n2\u0011\n+ (ρ−1)H\u0010δ\n2(ρ−1)\u0011\n.\nOtherwise, when δ⩾δmax, we have eT(∆ρ(τ), δ) = 2 ρH(1\nρ).\nProof. In case when τ=ρ−1, we have a sum of ρnterms equal to n, while (ρ−1)nterms are 0. Since every\nterm is nonnegative, these nnonzero terms must all be equal to 1. Hence, each vector is equivalent to a binary\nsequence of length r=ρnand weight n. Hence, the total ball size in this case is easily derived as |T(∆n,r,⌊δn⌋)|=\u0000r\nn\u0001P⌊δn/2⌋\ni=0\u0000n\ni\u0001\u0000r−n\ni\u0001\nand hence we have that eT(∆ρ(τ), δ) =ρH\u0010\n1\nρ\u0011\n+max 0⩽δ1⩽δ/2H(δ1)+(ρ−1)H(δ1/(ρ−1)).\nIt can be easily verified that eT(∆ρ(τ), δ)is maximum when δ1= min(ρ\nρ−1,δ\n2).\neT(∆ρ(τ), δ) =\n\nρH\u0010\n1\nρ\u0011\n+H\u0010\nδ\n2\u0011\n+ (ρ−1)H\u0010\nδ\n2(ρ−1)\u0011\n, δ⩽δmax,\n2ρH(1\nρ), δ ⩾δmax.\n■\nProposition 19. For fixed ρ⩾1, τ=ρ−1andδ⩾0, we have RGV(∆ρ(τ), δ) = 2Cap(∆ ρ(τ))−eT(∆ρ(τ), δ).\nIn Figure 1, all these curves are plotted for the case ρ= 2. Observe that both GV and GV-MR bounds improve\nthe GV lower bound corresponding to binary constant weight codes. Note also that the GV bound is positive only\nforδ <2(1 + ρ)/(2 +ρ) = 1 .5(see Proposition 9 and Theorem 10), while the GV-MR bound is positive for all\nδ <2(see Theorem 16).\nB. The Positive Simplex\nIn this subsection, we consider the code space\n∆+\nn,r=(\n(u1, u2, . . . , u r)∈Zr:ui⩾1,rX\ni=1ui=n)\n(28)15\nFig. 1: Bounds on the highest attainable code rate α(∆2, δ)for the standard simplex ( ρ= 2).\nof cardinality |∆+\nn,r|=\u0000n−1\nr−1\u0001\n. We are again interested in the regime n→ ∞ ,r=ρn, and we denote by ∆+\nρthe\ncorresponding family of simplices. The motivation for studying this space comes from the fact that designing codes\ncorrecting a given number of errors in run-preserving channels (i.e., channels that preserve the number of runs in\ninput sequences) such as the sticky-insertion channel is equivalent to designing codes in ∆+\nn,runder L1metric. In\nthis correspondence, the parameter nrepresents the length of the input sequence, the parameter rthe number of\nruns in that sequence, and uithe length of the i’th run.\nProposition 20. For fixed 0⩽ρ⩽1, we have\nCap(∆+\nρ)≜lim\nn→∞log\f\f∆+\nn,⌊ρn⌋\f\f\nn=H(ρ).\nOne immediately notices that ∆+\nn,r= (1, . . . , 1) + ∆ n−r,rand, hence, bounds for codes in ∆+\nn,rcan be directly\nobtained from the bounds for codes in ∆n,rgiven in the previous subsection. Nonetheless, we state some of the\nresults explicitly so that they can be easily accessed by those interested in coding problems in ∆+\nn,r. Additionally, we\nperform optimization of the obtained bounds over the parameter ρ, which, in the mentioned application, corresponds\nto deriving bounds on codes for the sticky insertion channel with no restrictions on the number of runs in input\nsequences. Namely, if we consider the code space ∆+\nn=Sn\nr=0∆+\nn,r, and given that the sticky-insertion channel is\nrun-preserving, implying that an optimal code in ∆+\nnis a union of optimal codes in ∆+\nn,rover all r, it follows that\nα(∆+, δ) = max\n0⩽ρ⩽1α(∆+\nρ, δ) (29)\nfor every δ.\n1) Evaluation of the GV Bound: or completeness, we state the following lemma to find the unique solution that\ncorresponds to the generating function of the number of pairs (u,v)withL1distance s, denoted by N(n1, n2, r, s) =\n|{(u,v)∈∆+\nn1,r×∆+\nn2,r:D(u,v) =s}|. The generating function of the quantity N(n1, n2, r, s)is given in the16\nfollowing lemma.\nLemma 21. F(x, y, z, w ) =G(x,y,z,w )\nH(x,y,z,w ), where G(x, y, z, w )is some multivariate polynomial and\nH(x, y, z, w ) = (1 −x2)(1−xz)−yx2(1 +xz).\nAs before, we are interested in the pairs residing in the simplex with the same weight n, dimension ρn, and\ndistance δn, thus the quantity of interest is N(n, n, ρn, δn ). By Theorem 1 and Proposition 2, with r=ρn, s =δn,\nwe need to solve the following system of equations:\nH= 0 andxHx\n2=yHy\nρ=zHz\nδ,where H= (1−x2)(1−xz)−yx2(1 +xz). (30)\nLemma 22. For fixed 0⩽ρ⩽1, the solution of (30) with respect to δis:\nx∗(δ) =r\n1−2ρ\n2−δ,\nz∗(δ) =p\nρ2+δ2−ρ\nx∗δ,\ny∗(δ) = 2p\nρ2+δ2−δ\n2−δ−2ρ.\nProof. The proof is the same as that of Lemma 8. ■\nApplying (16), we get\nlim\nn→∞logN(n, n, ρn, δn )\nn=−2 logx∗(δ)−ρlogy∗(δ)−δlogz∗(δ).\nThe total ball size with distance δnis|T(∆+\nρ, δn)|=Pδn\ns=0N(n, n, ρn, s ), implying that\neT(∆+\nρ, δ) = max\n0⩽δ1⩽δ−2 logx∗(δ1)−ρlogy∗(δ1)−δ1logz∗(δ1)\n=\n\n−2 logx∗(δ)−ρlogy∗(δ)−δlogz∗(δ), δ⩽δmax\n2H(ρ), δ ⩾δmax\nwhere δmax=2(1−ρ)\n2−ρ. Consequently, we have the following explicit formula for the asymptotic ball size.\nProposition 23. For fixed 0⩽ρ⩽1, setδmax= 2(1 −ρ)/(2−ρ). When 0⩽δ⩽δmax, we have\neT(∆+\nρ, δ) =−ρ+δlogδ−ρlog\u0000p\nρ2+δ2−δ\u0001\n−δlog\u0000p\nρ2+δ2−ρ\u0001\n+ (−1 +ρ+δ/2) log(2 −2ρ−δ) + (1 −δ/2) log(2 −δ).\nOtherwise, when δ⩾δmax, we have eT(∆+\nρ, δ) = 2H(ρ).\nTheorem 24. For fixed 0⩽ρ⩽1andδ⩾0, we have α(∆+\nρ, δ)⩾RGV(∆+\nρ, δ), where\nRGV(∆+\nρ, δ)≜2H(ρ)−eT(∆+\nρ, δ).\nWe next state the bound obtained by maximizing RGV(∆+\nρ, δ)overρ. As already mentioned, the resulting function\ncan be directly translated into the GV bound on optimal codes for the sticky insertion channel (having no constraints\non the number of runs in input sequences).17\nTheorem 25 (GV Bound for Positive Simplex) .For fixed δ⩾0, we have α(∆+, δ)⩾RGV(∆+, δ), where\nRGV(∆+, δ)≜2H(ρ)−eT(∆+\nρ, δ)\nandρ=3(2−δ)−√9δ2−4δ+4\n8.\nProof. Appendix C. ■\n2) Evaluation of Marcus and Roth’s Improvement of the GV Bound: Following the approach from [24], we\nintroduce an additional parameter, which we choose to be the number of vector coordinates with value 1, then\ndetermine the GV bound for this constrained space, and finally optimize the bound over all the values of the new\nparameter. In particular, we set ∆+\nn,r(p) ={(u1, u2, . . . , u r)∈Zr:ui⩾1,Pr\ni=0ui=n,|{i:ui= 1}|=p}. Note\nthat|∆+\nn,r(p)|=\u0000r\np\u0001\u0000n−r−1\nr−p−1\u0001\n. Allowing pto grow linearly with n, we denote p=τn.\nProposition 26. For fixed 0⩽ρ⩽1and0⩽τ⩽ρ, we have\nCap(∆+\nρ(τ))≜lim\nn→∞log\f\f∆+\nn,⌊ρn⌋(⌊τn⌋)\f\f\nn=ρH\u0010τ\nρ\u0011\n+ (1−ρ)H\u0010ρ−τ\n1−ρ\u0011\n.\nTo estimate the total ball size, we consider the number of pairs (u,v)withL1distance exactly s, denoted by\nN(n, n, r, s, p, p ) =|{(u,v)∈∆+\nn,r(p)×∆+\nn,r(p) :D(u,v) =s,|{i:ui= 1}|=p,|{i:vi= 1}|=p}|. The\ngenerating function of the quantity N(n, n, r, s, p, p )is given in the following lemma.\nLemma 27. F(x, y, z, w ) =G(x,y,z,w )\nH(x,y,z,w ), where G(x, y, z, w )is some multivariate polynomial and\nH(x, y, z, w ) = (1 −x2)(1−xz)−yx2(w2(1−xz)(1−x2) + 2wxz(1−x2) +x2(1 +xz)).\nAs before, the quantity of interest is N(n, n, ρn, δn, τn, τn ). From Theorem 1 and Proposition 2, we need to\nsolve the following system of equations.\nH= 0 andxHx\n2=yHy\nρ=zHz\nδ=wHw\n2τ(31)\nLetλ1=xzandλ2=x2.\nLemma 28. For fixed ρandτ, the solution of the (31) with respect to δis:\nx∗(δ) =p\nλ∗\n2,\nz∗(δ) =λ∗\n1\nx∗,\nw∗(δ) =λ∗\n2(δ(1−(λ∗\n1)2)−2λ∗\n1(ρ−τ))\nλ∗\n1(1−λ∗\n2)(2(ρ−τ)−δ(1−λ∗\n1)),\ny∗(δ) =(1−λ∗\n1)(1−λ∗\n2)\nλ∗\n2((w∗)2(1−λ∗\n1)(1−λ∗\n2) + 2w∗λ∗\n1(1−λ∗\n2) +λ∗\n2(1 +λ∗\n1))\nwhere λ∗\n1is the root of the equation\n(1−λ1)(δ(1−λ2\n1)−2λ1(ρ−τ))2(2(1−ρ)−2(ρ−τ)(1 + λ1)−δλ2\n1)\n+λ2\n1(1 +λ1)(2(ρ−τ)−δ(1−λ1))2(δ(1−λ2\n1)−2λ1(ρ−τ)−2τ) = 0 .18\nandλ∗\n2is given by\nλ∗\n2(δ) = 1−(1 +λ∗\n1)2(ρ−τ)−δ(1−λ∗\n1)\n(2(1−ρ)−δ).\nProof. The proof is similar to the proof of Lemma 14 and hence we skip the details. ■\nApplying (16), we have that\nlim\nn→∞logN(n, n, ρn, δn, τn, τn )\nn=−2 logx∗(δ)−ρlogy∗(δ)−δlogz∗(δ)−2τlogw∗(δ). (32)\nThe total ball size is |T(∆+\nρ(τ), δn)|=Pδn\ns=0N(n, n, ρn, s, τn, τn ), and hence\neT(∆+\nρ(τ), δ) = max\n0⩽δ1⩽δ−2 logx∗(δ1)−ρlogy∗(δ1)−δ1logz∗(δ1)−2τlogw∗(δ1)\n=−2 logx∗(δ)−ρlogy∗(δ)−δlogz∗(δ)−2τlogw∗(δ). (33)\nFinally, we optimize the GV bound over τand parameterize δas a function of λ∗\n1to obtain the GV-MR bound\ngiven in the following theorem.\nTheorem 29. For fixed 0⩽λ∗\n1⩽1, we have α(∆+\nρ, δ(λ∗\n1))⩾RMR(∆+\nρ, δ(λ∗\n1)), where\nRMR(∆+\nρ, δ(λ∗\n1))≜2Cap(∆+\nρ(τopt))−eT(∆+\nρ(τopt), δ(λ∗\n1))\nand\nτopt=ρ2\n1−λ∗\n1(1−ρ),\nδ(λ∗\n1) =2λ∗\n1ρ(1−ρ)\nρλ∗\n1+ (1−ρ)(1−(λ∗\n1)2).\nProof. Appendix D. ■\nAs with the GV bound, we state below the bound obtained by maximizing RMR(∆+\nρ, δ)overρ.\nTheorem 30 (GV Bound for Positive Simplex) .For fixed 0⩽λ∗\n1⩽1, we have α(∆+, δ(λ∗\n1))⩾RMR(∆+, δ(λ∗\n1)),\nwhere\nRMR(∆+, δ(λ∗\n1))≜2Cap(∆+\nρopt(τopt))−eT(∆+\nρopt(τopt), δ(λ∗\n1))\nand\nρopt=2p\n1−λ∗\n1\n3p\n1−λ∗\n1+p\n1 + 3 λ∗\n1,\nτopt=ρ2\nopt\n1−λ∗\n1(1−ρopt),\nδ(λ∗\n1) =2λ∗\n1ρopt(1−ρopt)\nρoptλ∗\n1+ (1−ρopt)(1−(λ∗\n1)2).\nProof. Appendix D. ■\nC. The Inverted Simplex\nIn this subsection, we consider the code space\n∇n,r=\b\nu= (u1, u2, . . . , u r)∈Zr: 1⩽u1< u 2<···< ur⩽n\t\n(34)19\nconsisting of vectors whose components are positive, strictly increasing, and not exceeding n. The motivation for\nstudying this space comes from the bit-shift channel, as well as some types of timing channels. Namely, codes\ncorrecting tshifts of 1’s are equivalently described as codes in ∇n,rhaving minimum L1distance >2t. In this\ncorrespondence, the parameter nrepresents the length of the input binary sequence, the parameter rits Hamming\nweight, and uithe position of the i’th1in that sequence. As the bit-shift channel does not alter the Hamming\nweight of the input sequence, one may without loss of generality consider codes for each weight rseparately. After\nderiving the lower bounds for ∇n,r, one can easily obtain the corresponding bounds for the case with no weight\nconstraints by performing maximization of the bounds over all possible values of r.\nThe set ∇n,ris also an r-dimensional simplex of cardinality |∇n,r|=\u0000n\nr\u0001\n. As before, we are interested in the\nasymptotic regime n→ ∞ ,r=ρn, and we denote by ∇ρthe family of simplices satisfying this relation.\nProposition 31. For fixed 0⩽ρ⩽1,\nCap(∇ρ)≜lim\nn→∞log|∇n,⌊ρn⌋|\nn=H(ρ).\nTo estimate the total ball size, consider the number of pairs (u,v)with L1distance exactly s, denoted by\nN(n1, n2, r, s) =|{(u,v)∈ ∇ n1,r× ∇ n2,r:D(u,v) =s}|, for which the following recursive relation holds.\nLemma 32.\nN(n1, n2, r, s) =X\ni⩾1N(n1−i, n2−i, r−1, s− |n2−n1|)\n+X\ni⩾1X\nj⩾1N(n1−i, n2−i−j, r−1, s− |n1−n2+j|)\n+X\ni⩾1X\nj⩾1N(n1−i−j, n2−i, r−1, s− |n1−n2−j|).\nProof. Letu= (u1, . . . , u r)∈ ∇ n1,randv= (v1, . . . , v r)∈ ∇ n2,r. We consider truncating the last coordinate ur\nandvr. Ifur=n1+ 1−i, vr=n2+ 1−i, we get the first sum where the distance is s−|ur−vr|=s−|n2−n1|.\nOtherwise, suppose ur=n1+ 1−iandvr=n2+ 1−i−j, for i⩾1andj⩾1. Here, the maximum\nvalue in any coordinate of the truncated vectors is n1−iandn2−i−j, respectively, and their distance is\ns− |ur−vr|=s− |n1−n2+j|. Hence, we get the second term. The last term is obtained similarly when\nur=n1−i−jandvr=n2−i. ■\nSince the total ball size is |T(∇n,ρn, δn)|=Pδn\ns=0N(n, n, ρn, s ), the recursive function of total ball size for\n∇n,ris the same as for ∆+\nn,r. Therefore, we have the same generating function and the same asymptotic total ball\nsize. Since the capacity expression for ∇n,ris also the same as for ∆+\nn,r, the resulting GV and GV-MR bounds are\nthe same as well (Theorems 25 and 30).\nIn Figure 2 we plot the GV and GV-MR lower bounds from Theorems 25 and 30, respectively, and compare\nthem against the lower bound on bit-shift error correcting codes given by Kolesnik and Krachkovsky in [11].\nTheorem 33 ([11, Theorem 2]) .For fixed 0⩽δ⩽1/2, we have that α(∇, δ)⩾RKK(∇, δ), where\nRKK(∇, δ)≜1−H(δ).20\nFig. 2: Lower bounds on the highest attainable code rate for the positive simplex and the inverted simplex.\nV. GV B OUND FOR THE HYPERCUBE\nIn this section, we consider the space\nZn\nq=\b\nu= (u1, u2, . . . , u n)∈Zn: 0⩽ui⩽q−1\t\n, (35)\na discrete hypercube of dimension nand cardinality |Zn\nq|=qn. Clearly, Cap(Zq)≜limn→∞1\nnlog|Zn\nq|= log q.\nA. Evaluation of the GV Bound\nTo estimate the ball size, we first consider the number of pairs (u,v)with L1distance exactly s, denoted by\nN(n, s) =|{(u,v)∈Zn\nq×Zn\nq:D(u,v) =s}|.\nLemma 34. The following recursion holds:\nN(n, s) =qN(n−1, s) + 2q−1X\nj=1(q−j)N(n−1, s−j).\nProof. Letu= (u1, . . . , u n)andv= (v1, . . . , v n)∈N(n, s). We consider truncating the last coordinate unand\nvn. Ifun=vn=ifori∈ {0,1, . . . , q −1}, we get the first sum where the distance remains the same. Otherwise,\nun=iandvn=i+jorun=i+jandvn=iforj∈ {1,2, . . . , q −1}andi∈ {0,1, . . . , q −j−1}. Here,\nthe distance decreases by |un−vn|=j. Since there are exactly (q−j)possible pairs with distance j, we get the\nsecond term. ■\nThe generating function of the bivariate sequence N(n, s), namely F(x, y) =P\nn,s⩾0N(n, s)xnys, is given in\nthe following lemma.21\nLemma 35. F(x, y) =1\nH(x,y), where\nH(x, y) = 1−x\u0012\nq+ 2q−1X\nj=1(q−j)yj\u0013\n.\nProof. We have\nF(x, y) =X\nn,s⩾0N(n, s)xnys\n=X\nn,s⩾0qN(n−1, s)xnys+X\nn,s⩾0q−1X\nj=12(q−j)N(n−1, s−j)xnys\n= 1 + xF(x, y)\u0012\nq+ 2q−1X\nj=1(q−j)yj\u0013\n.\n■\nAs before, the quantity of interest is N(n, δn). From Theorem 1, we need to solve the following system of\nequations:.\nH= 0 and xHx=yHy\nδ. (36)\nLemma 36. The solution of the (36) with respect to δis\nx∗(δ) =1\nq+ 2Pq−1\nj=1(q−j)(y∗)j,\nandy∗(δ)is the root of the equation\n2q−1X\nj=1(q−j)(j−δ)yj=qδ . (37)\nProof. From Lemma 35, we have that H= 1−x\u0000\nq+ 2Pq−1\nj=1(q−j)yj\u0001\n. Thus, differentiating Hwith respect to\nxandyand multiplying by xandyrespectively, we have that\nxHx=−x\u0012\nq+ 2q−1X\nj=1(q−j)yj\u0013\nyHy=−x\u0012\n2q−1X\nj=1(q−j)jyj\u0013\n.\nSince H= 1−x\u0000\nq+ 2Pq−1\nj=1(q−j)yj\u0001\n= 0, we have\nx(δ) =1\nq+ 2Pq−1\nj=1(q−j)yj.\nandxHx=−1. Furthermore, since δxH x=yHy, we have\nδ= 2x\u0012q−1X\nj=1(q−j)jyj\u0013\n. (38)\nBy substituting xin (38), we obtain the required expression. ■\nApplying (16), we have that22\nlim\nn→∞logN(n, δn)\nn=−logx∗(δ)−δlogy∗(δ). (39)\nThe total ball size is given by |T(Zn\nq, δn)|=Pδn\ns=0N(n, s). Hence, we have that\neT(Zq, δ) = max\n0⩽δ1⩽δ−logx∗(δ1)−δ1logy∗(δ1)\n=\n\n−logx∗(δ)−δlogy∗(δ), δ⩽δmax,\n2 logq , δ ⩾δmax.\nHere\nδmax=q2−1\n3q, (40)\nwhich is obtained by setting y= 1 in (37). We can now state the GV bound.\nTheorem 37 (GV bound for Hypercube) .For fixed δ⩾0, we have α(Zq, δ)⩾RGV(Zq, δ), where\nRGV(Zq, δ)≜2 logq−eT(Zq, δ).\nB. Evaluation of Marcus and Roth’s Improvement of the GV Bound\nIn this subsection, we introduce an additional parameter prepresenting the number of components with value\n0. In particular, we set Zn\nq(p) ={(u1, u2, . . . , u n)∈Zn: 0⩽ui⩽q−1,|{i:ui= 0}|=p}. Allowing p\nto grow linearly with nwe set p=τn, and we denote by Zq(τ)the corresponding family of spaces. Note that\n|Zn\nq(p)|=\u0000n\np\u0001\n(q−1)(n−p), and hence the capacity Cap(Zq(τ))is given by the following closed formula.\nProposition 38. For fixed 0⩽τ⩽1, we have that\nCap(Zq(τ))≜lim\nn→∞log|Zn\nq(⌊τn⌋)|\nn= (1−τ) log( q−1) +H(τ).\nWe perform the same analysis for total ball size to obtain Marcus and Roth’s improvement. Namely, we first\nconsider the number of pairs (u,v)withL1distance exactly s, denoted by N(n, s, p 1, p2) =|{(u,v)∈Zn\nq(p1)×\nZn\nq(p2) :D(u,v) =s,|{i:ui= 0}|=p1,|{i:vi= 0}|=p2}|.\nLemma 39. The following recursion holds:\nN(n, s, p 1, p2) =N(n−1, s, p 1−1, p2−1) +q−1X\nj=1N(n−1, s−j, p1−1, p2)\n+q−1X\nj=1N(n−1, s−j, p1, p2−1) + ( q−1)N(n−1, s, p 1, p2)\n+ 2q−1X\nj=1(q−1−j)N(n−1, s−j, p1, p2).\nProof. Letu= (u1, . . . , u n)andv= (v1, . . . , v n)∈N(n, s). We consider truncating the last coordinate unand\nvn. Ifun=vn= 0, we get the first term. If un=vn=ifori∈ {1, . . . , q −1}, we get the fourth term where the\ndistance remains the same. If un= 0 andvn=j, orun=jandvn= 0, forj∈ {1,2, . . . , q −1}, the distance\ndecreases by |un−vn|=jand we get the second and the third term, respectively. Finally, consider un=iand\nvn=i+j, orun=i+jandvn=i, forj∈ {1,2, . . . , q −1}andi∈ {1,2, . . . , q −j−1}. In these cases, the23\ndistance decreases by |un−vn|=j. Since there are exactly (q−1−j)such pairs with distance j, we get the fifth\nterm. ■\nAs before, the quantity of interest is N(n, s, p, p ). Therefore, we define the generating function F(x, y, w ) =P\nn,s,p⩾0N(n, s, p, p )xnysw2pwith three variables (see Proposition 2).\nLemma 40. F(x, y, w ) =1\nH(x,y,w ), where\nH(x, y) = 1−x\u0012\nw2+ (q−1) + 2 wq−1X\nj=1yj+ 2q−1X\nj=1(q−1−j)yj\u0013\n.\nProof.\nF(x, y, w ) =X\nn,s,p⩾0N(n, s, p, p )xnyswp\n=X\nn,s,p⩾0N(n−1, s, p−1, p−1)xnyswp+X\nn,s,p⩾0q−1X\nj=1N(n−1, s−j, p−1, p)xnyswp\n+X\nn,s,p⩾0q−1X\nj=1N(n−1, s−j, p, p−1)xnyswp+X\nn,s,p⩾0(q−1)N(n−1, s, p, p )xnyswp\n+X\nn,s,p⩾02q−1X\nj=1(q−1−j)N(n−1, s−j, p, p )xnyswp\n= 1 + xF(x, y)\u0012\nw2+ (q−1) + 2 wq−1X\nj=1yj+ 2q−1X\nj=1(q−1−j)yj\u0013\n.\n■\nHere, we are interested in N(n,⌊δn⌋,⌊τn⌋,⌊τn⌋). From Theorem 1 and Proposition 2, we need to solve the\nfollowing system of equations.\nH= 0 and xHx=yHy\nδ=wHw\n2τ. (41)\nLemma 41. The solution of the (41) with respect to δis\nx∗(δ, τ) =1\nw2+q−1 + 2 wPq−1\nj=1yj+ 2Pq−1\nj=1(q−1−j)yj,\nw∗(δ, τ) =δ(q−1) + 2Pq−1\nj=1(δ−j(1−τ))(q−1−j)yj\nPq−1\nj=1(2j(1−τ)−δ)yj.\nandy∗(δ, τ)is the root of the equation\nδw2+ 2wq−1X\nj=1(δ−j)yj+δ(q−1) + 2q−1X\nj=1(q−1−j)(δ−j)yj= 0. (42)\nProof. Appendix E. ■\nApplying (16), we have that\nlim\nn→∞logN(n,⌊δn⌋,⌊τn⌋,⌊τn⌋)\nn=−logx∗(δ, τ)−δlogy∗(δ, τ)−2τlogw∗(δ, τ). (43)24\nThe total ball size is |T(Zn\nq(⌊τn⌋),⌊δn⌋)|=P⌊δn⌋\ns=0N(n, s,⌊τn⌋,⌊τn⌋). Hence, we have that\neT(Zq(τ), δ) = max\n0⩽δ1⩽δ−logx∗(δ1, τ)−δ1logy∗(δ1, τ)−2τlogw∗(δ1, τ)\n=\n\n−logx∗(δ, τ)−δlogy∗(δ, τ)−2τlogw∗(δ, τ), δ⩽δmax\n2Cap( Zq(τ)), δ ⩾δmax\nwhere\nδmax=q(1−τ)\u0012\n1−(1−τ)(2q−1)\n3(q−1)\u0013\n. (44)\nWe have obtained δmaxby setting y= 1 in (42). We can now state the GV-MR bound.\nTheorem 42 (GV-MR bound for Hypercube) .For fixed δ⩾0, we have α(Zq, δ)⩾RMR(Zq, δ), where\nRMR(Zq, δ)≜2Cap( Zq(τ))−eT(Zq(τ), δ)\nandτis the solution of the equation\nτ(q−1)2−(1−τ)(q−1)−(q−1)q−1X\nj=1(y∗)j+ 2(1−τ)q−1X\nj=1j(y∗)j= 0. (45)\nProof. Appendix E. ■\nThe bound obtained in Theorem 42 is positive for δ < δ max, where\nδmax=3q(q−1)\n4(2q−1). (46)\nThis is seen by setting y= 1 in (45), which gives τopt= (q+ 1)/(4q−2), and then substituting τoptin (44).\nWe note that the standard GV bound (Theorem 37) is positive for δ <q2−1\n3q(see (40)). It is easy to show that the\nexpression in (46) is strictly larger thanq2−1\n3qfor every q >2.\nC. Numerical Results\nIn Figure 3, we plot the bounds obtained in Theorems 37 and 42 for quaternary alphabet ( q= 4). We note that, for\nthe GV bound, δmax= (q2−1)/3q= 5/4(see (40)), while for the GV-MR bound, δmax= 3q(q−1)/4(2q−1) = 9 /7\n(see (46)). For comparison purposes, we also plot the GV lower bound for codes in the related Lee distance .\nLower Bound using Lee distance: A simple upper bound on the total ball size |T(Zn\nq, d)|can be obtained by\nupper-bounding the volume of an L1-sphere around uby the volume of the Lee-sphere around u. Namely, the Lee\nmetric is a “modular version” of the L1metric, and a sphere with respect to the former always contains the sphere\nof the same radius with respect to the latter. Furthermore, the size of a Lee-sphere is independent of its center,\nand deriving the corresponding GV bound is straightforward. Here we give an example for q= 4 that is used in\nFigure 3.\nTheorem 43 (Lower Bound using Lee-metric) .For fixed 0⩽δ⩽1, we have\nα(Z4, δ)⩾(2−δ) log(2 −δ) +δlogδ.\nProof. The GV bound for Lee-metric codes in Zn\nqis well-known. Its asymptotic form for q= 4, namely (2−\nδ) log(2 −δ)+δlogδ, was given in [3, Theorem 6]. By what was said above, the highest attainable rate of L1-metric\ncodes in Zn\n4must also be lower-bounded by this function. ■25\nFig. 3: Lower bounds on the highest attainable code rate α(Z4, δ)for the family of hypercubes {Zn\n4}n.\nVI. C ONCLUSION\nObtaining nontrivial lower bounds on the rates of optimal codes in nonuniform spaces is generally a difficult\nproblem. 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Bose, “On efficient repetition error correcting codes,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT) ,\npp. 1012–1016, Austin, TX, USA, June 2010.\n[30] R. R. Varshamov, “Estimate of the number of signals in error correcting codes,” Doklady Akad. Nauk, SSSR , vol. 117, pp. 739–741,\n1957.\n[31] P. I. Vasil’ev, “On (d, k)-constrained bitshift correcting block codes,” in Proc. 2nd Int. Workshop on Algebraic and Combinatorial\nCoding Theory , pp. 215–219, 1990.28\nAPPENDIX A\nPROOF OF LEMMA 8\nProof. We need to find the positive solution of\nH= 0\nxHx\n2=yHy\nρ=zHz\nδ.(47)\nwhere H= (1−x2)(1−xz)−y(1 +xz).\nTake the partial derivatives as\nxHx\n2=−\u0000\n2x2(1−xz) +xz(1−x2)\u0001\n−yxz\n2, (i)\nzHz\nδ=−xz(1−x2)−yxz\nδ, (ii)\nyHy\nρ=−y(1 +xz)\nρ. (iii)\nSince H= 0, we substitute ywith\ny=(1−x2)(1−xz)\n(1 +xz). (48)\nTherefore, (i)–(iii) simplifies to\nxHx\n2=−(1 +xz)(2x2+xz−3x3z)−(1−x2)(1−xz)(xz)\n2(1 + xz)\n=−1\n2(1 + xz)h\n(2x2+xz−3x3z)(1 + xz)−(1−x2)(x2z2)i\n=−1\n1 +xzh\nx2(1−x2z2) +xz(1−x2)i\n, (iv)\nzHz\nδ=−xz(1−x2)\nδ−xz(1−x2)(1−xz)\nδ(1 +xz)\n=−2xz(1−x2)\nδ(1 +xz), (v)\nyHy=−(1−x2)(1−x2z2)\nρ(1 +xz). (vi)\nBy equating (v) and (vi), we have\n2xz(1−x2)\nδ=(1−x2z2)(1−x2)\nρ.\nSince xis positive, we have\n2xz\nδ=1−x2z2\nρ. (49)\nFrom (49), we have xz=−ρ+√ρ2+δ2\nδ. Lastly, we equate (iv) and (vi) to get\nx2(1−x2z2) +xz(1−x2) =(1−x2)(1−x2z2)\nρ.\nRecall from (49) that,\n1−x2z2=2ρ\nδxz.29\nTherefore, we have\n2ρ\nδ(xz)(x2) +xz(1−x2) =2xz(1−x2)\nδ.\nSolving this equation we get\nx=r\n1−2ρ\n2 + 2 ρ−δ. (50)\nTherefore, Lemma 8 results from (48), (49), and (50). ■\nWhen δ⩽δmax,eT(∆ρ, δ) =−2 logx−ρlogy−δlogz, where\nx=r\n1−2ρ\n2 + 2 ρ−δ, y = 2p\nρ2+δ2−δ\n2 + 2 ρ−δ, z =p\nρ2+δ2−ρ\nxδ.\nNote that δ=δmaxcorresponds to z∗= 1. Therefore, we have that\nδ2(2−δ) = (2 + 2 ρ−δ)\u0000p\nρ2+δ2−ρ\u00012.\nAsρ⩾0,0⩽δ⩽2, the unique solution is\nδmax=2(1 + ρ)\n2 +ρ. (51)\nAPPENDIX B\nPROOF OF LEMMA 14AND THEOREM 16\nProof of Lemma 14. We need to find the positive solution of\nH= 0\nxHx\n2=yHy\nρ=zHz\nδ=wHw\n2τ(52)\nwhere H= (1−x2)(1−xz)−y\u0000\nw2(1−x2)(1−xz) + 2wxz(1−x2) +x2(1 +xz)\u0001\n.\nComputing the partial derivatives gives\nH= (1−x2)(1−xz)−y\u0000\nw2(1−x2)(1−xz) + 2wxz (1−x2) +x2(1 +xz)\u0001\n, (i)\nxHx\n2=−2x2(1−xz)−xz(1−x2)−y\u0000\nw2((1−x2)(−xz)−2x2(1−xz)) + 2 w(xz(1−x2)−2x3z) + 2x2(1 +xz) +x3z\u0001\n2,(ii)\nzHz\nδ=−xz(1−x2)−y\u0000\n−w2xz(1−x2) + 2wxz (1−x2) +x3z\u0001\nδ, (iii)\nyHy\nρ=−y\u0000\nw2(1−x2)(1−xz) + 2wxz (1−x2) +x2(1 +xz)\u0001\nρ(iv)\nwH w\n2τ=−y\u0000\nw2(1−x2)(1−xz) +wxz (1−x2)\u0001\nτ(v)\nSetλ1=xz,λ2=x2, which transforms the set of equations to\nxHx\n2=−2λ2(1−λ1)−λ1(1−λ2)−y\u0000\nλ2((1−λ2)(−λ1)−2λ2(1−λ1)) + 2 w(λ1(1−λ2)−2λ1λ2) + 2λ2(1 +λ1) +λ1λ2\u0001\n2,\n(vi)\nzHz\nδ=−λ1(1−λ2)−y\u0000\n−w2λ1(1−λ2) + 2wλ1(1−λ2) +λ1λ2\u0001\nδ, (vii)\nyHy\nρ=−y\u0000\nw2(1−λ2)(1−λ1) + 2wλ1(1−λ2) +λ2(1 +λ1)\u0001\nρ, (viii)\nwH w\n2τ=−y\u0000\nw2(1−λ2)(1−λ1) +wλ1(1−λ2)\u0001\nτ. (ix)30\nThen, since H= 0, we substitute ywith\ny=(1−λ1)(1−λ2)\nw2(1−λ1)(1−λ2) + 2wλ1(1−λ2) +λ2(1 +λ1). (53)\nBy equating (vii), (viii) and (ix), we have\nλ2= 1−(1 +λ1)2(ρ−τ)−δ(1−λ1)\n2−δ(54)\nand\nw=λ2(δ(1−λ2\n1)−2λ1(ρ−τ))\nλ1(1−λ2)(2(ρ−τ)−δ(1−λ1)). (55)\nBy equating (vi), (vii) and substituting y, λ2andw, we have that λ∗\n1is the root of the equation\nδ2(1 +ρ)λ5\n1+ (−δ2(1 +τ) + 4δ(ρ−τ)(1 + ρ))λ4\n1+ (−δ2(ρ+ 2) + 2 δ(ρ−τ)(ρ−τ−2) + 4( ρ−τ)2(1 +ρ))λ3\n1\n+ (−δ2(ρ−2τ−2)−2δ(ρ−τ)(ρ+τ+ 2)−4(ρ−τ)2(1−ρ))λ2\n1+ (δ2−4δ(ρ−τ)(ρ−τ−1))λ1+δ2(ρ−τ−1) = 0 .\n(56)\nTherefore, Lemma 14 results from (53), (54), (55), and (56). ■\nProof of Theorem 16. Note that we already have the generating function\nF(x1, y1, w1) =X\nn⩾0|∆n,r(p)|xn\n1yr\n1wp\n1=1−x1\n(1−x1)−y1(w1(1−x1) +x1)\nthat counts the number of words in ∆n,r(p). Further, we have the unique solution w∗\n1=τ(1+τ−ρ)\n(ρ−τ)2from Theorem 1.\nThus, we apply Corollary 4 to obtain τopt, that is w∗\n1=w∗.\nTherefore, we have that\nρ(1 +τ−ρ)\n(ρ−τ)2=λ2(δ(1−λ2\n1)−2λ1(ρ−τ))\nλ1(1−λ2)(2(ρ−τ)−δ(1−λ1)). (57)\nWe substitute λ2from (54) in (57) and solve it together with (56) to get,\nτopt=ρ2\n1 +ρ−λ∗\n1,\nδ=2ρλ∗\n1\n1 +ρλ∗\n1−(λ∗\n1)2.\nNote that δ=δmax corresponds to z∗= 1, which implies that λ∗\n2= (λ∗\n1)2. Thus from (54), we have that\n1−(1 +λ∗\n1)2(ρ−τ)−δ(1−λ∗\n1)\n2−δ= (λ∗\n1)2, solving which we get\nτ=ρ+λ∗\n1−1.\nEquating it to τopt, we have that λ∗\n1= 1 and therefore δmax= 2. ■\nAPPENDIX C\nPROOF OF THEOREM 25\nIt is easy to show that F(x1, y1) =P\nn⩾0|∆+\nn,r|xn\n1yr\n1=1−x1\n1−x1−x1y1. Further, we have the unique solution\ny∗\n1=ρ\n1−ρfrom Theorem 1. Thus, we apply Corollary 4 to obtain ρopt, that is (y∗\n1)2=y∗, where y∗= 2√ρ2+δ2−δ\n2−δ−2ρ31\nfrom Lemma 22. Therefore, we have that\nρ2\n(1−ρ)2= 2p\nρ2+δ2−δ\n2−δ−2ρ.\nWe solve this equation to get\nρ=3(2−δ)−√\n9δ2−4δ+ 4\n8.\nAPPENDIX D\nPROOF OF THEOREM 29AND THEOREM 30\nProof of Theorem 29. The following generating function is easy to derive\nF(x1, y1, w1) =X\nn⩾0|∆+\nn,r(p)|xn\n1yr\n1wp\n1=1−x1\n1−x1−x1y1(w1(1−x1) +x1). (58)\nFurthermore, we have the unique solution w∗\n1=τ(1+τ−2ρ)\n(ρ−τ)2from Theorem 1. Thus, we apply Corollary 4 to obtain\nτopt, that is w∗\n1=w∗.\nNote that from Lemma 28, we have\nw∗(δ) =λ∗\n2(δ(1−(λ∗\n1)2)−2λ∗\n1(ρ−τ))\nλ∗\n1(1−λ∗\n2)(2(ρ−τ)−δ(1−λ∗\n1)), (i)\ny∗(δ) =(1−λ∗\n1)(1−λ∗\n2)\nλ∗\n2((w∗)2(1−λ∗\n1)(1−λ∗\n2) + 2w∗λ∗\n1(1−λ∗\n2) +λ∗\n2(1 +λ∗\n1)), (ii)\nλ∗\n2(δ) = 1−(1 +λ∗\n1)2(ρ−τ)−δ(1−λ∗\n1)\n(2(1−ρ)−δ)(iii)\nand\n(1−λ1)(δ(1−λ2\n1)−2λ1(ρ−τ))2(2(1−ρ)−2(ρ−τ)(1 + λ1)−δλ2\n1)\n+λ2\n1(1 +λ1)(2(ρ−τ)−δ(1−λ1))2(δ(1−λ2\n1)−2λ1(ρ−τ)−2τ) = 0 . (iv)\nTherefore, we have\nτ(1 +τ−2ρ)\n(ρ−τ)2=λ∗\n2(δ(1−(λ∗\n1)2)−2λ∗\n1(ρ−τ))\nλ∗\n1(1−λ∗\n2)(2(ρ−τ)−δ(1−λ∗\n1)). (59)\nWe substitute λ∗\n2from (iii) in (59) and then solve it together with (iv) to get,\nτopt=ρ2\n1−λ∗\n1(1−ρ), (v)\nδ(λ∗\n1) =2λ∗\n1ρ(1−ρ)\nρλ∗\n1+ (1−ρ)(1−(λ∗\n1)2). (vi)\nNote that we have δ=δmaxcorresponds to z∗= 1, which implies that λ∗\n2= (λ∗\n1)2. Thus from (iii), we have that\n1−(1 +λ∗\n1)2(ρ−τ)−δ(1−λ∗\n1)\n2(1−ρ)−δ= (λ∗\n1)2. Solving this, we get\nτ=ρ−(1−λ∗\n1)(1−ρ).\nand thus equating it by τopt, we have that λ∗\n1= 1 and therefore δmax= 2. ■\nProof of Theorem 30. We need to optimize the result obtained in Theorem 29 using Theorem 1 and Corollary 4.\nRecall the generating function (58). We have the unique solution y∗\n1=(1−ρ)(ρ−τ)2\nρ(1+τ−2ρ)2from Theorem 1. Thus, we32\napply Corollary 4 to obtain ρopt, that is y∗\n1=y∗. We have\n(1−ρ)(ρ−τ)2\nρ(1 +τ−2ρ)2=(1−λ∗\n1)(1−λ∗\n2)\nλ∗\n2((w∗)2(1−λ∗\n1)(1−λ∗\n2) + 2w∗λ∗\n1(1−λ∗\n2) +λ∗\n2(1 +λ∗\n1)). (60)\nNow, we first substitute λ∗\n2from (iii) in (60) and then we solve it by substituting δandτfrom (v) and (vi) to get,\nρopt=2p\n1−λ∗\n1\n3p\n1−λ∗\n1+p\n1 + 3 λ∗\n1. (vi)\n■\nAPPENDIX E\nPROOF OF LEMMA 41AND THEOREM 42\nProof of Lemma 41. We need to find the positive solution of\nH= 0\nxHx=yHy\nδ=wHw\n2τ(61)\nwhere H= 1−x\u0010\nw2+ (q−1) + 2 wPq−1\nj=1yj+ 2Pq−1\nj=1(q−1−j)yj\u0011\n.\nTake the partial derivatives as\nxHx=−x\u0012\nw2+ (q−1) + 2 wq−1X\nj=1yj+ 2q−1X\nj=1(q−1−j)yj\u0013\n=−1, (i)\nyHy\nδ=−x\nδ\u0012\n2wq−1X\nj=1jyj+ 2q−1X\nj=1(q−1−j)jyj\u0013\n, (ii)\nwHw\n2τ=−x\nτ\u0012\nw2+wq−1X\nj=1yj\u0013\n. (iii)\nSince H= 0, we substitute xwith\nx=1\nw2+ (q−1) + 2 wPq−1\nj=1yj+ 2Pq−1\nj=1(q−1−j)yj. (62)\nBy equating (i), (ii) and (iii), we have\n2wq−1X\nj=1jyj+ 2q−1X\nj=1(q−1−j)jyj=δ\u0012\nw2+ (q−1) + 2 wq−1X\nj=1yj+ 2q−1X\nj=1(q−1−j)yj\u0013\n,\nw2+wq−1X\nj=1yj=τ\u0012\nw2+ (q−1) + 2 wq−1X\nj=1yj+ 2q−1X\nj=1(q−1−j)yj\u0013\n,\nwhich simplifies to\nδw2+ 2wq−1X\nj=1(δ−j)yj+δ(q−1) + 2q−1X\nj=1(q−1−j)(δ−j)yj= 0, (iv)\n(1−τ)w2+ (1−2τ)wq−1X\nj=1yj−τ(q−1)−2τq−1X\nj=1(q−1−j)yj= 0. (v)33\nSolving (iv) and (v) together, we have\nw=δ(q−1) + 2Pq−1\nj=1(q−1−j)(δ−j(1−τ))yj\nPq−1\nj=1(2j(1−τ)−δ)yj. (63)\nTherefore, Lemma 41 results from (iv), (62) and (63). ■\nProof of Theorem 42. It is easy to obtain the generating function that counts the number of words in Zn\nq(p), namely\nF(x1, w1) =X\nn⩾0|Zn\nq(p)|xn\n1wp\n1=1\n1−x1(q−1 +w1).\nFurther, we have the unique solution w∗\n1=τ(q−1)\n1−τfrom Theorem 1. Thus, we apply Corollary 4 to obtain τopt, that\nisw∗\n1=w∗. We replace w∗byw∗\n1in (v) to get,\nτ(q−1)2−(1−τ)(q−1)−(q−1)q−1X\nj=1(y∗)j+ 2(1−τ)q−1X\nj=1j(y∗)j= 0. (64)\n■"    },    {        "title": "2402.14719v1.The_cosmological_constant_and_scale_hierarchies_with_emergent_gauge_symmetries.pdf",        "content": "arXiv:2402.14719v1  [hep-ph]  22 Feb 2024rsta.royalsocietypublishing.org\nResearch\nArticle submitted to journal\nSubject Areas:\nParticle physics, gauge symmetries,\ncosmology\nKeywords:\nCosmological constant, Emergent\ngauge symmetry, Hierarchy puzzle,\nHiggs boson, Dark matter\nAuthor for correspondence:\nSteven D. Bass\ne-mail: Steven.Bass@cern.chThe cosmological constant\nand scale hierarchies with\nemergent gauge symmetries\nSteven D. Bass1,2\n1Kitzbühel Centre for Physics, Kitzbühel Austria\n2Marian Smoluchowski Institute of Physics,\nJagiellonian University, Kraków, Poland\nMotivated by the stability of the electroweak Higgs\nvacuum we consider the possibility that the Standard\nModel might work up to large scales between about\n1010GeV and close to the Planck scale. A plausible\nscenario is an emergent Standard Model with gauge\nsymmetries originating in some topological like phase\ntransition deep in the ultraviolet. In this case the\ncosmological constant scale and neutrino masses\nshould be of similar size, suppressed by factor of the\nlarge scale of emergence. The key physics involves\na subtle interplay of Poincaré invariance, mass\ngeneration and renormalisation group invariance. The\nHiggs mass would be environmentally selected in\nconnection with vacuum stability. Consequences for\ndark matter scenarios are discussed.\n© The Authors. Published by the Royal Society under the terms of the\nCreative Commons Attribution License http://creativecom mons.org/licenses/\nby/4.0/, which permits unrestricted use, provided the orig inal author and\nsource are credited.2rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1. Introduction\nOur observations in particle physics and gravitation are de scribed by the Standard Model and\nGeneral Relativity. The Standard Model [ 1,2] works very well so far in all experiments from LHC\ncollider physics [ 3–6] to precision measurements of, e.g., the fine structure cons tant [ 7] and the\nelectron electric dipole moment [ 8] with so far no evidence for new particles or interactions in\nthe range of present experiments. The Standard Model is buil t on the local gauge symmetries\nof electroweak interactions and quantum chromodynamics, Q CD, with the gauge groups U(1) Y,\nSU(2)Land SU(3) c. Beyond the spin1\n2leptons and quarks plus spin-one gauge bosons, the Higgs\nboson with spin zero and mass 125 GeV discovered at CERN behav es very Standard Model like\nand completes the particle spectrum of the Standard Model [ 4,9].\nGeneral Relativity describes gravitation wherever it has b een tested with recent highlights\nincluding the discovery of gravitational waves [ 10] and black hole imaging [ 11]. In the laboratory\nNewton’s law works down to at least 52 µm [12]. General Relativity is also a gauge theory, under\nlocal transformations of the co-ordinate system [ 13,14].\nYet we know this is not the complete story. Some extra new phys ics is needed to understand\nissues with neutrinos (their masses, whether they might be D irac or Majorana particles, possible\nCP phases), baryogenesis (why there is more matter than anti matter), the absence of strong CP\nviolation, fermion families, dark energy (the acceleratin g expansion of the Universe) and dark\nmatter as well as primordial inflation.\nHow high in energy might the Standard Model work before one ru ns into new physics? The\nenergy scale of any new interactions is so far not known. Wher e do the gauge symmetries come\nfrom at a deeper level? What about the dynamics behind electr oweak symmetry breaking in the\nStandard Model? How should we understand the scale hierarch ies of particle physics with the\ncosmological constant scale 0.002 eV very much less than the electroweak scale 246 GeV which is\nitself very much less than the Planck scale MPl=1.2×1019GeV where quantum gravity effects\nmight apply?\nToday a vigorous programme of experiments and theory is prob ing the high energy and\nprecision frontiers together with cosmology looking for cr acks in our description of Nature\nprovided by the Standard Model and General Relativity. Idea s for new physics are discussed\nin the parallel theoretical contributions to this volume [ 15–17] with experiments discussed in\nRefs. [ 9,18–22].\nWhile so far not revealing evidence for new particles and int eractions, LHC data do suggest\nan intriguing result with stability of the Standard Model va cuum. If we assume no coupling to\nnew particles at higher energies and extrapolate the Standa rd Model into the deep ultraviolet\nusing renormalisation group, RG, evolution, then the vacuu m remains stable up to very high\nscales - at least 1010GeV and perhaps up to the Planck scale with vacuum stability v ery sensitive\nto the exact values of Standard Model parameters [ 23,24]. Might the Standard Model be more\nspecial than previously assumed? Here we consider this poss ibility. In particular, we discuss the\nidea that the particles and interactions of the Standard Mod el might be emergent below the scale\ncharacterising some topological like phase transition dee p in the ultraviolet [ 23,25–28]. In this\nscenario the Higgs mass is environmentally selected, conne cted to the stability of the vacuum,\nwith subtle connection between the infrared world of our exp eriments and the physics in the\nextreme ultraviolet. Interestingly, the cosmological con stant scale comes out similar to the size\nof tiny Majorana neutrino masses [ 29,30]. New global symmetry breaking interactions can occur\nin higher dimensional operators, suppressed by powers of th e large scale of emergence - in this\npaper taken about 1016GeV - which may have been active in the very early Universe and perhaps\nrelevant to cosmology. In the absence of larger multiplets a t mass dimension four, dark matter\nmight involve either primordial black holes or new physics c oupled through higher dimensional\noperators with small non-gravitational coupling to normal baryonic matter.\nThe plan of this paper is as follows. In Section 2we briefly describe ideas about the vacuum\nstability of the Standard Model. Section 3describes the phenomenology of a possible emergent3rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Standard Model including the cosmological constant and dar k matter scenarios. Sections 4and\n5are of more theoretical interest. Section 4discusses separate contributions to the cosmological\nconstant and how they might fit together to give a small value o f the cosmological constant.\nRelated issues with hierarchies of scales in particle physi cs including the Higgs mass are described\nin Section 5. Finally, in Section 6we summarise and conclude.\n2. Vacuum stability\nIf the Standard Model is evolved using RG up towards the Planc k scale assuming no coupling\nto extra particles, then one finds that the parameters measur ed in experiments and its ultraviolet\nbehaviour are strongly correlated.\nThe Standard Model couplings and particle masses are relate d. For the W and Z gauge bosons\nm2\nW=1\n4g2v2, m2\nZ=1\n4(g2+g′2)v2(2.1)\nwheregandg′are the SU(2) and U(1) electroweak couplings; vis the Higgs vacuum expectation\nvalue 246 GeV . The charged fermion masses are\nmf=yfv√\n2(f=quarksand charged leptons) (2.2)\nwhereyfare the Yukawa couplings. The Higgs mass is\nm2\nh=2λv2(2.3)\nwithλthe Higgs self-coupling.\nThe masses of the W and Z and also the quarks and charged lepton s come from the Brout-\nEnglert-Higgs, BEH, mechanism [ 31–36]. These particles come with gauge invariant coupling\nto a complex Higgs doublet field φwith potential V(φ)=1\n2m2φ∗φ+1\n4λ(φ∗φ)2andm2<0.\nThe Higgs field acquires a vacuum expectation value |φ|=v=/radicalbig\n−m2/λ=mh/√\n2λat the\nminimum of the potential implying spontaneous symmetry bre aking phenomena. Working in\nunitary gauge, of the four Higgs doublet components three Go ldstone states are absorbed as the\nlongitudinal components of the W and Z with the remaining sca lar degree of freedom being the\nmassive Higgs boson. Here spontaneous symmetry breaking is defined relative to the choice of\ngauge, e.g. the unitary gauge, with all gauge choices being p hysically equivalent [ 36].\nThe Standard Model relations Eqs. ( 2.1,2.2) have been tested to good precision about 10% for\nthe W and Z gauge bosons, the heavy top and bottom quarks, and t heτandµcharged leptons.\nThe measured coupling strengths scale as a function of the pa rticle masses just as predicted\nby the Standard Model – see [ 4,9]. For RG evolution one also needs input on the Higgs self-\ncoupling λ. In the absence of direct measurement, here we have to assume the Standard Model\nrelation connecting λto the Higgs mass, Eq. ( 2.3). Taking v=246 GeV and mh=125 GeV gives\nλ=m2\nh/2v2≈0.13as input at LHC laboratory scales. Measurement of λawaits future collider\nexperiments with precision about 50% expected from the high luminosity upgrade of the LHC\nand 5% precision requiring a new 100 TeV higher energy collid er [9].\nBesides ensuring gauge invariance with the massive W and Z ga uge bosons, the BEH\nmechanism with the measured Higgs mass also gives good high e nergy behaviour with\nperturbative unitarity [ 37–40] in scattering of the massive W and Z gauge bosons and a\nrenormalisable Standard Model [ 41–43]. Given the Standard Model couplings we can extrapolate\nthe theory up to the Planck scale using perturbative RG evolu tion. If we assume no coupling\nto undiscovered new particles, then it remains finite and wel l behaved with no Landau pole\nsingularities below the Planck scale. That is, the Standard Model is mathematically consistent\nup to the Planck scale.\nFurther, the Standard Model Higgs vacuum comes out very clos e to the border of stable\nand metastable, within about 1.3 σof being stable [ 24] – see also [ 23,44,45]. Vacuum stability\nis connected with the RG running of the Higgs self-coupling λ, with a stable vacuum for positive\ndefiniteλ. If this coupling crosses zero deep in the ultraviolet it can lead to vacuum metastability.4rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .410610810101012101410161018102010\n[GeV]µ00.20.40.60.81g'\ng\nsg\nty\nλ\n3104105106107108109101010\n[GeV]µ050010001500200025003000350040004500[GeV]Wm\nZm\nhm\ntm\nvev\nFigure 1. Left: Running of the Standard Model gauge couplings g,g′,gsfor the electroweak SU(2) and U(1) and colour\nSU(3), the top quark Yukawa coupling ytand Higgs self-coupling λ. (From left, the points describe the evolution of gs,\nyt,g,g′,λin descending order.) Right: Running MSmasses and the Higgs vev in the Standard Model. Uncertaintie s\nare calculated by varying all PDG values up and down by their r espective uncertainties. (In the printed black and white\nversion, the points from top describe the evolution of v,mt,mZ,mW,mh.) Figure taken from [ 30].\nWhether this happens is very sensitive to the values of the Hi ggs boson and top quark masses\nand to details of calculations of higher order radiative cor rections. Calculations are performed\nwith the Standard Model evolved up to the Planck scale with th e measured masses and couplings\nas input and using 3 loop RG, 2 loop matching plus pure QCD corr ections evaluated to 4 loops. In\ngeneral, a higher top quark mass tends to reduce λdeep in the ultraviolet whereas a larger Higgs\nmass tends to increase it. Sensitivity to QCD corrections in volvingαsmeans sensitivity also to\nthe numbers of colours and active flavours and the QCD scale Λqcd. Both electroweak and QCD\nphysics thus enter the vacuum stability calculations.\nThe Standard Model running couplings are shown in Fig. 1which displays calculations\n[30] performed using the C ++RG evolution package [ 46]. The QCD and electroweak SU(2)\ncouplings are asymptotically free, decaying logarithmica lly with increasing resolution, and the\nU(1) coupling is non-asymptotically free rising in the ultr aviolet. These couplings almost meet\nin the ultraviolet but don’t quite. The top quark Yukawa coup ling decreases with increasing\nresolution. With the measured top quark mass mtand QCD coupling αs, the Standard Model\nneeds a Higgs mass mhbigger than about 125 GeV to ensure vacuum stability, making the Higgs\nparticle discovered at CERN especially interesting. (The H iggs self coupling would generate a\nLandau pole singularity below MPlif the Higgs mass were about 30% larger with the measured\nvalue of mt, placing a perturbative upper bound on the possible Higgs ma ss [47].) To illustrate\nsensitivity to the top quark mass, if we set mh=125 GeV and vary the top quark mass in these\ncalculations, then the Higgs self-coupling λcrosses zero around 1010GeV with mt=173 GeV\n1and remains positive definite up to MPlwithmt=171 GeV . The recent and most accurate\nmeasurement of mtfrom CMS Run-2 data collected at√s=13 TeV is171.77±0.37GeV [ 48].\nTheoretical issues connecting the top quark Yukawa couplin g and the measured mass are\ndiscussed in [ 49].\nIn parallel to the running couplings, for illustration the r ight panel of Fig. 1shows the\nRG behaviour of the Standard Model MS masses and the Higgs vacuum expectation value v.\nThese are related to the running couplings through Eqs. ( 2.1)–(2.3). The Higgs mass mhevolves\nsmoothly with increasing scale µ. In Fig. 1the divergence of vatµ≈1010GeV corresponds to the\nscenario where λcrosses zero at this scale. With a stable vacuum λremains positive and vis finite\nwithout divergence.\nSummarising, if we assume that there are no couplings to extr a particles at higher energies,\nthen the Standard Model revealed by current experiments is s trongly correlated with its\n1Vacuum metastability would come with a new minimum in the eff ective Higgs potential not far below the Planck scale and\na vacuum lifetime ∼10600years, very much bigger than the present age of the Universe.5rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .behaviour in the extreme ultraviolet. This may be telling us something deeper about the origin of\nthe Standard Model. It is important to emphasise the large ex trapolations in these calculations\nwhen evolving the Standard Model couplings up to the Planck s cale. The existence of new\nphysics, even at the largest scales, can affect the vacuum st ability [ 50]. Modulo this caveat, within\nthe uncertainties on the top mass and radiative corrections , vacuum stability is certainly not\nexcluded up to the energies where Grand Unified Theories are s ometimes conjectured to apply\nor even up to the Planck scale [ 23,24]. The parameters of the Standard Model might thus be\nlinked to physics at these high scales. In particular, the Hi ggs mass might be environmentally\ndetermined, linked to the stability of the electroweak vacu um. The Higgs vacuum sitting \"close\nto the edge\" of stable and metastable may be pointing to possi ble new critical phenomena in\nthe ultraviolet [ 23,44,45]. There are ideas that near-criticality might act as an attr actor point in\nevolution of the higher energy phase, with analogous system s in Nature discussed in [ 45]. With\njust small changes in the Standard Model parameters the emer ging low energy theory would be\nvery different from the Standard Model.\nIf the Standard Model might really work up to these large scal es, then how might we\nunderstand the origin of gauge symmetries and the new physic s needed to the explain the open\npuzzles at the interface of particle physics and cosmology?\n3. Emergent gauge symmetries\nGauge symmetries can be \"emergent\" below a scale deep in the u ltraviolet.\nEmergence in physics occurs when a many-body system exhibit s collective behaviour in the\ninfrared that is qualitatively different from that of its mo re primordial constituents as probed\nin the ultraviolet [ 51,52]. For example, hadrons are emergent from quark gluon degree s of\nfreedom. As an everyday example of emergent symmetry, consi der a carpet which looks flat\nand translational invariant when looked at from a distance. Up close, e.g. as perceived by an ant\ncrawling on it, the carpet has structure and this translatio nal invariance is lost. The symmetry\nperceived in the infrared, e.g. by someone looking at it from a distance, “dissolves” in the\nultraviolet when the carpet is observed close up.\nEmergent gauge symmetries are seen in quantum many-body sys tems where they are\nassociated with topological phase transitions, e.g., phas e transitions that occur without an order\nparameter [ 53–59]. Here, the prototype is the Fermi-Hubbard model of strongl y correlated\nelectrons in an atomic lattice which exhibits spin-charge s eparation with the SU(2) spin becoming\ndynamical with an emergent gauge symmetry beyond the more fu ndamental QED [ 53,54]. A\nfurther example is the A-phase of superfluid3He [ 55,56]. Close to a Fermi point one finds\nemergent local gauge interactions with spin becoming dynam ical to internal observers. The quasi-\nparticles include gauge bosons and also emergent chiral fer mions, each with common limiting\nvelocity like with Lorentz invariance in the Standard Model , and an emergent metric as well as\nan analogue of the chiral anomaly. Emergent gauge symmetrie s are also believed to be important\nin high temperature superconductors [ 57].\nHow might emergent gauge symmetries work in particle physic s and might the Standard\nModel be emergent?\nConsider a statistical system near its critical point. The l ong range asymptote is a\nrenormalisable quantum field theory with properties descri bed by the renormalisation group\n[60,61]. If the spectrum includes J=1 excitations among the degree s of freedom in the low energy\nphase, then it is a gauge theory. Gauge symmetries would then be an emergent property of the\nlow energy phase and \"dissolve\" in some topological like pha se transition deep in the ultraviolet\n[23,25,28,62–65]. The quarks and leptons as well as the gauge bosons and Higgs boson would\nthen be the stable collective long-range excitations of som e (unknown) more primordial degrees\nof freedom that exist above the scale of emergence. The vacuu m of the low energy phase should\nbe stable below the scale of emergence.\nSuppose the Standard Model works like this. Then it behaves l ike an effective theory with\nthe theory at mass dimension four, D=4, supplemented by a tower of non-renormalisable6rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .higher dimensional operators suppressed by factors of the l arge scale of emergence. The\nglobal symmetries of the Standard Model at D=4 are constrained by gauge invariance and\nrenormalisability. The higher dimensional operators are l ess constrained and may exhibit extra\nglobal symmetry breaking [ 23,28,66–68]. Lepton number violation and tiny Majorana neutrino\nmasses may enter at D=5, suppressed by single power of the scale of emergence, via th e so\ncalled Weinberg operator [ 69]. One finds mν∼Λ2ew/MwithΛewthe electroweak scale and Mthe\nscale of emergence. Possible proton decays might occur at D=6suppressed by two powers of\nM[69,70]. New CP violation, needed for baryogenesis, might occur in Majorana phases at D=5\nas well as in new D=6+ operators [ 71].2\nIf one increases the energy much above the electroweak scale , then the physics becomes\nincreasingly symmetric with energies E≫Λewuntil we come within about 0.1% or so of the scale\nof emergence. Then new global symmetry violations in higher dimensional operators become\nimportant so the physics becomes increasingly chaotic unti l one goes through the phase transition\nassociated with the scale of emergence, with the physics abo ve this scale then described by new\ndegrees of freedom and perhaps new physical laws. This scena rio contrasts with unification\nmodels which involve maximum symmetry in the extreme ultrav iolet. The effect of higher\ndimensional operators might have been especially active in the very early Universe with energies\nE∼Mand be manifest through tiny numbers today.\nIn other ideas, emergent gauge symmetries can also appear th rough decoupling of gauge\nviolating terms in the RG at an infrared fixed point [ 74] and in connection with possible\nspontaneous breaking of Lorentz symmetry, SBLS [ 25,75–77]. In the former case, the coefficient of\nany local gauge symmetry violating terms blows up at the fixed point, in contrast to restoration\nof global symmetries where the coefficient of any symmetry vi olating term goes to zero at the\nfixed point. With SBLS, non observability of any Lorentz viol ating terms at D=4corresponds to\ngauge symmetries for vector fields like the photon. Possible Lorentz violation might be manifest\natO(Λ2ew/M2)with a preferred reference frame naturally identified with t he frame where the\ncosmic microwave background is locally at rest [ 25].\nWhen coupling to gravitation, this emergence picture gives a simple explanation of the\ncosmological constant. One also finds interesting constrai nts on possible dark matter scenarios.\n(a) Vacuum energy and the cosmological constant\nThe simplest explanation of the accelerating expansion of t he Universe is a cosmological constant\nΛin Einstein’s equations of General Relativity,\nRµν−1\n2gµνR=−8πG\nc4Tµν+Λgµν. (3.1)\nHereRµνis the Ricci tensor, Ris the Ricci scalar and Tµνis the energy-momentum tensor\nfor excitations above the vacuum; Gis Newton’s constant and cis the speed of light.3\nMatter including dark matter clumps together under normal g ravitational attraction whereas the\ncosmological constant is the same at all points in space-tim e. Einstein’s equations determine the\ngeodesics on which particles propagate in curved space-tim e in the presence of a gravitational\nsource.\nThe cosmological constant measures the vacuum energy densi ty perceived by gravitation,\nρvac=Λ×c4/(8πG), (3.2)\nwith associated scale µvac,ρvac=µ4vac. Astrophysics observations [ 79] tell us that Λ=1.088×\n10−56cm−2corresponding to\nρvac=(0.002 eV)4(3.3)\n2LHC data (so far) reveal no evidence for higher dimensional c orrelations divided by powers of a large mass scale below the\nfew TeV range [ 72,73].\n3Here we follow the convention with the Minkowski metric take n as diag[-1,+1,+1,+1] so that positive Λcorresponds to a\npositive energy density and a negative pressure [ 78].7rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .with a present period of accelerating expansion that began a bout five billion years ago when\nthe matter density of the expanding Universe fell below ρvac, which then took over as the main\ndriving term. So far, astrophysics data is consistent with d ark energy being a time independent\ncosmological constant with vacuum equation of state (energ y density = - pressure) [ 80].\nThe small value 0.002 eV in Eq. ( 3.3) is intriguing from the viewpoint of Standard Model\nquantum fields. In general, vacuum energy is sensitive to qua ntum fluctuations and potentials in\nthe vacuum with terms involving the much larger QCD and elect roweak scales ( ≈200MeV and\n246 GeV). If taken alone, these particle physics contributi ons give large curvature contributions\nwhen substituted into Einstein’s equations – inconsistent with the flat Universe we observe in\ncosmology. This issue for zero-point energies was noticed a lready in the early works [ 81,82] with\nspontaneous symmetry and vacuum potentials discussed in [ 35,83]. (The history of thinking on\nthis topic is reviewed in [ 84,85].) The net ρvaccorresponding to the cosmological constant also\ninvolves an extra gravitational contribution which may be d ynamical – for detailed discussion\nsee Section 4. Vacuum energy only becomes an observable when coupling to g ravitation via\nthe cosmological constant. Without gravitational couplin g only energy differences count, so in\nusual particle physics one is free to set the energy of the vac uum to zero - e.g., through normal\nordering (before considerations of spontaneous symmetry b reaking). The cosmological constant is\nan observable measurable through the accelerating expansi on of the Universe. Hence it should be\nparticle physics RG scale invariant. Individual contribut ions are, in general, RG scale dependent\n(so theorist/calculation dependent) with only the net cosm ological constant as the observable.\nWhy is the cosmological constant so small?\nThe cosmological constant is connected with the symmetries of the metric. With a finite\ncosmological constant Einstein’s equations have no vacuum solution where gµνis the constant\nMinkowski metric. That is, global space-time translationa l invariance (a subgroup of the group\nof general co-ordinate transformations) of the vacuum is br oken by a finite value of ρvac\n[78].4The reason is that a finite value of ρvacacts as a gravitational source which generates\na dynamical space-time with accelerating expansion for pos itiveρvac. Suppose the vacuum\nincluding condensates with finite vacuum expectation value s is space-time translational invariant\nand that flat space-time is consistent at mass dimension four , just as suggested by the success of\nthe Standard Model. With the Standard Model as an effective t heory emerging in the infrared, the\nlow-energy global symmetries including space-time transl ation invariance can be broken through\nadditional higher dimensional terms, suppressed by powers of the large scale of emergence M.\nThen the RG invariant scales Λqcdand electroweak Λewmight enter the cosmological constant\nwith the scale of the leading term suppressed by Λew/M (that is, ρvac∼(Λ2ew/M)4with one\nfactor of Λ2ew/Mfor each dimension of space-time) – see Refs. [ 29,30] and the early work [ 25,86].\nThis scenario, if manifest in nature, would explain why the c osmological constant scale 0.002 eV\nis similar to what we expect for the neutrino masses [ 87]5, which for Majorana neutrinos are\nthemselves linked to the dimension five Weinberg operator wi thmν∼Λ2ew/M[69]. That is,\nµvac∼mν∼Λ2\new/M. (3.4)\nHere one is taking the Standard Model as describing particle interactions at D=4 up to the\nlarge scale M. The cosmological constant would vanish at mass dimension f our. This vanishing\ncosmological constant contribution is equivalent to a reno rmalisation condition ρvac=0atD=4\nimposed by global space-time translational invariance of t he vacuum, even in the presence of\nQCD and Higgs condensates. The precision of global symmetri es in our experiments, e.g., lepton\nand baryon number conservation, tells us that in this scenar io the scale of emergence should be\ndeep in the ultraviolet, much above the Higgs and other Stand ard Model particle masses. Taking\nthe value µvac=0.002 eV from astrophysics together with Λew=246 GeV gives a value for M\nabout1016GeV .\n4SetTµν=0 to remove excitations above the vacuum. If the global Minkow ski metric is a solution, the on the left hand side\nof Eq. ( 3.1) there is no curvature, derivatives will vanish with R=0 so the cosmological constant will also vanish.\n5Assuming three species of neutrinos, the neutrino oscillat ion data constrains the largest mass squared difference to b e≈\n2×10−3eV2with the smaller one as (7.53±0.18)×10−5eV2[88].8rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The scale 1016GeV is within the range where the Higgs self-coupling λmight cross zero (if\nindeed it does) under RG evolution. It is also similar to the \" GUT scale\" that typically appears\nin unification models. If the Standard Model is emergent at a s cale where λis non-negative then\nits vacuum will be fully stable. Any perturbative extrapola tion of Standard Model degrees of\nfreedom above the scale of emergence would reach into an unph ysical region.\nWhile there is no evidence in present data for a time dependen t dark energy, the search\nfor possible time dependence as well as any deviations from G eneral Relativity is proceeding\nwith next generation cosmological surveys, e.g. with EUCLI D, the JWST, ELT, and the proposed\nEinstein telescope. As an example, de Sitter space is unstab le in the S−matrix formulation\nof quantum gravity with a quantum breaking effect and (slow) decay of the cosmological\nconstant [ 89,90]. Within the emergence scenario any time dependence might c orrespond either\nto a change in Λ2ew/M (withGandctaken as fixed) or to a vanishing cosmological constant\nbeyond leading order in 1/Mand dark energy instead corresponding to some new dynamics w ith\njust gravitational coupling, e.g., some BRST invariant coh erent state of gravitons as suggested\nin [91]. In the latter case, one then has the phenomenological issu e of explaining the similar sizes\nµvac∼mνobserved today.6\nCosmology observations of a spatially flat Universe [ 79] constrain the energy densities of\ndark energy and dark matter. For a flat Universe the density pa rameter Ω=8πGρ/3H2\n0=1\nwhereρis the sum over individual matter, radiation and dark energy density contributions.\nThe normal baryonic and cold dark matter, radiation and dark energy contributions measured\nby Planck are Ωb=0.049,Ωcdm=0.265(7) ,Ωγ=2.5×10−5,ΩΛ=0.685(7) with the Hubble\nconstant today H0=0.674(5) kms−1Mpc−1[79]7. The curvature density parameter comes out\nasΩk=1−Ω=k/˙a2=0.0007(19) withkthe curvature and athe scale factor appearing in\nthe Friedmann–Lemaitre–Robertson–Walker, FLRW, metric ( H0= ˙a/a witha=1 taken today)\nsuggesting a flat Universe. Observation of a spatially flat Un iverse together with measurements of\nthe dark matter contribution suggested the presence of a fini te cosmological constant [ 96] before\nthe discovery of accelerating expansion of the Universe thr ough supernova Sn1a observations\n[97,98]. Alternatively, with the cosmological constant scale ded uced using the emergent Standard\nModel arguments above, Eq. ( 3.4), flatness gives a constraint on the amount of extra dark matt er\nneeding extra theoretical understanding.\nThe observations of flatness, isotropy and homogeneity plus today the lack of magnetic\nmonopoles which might have been produced in the very early Un iverse motivate primordial\ninflation ideas [ 99,100] where initial exponential expansion is driven by an inflato n scalar field\nand slow-roll potential condition. In this picture the scal e of inflation is typically taken as around\n1016GeV , which is close to M, our scale of emergence. Perhaps the physics underlying infl ation\nand emergence might be connected. The size of the cosmologic al constant is further constrained\nby anthropic arguments, that if Λwere more than about ten times larger, galaxies would not hav e\nhad time to form [ 101].\n(b) Dark matter and higher dimensional couplings\nSome extra stable dark matter is suggested by studies of gala xies and galaxy clusters, gravitational\nlensing and the cosmic microwave background [ 102–104]. We see just a 5% contribution to the\nenergy budget of the Universe in visible matter; 68% is in dar k energy and 27% in dark matter\nobserved (so far) just through gravitational effects. The q uest to understand this dark matter has\ninspired vast experimental and theoretical activity with i deas including possible new types of\nelementary particles, primordial black holes as well as pos sible extensions of General Relativity.\nThe mass scale of possible new dark matter particles is so far not known from experiments\nwith ideas including new particles with masses ranging from 10−22eV up to 1015GeV – a range\nof 1046[102]. If the Standard Model does describe particle physics inte ractions at D=4up to\n6Other ideas relating the sizes of the cosmological constant scale and neutrino mass are discussed in Refs. [ 92–94].\n7This measurement of H0from the cosmic microwave background is in tension with rece nt late-time, low-redshift\nmeasurements [ 95] which give a value of H0about0.73kms−1Mpc−1– a present open puzzle in cosmology.9rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .the highest scales, then we do not expect lightest mass super symmetry particles or extra inert\nHiggs doublet states as sources of dark matter. With emergen ce small multiplets are preferred\nas collective excitations of the system that resides deep in the ultraviolet [ 23]. One might then\nlook to terms suppressed by powers of 1/M, whether new particle couplings or modified gravity\nscenarios, e.g. [ 105]. As a particle candidate, if present, pseudoscalar axions [106] would enter at\nD=5with masses and couplings to Standard Model particles suppr essed by a single power of a\nlarge axion \"decay constant\", which might be associated wit h any scale of emergence.8\nBesides possible new particles, primordial black holes are discussed as candidates for\ndark matter [ 113,114], and are consistent with General Relativity with coupling up toD=\n4. Suggestions include the black holes in the mass range obser ved with the LIGO-VIRGO\ngravitational wave measurements [ 115]. Primordial black holes formed in the early Universe\nbefore the time of stellar collapse origin might be detected in future gravity waves measurements\n[116].\nWhilst we search for evidence of possible new dark matter par ticles and/or black holes\nformed in the early Universe it is worthwhile also to conside r other observables that might\ngive clues to the nature of dark matter. Interesting informa tion may follow from studying the\nredshift dependence of correlations between dark matter an d galaxy properties, and also common\ntrajectories followed by particle signals from distant gal axies. One observes phenomenological\ncorrelations between galaxy rotation velocities and dark m atter [ 104] and correlations between\nthe masses of supermassive black holes at galaxy centres and their dark matter halos [ 117,118].\nThese correlations have so far been measured just for galaxi es at small redshifts. New windows\nof observation are now opening up with instruments such as, e .g., the JWST, ELT...\nThe baryonic version of of the Tully-Fisher relation connec ts the total baryonic mass in a galaxy\nMbarto the asymptotic rotation velocity v∞[119],\nv4\n∞=GMbara0, (3.5)\nwherea0is a phenomenological acceleration parameter, a0≈1.2×10−10ms−2to within 10-\n20% [ 120] obtained from galaxy rotation curves. Numerically 2πa0≈aH≈aΛwhereaH=\ncH0≈1.1×10−10ms−2andaΛ=c2/radicalbig\nΛ/3=cH∞≈0.9×10−10ms−2. HereH0is the Hubble\nparameter today, Λis the cosmological constant and H∞is the Hubble constant in the infinite\nfuture of ΛCDM after matter and radiation density terms have decayed ap proaching zero\nwith the cosmological constant taken as time independent. E q.(3.5) is sometimes taken as\nphenomenological evidence to hint that usual Newtonian gra vity might be modified, the so called\nMOND theories [ 121].9It may also be the net result that should come out from simulat ing\na microscopic theory of dark matter involving new dark matte r particles within usual General\nRelativity and connected to galaxy formation. This issue is a step beyond the simple fit to data\nassociated with Eq.( 3.5). Here we just note that aHandaΛhave different scaling behaviour\nin a1/M expansion with µvac∼Λ2ew/M:aHhas a mass dimension four component whereas\naΛ∼1/M2. Further, aHis time dependent through H0whereas aΛis time independent if we\ntake a time independent cosmological constant. It would be i nteresting to measure the redshift\ndependence of the phenomenological relation in Eq.( 3.5) through more precise observations of\ndistant galaxies: might the value of a0be time independent or its similarity with the Hubble\nconstant just a coincidence at low redshift? Possible redsh ift dependence of a0, including with\nΛCDM dynamics, is discussed in [ 122]. Ifa0were found to be redshift independent, then it\n8Axions are a candidate to explain the strong CP puzzle (why gl uon topological effects do not induce extra CP violation\nin QCD) [ 107–109] though this might also be associated with confinement relat ed phenomena [ 110]. QCD axion masses are\nusually taken from astrophysics and cosmology constraints between 1 µeV and 3 meV corresponding to an ultraviolet scale\nbetween 6×109and6×1012GeV. This is less than the 1016]GeV discussed here in connection with the cosmological\nconstant and neutrino masses. Possible relaxation of the ax ion bounds to include perhaps higher scales is discussed in\n[111]. Any axion Bose-Einstein condensate contribution would a pproximately scale as D=4 with the axion decay constant\nsuppression factor missing in the net condensate contribut ion [ 112].\n9These involve an extra asymptotic long-range gravitationa l acceleration g=a0rM\nrwithrM=(MG/a0)1\n2whereMis\nthe central mass.10rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .might hint at dark matter being associated with some higher d imensional coupling involving\nthe cosmological constant.\nAnother interesting constraint is that the gravitational e ffect of dark matter or any modified\ngravity scenarios is the same for all particles. Whatever da rk matter is, within present\nexperimental uncertainties it does not seem to affect the co mmon trajectories travelled by\ngravitational waves and by different particle species [ 102]. In multimessenger astronomy light\nand gravitational wave signals identified with the neutron s tar merger event GKB170817 were\nobserved to arrive very close to simultaneously constraini ng the speeds of gravity and light\nto be the same to within ∼10−15times the speed of light, meaning that they follow the\nsame geodesics [ 123]. Likewise, the initial neutrino multimessenger event Ice Cube-170922A with\nenergy 290 TeV [ 124] and the neutrino event Icecube-141209A with Eν=97.4±9.6TeV detected\nin coincidence with the blazar GB6 J1040+0617 in a phase of hi ghγ-ray activity [ 125] each\ncorrespond to the photon and neutrino speeds being the same t o within about 10−11of the speed\nof light assuming the neutrinos were emitted coincident wit h the blazar flaring periods.10\nDifferent particle trajectories might be induced in exotic modified gravity scenarios with\ndifferent metrics for gravitational waves coupling to Gene ral Relativity without dark matter\nand for normal matter coupling to General Relativity with da rk matter. Trajectories might\nalso be affected by any new direct matter to curvature coupli ngs which, if present, would\ninvolve an extended gravitational action involving a serie s inO(R/M2), e.g. involving terms\nlike1/M2RµνFµωFνωwithFµνthe electromagnetic field tensor. Such terms preserve the lo cal\ngravitational symmetries of invariance under local co-ord inate transformations but change the\ndynamics [ 126,127]. They modify the energy momentum tensor and equations of mo tion and can\nlead to particles not following geodesics, with the effect i n general different for different particles.\nIf present, these terms would contribute in regions of stron g gravitational curvature. In addition,\npossible violations of Lorentz invariance discussed in [ 25] might arise at O(µvac/M∼Λ2ew/M2)≈\n10−28.\n4. Structure of the cosmological constant\nSo far we have addressed the cosmological constant within th e framework of an emergent\nStandard Model. The tiny cosmological constant scale 0.002 eV is very much smaller than the\nelectroweak scale 246 GeV which is, in turn, very much smalle r than the Planck scale. In this\nSection and Section 5we discuss these scale hierarchies and how they connect with issues of\nrenormalisation and ultraviolet regularisation.\nThe cosmological constant through ρvacreceives contributions from the zero point energies,\nZPEs, of quantum field theory through vacuum-to-vacuum diag rams [ 128], any (dynamically\ngenerated) potential in the vacuum, e.g. induced by spontan eous symmetry breaking and Higgs\nand QCD condensates [ 35,83], and the renormalised version of a bare gravitational term ρΛ,\nviz. [ 78,129]\nρvac=ρzpe+ρpotential+ρΛ. (4.1)\nAs an observable the cosmological constant is renormalisat ion scale invariant. It is independent\nof how a theoretician might choose to calculate it,\nd\ndµ2ρvac=0. (4.2)\n(Here we take Newton’s constant Gas RG scale invariant with gravity treated as classical.)\nIndividual terms in Eq.( 4.1) contributing to ρvacare, however, renormalisation scale dependent\n10The gravitational wave event GKB170817 was observed in coin cidence with a short γ-ray burst, GRB 170817A at redshift\nz=0.008+0.002\n−0.003, with signals detected 2 seconds apart originating from a ne utron star merger event more than 108light\nyears away. The neutrino event IceCube-170922A was identifi ed in parallel with very high energy γ-rays from the blazar\nsource TXS 0506+056 at redshift z=0.3365±0.0010 with the blazar active within about ±14days of the 290 TeV neutrino\nevent. The statistically favoured joint blazar excitation and neutrino event IceCube-141209A was observed in a window of\nhighγ-ray activity from the blazar GB6 J1040+0617 at redshift z=0.7351±0.0045 which was brightest 4.5 days before the\nneutrino detection.11rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .as well as sensitive to large particle physics scales. The Hi ggs potential is RG scale dependent\nthrough the Higgs self-coupling, which determines the stab ility of the electroweak vacuum. The\nZPEs are discussed in detail below. This renormalisation sc ale dependence should cancel with\nthe gravitational term ρΛto give the scale invariant ρvac. The important question then is what is\nleft over or how the different terms combine. The small cosmo logical constant and net ρvacare\ndetermined by the symmetries of spacetime with Nature so muc h liking the Minkowski metric\nand a spatially flat Universe today. In any self-consistent d iscussion, each of the three terms in\nEq. ( 4.1) should be defined with respect to the same renormalisation s cheme. The net ρvacobeys\nthe vacuum equation of state, EoS.\nZPEs are a vacuum energy contribution induced by quantisati on and are an intrinsic part of\nquantum field theory. One sums over harmonic oscillator cont ributions. When we evaluate the\nZPEs, they come with an ultraviolet divergence requiring re gularisation and renormalisation.\nWorking in flat space-time the ZPE for a particle with mass mis\nρzpe=1\n2/summationdisplay\n{/planckover2pi1ω}=1\n2/planckover2pi1/summationdisplay\nparticlesgi/integraldisplaykmax\n0d3k\n(2π)3/radicalbig\nk2+m2. (4.3)\nHeremis the particle mass; gi=(−1)2j(2j+1)fis the degeneracy factor for a particle iof spin\nj, withgi>0for bosons and gi<0for fermions. The minus sign follows from the Pauli exclusio n\nprinciple and the anti-commutator relations for fermions. The factor fis 1 for bosons, 2 for each\ncharged lepton and 6 for each flavour of quark (2 charge factor s for the quark and antiquark, each\nwith 3 colours). The corresponding vacuum pressure is [ 130]\npzpe=1\n31\n2/planckover2pi1/summationdisplay\nparticlesgi/integraldisplaykmax\n0d3k\n(2π)3k2\n√\nk2+m2. (4.4)\nWhen evaluating the integrals in Eqs. ( 4.3) and ( 4.4) one finds that the equation of state is sensitive\nto the choice of ultraviolet regularisation. A Lorentz cova riant regularisation is needed to ensure\nthat the ZPEs satisfy the vacuum EoS ρ=−p.\nFor example, suppose we evaluate the ZPE using a (non covaria nt) brute force cut-off on the\ndivergent integral with some fixed kmax. Then the leading term in the ZPE, which is proportional\ntok4max, obeys instead the radiation equation of state ρ=3p, viz.\nρzpe/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nkmax=/planckover2pi1gik4max\n16π2/bracketleftbigg\n1+m2\nk2max+.../bracketrightbigg\n, pzpe/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nkmax=1\n3/planckover2pi1gik4max\n16π2/bracketleftbigg\n1−m2\nk2max+.../bracketrightbigg\n(4.5)\nwith quadratic terms behaving as ρ=−3pand just subleading logarithmic terms obeying the\nvacuum EoS [ 130]. Cosmology observations and local energy conservation re quires that the net\nvacuum energy contribution ρvacobeys the vacuum EoS [ 131]. While the leading term in Eq. ( 4.5)\nbehaves like an homogeneous sea of radiation, it is importan t to recall that ρzpecomes from\ncalculating vacuum-to-vacuum closed loop diagrams instea d of freely propagating photons like\nwhich dominated in the early Universe.\nThe vacuum equation of state ρ=−pis obtained with dimensional regularisation and minimal\nsubtraction, MS. Here, the ultraviolet divergence is controlled by analyti c continuation of the\ntheory including the dimensionality of loop integrals in th e complex plane and then taking the\nlimit that D→4[42]. One finds [ 130,132]\nρzpe=−pzpe=−/planckover2pi1gim4\n64π2/bracketleftbigg2\nǫ+3\n2−γ−ln/parenleftbiggm2\n4πµ2/parenrightbigg/bracketrightbigg\n+... (4.6)\nfor particles with mass m. HereD=4−ǫis the number of dimensions with divergence at D→\n4, ǫ→0andγ=0.577...is Euler’s constant. The renormalisation scale µenters also through the\nrunning mass and should decouple from the net ρvac. It enters the calculation of ρvacto keep12rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .the mass dimension of the cosmological constant fixed in Eq. ( 3.1) [132].11After subtracting out\nthe divergent pole term into a renormalisation counterterm , the remaining finite part with MS\nrenormalisation is (ρzpe=−pzpe)|MS=−/planckover2pi1gim4\n64π2/bracketleftbig3\n2−ln/parenleftbigm2\n4πµ2/parenrightbig/bracketrightbig\n.\nThe ZPE in Eq. ( 4.6) for each given particle is proportional to the particle’s m ass to the\nfourth power, thus vanishing for massless photons and gluon s. For the Standard Model particles\n(quarks, charged leptons, W, Z and Higgs) the non vanishing Z PEs are generated through\nthe BEH mechanism. Majorana neutrinos might get their mass t hrough the Weinberg operator\nmν∼Λ2ew/M with mass connected to the electroweak scale Λew. The ZPE would vanish for\npossible massless gravitons. For the Standard Model Higgs w ith mass squared m2\nh=2λv2, the\nHiggs boson ZPE develops an imaginary part if the Higgs self- coupling crosses zero through the\nm4\nhlnm2\nhterm in Eq. ( 4.6) signalling vacuum instability for negative λ,[lnλ=ln(−λ)−iπfor\nλ<0]– see Ref. [ 30].\nTo understand the interpretation of the gravity term ρΛhere, the net cosmological constant\nis the observable, not the individual terms in Eq. ( 4.1) which appear as intermediate steps in the\ncalculation. The cosmological constant term ρvacis RG scale invariant so the RG scale dependence\ncancels with the gravity term ρΛ, which also restores the symmetries of the low energy vacuum\nbelow the scale of emergence12. If one does wish to use the non-covariant brute force cut-of f to\ndefine the ZPEs, then one should compensate with a similar Lor entz violation in the ρΛterm to\nguarantee the correct vacuum EoS for the net ρvac.\nThe role of ρΛhere has analogy with the treatment of vacuum energy in quant um liquids in\ncondensed matter physics [ 133]. The Gibbs-Duhem relation for a quantum liquid at zero pres sure\nsets the corresponding vacuum energy density to zero. One fin ds cancellation from quasiparticle\nZPE quantum contributions up to the energy scale characteri sing the scale of emergence for the\nquantum liquid against the macroscopic degrees of freedom, e.g. atoms..., from above the cut-off\nfor the low-energy theory (the latter playing an analogous r ole toρΛ). A key difference between\nan emergent Standard Model and these low temperature conden sed matter systems is that in the\ncondensed matter systems we know the degrees of freedom both above and below the scale of\nemergence whereas Planck scale physics is not directly acce ssible to experiments.\nThe QCD phase transition in the early Universe, which occurr ed when the Universe was about\n10−5seconds old, involves a change in the free energy [ 134] but need not change the cosmological\nconstant if the change in the quark-gluon ZPE and QCD potenti al terms is compensated by a\nchange in the gravity term ρΛkeeping the symmetries of the metric held fixed, at least at le ading\norder in 1/M. This would correspond to a change in the ultraviolet comple tion of the effective\ntheory with a change in the degrees of freedom from quarks and gluons to hadrons.\nWe note an alternative treatment, [ 135] where a renormalisation prescription was introduced\nfor eliminating the m4terms in the ZPEs equivalent to cancelling the ZPE contribut ions in curved\nspace against the corresponding ZPE evaluated in Minkowski space. This prescription when\ntaken alone does not determine the value of the cosmological constant, which is here determined\nby the symmetries of the metric at different orders in the low energy expansion in powers of 1/M.\n11The full expression for the ZPE term in Eq. ( 4.3) after covariant dimensional regularisation is\nρvac=−pvac=−/planckover2pi1giµ4\n2(4π)(D−1)/2Γ(−D/2)\nΓ(−1/2)/parenleftbiggm\nµ/parenrightbiggD\n(4.7)\nwith Eq. ( 4.6) obtained in the limit ǫ→0. Eq. ( 4.7) respects the correct vacuum equation of state. The pole ter ms \"thrown\naway\" by the analytic dimensional continuation come with re sidue proportional to µ4for the pole at 1/(4−ǫ)andm2µ2\nfor the pole at 1/(2−ǫ).\n12Note the part similarity with the Chern-Simons current and t he axial anomaly. The Chern-Simons current restores gauge\ninvariance in the renormalised axial vector current and als o carries RG scale dependence. Here the ρΛterm is restoring the\nglobal symmetry of spacetime translation invariance of the vacuum. The Chern-Simons current in QCD involves gluon field s\nthat couple to the quark axial-vector vertex beyond simple t ree approximation, whereas here ρΛappears as a gravitational\nterm in General Relativity.13rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5. Scale hierarchies, symmetries and regularisation\nIn the last Section we discussed the appearance of different mass scales in the separate\ncontributions to the cosmological constant and the sensiti vity of the ZPEs equation of state to\nthe choice of regularisation used in evaluating it. In this S ection we extend this discussion to a\nmore general discussion of scale hierarchies including the Higgs mass naturalness puzzle.\nUltraviolet regularisation is an essential part in dealing with infinities, as a precursor to\nrenormalisation. Renormalisable quantum field theories in volve renormalisation counterterms\nparametrising our ignorance of deep ultraviolet physics (h ere including what is beyond the\nscale of emergence). Regularisation is needed for real life calculations in defining our notion of\n\"infinity\". However, regularisation is more than just a math ematical trick to isolate and control\nthe ultraviolet divergences and comes with real physical in put. It breaks some symmetries of\nthe classical theory which are not necessarily all restored in the continuum [ 136]. One has to\nbe sure that the regularisation and final calculation preser ves the fundamental ingredients of\ngauge invariance and Lorentz covariance (with any net viola tions of Lorentz invariance entering\nat most at O(Λ2ew/M2)within the emergence picture and classical symmetries pres erved as much\nas possible.) Beyond the issue with covariance and the ZPE eq uation of state discussed above,\nthere is also the famous example with the axial anomaly in tri angle diagrams with vertices\nbeing two vector currents and one axial vector current [ 137,138]. Gauge invariance at the vector\ncurrent vertices induces an anomalous term in the divergenc e of the singlet axial vector current\nwhich is missed in the brute force cut-off theory [ 139]. Among regulators, a brute force cut-off\non momentum integrals breaks Lorentz invariance. Dimensio nal regularisation involves analytic\ncontinuation into regulator dimensions. Here covariance a nd gauge invariance are respected,\nthe latter with suitable treatment of Dirac γ5matrices in the regulator dimensions to ensure\nthe axial anomaly [ 42]. The connection to physical resolution is less transparen t. One typically\nchooses the renormalisation scale with MS to minimise logarithms logm2/µ2in perturbative\ncalculations with corresponding RG evolution in µ(mdenotes a particle mass scale in the\ncalculation). Lattice regularisation violates translati onal and rotational invariance symmetries in\nthe continuum and gives issues with fermion doubling. In cal culations it is important to choose a\nconsistent regularisation and renormalisation procedure for all observables.\nTheoretically, the renormalised Higgs mass squared comes w ith the divergent counterterm\nm2\nhbare=m2\nhren+δm2\nh (5.1)\nwhere, at leading order, employing a cut-off in the Higgs sel f-energy diagrams give the quadratic\ndivergence\nδm2\nh=K2\n16π26\nv2/parenleftbigg\nm2\nh+m2\nZ+2m2\nW−4m2\nt/parenrightbigg\n=K2\n16π22/parenleftbigg9\n4g4+3\n4g′4+6λ−6y2\nt/parenrightbigg\n(5.2)\nHereKis an ultraviolet scale characterising the limit to where th e Standard Model should work.\nThe renormalised mass is the mass extracted from experiment s, being related by a perturbative\nexpansion to the particle pole mass. For a textbook discussi on of bare and renormalised mass\nparameters see [ 1,2]. In Eq. ( 5.2) we neglect contributions from lighter mass quarks; next-t o-\nleading order corrections are suppressed by factors of 1/(4π)2. The hierarchy or naturalness\npuzzle is why is m2\nhso much smaller than the Planck mass or any other large physic al scale\nthat might characterise new physics effects beyond the Stan dard Model?\nIf one instead evaluates the Higgs mass squared using dimens ional regularisation, the\nquadratic divergence in Eq. ( 5.2) corresponds to a pole at D=2[140]. With analytic continuation\nin the dimensions this pole is \"thrown away\" in the formal reg ularisation procedure and the\nresulting divergence comes out proportional to the 1/ǫpole term at D=4instead of the large\nultraviolet scale term K2. Results are available up to two loop calculations, see [ 141]. The leading\norder term with MSis\nδm2\nh=m2\nh1\n16π21\nǫ/parenleftbigg\n3y2\nt+6λ−9\n4g2−3\n4g′2/parenrightbigg\n+... (5.3)14rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Here the scale factor in δm2\nhis set by the Standard Model particle masses, which follows\non dimensional grounds since these are the only mass scales i n the calculation. The different\ncoefficients between Eqs.( 5.2) and ( 5.3) follow since the quadratic divergence in Eq. ( 5.2)\ncorresponds to a pole at D=2 whereas the coefficient in Eq. ( 5.3) is the pole at D=4 with\ndimensional regularisation.\nIf taken alone without coupling to extra particles, the Stan dard Model is self-consistent without\nany large scale hierarchy issue if we work in MS which involves no explicit large mass scale\nK[142]. Ultraviolet divergences can consistently be absorbed in renormalisation counterterms.\nHowever, usual effective theory arguments suggest that, in dependent of this, mass scales should\nbe close to the ultraviolet cut-off that defines the limit of t he effective theory unless some\nsymmetry of the Lagrangian like gauge invariance or chiral s ymmetry pushes them to zero. The\nHiggs mass and cosmological constant are not protected this way. Hence the issue whether and\nwhat new physics one needs to suppress them. In the scenario d iscussed here it is the dynamics of\nthe vacuum, its stability and spacetime symmetries with an e mergent Standard Model, that fix the\nsize of the Higgs mass and the cosmological constant with the infrared and ultraviolet connected\nthrough RG running of the Higgs self coupling λ. Vacuum stability involves a delicate fine tuning\nand conspiracy of Standard Model parameters.\nIt is interesting also to consider the theoretical issue whe ther the coefficient in Eq. ( 5.2) might\ncross zero and change sign. The running couplings in Eqs. ( 2.1)-(2.3) have different RG behaviour,\nsee Fig. 1, and come with different signs for bosons and fermions. Here the result is calculation\ndependent. If δm2\nhwere to cross zero – so called Veltman crossing – below the Pla nck scale and the\nscale of emergence, then this might be interpreted in terms o f a first order phase transition with\nelectroweak symmetry breaking below the crossing scale. Ab ove this scale the Standard Model\nwould enter a symmetric phase with the 4 Higgs states associa ted with the BEH Higgs doublet\nfield then carrying large mass of order the crossing scale and all other Standard Model particles\nmassless. This scenario was found in the calculations of [ 23] (with emergence taken at the Planck\nscale instead of 1016GeV) with a stable vacuum up to the Planck mass and Veltman cro ssing\naround1016GeV . If manifest in Nature, it would allow the Higgs to behave as the inflaton [ 143].\nIn other calculations crossing was found not below the Planc k mass in [ 30] and (with metastable\nvacuum) around 1020GeV in [ 144] and much above the Planck scale in [ 44] and [ 145].\nWe close mentioning possible similar phenomena between the scale hierarchy with\nelectroweak symmetry breaking, v≪MPl, and the ferromagnetic phase transition in condensed\nmatter physics. Below the phase transition the magnetisati on is small as the reduced temperature\n(T−Tc)/T→0−withTcthe critical temperature – that is, close to the phase transi tion. Whereas\nemergent gauge systems are associated with topological pha se transitions, Higgs phenomena\nis associated with spontaneous symmetry breaking defined wi th respect to a particular gauge\nchoice. Electroweak symmetry breaking might correspond to a Universe close to the phase\ntransition and near to the critical point [ 23].\n6. Conclusions\nThe main surprise from the LHC is that the Standard Model is wo rking so well. Perhaps the\nStandard Model is more special than previously assumed. If o ne assumes no coupling to other\nparticles then RG evolution suggests that the Standard Mode l with its measured parameters\nmay be working up to very high scales, perhaps up the scales 1016GeV where grand unified\ntheories have previously been proposed to work or even up to t he Planck scale. Crucial here is\nthe RG behaviour of the Higgs self coupling λwhich is very sensitive to a delicate interplay of\n\"low energy\" Standard Model parameters measured in laborat ory experiments. Small changes in\nsome of these parameters would lead to a very different theor y and to very different physics. The\ninfrared and ultraviolet limits of the Standard Model are th us strongly correlated.\nIf the Standard Model might really work up to these large scal es, then a plausible scenario is\nthat it might be emergent, with gauge symmetries \"dissolvin g\" in the ultraviolet and particles as\nthe stable long range excitations of some critical system th at resides close to the Planck scale. In15rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 00 00000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .this scenario, new physics most likely resides in higher dim ensional operator terms that become\nimportant at energies very close to the scale of emergence an d which may have been active in\nthe very early Universe. The cosmological constant comes ou t with a scale similar to what we\nexpect for tiny Majorana neutrino masses. In this picture da rk matter, if it is not primordial black\nholes, might be associated with axion-like particles or wit h new higher dimensional gravitational\ncouplings (beyond minimal General Relativity). Lorentz in variance might be broken at D≥6.\nThe question why the cosmological constant is so small then b ecomes the issue why Nature so\nlikes the Minkowski metric with connection to a spatially fla t Universe. The Higgs mass would\nbe environmentally selected in connection with vacuum stab ility, with the hierarchy puzzle a\ndelicate conspiracy of Standard Model parameters in the low energy phase. It is interesting to\nconsider possible connections between the scale of emergen ce and scale of inflation.\nExtra CP violation beyond the Standard Model Cabibbo-Kobay ashi-Maskawa matrix can arise\nthrough complex phases associated with Majorana neutrino m asses and through new D=6(+)\noperators. Baryon number violation can also occur at D=6with possible non-equilibrium at the\nphase transition associated with emergence and electrowea k symmetry breaking providing at\nleast qualitatively the conditions needed for baryogenesi s [146].\nThe Standard Model effective theory discussed here differs from usual discussions in particle\ntheory [ 147] where one supposes new interactions at D=4at higher energies above the range\nof present experiments. The effect of these interactions is integrated out with redefinition of the\nlow energy parameters as well as parametrised by higher dime nsional terms with mass scale\nrepresenting the scale of new physics, with new heavy partic le degrees of freedom liberated when\none goes through their production thresholds. In our emerge nce scenario the Standard Model is\ntaken to work at D=4up to the large scale of emergence. The higher dimensional te rms describe\nnew physics from the phase transition which produces the Sta ndard Model but unconstrained by\nrequirements of renormalisability at D=4.\nWith emergence open theoretical questions are the universa lity class for the Standard Model\nand any critical dimension for the topological-like phase t ransition that produces it. If the\nStandard Model might be emergent, then what about gravitati on? If the gauge symmetries of\nGeneral Relativity might be emergent at a scale below the Pla nck mass, then this would alleviate\nthe usual problems associated with quantising gravitation .\nThe similarity of the cosmological scale and values we expec t for light neutrino masses is\nintriguing, as is the similarity between the phenomenologi cal acceleration parameter a0and\naΛinvolving the cosmological constant. While these may be coi ncidences, these relations may\nbe hinting at some deeper connection between the infrared an d ultraviolet through higher\ndimensional terms in a low energy expansion in 1/Massociated with emergence. In the absence\nof signals of new particles one should at least treat such num erical similarities as possible clues\nand see where they might lead. Next generation experiments a t the high energy, precision and\ncosmology frontiers will teach us much about the deeper stru cture of Nature. If the Standard\nModel does indeed work up to the highest scales, then vacuum s tability might be pointing to\ndeep interconnections between the physics of the infrared a nd extreme ultraviolet.\nAcknowledgements. I thank M. Ackermann, F. Jegerlehner, J. Krysziak and G. Volo vik for stimulating\ndiscussions on the physics issues discussed in this paper. I thank the Alexander von Humboldt Foundation\nfor support for this meeting, the Humboldt Kolleg \"Clues to a Mysterious Universe - exploring the interface\nof particle, gravity and quantum physics\".\nReferences\n1. S. Pokorski. Gauge Field Theories , 2nd edition (Cambridge Univ. Press, 2000)\n2. J. C. Taylor, Gauge Theories of Weak Interactions (Cambridge Univ. Press, 1976)\n3. G. Altarelli, Collider Physics within the Standard Model : a Primer, [arXiv:1303.2842 [hep-ph]].\n4. S. D. Bass, A. De Roeck and M. Kado, The Higgs boson implicat ions and prospects for future\ndiscoveries, Nature Rev. Phys. 3(2021) 608. 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Monkewitza)\n´Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), CH-101 5, Lausanne,\nSwitzerland\n(Dated: 3 April 2024)\nThe difficulty of determining the slope of the famed logarithmic law in the mean\nvelocity profile in wall-bounded turbulent flows, the inverse of the K´ arm´ an “con-\nstant”κ, from direct numerical simulations (DNS) is discussed for channel fl ow.\nUnusual approaches, as well as the analysis of the standard log-in dicator function\nare considered and analyzed, including the identification of the highe r order linear\noverlap term. This leads to the conclusion that a definitive determina tion of the\nchannel flow κfrom DNS with an uncertainty of, say, 2-3% will require the residue\nof the mean stream-wise momentum equation in the simulations to be r educed by\nat least an order of magnitude.\nI. INTRODUCTION\nTraditionally, theK´ arm´ anparameter κinthelogarithmicoverlaplawofthemeanvelocity\nhas been determined in simple geometries, such as channel, pipe and fl at plate boundary\nlayer flows, from constant portions of the indicator function\nΞlog=y+dU+/dy+≡YdU+/dY (1)\nHere,y+andU+arethewallnormalcoordinateandstream-wisemeanvelocity,inne r-scaled\nwith friction velocity uτand kinematic viscosity ν, andY≡y+/Reτthe outer coordinate,\nwith Re τthe friction Reynolds number.\nSince von K´ arm´ an1,2and Millikan3, enormous efforts have been spent to determine the\nvalue ofκin the mean velocity overlap law\nU+\nOL= (1/κ) lny++B (2)\nand, nearly 100 years after its introduction, the turbulence comm unity can neither agree on\nwhether κis a universal constant or flow dependent nor on its value. The purp ose of the\npresent note is to offer some insight into why these questions have n ot yet been definitively\nresolved.\nThe study focusses on channel flow, for which a multitude of DNS da ta exist. The\nselection of data used for this study is summarized in table I, togeth er with the color code\nused in all the figures.\nFor channel flow, the stream-wise mean momentum equation is part icularly simple\n1−dU+/dy+=Y−∝angbracketle{tuv∝angbracketright+(3)\nwhich allows to evaluate Ξ logfrom either U+or∝angbracketle{tuv∝angbracketright+in sections III and IV.\nCombining equations (2) and (3) readily leads to\n∝angbracketle{tuv∝angbracketright+\nOL=Y−1+(κy+)−1, (4)\nwhich is the overlap between the 2-term inner and outer expansions of∝angbracketle{tuv∝angbracketright+up to order\nO(Reτ)−1(see for instance Afzal and Yajnik10for an early analysis). This is most easily\na)peter.monkewitz@epfl.ch2\nTABLE I. Channel DNS profiles considered in the present study , with the color scheme used in all\nfigures\nNo. Re τReference color\n#1 543 Lee & Moser4\n#2 640 Abe et al.5\n#3 944 Hoyas & Jim´ enez6\n#4 1001 Lee & Moser4\n#5 1020 Abe et al.5\n#6 1995 Lee & Moser4\n#7 2003 Hoyas & Jim´ enez6\n#8 3996 Kaneda & Yamamoto7\n#9 4179 Lozano-Dur´ an & Jimen´ ez8\n#10 5186 Lee & Moser4\n#11 7987 Kaneda & Yamamoto7\n#12 10049 Hoyas et al.9\n-1.2 -1 -0.8 -0.6 -0.4 -0.2 0¢uv ²+\nY10 -410 -310 -210 -11\nFIG. 1. Reynolds stress from the DNS #1, 4, 6, 10 and 12 of table I versus outer wall distance Y.\n• • •, corresponding fits (4) with κ= 0.417. Black − − −: limiting Reynolds stress ( −1+Y) at\nReτ=∞.\nseen by expressing the large y+expansion of the inner Reynolds stress, up to O(Re−1\nτ),\n∝angbracketle{tuv∝angbracketright+\nin(y+≫1) = 1−(κy+)−1+...−Re−1\nτ[y++...] and its outer counterpart ∝angbracketle{tuv∝angbracketright+\nout= 1−\nY−Re−1\nτ[(κY)−1+κ−1] in terms of the intermediate variable11η=y+Re−1/2\nτ=YRe+1/2\nτ.\nRealizing that the inner-outer overlap of U+and∝angbracketle{tuv∝angbracketright+must coincide and that ∝angbracketle{tuv∝angbracketright+\nOL\nmust be located around its minimum, Sreenivasan, known to his friend s as Sreeni, deduced\nthe scalingoftheminimum Reynoldsstressinseverallandmarkpublica tions, Sreenivasan12,\n13and Sreenivasan and Sahay14, fig. 4 as\n∝angbracketle{tuv∝angbracketright+\nmin=−1+(3.1±0.1)Re−1/2\nτ (5)\nFrom equation (4), one immediately deduces the location and value of the minimum\nReynolds stress as\ny+\nminRS=/parenleftbiggReτ\nκ/parenrightbigg1/2\nand∝angbracketle{tuv∝angbracketright+\nmin=−1+2\n(κReτ)1/2(6)\nComparing equations (5) and (6) shows that Sreeni’s estimate of κ= 0.416±6.5%, ob-\ntained from noisy Reynolds stress minima was amazingly close to prese nt day estimates (see\nfor instance Monkewitz and Nagib15, henceforth referred to as “MN2023”, and references\ntherein).3\nThe above analysis of the Reynolds stress minimum is supported by th e channel DNS of\nfigure 1 “within plotting accuracy”, except for the lowest Re τof 543. However, as will be\ndemonstrated in the following, this is not sufficient to “nail down” κto better than ±0.5%,\nwhich is believed necessary to definitively settle the question of its flo w dependence.\nII. THE NEAR-WALL REGION\nInafirststep, theconsistencyoftheTaylorexpansionsof[ Y−∝angbracketle{tuv∝angbracketright+]andof[1 −dU+/dy+]\nis examined in figure 2 for all the DNS of table I. The majority of the pr ofiles are seen to\nbe well fitted by the Taylor expansion, which must be anti-symmetric iny+.\nY−∝angbracketle{tuv∝angbracketright+≡1−dU+\ndy+=y+\nReτ+(t3y+)3−(t5y+)5+H.O.T. (7)\nwitht3= 0.109±3%, t5= 0.104±10%\nThe above coefficient t3is identical to the one found by Monkewitz and Nagib16in the zero\npressure gradient turbulent boundary layer, but differs from the 0.147 given by Panton17.\nThe difference stems from the fact that the Taylor expansion of Pa nton’s Reynolds stress\nlaw contains a ( y+)4term which violates the antisymmetry of [ Y−∝angbracketle{tuv∝angbracketright+] abouty+= 0.\nAs seen in figure 2, only the profiles #2 and #5 fall significantly below th e fit (7) for\n[1−dU+/dy+], but in view of the following cannot be dismissed outright as outliers. N ote\nalso that below y+≅2, [Y−∝angbracketle{tuv∝angbracketright+] is dominated by the exact linear term.\nFigure 2 also shows, that the two-term Taylor expansion (7) provid es an excellent fit of\nthe data up to around y+≈5. This range can be extended to beyond 10 by moving, in the\nspirit of the mean velocity Musker fit18, to the Pad´ e approximant\nY−∝angbracketle{tuv∝angbracketright+≡1−dU+\ndy+=y+\nReτ+(t3y+)3\n1+0.5(t3y+)2+(t3y+)3(8)\nwitht3= 0.109 as in equ. (7)\nEquation (8) may even serve as a rough composite fit, noting howev er that its asymptotic\nexpansion for y+≫1 is 1−(2×0.109y+)−1+..., corresponding to a K´ arm´ an parameter\nofκ= 0.218 .\nIII. MEAN VELOCITY OVERLAP PARAMETERS EVALUATED BY FITTING\nASYMPTOTIC EXPANSIONS TO [1−dU+/dy+]AND[Y−/angbracketleftuv/angbracketright+]\nAfter concluding in section II, that the Taylor expansions of [ Y− ∝angbracketle{tuv∝angbracketright+] and of [1 −\ndU+/dy+], extracted from the channel DNS of table I are, apart from minor question\nmarks, reasonably consistent, the more important question of th e K´ arm´ an parameter is\napproached. It can be extracted from either [1 −dU+/dy+] or [Y−∝angbracketle{tuv∝angbracketright+], which both have\nthe asymptotic expansion for large y+\n1−dU+/dy+≡Y−∝angbracketle{tuv∝angbracketright+→1−L\nReτ−1\nκy++O(y+)−3+1\nReτdW\ndYfory+≫1 (9)\nwhere [L/Reτ] is they+-derivative of the higher order linear term LYin the overlap of U+,\nmost recently discussed by MN2023, and W(Y) is a suitable wake function.\nThe asymptotic expansion (9) up to the third term, with L= 1.15 andκ= 0.417, as in\nMN2023, is shown in figure 3 and is seen to fit [1 −dU+/dy+] rather well for y+/greaterorapproxeql200. In\nparticular, the linear term in the overlap of U+, which leads to the L/Reτoffset in equation\n(9), is clearly visible in the data.\nA closer look, however, reveals a problem with fitting equation (9) to the data: As y+is\nreduced from its centerline value of Re τ, the sum of the first three terms of the asymptotic4\n0.1 1 10 100 \n1 - dU+/d y+Y-¢uv ²+\ny+1\n10 -1\n10 -2\n10 -31\n10 -1\n10 -2\n10 -3\n10 -4\n10 -5\nFIG. 2. Comparison of [ Y−/angbracketleftuv/angbracketright+] (left vertical axis) and [1 −dU+/dy+] (right vertical axis) versus\ny+, obtained from all the DNS of table I. Red •••, 1st two terms of Taylor expansion (7), evaluated\nfor profile #10; Orange •••, corresponding 3-term expansion; Aqua •••, Pad´ e approximant (8).\n0.95 0.96 0.97 0.98 0.99 11.01 1.02 \n0.93 0.94 0.95 0.96 0.97 0.98 0.99 1\n-áñ+1 -d+/d +\n+ 10                                             10 210 310 4\nFIG. 3. Enlarged view of the upper part of figure 2 for large y+. Red•••, basic outer asymptotic\nexpansion (1st three terms of equ. 9) with L= 1.15 andκ= 0.417, for Re τ= 104; Orange •••,\nsame for Re τ= 103.\nexpansion (9) is seen to first fall below the present DNS data of [1 −dU+/dy+] before\ncrossing them at y+≅50. This crossing prevents the asymptotic expansion (9) from bein g\ncontinued beyond the ( y+)−1term. To maintain the sum of the first three terms of (9)\nabove and to the left of the DNS data in figure 3 and to “leave room” f or the next term\n∝(y+)−3, the K´ arm´ an parameter κwould have to be increased beyond κ≈0.5, which is\nbeyond most estimates. A possible fix of the problem is to introduce a positive offset in\nthe log-law, which is compatible with its asymptotic character. Howev er, as long as the\nglaring discrepancy between [1 −dU+/dy+] and [Y− ∝angbracketle{tuv∝angbracketright+] is not clarified, there is no\npoint in perfecting the asymptotic expansion of [1 −dU+/dy+]. The above discrepancy is\nfurther analyzed in the next section IV, where it is magnified by lookin g at the log-indicator5\n23456\n;log-U\ny + 1                            10                            10 210 310 4(a) \n23456\n;log-uv \ny + 1                            10                           10 210 310 4(b) \nFIG. 4. Log indicator function for DNS #3, 4, 6, 7, 8, 10, 11, 12 of table I with color code given\nin the table. Profiles #4, 10 and 12 are highlighted by broken l ines (see text). Black · ··, leading\nTaylor term y+of Ξlog. (a) Ξ log−U=y+dU+/dy+; Aqua, pink and dark red · · ·, linear overlap\nterm 1.15Yof MN2023 for Re τ= 103,5.103and 104. (b) Ξ log−uv=y+[1−Y+/angbracketleftuv/angbracketright+]; Green ···,\nΞlog−uv= 2.3+2.10−3y+.\nfunction (1).\nIV. OVERLAP PARAMETERS EVALUATED FROM THE LOG INDICATOR FUNC TION\nAfter revealing the problems of determining overlap parameters by fitting asymptotic\nexpansions to [1 −dU+/dy+] and [Y−∝angbracketle{tuv∝angbracketright+] in section III, the standard method of looking\nfor constant portions of the log indicator function (1), i.e. for the K´ arm´ an “constant”, is\ninvestigated next.\nBoth variants of the indicator function, Ξ log−U≡y+dU+/dy+and Ξlog−uv≡y+[1−Y+\n∝angbracketle{tuv∝angbracketright+], are shown in figures 4(a) and (b), respectively. While the indicator functions Ξ log−U\nfor different Re τin figure 4(a) are clearly more consistent than the Ξ log−uvin figure 4(b),\nthetruechannel Ξ logmust be somewhere in between. In other words, extracting a reliab le\nvalue for κposes problems with both variants.\nThe indicator Ξ log−U(y+) of figure 4(a) is the one commonly shown in the literature.\nIt consists of an inner-scaled part, independent of Re τ, from which the outer-scaled parts\nemerge at Y≈0.2. These outer parts start out approximately linear, before going to zero\non the centerline. On the basis of DNS and experimental data, this lin ear overlap part for\nchannel flow has been fitted in MN2023 by 1 .15Y.\nFigure 4(b), on the other hand, shows Ξ log−uv=y+[1−Y+∝angbracketle{tuv∝angbracketright+] and reveals large\ndifferences relative to Ξ log−U, as well as between different DNS. The Ξ log−uvfor different\nReτand from different authors are seen to feather out already at y+≅30. Furthermore, all\nbut the DNS #12 approximately follow, beyond y+≈80, theinner-scaled linear function\n[2.3+0.002y+], the green dotted line in figure 4(b). Also, most Ξ log−uvrise much higher\nthan the corresponding Ξ log−U, before starting to drop to zero beyond Y≈0.4. The only\nexception is DNS #12 where Ξ log−uvshows no appreciable rise beyond y+≈80, which is\nmost likely due to an unusually small computational box with dimensions Lx= 2πhand\nLz=πh, which must increase the “organisation” of turbulence fluctuation s.\nDespite the reservations brought up above, it is of interest to fur ther analyze Ξ log−Uof\nfigure 4(a). In a first step, the fit L(Y) = 1.15Yof MN2023 for the outer-scaled part of\nΞlog−Uis refined for the present set of DNS profiles to\n/hatwideL(Y) =1.5\n50ln{1+e50(Y−0.09)→1.5Y−1.35 for e50(Y−0.09)≫1 (10)\nwhich represents a linear function branching off the inner part arou ndY= 0.09. It is noted\nin passing that, contrary to L(Y) = 1.15Y,/hatwideL(Y) becomes exponentially small for Y <0.096\n22.2 2.4 2.6 2.8 \n;log-U  -shifted linear    \ny + 10                                            10 210 310 4\nFIG. 5. Indicator function Ξ log−Uas in figure 4(a), minus the shifted linear function (10). The\nwake part beyond Y= 0.45 is shown as ···.\nand does therefore not have a match in the inner expansion of Ξ log−U.\nThe result ofsubtracting /hatwideL(Y) from Ξ log−Uis shownin figure5and is seento produce, for\nthe DNSconsidered,anexcellentcollapseoftheinnerpartsofΞ log−U, uptoY= 0.45,where\nthe wake contributions (indicated by dotted lines) become apprecia ble. The main result is\nthat, over the range of available Re τ’s, this inner part oscillates between 2.3 ( κ= 0.435)\nand 2.6 ( κ= 0.385) without an indication on whether it will eventually settle on a cons tant.\nThe difference to Ξ log−uv, and the close coincidence of the outer maximum of [Ξ log−U−\n/hatwideL(Y)] aty+≈400 (indicated by the grey arrow in figure 5) and the emerging outer\nmaximum19of∝angbracketle{tuu∝angbracketright+, gives rise to the speculation that the spatial oscillations may be due\nto a small, y+−dependent bias in the separation between mean and fluctuating str eam-wise\nvelocity. Pushing the speculation further, it is noted that, if plotte d versus ( y+)1/4(con-\nsistent with the Re−1/4\nτscaling of ∝angbracketle{tuu∝angbracketright+20–23), the period of the spatial oscillation in figure\n5 becomes approximately constant and the graph starts to look like the spatial version of\na simple under-damped step response in control theory, with the s tep being the sudden\nswitch-on of the no-slip condition at the wall.\nIn conclusion, no matter what the correct interpretation of figur es 4 and 5 will be, they\nillustrate why the turbulence community still cannot agree on K´ arm ´ an “constants”, even\nwithout considering the unusual indicator functions Ξ log−uvderived from the Reynolds\nstress∝angbracketle{tuv∝angbracketright+.\nV. CONCLUSIONS\nThe present extensive data analysis, using both standard and non -standard diagnostics,\nhas uncovered several previously unknown or neglected problems with the determination of\nchannel mean velocity overlap parameters, in particular with the slo pe of the log-law, i.e.\nthe K´ arm´ an parameter. Here, the focus has been on a clear des criptions of the problems\nand not on proposing still another set of parameters.\nIn section III the large differences, beyond y+=O(10), between the computed profiles\nof [1−dU+/dy+] and of [ Y− ∝angbracketle{tuv∝angbracketright+] have been documented. Both profiles have allowed\nto construct a reliable Taylor expansion about the wall. For large y+, on the other hand,\nfitting an asymptotic expansion to [1 −dU+/dy+] encounters problems and developing an\nasymptotic fit of [ Y−∝angbracketle{tuv∝angbracketright+] becomes virtually impossible due to inconsistent data. This is\nnot entirely surprising, as the evaluation of [1 −dU+/dy+] involves “only” the separation\nof the total stream-wise velocity into mean and fluctuating parts, which appears much less\nsensitive to the specific implementation of a DNS than determining the Reynolds stress.\nSection IV, devoted to the analysis of the log indicator function (eq u. 1), reinforces\nthe conclusions of section III, that the K´ arm´ an parameter κfor channel flow cannot be\ndetermined with the desired accuracyfrom presently availableDNS. Worse, all the standard7\nlog indicator functions Ξ log−U≡y+dU+/dy+, shown in figure 5, consistently exhibit, after\nsubtracting their outer-scaled part, an oscillation with a peak-to- peak amplitude of 0.3 , up\nto the highest available Re τ. This corresponds to κoscillating between 0.385 and 0.435 !\nSeveralinterpretationsofandspeculationsonthe oscillationsofΞ log−Umaybeconsidered\nfor future study, which will require massive computations. Three p ossible outcomes are\nenvisioned:\n- The oscillations are an artefact of the implementation of the DNS an d in a “perfect”\nDNS, the inner part of Ξ logwill monotonically approach κ−1.\n- In a DNS with significantly reduced residue of the mean momentum eq uation (3) and\nΞlog−U≅Ξlog−uv, the oscillations in figure 5 become more strongly damped and the\ndifferent indicators converge to a unique κ−1, specific to the channel.\n- It is finally conceivable that, even in DNS of extreme quality, which re spects Ξ log−U≅\nΞlog−uvat Reynolds numbers beyond 104, the oscillations persist. Such an outcome\nwould mean that the log law is only an approximate construct, a sacrile ge !\nTo answer these questions and resolve the problems of current DN S highlighted in the\npresent paper, the reduction of the residue of the momentum equ ation should have absolute\npriority. In some cases, high Reynolds numbers may also be desirable , but going much\nbeyond a Re τof 104does not appear useful, as the proper identification of higher orde r\nterms in the asymptotic expansions of ∝angbracketle{tuv∝angbracketright+and dU+/dy+requires the computational\nuncertainty to decrease as Re−1\nτ.\nACKNOWLEDGMENTS\nI first met Sreeni in the eighties, standing around a Helium jet in Yale’s Mason lab\ntogether with his student Paul Strykowski and discussing absolut e instability. Ever since,\nwe have kept in touch and I have always appreciated Sreeni’s openne ss and the priority of\nphysics in all the technical discussions with him.\n1T. von K´ arm´ an, “Mechanische ¨Ahnlichkeit und Turbulenz,” Nachr. Ges. Wiss. G¨ ottingen, Math. Phys.\nKlasse5, 58–76 (1930).\n2T. von K´ arm´ an, “Mechanical similitude and turbulence,” T ech. Rep. TM 611 (NASA, 1930).\n3Proc. 5th Int. Congr. Appl. Mech. (1938) wiley, NY.\n4M. Lee and R. D. Moser, “Direct numerical simulation of turbu lent channel flow up to Reτ= 5200,” J.\nFluid Mech. 774, 395–415 (2015).\n5H. Abe, H. Kawamura, and Y. Matsuo, “Surface heat-flux fluctua tions in a turbulent channel flow up to\nReτ=1020 with Pr=0.025 and 0.71,” International Journal of Hea t and Fluid Flow 25, 404–419 (2004).\n6S. Hoyas and J. Jim´ enez, “Scaling of the velocity fluctuatio ns in turbulent channels up to Reτ= 2003,”\nPhys. Fluids 18, 011702 (2006).\n7Y. Kaneda and Y. Yamamoto, “Velocity gradient statistics in turbulent shear flow: an extension of\nKolmogorov’s local equilibrium theory,” Journal of Fluid M echanics 929, A13 (2021).\n8A. Lozano-Dur´ an and J. Jim´ enez, “Effect of the computation al domain on direct numerical simulations\nof turbulent channels up to Reτ= 4200,” Phys. Fluids 26, 011702 (2014).\n9S. Hoyas, M. Oberlack, F. Alc´ antara- ´Avila, S. V. Kraheberger, and J. Laux, “Wall turbulence at hi gh\nfriction Reynolds numbers,” Phys. Rev. Fluids 7, 014602 (2022).\n10N. Afzal and K. Yajnik, “Analysis of turbulent pipe and chann el flows at moderately large reynolds\nnumber,” Journal of Fluid Mechanics 61, 23–31 (1973).\n11J. Kevorkian and J. D. Cole, Perturbation methods in applied mathematics (Springer, 1981).\n12K. R. Sreenivasan, “A unified view of the origin and morpholog y of the turbulent boundary layer struc-\nture,” in Turbulence Management and Relaminarization , edited by H. Liepmann and R. Narasimha\n(Springer-Verlag, 1987).\n13K. R. Sreenivasan, “The turbulent boundary layer,” in Frontiers in Experimental Fluid Mechanics , edited\nby M. Gad-el-Hak (Springer-Verlag, 1989).\n14K. R. Sreenivasan and A. Sahay, “The persistence of viscous e ffects in the overlap region and the mean\nvelocity in turbulent pipe and channel flows,” in Self-Sustaining Mechanisms of Wall Turbulence , edited\nby R. L. Panton (Comp. Mech. Publ., Southampton and Boston, 1 997) also available as arXiv:physics.flu-\ndyn/970801.8\n15P. A. Monkewitz and H. M. Nagib, “The hunt for the K´ arm´ an ‘co nstant’ revisited,”\nJournal of Fluid Mechanics 967, A15 (2023).\n16P. A. Monkewitz and H. M. Nagib, “Large Reynolds number asymp totics of the stream-wise normal stress\nin ZPG turbulent boundary layers,” J. Fluid Mech. 783, 474–503 (2015).\n17R. L. Panton, “A Reynolds Stress Function for Wall Layers,”\nJournal of Fluids Engineering 119, 325–330 (1997),https://asmedigitalcollection.asme.o rg/fluidsengineering/article-pdf/119/2/325/5900 \n18A. J. Musker, “Explicit expression for the smooth wall veloc ity distribution in a turbulent boundary\nlayer,” AIAA J. 17, 655–657 (1979).\n19P. A. Monkewitz, “Asymptotics of streamwise Reynolds stres s in wall turbulence,”\nJournal of Fluid Mechanics 931, A18 (2022).\n20X. Chen and K. R. Sreenivasan, “Reynolds number scaling of th e peak turbulence intensity in wall flows,”\nJournal of Fluid Mechanics 908, R3 (2021).\n21X. Chen and K. R. Sreenivasan, “Law of bounded dissipation an d its consequences in turbulent wall\nflows,” Journal of Fluid Mechanics 933, A20 (2022).\n22X. Chen and K. R. Sreenivasan, “Reynolds number asymptotics of wall-turbulence fluctuations,” arXiv\n2306.02438 , to appear in JFM (2023).\n23P. A. Monkewitz, “Reynolds number scaling and inner-outer o verlap of stream-wise Reynolds stress in\nwall turbulence,” arXiv 2307.00612 (2023)."    },    {        "title": "2402.14747v1.Inequality_indices_for_heterogeneous_systems__a_tool_for_failure_prediction.pdf",        "content": "Inequality indices for heterogeneous systems: a tool for failure prediction\nTarun Ram Kanuri1,∗, Subhadeep Roy2, Soumyajyoti Biswas1\n1Department of Physics, SRM University - AP, Andhra Pradesh 522240, India.\n2Department of Physics, BITS Pilani Hyderabad Campus, Secunderabad, Telangana 500078, India.∗\n(Dated: February 23, 2024)\nWe have numerically studied a mean-field fiber bundle model of fracture at a non-zero temperature\nand acted by a constant external tensile stress. The individual fibers fail (local damage) due to creep-\nlike dynamics that lead up to a catastrophic breakdown (global failure). We quantify the variations\nin sizes of the resulting avalanches by calculating the Lorenz function and two inequality indices –\nGini ( g) and Kolkata ( k) indices – derived from the Lorenz function. We show that the two indices\ncross just prior to the failure point when the dynamics goes through intermittent avalanches. For\na continuous failure dynamics (finite numbers of fibers breaking at each time step), the crossing\ndoes not happen. However, in that phase, the usual prediction method i.e., linear relation between\nthe time of minimum strain-rate and failure time, holds. The boundary between continuous and\nintermittent dynamics is very close to the boundary between crossing and non-crossing of the two\nindices in the temperature-stress phase space.\nINTRODUCTION\nFailure process in disordered solids [1] depends on en-\nvironment, such as pressure, temperature; driving condi-\ntion such as applied stress, strain rate; as well as struc-\ntural or material properties such as defects, micro-cracks,\nplasticity, etc. One important scenario in this regard is\nthat a material can break with time even if a constant\nstress, less than the critical value is applied on it - a\nphenomena widely know as the creep [2–4] in the ma-\nterial science community. Especially, the creep process\nshows a remarkable change as the temperature is tuned\ndue to the temperature-dependent dislocation motion,\nwhich becomes even easier when the thermal fluctuation\nis higher. A detailed study of the creep failure giving an\nestimation of creep time is very crucial for damage con-\ntrol regarding foundations like bridges, building blocks\nand other constructions where the material in constantly\nunder an external load. Apart from the obvious effect\nof temperature on the creep process, structural random-\nness and heterogeneity also plays an important role as\nthe creep takes place with the accumulation of plastic\nstrain and damages with time. The difficultly of han-\ndling the creep in heterogeneous solids leads to other ap-\nproaches, i.e, sub-critical crack growth dynamics [5] and\nphenomenological Voight’s model for precursory strain\n[6], where a time-dependent strain similar to experiments\nare observed assuming a heterogeneous feedback system.\nFor the present work, we will adopt a fiber bundle\nmodel [7], which is a statistical model lead by thresh-\nold activated dynamics [8]. The model is simple yet very\npowerful to simulate failure dynamics under different sce-\nnarios including a time-dependent creep process. Some\nstudies considered fluctuation in applied stress in form\nof thermal fluctuation [9]. In some studies, the failure\nevents are assumed to be probabilistic and not determin-\nistic in presence of a finite temperature [10]. Other stud-\nies involve complex rheological properties that depend onspecific rheological properties of fibers. For instance, the\nKelvin-Voigt rheology [12, 13], a nonlocal rheology [14] or\na damage rheology for each fiber [15]. Here, we consider\na ELS fiber bundle model but at a constant temperature\ninstead. A non-zero temperature will add an extra source\nof fluctuation, the thermal one, in addition to the fluctu-\nation in local strength we already have among the fibers.\nThough FBM has shown athermal creep [16] when the\napplied stress is closer to the critical value, a finite tem-\nperature makes the rupture events probabilistic, making\nthe failure process reminiscent of a thermal creep failure\nwhere a material can break under any stress no matter\nhow low it is. Such a probabilistic approach has been re-\nported in the in the literature exploring a time-dependent\nrelaxation dynamics [17]. For the present paper, we want\nto combine the probabilistic approach in FBM to mimic\ncreep with the inequality indices obtain from a generic\nform of the Lorenz function [18] to predict an upcoming\ncatastrophic failure.\nLorenz function L(p) has been introduced in the con-\ntext of wealth distribution and represents cumulative\nwealth fraction possessed by the pfraction of the pop-\nulation. For poorest and richest the L(p) function takes\nthe form 0 and 1 respectively; L(p) =pbeing the line\nof equality, showing equal distribution of wealth. The\ntwo most popular equality index in this context are as\nfollows: (i) Gini index (g) [19]: g= 0 corresponds to\nperfect equality, g= 0 for extreme inequality and (ii)\nKolkata index (k) [20]: k= 1/2 corresponds to perfect\nequality, g= 1 for extreme inequality. We get g=k, as\nper the law old Pareto [21] at an extremely competitive\nsituation between equality and inequality. In the present\npaper, we have used the same framework by replacing\nwealth by the avalanches during a failure process, where\nan avalanche of size scan occur with a probability p.arXiv:2402.14747v1  [cond-mat.stat-mech]  22 Feb 20242\nFIG. 1. The figure shows the variation of Gini index gand Kolkata index kas a function of temperature Tandσ= 0.2, 0.3,\n0.4, and 0.5. g-index and k-index compete with each other and becomes equal at 0.87. The threshold distribution is a delta\nfunction at 1.0.\nMODEL DESCRIPTION\nNumerically, we have studied a mean-field fiber bundle\nmodel (FBM) [7] at a non-zero temperature with an ex-\nternal load applied on it. A conventional FBM consists\nof a bunch (say L) of vertical fibers attached between\ntwo parallel bars (in case of 1d) or plates (in case of 2d),\nwhich are pulled apart with an external force Fcreating\na stress σ(=F/L) on each fiber. The disorder is intro-\nduced in the model through the fluctuation in strength\nfrom fiber to fiber. This is done by choosing the thresh-\nold stress of each fiber randomly from a distribution. For\nthe present work, we have chosen both the Uniform and\nthe Weibull distribution given as follows:\nρ(x) =\n\n1\n2∆for 1−∆≤x≤1 + ∆ ,\n0 elsewhere(1)\nρ(x) =k\nλ\u0010x\nλ\u0011k−1\ne(−x/λ)k(2)\nIn case of the Uniform distribution, the span ∆ of the\ndistribution is the measure of disorder while for Weibull,\nthe disorder strength is tuned by varying the shape ( k)\nand scale parameter ( λ).\nOnce the external stress crosses the threshold value\nfor a certain fiber, that fiber breaks irreversibly and the\nstress carried by the broken fiber is redistributed among\nFIG. 2. The figure shows the number of zeros versus the tem-\nperature for σ= 0.2 and a delta-distribution. As Tincreases,\nwe observe a decrease in the number of zeros and finally they\nsaturates at a low value at high temperature.\nthe rest of the model. In this paper, we adopted a equal-\nload sharing (ELS) scheme [22] where the stress after\neach rupture is redistributed equally among all surviv-\ning fibers. This is also the mean-field limit of the model.\nApart from ELS scheme there are other redistribution\nschemes that has been explored in the context of FBM:\n(i) local load sharing (LLS) scheme [23] where the stress\nis redistributed among the nearest neighbours only, (ii)\nmixed mode redistribution [24] where the stress redistri-\nbution depends on how far a fiber is from the broken one,3\nFIG. 3. The figure shows the variation of Gini index gand Kolkata index kas a function of temperature Tandσ= 0.3. As\na threshold distribution, we have used a uniform distribution spanning from 1 −∆ to 1 + ∆. The disorder strength ∆ is set\nto be 0.3, 0.4, 0.5, and 0.6. µandσcare respectively the mean strength and the critical stress for quasi-static lading. g-index\nandk-index compete with each other and becomes equal at 0.87.\netc.\nAs mentioned in the introduction, inclusion of temper-\nature makes the rupture events probabilistic depending\non the temperature as well as applied stress:\nP(σ, T) =σ\nσcexp\u0014\n−1\nT\u0010σc\nσ−1\u0011\u0015\n(3)\nwhere, σandσcare respectively the local stress and\nthreshold stress of a fiber. σcis either set to be either 1\n(equal for all fibers) or chosen from a random distribu-\ntion (say Uniform or Weibull). Due to such a probabilis-\ntic approach, a fiber can break even when σ < σ c. The\nrelaxation dynamics of the model goes as follows:\n1. At each time we chose a probability Pi(σ, T) for\na fiber iand compare it with a random number r\nchosen with uniform probability between 0 and 1.\n2. A fiber breaks irreversibly if Pi(σ, T)> rand the\nstress of the fiber is redistributed equally among all\nother surviving fibers increasing the value of σand\nhence the probability P(see Eq.3).\n3. In the next time step we chose another Pdepending\non temperature Tand the updated local stress σ\nand repeat the previous step.\n4. This process goes on until all fibers are broken sug-\ngesting global failure.We want to stress the following point here. For T= 0,\nP(σ, T) in equation 3 becomes zero suggesting no fibers\nwill break unless σ≈σc.\nANALYSIS\nConsider an equal load sharing FBM, where all fibers\nhave the same failure threshold σc. There is an external\nnoise T. Now, under a load σ, a fiber will break if σ≥\nσc. But even when σ < σ c, the fiber can break with a\nprobability given by Eq.3.\nNow, if a time step tis defined as one step of load re-\ndistribution, then the fraction of surviving fibers is given\nby\nU(t, σ, σ c, T) =σT\nσcln\u0014τ−t\nTexp (−1/T)+ 1\u0015\n,(4)\nwhere τis the failure time of the system, given by\nτ=Texp\u0012\n−1\nT\u0013h\nexp\u0010σc\nσT\u0011\n−1i\n. (5)\nThe magnitude of an avalanche, defined as the number\nof fibers failing in one time step, is then\nS(t) =\f\f\f\fdU\ndt\f\f\f\f=σT\nσc1\n(τ−t) +Texp (−1/T).(6)4\nThe Lorenz function is then\nL(p) =pτR\n0S(t)dt\nτR\n0S(t)dt. (7)\nThis on simplification gives,\nL(p) =ln|Texp (−1/T) +τ| −ln|Texp (−1/T) + (1 −p)τ|\nln|Texp (−1/T) +τ| −ln|Texp (−1/T)|\n(8)\nThe value of the Gini index gat the failure point can\nbe calculated from\ngf= 1−21Z\n0L(p)dp. (9)\nThis on simplification gives\ng= 1 + 2\"\nexp\u0010\n1−σc\nσT\u0011\n−1\nln\u0000\n1 +exp\u0000σc\nσT\u0001\n−1\u0001#\n.\n(10)\nThe value of the Kolkata index ( k) at the failure point\ncan be obtained from\n1−k=L(k), (11)\nwhich needs to be numerically solved.\nNUMERICAL RESULTS\nIn this section, we will discuss the numerical results\nthat will build towards the discussion as well as the main\nfinding of the present article. As already mentioned in\ntheModel Description section, the numerical simulation\nis done for a mean field fiber bundle model with system\nsize 105and 104realizations. For the first part of the\nnumerical results, we have considered that all fibers have\nthe same breaking threshold equals to unity while in the\nlater part Uniform and Weibull distributions are consid-\nered.\nFigure 1 the variation of the Gini index andKolkata\nindex as we vary the external temperature, which tunes\nthe strength of the thermal fluctuation. The distribution\nof the threshold values are chosen from a delta function\nwith it’s peak at 0.5. Due to the choice of the distri-\nbution, the fluctuation among the local threshold values\nis eliminated. Though, this does not eliminate the ther-\nmal fluctuation (equation 3) and the rupture events are\nstill probabilistic and not deterministic. The model is\nobserved under four different external stress values: 0.2,\n0.3, 0.4 and 0.5. The points corresponds to the numeri-\ncal results for a δ-distribution at 0.5. The dotted lines on\nthe other hand is the comparison of the numerical results\nwith our analytically observed g-index andk-index . The\nbehavior can be divided in two different regions:•Region I: This region is observed at a higher tem-\nperature where the thermal fluctuation is suffi-\nciently large. We observe the Gini index here to\nbe less than the k-index. Such behavior is evident\nthrough both analytical results as well as numerical\ndata.\n•Region II: This region takes place where the tem-\nperature is low and hence the thermal fluctuation\nis much less. The fiber rupture events are almost\ndeterministic here since the probabilistic element\nis negligible due to this very small temperature.\nHere the g−index overcomes k−index and becomes\nlarger.\nFIG. 4. Boundaries showing the separation of Region I and\nRegion II . Boundary 1 is drawn from the direct comparison\nofg−index and k−index. Boundary 2 is drawn where the\nnumber of zeroes disappears from the system dynamics.\nAsσincreases, we observe the crossover between region\nI and region II starts to take place at a lower tempera-\nture. Lets call this point T∗. We observe T∗≈0.38\nforσ= 0.2 while for σ= 0.5, we get T∗≈0.15.\nThough T∗changes with σ, the crossover takes place\nforg=k= 0.87 irrespective of the value of external\nstress. This universal value has been observed earlier\nin the context of other statistical systems as well, like\nBak-Tang-Wiesenfeld (BTW) [25], Manna model [26],\nEdwards-Wilkinson (EW) [27], interface dynamics based\non fiber bundle model [28], etc.\nNext we turn to find another way to understand the\ncrossover point when gbecomes more than k. Since the\ndynamics of the model is lead by the thermal fluctuation,\nthe probability that a certain fiber will break decreases\nas we go to lower temperature or lower applied stress.\nWhich means, at lower Tandσ, there will be many time\nsteps in-between where no rupture events happen. We\ncall this the zeroes of the model evolution. Figure 2\nshows the number of zeroes as a function of tempera-\nture when the external stress is kept at σ= 0.2. For\nlowTthere are many zeroes observed while it decreases5\nFIG. 5. The figure shows the variation of Gini index gand Kolkata index kas a function of temperature Tandσ= 0.3.\nAs a threshold distribution, we have used a Weibull distribution of scale parameter λand shape parameter k.µandσcare\nrespectively the mean and the critical stress for quasi-static lading. g-index and k-index compete with each other and becomes\nequal at 0.87. .\nand becomes gradually 0 after a certain temperature (say\nT′). Figure 3 shows the same result as figure 1 but for a\nuniform distribution spanning between 1 −∆ to 1 + ∆,\n∆ being the strength of disorder. Since the analysis be-\ncomes complicated due the fluctuation of strength from\nfiber to fiber (since its a distribution now instead of a\ndelta-function). To chose a certain σcin equation 10 and\n11, we adopted the following assumptions: (i) since its\na uniform distribution, we can assume the mean ( µ) to\nbe the critical value; (ii) we can drive the system in a\nquasi-static manner at zero temperature and found out\nthe average critical stress and use it.\nIn figure 3, we have kept the same external stress\nσ= 0.3 but changed the width of the uniform distri-\nbution, and hence the strength of disorder. We observe\na far better fit for g−index (equation 10) and k−index\n(equation 11) when we consider σcto be the critical stress\nfor quasi-static loading rather than the mean strength.\nAlso, quality of such fit decays as we go to higher disor-\nder strength (high ∆).\nWe want to argue here that, T∗where gbecomes\ngreater than k, and T′where number of zeroes itself be-\ncomes 0 has the same values. We will discuss this in de-\ntails next. Figure 4 shows this boundary between Region\nI and Region II, independently calculated from (i) in-\ntermittence behaviour of the system (study of fraction of\nzeroes ) and (ii) the interplay of gandk, respectively. Thepoints corresponds to the numerical results found from\ndirect simulation. The solid lines, on the other hand,\nare drawn from the following analytical approaches: (i)\nBoundary 1: This shows the ( σ, T) coordinates beyond\nwhich there are no zeroes in avalanches, suggesting that\nat least one fiber will break at each time step. This is\ndone by using the analytical results from the paper by\nPradhan et al [10]. (ii) Boundary 2: This boundary\nconsists of the coordinates below which govercomes k.\nThis boundary is obtained by numerically solving the\nequations 10 and 11. The bottom line this these bound-\naries, Boundary 1 orBoundary 2 , separates two regions\nI and II where the method of failure prediction is vastly\ndifferent. Region I , is the area above the boundary which\nis accessible by increasing either the temperature of the\nexternal stress. In this region, we observe fiber ruptures\nat each and every time step. The usual linear relation be-\ntween the time of minimum strain-rate and failure time\nholds here and the prediction of upcoming failure can\nbe done through this relation. On the other hand, for\nlowTandσ, we obtain Region II , where intermittent\nbehavior is observe during the rupture events and above\nlinear relationship does not hold good. However, in this\nregion the failure point can be predicted from the the\ncompetition of gandkindex and the Gini index is ob-\nserved to show higher value than k-index just before the\ncatastrophic failure.6\nFigure 5 finally shows the comparison of the analytical\nand numerical behavior of gandkindices for a Weibull\ndistribution of shape parameter kand scale parameter\nλ. This shows the same fact that the fitting is better by\nconsidering the critical stress for a quasi-brittle loading\nrather than the mean of the distribution. This strengthen\nour claim regarding the universality of our results irre-\nspective of the choice of the threshold distribution.\nDISCUSSION\nUnder a constant load, which is less than the critical\nload of a disordered solid, there are creep induced fail-\nures that lead to the eventual collapse of the system in\npresence of external noise e.g., temperature. The fiber\nbundle model of fracture reproduces essential features of\nsuch failure through a simple mean-field dynamics.\nHere we have shown that the resulting avalanches of\nthe creep failure can be quantified in terms of their vari-\nations in sizes. Particularly, a Lorenz function for the\navalanches under a constant load and temperature can\nbe analytically calculated for a delta-function threshold\ndistribution with it’s peak at 1. Consequently, the Gini\n(g) and Kolkata ( k) indices can also be derived (see Eq.\n(10) and (11)). Curiously, the time variations of gand\nk(cumulative measured of inequality until that time)\ncross each other just prior to the failure for sufficiently\nlow values of external load and thermal noise that en-\nsure an intermitten avalanche dynamics. For large load\nand/or thermal noise, the avalanche dynamics become\ncontinuous, where such crossing do not happen. In that\nregion, however, the linear relation between the time of\nstrain-rate minimum and failure time is well maintained.\nFor wides distributions of failure thresholds, for exam-\nple Weibull or uniform, the observations are qualitatively\nsimilar (see Figs. 3, 5). However, the analytical expres-\nsion for such cases could be done by approximating the\nthreshold distribution by its smallest value (for narrow\nthreshold distributions) or the average value (for wide\ndistributions).\nIn conclusion, for intermittent creep rupture dynam-\nics, the variations in sizes of the resulting avalanches grow\nwith time and provide a very reliable precursory signal for\nthe imminent failure in a wide range of the temperature-\nstress phase space, where traditional precursory indica-\ntions do not work.∗tarunram˙kanuri@srmap.edu.in;\nsubhadeep.r@hyderabad.bits-pilani.ac.in; soumyajy-\noti.b@srmap.edu.in\n[1] H. J. Herrmann and S. Roux, Statistical Models for the\nFracture of Disordered Media, North-Holland, Amster-\ndam (1990).\n[2] H. Nechad, A. Helmstetter, R. El Guerjouma, and D.\nSornette, J. Mech. Phys. Solids 53, 1099 (2005).\n[3] H. Nechad, A. Helmstetter, R. El Guerjouma, and D.\nSornette, Phys. Rev. Lett. 94, 045501 (2005).\n[4] S. Biswas, D. F. Castellanos, M. Zaiser, Sci. Rep. 10,\n16910 (2020).\n[5] I. G. Main, Geophys. J. Int. 142, 151 (2000).\n[6] B. Voight, Science 243, 200 (1989).\n[7] A. Hansen A, P. C. Hemmer, S. Pradhan. The Fibre Bun-\ndle Model. Germany: Wiley VCH (2015).\n[8] P. Ray and B. K. Chakrabarti, Solid State Commun. 53,\n477 (1985).\n[9] S. Ciliberto, A. Guarino, and R. Scorretti, Physica D 158,\n83(2001).\n[10] S. Pradhan, A. K. Chandra, and B. K. Chakrabarti,\nPhys. Rev. E 88, 012123, (2013).\n[11] B. K. Chakrabarti, S. Biswas, S. Pradhan. Front Phys 8,\n613392 (2021).\n[12] R. C. Hidalgo, F. Kun, and H. J. Herrmann, Phys. Rev.\nE65, 032502 (2002).\n[13] F. Kun, Y. Moreno, R. C. Hidalgo, and H. J. Herrmann,\nEurophys. Lett. 63, 347 (2003).\n[14] E. A. Jagla, Phys. Rev. E 83, 046119 (2011).\n[15] Z. Danku and F. Kun, Sci. Rep. 3, 2688 (2013).\n[16] S. Roy and T. Hatano, Phys. Rev. E 97, 062149 (2018).\n[17] S. Roy and T. Hatano, Phys. Rev. Research 2, 023104\n(2020).\n[18] M. O. Lorenz, Publication of the American Statistical\nAssociation 9, 209219 (1905).\n[19] C. Gini, Economics Journal 31, 124126 (1921).\n[20] A. Ghosh, N. Chattopadhyay and B. K. Chakrabarti,\nPhysica A: Statistical Mechanics and Applications 410,\n3034 (2014).\n[21] V. Pareto, A. N. Page, Translation of Manuale di econo-\nmia politica (Manual of political economy), A.M. Kelley\nPublishing, New York (1971).\n[22] T. Pierce, J. Text. Ind. 17, T355 (1926).\n[23] S. L. Phoenix, Adv Appl Probab. 11, 153 (1979).\n[24] S. Pradhan, B. K. Chakrabarti, and A. Hansen, Phys.\nRev. E 71, 036149 (2005).\n[25] P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59,\n381 (1987).\n[26] S. S. Manna, J. Phys. A: Math. Gen. 24, L363 (1991).\n[27] S. F. Edwards, D. R. Wilkinson, Proc. R. Soc. Lond. A\n381, 17 (1982).\n[28] S. Biswas, B. K. Chakrabarti, Phys. Rev. E 88, 042112\n(2013)."    },    {        "title": "2402.14791v2.Amplified_Amplitude_Estimation__Exploiting_Prior_Knowledge_to_Improve_Estimates_of_Expectation_Values.pdf",        "content": "Amplified Amplitude Estimation: Exploiting Prior Knowledge to Improve\nEstimates of Expectation Values\nSophia Simon,1,∗Matthias Degroote,2,†Nikolaj Moll,2\nRaffaele Santagati,2Michael Streif,2and Nathan Wiebe3, 4, 5, ‡\n1Department of Physics, University of Toronto, Canada\n2Quantum Lab, Boehringer Ingelheim, 55218 Ingelheim am Rhein, Germany\n3Department of Computer Science, University of Toronto, Canada\n4Pacific Northwest National Laboratory, Richland WA, USA\n5Canadian Institute for Advanced Studies, Toronto, Canada\nWe provide a method for estimating the expectation value of an operator that can utilize prior\nknowledge to accelerate the learning process on a quantum computer. Specifically, suppose we have\nan operator that can be expressed as a concise sum of projectors whose expectation values we know\na priori to be O(ϵ). In that case, we can estimate the expectation value of the entire operator\nwithin error ϵusing a number of quantum operations that scales as O(1/√ϵ). We then show how\nthis can be used to reduce the cost of learning a potential energy surface in quantum chemistry\napplications by exploiting information gained from the energy at nearby points. Furthermore,\nwe show, using Newton-Cotes methods, how these ideas can be exploited to learn the energy via\nintegration of derivatives that we can estimate using a priori knowledge. This allows us to reduce\nthe cost of energy estimation if the block-encodings of directional derivative operators have a smaller\nnormalization constant than the Hamiltonian of the system.\nI. INTRODUCTION\nQuantum simulation algorithms have progressed to the point where their asymptotic worst-case computa-\ntional complexity is essentially optimal [1–4]. Despite this and improved constant factor analyses [5, 6],\ncurrent resource estimates for industrially relevant applications such as FeMoco [7–9] indicate that solving\nthese problems with existing algorithms will likely remain out of reach for early fault-tolerant quantum com-\nputers let alone NISQ-era quantum computers. This suggests that further improvements may be needed to\nachieve a practical advantage.\nCrucially though, classical algorithms for quantum chemistry algorithms try to use prior knowledge as much\nas possible, while quantum algorithms tend to use such information less frequently. Cases where prior\nknowledge is used in quantum computing include state learning algorithms such as compressed sensing [10],\n“warm starts” in quantum machine learning [11], quantum machine learning with inductive biases [12–15]\nand even the phase estimation protocol in Shor’s factoring algorithm [16]. These examples exploit a special\nstructure that is known in the problem to surpass the performance of algorithms that do not possess this\nknowledge. Quantum simulation often uses prior knowledge in limited ways. In particular, in applications\nsuch as chemistry, we can often use classically tractable methods such as Hartree-Fock (HF) or Density\nFunctional Theory (DFT) to begin with an initial estimate of quantities of interest such as the ground state\nenergy [17, 18]. Existing quantum algorithms seldom depend strongly on this knowledge apart from its\nuse in providing an ansatz that has large overlap with the target eigenstate that we may wish to prepare\n(for problems that involve simulation of ground state energies or related quantities) [19, 20]. This is a\nsignificant drawback as in the most extreme case, perfect classical knowledge about the quantity one wishes\nto estimate eliminates the need for a quantum computing simulation, while in more realistic cases, it might\ndrastically reduce the cost of the simulation. Thus providing a new approach to simulation that properly\nutilizes this prior knowledge is a promising avenue if we wish to demonstrate a useful quantum advantage\n∗sophia.simon@mail.utoronto.ca\n†matthias.degroote@boehringer-ingelheim.com\n‡nathan.wiebe@utoronto.caarXiv:2402.14791v2  [quant-ph]  1 Mar 20242\non intermediate-term quantum computers which may not be able to execute the billions of gate operations\nneeded by existing algorithms [6, 8, 9, 21–24].\nIn this work, we take a step towards solving this problem by introducing an approach to learning expectation\nvalues that we call amplified amplitude estimation (AAE). The idea behind this technique is to use amplitude\namplification in concert with amplitude estimation to estimate probabilities that are a priori known to be\nsmall ( O(ϵ)) within error ϵusing a number of queries that scales only as O(1/√ϵ). We show how AAE can\nbe used to estimate expectation values of operators which do not satisfy the smallness assumption directly,\nassuming that we have some prior knowledge of the expectation values we wish to estimate. More specifically,\nwe use linear combinations of unitaries [25] together with techniques for implementing matrix square roots\nof positive semi-definite (PSD) operators to reduce the problem of estimating generic expectation values to\nthe case of estimating small probabilities. This allows us to extend the O(1/√ϵ) scaling beyond the case of\nsmall probability estimation.\nWe provide several applications for these ideas. First, we apply the idea to expectation value estimation\nof one-body operators for chemistry. Specifically, we show that this approach can yield a polynomial ad-\nvantage if the uncertainty provided is small relative to the mean value of the coefficients of the operator\nthat we wish to measure. We then discuss how to leverage knowledge of the expectation value for a given\nmolecular configuration to learn the expectation value for a nearby configuration at low cost, assuming that\nan appropriate smoothness condition on the coefficients in the decomposition of the one-body operator is\nsatisfied.\nOur second application involves estimating energy differences using Newton-Cotes formulas to integrate the\nenergy derivative. We show, under similar assumptions to the one-body operator case, that we can use\nthis approach to take a set of configurations for a molecular system and interpolate from them in order\nto estimate the energy at nearby points at a lower cost than you would ordinarily be able to do so using\nstandard approaches to estimating the energy of the system such as phase estimation.\nII. BACKGROUND\nIt is a common folk theorem in quantum metrology and quantum computing that classical strategies can be\nused to estimate a quantity within error ϵ(with high probability) using O(1/ϵ2) applications of a preparation\nunitary whereas coherent methods such as phase estimation can learn the same quantity using O(1/ϵ)\napplications. The latter scaling is often referred to as Heisenberg limited scaling, whereas the former is\ncalled shot noise limited scaling. It is perhaps less well appreciated that small or large probabilities can\nactually be estimated using quadratically better scaling than would otherwise be expected. Specifically,\nimagine that we have a state of the form√1−ϵ|0⟩+√ϵ|1⟩and that our goal is to learn the probability of\n|1⟩within error O(ϵ). Using the sample mean as an unbiased estimator of the probability, we have that the\nnumber of samples scales as\nNsamp∈O\u0012σ2\nϵ2\u0013\n=O\u0012ϵ(1−ϵ)\nϵ2\u0013\n=O\u00121\nϵ\u0013\n, (1)\nwhere σ2denotes the variance. This represents a quadratic advantage compared to the shot noise-limited\nscaling. The guarantee that the amplitude of |1⟩is small, i.e. O(√ϵ), is essential in this scaling and can\nbe considered prior knowledge. In this work, we show that the scaling for estimating the expectation value\nof an operator with prior knowledge can be further improved to O\u0010\n1√ϵ\u0011\nthrough amplitude amplification\ntogether with amplitude estimation.\nThe idea behind standard amplitude estimation [26] is intimately linked to that of amplitude amplification.\nSpecifically, amplitude amplification is a way of transducing a probability into a phase. After converting the\nprobability into a phase, the quantum phase estimation algorithm can be employed to learn the resultant\nphase. If we wish to compute the probability P=|⟨0|ψ⟩|2for some quantum state |ψ⟩, then we can estimate\nit by constructing a walk operator W=−(11−2|ψ⟩⟨ψ|)(11−2|0⟩⟨0|). The eigenvalues of Wthat are supported\nby|ψ⟩are then of the form e±iθwhere θ= sin−1(√\nP). Thus we can learn a probability Pby performing\nquantum phase estimation on Wto yield θfrom which we can estimate P= sin2(θ). The standard statement\nof amplitude estimation’s performance is given below.3\nTheorem 1 (Based on Theorem 12 in Brassard et al. [26]) .LetOψ:|0⟩ 7→ | ψ⟩andOϕ:|0⟩ 7→ | ϕ⟩\nbe invertible oracles and let R=11−2Oψ|0⟩⟨0|O†\nψ, be the reflection with respect to |ψ⟩. There exists an\nalgorithm that can estimate, for any state |ϕ⟩,⟨ϕ|(11−R)/2|ϕ⟩within error ϵand with a probability of\nsuccess greater than 8/π2while using a number of queries to OψandOϕthat is in O(1/ϵ).\nThis result is known to be optimal [1]. It is impossible to achieve better than an Ω(1 /ϵ) scaling for the phase\nestimation problem without making further assumptions. A key assumption behind this result is that no\nprior knowledge is given about the value of the phase in question. While the estimation of small probabilities\nprovides superior scaling, as shown above, its advantage is balanced by the assumptions of small probability\nand an equally small tolerance for the estimation error. It remains a question whether it is possible to\nfind a method that, under certain assumptions, can provide the O(1/√ϵ) scaling without requiring that the\nexpectation value that we wish to estimate is small. We address this question with a new technique called\namplified amplitude estimation, described in greater detail below. The general idea is as follows: if we have\na good estimate of the expectation value that we wish to improve, we can transform the question into the\nproblem of estimating the difference between the prior value we have and the true value we wish to know.\nIn this way, we have transformed the problem into the estimation of a small probability.\nIII. AMPLIFIED AMPLITUDE ESTIMATION\nImagine we know the probability of a measurement outcome within some small error. The question is how\nthis knowledge reduces the cost of probability estimation. In this section, we provide a way to leverage this\nknowledge by applying amplitude amplification on a small probability to boost it to a much larger probability\nif our prior knowledge rules out the possibility of over-rotation. We will later show how this can be applied\nto solve the problem of estimating the expectation values of operators.\nWe reframe the problem of estimating the overlap between two states of Theorem 1 in terms of the expectation\nvalue of a projector, ⟨ψ|Π|ψ⟩, i.e., Brassard et al. estimate ⟨ψ|Πgood|ψ⟩=⟨ψ|(11−R)/2|ψ⟩where Π goodis a\nprojector onto a “good” subspace and Ris a reflection operator [26]. For the case where the “good” subspace\nconsists of only a projector onto a state |ϕ⟩, the resulting probability is simply |⟨ψ|ϕ⟩|2. We can estimate the\ngood probability within error ϵusing their approach in O(1/ϵ) queries to the underlying state preparation\nunitaries, which is known to be optimal in the worst-case scenario.\nA fundamental difference compared to Brassard et al. [26] is that here we assume having access to prior\nknowledge in the form of an upper bound, P0, on the quantity we want to estimate, ⟨ψ|Π|ψ⟩. The new\nquantity we want to estimate is then the deviation from this upper bound, δ:=⟨ψ|Π|ψ⟩ −P0. This\nassumption of prior knowledge precludes the worst-case bounds discussed previously, which allows us, under\ncertain restrictions, to show better scaling than those results allow. More specifically, we use AAE to\nestimate δ, which then also yields an estimate of ⟨ψ|Π|ψ⟩. The number of queries AAE needs to estimate\nsuch a “good” or marked probability within error ϵunder these assumptions is given below.\nLemma 2 (Amplified Amplitude Estimation) .LetΠbe a projector and assume we have access to a unitary\noracle Oψsuch that Oψ|0⟩=|ψ⟩and a reflection oracle RΠsuch that Π := ( 11−RΠ)/2. We assume that\n⟨ψ|Π|ψ⟩=P0+δ, where P0= sin2(π/(2(2µ+ 1))) is a known upper bound on the quantity ⟨ψ|Π|ψ⟩which\nwe will interpret as a success probability. Furthermore, µ≥1is an integer and δ∈[−P0,0)is an unknown\nnegative number. Then there exists a quantum algorithm that can estimate δ, and hence ⟨ψ|Π|ψ⟩, within\nerror ϵ∈O(P2\n0/|δ|)⊆o(1)and with failure probability at most δ′using\nNqueries ∈O \nP1/2\n0log(1/δ′)\nϵ\u0012P0\n|δ|\u0013!\nqueries to RΠandOψ.\nProof. By assumption, we have that P0is a probability such that for some positive integer µ≥1\nsin−1(p\nP0) =π\n2(2µ+ 1), (2)4\nwhich is to say that the angle in the rotation carried out by a single application of amplitude amplification\nis at mostπ\n2(2µ+1). Further, we know that there exists a projector Π and a negative number δsuch that\n⟨ψ|Π|ψ⟩=P0+δ= sin2\u0012π\n2(2µ+ 1)\u0013\n+δ. (3)\nNext, note that the reflection operator Rψ:= (11−2|ψ⟩⟨ψ|) can be implemented using the unitary oracle Oψ:\nRψ=11−2Oψ|0⟩⟨0|O†\nψ. (4)\nThis allows us to implement amplitude amplification through the walk operator\nW:=−RψRΠ=−Oψ(11−2|0⟩⟨0|)O†\nψ(11−2Π) (5)\nusing O(1) queries to our oracles OψandRΠ[26] .\nNext, consider the boosted probability that would be found by applying Wµ, corresponding to µapplications\nofW, to the state |ψ⟩to amplify the probability given in Eq. (3). Using this amplified oracle is equivalent\nto constructing a new walk operator W′such that\nW′:=−\u0010\n11−2WµOψ|0⟩⟨0|O†\nψWµ†\u0011\n(11−2Π). (6)\nThe idea then is to perform amplitude estimation with the new walk operator W′. This operator,\nfrom the analysis of [26], has eigenvalues (within the two-dimensional subspace in question) of the form\nexp(±iarcsin√P1) where\nP1= sin2\u0010\n(2µ+ 1) sin−1(p\nP0+δ)\u0011\n= sin2\u0012π\n2\u0012sin−1(√P0+δ)\nsin−1(√P0)\u0013\u0013\n. (7)\nIf we are given ˆP1, anϵ′-accurate estimate of the probability, then we can solve for our estimate ˆδofδas a\nfunction of ϵ′:=P1−ˆP1. This expression and a series expansion for the quantity in terms of small ϵ′andδ\ncan be found using elementary algebra and is given below:\nˆδ=ˆ⟨ψ|Π|ψ⟩ −P0\n= sin2 \n2 sin−1(pˆP1) sin−1(√P0)\nπ!\n−P0\n= sin2\u00122 sin−1(√P1−ϵ′) sin−1(√P0)\nπ\u0013\n−P0\n=δ−8P0(1−P0) arcsin2(√P0)ϵ′\nπ2δ+O(ϵ′(1 +ϵ′/δ)). (8)\nHere we require P0<1 for the leading order term in the asymptotic series to have this scaling since for P0= 1\nthe arcsin and sine trivially invert each other, making this form inappropriate for a series expansion along\nwith causing the leading order term in the previous expansion to vanish. Note though that the condition\nP0<1 follows directly from Eq. (2) if µ≥1 since we then have that P0≤arcsin( π/6)2= 1/4. For this\nreason, we assume that µ≥1, which then automatically ensures that P0<1. From (8) we can obtain the\nerror in the estimated value of δ:\n|ˆδ−δ| ∈O\u0012P0(1−P0) arcsin2(√P0)ϵ′\n|δ|\u0013\n. (9)\nNote that we neglected the O(ϵ′(1 +ϵ′/δ)) term from Eq. (8) since ϵ′<ϵ′\nδ.\nNext, to understand the asymptotic behavior of this expression, it is useful to consider the case where P0≪1\nsuch that arcsin(√P0)∈Θ(√P0) and similarly, 1 −P0∈Θ(1). This leads to the conclusion that\n|ˆδ−δ| ∈O\u0012P2\n0ϵ′\n|δ|\u0013\n. (10)5\nAlgorithm 1: AmplifiedAmplitudeEstimation : Estimation of probabilities with prior knowledge\nInput: Reflection oracle RΠ= 11−2Π, unitary state preparation oracle Oψ:|0⟩ 7→ | ψ⟩, integer µ≥1,\nallowable estimation error ϵ, allowable failure probability δ′.\nOutput: An estimate of ⟨ψ|Π|ψ⟩.\n1. Construct the operator W′=−(11−2WµOψ|0⟩⟨0|O†\nψWµ†)(11−2Π)\nwhere W=−RψRΠandRψ=Oψ(1−2|0⟩⟨0|)O†\nψ;\n2. Prepare the state Oψ|0⟩;\n3. Apply a unitary implementation of an amplitude estimation protocol, UAE(ϵ, δ′), with W′to|0⟩a|ψ⟩where\nϵ, δ′are the requested error and failure probabilities for the protocol and |0⟩ais an ancilla register;\n4. Let ˆP1denote the value of the phase returned in the ancilla register at the end of the amplitude estimation\nprotocol;\n5.P0←sin2\u0010\nπ\n2(2µ+1)\u0011\n6. Classically compute and return sin2\u0012\n2 sin−1(√\nˆP1) sin−1(√P0)\nπ\u0013\nas our estimate of ⟨ψ|Π|ψ⟩.\nThus, if we wish to estimate δwithin error ϵ, it suffices to choose\nϵ′∈Θ\u0012ϵ|δ|\nP2\n0\u0013\n, (11)\nwhich implies that if we wish to perform amplitude estimation within error ϵand probability of failure\nsmaller than δ′, we will need a number of queries to the boosted walk operator W′that obeys [19, 26]\nNW′∈O\u0012P2\n0log(1/δ′)\nϵ|δ|\u0013\n. (12)\nThe factor of log(1 /δ′) comes from the need to repeat the amplitude estimation procedure log(1 /δ′) times\nto ensure that the failure probability is at most δ′. Further, taking Eq. (11) into account, the smallness\nassumptions leading to Eq. (8) are satisfied if we pick ϵ∈O(P2\n0/|δ|)⊆o(1). As there are O(µ) =O(1/√P0)\napplications of Wneeded for each application of the walk operator W′, we then find that\nNqueries =NW∈O(µNW′) =O \nP1/2\n0log(1/δ′)\nϵ\u0012P0\n|δ|\u0013!\n. (13)\nThis means that to leverage the prior information, we perform an estimation procedure not directly on the\nexpectation value but on the difference δbetween the expectation value and our (known) upper bound on\nit. The advantage arises only in those cases where our upper bound P0is only slightly larger than the\nvalue of δ. More specifically, we have an advantage over standard amplitude estimation if P3/2\n0∈o(|δ|).\nNote that above we assume that ϵ∈O(P2\n0/|δ|). However, if we have the more stringent assumption that\nϵ∈Θ(P2\n0/|δ|), then the scaling goes like\nNqueries ∈O\u0012log(1/δ′)√P0\u0013\n, (14)\nwhich demonstrates an advantage over what would be expected from Heisenberg limited scaling in this\ncontext. Put differently, if |δ| ∈Θ(P0) and P0∈Θ(ϵ), then we find Nqueries ∈O(1/√ϵ).\nA. Probability Estimation Using Fast-Square-Rootable Operators\nA challenge with amplified amplitude estimation is that we require the probability we wish to estimate to\nbe small. Usually, probability estimation relies on techniques like the SWAP or Hadamard test. However,6\nSymbol Meaning\n⟨ψ|Π|ψ⟩ Expectation value (EV) we want to estimate\nµ≥1 Integer which quantifies our prior knowledge\nP0 Known upper bound on the target EV: P0= sin2(π/(2(2µ+ 1)))\nδ <0 Defines how loose P0is:⟨ψ|Π|ψ]⟩=P0+δ\nϵ′:=P1−ˆP1Accuracy of the prior ˆP1\nϵ Error in the estimation of δ\nδ′Failure probability upper bound in estimation of δ\nˆδ ϵ-precise estimate of δ\nTABLE I. Summary of variables - δis the quantity we need to estimate.\nthe Hadamard test does not return the expectation value directly but will return (1 /2 + Re( ⟨ψ|U|ψ⟩)/2) for\nsome unitary Uencoding the operator whose expectation value we wish to estimate. For small expectation\nvalues, ⟨ψ|U|ψ⟩ ≪1, the amplitudes of the output state will be close to 1 /2 and our method from Lemma 2\ncannot boost the result. Instead, we require a setup that directly encodes the small expectation values into\nan amplitude. Below, we provide a method for computing small expectation values by exploiting ideas from\nlinear combinations of unitaries [25]. Our method relies on the decomposition of Ainto sums of matrices\nthat are “fast-square-rootable”, which means that there exists a constant query complexity algorithm for\npreparing the square root of the matrices given query access to the original matrix.\nLet us now clarify our access model for the operators and quantum states whose expectation values we wish\nto estimate.\nDefinition 3 (Access model) .Define Oψto be a unitary state preparation oracle such that Oψ|0⟩⊗n=|ψ⟩\nand the inverse, O†\nψ|ψ⟩=|0⟩⊗n, can be performed at the cost of a single query. Furthermore, let A=PJ−1\nj=0αjUjbe a unitary decomposition of a matrix A∈C2n×2nsuch that αj≥0andUj∈C2n×2nis unitary\nfor all j. Define the block-encoding operations for Aas follows:\nPREPARE A|0⟩|ψ⟩:=J−1X\nj=0√αj√α|j⟩|ψ⟩, SELECT A:=J−1X\nj=0|j⟩⟨j| ⊗Uj, (15)\nwhere α:=PJ−1\nj=0αj. Then the unitary\nUA:=PREPARE†\nASELECT APREPARE A (16)\nblock-encodes the matrix Avia(⟨0| ⊗11)UA(|0⟩ ⊗11) =A/α.\nThe following lemma shows how the expectation value of an operator can be encoded directly as a success\nprobability.\nLemma 4 (Expectation value as a success probability) .LetAjbe an operator such that Aj=P\nk≥1βjkΠjk\nwhere Πjkare projectors and βjk≥0. Each projector is of the form Πjk= (1−Rjk)/2where Rjkis a\nreflection operator. Let, for y∈ {0,1},\nSELECT π|k⟩|y⟩|ψ⟩|ϕ⟩=|k⟩Z|y⟩\u0010\nRy\njk⊗(|0⟩⟨k|+|k⟩⟨0|)\u0011\n|ψ⟩|ϕ⟩ (17)\nand\nPREPARE π|0⟩|0⟩=X\nky(βjk)1/4\n√\n2qP\nkp\nβjk|k⟩|y⟩ (18)\nbe two unitary quantum oracles. Then there exists a unitary circuit, Uπ, that can be implemented using a\nsingle query to each of PREPARE π,PREPARE†\nπandSELECT πsuch that for any state |ψ⟩\nTr(Uπ(|ψ⟩⟨ψ| ⊗ |0⟩⟨0|)U†\nπ(1⊗ |0⟩⟨0|)) =⟨ψ|Aj|ψ⟩\n\u0000P\nkp\nβjk\u00012.7\nProof. The result follows straightforwardly from the results of [25] and [27] but we provide a short proof here\nfor the interested reader. Specifically, it uses intuition that the square root can be block encoded as\nBj:=X\nk≥1p\nβjkΠjk⊗(|k⟩⟨0|+|0⟩⟨k|) (19)\nsince we then have that\nB2\nj=Aj⊗ |0⟩⟨0|+X\nkk′≥1p\nβjkp\nβjk′ΠjkΠjk′⊗ |k⟩⟨k′|. (20)\nFirst, let us choose Uπin the following manner:\nUπ:= (PREPARE π⊗1)†SELECT π(PREPARE π⊗1). (21)\nWe then have that\nUπ|0⟩|0⟩|ψ⟩|0⟩= (PREPARE π⊗1)†SELECT π(PREPARE π⊗1)|0⟩|0⟩|ψ⟩|0⟩\n= (PREPARE π⊗1)†SELECT πP\nky(βjk)1/4\n√\n2qP\nkp\nβjk|k⟩|y⟩|ψ⟩|0⟩\n= (PREPARE π⊗1)†P\nky(βjk)1/4\n√\n2qP\nkp\nβjk|k⟩Z|y⟩Ry\njk|ψ⟩|k⟩ (22)\n= (PREPARE π⊗1)†P\nky(βjk)1/4\n√\n2qP\nkp\nβjk|k⟩|y⟩(−Rjk)y|ψ⟩|k⟩.\nNext, with this expression in place, we have that the unnormalized quantum state after having measured\nthePREPARE ancilla register as |00⟩is given by\n(⟨00| ⊗1)(PREPARE π⊗1)†P\nky(βjk)1/4\n√\n2qP\nkyp\nβjk|k⟩|y⟩(−Rjk)y|ψ⟩|k⟩\n=X\nk′y′kyβ1/4\njkβ1/4\njk′\n2P\nkp\nβjk(⟨k′|⟨y′| ⊗1)(|k⟩|y⟩(−Rjk)y)|ψ⟩|k⟩\n=X\nkyp\nβjk\n2P\nkp\nβjk(−Rjk)y)|ψ⟩|k⟩. (23)\nThe probability of this occurring is by Born’s rule the sum of the squares of the amplitudes which yields\n\r\r\r\r\r\rX\nkyp\nβjk\n2P\nkp\nβjk(−Rjk)y|ψ⟩|k⟩\r\r\r\r\r\r2\n=X\nkβjk⟨ψ|(1−Rjk)|ψ⟩\n2\u0000P\nkp\nβjk\u00012=⟨ψ|Aj|ψ⟩\n\u0000P\nkp\nβjk\u00012. (24)\nThe above shows a straightforward experiment that can be performed to estimate the expectation value of\nany observable that is a convex combination of projection operators. This, unfortunately, is not general.\nWhile all operators can trivially be written as a sum of projectors, such a sum is not in general convex, and\nconvexity is required to ensure that the signs of the combinations used in the above lemma combine to form\nthe desired mean value. A natural tactic to deal with this is to express an arbitrary operator as a non-convex\ncombination of convex sums of projectors. We use this tactic in the following result, where we provide an\nend-to-end algorithm for estimating the expectation value of such a non-convex combination, given that we\nknow approximate expectation values for each of the convex sums.8\nTheorem 5 (Application of AAE to fast-square-rootable operators) .LetA=PJ−1\nj=0Aj=P\njkβjkΠjksuch\nthat for any j′, allβj′kare either non-negative or are all non-positive and Πjkare projectors. Assume we\nare provided a series of upper bounds P0(j)such that for all j\nP0(j) := sin2\u0012π\n2(2µj+ 1)\u0013\n=⟨ψ|Aj|ψ⟩\n\u0000P\nkp\nβjk\u00012−δj, (25)\nwith µj≥1being an integer and δj∈[−P0(j),0). Let ∥β∥1,1/2:=P\nj\u0000P\nkp\nβjk\u00012. Then there exists a\nquantum algorithm that, for any |ψ⟩andϵ∈O\u0000\n∥β∥1,1/2minj{P2\n0(j)/|δj|}\u0001\n⊆o(1), can estimate ⟨ψ|A|ψ⟩\nwithin error ϵand probability of failure at most δ′using a number of queries to SELECT πandPREPARE πthat\nscales as\nO P\njp\nP0(j)∥β∥1,1/2log(J/δ′)\nϵmax\nj\u0012P0(j)\n|δj|\u0013!\n.\nProof. The proof is an application of Lemma 4 and Lemma 2. The algorithm we propose for solving this\nproblem is simple. We loop over each of the Ajand measure each such operator within sufficient error to\nachieve our bounds. Specifically, if we wish to learn ⟨ψ|A|ψ⟩then it suffices to learn the individual estimates\n⟨ˆAj⟩of the ⟨ψ|Aj|ψ⟩such that\n|⟨ψ|A|ψ⟩ −X\nj⟨ˆAj⟩| ≤X\nj|⟨ψ|Aj|ψ⟩ − ⟨ ˆAj⟩| ≤ϵ. (26)\nLet us now choose the error tolerance for our estimate ⟨ˆAj⟩such that\n|⟨ψ|Aj|ψ⟩ − ⟨ ˆAj⟩|=(P\nkp\nβjk)2ϵ\nP\nj\u0000P\nkp\nβjk\u00012=:ϵj. (27)\nIt is clear that (27) implies (26). Next, as we have inherited the assumptions of Lemma 2, we can use\nthat algorithm to learn the probability that the method of Lemma 4 yields |00⟩in the ancilla register. Our\nexpectation value estimate is found from that success probability by multiplying the success probability by\u0000P\nkp\nβjk\u00012. Thus, we must aim to estimate the success probability within error ϵ′\njsuch that\nϵ′\nj≤ϵj\u0000P\nkp\nβjk\u00012=ϵ\nP\nj\u0000P\nkp\nβjk\u00012=:ϵ\n∥β∥1,1/2. (28)\nThen the cost of learning the probability within error ϵ′\njin terms of the number of queries made to our\noracles scales from Lemma 2 and our assumption that P2\n0(j)/|δj| ⊆o(1) as\nO p\nP0(j) log(1 /δ′′)\nϵ′\nj\u0012P0(j)\n|δj|\u0013!\n=O p\nP0(j)∥β∥1,1/2log(1/δ′′)\nϵ\u0012P0(j)\n|δj|\u0013!\n, (29)\nwhere δ′′is an upper bound on the failure probability and we chose ϵ′\njto saturate the upper bound in\nEq. (28). This process needs to be repeated for each Ajand so the total number of queries is the sum of\nthis result over all jwhich yields\nO P\njp\nP0(j)∥β∥1,1/2log(J/δ′)\nϵmax\nj\u0012P0(j)\n|δj|\u0013!\n.\nNote that the probability of error from the union bound is no longer bounded above by δ′′but instead\nbecomes Jδ′′. In order to ensure that the overall failure probability is at most δ′, it suffices to choose\nδ′′=δ′/Jfor each set which yields the above result.9\nThis shows that in the event that the expectation values tend to zero and under the assumption that we have\nsufficient prior knowledge to at least know the leading digit of the result, the cost of learning the expectation\nvalue of an operator, which is given as a sum of projectors, can be substantially lower than the cost of\nperforming na¨ ıve amplitude estimation which would scale asP\njkβjklog(1/δ′)/ϵ. Under the assumption that\nwe have constant relative uncertainty, max j(P0(j)/|δj|)∈O(1), we then expect an asymptotic advantage if\nX\njp\nP0(j)∈o\u0012∥β∥1,1\n∥β∥1,1/2\u0013\n. (30)\nNote that in general this condition will not be asymptotically attainable since ∥β∥1,1≤ ∥β∥1,1/2and so\nthe right-hand side of the above expression can vanish as the dimension of the problem increases. This\nunderscores the need to have applications where the probabilities that we wish to estimate are small.\nOur approach can be further generalized to compute a vector of expectation values straightforwardly. The\nadditional cost involved is simply a factor of mgreater where mis the number of distinct expectation\nvalues required. The cost of computing the vector of expectation values can be further reduced to a O(√m)\nscaling by using the methods of [28], which involves expressing the expectation value estimation as a gradient\nestimation problem which can be addressed from the results of [29, 30]. However, while such a process could\nbe used to find a vector of independent observables over a set of points, it cannot be used to compute\nthe values along a given path because of the way that prior knowledge is used in the amplified amplitude\nestimation algorithm. Specifically, the methods of [28] compute each component of the vector of expectation\nvalues at the same time in superposition, meaning that this approach does not use any continuity of the values\nat adjacent points. Because of these complications, we focus in the following on the case of interpolating\none-dimensional vectors while noting that similar results can be obtained for interpolating m-dimensional\nvectors.\nThe following corollary deals with the situation where we have prior knowledge of the expectation value\n⟨ψ|A|ψ⟩but the expectation value is too large for Lemma 2 or Theorem 5 to be directly applicable. The\nmain idea is to lower the expectation value of each of the operators Ajby exploiting prior knowledge to\nreduce the number of projectors to be evaluated on a quantum computer.\nCorollary 6 (Using prior knowledge to refine estimates of large expectation values) .Letϵ >0be an error\ntolerance and A=PJ−1\nj=0Aj=PJ−1\nj=0Pnj\nk=1βjkΠjkan operator such that for any j′, all βj′kare either\nnon-negative or are all non-positive and Πjkare projectors. Assume that for every jthere exists a subset\nIj⊆[nj]of indices and known positive numbers Cj,i≥eϵj, where\neϵj:=ϵ\n2P\ni∈Ijβj,i(P\nkp\nβjk)2\nP\nj(P\nkp\nβjk)2, (31)\nsuch that |Cj,i− ⟨ψ|Πj,i|ψ⟩| ≤eϵj.\nFurthermore, assume that we are provided with a series of upper bounds P0(j) = sin2(π/(2(2µj+ 1))) and\nintegers µj≥1such that\nPA′\nj:=P\nk̸∈Ijβj,k⟨ψ|Πj,k|ψ⟩\n(P\nk̸∈Ijp\nβji)2=P0(j) +δj (32)\nforδj∈[−P0(j),0). Let∥β∥1,1/2:=P\nj\u0000P\nkp\nβjk\u00012. Then there exists a quantum algorithm that, for any\n|ψ⟩andϵ∈O\u0000\n∥β∥1,1/2minj{P2\n0(j)/|δj|}\u0001\n⊆o(1), can estimate ⟨ψ|A|ψ⟩within error ϵand probability of\nfailure at most δ′using a number of queries to SELECT πandPREPARE πthat scales as\nO P\njp\nP0(j)∥β∥1,1/2log(J/δ′)\nϵmax\nj\u0012P0(j)\n|δj|\u0013!\n. (33)\nProof. The proof is similar to the proof of Theorem 5. As before, we wish to obtain an estimate ˆAof⟨ψ|A|ψ⟩\nsuch that\n|ˆA− ⟨ψ|A|ψ⟩| ≤ϵ. (34)10\nThe assumption PA′\nj=P\nk̸∈Ijβj,k⟨ψ|Πj,k|ψ⟩\n(P\nk̸∈Ij√\nβji)2=P0(j) +δjensures that ignoring the set of projectors {Πj,i}i∈Ij\nleads to a sufficiently low success probability PA′\njwhich can be estimated using Lemma 2. From this we can\nobtain an ϵ′\nj-precise estimate ˆA′\njofA′\nj:=P\nk̸∈Ijβj,k⟨ψ|Πj,k|ψ⟩. This then allows us to obtain an estimate\nˆAjforAjas follows:\nˆAj=ˆA′\nj+X\ni∈Ijβj,iCj,i. (35)\nThe error associated with ˆAjis bounded by\n|ˆAj− ⟨ψ|Aj|ψ⟩| ≤ϵ′\nj+eϵjX\ni∈Ijβj,i. (36)\nWe construct the estimate ˆAvia the estimates ˆAjof the individual terms Aj. Note that\n|ˆA− ⟨ψ|A|ψ⟩|=|X\njˆAj−X\nj⟨ψ|Aj|ψ⟩| ≤X\nj|ˆAj− ⟨ψ|Aj|ψ⟩|. (37)\nEnsuring that\n|ˆAj− ⟨ψ|Aj|ψ⟩| ≤(P\nkp\nβjk)2\nP\nj(P\nkp\nβjk)2ϵ=:ϵj (38)\nautomatically yields (34). This implies that it suffices to choose ϵ′\nj=ϵj/2 and\neϵj=ϵj\n2P\ni∈Ijβj,i. (39)\nNote that eϵjonly affects the cost of obtaining our prior knowledge but not the quantum cost associated with\nestimating the operators Aj. The quantum experiments yield estimates of the probabilities PA′\njrather than\nthe estimates for ˆA′\njdirectly. It suffices to estimate PA′\njwithin error\nϵPj:=ϵ′\nj\u0010P\nk̸∈Ijp\nβjk\u00112=ϵ\n2P\nj(P\nkp\nβjk)2(P\nkp\nβjk)2\n\u0010P\nk̸∈Ijp\nβjk\u00112≥ϵ\n2∥β∥1,1/2(40)\nto obtain an ϵ′\nj-precise estimate of ˆA′\nj. Thus, following the same argument as in the proof of Theorem 5,\nwe find that the number of queries to the unitaries used for implementing the projectors, PREPARE Πand\nSELECT Π, is in\nO P\njp\nP0(j)∥β∥1,1/2log(J/δ′)\nϵmax\nj\u0012P0(j)\n|δj|\u0013!\n, (41)\nwhere ∥β∥1,1/2=P\nj(P\nkp\nβjk)2.\nThe above corollary shows that we can get an asymptotic advantage over standard amplitude estimation even\nfor large expectation values as long as we have sufficient prior knowledge of the individual expectation values\nin the projector decomposition. This allows us to express a large expectation value that we wish to compute\nas a sequence of smaller ones, each of which we then estimate using amplified amplitude estimation to achieve\nan advantage relative to what would be possible with methods that do not use any prior information.11\nIV. APPLICATIONS\nThere are many different ways in which prior knowledge about the expectation value of an observable on\na wave function can be obtained. In what follows, we will explore two examples of how prior knowledge\ncan be exploited by amplified amplitude estimation to solve problems relevant to chemistry. However, the\napplications of AAE are potentially broader than the simulation of quantum systems as long as the conditions\non the expectation values and the prior knowledge are satisfied.\nThe first example we will consider involves using classical methods to obtain a low-cost a priori estimate of\nthe expectation value of a given observable and apply AAE to improve on such prior estimate. This is a\ncommon situation in simulations of molecular systems: often, a cheap mean-field solution such as Hartree-\nFock (HF) or Density Functional Theory (DFT) forms the basis for a more demanding correlated calculation.\nThe expectation values from these low-cost methods, together with a tight enough bound on their error, are\nthe two ingredients needed to apply our algorithm. Specifically, we will show how to use AAE to reduce\nthe cost of estimating expectation values of one-body fermionic operators, such as dipole moments for water\nclusters.\nThe second example focuses on estimating properties of a target quantum system by exploiting available prior\nknowledge regarding other quantum systems with similar parameters. For instance, in Markov Chain Monte\nCarlo dynamics, new molecular geometries are proposed by randomly changing the current coordinates.\nThese new geometries cannot differ too much in energy from the current system, or the detailed balance\nmechanism would reject them. The prior knowledge in this setting can be exploited to infer the energy\nfor the new geometry by using knowledge about the current energy. We can lower the cost even further\nby realizing that derivative operators may have a lower one-norm than the Hamiltonian operator. This\nmotivates us to consider the problem of computing the ground state energy of a molecular system within the\nBorn-Oppenheimer approximation by integrating the directional derivative of the energy using Newton-Cotes\nformulas.\nA. Fermionic operator estimation with prior knowledge from low-cost methods\nTo apply our prior results to the estimation of expectation values of fermionic operators, we first need to\nreduce the problem from one of measuring fermionic operators to a problem involving measuring projections.\nWe will demonstrate this for a general fermionic one-body operator\nA=X\npqApqa†\npaq, (42)\nwhere pandqlabel a set of spin orbitals such that a†\npis the fermionic creation operator acting on orbital pand\naqis the fermionic annihilation operator acting on orbital q. After using the Jordan-Wigner decomposition,\nwe have that\nA=X\np<q1\n2ApqXp⊗Zp+1···Zq−1⊗Xq+1\n2ApqYp⊗Zp+1···Zq−1⊗Yq+X\npApp1−Zp\n2. (43)\nThe Pauli products in this formula are reflection operators, each of which can be thought of as a projection\nonto a negative eigenspace added to a projection onto a positive eigenspace. Let us define, for example,\nΠ(±X)\np,q :=11pq±Xp⊗Zp+1···Zq−1⊗Xq\n2, (44)\nΠ(±Y)\np,q :=11pq±Yp⊗Zp+1···Zq−1⊗Yq\n2, (45)\nΠ(±Z)\np,p :=11pp±Zp\n2, (46)\nwhere the ±denotes the projector onto the positive and negative eigenspaces of the operator. This is an\nobvious choice that does not necessarily have the best properties for the AAE algorithm. Optimizing this12\nchoice might lead to better scaling of the algorithm. Using the fact that Π(+)+ Π(−)=1we then have that\n⟨ψ|A|ψ⟩=1\n2X\np<qApq⟨ψ|\u0010\nΠ(+X)\np,q−Π(−X)\np,q + Π(+Y)\np,q−Π(−Y)\np,q\u0011\n|ψ⟩+X\np⟨ψ|AppΠ(−Z)\np|ψ⟩\n=X\np<qApq⟨ψ|\u0010\nΠ(+X)\np,q + Π(+Y)\np,q\u0011\n|ψ⟩+X\npApp⟨ψ|Π(−Z)\np|ψ⟩ −X\np<qApq. (47)\nEven though the operator is now written as a sum of projectors, the sum is still not necessarily convex.\nIf terms are negative, then we can always group them separately from the positive terms and apply two\nseparate rounds of AAE. The above expectation value can then be decomposed as follows:\n⟨ψ|A|ψ⟩=X\nj=0,1X\np<qApq+ (−1)j|Apq|\n2⟨ψ|\u0010\nΠ(+X)\np,q + Π(+Y)\np,q\u0011\n|ψ⟩ (48)\n+X\nj=0,1X\npApp+ (−1)j|App|\n2⟨ψ|Π(−Z)\np|ψ⟩ −X\np<qApq\n=:X\njX\nkβjkΠjk−X\np<qApq, (49)\nwhere the last step is a relabeling of the two separate convex sums into a single sum over k. Further, the\nconstant offsetP\np<qApqis known a priori and so does not need to be considered for our purposes. This\nshows that we can use Theorem 5 to bound the query complexity of estimating ⟨ψ|A|ψ⟩.\nLet us now consider the example of estimating the expectation value of one-body operators for groups of\nwater molecules of increasing size. We will consider three types of operators: the x, y, z components of the\ndipole operator, the kinetic energy operator, and the one-body part of the Hamiltonian. The matrix elements\nof the components of the dipole operator are obtained as integrals over the molecular orbitals ϕpby\nDρ,pq=Z\nϕ∗\np(r)ρ ϕq(r) drforρ∈ {x, y, z}. (50)\nThe value of the dipole moment is often significant because it provides qualitative information about the\nstrength of intermolecular forces in the system, and the expectation value of the dipole operator in the\nground state will give an estimate of this dipole moment. Similarly, the kinetic energy operator K, with\nelements\nKpq=Z\nϕ∗\np(r)\u0012\n−1\n2∇2\u0013\nϕq(r) dr, (51)\nand the one-body Hamiltonian P, with elements\nPpq=Kpq+Z\nϕ∗\np(r)V(r)ϕq(r) dr, (52)\nwhere V(r) is the external potential due to attraction with the nuclei, are important operators that help\nassess the quality of a wave function.\nIn order to use amplified amplitude estimation to profitably estimate a quantity, it is important that\nthe requirements on the size of the perturbation ( δj∈[−P0(j),0)) and the prior knowledge ( P0(j)) s.t.\nϵ∈O(P2\n0(j)/|δj|)) imposed in Corollary 6 are met. Here we study these requirements numerically. In\nFigure 1, we explore the spread of δjfor the problem of estimating the dipole moment, the kinetic energy,\nand the single-particle Hamiltonian for a cluster of water molecules. Specifically, we plot the spread of the\nperturbation as a function of a growing water cluster for prior knowledge obtained either from RHF or DFT\nwith the B3LYP functional or low bond dimension DMRG. We use the projector decomposition of the single-\nparticle operators as described in Eq. (49). It is clear that different methods (depending on their accuracy)\nwill give different orders of magnitude for the δj: while RHF and DFT require learning a deviation of the\nsame order of magnitude, low bond dimension DMRG is more accurate, and the corresponding correction of\nthe prior knowledge is two orders of magnitude smaller. In all these cases, the value of δjthat is attainable is13\nsufficiently small so that we can profitably use this prior knowledge provided that P0(j)/|δj|is appropriately\nsmall. In particular, the difference between the DMRG calculations and the true values for the expectation\nvalues of the projectors differ on the order of 10−9for all data between 2 and 7 water molecules in our box,\nwhereas RHF and DFT calculations provide estimates that are accurate within 10−7and 10−6roughly. For\nRHF and DFT, we see evidence of polynomial decay in the difference as the number of water molecules\nincreases. The data is roughly consistent with a polynomial scaling of O((# waters)−2) for RHF and for\nDFT. In contrast, we see no evidence of a clear systematic trend for the data for the DMRG calculation\nresults at the scale of our computations.\nThe relative size of δjandP0(j) is a primary driver of the complexity of our method. Figure 2 addresses the\nquestion of relative error by examining the difference between the relative errors estimated using RHF, DFT\nand DMRG. Interestingly, the data shows a different scaling behavior between these results: For the first\ntwo methods, the ratio goes down as the water clusters grow, while for the last it goes up (at least up to the\nsizes we were able to simulate). For the DMRG, this means that the prior knowledge becomes of decreasing\nquality and the correction becomes more important as the water cluster grows. This is to be expected as\nthe bond dimension in the DMRG calculations is not growing with system size.\nWe find in this data that the relative error for these predictions is on the order of 10−3for the case of\nrestricted Hartree-Fock and DFT; whereas DMRG performs again substantially better giving errors on the\norder of 10−5. In these cases, the error is sufficiently small that we can reliably pick the δjsmall enough to\nensure that max( P0(j)/δj)≈1. We further see that in the case of DMRG the error seems to be increasing\nas we approach the thermodynamic limit. RHF seems to be decreasing at a rate that is approximately\nconsistent with an O((#waters)−1/4) scaling, but we see evidence of saturation in the DFT data. While\nthe trends in this data are difficult to divine, this provides evidence that advantages can be seen as long asP\njp\nP0(j)∈o(1).\nOne caveat in the above discussion is that we assume for generality that the quantum state |ψ⟩is provided\nby an oracle. However, in practice, particular methods must be considered to prepare such a state. In the\nfollowing, we will consider the case where |ψ⟩corresponds to the ground state of the system of interest. While\nmany methods can be considered for preparing an approximate ground state, adiabatic state preparation\nand phase estimation are two of the most popular methods for achieving a small preparation error. In both\ncases, the resources needed scale polynomially with the inverse eigenvalue gap. In the case of adiabatic state\npreparation, this is because of the dependence on the inverse gap. For phase estimation, the resource scaling\narises from the cost of measuring the energy with an error smaller than the gap. Here, we take the latter\napproach as it will often provide superior scaling with the eigenvalue gap, only tying the performance in\ncases where boundary cancellation methods are employed and we are in a regime where the evolution time\nis sufficiently long. Specifically, we use the results of [19], which we restate as the following lemma.\nLemma 7 (Ground state preparation, Ge et al. [19]) .For any ϵψ>0, assume we have access to a trivially\npreparable state |ψ′\n0⟩ ∈C2nthat has overlap at least |⟨ψ′\n0|ψ0⟩| ≥ | a0|>0with the target state |ψ0⟩, which\nis an eigenstate of a Hamiltonian H=P\nkαkUkfor unitary Uk∈C2n×2nwith coefficient 1-norm ∥α∥1.\nFurthermore, assume we have an eigenvalue estimate E′\n0of the eigenvalue E0associated with |ψ0⟩. We then\nfurther assume that |E0−E′\n0| ∈O(γ/log(|a0|ϵψ))where γis a lower bound on the spectral gap of H. Then\nthere exists a unitary Uψ0acting on b+nqubits such that\n1. We can perform a heralded measurement such that upon measuring zero on a qubit register,\n\f\f\f\f(⟨0|b⊗1)Uψ0|0⟩b|ψ′\n0⟩\n∥(⟨0|b⊗1)Uψ0|0⟩b|ψ′\n0⟩∥− |ψ0⟩\f\f\f\f\n2≤ϵψ.\n2. The probability of not measuring zero is at most 1−ϵ2\nψ.\n3.Uψ0can be implemented using a number of queries to the state preparation routine for |ψ′\n0⟩that scales\nas\neO\u0012log(1/ϵψ)\n|a0|\u001314\n2 3 4 5 6 7\n# waters0.01.02.03.04.05.06.07.0max(j)-min(j)\n×107\nDx RHF\nDy RHF\nDz RHF\nK RHF\nP RHF\n2 3 4 5 6 7\n# waters0.00.20.40.60.81.01.21.41.6max(j)-min(j)\n×106\nDx DFT\nDy DFT\nDz DFT\nK DFT\nP DFT\n2 3 4 5 6 7\n# waters0.00.51.01.52.02.5max(j)-min(j)\n×109\nDx DMRG(200)\nDy DMRG(200)\nDz DMRG(200)\nK DMRG(200)\nP DMRG(200)\nFIG. 1. Each plot shows the spread of δjfor five different single-particle operators versus the size of a growing\ncluster of water molecules when using prior information. The prior information is obtained from restricted Hartree-\nFock (RHF), density functional theory (DFT) with the B3LYP functional, and density matrix renormalization group\ntheory (DMRG) with a low bond dimension of 200, respectively. The ground truth to which this prior information\nis compared is obtained from DMRG with a high bond dimension of 750. The molecular properties are calculated in\nthe cc-pVDZ basis using an all-electron approach for RHF and DFT and an active space correlating 6 orbitals and\n8 electrons per water molecule for the DMRG calculations. The operators are the three components of the dipole\nmoment ( Dx,Dy, and Dz), the kinetic energy operator ( K) and the single-particle Hamiltonian ( P). The minimum\nand maximum values of δjare taken over the two projectors in the single-particle operator ( j= 0,1) and over an\nensemble of 26 geometries taken from a well-equilibrated 500 step molecular dynamics (MD) run at 300K taking 1fs\nsteps using the TIP3P water model. The decrease in value of the δjwith growing water cluster can be attributed to\nthe normalization of the projectors in a growing basis. The different behavior for the low bond dimension DMRG at\nsmall cluster sizes is due to the equivalence between low bond dimension DMRG and high bond dimension DMRG\nfor these systems. The prior knowledge in that case is essentially exact and the size of the perturbation to be learned\nis close to 0. The three components of the dipole operator should yield similar results since the water clusters should\nbe near-rotationally invariant over the course of the MD run. Finite size effects in the smaller clusters and the finite\nensemble of geometries result in some spread between the three components. MD calculations where performed with\nASE [31], HF and DFT calculations with PySCF [32] and DMRG calculations with Block2 [33, 34].\nand a number of queries to PREPARE HandSELECT Hthat scale as\neO\u0012∥α∥1\n|a0|γ\u0013\n.15\n2 3 4 5 6 7\n# waters0.20.40.60.81.01.2max(j/P0(j))-min(j/P0(j))\n×103\nDx RHF\nDy RHF\nDz RHF\nK RHF\nP RHF\n2 3 4 5 6 7\n# waters0.51.01.52.02.53.0max(j/P0(j))-min(j/P0(j))\n×103\nDx DFT\nDy DFT\nDz DFT\nK DFT\nP DFT\n2 3 4 5 6 7\n# waters0.01.02.03.04.0max(j/P0(j))-min(j/P0(j))\n×105\nDx DMRG(200)\nDy DMRG(200)\nDz DMRG(200)\nK DMRG(200)\nP DMRG(200)\nFIG. 2. Each plot shows the spread of δj/P0(j) for five different single-particle operators versus the size of a growing\ncluster of water molecules when using prior information. The prior information is obtained from restricted Hartree-\nFock (RHF), density functional theory (DFT) with the B3LYP functional, and density matrix renormalization group\ntheory (DMRG) with a low bond dimension of 200, respectively. The ground truth to which this prior information\nis compared is obtained from DMRG with a high bond dimension of 750. The molecular properties are calculated in\nthe cc-pVDZ basis using an all-electron approach for RHF and DFT and an active space correlating 6 orbitals and\n8 electrons per water molecule for the DMRG calculations. The operators are the three components of the dipole\nmoment ( Dx,Dy, and Dz), the kinetic energy operator ( K) and the single-particle Hamiltonian ( P). The minimum\nand maximum values of δjare taken over the two projectors in the single-particle operator ( j= 0,1) and over an\nensemble of 26 geometries taken from a well-equilibrated 500 step molecular dynamics (MD) run at 300K taking 1fs\nsteps using the TIP3P [35] water model.\nFor our problem, let us assume that we are provided an input ground state |ψ⟩and wish to estimate a one-\nbody operator ⟨ψ|A|ψ⟩given an a priori estimate of the operator that is provided by a classic approach such\nas density functional theory, Hartree-Fock theory or a low bond dimension tensor network approximation.\nOur aim is then to refine this knowledge into an improved estimate of the quantity using amplified amplitude\nestimation.\nThe following lemma shows how we can, under certain conditions, estimate ⟨ψ|A|ψ⟩using O(1/√ϵ) queries\nto the underlying PREPARE andSELECT unitaries.\nLemma 8 (Beating the Heisenberg limit with prior knowledge) .Letϵ > 0be an error tolerance and\nH=P\njαjUj, where Uj∈C2n×2nis unitary, be a Hamiltonian with ground state |ψ⟩and coefficient 1-norm\n∥α∥1. Let A=PNπ\nj=1βjΠjbe a projector decomposition of the operator Awith Πjbeing a projection16\noperator that is implemented via a reflection operator Rj. Assume we are provided with a subset I⊆[Nπ]\nof indices and non-negative numbers Ci≥eϵwhere i∈Iandeϵ:=ϵ\n2P\nj|βj|such that |Ci− ⟨ψ|Πi|ψ⟩| ≤eϵ.\nFurthermore, assume that we have a series of upper bounds P0(k) = sin2(π/(2(2µk+ 1))) fork∈[Nπ]\\I\nand integers µk≥1such that\nPk:=⟨ψ|Πk|ψ⟩=P0(k) +δk (53)\nwhere δ∈[−P0(k),0). Let ∥β∥1:=PNπ\nj=1|βj|. Then under the assumptions of Lemma 7 and if ϵ∈\nO\u0000\n∥β∥1mink{P2\n0(k)/|δk|}\u0001\n⊆o(1)andP0(k)∈O(ϵ)for all k∈[Nπ]\\I, we can learn ⟨ψ|A|ψ⟩within\nerror ϵand probability of failure at most δ′using a number of queries to SELECT H,SELECT Π,PREPARE Hand\nPREPARE Πthat scales as\neO\u0012N′\nπ∥α∥1∥β∥1log(N′\nπ/δ′)\nγ|a0|√ϵmax\nk̸∈I\u0012P0(k)\n|δk|\u0013\u0013\n, (54)\nwhere N′\nπ:=|[Nπ]\\I|.\nProof. Using Corollary 6 where now each Ajinvolves only a single projector Π j, we find that the number of\nqueries to PREPARE ΠandSELECT Πis in\nO P\nk̸∈Ip\nP0(k)∥β∥1log(N′\nπ/δ′)\nϵmax\nk̸∈I\u0012P0(k)\n|δk|\u0013!\n. (55)\nNext, we need to consider the cost of ground state preparation for the algorithm. Each state preparation can\nbe performed using quantum phase estimation with H=P\njαjUj, which is block-encoded using SELECT H\nand PREPARE H. We then use this process and a comparator to determine whether the energy is less than\nor equal to the threshold needed to determine whether the state is indeed the ground state. Then from\nLemma 7 the cost per query to the state preparation algorithm is ˜O(∥α∥1/|a0|γ). Thus, the overall number\nof queries is\neO P\nk̸∈Ip\nP0(k)∥α∥1∥β∥1log(N′\nπ/δ′)\nγ|a0|ϵmax\nk̸∈I\u0012P0(k)\n|δk|\u0013!\n⊆eO\u0012N′\nπ∥α∥1∥β∥1log(N′\nπ/δ′)\nγ|a0|√ϵmax\nk̸∈I\u0012P0(k)\n|δk|\u0013\u0013\n,(56)\nwhere we used the assumption that P0(k)∈O(ϵ).\nThe main idea used in the above lemma is to split the projector decomposition of Ainto two disjoint sets.\nThe first set consists of projectors Π ifor which we have eϵ-precise estimates Ciof their expectation values.\nImportantly, these estimates are sufficiently precise so that we do not need a quantum computer to refine\nthem. The other set contains the projectors whose expectation values are small enough such that their\nestimates can be refined via AAE.\nThe gate complexity of the above oracle queries depends strongly on the basis set chosen to represent the\nproblem. In general, for Gaussian basis sets which are commonly used in chemistry, the asymptotic behavior\nof the constants are difficult to assess. The worst-case scenario corresponds to the case where the Hamiltonian\ncoefficients as well as the dipole moment coefficients have no pattern. In this case, a lookup table is the\nmost natural way to load these elements in. Using a QROAM-based strategy, the work of [9] shows that this\ncan be attained with an eO\u0000\nN4\u0001\nscaling for the number of non-Clifford operations needed to implement the\nHamiltonian PREPARE circuit and an eO\u0000\nN2\u0001\nscaling for the dipole operator where Ndenotes the number\nof basis functions considered. Using similar tricks, the cost of the SELECT circuit using a Jordan-Wigner\nrepresentation for the operators can be shown to be sub-dominant to the cost of implementing PREPARE [9].\nThese observations combined lead us to the conclusion that these simulations are indeed efficient provided\nthat a polynomially large basis set is used.17\nThe remaining question is: given knowledge of the expectation value of an operator for one molecular\nconfiguration, how far can we extrapolate from that configuration before we violate the limits of the above\naccuracy bound? Sufficient conditions for the uncertainty can be found in our context by using bounds\nderived from analytic expressions for the derivatives of the eigenvalues and eigenvectors of the Hamiltonian.\nLet us consider a parameterized path on Hamiltonians H(s) with |ψ(s)⟩as the ground state at time s∈[0,1]\nand corresponding eigenvalue Eψ(s). Furthermore, let γ(s) be the spectral gap of H(s). Provided that\nγ(s)>0 and H(s) is differentiable for all s∈[0,1], we have that\n∥|ψ(1)⟩ − |ψ(0)⟩∥=\r\r\r\rZ1\n0∂s|ψ(s)⟩ds\r\r\r\r\n=\r\r\r\r\r\rZ1\n0X\nϕ̸=ψ|ϕ⟩⟨ϕ(s)|˙H(s)|ψ(s)⟩\nEψ(s)−Eϕ(s)ds\r\r\r\r\r\r\n≤max\nsvuutX\nϕ̸=ψ⟨ψ(s)|˙H(s)|ϕ(s)⟩⟨ϕ(s)|˙H(s)|ψ(s)⟩\n(Eϕ(s)−Eψ(s))2\n≤s\nmax s∥˙H(s)∥2\nminsγ(s)2=max s∥˙H(s)∥\nminsγ(s). (57)\nNext, let us show how this can be used to bound the difference in the expectation values of the projectors\nbetween different configurations. By the triangle inequality, we have that\n|⟨ψ(0)|Πj|ψ(0)⟩ − ⟨ψ(1)|Πj|ψ(1)⟩| ≤ |⟨ ψ(0)|Πj|ψ(0)⟩ − ⟨ψ(0)|Πj|ψ(1)⟩|+|⟨ψ(0)|Πj|ψ(1)⟩ − ⟨ψ(1)|Πj|ψ(1)⟩|.\n(58)\nNote that\n|⟨ψ(0)|Πj|ψ(0)⟩ − ⟨ψ(0)|Πj|ψ(1)⟩|=∥|ψ(0)⟩⟨ψ(0)|Πj|ψ0⟩ − |ψ(0)⟩⟨ψ(0)|Πj|ψ(1)⟩∥\n≤ ∥|ψ(0)⟩ − |ψ(1)⟩∥ ≤max s∥˙H(s)∥\nminsγ(s).(59)\nThe same holds for the second term, |⟨ψ(0)|Πj|ψ(1)⟩ − ⟨ψ(1)|Πj|ψ(1)⟩|. This implies that\n|⟨ψ(0)|Πj|ψ(0)⟩ − ⟨ψ(1)|Πj|ψ(1)⟩| ≤2max s∥˙H(s)∥\nminsγ(s). (60)\nRecall from Lemma 8 that we effectively split the projector decomposition of Ainto two disjoint sets.\nThe first set consists of projectors Π ifor which we have eϵ-precise estimates Ciof their expectation values.\nImportantly, these estimates need to be sufficiently precise so that we do not need a quantum computer to\nrefine them. For simplicity, we assume here that we have perfect knowledge of all the expectation values\nats= 0. The other set contains the projectors whose expectation values are small enough such that their\nestimates can be refined via AAE. As before, let us use kto index those projectors that are evaluated on a\nquantum computer and let P0(k) be an upper bound on ⟨ψ(0)|Πk|ψ(0)⟩. Furthermore, let P′\n0(k) be an upper\nbound on ⟨ψ(1)|Πk|ψ(1)⟩. Note that Eq. (60) implies that it suffices to have\nP′\n0(k)≤P0(k) + 2max s∥˙H(s)∥\nminsγ(s). (61)\nIn order to apply Lemma 8, we need to have that P′\n0(k)≤sin2(π/6) = 1 /4. This can be achieved by ensuring\nthat\nmax\ns∥˙H(s)∥ ≤2 min\nsγ(s)\u00121\n4−max\nkP0(k)\u0013\n. (62)\nFurthermore, we require\nmax\ns∥˙H(s)∥ ≤minsγ(s)ϵ\n4∥β∥1(63)18\nforeϵto satisfy the assumptions of Lemma 8. Overall, we thus require that\nmax\ns∥˙H(s)∥ ≤min\u001aminsγ(s)ϵ\n4∥β∥1,min\nsγ(s)\u00121\n4−max\nkP0(k)\u0013\u001b\n. (64)\nFrom this we can see that there exists a radius of Hamiltonians within which the expectation value of an\noperator can be extrapolated at low cost.\nB. Energy estimation through gradient integration\nTraditionally, energy estimation within the Born-Oppenheimer approximation in quantum simulation has\nfollowed a standard paradigm. Specifically, one begins by estimating the energy directly using phase esti-\nmation at a point or configuration of interest. The previous AAE-based approach can, of course, be used to\nprovide estimates of the energy based on a previously estimated value by measuring the expectation value\nof the Hamiltonian in a given quantum state. Here, we use a different approach that attempts to better use\nprior knowledge by integrating a directional derivative of the energy over a path that connects a known point\nto an unknown point. This approach, in essence, constructs a series of derivative estimates that are combined\ntogether to form a single estimate of the energy using a numerical integration formula. Further, under the\nassumption that the second derivatives of the energy are not large along the path, we can use the previous\npoint that we estimated to accelerate our estimation of the next point as per the previous discussion. How-\never, in principle this approach also has value outside of the framework of amplified amplitude estimation\nbecause it is not strictly speaking necessary to evaluate the small expectation values of the projectors that\narise in this approach.\nSpecifically, we construct our numerical integration formulas through a family of integrators known as\nNewton-Cotes formulas.\nLemma 9 (Newton-Cotes formulas) .Let∂xE(x)be the spatial derivative of a function Eand assume that\n∂xE(x)is analytic on [−7/3,13/3]and has an analytic continuation as a single valued regular function on\nthe contour in the complex plane C={z:z= 1 + 3 eiϕ+ 3−1e−iϕ}forϕ∈[0,2π). The (N+ 1)-point\nNewton-Cotes formula for odd N > 0provides an approximation of the form\nE(1)−E(−1) =NX\nk=0ℵk∂xE|x=xk+ ∆ N\nwhere xk=−1 + 2 k/N,ℵk=R1\n−1Q\ni̸=k(x−xi)Q\ni̸=k(xi−xk)dxand\n|∆N| ≤5\n3max\nz∈C|∂zE(z)|\u00123\n4\u0013N+1\nandNX\nk=0ℵk≤2(N+ 1).\nProof. From Theorem 1 of [36] we have that the error in the Newton-Cotes formula can be computed in\nterms of a contour integral. The contour is chosen to be an ellipse where the sum of the semi-axes is equal\ntoρ≥1. Here we take ρ= 3, although other choices are possible and will lead to different tradeoffs in the\nerror bounds between the norm of the integrand and the exponential dependence:\n|∆N| ≤4\n3ρ2+ 1\nρ2−1max\nz∈C∥∂zE(z)∥\u00122ρ\nρ2−1\u0013N+1\n≤5\n3max\nz∈C|∂zE(z)|\u00123\n4\u0013N+1\n. (65)\nFor us to use these expressions in LCU circuits, we need to also bound the sum of the coefficients, ℵk. First,\nnote that\nZ1\n−1Q\ni̸=k(x−xi)Q\ni̸=k(xk−xi)dx≤Z1\n−1\f\f\f\f\fQ\ni̸=k(xk−xi)Q\ni̸=k(xk−xi)\f\f\f\f\fdx≤2. (66)19\nThus,\nNX\nk=0ℵk=NX\nk=0Z1\n−1Q\ni̸=k(x−xi)Q\ni̸=k(xk−xi)dx≤2(N+ 1). (67)\nThe following result uses the above lemma to state a bound on the error that arises from the combination\nof using a finite number of interpolation points in the Newton-Cotes formula as well as the approximation\nerror in the values of the derivatives.\nCorollary 10 (Error bound for Newton-Cotes) .Under the assumptions of Lemma 9, we further have that\nif for each xkwe are given a real number F(xk)such that |F(xk)−∂xE|x=xk| ≤δ, then\n\f\f\f\f\fNX\nk=0ℵkF(xk)−[E(1)−E(−1)]\f\f\f\f\f≤ϵ (68)\nif\nN≥1\nlog(4/3)log\u001210 max z∈C|∂zE(z)|\n3ϵ\u0013\n−1 (69)\nand\nδ≤log(4/3)ϵ/\u0012\n4 log\u001210 max z∈C|∂zE(z)|\n3ϵ\u0013\u0013\n. (70)\nProof. The proof follows immediately from Lemma 9, using the triangle inequality and setting both sources\nof error to ϵ/2. Specifically,\n5\n3max\nz∈C|∂zE(z)|\u00123\n4\u0013N+1\n=ϵ/2 =⇒N=1\nlog(4/3)log(10 max\nz∈C|∂zE(z)|/3ϵ)−1. (71)\nNext, from the triangle inequality, we have that the error from approximating each of the derivatives within\nerror at most δis from (67)\n\f\f\f\f\fNX\nk=0ℵk(F(xk)−∂xE|x=xk)\f\f\f\f\f≤2(N+ 1)δ≤\u00122\nlog(4/3)log(10 max\nz∈C|∂zE(z)|/3ϵ)\u0013\nδ. (72)\nThe final result follows by setting this equal to ϵ/2 and solving for δ.\nNext, we will consider the problem of evaluating the energy difference between two nearby points using\nthis strategy. We will show how we can integrate the derivative of a quantity, such as energy, using a\nNewton-Cotes formula in order to compute the difference in that quantity between two points.\nTheorem 11 (Energy estimation with Newton-Cotes) .Assume the following:\n1. Let H: [−1,1]7→C2n×2nbe a mapping to Hermitian matrices such that H(x) =P\njαj(x)Ujfor\nunitary matrices Ujandαj(x)∈C. Furthermore, let γ(x)be a lower bound on the spectral gap of\nH(x).\n2. Let E:C7→Rbe a differentiable function such that Γ := max z∈C|∂zE(z)|and for any z∈C\nwe have that the analytic continuation of the energy onto the complex plane is written as E(z) =\n⟨ψ0(z)|H(z)|ψ0(z)⟩withE(z)∈R∀z∈[−1,1].\n3. Let G=PJ−1\nj=0Aj=PJ−1\nj=0Pnj\nk=1βjkΠjkbe the directional derivative operator for the energy such that\nfor parameter x,∂xE(x) =⟨ψ0(x)|G|ψ0(x)⟩at configuration xand|ψ0(x)⟩is an eigenstate of H(x)\nfor any x���[−1,1].20\n4. The assumptions of Corollary 6 and Lemma 7 hold for α(x)andγ(x)for all x∈[−1,1].\nThen there exists a quantum algorithm that takes the precise values of E(−1)and⟨ψ0(−1)|Πjk|ψ0(−1)⟩for\nallj, kand outputs an ϵ-approximate estimate of E(1)with probability of failure\nPf=ϵ2\nψ∈eo(max x∥α(x)∥1/minxγ(x))(73)\nusing a number of queries that scales as\neO \nlog(Γ) max x∥α(x)∥1P\njmax xp\nP0(j;x) max x∥β(x)∥1,1/2log(J/δ′)\nminxγ(x) min x|a0(x)|ϵmax\nj,x\u0012P0(j;x)\n|δj(x)|\u0013!\n.\nProof. First, we have from Corollary 10 that we can estimate the integral within error ϵusing N+ 1 points\nfor\nN∈O(log (Γ /ϵ)) (74)\nand such that each of the values that we require are computed within error O(ϵ/log(Γ /ϵ)). We can then learn\nthese N+ 1 values within the required error tolerance under the assumption that we have perfect knowledge\nof each value of ⟨ψ0(−1)|Πjk|ψ0(−1)⟩. Since we have assumed that the assumptions of Corollary 6 hold for\nallx∈[−1,1], we can apply the corollary to estimate the expectation value of the derivative at each of the\nNpoints in the interval since all of them are evaluated over [ −1,1]. Thus, by Corollary 6, we can compute\na value Dsuch that\n\f\f\f\fD −Z1\n−1⟨ψ0(x)|G(x)|ψ0(x)⟩dx\f\f\f\f≤ϵ. (75)\nDenoting the cost of preparing the ground state as Cost( PE), we can achieve this using a number of queries\nthat scales at most as\nNqueries ∈eO \nNCost( PE)P\njmax xp\nP0(j;x) max x∥β(x)∥1,1/2log(J/δ′)\nϵmax\nj,x\u0012P0(j;x)\n|δj(x)|\u0013!\n⊆eO \nlog(Γ)Cost( PE)P\njmax xp\nP0(j;x) max x∥β(x)∥1,1/2log(J/δ′)\nϵmax\nj,x\u0012P0(j;x)\n|δj(x)|\u0013!\n. (76)\nAsH=P\njαjUjand the spectral gap of the eigenstate in question is at least γ(x), the cost of performing\nphase estimation to project onto the correct energy eigenvalue is given by Lemma 7 to be\nCost( PE)∈eO\u0012max x∥α(x)∥1\nminx|a0(x)|minxγ(x)\u0013\n, (77)\nassuming that the acceptable probability of failure obeys ϵ2\nψ∈eo(max x∥α(x)∥1/minxγ(x)). By substituting (77)\ninto (76) we find that the required number of queries to HandGis in\neO \nlog(Γ) max x∥α(x)∥1P\njmax xp\nP0(j;x) max x∥β(x)∥1,1/2log(J/δ′)\nminxγ(x) min x|a0(x)|ϵmax\nj,x\u0012P0(j;x)\n|δj(x)|\u0013!\n. (78)\nAll that remains to show is that the estimate of Dprovides us an ϵ−approximation of the energy E(1) =\n⟨ψ0(1)|H(1)|ψ0(1)⟩atx= 1. Using the Hellmann-Feynman theorem and the fundamental theorem of\ncalculus, we find that\n⟨ψ0(1)|H(1)|ψ0(1)⟩=⟨ψ0(−1)|H(−1)|ψ0(−1)⟩+Z1\n−1\u0010\n∂x⟨ψ0(x)|H(x)|ψ0(x)⟩\u0011\ndx\n=⟨ψ0(−1)|H(−1)|ψ0(−1)⟩+Z1\n−1⟨ψ0(x)|\u0000\n∂xH(x)\u0001\n|ψ0(x)⟩dx\n=⟨ψ0(−1)|H(−1)|ψ0(−1)⟩+Z1\n−1⟨ψ0(x)|G(x)|ψ0(x)⟩dx.(79)21\nFrom the triangle inequality we then have that\n\f\f\f⟨ψ0(−1)|H(−1)|ψ0(−1)⟩+D − ⟨ ψ0(1)|H(1)|ψ0(1)⟩\f\f\f\n≤\f\f\f⟨ψ0(−1)|H(−1)|ψ0(−1)⟩+Z1\n−1⟨ψ0(x)|G(x)|ψ0(x)⟩dx− ⟨ψ0(1)|H(1)|ψ0(1)⟩\f\f\f\n+\f\f\fD −Z1\n−1⟨ψ0(x)|G(x)|ψ0(x)⟩dx\f\f\f≤ϵ\nand thus the claim follows.\nWe see from this result that, provided the Hamiltonian is sufficiently smooth and gapped and if prior\nknowledge is provided, the asymptotic scaling can be better than the initial scaling. In particular, there are\ntwo ways that this asymptotic scaling can be superior given that we are interested in finding the ground\nstate energy of a Hamiltonian H(1) that is close to the Hamiltonian H(−1) of an initial configuration that\nwe know well.\n1. If∥G∥is small then the value of ∥β∥1,1/2for the simulation will typically be small as well.\n2. If we have that ∥H(−1)−H(1)∥is small and the ground state is gapped, then we will have sufficient\nprior knowledge about P0(k) to estimate the expectation values of the individual projector terms\naccurately given that we know their values at H(−1).\nThis suggests that for each initial Hamiltonian there exists a ball of radius ηsuch that our method will\nbe able to outperform traditional phase estimation as ∥β∥1,1/2will shrink with the value of ηunder the\nassumption that the ground state is gapped. This is significant because it shows that if we have accurate\nprior knowledge about the energy and the expectation values of the projectors that comprise the directional\nderivative operator G, then amplified amplitude estimation can be used to provide a substantial advantage\nfor estimating the values of the energy in a neighborhood about this point. This shows that if we wish to learn\na potential energy landscape for chemistry applications, we do not need to learn every point accurately in a\nmesh. We can instead learn a few relevant points with great accuracy and then refine this prior knowledge to\ngive cheaper estimates of the expectation values for nearby molecular configurations to those that we have\nalready probed.\nV. CONCLUSION\nIn this work, we provide an approach that we call amplified amplitude estimation (AAE) which uses am-\nplitude amplification to increase the sensitivity of amplitude estimation. The main result is that we can\nestimate the expectation value of an operator within error ϵusing a number of queries that scale as\nNqueries ∈O \nP1/2\n0\nϵ\u0012P0\n|δ|\u0013!\n, (80)\nwhere P0is an upper bound on the expectation value and δmeasures the minimum gap between our estimate\nand the true value of the estimated probability. In cases where |δ| ∈Θ(P0) and P0∈Θ(ϵ) then we find\nNqueries ∈O(1/√ϵ). This shows that when estimating small probabilities, within error commensurate with\nthe values that we aim to estimate, we can achieve scaling that is better than what we would expect\nfrom the Heisenberg limit. Further, it is worth noting that this scaling does not simply arise from doing\namplitude estimation on a small probability, despite the fact that O(1/ϵ) estimation is classically possible in\nthe analogous limit. We find instead that the way we use amplitude amplification needs to be adapted using\nprior knowledge in order to be able to improve upon the classical results in such cases and further generalize\nthis to cases where we wish to estimate a sum of multiple expectation values.\nWe provide several ways in which amplified amplitude estimation can be used to provide an advantage\nfor quantum simulation algorithms. We show that this approach can provide an advantage for estimating\nexpectation values of fermionic observables based on prior knowledge of the expectation values of the Pauli\noperators that compose the Jordan-Wigner decomposition of the observables. We observe numerically, based22\non DFT and DMRG calculations, that existing classical methods can provide requisite levels of certainty\nabout such expectation values to make such a procedure profitable. We then conclude by discussing a\nnew approach for estimating energy differences for molecules by integrating the derivative of the energy\nusing Newton-Cotes formulas to perform the integration. These methods can, given sufficiently strong prior\nknowledge and appropriate continuity assumptions of the Hamiltonian, provide improved scaling for energy\nestimation.\nLooking forward, while our results show that prior information can be used to improve our ability to learn\nenergies and expectation values, there are a number of directions to explore beyond it. The first and most\nobvious question is whether these techniques can provide practical advantages for chemistry simulations.\nUnderstanding the practical range where these effects provide a substantial advantage is necessary to under-\nstand the extent to which these ideas can improve upon classical or quantum estimates or cheaply probe the\nenergy landscape of a system given a few points where the energy is known with high confidence. Further,\nit may be interesting to investigate whether it is possible to apply this technique to cases where we aim to\nlearn a vector of expectation values at cost that is sub-linear in the number of terms. Analogous results\nare possible using quantum gradient estimation [28] but in this setting it is not obvious how to use these\ntechniques to improve the estimates given the multitude of prior estimates needed.\nFrom a broader perspective, prior knowledge is still an under-utilized resource in quantum chemistry simu-\nlation algorithms and finding improved ways that perturbative estimates, interpolation techniques or better\nways of incorporating prior knowledge in phase estimation procedures promises to meaningfully reduce the\ncosts of simulation. It is our hope that these ideas, and ones related to it, will be instrumental in helping\nuseful applications of quantum chemistry simulation such as drug or catalyst design become a reality.\nACKNOWLEDGEMENTS\nSS acknowledges support from a Research Award from Google Inc., NSERC Discovery Grants, an On-\ntario Graduate Scholarship as well as support from Boehringer Ingelheim. 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Kambo, Error of the Newton-Cotes and Gauss-Legendre Quadrature Formulas, Mathematics of Computa-\ntion24, 261 (1970)."    },    {        "title": "2402.14794v1.Disjointness_of_inertial_KMS_states_and_the_role_of_Lorentz_symmetry_in_thermalization.pdf",        "content": "arXiv:2402.14794v1  [math-ph]  22 Feb 2024Disjointness of inertial KMS states and the\nrole of Lorentz symmetry in thermalization\nAlbert Georg Passegger∗1,2, Rainer Verch†1\n1Institute for Theoretical Physics, University of Leipzig, Germany\n2Max Planck Institute for Mathematics in the Sciences, Leipz ig, Germany\nDedicated to Detlev Buchholz on the occasion of his 80th birt hday\nAbstract\nFor any local, translation-covariant quantum field theory o n Minkowski spacetime we\nprove that two distinct primary states that are invariant un der the inertial time evolu-\ntions in different inertial reference frames and satisfy a ti melike cluster property called\nthe mixing property are disjoint, i.e. each state is not a per turbation of the other. These\nconditions are fulfilled by the inertial KMS states of the fre e scalar field, thus showing\nthat a state satisfying the KMS condition relative to one ref erence frame is far from\nthermal equilibrium relative to other frames. We review the property of return to equi-\nlibrium in open quantum systems theory and discuss the impli cations of disjointness on\nthe asymptotic behavior of detector systems coupled to stat es of a free massless scalar\nfield. We argue that a coupled system consisting of an Unruh-D eWitt detector moving\nwith constant velocity relative to the field in a thermal stat e, or an excitation thereof,\ncannot approach a KMS state at late times under generic condi tions. This leads to\nan illustration of the physical differences between heat bat hs in inertial systems and\nthe apparent “heat bath” of the Unruh effect from the viewpoin t of moving detectors.\nThe article also reviews, from a quantum field theoretical pe rspective, the quantum\ndynamical system of an Unruh-DeWitt detector coupled to a ma ssless scalar field in a\nKMS state relative to the rest frame of the detector.\n2020 Mathematics Subject Classification :46L60 ,81T05 ,82B10 ,82C10\n∗e-mail: albert_georg.passegger@uni-leipzig.de†e-mail: rainer.verch@uni-leipzig.de\n11. Introduction\nThermal equilibrium states in possibly infinitely extended quantum systems are characterized\nby the KMS (Kubo-Martin-Schwinger) condition [ Kub57 ,MS59 ,HHW67 ,BR97 ]. It can be\nobtained from the thermodynamic limit of Gibbs ensembles an d is expressed in the algebraic\nframework of quantum statistical mechanics as an analytici ty condition for state functionals.\nLetAbe the C∗-algebra of observables of a quantum system, and ω:A →Ca state, i.e.\na linear, positive, normalized expectation value function al corresponding to the preparation\nof the system in a certain ensemble. The KMS condition is form ally given by ω(Aαt(B)) =\nω(αt+iβ(B)A) for allA,B ∈ A andt∈R, whereαt(A) is the time evolution of Ain the\nHeisenberg picture, and the KMS parameter β > 0 represents the inverse temperature of\nthermal equilibrium described by ω. States satisfying the KMS condition, called KMS states,\nare characterized by properties that render them as faithfu l representatives of global thermal\nequilibrium, such as time invariance, perturbation stabil ity [HKT74 ], and a thermodynamic\ninterpretation [ KFGV77 ,PW78 ]. From a mathematical perspective, the KMS property has\na close connection to the theory of faithful normal states on von Neumann algebras, which\nis the topic of Tomita-Takesaki modular theory [ Tak70b ,BR87 ,Bor00 ]. A wide range of\napplications may be found in the monographs [ BR97 ,Haa96 ] and the references cited therein.\nKMS states have also been studied in the context of (relativi stic) quantum field theory.\nFor linear quantum fields on static spacetimes, KMS states an d their mixtures satisfy the\nmicrolocal spectrum condition (Hadamard condition) [ Rad96 ,SV00 ], a regularity criterion\non small scales that distinguishes physical states and is, u nder certain conditions, equivalent\nto the implementation of quantum energy inequalities for su ch states [ FV03 ].\nMimicking the intuitive measurement process in an experime ntal setup, thermal phenom-\nena in quantum field theory such as the Unruh [ Unr76 ,Ful73 ,Dav75 ,Sew82 ,CHM08 ] and\nHawking effect [ Haw74 ,Haw75 ,Wal75 ,FH90 ,Bac99 ] (see also [ Wal94 ,HW15 ]), which em-\nploy the KMS property in their description, may be investiga ted by coupling a “detector”\nsystem to the observables of the field. A simple detector mode l that is commonly considered\nin this context is the Unruh-DeWitt detector [ Unr76 ,DeW79 ,UW84 ,BD82 ], a point-like\n(or sufficiently spatially localized) non-relativistic qua ntum system with finitely many (com-\nmonly two) energy levels, whose coupling to the field is reali zed, for example, by a monopole\ninteraction. The process of measuring observables of a quan tum field using “probes” can be\nformulated in a model-independent, local and covariant way [FV20 ,FV23 ]. One expects that\nUnruh-DeWitt type detector models can be regarded as a speci al case of this general theory\nof quantum field measurement processes; first steps in that di rection were taken in [ FV20 ,\nSec. 5.3] (see also [ PGM22 ]).\nAn interesting question is how an Unruh-DeWitt detector sys tem responds to the state\nof a quantum field depending on the motion of the detector alon g some timelike worldline.\nThere are two physically different ways of observing what may be attributed to the system\n2as thermal behavior: recording thermal transition rates or probabilities between the energy\nlevels of the detector in time-dependent perturbation theo ry (“particle detector” excitation),\nand late time convergence of an initial state into a thermal e quilibrium state under the time\nevolution (“thermometer” equilibration), a process refer red to as thermalization .\nThe Unruh effect is a prominent instance in which such setups h ave been considered. The\noperational statement of the Unruh effect is that a linearly a nd uniformly accelerating de-\ntector with proper acceleration a>0 in Minkowski spacetime coupled to the Minkowski\nvacuum state of a quantum field theory detects the Unruh tempe rature a/2π. (A review of\nmultiple facets of what is called “Unruh effect” in the litera ture can be found in [ Ear11 ].) For\nan Unruh-DeWitt detector coupled to a scalar field one obtain s, in first order perturbation\ntheory, a transition rate corresponding to a Planck (black b ody) spectrum at the Unruh tem-\nperature [ Unr76 ,BD82 ,Wal94 ,CHM08 ,UW84 ,Tak86 ,LS08 ]. In this approach, thermality\nof a two-level Unruh-DeWitt detector can be expressed as the detailed balancing of the quo-\ntient of excitation and de-excitation probabilities (see, e.g., [ Tak86 ,FJL16 ,Per21 ], and the\nrecent [ LPTM23 ] for a discussion of peculiarities in higher dimensional de tector systems).\nThe asymptotic state behavior is a fundamentally different p roperty of the system. For\ninstance, [ Mou18 ] showed that the transition rate of a uniformly accelerated detector coupled\nto the Minkowski vacuum can be non-Planckian at early times, depending on the type of\ncoupling, but the Unruh effect is still observed in the sense t hat the asymptotic (reduced\ndensity matrix) state of the detector at late times is a Gibbs equilibrium state at the Unruh\ntemperature. Moreover, the shape of the response function o f the uniformly accelerated\ndetector coupled to the Minkowski vacuum can depend on the sp acetime dimension: For\nthe scalar field in odd spacetime dimensions the distributio n has a Fermi-Dirac form, as\nopposed to the Planck-type (Bose-Einstein) distribution t hat appears in even dimensions\n[Tak86 ,Unr86 ]. (By contrast, the response function of the detector coupl ed to the Dirac\nfield is Planckian in all spacetime dimensions [ LT16 ].) Despite this “inversion of statistics”,\nthe detector still thermalizes at late times [ DM06 ,ABGG21 ,Mou22 ].\nHence the asymptotic state behavior of detectors can be a rel evant indicator for thermality.\nThe behavior of Unruh-DeWitt detectors and more general ���pa rticle detectors” is a quite\nactive research area. A selection of recent investigations is given by [ MA17 ,Mou18 ] (open\nquantum systems analysis [ Dav76 ] of the reduced density matrix of uniformly accelerated\ndetectors), [ GJMT20 ] (Unruh-like effects for Unruh-DeWitt detectors along stat ionary accel-\nerated trajectories in Minkowski spacetime), and [ JM19 ,Per21 ] (detectors following timelike\nworldlines while being coupled to quantum fields in curved sp acetimes in KMS states relative\nto time evolution along the worldline); see the references t herein for further literature.\nA strong form of thermalization for the full detector-field s ystem (not just the detector\nalone) that will be the focus in our work has been considered b y De Bièvre & Merkli [ DM06 ]\nin the case of the Unruh effect: The open quantum system that is obtained from linearly\ncoupling a two-level detector (using a monopole) to the free scalar field satisfies the property\n3of “return to equilibrium”, an ergodicity property of quant um dynamical systems (see, e.g.,\n[JP96b ,JP97 ,BFS00 ,Pil06 ], and Section 5.1for more references). The coupled detector-field\nsystem converges to the KMS state corresponding to the Unruh temperature, which entails\nthe approach of the reduced state of the detector to the assoc iated Gibbs state. The result\ndoes hold not only if the initial state is given by the detecto r in the ground state and the\nfield in the Minkowski vacuum state, but also if the initial de tector-field state is an excitation\nof that. Therefore, the asymptotic state behavior of the cou pled system is stable in a quite\nstrong sense, as thermal equilibrium is reached from a whole folium of initial states.\nEven though the Unruh effect appears to hint towards a thermal interpretation for the\nMinkowski vacuum, one might ask if the Unruh temperature is a quantity that actually\npertains to a property of the quantum field as a heat bath of “Ri ndler particles” in the vacuum\nstate from the perspective of a uniformly accelerated obser ver (as claimed in, e.g., [ CHM08 ]).\nThe comparison with characteristics of conventional heat b aths in inertial reference frames\n(Lorentz frames) can be taken as a litmus test in that regard. A fundamental result against\nthe heat bath picture of the Unruh effect is obtained from the i nterpretation of KMS states.\nThe Unruh effect is related to the property that the Minkowski vacuum state is a KMS\nstate with KMS parameter 2 πwith respect to the dynamics given by the Lorentz boost\nisometries, which leave the right wedge of Minkowski spacet ime invariant [ Ful73 ,Unr76 ,\nBW75 ,BW76 ,Sew80 ,Sew82 ]. However, while the interpretation of inertial KMS states\nrepresenting thermal equilibrium at positive temperature in the sense of the zeroth and the\nsecond law is well-established [ KFGV77 ,PW78 ] (see also [ Haa96 ,BR97 ]), the inverse of the\nKMS parameter does not represent a temperature in non-inert ial situations [ BS13 ,Sol12 ]. It\nturns out that under the influence of external forces, such as the accelerating force in the\nUnruh effect, Unruh-DeWitt detectors cannot be used as relia ble “thermometers” for the\nquantum field [ BS13 ,BV15 ,BV16 ]. Still, it is remarkable that detector models can exhibit\nthermality under uniform acceleration (cf. the remarks in [ DM06 , Sec. 1]). Hence Unruh-\nDeWitt detectors are useful instruments to study conceptua l questions on idealized detector\nsystems coupled to quantum field theories.\nComplementary to the above results, this article treats the behavior of Unruh-DeWitt\ndetectors under the action of Lorentz boosts to reveal furth er differences between the Unruh\neffect and thermalization in inertial heat baths. It has been noted in [ SCD81 ] that the\nasymptotic behavior of a uniformly accelerated detector is preserved under change of the\nLorentz frame:\n“For example, the temperature depends on the acceleration o f the observer but\nis independent of his velocity. Thus two neighbouring obser vers with the same\nacceleration, but with velocities differing by nearly the ve locity of light would\nobserve the same temperature—there would be no Doppler transformation from\none temperature to the other. ” [ SCD81 , p. 334]\n4In other words, while the detector follows the orbit of a Lore ntz boost describing its uniformly\naccelerated motion and is therefore stationary relative to the Killing flow (in turn, station-\nary relative to the Minkowski vacuum to which it couples), ch anging the Lorentz frame by\nboosting to a different velocity relative to an inertial obse rver does not affect the asymptotic\ntemperature registered by the detector.\nThe goal of this article is to substantiate that statement at the level of asymptotic state\nproperties. We discuss the implications on the thermalizat ion of coupled detector-field sys-\ntems (in the spirit of [ DM06 ]) by contrasting the thermalization in the Unruh effect with an\ninertially moving detector coupled to an inertial heat bath , which further reveals the innate\ndifferences between these two physical scenarios.\nFor our discussion we prove the following algebraic propert y of states: Let σandωbe two\ndistinct states of a local quantum field theory on (1 + d)-dimensional Minkowski spacetime\nthat are invariant under the time translations in different i nertial frames (time evolution\nalong two timelike vectors that are related by a Lorentz boos t). Thenσandωare not in\nthe same folium of states, i.e. they are not a perturbation of one another, if they satisfy\ncertain physically reasonable properties: primarity, whi ch formalizes a “pure phase”, and an\nasymptotic timelike cluster property called the mixing pro perty. We express the statement\nin the theory of operator algebras by the property that the GN S (Gelfand-Naimark-Segal)\nrepresentations associated to the two states are disjoint. The algebraic conditions of primarity\nand mixing are characterized by spectral properties of the g enerator of the time evolution,\nwhich are shown to be fulfilled by the inertial KMS states of th e free scalar quantum field on\n(1 + 3)-dimensional Minkowski spacetime. Hence two KMS stat es with respect to different\ninertial frames are disjoint.\nThe result can be expressed as Lorentz symmetry breaking in i nertial KMS states (cf.\n[Oji86 ]), which we formulate explicitly by showing that Lorentz bo osts are not implementable\nby unitary operators in the GNS representation of general pr imary, mixing states. One can\ninterpret this as the formalization of the observation that there is no Lorentz transformation\nlaw relating temperatures relative to different inertial fr ames [ LM96 ,LM04 ] (see also [ Sew08 ,\nSew09 ]).\nThe disjointness of inertial KMS states strengthens the res ult by Sewell [ Sew08 ,Sew09 ]\nthat a state of a quantum field theory which satisfies the KMS co ndition with respect to time\ntranslations in one inertial frame cannot be a KMS state with respect to time translations\nin a different, Lorentz boosted frame. Sewell concluded that a detector moving relative to\nan inertial heat bath with constant non-zero velocity does n ot approach a KMS state, since\nthe state of the heat bath does not satisfy the KMS condition r elative to the rest frame\nof the detector. For a heat bath given by a massless scalar fiel d, this is supported by the\nnon-thermal transition rate resulting from the Doppler shi fted quanta that are registered by\nthe detector [ CM95a ,CM95b ]; changing to an inertial frame that is moving relative to th e\nrest frame of the heat bath thus has an influence on the excitat ion spectrum of the detector.\n5In our work we present arguments corroborating these result s for the full, coupled detector-\nfield system in the Hamiltonian approach to open quantum syst ems theory [ AJP06 ]. As the\ninitial state of the heat bath is far from equilibrium with re spect to the notion of thermal\nequilibrium determined by the rest frame of the inertially m oving detector, it is suggested\nthat the coupled detector-field system does not approach a KM S state for generic couplings.\nWe motivate this conclusion by combining the disjointness o f inertial KMS states with the\n“return to equilibrium” property of an Unruh-DeWitt detect or coupled to an inertial heat\nbath (massless scalar field) in its rest frame. By contrast, t he Minkowski vacuum state is\nPoincaré-invariant, which implies that an accelerated det ector will return to equilibrium in\nthe sense of [ DM06 ] in every boosted Lorentz frame. Our discussion thereby pla ces the above\nstatement from [ SCD81 ] in a broader context and illustrates that the uniformly acc elerated\ndetector in the Unruh effect does not behave as if it was couple d to a heat bath.\nThis work is structured as follows. In Section 2we introduce necessary notions from\nalgebraic quantum field theory and quantum statistical mech anics, in particular primarity and\nthe mixing property, with complementary material being col lected in Appendix A. Section\n3contains the disjointness results for invariant states rel ative to different inertial reference\nframes, first for general primary, mixing states (Theorem 3.4), then applied to inertial KMS\nstates of the free scalar field (Theorem 3.6). The formulation in terms of the breakdown\nof Lorentz symmetry in these states is presented in Section 3.4. Since similar results have\nbeen obtained before, we discuss the relation to the literat ure, first and foremost [ HT70 ], in\nSection 4. In Section 5the suggested implications of disjointness on the asymptot ic state\nproperties of Unruh-DeWitt detectors are discussed. After an introduction to the algebraic\nformalization of thermalization, we discuss the return to e quilibrium property for a detector\nthat rests relative to a heat bath of a massless scalar field (S ection 5.3). The necessary\nfield theoretic formalism and a useful implementation of the GNS representation for KMS\nstates of the massless scalar field (the Jakšić-Pillet glued representation) are compiled in two\nappendices B&C, which shows the connection to literature on return to equil ibrium for spin-\nboson models. If the detector moves inertially relative to s uch a heat bath, our disjointness\nresult indicates that the coupled detector-field system wil l not thermalize (Section 5.4). In\nSection 5.5we compare the results with the Unruh effect. An outlook and co ncluding remarks\npointing out open questions are given in Section 6.\nConventions We use the “mostly minus” metric signature (+ ,−,..., −), and physical units\nwhere the vacuum speed of light, reduced Planck constant, an d Boltzmann constant are set to\n1. By default or assumption, all Hilbert spaces are complex a nd separable, operator algebras\nare unital (with unit element 1), and morphisms between them are unit-preserving.\n62. Setting and preliminaries\nIn this section we introduce the setting and concepts needed in this work.\n2.1. Algebraic quantum field theory\nWe consider a local quantum field theory on Minkowski spaceti me, described in the model-\nindependent algebraic framework of Haag & Kastler [ HK64 ,Haa96 ,Bor96 ] (see also [ Fre15 ]\nfor a concise review). The quantum field theory is given by a so -called “net” of C∗-algebras:\nTo every open, bounded, causally convex subset O⊂Mof (1 +d)-dimensional Minkowski\nspacetime1M=R1,d, referred to as region , one associates a unital C∗-algebra A(O), called\nalocal observable algebra , subject to the following physically motivated assumption s.\n(A)Isotony : IfO1⊂O2then A(O1)⊂ A(O2) (i.e. there is a∗-monomorphism A(O1)֒→\nA(O2), and A(O1) is identified with the image under this map).\nThe C∗-algebra A=/uniontext\nOA(O) of the theory, called the quasi-local algebra , is the norm-closed\ndirect limit of the family {A(O)}O⊂M, whereOruns over the direct system of all regions in\nM(with set inclusion relation). The C∗-algebras A(O) can be viewed as∗-subalgebras of A.\n(B)Locality (Einstein causality): If O1⊂O⊥, whereO⊥is the causal complement of O,\nthen [ A(O1),A(O)] ={0}. This means that operators associated to spacelike separat ed\nregions commute.\n(C)Translation covariance : The additive group of spacetime translations T:= (R1+d,+)\nis represented on Aas a group of∗-automorphisms {δx}x∈R1+dsuch that δx(A(O)) =\nA(O+x) for all regions O⊂Mandx∈R1+d, whereO+x:={y+x:y∈O}.\nCovariance under spacetime translations is sufficient for ou r main statements on the dis-\njointness of states, where we will be interested in∗-automorphisms {δtu}t∈Rrepresenting the\ntime evolution along the time direction of an inertial frame prescribed by a future-directed\ntimelike unit vector u. Assumption (C) encompasses theories that are covariant un der the\nfull symmetry group of Minkowski spacetime, the proper orth ochronous Poincaré group P↑\n+,\nof which Tforms an abelian subgroup. This stronger form of covariance will be needed in\nthe discussion of Lorentz symmetry breaking and the applica tion to the thermalization of\ndetector systems.\n(C’) Poincaré covariance : The proper orthochronous Poincaré group P↑\n+ofMis repre-\nsented on Aas a group of∗-automorphisms {δ(x,Λ)}(x,Λ)∈P↑\n+such that δ(x,Λ)(A(O)) =\nA(ΛO+x) for all regions O⊂Mand (x,Λ)∈P↑\n+, whereΛO+x:={Λy+x:y∈O}.\n1Mwill be used to denote Minkowski spacetime as a manifold, but also the underlying set (which can be\nidentified with R1+d) without notational distinction.\n7Here we identified the group elements of the semidirect produ ctP↑\n+=T⋊L↑\n+with pairs\n(x,Λ) for vectors x∈R1+dof spacetime translations and proper orthochronous Lorent z\ntransformations Λ∈L↑\n+onM, with the usual group law ( x,Λ)·(x′,Λ′) = (x+Λx′,ΛΛ′).\n(An introduction to these groups can be found in, e.g., [ SU01 ].) The∗-automorphisms of\nspacetime translations from assumption (C) are obtained fr om (C’) via δx=δ(x,1)forx∈\nR1+d. The representation of Lorentz transformations on Awill be denoted by δΛ:=δ(0,Λ).\nFrom the group homomorphism property of ( x,Λ)/ma√sto→δ(x,Λ)and the group law of P↑\n+we get\nδΛ−1= (δΛ)−1and the relation\nδΛ◦δx=δΛx◦δΛfor all (x,Λ)∈P↑\n+. (2.1)\nOther typical assumptions on local quantum field theories, l ike the spectrum condition,\nnuclearity, or a dynamical law (time slice axiom) [ Haa96 ,FV15 ], will not be needed in this\nwork. The axioms (A)–(C’) fit into the general underlying str ucture of quantum field theories\non globally hyperbolic spacetimes [ BFV03 ,FV15 ] (with covariance being a consequence of\nfunctoriality of locally covariant quantum field theory) an d can be shown to be fulfilled, for\nexample, for the free scalar (Klein-Gordon) field [ Bor96 ,FR20 ,Dim80 ] and for free Dirac\n(spin1\n2) and Proca (spin 1) fields [ Bor96 ,Dim82 ,San10 ,Dap11 ].\n2.2. Quantum dynamical systems and invariant states\nFor the reader’s convenience and to fix notation we summarize some terminology and results\nfrom quantum statistical mechanics. Basic familiarity wit h operator algebras will be assumed.\nAdditional material is put into Appendix A. For a thorough exposition, including proofs of\nall statements, we refer to [ BR87 ,BR97 ,Emc72 ,Haa96 ,Pil06 ].\nThroughout this section, Ais a complex C∗-algebra, B(H) is the C∗-algebra of bounded\noperators on a Hilbert space H, and von Neumann algebras, i.e.∗-subalgebras of some B(H)\nclosed in weak operator topology, will be denoted M. A detailed discussion of operator\ntopologies can be found in [ Emc72 , Sec. 2.1.e], [ BR87 , Sec. 2.4.1], and [ Sak98 , Sec.s 1.7, 1.8,\n1.15]. Here we just recall that the weak operator topology onB(H) is the weakest (coarsest)\ntopology such that /a\\}bracketle{tv,(·)w/a\\}bracketri}ht:B(H)→Cis continuous for every v,w∈ H, so a net (Ai)i∈Iin\nB(H) converges to A∈ B(H) in weak operator topology if and only if lim i∈I/a\\}bracketle{tv,Aiw/a\\}bracketri}ht=/a\\}bracketle{tv,Aw /a\\}bracketri}ht\nfor allv,w∈ H in the standard topology of C. The ultraweak (also called weak-∗orσ-weak)\ntopology onB(H) is the weakest topology such that the functionals tr( ρ(·)) :B(H)→Care\ncontinuous for every trace class operator ρonH. The ultraweak topology is stronger (finer)\nthan the weak operator topology, the topologies agree on nor m-bounded subsets, and the\nclosures of a∗-subalgebra of B(H) in weak operator and ultraweak topology coincide.\nAlgebraic states are positive, normalized, linear functio nals on an operator algebra. Any\nstate can be represented as a vector in a Hilbert space.\n8Proposition 2.1 (GNS representation [ BR87 , Thm. 2.3.16])\nFor every state ω:A →Cthere is a∗-representation πω:A → B (Hω)on a Hilbert space\nHωand a unit vector Ωω∈ Hωsuch thatω=/a\\}bracketle{tΩω,πω(·)Ωω/a\\}bracketri}htandπω(A)Ωωis norm dense\ninHω. The triple (πω,Hω,Ωω), orπωfor short, is called the GNS (Gelfand-Naimark-Segal)\nrepresentation ofω. It is unique up to unitary equivalence.\nDefinition 2.2 (Commutant, cyclic, separating, normal, fai thful)\nThecommutant of a∗-subalgebra R ⊆ B (H) isR′:={X∈ B(H) :XY=YXfor allY∈\nR} ⊆ B (H). If M ⊆ B (H) is a von Neumann algebra, a vector Ω ∈ H iscyclic forMif\nMΩ is dense in H, and separating ifX∈ M andXΩ = 0 implies X= 0. A state ϕonMis\nnormal if it is given by a density operator, i.e. ϕ= tr(ρ(·)) for a self-adjoint, positive trace\nclass operator ρonHwith unit trace. A normal state ϕisfaithful ifϕ(X∗X) = 0 implies\nX= 0.\nA vector Ω ∈ H is separating for Mif and only if it is cyclic for M′. Ifϕ=/a\\}bracketle{tΨϕ,(·)Ψϕ/a\\}bracketri}htis\na vector state for a unit vector Ψ ϕ∈ H, thenϕis faithful if and only if Ψ ϕis separating for\nM.\nDefinition 2.3 (Induced von Neumann algebra)\nLetωbe a state on Awith GNS representation ( πω,Hω,Ωω). The von Neumann algebra\ninduced by ωisMω:=πω(A)′′⊆ B(Hω), where ( ·)′′= ((·)′)′is the bicommutant.\nThe bicommutant of a∗-subalgebra of some B(H) coincides with the closure in weak operator\ntopology by von Neumann’s bicommutant theorem (see [ BR87 , Thm. 2.4.11]).\nDefinition 2.4 (Folium, quasi-equivalence, disjointness)\nLetωbe a state on Awith GNS representation πω. The folium Fol(ω) ofωis the collection\nof density operator states tr( ρπω(·)), called states normal toω. Two states σ,ωonAare\nquasi-equivalent if Fol(σ) = Fol(ω), and disjoint if Fol(σ)∩Fol(ω) =∅.\nThe folium Fol( ω) ofωcan be characterized as the norm closure of the convex hull of states\nω(B∗(·)B)/ω(B∗B) forB∈ A such thatω(B∗B)/\\e}atio\\slash= 0, which are interpreted as “perturba-\ntions” or “excitations” of ω. A stateσis in the folium Fol( ω) if and only if there is a unique\nnormal state ˜ σonMω=πω(A)′′such thatσ= ˜σ◦πω. This establishes an isomorphism\nbetween Fol( ω) and normal states on Mω(see the discussion preceding [ Pil06 , Def. 2.20]).\nDefinition 2.5 (Quantum & W∗-dynamical system, invariant state)\nAquantum dynamical system (A,α) is a pair consisting of a C∗-algebra Aand a one-\nparameter group α={αt}t∈Rof∗-automorphisms on A, called the dynamics onA. If\nM ⊆ B (H) is a von Neumann algebra equipped with dynamics γ={γt}t∈Rthat is ultra-\nweakly continuous, i.e. t/ma√sto→tr(ργt(X)) is continuous for all positive trace class operators ρ\nonHandX∈ M , the quantum dynamical system ( M,γ) is called a W∗-dynamical system\nwith W∗-dynamicsγ. A stateωon (A,α) is called invariant under α(orα-invariant ) if\nω◦αt=ωfor allt∈R.\n9The one-parameter group αis usually interpreted as “time evolution” in the Heisenber g pic-\nture, acting on the system with “observable algebra” A. In general we do not demand any\ncontinuity for t/ma√sto→αt. A particular family of invariant states on a quantum dynami cal sys-\ntem ( A,α) is given by ( α,β)-KMS (Kubo-Martin-Schwinger) states [Kub57 ,MS59 ,HHW67 ,\nBR97 ], also referred to as β-KMS states on ( A,α) forβ > 0, which are used to describe\nquantum systems in thermal equilibrium. The precise definit ion of the KMS condition satis-\nfied by these states is given in Definition A.1, and we refer to Section 1for more conceptual\nexplanations.\nThe following proposition is compiled from [ Pil06 , Lemma 4.26], [ KM15 , Prop. 5.1.17], and\n[Tak70b , Thm. 13.3] (see also [ BR97 , Cor. 5.3.4] for the special case of strongly continuous\ndynamics).\nProposition 2.6 (Induced W∗-dynamical system, Liouvillean)\nLetωbe anα-invariant state on a quantum dynamical system (A,α)with GNS representation\n(πω,Hω,Ωω). Then there is a one-parameter group {Uω(t)}t∈Rof unitary operators on Hω,\nuniquely determined by\nπω(αt(A)) =Uω(t)πω(A)Uω(t)−1, Uω(t)Ωω= Ωωfor allA∈ A, t∈R, (2.2)\nwhich uniquely defines a one-parameter group γ={γt}t∈Rof∗-automorphisms\nγt=Uω(t)(·)Uω(t)−1\nonMω=πω(A)′′. The GNS vector state /a\\}bracketle{tΩω,(·)Ωω/a\\}bracketri}ht, which represents ωonMω, is a\nγ-invariant state.\n•The one-parameter group {Uω(t)}t∈Ris strongly continuous if and only if\nR∋t/ma√sto→ω(A∗αt(A))∈Cis continuous for all A∈ A. (2.3)\nIn this case, γis generated by a unique (possibly unbounded) self-adjoint operatorLon\nHωby Stone’s theorem, i.e. γt= eiLt(·)e−iLtfor allt∈R, such that, by Eq. (2.2),\nγt(πω(A)) =πω(αt(A)) = eiLtπω(A)e−iLtfor allA∈ A, t∈R, L Ωω= 0,(2.4)\nandγis a W∗-dynamics on Mω. The operator Lis called the Liouvillean of the W∗-\ndynamical system (Mω,γ)induced by ω.\n•Ifωis aβ-KMS state on (A,α)forβ∈R\\ {0}, thenγis generated by a Liouvillean\nas above, and /a\\}bracketle{tΩω,(·)Ωω/a\\}bracketri}htis a faithful, normal β-KMS state on the W∗-dynamical system\n(Mω,γ). In particular, the GNS vector Ωωof the KMS state ωis cyclic and separating\nforMω.\n10The operator Lis also referred to as the “ ω-Liouvillean” or “Ω ω-Liouvillean” in the theory\nof standard representations [ DJP03 ,Pil06 ]. For systems at positive temperature the Liou-\nvillean is sometimes also called the “thermal Hamiltonian” [BFS00 ]. At zero temperature\none typically writes Hinstead ofLand refers to it as the “Hamiltonian”; if H≥0, the\nstateωis called a ground state [San13 ,KM15 ]. The continuity condition ( 2.3) is fulfilled,\nfor instance, if αis strongly continuous, i.e. lim t→0/bardblαt(A)−A/bardbl= 0 for all A∈ A (for any\ninvariant state ω), ifωis a KMS state (for any dynamics α, cf. Definition A.1), or if ( A,α)\nis a W∗-dynamical system and ωis a normal invariant state (cf. [ Pil06 , Sec. 4.4]).\n2.3. Primarity and mixing property\nTwo main assumptions for the states we consider in our theore ms are primarity [ BR87 ,KR97 ,\nHaa96 ] and the (strong) mixing property [ BR87 ,JP96b ,Pil06 ]. We introduce these no-\ntions, discuss their interpretation, and present sufficient criteria for their validity that can be\nchecked in concrete models. Let ( A,α) be a quantum dynamical system.\nDefinition 2.7 (Primary state)\nLetωbe a state on Awith GNS representation ( πω,Hω,Ωω). The state is called primary (or\na “factor state”) if the induced von Neumann algebra Mω=πω(A)′′⊆ B(Hω) is a factor , i.e.\nhas trivial center Mω∩ Mω′=C1.\nPrimary states are precisely those states that do not posses s a non-trivial decomposition into\ndisjoint states. An important property of primary states is their dichotomy with regards\nto their folia: Two primary states are either disjoint or qua si-equivalent (see [ KR97 , Prop.\n10.3.12 (ii)]).\nFor strongly continuous dynamics α, i.e. when t/ma√sto→αt(A) is norm-continuous in Afor all\nA∈ A, the convex set K(α,β)of (α,β)-KMS states on Ais a weak-∗compact subset of the\nset of states, and an ( α,β)-KMS state is primary if and only if it is an extremal point of\nK(α,β)[BR97 , Thm. 5.3.30]. In that case, primary ( α,β)-KMS states are those which are not\na mixture of other ( α,β)-KMS states, and thus describe pure thermodynamic phases; see\n[Haa96 , Sec. V.1.5] for a detailed discussion. Without the continu ity assumption on α(in\nparticular for W∗-dynamics), K(α,β)is not necessarily weak-∗compact and may lack extremal\npoints [ TW73 , Sec. §4] (see also the comments in [ BR97 , p. 118]).\nWithout any assumption on the continuity of the dynamics, a s ufficient, spectral condition\nfor the primarity of a KMS state is given by the simplicity of t he zero eigenvalue of the\nLiouvillean, which signifies the uniqueness (up to a phase) o f the invariant vector under the\ninduced W∗-dynamics (see Proposition A.4). For completeness we include a proof in the\nappendix (Proposition A.5).\nProposition 2.8 (Criterion for primarity of KMS states)\nLetωbe a KMS state on (A,α)with GNS representation (πω,Hω,Ωω)and Liouvillean L\n11generating the dynamics γof the induced W∗-dynamical system (Mω,γ). IfLhas a simple\neigenvalue at 0, i.e. kerL=CΩω, thenωis primary.\nNow we turn to the cluster property that will be of interest in this work.\nDefinition 2.9 (Mixing property)\nAnα-invariant state ωon (A,α) is called mixing , or the triple ( A,α,ω )satisfies the mixing\nproperty , if\nlim\nt→∞ω(Bαt(A)) =ω(A)ω(B) for allA,B ∈ A.\nThe mixing property is one of many cluster properties, i.e. a symptotic factorization properties\nfor expectation values of states under some dynamics (see, e .g., [BR87 , Sec. 4.3] and the\nintroductory lecture notes [ Ver69 ]). In thermal quantum field theory this property appears\nas the “time-clustering” of correlation functions [ BB96 ]. Variants of the mixing property for\nGalilei-invariant fermionic systems have been investigat ed in [ NT89 ] (see also [ NT91 ]). For\na local quantum field theory on Minkowski spacetime it has bee n shown in [ JNW10 ] that\ncertain states can be weakly approximated by states that sat isfy weak asymptotic abelianness,\na property closely related to the mixing property (see at the end of this section).\nThe mixing property is the quantum version of a similar prope rty for classical ergodic\nsystems [ AA68 ], where it constitutes one level of randomness in a whole hie rarchy [ BFK06 ].\nErgodicity properties of invariant states on quantum dynam ical systems can be characterized\nby spectral properties of the generator of the dynamics, a co rrespondence referred to as\n“quantum Koopmanism” [ JP96b ,Pil06 ] due to the analogy to Koopman’s classical results\n[Koo31 ] (see [ AA68 , Sec. §9] and also [ Pil06 , Sec. 3.2], [ RS72 , Sec. VII.4]).\nProposition 2.10 (Quantum Koopman mixing criterion)\nLetωbe anα-invariant state on (A,α)with GNS representation (πω,Hω,Ωω), such that Ωω\nis separating for Mω=πω(A)′′(i.e., /a\\}bracketle{tΩω,(·)Ωω/a\\}bracketri}htis faithful) and ωsatisfies the continuity\ncondition (2.3). LetLbe the Liouvillean of the induced W∗-dynamical system (Mω,γ). If the\nspectrum of Lis absolutely continuous apart from a simple eigenvalue at 0, i.e. kerL=CΩω,\nthen limt→∞eiLt=|Ωω/a\\}bracketri}ht/a\\}bracketle{tΩω|in weak operator topology and thus (A,α,ω )satisfies the mixing\nproperty.\nThe proof can be found in [ JP96b , Sec. 4], among the spectral characterizations of other\nergodicity properties; see also [ BFS00 , Lemma III.1] and [ Pil06 , Cor. 4.66]. In [ JP96b , Def.\n4.1] the mixing property is formulated for the Liouvillean i n the GNS representation of a\nfaithful, normal, invariant state on a W∗-dynamical system. In Proposition 2.10, the GNS\nrepresentation of the faithful, normal, γ-invariant state /a\\}bracketle{tΩω,(·)Ωω/a\\}bracketri}htonMω⊆ B(Hω) is the\nidentity morphism on Mω, hence the mixing property for ( Mω,γ,/a\\}bracketle{tΩω,(·)Ωω/a\\}bracketri}ht), and thereby\nfor (A,α,ω ), is obtained from [ JP96b , Thm. 4.2 & Cor. 4.3] by setting π≡id.\n12We mention that the faithfulness of the state /a\\}bracketle{tΩω,(·)Ωω/a\\}bracketri}htthat extends ωtoMωis essential\nfor this spectral characterization of the mixing property. Ifωis aβ-KMS state for some β >0,\nfaithfulness is guaranteed by Proposition 2.6, and the other conditions in Proposition 2.10\nare met as well. In the “zero temperature”, ground state limi tβ→ ∞ faithfulness is lost (see\n[RW85 , Prop. 2]), and Proposition 2.10is not valid for such states (see Remark 1 in [ JP96b ,\nSec. 4]). However, there is a generalization that applies to possibly non-faithful invariant\nstates and concerns the spectral properties of a certain res triction of the Liouvillean [ Pil06 ,\nThm. 4.65].\nFor KMS states the mixing property is closely related to the s tability under some perturba-\ntion of the dynamics [ HKT74 ,BKR78 ]. Most important for our purposes, the mixing property\nof KMS states is equivalent to the “return to equilibrium” pr operty (see [ JP96b , Thm. 4.2]),\nto be discussed later in Section 5.1in the context of thermalization of Unruh-DeWitt detec-\ntors coupled to KMS states of a quantum field. Under the condit ions of Proposition 2.10, it\nholds kerL=CΩωif and only if ( A,α,ω ) satisfies return to equilibrium in the ergodic mean\n(see Definition 5.1) [JP96b , Thm. 4.2], which is weaker than mixing (respectively retur n to\nequilibrium).\nA cluster property that is closely related to mixing is weak a symptotic abelianness [ DKKR67 ,\nDKS69 ,BR87 ] (here we use the formulation appearing in, e.g., [ NT89 ,NT91 ,JNW10 ]). The\ntriple ( A,α,ω ) isweakly asymptotically abelian for anα-invariant state ωif\nlim\nt→∞ω(C[αt(A),B]C′) = 0 for all A,B,C,C′∈ A, (2.5)\nor, equivalently, lim t→∞[πω(αt(A)),πω(B)] = 0 for all A,B ∈ A in weak operator topology\nunder the GNS representation πωofω. Ifωis primary and has a separating GNS vector,\nthen ( A,α,ω ) is mixing if and only if it is weakly asymptotically abelian (see Lemma A.6, cf.\n[NT91 ] or [ BR87 , p. 403]). In our main Theorem 3.4the mixing property can therefore be\nreplaced by weak asymptotic abelianness. Nevertheless, th e proofs will be formulated using\nthe mixing property, which has a useful spectral characteri zation by Proposition 2.10.\n3. Disjointness of invariant states in different inertial\nframes\nIn this section we present our main results on the disjointne ss of states with respect to\ndifferent inertial reference frames. For the proofs we will n eed some results regarding limits\nof∗-automorphism groups in quantum dynamical systems and alge braic quantum field theory,\nwhich are discussed in Section 3.1. After that, we prove the disjointness of primary, mixing\nstates in Section 3.2. In Section 3.3we show that the result applies to states of the free scalar\nfield satisfying the KMS condition with respect to inertial t ime translations.\n133.1. Three lemmata on limits\nA cluster point (accumulation point) of a net (Moore-Smith s equence) in a topological space\nis a point for which there exists a subnet converging to it. Th ere are several non-equivalent\ndefinitions for the notion of a subnet [ Sch97 , 7.14 ff.]. For the sake of a concrete definition\nwe may consider Kelley subnets: Let ( I,⊳I),(J,⊳J) be directed sets, and ( ai)i∈Ia net in a\ntopological space. A (Kelley) subnet of (ai)i∈Iis given by ( af(j))j∈J, wheref:J→Iis a\nmap such that for every i∈Ithere exists j0∈Jso thati⊳If(j) whenever j0⊳Jj. It is\neasy to see that if a net converges to a point, so does every sub net. Moreover, in a compact\nspace every net has a cluster point. These fundamental prope rties are shared by any of the\ncommon definitions of a subnet, as explained in [ Sch97 , 7.19, 15.38, 17.2].\nThe first lemma is a consequence of the mixing property (Defini tion2.9). The result and\nvariants of it have been presented before in different contex ts, see for example [ HKT74 ,NT91 ,\nBV15 ]. For systems with “discrete” dynamics given by a single non -trivial∗-automorphism\nan analogous statement is proven in [ NTW88 ] (see also [ Thi92 , Prop. (3.1)]).\nLemma 3.1\nLetωbe a mixing state on a quantum dynamical system (A,α)with a GNS representation\n(πω,Hω,Ωω)such that Ωωis separating for Mω=πω(A)′′. Then\nlim\nt→±∞πω(αt(A)) =ω(A)1for allA∈ A\nin weak operator topology.\nProof.\nLetA∈ A. We first prove the statement for the limit t→ ∞ . The net ( πω(αt(A)))t∈Ris\nnorm-bounded by /bardblπω(αt(A))/bardbl ≤ /bardblαt(A)/bardbl=/bardblA/bardbl(as∗-homomorphisms between C∗-algebras\nare norm-decreasing, and∗-automorphisms are isometric), thus it is contained in a nor m-\nclosed ball in Mω, which is compact in ultraweak (weak-∗) topology by the Banach-Alaoglu\ntheorem. Hence the net has a cluster point X∈ Mωwith respect to weak operator topology,\ni.e. there exists some directed set Iand a map f:I→Rdefining a subnet ( πω(αf(i)(A)))i∈I\nthat converges to Xin weak operator topology. In particular,\nlim\ni∈I/a\\}bracketle{tπω(B)Ωω,πω(αf(i)(A))Ωω/a\\}bracketri}ht=/a\\}bracketle{tπω(B)Ωω,XΩω/a\\}bracketri}ht\nfor allB∈ A. On the other hand, the mixing property of ωimplies that\nlim\nt→∞/a\\}bracketle{tπω(B)Ωω,πω(αt(A))Ωω/a\\}bracketri}ht=ω(A)ω(B∗)≡ /a\\}bracketle{tπω(B)Ωω,ω(A)1Ωω/a\\}bracketri}ht\nfor allB∈ A. Since ( /a\\}bracketle{tπω(B)Ωω,πω(αf(i)(A))Ωω/a\\}bracketri}ht)i∈Iis a subnet of the convergent net\n(/a\\}bracketle{tπω(B)Ωω,πω(αt(A))Ωω/a\\}bracketri}ht)t∈Rthe limits must coincide, i.e. /a\\}bracketle{tπω(B)Ωω,(X−ω(A)1)Ωω/a\\}bracketri}ht= 0.\nThis holds for all B∈ A, and Ωωis cyclic, so ( X−ω(A)1)Ωω= 0, which implies X=ω(A)1\n14by the assumption that Ω ωis separating. As this applies to all cluster points, ω(A)1is the\nunique cluster point of ( πω(αt(A)))t∈R, which is a net in a weakly compact subspace and thus\nhas to converge to ω(A)1.\nFor the limit t→ −∞ we show the equivalent statement that lim t→∞πω(α−t(A)) =ω(A)1\nin weak operator topology for A∈ A. Notice that the mixing property is equivalent to\nlimt→∞ω(α−t(A)B) =ω(A)ω(B) for allA,B ∈ A, sinceω(α−t(A)B) =ω(Aαt(B)) by the\nα-invariance of ω. The proof therefore proceeds in a similar way as before.\n/square\nOur second lemma concerns dynamical systems of von Neumann a lgebras. A similar result\nfor KMS states is [ HT70 , Thm. 4] (see the discussion in Section 4). We formulate the\nstatement in a slightly more general form than needed for the proof of our main theorem.\nFrom now on we choose to write the concatenation of automorph isms as juxtaposition to\nsimplify notation. Furthermore, two∗-automorphism groups {αt}t∈Rand{α′\nt}t∈Rare said to\ncommute if αtα′\ns=α′\nsαtfor alls,t∈R.\nLemma 3.2\nLetM ⊆ B (H)be a von Neumann algebra of operators on a Hilbert space H. Letϕ=\n/a\\}bracketle{tΨϕ,(·)Ψϕ/a\\}bracketri}htandψ=/a\\}bracketle{tΨψ,(·)Ψψ/a\\}bracketri}htforΨϕ,Ψψ∈ H be vector states on Mthat are invariant\nunder commuting one-parameter∗-automorphism groups γϕ={γϕ\nt}t∈Randγψ={γψ\nt}t∈R,\nrespectively. Assume that there is a∗-subalgebra N ⊆ M with N′′=Msuch that for every\nX∈ N the nets (γϕ\nt(X))t∈R,(γψ\nt(X))t∈R, respectively (γψ\n−tγϕ\nt(X))t∈R, have cluster points in\nC1in weak operator topology as t→ ±∞ , respectively t→ ∞ . Then limt→±∞γϕ\nt(X) =\nϕ(X)1andlimt→±∞γψ\nt(X) =ψ(X)1for allX∈ N in weak operator topology, and ϕ=ψ\nonM.\nProof.\nFixX∈ N. The existence of cluster points for the stated nets is guara nteed from the outset\nby the Banach-Alaoglu theorem; for example, the net ( γϕ\nt(X))t∈Rover the directed set ( R,≤)\n(i.e. we consider the limit t→ ∞ ) is norm-bounded by /bardblγϕ\nt(X)/bardbl=/bardblX/bardbl(as∗-automorphisms\nare isometric). The assumption is that the cluster points ar e multiples of the identity, so\nthere exists some directed set Iand a map f:I→Rdefining a subnet ( γϕ\nf(i)(X))i∈Iof\n(γϕ\nt(X))t∈Rsuch that lim i∈I/a\\}bracketle{tv,γϕ\nf(i)(X)w/a\\}bracketri}ht=cϕ\nX/a\\}bracketle{tv,w/a\\}bracketri}htfor somecϕ\nX∈Cfor allv,w∈ H. In\nparticular, we get cϕ\nX= limi∈I/a\\}bracketle{tΨϕ,γϕ\nf(i)(X)Ψϕ/a\\}bracketri}ht=ϕ(X) asϕisγϕ-invariant. This shows that\nϕ(X)1is the unique cluster point and therefore (as in the proof of L emma 3.1) (γϕ\nt(X))t∈R\nconverges to ϕ(X)1. Similarly, the net ( γψ\nt(X))t∈Rconverges to ψ(X)1in weak operator\ntopology. The proof for the limit t→ −∞ is identical.\nForX∈ N the net (γψ\n−tγϕ\nt(X))t∈Rhas a cluster point cψϕ\nX1for somecψϕ\nX∈C. Let\n(γψ\n−g(j)γϕ\ng(j)(X))j∈Jbe a subnet (given by a directed set Jand a map g:J→R) converging\ntocψϕ\nX1. Then\ncψϕ\nX= lim\nj∈J/a\\}bracketle{tΨψ,γψ\n−g(j)γϕ\ng(j)(X)Ψψ/a\\}bracketri}ht= lim\nj∈J/a\\}bracketle{tΨψ,γϕ\ng(j)(X)Ψψ/a\\}bracketri}ht=ϕ(X)\n15byγψ-invariance of ψand the weak operator limit properties (note that ( γϕ\ng(j)(X))j∈Jis a\nsubnet of the convergent net ( γϕ\nt(X))t∈R). On the other hand,\ncψϕ\nX= lim\nj∈J/a\\}bracketle{tΨϕ,γψ\n−g(j)γϕ\ng(j)(X)Ψϕ/a\\}bracketri}ht= lim\nj∈J/a\\}bracketle{tΨϕ,γψ\n−g(j)(X)Ψϕ/a\\}bracketri}ht=ψ(X),\nusing that γϕandγψcommute,ϕisγϕ-invariant, and the limit properties. One therefore\nconcludes that ϕ(X) =cψϕ\nX=ψ(X) for allX∈ N. Since N ⊆ M is ultraweakly dense\nby von Neumann’s bicommutant theorem [ BR87 , Thm. 2.4.11], and the states ϕandψare\nnormal, they are already uniquely determined on Nand soϕ↾N=ψ↾Nimpliesϕ=ψ.\n/square\nFinally, we state (without proof) a cluster property regard ing the asymptotic commutativ-\nity of observables under spacelike translations.\nLemma 3.3 ([ Bor96 , Lemma IV.4.2])\nLetAbe the quasi-local algebra of an algebraic quantum field theo ry on (1 +d)-dimensional\nMinkowski spacetime M∼=R1+d, satisfying isotony, locality and translation covariance (as-\nsumptions (A)–(C) in Section 2.1), where spacetime translations are represented on Aby\n∗-automorphisms {δx}x∈R1+d. Letπ:A → B (H)be a∗-representation of Aon a Hilbert space\nH, with induced von Neumann algebra M:=π(A)′′. Then for every A∈ A,X′∈(M∩M′)′,\nand spacelike vector a∈R1+d, one has\nlim\nt→∞[π(δta(A)),X′] = 0\nin weak operator topology.\n3.2. Disjointness of primary states\nConsider a quantum field theory on (1 + d)-dimensional Minkowski spacetime M=R1,d,\ngiven by a family {A(O)}O⊂Mof C∗-algebras satisfying (A)–(C) as introduced in Section\n2.1, with quasi-local algebra A. We want to compare states that are invariant according to\ndifferent inertial observers. Each observer measures time r elative to the respective inertial\nrest frame of reference (Lorentz frame). The worldline of an inertial observer located at some\npointx∈Mat time 0 is given by t/ma√sto→x+tu, where uis a future-directed timelike unit vector\nspecifying the time direction of the observer’s reference f rame, which we also call uto keep\nthe notation simple. Time evolution by the amount t∈Rrelative to uis therefore described\nby spacetime translation with tu. By assumption (C) there is a one-parameter group of\n∗-automorphisms {δtu}t∈RonAdescribing the time evolution relative to ufor observables.\nLetuandwbe two linear independent future-directed timelike unit ve ctors, specifying the\ntime directions of two inertial reference frames that are re lated by a non-trivial Lorentz boost.\n(Without loss of generality one may realize these reference frames using the isomorphism\n16M∼=R1+dwith coordinates ( x0,x1,x2,...,xd) onR1+dadapted to the frame u, such that\nu= (1,0,0,..., 0) and w= (1 −v2\nrel)−1/2(1,−vrel,0,..., 0) for relative velocity −1<v rel<1,\nvrel/\\e}atio\\slash= 0, along the x1-axis.) The time translations relative to uandware represented by\ncommuting one-parameter groups α(u)={α(u)\nt}t∈Randα(w)={α(w)\nt}t∈Rof∗-automorphisms\nα(u)\nt=δtu, α(w)\nt=δtw (3.1)\nonA, respectively. The proof of the following theorem relies on the three lemmata presented\nin the previous section.\nTheorem 3.4 (Disjointness of primary, mixing states)\nLetσ,ωbe two distinct states on Athat are invariant under the commuting one-parameter\n∗-automorphism groups α(u),α(w), respectively. Assume that σ,ωare primary, (A,α(u),σ)\nand(A,α(w),ω)have the mixing property, and the GNS vectors of σ,ωare separating for the\nrespective induced von Neumann algebras. Then σandωare disjoint.\nProof.\nSinceσandωare primary, they are either disjoint or quasi-equivalent ( see [KR97 , Prop.\n10.3.12 (ii)]). Hence assume, on the contrary, that σandωare quasi-equivalent.\nLet (πσ,Hσ,Ωσ) and (πω,Hω,Ωω) be the GNS representations of σandω, respectively, and\ndenote the induced von Neumann algebras by Mσ:=πσ(A)′′⊆ B(Hσ) and Mω:=πω(A)′′⊆\nB(Hω). Then there exists a∗-isomorphism Θ : Mω→ Mσsuch that Θ ◦πω=πσ[BR87 , Thm.\n2.4.26]. By [ KR97 , Thm. 7.2.9], a∗-isomorphism between von Neumann algebras with cyclic\nand separating vectors is unitarily implemented, so quasi- equivalent states with separating\nGNS vectors are unitarily equivalent. Hence there is a unita ryU:Hω→ Hσsuch that\nΘ =U(·)U−1andπσ=Uπω(·)U−1.\nDefine the states\n˜σ=/a\\}bracketle{tU−1Ωσ,(·)U−1Ωσ/a\\}bracketri}ht,˜ω=/a\\}bracketle{tΩω,(·)Ωω/a\\}bracketri}ht\nonMω, which represent σandωin the sense that σ= ˜σ◦πωandω= ˜ω◦πωonA. SinceU\nis unitary,U−1Ωσis cyclic and separating for Mω, and the GNS representation of σis given\nby (πω,Hω,U−1Ωσ).\nThe induced one-parameter∗-automorphism groups γ(u)andγ(w)onMωare defined by\nπω◦α(u)\nt=γ(u)\nt◦πω(by transporting the induced dynamics on MσtoMωvia Θ) and\nπω◦α(w)\nt=γ(w)\nt◦πω. They commute, i.e. γ(u)\ntγ(w)\ns=γ(w)\nsγ(u)\ntfor alls,t∈R, becauseα(u)and\nα(w)commute, and the states ˜ σand ˜ωare invariant under γ(u)andγ(w), respectively.\nLetX=πω(A)∈πω(A). Thenγ(u)\nt(X) =πω(α(u)\nt(A)) andγ(w)\nt(X) =πω(α(w)\nt(A)),\nand sinceσandωare mixing and have separating GNS vectors, Lemma 3.1implies that\nlimt→±∞γ(u)\nt(X) = ˜σ(X)1and limt→±∞γ(w)\nt(X) = ˜ω(X)1. Furthermore, as uandware\nlinear independent future-directed timelike vectors, α(u,w)\nt :=α(w)\n−tα(u)\nt=δt(u−w), which is\n17represented under πωbyγ(u,w)\nt:=γ(w)\n−tγ(u)\ntonMω, represents a translation along the spacelike\nvector u−wfort∈R. The net ( γ(u,w)\nt(X))t∈R, whereγ(u,w)\nt(X) =πω(α(u,w)\nt(A)), is norm-\nbounded, so the Banach-Alaoglu theorem provides the existe nce of a weak cluster point Y,\nwhich is the limit of a subnet ( πω(α(u,w)\nf(i)(A)))i∈Ifor some directed set Iand mapf:I→R.\nSpacetime translations in spacelike directions act asympt otically abelian by Lemma 3.3, thus\nlimt→∞[πω(α(u,w)\nt(A)),X′] = 0 in weak operator topology for every X′∈(Mω∩ Mω′)′=\nB(Hω), where we use the primarity of ω. For the subnet this implies\n[Y,X′] = lim\ni∈I[πω(α(u,w)\nf(i)(A)),X′] = 0 for all X′∈ B(Hω)\nin weak operator topology, and therefore Y∈C1. Hence, for every X∈πω(A), the net\n(γ(u,w)\nt(X))t∈Rhas a cluster point in weak operator topology that lies in C1.\nNow we can apply Lemma 3.2(forM ≡ M ω,N ≡πω(A),ϕ≡˜σ,ψ≡˜ω,γϕ≡γ(u),\nγψ≡γ(w)), which gives σ= ˜σ◦πω= ˜ω◦πω=ω, a contradiction to the assumption that σ,ω\nare distinct states on A.\n/square\n3.3. Disjointness of KMS states of the free scalar field\nThe disjointness result proven in Theorem 3.4applies, in particular, to inertial KMS states sat-\nisfying the assumptions of primarity and mixing. As an examp le, we consider the KMS states\nof the free real scalar (Klein-Gordon) field of mass m≥0 on four-dimensional Minkowski\nspacetime M=R1,3, given by a net {AF(O)}O⊂Mof C∗-algebras satisfying assumptions\n(A)–(C). The construction of the theory and its inertial KMS states in terms of one-particle\nstructures is summarized in Appendix B, and we use the notation introduced there.\nThe KMS states are defined in Appendix B, without loss of generality, using coordinates\nsuch that the time direction is (1 ,0) (Eq. ( B.1)). The free scalar field on Malso satisfies the\nPoincaré covariance condition (C’) of Section 2.1. By applying Lorentz boosts one obtains\nKMS states with respect to time translations in different ine rtial frames. If ωF,βis the unique\nβ-KMS state in a frame u, andΛ∈L↑\n+is a boost represented on Aby the∗-automorphism δΛ,\nthenωF,β◦(δΛ)−1is the unique β-KMS state with respect to time translations δΛ◦δtu◦(δΛ)−1=\nδtΛuin the frame Λu(cf. Eq. ( 2.1) and Definition A.1). A consequence of the disjointness\nof inertial KMS states presented in Theorem 3.6is that the automorphisms δΛrelating the\nKMS states of different inertial frames are not implemented a s unitary operators in the GNS\nrepresentation (see also Section 3.4). Note also that the proof of Theorem 3.6only makes use\nof the translation covariance condition (C) (by virtue of Th eorem 3.4); Poincaré covariance\n(C’) is not needed.\nAs a side remark, we also mention that in relativistic field th eory the natural concept for\nKMS states is the relativistic KMS condition, first introduc ed in [ BB94 ], as it incorporates\nthe time direction of the reference frame under considerati on. The analyticity requirements\n18in the relativistic KMS condition are stronger than in the re gular KMS condition, but they\nare satisfied by the KMS states of the free scalar field [ BB94 ].\nLet us now check that the conditions put forward in Theorem 3.4, primarity and mixing,\nare satisfied by the inertial KMS states of the scalar field. We do this by employing the\ncriteria from Propositions 2.8&2.10, which demonstrates the usage of these general results.\nLemma 3.5 (KMS states of the free scalar field are primary and m ixing)\nLetωbe a KMS state on the quasi-local algebra AFof the free scalar field on four-dimensional\nMinkowski spacetime Mwith respect to the∗-automorphism group αof time translations in\nan inertial reference frame. Then ωis primary, and (AF,α,ω )satisfies the mixing property.\nProof.\nAs in Appendix Bwe choose Minkowski coordinates ( t,x) onM∼=R4such that the time\ndirection of the reference frame is given by (1 ,0) with dynamics αF(Eq. ( B.1)), and we use\nthe setup presented in that appendix, with ω=ωF,βdefined by Eq. ( B.8) (for anyβ >0), and\nm≥0. The Schrödinger operator −∆+m2onL2(R3,d3x) has the spectrum [ m2,∞) which is\nabsolutely continuous (see, e.g., [ Tes14 ]), hence the self-adjoint operator h0= (−∆ +m2)1/2\nhas purely absolutely continuous spectrum [ m,∞) by spectral calculus. The one-particle\nHamiltonian h=h0⊕ −h0onK=L2(R3,d3x)⊕L2(R3,d3x) (Eq. ( B.6)) also has purely\nabsolutely continuous spectrum. In the Fock space GNS repre sentation of ω, the Liouvillean\nis given by L= dΓ( h) (Eq. ( B.10)), which, by [ Ara18 , Thm. 5.3], has the spectrum Rthat is\nabsolutely continuous except for a simple (embedded) eigen value at 0 corresponding to the\nFock vacuum vector. Primarity and mixing hence follow from P ropositions 2.8&2.10.\n/square\nA complementary elaboration of the mixing property in terms of one-particle structures is\ngiven in Appendix B.3.\nTheorem 3.6 (Disjointness of inertial KMS states of the free scalar field)\nLetσ,ωbe two KMS states on the quasi-local algebra AFof the free scalar field on Mwith\nrespect to the commuting inertial time translation∗-automorphism groups α(u),α(w)associated\nto two inertial frames with linear independent time directi ons specified by u,w, respectively.\nThenσandωare disjoint.\nProof.\nLemma 3.5shows that the KMS states σ,ωare primary and ( AF,α(u),σ) and ( AF,α(w),ω)\nsatisfy the mixing property. Being KMS states, the GNS vecto rs ofσ,ωare separating for\nthe respective induced von Neumann algebras (see Propositi on2.6). Furthermore, σ/\\e}atio\\slash=ωdue\nto [Sew08 , Prop. 3.1]. The statement therefore follows from Theorem 3.4.\n/square\nSewell’s result [ Sew08 , Prop. 3.1] shows that an inertial KMS state of a local quantu m field\ntheory with respect to time translations along a time direct ionucannot satisfy the KMS\n19condition with respect to time translations in any other ine rtial frame with time direction\nw, irrespective of the KMS parameters (see also [ Sew09 ]). The above theorem extends this\nresult: A KMS state with respect to uis not only non-KMS with respect to w, but it is not\neven “close to equilibrium” in the sense of being in the foliu m of a KMS state relative to\nw. We remark that the degree of “non-equilibrium” of a KMS stat e viewed from a moving\ninertial frame can be quantified by the maximum amount of work that can be performed in\na cyclic process, which is formalized using the property of “ semipassivity” [ Kuc02 ].\n[Sew08 , Prop. 3.1] relies on the rather mild assumption that the aut omorphism group of\nspacetime translations acts non-trivially in the GNS repre sentation of the state, i.e. if ω\nis a state on the quasi-local algebra Aof the theory and x∈R4is non-zero, then there\nexists some A∈ A andt∈Rsuch thatπω(δtx(A))/\\e}atio\\slash=πω(A) (see [ Sew08 , Def. 3.1]). In the\nproof, which proceeds by contradiction, this assumption is specifically used to exclude the\ntrivial action of spacetime translations in a spacelike dir ection obtained from (up to positive\nprefactors) the difference of the time directions of the two r eference frames (cf. [ Sew08 , Eq.\n(3.15)]). Excluding trivial cases, non-triviality of tran slations in spacelike directions always\nholds for primary states. Indeed, let ωbe a primary state on Aanda∈R4a spacelike\nvector, and suppose that πω(δta(A)) =πω(A) for allA∈ A and allt∈R. For fixed A∈ A,\nprimarity of ωand Lemma 3.3imply that the net ( πω(δta(A)))t∈Rhas a cluster point in weak\noperator topology lying in C1(using the same argument as in the proof of Theorem 3.4).\nBy assumption the corresponding subnet converging to that c luster point is a constant net\nwith value πω(A), henceπω(A)∈C1. As this holds for all A∈ A one concludes that the\ntranslations in a spacelike direction in the GNS representa tion of a primary state ωcan only\nbe trivial if πω(A)′′=C1.\n3.4. Lorentz boosts are not unitarily implemented\nTemperature is a quantity to be properly assigned to a system in its rest frame, a conclusion\nthat has been drawn from different perspectives in, e.g., [ CM95b ,LM96 ,LM04 ,Sew08 ,Sew09 ,\nSew10 ] (see also earlier literature cited therein). The distingu ished nature of the rest frame\nfor thermal states is also emphasized in [ BS13 ,BV15 ,BV16 ] in the analysis of temperature\nmeasurements in accelerated systems. By contrast, the Mink owski vacuum state is invariant\nunder the Poincaré group (see Eq. ( 4.1)) and therefore serves as a ground state in every\ninertial reference frame.\nThese observations express the spontaneous breakdown of Lo rentz symmetry in KMS states\n(cf. [Haa96 , Sec. V.1.5], and [ Oji04 ,Oji05 ] for a general, abstract discussion of broken sym-\nmetries in quantum systems): Inertial KMS states are not inv ariant under Lorentz boosts,\nbecause boosts do not commute with inertial time translatio ns; therefore, boosts cannot be\nimplemented by unitary operators in the GNS representation such that the GNS vector is\ninvariant [ Oji86 ]. The disjointness of primary states relative to different i nertial reference\n20frames allows to show that a unitary implementation of Loren tz boosts in these states is\ngenerally prohibited, regardless of the failure of invaria nce, i.e. even when permitting unitary\nimplementations that do not necessarily leave the GNS vecto r invariant. A similar result for\nautomorphisms commuting with the dynamics of a KMS state is p resented in [ Nar77 ].\nDefinition 3.7 (Unitary implementation)\nLetωbe a state on a C∗-algebra Awith GNS representation ( πω,Hω,Ωω). A∗-automorphism\nκonAis said to be unitarily implemented in the GNS representation of ωif there exists\na unitary operator UonHωsuch thatπω(κ(A)) =Uπω(A)U−1for allA∈ A. A group\nthat is represented on Aby a group of∗-automorphisms is unitarily implemented in ωif\nevery representing∗-automorphism is unitarily implemented in the GNS represen tation ofω\n(resulting in a one-parameter group of unitary operators).\nWe again consider a quantum field theory {A(O)}O⊂MonMsatisfying (A)–(C) stated in\nSection 2. In addition, we assume the stronger covariance condition ( C’) under the proper\northochronous Poincaré group. The proper orthochronous Lo rentz group L↑\n+is represented\non the quasi-local algebra Aby∗-automorphisms {δΛ}Λ∈L↑\n+. We say that a state ωis Lorentz\nboost invariant if ω◦δΛ=ωfor all boosts Λ∈L↑\n+, in which case every δΛis implemented by\na unique unitary operator UΛon the GNS Hilbert space that leaves the GNS vector invariant ,\ni.e.UΛΩω= Ωω(see [ BR87 , Cor. 2.3.17]). As mentioned above, [ Oji86 ] showed that inertial\nKMS states are never Lorentz boost invariant. The following result is a corollary of Theorem\n3.4.\nCorollary 3.8 (Lorentz boosts are not unitarily implemente d)\nL↑\n+is not unitarily implemented in states on Athat are primary, mixing (with respect to\ntime translations in some inertial frame), have a separatin g GNS vector, and are not Lorentz\nboost invariant. In particular, this applies to the quasi-f ree inertial KMS states of the free\nscalar field on M.\nProof.\nLetωbe a state on Awith GNS representation ( πω,Hω,Ωω) and separating Ω ω. Assume that\nωis primary, invariant under the one-parameter group α(w)={α(w)\nt}t∈Rof∗-automorphisms\ngiven byα(w)\nt=δtwfor some inertial frame with time direction (future-direct ed timelike unit\nvector) w, and ( A,α(w),ω) is mixing. Let Λ∈L↑\n+be a Lorentz boost between the (linear\nindependent) time directions u=Λ−1wandwsuch thatω◦δΛ/\\e}atio\\slash=ω. Assume, on the contrary,\nthatL↑\n+is unitarily implemented in ω. Then there exists a unitary operator UΛonHωsuch\nthatπω(δΛ(A)) =UΛπω(A)(UΛ)−1for allA∈ A. The state σ:=ω◦δΛonAis invariant\nunderα(u)={α(u)\nt}t∈Rforα(u)\nt=δtuand has the GNS representation ( πω,Hω,(UΛ)−1Ωω),\nthusσis also primary with separating GNS vector. The mixing prope rty of ( A,α(w),ω)\nimplies that ( A,α(u),σ) is mixing, for if A,B ∈ Awe have\nlim\nt→∞σ(Bα(u)\nt(A)) = lim\nt→∞ω/parenleftBig\nδΛ(B)α(w)\nt(δΛ(A))/parenrightBig\n=ω(δΛ(A))ω(δΛ(B))≡σ(A)σ(B).\n21Under these conditions Theorem 3.4tells us that σandωhave to be disjoint. But σis\nunitarily equivalent (in particular, quasi-equivalent) t oωby construction, which proves the\nclaim. The last sentence of the corollary follows from Theor em3.6.\n/square\n4. Disjointness: Discussion and review of literature\nIn this section some consequences of our results are present ed and related literature is re-\nviewed.\n4.1. Comments on disjointness\n• Theorem 3.4establishes a superselection rule in that inertial referen ce frames partition the\ntheory into different sectors given by quasi-equivalence cl asses (folia) of primary, mixing\nstates (cf. [ Oji04 ,Oji05 ]).\n• The disjointness of inertial KMS states represents a globa l property as it concerns states\non the quasi-local algebra of the theory. Restricted to any l ocal observable algebra of\nthe theory, KMS states are quasi-equivalent thanks to their Hadamard property [ Ver94 ,\nRad96 ,SV00 ]. In view of Theorem 3.4this also applies to primary, mixing Hadamard\nstates.\n• The disjointness result for KMS states is independent of th e KMS parameters. Even if\nthe two KMS states in Theorem 3.6have the same KMS parameter β < ∞(inverse\ntemperature relative to the respective frame), they are dis joint. This is no longer the case\nwhen taking the limit β→ ∞ in the weak-∗sense, since the KMS states in any frame\napproach the Poincaré-invariant Minkowski vacuum state ωM:\nωM◦δ(x,Λ)=ωMfor all (x,Λ)∈P↑\n+ (4.1)\nThe Minkowski vacuum is therefore a ground state for every in ertial observer [ Haa96 ] (it\nis the unique quasi-free pure state that is P↑\n+-invariant, see [ KM15 , Prop. 5.2.33]).\n• Complementary to the disjointness with respect to differen t inertial reference frames (irre-\nspective of the KMS parameter), two KMS states with respect t o the same dynamics but\nwith different KMS parameters are disjoint as well, provided the von Neumann algebra\ninduced by one of the states is of type III [ Tak70a ] (see also [ BR97 , Thm. 5.3.35]). The\ntype III property (see [ Haa96 , Sec. V.2.4] and references cited there) is typically fulfil led\nby the local and global von Neumann algebras induced by quasi -free Hadamard states\nof a local quantum field [ Yng05 ]. For the inertial KMS states (of finite temperature) of\nthe free scalar field on Minkowski spacetime the type III prop erty of the induced von\n22Neumann algebra can be obtained from the characterization p roven in [ Hug67 ], using that\nthe von Neumann algebra is a factor (Lemma 3.5). (From the spectral properties of the\nLiouvillean one can even prove that the type is III 1using the classification of factors due\nto Connes [ Con73 ], cf. [ Kay85a , Thm. 1.5].) Some other results on the disjointness of\nstates on a single dynamical system are given in [ MH80 ] (disjointness of primary β-KMS\nstates with different chemical potentials for every β), [BR97 , Thm. 5.3.30] (disjointness of\ndistinct primary β-KMS states with respect to strongly continuous dynamics fo r everyβ),\nand [ BR87 , Thm. 4.3.19] (disjointness of distinct “centrally ergodi c” states with respect\nto a representation of a group by∗-automorphisms).\n• Corollary 3.8can be compared to a similar situation in supersymmetric the ories. Given\na (graded) C∗-dynamical system with an asymptotically abelian time evol ution that com-\nmutes with a supersymmetry transformation, it has been show n in [ BO97 ,Buc00 ] that\nany time-invariant state for which supersymmetry can be imp lemented by operators on\nthe GNS Hilbert space (called a “super-regular” state) has t o be an excitation of a ground\nstate. Consequently, supersymmetry is not implemented in t he GNS representation of\nKMS states and their mixtures (the “spontaneous collapse of supersymmetry” in thermal\nstates [ BO97 ]). Similarly, the Lorentz group is unitarily implemented i n the Minkowski\nvacuum (and excitations or perturbations thereof, i.e. sta tes in the folium), since it is P↑\n+-\ninvariant (Eq. ( 4.1)). KMS states, on the other hand, single out a particular ref erence\nframe, which leads to the breakdown of Lorentz symmetry as we ll as supersymmetry.\n4.2. Comparison with Herman & Takesaki (1970)\nA statement that is closely related to our result has been pre viously considered by Herman\n& Takesaki [ HT70 ].\nProposition 4.1 ([ HT70 , Sec. §1, Thm. 4])\nIfσ,ωare two quasi-equivalent states on a C∗-algebra Asatisfying the (β=−1)-KMS con-\ndition with respect to commuting one-parameter∗-automorphism groups, such that\nFω:={X∈ Mω:τω\nt(X) =Xfor allt∈R}=C1,\nwhereτωis the modular automorphism group associated to ωby Tomita-Takesaki theory, then\nσ=ω.\nFrom this result one can obtain the disjointness of primary, mixing KMS states with respect\nto time translations in different inertial frames, or even an y commuting one-parameter∗-\nautomorphism groups of dynamics. To retain the specific phys ical setup of our theorems,\nwe again consider commuting inertial time translation∗-automorphism groups α(u),α(w)as-\nsociated to distinct reference frames u,w, respectively. Let σ,ωbe quasi-equivalent, primary,\nmixing KMS states with respect to α(u),α(w), respectively. If σhas the KMS parameter β,\n23then it is a ( −1)-KMS state with respect to the rescaled dynamics /tildewideα(u):={α(u)\n−βt}t∈R. Simi-\nlarly,ωcan be viewed as ( −1)-KMS state with respect to some rescaled dynamics /tildewideα(w)of time\ntranslations in the frame w. The assumed quasi-equivalence allows to represent both σand\nωand the dynamics /tildewideα(u),/tildewideα(w)on the induced von Neumann algebra Mωofω(cf. the proof of\nTheorem 3.4). The KMS property implies that the GNS vector Ω ωis cyclic and separating,\nhence there are Tomita-Takesaki modular objects associate d to the pair ( Mω,Ωω) such that\nthe corresponding modular automorphism group τωequals the W∗-dynamics induced by /tildewideα(w)\n(see Propositions A.2&A.3). The mixing property for ( Mω,τω,/a\\}bracketle{tΩω,(·)Ωω/a\\}bracketri}ht) and Lemma\n3.1show thatFω⊆C1for the fixed point subalgebra Fωdefined above, thus Fω=C1and\n[HT70 , Sec. §1, Thm. 4] implies σ=ω. Therefore, if σandωare supposed to be distinct\nstates, primarity implies that they have to be disjoint. For the free scalar field this gives an\nalternative proof of Theorem 3.6.\nHowever, our theorem 3.6is formulated as an application of Theorem 3.4, which shows the\ndisjointness of any two primary states of an isotonous, loca l, translation-covariant algebraic\nquantum field theory with respect to inertial time translati ons, provided the mixing property\nis fulfilled. We point out sufficient conditions for these prop erties, which can be verified\nin concrete examples. Explicit properties of the modular au tomorphism group, or some\nrelation between modular automorphisms and the dynamics, a re not required. This sets our\nproof apart from the above argumentation based on [ HT70 ], which applies to any commuting\ndynamics, but necessarily relies on the KMS property of the i nvolved states. Let us explain\nthis point in more detail by summarizing the proof of [ HT70 , Sec. §1, Thm. 4]:\n• The dynamics of the KMS states σ,ωextend to the respective modular automorphism\ngroups on the induced von Neumann algebras. Since the states are assumed to be quasi-\nequivalent, σcan be implemented on the induced von Neumann algebra Mωofω, and\nthe modular automorphism group of σcan be transported to Mω. By assumption, the\nresulting modular automorphism groups on Mωcommute.\n• The commutativity implies the invariance of each state und er the modular automorphism\ngroup of the other (see [ HT70 , Sec. §1, Thm. 2]). Hence there is a self-adjoint operator\nH > 0 affiliated with Fωso thatσis represented on Mωby the vector HΩω(see [ Tak70b ,\nThm. 15.2]), which implies σ=ωasFω=C1.\nThe proof of [ HT70 , Sec. §1, Thm. 4] rests on the fact that for the two quasi-equi valent\nKMS states the dynamics are implemented on the von Neumann al gebra level by commuting\nmodular automorphism groups. This cannot be guaranteed if w e instead start with quasi-\nequivalent, primary, mixing states σ,ωwith separating GNS vectors, not necessarily satisfying\nthe KMS condition, like in the proof of Theorem 3.4. These states induce faithful normal\nstates on Mωwhich are KMS states with respect to the corresponding modul ar automorphism\ngroups (cf. Proposition A.2). But the induced W∗-dynamics for σ,ωonMωdo not have to\nbe related to their modular automorphism groups (as would be the case for KMS states by\n24Proposition A.3), and while the induced dynamics of each of the states on Mωcommutes\nwith its respective modular automorphism group (see [ HT70 , Sec. §1, Thm. 1] or the general\nresult [ SW70 , Thm. 2.4]), and the induced dynamics commute by assumption , there is no\nreason for the two modular automorphism groups themselves t o commute.\nTheorem 3.4thereby provides a more general result on the disjointness w ith respect to\ndifferent inertial reference frames, and we demonstrate the application to KMS states in the\ncase of the scalar field. Our proof uses Lemma 3.2, which can be interpreted as an analog\nto the statement of Herman & Takesaki. The “tradeoff” for not r elying on the KMS condi-\ntion is the assumed weak cluster point property for the combi ned dynamics ( γψ\n−tγϕ\nt(X))t∈R,\nwhich is (γ(u,w)\nt(X))t∈Rin the proof of Theorem 3.4. However, as shown in the latter proof,\nthis property is always satisfied for inertial time translat ions in a local translation-covariant\nquantum field theory due to spacelike asymptotic abeliannes s of the dynamics (Lemma 3.3).\nIt might be worth investigating if the assumptions in Lemma 3.2can be relaxed or modified\n(e.g. by combining with the results in [ HT70 ]) to cover dynamics other than inertial time\ntranslations.\n5. (Non-)Thermalization of Unruh-DeWitt detectors\nIn this section we want to relate the disjointness result to a symptotic equilibrium properties\nof Unruh-DeWitt detectors [ Unr76 ,DeW79 ], i.e. point-like non-relativistic two-level quantum\nsystems, that interact with a quantum field while being in ine rtial or uniformly accelerating\nmotion. Inspired by the approach in [ DM06 ] we cast the model of a detector coupled to\nand moving inertially relative to a thermal bath in the frame work of open quantum systems\nand argue that the disjointness result prohibits thermaliz ation for a whole folium of initial\ndetector-field states.\n5.1. Thermalization property and return to equilibrium\nConsider a quantum dynamical system ( A,α), given by a C∗-algebra Aand a one-parameter\ngroupα={αt}t∈Rof∗-automorphisms on it. Starting from a state representing th e initial\n(timet= 0) configuration of ( A,α), does the system eventually reach thermal equilibrium if\none waits long enough? The following definition formalizes t his property (see, e.g., [ JP96b ,\nBFS00 ], and further references in the subsequent discussion).\nDefinition 5.1 (Thermalization and return to equilibrium)\nLetωbe a KMS state on a quantum dynamical system ( A,α), andω′a state on Areferred\nto as the initial state . (If ( A,α) is a W∗-dynamical system, assume that ω,ω′are normal.)\nThen ( A,α)thermalizes toω, or ( A,α,ω ;ω′) satisfies the thermalization property , if\nlim\nt→∞ω′(αt(A)) =ω(A) for allA∈ A. (5.1)\n25The quantum dynamical system ( A,α)returns to the equilibrium state ω, or (A,α,ω ) satisfies\ntheRTE (return to equilibrium) property , if (A,α,ω ;ω′) satisfies the thermalization property\nfor all initial states ω′∈Fol(ω). If the convergence in Eq. ( 5.1) is replaced by ergodic mean\nconvergence (a continuous generalization of convergence i n the Cesàro mean),\nlim\nt→∞1\ntt/integraldisplay\n0ω′(αt′(A)) dt′=ω(A) for allA∈ A,\nthen one refers to thermalization and RTE in the ergodic mean .\nThermalization and RTE are properties that are stronger tha n their ergodic mean coun-\nterparts. The subsequent discussion mainly focuses on thes e strong notions, although the\ninterpretation and references also apply to the weaker noti ons in the ergodic mean.\nThe thermalization property for ( A,α,ω ;ω′) means that if the initial state of the dynamical\nsystem ( A,α) is prescribed by ω′, the expectation values of observables under the dynamics\nαapproach the expectation values in the thermal equilibrium stateωat asymptotically late\ntimes. The considered systems encompass observable algebr as that can be proper C∗-algebras\nas well as, more specifically, von Neumann algebras. We put no restrictions on the allowed\nfamily of initial states; in principle, ω′could be non-invariant or even “far from equilibrium”\n(see the following sections). Whether thermalization to so me KMS state is even conceivable in\nsuch situations depends on the given quantum dynamical syst em. Thermalization problems\nhave been considered for certain closed, finite-dimensiona l quantum systems, for which we\nrefer to the reviews [ EFG15 ,GE16 ] with their comprehensive lists of references.\nIn this work we are interested in a particular class of open qu antum systems [ Dav76 ,\nAJP06 ] that arise from coupling an Unruh-DeWitt detector (a “smal l”, finite-dimensional\nquantum system) to a relativistic quantum field (an infinitel y extended “reservoir”) at positive\ntemperature. This falls into the category of so-called Paul i-Fierz models (see, e.g., [ DG99 ,\nDJ01 ,DJ03 ,Møl14 ] and references therein, and [ PF38 ] for the historical origin). The coupling\nis introduced by a suitable perturbation of the dynamics of t he uncoupled (combined, tensor\nproduct) system. One would expect that, at least for sufficien tly weak and analytically\n“regular” coupling, the coupled system thermalizes to an eq uilibrium state at the “reservoir”\ntemperature under dissipative dynamics, as long as the init ial state is sufficiently close to\nthermal equilibrium. This asymptotic dynamical stability is, in fact, a paradigmatic property\nof open quantum systems (cf. the “Quasitheorem” in the intro ductory section of [ DJ03 ]),\nput on mathematical grounds by the RTE property: All states t hat are perturbations of (i.e.\nnormal to) a KMS state (cf. below Definition 2.4) will return to that equilibrium at late times.\nWe do not attempt to review the extensive literature on the to pic of RTE. A non-exhaustive\nlist of works on RTE and proofs of the validity in various mode ls and under different conditions\nis [BR97 ,JP96a ,JP96b ,BFS00 ,DJ03 ,Mer01 ,FM04a ,Mer05 ,DM06 ,KMS14 ,MS15 ], with\nearly works [ LR72 ,Rob73 ] and reviews [ JP97 ,Der07 ].\n26We mention that, in contrast to Definition 5.1, the term “thermalization” has been used in\n[MSB08 ] to describe the asymptotically thermal behavior of the sma ll system alone, with the\nreservoir being traced out. Recently the RTE property has al so been studied in the context\nof perturbative (interacting) quantum field theory [ DFP18 ,GPS23 ].\nStarting from Definition 5.1, consider a quantum dynamical system ( A,α) and a KMS\nstateω(assumed to be normal if ( A,α) is a W∗-dynamical system). Let ( Mω,γ) be the\ninduced W∗-dynamical system obtained from the GNS representation ( πω,Hω,Ωω) ofω, as\nsketched in Proposition 2.6. There is a close relation between the RTE property for ( A,α,ω )\nand spectral properties of the Liouvillean L(Proposition 2.6) generating the dynamics by\nγt(X) = eiLtXe−iLtforX∈ Mω,t∈R. As shown in [ JP96b , Thm. 4.2], RTE is equivalent\nto mixing (Definition 2.9), and Proposition 2.10gives a criterion for RTE. As a reminder we\nrecall it here, including the characterization of RTE in the ergodic mean. A short proof for\nthe RTE property is discussed in [ DM06 , Lemma 3.6]. Applications of this characterization\ncan be found in the mentioned references.\nProposition 5.2 (Quantum Koopmanism for RTE [ JP96b ,JP97 ,BFS00 ,Pil06 ])\n(A,α,ω )satisfies the RTE property in the ergodic mean if and only if kerL=CΩω, i.e. the\nLiouvillean Lhas a simple eigenvalue at 0corresponding to the GNS vector Ωωofω. If the\nspectrum of Lis absolutely continuous apart from a simple eigenvalue at 0, then (A,α,ω )\nsatisfies the RTE property.\nIn particular, this yields a spectral characterization for the (ergodic mean) RTE property of a\nW∗-dynamical system ( M,γ) with a normal KMS state /a\\}bracketle{tΩ,(·)Ω/a\\}bracketri}htwith cyclic and separating\nvector Ω.\nThe definition of (ergodic mean) RTE, the equivalence betwee n RTE and the mixing prop-\nerty, and the spectral characterization in Proposition 5.2naturally extend to any invariant\nstate that induces a W∗-dynamical system with a Liouvillean and a separating GNS ve ctor\n[JP96b ] (see the assumptions of Proposition 2.10). Since this section concerns KMS states,\nRTE will be viewed as “return to thermal equilibrium”, thereby justifying our restriction to\nKMS states in Definition 5.1.\n5.2. General setup\nIn the following sections we want to look at the asymptotic st ate behavior (thermalization\nand RTE) of an Unruh-DeWitt detector that interacts with a qu antum field in different\nstates. For this we need to define the corresponding quantum d ynamical system, which\nshould represent the combined system consisting of a finite- dimensional quantum system and\na quantum field, with dynamics describing the time evolution of the combined system under\nsome interaction coupling. For the sake of simplicity we wil l consider a two-level quantum\nsystem coupled to the free massless scalar field. The general setup, which is motivated by\n[DM06 ], is as follows:\n27Unruh-DeWitt detector system (AD,αD) This will be the “probe” in our setup. The C∗-\nalgebra is AD=B(C2), andαD={αD,t}t∈Ris the one-parameter group of∗-automorphisms\nαD,t= eiHDt(·)e−iHDt, HD:= diag(E,0)\nfor the Hamiltonian HDonC2withE > 0. This describes a two-level system (qubit) with\nground state at zero energy and excited state at the detector excitation energy E > 0. The\nparametert∈Rstands for the proper time of the detector, whose physical in terpretation\n— the stationary, timelike trajectory that the detector tra vels along — is determined by\nthe time evolution of the quantum field with which the detecto r system is combined. In\nour case,tcoincides with the time parameter in some inertial referenc e frame (see the next\nparagraph). Hence the data of this model exclusively encomp asses the detector’s internal\ndegrees of freedom (cf. [ Tak86 , Sec. §3.1], and the comment below [ DM06 , Eq. (13)]).\nQuantum field system (AF,αF) This will be the “heat bath” or “reservoir” in our setup.\nThe C∗-algebra AFis the quasi-local algebra of the free massless scalar quant um field on four-\ndimensional Minkowski spacetime M=R1,3, obtained as the Weyl algebra over the classical\nphase space of Klein-Gordon theory, as discussed in Appendi xB. For a uniform notation we\nattach the index “ F” to objects related to the field. The dynamics αF={αF,t}t∈Rdescribing\nthe field’s inertial time evolution relative to some inertia l reference frame can be defined by\n∗-automorphisms αF,tonAFas in Eq. ( B.1), where the coordinates are chosen such that the\ntime direction of the reference frame is (1 ,0).\nCoupled detector-field system The C∗-algebra describing the combination of detector and\nquantum field is the tensor product AD⊗ AF. The dynamics of the uncoupled system, with\ndetector and quantum field evolving independently without i nteraction between them, is\ngiven byαD⊗αF={αD,t⊗αF,t}t∈R. The dynamics of the coupled system on AD⊗ AFis\nintroduced, in principle, by “perturbing” the uncoupled dy namicsαD⊗αF, parametrized by a\ncoupling parameter λ∈Rsuch thatαD⊗αFis recovered when λ→0. The resulting coupled\ndynamics describes the time evolution of the detector-field system under the influence of an\ninteraction between detector and quantum field. As we will di scuss in the next section, the\nconstruction of the coupled dynamics can be performed rigor ously in the GNS representation\nof a quasi-free reference state for a monopole interaction, i.e. a linear coupling of the detector’s\nmonopole moment to a quantum field operator smeared with some coupling test function.\n5.3. RTE of a detector at rest in a heat bath\nSuppose the Unruh-DeWitt detector interacts with the quant um field and follows the world-\nlineR∋t/ma√sto→tu, the time direction of an inertial reference frame specified by a future-directed\ntimelike unit vector u. Let the quantum field be in the quasi-free β-KMS state ωF,βonAF\n28with respect to the time translation relative to u, so the field is in thermal equilibrium relative\nto the rest frame of the detector. The definition of ωF,βis given in the Appendices B&C,\nand as done there and also earlier we choose Minkowski coordi nates such that u= (1,0).\nOne expects that, for some sufficiently “benign” coupling bet ween the detector and the field,\nthe coupled system asymptotically approaches a β-KMS state in the sense of Definition 5.1.\nFor the mathematical analysis it is beneficial to consider th e W∗-dynamical systems induced\nby the GNS representation of certain reference states. The f ormal constructions are moti-\nvated by [ DM06 ] and follow the typical setup of Pauli-Fierz models (see als o [Mer07 ] for an\noverview, and further references in the discussion below Pr oposition 5.3).\nForβ >0 and the detector Hamiltonian HD= diag(E,0),E > 0, let\nωD,β:=tr(e−βHD(·))\ntr e−βHD\nbe the corresponding Gibbs equilibrium state at temperatur eβ−1, which is the unique ( αD,β)-\nKMS state on AD=B(C2). The GNS representation of ωD,βcan be given as ( πD,HD,ΩD,β),\nwhere\nHD=C2⊗C2, πD= (·)⊗12,ΩD,β=1√\n1 + e−βE(e−βE/2v+⊗v++v−⊗v−) (5.2)\nfor the identity matrix 12:= diag(1,1) onC2and the standard basis v+= (1,0)T,v−= (0,1)T\nofC2(the eigenbasis of HD).\nFor the GNS representation of ωF,β,β >0, the unique ( αF,β)-KMS state on AFdefined\nin Appendix B, we use the Jakšić-Pillet glued representation discussed i n Appendix C: The\nGNS representation ( πF,β,F(HF),ΩF) is realized as Weyl operators on the symmetric Fock\nspace over the one-particle Hilbert space HF:=L2(R×S2,dsdΩ), with Fock vacuum Ω F\n(Lemma C.1). The von Neumann algebra induced by the GNS representation of theβ-KMS\nstateωD,β⊗ωF,βon the uncoupled system ( AD⊗ AF,αD⊗αF) hence is given by\nMβ:=πD(AD)′′⊗πF,β(AF)′′⊂ B(H) (5.3)\nfor\nH:=HD⊗ F(HF) =C2⊗C2⊗ F/parenleftBig\nL2(R×S2,dsdΩ)/parenrightBig\n.\nThe uncoupled dynamics αD⊗αFis represented on Mβby a one-parameter group γ0of\n∗-automorphisms\nγ0,t:= eiL0t(·)e−iL0t\n29fort∈R, generated by the sum L0of the Liouvilleans for the detector and field:\nL0:=LD⊗1F(HF)+1HD⊗LF, LD:=HD⊗12−12⊗HD, LF= dΓ(s)\nThe operator dΓ( s) is the second quantization of the self-adjoint operator of multiplication\nwith the function ( s,Ω)/ma√sto→sonHF(see Lemma C.1). The operator L0is self-adjoint on\nHD⊗dom(dΓ(s)) (see again Lemma C.1) and satisfies\nL0Ω0,β= 0,Ω0,β:= ΩD,β⊗ΩF,\nbecauseLDΩD,β= 0 andLFΩF= 0. The W∗-dynamical system ( Mβ,γ0) with Liouvillean L0\ndescribes the uncoupled detector-field system in the GNS rep resentation ( πD⊗πF,β,H,Ω0,β)\nof the (αD⊗αF,β)-KMS state ωD,β⊗ωF,β(cf. Proposition 2.6).\nThe dynamics of the coupled system is implemented by perturb ing the Liouvillean L0of\nthe uncoupled dynamics. Define\nLλ,β:=L0+λVβ, (5.4)\nwhereλ∈Ris a coupling parameter, and Vβis some interaction term on Hencoding\nthe specifics of the coupling between detector and quantum fie ld. The operator Vβis, in\ngeneral, unbounded but affiliated with the von Neumann algebr aMβ. Here we consider a\nmonopole coupling, which has been frequently used to study t he transition rates of Unruh-\nDeWitt detectors in quantum field theory [ BD82 ,UW84 ,Tak86 ,DM06 ,LS08 ,FJL16 ], and\nthe interaction between toy atom systems and radiation field s [LCD+87,JP96a ,JP96b ,JP97 ,\nMer01 ,FM04a ,FM04b ,MSB08 ] (cf. the spin-boson model discussed later in this section) . It\nis a linear interaction of the form\nVβ:=Iβ−JIβJ, Iβ:=G⊗12⊗(a(hβ) +a∗(hβ)) (5.5)\nfor a self-adjoint operator G∈ B(C2) representing the monopole moment operator of the\ndetector, and creation and annihilation operators a∗(·),a(·) on the Fock space F(HF) asso-\nciated to the “form factor” hβ∈ HFdefined in Eq. ( C.2) for a test function h∈C∞\nc(M;R).\nThe operator Gis chosen to be a multiple of the first Pauli matrix. We remark t hat non-\nvanishing off-diagonal components of Gdetermine dissipative interaction in the spin-boson\nmodel [ LCD+87,JP97 ], while diagonal Gyield an interaction for which thermalization does\nnot occur due to the suppression of energy exchange (see [ MSB08 ] for details and references).\nThe operator\nJ=JD⊗JF\nis the modular conjugation associated to ( Mβ,Ω0,β), whereJDis the modular conjugation\n30associated to ( πD(AD)′′,ΩD,β) given byJD(v1⊗v2) =v2⊗v1onHD(and continuously and\nanti-linearly extended, with the bar denoting the complex c onjugation of components in the\nbasis of C2), andJFis the modular conjugation on F(HF) associated to ( πF,β(AF)′′,ΩF)\ndefined in Eq. ( C.3). Therefore, by Eq. ( C.4),\nVβ=G⊗12⊗(a(hβ) +a∗(hβ))−12⊗G⊗(a(e−βs/2hβ) +a∗(e−βs/2hβ))\nforG∈ B(C2) obtained by complex conjugating the components of Gin the standard basis.\nUsing the Glimm-Jaffe-Nelson commutator theorem [ RS75 , Thm. X.37] one proves that\nL0+λIβandL0+λVβare essentially self-adjoint on dom( L0)∩dom(Iβ) and dom( L0)∩\ndom(Iβ)∩dom(JIβJ), respectively (along the lines of, e.g., [ JP96a , Prop. 2.1, Lemma 3.2]).\nGeneral results on perturbations of Liouvilleans, which ar e presented in [ DJP03 , Thm. 3.3 &\n3.5], show that Lλ,β(which henceforth stands for the self-adjoint closure of th e expression in\nEq. ( 5.4)) generates a W∗-dynamicsγλ={γλ,t}t∈RonMβvia\nγλ,t:= eiLλ,βt(·)e−iLλ,βt. (5.6)\nThe pair ( Mβ,γλ) is the W∗-dynamical system describing the coupled detector-field sy stem,\nand one recovers the uncoupled system with Liouvillean L0in the limit λ→0.\nOne can show that Ω 0,β∈dom(e−β(L0+λIβ)/2) for allλ∈Rby a Dyson series expansion\nof the exponential and standard Fock space estimates (adapt ing [DM06 , Prop. 3.4], see also\n[BFS00 , Sec. III. E], [ Mer07 , Eq. (95) ff.]). Araki’s perturbation theory of KMS states\n[Ara73 ,BR97 ,FNV88 ,BFS00 ], which is developed for unbounded perturbations in [ DJP03 ],\nthen allows to construct, for any λ∈Randβ >0, a unique ( γλ,β)-KMS state on Mβgiven\nby\nϕλ,β:=/a\\}bracketle{tΩλ,β,(·) Ωλ,β/a\\}bracketri}ht,Ωλ,β:=e−β(L0+λIβ)/2Ω0,β\n/bardble−β(L0+λIβ)/2Ω0,β/bardbl∈ H. (5.7)\nThe vector Ω λ,βis cyclic and separating for Mβand approaches (in the norm) the GNS\nvector Ω 0,βof theβ-KMS state ωD,β⊗ωF,βof the uncoupled system as λ→0 [DJP03 , Thm.\n5.5]. In this sense, the KMS state for the coupled system is a p erturbation of the uncoupled\none, and these states are “close” to each other as both are rep resented by normal states on\nMβ. We note that L0+λIβwould also generate γλ(Eq. ( 5.6)). However, adding the term\n−JIβJ(see Eq. ( 5.5)), which is affiliated with the commutant Mβ′, results in the generator\nLλ,βwith the property (see [ DJP03 , Sec. 5.6] or [ Mer20 , Eq. (3.30) ff.])\nLλ,βΩλ,β= 0,\nsoLλ,βis the (so-called “standard” [ DJP03 ]) Liouvillean of ( Mβ,γλ) viewed as the W∗-\ndynamical system induced by ϕλ,β.\n31If the coupling parameter is small enough and the interactio n satisfies sufficient regularity\nconditions, the coupled system will exhibit RTE.\nProposition 5.3 (RTE for detector at rest in a heat bath)\nFor everyβ >0, there exists λ0(β)>0such that (Mβ,γλ,ϕλ,β)satisfies the RTE property\nfor0<|λ|<λ 0(β).\nThe proof makes use of “quantum Koopmanism” (Proposition 5.2). For sufficiently small\n|λ|>0 one shows that ker Lλ,β=CΩλ,βand that the rest of the spectrum of Lλ,βon{Ωλ,β}⊥\nis absolutely continuous. Since the spectrum of LD=HD⊗12−12⊗HD(forHD= diag(E,0))\nconsists of the eigenvalues 0 ,±E, where 0 is doubly degenerate, and LF= dΓ(s) has a simple\nzero eigenvalue embedded in an otherwise absolutely contin uous spectrum on R, it follows\nthat the Liouvillean L0of the uncoupled system has a degenerate kernel. One therefo re\nproves that the zero eigenvalue loses its degeneracy under t he interaction. (The instability\nof the eigenvalues ±Efollows from [ JP01 ]: If the Liouvillean of a KMS state has a simple\nzero eigenvalue, then there do not exist any non-zero eigenv alues.)\nThe representation introduced in this section and in the app endices enables the identi-\nfication of our setup with the representation of models that a ppear in the literature. In\nfact, our setup is analogous to [ Mer20 , Sec.s 3.2 & 3.3] or [ Mer22 , Sec.s 2.1 & 2.2], where\nthe “purification” representation of an N-level system coupled to a thermal gas of massless\nbosons is summarized. In the spin-boson model, which corres ponds toN= 2, a spin-1\n2\nor qubit system (which could be seen as an idealization of an a tom or molecule confined\nin a way that there are effectively only two discrete energy le vels) interacts with a reser-\nvoir of massless bosons described by a scalar field (which cou ld be a proxy for a phonon\nor photon field, disregarding polarization degrees of freed om [FM04a ,FM04b ]); see, e.g.,\n[LCD+87,FNV88 ,HS95a ,HS95b ,JP96b ,JP97 ,JP96a ,BFS00 ,FM04a ,Mer22 ] (and the ref-\nerences therein), where the last four works consider more ge neral “small” systems with finitely\nmany (N < ∞) energy levels. The massless scalar field in our setup is give n in a Jakšić-Pillet\nglued representation of Araki-Woods type (cf. Appendix C), similar to the thermal reservoir\nin spin-boson models, for which the required spectral prope rties of the coupled Liouvillean\nhave been derived in [ JP96b ,BFS00 ,DJ03 ,Mer01 ,FM04a ] using Mourre theory or spec-\ntral deformation techniques. We refer to these works for the full, technically involved proof\nof the spectral properties leading to Proposition 5.3, including a derivation of the required\n“Fermi Golden Rule” [ JP96a ,BFS00 ,DJ03 ,DF06 ], which controls the (partial) instability\nof eigenvalues respectively resonances under the interact ion at order λ2, and conditions on\nthe “form factor” hβ∈ HF(Eq. ( C.2)) appearing in the interaction Vβ(Eq. ( 5.5)). A similar\nderivation for the Unruh effect using spectral deformation t heory can be found in [ DM06 ],\nwhich is summarized in [ Mer07 , Sec. 2.4].\nWe also want to mention the recent article [ TG24 ], which considers the relation between\nthe spin-boson model and a gapless Unruh-DeWitt detector co upled to a scalar field and dis-\n32cusses the construction of interacting ground states for th is model under sufficient regularity\nconditions on the coupling function.\nProposition 5.3means that the coupled system of an Unruh-DeWitt detector an d a heat\nbath relative to its rest frame thermalizes, as long as the in itial state is sufficiently close\nto equilibrium. In particular, the reduced state of the dete ctor asymptotically approaches\nϕλ,β((·)⊗1), which at lowest order in the coupling parameter λis the Gibbs equilibrium\nstate (with respect to the free detector dynamics generated byLD) at temperature β−1, i.e.\nlim\nt→∞ϕ′(γλ,t((A⊗12)⊗1)) =/a\\}bracketle{tΩD,β,(A⊗12)ΩD,β/a\\}bracketri}ht+O(λ)≡tr(e−βHD(A))\ntr e−βHD+O(λ) (5.8)\nfor all normal (density operator) states ϕ′onMβand detector observables A⊗12∈\nπD(AD)′′=B(C2)⊗12, as Ωλ,β= ΩD,β⊗ΩF+O(λ) by the perturbation theory of KMS\nstates [ DJP03 ].\nThe admissible initial states in the RTE property lie in the f olium ofϕλ,β, which consists\nof the normal states on Mβand therefore are extensions of states in the folium of the\nuncoupled KMS state ωD,β⊗ωF,βonAD⊗AF(see Eq. ( 5.3)). The initial state reflecting the\nphysical setting described at the beginning of this section is the product state given by an\narbitrary detector density matrix state2and theβ-KMS state /a\\}bracketle{tΩF,(·)ΩF/a\\}bracketri}htof the quantum\nfield (representing ωF,βonπF,β(AF)′′). We emphasize that the initial state does not have to\nbe of product form, but could also be a normal state on Mβthat is correlated (entangled)\nacross detector and field. This “folium invariance” of the as ymptotic state behavior, which\nis manifest in the RTE property, is a strong ergodic feature o f the system.\n5.4. Detector moving inertially relative to a heat bath\nAs before, assume that the worldline of the Unruh-DeWitt det ector is given by R∋t/ma√sto→tuin\ncoordinates where the time direction is u= (1,0). The detector and quantum field dynamical\nsystems ( AD,αD) and ( AF,αF) are defined as in the previous section, with the dynamics\nαD⊗αF={αD,t⊗αF,t}t∈Rimplementing the time evolution of the combined detector-fi eld\nsystem in the time coordinate tof the detector’s rest frame u.\nLetΛ∈L↑\n+be a Lorentz boost mediating between uand a different inertial frame wwith\nconstant non-zero velocity relative to u, such that w=Λu(cf. the beginning of Section 3.2).\nThe boost Λis represented by a∗-automorphism δΛonAF. Forβ′>0, consider\nω(w)\nF,β′=ωF,β′◦δΛ−1, (5.9)\n2Every density matrix state on ADis represented by a cyclic vector in HDand therefore all detector states\nlie in the same folium. For if ρ=/summationtext\nj∈{1,2}̺j|ψj/a\\}bracketri}ht/a\\}bracketle{tψj|is a density matrix on C2, diagonalized in a basis\n{ψ1,ψ2}of eigenvectors with eigenvalues ̺1,̺2≥0, the GNS representation of tr( ρ(·)) is (πD,HD,Ωρ) with\nΩρ=/summationtext\nj∈{1,2}√̺jψj⊗ψj, where the overline denotes the complex conjugation of vect or components in the\neigenbasis of HD. Forρβ= (tr e−βH D)−1e−βH Dwe obtain Ω ρβ= Ω D,β(Eq. ( 5.2)).\n33which is the unique β′-KMS state on AFwith respect to the dynamics α(w)\nF={α(w)\nF,t}t∈R\nrelative to wgiven byα(w)\nF,t=δΛ◦αF,t◦δΛ−1. For any KMS parameters β,β′>0, the\ntwo KMS states ωF,β,ω(w)\nF,β′are distinct [ Sew08 , Prop. 3.1] (see also the paragraph below\nTheorem 3.6). We continue to denote states and dynamics relative to the d etector’s rest\nframe uwithout any index, while quantities relative to ware decorated accordingly. The von\nNeumann algebra induced by ωD,β⊗ωF,βis again denoted Mβ(see Eq. ( 5.3)).\nTheorem 5.4\nLetωDbe any detector state on AD, and letβ′>0be fixed. Any state in Fol(ωD⊗ω(w)\nF,β′)is\nfar from equilibrium relative to the rest frame uof the detector, that is, it is not normal to\nany(αD⊗αF,β)-KMS state ωD,β⊗ωF,βonAD⊗ AFforβ >0.\nProof.\nFrom Theorem 3.6we can conclude that\nFol(ω(w)\nF,β′)∩/uniondisplay\nβ>0Fol(ωF,β) =∅ (5.10)\n(in particular, δΛ−1in Eq. ( 5.9) cannot be unitarily implemented in the GNS representation\nofωF,β′, cf. Corollary 3.8). Hence, for any β >0, there is no normal state representing any\nstate in Fol( ωD⊗ω(w)\nF,β′) onMβ.\n/square\nTheorem 5.4suggests that the detector will not asymptotically approac h a thermal equi-\nlibrium state (in proper time of the rest frame u) if it is coupled to the massless scalar\nfield prepared in the KMS state ω(w)\nF,β′(viewed as thermal equilibrium state relative to the\nframe wmoving inertially relative to u) or any perturbation of it. More concretely, let\nω′∈Fol(ωD⊗ω(w)\nF,β′) be an initial state of the combined detector-field system (f or any detec-\ntor stateωD), and suppose that the detector couples to the quantum field v ia some interaction.\nThe corresponding dynamics should describe the time evolut ion relative to u, as the coupled\nsystem is supposed to evolve along the worldline of the detec tor and we ask about thermality\nrelative to the detector’s rest frame. Since the initial sta te is not represented as a normal\nstate on Mβ(for anyβ > 0), we cannot directly adopt the setup from Section 5.3. It is\ntherefore suggested to treat the problem on a representatio n-independent C∗-algebraic level.\nAssume that, for λ∈R, there exists a one-parameter group of∗-automorphisms αλ=\n{αλ,t}t∈Rdescribing the coupled dynamics on AD⊗ AFrelative to u, such that\n(i) one recovers αD⊗αFasλ→0,\n(ii) there exists, for any β >0, a unique ( αλ,β)-KMS state ωλ,βonAD⊗ AFdepending\ncontinuously on λ∈Rsuch that, in the weak sense, lim λ→0ωλ,β=ωD,β⊗ωF,β, and\nFol(ωλ,β) = Fol(ωD,β⊗ωF,β).\n34Assumptions (i) and (ii) replicate the typical properties o f the coupled dynamics and corre-\nsponding KMS states based on the general framework of pertur bation theory of dynamics\nand KMS states [ DJP03 ]. In particular, the hypothetical KMS state ωλ,βshould be a per-\nturbation of the uncoupled KMS state ωD,β⊗ωF,βrelative to the detector’s rest frame u.\nThe coupled system with initial state ω′∈Fol(ωD⊗ω(w)\nF,β′) approaches a thermal equilibrium\nstate with respect to the rest frame uof the detector, if ( AD⊗ AF,αλ,ωλ,β;ω′) satisfies the\nthermalization property for some β >0 and sufficiently small |λ|>0. But Eq. ( 5.10) tells us\nthat all quantum field states in Fol( ω(w)\nF,β′) are as far as possible from any thermal equilibrium\nstate relative to u, and thusω′/∈Fol(ωλ,β) for anyβ >0. Under these conditions the coupled\ndetector-field system will generally not satisfy the therma lization property (which, in turn,\nwould imply the asymptotic thermality of the detector). The system would have to leave its\ninitial folium under a sufficiently strong and finely tuned int eraction in order to settle in a\nlimit state that would be deemed thermal in the reference fra me of the detector.\nIt is important to note that this discussion provides eviden ce, but not a definite proof\nof non-thermalization for a whole folium of initial states. Under special conditions, the\ncoupled detector-field system could still end up in a thermal equilibrium state or at least\nsome invariant state (cf. the final paragraph of Section 5.1) relative to the detector’s rest\nframe u. The possibility for such equilibration remains an open pro blem that will not be\ntreated here (see Section 6for an outlook). Our result shows that this problem is not one\nofreturn to equilibrium and thus is not covered by quantum Koopmanism in the form of\nProposition 5.2.\nFinally, let us note that a favorable choice for coupled dyna micsαλsatisfying, at least\nformally, assumptions (i) and (ii) would be a one-parameter group of∗-automorphisms that\nis generated, for any β >0, by the Liouvillean Lλ,β(Eq. ( 5.4)) with monopole coupling Vβ\n(Eq. ( 5.5)) in the GNS representation of ωD,β⊗ωF,β, so the coupled dynamics is represented\nbyγλ={γλ,t}t∈Rdefined in Eq. ( 5.6) on the von Neumann algebra Mβ(Eq. ( 5.3)). Consider\nαλinducingγλin this sense. Then one can assume that ωλ,βin assumption (ii) has the GNS\nrepresentation ( πD⊗πF,β,H,Ωλ,β), so it is represented on Mβby the coupled equilibrium\nstateϕλ,βdefined in Eq. ( 5.7). Still, the RTE property of ( Mβ,γλ,ϕλ,β), which holds for\neveryβ >0 and sufficiently small |λ|>0 by Proposition 5.3, cannot imply thermalization in\nthe present situation due to the above argument.\nWe however remark that a one-parameter group αλof∗-automorphisms inducing γλon\nMβ(β >0) can only be constructed in a regularized form. This has bee n shown in [ FM04b ]\nfor the interaction between an atom (with a single electron a nd a static nucleus) and black\nbody radiation. The idea is to consider a “time-averaged” C∗-algebra for the field on which\nthe dynamics is represented by a strongly continuous one-pa rameter group (cf. [ FV03 ] for\na similar construction). On the tensor product C∗-algebra of the combined detector-field\n35system one can define a regularized interaction operator, fr om which the coupled dynamics\nis obtained via a Dyson series. In the GNS representation of t he uncoupled β-KMS state the\ndynamics is represented by a one-parameter group of∗-automorphisms on Mβthat weakly\napproachγλ,t= eiLλ,βt(·)e−iLλ,βtas the regularization is removed; see [ FM04b , Sec. 2.1] for\ndetails.\n5.5. Comparison with the Unruh effect\nTheorem 5.4allows to highlight the difference to the behavior of an Unruh -DeWitt detector\nthat is uniformly accelerating through Minkowski spacetim eM∼=R4and is coupled to the\nMinkowski vacuum state of the free scalar field. Such a detect or moves along a hyperbolic\nworldline in the (right) Rindler wedge W:={(x0,x1,x2,x3)∈M:|x0|< x 1}, which\nis a submanifold of Mequipped with the induced metric and causal structure (see, e.g.,\n[Rin77 ]). If the detector is coupled to the Minkowski vacuum and fol lows the orbit of a\nLorentz boost describing its uniformly accelerated motion with proper acceleration a>0,\nthen it will converge (in proper time) to a thermal equilibri um state with Unruh temperature\na/2π. This is the Unruh effect [ Unr76 ] (see Section 1for more comments and references),\nproven rigorously in [ DM06 ] in terms of return to equilibrium of the coupled detector-fi eld\nsystem.\nHence the detector appears to respond as if it was coupled to a heat bath at temperature\na/2π. Does this mean that there is, from the viewpoint of the accel erated observer, an Unruh\n“heat bath” or a “thermal gas” of Rindler quanta (relative to Fulling-Rindler quantization of\nthe scalar field on the Rindler wedge [ Ful73 ,Wal94 ,CHM08 ])? On the observable algebra\nassociated to W, the Minkowski vacuum state is a 2 π-KMS state with respect to time trans-\nlations given by the Killing flow of Lorentz boosts (respecti vely 2π/a-KMS in proper time\nparametrization) [ Ful73 ,Unr76 ,BW75 ,BW76 ,Sew80 ,Sew82 ]. Although it might be tempt-\ning to view this KMS parameter as the inverse temperature of t he Minkowski vacuum relative\nto an accelerating reference frame, the physical interpret ation needs to be distinguished from\ninertial KMS states, i.e. KMS states with respect to ordinar y time evolution relative to an\ninertial reference frame.\nThe interpretation of the latter as global thermal equilibr ium states of inertial heat baths\nat positive temperature is adopted from KMS states of genera l quantum systems [ Haa96 ,\nBR97 ]. They represent systems that are in thermal equilibrium in the sense of the zeroth law\n[KFGV77 ] (describing equilibration and coexistence under thermal contact, see also [ Roo72 ])\nand the second law [ PW78 ] (KMS states are “passive”, meaning that energy extraction\nvia cyclic processes is prohibited, see also [ Kuc02 ]). In that case, the inverse of the KMS\nparameter specifies an empirical and absolute temperature s cale of the system.\nOn the other hand, as mentioned in Section 1, this correspondence fails in non-inertial\nsituations. For the Unruh effect this has been demonstrated i n [BS13 ] based on a concept\n36of local thermality for non-equilibrium states [ BOR02 ] (which has been developed for curved\nspacetimes in the framework of locally covariant quantum fie ld theory [ BS07 ,Sol10 ,Sol12 ]\nand has connections to quantum energy inequalities [ SV08 ] and cosmology [ Ver12 ]). In the\nformalism of [ BOR02 ], states are compared with global thermal equilibrium stat es via the\nexpectation values of a class of “local thermal observables ” at any spacetime point. As a result,\nthe Minkowski vacuum turns out to have zero temperature rela tive to every (inertial or non-\ninertial) observer [ BS13 ], and there is no heat bath or thermal gas of “Rindler quanta” in the\nMinkowski vacuum [ BV15 ,BV16 ]. The thermalization in the Unruh effect can be attributed\nto quantum excitations that are created from the vacuum by th e coupling and are picked\nup by the detector, which responds in an ergodic fashion; the picture is that work from the\naccelerating force is effectively converted into heat in a di ssipative process, similar to friction\n[BV15 ,BV16 ]. Consequently, Unruh-DeWitt detectors are not adequate t hermometers under\nthe influence of external forces. In the words of [ BV15 ], “instead of indicating the temperature\nof the vacuum, the probe indicates its own temperature at asy mptotic times” .\nOur analysis highlights yet another aspect of the difference to the inertial case. The proper\northochronous Lorentz group L↑\n+is unitarily implemented in the Minkowski vacuum state,\nwhich is invariant under Lorentz boosts (see Eq. ( 4.1)), in contrast to inertial KMS states (cf.\nCorollary 3.8). In the Unruh effect, the uniformly accelerated detector is stationary relative\nto the Killing flow induced by the Lorentz boost isometries of the Rindler wedge, and the\nMinkowski vacuum state of the field is a 2 π-KMS state with respect to boosts and thus is\ninvariant under the same flow. Hence the Unruh effect is observ ed in any Lorentz frame,\nand the accelerated detector does not measure any Doppler sh ifts in the vacuum, as noted\nin [SCD81 ] (see the quote in Section 1). On the other hand, a detector that moves inertially\nrelative to the thermal bath of a quantum field interacts with an inertial KMS state that is far\nfrom equilibrium with respect to the time evolution along th e trajectory of the detector. In\nthis sense, the breakdown of Lorentz symmetry can be observe d in the asymptotic behavior\nof detectors, which distinguishes the alleged Unruh “heat b ath” from a proper inertial heat\nbath.\n6. Conclusions and outlook\nIn this article we showed that in a local, translation-covar iant quantum field theory distinct\nprimary, mixing states (with separating GNS vectors) relat ive to two different inertial refer-\nence frames are disjoint. The proof uses the algebraic frame work of quantum field theory and\nquantum statistical mechanics. From this result we obtaine d a proof of the disjointness of\ninertial KMS states of the free scalar field, and the Lorentz s ymmetry breaking in these states.\nAs these statements are not entirely new, we summarized and r eviewed previous results on\nthe subject and compared them with our presentation.\n37A motivation for our analysis is the clarification of the beha vior of detector systems in iner-\ntial and accelerated motion. The disjointness result suppo rts the conclusion that the coupled\nsystem consisting of an Unruh-DeWitt detector moving with c onstant velocity relative to an\ninertial heat bath does not thermalize. The fact that the ini tial state is far from equilibrium\nrelative to the detector’s rest frame indicates that the sta te of the coupled system will not\napproach a KMS state at asymptotically late times. In partic ular, Lorentz boosts are not\nunitarily implementable in the GNS representation of KMS st ates with respect to inertial\ntime translations, while the Minkowski vacuum state is inva riant under Lorentz boosts. Our\ndiscussion complements earlier results and elaborates the statement in [ SCD81 ] that a uni-\nformly accelerated detector thermalizes to the Unruh tempe rature in every boosted Lorentz\nframe, which illustrates that the Unruh effect has to be disti nguished from the interaction\nwith a relativistic thermal gas in an inertial frame.\nRelated conclusions have been reached before. At the level o f perturbation theory Costa &\nMatsas [ CM95a ,CM95b ] showed that an inertially moving detector does not attain t hermal\ntransition rates from the heat bath of a massless scalar field at asymptotically late times\ndue to the Doppler shift of quanta from the field. The transiti on rate is a function of the\nrelative velocity and the detector’s energy gap, and is not o f Planck (black body) form. Based\non these results Landsberg & Matsas [ LM96 ,LM04 ] concluded that there exists no Lorentz\ntransformation for temperature from the frame of the bath to the rest frame of the detector.\nA well-defined temperature, obtained from the transition ra te corresponding to a Planck\nspectrum, only exists for a thermal bath relative to its own r est frame. This has also been\ncorroborated by Sewell in [ Sew08 ,Sew09 ], where it is shown that KMS states in one inertial\nreference frame are non-KMS with respect to an inertial fram e moving relative to it. For\nan Unruh-DeWitt detector coupled to and moving relative to a n inertial thermal reservoir it\nis reasonable to expect that it does not converge to a KMS stat e at late times: The KMS\ncondition characterizes systems that are thermal in the sen se of the zeroth law [ KFGV77 ],\nbut the detector is coupled to a reservoir that is not in a KMS s tate relative to the detector’s\nrest frame [ Sew08 ].\nThese results suggest that the moving detector does not reac h a thermal equilibrium state\nat late times. However, as emphasized in Section 1, it is important to note that recording\nexcitation spectra is a physical procedure that differs from observing the thermalization in\nthe state of the detector or the coupled detector-field syste m. The system could be driven\ninto thermal equilibrium, even though the reservoir is not t hermal from the viewpoint of the\ndetector and the detector transition rate does not correspo nd to a Planck spectrum. On a\nrelated note, it is difficult to exclude thermalization on the basis of thermodynamic properties\nalone; after all, the asymptotic temperature of a thermaliz ing detector is not necessarily a\npointer for the local temperature of the quantum field it coup les to, as can be observed in\nnon-inertial situations such as the Unruh effect (cf. Sectio n5.5). Moreover, we can expect\nthat the asymptotic behavior of the full, coupled system wil l depend on the details of the\n38interaction between detector and quantum field.\nFor the moving detector our Theorem 5.4, which is based on the disjointness of KMS states\nrelative to different inertial frames, provides strong indi cation for the non-thermalization\nunder generic conditions. However, as we noted in Section 5.4, thermalization of the coupled\nsystem under specially fine-tuned, coupling-dependent con ditions cannot be excluded. In\nlight of the above comments, the question arises to which deg ree deviations from a thermal\ntransition rate are “visible” in the asymptotic state behav ior. The evolution of the reduced\ndensity matrix of the Unruh-DeWitt detector (i.e. taking th e partial trace over the field part),\nexpressed in an open quantum systems approach by a master equ ation, has been shown in\n[PA20 ] to reflect the non-Planckian response derived by Costa & Mat sas [CM95b ]. However,\nthe relation to non-thermalization of the full detector-fie ld system in the context of our work is\nnot obvious, especially when the invariance of the asymptot ic state behavior under the choice\nof the (not necessarily factorizing) initial state from its folium, as suggested by Theorem 5.4,\nis to be retained. We save these investigations for future wo rk.\nWe also propose a possible connection to non-equilibrium st eady states (NESS) [ Rue00 ,\nJP02a ,JP02b ,MMS07 ]. A detector moving inertially relative to a heat bath of a ma ssless\nscalar field observes a heat current with a Doppler shift depe nding on the relative velocity,\nas shown in [ CM95b ] for the transition rate. The master equation for the reduce d density\nmatrix reveals the effect of a continuum of heat baths interac ting with the detector in its\nrest frame, corresponding to temperatures ranging between the red- and blue-shifted values\nof the field temperature [ PA20 ]. In future work we will pursue the question whether the stat e\nof the quantum field is a NESS, as defined in [ HV18 ] for the scalar field, from the perspective\nof the detector’s rest frame.\n•\nAcknowledgments The authors would like to express their gratitude to Detlev B uchholz,\nChristopher J. Fewster, and Marco Merkli for illuminating d iscussions. A.G.P. thanks the\nIMPRS and MPI for Mathematics in the Sciences, Leipzig, wher e most parts of this work\nhave been carried out, for support and hospitality.\n•\n39A. More on quantum dynamics and operator algebras\nIn this appendix we compile some additional material on quan tum statistical mechanics and\ndynamical systems, continuing Section 2.2and using the notation introduced there.\nDefinition A.1 (KMS state)\nA stateωon a quantum dynamical system ( A,α) is called a ( α,β)-KMS state (orβ-KMS\nstate with respect to α, orβ-KMS state on ( A,α)) forβ∈R\\ {0}, the KMS parameter ,\nif for every A,B ∈ A there exists a function z/ma√sto→Fβ\nA,B(z) that is bounded and continuous\non{z∈C: 0≤sgn(β) Im(z)≤ |β|}and analytic on the interior of this strip, such that\nFβ\nA,B(t) =ω(Aαt(B)) andFβ\nA,B(t+iβ) =ω(αt(B)A) for allt∈R.\nProposition A.2 (Tomita-Takesaki modular theory [ Tak70b ,BR87 ,Sum06 ])\nLetM ⊆ B (H)be a von Neumann algebra on a Hilbert space Hwith cyclic and separating\nunit vector Ω∈ H. The Tomita-Takesaki modular objects associated to the pair (M,Ω)\nare given by the modular conjugation J, which is an anti-unitary involution on H, and the\nmodular operator ∆, which is positive and self-adjoint on H, uniquely determined by the\nproperty\nJ∆1/2XΩ =X∗Ωfor allX∈ M.\nThe modular objects satisfy J=J∗=J−1,JΩ =∆Ω = Ω ,JMJ=M′, and∆itM∆−it=\nMfor allt∈R. The modular automorphism group is the one-parameter group of∗-\nautomorphisms ∆it(·)∆−it,t∈R, on M. The faithful vector state /a\\}bracketle{tΩ,(·)Ω/a\\}bracketri}htis a KMS\nstate on Mwith KMS parameter −1with respect to modular automorphisms.\nThe significance of modular theory in the context of quantum fi eld theory is discussed in\n[Bor00 ].\nIfωis a KMS state, its GNS vector is cyclic and separating for the induced von Neumann\nalgebra (Proposition 2.6), so one can associate Tomita-Takesaki modular objects to i t. The\nfollowing relation between the modular group and the dynami cs of the KMS state is shown\nin, e.g., [ BR97 , Thm. 5.3.10] (see also [ Sum06 , Thm. 2.2]).\nProposition A.3 (Modular theory for KMS states)\nLetωbe aβ-KMS state on a quantum dynamical system (A,α)with GNS representation\n(πω,Hω,Ωω), inducing the W∗-dynamical system (Mω,γ)with Liouvillean L. LetJ,∆ be\nthe Tomita-Takesaki modular objects associated to (Mω,Ωω). Then log∆=−βL, i.e. the\nmodular automorphism group coincides with the dynamics γup to rescaling:\n∆it(·)∆−it=γ−βtfor allt∈R.\nWe state a useful characterization of the collection of inva riant states on a W∗-dynamical\nsystem in terms of the Liouvillean (Proposition 2.6), following [ DJP03 , Sec. 2.10].\n40Proposition A.4 (Characterization of normal invariant sta tes)\nLetωbe a state on a C∗-algebra Awith GNS representation (πω,Hω,Ωω), such that Ωωis\nseparating for Mω=πω(A)′′. LetJbe the modular conjugation associated to (Mω,Ωω). The\nnatural cone associated to (Mω,Ωω)is defined as\nH+\nω:={XJX Ωω:X∈ Mω} ⊂ Hω.\nFor every normal state ϕonMωthere is a unique unit vector ˆΩϕ∈ H+\nωsuch thatϕ=\n/a\\}bracketle{tˆΩϕ,(·)ˆΩϕ/a\\}bracketri}ht. The vector ˆΩϕis cyclic for Mωif and only if it is separating. If ωis anα-\ninvariant state on (A,α)and induces a W∗-dynamical system (Mω,γ)with Liouvillean L,\nthen such a normal state ϕonMωisγ-invariant if and only if ˆΩϕ∈kerL. This provides a\nbijection between kerL∩ H+\nωand the collection of normal, γ-invariant states on Mω.\nThe result is part of the rich theory of “standard forms” of vo n Neumann algebras, discussed\nin [BR87 , Sec. 2.5.4] and [ DJP03 ] (see also the original literature cited therein). Notice t hat\nΩω∈ H+\nωandLΩω= 0 by Proposition 2.6, hence 0 is a simple (non-degenerate) eigenvalue\nofLif and only if the normal state /a\\}bracketle{tΩω,(·)Ωω/a\\}bracketri}htextendingωtoMωis the unique normal,\nγ-invariant state on Mω.\nWe present a proof of Proposition 2.8that is based on the concepts introduced in Section\n2.2and this appendix as well as the equivalence between the uniq ueness of KMS states on a\nvon Neumann algebra and factoriality shown in [ BR97 , Prop. 5.3.29]. Another proof can be\nfound in [ Kay85b , Thm. 1] for “double KMS states” (which could be rewritten to apply to\nour setup).\nProposition A.5 (Criterion for primarity of KMS states)\nLetωbe a KMS state on (A,α)with GNS representation (πω,Hω,Ωω)and Liouvillean L\ngenerating the dynamics γof the induced W∗-dynamical system (Mω,γ). IfLhas a simple\neigenvalue at 0, i.e. kerL=CΩω, thenωis primary.\nProof.\nAssume that ωis an (α,β)-KMS state for β >0, and let H+\nωbe the natural cone associated to\n(Mω,Ωω). Due to the bijection between ker L∩ H+\nωand the collection of normal, γ-invariant\nstates (Proposition A.4), the assumption ker L=CΩωimplies that /a\\}bracketle{tΩω,(·)Ωω/a\\}bracketri}htis the unique\nnormal,γ-invariant state on Mω. That state is a ( γ,β)-KMS state by Proposition 2.6,\nthus a ( −1)-KMS state with respect to the rescaled dynamics {γ−βt}t∈R. Tomita-Takesaki\ntheory (Proposition A.3) tells us that the latter coincides with the modular automor phism\ngroup. Hence /a\\}bracketle{tΩω,(·)Ωω/a\\}bracketri}htis the unique ( −1)-KMS state on Mωwith respect to modular\nautomorphisms. By [ BR97 , Prop. 5.3.29] this uniqueness is equivalent to the factor p roperty\nofMω.\n/square\nThe following lemma shows the claim at the end of Section 2.3that the mixing property\n(Def. ( 2.9)) is equivalent to weak asymptotic abelianness (defined in E q. (2.5)) for states that\n41are primary and have a separating GNS vector. This and simila r statements have appeared\nbefore in, e.g., [ NT91 ,NTW88 ,Thi92 ] (see also [ BR87 , p. 403]).\nLemma A.6 (Mixing vs. weak asymptotic abelianness)\nLetωbe anα-invariant state on a quantum dynamical system (A,α)with GNS representation\n(πω,Hω,Ωω)and induced von Neumann algebra Mω=πω(A)′′. If Ωωis separating for\nMωand(A,α,ω )satisfies the mixing property, then it is weakly asymptotica lly abelian, i.e.\nlimt→∞[πω(αt(A)),πω(B)] = 0 for allA,B ∈ A in weak operator topology. If ωis primary,\nthen weak asymptotic abelianness implies mixing.\nProof.\nFor the first statement, Lemma 3.1shows that lim t→∞πω(αt(A)) =ω(A)1in weak operator\ntopology for all A∈ A, which implies weak asymptotic abelianness. For the second one\nnotices that, for every A∈ A, the net (πω(αt(A)))t∈Rhas weak cluster points in the center of\nMωdue to weak asymptotic abelianness, which is C1by primarity. Using arguments similar\nto the ones in the proof of Lemma 3.2, theα-invariance of ωshows that lim t→∞πω(αt(A)) =\nω(A)1in weak operator topology for every A∈ A, and the mixing property for ωfollows (cf.\n[NT91 ]).\n/square\nWe close this section by noting that the mixing property by it self (without the assumption\nin Lemma A.6that the GNS vector is separating) merely implies lim\nt→∞ω([αt(A),B]) = 0 for\nallA,B ∈ A, which is slightly weaker than weak asymptotic abelianness .\nB. Inertial KMS states of the free scalar field\nIn this appendix we summarize the theory of the free scalar fie ld of mass m≥0 on four-\ndimensional Minkowski spacetime M=R1,3using Weyl quantization and define the quasi-\nfree KMS states of the theory. For details on the constructio n of the free scalar field on\nglobally hyperbolic spacetimes in terms of Weyl algebras we refer to [ Dim80 ,FR20 ] (see\nalso the summary in [ Ver93 , Sec. 2]). The construction of quasi-free KMS states follow s\nthe formalism of one-particle structures [ Kay78 ,Kay85a ,Kay85b ,Kay85c ,DK87 ,KW91 ].\nThis is a more abstract and elaborate approach compared to tr aditional specification of the\nthermal two-point function on a Fock space, but the implemen tation of the dynamics in the\nGNS representation is rather simple and concise on a formal l evel, which makes it well-suited\nfor the purposes of this work.\nB.1. Quantum field\nLetP:=/square+m2be the Klein-Gordon operator for mass m≥0 onC∞\nc(M;R), the space of\nreal-valued, smooth, compactly supported functions on Min kowski spacetime M, where/squareis\n42the d’Alembert (wave) operator on M. Let\n[ [·] ] :C∞\nc(M;R)→C∞\nc(M;R)/PC∞\nc(M;R) =:CM\nbe the canonical quotient map. Let EP:C∞\nc(M;R)→C∞(M;R) be the advanced-minus-\nretarded Green operator associated to P(also called Pauli-Jordan or causal propagator)\n[KM15 ,BGP07 ]. It has the property that ker EP=PC∞\nc(M;R), and the map CM→\nC∞(M;R) induced by EPis a well-defined R-linear isomorphism onto the space of real-valued\nsmooth solutions of the Klein-Gordon equation (i.e. elemen ts of kerP ֒→C∞(M;R)) with\ncompactly supported Cauchy data. The bilinear form ([ [ f] ],[ [g] ])/ma√sto→ E([ [f] ],[ [g] ]) :=/integraltext\nMfEPg\ndefines a real-valued symplectic form on CM.\nThe C∗-algebra AFis defined as the Weyl (CCR) algebra over the symplectic space\n(CM,E), which represents the classical phase space. It is generat ed by elements W([ [f] ]),\nf∈C∞\nc(M;R), such that W(−[ [f] ]) =W([ [f] ])∗andW([ [f] ])W([ [g] ]) = e−i\n2E([ [f] ],[ [g] ])W([ [f+g] ])\nfor allf,g∈C∞\nc(M;R), which determines AFuniquely up to∗-isomorphism [ BR97 ,BGP07 ,\nPet90 ,Dim80 ]. The local algebras AF(O) for regions (open, bounded, causally convex sub-\nsets)O⊂Mare the C∗-subalgebras of AFthat are generated by elements W([ [f] ]) for test\nfunctionsf∈C∞\nc(O;R) compactly supported in O. The net {AF(O)}O⊂Mof C∗-algebras\nsatisfies the axioms (A)–(C’) from Section 2.1[Dim80 ] (see also [ FR20 ] and [ Bor96 , Sec. I.2]).\n(The axioms correspond to Axiom 1, 3, 5 for the free scalar fiel d on a globally hyperbolic\nspacetime in [ Dim80 ], with the covariance Axiom 5 implying Poincaré covariance for the field\non Minkowski spacetime.) The algebra AFis the associated quasi-local algebra (see [ BR97 ,\nProp. 5.2.10]).\nThe time translations in an inertial reference frame uare implemented as a∗-automorphism\ngroupα(u)onAF, such that α(u)\nt(AF(O)) = AF(O+tu) for allt∈Rand regions O⊂M.\nWithout loss of generality, as we only consider a single fixed reference frame in this appendix,\nwe will employ the isomorphism M∼=R4and coordinates ( x0,x1,x2,x3)≡(t,x) onR4such\nthat u= (1,0). In the chosen coordinates the Klein-Gordon operator is ex pressed in the\nfamiliar form P=∂2\nt−∆ +m2, where ∆ is the Laplace operator on R3. The dynamics\nα(u)≡αF={αF,t}t∈Ris given by (cf. [ BR97 , Thm. 5.2.8])\nαF,t(W([ [f] ])) =W([ [f◦T−t] ]) (B.1)\non generators of AFfor the time translation Tt:M→M, (t′,x)/ma√sto→(t+t′,x) (action of the\nKilling flow along ∂t). The pair ( AF,αF) is the quantum dynamical system (Definition 2.5)\nof the free scalar field with respect to inertial time transla tions along u. Note that t/ma√sto→αF,t\nis not strongly continuous since the C∗-norm distance of distinct Weyl algebra elements is\nalways 2 (see, e.g., [ BGP07 , Prop. 4.2.3]).\n43B.2. KMS states\nThe KMS states on AFwith respect to inertial time translations αFare quasi-free, i.e.\nthey are completely determined by their two-point function , with all odd n-point func-\ntions vanishing (see, e.g., [ Pet90 ,KW91 ,KM15 ]). Given a symmetric R-bilinear form\nµ:CM×CM→RonCMsatisfying the inequality |E([ [f] ],[ [g] ])|2≤4µ([ [f] ],[ [f] ])µ([ [g] ],[ [g] ])\nfor allf,g∈C∞\nc(M;R), one can define a quasi-free state by ω(W([ [f] ])) := e−1\n2µ([ [f] ],[ [f] ])(ex-\ntended linearly and continuously to all of AF), with two-point function given by ( f,g)/ma√sto→\nµ([ [f] ],[ [g] ]) +i\n2E([ [f] ],[ [g] ]) [KW91 ]. Quasi-free states are equivalently characterized by the ir\nso-called one-particle structure on some Hilbert space, th at is, a pair ( k,H) consisting of a\nsymplectic map k: (CM,E)→(H,2 Im/a\\}bracketle{t ·,· /a\\}bracketri}htH) to a “one-particle Hilbert space” Hsuch\nthat ran k+irank⊂ H is dense. Given a one-particle structure ( k,H), the bilinear form\nµdefined by µ([ [f] ],[ [g] ]) = Re /a\\}bracketle{tk[ [f] ],k[ [g] ]/a\\}bracketri}htHsatisfies the above inequality and induces the\nquasi-free state ω; conversely, given a bilinear form µsatisfying this inequality, there exists\na one-particle structure ( k,H), unique up to unitary equivalence, such that µarises this way\n[Kay85b ,Kay85c ,KW91 ] (see also [ Wal94 ] for a general discussion).\nFor the construction of the inertial KMS states on AFthe necessary ingredients in terms\nof one-particle structures can be found in [ KW91 , Sec. 3.2]. The explicit construction is\npresented in [ DK87 , Sec.s 7.1 & A4]. Here we only summarize the most important po ints.\nLet Σ := {0} ×R3∼=R3be thet= 0 Cauchy surface in M. Forf∈C∞\nc(M;R), let\nf\n/tildewide1:=EPf↾Σ∈C∞\nc(Σ;R), f\n/tildewide2:=∂tEPf↾Σ∈C∞\nc(Σ;R)\nbe the pair of Cauchy data on Σ of the solution EPfto the Klein-Gordon equation corre-\nsponding to funder the isomorphism of CMwith the space of real-valued smooth solutions\nmentioned in the previous section. Due to the well-posednes s and uniqueness of the Cauchy\nproblem for the Klein-Gordon equation, solutions and Cauch y data can be identified (cf.\n[BGP07 , Thm. 3.2.11]), and thus CM∼=C∞\nc(Σ;R)⊕C∞\nc(Σ;R).\nLet\nG:=L2(Σ)≡L2(R3,d3x).\nThe operator −∆ +m2is essentially self-adjoint on C∞\nc(Σ;C)⊂ G (see, e.g., [ Che73 ]), and\nwe denote its self-adjoint closure by the same symbol. The R-linear symplectic map g:\n(CM,E)→(G,2 Im/a\\}bracketle{t ·,· /a\\}bracketri}htG) defined by\ng[ [f] ] :=1√\n2/parenleftBig\nh1/2\n0f\n/tildewide1+ih−1/2\n0f\n/tildewide2/parenrightBig\n,h0:= (−∆ +m2)1/2(B.2)\nis part of the so-called ground one-particle structure ( g,G,eih0t) for the Minkowski vac-\nuum state with one-particle Hamiltonian h0. We will not go into the details here and in-\n44stead refer to [ Kay85a , Sec.s 1 & 2] (in particular Eq. (2.8) ff.) and [ DK87 , Sec.s 7.1 &\nA4]. On generators of AF, the Minkowski vacuum state ωF,∞is uniquely determined by\nωF,∞(W([ [f] ])) := e−1\n2/bardblg[ [f] ]/bardbl2\nGforf∈C∞\nc(M;R). The map ghas the properties (see [ DK87 ,\nSec. A4])\nran(g) =G,ran(g)⊂dom( h−1/2\n0), (B.3)\nwhere the first is equivalent to the statement that ωF,∞is a pure state by [ KW91 , Lemma\nA.2], and the second property, called “regularity” in [ Kay85a ,Kay85c ], is essential to the\nconstruction of thermal states.\nThe definition of the KMS states is based on a standard constru ction presented in [ Kay85c ,\n§3.1] and summarized in [ KW91 , p. 102 & Lemma 6.2]. For β >0, theR-linear symplectic\nmap kβ: (CM,E)→(K,2 Im/a\\}bracketle{t ·,· /a\\}bracketri}htK) on the one-particle Hilbert space\nK:=G ⊕ G (B.4)\nis defined by\nkβ[ [f] ] :=/parenleftbigg/radicalBig\n1 +µβ(h0)g[ [f] ]/parenrightbigg\n⊕/parenleftbigg/radicalBig\nµβ(h0)Cg[ [f] ]/parenrightbigg\n≡g[ [f] ]√\n1−e−βh0⊕Cg[ [f] ]√\neβh0−1(B.5)\nusing spectral calculus, where µβ:R>→R>,x/ma√sto→(eβx−1)−1, and the anti-unitary map\nC:G → G is the complex conjugation. The one-particle Hamiltonian o nKis given by the\nself-adjoint operator (on the usual domain)\nh=h0⊕ −h0. (B.6)\nIt implements the time translations on K, that is,\neiht◦kβ=kβ◦(Tt)∗ (B.7)\nonCM, where eiht= eih0t⊕e−ih0tas an operator on K, and (Tt)∗([ [f] ]) := [ [f◦T−t] ] for\nf∈C∞\nc(M;R) andTt:M→Mthe time translation as defined below Eq. ( B.1). The triple\n(kβ,K,eiht) is the KMS one-particle structure (cf. [ Kay85c ] for the general definition) of the\nunique quasi-free ( αF,β)-KMS state ωF,βonAF, uniquely determined on generators by\nωF,β(W([ [f] ])) := e−1\n2/bardblkβ[ [f] ]/bardbl2\nK≡e−1\n2/angbracketleftg[ [f] ],coth(βh0/2)g[ [f] ]/angbracketrightG(B.8)\nforf∈C∞\nc(M;R). The definition requires the regularity condition in ( B.3), see [ Kay85c ,\n§A.2]. The final expression follows from the definition of kβand 1+2µβ(h0) = coth(βh0/2) (cf.\n[Kay85c , Eq. (3.2)], [ DK87 , Eq. (A4.3)]) and reveals the similarity to the generating f unction\nfor the canonical state of the thermal (infinitely extended) free Bose gas in the Araki-Woods\n45representation [ AW63 ] (see also [ Can73 ,JP96a ,JP97 ,Mer06 ], and the comment on top of p.\n1021 in [ Kay85c ]), whereµβ(ν) = (eβν−1)−1is the Planckian momentum density distribution\nof the gas at temperature β−1with single boson energy ν.\nThe GNS representation of ωF,βcan be expressed via Weyl operators on a Fock space (see\n[KW91 , p. 76 & Lemma A.2]): It is given by ( πβ,F(K),ΩF), where F(K) :=C⊕/circleplustext∞\nn=1K⊙n\nis the symmetric (bosonic) Fock space over the one-particle space Kwith symmetric tensor\nproduct ⊙and Fock vacuum vector Ω F:= 1⊕0⊕0⊕..., and\nπβ(W([ [f] ])) := ei(a(kβ[ [f] ])+a∗(kβ[ [f] ]))=:WF(kβ[ [f] ]) (B.9)\nforf∈C∞\nc(M;R), in which the self-adjoint closure is implied in the expone nt, anda(·)\nanda∗(·) denote the annihilation and creation operators on F(K), respectively. (For a\ncomprehensive treatment of the theory of Fock spaces we refe r to [ Ara18 ].)\nUsing Eq.s ( B.1), (B.7), (B.9) one finds that, on Weyl algebra generators of AF,\nπβ(αF,t(W([ [f] ]))) =πβ(W([ [f◦T−t] ])) =WF(kβ[ [f◦T−t] ]) =WF(eihtkβ[ [f] ]) =\n= Γ(eiht)WF(kβ[ [f] ])Γ(eiht)−1= eidΓ(h)tπβ(W([ [f] ]))e−idΓ(h)t,\nwhere Γ(eiht) =/circleplustext∞\nn=0(eiht)⊙n= eidΓ(h)tfor the second quantization map dΓ, and the step\nfrom the first to the second line follows from [ Ara18 , Thm. 5.30]. This extends to AFby\nlinearity and continuity, and it holds dΓ( h)ΩF= 0. Using Proposition 2.6, the Liouvillean\nLgenerating the W∗-dynamics on the von Neumann algebra πβ(AF)′′induced by ωF,βis\ntherefore given by the second quantization\nL= dΓ( h) (B.10)\nof the one-particle Hamiltonian h.\nThe “exponential law” for Fock spaces, F(K)∼=F(G)⊗ F(G), allows to express the Fock\nspace over K=G ⊕ G as a tensor product of Fock spaces over G(see [ Ara18 , Thm. 5.38]).\nThe tensor product Fock space is frequently used for the GNS H ilbert space of the thermal\nmassless Bose gas in the Araki-Woods representation, see, e .g., [JP96a ,MSB08 ,Mer20 ] and\nthe overview in [ Mer07 , Sec. 2.2]. We will not make use of this representation here.\nB.3. Comments on the mixing property\nIn Lemma 3.5in the main text we discussed the mixing property for ( AF,αF,ω) for any\ninertial KMS state ω=ωF,βusing the spectral characterization of Proposition 2.10. Let us\nillustrate the connection to spectral properties of the Lio uvillean using the explicit definition\nof the mixing property (Definition 2.9) and the one-particle structure formalism (see also\n[Mer06 , Sec. 4.5] for a similar discussion for equilibrium states o f the free Bose gas). Recall\n46from the proof of Lemma 3.5thatL= dΓ( h) (Eq. ( B.10)) has an absolutely continuous\nspectrum Rwith a simple embedded eigenvalue at 0 corresponding to Ω F. For the mixing\nproperty, it suffices to prove that lim t→∞ω(W([ [g] ])αF,tW([ [f] ])) =ω(W([ [f] ]))ω(W([ [g] ])) for\nallf,g∈C∞\nc(M;R); this then extends by linearity to the span of all Weyl gener ators, which\nis dense in AF, yielding the desired mixing property. By the Weyl relation s forWF(Eq.\n(B.9)) and the definition of ω(Eq. ( B.8)),\nω(W([ [g] ])αF,tW([ [f] ])) = /a\\}bracketle{tΩF,WF(kβ[ [g] ])WF(kβ[ [f◦T−t] ])ΩF/a\\}bracketri}ht=\n= e−iIm/angbracketleftkβ[ [g] ],kβ[ [f◦T−t] ]/angbracketrightK/a\\}bracketle{tΩF,WF(kβ[ [g+f◦T−t] ])ΩF/a\\}bracketri}ht=\n= e−/angbracketleftkβ[ [g] ],kβ[ [f◦T−t] ]/angbracketrightK·ω(W([ [f] ]))ω(W([ [g] ])),\nwhere in the last step it is used that /a\\}bracketle{tΩF,WF(kβ[ [g+f◦T−t] ])ΩF/a\\}bracketri}ht= e−1\n2/bardblkβ[ [g+f◦T−t] ]/bardbl2\nKand\n/bardblkβ[ [g+f◦T−t] ]/bardbl2\nK=/bardblkβ[ [g] ]/bardbl2\nK+/bardblkβ[ [f] ]/bardbl2\nK+2 Re /a\\}bracketle{tkβ[ [g] ],kβ[ [f◦T−t] ]/a\\}bracketri}htK, since /bardblkβ[ [f◦T−t] ]/bardbl2\nK=\n/bardbleihtkβ[ [f] ]/bardbl2\nK=/bardblkβ[ [f] ]/bardbl2\nKby Eq. ( B.7).\nHence we have to show that lim t→∞W2(g,f◦T−t) = 0, where W2(f,g) :=/a\\}bracketle{tkβ[ [f] ],kβ[ [g] ]/a\\}bracketri}htK=\n/a\\}bracketle{tΩF,Q(f)Q(g)ΩF/a\\}bracketri}htdefines the smeared two-point (Wightman) function of ωforf,g∈C∞\nc(M;R)\nand self-adjoint field operator Q(g) :=a(kβ[ [g] ]) +a∗(kβ[ [g] ]) (the closure of this expression is\nimplied) on the Fock space F(K). The desired decay of the two-point function at timelike\nseparations is a manifestation of Huygens’ principle [ Haa96 ]. The absolute continuity of the\nspectrum of L= dΓ( h) on{ΩF}⊥implies that lim t→∞eiLt=|ΩF/a\\}bracketri}ht/a\\}bracketle{tΩF|in the weak operator\nsense (as a consequence of the Riemann-Lebesgue lemma), and therefore\nlim\nt→∞W2(g,f◦T−t) = lim\nt→∞/a\\}bracketle{tΩF,Q(g)eiLtQ(f)ΩF/a\\}bracketri}ht=/a\\}bracketle{tΩF,Q(g)ΩF/a\\}bracketri}ht · /a\\}bracketle{tΩF,Q(f)ΩF/a\\}bracketri}ht= 0\nsince /a\\}bracketle{tΩF,Q(f)ΩF/a\\}bracketri}ht=/a\\}bracketle{tΩF,a∗(kβ[ [f] ])ΩF/a\\}bracketri}ht= 0.\nC. Jakšić-Pillet glued representation\nConsider the free scalar field of mass m= 0, using the setup of Appendix B. We present\na unitary transformation (isometric isomorphism) of the fie ld representation in which the\nLiouvillean L= dΓ( h) (Eq. ( B.10)) is given by the second quantization of a simple mul-\ntiplication operator. The main part of the construction is d ue to [ JP96a ] and is therefore\nknown as “Jakšić-Pillet gluing” . We first unitarily transfo rm the KMS one-particle structure\n(kβ,K,eiht) (Appendix B), from which a unitarily equivalent GNS representation of ωF,βis\nobtained.\nLetˆ∆ := −∆, so we have h0=ˆ∆1/2in Eq. ( B.2) and h=ˆ∆1/2⊕ −ˆ∆1/2in Eq. ( B.6). The\noperator ˆ∆ is positive, unbounded, densely defined and self-adjoint o nG=L2(R3,d3x) with\n47spec ˆ∆ = [0,∞). Its spectral representation is provided by the Fourier tr ansform\nF:L2(R3,d3x)→L2(R3,d3k),F[g](k) :=1\n(2π)3/2/integraldisplay\nR3e−ixkg(x) d3x,\nwhich is a unitary operator by Plancherel’s theorem. For any Borel subset X⊆R3(equipped\nwith the Lebesgue measure) and measurable function f:X→Rwe define the self-adjoint\nmultiplication operator MfbyMfg:=f·gon the dense domain dom( Mf) :={g∈L2(X) :\nf·g∈L2(X)}. To shorten the notation we will write the image of fin the index. For\nexample, for f:R3→R,k/ma√sto→ /bardblk/bardbl2we writeM/bardblk/bardbl2to denote the multiplication operator Mf\nonL2(R3,d3k). It holds F[ˆ∆g](k) =/bardblk/bardbl2F[g](k) for allg∈dom(∆) ⊂ Gand dom(M/bardblk/bardbl2) =\nFdom(∆). By spectral calculus,\nFh0F−1=M/bardblk/bardbl\non the natural domain. Define\n/tildewideF:=F⊕CFC:K=G ⊕ G → FG ⊕FG, g ⊕g′/ma√sto→F[g]⊕F[g′],\nwhere the map Cis the natural complex conjugation operator, which is an ant i-unitary in-\nvolution on the respective L2-spaces whose action on L2-functions is denoted by an overline.\nThe maps CFCand/tildewideFare unitary, and we immediately obtain that\n/tildewideFh/tildewideF−1=Fˆ∆1/2F−1⊕ −CFCˆ∆1/2CF−1C=M/bardblk/bardbl⊕ −M/bardblk/bardbl,\nwhere we used that [ C,ˆ∆] = 0 and [ C,M/bardblk/bardbl] = 0.\nLet\nHF:=L2(R×S2,dsdΩ),\nwhereΩis the standard Riemannian metric on the unit 2-sphere S2⊂R3. There is an\nisomorphism FG=L2(R3,d3k)→L2(R>×S2,q2dqdΩ) induced by introducing spherical\ncoordinates on R3, denotedf/ma√sto→f◦. In particular, ( M/bardblk/bardblf)◦(q,Ω) =qf◦(q,Ω) almost\neverywhere for f∈dom(M/bardblk/bardbl)⊂L2(R3,d3k). Forζ∈R, define (almost everywhere)\nJPζ:FG ⊕FG → HF,JPζ[f⊕g](s,Ω) :=s\n\nf◦(s,Ω) fors>0\neiζg◦(−s,Ω) fors<0,\ncalled the Jakšić-Pillet gluing map [ JP96a ,DJ03 ] (see also [ MSB08 , Appendix A]). The map\n48JPζis unitary, and it holds\nJPζ[M/bardblk/bardblf⊕ −M/bardblk/bardblg](s,Ω) =s\n\nsf◦(s,Ω) for s>0\n−eiζ(−s)g◦(−s,Ω) fors<0= (MsJPζ[f⊕g])(s,Ω)\nalmost everywhere, where Msis the operator of multiplication with the function ( s,Ω)/ma√sto→s\nonHF. We therefore obtain a unitary map\nTζ:=JPζ◦/tildewideF:K → HF\nsuch that h=ˆ∆1/2⊕ −ˆ∆1/2(Eq. ( B.6)) is represented by the multiplication operator\nTζhT−1\nζ=Ms (C.1)\nonTζdom( h) = dom(Ms)⊂ HF.\nThe unitary map Tζcan be used to unitarily transform the one-particle structu re (kβ,K,eiht)\nofωF,β(defined in Eq.s ( B.4), (B.5), (B.6)). For allβ >0 andζ∈R, the triple\n(Kβ,HF,eiMst),Kβ:=Tζ◦kβ: (CM,E)→ HF\nis aβ-KMS one-particle structure such that ωF,β(W([ [f] ])) = e−1\n2/bardblKβ[ [f] ]/bardbl2\nHFforf∈C∞\nc(M;R)\n(cf. Eq. ( B.8)). The KMS one-particle structures describe the same KMS st ateωF,βsince\n/bardblKβ[ [f] ]/bardblHF=/bardblkβ[ [f] ]/bardblKfor allf∈C∞\nc(M;R) by unitarity of Tζ(cf. the uniqueness result\n[Kay85c , Thm. 1b]). The definitions, functional calculus, and the fa ct that Ccommutes with\nˆ∆ and ( ·)◦imply that\nfβ(s,Ω) := ( Kβ[ [f] ])(s,Ω) =s\n\n/radicalBig\n1 +µβ(s)F[g[ [f] ]]◦(s,Ω) fors>0\neiζ/radicalBig\nµβ(−s)F[g[ [f] ]]◦(−s,Ω) fors<0=\n=/radicalBiggs\n1−e−βs|s|1/2\n\nF[g[ [f] ]]◦(s,Ω) for s>0\n−eiζF[g[ [f] ]]◦(−s,Ω) fors<0(C.2)\nalmost everywhere for all f∈C∞\nc(M;R), where gis defined in Eq. ( B.2), andµβis given\nbelow Eq. ( B.5).\nFrom this unitarily transformed one-particle structure, t he GNS representation of ωF,βis\nnow obtained as in Eq. ( B.9) (using [ KW91 , p. 76 & Lemma A.2]):\nLemma C.1\nThe GNS representation of ωF,βis given by (πF,β,F(HF),ΩF)for the symmetric Fock space\nF(HF)with Fock vacuum vector3ΩF, andπF,βis defined analogous to Eq. (B.9)(replacing kβ\nwithKβ). By construction, this GNS representation is unitarily eq uivalent to (πβ,F(K),ΩF).\n3Here we slightly abuse notation by using the same symbol as in Appendix B.\n49The Liouvillean LFgenerating the dynamics αF(Eq. (B.1)) in this representation is unitarily\nequivalent to Eq. (B.10), and is given by the second-quantized multiplication opera tor\nLF= dΓ(Ms),\nwhich is commonly denoted dΓ(s)and is self-adjoint on the natural domain dom(dΓ(s))(see\n[Ara18 , Thm. 5.2]).\nThe same result is also obtained if we first transform the one- particle structure ( g,G,eih0t)\nfor the Minkowski vacuum state to ( F◦g,FG,eiM/bardblk/bardblt) usingF, then perform the construction\nof the KMS one-particle structure for this unitarily transf ormed ground one-particle structure\nfollowing mutatis mutandis Eq. ( B.4) ff., and finally apply the gluing map JPζ. Comparing\nwith, e.g., the summaries in [ Mer20 , Sec.s 3.2 & 3.3] or [ Mer22 , Sec.s 2.1 & 2.2], we see that\nthe GNS representation of the quasi-free KMS state ωF,βof the free massless scalar field\nformally coincides with the Jakšić-Pillet glued Araki-Woo ds representation [ AW63 ,JP96a ]\nof a thermal massless Bose gas, which is a common choice for th e reservoir in the study of\nreturn to equilibrium (see Section 5.1). As a complex-valued function, fβappears in the\ninteraction term of the generator of dynamics for the couple d system (see Eq. ( 5.5)). The\nphase parameter ζ∈Ris chosen such that the two parts of fβfit together at s= 0 in a\ncontinuous, sufficiently regular way [ FM04a ,Mer20 ].\nSince the Fock vacuum vector Ω Frepresents a KMS state, there exists a Tomita-Takesaki\nmodular conjugation JFonF(HF) associated to the pair ( πF,β(AF)′′,ΩF) (Propositions A.2\n&A.3). It is uniquely determined by the relations JFeiLFtJF= eiLFtand\nJFe−βLF/2πF,β(W([ [f] ]))ΩF=πF,β(W([ [f] ]))∗ΩF\nfor allf∈C∞\nc(M;R), which extends to πF,β(AF)′′and implies, together with the fact that\nπF,β(AF)′′ΩFis a core for e−βLF/2, that such JFindeed coincides with the Tomita-Takesaki\nmodular conjugation (see [ KW91 , Eq.s (3.47)–(3.48)] and the proof of [ Kay85b , Thm. 1]).\nFrom the one-particle formalism [ Kay85a ,Kay85b ,Kay85c ,KW91 ] one obtains\nJF= Γ(jF), (C.3)\nwherejFis the anti-unitary involution on HFgiven almost everywhere by\njF(f)(s,Ω) =−eiζf(−s,Ω)\n(recall that the GNS representation πF,βdepends on the parameter ζ∈Rappearing in the\ndefinition of Tζ). A direct calculation using Eq. ( C.2) and the identity µβ(s) = e−βs(1+µβ(s)),\n50s∈R\\ {0}, shows that, almost everywhere,\n(jFfβ)(s,Ω) = e−βs/2fβ(s,Ω). (C.4)\n•\n51References\n[Ara18] A. 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We applied the method of finite-part integration [Galapon E.A\nProc.R.Soc A 473, 20160567(2017)] to evaluate in closed-form the exact one-\nloop integral representations of the Heisenberg-Euler Lagrangian from QED for\na constant electric field and electric-like self-dual background. We also devise a\nprescription based on the finite-part integration of the Cauchy principal value\nintegral to sum and extrapolate the non-alternating divergent weak-field ex-\npansions of the Heisenberg-Euler Lagrangians to recover the non-perturbative\nSchwinger effect well into the strong field limit.\n1.Introduction\nEmploying perturbation theory (PT) for approximating some function or a finite\nphysical observable, F(κ), around some small real perturbing parameter, κ, often\nleads to a divergent power series expansion which represents the exact solution only\nin the asymptotic sense [1, 2, 3, 4],\n(1) F(κ)∼∞X\nn=0anκn, κ →0+.\nAn optimally truncated asymptotic power series expansion rapidly delivers exper-\nimentally relevant approximations to F(κ) near the perturbative regime κ→0+.\nVarious summability methods and extrapolation procedures [5, 6] such as Borel\nsummation and Pad´ e approximants are then required to combine the information\nE-mail addresses :eagalapon@up.edu.ph .\nDate : February 26, 2024.\n1arXiv:2402.14839v1  [math-ph]  19 Feb 20242 GENERALIZED STIELTJES TRANSFORM\ncontained within a finite string of the coefficients, an, with the known analytic\nproperties of F(κ) to piece together a reliable reconstruction of the exact solution\nin the nonperturbative regime, κ→ ∞ , along the real line.\nComplications arise when the exact solution possesses elements that are invisible\nto any finite order of perturbation theory. Borel analysis of the PT series in this\ncase indicate that nonperturbative ambiguities relate to the factorial divergence\nand a non-alternating sign character of the expansion coefficients, an[7, 8, 9]. In\nparticular, nonperturbative contributions appear to arise from the poles of the Borel\ntransform especially those situated along the real axis of the complex Borel plane\nwhich render the Borel-Laplace integral ill-defined in the conventional sense [10, 11].\nThe systematic treatment and the recovery of nonperturbative ambiguities arising\nfrom Borel nonsummability of PT series expansions is a subject in the theory of\nresurgence [12, 13, 14].\nParadigmatic examples of nonperturbative effects occur in singular eigenvalue\nproblems in single-particle quantum mechanics [7, 15, 16, 17, 18, 19, 20, 21]. Per-\nturbing potentials such as in the quartic anharmonic oscillator with a negative\ncoupling, the cubic oscillator, and the Stark Hamiltonian, impart an exponentially\nsuppressed imaginary part to the perturbed energy which characterizes the decay\nof the resulting metastable states via quantum tunneling process. Unlike the real\npart of the energy, the imaginary part cannot be recovered from a straightforward\nsummation of the nonalternating divergent weak-coupling PT expansion.\nThe standard approach for carrying out the resummation of divergent nonalter-\nnating PT expansions is the Pad´ e-Borel technique [21, 22, 23, 24, 25, 26]. Pad´ e\napproximants are used to perform the analytic continuation of the Borel transform\nof (1) to a neighborhood of the positive real axis. They are also used to simulate\nthe singularity structure of the Borel transform such as the location of the poles\nand generate their contributions. Other functions such as the Gauss hypergeomet-\nric function, 2F1, and the Meijer G-function are also used in place of the Pad´ e\napproximants to accommodate more complicated singularity structures such as the\npresence of branch cuts [27, 28].\nNone of these prescriptions however allow for the incorporation of the known spe-\ncific leading-order behavior of the exact solution in the opposite regime, F(κ→ ∞ ),\nalong the real line which is the key for an extrapolation procedure along this di-\nrection [29, 30, 31, 32, 33, 34]. Here, we devise a resummation scheme to recover\nnonperturbative information from the real nonalternating coefficients, an>0, of\nthe PT expansion (1) and simultaneously provide a means for a constrained ex-\ntrapolation to the opposite regime κ→+∞along the real line.\nTo demonstrate the prescription, we will recover the Schwinger effect, a well-\nknown example of nonperturbative effect in QED [35], from the nonalternating\nweak-field PT expansion of the Heisenberg-Euler Lagrangian for a constant electric\nand self-dual electric-like background well into the strong field limit. The results\npresented in this paper complement those derived in [36] for the case of a constant\nmagnetic and a magnetic-like self-dual background where the Heisenberg-Euler La-\ngrangian is real-valued and the weak-field PT expansion is alternating and Borel\nsummable.\nWe will also derive a closed form for the complex-valued Heisenberg-Euler La-\ngrangian in both scalar and spinor QED by evaluating the respective exact integral\nrepresentations using the method of finite-part integration [37]. Central to thisGENERALIZED STIELTJES TRANSFORM 3\nmethod is the novel contour integral representations of Hadamard’s finite part and\nits equivalence to two other formulations for which we give a brief discussion in sec-\ntion 2. We then carry out the derivation of the closed-form for the complex-valued\nHeisenberg-Euler Lagrangian in section 3.\nIn section 4, we carry out the summation and constrained extrapolation to the\nstrong electric field regime of the divergent nonalternating weak-field PT expansion\nfor the Heisenberg-Euler Lagrangian. The initial step is to map the expansion\ncoefficients, an, to the positive-power moments of some positive function ρ(x),an=\nµ2n=R∞\n0x2nρ(x)dxso that we sum the divergent non-alternating PT expansion\n(1) formally to,\nH(κ)\nκ=Z∞\n0ρ(x)\n1−κx2dx∼∞X\nn=0µ2nκn, κ →0+. (2)\nThe integral in equation (2) is ill-defined due to the pole x= 1/√κsituated\nalong the integration path. In section 4, we regularize this integral by a suitable\ndeformation of the contour to circumvent the pole and recover the nonperturbative\nimaginary part from its residue contribution. This will yield\nHs(κ) = PVZ∞\n0ρs(x)\n1/κ−x2dx±πi√κ\n2ρs\u00121√κ\u0013\n, (3)\nwhere the sign corresponds to the direction in which the pole is evaded. Evaluating\nthe Cauchy principal value integral in the opposite asymptotic regime by expanding\n(1−κx2)−1in powers of 1 /(κx2) and a term-wise integration is carried out leads\nto an expansion with terms that diverge individually,\nPVZ∞\n0ρs(x)\n1/κ−x2dx→ −∞X\nk=0µ−(2n+2)\nκn, (4)\nwhere the negative-power moments, µ−(2n+2)=R∞\n0ρ(x)/x2n+2dx, n= 0,1. . .are\ndivergent integrals for sufficiently large n.\nWe overcome this problem by interpreting the divergent negative-power mo-\nments, µ−(2n+2), as Hadamard’s finite part and use its complex contour integral\nrepresentation to arrive at a convergent expansion in inverse powers of κplus a\ncorrection term that enables us to incorporate the known leading-order behavior\nof the complex-valued Heisenberg-Euler Lagrangian in the strong-field limit. This\nleading strong field behavior will also dictate the form of the the reconstruction of\nthe unknown function, ρ(x), from the positive-power moments, µ2n.\nFinally in section 5, we summarize our results and provide a possible direction\nin which to improve the efficacy of the summation and extrapolation prescription.\n2.Hadamard’s Finite Part\nIn this section, we give a concise discussion on the computation of Hadamard’s\nfinite part of divergent integrals with a pole singularity at the origin,\n(5)Za\n0f(x)\nxmdx, m = 1,2, . . . , a > 0,\nwhere f(x) is analytic at the origin and f(0)̸= 0. Consistent with the expedi-\nent canonical definition [38], the Hadamard’s finite part may be formulated more\nrigorously as a complex contour integral [37, 39] or alternatively as a regularized4 GENERALIZED STIELTJES TRANSFORM\nlimit at the poles of Mellin transform integrals [40]. The equivalence of these dual\nrepresentations is central to the results we present here and their applications.\n2.1.Canonical Representation. The canonical representation of the finite part\nof the divergent integral (5) is obtained by introducing an arbitrarily small cut-off\nparameter ϵ, 0< ϵ < a , to replace the offending non-integrable origin. The resulting\nconvergent integral is grouped into two sets of terms\n(6)Za\nϵf(x)\nxmdx=Cϵ+Dϵ,\nwhere Cϵis the group of terms that possesses a finite limit as ϵ→0, while Dϵ\ndiverges in the same limit and consists of terms in inverse powers of ϵand ln ϵ.\nThe finite part of the divergent integral is then defined uniquely by dropping the\ndiverging group of terms Dϵ, leaving only the limit of Cϵand assigning the limit as\nthe value of the divergent integral,\n\\ \\Za\n0f(x)\nxmdx= lim\nϵ→0Cϵ. (7)\nThe upper limit acan be also be taken to infinity provided f(x)x−mis integrable\nat infinity, in which case,\n\\ \\Z∞\n0f(x)\nxmdx= lim\na→∞\\ \\Za\n0f(x)\nxmdx. (8)\n2.2.Contour Integral Representation. A rigorous formulation of the Hadamard’s\nfinite part as a complex contour integral may also be derived from the following\nform of equation (6)\n\\ \\Za\n0f(x)\nxmdx= lim\nϵ→0\u0014Za\nϵf(x)\nxmdx−Dϵ\u0015\n. (9)\nThis contour integral representation is given in the following lemma. The full\nderivation is given as a proof of Theorem 2.2 in [37].\nLemma 2.1. Let the complex extension, f(z), off(x), be analytic in the interval\n[0, a]. Iff(0)̸= 0, then\n(10) \\ \\Za\n0f(x)\nxmdx=1\n2πiZ\nCf(z)\nzm(logz−πi) dz, m = 1,2. . .\nwhere logzis the complex logarithm whose branch cut is the positive real axis and\nCis the contour straddling the branch cut of logzstarting from aand ending at a\nitself, as depicted in figure 1. The contour Cdoes not enclose any pole of f(z).\n2.3.Regularized Limit Representation. Another equivalent formulation of the\nHadamard’s finite part is facilitated by the concept of the regularized limit intro-\nduced in [40]. We quote here some of the relevant results.\nLetw(z) be a function of the complex variable zthat is analytic at some domain\nD⊆Candz0be an isolated singularity of w(z) inD, then we can define the\ndeleted neighborhood δz0=D(r, z0)\\z0where D(r, z0) is an open disk of radius r\ncentered at z0such that the function w(z) admits the Laurent series expansion\nw(z) =∞X\nn=−∞an(z−z0)n, (11)GENERALIZED STIELTJES TRANSFORM 5\nwhere the coefficients anare given by\nan=1\n2πiI\n|z−z0|=r′w(z)\n(z−z0)n+1dz (12)\nfor any r′< r. The radius ris bounded by the distance of the nearest singularity\nofw(z) from z0. The regularized limit of the function w(z) atz=z0is defined as\nfollows.\nDefinition 2.1. Letz0be an interior point in the domain D of w(z). The regu-\nlarized limit of w(z)asz→z0, denoted by\n(13)×\nlim\nz→z0w(z),\nis the coefficient a0in the Laurent series expansion (11)ofw(z)in a deleted neigh-\nborhood of z0.\nIn the case when the function w(z) can be rationalized, that is written in the\nform w(z) =h(z)/g(z) where h(z) and g(z) are both analytic at z0, and z0is a\nsimple zero of g(z) while h(z0)̸= 0, then the regularized limit is computed as,\n×\nlim\nz→z0h(z)\ng(z)=h′(z0)\ng′(z0)−h(z0)g′′(z0))\n2(g′(z0))2. (14)\nFor the case when g′′(z0) = 0, this result reduces to a form similar to the L’Hospital’s\nrule,\n×\nlim\nz→z0h(z)\ng(z)= lim\nz→z0h′(z)\ng′(z). (15)\nThe derivation of this result is given in the proof of Corollary 3.1 in [40].\nThe finite part of the divergent integral (5) can be extracted from the analytic\ncontinuation, M∗[f(x);s], to the whole complex splane of the Mellin transform,\nM[f(x);s] =Z∞\n0xs−1f(x)dx, (16)\nprovided there is a non-trivial strip of analyticity of the Mellin integral. The case\nwhen the upper limit is a finite afollows from equation (16) by considering f(x) =\ng(x)Θ(a−x), where Θ( x) is the Heaviside step function. Furthermore, if f(x) is\nanalytic at x= 0, the Mellin transform M[f(x);s] has at most simple poles along\nthe real line.\nFor a positive integer m, ifs= 1−mis a pole of M∗[f(x);s], then the finite\npart integral (5) is given by the regularized limit of M∗[f(x);s] ats= 1−m,\n\\ \\Z∞\n0f(x)\nxmdx=×\nlim\ns→1−mM∗[f(x);s]. (17)\nIn most situations, the analytic continuation can be written in rational form,\nM∗[f(x);s] =h(s)/g(s), where h(1−m)̸= 0 and g(1−m) = 0. In this case,\nit may be possible to choose the rationalization such that g′′(1−m) = 0 so that\nwhen s= 1−mis a simple pole of M∗[f(x);s] =h(s)/g(s) the calculation of the\nregularized limit at s= 1−mreduces to equation (15).\nBoth the canonical (7) and regularized limit (17) representations are used pri-\nmarily for computing the value of the Hadamard’s finite part explicitly. We demon-\nstrate the foregoing discussion by computing the finite part integral (17) for the\ncasef(x) =e−bxforb >0. The finite part integral is the regularized limit at the6 GENERALIZED STIELTJES TRANSFORM\nFigure 1. The contour C used in the representation (10) for the\nHadamard’s finite part. The contour C excludes any of the poles\noff(z). The upper limit acan be infinite. ϵis a small positive\nparameter.\npoles of the analytic continuation of the following Mellin transform integral [41, p\n20],\nM\u0002\ne−bx;s\u0003\n=Z∞\n0xs−1e−bxdx (18)\n=b−sΓ(s), Reb,Res >0. (19)\nThe analytic continuation of the Mellin transform to the whole complex plane is\nsimply the right-hand side of equation (19),\nM∗\u0002\ne−bx;s\u0003\n=b−sΓ(s). (20)\nHence, the finite part integral is\n\\ \\Z∞\n0e−bx\nxmdx=×\nlim\ns→1−mb−sΓ(s) =×\nlim\ns→1−mπb−s\nsin(πs)Γ(1−s). (21)\nwhere we used the reflection formula Γ( s) =π/sin(πs)Γ(1−s). We then let g(s) =\nsin(πs) and h(s) =πb−s/Γ(1−s) so that from equation (15), the regularized limit\nevaluates to\n\\ \\Z∞\n0e−bx\nxmdx=×\nlim\ns→1−mh(s)\ng(s)= lim\ns→1−mh′(s)\ng′(s). (22)\nHence substituting g′(s) =πcos(πs) and\nh′(s) =−πb−s\nΓ(1−s)(logb−ψ(1−s)) (23)\nand computing the limit in equation (22) we obtain for the case when bis real and\npositive,\n(24) \\ \\Z∞\n0e−bx\nxmdx=(−1)mbm−1\n(m−1)!(lnb−ψ(m)), m = 1,2. . . ,\nwhere ψ(z) is the digamma function. This is consistent with the result in [37, eq\n3.15] obtained after a lengthy calculation using the canonical definition (7). For\nthe case when bis imaginary, b=−iafor any real a,\n\\ \\Z∞\n0eiax\nxmdx=−(ia)m−1\n(m−1)!\u0012\nln|a| −iπ\n2sgn(a)−ψ(m)\u0013\n, m = 1,2. . . (25)GENERALIZED STIELTJES TRANSFORM 7\nIn this particular case, the computation of the finite part integral (24) as a reg-\nularized limit is more convenient. In general however, especially for cases when the\nMellin transform M[f(x);s] does not exist for some function f(x), the Hadamard’s\nfinite part integral can always be obtained from canonical definition (7).\n3.Finite-Part Integration\nAn important application of the contour integral representation (10) is a tech-\nnique known as finite-part integration [37, 42, 43, 44] and relies on the equivalence\namong the three different representations discussed above. The relevant procedure\nemployed here is a special case of the more general result [45, eq 64]. We apply\nit here to give a rigorous derivation of the closed-form for the Heisenberg-Euler\nLagrangian from which exact values can be computed. In the next section, we will\nuse these values to check the results of a summation and extrapolation procedure\napplied on the divergent weak-field expansions of the Heisenberg-Euler Lagrangian.\n3.1.The Heisenberg-Euler Lagrangian. The Heisenberg-Euler Lagrangian is a\nnonlinear correction to the Maxwell Lagrangian resulting from the interaction of a\nvacuum of charged particles of mass mand spin swith an external electromagnetic\nfield [35, 46, 47, 48, 49, 50, 51]. In the one-loop order and for the case when the\nbackground field is constant and uniform, it depends on the invariant quantity β=\ne2(B2−E2)/m4where eis the electron charge. For a purely electric background so\nthatβ=−κ, κ=e2E2/m4, it is a complex-valued quantity and, for spin-0 particles,\nit admits the following integral representation in natural units ( h=c= 1),\nL0(κ) =m4\n16π2f0(κ), f0(κ) =−Z∞\n0e−iτ\nτ3\u0014√κ τcsch\u0000√κτ\u0001\n−1 +κτ2\n6\u0015\ndτ. (26)\nIn [36], we applied the method of finite-part integration on the corresponding\nrepresentation in the case of a constant magnetic field background in which case\nthe Heisenberg-Euler Lagrangian is real-valued. Similar to the magnetic case, the\nrepresentation (26) is a finite sum of divergent integrals with pole singularities at\nthe origin.\nThe initial step is to bring the integration into the complex plane with the use\nof the contour C in the representation (10) for the Hadamard’s finite part,\nZ\nCg(z)\nz3logzdz, g (z) =e−iz\u0014√κ zcsch\u0000√κz\u0001\n−1 +κz2\n6\u0015\n, (27)\nwhere the contour C is given in figure 1 and acan be taken to infinity. The poles\ndue to g(z) along the imaginary axis are exterior to C . We deform the contour C\nto an equivalent contour C′so that\nZ\nC′g(z)\nz3logzdz= 2πiZa\nϵg(x)\nx3dx+Z\nCϵg(z)\nz3logzdz, (28)\nwhere C ϵis the circular contour about the origin. In the limit as ϵ→0, the integral\nalong the circular contour C ϵvanishes so that equation (28) reduces to\nZa\n0g(x)\nx3dx=1\n2πiZ\nCg(z)\nz3logzdz. (29)8 GENERALIZED STIELTJES TRANSFORM\nWe then add a zero term,\n−πi\n2πiZ\nCg(z)\nz3dz=−πiX\nRes\u0014g(z)\nz3\u0015\n= 0, (30)\nto the right-hand side of equation (29) so that,\nZa\n0g(x)\nx3dx=1\n2πiZ\nCg(z)\nz3(logz−πi) dz=\\ \\Za\n0g(x)\nx3dx (31)\nwhere we used the contour integral representation (10) of the Hadamard’s finite\npart. Hence, taking the limit a→ ∞ , the integral in the exact representation (26)\nevaluates to\nf0(κ) =−Z∞\n0e−iτ\nτ3\u0014√κ τcsch\u0000√κτ\u0001\n−1 +κτ2\n6\u0015\ndτ (32)\n=−√κ\\ \\Z∞\n0e−iτcsch(√κτ)\nτ2dτ+\\ \\Z∞\n0e−iτ\nτ3dτ−κ\n6\\ \\Z∞\n0e−iτ\nτdτ. (33)\nIn effect, the result (33) follows directly from a term-by-term integration followed\nby the immediate regularization of the divergent terms as Hadamard’s finite part.\nIn section 4, we will demonstrate that this procedure will generally result to missing\nterms especially when one performs a term-by-term integration involving an infinite\nnumber of divergent integrals.\nThe finite part integrals in right-hand side of equation (33) can be computed\nfrom the following Mellin transform integral [41, p 34],\nM\u0002\ne−aτcsch( b τ);s\u0003\n=Z∞\n0τs−1e−aτcsch( b τ)dτ\n=21−s\nbsΓ(s)ζ\u0012\ns,a+b\n2b\u0013\n, Rea >−|Reb|; Res >1, (34)\nwhere ζ(z, ν) is the Hurwitz zeta function. The finite part integral is the regularized\nlimit at the pole s=−1 of the analytic continuation of the Mellin transform to the\nwhole complex plane which is given simply by the right-hand side of equation (34),\nM∗\u0002\ne−aτcsch( b τ);s\u0003\n=21−s\nbsΓ(s)ζ\u0012\ns,a+b\n2b\u0013\n. (35)\nSo that\n\\ \\Z∞\n0e−iτcsch(√κτ)\nτ2dτ=×\nlim\ns→−1M∗\u0002\ne−iτcsch\u0000√κτ\u0001\n;s\u0003\n(36)\n=×\nlim\ns→−12π(2√κ)−s\nsin(πs)Γ(1−s)ζ\u0012\ns,√κ+i\n2√κ\u0013\n, (37)\nwhere we made use of the reflection formula for Γ( s) =π/sin(πs)Γ(1−s). We then\nrationalize the analytic continuation (37) as,\nM∗\u0002\ne−iτcsch\u0000√κ τ\u0001\n;s\u0003\n=h(s)\ng(s), (38)\nwhere\nh(s) =2π(2√κ)−s\nΓ(1−s)ζ\u0012\ns,√κ+i\n2√κ\u0013\n, g (s) = sin( πs). (39)GENERALIZED STIELTJES TRANSFORM 9\nHence, from the formula (15), the regularized limit is given by\n\\ \\Z∞\n0e−iτcsch(√κτ)\nτ2dτ=×\nlim\ns→−1h(s)\ng(s)= lim\ns→−1h′(s)\ng′(s). (40)\nwhere g′(s) =πcos(πs) and\nh′(s) =2(2√κ)−sπ\nΓ(1−s)\u0014\u0000\nψ(1−s)−ln\u0000\n2√κ\u0001\u0001\nζ\u0012\ns,√κ+i\n2√κ\u0013\n+ζ(1,0)\u0012\ns,√κ+i\n2√κ\u0013\u0015\n, (41)\nso that,\n\\ \\Z∞\n0e−iτcsch(√κτ)\nτ2dτ=−4√κ\u0014\u0000\nψ(2)−ln\u0000\n2√κ\u0001\u0001\nζ\u0012\n−1,√κ+i\n2√κ\u0013\n+ζ(1,0)\u0012\n−1,√κ+i\n2√κ\u0013\u0015\n. (42)\nThe other finite part integrals in the right-hand side of equation (33) are special\ncases of the result (25). Substituting the finite-part integral (42) to equation (33),\nwe obtain\nf0(κ) =κ\n6−1\n4−\u0012κ\n6+1\n2\u0013\nln\u0000\n2√κ\u0001\n+i\u0010π\n4+κπ\n12\u0011\n+ 4κ ζ(1,0)\u0012\n−1,√κ+i\n2√κ\u0013\n(43)\nwhere we’ve made use of the formula [61]\nζ(−1, ν) =��1\n12+ν\n2−ν2\n2. (44)\nThe real and imaginary parts are then given by\nRef0(κ) =κ\n6−1\n4−\u0012κ\n6+1\n2\u0013\nln\u0000\n2√κ\u0001\n+ 4κRe\u0014\nζ(1,0)\u0012\n−1,√κ+i\n2√κ\u0013\u0015\n(45)\nand\nImf0(κ) =π\n4+κπ\n12+ 4κIm\u0014\nζ(1,0)\u0012\n−1,√κ+i\n2√κ\u0013\u0015\n. (46)\nAs suggested in [47], a result numerically consistent with (43) can be obtained\nby a suitable analytic continuation, B→ − iE, of the corresponding closed-form\n[36, eq 42] for the case of a magnetic background. This procedure must be care-\nfully implemented by choosing an appropriate branch for the complex logarithmic\nfunction and the Hurwitz zeta functions appearing in [36, eq 42]. In particular, we\nfind that the branch log( iy) = ln y+iπ/2 is compatible with Mathematica 13.2 ’s\nimplementation for ζ(−1, x+iy) and ζ(1,0)(−1, x+iy) for any x, y > 0, to give a\nnumerical value consistent with that computed from (43).10 GENERALIZED STIELTJES TRANSFORM\n3.2.Spin-1\n2.Similarly in the case of spinor QED, the Heisenberg-Euler Lagrangian\nfor a purely electric background is given by the exact integral representation\nL1\n2(κ) =m4\n8π2f1\n2(κ), f 1\n2(κ) =Z∞\n0e−i τ\nτ3\u0014√κ τcoth\u0000√κ τ\u0001\n−1−κ τ2\n3\u0015\ndτ. (47)\nEmploying finite-part integration as in the previous case,\nf1\n2(κ) =√κ\\ \\Z∞\n0e−i τcoth (√κ τ)\nτ2dτ− \\ \\Z∞\n0e−i τ\nτ3dτ−κ\n3\\ \\Z∞\n0e−i τ\nτdτ. (48)\nThe first finite part integral on the right-hand side of equation (48) is the regu-\nlarized limit of the analytic continuation to the whole complex plane of the Mellin\ntransform integral [41, p. 34]. The procedure is similar to that performed in equa-\ntion (42) and the result is given by\n\\ \\Z∞\n0e−i τcoth (√κ τ)\nτ2dτ=i(1−γ) +π\n2+ 2√κ(2(γ−1) + ln(4 κ))ζ\u0012\n−1,i\n2√κ\u0013\n−4√κ ζ(1,0)\u0012\n−1,i\n2√κ\u0013\n, (49)\nwhere γ=−ψ(1) is the Euler-Mascheroni constant. The other finite part integrals\nin the right-hand side of equation (48) are again special cases of equation (25) so\nthat\nf1\n2(κ) =1\n4+π√κ\n2+κ\n3−κln (4κ)\n6+ln (4κ)\n4−4κ ζ(1,0)\u0012\n−1,i\n2√κ\u0013\n+i\u0012\n−π\n4+κπ\n6+√κln (4κ)\n2\u0013\n. (50)\nwhere we again made use of the property (44). The result (50) is also numerically\nequal to that obtained by an appropriate analytic continuation, B→ −iE, of the\ncorresponding closed-form [46, eq 1.47] for the Heisenberg-Euler Lagrangian in the\nmagnetic case. We again chose log( iy) = ln y+iπ/2 for any real y >0.\n3.3.Self-dual electromagnetic background. In the case when the electromag-\nnetic background is self-dual (SD) [46, 53, 54, 55, 56] satisfying\n(51) Fµν=eFµν≡1\n2ϵµνρλFρλ,\nthe Heisenberg-Euler Lagrangian describing a charged scalar particle is given by\nLSD(β) =e2F2\n4π2fSD(β), fSD(β) =1\n4Z∞\n0dτ\nτ\u00121\nsinh2τ−1\nτ2+1\n3\u0013\ne−2τ/√β(52)\nwhere F2=1\n4FµνFµνand the natural dimensionless parameter√β= 2eF/m2.\nThe integral, fSD(β), also appears in the effective Lagrangian in the case of spinor\nQED. It is also relevant in the study of nonperturbative phenomena in topological\nstring theory and in the c= 1 string theory at self-dual radius [53]. A weak-field\nexpansion for fSD(β) is given by [56],\nfSD(β)∼β∞X\nn=0−(−1)nB2n+4\n(2n+ 2)(2 n+ 4)(−β)n, β →0. (53)GENERALIZED STIELTJES TRANSFORM 11\nFor a real or magnetic-like background F,L0(β) is real and PT expansion (53) is\nalternating. For the case when the background is imaginary, F=−iFor electric-\nlike, the Heisenberg-Euler Lagrangian is complex-valued and is given by,\nLSD(κ) =e2F2\n4π2fSD(κ), fSD(κ) =−1\n4Z∞\n0dτ\nτ\u00121\nsinh2τ−1\nτ2+1\n3\u0013\ne−2iτ/√κ.(54)\nwhere√κ= 2eF/m2. As in the previous cases, a closed-form for the integral\nfSD(κ) can be derived by finite-part integration,\nfSD(κ) =−1\n4\\ \\Z∞\n0e−2iτ/√κ\nτsinh2τdτ+1\n4\\ \\Z∞\n0e−2iτ/√κ\nτ3dτ−1\n12\\ \\Z∞\n0e−2iτ/√κ\nτdτ. (55)\nThe finite part integral in the first term is computed as a regularized limit of the\nanalytic continuation of the Mellin transform integral [41, p 29, eq 10] so that\n1\n4\\ \\Z∞\n0e−2iτ/√κ\nτsinh2τdτ= (ψ(1)−ln 2)\u0012\nζ\u0012\n−1,i√κ\u0013\n−i√κζ\u0012\n0,i√κ\u0013\u0013\n+ζ(1,0)\u0012\n−1,i√κ\u0013\n−i√κζ(1,0)\u0012\n0,i√κ\u0013\n. (56)\nThe other finite part integrals in the right-hand side of equation (55) are again\nspecial cases of the finite part integral given in equation (25). We then use the\nfollowing results for the Hurwitz zeta function\nζ\u0012\n0,i√κ\u0013\n=1\n2−i√κ, ζ\u0012\n−1,i√κ\u0013\n=1\n2κ+i\n2√κ−1\n12(57)\nto simplify the result and obtain,\nfSD(κ) =i√κζ(1,0)\u0012\n0,i√κ\u0013\n−ζ(1,0)\u0012\n−1,i√κ\u0013\n(58)\n−1\n4κ(lnκ+ 3)−lnκ\n24+πi\u00121\n24+1\n4κ\u0013\n.\nAs in the previous cases, the result (58) gives consistent numerical values by per-\nforming a suitable analytic continuation, F → − iF, on the corresponding closed-\nform for a real magnetic-like background in [36] which we also derived using finite-\npart integration.\n4.Summation and extrapolation of the weak-field expansion for the\nHeisenberg-Euler Lagrangian\nWe now discuss the main result of this paper which is the application of the\nmethod of finite-part integration on the resummation of divergent nonalternat-\ning PT expansion. In particular, we will piece together a convergent extrapolant\nthat enables us to compute the complex-valued Heisenberg-Euler Lagrangian in\nthe strong electric field limit from the real coefficients of the divergent weak-field\nexpansion.\nIn the case of the scalar QED, a formal representation is obtained from the exact\nrepresentation (26) by a rotation, iτ=t, so that formally [46],\nf0(κ) =Z∞\n0e−t\nt3\u0012√κ tcsc\u0000√κt\u0001\n−1−κt\n6\u0013\ndt, κ =e2E2\nm4. (59)12 GENERALIZED STIELTJES TRANSFORM\nFigure 2. The contours of integration, C+and C−used in the\nprescription for evaluating the integral in (71). They correspond\nto two directions with which the half-circle about the pole C ϵis\ntraversed.\nThen a divergent weak-field PT expansion is obtained by replacing the factor in\nthe parenthesis by an expansion about κ= 0 and carrying out a term-by-term\nintegration that yields,\nf0(κ)∼κ2∞X\nn=0(−1)nB2n+422n+4\n(2n+ 4)(2 n+ 3)(2 n+ 2)κn, κ →0+, (60)\nwhere the constants B2nare in terms of the Bernoulli numbers, B2n,\nB2n=\u0000\n1−22n−1\u0001\n22n−1B2n. (61)\nThe imaginary part comes from the contributions of the poles along the integra-\ntion path in the real line in the formal representation (59). A convergent weak-field\nexpansion can be obtained from (59) using the method of residues. The result is\ngiven by [46, 51],\nImf0(κ) =κ\nπ∞X\nn=1(−1)n+1\nn2exp\u0012\n−nπ√κ\u0013\n. (62)\nSimilarly for the case of spinor QED, the Heisenberg-Euler Lagrangian (47) as-\nsumes the following formal representation after the rotation iτ=t,\nf1\n2(κ) =−Z∞\n0e−t\nt3\u0012√κ tcot\u0000√κ t\u0001\n−1 +κ t2\n3\u0013\ndt, (63)\nfrom which a divergent weak-field expansion for the real part is obtained,\nf1\n2(κ)∼κ2∞X\nn=0(−1)n+1B2n+422n+4\n(2n+ 4)(2 n+ 3)(2 n+ 2)κn. (64)\nFurthermore, a convergent expansion for the imaginary part, Im f1\n2(κ), can be de-\nrived similarly from the formal representation (63) using the method of residue\n[46],\nImf1\n2(κ) =κ\nπ∞X\nn=11\nn2exp\u0012\n−nπ√κ\u0013\n. (65)\nIn both spin cases, the coefficients of the weak-field expansions (60) and (64)\npossess a leading a(s)\n2n∼(2n)! growth as n→ ∞ . We present in table 1, the partial\nsums of the expansion (60) to compute Re f0(κ) for some values of the parameter κ.\nThe exponentially vanishing imaginary parts (62) and (65) of the Heisenberg-EulerGENERALIZED STIELTJES TRANSFORM 13\nd κ = 0.01 κ= 0.1 κ= 0.2\n1 1 .9567460317460(10−6) 2.06746032(10−4) 8.7619(10−4)\n3 1 .9570087692400(10−8) 2.10341210(10−4) 9.509187109(10−4)\n5 1 .9570096417862(10−6) 2.12001352(10−4) 1.1131746(10−3)\n9 1 .9570096596544(10−6) 2.52547209(10−4) 6.88452(10−2)\n20 1 .9570096601688(10−6) 9.50638258(105)\n50 4 .1365574383679(104)\nExact 1 .9570096599269(10−6) 2.10778565(10−4) 9.19523360907(10−4)\nTable 1: Convergence of the perturbative expansion (60) for the real part of f0(κ)\nin the Heisenberg-Euler Lagrangian in the case of a scalar QED. The exact result\nis computed from the closed-form (45).\nLagrangian in both cases are invisible up to any finite order of the corresponding\nweak-field perturbation expansions\nFurthermore, in the purely magnetic case, β= (eB)2/m4, the Heisenberg-Euler\nLagrangians possess the following leading behavior in the strong-magnetic field\nregime [46, eqs 1.63 and 1.54],\nf0(β)∼βlnβ\n12+ln 2\n6β+. . .andf1\n2(β)∼βlnβ\n6+ln 2\n3β+. . . (66)\nBy analytic continuation, B→ −iE, the real and imaginary parts of the Heisenberg-\nEuler Lagrangians in a purely electric field background will exhibit the following\nleading-order behaviors in the strong electric field regime [47],\nRefs(κ)∼κlnκ+κ, Imfs(κ)∼κ, κ → ∞ . (67)\nOn the basis of this information, we map the nonalternating coefficients, a(s)\nn,\nin the divergent expansions (60) and (64) to the positive-power moments, µ(s)\n2n, of\nsome positive function ρs(x),\na(s)\nn=µ(s)\n2n=Z∞\n0x2nρs(x)dx, n = 0,1, . . . (68)\nThis particular mapping is motivated by the leading (2 n)! growth of the expansion\ncoefficients, a(s)\nn, and in anticipation of the leading exponential behavior of the\nreconstruction for ρs(x) from the moments µ2nwhich we will carry out later in the\nsection.\nHence, the divergent expansions for fs(κ) can be summed formally as\nfs(κ) =κ2∞X\nn=0Z∞\n0x2nρs(x)(κ)ndx=κ2Z∞\n0ρs(x) ∞X\nn=0(κx2)n!\ndx\n=κ2Z∞\n0ρs(x)\n1−κx2dx=κH(κ). (69)\nwhich is in terms of the formal integral\nHs(κ) =Z∞\n0ρs(x)\n1/κ−x2dx. (70)\nThe integration can be carried out by a suitable prescription for the deformation\nof the contour. Two such prescriptions are shown in figure 2. In the limit as ϵ→0,14 GENERALIZED STIELTJES TRANSFORM\nwe obtain\nHs(κ) = PVZ∞\n0ρs(x)\n1/κ−x2dx±iπ√κ\n2ρs\u00121√κ\u0013\n. (71)\nwhere the plus(minus) sign corresponds to performing the integration along the\npath C+(−). Yet another possibility is to deform the contour in a manner that\nrenders the imaginary part in (71) extraneous so that H(κ) is real-valued and is\ngiven by the Cauchy principal value integral. In general, the manner in which this\nis done is unclear from perturbation theory alone and a separate prescription must\nbe provided to clear this nonperturbative ambiguity. In the case of the Heisenberg-\nEuler Lagrangian, the imaginary part gives the particle-antiparticle pair production\nrate, Γ prod(κ) = 2Im ( fs(κ)), so that the appropriate contour is C+.\nFinally, the Cauchy principal value in the right-hand side of equation (71) can be\ncomputed explicitly using finite-part integration. This allows us to incorporate the\nstrong field constraint (67). This is given in the following theorem. An independent\nbut equivalent derivation is also given in [52].\nTheorem 4.1. Let the complex extension, ρ(z), of the real-valued function ρ(x)\nalong the real line, be entire, then\nPVZ∞\n0ρ(x)\n1/κ−x2dx=−∞X\nn=0µ−(2n+2)\nκn−∆(κ), (72)\nwhere the term ∆(κ)is given by\n∆(κ) =√κ\n2ln\u0000√κ\u0001\u0012\nρ\u00121√κ\u0013\n−ρ\u0012\n−1√κ\u0013\u0013\n. (73)\nandµ−(2k+2), the divergent negative-power moments of ρ(x), are Hadamard’s finite\npart integrals,\n(74) µ−(2n+2)=\\ \\Z∞\n0ρ(x)\nx2n+2dx.\nProof. Deform the contour C to C′as shown in figure 3 and perform the following\ncontour integration,\nZ\nC′ρ(z)\n1/κ−z2logzdz= 2πiPVZa\n0ρ(x)\n1/κ−x2dx+Z\nCϵ1ρ(z)\n1/κ−z2logzdz\n+Z\nCϵ2ρ(z)\n1/κ−z2logzdz, (75)\nwhere the contour integral around the circular loop C ϵvanishes as ϵ→0 and the\nfirst term of the right-hand side of equation (75) containing the desired principal\nvalue integral is the sum of the integrals along the segments labelled 1 ,2,5 and 6\nin the limit as ϵ, ϵ2→0. The remaining integrals in the right-hand side of equation\n(75) are evaluated as\nlim\nϵ2→0Z\nCϵ2ρ(z)\n1/κ−z2logzdz=−(2πi)\u0014√κ\n2ρ\u00121√κ\u0013\nln\u00121√κ\u0013\n+πiRes\u0014ρ(z)\nz2−1/κ;z=1√κ\u0015\u0015\n. (76)GENERALIZED STIELTJES TRANSFORM 15\nFigure 3. The contour of integration used for the evaluation of\nthe principal value integral (72). Poles located at z=±1/√κare\nin the interior of the contour C.\nand\nlim\nϵ1→0Z\nCϵ1ρ(z)\n1/κ−z2logzdz= (2πi)\u0014√κ\n2ρ\u0012\n−1√κ\u0013\nln\u00121√κ\u0013\n−πiRes\u0014ρ(z)\nz2−1/κ;z=−1√κ\u0015\u0015\n. (77)\nSubstituting these to equation (75) and solving for the principal value integral\nwe obtain,\nPVZa\n0ρ(x)\n1/κ−x2dx=1\n2πiZ\nCρ(z)(log z−πi)\n1/κ−z2dz\n+√κ\n2ln\u00121√κ\u0013\u0012\nρ\u00121√κ\u0013\n−ρ\u0012\n−1√κ\u0013\u0013\n, (78)\nwhere we made use of\nπiX\nRes\u0014ρ(z)\nz2−1/κ;z=±1√κ\u0015\n=−πi\n2πiZ\nCρ(z)\n1/κ−z2dz. (79)\nIn the first term in the right-hand side of equation (78), we expand (1 /κ−z2)−1\nin powers of 1 /(κz2) and implement a term-by-term integration,\n1\n2πiZ\nCρ(z)(log z−πi)\n1/κ−z2dz=−∞X\nn=01\nκn1\n2πiZ\nCρ(z)(log z−πi)\nz2n+2dz (80)\n=−∞X\nn=01\nκn\\ \\Za\n0ρ(x)\nx2n+2dx,|z|>1√κ(81)\nwhere in equation (81), we used the contour integral representation (10) of the\nHadamard’s finite part integral. Substituting equation (81) for the first term of the\nright-hand side of equation (78) and taking the limit as a→ ∞ ,\nlim\na→∞PVZa\n0ρ(x)\n1/κ−x2dx=−∞X\nn=01\nκnlim\na→∞\\ \\Za\n0ρ(x)\nx2n+2dx\n+√κ\n2ln\u00121√κ\u0013\u0012\nρ\u00121√κ\u0013\n−ρ\u0012\n−1√κ\u0013\u0013\n, (82)\nThis proves the result (72). □16 GENERALIZED STIELTJES TRANSFORM\nIn the purely magnetic case, κ→ − β, β =e2B2/m4, so that the weak field\nexpansions (60) and (64) are alternating, the prescription (68) sums the divergent\nexpansions to a generalized Stieltjes integral, fs(β) =βSs(β), which we also eval-\nuated exactly in [36] using finite-part integration to give\nSs(β) =Z∞\n0ρs(x)\n1/β+x2dx=∞X\nn=0(−1)n\nβnµ(s)\n−(2n+2)+ Λ(β), (83)\nwhere the term Λ( β) is given by\nΛ(β) =π√β\n4\u0012\nρ\u0012i√β\u0013\n+ρ\u0012−i√β\u0013\u0013\n+√βlnβ\n4i\u0012\nρ\u0012i√β\u0013\n−ρ\u0012−i√β\u0013\u0013\n. (84)\nThe regularization carried out in equation (71) for the formal integral Hs(κ) can\nthen be recovered naturally by performing an analytic continuation, Ss(β→ −κ),\non equation (83) to the branch cut of Ss(β) along the negative real line in the\ncomplex- βplane. The sign in the second term of the right hand side of (71) now\ncorresponds to whether the branch cut is approached from above (+) or below ( −).\nIn the case of the Heisenberg-Euler Lagrangian, the approach should come from\nabove to yield a positive imaginary part.\nThe expansion in the first term of the right-hand side of (72) can be shown to\nbe absolutely convergent (see the proof of Theorem 3.1 in [52]). This convergence\nand the existence of the limit (8) justify interchanging the limit operation with\nthe infinite sum in equation (82). Meanwhile, the term ∆( κ) is missed by merely\nperforming a formal term-by-term integration followed by an immediate regular-\nization of the divergent integrals by Hadamard’s finite part. This term provides\nthe dominant behavior of the principal value integral (72) as κ→ ∞ . It originates\nfrom contributions of the poles z=±1/√κinterior to the contour C in figure 3\nwhich results from the uniformity condition imposed in equation (81).\nBy contrast, the finite-part integration (33) of the integral representation for the\nHeisenberg-Euler Lagrangian involves term-by-term integration over a finite number\nof divergent integrals so that no uniformity condition is imposed and consequently,\nthe contour C in figure 1 excludes any of the poles of the integrand along the\nimaginary axis.\nThus, we arrive at a convergent extrapolant for the complex-valued Heisenberg-\nEuler Lagrangian fs(κ),\n(85) fs(κ) =−∞X\nn=0µ(s)\n−(2n+2)\nκn−1−κ∆(κ) +iπ\n2κ3/2ρs\u00121√κ\u0013\n,\nconstructed from the real coefficients of the corresponding divergent non-alternating\nweak-field expansion expansions (60) and (64).\nWe can now incorporate the known leading-order behavior (67) into the expan-\nsion (85) through the term ∆( κ) which is given in (73). To this end, we require the\nreconstruction of the function ρs(x) from its positive-power moments µ(s)\n2nto be of\nthe form ρs(x) =xgs(x) where g(0)̸= 0 and has an entire complex extension g(z).\nAt the same time, we choose a reconstruction for g(x) that can simulate the lead-\ning large-order behavior of the expansion coefficients, a(s)\nn∼(2n)!. This is done by\nexpanding g(x) as a generalized Fourier series expansion in terms of the LaguerreGENERALIZED STIELTJES TRANSFORM 17\nMoments κ= 1 κ= 4\n200 0 .020901957 + i0.01366231 0 .1021807 + i0.25054011\n500 0 .020943108 + i0.013610668 0 .1022391 + i0.25203733\n800 0 .020942796 + i0.013610094 0 .1022458 + i0.25205037\n1000 0 .020943004 + i0.013609780 0 .1022316 + i0.25205760\n1500 0 .020942962 + i0.013609607 0 .1022266 + i0.25206534\n2000 0 .020942968 + i0.013609603 0 .1022263 + i0.25206487\nP999\n1000(κ) 0.040044268 −0.296120\nP49\n50(κ) 0 .034964576 0 .2379596\nδ499(κ) 0 .012249109 0 .2716414\nδ50(κ) 0 .029247263 1 .6506276\nExact 0 .020942969 + i0.013609598 0 .1022258 + i0.25206464\nMoments κ= 0.2\n200 9 .1916603636(10−4) +i5.619833452(10−5)\n500 9 .1952503862(10−4) +i5.662175099(10−5)\n800 9 .1952378811(10−4) +i5.661776998(10−5)\n1000 9 .1952367536(10−4) +i5.661682562(10−5)\n1500 9 .1952336756(10−4) +i5.661679405(10−5)\n2000 9 .1952336115(10−4) +i5.661679403(10−5)\nP999\n1000(κ) 9.6404911601(10−4)\nP49\n50(κ) 1 .0211638800(10−3)\nδ499(κ) 1 .0414341101(10−3)\nδ50(κ) 1 .0248313456(10−3)\nExact 9 .1952336091(10−4) +i5.661679704(10−5)\nMoments κ= 10−2\n200 1 .95700974450299352344(10−6) +i1.63041311(10−13)\n500 1 .95700965990567802111(10−6) +i1.44973855(10−16)\n800 1 .95700965992664090074(10−6) +i7.13212804(10−17)\n1000 1 .95700965992690633791(10−6) +i7.23822515(10−17)\n1500 1 .95700965992690633791(10−6) +i7.22914778(10−17)\n2000 1 .95700965992690634640(10−6) +i7.22913837(10−17)\nP999\n1000(κ) 1.95700965992575734245(10−6)\nP49\n50(κ) 1 .95700965279330838602(10−6)\nδ499(κ) 1 .95700965958823661533(10−6)\nδ50(κ) 1 .95700965988694044016(10−6)\nExact 1 .95700965992690635033(10−6) +i7.22913923(10−17)\nTable 2: Convergence of the extrapolant (85) constructed from the divergent ex-\npansion (60) for f0(κ).PN\nM(κ) is a Pade approximant constructed using N+M+ 1\nof the coefficients of the divergent expansion (60) at 3000-digit working precision\nwhile δn(κ) from [6, eq 4] is constructed using n+ 1 coefficients. The exact result\nis computed from the closed-form (43).\npolynomials, Lm(x) [58]\n(86) gs(x) =∞X\nm=0c(s)\nmψm(x), ψm(x) =e−x/2Lm(x).18 GENERALIZED STIELTJES TRANSFORM\nMoments κ= 1017κ= 1019\n200 −3.08858(1017) +i2.70604(1016)−3.4853(1019) +i2.70604(1018)\n500 −2.99011(1017) +i2.61264(1016)−3.3731(1019) +i2.61264(1018)\n800 −2.98975(1017) +i2.61232(1016)−3.3727(1019) +i2.61232(1018)\n1000 −2.99338(1017) +i2.61588(1016)−3.3768(1019) +i2.61588(1018)\n1500 −2.99637(1017) +i2.61883(1016)−3.3803(1019) +i2.61883(1018)\n2000 −2.99599(1017) +i2.61845(1016)−3.3798(1019) +i2.61845(1018)\nP999\n1000(κ)−1.5151(1016) −1.5151(1018)\nP49\n50(κ)−1.0723(1016) −1.0723(1018)\nδ499(κ) 1 .013(1025) 1 .013(1029)\nδ50(κ)−4.9399(1026) −4.9399(1030)\nExact −2.99554(1017) +i2.61799(1016)−3.3793(1019) +i2.61799(1018)\nMoments κ= 108κ= 1012\n200 −1.30356(108) +i2.70526(107)−2.0969(1012) +i2.70604(1011)\n500 −1.26671(108) +i2.61196(107)−2.0327(1012) +i2.61264(1011)\n800 −1.26656(108) +i2.61163(107)−2.0324(1012) +i2.61232(1011)\n1000 −1.26784(108) +i2.61519(107)−2.0348(1012) +i2.61588(1011)\n1500 −1.26888(108) +i2.61814(107)−2.0367(1012) +i2.61883(1011)\n2000 −1.26875(108) +i2.61776(107)−2.0364(1012) +i2.61845(1011)\nP999\n1000(κ)−1.5151(106) −1.5151(1011)\nP49\n50(κ)−1.0723(107) −1.0723(1011)\nδ499(κ)−7.4396(107) 1 .012(1015)\nδ50(κ)−5.3991(108) −4.93990(1016)\nExact −1.26859(108) +i2.61730(107)−2.0361(1012) +i2.61799(1011)\nMoments κ= 10 κ= 100 κ= 103\n200 0 .046028 + i1.080516 −12.8842 + i20.29965 −314.188 + i247.16\n500 0 .038571 + i1.084611 −13.0081 + i19.98976 −311.021 + i240.37\n800 0 .038509 + i1.084658 −13.0086 + i19.98748 −310.996 + i240.33\n1000 0 .038462 + i1.084547 −13.0022 + i19.99103 −311.018 + i240.53\n1500 0 .038408 + i1.084494 −12.9973 + i19.99268 −311.021 + i240.69\n2000 0 .038414 + i1.084493 −12.9977 + i19.99272 −311.024 + i240.67\nP999\n1000(κ) 0.948117 −18.2559 −154.067\nP49\n50(κ)−13.0793 −11.7271 −108.15\nδ499(κ)−0.59973 −11.7680 7542 .32\nδ50(κ)−4.67879 207 .549 −9595.11\nExact 0 .038419 + i1.084491 −12.9983 + i19.99284 −311.028 + i240.65\nTable 3: Convergence of the extrapolant (85) for f0(κ) in the Heisenberg-Euler La-\ngrangian for scalar QED in the strong-field regimes. PN\nM(κ) is a Pad´ e approximant\nconstructed from the divergent expansion (60) at 3000-digit precision. δn(κ) is from\n[6, eq 4]. The exact result is computed from the closed-form (43).\nThe basis functions ψm(x) obey the orthonormality relation [59],\n(87)Z∞\n0\u0010\ne−x/2Lm(x)\u0011 \u0010\ne−x/2Ln(x)\u0011\ndx=δn,mGENERALIZED STIELTJES TRANSFORM 19\nMoments κ= 10 κ= 100 κ= 103\n200 0 .19071 + i1.34160 15 .487 + i28.28061 −421.94 + i358.26\n500 0 .22397 + i1.32149 14 .876 + i29.713812 −436.49 + i390.45\n800 0 .22501 + i1.31358 14 .573 + i29.997207 −438.88 + i400.86\n1000 0 .22414 + i1.31153 14 .455 + i30.063048 −439.29 + i404.59\n1500 0 .22225 + i1.30998 14 .287 + i30.110882 −439.28 + i409.83\n2000 0 .22129 + i1.30993 14 .204 + i30.109188 −438.88 + i412.54\nP999\n1000(κ) 1.07421 −25.98 −212.91\nP49\n50(κ) 32 .5797 −15.902 −144.89\nδ499(κ)−0.550 −12.260 13947 .7\nδ50(κ)−5.439 285 .07 3 .34(105)\nExact 0 .22085 + i1.31070 14 .104 + i30.009628 −434.46 + i418.17\nMoments κ= 1 κ= 4\n200 0 .024395556 + i0.013677328 0 .152770 + i0.286401\n500 0 .024230912 + i0.013910335 0 .152166 + i0.279692\n800 0 .024264476 + i0.013916726 0 .151036 + i0.279399\n1000 0 .024268321 + i0.013910915 0 .150773 + i0.279533\n1500 0 .024266191 + i0.013906063 0 .150646 + i0.279794\n2000 0 .024264845 + i0.013906401 0 .150688 + i0.279886\nP999\n1000(κ) 0.041389591 −0.41360\nP49\n50(κ) 0 .036873788 0 .265364\nδ499(κ) 0 .015381186 0 .153812\nδ50(κ) 0 .032749453 1 .877584\nExact 0 .024264996 + i0.013906949 0 .150756 + i0.279879\nMoments κ= 0.1 κ= 0.2\n200 2 .38937659(10−4) +i1.383679(10−6) 1.03560108(10−3) +i5.82982(10−5)\n500 2 .38980630(10−4) +i1.541857(10−6) 1.03412967(10−3) +i5.66067(10−5)\n800 2 .38975131(10−4) +i1.542869(10−6) 1.03412967(10−3) +i5.66579(10−5)\n1000 2 .38975709(10−4) +i1.543011(10−6) 1.03424568(10−3) +i5.66405(10−5)\n1500 2 .38975688(10−4) +i1.542813(10−6) 1.03423770(10−3) +i5.66416(10−5)\n2000 2 .38975690(10−4) +i1.542835(10−6) 1.03423876(10−3) +i5.66422(10−5)\nP999\n1000(κ) 2.39223743(10−4) 1 .07642628(10−3)\nP49\n50(κ) 2 .39255275(10−4) 1 .12984713(10−3)\nδ499(κ) 2 .40535408(10−4) 1 .15620371(10−3)\nδ50(κ) 2 .39892152(10−4) 1 .13959410(10−3)\nExact 2 .38975692(10−4) +i1.542832(10−6) 1.03423873(10−3) +i5.66420(10−5)\nTable 4: Convergence of the extrapolant (85) constructed from the divergent ex-\npansion (64) for f1\n2(κ).PN\nM(κ) is a Pade approximant constructed using N+M+1\nof the coefficients of the divergent expansion (64) at 3000-digit working precision\nwhile δn(κ) from [6, eq 4] is constructed using n+ 1 coefficients. The exact result\nis computed from the closed-form (50).\nand the Laguerre polynomials are given by [60],\n(88) Lm(x) =mX\nk=0(−m)kxk\n(k!)2=m!mX\nk=0(−x)k\n(k!)2(m−k)!.20 GENERALIZED STIELTJES TRANSFORM\nSo that the reconstruction of gs(x) takes the form,\ngs(x) =e−x/2∞X\nm=0c(s)\nmm!mX\nk=0(−x)k\n(k!)2(m−k)!, (89)\nThe first d+1 expansion coefficients c(s)\nmare then computed by imposing the condi-\ntion (68). This results to a system of linear equations for the first d+ 1 coefficients\na(s)\nn+2=dX\nm=0c(s)\nmP(n, m) (90)\nwhere the matrix is given by\n(91) P(n, m) =m!22n+2mX\nk=0(−2)k(2n+k+ 1)!\n(k!)2(m−k)!.\nWe then substitute the reconstruction ρs(x) =xgs(x) where g(x) is given in (89)\nto the expansion (85) so that the first term evaluates to\n∞X\nk=0µ(s)\n−(2n+2)\nκn−1=⌊d−1\n2⌋X\nn=01\nκn−1(In+Jn+Ln) +∞X\nn=⌊d−1\n2⌋+1Mn\nκn−1, (92)\nwhere ⌊x⌋is the floor function and\n(93) In=2nX\nm=0c(s)\nmm!mX\nl=0(−1)l\n(l!)2(m−l)!\\ \\Z∞\n0e−x/2\nx2n+1−ldx,\n(94) Jn=dX\nm=2n+1c(s)\nmm!2nX\nl=0(−1)l\n(l!)2(m−l)!\\ \\Z∞\n0e−x/2\nx2n+1−ldx,\n(95) Ln=dX\nm=2n+1c(s)\nmm!mX\nl=2n+1(−1)l(l−2n−1)! 2l−2n\n(l!)2(m−l)!,\nand\n(96) Mn=dX\nm=0c(s)\nmm!mX\nl=0(−1)l\n(l!)2(m−l)!\\ \\Z∞\n0e−x/2\nx2n+1−ldx.\nThe finite part integrals appearing in theses terms are given by equation (24)\n(97) \\ \\Z∞\n0e−x/2\nx2n+1−ldx=(−1)1−l\u00001\n2\u00012n−l\n(2n−l)!\u0012\nln\u00121\n2\u0013\n−ψ(2n+ 1−l)\u0013\n.\nThe convergence of the extrapolant (85) across a wide range of field strengths\nis summarized in tables 2 and 3 for the spin-0 case and in table 4 for spin-1\n2. The\nresult presented along each row is computed by adding up to k= 2dterms of\nthe convergent expansion in equation (92), where d+ 1 is the number of moments\nused. As a rule, the working precision in digits at which we carry out the compu-\ntation equals the number of moments, a(s)\nn+2, used as inputs in the system of linear\nequations (90).GENERALIZED STIELTJES TRANSFORM 21\nWe also included the results of the Pad´ e approximants, PN\nM(κ), computed from\nthe non-alternating divergent expansions (60) and (64) using N+M+ 1 expan-\nsion coefficients and the non-linear sequence transformation δn(κ) from [6, eq 4]\nconstructed using n+ 1 coefficients. Regardless of the orders at which these ap-\nproximants are constructed, both become unreliable at summing and extrapolating\nthe non-alternating weak-field expansions for the Heisenberg-Euler Lagrangian be-\nyond perturbations of order κ= 10−1. This can be traced to the inability of these\nmethods to incorporate the precise logarithmic strong-field behaviors (67) of the\nHeisenberg-Euler Lagrangian. The Pad´ e approximant for instance exhibits an in-\nteger power leading-order behavior, PN\nM(κ)∼κN−Masκ→ ∞ . More importantly,\non their own, they do not offer any means by which the non-perturbative imaginary\nparts of the Heisenberg-Euler Lagrangians can be recovered from a finite collection\nof the real coefficients of the corresponding divergent expansion.\n4.1.Self-dual electromagnetic background. The rotation, iτ=t, in the in-\ntegral fSD(κ) in the representation (54) for the case of the electric-like self-dual\nbackground leads to the formal representation,\nfSD(κ) =1\n4Z∞\n0dt\nt\u00121\nsin2t−1\nt2−1\n3\u0013\ne−2t/√κ,√κ=2eF\nm2(98)\n∼κ∞X\nn=0−(−1)nB2n+4\n(2n+ 2)(2 n+ 4)κn, κ →0. (99)\nwhich is just the non-alternating version of the expansion (53) for a magnetic-like\nbackground. The coefficients also exhibit a (2 n)! large-order growth. As in the\nprevious cases, the imaginary part of fSD(κ) comes from the contributions of the\npoles along the real line in the formal representation (98). This is obtained using\nthe method of residue and the result is given by [53, eq 2.22]\nImfSD(κ) =1\n4π√κ∞X\nn=1\u00122π\nn+√κ\nn2\u0013\nexp\u0012\n−2πn√κ\u0013\n. (100)\nIn the strong field-limit, Re fSD(κ) exhibits the following strong field behavior,\nRefSD(κ)∼lnκ, κ → ∞ . (101)\nAs in the previous cases, we recover the complex-valued function fSD(κ) from\nthe divergent non-alternating PT expansion (99) by mapping the first d+ 1 PT\ncoefficients in (99) into the positive-power moments, µ2n, of some positive function\nρ(x),\n(102)(−1)n+1B2n+4\n(2n+ 2)(2 n+ 4)=µ2n=Z∞\n0x2nρ(x)dx, n = 0,1, . . . , d\nSo that we sum the PT expansion (99) formally as\nfSD(κ) =κZ∞\n0ρ(x)\n1−κx2dx=H(κ), (103)\nso that from equations (71) and (72),\nfSD(κ) =−∞X\nn=0µ−(2n+2)\nκn−∆(κ) +iπ\n2√κρ\u00121√κ\u0013\n. (104)22 GENERALIZED STIELTJES TRANSFORM\nMoments κ= 105κ= 1014\n200 −0.31855198 + i0.132501 −1.204100 + i0.1342523\n500 −0.32024269 + i0.132899 −1.109493 + i0.1346100\n800 −0.31260684 + i0.128091 −1.167379 + i0.1295846\n1000 −0.31251296 + i0.128031 −1.166875 + i0.1295226\n1500 −0.31414365 + i0.129213 −1.176806 + i0.1307814\n2000 −0.31453749 + i0.129518 −1.179359 + i0.1311091\nP999\n1000(κ)−0.089304411 −0.089257\nP49\n50(κ)−0.063486267 −0.063470\nδ499(κ)−0.287217551 −1.16(104)\nδ50(κ)−0.253811651 −9.77(106)\nExact −0.314312480 + i0.129326 −1.177753 + i0.1308996\nMoments κ= 10 κ= 100 κ= 103\n200 0 .0126095 + i0.0342766 −0.036638 + i0.0882078 −0.123834 + i0.11768\n500 0 .0130044 + i0.0344001 −0.036929 + i0.089046 −0.124863 + i0.11835\n800 0 .0129595 + i0.0346459 −0.037837 + i0.088152 −0.124096 + i0.11518\n1000 0 .0129595 + i0.0346490 −0.037849 + i0.088141 −0.124088 + i0.11515\n1500 0 .0129362 + i0.0346304 −0.037643 + i0.088198 −0.124084 + i0.11579\n2000 0 .0129308 + i0.0346297 −0.037597 + i0.088199 −0.124066 + i0.11593\nP999\n1000(κ) 0.1619292 −0.219567 −0.094266\nP49\n50(κ) 0 .0162103 −0.086270 −0.065150\nδ499(κ) 0 .0393482 −0.050233 −0.179900\nδ50(κ) 0 .0101395 −0.214682 −0.086707\nExact 0 .0129323 + i0.0346283 −0.037618 + i0.0882075 −0.124091 + i0.11586\nMoments κ= 1 κ= 4\n200 0 .00543906 + i0.001079397 0 .01392382 + i0.014651352\n500 0 .00537742 + i0.001086957 0 .01408884 + i0.014482129\n800 0 .00536667 + i0.001083803 0 .01417685 + i0.014511208\n1000 0 .00536653 + i0.001083856 0 .01417803 + i0.014511206\n1500 0 .00536625 + i0.001083278 0 .01417434 + i0.014519223\n2000 0 .00536617 + i0.001083284 0 .01417479 + i0.014520612\nP999\n1000(κ) 0.00527042 0 .12226761\nP49\n50(κ) 0 .00490116 0 .06311474\nδ499(κ) 0 .03133799 0 .01689188\nδ50(κ) 0 .00611339 0 .05512124\nExact 0 .00536616 + i0.001083270 0 .01417435 + i0.014520598\nTable 5: Convergence of the extrapolant (104) constructed from the divergent ex-\npansion (99) for fSD(κ) in the Heisenberg-Euler Lagrangian for the case of a self-\ndual background. PN\nM(κ) is a Pade approximant constructed from the divergent\nexpansion (99) at 3000-digit precision. While is δn(κ) from [6, eq 4]. The exact\nresult is computed from the closed-form (58).\nWe again accommodate the leading-order behavior (101) through the term ∆( κ)\nby requiring the reconstruction of the positive function ρ(x) in the form ρ(x) =\nxg(x) where g(x) is again given by (89) and the reconstruction scheme proceeds as\nin the previous cases. The first term in the right-hand side of the expansion (104),\nis computed in exactly the same way as in equation (92) in the previous examples.GENERALIZED STIELTJES TRANSFORM 23\nThe convergence of the extrapolant (104) is summarized in table (5) across a wide\nrange of field strengths. As in the previous cases, (104) successfully reproduces the\nfirst several digits of the complex-valued Heisenberg-Euler Lagrangian from the real\ncoefficients of the nonalternating divergent weak field expansion (99) well into the\nstrong-field limit. Both approximants PN\nM(κ), and δn(κ) also fail at summing or\nextrapolating the divergent non-alternating expansion (99).\n5.Conclusion\nIn this paper, we proposed a prescription based on the method of finite-part\nintegration to sum divergent PT series expansions with coefficients that we map\nto the positive-power moments, µ2n=R∞\n0x2nρ(x)dx, of some positive function\nρ(x). We applied the summation and extrapolation procedure on the divergent\nnon-alternating weak-field expansions of the exact integral representations for var-\nious Heisenberg-Euler Lagrangians from QED for electric and electric-like self-dual\nbackground. In each of these examples, the procedure allowed us to transform the\nweak-field expansion into a novel convergent expansion in inverse powers of per-\nturbation parameter plus a correction term that led us to incorporate the known\nlogarithmic leading-order behavior in the strong-field regime. The prescription also\nrecovered the non-perturbative imaginary parts from the real coefficients of the di-\nvergent expansions. This enabled us to construct extrapolants which can be used\nacross a wide range of values for the field strength with considerable accuracy. Fur-\nthermore, we also showed how the method of finite-part integration can be used\nto evaluate in closed-form the exact integral representations of the complex-valued\nHeisenberg-Euler Lagrangians.\nThe second and third terms in the extrapolants (85) which simulate the leading\nbehaviours (67) contain the sampling of the reconstruction ρ(x) near the origin as\nκ→ ∞ . In this regard, the resummation prescription might be improved by using\na more parsimonious prescription for solving the underlying Stieltjes moment prob-\nlem with a better point-wise convergence near the origin. A promising alternative\nproposed in [62] is an information-theoretic approach and is based on the maxi-\nmization of the Shannon entropy functional. This could enhance the applicability\nof the resummation procedure on problems with only a few accessible perturbative\ncoefficients.\nAcknowledgments\nWe acknowledge the Computing and Archiving Research Environment (COARE)\nof the Department of Science and Technology’s Advanced Science and Technology\nInstitute (DOST-ASTI) for providing access to their High-Performance Computing\n(HPC) facility. This work is funded by the University of the Philippines System\nthrough the Enhanced Creative Work Research Grant (ECWRG 2019-05-R). 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A. Gómez-Muñoz1,2, D. A. García-Hernández1,2, R. Barzaga1,2, A. Manchado1,2,3, and T. Huertas-Roldán1,2\n1Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain\ne-mail: magm@iac.es, agarcia@iac.es\n2Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain\n3Consejo Superior de Investigaciones Científicas (CSIC), 28006 Madrid, Spain\nReceived 22 December 2023; Accepted 05 February 2024\nABSTRACT\nFullerenes have been observed in several astronomical objects since the discovery of C 60in the mid-infrared (mid-IR) spectrum of\nthe planetary nebula (PN) Tc 1. It has been suggested that the carriers of the broad unidentified infrared (UIR) plateau features, such\nas the 9−13µm emission feature (12 µm hereafter), may be related to the formation of fullerenes. In particular, their carriers have\nbeen suggested to be mixed aromatic or aliphatic hydrocarbons such as hydrogenated amorphous carbon (HAC-like hereafter) grains.\nFor this study, we modeled the mid-IR emission of the C 60-PN Tc 1 with a photoionization code, including for the first time the\nlaboratory optical constants ( nandkindices) of HAC-like dust at 300 K. Interestingly, we find that the broad 12 µm plateau feature\nin Tc 1 is well reproduced by using a distribution of canonical HAC grains, while at the same time they provide an important fraction\nof the IR dust continuum emission and are consistent with the other UIR features observed (e.g., the broad 6 −9µm plateau feature).\nThis finding suggests that HAC-like grains may be possible carriers of the 12 µm plateau feature, being likely related to the fullerene\nformation mechanism in PNe. More laboratory experiments, to obtain the optical constants of HAC-like dust with several structures\nor a composition at di fferent physical conditions, are strongly encouraged – that is, in order to extend this pilot study to more fullerene\nPNe, and to unveil the details of fullerene formation and of the potential carriers of the elusive UIR plateau features.\nKey words. astrochemistry – circumstellar matter – infrared: stars — planetary nebulae: general – stars: AGB and post-AGB\n1. Introduction\nPlanetary nebulae (PNe) represent the late stages in the evolu-\ntion of low- and intermediate-mass stars (LIM; ∼1−8 M⊙), the\nmajority of stars in the Universe. On their way to the PN phase,\nLIM stars experience a strong mass loss during the preceding\nasymptotic giant branch (AGB; see e.g., Herwig 2005, for a re-\nview) phase and they chemically enrich the surrounding inter-\nstellar medium (ISM). The AGB stars are also major suppliers\nof dust grains and molecular species that are routinely seen in\nthe local Universe (e.g., Ferrarotti & Gail 2006), which makes\nthem (and their subsequent evolutionary stages) very important\nfor our fundamental understanding of the enrichment and chem-\nical composition of the ISM.\nFullerenes, among the most resistant and stable three-\ndimensional molecules that are only formed by C atoms, such as\nC60(first discovered in the laboratory by Kroto et al. 1985), were\nsupposed to be widely spread in space for decades. The presence\nof fullerenes in astrophysical environments was under debate un-\ntil the infrared (IR) signatures of the C 60and C 70fullerenes were\nunambiguously detected in the Spitzer mid-IR spectrum of the\nyoung PN Tc 1 (Cami et al. 2010). After the first detection of\nfullerenes in a PN, the presence of C 60was found in several other\nastrophysical objects such as reflection nebulae (Sellgren et al.\n2010), additional PNe (García-Hernández et al. 2010, 2011a,\n2012; Otsuka et al. 2014), peculiar R Coronae Borealis (RCB)\nstars (García-Hernández et al. 2011b), post-AGB stars (Zhang &\nKwok 2011; Gielen et al. 2011), and Herbig Ae /Be stars (Arun\net al. 2023), among others, by its four strongest mid-IR emissionfeatures at∼7.0, 8.5, 17.4, and 18.9 µm. However, the formation\nroute of fullerenes in such H-rich objects, similar to the major-\nity of PNe, is not well understood. Nowadays, the most suitable\nroutes for the formation of fullerenes are1: i) the photochemi-\ncal processing of hydrogenated amorphous carbon (HAC) grains\nor similar mixed aromatic or aliphatic hydrocarbons (i.e., same\nchemical composition but a di fferent internal structure; here-\nafter HAC-like; García-Hernández et al. 2010) and ii) the photo-\nchemical processing of large polycyclic aromatic hydrocarbons\n(PAHs; Berné & Tielens 2012; Murga et al. 2022).\nThe C 60fullerenes are mainly detected toward PNe, and\nfullerene PNe are very young low-excitation ( Teff∼30 000–\n45 000 K) C-rich objects, which evolved from low-mass\n(∼1−3 M⊙) progenitors (García-Hernández et al. 2012; Otsuka\net al. 2014). Remarkably, the IR Spitzer spectra ( ∼5-38µm) of\nthe fullerene-rich PNe are dominated by aliphatic hydrocarbon-\nrich dust emission features (superimposed on the underlying and\nfeatureless dust continuum emission), showing several broad\nunidentified IR (UIR) plateau emission features at 6 −9 (here-\nafter 7µm), 9−13 (hereafter 12 µm), 15−20, and 25−35µm (e.g.,\nGarcía-Hernández et al. 2012). Characteristic aliphatic discrete\n1We note that, more recently, Bernal et al. (2019) proposed the shock\nheating and ion bombardment induced processing of SiC grains as an al-\nternative route toward fullerenes in the ISM, but such a strong energetic\nprocess seems to be unlikely to happen in the circumstellar environ-\nments of fullerene PNe where the atomic nebular emission lines can be\nwell explained by photoionization and shocks seem to be unimportant\nfor fullerene formation (see e.g., García-Hernández et al. 2012).\nArticle number, page 1 of 7arXiv:2402.14943v1  [astro-ph.SR]  22 Feb 2024A&A proofs: manuscript no. main\nfeatures are located at 3.4 and 6.9 µm, arising from symmetric\nand asymmetric C-H stretching and bending modes of methyl\nand methylene groups attached to aromatic rings, respectively\n(see e.g., Kwok & Zhang 2011). The identification of the carri-\ners of the broad (and discrete) UIR emission features widely ob-\nserved in space (toward Solar System bodies, circumstellar en-\nvelopes of PNe, di ffuse ISM, and remote galaxies, among others)\nis a long-standing problem in astrophysics (see e.g., Kwok 2016,\nfor a review). Thus, the potential identification of the carrier(s)\nof any of the broad UIR plateau emission features in fullerene\nPNe would impact very di fferent astronomical fields.\nNano- and micro-sized particles in the ISM and circumstellar\nenvironment play an important role in astrophysical processes\nthat lead to the formation of such exotic UIR emissions seen\nin the spectra of C-rich PNe and other astrophysical objects.\nThe optical constants, which are the complex refractive indices\nor the complex dielectric function, of di fferent materials (e.g.,\ngraphite, silicates, SiC, MgS, and HAC) are commonly used, in\nconjunction with radiative transfer codes, to explain the origin of\nthe IR emission in astronomical spectra (e.g., HACs, soots, SiC,\nand other hydrocarbon materials; see Bernard-Salas et al. 2012;\nGavilan et al. 2016; Gómez-Llanos et al. 2018; Dubosq et al.\n2023, respectively). Fortunately, the latter can be done thanks to\nthe databases2dedicated to compile optical properties of analog\nmaterials of cosmic dust in the wavelength range from the UV to\nthe far-IR.\nIn particular, the broad UIR plateau emission feature at\n12µm was first observed in the low resolution spectrometer of\nthe IR astronomical satellite (LRS IRAS) spectra by Cohen et al.\n(1985), and its carrier’s identification has been a long outstand-\ning problem. Cohen et al. (1985) attributed the carrier to PAHs,\nwhile Blanco et al. (1988) and Buss et al. (1990) attributed both\nthe 7 and 12 µm plateau features to a mixture of large PAHs and\nHAC grains. Later, Kwok et al. (2001) specifically suggested that\nthe 7 and 12 µm plateaus are superpositions of in-plane and out-\nof-plane bending modes of aliphatic side groups. Despite these\nearlier suggestions, the broad UIR emission at 12 µm has been\ngenerally attributed, in the literature, to SiC ( α-SiC; e.g., Speck\net al. 2009; Jones et al. 2023). García-Hernández (2012) argued\nagainst the SiC identification of this feature in fullerene PNe be-\ncause the spectral characteristics (position and shape) (see Table\n4 in García-Hernández et al. 2012) of the 9 −13µm emission in\nC60-PNe significantly di ffer from those of the SiC 11.5 µm fea-\nture seen in the preceding AGB phase. This may suggest that the\ncarrier could be di fferent in AGBs and in fullerene PNe or even\nthat the identification of SiC in AGB stars could be revisited.\nThe PN Tc 1 is the prototypical fullerene-rich PN (e.g., Cami\net al. 2010; Aleman et al. 2019), being one of the best objects\nto study because of its simple round-elliptical morphology, the\navailable high-quality and high-resolution IR Spitzer spectra,\nand the lack of PAH emission at 6.2, 7.7, 8.6, and 11.3 µm (Cami\net al. 2010), which could strongly contribute to the fullerene and\nbroad (and discrete) UIR emission features. In this Letter, we\npresent an analysis of the high-quality Spitzer spectrum of PN\nTc 1 in comparison with a photoionization model that includes,\nfor the first time, the emission of HAC-like dust grains as mea-\nsured in laboratory, showing that HAC-like grains are a possible\nexplanation, di fferent from the often assumed SiC, for the broad\n12µm plateau emission feature.\n2For example, the Jena database ( https://www2.mpia-hd.mpg.\nde/HJPDOC/index.php ) and the refractive index database ( https:\n//refractiveindex.info/ ).2. Infrared spectrum of the C 60-PN Tc 1\nWe downloaded all the available mid-IR spectra from the\nSpitzer IR Spectrograph (IRS) (Werner et al. 2004; Houck\net al. 2004) Heritage archive of the PN Tc 1 for which high-\nquality – a signal-to-noise ratio (S /N)>50 – short-low (SL;\n5.2−14.5µm), short-high (SH; 9.9 −19.6µm), and long-high\n(LH; 18.7−37.2µm) spectra were available (Program Identifier\n3633) (see below and Perea-Calderón et al. 2009, for more data\nreduction details). We used the package smart3(v8.2.9; Higdon\net al. 2004) to process the extracted wavelength and flux cali-\nbrated 1D spectra for each nod position, that is, to clean for bad\ndata points, spurious jumps, and glitches and to combine and\nmerge into one single spectrum per module (SL, SH, and LH). In\norder to get the final merged spectrum, we scaled the flux density\nof the SH spectra to match the overlapping wavelength region in\nthe LH. Then, we scaled the SL spectra to match the overlap-\nping wavelength region to the SH spectra. Finally, we scaled the\nflux density of the resulted combined spectra to that of the Wide-\nfield IR Survey Explorer (WISE) band 4 ( λc=22µm) photomet-\nric value (similarly to what was done in Otsuka et al. 2014). The\nresulting final Tc 1 Spitzer IR spectrum is shown in Fig. 1, where\nthe emission features such as the C 60and C 70IR bands, and the\nbroad 7 and 12 µm plateau emission features are indicated.\n3. Cloudy simulations\nTo the best of our knowledge, the only previous attempt to model\nthe 7 and 12 µm plateau emission features seen in the Spitzer\nspectrum of the C 60-PN Tc 1 (Fig. 1) was made by Bernard-Salas\net al. (2012). They used the Jones (2012) theoretical absorp-\ntion coe fficients ( Qabs) of 3 nm HAC particles with a di fferent H\ncontent (H /C=0.23, 0.29, and 0.35) in conjunction with a 200 K\nblackbody (see Fig.7 in Bernard-Salas et al. 2012). They could\nreproduce the 7 µm plateau emission feature – as seen in the Tc\n1 continuum-subtracted Spitzer spectrum – with some degree of\nsuccess, depending on the atom hydrogen fractions of the HAC\nparticles. However, they could not reproduce the Tc 1 12 µm\nplateau emission feature. In particular, a too strong 11.3 µm fea-\nture was theoretically predicted for all H /C ratios, and they did\nnot study or analyze the HAC contribution to the IR dust contin-\nuum emission. Bernard-Salas et al. (2012) concluded that their\napproximate reproduction of the 7 µm plateau emission feature\nmay imply that fullerenes are formed by photochemical process-\ning of HACs. Below we explain how we used, for the first time,\na detailed photoionization model of Tc 1 together with the labo-\nratory optical constants of HAC-like dust in order to model the\n7 and 12µm plateau features and the underlying dust continuum\nemission.\n3.1. Modeling the 12 µm plateau emission\nWe used the stellar parameters and nebular abundances predicted\nin the detailed photoionization model developed by Aleman et al.\n(2019)4(without graphite as the IR dust continuum source; see\nbelow) as input in cloudy (v22.02; Ferland et al. 2017) to model\nthe mid-IR emission of the C 60-PN Tc 1 by using, for the first\n3SMART was developed by the IRS Team at Cornell University and\nis available through the Spitzer Science Center at Caltech.\n4Aleman et al. (2019) developed a detailed photoionization model to\nobtain accurate stellar and physical parameters of the star and the neb-\nula, respectively, and in which they assumed graphite in order to model\nthe IR dust continuum emission. It is available at https://github.\ncom/Morisset/Tc1\nArticle number, page 2 of 7Gómez-Muñoz et al.: HAC-like grains as a possible carrier of the 12 µm plateau emission\n6 8 10 12 14 16 18 20\nWavelength [m]\n0.000.250.500.751.001.251.501.752.00Flux [1017 W cm2 m1]\n7.0 m\n6 - 9 m\n12 m\n9 - 13 m\nC60\nC70PN T c 1\nCLOUDY model + HAC dust\nFig. 1. Spitzer mid-IR spectrum of the PN Tc 1 (black line) compared with the best photoionization model spectrum including HAC-like dust\ngrains (green line). The C 60(vertical dashed blue line) and C 70(vertical dashed red line) emission bands and UIR plateau emission features are\nindicated. We note that the narrow emission features seen in the best model spectrum (green) are just atomic nebular emission lines, whose detailed\nmodeling is out of the scope of the present work.\ntime, the laboratory optical constants of HAC-like dust. We note\nthat a change in the dust composition does not significantly\nchange the emission line and H βfluxes predicted in the Ale-\nman et al. model (a maximum flux di fference of∼5% is ob-\ntained), and so the stellar and nebular parameters and chemical\nabundances are una ffected. The laboratory optical constants, that\nis, the complex refractive indices nandkof HAC-like particles\nwere obtained from Prof. W. Duley (private communication; see\nDuley 1984, for a detailed description of the laboratory measure-\nments), covering a spectral range from 0.1 to 17 µm with a gap\nbetween 0.65 to 2.5 µm due to a limitation of the laboratory in-\nstrumentation to measure over this wavelength range (Fig. 2).\nThe former are the only available laboratory optical constants,\nprepared under simulated interstellar conditions, covering a suf-\nficient range from the UV to the mid-IR. The laboratory nandk\nvalues were measured in HAC thin films, ∼50−200 nm thick-\nness, deposited at 300 K and with slightly di fferent hydrogen\nconcentrations (H /C∼0.3–0.4; Duley et al. 1998; Duley & Hu\n2012). However, it should be noted that it is mostly the kindex\nthat is sensitive to the H content; this is because the concentra-\ntion of CH and CH nin the HAC samples determines the IR k. For\nsimplicity, we adopted the mean nandkvalues (presumably with\nH/C∼0.35), which are shown in Fig. 2. Because cloudy needs a\nminimum wavelength mesh ( ∼0.001–96µm), an extension of the\nnandkvalues to lower and upper wavelengths was carried out\nin a similar way as in Gómez-Llanos et al. (2018); that is, we\nextrapolated the nvalue to lower wavelengths on the basis of the\nBE1 amorphous carbon included in cloudy (Rouleau & Martin\n1991), and for kwe extrapolated using a power law of slope 2,\nwhereas for higher wavelengths we extrapolated nas a constant\nvalue and kusing a power law of slope −1. For the gap between\n0.65 to 2.5µm, a linear extrapolation was used.\nFor the model, we compiled the HAC optical constants with\nthegrains command inside cloudy , using a canonical ISM\ngrain distribution ( amin=0.001µm to amax=0.25µm; 1−250 nm5,\ndistributed in ten grain size logarithmic steps following a power\n5We note that equally good fits were obtained for di fferent grain size\ndistributions such as very small grains (VSG) or nanograins (1-15 nm),\nonly big grains (15-250 nm), or the selected canonical ISM grain distri-\nbution (1-250 nm).of−3.5), and generated the opacities (with the integrated Mie\ncode for spherical grains; see van Hoof et al. 2004) needed for\nthecloudy simulations. We note that cloudy takes into account\nthe molar mass of the HAC (12.36 g mol−1), the band gap en-\nergy ( Eex\ngap=1.27 eV; Duley & Hu 2012), and the density of\nthe material (1.5 g cm−3) (see Duley et al. 1998; Duley & Hu\n2012). A reduced χ2\nredminimization was carried out for a grid\nofcloudy models – by varying the HAC grains’ relative con-\ntent – to find a convergence between the model and the observed\nspectrum of Tc 1 between 10 and 14 µm (485 spectral points),\nyielding aχ2\nred≃1.6 assuming a 10% error in the observations.\nThis resulted in a C abundance (in units of log(C /H)) of−4.60\nfor the HAC-like grains, which is ∼3% of the total C abundance\n(gas+dust phase abundance; C ≃−3.08), a mean dust tempera-\nture across the nebula of ∼215 K, and a dust-to-gas (D /G) ratio\nof 2.1×10−4.\nFigure 1 also shows the best cloudy model (green line) ob-\ntained from the χ2\nredanalysis applied to the observed spectrum of\nTc 1. As it can be seen, the HAC-like grains at 300 K6provide a\nvery good fit to the broad 12 µm plateau feature in Tc 1, while at\nthe same time they fully reproduce the IR dust continuum emis-\nsion from 9 to 14 µm. For comparison, α-SiC dust grains were\nalso used to similarly model the mid-IR spectrum of Tc 1, which\nresulted in a worst fit to the 12 µm feature (see Appendix A).\nInterestingly, it turns out that HAC-like grains may be an alter-\nnative explanation of the broad 12 µm feature because it is very\nwell reproduced with a simple spherical grain size distribution\nusing the laboratory HAC optical constants at a deposition tem-\nperature close to the fullerene-dust temperature in Tc 1.\nIn addition, there is a small contribution to the observed 7 µm\nplateau feature, which is also clearly seen in the model spectrum.\nHowever, we still would need to include other emitters and /or an\nadditional dust component to reproduce the 7 µm plateau feature\n(see below) in addition to the shape and rise of the underlying\ndust continuum emission at 5 −9 and 14−17µm, respectively.\n6We note that this value is close to the circumstellar dust grains’ equi-\nlibrium temperature and the excitation temperature ( ∼330 K) of the C 60\nemission in Tc 1 as derived by Cami et al. (2010), who assumed that\nC60is attached to the dust grains.\nArticle number, page 3 of 7A&A proofs: manuscript no. main\n0.1 1.0 10\nWavelength [m]\n1.601.651.701.751.801.851.90n\n0.010.11.0\nk\nFig. 2. Laboratory average values of the complex refractive indices n\nandk(black and purple lines, respectively) of HAC-like particles at\n300 K and with a hydrogen concentration of H /C≃0.35 (Prof. W. Duley,\nprivate communication; see text for more details). The errors of the\ncomplex refractive indices were measured to be ∼20% (see e.g., Duley\n1984).\n3.2. Modeling the 12 µm plateau emission and dust\ncontinuum\nThus, in conjunction with the HAC-like grains as above, we\nadded graphite dust grains separately as a possible additional\ndust continuum component (as assumed by Aleman et al. (2019))\nand we again applied a χ2\nredanalysis to a grid of cloudy mod-\nels in order to find the best fit by varying the HAC grains and\ngraphite relative contents in the 9 to 16 µm range, yielding a\nvalue ofχ2\nred≃1.3. This resulted in C abundances of −3.74 and\n−4.90, which correspond to ∼18.7% and∼1.3% of the total C\nabundance (gas +dust phase) as well as D /G of 1.5×10−3and\n1.1×10−4for graphite and HAC grains, respectively.\nWe note that other carbon materials such as amorphous car-\nbon (AC), generally used in the literature to model the IR dust\ncontinuum emission, provide worst fits because AC marginally\ncontribute below 14 µm. Figure 3 displays the best photoioniza-\ntion model spectra for HAC-like grains and graphite, overplot-\nted on the Tc 1 Spitzer spectrum; the HAC-like and graphite IR\nemission contributions are also shown separately for compari-\nson. This way, we can reproduce the Tc 1 IR spectrum well from\n∼5 to 17µm (both the 12 µm plateau feature and the dust con-\ntinuum) with the HAC-like grains providing an important frac-\ntion (∼20-40%) of the dust continuum emission, as evaluated\nfrom the integrated IR fluxes due to only graphite and HACs\nwith respect to the total integrated IR continuum flux at three\ndifferent wavelengths (5.6, 9.6, and 15.2 µm) representative of\nthe dust continuum emission. Again, for comparison, α-SiC dust\ngrains in conjunction with graphite dust grains were also used\nto similarly model the mid-IR spectrum of Tc 1, resulting in a\nworst fit of the red wing of the broad 12 µm feature (see the Ap-\npendix A). Finally, we stress that preliminary photoionization\nmodels with HAC-like grains deposited at 300 K also provide\nacceptable good fits to the 12 µm plateau feature (and dust con-\ntinuum) in another fullerene PNe such as IC 418 and SMC 16,when playing with the model parameters (e.g., grain size and\nshape distribution; Gómez-Muñoz et al., in prep.).\n4. Astrophysical implications\nTo the best of our knowledge, this is the first time that HAC\ngrains convincingly reproduce the 12 µm plateau emission fea-\nture in a fullerene PN (Tc 1), posing doubts on the generally\naccepted SiC identification in the literature as the main carrier\nof this feature. Remarkably, the HAC grains naturally provide\nan important fraction of the IR dust continuum emission and are\nconsistent with the other UIR features observed such as the 7 µm\nplateau feature (see below). From our cloudy models, we deter-\nmined that∼1.3–3.0% (with and without graphite, respectively)\nof the C in Tc 1 is in the form of HAC-like dust. These values\nare comparable to the amount of C that is estimated to be in the\nform of fullerenes, such as C 60, in Tc 1 (∼1.5%; see Cami et al.\n2010).\nThe laboratory HAC-like grains (H /C∼0.35) used in our\nmodels (Duley & Hu 2012) mainly contain aliphatic car-\nbon chains (sp3content), which represent 72% (i.e., aro-\nmatic /aliphatic ratio of∼0.4) of the chemical composition, but\nalso polyyne (-C≡C-)nand cumulenic (-C =C-)nchains. Accord-\ning to Duley & Hu (2012), these HAC-like grains can have\nmore aromatic rings than aromatic chains. In fact, they sug-\ngest that above 10 µm, the features can be caused by a variety\nof vibrational modes associated with ring deformation and with\nCH ngroups (i.e., out-of-plane CH vibrations). In particular, the\nspectral range of the broad 12 µm plateau emission may con-\ntain the features produced by small aromatic rings (pentagons\nand hexagons) with side chains such as dimethylnaphthalene or\ndimethylphenanthrene, but also those produced by solo, duo CH,\nand CH ngroups as components of the hydrocarbon matrix (Du-\nley & Hu 2012). Clearly, the broadening of the 12 µm plateau\nemission is the result of this coexistence of small aromatic rings\nand hydrocarbon chains; that is, the mixed aromatic and aliphatic\nhydrocarbon composition inside the HAC-like material is the key\nbehind the broadening of the 12 µm feature. We remark that the\nonly other laboratory optical constants available to us correspond\nto those measured in HAC films deposited at a lower temperature\nof 77 K (Prof. W. Duley; priv. comm.) and with slightly higher H\nconcentrations (H /C∼0.5–0.6), which have fewer aromatic rings\nin its composition (Duley et al. 1998). The 12 µm plateau feature\nin Tc 1 cannot be reproduced with the 77 K HAC-like sample\n(i.e., it is absent; Gómez-Muñoz et al., in prep.), suggesting that\nthe aromatic ring /chain ratio, rather than the aromatic /aliphatic\nratio, is a key parameter for the formation of the 12 µm plateau\nfeature.\nThe presence of HAC-like grains in conjunction with\nfullerenes have been previously proposed in the literature as\nproof of the photochemical processing of HAC-like grains for\nfullerene formation (see e.g., García-Hernández et al. 2010,\n2011b,a, 2012; Bernard-Salas et al. 2012). Naturally, our find-\ning in Tc 1 suggests that the formation of fullerenes in PNe cir-\ncumstellar environments may be related to the photochemical\nprocessing of HAC-like grains. However, the only experimental\ndata on the decomposition of HAC-like grains are those from\nScott et al. (1997), where they performed laser ablation (UV\nradiation-induced decomposition) of HAC films and found PAHs\nand fullerenes, among other species, but the chemical reactions\nare presently unknown. The lower aromatic content compared\nto more dehydrogenated hydrocarbons in our HAC sample sug-\ngest that the main process to generate the precursors to form C 60,\nfor example, the dehydrogenation of the HAC, is through radia-\nArticle number, page 4 of 7Gómez-Muñoz et al.: HAC-like grains as a possible carrier of the 12 µm plateau emission\n6 8 10 12 14 16 18 20\nWavelength [m]\n0.000.250.500.751.001.251.501.752.00Flux [1017 W cm2 m1]\nPN T c 1\nCLOUDY model + HAC+graphite dust\nCLOUDY model + graphite only\nCLOUDY model + HAC only\nFig. 3. Spitzer mid-IR spectrum of the PN Tc 1 (black line) compared with the best photoionization model spectra for HAC-like (dashed gray line)\nand graphite (dashed magenta line) dust grains separately. The best photoionization model including HAC-like grains in conjunction with graphite\ndust grains (solid green line) is also shown for comparison.\ntive processing as aliphatic hydrocarbons have been proven to\nbe efficiently dehydrogenated in the ISM by UV radiation (e.g.,\nMuñoz Caro et al. 2001). We speculate that the presence of some\nsmall aromatic rings in our 300 K HAC-like sample might act\nas templates or catalysts for the formation of more complex C n\nstructures when the material is subjected to photochemical pro-\ncessing. More sophisticated laboratory experiments on the pho-\ntochemical transformation of HAC-like grains at circumstellar\nconditions would be needed to unveil the details of fullerene for-\nmation via chemical reaction routes.\nAnother obvious implication of our work involves the car-\nriers of the 7 µm plateau emission feature seen in the fullerene\nPN Tc 1. Our findings also suggest that HAC-like grains, with\nH/C=0.35 and mainly composed by aliphatic carbon chains\n(72%; see above) with some small aromatic rings, are not a\nmain contributor to such a 7 µm emission complex and that other\nstill unidentified species should be the carriers. Fullerene-based\nspecies are natural candidate species as potential additional emit-\nters at these wavelengths in fullerene PNe. Very recent quantum-\nchemistry simulations of metallofullerenes (neutral and charged\nendohedral and exohedral species) do indeed demonstrate that\nthey strongly emit at 6 −9µm (Barzaga et al. 2023a,b; Hou et al.\n2023) with no significant contribution in the 9 −13µm spectral\nregion.\n5. Concluding remarks\nThis pilot work leads us to conclude that HAC-like grains are\na convincing alternative explanation (i.e., di fferent from the of-\nten assumed SiC) for the broad 12 µm plateau emission feature\nseen in the fullerene PN Tc 1. In addition, our work suggests that\nHAC-like grains may be possible carriers of the 12 µm plateau\nfeature, being likely related to the fullerene formation process\n(possibly as fullerene precursors) in the circumstellar environ-\nment of PNe.\nInterestingly, HAC-like grains can also reproduce the far-\nultraviolet (FUV) rise seen in the extinction curve (Gavilan et al.\n2016, 2017) and very recently detected also in the International\nUltraviolet Explorer (IUE) UV spectrum of Tc 1 (Gómez-Muñoz\net al. 2024). In this particular case, Gómez-Muñoz et al. (2024)\ncould reproduce the FUV rise toward Tc 1 very well by usingHAC-like nanograins (1-15 nm). Although other carbonaceous\nmaterials such as hydrogenated carbon clusters and large PAHs,\npossibly among others, may also explain the FUV rise in extinc-\ntion curves (see e.g., Dubosq et al. 2023; Lin et al. 2023); such\nan independent finding would be consistent with the results pre-\nsented here.\nYet, we do not fully understand the exact chemical pathways\nfrom large HAC-like dust grains toward the most stable fullerene\nmolecules as well as the origin of the other broad plateau emis-\nsion features seen in fullerene PNe, such as 6 −9, 15−20, and\n25−35µm. The James Webb Space Telescope o ffers the oppor-\ntunity to observe these broad features in fullerene PNe at a much\nhigher sensitivity and spectral resolution than Spitzer , permitting\nthe variations in their positions and shapes to be studied precisely\n(even resolving a possible substructure) and providing new ad-\nditional constraints for future modeling studies. Undoubtedly,\nmore laboratory e fforts are needed in order to definitively reveal\nthe details of fullerene formation in PNe circumstellar environ-\nments as well as the carriers of the elusive UIR plateau features.\nOur work strongly encourages more laboratory experiments to\nobtain the refractive indices nandk(over a large wavelength\nrange, from the UV to the far-IR) of HAC-like dust grains with\nseveral structures and a composition (e.g., H /C ratios) as well as\nat different physical conditions (e.g., temperatures or aromatic\nring/chain ratios). Such laboratory measurements (not presently\navailable) would be used to expand the pilot novel work pre-\nsented here for Tc 1 to a larger sample of fullerene PNe.\nAcknowledgements. We kindly acknowledge Prof. Walt Duley for supplying us\nwith the laboratory optical constants of HAC-like particles and Dr. Christophe\nMorisset for earlier work and discussions on the IR spectral modeling, in-\ncluding HAC-like dust, with the photoionization code cloudy . We acknowl-\nedge the support from the State Research Agency (AEI) of the Spanish Min-\nistry of Science and Innovation (MCIN) under grant PID2020-115758GB-\nI00/AEI/10.13039 /501100011033. This article is based upon work from Eu-\nropean Cooperation in Science and Technology (COST) Action NanoSpace,\nCA21126, supported by COST. This work is based on observations made with\nthe Spitzer Space Telescope, which was operated by the Jet Propulsion Labora-\ntory, California Institute of Technology under a contract with NASA.\nReferences\nAleman, I., Leal-Ferreira, M. 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J., et al. 2004, ApJS, 154, 1\nZhang, Y . & Kwok, S. 2011, ApJ, 730, 126\nArticle number, page 6 of 7Gómez-Muñoz et al.: HAC-like grains as a possible carrier of the 12 µm plateau emission\nAppendix A: Modeling the Tc 1 mid-IR spectrum\nusing α-SiC grains\nFor comparison, Fig. A.1 shows the mid-IR spectrum of Tc 1\ntogether with the best cloudy model spectrum obtained using\nα-SiC grains (as in Gómez-Llanos et al. 2018, for the case of\nthe PN IC 418) with the same shape (spherical) and grain size\ndistribution as when using HAC-like grains. As can be seen from\nthe figure, when using only α-SiC (top panel), the best model\nspectrum only fits the weak 11.3 µm feature seen in Tc 1 and not\nthe entire broad 12 µm feature. On the other hand, when using\nα-SiC+graphite (bottom panel), the dust continuum is very well\nfitted but the red wing of the broad 12 µm emission is poorly\nreproduced.\nWe remark that it is already well known that much better fits\nwithα-SiC can be obtained when playing with very di fferent ex-\ntreme shapes and grain size distributions (e.g., the case of PN\nIC 418; Gómez-Llanos et al. 2018) but here we are exploring if\nthere is any other contribution to or explanation for the broad\n12µm plateau feature in PNe with fullerene-dominated IR spec-\ntra such as Tc 1, that is, di fferent from the often assumed SiC.\nIt is important to mention that the 12 µm feature in IC 418 has\ndifferent characteristics (position, width, and emission strength)\nthan in Tc 1 (Otsuka et al. 2014). Gómez-Llanos et al. (2018)\nneeded to use very di fferent shapes (spheres and five extreme\nellipsoid types) and grain size distributions along the nebula in\norder to fit the red wing of the 12 µm feature with α-SiC. Also,\nthe IR spectrum of IC 418 is contaminated by other species (e.g.,\nPAHs), displaying weaker C 60bands and a di fferent fullerene\nand dust temperature than Tc 1.\n6 8 10 12 14 16 18 20\nWavelength [m]\n0.000.250.500.751.001.251.501.752.00Flux [1017 W cm2 m1]\n7.0 m\n6 - 9 m\n12 m\n10 - 13 m\nC60\nC70PN T c 1\nCLOUDY model + SiC dust\n6 8 10 12 14 16 18 20\nWavelength [m]\n0.000.250.500.751.001.251.501.752.00Flux [1017 W cm2 m1]\nPN T c 1\nCLOUDY model + SiC + graphite dustFig. A.1. Spitzer mid-IR spectrum of the PN Tc 1 (black line) com-\npared with the best photoionization model spectrum including α-SiC\ndust grains only (green line ) are shown (top panel). The C 60(blue line)\nand C 70(red line) emission bands and UIR plateau emission features\nare indicated. We note that the narrow emission features seen in the\nbest model spectrum (green line) are just atomic nebular emission lines,\nwhose detailed modeling is out of the scope of the present work. The\nbest photoionization model including α-SiC grains in conjunction with\ngraphite dust grains (green line) is also shown for comparison (bottom\npanel).\nArticle number, page 7 of 7"    },    {        "title": "2402.14985v1.Nonsmooth_Nonparametric_Regression_via_Fractional_Laplacian_Eigenmaps.pdf",        "content": "Nonsmooth Nonparametric Regression via\nFractional Laplacian Eigenmaps\nZhaoyang Shi∗Krishnakumar Balasubramanian†Wolfgang Polonik‡\nDepartment of Statistics, University of California, Davis\nAbstract\nWe develop nonparametric regression methods for the case when the true regression function\nis not necessarily smooth. More specifically, our approach is using the fractional Laplacian\nand is designed to handle the case when the true regression function lies in an L2-fractional\nSobolev space with order s∈(0,1). This function class is a Hilbert space lying between the\nspace of square-integrable functions and the first-order Sobolev space consisting of differentiable\nfunctions. It contains fractional power functions, piecewise constant or polynomial functions\nand bump function as canonical examples. For the proposed approach, we prove upper bounds\non the in-sample mean-squared estimation error of order n−2s\n2s+d, where dis the dimension, s\nis the aforementioned order parameter and nis the number of observations. We also provide\npreliminary empirical results validating the practical performance of the developed estimators.\n1 Introduction\nLaplacian based nonparametric regression is a widely used approach in machine learning that\nleverages the Laplacian Eigenmaps algorithm to perform regression tasks without relying on explicit\nparametric models. The nonparametric nature of the approach makes it flexible and adaptable\nto data generating process without imposing strict assumptions about the functional form of the\nrelationship between the response and the covariates. Existing theoretical studies of this approach\nare restricted to establishing minimax rates of convergence and adaptivity properties when the true\nregression function lies in Sobolev spaces; see Section 1.1 for details. Such spaces are inherently\nsmooth in nature and exclude important function classes in nonparametric statistics, such as\npiecewise constant or polynomial functions, bump functions and other such nonsmooth function\nclasses.\nIn this work, using the framework of fractional Laplacians, we propose a novel approach\ncalled Principal Component Regression using Fractional Laplacian Eigenmaps (PCR-FLE) for\nnonsmooth and nonparametric regression. The PCR-FLE algorithm generalizes the PCR-LE\nalgorithm by Green et al. (2023) and the PCR-WLE algorithm by Shi et al. (2024), and is designed\nto naturally handle the case when the true regression function lies in an L2-fractional Sobolev space\nHs(X) (see Definition 2.1). Specifically, consider the following regression model, Yi=f(Xi) +εi, for\ni= 1, . . . , n, where f:X →R,f∈Hs(X) for s∈(0,1),Xii.i.d.∼g, where gis a density on X ⊂Rd,\nandεii.i.d.∼N(0,1) is the noise (independent of the Xi’s). The goal is to estimate the regression\n∗zysshi@ucdavis.edu\n†kbala@ucdavis.edu\n‡wpolonik@ucdavis.edu\n1arXiv:2402.14985v1  [math.ST]  22 Feb 2024function fgiven pairs of observations ( X1, Y1), . . . , (Xn, Yn). The proposed PCR-FLE algorithm\nproceeds by estimating the eigenvalues and the eigenfunction of the fractional Laplacian operator\n(see(6)) based on the eigenvalues and the eigenvectors of the ϵ-graph constructed from the samples\n{Xi}n\ni=1, and projecting the response vector onto the top- Keigenvectors.\nFor this procedure, we make the following technical contributions in this work:\n•In Theorem 3.1, we establish upper bounds on the estimation error of the PCR-FLE algorithm\nin a nonparametric regression setting, when the true regression function lies in fractional\nSobolev spaces of order s∈(0,1). Such spaces include piecewise constant, polynomial function,\nbump functions and other nonsmooth function classes that are of practical importance.\n•As a part of the proof of our main results in Theorem 3.1, we derive a concentration inequality\nof the discrete fractional Sobolev seminorm/energy to its continuum, which serves as an\nimportant quantity for many fractional Laplacian based machine learning algorithms.\nWe provide preliminary simulations of the proposed approach validating the performance of PCR-\nFLE algorithm in estimating various nonsmooth functions in Section 4. Our contributions underscore\nthe importance of employing fractional graph Laplacians in nonsmooth nonparametric regression\nand lay a strong statistical groundwork for this technique. The theoretical results characterize the\ndependability and efficacy of this approach.\n1.1 Literature review\nGraph Laplacians find extensive use in various data science applications, including feature learning\nand spectral clustering (Weiss, 1999; Shi and Malik, 2000; Ng et al., 2001; von Luxburg, 2007).\nThey are employed for tasks such as extracting heat kernel signatures for shape analysis (Sun et al.,\n2009; Andreux et al., 2015; Dunson et al., 2021), reinforcement learning (Mahadevan and Maggioni,\n2007; Wu et al., 2019), and dimensionality reduction (Belkin and Niyogi, 2003; Coifman and Lafon,\n2006), among others. Additional discussions can be found in works by Belkin et al. (2006); Wang\net al. (2015); Chun et al. (2016); Hacquard et al. (2022).\nSeveral papers in the recent past have focused on obtaining theoretical rates of convergence in\nthe context of Laplacian operator estimation and related eigenvalue and/or eigenfunction estimation.\nPointwise consistency under ϵ-graphs have been studied by Belkin and Niyogi (2005); Hein et al.\n(2005); Gin´ e and Koltchinskii (2006); Hein et al. (2007); see references therein for more related\nworks. Furthermore, Trillos and Slepˇ cev (2018) derives consistency properties of spectral clustering\nmethods through studying the above spectral convergence with no specific error estimates. Following\nthem, Shi (2015); Trillos et al. (2020); Calder and Trillos (2022) derived rates of convergences of\nLaplacian eigenvalues and eigenvectors to population counterparts with explicit error estimates\nunder both ϵ-graphs and k-NN graph. Recently, Hoffmann et al. (2022) developed a framework for\nextending the above convergence results to a general Laplacian family, the weighted Laplacians\nand Shi et al. (2024) provided additional theoretical results on the convergence of the weighted\nLaplacians. Green et al. (2021) considered Laplacian smoothing estimation and showed its minimax\noptimal rates in low-dimensional space. Green et al. (2023) proposed the principal components\nregression with the Laplacian eigenmaps (PCR-LE) algorithm that achieves minimax optimal rates\nof nonparametric regression under uniform design. The PCR-LE algorithm was later generalized by\nShi et al. (2024) to the weighted Laplacians that include other commonly applied Laplacians such\nas the normalized Laplacian and the random walk Laplacian. Moreover, from a methodological\nperspective, Rice (1984) investigated spectral series regression on Sobolev spaces, and Trillos et al.\n(2022) applied the graph Poly-Laplacian smoothing to the regression problem.\n2The literature on both theoretical analysis and statistical applications fractional (graph) Lapla-\ncians is still in its infancy. Antil et al. (2021) extended the standard diffusion maps algorithm to\nthe fractional setting that involve the use of a non-local kernel. Fractional Laplacian regularization\nwas applied in Antil et al. (2020) to study tomographic reconstruction. Dunlop et al. (2020) studied\nlarge graph limits of semi-supervised learning problems via powers of graph Laplacian, including\nfractional Laplacians and provided their consistency guarantee in terms of Γ-convergence. Based\non it, semi-supervised Learning with finite labels was explored in Weihs and Thorpe (2023) via\nminimizing the fractional Sobolev seminorm/energy and consistency of such approach was provided\nthrough showing the Γ-convergence.\nThere is an extensive literature on nonparametric statistics on estimating piecewise constant or\npolynomial functions; see, for example, Chaudhuri et al. (1994); Donoho (1997); Scott and Nowak\n(2006); Tibshirani (2014) and references therein for a sampling of such work. While most of this work\nfocuses on the one or two dimensional setting, Chatterjee and Goswami (2021) recently considered\nthe multivariate setting and established adaptive rates. Many of these contributions either consider\nfixed (lattice-based) designs or axis-aligned partitions. Under appropriate boundary conditions on\nthe shape of the partition cells (not necessarily axis-aligned), piecewise constant or polynomial\nfunctions belong to fractional Sobolev spaces. Furthermore, rates of estimation in the case of H¨ older\nand Lipschitz functions are well-studied (Gy¨ orfi et al., 2002; Tsybakov, 2008). In particular, H¨ older\nfunctions on bounded domains (which is typically needed in statistical estimation contexts) belong\nto fractional Sobolev spaces. The inclusion in the other direction is more complicated, and we\nrefer to Rybalko (2023) for the state-of-the art results in one-dimension. Estimating functions with\nbounded variation, in both one and multidimensional settings is also considered in the literature;\nsee, for example, Mammen and Van De Geer (1997); Koenker and Mizera (2004); Sadhanala\net al. (2016); H¨ utter and Rigollet (2016); Sadhanala et al. (2017) and references therein for some\nrepresentative works. In particular, a recent work by Hu et al. (2022) considered the random design\nsetting in the multivariate case and established minimax rates. More technical details regarding\nthe relationship between function spaces of bounded variation and fractional Sobolev spaces are\nprovided in Section 2.4.\n2 Preliminaries and Methodology\n2.1 Laplacian matrices based on ϵ-neighborhood graphs\nFor i.i.d data X1, . . . , X nfrom a distribution Gsuported on X ⊆Rdwith the density g, the\nϵ-neighborhood graph is defined by setting the vertex set as {X1, . . . , X n}and putting an edge\nonXi∼Xjprovided that ∥Xi−Xj∥ ≤ϵwith the weights that depend inversely on the distance\nbetween the vertices connected by them:\nwϵ\ni,j:=η\u0012∥Xi−Xj∥\nϵ\u0013\n, i, j = 1, . . . , n, (1)\nwhere ∥ · ∥denotes the standard Euclidean norm. Here η≥0 is a non-increasing kernel function\nandϵis the bandwidth parameter.\nThe adjacency matrix is then defined as W= (wϵ\ni,j)i,j=1,...,nand the degree matrix D=\n(dij)i,j=1,...,nis then given by a diagonal matrix with the i-th diagonal element as di:=Pn\nj=1wϵ\ni,jfor\ni= 1, . . . , n . The associated (unnormalized) graph Laplacian is a matrix on the ϵ-graph defined as:\nLn,ϵ:=1\nnϵd+2(D−W), (2)\n3where 1 /(nϵd+2) is a scaling factor to ensure a stable limit. For u∈Rn, the i-th coordinate of the\nvector Ln,ϵuis given by\n(Ln,ϵu)i=1\nnϵd+2nX\nj=1wϵ\ni,j(ui−uj). (3)\nIt is well known that the (unnormalized) graph Laplacian (2)is self-adjoint with respect to the\nEuclidean inner product ⟨·,·⟩. We denote by the scaled Euclidean inner product ⟨·,·⟩n:=n−1⟨·,·⟩\nand write its corresponding scaled norm as ∥ · ∥ n.\nWhile we restrict our attention here to the unnormalized graph Laplacian, other forms of\nthe graph Laplacians are also widely used in machine learning tasks; some examples include the\nnormalized Laplacian, the random walk Laplacian and a larger family of the weighted graph\nLaplacians (Hoffmann et al., 2022; Shi et al., 2024) (which includes the normalized Laplacian and\nrandom walk Laplacian as special cases). It is possible to extend the procedure proposed in this\npaper to the class of weighted graph Laplacians, as done in Shi et al. (2024). We leave a detailed\nanalysis of the merits of such an extension for future work.\n2.2 Principal Component Regression via Fractional Laplacian-Eigenmap\nThe eigenmap was first proposed in Belkin and Niyogi (2003) to deal with nonlinear dimensionality\nreduction and data representation. Recently, Green et al. (2023); Shi et al. (2024) established\nminimax optimal rates of nonparametric regression via eigenmap on the (weighted) Laplacian. Here,\nwe propose the following principal components regression with the fractional Laplacian eigenmaps\n(PCR-FLE) algorithm based on the fractional Laplacian matrix Ls\nn,ϵfor 0 < s < 1:\n(1)For a given parameter ϵ >0 and a kernel function η, construct the ϵ-graph according to\nSection 2.1.\n(2)Compute the fractional Laplacian matrix Ls\nn,ϵbased on (2)via its eigen-decomposition\nLs\nn,ϵ=Pn\ni=1λs\nivivT\ni, where ( λi, vi) are the eigenpairs with eigenvalues 0 = λ1≤. . .≤λnin\nan ascending order and eigenvectors normalized to satisfy ∥vi∥n= 1, for i= 1, . . . , n .\n(3)Project the response vector Y= (Y1, . . . , Y n)Tonto the space spanned by the first Keigenvec-\ntors, i.e., denote by VK∈Rn×Kthe matrix with j-th column as VK,j=vjforj= 1, . . . , K\nand define\nˆf:=VKVT\nKY, (4)\nas the estimator.\nWe remark here that the entries of the vector ˆfare the in-sample values of the estimator of the\nregression function f. Intuitively speaking, PCR-FLE algorithm can be regarded as a PCR variant\nby substituting the sample covariance matrix with the fraction Laplacian matrix Ls\nn,ϵ, for 0 < s < 1.\nThe spotlight of PCR-FLE, however, lies in its capacity to learn nonsmooth functions by the\nfractional Laplacian compared to the sample covariance matrix or the (weighted) Laplacian.\n2.3 Fractional Laplacian operator and fractional Sobolev spaces\nIn this section, we introduce the functional space that we consider for our analysis for where the true\nregression function flies, the fractional Sobolev space, which includes many nonsmooth functions\nthat are of great interest in practice.\n4Definition 2.1. For any 0 < s < 1, the L2-fractional Sobolev space Hs(X) is defined as:\nn\nu∈L2(X) :Z\nX×X|u(x)−u(y)|2\n|x−y|d+2sdxdy < ∞o\n,\nwhere L2(X) :=\b\nu:R\nXu2(x)dx <∞\t\n.\nConsequently, the fractional Sobolev space is an intermediary space between L2(X) and the\nfirst-order Sobolev space H1(X) (consisting of differentiable functions) with the fractional Sobolev\nseminorm:\n|u|Hs(X):=\u0012Z\nX×X|u(x)−u(y)|2\n|x−y|d+2sdxdy\u00131\n2\n,\nand the fractional Sobolev norm:\n∥u∥Hs(X):=\u0012Z\nXu2(x)dx+|u|2\nHs(X)\u00131\n2\n.\nForM > 0 and 0 < s < 1, the class of all functions usuch that ∥u∥Hs(X)≤Mis called a fractional\nSobolev ball denoted by Hs(X;M) of radius M.\nThe above definition of the fractional Sobolev space Hs(X) is also linked with the following\nspectrally defined fractional Sobolev space:\nHs(X) :=(\nu∈L2(X) :∞X\ni=1Λs\nia2\ni<∞)\n, (5)\nwhere ai:=⟨u, ϕi⟩, fori≥1 and {(Λi, ϕi)}∞\ni=1are the eigenpairs of the Laplace–Beltrami operator\nLsuch that for i≥1:\nLϕi= Λ iϕiwith∂\n∂nϕi= 0,on∂X,\nwhere nstands for the outer normal vector and Lu=−div(∇u). Note also that the continuum\nlimit of (2)corresponds to Las long as gis uniform. In this discussion, we stick to the case of\nuniform gfor simplicity and remark that the connection holds for a general class of densities.\nIt has been emphasized in Dunlop et al. (2020) that Hs(X),→Hs(X)1. The above representation\nof the fractional Sobolev space is more related to the spectral series regression (see Rice (1984);\nGreen et al. (2023)) and semi-supervised learning for missing labels (see Weihs and Thorpe (2023)).\nMoreover, the fractional Sobolev space Hs(X) is naturally related to the fractional Laplacian\noperator Lsfor 0 < s < 1. Readers are referred to Di Nezza et al. (2012) for more details. Here,\ngiven a function uin the Schwartz space of rapidly decaying C∞\nc(X)2functions, the fractional\nLaplacian is defined as\nLsu(x) =cn,sP.V.Z\nRdu(x)−u(y)\n|x−y|d+2sdy, (6)\nwhere ‘P.V.’ stands for the Cauchy Principle Value and cn,s:=s22sΓ((d+ 2s)/2)/Γ(1−s). Di Nezza\net al. (2012, Proposition 3.6) show the following relationship between their norms:\n|u|2\nHs(Rd)= 2c−1\nn,s∥Lsu∥2\nL2(Rd).\n5Figure 1: Examples of functions that lie in a Fractional Sobolev spaces. The function f1,f2,f3and\nf4are as defined in (7), (8), (9) and (10) respectively.\nWe end this section by providing some examples of nonsmooth functions that are of interest in\nnonparametric statistics while not covered by the integer-indexed Sobolev space.\nExample 1: Power functions. It has to be noted that the Sobolev space Hs(X) for s∈N+\nnot only requires the functions to be s-times (weakly) differentiable but the derivatives have to be\nsquare-integrable as well. Let’s consider the following function:\nf1(x) :=|x|α,0< α < 1, (7)\non (−1,1). Obviously, f1is not (weakly) differentiable at 0. Furthermore, note thatR1\n0x2(α−1)dx=\n∞, for 0 < α≤1/2. Therefore, it doesn’t belong to any integer-indexed Sobolev space when\n0< α < 1. However, let us consider\n|f1|Hs((−1,1))=\u0012Z1\n−1Z1\n−1||x|α− |y|α|2\n|x−y|1+2sdx dy\u00131/2\n,\nfor 0 < α < 1 and 0 < s < 1. Note that the singularity is at x=y. Furthermore, we have\n|x|α− |y|α∼α|y|α−1(x−y) asx→yfory̸= 0 andR1\n0x−pdx <∞forp <1. Hence, the fractional\nSobolev seminorm |f1|Hα((−1,1))<∞,for 0 < s < 1 and 1 /2≤α <1. In summary, the function\nhence belongs to the fractional Sobolev space Hs((−1,1)) for 0 < s < 1 when 1 /2≤α <1.\n1,→stands for continuous embedding.\n2the function uis compactly supported on X\n6Example 2. Piecewise constant functions. Now, consider the following function on [0 ,1]:\nfpc(x) =(\n1,0< x≤1/2,\n0,1/2< x < 1.\nClearly, on the same support, fpc(x) =fpc(y) for 0 < x, y ≤1/2 or 1 /2< x, y < 1. It then suffices\nto only consider the following integral:\nZ1\n2\n0Z1\n1\n21\n|x−y|1+2sdx dy,\nfor 0 < s < 1. AsR1\n0x−pdx <∞forp <1, it is finite for 0 < s < 1/2. Then, the fractional Sobolev\nseminorm is finite for 0 < s < 1/2, which implies fpc(x) belongs to the fractional Sobolev space\nHs([0,1]) for 0 < s < 1/2.\nA generalization of the above example is the piecewise constant functions or blocks (see Donoho\nand Johnstone (1994) for more details) denoted by f2(x), where the function is constant on multiple\nintervals that form a partition of X. For instance,\nf2(x) =\n\n1,0< x≤1\n0.5,1< x≤2\n2,2< x≤3\n−2.5,3< x < 5.(8)\nThe piecewise constant functions/blocks belong to the fractional Sobolev space Hs(X) for 0 < s <\n1/2.\nExample 3. Piecewise polynomial functions. The piecewise polynomial functions extend\nthe blocks above by putting a polynomial function with degree at most r≥0 on each interval partition\nof the support X. Furthermore, since we are considering nonsmooth functions, discontinuities at\nthe boundaries of each interval are allowed here. According to the boundedness of the fractional\nSobolev seminorms of the power functions f1(x) and the blocks f2(x), the piecewise polynomial\nfunctions belong to the fractional Sobolev space Hs(X) for 0 < s < 1/2 and all r∈Nwhen Xis\nconsidered to be an open, connected and bounded subset of R(see Section 3.1). For example,\nf3(x) =\n\nx,0< x≤1\n2x2+ 2,1< x≤2\n−x+ 2,2< x≤3\n0.2x3−2x−4,3< x < 5.(9)\nIn general dimension d≥1, the above arguments can be generalized to imply that the piecewise\npolynomial functions (including the piecewise constant functions) belong to the fractional Sobolev\nspace when each partition of the support Xis connected and bounded and the boundary of each\npartition is a lower dimensional space, which allows non-axis aligned partitions.\nExample 4: Bumps functions The (multiple) bumps are functions that decay fast from each\npeak of the bumps. In signal processing, the bumps with polynomial decay or exponential decay are\ncommonly considered. For example,\nf4(x) :=JX\nj=1hjK\u0012t−tj\nwj\u0013\n, (10)\n7where K(|t|) := (1 + |t|)−4, and {tj, hj, wj}J\nj=1are parameters that tjare the locations of each\npeak and hjare the peak values. Due to the polynomial/exponential decay, the bumps f4(x) also\nbelong to the fractional Sobolev space Hs(X) for 0 < s < 1/2 when Xis considered to be an open,\nconnected and bounded subset of R(see Section 3.1).\nWe emphasize that the the nonsmooth functions presented above (including their extensions\ninRd) are representative of spatially variable functions arising in imaging, spectroscopy and other\nsignal processing applications that are of considerable practical importance. We refer to Donoho and\nJohnstone (1994); Boudraa et al. (2004); Liu et al. (2016); Sardy et al. (2000, 2001) for additional\nexposition of the examples.\n2.4 Relationship to other function classes\nFirst note that according to the definition of the fractional Sobolev space (i.e., Definition 2.1),\nbounded H¨ older functions of order α >0 on bounded domains belong to the fractional Sobolev\nspace for 0 < s < (α∧1).\nHu et al. (2022) considered nonparametric estimation with general measure-based bounded\ntotal variation class, with finite L∞norms. For this class, they developed minimax estimators.\nIn particular, the total variation norm was with respect to the L1norm of the weak derivatives.\nMoreover, Fang et al. (2021) investigated a similar class: the bounded variation in the sense of\nHardy-Krause. The fractional Sobolev space that we focus on in this work are set to be a subspace\nofL2space, the corresponding norms are function-value based and do not require weak derivatives\nto exist. Furthermore, the class of fractional Sobolev spaces can be considered with respect any\nLpspace for p≥1 (see Di Nezza et al. (2012) for the definition). In general, both bounded\nvariation functional space and fractional Sobolev space can characterise nonsmooth functions. When\nspecializing in the indicator functions, the bounded variation functional space only contains such\nfunctions with locally finite perimeter for the support. The exact inclusion relationships between\nthe spaces of bounded variation functions (with respect to the weak derivatives) and L1orL2based\nfractional Sobolev spaces are not well-explored in the literature, to the best of our knowledge.\nRockova and Rousseau (2021) studied Bayesian estimators when the truth lies in the set of locally\nH¨ older functions with finite L∞norm. When considering the bounded support X, locally H¨ older\nfunctions belong to the fractional Sobolev space. However, in general, the H¨ older functions include\nfunctions that may not even be in L1orL2. Imaizumi and Fukumizu (2019) applied deep neural\nnetworks to learn a class of nonsmooth functions that are piecewise H¨ older. Similar to locally H¨ older\nfunctions, when considering the bounded support X, bounded piecewise H¨ older functions belong to\nthe fractional Sobolev space while in general, the former one allows functions not necessarily in L1\norL2.\nIntuitively speaking, when characterising nonsmooth functions, compared to the aforementioned\nfunctional spaces, the fractional Sobolev space tends to allow ‘worse’ local non-smoothness while\nrequiring ��better’ global smoothness (in L1orL2).\n3 Theoretical Result\nBefore stating our assumptions and results, we introduce some conventions. For two real-valued\nquantities, A, B, the notation A≲Bmeans that there exists a constant C > 0 not depending\nonf,Mornsuch that A≤CBandA≍Bstands for A≲BandB≲A. Also, applying the\nscaled Euclidean norm ∥ · ∥ nor the corresponding scaled dot-product ⟨·,·⟩nto a function f, is to be\nunderstood as applying it to the vector in-sample evaluations ( f(X1), . . . , f (Xn)) of the function.\n83.1 Assumptions\nWe list the following major assumptions needed for the sampling distribution/density and the kernel\nη.\n(A1) The distribution Gis supported on X, which is an open, connected, and bounded subset of\nRdwith Lipschitz boundary.\n(A2) The distribution Ghas a density gonXsuch that\n0< gmin≤g(x)≤gmax<∞,for all x∈ X,\nfor some gmin, gmax>0. Additionally, gis Lipschitz on Xwith Lipschitz constant Lg>0.\n(A3) The kernel ηis a non-negative, monotonically non-decreasing function supported on the\ninterval [0 ,1] and its restriction on [0 ,1] is Lipschitz and for convenience, we assume η(1/2)>0\nand define\nσ0:=Z\nRmη(∥x∥)dx, σ 1:=1\ndZ\nRm∥y∥2η(∥y∥)dy.\nWithout loss of generality, we will assume σ0= 1 from now on.\nAssumptions ( A1) and ( A2) are mild and standard assumptions on the density function in the\nfield of graph Laplacians, which are also made in Green et al. (2023); Shi et al. (2024); Trillos et al.\n(2020). Assumption ( A3) is a standard normalization condition made on the smoothing kernel; see\nTrillos et al. (2020) for more details. The requirement that ηis compactly supported is purely due\nto our proof technique. While it is in principle possible to generalize it for non-compact kernels as\nlong as the tails decay relatively fast including the Gaussian kernel, that would require obtaining\nerror bounds on extra terms on the tail, which is beyond the scope of this paper.\n3.2 Estimation error of PCR-FLE algorithm\nTheorem 3.1. Let Assumptions (A1)-(A3)hold, and further assume f∈Hs(X;M)for0< s < 1\nandM > 0. Suppose there exist constants c0, C0>0such that\nc0\u0012logn\nn\u00131\nd\n≤ϵ≤C0K−1\nd,\nwith\nK= minn\n⌊(M2n)d\n2s+d⌋ ∨1, no\n. (11)\nThen, there exist constants c, C > 0not depending on f, M ornsuch that for nlarge enough, the\nestimator ˆfdefined in (4)satisfies:\n∥ˆf−f∥2\nn≤C\b\u0000\nM2(M2n)−2s\n2s+d∧1\u0001\n∨n−1\t\n,\nwith probability at least 1−Cn2e−cnϵd+4−Cne−cn−Cne−cnϵd−e−K.\n9Remark 3.2. Theorems 3.1 implies that the PCR-FLE algorithm achieves an upper bound of rates\nn−2s/(2s+d)with respect to the fractional Sobolev spaces Hs(X) for 0 < s < 1 with high probability,\nprovided that n−1/2≲M≲ns/d. Recall that the minimax optimal rates for the integer-valued\nSobolev space Hs(X) (s∈N+) is given by M2(M2n)−2s\n2s+din Gy¨ orfi et al. (2002); Wasserman\n(2006); Tsybakov (2008). While allowing s→1−, it is consistent with the above rates for the\nfirst-order Sobolev space H1(X).\nRemark 3.3. Under a fixed-design setup (i.e., a regular lattice/grid), Chatterjee and Goswami\n(2021) considered optimal regression tree (OPT) and showed that the finite sample risk of ORT is\nalways bounded byC(r)klogN\nNfor some constant C(r)>0 and N=cdfor some grid size c >0 when\nthe regression function is piecewise polynomial of degree ron some reasonably regular axis-aligned\nrectangular partition of the domain with at most krectangles. While such piecewise polynomial\nregression function belongs to the fractional Sobolev space, our bound in Theorem 3.1 is valid for a\nlarger family of nonsmooth functions and allows random design set-up.\nRemark 3.4. A phase transition in the fractional Sobolev space Hs(X) was discussed in Dunlop\net al. (2020, Lemma 4) that the regularity of the fractional Sobolev space depends on s < d/ 2\nors > d/ 2 (when s < d/ 2,Hs(X) cannot even embed continuously into the space of continuous\nfunctions C0(X)). However, it should be noted that Theorem 3.1 does not require the condition\ns > d/ 2 regardless of the phase transition.\nRemark 3.5. The lower bound for ϵmakes sure that with this smallest radius, the resulting graph\nwill still be connected with high probability and the upper bound for ϵensures the eigenvalue of the\ngraph Laplacian to be of the same order as its continuum version, the eigenvalue of the Laplacian\noperator (Weyl’s law). The condition on Kis set to trade-off bias and variance.\nRemark 3.6. Antil et al. (2020) considered nonparametric regression via fractional Laplacian\nregularization. However, no convergence rates of any kind were investigated there. On the other hand,\nit has been discussed and emphasized in Green et al. (2021, 2023) that Laplacian regularization\nusually achieves worse minimax rates of convergence especially in high-dimensional space Rd\ncompared to Laplacian eigenmaps.\n4 Numerical Experiments\nIn this section, we empirically demonstrate the performance of the PCR-FLE algorithm in Section\n2.2 for learning nonsmooth regression functions. All experiments are done in Python and the codes\ncan be found in the following link:\nhttps://github.com/10Mzys/Fractional_Laplacian_Nonsmooth_Regression/tree/main .\nIn our experiments, we stick to considering those functions that are of practical importance as\nintroduced in Section 2.3. For simplicity, we set the design distribution Gas the uniform distribution\nand examine the piecewise polynomial (including piecewise constant/the blocks) functions as the\ntrue regression function. For the construction of graph Laplacian, we pick a truncated Gaussian\nkernel. Unless otherwise stated, all tuning parameters are set as the optimal values according to\ngrid search and each experiment is averaged over 200 repetitions.\nWe now consider the mean squared error of the PCR-FLE estimation on the nonsmooth functions:\nthe piecewise constant function and the piecewise polynomial function ( f2(x) and f3(x) respectively\nin Figure 1). Due to relatively rapid rate of convergence, the sample size nis set to vary from 500\nto 1000. In Figure 2, we present the fitted regression function by PCR-FLE, visually. In Figure 3\n10Figure 2: The red line shows the true regression function. The blue line shows the average of the\nregression fit by PCR-FLE estimation condition on a generation of uniform sample on [0 ,5]:n= 100\n(left) and n= 1000 (right).\n(log-logscale), we show the in-sample mean squared errors of both estimators as a function of the\nsample size n. We see that both estimators have mean squared error converging to 0 roughly at our\ntheoretical rate in Theorem 3.1 while this provides a high probability upper bound.\n5 Discussion\nWe proposed and analyzed the PCR-FLE algorithm for performing nonparametric regression when\nthe true function lies in the L2-fractional Sobolev space, Hs(X, M). The approach is computational\nefficient and it involves computing the top- Keigenvalues and eigenvectors of size n×ngraph\nLaplacian matrix. Under a random design setting, we established rates of convergence of order\nn−2s\n2s+d, where nis the number of observations. There are several avenues for future works:\n•It is important to establish minimax lower bounds for estimation when the truth lies in\nfractional Sobolev spaces, under the random design setting, to evaluate worst-case performance\nof the PCR-FLE algorithm.\n•Our current results require knowledge of sandMin setting the bandwidth parameter ϵand\nthe number of eigenvalues K. It is interesting to develop estimators that are adaptive to the\nchoice of sandM, by extending the recent results in Shi et al. (2024).\n•It is interesting to go beyond L2-fractional Sobolev spaces and consider L1-fractional Sobolev\nspaces which allow for richer class of nonsmooth true functions.\n11Figure 3: In-sample mean squared error of PCR-FLE as a function of the sample size n. Each\nsubplot is on the log-logscale. The blue line presents the empirical error by PCR-FLE. 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Springer Science & Business Media, 2006. (Cited\non page 10.)\nAdrien Weihs and Matthew Thorpe. Consistency of Fractional Graph-Laplacian Regularization\nin Semi-Supervised Learning with Finite Labels. arXiv:2303.07818 , 2023. (Cited on pages 3, 5, 17,\nand 20.)\nYair Weiss. Segmentation using eigenvectors: A unifying view. In Proceedings of the Seventh IEEE\nInternational Conference on Computer Vision , volume 2, pages 975–982. IEEE, 1999. (Cited on\npage 2.)\nYifan Wu, George Tucker, and Ofir Nachum. The Laplacian in RL: Learning Representations with\nEfficient Approximations. In International Conference on Learning Representations , 2019. (Cited\non page 2.)\nA Proof of Theorem 3.1\nIn this section, we will prove Theorem 3.1. To this end, we first present some auxiliary lemmas. In\nthe following, Cstands for positive constants that may change from line to line but do not depend\nonnorM.\nLemma A.1 (Weyl’s Law) .Suppose Assumptions (A1) and (A2) hold. There exist constants\nc, C > 0such that\nck2\nd≤Λk≤Ck2\nd,for all k≥1.\nHere, Λkis the k-th eigenvalue of the weighted Laplacian operator Lgin the ascending order, where\nLgu:=−1\n2gdiv(g2∇u).\nThe Weyl’s law is a standard result in operator analysis. We refer interested readers to Dunlop\net al. (2020, Lemma 7.10) for a detailed proof.\nLemma A.2 (Lemma 2 in Green et al. (2021)) .There exist constants C1, C2, C3, C4, C5>0such\nthat for nlarge enough and C1(logn/n)1\nd≤ϵ≤C2, with probability at least 1−C3ne−C3nϵd, it holds\nthat\nC4\u0010\nk2\nd∧ϵ−2\u0011\n≤λk≤C5\u0010\nk2\nd∧ϵ−2\u0011\n,for2≤k≤n.\nThe above result actually is different from similar results established in Burago et al. (2015);\nTrillos et al. (2020); Calder and Trillos (2022), who establish similar results for manifolds without\nboundary. We refer to Green et al. (2023, Appendix D) for a more detailed discussion.\nWe are now in the position to prove the main Theorem 3.1.\n16Proof of Theorem 3.1. Below, Edenotes the conditional expectation, given X1, . . . , X n.Also recall\nthat the scaled Euclidean norm ∥·∥n(or the corresponding dot-product ⟨·,·⟩n) applied to a function\nf, means that it is applied to the vector of in-sample evaluations ( f(X1), . . . , f (Xn)) of the function.\nFirst note that by Cauchy-Schwarz inequality, we have:\n∥ˆf−f∥2\nn≤2(∥Eˆf−f∥2\nn+∥ˆf−Eˆf∥2\nn).\nThen, according to PCR-FLE algorithm in Section 2.2, we obtain\n∥Eˆf−f∥2\nn=nX\nk=K+1⟨vk, f⟩2\nn≤⟨Ls\nn,ϵf, f⟩n\nλs\nK+1, (12)\nand\n∥ˆf−Eˆf∥2\nn=KX\nk=1⟨vk, ε⟩2\nn,\nwhere ε:= (ε1, . . . , ε n)T.\nNow, note that if K= 0, ˆf= 0 then ∥ˆf−Eˆf∥2\nn= 0. We then focus on the case when K≥1.\nSince⟨vk, ε⟩nis normally distributed with 0 mean and variance\nVar⟨vk, ε⟩n=1\nn2Var⟨vk, ε⟩=1\nn, (13)\nas⟨vk, vk⟩=n. Then, we obtain:\n∥ˆf−Eˆf∥2\nn=1\nnKX\nk=1(√n⟨vk, ε⟩n)2d=1\nnKX\nk=1Z2\nk,\nwhere {Zk}K\nk=1are i.i.d. standard normal by the orthonormality of the eigenvectors.\nAccording to an exponential concentration inequality for chi-square distributions from Laurent\nand Massart (2000), we have\nP \n∥ˆf−Eˆf∥2\nn≥K\nn+ 2√\nK\nn√\nt+ 2t\nn!\n≤e−t. (14)\nWith (12) and (14), it yields that\n∥ˆf−f∥2\nn≤⟨Ls\nn,ϵf, f⟩n\nλs\nK+1+K\nn, (15)\nwith probability at least 1 −e−Kif 1≤K≤n. Of course, when K= 0, (15) holds.\nNow, it remains to bound the empirical fractional Sobolev seminorm ⟨Ls\nn,ϵf, f⟩nand the graph\nLaplacian eigenvalue λK+1(and its power λs\nK+1) for 0 < s < 1. We first focus on the empirical\nfractional Sobolev seminorm ⟨Ls\nn,ϵf, f⟩nfor 0 < s < 1. Note that Weihs and Thorpe (2023); Dunlop\net al. (2020); Trillos and Slepˇ cev (2018) showed Γ-convergence of the above empirical fractional\nSobolev seminorm to its continuum in (5). However, Γ-convergence does not fully suffice for our\npurpose. Instead, we will adapt the proof procedures applied in Calder and Trillos (2022); Green\n17et al. (2021, 2023) for our situation, as we describe next.\nFirst note that according to the eigendecomposition of Ln,ϵ, we obtain:\n⟨Ls\nn,ϵf, f⟩n=nX\ni=1λs\ni⟨f, vi⟩2\nn, (16)\nand by Lemma A.1 and Lemma A.2, we have that\nλi≲Λi,\nfor 1≤i≤nwith probability at least 1 −Cne−cnϵd. We now focus on the eigenvectors {vi}n\ni=1. For\na given eigenvalue Λ >0 ofLg, assume Λ = Λ i+1=. . .= Λ i+kfor some iandk, where kis the\nmultiplicity of Λ. We then define the eigenvalue gap of Λ as:\nγΛ:=1\n2(|Λ−Λi| ∧ |Λ−Λi+k+1|). (17)\nNow, according to Green et al. (2021, Proof of Theorem 5 of Supplementary Material) there\nexists a constant A > 2, such that for θ,˜δ >0 as defined in Green et al. (2021, Supplementary\nMaterial, Equation (A20)) and ϵsmall enough, such that\n1−A\u0010\nϵ√\nΛ +θ+˜δ\u0011\n≥1\n2,\nwe have the following: For Λ = Λ l(l-th eigenvalue of Lgin the ascending order) satisfying the above,\nwe have with probability at least 1 −Cne−cnθ2˜δd,\naλl≤σ1Λl≤Aλl, (18)\nwhere ais given in Green et al. (2021, Proof of Theorem 5, Supplementary Material) with a−1≥2.\nHence, we have with probability at least 1 −Cne−cnθ2˜δd, that\n|λl−σ1Λl| ≤((a−1−1)∨(A−1))σ1Λl≤CγΛl. (19)\nLetSbe the subspace of l2spanned by the eigenvectors of Ln,ϵassociated to the eigenvalues\nλi+1, . . . , λ i+r. In the following, we will establish the bound on the eigenfunctions/eigenvectors of\nΛjandλjrespectively for j=i+ 1, . . . , i +r. Denote by PSthe orthogonal projection (with respect\nto⟨·,·⟩n) onto SandP⊥\nSthe orthogonal projection onto the orthogonal complement of S. Let hbe\nthe eigenfunction of Lgcorresponding to the eigenvalue Λ, i.e., Lgh= Λh. Considering restriction\nofhonX1, . . . , X n, we have\nP⊥\nSLgh= ΛP⊥\nSh= ΛX\nj̸=i+1,...,i+r⟨h, vj⟩nvj,\nwhere recall that {vj}n\nj=1are the set of the orthonormal basis of eigenvectors of Ln,ϵwith respect to\nλ1, . . . , λ n. Similarly, we have (again, restrict honX1, . . . , X n):\nP⊥\nSLn,ϵh=X\nj̸=i+1,...,i+rλj⟨h, vj⟩nvj.\n18Combining the two results above, we obtain:\nmin{|σ1Λ−λi|,|σ1Λ−λi+r+1|}∥P⊥\nSh∥n\n≤ ∥P⊥\nw,S(Lw,n,ϵh−σ1Lgh)∥n\n≤ ∥Ln,ϵh−σ1Lgh∥n,\nwhere σ1is defined in Section 3.1 (see Assumption (A3)).\nOn the other hand, according to (17) and (19), we have\nmin{|σ1λ−λi(Ln,w,ϵ)|,|σ1λ−λi+r+1(Lw,n,ϵ)|} ≥σ1CγΛ.\nThen, we obtain:\n∥P⊥\nSh∥n=∥h−PSh∥n≤1\nσ1CγΛ∥Ln,ϵh−σ1Lgh∥n. (20)\nNow, we divide the above norm ∥ · ∥ ninto two parts by Cauchy’s inequality:\n∥Ln,ϵh−σ1Lgh∥n≲∥Ln,ϵh−σ1Lgh∥n,Xϵ+∥Ln,ϵh−σ1Lgh∥n,∂Xϵ,\nwhere we write X=Xϵ⊔∂Xϵ, where for any x∈ Xtϵ,Bx(ϵ)⊂ X and∂ϵXas its complement within\nXconsisting of points ‘close’ to the boundary. According to Calder and Trillos (2022, Theorem 3.3),\nit follows that if h1, . . . , h kis an orthonormal basis for the eigenspace of eigenfunctions of Lgwith\nrespect to eigenvalue Λ, then with probability at least 1 −2kne−Cnϵd+4,\n∥Ln,ϵhj−σ1Lghj∥n,Xϵ≤Cϵ, 1≤j≤k.\nOn the other hand, near the boundary, by setting k= 1 and s= 3 (since all hjat least belongs to\nC3(X)) in Green et al. (2023, Lemma 5), it yields that almost surely,\n∥Ln,ϵhj∥n,∂Xϵ≤Cϵ, 1≤j≤k,\nwith∥σ1Lghj∥n,∂Xϵ≤Cϵfor 1≤j≤since all hjat least belongs to C3(X).\nPutting the bounds in the interior and near the boundary together, we conclude: with probability\nat least 1 −2kne−Cnϵd+4,\n∥Ln,ϵhj−σ1Lghj∥n≤Cϵ, 1≤j≤k.\nNow, combining the above result with (20), it follows that with probability at least 1 −\n2kne−Cnϵd+4−Cne−cnθ2˜δd, we can find an orthonormal set ˜h1, . . . , ˜hkof spanning Ssuch that\n∥hj−˜hj∥n≤Cϵ.\nHere, recall that {hj}k\nj=1is an orthonormal basis for the eigenspace of eigenfunctions of Lgwith\nrespect to eigenvalue Λ = Λ land{˜hj}k\nj=1is an orthonormal set spanning S, i.e., the eigenspace of\neigenvectors of Ln,ϵwith respect to the eigenvalue λ=λl. These two sets of functions/vectors are\nclose in ∥ · ∥ nnorm by Cϵ. Therefore, with ϕias the projection i-th eigenfunction into Rnvia the\ntransportation map ˜Tdefined in Green et al. (2021, Proposition 3), we have: with probability at\nleast 1 −2kne−Cnϵd+4−Cne−cnθ2˜δd,\n∥vi−ϕi∥n≤Cϵ.\n19Then, plugging the above approximation in (16)with Weihs and Thorpe (2023, Proposition 4.21),\nwe obtain:\n⟨Ls\nn,ϵf, f⟩n=nX\ni=1λs\ni⟨f, vi⟩2\nn≲nX\ni=1λs\ni⟨f, vi−ϕi⟩2\nn+nX\ni=1λs\ni⟨f, ϕi⟩2\nn\n≲C \nϵ+nX\ni=1λs\ni⟨f, ϕi⟩2\nn!\n. (21)\nNow recall (5). For nlarge enough (or equivalently ϵsmall enough) we have\n⟨Ls\nn,ϵf, f⟩n≲CnX\ni=1Λs\ni⟨f, ϕi⟩2\nn≤CM2.\nwith probability at least 1 −Cn2e−Cnϵd+4−Cne−Cn, where we have used (18) above.\nFurthermore, according to Lemma A.2, we have for 0 < s < 1,\n\u0010\nk2s\nd∧ϵ−2s\u0011\n≲λs\nk≲\u0010\nk2s\nd∧ϵ−2s\u0011\n, (22)\nfor 1≤k≤n(since the case k= 1 can be bounded alone), with probability at least 1 −Cne−Cnϵd.\nNow, we are ready to proceed based on (15):\n∥ˆf−f∥2\nn≤⟨Ls\nn,ϵf, f⟩n\nλs\nK+1+K\nn,\nwith probability at least 1 −e−Kif 1≤K≤n. According to (21)and(22), we have with probability\nat least 1 −Cn2e−Cnϵd+4−Cne−Cn−Cne−cnϵd−e−Kandnlarge enough:\n∥ˆf−f∥2\nn≲M2\n(K+ 1)2s/d∧ϵ−2s+K\nn.\nFurthermore, based on the assumption ϵ≲K−1/d, the above inequality becomes:\n∥ˆf−f∥2\nn≲M2(K+ 1)−2s/d+K\nn. (23)\nThe choice K=⌊M2n⌋d/(2s+d)balances the two terms on the right-hand side, and we obtain\n∥ˆf−f∥2\nn≲M2(M2n)−2s/(2s+d), (24)\nwith probability at least 1 −Cn2e−Cnϵd+4−Cne−Cn−Cne−cnϵd−e−K.\nIfM2< n−1, we can take K= 1 and obtain from (23) that:\n∥ˆf−f∥2\nn≲1\nn.\nIfM > ns/d, we take K=nand in this case, we actually have ˆf(Xi) =Yifori= 1, . . . , n and\n∥ˆf−f∥2\nn=1\nnnX\ni=1ε2\ni≲C,\nwith probability at least 1 −e−nfor some constant C. Combining all above cases depending on\nchoices of K, it yields that bound in Theorem 3.1.\n20"    },    {        "title": "2402.14999v1.Wave_turbulence__A_solvable_problem_applied_to_Navier_Stokes.pdf",        "content": "SciPost Physics Lecture Notes Submission\nWave turbulence\nA solvable problem applied to Navier-Stokes\nSébastien Galtier\nUniversité Paris-Saclay\nLaboratoire de Physique des Plasmas, École polytechnique, 91128 Palaiseau, France\nsebastien.galtier@lpp.polytechnique.fr\nFebruary 26, 2024\nAbstract\nWave turbulence and eddy turbulence are the two regimes that we may encounter in\nnature. The attention of fluid mechanics being mainly focused on incompressible hy-\ndrodynamics, it is usually the second regime that is treated in books, whereas waves\nare often present in geophysics and astrophysics. In these lecture notes, I present the\ntheory of wave turbulence which is free from the closure problem encountered in eddy\nturbulence. Basically, the wave amplitude is introduced in a multiple time scale method\nas a small parameter to derive the so-called kinetic equations from which exact results\ncan be obtained (power-law spectra, direction of the cascade, Kolmogorov’s constant)\nand compared with the data. Two hydrodynamic applications are considered with cap-\nillary waves and inertial waves, the first leading to isotropic turbulence and the second\nto anisotropic turbulence.\nContents\n1 Introduction to wave turbulence: origin and applications 3\n1.1 Resonant wave interactions 4\n1.2 Multiple time scale method 4\n1.3 Kolmogorov-Zakharov spectrum 5\n1.4 Applications of wave turbulence 6\n2 Duffing’s equation: a pedagogical model 8\n2.1 Classical perturbation method 8\n2.2 Multiple time scale method 9\n3 Sequential time closures for three-wave interactions 11\n3.1 Fundamental equation 11\n3.2 Dispersion relation and resonance 12\n3.3 Multiple time scale method 13\n3.4 First asymptotic closure at time T1 15\n3.5 Second asymptotic closure at time T2 16\n3.6 Conclusion 19\n4 Applications to Navier-Stokes equations 20\n4.1 Capillary wave turbulence 20\n4.2 Inertial wave turbulence 20\n1arXiv:2402.14999v1  [physics.flu-dyn]  22 Feb 2024SciPost Physics Lecture Notes Submission\n5 Kolmogorov-Zakharov spectrum 20\n5.1 A toy model 20\n5.2 Applications to Navier-Stokes 23\nA Appendix: sequential time closures for four-wave interactions 23\nA.1 Fundamental equation 23\nA.2 Multiple time scale method 24\nA.3 First asymptotic closure at time T1 25\nA.4 Second asymptotic closure at time T2 26\nA.5 Third asymptotic closure at time T3 26\nA.6 Fourth asymptotic closure at time T4 27\nReferences 33\nFigure 1: Gravity and capillary wave turbulence.\nWave turbulence offers the possibility of a deep understanding of physical systems com-\nposed of an ensemble of random waves interacting nonlinearly. The reason for this is, first\nof all, the possibility of analytically deriving a set of integro-differential equations for spectral\ncumulants – the so-called kinetic equations – which are free from the closure problem classi-\ncally encountered in eddy turbulence. To achieve this natural asymptotic closure, the wave\namplitude is used as a small parameter. Second, exact solutions can be found from the kinetic\nequations. In addition to the usual thermodynamic solutions, the kinetic equations have finite\nflux solutions that capture the flow of conserved densities from sources to sinks. Third, these\nexact solutions correspond to power-law spectra that can be compared to the data. The num-\nber of experiments, observations, and diagnostics has increased dramatically over the past two\ndecades, and now, with direct numerical simulations, wave turbulence has become a leading\nfield in turbulence from which new fundamental questions have arisen [1].\nThe Navier-Stokes equations are fundamental in turbulence because it is from them and\nfrom laboratory experiments based on water or air that the first theoretical advances (con-\ncepts, exact laws) have been made. This is a reason why the literature on turbulence focuses\nmainly – if not entirely – on incompressible hydrodynamics, thus excluding a whole series\nof applications where waves make a major contribution to the dynamics. Yet turbulent sys-\ntems containing waves are very numerous, and incompressible hydrodynamics (pure eddy\nturbulence) seems to be quite singular. In essence, wave turbulence concerns only systems in\nwhich waves can be excited, which excludes a priori the classical Navier-Stokes equations, the\narchetypal system of turbulence. However, wave turbulence can also be found in incompress-\n2SciPost Physics Lecture Notes Submission\nible hydrodynamics if one modifies this system to include, for example, surface (gravity and\ncapillary) waves, or if one adds the Coriolis force which introduces inertial waves.\nA priori, we can distinguish two regimes in wave turbulence: the one where the waves\nare of weak amplitude and the one where they are not. The regime that concerns us here is\nthe first case called weak wave turbulence. I will follow the usual practice and simply write\n’wave turbulence’ to avoid thinking that weak (wave) turbulence also means small Reynolds\nnumber: indeed, as for eddy turbulence, the Reynolds number is not limited or, in other words,\nit is only limited by the effect of viscosity and the size of the experiment. The existence of a\nsmall parameter – the amplitude of the wave – allows a systematic approach to the problem.\nThe general idea is that the weak amplitude of the waves makes it possible to distinguish the\ntemporal evolution of the amplitude from that of the phase, the first evolving more slowly\nthan the second. This separation of the time scales can be formally written using a multiple\ntime scale method. This is the condition applied to obtain a uniform asymptotic development\nwhich leads to a natural closure of the hierarchy of equations and finally to the so-called kinetic\nequations.\nIn these lecture notes, I will introduce the multiple time scale method to an archetypal\nwave equation that involves three-wave interactions. I will show in detail how the kinetic\nequation can emerge from this development. This is a technical part that has rarely (if never)\nbeen described in detail since the seminal paper [2]. A brief history will allow us to understand\nhow the theory of wave turbulence was built. For pedagogical reasons, the multiple time scale\nmethod will first be applied to a simple example: the Duffing model. We will see that the main\nproblem is the existence of secular terms proportional to time which can break the uniformity\nof the development. This is precisely the condition applied to eliminate them that leads to\nthe kinetic equations of wave turbulence. I will examine two examples from the modified\nNavier-Stokes equations: capillary wave turbulence and inertial wave turbulence involving\nthree-wave interactions. The first implies isotropy while the second leads to anisotropy, and\nthus to a new difficulty. Finally, using a toy model, I will show how the Kolmogogov-Zakharov\nspectrum can be derived.\n1 Introduction to wave turbulence: origin and applications\nIt is essentially in the field of oceanography and the study of surface waves that we find the\nbeginnings of the theory of wave turbulence [3]. This work began in the late 1950s, at a time\nwhen the surface waves were treated essentially in a linear manner, the spectra as a superposi-\ntion of linear waves and non-linear effects restricted, for example, to periodic wave distortion\nphenomena described for a long time by Stokes [4]. There are two kinds of surface waves:\ngravity waves – that is, waves on the ocean (see Figure 1), and at smaller scale capillary waves.\nThe Navier-Stokes equations, modified by the force of gravity, are the theoretical framework\nfrom which the first results emerged. To simplify the problem, the movements are generally\nassumed to be irrotational: this corresponds to an air-water interface disturbed by a unidirec-\ntional blowing wind – a typical condition encountered in the open sea. The problem is then\nreduced to Bernoulli’s equation applied to the free surface of the fluid to which is added a\nLagrangian equation to describe the deformation of the fluid surface and another to take into\naccount the additional hypothesis called deep water.\n3SciPost Physics Lecture Notes Submission\n1.1 Resonant wave interactions\nWork in the late 1950s and early 1960s led to a major theoretical breakthrough with the dis-\ncovery of the existence of resonant interactions1between non-linear waves of weak amplitude.\nThe waves in question are of course gravity waves. The idea has its origin in the work of [7],[8]\nand[9]2who tried to understand how the non-linear interactions between gravity waves could\nredistribute the energy initially present in these modes. The underlying idea is that in the ini-\ntial phase of the development of turbulence it is the resonant interactions that provide the\ndominant mechanism for transferring energy from one wave to another. The ultimate phase\nof development had already been studied previously by [10]who, using dimensional analysis,\nwas able to make a spectral prediction for strong (wave) turbulence which is still in use today\n(the so-called critical balance).\nAfter long calculations based on a classical perturbative development, it was possible to\ndemonstrate that the energy redistribution was generally negligible for interactions involving\nless than four waves, but that it became important for four waves if the following resonance\nrelationship was satisfied\u001a\nk1+k2=k3+k,\nω1+ω2=ω3+ω,(1)\nwith the dispersion relation ω2=g|k|(gbeing the acceleration of gravity). The experimental\ndemonstration of the existence of resonant interactions was then published by [11]and[12].\nAlthough the existence of resonant wave interactions appeared to be real, their interpretation\nwas then subject to debate because the energy exchange between resonant waves also led to a\nlinear growth (in time) – the secular terms – of the amplitude of the waves ultimately breaking\nthe hypothesis underlying the existence of resonant interactions, i.e. the presence of waves of\nweak amplitude; in this case, the perturbation is said to be non-uniform. It was therefore not\nclear whether this energy transfer could be really effective.\nAn answer to this question was first proposed statistically by [9]who considered a random\nset of gravity waves. The hypothesis of weak non-linearities was then used to justify another\nhypothesis, that of a Gaussian statistics which greatly simplifies the calculations by eliminating\nin particular moments of odd order (see also [13]).\n1.2 Multiple time scale method\nA second major theoretical breakthrough came with Benney’s work which showed that the\nproblem of wave turbulence is similar to that of weakly coupled oscillators in mechanics [14].\nWith a new mathematical technique for the analysis of dispersive wave packets involving two\ntime scales, [15]obtained a relatively simple form for the (discrete and non-statistical) equa-\ntions which govern the temporal evolution of resonant modes. In this approach, the secular\nterms disappear: they are somehow absorbed by the slow variation of the wave amplitude.\nThis work shows, in a simple way, how the exchange of energy between four gravity waves is\ncarried out: this exchange takes place while conserving energy. Here we recover a property of\nstrong turbulence, that of detailed conservation.\nThis approach paved the way for the use of a new mathematical technique called multi-\nple time scale method [16–18 ], which made it possible to demonstrate that the (continuous)\nstatistical equations of wave turbulence have a natural closure due to the separation of scales\nin time. As shown by [2]for quadratic non-linearities and [19]for cubic non-linearities, the\n1Note that earlier studies on three- and four-wave resonance had already been carried out, for example, in the\nfield of thermal conduction in crystals [5]. See also [6]who wrote the first kinetic equation: it was for four-wave\ninteractions in the context of the electron theory of conductivity.\n2K. Hasselmann was awarded the Nobel prize in Physics in 2021 ’for the physical modelling of Earth’s climate,\nquantifying variability and reliably predicting global warming’.\n4SciPost Physics Lecture Notes Submission\nequations of wave turbulence are asymptotically valid over long periods of time and do not\nrequire the statistical assumption of Gaussianity made by [9]. The method of multiple time\nscales offers a systematic and consistent theoretical framework in which the procedure expan-\nsion allows in principle to determine the slow rate of change of the amplitude of the waves at\nany order inε[20, 21 ]. The so-called kinetic equation of gravity wave turbulence then takes\nthe following (schematic) integro-differential form in three-dimension\n∂N(k)\n∂t=ε4Z\nR9S(k,k1,k2,k3)[N1N2(N+N3)−N3N(N1+N2)]δ(ω1+ω2−ω3−ω)\nδ(k1+k2−k3−k)dk1dk2dk3, (2)\nwith N≡N(k) =E(k)/ωthe wave action spectrum3andE(k)the energy spectrum. The Dirac\nfunctions translate the resonance condition (1) discussed above. The presence of the ε4factor\nmeans that the time scale (normalized to the wave period ∼1/ω) on which the spectra are\nmodified by the non-linear dynamics is of the order of O(1/ε4). Therefore, it is a relatively\nslow process.\n1.3 Kolmogorov-Zakharov spectrum\nParallel to the work carried out in the Western world, major theoretical advances were also\nmade in the East. From the beginning of the 1960s, the Soviet school became interested in\nwave turbulence mainly through plasma physics [22, 23 ]from which certain notations and\nvocabulary were borrowed (one speaks, for example, of kinetic equation or collision integral).\nIn passing, it is curious to note that this work was carried out simultaneously by the two parts\nof the world without much communication between them. In particular, by the assumption\nknown as random phase approximation, kinetic equations of wave turbulence were proposed\nin a primitive form by [24]for a problem of plasma physics, then in a modern form by Zakharov\nand his collaborators [25–28 ]. This work is generally based on a Hamiltonian approach to the\nproblem, whereas it is the Eulerian approach which was mainly used in the West.\nThe random phase approximation leads, in practice, to the same kinetic equations as by\nthe method of multiple time scales where the random phase occurs naturally dynamically (and\ntherefore is not assumed). From these integro-differential equations, a major breakthrough\nwas achieved with the discovery of a conformal transformation to extract from the non-linear\nkinetic equations the exact power law solutions. This transformation – now called the Za-\nkharov transformation4– was first proposed for capillary wave turbulence involving triadic\ninteractions [26, 27 ], then for Langmuir wave turbulence where the interactions are quar-\ntic[28, 29 ]. There are two types of solution: the zero flux solution (which was the regime\nstudied by [6]) and the non-zero constant flux solution. The first case corresponds to the ther-\nmodynamic solution (constant entropy) and the second to the Kolmogorov-Zakharov spec-\ntrum, which is the most interesting solution because it is non-trivial and corresponds to a\ncascade.\n3By analogy with plasma physics, wave action is often associated with particles. This is a quantity that is\nsometimes conserved in four-wave processes. This is the case for gravity wave turbulence.\n4Zakharov was a student of Sagdeev. He defended his PhD thesis on surface waves in 1966 with several funda-\nmental results in wave turbulence to his credit, such as the discovery of exact solutions to the kinetic equations.\nThis discovery is reported in the article by [25]in which the author first verified that the collision integral (ob-\ntained from a relatively simple ad hoc model of three-wave interactions) tended towards ±∞ for two power law\nexponents. He then demonstrated that the solution (the energy spectrum) associated with the index exactly in the\nmiddle of this interval cancels non-trivially the collision integral: Zakharov had just discovered an exact stationary\nsolution. The discovery of the so-called Zakharov transformation came shortly after [26].\n5SciPost Physics Lecture Notes Submission\n1.4 Applications of wave turbulence\nThere are many examples of application of wave turbulence. Below is a non-exhaustive list of\napplications with some references.\n•Surface wave: These are the first applications of wave turbulence. They include capillary\nwaves and gravity waves. The former involve three-wave interactions, the theory of which\nwas published in English by [27]in the deep water limit. Note that the shallow water limit is\nalso the subject of studies (see for example [30]). Gravity wave turbulence is a problem that\nneeds to be addressed at the level of four waves [9,31]. This regime has been well reproduced\nin the laboratory or by direct numerical simulations (see eg. [32, 33 ]). Its detection at sea is\nmore difficult but not impossible (see eg. [34, 35 ]). The two subjects being linked, several\nexperiments deal with the interaction between gravity and capillary waves. All in all, it is a\nsubject that is still very much under study (see the review by [36]).\n•Internal gravity wave: These waves are a variant of the previous ones in the sense that\nwe are interested here in gravity waves in oceans. These waves contribute dynamically to\nthe turbulent transport of heat, which is important to understand to properly evaluate the\nimpact of oceans on the climate [37]. Internal gravity wave turbulence is an anisotropic three-\nwave problem for which an Eulerian theory has been developed by [38]as well as several\nexperiments (see eg. [39]).\n•Inertial wave: This is the closest example to the standard case of eddy turbulence in the\nsense that the equations are those of Navier-Stokes which are modified simply by adding the\nCoriolis force. Inertial wave turbulence is a three-wave anisotropic problem whose theory\nhas been published by [40]. Several experiments have been made to study this regime (see\ne.g.[41]). This example will be discussed later.\n•Rossby wave: These waves appear in a situation of differential rotation. They are used\nin the modeling of planetary atmospheres, the interiors of gaseous planets and stars; these\nare often referred to as planetary waves [42]. In wave turbulence, the dynamics is driven by\nthree-wave processes with a dominance of non-local interactions [43,44 ].\n•Plasma wave: As mentioned above, we find the beginnings of wave turbulence in the field\nof plasma physics [22, 23 ]. In this very vast domain, waves are legion. Incompressible mag-\nnetohydrodynamics (MHD) is an important example of application because it is a model often\nused in astrophysics. We talk about Alfvén wave turbulence which involves three-wave inter-\nactions. The theory was published by [45]: it is a case where the anisotropy is so strong that\nthe cascade is completely inhibited in the direction of the strong applied magnetic field.\n•Geodynamo wave: Geodynamo waves are defined as waves present in the Earth’s outer\nliquid core. These waves take part in the dynamo effect, i.e. the physical mechanism that\nmaintains the Earth magnetic field [46]. These are magnetostrophic waves and inertial waves.\nThe three-wave theory models a homogeneous medium with a small Rossby number [47]: in\nthis framework, an anisotropic inverse cascade of hybrid helicity is predicted, which could be\nat the origin of the regeneration of the magnetic field on a large scale [48].\n•Acoustic wave: Acoustic wave turbulence is a priori driven by three-wave interactions but\nthese waves are semi-dispersive. This is a critical situation for the application of wave turbu-\nlence because the uniformity of the development is not guaranteed. The first works on the\nsubject date back to the early 1970s [49]. The way in which the asymptotic is subtly modified\nis discussed by [50], then by [51]. It can be shown for this regime that energy is at best redis-\ntributed according to rays (in three-dimensional Fourier space). Note that a similar situation\ncan be found in compressible MHD [52].\n6SciPost Physics Lecture Notes Submission\n•Elastic wave: It is a turbulence produced by a thin elastic plate or by the introduction\nof long polymer molecules in a liquid [53]. In the former case, we are in a very different\nsituation from the traditional turbulence produced by a fluid. The vibrations of a plate can\nin theory produce weak or strong wave turbulence, depending on the forcing. The theory of\nwave turbulence has been published by [54]: it is a four wave problem characterized by a direct\nenergy cascade. Although the wave action is not conserved in this problem, direct numerical\nsimulations show an inverse cascade which seems to be established in an explosive way [55].\nLaboratory experiments using steel plates have been carried out in order to reproduce with\nvaried success the theoretical predictions [56–59 ].\n•Optical wave: Wave turbulence is also found in the field of non-linear optics. The multi-\ndimensional non-linear Schrödinger equation can be used to describe the evolution of quasi-\nmonochromatic plane wave envelopes [60]. The theory is explained by [61]: this wave turbu-\nlence is governed by four-wave interactions and an intermittency mechanism characterized by\na collapse phenomenon in physical space can occur. In the case of a reduction of the problem\nto one dimension, it can be shown that the dominant resonant interactions imply six waves,\ninvolving a much slower non-linear dynamics [62]. All this work is part of the study of non-\nlinear effects on the propagation of incoherent optical beams [63,64 ].\n•Quantum turbulence: This subject is very close to the previous one in the sense that the\nmodel used is a variant of the non-linear Schrödinger equation: by changing the sign of the\nnon-linear term, the interaction becomes repulsive and the physics of turbulence is modified.\nThe associated equation – also called the Gross-Pitaevskii equation – describes a Bose gas at\nvery low temperatures. Note that the emergence of quantum turbulence in an oscillating Bose-\nEnstein condensate has been experimentally demonstrated [65]. The wave turbulence regime\nis described by [61]: it is shown that four-wave interactions conserve energy and wave action.\nThe inverse cascade associated with the latter leads to the formation of a stable condensate,\ni.e. the accumulation of wave action in the k=0 mode. Direct numerical simulations in\ntwo[66,67 ]and three [68,69 ]dimensions illustrate this regime.\n•Kelvin wave: Kelvin waves are studied in the context of superfluids whose temperature is\nclose to the absolute zero. These waves can propagate along filaments of vorticity and modify\nthe dynamics of turbulence [70, 71 ]. Kelvin’s theory of wave turbulence has been proposed\nby[72]: it involves six-wave processes that conserve both energy and wave action. As these\ncascade processes are extremely slow, it is particularly interesting to consider a local model of\nnon-linear diffusion [73].\n•Gravitational wave: This example illustrates the wide range of possible applications of wave\nturbulence. It is about cosmology and more precisely the birth of the Universe: primordial\ngravitational waves could be at the origin of the inflation phase of the Universe, a question\nwhich is still open to this day and which touches the limits of our knowledge [74]. The theory\nhas been published by [75,76 ]: the dynamics is governed by four-wave processes sufficiently\nsymmetric to conserve wave action. The latter is characterized by an explosive inverse cascade.\nIt is interesting to note that this regime is close to that of elastic waves in the strong tension\nlimit as shown by the theoretical, numerical and experimental work of [77].\n7SciPost Physics Lecture Notes Submission\n2 Duffing’s equation: a pedagogical model\n2.1 Classical perturbation method\nThe aim of this section is to discuss on a pedagogical model the idea of the multiple time scale\nmethod through a toy model, the Duffing equation\nd2f\ndt2+f=−εf3, (3)\nwhereεis a small parameter (0 <ε≪1) which measures the intensity of the non-linearity\nandfis a function of time only. The main solution to this equation is a harmonic oscillator.\nThis solution will, however, be slightly modified over time by the presence of the small non-\nlinear perturbation (right hand side term). Let us first use the classical perturbation theory\nand introduce the following development\nf=+∞X\nn=0εnfn. (4)\nWe then obtain an infinite system of equations which, for the first three orders, can be written\nO(ε0):d2f0\ndt2+f0=0 , (5a)\nO(ε1):d2f1\ndt2+f1=−f3\n0, (5b)\nO(ε2):d2f2\ndt2+f2=−3f2\n0f1. (5c)\nWe can see that the solution to a given order will affect the solution to the higher order. The\nfirst solution is trivially f0=Acos(t+φ). By introducing this one in the second equation, we\ncan deduce the exact solution for f1. If we stop at this order, the solution reads\nf(t) =Acos(t+φ) +εA3\n−3\n8tsin(t+φ) +1\n32cos(3t+3φ)\n+O(ε2). (6)\nAs a first approximation, this system behaves well as a harmonic oscillator. The condition\nto be checked is that the non-linear perturbation remains of weak amplitude and the time\nconsidered relatively short. On the other hand, over long periods of the order of O(1/ε), this\nsolution is modified in a non-negligible way. The term which is at the origin of it, in εt, is said\nto be secular. For even longer periods of time the development diverges; this divergence is all\nthe more so as certain secular terms of a higher order reinforce it. Then, the uniformity of the\ndevelopment is broken.\nWe can pursue the analysis to evaluate the period of oscillations. To do this, the Duffing\nequation, which has been previously multiplied by the derivative of f, is integrated\n1\n2d f\ndt2\n+1\n2f2+1\n4εf4=E, (7)\nwhere Eis a constant. One obtains\ndt=d fq\n2E−f2−1\n2εf4. (8)\n8SciPost Physics Lecture Notes Submission\nThis equation can be integrated over a period T. By posing f=asinθ, we obtain the following\nexpression which is exact in its integral form\nT=4Zπ/2\n0dθq\n1+εa2(1−1\n2cos2θ)=2π\n1−3\n8εa2+O(ε2a4)\n. (9)\nA Taylor expansion is used to evaluate the integral. It is found that the 2 πperiod of the\nharmonic oscillator is corrected by the non-linear perturbation. The smaller the non-linearity,\nthe smaller the correction.\nWhat information is useful for our problem? This very simple system illustrates schemat-\nically the problems that we can encounter in a classical perturbation development. Such a\ndevelopment should allow us, in principle, to follow the evolution of the dynamics of wave\nturbulence at various orders in ε. The main problem is to ensure that the development remains\nneat (or uniform). In other words, this means making sure that no secular term of the order n\ncomes to interfere in the dynamics at an order m, such as m<n, in order not to find oneself in\nthe same situation as [9], i.e. in the presence of a secular term. This type of term constitutes\nan obstacle to the closure of the hierarchy of equations. The more sophisticated method of\nmultiple time scales [16]will allow us to solve this problem.\n2.2 Multiple time scale method\nLet us take the Duffing equation again and introduce the following independent time variables\nTn≡εntwith n=0, 1, 2, ... . (10)\nThe power series development of the variable fis then written ( t≡T0)\nf=+∞X\nn=0εnfn(t,T1,T2, ...), (11)\nand the time derivative becomes a sum of partial derivatives\nd\ndt=+∞X\nn=0εn∂\n∂Tn. (12)\nThe introduction of relations (10)–(12) into the Duffing equation gives\n∂\n∂t+ε∂\n∂T1+ε2∂\n∂T2+...2\n(f0+εf1+ε2f2+...) + (f0+εf1+ε2f2+...) =\n−ε(f0+εf1+ε2f2+...)3. (13)\nWe obtain an infinite system of equations which, for the first two orders, can be written\nO(ε0):∂2f0\n∂t2+f0=0 , (14a)\nO(ε1):∂2f1\n∂t2+f1+2∂2f0\n∂t∂T1=−f3\n0. (14b)\nWe can see the presence of a new term in the time evolution equation of f1whose importance\nemerges over long times. We look for a solution of the form\nf0=A(T1,T2, ...)cos(t+φ(T1,T2, ...)), (15)\n9SciPost Physics Lecture Notes Submission\nwith an amplitude Aand a phaseφwhich can vary slowly over time (hence the dependence\ninT1,T2, etc). The introduction of previous expression in (14b) gives the relation\n∂2f1\n∂t2+f1=2∂A\n∂T1sin(t+φ) +\n2A∂φ\n∂T1−3\n4A3\ncos(t+φ)−A3\n4cos(3(t+φ)). (16)\nTo ensure that the f1solution does not include a secular term, the following conditions must\nbe imposed\n∂A\n∂T1=0 and∂φ\n∂T1=3\n8A2, (17)\nwhich givesφ= (3/8)A2εt+θ. The solution is then written\nf(t) =Acos\nt+3\n8A2εt+θ\n+εA3\n32cos\n3t+9\n8A2εt+3θ\n+O(ε2), (18)\nwith A(T2,T3, ...)andθ(T2,T3, ...). In the order of the truncation, we can therefore consider\nthatAandθare constant. Note that this development is compatible with the previous one (6)\nfort≤O(ε). This example illustrates the fact that we can achieve a uniform development in\na systematic way using the multiple time scale method.\nFigure 2: Variation of f(t)(solid line) obtained by a numerical simulation of the\nDuffing equation with ε=0.01. The solutions (6) and (18) are plotted in red dotted\nand black dashed lines, respectively.\nIn Figure 2 we show the result of a numerical simulation of the Duffing equation with\nε=0.01. The function f(t)is plotted in solid line. At the top, we see the evolution for\n10SciPost Physics Lecture Notes Submission\nt∈[0, 200 ]and at the bottom, we display an enlargement of the end of the simulation. The\nmathematical solutions (6) and (18) are plotted in dotted and dashed lines, respectively. The\ndivergence with tof the solution (6) whose origin is the secular term can be seen. As expected,\nthis divergence occurs over a time tof the order of O(1/ε)(the reference time is the wave\nperiod). On the other hand, the solution (18) remains close to the real solution until the final\ntime.\n3 Sequential time closures for three-wave interactions\n3.1 Fundamental equation\nWe present here a detailed derivation of the kinetic equation for three-wave interactions, a\nsituation of interest for the Navier-Stokes equations when capillary waves or inertial waves\nare involved, two examples that will be used to illustrate the theory of wave turbulence.\nConsider the following inviscid model\n∂u(x,t)\n∂t=L(u) +εN(u,u), (19)\nwhere uis a random variable such that 〈u〉=0 (with〈〉an ensemble average), Lis a linear\noperator which guarantees that the waves are linear solutions to the problem, and Nis a\nnon-linear quadratic operator (thus involving a priori triadic interactions). The coefficient ε\nis a small parameter (0 <ε≪1) which measures the amplitude of the non-linearities (it is\noriginally a measure of the wave amplitude). The Fourier transform in d-dimension is\nˆu(k,t)≡1\n(2π)dZ\nRdu(x,t)e−ik·xdx. (20)\nNote that we consider a continuous medium which can lead to mathematical difficulties con-\nnected with infinite dimensional phase spaces. For this reason, it is preferable to assume a\nvariable spatially periodic over a box of finite size L. However, in the derivation of the ki-\nnetic equation, the limit L→+∞is finally taken (before the long time limit, or equivalently\nthe limitε→0). As both approaches lead to the same kinetic equation, for simplicity, we\nanticipate this result and follow the original approach of [2].\nThe Fourier transform of equation (19) takes the following schematic form when the canon-\nical variables are introduced\n∂\n∂t+iskωk\nAsk(k,t) =εX\nspsqZ\nR2dH−kpqAsp(p,t)Asq(q,t)δ(k−p−q)dpdq, (21)\nwith si=±the directional polarity. On the left hand side, we find ωkwhich is fixed by the\ndispersion relation (we assume 0 <ωk∝|k|xwith x>1) while on the right hand side\nHkpqis an operator which depends on the shape of the non-linearities and is symmetrical in\npandq(ie.Hkqp=Hkpq). We recall that the presence of the Dirac function originates in\nthe convolution product, and that the quadratic nature of the non-linearities leads to triadic\ninteractions.\nEquation (21) is simplified by making the following change of variables (the writing is\nsimplified in passing)\nAsk(k,t) =ask(k,t)e−iskωkt=ask\nke−iskωkt. (22)\nIn the interaction representation, we obtain the fundamental equation\n∂ask\nk\n∂t=εX\nspsqZ\nR2dH−kpqasp\npasq\nqeiΩk,pqtδk,pqdpdq, (23)\n11SciPost Physics Lecture Notes Submission\nFigure 3: Resonance condition for a dispersion relation of the type ωk∼|k|xwith\nx>1 (left) and 0<x<1 (right).\nwith by definition δk,pq≡δ(k−p−q)andΩk,pq≡skωk−spωp−sqωq. Additional properties can\nbe established if first we use the relation Ask(k) = (A−sk(−k))∗, with∗the complex conjugate:\nthen, we have necessarily Hkpq=H∗\n−k−p−q. As mentioned above, we consider only zero mean\nfluctuations in the physical space, which means with our notation ask(k=0,t) =0 (below,\nwe will assume statistical spatial homogeneity) and therefore, H0pq=0. Finally, if Hkpqis\npurely imaginary, we have H∗\nkpq=−Hkpq. This property may be used to simplify the kinetic\nequations.\nEquation (23) highlights the temporal evolution of the wave amplitude: this evolution is\na priori relatively slow since it induces a non-linear term proportional to ε. We will find this\nproperty with the multiple time scale method. The presence of the complex exponential is\nfundamental for the asymptotic closure: since we are interested in the dynamics over a long\ntime with respect to the wave period, the contribution of this exponential is essentially zero.\nOnly some terms will survive: those for which Ωk,pq=0. With the condition imposed by the\nDirac, we obtain the resonance condition\n\u001a\nk=p+q,\nskωk=spωp+sqωq.(24)\n3.2 Dispersion relation and resonance\nThe triadic resonance condition (24) does not always have a solution. To realize this, we\ncan represent this condition geometrically in the two-dimensional and isotropic case: Figure\n3 shows a (axisymmetric) dispersion relation of the type ωk∼|k|x, with x>1 (left) and\n0<x<1 (right). The solutions of the resonance condition (24) correspond to the intersection\nbetween the surface ω(p)(which is identified with that of ω(k)) and the surface ω(q). We\nfind that this intersection exists only for a convex dispersion relation ( x>1). The concave\ncase corresponding to 0 <x<1 is the one encountered a priori with gravity waves. Note that\nfor anisotropic problems like inertial wave turbulence this simple picture does not work.\n12SciPost Physics Lecture Notes Submission\nThe special case x=1 is that of non-dispersive (or semi-dispersive) waves. We can easily\nsee that the only possible solution is that where all three wave vectors are aligned. Although\na solution to the triadic resonance condition exists, an asymptotic closure of the hierarchy of\nequations is not guaranteed. Physically, we can easily understand this problem: two semi-\ndispersive waves propagate at the same speed, therefore, if they initially overlap their interac-\ntions will be strong, otherwise they will never interact. Using the multiple time scale method\npresented in the next section, it can be shown that, in one dimension, a contribution from\nnon-linear terms is possible on a time scale of the order of T1while the closure is made on\na longer time scale in T2[2]. The appearance of these secular contributions requires a new\nanalysis to suppress them and justify the application of the wave turbulence theory. This situ-\nation can be found in acoustic wave turbulence or in fast magneto-acoustic wave turbulence\nin three dimensions [52].\nNote that for four-wave interactions, the situation is less constraining because new possi-\nbilities exist. For example, gravity waves can be analyzed at this order. In the case of non-\ndispersive waves, it may be possible to develop a theory of wave turbulence because the as-\nsociated four wave vectors are not necessarily collinear. An example is given by gravitational\nwaves for which the dispersion relation is ωk=ck, with cthe speed of light [75].\n3.3 Multiple time scale method\nFollowing our analysis of the Duffing equation, we wish to make a development of equation\n(23) following the method of multiple time scales [16]. We introduce a sequence of time\nscales, T0,T1,T2, ..., which will be treated as independent variables, with\nT0≡t,T1≡εt,T2≡ε2t, ... . (25)\nThe variation of the wave amplitude with T1andT2represents the slow variation that we\nwish to extract. Note that it is because of the weak dependence in tofT1,T2, ..., that all these\nvariables can be treated (it is an approximation) as independent. (The smaller εis, the better\nthe approximation.) We obtain ( T0being replaced by t)\n∂\n∂t+ε∂\n∂T1+ε2∂\n∂T2+...\nask\nk=εX\nspsqZ\nR2dH−kpqasp\npasq\nqeiΩk,pqtδk,pqdpdq. (26)\nThe variable ask\nkmust also be expanded to the power of εand make the various scales appear\nin time\nask\nk=+∞X\nn=0εnask\nk,n(t,T1,T2, ...) =ask\nk,0+εask\nk,1+ε2ask\nk,2+... . (27)\nExpression (27) is then introduced into the fundamental equation (26). We obtain for the first\nthree terms\n∂ask\nk,0\n∂t=0 , (28a)\n∂ask\nk,1\n∂t=−∂ask\nk,0\n∂T1+X\nspsqZ\nR2dH−kpqasp\np,0asq\nq,0eiΩk,pqtδk,pqdpdq, (28b)\n∂ask\nk,2\n∂t=−∂ask\nk,1\n∂T1−∂ask\nk,0\n∂T2+X\nspsqZ\nR2dH−kpq\nasp\np,1asq\nq,0+asp\np,0asq\nq,1\neiΩk,pqtδk,pqdpdq. (28c)\n13SciPost Physics Lecture Notes Submission\nTo lighten the writing, the time dependency of the variables has been omitted. After integration\nont, one finds\nask\nk,0=ask\nk,0(T1,T2, ...), (29a)\nask\nk,1=−t∂ask\nk,0\n∂T1+bsk\nk,1, (29b)\nask\nk,2=t2\n2∂2ask\nk,0\n∂T2\n1−t∂ask\nk,0\n∂T2−Zt\n0∂bsk\nk,1\n∂T1dt+˜bsk\nk,2, (29c)\nwith by definition\nbsk\nk,1≡X\nspsqZ\nR2dH−kpqasp\np,0asq\nq,0∆(Ωk,pq)δk,pqdpdq, (30)\n∆(X)≡Zt\n0eiX tdt=eiX t−1\niX, (31)\nand\n˜bsk\nk,2≡X\nspsqZ\nR2dH−kpqZt\n0\nasp\np,1asq\nq,0+asp\np,0asq\nq,1\neiΩk,pqtdtδk,pqdpdq. (32)\nThe previous equation is modified when expression (29b) is introduced; one finds\n˜bsk\nk,2=−X\nspsqZ\nR2dH−kpq∂(asp\np,0asq\nq,0)\n∂T1\u0012Zt\n0teiΩk,pqtdt\u0013\nδk,pqdpdq\n+X\nspsqZ\nR2dH−kpq\u0012Zt\n0(bsp\np,1asq\nq,0+bsq\nq,1asp\np,0)eiΩk,pqtdt\u0013\nδk,pqdpdq. (33)\nExpression (29c) becomes\nask\nk,2=t2\n2∂2ask\nk,0\n∂T2\n1−t∂ask\nk,0\n∂T2+bsk\nk,2, (34)\nwith\nbsk\nk,2=−X\nspsqZ\nR2dH−kpq∂(asp\np,0asq\nq,0)\n∂T1\u0012Zt\n0\u0002\n∆(Ωk,pq) +teiΩk,pqt\u0003\ndt\u0013\nδk,pqdpdq\n+2X\nspsqsp′sq′Z\nR4dH−kpqH−pp′q′asp′\np′,0asq′\nq′,0asq\nq,0\u0012Zt\n0∆(Ωp,p′q′)eiΩk,pqtdt\u0013\nδk,pqδp,p′q′dpdqdp′dq′.\nThe time integrals give the relations\nZt\n0\u0002\n∆(Ωk,pq) +teiΩk,pqt\u0003\ndt =t∆(Ωk,pq), (35)\nZt\n0∆(Ωp,p′q′)eiΩk,pqtdt =∆(Ωk,p′q′q)−∆(Ωk,pq)\ni(Ωk,p′q′q−Ωk,pq), (36)\nthat will be used below in the long time limit.\n14SciPost Physics Lecture Notes Submission\n3.4 First asymptotic closure at time T1\nWith the previous definitions, the perturbative expansion of the second-order moment writes\n〈ask\nkask′\nk′〉=〈(ask\nk,0+εask\nk,1+ε2ask\nk,2+...)(ask′\nk′,0+εask′\nk′,1+ε2ask′\nk′,2+...)〉 (37)\n=〈ask\nk,0ask′\nk′,0〉+ε〈ask\nk,0ask′\nk′,1+ask\nk,1ask′\nk′,0〉+ε2〈ask\nk,0ask′\nk′,2+ask\nk,1ask′\nk′,1+ask\nk,2ask′\nk′,0〉+...\nWe shall assume that this turbulence is statistically homogeneous. In this case, the second-\norder moment can be written in term of second-order cumulant qsksk′(k,k′)≡qsksk′\nksuch that\n〈ask\nkask′\nk′〉=qsksk′\nkδ(k+k′), (38)\nwhere the presence of δ(k+k′)is the consequence of the statistical homogeneity [1]. We\nalso assume – this is the basic idea of the method – that the second-order moment (in fact,\nthe coefficient qsksk′\nkin front of the Dirac function) in the left hand side remains bounded\nat all time [2]. As an example, we can think of the energy spectrum which, as we know,\nremains physically bounded. Therefore, the contributions on the right hand side must also\nbe bounded (in the sense introduced above). We will see that secular terms can appear at\ndifferent orders in ε; this leads to certain conditions to cancel their contributions in order to\nkeep the development uniform in time. As we shall see, at order O(ε2)this condition leads to\nthe so-called kinetic equations.\nAt order O(ε0), we have the contribution of 〈ask\nk,0ask′\nk′,0〉which will therefore be assumed to\nbe bounded at all times.\nAt order O(ε1), we have the contribution\n〈ask\nk,0ask′\nk′,1+ask\nk,1ask′\nk′,0〉=®\nask\nk,0\n−t∂ask′\nk′,0\n∂T1+bsk′\nk′,1\n+\n−t∂ask\nk,0\n∂T1+bsk\nk,1\nask′\nk′,0¸\n=−t∂\n∂T1〈ask\nk,0ask′\nk′,0〉+〈ask\nk,0bsk′\nk′,1+bsk\nk,1ask′\nk′,0〉. (39)\nThe first term on the right hand side gives a secular contribution proportional to t. For the\nsecond term, we have\nX\nspsqZ\nR2dH−kpq〈ask\nk,0asp\np,0asq\nq,0〉∆(Ωk,pq)δk,pqdpdq. (40)\nThe long time limit ( t≫1/ω) of this oscillating integral will be given by the Riemann-\nLebesgue lemma (the proof requires to use of generalized functions)\n∆(X) =eiX t−1\niXt→+∞−−−−−→πδ(X) +iP1\nX\n, (41)\nwhere Pis the Cauchy principal value of the integral. Therefore, the long time limit of expres-\nsion (40) gives no secular contribution. The same conclusion is obtained for the third term of\nexpression (39). Also, the condition to cancel the unique secular term is\n∂〈ask\nk,0ask′\nk′,0〉\n∂T1=0 , (42)\nwhich means that the second-order moment does not evolve over a time scale T1. As will be\nseen later, a turbulent cascade is only expected on a time scale T2.\n15SciPost Physics Lecture Notes Submission\n3.5 Second asymptotic closure at time T2\nThe analysis continues at order O(ε2). With expression (34), the next contribution reads\n¬\nask\nk,1ask′\nk′,1+ask\nk,0ask′\nk′,2+as\nk,2ask′\nk′,0¶\n=®\n−t∂ask\nk,0\n∂T1+bsk\nk,1\n−t∂ask′\nk′,0\n∂T1+bsk′\nk′,1¸\n(43)\n+®\nask\nk,0\nt2\n2∂2ask′\nk′,0\n∂T2\n1−t∂ask′\nk′,0\n∂T2+bsk′\nk′,2\n+ask′\nk′,0\nt2\n2∂2ask\nk,0\n∂T2\n1−t∂ask\nk,0\n∂T2+bsk\nk,2¸\n,\nwhich gives after development and simplifications\n¬\nask\nk,1ask′\nk′,1+ask\nk,0ask′\nk′,2+ask\nk,2ask′\nk′,0¶\n=t2\n2∂2〈ask\nk,0ask′\nk′,0〉\n∂T2\n1−t∂〈ask\nk,0ask′\nk′,0〉\n∂T2+¬\nbsk\nk,1bsk′\nk′,1¶\n(44)\n−t®∂ask\nk,0\n∂T1bsk′\nk′,1+bsk\nk,1∂ask′\nk′,0\n∂T1¸\n+¬\nask\nk,0bsk′\nk′,2+ask′\nk′,0bsk\nk,2¶\n.\nThe first term on the right hand side cancels over the long time as required by the first asymp-\ntotic closure. The second term gives a secular contribution. The other three terms can poten-\ntially give a secular contribution: it is obvious for the fourth term and non-trivial for the third\nand fifth terms which require further development.\nThe third term on the right hand side writes\n¬\nbsk\nk,1bsk′\nk′,1¶\n=X\nspsqsp′sq′Z\nR4dH−kpqH−k′p′q′〈asp\np,0asq\nq,0asp′\np′,0asq′\nq′,0〉∆(Ωk,pq)∆(Ωk′,p′q′)\nδk,pqδk′,p′q′dpdqdp′dq′, (45)\nHere again, the theory of generalized functions gives us the long time behavior of this oscil-\nlating integral (with the Poincaré-Bertrand formula)\n∆(X)∆(−X)t→+∞−−−−−→ 2πtδ(X) +2P1\nX∂\n∂X. (46)\nTherefore, a secular contribution involving a Dirac function is possible. The fourth-order mo-\nment in expression (45) can be decomposed into products of second-order cumulant plus a\nfourth-order cumulant such that (the statistical homogeneity is used as well as 〈ask\nk,0〉=0)\n〈asp\np,0asq\nq,0asp′\np′,0asq′\nq′,0〉=qspsqsp′sq′\npqp′,0δ(p+q+p′+q′) +qspsq\np,0qsp′sq′\np′,0δ(p+q)δ(p′+q′)(47)\n+qspsp′\np,0qsqsq′\nq,0δ(p+p′)δ(q+q′) +qspsq′\np,0qsqsp′\nq,0δ(p+q′)δ(q+p′).\nNote that according to expression (37) by homogeneity we also have the relation k=−k′. We\nobtain\n¬\nbsk\nk,1bsk′\nk′,1¶\n=X\nspsqsp′sq′Z\nR4dH−kpqH−k′p′q′\nqspsqsp′sq′\npqp′,0δ(p+q+p′+q′) (48)\n+qspsq\np,0qsp′sq′\np′,0δ(p+q)δ(p′+q′) +qspsp′\np,0qsqsq′\nq,0δ(p+p′)δ(q+q′)\n+qspsq′\np,0qsqsp′\nq,0δ(p+q′)δ(q+p′)\n∆(Ωk,pq)∆(Ωk′,p′q′)δk,pqδk′,p′q′dpdqdp′dq′.\nWe are looking for secular contributions. In the second line, the first term does not contribute\nsince it imposes k=0which cancels Hkpqbut the second term can contribute when the\n16SciPost Physics Lecture Notes Submission\nconditions sp=−sp′andsq=−sq′are satisfied. Likewise, in the third line a contribution is\npossible when sp=−sq′andsq=−sp′. There is no contribution from the first line, which\nmeans that the situation is the same as if the distribution were Gaussian (however, we do not\nmake this assumption). Finally, in the long time limit, the secular contribution Ct¬\nbsk\nk,1bsk′\nk′,1¶\nis\nCt¬\nbsk\nk,1bsk′\nk′,1¶\n=4πtX\nspsqZ\nR2dH−kpqHk−p−qqsp−sp\np,0qsq−sq\nq,0δ(Ωk,pq)δk,pqδkk′dpdq\n=4πtX\nspsqZ\nR2d|H−kpq|2qsp−sp\np,0qsq−sq\nq,0δ(Ωk,pq)δk,pqδkk′dpdq. (49)\nThe fourth term on the right hand side of equation (44) does not contribute over the long\ntime because it depends on the T1derivative. The proof is given by a new relation involving\nthe n-order moments which can be written\n〈ask\nkask′\nk′ask′′\nk′′...〉=〈ask\nk,0ask′\nk′,0ask′′\nk′′,0...〉 (50)\n+ε〈ask\nk,1ask′\nk′,0ask′′\nk′′,0...+ask\nk,0ask′\nk′,1ask′′\nk′′,0...+ask\nk,0ask′\nk′,0ask′′\nk′′,1....+....〉\n+ε2〈....〉+...\nAs for the second-order moment, we demand that the moments of order n are bounded. At\norderO(ε), we obtain the relation\n〈ask\nk,1ask′\nk′,0ask′′\nk′′,0...+ask\nk,0ask′\nk′,1ask′′\nk′′,0...+ask\nk,0ask′\nk′,0ask′′\nk′′,1...+...〉 (51)\n=®\n−t∂ask\nk,0\n∂T1+bsk\nk,1\nask′\nk′,0ask′′\nk′′,0...+ask\nk,0\n−t∂ask′\nk′,0\n∂T1+bsk′\nk′,1\nask′′\nk′′,0...+...¸\n=−t∂〈ask\nk,0ask′\nk′,0ask′′\nk′′,0...〉\n∂T1+〈bsk\nk,1ask′\nk′,0ask′′\nk′′,0...〉+〈ask\nk,0bsk′\nk′,1ask′′\nk′′,0...〉+...\nWe see that only the first term of the last line gives a secular contribution over the long time,\nwhich means that we have to satisfy the condition\n∂〈ask\nk,0ask′\nk′,0ask′′\nk′′,0...〉\n∂T1=0 , (52)\nat any order n. Therefore, the probability density function does not depend on T1and we can\nassume that the variable itself does not depend on T1. It is a mild hypothesis because it is\ndifficult to imagine a turbulent system where everything fluctuates, and in which it would be\npossible to have a T1dependence for the amplitude whereas the probability density function\nhas no such dependence. This shows that the fourth term on the right hand side of equation\n(44) does not contribute in the long time limit.\nThe last term of equation (44) writes\n¬\nask\nk,0bsk′\nk′,2+ask′\nk′,0bsk\nk,2¶\n= (53)\nX\nspsqsp′sq′Z\nR4d2H−k′pqH−pp′q′¬\nask\nk,0asp′\np′,0asq′\nq′,0asq\nq,0¶\u0012∆(Ωk′,p′q′q)−∆(Ωk′,pq)\ni(Ωk′,p′q′q−Ωk′,pq)\u0013\nδk′,pqδp,p′q′dpdqdp′dq′\n+X\nspsqsp′sq′Z\nR4d2H−kpqH−pp′q′¬\nask′\nk′,0asp′\np′,0asq′\nq′,0asq\nq,0¶\u0012∆(Ωk,p′q′q)−∆(Ωk,pq)\ni(Ωk,p′q′q−Ωk,pq)\u0013\nδk,pqδp,p′q′dpdqdp′dq′.\n17SciPost Physics Lecture Notes Submission\nThe secular contributions of these oscillating integrals will be given by the theory of general-\nized functions with the relation\n∆(X)−∆(0)\niXt→+∞−−−−−→πtδ(X) +itP1\nX\n. (54)\nWe also need to use the following development (and its symmetric in k′)\n〈ask\nk,0asp′\np′,0asq′\nq′,0asq\nq,0〉=qsksp′sq′sq\nkp′q′,0δ(k+p′+q′+q) +qsksq\nk,0qsp′sq′\np′,0δ(k+q)δ(p′+q′)(55)\n+qsksq′\nk,0qsp′sq\np′,0δ(k+q′)δ(p′+q) +qsksp′\nk,0qsq′sq\nq′,0δ(k+p′)δ(q′+q).\nIn the right hand side of expression (55), the first two terms do not give a secular contribution,\nhowever, the last two terms give a contribution when the following conditions are satisfied,\nnamely sk=−sq′,sp′=−sqandsk=−sp′,sq′=−sq, respectively. After substitution and\nsimplification, we obtain the secular contributions in the long time limit\nCt¬\nask\nk,0bsk′\nk′,2+ask′\nk′,0bsk\nk,2¶\n= (56)\n+4tX\nspsqZ\nR2dH−k′pqH−p−qk′qsksk′\nk,0qsq−sq\nq,0\u0012\nπδ(Ωk′,pq) +iP\u0012\n1\nΩk′,pq\u0013\u0013\nδk′,pqδkk′dpdq\n+4tX\nspsqZ\nR2dH−kpqH−p−qkqsksk′\nk,0qsq−sq\nq,0\u0012\nπδ(Ωk,pq) +iP\u0012\n1\nΩk,pq\u0013\u0013\nδk,pqδkk′dpdq\n=8πtX\nspsqZ\nR2dH−kpqH−p−qkqsksk′\nk,0qsq−sq\nq,0δ(Ωk,pq)δk,pqδkk′dpdq.\nThe last writing is obtained by using the property H∗\n−kpq=H−kpqand the symmetry in p→−p\nandq→−q.\nIf we impose the nullity of the sum of the different secular contributions at time T2, we\nfind the condition\n∂〈ask\nk,0ask′\nk′,0〉\n∂T2=∂qsksk′\nk,0\n∂T2δkk′ (57)\n=4πX\nspsqZ\nR2d|H−kpq|2qsp−sp\np,0qsq−sq\nq,0δ(Ωk,pq)δk,pqδkk′dpdq\n+8πX\nspsqZ\nR2dH−kpqH−p−qkqsksk′\nk,0qsq−sq\nq,0δ(Ωk,pq)δk,pqδkk′dpdq.\nAfter integrating over k′, using the property that qsksk′\nk,0is real and thus sk=−sk′and introducing\nNsk\nk≡qsk−sk\nk,0, we end up with (some simple manipulations are also used to symmetrize the\nequation)\n∂Nsk\nk\n∂t= (58)\n4πε2X\nspsqZ\nR2dHkpq\nH−k−p−q\nNsk\nk+H−p−q−k\nNsp\np+H−q−p−k\nNsq\nq\nNsk\nkNsp\npNsq\nqδ(Ωkpq)δkpqdpdq,\nwhich is the kinetic equation for three-wave interactions.\n18SciPost Physics Lecture Notes Submission\n3.6 Conclusion\nThe derivation of the kinetic equation (58) does not involve any closure assumption such as\nthe quasi-Gaussianity and the random phase appears naturally dynamically in wave turbu-\nlence. The kinetic equation is obtained by using a sequential asymptotic closures at times T1\nandT2. This method consists in imposing the nullity of secular terms (proportional to t) which\nemerge over long times in order to guarantee a bounded value for the associated moments.\nThese secular terms do not involve fourth-order cumulants but only products of second-order\ncumulants. Note, however, that the derivation made, although systematic, says nothing about\nthe remaining terms (not used to obtain the kinetic equation) in the small εperturbation ex-\npansion, the implicit conjecture being that they are subdominant. The proof of this conjecture\nremains a mathematical challenge [78].\nIn summary, it can be said that wave turbulence for three-wave interactions is characterized\nby a dynamics on two time scales. On short timescales, of the order of the wave period, there\nis a phase mixing which leads, due to the dispersive nature of the waves, to the decoupling\nof the correlations if initially present and to a statistics close to Gaussianity, as expected from\nthe central limit theorem. On a longer time scale, the non-linear coupling – weak at short\ntimes – becomes non-negligible because of the resonance mechanism. This coupling leads to\na regeneration of the cumulants via the product of lower order cumulants. It is these terms\nthat are at the origin of the energy transfer mechanism.\nIn this derivation, the system studied is assumed to be of infinite size and can therefore\nbe treated as continuous. Note that numerical simulation with its grid of points escapes this\ndescription. Effects (freezing of the cascade) linked to the discretization of the Fourier space\ncan appear because the resonance conditions are a priori more difficult to satisfy (see for\nexample [79]for capillary waves). In theory, the weaker the non-linearities, the stronger\nthese effects are. For example, in inertial wave turbulence [80]has shown that discretization\neffects become preponderant when the Rossby number, Ro, is smaller than 10−3. Beyond this\nvalue, but still for a small Ro≪1, these effects are negligible because of the quasi-resonances\nwhich, together with the resonances, contribute to the transfer of energy.\nIn nature we often encounter three-wave problems. As mentioned above, sometimes\nhigher-level problems (up to six waves) can occur. Despite the diversity of situations, the\nkinetic equation takes a universal form in the sense that it is written (at leading order)\n∂Nsk\nk\n∂t=ε2n−4T(k), (59)\nforn-wave interactions, with Tthe transfer function (or collision integral). The higher the\ndegree of interaction, the longer the transfer time τtr. This time is characterized by the relation\nωτtr∼O(1/ε2n−4). Since the characteristic time of the waves is τω∼1/ωand the small\nparameter is none other than the ratio between this time and the non-linear time τN L, we\narrive at the following phenomenological expression for the transfer time\nτtr∼τω\nε2n−4∼τ2n−4\nN L\nτ2n−5\nω. (60)\nTherefore, for three-wave interactions, we can use the expression: τtr∼ωτ2\nN L.\n19SciPost Physics Lecture Notes Submission\n4 Applications to Navier-Stokes equations\n4.1 Capillary wave turbulence\nTogether with gravity waves, capillary waves are the main surface waves encountered in na-\nture. The latter have an advantage over the former in that they are easier to treat analytically in\nthe non-linear regime since they involves three-wave interactions. Following the multiple time\nscale method, we obtain the following kinetic equation for capillary wave turbulence [27,81 ]\n∂Nsk\nk\n∂t=4πε2X\nspsqZ\nR4|Hkpq|2spsq\nspsq\nNsk\nk+sksq\nNsp\np+sksp\nNsq\nq\nNsk\nkNsp\npNsq\nqδ(Ωkpq)δkpqdpdq, (61)\nwith\n|Hkpq|2=pσ\n8\u0014\nsk(p·q+pq)\u0010pq\nk\u00111/4\n+sp(k·q+kq)qk\np1/4\n+sq(k·p+kp)pk\nq1/4\u00152\n,(62)\nwhereωk=pσk3/2andσ=γ/ρwater, withγthe surface tension coefficient (for the air-water\ninterfaceγ≃0.07 N /m) andρwater the mass density of the water.\n4.2 Inertial wave turbulence\nInertial wave turbulence is the regime reached by an incompressible fluid subjected to a rapid\nand uniform rotation within the limit of a large Reynolds number. At first sight, this situation\nseems similar to eddy turbulence since it simply involves adding a well-known force to the\nNavier-Stokes equations, namely the Coriolis force. However, there are two major differences:\nwaves exist and the spherical symmetry is broken by the presence of a privileged axis, that\nof rotation. This anisotropy makes the study of inertial wave turbulence more complex than\nthat of capillary waves. Nevertheless, the multiple time scale method can be applied and the\nfollowing anisotropic kinetic equation can be obtained in the limit k⊥≫k∥[40]\n∂Ns\nk\n∂t=πε2\n4sωkX\nspsqZsinθk\nk⊥2\n(spp⊥−sqq⊥)2(sk⊥+spp⊥+sqq⊥)2(63)\n×\u0002\nsωkNsp\npNsq\nq+spωpNs\nkNsq\nq+sqωqNs\nkNsp\np\u0003\nδ(Ωkpq)δkpqdpdq,\nwithωk=2Ω0k∥/k⊥,e∥(|e∥|=1) the axis of rotation, Ω0the constant rotating rate and θk\nthe opposite angle to k⊥in the triangle k⊥=p⊥+q⊥.\n5 Kolmogorov-Zakharov spectrum\n5.1 A toy model\nExact stationary solutions with a non-zero flux can be found from the previous kinetic equa-\ntions. We usually called these solutions the Kolmogorov-Zakharov spectra. To expose the\ntechnique to get such a solution, which implies the Zakharov transformation, I will introduce\na simple kinetic equation that must be seen as a toy model because it does not correspond to\nany physical system. This kinetic equation is the following\n∂Ek\n∂t=X\nskspsqZ\n∆skωk\u0002\nskEpEq+spEkEq+sqEkEp\u0003\nδ(skωk+spωp+sqωq)dpdq , (64)\n20SciPost Physics Lecture Notes Submission\nxpxq\n110\nxpxq\n110\nFigure 4: The isotropic kinetic equation (64) involves triadic interactions limited to\nthe infinitely long gray band. The Zakharov transformations (TZ1 and TZ2 are in\nthe right and left panels, respectively) consists in the exchange of the four regions\nseparated by dashes.\nwithωk=k2. The kinetic equation is isotropic and Ekis the one dimensional energy spectrum.\nAlso, the integration is limited to a two-dimensional region ( ∆) where the triadic relation is\nsatisfied (see Figure 4). Note that equation (64) verifies the energy conservation.\nThe first obvious solution that appears is the thermodynamic spectrum, Ek=Ak−2. Indeed,\nthe substitution of this solution into (64) gives trivially\n∂Ek\n∂t=X\nskspsqZ\n∆skωk\u0002\nskk2+spp2+sqq2\u0003A2\n(kpq)3δ(skωk+spωp+sqωq)dpdq =0 . (65)\nThe second solution is less trivial to obtain. We need to apply the Zakharov transformation.\nFirst, we need to write the previous equation in terms of adimensional wavenumbers. Let us\nintroduceξp≡p/kandξq≡q/k; then, we obtain\n∂Ek\n∂t=A2k2+2xX\nskspsqZ\n∆sk\nskξx\npξx\nq+spξx\nq+sqξx\np\nδ(sk+spξ2\np+sqξ2\nq)dξpdξq, (66)\nwhere we have also introduced Ek=Akxwith Aest positive quantity. The Zakharov transfor-\nmations correspond to the following change of variables\nξp→1\nξp,ξq→ξq\nξp,(TZ1) (67)\nand\nξp→ξp\nξq,ξq→1\nξq.(TZ2) (68)\nThe meaning of these transformations is illustrated in Figure 4. The method consists of splitting\nthe integral (66) into three equal contributions, applying the Zakharov transformation TZ1 to\n21SciPost Physics Lecture Notes Submission\nthe second and TZ2 to the third, leaving the first integral intact. We then combine the different\ncontributions and, after a few manipulations, obtain the following expression\n∂Ek\n∂t=A2\n3k2+2xX\nskspsqZ\n∆ξx\nqξx\nq\nsk+spξ−x\np+sqξ−x\nq\nsk+spξ−2x−1\np+sqξ−2x−1\nq\nδ(sk+spξ2\np+sqξ2\nq)dξpdξq. (69)\nTwo solutions appear, x=−2 and x=−3/2. The first corresponds to the thermodynamic spec-\ntrum (as mentioned above) for which the energy flux is zero and the second to the Kolmogorov-\nZakharov spectrum with a non-zero energy flux. These properties can be obtained by intro-\nducing the isotropic energy flux Πkas follows\n∂Ek\n∂t=−∂Πk\n∂k=A2\n3k2+2xI(x). (70)\nThis leads to the expression\nΠk(x) =−A2\n3k3+2x\n3+2xI(x). (71)\nWe can immediately see that x=−2 corresponds to a zero flux since I(−2) =0. On the other\nhand, the value x=−3/2 requires a little calculation since Iand the denominator are both\nzero. Using L’Hospital’s rule, we find\nϵ≡Πk(−3/2) =−A2\n3lim\nx→−3/2I(x)\n3+2x=A2\n3J, (72)\nwith\nJ≡− lim\nx→−3/2∂I(x)\n∂x=X\nskspsqZ\n∆ξ−3/2\nqξ−3/2\nq\nsk+spξ3/2\np+sqξ3/2\nq\nspξ2\npln(ξp) +sqξ2\nqln(ξq)\nδ(sk+spξ2\np+sqξ2\nq)dξpdξq. (73)\nThe direction of the cascade can be obtained numerically from this analytical expression. A\npositive value for Jmeans a direct energy cascade. The value of Jcan finally lead to an\nestimate of the Kolmogorov constant CKwith\nEk=CKpϵk−3/2and CK=vt3\nJ. (74)\nThe last task is to check the validity of the Kolmogorov-Zakharov spectrum. We require this\nsolution (the inertial range) to be independent of the largest and smallest scales. This is a\nquestion of convergence of the collisional integral (66). Solving this exercise gives a condition\nx1<x<x2which can prove that our solution x=−3/2 is physical if it is inside the domain\nof convergence.\nIn conclusion, we can say that wave turbulence is a solvable problem. Indeed, we have seen\nfrom this toy model that in principle we can determine all the properties of wave turbulence\nby giving the exact solutions, the direction of the cascade, evaluating the Kolmogorov constant\nand showing the existence of an inertial range. All these properties cannot be found in eddy\nturbulence, even using the exact Kolmogorov laws.\n22SciPost Physics Lecture Notes Submission\n5.2 Applications to Navier-Stokes\nFor capillary wave turbulence the isotropic Kolmogorov-Zakharov spectrum is [27]\nEk=Ak−7/4. (75)\nIt is an isotropic spectrum well observed experimentally (see the review [36]).\nFor inertial wave turbulence the problem is more complex because turbulence is anisotropic\nwith an energy cascade mainly in the direction transverse to the rotation axis. In this case, we\nhave to use a generalization of the Zakharov transformation. This leads to the axisymmetric\nKolmogorov-Zakharov spectrum [40]\nE(k⊥,k∥) =Ak−5/2\n⊥k−1/2\n∥. (76)\nNote that the convergence properties and the Kolmogorov constant have been studied only\nrecently [82]. This spectrum is observed experimentally and numerically [41,83,84 ].\nA Appendix: sequential time closures for four-wave interactions\nA.1 Fundamental equation\nIn this Appendix, we will present a detailed derivation of the kinetic equation for four-wave\ninteractions when we have only cubic nonlinearities. This situation is interesting for example\nfor gravitational waves [75]. Consider the following model without dissipation\n∂u(x,t)\n∂t=L(u) +ε2N(u,u,u), (77)\nwhere uis a random variable such that 〈u〉=0,Lis a linear operator which guarantees that the\nwaves are linear solutions to the problem, and Nis a non-linear cubic operator (thus involving\nquartic interactions). The coefficient εis a small parameter (0 <ε≪1) which measures the\namplitude of the non-linearities (it is originally a measure of the wave amplitude). The Fourier\ntransform of equation (77) takes the following schematic form when the canonical variables\nare introduced\n∂\n∂t+iskωk\nAsk(k,t) =ε2X\nspsqsrZ\nR3dH−kpqrAsp(p,t)Asq(q,t)Asr(r,t)δ(k−p−q−r)dpdqdr.\n(78)\nHere,Hkpqris an operator symmetric in p,qandr. As for three-wave interactions, we con-\nsider a continuous medium which can lead to mathematical difficulties connected with infinite\ndimensional phase spaces. For this reason, it is preferable to assume a variable spatially peri-\nodic over a box of finite size L. However, in the derivation of the kinetic equation, the limit\nL→+∞is finally taken (before the long time limit, or equivalently the limit ε→0). As\nboth approaches lead to the same kinetic equation, for simplicity, we anticipate this result and\nfollow the original approach of [31].\nEquation (78) is simplified by making the following change of variables\nAsk(k,t) =ask(k,t)e−iskωkt=ask\nke−iskωkt. (79)\nIn the interaction representation, we obtain the fundamental equation\n∂ask\nk\n∂t=ε2X\nspsqsrZ\nR3dH−kpqrasp\npasq\nqasr\nreiΩk,pqrtδk,pqrdpdqdr, (80)\n23SciPost Physics Lecture Notes Submission\nwith by definition δk,pqr≡δ(k−p−q−r)andΩk,pqr≡skωk−spωp−sqωq−srωr. An\nadditional property can be established if we use the relation Ask(k) = ( A−sk(−k))∗, with∗\nthe complex conjugate: then, we have necessarily Hkpqr =H∗\n−k−p−q−r. Finally, Hkpqrbeing\npurely imaginary, we have H∗\nkpqr=−Hkpqr. This property may be used to simplify the kinetic\nequations.\nEquation (80) highlights the temporal evolution of the amplitude of the wave: this evolu-\ntion is a priori relatively slow since it induces a non-linear term proportional to ε2, which is\nweaker than for three-wave interactions. The presence of the complex exponential is funda-\nmental for the asymptotic closure: since we are interested in the dynamics over a long time\nwith respect to the period of the waves, the contribution of this exponential is essentially zero.\nOnly some terms will survive: those for which Ωk,pqr=0. With the condition imposed by the\nDirac, we obtain the resonance condition (whose solutions are discussed eg. in [1])\n\u001a\nk=p+q+r,\nskωk=spωp+sqωq+srωr.(81)\nA.2 Multiple time scale method\nFollowing our analysis for three-wave interactions, we use the method of multiple time scales\nand introduce a sequence of time scales, T0,T1,T2, ..., which will be treated as independent\nvariables, with\nT0≡t,T1≡εt,T2≡ε2t, ... . (82)\nThe variation of the wave amplitude with T1,T2,T3andT4represents the slow variation that\nwe wish to extract. We obtain ( T0being replaced by t)\n∂\n∂t+ε∂\n∂T1+ε2∂\n∂T2+...\nask\nk=ε2X\nspsqsrZ\nR3dH−kpqrasp\npasq\nqasr\nreiΩk,pqrtδk,pqrdpdqdr. (83)\nThe variable ask\nkmust also be expanded to the power of εand make the various scales appear\nin time\nask\nk=+∞X\nn=0εnask\nk,n(t,T1,T2, ...) =ask\nk,0+εask\nk,1+ε2ask\nk,2+... . (84)\nThe previous expression is then introduced into the fundamental equation (83). We obtain for\nthe first five terms\n∂ask\nk,0\n∂t=0 , (85a)\n∂ask\nk,1\n∂t=−∂ask\nk,0\n∂T1, (85b)\n∂ask\nk,2\n∂t=−∂ask\nk,1\n∂T1−∂ask\nk,0\n∂T2+bsk\nk,2, (85c)\n∂ask\nk,3\n∂t=−∂ask\nk,2\n∂T1−∂ask\nk,1\n∂T2−∂ask\nk,0\n∂T3+bsk\nk,3(85d)\n∂ask\nk,4\n∂t=−∂ask\nk,3\n∂T1−∂ask\nk,2\n∂T2−∂ask\nk,1\n∂T3−∂ask\nk,0\n∂T4+bsk\nk,4. (85e)\nwith\nbsk\nk,2=X\nspsqsrZ\nR3dH−kpqrasp\np,0asq\nq,0asr\nr,0eiΩk,pqrtδk,pqrdpdqdr, (86)\n24SciPost Physics Lecture Notes Submission\nbsk\nk,3=X\nspsqsrZ\nR3dH−kpqr\nasp\np,1asq\nq,0asr\nr,0+asp\np,0asq\nq,1asr\nr,0+asp\np,0asq\nq,0asr\nr,1\neiΩk,pqrtδk,pqrdpdqdr(87)\nbsk\nk,4=X\nspsqsrZ\nR3dH−kpqr\nasp\np,2asq\nq,0asr\nr,0+asp\np,0asq\nq,2asr\nr,0+asp\np,0asq\nq,0asr\nr,2+asp\np,1asq\nq,1asr\nr,0(88)\n+asp\np,1asq\nq,0asr\nr,1+asp\np,0asq\nq,1asr\nr,1\neiΩk,pqrtδk,pqrdpdqdr.\nAfter integration over t, one finds with some simplifications\nask\nk,0=ask\nk,0(T1,T2, ...), (89a)\nask\nk,1=−t∂ask\nk,0\n∂T1, (89b)\nask\nk,2=t2\n2∂2ask\nk,0\n∂T2\n1−t∂ask\nk,0\n∂T2+Zt\n0bsk\nk,2dt, (89c)\nask\nk,3=−t3\n6∂3ask\nk,0\n∂T3\n1+t2∂2ask\nk,0\n∂T1∂T2−t∂ask\nk,0\n∂T3+Zt\n0bsk\nk,3dt−Zt\n0Zt\n0∂bsk\nk,2\n∂T1dtdt , (89d)\nask\nk,4=t4\n24∂4ask\nk,0\n∂T4\n1−t3\n2∂3ask\nk,0\n∂T2\n1∂T2+t2∂2ask\nk,0\n∂T1∂T3+t2\n2∂2ask\nk,0\n∂T2\n2−t∂ask\nk,0\n∂T4+Zt\n0bsk\nk,4dt (89e)\n−Zt\n0∂bsk\nk,2\n∂T1dt−Zt\n0Zt\n0∂bsk\nk,2\n∂T2dtdt +Zt\n0Zt\n0Zt\n0∂2bsk\nk,2\n∂T2\n1dtdtdt .\nA.3 First asymptotic closure at time T1\nWith the previous definitions, the perturbative expansion of the second-order moment writes\n〈ask\nkask′\nk′〉=〈(ask\nk,0+εask\nk,1+ε2ask\nk,2+...)(ask′\nk′,0+εask′\nk′,1+ε2ask′\nk′,2+...)〉 (90)\n=〈ask\nk,0ask′\nk′,0〉+ε〈ask\nk,0ask′\nk′,1+ask\nk,1ask′\nk′,0〉+ε2〈ask\nk,0ask′\nk′,2+ask\nk,1ask′\nk′,1+ask\nk,2ask′\nk′,0〉+...\nWe assume a turbulence statistically homogeneous, then we have qsksk′(k,k′)≡qsksk′\nksuch that\n〈ask\nkask′\nk′〉=qsksk′\nkδ(k+k′). (91)\nAs with three-wave interactions, we assume that the second-order moment on the left hand\nside (in fact, the coefficient qsksk′\nkin front of the Dirac function) remains bounded at all time,\nand so the contributions from the right hand side must also be bounded. At order O(ε0), we\nhave the contribution of 〈ask\nk,0ask′\nk′,0〉which will be assumed to be bounded at all times.\nAt order O(ε1), we find\n〈ask\nk,0ask′\nk′,1+ask\nk,1ask′\nk′,0〉=−t∂\n∂T1〈ask\nk,0ask′\nk′,0〉. (92)\nThis unique term is secular which means that over the long time limit we have to impose\n∂\n∂T1〈ask\nk,0ask′\nk′,0〉=0 . (93)\nThis result can be easily generalized to any nth-order moments. Therefore, the probability\ndensity function does not depend on T1and we can assume that the variable itself does not\ndepend on T1; we have at most\nask\nk,0=ask\nk,0(T2,T3, ...). (94)\n25SciPost Physics Lecture Notes Submission\nThis result drastically simplifies the previous expressions, which read\nask\nk,1=0 , (95a)\nask\nk,2=−t∂ask\nk,0\n∂T2+Zt\n0bsk\nk,2dt, (95b)\nask\nk,3=−t∂ask\nk,0\n∂T3+Zt\n0bsk\nk,3dt, (95c)\nask\nk,4=t2\n2∂2ask\nk,0\n∂T2\n2−t∂ask\nk,0\n∂T4+Zt\n0bsk\nk,4dt−Zt\n0Zt\n0∂bsk\nk,2\n∂T2dtdt . (95d)\nA.4 Second asymptotic closure at time T2\nAt order O(ε2), we find\n¬\nask\nk,0ask′\nk′,2+ask\nk,2ask′\nk′,0¶\n=®\nask\nk,0\n−t∂ask′\nk′,0\n∂T2+Zt\n0bsk′\nk′,2dt\n+\n−t∂ask\nk,0\n∂T2+Zt\n0bsk\nk,2dt\nask′\nk′,0¸\n=−t∂\n∂T2〈ask\nk,0ask′\nk′,0〉+\u001c\nask\nk,0Zt\n0bsk′\nk′,2dt\u001d\n+\u001c\nask′\nk′,0Zt\n0bsk\nk,2dt\u001d\n, (96)\nwith for the second term on the right hand side\n\u001c\nask\nk,0Zt\n0bsk′\nk′,2dt\u001d\n=X\nspsqsrZ\nR3dH−k′pqr〈asp\np,0asq\nq,0asr\nr,0ask\nk,0〉∆(Ωk′,pqr)δk′,pqrdpdqdr. (97)\nThis term may produce a secular contribution in the long time (recall that ∆(0) =tand that\nHkpqris purely imaginary) but it is exactly compensated by the third term on the right hand\nside. Therefore, to keep the moment bounded over the long time, we impose\n∂\n∂T2〈ask\nk,0ask′\nk′,0〉=0 . (98)\nA.5 Third asymptotic closure at time T3\nAt next order O(ε3), we find\n¬\nask\nk,0ask′\nk′,3+ask\nk,3ask′\nk′,0¶\n=®\nask\nk,0\n−t∂ask′\nk′,0\n∂T3+Zt\n0bsk′\nk′,3dt\n+\n−t∂ask\nk,0\n∂T3+Zt\n0bsk\nk,3dt\nask′\nk′,0¸\n=−t∂\n∂T3〈ask\nk,0ask′\nk′,0〉+\u001c\nask\nk,0Zt\n0bsk′\nk′,3dt+ask′\nk′,0Zt\n0bsk\nk,3dt\u001d\n, (99)\nbut according to the first closure, bsk\nk,3=0. Therefore, we find the condition over the long time\n∂\n∂T3〈ask\nk,0ask′\nk′,0〉=0 . (100)\nAll these results mean that the dynamics is slower than T3. As we will see later, the kinetic\nequation for four-wave interactions can be derived to the next order.\n26SciPost Physics Lecture Notes Submission\nA.6 Fourth asymptotic closure at time T4\nAt order O(ε4), we have\n¬\nask\nk,0ask′\nk′,4+ask\nk,2ask′\nk′,2+ask\nk,4ask′\nk′,0¶\n= (101)\n®\nask\nk,0\nt2\n2∂2ask′\nk′,0\n∂T2\n2−t∂ask′\nk′,0\n∂T4+Zt\n0bsk′\nk′,4dt−Zt\n0Zt\n0∂bsk′\nk′,2\n∂T2dtdt¸\n+®\nask′\nk′,0\nt2\n2∂2ask\nk,0\n∂T2\n2−t∂ask\nk,0\n∂T4+Zt\n0bsk\nk,4dt−Zt\n0Zt\n0∂bsk\nk,2\n∂T2dtdt¸\n+®\n−t∂ask\nk,0\n∂T2+Zt\n0bsk\nk,2dt\n−t∂ask′\nk′,0\n∂T2+Zt\n0bsk′\nk′,2dt¸\n,\nwhich gives after development and simplifications\n¬\nask\nk,0ask′\nk′,4+ask\nk,2ask′\nk′,2+ask\nk,4ask′\nk′,0¶\n=t2\n2∂2\n∂T2\n2¬\nask\nk,0ask′\nk′,0¶\n−t∂\n∂T4¬\nask\nk,0ask′\nk′,0¶\n(102)\n+\u001c\nask\nk,0Zt\n0bsk′\nk′,4dt\u001d\n+\u001c\nask′\nk′,0Zt\n0bsk\nk,4dt\u001d\n−®\nask\nk,0Zt\n0Zt\n0∂bsk′\nk′,2\n∂T2dtdt¸\n−®\nask′\nk′,0Zt\n0Zt\n0∂bsk\nk,2\n∂T2dtdt¸\n−t®∂ask\nk,0\n∂T2Zt\n0bsk′\nk′,2dt¸\n−t®∂ask′\nk′,0\n∂T2Zt\n0bsk\nk,2dt¸\n+\u001cZt\n0bsk\nk,2dtZt\n0bsk′\nk′,2dt\u001d\n.\nAccording to the second closure, the first term on the right hand side is null. The other terms\nmay give secular contributions. The second term on the right hand side is trivially secular. The\nthird term can be written \u001c\nask\nk,0Zt\n0bsk′\nk′,4dt\u001d\n= (103)\nX\nspsqsrZ\nR3dH−k′pqrZt\n0¬\nask\nk,0asp\np,2asq\nq,0asr\nr,0+ask\nk,0asp\np,0asq\nq,2asr\nr,0+ask\nk,0asp\np,0asq\nq,0asr\nr,2¶\neiΩk′,pqrtdtδk′,pqrdpdqdr,\nwith\nask\nk,2=−t∂ask\nk,0\n∂T2+X\nspsqsrZ\nR3dH−kpqrasp\np,0asq\nq,0asr\nr,0∆(Ωk,pqr)δk,pqrdpdqdr. (104)\nWe find\n\u001c\nask\nk,0Zt\n0bsk′\nk′,4dt\u001d\n=−X\nspsqsrZ\nR3dH−k′pqr*\nask\nk,0∂(asp\np,0asq\nq,0asr\nr,0)\n∂T2+\nW(Ωk′,pqr)δk′,pqrdpdqdr\n+X\nspsqsrsp′sq′sr′Z\nR6d3H−k′pqrH−pp′q′r′¬\nask\nk,0asq\nq,0asr\nr,0asp′\np′,0asq′\nq′,0asr′\nr′,0¶\nX(Ωk′,p′q′r′qr,Ωk′,pqr)\nδp,p′q′r′δk′,pqrdp′dq′dr′dpdqdr (105)\nwhere\nW(X)≡Zt\n0teiX tdt=teiX t\niX−∆(X)\nix, (106)\nand by definition\nX(X,Y)≡Zt\n0∆(X−Y)eiY tdt=∆(X)−∆(Y)\ni(X−Y), (107)\n27SciPost Physics Lecture Notes Submission\nwhose long time limit is given by the theory of generalized functions (here we consider X̸=0)\nX(X, 0)t→+∞−−−−−→πtδ(X) +itP1\nX\n. (108)\nThe fifth term on the right hand side of expression (102) reads\n®\nask\nk,0Zt\n0Zt\n0∂bsk′\nk′,2\n∂T2dtdt¸\n= (109)\nX\nspsqsrZ\nR3dH−k′pqr*\nask\nk,0∂(asp\np,0asq\nq,0asr\nr,0)\n∂T2+\nY(Ωk′,pqr)δk′,pqrdpdqdr,\nwith by definition\nY(X)≡Zt\n0∆(X)dt=∆(X)−t\niX. (110)\nThe seventh term on the right hand side of expression (102) reads\nt®∂ask\nk,0\n∂T2Zt\n0bsk′\nk′,2dt¸\n=tX\nspsqsrZ\nR3dH−k′pqr®∂ask\nk,0\n∂T2asp\np,0asq\nq,0asr\nr,0¸\n∆(Ωk′,pqr)δk′,pqrdpdqdr\n(111)\nAnd the last term on the right hand side of expression (102) reads\n\u001cZt\n0bsk\nk,2dtZt\n0bsk′\nk′,2dt\u001d\n=X\nspsqsrsp′sq′sr′Z\nR6dH−kpqrH−k′p′q′r′¬\nasp\np,0asq\nq,0asr\nr,0asp′\np′,0asq′\nq′,0asr′\nr′,0¶\n∆(Ωk,pqr)∆(Ωk′,p′q′r′)δk,pqrδk′,p′q′r′dpdqdrdp′dq′dr′. (112)\nWe recall long time limit ( X̸=0)\n∆(X)∆(−X)t→+∞−−−−−→ 2πtδ(X) +2P1\nX∂\n∂X. (113)\nThe addition of all these terms gives after simplifications\n¬\nask\nk,0ask′\nk′,4+ask\nk,2ask′\nk′,2+ask\nk,4ask′\nk′,0¶\n=−t∂\n∂T4¬\nask\nk,0ask′\nk′,0¶\n(114)\n−tX\nspsqsrZ\nR3dH−k′pqr∂¬\nask\nk,0asp\np,0asq\nq,0asr\nr,0¶\n∂T2∆(Ωk′,pqr)δk′,pqrdpdqdr\n−tX\nspsqsrZ\nR3dH−kpqr∂¬\nask′\nk′,0asp\np,0asq\nq,0asr\nr,0¶\n∂T2∆(Ωk,pqr)δk,pqrdpdqdr\n+X\nspsqsrsp′sq′sr′Z\nR6d3H−k′pqrH−pp′q′r′¬\nask\nk,0asq\nq,0asr\nr,0asp′\np′,0asq′\nq′,0asr′\nr′,0¶\nX(Ωk′,p′q′r′qr,Ωk′,pqr)\nδp,p′q′r′δk′,pqrdp′dq′dr′dpdqdr\n+X\nspsqsrsp′sq′sr′Z\nR6d3H−kpqrH−pp′q′r′¬\nask′\nk′,0asq\nq,0asr\nr,0asp′\np′,0asq′\nq′,0asr′\nr′,0¶\nX(Ωk,p′q′r′qr,Ωk,pqr)\nδp,p′q′r′δk,pqrdp′dq′dr′dpdqdr\n+X\nspsqsrsp′sq′sr′Z\nR6dH−kpqrH−k′p′q′r′¬\nasp\np,0asq\nq,0asr\nr,0asp′\np′,0asq′\nq′,0asr′\nr′,0¶\n∆(Ωk,pqr)∆(Ωk′,p′q′r′)\nδk,pqrδk′,p′q′r′dpdqdrdp′dq′dr′.\n28SciPost Physics Lecture Notes Submission\nTo further simplify the previous expression one needs to consider the fourth-order moment.\nWith ask\nk,1=0, we find\n〈ask\nkask′\nk′ask′′\nk′′ask′′′\nk′′′〉=〈(ask\nk,0+ε2ask\nk,2+...)(ask′\nk′,0+ε2ask′\nk′,2+...)...〉 (115)\n=〈ask\nk,0ask′\nk′,0...〉+ε2〈ask\nk,2ask′\nk′,0...+ask\nk,0ask′\nk′,2...+...〉+...\nAs before, we assume that the left hand side is bounded, which means that at each order the\ncontribution from the right hand side must also be bounded. At order O(ε2)we have\n〈ask\nk,2ask′\nk′,0ask′′\nk′′,0ask′′′\nk′′′,0+ask\nk,0ask′\nk′,2ask′′\nk′′,0ask′′′\nk′′′,0+...〉= (116)\n®\n−t∂ask\nk,0\n∂T2+Zt\n0bsk\nk,2dt\nask′\nk′,0ask′′\nk′′,0ask′′′\nk′′′,0+ask\nk,0\n−t∂ask′\nk′,0\n∂T2+Zt\n0bsk′\nk′,2dt\nask′′\nk′′,0ask′′′\nk′′′,0+...¸\n=−t∂¬\nask\nk,0ask′\nk′,0ask′′\nk′′,0ask′′′\nk′′′,0¶\n∂T2\n+X\nspsqsrZ\nR3dH−kpqr¬\nasp\np,0asq\nq,0asr\nr,0ask′\nk′,0ask′′\nk′′,0ask′′′\nk′′′,0¶\n∆(Ωk,pqr)δk,pqrdpdqdr+... .\nSecular contributions (with ∆(0) =t) can emerge from terms involving sixth-order moments\nwhich must be cancelled with the first term in the right hand side. We obtain three types of\ncontribution, namely\n∂¬\nask\nk,0ask′\nk′,0ask′′\nk′′,0ask′′′\nk′′′,0¶\n∂T2=X\nspZ\nRdH−kp−pkqsp−sp\np,0¬\nask\nk,0ask′\nk′,0ask′′\nk′′,0ask′′′\nk′′′,0¶\ndp (117)\n+X\nspZ\nRdH−kpk−pqsp−sp\np,0¬\nask\nk,0ask′\nk′,0ask′′\nk′′,0ask′′′\nk′′′,0¶\ndp+X\nsqZ\nRdH−kkq−qqsq−sq\nq,0¬\nask\nk,0ask′\nk′,0ask′′\nk′′,0ask′′′\nk′′′,0¶\ndq+....\nUsing the symmetries of Hkpqr, we eventually find\n∂¬\nask\nk,0ask′\nk′,0ask′′\nk′′,0ask′′′\nk′′′,0¶\n∂T2= (118)\n3¬\nask\nk,0ask′\nk′,0ask′′\nk′′,0ask′′′\nk′′′,0¶X\nspZ\nRd\u0000\nH−kk−pp+H−k′k′−pp+H−k′′k′′−pp+H−k′′′k′′′′−pp\u0001\nqsp−sp\np,0dp.\nFrom the second closure, we know that the second-order moment does not depend on T2,\nand therefore the left hand side can be reduced to a fourth-order cumulant. On the right\nhand side, we see that the decomposition in terms of second-order moments does not give\nany contribution because of the pairwise cancellation of the Hcoefficients. Therefore, this\nequation can be reduced to (with k+k′+k′′+k′′′=0)\n∂qsksk′sk′′sk′′′\nkk′k′′\n∂T2= (119)\n3qsksk′sk′′sk′′′\nkk′k′′X\nspZ\nRd\u0000\nH−kk−pp+H−k′k′−pp+H−k′′k′′−pp+H−k′′′k′′′′−pp\u0001\nqsp−sp\np,0dp\nwhose exact solution is\nqsksk′sk′′sk′′′\nkk′k′′ =q0e3T2P\nspR\nRd(H−kk−pp+H−k′k′−pp+H−k′′k′′−pp+H−k′′′k′′′′−pp)qsp−sp\np,0dp, (120)\n29SciPost Physics Lecture Notes Submission\nwith q0=qsksk′sk′′sk′′′\nkk′k′′ (t=0)the value of the fourth-order cumulant at t=0. Also, if the fourth-\norder cumulant is zero initially (if we assume a Gaussian statistics), it will remain so at time\nT2, and there is therefore no contribution from the derivatives in T2in expression (114). This\nproperty may be linked to the propagation of chaos discussed by mathematicians [85]. Note\nthat the assumption of random phase approximation is often made for the derivation of the\nkinetic equation [86]. We can also note that the solution (120) corresponds to an oscillating\nterm (we recall that Hkpqris purely imaginary and qsp−sp\np,0is real) and keeping this term in the\nkinetic equation will not lead to a contribution at time T4. Therefore, we obtain\n¬\nask\nk,0ask′\nk′,4+ask\nk,2ask′\nk′,2+ask\nk,4ask′\nk′,0¶\n=−t∂\n∂T4¬\nask\nk,0ask′\nk′,0¶\n(121)\n+X\nspsqsrsp′sq′sr′Z\nR6dH−kpqrH−k′p′q′r′¬\nasp\np,0asq\nq,0asr\nr,0asp′\np′,0asq′\nq′,0asr′\nr′,0¶\n∆(Ωk,pqr)∆(Ωk′,p′q′r′)\nδk,pqrδk′,p′q′r′dpdqdrdp′dq′dr′\n+3X\nspsqsrsp′sq′sr′Z\nR6dH−k′pqrH−pp′q′r′¬\nask\nk,0asq\nq,0asr\nr,0asp′\np′,0asq′\nq′,0asr′\nr′,0¶\nX(Ωk′,p′q′r′qr,Ωk′,pqr)\nδp,p′q′r′δk′,pqrdp′dq′dr′dpdqdr\n+3X\nspsqsrsp′sq′sr′Z\nR6dH−kpqrH−pp′q′r′¬\nask′\nk′,0asq\nq,0asr\nr,0asp′\np′,0asq′\nq′,0asr′\nr′,0¶\nX(Ωk,p′q′r′qr,Ωk,pqr)\nδp,p′q′r′δk,pqrdp′dq′dr′dpdqdr,\nfrom which it remains to find the secular contributions. To do this, it is necessary to develop\nthe sixth-order moments into the product of the lower order moments plus the cumulant.\nLet us illustrate this with the second term on the right hand side of the previous expression.\nWe have 41 terms which are written (to simplify the writing, we keep the moment notation\nof the fourth-order cumulant; we only use the cumulant notation for the sixth-order cumulant\nfor which there is an ambiguity)\n〈asp\np,0asq\nq,0asr\nr,0asp′\np′,0asq′\nq′,0asr′\nr′,0〉= (122)\n〈asp\np,0asq\nq,0〉〈asr\nr,0asp′\np′,0〉〈asq′\nq′,0asr′\nr′,0〉+〈asp\np,0asr\nr,0〉〈asq\nq,0asp′\np′,0〉〈asq′\nq′,0asr′\nr′,0〉+〈asp\np,0asp′\np′,0〉〈asq\nq,0asr\nr,0〉〈asq′\nq′,0asr′\nr′,0〉\n+〈asp\np,0asq′\nq′,0〉〈asq\nq,0asr\nr,0〉〈asp′\np′,0asr′\nr′,0〉+〈asp\np,0asr′\nr′,0〉〈asq\nq,0asr\nr,0〉〈asp′\np′,0asq′\nq′,0〉+〈asp\np,0asq\nq,0〉〈asr\nr,0asq′\nq′,0〉〈asp′\np′,0asr′\nr′,0〉\n+〈asp\np,0asq\nq,0〉〈asr\nr,0asr′\nr′,0〉〈asp′\np′,0asq′\nq′,0〉+〈asp\np,0asr\nr,0〉〈asq\nq,0asr′\nr′,0〉〈asp′\np′,0asq′\nq′,0〉+〈asp\np,0asr\nr,0〉〈asq\nq,0asq′\nq′,0〉〈asp′\np′,0asr′\nr′,0〉\n+〈asp\np,0asp′\np′,0〉〈asq\nq,0asq′\nq′,0〉〈asr\nr,0asr′\nr′,0〉+〈asp\np,0asp′\np′,0〉〈asq\nq,0asr′\nr′,0〉〈asr\nr,0asq′\nq′,0〉+〈asp\np,0asq′\nq′,0〉〈asq\nq,0asp′\np′,0〉〈asr\nr,0asr′\nr′,0〉\n+〈asp\np,0asq′\nq′,0〉〈asq\nq,0asp′\np′,0〉〈asr\nr,0asp′\np′,0〉+〈asp\np,0asr′\nr′,0〉〈asq\nq,0asp′\np′,0〉〈asr\nr,0asq′\nq′,0〉+〈asp\np,0asr′\nr′,0〉〈asq\nq,0asq′\nq′,0〉〈asr\nr,0asr′\nr′,0〉\n+〈asp\np,0asq\nq,0asr\nr,0〉〈asp′\np′,0asq′\nq′,0asr′\nr′,0〉+〈asp\np,0asq\nq,0asp′\np′,0〉〈asr\nr,0asq′\nq′,0asr′\nr′,0〉+〈asp\np,0asq\nq,0asq′\nq′,0〉〈asr\nr,0asp′\np′,0asr′\nr′,0〉\n+〈asp\np,0asq\nq,0asr′\nr′,0〉〈asr\nr,0asp′\np′,0asq′\nq′,0〉+〈asp\np,0asp′\np′,0asr\nr,0〉〈asq\nq,0asq′\nq′,0asr′\nr′,0〉+〈asp\np,0asq′\nq′,0asr′\nr′,0〉〈asp′\np′,0asq′\nq′,0asr\nr,0〉\n+〈asp′\np′,0asq\nq,0asr\nr,0〉〈asp\np,0asq′\nq′,0asr′\nr′,0〉+〈asq′\nq′,0asq\nq,0asr\nr,0〉〈asp′\np′,0asp\np,0asr′\nr′,0〉+〈asr′\nr′,0asq\nq,0asr\nr,0〉〈asp′\np′,0asq′\nq′,0asp\np,0〉\n+〈asp\np,0asr′\nr′,0asr\nr,0〉〈asq\nq,0asq′\nq′,0asp′\np′,0〉\n+〈asp\np,0asq\nq,0〉〈asr\nr,0asp′\np′,0asq′\nq′,0asr′\nr′,0〉+〈asp\np,0asr\nr,0〉〈asq\nq,0asp′\np′,0asq′\nq′,0asr′\nr′,0〉+〈asp\np,0asp′\np′,0〉〈asr\nr,0asq\nq,0asq′\nq′,0asr′\nr′,0〉\n+〈asp\np,0asq′\nq′,0〉〈asr\nr,0asq\nq,0asp′\np′,0asr′\nr′,0〉+〈asp\np,0asr′\nr′,0〉〈asq\nq,0asr\nr,0asp′\np′,0asq′\nq′,0〉+〈asr\nr,0asq\nq,0〉〈asp\np,0asp′\np′,0asq′\nq′,0asr′\nr′,0〉\n30SciPost Physics Lecture Notes Submission\n+〈asp′\np′,0asq\nq,0〉〈asr\nr,0asp\np,0asq′\nq′,0asr′\nr′,0〉+〈asq\nq,0asq′\nq′,0〉〈asr\nr,0as\np,0asp′\np′,0asr′\nr′,0〉+〈asq\nq,0asr′\nr′,0〉〈asp\np,0asr\nr,0asp′\np′,0asq′\nq′,0〉\n+〈asr\nr,0asp′\np′,0〉〈asp\np,0asq\nq,0asq′\nq′,0asr′\nr′,0〉+〈asr\nr,0asq′\nq′,0〉〈asp\np,0asq\nq,0asp′\np′,0asr′\nr′,0〉+〈asr\nr,0asr′\nr′,0〉〈asp\np,0asq\nq,0asp′\np′,0asq′\nq′,0〉\n+〈asp′\np′,0asq′\nq′,0〉〈asp\np,0asq\nq,0asr\nr,0asr′\nr′,0〉+〈asp′\np′,0asr′\nr′,0〉〈asp\np,0asq\nq,0asp′\np′,0asr\nr,0〉+〈asq′\nq′,0asr′\nr′,0〉〈asp\np,0asq\nq,0asp′\np′,0asr\nr,0〉\n+qspsqsrsp′sq′sr′\npqrp′q′δ(p+q+r+p′+q′+r′).\nThe same form is obtained for the third and fourth terms of expression (121).\nWe will consider the super-secular contributions ∝t2that can emerge because of the\nrelation ∆(0)∆(0) =t2. The first nine terms give the same super-secular contributions; after\nsimplification one finds\n9t2X\nspsqZ\nR2dHkkp−pH−k−kq−qqsk\nkqsp\npqsq\nqδkk′dpdq. (123)\nThe third term in expression (121) gives the following super-secular contributions (due to\nX(0, 0) =t2/2)\n9t2\n2X\nspsqZ\nR2dHkkp−pHkkq−qqsk\nkqsp\npqsq\nqδkk′dpdq. (124)\nFinally, the fourth term in expression (121) gives for the same reason the same super-secular\ncontributions as above. The addition of the three super-secular contributions gives zero be-\ncause of the relation H−k−kq−q=−Hkk−qq=−Hkkq−q.\nNext, we will consider the (classical) secular terms ∝t. The first nine terms of expression\n(122) give no contribution. The next six terms give the same contribution, the addition of\nwhich gives\n12πtX\nspsqsrZ\nR3dHkpqrH−k−p−q−rqsp\npqsq\nqqsr\nrδk,pqrδ(Ωk,pqr)dpdqdrδkk′. (125)\nThe product of the third-order cumulant give no secular contribution, but the six products\nof the second-order cumulant with fourth-order cumulant give a secular contribution. We find\nthe following expression\n3tX\nspsp′sq′sr′Z\nR4dHkkp−pH−kp′q′r′qsp\npqsrsp′sq′sr′\np′q′r′δ−k,p′q′r′∆(Ω−k,p′q′r′)dpdp′dq′dr′δkk′(126)\n+3tX\nspsqsrsp′Z\nR4dHkpqrH−k−kp′−p′qsp′\np′qspsqsrsr′\npqrδk,pqr∆(Ωk,pqr)dpdqdrdp′δkk′,\nfor which the long time limit is given by the Riemann-Lebesgue lemma. Finally, the sixth-order\ncumulant gives no secular contribution.\nThe third term in expression (121) gives secular contributions. First of all, they come from\nproducts of second-order cumulants; we obtain in the long time limit\n18tX\nspsqsrZ\nR3dHkpqrHpk−q−rqsk\nkqsq\nqqsr\nrδk,pqr\u0012\nπδ(Ωk,pqr)−iP\u0012\n1\nΩk,pqr\u0013\u0013\ndpdqdrδkk′.\nThe other contributions come from products of second-order cumulant with fourth-order cu-\nmulant; we have\n6X\nspsp′sq′sr′Z\nR4dHkkp−pHpp′q′r′qsk\nkqsrsp′sq′sr′\np′q′r′δp,p′q′r′X(Ωp,p′q′r′, 0)dpdp′dq′dr′δkk′(127)\n+3X\nsrsp′sq′sr′Z\nR4dHkkr−rHkp′q′r′qsr\nrqsk′sp′sq′sr′\np′q′r′δk,p′q′r′X(Ωk,p′q′r′, 0)drdp′dq′dr′δkk′,\n31SciPost Physics Lecture Notes Submission\nwith the long time limit\nX(X, 0)t→+∞−−−−−→πtδ(X) +itP1\nX\n. (128)\nThe fourth term in expression (121) gives secular contributions similar to the third term (we\nonly need to exchange kandk′).\nThen, we obtain the following condition to cancel the contribution of secular terms in the\nlong time limit\n∂qsk\nk\n∂T4=12πX\nspsqsrZ\nR3dHkpqrH−k−p−q−rqsp\npqsq\nqqsr\nrδk,pqrδ(Ωk,pqr)dpdqdr (129)\n+3X\nspsp′sq′sr′Z\nR4dHkkp−pH−kp′q′r′qsp\npqsrsp′sq′sr′\np′q′r′δ−k,p′q′r′\u0012\nπδ(Ω−k,p′q′r′)−iP\u0012\n1\nΩ−k,p′q′r′\u0013\u0013\ndpdp′dq′dr′\n+3X\nspsqsrsp′Z\nR4dHkpqrH−k−kp′−p′qsp′\np′qspsqsrsr′\npqrδk,pqr\u0012\nπδ(Ωk,pqr)−iP\u0012\n1\nΩk,pqr\u0013\u0013\ndpdqdrdp′\n+18X\nspsqsrZ\nR3dHkpqrHpk−q−rqsk\nkqsq\nqqsr\nrδk,pqr\u0012\nπδ(Ωk,pqr)−iP\u0012\n1\nΩk,pqr\u0013\u0013\ndpdqdr\n+6X\nspsp′sq′sr′Z\nR4dHkkp−pHpp′q′r′qsk\nkqsrsp′sq′sr′\np′q′r′δp,p′q′r′\u0012\nπδ(Ωp,p′q′r′)−iP\u0012\n1\nΩp,p′q′r′\u0013\u0013\ndpdp′dq′dr′\n+3X\nsrsp′sq′sr′Z\nR4dHkkr−rHkp′q′r′qsr\nrqsk′sp′sq′sr′\np′q′r′δk,p′q′r′\u0012\nπδ(Ωk,p′q′r′)−iP\u0012\n1\nΩk,p′q′r′\u0013\u0013\ndrdp′dq′dr′\n+18X\nspsqsrZ\nR3dH−kpqrHp−k−q−rqsk\nkqsq\nqqsr\nrδ−k,pqr\u0012\nπδ(Ω−k,pqr)−iP\u0012\n1\nΩ−k,pqr\u0013\u0013\ndpdqdr\n+6X\nspsp′sq′sr′Z\nR4dH−k−kp−pHpp′q′r′qsk\nkqsrsp′sq′sr′\np′q′r′δp,p′q′r′\u0012\nπδ(Ωp,p′q′r′)−iP\u0012\n1\nΩp,p′q′r′\u0013\u0013\ndpdp′dq′dr′\n+3X\nsrsp′sq′sr′Z\nR4dH−k−kr−rH−kp′q′r′qsr\nrq−sksp′sq′sr′\np′q′r′δk,p′q′r′\u0012\nπδ(Ω−k,p′q′r′)−iP\u0012\n1\nΩ−k,p′q′r′\u0013\u0013\ndrdp′dq′dr′.\nAt this level, symmetries are very useful. First, we can show that the contributions of the\nproducts between second- and third-order cumulants cancel out exactly. Then, the remaining\nterms can be simplified using the symmetries of Hkpqrand by playing with the dummy variables\nof integration. Finally, we find\n∂qsk\nk\n∂T4=12πX\nspsqsrZ\nR3dHkpqrH−k−p−q−rqsp\npqsq\nqqsr\nrδk,pqrδ(Ωk,pqr)dpdqdr\n+36πX\nspsqsrZ\nR3dHkpqrHpk−q−rqsk\nkqsq\nqqsr\nrδk,pqrδ(Ωk,pqr)dpdqdr.\nA last simplification can be made by splitting the second term on the right hand side in three\nsymmetrical contributions. We then obtain the kinetic equation for four-wave interactions\n(with T4=ε4t)\n∂qsk\nk\n∂t=12ε4πX\nspsqsrZ\nR3dHkpqr\n−Hkpqr\nqsk\nk+Hpk−q−r\nqsp\np+Hpk−p−r\nqsq\nq+Hrk−p−q\nqsr\nr\nqsk\nkqsp\npqsq\nqqsr\nr\nδk,pqrδ(Ωk,pqr)dpdqdr. (130)\n32SciPost Physics Lecture Notes Submission\nThis equation describes wave turbulence at the level of four-wave interactions. 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In order to apply\nGCI in a sensible manner it is shown in [3, Lemma 3.1] that any c entred\nGaussian measure µon a finite dimensional, real normed space Ω can be\nwritten as a mixture of two measures νgoodandνbad.νgoodis supported\non an (essentially) compact set and is dominated by µin the sense defined\nbelow. Crucially, νbadis dominated by a dilated version of µ, allowing for\neasy error estimates on the part of µwhere large deviations are possible.\nIn this note we present a slightly more general, quantitativ e version of this\nmixture decomposition.\nFor two probability measures νandµwe write ν∝√recedesequalµwheneverdν\ndµis the\nfinite product of non-negative, symmetric quasi-concave fu nctions or the\nuniformly bounded limit of such functions. The relation ∝√recedesequalis interesting be-\ncause whenever µis centred Gaussian it is a consequence of GCI that\nν(f)≤µ(f)\nforfquasi-convex and bounded. For more information on this form of\nGaussian domination see e.g. [3].\nDefine for each Borel set B∈ B(Ω) and n≥1\nµ×n(B) :=µ(B/n).\nE-mail address :tobias.schmidt@tu-darmstadt.de .\n12 EXPLICIT BOUNDS FOR A GAUSSIAN DECOMPOSITION LEMMA OF SELL KE\nIt is easily verified that whenever µis standard normal on Rd, then\ndµ\ndµ×n∝exp/parenleftbigg\n−n2−1\n2n2∝bardblx∝bardbl2/parenrightbigg\n.\nIn the following, Kcdenotes the complement of K.\nLemma 1. Letµbe a centered Gaussian measure on (Ω,B(Ω)). Further-\nmore, let K∈ B(Ω)be a symmetric, closed convex set with µ(Kc)≤δ <0.5\nand\nπ/parenleftbig\n2log(δ−1)+8/parenrightbig\n≤δ−1.\nThen, there exist two probability measures νgood,νbadandδ′>0such that\nµ= (1−δ′)νgood+δ′νbad.\nMoreover, for n >1,C:= 32n2\nn2−1it holds that\nR.1δ′≤2δ,\nR.2supp(νgood)⊆CK,\nR.3νgood∝√recedesequalµ,\nR.4νbad∝√recedesequalµ×n.\nWe briefly compare the results from Lemma 1 with the original s tatement\n[3, Lemma 3.1]. The main difference is that the original statem ent shows\nthe existence of a constant C, whereas we provide exact values for δsmall\nenough. Moreover, we allow arbitrary dilations of µto dominate νbad. Com-\npared to the original result we only show δ′≤2δ, but the reader will easily\nverify that the additional factor of 2 can be omitted by incre asing the value\nofCaccordingly. The same arguments can be used to trade the requ ire-\nment\nπ/parenleftbig\n2log(δ−1)+8/parenrightbig\n≤δ−1\nfor justδ <0.5 (this condition suffices to always find a centred ball B(ǫ) of\nsome radius ǫ >0 contained in K), but paying the price of increasing C.\nIndeed, scaling Kby an additional factor ˜ cuntilµ((˜cK)c) is small enough\nfor the condition in the last display to hold suffices.\nWe follow the original proof provided in [3] for the most part and only\nchange the arguments whenever it is more convenient with res pect toC.\nFor example, we do not provide a new argument to show R.4, but to show\nR.1we refrain from using the isoperimetric inequality in order to keep track\nofC.\nProof of Lemma 1. W.l.o.g. we may assume that Ω = Rdwithd= dim(Ω)\nand that µis standard normal on Rd. We first claim that\nB(/radicalbig\nlog(δ−1))⊆K. (1)EXPLICIT BOUNDS FOR A GAUSSIAN DECOMPOSITION LEMMA OF SELLK E 3\nTo show this, recall that the standard normal on Rdis rotation-invariant.\nAssume that there is a point x∈Kcthat isc:=/radicalbig\nlog(δ−1)-close to the\norigin (i.e., the biggest radius ball we could fit into Kwould be exactly\nthat). Let Gbe the hyperplane separating {x}andK. By rotating G\nwe may w.l.o.g. assume that it lies parallel above the d-th basis axis with\ndistance c. It follows that, using the convention that erfc(0) = 1,\nµ(Kc)≥(2π)−1/2∞/integraldisplay\ncexp/parenleftbigg\n−1\n2x2/parenrightbigg\ndx=1\n2erfc(c/√\n2).\nNow, estimating the erfc term with a well-known lower bound ( see e.g. [1,\ninequality 7.1.13])\n1\n2erfc(c/√\n2)>1√πexp/parenleftbig\n−1\n2log(δ−1)/parenrightbig\n/radicalBig\nlog(δ−1)\n2+/radicalBig\nlog(δ−1)\n2+2\n≥1√πδ1/2\n2/radicalBig\nlog(δ−1)\n2+2=: (∗).\nAs/radicalbig\nπ(2log(δ−1)+8)≤δ−1/2\nby assumption,\n(∗)≥δ1/2\nδ−1/2=δ.\nLooking back, we have shown that\nµ(Kc)> δ.\nThe conclusion is that a set with µ(K)≥1−δmust contain all points in\nB(/radicalbig\nlog(δ−1)) as erfc( ·) is non-increasing. This shows (1).\nWe denote\nR:=/radicalbig\nlog(δ−1), C:=8n2\nn2−1>1\nand define\nd(x) :=d(x,K).\nWith this notation the measure split is given by\nνbad(dx)∝e−σ(d(x))µ(dx),\nνgood(dx)∝1−e−σ(d(x))µ(dx),(2)\nwhereσ: [0,∞]→[0,R2] is anyC2\nbfunction satisfying\n(1)σis non-increasing,\n(2)σ(x) =R2forx∈[0,1],\n(3)σ(x) = 0 for x≥3CR,4 EXPLICIT BOUNDS FOR A GAUSSIAN DECOMPOSITION LEMMA OF SELL KE\n(4)|σ(x)′| ≤(x−1)\nCforx≥1,\n(5)|σ(x)′′| ≤1\nCforx≥0.\nSuch a function can be constructed explicitly (see [3, Lemma 3.1]). We now\nshow that all claimed properties are fulfilled by the measure s in (2).\nTo show R.1simply estimate\nδ′=E/bracketleftBig\ne−σ(d(x))/bracketrightBig\n≤exp(−R2)+1−µ(K)\n≤δ+δ\n= 2δ.\nForR.2, fix˜C >1 and note that d(x)≥(˜C−1)Rwhenever x /∈˜CK, which\nshows that d(x)≥3CRwhenever x /∈4CK. This implies that\nsupp(νgood)⊆4CK.\nBy definition Kis symmetric and convex, so it holds thatdνgood\ndµis (QC) and\ntherefore R.3is fulfilled.\nWe finally show that\ndνbad\ndµ×n∝exp/parenleftbigg\n−σ(d(x))−n2−1\n2n2∝bardblx∝bardbl2/parenrightbigg\n(3)\nis log-concave. The required symmetry such that the density is (QC) is\nevident.\nTo show convexity of\nx∝ma√sto→σ(d(x))+n2−1\n2n2∝bardblx∝bardbl2\ntakex,z∈Rdand set for some p∈(0,1)\ny:=px+(1−p)z.\nNote that\nσ(d(x))≥σ(∝bardblx−P(y)∝bardbl).\nOf course, the same holds whenever xis replaced by z. Using this inequality\nand rearranging\nσ(d(y))+n2−1\n2n2∝bardbly∝bardbl2≤p/parenleftbigg\nσ(d(x))+n2−1\n2n2∝bardblx∝bardbl2/parenrightbigg\n+(1−p)/parenleftbigg\nσ(d(z)) +n2−1\n2n2∝bardblz∝bardbl2/parenrightbigg\n,EXPLICIT BOUNDS FOR A GAUSSIAN DECOMPOSITION LEMMA OF SELLK E 5\nthe claim is proven whenever\npσ(∝bardblx−P(y)∝bardbl)+(1−p)σ(∝bardblz−P(y)∝bardbl)−σ(d(y))\n≥ −n2−1\n2n2/parenleftbig\np∝bardblx∝bardbl2+(1−p)∝bardblz∝bardbl2−∝bardbly∝bardbl2/parenrightbig\n=−p(1−p)∝bardblx−z∝bardbl2n2−1\n2n2(4)\nis shown. Denote theline connecting xandzbyxzandleto=p′x+(1−p′)z\nbe the projection of P(y) ontoxz. Moreover, let wx:=−∝bardblx−o∝bardbland\nwz:=∝bardblz−o∝bardbl.Clearly|wx−wz|=∝bardblx−z∝bardbl. By writing out the definitions,\npwx+(1−p)wz= (p′−p)∝bardblx−z∝bardbl.\nIt follows with wy:=∝bardbly−o∝bardbl=|p′−p|∝bardblx−z∝bardblthat\nw2\ny= (pwx+(1−p)wz)2.\nDefinea:=∝bardblo−P(y)∝bardbland assume w.l.o.g. that a >0. If this is not the\ncase, do a limiting procedure in equation (6) by using the con tinuity of σ.\nThen,\nd(x,P(y))2=a2+w2\nx,\nd(y,P(y))2=a2+(pwx+(1−p)wz)2,\nd(z,P(y))2=a2+w2\nz.\nDefine\nf(w) :=σ/parenleftBig/radicalbig\na2+w2/parenrightBig\n.\nUsing the chain rule,\nf′(w) =w√\na2+w2σ′/parenleftBig/radicalbig\na2+w2/parenrightBig\n,\nf′′(w) =w2\na2+w2σ′′/parenleftBig/radicalbig\na2+w2/parenrightBig\n+a2\n(a2+w2)3/2σ′/parenleftBig/radicalbig\na2+w2/parenrightBig\n.\nVia property 5 of σ,\nw2\na2+w2/vextendsingle/vextendsingle/vextendsingleσ′′/parenleftBig/radicalbig\na2+w2/parenrightBig/vextendsingle/vextendsingle/vextendsingle≤1\nC.\nThe fourth property allows to bound\na2\n(a2+w2)3/2/vextendsingle/vextendsingle/vextendsingleσ′/parenleftBig/radicalbig\na2+w2/parenrightBig/vextendsingle/vextendsingle/vextendsingle=a2\na2+w2/vextendsingle/vextendsingle/vextendsingleσ′/parenleftBig√\na2+w2/parenrightBig/vextendsingle/vextendsingle/vextendsingle\n√\na2+w2≤1\nC.\nIn total one obtains\nsup\nw≥0|f′′(w)| ≤2\nC. (5)\nThe introduction of flets us rewrite the left-hand-side of equation (4) to\npf(wx)+(1−p)f(wz)−f(pwx+(1−p)wz). (6)6 EXPLICIT BOUNDS FOR A GAUSSIAN DECOMPOSITION LEMMA OF SELL KE\nA simple Taylor expansion gives (6) ≥ −2p(1−p)(wx−wz)2sup\nw≥0|f′′(w)|.\nApplying estimate (5) yields\npf(wx)+(1−p)f(wz)−f(pwx+(1−p)wz)≥ −p(1−p)4∝bardblx−z∝bardbl2\nC.\nBy choice of Cinequality (4) is fulfilled, which shows the claim. /square\nAcknowledgement: we thank Volker Betz and Mark Sellke for helpful\nremarks. The author was partially supported by the german re search asso-\nciation (DFG) grant 535662048.\nReferences\n[1] M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions with\nFormulas, Graphs and Mathematical Tables. US Government printing office , 1968.\n[2] T. Royen. A simple proof of the Gaussian correlation conj ecture extended to\nmultivariate gamma distributions Far East Journal of Theoretical Statistics , Vol 48,\nIssue 2, 139–145, 2014.\n[3] M. Sellke. Almost Quartic Lower Bound for the Fr¨ ohlich P olaron’s Effective Mass\nvia Gaussian Domination arXiv:2212.14023 [math-ph]"    },    {        "title": "2402.15015v1.Turbulence_unsteadiness_drives_extreme_clustering.pdf",        "content": "Turbulence unsteadiness drives extreme clustering\nF. Zapata,1, 2,∗S. Angriman,1, 2A. Ferran,3, 4P. Cobelli,1, 2M. Obligado,3and P.D. Mininni1, 2\n1Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales,\nDepartamento de F ´ısica, Ciudad Universitaria, 1428 Buenos Aires, Argentina,\n2CONICET - Universidad de Buenos Aires, Instituto de F ´ısica Interdisciplinaria\ny Aplicada (INFINA), Ciudad Universitaria, 1428 Buenos Aires, Argentina.\n3Universit ´e Grenoble Alpes, CNRS, Grenoble-INP , LEGI, F-38000 Grenoble, France\n4Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195-2600, USA\nWe show that the unsteadiness of turbulence has a drastic effect on turbulence parameters and in particle clus-\nter formation. To this end we use direct numerical simulations of particle laden flows with a steady forcing that\ngenerates an unsteady large-scale flow. Particle clustering correlates with the instantaneous Taylor-based flow\nReynolds number, and anti-correlates with its instantaneous turbulent energy dissipation constant. A dimen-\nsional argument for these correlations is presented. In natural flows, unsteadiness can result in extreme particle\nclustering, which is stronger than the clustering expected from averaged inertial turbulence effects.\nOne of the most counterintuitive effects of turbulence is\nthat, when a fluid is loaded with inertial particles, the flow\ncan segregate the particles instead of mixing them [1, 2]. This\nphenomenon, which results in the formation of clusters with\nenhanced particle density, is relevant in volcanic clouds [3, 4],\nto explain cloud formation [5] and electrification [6], in other\ngeophysical and natural contexts [7], and for industrial appli-\ncations. In homogeneous and isotropic turbulence, two mech-\nanisms govern the formation of clusters: particles with small\ninertia are expelled out of vortices [8], while particles with\nlarge inertia accumulate near points with zero net forces [9].\nTurbulence is an out-of-equilibrium phenomenon that is of-\nten studied in the statistical steady state, i.e., when external\nforces and dissipation balance in such a way that the system\nhas well defined time averages. However, in many natural\nand industrial systems this is not the case. Out-of-equilibrium\nsystems can fluctuate randomly between two or more states,\nin such a way that time averages never converge [10]. Un-\nsteadiness affects energy dissipation rates and the flow spec-\ntral properties [11]. This in turn has an effect in the mixing\nand transport of particles. As an example, it has been reported\nthat motile particles such as phytoplankton can change their\ndirection of migration in response to overturning events asso-\nciated to the turbulent flow in which they move [12].\nWhat is the effect of unsteadiness in passive particles’ clus-\nter formation? And is the formation and evolution of clusters\nin realistic flows driven by turbulence, by the flow unsteadi-\nness, or by a combination of both? Here we show that the\nnaturally occurring modulation of out-of-equilibrium systems\nin time has a drastic effect on turbulence parameters and in\nparticle cluster formation. Moreover, we show that cluster-\ning correlates with a small time delay with the instantaneous\nTaylor-based Reynolds number of the flow, and anticorrelates\nwith its instantaneous turbulent energy dissipation rate. Hys-\nteresis is present in this process, indicating the particles pre-\nserve a memory of previous states. We present a dimensional\nargument that considers this phenomenon as a change in the\nparticles’ effective inertia (measured by the Stokes number)\ndepending on the flow state. This result allows for estimation\nof turbulent parameters from particles measurements, and in-dicates that flow unsteadiness must be considered in the study\nof many multiphase flows.\nWe performed direct numerical simulations (DNSs) of the\nincompressible Navier-Stokes equation\n∂tu+u·∇u=−∇p+ν∇2u+F, (1)\nwhere uis the solenoidal fluid velocity field ( ∇·u=0),\npis the pressure per unit mass density, νis the kinematic\nviscosity, and Fis an external volumetric mechanical forc-\ning. Equations are written in dimensionless units based on a\nunit length L0and a unit velocity U0, and solved in a three-\ndimensional 2 πL0-periodic cubic box with a parallel pseudo-\nspectral method using the GHOST code [13, 14]. Spatial res-\nolutions of N3=5123, 7683, and 10243grid points were used,\nyielding increasingly larger Reynolds numbers with kinematic\nviscosities respectively of ν512=1.1×10−3L0U0,ν768=\n6.7×10−4L0U0, and ν1024=4.6×10−4L0U0. The exter-\nnal forcing Fgenerates large-scale periodic counter-rotating\ncolumns (in the following abbreviated as CRC). It was used\nbefore to study unsteadiness in [15], and is given by\nF=F0[sin(x)cos(y)ˆx−cos(x)sin(y)ˆy] (2)\nThis forcing corresponds to an array of four counter-rotating\nvortices in the xyplane, with translational symmetry in z.\nThe first columnar vortex occupies the volume [0,πL0)×\n[0,πL0)×[0,2πL0), and is separated from the others by two\nvertical shear layers in the middle of the domain, aligned re-\nspectively with the xyandxzplanes.\nWe also performed DNSs of homogeneous and isotropic\nturbulence (HIT) with random forcing, to compare against the\nCRC runs, following the same procedures used for the CRC\nforcing and using N3=7683and 10243grid points. In these\nsimulations the flow was sustained using a forcing with fixed\namplitude and random phases, which were slowly evolved in\ntime with a correlation time of 0 .5 large-scale eddy turnover\ntimes to prevent the development of a mean flow. The forcing\nwas applied at the lowest wave numbers, resulting in an inte-\ngral length scale L≈1.1L0. The kinematic viscosities were\nν768=3.1×10−4L0U0andν1024=2.1×10−4L0U0. A DNSarXiv:2402.15015v1  [physics.flu-dyn]  22 Feb 20242\n300 400 500 600 700 800\nRλ0.100.140.180.22C/epsilon1−1 N3= 5123\nN3= 7683\nN3= 10243\n250 500 750\nRλ2×1034×1036×103\nC/epsilon1/√Re0 −1\nFIG. 1. Instantaneous value of Cεas a function of Rλfor the CRC\nflow at three different spatial resolutions. Each resolution corre-\nsponds to a different viscosity and to a different averaged Reynolds\nnumber. Inset: Cεcompensated by√Re0as a function of Rλ.\nsimilar to HIT, but with time dependent forcing amplitude to\nsynthetically generate unsteadiness, is discussed in [16]. All\nsimulations have κη>1, where κ=N/3 is the largest re-\nsolved wave number, η= (ν3/ε)1/4is the dissipation scale,\nandεis the energy dissipation rate.\nIn all simulations we integrated a simple model of one way\ncoupled and heavy point particles with equation of motion\n˙xp=v(t),˙v=1\nτp[u(xp,t)−v(t)], (3)\nwhere u(xp,t)is the fluid velocity at the particle position xp\nat time t, and v(t)andτpare respectively the particle velocity\nand the particle Stokes time. In each run different sets of par-\nticles were added, each with 106particles and with different\nvalues of τp. The Stokes numbers of these sets, St =τp/τη\n(where τη= (ν/ε)1/2is the Kolmogorov dissipation time of\nthe flow), were St =3 and 8 for all flows and all spatial reso-\nlutions considered. A third set with St =14 was also evolved\nonly in the simulations with 10243grid points. For the CRC\nruns, τηand St are the time average over very long times.\nThe overall dynamics of the flows is as follows. While the\nHIT simulations display fluctuations in global quantities with\na correlation time proportional to the integral turnover time,\nthe CRC runs display distinct dynamics. Large excursions\nin the energy dissipation and other global quantities are ob-\nserved, resulting from the flow transitioning from two states:\none in which the large-scale columns can be clearly recog-\nnized (e.g., by direct inspection of the instantaneous spatial\ndistribution of particles), and one in which the columns be-\ncome unstable and the system displays a more homogeneous\nstate. A movie of this time evolution can be seen in [16].\nAn out-of-equilibrium dissipation law has been reported in\na variety of unsteady turbulent flows [17, 18], such that\nCε∼√Re0\nRλ, (4)\n10−210−1100\nV=V//angbracketleftV/angbracketright10−210−1100PDF(a) CRC St = 3\nCRC St = 8HIT St = 3\nHIT St = 8\n10−210−1100\nV=V//angbracketleftV/angbracketright(b) CRC St = 3\nCRC St = 8\nCRC St = 14HIT St = 3\nHIT St = 8\nHIT St = 14FIG. 2. Probability density functions (PDFs) of the normalized\nV onoro ¨ı volumes V=V/⟨V⟩of inertial particles for CRC and HIT\nsimulations, for (a) N3=7683, and (b) N3=10243. For CRC forc-\ning, the PDFs are time-averaged over long times. A Random Poisson\nProcess (RPP) is indicated by the dashed line.\nwhere Re 0=u0l0/νis a global Reynolds number based on\nthe initial r.m.s. flow velocity u0and the initial integral length\nscale l0, and Rλ=u(t)λ(t)/νis the local-in-time Reynolds\nnumber based on the instantaneous Taylor length scale λ(t)\nand r.m.s. turbulent velocity u(t).Cεis given by the energy\ndissipation rate as ε(t) =Cεu3(t)/L(t), where L(t)is the in-\nstantaneous flow integral scale. For a turbulent steady state\n(e.g., in HIT) this relation reduces to the well-known dissi-\npation law ε=Cεu3/Lthat states that the energy dissipation\nrate is governed by the large-scale energy flux towards smaller\nscales. In this sense, in unsteady flows Cε(t)provides a mea-\nsure of temporal scale-by-scale energy imbalance.\nIn Fig. 1 we see that the CRC flows display excursions com-\npatible with Eq. (4) as Cεis inversely proportional to Rλ. In\nthe inset we show Cεcompensated by the square root of the\nreference Reynolds number Re 0; note how all curves from\nflows with different viscosities collapse. For a given viscosity\n(e.g., for N3=5123)RλandCεchange in time by factors of 2,\nwith a typical time scale of the excursions of 10 ⟨T⟩to 20⟨T⟩,\nwhere ⟨T⟩is the mean large-scale eddy turnover time (see de-\ntails below). These excursions, as well as their characteristic\ntime scale, are much larger than those associated to the fluctu-\nations in simulations of HIT (which take place in time scales\nof the order of the turnover time). In spite of these differ-\nences, the instantaneous energy spectrum of the CRC simula-\ntions still displays Kolmogorov scaling (not shown).\nWe want to know if these excursions in the dissipation and\nin the Taylor-based Reynolds number, similar to those re-\nported in other unsteady turbulent flows [15, 17, 18], affect\nparticle cluster formation and time evolution. To estimate the\namount of clustering in the different simulations we calculated\nthe three dimensional V orono ¨ı tessellation of the particles as a\nfunction of time. V orono ¨ı diagrams have proven to be a pow-\nerful tool to study particle clustering [19, 20]. The V orono ¨ı\ncell associated to a given particle at a certain time is defined\nas the set of points closer to that particle than to any other\nparticle. The volumes of the V orono ¨ı cells Vwere normal-3\n0 20 40 60 80 100\nt//angbracketleftT/angbracketright2.55.07.5σV(d)300500700Rλ(c)0.51.01.5/epsilon1(b)0.10.20.3C/epsilon1(a) N3= 5123St = 3\nN3= 7683St = 3\nN3= 10243St = 3N3= 512 St = 8\nN3= 768 St = 8\nN3= 1024 St = 8\n−5 0 5\nτ//angbracketleftT/angbracketright0.51fcorr\nFIG. 3. Time series of (a) Cε, (b)ε, (c)Rλ, and (d) σVfor CRC runs\nwith St =3 and 8. Time is normalized by the mean turnover time\n⟨T⟩, and σRPPis indicated by the dotted grey line in panel (d). Inset:\ncross-correlation of σVandRλas a function of the time lag τ.\nized by the mean volume of all cells ⟨V⟩, to define normalized\nvolumes V=V/⟨V⟩. Figure 2 shows the time-averaged prob-\nability density functions (PDFs) of the normalized volumes,\ncompared against the PDF resulting from a random Poisson\nprocess (herein RPP, corresponding to a homogeneous distri-\nbution of particles [21]) which is shown as a reference. The\nstronger the tails of the PDFs compared against the RPP (i.e.,\nthe excess of probability for small Vcorresponding to an ex-\ncess of clusterized small volumes, or for large Vcorrespond-\ning to large voids), and the larger the standard deviation of the\nPDFs, σV(compared to the RPP which has a standard devia-\ntion of σRPP≈0.42), indicate enhanced clustering. Positions\nof the maxima also change as variables in the PDFs are nor-\nmalized to obtain mean volumes occupied per particle.\nAs shown in Fig. 2, the CRC runs present stronger clus-\ntering than HIT, when comparing cases with the same Stokes\nnumber. As St increases (at fixed spatial resolution) clustering\ndiminishes, indicating that particles with more inertia cluster\nless both in CRC and HIT flows (results presented here are\nfor St≳1, as for St →0 and →∞particles do not cluster;\nnote that in the CRC flow for St =14 clustering is similar to\n246σV(a)\n300 400 500 600 700 800\nRλ246σV(b)\nN3= 5123\nN3= 7683\nN3= 10243FIG. 4. Standard deviation of the V orono ¨ı volumes, σV, as a function\nofRλfor CRC runs, differentiating branches in which Rλincreases\n(solid lines) and decreases (dashed lines), for (a) St =3, and (b) 8.\nRed arrows indicate the direction of time evolution.\nSt=8 and less than for 3). Differences between the CRC flow\nand HIT could in principle be associated with the presence of\na large-scale flow in the CRC runs, but this is not sufficient\nto explain the observed enhancement in clustering. In other\nturbulent flows with a steady large-scale circulation, particle\nclustering was observed to be closer to that of HIT [22]. The\nreason for the stronger clustering here becomes more clear\nwhen inspecting the instantaneous PDFs of V. Note that all\nPDFs in Fig. 2 are averaged over a time window of ≈5⟨T⟩,\nusing 106V orono ¨ı volumes in each snapshot with a cadence\nof at least 0 .04⟨T⟩. But while the PDFs of Vin HIT are sta-\ntionary, the PDFs in the CRC runs are not (see a movie with\nthe PDFs as a function of time in [16]).\nThe instantaneous PDFs of the CRC flow vary significantly\nin time, and the time averaged PDFs showed in Fig. 2 alone\nare not representative of the actual level of particle clustering.\nTo study how particle clustering is affected by the flow un-\nsteadiness, we calculated the time evolution of Cε,ε,Rλ, and\nthe standard deviation of the V orono ¨ı volumes, σV. Results\nare shown in Fig. 3. Particles are injected when the flows are\nalready in a fully developed turbulent regime, at a time arbi-\ntrarily labeled as t=0, and at random positions in space (note\nthat at t=0,σV=σRPPin all cases). After a short transient,\nparticles form clusters as indicated by σV>σRPP. It has been\nreported before [15] that this flow displays irregular behavior\nof large-scale quantities. Our results for CεandRλare com-\npatible with this observation: both quantities display large ex-\ncursions with a characteristic time scale much larger than the\nintegral turnover time. Fluctuations in Cεhave a correlation4\n0 1 2 3 4 5 6 7 8\nσV0.00.51.01.5PDFHIT\nCRC\nFIG. 5. PDFs of σVfor HIT and CRC runs with N3=7683and for\nparticles with St=8. Note the larger values of σV(i.e., enhanced\nclustering) in the CRC run, as well as the larger dispersion.\ntime between 10 ⟨T⟩to 20⟨T⟩, and result in fluctuations of the\nflow dissipation rate εwith a similar characteristic time. More\nsurprising are the fluctuations in σV, which can vary between\n≈2.5 to values larger than 6, indicating strong variations in\nthe level of particle clustering as the flow evolves.\nComparing the four quantities in Fig. 3 we see that when\nCεandεdisplay a local minimum, RλandσVdisplay lo-\ncal maxima (i.e., turbulence becomes stronger and particles\ncluster more). To quantify the correlation between cluster-\ning and Rλ, the cross-correlation function fcorrofσVandRλ\nis shown in an inset in Fig. 3, for the simulation with 5125\ngrid points as this simulation has the longest integration in\ntime. The maximum cross-correlation is reached for time in-\ncrements |τ|/⟨T⟩≲1 (negative increments are associated to\nparticles clustering during the growth of Rλ). Similar results\nare obtained when the cross-correlation is computed with Cε.\nThus, CεorRλandσVare correlated with a time lag that is\nproportional to the large-scale eddy turnover time. The exis-\ntence of this time lag suggests that the response of the particles\nto changes in flow properties may display hysteresis.\nFigure 4 shows σVas a function of Rλin CRC runs, for par-\nticles with St =3 and 8, and for different viscosities and spa-\ntial resolutions. Time intervals with increasing Rλare marked\nwith solid lines, while intervals with decreasing Rλare in-\ndicated with dashed lines. The figure indicates a Reynolds\nnumber dependence, and further confirms the correlation be-\ntween these quantities and the existence of a time lag with a\nhysteresis cycle superposed over the strong fluctuations.\nAs previously mentioned, this level of particle clustering\nand its fluctuations are stronger than in HIT. Figure 5 com-\npares the PDFs of σVfor CRC forcing and for HIT, in the case\nwith N3=7683and St =8. The two flows display distinct\nvalues of σV. In HIT σVtakes values between 0 .5 and 1 .8,\nwhile in the CRC flow values go from 0 .5 to 7 .4, reaching its\nmaximum value around ≈4.5. Not only is σVon average sig-\nnificantly larger in the CRC flow (i.e., particles cluster more),\nbut the dispersion in the values of σVis also much larger, in-cluding instances (albeit less probable) in which the clustering\nis similar to that found in HIT, as the ones captured by the left\ntail of the PDF. The dispersion in σV, and the correlation with\nRλ, confirm that the extreme clustering observed in this flow\nis associated to its unsteady dynamics (see also [16]).\nHow does the out-of-equilibrium dynamics of the flow af-\nfect the particle behavior? We can consider first the situation\nin which turbulence is in a scale-by-scale steady state. Under\nthese conditions, Re 0=u0l0/ν≈uL/ν(where u=⟨u(t)⟩is\nthe time average of the r.m.s. flow velocity, and L=⟨L(t)⟩is\nthe averaged flow integral scale). The energy dissipation rate\nis also ε≈u3/L. For a small spherical particle we can write\nthe Stokes time as τp=2a2ρp/(9ρfν)(where ais the parti-\ncle radius, ρpis the particle density, and ρfis the fluid den-\nsity). Then St =τp/τη= (2/9)(ρp/ρf)(a/L)2Re3/2\n0. Thus,\nthe Stokes number of the particles (and as a result, the sensi-\ntivity of the particles to flow fluctuations at different scales)\nis fixed given a particle radius, a mass density ratio, and a\nflow Reynolds number. However, when scale-by-scale steadi-\nness is broken, we must use ε=CεU3/L, which using Eq. (4)\nresults in St ∼(2/9)(ρp/ρf)(a/L)2Re7/4\n0/R1/2\nλ. The sensi-\ntivity of the particles to flow fluctuations thus changes de-\npending on the instantaneous dissipation (or the Taylor-based\nReynolds number). Equivalently, we could interpret this ex-\npression as the effective ratio of the particle size to the flow\nscale being replaced by a/(LR1/4\nλ): the integral scale of the\nflow “seen” by the particle depends on Rλ. Perhaps counter-\nintuitively, for larger Rλthe particles become more sensitive\nto the larger scale eddies, which results in stronger clustering\nas can be seen in the movie in [16]. When Rλis smaller ( Cε\nlarger) the effective Stokes number of the particles is larger\nand the spatial distribution of particles is more homogeneous,\nwith clustering closer to that observed in inertial clustering in\nHIT. When Rλis larger ( Cεsmaller), St is smaller, and parti-\ncles accumulate outside the large-scale columnar vortices in a\nphenomenon reminiscent of turbophoresis (i.e., expelled from\nthe vortices [8]), resulting in extreme clusterization. As dis-\ncussed previously, this argument can hold for St larger than 1\nbut in its vicinity, as considered in our simulations.\nNatural flows are unsteady: volcanic and other sources of\nparticulate material pulsate and oscillate in time, convection\nin clouds is inhomogeneous, and atmospheric turbulence in\ngeneral is bursty [23]. The results presented here show that\nflow unsteadiness, which in this case occurs naturally as a\nsteady forcing is used, can drastically enhance particle cluster-\ning, well above what previous studies have reported for steady\nstate homogeneous and isotropic turbulence. Thus this effect,\nwhich has been neglected so far, can change estimations of\nclustering and particle aggregation for many systems. As an\nexample, estimations of collision frequencies between parti-\ncles are proportional to n2∼ V−2(where nis the particle den-\nsity), and changes in the volume per particle Vaffect the num-\nber of collisions. The results also open the door to the estima-\ntion of instantaneous turbulence parameters from direct obser-\nvations of particle aggregation (albeit taking into account that5\nthe dependence on Re 0could be different for different flows),\nand can be useful for the study of other out-of-equilibrium\nunsteady systems.\nThis work has been partially supported by ECOS-\nSud Project No. A18ST04. F.Z., S.A., P.C., and\nP.D.M. acknowledge financial support from UBACyT Grant\nNo. 20020170100508BA and PICT Grant No. 2018-4298.\nA.F. and M.O. acknowledge the LabEx Tec21 (Investisse-\nments d’Avenir - Grant Agreement # ANR-11-LABX-0030).\n∗Corresponding author:fnzapata@df.uba.ar\n[1] L. Brandt and F. Coletti, Annual Review of Fluid Mechanics\n54, 159 (2022).\n[2] M. Obligado, T. Teitelbaum, A. Cartellier, P. D. Mininni, and\nM. Bourgoin, Journal of Turbulence 15, 293 (2014).\n[3] T. Esposti Ongaro and M. Cerminara, Journal of V olcanology\nand Geothermal Research 326, 127 (2016).\n[4] M. Ichihara, P. D. Mininni, S. Ravichandran, C. Cimarelli,\nand C. Vagasky, Communications Earth & Environment 4, 417\n(2023).\n[5] R. A. Shaw, Annual Review of Fluid Mechanics 35, 183 (2003).\n[6] M. Di Renzo and J. Urzay, Nature Communications 9, 1676\n(2018).\n[7] R. E. Breier, C. C. Lalescu, D. Waas, M. Wilczek, and M. G.\nMazza, Proceedings of the National Academy of Sciences 115,\n12112 (2018).\n[8] L.-P. Wang and M. R. Maxey, Journal of Fluid Mechanics 256,27 (1993).\n[9] S. Coleman and J. Vassilicos, Physics of Fluids 21, 113301\n(2009).\n[10] A. de la Torre and J. Burguete, Physical Review Letters 99,\n054101 (2007).\n[11] S. Berti, G. Boffetta, and S. Musacchio, Physical Review Fluids\n8, 054601 (2023).\n[12] A. Sengupta, F. Carrara, and R. Stocker, Nature 543, 555\n(2017).\n[13] P. D. Mininni, D. Rosenberg, R. Reddy, and A. Pouquet, Paral-\nlel Computing 37, 316 (2011).\n[14] D. Rosenberg, P. D. Mininni, R. Reddy, and A. Pouquet, Atmo-\nsphere 11, 178 (2020).\n[15] S. Goto and J. Vassilicos, Physics Letters A 379, 1144 (2015).\n[16] For the analysis of a third forcing, that synthetically generates\nunsteadiness in an isotropic and homogeneous set-up, and for\na movie showing the time evolution of the CRC flow and the\nstatistics of the particles, see the Supplemental Material.\n[17] P. Valente, R. Onishi, and C. da Silva, Physical Review E 90,\n023003 (2014).\n[18] J. C. Vassilicos, Atmosphere 14, 736 (2023).\n[19] R. Monchaux, M. Bourgoin, and A. Cartellier, Physics of Fluids\n22(2010).\n[20] R. Monchaux, M. Bourgoin, and A. Cartellier, International\nJournal of Multiphase Flow 40, 1 (2012).\n[21] M. Uhlmann, Physica A: Statistical Mechanics and its Applica-\ntions 555, 124618 (2020).\n[22] S. Angriman, A. Ferran, F. Zapata, P. J. Cobelli, M. Obligado,\nand P. D. Mininni, Journal of Fluid Mechanics 948, A30 (2022).\n[23] P. R. Imazio, P. D. Mininni, A. Godoy, N. Rivaben, and\nA. D ¨ornbrack, Journal of Geophysical Research: Atmospheres\n128, e2022JD037491 (2023).Supplemental Material: Turbulence unsteadiness drives extreme clustering\nF. Zapata,1, 2S. Angriman,1, 2A. Ferran,3, 4P. Cobelli,1, 2M. Obligado,3and P.D. Mininni1, 2\n1Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales,\nDepartamento de F ´ısica, Ciudad Universitaria, 1428 Buenos Aires, Argentina,\n2CONICET - Universidad de Buenos Aires, Instituto de F ´ısica Interdisciplinaria\ny Aplicada (INFINA), Ciudad Universitaria, 1428 Buenos Aires, Argentina.\n3Universit ´e Grenoble Alpes, CNRS, Grenoble-INP , LEGI, F-38000 Grenoble, France\n4Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195-2600, USA\nSIMULATIONS OF TIME-MODULATED HIT\nThe CRC forcing generates a flow which, besides being unsteady, also has an inhomogeneous and anisotropic large-scale\ncomponent. Thus, its extreme clustering could be influenced by these flow properties instead of being driven by turbulence\nunsteadiness. To help disentangle the effects of the large-scale flow from those caused by the modulation of the energy dissipation\nrate in particle clustering, we performed another DNS with a third type of forcing. In this DNS a random forcing was initially\ngenerated, exciting the same (isotropic) Fourier modes as those used in HIT. But instead of evolving the phases in time, the\nglobal amplitude of the forcing was modulated as\nF0(t) =A0cos(f0t)α, (1)\nwhere A0and f0are respectively the initial forcing amplitude, and the modulation frequency. Their values were chosen to\nhave fluctuations in εthat resemble those in the CRC flow. The parameters used were A0=3.5U2\n0/L0,f0= (π/15)U0/L0, and\nα=100. These values result in strong peaks in the forcing amplitude that repeat periodically, and that synthetically generate\na modulation of ε,Cε, and Rλ, but with a more homogeneous and isotropic forcing. We label this run as modulated HIT\n(mHIT) in the following. The simulation was made using a resolution of 5123grid points and with a kinematic viscosity\nνmHIT =1.1×10−3L0U0. As in all other simulations, this DNS has κη>1 at all times, where κ=512/3, and a mean integral\nlength scale L≈1L0.\nFigure 1 shows the instantaneous value of Cεas a function of Rλfor mHIT. Note that the out-of-equilibrium dissipation law\nreported for other unsteady turbulent flows and found in the CRC flow is also compatible with this flow, with Cε∼√Re0/Rλ.\nFollowing the same procedures as in the other simulations in this study, 106particles with St=8 were injected when the\nflow reached a fully developed turbulent regime, at a time arbitrarily labeled as t=0, and at random positions in space. The\ninstantaneous PDFs of the normalized V orono ¨ı volumes were calculated for all particles positions at each time. Time series of\nCε,ε,Rλ, and σVare shown in Fig. 2. As in the other flows, in mHIT we also find that σVis correlated (anticorrelated) with Rλ\n(Cεandε). Values for mHIT display strong fluctuations around a mean, as in the CRC flow. Fluctuations in Cεandεare directly\ncaused by the amplitude modulation of the forcing, but σVstill responds to the fluctuations in the other quantities. The mean\nvalue of σνis closer to that in HIT, but large excursions reaching σν≳4 can be seen. This indicates that extreme clustering is\n300 500 700 900\nRλ0.050.100.150.200.250.30C/epsilon1\n−1\nFIG. 1. Instantaneous value of Cεas a function of Rλin the mHIT simulation. A slope of −1 is indicated as a reference.arXiv:2402.15015v1  [physics.flu-dyn]  22 Feb 20242\n0 10 20 30 40 50 60 70\nt//angbracketleftT/angbracketright2.55.0σV(d)\n5001000Rλ(c)24/epsilon1\n(b)0.10.2C/epsilon1(a)\nFIG. 2. Time series of (a) Cε, (b)ε, (c) Rλ, and (d) σVfor mHIT with particles with St =8. Time is normalized by the mean turnover time\n⟨T⟩, and σRPPis indicated as a reference by the dotted grey line in panel (d)\n300 400 500 600 700 800 900\nRλ1.52.02.53.03.54.0σV\nFIG. 3. Standard deviation of the V oronoi volumes, σV, as a function of Rλfor the mHIT simulation, differenciating branches in which Rλ\nincreases (solid lines) or decreases (dashed lines). Red arrows indicate the direction of time evolution.\nmostly associated to the flow unsteadiness, while large-scale flow anisotropies and inhomogeneities can further increase particle\nclustering. Finally, Fig. 3 shows σVas a function of Rλ, confirming the presence of a hysteresis cycle and displaying a behavior\nsimilar to that observed in the CRC flow at different values of Rλ.\nTIME EVOLUTION OF THE CRC FLOW AND OF PARTICLES STATISTICS\nA movie with particles positions and the time evolution of several quantities, for the CRC simulation with 5123grid points\nand with particles with St =3, is available as supplemental material. The top panel shows σVandCεas a function of time,\nthe bottom left panel shows instantaneous probability density functions (PDFs) of the normalized V orono ¨ı volumes (a random\nPoisson process is indicated as a reference by the dashed line), and the top right panel shows particles in the xzplane for a slice of\nthe box centered around y= (π/2)L0. Note the accumulation of particles when σVbecomes larger, and the more homogeneous\ndistribution when σVbecomes smaller."    },    {        "title": "2402.15028v1.The__3k_4__Theorem_modulo_a_Prime__High_Density_for__A_B_.pdf",        "content": "arXiv:2402.15028v1  [math.NT]  23 Feb 2024THE3k−4THEOREM MODULO A PRIME: HIGH DENSITY FOR A+B\nDAVID J. GRYNKIEWICZ\nAbstract. The 3k−4 Theorem for Zasserts that, if A, B⊆Zare finite, nonempty subsets\nwith|A| ≥ |B|and|A+B|=|A|+|B|+r <|A|+ 2|B| −3, then there exist arithmetic\nprogressions PAandPBof common difference such that X⊆PXwith|PX| ≤ |X|+r+1 for\nallX∈ {A,B}. There is much partial progress extending this result to Z/pZwithp≥2 prime.\nHere, among other results, we begin by showing that, if A, B⊆G=Z/pZare nonempty\nsubsets with |A| ≥ |B|,A+B/\\e}atio\\slash=G,|A+B|=|A|+|B|+r≤ |A|+ 1.0527|B| −3, and\n|A+B| ≤ |A|+|B| −9(r+ 3), then there exist arithmetic progressions PA,PBandPCof\ncommon difference such that X⊆PXwith|PX| ≤ |X|+r+ 1 for all X∈ {A,B,C}, where\nC=−G\\(A+B). This gives a rare high density version of the 3 k−4 Theorem for general\nsumsets A+Band is the first instance with tangible (rather than effective ly existential) values\nfor the constants for general sumsets A+Bwithout also imposing additional constraints on the\nrelative sizes of |A|and|B|. The ideal conjectured density restriction under which a ve rsion of\nthe 3k−4 Theorem modulo pis expected is |A+B| ≤p−(r+3). In part by utilizing the above\nresult as well as several other recent advances, we are able t oextendmethods of Serra and Z´ emor\nto give a version valid under this ideal density constraint. We show that, if A, B⊆G=Z/pZ\nare nonempty subsets with |A| ≥ |B|,A+B/\\e}atio\\slash=G,|A+B|=|A|+|B|+r≤ |A|+1.01|B|−3,\nand|A+B| ≤ |A|+|B|−(r+3), then there exist arithmetic progressions PA,PBandPCof\ncommon difference such that X⊆PXwith|PX| ≤ |X|+r+ 1 for all X∈ {A,B,C}, where\nC=−G\\(A+B). This notably improves upon the original result of Serra an d Z´ emor [31], who\ntreated the case A+A, required pbe sufficiently large, and needed the much more restrictive\nsmall doubling hypothesis |A+A| ≤ |A|+1.0001|A|.\n1.introduction\nLetGbe an abelian group. For subsets A, B⊆G, their sumset and difference set are\nA+B={a+b:a∈A, b∈B}andA−B={a−b:a∈A, b∈B}.\nWhenA=B, we write 2 A=A+A. WhenB={x}is a single element, we write {x}+A=\nx+A=A+xfor the translate of Abyx. We let\nAc=G\\A\ndenote the complement of A. For a nonempty subset A⊆Z, we let gcd( A) denote the greatest\ncommon divisor of all elements in A. We often use [ m,n] ={x∈Z:m≤x≤n}to denote the\n2020Mathematics Subject Classification. 11P70, 11B75, 11B25.\nKey words and phrases. Freiman, 3 k−4 Theorem, sumset, isoperimetric method, atom, arithmetic progression.\n12 DAVID J. GRYNKIEWICZ\ndiscrete interval between mandn, for anym,n∈R, with context making it clear when [ m,n]\nshould instead be considered as an interval of real numbers o r an interval modulo p(and thus\nas a subset of Z/pZ).\nThe study of sumsets is a core topic in Additive Combinatoric s. The following is a modern\nversion of the classical Freiman 3 k−4 Theorem. It says that an integer sumset A+B⊆Zwith\nvery small sumset must have AandBcontained in short arithmetic progressions with A+B\ncontaining a long length arithmetic progression, all with t he same common difference. Notable\nis the fact that all bounds, including those for the progress ionsPA,PBandPA+Bas well as the\nhypothesis (1) are tight, and that the result holds true for d istinct summands A/\\⌉}atio\\slash=B(including\nthe commonality of the difference) rather than only treating t he caseA=B. Having either of\nthese desirable features is rare in this area of Additive Com binatorics, and they are achieved\nat the cost of imposing a strong small doubling restriction o n the sumset |A+B|that, when\nA=Bwith|A|=k, amounts to |A+B| ≤3k−4. Hence the name.\nTheorem 1.1 (3k−4 Theorem) .LetA, B⊆Zbe finite and nonempty with |A| ≥ |B|and\n(1) |A+B|=|A|+|B|+r≤ |A|+2|B|−3−δ,\nwhereδ= 1ifA=x+Bfor somex∈Zand otherwise δ= 0. Then there are arithmetic\nprogressions PA, PB, PA+B⊆Zhaving common difference d:= gcd(A−A+B−B)such that\nX⊆PXand|PX| ≤ |A|+r+1,for allX∈ {A,B}, while\nPA+B⊆A+Band|PC| ≥ |A|+|B|−1.\nWithout the conclusions regarding PA+Bor the refinement involving δ, Theorem 1.1 may be\nderived from an old result of Freiman [8] (see the arguments f rom [33]) with him having treated\nthe caseA=Bin a separate earlier work [11]. An alternative (and now stan dard) proof was\ngiven by Lev and Smelianksy [24], who derive a variation wher eBis assumed to be the set with\nsmaller diameter rather than smaller cardinality. The conc lusion regarding PA+Bwas derived\nby Freiman for A=Bin [7] with the case A/\\⌉}atio\\slash=Btreated in [1]. The modern text [13] collects\nall these versions together into one source and includes the added refinement involving δ. The\ntightness of all bounds in Theorem 1.1 can be seen by consider ing the following examples.\nExample 1. Take\nA={0,2,...,2(r+1)}∪[2r+3,m] andB={0,2,...,2(r+1)}∪[2r+3,n],\nwherer≥ −1,m≥n≥2r+3 andm≥2r+4. ThenA+B={0,2,...,2(r+1)}∪[2r+3,m+n]\nwith|A|=m−r≥r+4,|A| ≥ |B|=n−r≥r+3 and |A+B|=m+n−r=|A|+|B|+r,\nwhilePA= [0,m],PB= [0,n] andPC= [2r+2,m+n]. This shows all bounds in the conclusion\nof Theorem 1.1 can hold with equality simultaneously.\nExample 2. Letr≥0 and letA=B=P1∪P2be the union of two arithmetic progressions\nwith common difference such that |A|=r+3 and max P1is much smaller than min P2. ThenTHE 3k−4 THEOREM MODULO A PRIME 3\nA+B=A+Awith|A+A|= 3|A| −3 = 2|A|+r,A=Band|A|=|B|=r+3, and the\nconclusion of Theorem 1.1 cannot hold.\nExample 3. TakeB= [0,r+1] withr≥ −1 and takeA=P1∪P2to be a union of two intervals\nwith|A| ≥max{|B|,3}and maxP1much smaller than min P2. Then|A+B|=|A|+2|B|−2 =\n|A|+|B|+rwith|A| ≥ |B|=r+2, and the conclusion of Theorem 1.1 cannot hold.\nTheorem 1.1 is valid for G=Z. It is a general principle that, so long as |A|and|B|are\nextremely small [21] [13, Chapter 20], then a sumset A+Bin a finite abelian group Ghas the\nsame behaviour as that of an integer sumset. Thus one can expe ct a version of Theorem 1.1\nto hold in a finite abelian group Gprovided the sets involved are not too large. In this paper,\nwe are concerned with such extensions for G=Z/pZwithpprime. The question of extending\nTheorem 1.1 to Z/pZhas received considerable attention. In order to help orien t ourselves, we\nbegin with the following conjecture representing the curre nt best-case scenario that could hold\ninZ/pZ. We remark that the definitions of when δB= 1 orδC= 1 are still tentative below.\nThe reader who wishes to ignore the complicated definitions o fδCandδBcan simply take both\nthese constants equal to the constant δ, whose value is 1 when one of the sets A,BorCis a\ntranslate of another of these sets and otherwise equals 0.\nConjecture 1.2. LetG=Z/pZwithp≥2prime, letA, B⊆Gbe nonempty subsets with\nA+B/\\⌉}atio\\slash=G, and setC=−G\\(A+B). Suppose |A| ≥ |B|and\n|A+B|=|A|+|B|+r≤min{|A|+2|B|−3−δB, p−r−3−δC},\nwhere, forX∈ {A,B,C}with(X,Y,Z)a permutation of (A,B,C),\nδX=\n\n1ifr≥0andXis a translate of either YorZ\n1ifr≥2,YandZare translates of each other, and |Y|=|Z|=r+4\n0otherwise.\nThen there are arithmetic progressions PA, PB, PC⊆Gall having common difference such that\nX⊆PXand|PX| ≤ |X|+r+1for allX∈ {A,B,C}.\nThereadermaynoticethattheconclusioninConjecture1.2h asamoresymmetricformulation\nin terms of complementation than that of Theorem 1.1. Howeve r, setting\nPA+B= (−PC)c=−(PC)c=−G\\PC,\nit follows that PA+Bis also an arithmetic progression with the same common differe nce asPC,\nand that the conclusion C⊆PCwith|PC| ≤ |C|+r+1 is equivalent to PA+B⊆A+Bwith\n|PA+B|=p−|PC| ≥p−|C|−r−1 =|A+B|−r−1 =|A|+|B|−1, which is what is stated\nin Theorem 1.1. To explain the additional density restricti on|A+B| ≤p−r−3−δCgiven in\nConjecture 1.2, consider the following examples.4 DAVID J. GRYNKIEWICZ\nExample 4. Fixr≥ −1, letC= [0,r+ 1]⊆Zand letB=P1∪P2⊆Zbe a union of two\nintervals with |B| ≥max{r+2,3}, minP1= 0andmin P2−maxP1=Mforsomelarge constant\nM≥r+3. Then |B| ≥ |C|and|B+C|=|B|+2|C|−2 =|B|+|C|+rwithPB= [0,maxP2]\nthe minimal arithmetic progression containing Band|PB|=|B|+M−1, which is unbounded\nwith respect to |B|orr. Fix any large prime pand now consider BandCas subsets of\nG=Z/pZ, so we consider the original sets modulo p. Forpsufficiently large, the set B⊆Gstill\ncan be covered no more effectively by an arithmetic progressio n than by using the progression\nPBconsidered modulo p, meaning the conclusion of Conjecture 1.2 fails by an arbitr arily large\nmargin for B. LetA=−(B+C)c. Then|A| ≥ |B|(forpsufficiently large). It is easily noted\nthatB+(C∪{x})/\\⌉}atio\\slash=B+Cfor allx∈Cc(forpsufficiently large). Thus Vosper Duality (stated\nin Proposition 2.5) ensures that A+B=−Ccwith|A+B|=|A|+|B|+r, and nowA+Bis\nan example for which the conclusions of Conjecture 1.2 fail w ith|A+B|=|Cc|=p−r−2 and\n|A| ≥ |B| ≥ |C|=r+2.\nExample 5. Fixr≥0 and letB=P1∪P2⊆Zbe a union of two intervals with |B|=r+3,\nminP1= 0 and min P2−maxP1=Mfor some large constant M. Then|B+B|= 3|B| −\n3 = 2|B|+rwithPB= [0,maxP2] the minimal arithmetic progression containing Band\n|PB|=|B|+M−1, which is unbounded with respect to |B|=r+ 3. Fix any large prime\npand now consider Bas a subset of G=Z/pZ. Forpsufficiently large, the set B⊆Gstill\ncan be covered no more effectively by an arithmetic progressio n than by using the progression\nPBconsidered modulo p, meaning the conclusion of Conjecture 1.2 fails by an arbitr arily large\nmargin forB. LetA=−(2B)c. It is easily noted that B+(B∪{x})/\\⌉}atio\\slash=B+Bfor allx∈Bc(for\npsufficiently large). Thus Vosper duality (stated in Proposit ion 2.5) ensures that A+B=−Bc\nwith|A+B|=|A|+|B|+r=|A|+2|B|−3 and|A| ≥ |B|(forpsufficiently large), and now\nA+Bis a counter-example to Conjecture 1.2 holding with |A+B|=|Bc|=p−|B|=p−r−3,\nB=Cand|A| ≥ |B|=|C|=r+3.\nExample 6. Fixr≥2 withp= 4r+11 prime and let A=B= [0,r+1]∪{r+3,2r+6}=\n[0,r+ 1]∪ {p+1\n4,p+1\n2}considered modulo p(so as subsets of Z/pZ). Then |A|=r+ 4 and\nA+B= 2A= [0,2r+4]∪[2r+6,3r+7]∪{3r+9}with|A+A|= 3r+8 = 3|A|−4 = 2|A|+r=\np−r−3. By Proposition 2.14, the minimal length of an arithmetic p rogressionPAcontaining\nAis 2r+7 =|A|+r+3, meaning A+B=A+Afails to satisfy the conclusion of Conjecture\n1.2 with |A+A|= 3|A|−4 = 2|A|+r=p−r−3,A=B,|A|=|B|=r+4 and |C|=r+3.\nUnlike Examples 1–5, Example 6 was at first missed. Its discov ery in [5] led to the slightly\nmodified Conjecture 1.2 versus earlier versions of the conje cture (e.g., [31]). Since the details\nshowing Example 6 is a counterexample were not given in [5], w e have included them in Proposi-\ntion 2.14 below. Worth noting, this construction requires r≥2, and the failure in the conclusion\nof Conjecture 1.2 is only off by 2, rather than an unbounded amo unt as for the other examples.THE 3k−4 THEOREM MODULO A PRIME 5\nFor the purpose of overviewing previous progress towards Co njecture 1.2, consider the follow-\ning general setup. Let G=Z/pZwithp≥2 prime and let A, B⊆Gbe nonempty subsets with\nA+B/\\⌉}atio\\slash=G,|A| ≥ |B|and\n(2) |A+B|=|A|+|B|+r≤ |A|+(1+α)|B|−3,\nwhereα∈(0,1] is a real number. Since A+B/\\⌉}atio\\slash=G, the Cauchy-Davenport Theorem (Theorem\n2.2) ensures r≥ −1. Asα→1, the bound (2) becomes that from Conjecture 1.2. Note, it is\nequivalent to r≤α|B|−3 and also to |B| ≥α−1(r+3). Smaller values of αthen correspond to\na stronger small doubling hypothesis. Sometimes, the hypot hesis (2) is stated without the −3,\nso in the form |A+B| ≤ |A|+(1+α)|B|. Particularly in older work, the conclusion regarding\nPCis often omitted. In all such cases, rectification arguments (see Proposition 2.11) can be\nused to deduce the desired conclusion regarding PCfrom the conclusions regarding PAandPB\ncombined with Theorem 1.1. As such, there is no need to distin guish which results explicitly\nstate a conclusion regarding PC, and which do not.\nMost partial progress towards Conjecture 1.2 involves achi eving the desired conclusions in\nConjecture 1.2 by imposing a stronger small doubling hypoth esisr≤α|B| −3 orr≤α|B|\ntogether with an additional density assumption ensuring th e sets in question are not too large.\nThere are three major variations for the latter restriction . The first style has the form |A| ≤cp\nfor some real constant c>0. Thesecond, closely related, has the form |A+B| ≤cpfor some real\nconstantc>0. Note that |B| ≤ |A| ≤cpimplies|A+B|<|A|+(1+α)|B|−3≤(2+α)cp−3,so\na restriction of the first type implies one of the second type. Finally, in some cases, the density\nrestriction on |A+B|or|A|is shifted to a relative one between randp, thus having the form\nr≤cpfor some real constant c >0. This removes unnecessary density restrictions from A,\nBorA+Band instead shifts them to the parameter r. Of course, if the upper bound in (2)\nholds with equality, so for A+Bhaving small doubling constant close to the value given by α,\nthen|B| ≈α−1(r+3)≤α−1(cp+3) is implied, which is again a density restriction of the fir st\ntype, though only on the smaller set B. However, the value ccan be held constant as α→0, so\nsufficiently small values of α(relative to the fixed constant c) lead only to trivial constraints on\n|B|. Indeed, we always have |B| ≤p−1\n2. Otherwise |A| ≥ |B| ≥p+1\n2and the Cauchy-Davenport\nTheorem (Theorem 2.2) imply A+B=G, contrary to hypothesis. This means, once α≤2c,\nthen the bound r≤α|B|<αp\n2≤cpis guaranteed to hold, so a restriction of the form r≤cp\nincludes the case of optimal density restrictions for suffici ently small values of α >0, as well\nas including cases with αlarger, even up to 1, having increasingly stronger density r estrictions\nimposed on B.\nIdeally, one would like to minimize the amount of unnecessar y density restrictions while at the\nsame time allowing αto be as close to 1 as possible. These are competing goals, wit h improved\nestimates for one coming at the cost of the estimates for the o ther. The range of results can\ngenerally be grouped as follows.6 DAVID J. GRYNKIEWICZ\nSmallr.Whenris very small, results more general than Conjecture 1.2 are k nown, often\nvalid in more arbitrary groups Gor fully characterizing AandB. Along these lines, we always\nhaver≥ −1 by the Cauchy-Davenport Theorem (Theorem 2.2). When r=−1, Vosper’s\nTheorem [35, Theorem 8.1] [13] implies that Conjecture 1.2 h olds without restriction. When\nr= 0, Hamidoune and Rodseth [16] established Conjecture 1.2 u nder the additional assumption\n7≤ |A|+|B|=|A+B| ≤p−4. A more general result from [14] fully characterizes all AandB\nwith|A+B|=|A|+|B|in all cases. In particular, Conjecture 1.2 is true without r estriction for\nr= 0. (To derive Conjecture 1.2 from [14, Theorem 4.1], note th at the hypotheses of Conjecture\n1.2 ensure |A|,|B|,|C| ≥3, and we must have a strict inequality in one of these estimat es as\np= 9 is not prime. Using Vosper duality (Proposition 2.5) allo ws one to w.l.o.g. assume |C| ≥4,\nand then applying [14, Theorem 4.1] reduces consideration t o Vosper’s Theorem). This allows us\nto assumer≥1 whenever trying to establish a case in Conjecture 1.2. When r= 1, Hamidoune,\nSerra and Z´ emor [17] established Conjecture 1.2 additiona lly assuming |A|,|C| ≥5,|B| ≥4\nandp≥53.\nLow Density. Ifoneiswillingtosacrificeall restraintwithregardtothe densityassumptions,\nit is possible to achieve the maximal value α= 1. Along these lines, the rectification results\nof Bilu, Lev and Ruzsa from [2, Theorem 3.1] combined with The orem 1.1 ensure Conjecture\n1.2 is true additionally assuming |A∪B| ≤log4p. These were later improved as a consequence\nof a result of Lev [21] (combined with [13, Theorem 20.2]) to |A∪B| ≤ ⌈log2p⌉(see Theorem\n2.12). When A=B, taking into account that A+Ahas small sumset, alternative rectification\narguments of Green and Ruzsa [12] combined with Theorem 1.1 e nsure Conjecture 1.2 is true\nassumingA=Bwith|A|< cp, wherec= (1/96)108. For general sumsets A+B, the added\nassumption r≤cp−1.2 suffices, where c= 1.4·10−63951[13, Theoreom 21.8] (note there\nis an error in the explicit statement of the effectively existe ntial constant cin [13, Theoreom\n21.8] owing to a typographic error when stating the rectifica tion threshold of Green and Ruzsa).\nFinally, for smaller r, there is a similar result [13, Theorem 21.7] showing the log arithmic bound\nr+2≤5\n16⌈log2p⌉suffices.\nMid-Range Density. In this range, a balanced approach is taken aiming for tangib le (rather\nthan effectively existential) estimates for both the small do ubling and density constraints simul-\ntaneously. The original result of Freiman [10] [9] [25] is th e prototype here. It shows Conjecture\n1.2 true for A=Bassuming a flexible pair of assumptions r≤α|A| −3 and|A| ≤cp, with\nthe values of candαdependent on each other. Despite the variation allowed, the choice of\nassumptions |2A| ≤2.4|A|−3 and|A| ≤p\n35fixed in [25, Theorem 2.11] is now emblematic with\nthe result itself. Rodseth [26] improved Freiman’s estimat es to|2A| ≤2.4|A|−3 and|A| ≤p\n10.7.\nA range of similar results based on these methods are present ed in [13, Chapter 19] with the\ndensity assumptions of the form |2A| ≤cpwithc≤1\n2. Candela, Serra and Spiegel showed\n|2A| ≤2.48|A| −7 and|A|< p/1010[5] suffices for A=B. Lev and Shkredov [23] showed\n|2A|<2.59|A| −3 and 101 ≤ |A|<0.0045psuffices for A=B, and that |A−A|<2.6|A| −3THE 3k−4 THEOREM MODULO A PRIME 7\nand|A|<0.0045psuffices for A=−B. Lev and Serra [22] showed |2A|<2.7652|A| −3 and\n10≤ |A|<1.25×10−6psuffices for A=B. All these results are for A=BorA=−B. For\ngeneral sumsets A+B, there are results of this type given in [13, Chapter 19] prov ided|A|\nand|B|are very close in size. For example, |B| ≤ |A| ≤4\n3,|A+B| ≤ |A|+ 1.05|B| −3 and\n|A+B| ≤0.0044p. Huicochea achieved similar results with more relaxed cons traints on the rel-\native sizes of |A|and|B|, e.g., 103|A|2\n3≤ |B| ≤ |A|,|A+B|<|A|+1.03|B|and|A|<0.0045p\n[18], with improved bounds when |A|/|B|is large.\nHigh Density. Many results in the mid-range density range have a bias towar ds focussing\non the bound r≤α|B| −3 while simultaneously having a “reasonable” density restr iction.\nHowever, applications often require working with high dens ity sets. An alternative focus is to\nachieve as a high a density as possible while simultaneously having “reasonable” small doubling\nconstraints. The work of Candela, Gonz´ alez-S´ anchez and G rynkiewicz [4] is the example in this\nrange. There, Conjecture 1.2 is established for A=Bwhen|2A| ≤2.136|A|−3 and|2A| ≤3\n4p,\nwith improved estimates for the small doubling assumption w hen|A|is close to1\n3p.\nOptimal Density. The goal here is to sacrifice restraint with αin order to achieve density\nconstraints of the optimal conjectured order of magnitude |A+B| ≤p−O(r). Per earlier\ndiscussion, results with relative constraints between randpyield results with no unnecessary\ndensity restrictions (for very small values of α). Thus the two results mentioned in the low\ndensity section, of the form r≤cp−1.2 andr+2≤5\n16⌈log2p⌉, also yield optimal density results\nfor general sumsets A+Bforverysmallα. However, for A=B, the gold standard goes to Serra\nand Z´ emor [31] who showed Conjecture 1.2 to be true needing o nly the additional assumptions\n|2A| ≤2.0001|A|,p >294and|A| ≥4 (this condition ensures that either |2A| ≤3|A| −4 or\nthat the case r≤0 is applicable).\nThe focus of this paper regarding Conjecture 1.2 is on genera l sumsetsA+Bin the high\nand optimal density ranges, where there are currently noresults having tangible estimates for\nthe small doubling constraint. Indeed, without constraint s on the relative sizes of |A|and|B|,\nthere are no results of any sort having tangible estimates fo r the small doubling constraint for\ngeneral sumsets A+B. The high density result from [4] was achieved by means of the original\nexponential sum argument of Freiman (using Rodseth’s impro ved handling of the exponential\nsum calculations) combined with a new usage of Vosper Dualit y to reduce the density restriction\nimposed by the original combinatorial reduction portion of the argument. We will show (in\nProposition 3.2) how this Vosper Duality argument can beada pted to general sumsets A+Band\nthen combine it with a new adaptation of exponential sum argu ments to general sumsets A+B.\nThis will give us a result (Theorem 3.3) with tangible estima tes for the small doubling constant\nassuming the sizes of |A|and|B|are very close. Combining this with isoperimetric reductio n\nmethods will then allow us to remove most density restrictio ns as well as the restrictions on the\nsizes of|A|and|B|, yielding the following result. Note Theorem 1.3 will be der ived as a simple\nconsequence of Theorem 3.5, whose statement is slightly mor e technical but more general.8 DAVID J. GRYNKIEWICZ\nTheorem 1.3. LetG=Z/pZwithp≥2a prime, let A, B⊆Gbe nonempty subsets with\nA+B/\\⌉}atio\\slash=G, and setC=−G\\(A+B). Suppose |A| ≥ |B|and\n|A+B|=|A|+|B|+r≤min{|A|+1.0527|B| −3, p−9(r+3)}.\nThen there are arithmetic progressions PA, PB, PC⊆Gall having common difference such that\nX⊆PXand|PX| ≤ |X|+r+1for allX∈ {A,B,C}.\nAny density assumption of the form |A+B| ≤p−c(r+3), for some c≥1, can be viewed as\na version of Conjecture 1.2 valid with an additional (1 −ǫ)-density restriction on A+B. Indeed,\nfor anyǫ >0, there then exists some α >0, namelyα=2ǫ\nc, such that Conjecture 1.2 holds\nunder the additional assumptions\n|A+B| ≤ |A|+(1+α)|B|−3 =|A|+/parenleftbig\n1+2ǫ\nc/parenrightbig\n|B|−3 and |A+B| ≤(1−ǫ)p.\nThis can be seen since A+B/\\⌉}atio\\slash=Gand|A| ≥ |B|ensures|B| ≤p\n2, while|A+B|=|A|+|B|+r≤\n|A|+ (1 +α)|B| −3 is equivalent to r+ 3≤α|B|. Thus|A+B| ≤(1−ǫ)p=p−cα\n2p≤\np−cα|B| ≤p−c(r+3), allowing the result of the form |A+B| ≤p−c(r+3) to be applied. As\nsuch, Theorem 1.3 allows densities arbitrarily close to 1 at the cost of an increasingly stronger\nsmall doubling hypothesis α≤2\n9ǫ(forǫ≤.23715).\nFor those that would prefer the −3 be removed from the small sumset hypothesis in Theorem\n1.3, this can be derived from Theorem 1.3, for sufficiently lar gep, by imposing a slightly more\nrestrictive small doubling hypothesis. For instance, if on e assumes |A| ≥ |B| ≥3 +δ, where\nδ= 1 ifBis a translate of Aand otherwise δ= 0, and that\n(3) |A+B|=|A|+|B|+r≤min{|A|+1.05|B|, p−9(r+3)},\nthen Theorem 1.3 can be applied except when 3 >(.0027)|B| ≥(.0027)(.05)−1r, implying\nr≤55. However, as mentioned in the description of results for l ow density, Conjecture 1.2\nis known when r+ 2≤5\n16⌈log2p⌉. Thus for p >2182, the cases r≤55 are known, meaning\nConjecture 1.2 holds when additionally assuming |B| ≥3+δ, (3) andp>2182(the hypothesis\n|B| ≥3+δis needed to ensure (3) implies either |A+B| ≤ |A|+2|B|−4 or that the mentioned\nr≤0 case in Conjecture 1.2 is available) .\nIf one wishes to remove the −3 from the small sumset hypothesis in Theorem 1.3 and have\na result valid without an additional assumption that pbe large, this can also be derived from\nTheorem 1.3 by imposing a more restrictive small doubling hy pothesis. For instance, if one\nassumes |A| ≥ |B| ≥3+δas before and that\n(4) |A+B|=|A|+|B|+r≤min{|A|+1.021|B|, p−9(r+3)},\nthen Theorem 1.3 can be applied except when 3 >(.0317)|B| ≥(.0317)(.021)−1r, implying\nr≤1. If|B| ≤4, then (4) implies r≤0, in which case Conjecture 1.2 is known. As mentioned\nin the description of results for small r, Conjecture 1.2 is also known when r= 1 providedTHE 3k−4 THEOREM MODULO A PRIME 9\n|B| ≥5,|A+B| ≤p−5 andp≥53. However, for r= 1, we have |B| ≥(.021)−1r >47 with\n2·47+1≤ |A|+|B|+r=|A+B| ≤p−9(r+3) =p−36, ensuring p≥131, so that all extra\nconditions needed for the case r= 1 hold. This means Conjecture 1.2 holds when additionally\nassuming |B| ≥3+δand (4).\nTheorem 1.3 has a similar order of magnitude |A+B| ≤p−O(r) density restriction as that\nfor the result of Serra and Z´ emor [31]. However, Theorem 1.3 is valid with a significantly better\nsmall doubling constant 0 .0527 as opposed to 0 .0001, allows for general sumsets A+B, and does\nnot require an added restriction that pbe sufficiently large. Unfortunately, it does not achieve\noptimal density like the result of Serra and Z´ emor. Using Pr oposition 3.2, the methods of Lev\nand Shkredov [23] easily adapt to yield a flexible high densit y version of their result, stated in\nTheorem 3.8. In the final portion of the paper, we utilize Theo rems 3.5 and Theorem 3.8, as well\nas several other recent advances [19] [23] [13] [15], in orde r to extend and improve the methods\nof Serra and Z´ emor [31], resulting in Theorem 1.4. Like the r esult of Serra and Z´ emor [31], this\ngives a version of the 3 k−4 Theorem in Z/pZvalid in the ideal density range. However, the\nresult here is valid for general sumsets A+Brather than only A+A, does not require pbe\nsufficiently large, and has a noticeably improved small doubl ing constant 0 .01 versus 0.0001. As\nwith Theorem 1.3, the −3 can also be removed by imposing a more stringent small doubl ing\nhypothesis, replacing the bound |A+B| ≤ |A|+1.01|B|−3 by the bounds\n|A+B| ≤ |A|+1.004|B|and|B| ≥3+δ,\nwhereδ= 1 ifBis a translate of either Aor−G\\(A+B) and otherwise δ= 0, or alternatively\nby replacing the bound |A+B| ≤ |A|+1.01|B|−3 by the bounds\n|A+B| ≤ |A|+1.009|B|,|B| ≥3+δandp>289.\nTheorem 1.4. LetG=Z/pZwithp≥2prime, letA, B⊆Gbe nonempty subsets with\nA+B/\\⌉}atio\\slash=G, and setC=−G\\(A+B). Suppose |A| ≥ |B|and\n|A+B|=|A|+|B|+r≤min{|A|+1.01|B|−3, p−r−3}.\nThen there are arithmetic progressions PA, PB, PC⊆Gall having common difference such that\nX⊆PXand|PX| ≤ |X|+r+1for allX∈ {A,B,C}.\n2.Preliminaries\nWe overview the basic background and results needed for the p roofs. Let Gbe an abelian\ngroup and let A,B⊆G. For a subset I⊆Zandd∈G, we let\nI·d={xd:x∈I} ⊆G.\nWe letH(A) ={x∈G:x+A=A} ≤Gdenote the stabilizer of A, which is a subgroup of G.\nForx∈Z, let exp(x) =e2πix∈C,whereiis the square root of −1, which is a complex point\ncontained on the unit circle whose value depends only on xmodulo 1. An open half-arc of the10 DAVID J. GRYNKIEWICZ\ncomplex unit circle is a set of the form C={exp(x) :x∈(u,u+1\n2)}for someu∈R. Given\na complex number z∈C, we letzdenote its complex conjugate. For p≥2 prime, the value of\nexp(a/p) depends only on the value of amodulop, making exp( a/p) will defined for a∈Z/pZ.\nGiven a subset A⊆Z/pZandx∈Z, we letSA(x) =/summationtext\na∈Aexp(ax/p). See [13, Chapter 19] for\nbasic results connecting the exponential sums SA(x) with sumsets. In particular, the following\nbasic lemma of Freiman (see [13, Lemma 19.1]) provides a cruc ial link between exponential sums\nand Additive Combinatorics.\nLemma 2.1. Ifz1,...,zN∈Cis a sequence of points lying on the complex unit circle such t hat\nevery open half-arc contains at most nof the terms zifori∈[1,N], then|N/summationtext\ni=1zi| ≤2n−N.\nBasic Sumset Results. WhenG=Z/pZwithp≥2 prime, the basic lower bound for sumsets\nis the Cauchy-Davenport Theorem. See [13, Theorem 6.2].\nTheorem 2.2 (Cauchy-Davenport Theorem) .LetG=Z/pZwithp≥2prime and let A,B⊆G\nbe nonempty. Then |A+B| ≥min{p,|A|+|B|−1}.\nThe generalization to more general abelian groups is Kneser ’s Theorem [13, Theorem 6.1] [25,\nTheorem 4.1] [34, Theorem 5.5].\nTheorem 2.3 (Kneser’s Theorem) .LetGbe an abelian group and let A,B⊆Gbe finite and\nnonempty. Then |A+B| ≥ |H+A|+|H+B|−|H|, whereH=H(A+B).\nIfGis a finite abelian group and A⊆G, then\n(5) ( A−Ac)c=H(A).\nIndeed, it is immediate that 0 /∈A−Ac, and then, since |A|+|Ac|=|G|, Kneser’s Theorem\nimplies that A−Ac=G\\H, whereH=H(A−Ac). To seeH=H(A), note that if x∈Awith\nx+H/notsubseteqlA, thenx+Hintersects both AandAcwith|(x+H)\\A|+|(x+H)\\Ac|=|H|,\nwhence|A−Ac| ≥ |H+A|+|H+Ac|−|H| ≥(|A|+|Ac|+|H|)−|H|=|G|, contradicting that\n0/∈A−Ac. ThusH(A)≤H=H(A−Ac) with the reverse inclusion trivial.\nThe following are Petridis’ version of the Ruzsa-Pl¨ unneck e Bounds [28, Theorem 1.6.1]. The-\norem 2.4.1 is [30, Theorem 1.5] and Theorem 2.4.2 is [30, Theo rem 3.1]. In all cases, the explicit\ndefinition of A′(as given below) is not found in the statement of the theorems from [30] but is\nhowA′is defined in the proof of these theorems.\nTheorem 2.4 (Petridis’ Ruzsa-Pl¨ unnecke Bounds) .LetGbe an abelian group, let A, B⊆G\nbe finite, nonempty subsets and let A′⊆Abe a nonempty subset attaining the minimum\nα:= min/braceleftBig|A′+B|\n|A′|:∅ /\\⌉}atio\\slash=A′⊆A/bracerightBig\n≤|A+B|\n|A|.\nThen the following all hold.THE 3k−4 THEOREM MODULO A PRIME 11\n1.|C+A′+B| ≤α|C+A′|for all finite subsets C⊆G.\n2.|A′+nB| ≤αn|A′| ≤αn|A|for alln≥0.\nAdditive Isomorphism. Ford∈G, we let\nℓd(A) be the minimal length of an arithmetic progression with diffe rencedcontainingA.\nNoteℓd(A) =∞if no such progression exists. Let GandG′be abelian groups and let A,B⊆G\nbe finite and nonempty. A Freiman homomorphism is a mapϕ:A+B→G′for which there\nare coordinate maps ϕA:A→G′andϕB:B→G′such thatϕ(a+b) =ϕA(a) +ϕB(b) for\nalla∈Aandb∈B. An injective Freiman homomorphism ϕ:A+B→G′defines a Freiman\nisomorphism ϕ:A+B→A′+B′⊆G′, whereA′=ϕA(A) andB′=ϕB(B), in which case we\nsay the sumsets A+B⊆GandA′+B′⊆G′areisomorphic , denotedA+B∼=A′+B′. There\nis often no loss of generality to replace AandBwith appropriate translates so that 0 ∈A∩B.\nLikewise for A′andB′. If 0∈A∩BandϕA(0) =ϕB(0) = 0, we call ϕanormalized Freiman\nhomomorphism. If A+B∼=A′+B′and 0∈A∩B, then replacing A′andB′by appropriate\ntranslates with 0 ∈A′∩B′(namely −ψA(0) +A′and−ψB(0) +B′), there is a normalized\nFreiman isomorphism ψ:A+B→A′+B′. Since any normalized Freiman homomorphism must\nhave its coordinate maps agree on any element common to their domains, this ensures that\nA, B⊆Gare translates of each other in Gif and only if A′,B′⊆G′are translate of each other\ninG′, for isomorphic sumsets A+B∼=A′+B′. Any group homomorphism ϕ:G→G′restricts\nto a Freiman homomorphism of A+Bwhich is an isomorphism when the restriction ϕ|A+Bis\ninjective, and is normalized when 0 ∈A∩B. Every sumset A′+B′⊆G′with 0∈A′∩B′has\na universal ambient group Gcontaining an isomorphic sumset A+Bfor which any normalized\nFreiman homomorphism ϕ:A+B→A′+B′can be extended to a group homomorphism\nϕ:G→G′. See [13, Chapter 20] [34, Chapter 5.5] for a fuller discussi on of the basic theory\nof Freiman homomorphisms, including the details mentioned above (the latter reference only\ntreatsA=B).\nAn example of when a well-behaved isomorphism exists is as fo llows. We say that the sumset\nA′+B′⊆G′rectifies if there is some d∈G′such that\n(6) ℓd(A′)+ℓd(B′)≤ord(d)+1.\nIn such case, A′⊆a0+ [0,M−1]·dandB′⊆b0+ [0,N−1]·dfor somea0,b0∈G′and\nM,N≥1 withM+N≤ord(d)+1, and then the maps ϕA:Z→G′andϕB:Z→G′given by\nϕA(x) =a0+xdandϕB(y) =b0+ydinduce a Freiman isomorphism ϕ:A+B⊆Z→A′+B′,\nwhereA=ϕ−1\nA(A′)∩[0,n−1]⊆ZandB=ϕ−1\nB(B′)∩[0,n−1]⊆Zwithn= ord(d).\nVosper Duality and Additive Trios. We continue with the rather crucial notion of Vosper\nDuality (in basic form, this can be traced to [35]), which we w ill afterwards formulate in the\nmore abstract terms of additive trios (introduced in [3]). I f ({x}∪A)+B/\\⌉}atio\\slash=A+Bfor allx∈Ac,12 DAVID J. GRYNKIEWICZ\nthen we say that Aissaturated in the sumset A+B. The following basic observation will be\ncrucial. Its proof is straightforward. See [13, Lemma 7.2].\nProposition 2.5 (Vosper Duality) .LetGbe an abelian group and let A, B⊆Gbe subsets.\nThen\n−(A+B)c+B⊆ −Ac\nwith equality holding if and only if Ais saturated in A+B. When this is the case with |A+B|=\n|A|+|B|+randGfinite, then |−(A+B)c+B|=|(A+B)c|+|B|+rwith−(A+B)csaturated\nin−(A+B)c+B.\nAnadditive trio in a finite abelian group Gis a triple T= (A,B,C) withA, B, C ⊆G\nnonempty subsets satisfying A+B+C/\\⌉}atio\\slash=G. Thusz /∈A+B+Cfor somez∈G. In such case,\n(7)A⊆z−(B+C)c, B⊆z−(A+C)candC⊆z−(A+B)c,for anyz∈(A+B+C)c.\nWe say that Aissaturated in the trio Tif (A∪{x})+B+C=Gfor everyx∈Ac, withB\nandCbeing saturated defined analogously. The trio Tissaturated ifA,BandCare each\nsaturated in T. The setz−(B+C)cis precisely the set of those elements x∈Gfor which\nz /∈(A∪ {x}) +B+C. From this observation, it readily follows that Ais saturated in Tif\nand only if A=z−(B+C)cfor allz∈(A+B+C)c. Likewise, equality holding in the\nother inclusions in (7) are equivalent to BorCbeing saturated in T. As a consequence, if Ais\nsaturated in T, then (5) implies A+B+C=G\\(z+H) withH=H(A) =H(B+C). For an\nadditive trio, we define\nr=r(T) =|G|−|A|−|B|−|C|.\nWhenG=Z/pZ, the condition A+B+C/\\⌉}atio\\slash=Gtogether with the Cauchy-Davenport Theorem\n(Theorem 2.2) ensures |B+C| ≥ |B|+|C|−1, in which case (7) implies |A| ≤ |G|−|B+C| ≤\n|G|−|B|−|C|+1, meaning r=r(T)≥ −1. Note (A,B,−(A+B)c) is always an additive trio\nwith\nr/parenleftbig\nA,B,−(A+B)c/parenrightbig\n=rwhere|A+B|=|A|+|B|+r,\nso long asAandBare nonempty with A+B/\\⌉}atio\\slash=G, as exhibited by the element z= 0.\nIfT= (A,B,C) is not saturated, we can successively saturate the sets by r eplacingCby\nC′=z−(A+B)c, for anyz∈(A+B+C)c, then replace Bwith the maximal set B′such that\nB⊆B′andA+B′=A+B. NoteB′exists since A+Bis finite. In view of C′=z−(A+B)c,\nA+Bis the maximal subset such that A+B+C′= (A+B)+(z−(A+B)c) =G\\(z+H),\nwhereH=H(A+B), which makes B′⊆Gthe maximal subset with A+B′+C′/\\⌉}atio\\slash=G, while\nC′=z−(A+B)c=z−(A+B′)c. It follows that B′andC′are saturated in the trio ( A,B′,C′).\nFinally, replacing Awith the maximal subset A′such thatA⊆A′andA′+B′=A+B′=A+B,\nwe likewise obtain a saturated trio T′= (A′,B′,C′) withA⊆A′,B⊆B′andC⊆C′(the\nlatter in view of (7)), which we refer to as a successive satur ation ofT. Such a process can be\ndone using any z∈(A+B+C)cand any ordering of the sets A,BandC.THE 3k−4 THEOREM MODULO A PRIME 13\nThe hypotheses of Conjecture 1.2 ensure that T= (A,B,C) is an additive trio in G=Z/pZ\nwith\n(8) r=r(T)≥ −1 and |X| ≥r+3+δXfor everyX∈ {A,B,C},\nwhereA,B,C,randδXare all as defined in Conjecture 1.2. Conversely, given any ad ditive trio\nT= (A,B,C) withr(A,B,C)≤r, wecan passtoasaturated trio( A′,B′,C′) asdescribedabove\nand translate to w.l.o.g. assume 0 /∈A′+B′+C′. ThenA′+B′=−(C′)c,A′+C′=−(B′)c,\nB′+C′=−(A′)candr(A′,B′,C′)≤r−∆, where ∆ = ( |A′|−|A|)+(|B′|−|B|)+(|C′|−|C|).\nMoreover anyarithmeticprogression PXcontainingX′∈ {A′,B′,C′}with|PX| ≤ |X′|+r−∆+1\nalso contains the corresponding X∈ {A,B,C}with|PX| ≤ |X|+r+1. Thus the hypotheses\nin (8) are the more symmetric equivalent formulation for the hypotheses in Conjecture 1.2.\nIsoperimetric Machinery. The Ruzsa-Pl¨ unnecke bounds involve multiplicative expan sion,\nconsidering a subset X⊆Athat minimizes|X+B|\n|X|. The isoperimetric method instead considers\nthe additive expansion |X+B|−|X|. The basic setup is as follows. Let Gbe an abelian group\nand letB⊆Gbe a finite, nonempty subset. For an integer k≥1, we say that Bisk-separable\nif there is a finite subset X⊆Gwith|X| ≥kand|(X+B)c| ≥k. When this is the case, we\ndefine\n(9) κk(B) = min{|X+B|−|X|:X⊆G,|X| ≥k,|(X+B)c| ≥k}.\nA setX⊆Gwith|X|,|(X+B)c| ≥kattaining theminimumin(9) is called a k-fragment ofB,\nand a minimal cardinality k-fragment is called a k-atom. We letαk(B) denote the cardinality\nof ak-atom ofB. By its definition, we have\n(10) |Y+B| ≥min{|G|−k+1,|Y|+κk(B)}\nfor any subset Y⊆Gwith|Y| ≥k. IfXis ak-atom forB, then (by [13, Lemma 21.1])\n(11) H(X) =H(X+B).\nLettingH=H(X) andk′=⌈k/|H|⌉, it then follows (by [13, Proposition 21.5]) that\n(12) Bisk′-separable with Xak′-atom forB,\nwherexdenotes the image of x∈GmoduloH. The fundamental property of atoms (originally\ndue to Hamidoune) is the following (see [13, Theorem 21.1]).\nTheorem 2.6 (Fundamental Theorem for Atoms) .LetGbe an abelian group, let k≥1, let\nB⊆Gbe a finite, nonempty subset, and suppose Bisk-separable. Let Xbe ak-atom ofBand\nletFbe ak-fragment of B. If|X∩F| ≥k, thenX⊆F. In particular, two distinct k-atoms\nintersect in at most k−1elements.14 DAVID J. GRYNKIEWICZ\nThe following is a useful upper bound for the size of a 2-atom [ 15, Theorem 8]. Note, if\n/a\\}⌊ra⌋k⌉tl⌉{tB−B/a\\}⌊ra⌋k⌉tri}ht=His a proper, finite subgroup of G, then the 2-atoms of Bare precisely the H-cosets\ninG, so the hypotheses of Theorem 2.7 ensure /a\\}⌊ra⌋k⌉tl⌉{tB−B/a\\}⌊ra⌋k⌉tri}htis either infinite or equal to G. This\nensures that the case /a\\}⌊ra⌋k⌉tl⌉{tB−B/a\\}⌊ra⌋k⌉tri}ht=Gimplies the general case stated in Theorem 2.7.\nTheorem 2.7. LetGbe an abelian group and let B⊆Gbe a finite 2-separable set with r:=\nκ2(B)−|B|. SupposeBhas a2-atom that is not a coset. Then α2(B)≤r+3.\nWhenG=Z/pZwith|B|not extremely large, we can improve the bound in Theorem 2.7.\nTheorem 2.8. LetG=Z/pZwithp≥2prime, and let B⊆Gwith|B| ≥2. SupposeBis\n2-separable and let r≥κ2(B)−|B|be an integer.\n1.If|B| ≤p−3r−4,thenα2(B)≤2+√2r+2.\n2.If|B| ≤p−3r−5andr≥1, thenα2(B)≤1\n2+/radicalBig\n2r+17\n4.\nProof.Part 1 is [13, Theorem 21.6]. Note r≥r′:=κ2(B)−|B| ≥ −1 follows from the Cauchy-\nDavenport Theorem (Theorem 2.2). Since r≥1, the bound given in Part 2 is always at least 3.\nAs a result, if r′≤0, then Part 1 (applied with r=r′) yields the desired bound for Part 2. This\nallows us to assume r′=r. Then the proof of Part 1 given in [13, Theorem 21.5] also deri ves\nPart 2. Note—in the notation of [13, Theorem 21.5], where the irrequals ourr+1—that the\nassumptions |B| ≤p−3r−1,|X| ≥5 andr≥3 derived at the beginning of the proof get\nreplaced by |B| ≤p−3r−2,|X| ≥4 andr≥2, leading with bare minimal modification in the\ncalculations to the conclusion |X| ≤1\n2+/radicalBig\n2r+9\n4stated at the end of the proof, which replacing\nrbyr+1 gives the bound stated here. /square\nPer the arguments in [31, Lemma 2.3], we can obtain the follow ing upper bound for the size\nof ak-atom in terms of the size of a 2-atom. The result in [31, Lemma 2.3] was only given for\nG=Z/pZwithpprime.\nTheorem 2.9. LetGbe a finite abelian group, let k≥2, and letB⊆Gbe ak-separable subset\nwithr=κk(B)−|B|and/a\\}⌊ra⌋k⌉tl⌉{tB−B/a\\}⌊ra⌋k⌉tri}ht=G. SupposeBhas ak-atomXthat is not a coset. Then\nαk(B) =|X| ≤ ⌈k/|H|⌉·|H|−|H|+r+|H|α2(X)≤k−1+r+|H|α2(X),\nwhereH=H(X)andX⊆G/Hdenotes the image of XmoduloH.\nProof.LetXbe ak-atom ofB, letH=H(X) and letx∈G/Hforx∈Gdenote the image of\nxmoduloH. By hypothesis, |X+B|=|X|+|B|+r≤ |G|−k. Note|X| ≥k≥2 ensures that\nGis nontrivial, which combined with the hypothesis /a\\}⌊ra⌋k⌉tl⌉{tB−B/a\\}⌊ra⌋k⌉tri}ht=Gforces|B| ≥2. Sincek≥2\nand|B| ≥2, it follows that Xis 2-separable with\nκ2(X)≤ |X+B|−|B|=|X|+r.THE 3k−4 THEOREM MODULO A PRIME 15\nLetYbe a 2-atom of Xtranslated so that 0 ∈Y. Since|Y| ≥2, letd∈Ybe any nonzero\nelement. Thus\n(13) |X∪(d+X)|=|{0,d}+X| ≤ |Y+X|=κ2(X)+|Y| ≤ |X|+|Y|+r.\nSinceXandd+Xare bothk-atoms ofB, it follows from Theorem 2.6 that either d+X=X\nor|(d+X)∩X| ≤k−1.\nIfH=H(X) is trivial, the latter case must hold (as d+X=Xwithd/\\⌉}atio\\slash= 0 ensures that\nd∈HwithHnontrivial). Thus |X∪(d+X)|= 2|X| −|X∩(d+X)| ≥2|X| −k+1, which\ncombined with (13) yields the desired bound αk(B) =|X| ≤k−1+r+|Y|=k−1+r+α2(X) =\nk−1+r+|H|α2(X). Therefore we may instead assume His nontrivial.\nBy (11), we have H(X+B) =H(X) =H. Letk′=⌈k/|H|⌉. By Kneser’s Theorem (Theorem\n2.3),|G|−k−|H| ≥ |X+B|−|X| ≥ |H+B|−|H|. As result, if |H| ≥k, thenHis ak-fragment\nofBwith|X+B|− |X|=|H+B|− |H|, and then, since |X| ≥ |H|andH(X) =HwithX\nak-atom ofB, it follows that Xis anH-coset, contrary to hypothesis. Therefore we instead\nconclude that |H|< k, whencek′≥2. By (12), Bisk′-separable in G/HwithXak′-atom\nthat is not a coset (since k′≥2 andH=H(X)) and\nαk(B) =|X|=|H+X|=|H||X|=|H|αk′(B).\nThus, since H(X) is trivial with k′≥2, we can apply the argument of the previous paragraph to\nBusingthek′-atomXtoconcludethat αk′(B)≤k′−1+r′+α2(X), wherer′=|X+B|\n|H|−|X|−|B|.\nConsequently, r′·|H| ≤ |X+B|−|X|−|B|=rand\nαk(B) =|H|αk′(B)≤/parenleftbig\nk′−1+r′+α2(X)/parenrightbig\n|H|=⌈k/|H|⌉·|H|−|H|+r+|H|α2(X),\nas desired. /square\nCorollary 2.10. LetG=Z/pZwithp≥2prime, letk≥2, and letB⊆Gbe ak-separable\nsubset with r≥κk(B)−|B|and|B| ≥2.\n1.αk(B)≤k+2r+2\n2.Ifk≤p−5r−6, thenαk(B)≤k+r+1+√2r+2\n3.Ifk≤p−5r−7andr≥1, thenαk(B)≤k+r−1\n2+/radicalBig\n2r+17\n4\nProof.SinceBisk-separable with k≥2, it is also 2-separable. Let Xbe a 2-atom of B, which\ncannot be a coset and must have H(X) trivial, both since G=Z/pZhas no proper, nontrivial\nsubgroups. Thus Theorem 2.9 yields\n(14) αk(B)≤k−1+r+α2(X).\nSincek≥2and|B| ≥2, it follows that Xis2-separable with κ2(X)−|X| ≤ |X+B|−|B|−|X|=\nκ2(B)− |B| ≤r. Likewise, any 2-atom of Xcannot be a coset. Thus Theorem 2.7 implies\nα2(X)≤r+ 3, which combined with (14) yields Part 1. As a result, |X| ≤k+ 2r+ 2.\nConsequently, if k≤p−5r−6, then|X| ≤k+ 2r+ 2≤p−3r−4, in which case Theorem16 DAVID J. GRYNKIEWICZ\n2.8.1 implies that α2(X)≤2+√2r+2, which combined with (14) yields Part 2. Likewise, if\nk≤p−5r−7 andr≥1, then|X| ≤k+ 2r+ 2≤p−3r−5, in which case Theorem 2.8.2\nimplies that α2(X)≤1\n2+/radicalBig\n2r+17\n4, which combined with (14) yields Part 3. /square\nResults Regarding the 3k−4Theorem. Thefollowing propositionshowshow theconclusion\nof Conjecture 1.2 is equivalent to various statements regar ding a sumset X+Ybeing isomorphic\nto a sumset of subsets of the integers.\nProposition 2.11. LetG=Z/pZwithp≥2prime, let T= (A,B,C)be an additive trio from\nGwithr(T) =rand\n|X| ≥r+3+δXfor allX∈ {A,B,C},\nwhereδX= 1ifX∈ {A,B,C}is a translate of one of the other two sets from A,BandC, and\notherwiseδX= 0. Then the following are equivalent.\n1.There is some nonzero d∈Gsuch thatℓd(X)≤ |X|+r+1for allX∈ {A,B,C}.\n2.There is some nonzero d∈Gsuch thatℓd(X) +ℓd(Y)≤p+ 1for every (X,Y)∈\n{(A,B),(B,C),(C,A)}.\n3.There is some nonzero d∈Gsuch thatℓd(X) +ℓd(Y)≤p+ 1for some (X,Y)∈\n{(A,B),(B,C),(C,A)}.\n4.The sumset X+Yis isomorphic to a sumset of subsets from Zfor every (X,Y)∈\n{(A,B),(B,C),(C,A)}.\n5.The sumset X+Yis isomorphic to a sumset of subsets from Zfor some (X,Y)∈\n{(A,B),(B,C),(C,A)}.\nIn case Part 1 holds, then the dfor Part 2 may be taken to be the dfrom Part 1. If Part 3 holds\nand eitherℓd(X)≤2|X|−2ord∈Gis chosen such that ℓd(X)+ℓd(Y)is minimal, then Parts\n1, 2 and 3 all hold using this d.\nProof.1.⇒2.Let(X,Y)∈ {(A,B),(B,C),(C,A)}bearbitraryandlet Zberemainingsetfrom\nthe trio. Then Part 1 and the definition of r(T) imply that ℓd(X)+ℓd(Y)≤ |X|+|Y|+2r+2 =\np− |Z|+r+ 2≤p−1, yielding 2 (with the same d). The implication 2 .⇒3.is trivial.\nThe implications 2 .⇒4.and 3.⇒5.follow from the discussion in the Additive Isomorphism\nSubsection. The implication 4 .⇒5.is trivial. To show the equivalence of Parts 1 through 5, it\nremains to show 5 .⇒1.\nSupposew.l.o.g. that A+B∼=A′+B′forsomeA′,B′⊆Z. Letδ= 1ifA′andB′aretranslates\nof each other andotherwiselet δ= 0. Asnoted intheAdditive IsomorphismSubsection, we have\nδ= 1 if and only if AandBare translates of each other, which is equivalent to δA=δB= 1 by\ndefinition of δX. By an appropriatechoice of translations (of A,B,C,A′andB′), we can w.l.o.g.\nassume 0 ∈A∩Band 0∈A′∩B′with 0 taken to 0 by the isomorphism ϕ:A′+B′→A+B,THE 3k−4 THEOREM MODULO A PRIME 17\nand that 0 /∈A+B+C, whence (7) and our hypotheses imply\nC⊆ −(A+B)c,and (15)\n|A+B| ≤p−|C|=|A|+|B|+r≤ |A|+|B|−3+min{|A|−δA,|B|−δB}. (16)\nBy replacing Zby/a\\}⌊ra⌋k⌉tl⌉{tA′−A′+B′−B′/a\\}⌊ra⌋k⌉tri}ht, we can w.l.o.g. assume gcd( A′−A′+B′−B′) = 1. Since\nδ= 1 if and only if δA=δB= 1, (16) allows us to apply Theorem [13, Theorem 20.2] to A′+B′\nto conclude that the universal ambient group of A′+B′isZ. Thus, replacing A′+B′by the\nisomorphic sumset given in the definition of a universal ambi ent group (this amounts to applying\nan appropriate automorphism of ZtoA′andB′and then translating as need be), it follows that\nthe normalized Freiman isomorphism ϕ:A′+B′→A+Bextends to a group homomorphism\nϕ:Z→G, say withϕ(1) =d∈G. Sinceδ= 1 if and only if δA=δB= 1, (16) also allows us to\napply Theorem 1.1 to A′+B′to find arithmetic progressions P′\nA,P′\nBandP′\nA+Bwith difference 1\nsuch thatA′⊆P′\nAand|P′\nA| ≤ |A|+r+1,B′⊆P′\nBand|P′\nB| ≤ |B|+r+1, andP′\nA+B⊆A′+B′\nand|P′\nA+B| ≥ |A|+|B|−1. LetPX=ϕ(P′\nX) forX∈ {A,B,A+C}. ThenA⊆PAandB⊆PB\nwith|PA| ≤ |A|+r+1 and|PB| ≤ |B|+r+1 whilePA+B⊆A+Bwith|PA+B| ≥ |A|+|B|−1.\nSetPC=−(PA+B)c. ThenPA+B⊆A+BimpliesC⊆ −(A+B)c⊆ −(PA+B)c=PC(in view\nof (15)) with |PC|=p−|PA+B| ≤p+1−|A|−|B|=|C|+r+1 (by definition of r(T)). The\nprogressions PA,PBandPCeach have difference ϕ(1) =dand show that ℓd(X)≤ |X|+r+1\nfor everyX∈ {A,B,C}, yielding 1.\nFinally, itremainstoshowtheimplication3 .⇒1.allowsthedfromPart3tobeusedinPart1,\nassumingtheadditional constraints. Tothisend, supposew .l.o.g. thatℓd(A)+ℓd(B)≤p+1. Let\nPA, PB⊆Gbe minimal length arithmetic progressions with difference dthat contain AandB,\nrespectively. PerthediscussionfromtheAdditiveIsomorp hismSubsection, wecanthustranslate\nAandB(so that the initial terms in PAandPBare 0) and find subsets A′,B′⊆[0,p−1]⊆Z\nsuch thatA′+B′∼=A+Bvia the homomorphism ϕ:Z→Ggiven byϕ(1) =d∈G. If\nℓd(X)≤2|X| −2, the Pigeonhole Principle ensures that there is a pair of co nsecutive terms\nfromPAcontained in A, in which case there is a pair of consecutive terms in A′with difference\n1. Thus gcd( A′−A′) = 1, implying gcd( A′−A′+B′−B′) = 1. On the other hand, if\ngcd(A′−A′+B′−B′) =n≥2, thenℓnd(A)+ℓnd(B)<ℓd(A)+ℓd(B) (since|A|,|B| ≥r+3≥2).\nAs such, regardless of whether d∈Ghasℓd(X)≤2|X|−2 orℓd(A)+ℓd(B) minimal, weconclude\nin both cases that gcd( A′−A′+B′−B′) = 1. The proof of the implication 5 .⇒1.now shows\nthat Part 1 holds using d. /square\nWe will also need the following small rversion of Conjecture 1.2, which follows by combining\nthe main result from [21] with Theorem 1.1 and using [13, Theo rem 20.2] to ensure the Freiman\nisomorphism given by [21] is well-behaved.18 DAVID J. GRYNKIEWICZ\nTheorem 2.12. LetG=Z/pZwithp≥2a prime, let A, B⊆Gbe nonempty subsets with\nA+B/\\⌉}atio\\slash=G. SetC=−(A+B)c. Suppose |A| ≥ |B|,|A∪B| ≤ ⌈log2p⌉and\n|A+B|=|A|+|B|+r≤ |A|+2|B|−3−δ,\nwhereδ= 1for ifAandBare translates of each other and otherwise δ= 0. Then there are\narithmetic progressions PA, PB, PC⊆Gall having common difference such that X⊆PXand\n|PX| ≤ |X|+r+1for allX∈ {A,B,C}.\nProof.Since|A∪B| ≤ ⌈log2|A|⌉, it follows from [21] (see also [13, Theorem 20.4]) that A+Bis\nisomorphic to sumset of subsets of the integers, say A+B∼=A′+B′withA′, B′⊆Z. Then, as\nin the proof of Proposition 2.11, we can apply [13, Theorem 20 .2] to conclude that the universal\nambient group of A′+B′isZ, allowing us to w.l.o.g. assume (translating sets appropri ately)\nthat thereis a Freiman isomorphism ϕ:A′+B′→A+Bthat extends toa grouphomomorphism\nϕ:Z→G, say withϕ(1) =d∈G. Applying Theorem 1.1 to A′+B′and applying ϕto the\nresulting progressions then yields the needed arithmetic p rogressions with difference dusing the\nsame argument as for Proposition 2.11. /square\nThe following is a very useful result of Huicochea [19] that s hows that, under some additional\nconstraints, the weaker conclusion that one of the sets X∈ {A,B,C}is contained in a short\nprogression is also equivalent to the statements given in Pr oposition 2.11. This means it is often\nsufficient to cover only one of the three sets in an additive tri o by a short length arithmetic\nprogression in order to establish the desired conclusion in Conjecture 1.2.\nTheorem 2.13. LetG=Z/pZwithp≥2prime, let (A,B,C)be an additive trio from G, and\nletr, h∈Zbe integers.\n1.Ifℓd(A)≤ |A|+hfor somed∈G\\ {0},r(A,B,C)≤r,|A| ≥r+ 3 +hand\n|B| ≥r+3+2hwith strict inequality in one of these estimates, and |C| ≥r+3, then\nℓd(A)≤ |A|+r+1,ℓd(B)≤ |B|+r+1andℓd(C)≤ |C|+r+1.\n2.Ifℓd(A)≤ |A|+hfor somed∈G\\{0},r(A,B,C)≤h−1,h+2≤ |A| ≤max{|B|,|C|},\n|B|,|C| ≥h+3and35h+10≤p, thenℓd(B)≤ |B|+handℓd(C)≤ |C|+h.\nProof.1. The proof is identical to that given for [19, Theorem 1.2]. One has only to use\nProposition 2.11 rather than [19, Corollary 2.3] and modify the calculations slightly to account\nforh(the result in [19] is only stated for h=r+1). Note in the derivation of (16) in [19], the\ncited equation (8) should be (11), and that Proposition 2.11 is not available for the case h= 0,\nwhenAis itself an arithmetic progression, if two of the three sets A,BandChave sizer+3\nand are translates of each other. This case should be verified separately. However, if the desired\nconclusion fails when Ais an arithmetic progression, then either |A+B| ≥ |A|+|B|+r+ 1\nor|A+C| ≥ |A|+|C|+r+ 1 can be derived (since A+B/\\⌉}atio\\slash=GandA+C/\\⌉}atio\\slash=G), leading\nto a contradiction with r(A,B,C)≤rvia (7). Also, when using the implication 3 .⇒1.in\nProposition 2.11, the hypothesis |A| ≥r+3+hensuresℓd(A)≤ |A|+h≤2|A|−r−3≤2|A|−2.THE 3k−4 THEOREM MODULO A PRIME 19\n2. This is [19, Theorem 1.3] (with the notation altered via th e substitution h=r+1). /square\nWe conclude the section with the details for the constructio n from [5].\nProposition 2.14. Letn= 4r+ 11withr≥1, letG=Z/nZand letA= [0,r+ 1]∪ {r+\n3,2r+6} ⊆G. Then the minimal length of an arithmetic progression contai ningAis2r+7.\nProof.Note this is equivalent to showing that the maximal length of an arithmetic progression\ncontained in\n(17) Ac={r+2}∪[r+4,2r+5]∪[2r+7,4r+10]\nis 2r+4 =n−(2r+7). To this end, let P⊆Acbe a maximal length arithmetic progression, say\nwith difference d∈[1,n−1\n2]. Ifd= 1, then |P|= max{1,r+2,2r+4}= 2r+4 (sincer≥ −1),\nas desired.\nIfd∈[2,r+2], then 4 r+10+d=n−1+d<n+r+2, so any maximal length progression\nPwith difference dcontained in (17) (considered as a subset of Z/nZ) must also be such a\nmaximal length progression contained in (17) when consider ed instead as a subset of Z. When\nd= 2, to maximize its length, Pmust then contain both 2 r+5 and 2r+7, ensuring 4 r+10/∈P,\nand thusPcan be no larger than all odd numbers contained in [ r+ 2,4r+ 9]⊆Z. Thus\n|P| ≤3r+9\n2≤2r+ 4 (since r≥1), as desired. If d∈[3,r+ 2], then |P|is no larger than\nthe largest arithmetic progression with difference dcontained in [ r+ 2,4r+ 10]⊆Z. Thus\n|P| ≤ ⌈3r+9\nd⌉ ≤ ⌈3r+9\n3⌉=r+3≤2r+4 (sincer≥ −1), as desired.\nIfn=r+3 =n+1\n4, then the argument for d≤r+ 2 again yields the desired bound unless\n4r+10∈P. Assumingthisisso, wehave4 r+10+d=r+2∈P, followed by r+2+d= 2r+5∈P,\nfollowed by2 r+5+d= 3r+8∈P. Butthen, since3 r+8+d= 4r+11 = 0/∈Ac, weconcludethat\n3r+8 is the final term in P. The term preceding 4 r+10∈Pmust be 4r+10−d= 3r+7∈P,\npreceded by 3 r+ 7−d= 2r+ 4∈P. Then, since 2 r+ 4−d=r+ 1/∈Ac, it follows that\n2r+ 4 is the first term in P, whenceP={2r+ 4,3r+ 7,4r+ 10,r+ 2,2r+ 5,3r+ 8}with\n|P|= 6≤2r+4 (in view of r≥1), as desired.\nIfd=n−1\n2= 2r+5, thenPmust be a translate of a set of the form [1 ,ℓ]∪/parenleftbign−1\n2+[1,ℓ]/parenrightbig\nor\nof the form [1 ,ℓ]∪/parenleftbign−1\n2+[2,ℓ]/parenrightbig\n, whereℓ≥1. In the former case, Acmust contain two disjoint\nintervals of length ℓ, whence (17) ensures ℓ≤r+2 and |P|= 2ℓ≤2r+4, as desired. In the\nlater case,Acmust contain two disjoint intervals, one of length ℓand one of length ℓ−1. Thus\n(17) ensures ℓ−1≤r+ 2. If this inequality is strict, then |P|= 2ℓ−1≤2r+ 3, as desired.\nOtherwise, ℓ−1 =r+2≥2, which is only possible if P=/parenleftbign+1\n2+[r+3,2r+5]/parenrightbig\n∪[r+4,2r+5] =\n[r+4,2r+5]∪[3r+9,4r+11]. However this yields the contradiction 0 ∈P⊆Ac. It remains\nto consider d∈[r+4,2r+4].\nLet\nX= [0,r+1]∪{r+3}.20 DAVID J. GRYNKIEWICZ\nNowXis disjoint from Acand (−2d+X)∩(−d+X) =∅, the latter holding else d∈X−X⊆\nn+ [−(r+ 3),r+ 3] = [3r+ 8,n+r+ 3], which contradicts that d∈[r+ 4,2r+ 4]. Since\nd∈[r+ 4,2r+ 4], we have −d+X⊆[2r+ 7,4r+ 10]⊆Ac. SinceXis disjoint from Ac, it\nfollows that any element from −d+Xcontained in Pmust be the last term in the progression P,\nensuringat most oneof thetermsfrom −d+Xlies inP. As aresult, |P| ≤ |Ac|−|X|+1 = 2r+5.\nIf this inequality is strict, the desired bound on |P|follows. Assuming otherwise, we conclude\nthat\nP=Ac\\(−d+X′)\nfor someX′⊆Xwith|X′|=|X| −1 =r+ 2. Moreover, the unique term from the set\n(−d+X)\\(−d+X′)⊆[2r+7,4r+10] must be the last term in P. This ensures that −2d+X′\nis disjoint from Ac. Otherwise, any term x∈(−2d+X′)∩Acmust be contained in P(since\nP=Ac\\(−d+X′) with−2d+X′and−d+X′disjoint), and then x+d∈Pfollows (asxis not\nthe last term in P), contradicting that x+d∈ −d+X′is disjoint from Pby definition. Since\nr≥1, it follows that either −2d+X′⊆ −2d+([0,r+1]∪{r+3}) contains two consecutive terms\nor else equals a three term arithmetic progression with differ ence 2. Either way, it follows from\n(17) that the only way −2d+X′can be disjoint from Acis if−2d+X′⊆[0,r+3]. However,\nsinceX= [0,r+1]∪{r+3}with|X′|=|X| −1, this is only possible if 2 d∈[−2,1] modn,\ncontradicting that 2 d∈[2r+8,8r+8]⊆[2,n−3]. /square\n3.Density 1−ǫ\nA main contribution of [4] was an argument using Vosper duali ty to allow the combinatorial\nportion of Freiman’s original argument [10] [9] [25] [26] to work at higher densities for A+A.\nGetting such arguments to also work for A+B, which we will accomplish in Proposition 3.2,\nis the goal of the first part of this section. We begin with a sim ple consequence of the Cauchy-\nDavenport Theorem.\nLemma 3.1. Letpbe a prime and let X, Z⊆Z/pZbe subsets. Suppose Xand/intersectiontext\nx∈X(x+Z)\nare nonempty and Z⊂Z/pZis proper. Then\n|/intersectiondisplay\nx∈X(x+Z)| ≤ |Z|−|X|+1.\nProof.LetY=/intersectiontext\nx∈X(x+Z). ThenXandYare nonempty by hypothesis. Observe that\nY−X⊆Z, for ifx∈Xandy∈Y=/intersectiontext\nx∈X(x+Z), theny=x+zfor somez∈Z, whence\ny−x=x+z−x=z∈Z. Thus, since Z⊂Z/pZis a proper subset and XandYare\nnonempty, we can apply the Cauchy-Davenport Theorem (Theor em 2.2) toY−Xto conclude\n|X|+|Y|−1≤ |Y−X| ≤ |Z|, yielding the desired upper bound for |Y|=|/intersectiontext\nx∈X(x+Z)|./square\nProposition 3.2. Letpbe prime, let A, B⊆Z/pZbe nonempty subsets with\n|A+B|=|A|+|B|+r≤3\n4(p+1)andp≥4r+9,THE 3k−4 THEOREM MODULO A PRIME 21\nand setC=−(A+B)c. Suppose there exist subsets A′⊆AandB′⊆BorA′⊆BandB′⊆A\nsuch thatA′+B′rectifies, |B′| ≤ |A′|and\n(18) |A′|+2|B′|−4≥ |A+B|.\nThen there are arithmetic progressions PA, PB, PC⊆Gall having common difference such that\nX⊆PXand|PX| ≤ |X|+r+1for allX∈ {A,B,C}\nProof.We can w.l.o.g. assume |B| ≤ |A|and that |A′|+|B′|is maximal among all subsets A′\nandB′satisfying the hypothesis of the proposition. Since A′+B′rectifies, let g∈Z/pZbe a\nnonzero difference with ℓg(A′)+ℓg(B′)≤p+1 minimal. Let\n{X, Y}={A, B}withA′⊆XandB′⊆Y.\nNote|B′| ≤ |A′|andA′⊆Xensure|A′| ≤ |X| ≤ |A|and|B′| ≤ |A′| ≤ |X|, which combined\nwithB′⊆Ygives us|B′| ≤min{|X|,|Y|}=|B|. In summary,\n|A′| ≤ |A|and|B′| ≤ |B|.\nBy hypothesis, |X| ≤ |A| ≤ |A+B| ≤ |A′|+2|B′|−4≤ |X|+2|B′|−4, ensuring |A′| ≥ |B′| ≥2.\nCombinedwiththefactthat A′+B′rectifies, wefindthat p+1≥ℓg(A′)+ℓg(B′)≥ |A′|+|B′| ≥4,\nensuringp≥3 is odd. If p= 3, then, since A′+B′rectifies, it follows that |A′|=|B′|= 2,\nwhence|A+B| ≤ |A′|+2|B′| −4 = 2< p. Forp >3, we also have |A+B| ≤3\n4(p+ 1)< p.\nThusA+B/\\⌉}atio\\slash=Gin all cases. Now |A|+|B|+r=|A+B| ≤ |A′|+2|B′|−4≤ |A|+2|B|−4,\nensuring\n(19) |A| ≥ |B| ≥r+4 and |A+B|=|A|+|B|+r≥3r+8.\nSet\n|A′+B′|=|A′|+|B′|+r′.\nTranslating AandBappropriately, we can assume\n(20) A′⊆[0,m]·g, B′⊆[0,n]·gandA′+B′⊆[0,m+n]·g\nwith 0, mg∈A′, 0, ng∈B′, andm+n≤p−1. Thusℓg(A′) =m+1 andℓg(B′) =n+1. Since\nA′+B′rectifies, it follows that the map ϕ:Z/pZ→[0,p−1]⊆Z, defined by ϕ(sg) =sfor\ns∈[0,p−1], gives a Freiman isomorphism of A′+B′with the integer sumset ϕ(A′)+ϕ(B′)⊆Z.\nObserve that\n(21) gcd/parenleftbig\nϕ(A′)−ϕ(A′)+ϕ(B′)−ϕ(B′)/parenrightbig\n= 1,\nsince ifϕ(A′) +ϕ(B′) were contained in an arithmetic progression with difference d≥2, then\nthis would also be the case for ϕ(A′) andϕ(B′), and then ℓdg(A′) +ℓdg(B′)< ℓg(A′)+ℓg(B′)\nwould follow in view of |A′| ≥ |B′| ≥2, contradicting the minimality of ℓg(A′)+ℓg(B′) forg.\nIn view of (18) and |B′| ≤ |A′|, we have |A′+B′| ≤ |A+B| ≤ |A′|+|B′|+min{|A′|,|B′|}−4.\nThus, in view of (21), we can apply the 3 k−4 Theorem (Theorem 1.1) to the isomorphic sumset22 DAVID J. GRYNKIEWICZ\nϕ(A′)+ϕ(B′). Then, letting PA= [0,m]·g,PB= [0,n]·gandPA+B⊆A′+B′be the resulting\narithmetic progressions with common difference g, we conclude that |PA+B| ≥ |A′|+|B′|−1,\n(22) |PA\\A′| ≤r′+1 and |PB\\B′| ≤r′+1.\nIfA′=XandB′=Y, then the original sumset A+B=X+Yrectifies,r′=rand the\nproposition follows with the progressions PA,PBand−(PA+B)cjust defined (interchanging PA\nandPBifX=B). Therefore we can assume otherwise:\n(23) Y\\B′/\\⌉}atio\\slash=∅orX\\A′/\\⌉}atio\\slash=∅.\nLet ∆≥0 be the integer such that\n|A+B|=|A′+B′|+∆,\nin which case\n(24) r′=|A\\A′|+|B\\B′|+r−∆.\nSince|A′|+|B′|+r′=|A′+B′|=|A+B|−∆, it follows from (18) and |B′| ≤ |A′|that\n(25) r′≤ |B′|−4−∆ and r′≤ |A′|−4−∆.\nAveraging both bounds above together with the bound (24), an d recalling that |A+B|=\n|A|+|B|+r, we obtain\n(26) r′≤1\n3|A+B|−8\n3−∆.\nStep A. |−A′+(A′+Y)c| ≤ |(A′+Y)c|+2|A′|−4.\nProof.If Step A fails, then it follows from p−|A+B|=|(A+B)c|=|(X+Y)c| ≤ |(A′+Y)c|\nand Proposition 2.5 that\np−|A+B|+2|A′|−3≤ |(A′+Y)c|+2|A′|−3≤ |−A′+(A′+Y)c| ≤ |Yc|=p−|Y|.\nThus (18), |B′| ≤ |Y|and|B′| ≤ |A′|imply\n|Y|+2|A′|−3≤ |A+B| ≤ |A′|+2|B′|−4≤ |Y|+2|A′|−4,\nwhich is not possible. /square\nStep B. |−B′+(X+B′)c| ≤ |(X+B′)c|+2|B′|−4.\nProof.If Step B fails, then from p− |A+B|=|(A+B)c|=|(X+Y)c| ≤ |(X+B′)c|and\nProposition 2.5 it follows that\np−|A+B|+2|B′|−3≤ |(X+B′)c|+2|B′|−3≤ |−B′+(X+B′)c| ≤ |Xc|=p−|X|.\nThus (18) and |A′| ≤ |X|imply\n|X|+2|B′|−3≤ |A+B| ≤ |A′|+2|B′|−4≤ |X|+2|B′|−4,\nwhich is not possible. /squareTHE 3k−4 THEOREM MODULO A PRIME 23\nStep C. If|A′|+|Y| ≥ |B′|+|X|, then|−A′+(A′+Y)c| ≤ |A′|+2|(A′+Y)c|−3.\nProof.Note 2p−2|A+B|= 2|(A+B)c|= 2|(X+Y)c| ≤2|(A′+Y)c|, while Proposition 2.5\nimplies|−A′+(A′+Y)c| ≤ |Yc|. As a result, if Step C fails, we obtain\n|A′|+2p−2|A+B|−2≤ |A′|+2|(A′+Y)c|−2≤ |−A′+(A′+Y)c| ≤ |Yc|=p−|Y|.\nCollecting terms in the above inequality, multiplying by 2, and using the hypothesis of Step C,\nwe obtain\n2p≤4|A+B|−2|Y|−2|A′|+4≤4|A+B|−|X|−|Y|−|A′|−|B′|+4.\nUsing the identities |X|+|Y|=|A|+|B|=|A+B| −rand|A′|+|B′|=|A′+B′| −r′=\n|A+B|−r′−∆ in the above estimate, and then applying the estimate (26), yields\n(27) 2 p≤2|A+B|+r+r′+∆+4 ≤7\n3|A+B|+r+4\n3.\nBy hypothesis, |A+B| ≤3p+3\n4andp≥4r+9. Since p≥3 is odd, strict inequality in the\nfirst estimate together with (27) yields3p+1\n4≥ |A+B| ≥6p−3r−4\n7, implyingp≤4r+7, which is\ncontrary to hypothesis. Therefore we can instead assume equ ality holds: |A+B|=3p+3\n4. As a\nresult,p≡3 mod 4, in which case the hypothesis p≥4r+9 improves to p≥4r+11. In such\ncase, (27) yields3p+3\n4=|A+B| ≥6p−3r−4\n7, implyingp≤4r+11 (recall p≡3 mod 4). Thus\np= 4r+11 and |A+B|=3\n4(p+1) = 3(r+3) = 3r+9.\nSince(19)implies3 r+9 =|A+B|=|A|+|B|+rwith|A| ≥ |B| ≥r+4, itfollows that |B|=r+4,\n|A|=r+5 and|A|+2|B|−4 =|A+B|. Hence, since |A+B| ≤ |A′|+2|B′|−4≤ |A|+2|B|−4,\nit follows that |A′|=|A|and|B′|=|B|. But then A′=XandB′=Y, contradicting (23). /square\nStep D. If|B′|+|X| ≥ |A′|+|Y|, then|−B′+(X+B′)c| ≤ |B′|+2|(X+B′)c|−3.\nProof.Note 2p−2|A+B|= 2|(A+B)c|= 2|(X+Y)c| ≤2|(X+B′)c|, while Proposition 2.5\nimplies|−B′+(X+B′)c| ≤ |Xc|. As a result, if Step D fails, we obtain\n|B′|+2p−2|A+B|−2≤ |B′|+2|(X+B′)c|−2≤ |−B′+(X+B′)c| ≤ |Xc|=p−|X|.\nCollecting terms in the above inequality, multiplying by 2, and using the hypothesis of Step D,\nwe obtain\n2p≤4|A+B|−2|X|−2|B′|+4≤4|A+B|−|X|−|Y|−|A′|−|B′|+4,\nand we now obtain the same contradiction as at the end of Step C . /square\nByourapplication ofthe3 k−4 Theorem(Theorem1.1)to ϕ(A′)+ϕ(B′), weknowthat A′+B′\ncontains an arithmetic progression PA+Bwith difference gand length |PA+B| ≥ |A′|+|B′|−1,\nwhich implies\n(28) ℓg/parenleftbig\n(A′+B′)c/parenrightbig\n≤p−|A′|−|B′|+1.24 DAVID J. GRYNKIEWICZ\nFrom (22) and (25), we obtain\n(29) ℓg(−A′) =ℓg(A′)≤ |PA| ≤ |A′|+r′+1≤ |A′|+|B′|−3.\nCombined with (28), it follows that ℓg(−A′)+ℓg/parenleftbig\n(A′+B′)c/parenrightbig\n≤p−2, ensuring −A′+(A′+B′)c\nrectifies via the difference g. Since (A′+Y)c⊆(A′+B′)c, it follows that −A′+(A′+Y)calso\nrectifies via the difference g.\nFrom (22) and (25), we obtain\n(30) ℓg(−B′) =ℓg(B′)≤ |PB| ≤ |B′|+r′+1≤ |A′|+|B′|−3.\nCombined with (28), it follows that ℓg(−B′)+ℓg/parenleftbig\n(A′+B′)c/parenrightbig\n≤p−2, ensuring −B′+(A′+B′)c\nrectifies via the difference g. Since (X+B′)c⊆(A′+B′)c, it follows that −B′+(X+B′)calso\nrectifies via the difference g.\nBy our application of the 3 k−4 Theorem (Theorem 1.1) to ϕ(A′)+ϕ(B′), we know ϕ(A′) is\ncontained in thearithmetic progression ϕ(PA) = [0,m] with difference 1 and length |PA| ≤ |A′|+\nr′+1, with the latter inequality by (22). Moreover, (25) ensur es thatr′+1≤ |B′|−3≤ |A′|−3,\nin which case ϕ(A′) contains at least 2 consecutive elements. Hence\n(31) gcd/parenleftbig\nϕ(A′)−ϕ(A′)/parenrightbig\n= 1.\nLikewise, by our application of the 3 k−4 Theorem (Theorem 1.1) to ϕ(A′)+ϕ(B′), we know\nϕ(B′) is contained in the arithmetic progression ϕ(PB) = [0,n] with difference 1 and length\n|PB| ≤ |B′|+r′+1, withthelatterinequalityby (22). Moreover, (25)ensur esthatr′+1≤ |B′|−3,\nin which case ϕ(B′) contains at least 2 consecutive elements. Hence\n(32) gcd/parenleftbig\nϕ(B′)−ϕ(B′)/parenrightbig\n= 1.\nWe divide the remainder of the proof into two short, and nearl y identical, subcases.\nCASE 1: |A′|+|Y|>|B′|+|X|.\nSince−A′+ (A′+Y)crectifies via the difference g, we see that it is isomorphic to the\nsumsetϕ(m−A′) +ϕ(−x+ (A′+Y)c), wherex∈Z/pZis the first term in the minimal\nlength arithmetic progression with difference gcontaining ( A′+Y)c. Hence, in view of (31),\nStep A, the case hypothesis and Step C, we can apply the 3 k−4 Theorem (Theorem 1.1) to the\nisomorphic sumset ϕ(m−A′)+ϕ(x+(A′+Y)c) and thereby conclude that there is an arithmetic\nprogression P⊆ −A′+ (A′+Y)cwith difference gand length |P| ≥ |A′|+|(A′+Y)c| −1≥\n|A′|+|(X+Y)c|−1 =p−|A+B|+|A′|−1. Consequently, since Proposition 2.5 ensures that\nP⊆ −A′+(A′+Y)c⊆Yc, it follows that\nℓg(Y)≤ |A+B|−|A′|+1.\nCombined with (29), we find that\n(33) ℓg(A′)+ℓg(Y)≤ |A+B|+r′+2.THE 3k−4 THEOREM MODULO A PRIME 25\nIfA′+Ydoes not rectify, then (33) and (26) imply p≤ |A+B|+r′≤4\n3|A+B|−8\n3, whence\n|A+B| ≥3\n4p+2,\ncontrary to hypothesis. Therefore we instead conclude that A′+Yrectifies. Consequently,\nA′+Y′also rectifies for any subset Y′withB′⊆Y′⊆Y. As a result, if |B′|<|Y|, then there\nis subsetY′withB⊆Y′⊆Yand|Y′|=|B′|+1 such that A′+Y′rectifies. If |B′|<|A′|, then\n|Y′|=|B′|+1≤ |A′|and|A′|+2|Y′|−4≥ |A′|+2|B′|−2≥ |A+B|, which would contradict\nthe maximality of |A′|+|B′|. If|B′|=|A′|, then|Y′|=|B′|+1>|A′|and|Y′|+2|A′| −4 =\n|A′|+2|B′|−3≥ |A+B|, also contradicting the maximality of |A′|+|B′|. We are left to conclude\nthat|B′|=|Y|. But now the case hypothesis yields |A′|+|Y|>|B′|+|X|=|Y|+|X|, yielding\nthe contradiction |A′|>|X|(asA′⊆X).\nCASE 2: |B′|+|X| ≥ |A′|+|Y|.\nThis subcase is nearly identical with the last, though there are a few small differences. Since\n−B′+(X+B′)crectifies via the difference g, it is isomorphic to ϕ(n−B′)+ϕ(−y+(X+B′)c),\nwherey∈Z/pZis the first term in the minimal length arithmetic progressio n with difference g\ncontaining ( X+B′)c. Hence, in view of (32), Step B, the case hypothesis and Step D , we can\napply the 3 k−4 Theorem (Theorem 1.1) to the isomorphic sumset ϕ(n−B′)+ϕ(y+(X+B′)c)\nand thereby concludethat thereis an arithmetic progressio nP⊆ −B′+(X+B′)cwithdifference\ngand length |P| ≥ |B′|+|(X+B′)c| −1≥ |B′|+|(X+Y)c| −1 =p− |A+B|+|B′| −1.\nConsequently, since Proposition 2.5 ensures that P⊆ −B′+(X+B′)c⊆Xc, it follows that\nℓg(X)≤ |A+B|−|B′|+1.\nCombined with (30), we find that\n(34) ℓg(B′)+ℓg(X)≤ |A+B|+r′+2.\nIfX+B′does not rectify, then (34) and (26) imply p≤ |A+B|+r′≤4\n3|A+B|−8\n3, whence\n|A+B| ≥3\n4p+2, contrary to hypothesis. Therefore we instead conclude t hatX+B′rectifies,\nin which case X+B′contradicts the maximality of |A′|+|B′|unlessA′=X. However, A′=X\nimplies|A′|=|X|, which combined with the case hypothesis yields |B′| ≥ |Y|. SinceB′⊆Y,\nthis in turn forces B′=Y. But nowA′=XandB′=Y, contradicting (23), which completes\nCASE 2, and the proof as well. /square\nWith Proposition 3.2 in hand, the method of Freiman [10] [9] [ 25] [26] (used for A+A) can\nnow be pushed to higher densities for A+B, as was done in [4] for A+A. However, to deal\nwith general sumsets A+Brather than A+A, we will need to use a new variation for how the\nexponential sums are estimated in the argument. For followi ng the numerical calculations, both\nhere and later, the reader may wish to use an algebraic comput ation system since they can be\nvery tedious to do by hand.26 DAVID J. GRYNKIEWICZ\nTheorem 3.3. LetG=Z/pZwithp≥2a prime, let β∈[0.731,1]andα∈(0,0.212]be real\nnumbers, let A, B⊆Gbe nonempty subsets, and set C=−(A+B)c. Suppose\n|A+B|=|A|+|B|+r≤ |A|+(1+α)|B|−3andβ|A| ≤ |B| ≤ |A| ≤min{c1,cβ}p,\nwherec1=5−18α−24α2−8α3\n14−9α−24α2−8α3andcβ=−4−(1+12α)β+3(5+α−4α2)β2+(−1+3α−4α3)β3−4(1+α)3β4\n−4+11β2+11β3−4β5−3αβ(4−5β2+4β4)−12α2(β2+β5)−4α3(β3+β5).\nThen there are arithmetic progressions PA, PB, PC⊆Gall having common difference such that\nX⊆PXand|PX| ≤ |X|+r+1for allX∈ {A,B,C}.\nProof.The real roots/singularities of c1areα∼=0.2129,−2,−1.56 and 0.56. Thusc1∈(0,5\n14]\nfor allα∈(0,0.212]. Consider the denominator of cβ:\n−4+11β2+11β3−4β5−3αβ(4−5β2+4β4)−12α2(β2+β5)−4α3(β3+β5).\nCalculating successive derivatives with respect to α, it follows that all partial derivatives with\nrespect toαare non-positive, for α≥0 andβ >0. Hence, since α≤0.212, it follows that the\ndenominator of cβis at least the value obtained underthe substitution α= 0.212, which is a 5-th\ndegree polynomial whose only real roots are β≈ −0.825, 0.562 and 1.6199, from which we can\nderive that the denominator of cβis positive for all α∈(0,0.212] andβ∈[0.731,1]. Thus the\nhypothesis 0 <|A| ≤cβpensures that both cβand the numerator of cβare positive. We have\n|A+B|<(2 +α)|A| ≤(2 +α)c1p=(5−18α−24α2−8α3)(2+α)\n14−9α−24α2−8α3p. The derivative of the coefficient\nofpequals2(−43−128α−60α2+64α3+32α4)\n(−7+8α+8α2)2, which has roots/discontinuities for α≈ −1.56, 0.56,\n−2.139,−0.8455,−0.5 and 1.48 with the derivative negative for α∈[0,0.5]. Thus |A+B| ≤\n(5−18α−24α2−8α3)(2+α)\n14−9α−24α2−8α3p≤10\n14p<0.715p.As a result,\n(35) |A+B| ≤3\n4p.\nIn particular, A+B/\\⌉}atio\\slash=G, so the Cauchy-Davenport Theorem (Theorem 2.2) implies r≥ −1.\nBy hypothesis, α<1 and\nr≤α|B|−3<|B|−3.\nThus|A| ≥ |B| ≥r+4, whence 3 r+8≤ |A|+|B|+r=|A+B| ≤3\n4p, in turn implying that\n(36) p≥4r+11 and r≤1\n4p−2.75.\nLet\nℓ=|A+B|=|A|+|B|+r, a=|A|andb=|B|.\nIn view of (35) and (36), we can apply Proposition 3.2 and ther eby complete the proof unless\nevery pair of subsets A′⊆AandB′⊆Bsuch thatA′+B′rectifies has\n|A′|+|B′|+min{|A′|,|B′|} ≤ |A+B|+3 =ℓ+3,\nin turn implying\n(37) min {|A′|,|B′|} ≤1\n3ℓ+1.THE 3k−4 THEOREM MODULO A PRIME 27\nFor the rest of this proof, let us identify Z/pZwith the set of integers [0 ,p−1] with addition\nmodp. Then, for every X⊆Z/pZandd∈Z/pZ, we haveSX(d) =/summationtext\nx∈Xexp(dx/p)∈C.\nClaim A: For everyd∈[1,p−1], either |SA(d)| ≤γA:=2ℓ+6−3|A|\n3≤1\n3/parenleftbig\n2αb+ 2b−a/parenrightbig\nor\n|SB(d)| ≤γB:=2ℓ+6−3|B|\n3≤1\n3/parenleftbig\n2αb+2a−b/parenrightbig\n.\nProof.Letd∈[1,p−1] be arbitrary. For u, v∈[0,1), consider the open half-arcs Cu=\n{exp(x) :x∈(u,u+1\n2)}andCv={exp(x) :x∈(v,v+1\n2)}in the unit circle in C. Let\nA′={x∈A: exp(dx/p)∈Cu}andB′={x∈B: exp(dx/p)∈Cv}, and choose u, v∈[0,1)\nto maximize |A′|and|B′|. Since the set of p-th roots of unity contained in CuorCvcorrespond\nto an arithmetic progression of difference 1 in Z/pZ, it is clear that, for d∗the multiplicative\ninverse ofdmodulop, we haveℓd∗(A′)≤p+1\n2andℓd∗(B′)≤p+1\n2. Hence the sumset A′+B′\nrectifies. Then (37) implies that min {|A′|,|B′|} ≤1\n3ℓ+1. If|B′| ≤ |A′|, then the maximality of\nvtogether with Lemma 2.1 implies |SB(d)| ≤2(1\n3ℓ+1)−|B|. If|A′| ≤ |B′|, then the maximality\nofutogether with Lemma 2.1 implies |SA(d)| ≤2(1\n3ℓ+1)−|A|. In either case, one of the desired\nbounds follows. /square\nIn view of Claim A, we can partition [1 ,p−1] =IA∪IBsuch that |SA(d)| ≤γA=2ℓ+6−3|A|\n3for\nd∈IA, and|SB(d)| ≤γB=2ℓ+6−3|B|\n3ford∈IB. Moreover, set Cd=BandDd=Aifd∈IA,\nandCd=AandDd=Bford∈IB. To complete the proof, we now exploit Claim A to obtain\na contradiction, using in particular the following manipul ations which are a new variation on\nfourier estimates used in Additive Combinatorics (e.g. [13 , Corollary 19.1] [13, pp. 290–291]).\nBy Fourier inversion, the triangle inequality, the fact tha tSA(0) =|A|=a,SB(0) =|B|=b\nandSA+B(0) =ℓ, and the Cauchy-Schwarz Inequality, we have\nabp=p−1/summationdisplay\nx=0SA(x)SB(x)SA+B(x) =abℓ+p−1/summationdisplay\nx=1SA(x)SB(x)SA+B(x)\n≤abℓ+p−1/summationdisplay\nx=1|SA(x)||SB(x)||SA+B(x)| ≤abℓ+p−1/summationdisplay\nx=1γDx|SCx(x)||SA+B(x)|\n≤abℓ+/parenleftBigp−1/summationdisplay\nx=1γ2\nDx|SCx(x)|2/parenrightBig1/2/parenleftBigp−1/summationdisplay\nx=1|SA+B(x)|2/parenrightBig1/2\n≤abℓ+/parenleftBig\nγ2\nBp−1/summationdisplay\nx=1|SA(x)|2+γ2\nAp−1/summationdisplay\nx=1|SB(x)|2/parenrightBig1/2/parenleftBigp−1/summationdisplay\nx=1|SA+B(x)|2/parenrightBig1/2\n=abℓ+/parenleftbig\nγ2\nB(ap−a2)+γ2\nA(bp−b2)/parenrightbig1/2(ℓp−ℓ2)1/2.\nThusa2b2(p−ℓ)2≤/parenleftbig\nγ2\nBa(p−a)+γ2\nAb(p−b)/parenrightbig\nℓ(p−ℓ),in turn implying\n(38) a2b2(p−ℓ)−/parenleftbig\nγ2\nBa(p−a)+γ2\nAb(p−b)/parenrightbig\nℓ≤0.\nIt remains to derive an estimate for p/afrom (38) that is contrary to our hypotheses.28 DAVID J. GRYNKIEWICZ\nLet\nx=b\naandz=p\na.\nNote\n0.731≤β≤x≤1 andz≥max{c−1\n1,c−1\nβ}.\nMultiplying (38) by 9 /a5, applying the estimate ℓ=|A+B|<|A|+(1+α)|B|=a+(1+α)b,\nand using the estimates for γAandγBgiven in Step A, we obtain\n(39) 9x2(z−1−x−αx)−(1+x+αx)/parenleftBig\n(2αx+2−x)2(z−1)+(2αx+2x−1)2x(z−x)/parenrightBig\n<0.\nLetf(x,z,α) denote the left hand side of (39). Now\n∂2\n∂x2f= (30−6x−48x2+6α−24α2+18αx−24α3x−144αx2−144α2x2−48α3x2)z\n−22−66x+80x3+24α2−90αx+24α3x+240αx3+240α2x3+80α3x3.\nSince 0.731≤β≤x≤1 andα>0, we have 30 −6x−48x2<0 and 6α+18αx−144αx2≤0,\nso the coefficient of zabove is negative Thus, using the hypothesis z≥c−1\n1≥14\n5= 2.8, we find\nthat\n∂2\n∂x2f≤62−82.8x−134.4x2+80x3+α(16.8−39.6x−403.2x2+240x3)\n+α2(−43.2−403.2x2+240x3)+α3(−43.2x−134.4x2+80x3)\nSince 0.731≤β≤x≤1, it follows that each coefficient of aterm αiabove (fori≥1) is negative,\nwhence applying the estimate α≥0 yields∂2\n∂x2f≤62−82.8x−134.4x2+80x3<0, with latter\ninequality in view of 0 .5<x<2. Thus∂2\n∂x2f <0, ensuring that the value of f(x,z,α) will be\nminimized at an extremal value for x, and thus (39) holds either with x= 1 orx=β≥0.731.\nSuppose (39) hold with x= 1. Then (39) yields\n−14+9α+24α2+8α3+(5−18α−24α2−8α3)z <0.\nSinceα≤0.212, itfollowsthatthecoefficient of zaboveispositive, whence z<14−9α−24α2−8α3\n5−18α−24α2−8α3=\nc−1\n1, contrary to hypothesis. So we may now instead assume (39) ho lds withx=β≥0.731. In\nthis case, (39) instead implies z <c−1\nβ, again contrary to hypothesis, completing the proof. /square\nTo gain a feel for possible values of the constants allowed in Theorem 3.3, we state three\nexplicit cases. If one compares with the recent work of Huico chea [18] mentioned in the intro-\nduction, here we obtain better constants at the expense of |A|and|B|being required to be\nmuch closer in size. Since we will later be able to remove such restrictions using isoperimetric\narguments, these added restrictions are less of a concern wi th Theorem 3.3 an intermediary step\nfor us.THE 3k−4 THEOREM MODULO A PRIME 29\nCorollary 3.4. LetG=Z/pZwithp≥2a prime, let A, B⊆Gbe nonempty subsets, and set\nC=−(A+B)c. Suppose either\n|A+B|=|A|+|B|+r≤ |A|+1.021|B|−3and4\n5|A| ≤ |B| ≤ |A| ≤p\n3,or\n|A+B|=|A|+|B|+r≤ |A|+1.105|B|−3and4\n5|A| ≤ |B| ≤ |A| ≤p\n5, or\n|A+B|=|A|+|B|+r≤ |A|+(1+1\n9)|B|−3and0.8484|A| ≤ |B| ≤ |A| ≤1963\n9253p.\nThen there are arithmetic progressions PA, PB, PC⊆Gall having common difference such that\nX⊆PXand|PX| ≤ |X|+r+1for allX∈ {A,B,C}.\nProof.Thisfollows from Theorem 3.3 appliedwith α= 0.021 andβ=.8, appliedwith α= 0.105\nandβ=.8, and applied with α=1\n9andβ= 0.8484 /square\nOur next step is to use isoperimetric machinery to remove mos t size and density restrictions\nfrom Theorem 3.3, resulting in the following theorem.\nTheorem 3.5. LetG=Z/pZwithp≥2a prime, let A, B⊆Gbe nonempty subsets with\nA+B/\\⌉}atio\\slash=G, and setC=−(A+B)c. Suppose |A| ≥ |B|,9253\n1963/parenleftBig\n10r+26.5 +/radicalBig\n2r+17\n4/parenrightBig\n≤p,\nand\n|A+B|=|A|+|B|+r≤min{|A|+/parenleftbig\n1+1\n9/parenrightbig\n|B|−3, p−9(r+3)}.\nThen there are arithmetic progressions PA, PB, PC⊆Gall having common difference such that\nX⊆PXand|PX| ≤ |X|+r+1for allX∈ {A,B,C}.\nProof.SinceA+B/\\⌉}atio\\slash=G, the Cauchy-Davenport Theorem (Theorem 2.2) ensures that r≥ −1,\nwhence|A+B| ≤p−9(r+3)≤p−18. By hypothesis, we have\n|A|,|B|,|C| ≥9(r+3)≥r+19≥18.\nAs a result, if r≤0, then Conjecture 1.2 holds for A+B(as mentioned in the Small rSubsection\nof the Introduction), and the desired conclusion follows. T herefore we can assume r≥1. By\nhypothesis, we have\n(40) p≥9253\n1963/parenleftBig\n10r+26.5+/radicalbigg\n2r+17\n4/parenrightBig\n≥14r+34.\nSince|A|,|C| ≥9(r+3), itfollowsthat Bis9(r+3)-separablewith κ9(r+3)(B)≤ |A+B|−|A|=\n|B|+r. LetXbe a 9(r+3)-atom of B. Then\n(41) |X+B|−|X|=κ9(r+3)(B)≤ |B|+rand|X+B| ≤ |X|+|B|+r.\nIn view of (40) and r≥1, we can apply Corollary 2.10.3 to conclude that\n(42) |X|=α9(r+3)(B)≤10r+26.5+/radicalbigg\n2r+17\n4≤1963\n9253p,30 DAVID J. GRYNKIEWICZ\nwith the latter inequality holding by hypothesis. Since Bis 9(r+3)-separable with Xa 9(r+3)-\natom ofB, it follows by definition of an atom that |X+B| ≤p−9(r+3). Thus, since |B| ≥\n9(r+3), it follows that Xis also 9(r+3)-separable with κ9(r+3)(X)≤ |X+B|−|B| ≤ |X|+r.\nLetYbe a 9(r+3)-atom of X. Then\n(43) |X+Y|−|Y| ≤κ9(r+3)(X)≤ |X|+rand|X+Y| ≤ |X|+|Y|+r.\nSinceXandYare 9(r+3)-atoms, we have\n(44) |X|,|Y| ≥9(r+3) and |X+Y| ≤p−9(r+3),\nand Corollary 2.10.3 again gives\n(45) |Y|=α9(r+3)(X)≤10r+26.5+/radicalbigg\n2r+17\n4≤1963\n9253p,\nwith the latter inequality holding by hypothesis.\nIn view of (43) and (44), we have\n(46) |X+Y| ≤ |X|+|Y|+r≤min/braceleftBig\n|X|+|Y|+1\n9min{|X|,|Y|}−3, p−9(r+3)/bracerightBig\n.\nIn view of (42), (45) and (44), we have\n(47)min{|X|,|Y|}\nmax{|X|,|Y|}≥9r+27\n10r+26.5+/radicalBig\n2r+17\n4≥0.8877,\nwith the latter inequality following by a routine calculus m inimization question showing that\nthe above expression is minimized for r= 24. Now (42), (45), (46) and (47) allow us to apply\nCorollary 3.4 to X+Yto conclude that\n(48) ℓd(X)≤ |X|+r+1 and ℓd(Y)≤ |Y|+r+1\nfor some nonzero d∈G. We have |B|,|X| ≥9(r+3)≥3r+6 and |X+B| ≤p−9(r+3)≤\np−(3r+6), so (41) and (48) allow us to apply Theorem 2.13.1 (with h=r+1) to the additive\ntrio (X,B,−(X+B)c) to conclude that\nℓd(B)≤ |B|+r+1.\nWe again have |A|,|B|,|C| ≥9(r+3)≥3r+6, so this allows us to apply Theorem 2.13.1 (with\nh=r+1) to the additive trio ( B,A,C) to conclude that ℓd(A)≤ |A|+r+1,ℓd(B)≤ |B|+r+1\nandℓd(C)≤ |C|+r+1, which is the desired conclusion of thetheorem, completi ng the proof. /square\nLemma 3.6. Letr≥ −1be an integer. Then/radicalBig\n2r+17\n4≤10\n3+0.16797r.\nProof.The desired inequality is true for r∈ {−1,0}, so we can assume r≥1 and seek to\nmaximize the expression/radicalBig\n2r+17\n4−10\n3\nrforr≥1 an integer. The derivative of this expression is\n−51−12r+20√17+8r\n6r2√17+8r, which is positive for r= 1 and negative for sufficiently large rhaving a uniqueTHE 3k−4 THEOREM MODULO A PRIME 31\nroot (larger than 1) at r≈15.5923, with the expression then maximized when r= 16. Thus/radicalBig\n2r+17\n4−10\n3\nr≤/radicalBig\n32+17\n4−10\n3\n16≤0.16797, as desired. /square\nRemoving the unwanted hypothesis9253\n1963/parenleftBig\n10r+ 26.5 +/radicalBig\n2r+17\n4/parenrightBig\n≤pfrom Theorem 3.5 is\nsimply a matter of imposing a sufficiently stronger small sums et hypothesis that implicitly forces\nthis needed hypothesis. Thus we now proceed with the proof of Theorem 1.3.\nProof of Theorem 1.3. If we can show9253\n1963/parenleftBig\n10r+ 26.5 +/radicalBig\n2r+17\n4/parenrightBig\n≤pis implied by the\nhypotheses of Theorem 1.3, then we can apply Theorem 3.5 to A+Bto complete the proof.\nThus, in view of Lemma 3.6, it suffices to show\n(49)9253\n1963/parenleftBig\n10.16797r+26.5+10\n3/parenrightBig\n≤p.\nBy hypothesis of Theorem 1.3, we have 2( r+3)α−1+r≤ |A|+|B|+r=|A+B| ≤p−9(r+3),\nwhereα= 0.0527, implying\n(50) (2 α−1+10)r+(27+6α−1)≤p.\nSince (2α−1+ 10)≥9253\n1963·10.16797 and (27 + 6 α−1)≥9253\n1963/parenleftBig\n26.5 +10\n3/parenrightBig\n, (50) implies (49), as\nneeded. /square\nTheorem1.3isnottheonlyvariationderivablefromTheorem 3.5. Forinstance, thehypothesis\n9253\n1963/parenleftBig\n10r+26.5+/radicalBig\n2r+17\n4/parenrightBig\n≤pcan also be removed by increasing the density restriction.\nCorollary 3.7. LetG=Z/pZwithp≥2a prime, let A, B⊆Gbe nonempty subsets with\nA+B/\\⌉}atio\\slash=G, and setC=−G\\(A+B). Suppose |A| ≥ |B|and\n|A+B|=|A|+|B|+r≤min{|A|+1\n9|B|−3, p−29(r+3)}.\nThen there are arithmetic progressions PA, PB, PC⊆Gall having common difference such that\nX⊆PXand|PX| ≤ |X|+r+1for allX∈ {A,B,C}.\nProof.As in the proof Theorem 1.3, it suffices to show (49) holds as the n Theorem 3.5 can\nbe applied to A+Bto complete the proof. By hypothesis, we have 18( r+ 3) +r≤ |A|+\n|B|+r=|A+B| ≤p−29(r+ 3), implying 48 r+ 141≤p. Since 48 ≥9253\n1963·10.16797 and\n141≥9253\n1963/parenleftBig\n26.5+10\n3/parenrightBig\n, this implies (49), as needed. /square\nWe would like to use the improved mid-range density result fr om [23] in the proof of Theorem\n1.4. Unfortunately, as stated, it includes the hypothesis |A| ≥101 in order to yield a slightly\nbetter small doubling constraint α <0.59|B| −3. This makes their result unusable for our\npurposes since we will need to apply it to an auxiliary set Xthat may be much smaller than A.\nHowever, it is a simple matter to vary the calculations at the end of their proof to obtain the\nfollowing flexible version of Lev and Shkredov’s result vali d for smaller sets. While the proof\nonly requires K≤3, we need c >0 for the result to be non-void, which requires K/lessorsimilar2.595.32 DAVID J. GRYNKIEWICZ\nAdditionally, by using Proposition 3.2 and Lemma 2.1 instea d of [21, Lemma 2], we extend the\narguments of Lev and Shkredov to higher densities |A| ≤3\n4Kprather than |A|<1\n12p. Note, the\nmethods of [4] (based on the original fourier sum estimates o f Freiman and Rodseth) still yield\nbetter constants at high densities (roughly once K≤2.349 and |A+A| ≥0.364p, fors= 7),\nso the improvements here are of principal interest for the hi gher mid-range densities we will\nencounter in the proof of Theorem 1.4.\nTheorem 3.8. LetG=Z/pZwithp≥2a prime, let A⊆Gbe a nonempty subset with\n2A/\\⌉}atio\\slash=G, and setC=−G\\(2A). LetK∈(2,3)be a real number, let s≥K2and let\nc=−27K+9K2+s(9+9K−9K2+12K3−4K4)\n4s(3−K)K4.\nSupposes≤ |A| ≤cp,|2A|= 2|A|+r≤3\n4pand|2A|<K|A|−3. Then there are arithmetic\nprogressions PA, PC⊆Gboth having common difference such that X⊆PXand|PX| ≤ |X|+r+1\nfor allX∈ {A,C}.\nProof.The proof is nearly identical to [23, Theorem 4] apart from th e usage of Proposition\n3.2 and Lemma 2.1 in place of [23, Corollary 1]. We sketch the m inor differences. View-\ningcKas a function of sandK, it has positive derivative with respect to swithcK→\n9+9K−9K2+12K3−4K4\n4(3−K)K4<1 (as 2< K < 3). This show that 1 −cK >0 always holds. Set\nK′:=|2A|\n|A|<K. While their notation is different from ours (their ˆA(χ) forχ∈ˆFpare the same\nas ourSA(x) forx∈[0,p−1]), defining ηby max{|SA(x)|:x∈[1,p−1]}=η|A|agrees with\nhowηis defined in [23, Proof Theorem 4]. Since the derivative (wit h respect to K′) of1\nK′+K′\n|A|\nis1\n|A|−1\n(K′)2<1\n|A|−1\nK2≤0 (in view of |A| ≥s≥K2), the fourier calculation in [23, Proof\nTheorem 4] ending with the estimate ≥/parenleftBig\n1+1\nK′−3−K′\n|A|/parenrightBig\n|A|3is made smaller by replacing K′\nwith its upper bound K, allowing us to assume K=K′for the numerical calculations from that\npoint onward. The inequality\n(51)η2≥1\nK(1−cK)/parenleftBig\n1+1\nK−3−K\n|A|−cK2/parenrightBig\n≥1\nK(1−cK)/parenleftBig\n1+1\nK−3−K\ns−cK2/parenrightBig\nfollows by the authors’ arguments, with the latter inequali ty holding since K≤3 allows us to\napply the hypothesis |A| ≥s.\nLetA′⊆Abe a maximal cardinality subset such that A′+A′rectifies. Let n=|A′|. We\nhave|2A| ≤3\n4pand 2|A|+r=|2A|< K|A| −3 by hypothesis, whence r <(K−2)|A| −3.\nThus3\n4p≥ |2A|= 2|A|+r >2(r+3)\nK−2+r≥3r+ 6, implying 4 r+ 9≤p. This gives us the\nhypotheses of Proposition 3.2. As a result, if 3 n−4≥ |2A|, then Proposition 3.2 yields the\ndesired conclusion. Therefore we can instead assume n≤1\n3(|2A|+3)<1\n3K|A|. This ensures,\nfor anyx∈[1,p−1], that less than1\n3K|A|of the terms in the fourier sum SA(x) lies in an open\nhalf arc. Otherwise the inverse of xmodulopwill be the difference of an arithmetic progression\nof length at mostp+1\n2containing at least1\n3K|A|elements of A(as argued in Claim A of theTHE 3k−4 THEOREM MODULO A PRIME 33\nproof of Theorem 3.3), contrary to what was just concluded. A pplying Lemma 2.1, we conclude\nthatη|A|<2\n3K|A| − |A|, whenceη <2\n3K−1. Applying this estimate in (51) yields a linear\ninequality in cthat (in view of 1 −cK >0) impliesc>−27K+9K2+s(9+9K−9K2+12K3−4K4)\n4s(3−K)K4 after\nrearranging all expressions, which is contrary to hypothes is. /square\n4.Optimal Density\nThe final section is devoted to the proof of Theorem 1.4. We tak e the general approach of [31]\nas a broad outline, though we modify and add to the arguments a nd incorporate several new\nadvances, including our Theorem 1.3, the reduction result o f Huichochea [19], the mid-range\ndensity result of Lev and Shkredov [23] as modified in Theorem 3.8, and the improved bounds\nfor 2-atoms given in [13] [15]. The use of [31, Lemma 4.5] is re placed by an alternative reduction\nargument utilizing Theorem 2.13.\nProof of Theorem 1.4. SinceA+B/\\⌉}atio\\slash=G, we haver≥ −1 by the Cauchy-Davenport Theorem\n(Theorem 2.2). Let\nγ= 0.01\nIn view of the hypotheses, we have\n(52) |A| ≥ |B| ≥γ−1(r+3) = 100( r+3) and |C| ≥r+3.\nIfr≤0, then (as mentioned in the Small rSubsection of the Introduction), Conjecture 1.2\nis known yielding the desired result. If r= 1, then (52) ensures |A| ≥ |B| ≥6 and|C| ≥4,\nin which case Conjecture 1.2 holds for A+C(as mentioned in the Small rSubsection of the\nIntroduction), yielding the desired result for A+Bas well (via Proposition 2.5). Therefore we\ncan assume\nr≥2.\nOur hypotheses ensure 2 γ−1(r+3)+r≤ |A|+|B|+r=|A+B| ≤p−r−3, whence\n(53) p≥2(γ−1+1)r+6γ−1+3≥202r+603≥1007,\nwith the final inequality in view of r≥2.\nSince|A|,|B|,|C| ≥r+3, it follows that Ais (r+3)-separable with κr+3(A)≤ |A+B|−|B| ≤\n|A|+r. LetX⊆Gbe an (r+ 3)-atom for A. Then|X| ≥r+ 3,|X+A| ≤p−r−3 and\n|A+X|−|X|=κr+3(A)≤ |A|+r, implying\n(54) |A+X| ≤ |A|+|X|+r.\nSincer≥1, (53) allows us to apply Corollary 2.10 yielding\n(55) r+3≤ |X| ≤/floorleftBig\n2r+2.5+/radicalbigg\n2r+17\n4/floorrightBig\n.34 DAVID J. GRYNKIEWICZ\nBy calculating its derivative, the expression3r+2.5+/radicalBig\n2r+17\n4\nr+3is maximized when r= 84 via basic\ncalculus. If r∈[2,79]∪[89,∞), then this expression is maximized at r= 89, which combined\nwith (55) yields\n(56) |X|+r≤283\n92(r+3).\nTaking advantage of the fact that |X|is an integer, it can be manually checked that (55) yields\n(56) forr∈[80,89] as well, meaning (56) holds in general. Similar analysis (checking the values\nr∈[19,25] manually for the first case below), shows that\n(57) |X|+r≤3.143(r+2),|X|+r≤3.1(r+2.5),|X|+r≤3.0885(r+2.7).\nUsing (54) and the Ruzsa-Pl¨ unnecke Bounds (Theorem 2.4), w e find a nonempty subset\nA′⊆Asuch that\n(58) |A′+nX| ≤/parenleftBig|A+X|\n|A|/parenrightBign\n|A′| ≤/parenleftBig\n1+|X|+r\n|A|/parenrightBign\n|A′|for alln≥1.\nIn particular, letting\n1−ǫ=|A′|/|A|,\nwe haveǫ∈[0,1) and\n(59) |A′+X| ≤ |A′|+(1−ǫ)(|X|+r) =|A′|+|X|+r−ǫ(|X|+r).\nSince|A′+X| ≤ |A+X| ≤p−r−3< p, the Cauchy-Davenport Theorem (Theorem 2.2)\nensures|A′+X| ≥ |A′|+|X| −1, which combined with (59) and 2 ≤r≤ |X| −3 forces\n−1≤r−ǫ(|X|+r)≤ |X|−3−ǫ(2|X|−3), in turn implying ǫ≤|X|−2\n2|X|−3<1\n2. Thus (from (52))\n(60) 0 ≤ǫ<1\n2and|A′|>1\n2|A| ≥1\n2γ−1(r+3) = 50r+150.\nIn view of (52) and (56), we have\n(61)|X|+r\n|A|≤|X|+r\nγ−1(r+3)≤283\n92γ≤0.0308.\nWe claim that\n(62) (1+ z)n≤1+nz+0.466n2z2= 1+nz(1+0.466nz)\nfor any non-negative real number z≤0.0313 and any positive integer n≤8. Indeed, for n= 2,\nwe have 1+ nz+0.466n2z2−(1+z)n= 0.864z2≥0, while for n≥3 the smallest positive root\nof 1+nz+0.466n2z2−(1+z)n, forn∈[3,8], is always greater than 0 .0313 with this expression\npositive for 0 <z≤0.0313, establishing (62).THE 3k−4 THEOREM MODULO A PRIME 35\nNow (61) allows us to apply (62) with z=|X|+r\n|A|, which combined with (58), (61) and γ= 0.01\nyields\n|A′+nX| ≤ |A′|+(|A′|/|A|)n(|X|+r)/parenleftBig\n1+0.466·n/parenleftbig|X|+r\n|A|/parenrightbig/parenrightBig\n≤ |A′|+(|A′|/|A|)n(|X|+r)/parenleftBig\n1+0.01434·n/parenrightBig\nfor alln∈[2,8]. (63)\nBy hypothesis, |A|+γ−1(r+3)+r≤ |A|+|B|+r=|A+B| ≤p−r−3, implying\n(64) p−|A| ≥(γ−1+1)(r+3)+r= 102r+303.\nUsing the estimates |A′| ≤ |A|, (56) and (64) in (63), we find that\n|A′+8X| ≤ |A|+8283\n92(r+3)(1+0.01434·8)\n≤ |A|+27.432(r+3)≤p−102r−303+27.432(r+3)≤p−5.\nThis allows us to apply the Cauchy-Davenport Theorem (Theor em 2.2) to A′+ 8X(or any\nsub-sumset later in the proof). Doing so, we see that (63) imp lies\n(65) |nX| ≤(|A′|/|A|)n(|X|+r)/parenleftBig\n1+0.01434·n/parenrightBig\n+1 for all n∈[2,8].\nBy calculating its derivative, basic calculus shows that 26 .5−0.246719r′+/radicalBig\n2r′+17\n4is\nmaximized (over all integers r′≥ −1) forr′= 6, whence 26 .5 +/radicalBig\n2r′+17\n4≤0.246719r′+\n29.05082. Hence\n(66)9253\n1963/parenleftBig\n10r′+26.5+/radicalbigg\n2r′+17\n4/parenrightBig\n≤48.3r′+136.937 for all integers r′≥ −1.\nClaim A: Ifℓd(A′)≤ |A′|+48.9r+145.3 for some nonzero d∈G, then all desired conclusions\nin Theorem 1.4 follow.\nProof of Claim A. Let|A′+X|=|A′|+|X|+r′, leth= 48.9r+145.3, and letC′=−(A′+X)c.\nBy (59), we have r′≤r. By (60), we have |A′|>50r+ 150≥r+ 4 +h. By (55), we have\n|X| ≥r+3. By (54), (56) and (64), we have\n(67) |A′+X| ≤ |A+X| ≤ |A|+|X|+r≤p−102r−303+283\n92(r+3),\nwhence|C′| ≥98.9r+293.7≥2h+r+3. Thus we can apply Theorem 2.13.1 to the additive trio\n(A′,C′,X) (with the hin Theorem 2.13 taken to be ⌊h⌋) to conclude that ℓd(X)≤ |X|+r+1.\nLetC′′=−(A+X)c. By (67), We have |A+X| ≤ |A|+|X|+r≤p−98.9r−293.7<p−(r+3),\nwhence|C′′| ≥r+4. By (52), (56) and (55), we have |A| ≥100(r+3)>283\n92(r+3)≥ |X| ≥r+3.\nBy (53), we have p≥35r+45. Thuswe can apply Theorem 2.13.2 to the additive trio ( X,A,C′′)\n(withh=r+1) to conclude that ℓd(A)≤ |A|+r+1.36 DAVID J. GRYNKIEWICZ\nBy (52), we have |A| ≥ |B| ≥100(r+3)≥3r+6 and |C|=p−|A+B| ≥r+3. Thus we\ncan apply Theorem 2.13.1 (with h=r+ 1) to the additive trio ( A,B,C) to yield all desired\nconclusions for Theorem 1.4. /square\nClaim B: If 1≤n��8 andℓd(nX)≤2|nX|−2 for some nonzero d∈G, thenℓd(2X)<p\n2.\nProof of Claim B. By (65) and (56), we have\n|nX| ≤ |8X| ≤27.44(r+3)+1 = 27 .44r+83.32.\nThus, ifn≥2, then the hypothesis of Claim B together with (53) yields ℓd(2X)≤ℓd(nX)≤\n2|nX|−2≤54.88r+164.64<p\n2, as desired. If n= 1, then the hypothesis of Claim B together\nwith (56) and (53) yields ℓd(2X)<2ℓd(X)≤4|X|−4≤8.31r+33<p\n2, as desired. /square\nClaim C:\n|2X|<(0.9)(|A′|/|A|)/parenleftbig\n2.0574(|X|+r)+3/parenrightbig\n|4X|<(0.9)(|A′|/|A|)/parenleftbig\n4.2295(|X|+r)+3/parenrightbig\n,and\n|8X|<(0.9)(|A′|/|A|)/parenleftbig\n8.9178(|X|+r)+3/parenrightbig\n.\nProof.By (65), we have\n(68) |8X| ≤(|A′|/|A|)8.9178(|X|+r)+1.\nIn view of (68), (56), and (60), it follows that\n|8X| ≤27.44(r+3)+1<|A′|. (69)\nLetr′be defined by |A′+8X|=|A′|+|8X|+r′.\nSuppose\nr′≤1\n9|8X|−3.\nThen (69) implies\n(70) r′≤3.05r+6.26,\nand then (66) implies\n9253\n1963/parenleftBig\n10r′+26.5+/radicalbigg\n2r′+17\n4/parenrightBig\n≤48.3r′+136.937≤147.4r+439.3<p,\nwith the final inequality by (53). By (69), (64) and (70), we ha ve|A′+8X|=|A′|+|8X|+r′≤\n|A|+27.44(r+3)+1+r′≤p−102r−303+27.44(r+3)+1+r′=p−74.56r−219.68+r′≤\np−23.44r′−66.64≤p−9(r′+3). Thus we can apply Theorem 3.5 to A′+ 8Xto conclude\nvia (70) that ℓd(A′)≤ |A′|+r′+ 1≤ |A′|+ 3.05r+ 7.26 for some nonzero d∈G, and then\nClaim A completes the proof. So we instead conclude that r′>1\n9|8X|−3, meaning |A′+8X|>\n|A′|+10\n9|8X|−3. Combined with (63), we now have |8X|<(0.9)(|A′|/|A|)/parenleftbig\n8.9178(|X|+r)+3/parenrightbig\n,\nas desired.THE 3k−4 THEOREM MODULO A PRIME 37\nWe have |4X| ≤(|A′|/|A|)4.2295(|X|+r) + 1 and |2X| ≤(|A′|/|A|)2.0574(|X|+r) + 1 by\n(65). Thus, since |2X| ≤ |4X| ≤ |8X|, the above arguments can also be applied to 2 Xand 4X\nto conclude that |A′+4X|>|A′|+10\n9|4X|−3 and|A′+2X|>|A′|+10\n9|2X|−3, yielding the\nstated bounds for |4X|and|2X|. /square\nClaim D: ℓd(2X)<p\n2for some nonzero d∈G.\nProof of Claim D. Assume by contradiction that\n(71) ℓd(2X)≥p\n2for all nonzero d∈G.\nBy Claim C, (56), (57) and (53), we have\n|X| ≤3.143(r+2)−r≤2.143r+6.286<0.01061p,\n|2X| ≤(0.9)(2.0574)(3.1)(r+2.5)+2.7≤5.741r+17.051<0.02843p,\n|4X| ≤(0.9)(4.2295)(3.0885)(r+2.7)+2.7≤11.757r+34.443<0.05821p,\n|8X| ≤(0.9)(8.9178)283\n92(r+3)+2.7≤24.689r+76.767<0.128p. (72)\nWe handle three cases.\nCASE D1: |X| ≥11 and|X| ≥r+4.\nIn this case, (72) allows us to apply Theorem 3.8 with s= 11 toX+X, 2X+2Xand 4X+4X\nusing the values K= 2.572 andc >0.0111,K= 2.552 andc >0.0289, andK= 2.515 and\nc>0.0588, respectively. Thus, in view of Claim B and (71), it foll ows that |2X| ≥2.572|X|−3,\n|4X| ≥2.552|2X|−3 and|8X| ≥2.515|4X|−3. Iterating these bounds and comparing with the\nbound from Claim C combined with the case hypothesis r≤ |X|−4, we find\n16.5|X| −29.8<|8X| ≤16.06|X| −29.4,\nwhich contradicts that |X| ≥2.\nCASE D2: |X| ≤10.\nInthis case, wehave |X| ≤ ⌈log2p⌉by (53). Thuswe can apply Theorem 2.12 to X+X, which\nin view of Claim B and(71) implies |2X| ≥3|X|−3. We have |2X| ≥3|X|−3≥3(r+3)−3≥12\n(by (55)). Thus, as in CASE D1, we can assume |4X| ≥2.552|2X|−3 and|8X| ≥2.515|4X|−3.\nIterating these boundsand comparing with the boundfrom Cla im C combined with the estimate\nr≤ |X|−3 (from (55)), we find\n19.254|X| −29.8<|8X| ≤16.06|X| −21.378,\nwhich implies |X| ≤2, contradicting that |X| ≥r+3≥5 (by (55)).\nCASE D3: |X|=r+3 andr≥8.38 DAVID J. GRYNKIEWICZ\nIn this case, by Claim C and (53), we have\n|X|=r+3<0.00498p\n|2X| ≤3.704r+8.255<0.0184p\n|4X| ≤7.614r+14.12<0.0377p\n|8X| ≤16.053r+26.78<0.0795p.\nThis allows us to apply Theorem 3.8 with s= 11 toX+X, 2X+ 2Xand 4X+ 4Xusing\nthe values K= 2.578 andc >0.00561,K= 2.564 andc >0.01845, and K= 2.541 and\nc>0.0382, respectively. Thus, in view of Claim B and (71), it foll ows that |2X| ≥2.578|X|−3,\n|4X| ≥2.564|2X|−3 and|8X| ≥2.541|4X|−3. Iterating these bounds and comparing with the\nbound from Claim C combined with the case hypothesis r=|X|−3, we find\n(73) 16 .795|X| −30.169<|8X| ≤16.06|X| −21.378,\nwhich in view of |X|=r+3≥11 is only possible if r= 8 and |X|= 11. For this case, we have\n|2X| ≥ ⌈2.578|X| −3⌉= 26,|4X| ≥ ⌈2.564|2X| −3⌉ ≥64 and|8X| ≥ ⌈2.541|4X| −3⌉ ≥160,\nwhile (73) implies |8X| ≤155, contradicting this to complete Claim D. /square\nIn view of Claim D, we have ℓd(X)≤ℓd(2X)<p\n2for some nonzero d∈G, ensuring that both\nX+Xand 2X+2Xrectify. This will allow us to apply Theorem 1.1 to X+Xand 2X+2X\nas done when proving Theorem 3.3.\nClaim E: |2X| ≥3|X|−3.\nProof of Claim E. Let\nh=|X|−2\nand letC′=−(A+X)c. Note that\n(74) r≤ |X|−3 =h−1\nby (55). Assume by contradiction that |2X| ≤3|X|−4. Then, since X+Xrectifies, Theorem\n1.1 implies that\nℓd(X)≤ |X|+(|X|−3) =|X|+h−1≤ |X|+h,\nfor some nonzero d∈G. By (54) and (74), we have |A+X| ≤ |A|+|X|+h−1 and|X|=h+2.\nBy (52) and (56), we have |A| ≥100(r+3)≥100·92\n283(|X|+r) = 100·92\n283(h+2+r)≥h+4.\nBy (54), (64) and (56), we have |A+X|+h+ 3≤ |A|+ 2|X|+r+ 1≤ |A|+ 2(|X|+r)≤\np−102r−303 +283\n46(r+ 3)< p, ensuring that |C′| ≥h+ 3. By (56) and (53), we have\n35h+10 = 35 |X| −60≤35·283\n92(r+ 3)−60≤108r+263< p. Thus we can apply Theorem\n2.13.2 to the additive trio ( X,A,C′) (withh=|X|−2) to conclude that\nℓd(A)≤ |A|+h=|A|+|X|−2.THE 3k−4 THEOREM MODULO A PRIME 39\nBy hypothesis, we have |A+B|=|A|+|B|+r. By (52) and (56), we have |A|,|B| ≥\n100(r+3)≥283\n46(r+3)>r+2|X|=r+4+2h. By (52), we have |C| ≥r+3. This allows us to\napply Theorem 2.13.1 to the additive trio ( A,B,C) to yield all desired conclusions for Theorem\n1.4. So we can now instead assume |2X| ≥3|X|−3, as claimed. /square\nLetC′=−(A′+2X)c. If|4X| ≥3|2X|−3, then Claims E and C together with r≤ |X|−3\nimply that 9 |X|−12≤ |4X| ≤(0.9)(4.2295(2|X|−3)+3)≤7.6131|X|−8.719, implying |X| ≤2,\nwhich contradicts that |X| ≥r+3 = 5 by (55). Therefore we must instead have\n(75) |4X| ≤3|2X|−4.\nAs a result, since 2 X+ 2Xrectifies, we can apply Theorem 1.1 to 2 X+ 2Xto conclude via\nClaim C and (56) that\n(76) ℓd(2X)≤ |2X|+hwhereh:=|2X|−3≤5.696r+16.788.\nIf|A′+ 2X| ≥ |A′|+|2X|+h, then combining this with Claim E, (63) and r≤ |X| −3 (by\n(55)) yields |A′|+6|X| −9≤ |A′|+|2X|+h≤ |A′+2X| ≤ |A′|+4.115|X| −6.172, implying\n|X| ≤1, which contradicts that |X| ≥r+ 3≥5 by (55). Therefore we instead conclude that\n|A′+2X| ≤ |A′|+|2X|+h−1. By definition of h, we have |2X|=h+3. By (60) and (76),\nwe have |A′|>50r+ 150≥5.696r+ 20.788≥h+ 4. By (63), (56), (76) and (64), we have\n|A′+2X|+h+4≤ |A′|+6.329r+22.986+h≤ |A|+12.025r+39.774≤p−89.975r−263.226<p,\nensuring |C′| ≥h+4. Finally, in view of (76) and (53), we have 35 h+10≤199.36r+597.58<p.\nThus we can apply Theorem 2.13.2 to the additive trio (2 X,A′,C′) to conclude that ℓd(A′)≤\n|A′|+h≤ |A′|+ 5.696r+ 16.788, with the latter inequality by (76). This allows us to app ly\nClaim A to complete the proof. /square\nReferences\n[1] I. Bardaji and D. J. Grynkiewicz, Long arithmetic progre ssions in small sumsets, Integers 10 (2010), A28,\n335–350.\n[2] Y. F. Bilu, V. F. Lev and I. Z. Ruzsa, Rectification princip les in additive number theory, Discrete Comput.\nGeom.19 (1998), no. 3, Special Issue, 343–353.\n[3] T. Boothby, M. DeVos, and A. Montejano, A new proof of Kemp erman’s Theorem, Integers 15 (2015), A15.\n[4] P. Candela, D. Gonz´ alez-S´ anchez, and D. J. 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Math.\nMonogr., Vol 37 American Mathematical Society, Providence , RI, 1973. vii+108 pp.\n[10] G. A. Fremain, Inverse problems in additive number theo ry. Addition of sets of residues modulo a\nprime.(Russian) Dokl. Akad. Nauk SSSR 141 (1961), 571–573.\n[11] G. A. Freiman, The addition of finite sets I, Izv. Vyss. Ucebn. Zaved. Matematica 6 (1959), no. 13, 202–213.\n[12] B. Green and I. Z. Ruzsa, Sets with small sumset and recti fication, Bull. London Math. Soc. 38 (2006), no.\n1, 43–52.\n[13] D. J. Grynkiewicz, Structural Additive Theory , Developments in Mathematis 30, Springer, 2013.\n[14] D. J. Grynkiewicz, A step beyond Kemperman’s structure theorem, Mathematika 55 (2009), no. 1–2, 67–114.\n[15] Y. O. Hamidoune, Hyperatoms applied to the critical pai r theory, Combinatorica 39 (2019), 1281–1315\n[16] Y. O. Hamidoiune and Ø. Rødseth, An inverse theorem mod p,Acta Arithmetica 92 (2000), 251–262.\n[17] Y. O. Hamidoune, O. Serra, and G. Z´ emor, On the critical pair theory in Z/pZ,Acta Arith. 121 (2006), no.\n2 , 99–115.\n[18] M. Huicochea, A Freiman’s 2.4 theorem-type result for d ifferent subsets, Acta Math. Hungar. 166 (2022),\n393–406.\n[19] M. Huicochea, Inversetheorems in Z/pZwhen asubset is contained in a short arithmetic progression ,Integers\n17 (2017), Paper No. A37, 12 pp.\n[20] V. F. Lev, Small doubling in cyclic groups, J. Number Theory 243 (2023), 561–614.\n[21] V. F. Lev, The rectifiability threshold in abelian group s,Combinatorica 28 (2008), no. 4, 491–497.\n[22] V. Lev and O. Serra, Towards 3 n−4 in groups of prime order, Elect. J. Combin. 30 (2023), no. 2., Paper\nNo. 2.48.\n[23] V. Lev and I. D. Shkredov, Small doubling in prime-order groups: from 2.4 to 2.6, J. Number Theory 217\n(2020), 278–291.\n[24] V. Lev and P. Smeliansky, On addition of two distinct set s of integers, Acta Arith. 70 (1995), no.1, 85–91.\n[25] M. Nathanson, Additive Number Theory: Inverse Problems and the Geometry o f Sumsets , Graduate Texts\nin Mathematics 165, Springer-Verlag, New York, 1996.\n[26] Ø. J. Rødseth, On Freiman’s 2.4-Theorem, Skr. K. Nor. Vidensk. Selsk. (2006), no. 4, p. 11–18\n[27] I. Z. Ruzsa, An application of graph theory to additive n umber theory, Scientia, Ser. A , 3 (1989), 97–109.\n[28] I. Ruzsa, Sumsets and Structure, in Combinatorial Number Theory and Additive Group Theory , Birkh¨ auser\n(2000), Basel.\n[29] Y. O. Hamidoune and Ø. J. Rødseth, An inverse theorem mod p,Acta Arith. 92 (2000), no. 3, 251–262.\n[30] G. Petridis, New Proofs of Pl¨ unnecke-type Estimates f or Product Sets in Groups, Combinatorica 32 (2012),\n721–733.\n[31] O.SerraandG. Z´ emor, Large setswithsmall doublingmo dulopare wellcoveredbyanarithmetic progression,\nAnn. Inst. Fourier (Grenoble) 59 (2009), no. 5, 2043–2060.\n[32] O. Serra and G. Z´ emor, On a generalization of a Theorem b y Vosper, Integers 0 (2000), Paper A10, 10 pp.\n[33] Y. Stanchescu, On addition of two distinct sets of integ ers,Acta Arith. 75 (1996), no.2, 191–194.\n[34] T. Tao and V. Vu, Additive Combinatorics , Cambridge Studies in Advanced Mathematics 105, Cambridge\nUniversity Press (2006).\n[35] A. G. Vosper, The critical pairs of subsets of agroup of p rime order, J. London Math. Soc. 31 (1956), 200–205.\nDepartment of Mathematical Sciences, University of Memphi s, Memphis, TN 38152, USA\nEmail address :diambri@hotmail.com"    },    {        "title": "2402.15099v1.Understanding_the_radio_luminosity_function_of_star_forming_galaxies_and_its_cosmological_evolution.pdf",        "content": "MNRAS 000, 1–18 (2024) Preprint 26 February 2024 Compiled using MNRAS L ATEX style file v3.0\nUnderstanding the radio luminosity function of\nstar-forming galaxies and its cosmological evolution\nCharles Jose1⋆, Luke Chamandy2,3†, Anvar Shukurov4‡, Kandaswamy Subramanian5,6,\nLuiz Felippe S. Rodrigues7and Carlton M. Baugh8\n1Department of Physics, CUSAT, Cochin, 682022, India\n2National Institute of Science Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar 752050, Odisha, India\n3Department of Physics and Astronomy, University of Rochester, Rochester NY 14627, USA\n4School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK\n5IUCAA, Post Bag 4, Ganeshkhind, Pune 411007, India\n6Department of Physics, Ashoka University, Rajiv Gandhi Education City, Rai, Sonipat 131029, Haryana, India\n7HAL24K Agri, Uitmeentsestraat 19, 6987 CX Giesbeek, Netherlands\n8Institute for Computational Cosmology, Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nWe explore the redshift evolution of the radio luminosity function (RLF) of star-forming galaxies using galform , a\nsemi-analytic model of galaxy formation (SAMGF) and magnetizer , a dynamo model of the magnetic field evolving\nin a galaxy through its formation history. Assuming energy equipartition between the magnetic field and cosmic rays,\nwe derive the synchrotron luminosity of each sample galaxy. In a model where the turbulent speed is correlated with\nthe star formation rate, the RLF is in fair agreement with observations in the redshift range 0 ≤z≤2. At larger\nredshifts, the structure of galaxies, their interstellar matter and turbulence appear to be rather different from those\natz≲2, so that the turbulence and magnetic field models applicable at low redshifts become inadequate. The strong\nredshift evolution of the RLF at 0 ≤z≤2 can be attributed to an increased number, at high redshift, of galaxies with\nlarge disc volumes and strong magnetic fields. On the other hand, in models where the turbulent speed in galaxies is\na constant or assumed to be an explicit function of the redshift, the redshift evolution of the RLF is poorly captured.\nThe evolution of the interstellar turbulence and outflow parameters appear to be major (but not the only) drivers of\nthe RLF changes. We find that both the small- and large-scale magnetic fields contribute significantly to the RLF but\nthe small-scale field dominates at high redshifts. Polarisation observations will therefore be important to distinguish\nthese two components and understand better the evolution of galaxies and their nonthermal constituents.\nKey words: galaxies: evolution – galaxies: spiral – radio continuum: galaxies – galaxies: luminosity function –\ngalaxies: magnetic fields – dynamo\n1 INTRODUCTION\nThe galaxy luminosity function (LF) – the comoving num-\nber density of galaxies as a function of their magnitude at\na given wavelength – is one of the primary observables used\nto probe the physics of formation and evolution of galax-\nies. The observed LFs and their redshift evolution have been\nmeasured, to various degrees of detail and reliability, for a\nwide range of wavelengths from the UV to the radio. When\nassessed in the framework of a theoretical model of galaxy\nformation, the LFs provide key insights into the evolution of\nstar formation and feedback processes, the nature of and con-\nditions in the ISM (consisting of gas, dust, magnetic fields,\n⋆charles.jose@cusat.ac.in\n†lchamandy@niser.ac.in\n‡anvar.shukurov@ncl.ac.uk)and cosmic rays), and dynamical processes such as galaxy\nmergers. Two major approaches used to model galaxy for-\nmation are: (i) semi-analytic models (SAMGFs) that account\nfor the complex baryonic processes in evolving dark matter\n(DM) haloes derived from N-body simulations using a com-\nbination of analytic approximations and numerical prescrip-\ntions, (ii) hydrodynamic or magnetohydrodynamic (MHD)\nsimulations based on fundamental dynamical equations, that\ninclude subgrid models to account for unresolved physical\nprocesses(often similar to those used in SAMGFs). There are\nseveral galaxy formation models (Trayford et al. 2015; Trˇ cka\net al. 2022; Croton et al. 2006; Samui et al. 2007; Jose et al.\n2013; Lacey et al. 2016) that interpret the ultra-violet (UV)\n(Wyder et al. 2005; Page et al. 2021), optical (Blanton et al.\n2003; Loveday et al. 2012) and infrared (IR) (Dunne et al.\n2011; Gruppioni et al. 2013; Marchetti et al. 2016) LFs and\nshed light on the evolution of obscured star formation rate\n©2024 The AuthorsarXiv:2402.15099v1  [astro-ph.GA]  23 Feb 20242C. Jose et al.\n(SFR), stellar mass ( M∗), ISM, dust and feedback processes\nin galaxy populations.\nComplementary to the UV, optical and IR LFs, is the ra-\ndio LF (RLF) of star-forming galaxies (SFGs) at the emis-\nsion frequency 1 .4 GHz obtained with the Very Large Ar-\nray (VLA) and the Westerbork Synthesis Radio Telescope\n(WSRT) (Adams & van Leeuwen 2019; Condon et al. 1998).\nRecent estimates of RLFs of SFGs at 1 .4 GHz include those of\nSadler et al. (2002); Condon et al. (2002); Best et al. (2005);\nMauch & Sadler (2007); Padovani et al. (2011) and Con-\ndon et al. (2019) for the local Universe and from Smolˇ ci´ c\net al. (2009); Pracy et al. (2016); Novak et al. (2017); Bon-\nato et al. (2021); Malefahlo et al. (2022); Enia et al. (2022)\nand van der Vlugt et al. (2022) for redshifts up to z≃4.5.\nMoreover, several ongoing and upcoming deep surveys using\npresent and next-generation radio telescopes, including the\nSquare Kilometre Array (SKA), will improve our knowledge\nof the RLFs of SFGs to higher redshifts and fainter luminosi-\nties with unprecedented accuracy (Norris et al. 2013; Adams\n& van Leeuwen 2019). For example, deep surveys using SKA\nPhase-I will detect sub- µJy SFGs up to z≃6 which will help\nto measure RLFs down to luminosities more than an order\nof magnitude lower than what is available now (Jarvis et al.\n2015).\nSynchrotron emission dominates the radio continuum at\n1.4 GHz (Ginzburg & Syrovatskii 1965), so models of the\ngalactic magnetic field and cosmic ray propagation should\nbe the starting points in a prediction of the RLF. MHD sim-\nulations of evolving galaxies (e.g., Liu et al. 2022; Pfrommer\net al. 2022) cannot provide the statistically significant galaxy\nsamples needed to compute RLFs. Moreover, limited spa-\ntial resolution prevents such simulations from probing scales\nas small as 1–100 pc, which is crucial for capturing dynamo\nprocesses at the turbulent and global scales, and hence re-\nalistic magnetic fields and, consequently, any realistic distri-\nbution of cosmic rays controlled by them (see section 13.14\nof Shukurov & Subramanian 2021, for a review). Turbulent\nmagnetic fields, produced by the fluctuation dynamo, super-\nnova shock fronts, and tangling of the large-scale magnetic\nfield, require a spatial resolution of order 1 pc to be real-\nistically reproduced (Gent et al. 2021, 2023), whereas the\nmean-field dynamo, that generates large-scale galactic mag-\nnetic fields, relies, apart from overall rotation, on the den-\nsity stratification at scales of order 100 pc (chapter 11 of\nShukurov & Subramanian 2021). MHD simulations at an ad-\nequate resolution are only available for local ISM regions of\norder a few kiloparsecs in size. (e.g. Gressel et al. 2008; Gent\net al. 2013b,a; Hollins et al. 2017; Gent et al. 2024), and sub-\ngrid dynamo models are not yet available despite an effort to\nrelate the statistical properties of magnetic fields in nearby\ngalaxies to theories (Van Eck et al. 2015; Chamandy et al.\n2016; see Beck et al. 2019, for a review).\nTherefore, at present, combining SAMGFs with a model for\nthe evolution of magnetic fields and cosmic rays is arguably\nthe only viable theoretical approach for modelling RLFs.\nThe earliest effort towards incorporating galactic dynamo\ntheory into a hierarchical model for galaxy formation in cold\nDM cosmology to produce galactic magnetic fields in a cos-\nmological volume-sized sample of galaxies was by Rodrigues\net al. (2015). Rodrigues et al. (2019) (hereafter R19) extended\nthis model into a computational framework called magne-\ntizer (Rodrigues & Chamandy 2020) which simulates mag-netic fields in individual galaxies as they evolve. In particular,\nthese authors coupled the magnetizer with the output of\nthegalform SAMGF of Lacey et al. (2016) and Gonzalez-\nPerez et al. (2014) to obtain simulated magnetic fields for a\nlarge sample of galaxies in the redshift range 0 ≤z≤6 and\ncompared the predictions of mean-field strengths and pitch\nangles with observations of local galaxies, finding reasonable\nagreement.\nThe present work extends the model of R19, to derive the\nRLF of SFGs at the rest-frame frequency of 1 .4 GHz over\ncosmic history. We calculate the total radio luminosity due\nto the synchrotron emission based on our magnetic field and\ncosmic ray models. Any significant differences of this work\nfrom the model of R19 are mentioned in the text.\nKey quantities that affect the amplification and sustenance\nof galactic magnetic fields are the gas density and root-mean-\nsquare (rms) turbulent speed, as well as the galactic rotation\nand stratification (the gas scale height). Some observational\nstudies indicate that the gas velocity dispersion in galaxies\n(which includes a contribution of the turbulence) is correlated\nwith the SFR surface density (SFRD) (Green et al. 2010;\nLehnert et al. 2013; Moiseev et al. 2015) while other authors\nfind a correlation with the global SFR (Genzel et al. 2011;\nVaridel et al. 2016, 2020; Zhou et al. 2017; ¨Ubler et al. 2019a).\nWe investigate how the predicted shape and redshift evolu-\ntion of the RLF are affected by various phenomenological ISM\nturbulence models and compare the results with observations.\nBy contrast, past works have probed the radio luminosity of\nisolated template galaxies and used the radio-FIR correlation\nto deduce the RLF (e.g. Werhahn et al. 2021; Pfrommer et al.\n2022; Vollmer et al. 2022; Schober et al. 2023).\nThe RLFs obtained from observations contain contribu-\ntions from both active galactic nuclei (AGN) and SFGs\n(Mauch & Sadler 2007; van der Vlugt et al. 2022). The phys-\nical nature and processes responsible for the radio emission\nin these objects are rather different and should be consid-\nered separately. These contributions can be separated, with\nvarying degrees of confidence, using multi-wavelength obser-\nvations to estimate the RLF of each. In this work, we model\nthe RLF of star-forming disc galaxies only.\nThis paper is organized as follows. In Section 2, we discuss\nour models to compute the RLF of galaxies at 1 .4 GHz. We\npresent the predictions for the magnetic field in Section 3,\nand for the RLF in Section 4 along with a comparison with\nobservational data and a discussion of the roles of various\nphysical processes involved. Our results are put into a wider\ncontext in Section 5 and conclusions are formulated in Sec-\ntion 6. Throughout this work, we use cosmological parame-\nters based on seven-year WMAP data (Komatsu et al. 2011),\nwhich are also used in the galform model of Lacey et al.\n(2016). We use the same parameter values in magnetizer as\nR19 unless otherwise stated.\n2 THE MODEL\nThe RLFs presented in this work are the products of a three-\nlevel model. In the first level, the output of the galform\nSAMGF of Lacey et al. (2016) is used to derive the proper-\nties of a representative galaxy population of about 7 ×105\ndisc galaxies in a comoving volume of 6 ×106Mpc3at each\nredshift in the redshift range 0 ≤z≤6 (corresponding to\nMNRAS 000, 1–18 (2024)The RLF of star-forming galaxies and its cosmological evolution 3\n3.5% of the total volume of the N-body simulation used to\nprovide halo merger histories), with stellar masses between\nabout 108M⊙and 1012M⊙. In the second level, magnetizer\nis used to couple the galactic dynamo theory and the galaxy\nformation model to predict the evolution of both turbulent\nand global magnetic fields as a function of time and galacto-\ncentric radius in each galaxy. To avoid excessive and unnec-\nessary complications, all galaxy properties are assumed to be\naxisymmetric; the dominance of axially symmetric large-scale\nmagnetic fields in nearby galaxies is well established (e.g.,\nBeck et al. 2019). In the final third stage, the synchrotron\nemission at 1 .4 GHz is computed for each galaxy to derive\nthe RLF over a wide redshift range.\n2.1 Galaxy formation model\nThegalform version that we use (Lacey et al. 2016, see also\nCole et al. 2000; Baugh et al. 2005; Bower et al. 2006) incorpo-\nrates theoretically and empirically motivated models for com-\nplex physical processes (star formation, supernova, and AGN\nfeedback, various dynamical processes, the evolution of the\ngas, stars, and dust, etc.) that baryons undergo in DM haloes.\nThe assembly histories of haloes are described by halo merger\ntrees extracted from N-body simulations (Guo et al. 2013).\nThe model successfully reproduces the observed K-band,\noptical, near-infrared, and far-UV luminosity functions of\nSFGs, far-IR number counts and H iand stellar mass func-\ntions over a wide range of redshifts along with the Tully–\nFisher, metallicity–luminosity and size–luminosity relations\natz= 0. Nearly all radio-bright SFGs are spirals (Sadler\net al. 2002; Condon et al. 2002). Therefore, as in R19, we\nonly consider disc galaxies in the output of galform , which\nare defined as galaxies where the bulge mass is less than half\nthe total stellar mass. Several ISM parameters for magne-\ntizer are derived from the galform output in the same way\nas in R19 as briefly discussed below, but we refer the reader\nto that paper for further details.\n2.1.1 Derived galactic quantities\nTo obtain the RLF, we need to deduce the distribution of the\ngas density and the magnetic field along the galactocentric\ndistance rand distance Zfrom the midplane in each galaxy,\nas well as its rotation curve and the gas scale height.\nThe galaxy rotation curve, V(r), is computed by assum-\ning that galaxies have a thin stellar disc with an exponen-\ntial surface mass density profile, a bulge with the Hernquist\n(1990) profile and a DM halo with an adiabatically con-\ntracted Navarro–Frenk–White density profile (Navarro et al.\n1997). The rotation curve is then used to determine the\ngalaxy angular velocity, Ω( r), and the rotational shear rate,\nS(r) =rdΩ/dr.\nAs in R19, we adjust the half-mass radius r1/2of the gal-\nform galaxies to reproduce the observed stellar mass– r1/2\nrelation of Lange et al. (2016) and rd= 2.7r1/2is adopted as\nthe maximum gas disc radius for computing quantities like\nthe volume-averaged magnetic field strength.\nThe surface mass densities of the stars and gas in the disc\nare both assumed to have an exponential radial profile withthe scale length rs=r1/2/a, where a≈1.68.1As in R19, the\ndisc gas mass is separated into diffuse and ‘molecular’ phases.\nThe ratio of the surface densities of the diffuse and molecular\ngas components is computed using the empirical relation from\nBlitz & Rosolowsky (2004, 2006). The density distributions\nof diffuse gas, molecular gas and stars perpendicular to the\ngalactic disc are assumed to be exponential with different\nscale heights. Thus, for example, for the diffuse gas density\nwe have\nρd(r, Z) =ρd0(r)e−|Z|/hd(r), (1)\nwhere Zis the distance from the galaxy midplane, ρd0(r) is\nthe midplane density and hd(r) is the density scale height,\nboth functions of the galactocentric radius r, determined as-\nsuming that the ISM is in hydrostatic equilibrium with a total\npressure that includes the thermal, turbulent, magnetic and\ncosmic-ray contributions. We use empirical relations for the\nscale heights of the molecular gas (0 .032rs) and stars (0 .1rs)\n(Kregel et al. 2002; Licquia & Newman 2016).\n2.1.2 Interstellar turbulence\nGalactic magnetic fields are amplified and sustained by a\nturbulent dynamo. Thus, magnetic field models depend on\ncertain statistical properties of the turbulence, like the cor-\nrelation length l, correlation time τ, and rms turbulent ve-\nlocity v. As in R19, we adopt l(r) = min[100 pc , hd(r)] and\nτ=l/v(for estimates of interstellar turbulence parameters,\nsee Chamandy & Shukurov 2020).\nThus far, the magnetizer model described is the same as\nthat of R19. However, R19 assumed that the rms turbulent\nspeed is v= 10 km s−1, close to the sound speed in the warm\ngas and independent of the redshift and SFR. However, the\ngas velocity dispersion (which includes an uncertain contribu-\ntion from the turbulence) appears to depend on the intensity\nof star formation (see Fig. 1 and the text below). Given this\nuncertainty, we considered three alternative prescriptions for\ndetermining the turbulent speed.\nInterstellar turbulence has various drivers, including star\nformation and the associated supernova activity, gravita-\ntional instability (Krumholz & Burkhart 2016; Krumholz\net al. 2018), cosmological gas accretion, and galaxy merg-\ners (Jim´ enez et al. 2023; Ginzburg et al. 2022). Some models\n(e.g., Krumholz et al. 2018; Ginzburg et al. 2022) predict\nthat the turbulent speed is independent of the SFR when\nSFR <SFR 0with a certain threshold SFR 0and increases\nwith the SFR otherwise. The velocity dispersion data, con-\nsistent with more recent observations (Varidel et al. 2020; Yu\net al. 2021), are shown in the upper panel of Fig. 1, and we\nuse the form\nv=v0(\n1, if SFR ≤SFR 0,\n(SFR /SFR 0)c,otherwise ,(2)\n1To obtain a, we note that M(r) = 2 πΣ0r2\nsRr/rs\n0xe−xdxfor the\nmass of either stars or gas within the cylindrical radius r, where Σ 0\nis the surface density at r= 0. As noted in the text, the scale length\nrsis assumed to be the same for both stars and total gas.SinceR∞\n0xe−xdx= 1, we have M∞≡limr→∞M(r) = 2 πΣ0r2\ns.Now,\nM(r1/2) =M∞/2 by definition, which leads toRa\n0xe−xdx= 1/2\nandafollows.\nMNRAS 000, 1–18 (2024)4C. Jose et al.\nFigure 1. Top panel: the observed three-dimensional velocity dis-\npersion as a function of SFR (Krumholz et al. 2018 and references\ntherein, circles) and the best-fitting curves of the form of equation\n(2) to the median (solid/green) and the 16th (dash-dot/red) and\n84th (dashed black) percentiles. The dotted blue line corresponds\ntov= 25 km s−1. Bottom panel: The turbulent speed of the cold\ninterstellar gas ( T≤10 K) as a function of redshift from the EA-\nGLE cosmological simulations for z≤4 (solid line: Jim´ enez et al.\n2023), extrapolated to z >4 as a constant (dotted).\nwith certain constants v0andc. As suggested by fig. 3 of Yu\net al. (2021), we adopt SFR 0= 3 M ⊙yr−1and estimate v0=\n25 km s−1andc= 0.50 by fitting the median of the relation\nshown in Fig. 1. The fitted dependence is shown as a solid\ncurve in Fig. 1, where the one-dimensional velocity dispersion\nσhas been converted to the three-dimensional speed v=\nσ√\n3 assuming the turbulence to be isotropic. Further, vis\nassumed to be independent of rfor simplicity.\nThe data points in Fig. 1 have a large scatter. To account\nfor this, we fit equation (2) to the 16thand 84thpercentiles of\nvas a function of SFR, to obtain v0= 17 km s−1andc= 0.45\nfor the 16thpercentile and v0= 40 km s−1andc= 0.55 for\nthe 84thpercentile, also shown in Fig. 1. These measures of\nthe scatter are used to estimate the degree of uncertainty of\nthe predicted luminosity functions. We note that the SFR-\nvrelation is assumed to be independent of redshift in this\nmodel, which we adopt as the fiducial model.\nThe turbulent speed of v0= 25 km s−1can be understood\nas the volume average of the rms turbulent speed in the multi-\nphase ISM. As an illustration, if the turbulence is transonic\nFigure 2. Top panel: the best-fitting dependence of the ob-\nserved SFRD on the redshift from Hopkins & Beacom (2006)\n(dashed/blue) and the SFRD from Lacey et al. (2016) (solid/red).\nBottom panel: the ratio of the SFRDs shown in the top panel.\nin both warm and hot phases, and their respective fractional\nvolumes and sound speeds are fw= 0.9,cw= 10 km s−1\nandfh= 0.1,ch= 100 km s−1, the volume average follows\nas (fwc2\nw+fhc2\nh)1/2≈33 km s−1. We note, however, that\nthe fractional volumes of the ISM phases are likely to vary\nbetween galaxies with different SFRs and between various\nlocations within a given galaxy.\nAs an alternative, we consider a model with a constant\nturbulent speed v= 25 km s−1suggested at lower SFRs by\nFig. 1, independently of the SFR and redshift (whereas R19\nused v= 10 km s−1). For the gas number density 0 .1 cm−3\nof the warm interstellar gas, the magnetic field strength at\nenergy equipartition with the turbulence is then about 4 µG,\nclose to the strength of the random magnetic field observed in\nnearby spiral galaxies (see Section 2.2.1). The increase in the\nvelocity dispersion at large SFR in Fig. 1 can be interpreted\nto arise partly from chaotic galactic fountains and winds,\nwhich are especially vigorous in galaxies with high SFR. This\nmodel is called v25-R19 below.\nThe velocity dispersion is observed to be correlated with\nother parameters in addition to the SFR. These include\nthe gas fraction, DM halo mass and the stellar mass (e.g.,\nKrumholz et al. 2018; Ginzburg et al. 2022). This implies\nthat the SFR– vcorrelation evolves with the redshift, which\nhas been verified by the observations of ¨Ubler et al. (2019a)\nand also found by Jim´ enez et al. (2023) using the EAGLE hy-\ndrodynamic simulations (Schaye et al. 2015). Therefore, we\nalso consider a third model, referred to as v(z)-J23, where v\nis an explicit function of z, as given by the REF-L100 model\nof Jim´ enez et al. (2023) and plotted in the bottom panel of\nFig. 1. In this model, vdoes not depend on the SFR explicitly.\n2.1.3 Correction to the SFR from used in galform\nIn the version of galform described in Lacey et al. (2016),\nstars form out of cold gas in the disc and spheroid, and\nthe corresponding SFR is calculated using the prescription\nof Blitz & Rosolowsky (2006). The comoving SFR density\n(SFRD), which is the SFR per unit comoving volume, from\nMNRAS 000, 1–18 (2024)The RLF of star-forming galaxies and its cosmological evolution 5\nLacey et al. (2016) is shown as a function of redshift in the\ntop panel of Fig. 2. Also shown in the figure is the best fit to\nthe observed redshift evolution of the SFRD given by Hop-\nkins & Beacom (2006). Both the theoretical and observed\nSFRDs are computed after assuming the IMF of Kennicutt\n(1983) (see section 6.3 of Lacey et al. 2016, for details). It\nis clear from Fig. 2 that, while observations and predictions\nagree quite well at z= 0, the predicted SFRD lies below the\nfit to the observations at all redshifts, with their ratio shown\nin the lower panel. To account for this discrepancy and rec-\noncile the SFR with that observed, we multiply the SFR of\ngalaxies from galform at a given redshift with this ratio,\ntaken at the same redshift.\n2.2 Magnetic field\nLarge-scale (mean) Band small-scale (random) bmag-\nnetic fields contribute comparably to the galactic synchrotron\nemission (here and elsewhere, a bar above a variable denotes\nensemble or volume average). In nearby galaxies, the ratio\nof their mean energy densities, b2/B2, often significantly ex-\nceeds unity (Beck 2015a; Beck et al. 2019), but this is sub-\nject to a range of assumptions involved in the interpretation\nof Faraday rotation and synchrotron observations, including\nthe assumption of energy equipartition between cosmic rays\nand magnetic fields. Beyond semi-quantitative observational\nand theoretical estimates, the absolute and relative strengths\nof the two parts of an interstellar magnetic field remain some-\nwhat uncertain.\nA few results of the mean-field dynamo theory used in the\ntext are presented in Appendix A; further details and ref-\nerences can be found in Shukurov & Subramanian (2021),\namong a number of other books and reviews (e.g., Branden-\nburg & Subramanian 2005).\n2.2.1 The random (small-scale) field\nRandom (turbulent) interstellar magnetic fields are amplified\nby the fluctuation (or small-scale) dynamo, the tangling of\nthe large-scale (or mean) magnetic field by turbulence and\ncompression, in particular at random shock fronts.\nThe fluctuation dynamo amplifies the random component\nof the magnetic field on time scales comparable to the turbu-\nlent eddy turnover time. Order-of-magnitude estimates sug-\ngest that the random field reaches energy equipartition with\nturbulence after a time of the order of 10 Myr (e.g., Beck et al.\n1994, and sections 6.7 and 13.3 of Shukurov & Subramanian\n2021). Since this time is short compared to the time scales\non which galaxies evolve, we assume that at any given time\nand position within the galaxy the random field has already\nsaturated and its rms strength is proportional to Beq, the\nfield strength corresponding to equipartition with the local\nturbulent kinetic energy density,\nbrms=fbBeq, (3)\nwhere\nBeq= (4πρ)1/2v , (4)\nρis the gas density (a function of randZas well as of the\nredshift) and fbis a constant.\nThe value of fbremains uncertain. The fluctuation dynamoproduces intense magnetic ropes and filaments that do not fill\nthe volume and the rms field strength depends on both the\nfield strength within the filaments and their fractional vol-\nume. Most existing simulations of the fluctuation dynamo\nsuggest fb≃0.5 (Brandenburg & Subramanian 2005, see\nalso section 13.3 of Shukurov & Subramanian 2021, and ref-\nerences therein) in incompressible turbulence and even lower\nin supersonic flows (e.g., Federrath et al. 2014). However,\nthe simulations are severely limited to modest values of the\nkinetic and magnetic Reynolds numbers, while fbcan de-\npend on them and is likely to do so. Much stronger, super-\nequipartition magnetic fields with fb>1 cannot be excluded.\nFor example, this would be the case as long as the fluctuation\ndynamo produces locally helical magnetic fields thus dimin-\nishing the associated Lorentz force and their back-reaction\non the velocity field (see section 4.2 of Belyanin et al. 1994,\nfor a heuristic model). The ISM multi-phase structure further\ncomplicates the action of the fluctuation dynamo in galaxies\n(Gent et al. 2023).\nThe tangling of the mean (large-scale) magnetic field by\nthe turbulence (an integral part of the mean-field dynamo ac-\ntion) is thought to produce a volume-filling random magnetic\nfield of a strength comparable to that of the mean-field (sec-\ntion 13.3 of Shukurov & Subramanian 2021). Since the mean\nmagnetic field can be stronger than Beqif the mean-field dy-\nnamo is strong, e.g., in the central parts of galaxies where\nthe differential rotation is especially strong (section 13.7 of\nShukurov & Subramanian 2021), this may lead to fb>1.\nCompression of magnetic fields at random shock fronts\ndriven by supernovae can produce very strong local random\nmagnetic fields but their fractional volume is likely to be\nsmall; nevertheless, the resulting rms field strength can cor-\nrespond to fb≃1 (Bykov & Toptygin 1987).\nWe adopt fb= 1, whereas R19 used 0 .5. As the random\nfield is taken to be isotropic, b2r=b2\nϕ=b2\nZ=b2\nrms/3 for the\ncylindrical field components.\n2.2.2 The mean (large-scale) field\nThe mean-field dynamo, by contrast, can amplify a large-\nscale magnetic field Bon length scales of the order of a kilo-\nparsec or larger, on time scales that depend on the galactic\nrotation period, velocity shear due to the differential rota-\ntion and the time scale of the turbulent magnetic diffusion\nacross the gas layer. The relevant parameters can be com-\nbined into the dynamo number of equation (A2). The ampli-\nfication time of the mean field ranges from a fraction of a gi-\ngayear in the inner parts of spiral galaxies to a few gigayears\nin the outer parts. We simulate the evolution of the mean\ncomponent of the magnetic field in galactic discs by solving\nnumerically the mean induction equation supplemented by a\ndynamical equation that models the non-linear back-reaction\nfrom the Lorentz force on the velocity field that quenches\nthe dynamo to establish a stationary magnetic field. The\nresulting strength of the large-scale magnetic field is pro-\nportional to Beqand also depends on the dynamo number\nand other parameters (Shukurov & Subramanian 2021). The\nequations solved and the numerical implementation can be\nfound in R19, and they rely on the thin-disc approximation\nin the mean-field dynamo equations applicable for h/r≲0.1\n(Baryshnikova et al. 1987). The upper left panel of Fig. 3\nshows that this ratio is less than 0 .4 at the half-mass radius\nMNRAS 000, 1–18 (2024)6C. Jose et al.\nfor the vast majority of galaxies, and even smaller for the\ngalaxies in the larger stellar mass range and at a high red-\nshift. To ensure that the solutions of the dynamo equations\nare computed as fast as required with a large sample of galax-\nies, we employ well-tested (chapters 11 and 13 of Shukurov &\nSubramanian 2021, see also Appendix A) approximations ap-\nplicable to a thin disc (R19 and references therein), including\nthe no- Zapproximation, which replaces Z(the distance from\nthe midplane) derivatives with divisions by the appropriately\nscaled gas scale height, allowing the equations to depend on\nonly one spatial dimension ( r).\nThe initial (seed) magnetic field required to launch the\nmean-field dynamo is the large-scale component of the ran-\ndom magnetic field produced by the fluctuation dynamo: it\ndoes not vanish when the random field is averaged over a\nfinite volume of the galactic disc. This provides an initial\nkiloparsec-scale random magnetic field of the order of 10−9G\nin strength, stronger than any plausible primordial magnetic\nfield (Ruzmaikin et al. 1988; Poezd et al. 1993; Bhat et al.\n2016; Gent et al. 2024; see chapter 13 of Shukurov & Subra-\nmanian 2021, for a review). Thus, we used a Gaussian random\nvariable with zero mean and standard deviation 10−2Beq(r)\nas the seed for each mean magnetic field component ( Brand\nBϕ) at each radius. This seed is somewhat larger than that\nused in R19; however, we use the same prescription as R19 for\npreventing the mean field from decaying to very low values\nwhen the dynamo is subcritical.\nWhen an evolving galaxy experiences a major merger\n(where the ratio of the combined stellar and cold gas masses\ninvolved exceeds 0 .3), the mean magnetic field is reset to\nthe seed field assuming conservatively that such a merger de-\nstroys the large-scale field or distorts it to such an extent\nthat it is so different from an eigenfunction of the dynamo\nequations that most of it decay rapidly because of the turbu-\nlent diffusion and the dynamo starts again with a weak seed\nmagnetic field described above.\n2.2.3 Field distribution across the disc\nThe procedures described above specify the variation of the\nrandom and mean magnetic fields along the galactocentric\ndistance in each galaxy of the sample, but the field distribu-\ntion across the gas layer (in Z) also has to be specified. In\nour fiducial model, the scale heights, hB, of both the random\nand the mean magnetic field are assumed to be the same as\nthat of the diffuse gas density hd. Thus, for example, the\ndistribution in Zof the mean field is adopted as\nB(r, Z) =BM(r)e−|Z|/hB, (5)\nwhere hB=hdandBM(r) is the solution of the mean-field\ndynamo equations produced using magnetizer .\nTo provide a simple measure of the magnetic field strength,\nwe use its strength B0corresponding to the volume-averaged\ntotal magnetic energy density,\nB0=\"Rrd\n0B2hdrdrRrd\n0hdrdr#1/2\n, (6)\nfor the total magnetic field B(where B≡ |B|) and similarly\nB0for its mean part and b0for the random part. We also\nintroduce a similar quantity for the equipartition magneticfield of equation (4),\nBeq,0=\"Rrd\n0B2\neqhdrdrRrd\n0hdrdr#1/2\n. (7)\n2.3 The synchrotron emission\nThe mid-radio (1–10 GHz) continuum emission of an SFG\nconsists of the non-thermal (synchrotron) emission due to\nrelativistic (cosmic ray) electrons and the free–free emission\nfrom thermal electrons. The radio emission at 1 .4 GHz is dom-\ninated by the synchrotron, with the thermal emission con-\ntributing about 10 per cent (Tabatabaei et al. 2017), and we\nneglect the thermal contribution to the RLF (see Schober\net al. 2023 for a discussion of the thermal contribution at\nother frequencies).\nWe assume that the relativistic electrons have an isotropic\nenergy spectrum\nN(E) dE=KEE−sdE , (8)\nwith a factor KE, where N(E) dEis the number density of\nelectrons with energy between EandE+ dEandsis the\nspectral index. The synchrotron emissivity (the energy emit-\nted by the unit volume of the source per unit time and unit\nfrequency interval within the unit solid angle) in a homoge-\nneous magnetic field Bis then given by (e.g., Ginzburg &\nSyrovatskii 1964; Rybicki & Lightman 1979)\nε=KEa(s)e3\nmec2\u00123e\n4πm3ec5\u0013(s−1)/2\nB(s+1)/2\n⊥ ν−(s−1)/2,(9)\nwhere we use the standard notation for physical constants, ν\nis the emission frequency (= 1 .4 GHz in our case), B⊥is the\ntotal magnetic field strength in the sky plane perpendicular\nto the line of sight, and\na(s) =√\n3\n4π(s+ 1)Γ\u00123s−1\n12\u0013\nΓ\u00123s+ 19\n12\u0013\n, (10)\nwhere Γ is the gamma-function (See chapter 3 of Shukurov\n& Subramanian 2021 for details). This expression is applica-\nble to a partially ordered magnetic field, B=B+bin our\ncase, unless the field is inhomogeneous on very small scales:\nthe synchrotron emission pulse from a relativistic electron is\nformed over a distance of order rB/γ≃109cm(B/1µG)−1\n(where rBis the electron Larmor radius and γis the particle\nLorentz factor), and magnetic fluctuations at larger scales do\nnot affect the applicability of equation (9). We adopt s= 3\nfor the electron spectrum, close to the observed value (Biss-\nchoff & Potgieter 2014; Bisschoff et al. 2019). For s= 3,\nthe statistically averaged value of εinvolves the average of\nb2\n⊥, a quantity predictable theoretically as discussed in Sec-\ntion 2.2.1.\nTo derive the synchrotron luminosity, the coefficient KEin\nequation (8) has to be determined. An often-adopted assump-\ntion regarding the energy density of cosmic rays (dominated\nby cosmic ray protons) is their energy equipartition with the\nmagnetic field,\nκcrZE2\nE1EN(E) dE=B2\n8π, (11)\nwhere κcris the ratio of the energy densities of the relativistic\nprotons and electrons. For E2→ ∞ , this yields KE= (s−\nMNRAS 000, 1–18 (2024)The RLF of star-forming galaxies and its cosmological evolution 7\nFigure 3. The probability distributions of the relative gas disc thickness hd(r1/2)/r1/2(upper left), volume-averaged mean (upper right)\nand random (lower left) magnetic field strengths, and the ratio of B0toBeq,0(lower right). The magnetic field parameters presented are\ndefined in equations (6) and (7). At each redshift, the galaxies are separated into two stellar mass bins defined in the legend. The area\nunder each curve is proportional to the number of spiral galaxies in the given mass bin at the given redshift.\nFigure 4. The galaxy number density (in comoving coordinates) as a function of various volume-averaged magnetic field properties\ndefined in equations (6) and (7) for the mean (solid) and random (dotted) fields (upper panel) and the ratio of B0toBeq,0(lower panel)\nfor various redshifts.\nMNRAS 000, 1–18 (2024)8C. Jose et al.\n2)B2Es−2\n1/(8πκcr) and, for s= 3 and κcr= 100 (e.g., Beck\n& Krause 2005), we have\nKE≃6.3×10−19particles\ncm3erg−2\u0012B\n1µG\u00132E1\n1 GeV. (12)\nForE1, we adopt the energy above which the electron spec-\ntrum measured in the Solar vicinity has the spectral index\ns≈3,E1≈8 GeV (see fig. 10.10 of Shukurov & Subrama-\nnian 2021, and references therein). Since cosmic rays have\na very high diffusivity (e.g., Longair 1994), we assume that\ntheir energy density depends on the spatially averaged mag-\nnetic field and use B2=B2+b2\nrmsin equation (12).\nThe assumption of the local energy equipartition between\ncosmic rays and magnetic fields lacks any general and con-\nvincing evidence. Both the observed synchrotron intensity\nfluctuations in galaxies (Stepanov et al. 2014) and simula-\ntions of cosmic ray propagation rather suggest a weak anti-\ncorrelation between their energy densities (Tharakkal et al.\n2023c,b,a). However, there are no readily available, simple\nalternative approaches to estimate the energy density of cos-\nmic rays in galaxies. For the exploratory study of this paper,\nwe adopt the equipartition assumption but a refinement of\nour model would include an estimate based on the SFR and\na cosmic ray propagation model.\nUsing the large-scale and small-scale magnetic fields com-\nputed for each galaxy, we derive B⊥and obtain the syn-\nchrotron luminosity assuming that the galaxies in the sample\nhave random orientations with respect to the line of sight, (we\nuse the uniform distribution of cos i, where iis the inclination\nangle of a galaxy to the line of sight),\nL=Z\ndΩZ\nVε(r) d3r\n= 8πZrd\n0rdrZ2π\n0dϕZhd\n0dZ ε(r, ϕ, Z ), (13)\nwhere Ω is the solid angle. The method for computing the\nintegral in equation (13) is described in Appendix B.\n3 MAGNETIC FIELDS IN EVOLVING\nGALAXIES\nThe top panel of Fig. 4 shows the number density of galaxies\nin the fiducial model as a function of the volume-averaged\nfield strengths of the mean and random magnetic fields, B0\nandb0defined in equation (6) in each galaxy at various red-\nshifts. At z= 0, galaxies with 1 µG≲b0≲5µG are more\ncommon than galaxies with B0in the same range, while\ngalaxies with B0<0.4µG or B0>10µG are far more com-\nmon than those with b0in that range. With redshift, these\nlimits shift to higher values of magnetic field strength. At\nlower redshifts, the number density of galaxies as a func-\ntion of b0has two distinct maxima. The distribution of B0is\nwider than that of b0because brmsis directly related to Beq\nthrough equation (4), whereas Bdepends on other galactic\nparameters as well. The lower panel of Fig. 4 shows the num-\nber density of galaxies as a function of B0/Beq,0and since\nbrms=Beqforfb= 1 (Section 2.2.1) and hence b0=Beq,0, it\nis clear that most galaxies in our model have a stronger ran-\ndom magnetic field compared to the mean field. As a larger\nvolume-averaged field strength typically results in a highersynchrotron luminosity, magnetic fields in radio-bright galax-\nies are predominantly random, especially at high redshift (see\nalso Section 4).\nFigure 3 shows the evolution of the probability density of\nthe mean and random magnetic field components in the red-\nshift range 0 < z < 3 with galaxies grouped into two stellar\nmass bins, 8 .5≤log(M∗/M⊙)<10.5 and log( M∗/M⊙)≥\n10.5. The probability density is normalized such that the to-\ntal probability over all redshifts is unity in each mass bin,\nseparately. This allows us to visualize the relative number of\ngalaxies with a given magnetic field both at each redshift and\nbetween different redshifts. The highest mass bin contains\nabout 3% of the total number of galaxies in our sample, and\nthe number of galaxies in a given mass bin changes with z\nowing to star formation, mergers, etc. The probability dis-\ntribution of B0has a single peak, with most galaxies having\nB0>0.1µG at all redshifts. This is different from the results\nof R19, their fig. 6, where a significant fraction of galaxies\nhave very weak magnetic fields of the order of 10−4µG (the\nstrength of the seed field) because the mean-field dynamo is\nsubcritical in some galaxies, particularly at higher redshifts.\nThe mean-field dynamo is significantly stronger in our fidu-\ncial model because of the higher turbulent speed v. Stronger\nturbulent flows promote the mean-field amplification despite\nthe increased turbulent diffusion because the dynamo number\nin the kinematic regime scales as h2\nd/v2(see Appendix A) and\nhdincreases with vfaster than linearly. Indeed, under hydro-\nstatic equilibrium, the disc scale height is proportional to the\ngas pressure, which includes the turbulent contribution ρv2,\nsohdincreases with vapproximately as v2, and therefore,\nthe dynamo number increases with v.\nFigures 4 and 3 also show that the mean magnetic field\nacross a sample of galaxies decreases with time because of the\ndepletion of the interstellar gas (although individual galaxies\nmay have monotonically increasing field strength depending\non their formation history), which is consistent with the re-\nsults of R19 and Rodrigues et al. (2015). We also find, as\nR19 do, that the mean field strength shows a stronger scat-\nter for galaxies with higher stellar masses. However, a new\nfeature of the present model is that galaxies with larger stel-\nlar masses have stronger mean magnetic fields, on average,\nwhereas in R19 there was less of a distinction. Galaxies with\nhigher stellar mass generally have higher SFR, as seen in\nthe right-hand panels of Fig. 5, where we show the scatter\nplot of M∗versus SFR, with data points coloured according\nto the volume-averaged total magnetic field strength. The\ncorrelation between the SFR and M∗(the so-called galaxy\nmain sequence) is consistent with several observational stud-\nies and galaxy formation models (Dav´ e 2008; Davies et al.\n2017; Bauer et al. 2011; Whitaker et al. 2014; Sparre et al.\n2015; Katsianis et al. 2016; Cowley et al. 2017). In our fiducial\nmodel, vincreases with SFR. The strength of the saturated\nmean field increases with vbecause of the increased dynamo\nnumber and the enhancement of Beqof equation (4). This ex-\nplains the increase of B0with the stellar mass seen in Fig. 3.\nThe increase of the total magnetic field strength B0with the\nstellar mass, evident in Fig. 5, is explained by the increase of\nboth B0andb0(≡Beq,0) with M∗.\nFigure 3 also shows the probability density of the volume-\naveraged random magnetic field strength b0. Similarly to B0,\ngalaxies of a higher stellar mass have larger b0. This can be\nexplained by the higher SFR of such galaxies, hence higher v\nMNRAS 000, 1–18 (2024)The RLF of star-forming galaxies and its cosmological evolution 9\naccording to equation (2) and larger Beqaccording to equa-\ntion (4) (we have confirmed that the correlation of ρwith M∗\nis very weak). The distribution of b0is bimodal for low-mass\ngalaxies, but not for galaxies of a high mass, for reasons not\nyet quite clear to us.\nThe lower panel of Fig. 3 shows the distribution of the\nratio of the characteristic field strengths B0andBeq,0in-\ntroduced in equations (6) and (7). The ratio B0/Beq,0tells\nus how the volume-averaged magnetic energy density of a\ngalaxy B2\n0/8πcompares to its averaged turbulent energy den-\nsityB2\neq,0/8πand how close is the mean-field dynamo to the\nsaturated state.\nThe ratio B0/Beq,0is sensitive to the distribution of the\nmagnetic field within the disc. The energy density of the tur-\nbulence, given by B2\neq/8πfrom equation (4), decreases with\nthe galactocentric radius rbecause of the reduction in the\ngas density ρalone as we assume that the turbulent speed\nvis independent of r. The strength of the mean magnetic\nfield, while generally decreasing with r, can have a different\nradial distribution as it depends, in particular, on the angular\nvelocity and shear rate of the galactic rotation. As an illus-\ntration, consider the radial dependencies B=˜Bexp(−r/rB),\nBeq=˜Beqexp(−r/req) and hd=˜hder/rhwith certain length\nscales rB,reqandrhfor 0≤r <∞(and the variables with\nthe tilde denoting the central values, those at r= 0), so that\nB0\nBeq,0=˜B\n˜Beq\f\f\f\f2rh/req−1\n2rh/rB−1\f\f\f\f. (14)\nAt all redshifts, the probability distributions of B0/Beq,0\nhave extended tails, especially in the case of massive galaxies,\nwhere this ratio exceeds unity (up to three) suggesting that\nmost galaxies have ˜B0/˜Beq≳1 and/or rB≳req. The mean\nmagnetic field strength likely exceeds Beqin the central parts\nof galaxies where the dynamo is the strongest. A very slow\ndecrease in the strength of the mean magnetic field with r,\nrB≫req, has been observed in nearby galaxies (section 4.2 of\nBeck 2015b, and references therein). A large radial extent of\nmagnetised region in spiral galaxies is explained by the slow\ndecrease of the dynamo number with rbecause of the disc\nflaring which compensates for the relatively slow decrease of\nthe rotation rate in a flat rotation curve, as can be seen from\nequation (A3). The radial extent of the magnetised region is\ncontrolled by the decrease of the gas density and Beqwith r.\n4 THE RADIO LUMINOSITY FUNCTION\nFigure 6 shows the variation with the redshift of the RLF\nΦ(L1.4) of SFGs at the rest-frame frequency 1 .4 GHz, where\nΦ(L1.4) is the number of galaxies per unit comoving volume\nper decade in luminosity, at luminosity L1.4. For compari-\nson, we note that the Milky Way luminosity at 1 .4 GHz is\nL0= 3×1021W Hz−1(Berkhuijsen 1984). The predictions\nare computed at the redshifts (shown on each panel) closest\nto the median of the redshift bins utilized for RLF measure-\nments by Condon et al. (2002) and van der Vlugt et al. (2022)\nwhere simulation snapshots are available. We show the RLFs\nobtained from two models described in Section 2.1.2: the fidu-\ncial model (where the turbulent speed vdepends on the SFR)\nand the model v25-R19 where v= 25 km s−1in all galaxies.The observed luminosity functions are from Condon et al.\n(2002, 2019) and Mauch & Sadler (2007) for the local Uni-\nverse and Novak et al. (2017) and van der Vlugt et al. (2022)\nfor the higher redshifts.\nThe agreement of the fiducial model with the data is quite\nsatisfactory, if not perfect, over a wide redshift range. In par-\nticular, for z≲1.2, the high-luminosity end of the RLF is\nreproduced remarkably well. Although the median luminos-\nity is under-predicted at higher redshifts for all values of L1.4,\nthe model and observations still marginally agree for z≲2,\nincluding the changes in the slope around L1.4= 1023W Hz−1\nforz≲0.5. Moreover, when extrapolated to higher luminosi-\nties, the median outcome of the fiducial model is in agree-\nment with the data point at the highest luminosity at small\nz(our galaxy sample is not large enough to probe such bright,\nrare galaxies at certain redshifts including z≈0). How-\never, the number density of galaxies in the luminosity range\n20.5≤log(L1.4/W Hz−1)≤21.5 in the local Universe is sig-\nnificantly higher than what is observed. Also, for z >2, the\nmodel number densities are systematically and significantly\nsmaller than those observed, and the discrepancy between\nthe predictions and the data increases with redshift.\n4.1 The role of the turbulent speed\nThe turbulent speed vis among the model parameters that\naffect directly the RLF because the strength of the magnetic\nfields on both large and small scales increases in proportion\nto the turbulent kinetic energy via their dependence on Beq\nthrough equation (3), and the large-scale field strength has\nadditional dependence on vvia the αeffect and turbulent\ndiffusion. The models v25-R19 and v(z)-J23 are designed to\nexplore the role of this parameter.\nApart from the fiducial model results, Fig. 6 also shows\nthe prediction of the v25-R19 model (dotted black) which has\nthe constant turbulent speed v= 25 km s−1in all galaxies,\nin contrast to the fiducial model where vis larger in galax-\nies with SFR ≥3 M⊙yr−1(Section 2.1.2). The two models\ndiffer little in the predicted RLFs at low luminosities, espe-\ncially at low redshifts where SFR <3 M⊙yr−1and hence\nv= 25 km s−1in most galaxies. However, the increase in the\nSFR up to z≃2 makes the v25-R19 model predictions differ\nfrom those of the fiducial model, and they become less and\nless adequate with increasing z, severely under-predicting the\nnumber density of galaxies at the high luminosity end. Even\nif the overall luminosity were increased by a factor by adjust-\ning model parameters (as discussed in Section 5), the shape\nof the v25-R19 RLF does not match the data as well as the\nfiducial model.\nFigure 7 compares the fiducial model with the model v(z)-\nJ23 (dotted black), where vis an explicit function of the\nredshift taken from the bottom panel of Fig. 1. Neither the\noverall shape nor the magnitude of the resulting RLFs agree\nwith the observations: this model is inferior to the fiducial\none.\nIn fact, neither the fiducial nor the v(z)-J23 model is fully\njustified because model v(z)-J23 does not capture the SFR-\ndependence of turbulence at constant z, while the fiducial\nmodel does not allow for such effects as the gravitational in-\nstability (Krumholz et al. 2018) and gas accretion and merg-\ners (Ginzburg et al. 2022) which are likely to result in an\nadditional dependence of the turbulent intensity on redshift\nMNRAS 000, 1–18 (2024)10 C. Jose et al.\nFigure 5. Left: the scatter plots of the colour-coded radio luminosity log( L1.4) of 103randomly selected galaxies with L1.4≥1021WHz−1\natz= 0.0,0.5,1.0 and 1 .5 versus the fourth power of the volume-averaged magnetic field strength and the volume of the emission region\n(disc volume). Right: the colour-coded volume-averaged magnetic field strength of galaxies shown in the left-hand panels versus their star\nformation rate and stellar mass.\nindependently of the SFR. A more realistic model for turbu-\nlence will be among the refinements of our model to follow.\n4.2 Relative importance of the random and mean\nmagnetic fields\nUnder the assumption of energy equipartition between cos-\nmic rays and magnetic field adopted here, the contributions\nof the mean and random magnetic fields to the RLF are notadditive because the radio luminosity includes the integrals of\n(B⊥·b⊥),B2b2\nrmsandB2\n⊥b2\nrms. Therefore, to assess the signif-\nicance of the mean and random magnetic fields, we consider\nresults obtained when one of them is ignored. In Fig. 7, we\nshow the predictions with only the random field (dash-dotted\ncyan line) or the mean field (dashed blue line) retained. Both\nrandom and mean fields are clearly important. However, the\nRLF that relies solely on the mean magnetic field decreases\nwith the luminosity too rapidly and by z≈1 the contribu-\nMNRAS 000, 1–18 (2024)The RLF of star-forming galaxies and its cosmological evolution 11\nFigure 6. The 1.4-GHz radio luminosity function of star-forming galaxies at various redshifts zspecified within each frame. The median\noutcome of the fiducial model is shown as a red solid line with the shaded region representing the 16–84 percentile range of the turbulent\nspeed shown in Fig. 1. The v25-R19 model (with the constant turbulent speed v= 25 km s−1) is represented with the dotted black line.\nThe observational data are taken from Condon et al. (2002) (black crosses), Condon et al. (2019) (blue crosses), Mauch & Sadler (2007)\n(red circles), Novak et al. (2017) (black circles) and van der Vlugt et al. (2022) (green triangles).\ntion from the random field dominates that of the mean-field\nat least at those high luminosities where observational data\nare available. On the other hand, at z≈0, we obtain a sig-\nnificantly better fit to the data by neglecting the random\nfield and including only the mean field, particularly at low\nluminosities where the fiducial model over-predicts the RLF.\nWe stress that the relation between the contributions of the\nrandom and mean magnetic field components to the RLF de-pends on poorly constrained parameters like fBfor the ran-\ndom field and the magnitude of the mean helicity of interstel-\nlar turbulence for the mean field. Since the radio luminosity\nscales as the fourth power of the magnetic field (Section 4),\nthe relative influence of the random and mean magnetic fields\nis rather sensitive to such parameters. Clearly, the same mag-\nnitude of the luminosity function could be obtained by chang-\ning the parameter values so as to boost one field component\nMNRAS 000, 1–18 (2024)12 C. Jose et al.\nFigure 7. As Fig. 6 but for the fiducial model, presented as in Fig. 6, compared with the v(z)-J23 model (black dotted) where the turbulent\nspeed is an explicit function of the redshift. The dashed blue and dash-dotted cyan lines show the contributions to the synchrotron\nluminosity solely due to the mean and random magnetic fields, respectively, in the fiducial model.\nand reduce the other. Therefore, the relative importance of\nmean and random field is not well-constrained by the model\nand for realistic parameter values, neither can be neglected.\nThe current interpretations of the observations of the syn-\nchrotron emission in nearby galaxies suggest that the energy\ndensity of the random magnetic field exceeds that of the\nlarge-scale field by a factor of three or even larger (Beck et al.\n2019). However, the interpretations rely on the assumption\nof the local energy equipartition between cosmic rays and\nmagnetic fields, and so are model-dependent.To conclude, there are several plausible reasons for the ap-\nparent complexity in the relative roles of the mean and ran-\ndom magnetic fields at different redshifts. It is likely that a\nbetter model for cosmic rays is necessary (e.g. because their\nnumber density might be more sensitive to the mean mag-\nnetic field than to its random part). Another reason could be\nour limited understanding of the interaction of the fluctuation\nand mean-field dynamos (chapter 8 of Shukurov & Subrama-\nnian 2021). Both types of dynamo action are sensitive to the\nmulti-phase ISM structure but in a poorly understood man-\nMNRAS 000, 1–18 (2024)The RLF of star-forming galaxies and its cosmological evolution 13\nner while it is clear that the ISM structure depends on both\nthe SFR and redshift. Remarkably, the RLF can shed light\non rather subtle aspects of both the ISM structure and galac-\ntic dynamos. The relative importance of mean and random\nfields thus deserves careful analysis beyond the scope of this\npaper.\n4.3 The redshift evolution of the RLF\nThe redshift evolution of the observed RLF for bright star-\nforming galaxies ( L1.4≳1021W Hz−1) is discussed by Novak\net al. (2017) and van der Vlugt et al. (2022, 2023); their ob-\nservational data are shown in Figs 6 and 7 together with our\nresults. Both the fiducial model (where the turbulent speed\nincreases with the SFR) and the observations agree that the\nrange of the radio luminosities extends to larger L1.4asz\nincreases. The galaxy luminosity functions observed in far\ninfrared (Gruppioni et al. 2013; Koprowski et al. 2017; Lim\net al. 2020) and K-band (Cirasuolo et al. 2010; Mortlock et al.\n2017) evolve similarly with the redshift. However, the RLF\nshows a slightly stronger evolution compared to that in the\ninfrared as the infrared-to-radio luminosity ratio is observed\nto decrease mildly with increasing redshift (Delhaize et al.\n2017; Calistro Rivera et al. 2017; Magnelli et al. 2015; Ivison\net al. 2010).\nBoth the size of the emission region and the strength of the\nmagnetic field contribute to the increase of the synchrotron\nluminosity with z. This is illustrated in the left-hand panels\nof Fig. 5, which presents the colour-gradient scatter plot of\nthe radio luminosity of a thousand randomly selected galax-\nies with L1.4≥1021WHz−1from our sample at different red-\nshifts as a function of their disc volume and B4\n0(asL1.4∝B4\nunder the assumption of energy equipartition between cosmic\nrays and magnetic fields). It is clear that the brightest galax-\nies typically have both large emitting volume and volume-\naveraged magnetic field strength. For example, the volume-\naveraged magnetic field strength of the brightest galaxies at\nz= 1.5 is about a factor of two larger than at z= 0\nCertain factors that contribute to stronger magnetic fields\nat high redshifts are clarified in the right-hand panels of\nFig. 5. Galaxies with larger M∗tend to contain stronger mag-\nnetic fields, for reasons explained in Section 3. The SFR den-\nsity has a maximum at z≃1.0 (Fig. 2), indicating high SFR\nin galaxies at high redshifts. This results in a higher turbu-\nlent speed in galaxies via equation 2 leading to a systematic\nincrease of the magnetic field strength with zat 0≲z≲1.5.\nThus, our fiducial model finds a shift to higher luminosities\nwith increasing z, in agreement with the observations up to\nz≈1.8. However, as mentioned at the beginning of Section 4,\nthez-dependent shift in the model RLF is weaker than that\ninferred from observations. The median predictions of the\nmodel seem to underestimate the observed RLF at z≳1,\nand the difference between them increases with increasing z,\nand at z≳2 they clearly disagree for reasons discussed in\nthe next section.\n5 DISCUSSION\nOur goal in this paper is, firstly, to explore the applicability\nof the theory and models of galactic magnetic fields to youngand evolving galaxies and, secondly, to identify the most im-\nportant galactic properties and parameters that can be de-\nduced and understood with the help of their radio luminosity\nfunction. The synthetic RLFs of star-forming galaxies ob-\ntained from their modelled synchrotron emission at 1 .4 GHz\nare in broad agreement with the available observations, re-\nproducing the number density of galaxies in the luminosity\nrange 1019≲L1.4≲1025W Hz−1(3×10−3to 3×103in\nterms of the Milky Way luminosity at 1 .4 GHz – Berkhuijsen\n1984) in the redshift range 0 ≲z≲2, within the uncertainties\nin the turbulent speed discussed in Section 2.1.2. Moreover,\nthe form (e.g., changes in the slope) of the RLF agrees well\nwith what is observed. We do not consider z≳2, because the\nsynchrotron luminosity predicted by the model becomes sig-\nnificantly lower than what is observed, continuing the trend\nwith zalready visible in Fig. 6.\nWe have made no persistent attempt to achieve any better\nagreement with the observational results even though this\nwould be possible given that many properties of the ISM\nin evolving galaxies leave much freedom for adjustment. We\navoid making heuristic assumptions that do not have explicit\nobservational or theoretical justification.\nWe have identified the turbulent speed vas one of the\npredominant factors affecting the RLF. Our results firmly\nindicate that vmust be an increasing function of the star\nformation rate for the SFR exceeding a certain threshold\n(here adopted as SFR 0= 3 M ⊙yr−1), and the model with\nv= const ( v25-R19) cannot explain the increases in the\ngalactic luminosity with zeven though its results are close to\nthose with the SFR-dependent turbulent speed (the fiducial\nmodel) at z≈0 (Fig. 6).\nWe also find that the shape and redshift-dependence of the\nRLF cannot be reproduced by making vdepend explicitly on\nzinstead of on the SFR. This is elucidated by the model v(z)-\nJ23, which predicts a decrease of the RLF with luminosity\nthat is too steep, failing badly to reproduce observations.\nWe appreciate, however, that the variation of the turbulent\nspeed with the SFR adopted in the fiducial model may just\nserve as a proxy for other effects discussed below.\n5.1 Interstellar magnetic fields\nThe energy density of the random magnetic field is propor-\ntional to the energy density of the turbulence with the pro-\nportionality factor f2\nbof equation (3) only known to be of\norder unity. Adopting fb= 1, we find a reasonable agree-\nment between the predicted RLF and observations at the\nredshifts z≲1.2 (see Fig. 6). Our knowledge of the strength\nand structure of turbulent magnetic fields largely relies on\nnumerical simulations severely limited to modest Reynolds\nnumbers. For example, adopting fb= 1.5 improves signifi-\ncantly the agreement at higher redshifts. Values of fbexceed-\ning unity can be due to a stronger magnetic helicity since\npartially helical fields are close to the force-free state, so\ntheir back-reaction on the flow is reduced, which allows them\nto have a higher energy density. Thus, our model might fit\nthe observations much better, had we assumed that fbin-\ncreases slightly with z. Moreover, fbcan well depend on var-\nious galactic parameters and be sensitive to the multi-phase\nISM structure rather than be a constant or a simple func-\ntion of z. Our results highlight the need to understand bet-\nter how random magnetic fields are maintained in galaxies.\nMNRAS 000, 1–18 (2024)14 C. Jose et al.\nThe theory of the mean magnetic fields is developed better\nalthough many aspects of the action of the mean-field dy-\nnamo in the multi-phase, evolving ISM remain speculative.\nSince the energy density of both the turbulent and large-scale\nmagnetic fields depends on the turbulent energy density, the\ngas density and turbulent speed are identified as major fac-\ntors affecting the RLF. However, the mean magnetic field has\na more complicated dependence on the galactic parameters\nthan its random part, including the differential rotation and\nthe gaseous disc thickness in addition to the turbulent energy\ndensity. The energy density of the mean magnetic field (un-\nlike the random field in our model) can exceed the turbulent\nenergy density. As a result, the total magnetic energy den-\nsity in a significant fraction of galaxies (especially massive\nones) exceeds the turbulent energy density (see also fig. 6 of\nRodrigues et al. 2019). The RLF in polarised emission would\nhelp to isolate the contribution of the mean magnetic field\nto the synchrotron luminosity and thus assess rather subtle\nproperties of evolving galaxies. Our predictions for the po-\nlarised RLF will be published elsewhere.\n5.2 Galactic structure\nThe predictions of our fiducial model, which assumes that\nthe structure of a galaxy is broadly similar to a spiral galaxy\nin the local Universe, are reasonable until z≈2, the red-\nshift at which galaxies appear to develop persistent thin discs\ndominated by rotation similar to those at z≈0 (Ginzburg\net al. 2022; Jim´ enez et al. 2023, and references therein). Our\nmodels implicitly assume that the disc is formed instanta-\nneously whereas the history of the disc formation can affect\nthe outcome of the evolution in the strongly nonlinear galac-\ntic systems. Certain effects of enhanced star formation on the\ngalactic structure and environment, such as outflows (foun-\ntains and winds) and the emergence of radio haloes, are not\ncaptured by our model which only includes magnetic fields\nin galactic discs. Such effects can affect high-redshift galaxies\nmore profoundly and may help to explain why our model does\nworse at matching observations as zincreases. For example,\ngalaxies with a high SFR may have radio haloes with vertical\nextent 5–10 kpc and magnetic field strength comparable to\nthat in the disc. Thus, future improvements to the magnetic\nfield model should include the radio halo and its contribution\nto the synchrotron luminosity.\n5.3 Cosmic rays\nTo obtain the number density of cosmic ray electrons, we have\nadopted a widely used (albeit poorly justified) assumption\nof energy equipartition between cosmic rays and magnetic\nfields. However, the energy density of cosmic rays (including\nthe number density of relativistic electrons) is likely to be\nan increasing function of the supernova rate directly related\nto the SFR. We shall explore elsewhere an alternative model\nwhere the abundance of cosmic rays is controlled by their\nsources and propagation in the evolving magnetic field.\n5.4 Interstellar turbulence and galactic outflows\nIt appears that the mechanisms of magnetic field genera-\ntion, and the ISM structure on which they rely, experiencea significant change at z≃2. This is suggested, in par-\nticular, by the increase in the gas velocity dispersion with\nthe SFR (Fig. 1) above the sound speeds in the diffuse ISM\nphases, cs≃10 km s−1in the warm gas and cs≃130 km s−1\nin the hot phase. Meanwhile, simulations of the supernova-\ndriven multi-phase ISM show that the fractional volume of\nthe hot gas at the galactic midplane increases only slightly\n(as SFR0.36assuming that the supernova rate is proportional\nto the SFR) from about 0.2 to 0.28 when the supernova rate\nincreases by a factor of four in comparison with the Milky\nWay rate and does not exceed 0.5 when the supernova rate\nis 16 times the local rate, while the abundance of the cooler\nphases reduces as the supernova rate increases (de Avillez\n& Breitschwerdt 2004; Breitschwerdt & de Avillez 2021, and\nreferences therein). The main consequence of the increase in\nthe SFR is a more vigorous outflow of the hot gas in the form\nof the galactic fountain or wind. Although the off-planar gas\nis hot and highly ionised, significant amounts of neutral hy-\ndrogen are present, entrained from the disc, or cooled down\nin situ ). Its H ivelocity dispersion is about 60 km s−1in the\nMilky Way (table 1 of Kalberla 2003) while the velocity dis-\npersion of the hot ionised gas is 75 km s−1(section 8.3 of\nSavage et al. 1997). A review of relevant observations and\nfurther references can be found in section 10.2.2 of Shukurov\n& Subramanian (2021). Meanwhile, the turbulent velocity in\nthe disc hardly changes as the supernova rate varies by a fac-\ntor 512, remaining close to the sound speed in the warm gas\nwhile the H iline width is larger than the (one-dimensional)\nturbulent velocity dispersion (Joung et al. 2009, see also Dib\net al. 2006). Therefore, an increase of the galactic outflow\nintensity and an enhanced contribution of turbulence in the\noff-planar gas appear to be the dominant effects of the in-\ncreased SFR on a spiral galaxy. In this context, the increase\nof the turbulent speed with the SFR adopted in the fidu-\ncial model can be interpreted as a reflection of the increasing\nimportance of the off-planar gas with its more intense turbu-\nlence.\nSimilarly to R19, we assume that the rms turbulent speed\nis independent of the galactocentric distance r. Some obser-\nvations of the H ivelocity dispersion in spiral galaxies show\nthat it decreases with the galactocentric radius rfrom often\nsupersonic values in the inner parts of galaxies (Boulanger\n& Viallefond 1992; Petric & Rupen 2007; Tamburro et al.\n2009; Mogotsi et al. 2016). On the other hand, there is signif-\nicant observational evidence for a very weak variation of the\nHivelocity dispersion with rin the discs of spiral galaxies\n(Dickey et al. 1990; Blitz & Spergel 1991, Section 12.2.3 of\nKamphuis 1993). It is plausible that the turbulent speed is in-\ndeed independent of the galactocentric radius and the spatial\nvariation of the gas velocity dispersion is due to variations\nin the outflow intensity. This aspect of the model requires\nfurther analysis.\nThe models and observational estimates of the H iand H α\nvelocity dispersions discussed in Section 2.1.2 are often in-\nterpreted as suggesting strongly supersonic turbulence, espe-\ncially at high SFR. However, the turbulence in the diffuse\n(warm and hot) interstellar gas is likely to be transonic (or\nsubsonic, depending on the kinetic energy injection rate) be-\ncause of the self-regulation of the supersonic turbulence (Sec-\ntion 2.10.2 of Shukurov & Subramanian 2021). The kinetic\nenergy of supersonic turbulence efficiently dissipates at shock\nfronts to heat the gas until an equilibrium state is reached in\nMNRAS 000, 1–18 (2024)The RLF of star-forming galaxies and its cosmological evolution 15\nwhich the turbulent speed is comparable to the sound speed.\nThus, turbulent flows with sufficiently strong energy injec-\ntion rates are likely to be transonic. Such a self-regulation\nof a turbulent system can be affected by radiative cooling\nwhich can prevent the gas heating by removing the dissipated\nturbulent energy via radiation (Enrique V´ azquez-Semadeni,\nprivate communication). The cooling is especially efficient in\ndense regions (like molecular clouds which occupy a negli-\ngible fraction of the disc volume), where supersonic turbu-\nlence can be maintained, but perhaps not in the warm and\nhot ISM phases which dominate the radio luminosity. The\nself-regulation of supersonic turbulence is not included in the\nturbulence models of Krumholz et al. (2018) and Ginzburg\net al. (2022), who assume that the disc is marginally sta-\nble concerning the gravitational instability at all times and\nconsider only the energy balance of turbulence driven by su-\npernovae and accretion flows onto and through the disc. The\nsample of SFGs where Varidel et al. (2020) measured the ver-\ntical velocity dispersion in the H αspectral line deliberately\nincludes only nearly face-on galaxies (0 < i < 60◦for the\ninclination angle), so not only the turbulence in the galactic\ndiscs but also outflows and turbulence of the off-planar gas\nare likely to contribute to the velocity dispersion obtained.\nThe velocity dispersion may increase faster with redshift\nthan expected from the SFR–redshift dependence alone (Wis-\nnioski et al. 2015; ¨Ubler et al. 2019b), so a better understand-\ning of the interstellar turbulence and the role of the evolving\nmulti-phase ISM structure appears to be essential for further\nprogress in modelling the RLF.\n6 CONCLUSIONS\nThe magnetic field and dynamo models presented here lead\nto a satisfactory agreement, within uncertainties, of the pre-\ndicted RLF of star-forming galaxies with observations up to\nz≃2. At higher redshifts, although the form of the theoret-\nical luminosity function is similar to what is observed, the\ntheory predicts a smaller number of galaxies of high luminos-\nity, and the discrepancy increases with the redshift. While\nthe observational data at higher redshifts still lack galaxies\nof faint and moderate luminosity, our findings enable us to\nidentify a range of effects that need to be understood and\nincluded to improve the agreement with the data.\nWe have identified the strength of interstellar turbulence as\none of the major factors affecting the RLF. A better under-\nstanding of the interstellar turbulence, primarily the turbu-\nlent speed (which makes only a limited and uncertain contri-\nbution to the observed spectral line widths) at high redshifts\nand the effects of the evolving multi-phase structure of the\ninterstellar medium on magnetic fields and cosmic rays are\nrequired to improve our models.\nModels presented here are based on a well-tested galaxy\nformation model; however, we rely on an implicit assump-\ntion that the overall structure of the interstellar medium does\nnot change as the galaxies evolve. This is an oversimplifica-\ntion but little is known about some aspects of galactic evo-\nlution which are relevant to magnetic fields and cosmic rays\n(the turbulent parameters, gas disc thickness, the multi-phase\nISM structure, etc.). It is then not surprising that our model\nbecomes inapplicable beyond the redshift z≃2 at which\ngalaxies undergo a significant structural change, the develop-ment of pronounced gas discs in particular. Among distinct\nstructural features of young galaxies, important for their non-\nthermal constituents, might be widespread radio haloes as-\nsociated with stronger star formation and correspondingly\nvigorous galactic fountain flows. Our limited knowledge of\nthe properties of turbulent magnetic fields, especially in the\nevolving multi-phase ISM, is another question that requires\na better answer.\nWe have shown that the galactic luminosity function in\nthe total radio intensity is a sensitive indicator of the state\nof the interstellar gas, especially its turbulence, density, and\ngalactic fountains and winds. Our results are consistent with\nthe notion that the structure of star-forming galaxies, and\ntheir magnetic fields, are different at z≲2 and the higher\nredshifts. The luminosity function in polarised radio emission\nwould provide even richer information on galactic rotation,\ngas stratification, and anisotropy of interstellar turbulence.\nACKNOWLEDGEMENTS\nCJ is supported by the University Grant Commission, In-\ndia through a BSR start-up grant (F.30-463/2019(BSR)) and\nthe Rashtriya Uchchatar Shiksha Abhiyan (RUSA) scheme\n(No.CUSAT/PL(UGC).A1/2314/2023, No:T3A). CJ also ac-\nknowledges the utilization of the high-performance cluster at\nIUCAA (Pune, India) facilitated through the associateship\nprogram. CMB is supported by the UK Science and Technol-\nogy Funding Council (STFC) through grants ST/T000244/1\nand ST/X001075/1.\nDATA AVAILABILITY STATEMENT\nThe model data presented in this paper may be available\nupon request from the lead author.\nREFERENCES\nAdams E. A. K., van Leeuwen J., 2019, Nature Astronomy, 3, 188\nBaryshnikova I., Shukurov A., Ruzmaikin A., Sokoloff D. D., 1987,\nA&A, 177, 27\nBauer A. E., Conselice C. 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A., R¨ ottgering H., 2022, ApJ,\n941, 10\nvan der Vlugt D., et al., 2023, ApJ, 951, 131\nAPPENDIX A: GALACTIC MEAN-FIELD\nDYNAMO\nThe interstellar magnetic field can be separated into mean\nand random components,\nB=B+b, (A1)\nwhere the overbar represents ensemble or volume averaging.\nThe mean magnetic field is amplified by the joint inductive\naction of the galactic differential rotation (at an angular ve-\nlocity ω) and helicity of interstellar turbulence (the α-effect)\nin what is known as the αω-dynamo (Shukurov & Subra-\nmanian 2021, and references therein). The intensity of these\neffects relative to the turbulent magnetic diffusion, which de-\nstroys the mean magnetic field by tangling it, is quantified\nby the dimensionless dynamo number,\nD=αSh3\nd\nβ2, (A2)\nˆZ\nφ= 0ˆn\niˆn\nˆnd\nˆnd (r= 0)(r,φ)Figure B1. The schematic diagram of a galaxy disc plane showing\nthe line of sight unit vectors bnpassing through r= 0 and an arbi-\ntrary location ( r, ϕ) as red arrows. The vector bndis the projection\nofbnonto the disc plane.\nwhere αis the magnitude of the αeffect, βis the turbu-\nlent magnetic diffusivity and S=rdω/dr. The induction\neffects that amplify the magnetic field can overcome the de-\nstructive action of the turbulent diffusion if the magnitude\nof the dynamo number exceeds a certain critical value |Dcr|,\n|D|>|Dcr|, and we note that both D < 0 and Dcr<0 be-\ncause S <0 in most parts of galactic discs. In a thin, rotating\ngas layer, Dcr≈ −10. Using the estimates α≃l2\n0ω/h dand\nβ≃1\n3l0v, where l0andvare the turbulent scale and speed,\nthe dynamo number can be expressed in terms of directly\nobservable galactic parameters,\nD≃9ωSh2\nd\nv2. (A3)\nThe magnetic field thus generated in a thin disc has the\nquadrupolar parity with respect to the galactic midplane\nZ= 0: Br(−Z) =Br(Z),Bϕ(−Z) =Bϕ(Z) and Bz(−Z) =\n−Bz(Z) in cylindrical coordinates ( r, ϕ, Z ).\nAs the magnetic field strength increases to become compa-\nrable to Beq, the Lorentz force becomes comparable to the\nCoriolis and other forces, and the field amplification slows\ndown until the dynamo reaches a saturated, steady state\nwhere the magnetic field strength is estimated as (Chamandy\net al. 2014, and sections 12.3 and 13.7.3 of Shukurov & Sub-\nramanian 2021)\nB2≃π2B2\neq\u0012l\nhd\u00132\u0012D\nDcr−1\u0013\nRκ. (A4)\nwhere Rκis a parameter of order unity, and we have set the\noutflow speed to zero. magnetizer includes the nonlinearity\nassociated with the magnetic helicity balance and solves for\nthe equivalent strength of the magnetic field as a function of\nthe galactocentric radius rusing the so-called no- Zapprox-\nimation applicable to a thin disc, hd/r≲0.1. The magnetic\nfield components in this solution can be thought of as the\nmidplane values if the magnetic field distribution in Zis ex-\nponential (and the average values otherwise).\nMNRAS 000, 1–18 (2024)18 C. Jose et al.\nAPPENDIX B: SYNCHROTRON LUMINOSITY\nThe synchrotron luminosity given in equations (9) and (13) is\nproportional to the integralR\nVKEB2\n⊥d3r′,where KE∝B2\nunder the assumption of the energy equipartition between\ncosmic rays and the magnetic field. magnetizer solves the\nmean-field dynamo equations for the cylindrical components\nof the axisymmetric mean magnetic field B= (Br,Bϕ,Bz)\nfor each galaxy in the sample as a function of the distance to\nthe galactic centre rin the galactic disc plane (corresponding\ntoZ= 0), and we augment the solution assuming its expo-\nnential distribution across the disc (in |Z|) with the scale\nheight hB(Section 2.2.3). The random magnetic field is as-\nsumed to have the same scale height, with the strength ob-\ntained from the turbulent energy density (Section 2.2.1).\nThe synchrotron luminosity depends on the inclination an-\nglei(0≤i≤π) between the observer’s line of sight (LoS)\nand the Z-axis (perpendicular to the galactic disc). The unit\nvector along any LoS ( bn) is resolved into nd=bndsinialong\nthe galactic disc plane and nZ=bnZcosialong the Z-axis\n(see Fig. B1). The ϕ= 0 direction is determined by the di-\nrection of bndas it passes through r= 0. Hence, the rand\nϕcomponents of bndfor an arbitrary LoS passing through a\nlocation ( r, ϕ,0) in the disc plane are cos ϕand−sinϕand\ncorrespondingly, the components of the unit vector bnpassing\nthrough ( r, ϕ,0) are\nbn≡(nr, nϕ, nZ) = (sin icosϕ,−sinisinϕ,cosi), (B1)\nwhich is used to derive the magnetic field component in the\nsky plane B⊥=B−bn(B·bn).\nAssuming that the turbulent magnetic field bis isotropic,\nb2\n⊥=2\n3b2\nrms, and brmsis estimated in Section 2.2.1. Since\nwe have adopted KE∝B2=B2+b2\nrms, the synchrotron\nluminosity of a galaxy is proportional to the following integral\nover the volume Vof the magnetised region:\nJ=Z\nV(B2+b2\nrms)(B2\n⊥+2\n3b2\nrms) d3r′. (B2)\nWhen both parts of the magnetic field have an exponential\ndistribution in Zwith a scale height hBand integration over\nZis extended over |Z|<∞, we have\nJ=1\n2Zrd\n0r′dr′Z2π\n0dϕ hB(r′)(B2+b2\nrms)(B2\n⊥+2\n3b2\nrms),\n(B3)\nwhere we adopt hB=hd;hdandrdare defined in Sec-\ntions 2.1.1 and 2.2.3 and B⊥depends on ϕvia the unit\nLoS vector (B1). We note that the scale height of the mag-\nnetic field can be larger than that of the gas. For example,\nhB= 2hdforB∝Beq∝ρ1/2. Such an increase in hBwould\nhave led to an increase in the RLF by a factor of two, so the\nvalues of the RLF presented are conservative.\nAPPENDIX C: TESTING THE SYNCHROTRON\nEMISSION MODEL: M51\nTo verify the evaluation of the synchrotron luminosity, we\napply the procedure described above to the nearby spiral\ngalaxy M51. The radio emission of this galaxy is thoroughly\nexplored and interpreted in terms of the galactic magnetic\nfield by Fletcher et al. (2011). To derive the magnetic fieldstrength, these authors use the assumption of the local en-\nergy equipartition between cosmic rays and magnetic fields,\nsimilar to equation (11). The total magnetic field strength\nderived from the radio intensity at the wavelength λ= 6 cm\n(ν= 5 GHz in terms of the frequency) assuming the path\nlength of S= 1 kpc through the synchrotron-emitting re-\ngion is 30 µG in the central part of the galaxy, 20–25 µG\nin the spiral arms and 15–20 µG between the arms in the\nmain part within 5 kpc of the centre (section 4.1 and fig. 8\nof Fletcher et al. 2011). The mean magnetic field strength\nderived from the Faraday rotation does not exceed 3 µG\n(section 6.2 of Fletcher et al. 2011). The intensity of the ra-\ndio emission at λ= 6 cm for distances 1 .6< r < 4.8 kpc from\nthe centre, given in Table 2 of Fletcher et al. (2011), ranges\nfrom 0 .5 mJy/beam between the arms to 1 .1 mJy/beam\nwithin them, with a beam of W= 15 arcsec in diameter. It\nincludes the thermal emission, and its fraction at λ= 6 cm is\nestimated as 25 per cent (section 4.1 of Fletcher et al. 2011).\nOur goal is to test our choice of various parameters in-\nvolved in the calculation of the synchrotron intensity rather\nthan to achieve a precise agreement with the observations of\nM51. Therefore we only include isotropic and homogeneous\nmagnetic field (with no Zdependence) with a scale-height\nof 0.5 kpc (half the path length of the synchrotron-emitting\nregion) that is independent of the distance from the galactic\ncentre. The rest of the parameters are the same as in the\nfiducial model presented in this paper. The synchrotron in-\ntensity is derived using equations (9) and (13) with b2\n⊥=2\n3b2\nneglecting both the anisotropy of the random field and the\nmean magnetic field since b4≫B4in M51; this makes our\nestimate of the radio intensity quite conservative.\nThe total synchrotron intensity for the solid angle π(W/2)2\nof a flat beam is then\nI(ν) =π\u0012W\n2\u00132L(ν)\n4πϵth, (C1)\nwhere the factor ϵth= 1.25 accounts for the contribution of\nthe thermal radio emission. The total synchrotron intensity\nfrom (C1) at ν= 5 GHz in the main part of the disc of\nM51 within r= 5 kpc for a field strength of 25 µG, and a\nflat beam of W= 15 arcsec in diameter is 0 .7 mJy/beam,\nwhich is in good agreement with Fletcher et al. (2011), who\nmeasures the same to be 0 .8 mJy/beam. If we instead use the\nmagnetic field strength of 20 µG in our model, the resulting\nsynchrotron intensity is 0 .3 mJy/beam.\nThis paper has been typeset from a T EX/LATEX file prepared by\nthe author.\nMNRAS 000, 1–18 (2024)"    },    {        "title": "2402.15148v1.Stability_of_a_dispersion_of_elongated_particles_embedded_in_a_viscous_membrane.pdf",        "content": "arXiv:2402.15148v1  [physics.flu-dyn]  23 Feb 20241\nStability of a dispersion of elongated particles\nembedded in a viscous membrane\nHarishankar Manikantan †\nDepartment of Chemical Engineering, University of Califor nia, Davis, CA 95616, USA\nWe develop a mean-field model to examine the stability of a ‘qu asi-2D suspension’ of\nelongated particles embedded within a viscous membrane. Th is geometry represents several\nbiological and synthetic settings, and we reveal mechanism s by which the anisotropic mobility\nof particles interacts with long-ranged viscous membrane h ydrodynamics. We first show that\na system of slender rod-like particles driven by a constant f orce is unstable to perturbations in\nconcentration – much like sedimentation in analogous 3D sus pensions – so long as membrane\nviscous stresses dominate. However, increasing the contri bution of viscous stresses from\nthe surrounding 3D fluid(s) suppresses such an instability. We then tie this result to the\nhydrodynamic disturbances generated by each particle in th e plane of the membrane and\nshow that enhancing subphase viscous contributions genera tes extensional fields that orient\nneighboring particles in a manner that draws them apart. The balance of flux of particles\naggregating versus separating then leads to a wave number se lection in the mean-field model.\n1. Introduction\nLipid molecules that comprise the cell membrane are free to fl ow within the 2D surface\nthat represents the bilayer, but exchange momentum with the adjacent 3D medium. Insoluble\nsurfactant monolayers and self-assembled polymer or nanop article layers have similar flow\nphysics, driven by this unique ‘quasi-2D’ nature of momentu m transport. In their seminal\nwork, Saffman & Delbr¨ uck (1975) approximated the membrane as a thin Newtonian fluid\nlayer sandwiched between and coupled to Stokes flow in adjace nt 3D fluid phases. Proteins,\nion channels, molecular motors, or synthetic particles are embedded in and constrained to\nmove within this 2D layer. Such a description reveals a subtl e transition from 2D to 3D\nhydrodynamics: the interface decouples from the bulk fluid i n the ‘membrane-dominant’\nlimit when surface viscous stresses far exceed the traction from the surrounding 3D fluid.\nThis limit is described by the 2D Stokes equation and has no so lution for the steady\ntranslation of a cylinder: a consequence of the Stokes parad ox (Manikantan & Squires 2020 ).\nHowever, Saffman (1976) recognized that subphase viscous stresses eventually cat ch up with\nsurface viscous stresses beyond a critical length scale, ul timately regularizing the divergence\ninherent to 2D Stokes flow. Building on this framework, the hy drodynamics of disks\n(Hughes et al. 1981;Stone & Ajdari 1998 ), spheres ( Danov et al. 2000;Fischer et al. 2006),\nrods ( Levine et al. 2004;Fischer 2004 ) and ellipsoids ( Stone & Masoud 2015 ) embedded in\nsurface viscous interfaces are now firmly established, and w idely employed in quantifying\nlateral diffusion in membranes, monolayers, and biofilms.\n†Email address for correspondence: hmanikantan@ucdavis.e du2\nMost of these efforts, however, address flow around single par ticles. Correlated diffusion\nand collective effects become relevant in biological membra nes that contain a high con-\ncentration of proteins ( Bussell et al. 1995;Oppenheimer & Diamant 2009 ). Recent efforts\nhave highlighted the role of long-ranged hydrodynamics in a ggregating and assembling\nactive ( Oppenheimer et al. 2019;Manikantan 2020 ) and driven ( Vig & Manikantan 2023 )\nmembrane-bound point particles. While a point-particle de scription offers useful insight\ninto this complex problem, real particles have a finite size a nd orientability with anisotropic\nhydrodynamic mobilities ( Levine et al. 2004;Fischer 2004 ) that lead to nontrivial dynamics\n(Camley & Brown 2013 ;Shiet al. 2022). Cellular processes like signaling, trafficking,\nand curvature sensing are associated with the transport and reorganization of filamentous\nproteins and rod-like domains ( Simunovic et al. 2013) on the plasma membrane. Recent\nadvances also enable synthetic assembly of catalytic ( Dhar et al. 2006) and DNA origami-\nbased ( Khmelinskaia et al. 2021) nanorods embedded in monolayers and membranes, yet no\ndescription of their collective surface viscous interacti ons yet exist. We aim to address this\ngap by borrowing fluid mechanical insights from analogous wo rk on the dynamics of 3D\nsuspensions of settling particles ( Koch & Shaqfeh 1989 ).\nIn§2we develop a mean-field description of a dilute quasi-2D susp ension of driven\nmembrane-attached slender particles by coupling their ani sotropic mobilities to long-ranged\ninterfacial viscous hydrodynamics. In §3, we analyze the linear stability of such a system\nto concentration perturbations, revealing signatures of t he resolution of the Stokes paradox\nin collective dynamics. In §4, we connect straining fields set up around driven particles t o\nsuspension stability and reveal mechanisms for a length sca le selection in this problem.\n2. Theoretical formulation\n2.1.Mean-field description\nThe geometry of our system is shown in figure 1. Slender rod-like particles of length /u1D43Fand\ncharacteristic thickness /u1D44E(with/u1D44E≪/u1D43F), each with a unit orientation vector /u1D491=(cos/u1D703,sin/u1D703),\nare embedded within a 2D viscous layer atop a 3D subphase. In a mean-field description,\nwe define a probability distribution /u1D713(/u1D499,/u1D491,/u1D461)such that the local concentration /u1D450(/u1D499,/u1D461)is\nobtained by integrating /u1D713across all possible orientations /u1D491:\n/u1D450(/u1D499,/u1D461)=/uni222B.dsp\n/u1D713(/u1D499,/u1D491,/u1D461)d/u1D491=/uni222B.dsp2/u1D70B\n0/u1D713(/u1D499,/u1D491,/u1D461)d/u1D703. (2.1)\nWe will also define /u1D45Bas the number density of particles:\n/u1D45B=1\n/u1D434/uni222B.dsp\n/u1D434/u1D450(/u1D499,/u1D461)d/u1D499, (2.2)\nwhere/u1D434is the area of the membrane. Rods are constrained to translat e and rotate in the plane\nof the membrane. Conservation of particles is then expresse d by\n/u1D715/u1D713\n/u1D715/u1D461+∇/u1D460·(∝dotacc/u1D499/u1D713) +∇/u1D45D·(∝dotacc/u1D491/u1D713)=0, (2.3)\nwhere∝dotacc/u1D499and∝dotacc/u1D491are translational and rotational velocities that capture p robability flux. The\nsurface gradient operator in the plane of the membrane is defi ned as∇/u1D460=(I−/u1D48F/u1D48F)·∇where\n/u1D48Fis the local normal to the 2D manifold that represents the mem brane, and the orientational\ngradient operator simplifies to\n∇/u1D45D=(I−/u1D491/u1D491)·/u1D715\n/u1D715/u1D491=ˆ/u1D73D/u1D715\n/u1D715/u1D703. (2.4)3\nFigure 1: System geometry: elongated particles ( /u1D43F≫/u1D44E) are embedded within an infinitesimally thin 2D\nviscous layer atop a 3D fluid subphase, and are all driven in th e membrane plane by an external force /u1D46D.\nIn this initial work, we will restrict ourselves to planar me mbranes at /u1D467=0, and so /u1D48F=ˆ/u1D49B.\n2.2.Micromechanical model\nWe will explore the response of such a quasi-2D suspension of elongated particles to an\nexternally applied force. For simplicity, the force /u1D46Dwill be taken to be a constant and acting in\nthe same direction on all particles, akin to sedimentation i n 3D suspensions ( Koch & Shaqfeh\n1989). In membranes, this might represent driven or anchored col lections of trans-membrane\nproteins in the context of cell motility ( Fogelson & Mogilner 2014 ). The translational velocity\nin (2.3) then has contributions from the self-mobility of each part icle, and from advection\ndue to the disturbance field generated by neighboring rods:\n∝dotacc/u1D499=/u1D496/u1D460+/u1D496/u1D451. (2.5)\nHere,/u1D496/u1D460is the local response of a rod-like particle to a constant for ce/u1D46D:\n/u1D496/u1D460=/u1D707⊥(I−/u1D491/u1D491)·/u1D46D+/u1D707∝ba∇dbl/u1D491/u1D491·/u1D46D, (2.6)\nwith hydrodynamic mobilities /u1D707⊥and/u1D707∝ba∇dblfor translation in directions perpendicular and\nparallel to /u1D491, respectively ( Fischer 2004 ;Levine et al. 2004).\nPhospholipids that make up biological membranes and monola yers are insoluble and well\napproximated as 2D incompressible fluids ( Manikantan & Squires 2020 ). The disturbance\nfield/u1D496/u1D451then solves the Boussinesq-Scriven equation ( Scriven 1960 ) for the stress balance\nwithin a viscous incompressible interface of surface visco sity/u1D702/u1D460coupled to 3D Stokes flow\nin the adjacent bulk fluid of viscosity /u1D702and forced by /u1D46Ddistributed at concentration /u1D450(/u1D499,/u1D461):\n∇/u1D460/u1D6F1=/u1D702/u1D460∇2\n/u1D460/u1D496/u1D451−/u1D702/u1D715/u1D497\n/u1D715/u1D467/barex/barex/barex/barex\n/u1D467=0+/u1D46D/u1D450(/u1D499,/u1D461),∇/u1D460·/u1D496/u1D451=0. (2.7)\nHere,/u1D6F1is the surface pressure. Membrane flow is coupled to the 3D flow field/u1D497in the adjacent\nsemi-infinite ( /u1D467from 0 to −∞) subphase via the traction term and a no-slip condition /u1D496=/u1D497\nat/u1D467=0. This framework can be generalized to accommodate membran e curvature and to\nthe presence of 3D fluids on either side of the membrane ( Shiet al. 2022,2024).\nFinally, the rotational flux in ( 2.3) captures reorientation due to non-uniform disturbance\nfields. For slender rod-like particles, this is given by Jeffe ry’s equation ( Jeffery 1922 ):\n∝dotacc/u1D491=(I−/u1D491/u1D491)·∇/u1D460/u1D496/u1D451·/u1D491. (2.8)\nThermal noise may contribute additional terms to ∝dotacc/u1D499and∝dotacc/u1D491, or equivalently as diffusive terms\nin the conservation equation ( 2.3). We ignore diffusion in what follows to highlight the role\nof hydrodynamics alone: however, a diffusive flux is straight forward to add to the analysis\nbelow, and diffusion always acts to suppress large wave numbe r fluctuations.4\n3. Linear stability\nWe wish to examine the linear stability of a dilute quasi-2D s uspension that is homogeneous\nin concentration and isotropic in orientational distribut ion. The disturbance field associated\nwith such a base state is /u1D496/u1D451=0. We perturb the probability distribution around this base\nstate as/u1D713(/u1D499,/u1D491,/u1D461)=/u1D45B[/u1D7130+/u1D716/u1D713′(/u1D499,/u1D491,/u1D461)]where/u1D7130=(2/u1D70B)−1,/u1D496/u1D451=/u1D716/u1D496′\n/u1D451and∝dotacc/u1D491/u1D451=/u1D716∝dotacc/u1D491′\n/u1D451. The\ncorresponding local concentration is /u1D450=/u1D45B[1+/u1D716/u1D450′]. Plugging perturbed quantities into ( 2.3)\nand recognizing the isotropic base state as well as 2D incomp ressibility of the membrane\nfluid field ( ∇/u1D460·/u1D496=0) gives the probability conservation equation at /u1D442(/u1D716):\n/u1D715/u1D713′\n/u1D715/u1D461+∇/u1D460/u1D713′·/u1D496/u1D460+/u1D7130∇/u1D45D·∝dotacc/u1D491′\n/u1D451=0 (3.1)\nWe then impose normal modes of the form\n/u1D713′(/u1D499,/u1D491,/u1D461)=˜/u1D713(/u1D48C,/u1D491,/u1D714)/u1D452/u1D456(/u1D48C·/u1D499−/u1D714/u1D461), (3.2)\nso that ˜/u1D450(/u1D48C,/u1D714)=∫˜/u1D713d/u1D491is the associated local concentration. The associated hydr odynamic\ndisturbance field can be obtained by solving ( 2.7) for the corresponding Fourier transform of\nthe disturbance velocity ( Manikantan & Squires 2020 ):\n˜/u1D496/u1D451(/u1D48C,/u1D491,/u1D714)=(I−ˆ/u1D48Cˆ/u1D48C)·/u1D45B/u1D46D˜/u1D450\n/u1D4582/u1D702/u1D460+/u1D458/u1D702, (3.3)\nwhere/u1D458=|/u1D48C|andˆ/u1D48C=/u1D48C//u1D458is a unit vector. Then, using Jeffery’s equation ( 2.8), we find\n˜∝dotacc/u1D491/u1D451=/u1D456(/u1D491·/u1D48C)(I−/u1D491/u1D491)·(I−ˆ/u1D48Cˆ/u1D48C)·/u1D45B/u1D46D˜/u1D450\n/u1D4582/u1D702/u1D460+/u1D458/u1D702⇒∇/u1D45D·˜∝dotacc/u1D491/u1D451=−2/u1D456(/u1D491·/u1D48C)/u1D491·(I−ˆ/u1D48Cˆ/u1D48C)·/u1D45B/u1D46D˜/u1D450\n/u1D4582/u1D702/u1D460+/u1D458/u1D702.\n(3.4)\nPlugging ( 3.2) and ( 3.4) into ( 3.1) gives\n(−/u1D456/u1D714+/u1D456/u1D48C·/u1D496/u1D460)˜/u1D713−2/u1D456/u1D7130(/u1D491·/u1D48C)/u1D491·(I−ˆ/u1D48Cˆ/u1D48C)·/u1D45B/u1D46D˜/u1D450\n/u1D4582/u1D702/u1D460+/u1D458/u1D702=0, (3.5)\nwhere/u1D496/u1D460follows ( 2.6).\nWe now non-dimensionalize over characteristic values of /u1D43Ffor length, /u1D439for force, and\n/u1D439/(4/u1D70B/u1D702/u1D460)for velocities. We will take the direction of the force to be ˆ/u1D487so that/u1D46D=/u1D439ˆ/u1D487. Then,\n/parenleftBig\n−/u1D714+/u1D48C·/bracketleftbig\n/u1D707∝ba∇dbl/u1D491/u1D491+/u1D707⊥(I−/u1D491/u1D491)/bracketrightbig\n·ˆ/u1D487/parenrightBig\n˜/u1D713−8/u1D441/u1D7130ℓ(/u1D491·/u1D48C)/u1D491·(I−ˆ/u1D48Cˆ/u1D48C)·ˆ/u1D487˜/u1D450\n/u1D4582ℓ+/u1D458=0,(3.6)\nwhereℓ=/u1D702/u1D460//u1D702/u1D43Fis the dimensionless Saffman-Delbr¨ uck length. Alternativ ely,ℓcan be\ninterpreted as a Boussinesq number ( Manikantan & Squires 2020 ) which characterizes\nsurface viscous stresses relative to 3D viscous stresses fr om the adjacent subphase fluid over\nthe length /u1D43F. Momentum transport is membrane-dominated when ℓ→ ∞ and subphase-\ndominated when ℓ→0./u1D441=/u1D45B/u1D70B/u1D43F2is an effective area fraction occupied by particles. The\nlong-ranged description is valid only in the dilute regime, and so/u1D441is at most /u1D442(1).\nTo proceed, we simplify our problem by considering only pert urbations in concentration\nalong the direction perpendicular to the external forces: /u1D48C·ˆ/u1D487=0. This is motivated by the\nfact that these transverse perturbations are the most unsta ble in analogous 3D sedimentation\nproblems ( Koch & Shaqfeh 1989 ) and driven 2D suspensions ( Vig & Manikantan 2023 ).\nSimplifying with this assumption and rearranging gives\n˜/u1D713=/parenleftbigg8/u1D441/u1D7130ℓ˜/u1D450\n/u1D4582ℓ+/u1D458/parenrightbigg(/u1D491·/u1D48C)(/u1D491·ˆ/u1D487)\n(/u1D707∝ba∇dbl−/u1D707⊥)(/u1D491·/u1D48C)(/u1D491·ˆ/u1D487) −/u1D714. (3.7)5\nFigure 2: (a) Growth rates /u1D70Eof unstable modes at various ℓ=/u1D702/u1D460//u1D702/u1D43Ffor a dimensionless number density\nof/u1D441=/u1D45B/u1D70B/u1D43F2=1. (b) Maximum growth rate /u1D70E/u1D45A, and (c) most unstable wave number /u1D458/u1D45Aand largest wave\nnumber/u1D4580as a function of ℓ. Dashed lines in (c) are asymptotic limits from ( 3.14).\nIntegrating ( 3.7) across all orientations while noting that ˜ /u1D450(/u1D48C,/u1D714)=∫˜/u1D713d/u1D491eliminates ˜/u1D713to\nreveal an implicit relationship for /u1D714(/u1D458):\n/uni222B.dsp(/u1D491·/u1D48C)(/u1D491·ˆ/u1D487)\n(/u1D707∝ba∇dbl−/u1D707⊥)(/u1D491·/u1D48C)(/u1D491·ˆ/u1D487) −/u1D714d/u1D491=/u1D4582ℓ+/u1D458\n8/u1D441/u1D7130ℓ. (3.8)\nThe mobility coefficients /u1D707∝ba∇dbland/u1D707⊥are functions of ℓ, and/u1D7130=(2/u1D70B)−1is the isotropic base\nstate distribution. For a given area fraction /u1D441, (3.8) thus represents a family of dispersion\nrelations parametrized by the Saffman-Delbr¨ uck length ℓ.\nIn a 2D Cartesian plane representing the membrane, we can tak eˆ/u1D487=−ˆ/u1D49Aandˆ/u1D48C=ˆ/u1D499\nwithout loss of generality. Simplifying ( 3.8) using/u1D491=(cos/u1D703,sin/u1D703)then gives\n1\n2/u1D70B/uni222B.dsp2/u1D70B\n0/u1D458sin/u1D703cos/u1D703\n(/u1D707∝ba∇dbl−/u1D707⊥)/u1D458sin/u1D703cos/u1D703+/u1D714d/u1D703=/u1D4582ℓ+/u1D458\n8/u1D441ℓ, (3.9)\nwhich is an implicit integral equation for the dispersion re lation/u1D458(/u1D714). For normal modes of\nthe form exp [/u1D456(/u1D48C·/u1D499−/u1D714/u1D461)], perturbations are unstable if the imaginary part of /u1D714is positive.\nFigure 2(a) shows the growth rate /u1D70E=Im[/u1D714]as a function of wave number obtained by\nnumerically evaluating roots of ( 3.9) with a secant method. The ℓ→ ∞ limit is reminiscent\nof analogous 3D suspensions, where the system-spanning mod e corresponding to /u1D458=0 is\nthe most unstable (see Koch & Shaqfeh 1989 ;Manikantan et al. 2014). To obtain the growth\nrate at/u1D458=0 analytically, we expand the integrand in ( 3.9) as a Taylor series in /u1D458:\n/u1D458\n2/u1D70B/u1D714/uni222B.dsp2/u1D70B\n0sin/u1D703cos/u1D703/bracketleftbigg\n1−(/u1D707∝ba∇dbl−/u1D707⊥)/u1D458\n/u1D714sin/u1D703cos/u1D703+.../bracketrightbigg\nd/u1D703=/u1D4582ℓ+/u1D458\n8/u1D441ℓ. (3.10)\nIntegrating term by term and simplifying gives\n−(/u1D707∝ba∇dbl−/u1D707⊥)/u1D4582\n/u1D7142+/u1D442(/u1D4584)=/u1D4582ℓ+/u1D458\n/u1D441ℓ. (3.11)\nThe dimensionless mobility difference is /u1D707∝ba∇dbl−/u1D707⊥→1 (Fischer 2004 ;Levine et al. 2004) in\nthe membrane-dominated regime ( ℓ→ ∞ ), which gives /u1D7142=−/u1D441+/u1D442(/u1D4582)or\n/u1D70E(/u1D458→0,ℓ→ ∞)=√\n/u1D441. (3.12)6\nIn fact, this turns out to be the maximum value of the growth ra te across all /u1D458andℓ.\nIncreasing subphase viscous stresses (decreasing ℓ) always acts to suppress the instability.\nNotably, the /u1D458=0 mode is stable for finite /u1D702, reflecting a boundary layer at infinity that\nregularizes the singularity in 2D Stokes by accounting for s ubphase viscous stresses ( Saffman\n1976). For finite ℓ, (3.11) gives/u1D7142∼ −/u1D458/u1D441ℓ(/u1D707∝ba∇dbl−/u1D707⊥)or/u1D70E=Im[/u1D714] ∝√\n/u1D458. In other words,\nthe growth rate goes to zero as /u1D458→0 for all values of ℓexcept in the limit ℓ→ ∞ . The\nmost unstable mode then corresponds to a finite value of /u1D458which suggests a mechanism for\nwave number selection. The most unstable wave number /u1D458/u1D45Acan be determined by taking the\nderivative of ( 3.9) with respect to /u1D458and numerically solving the implicit integral equation\ncorresponding to /u1D715/u1D714//u1D715/u1D458=0. Figure 2(b,c) shows /u1D458/u1D45Aand the corresponding growth rate /u1D70E/u1D45A\nas a function of ℓ. Note that the maximum growth rate /u1D70E/u1D45Aasymptotes to the ℓ→ ∞ limit as\npredicted by ( 3.12), and monotonically decreases upon decreasing ℓ. The wave number /u1D458/u1D45A\ncorresponding to fastest growth of perturbations, however , varies non-monotonically with ℓ\nand peaks at ℓ=/u1D442(1): in§4we propose a mechanism of instability that clarifies the leng th\nscale selection underlying /u1D458/u1D45Aand its dependence on ℓ.\nFinally, a key parameter is the largest unstable wave number /u1D4580which reveals the length\nscale of shortest unstable modes. Setting /u1D714=0 in ( 3.9) gives\n/u1D4582\n0ℓ+/u1D4580=8/u1D441ℓ\n/u1D707∝ba∇dbl−/u1D707⊥. (3.13)\nSolving this quadratic equation for positive /u1D4580yields the curve in figure 2(c). Note that this\n/u1D714=0 limit is only valid for /u1D458≠0. The range of unstable modes increases monotonically\nwithℓ. The high and low ℓlimits can be found analytically: /u1D707∝ba∇dbl−/u1D707⊥asymptotes to 1 and\n4ℓ[ln(0.48ℓ) −1]in these limits respectively ( Fischer 2004 ;Levine et al. 2004) giving\n/u1D4580(ℓ→ ∞)=√\n8/u1D441, /u1D458 0(ℓ→0)=2/u1D441\n−log(ℓ) −1.73. (3.14)\nThese are shown as the dashed curves in figure 2(c).\n4. Instability mechanism and length scale selection\nThe mechanism of instability can be understood by examining the disturbance flow created\nin the plane of the membrane by a point force corresponding to a reference particle, and the\neffect of this flow on realigning neighboring particles. As al l particles are driven in the same\ndirection, and since alignment dictates direction of mobil ity given a force, such a picture\nhelps identify microstructural configurations that favor a ggregation or separation.\nThe dimensionless velocity at a point /u1D493in the plane of the membrane due to a point force\nat the origin is ( Fischer 2004 ;Levine et al. 2004;Manikantan & Squires 2020 )\n/u1D496(/u1D493)=/u1D70B/bracketleftbigg/parenleftbigg/u1D43B1(/u1D451)\n/u1D451−2\n/u1D70B/u1D4512−/u1D44C0(/u1D451) +/u1D44C2(/u1D451)\n2/parenrightbigg/u1D493/u1D493\n/u1D45F2+\n+/parenleftbigg\n/u1D43B0(/u1D451) −/u1D43B1(/u1D451)\n/u1D451+2\n/u1D70B/u1D4512−/u1D44C0(/u1D451) −/u1D44C2(/u1D451)\n2/parenrightbigg/parenleftBig\nI−/u1D493/u1D493\n/u1D45F2/parenrightBig/bracketrightbigg\n·ˆ/u1D487,(4.1)\nwhere/u1D45F=|/u1D493|is the dimensionless distance, and /u1D451=/u1D45F/ℓ. Here,/u1D43B/u1D708and/u1D44C/u1D708are Struve and\nBessel functions of the second kind or order /u1D708, respectively. Note that ℓis renormalized by\nthe distance /u1D45Fin the far-field description: in other words, at constant /u1D702/u1D460and/u1D702, the system\nswitches from membrane-dominated to subphase-dominated a t large enough length scales.\nThe velocity field in the plane of the membrane from ( 4.1) asymptotes to that corresponding7\n(a)ℓ≫r (b)ℓ≪r\nFigure 3: Mechanism of the instability in terms of the distur bance due to a point-force (at the center) on\nthe orientation of elongated particles at a distance /u1D45F. Gray streamlines represent the in-plane velocity /u1D496.\nThe magenta extensional streamlines around each neighbori ng particle depict the symmetric part of the\nlocal, linear shear field. For ℓ≫/u1D45F(membrane-dominated regime), neighbors always reorient s uch that they\nare drawn towards the point force, thereby increasing parti cle density. By contrast, at much longer length\nscales/u1D45F≫ℓ(subphase-dominated regime), particles that fall within t he shaded region are drawn away from\nthe point force, reducing particle density and stabilizing the suspension. For a fixed /u1D702/u1D460and/u1D702, the system\ntransitions continuously from membrane-dominated to subp hase-dominated at large enough distances.\nto the 2D stokeslet in the membrane-dominated regime ( ℓ≫/u1D45F):\n/u1D496(/u1D493,ℓ≫/u1D45F) →/bracketleftBig\n−log/parenleftBig/u1D45F\nℓ/parenrightBig\nI+/u1D493/u1D493\n/u1D45F2/bracketrightBig\n·ˆ/u1D487. (4.2)\nThe mechanism of the instability is tied to the reorientatio n of neighboring particles when\nplaced in such a disturbance field. Specifically, rod-like pa rticles align with the extensional\naxis of the local flow. The rate-of-strain tensor in the membr ane-dominated case is\nE(ℓ≫/u1D45F)=∇/u1D460/u1D496/u1D451+∇/u1D460/u1D496/u1D447\n/u1D451\n2=(/u1D493·ˆ/u1D487)I\n/u1D45F2−2/u1D493/u1D493/u1D493·ˆ/u1D487\n/u1D45F4. (4.3)\nUsing ˆ/u1D487=−ˆ/u1D49Aand/u1D493=/u1D45Fˆ/u1D493=(/u1D45Fcos/u1D719,/u1D45Fsin/u1D719)where/u1D719is measured relative to the positive /u1D465\ndirection diagonalizes the rate-of-strain tensor in the (ˆ/u1D493,ˆ/u1D753)basis as\nE(ℓ≫/u1D45F)=sin/u1D719\n/u1D45F/bracketleftbigg\n1 0\n0−1/bracketrightbigg\n. (4.4)\nThe eigenvectors of Eare the principal directions of extension. We see from ( 4.4) that the\nextensional axis is along ˆ /u1D493(when/u1D719 > 0) or perpendicular to ˆ /u1D493(when/u1D719 < 0) as shown\nin figure 3(a). Elongated particles thus always align in a direction th at draws them towards\nthe reference particle. This is analogous to the classic 3-D instability in a suspension of\nrod (see Koch & Shaqfeh 1989 ). Note also from ( 4.4) that the eigenvalues of E(ℓ≫/u1D45F),\ncorresponding to the local rates of extension /u1D706ext, peak at/u1D719=±1\n2/u1D70Band vanish at /u1D719=0: rods\non the same horizontal line thus do not reorient in response t o the disturbance due to the\nother.\nThe in-plane extensional field surrounding a point force cha nges dramatically upon\nincreasing the contribution from the subphase (decreasing ℓ). The membrane velocity field8\nFigure 4: (a) Principal axes of stretch /u1D703extcorresponding to preferred orientation of a neighboring pa rticle\nas a function of azimuthal position /u1D719around a reference particle for ℓ//u1D45F=0.1 (dotted line), 1 (dash-dot),\nand 10 (dashed line). The shaded patch denotes the set of orie ntations that draws particles away from each\nother: this region is not accessed in the limit of ℓ→ ∞ (solid line), whereas ℓ→0 maximizes this window\nfor a range of relative positions |/u1D719|<|/u1D719/u1D450|=sin−1(1/√\n6). (b) Principal rates of stretch /u1D706extcorresponding\nto cases shown in (b). (c) /u1D706extdoes not vanish at /u1D719=0 for finite ℓ, and peaks when /u1D45F∼ℓ.\nin the surface-inviscid limit ( ℓ≪/u1D45F) is\n/u1D496(/u1D493,ℓ≪/u1D45F) →2ℓ/u1D493/u1D493\n/u1D45F3·ˆ/u1D487. (4.5)\nNote that ( 4.5) is not simply a 2-D slice of the flow due to a point force in the b ulk 3-D fluid:\nsurface incompressibility requires that /u1D496be divergence-free in the plane of the membrane.\nThe rate-of-strain tensor is no longer easily diagonalizab le in the(ˆ/u1D493,ˆ/u1D753)basis; however, E\nstill reveals key features when expressed in the (ˆ/u1D499,ˆ/u1D49A)basis:\nE(ℓ≪/u1D45F)=2ℓ\n/u1D45F2/bracketleftbigg\nsin/u1D719(3 cos2/u1D719−1)cos/u1D719(3 sin2/u1D719−1\n2)\ncos/u1D719(3 sin2/u1D719−1\n2)sin/u1D719(3 sin2/u1D719−2)/bracketrightbigg\n. (4.6)\nFirst, principal rates of extension scale with ℓ, consistent with a weakening instability as seen\nin§3. Additionally, neighboring particles along the wave vecto r (on the same horizontal line\nas the point force, or along /u1D719=0) experience a non-zero rate of extension along a principal\ndirection /u1D703ext=3/u1D70B/4 (figure 4). Coupled to the anisotropic mobility of rods, local extens ion\ntherefore reorients rods in a manner that draws them away fro m the reference particle if\ninitially placed in a region near /u1D719=0, which acts as a stabilizing mechanism. Note also\nfrom ( 4.6) that Eis diagonal in the (ˆ/u1D499,ˆ/u1D49A)basis when 3 sin2/u1D719=1\n2: i.e., for a critical angle\n/u1D719/u1D450=±sin−1(1/√\n6). This corresponds to principal axes that are aligned with ˆ /u1D499or ˆ/u1D49Aand\nreveals a transition from configurations that draw neighbor ing particles towards the point\nforce (when |/u1D719|>|/u1D719/u1D450|) or away from it. This also suggests a mechanism for wave numb er\nselection as seen in the stability analysis in §3: the critical length scale emerges from a9\nbalance of fluxes of particles driven out from within the regi on|/u1D719|<|/u1D719/u1D450|versus drawn in\nfrom regions where |/u1D719|>|/u1D719/u1D450|.\nPrincipal directions of extension for arbitrary ℓ, obtained by numerically evaluating the\neigenvectors of Efrom the general velocity field in ( 4.1), are shown in figure 4(a). This further\nclarifies the emergence of a ‘stabilizing region’ in terms of preferred orientations for finite\nℓ. Particles placed in this region are drawn away from the refe rence particle, contributing\nto a suppression of the instability. The ℓ//u1D45F→ ∞ limit avoids this region entirely as seen\nmechanistically in figure 3and quantitatively as a discontinuity in /u1D703ext(/u1D719)in figure 4(a). This\nreflects the singularity of 2D stokes flow at infinity, and the s ubphase viscous contribution\nintroduces a boundary layer that regularizes this disconti nuity. In doing so, /u1D703exttakes on\nvalues that fall within the shaded patch in figure 4(a) that represents a stabilizing region,\nwhere particle orientations are such that they are drawn awa y from the reference particle.\nThis mechanism also explains the stabilization of the /u1D458=0 mode for finite ℓin§3: for\nany/u1D702and/u1D702/u1D460(and since /u1D48Cis parallel to ˆ /u1D499for transverse perturbations), particles along the\nhorizontal line ( /u1D719=0) will always fall in the stabilizing region at a large enoug h value of /u1D45F.\nThe associated eigenvalues of Eare local rates of extension /u1D706ext: while these principal\nrates of extension always peak at /u1D719=±1\n2/u1D70Band their magnitudes decrease upon decreasing\nℓ, they do not always vanish at /u1D719=0 (figure 4b,c). Indeed, /u1D706ext(/u1D45F,/u1D719=0)peaks when /u1D45F∼ℓ.\nFor a pair of particles placed in the same horizontal line, th e reorientation that favors particle\nseparation is thus strongest at distances comparable to the Saffman-Delbr¨ uck length.\n5. Conclusion\nIn this paper, we developed a mean-field model to examine hydr odynamic collective modes\nof elongated particles embedded in viscous membranes or mon olayers. We focused on a\nlinear stability analysis around a homogeneous and isotrop ic ‘quasi 2D suspension’ of such\nparticles. We discover a unique mechanism for stabilizatio n of such a system, reflecting\naspects of the Stokes paradox in 2D viscous flows and its regul arization via 3D subphase\nstresses. We then tie this mechanism to the interplay betwee n anisotropic particle mobility\nand long-ranged hydrodynamics in the plane of the membrane.\nMore generally, this mean-field framework opens a rich avenu e of fluid dynamical problems\nand tools to examine microstructure on biological or synthe tic membranes. The method can\nbe readily adapted to membranes with curvature using modifie d Green’s functions ( Shiet al.\n2022,2024), to active particles on membranes via a stresslet solution (Manikantan 2020 ;\nOppenheimer et al. 2019), and to systems with weakly non-Newtonian membrane rheolo gy\nusing tools such as the Lorentz reciprocal theorem ( Vig & Manikantan 2023 ). We have\nillustrated the ‘deep’ subphase limit: confining the 3D fluid has been shown to amplify\nbulk stresses ( Camley & Brown 2013 ;Shiet al. 2024) and modify membrane flow fields in\nmanners that favor aggregation ( Manikantan 2020 ). 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